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Past Board Exam Problems in Advanced Engineering Mathematics CE Board Exam May 1994 The expression 3 + j4 is a complex number. Compute its absolute value. A. 4 B. 5 C. 6 D. 7 CE Board Exam November 1996 Compute the value of x bydeterminant =####### 4 1 2 3####### 2 0 2 1x 10 3 0 1 14 2 4 5A. - B. - ...

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Past Boa Board rd Exam Problems in Advance Advanced d Engineering Mathematics 1.

2.

CE Board Exa Exam m May 1994 The expression 3 + j4 is a complex number. Compute its absolute value. A. 4 B. 5 C. 6 D. 7 CE Board Exa Exam m November 1996 Compute the value of x determinant 4 - 1 2 3 2 0 2 1 x= 10 3 0 1 14 2 4 5 A. B. C. D.

3.

4.

CE Board Exa Exam m May 1996 1 2 Element of matrix B = 0 - 5 Element of matrix C =

3 6 4 1

Find the elements of the product of the two matrices, matrix BC. 11 8 answer A. - 20 - 5

5.

EE Board Exa Exam m April 1997 Write in the form a + jb the expression j3217 – j427 + j18 A. 1 + j2 B. 1 – j C. -1 + j2 D. 1 + j

7.

EE Board Exa Exam m October 1993 Write the polar form of the vector 3 + j4. A. 6 cis 53.1 deg B. 10 cis 53.1 deg C. 5 cis 53.1 deg D. 8 cis 53.1 deg

8.

EE Board Exa Exam m April 1995 Simplify (3 – j)2 – 7(3 – j) + 10. A. -(3 + j) B. 3 + j C. 3 – j D. -(3 – j)

9.

EE Board Exa Exam m April 1996 If A = 40ej120° , B = 20 cis(-40), C = 26.46 + j0, solve for A + B + C. A. 27.7 cis(45°) B. 35.1 cis(45°) C. 30.8 cis(45°) D. 33.4 cis(45°)

-32 -28 16 52

-4, 6 -4, 2 -4, -2 -4, -6

B.

- 11 8 19 5

C.

- 10 9 - 19 6

D.

- 11 9 - 20 - 4

EE Board Exa Exam m April 1997 Simplify: j29 + j21 + j A. j3 B. 1 – j C. 1 + j

j2

6.

by

CE Board Exa Exam m November 1997 Given the matrix equation, solve for x and y.  1 1  x   2   3 2  y  =  0      A. B. C. D.

D.

10. EE Board Exam October 1 1997 997 What is j4 cube times j2 square? A. -j8 B. j8 C. -8 D. -j28 11. EE Board Exam April 19 1997 97 What is the simplified complex expression of (4.33 + j2.5) square? A. 12.5 + j21.65 B. 20 + j20 C. 15 + j20 D. 21.65 + j12.5 12. EE Board Exam November 1 1997 997 Find the principal 5th root of [50(cos 150° + j sin 150°)]. A. 1.9 + j1.1 B. 3.26 – j2.1 C. 2.87 + j2.1 D. 2.25 – j1.2 13. EE Board Exam October 1 1997 997 What is the quotient when 4 + j8 is divided by j3? A. 8 – j4 B. 8 + j4 C. -8 + j4 D. -8 – j4

14. EE Board Exam October 1 1997 997 If A = -2 – j3 and B = 3 + j4, what is A/B? 18 - j A. 25 -18 - j answer B. 25 -18 + j C. 25 18 + j D. 25 Boar oar oard d Ex Exam am Octo October ber 1 1997 997 15. EE B 4 + j3 Rationalize 2− j A. 1 + j2 11 + j10 B. 5 5 + j2 C. 5 D. 2 + j2 16. EE B Boar oar oard d Ex Exam am Octo October ber 1 1997 997 (2 + j3)(5 − j) Simplify (3 − j2)2 A. (221 – j91)/169 B. (21 + j52)/13 C. (-7 + j17)/13 D. (-90 + j220)/169

17. EE Bo Board ard Ex Exam am Ap April ril 1 1996 996 What is the simplified expression of 6 + j2.5 ? the complex number 3 + j4 A. B. C. D.

