Math-pre-board-exam-set-a-key.pdf

Republic of the Philippines Review Finest Inc. Board of Electronics Engineering ECE Licensure Examination – Pre-Board Examination

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ENGINEERING MATHEMATICS

SET A

INSTRUCTION: Select the correct answer for each of the following questions. Mark only one answer for each item by shading the box corresponding to the letter of your choice on the answer sheet provided. STRICTLY NO ERASURES ALLOWED. Use pencil no. 2 only. NOTE:

Whenever you come across a caret (^) sign, it means exponentiation. Ex. x^2 means x2; Pi=3.1416.

1.

If the circumference of a circle is 1, what is its area? A. 0.08* C. 1.27 B. 0.79 D. 3.14

2.

Family of curves that intersect another family of curves at right angles is called A. orthogonal* C. radial B. orthotomic D. cissoids

3.

What is the arithmetic mean of 4x – 2, x + 2, 2x + 3 and x + 1? A. 2x – 1 C. 2x + 1* B. 2x D. x + 2

4.

Two towers A and B are placed 100m apart horizontally. The height of A is 40 m and that of B is 30 m. At what distance vertically above the ground will the intersection of the lines forming the angles of elevation of the two towers are observed from the bases of the towers A and B respectively. A. 12.33 m C. 13.66 m B. 17.14 m* D. 18.15 m

5.

The vertex of the parabola 4y = x^2 – 6x + 21 is located at: A. (2,3) C. (3,3)* B. (-3,3) D. (4,-4)

6.

If point A(3,5) is located on a circle in the coordinate plane, and the center of the circle is the origin, which of the following points must lie outside this circle? A. (4.5, 4.0)* C. (2.5, 4.5) B. (1.5, 5.5) D. (4.0, 4.0)

7.

If function has continuous partial derivative, its differential is called: A. general differential C. singular differential B. total differential* D. autonomous differential

8.

A bag of jellybeans contains 8 black beans, 10 green beans, 3 yellow beans, and 9 orange beans. What is the probability of selecting either a yellow or an orange bean? A. 1/10 C. 4/15 B. 2/5* D. 3/10

9.

Equation that can be transformed into a separable equation by a change of variables is called: A. homogeneous* C. linear B. non homogeneous D. non linear

10.

Ordinary Differential Equation that doesn't show independent variable let x, explicitly is called A. singular C. partial B. autonomous* D. general

11.

What is the least positive integer that is divisible by both 2 and 5 and leaves a remainder of 2 when it is divided by 7? A. 20 C. 50 B. 30* D. 65

12.

One liter of water is evaporated from six liters of a solution containing 5% sugar. What is the percentage of sugar in the remaining solution? A. 5.5% C. 6%* B. 7.5% D. 8%

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ENGINEERING MATHEMATICS

SET A

13.

Find the arclength of the curve 3y = 4x between the points (3, 4) and (9, 12). A. 8 C. 10* B. 9 D. 12

14.

Find the Laplace transform of e^(t)∙sinh(2t). A. 3/(s^2 – 2s – 4) C. 3/(s^2 – 2s – 3) B. 2/(s^2 – 2s - 4) D. 2/(s^2 – 2s - 3)*

15.

The White-Bright Toothbrush Company hired 30 new employees. This hiring increased the company’s total workforce by 5%. How many employees now work at White-Bright? A. 530 C. 605 B. 600 D. 630*

16.

If the sum of two angles is 360°, they said to be: A. complementary C. explementary* B. supplementary D. none of the above

17.

Method which gives the solution in power series, multiplied by logarithmic term or fractional power is called A. Frobenus method* C. Logarithmic method B. Laplace method D. Linear method

18.

Evaluate ‖14.2‖ + ⌈−3.6⌉ A. 10 B. 11*

C. 12 D. none of the above

19.

If the radius of a sphere is increased by 10%, the volume increases by: A. 27% C. 33%* B. 30% D. 42%

20.

Points (√2, 4), (6, -√3) and C are collinear. If B is the midpoint of line segment AC, approximately what are the (x, y) coordinates of point C? A. (3.71, 1.13) C. (10.59, 5.73) B. (3.71, 5.73) D. (10.59, –7.46)*

21.

Consider the series S_n = 1, -1, +1, -1, +1, + -… If n is even, the sum is zero and if n is odd, the sum is 1. What do you call this kind of infinite series? A. Oscillating series* C. Bilateral Series B. Geometric series D. Di-valued Series

22.

