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1 Republic of the Philippines PROFESSIONAL REGULATION COMMISSION Manila BOARD OF ELECTRONICS ENGINEERING ELECTRONICS ENGINEERING Licensure Examination Saturday, September 2, 2023 08:00 am – 12:00 noon --------------------------------------------------------------------------------------------------------- MATHEMATICS SET A INSTRUCTION: Do not write anything on this questionnaire. Select the correct answer for each of the following questions. Mark only one answer for each for each item by shading the box corresponding to the letter of your choice on the answer sheet provided. STRICTLY NO ERASURES ALLOWED. NOTE: Whenever you come across with caret (^) sign, it means exponentiation. Ex. x^2 means x2; (x+y)^(x-z) means (x+y)x-z, x_3 means x3 (x+y)_(x-z) means (x+y)x-z. Pi = 3.1416 MULTIPLE CHOICE 1. The minimum and maximum eigenvalues of the matrix [ 1 1 3 1 5 1 3 1 1 ] are -2 and 6, respectively. What is the other eigenvalue? A. 5 B. 3 C. 1 D. -1 2. The degree of the differential equation d 2 x dt 2 +2x 3=0 is A. 0 B. 1 C. 2 D. 3 3. If the values of α and β are constants such that the following simultaneous equations have an infinite number of solutions: x+y+z = 5; x+2y+3z=9; x+3y+αz = β, find β-α A. 13 B. 5 C. 8 D. 9 4. The following equation needs to be numerically solved using Newton-Raphson method: x^3 + 4x – 9 = 0. The iterative equation for this purpose is (k indicates the iteration level) A. xk+1= 3xk 2+4 2xk 3+9 B. xk+1= 4xk 2+3 9xk 2+2 C. xk+1=2xk 3+9 − 3xk 2+4 D. xk+1= 2xk 3+9 3xk 2+4 5. Evaluate: ∫ sin t t dt ∞ 0 . A. π B. π/4 C. π/3 D. π/2 6. If the standard deviation of the voltage levels in an electronic circuit is 8.8 mV, and the mean voltage level is 33 mV, calculate the coefficient of variation in voltage levels. A. 0.1517 B. 0.1867 C. 0.2667 D. 0.3646 7. For function e-x, the linear approximation around x = 2 is A. 1 – x B. (3-x)e-2 C. [3 + 2√2 – (1+√2)x]e-2 D. e-2 8. If the Laplace transform of a signal y(t) is Y(s) = 1/[s(s-1)], then its final value is equal to A. -1 B. 0 C. 1 D. unbounded 9. An examination consists of two papers, Paper 1 and Paper 2. The probability of failing Paper 1 is 0.3 and that in Paper 2 is 0.2. Given that a student has failed in Paper 2, the probability of failing in Paper 1 is 0.6. Determine the probability of a student failing in either paper. A. 0.12 B. 0.18 C. 0.38 D. 0.06 10. A sequence x(n) with the z-transform X(z) = z4 + z2 – 2z + 2 – 3z-4 is applied as an input to a linear time-invariant system with the impulse response h(n) = 2δ(n-3) where δ(n) is a unit step sequence. Determine y(4). A. -6 B. 2 C. 0 D. 4 11. The value of ∫_0^π sin x dx when evaluated using trapezoidal method with 8 equal intervals, to 5 significant digits is A. 2.0000 B. 1.9742 C. 1.6232 D. 3.4512 12. The initial condition for which the following equation: (x 2+2x) dy dx =2(x+1)y has infinitely many solutions is A. y(0) = 5 B. y(0) = 1 C. y(2) = 1 D. y(-2) = 0 13. The directional derivative of f = 1⁄2 √x2+y2 at (1,1) in the direction <1, -1> is A. 1/√2 B. √2 C. 2 D. 0 14. Consider the series xn+1 = xn/2 + 9/8xn, x0 = 0.5 obtained from the Newton-Raphson method. The series converges to A. √2 B. 1.5 C. 1.6 D. 1.4 15. The general solution of d2y/dx2 + y = 0 is A. y = P sin2x B. y = P sin x C. y = P cos x D. y = P cos x + Q sin x 16. The value of ∫_0^3∫_0^x (6 – x – y )dydx is A. 54 B. 40.5 C. 27 D. 27/2 17. Three values of x and y are to be fitted in a straight line in the form of y = a + bx by the method of least squares. Given Σx = 6, Σy = 21, Σx2 = 14, and Σxy = 46, determine the values of a and b. A. 2, 3 B. 3, 2 C. 4,-2 D. -2, 4 18. The inner (dot) product of two vectors P and Q is zero. The angle (degrees) between the two vectors is A. 0 B. 30 C. 90 D. 120 19. Which of the following functions would have only odd powers of x in its Maclaurin series expansion?