-0.32 + j0.66 1.12 – j0.66 0.32 - j0.66 -1.75 + j1.03

Board ard Ex Exam am Ap April ril 1 1997 997 18. EE Bo Perform the operation: 4(cos 60° + j sin 60°) divided by 2(cos 30° + j sin 30°)] in rectangular coordinates. A. square root of 3 – j2 B. square root of 3 – j C. square root of 3 + j D. square root of 3 + j2 Board ard Ex Exam am JJune une 19 1990 90 19. EE Bo 50 + j35 Find the quotient of . 8 + j5 A. B. C. D.

6.47 cis (3°) 4.47 cis (3°) 7.47 cis (30°) 2.47 cis (53°)

20. EE Bo Board ard Ex Exam am Ma March rch 1998 Three vectors A, B and C are related as follows: A/B = 2 at180°, A + C = -5 + j15, C = conjugate of B. Find A. A. 5 – j5 B. -10 + j10 C. 10 – j10 D. 15 + j15 Board ard Ex Exam am Ap April ril 1 1999 999 21. EE Bo  π Evaluate cosh j   4 A. 0.707 B. 1.41 + j0.866 C. 0.5 + j0.707 D. j0.707 22. EE Bo Board ard Ex Exam am Ap April ril 1 1999 999  π Evaluate tanh j   3 A. B. C. D.

0.5 + j1.732 j0.866 j1.732 0.5 + j0.866

Board ard Ex Exam am Ap April ril 1 1999 999 23. EE Bo Evaluate ln (2 + j3). A. 1.34 + j0.32 B. 2.54 + j0.866 C. 2.23 + j0.21 D. 1.28 + j0.98 24. EE Bo Board ard Ex Exam am Oc Octobe tobe toberr 199 1997 7 Evaluate the terms at t = 1 of the Fourier series 2ej10πt + 2e-j10πt A. 2 + j B. 2 C. 4 D. 2 + j2 25. EE Bo Board ard Ex Exam am Ma March rch 1998 Given the following series: 3 5 x x sinx = x + + .... 3! 5! x2 x 4 + + .... cos x = 12! 4! x2 x3 x + + .... e = 1+ x + 2! 3! What relation can you draw from these series? A. ex = cos x + sin x B. ejx = cos x + jsin x C. ejx = jcos x + sin x D. jex = icos x + jsin x 26. EE Bo Board ard Ex Exam am Oc Octobe tobe toberr 199 1997 7

One term of a Fourier series in cosine form is 10cos 40πt. Write it in exponential form. A. 5ej40πt B. 5ej40πt + 5e-j40πt C. 10e-j40πt D. 10ej40πt 27. EE Bo Board ard Ex Exam am Ap April ril 1 1997 997 Evaluate the determinant 1 2 3

4 2 5 0

C.

-

B. C. D.

E xam equat x+y+z=2 3x – y – 2z = 4 5x – 2y + 3z = -7 Solve for y by determinants A. 1 B. -2 C. 3 D. 0

2 3 0 5

answer

1 7 2 0

3 1 5 7

 −2 −1  0 2   3 2    0 − 1    −2 0  −  answer  0 − 1

33. EE Boa Board rd E Exam xam Oc October tober 1997 If a 3 x 3 matrix and its inverse are multiplied, write the product.  1 0 0 A.  0 1 0 answer  0 0 1 B.

30. EE Boa Board rd E Exam xam A April pril 19 1997 97 Solve the equations by Cramer’s Rule 2x – y + 3z = -3 3x + 3y – z = 10 -x – y + z = -4 A. (2, 1, -1) B. (2, -1, 1) C. (1, 2, -1) D. (-1, -2, 1) Board rd E Exam xam Oc October tober 1997 31. EE Boa 2 3 1 If A = - 1 2 4 , what is cofactor of third

-

cofactor with the first row, second column element?  3 2 A. −    0 − 1

1 - 2 2 - 3 3 - 4 - 3 - 4 489 389 326 452

0 5 7 the second row, element?

