Math Preboard

PREBOARD EXAMINATION IN MATHEMATICS 1. Pikachu, Mudkip and Cynadaquill have different chances of hitting the target. Pikachu can hit the target ¾ of the time, Mudkip can hit the target 2/3 of the time and Cyndaquill can hit the target ½ of the time. They all go for the target simultaneously. What is the probability that at least one of them will hit the target? A. 23/24 B. 1/24 C. 1/4 D.3/4 2. Maria loves chocolate milk so she visits a plant every day. However, each day the plant has a 2/3 chance of bottling chocolate milk. What is the probability that the bottling plant bottles chocolate milk exactly 4 of the 5 days Maria visits? A. 0.329 B. 0.923 C. 0.239 D. 0.392 3. What is the probability of getting 9 exactly once in three throws of a pair of dice? A. 0.561 B. 0.862 C. 0.263 D. 0.751 4. We flip a fair coin 10 times. What is the probability that we get heads in at least 8 of the 10 flips? A. 45/1024 B. 7/128 C. 47/1024 D. 5/128 5. The probabilities are 0.4, 0.2, 0.3 and 0.1 respectively, that a delegate to a certain convention arrived by air, bus, automobile, or train. What is the probability that among 9 delegates randomly selected at this convention, 3 arrived by air, 3 arrived by bus, 1 arrived by automobile, and 2 arrived by train? A. 0.77% B. 7.7% C. 0.27% D. 2.7% 6. Four light bulbs are chosen at random from 20 bulbs of which 5 are defective. Find the probability that exactly one is defective? A. 0.2787 B. 0.0493 C. 0.1897 D. 0.4696 7. What is the probability that of 5 cards dealt from a well- shuffled deck, 3 will be hearts and two spades? A. 143/16660 B. 29/4560 C. 253/16660 D. 43/44560 8. In a class of 28 students, the teacher selects four people at random to participate in a geography contest. What is the probability that this group of four students includes at least two of the top three geography students in the class? Express your answer in as a common fraction. A. 37/819 B. 73/819 C. 73/918 D. 37/918 9. During a laboratory experiment, the average number of radioactive particles passing through a counter in 1 millisecond is 4. What is the probability that 6 particles enter the counter in a given millisecond? A. 0.2041 B. 0.4102 C. 0.4012 D. 0.1042 10.Your neighbor has two children. If you know that one of his children’s name is Peter, what is the probability that Peter’s siblings is also a boy? A. ½ B. 1/3 C. ¼ D. 1/5 11.Urn 1 contains 5 white balls and 7 black balls. Urn 2 contains 3 whites and 12 black. A fair coin is flipped; if it is Heads a ball is drawn from Urn 1, and if it is Tails, a ball is drawn from Urn 2. Supposed that this experiment is done and you learn that a white ball was selected. What is the probability that this ball was in fact taken from Urn 2? A. 12/37 B. 12/25 C. 13/44 D. 31/47 12.In a certain assembly plant, three machines, B1, B2 and B3 make 30%, 45%, and 25%, respectively, of the products. It is known from past experience that 2%, 3%, and 2% of the product made by each machine, respectively, are defective. Now, suppose that a finished product is randomly selected and is found defective. What is the probability that it is made by B3? A. 10/49 B. 12/25 C. 13/44 D. 31/47 13.One half percent of the population has a particular disease. A test is developed for the disease. The test gives a false positive 3% of the time and a false negative 2% of the time, Tsupul(a random person) just got the bad news that the test came back positive; what is the probability that Tsupul has the disease? A. 12% B. 14% C. 16% D. 21%