Kyla, Jericho and Trixi take turns flipping a coin in their respective order. The first one to flip head wins. What is the probability that Trixi will win? A. 1/8 C. 1/4 B. 1/7* D. 1/3

23.

In the Figure shown, the radius of the circles is 1. What is the perimeter of the shaded part of the figure? A. 4π/3* B. π C. 2π/3 D. π/3

24.

On a recent chemistry test, the average (arithmetic mean) score among 5 students was 83, where the lowest and highest possible scores were 0 and 100, respectively. If the teacher decides to increase each student’s score by 2 points, and if none of the students originally scored more than 98, which of the following must be true? I. After the scores are increased, the average score is 85. II. When the scores are increased, the difference between the highest and lowest scores increases. III. After the increase, all 5 scores are greater than or equal to 25. A. I only C. I and II only B. II only D. I and III only*

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ENGINEERING MATHEMATICS

SET A

25.

In an exam, two reasoning problems, 1 and 2, are asked. 35% students solved problem 1 and 15% students solved both the problems. How many students who solved the first problem will also solve the second one? A. 0.15 C. 0.43* B. 0.35 D. 0.05

26.

Evaluate the derivative of e^(2x)/x. A. (xe^(2x)-e^(2x))/x^2 B. (2xe^(2x)-e^(2x))/x^2*

C. (e^(2x)-2xe^(2x))/x^2 D. (2e^(2x)-2e^(2x))/x^2

27.

Suppose that A is a 3×3 matrix and det(A)= −3. What is the det(4A)? A. -3/4 C. -12 B. -3 D. -192*

28.

If an empty rectangular water tank that has dimensions 100 centimeters, 20 centimeters, and 40 centimeters is to be filled using a right cylindrical bucket with a base radius of 9 centimeters and a height of 20 centimeters, approximately how many buckets of water will it take to fill the tank? A. 14 C. 18 B. 16* D. 20

29.

What is the area of a triangle with vertices (1,1), (3,1), and (5,7)? A. 6* C. 9 B. 7 D. 10

30.

Find the value of the limit: 𝑥2

A. 0 B. 1* 31.

∫ sec 2 𝑡 𝑑𝑡 lim 0 𝑥→0 𝑥 sin(𝑥)

C. 3 D. 4

Solve for the value of ‘x’ in the expression:

(𝑥 2 − 5𝑥 + 5)𝑥

2 −9𝑥+20

=1

A. 1, 1, 4 and 5 B. 1, 4, 4 and 5*

C. 1, 4, 5 and 5 D. none of the above

32.

Find the Laplace transform of the function f(t) = [(cos(4t)–cos(5t))/t]. A. 0.125×ln[(s^2 + 25)/(s^ + 16)] C. 0.5×ln[(s^2 + 25)/(s^ + 16)]* B. 0.25×ln[(s^2 + 25)/(s^ + 16)] D. ln[(s^2 + 25)/(s^ + 16)]

33.

Two dice are rolled, find the probability that the sum is less than 13. A. 0 C. 3/4 B. 1/2 D. 1*

34.

What is the last term in the expansion (2x + 3y)^4? A. y^4 C. 27y^4 B. 9y^4 D. 81y^4*

35.

Ann has 6 dice and rolls to see if at least one of them comes up six. Bob has 12 dice and rolls hoping for two or more to come up six. Who has a better chance of succeeding? A. Ann* C. they have equal probabilities B. Bob D. indeterminable

36.

Evaluate: 3𝑥 2 + 1 ∫3 𝑑𝑥 √(2𝑥 3 + 2𝑥 + 1)2 A. 3/2×(2x^3 + 2x^2 - 2x + 1)^(1/3) + C B. 1/2×(2x^3 + 2x^2 - 2x + 1)^(1/3) + C

37.

C. 3/2×(2x^3 + 2x + 1)^(1/3) + C* D. 1/2×(2x^3 + 2x + 1)^(1/3) + C

If i = √(-1), for which of the following values of n does i^n + (–i)^n have a positive value? A. 23 C. 25 B. 24* D. 26

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ENGINEERING MATHEMATICS

SET A

38.

The value of the integral: ∫x(1 – x)^n dx from 0 to 1. A. 1/(n + 1) C. 1/(n + 1) – 1/(n + 2)* B. 1/(n + 2) D. 1/(n + 1) + 1/(n + 2)

39.

In the increasing arithmetic sequence, u_1, u_2 … u_n , the common difference is an integer. If u_3 = 4 and u_(n−2) = 9, find u_n + 2n. A. 27 C. 33 B. 31* D. 37

40.