0 5

Board rd E Exam xam Oc October tober 1997 32. EE Boa  3 1 2 If A =  −2 −1 0  , what is the  0 2 −1

Board rd E Exam xam A April pril 19 1997 97 28. EE Boa Evaluate the determinant 2 14 3 1 1 5 -1 3

A. B. C. D.

2 3

B.

D.

- 2 - 1 - 2 3 1 4 A. B. C. D.

A.

column

C.

D.

0  0  0 0  0  1

0 0  0 0 0 0 0 1  1 0 0 0

 1 1 1    1 1 1  1 1 1

34. EE Boa Board rd E Exam xam A April pril 19 1996 96  1 −1 2    If matrix  2 1 3  is multiplied by  0 − 1 1 x  x   y  is equal to zero, then matrix  y       z   z  is

A. B. C. D.

3 1 0 -2

C.

35. EE Boa Board rd Ex Exam am Oc October tober 1997 Given: 4 5 0 1 0 0 A= 6 7 3 B= 0 1 0 , 1 2 5 0 0 1 What is A times B equal to? 4 0 0 A. 0 7 0 0 0 5 B.

0 0 0 0 7 0 6 7 0 8 9 4 2 3 5 4 5 0

D.

6 7 3 answer 1 2 5

36. EE Boa Board rd Ex Exam am Ap April ril 19 1997 97 2 1 - 1 2 + 2 Matrix = Matrix - 1 3 1 1 A.

-2 4 Matrix 2 2

B.

Matrix

C.

Matrix

D.

Matrix

-1 2 1 1 2

Board rd E Exam xam A April pril 19 1997 97 38. EE Boa What is the inverse Laplace transform of k divided by [(s square) + (k square)]? A. cos kt B. sin kt C. (e exponent kt) D. 1.00

C. D.

1

-1 3 0 5 1 5

answer

Board rd Ex Exam am Oc October tober 1997 37. EE Boa 3 1 2   Transpose the matrix − 2 − 1 0   0 2 − 1 A.

−1 2 0  0  −1 −2    2 1 3 

B.

3 − 2 0  1 − 1 2  answer   2 0 − 1

Board rd E Exam xam A April pril 19 1997 97 40. EE Boa Find the Laplace transform of 2/(s + 1) – 4/(s + 3). A. 2 e(exp -t) – 4 e(exp -3t) B. e(exp -2t) + e(exp -3t) C. e(exp -2t) – e(exp -3t) D. [2 e(exp -t)][1 – 2 e(exp -3t)] 41. EE Boa Board rd E Exam xam Ma March rch 1998 Determine the inverse Laplace 200 transform of I(s) = 2 s −50s +10625 A. i(t) = 2e-25t sin 100t B. i(t) = 2te-25t sin 100t C. i(t) = 2e-25t cos 100t D. i(t) = 2te-25t cos 100t 42. EE Boa Board rd E Exam xam A April pril 19 1997 97 The inverse Laplace transform of s/[(s square) + (w square)] is A. sin wt B. w C. e exponent wt D. cos wt 43. ECE Bo Board ard E Exam xam A April pril 1 1999 999 Simplify the expression j 1997 + j1999 . A. 0 B. -j C. 1 + j D. 1 – j Board ard E Exam xam N Nove ove ovember mber 1998 44. ECE Bo Find the value of (1 + j)5 A. 1 – j B. -4(1 + j)

1+j 4(1 + j)

45. ECE Bo Board ard E Exam xam N Nove ove ovember mber 1991 Evaluate the determinant 1 6 0 4 2 7 0 5 3 A. B. C. D.

110 -101 101 -110

Board ard Exam Ap April ril 19 1997 97 46. ME Bo Evaluate the value of

−10

multiplied by −7 . A. j

39. EE Boa Board rd E Exam xam A April pril 19 1995, 95, April 1997 The Laplace transform of cos wt is A. s/[(s square) + (w square] B. w/[(s square) + (w square] C. w/(s + w) D. s/(s + w)

1 0 0 C.

D.

3 1 2    1 2 −1 −2 −1 0  1 3 2   −1 −2 0   2 2 −1

B.

70 answer

C.

- 70

D.

