Ee Preboard Math 2012 2018

AUGUST 2012

MATHEMATICS PRE-BOARD EXAM

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

1. The equation y^2 = cx is the general equation of: A. y’ = 2y/x B. y’ = 2x/y C. y’ = y/2x D. y’ = x/2y SOLUTION: 0 = x(2yy ′ ) − y 2 )/x^2

y 2 = cx c=

y2

y 2 = 2xyy′

x y2

y ′ = 2xy = y/2x

Differentiate:

2. A line segment joining two points on a circle is called: A. arc B. tangent C. sector

D. chord

3. Sand is pouring to form a conical pile such that its altitude is always twice its radius. If the volume of a conical pile is increasing at the rate of 25 pi cu.ft/min, how fast is the radius is increasing when the radius is 5 feet? A. 0.5 ft/min B. 0.5 pi ft/min C. 5 ft/min D. 5 pi ft/min SOLUTION: h = 2r, r = 5ft Vcone =

1

1

2

πr²h = 3 πr 2 (2r) = 3 πr³ 3

2

dr

25ft³ = 3 π , 3πr² dt dr

25π = 2π(5)² dt dr dt

25π

= 2π(25) = 0.5 ft/min

4. Evaluate ʃ ʃ 2r²sin Ө dr dӨ, 0 > r >sin Ө, > Ө > pi/2 A. pi/2 B. pi/8 C. pi/24

D. pi/48

SOLUTION: π

sin θ

∫02 ∫0

π

2r² sin θ cos ²θ drdθ

π

sin θ

= ∫02 ∫0 π

2

2

2r² dr sin θ cos ²θ dθ sin θ

= ∫02 3 r² ∫0

sin θ cos ²θ dθ

2

=∫02 3 (sin θ)³ sin θ cos ²θ dθ π

= 3 ∫02 sin4 θ cos²θ dθ 2

(3)(1)(1) π

= 3 [(6)(4)(2)] 2 =

π 48

5. A shopkeeper offers a 25% discount on the marked price on an item. In order to now cost $ 48, what should the marked price be? A. $ 12 C. $ 60 B. $ 36 D. $ 64 SOLUTION: 48 = (1 − 0.25)X 48

x = 0.75 = $ 64

6. An observer wishes to determine the height of a tower. He takes sights at the top of the tower from A to B, which are 50 ft. apart, at the same elevation on a direct line with the tower. The vertical angle at point A is 30 degrees and at point B is 40 degrees. What is the height of the tower? A. 85.60 ft B. 143.97 ft C. 110.29 ft D. 92.54 ft SOLUTION: β = 180 − 40 = 140° α = 180 − 30 − 140 = 10° 50

x

= sin 30 ; x = 143.969621 sin θ h = 143.969621 sin(40) = 92.54 ft 7. A tangent to a conic is a line A. which is parallel to the normal B. which touches the conic at only one point C. which passed inside the conic D. all of the above 8. Find the area of the triangle which the line 2x – 3y + 6 = 0 forms with the coordinate axes. A. 3 B. 4 C. 5 D. 2 SOLUTION: 2x − 3(0) + 6 = 0 x=

−6 2

= −3

2(0) − 3y + 6 = 0

6

y=3=2 1

A = 2 (3)(2) = 3 sq. units

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

9. Find the general solution of (D² - D + 2)y = 0 A. y = e^x/2 (C1 sin sqrt. 7/2 x + C2 cos sqrt. 7/2 x) B. y = e^x/2 (C1 sin sqrt. 7/2 x - C2 cos sqrt. 7/2 x) C. y = e^x/2 (C1 cos sqrt. 7/2 x + C2 sin sqrt. 7/2 x) D. y = e^x/2 (C1 cos sqrt. 7/2 x - C2 sin sqrt. 7/2 x) SOLUTION: (D2 − D + 2)y = 0

1

−7

7

m − 2 = √ 4 = √2 i

m² − m + 2 = 0 1

1 2

m= +

7

2

(m − 2) + 4 = 0

√7 i 2

𝐲 = 𝐞𝐀𝐱 (𝐂𝟏 𝐜𝐨𝐬𝐁𝐱 + 𝐂𝟐 𝐬𝐢𝐧𝐁𝐱)

10. If 10 is subtracted from the opposite of a number, the difference is 5. What is the number? A. 5 B.15 C.-5 D. -15 SOLUTION: x - 10 = 5 Opposite of x – 10 = 5 15 – 10 = 5 ∴ −5 11. If y = 5 – x, find x when y = 7 A. 12 B.-12

C. 2

D. -2

SOLUTION: y = 5 – x, find x when y = 7 7=5–x x = -7 + 5 = −2 12. A ranch has a cattle and horses in a ratio of 9:5. If there are 80 more head of cattle than horses, how many animals are on the ranch? A.140 B. 168 C. 238 D. 280 SOLUTION: Cattle → x Horses → y

x y

9

= 5 ; x = y + 80

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y=

5x

y = 180 − 80 = 100

9

x + y = 180 + 100 = 280

Substitute: 5x 9

+ 80 = 180

13. Martin bought 3 pairs of shoes at P240 each pair and 3 pieces of t-shirts at P300 each. How much did he spent? A. P720 B. P900 C. P22,500 D. P 1,620 SOLUTION: 3(240) + 3(300) = P1,620 14. Find the standard equation of the circle with the center at (1,3) and tangent to the line 5x – 12y -8 =0. A. (x-1)2 + (y-3)2 = 8 C. (x-1)2 + (y-3)2 = 9 2 2 B. (x-1) + (y-3) = 12 D. (x-1)2 + (y-3)2 = 23 SOLUTION: 5x -12y – 8 = 0, center of the circle C (1,3) d=r=

5(1)−12(3)−8 |√5²+12²|

=3

(x – h)² + (y – r)² = r (x − 1)2 + (y − 3)2 = 9 15. Find the volume of the solid formed by revolving the area bounded by the curve y 2 = (x3)(1x) in the first quadrant about x-axis. A. 0.137 B. 0.147 C. 0.157 D.0.167 SOLUTION: y 2 = (x 3 )(x − 1)

LR = 4

y² = (x 3 − x 4 )

π ∫0 (x 3 − x 4 ) dx = 0.157

1

a=1

16. In the pile of logs, each layer contains one more log than the layer above and the top contains just one log. If there are 105 logs in the pile, how many layers are there?

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A. 11

B. 12

C. 13

D. 14

SOLUTION: Sn =

n [a + (n − 1)d] 2 1

Sn = 105 n=1

a1 = 1

n

105 = 2 [2(1) + (n − 1)(1)] ∴ n =

a2 = 2

14 layers

17. A wall 8 feet high is 3.375 feet from a house. Find the shortest ladder that will reach from the ground to the house when leaning over the wall. A. 16.526 ft B. 15.625 ft C. 14.625 ft D. 17.525 ft SOLUTION: 2

2

2

2

2

L3 = h3 + x 3 2

L3 = 83 + 3.3753 ∴ L = 15.625 ft 18. If f(x) = 10x + 1, then f(x+1) is equal to A. 10(10x ) B. 9(10x)

C. 1

D. 9(10x+1)

SOLUTION: if f(x) = 10x + 1, then f(x + 1) − f(x) =?

then f(1 + 1) − f(1) = 10 − 12 = 90

let x = 1

test from the choices, set x = 1

f(1) = 101 + 1 = 11

b = 9(101 ) = 90 ∴ 9(10x )

f(1 + 1) = 101+1 + 1 = 101

19. A particle moves on a straight line with a velocity v = (4 – 2t)3 at time t. Find the distance traveled from t = 0 to t = 3. A. 32 B. 36 C. 34 D. 30 SOLUTION: V = dx/dt dx = Vdt 3

∫ dx = ∫0 (4 − 2t)3 dt = 30

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

20. The area enclosed by the ellipse 4x2 + 9y2 = 36 is revolved about the line x = 3, what is the volume generated? A. 370.3 B. 360.1 C. 355.3 D. 365.10 SOLUTION: [4x² + 9y² = 36] x² 3²

1 36

y²

+ 2² = 1

V = AC, A = πab, C = 2πR V = π(3)(2)(2π)(3) = 355.31 21. If the vertex of y = 2x2 + 4x + 5 will be shifted 3 units to the left and 2 units downward, what will be the new location of the vertex? A. (-2, 1) B. (-5, -1) C. (-3,1) D. (-4,1) SOLUTION: 1 2 y 5 x² + 2x − + = 0

[y = 2x² + 4x + 5] 2

2

y

5

y

3

(x + 1)2 − + − 1 = 0 2 2

y

3

(x + 1)2 = + 2 2 1

(x + 1)2 = (y − 3) 2 C(−1,3) ∴ (−4, 1)is the answer

(x + 1)2 − + = 0 2 2 22. A coat of paint of thickness 0.01 inch is applied to the faces of a cube whose edge is 10 inches, thereby producing a slightly larger cube. Estimate the number of cubic inches of paint used. A. 4 B. 6 C. 3 D. 5 SOLUTION: V = s²

Vpoint = |Vold − Vnew |

Snew = 10 + (0.01x2) = 10.02

= 1006.01 − 1000 = 6.01in³ ≅ 6

Vold = 10³ = 1000 in³ Vnew = 10.02³ = 1006.01 in³

23. Find the mass of lamina in the given region and density function: π D[(x, y)], 0 ≤ x ≤ 2 , 0 ≤ y ≤ cos x and ρ = 7x

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A. 2

B. 3

C. 4

D. 5

24. Find the area of the region bounded by the curves y = x2 – 4x and x + y = 0 A. 4.5 B. 5.5 C. 6 D. 5 SOLUTION: x 2 − 4x = y ,

x+y= 0,

LR = 4

y = −x

(x − 2)2 = y + 4

V(2, −4)

a=1

A = ∫0 (−x − x 2 + 4x)dx = 4.5

3

25. A conic section whose eccentricity is less than one is known as: A. circle B. parabola C. hyperbola

D. ellipse

26. The plate number of a vehicle consists of 5-alphanumeric sequence is arranged such that the first 2 characters are alphabet and the remaining 3 are digits. How many arrangements are possible if the first character is a vowel and repetitions are not allowed? A. 90 B. 900 C. 9,000 D. 90,000 SOLUTION: Vowel = a , e , i , o , u = 5 ; =(5)(25)(10)(9)(8) = 90,000 27. The axis of the hyperbola, which is parallel to its directrices, is known as: A. conjugate axis B. transverse axis C. major axis D. minor axis 28. The minute hand of a clock is 8 units long. What is the distance traveled by the tip of the minute hand in 75 minutes. A. 10pi B. 20pi C. 25pi D. 40pi SOLUTION: 1 min = 6° 6°

π

75 min (1min) = 450° (180) = s = rθ = 8 x

5π 2

5π 2

= 20π

29. Find k so that A = (3, -2) and B = (1, k) are perpendicular. A. 2 B. 3 C. 1/2

D. 3/2

SOLUTION: mA =

0+2 2 =− 0−3 3

−1

3

mB = m = 2 A

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0−k

mB = 2 = 0−1

−2k = −3 = k =

3 2

30. The probability of a defect of a collection of bolts is 5%. If a man picks 2 bolts, what is the probability that does not pick 2 defective bolts? A. 0.950 B. 0.9975 C. 0.0025 D. 0.9025 SOLUTION: 1 − (0.05)(0.05) = 0.9975 1

31. If f(x) = x−2 ,(f·g)’*(1) = 6 and g’(1) = -1, then g(1) = A. -7 B. -5 C. 5

D. 7

32. 3 randomly chosen senior high school students were administered a drug test. Each student was evaluated as positive to the drug test (P) or negative to the drug test (N). Assume the possible combinations of the 3 students drug test evaluation as PPP, PPN, PNP, NPP, PNN, NPN, NNP, NNN. Assuming each possible combination is equally likely, what is the probability that at least 1 student gets a negative result? A. 1/8 B. 1/2 C. 7/8 D. ¼ SOLUTION: no. s of N → 12 total outcomes → 24

3 students → 1 − (0.5)(0.5)(0.5) = 7/8

12

∴ 24 = 0.5 possible

33. The tangent line to the function h(x) at (6, -1) intercepts the y-axis at y = 4. Find h’ (6). A. -1/6 B. -2/3 C. -4/5 D. -5/6 SOLUTION: 6=

−1 − 4 x

x = −5/6

34. The cable of a suspension bridge hangs in the form if a parabola when the load is uniformly distributed horizontally. The distance between two towers is 150m, the points of the cable on the towers are 22 m above the roadway, and the lowest point on the cables is 7 m above the roadway. Find the vertical distance to the cable from a point in the roadway15 m from the foot of a tower.

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A. 16.6 m

B. 9.6 m

C. 12.8 m

D. 18.8 m

SOLUTION:

1

22 = a(0)2 + b(0) + c → eq. 1 7 = a(75)2 + b(75) + c → eq. 2

∴ the parabola equation is ∶ y

2

22 = a(150) + b(150) + c → eq. 3 =(

from eq. 1, c = 22, substitute it from eq 2 and 3

2

a = 375 , b = − 5 1 2 ) x 2 − x + 22 375 5

the point of the parabola is (15, y)

5625a + 75b = −15 → eq. 2

plugging x = 15

22500a + 150b = 0 → eq. 3

1 2 y=( ) (152 ) − (15) + 22 375 5 = 16.6m

solving the equations gives the value of:

35. In how many ways different orders may 5 persons be seated in a row? A. 80 B. 100 C. 120 D. 160 SOLUTION: 5! = 5 x 4 x 3 x 2 x 1 = 120 36. The symbol “/” used in division is called. A. modulus B. minus

C. solidus

D. obelus

37. Find the area of one loop r2 = 16 sin 2theta. A. 16 B. 8

C. 4

D. 6

SOLUTION: r² = 16 sin θ 1

π

= ∫02 16 sin 2θdθ = 8 2 38. Find the centroid of the upper half of the circle x2 + y2 = 9. A. (0, 3/pi) B. (0, 4/𝐩𝐢) C. (0, 5/pi) SOLUTION: x 2 + y 2 = 32 → r h = 0, k = 0, r =3

D.(0, 6/pi)

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS 4r

y = 3π (centroid) y=

4(3) 3π

4

=π 4

x = 0 ∴ (0, π)

39. In polar coordinate system, the distance from a point to the pole is known as A. polar angle C. radius vector B. x-coordinate D.y-coordinate 40. The number that is subtracted in subtraction. A. minuend C. dividend B. subtrahend D. quotient 41. In how many ways can a person choose 1 or more of a 4 electrical appliances? A. 12 B. 13 C. 14 D. 15 SOLUTION: c = 2n − 1 = 24 − 1 = 15 ways 42. The surface area of a spherical segment. A. lune B. Zone

C. Wedge

D. sector

43. A particle has a position vector (2cos2t, 1+3sint). What is the speed of the particle at time t = pi/4? A. 1.879 B. 4.5 C. 5.427 D. 7.245 SOLUTION: (2cos2t, 1 + 3sint) Dx =

dv dt

v = √dx 2 + dy 2

(2cos2t)dy =

dv (1 + 3sint) dt dx = −2sin(2) dx = −4sin2t t =

v= 2

dy = 3cost π 4

2

√(9 − 4sin (π)) + (3cos π) 4 4 v = 4.528

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44. If the equation is unchanged by the substitution of –x for x, its curve is symmetric with respect to the A. y-axis C. origin B. x-axis D. line 45 degrees with the axis 45. Find the number of sides of a regular polygon if each interior angle measures 108 degrees. A. 7 B. 8 C.5 D. 6 SOLUTION: (n−2)(180) n

= 108

n= 5 46. The integer part of common logarithm is called the________. A. radicand B. root C. characteristic

47. The constant “e” is named in honor of: A. Euler B. Eigen

D. mantissa

C. Euclid

D. Einstein

48. A man rows upstream and back in 12 hours. If the rate of the current is 1.5 kph and that of the man in still water is 4 kph, what was time spent downstream? A. 1.75 hrs B. 2.75 hrs. C. 3.75 hrs D. 4.75 hrs SOLUTION: S

T = Tup + Tdown C = 1.5kph, v = 4kph

S

S

T = 2.5 + 5.5 = 20.625 km

Tdown =? Tdown =

S = vt S

S

Tdown = V+C = 5.5

20.625 5.5

= 3.75 hrs

S

Tup = V−C = 2.5

49. The probability that A can solve a given problem is 4/5, that B can solve it is 2/3, and that C can solve it is 3/7. If all three try, compute the probability that the problem will be solved. A. 101/105 B. 102/105 C. 103/105 D. 104/105 SOLUTION: 4

2

3

101

5

3

7

105

1 − (1 − ) x (1 − ) x (1 − ) =

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50. A steel ball at 110 deg C cools in 8 min to 90 deg c in a room at 30 deg C. Find the temperature of the ball after 20 minutes. A. 58.97 °C B. 68.97 °C C. 78.97 °C D. 88.97 °C SOLUTION: t1 t2

=

Tb1 −Tm ) Tb0 −Tm Tb2 −Tm ln( ) Tb0 −Tm

ln(

8

= 20 =

90−30 ) 110−30 Tb2 −30 ln( ) 110−30

ln(

Tb2 = 68.97℃ 51. A freight train starts from Los Angeles and head for Chicago at 40 mph. Two hours later passenger train leaves the same station for Chicago traveling at 60 mph. How long will it be before the passenger train overtakes the freight train? A. 3 hrs B. 4 hrs C. 5 hrs D. 6 hrs SOLUTION: S = vt Sft = (40)(20) = 80 miles Spt = (80 + Sft Vpt = 80 + 40(Vft )(t) 60(t) = 80 + 40(t) T = 4 hrs 52. Given the triangle ABC in which A = 30 deg 30 min, b = 100 m and c = 200 m. Find the length of the side a. A. 124.64 m B. 142.24 m C. 130.50 m D. 103.00 SOLUTION: a = √200² + 100² − 2(200)(100) cos(30°30´) a = 124.64 m 53. Lines that intersect in a point are called______. A. Skew lines B. Intersecting lines C. Agonic lines D. Coincident lines 54. Find the average rate of change of the area of a square with respect to its side x as x changes from 4 to 7. A. 14 B. 6 C. 17 D. 11 SOLUTION:

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A = s2 da = 2sds da ds

Vave = Vave =

= 2s

Vo+Vf 2 [(2)(4)−(2)(7)] 2

= 11

55. If the distance x from the point of departure at time t is defined by the equation x = -16t2 + 5000t + 5000, what is the initial velocity A. 20000 B. 5000 C. 0 D. 3000 SOLUTION: x − 16t 2 + 5000t + 500 x´ = −32t + 5000, @t = 0 x´ = −32(0) + 500 = 5000

56. What conic section is represented by 2x2 + y2 – 8x + 4y = 16? A. parabola B. ellipse C. hyperbola

D. circle

57. If 9 ounces of cereal will feed 2 adults or 3 children, then 90 ounces of cereal, eaten at the same rate, will feed 8 adults and how many children? A. 8 B. 12 C.15 D. 18 SOLUTION: rate of children and adult 9oz 2

= 4.5 oz/adult

9oz 3

= 3oz/children

formulate an equation: (8)(4.5) + (x)(3) = 90 x = 18 children

58. Mary is twice as old as Helen. If 8 is subtracted from Helen’s age and 4 is added to Mary’s age, Mary will then be four times as old as Helen. How old is Helen now? A. 24 B. 36 C. 18 D. 16

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SOLUTION: mary = x helen = y mary(x) = 2y if y − 8, x + 4, then x = 4y find y =? (x + 4) = 4(y − 8) (x + 4) = 4y − 32 4y − x = 36 → eq. 1

x = 2y

Substitute: 4y − 2y = 36 36 y = 2 = 18

59. A point on the curve where the second the derivative of a function is equal to zero is called. A. maxima B. minima C. point of inflection D. point of intersection 60. Find the area of the triangle whose sides are 25, 39, and 40. A. 46 B. 684 C. 486 SOLUTION:

D. 864

a = 25, b = 39, c = 40 A = √s(s − a)(s − b)(s − c) s=

a+b+c 2

=

25+39+40 2

= 52

A = √52(52 − 25)(52 − 39)(52 − 40) = 468 sq. units

61. A/An_______triangle is a triangle having three unequal sides. A. oblique B. scalene C. equilateral D. isosceles 62. Find the length of the arc of 6xy = x4 + 3 from x = 1 to x = 2. A. 1.34 B. 1.63 C. 1.42 SOLUTION: y=

x4 +3

D. 1.78

[(24x 2 s = ∫1 √1 + (

6x

vdu−udv

s = 1.42

v2 dy

= dx

[(6x)(4x3 )−(x4 +3)(6)] 36x2 2

2 dy s = ∫1 √1 + (dx)

63. Give the degree measure of angle 3pi/5 radians. A. 108 B. 120 C. 105

D. 136

4 −(6)(x4 +3)]

36x2

2

) dx

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SOLUTION:

3π

rad 5

180°

= π rad = 108°

64. What do you call a radical expressing an irrational number? A. surd B. radix C. complex number 65. Find the derivative of the function f(x) = (2x – 3x)2. A. 2x - 4 B. 2x - 3 C. 6x - 8 SOLUTION:

D. index

D. 8x -12

f(x) = (2x − 3)² x´ = 2(2x − 3)(2) = 4(2x − 3) = 8x − 12 66. What is the length of the line with a slope of 4/3 from a point (6, 4) to the y- axis? A. 10 B. 25 C. 50 D. 75 SOLUTION: 4

y−4

m = 3 = 0−6 y = −4 d = √(−4 − 4)2 + (0 − 6)² d = 10 67. The inclination of the line determine by the points (4, 0) and (5√3) is A. 30 degrees B. 45 degrees C. 60 degrees D. 90 degrees SOLUTION: P = (4,0) and P(5, √3) θ = tan−1 m =m=

√3−0 5−4

= √3

θ = tan−1 (√3) = 60° 68. A sequence of numbers where the succeeding term is greater than the preceding term is called: A. dissonant resonance C. Isometric series B. convergent series D.divergent series

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69. Find the value of x for which y = 4 + 3x – 3x3 will have a maximum value. A. 0 B. -3 C. -2 D. 1 SOLUTION: dy dx

= 4 + 3x − x³ = 3 − 3x 2 = 0 3

x = √3 = 1 70. How many cubic meters is 500 gallons of liquid? A. 4.8927 B. 3.0927 C. 2.8927 SOLUTION: 1 gal = 3.78 li ∶ 500 gal x

3.785li 1gal

x

1m³ 10³li

D. 1.8927

= 1.8925 ≈ 1.8927 m³

71. A certain radioactive substance has a half-life of 3 years. If 10 grams are present initially, how much of the substance remains after 9 years? A. 1.50 grams B. 1.25 grams C. 2.50 grams D. 1.75 grams SOLUTION: t1 t2

=

q ln 1

Q0 q ln 2 Q0

3

∴9=

(0.5Q0) Q0 q ln 2 100

ln

= q 2 = 1.25 sq. units

72. A statement of the truth of which is admitted without proof is called: A. an axiom B. a postulate C. a theorem D. a corollary 73. A rectangular trough is 8 feet long, 2 feet across the top and 4 feet deep. If water flows in at a rate of 2 ft3/min, how fast is the surface rising when the water is 1 ft deep? A. ¼ ft/min B. ½ ft.min C. 1/8 ft/min D. 1/6 ft/min SOLUTION: V = LWH dv dt

= (8)(2)(4)H′

2 = (8)(2)(4)H′ 2 16

= H′ =

1 8

ft/min

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74. Find the point(s) on the graph of y = x2 at which the tangent line is parallel to the line y = 6x -1. A. (3, 17) B. (3, 9) C. (1, 2) D. (2, 4) SOLUTION: y1 ´ = 2x y2 ´ = 6 since tangent, the M or slope are equal y1 ´ = y2 ´ 2x = 6 y=3 y = x² ; y = 3² = 9 = P(3,9) 75. How many petals are three in the rose curve r = 3 cos 5theta? A. 5 B. 10 C. 15

D. 6

SOLUTION: r = cos5θ ↓ odd ∴ n = 5 76. Find the acute angle between the vectors z1 = 3 – 4i and z2 = -4 + 3i. A. 17 deg 17 min C. 15 deg 15 min B. 16 deg 16 min D. 18 deg 18 min SOLUTION: Z1 = 3 − 4i = 5∠ − 53.13

= 143.13 + 53.13 = 196.26

Z2 = −4 + 3i = 5∠143.13

θ = 196.26 − 180

ZT = Z2 − Z1 (5∠143.13) − (5∠ − 53.13)

77. If z1 =1 – i and z2 = -2 + 4i evaluate z12 + 2z1 – 3. A. -1 + 4i B. 1 - 4i C. -1 – 4i SOLUTION: z1 = 1 − i → √2 < −45 solve for Z1 ² + 2Z1 − 3

θ = 16.26 = 16°16′

D. 1 + 4i

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

√2 < −45)² + 2√2 < −45) − 3 = −1 − 4i

78. A motorboat moves in the direction N 40 deg E for 3 hours at 20 mph. How far north does it travel? A. 58 mi B. 60 mi C. 46 mi D. 32 mi SOLUTION: 3hrs @ 20mph S1 = vt = (20)(3) = 60 miles S2 = 60 cos 40 = 45.96 ≈ 46 miles 79. Find the value of 4 sinh(pi i/3) A. 2i (sqrt. of 3) B. 4i (sqrt. of 3) SOLUTION:

C. i (sqrt. of 3)

D. 3i (sqrt. of 3)

πi 4sinh( ) 3 π i4 sin ( ) = 2√3i 3

80. Find the upper quartile in the set (0, 1, 3, 4) A. 0.5 B. 0.25 C. 2 SOLUTION:

D. 3.5

𝟎 , 𝟏, 𝟑 , 𝟒 𝟎.̌𝟓 𝟏.̌𝟓 𝟑.̌𝟓 → 𝐮𝐩𝐩𝐞𝐫 𝐪𝐮𝐚𝐫𝐭𝐢𝐥𝐞 = 𝟑. 𝟓

81. In debate on two issues among 32 people, 16 agreed with the first issue, 10 agreed with the second issue and of these 7 agreed with both. What is the probability of selecting a person at random who did not agree with either issue? A. 1/32 B. 13/32 C. 3/8 D. 3/10 SOLUTION: 32 people

2nd issue = 10 − 7 = 3

1st issue → (16 Agreed), (7 agreed) 2nd issue → (10 Agreed), (Both) 1st issue = 16 − 7 = 9 both = 7

19 agreed 32 − 19 = 13 disagreed ∴

13 32

82. From the top of the lighthouse, 120 m above the sea, the angle of depression of a boat is 15 degrees. How far is the boat from the lighthouse? A. 448 m B. 428 m C. 458 m D. 498 m SOLUTION: 120

x = tan15 = 447.85 ≈ 448m 83. The cross section of a certain trough are inverted isosceles triangles with height 6 ft and base 4 ft. Suppose the trough contains water to a depth of 3 ft. Find the total fluid force on one end. A. 187.2 lb B. 178.2 lb C. 192.4 lb D. 129.4 lb SOLUTION: F = γh F = 624(3)[ib. ft 3 ][3ft] F = 187.2 lb/ft 2 84. Two lines are not coplanar. A. Parallel lines B. Skew lines C. Secant lines

D. Straight lines

2

85. Find the inverse Laplace transform of − s−3. A. 2 e-3t B. 2e3t

C. 3e-2t

D. 3e2t

SOLUTION: 2

Inverse Laplace of {s−3} 1

1

= 2 [s−3] = e±at = s∓a 2

Inverse laplace of {s−3} = 2e3t 86. Find the length of the latus rectum of the curve rcos2 theta – 4cos theta = 16sin theta. A. 4 B. 16 C. 12 D. 18 SOLUTION: [rcos²θ − 4cosθ = 16sinθ] rcos²θ = 16sinθ + 4cosθ 1

rcos²θ = 16sinθ + 4 cosθ ↓ 4a → LR ∴ LR = 16

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

87. A quadrilateral with no pair of parallel sides. A. Trapezoid B. Trapezium

C. Rhombus

D. Rhomboid

88. Find the equation of the line tangent to the curve y = x3 – 6x2 + 5x + 2 at its point of inflection. A. 7x – y B. -7x + y = 0 C. 7x +y = 10 D. -7x – y = 10 SOLUTION: 3(2)2 − 12(2) + 5 = −7 → m

y = x³ − 6x 2 + 5x + 2 y ′ = 3x² − 12x + 5

y − y1 −= m(x − x1

y" = 6x − 12 = 0

y + 4 = −7(x − 2)

y = (2)3 − 6(2)2 + 5(2) +

y + 4 = −7x + 14

2

P. O. I. (2, −4)

7x + y = 10

y ′ = 3x² − 12x + 5 = m

y = −4

; x=0

89. Find the area of the polygon with vertices at 2 + 3i, 3 + i, -2 – 4i, -1 + 2i. A. 47/5 B. 47/2 C. 45/2 D.45/4 SOLUTION: 1 2 1 2 1 2

1

(3.16)(3.61)sin(37.28) + (3.61)(2.24)sin(60.26) + 2 1

(2.24)(4.12)sin(77.47) + (4.12)(4.47)sin(49.39) + 2

(4.47)(3.16)sin(116.5718.43) =

47 2

or 23.50 sq units

90. Find the radius of curvature of y = x3 at x =1. A. 5.27 B. 4.27 C. 6.27 SOLUTION: R =? y = x3 @ x = 1 R=

[1+(y′)²]3/2 y"

y ′ = 3x² = 3(1)2 = 3 y" = 6x = 6(1) = 6 R=

[1 + (3)²]3/2 = 5.27 6

D. 7.27

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

91. Determine the probability of throwing a total of 8 in a single throw with two dice, each of whose faces is numbered from 1 to 6. A. 1/3 B. 1/18 C. 5/36 D. 2/9 92. Find the distance between the point (3, 2, -1) and the plane 7x – 6y + 6z + 8 = 0. A. 1 B. 2 C. 3 D. 4 SOLUTION: d=

7(3)−6(2)+6(−1)+8

= 1

1√7²+6²+6²

93. How many parallelograms are formed by a set of 4 parallel lines intersecting another set of 7 parallel lines? A. 123 B. 124 C. 125 D. 126 SOLUTION: 𝐦(𝐦−𝟏)𝐧(𝐧−𝟏) 𝟒 [𝟕(𝟕−𝟏)(𝟒)(𝟒−𝟏)] 𝟒

= 𝟏𝟐𝟔

94. The graphical representation of the cumulative frequency distribution in a set of statistical data is called: A. Ogive B. Histogram C. Frequency polyhedron D. mass diagram 95. Find the area bounded by the curve defined by the equation x2 = 8y and its latus rectum. A. 11/3 SOLUTION:

B. 32/3

C. 16/3

D. 22/3

x² = 8y 8

a = 4 = 2, LR = 8 4

x²

A = ∫−4 (2 − 8 ) dx A=

32 3

sq. units

96. Evaluate lim (i z 4 + 3z² − 10i) z→2i

A. -12 +6i SOLUTION: lim(i z 4 + 3z² − 10i)

z→2i

B. 12 - 6i

C. 12 +6i

D. -12 – 6i

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

= i(2i)4 + 3(2i)2 − 10i = i(24 i4 ) + 3(22 i2 ) − 10i = 16i − 12 − 10i = −12 + 16i

97. Naperian logarithm have a base of A. 3.1416 B. 2.171828 SOLUTION:

C. 10

D. 2.71828

e = 2.71828 98. If an aviator flies around the world at a distance 2km above the equator, how many more km will he travel than a person who travels along the equator? A. 12.6 km B. 16.2 km C. 15.8 km D. 18.5 km SOLUTION: 1 rev = 2π (2km)(2π) = 4π = 12.566 or 12.6 km 99. Find the volume of a spherical whose central angle is pi/5 radians on a sphere of radius 6 cm. A. 90.48 cu. cm B. 86.40 cu. cm C. 78.46 cu. cm D. 62.48 cu. cm SOLUTION: θ= V=

π 5

rad , r = 6 cm

πr³θ 270

Vwedge =

π 180 ) 5 π

π(6)3 ∙( x 270

= 90.48 cu. cm 100. What is the coefficient of the (x -1)3 term in the Taylor series expansion of f(x) = lnx expanded about x = 1? A. 1/6 B. 1/4 C. 1/3 D. 1/2

MARCH 2013 ELECTRICAL ENGINEERING PRE-BOARD EXAM

1. If the man sleeps from 6:48 PM up to 7:30 AM. The number of hours and minutes he sleeps is. A. 11 hrs and 42 min B. 12 hrs and 42 min

C. 13 hrs and 42 min D. 10 hrs and 42 min

Solution: 6:48PM – 7:30AM = 12hrs and 42mins

2. The price of a ballpen rises from Php 4.00 to Php 12.00. What is the percent increase in.price? A. 100 percent C. 150 percent B. 120 percent D. 200 percent Solution: 8

12 - 4 = 8; 4 x 100% = 200% 𝜋𝑥

3. Evaluate: lim(2 − 𝑥)^tan( 2 ). 𝑥→1

A. e^(2/pi)

B. e^(pi/2)

C. e^(2pi)

D. 0

Solution 𝜋(0.09999)

180

2

𝜋

(2– 0.09999) tan(

)(

) = 1.89 or e2/𝝅

4. Thirty is 40 percent of what number? A. 60

B. 70

C. 75

Solution: 30 = 40% (X) X = 75

D. 80

5. Roll a pair of dice. What is the probability that the sum of two numbers is 11? A. 1/36 B. 1/9 C. 1/18 D. 1/20 Solution Pair of dice = 2 Possible rolls = 36 Two ways to roll 11 = (5,6) (6,5) 2 𝟏 = 36 𝟏𝟖 6. If the logarithm of MN is 6 and the logarithm of N/M is 2, find the logarithm of M. A. 2 B. 3 C. 4 D. 6 Solution: Log N = 6 -Log M 6 – 2(LogM) = 2 -2Log M = 2- 6 LogM = 2 7. The mean duration of television commercials on a given network is 75 seconds, with a standard deviation of 20 seconds. Assume that duration time are approximately normally distributed. What is the approximate probability that a commercial will last less than 35 seconds? A. 0.055 B. 0.025 C. 0.045 D. 0.035 8. In how many ways can 5 people be lined up if two particular people refuse to follow each other? A. 52 B. 62 C. 72 D. 82 Solution: 5! – 2(4!) = 72

9. Which of the following is not included? A. 0.60

B. 60%

10. Which of the following is not included?

C. 0.06

D. 3/5

A. 0.60 B. 60%

C. 0.06

11. The area of the circle A. 32.25 in B. 33.52 in Solution: 89.24 = 𝜋 𝑟2 R = 5.3351

;

is

D. 3/5

89.42 sq. C. 35.33 in

in.

What is its D. 35.55 in

circumference?

C = 2𝜋 (5.3351) = 33.52in

12. If a truck parks in at 1 PM in a parking lot and leaves at 4 PM. Find the number of hours it stayed at the parking lot A. 1

B. 2

C. 3

D. 4

C. 4

D. 2

13. If (x+3): 10=(3x-2): 8, find 2x-1. A. 1

B. 3

Solution: ((x+3))/10 = ((3x-2))/8 8x+24 = 30x-20 30x-8x = 24+20

22x = 44 x=2 2(2) – 1 = 3

14. Evaluate the Laplace transform of t^n A. n!/s^n C. n!/2s^n

B. n!/s^(n+1) D. n!/2s^(n+1)

Solution: 𝑡 𝑛 𝑒 −𝑠𝑡

∞

Laplace {t^n} = ∫0 𝑡 𝑛 𝑒 −𝑠𝑡 𝑑𝑡 = Let du = ntn-1 ; v = Laplace

{tn}

=

Laplace {tn} =

0-0n 𝑛 𝑠

Laplace {tn} =

−𝑠𝑒 −𝑠𝑡 𝑠 𝑛

𝑠

∞

+ 𝑠 ∫0 𝑡 𝑛−1 𝑒 −𝑠𝑡 𝑑𝑡

ւ [𝑡 𝑛−1 ] ; 𝑡 1 = 𝐧! 𝐬 𝐧+𝟏

1 𝑠2

∞

− ∫0 𝑛𝑡 𝑛−1 −

then s>0

𝑠𝑒 −𝑠𝑡 𝑠

𝑑𝑡

15. Find the volume generated by rotating a circle x^2+y^2+6x+4y+12=0 about the y-axis. A. 58.24

B. 62.33

C. 78.62

Solution: x2+y2+6x+4y+12=0 (x2+6x+9) +(y2+4y+4) =-12+9+4 (x + 3)2 + (y + 2)2 = 1 (x - h)2 + (y - k)2 = r2 r=1 C = (-3, -2)

D. 59.22

By inspection: d = 3 Using second proposition of Pappus V = A x 2πd V = π (1)2 x 2π (3) V = 59.22 cubic units

16.Determine all the values of 1^sqrt. of 2. A. sin (sqrt. of 2 kpi) + icos (sqrt. of 2 kpi) B. cos (sqrt. of 2 kpi) + isin (sqrt. of 2 kpi) C. sin (2sqrt. of 2 kpi) + icos (2sqrt. of 2 kpi) D. cos (2sqrt. of 2 kpi) + isin (2sqrt. of 2 kpi) 17. The slope of the curve y^2-xy-3x=1 at the point (0, -1) is A. -1 B. -2 C. 1 D. 2 Solution: 𝑦+3

Y1 = 2𝑦−𝑥 =

−1+3 2(−1)−0

= -1

18. Express Ten million forty-three thousand seven hundred seventy-one. A. 10,403,771 C. 10,430,771

B. 10,433,771 D. 10,043,771

19. Find the length of the curve r = 8 sin theta. A. 8

B. 4

C. 8 pi

D. 4 pi

Solution: r2 = 64sin2ϴ 𝑑𝑟 (𝑑𝛳)2 = 64cos2ϴ 𝑏

Note: sin2ϴ + cos2ϴ = 1 π L = ∫0 √64(1) 𝑑𝜃 π

𝑑𝑟

L = ∫𝑎 √𝑟 2 + (𝑑𝛳)2 𝑑𝜃

L = ∫0 8 𝑑𝜃 = 8 [ϴ − ϴ]

π

L = ∫0 √64𝑠𝑖𝑛2 ϴ + 64𝑐𝑜𝑠 2 ϴ 𝑑𝜃

L = 8 [π − 0] = 8𝛑

20. A pole which leans 11 degrees from the vertical toward the sun cast a shadow 12 m long when the angle of elevation of the sun is 40 degrees. Find the length of the pole. A. 15.26 m B. 14.26 m C. 13.26 m D. 12.26 m Solution: x = 90 + 11 = 101

Sine Law

B = 180 – (101 +40)

𝑆𝑖𝑛(40)

B= 39

X = 12.26m

𝑋

=

𝑆𝑖𝑛(39) 12

21. How long is the latus rectum of the ellipse whose equation is 9x^2+16y^2-576=0? A. 7 B. 9 C. 10 D. 15 Solution 𝑥2 64

+

𝑦2 36

=1

L.R =

; 2(6)2 8

a=8 b=6

=9

22. If the initial and final temperatures of an object are 97.2 and 99 deg F respectively, find the change in temperature. A. 1.7 deg F B. 1.8 deg F C. 1.9 deg F D. 1.6 deg F Solution: 99 – 97.2 = 1.8 F

23. A rectangular plate 6 m by 8 m is submerge vertically in a water. Find the force on one face if the shroter side is uppermost and lies in the surface of the liquid. A. 941.76 kN C. 3,767.04 kN

B. 1,883.52 kN D. 470.88 kN

Solution: F = (8)(6)(4)(9.81) = 1,883.52 24. Find the area enclosed by the loop y^2 = x(x-1) ^2 A. 8/15

B. 8/17

C. 7/15

D. 7/17

25. The GCF of two numbers is 34, and their LCM is 4284. If one of the number is 204, the other number is A.714 B. 716 C. 2124 D. 3125 Solution: Other Number =

(34)(4284) 204

= 714

26. Jonas, star player of Adamson University has free throw shooting of 83%. The game is tied at 87-87. He is fouled and given 2 free throws. What is the probability that the game will go overtime? A. 0.3111

B. 0.6889

C. 0.0289

D. 0.9711

27. Find the work done in moving an object along a vector a = 31 + 4j if the force applied is b = 21 + j. A. 8

B. 9

C. 10

D. 12

Solution: A = 3i +4j B = 2i+j

;

w = (3) (20) +(4)(1) w = 10

28. If 3z + 5 = 7z-7. Find Z A. 3 B. 5 C. 7 Solution: 3z -7z = -7 – 5 −4𝑧 −4

=

−12 −4

Z=3

D. 9

29. Where does the normal line of the curve y = x - x^2 at the point (1,0) intersect the curve a second time? A. (-3, -12) 30. Simplify

B. (0,0)

C. (-2, -6)

D. (-1, -2)

C. csc2x

D. cot2x

1+tan2 𝑥 1+cot2 𝑥

A. sec2x B. tan2x Solution: 1+tan2 𝑥 1+cot2 𝑥

sec2 𝑥

1

= csc2 𝑥 = cos2 𝑥 =

sin2 𝑥 1

= tan2X

31. Jodi wishes to use 100 feet of fencing to enclose a rectangular garden. Determine the maximum possible area of her garden. A. 850 sq. ft. C. 625 sq. ft.

B. 1250 sq. ft. D. 1650 sq. ft.

32. Simplify 1/(csc x + 1) + 1/(csc x – 1). A. 2 sec x tan x B. 2 csc x cot x C. 2 sec x D. 2 csc x Solution: 2𝑐𝑠𝑐𝑥

2𝑐𝑠𝑐𝑥

2

sin2 𝑥

1/(csc x + 1) + 1/(csc x – 1) = 𝑐𝑠𝑐𝑥−1 = cot2 𝑥 = 𝑠𝑖𝑛𝑥 ∗ cos2 𝑥 = 2secx tanx 33. A certain chemical decomposes exponentially. Assume that 200 grams becomes 50 grams in 1 hour. How much will remain after 3 hours? A. 1.50 grams B. 6.25 grams C. 4.275 grams D. 3.125 grams 34. The locus of a point that moves so that the sum of its distances between two fixed points is constant called: A. a parabola B. a circle C. an elipse D. a hyperbola

35. Michael’s age is seven-tenths of Richard’s age. In four years Michael’s age will be eightelevenths of Richard’s age. How old is Michael? A. 26 yrs. B. 28 yrs.

C. 40 yrs.

D. 48 yrs.

Solution: 7 10

8

x +4 = 11 (x+4)

X = 40 36. A

conic

7

; 10 (40) = 28 section

whose

eccentricity

is

equal

to

one

(1)

is

known

as:

A. a parabola B. an elipse C. a circle D. a hyperbola 37. The angle of a sector is 30 degrees and the radius is 15 cm. What is the area of a sector? A. 59.8 sq. cm. C. 89.5 sq. cm.

B. 58.9 sq. cm. D. 85.9 sq. cm.

Solution: 1

𝜋

A sector = 2 (15)2 (30)( 180) = 58.90 38. In a conic section, if the eccentricity is greater than (1), the locus is: A. a parabola

B. an elipse

C. a circle

39. If f’(x) = sin x and f(pi) = 3, then f(x) = A. 4 + cos x C. 2 – cos x

B. 3 + cos x D. 4 – cos x

D. a hyperbola

40. Two stones are 1 mile apart and are of the same level as the foot of a hill. The angles of depression of the two stones viewed from the top of the hill are 5 degrees and 15 degrees respectively. Find the height of the hill. A. 109.1 m

B. 209.1 m

C. 409.1 m

D. 309.1 m

Solution: 1 mile = 1609.75m ℎ

Tan 15 = 1609.75+𝑋 = eq.1 ℎ

Tan 15 = 𝑥

H = xtan15 = eq. 2 (1606.75+x) tan15 = xtan15 X = 780.425m H = 780.425 (tan15) = 209.11m 41. What is the equation of the line, in the xy-plane, passing through the point (6, 4) and parallel to the line with parametric equations x = 5t + 4 and y = t – 7? A. 5y – x = 14 C. 5y – 4x = -4

B. 5x – y = 26 D. 5x – 4y = 14

42. Evaluate (8+7i) ^2 B. 15 – 112i D. -15 – 112i

A. 15 + 112i C. -15 + 112i Solution:

(8+7i)(8+7i) = 15 + 112i 43. How far is the directrix of the parabola (x-4)^2 = -8(y-2) from the x-axis? A. 2 Solution:

B. 3

1

y = − 8 (𝑥 − 4)2 + 2 1

Where: a = − 8 , b = 1, c = 0 y=k–p 𝑦=

4𝑎𝑐−𝑏 2 −1 4𝑎

y =4

C. 4

D. 1

44. A weight W is attached to a rope 21 ft long which passes through a pulley at P, 12 ft above the ground. The other end of the rope is attached to a truck at a point A, 3 ft above the ground. If the truck moves off at the rate of 10ft/sec, how fast is the weight rising when it is 7 ft above the ground? A. 9.56 ft/sec C. 8.27 ft/sec

B. 7.82 ft/sec D. 6.25 ft/sec

45. The first farm of GP is 160 and the common ratio is 3/2. How many consecutive terms must be taken to give a sum of 2110? A. 5

B. 6

C. 7

D. 8

Solution: 2𝑛

2110 =

160( 1− ) 3 1−3 2

n=5

46. Steve earned a 96% on his first math test, a 74% his second test, and 85% on 3 tests average. What is his third test? A. 82%

B. 91%

C. 87%

D. 85%

Solution: 0.96+0.74+𝑋 3

= 0.85

X = 0.85 * 100 = 85% 47. The base radius of a right circular cone is 4 m while its slant height is 10 m. What is the surface area? A. 124.8 sq. m. C. 226.8 sq. m.

B. 128.6 sq. m. D.125.7 sq. m

Solution: Surface area = 𝜋 (4)(10) = 40𝜋 or 125.66 m2

48. Ian remodel a kitchen in 20 hrs and Jack in 15 hours. If they work together, how many hours to remodel the kitchen? A. 8.6

B. 7.5

C. 5.6

D. 12

Solution:

1

1

+ 15 = 20

1 t

T = 8.6hrs 49. If 15% of the bolts produced by a machine will be defective, determine the probability that out of 5 bolts chosen at random, at most 2 bolts will be defective. A. 0.9754 B. 0.9744

C. 0.9734

D. 0.9724

Solution: 1 – 0.15 = 0.85 P (0) = 0.852 = 0.04437 P (1) = (5) (0.15) (0.85)4 = 0.3915 1

P(2) = (2) (5) (4)(0.15)2(0.85)3 = 0.138178 P (0 or 1 or 2) = 0.9734 50. Find the average rate of the area of a square with respect to its side x as x changes from 4 to 7. A. 9 B. 3 C. 11 D. 18 51. The equations for two lines are 3y – 2x = 6 and 3x + ky = -7. For what value of k will the two lines be parallel? A. -9/2

B. 9/2

C. -7/3

Solution: x2/y2= x1/y1 -3/k= 2/3 k = -9/2 = A.

D. 7/3

52. 5pi/18 rad is how many deg? A. 60

B. 50

C. 30

D. 90

Solution: 5 180 𝜋

(

𝜋

) = 50 deg

53. Find the point of infection of the curve y = x^3 + 3x^2 – 1. A. (-1, 1) B. (-2, 3) C. (0, -10) D. (-3, -1) Solution: Y1 = 3x2 + 6x Y2 = 6x + 6 X = -1

y = (-1)3 + 3(1)2 -1 y=1 P (-1,1)

54. A fair coin is tossed three times. Find the probability that there will appear three heads. A. 1/4

B. 1/2

C. 1/8

D. 1/6

Solution: You have a fair coin: this means it has a 50% chance of landing heads up and a 50% chance of landing tails up. pH=pT=1/2 pHxpTxpH=1/2×1/2×1/2 = 1/8 = C. 1

1

𝟏

P3H = C(3,3) (2)3 (2)3-3 = 𝟖 55. A spherical balloon inflated with r = 3(cube root of t) as t is greater than zero and t is less than equal or equal to 10. Find the rate of change of volume in cubic cm at t = 8. A. 37.70 Solution:

B. 150.80

C. 113.10

r= 3 (t) 1/3 ; @ t=8: r= 3 (8) 1/3 = 6 r’= 3 (1/3) t -2/3 ; @ t=8; r’= 8-2/3 = ¼ v= 4/3pi r3 v’= 4pi r2r’ = 4pi (6)2(1/4) v’= 113.10 = C.

D. 75.40

56. Joe and his dad are bricklayers. Joe can lay bricks for a well in 5 days. With his father’s help, he can build it in 2 days. How long would it take his father to build it alone? A. 3-1/4 days C. 2-1/3 days Solution: 1

B.3-1/3 days D.2 -2/3 days

1

2((5 + 𝑥)) = 1

x = 3.33 = 3 -

𝟏 𝟑

days

57. Find x so that the line containing (x, 5) and (3, -4) has a slope of 3. A. 3 B. 4 C. 5 D. 6 Solution: 3=

5−(−4) 𝑋−3

;x=6

58. Find the length of the chord of a circle of radius 20 cm subtended by a central angle of 150 degrees. A. 49 cm B. 42 cm C. 39 cm D. 36 cm Solution: COSINE LAW C = √202 + 202 − 2 (20)(20)cos(15) C = 38.64 or 39 59. Find the area of the ellipse 4x^2 + 9y^2 =36. A. 15.71

B. 18.85

C. 21.99

Solution: A = 2 and b = 3 A = 𝜋 (2) (3) = 18.85

D. 25.13

60. Convert Cartesian coordinates (9, -9, 2) into cylindrical coordinates. A. (-9sqrt. of 2, pi/4, 2) B. (9sqrt. of 2, pi/4, 2) C. (-9sqrt. of 2, 7pi/4, 2) D. (9sqrt. of 2, 7pi/4, 2) Solution: X = r = √92 + −92 = 9√𝟐 Y= tan-1 (

−9 9

𝟏

)=-𝟒𝝅

Z=2 Rectangular Coordinates: 9, -9, 2 r = sqrt(x2+y2) r = sqrt((9)2+(-9)2) r = 9 sqrt 2 Ɵ = tan-1 (y/x) Ɵ = tan-1 (-9/9) Ɵ = -45 = -45+360 = 315 degrees = 7pi/4 rad z=2 Cylindrical Coordinates (9sqrt. of 2, 7pi/4, 2) = D. 61. The area of a square is 32 square feet. Find the perimeter of the square. A. 27. 71 feet B. 55. 43 feet C. 45. 25 feet D. 22.63 feet Solution: √𝟑𝟐 = √𝒂𝟐 a = 4 √2

P = 4(4√2 ) = 22.63

62. If cos theta = -3/4 and tan theta is negative, the value of sin theta is A. -4/5 B. – (sqrt. of 7)/4 C. (4 sqrt. of 7)/7 D. (sqrt. of 7)/4 Solution: 3

𝜽 = cos-1 ( - 4 ) = 2.42

; sin𝜃 = sin (2.42) = 0.66 or

√7 4

63. What is the numerical coefficient of the term containing x^3y^2 in the expansion of (x+2) ^5? A. 10 B. 20 C. 40 D. 80 Solution: 5c(x)(1)5-x (2)x = 5c(2)(1)3 (2)2 = 40 64. Find the area bounded by y = 6x – x^2 and y = x^2 -4x. A. 125/3

B. 125/2

C. 100/3

D. 100/9

Solution: 5

∫0 ( 𝑥 − 2𝑥 2 + 10𝑥 ) dx

6x – x2 = x2 – 4x X2 – 10x = 0

-

2 (5)3 3

+

10 (5)2 2

X = 0 and (x-5) =0 X=5

= 41.67 or

𝟏𝟐𝟓 𝟑

65. Find the second derivative of y = x ln x. A. x

B. 1/x

C. 1

D. x squared

C.92.8

D. 98.2

Solution: 1

Y1 = x ( 𝑥 ) + ln x 𝟏

Y2 = 0 + 𝒙 66. What is 30% of 293? A. 87.9

B. 89.7

Solution: (293) (0.30) = 87.9

67. The height (in feet) at any time t (in seconds) of a projectile thrown vertically is h(t) = 16t^2 + 256t. What is the projectile’s average velocity for the first 5 seconds of travel? A. 48 fps

B. 96 fps

C. 176 fps

D. 192 fps

Solution: H(t) =

16 (5)2 +256 (5) 5

= 176 fps

68. Find the general solution of y” + 6y’ + 9y = x+ 1. A. y = (C1x + C2x2) e-3x + 1/27 + x/9 C. y = (C1x + C2x2) e3x + 1/27 + x/9 B. y = (C1 + C2x) e-3x + 1/27 + x/9 D. y = (C1 + C2x) e3x + 1/27 + x/9 69. For a complex number z = 3 + j4 the modulus is A. 3 B. 4 C. 5

D. 6

Solution: X = √𝑎2 + 𝑏 2 = √32 + 42 = 5 70. Evaluate lim

sqrt.of (x2 −9)

x →3

2𝑥−6

A. 3

B. 0

C. infinity

D. Undefined

Solution: 2𝑥 2√(𝑥 2 −9) (2)

=∞

71. The probability that a man, age 60, will survive to age 70 is 0.80 the probability that a woman of the same age will live up to age 70 is 0.90. What is the probability that only one of the survives? A. 0.72 B. 0.26 C. 0.28 D. 0.0

72. Simplify 1(sec theta -1) + 1/ (sec theta + 1). A. 2 sec theta tan theta C. 2 sec theta B. 2 csc theta cot theta D. 2 csc theta Solution: 1 sec2 𝜃−1

=

2 𝑐𝑜𝑠𝜃 tan2 𝜃

=

2 𝑐𝑜𝑠𝜃

*

cos2 𝜃 sin2 𝜃

= 2csc𝜽 𝒄𝒐𝒕𝜽

73. Find the base of an isosceles triangle whose vertical angle is 65 degrees and whose equal sides are 415 cm. A. 530 cm

B. 464 cm

C. 350 cm

Solution: Cosine Law B = (415)2 (415)2 -2(415) (415) cos65 B = 446 74. Find the general solution of y” + 10y = 0. A. y = C1 cos (sqrt. of 10x) + C2 sin (sqrt. of 10x) B. y = C1 cos (sqrt. of 5x) + C2 sin (sqrt. of 5x) C. y = C cos (sqrt. of 10x) D. y = C sin (sqrt. of 10x) 75. Evaluate the inverse Laplace transform of 6 over (s^2 + 4). A. 3 sin 2t C. 3 sinh 2t

B. 3 cos 2t D. 3 cosh 2t

Solution: 6 𝑠 2 +4

6

1

2

𝑠 2 +22

= ∫

=

𝑏 𝑠 2 +𝑏2

= 3sin2t

D. 446 cm

76. Evaluate L {sin t cos t} A. 1/2 (s^2 + 4) C. 1/ (s^2 + 1)

B. 1/ (s^2 + 4) D. 1/2 (s^2 + 1)

Solution:

L ( sint cost) =(𝑠2

1 +1 )2

=

𝟏 𝒔𝟐 +𝟒

77. Determine the moment of inertia of the area enclosed by the curved x^2 + y^2 = 36 with respect to the line y = 8. A. 8628 B. 8256 C. 7642 D. 7864 78. A man sleeps on Monday, Tuesday, Wednesday, Thursday and Friday for 8, 6, 7, 4, and 5 hours, respectively. Find the number of hours he slept for 5 days. A. 35 B. 31 C. 30 D. 25 Solution: 8 + 6 + 7 + 4 + 5 = 30 79. Find A fir which y = Ae^x will satisfy y” - 2y’ = 4e^x. A. -1 B. -2 C. -3 Solution: Aex -2 (Aex ) – Aex = 4ex Aex (1- 2- 1 ) = 4ex A=-2 80. Simplify 1/csc2 theta. A. sin2 theta C. cot2 theta

B. cos2 theta D. tan2 theta

Sin2𝜃 =

1 csc2 𝜃

=

1 1 sin2 𝜃

D. -4

81. Timothy leaves home for Legaspi City 400 miles away. After 2 hours, he has to reduce his speed by 20 mph due to rain. If he takes 1 hour for lunch and gas and reaches Legaspi City 9 hours after left home, what was his initial speed? A. 63 mph B. 62 mph C. 65mph D. 64 mph 82. How many arrangements of the letters in the word “VOLTAGE” begin with a vowel and end with a consonant? A. 1490 B.1440 C.1460 D.1450 Solution: 3! (4!) (10) = 1440 83. An airplane flying with the wind, took 2 hours to travel 1000 km and 2.5 hours in flying back. What was the wind velocity in kph? A. 50

B. 60

Solution: 100 2

–x=

C. 70

1000 2.5

D. 40

+x

X = 50 mph 84. A woman is paid $ 20 for each day she works and the forfeits $ 5 for each day she is idle. At the end of 25 days she nets $ 450. How many days did she work? A. 21

B. 22

C.23

D.24

Solution: P/day = $20 – 5 = $15 20x – 5 = 450 X = 22.75 or 23days 85. Find the centroid if the solid formed by revolving about x = 2 bounded by y = x^3, X = 2 and y = 0. A. (2, 10/30) B. (2, 10/7) C. (2, 10/9) D. (2, 10) 86. What is the lowest common factor of 10 and 32? A. 320 B. 2 C. 180

D. 90

87. The positive value of k which make 4x^2 – 4kx + 4k + 5 a perfect square trinomial is A. 6 B. 5 C. 4 D. 3 88. A tree is broken over by a windstorm. The tree was 90 feet high and the top of the tree is 25 feet from the foot of the tree. What is the height of the standing part of the tree? A. 48.47 ft B. 41.53 ft C. 45.69 ft D. 44.31 ft 89. The Rotary Club and the Jaycee Club had a joint party. 120 members of the Rotary Club and 100 members of the Jaycees Club also attended but 30 of those attended are members of both clubs. How many persons attended the party? A. 190

B. 220

C. 250

D. 150

Solution: 120 -x + x + 100 – x = 30 X = 190 90. If sin 3A = cos 6B, then A. A + B = 90 deg B. A + 2B = 30 deg

C. A + B = 180 deg D. A +2B = 60 deg

Solution: Cos6B = sin (30 – 6B) Sin3A = Sin (90 – 6B) 3𝐴 3

=

90−6𝐵 3

A = 30 – 2B or A +2B = 30 91. MCM is equivalent to what number? A. 1000 B. 2000 Solution: M = 1000 C= 100 MCM = 1000 + (1000-100) = 1900

C. 1800

D.1900

92. What is the discriminant of the equation 5x^2 – 6x + 1 = 0? A. 12 B. 20 C. 16

D. 18

Solution: a=5 b = -6 c=1 D = (-6)2 – 4(5)(1) = 16 93. The number of ways can 3 nurses and 4 engineers be seated in a bench with the nurses seated together is A. 144 B.258 C. 720 D. 450 Solution: N = Total no. of ways N = (3!)(4!)(No. of patterns) N = (3!)(4!)(5) N = 720 ways 94. Find the distance from the plane 2x + y – 2z + 8 = 0 to the point (-1, 2, 3). A. 1/3

B. 2/3

C. 4/3

D. 5/3

Solution:

D=

2(−1)+(2)−(2)(3)+8 √22 +12 +22

=

2 √9

=

𝟐 𝟑

95. Find the value of x if log x base 12 = 2. A. 144

B. 414

C. 524

D. 425

C. -5

D. -1

Log12 x = 2 X = 122 = 144 96. If f(x) = x^3 – 2x – 1, then f (-2) = A. -17

B. -13

Solution: X3 – 2x – 1 = 0 F (-2) = (-2)3 – 2(-2) -1 = - 5

97. A particle moves along a line with acceleration 2 + 6t at time t. When t = 0, its velocity equals 3 and it is at position s = 2. When t =1, it is at position s = A. 2 B. 5 C. 6 D. 7 Solution: @t = 0 A = 2 +6(0) A=2

@t = 1 A = 2 + 6(1) A=8

at = 10 S = 10 - 3 = 7

98. The edge of a cube has length 10 in., with a possible error of 1 %. The possible error, in cubic inches, in the volume of cube is A. 3 B. 1 C. 10 D. 30 Solution: v = s3 dv/ds = 3s2 dv/v = (3s2ds)/s3 =3 99. What is the rate of change of the area if an equilateral triangle with respect to its side s when s = 2? A. 0.43 B. 0.50 C.10 D. 1.73 Solution: A=

1 4

s2 √3 ;

𝑑𝑎 𝑑𝑠

=

1 2

s √3

@s=2 𝑑𝑎 𝑑𝑠

100.