What is the base of the numeration system in which 121 represents the same number as the decimal number 81? A. 7 C. 9 B. 8* D. 6

41. An urn contains 4 green balls and 6 blue balls. A second urn contains 16 green and N blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is 0.58. Find the value of N. A. 114 C. 134 B. 124 D. 144* 42.

If f(x) = log((1+x)/(1-x)), then f(a) + f(b) is equal to: A. f((a+b)/(1-ab)) C. f((a-b)/(1-ab)) B. f((a+b)/(1+ab))* D. f((a-b)/(1+ab))

43.

Find the quadratic mean of {1.3, 1.5, 1.7, 1.0, 1.1} A. 9.04 C. 1.32 B. 3 D. 1.34*

44.

−2 Let A = [ 0 0

1 3 0

4 7]. Find the values of λ such that det(A -λI)= 0. 1

A. -2, 1, 3* B. -2, 1, 4

C. -2, 0, 3 D. -2, 0, 4

45.

Two identical spheres of radius 6 intersect so that the distance between their centers is 10. The point of intersection of the two spheres form a circle. What is the area of this circle? A. 5π C. 9π B. 7π D. 11π*

46.

The parabola with the equation y = 4x – 0.5x^2 has how many points with (x, y) coordinates that are both positive integers? A. 3* C. 8 B. 4 D. infinitely many

47.

Find the inverse laplace transform of the function F(s) = (8 – 3s + s^2)/s^3. A. 4t^2 + 3t + 1 C. 4t^3 + 3t^2 + 2t - 1 B. 4t^2 – 3t + 1* D. 4t^3 – 3t^2 – 2t + 1

48.

Find the area bounded by the curve x = y^2 + 2y and the line x = 3. A. 23/2 sq. units C. 32/2 sq. units B. 23/3 sq. units D. 32/3 sq. units*

49.

If ax + by = p, bx – ay = q and a^2 + b^2 = 1, then x^2 + y^2 = A. pq C. p + q B. p^2 + q^2* D. p^2 – q^2

50.

Find the Laplace transform of the piecewise function: 𝑔(𝑥) = { A. e^(-s)/s^3 B. 2e^(-s)/s^3*

(𝑡 − 1)2 0

𝑡>1 𝑡<1 C. e^(-2s)/s^3 D. 2e^(-2s)/s^3

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ENGINEERING MATHEMATICS

SET A

51.

Each of the three circles is tangent to the other two, and each side of the circumscribing equilateral triangle is tangent to two of the circles. If the length of one side of the triangle is x, what is the radius of one of the circles? 𝑥 𝑥 A. C. 1+2√3 1+√3 𝑥 2𝑥 B. * D. 2+2√3 1+2√3

52.

If f(x) = x/(x-1), then f(a)/f(a+1) is equal to: A. f(-a) C. f(a2)* B. f(1/a) D. f(a/(a-1))

53.

A, B, C are three sets such that n(A)=25, n(B)=20, n(C)=27, n(A∩B)=5, n(B∩C)=7 and A∩C=ϕ then n(A∪B∪C) is A. 60* C. 67 B. 65 D. 72

54.

Find the determinant of the matrix: 1 [2 3

−3 1 −2

−4 0] 5

A. 36 B. 54

C. 63* D. 72

55.

A proposition by which at any case the output is sometimes true and sometimes false. A. Tautology C. Contradiction B. Contingency* D. Implication

56.

Determine the frequency domain representation of [e^(4t) - e^(-3t)]/t. A. ln[(s+3)/(s+4)] C. ln[(s-3)/(s+4)] B. ln[(s+3)/(s-4)]* D. ln[(s-3)/(s-4)]

57.

An elliptical plot of garden has a semi-major axis of 6m and a semi-minor axis of 4.8meters. If these are increased by 0.15m each, find by differential equations the increase in area of the garden in sq.m. A. 0.62π C. 2.62π B. 1.62π* D. 2.62π

58.

1 What is the size of the matrix [12 13 A. 3×4* B. 4×3

59.

60.

2 3 14

3 4 1

4 1]? 2 C. either A or B D. none of the above

A cube of volume 216 cubic inches is inscribed in a sphere. What is the surface area of the sphere? A. 72π in^2 C. 96π in^2 B. 81π in^2 D. 108π in^2* The value of A. 0 B. 1

𝜋 2

∫0

𝑑𝑥 1+tan3 𝑥

is: C. π/2 D. π/4*

61.