2 A. sin(x2) B. sin(x3) C. cos(x2) D. cos(x3) 20. Step responses of a set of three second-order underdamped systems all have the same percentage overshoot. Which of the following diagrams represents the poles of the three systems? A. s-plane 1 B. s-plane 2 C. s-plane 3 D. s-plane 3 21. Evaluate the integral ∮ F ⋅ dl given F = <-x2y, xy2> in the region C if C is a circle with radius 2 centered at the origin. A. 2π B. 4π C. 8π D. 16π 22. What will happen to the confidence interval if the number of samples will decrease? A. the confidence interval will become wider B. the confidence interval will become narrower C. the confidence interval is not affected by the number of samples D. the confidence interval will remain the same 23. What is the equation used for solving the autocorrelation of a given signal, x(t)? A. R(τ) = ∫ x(t)x(t − τ)dt ∞ 0 B. R(τ) = ∫ x(t)x(τ − t)dt ∞ 0 C. R(τ) = ∫ x(t)x(τ − t)dt ∞ −∞ D. R(τ) = ∫ x(t)x(t − τ)dt ∞ −∞ 24. X is a uniformly distributed random variable that takes values between 0 and 1. The value of E(X3) will be A. 0 B. 1/8 C. 1/4 D. 1/2 25. Given vector A = 2i – 3j + 4k and B = 3i – Xj + 2k. What should be the value of X such that vector A and B are perpendicular. A. 11/3 B. -14/3 C. 13/3 D. -13/3 26. This states that the circulation of a vector field A around a closed path L is equal to the surface integral of the curl A over the open surface S bounded by L, provided A and ∇×A are continuous on S. A. Divergence theorem B. Green’s thereom C. Stokes’ theorem D. Fundamental theorem of calculus 27. This theorem relates the flux of a vector field F across a closed surface S to the volume integral of the divergence of F over the volume V bounded by S. A. Divergence theorem B. Green’s thereom C. Stokes’ theorem D. Fundamental theorem of calculus 28. Let f(x) be defined as: f(x)= { x 2+3x+2 x≠-2 k x=2 , For what value of k is f(x) continuous? A. -4 B. 0 C. 6 D. 2 29. Which of the following statements is true about a function that is continuous at a point c? A. The function is always differentiable at c. B. The limit of the function as x approaches c does not exist. C. The function may have a sharp corner at c. D. The function must have a horizontal tangent line at c. 30. Given Q = <2, -1, 2> and R = <2, -3, 1>, solve for a unit vector perpendicular to both Q and R. A. u⊥ = <-0.75, 0.30, -0.60> C. u⊥ = <0.57, 0.30, -0.60> B. u⊥ = <-0.57, 0.30, -0.60> D. u⊥ = <0.75, 0.30, -0.60> 31. Given y = x^2 + 2x + 10, the value of dy/dx|x=1 is equal to: A. 0 B. 4 C. 12 D. 13 32. Which of the following discrete-time systems is time invariant? A. y(n) = nx(n) B. y(n) = x(-n) C. y(n) = x(3n) D. y(n) = x(n-3) 33. The Bode asymptotic plot of a transfer function is given below. In the frequency range shown, the transfer function has A. 3 poles and 1 zero C. 1 pole and 2 zeros B. 2 poles and 1 zero D. 2 poles and 2 zeros 34. Consider the differential equation dy/dx = 1 + y2. Which of the following can be a particular solution of this differential equation? A. y = tan(x+3) B. y = tan(x)+3 C. x = tan(y+3) D. x = tan(y)+3 35. Consider the function y = x^2 -6x + 9. The maximum value of y obtained when x varies over the interval 2 to 5 is A. 1 B. 3 C. 4 D. 9 36. Consider the function f(x)=x^3−3x^2−4x+12. Solve for the point of inflection. A. (-2, 0) B. (-1, 12) C. (0, 12) D. (1,6) 37. Consider a discrete-time LTI system with input x(n) = δ(n)+ δ(n-1) and the impulse response h(n) = δ(n) - δ(n-1). The output of the system will be A. δ(n-1)+ δ(n-2) B. δ(n)- δ(n-2) C. δ(n)- δ(n-1) D. δ(n)+ δ(n-1)- δ(n-2) 38. A unity feedback system has open-loop transfer function G(s) = 100/(s(s+p)). The time at which the response to a unit step input reaches peak at π/8 seconds. Determine the damping ratio and the value of p for the closed loop system. A. 0.4, 8 B. 0.6, 12 C. 0.8, 16 D. 1, 20