=

1 2

(2)(√3 ) = √𝟑 or 1.73

If ∫ ˥ f(x)dx = 4 and ∫ ˥ g(x)dx = 2, find ∫ ˥ [3f(x) + 2g (x) + 1]dx. A. 22

B. 23

C. 24

D. 25

Solution: 7

7

7

∫1 𝑓(𝑥)𝑑𝑥 = 4 ∫1 𝑔(𝑥)𝑑𝑥 = 2 ∫1 (3𝑓(𝑥) + 2𝑔(𝑥) + 1)𝑑𝑥 = 4 = 3(4) + (2)(2) + (7-1) = 22

AUGUST 2013

1

1

1. Simplify (csc 𝑥+1) + (csc 𝑥−1) A. 2 sec x tan x

B. 2 csc x cot x

C. 2 sec x

D. 2 csc x

Solution: (𝑐𝑠𝑐𝑥−1)+(𝑐𝑠𝑐𝑥+1) (𝑐𝑠𝑐𝑥+1)(𝑐𝑠𝑐𝑥−1)

=

2𝑐𝑠𝑐𝑥 𝑐𝑠𝑐 2 𝑥−1

=

2𝑐𝑠𝑐𝑥 𝑐𝑜𝑡 2 𝑥

2𝑐𝑠𝑐𝑥

=( 𝑐𝑜𝑠2𝑥 ) = 𝑠𝑖𝑛2 𝑥

2𝑠𝑖𝑛𝑥 𝑐𝑜𝑠 2 𝑥

= 2(

𝑠𝑖𝑛𝑥

𝑐𝑜𝑠𝑥

)(

1

𝑐𝑜𝑠 2 𝑥

)=

𝟐 𝐬𝐞𝐜 𝒙 𝐭𝐚𝐧 𝒙 2. A bus leaves Manila at 12NN for Baguio 250 km away, traveling an average of 55 kph. At the same time, another bus leaves Baguio for Manila traveling 65kph. At what distance from Manila they will meet? A. 135.42 km B. 114.56km C. 129.24km D. 181.35km Solution: T R x 55 x 65 55x + 65x = 250

D 55x 65x D = TR

120x = 250

D = (2.0833)(55)

x = 2.0833

D = 114.56 km

3. Simplify (cos β -1)(cos β+1) A. -1/sin2β B. -1/cos2β

C. -1/csc2β

D. -1/sec2β

C. 2 csc x

D, 2 sin x

Solution: cos2β – 1 (1/ csc2 β)2 – 1 = -1/ csc2β 4. Simplify 1/(csc x + cot x) + 1 /(csc x – cot x). A. 2 cos x B. 2 sec x Solution: 𝑐𝑠𝑐𝑥−𝑐𝑜𝑡𝑥+𝑐𝑠𝑐𝑥+𝑐𝑜𝑡𝑥 (𝑐𝑠𝑐𝑥+𝑐𝑜𝑡𝑥)(𝑐𝑠𝑐𝑥−𝑐𝑜𝑡𝑥)

=

2𝑐𝑠𝑐𝑥 𝑐𝑠𝑐 2 𝑥−𝑐𝑜𝑡 2 𝑥

=

2𝑐𝑠𝑐𝑥 1 1 − 𝑠𝑖𝑛2 𝑥 𝑡𝑎𝑛2 𝑥

= 2cscx

5. From past experience, it is known 90% of one year old children can distinguish their mother’s voice from the voice of a similar sounding female. A random sample of 20 one year’s old are given this voice recognize test. Find the probability that all 20 children recognize their mother’s voice. A. 0.122 B. 1.500 C. 1.200 D. 0.222 Let X - number of children who recognize their mother’s voice X has Binomial distribution (n=20, p= 0.90) E(X)=m= np= 20* 0.90=18 P(x = 20) = P(x ≤ 20) – P(x ≤ 19) = = 1 – 0.878 = 0.122 6. Find the differential equation of the family of lines passing through the origin. A. xdx – ydy = 0 C. xdx – ydy = 0 B. xdy – ydx = 0

D. ydx – xdy = 0

Solution Let y = mx be the family of lines through origin. Therefore, dy dx = m Eliminating m, x dy – ydx = 0. 7. A chord passing through the focus of the parabola y2 = 8x has one end at the point (8, 8). Where is the other end of the chord? A. (1/2, 2) B. (-1/2, -2) C. (-1/2, 2) D. (1/2, -2) 8. Find the radius of the circle inscribed in the triangle determined by the line 2

y= x+4, y= -x -4, and y = 7x + 2. A. 2.29

B. 0.24

C. 1.57

D. 0.35

9. What would happen to the volume of a sphere if the radius is tripled? A. Multiplied by 3 C. Multiplied by 27 B. Multiplied by 9 D. Multiplied by 6 Solution: V1/V2 = (r1/ r2)3 = (r1/ 3r1)3 Therefore: V2= 27V1 10. Six non- parallel lines are drawn in a plan. What is the maximum number of point of intersection of these lines? A. 20 B. 12 C. 8 D. 15

Solution: 𝑁(𝑁−1) 2

=

N=6 6(6−1) 2

= 15

11. In a triangle ABC where AC=4 and angle ACB=90 degrees, an altitude t is drawn from C to the hypotenuse. If t = 1, what is the area of the triangle ABC? A. 1.82 B. 1.78 C. 2.07 D. 2.28 Solution: Using sine law: (4/sin45) = (x/sin90) X=AB=4.2 Side CB= sq.rt of (4.2^2-4^2) CB= 1.289 Area=(1/2)(b)(h)sin theta = (1/2)(1.289)(4)sin90 = 2.07 12. In a 15 multiple choice test questions, with five possible choices if which only on is correct, what is the standard deviation of getting a correct answer? A.1.55 B. 1.65 C. 1.42 D. 1.72 Solution: square root of [15×(1/5)×(4/5)] = 1.55 13. What is the area bounded by the curve y = tan 2 x and the lines y = 0 and x = pi/2? A. 0 B. infinity C. 1 D. Ɵ

14. What is the power series of (e^x)/(1-x) about x = 0? A. 1-2x+(5/2)x^2-(8/3)x^3 C. 2x-(5/2)x^2+(8/3)x^3 B. 1+2x+(5/2)x^2+(8/3)x^3 D. 2x+(5/2)x^2+(8/3)x^3 Solution: CnX^n = Co+C1X+C2X^2+....CnX^n = 1+C1(X-0)+C2(X-0)^2+C3(X-0)^3 = 1+2X+(5/2)X^2+(8/3)X^3

15. What is the vector which is orthogonal both to 9i + 9j and 9l + 9k? A. 81l + 81j – 81k C.81l - 81j + 81k B. 81l – 81j – 81k D.81l+81j – 81k

16. 24 is 75 percent of what number? A. 16 B. 40 Solution: 32×0.75 =24 Therefore 24 is 75 percent of 32 Ans. =32

C. 36

D. 32

17. Evaluate lim (x^2-4)/(x-4), when X is approaches to 4. A. 4 B. 2 C. 16 Solution: (x^2-4)/(x-4) The derivative of the numerator is 2x The derivative of the denominator is 1 Therefore, 2x/1 =2(4)/1 =8

D. 8

18. If sin A = and cot B = 4, both in Quadrant III, the value of sin ( A + B) is A. -0.844 B. 0.844 C. -0.922

D. 0.922

Solution: sin( A + B ) = (-4/5) (4/) + ( 3/5 ) (1/) = 0.922 19. A fence of 100 m perimeter such that its width is 6m less than thrice its length. Find the width? A. 28 m B. 14 m C. 36 m D. 40 m Solution: P=100m W=3L-6 P=2(W+L) 100=2(3L-6+L) L=14 Therefore, W=3(14)-6 W=36 20. Evaluate log (2 – 5i) A. 0.7 – 0.5i

B. -0.7 + 0.5i

C. 0.7 + 0.5i

D. -0.5 – 0.7i

21. An air balloon flying vertically upward at constant speed is situated 150m horizontally from an observer. After one minute, it is found that the angle of elevation from the observer is 28 deg 50 min. what will be then the angle of elevation after 3 minutes from its initial position? A. 48 deg B. 56 deg C. 61 deg D. 50 deg 22. If m is jointly proportional to G and x, where a,b,c and d are constant. Therefore. A. M = aG + bx C. m = aG B. m = aGz D. m = bG 23. In how many ways can a student going to abroad accompanied by 3 teachers selecting from 6 teachers? A. 16 B. 24 C. 20 D. 12 Solution: Permutation Using calculator(6-shift-divide sign(nCr)-3) 6C3=20 24.If a man travels 1 km north, 3 km west, 5 km south, and 7 km east, what is his resultant displacement vector? A. 5.667 km, 45 deg above + x-axis C. 5.667 km, 225 deg above + x-axis B. 5.667 km, 45 deg above – x-axis D. 5.667 km, 225 deg above – x-axis Solution: N 3km

W

E

1km 5km Resultant vector b

S 7km a=7km-3km=4km b=5km-1km=4km c=? resultant vector Using Pythagorean theorem C2=42+42 =5.6568 km, 225 deg above – X axis

25. What is the general solution of (D4 – 1) y(t) = 0?

a

A. y = c1Ɵt + c2Ɵ-t +c3 cost + c4 sint B. y = c1Ɵt + c2Ɵt +c3 Ɵ-t + c4t Ɵ-t

C. y = c1Ɵt + c2Ɵ-t D. y = c1Ɵt + c2tƟt

Solution: It is a homogeneous linear differential equation of IV order with constant coefficients. The corresponding auxiliary equation is m4 + 1 = 0, whose roots are the four complex 4th roots (-1) = cost + isint 26. Marsha is 10 years older than John, who is 16 years old. How old is Marsha? A. 24 yrs. B. 26 yrs. C. 6 yrs. D. 12 yrs. Solution: Marsha: 10 + age of john (x) John(x): 16 y.o Marsha = 10 + 16 = 26 yrs. 27. Seven times a number x increased by 2 is expressed as A. 7(x + 20 B. 2x + 7 C. 7x + 2

D. 2(x + 7)

28. The plane rectangular coordinate system is divided into four parts which are known as: A. octants B. quadrants C. axis D. coordinates 29. A student already finished 70% of his homework in 42 minutes. How many minutes does she still have to work? A. 18 B. 15 C. 20 D. 24 Solution: Equation; 0.70 x total time(t) = 42min Total time(t) = 60min 60 – 42 = 18min 30. In how many ways can 5 people be lined up to get on a bus, if a certain 2 persons refuse to follow each other? A. 36 B. 48 C. 96 D. 72 Solution: Using calculator 3!(3)(4)= 72 31. Water is being pumped into a conical tank at the rate of 12 cu.ft/min. The height of the tank is 10 ft and its radius is 5ft. How fast is the water level rising when the water height is 6ft? A. 2/3 pi ft/min

B. 3/2 pi ft/min

C. ¾ pi ft/min

D. 4/3 pi ft/min

32. Write the equation of the line with x-intercept a = -1, and y intercept b = 8 A. 8x + y – 8 = 0

C. 8x + y + 8 = 0

B. 8x – y + 8 = 0

D. 8x – y – 8 = 0

Solution: x/a+y/b=1

8x – y = -8

x / -1 + y / 8 = 1

8x – y + 8 = 0

33. In a single throw of pair of dice. Find the probability that sum is 11. A. 1/12

B. 1/16

C.

1/36

D. 1/18

Solution: P = no. of successful trials / total no. of trials Total no. of trials = 36

P = 2 / 36

No. of trials w/ sum 11 = 2

P = 1 /18

34. Find the area bounded by one arch of the companions to the cycloid x = a theta, y = a (1cos theta) and the y-axis. A. 2pi a^2

B. 4pi a^2

C. pi a^2

D. 3pi a^2

35. A rectangular plate 6m by 8m is submerged vertically in a water. Find the force on one face if the shorter side is uppermost and lies in the surface of the liquid. A. 941.76 kN

B. 1,583.52 kN

C. 3,767.04 kN

D. 470.88 kN

36. Michael is four times as old as his son Carlos. If Michael was 18 years old when Carlos was born, how old is Michael now? A. 36 yrs.

B. 20 yrs.

C. 24 yrs.

D. 32 yrs.

Solution: Given:

Then

x + 4x = x + x + 18

X – Carlo’s age

x – Carlo’s age was born

5x = 2x + 18

4x – Michael’s age

x + 18 – Michael’s age

x=6

Substitute value of x=6 to x + 18: x + 18 = 24 yrs. 37. In polar coordinate system the distance from a point to the pole is known as: A. polar angle

C. X-coordinates

B. radius vector

D. Y-coordinates

38. A certain man sold his ballot at Php 1.13 per piece. If there 100 balots sold all in all, how much is his total collection? A. Php 113.00

B. Php 115.00

C. Php 112.00

D. 116.00

Solution: X = 1.13(100) = Php 113.00

39. A certain population of bacteria grows such that its rate of change is always proportional to the amount present. It doubles in 2 years. If in 3 years there are 20,000 of bacteria present, how much is present initially? A. 9,071

B. 10.071

C. 7,071

D. 8,071

Solution: 1

Q=2𝑄𝑜

Q = 𝑄𝑜 22𝑡

Q = 𝑄𝑜 𝑒 𝑟𝑡

20000 = 𝑄𝑜 (2)3/2

2𝑄𝑜 = 𝑄𝑜 𝑒 2𝑟

𝑄𝑜 = 20000 / (2)3/2

2 = (𝑒 2𝑟 )1/2

𝑸𝒐 = 7,071

𝑒 𝑟 = 21/2

40. In throwing a pair of dice, what is the probability of getting of 5? A. 1/36

B. 1/9

C. 1/16

D. 1/6

Solution: P = no. of successful trials / total no. of trials Total no. of trials = 36

P = 4 / 36

No. of trials 5 = 4

P=1/9

41. What is the distance between at any point P(x ,y) on the ellipse b2x2 + a2y2 = a2b2 to its focus. A. by ±ax B. b ± ay C. ay ± bx D. a ± ex

42. Calculate the eccentricity of an ellipse whose major axis and latus rectum has length of 10 and 32/5, respectively. A. 0.4 B. 0.5 C. 0.8 D. 0.6 43. Evaluate (3 + j4)(3 – j4) A. 9 – j16 B. 9 + j16 Solution:

C. 25

D. 36

9-j12+j12-j216 = 9+16 = 25 44. What is the area bounded between y = 6x^2 and y = x^2 + 7? A. 9 B. 10 C. 11 Solution: x2

+7=

6x2

D. 12

7 5

√

∫− 7(𝑥 2 + 7) − (6𝑥 2 )𝑑𝑥 √

5

x2 - 6x2 +7 = 0

= 11

7 = 5x2 7

±√5 = x

45. Two vertical poles are 10 m apart. The poles are 5 m and 8 m, respectively. They are to be stayed by guy wires fastened to a single stake on the ground and attached to the tops of the poles. Where should the stake be placed to use the least amount of wire? A. 6.15 m from 5 m pole C. 6.51 m from 5 m pole

B. 6.15 m from 8 m pole D. 6.51 m from 8 m pole Solution: x = ab/ b + c a – x = 10 – 3.85 x = 10(5)/ 8 + 8 = 6.15m from 8m pole x = 3.58 46. A and B are points on circle Q such that triangle AQB is equilateral. If AB = 12, find the length of arc AB. A. 15.71 B. 9.42 C. 12.57 D. 18.85 47. The area under the portion of the curve y = cosx from x = 0 to x = pi/2 is revolved about the x-axis. Find the volume of the solid generated. A. 2.47 B. 2.74 C. 3.28 D. 3.82 48. Find the length of arc of r = 2/(1 +costheta) from theta = 0 to theta = pi/2. A. 2.64 B. 3.22 C. 2.88 D. 3.49 49. Find the equation of the straight line which passes through the point (6, -3) and with an angle of inclination of 45 degrees. A. x + y = 3 B. 4x – y =27 C. x- 2y = 12 D. x – y = 9 Solution: m = tan Ɵ (y-y1) = m (x-x1) = tan 45

(y+3)=1(x-6)

m=1

x–y=9

50. The equation of the directrix of the y^2 = 6x is A. 2x – 3 = 0 B. 2x + 3 = 0 C. 3x – 2 = 0 Solution: 4a = 6

(x + 3/2 = 0)2

a = 3/2

2x + 3 = 0

51. Find the area bounded by r = 4(sq.rt. of cos 2 theta). A. 16 B. 8 C. 4 Solution: 𝜋 𝜋 (𝑟 = 4√𝑐𝑜𝑠2𝜃)2 −4 < 𝜃 < 4 𝑟 2 = 16𝑐𝑜𝑠2𝜃 1 𝐴 = 2 ∫ 𝑟 2 𝑑𝑟 𝐴=

1 2

𝜋 4 𝜋 − 4

∫ 16𝑐𝑜𝑠2𝜃𝑑𝜃 𝜋

𝐴 = 8 ∫ 4𝜋 𝑐𝑜𝑠2𝜃 2𝑑𝜃 − 4

D. 3x + 2 = 0

D. 12

𝜋

𝐴 = 4 ∫ 4𝜋 𝑐𝑜𝑠2𝜃 2𝑑𝜃 −

4

𝜋 4

𝐴 = 4 sin 2𝜃]

−

𝜋

𝜋

𝜋 4

𝜋

= 4 sin(2 4 ) − 2 sin(2 (− 4 )) 𝜋

𝐴 = 4 sin( 4 ) − 4 sin (− 4 ) = 4(1) − 4(−) 𝑨=𝟖 52. In an arithmetic progression whose first term is 5, the sum of 8 terms is 208. Find the common difference. A. 3 B. 4 C. 5 D. 6 Solution: 𝑛 𝑆 = 2 [2𝑎1 + (𝑛 − 1)𝑑] 8

208 = 2 [2(5) + (8 − 1)𝑑] 𝒅=6 53. If 3x = 7y, then 3x2/7y2 = ? A. 1 B. 3/7 Solution: 𝟕𝒚 𝒙= 𝟑 𝟑𝒙𝟐 𝟕𝒚𝟐

=

𝟕𝒚 𝟑 𝟕𝒚𝟐

𝟑( )𝟐

=

𝟐𝟏 𝟗

C. 7/3

D. 49/9

𝟕

=𝟑

54. What is the area of the ellipse whose eccentricity is 0.60 and whose major axis has a length of 6? A. 40.21 B. 41.20 C. 42.10 D. 40.12 Solution: 2𝑎 = 6 𝑎=3 𝑐 𝑒=𝑎 𝑐 = .6 ∗ 3 = 1.8 𝑏 = √𝑎2 − 𝑐 2 𝑏 = √32 − 1.82 = 2.4 𝐴 = 𝜋𝑎𝑏 𝐴 = 𝜋(3)(2.4) 𝐴 = 22.61 55. Tickets to the school play sold at $4 each for adults and $1.50 each for children. If there were four times as many adult’s tickets sold as children’s tickets, and the total were $3500. How many children’s tickets were sold? A. 160 B. 180 C. 200 D. 240 Solution:

𝑥 = 𝑎𝑑𝑢𝑙𝑡 ; 𝑦 = 𝑐ℎ𝑖𝑙𝑑𝑟𝑒𝑛 𝑦 = 1100 − 𝑥 4𝑥 + 𝑦 = 3500 4𝑥 + (1100 − 𝑥) = 3500 𝑥 = 800 𝑦 = 1100 − 800 𝑦 = 300 4(800) + 300 = 3500 300 = 𝟐𝟎𝟎 1.5 56. If the line kx + 3y + 8 = 0 has a slope of 2/3, determine k. A. -3 B. -2 C. 3 Solution: 3𝑦 = −𝑘𝑥 − 8 −𝑘𝑥−8 𝑦= 3 −𝑘𝑥

𝑦= 𝑚= 2

3 −𝑘

D. 2

8

−3

3 −𝑘

= 3 𝒌 = −𝟐 3

57. The Rotary Club and the Jaycees Club had a joint party. 120 members of the Rotary Club attended and 100 members of the Jaycees Club also attended but 30 of those who attended are members of both parts. How many persons attended the party? A. 190 B. 220 C. 250 D. 150 Solution: 120 + 100 = 220 220 − 30 = 𝟏𝟗𝟎 58. Find the value of k for which the graph of y = x^3 + kx^2 + 4 will have an inflection point at x = -1. A. 3 B. 4 C. 2 D. 1 Solution: 𝑦 ′ = 3𝑥 2 + 𝑘𝑥+ 0 𝑦 ′′ = 6𝑥 + 2𝑘 2𝑘 = −6𝑥 𝑘 = −3𝑥 𝑘 = −3(−1) 𝒌=𝟑

59. Solve for x if log4x = 5. A. 2048 B. 256 Solution: 45 = 𝑥

C. 625

D. 1024

𝒙 = 𝟏𝟎𝟐𝟒 60. An observer wishes to determine the height of a tower. He takes sights at the top of the tower from A and B, which are 50 ft apart at the same elevation on a direct line with the tower. The vertical angle at point A is 30 degrees and at point B is 40 degrees. What is the height of the tower? A. 85.60 ft B. 143.97 ft C. 110.29 ft D.92.54 ft Solution: ℎ 𝑡𝑎𝑛 𝜃 = 𝑥 ℎ

ℎ

𝑡𝑎𝑛 30 = (50+𝑥)

𝑡𝑎𝑛 40 = 𝑥

𝑡𝑎𝑛 30(50 + 𝑥) = 𝑡𝑎𝑛 40𝑥 50+𝑥 𝑥

𝑡𝑎𝑛40

= 𝑡𝑎𝑛30

𝑥 = 110.29 ℎ = 𝑡𝑎𝑛 40𝑥 ℎ = 𝑡𝑎𝑛 40(110.29) 𝒉 = 𝟗𝟐. 𝟓𝟒 𝒇𝒕.

62. If four babies are born per minute, how many babies are born in one hour? A. 230 B. 250 C. 240 D. 260 𝑀=

4 𝑚𝑖𝑛 𝑥 1 min 𝑥 60 = 𝟐𝟒𝟎 𝒃𝒂𝒃𝒊𝒆𝒔 min ℎ𝑟

63. What was the marked price of a shirt that sells at P 225 after a discount of 25%? A. P 280 B. P 300 C. P 320 D. P 340 x - 0.25 x = 225 x = P300

64. Which number is divisible by both 3 and 5? A. 275 B. 445 C. 870

D. 955

870 3 870 5

= 290 = 174 : (3, 5)

65. If s = t^2 – t^3, find the velocity when the acceleration is zero A. 1/4 B. 1/2 C. 1/3

D. 1/6

S = t2 – t1 find when a = 0 𝑣= 𝑎=

ds

= 2t - 3t2

dt ds" d"t

@ a=0

= 2 – 6t 1

a = 2 – 6t

1

𝑣 = 2 (3) − 3 (3)² 𝑣=

𝒕 =

𝟏

𝟏 𝟑

𝟑

66. Find k so that A = (3, -2) and B = (1, k) are parallel A. 3/2 B. -3/2 C. 2/3 A = ( 3, -2) B = ( 1 , k ) 1 𝑘 = −2 3 k=

D. -2/3

an parallel

−𝟐 𝟑

67. A lady gives a dinner party for six guest. In how many may they be selected from among 10 friends? A. 110 B. 220 C. 105 D. 210 r=6 n = 10 P = 10 C6 =

𝟏𝟎! (𝟏𝟎−𝟔)!(𝟔)!

= 𝟐𝟏𝟎 𝒘𝒂𝒚𝒔

68. A wheel 4 ft in diameter is rotating at 80 rpm. Find the distance (in ft) traveled by a point on the rim in 1 s. 9.8 ft B. 19.6 ft C. 16.8 ft D. 18.6 ft d = 4ft

v = 80

𝐬𝐞𝐜 𝐦𝐢𝐧

𝟏𝒎𝒊𝒏

𝟒

𝒙 𝟔𝟎𝒔𝒆𝒄 = 𝟑 𝒓𝒑𝒔

4

s = cv = = 𝜋(4ft) (3 𝑟𝑝𝑠)( 1 𝑠𝑒𝑐) s = 16.76 ft

69. If f(x) = 6x – 2 and g(x) = 4x + 3, then f(g(2)) = ____? 52 B. 53 C. 50 f (6x) = 5x -2 g(x) = 4x+3 g (2) = 4 (2) + 3 = 11 f(g(2)) = 5 (11) – 2 = 53 f (g(2)) = 53

D. 56

find f(g (2) = ________

70. From the top of lighthouse, 120 ft above the sea, the angle of depression of a boat is 15 degrees. How far is the boat from the lighthouse? A. 444 ft B. 333 ft C. 222 ft D. 555 ft h = 120ft θ = 1s ℎ tan (15) = 𝑑 𝒅=

𝟏𝟐𝟎

𝟎.𝟐𝟕

= 𝟒𝟒𝟒. 𝟒𝟒 𝒇𝒕

71. If 8 men take 12 days to assemble 16 machines, how many days will it take 15 men to assemble 50 machines? 16 B. 24 C. 16 D. 20 𝑟𝑎𝑡𝑒 = 𝟏𝟔 𝟖 (𝟏𝟐)

𝒎𝒂𝒄𝒉𝒊𝒏𝒆 𝒎𝒆𝒏 𝒙 𝒅𝒂𝒚𝒔 𝟓𝟎

= 𝟏𝟓(𝒙)

X = 20 days

72. Find the coordinate of the highest point of the curve x = 90t, y = 96t – 16t^2. A. (288, 144) B. (144, 288) C. (288, -144) D.(-144, 288) x = 96t y = 96t – 16t2

dy = 96 – 32t dx = 96

𝒅𝒚

= 96 − 𝒅𝒙 t=3

𝟑𝟐𝒕 𝟗𝟔

=𝟎

x = 96 (3) = 288 y = 96 (3) – 6 (32) = 144 288, 144

73. The vertex of parabola y = (x – 1)^2 + 2 is _____. (-1, 2) B. (1, 2) C. (1, -2)

D. (-1, -2)

y = ( x-1) 2 + 2 ( x-1) 2 =y-2 V ( 1,2)

74. Two angles measuring p deg and q are complementary. If 3p – 2q = 40 deg, then the smaller angle measures 40 deg B. 44 deg C. 46 deg D. 60 deg

p and q are complementary q = 90 – p 3p -2 (90-p) = 40 p = 44° q = 90 – 44 = 46° smaller angle is 44°

75. In an ellipse, a chord which contains a focus and is in a line perpendicular to the major axis is a: A. latus rectum C. focal width B. minor axis D. conjugate axis

76. Determine the rate of a woman rowing in still water and the rate of the river current, if it takes her 2 hours to row 9 miles with the current and 6 hours to return against the current. 1 mph B. 2 mph C. 3 mph D. 4 mph d1 = d 2

d = 9 miles

V 1 t1 = V 2 t2 ( V1 + VR) (2) = ( V – VR ) 6 V + VR = 3 ( V – VR) 2V – 4VR = 0 V = 2VR VR = ½ V

2 (V + VR ) = 9 2V + 2VR = 9 2V + ½ V = 9 3V = 9 V = 3 mph

77. If f(x) = sin x and f(pi) = 3, then f(x) = 4 + cos x B. 3 + cos x C. 2 – cos x f(x) = sin x f (x) = 2- c0s x

f(x) = 3

D. 4 – cos x

then f(x) = ?

78. What is the value of the circumference of a circle at the instant when the radius is increasing at 1/6 the rate the area is increasing? A. 3 B. 3/pi C. 6 D. 6/pi C= 2 𝜋𝑟 Error question 79. A ball is thrown from the top of a 1200-foot building. The position function expressing the height h of the ball above the ground at any time t is given as h(t) = -16t^2 – 10t + 1200. Find the average velocity for the first 6 seconds of travel. A. -202 ft/sec B. -106 ft/sec C. -96 ft/sec D. -74 ft/sec h (t) = - 16t2 – 10t + 1200 @t=6 h = -16(6) 2 – 10 (t) + 1200 = 564 @ t= 0 h = 1200 𝐻₁−𝐻₂ = 564−1200 = -106ft/sec VA = t1 +t₂ 6−0

−1

80. ∫−2 |𝑥 3 |𝑑𝑥 = A. -7/8 −1

∫−2 |𝑥 3 |𝑑𝑥 = 1 4

B. 7/8 x⁴ 4

D. 16/4

−1

|𝑥 3 | ∫−2

[(−2)4 − (−1)4 ] =

=

C. -15/4

1 4

[16 − 1]

𝟏𝟓 𝟒

81. The distance covered by an object falling freely rest varies directly as the square of the time of falling. If an object falls 144 ft in 3 sec, how far will it fall in 10 sec? A. 1200 ft B. 1600 ft C. 1800 ft D. 1400 ft 82. For what values(s) of x will the tangent lines to f(x0 + ln x and g(x) = 2x^2 be parallel? A. 0 B. 1/4 C. 1/2 D. ±1/2

83. What kind of graph has r =2 sec theta? A. Straight line B. parabola

C. ellipse

D. hyperbola

84. The probability of A’s winning a game chess against B is 1/3. What is the probability that A will win at least 1 of a total 3 games? A. 11/27 B. 6/27 C. 19/27 D. 16/27 85. If f(x) = 2^(x^3 + 1), then to the nearest thousandth f(1) = A. 2.000 B. 2.773 C. 4.000 D. 8.318 𝑎 𝑎 86. If line function f is even and ∫0 𝑓(𝑥)𝑑𝑥 = 5𝑚 − 1, then ∫−𝑎 𝑓(𝑥)𝑑𝑥 = A. 0 B. 10m – 2 C. 10m – 1 D. 10m

87. What is the slope of the line through (-1, 2) and (4, -3)? A. 1 B. -1 C. 2

D. -2

88. The axis of the hyperbola through its foci is known as: A. Conjugate axis B. major axis C. transverse axis D. minor axis 89. Determine a point of inflection for the graph of y = x^3 + 6x^2 A. (-2, 16) B. (0, 0) C. (-1, 5) D. (2, 32) Solution: yI = 3x2 + 12x x = -2 yII = 6x + 12 y = (-2)3 + 6(-2)2 6x = -12 y = 16 POI = (-2, 16) 90. Clarify the graph of the equation x^2 + xy + y^2 – 6 = 0. A. circle B. parabola C. ellipse D. hyperbola

91. What is the coefficient of the (x – 1)^3 term in the Taylor series expansion of f(x) = ln x expanded about x = 1? A. 1/6 Solution: f(x) = ln(x) f’(x) = 1/x f’’(x) = - 1/x2 f’’’ (x) = 2/x3

B.1/4 f(1) = 0 f’(1) = 1 f’’(1) = -1 f’’’ (1) = 2

C. 1/3

D. ½

ln(x) = 0 + 1 (x - 1)’ - 1(x - 1)2 / 2! + 2(x - 1)3 / 3! ln(x) = (x-1)3 / 3 ln(x) = (2-1)3 / 3 = 1/3

92. If x varies directly as y and inversely as z, and x = 14, when y = 7 and z = 2, find x when y = 16 and z = 4. A. 4 B. 14 C. 8 D. 16

Solution: X=kxY/Z Where: k is constant and when x = 14, y = 7 & z = 2 14 = k x 7/2 k=4 X = 4 x Y/Z Where: y = 16 & z = 4 X = 4 x 16/4 = 16

93. Solve the differential equation A. y = cx Solution:

𝑑𝑦 𝑑𝑥 1

𝑦

+𝑥 =2

B. y = 𝑥 + 𝑐

C. y = 3x + c

𝑐

D. y = x + 𝑥

94. In triangle ABC, AB = 40 m, BC = 60 m and AC = 80m. How far from a will the other end of the bisector angle B located along the line AC? A. 40 B. 32 C. 38 D. 35 Solution: X/AC-X = AB/BC X/80-X = 40/60 60X = 40 (80-X) 60X + 40X = 3200 100X/100 = 3200/100 X =32 95. What amount should an employee receive a bonus so that she would net $500 after deducting 30% from taxes? A. $ 714.29 B. $814.93 C. $ 624.89 D. $ 538.62 Solution: 96 A rectangular trough us 8ft long, 2ft across the top, and 4 ft deep. If water flows in at a rate of 2 cu. Ft per min. how fast is the surface rising when the water is 1ft deep ? A. 1/4 ft/min B. 1/6 ft/min C. 1/3 ft/min D. 1/5 ft/min Solution: Volume of water : V = ½ (xy)(8) = 4xy

By similar triangle : x/y = 2/4 x=½y Y= 8(1/2y)y = 4y2 dv/dt = 8y dy/dt When y=1ft 2ft3/min = 8 (1) dy/dt 2 / 8 = dy/dt dy/dt = ¼ ft/min 97. If the parabola y = x^2 + C is tangent to the line y = 4x + 3, find the value of C. A. 4 B. 7 C. 6 D. 5 Solution: y = x2 + c y = 4x + 3 4x + 3 = x2 + c x2 - 4x + (c - 3) = 0 √b2-4ac = 0 √(-4)2 - 4(1)(c-3) = 0 √16 - (4c-12) = 0 Squared both sides: √ 16 - (4c-12) = √0 16 - 4c + 12 = 0 -4c = 28 -4c / - 4 = 28 / 4 C=7 98. A parabola having its axis along the x-axis passes through (-3, 6). Compute the length of latus rectum if the vertex is at the origin. A. 12 B. 8 C. 6 D. 10 Solution: Formula 4p 4(3) 12 99. If the average value of the function f(x) = 2x^2 on the interval (0, c) is 6, then c = A. 2 B. 3 C. 4 D. 5 Solution: 100. Find the volume of the tetrahedron bounded by the coordinate planes and the plane z = 6 – 2x + 3y. A. 4 B. 5 C. 6 D. 3 Solution:

MARCH 2014

1.

What is the differential equation of the family of parabolas having their vertices at the origin and their foci on the x-axis? A. 2xdy – ydx = 0 C. 2ydx –xdy = 0 B. ydx + ydx = 0 D. dy/dx – x = 0

SOLUTIONS: 𝑦 2 = 4𝑎𝑥 4𝑎 =

𝑦2 𝑥

Differentiating 0=

𝑥(2𝑥𝑦𝑑𝑦)−𝑦 2 𝑑𝑥 𝑥2

[0 = 2𝑥𝑦𝑑𝑦 − 𝑦 2 𝑑𝑥]

1 𝑦

.𝑶 = 𝟐𝒙𝒅𝒚 − 𝒚𝒅𝒙 2. Find the rthogonal trajectories of the family of parabolas y^2 = 2x + C. A. y = Ce^x B. y = Ce^(-x) C. y = Ce^(2x) D. y = Ce^(-2x) SOLUTIONS: 𝑦 2 = 2𝑥 + 𝐶 2𝑦 𝑑𝑦 𝑑𝑥

𝑑𝑦 𝑑𝑥

=2 1

=𝑦

Slope of orthogonal trajectories 𝑑𝑦 𝑑𝑥

1

𝑑𝑥

= − 𝑑𝑦 = − 𝑑𝑦 𝑑𝑥

Subs. 𝑑𝑦 𝑑𝑥

= −𝑦

𝑑𝑦 = − ∫ 𝑑𝑥 𝑦 ln 𝑦 = −𝑥 + 𝑐 𝑒 ln 𝑦 + 𝑒 −𝑥+𝑐 𝑦 = 𝑒 −𝑥 (𝑒 𝑐 ) 𝐲 = 𝐂𝐞−𝐱 ∫

3. A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is 30 ft. If the distance across the top of the mirror is 64 in., how deep is the mirror of the center? A. 32/45 in. B. 30/43 in. C. 32/47 in. D. 35/46 in.

SOLUTIONS: 1. 𝑥 2 = −4𝑎𝑦 𝑉𝑡𝑜𝐹 = 𝑎 = 30𝑓𝑡 = 360 𝑖𝑛 𝐿𝑅 = 4𝑎 = 1440 𝑥 2 = 4𝑎𝑦 322 = 1440𝑦 𝟑𝟐𝟐

𝟑𝟐

𝒚 = 𝟏𝟒𝟒𝟎 = 𝟒𝟓 𝐢𝐧 4. Simplify (1 – tan2x) / (1 + tan2x) A. sin 2x B. cos 2x SOLUTIONS: 1−tan2 𝑥 1+tan2 𝑥

=

1−tan2 𝑥 sec2 𝑥

1

= sec2 𝑥 −

C. sin x

D. cos x

C. n!/s^(n-1)

D. n!/s^(n+2)

sin2 𝑥 cos2 𝑥 sec2 𝑥

sin2 𝑥 (cos 2 𝑥) = cos 𝑥 − 2 cos 𝑥 = cos 2 𝑥 − sin2 𝑥 = 𝐜𝐨𝐬 𝟐𝒙 2

5. Evaluate L { t^n }. A. n!/s^n SOLUTIONS:

B. n!/s^(n+1)

𝒏!

.∫(𝒕𝒏 ) = 𝑺𝒏+𝟏

6. Simplify 12 cis 45 deg + 3 cis 15 deg. A. 2 + j B. sqrt. of 3 + j2 SOLUTIONS:

C. 2 sqrt. Of 3 + j2 D. 1 + j2

12cis45/3cis15 =

12 3

𝑐𝑖𝑠(45 − 15)

= 𝟐√𝟑 + 𝒋𝟐 arcsin 9𝑥

7. Evaluate lim ( 𝑥→0

A. 9/2 SOLUTIONS:

2𝑥

) B. π

C. ∞

D. -∞

log 𝑥=0

sin−1 𝑎𝑥 2𝑥

log x=0

sin−1 (9) 0.0001 2(0.0001) 𝟗

𝟒. 𝟓 𝒐𝒓 𝟓

8. Find the area of the lemniscate r2 = a2cos2θ A. a2 B. a C. 2a

D. a3

SOLUTIONS: 𝑟 2 = 𝑎2 cos 2𝜃 1

𝜃2

𝐴 = 2 ∫𝜃1 𝑟 2 𝑑𝜃 1

𝜋

𝐴 = [2 ∫04 cos 2𝜃𝑑𝜃] 𝑨 = 𝒂𝟐

9. Find the area bounded by the parabola sqrt. of x + sqrt. of y = sqrt. of a and the line x + y = a. A. a2 B. a2/2 C. a2/4 D. a2/3 SOLUTIONS: √𝑥 + √4 = √𝑎 𝑎𝑠𝑠𝑢𝑚𝑒𝑎 = 1 2

(√𝑦 = 1 − √𝑥) 𝑥 + 𝑦 = 1 2 𝑦 = (1 − 𝑥) 𝑦 = 1 − 𝑥 2

1

𝐴 = ∫0 (1 − 𝑥) − (1 − √𝑥) 𝑑𝑥 𝑨 = 𝟎. 𝟑𝟑𝟑 𝒐𝒓

𝒂𝟐 𝟑

10. Ben is two years away from being twice Ellen’s age. The sum of twice Ben’s age and thrice Ellen’s age is 66. Find Ben’s age now. A. 19 B. 20 C. 16 D. 21 SOLUTIONS: 2𝑥 + 3(2𝑥) = 66 X=8.25

Age of ben =2X =2(8.25) = 16.5 11. What percentage of the volume of a cone is the maximum volume right circular cylinder that can be inscribed in it? A. 24% B. 34% C. 44% D. 54% 12. A balloon rising vertically, 150 m from an observer. At exactly 1 min, the angle of elevation is 29 deg 28 min. How fast is the balloon using at that instant? A. 104m/min B. 102m/min C. 106m/min D. 108m/min 13. A conic section whose eccentricity is less than one (1) is known as: A. a parabola B. an ellipse C. a circle D. a hyperbola 14. A tangent to a conic is a line A. which is parallel to the normal B. which touches the conic at only one point C. which passes inside the conic D. all of the above 15. A die and a coin are tossed. What is the probability that a three and a head will appear? A. 1/4 B. 1/2 C. 2/3 D.1/12 5 5 16. Find the integral of 12sin xcos xdx if lower limit = 0 and upper limit = pi/2. A. 0.8 B.0.6 C.0.2 D.0.4 17. 12 oz of chocolate is added to 10 oz of flavoring is equivalent to A.1 lb and 8 oz B. 1 lb and 6 oz C.1 lb and 4 oz D.1 lb and 10 oz 18. The Ford company increased its assets price from 22 to 29 pesos. What is the percentage of increase? A.24.14% B.31.82% C.41.24% D.28.31% 19. Find the area bounded by outside the first curve and inside the second curve, r = 5, r = 10sinθ A. 47.83 B.34.68 C.73.68 D.54.25 20. In two intersecting lines, the angles opposite to each other are termed as: A. opposite angles C. horizontal angles B. vertical angles D. inscribed angles Soln.B = vertical angles 21. The area in the second quadrant of the circle x^2 + y^2 = 36 is revolved about the line y + 10 = 0. What is the volume generated? A. 2932 c.u. B. 2392 c.u. C. 2229 c.u. D. 2292 c.u.

22. A cardboard 20 in x 20 in is to be formed into a box by cutting four equal squares and folding the edges. Find the volume of the largest box. A.592 cu.in. B.529 cu.in. C.696 cu.in. D.689 cu.in.

Soln. V =(20-2x)(20-2x)(x) V=(400 – 40x – 40x +4x^2) x V = 400x – 80 X^2 + 4x^3 𝑑𝑉 𝑑𝑥

= 12x2 – 160x + 400 = 0

X1 = 10 ---reject X2 = 3.33 ---accept Subs. V= ((20-2)(3.33))(20-2(3.33))(3.33) V = 592 cu. in 23. A retailer bought a number of ball pens for P90 and sold all but 3 at a profit P2 per ball pen. With the total amount received she could buy 15 more ball pens than before. Find the cost per ball pen. A. P2 B. P3 C.P4 D.P5 24. What is –i^i? A.4.81 B.-4.81 C.0.21 D.-0.21 25. A balloon travel upwards 6m, North and 8m, East. What is the distance traveled from the starting point? A. 7 B. 10 C.14 D. 20 Soln. x=8 y=6 d=? d=√𝑥 2 + 𝑦 2 d = √82 + 62 = 10 26. What do you call the integral divided by the difference of the abscissa? A. average value C. abscissa value B. mean value D. integral value ANSWER: A.average value

27. Water is running out of a conical funnel at the rate of 1 cubic inch per sec. If the radius of the base of the funnel is 4 in. and the altitude is 8 in., find the rate at which the water level is dropping when it is 2 in. from the top. ` A. -1/pi in./sec B. -2/pi in./sec C. -1/9pi in./secD.-2/9pi in./sec Soln. 1

V = 3 𝜋𝑟 2 ℎ 𝑅 𝑟 1 = = 𝐻 ℎ 2 𝜋 ℎ 2

𝜋

3

12

V= ( ) ℎ= 2

;𝑟 =

ℎ 2

ℎ3 𝑑𝑉 3𝜋 2 𝑑ℎ = ℎ 𝑑𝑡 12 𝑑𝑡 −1 =

3𝜋 𝑑ℎ (2)2 12 𝑑𝑡

𝑑ℎ 𝟏 = − 𝝅/𝒔𝒆𝒄 𝑑𝑡 𝟗

28. How many inches is 4 feet? A. 36 B. 48

C. 12

D. 56

4ft x 12inch / 1ft = 48inch 29. A rectangular trough is 8 ft. long, 2 ft. across the top, and 4 ft. deep. If water flows in at a rate of 2 cu. ft./min., how fast is the surface rising when the water is 1 ft. deep? A. 1/5 ft./min B. 1/8 ft./min C. 1/6 ft./min D. 1/16 ft./min

Soln. V = (8)(2)(1)h 𝑑𝑉 𝑑ℎ = 16 𝑑𝑡 𝑑𝑡 𝑑ℎ 𝟏 = 𝒇𝒕/𝒎𝒊𝒏 𝑑𝑡 𝟖

30. Five tables and eight chairs cost $115; three tables and five chairs cost $70. Determine the total cost of each table.

A. $15

B. $30

C. $25

D. $20

Soln. 5 tables + 8 chairs = 115 3 tables + 5 chairs = 70 (5T + 8C = 115) 5 (3T + 5C = 70 ) -8 T=15 31. Find the 16th term of the arithmetic sequence; 4, 7, 10,…….. A. 47 B. 46 C. 49 32. Find the slope of the line through the points (-2, 5) and (7, 1). A. 9/4 B. -9/4 C. 4/9

D. 48 D. -4/9

33. For what value of k will the line kx +5y = 2k have a y-intercept 4? A. 8 B. 7 C. 9 D.10 34. If a bug moves a distance of 3pi cm along a circular arc and if this arc subtends a central angle of 45 degrees, what is the radius of the circle? A. 8 B. 12 C. 14 D. 16 35. Two vertices of a rectangle are on the positive x-axis. The other two vertices are on the lines y = 4x and y = -5x + 6. What is the maximum possible area of the rectangle? A.2/5 B.5/2 C.5/4 D. 4/5 36. Find the length of the arc of 6xy = x^4 + 3 from x = 1 to x = 2. A.12/17 B.17/12 C.10/17 D.17/10 37. A certain radioactive substance has half-life of 3 years. If 10 grams are present initially, how much of the substance remain after 9 years? A.2.50g B.5.20g C. 1.25g D.10.20g 38. A cubical box is tobuilt so that it holds 125 cu. cm. How precisely should the edge be made so that the volume will be correct to within 3 cu. cm.? A.0.02 B.0.03 C.0.01 D.0.04 39. Find the eccentricity of the ellipse when the length of its latus rectum is 2/3 of the length of its major axis. A.0.62 B. 0.64 C.0.58 D.0.56 40. Find k so that A = <3, -2> and B =<1, k> are perpendicular. A. 2/3 B.3/2 C.5/3 D.3/5 41. Find the moment of inertia of the area bounded by the curve x^2 = 8y, the line x = 4 and the x-axis on the first quadrant with respect to y-axis. A.25.6 B. 21.8 C.31.6 D.36.4 42. Find the force on one face of a right triangle of sides 4m and altitude of 3m. The altitude is submerged vertically with the 4m side in the surface.

A.62.64 kN

B.58.86 kN

C.66.27 kN

D.53.22 kN

43. In how many ways can 6 people be seated in a row of 9 seats? A. 30,240 B. 30,420 C.60,840 D. 60,480 SOLUTIONS: 9P6 = 60,480 44. The arc of a sector is 9 units and its radius is 3 units. What is the area of the sector? A.12.5 B.13.5 C.14.5 D.15.5 SOLUTIONS: 1

A = 2 𝑟𝐶 1

A = 2 (3)(9) A = 13.5 45. The sides of a triangle are 195, 157, and 210, respectively. What is the area of the triangle? A.73,250 B.10,250 C.14,586 D.11,260 SOLUTIONS: S=

195+157+210 2

= 281

A = √281 (281 − 195(281 − 157)(281 − 210) A = 14586.21 46. A box contains 9 red balls and 6 blue balls. If two balls are drawn in succession, what is the probability that one of them is red and the other is blue? A.18/35 B.18/37 C.16/35 D.16/37 47. A car goes 14 kph faster than a truck and requires 2 hours and 20 minutes less time to travel 300 km. Find the rate of the car. A.40 kph B.50 kph C.60 kph D.70 kph 48. Find the slope of the line defined by y – x = 5. A.1 B.1/4 C.-1/2 SOLUTIONS: y = mx+b y–x=5

D.5

y=x+5 by inspection, the slope is equal to 1 49. The probability of John’s winning whenever he plays a certain game is 1/3. If he plays 4 times, find the probability that he wins just twice. A.0.2963 B.0.2936 C.0.2693 D.0.2639 SOLUTIONS: nCrpq n=4 , r=2 , therefore :

p = 1/3 q = 2/3

1 2 2 2

4C2 x (3) (3) = 0.2963 50. A man row upstream and back in 12 hours. If the rate of the current is 1.5 kph and that of the man in still water is 4 kph, what was the time spent downstream? A.1.75 hr B.2.75 hr C.3.75 hr D. 4.75 hr SOLUTIONS: d=d (V + c)(t) = (V – c)(t) (4 + 1.5)(x) = (4 – 1.5)(12 - x) x = 3.75 hrs 51. If cot A = -24/7 and A is in the 2nd quadrant, find sin 2A. A.336/625 B.-336/625 C.363/625 SOLUTION: Cot A = 1 tan 𝐴

=

D. -363/625

−24

7 −24 7 7

tan A = −24 7

A = tan−1 (−24) A = -16.260 sin 2A = sin (2x = 16.250) =

−336 625

52. The volume of a square pyramid is 384 cu. cm. Its altitude is 8 cm. How long is an edge of the base? A.11 B.12 C.13 D.14

SOLUTION: V = 384 cm^3

h= 8 cm

1

V = 3Abh 1

384 = 3Ab (8) Ab = 144 A = a2 √𝑎2 = √𝐴 = √144 = 12 53. The radius of the circle x^2 + y^2 – 6x + 4y – 3 = 0 is A.3 B.4 C.5 SOLUTION: 𝑥 2 + 𝑦 2 − 6𝑥 + 4𝑦 − 3 = 0 (𝑥 2 − 6𝑥 + 9) + (𝑦 2 + 4𝑦 + 4) (𝑥 + 3)2 + (𝑦 + 2)2 = 16 = 42

D.6

54. If the planes 5x – 6y - 7z = 0 and 3nx + 2y – mz +1 = 0 A.-2/3 B. -4/3 C.-5/3 D.-7/3 55. If the equation of the directrix of the parabola is x – 5 = 0 and its focus is at (1, 0), find the length of its latus rectum. A.6 B.8 C.10 D.12 SOLUTION: d=x–5=0 d=5 f(1,0) = a = 1 LR = 2a d=F 2a = 4 a=2 LR = 2a = 8 56. If tan A = 1/3 and cot B = 4, find tan (A + B). A. 11/7 B. 7/11 C. 7/12 D. 12/7 SOLUTION: 1

A = tan−1 ( 3) = 18.43 1

B = tan−1 ( 4) = 14.04 tan (18.43 + 14.04) = 0.636 7

= 11 57. A club of 40 executives, 33 like to smoke Marlboro, and 20 like to smoke Philip Morris. How many like both? A. 13 B. 10 C. 11 D. 12 SOLUTION: (33 - x) + x + (20 - x) = 40

x=13 58. The area of the rhombus is 264 sq. cm. If one of the diagonals is 24 cm long, find the length of the other diagonal. A. 22 B. 20 C. 26 D. 28 SOLUTION: A=

1 2

d1 d2 1

264 = 2 (26) d2 d 2 = 22 cm

59. How many sides have a polygon if the sum of the interior angles is 1080 degrees? A. 5 B. 6 C. 7 D. 8 SOLUTION: S = (n - 2)(180) 1080 = (n - 2)(180) n=8 60. The line segment connecting (x, 6) and (9, y) is bisected by the point (7, 3). Find the value of x and y. A. 5, 0 B. 4, 0 C. 5, 2 D.4,1 SOLUTION: Let Xm and Ym the coordinates of the midpoint 𝑋1+𝑋2 𝑌1+𝑌2 Xm = Ym = 2 2

7=

𝑥+9

3=

2

6+𝑦 2

x=5 y=0 61. What is the height of the parabolic arch which has span of 48 ft. and having a height of 20 ft. at a distance of 16 ft. from the center of the span? A. 30 ft. B. 40 ft. C. 36 ft. D.34ft. SOLUTION:

62. Determine B such that 3x + 2y – 7 = 0 is perpendicular to 2x –By + 2 =0. A. 2 B. 3 C. 4 D.5 SOLUTION: y=

−3 2

𝑥+

m1 = - 3/2

7 2

y=

−2 𝐵

𝑥+

m2 = -2/B

2 𝐵

Since perpendicular, m2 = - 1/m1 −2 𝐵

=

1

−3 2

=3

63. The value of x + y in the expression 3 + xi = y + 2i is; A. 5 B. 1 C. 2 SOLUTION: 64. If sin3A = cos6B then: A. A + B = 180 deg B. A + 2B = 30 deg SOLUTION: Sin 3A = cos 6B

D.3

C. A - 2B = 30 deg D. A + B = 30 deg

Sin 3A = sin (90 – 6B) 3A = 90 – 6B (3A + 6B = 90) 1/3 A + 2B = 30 65. What is the area between y = 0, y = 3x^2, and x = 2? A. 8 B. 12 C. 24 SOLUTION:

D.6

2 2 𝑦𝑑𝑦 = 3∫ 𝑥 2 𝑑𝑥 0 0 𝑥3 =3 3

A=∫

= x3 = (2)3 = 8 66. The volume of the sphere is 36pi cu. m. The surface area of this sphere in sq. m is: A. 36pi B. 24pi C. 18pi D. 12pi SOLUTION: Vs = 36π 4

V = 3 𝜋𝑟 3 36π = 4/3πr3 r=3 As = 4πr2 As = 4π(3)2 = 36π m2 67. The vertex of the parabola y^2 – 2x + 6y + 3 = 0 is at:

A. (-3, 3) B. (3, 3) SOLUTION: 𝑦 2 − 2𝑥 + 6𝑦 + 3 = 0 𝑦 2 + 6𝑦 + 9 = 2𝑥 − 3 (𝑦 + 3)2 = 2𝑥 − 3 + 9 (𝑦 + 3)2 = 2𝑥 + 6 (𝑦 + 3)2 = 2(𝑥 + 3) (𝑦 − 𝑘)2 = 4𝑎(𝑥 − ℎ) = - 3, - 3

C. (3, -3)

D.