14.Find the particular solution of the curve which passes through the point (1,2) if the equation of its slope is dy/dx =2x-3. A. y= 2x2-3x +4 B. y=x2-2x+6 C. y= 3x2-3x+4 D. y=x2-3x+4 15.Find the general solution to the differential equation: cos 3xdy-dx=0 A. 2y= sec x tan x + ln (sec x + tan x) +C B. 2y= sec x tan x - ln (sec x + tan x) +C C. y= 2 sec x tan x + ln (sec x + tan x) +C D. y= sec x tan x + 2 ln (sec x + tan x) +C 16.The solution of 2xydx + (x2 +1)dy =0 A. xy2 + 2y = C B. 3xy -5y =C C. 2xy + y2 =C D. x2y +y =C 17.Solve the differential equation: x (y+1)dx + (x2-1) dy = 0 If y=2 when x =2, determine y when x=4. A. 0.376 B. 0.311 C. 0.342 D. 0.282 18.Determine the solution to the differential equation; (xy + y2)dx- x2dy=0 If y=1 when x=1. A. X=exp(1-y/x) B. X=exp(1-x/y) C. X=exp(1-x/y) D. Y=exp(1-x/y) 19.Which of the following is an exact DE? A. (x2 + 1)dx -2xydy=0 B. 2xydy + (2+x2) dy =0 C. xdy + (3x-2y)dy =0 D. x2ydy –ydx =0 20.Solve (x+y)dy=(x-y)dx A. x2 + y2 = C B. x2-2xy + y2 =C C. x2+2xy + y2 =C D. x2-2xy - y2 =C 21.Solve the differential equation: y’- (3y/x) =x3; y(1)=4 Find the particular solution. A. y= x3+x2 B. y= x4+3x3 C. y= x4+3x2 D. y= 3x3+x2 22.A certain piece of dubious information about phenyl ethylamine in the drinking water began to spread one day in a city with the population of 100,000. Within a week, 10,000 had heard the rumor. Assume that the rate of the increase of the number who have heard the rumor is proportional to the number who haven’t heard it. How long will be until half the population of the city has heard the rumor? A. 49 days B. 42 days C. 46 days D. 44 days

23.A 400 gallon tank initially contains 100 gal of brine containing 50 lb of salt. Brine containing 1 lb/gal enters the tank at a rate of 5 gal/sec and flows out a rate of 3 gal/sec, how much salt will the tank contain when it is full of brine? A. 393.75 lbs B. 375.25 lbs C. 379.96 lbs D. 493.75 lbs 24.A tank contains 200 L of fresh water. Brine containing 2kg/L of salt enters the tank at the rate of 4L/min. The mixture is kept uniform by stirring, run outs at 3L/min. Find the amount of salt in the tank after 30 mins. A. 196.99 kg B. 186.50 kg C. 312.69kg D. 234.28 kg 25.It is now between 2 and 3 o’clock. In what time will the minute hand and hour hand be perpendicular for the first time? a. 2:27.27 b. 2:31.35 c. 2:25.31 d. 2:29.57 26.It is between 3 and 4 o’clock. In 10 minutes, the minute hand will be as much after the hour hand as it is now behind it. What is the time? a. 3:10.36 b. 3:11.36 c. 3:12.36 d. 3:13.36 27.If a☺b=(a2b-2a2)/(3ab-6b2), find a☺a. a. (2-b)/3 b. (2-a)/3 c. (3-a)/2 d. (3-b)/2 28.Find the center of the conic whose equation is 13x 2+10xy+13y2+6x-42y-27=0. a. (-1,-2) b. (1,2) a. (-1,2) a. (1,-2) 29.A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. What is the probability that the bug moves to starting position on its tenth move? a. 341/1024 b. 21/64 c. 85/256 d. 171/512 30.Let E, D, G and M be the vertices of a regular tetrahedron. An ant starting from vertex E, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Find the probability that the ant will be on vertex E on its seventh move. a. 547/2187 b. 182/729 c. 61/243 d. 20/81 31.Find the asymptotes of the curve 4x2-y2=49. a. 2x-y=0 and 2x+y=0 b. 4x-y=2 and 4x+y=2