17

47. A. B. C. D. 48. A. B. C. D. 49. A. B. C. D. 50. A. B. C. D. Past Boar Board d Exam Proble Problems ms in Algebra 51. CE Boa Board rd Ex Exam am Ma May y 199 1997 7 Find the value of w in the following equations 3x – 2y + w = 11 x + 5y – 2w = -9 2x + y – 3w = -6 A. 3

B. C. D.

2 4 -2

Board rd Ex Exam am Ma May y 199 1996 6 52. CE Boa Find the value of A in the equation: x 2 + 4x + 10 3

2

x + 2x + 5x A. B. C. D.

-2 1/2 -1/2 2

=

Board rd Ex Exam am No Novemb vemb vember er 1997 58. CE Boa A B(2x + 2) C + 2 + 2 Evaluate the log6 845 = x. x x + 2x + 5 x + 2x + 5 A. 3.76 B. 5.84 C. 4.48 D. 2.98

53. CE Boa Board rd Ex Exam am No Novemb vemb vember er 1991 Solve for x in the given equation. 4 3

8 2 8x = 2

A. B. C. D.

4 2 3 5

54. CE Boa Board rd Ex Exam am No Novemb vemb vember er 1997 Find the remainder if we divide 4y3 + 18y 2 + 8y – 4 by 2y + 3. A. 10 B. 11 C. 15 D. 13 55. CE Boa Board rd Ex Exam am No Novemb vemb vember er 1993 A 400-mm φ pipe can fill a tank alone in 5 hours and another 600-mm φ pipe can fill the tank alone in 4 hours. A drain pipe 300-mm φ can empty the tank in 20 hours. With all the three pipes open, how long will it take to fill the tank? A. 2.00 hours B. 2.50 hours C. 2.25 hours D. 2.75 hours 56. CE Boa Board rd Ex Exam am No Novemb vemb vember er 1996 Find the 6th term of the expansion of 16 æ çç 1 - 3ö÷ ÷ çè 2a ø÷

A.

-

B.

-

C.

-

D.

-

Find the value of log8 48. A. 1.86 B. 1.68 C. 1.78 D. 1.98

66939 256a11 66339 128a 11 33669

answer

256a11 39396 128a 11

November er 1993, 57. CE Board Exam Novemb ECE No Novemb vemb vember er 19 1993 93

59. CE Board Exam Novemb November er 1992, May 1 1994 994 If loga 10 = 0.25, what is the value of log10 a? A. 2 B. 4 C. 6 D. 8 60. CE Boa Board rd Ex Exam am No Novemb vemb vember er 1993 It takes Butch twice as long as it takes Dan to do a certain piece of work. Working together they can do the work in 6 days. How long would it take Dan to do it alone? A. 9 days B. 10 days C. 11 days D. 12 days Board rd Ex Exam am No Novemb vemb vember er 1994 61. CE Boa An airplane flying with the wind, took 2 hrs to travel 1000 km and 2.5 hrs in flying back. What was the wind velocity in kph? A. 50 kph B. 60 kph C. 70 kph D. 40 kph 62. CE Boa Board rd Ex Exam am Ma May y 199 1998 8 A boat travels downstream in 2/3 of the time as it goes going upstream. If the velocity of the river’s current is 8 kph, determine the velocity of the boat in still water. A. 40 kph B. 50 kph C. 30 kph D. 60 kph Board rd Ex Exam am Ma May y 199 1995 5 63. CE Boa In how many minutes after 2 o’clock will the hands of the clock extend in opposite directions for the first time? A. 42.4 minutes B. 42.8 minutes C. 43.2 minutes D. 43.6 minutes