(-3,

-3)

68. Add the following and express in meters: 3 m + 2 cm + 70 mm A. 2.90 m B. 3.14 m C. 3.12 m D.3.09m SOLUTION: 3+(𝑐𝑚 𝑥

1𝑐𝑚 100𝑐𝑚

) + (70𝑚𝑚 𝑥

1𝑚 1000𝑚𝑚

)

= 3.09m 69. A store advertised on sale at 20 percent off. The sale price was $76. What was the original price? A. $95 B. $96 C. $97 D.$98 SOLUTION: 76 = .80(x) X = 95 70. Find the equation of the straight line which passes through the point (6, -3) and with an angle of inclination of 45 degrees. A. x + y = 8 B. x – y = 8 C. x + y = 9 D. x – y = 9 SOLUTION: n = tan 𝜃 n = tanus = 1 y + 3 = 1(x-6) y=x–6–3 x–y=9 71. A freight train starts from Los Angeles and heads for Chicago at 40 mph. Two hours later a passenger train leaves the same station for Chicago traveling at 60 mph. How long will it be before the passenger train overtakes the freight train? A. 3 hrs. B. 5 hrs. C. 4 hrs. D. 6 hrs. SOLUTION: Time Rate distance

X 40 CHI-Lo

40x x-2

60

60(x-2)

d1 = d2 40x =60(x-2) x=6 x -2 = 6 – 2 = 4 hrs 72. The number of board feet in a plank 3 inches thick, 1 ft. wide, and 20 ft. long is: A. 30 B. 60 C. 120 D. 90 SOLUTION: V= 3(1)(20) = 60 inch

73. Boyles’s law states that when a gas is compressed at constant temperature, the product of its pressure and volume remains constant. If the pressure gas is 80 lb/sq.in. when the volume is 40 cu.in., find the rate of change of pressure with respect to volume when the volume is 20 cu.in. A. -8 B. -10 C. -6 D.-9 SOLUTION: 74. Find the average rate of change of the area of a square with respect to its side x as x changes from 4 to 7. A. 8 B. 11 C. 6 D. 21 SOLUTION: A= x^2 Limits 7-4=3 lim A’= 2x = 2(3) = 6 75. How many cubic feet is equivalent to 100 gallons of water? A. 74.80 B. 1.337 C. 13.37 SOLUTION: 100L = 1m 3

D. 133.7

1m = 3.28 ft 1L = 0.2642 gal 1𝐿

1𝐿3

100 gal = 0.2642 𝐿 1000𝐿 𝐿 (

3.18 𝐿𝐿 3 1𝐿

) = 13.37

76. A merchant purchased two lots of shoes. One lot he purchased for $32 per pair and the second lot he purchased for $40 per pair. There were 50 pairs in

the first lot. How many pairs in the second lot if he sold them all at $60 per pair and made a gain of $2800 on the entire transaction? A. 50 B. 40 C. 70 D. 60 SOLUTION: PB=50(32)=1600 2800 = 1400 + PR2

PS= 60(50)=3000 PR=PS –PR = 3000-1600 = 1400

PRT = PR1 + PR2

PR2 = 1400 PB=40(Y) PS= 60(Y) PR=PS –PR 1400 = 40(Y) -40(Y) 1400 = 20Y Y = 70

77. The diagonal of a face of a cube is 10 ft. The total area of the cube is A. 300 sq. ft. B. 150 sq. ft. C. 100 sq. ft. D. 200 sq. ft. SOLUTION: √2𝐿 = 10 5

a = √2 total Area = 6𝐿2 A total = 300 ft2 78. A ship is sailing due east when a light is observed bearing N 62 deg 10 min E. After the ship has traveled 2250 m, the light bears N 48 deg 25 min E. If the course is continued, how close will the ship approach the light? A. 2394 m B. 2934 m C. 2863 m D. 1683 m SOLUTION: 79. If f(x) = 1/(x – 2), (f g)’(1) = 6 and g’(1) = -1, then g(1) = A.-7 B. -5 C. 5 D. 7 SOLUTION: f(1)g’(1)+g(1)f’(1)=6 −1

(-1)(-1) + g(1) ((1−2)^2)=6

g(1) (-1) = 5 g(1) = -5 80. Find the work done by the force F = 3i + 10j newtons in moving an object 10 meters north. A.104 40 J B. 100 J C.106 J D. 108.60 J SOLUTION: F = 3 + j10 d = 10m W = Fd W = 10j(10) = 100 cis 90 81. The volume of a frustum of a cone is 1176pi cu.m. If the radius of the lower base is 10m and the altitude is 18m, compute the lateral area of the frustum of a cone A.295pi sq. m. B. 691pi sq. m. C.194pi sq. m. D. 209pi sq. m.

82. In an ellipse, a chord which contains a focus and is in a line perpendicular to the major axis is a: A.latus rectum B. minor axis C. focal width D. major axis 83. With 17 consonant and 5 vowels, how many words of four letters can be four letters can be formed having 2 different vowels in the middle and 1 consonant (repeated or different) at each end? A.5780 B. 5785 C. 5790 D. 5795 2 84. Evaluate tan (j0.78). A.0.653 B.-0.653 C.0.426 D. -0.426 85. A particle moves along a line with velocity v = 3t^2 – 6t. The total distance traveled from t = 0 to t = 3 equals A.8 B. 4 C. 2 D. 16 86. An observer at sea is 30 ft. above the surface of the water. How much of the ocean can he sea? A.124.60 sq. mi. C. 154.90 sq. mi. B.142.80 sq. mi. D. 132.70 sq. mi. 87. There are three consecutive integers. The sum of the smallest and the largest is 36. Find the largest number. A.17 B. 18 C.19 D. 20 88. If y = sqrt. of (3 – 2x), find y. A.1/sqrt. of (3 – 2x) C. 2/sqrt. of (3 – 2x) B. -1/sqrt. of (3 – 2x) D. -2/sqrt. of (3 – 2x) 89. The logarithm of MN is 6 and the logarithm of N/M is 2, find the value of logarithm of N. A.3 B. 4 C. 5 D.6 90. A woman is paid $20 for each day she works and forfeits $5 for each day she is idle. At the end of 25 days she nets $450. How many days did she work? A.21 days B. 22 days C. 23 days D.24 days 91. Francis runs 600 yards in one minute. What is his rate in feet per second? A.25 B. 30 C.35 D.40 92. For a complex number z = 3 + j4 the modulus is: A.3 B. 4 C. 5 D. 6 93. Which of the following is an exact DE? A. (x^2 + 1)dx – xydy = 0 C. 2xydx + (2 + x^2)dy = 0 B. xdy + (3x – 2y)dy = 0 D. x^2 ydy – ydx = 0 94. There are 8 different colors, 3 of which are red, blue and green. In how many ways can 5 colors be selected out of the 8 colors if red and blue are always included but green is excluded? A.12 B.11 C. 10 D.9 95. Five cards are drawn from a pack of 52 well – shuffled cards. Find the probability that 3 are 10’s and 2 are queens.

A. 1/32 96. If

7 ∫1 𝑓(𝑥)𝑑𝑥

B. 1/108,290 = 4 and

7 ∫1 𝑔(𝑥)𝑑𝑥

= 2, find

C. 1/54,350 7 ∫1 [3𝑓(𝑥)

D.1/649,740

+ 2𝑔(𝑥) + 1]𝑑𝑥.

A. 23 B. 22 C. 25 D. 24 97. When the ellipse is rotated about its longer axis, the ellipsoid is A. spheroid B. oblate C. prolate D. paraboloid 98. If the distance between points A(2, 10, 4) and B(8, 3, z) is 9.434, what is the value of z? A. 4 B. 3 C. 6 D. 5 99. A line with equation y = mx + b passes through (-1/3, -6) and (2, 1). Find the value of m. A. 1 B. 3 C. 4 D. 2 100. For the formula R = E/C, find the maximum error if C = 20 with possible error 0.1 and E = 120 with a possible error of 0.05. A. 0.0325 B. 0.0275 C. 0.0235 D. 0.0572

SOLUTIONS: 1. 𝑦 2 = 4𝑎𝑥 4𝑎 =

5.

𝑦2 𝑥

Differentiating 0=

=

𝑥2

12 3

𝑐𝑖𝑠(45 − 15)

1

= 2√3 + 𝑗2

𝑦

0 = 2𝑥𝑑𝑦 − 𝑦𝑑𝑥

log 𝑥=0

7.

2. 𝑦 2 = 2𝑥 + 𝐶

sin−1 𝑎𝑥

2𝑥 sin−1 (9) 0.0001

log x=0

𝑑𝑦

2(0.0001) 9

2𝑦 𝑑𝑥 = 2 𝑑𝑥

12cis45/3cis15

6.

𝑥(2𝑥𝑦𝑑𝑦)−𝑦 2 𝑑𝑥

[0 = 2𝑥𝑦𝑑𝑦 − 𝑦 2 𝑑𝑥]

𝑑𝑦

𝑛!

∫(𝑡 𝑛 ) = 𝑆𝑛+1

4.5 𝑜𝑟 5

1

=𝑦 𝑟 2 = 𝑎2 cos 2𝜃

Slope of orthogonal trajectories

8.

𝑑𝑦

𝐴 = 2 ∫𝜃1 𝑟 2 𝑑𝜃

𝑑𝑥

1

𝑑𝑥

1

= − 𝑑𝑦 = − 𝑑𝑦 𝑑𝑥

1

𝑑𝑥

𝜋

𝐴 = [2 ∫04 cos 2𝜃𝑑𝜃]

Subs. 𝑑𝑦

𝜃2

𝐴 = 𝑎2

= −𝑦

𝑑𝑦 = − ∫ 𝑑𝑥 𝑦 ln 𝑦 = −𝑥 + 𝑐 𝐿ln 𝐿 + 𝐿−𝐿+𝐿 ∫

9.

√𝑥 + √4 = √𝑎 𝐿𝐿𝐿𝐿𝐿𝐿𝐿 = 1 2

𝐿 = 𝐿−𝐿 (𝐿𝐿 ) 𝐿=1 𝐿 = 𝐿𝐿−𝐿

(√𝐿 = 1 − √𝐿) 𝐿 + 𝐿 = (1 − 𝐿)2 𝐿 = 1 − 𝐿 𝐿= 2

1

∫0 (1 − 𝐿) − (1 − √𝐿) 𝐿𝐿 3. 𝐿2 = −4𝐿𝐿

𝐿 = 0.333 𝐿𝐿

𝐿 𝐿𝐿 𝐿 = 𝐿 = 30𝐿𝐿 = 360 𝐿𝐿 𝐿𝐿 = 4𝐿 = 1440 𝐿2 = 4𝐿𝐿 322 = 1440𝐿 4 9

10. 11.

𝐿2 3

2𝐿 + 3(2𝐿) = 66

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 =

𝐿𝐿𝐿𝐿𝐿 322

32

𝐿 = 1440 = 45 𝐿𝐿 44.44%

0.4444 𝐿𝐿𝐿𝐿𝐿 =

4.

1−tan2 𝐿 1+tan2 𝐿

=

1−tan2 𝐿

1

= sec2 𝐿 −

sec2 𝐿

sin2 𝐿 cos2 𝐿 sec2 𝐿

sin2 𝐿 = cos 𝐿 − (cos2 𝐿) cos2 𝐿 = cos2 𝐿 − sin2 𝐿 = cos 2𝐿 2

12. @𝐿 = 1 𝐿𝐿𝐿, 𝐿 = 29.28 𝐿𝐿

29.28

% 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 =

= 1 𝐿𝐿𝐿 𝐿𝐿 29.22 22

18.

𝐿 100 = 31.82 % 29+

𝐿𝐿

28

60 = (1 𝐿𝐿𝐿 )𝐿 𝐿𝐿

𝐿𝐿 𝐿𝐿

= 0.5143

10 sin 𝐿

𝐿

𝐿𝐿𝐿 180

𝐿 180 1

19.

r= 10sin𝐿

r=s

𝐿 6

𝐿=

/𝐿𝐿𝐿

sin 𝐿 = 2

tan 𝐿 =

𝐿

𝐿 2 𝐿 6

1

A = [2] ∫ (10 sin 𝐿)2 −

190

𝐿2 )𝐿𝐿) 𝐿

𝐿(tan 𝐿) = 𝐿(190) 𝐿𝐿

sec2 𝐿 𝐿𝐿 = 𝐿𝐿 𝐿𝐿 𝐿𝐿 𝐿𝐿 𝐿

𝐿𝐿 𝐿𝐿

190 1

= 190 (cos2 29.28) (0.5143)

= 47.83 sq.u. 20.

B = vertical angles

21.

Solving Y.. 𝐿

= 102 𝐿/𝐿𝐿𝐿

𝐿𝐿 = ∫ 𝐿𝐿 𝐿 2

𝐿

= ∫0 𝐿𝐿𝐿 . 2

𝐿

∫ 𝐿2 𝐿𝐿 2 𝐿

B = e<1 = ellipse

14.

B = tangent to comic line is a

=

Line touches the comic at only one point.

= 2 [𝐿2 𝐿 −

1 4

15.

𝐿𝐿2 𝐿 =

1 2

[ 𝐿3 ]

2 𝐿

=

1

13.

1

𝐿

∫ (𝐿2 − 𝐿2 )𝐿𝐿 2 0 1

𝐿3 3

]

1

𝐿=6 𝐿

1

= 2

1

1

𝐿

(4 𝐿𝐿 = 3 ) 4

12

𝐿𝐿 =

4𝐿 3

@ x2 + y2 = 36

16. 𝐿 2

5 ∫0 12 sin 𝐿 cos2 𝐿 𝐿𝐿 pappus

r=6

By second Proposition of 𝐿 = 𝐿2𝐿𝐿

Wallis Formula

sub

r=6 1

(4)(2)(4)(2)

12 = [(10)(8)(6)(4)(2)] = 5 𝐿𝐿 0.2 1 4

𝐿=

𝐿(6)2 𝐿 2𝐿(𝐿 + 10) V = 2228.918 cu.units

17.

25. 22 𝐿𝐿 𝐿

28.30 𝐿 1 𝐿𝐿 1𝐿𝐿

𝐿

0.37𝐿𝐿 = 2.2𝐿𝐿 𝐿

1 𝐿𝐿 1000𝐿 1000𝐿 1𝐿𝐿

𝐿

2.2 𝐿𝐿 1 𝐿𝐿

= 0.37𝐿𝐿

1 𝐿𝐿

𝐿 28.35𝐿 = 6 𝐿𝐿

Therefore: 1 lb and 6 oz 22. V =(20-2x)(20-2x)(x)

26.

V=(400 – 40x – 40x +4x^2) x V = 400x – 80 X^2 + 4x^3

A = Average Value 27.

𝐿𝐿

1

V = 3 𝐿𝐿2 𝐿

= 12x2 – 160x + 400 = 0 𝐿𝐿

𝐿

X1 = 10 ---reject

𝐿

𝐿

1

=𝐿=2

𝐿 𝐿 2

;𝐿 =

𝐿 2

𝐿

V = 3 ( 2 ) 𝐿 = 12 𝐿3

X2 = 3.33 ---accept

𝐿𝐿

Subs.

𝐿𝐿

=

3𝐿 12

𝐿𝐿

𝐿2 𝐿𝐿

−1 =

V= ((20-2)(3.33))(20-2(3.33))(3.33)

𝐿𝐿

V = 592 cu. In

𝐿𝐿

3𝐿

(2)2

12

𝐿𝐿 𝐿𝐿

1

= − 9 𝐿/𝐿𝐿𝐿

23. 28. 4ft x

12 𝐿𝐿𝐿𝐿𝐿𝐿 1 𝐿𝐿

= 48 inches

29. 24.

V = 16 h 𝐿𝐿 𝐿𝐿 𝐿𝐿 𝐿𝐿

30.

𝐿𝐿

= 16 𝐿𝐿

36. 5 tables + 8 chairs = 115 3 tables + 5 chairs = 70 (5T + 8C = 115) 5 (3T + 5C = 70 ) -8 T= 15

31. A15= ? l d =3 a =4 A15 = A1 + (n-1) d A15 = 4 + (15 – 3)(3) = 49

37. HL= 3yrs

32. m=

𝐿2− 𝐿1 𝐿2−𝐿1

1−5

= 7+2 =

−4 9

33. Kx + 5y = 2k K =? @ y=4 @ x=0 K(0) + 5y = 2k 5y = 2k @

38.

y=4 5 (4) = 2k K = 20/2 = 10

34. C= r𝐿 R=

3𝐿 45 𝐿

𝐿 180

= 12 cm 39.

35.

1

= 8 𝐿𝐿/𝐿𝐿𝐿

40.

45. S=

195+157+210 2

= 281

A= √𝐿 (𝐿 − 𝐿)(𝐿 − 𝐿)(𝐿 − 𝐿) A= √281 (281 − 195(281 − 157)(281 − 210) A = 14586.21 46.

41.

47.

42.

48. y = mx + b y=x+s m =1 49. nCrpq n=4 , r=2 , therefore :

p = 1/3 q = 2/3

1 2 2 2

4C2 x (3) (3) = 0.2963

43. 9P6 = 60,480

50. d=d (V + c)(t) = (V – c)(t) (4 + 1.5)(x) = (4 –

44. S = r𝐿 , 𝐿 = 9/3 1.5)(12 - x) 1

A = 2 𝐿2 𝐿 1

x = 3.75 hrs

9

= 2 (3)2 (3) = 13.5

51.

55. Cot A = 1 tan 𝐿

=

−24

d=x–5=0 d=5

7

−24

f(1,0) = a = 1

7 7

tan A = −24 A = tan−1 (

LR = 2a 7 −24

)

d=F

A = -16.260

a=2

sin 2A = sin (2x = 16.250)

LR = 2a = 8

=

2a = 4

−336 625

56. 1

A = tan−1 ( 3) = 18.43

52. V = 384 cm^3

1

B = tan−1 ( 4) = 14.04

h= 8 cm

1

V = 3Abh

tan (18.43 + 14.04) =

0.636 1

7

= 11

384 = 3Ab (8) Ab = 144 A = a2 √𝐿2 = √𝐿 = √144 = 12

57. (33 - x) + x + (20 - x) = 40 X=B

53. 𝐿2 + 𝐿2 − 6𝐿 + 4𝐿 − 3 = 0

58.

(𝐿2 − 6𝐿 + 9) + (𝐿2 + 4𝐿 + 4)

A=

(𝐿 + 3)2 + (𝐿 + 2)2 = 16 = 42

264 = 2 (26) d2

1 2

d1 d2 1

54.

d 2 = 22 cm

59.

63. S = (n - 2)(180) 1080 = (n - 2)(180) n=8

60. 64. Sin 3A = cos 6B Sin 3A = sin (90 – 6B) 3A = 90 – 6B (3A + 6B = 90) 1/3 A + 2B = 30

61. 2

65. 𝐿 = 0 , 𝐿 = 3𝐿2 , 𝐿 = 𝐿2 =

𝐿 3

𝐿

𝐿 = √3

𝐿

2

(2 = √3 )

y = 12 12

𝐿

A = ∫0 (2 − √ 3 ) 𝐿𝐿 = 8

66. Vs = 36π 4

V = 3 𝐿𝐿3

36π = 4/3πr3 As = 4πr2`

62. 3x + 2y – 7 = 0

r=3

As = 4π (3)2 = 36π

2x – By + 2 = 0 Ax + By – C1 = 0 If perpendicular Bx – Ay + C2 = 0 Therefore 2x – 3y + 2 = 0

67.

71. 𝐿2 − 2𝐿 + 6𝐿 + 3 = 0 distance

Lachi Time Rate

𝐿2 + 6𝐿 + 9 = 2𝐿 − 3

X

(𝐿 + 3)2 = 2𝐿 − 3 + 9 60(x-2)

CHI-La

(𝐿 + 3)2 = 2𝐿 + 6

d1 = d2

(𝐿 + 3)2 = 2(𝐿 + 3)

40x =60(x-2)

(𝐿 − 𝐿)2 = 4𝐿(𝐿 − 𝐿)

40

40x

x-2

60

x=6

x -2 = 6 – 2 = 4 hrs

V(h,k) = -3 , -3 72.

68.

V= 3(1)(20) = 60 inch 3+(𝐿𝐿 𝐿

1𝐿𝐿 100𝐿𝐿

) + (70𝐿𝐿 𝐿

1𝐿 1000𝐿𝐿

)

73.

= 3.09m

69. 76 = 80(x) X = 95

70.

74. n = tan 𝐿 n = tanus = 1 y + 3 = 1(x-6) y=x–6–3 x–y=9

75. 100L = 1m 3 1m = 3.28 ft 1L = 0.2642 gal 100 gal = 1𝐿 0.2642

1𝐿3

𝐿 1000𝐿 𝐿 (

3.18 𝐿𝐿 3 1𝐿

)

= 13.37

76.

79.

80. 77.

F = 3 + j10 √2𝐿 = 10 5

a = √2

d = 10m W = Fd

total Area = 6𝐿2

W = 10j(10) = 100 cis 90

A total = 300 ft2 81. V = 1176πm3

78.

𝐿

V = 3 [𝐿1 + 𝐿2 + √𝐿1𝐿2] 1176π = √𝐿𝐿2 (10)2

18 3

[𝐿𝐿2 + 𝐿(10)2 ]+

r=6 𝐿2 = (10 − 6)2 + (18)2 𝐿 = 2√85 2√85

AL =

2

(2π(6) + 2π(10))

AL = 926.77 m2

79. 82. A

83.

88. Y = √(3 − 2𝐿) 1

𝐿 = (3 − 2𝐿)2 −1

1

Y’ = 2 (3 − 2𝐿) 2 (−2) y’ = y= 84.

1 (3−2𝐿)^

1 2

1 √((3−2𝐿)

89. tan(𝐿0.78)2 =

sin(𝐿 0.78)2

log MN =6

cos(𝐿0.78)

= -0.426

log M/N = 2

85.

log M + log N = 6 Log MN – log M = 2 M+N=6 -M + N =2 M =2

N=4

90.

86. 1 mile = 1.609344km=1609.344m 1𝐿

30 ft x 3.28 𝐿𝐿 𝐿 600 𝐿𝐿 𝐿𝐿𝐿

1 𝐿𝐿𝐿

𝐿 60 𝐿𝐿𝐿 𝐿

1𝐿𝐿𝐿𝐿

91.

=5.68x10^-3

1609.344𝐿 0.914𝐿 3.28𝐿𝐿 1𝐿𝐿

𝐿

1𝐿

=30 ft/sec 87.

92. x + x + 2 =36

z = 3 + j4

x = 17 –small

z = 5 cis 53.13

x + 2 = 19 - largest

z=5

93.

99. C = 2xydx + (2 + x^2)dy = 0

y = mx + b

94

m=

1+6 2+

m=3

100.

95.

96.

1 3

97. C 98. 9.434 = √(8 − 2)2 + (3 − 10)2 + (2 − 4)2 Z= 6

ANSWER KEY 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.

A B A B B C A A D C C B B B D C B B A B C A B D C A C B B A C D D B D B C D C B A B

51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92.

B B B D B B A A D A C B A A B A D D A D C B A B C C A B B B A A A D A B C B B C B C

43. 44. 45. 46. 47. 48. 49. 50.

D B C A B A A C

93. 94. 95. 96. 97. 98. 99. 100.

C C B B C C B A

AUGUST 2014

1. What percentage of the volume of a cone is the maximum right circular cylinder that can be inscribed in it? A. 24%

B. 34%

C. 44%

D. 54%

2. A railroad curve is to be laid out on a circle. What radius should be used if the track is to change direction by 30 degrees in a distance of 300 m? A. 566 m

B. 592 m

C. 573 m

D. 556 m

Solution: 30C = 300x360; C = 3600 = 2πr;

r = 𝟓𝟕𝟐. 𝟗𝟔 𝐦 (C)

3. Express in polar form: -3-4i 𝟒

𝟒

A. 𝟓𝐞−𝐢(𝐩𝐢+𝐭𝐚𝐧−𝟏𝟑)

C. √𝟓𝐞−𝐢(𝐩𝐢+𝐭𝐚𝐧−𝟏𝟑)

𝟒

𝟒

B. 𝟓𝐞𝐢(𝐩𝐢+𝐭𝐚𝐧−𝟏𝟑)

D. √𝟓𝐞𝐢(𝐩𝐢+𝐭𝐚𝐧−𝟏𝟑)

4. Find the values of z for which 𝒆𝟒𝒛 = 𝒊. A. 1/6 pi i + ½ kpi i

C. 1/8 pi i + ½ kpi i

B. -1/6 pi i + ½ kpi i

D. -1/8 pi i + ½ kpi i

5. A ladder leans against the side of a building with its foot 12 ft. from the building. How long is the ladder if it makes of 70 degrees with the ground? A. 32 ft

B. 33 ft

C. 34 ft

D. 35 ft

Solution: cos70° =

12 x

;

𝐱 = 𝟑𝟓. 𝟎𝟖 𝐟𝐭 (𝐃)

6. If the 5th term in arithmetic progression is 17 and the 3rd is 10, what is the 8th term? A. 27.5

B. 24.5

C. 36

D. 38

Solution: 𝐀 𝐧 = 𝐀 𝟏 + (𝐧 − 𝟏)𝐝; 𝐀 𝟓 = 𝐀 𝟑 + (𝟐 − 𝟏)𝐝; 𝟏𝟕 = 𝟏𝟎 + (𝟏)𝐝; 𝐝 = 𝟑. 𝟓𝐀 𝐧 = 𝐀 𝐦 + (𝐧 − 𝐦)𝐝; 𝐀 𝟖 = 𝟏𝟕 + (𝟖 − 𝟓)(𝟑. 𝟓) = 𝟐𝟕. 𝟓 (𝐀) 7. A balloon is released of eye level and rises at the rate of 5 ft/s. An observer 50 ft away watches the balloon rise. How fast is the angle of elevation measuring 6 seconds after the moment of release?

A. 0.007 rad/s

B. 0.07 rad/s

C. 0.008 rad/s

D. 0.08 rad/s

Solution: 𝐝𝛉 𝐝𝐭

=

𝟏 𝐭𝐚𝐧−𝟏 𝟏𝟎 𝐭

=

𝟏 𝟏𝟎 𝟏 𝟐 𝟏+( 𝐭) 𝟏𝟎

=

𝟏 𝟏𝟎 𝟐 𝟏 𝟏+( (𝟖)) 𝟏𝟎

𝐫𝐚𝐝

= 𝟎. 𝟎𝟕𝟑𝟓 𝐬𝐞𝐜 (𝑩)

8. If cos z = 2, find cos 3z. A. 7

B. 17

C. 27

D. 37

9. Points (6,-2) and (a,6) are on a line with a slope of 4/3. What is the value of a? A. -2

B. 4.5

C. 9

D. 12

Solution: 𝐦=

𝐲𝟐 − 𝐲𝟏 ; 𝐱𝟐 − 𝐱𝟏

𝟒 𝟔+𝟐 = ; 𝐱 = 𝟏𝟐 (𝐃) 𝟑 𝐱−𝟔

10. The foci of an ellipse are on the points (4,0) and (-4,0) and its eccentricity is 2/3. Find the equation of the ellipse. A. x^2/36 + y^2/20 = 1

C. x^2/20 + y^2/16 = 1

B. x^2/20 + y^2/36 = 1

D. x^2/16 + y^2/20 = 1

11. The plate number of a vehicle consists of 5 alphanumeric sequence is arranged such that the first 2 characters are alphabet and the remaining are 3 digits. How many arrangements are possible if the first character is a vowel and repetition is not allowed? A. 90

B. 900

C. 9,000

D. 90,000

Vowel – Letter – Digit – Digit – Digit and repetition is not allowed 5 X 25

X 10

X

9

X

8 = 90,000

12. One end of a 32-meter ladder resting on a horizontal plane leans on a vertical wall. Assume the foot of the ladder to push towards the wall at the rate of 2 meters per minute. When will be the top and bottom of the ladder move at the same rate? A. 30.4 m

B. 22.6 m

C. 17.75 m

D. 26.6 m

13. A triangle is inscribed in a circle of radius 10. If two angles are 70 degrees and 50 degrees, find the length of the side opposite to the third angle.

A. 15.32

B. 16.32

C. 17.32

D. 18.365

Third angle = 180˚- 70˚- 50˚ = 60˚ The central angle intercepting the same chord is 120˚. Joining the endpoints of the chord to the circle center results in an isosceles triangle have equal legs 10 and odd angle 120˚. Let x= bet the length of the chord. By Cosine Law: 𝑥 2 = 102 +102 -2(10)(10) cos(120˚) 𝑥 2 = 300 𝑥 = √300 𝑥 = 10√3 or 17.32 14 Find the volume generated by revolving the area cut off from the parabola y=4xx^2 by the axis about the line y=6. A. 295 B. 340 C. 286 D. 362 The outer radius, R, is the difference between y=6 and the x-axis (y=0) or 6. The inner radius r, is the difference between y=6 y= 4𝑥 − 𝑥 2 ; at x= 0, x=4 The volume, V is given by the integral 4

v=𝜋 ∫0 ((6)2 − (6 − (4𝑥 − 𝑥 2 ))²)𝑑𝑥 = 294.89 ≈ 𝟐𝟗𝟓 cubic unit 15. The axis of the hyperbola through its foci is known as: A. conjugate axis B. transverse axis C. major axis D. minor axis 16. From past experience it is known 90% of one year old children can distinguish their mother’s voice of similar sounding female. A random sample of 20 one year’s old given this voice recognize test. Find the probability that all children recognize mother’s voice. A. 0.122 B. 0.500 C. 1.200 D. 0.222 The probability that all 20 will recognize Mom is p=.90; 𝑛 = 20 𝑝𝑛 =. 9020 = 𝟎. 𝟏𝟐𝟐 17. If the equation of the directrix of a parabola is x-5=0 and its focus is at (1,0), find the length of its latus rectum. A. 6

B. 8

C. 10

D. 12

18. Describe the locus represented by |𝒛 − 𝒊| = 𝟐. A. circle

B. parabola

C. ellipse

D. hyperbola

19. Evaluate lim (( z – 1 – I )/( z2 -2z+2))

2

z―›1+i A. ¼

B. -1/4

C. ½

D. -1/2

20. Nanette has a ribbon with a length of 13.4 m and divided it by 4. What is the length of each part? A. 3.35 m

B. 3.25 m

C. 3.15 m

D. 3.45m

Length of ribbon = 13.4m and divided by 4 Length of each part =

𝟏𝟑.𝟒 𝟒

= 𝟑. 𝟑𝟓𝒎

21. Simplify 1 (csc x + cot x) 1(csc x – cot x). A. 2 cos x

B. 2 sec x

C. 2 csc x

D. 2 sin x

SOLUTION: = 1/ (1/sinx − cosx/sinx) + 1/ (1/cscx + cosx/sinx) = 1/ 1−cosx/sinx + 1/ 1+cosx/sinx = Sinx/ 1−cosx + sinx/ 1+cosx = (sinx(1+cosx)+sinx(1−cosx)) / (1−cosx)(1+cosx) = (sinx+sinxcosx+sinx−sinxcosx) /1−cos2x = (sinx+sinx) / sin2x = 2sinx / sin2x = 2 / sinx = 2cscx 22. If the area of a sector of a circle is 248 sq. m and the central angle is 135 degrees. Find the diameter of the circle. A. 29 m

B. 26 m

C. 32 m

D. 39 m

SOLUTION: qπr 135(π)(r) 1 1 135πr L= = ; Asector = rL = r ( ) ; r = 14.51; d = 29.01 m (A) 180 180 2 2 180

23. In how many ways can two lines intersect from given 6 lines? A. 14

B. 15

C. 16

D. 17

SOLUTION: n(n-1)/2= 6(6-1)/2 = 15 24. Find the half line of a radioactive substance if 20 percent of it disappears in 40 years. A. 123.25 yrs.

B. 124.25 yrs.

C. 125.25 yrs.

D. 126.25 yrs

25. Find the area of curvature of 𝐲 = 𝐞𝐱 − 𝟐𝐱 at the point (0,1). A. 2.91

B. 2.83

C. 2.72

D. 2.63

26. 3 randomly chose high school students were administered a drug test. Each student was evaluate as positive to the drug test (P) or negative to the drug test (N). Assume the possible combination of the 3 students drug test evaluation as PPP, PPN, PNP, NPN, NNP, NNN. Assume the possible combination is equally likely and knowing that 1 student get a negative results, what is the probability that all 3 students get a negative result? A. 1/8

B. 1/7

C. 7/8

D. ¼

27. A bridge is 1.4 kilometers long. A bus 10 meters long is crossing the bridge at 30 kph. How many minutes will it take the bus to completely cross the bridge? A. 1.82 min

B. 2.82 min

C. 3.82 min

D. 4.82 min

28. Find the fifth term of the sequence 16, 4, 1, -1/4,… A. 4

B. 16

C. ¼

D. 1/16

SOLUTION: 𝟏

𝟏

𝐀 𝐧 = 𝐀 𝟏 𝐫 𝐧−𝟏 = 𝟏𝟔(𝟒)𝟓−𝟏 = 𝟏𝟔 (D) 29. Find the area of the three-leaved rose r = 2 sin 2 theta. A. pi

B. 2 pi

C. 3 pi

D. 4 pi

SOLUTION: 𝐀 = 𝐩𝐢

𝐚𝟐 𝟒

= 𝐩𝐢

𝟐𝟐 𝟒

= 𝐩𝐢 (𝑨)

30. Evaluate lim (x-6) tan (pix/12) x―›6 A. -3.82

60

B. 0

C. -1.91

D. -2.64

31. What is the area of an isosceles triangle whose base is 10 and its base angle is degrees? A. 25 (sqrt of 3) B. 50 (sqrt of 3)

C. 25

D. 50

Solution: A =½ a^2 sin 60 A = ½ (10) (10) sin 60 A=25 square root of 3 32. If y = 2x + sin 2x, what is the value of x so that y' =0? 3 pi/2

B. pi/2

C. pi/3

D. 2 pi/3

Solution: Y= 2x = sin 2x x = ? y= 0 Y = 2x + sin 2x Y^1 = 2+2 cos 2x 0 =2 + 2 cos 2x choose X = pie/2 Then @ rad mode 0 = 2 + 2 cos (2x pie/2) 0=0 =pi/2 33. What is the vector length 2 and direction 150 degrees in the form ai + bj. 1.73i + j

B. -1.73i – j

C.1.73i - j

D. -1.73i + j

Solution: @ COMPLEX MODE Z = 2 angle 150 Z = -1.73i plus j 34. If a man works at an average speed of 4 kph, what is the time consume to reach 250 m. 0.25 min

B. 2.50 min

Solution: V= s/t T =0.25 km/4km/hr x hr/60min

C. 3.75 min

D. 4.25 min

T = 3.75 min

35. N engineers and N nurses, if two engineers are replaced by nurses, 51% of the engineers and nurses are nurses. Find N.

100

B. 110

C.50

D.200

Solution: N ENGINEERS, N NURSES 0.51 (2x) = x + 2 1.02x = x + 2 X = 100 36. A house has assessed value of P 720,000.00 worth which is 60% of the market value. If the tax is P 3.00 for P 1,000.00 market value, how much is the tax? P 3, 200.00 B. P 3,800.00

C. P 3,600.00

D. 3,400.00

Solution: ASSESSED VALUE =720,000 Tax = 3 FOR EVERY 3 Php of Market Value X =market value = 0.6 of as V. 720,000 = 0.6 (x) X =1,200,000/1000 x3 X = 3600Php 37. 1/6 is what percent of ¾? 37.5

B. 66.67

C. 50

Solution: ½ =x(3/4)

D.75

X =66.7%

38. In a hotel it is known that 20% of the total reservation will be cancelled in the last minute. What is the probability that out of 15 reservatons there will be more than 8 but less than 12 cancelled? 0.00784

B. 0.0784

C. 0.000784

D. 0.784

Solution: N = 9,10,11 Pr = n Cr (p) ^(q)^n-r N =15 reservations Pa =15 C9 (0.20)^9 (0.8)^15-9= 6.718x 10^-4 Pb =15 C10 (0.20)^10 (0.8)^15-10= 1x 10^-4 Pc =15C 11(0.20)^11 (0.8)^15-11 =1.1145 x 10 ^ -5 Pt =Pa +Pb +Pc Pt =0.000784 39. If 16 is more than 4x, find x. 1.4

B. 3

C. 12

D. 5

Solution: 16 =4 + 4x X =12/4 X =3 40. Locate the midpoint of the line segment joining point 1(2,15,4)and point 2 (6,3,12) A. (4,9,4) B. (4,-9,4) C. (4,94) D. (-4,9,4) Solution:P1(2,15,4)

P2(6,3,-12)

MP(2 + 6/2,15 +3/2, 4-12/2) MP(4,9,-4) 41. A conic section whose eccentricity is greater than one (1) is known as? A. A parabola

B. an ellipse

C. a circle

D. a hyperbola

42. Find the distance travelled by the tip of a pendulum if the distance of the first swing is 8 cm and the distance of each succeeding is 0.75 of the distance of the previous swing. A. 32 cm

B. 28 cm

C. 27 cm

D. 30 cm

43. Describe the locus represented by the curve |𝒛 + 𝟐𝒊| + |𝒛 − 𝟐𝒊| = 𝟔. A. circle

B. parabola

C. ellipse

D. hyperbola

44. Find the area bounded by the curve 𝒚𝟐 = 𝟑𝒙 − 𝟑 and the line x = 4. A. 10

B. 16

C. 15

D. 12

Solution: 𝐲 𝟐 = 𝟑𝐱 − 𝟑 = 𝟑(𝟒) − 𝟑 = 𝟗; 𝐲 = 𝟑 & 𝐲 = −𝟑 𝟑 𝐲𝟐 + 𝟑 ∫ ( ) 𝐝𝐲 = 𝟏𝟐 (𝑫) 𝟑 −𝟑

45. Helium is escaping a spherical balloon at the rate of 2 cm3 /min. When the surface area is shrinking at the rate of 1/3 cm2 /min, find the radius of the spherical balloon. A. 14 cm

B. 12 cm

C. 16 cm

D. 8 cm

46. What is the maximum area of the rectangle whose base is on the z-axis and whose upper two vertices lie on the parabola 𝒚𝟐 = 𝟏𝟐 − 𝒙𝟐 . A. 30

B. 32

Solution: A(x)=2x(12−x2) A(x)=24x−2x3A(x)=24x−2x3

C. 36

D. 40

A′(x)=24−6x2.A′(x)=24−6x2. Solving A′(x)=0A′(x)=0 gives x=2x=2 and A(x)=2⋅2⋅(12−22)=2⋅2⋅8=32A(x)=2⋅2⋅(12−2 2)=2⋅2⋅8=32 47. A car racer covers 225 km in 2.5 hrs. How far can he go in 1.75 hrs? A. 267.5 km

B.168.75 km

C. 394 km

D. 157.5 km

Solution: 1. 225km/2.5hrs =x/1.75hrs X=157.5 (D) 48. Find the area of the triangle with vertices A (0,1), B (5,3), and C (-2,-2) A. 19

B. 19/2

C. 15

D. 15/2

49. What is the sum of coefficients of the expansion of (𝟐𝒙 − 𝟏)𝟐𝟎 ? A. 0

B. 1

C. 2

D. 3

Solution: ((2)(1)-1)20-(-1)20=0 50. The parabola defined by the equation 𝟑𝒚𝟐 + 𝟒𝒙 = 𝟎 opens ____________. A. upward

B. downward

C. to the left

D. to the right

51. How many tiles 10 cm on a side are needed to cover a rectangular wall 3 m by 4 m? A. 1500 Solution:

B. 1000

C. 1200

D. 1600

𝐀 𝐫𝐞𝐜 = 𝟑𝐱𝟒 = 𝟏𝟐 = 𝐱𝐀 𝐬𝐦𝐚𝐥𝐥 𝐫𝐞𝐜 ; 𝐱 = 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐬𝐦𝐚𝐥𝐥 𝐫𝐞𝐜; 𝐱=

𝐀 𝐫𝐞𝐜 𝐀 𝐬𝐦𝐚𝐥𝐥 𝐫𝐞𝐜

=

𝟏𝟐𝐦𝟐 = 𝟏𝟐𝟎𝟎 (𝐂) 𝟏𝐦 𝟏𝟎 𝐜𝐦𝐱 𝟏𝟎𝟎 𝐜𝐦

52. Find the equation of the line whose slope is -3 and the x-intercept is 5. A. 𝒚 = −𝟑𝒙 + 𝟓

B. 𝟑𝒙 − 𝒚 = 𝟓

B. C. 𝒚 = 𝟑𝒙 + 𝟏𝟓

D. 𝟑𝒙 + 𝒚 = 𝟏𝟓

Solution 𝐲 = 𝐦(𝐱 − 𝐱 𝟏 ) = −𝟑(𝐱 − 𝟓) = −𝟑𝐱 + 𝟏𝟓 = 𝐲 𝐨𝐫 𝟑𝐱 + 𝐲 = 𝟏𝟓 (D) 53. In how many ways can the letters of the word “CHACHA” be arranged by taking the letters all at a time? A. 120

B. 720

C. 85

D. 90

54. Find the equation of the horizontal line though (-4,3). A. x = 4

B. x = -4

C. y = 3

D. y = -3

C. -36

D. -28

Solution (4,3) X=-4 vertical line Y=3 horizontal line (C) 55. If g(x) = 9f(x) and f(-6), find g’(-6) A. -54

B. -40

Solution g(x)=9f(x);f(-6)=-6,find g’(-6) g’(-6) =9f(x) g’(-6) =9f(-6) =-54 (C) 56. Determine k so that the points A (7,3), B (-1,0), and C (k,-2) are the vertices of a right triangle with right angle at B. A. -1

B. 1

C. -1/4

D. ¼

57. The radius of the circle 𝒙𝟐 + 𝒚𝟐 + 𝟒𝒙 − 𝟔𝒚 − 𝟑 = 𝟎 is _______ A. 2 B. 3 C. 4 𝟐 𝟐 Solution: 𝒙 + 𝒚 + 𝟒𝒙 − 𝟔𝒚 − 𝟑 = 𝟎 𝒙𝟐 + 𝟒𝒙 + 𝟒 𝒚𝟐 − 𝟔𝒚 + 𝟗 = 𝟑 + 𝟒 + 𝟗 (𝒙 + 𝟐)𝟐 + (𝒚 − 𝟑)𝟐 = 𝟒𝟐 (𝒙 + 𝒃)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐 𝒓 = 𝟒 (C)

D. 5

58. If the logarithm of MN is 6 and the logarithm N/M is 2, find the logarithm of N. A. 3

B. 4

C. 5

D. 6

𝐥𝐨𝐠 𝐌𝐍 = 𝐥𝐨𝐠 𝐌 + 𝐥𝐨𝐠 𝐍 = 𝟔; 𝐥𝐨𝐠 𝐌 = 𝟔 − 𝐥𝐨𝐠 𝐍 𝐥𝐨𝐠 𝐍 𝐥𝐨𝐠 𝐍 − 𝐌 = = 𝟐; 𝐥𝐞𝐭 𝐱 = 𝐥𝐨𝐠 𝐍; 𝐥𝐨𝐠 𝐌 𝐱 = 𝟒 (𝐁)

Solution

𝐥𝐨𝐠 𝐍

𝐱

= 𝟔−𝐱 = 𝟐; 𝐥𝐨𝐠 𝐌

59. If 4 electricians earn x pesos in 7 days, how much can 14 carpenters paid of the same rate, earn in12 days? A. 3x

B. 4x

C. 5x

D. 6X

60. Write the differential equation of the family of circle with center at the origin. A. 𝐱𝐝𝐲 + 𝐲𝐝𝐱 = 𝟎

B. 𝐱𝐝𝐲 − 𝐲𝐝𝐱 = 𝟎

C. 𝐱𝐝𝐱 + 𝐲𝐝𝐲 = 𝟎

D. 𝐱𝐝𝐲 − 𝐲𝐝𝐱 = 𝟎

No Answer! 61. Find the volume of a spherical segment, the radii of whose bases are 4 m and 5 m respectively with an altitude of 6 m. A. 159 pi

B. 165 pi

C. 150 pi

D. 145 pi

62. A taxpayer’s state and the federal income taxes plus an inheritance tax totaled $ 14,270. His California state income tax was $ 5,780 less than his federal tax. His inheritance tax was $ 2, 750. How much did he pay in state tax? A. $ 8,560

B. $ 2,870

C. $ 8,650

D. $ 2,780

𝐱 = 𝐬𝐭𝐚𝐭𝐞 𝐢𝐧𝐜𝐨𝐦𝐞 𝐭𝐚𝐱; 𝐲 = 𝐟𝐞𝐝𝐞𝐫𝐚𝐥 𝐢𝐧𝐜𝐨𝐦𝐞 𝐭𝐚𝐱; 𝐳 = 𝐢𝐧𝐡𝐞𝐫𝐢𝐭𝐚𝐧𝐜𝐞 𝐱 + 𝐲 + 𝐳 = 𝟏𝟒, 𝟐𝟕𝟎; 𝐱 = 𝐲 − 𝟓, 𝟕𝟖𝟎; 𝐳 = 𝟐, 𝟕𝟓𝟎 𝟏𝟒, 𝟐𝟕𝟎 = (𝐲 − 𝟓, 𝟕𝟖𝟎) + 𝐲 + 𝟐, 𝟕𝟓𝟎; 𝐲 = 𝟖𝟔𝟓𝟎; 𝐱 = 𝟖, 𝟔𝟓𝟎 − 𝟓, 𝟕𝟖𝟎 = $ 𝟐, 𝟖𝟕𝟎 (B) Solution:

63. The first term of a geometric sequence is 160 and the common ratio is 3/2. How many consecutive terms must be taken to give a sum of 2110? A. 3 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏: 𝐒𝐧 =

B. 4 𝐚𝟏 (𝐫 𝐧 −𝟏) 𝐫−𝟏

=

C. 5 𝟑𝐗 −𝟏) 𝟐

𝟏𝟔𝟎(

𝟑 −𝟏 𝟐

;

D. 6

𝐗 = 𝟓. 𝟎𝟖 (C)

64. The total area of a cube is 150 sq. in. A diagonal of the cube is ______ in. A. 5(sqrt of 2)

B. 4(sqrt of 3)

C. 5(sqrt of 3)

D. 4(sqrt of 2)

Solution: 𝐀 𝐜𝐮𝐛𝐞 = 𝟔𝐚𝟐 ;

𝟏𝟓𝟎

𝐚=√

𝟔

= 𝟓;

𝐝 = 𝐚√𝟑 = 𝟓√𝟑 (C)

65. In triangle ABC, sin (A+B) = 3/5. What is the value of sin C? A. 2/5

B. 2/3

C. 3/5

D. ½

66. Find the slope of the curve whose parametric equations are x = -1 +t and y=2t. A. 2

B. 3

C. 1

D. 4

67. Find the length of the latus rectum of the ellipse 25x^2 + 9^2 – 300x – 144y + 1251 = 0. A. 3.4

B. 3.2

C. 3.6

D. 3.0

68. A triangular trough whose the edges are 5, 5, and 8 m long is place vertically in water with its longest edege uppermost, horizontal, and 3 m below the water level. Calculate the force on a side of the plate. A.235.2 kN

B. 470.4 kN

C. 940.8 kN

D. 1,881.6 Kn

69. Find the area of the ellipse whose eccentricity is 4/5 and whose major axis is 10. A. 12 pi

B. 13 pi

C. 14 pi

D. 15 pi

70. Find the average rate of change of the area of a square with respect to its side x as x changes from 4 to 7. A. 14

B. 11

C. 12

Solution: 𝐫𝐚𝐭𝐞 𝐨𝐟 𝐜𝐡𝐚𝐧𝐠𝐞 =

∆𝐀 ∆𝐱

=

𝟕𝟐 −𝟒𝟐 𝟕−𝟒

D. 13

= 𝟏𝟏 (𝐁)

71. Find the moment of inertia with respect to the y-axis of the area bounded by y = x^2 and y = 2x. A. 11/5

B. 9/5

C. 7/3

D. 8/5

72. Find the length of the arc of r = 4 sin u from u = 0 to u = pi/2. A. pi 𝐫 = 𝟒𝐬𝐢𝐧𝐮;

B. 2 pi

C. 3 pi 𝛑

𝐫 = 𝟒; 𝛉 = 𝟐 ;

𝛑

𝐒 = 𝐫𝛉;

D. 4 pi 𝛑

∫𝟎𝟐 𝟒𝐬𝐢𝐧 𝐮 𝐫 = 𝟒 𝟐 ;

𝐒 = 𝟐𝛑 (B)

73. What is the angle between -2.5 + j4.33 and 4.33 – j2.5? A. 0 deg

B. 30 deg

C. 120 deg

D. 150 deg

−𝟐. 𝟓 + 𝐣. 𝟑𝟑 = 𝟓∠𝟏𝟐𝟎;

𝟏𝟐𝟎 − (−𝟑𝟎) = 𝟏𝟓𝟎 (D)

74. If A = (2, 4) and B = (4,3), find |𝟕𝑨 − 𝑩|. A. sq. rt. of 21

B. sq. rt. of 1061

B. C. sq. rt. of 41

D. sq. rt. of 949

75. Find the initial poin of v = -3i + j +2k if the terminal point is (5, 0, -1). A. (8,1, -3)

B. (8, -1, 3)

C. (8,-1,-3)

D. (8,1,3)

76. What is the laplace transform of 1/sqrt of t? A. (sqrt of pi)/s^2

B. (sqrt of pi)/s

C. pi/sqrt of s

D. sqrt of (pi/s)

Solution: Pi (n+1)/ s^n+1 = sqrt (pi/s) 77. A pair of dice is tossed. Find the probability of getting at most a total of 5. A. 5/9

B. 5/16

C. 5/18

D. 5/36

Solution: PE= E/S =5(2)/6^2 =10/36 = 5/18 78. On a day when the temperature is 30 deg C. a cool drink is taken from a refrigerator whose temperature is 5 deg. C. If the temperature of the drink is 20 deg C after 10 minutes, what will its temperature be after 20 minutes? A. 21 deg C

B. 24 deg C

C. 28 deg C

D. 26 deg C

Solution: MODE 3,5 INPUT: X Y 0 30 10 20+5 AC, 20, SHIFT 1,7,5,5 = 20.83 =25.83 + 5 = 26 79. The positive value of k which will make 4x^2 – 4kx + 4k +5 a perfect square trinomial is A. 6

B. 5 4x^2 – 4(5)x + 4(5) + 5

C. 4

D. 3

4x^2 – 20x + 25 Therefore = 5 80. A stone advertises a 20 percent-off sale. If an article is marked for the sale at $24.48, what is the regular price? A. $30.60

B. $34.80

C. $36.55

D. $28.65\

Solution: 𝟐𝟒. 𝟒𝟖 = 𝐱 + 𝟎. 𝟐𝟎𝐱;

𝐱 = 𝟑𝟎. 𝟔 (𝐀)

81. For a given arithmetic series the sum of the first 50 terms is 200, the sum of the next 50 terms is 2700. The first term of the series is: A. -12.2

B. -21.5

C. -20.5

D. -25.2

The sum of the first n terms of an arithmetic sequence is given by: . . S(n) .= .(n/2)[2a + (n-1)d] The sum of the first 50 terms is 200. . . S(50) = (50/2)[2a + 49d] = 200 → 2a + 49d = 8 [1] The sum of the next 50: (sum of the first 100) - (sum of the first 50) . . (100/2)[2a + 99d] - 200 Hence, we have: .50[2a + 99d] - 200 = 2700 → 2a + 99d = 58 [2] Subtract [1] from [2]: .50d = 50 → d = 1 Substitute into [1]: .2a + 49(1) = 8 → a = -41/2 82. The total area of a cube is 150 sq. in. A diagonal of the cube is: A. 4 in 𝐀 𝐜𝐮𝐛𝐞 = 𝟔𝐚𝟐 ;

B. 5 in 𝟏𝟓𝟎

𝐚=√

𝟔

C. 7.07 in = 𝟓;

D. 8.66 in

𝐝 = 𝐚√𝟑 = 𝟓√𝟑 = 𝟖. 𝟔𝟔 (D)

83. A tree is broken over by a windstorm. The tree was 90 feet high and the top of the tree is 25 feet from the foot of tree. What is the height of the standing part of the tree? A. 48.47 ft

B. 41.53 ft

C. 45.69 ft

Soln: 𝟐𝟓 = √(𝟗𝟎 − 𝒙)𝟐 − (𝒙)𝟐

D. 44.31 ft

√𝟖𝟏𝟎𝟎 − 𝟏𝟖𝟎𝒙 + 𝒙𝟐 − 𝒙𝟐 (𝟐𝟓)𝟐 = √𝟖𝟏𝟎𝟎 − 𝟏𝟖𝟎𝒙 𝟔𝟐𝟓 = 𝟖𝟏𝟎𝟎 − 𝟏𝟖𝟎𝒙 𝟏𝟖𝟎𝒙 𝟖𝟏𝟎𝟎 − 𝟔𝟐𝟓 = 𝟏𝟖𝟎 𝟏𝟖𝟎 𝒙 = 𝟒𝟏. 𝟓𝟑 84. Goods cost a merchant & 72. At what price should he mark them so that he may sell them at a discount of 10% from his marked price and still make a profit of 20% on the selling price? A. $ 150

B. $ 200

C. $ 100

D. $ 250

Solution: 𝐱 = 𝟕𝟐 + 𝟕𝟐(𝟎. 𝟏𝟎 + 𝟎. 𝟏𝟎 + 𝟎. 𝟐𝟎) = 𝟏𝟎𝟎. 𝟖 (C) 85. An edge of the base of a regular hexagonal prism is 4 in. and a lateral edge is 9 in. Find the lateral area of the prism. A. 216 sq. in.