c. y-2x=0 and y+2x=0 d. y-4x=2 and y+4x=2 32.What is the equation of the upward asymptote of the hyperbola (x-2) 2/9 – (y+4)2/16 = 1? A. 4x+3y-20=0 B. 3x+4y+20=0 C. 4x-3y-20=0 D. 3x-4y-20=0 33.The second degree equation 19x2 + 6xy + 11y2 + 20x – 60y + 80 = 0 represents a conic. To remove the xy – terms, we rotate the coordinate axes through an angle of a. 16.400 b. 17.410 c. 18.430 d. 19.450 34.Find the maximum area of a rectangle that can be inscribed in an ellipse with semi-major axis of 4 units and minor axis of 6 units. a. 24 sq. units b. 12 sq. units c. 24pi sq. units d. 12pi sq. units 35.The curve has a parametric equation of x=4cosθ and y=3sinθ. Find its area. a. 24 sq. units b. 12 sq. units c. 24pi sq. units d. 12pi sq. units 36.Solve for the solution of the differential equation y’’-5y’+6y=8 a. y = C1e3x + C2e2x + 8/6 b. y = C1e-3x + C2e-2x + 8/6 c. y = C1e3x + C2e2x + 8 d. y = C1e-3x + C2e-2x + 8 37.Solve for the solution of the differential equation y’’+2y’=5 a. y = C1e2x + C2 + 5x/2 b. y = C1e-2x + C2 + 5x/2 c. y = C1e2x + C2 + 5/2 d. y = C1e-2x + C2 + 5/2 38.Solve for the particular solution of the differential equation (D 2+4D+3)y=e2x a. y = xe2x/4 b. y = xe2x/15 c. y = e2x/15 d. y = e2x/4 39.Solve for the particular solution of the differential equation d 2y/dx2-16y=e4x a. y = e4x/4 b. y = e4x/8 c. y = xe4x/4 d. y = xe4x/8 40.Solve for the particular solution of the differential equation (D 2+3D+2)y=3sin2x a. y = -3/20 sin2x – 9/20 cos2x b. y = -9/20 sin2x – 3/20 cos2x c. y = 3/20 sin2x + 9/20 cos2x d. y = 9/20 sin2x + 3/20 cos2x 41.Solve for the solution of the differential equation (D 2+4)y=5cos2x a. y = C1cos2x + C2sin2x – (5)xsin2x b. y = C1cos2x + C2sin2x – (5/4)xsin2x c. y = C1cos2x + C2sin2x – (5)sin2x d. y = C1cos2x + C2sin2x – (5/4)sin2x 42.Solve for the particular solution of the differential equation (D 2-6D+5)y=e2xsin3x a. y = e2x(1/30sin3x + 1/15cos3x)

b. y = e2x(1/15sin3x + 1/30cos3x) c. y = e2x(-1/15sin3x + 1/30cos3x) d. y = e2x(-1/30sin3x + 1/15cos3x) 43.Solve for the particular solution of the differential equation y’’-y’-2y=4x 2 a. y = 2x2+2x-3 b. y = 2x2-2x-3 c. y = -2x2-2x-3 d. y = -2x2+2x-3 44.Suppose that f and g are differentiable functions and g(x)=xf(x 2+4). If f(8)=20, f’(8)=-3, what is g’(2)? a. 13 b. 14 c. 15 d. 18 45.Find the total number of set relations between A={1, 2} and B={x, y, z}. a. 16 b. 32 c. 64 d. 128 46.The functions f(t)=2t2 and f(t)=t4 are a. Linearly independent b. Linearly dependent c. Inconsistent d. consistent 47.Determine the Laplacian of f=3+cos(x/2)sin(y/2) a. -1/2 cos(x/2)sin(y/2) a. 1/2 cos(x/2)sin(y/2) a. 1/2 + cos(x/2)sin(y/2) a. -1/2 + cos(x/2)sin(y/2) 48.Determine the Jacobian of x=rsinθ and y=rcosθ a. -r b. rcosθ c. rsinθ d. r2sinθcosθ 2 49.∂ z/∂x2 - 5∂2z/∂y2 + 3∂z/∂x - 4∂z/∂y = 6 is a kind of a. Heat equation b. Laplace equation c. Parabolic equation d. Wave equation 50.A statement that can be true or false a. Lemma b. Proposition c. Corollary d. Theorem 51.No matter what the combinations or values, the functions will always be true a. Tautology b. Contingency c. Contradiction d. Fallacy 52.A proposition regarded as self-evidently TRUE WITHOUT PROOF. A. Axiom B. Conjecture C. Hypothesis D. Porism 53.A statement also known as an axiom. A. Proof

B. Corollary C. Conjecture D. Postulate 54.A statement that can be demonstrated TO BE TRUE by accepted mathematical operations and arguments. A. Proposition B. Conjecture C. Hypothesis D. Theorem 55.A rigorous mathematical argument that unequivocally demonstrates the truth of a given proposition. A. Principle B. Conclusion C. Proof D. Hypothesis 56.A LOOSE term for a true statement that may be a postulate, theorem, etc. A. Principle B. Conclusion C. Proof D. Hypothesis 57.A proposition that is CONsistent with known data, but has neither been verified nor shown to be false. A. Lemma B. Paradox C. Proof D. CONjecture 58.A statement that is to be proven. A. Proposition B. Paradox C. Conclusion D. Proof 59.An immediate consequence of a result already proven. A. Axiom B. Conjecture C. Conclusion D. Corollary 60.A/An _________ is a statement that can EITHER be true or false. A. conclusion B. permutation C. involution D. proposition 61.Possible RESULTS from an experiment. A. Events B. Smoke C. Outcomes D. Means 62.The probability of one or both of two events occurring is: A. P(A or B) = P(A) + P(B) – P(A and B) B. P(A or B) = P(A) + P(B) C. P(A and B) = P(A) x P(B) D. P(A and B) = P(A) x P(B) + P(A or B) 63.If two events are MUTUALLY EXCLUSIVE, then the probability of one or the other occurring is: A. P(A or B) = P(A) + P(B) – P(A and B) B. P(A or B) = P(A) + P(B) C. P(A and B) = P(A) x P(B)