The domain of definition of the function f(x) = √(log((5x-x2)/3)) is: A. [1,4]* C. (0,5) B. (1,4) D. [0,5]

62.

Evaluate the Laplace of 5e^(2t) – 4cos(3t). A. 5/(s - 2) – 4/(s^2 + 3) C. 5/(s - 2) – 4/(s^2 + 9) B. 5/(s - 2) – 4s/(s^2 + 3) D. 5/(s - 2) – 4s/(s^2 + 9)*

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ENGINEERING MATHEMATICS

SET A

63.

It is a curve that is generated by a point on the circumference of a circle as it rolls along a straight line. A. Paraboloid C. Cycloid* B. Ellipsoid D. Hyperboloid

64.

In the figure the line segment AP is tangent to the circle and has length 12. Find the length of the line segment CD given that DP has length 8. A. 8 B. 9 C. 10* D. 11

65.

You are off to soccer, and want to be the Goalkeeper, but that depends who is the Coach today: With Coach Sam the probability of being Goalkeeper is 0.5, with Coach Alex the probability of being Goalkeeper is 0.3. Sam is the Coach more often about 60% of all the games. What is the probability you will be a Goalkeeper today? A. 36% C. 53% B. 42%* D. 67%

66.

Which of the following differential equation is hyperbolic? A. Poisson equation C. Diffusion equation B. Heat equation D. Wave equation*

67.

Find L^(-1) = (s - 5)/(s^2 + s - 6) A. 8e^(-3t) – 3e^(2t) B. 0.5×(8e^(-3t) – 3e^(2t))

C. 0.25×(8e^(-3t) – 3e^(2t)) D. 0.2×(8e^(-3t) – 3e^(2t))*

68.

Six consecutive integers are written on a blackboard. When one of them is erased, the sum of the remaining five is 1999. What number was erased? A. 396* C. 426 B. 412 D. 472

69.

Find the 1000th derivative of sin(x). A. sin x* B. cos x

C. –sin x D. –cos x

70.

What conic section is represented by the equation x^2 + y^2 – 3x + 4y - 5xy = 12? A. Circle C. Parabola B. Ellipse D. Hyperbola*

71.

If f(x) = –x^2 – 3 and g(x) = 3 – x^2, what is the value of f(f(g(7)))? A. –46 C. –73207 B. –2119 D. –4490164*

72.

The smallest of n consecutive integers is also n. If their average is 64, what is n? A. 43* C. 45 B. 44 D. 46

73.

Find the value/s of x for the function x^3 – 6x^2 + 9x + 2 from [0, 4] that will satisfy the Mean Value Theorem. A. 0.85 C. 0.85 and 3.15* B. 3.15 D. nothing satisfies

74. Evaluate the differential equation: (2xy – 3x^2)dx + (x^2 + y)dy = 0 A. xy – x^3 + y^2 = c C. x^2(y^2) – x^3 + 1/2y^2 = c B. xy^2 – x^3 + 1/2y = c D. yx^2 – x^2 + 1/2y^2 = c* 75.

If be A. B.

15 men can build a wall 108 meters long in 6 days, what length of similar wall can built by 25 men in 3 days? 90 meters* C. 108 meters 75 meters D. 96 meters

76.

Find the Laplace transform of e^(2t)∙t^3. A. 3/(s - 2)^4 B. 6/(s - 2)^4*

C. 3/(s - 2)^3 D. 6/(s - 2)^3

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ENGINEERING MATHEMATICS

SET A

77.

Determine the limit of the function √𝑥 − √𝑥 − √𝑥 + √𝑥 A. -1* C. 1 B. 0 D. ∞

78.

Which of the following algebraic curves is not a 3rd degree curve? A. Folium of Descartes C. Conchoid of de Sluze B. Witch of Agnesi D. Kampyle of Eudoxus*

79.

Evaluate the integral of (sin(x) + cos(x))^2/sqrt(1 + sin(2x)) on the first quadrant. A. -1 C. 1 B. 0 D. 2*

80.

Find the Laplace transform of the piecewise function: 𝑔(𝑥) = {

(𝑡 − 2)3 0

𝑡>2 𝑡<2

A. 3e^(-s)/s^4 B. 6e^(-s)/s^4*

C. 3e^(-2s)/s^4 D. 6e^(-2s)/s^4*

81.

It is a line that intersects with a given curve at two or more points. A. tangent line C. cotangent line B. secant line D. cosecant line

82.