B. 299 sq. in.

C. 206 sq. in.

D. 288 sq. in.

Solution: B= 4in h= 9 s=6 𝑨 = 𝟒 ∗ 𝟔 ∗ 𝟗 = 𝟐𝟏𝟔 𝒔𝒒𝒎. 𝒊𝒏.

86. In a potato race, 8 potatoes are place 6 ft apart on a straight line, the first being 6 ft from the basket. A contestant starts from the basket and puts one potato at a time into the basket. Find the total distance must run in order to finish the race. A. 423 ft B. 432 ft C. 428 ft D. 436 ft Solution: a1=6x2=12, n=8, d=12, an=a1+(n-1)d an=12+(8-1)(12)= 96 𝑛

S = ( 2) ∗ (𝑎1 + 𝑎𝑛) 8

S = (2) ∗ (12 + 96) S = 432 ft

87. Given that sin theta = 3/5 and theta is acute, find cos 2theta. A. -7/25

B. -4/5

C. 7/25

D. 4/25

Solution: 𝟑 ) 𝐬𝐢𝐧−𝟏 𝟓

𝐬𝐢𝐧−𝟏 (𝒔𝒊𝒏𝒕𝒉𝒆𝒕𝒂 =

𝒕𝒉𝒆𝒕𝒂 = 𝐬𝐢𝐧−𝟏 𝟑⁄𝟓 𝐜𝐨𝐬 𝟐 𝐬𝐢𝐧−𝟏

𝟑 𝟓

= 𝟕/𝟐𝟓 88. A side and a diagonal of a parallelogram are 12 inches and 19 inches, respectively. The angle between the diagonals, opposite the given side is 124 degrees. Find the length of the other diagonal. A. 7.48 in

B. 7.84 in

C. 8.47 in

D. 8.74 in

Solution: 𝟏𝟐 𝟗. 𝟓 = ; 𝛃 = 𝟒𝟏. 𝟎𝟐° 𝐬𝐢𝐧𝟏𝟐𝟒 𝐬𝐢𝐧𝛃 𝛂 = 𝟏𝟖𝟎° − 𝛃 − 𝟏𝟐𝟒° = 𝟏𝟖𝟎° − 𝟒𝟏. 𝟎𝟐° − 𝟏𝟐𝟒° = 𝟏𝟒. 𝟗𝟖° 𝐛𝐲 𝐜𝐨𝐬𝐢𝐧𝐞 𝐥𝐚𝐰: 𝐚 = √𝟗. 𝟓𝟐 + 𝟏𝟐𝟐 − 𝟐(𝟗. 𝟓)(𝟏𝟐)𝐜𝐨𝐬𝟏𝟒. 𝟗𝟖° = 𝟑. 𝟕𝟒𝟏𝟒 𝟐𝐧𝐝 𝐝𝐢𝐚𝐠𝐨𝐧𝐚𝐥 = 𝟐𝐚 = 𝟐(𝟑. 𝟕𝟒𝟏𝟒) = 𝟕. 𝟒𝟖𝟑 𝐢𝐧𝐜𝐡𝐞𝐬 (𝐀) 𝐛𝐲 𝐬𝐢𝐧𝐞 𝐥𝐚𝐰:

89. A window in Mr. Royce’s house is stuck. He takes an 8-inch screwdriver to pry open the window. If the screwdriver rests on the still (fulcrum) 3 inches from the window and Mr. Royce has to exert a force of 10 pounds on the other end to pry open the window, how much force was the window exerting? A. 12-2/3

B. 14-2/3

C. 18-2/3

D. 16-2/3

90. A boat, propelled to move at 25 mi/hr in still water, travels 4.2 mi against the river current in the same time that it can travel 5.8 mi with the current. Find the speed of the current in mi/hr. A. 4

B. 5

C. 3

D. 2

91. An open-top cylindrical tank is made of metal sheet having an area of 43.82 square meter. If the diameter is 2/3 the height, what is the height of the tank?

A. 3.24 m Solution:

B. 2.43 m

C. 4.23 m

D. 5.23 m

𝐴 = 43.82 𝑠𝑞. 𝑚 2 𝑑= ℎ 3 𝑑 = 2𝑟 2 1 2𝑟 = ℎ , 𝑟 = ℎ 3 3 𝐴 = 2𝜋𝑟ℎ 1 43.82 = 2𝜋 ( ℎ) (ℎ) 3 𝒉 = 𝟒. 𝟓𝟕 92. How much water must be added to 8 gallons of 80% boric solution to reduce it to a 50% solution? A. 4 gal

B. 4-4/5 gal

C. 5 gal

Solution: 𝑙𝑒𝑡 𝑥 = 𝑎𝑛𝑜𝑢𝑛𝑡 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑡𝑜 𝑏𝑒 𝑎𝑑𝑑𝑒𝑑

+

80%

=

x

50%

D. 5-3/5 gal

8 gal.

8+𝑥 .8(8) + 0(x)= .70(8+𝑥) 𝟒

x= 𝟒 𝟓 93. The line y = 3x + b passes through the point (2, 4). Find the b. A. 2 Solution:

B. 10

C. -2

D. -10

𝑦 = 3𝑥 + 𝑏 4= 3×2 +𝑏 𝑏 = −2 94. The simplest form of in (𝒆𝟑𝒙 ) is ______ A. 3 Solution:

B. 𝒆𝒙

C. e

D.3x

𝑙𝑛(33𝑥 ) ln(𝑒 𝑛 ) 𝑙𝑛(𝑒 3𝑥 ) = 𝟑𝐱

95. Thirty degrees is how many radius? A. pi/3 Solution:

B. pi/6

C. pi/4

D.

pi/2

𝜋

30°(180°) 𝝅 𝟏𝟖𝟎 96. If the measure of one angle of a regular polygon is 135 degrees, then the number of sides of that polygon is ______. A. 4

B. 6

C. 8

D. 9

C. 1/s-1

D. 1/s+1

97. What is the Laplace transform of 𝒆−𝟐𝒕 A. 1/s-2 Solution:

B. 1/s+2 𝑒 −2𝑡 𝐿[𝑒 ±𝑎𝑡 ] =

1 𝑠∓𝑎

𝟏 𝒔+𝟐 98. The area in the second quadrant of the circle 𝒙𝟐 + 𝒚𝟐 = 𝟑𝟔 is revolved about the line y+10=0. What is the volume?

A. 228.63

B. 2228.83

C. 2233.43

D. 2208.53

99. The average of six scores is 83. If the highest score is removed, the average of the remaining scores is 81.2. Find the highest score. A. 91

B. 92

C. 93

D. 94

Solution: 6 𝑥 83 = 498 5 𝑥 81.2 = 406 498 − 406 = 𝟗𝟐

100. The sum of the base and altitude of an isosceles triangle is 36 cm. Find the altitude of the triangle if its area is to be a maximum. A. 18 cm Solution:

B. 16 cm

C. 9 cm

1 (𝑏 + 𝑏2 )ℎ 2 1 𝑏1 + 𝑏2 + ℎ = 36 1 𝐴 = (36 − ℎ)ℎ 2 ℎ2 𝐴 = 18 − 2 Taking the derivative 𝑑𝐴 = 18 − ℎ 𝑑ℎ 𝒉=𝟏 𝐴=

D. 17 cm

MARCH 2015

1. Sand is pouring to from a conical pile such that its radius is always twice its height. If the volume of a conical pile is increasing at the rate of 2 cu. m/sec. how fast is the height is increasing when the height is 4m? A. 1/16pi m/s B. 1/32 pi m/s C. 1/64 pi m/s D. 1/8 pi m/s SOLUTION V=(лr^2h)/3 dv/dt=2 r=2h r=Rh/H V=(лr^2h)/3 V=лR^2h^3/3H^2 dv/dt=лR^2h^2/H^2 2=лR^2r^2H^2/H^2R^2 2=л(2(4))^2 h=1/32л 2. A triangular corner lot has perpendicular sides of lengths 90 m and 60 m. find the dimension of the largest rectangular building that can be constructed on the lot with sides parallel to the streets. A. 30 m x 30 m B. 24 m x 24 m C. 25 m x 40 m D. 45m x 30 m 3. Joy is 10% taller than Joseph and Joseph is 10% taller than Tom. How many percent is Joy taller than Tom? A. 18% B. 20% C. 21% D. 23% SOLUTION Tom=10% + 10% =20% 4. What is the length of the shortest line segment in the first quadrant drawn tangent to the ellipse b2 x2 + a2 y2 = a2 b2 and meeting to the coordinate axes? A. a/b B. a + b C. ab D. b/a U 5. What is the area of largest rectangle that can be inscribed in an ellipse with equation 4x^2+y^2=4? A. 3 B. 4 C. 2 D. 1 6. A company hires 30 new employees today. It increases their workforce by 5%. How many workers now? A. 610 B. 600 C. 630 D. 620 SOLUTION 30/.05=workers workers=600 workers now=600 +30 =630

7. Find the radius of the circle inscribed in the triangle determined by the lines y=x+4, y=-x-4 and y=7x-2. A.5/sqrt of 2 B. 5(2sqrt of 2) C. 3/R D. 3/(2sqrt of 2) 8. What is the ratio of the surface area of a sphere to its volume? A. 5/R B. 4/R C. 3/R D. 2/R SOLUTION ratio=4лr^2/4лr^3/3 ratio= 3/r 9. Using original diameter, d, what is the new diameter when the volume of the sphere is increased 8 times? A. 2d B.3d C.4d D. 5d SOLUTION V= eight times 8=4л(d/2)^3/3 48/ л=d^3 2(6/ л)^1/3=d V= original 6/ л=d^3 (6/ л)^1/3=d There 2d 10. In a hotel it is known that 20% of the total reservation will be cancelled in the last minute. What is the probability that there will be less than 2 reservations cancelled out of 4 reservations? A. 0.6498 B. 0.5629 C. 0.3928 D. 0.4096 11. Find the area of the region inside the triangle with vertices (1,1), (3,2) and (2,4) A. 5/2 B. 3/2 C. ½ D. 7/2 SOLUTION 11 3 3 2 2 1 A = 2[ + + ] 1 2 2 4 4 1 1

A = 2 [(2 − 3) + (12 − 4) + (2 − 4)] 𝟓

A = 𝟐 or 2.5 square units

12. The three sides of trapezoid are each 10m long. How long must the fourth side be to make the area a maximum? A. 20m

B. 50m

𝑐 2 = 𝑎2 + 𝑏 2

C. 52m

102 = ℎ2 + 𝑥 2

D. 45m

100 = ℎ2 + 𝑥 2

Base 2 = 2X + 10 1

A = 2 (ℎ)(𝑏𝑎𝑠𝑒1 + 𝑏𝑎𝑠𝑒2) 1

A = 2 (100 − 𝑥 2 )(10 + 2𝑋 + 10) 1

A = 2 (√100 − 𝑥 2 )(2𝑋 + 20) 1

A = 2 (2)(𝑋 + 10)(√100 − 𝑥 2 ) A = (𝑋 + 10)(√100 − 𝑥 2 )

Maximum area 𝑑𝐴 𝑑𝑥 𝑑𝐴 𝑑𝑥 𝑑𝐴 𝑑𝑥

𝑑

= (√100 − 𝑥 2 )(1) + (𝑋 + 10) ∗ 𝑑𝑥

(√100− 𝑥2

2√100− 𝑥 2

= (√100 − 𝑥 2 ) + (𝑋 + 10) ∗ = (√100 − 𝑥 2 ) −

(√100 − 𝑥 2 ) =

𝑥 2 + 10𝑥 √100− 𝑥2

𝑥 2 + 10𝑥 √100 − 𝑥 2

100 − 𝑥 2 = 𝑥 2 + 10𝑥 2𝑥 2 + 10𝑥 + 100

−2𝑥 2√100− 𝑥 2

=0

=0

=0

ℎ = √𝑥 2 − 100

𝑥 = 5 , 𝑥 = −10 Base 2 = 2(5) +10 Base 2 = 20

13. Simplify i^39 A.1 B. -1 SOLUTION 𝑖 39 = [(𝑖 2 )19 ](𝑖) 𝑖 39 = (−1) ∗ 𝑖 𝒊𝟑𝟗 = 𝒊

D. –i

C. i

14. What is the value of x in Arc tan 3x+Arc tan2x= 45deg? A. -1/6 and 1 B. 1/6 and -1 C. 1/6 SOLUTION Arc tan 3x+Arc tan2x= 45 tan[Arc tan (6X) = 45 ] 6X = 1 X = 1/6

D. -1

15. Find the moment of inertia of the area bounded by the parabola y^2=4x and the line x=1 with respect to the x-axis A. 2.133 B. 1.333 C. 3.333 D. -1 SOLUTION 𝑦 2 = 4𝑥 @ x = 1 𝑦 2 = 4(1) y = 2, y = -2 therefore the parabola intersect at points (1,2) and (1,-2) 2

2

Ix = 2 ∫0 𝑦 2 (𝑥𝑑𝑦) = 2 ∫0 𝑦 2 (𝑦𝑙 − 𝑦𝑝)𝑑𝑦 2

Ix = 2∫0 𝑦 2 (1 − 𝑦3

Ix = 2 [ 3 − 23

Ix = 2( 3 − Ix = 2.133

𝑦5

𝑦2 4

] 0 to 2

4(5) 25

4(20)

2

𝑑𝑦) = 2 ∫0 (𝑦 2 −

𝑦4 4

) 𝑑y

16. The cost per hour of running a boat is proportional to the cub of the speed of the boat. At what speed will the boat run against a current of 8 kph in order to go a given distance most economically? A. 15kph B. 14kph C. 13kph D. 12kph SOLUTION Let C = cost per hour x = Speed of the boat Ct = total cost C = kx3 eq 1 t=

𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑠𝑝𝑒𝑒𝑑

𝑑

= 𝑥−8 eq 2

Ct = Ct Substitute eq 1, 2 and 3 𝑑

Ct = kx3(𝑥−8) (x − 8)(3kdx 3 ) − kdx 3 (1) 𝑑Ct = 𝑑𝑥 ( x − 8)2 ( x – 8)(3x2) = x3 3x3 – 24x2 = x3 2x3= 24x2 x = 12 kph 17. What is the unit vector which is orthogonal both to 9i+9j and 9i+9k? A. i/sqrt3+i/sqrt3+k/sqrt3 B. i/sqrt3+jsqrt3+k/3 C. i/sqrt3-jsqrt3ksqrt3 18. A train running at 60 kph decelerated at 2.5m/min2 for 12 minutes. Find the distance traveled in km within the period. A. 1.182 B. 11.82 C. 1.812 D. 2.282 SOLUTION 1

S = VIt + 2 𝑎𝑡 2 S=

60𝑘𝑚 ℎ𝑟

1ℎ𝑟

(60𝑚𝑖𝑛) (12) +

1 2

(2.5)(12)2

S = 11820 meters or 11.820 KM 19. A conic section whose eccentricity is less than one (1) is known as: A. A parabola B. an ellipse C. a circle D. a hyperbola

20. A transmitter with a height of 15m is located on top of a mountain whis is 3.0 km high. What is the farthest distance on the surface of the earth that can be seen from the top of the mountain? Take the radius if the earth to be 6400 km. A. 225km B.152km C.196km D. 205km SOLUTION d =√ℎ2 + 2𝑅ℎ d = √3.0152 + 2(3.015)(6400) d = 196.471 ≅ 196𝑘𝑚

21. A political scientist asked a group of people how they felt about two political policy statements. Each person was to respond A (agree) (N) neutral or (D) disagree to each NN, NA, DD, DN, DA, AA, AD and AN. Assuming each response combination is equally likely, what is the probability that the person being interviewed agrees with exactly one of the political policy statements? A. 1/9 B. 2/5 C. 2/9 C. 4/9 SOLUTION AA, AN, AD, NN, NA, ND, DD,DA &DN P(A) = 4/9 22. Evaluate Laplace transform of t^n A. n!/s^n B. n!/s^(n+1)

C. n!/s^(n-1)

D. n! s^(n+2)

23. Find the area of a quadrilateral having vertices at (2,-1), (4.3), (-1,2) and (-3,2) A. 16 B. 18 C. 17 D. 14 SOLUTION AB = square root of ((4-2)^2) + (3+1)^2)) = 4.47 BC = square root of ((-1-4)^2) + (2-3)^2)) = 5.1 CD = square root of ((-3+1)^2) + (-2-2)^2)) = 4.47 DA = square root of ((2+3)^2) + (-1+2)^2)) = 5.1 SA=(1/2) times square root of (AB+BC+CD+DA) SA=(1/2) times square root of (4.47+5.1+4.47+5.1) SA=18 sq. unit 24. In a 15 multiple choice test questions with five possible choices of which only one is correct, what is the standard deviation of getting a correct answer? A. 1.55

B. 1.07

C. 1.50

D. 1.65

25. In polar coordinate system the distance from a point to the pole is known as: A. Polar angle B. radius vector

C. x- coordinate

26. Evaluate Laplace transform of cos2kt. A. s/s(s2 -2k2 ) B. s/(s2+2k2) SOLUTION Cosbt = s^2/(s^2+b^2) b=2k cos2kt = s^2/(s^2+4k^2) 27. Find the power series of tan-1 (t2) A. T2+t6/2 +t12/6 +t24/12+… B. T2 - t6/2 + t12/6 – t24 /12+… 28. Simplify (1+tanx)/(1-tanx) A. Sec x + tan x B. cos x + tan x

C. s/(s2-4k2)

D. y-coordinate

D. s/(s2+4k2)

C. t2+t6/3+t10/5+t14/7+… D. t2-t6/3+t10/5-t14/7+…

C. cos 2x+ tan 2x

29. Evaluate lim x+4/x-4 as x approaches to infinity A. 1 B. 0 C. 2 SOLUTION: Lim ((x+4)/(x-4)) Lim ((infinite+4)/(infinite-4)) = indeterminate Lim (1/1)=1 30. It represents the distance of a point from the y-axis A. Ordinate B. coordinate C. abscissa

D. sec 2x+tan 2x

D. infinite

D.polar distance

31. A and B can do piece of work is 5 days, B and C in 4 days while A and C in 2.5days in how many days can all of them do the work together? A. 40/11 B. 30//11 C. 30/17 D. 40/17 SOLUTION 2/A + 2/B + 2/C =x/0.825 X = 1/0.425 X = 40 / 17 32. Chona the golden retriever gained 5.1 pounds is one month. She weights 65.1 pounds now. What is the percent weight gain of Chona in one month? A. 7.3% B. 8.2% D. 7.8% D. 8.5% SOLUTION Weight gain = 5.1 Present weight = 65.51 65.51- 5.1 = x

X = 60.41 % of weight gain = (weight gained / original weight) x 100 % = (5.1 / 60.42) x 100 % = 8.44 % 33. What is the center and radius of a circle with an equation x2+y2-1/4x1/4y=1/64? A. C (1,1/2) R=4 B. C(1,1), R=sqrt5/9 C. C(1/2-1/2 R=sqrt2/5 D. C (1/8, 1/8) R=sqrt 3 SOLUTION X^2 + y^2 – 1/4x – 1/4y = 1/64 X^2-1/4x + y^2-1/4y = 1/64 (x^2 – 1/4x + 1/64) + (y^2 -1/4y+1/64) = 1/64 +1/64 + 1/64 (x – 1/80)^2 + (y-1/8)^2 = 3/64 C ( 1/8 , 1/8) r= sqrt.of 3/8 34. A machine only accepts quarters. A bar of candy cost 25c a pack of peanuts cost 50c and the bottle of coke cost 75c. If Marie bought 2 candy bars a pack of peanut and a bottle of coke how many quarters did she pay? A. 5 B. 6 C.7 D. 8 SOLUTION 50c = 2quarter 50c = 2quarter 75c = 3quarter Total number of quarters is 2 + 2 + 3 = 7 quarters 35. Solve for x and y in xy +8+j (x2y+y)=4x+4+j(xy2+x) A. 2, 2 B. 2,3 C. 3,1 SOLUTION Real number : xy + 8 = 4x + 4 Imaginary number: x^2y + y = xy^2 = x Get y @ eq. of real no. Y= 4x-4 / x X^2y + y = xy^2+y X^2(4x-4)/x) + 4x-4 / x = x(4x-4 / x)2 +x X= 2 Y= 4x-4 / x = 4x-4 / x = 2 x,y (2,2)

D. 3,4

36. There are a set of triplets. If there are 11 generations how many ancestors do they have if duplication is not allowed? A. 4095

B. 4065

C.59,049

D. 265,719

37. Carmela and Marian were hired on a summer job. Each of them work 15 hours a week. Carmela was absent for one week and Marian has to take her shift. If they work for 8 weeks, what is the total number of hours did Marian works? A. 120 B. 135 C. 67.5 D. 60 SOLUTION

15(8) +15 = 135 hrs. 38. From the top of a building the angle of depression of the floor of a pole is 48 deg 10min. from the foot of a building the angle of elevation of the top is 18 deg 50 min, both building and pole are on a level ground. If the height of a pole is 4m, how high is the building? A. 13.10m B. 12.10 C. 10.90 D. 11.60 SOLUTION TanƟ =h/x h=height of building/pole x= distance between tan (18°50°) = 4 / x x = 11.73 x=11.73 tan (48°10°) = h / x h = 11.73 (tan 48°10°) h= 13.10m 39. The towers of a parabolic suspension brindge 300m long are 60 m high and the lowest point of a cable is 20m above the roadway. Find the vertical distance from the roadway to the cable at 100m from the center A. 17.78 B.37.78 C.12.86 D. 32.86 SOLUTION Y= ax^2+bx+c @(0,20) lowest point of the cable

20= a(0)^2 + b(0) +c 20= c Solving a and b @ P (150 , 60) 60= a(150)^2 + b(150) + 20 60-20 = 150^2 a + 150b 40 = 150 (150a +b) 4/15 = 150 a+b 4/15 – 150a = b eq.1 @ P(-150, 60) 60 = a(-150)^2 + b(-150) + 20 60-20 = (-150)^2 a + (-150)b 40 = -150 (b-150a) -4/15 = b-150a Subst 1 to 2 -4/15 = 4/15 -150a – 150a 300a = 8/15 A = 2/1125 B= 4/5 – 150(2/1125) B= 4/5 – 4/5 B= 0 @ x=100 find y=? Y = ax^2 +bx + c Y= 2/1125(100)^2 + 0(100) + 20 Y =37.78 40. Find the centroid of the plane area bounded by the parabola y=4-x^2 and the x-axis A. (0 3/2) B. (0,1) C. (0 , 12/5) D. (0,8/5) SOLUTION Y = 4-x^2 when x=0 Y= 4 When y= 0 X^2 = 4 X= +- (2) 2

A= ∫−2(4 − 𝑋^2)𝑑𝑥 A= 32/3 , 𝑥̅ =0 A𝑦̅ = ∫ 𝑦𝑐 𝑑𝐴 2

32/3𝑦̅ = ∫−2(4 − 𝑋 2 )/2)𝑑𝑥 (4 − 𝑥 2 )𝑑𝑦)

8

𝑦̅ = 5 C ( 0 , 8/5) 41. Evaluate the double integral 1/(x-y) dxdy with inner bounds of 2y to 3y and outer bounds of 0.2. A. Ln3 B. ln4 C. ln2 D ln8

42. Write the equation of a line with x-intercepts a=8 and y intercept b=-1 A. 8x+y-8=0 B. 8x-y+8=0 C. 8x+y+8=0 D. 8x-y-8=0

43. Solver for x; 125x-5=5x-4 A. 21/2 B. 15/2

C. 17/2

D. 19/2

44. Find the ration of the surface area of a cube to its volume if the side is s. A. 21/2 B. 15/2 C. 17/2 D.5/s

45. Solve the equation y” = y/2x A. Y^2=cx^3 B. y=cx^2

C. y^2=cx

D. y=cx

46. The sum of the first 7 terms of an A.P is 98 and the sum of the first 12 terms is 288. Find the sum of the first 20 terms A. 980 B. 800 C. 880 D. 980

47. When the sun is 20 degrees above the horizon, how long is the shadow cast by a building 150 ft high? A. 550 ft B. 580ft C. 405ft D. 450ft

48. A central angel of a circle of radius 30 in intercepts an arc of 6 in is how many radian? A. 1/3 B. 1/5 C. ¼ D. ½

49. A, B and C work independently on a problem. If the respective probabilities that they will solve it are ½, 1/3, 2/5 find the probability that the problem will be solved. A. 1/5 B. 2/5 C. 3/5 D. 4/5

50. A car goes 14kph faster than a truck and requires 2 hours and 20 minutes less time to travel 300km. Find the rate of the car. A. 40kph B. 50kph C. 60kph D. 70kph

51. Find the area bounded by 𝑥 = 2𝑦 − 𝑦 2 and the y-axis. A. 4/3

B. 5/3

C. 2/3

D. 1/3

Solution: Get the limit. Let X = 0; 0 = 2𝑦 − 𝑦 2 𝑦 2 − 2𝑦 = 0 −𝑏±√𝑏 2 −4𝑎𝑐 2𝑎

→

2±√−22 −4(1)(0) 2(1)

= (2,0)

2

A = ∫0 ( 2𝑦 − 𝑦 2 ) 𝑑𝑦 2

A = 2 – 2y∫0 A = 4/3

52. A steel ball at 120 deg C cools in 20 minutes to 80 deg C in a room at 25 deg C. Find the temperature of the ball after half an hour. A. 40.96 deg C Solution:

𝑑𝑇 𝑑𝑡

B. 45.96 deg C

C. 66.85 deg C

= −𝑘(𝑡 − 25) ∫ 𝑑𝑇⁄𝑇 − 25 = −𝑘 ∫ 𝑑𝑡 Ln(T-25) = -kt +C 𝑒 ln(𝑇−25) = 𝑒 −𝑘𝑡 𝑒 𝐶 𝑇 − 25 = 𝑒 −𝑘𝑡 𝑒 𝐶 𝑇 = 𝐶𝑒 −𝑘𝑡 + 25, solve for C and k 120 = 𝐶𝑒 0 + 25 120 = C + 25 C = 95, solve for k 80 = 95𝑒 −𝑘(20) + 25 55 = 95𝑒 −20𝑘 11 19

= 𝑒 −20𝑘 11

Ln ( ) = 𝑙𝑛𝑒 −20𝑘 19

D. 55.96 deg C

11

Ln(19) = −20𝑘 K = 0.027, then solve for the temperature after 30 min. 𝑇 = 95𝑒 −0.027(30) + 25 T = 67 deg C ≅ 66.85 deg C 53. If the line kx+3y+8=0 has a slope of 2/3, determine k. A. -3 Solution:

B. -2

C. 3

D. 2

Kx+3y+8=0 3y = kx + 8 −3𝑦 −3

𝑦=− 2

𝑘𝑥

8

= −3 + −3 𝑘𝑥 3

8

𝑘

− 3; - 3 = 𝑠𝑙𝑜𝑝𝑒

𝑘

= − 3, therefor k = - 2 3 54.Find the numerical coefficient of the term involving 𝑋 20 of (3𝑥𝑦 2 − 𝑥 4 )3 without expanding. A. 21.402

B. 22.104

C. 20.412

D. 23.214

Solution:

55. A rock is dropped down a well that is 256 feet deep. When will it hit the bottom of the well? A. 1 sec Solution:

B. 2 sec

C. 3 sec

D. 4 sec

256ft(1m/3.281ft) = ½(9.8 m/sec^2)t^2 t^2= 78.02m/2(9.8m/sec^2) t = 4 sec

56. if the side of a cube is measured with an error of at most 3 percent, estimate error in the volume of the cube. A. 3 percent Solution:

B. 6 percent

C. 9 percent

let x be the side of the cube

D. 12 percent

V = x^3 V = 1+0.03x^3 V = (1.03x)^3 V = 1.092727x^3, subtracting the volume it is supposed to have, We have an error of 0.092727 or 9.27 percent 57. Find the k so that A = <3, -2> and B = <1, k > are parallel. A. 2/3

B. -2/3

C. 3/2

D. -3/2

Solutions: 58. Find the slope of the curve x =3t, y = 9t^2 – 3t when t=1. A. 4 Solutions:

B. 5

C. 6

D. 3

x = 3t Dx = 3; Dy= 18t-3 Y’ = dy/dx = (18t-3)/3 Y’ = [18(1)-3]/3 = 5

59. Find the area of a triangle having vertices at -4-i, 1+2i, 4-3i A. 15

B. 16

C. 17

D. 18

60. Find the acute triangle between the vectors 𝑧1 = 3 − 4𝑖 and 𝑧2 = 4 + 3𝑖. A. 18 deg 18 min

B. 15 deg 15 min

C. 17 deg 17 min

D. 16 deg 16 min

61. A chord is 36 cm long and its midpoint is 36 cm from the midpoint of the longer arc. Find the radius of the circle. Solution: R^2 = (36-R)^2 + (36/2)^2 R = 22.5 62. Sarah leaves seattle for New York in her car, averaging 80 mph across open country. One hour later a plane leaves seattle for New York following the same route and flying 400 mph. How long it be before the plane overtakes the car?

Solution: 80(x-1) = 400(x) T = 1/4 hrs

63. What is the length of the latus rectum of the parabola x^2 = -16y Solution: X^2 = -16y LR=4a=16 64. Mr. Santos owns a jewelry store. He marks up all merchandise 50 percent of cost. If he sells a diamond ring for P15,000, what did he pay the wholesaler for it? Solution: 15,000 = x(1+0.05) X = 10,000 65. What is the equation of the normal to the curve X^2 +y^2 = 25 at (4,3)? Solution: X^2 + y^2 = 25 2x + 2y = 0 Y’ = -2x/2y = -x/y M1 = dy/dx = -4/3 M2 = -1/(-4/3) = 3/4 Y = 3/4x 4y = 3x 3x - 4y = 0

P(4,3)

66. for what values of X is I x-3 I = 1? Solution: I 4-3 I = 1 I 2-3 I = 1 = 2,4 67. If 3x = 4y then 4y^2/3x^2 is equal to: Solution: 3x = 4y then 4y^2/3x^2 = ? 4y.y’/3x^2 - 3x( 3y/4 )/3x^2 = 3/4

68. A wall is 15 ft high and 10 ft from a house. Find the length of the shortest ladder which will just touch the top of the wall and reach a window 20.5 ft above the ground. Solution: Tan Ɵ = 20.5 Xa 20.5/10+Xb = 15/Xb Xb = 27.27 L^2/3 = (20.15)^2/3 + 10^2/3 L = 42.25

Tan Ɵ = 15 Xb

69. A bag contains 3 white and 5 red balls. If two balls are drawn at random, find the probability that both are white. Solution: X= 4t , t = 1 y = t^2 -1 y’ = 2t Y” = 2 T = x/4 x’ =4 R = {[(x’)^2 + (y’)^2]^3/2} / I x’y” - y’x” I R = {[(4)^2 + (2(1))^2]^3/2} / I 4(2) - (2x1)(0) I R = 5√5 70. Determine the eccentricity of the hyberbola xy = 8 Solution: xy = 8 x=y x^2 = 8 X = √8 /2 = 1.414 71. Which term of the arithmetic sequence 2, 5, 8, . . . is equal to 227? Sol. An=A1+(n-1)d

An=227, A1=2, d=3

227=2+(n-1)3 n=76 72. Name the type of graph represented by x2-4y2-10x-8y=0

Hyperbola 73. If logx3=1/4, then x= Sol. log3/logx=0.25

101.908=x

0.477=0.25logx

x=81

74. If f(x0=-x2, then f(x+1)= Sol. –(x+1)2=-(x2+2x+2-2)=-x2-2x 75. If this graph of y=(x-2)2-3 is translated 5 units up and 2 units down to the right, then the equation of the graph obtained is given by Sol. (x+h)2=4a(y+k)

y-5=(x-2-2)2-3

y=(x-4)2+2

76. Which one is not a root of the fourth root of unity? i/sqrt 2 77. Find the area of the largest circle which can be cut from a square of edge 4 in. Sol. A=piR2

R=2

dA=2pi(2)=12.57

78. If I=(-1)1/2, find the value of i36 Sol. (-1)36/2=(-1)18=1 79. If cot B=5/2, find sin B sqrt of 52+42=sqrt of 29

Sol. cot B=cos B/sin B

sin B=22/sqrt of 29

80. A man is 1.6 m tall casts a shadow 4 m long. Nearby, a flagpole casts a shadow 18 m long. How high is the flagpole? Sol. x:1.6=18:4

4x=28.8

x=7.2 m

81. The rotary Club and the Jaycees Club had a joint party 120 members of the rotary Club and 100 members of the Jaycees Club also attended but 30 of those attended are members of both clubs. How many Jaycees attended the party? A. 150

B.250

C. 190

SOLUTION 220-30=100-X X=290, but Jaycees have 100 members 290-100=190

D. 22

82. Find the work done by a force F= -2j( pounds) applied to a point that moves on a line from (1, 3) to (4, 7).Assume that distance is measure in feet. A. 8ft.lb

B. -10ft.lb

C. -12 ft.lb

D. 15ft.lb

SOLUTION Given: distance= [(1-3)(4-7)]j Force= (-2j) work done= F.d Required: work work done= F.d d= [(1-3)(4-7)]j=-6j work done= ( -2j)(-6j) Work done= -12ft.lb 83. A particle has a position vector<2 cos2t, 1-3sint>. What is the speed of the particle at time t= pi/4? A. 5.427

B. 7.245

C. 1.879

D. 4.528

SOLUTION 84. Evaluate Γ(-3/2) 2

3

A. 3(sqrt.of pi)

B. 4( sqrt of pi)

SOLUTION n Γ n= Γ( n-1) −3

−3

3

( 2 )Γ( 2 )= Γ((2+1) [-3/2Γ -3/2= Γ( -1/2)] -2/3 3

2

1

Γ2=-3 Γ(− 2) 1

1

1

=-2Γ(-2)=Γ⟦− 2 + 1⟧-2 −3

=Γ( 2 ) = 𝟒

=𝟑π√𝟐

−2 3

(−2)Γ

1 2

1

C. 2(sqrt of pi)

𝟒

D. 𝟑(sqrt of pi)

85. Evaluate tan2(j 0.78) A. 0.533

B. -0.653

C. 0.426

D. -0.426

SOLUTION 86. A store advertised dresses on sale at 20 percent off. The sale price $76. What was the original price of the dress? A. $95

B. $60.80

C. $ 59

D. $80.60

SOLUTION X- 76= 0.20(X) X= 95 87. A woman is paid $20 for each day she works and forfeits $5 for each day she is idle. At the end of 25 days and nets $450. How many days did she work? A. 20

B. 21

C. 22

D. 23

SOLUTION 450pesos/ 20pesos/day = 23 days 88. What do you call a radical expressing an irrational number? A. Surd

B. Radix

C. Complex number

D. Index

SOLUTION 89. The arc of a sector is 9 units and its radius is 3 units. What is the area at the sector in square units? A. 12.5

B. 13.5

C. 14.5

D. 15.58

SOLUTION A=1/2rC A= 1/2(3)(9) A=13.5 sq.units 90. The base radius of a right circular cone is 4m while is slant height is 10m. What is the surface area?

A. 127.5 sq.m

B. 125.7 sq.m

C. 139.5 sq.m

D. 135.9sq.m

SOLUTION C= 2πr C=2π (4) C=25.13 𝐶𝐿

25.13(10)

A= 2 =

2

A=125.65 sq.m

91. A line with equation y=mx+b passes through (-1/3, -6) and (2,1). Find the value of m. A. 1

B. 3

C. 4 𝑚=

D. 2

𝑦2 − 𝑦1 1+6 = =𝟑 𝑥2 − 𝑥1 2 + 1/3

92. The vertical end of a water trough is an isosceles triangle with width of 6 feet and depth of 3 feet. Find the force on one end when the trough field with water. A. 638 lbs

B. 683 lbs

C. 562 lbs

D. 526 lbs

3 6 𝐹 = ∫ 62.4 (6 − 𝑋) 𝑋𝑑𝑋 = 𝟓𝟔𝟏. 𝟔 𝒐𝒓 𝟓𝟔𝟐 3 0

93. A lady gives a dinner party for six guests. In how many ways the be selected from among 10 friends? A. 110

B. 220

C. 105

D. 210

10C6= 210 Note: direct in calculator 94. For a complex number Z=2=2(sqrt of 3) i. The modulus is. A. 2

B. 3

C. 4

D. 5

2

𝑟 = √22 + 2√3 = 4 95.Which of the following has no middle term? A. (x-2y)6

B. (x+y)8

C. (x-y)5

D. (x+2y)4

96. A sports car 2 m long overtakes a 12 m van which is traveling at the rat of 36 kph. How fast must the car travel to overtake the van in 3 seconds if their rear ends are aligned initially? A. 46 kph

B. 47 kph

C. 48 kph

D. 49 kph

16 12 = = 𝑥 = 48 𝑋 36 97. A tank in an ice plant is to contains 3,000 liters of brine. It is constructed to be 4 m long and 1.5 wide. Find the height of the tank. A. 0.3

B. 0.4 𝑉 = 𝐿𝑥𝑊𝑥𝐻;

C. 0.5 3𝑚3 = 4𝑥1.5𝑥𝐻;

𝐻=

D. 0.6 3 = 0.5 4 ∗ 1.5

98. The eccentricity of the hyperbola having the rectangular equation 3x 2-4y224x+16y+20=0 is A. 1.12

B. 1.22

C. 1.32

D. 1.42

3𝑥 2 − 4𝑦 − 24𝑥 + 16𝑦 + 20 = 0 3(𝑥 2 − 8𝑥 = 16) − 4(𝑦 2-16y+64) =16+64-20 (𝑥 2 − 4)2 (𝑦 2 − 8)2 − = 60 4 3 𝑒=

𝑐 ; 𝑐 = √4 + 3 ; 𝑐 = 1.32 𝑎

99.Find the equation of the parabola whose vertex is the origin and whose foucus is the point (0,2) A. x2=10y

B. x2=8y

C. x2=-10y

D. x2=-8y

100. Find the equation of the family of curves at every point which the tangent line has a slope of 2y. A. x=Cey

B. y=Cex

C. x= Ce2y

D.y=Ce2y

AUGUST 2015

1. Given a conic section, if B2 - 4AC = 0, it is called? A. Circle B. Parabola C. Hyperbola

D. Ellipse

2. Given a conic section, if B2 - 4AC > 0, it is called? A. Circle B. Parabola C. Hyperbola

D. Ellipse

3. Describe and graph the locus represented by lm{z4} = 4. A. Circle B. Parabola C. Hyperbola

D. Ellipse

4. A tangent to conic is a line A. which is parallel to the normal B. which touches the conic at only one point C. which passes inside the conic D. all of the above 5. All circle having the same center but with unequal radii are called A. encircle B. tangent circles C. concyclic D. concentric circles 6. If z = 6eiπ/3, evaluate |eiz|, A. e-3(sqrt. of 3) B. e3(sqrt. of 3)

C. e-2(sqrt. of 2)

D. e2(sqrt. of 2)

7. Simply (cosβ - 1)(cosβ + 1) A. -1/sin2 β B. -1/cos2 β

C. -1/csc2 β

D. -1/sec2 β

8. Find the height of a right circular cylinder of maximum volume which can be inscribed in a sphere of radius 10 cm. A. 11.55 cm B. 14.55 cm C. 12.55 cm D. 18.55 cm 9. A bus leaves Manila at 12 NN for Baguio 250 km away, traveling an average of 55 kph. At the same time, another bus leaves Baguio for Manila traveling 65 kph. At what distance from manila they will meet? A. 135.42 km B. 114.58 km C. 129.24 km D. 120.76 km 10. A waiter earned tips for a total of $17 for 4 consecutive days. How much he earned per day? A. $4.25 B. $4.50 C. $3.25 D. $3.50 11. What is the value of x in Arctan 2x + Arctan x = pi/4 ?

A. 0.28 and -1.78

B. -0.28 and 1.78

C. 0.28

12. The length of the latus rectum of the parabola y2 = 4px is: A. 4p B. 2p C. p

D. -1.78

D.-4p

13. A post office can accept for mailing only if the sum of its length and its girth (the circumference of its cross section) is at most 100 in. What is the maximum volume of a rectangular box with square cross section that can be mailed? A. 5432.32in3 B. 1845.24in3 C. 2592.25in3 D. 9259.26in3 14. Water is running out of a conical funnel at the rate of 1 cu. In/sec. If the radius of the base of the funnel is 4 in. and the altitude is 8 in, find the rate at which the water level is dropping when it is 2 in. from the top. A. -1/9pi in/sec B. -1/2pi in/sec C. 1/2pi in/sec D. 1/9pi in/sec 15. A ball is dropped from a height of 18m. On each rebound it rises 2/3 of the height from which it last fell. What distance has it traveled at the instant it strikes the ground for the 5th time? A. 37.89 m B. 73.89 m C. 75.78m D. 57.78 m 16. 3 randomly chosen senior high school students was administered a drug test. Each student was evaluated as positive to the drug test (P) or negative (N). Assume the possible combinations of the three student’s drug test evaluation as PPP, PNP, NPN, NNP, NNN. Assuming each possible combination is equally likely, what is the probability that all 3 students get positive results? A. 1/8 B. 3/4 C. 1/4 D. 1/2 17. The cost per hour of the running the boat is proportional to the cube of the speed of the boat. At what speed will the boat run against a current of 4 kph in order to go a given distance most economically? A. 6 kph B. 12 kph C. 20 kph D. 24 kph 18. Ben is two years away from being twice Ellen’s age. The sum of Ben’s age and thrice Ellen’s age is 66. Find Ben’s age now. A. 19 B. 20 C. 18 D. 21 19. The cable of suspension bridge hangs in the form of a parabola when the load is uniformly distributed horizontally. The distance between towers is 150 m, the points of the cable on the towers are 22m above the roadway, and the lowest point on the cable is7 m above the roadway. Find the vertical distance to the cable form a point in the roadways 15m from the foot of a tower.

A. 16.6 m

B. 9.6 m

C. 12.8 m

D.18.8 m

20. If z is directly proportional to x and inversely proportional to the square of y and that y= 2 when z=4 and x= 2. Find the value of z when x= 3 and y=4. A. 2/3 B. 3/2 C.3/4 D.4/3 21. Find a∙b if lal = 26 and lbl =17 and the angle between them is pi/3. A. 221 B. 212 C. 383 D.338 22. The side of a square is 5 cm less than the side of the other square. If the difference of their areas is 55cm2, what is the side of the smaller square? A. 3 B. 4 C. 5 D. 6 23. The area bounded by the curve y2= 12x and the line x= 3 is revolved about the line x= 3. What is the volume generated? A. 186 B. 179 C. 181 D. 184 24. Evaluate the integral of (sinx) raised to the 6th power and the limits from 0 to pi/2. A. 0.49087 B. 0.48907 C. 0.96402 D. 0.94624

25. How many ounces will she make to serve 25 half-cup? A. 25 B. 50 C. 12.5

D. 75

26. Two engineers facing each other with a distance of 5km from each other, the angles of elevation of the balloon from the two engineers are 56 degrees and 58 degrees, respectively. What is the distance of the balloon from the two engineers? A. 4.46 km, 4.54km B. 4.64, 4.45km C. 4.64km, 4.54km D.4.46km, 4.45km 27. Evaluate the line integral from (0,0) to (1,1) .∫[√𝑦𝑑𝑥 + (𝑥 − 𝑦)𝑑𝑦] A. 5/3 B.4/3

C. 2/3

28. Find the area of the triangle having vertices at -4-I, 1+2i, 4-3i. A. 15 B. 16 C. 17

D. 1/3

D. 18

29. How many even numbers of three digits each can be made with the digits 0,2,3,5,7,8,9 if no digit is repeated? A. 102 B. 126 C. 80 D. 90 30. What is the angle subtended in mils of arc length of 10 yards in a circle of radius 5000 yards? A. 1.02 B. 2.40 C. 4.02 D. 2.04

31. How many 5 poker hands are there in a standard deck of cards? A. 2,598,960 B. 2,958,960 C. 2,429,955

D. 2,942,955

32. In delivery of 14 transformers, 4 of which are defective, how many ways those in 5 transformers at least 2 are defective? A. 940 B. 920 C. 900 D. 910 33. A point is chosen at random inside the circle of diameter 8 in. What is the probability that it is at least 1.5 in away from the center of the circle? A. 53/64 B. 55/64 C. 52/64 D. 56/64 34. A student did not study for his upcoming examination on which is 15 multiple choice test questions, with five possible choices of which only one is correct, what is the expected number of correct answers he can get? A. 2 B. 3 C. 4 D. 5 35. Evaluate (1+i) raised to (1-i). A. 2.82+i1.32 B. 2.82-i1.32 C. -2.82-j1.32 D. -2.82+i1.32 36. A boy, 1.20m tall, is walking directly away from the lamp post at the rate of 0.90 m/sec. If the lamp is 6m above the ground, find the rate at which his shadow is lengthening. A. 2.25 m/sec B. 0.225 m/sec C. 1.125 m/sec D. 0.235 m/sec 37. A painter needs to find the area of the gable end of the house. What is the area of the gable if it is a triangle with two sides of 42.0 ft. that meet at a 105 degrees angle? A. 852 sq. ft. B. 825 sq. ft. C. 892 sq. ft. D. 829 sq. ft. 38. A sector of a circle has a central angle of 50 degrees and an area of 605 sq. cm. Find the radius of the circle A. 34.6 cm B. 36.4 cm C. 37.2 cm D. 32.7 cm 39. If f(x) = sin x and f(𝜋) = 3, then f(x) = A. 4+cos x B. 3+cos x 40. If f(x) = 32x, then f(x) = A. 2(32x) B. 62x

C. 2-cos x

C. 9(ln6)

D. 4-cos x

D. 9(ln9)

41. Find the slope of the line tangent to 3y2 - 2x2 = 5xy at the point (1,2). A. -1 B.-2 C. 1 D.2 42. The volume V in3 of unmelted ice remaining from the melting ice cube after t seconds is given by V(t)=2000-40t+0.2t2. How fast is the volume changing when t= 40 seconds?

A.-26 in3 /sec

B. -24in3 /sec

C. -20in3 /sec

D. -8in3 /sec

43. The radius of a circle is measured to be 3 cm correct to within 0.02 cm. Estimate the propagated error in the area of the circle. A. 0.183 cm B. 0.213 cm C. 0.285 cm D. 0.377 cm 44. What is the area within the curve r2 = 16cos𝜃. A. 26 B. 28 C. 30

D. 32

45. A solid is formed by revolving about the axis, the area bounded by the curve x3 = y, the y-axis and the line y = 8. Find its centroid. A. (0, 4.75) B. (0, 4) C. (0, 5.25) D. (0, 5) 46. Find the area in the first quadrant that is enclosed by y = sin 3x and the x-axis from x = 0 the first x-intercept on the positive x-axis. A. -1/4 B. 2/3 C. 1 D.2 47. Let f(x) = x3 + x + 4 and let g(x) = f-1 (x). Find g’(6) A. -1/4 B. -4 C. 1/4

D. 4

48. 2 gallons is how many quartz? A. 2 B. 4

D. 8

C. 6

49. A recipe calls for 1 cup of milk for every 2-1/2 cups of flour to make a cake that would feed 6 people. How many cups of both flour and milk need to be measured to make a similar cake for 8 people? A. 1-1/3 B. 2-1/3 C. 1-1/2 D.2-1/2 50. Find the vertex of the parabola y2 - 8x + 6y + 1 = 0 A. (3, -1) B. (-3, 1) C. (3, 1)

D. (-3,-1)

51. Find the volume of a cone to be constructed from a sector having a diameter of 72 cm and a central angle of 150 degrees. A. 7711.82 B. 5533.32 C. 6622.44 D. 8866.44 52. A and B are points on the opposite sides of a certain body of water. Another point C is located such that AC= 200 meters, BC= 160 meters and angle BAC= 50 degrees. Find the length of AB. A. 164.67 m B. 174.67 m C. 184.67 m D.194.67 m 53. Find the area of the ellipse 4x2 + 9y2 = 36. A. 15.71 B. 18.85 C. 12.57 D. 21.99 54. A couple plans to have 7 children. Find the probability of having at least one boy.