D. P(A and B) = P(A) x P(B) + P(A or B) 64.A collection of 33 coins, consisting of nickels, dimes, and quarters, has a value of $3.30. If there are three times as many nickels as quarters, and onehalf as many dimes as nickels, how many dimes are there? A. 6 B. 9 C. 10 D. 18 65.The derivative with respect to x of 2 Cos² (x² + 2). a. -9 Sin(x² + 2) Cos (x² + 2) b. -8x Sin(x² + 2) Cos (x² + 2)) c. -8x Sin(x² + 2) Cos (x² -2)) d. -12Sin(x² + 2) Cos (x² + 2))

66.Find the derivative of

3( x  1) ( x  1)  x x2 2

b.

3( x  1) ( x  1)  x x2 2

c.

( x  1) 3 x

.

3

b.

3

d.

3( x  1) 2 (2 x  1) 3  x x2  3( x  1) 2 ( x  1) 3  x x2

67.Compute the first derivative equation? a. c.

4.2 cos 2 x e Sin 2 x 4.32  2 x e

b.

Sin 2 x

d.

y  2.16 e sin 2 x.

3.42 cos 2 x e Sin 2 x 4.32 cos 2 x e Sin 2 x

68.If y = ax³ + bx² and the point of inflection is at (2, 8), what is the value of b? a. 9 b. 6 c. 3 d. 8 69.A printed page must contain 60 sq. m. of printed material. There are to be margin of 5 cm. on either side and margins of 3 cm. on top and bottom. How long should the printed lines be in order to minimize the amount of paper used? a. 10 b. 11 c. 12 d. 13 70.A school sponsored trip will cost each student 15 pesos if not more than 150 students make the trip, however the cost per student in excess of 150. How many students should make the trip in order for the school to receive the largest group income? A. 222 b. 225 c. 223 d. 1144 71.Two houses A and B are to be connected to a water main 1200 m. long and it is desired to tap the main in one place only. If A and B are 300 m. and 900 m. from the main pipe respectively. Where the pipe must be tapped to use the least pipe. a. 300 m b. 400 m c. 500 m d. 600 m 72.The angle between the sides of a triangle 8 cm. and 12 cm. long is changing causing the area to change at a constant rate of 45 m²/min. At what rate is the angle changing in rad/min when the angle is 30°. a. 1.08 rad/min b. 1.8 rad/min c. 8 rad/min d. 1 rad/min 73.A man is 1.8 m tall walks away from a lamp post 4 m. high at a speed of 1.5 m/s. How much does the end of his shadow move with respect to the lamp post. a. 2.3 m/s b. 2.73 m/s c. 2.7 m/s d. 73 m/s 74.Find the area bounded by the curve 5y² = 16x and the curve y² = 8x – 24. a. 12 sq. units b. 14 sq units c. 16 sq.units d. 5 sq.units 75.Find the area bounded by the curve y² = 4x and the line 2x + y = 4. a. 10 sq units b. 9 sq units c. 12 sq units d. 2 sq units 76.Find the area of the region in the first quadrant bounded by the curves y = Sin x, y = Cos x and the y axis. a. 0.41 sq.unitsb. 0.234 sq.units c. 0.414 sq.units d. 234 sq.units

x1  0 to x 2  2

77.Find the surface area of the portion of the curve x² + y² = 4 from when it is revolved about the y axis. a. 8π b. 6π c. 5π d. 9π 78.Compute the surface area generated when the first quadrant portion of the x² - 4y + 8 = 0