64. CE Boa Board rd Ex Exam am No Novemb vemb vember er 1995 In how many minutes after 7 o’clock will the hands be directly opposite each other for the first time? A. 5.22 minutes B. 5.33 minutes C. 5.46 minutes D. 5.54 minutes Board rd Ex Exam am Ma May y 199 1997 7 65. CE Boa What time after 3 o’clock will the hands of the clock be together for the first time? A. 3:02.30 B. 3:17.37 C. 3:14.32 D. 3:16.36 66. CE B oard Exa Exam m Ma May y 19 1993, 93, April 2004 Given that “w” varies directly as the product of “x” and “y” and inversely as the square of “z” and that w = 4 when x = 2, y = 6 and z = 3. Find the value of “w” when x = 1, y = 4 and z = 2. A. 3 B. 4 C. 5 D. 6 67. CE B oard Exam M ay 1 993, May 1994 1994,, Nove November mber 1994 How many terms of the progression 3, 5, 7, … must be taken in order that their sum will be 2600? A. 48 B. 49 C. 50 D. 51 68. CE Boa Board rd Ex Exam am Ma May y 199 1995 5 What is the sum of the progression 4, 9, 14, 19… up to the 20th term? A. 1030 B. 1035 C. 1040 D. 1045 69. CE Boa Board rd Ex Exam am Ma May y 199 1998 8 Determine the sum of the progression if there are 7 arithmetic means between 3 and 35. A. 171 B. 182 C. 232 D. 216 Board rd Ex Exam am Ma May y 199 1991 1 70. CE Boa In the “Gulf War” in the Middle East, the allied forces captures 6400 of Saddam’s soldiers and with provisions on hand it will last for 216

meals while feeding 3 meals per day. The provision lasted 9 more days because of daily deaths. At an average, how many people died per day? A. 15 B. 16 C. 17 D. 18 71. CE Boa Board rd Ex Exam am No Novemb vemb vember er 1993 The 3rd term of a harmonic progression is 15 and the 9th term is 6. Find the 11th term. A. 4 B. 5 C. 6 D. 7 Board rd Ex Exam am Ma May y 199 1995 5 72. CE Boa The numbers 28, x + 2, 112 form a geometric progression. What is the 10 th term? A. 14336 B. 13463 C. 16433 D. 16344 Board rd Ex Exam am No Novemb vemb vember er 1994 73. CE Boa In a benefit show, a number of wealthy men agreed that the first one to arrive would pay 10 centavos to enter and each later arrive would pay twice as much as the preceding man. The total amount collected from all of them was Php104,857.50. How many wealthy men paid? A. 18 B. 19 C. 20 D. 21 Board rd Ex Exam am Ma May y 199 1998 8 74. CE Boa Find the sum of 1, -1/5, 1/25, … A. 5/6 B. 2/3 C. 0.84 D. 0.72 75. CE Boa Board rd Ex Exam am Ma May y 199 1992 2 To conserve energy due to present energy crisis, the Meralco tried to readjust their charges to electrical energy users who consume more 2000 kW-hrs. For the first 100 kW-hr, they changed 40 centavos and increasing at a constant rate more than the preceding one until the fifth 100 kW-hr, the charge is 76 centavos. How much is the average charge for the electrical energy per 100 kW-hr?

A. B. C. D.

58 centavos 60 centavos 62 centavos 64 centavos

Board rd E Exam xam Oc October tober 1992 76. EE Boa Find the value of x x + 1 2x + = 47 - 2x 3 4 A. 16.47 B. 12.84 C. 18.27 D. 20.17

B. C. D.

in

Board rd E Exam xam Oc October tober 1991 77. EE Boa Find the value of x in the equations: éA A ù é3A 4Aù ú= A 10 ê + ú= A 2ê êx ú êx y yú ë û ë û A. 50/9 B. 80/9 C. 70/9 D. 60/9 78. EE Boa Board rd E Exam xam Oc October tober 1997 Find the values of x and y from the equations: x – 4y + 2 = 0 2x + y – 4 = 0 A. 11/7, -5/7 B. 14/9, 8/9 C. 4/9, 8/9 D. 3/2, 5/3 Board rd E Exam xam Oc October tober 1993 79. EE Boa Solve for the value of x. 2x – y + z = 6 x – 3y – 2z = 13 2x – 3y – 3z = 16 A. 4 B. 3 C. 2 D. 1 Board rd E Exam xam A April pril 19 1997 97 80. EE Boa Multiply (2x + 5y)(5x – 2y) A. 10x 2 – 21xy + 10y 2 B. -10x2 + 21xy + 10y2 C. 10x2 + 21xy – 10y 2 D. -10x2 – 21xy – 10y2 81. EE Boa Board rd E Exam xam Ma March rch 1998 Determine the sum of the positive valued solution to the simultaneous equations: xy = 15, yz = 35; xz = 21. A. 15 B. 13 C. 17 D. 19 82. EE Boa Board rd E Exam xam Oc October tober 1997 If f(x) = 2x 2 + 2x + 4, what is f(2)? A. 4x + 2