A. 0.1429

B. 0.1667

C. 0.9922

D. 0.8571

55. A person has 2 parents, 4 grandparents, 8 great grandparents and soon. How many ancestors during the 15 generations preceding his own, assuming no duplication? A. 131070 B. 65534 C. 32766 D. 16383 56. A vendor buys an apple for Php 10 and sells it for Php 15. What percent of the selling price of apple is the vendor’s profit? A. 50 B. 33.33 C. 25 D. 66.67 57. What is the numerical coefficient of the term next to 240x2y2? A. 220 B. 240 C. 320

D. 340

58. Determine the sum of the first 12 terms of the arithmetic sequence: 3, 8, 13,.. A. 366 B. 363 C. 379 D. 397 59. In how many ways can 5 letters be mailed if there are 3 mailboxes available? A. 60 B. 80 C. 243 D. 326 60. James is 20 years old and john is 5 years old. In how many years will James be twice as old as john? A. 15 B. 10 C. 12 D. 8 61. The diagonal of square is 6 cm. Find its area. A. 18 B. 24 C. 28

D. 16

62. If cos A = 4/5 and angle A is not in Quadrant I, what is the value of sin A? A. 0.6 B. -0.6 C. 0.75 D. -.75 63. Find the area of a circle inscribed in a rhombus whose perimeter is 100 in. and whose longer diagonal is 40 in. A. 116 pi in2 B. 128 pi in2 C. 144 pi in2 D. 188 pi in2 64. A ranger’s tower is located 44 m from a tall tree. From the top of the tower, the angle of elevation to the top of the tree is 28 degrees, and the angle of depression to the base of the tree is 36 degrees. How tall is the tree? A. 48 m B. 62 m C. 55 m D. 99 m 65. In an ellipse, a chord which contains a focus and is in line perpendicular to the major axis is a: A. latus rectum B. minor C. focal width

D. Conjugate axis 66. Find the force on one end of a parabolic trough full of water, if depth is 2ft, and with across the top is 2 ft. Use 𝜔 = 62.5 lb/ft3 A. 125 lbs B. 133.33 lbs C. 200 lbs D. 208.33 lbs 67. Find the Laplace transform of f(t)= e raised to (3t+1). A. e/(s+3) B. e/(s-3) C. e/(s2 + 3)

D. e/(s2 - 3)

68. If the half-life of a substance is 1,200 years, find the percentage that remains after 240 years. A. 76% B. 77% C. 87% D. 97% 69. Robin flies to San Francisco from Santa Barbara in 3 hours. He flies back in 2 hours. If wind was blowing from north at velocity of 40 mph going, but changed to 20 mph from the north returning, what was the airspeed of the plane? A. 140 mph B. 150 mph C. 160 mph D. 170 mph 70. A tree is broken over by a windstorm. The tree was 90 feet high and the top of the tree is 25 feet from the foot of the tree. What is the height of the standing part of the tree? A. 48.47 ft. B. 41.53 ft. C. 45.69 ft. D. 44.31 ft. 71. In a frustum of cone of revolution the radius of the lower base is 11 in, the radius of the upper base is 5 in, and the altitude is 8 in. Find the total area in square inches. A. 80pi B. 160pi C. 226pi D. 306pi 72. A cask containing 20 gallons of wine emptied on one-fifth of its content and then is filled with water, if this is done 6 times, how many gallons of wine remain in the cask? A. 5.242 B. 5.010 C. 5.343 D. 5.121 73. Goods cost a merchant $ 72. At what price should he mark them so that he may sell them at a discount of 10% from his mark price and still make a profit of 20% on the selling price? A. $ 150 B. $ 200 C. $ 100 D. $ 250 74. Determine the length of the latus rectum of the curve r= 4(1-sin theta). A. 6 B. 9 C. 8 D. 7 75. Find the radius of the curvature of r= tan theta at theta= 3pi/4. A. sqrt. of 3 B. sqrt. of 5 C. sqrt. of 6

D. sqrt. of 2

76. Given A= 5i+3j and B=2i+kj where k is a scalar, find k such that A and B are parallel.

A. 3/5

B. 3

C. 6/5

D. 6

77. What is the x-intercept of the line whose parametric equations are x= 2t -1 and y= 6t+11? A. -2/3 B. -5/3 C. -7/3 D. -14/3 78. What is the coefficient of the (X-1)3 term in the Taylor series expansion of f(x)= lnx expanded about x= 1? A. 1/6 B. 1/4 C. 1/3 D. 1/2 79. The position of a particle moving along the x-axis at any time t is given by x(t)= 2t3 - 4t2 + 2t - 1. What is the slowest velocity achieved by the particle? A. 17/4 B. 3 C. -2/3 D. -3/2 80. For what value of k will the line kx +5y= 2k have y-intercept 4? A. 8 B. 9 C. 10 81. Find the circumference of the circle x2+y2-12x+10y+15=0 A. 75.40 B. 57.40 C. 96.12

D. 11

D. 69.12

82. Find the slope of the curve x=t2+et, y=t+et. At the point (1,1). A. 1 B. 2 C. 3 83. Which of the following is true? A. sin(-θ)=sin θ B. tan(-θ)=tan θ C. cos(-θ)=cos θ

D. 4 D. csc(-θ)=csc θ

84. The hypotenuse of a right triangle is 34 cm. Find the length of the two legs, if one leg is 14 cm longer than the other. A. 15 and 29 B. 16 and 30 C. 18 and 32 D. 17 and 31 85. John’s factory has 60 workers. If 4 out of 5 workers are married, how many workers are not married? A. 12 workers B. 24 workers C. 48 workers D. 60 workers 86. Find the equation of the line whose slope is-3 and the x-intercept is 5. A. y= -3x+5

B. 3x-y=5

C. 3x+y=15

D. y=3x+15

87. The positive value of k which will make 4x2-4kx+4k+5 a perfect square trinomial is A. 6 B. 5 C. 4 D. 3 88. If ln x=2 and ln y= 3, find ln(x3/y1/2). A. 3.5 B. 4.5

C. 2.5

D. 1.5

89. If 3x3y= 27 and 2x + y=5, find x. A. 3 B. 4

C. 2

D. 1

90. The area of a circle is six time its circumference. What is the radius of the circle? A. 10 B. 11 C. 12 D. 13 91. Twelve round holes are bored through a piece of steel plate. Their centers are equally spaced on the circumference of a circle 18 cm in diameter. What is the difference between the centers of two consecutive holes? A. 4.71 cm B. 4.66 cm C. 4.32 cm D. 4.55 cm 92. What is the minimum possible perimeter for a rectangle whose area is 100 sq. in? A. 50 in. B. 60 in. C. 30 in. D. 40 in. 93. Find the work done by the force of F= 3i + 10j newton’s in moving an object 10 meters north. A. 104.40J B. 100J C. 106J D. 108.60J 94. Find the abscissa of a point having an ordinate of 4 of a line that has a y-intercept of 8 and slope of 2. A. -2 B. +2 C. -3 D. +3 95. Find arch of an underpass semi-ellipse 60ft wide and 20ft high. Find the clearance at the edge of a lane if the edge is 20 ft. from the middle. A. 18.2 ft. B. 12.8 ft. C. 14.9 ft. D. 16.8 ft. 96. Find the moment of inertia with respect to the y-axis of the first-quadrant area bounded by the parabola x2= 4y and the line y=x. A. 34/5 B. 24/5 C. 54/5 D. 65/5 97. What is the length of the transverse axis of the hyperbola whose equation is 9y216x2=144? A. 6 B. 9 C. 8 D. 7 98. Find the mass of lamina in the given region and density function: pi D[(x, y)], 0 ≤ x ≤ , o ≤ y ≤ cosx and ρ = 7x 2 A. 2 B. 3 C. 4

D. 5

99. How many cubic inches of lumber does a stick contain if it is 4 in. by 4 in. at one end, 2 in. by 2 in. at the other end, and 16ft long? A. 1729 B. 1927 C. 1972 D. 1792

100. A goat is tied to a corner of 30ft by 35ft building. If the rope is 40ft and the goat can reach 1ft farther than the rope length, what is the maximum area the goat can cover? A. 4840.07 B. 4084.07 C. 4804.07 D. 4408.07 SOLUTION: 1. Parabola 2. Hyperbola 3. Hyperbola 4. which touches the conic at only one point 5. concentric circles 7. cos2 β − 1 sin2 β + cos 2 β = 1 cos 2 β − 1 = −sin2 β −𝟏 = −sin2 β 𝐜𝐬𝐜 𝟐 𝛃

9. 𝑑 = 𝑟1 𝑡 + 𝑟2 𝑡 250 = 55𝑡 + 65𝑡 𝑡 = 2.0833 𝑑 = 𝑟1 𝑡 = 55(2.0833) = 𝟏𝟏𝟒. 𝟓𝟖 $17

10.4 days = $𝟒. 𝟐𝟓 11. Using calculator (Radian mode) tan−1(2x) + tan−1(x) = 𝑆ℎ𝑖𝑓𝑡 𝑠𝑜𝑙𝑣𝑒 𝑋 = 𝟎. 𝟐𝟖 12. 𝐿𝑅 = 𝟒𝒑 18. by inspection: 2(𝑋) + 3(10) = 66; 𝑥 = 𝟏𝟖

π 4

𝑥

20. 𝑧 = 𝑦 2 𝑧1 𝑦1 2 𝑧2 𝑦2 2 = 𝑥1 𝑥2 4(2)2 𝑧2 (4)2 𝟑 = ∴ 𝑧2 = 2 3 𝟐

𝜋

𝜋

21. 𝐴𝐵 cos ( 3 ) = 26(17) cos ( 3 ) = 𝟐𝟐𝟏 22. by inspection: 82 − 𝑋 2 = 55; 𝑥 = 𝟑 𝑥2

23. 12 = 3; shift solve -99 = -6; shift solve 99 = 6 6

𝐴 = 𝜋∫ | −6

𝑥2 − 3| 𝑑𝑥 = 𝟏𝟖𝟎. 𝟗𝟔 12

5(3)(1) 𝜋

24. 6(4)(2) (2 ) = 𝟎. 𝟒𝟗𝟎𝟖𝟕 25. (90 − 56) + (90 − 58) = 66 5 𝑎 = ; 𝑎 = 𝟒. 𝟔𝟒 sin(66) sin(58) 5 𝑏 = ; 𝑏 = 𝟒. 𝟓𝟒 sin(66) sin(56) 29. Case I: not including 0 4 × 5 × 2 = 40 Case II: including 0 5 × 1 × 2 = 10 5 × 6 × 1 = 30 ∴ 40 + 10 + 30 = 𝟖𝟎 31. Using Calculator: 𝑛𝐶𝑟 = 52𝐶5 = 𝟐𝟓𝟗𝟖𝟗𝟔𝟎 1

38. 𝐴 = 2 𝑟 2 𝜃

𝜋 ) = 0.8727 180°

50° (

1 605 = 𝑟 2 (0.8727) 2 𝑟 = 𝟑𝟕. 𝟐 40. 𝑓(𝑥) = 32𝑥 = 9𝑥 𝑑(9𝑥 ) = 𝑎𝑢 𝑙𝑛𝑎𝑑𝑢 = 𝟗𝒙 𝒍𝒏𝟗 44. 𝑟 2 = 𝑘 cos(𝜃) = 16 cos(𝜃) ; 𝐴 = 2𝑘 = 2(16) = 𝟑𝟐 4 𝑞𝑢𝑎𝑡𝑠

48. 2𝑔𝑎𝑙𝑙𝑜𝑛𝑠 × 1 𝑔𝑎𝑙𝑙𝑜𝑛 = 𝟖 𝒒𝒖𝒂𝒓𝒕𝒔 50. 𝑦 2 − 8𝑥 + 6𝑦 + 1 = 0 𝑦 2 + 6𝑦 + 9 = 8𝑥 − 1 + 9 (𝑦 + 3)2 = 8(𝑥 + 1) 𝑽(−𝟏, −𝟑) 160

200

52. sin(50) = sin(𝐶) ; 𝐶 = 73.25 90 − 50 = 40 73.25 − 40 = 33.25 𝐴 = 90 − 33.25 = 56.75 160 𝑎 = ; 𝑎 = 𝟏𝟕𝟒. 𝟔𝟕 sin(50) sin(56.75) 53. 4𝑥 2 + 9𝑦 2 = 36 4𝑥 2 9𝑦 2 36 + = 36 36 36 𝑥2 𝑦2 + =1 32 22 𝐴 = 𝜋𝑎𝑏 = 𝜋(3)(2) = 𝟏𝟖. 𝟖𝟓 56.

15−10 15

× 100 = 𝟑𝟑. 𝟑𝟑%

𝐴𝐵

57. 𝐷 = 𝐶+1 =

240(4) 2+1

= 𝟑𝟐𝟎

58. 𝑑 = 𝑎2 − 𝑎1 = 8 − 3 = 5 𝑎𝑛 = 𝑎𝑚 + (𝑛 − 𝑚)𝑑 = 3 + (12 − 1)(5) = 58 𝑆=

𝑛 12 (𝑎1 + 𝑎𝑛 ) = (3 + 58) = 𝟑𝟔𝟔 2 2

60. 20 + 𝑋 = 2(5 + 𝑋); 𝑋 = 𝟏𝟎 61. 𝑑 = 𝑎√2; 6 = 𝑎√2; 𝑎 = 3√2 𝐴 = 𝑎2 = (3√2)2 = 𝟏𝟖 4

62. 𝐴 = cos −1 (5) = 36.87 𝑏𝑢𝑡 𝑖𝑠 𝑛𝑜𝑡 𝑖𝑛 𝑄𝑢𝑎𝑑𝑟𝑎𝑛𝑡 𝐼 𝑠𝑜 𝐴 𝑖𝑠 − 36.87 sin(− − 36.87) = −𝟎. 𝟔 65. Latus Rectum 68. 𝑞1 = 𝑄0 𝑒

1 ln( ) 2 )(𝑡) ℎ𝑙

(

= 240𝑒

1 ln( ) 2 )(240) 1200

(

= 208.93

208.93 × 100 = 𝟖𝟕. 𝟎𝟓% 240 69. 3(𝑋 − 40) = 2(𝑋 + 20); 𝑋 = 𝟏𝟔𝟎 77. 𝑥 = 2𝑡 − 1 𝑒𝑞. 1; 𝑦 = 6𝑡 + 11 𝑒𝑞. 2; 𝑡 = 𝑠𝑢𝑏𝑡. 𝑒𝑞. 3 𝑡𝑜 1: 𝑥 = 2 (

𝑦−11

2𝑦

6

6

)−1=

−

22 6

[𝑥 =

𝑦−11 6

𝑒𝑞. 3

−1

2𝑦 22 − − 1] 6 6 6

6𝑥 = 2𝑦 − 22 − 6 = 2𝑦 − 28 6𝑥 = 2(0) − 28 ∴ 𝑥 = 78. 𝑓(𝑥) = 𝑓(𝑎) + 𝑓 ′ (𝑎)

(𝑥−𝑎) 1!

+ 𝑓′′(𝑎)

(𝑥−𝑎)2 2!

+ 𝑓′′′(𝑎)

−28 −𝟏𝟒 = 6 𝟑 (𝑥−𝑎)3 3!

𝑓(𝑥) = 𝑙𝑛𝑥 = ln(1) = 0

𝑓 ′ (𝑥) = 𝑓 ′′ (𝑥) = −

1 1 = − 2 = −1 2 𝑥 1

𝑓 ′′′ (𝑥) = 𝑓(𝑥) = 0(1) + 1(1)

1 1 = =1 𝑥 1

2 2 = 3=2 3 𝑥 1

(𝑥 − 1) (𝑥 − 1)2 (𝑥 − 1)3 − 1(1) + 2(1) 1! 2! 3!

𝟏 𝑓(𝑥) = 0 + (𝑥 − 1) − (𝑥 − 1)2 + (𝑥 − 1)3 𝟑 80. by inspection using inserting the choices k𝑥 + 5𝑦 = 2k 10𝑥 + 5𝑦 = 2(10) 10 5 20 𝑥+ 𝑦= 20 20 20 𝑥 𝑦 + = 1; ∴ 𝑘 = 𝟏𝟎 2 4 81. 𝑥 2 + 𝑦 2 − 12𝑥 + 10𝑦 + 15 = 0 (𝑥 2 − 12𝑥 + 36) + (𝑦 2 + 10𝑦 + 25) = −15 + 36 + 25 2

(𝑥 − 6)2 + (𝑦 + 5)2 = √46 𝐶 = 2𝜋𝑟 = 2𝜋 = 𝟒𝟐. 𝟔𝟏

83. Trigonometry Identities (Negative Relations): 𝐜𝐨𝐬(−𝜽) = 𝐜𝐨𝐬 𝜽 84. by inspection 𝑐 = √𝑎2 + 𝑏 2 = √162 + 302 = 34; 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑎 = 𝟏𝟔 𝑎𝑛𝑑 𝑏 = 𝟑𝟎 85.

60 5

= 𝟏𝟐

87. by inspection 4𝑥 2 − 4𝑘𝑥 + 4𝑘 + 5 = 4𝑥 2 − 4(5)𝑥 + 4(5) + 5 = 4𝑥 2 − 20𝑥 + 25 mode 5 − 3 ∶ roots x = 90. 𝐴 = 6𝐶; 𝜋𝑟 2 = 6(2𝜋𝑟 2 ); 𝑟 = 12

5 ∴ 𝑘=𝟓 2

97. 9𝑦 2 − 16𝑥 2 = 144 9 2 16 2 144 𝑥 − 𝑦 = 144 144 144 𝑥2 𝑦2 − =1 42 32 𝑇𝐴 = 2𝑎 = 2(4) = 𝟖

MARCH 2016

1. Given a conic section, if B2-4AC=0, it is called? A. circle

B. parabola

C. hyperbola

D. ellipse

C. hyperbola

D. ellipse

2. Give a conic section, if B2-4AC >0 it is called? A. Circle

B. parabola

3. A conic section whose eccentricity is equal to one is known as A. A parabola

B. an ellipse

C. a circle D. a hyperbola

4. A length of the latus rectum of the parabola y2 = 4px is A. 4p

B. 2p

C. p

D. -4p

Solution: LR= 4P

5. Two engineers facing each other with a distance of 5km from each other, the angles of elevation of the balloon from the two engineers are 56 degrees and 58 degrees, respectively. What is the distance of the balloon from the two engineers?

A. 4.45km,4.54km

C.4.64km,4.54km

B. 4.54km,4.45km

D. 4.46km,4.45km

Solution: (90-56) + (90-58) = 66 5/sin(66) = a/sin(58) = 4.64km 5/sin(66) = b/sin(56) = 4.54km

6. Joy is 10% taller than joseph is 10% taller than Tom. How many percent is Joy taller than Tom? A. 18%

B. 20%

C. 21%

D. 23%

Solution: JOY = JOSEPH (1+.10) JOSEPH = TOM (1+.10) JOY [TOM (1+.10)] (1+.10) JOY = TOM (1+.10)2 JOY = TOM (1+.21) .21 = 21%

7. In a hotel it is known than 20% of the total reservation will be cancelled in the last minute. What is the probability that these will be fewer than 2 reservations cancelled out of 4 reservations? A. 0.6498

B. 0.5629

C. 0.3928

D. 0.8192

Solution: Probability = 4*0.2*0.83 = 0.4096

8. Find the area of the region inside the triangle with vertices (1,1),(3,2), and (2,4) A. 5/2

B. 3/2

C. ½

D. 7/2

Solution: 111 A= ½ {3 2 1 = 5/2 Ans.

41 9. The cost per hour of running a boat is proportional to the cube of the speed of the boat. At what speed will the boat run against a current of 8kph in order to go a given distance most economically? A. 15kph

Solution:

B. 14kph

C. 13kph

D. 12kph

Let c = cost per hour X = speed of motor boat C1 = total cost C =kx3 Where: k = proportionality constant t = d/x-8 Ct = Ct C1 = kx3 (d/x-8) dCt/dx = (x-8)(3kdx2)-kdx3(1)/(x-8)2 = 0 (x-8)(3x2)= x3 3x3-24x2= x3 2x3 = 24x2 X = 12kph

10. What is the unit vector which is orthogonal both to 9i+9j and 9i+9k? A.

B.

Solution: 9i+9j

C.

D.

9i+9k

A x B = (9i+9j)*( 9i+9k) = 81(i+j)(i+k) = 81(i-j-k) = (a x b)/ a x b = 81(i-j-k)/81 sqr. Rt. Of 3 = (i-j-k)/ sqr. Rt. Of 3 =

Ans.

11. In polar coordinate system the distance from a point to the pole is known as

A. Polar angle B. radius vector

C. x-coordinate D. y-coordinate

12. N engineers and N nurses. If two engineers are replaced by nurses, 51% of the engineers and nurses are nurses. Find N A. 100

B. 110

C. 50

D. 200

Solution: { 0.51 [ (N-s) + (N+2)] = N+2 } = 100 Ans. 13. If sinA=

and cotB= 4, both in Quadrant III, the value of sin (A+B) is

A. -0.844

B. 0.844

C. -0.922

D. 0.922

14. Two stores are 1 mile apart and are of the same level as the foot of the hill. The angles of depression of the two stores viewed from the top of the hill are 5 degrees and 15 degrees respectively. Find the height of the hill A. 109.01m

B. 209.01m

C. 409.01m

D. 309.01

Solution: Tan 5 = Tan 15 =

; X= 109.01m Ans

15. A fair coin is tossed three times and it appeared always exactly three heads. Find the probability in a single toss it will appear head. A. ½

B. ¼

C. 1/6

Solution: 1

1

#Flip = (# 𝑜𝑓 𝐻𝑒𝑎𝑑)(#𝑜𝑓 𝑇𝑎𝑖𝑙 ) Since 1 coin = 2 outcome

D. 1/16

𝟏 𝟐

#Flip =( ) 16. The product of the slopes of any two straight lines is negative 1, one of these lines are said to be A. Perpendicular

B. parallel

C. non intersecting D. skew

17. When two lines are perpendicular, the slope of one is

A. B. C. D.

Equal to the negative of the other equal to the other equal to the negative reciprocal of the other equal to the reciprocal of the other

18. A statistic department is contacting alumni by telephone asking for donations to help fund a new computer laboratory. Past history shows that 80% of the alumni contacted in this manner will make a contribution of at least P50, 000. A random sample of 20 alumni is selected. What is the probability that more than 15 alumni will make a contribution of at least P50.00? A. 0.4214

B. 0.5890

C. 0.6296

D. 0.3018

19. If z1 =1-i , z2= -2+4i, z3= sqrt of 3-2i, evaluate Re(2z13+3z22-5z32) A. 35

B. 35i

Solution: 𝑧3 = √3 − 2𝑖 = 3 − 2𝑖

20. Simplify (1-tan theta) / (1+tan theta)

A. B. C. D.

(cos theta+ sin theta)/(cos theta- sin theta) Cos theta/(cos theta-sin theta) (cos theta-sin theta)/(cos theta+sin theta) Sin theta/ (cos theta+sin theta)

C. -35

D. -35i

Solution: Assume the value of 𝜃 is = 30 1−tan 𝜃 1−tan 30 = = 1+ tan 𝜃 1+ tan 30

2 − √3

Then troubleshoot the choices, A. B.

cos 𝜃+ sin 𝜃 cos 30+ sin 30 = =2 + √3 cos 𝜃−sin 𝜃 cos 30−sin 30 cos 𝜃 cos 30 3+√3 = = 2 cos 𝜃−sin 𝜃 cos 30−sin 30

cos 𝜃− sin 𝜃 cos 30− sin 30

C. cos 𝜃+sin 𝜃 = cos 30+sin 30 =𝟐 − √𝟑 𝑨𝒏𝒔. sin 𝜃

sin 30

D. cos 𝜃+sin 𝜃=cos 30+sin 30=

−1+√3 2

21. A sinking ship makes a distance signal seen by three observers all 20m inland from the shore. First observer is perpendicular to the ship, second observer 100m to the right of the first observer and the third observer is 125m to the right of the first observer. How far is the ship from the shore? A. 60m

B. 80m

C. 100m

D. 136.2m

22. A die and a coin are tossed. What is the probability that a three and a head will appear? A. ¼

B. ½

C. 2/3

D. 1/12

Solution: Probability of the die= 1/6 Probability of the coin= 1/2 Total Probability = (1/6)(1/2)= 1/12

23. A tangent to a conic is a line

A. B. C. D.

Which is parallel to the normal Which touches the conic at only one point Which passes inside the conic All of the above

24. If tanA=1/3 and cotB=4 find tan(A+B) A. 11/7

B. 7/11

C. 7/12

D. 12/7

Solution: tan (A + B)= (tanA + tanB)/(1-tanAtanB) tan A= 1/3 cot B= 4 ; it is also equal to tanB= 1/4 Substitute: tan (A+B)= (.3333+.25)/(1-(.3333)(.25)) =7/11

25. What would happen to the volume of a sphere if the radius is tripled?

A. Multiplied by 3 B. multiply by 9

C. multiply by 27 D. multiply by 6

26. A container is in the form of a right circular cylinder with an altitude of 6in and a radius of 2in. If an asbestos of 1in thick is inserted inside the container along its lateral surface, find the volume capacity of the container. A. 12.57 cu. in

B. 12.75 cu. in

C. 18.58 cu. in

D. 18.85 cu. in

Solution: Asbestos is placed inside, the thickness of it will be subtracted to the radius since it serves as an inside coating. V= pi(r^3)(h) = pi(1^3)(6) = 18.85 cu.in.

27. Is it convergent or divergent? If convergent, what is the limit?

A. Convergent, pi/2 B. divergent

C. convergent, pi D. convergent, pi/4

28. If the sides of a right triangle is in arithmetic progression, what is the ratio of its sides?

A. 1,2,3

B. 4,5,6

C. 3,4,5

D. 2,3,4

Solution: Since right triangle, it must satisfy the Pythagorean's theorem

29. What is the area bounded by the parabola x2 = 8y and its latus rectum? A. 54/3 s.u.

B. 8/3 s.u.

C. 16/3 s.u.

D. 31/3 s.u.

Solution: Latus rectum= 8 So that we will choose limits (-4,4) then came up with: Integral of (x^2/8)dx with limits -4 to 4 = 16/3 s.u.

30. Find the general solution if y’’+10y=0 A. y = 𝐶1 cos(𝐶𝐶𝐶𝐶. 𝐶𝐶 10) 𝐶 + 𝐶2 sin(𝐶𝐶𝐶𝐶. 𝐶𝐶 10) 𝐶 𝐶𝐶 5) 𝐶

B. y = 𝐶1 cos(𝐶𝐶𝐶𝐶. 𝐶𝐶 5) 𝐶 + 𝐶2 sin(𝐶𝐶𝐶𝐶.

C. y = 𝐶 cos(𝐶𝐶𝐶𝐶. 𝐶𝐶 10) 𝐶

D. y = 𝐶 sin(𝐶𝐶𝐶𝐶. 𝐶𝐶 10) 𝐶

Solution: Case 3 of Conjugate Complex Roots D²y + 10 =0 dx²

y = e^ax ( C1 cos bx + C2 sin bx)

( D²+10 )y =0

y = e^0x ( C1 cos √10x + C2 sin √10x )

m² + 10 = 0

y = C1 cos √10x + C2 sin √10x Ans.

m= + √10

31. The volume of a cube becomes three times when its edge is increased by 1inch. What is the edge of a cube? A. 2.62

B. 2.26

C. 3.26

D. 3.62

Solution: when edge increased by 1 inch 3V = ( a+1 )³ 3dV = 3 (a+1)² 3(3a²) = 3a² + 6a +3 6a² - 6a – 3 =0 (a-1.366) (a+0.366) = 0 A= 1.366+1 = 2.366 Ans.

V= a³ Dv = 3a²

32. The areas if a regular pentagon and a regular hexagon are equal to 12 sq.cm. What is the difference between their perimeters? A. 0.02

B. 0.03

C. 0.2

D. 0.3

Solution: Area of Pentagon

Area of Hexagon

12=¼(5b²cot 180/5)

12=¼(6b²cot 180/6)

b = 2.641 inch

b = 2.149 inch

Perimeter P = nb P = 5(2.641) =13.205

= 13.205 – 12.894 = 0.311 Ans.

P = 6(2.149) = 12.894

33. Evaluate limxA. 4

B. 6

C. 8

D. 16

Solution: Apply L’Hospital’s rule x²-4 = 2x = 2(2) = 4 Ans. x-2 1 1 34. The length of a rectangle is seven times of its width. If its perimeter is 72cm, find its width A. 3

B. 3.5

C. 4

D. 15

Solution: P= 2(w+L) W -72= 2(w+7w) W= 4.8 Ans 35. A family’s electricity bill averages $80 a month for seven months of the year and $20 a month for the rest of the year. If the family’s bill were averaged over the entire year, what would the monthly bill be? A. $45

B. $50

C. $55

D. $60

Solution:

= 55 Ans. 36. In order to pass a certain exam, candidates must answer 70% of the last questions correctly. If there are 70 questions on the exam, how many questions be answered correctly in order to pass A. 46

B. 52

Solution: (70)(70%) = 49 Ans.

C. 56

D. 60

37. A firefighter determines that the length of hose needed to reach a particular building is 131m. If the available hoses are 47m long, how many sections of hose when connected together will it takes to reach the building? A. 3

B. 4

C. 5

D. 6

Solution: 141/47 = 3 Ans.

38. If the average person throws away 38.6 pounds of trash every day, how much trash would the average person throw away in one week? A. 270.2 pounds B. 207.2 pounds

C. 290.6 pounds

D. 209.6 pounds

Solution: 38.6 x 7 = 270.2 Ans.

39. If the csc2∅= 1+x, find cot2∅ A. X

B. 1 + x

C. 1 – x

D. 𝑥2

40. A runner runs a circular track and a set of data is recorded: Time

Distance

68 sec-----------------

400m

114 sec ----------------

600m

168 sec ----------------

800m

209 sec ----------------

1000m

256 sec ----------------

1200m

322 sec ----------------

1400m

What is the average velocity from 68 sec to 168 sec? A. 3 𝑚/𝑠2

B. 4 𝑚/𝑠2

C. 8 𝑚/𝑠2

D. 𝟔 𝒎/𝒔𝟐

Solution:

Vave

6 Ans. ?

A. ½

B. ¼

C. 2/5

D. 5/2

Solution: (2/3 – 1/4)

= 5/12= 2/5 Ans

(3/8 + 1/2 + 1/6)

25/24

42. Water is flowing into a conical vessel 10ft high and 2ft radius at the rate of 50 cu. Ft per minute. If the deep of the water is 6ft, how fast is the radius increasing? A. 2.12 ft/min

B. 12 ft/min

C. 2.21 ft/min

D. 11 ft/min

Solution: V = 1/3

r²h

Dv = (1/3)

2r h dr

Dr = 50 ft³ /min Dv

(2/3)

= 1.936 ft/min

(2)(10)

43. A steel grinder 8m long is moved on rollers along a passageway 4m wide and into a corridor at right angles with the passageway. Neglecting the width of the girder, how wide must the corridor be? A. 3.6 m

B. 1.4 m

C. 1.8 m

D. 2.8 m

Solution:

44. If in the Fourier series of a periodic function, the coefficient a0 is zero, it means that the function has

A. Odd symmetry B. Even quarter-wave symmetry

C. odd-quarter wave symmetry D. any of the above

45. What is the general solution of (D4-1) y (t) = 0?

A. 𝐶 = 𝐶1𝐶𝐶 + 𝐶2𝐶−𝐶 + 𝐶3𝐶𝐶𝐶𝐶 + 𝐶4𝐶𝐶𝐶𝐶

C. 𝐶 = 𝐶1𝐶𝐶 + 𝐶2𝐶−𝐶

B. 𝐶 = 𝐶1𝐶𝐶 + 𝐶2𝐶−𝐶 + 𝐶3𝐶𝐶𝐶 + 𝐶4𝐶𝐶−𝐶

D. 𝐶 = 𝐶1𝐶𝐶 + 𝐶2𝐶𝐶−𝐶

Solution:

46. Remy earns P10 an hour for walking the neighbor’s dog. Today she can only walk the dog for 45. How much will Remy make today? A. P10.00 Solution:

B. P7.25

C. P7.60

D. P6.75

47. When a baby born the weighs 8 lbs. and 12 oz. After two weeks during his checkup he gains 8 oz. What is his weight now in lbs. and oz.?

A. B. C. D.

8 lbs. and 10 oz. 9 lbs. and 4 oz. 9 lbs. and 2 oz. 10 lbs. and 4 oz.

Solution:

48. An equation of the form A. An inequality

B. an equality

is C. a proportion

D. a ratio

49. Michael’s favorite cake recipe calls for 0.75 pounds of flour, he has a 5 pound bag. He wants to make several cakes for the school bake sale. How many cakes can he make? A. 5

B. 6

C. 7

D. 8

Solution:

50. Simplify (1+tan2x) / (1-tan2x) A. Sin 2x

B. Cos 2x

C. Csc 2x

D. Sec 2x

Solution:

51.

52. 53. Find the general solution of y’’+10y’+41y=0

A. 𝐶 = 𝐶−5 (𝐶1𝐶𝐶𝐶4𝐶 + 𝐶2𝐶𝐶𝐶4𝐶) C. 𝐶 = 𝐶−4(𝐶1𝐶𝐶𝐶5𝐶 + 𝐶2𝐶𝐶𝐶5𝐶) B. 𝐶 = 𝐶5(𝐶1𝐶𝐶𝐶4𝐶 + 𝐶2𝐶𝐶𝐶4𝐶)

D. 𝐶 = 𝐶4𝐶(𝐶1𝐶𝐶𝐶5𝐶 + 𝐶2𝐶𝐶𝐶5𝐶)

Solution:

54. Find the general solution of y’+

A. 𝐶2 + 2𝐶2 = 𝐶

C. 𝐶2 − 2𝐶2 = 𝐶

B. 𝐶2 + 𝐶2 = 𝐶

D. 𝐶2 − 𝐶2 = 𝐶

Solution:

55. Find the general solution of y’’-4y’+10y=sin x

A. B. C. D. 56. Find the equation of the line that passes through (1,3) and tangent to the curve y= 𝑥

A. 4x+y-7=0

B. 24x+y-27=0

C. 4x-y+7=0

D. 24x-y+27=0

57. The ceiling in a hallway 10m wide is in the shape of a semi-ellipse and is 9m high in the center and 5m high at the side walls. Find the height of the ceiling 2m from either wall. A. 11.7 m

B. 8.4 m

Solution: 2m from the wall =3m from center 𝑥 𝑦 ( )2 + ( )2 5 5 𝑥2 𝑦2 + 9 =1 25 𝑦2 𝑥2 = 1 − 9 25

C. 6.4 m

D. 17.5 m

the origin is 5m high @ side wall

=1

y= 2.4m + 5= 7.4 T.S: origin is 6m high @ side wall y= 2.4 + 6 = 8.4 m Ans.

32

𝑦 = √9(1 − 25) Y=2.4 m 58. If in the Fourier series of a periodic function, the coefficient a0=0 and a=0, then it must be having _____ symmetry. A. Odd B. Odd-quarter wave C. Even D. Either A or B

59. If the Fourier coefficient b0 of a periodic function is zero then it must possess ______ symmetry. A. Even

B. Even-quarter-wave

C. Odd

D. Either A or B

60. Find the area of the region between the x-axis and y=(x-1)2 from x=0 to x=2 A. 1/3 Solution:

B. 2/3

C. ½

D. ¼

1

2

∫0 (𝑥2 − 2𝑥 + 1 − 0)𝑑𝑥 − ∫1 0 − (𝑥2 − 2𝑥 + 1)𝑑𝑥 = 𝟐/𝟑

61. Find the slope of the line through the points (-2,5) and (7,1) A. 4/9

B. -4/9

C. 9/4

D. ¼

Solution: m=

62. A train is moving at the rate of 8mi/h along a piece of circular track of radius 2500 ft. Through what angle does it turn in 1min? A. 16 deg. 8 min.

C. 18 deg. 9 min.

B. 15 deg. 6 min.

D. 17 deg. 10 min.

Solution: 8 mi/hr = 8(5280)/60 ft/min = 704 ft/min

The train passes over an arc of length s =704 ft in 1 min.

Then s/r = 704/2500 = 0.2816 rad or

16°8' Ans.

63. An artist wishes to make a sign in the shape of an isosceles triangle with a 42 degrees vertex angle and a base of 18m. What is the area of the sign? A. 109 sq. m

B. 209 sq. m

C. 112 sq. m

D. 211 sq. m

Solution: tan = opp/adj tan= opp/9 opp= height = 9tan(69) = 23.4 A=1/2bh A=1/2(18)(24.4) =

211 sq. m Ans.

64. If x2-y2=1 find y’’’

A. −2𝐶/𝐶5

B. 2𝐶/𝐶5

C. −𝐶/𝐶4

D. 𝐶/𝐶4

Solution:

65. A second hand scientific calculator was sold to Michael for P600. The original price of the item was P800. How many percent discount was given to him? A. 25

B. 35

Solution: Discount = 800-800(X%)= 600 Discount = 25%

C. 40

D. 20

66. Find the volume of a cube if its total surface area is 54 sq. cm. A. 21 cu. m

B. 30 cu. m

C. 27 cu. m

D. 54 cu. m

Solution: S =6a^2 54 = 6(a)^2 a=3 V = a^3 V = 3^3 = 27cu.m

67. A girl is flying a kite which is at a height of 120ft. The wind is carrying the kite horizontally away from the girl at a speed of 10ft/sec. How fast must be kite sizing be let out when the sizing is 150 feet long? A. 4 ft./s Solution:

B. 5 ft./s

D. 6 ft./s

C. 8 ft./s

Solution:

6-3(-15)+12 = 63 6.3 =6.3 X = 6 Ans. 70. Robert has 50 coins all in nickels and dimes amounting to $3.50. How many nickels does he have? A. 20 Solution: 0.05 + 0.01d = 3:5 n = 30

B. 30

C. 15

D. 35

d = 20 Ans. 71. The equation of the folium of Descartes is x2+y2=34xy. Find the area enclosed by the loop A.

B.

C.

D.

72. Find the acute angle of intersection of the curves x2+y2=5 and x2-y26x=15

֯

A. 53.14

B. 52.13

֯

C. 36.86

֯

D. 37.87 ֯

73. For what value of k will the line kx+5y=2k have y-intercept 4? A. -10

B. 10

C. 9

D. -9

Solution: Kx+5y=2k; y-int =4 5y = -kx + 2k; When x= 0, y= 4 5y= -k(0) + 2k

y= 4(5) = 2k K =10 Ans.

74. Find the volume formed by revolving the triangle whose vertices are (1,1),(2,4) and (3,1) about the line 2x-5y=10 A. 49

B. 94

C. 65

D. 56

75. A tank contains 760 liters of fresh water. Brine containing 2.5N/liter of salt enters the tank at 15 liter/min, and the mixture kept uniform by stirring runs out at 10liters/min. Find the amount of salt in the tank after 30 minutes? A. 1028.32 N

B. 649.52 N

C. 949.75 N

D. 864.88 N

Solution: Ds/Dt = (2.5)(15) – [s(10)/v] V= 760+(15-10t) V= 760 + 5t Ds/Dt = 37.5 – [s(10)/760+5t] (Ds/Dt) + [s(10)/760+5t] = 37.5 Integrating Factor (IF) = e^(10 integral of (dt/760+5t)) IF = e^[ln(760+5t)]^2 IF = (760+5t)^2 IF(s) = C integral of IF + C (760+5t)^2 = integral of (760+5t)^2 + C [(760+5t)^2 = (1/15) (760+5t)^3 + C @ t= 0; s=0 C= -1097440000 @ C= -7097440000; t=30 mins S=949.749 N = 949.75 N (ANS) 76. Find the volume of the solid generated when the region bounded by y=x2-4x+6 and y=x+2 is revolved about the x-axis A. 100.89 B. 104.60 C. 103.04 D. 101.79 Solution: y = X2-4x +6 y = x +2 revolved in x – axis x+2 = 9x2 -4x + 6 x2- 5x – 4 =0 (x-4) (x-1) =0 X =4 & 1 y = 4+2 = 6 y = 1+2 = 3 intersection (4,6) & (1,3) v= v = 101.79 X = 101.79 Ans.

77. The rate at which a body cools is proportional to the difference in temperature between it and the surrounding atmosphere. If in air at 60 deg. C a body cools from 90 deg. C to 80 deg. C in 10min, find its temperature 10 minutes later? A. 80 deg. C

B. 73.3 deg. C

C. 90 deg. C

D. 64.4 deg. C

Solution: Tb @ 20mins = 73.33oC 30 – 60 = (90 – 60) ek(10) 2

ln

K=

3

10

= 0.0405

Tb – 60 = (90-60)e-( 0.0405)(20) = 73.33 Ans. 78. A sector of a circle has a central angle of 80 degrees and radius of 5m. What is the area of the sector? A. 16.5 sq. m

B. 17.5 sq. m

C. 15.8 sq. m

D. 18.8 sq. m

Solution: 2

A=

= 25π = 78.534 (

Aa= 17.4533 Ans. 79. A grocer bought a number of cans of corn for $14.40. Later the price increased 2 cents a can and as a result she received 24 fewer cans for the same amount of money. How many cans were in his first purchase? A. 142

B. 140

Solution: xy=14.40 (y+0.02)(x-24)=14.40 x=?

y=

C. 144

D. 143

24) = 14.40 ; x=144 80. Find the area inside the cardiod r=1+costheta and outside the circle r=1. A. 2.79

B. 2.97

C. 3.98

D. 3.89

Solution: A = (1/2

A= 2.79 Ans.

81. If 2log4x-log49=2, find the value of x A. 10

B. 12

C. 11

D. 9

Solution: 2log4x-log49 = 2 2log4x = 2-log 49 X = 12 Ans. 82. If 7 coins are tossed together in how many ways can they fall with at most 3 heads? A. 63

B. 64

C. 65

D. 62

Solution: (7C3)+(7C2)+(7C1) = 63

83. The eccentricity if the hyperbola having the rectangular equation 3x24y2-24x+16y+20=0 A. 1.12

B. 1.22

C. 1.32

D. 1.42

Solution: 3x2-4y2-24x+16y+20=0 Ax2-Cy2+Dx+Ey+F=0 e=c/a

or

e=a/d

3x2-24x -4y2+16y = -20 3(x2-8x+16)-4(y2-4y+4)=-20+16(3)+4(-4) 3(x2-8x+16)-4(y2-4y+4)=12 (x-4)2/4 – (y-2)2 = 1 STANDARD EQUATION a=sq.4 b=sq.3 c=sq (a2+b2) c=sq(22+(sq.3)2) c= sq.7 e=c/a = sq.7/2 = 1.32

84. Find the slope of the tangent line to the parabola y2=4x+1 at the point (2,3) A. 1/3

B. 2/3 C. ¼

Solution: y= y= y’

y

’

D. ¾

85. If x=3t-1, y=1-3t^2, find d^2y/dx^2 A. -1/3

B. -2/3

C. -1

D. -4/3

Solution: y = 1-3t^2 t = (x+1/3) y = 1-3(x+3/3)^2 y= 1-(x^2/3)-(2x/3)-(1/3) yI = 0 – (2x/3) – (2/3) -0 yII = -2/3 ans.

86. Find the equation of the line through the point (3,4) which cuts from the first quadrant a triangle of maximum are

A. 4x+3y-24=0 B. 4x-3y+24=0

C. 3x+4y-25=0 D. 3x-4y+25=0

Solution: y=m=rise/run=4/-3 y-y1=m(x-x1) y-4=4/-3 (x-3) (-3)(y-4)=4(x-3) -3y+12 = 4x-12 4x+3y-24=0 ans. 87. Find the moment of inertia with respect to the y-axis of the plane area between the parabola y=2-x^2 and the x-axis A. 243/5

B. 234/5

C. 342/5

D. 324/5

88. A man drives 500ft along a road which is inclined 20 degrees to the horizontal. How high above the starting point is he? A. 171 ft.

B. 182 ft.

C. 470 ft.

D. 162 ft

Solution: tan Ø = h/500 h= 500 tan(20) h= 181.985ft = 182 ft Ans. . 89. An angle is 30 degrees more than one-half its complement. Find the angle

A. 20 degrees B. 50 degrees

C. 60 degrees D. 75 degrees

Solution: Complementary Angle = 90 degrees An angle is 30 degree more than one half its complement Angle = 45 + 30 = 75 degrees Ans. 90. How many ways can 5 keys be placed on a key ring? A. 8

B. 12

C. 20

D. 24

Solution: nPn= ( n-1 )! = ( 5-1 )! = 24

91. What is the diameter of a sphere for which its volume is equal to its surface area? A. 4

B. 6

Solution: V= 4/3πr^3 A= 4πr^2

C. 5

D. 7

r = d/2 4/3π(d/2)^3 = 4π(d/2)^2 d/24 = 1/4 d = 24/4 d=6

92. Find the area of the triangle whose vertices are A(4,2,3) B(7,-1,4) and (3,-4,6) A. sqrt of 156

B. sqrt of 155

C. 13.5

D. 15.5

93. If the second term of a geometric progression is 6 and the fourth term is 64. How many terms must be taken for their sum to equal 242? A. 4

B. 6

C. 5

Solution: a1 r = a2 a1 r r^2 = a4 6r^2 = 64 r = 3.2699 a1 = 6/3.26599 a1 = 1.83712 Sn = a1 (1 – r^n-1) / (r – 1) 242 = 1.83712 [ 1 – ( 3.26599^n-1 ) / (1-3.26599) ] -298.494 = 1- 3.26599^n-1 3.26599^n-1 = 299.494 n-1 log(3.26599) = log(299.424) n = 5.81 = 5

D. 7

94. Convert the point (r, , Φ) = (10, pi/2, 0) from spherical to Cartesian coordinates A. (10, 0, 1)

B. (10, 1, 1)

C. (10, 1, 0)

D. (10, 0, 0)

Solution: ( r , α , Φ ) = ( 10. π/2, 0 ) x = r sin α cos Φ = 10 (1) (1) x = 10 y = r sin α sin Φ =(10)(1)(0) y=0 z = r cos α = (10)(0) z=0 ( 10, 0, 0 )

95. The probability that A can solve a given problem is 4/5 that B can solve it is 2/3 and that C can solve it is 3/7. If all three try, compute the probability that the problem will be solved. A. 101/305

B. 101/105

C. 102/305

D. 102/105

96. A club of 40 executives, 33 like to smoke Marlboro, and 20 like to smoke Philip Morris. How many like to smoke Philip Morris only? A. 33

B. 13

C. 20

Solution:

(33-x) + x + (20-x) = 40 33 + 20 – x = 40

D. 7

x = 13 From equation of Philip Morris 20 – 13 = 7

ans.

97. Find the value of 4 sinh (pi i/3)

A. -2i (sqrt of 3) B. 2i (sqrt of 3)

C. -4i (sqrt of 3) D. 4i (sqrt of 3)

Solution: 4sinh (πi/3) = 4sinh i (π/3) 4sinh i (π/3) = 4isin (π/3) * (π/180) = 2i (sqrt. of 3) Ans. 98. An equilateral triangle has an altitude of 5(sqrt of 3) cm long. Find the area in sq. m.

A. 5(sqrt of 3) B. 25(sqrt of 3)

C. 100(sqrt of 3) D. 50(sqrt of 3)

Solution: A = ½ bh b = 5 sqrt. of 3 /tan(60) = 5 cm A = 2(½ (5 cm)(5 sqrt. of 3 cm)) A = 25 (sqrt. of 3)

“troubleshoot sq. m to cm”

99. The line y = 3x + b passes through the point (2, 4). Find b. A. 2

B. 10

Solution: y = 3x + b Substitute (2,4) 4 = 3(2) + b b=4–6 b = -2

C. -2

D. 10

100. If f(x) = sinx and f(pi) = 3 then f(x) = A. 4 + cosx

B. 3 + cosx

Solution: “troubleshoot f(x) to f’(x)” ∫ 𝑓 ′ (𝑥) = ∫ sin 𝑥 f(x) = - cos x + c ( g.e.) solve for c : f(pi) = - cos (pi) + c 3=1+c c=2 from g.e. : f (x) = - cos x + 2 or 2 – cosx

C. 2 – cosx

D. 4 – cosx

AUGUST 2016

PRE-BOARD EXAM IN MATHEMATICS

1. What is the area of the largest rectangle that can be inscribed in an ellipse with equation 4x˄2+y˄2=4? A. 3 B. 4 C. 2 D. 1 SOLUTION: 4𝑥 2 + 𝑦 2 = 4 [4𝑥 2 𝑥2 +

Ellipse: 1

2

+ 𝑦 = 4 ] (4) 𝑦2

=1 9 1 𝑥= = √2 √2 6 2 𝑦= = √2 √2 𝐴 = (𝑥)(𝑦) 1 2 𝐴 = ( ) ( ) = 1 sq. units Ans. 𝐃 √2 √2 4

𝑥2 𝑎2

𝑦2

+ 𝑏2 = 1

𝑎2 = 1 , 𝑏 2 = 4 , 𝑏 = 2

2. Sand is pouring to form a conical pile such that its altitude is always twice its radius. If the volume of a conical pile is increasing at rate of 25pi cu. Ft/min, how fast is the radius is increasing when the radius is 5 feet? A. 0.5 ft./min B. 0.5pi ft./min C. 5ft./min D. 5pi ft./min SOLUTION: 𝑉= 𝑉= 𝑉=

𝜋

r² h 3 𝜋

3 2𝜋 3

𝑟 2 (2𝑟) 𝑟3

𝑑𝑣 𝑑𝑡

2𝜋

=

3

𝑑𝑟

(3)(𝑟 2 )( ) 𝑑𝑡

𝑑𝑟

25𝜋 = 2𝜋(5)2 𝑑𝑡 𝑑𝑟 𝑑𝑡

=

1 2

ft per minute Ans. 𝐀

3. An air balloon flying vertically upward at constant speed is suited 150m horizontally from an observer. After one minute, it is found that the angle of elevation from the observer is 28 deg 59 min. what will be then the angle of elevation after 3 minutes from its initial position? A. 63 deg 24 min B. 58 deg 58 min C. 28 deg 54 min D.14 deg 07 min SOLUTION:

@𝑡 = 1 𝑚𝑖𝑛 𝜃 = 28′ 59′′ ℎ(1) 𝑡𝑎𝑛 𝜃 = = 130𝑡𝑎𝑛 (28′ 59′′ ) 150𝑚 ℎ = 83.089 𝑚 𝛥ℎ ℎ(1) 83.089 𝑉= = = = 83.089 𝛥𝑡 1𝑚𝑖𝑛 1𝑚𝑖𝑛

θ

@𝑡 = 3𝑚𝑖𝑛𝑠 ℎ(3) = 𝑣𝑡 83.089 = = 249.268 𝑚 3𝑚𝑖𝑛𝑠

𝜃 = 𝑡𝑎𝑛−1 (

249.268 ) = 58′ 57"Ans. 𝐁 150

4. A machine only accepts quarters. A bar of candy cost 25ȼ, a pack of peanuts cost 50ȼ and a bottle of a coke cost 75ȼ. If Marie bought 2 candy bars, a pack of peanuts and a bottle of coke, how many quarters did she pay? A. 5 B. 6 C. 7 D. 8 SOLUTION: 𝐶𝑎𝑛𝑑𝑦 = 25ȼ 𝑃𝑒𝑎𝑛𝑢𝑡𝑠 = 50ȼ 𝐶𝑜𝑘𝑒 = 75ȼ 2 Candy + 1 peanut + 1 coke 2(25) + 75 + 50 = 175ȼ Note: 1ȼ = 0.04 quarters 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 175(0.04) = 7 quarters Ans. 𝐂 5. A ball is dropped from a height of 18 m. On each rebound it rises 2/3 of the height from which it last fell. What is the total distance it travels in coming to rest? A. 80m B. 90m C. 72 m D. 86 m SOLUTION: 2

𝑎𝑟 = (3) (18) = 12 𝑎2 =

2 3

2

[ 3 (18)] = 8

𝑠4 = 𝑎1 [ 𝑠4 = [

1−𝑟 𝑛 1−𝑟

2 4 3 2 1− 3

1−( )

]

] = 28.89

𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑡𝑜𝑡𝑎𝑙 = 18 + 2 (28.89 ) = 75.78 ≈ 72m Ans. 𝐂 6. Evaluate lim (𝑥 + 4 sin 𝑥) 𝑥→13𝑝𝑖

A. 2

B. 1

C. -1

D. 0

SOLUTION:

lim sin(𝑥 + 4 sin 𝑥 )

𝑥→13𝜋

cos (x + 4cos x) cos (1 + 4 cos(13𝜋)) = 1 Ans. B 7. Find the length of the arc of the parabola x˄2=4y from x= ˗2 to x= 2. A. 4.2 B. 4.6 C. 4.9 D. 5.2 SOLUTION: 1

S= 2 √16ℎ² + 𝑏² +

𝑏² 8ℎ

ln √4ℎ +

√16ℎ²+𝑏² 𝑏

2

1 4² 16(1) +4² S= 2 16(1)² + 4² + 8(1) ln √4(1) + √ 4

S= 4.6 units Ans. B 8. Find the coordinates of the centroid of the plane area bounded by the parabola y=4 x˄2 and the x-axis. A. (0,1.5) B. (0,1) C. (0,2) D. (0, 1.6) SOLUTION:

𝑥′ = 0 2

𝑦′ = 5 ℎ

2

𝑦 ′ = (5) (4)

(0, 1.6) Ans. 𝐃

𝑦 ′ = 1.6

9. In how many ways can you pick 3 dogs from a pack of 7 dogs? A. 32 B. 35 C. 30 SOLUTION:

=

𝑃! (𝑃 − 𝑛)! 𝑛!

=

7! (7 − 3)! 3!

D. 36

= 35 ways Ans. 𝐁

10. In how many ways can 4 coins be tossed? A. 8 B. 12

C. 16

D. 20

SOLUTION:

2 faces of coin and 4 coins 2 x 2 x 2 x 2= 16 Ans. C

11. Which of the following is not multiple of 11? A. 957 B. 221 C. 122

D. 1111

SOLUTION: 221 11

= 20.09 Therefore; 221 Ans. B

12. A certain rope is divided into 8 m, 7 m, 5 m. What is the percentage of 5 m with the original length? A. 20 B. 15 C. 10 D. 25 SOLUTION: 8+7+5=20m 20m x %= 5m=25% Ans. D

13. Nannette has a ribbon with a length of 13.4 m and divided it by 4. What is the length of each part? A. 3.35 m B. 3.25 m C. 3.15 m D. 3.45 m SOLUTION: 13.4m 4

= 𝟑. 𝟑𝟓𝒎 Ans A.

14. The area in the second quadrant of the circle x˄2 + y˄2 = 36 is revolved about the line y+ 10 = 0. What is the volume generated? A. 2208.53 B. 2218.33 C. 2228.83 D. 2233.48 SOLUTION: 𝑥 2 + 𝑦 2 = 36 𝑥 2 + 𝑦 2 = 62 (0,0)𝑟 = 6 𝑦 + 10 = 0

𝑦 = −10 𝑉 = 𝐴(2𝜋𝐶 ′ ) 𝜋 416 𝑉 = ( (6)2 ) (2𝜋) (10 + ) 4 3𝜋 𝑽 = 𝟐𝟐𝟐𝟖. 𝟗𝟏𝟖 𝑨𝒏𝒔. 𝑪 15. It represents the distance of a point from the y-axis. A. Abscissa B. Ordinate C. Coordinate D. Polar distance 16. In polar coordinate system, the polar angle is negative when; A. Measured counterclockwise C. measured at the terminal side of ϴ B. Measured clockwise D. none of these 17. A coin is tossed in times. If it is expected that 7 heads will occur, how many times the coin is tossed? A. 12 B. 14 C. 16 D. 10 SOLUTION: One result in 2 sides of coin in every toss = ½ 7 heads in every tossed coin = 7/x Equating the equation: ½ = 7/x X = 14 18. A long piece of galvanized iron 60 cm wide is to be made into a trough by bending up two sides. Find the width of the sides of the base if the carrying capacity is maximum? A. 30 B. 20 C. 40 D. 50 SOLUTION: 1

𝐴 = 𝑏ℎ

𝐴 = (𝑏1 + 𝑏2)ℎ

𝐴 = (60 − 2𝑥)(𝑥)

𝐴=

𝐴 = 60𝑥 − 2𝑥

2

𝑑𝐴 = 60 − 4𝑥 = 0 𝑑𝑥 𝑥 = 15

𝑏 = 60 − 2𝑥 = 60 − 2(15) 𝑏 = 30

2 1 2

(20 + 40)(10√3)

𝐴 = 519.6𝑐𝑚2 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝐴 𝑖𝑠 𝑛𝑜𝑡 @ 𝒃 = 𝟐𝟎𝒄𝒎

𝐴 = 30(15) 𝐴 = 450

19. Totoy is 5 ft. 11 in. Nancy is 6 ft. 5 in. What is the difference in their height? A. 5 in B. 6 in C. 7 in D. 8 in SOLUTION: 5ft.and 11 in 5ft x

12 𝑖𝑛 1 𝑓𝑡

= 60in

60 in + 11 in = 71in 77in – 71 in = 6 in Ans. B

20. 5 years-old Tomas can tie his shoelace in 1.5 min and his right shoelace in 1.6 min. How long will it take him to tie both shoe lace? A. 2.9 min B. 3 min C. 3.1 min D. 3.2 min SOLUTION: 𝐿𝑠ℎ𝑜𝑒 = 1.5𝑚𝑖𝑛 𝑅𝑠ℎ𝑜𝑒 = 1.6𝑚𝑖𝑛 1.5min + 1.6 min = 3.1 min Ans. C

21. The area enclosed by the ellipse 4x˄2+9y˄2 = 36 is revolved about the line x = 3, what is the volume generated? A. 370.3 B. 360.1 C. 355.3 D. 365.1 Solution: 4x²+ 9y² = 36 , x = 3

r=3-x

1

y=±2√1 −

[4x² + 9y² = 36}36 𝑥² 9

+

𝑦² 4

=1

𝑥2 9

therefore: h = (2√1 −

𝑥2

) − (−2√1 − 9

𝑥2 9

)

h=4√1 −

𝑥2 9

𝑏

V=2π∫𝑎 𝑟ℎ𝑑𝑥 3

V=2π∫−3(3 − x)(4√1 −

𝑥2 9

)𝑑𝑥

V = 355.3 cu. units 22. The equation y² = cx is the general solution of A. y’= 2y/x B. y’= 2x/y C. y’= y/2x

D. y’= x/2y

Solution: y²=cx 2y y´= x

y´=

𝒙 𝟐𝒚

23. Solve the differential equation y’=y/2x. A. y= cx B. y˄2= cx Solution:

C. y= cx˄2

D. y˄3= cx

𝑦

y´= 2𝑥

𝑑𝑦

𝑦

[𝑑𝑥 =2𝑥] ʃ

𝑑𝑦 𝑦

𝑑𝑥 𝑦

𝑑𝑥

=ʃ 2𝑥

𝑙𝑛 𝑦 = 2 𝑙𝑛 𝑥 + 𝑐 𝑙𝑛 𝑦 = 𝑙𝑛 𝑥² + 𝑐 𝑦 𝑒 𝑙𝑛 (𝑥 2 ) = 𝑒 ln 𝑐 𝑦

𝑥²

=c

𝒚 = 𝒄𝒙²

24. In a school, 30 percent of students are involved in athletics. 15 percent of these play football. What percent of the student in the school play football? A. 4.5 B. 15 C. 5.4 D. 30 Solution: A = 0.35 → 5 =

A 0.03

𝐹 = 0.15𝐴 𝐹 15𝐴 = 0. 𝑥100 𝐴 3 0.3 = (0.3)(0.15)𝑥100 = 𝟒. 𝟓%

25. Find the point along the line x = y = z that is equidistant from (3, 0, 5) and (1, -1, 4). A. (1, 1, 1) B. (2, 2, 2) C. (3, 3, 3) D. (4, 4, 4) Solution: d = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 + (𝑧2 − 𝑧1 )2 d = √(2 − 3)2 + (2 − 0)2 + (2 − 5)2 = √14 d = √(2 − 1)2 + (2 − (−1))2 + (2 − 4)2 = √14 answer: (2, 2, 2) 26. Which of the following is divisible by 6? A. 792 B. 794

C. 790

D. 796

Solution: 792 6

= 132 Therefore; 792 is divisible by 6

27. The cost of operating a vehicle is given by C(x) = 0.25x + 1600, where x is in miles. If Jam just bought a vehicle and plan to spend between P5350 to P5600. Find the range of distance she can travel. A.14000 to 15000 B. 15000 to 16000 C. 16000 to 17000 D. 13000 to 14000 Solution: C(x) = 0.25x + 1600 C(x) = 5350 - 5600 C(x) = 0.25x + 1600 5350 = 0.25x + 1600 = 15000 5600 = 0.25x + 1600 = 16000

28. A 20-ft lamp casts a 25 ft. shadow. At the same time, a nearby building casts a 50 ft. shadow. How tall is the building? A. 20 ft. B. 40 ft. C. 60 ft. D. 80 ft.