The triangle with vertices (-3, 4), (x, 3) and (5, 7) has an area of 11.5 sq. units. Find the positive value of x. A. 5 C. 2* B. 4 D. 3

83.

What is the power series representation of ln[(1+x)/(1-x)]? A. x + x^3/3 + x^5/5 + x^7/7 + … C. 0.5×(x + x^3/3 + x^5/5 + x^7/7 + …) B. 2(x + x^3/3 + x^5/5 + x^7/7 + …)* D. –(x + x^3/3 + x^5/5 + x^7/7 + …)

84.

Evaluate: 𝜋/2

∫

sin4 𝜃 cos 3 𝜃 𝑑𝜃

0

A. 5π/32 B. 16/35

C. 2/35* D. 6/35

85.

This theorem states that: “The number of common points of two such curves is at most equal to the product of their degrees.” A. Cramer’s theorem C. Klein quartic theorem B. Weber’s theorem D. Bézout's theorem*

86.

Find the arc length of the curve y = ln(cos(x)), 0 < x < π/3. A. ln(1 + √2) C. ln(1 + √3) B. ln(2 + √2) D. ln(2 + √3)*

87.

For what values of ‘c’ does the function x^4 - 16x^2 + 2 will satisfy the conditions of the M.V.T. from -1 to 3. A. -3, 0.38 C. -3, 2.62 B. 0.38, 2.62* D. -3, 0.38 and 2.62

88.

A certain quadratic function has its product of roots equal to the sum of its roots. Given that f(3) = 39 and f(4) = 66, what are the roots of the function? A. 1 +/- √3 C. 2 +/- √3 B. -1 +/- √3 * D. -2 +/- √3

89.

If x^2 + ax + bx + ab = 0, and x + b = 2, then x + a = A. –8 C. –2 B. –4 D. 0*

90.

A 4-degree curve whose cartesian expression is: x^2(x^2 + y^2) = a^2(y^2) A. Right strophoid C. Cissoid of Diocles B. Kappa curve* D. Trident curve

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ENGINEERING MATHEMATICS 91.

SET A

Find the divergence of the vector F = (x^2)y_i + xyz_j – (x^2)(y^2)_k. A. xy + xz C. xy - 2xz B. 2xy + xz* D. 2xy + 2xz

92. A bag contains red balls and white balls. If five balls are to be pulled from the bag, with replacement, the probability of getting exactly three red balls is 32 times the probability of getting exactly one red ball. What percent of the balls originally in the bag are red? A. 1/5 C. 3/5 B. 2/5 D. 4/5* 93.

The blood groups of 200 people is distributed as follows: 50 have type A blood, 65 have B blood type, 70 have O blood type and 15 have type AB blood. If a person from this group is selected at random, what is the probability that this person has O blood type? A. 0.20 C. 0.30 B. 0.25 D. 0.35*

94. Which A. It B. It C. It D. It 95.

is is is is is

true about the differential equation xy’’+ 2y’+ y^2 = xe^x? a 2nd order linear DE with degree of 2. a 2nd order non-linear DE with degree of 2. a 2nd order linear DE with degree of 1. a 2nd order non-linear DE with degree of 1.*

If 3cos(x) = 4sin(x), what is 50sin(2x)? A. 24 B. 48*

C. 72 D. 96

96. Terence Tao is playing rock-paper-scissors. Because his mental energy is focused on solving the twin primes conjecture, he uses the following very simple strategy: i. He plays rock first. ii. On each subsequent turn, he plays a different move each with probability 1/2. More probably, what would be his 5th move? A. Rock* C. Scissor B. Paper D. All moves will have equal probabilities 97.

For the given matrix A below, which of the following statement is true? 1 𝐴 = [1 3

2 0 4

A. A is an idempotent matrix B. A is not invertible*

3 −1] 5 C. A is a unitary matrix D. A is a symmetric matrix

98.

In how many ways can 5 persons be seated in an automobile having places for 2 in the front seat and 3 in the back if only 2 of them can drive and one of the others insisting on riding in the back? A. 36* C. 12 B. 24 D. 30

99.

A line has an equation x+5y-4=0. If the line x-ky-17=0 makes an angle of 45 degrees counter clockwise from the line x+5y-4=0, find the value of k. A. 2/3 C. 3/2* B. -2/3 D. -3/2

100. Find the integrating factor of the differential equation: (x^2 + xy)dy = -(3xy + y^2)dx A. 1/x C. ln(x) B. x* D. x^2

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