3 39. In Taylor series expansion of ex about x = 2, the coefficient of (x-2)4 is A. 1/4! B. 24/4! C. e2/4! D. e4/4! 40. Consider a function f(x, y) = x^2 + xy + y^3. What is the expression for the partial derivative of f with respect to x, (∂f/∂x)? A. 2x + y^2 B. 2x + y C. x + 3y^2 D. 2x + 3y^2 41. The divergence of the vector field is A. 0 B. 1 C. 2 D. 3 42. Consider the shaded triangular region P shown in the figure. What is ∫∫Pxydxdy? A. 2/9 B. 1/6 C. 7/16 D. 1 43. Suppose we have an integral ∫-a a f(x)dx, where f(x) is an odd function. Which of the following statements is true? A. The integral is equal to F(a) where F(x) is the antiderivative of f(x). B. The integral is equal to zero. C. The integral is equal to 2F(a) where F(x) is the antiderivative of f(x). D. The integral is equal to F(-a) where F(x) is the antiderivative of f(x). 44. The integral ∮ f(z)dz evaluated around the unit circle on the complex plane for f(z) = cos(z)/z is: A. 0 B. 2πi C. 4πi D. -2πi 45. The length of the curve y = 2/3 x3/2 between x = 0 and x = 1 is: A. 0.27 B. 0.67 C. 1 D. 1.22 46. Let f = yx, what is ∂ 2f ∂x∂y at x = 2 and y = 1? A. 0 B. ln 2 C. 1 D. 1/ln(2) 47. The product structure of assembly P is shown in the figure. Estimated demand for end product P is as follows: Week 1 2 3 4 5 6 Demand 1000 1000 1000 1000 1200 1200 Ignore lead times for assembly and sub-assembly. Production capacity (per week) for component R is the bottleneck operation. Starting with zero inventory, the smallest capacity that will ensure a feasible operation plan up to week 6 is A. 1000 B. 1200 C. 2200 D. 2400 48. Determine the number of treatments in a one-way ANOVA experiment if the mean square between groups is 4 and the sum of square beween groups is 8. A. 2 B. 3 C. 4 D. 5 49. During a one-way ANOVA study with 5 treatments, the degrees of freedom total was found to be 49. If each treatment had an equal number of observations, what is the total number of observations in the study? A. 45 B. 50 C. 55 D. 60 50. For scalar function f = x^2+3y^2+2z^2, the gradient at the point P(1, 2, -1) is A. <2, 6, 4> B. <2, 12, -4> C. <2, 12, 4> D. √56 51. Given the parametric equations: x(t) = t 2 + 2t y(t) = t 3 − t , Find the value of d 2y dx2 when t = 1. A. 3/2 B. 5/16 C. 3/8 D. 7/12 52. The analytic function f(z) = (z-1)/(z^2+1) has singularities at A. 1 and -1 B. 1 and i C. 1 and -i D. i and -i 53. If the function f is integrable on the closed interval [a, b], then the _________ of f on [a, b] is given by: A. theoretical value B. sag value C. instantaneous value D. average value 54. Solve for the Laplace transform of f(t) = t cosωt for t > 0. A. s 2+ω2 (s+w)2 B. s 2−ω2 (s 2+ω2)2 C. s 2−ω2 2(s 2+ω2)3 D. 2ws (s 2+ω2)2 55. Solve for the Laplace transform of f(t) = t sinωt for t > 0. A. s 2−ω2 (s 2+ω2)2 B. s 2+ω2 (s+w)2 C. s 2−ω2 2(s 2+ω2)3 D. 2ws (s 2+ω2)2 56. Solution of the differential equation 3ydy/dx + 2x = 0 represents a family of A. circles B. ellipses C. parabolas D. hyperbolas 57. An electronics engineer is testing the signal strength of a new communication device. After taking 100 samples, she calculates a mean signal strength of 55 dB with a standard deviation of 5 dB. If she wants to construct a 95% confidence interval for the mean signal strength of the device, which of the following intervals is most likely correct? A. (53.70, 56.30) dB B. (50.00, 60.00) dB C. (52.45, 57.55) dB D. (54.02, 55.98) dB 58. When do we use the t-distribution in solving confidence intervals? A. When the sample size is large (>30) and the population standard deviation is known. B. When the sample size is small (<30) and the population standard deviation is unknown. C. When the population is normally distributed and the sample standard deviation is known. D. Whenever the sample mean is greater than the population mean. 59. In hypothesis testing, when doing two-tailed tests and the level of signficance is 0.05, determine the confidence level. A. 90% B. 95% C. 5% D. 10%