Solution: 20ft

20

ϴ = ϴ 𝑡𝑎𝑛−1 = 25 = 38.66 ͦ

25ft

H

𝐻

ϴ = ϴ tan(38.66 ͦ) = 25 = 𝟒𝟎𝒇𝒕

50ft

29. Three circle of radii 3, 4, and 5 inches, respectively, are tangent to each other externally. Find the largest angle of a triangle found by joining the centers of the circle. A. 72.6 degrees B. 75.1 degrees C. 73.4 degrees D. 73.5 degrees Solution: 7+8+9 = 12 2 𝐴 = √(12)(12 − 7)(12 − 8)(12 − 9) 𝐴 = 26.83 𝑆=

1 2

𝐴 = 𝑎𝑏𝑠𝑖𝑛 1 26.83 + (7)(8)𝑠𝑖𝑛𝜃 2 𝜽 = 𝟕𝟑. 𝟑𝟖

30. Simplify the expression cos²ϴ-sin²ϴ A. cos 2ϴ B. sin 2ϴ

C. sin 2ϴ

D. sec 2ϴ

Solution: cos²ϴ - sin² ϴ = cos 2ϴ 31. csc 520º =? A. Cos 20º

B. csc 20º

C. sin 20º

D. sec 20º

B. SOLUTION: 1

csc 520 ͦ = sin 520 ͦ 1

2.92= sin 520ͦ 1 sin 20 ͦ

= 2.92

Therefore; CSC 20 ͦ = 1/ sin 32. Simplify x/(x – y) + y/(y –x). A. -1 B. 1

C. x 𝑥 𝑥 + 𝑥−𝑦 𝑦−𝑥

D. y

𝑥(𝑦−𝑥)+𝑦 ( 𝑥−𝑦) (𝑥−𝑦)(𝑦−𝑥) 𝑥𝑦−𝑥 2 +𝑦(𝑥−𝑦) 𝑥𝑦−𝑥 2 +𝑥𝑡−𝑦 2 −𝒙𝟐 −𝒚² −𝒙𝟐 −𝒚²

=𝟏

cos 𝐴

33. Simplify 1−sin 𝐴 − tan 𝐴. A. csc A B. sec A C. sin A Assume A=30 @Radmode Trial and Error

D. cos A

(Cos 30/1-sin30)-Tan30=6.48 Sec 30= (1/cos 30) = 6.48 =Sec 30

34. Find the minimum distance from the point (4, 2) to the parabola y² = 8x. A. 3 sqrt. of 3 B. 2 sqrt. of 3 C. 3 sqrt. of 2 D. 2 sqrt. of 2 SOLUTION: LR=8 a=2 x=2

y=2 2 2 d=√2 + 2

=2√𝟐

35. From the past experience, it is known 90 percent of one year old children can distinguish their mother’s voice of a similar sounding female. A random sample of one year’s old are given this voice recognize test. Find the standard deviation that all 20 children recognize their mother’s voice? A. 0.12 B. 1.34 C. 0.88 D. 1.43 = √ℎ𝑝𝑞 = √20(0.9)(1 − 0.9) = 𝟏. 𝟑𝟒 36. An equilateral triangle is inscribed in the parabola x² = 8y such that one of its vertices is at the origin. Find the length of the side of the triangle. A. 22.51 B. 24.25 C. 25.98 D. 27.71 𝑎√3 ℎ= 2 √3 𝑦 = 2𝑥 ( ) 2 𝑥2 = 8√3 8 𝑥 = 8√3 9 = 2𝑥 = 2(8√3) = 𝟐𝟕. 𝟕𝟏 37. Mary’s father is four time as old as Mary. Five years ago he was seven times as old. How old is Mary now? A. 8 B. 9 C. 11 D.10 Mary = x-5 Father = 4x-5 7(x-5) = 4x-5

X= 10 Ans. D 38. The lateral area of a right circular cylinder is 77 sq. cm. and its volume is 231 cu. cm. Find its radius. A. 4 cm B. 5 cm C. 6 cmD. 7 cm 2 2 A =77 cm V = ᴨr h 71/2ᴨr = 231/ᴨr2 V = 231 cm2 HA = HV A = 2ᴨrh A/2ᴨr = V/ᴨr2 r= 6 Ans. C 2 A/2ᴨr = V/ᴨr

39. A weight of 60 pounds rest on the end of an 8-foot lever and is 3 feet from the fulcrum. What weight must be placed on the other end of the lever to balanced 60 pound weight? A. 36 pounds B. 32 pounds C. 40 pounds D. 42 pounds 5x = 60 (3) 5x= 180 X = 36 lbs. 40. The average of six scores is 83. If the highest score is removed, the average of the remaining scores is 81.2. Find the highest score. A. 91 B. 92 C. 93 D. 94 Solution : (81.2x5)+x/6 = 92

41. A point moves on the hyperbola x²- 4y² = 36 in such a way that the x-coordinate increase at a constant rate of 20 unit per second. How fast is the y-coordinate changing at a point (10, 4)? A. 30 units/sec C. 30 units/sec B. 30 units/sec D. 30 units/sec SOLUTION:

x 2 − 4y 2 = 36 2𝑥𝑑𝑥 8𝑦𝑑𝑦 − =0 𝑑𝑡 𝑑𝑡

𝑑𝑦 𝑥 𝑑𝑥 = 𝑑𝑡 4𝑦 𝑑𝑡 10 (20) = 12 = 4(4) 𝐵𝑦 𝑇𝑟𝑜𝑢𝑏𝑙𝑒𝑠ℎ𝑜𝑜𝑡𝑖𝑛𝑔: 𝟏𝟎 (𝟏𝟐) = 𝟑𝟎𝒖𝒏𝒊𝒕/𝒔𝒆𝒄 𝟒 42. If the tangent of angle A is equal to the square root of 3, angle A in the 3rd quadrant, find the square of the tangent A/2. A. 2 B. 3 C. 4 D. 5 SOLUTION:

tan A = √3 A = tan-1(√3) A = 180 – 60 = 120 [ tan (A/2) ]2 = [ tan(120/2) ] = (√3) = 3 43. A stone, projected vertically upward with initial velocity 112 ft./sec, moves according to s = 112t – 16t², where s is the distance from the starting point. Compute the greatest height reached. A. 196 ft. B. 100 ft. C. 96 ft. D. 216 ft. SOLUTION:

dS = 112t- 16t2 dS/dt = 112-32(t) = 0 @ t = 3.5s S = 122 (3.5) -16 (3.5)2 = 196ft 44.) A cylinder of radius 3 is cut through the center of the base by a plane making an angle of 45 degrees with the base. Find the volume cut off. A. 15 B. 16 C. 17 D. 18 SOLUTION:

h 𝑉 = (𝐴12 + 𝐴13 + 𝐴14) 6 3 1 = [(0 + 0 + 4) ( ) (3)(3)] 3 2 𝑽 = 𝟏𝟖 𝒄𝒖𝒃𝒊𝒄 𝒖𝒏𝒊𝒕 45.) Find the diameter of a circle with the center at (2, 3) and passing through the point (-1, 5).

A. 3.6

B. 7.2

C.13

D. 16

SOLUTION:

(x-h) 2 + (y-k) 2= r 2 (-1-2) 2+ (5-3) 2=r 2 √13 = √r 2 r = √13 d = 2(r) = 2 (√13) = 7.21 46.) Find the value of x for which the tangent to y = 4x-x² is parallel to the x-axis. A. 2 B. -1 C. 1 D. -2 SOLUTION:

y = 4x – x2 y = x2 – 4x y = (x – 2)2

if y = 0

Therefore, x= -2 47. Find the surface area generated by rotating the parabolic arc about the x-axis from x = 0 to x = 1. A. 5.33 B. 4.98 C. 5.73 D. 4.73 SOLUTION:

𝑦 = 𝑥2 =

𝑑𝑥 𝑑𝑦

= 2𝑥

𝑆 = ∫ 2𝜋𝑟 𝑑𝑠q7’ 1

= ∫ 2𝜋𝑦√1 + ( 0

𝑑𝑦 2 ) 𝑑𝑥 𝑑𝑥

1

∫ 2𝜋𝑥 2 √1 + (2𝑥)2 𝑑𝑥 = 𝟓. 𝟐𝟕𝟗 ≈ 𝟓. 𝟑𝟑 0

48. A group of students plan to pay equal amount in hiring a vehicle for an excursion trip at a cost of P 6, 000. However, by adding 2 more students to the original group, the cost of each student will be reduced by P 150. Find the number of each students in the original group. A. 10 B. 9 C. 8 D. 7 SOLUTION:

6000 / 8 = 750

6000 / 10 = 600 750 – 600 = 150 Therefore, 8 is the no. of students in original group 49. What is the allowable error in measuring the edge of the cube that is intended to hold 8 cu. m., if the error of the computed volume is not to exceed 0.03 cu.m. A. 0.002 B. 0.003 C. 0.0025 D. 0.001 SOLUTION:

3 V=E 𝑑𝑉 = 3𝐸 2 𝑑𝐸 𝑑𝑉

𝑑𝐸 =

3𝐸 2

3

E = √8 = 2

=

0.03 3 ×22

= 0.0025

50. Find the value of x for which y = 2x³- 9x² + 12x – 2 has a maximum value. A. 1 B. 2 C. -1 D. -2 SOLUTION:

y = 2x3- 9x2+12x – 2 y’= 6x2 – 18x + 12 = 0 By Quadratic Formula [mode, 5, 3] x = 1, x=2 51. At a height of 23,240 ft., a pilot of an airplane measures the angle of depression of a light at an airport as 28 deg 45 min. How far is he from the light? A. 20,330 ft. B. 26,510 ft. C. 11, 180 ft. D. 48, 330 ft. Solution 23240 Sin Ɵ = 𝑦 Sin( 28′45′ ) =

23240 𝑦

y= 48137ft or 48,330f 52. A substance decreases at a rate which is inversely proportional to the amount present. If 12 units of the substance are present initially and 8 units are present after 2 days, how long will it take the substance to disappear? A. 1.6 days B. 2.6 days C. 3.6 days D.4.6 days

53. A tower 150m high is situated at the top of a hill. At a point 650m down the hill, the angle between the surface of the hill and the line of sight to the top of the tower is 12 deg 30 min. Find the inclination of the hill to a horizontal plane. A. 7 deg 50 min B. 20 deg 20 min C. 77 deg 30 min Solution By Sine Law

D. 12 deg 55 min sin(12°30′) sin 𝐶 = 150 650

C=69.70° Answer=90° − 69.70° − 12°30′= 7°48’ ≅7°50’ 54. A telephone company has a profit of $80 per telephone when the number of telephones in exchange is not over 10,000. The profit per telephone decreases by $0.40 for each telephone over 10, 000. Find the numbers of telephone that will yield the largest possible profit. A. 13,000 B. 14,000 C. 15,000 D. 16,000 55. Find the work done in moving an object along the vector a = 3i + 4j if the force applied is b = 2i +j. A. 11.2 B. 10 C. 12.6 D. 9 Solution A=3i+4j , B=2i+j 5+√5=11.2 √32 + 42 =5 2 2 √2 + 1 =√5

56. A man is paid P 1, 800 for each day he works and forfeits P 300 for each day he is idle. If at the end of 40 days, he nets P 53, 100, how many days was he idle? A. 6 B. 7 C. 8 D. 9 Solutions let X number of days he idle 40-X number of days he work 1800(40-X)-300X=53100 X=9 57. By stringing together 9 differently color beads, how many different bracelets can be made? A. 362,880 B. 20,160 C. 40,320 D. 181,440 Solutions (9!)=362,880 58. In a circle of diameter 26 cm, a chord 10 cm in length is drawn. How far is the chord from the center of the circle?

A. 5 cm Solution D=26cm L=10cm

B. 12 cm

C. 13 cm

D. 24 cm

√132 + 52 = 12

59. Find the slope of the line passing through the pair of points (-2, 0) and (3, 1). A. 1/3 B. 1/4 C. 1/6 D. 1/5 Solutions 𝒚𝟐−𝒚 1−0 𝟏 𝒎 = 𝑿 −𝑿𝟏 =3+2= 𝟓 𝟐

𝟏

60. Find the inverse of the function f(x) = sqrt. of (2x – 3). A.sqrt. of (2y-3) B. 1/ sqrt. Of (2x-3) ½(x2+3) D. ½ (y2+3) Solutions F(x)=√2𝑋 − 3 Y=√2𝑋 − 3 𝑦 2 = 2𝑥 − 3 𝑦2 − 3 =𝑥 2 𝟏 𝟑 𝑿 = 𝒚𝟐 + 𝟐 𝟐 61. If f (3) =7, f’ (3) = -2, g (3) =6 and g’ (3) = -10, find the (g/f)’ (3). A. -82/49 B. -49/82 C. -49/58 D. -58/49

C.

SOLUTION:

f ( g’ ) – g (f’) / f 2 (3) = 7(-10) – 6 (-2) / 72 = -70 + 2 /49 = -58/49 62. The length of the median drawn the hypotenuse of a right triangle is 12 inches. Find the length of the hypotenuse. A. 24 in B. 20 in C. 23 in D. 25 in SOLUTION:

H = 12 + 12 = 24 63. Find the derivative of the function y = 3/(x²+ 1). A. 6x/(x2+1)2 B.6x(x2+1)2 C. -6x/(x2+1)2 SOLUTION:

y = 3/(x²+ 1) y = 3(x²+ 1)-1 y’ = 3(x²+ 1)-2 (2x)

D.-6x(x2+1)2

−

=

𝟔 𝟐

(𝐗

+𝟏)

𝟐

64. A passenger in a helicopter shines a light on a car stranded 45 ft from a point just below the helicopter is hovering at 85 ft, what is the angle of depression from the light source to the car? A. 82 degrees B. 80 degrees C. 60 degrees D.62 degrees SOLUTION: 85

𝜃 = tan−1 (45) = 𝟔𝟐. 𝟏𝟎 65. Find the area bounded by the curve r = 8 cos ѳ. A. 50.27 B.12.57 C. 8

D. 67.02

SOLUTION:

A = (ᴨ / 4) (a2) A = (ᴨ / 4) (42) = 12.57 sq.units 66. If 2log4x – log49 = 2, find the value of x. A. 10 B. 12

C. 11

D. 9

C. 2.65

D. 265i

SOLUTION:

2log4x – log49 = 2 Solving x, X= 12

67. Find the value of 2 cos (pii/4). A. 1.41 B. 1.41i SOLUTION:

ᴨ / 4 x 180 / ᴨ = 45 degrees 2 cos (45) = √2 = 1.414 68. A pole is on top of a building. At a point 240 meters from the base of the building, the angle of elevation of the base and top of the pole are 42 degrees and 44 degrees respectively. Find the height of the pole. A. 15.8m B. 18.5m C. 16.9m D. 19.6m SOLUTION:

Base to top 𝑜

tan ᴓ = 𝑎

Base to pole tan(44) =

216+𝑥 240

ℎ

tan(42) = 240

x= 15.8 m

h = 216 m. 69. The volume of a hemisphere of radius 2 m is A.14.67 cu.m B.67.04cu.m C.16.76cu.m

D.33.53cu.m

SOLUTION:

𝑉=

2 3

𝜋𝑟 3 =

2 3

𝜋(2)3 = 16.76 m3

70. Five scores and 4 years is equivalent to how many years? A. 49 B. 29 C. 54

D. 104

SOLUTION:

5 scores and 4 years Scores = 20 years 5 scores = 100 years

100 years + 4 years = 104 years

71. Find the equation of one of the asymptotes of the hyperbola 𝑥 2 − 4𝑦 2 − 6𝑥 − 8𝑦 + 1 = 0. A. x – 2y – 5 = 0 B. x – 2y + 5 = 0 C. x – 2y – 1 = 0 D. x – 2y + 1 = 0 72. The wheel of a truck is turning at 6 rps. The wheel s 4 ft in diameter. Find the linear velocity iin fps point on the rim of the wheel. A.75.4 B.57.4 C.150.8 D.105.8 Solution: 2𝜋 ῳ = (6 rps)(1𝑟𝑒𝑣) = 37.7 rad/sec d = 4ft; r = 2ft v = rῳ = (2ft)(37.7 rad/sec) = 75.4 ft/sec 73. Solve the inequality 3 – 2x < 4x -5. A. x < 4/3 B. x > 4/3 Solution: -4x-2x < -3-5 -6x < -8 x < 4/3

C. x < ¾

D. x > ¾

74. The polynomial 𝑥 2 + 4𝑥 + 4 is the area of a square floor. What is the length of its side? A. x + 2 B. x – 2 C. x + 1 D. x – 1 75. If there are 2 computers for every 4 students, how many computers are needed .For 60 students? A.24 B.26 C.30 D.32 Solution: 2:4 = 60:x

4x = (60)(2) X = 30 76. From Pagasa island in the Spratlys, two helicopters travel to two different islands.One helicopter travels 185 km N 65 deg E to island A and the other travels at S 25 deg E for 120 km to island B. What is the distance between the two islands? A. 198.5 km B. 187.3 C. 235.2 D. 202.5 Solution: Ɵ1 = 65° Ɵ2 = 25° ƟT = 90° AB = √(185)2 + (120)² = 202.5 km 77. If x = y + 2, what is the value of (𝑥 − 𝑦)4 ? A. 10 B. 16 C. 18 Solution: (y + 2 – y)4 = ? 24 = 16

D. 24

78. An equilateral triangle has sides of 8 inches. What us the height? A. 6.32 in B. 6.93 in C. 5.66 in D. 6.56 in Solution: s = 8/2 = 4 h = √(8)2 − (4)2 = 6.93in 79. If in the Fourier series of a periodic function, the coefficient 𝑎0 = 0 and 𝑎𝑛 = 0, then It must be having ____________ symmetry. A. odd B. odd quarter-wave C. even D. either A or B 80. Find the area of the triangle whose vertices are (4,2,3), (7,-2,4) and (3,-4, 6). A. 15.3 B. 13.5 C. 12.54 D. 12.45 81. Find the moment of inertia of the area bounded by the curve x^2=8y, the line x =4 and the x-axis on the first quadrant with respect to y-axis. A. 25.6

B. 21.8

4

SOLUTION: 𝑥 2 = 8𝑦 Subs. x= 4 42 = 8𝑦 16 =𝑦 8

C. 31.6 𝑥2

𝐼𝑦 = ∫0 𝑥 2 ( 8 ) 𝑑𝑥 𝑰𝒚 = 𝟐𝟓. 𝟔

D. 36.4

𝑦=2

82. If 8 oranges cost Php 96, how much do 1 dozen at the same rate? A. Php 144

B. Php 124

C. Php 148

D. Php 168

SOLUTION: 8𝑥 = 96 8𝑥 8

=

1 𝐷𝑜𝑧𝑒𝑛 = 12

96

𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 12 𝑥 12 = 𝑷𝒉𝒑 𝟏𝟒𝟒

8

𝑥 = 12

83. A particle moves in simple harmonic in accordance with the equation s = 3sin 8pit + 4cos 8 pit, where s and t are expressed in feet and seconds, respectively. What is the amplitude of its motion? A. 3ft

B. 4ft

C. 5ft

D. 8ft

84. If 𝑍1 = 1 – 𝑖 and 𝑍2 = −2 + 4𝑖, evaluate 𝑍12 + 2𝑍1 – 3. A. -1 +4i

B. 1 – 4i

C. 1 + 4i

D. -1 – 4i

SOLUTION: (1 − 𝑖)2 + 2 (1 − 𝑖) − 3 = 0

𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝟏 + 𝟒𝒊 = 𝟎

1 − 2𝑖 + 𝑖 2 + 2 − 2𝑖 − 3 = 0 𝑖 2 = 4𝑖 −1 = 4𝑖

85. Identify the property of real numbers being illustrated: x + (y + z) = (x + y) + z A. Commutative Property of Addition

B. Commutative Property of Multiplication C. Associative Property of Addition D. Associative Property of Multiplication

86. The distance between -9 and 19 on the number line is A. 28

B. -28

C. 10

D. -10

SOLUTION: 19 + 9 = 𝟐𝟖

a

a

87. If the function f is odd and ∫0 f(x)dx = 5m − 1, then ∫−a f(x)dx =? B. 10m – 2

A. 0

C. 10m – 1

D. 10m

88. Find the mass of a 1.5-m rod whose density varies linearly from 3.5 kg/m from end to end A. 3.5 kg

B. 2.5kg

C. 4.5kg

D. 5.0kg

SOLUTION: 𝑘𝑔

𝑚=

𝑘𝑔

(2.5 𝑚 +3.5 𝑚 )(1.5 𝑚) 2

= 𝟒. 𝟓 𝒌𝒈

89. Find the area bounded by the parabola 𝑦 = 𝑥 2 , the tangent line to the parabola at the point (2, 4) and the x axis. A. 9/2

B. 8/3

C. 8/5

D. 9/4

SOLUTION: 2

𝐴 = ∫0 𝑥 2 𝑑𝑥

𝑨 = 𝟖/𝟑

90. Find the coordinates of an object that has been displaced from the point (-4, 9) by the vector (4i – 5j)

A. (0, 4)

B. (0, -4)

C. (4, 0)

D. (-4, 0)

SOLUTION:

𝑚=

𝑉𝐸𝐶𝑇𝑂𝑅: (4 − 5𝑖) = 6.40 < −51.34 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑋 𝑎𝑛𝑑 𝑌 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟. cos(−51.34) =

𝑥

−5 4

=

𝑋2 − 𝑋1 𝑦−9 𝑥+4

(𝑥 + 4) =

6.40

𝑥=4

41 =

sin(−51.34) =

𝑌2 − 𝑌1

𝑦

(𝑦−9) 4 −5

[ (𝑦−4)4 ]2 −5

+ (𝑦 − 9)2

𝒚=𝟒

6.40

𝑦 = −5

41 = (𝑥 + 4)2 + (4 − 9)2

𝑑 = √(𝑥 + 4)2 + (𝑦 − 9)2

𝒙 = 𝟎 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒,

𝑑 = √41 𝑜𝑟 6.40

𝒕𝒉𝒆 𝒄𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔 𝒂𝒓𝒆 ( 𝟎 , 𝟒).

91. Find the major axis of the ellipse x2 +4y2 -2x – 8y + 1 = 0. A. 2 B. 10 C. 4

D. 6

.SOLUTION: 𝑥 2 − 2𝑥 + 4𝑦 2 + 8𝑦 = −1 (𝑥 2 − 2𝑥 + 1) + 4(𝑦 2 − 2𝑦 + 1) = −1 + 1 + 4 1

(x -1)2 + 4(y-1)2 = 4) (4) (𝑥−1)2 4

+

(𝑦−1)2 1

=1

𝑎2 = 4 = 22 a=2

92. A car travels 90kph. What is its speed in meter per second? A. 43 B. 30 C. 25 SOLUTION:

D. 50

1000𝑚

90 kph (

1𝑘𝑚

1ℎ𝑟

) (3600𝑠) = 𝟐𝟓𝒎/𝒔

93. The vertices of the base of the isosceles triangle are (1, -2) and (1, 4). If the third vertex lies on the line 4x + 3y = 12. Find the area of the triangle. A. 8 B. 15 C. 12 D. 10 SOLUTION:

94. Assume that f is a liner function. If f(4) = 10 and f(7) = 24, find f(100). A.98 B. 144 C. 576 D.458 Solution: f (4) = 10 ; (4,10) f (7) = 24 ; (7,24) m= y2 – y1/ x2 – x1 = (24-10)/ (4-3) m= 14/3

using point slope form: f(x) – 10 =m(x-4) f(x) = f(x) =

14 3

(x-4) -10

14

26

3

3

x-

f(100) =

14 3

(100) -

26 3

f(100) = 458

95. The line y = 3x +b passes thru the point (2, 4). Find b. A. 2 B. 10 C. -2

D. -10

SOLUTION:

y = 3x + b, x = 2 ; y = 4 4 = 3(2) + b b = 4 – 6 = -2

96. How far is the directrix of the parabola (x - 4)2 = -8(y - 2) from the x-axis? A. 2 B. 3 C. 4 D. 1 SOLUTION:

(x – 4)2 = -8(y – 2) 2x – 8 = -8y +16 2x + 8y = 8 97. Find the second derivate of y = xlnx. A. x B. 1/x

C. 1

D. x2

SOLUTION: 𝑦 ′ = 𝑢𝑑𝑣 + 𝑣𝑑𝑢 1

= (x)(𝑥) + ln 𝑥(1) = 1 + lnx 𝑑

Since, 𝑑𝑥 = 𝑦" =

𝑑𝑢 𝑑𝑥

𝑢

, then :

𝟏 𝒙

98. Find the point where the normal to y = x + x1/2 at (4, 6) crosses the y-axis. A. 5.75 B. 9.2 C. 23 D. 11 Solution: y=x+(square root of x) y ' = 1 + 1/[2*sqrt(x)]

At (4, 6) slope of tangent = 1 + 1/[2*sqrt(4)] = 5/4 Slope of normal = -4/5 Normal line is: y - 6 = (-4/5)(x - 4) y = (-4/5)x + 16/5 + 30/5 y = (46 - 4x) / 5 Crossing y-axis means x = 0 y = (46 - 0) / 5 = 46/5 (x, y) = (0, 46/5 or 9.2)

99. There are four geometric mean between 3 and 729. Find the sum of the geometric progression. A. 1092 B. 1094 C. 1082 D. 1084 .SOLUTION: 3_,_,_,_,729 𝑎𝑡 = 𝑎1(𝑟)𝑛 729 = 3(𝑟)5 𝑟=3 𝑛(1) = (3)(3) = 9 𝑛(2) = (3)(3)2 = 27 𝑛(3) = (3)(3)3 = 81 𝑛(4) = (3)(3)4 = 243 𝑛(𝑡) = 3 + 9 + 27 + 81 + 243 + 729 𝒏(𝒕) = 𝟏𝟎𝟗𝟐

100. Find the area of a circle inscribed in a rhombus whose perimeter is 100 inches and whose longer diagonal is 40 inches. A. 364. 43 sq. in C. 452. 39 sq. in. B.590. 62 sq. in. D. 389. 56 sq. in. Solution: the perimeter is 100 the sides are 25 each. The diagonals are perpendicular and meet in the center of the circle. There is a right triangle with a side of 40/2=20 and hypotenuse of 25. 20^2+b^2=25^2 400+b^2=625 b^2=225 b=15 (15*20)/2=150 sq cm Area of triangle. Draw a line from the right angle, which is also the center of the circle, perpendicular to the hypotenuse and label it x. 25x/2=150 25x=300 x=12 radius of the circle. pi*12^2=144pi sq cm area of the circle

MARCH 2017

1. The rectangular coordinate system in space is divided into eight compartments, which are known as:

A. quadrants

B. octants

C. axis D. coordinate

2. What is the value of x in Arctan 3x + Arctan 2x = 45 degrees? A. -1/6 and 1

B. 1/6 and -1

C. 1/6

D. -1

SOLUTION:

Arctan 3x + Arctan 2x = 45° 𝑡𝑎𝑛−1 (3𝑥) + 𝑡𝑎𝑛−1 (2𝑥) = 45° tan 𝐴 + 𝐵 =

𝑡𝑎𝑛 3𝑥 + 𝑡𝑎𝑛 2𝑥 = 𝑡𝑎𝑛45° 1 − 𝑡𝑎𝑛 3𝑥 𝑡𝑎𝑛2𝑥 3𝑥 + 2𝑥 =1 1 − 3𝑥(2𝑥) 5𝑥 =1 1 − 6𝑥 2 5𝑥 = 1 − 6𝑥 2

6𝑥 2 + 5𝑥 − 1 = 0 ( 6𝑥 − 1 )(𝑥 + 1 ) = 0 𝒙=

𝟏 ; 𝑥 = −1 𝟔

3. In delivery of 14 transformers, 4 of which are defective, how many ways those in 5 transformers at least 2 are defective?

A. 940 SOLUTION:

𝑛𝑪𝑟

B. 920

C. 900

D. 910

(14𝑪10) − (14𝑪2) = 𝟗𝟏𝟎

4. Sand is pouring to form a conical pile such that its altitude is always twice its radius. If the volume of a conical pile is increasing at a rate of 25 pi cu. ft./min, how fast is the radius is increasing when the radius is 5 feet?

A. 0.5 ft/min

B. 0.5pi ft/min

C. 5 ft/min

D. 5pi ft/min

SOLUTION: 𝑉= ℎ = 2𝑟

= 𝑓𝑡 3 𝑚𝑖𝑛

𝑑𝑣 𝑑𝑡

= 25𝜋

𝑑𝑟 𝑑𝑡

= ? 𝑎𝑡 𝑟 = 5

1 2

= 𝑑𝑣 𝑑𝑡

𝟏 𝜋𝑟 2 ℎ 𝟐

𝜋(𝑟 2 )(2𝑟) 2 3

𝜋𝑟 3

= 2𝜋𝑟 2

𝑑𝑟 𝑑𝑡

25𝜋 = 2𝜋(5)2

𝑑𝑟 𝑑𝑡

𝒅𝒓 𝒇𝒕 = 𝟎. 𝟓 𝒅𝒕 𝒎𝒊𝒏

𝑥+4

5. Evaluate lim 𝑥−4 as x approaches to infinity. A. 1

B. 0

C. 2

D. infinite

SOLUTION: Using, 𝑥 = 109 𝑥+4 (10)9 + 4 = = 𝟏 𝑥→∞𝑥 − 4 (10)9 − 4 lim

6. Describe the locus represented by the equation |𝑧 − 1| = 2>

A. circle

B. ellipse

C. parabola

D. hyperbola

7. An air balloon flying vertically upward at constant speed is situated 150 m horizontally from an observer. After one minute, it is found that the angle of elevation from the observer is 28 deg 59 min. What will be then the angle of elevation after 3 minutes from its initial position?

A. 63 deg 24 min B. 58 deg 58 min C. 28 deg 54 mi D. 14 deg 07 min SOLUTION:

height in 1 min. ℎ = tan(29° 59` )(150) = 86.54 𝑚

height in 3 mins. ℎ = 3 (86.54) = 259.63 𝑚

h

tan 𝜃 =

259.63 259.63 ; 𝜃 = 𝑡𝑎𝑛−1 ( ) 150 150

150m 𝜽 = 𝟓𝟗° 𝟓𝟖` 8. In how many ways can you pick 3 dogs from a pack of 7 dogs?

A. 32

B. 35

C. 30

D. 36

SOLUTION:

nCr = 7C3 = 35

9. Find the volume (in cubic units) generated by rotating a circle X2 + y2 + 6x + 4y + 12 = 0 about the y-axis.

A. 47.23

B. 59.22

C. 62.11

D. 39.48

SOLUTION: 𝑥 2 + 𝑦 2 + 6𝑥 + 4𝑦 + 12 = 0 (𝑥 2 + 6𝑥 + 9 ) + (𝑦 2 + 4𝑦 + 4 ) = −12 + 9 + 4 ( 𝑥 + 3 )2 + (𝑦 + 2 )2 = 1

𝑉 = 𝐴𝐶 = 𝜋𝑟 2 (2𝜋𝑟) 𝑉 = 𝜋(1)2 (2𝜋(3)) = 𝟓𝟗. 𝟐𝟐 𝒄𝒖. 𝒖𝒏𝒊𝒕𝒔

10.Peter can paint a room in 2 hrs and John can paint the same room in 1.5 hrs. How long can they do it together in minutes?

A. 0.8571

B. 51.43

C. 1.1667

D. 70

SOLUTION: 𝑟1 𝑡 + 𝑟2 𝑡 = 𝐴 1 1 𝑡+ 𝑡=1 2 1.5 1 1 1 + = 2 1.5 𝑡 6𝑜𝑚𝑖𝑛 𝑡 = 0.875 ℎ𝑟𝑠 ( ) = 𝟓𝟏. 𝟒𝟑 𝒎𝒊𝒏. 1ℎ𝑟

11. Solve the differential equation 7yy’ = 5x. A. 7x2 + 5y2 = C

SOLUTION:

B. 5x2 + 7y2 = C

C. 7x2 - 5y2 = C

D. 5x2 - 7y2 = C

7yy' 7y dy ∫ 7y {(7/2)y² = (5/2)x² + C}1/2 5x2 - 7y2 = C

dy

= = =

5x ∫

5x

5x dx dx

12. A cylindrical container open at the top with minimum surface area at a given volume. What is the relationship of its radius to height? A. radius = height B. radius = 2height C. radius = height/2 D. radius = 3height

13. A water tank is shaped in such a way that the volume of water in the tank is V = 2y3/2cu. in. when its depth is y inches. If water flows out through a hole at the bottom of the tank at the rate of 3(sqrt. Of y) cu. in/min. At what rate does the water level in the tank fall? A. 11 in/min

B. 1 in/min

C. 0.11 in/min

D. 1/11 in/min

SOLUTION: 14. A family’s electricity bill averages $80 a month for seven months of the year and $20 a month for the rest of the year. If the family’s bill were averaged over the entire year, what would the monthly bill be? A. $45

B. $50

C. $55

D. $60

SOLUTION:

12months = 560 + 100 = 660 × 1/12 = $55

15. When a baby born he weighs 8 lbs and 12 oz. After two weeks during his check-up he gains 6 oz. What is his weight now in lbs and oz? A. 8 lbs and 10 oz B. 9 lbs and 4 oz 8 lbs and 12 oz. SOLUTION:

C. 9lbs and 2 oz D. 10 lbs and 4 oz

12 𝑜𝑧 × 0

.0625𝑙𝑏𝑠

= 0.75 𝑙𝑏𝑠 𝑜𝑧

0.0625𝑙𝑏𝑠 8 𝑜𝑧 × 𝑜𝑧

𝑡𝑜𝑡𝑎𝑙 = 9.25 𝑙𝑏𝑠 𝑜𝑟 9𝑙𝑏𝑠 𝑎𝑛𝑑 4 𝑜𝑧

= 0.5 𝑙𝑏𝑠

8 + 0.75 + 0.5 = 9.25 𝑙𝑏𝑠 16. A given function f(t) can be represented by a Fourier series if it A. is periodic B. is singled valued C. is periodic, single valued and has a finite number of maxima and minima in any one period D.has a finite number of maxima and minima in any one period

17.A periodic waveform possessing half-wave symmetry has no A. even harmonics

B. odd harmonic

C. sine terms

D. cosine terms

18. N engineers an N nurses. If two engineers are replaced by nurses, 51 percent of the engineers and nurses are nurses. Find N, A. 102

B. 100

C. 55

SOLUTION: 0.51 (N+N) = N+2 N = 100 19.If f(x) = 10^x + 1, then f(x + 1) – f(x) is equal to A. 10(10^ + 1) SOLUTION:

f(x) = 10^x + 1 f(x + 1) – f(x) = ?

B. 9(10^x)

C. 1

D. 9(10^x + 1)

D. 110

f(x+1) = 10^x + 1 = 10^x . 10 + 1 f(x + 1) – f(x) = 10 . 10^x – (10^x + 1) = 10 . 10^x – 10^x -1 = 9(10^x)

20. There is a vector v = 7j, another vector u starts from the origin with a magnitude of 5 rotates in the xy plane. Find the maximum magnitude of u x v. A. 24

B. 70

C. 12

D. 35

21. Find the coordinates of the centroid of the plane area bounded by the parabola y = 4 + x 2 and the xaxis A. (0, 1.5)

B. (0, 1)

C. (0, 2)

D. (0, 1.6)

22. A long piece of galvanized iron 60 cm wide is to be made into a trough by bending up two sides. Find the width of the base if the carrying capacity is a maximum. A. 30

B. 20

C. 40

D. 50

23. The price of gas increased by 10 percent. A consumer reacts by decreasing his consumption by 10 percent. How does his total spending change? A. increase 1 percent

B. decrease 1 percent

C. no change D. decrease 1.5 percent SOLUTION: (a + b + ab)/100 a=10%, b,-10%

(10+(-10)+(10)(-10))/100 =-1%(negative sign shows a decrease)

24. An audience of 450 persons is seated in rows having the same number of persons in each row. If 3 more persons seat in each row, it would require 5 rows less to seat the audience. How many rows? A. 27

B. 32

C. 24

SOLUTION: r - rows; n - number of persons 450 = rn = (r-5)(n+3)

D. 30

rn = rn - 5n + 3r - 15 n = (3r-15)/5 450 = r*(3r-15)/5 750 = r² - 5r r²-5r-750=0 (r-30)(r+25)=0 then r=30 25. The volume of a cube becomes three times when its edge is increased by 1 inch. What is the edge of a cube? A. 2.62

B. 2.26

C. 3.26

D. 3.62

SOLUTION: a3 = V; (a+1)3 = 3V ; (a+1)3=3a3; a = 2.26

26. What is the angle of the sun above the horizon, when the building 150 ft high cast a shadow of 405 ft? A. 21.74

B. 68.26 deg

C. 20.32 deg

D. 69.68 deg

SOLUTION: Arc tan(105/405)=20.32 27. Water ir running out of a conical tunnel at the rate of 1 cu. in/sec. If the radius of the base of the tunnel is 4 in and the altitude is 8 in, find the rate at which the water level is dropping when it is 2 in from the top. A. -1/9pi in/sec B. -1/2pi in/sec

C. 1/2pi in/sec

SOLUTION: dv/dt=1in^3/sec 2r=h V=(1/3)pi*r^2h =(1/3)pi*(h^2/4)h =(1/3)pi*(h^3/4) =(1/12)pi(3)h^2 (dh/dt) 1in^3/sec = (1/4)pi(6)^2(dh/dt)

dh/dt= 1/9pi in/sec

D. 1/9pi in/sec

28. A statistic department is contacting alumni by telephone asking for donations to help fund a new computer laboratory. Past history shows that 80% of the alumni contacted in this manner will make a contribution of at least P50.00. A random sample of 20 alumni is selected. What is the probability that between 14 to 18 alumni will make a contribution of at least P50.00? A. 0.421 B. 0.589

C. 0.844

D. 0.301

29. Jun rows has banca a river at 4 km/hr. What is the width of the river if he goes at a point 1/3 km. A. 5.33 km

B. 2.25 km

C. 34.25 km

D. 2.44

30. Find the volume generated by revolving about the x-axis, the area bounded by = cosh x from x = 0 to x = 1. A. 5.34

B. 3.54

C. 4.42

31. Evaluate the integral of xsinxcosxdx 1 1 1 A. - 4 xcos2x + C C. - 4 xsin2x + 8 xcos2x + C B.

1 8

xsin2x + C

the curve y

D. 2.44

𝟏 𝟒

𝟏 𝟖

D. - xcos2x + sin2x + C

Solutions: (Sin(2x)=2sin(x)cos(x)) *½ = ½ sin(2x)=sin(x)cos(x) = (1/2) x sin(2x)dx udv = uv - vdu = ((½)x) (-(½)cos(2x) + (½)cos(2x) (½)dx = -(¼)xcos(2x) + (¼) (½)sin(2x) + C = -(¼)xcos2x + (1/8)sin(2x) + C

u = (½)x du = (½)dx

v = -(½)cos(2x) dv = sin(2x)dx

32.A cross-section of a trough is a semi-ellipse with width at the top 18 cm and depth 12 cm. The trough is filled with water to a depth of 8 cm. Find the width at a surface of the water. A. 5√2 cm B. 𝟏𝟐√𝟐 cm C. 7√2 cm D. 6√2 cm Solutions: Standard form : x2/81 + y2/144 = 1 Major axis: 24 = 2a ; a = 12 ; a2 = 144 Minor axis: 18 = 2b ; b = 9 ; b2 = 81 (x,y) = (x,-4) X2/81 + -42/144 = 1 x2/81 = 1 – 16/144

x2 = 81*128/144 x = (3/4)*(square root of 128) = 8.48 width of surface water = 2x = 16.97

33.Simplify cos2x + sin2x + tan2x A. cos2x B. sin2x

C. sec2x

D. csc2x

Solutions: = sin2x + cos2x = 1 = 1 + tan2x = sec2x 34.What is the general solution of (D2 + 2)y(t) = 0? A. y = C1cos2t + C2sin2t B. y = C1sin2t + C2cos2t 35.What is the distance between the lines.

A. √6

B. 5

C. C1cos√𝟐t + C2sin√𝟐t D. C1sin√2t + C2cos√2t 𝑥 1 90

C. √ 7

D.

90 7

36. What is a so that the points (-2, -1, -3), (-1, 0, -1) and (a, b, 3) are in straight line? A. 2 B. 4 C. 3 D. 1 37 Find the volume generated when the area bounded by y = 2x – x and y = (x – 1)2 is revolved about the x-axis A. 2.34 B. 3.34 C. 4.43 D. 1.34 38. Find the centroid of a semi-ellipse given the area of semi-ellipse as A = ab and volume of 4

the ellipsed as V = 3 𝜋ab2 A. 2b/3𝜋 B. b/2𝜋

C. 4b/3𝝅

39. How many 5 poker hands are there in a standard deck of cards? A. 2,595,960 B. 2,959,960 C. 2,429,956

D. 3b/4𝜋

D. 2,942,955

Solutions: C = n!/k! / (n-k)! = 52!/5! / (52-5)! = 2,598,960 40. A biker is 30 km away from his home, he travel 10 km and rest for 30 mins. He travel the rest of the distance 2kph faster. What is his original speed? A. 7 kph B. 10 kph C. 8 kph D. 12 kph

41. Cup A = fulll, cup B = full, cup C = cups, what is left in the cup? A. 1/2

B. 3/4

C. 1/4

full, cup D =

17

full. If the 4th cup is used to fill the three

D. 19/36

Solution: A= 1-5/9=4/9 B= 1-5/6=1/6 C= 1-11/12=1/2

Total: 25/36

D=17/18 – 25/36 = 1/4

42. What percent of 500 is 750% A.50 B. 175

C. 57

D. 125

SOLUTION: (750)(100)/500= 150 or 125

43. Using power series expansion about 0, find cosx by differentiating from sinx A. 1- (x^2/2!)+(x^4/4!)-(x^6/6!)+ C. 1-(x^3/3!)+(x^5/5!)-(x^7/7!)+

B.x-(x^2/2!)+(x^4/4!)-(x^5/5!)+ D.x-(x^3/3!)+(x^5/5!)-(x^7/7!)+

44. Find the area bounded by y = √4 A. 7.8

B 6.7

𝑥in the first quadrant and the lines x = and x= 3 C. 5.5

D. 6.5

C. 24 x 10

D. 2.4 x 105

45. Express 2,400,000 in scientific notation A. 2.4 x 10

B. 2.4 x 106

SOLUTION: 2.40000x106

46.An interior designer has to design two offices, each office containing 1 table, 1 chair, 1 mirror, 2 cabinets. A supplier gives him options between 4 tables, 5 chairs, 5 mirrors and 10 cabinets. In how many ways can he design the offices assuming there is no repetition? A. 14100

B. 2400

C. 21600

D. 1740

47. What is the equation of a circle that passes through the vertex and the points of latus rectum of y 2 = x

A. x2 + y2 + 4x + 2y = 0

C. x2 + y2 + 4y +

2x = 0

B. x2 + y2 + 10x = 0

D. x2 + y2 - 10x =0

48.Find the power series expansion of ln (1 – x)

A. 1 + x + (x^2)/2 + (x^3)/3 + B. -1 – x – (x^2)/2 – (x^3)/3 -

C. x + (x^2)/2 + (x^3)/3 + (x^4)/4 + D. –x –(x^2)/2 – (x^3)/3 – (x^4)/4 –

49. Evaluate 10(-20j) + 4(-4j) A. 20 B. 20j

C. -20

D. -20j

C.4/7

D.7/4

50. Evaluate 1= 1/(1+1/1+7) A.15/7 B.13/15

51. The value of all the quarters and dimes in a parking meter is $18. There are twice as many quarters as dimes. What is the total number of dimes in the parking meter? A. 40 B. 20 C. 60 D. 80 x+y=18

2x+y=19

x=18-x

2x+18+x=18

3x=18 X=6

52. A ball is dropped from height of 12 m and it rebounds ½ of the distance it falls. If it continues to fall and rebound in this way, how far will it travel before coming to rest? B. 30 m C. 48 m D. 60 m

A. 36 m

53. At t = o, a particle starts at rest and moves along a line in such a way that at time t its acceleration is 24t2 feet per second per second. Through how many feet does the particle move during the first 2 seconds? A. 32 B. 48 C. 64 D. 96 SOLUTION: S = wot+at = 0+24(2) = 48 ft.

54. If a trip takes 4 hours at an average speed of 55 miles per hour, which of the following is closest to the time the same trip would take at an average speed of 65 miles per hour? A. 3.0 hours B. 3.4 hours C. 3.8 hours D. 4.1 hours SOLUTION: V1t1 = V2t2; t2 = 55 65

(4)

= 𝟑. 𝟒 𝒉𝒓𝒔

55. A laboratory has a 75-gram sample of radioactive materials. The half-life of the material. The half life on the material is 10 days. What is the mass of the laboratory’s sample remaining after 30 days? A. 9,375 grams B. 11.25 grams C. 12.5 grams D. 22.5 grams SOLUTION: 𝑑𝑜

𝑑𝑄 = ∫𝑘𝑑𝑡; 𝑄 = 𝑘𝑡 + 𝑐; 𝑙𝑛75 = 𝑘(0) + 𝑐; 𝑐 = 𝟒. 𝟑𝟐 𝑄

= 𝑘𝑄; ∫ 𝑑𝑡

56. The unit normal to the plane 2x + y + 2z = 6 can be expressed in the vector form as A. i3 + j2 +k2 B. i2/3 + j1/3 + k2/3 C. i1/3 + j1/2 + k1/2 D. i2/3 + j1/3 + k1/3 57.

(ln e2x) is 𝑑𝑥 𝑑

A.

1 2𝑥

𝑒

B.

2 2𝑥

C. 2x 𝑒

D. 2

58. Determine where, if anywhere, the tangent line to f(x) = x3 – 5x2 + x is parallel to the line y = 4x + 23 A. x = 3.61

B. x = 3.23

C. x = 3 D. x = 3.43

59. Which of the following is equivalent to the expression below? (x2 – 3x + 1) – (4x – 2)

A. x2 – 7x – 1

B. x2 – 7x + 3

C. -3x2 – 7x + 3 D. x2 + 12x+ 2

SOLUTION : (x2-2x+1)-(4x-3)=0; x2-7x+3=0

60. For what value of k will x + have a relative maximum at x = -2? 𝑥

A. -4

B. -2

C. 2

D. 4

SOLUTION: x-k/x=0

; x=-2

-2-K/-2=0; k=4

61. When the area in sq. units of an expanding circle is increasing twice as fast as its radius in linear units, the radius is

A. 1/4 𝜋𝝅

C. 1 1/4

B. 0

D. 1

62. If the function f is defined by f(x)= f(0) = x5 – 1, then f-1, the inverse function of f, is defined by f-1(x) =

A.

B.

C.

D.

SOLUTION: f (0) = x5 – 1 =

f (x) = f-1(x) =

63. A school has 5 divisions in a class IX having 60, 50, 55, 62, and 58 students. Mean marks obtained in a History test were 56, 64, 72, 63 and 50 by each division respectively. What is overall average of the marks per student?

A. 56.8

B. 58.2 C. 62.4 D. 60.8

SOLUTION: Overall average = [56 + 56 + 64 + 72 + 63 + 50] ÷ 5 = 61 ≈ 60.8

64. The number n of ways that an organization consisting of twenty-six members can elect a president, treasury, and secretary (assuming no reason is elected to more than one position) is A. 15600 B. 15400 C. 15200 D. 15000 SOLUTION: 26!/(26-3)! = 15600

65. Find the equation of the line that passes through (3, -8) and is parallel to 2x + 3y = 2

A. 2x + 3y = -18

B. 2x + 3y = 30 C. 2x + 3y = -30 D. 2x + 3y = 18

SOLUTION: 2x+3y=2; (3,8) [3y= -2x+2] 1/3 y= -2x/3 + 2/3 Y= mx + b m= - 2/3 y - y1= m (x-x1) [y – 8 = - 2/3 (x-3)] 3 3y

+

2x

=

30

or

2x+

3y

=30

66. Find the center of the circle x2 + y2 + 16x + 20y + 155 = 0. A. (-8, -10) B. (8, 10) C. (8, -10) D. (-8, 10) SOLUTION: x2 +y2+16x+20y+155=0 (x2 +16x) + (y2-120y) =-155 (X2 + 16x + 64) + (y2 -120y + 100) = -155 + 64 + 100 (x+8)2 + (y+10)2 = 9 X= -8; y= -10 or h=-8 k=-10 P (-8,-10)

67. In how many ways can 5 red and 4 white balls be drawn from a bag containing 10 red and 8 white balls?

A. 11760 B. 17640

C. 48620

D. none of these

SOLUTION: 10!/(10-5)! + 8!/(8-4)! = 31920

68. The area of a right triangle is 50. One of its angles is 45°. Find the hypothenuse of the triangle A. 10 B. C. 10 D. 10 SOLUTION:

A=50 A=1/2 bh = 1/2 (h/sinǾ)(h)

1

Ǿ=45 sinǾ=h/b b=h/sinǾ 1

h=

69. Each side of the square pyramid is 10inches. The slant height, H, of this pyramid measures 12 in. What is the area in square inches, of the base of the pyramid? A. 100 B. 144 C. 120 D. 240 SOLUTION:

Ab= S2 =102 =100 sq. inches tan25°+tan 50°

70. Find the exact value of

1−tan 25° tan 50°

A. 1.732

B. 3.732

C. 2.732

SOLUTION:

= 𝟑. 𝟕𝟑𝟐

71. Which term of the arithmetic sequence 2, 5, 8, … is equal to 227? A. 74

B. 75

C. 76

D. 77

SOLUTION : An = A1 + (n -1 ) d 227 = 2 + (n -1) 3 n = 76

72. Name the type of graph represented by x2 – 4y2 – 10x – 8y + = 0 A. circle

B. parabola

C. ellipse

D. hyperbola

D. 0.732

73. If logx 3 = ¼, then x = A. 81

B.1/81 C. 3

D. 9

SOLUTION : logx 3 = log 3 / log x log 3 / log x = 1/4 log 3 (4) = log x (1) x = 81 74. If f(x0 = -x2, then f(x + 1) = A. –x2 + 1

B. –x2 C. –x2 – 2x

D. –x2 – 2x – 2

75.cIf this graph of y = (x – 2)2 – 3 is translated 5 units up and 2 units to the right, then the equation of the graph obtained is given by A. y = x2 + 2 B. y = (x-2)2 + 5 C. y = (x + 2)2 + 2

D. y = (x – 4)2 + 2

76. Which one is not a root of the fourth root of unity? A. I

B. 1

C. i/√𝟐 D. –i

77.Find the area of the largest circle which can be cut from a square of edge 4 in. A. 12.57

B. 3.43 C. 50.27

D. 16

SOLUTION : A=πd2/4 = π(4)2/4 = 12.57 in2 78. If I = (-1)1/2, find the value of i36 A. 0

B. I

C. –I

D. 1

SOLUTION : i^n = n/4 0.25 = i 0.50 = -1 0.75 = -i 1.00 = 1 therefore 36 / 4 = 9 Since 9 is a whole number i^36 = 1

79. If cot B = 5/2, find sin B A.

/5

B.

C.