x1  0 to x 2  2

from is revolved about the y axis. a. 20.34 b. 30.64 c. 40.45 d. 50.78 79.Find the centroid of the area bounded by the curve y = 4 - x² the line x = 1 and the coordinate axes. a. 1.85 b. 1.96 c. 1.77 d. 1.88 80.Given the area in the first quadrant bounded by x² = 8y, the line y – 2 and the y – axis. What is the volume generated this area is revolved about the line y – 2 = 0? a. 28.81 cu. units b. 2.81 cu. units c. 28.1 cu. units d. 21 cu. units 81.Given the area in the first quadrant bounded by x² = 8y, the line x = 4 and the x axis. What is the volume generated by revolving this area about the y axis. a. 50.34 b. 34.50 c. 50.265 d. 265.50 82.Given the area in the first quadrant bounded by x² = 8y, the line y – 2 = 0 and the y axis. What is the volume generated when this area is revolved about the x axis? a. 41.20 cu. units b. 40.21 cu. units c. 2.18 cu. units d. 2.1 cu. units 83.Find the moment of inertia of the area bounded by the curve x² = 4y, the line y = 1 and the y axis on the first quadrant with respect to x axis. a. 7/4 b. 4/7 c. 6/4 d. 4/6 84.A right circular cylindrical tank of radius 2 m. and a height 8 m. is full of water. Find the work done in pumping the water to the top of the tank. Assume water to weight 9810 N/m³. 3589 kN. m b. 4546 kN. m c. 3945 kN.m d. 3954 kN.m 85.A tank is full of oil (density = 50 pcf). It has a diameter of 10 ft. find the work done in ft – lb in pumping all the liquid out of the top of the tank if the tank is a hemispherical tank. a. 24555 ft – lb b. 45556 ft – lb c. 3456 ft- lb d. 24544 ft - lb 1 y

x 86.Evaluate a. ½

2

dxdy

1 0

. b. 1/13 2

2

4 x 4 x  y

0

0

  87.Evaluate 4 b. 8π

2

 0

c. 1/14

dxdydx .

b. 9π

c. 3π

88.Determine the inverse Laplace transform of

11e 10t  11e 20t

d. 1/12

2

12e 10t  12e 20t

d. 6π

100 ( S  10)( S  20)

10e 10t  10e 20t

.

1e 10t  1e 20t

c. b. c. d. 89.Of 300 students, 100 are currently enrolled in mathematics and 80 are currently enrolled in Physics. These enrolment figures include 30 students whose are enrolled in both subjects. What is the probability that a randomly chosen student will be enrolled in either Mathematics or Physics? a. 0.45 b. 0.50 c. 0.55 d. 0.60

10

90.Given: sec 2θ = and 2θ in quadrant IV. Find: cos 4θ a. -0.60 b. -0.70 c. -0.80 d. -0.90 91.Find the height of a tree if the angle of elevations of its top changes from 20 degrees to 40 degrees as the observer advances 23 m toward its base. a. 138.5 m b. 148.5 m c. 158.5 m d. 159.5 m 92.A tower standing on level ground is due north of point A and due east of point B. At A and B, the angles of elevation of the top of the tower are 60 degrees and 45 degrees respectively. If AB = 20, find the height of the tower. a. 18.32 m b. 17.32 m c. 16.32 m d. 15.32 m 93.Find the area of a regular five pointed star that is inscribed in a circle of the radius 10. a. 121.62 b. 112.26 c. 122.16 d. 126.21 94.Compute the difference between the perimeters of a regular pentagon and a regular hexagon if the area of each is 12. a. 0.31 b. 0.21 c. 0.41 d. 0.51 95.The area of the sector of a circle having a central angle of 60 degrees is 24 perimeter of the sector. a. 34.4 b. 35.5 c. 36.6 d. 37.7



. Find the

96.Find the volume of a regular tetrahedron whose edges are each equal to 6. a.

16 2

b.

17 2

c.

18 2

d.

19 2

97.The lateral area of a regular pyramid is 514.5 and the slant height is 42. Find the perimeter of the base. a. 24.5 b. 26.5 c. 22.5 d. 28.5 98.A wedge is cut from a circular tree whose diameter is 2 m by a horizontal plane up to the vertical axis and another cutting plane which is inclined at 45 degree from the previous plane. The volume of the wedge is a. ¼ b. ½ c. 2/3 d. ¾ 99.A parabolic arch has a span of 20 m and a maximum height of 15 m. How high is the arch 4 m from the center of the span? a. 10.6 m b. 11.6 m c. 12.6 m d. 13.6 m 100. The earth’s orbit is an ellipse with eccentricity 1/60. If the semi major of the orbit is 93 M miles and the sum is at one of the foci, what is the shortest distance between earth and the sun? a. 89.43 M mi b. 90.44 M mi c. 91.45 M mi d. 92.46 M mi