4 60. The Fourier series of a real periodic function has only: i) cosine terms if it is even ii) sine terms if it is even iii) cosine terms if it is odd iv) sine terms if it is odd A) i and ii only B. i and iii only C. ii and iii only D. ii and iv only 61. At point of inflection A. f’’(x) is negative B. f’’(x) is zero C. f’’(x) is positive D. All of these are correct. 62. The first two rows of Routh’s tabulation of a third-order equation are as follows: s3 2 2 s2 4 4 This means there are A. two roots at s = ±j and one root in the right half s-plane B. two roots at s = ±j2 and one root in the left half s-plane C. two roots at s = ±j2 and one root in the right half s-plane D. two roots at s = ±j and one root in the left half s-plane 63. If f(x) = excos(x2), find f’’(x). A. excos(x2) - 2exsin(x2) - 4x2excos(x2) - 4xexsin(x2) B. excos(x2) + 2exsin(x2) - 4x2excos(x2) + 4xexsin(x2) C. excos(x2) + 2exsin(x2) + 4x2excos(x2) + 4xexsin(x2) D. excos(x2) - 2exsin(x2) + 4x2excos(x2) - 4xexsin(x2) 64. The derivative of sec-1(-1/(2x^2-1)) w.r.t. √(1-x^2) at x = 1/2 is A. 3 B. -2 C. 1 D. -4 65. Given: x = t^2 + t and y = 2t – 1, solve for d 2 y dx2 in terms of the parameter ‘t’. A. -4/(t2 + t)3 B. -4/(2t + 1)3 C. 2/(t2 + t)3 D. 2/(2t + 1)3 66. The result of an indefinite integral always produces __________. A. a residue B. a limit C. an arbitrary constant D. all of these 67. Solve for h’(x) given that h(x) = ∫_1^(e^x)(ln(t)dt). A. xe^x B. ln x C. ln ex D. ex 68. In using integration by parts, it is preferred that the integral ∫ ln(x)sin(x)dx be solved by setting u as _______. A. sin(x) B. ln(x) C. 1 D. ln(x)sin(x). 69. The value of the integral I = 1 √2π ∫ exp ( −x 2 8 )dx ∞ 0 is A. 1 B. 2 C. π D. 2π 70. Find d2y/dx2 if x = t3 + t and y = t5 + 1. A. 5t4/(3t2 + 1) B. (30t5 + 20t3)/(3t2+1)3 C. 5t4 D. 3t2 + 1 71. Two random variables X and Y are distributed according to: The probability P(X + Y ≤ 1) is ________ A. 0.25 B. 0.33 C. 0.50 D. 0.75 72. In Mason's gain rule, the term "non-touching loops" stands for A. two or more loops sharing a node B. loops that do not share a node with the forward path C. any path or loop without common nodes D. the products of loop gains taken two at a time. 73. For the block diagram shown in the figure, determine the transfer function Y(s)/R(s). A. (2s+3)/(s+1) B. (3s+2)/(s-1) C. (s+1)/(3s+2) D. (3s+2)/(s+1) 74. The signal flow graph for a system is given below. The transfer function Y(s)/U(s) for this system is: A. s+1 s 2+6s+2 B. s+1 5s 2+6s+2 C. s+1 s 2+4s+2 D. 1 5s 2+6s+2 75. Solve for the convolution: x(t) = 1/√t and h(t) = t2. A. y(t) = 16 15 t −5/2 B. y(t) = 8 5 t −3/2 C. y(t) = 16 15 t 5/2 D. y(t) = 8 5 t 3/2 76. Find the sum of the first three terms of the Maclaurin series expansion of f(x) = (4-x)-1/2 A. 1/2 - x/16 + 3x^2/256 C. 1/2 + x/16 + 3x^2/256 B. 1/2 + x^2/16 + 3x^4/256 D. 1/2 – x^2/16 + 3x^4/256 77. For two independent variables, linear, second-order partial differential equations (PDE) can be expressed in the following general form: A ∂ 2 u ∂x2 +B ∂ 2 u ∂x ∂y +C ∂ 2 u ∂y2 +D ∂u ∂x +E ∂u ∂y +F(x,y)=0 where A, B, and C are functions of x and y and D, E and F is a function of x, y, u, ∂u/∂x, and ∂u/∂y. If B2 – 4AC = 0, then the above PDE can be classified as: A. Elliptic B. Parabolic C. Hyperbolic D. Conic 78. Given z = (x-a)^2 + (y-b)^2, form the corresponding PDE through elimination of arbitrary constants.