/2

D. 2/

SOLUTION : B=cot-1(5/2) = 0.38; sin(0.38)=

80. A man 1.60 m tall casts a shadow 4 m long. Nearby, a flagpole casts a shadow 18 m long. How high is the flagpole? 6.4 m

B. 7.2 m

C. 4.5 m

D. 11.25 m

SOLUTION L^2/3 = X^2/3 + Y^2/3 L^2 = ( 4^2/3 + 1.6^2/3) ^3 L^2 = 58.765 L = (58.765)^1/2 L = 7.66 m

81.If Z1= 1-I, Z2= -2 + 4i, Z

B. B. 7.2 m

2i, Evaluate Z12+2z1-3. C. 4.5 m

D. 11.25 mi+ z2+2z-3; (1-i)2+2(1-i)-3=0 = -1-4i

82.A box contains 20 balls, 10 white, 7 blue, 3 red. What is the probability that a ball drawn at random is red? A. 3/20

B. 10/20

C. 7/20

D. 13/20

Solution: 3 12 83. What is the probability of a three with a single die exactly 4 times out of 5 trials? 𝑃=

A. 25/776

B. 125/3888

C. 625/3888

D. 1/7776

84. A man is on a wharf 4 m above the water surface. He pulls in a rope to which is attached a coat at the rate of 2 m/sec. How fast is the angle between the rope and the water surface changing when there are 20 m of rope out? A. 0.804 rad/sec B. 0.0408 rad/sec C. 0.0402 rad/sec D. 0.0204 rad/sec

85. Find the area of the largest rectangle that can be inscribed in the ellipse 25x^2 + 16x^2 = 400 A. 30

B. 40

C. 10

D. 20

SOLUTION: 25𝑥 2 + 16𝑦 2 = 400 𝑥 2 𝑦 2 ; + = 1 ; 𝑎 = √16 = 4 ; 𝑏 = √25 = 5 400 16 25 𝐴 = (4)(5) = 20

86. From the given values of A and B, find the vector cross product of A and B if: A=2i – k B= j A. 5i+2k

B. 4i-2k

C.3i-4j +2k

D. 3i-2j

87. The area of a lune is 30 sq. m. If the area of the sphere is 120sq. m. What is the angle of the lune? A. 80 degree

B. 90 degree

C. 120 degree D. 60 degree

88. If tan x = ½, tan y = 1/3, what is the value of tan (x + y)? A. 1

B. 2/3

C. 2

D. ½

SOLUTION 1

1

tan (tan−1 2 + tan−1 3) = 1 89. Determine the distance between the foci of the curve 9x^2 + 18x + 25y^2 – 100y = 116 A. 8

B. 10

C. 12

D. 6

SOLUTION 9𝑥 2 + 18𝑥 + 25𝑦 2 − 100𝑦 + 9 + 100 = 225 (𝑥 + 1)2 (𝑦 − 2)2 ; + =1 225 52 32

𝑎 = 5;𝑏 = 3 𝑓𝑜𝑐𝑖 𝑡𝑜 𝑓𝑜𝑐𝑖 = 2(√52 + 32 = 8

90. Using synthetic division, compute the remainder if we divide 2x^3 + x^2 = 18x + 7 by x -2 A. -9

B. -8

C. 7

D. 6

SOLUTION (x-2) 2 2

1 4 5

-18 10 -8

17 -16 -9

91. The force required to stretch a spring is proportional to the elongation. If 24 N stretches a spring 3 mm, find the force required to stretch a spring 2 mm. A. 16

B. 18

C. 14

D.12

SOLUTION: F= (24x 2mm)/ 3mm = 16N

92. A is 3 times as old as B. Three years ago, A is four times as old as B. Find the sum of their ages. A. 30 Solution: 4( X-3)- (X-3)=3X-X 4X- 12- X+3= 2X 3X-9= 2X X= 9 B=X = 9 A=3X= 27 B+A= 9+27 = 36

B. 36

C. 26

D. 28

93. The area of a rhombus is 264 sq. cm. If one of the diagonals is 24 cm long, find the length of the other diagonal. A. 22

B. 20

C. 26

D. 28

SOLUTION: A= 1/2(D1D2) 264= 1/2(24xD2) D2= 22 94. In a triangle ABC, angle A= 60 degree and angle B =45 degree. What is the ratio of side BC to side AC? A. 1:22

B. 1:36

C. 1:48

D. 1:19

95. Solve the equation cos^2 A= 1 – cos^A. A. 45o, 315o

B. 45o,225o

C. 45o,135o

96. Find the distance from the point (6, -2) to the line 3x + 4y + 10 = 0. A.4

B. 5.

C. 6.

D. 7

97. If y = tanh x, find dy/dx : A. sech^2 x

B. csch^2 x

C. sinh^2 x

D. tanh^2 x

Solution: Y=tanh X Y'= sech^2 X 98. What number exceeds its square by the maximum? A. 1

B. ½

C. 1/3.

D. 1/4

SOLUTION: D= X-X^2 D=-X^2+X D=-(X^2-X) D=-(X^2-X+(1/2)^2+(1/2)^2 D=-(X-1/2)^2-1/4 X= 1/2 99. Find the derivative of x^-8

D. . 45o,225o

A. -8x^-9.

B. -8x^-7

C. x^-9

D. 0

SOLUTION: F(x)= X^-8 F'(x)=-8x^-9 100. Solve for x : X = (0.125)^-4/3 100. Solve for x : X = (0.125)^-4/3 A. 8

B. 4

SOLUTION: X= (0.125)^-4/3 X= 16

C. 16

D.2

AUGUST 2017

1. What is the value of x in Arctan 3x + Arctan 2x = 45 degrees? SOLUTION .tan−1(3𝑥) + tan−1(2𝑥) = 45 tan^-1 (3x)(2x) = 45 tan^-1 (6x) = 45 x = tan45/6 x= 1/6 ANSWER: C.1/6 2. Find the volume (in cubic units) generated by rotating circle 𝑥 2 + 𝑦 2 + 6𝑥 + 4𝑦 + 12 = 0 about the y axis. SOLUTION : 𝑥 2 + 𝑦 2 + 6𝑥 + 4𝑦 + 12 = 4 𝑣 𝑠𝑝ℎ𝑒𝑟𝑒 = 𝜋𝑟 3 3 (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2 𝑥 2 + 6𝑥 + 𝑦 2 + 4𝑦 = −12 + 9 + 4 (𝑥 2 + 6𝑥 + 9) + (𝑦 2 + 4𝑦 + 4) = 5 (𝑥 + 3)2 + (𝑦 + 2)2 = 1 (𝒉, 𝒌) = (−𝟑, −𝟐) V = A2 𝜋D V = 𝜋𝑟^2(2 𝜋)𝐷 V = 𝜋)(1)^2(2𝜋)(3) V = 59.22 cu.units ANSWER: B. 59.22 3. If i= (-1)^1/2 find the value i^30 A.1 B.-1

C.-I

D.i

4. Solve the equation cos^2, A=1-cos^2 A SOLUTION cos2𝐴 = 1 − 𝑎𝑠 2𝐴 = 2𝑎𝑠 2𝐴 = 1 𝑨𝑺𝟐 = ½ ANSWER: A. 45֯,315 5. Find the change in volume of a sphere if you increase the radius from 2 to 2.05 units. SOLUTION 4 Vsphere =3 𝜋 3 = 4/3 π(2)^3 = 33. 51 = 4/3 π2.0503

= 36.09 ΔVsphere = 36.09-33.51 = 2.58 ANSWER: A.2.51 6. What is the general solution of (D^4 – 1)y(t) = 0? A. 𝐲 = 𝐂𝟏𝐞𝐭 + 𝐂𝟐𝐞−𝐭 + 𝐂𝟑 𝐜𝐨𝐬 𝐭 + 𝐂𝟒 𝐬𝐢𝐧 𝐭 C. y = C1et + C2e−t t −t t −t B. y = C1e + C2e + C3te + C4te D. y = C1et + C2e−t SOLUTION 𝒚 = 𝑪𝟏𝒆𝒕 + 𝑪𝟐𝒆−𝒕 + 𝑪𝟑 𝐜𝐨𝐬 𝒕 + 𝑪𝟒 𝐬𝐢𝐧 𝒕 7. What percentage of the volume of a cone is the maximum right circular cylinder that can be inscribed in it? Answer: C.44 percent 8. if e^2x-3e^x + 2 = 0 , find x. SOLUTION e^2x-3e^x + 2 = 0 lne^2x – 3ln^ex=-ln2 2x-3x=-ln2 X(2-3) = -ln2 X = -ln2/-1 X= ln2 ANSWER: A. ln2 9. On a cortain day the nurses at a hospital worked the following number of hours; nurse howard worked 8 hrs, nurse pease worked 10hrs, nurse campbell worked 9 hrs, nurse grace worked 8 hrs, nurse mccarthy worked 7 hrs, and nurse murphy worked 12 hrs. What is the average number of hrs per nurse on this day? SOLUTION Howard = 8hours Pease = 9 hours Campbell = 9 hours

Ave= summation of number of hours/number of nurse

Grace = 8 hours

= 8+10+9+8+7+12/6 = 9

ANSWER: C. 9 10. Joy is 10 percent taller than joseph and joseph is 10 percent taller than tom. How many percent is joy taller than tom? SOLUTION Joy + 10% than joseph

Joseph + 10% than tom 10%+10% = 20% ANSWER: 20% B 11. An army food supply truck can carry 3 tons. A breakfast ration weights 12 ounces, and the other two daily meals weigh 18 ounces each assuming each soldier gets 3 meals per day, on a ten day trip how many soldiers can be supplied by one truck? SOLUTION 1 ounce = 28.34g 1 ton = 100kg 3tons/day=0.3tons/1day 12 ounce +18+18 48 ounces/day (48 ounces/day)(28.34g/1ounce)(1kg/1000g)(1ton/100kg) =1.36x10^3 =(0.3tons/day)/(1.36x10^3ton/day/soldiers) =220 soldiers ANSWER: C. 200 soldiers 12. Find the area enclose in the second and third quadrants by the curve x=t -1, y= 5t^3(t^2-1) SOLUTION ANSWER: B. 8/7 13.csc520֯=? SOLUTION Csc 520 = csc (520 – 360) Csc 520 = csc 160 Csc 160 = Csc (180 – 160) Csc 16 = csc 20 Csc 520 = csc 20 ANSWER: B.

csc20

14. From past experience it is known 90 percent of one year old children can distinguish their mothers voice of a similar sounding female. A random sample of one years old are given this voice recognize test. Find the probability that atleast 3 children did not recognize their mothers voice. SOLUTION

0.9/3 = 0.3 ANSWER: B. 0.323 15. ln y = mx + b what is m? ANSWER: A. slope 16.Find the area bounded by the parabola sqrt of x + sqrt of a and the line x + y = a SOLUTION 𝑋2 A= ∫𝑋1 (𝑌𝑐 − 𝑌𝑙) 2

1

A= ∫2 (𝑋 − 1) − (1 − √2) 𝑑𝑥 1

A =0.8333 ≈ 3 Since a = 1 A=

𝒂𝟐 𝟑

D. a^2/3 17. What is the integral of cosxe ^sinx dx SOLUTION = ∫ 𝑐𝑜𝑠𝑥𝑒 𝑠𝑖𝑛𝑋 𝑑𝑥 u = sinx du= cosxdx ∫ 𝑒 𝑠𝑖𝑛𝑥 (𝑐𝑜𝑠𝑥)𝑑𝑥 Let u =sinx u = cosxdx ∫ 𝑒 𝑢 𝑑𝑢 = 𝑒 𝑢 + 𝐶 = 𝒆𝒔𝒊𝒏𝒙 + 𝑪 ANSWER: B. 𝒆𝒔𝒊𝒏𝒙 + 𝑪 18. The geometric mean and the arithmetic mean of number is 0 and 10 respectively what is the harmonic mean? SOLUTION AM = a + b AM = 10 =

𝐺𝑀2 +𝑏) 𝑏

(

2

82 ( +𝑏) 𝑏

2

b=4 𝐺𝑀2

a=

𝑏 82

= 4 a = 16 HM =

𝑛 1 1 + 𝑎 𝑏

HM = 1 4

2 +

1 16

HM = 6. 4

ANSWER: C. 6.4 19. In how many ways can four coins be tossed once? SOLUTION n = 4 coins N = 2𝑛 N = 24 N = 16 ANSWER: B. 16

20. A statue 3 m high is standing on a base of 4m high. If an observers eye is 1.5 m above the ground how far should he stand from the base in order that the angle subtended by the statue is a maximum? SOLUTION X=√𝐻1𝐻2 =√(3)(4) X=3.71m ANSWER: C. 3.71 21. What is the number in the series below? 3, 16, 6, 12, 12, 6, SOLUTION 3, 16, (3x2), (16-2^2), (3x2^2), (16-2^3), (3x2^3) =(3x2^2) =24 ANSWER: D. 24 22. A man who is on diet losses 24 lb in 3 months 16 lb in the next 3 months and so on for a long time. What is the maximum total weight loss? A. 72 B. 64 C. 54 D. 81

23. What is the slope of the linear equation 3y-x=9? SOLUTION 3y-x=9 3y=x+9 y=1/3x+(9)(1/3) m=1/3 ANSWER: A. 1/3 24. Each of the following figures has exactly two pairs of parallel sides except a A. parallelogram B.rhombus C. trapezoid D. square 25. A points A and B are 100 m apart and are of the same elevation as the foot of the building. The angles of elevation of the top of the building from points A and B are 21 degrees and 32 respectively. How far is A from the building? SOLUTION

ℎ

Tan32=

Tan21=

𝑥

ℎ

100+𝑥

𝑥𝑡𝑎𝑛32

Tan21=

100+𝑥

X=159.276 100+x= 100+159.276 =259.28m

ANSWER: A. 259.28 26. What is the area in sq.m.of the zone of a spherical segment having a volume of 1470.265 cu.m if the diameter of the sphere is 30m. A. 655.487 B. 565.487 C. 756.847 D. 465.748 SOLUTION A=2 πrh V=

πh2

1470.265=

(3𝑟 − ℎ)

3 πh2 3

(3(15) − ℎ)

h=6 A=2 πrh =2π(15)(6) A= 565.487 sq. m ANSWER: B. 565.487 27. Which of the following numbers can be divided evenly by 19? SOLUTION 𝟕𝟔 =𝟒 𝟏𝟗

ANSWER: C. 76 28. Where is the center of the circle x^2 + y^2 -10x + 4y – 196 = 0 SOLUTION 𝑋 2 − 10𝑋 + 25 + 𝑌 2 + 4𝑌 + 4 = 196 + 25 + 4 (𝑋 − 5)2 + (𝑌 + 2)2 = 225 𝑪(𝟓, −𝟐) ANSWER: D. (5,-2) 29. Two ships leave from a port. Ship A sails west for 300 miles and ship B sails north 400 miles. How far apart are the ships after their trips? SOLUTION 𝑆 = √𝑎2 +𝑏 2 𝑆 = √3002 +4002 𝑺 = 𝟓𝟎𝟎 𝒎𝒊 ANSWER: C. 500 miles 30. if the radius of a sphere is increasing at the constant rate of 3m per second how fast is the volume changing when the surface area is 10 sq.mm? SOLUTION 3m/s x 10 𝑚𝑚2 =30 cu. mm per sec ANSWER: C. 30 cu. mm per sec

31. The sum of the base and altitude of an isosceles triangle is 36cm. Find the altitude of the ttriangle if its area is to be a maximum. SOLUTION: x + y = 36 x = 36 - y 1 A= 2 bh 1

A = 2 ( 36 − 𝑦 )𝑦 1

A= 2 ( 36 -𝑦 2 ) 𝑦2

A = 18 − 2 0 = 18 - y y = 18 ANSWER: C 18cm

32. An insurance policy pays 80 percent of the first P20,000 of a certain patients medical expenses, 60 percent of the next P40,000 and 40 percent of the P40,000 after that. If the patients total medical bill is P92,000 how much will the policy pay? ANSWER: C. 52,800 33. A scientist found 12mg of radioactive isotope is a soil sample. After 2 hours, only 8.2 mg of the isotope remained. Determine the half life of the isotope? SOLUTION: 𝑙𝑛𝑥1 𝑡 = 𝑡1 𝑙𝑛𝑥 2

2

x1 = 12 mg t2 = ? 8.2 12 6 𝑙𝑛 12

𝑙𝑛

=

2 𝑥

x = 3.64 hrs. ANSWER: C 3.64hrs 34. find the area bounded the curves r = 2cosѲ and r = 4cosѲ. A. 6.28 B. 9.42 C. 12.57 D. 15.72 35. Give the degree measure of angke 3pi/5 A. 150 degrees B. 106 degrees C. 160 degrees D. 108 degrees SOLUTION: 3𝜋 180 . 5 ∗ 𝜋 = 𝟏𝟎𝟖 𝒅𝒆𝒈 ANSWER: D 108 deg 36. What is the median of the following group numbers? 1412 20 22 14 16 SOLUTION: 1

M = 2 ( 14 +16) = 15 ANSWER: C 15 37. For what value of k will the line kx + 5y = 2k hace slope 3? SOLUTION: K(3) + 5(3) = 2k k= -15 ANSWER: D. -15 38. The cross product of vector A=4i + 2j with vector B=0. The dot product A·B=30, Find B. ANSWER: A. 6i+3j 39. Find the length of the curve r = (1 – cos Ѳ). ANSWER: D. 32 40. Find the equation of the curve that passes through (4,-2) and cuts at right angles every curve of the family 𝑦 2 = 𝐶𝑥 3 ANSWER: C.𝟐𝒙𝟐 + 𝟑𝒚𝟐 = 𝟒𝟒 41.Find the area of circle with center at (1,3) and tangent to the line 5x – 12y – 8 = 0.

SOLUTION √52 + (−12)2 A= π𝑟 2 =π(3)2 = 𝟐𝟖. 𝟐𝟕 ANSWER: B. 28.27 42. If a flat circular plate of radius r = 2 m is submerged horizontally in water so that the top surface is at a depth of 3m, then the force on the top surface of the plate is SOLUTION F= WhA =w = 9810N F=(9810)(3)(𝜋(2)2 ) F=369828.29N =369,829.15N ANSWER: A. 369,829.15N 43. A hemispherical tank with a diameter of 8 ft is full of water find the work done in ft-lb in pumping all the liquid out of the top of the tank. B. 12,546 𝑑2 𝑦

44. If 𝑥 = 3𝑡 − 1 , 𝑦 = 1 − 3𝑡 , 𝑓𝑖𝑛𝑑 𝑑𝑥 2 SOLUTION x = 3 + 1 , y = 1-3𝑡 2 𝑥

1

Y= 1-3 (3 + 3)2

2 2 𝑦′ = − 𝑥 − 3 3 𝑦" = −2/3

2

𝑦 =1− 𝑦 = 1−

𝑥 2 1 − − 3 3 6

ANSWER: B. -2/3

𝑥2 2 5 − 𝑥+ 3 3 6

45. if sin3A = cos 6B then: A+2B = 30 deg 46. It takes a typing student 0.75 seconds to type one word. At this rate, how many words can the student type in 60 seconds? SOLUTION 0.75𝑠𝑒𝑐 𝑠𝑒𝑐 = 60 1 𝑥

X = 80 ANSWER: D. 80 47. A chord, 6 inches long from the center of a circle. Find the length of the radius of the circle. SOLUTION chord = 16 in 16 2

r=√62 + ( 2 ) = 𝟏𝟎 𝒊𝒏 ANSWER: D. 10 in 48. A train is moving at the rate of 8 mph along a piece of circular track of radius 2500 Through what angle does it turn in 1 min? SOLUTION 180 𝑚 1ℎ𝑟 Ѳ=0.2816 * 𝜋 . 80 ∗ ℎ

60 𝑚𝑖𝑛

=16֯18֯

=1.33m/in = 704 ft / min 704𝑓𝑡 𝑚𝑖𝑛

S=rѲ 2500𝑓𝑡 = Ѳ ANSWER: A. 16 deg 8 49. The diagonal of a face of a cube is 10 ft. The total area of the cube is SOLUTION d= 10ft d= √3a

10

A= 6𝑎2 = 6( )2 = 𝟑𝟎𝟎𝒇𝒕𝟐 √3

ANSWER: D. 300 sq.ft 50. The volume of the sphere is 36 pi cu. m. The surface area of this sphere in sq.m. is: SOLUTION .v= 36 π𝑚3 4

A = 4𝜋𝑟 2 𝑉 = 3 𝜋𝑟 3 , 𝑟 = 3 A= 44𝜋(3)2 A= 36 π ANSWER: B. 36pi 51. Which of the following is an exact DE? ƔM ƔN SOLUTION:exact D.E Ɣy = Ɣx = 1 (2𝑥𝑦)𝑑𝑥 + ( 2 + 𝑥 2 )𝑑𝑦 = 0

𝑀 = 2𝑥𝑦, 𝑁 = 2 + 𝑥 2 ƔM 2𝑥𝑦 = = 2𝑥 Ɣy 𝑦 Ɣ𝐍 𝟐 ∗ 𝒙𝟐 𝟐𝒙𝟐 = = = 𝟐𝒙 Ɣ𝐱 Ɣ𝐱 𝒙 𝟐𝒙𝒚𝒅𝒙 + (𝟐 + 𝒙𝟐 )𝒅𝒚 = 𝟎 ANSWER: C. 𝟐𝒙𝒚𝒅𝒙 + (𝟐 + 𝒙𝟐 )𝒅𝒚 = 𝟎 52. Find the value of 4sinh(pi i/3) 𝜋 SOLUTION: 4sinh (3 𝑖) sinhjѲ= jsinѲ =4jsinѲ 𝜋

=4jsin(3 ∗ 180/𝜋) 4𝑗√3

=

2

=2i√3 ANSWER:B. 2i(sqrt. of 3) 53. Find the coordinates of an object that has been displaced from the point (-4, 9) by the vector 4i-5j). A. (0,4) B. (0,-4) C. (4,0) D. (-4,0) 54. Find the work done in moving an object along a vector r= 3i + 2j - 5k if the applied force F = 2i – j – k. SOLUTION:. r= 3i +2j -5k F= 2i-j-k

(3·2)i = 6 (2·-1)i = -2 (-5·-1)k = 5 6+(-2)+5=9 ANSWER: A. 9 55. Find the value of k for which the line 2x + ky = 6 is parallel to the y-axis. SOLUTION: 2(3-KY) + KY = 6 6-2KY + KY = 6 -2KY + KY = 6

K=6 ANSWER: A. k=0 56. Find the area inside one petal of the four leaved rose r = sin2theta. SOULITON:rsin2Ѳ 𝜋/2

A= ∫0 𝜋

𝜋𝑟

𝜋

=2 ∫02 (𝑠𝑖𝑛2Ѳ(2)𝑑Ѳ) 𝜋

𝜋

=2 (− cos ( 2 ) − cos(0)) 𝜋

=2 (1) 𝝅

=𝟐

ANSWER:D. pi/8 57. Which of the following is a vector? A. kinetic energy B. electric field intensity C. entropy D. work 58. In how many ways can 6 people be lined up to get on a bus if certain 3 persons refuse to follow each other? SOLUTION:. 6P3 =120 ways ANSWER:D. 480 59. The bases of a frustum of a pyramid are 18cm by 18cm and 10cm by 10cm. Its lateral area is 448 sq. cm. what is the altitude of the frustum? ANSWER:B. 6.93cm 60. A store advertises a 20 percent off sale. If an article marked for sale at $24.48, what is the regular price? SOLUTION:20 % discount $24.48 discounted price $24.48 = x – 20% (x) X = $30.60 ANSWER:C. $30.60

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2017 MATHEMATICS

61. If the area of the equilateral triangle is 4 (sqrt. of 3), find the perimeter. SOLUTION: . A= 4√3 A= √s (s-x)(s-x)(s-x) S= 4√3 = √( x=4

𝑥+𝑥+𝑥 2

𝑥+𝑥+𝑥 2

)(

𝑥+𝑥+𝑥 2

− 𝑥))3

P= x+x+xP=12

ANSWER: B. 12 62. Dave is 46 yrs old. Twice as old as rave. How old is rave? SOLUTION: D=46 yrs R=2X 2x=46 X = 23yrs old ANSWER: C. 23 yrs 63. The angles of elevation of the top of a tower at two points 30 m and 80 m from the foot of the tower, on a horizontal line are complementary. What is the height of the tower? SOLUTION: A+B = 90 A= 90-B 𝐻

tanѲ=80

𝐻

B =tan−1 30 equation no. 2 𝐻

𝐻

Tan 90- (tan−1(30)) = 80 H= 49m

𝐻

tan(90-B)=80 equation no. 1 ANSWER: C. 49m 64.A large tank filled with 500 gallons of pure water. Brine containing 2 pounds of salt per gallon is pumped into the tank at a rate of 5 gal/min. The well-mixed solution is pumped out at the same rate. What is the concentration of the solution in the tank at t = 5 min?

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2017 MATHEMATICS

ANSWER: C. 0.0795 lb/ gal 65. The intensity I of light at a depth of x meters below the surface of a lake satisfies the differential dldx = (-1/4)I. At what depth will the intensity be 1 percent of thtat at the surface? ANSWER: B. 2.29m 66. What is the discriminant of the equation4𝑥 2 = 8𝑥 − 5? ANSWER: B. -16 67. Find the percentage error in the area of a square of side s caused by increasing the side by 1 percent. ANSWER: B. 2 percent 68. What is the height of a right circular cone having a slant height of 3.162 m and base diameter of 2 m? SOLUTION: H=√(3.162)2 − (1)2 H= 3m ANSWER: C. 3 69. In how many orders can 7 different pictures be hung in a row so that 1 specified picture is at the center? SOLUTION: 6i = 720 ways ANSWER: D. 720 70. What is the x-intercept of the line passing through (1,4) and (4,1)? ANSWER: B. 5 71. One ball is drawn at random from a box containing 3 red balls, 2 white balls, and 4 blue balls. Determine the probability that is not red. SOLUTION # 𝑜𝑓𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑓𝑢𝑙 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑃= # 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 6 𝑃= 9 𝟐 𝑷= 𝟑 ANSWER: B. 2/3 72. An airplane flying with the wind took 2 hours to travel 1000 km and 2.5 hours flying back. What was the wind velocity in kph?

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2017 MATHEMATICS

SOLUTION 𝑆 = 𝑉𝑡 1000 = (𝑉𝑎 + 𝑉𝑤)2 𝑉𝑎 = 500 − 𝑉𝑤 1000 = (𝑉𝑎 − 𝑉𝑤)2.5 𝑉𝑎 = 400 + 𝑉𝑤 500 − 𝑉𝑤 = 400 + 𝑉𝑤 2𝑉𝑤 = 100 𝑽𝒘 = 𝟓𝟎 ANSWER: A. 50 73. In how many ways can a person choose 1 or more of 4 electrical appliances? SOLUTION 𝑁 = 𝑛𝐶𝑟 𝑁 = 4𝐶1 + 4𝐶2 + 4𝐶3 + 4𝐶4 𝑵 = 𝟏𝟓 ANSWER: A. 15 74. What are the third proportional to y/x and 1/x? SOLUTION 𝑎 𝑐 1 1 = (𝑥 ) (𝑥) 𝑏 𝑑 𝑑= 𝑦 𝑐𝑏 𝑑= 𝑥 𝑎 𝟏 𝒅= 𝒙𝒚 ANSWER: C. 1/xy 75. If 7 coins are tossed together, in how many ways can they fall with most three heads? SOLUTION 𝑁 = 𝑛𝐶𝑟 𝑁 = 7𝐶3 + 7𝐶2 + 7𝐶1 + 7𝐶0 𝑵 = 𝟔𝟒 ANSWER: B. 64 76. If y = ln (sec x tan x). find dy/dx. A. cot x B. cos x C. csc x D. sec x SOLUTION 1 (𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥) 𝑦′ = 𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥 1 (𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥 + 𝑠𝑒𝑐 2 𝑥) 𝑦′ = 𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥 1 (𝑠𝑒𝑐𝑥(𝑡𝑎𝑛𝑥 + 𝑠𝑒𝑐𝑥)) 𝑦′ = 𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥 𝒚′ = 𝒔𝒆𝒄𝒙 ANSWER: D. sec x

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2017 MATHEMATICS

77. A rubber ball is made to all from height of 50 ft and is observed to rebound 2/3 of the distance it falls. How far will the ball travel before coming to rest if the ball continues to fall in this manner? SOLUTION a1= 50 x 2/3 = 33.33 𝑎1

33.33

S=1−𝑟=1−2/3 = 100 St= 50 +(2)(100) St=250ft ANSWER: A. 250 78. In a class of 40 students, 27 like calculus and 25 like Chemistry. How many like calculus only? SOLUTION 40 students, 27 like cal, 25 like chem 40 = x + 25 X=15 ANSWER: B. 15 79. Simplify (cos θ / sin θ + 1 ) + tan θ SOLUTION 𝑐𝑜𝑠 𝜃 + 𝑠𝑖𝑛2 𝜃 + 𝑠𝑖𝑛𝜃 = 𝑐𝑜𝑠𝜃(𝑠𝑖𝑛𝜃 + 1) 𝑠𝑖𝑛𝜃 + 1 = 𝑐𝑜𝑠𝜃(𝑠𝑖𝑛𝜃 + 1) ANSWER: A. sec 2

1 𝑐𝑜𝑠𝜃 = 𝒔𝒆𝒄𝜽 =

80. What kind of graph is r = 2 sec θ? A. straight line B. parabola

C. ellipse

D. hypebola

81. Find the inclination of the line passing through (5,3) and (10,7) SOLUTION: p1(-5,3) p2(10,7) Tan theta = (y2-y1) / (x2-x1) = (7-3) / 10-(-5) = 14.92 degrees ANSWER: B. 14.93֯ 82. An ellipse has an eccentricity of 1/3. Find the distance between the two directrix if the distance between the foci us 4.

SOLUTION: 2ae=3.

distance between the directrix =2a/e.

2a*1/3=3.

2*4.5/1/3

2a/3 =3.

=9/1/3

2a=9.

=9*3=27

a=9/2=4.5 ANSWER: A.36 83. Find the value of sin (arc cos 15/17). SOLUTION: . Call x the arc whose cosx=1517. Find sin x. sin2x=1−cos2x=1−225289=64289. sinx= ±8/17 ANSWER: D. 8/17 84. Find the area of the triangle having vertices at -4 -I, 1 +2i, 4-3i. SOLUTION: (-4 –I)(1 +2i)(4-3i)/2 = 17 ANSWER: C. 17 85. Find the location of the focus of the parabola 𝑥 2 + 4𝑦 − 4𝑥 − 8 = 0. SOLUTION: x^2-4x+2^2 = -4y + 8 +2^2 (x-2)^2 = -4(y-3) (x-h)^2 = -4a(y-k) A=1 therefore, focus is (-2,-2) ANSWER: D.(-2,-2) 86. What conic section is2𝑥 2 − 8𝑥𝑦 + 4𝑥 = 12? A. hyperbola B. ellipse C. parabola

D. circle

87. A man bought 5 tickets in a lottery for aprize of P 2,000.00. If there are total 400 tickets, what is his mathematical expectation? SOLUTION: . 5/400 = x/2000 ; x = 25 ANSWER: A. P25.00 88. In what quadrants will Ѳ be terminated if cos Ѳis negative?

SOLUTION: Quadrant II, the x direction is negative, and both cosine and tangent become negative Quadrant III, sine and cosine are negative Therefore 2,3 ANSWER: B. 2,3 89. For what value of the constant k is the lie x + y = k normal to the curve 𝑦 = 𝑥 2 SOLUTION: So the slope of the normal is -1, which means that the slope of the tangent is 1. dy/dx = 2x Find out where the slope is 1: 2x = 1 --> x = 1/2 So we have the coordinates (1/2, 1/4). So eqn of normal is: y - 1/4 = -1*(x - 1/2) y =-x + 1/2 + 1/4 k = 1/2 + 1/4 = 3/4

ANSWER: A. 3/4 90. Any number divided by infinity is equal to: A. 1 B. infinity C. zero D. indeterminate 91. The points Z1,Z2,Z3,Z4 in the complex plane are vertices of parallelogram taken in order if and only if SOLUTION . (z1 + z3)/2 = (z2 + z4)/2 Therefore z1 + z3 = z2 + z4 ANSWER:C. Z1+ Z3 = Z2 + Z4 92. If the points (-1,-1,2),(2,m,5) and (3,11,6) are collinear, find the value of m. SOLUTION AB = (2 + 1)i + (m + 1) j + (5-2)k = 3i + (m+1)j + 3k And AC = (3+1)I + (11+1)j + (6-2)k = 4i + 12j + 4k ( 3i + (m+1)j = λ ( 4i + 12j + 4k ) 3 = 4 λ and m + 1 = 12 λ And m=8

ANSWER: A. 8

93. Infinity minus infinity is: A. infinity B. zero

C. indeterminate

D. none of these

94. If in the fourier series of a periodic function, the coefficient aჿ = 0 and aⁿ = 0, then it must be having ____________ symmetry. A. odd B. odd quarter wave C. even D. either A or B 95. Tickets number 1 to 20 are mixed up then and then a ticket is drawn has a number which is a multiple of 3 or 5? SOLUTION Here, S = {1, 2, 3, 4, ...., 19, 20}. Let E = event of getting a multiple of 3 or 5 = {3, 6 , 9, 12, 15, 18, 5, 10, 20}. P(E) = n(E)/n(S) = 9/20. ANSWER: D. 9/20 96. A car travels 90 kph. What is its speed in meter per second? SOLUTION: 90 km/hr x 1000 meter/1 km 1 hr/3600 sec. = 25 ANSWER: C. 25 97. The line y = 3x = b passes through the point (2,4) Find b. SOLUTION:

(4)=3(2)-b therefore b= -2 ANSWER: C. -2 98.If y = tanh x, find dy/dx: A. 𝐬𝐞𝐜 𝟐 𝒙 B. csc 2 𝑥

C. sin2 𝑥

D. tan2 𝑥

99.From the given values A and B, find the vector cross product of A and B, if: A=2i – 5k, B=j SOLUTION: (2i – 5k)(j) = 5i + 2k ANSWER: A. 5i + 2k 100. If a place on the earth is 12 degrees south of the equator, find its distance in nautical miles from the north pole. SOLUTION: theta = 90+12 = 10^2 102 degrees x 60 min/ 1 degrees = ln m/1 min = 6,120 nautical miles ANSWER: D. 6,120

MARCH 2018

1. A tangent to a conic is line A. which is parallel to the normal

C. which passes inside the conic

B. which touches the conic at only one point D. All of the above

2. Simplify 1/(csc x+1) + 1/(csc x -1) A. 2 sec x tan x B. 2 csc x cot x

C. 2 sec x

D. 2 csc x

Solution: 1/(csc x+1) + 1/(csc x -1) .csc²-1 csc x -1 + csc x +1 = 2 csc x 3. Find the coordinates of the centroid of the plane are bounded by the parabola y= 4-x² and the x-axis A. (0 , 1.5)

B. (0 , 1)

Solution: y=4-x² ; x²= -1(y-4) at y=0 x=+/-2 lower limit =0 upper limit = 2 A=2/3bh = 2/3(4)(4)= 32/3 Ay= 2 ∫ ydx (y/2) 32/3 (y)= 2 ∫ (4-x²)²dx y=1.6 x=0 therefore centroid is (0, 1.6)

C. (0 , 2)

D. (0 , 1.6)

4. Evaluate Γ(-3/2) A. -2(sqrt of pi)/3

B. 2(sqrt of pi)/3

C. -4(sqrt of pi)/3

D. 4(sqrt of pi)/3

Solution: Γ-3/2 = Γ(-3/2 +1)/(-3/2) = Γ (-1/2)/(-3/2) = -2/3(-2√π) = 4/3√π 5. Ben is two years away from being twice Ellen’s age. The sum of twice Ben’s age and Thrice Ellen’s age is 66. Find Ben’s age now. A. 19

B. 20

C. 18

D.21

Solution: x + 2 = 2y ; x= 2y-2 2x + 3y = 66 2(2y-2) + 3y = 66 Y=10 X= 2(10) - 2 = 18 6. Find the area bounded by the outside the first curve and inside the second curve r=5, r=10sin theta A. 47.83

B. 34.68

C. 73.68

Solution: Area = ½(5²) π – area of sector1 – area of sector2 area of sector1 = 1/2(5²)(π/3)= 25 π /6 lower limit = π/3 upper limit = π/2 area of sector2= ½ ∫ (10cos Θ)² = 25 π /6 – 25/4 √3

D. 54.26

Area= ½(5²) π - 25 π /6 - 25 π /6 – 25/4 √3 = 47.83 7. In polar coordinate system, the polar angle is negative when A. measured counterclockwise C. measured at the terminal side of theta B. measured clockwise

D. none of these

8. A balloon rising vertically 150m from and observer. At exactly 1min, the angle of elevation is 29 deg 28min. How fast is the balloon rising at that instant? A. 104 m/min

B. 102 m/min

C.106 m/min

D. 108 m/min

Solution: y= 150 tan Θ dy/dt = 150 sec²Θ dΘ/dt Θ=29deg 28min = 0.5143rad dΘ/dt = Θ/t = 0.5143/1 = 0.5143 rad/min dy/dt = 150 sec²(29deg 28min)(0.5143 rad/min) = 101.77 m/min = 102 m/min 9. When the ellipse is rotated about its longer axis, the ellipsoid is A. spheroid

B. oblate

C. prolate

D. paraboloid

10. For the formula R= E/C, find the maximum error if C= 20 with possible error 0.1 and E= 120 with a possible error of 0.05 A. 0.0325

B. 0.0275

Solution: dR = 1/C dE – E/C² dC

C. 0.0235

D. 0.0572

dR = 1/20(0.05) – 120/20² (-0.01) = 0.0325 11. The probability that a married man watches a certain television show is 0.4 and the probability that a married woman watches the show is 0.5. The probability that a man watches the show, given that his wife does is 0.7. Find the probability that a wife watches the show given that her husband does. A. 0.875

B. 0.745

C. 0.635

D. 0.925

Solution: Let :

M – the event that the man watch the show W - the event that the woman watch the show

Given : P(m) = o.4 P(w) = 0.5 P(m/w) = 0.7 Solution : P(m or w) = P(w)*P(m/w) = 0.5 x 0.7 = 0.35 P(w/m) = P(w or m)/P(m) = P(m or w)/P(m) = 0.35/0.4 = 0.875 12. Four friends took the EE Board exam, each with a probability 0.6 passing the said exam. Find the probability that at least one of them will pass the exam. A. 0.7494

B. 0.7449

C. 0.9744

Solution: Let: x – probability of passing the said exam Y – probability that at least one of them will pass the exam. Z – probability that fail the exam Given: x = 0.6 z = 1 – x = 1 – 0.6 = 0.4 Y = 1 – 0.44 = 0.9744 13. Evaluate lim ( sin-19x )/2x , when x = 0.

D. 0.9474

A. 9/2

B.π

C. ∞

D. - ∞

Solution: Let: x = 0.0000001 ( sin-19x )/2x = ( sin-19(.0000001) )/2(0.0000001) = 4.5 or 9/2 Note : Set in radian mode

14. A sequence of numbers where the succeeding term is greater than the preceding term is called. A. Dissonant series

B. Convergent series

C. Isometric series

D. Divergent series

15. Find the initial point of v = <-3,1,2> if the terminal point is <5,0,-1> A. <8,1,-3>

B. <8,-1,-3>

C. <-8,1,3>

D. <-8,-1,3>

Solution: Given : <-3,1,2> , <5,0,-1> ( 5 – (-3), 0 – (1), -1 –(2)) = ( 8,-1,-3 )

16. What do you call the integral divided by difference of the abscissa? A. Average value

B. Mean value C. Abscissa value

D. Integral value

17. Solve (D2-3D+2)y=4x A. c1ex + c2e2x B. c1ex + c2e2x + 2 Solution: (D2-3D+2)y=4x (D – 1)(D – 2), Therefore D1 = 1, D2 = 2

C. c1ex + c2e2x + 3 D. c1ex + c2e2x + 2x + 3

Yc = c1eD1x + c2eD2x = c1ex + c2e2x Yp = Ax + B Yp’ = A Yp’’ = 0 Subst. to equation, O – 3(A) +2(Ax + B) = 4x @ x : 2A = 4 A=2 @ k : -3A + 2B = 0 B=3 Yp = 2x + 3 Y = Yc + Yp = c1ex + c2e2x + 2x + 3 18. Find the second derivative of the function y=5x3 +2x + 1 A. 2x

B. x

C. 30x

D. 24x

Solution: Given : y = 5x3 +2x + 1 y’ = 15x2 +2 y’’ = 30x

19. Three circle of radai 3, 4, and 5 inches respectively, are tangent to each other extremely. Find the largest angle of a triangle found by joining the center of the circles. A. 72.6 degrees B. 75.1 degrees C. 73.4 degrees D. 73.5 degrees

Solution: Given: r1 = 3, r2 = 4, r3 = 5 sides of a triangle are 7, 8, 9

S = ( 7 + 8 + 9 )/2 = 12 A = √(s(s-7)(s-8)(s-9)) = 26.83 sq. unit Angle 1:

26.83 = (1/2)(7)(8)sinƟ Ɵ = 73.4 deg

Angle 2:

26.83 = (1/2)(7)(9)sinƟ Ɵ = 58.4 deg

Angle 3:

26.83 = (1/2)(9)(8)sinƟ Ɵ = 48.18 deg Therefore: Ɵ = 73.4 deg is the highest

20. A reflecting telescopes has a parabolic mirror for witch the distance from the vertex to the focus is 30 ft. If the distance across the top of the mirror is 64 in, how deep is the mirror of the center? A. 32/45 in

B. 30/43 in

C. 32/47 in

D. 35/46 in

Solution: Given: x = 64/2 = 32 , p = 30x12 = 360 at origin at the center X2 = 4py y = x2/4p = 322/4(360) = 32/45 in 21. An observer wishes to determine the height of a tower. He takes sights at the top of the tower from A and B, which are 50 ft apart at the same elevation on a direct line with the tower. The vertical angle at point A is 30 degrees and at point B is 40 degrees. What is the height of the tower? A. 85.60 ft C. 110.29 ft B. 143.97 ft D. 95.24 ft

Solution: Tan40=h/x X=h/tan40 - eq 1 Tan30= h/50+x X=h/tan30 - eq 2 Equate 1 and 2

h/tan40 = h/tan 30 h=95.24ft

22. The average of six scores is 83. If the highest score is removed, the average of the remaining scores is 81.2. Find the highest score. A. 91 C. 93 B. 92 D. 94 Solution: Given: 𝐴𝑣𝑒 𝑜𝑓 𝑠𝑖𝑥 𝑠𝑐𝑜𝑟𝑒𝑠 = 83

𝐴𝑣𝑒 𝑜𝑓 𝑓𝑖𝑣𝑒 𝑠𝑐𝑜𝑟𝑒𝑠 = 81.2

Find: Highest score 𝑥 = 83 6 𝑥 = 498 𝑥−𝑦 = 81.2 5 𝑥 − 𝑦 = 407.5 𝑦 = 498 − 407.5 𝒚 = 𝟗𝟎. 𝟓 𝒐𝒓 𝟗𝟏 23. A coat of paint of thickness 0.01 inch is applied to the faces of a cube whose edges is 10 inches, thereby producing a slightly larger cube. Estimate the number of cubic inches of paint used. A. 3 B. 6 C. 2 D. 4 Solution:

V=x3 Dv=3x2dx Dv=3(10)2(0.01) Dv=3 24. The area in the second quadrant of the circle x^2+y^2=36 is revolved about the line y+10=0. What is the volume generated? A. 2932 c.u C. 2229 c.u B. 2392 c.u D. 2292 c.u Solution:

y’=4r/3π y’=4(6)/3π Second prop of pappus V=Ax2πxd’ =1/4(πr2)(2π)(10+y’) V=2228.83 cubic units

25. Find the equation of the parabola whose vertex is the origin and whose directrix is the line x=4 A. y^2=16 B. y^2=-16x C. x^2=16y D. x^2=-16y Solution: a=4 y =-4ax y2=-4(4)x y2=-16x 2

26. A solid has a circular base of radius 4 units. Find the volume of the solid if every plane section perpendicular to a fixed diameter is an equilateral triangle. A. 147.80 B. 256 C.148.96 D. 86 Solution: 2r=d 2(4)=d D=8 d/6(0+4am+0)=V d/6(0+4(Sqrt of ¾ a2)+0)=V V=147.80 cubic units

27. From past experience, it is known 90% of one year old children can distinguish their mother’s voice from the voice of a similar sounding female. A random sample of 20 one year’s old are given this voice recognize test. Find the probability that all 20 children recognize their mother’s voice. A. 0.122 B. 0.500 C. 1.200 D. 0.222 Solution: .9022=0.122

28. If Jose is is 10% taller than Pedro and Pedro is 10% taller than Mario, then Jose taller than Mario by _______%. A. 18 B.20 C.21 D.23 Solution: Jose Pedro Mario

1.1(1.1x) 1.1x x 1.1(1.1x)-x=0.21x= 21%

29. The area of circle is six times it’s circumference. What is the radius of the circle? A. 10 B. 11 C. 12 D. 13 Solution: (πr2)= 6(2πr) r=6 30. Find the orthogonal trajectories of the family of parabolas y^2=2x+C A. y=Ce^x C.y=Ce^(2x) B.y=Ce^(-x) D.y=Ce^(-2x) Solution: Y2=2X+C 2ydy=2dx+0 dy/dx= 2/2y dy/dx=1/y dy/dx=-dx/dy dy/dx=-y ∫ 𝑑𝑦/𝑦 = − ∫ 𝑑𝑥 Lny=-x+c e^lny=e^-x+c

y=ce-x

31. A pole which lean 11 degrees from the vertical toward the sun cast a shadow 12m long when the angle of the elevation of the sun is 40 degrees. Find the length of the pole. A. 15.26 m

B. 14.26 m

C. 13.26 m

D. 12.26 m

Solution: X= 180 - 40 – 90 – 11=39 Z

=

12

Sin40

= 12.26

sin39

32. A tree stands vertically on a sloping hillside. At a distance of 16 m down the hill, the tree subtends an angle of 34 degrees. If the inclination of the hill is 20 degrees. Find the height of the tree. A. 12.5 m

B. 13.4 m

C. 14.3 m

=

=14.3m

D. 15.2 m

Solution: 16 Sin56

h sin14

33. Robin flies to San Francisco from Santa Barbara in 3 hours. He flies back in 2 hours. If the wind are blowing from the north at the velocity of 40mph going but changed in 20mph from north returning. What was the air speed of the plane.

A. 140mph

B. 150mph

C. 160mph

D. 170mph

Solution: (x-3) (40) = (x+2)(20) 40x-120 = 20x+40 x= 40+ 120 x= 160mph 34. What would happen in the volume of a sphere if the radius is tripled? A. Multiplied by 3

B. Multiplied by 9

C. Multiplied by 27

D. Multiplied by 6

Solution: V=4/3 πr^3 V(3) = 4/5π(3)^3 = 4/3π = 27

therefore: multiplied by 27

35 The distance between the center of the 3 circles which are mutually tangent each other are 10,12, and 14 units. Find the area of the largest circle. A. 72pi

B. 64pi

C. 23pi

Solution: A= πr^2 A= π(8)^2

=64π

D. 16pi

36. What is the vector which is orthogonal both to 9i + 9j and 9i + 9k? A. 81i+ 81j – 81k

B. 81i - 81j – 81k

C. 81i - 81j + 81k

D. 81 i+ 81j + 81k

Solution: (9i + 9j) (9i + 9k) = 81(i + j) (i + k) = 81 (i –j –k) = 81 -81i -81k 37. Good costs in merchants P72 at what price should he mark them so that he may sell the at discount of P10 form marked price and still make a profit of 20% on the selling price? A. P150

B. P200

C. P100

Capital

P72

Worked price

X

D. P250

Solution:

Selling price

0.20X

Profit

0.20(0.90)

Profit = Income + capital = 0.20 (0.90) = 0.90X-72

X= 100

38. A ranch has cattle and horses in a ratio of 9.5. If there are 80 more heads of a cattle than horses. How many animals are on the ranch? Solution: (9/5) = (x+80/x) 9x = 5(x+80) 9x-5x = 400 X = 100 + 80 Y = 100 Total= 180+100 = 280 A. 140

B. 150

C. 238

D. 280

39. A group of students plan to pay equal amount in hiring a vehicle for an excursion trip at a cost of P6000. However, by adding two more students to the original group, th cost of each student will be reduced by P150. Find the number of students in the original group. A. 10

B. 9

C. 8

D. 7

Solution: 6000= n (2a1 + (n-1) ( 600 ) 2

n= 8 n

40. The volume of the sphere is 36pi cu.m. The surface area of the sphere in sq. m is. A. 36pi

B. 24pi

C. 18pi

D. 12pi

Solution: V = 4/3 πr^3 36π = 4/3 πr^3 r=3 A = 4π(3)^2 A = 36π 41. The logarithm of MN is 6 and the logarithm of N/M is 2. Find the value of logarithm of N. A. 3

B. 4

C. 5

D. 6

Solution: Given: log 𝑀𝑁 = 6 𝑁

log 𝑀 = 2 log 𝑀 + log 𝑁 = 6 log 𝑀 = 6 − log 𝑁 → ① 𝑁

log 𝑀 = 2 log 𝑁 − log 𝑀 = 2 log 𝑁 − 2 = log 𝑀 → ② Equate 1 & 2 6 − log 𝑁 = log 𝑁 − 2 2 log 𝑁 = 8 𝐥𝐨𝐠 𝑵 = 𝟒 42. Peter can paint a room for 2 hrs and John can paint the same room in 1.5 hrs. How long can they do it together in minutes?

A. 0.8571

B. 51.43

C. 1.1667

D. 70

Solution: Given: Peter = John =

1 𝑟𝑜𝑜𝑚 2 ℎ𝑟𝑠

1 𝑟𝑜𝑜𝑚 1.5 ℎ𝑟𝑠

=

=

1 2 1 1.5

1 1 1 + = 2 1.5 𝑥 60 𝑚𝑖𝑛𝑠 ) 1ℎ𝑟

𝑥 = 0.86 ℎ𝑟𝑠 (

= 53.43 mins.

43. An airplane has an airspeed of 210 mph the bearing of N 30deg E a wind is blowing due west at 30 mph. Find its ground speed rounded to the nearest degree. A. 201

B. 187

C. 197

D. 175

Solution: 𝐻𝑜𝑟𝑖𝑧𝑎𝑜𝑛𝑡𝑎𝑙 ∶ 30 sin 30 = −15 𝑚𝑝ℎ 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 ∶ 30 cos 30 = 25.98 𝑚𝑝ℎ 𝑝𝑙𝑎𝑛𝑒 𝑠𝑝𝑒𝑒𝑑 = 210 𝑚𝑝ℎ ∑ 𝐹𝐻 = 210 + (−15) = 195 ∑ 𝐹𝑉 = 25.98 ∑ 𝑅𝑒𝑠𝑢𝑙𝑡𝑎𝑛𝑡 = √(195)2 + (25.98)2 = 𝟏𝟗𝟔. 𝟕 𝒎𝒑𝒉 ≈ 𝟏𝟗𝟕 𝒎𝒑𝒉

44. Find the area of a regular hexagon circumscribing a circle with an area of 289pi sq. cm. A. 2,002 sq. cm. B. 1,001 sq. cm. C. 550 sq. cm. D. 328 sq. cm.

Solution: Given:

𝐴2 = 289 𝜋 𝑐𝑚2 𝜋𝑟 2 = 289 𝜋 r = 17 𝐴 = 𝑛𝑟 2 tan

180 6

𝐴 = 6(17)2 tan

180 6

= 𝟏, 𝟎𝟎𝟏 𝒄𝒎𝟐

45. If y = 4cosx + sin2x, what is the slope of the curve when x = 2? A. -2.21

B. -4.94

C. -3.25

D. 2.21

Solution:

Given: y = 4cosx + sin2x, x=2 rad 𝑦 ′ = 4(−sin 𝑥) + 2 cos 2𝑥 = 2 cos 2𝑥 − 4 sin 𝑥 @ 𝑥 = 2 𝑟𝑎𝑑 180 )) − 𝜋

𝑦 ′ = 2 cos 2 (2 (

180 )) 𝜋

4 sin (2 (

𝑦 ′ = 2 cos 229.183 − 4 sin 114.591 𝒚′ = −𝟒. 𝟗𝟒

46. A rectangular plate of 6 m by 8 m is submerged vertically in a water. Find the force on one face if the shorter side is uppermost and lies in the surface of the liquid. A. 941.76 kN

Solution: Given:

B. 1,883.52 kN C. 3,767.04 kN D. 470.88 Kn

6m x 8m ℎ ℎ̅ = 2 + 6 =

8 + 2

6 = 10

𝐹 = (𝐷𝐻2 0)(ℎ̅)(𝐴) = (981)(10)(6(8)) 𝑭 = 𝟒𝟕𝟎. 𝟖𝟖 𝒌𝑵

47. A steel ball at 120 deg C cools in 20 minutes to 80 deg C in a room at 25 deg C. Find the temperature of the ball after half an hour. A. 40.96 deg C B. 45.96 deg C C. 66.85 deg C D. 55.96 deg C

Solution: 𝑻𝒕 − 𝑻𝒔 = (𝑻𝒐 − 𝑻𝒔 )𝒆−𝒌𝒕 80 − 25 = (120 − 25)𝑒 −𝑘(20) 50 = 95 (−20𝑘) ln 𝑒 𝑘 = 0.02733 @𝑡 =0 𝑇𝑡 − 25 = (120 − 25)𝑒 −0.02733(30) 𝑻𝒕 = 𝟔𝟔. 𝟖𝟓 ℃

10

48. Evaluate the inverse Laplace transform of 𝑠+50 A. 10𝑒 −5𝑡

Solution: 𝟏𝟎

𝓛−𝟏 (𝒔+𝟓𝟎) = 𝟏𝟎𝒆−𝟓𝟎𝒕

B. 10𝑒 −𝑡

C. 𝟏𝟎𝒆−𝟓𝟎𝒕

D. 10𝑡𝑒 −50𝑡

49. In a printed circuit board may be purchased from 5 suppliers in how many ways can 3 suppliers can be chosen from the 5? A. 20

B. 5

C. 10

D. 68

C. 4

D. 8

Solution: 5C3

5!

= 3!(5−3)! = 𝟏𝟎

50. Find the length of the vector (2, 4, 4). A. 5

B. 6

Solution: |𝒂 ̅ | = √𝒂𝟐 + 𝒃𝟐 + 𝒄𝟐 = √22 + 42 + 42 |𝒂 ̅| = 𝟔

51. What is the perimeter of a regular 15-sided polygon inscribed in a circle with radius 10 cm? A. 63.77 cm B. 62.37 cm C. 64.52 cm D. 68.48 cm Solution: 𝑃 = 2𝑛𝑟 𝑠𝑖𝑛

180 𝑛

𝑃 = 2(15)(10)sin

180 15

= 𝟔𝟐. 𝟑𝟕𝒄𝒎

52. Find the area bounded by the curve (y square) – 3x + 3 = 0 and x = 4. A. 12 B. 9 C. 16 D. 8 Solution: 𝑦 2 − 3𝑥 + 3 = 0 ⟺ 𝑥 =

𝑦 2⁄ 3+1

𝑥=4 Intersection points are 𝑦 2 − 3(4) + 3 = 0 ⟺ 𝑦 = ±√9 𝑦 = ±√9 ⟺ 𝑦 = ±3 3 3 𝑦2 𝑦2 ∫−3 4 − ( ⁄3 + 1) 𝑑𝑦 ⟺ ∫−3 3 − ⁄3 𝑑𝑦

3

3

∫−3 3 −

𝑦 2⁄ 𝑦 3⁄ 3 𝑑𝑦 ⟺ [3𝑦 − 9]−3 3

[3𝑦 −

3

𝑦 3⁄ 𝑦 3⁄ ⟺ 2 − ] [3𝑦 9 −3 9] 0 3

𝑦3 2 [3𝑦 − ⁄9] = 𝟏𝟐 0 53. A circle with a radius of 10 cm is revolved about a line tangent to it. Find the volume generated. A. 19, 739 𝑐𝑚3 B. 17, 843 𝑐𝑚3 C. 1193.24 𝑐𝑚3 D. 1295.36 𝑐𝑚3 Solution: 54. An inscribed angle is 𝜋⁄4 radian, and the chord of the circle subtended by the angle is 12√2 cm. Find the radius of the circle. A. 10 cm B. 12 cm C. 14 cm D. 16 cm

Solution: ∝= 𝜋⁄4

12√2⁄ = 6√2 𝑠𝑖𝑛 𝜋 = 6√2 2 4 𝑟

ɵ = (2) ∝= 𝜋⁄2

𝑠𝑖𝑛 ∝=

𝒓 = 𝟏𝟐𝒄𝒎

6√2 𝑟

55. In Jones family, each daughter has as many brothers as sisters and each son has three times as many sisters as brothers. How many daughters and sons are there in the Jones family? A. 3, 2 B. 4, 2 C. 5, 2 D. 6, 3 Solution: 𝐺 = 𝑛𝑜. 𝑜𝑓 𝑠𝑖𝑠𝑡𝑒𝑟𝑠 𝐺−1=𝐵 3(𝐵 − 1) = 𝐺

𝐵 = 𝑛𝑜 𝑜𝑓 𝑏𝑟𝑜𝑡ℎ𝑒𝑟𝑠 3(𝐵 − 1) − 1 = 𝐵 3(2 − 1) = 𝐺

56. find th bounded by 𝑦 = 8 − 𝑥 3 , the x-axis and the y-axis. A. 14 B. 10 C. 16

𝑩=𝟐 𝑮=𝟑

D. 12

Solution:

57. Find the area of the square with a diagonal of 15 cm. A. 225 𝑐𝑚2 B. 115.5𝑐𝑚2 C. 112.5 𝒄𝒎𝟐

D. 121.5 𝑐𝑚2

Solution: 1

𝐴 = 2 𝑑2 1

𝐴 = 2 (15)2 = 𝟏𝟏𝟐. 𝟓 𝒄𝒎𝟐 58. Find the greatest area of a rectangle inscribed in a given parabola 𝑦 = 16 − 𝑥 2 and the x-axis. A. 24.63 s.u. B. 49.27 s.u. C. 98.53 s.u. D. 46.87 s.u. Solution: 4√3 2 ) 3

𝑦 = 16 − (

A = LW W = 𝑦 = 16 − 𝑥

= 32⁄3

2 4√3 32 )( ⁄3) 3

𝐴(𝑥) = 2𝑥(16 − 𝑥 2 ) = 32𝑥 − 2𝑥 3

𝐴 = 2(

𝑑𝐴 𝑑𝑥

𝑨 = 𝟒𝟗. 𝟐𝟕 𝒔. 𝒖.

= 32 − 6𝑥 2 = 0

=± 59. Evaluate Laplace transform of 𝑡 2 . A. 2⁄𝑠 B. 1⁄𝑠 2

C. 𝟐⁄𝒔𝟑

4√3 3

D. 1⁄𝑠

Solution: 𝑡𝑛 = 𝑡2 =

𝑛! 𝑠𝑛+1 2! = 𝑠2+1

𝟐⁄𝒔𝟑

60. Two circles of different radii are concentric. If the length of the chord of the larger circle that is tangent to the smaller circle is 40 cm, find the difference in area of the two circles. A. 350π sq. cm B. 400π sq. cm C. 500π sq. cm D. 550π sq. cm Solution: ɵ = 180⁄3 = 60 ∝= 60⁄2 = 30 𝑟 = 20 𝑡𝑎𝑛(30) = 20√3⁄3 𝑅 = √202 + (20√3⁄3)2 = 40√3⁄3 𝐴𝐵𝑂 = 𝜋(40√3⁄3)2 = 1600⁄3 𝜋 𝐴𝑆𝑂 = 𝜋(20√3⁄3)2 = 400⁄3 𝜋

𝐴𝐵𝑂 − 𝐴𝑆𝑂 = 1600⁄3 𝜋 − 400⁄3 𝜋 = 𝟒𝟎𝟎𝝅 𝒔𝒒. 𝒄𝒎 61. Solve dy/dx = 4y divided by x(y-3) A. 𝑥 3 𝑦 4 = 𝐶𝑒 𝑦 B. 𝐱 𝟒 𝐲 𝟑 = 𝐂𝐞𝐲 C. 𝑥 4 𝑦 2 = 𝐶𝑒 𝑦 D. 𝑥 3 𝑦 2 = 𝐶𝑒 𝑦 Solution: 𝑑𝑦 4𝑦 [ = ] 𝑥(𝑦 − 3) 𝑑𝑥 𝑥(𝑦 − 3)

= 𝑥(𝑦 − 3)

𝑑𝑦

= [𝑥(𝑦 − 3) =

(𝑦−3) 𝑦 𝑦

= 4𝑦

𝑑𝑥 𝑑𝑦 𝑑𝑥

𝑑𝑦 =

4 𝑥

= 4𝑦]

𝑑𝑥 𝑥𝑦

𝑑𝑥

3

4

3

4

= ( − ) 𝑑𝑦 = 𝑑𝑥 𝑦 𝑦 𝑥 =(1 − ) 𝑑𝑦 = 𝑦

𝑥

𝑑𝑥

3

4

=∫ (1 − ) 𝑑𝑦 = ∫ 𝑑𝑥 𝑦 𝑥 y−3 ln(𝑦) = 4ln(𝑥) + 𝐶 y + C’ = 4 ln(𝑥) + 3ln(𝑦) 𝑦 + 𝐶 ′ = ln(𝑥 4 ) + ln(𝑦 3 ) 𝑦 + 𝐶 ′ = ln(𝑥 4 )(𝑦 3 ) 𝑒 𝑦+𝐶 = 𝑒 ln(𝑥 𝑪𝒆𝒚 = 𝒙𝟒 𝒚𝟑

4 )(𝑦 3 )

62. The towers of a 60 meter parabolic suspension bridge are 15 m high and the lowest point of the cable is 3 m above the roadway. Find the vertical distance from the roadway to the cable at 15 m from the center. A. 3 𝑚 B. 5 𝑚 C. 𝟔 𝒎 D. 8 𝑚 Solution: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 𝑦 x =0, y = 3 x = -30, y = 15 x = +30, y = 15 @ x = 0; y =15 𝑎(0)2 + 𝑏(0) + 𝑐 = 3

𝑐=3 @ x = - 30; y =15 −302 𝑎 − 30𝑏 + 3 = 15 900𝑎 − 30𝑏 + 3 = 15 → 𝑒𝑞𝑛 1 @ x = +30; y =15 302 𝑎 + 30𝑏 + 3 = 15 900𝑎 + 30𝑏 + 3 = 15 → 𝑒𝑞𝑛 2 𝐴𝑑𝑑 𝐸𝑞𝑛 1 𝑎𝑛𝑑 𝐸𝑞𝑛 2 900𝑎 − 30𝑏 + 3 = 15 + 900𝑎 + 30𝑏 + 3 = 15 1800𝑎 + 0 + 6 = 30 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑎 1800𝑎 = 30 − 6 1800𝑎 = 24 24 𝑎= 1800 𝑎 = 0.01333 Solve for x 𝑥 = 30 − 15 𝑥 = 15 Solve for y

𝑦 = 0.01333𝑥 2 + 3 𝑦 = 0.01333(15) + 3 𝑦 = 5.99 ≈ 𝟔𝒎 63. A target with a black circular center and a white ring of uniform width is to be made. If the radius of the center is to be 3 cm, how wide should the ring be so that the area of the ring is the same as the area of the center? A. 1.232 𝑐𝑚 B. 1.263 𝑐𝑚 C. 1,252 𝑐𝑚 D. 1.243 𝑐𝑚

SOLUTION: 64. Evaluate 0.9 + 0.92 + 0.93 + ⋯ + 0.9𝑛 A. 9 B. 8

C. 7

D. 6

SOLUTION: 65. Which of the following is a prime number? A. 97 B. 91 C. 133 Solution: Prime numbers 2, 3, 5, 7, 9, 11 … @ 91 =√91 = 9.53 Divide 91 by prime numbers less than the √91 91 = 13 → 𝑛𝑜𝑡 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟! 7 @ 133 =√133 = 11.53 Divide 133 by prime numbers less than the √133 133 = 19 → 𝑛𝑜𝑡 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟! 7 @ 119 =√119 = 10.91 Divide 119 by prime numbers less than the √119 119 = 17 → 𝑛𝑜𝑡 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟! 7

D. 119

𝐵𝑦 𝐸𝑙𝑖𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛 Answer is 97 66. Find the sum of the interior angle of a regular hexagon? A. 810° B. 540° C. 𝟕𝟐𝟎° D. 630° Solution: Formula: Sum of Interior angle = (𝑛 − 2)180° Regular hexagon; 6 sides, 6 angles 𝑛=6 (6 − 2)180° = 𝟕𝟐𝟎°

67. From a hill 600 ft high, the angles of depression to the bases in opposite directions are 42° and 19° 23′ respectively, Find the length of the proposed tunnel through the bases. A. 2,589.15 ft B. 2,371.74 ft C. 2590.05 ft D. 1592.20 ft 𝛼= 19°23’

𝜃 = 42°

Solution:

600 𝑓𝑡 A

B

𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡𝑢𝑛𝑛𝑒𝑙 = 𝐴 + 𝐵 6𝑜𝑜 ; 𝐴 600 = tan 42°

𝑡𝑎𝑛 𝜙 = 𝐴=

𝑡𝑎𝑛 𝛼 = 𝐵=

666.37𝑓𝑡

6𝑜𝑜 ; 𝐵

600 = 1705.38𝑓𝑡 tan 19°23′

𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡𝑢𝑛𝑛𝑒𝑙 = 𝐴 + 𝐵 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡𝑢𝑛𝑛𝑒𝑙 = 666.37𝑓𝑡 + 1705.38𝑓𝑡 =𝟐𝟑𝟕𝟏. 𝟕𝟓 𝒇𝒕.

68. Find the distance of the directrix from the center of an ellipse if its major axis is 10 and its minor axis is 8. A. 8.1 B. 8.3 C. 8.5 D. 8.7 Given: 𝑀𝑎𝑗𝑜𝑟 𝑎𝑥𝑖𝑠 = 𝑎 = 10 𝑀𝑖𝑛𝑜𝑟 𝑎𝑥𝑖𝑠 = 𝑏 = 8

𝐹𝑜𝑐𝑖 = 𝑐 𝐷𝑖𝑟𝑒𝑐𝑡𝑟𝑖𝑥 = ? 𝑑=

𝑎2 ; 𝑐

𝑐 = √𝑎2 − 𝑏 2

Solution: 𝑐 = √102 − 82 = 6 102 𝑑= = 16.67 6 16.67 = 𝟖. 𝟑 2 69. If the logarithm of MN is 6 and the logarithm of M/N is 2, find the logarithm of N A. 2 B. 3 C. 4 D. 5

Solution: 70. Two buildings with flat roofs are 60 m apart. From the roof of the shorter building 40 m in height, the angle of elevation to the edge of the roof of the taller building is 40°. How high is the taller building? A. 60 m B. 70 m C. 80 m D. 90 m tan 40 =

x

𝑥 60

40 = 𝜃

𝑥 = (tan 40)(60) 𝑥 = 50 𝐻𝑡𝑎𝑙𝑙 𝑏𝑢𝑖𝑙𝑑𝑖𝑛𝑔 = 40 + 50 = 𝟗𝟎 𝒎

40m

60m

71. Three ships are situated as follows A is 225 mi due north of C, and B is 375 mi due to east of C. What is the bearing of B from A? A. N 56° E

B. S 56° E

C. N 59° E

D. S 59° E

Solution: 𝐭𝐚𝐧 θ =

225 375

225

= tan−1 375 = 30. 96

θ = 90° − 30. 96° = 𝟓𝟗. 𝟎𝟒 ∴ Bearing of B from A is 𝐒 𝟓𝟗° 𝐄

72. The longest diagonal of a cube is 6 cm. The total area of the cube is A. 32√2 sq. m

B. 72 sq. m

C. 24√2 sq. m

D. 36 sq. m

Solution: 𝐴𝑆 = 6 𝑎2 𝑑 = √3 𝑎 𝑎=

𝑑 √3

6 √3

=

= 2√3

𝐴𝑆 = 6 (2√3)2 = 𝟕𝟐 𝒎𝟐 73. A support wire is anchored 12 m up from the base of a flagpole and the wire makes a 15° angle with the ground. How long is the wire? A. 12 m

B. 92 m

C. 46 m

D. 24 m

Solution: Tan 15° =

12 𝑎𝑑𝑗

𝑎𝑑𝑗 = 44. 78 𝑚 𝑐 = √44. 782 + 122 = 𝟒𝟔. 𝟑𝟓 𝒎 ∴ 𝑤𝑖𝑟𝑒 𝑖𝑠 𝟒𝟔 𝒎 𝑙𝑜𝑛𝑔

74. A motorboat weighs 32000 lb and its motor provides a thrust of 5000 lb. Assume that the water resistance is 𝑑𝑣 100 pounds for each foot per second of the speed v of the boat. Then 1000 𝑑𝑡 = 5000 – 100 v. If the boats starts from the rest, what is the maximum velocity that it can attain? A. 20 ft/s

B. 25 ft/s

C. 40 ft/s

Solution: 1000

𝑑𝑣 𝑑𝑡

= 5000 − 100 𝑣

1000

𝑑𝑣 𝑑𝑡

= 100(50 − 𝑣)

10

𝑑𝑣 𝑑𝑡 𝑑𝑣

= (50 − 𝑣)

∫ (50−𝑣) =

1 10

∫ 𝑑𝑡

𝑛𝑜𝑤 𝑢𝑠𝑒 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑤 = 50 − 𝑣

D. 50 ft/s

𝑑𝑣 𝑤

=

1 10

∫ 𝑑𝑡

− ln 𝑤 =

𝑡 10

+𝐶

−∫

𝑡

ln 𝑤 = − 10 − 𝐶 𝑡

ln( 50 − 𝑣) = − 10 − 𝐶 −𝑡

50 − 𝑣 = 𝐶1 𝑒 10 𝑠𝑖𝑛𝑐𝑒 𝑣0 = 0 𝑡ℎ𝑒𝑛 0

50 − 0 = 𝐶1 𝑒 10 50 − 0 = 𝐶1 = 50 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝐶 𝑖𝑛 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛, 𝑤𝑒 𝑔𝑒𝑡 1

50 − 𝑣 = 50𝑒 10 1

𝑣(𝑡) = 50 − 50𝑒 −10 1

𝑣(𝑡) = 50(1 − 𝑒 −10 ) 𝒗𝒎𝒂𝒙 = 𝟓𝟎 𝒇𝒕/𝒔 75. The base of an isosceles triangle is 20.4 and the base angles are 48°40’. Find the altitude of the triangle A. 11.6

B. 10.8

C. 12.7

D. 9.5

Solution: tan 48°40′ =

10.24 𝑎𝑑𝑗

𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 = 𝟏𝟏. 𝟓𝟗 𝒐𝒓 𝟏𝟏. 𝟔

76. Find the exact value of sec (-pi/6) A. 3/√2

B. 1/√2

Solution: 𝟏 𝝅 𝟔

𝐜𝐨𝐬−

=

𝟏 √𝟑 𝟐

=

𝟐 √𝟑

C. 3/√6

D. 2/√3

77. A snack machine accepts only quarters. Candy bars cost 25₵ packages of peanuts cost 75₵ and cans of cola cost 50₵. How many quarters are needed to buy two candy bars, one package of peanuts and one can of cola? A. 8

B. 7

C. 6

D. 5

Solution:

78. A ball is dropped from a height of 18 m. On each rebound it rises 2/3 of the height from which it last fell. What is the total distance it travels in coming to rest? A. 80 m

B. 90 m

C. 72 m

D. 86 m

Solution:

79. Find the work done in moving an object along the vector a=3i + 4j if the force applied is b= 2i + j A. 11.2

B. 10

C.12.6

D. 9

Solution: 𝑊 = 𝐹 𝑥 𝑣 = ( 3𝑖 + 4𝑗 )( 2𝑖 + 𝑗) = 𝟏𝟎

80. By stringing together 9 differently colored beads. How many different bracelets can be made? A. 362, 880

B. 20, 160

C. 40, 320

D. 181, 440

Solution: (9−1)! 2

= 𝟐𝟎, 𝟏𝟔𝟎

81. Find the derivative of the function y=3/(x2 +1). A. 6x/(x2 +1)2 B. 6x(x2 +1)2 C. -6x/(x2 +1)2 D. -6x(x2 +1)2

Solution:

𝑦= 𝑦′ = 𝑦′ =

3 𝑥 2 +1

𝑢

=𝑣

𝑣𝑑𝑢−𝑢𝑑𝑣 𝑣2

(𝑥 2 + 1)(0) − (3)(2𝑥) (𝑥 2 + 1)2 𝟐

∴ 𝒚′ = −𝟔𝒙/(𝒙𝟐 + 𝟏)

82. If 8 oranges cost Php 96, how much do 1 dozen cost at the same rate? A. Php 144 B. Php 124 C. Php 148 D. Php 168 Solution: 𝑅𝑎𝑡𝑒 =

𝑃ℎ𝑝 96 = 𝑃ℎ𝑝 12/𝑜𝑟𝑎𝑛𝑔𝑒 8 𝑜𝑟𝑎𝑛𝑔𝑒𝑠

1 𝑑𝑜𝑧𝑒𝑛 = 12 𝑝𝑖𝑒𝑐𝑒𝑠 @ 1 𝑑𝑜𝑧𝑒𝑛 ∶ 𝑐𝑜𝑠𝑡 = 𝑃ℎ𝑝

12 𝑥 𝑜𝑟𝑎𝑛𝑔𝑒

12 𝑜𝑟𝑎𝑛𝑔𝑒𝑠\

𝑐𝑜𝑠𝑡 = 𝑃ℎ𝑝 144 83. What is the slope of the linear equation 3y-x=9? A. 1/3 B. -3 C. 3 D. 9 Solution: 3𝑦 − 𝑥 = 9 3𝑦 = 𝑥 + 9 𝑦=

𝑥+9 3

𝑦′ = 𝑦′ =

𝑢

=𝑣

𝑣𝑑𝑢−𝑢𝑑𝑣 𝑣2

3(1) − (𝑥 + 9)(0) 32

𝟏

∴ 𝒚′ = 𝒎 = 𝒔𝒍𝒐𝒑𝒆 = 𝟑 84. Points A and B are 100 m apart and are of the same elevation as the foot of the building. The angles of elevation of the top of the building from points A and B are 21 degrees and 32 degrees respectively. How far is A from the building? A. 259.28 m B. 265.42 m C. 271.62 m D. 277.92 m

Solution: 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

tan(𝛳) = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 85. Give the degree measure of the angle 3pi/5. A. 150 degrees B. 106 degrees C. 160 degrees D. 108 degrees Solution 3𝜋 180 𝑥( ) = 108 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 5 𝜋 86. For what value of k will the line kx+5y=2k have slope 3? A. 5 B. -5 C. 15 D. -15 Solution: 𝑘𝑥 + 5𝑦 = 2𝑘 5𝑦 = 2𝑘 − 𝑘𝑥 2𝑘 − 𝑘𝑥 𝑢 𝑦= = 5 𝑣 𝑣𝑑𝑢 − 𝑢𝑑𝑣 𝑦′ = 𝑣2 5(−𝑘) − (2𝑘 − 𝑘𝑥)(0) 𝑦′ = 52 −𝑘 ′ 𝑦 = 5

−𝑘 5 ∴ 𝒌 = −𝟏𝟓 3=

87. The cross product of vector A=4i+2j with vector B=0. The dot product A B=30. Find B. A. 6i+3j B. 6i-3j C. 3i+6j D. 3i-6j

Solution: 𝑥𝑖 + 𝑦𝑗 =? 𝑢𝑠𝑖𝑛𝑔 𝑐𝑟𝑜𝑠𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 42 | |=0 𝑥𝑦 4𝑦 − 2𝑥 = 0 𝑒𝑞. 1 𝑢𝑠𝑖𝑛𝑔 𝑑𝑜𝑡 𝑝𝑟𝑜𝑑𝑢𝑢𝑐𝑡 4𝑥 + 2𝑦 = 30 𝑒𝑞. 2

𝑢𝑠𝑖𝑛𝑔 𝑒𝑞. 1 2𝑥 𝑒𝑞. 3 4 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑒𝑞. 3 𝑡𝑜 𝑒𝑞. 2 𝑦=

2𝑥

4𝑥 + 2 ( 4 ) = 30, 𝑥 = 6 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑥 𝑡𝑜 𝑒𝑞. 3 𝑦=

2𝑥 4

=

2(6) 4

=3 ∴ 6𝑖 + 3𝑗

88. What is the discriminant of the equation 4x2=8x-5? A. 8 B. -16 C. 16 D. -8 Solution: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 4𝑥 2 − 8𝑥 + 5 = 0 𝑑𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑛𝑡 = 𝑏 2 − 4𝑎𝑐 = (−8)2 − 4(4)(5) ∴ 𝒅𝒊𝒔𝒄𝒓𝒊𝒎𝒊𝒏𝒂𝒏𝒕 = −𝟏𝟔 89. Find the slope of the curve y=x+2(x raised to -1) at (2,3) A. 2 B. ½ C. 1 D. ¼ Solution: 𝑦 = 𝑥 + 2𝑥 −1 𝑚 = 𝑦 ′ = 1 − 2𝑥 −2 𝟏 𝟐 90. A wheel 4 ft. in diameter is rotating at 80 r/min. Find the distance (in ft.) travelled by a point on the rim in 1s. A. 18.6 B. 16.8 C. 17.8 D. 18.7 ∴ 𝑚 = 𝑦 ′ = 1 − 2(2)−2 =

Solution: ῳ = 80𝑟𝑝𝑚 =

𝑟=

80𝑟𝑒𝑣 2𝑝𝑖 𝑟𝑎𝑑 1 𝑚𝑖𝑛 𝑥 1 rev 𝑥 60s min

=

8𝑝𝑖 𝑟𝑎𝑑 3 𝑠

𝑑 4 = = 2 𝑓𝑡. 2 2

∴ 𝑠 = ῳ𝑡𝑟 =

8𝑝𝑖 (1)(2) = 𝟏𝟔. 𝟖 𝒇𝒕. 3

91. A toll road averages 300,000 cars a day when the toll is $2.00 per car. A study has shown that for each 10-cent increase in the toll, 10,000 fewer cars will use the road each day. What toll will maximize the revenue? A. $2.25 B. $2.75 C. $3.00 D. $2.50 Solution: Let: n = cars

P = price

R = revenue

n= no. of increment

n = 300,000 – 10,000x P = 2.00 + 0.10x R = nP R = (300,000 – 10,000x)(2.00 + 0.10x) R = 600,000 + 30,000x – 20,000x – 1000x2

𝑑𝑅 𝑑𝑥 𝑑𝑅 𝑑𝑥

Substitute x, n = 300,000 – 10,000(5) n = 250,000

= -1000x2 + 10,000x + 600,000

P = 2.00 + 010(5)

= -2,000x + 10,000

P = $2.50

-2,000x + 10,000 = 0 x=5

92. Find the equation of the line determined by points A(5, -2/3) and (1/2, -2) A. 8x + y = 58 B. 8x + 27y = 58 C. 8x – 27y = 58 D. x – 2y = 58 Solution: m= m= m=

𝑌2−𝑌1 𝑋2−𝑋1 −2+2/3 1 −5 2

8 27

(y – y1) = m (x – x1) (y + 2/3) = [(y+2/3) =

8 27 8 27

(x – 5) x-

40 27

] 27

8x – 27y = 58

93. Find the eccentricity of a hyperbola whose transverse and conjugate axes are equal in length. A. √𝟐 Solution:

B. √3

C. 2 √2

D. 2 √3

(x2/a2) – (y2/b2) = 1

e=

√𝑎2 +𝑏2 𝑎 a=b

e=

√2𝑎2

e=a

𝑎 √2

𝑎 e = √𝟐

94. For what values of x is |x-3| = 1? A. 4 B. 2

C. 2, 4

D. -2, -4

Solution: By inspection and substituting all the given in the equation: |x-3| = 1 |x-3| = 1 |2-3| = 1 |4-3| = 1 |1|= 1 |1|= 1 95. Susan’s age in 20 years will be the same as Thelma’s age now. Ten years from now, Thelma’s age will be twice Susan’s. What is the present age of Susan? A. 45 B. 40 C. 50 D. 30 Solution:

Thelma Susan

PRESENT X x + 20

FUTURE 2(x + 10) (x + 20) + 10

2(x + 10) = (x + 20) + 10 2x + 20 = x + 30 x = 10 Substitute: 10 + 20 = 30 years old 96. The circumference of a great circle of a sphere is 18𝜋 m. Find the volume of the sphere. A. 3053.6 cu. m B. 3043.6 cu. m C. 3033.6 cu. m D. 3023.6 cu. m

Solution: 4

Vsphere= 3 𝜋𝑟 3

C = 2𝜋r

4

18𝜋 = 2𝜋r

= 3 𝜋 (93)

r = 9m

= 3053.6 m3

97. What is the Laplace transform of f(t) = cosh at? A. a/(s squared + a squared) C. s/(s squared + a squared) B. a/(s squared – a squared) D. s/(s squared – a squared) 98. Tom inherited two different stocks whose yearly income was Php 2,100. The total appraised value of the stocks was Php 40,000, one was paying 4% and one 6% per year. What was the value of the stock paying 6%? A. 27,000 B. 23,000 C. 25,000 D. 24,000 Solution: Let x = stock of value (40,000 – x) = Appraisal value 0.06x + 0.04(40,000 – x) = 2,100 x = 25,000 99. Joe and his dad are bricklayers. Joe can lay bricks for a wall in 5 days. With his father’s help, he can build it in 2 days. How long would it take his father to build it alone? A. 3-1/4 hrs B. 3-1/3 hrs C. 2-1/3 hrs D. 2-2/3 hrs Solution: [

1 5

+

1 𝑥

=

1 2

]10x

2x + 10 = 5x x=

10 3

days or 3

𝟏 𝟑

hrs

100. Find the nth term of the arithmetic sequence 11, 2, -7. A. -6n + 12 B. -9n + 20 C. –n + 24 Solution: d= a2 – a1= (2) – (11) = -9

D. -2n + 8

a3= a3 + (n-3)d = (-7) + (n-3)-9 = -7 - 9n + 27 = -9n + 20

JULY 2018

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION JULY 2018 PROFESSIONAL MATHEMATICS SUBJECTS

1. Joseph gave ¼ of his candies to Joy and Joy gave 1/5 of what she got to Tim. If Tim received 2 candies, how many candies did Joseph have originally? A. 30 B. 20 C. 50 D. 40 2. What conic section is described by the equation 4x2-y2+8x+4y=15? A. parabola B. hyperbola C. circle D. ellipse 3. Find the maximum area of a rectangle which can be inscribed in an ellipse having the equation x2 + 4y2 = 4 A. 4 B. 3 C. 2 D. 5

4. If the general equation of the conic Ax2 + Bxy +Cy2 + Dx + Ey +F = 0. If B2 –AC>0 the equation describes is _____________. A. ellipse B. hyperbola C. parabola D. circle 5. Determine the equation that expressed that G is proportional to x and inversely proportional to C and z. Symbols a, b, and c are constants. 𝑐𝑘

A. G= 𝐺𝐺

𝑎

B. G = 𝑏𝑐

𝒄𝒌

C. G = 𝒛𝑪

𝑏𝑐

D. G = 𝑧𝐾

6. The chord passing through the focus of the parabola and is perpendicular to its axis is termed as A. axis B. latus rectum C. directrix D. translated axis

7. What’s the equation of the hyperbola with focus at (-3 -3√13 , 1) asymptotes intersecting at (-3, 1) and one asymptotes passing thru the point (1, 7)? A. 4x2- 9y2 + 54x + 8y - 247 = 0 C. 9x2- 4y2 + 54x + 8y - 247 = 0 B. 4x2+ 9y2 + 54x - 8y + 284 = 0 D. 9x2 + 9y2 + 54x - 8y + 284 = 0 8. Find the ratio of the sides of triangle if its sides form an arithmetic progression and one of the angles is 90 degrees. A. 4 : 5 : 6 B. 1 : 2 : 3 C. 3 : 4 : 5 D. 2 : 3 : 4 Sol’n: Let a = first term d= common difference (a-d) , a , (a+d) By Pythagorean Theorem, (a-d)2 + a2 = (a+d)2 a2- 2ad + d2 + a2 = a2 + 2ad + d2 a2-4ad = 0 a(a-4d) = 0 a= 0 a-4d = 0 a= 4d (4d-d) , 4d , (4d+d) 3d, 4d, 5d 3 :4:5 9. The area enclosed by the ellipse 4x2 + 9y2 = 36 is revolved about the line x = 3, what is the volume generated? A. 370.3 B. 360.1 C. 355.3 D. 365.1 Sol’n: (4x2 + 9y2 = 36) 1/36 x2/ 9 + y2/ 4 = 1 a= √9 = 3 b= √4 = 2

v = ac = (𝜋𝑎𝑏)(2𝜋𝑎) = 2𝜋2a2b =2𝜋2(3)2(2)

v = 355.3 10. The polynomial x2 + 4x + 4 is the area of a square floor. What is the length of its side? A. x + 2 B. x – 2 C. x + 1 D. x – 1 Sol’n: A = x2 + 4x + 4 A = (x+2) (x+2) = (x+2)2 Asquare = s2 s = x+2 11. Given a conic section, if B2 – AC = 0, it is called? A. circle B. parabola C. hyperbola

D. ellipse

12. Find the height of a right circular cylinder of maximum volume which can be inscribed in a sphere of radius 10cm. A. 11.55 cm B. 14.55 cm C. 12.55 cm D. 18.55 cm Sol’n: ℎ

R2= r2 + (2)2 ℎ

r2 = R2- (2)2 ℎ

r2 = 102- (4)2 v= 𝜋 r2 h ℎ

v= 𝜋(102- (4)2)(h) 𝜋ℎ3

=100 𝜋h -

4

dv = 100 𝜋 0 = 100 𝜋 100 𝜋 = 4

4 3𝜋ℎ2

3𝜋ℎ2 4

100 (3) = h2

3𝜋ℎ2

4

400/ 3 = h2 400 h = √ 3 = 11.55 cm 13. The length of the latus rectum of the parabola y = 4px2 is: A. 4p B. 2p C. p

D. -4p

Sol’n: y = 4px2 LR = 4a = 4p 14. The area bounded by the curve y2 = 12x and the line x = 3 is revolved about the line x = 3. What is the volume generated? A. 186 B. 179 C. 181 D. 184 Sol’n: r= xr – xl ∫ 𝑑𝑣 = ∫ 𝜋𝑟3 dh 6

𝑣 = 𝜋 ∫−6 v=

288𝜋 5

(3 −

𝑦ˆ2 12

) 𝑑𝑦

𝑜𝑟 𝟏𝟖𝟏

15. What is the length of the shortest line segment in the first quadrant drawn tangent to the ellipse b2x2 + a2y2 = a2b2 and meeting to the coordinates axes? A. a/b B. a + b C. ab D. b/a 16. Find the radius of the circle inscribed in the triangle determined by the lines y=x+4, y= x-4 and y = 7x-2. 5

𝟓

A. √2

B. 𝟐√𝟐

Sol’n: Radius of Circle y=x+4 ; y= -x-4 ; y = 7x-2 

Solve for 1st pt., y= x+4 ; x= -4-y y= (-4-y) + 4 y=0

3

C. √2

3

D. 2√2

x= -4-0 x= -4 (-4, 0) 

Solve for 2nd pt., y=x+4 y+2 y=7x-2 ; x = 7 y= x=



y+2

+ 4 ; y=5

7 5+2

+4 =1

7

(1, 5) Solve for 3rd pt., y= -x-4 y+2 y= 7x-2 ; x= 7 y+2

y= − ( x=

7

) − 4; y= -15 / 4

−15 +2 4

7

; x= -1/4

(-1/4 , -15 / 4 )

𝑥1 A= 2 ( 𝑦1 1

−4 1 𝑥2 𝑥3 𝑥1 1 )=2( 𝑦2 𝑦3 𝑦1 0 5

1

= 2 |(−20 −

15 4

5

− 0) − (0 − 4 + 15)|

A= 75/4

 

Find the perimeter: Side between (-4, 0) and (1,5) d= √(−1 + 4)2 + (5 − 0)2 = 5√2 Side between (-4, 0) and (-1/4 , -15 / 4 ) 1

d= √(− 4 + 4)2 + (− 

15 4

− 0)2 =

15√2 4

Side between (1, 5) and (-1/4 , -15 / 4 ) 1

d= √(− 4 − 1)2 + (− P=5√2 +

15√2 4

+

15 4

25√2 4

− 5)2 =

= 15√2

25√2 4

−1 4 −15 4

−4 ) 0

One-half of the perimeters =

15√2 2

Radius of inscribed circle in a triangle =

75/4 15√2 2

𝟓

= 𝟐√𝟐 17. Find the moment of inertia of the area bounded by the parabola y2=4x and the line x=1, with respect to the x-axis. A. 2.133 B. 1.333 C. 3.333 D. 4.133 Sol’n: y2=4x, x=1 y = yR – yL y = 1- y2 / 4 𝑏

Ix = ∫𝑎 𝑟2dA 2

(1− y2 / 4)dy

=∫−2 𝑦2 𝑑𝐴 Ix = 32 / 15 or 2.133 18. What is the unit vector which is orthogonal both to 9i + 9j and 9i+9k? 𝑖

𝑗

𝑘

A. √3 + √3 + √3

𝑖

𝑗

𝑘

B. 3 + 3 + 3

Sol’n: a=9i + 9j ; (i, j, k) ; (9, 9, 0) b=9i+9k ; (i, j, k) ; (9, 0, 9) By determinants, 𝑖 𝑗 9 9 9 0

𝑘 9 0 9 0 9 9 )–j( )+𝑘( ) 0= i( 0 9 9 9 9 0 9 = i( 81 -0 ) – j ( 81- 0) + k (0-81) = i( 81) – j ( 81) + k (-81)

𝒊

𝒋

𝒌

C. √𝟑 - √𝟑 − √𝟑

𝑖

𝑗

𝑘

D. 3 - 3 - 3

Solving for modulus, = √81ˆ2 + (−81)ˆ2 + 81ˆ2 =81 √3 The unit vector is, 1

=81 √3 (81i-81j-81k) 81𝑖

=81 √3 −

19.

81𝑗 81 √3

81𝑘

𝒊

𝒋

𝒌

− 81 √3 = √𝟑 - √𝟑 − √𝟑

Express in polar form: -3 -4i 4

C. √5eˆ-𝑖(𝜋 + tan−1 3)

4

4

D. √5eˆ𝒊(𝝅 + 𝐭𝐚𝐧−𝟏 𝟑)

A. 5eˆ-𝑖(𝜋 + tan−1 3)

𝟒

B. 5eˆ𝑖(𝜋 + tan−1 3)

20. The axis of the hyperbola through its foci is known as: A. conjugate axis B. transverse axis C. major axis

D. minor axis

21. Describe the locus represented by l z+2i l + l z-2i l = 6. A. circle B. parabola C. ellipse

D. hyperbola

22. If the radius of the sphere is increased by a factor of 3, by what factor does the volume of the sphere change? A. 9 B. 18 C. 27 D. 54 Sol’n: V = 4/3 𝜋𝑟3 = k 𝑟3 r2 = 3r1 v2 / v1 = r23 / r13 = 33 r13 / 𝑟3 = 27 23. Evaluate the ∫(7x 3 − 4x 2 )dx. A.

7x4 4

+

4𝑥 2 3

+𝐶

B.

7x4 4

−

4𝑥 2 3

+𝐶

C.

7x4 4

+

4𝑥 3 3

+𝐶

24. Describe the locus represented by l z-3 l – l z+3 l = 4. A. ellipse B. circle C. hyperbola

D.

𝟕𝐱 𝟒 𝟒

−

𝟒𝒙𝟑 𝟑

D. parabola

+𝑪

25. Melissa is 4 times as old as Jun. Pat is 5 years older than Melissa. If Jim is y, how old is Pat? A. 4y + 5 B. y + 5 C. 5y + 4 D. 4 + 5y Sol’n: Melissa – 4y Jim – y Pat – 4y + 5 Therefore, Pat = 4y + 5 26. A conic section whose eccentricity, is less than one is known as: A. a parabola B. an ellipse C. a circle D. a hyperbola 27. Two lines passing through the point (2,3) make an angle of 45 degrees with each other. If the pipe of one of the lines is 2, find the slope of the other. A. -2 B. -1 C. -3 D. 0 Sol’n: (2,3) 𝜃 = 45 m1= 2 Tan 𝜃 = m2 –m1 / 1+ m2 m1 Tan 45 = m2 –2 / 1+ m2 (2) m2 = -3 28. From the top of a building the angle of depression of the foot of a pole is 48 deg 10 min. From the foot of a building the angle of elevation of the top of a pole is 18 deg 50min. Both building and pole are on a level ground. If the height of a pole is 4m, how high is the building? A. 13.10m B. 12.10m C. 10.90m D. 11.60m Sol’n: Tan 𝜃 = y / x x= y / Tan 𝜃 = 4 / tan 18°50’ x= 12.13

Tan 𝜃 = x / h h = x / tan 𝜃 = 12.13 / tan 48° 10’ h = 10.90m

29. The locus of a point which moves so that the sum of its distances between two fixed points is constant is called A. ellipse B. parabola C. circle D. hyperbola 30. Totoy is 5 feet 11 inches tall and Nancy is 6 feet 5 inches tall. How much taller is Nancy than Totoy? A. 1 foot 7 inches B. 1 foot C. 7 inches D. 6 inches Sol’n: h2 = 5’ 11’’ = 5.917 h2 = 6’ 5’’ = 6.417 = h2- h2 = 6.417 - 5.917 12𝑖𝑛

= 0.5ft ( 1𝑓𝑡 ) = 6 inches 31. If log64 x = 3/2, find x. A. 512 B. 521

C. 253

D. 258

Sol’n: 3

log64 x = 2 𝑙𝑜𝑔𝑥

𝑙𝑜𝑔64

3

3

logx = 2 log64 ; x = 64ˆ3/2= 512

=2

32. What is the product of -9p3r and 2p-3r? A. 18p4r + 27p6r2 B. -18p4r + 27p3r2 C. 18p2r + 27p2r3

D. -18p2r + 27p2r3

Sol’n: = (-9p3r) (2p-3r) = 18p4r + 27p3r2 𝑥2

33. Evaluate ∫ √𝑥 2 +25dx , using trigonometric substitution x = 5 tan 𝜃. 𝟏

𝑨. 𝟑 (𝒙𝟐 + 𝟐𝟓)3/2 – 25(𝒙𝟐 + 𝟐𝟓)1/2 + C 1

𝐵. 3 (𝑥 2 + 25)3/2 + 25(𝑥 2 + 25)1/2 + C 𝐶.

25 3

(𝑥 2 + 25)3/2 – 25(𝑥 2 + 25)1/2 + C

𝐷.

25 3

(𝑥 2 + 25)3/2 + 25(𝑥 2 + 25)1/2 + C

Sol’n: = 125∫

(sin ˆ3 𝜃 / cosˆ3 𝜃) ( 1 / cosˆ2 𝜃 ) (1 /cos𝜃 )

d𝜃

(sin ˆ3 𝜃)

= 125∫ (cosˆ4 𝜃 ) d 𝜃 = 125∫

(sin ˆ3 𝜃) (cosˆ4 𝜃 )

sin 𝜃 d 𝜃

u = cos 𝜃 , du= - sin 𝜃 d 𝜃 = 125∫ − = 125∫

(1−cos ˆ2 ) (− sin 𝜃 )d 𝜃 (cosˆ4 𝜃 )

−1+𝑢ˆ2 𝑢ˆ4 1

d𝑢 1

= 125 (3𝑢ˆ3 - 𝑢 + 𝐶) 1

1

= 125 (3𝑐𝑜𝑠ˆ3θ - 𝑐𝑜𝑠𝜃 + 𝐶) 1

= 125 (3 sec ˆ3 𝜃 − 𝑠𝑒𝑐𝜃 + 𝐶) 1

= 125 (3 (√𝑡𝑎𝑛𝜃 + 1) ˆ3 − √𝑡𝑎𝑛𝜃 + 1 + 𝐶) 1

= 3 5ˆ3 (√𝑡𝑎𝑛𝜃 + 1)ˆ3 − 25 (5) √𝑡𝑎𝑛𝜃 + 1 + 𝐶 ) =

𝟏 𝟑

(𝒙𝟐 + 𝟐𝟓)3/2 – 25(𝒙𝟐 + 𝟐𝟓)1/2 + C

34. Michael’s favorite cake recipe calls for 0.75 pounds of flour, he has a 5 pound bag. He wants to make several cakes for the school bake sale. How many cakes can he make? A. 5

B. 6

C. 7

Solution:

Five pounds of flour divided by .75 equals

= 6.6666

D. 8

Michael can make 6 cakes. 35. Find the minimum amount of tin sheet that can be made into a closed cylinder having a volume of 108 cu. Inches in square inches. A. 125

B. 137

C. 150

D. 120

Solution: V = 108 cu. in, V = 𝜋𝑟 2 h h=r 𝑣 = 𝜋𝑟 3 100 = 𝜋𝑟 3 3

100

r= √

𝜋

= 3.17 𝑖𝑛.

AT = 2𝜋𝑟ℎ + 2𝜋𝑟 2 = 2𝜋𝑟 2 + 2𝜋𝑟 2 = 2𝜋(3.17)2 + 2𝜋(3.17)2 = 126. 28 𝑖𝑛.2 = 125 𝒊𝒏.𝟐 36. A chord of a circle 10 ft. in diameter is increasing at the rate of 1 ft/s. Find the rate of change on the smaller arc subtended by the chord when the cord is 8 ft. long. A. 5/2 ft/min.

B. 2/5 ft/min.

C. 5/3 ft/min.

D. 3/5 ft/min.

37. Find the centroid of a semicircular area of radius a. A. 2a/π

B. 4a/π

C. 2a/3π

D. 4a/3π

38. An equilateral triangle with side “a” is revolved about its altitude. Find the volume of the solid generated. A. 0.32a3

B. 0.23a3

C. 0.41a3

D. 0.14a3

39. If the area bounded by the parabolas y=x2-C2 and y=C2-x2 is 576 square units, find the value of C. A. 5

B. 6

C. 7

D. 8

40. Solve y”-5y’+4y = sin 3x. 1

A. y= 25 (3 cos 3𝑥 − sin 3𝑥) + 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 4𝑥 1

B. y= 25 (3 sin 3𝑥 − cos 3𝑥) + 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 4𝑥 𝟏

C. y= 𝟓𝟎 (𝟑 𝐜𝐨𝐬 𝟑𝒙 − 𝐬𝐢𝐧 𝟑𝒙) + 𝑪𝟏 𝒆𝒙 + 𝑪𝟐 𝒆𝟒𝒙 1

D. y= 50 (3 sin 3𝑥 − cos 3𝑥) + 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 4𝑥 41. A car is travelling at a rate of 36 m/s towards a statue of height 6m. What is the rate of change of a distance of the car towards the top of the statue when it is 8m from the statue? A. 32.4 m/s

B. 39.6 m/s

Solution: S2=s12 + 62 S2= (36t) 2 + 36 S2 = 1296t2 + 36 Differentiate 𝑑𝑠 2s𝑑𝑡 = 2592𝑡 𝑑𝑠

2592𝑡

= 2𝑠 @ S1 = 8m 8 = 36 t t = 0.222 sec. @ t = 0.222 sec. S= √1296t 2 + 36 = √1296 (0.222)2 + 36 𝑑𝑡

S= 9.99 m Ds/ dt = 2592t / 2s @ S= 9.99 m

@ t = 0.222 sec

dS/ dt = 2592 (0.222) / 2(9.99)

C. 26.6 m/s

D. 28.8 m/s

dS / dt = 28.8 m/s 42. A fencing is limited to 20 ft. length. What is the maximum rectangular area that can be fenced in using two perpendicular corner sides of an existing wall? A. 120

B. 100

Solution: x+y=20 y = 20-x A = xy Subs. Y A = x (20-x) A = 20x – x2 Differentiate: 𝑑𝐴 = 20 − 2𝑥 𝑑𝑥 0 = 20-2x X = 10 ft.

C. 140

D. 190

y = 20-x y = 20 - 10 y = 10 ft.

A = (10) (10) A = 100 ft.2

43. Evaluate Laplace transform of t cos kt. A. s2/(s2+k2)2

B. k2/(s2+k2)2

C. (-s2+k2)/(s2+k2)2 D. (s2-k2)/(s2+k2)2

Solution: (𝐬𝟐−𝐤𝟐)

ℒ(t cos kt) = (𝐬𝟐+𝐤𝟐)𝟐 44. Carmela and Marian got summer jobs at the ice cream shop and were supposed to work 15 hours per week each for 8 weeks. During that time Marian was ill for one week and Carmela took her shifts. How many hours did Carmela work during the 8 weeks? A. 120

B. 135

Solution: Total hours in 8 weeks 15 ℎ𝑜𝑢𝑟𝑠 𝑤𝑒𝑒𝑘

𝑥 8 𝑤𝑒𝑒𝑘𝑠 = 120 ℎ𝑜𝑢𝑟𝑠

C. 150

D. 185

Total hours Carmela works when Marian was ill for 1 week 120 hours + 15 hours = 135 hours 45. Manuelita had 75 stuffed animals. Her grandmother gave 15 of them to her. What percentage of the stuffed animals did her grandmother give her? A. 20%

B. 15%

C. 25%

D. 10%

Solution: 75 15

=5

100% / 5 = 20 % 46. Find the coordinates of an object that has been displaced from the point (-4,9) by the vector 4i-5i. A. (0,4)

B. (0,-4)

C. (4,0)

D. (-4,0)

Solution: P( -4, 9) Vector (4i -5i) = P ( 4, -5) X = -4 + 4 = 0 X = 9 + (-5) = 4 P (0, 4) 47. A triangle has two congruent sides and the measured of one angle 40 degrees. Which of the following types of triangle is it? A. Isosceles

B. equilateral

C. right

D. scalene

B. 48. The parabola defined by the equation 3y2+4x=0 opens ___________. A. Upward

B. downward

C. to the left

D. to the right

49. If a place on the earth is 12 degrees south of the equator, find its distance in nautical miles from the North Pole.

A. 6,021

B. 6,102

C. 6,210

D. 6,120

Solution: R = 3959 Statute Miles 𝜋

Θ = 102° (180°) = S = r𝜃 = (3959)(

17𝜋 30 17𝜋 30

S = 17047.95 SM (

)

5280 𝑓𝑡. 15 𝑀

)(

1 𝑁𝑀

6080 𝑓𝑡.

)

S = 6120 NM

50. If the standard deviation of a population is 9, the population variance is. A. 9

B. 3

C. 21

D. 81

C. tan2 𝜃

D. cos2 𝜃

Solution: σ=9 σ = √𝑣 v = σ2 = 9 2 v = 81 51. Simply the equation 𝑠𝑖𝑛2 𝜃 (1 + 𝑐𝑜𝑠 2 𝜃). A. 1

B. sin2 𝜃

52. What is the complement of a 60 degree angle? A. 120 degrees

B. 30 degrees

Solution : Complementary 𝜃 = 90 ° 90° = 𝜃1 + 𝜃2 𝜃2 = 90°- 𝜃1 = 90° - 60° 𝜽2 = 30

C. 40 degrees

D. 20 degrees

53. If 2xy-y2=3, find y” A. 2/(x-y)4

B. -2/(x-y)4

C. 3/(x-y)3

D. 1/(2-x)

54. The Rotary Club and the Jaycee Club had a joint party, 120 members of the Rotary Club and 100 members of the Jaycees Club also attended but 30 of those attended are members of both clubs. How many persons attended the party? A. 220

B. 190

C. 150

D. 250

55. Two numbers have a harmonic mean of 9 and a geometric mean of 6. Determine the arithmetic mean. A. ¼

B. 4

C. 1/9

D. 9

Solution: HM = 9 GM = 6 GM2 = (HM)(AM) AM =

𝐺𝑀2 𝐻𝑀

=

62 9

=4

56. Find the force on one force of a right triangle of sides 4m and altitude of 3m. The altitude is submerged vertically with the 4m side in the surface. A. 58.86 kN

B. 62.64 kN

Solution: W(0) = 4 m W(3) = 0 0−4 3−0

=

−4 3 4

W (h) = 4 - 3 ℎ F = ∫ 𝛾𝐻2𝑂 ℎ 𝑤(ℎ) 𝑑ℎ 3

F = ∫0 (9810)(ℎ) (4 −

4 3

ℎ ) 𝑑ℎ

C. 53.22 kN

D. 66.67 kN

3

F = (9810) ∫0 (4ℎ −

4 3

𝑥 2 ) 𝑑ℎ

= 58.86 kN

57. An airplane flying with the wind, took 2 hours to travel 1000 km and 2.5hours in flying back. What was the wind velocity in kph? A. 40

B. 50

C. 60

D. 70

Solution: V1 – V2

t2 = 2.5 hours

V1 = Airplane Ve;ocity V2 = Wind velocity D = Vt

; V = D/t

@ flying with wind V1 + V2 = 1000/2 = 500 @ flying bak V1 – V2 = 1000/ 25

= 400

(V1 + V2) - (V1 - V2) = 500-400 V1 + V 2 - V1 + V 2

= 100

V2 = 50kph 58. In how many ways can 6 people be lined up to get on a bus, if certain 3 persons insist on following each other? A. 72

B. 144

C. 480

D. 120

C. 2

D. 1

Solution: (4 !) (3 !) = 144 59. If 3x3y=27 and 2x+y=5, find x. A. 3

B. 4

60. Find the work done in moving an object along a vector a= 3i + 4i if the force applied is b = 2i + i. A. 8

B.9

C. 10

D. 12

Solution: d = a = 3i + 4i F = b = 2i + i W=Fxd Using dot product W = (a1)(b1) + (a2) (b2) = (3)(2) + (4)(1) W = 10 61. If the line 3x-ky-8 = 0 passes through the point (-2,4), then k is equal to A.-7/2

B. -5/2

C. -3/2

D. -1/2

Solution: 3x –ky – 8 = 0 @ (-2,4)

k=?

3 (-2) – k (4) – 8 = 0 𝟕

k = −𝟐 62. What is the allowable error in measuring the edge of the cube that is intended to hold 8 cu. M. of the error of the computed volume is not to exceed 0.03 cu. m? A. 0.002 Solution:

B. 0.003

C. 0.0025

D. 0.001

3

3

Edge = √𝑣 = √8

=2

dv = 3E2dE 𝑑𝑣

0.03 (3)(2)2

dE = 3𝐸2 = dE = 0.0025

63. A man can do a job in 8 days. After the man has worked for 3 days, his son joins him together they complete the job in 3 more days. How long will it take the son to do job alone? A. 12 days

B. 10 days

C. 13 days

D. 11 days

Solution: Let x = For son Man = 1/8 1

Son = 1/x 1

1

3 (8) + 3 (𝑥 + 8) = 1 3

8x (8 +

3 𝑥

+

3 8

= 1) 8𝑥

3x + 24 + 3x = 8x 6X + 24 + 8X X = 12 days 64. The probability that a randomly chosen safes prospects will make a purchase is 0.18. If a salesman calls on 5 prospects, what is the probability that the salesmen will make exactly 3 sales? A. 0.0392

B. 0.0239

Solution: ( 5 C3 ) ( 0.18 )3 (1 – 0.18 )2

C. 0.0329

D. 0.0293

X = 0.0392 5

65. If 𝑠𝑒𝑐 2 𝐴 = 2 , 𝑡ℎ𝑒𝑛 1 − 𝑠𝑖𝑛2 𝐴 = A. 0.20

B. 0.30

C. 0.40

D. 0.50

Solution: 5

𝑠𝑒𝑐 2 𝐴 = 2

------- 1

1 − 𝑠𝑖𝑛2 𝐴 = 𝑐𝑜𝑠 2 𝐴 𝑠𝑒𝑐 2 𝐴 + 𝑐𝑜𝑠 2 𝐴 cos 𝐴 =

1 𝑠𝑒𝑐𝐴

--------- 2

=1

= 𝑐𝑜𝑠 2 𝐴 =

1 − 𝑠𝑖𝑛2 𝐴 =

1 𝑠𝑒𝑐 2 𝐴

=

1 5 2

1 𝑠𝑒𝑐 2 𝐴

= 𝟎. 𝟒𝟎

66. What is the angle between the diagonal of a cube and one of its edges? A. 44.74°

B. 54.74°

C. 64.74°

D. 74.74°

Solution: A (1, 1, 1) B (0, 0, 1) Cos𝜃 =

(𝑎)(𝑏) 𝑙𝑎𝑙𝑙𝑏𝑙

= cos −1(

(1,1,1)(0,0,1) √3

)

𝜽 = 𝟓𝟒. 𝟕 °

67. The line 3x-4y=5 is perpendicular to the line A. 3x-4y=1 B. 4x-3y=1 Solution : 3x-4y=5 4y= 3x-5

C. 4x+3y=3

D. 3x+4y=0

3𝑥−5

𝑦=

4 3

𝑦 = 4 (𝑥 −

20 3

)

@ perpendicular 1

𝑚2 = − 𝑚1 =

−1 3 4

4

= −3

𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1) 4

𝑦 − 𝑘 = − (𝑥 − ℎ) 3

3𝑦 − 3ℎ = −4𝑥 + 4ℎ 4𝑥 + 3𝑦 = (3𝑘 + 4ℎ) 𝟒𝒙 + 𝟑𝒚 = 𝟑 68. If the plane 3x+2y-3x=0 is perpendicular to the plane 9x-3ky+y-t=0 A. 2 B. -2 C. 3 D. -3 Solution : 3x + 2y -3z = 0 9x – 3ky + 5zy =0 For parallel 𝐴 𝐵 3 2

𝐹

=𝐺 =

9 −3𝑘

K= -2 69. A solid has a circular base of radius r. Find the volume of the solid if every plane section perpendicular to z fixed diameter is in semicircle. A.1.20r3

B. 2.09r3

C. 2.51r3

D. 4.10r3

70. Find the y-intercept of the line given by the equation 7x+4y=8 A. 2 Solution :

B. -2

C. 3

D. -3

7x+4y=8 𝑦 = 𝑚𝑥 + 𝑏 4𝑦 4

=

8−7𝑥 4

𝑦= −

7𝑥 4

+2

b=2 71. Find the area inside the cardioid r=1+cos ϴ and outside the circle r=1. A. 2.97

B. 2.79

C. 2.85

D. 2.58

72. A person had a rectangular-shaped garden with sides of lengths 16 feet and 9 feet. The garden was changed into square design with the same area as the original rectangularshaped garden. How many feet in length are each sides of the new square-shape garden. A. 7

B. 9

C. 12

D. 16

C. 32 inches

D. 85 cm

Solution: Δ =(16)(9) = 144 sq.ft = √144 = 12 73. which of the following rope length is longest? A. 1 meter

B. 1 yard

74. Martin , a motel housekeeper, has finished cleaning about 40% of the 32 rooms he's been assigned. About how many more rooms does he have left to clean? A. 29

B. 25

Solution: Room left to clean = 60% (30) = 19 Room

C. 21

D. 19

75. A horse tied to a post with twenty-foot rope. What is the longest path that the horse can walk? A. 20 feet

B. 40 feet

C. 62.83 feet

D.125.66feet

76. Doming wants to know the height of a telephone pole. He measures his shadow, which is 3 feet long , and the pole's shadow, whcih 10 feet long . Domingo's height is 6 feet. How tall is the pole ? A. 40 ft

B. 30 ft

C. 20 ft

D. 10 ft

77. A weight of 60 pounds rests on the end of an 8-foot lever and is 3 feet from the fulcrum. What weight must be placed on the other and of the lever to balance the 60 pound weight. A. 36pounds

B. 32pounds

C. 40pounds

D. 46pounds

Solution : 5x =60(3) =180 X= 36 78. A number is 1 more than twice another. Their squares differ by 176. What is the larger number? A. 9

B. 7

C. 15

D. 16

79. The sides of a right triangle is in arithmetic progression whose common difference is 6cm. Its area is A. 216sq.cm

B. 270sq.cm

C. 360sq.cm

D. 144sq.cm

Solution : A.P. (x)

(x+12) ^2 = x^2 + (x+6)^2

(x+6)

X^2+24x+144 = x^2 + x^2 +12x +36

(x+12) ---- hypo

X^2 – 12x – 108 = 0

C^2 = A^2 + B^2

X^2 – 18x + 6x – 108 = 0 X(x-18) + 6(x-18)=0 (x-18)(x+6)=0

X=-6 or 18 Area = ½ (18)(24) = 216

80. A tank has 100 liters of brine with 40 N dissolved salt. Pure water enters the tank at the rate of 2 liters per minute abd the resulting mixture leaves the tank at the same rate. When will the concentration in the tank be 0.20 N/L A. 24.6min

B. 34.7min

C. 44.8min

D. 54.9min