16 x2 + x + 2 8

83. EE Boa Board rd E Exam xam A April pril 19 1997 97 If n is any positive integer, then (n – 1)(n – 2)(n – 3) … (3)(2)(1) = A. e(n – 1) B. (n – 1)! C. n! D. (n – 1)n March h 84. EE Board Exam April 1996, Marc 1998 The polynomial x3 + 4x2 – 3x + 8 is divided by x – 5, the remainder is A. 175 B. 140 C. 218 D. 200 Board rd E Exam xam Oc October tober 1993 85. EE Boa In the equation x2 + x = 0, one root is equal to A. 1 B. 5 C. 1/4 D. none of the above 86. EE Boa Board rd E Exam xam Oc October tober 1997 Find the values of x in the equation 24x 2 + 5x – 1 = 0 A. (1/6, 1) B. (1/6, 1/5) C. (1/2, 1/5) D. (1/8, -1/3) 87. EE Boa Board rd E Exam xam Oc October tober 1990 Determine k so that the equation 4x2 + kx + 1 = 0 will have just one real solution. A. 3 B. 4 C. 5 D. 6 88. EE Boa Board rd E Exam xam Oc October tober 1992 Given: log 6 + x log 4 = log 4 + log (32 + 4x). Find x. A. 2 B. 3 C. 4 D. 6 Board rd E Exam xam A April pril 19 1997 97 89. EE Boa The sum of Kim’s and Kevin’s ages is 18. In 3 years, Kim will be twice as old as Kevin. What are their ages now? A. 4, 14 B. 5, 13 C. 7, 11 D. 6, 12

90. EE Boa Board rd Ex Exam am Ap April ril 19 1996 96 A and B can do a piece of work in 42 days, B and C in 31 days and C and A in 20 days. In how many days can all of them do the work together? A. 19 B. 17 C. 21 D. 15

96. EE Boa Board rd E Exam xam A April pril 19 1993 93 If eight is added to the product of nine and the numerical number, the sum is seventy-one. Find the unknown number. A. 5 B. 6 C. 7 D. 8

Board rd E Exam xam Oc October tober 1997 91. EE Boa Ten liters of 25% salt solution and 15 liters of 35% salt solution are poured into a drum originally containing 30 liters of 10% salt solution. What is the per cent concentration of salt in the mixture? A. 19.55% B. 22.15% C. 27.05% D. 25.72%

Board rd E Exam xam A April pril 19 1997 97 97. EE Boa A train, an hour after starting, meets with an accident which detains it an hour after which it proceeds at 3/5 of its former rate and arrives three hour after time; but had the accident happened 50 miles farther on the line, it would have arrived one and one-half sooner. Find the length of the journey. A. 910/9 miles B. 800/9 miles C. 920/9 miles D. 850/9 miles

92. EE Boa Board rd Ex Exam am Oc October tober 1994 If a two digit number has x for its unit’s digit and y for its ten’s digit, represent the number. A. 10x + y B. 10y + x C. xy D. none of these 93. EE Boa Board rd Ex Exam am Oc October tober 1994 One number is 5 less than the other. If their sum is 135, what are the numbers? A. 85, 50 B. 80, 55 C. 70, 65 D. 75, 60 94. EE Boa Board rd Ex Exam am Ap April ril 19 1997 97 A jogger starts a course at a steady state of 8 kph. Five minutes later, a second jogger starts the same course at 10 kph. How long will it take the second jogger to catch the first? A. 20 min B. 21 min C. 22 min D. 18 min 95. EE Boa Board rd Ex Exam am Ap April ril 19 1997 97 A boat man rows to a place 4.8 miles with the stream and back in 14 hours, but finds that he can row 14 miles with the stream in the same time as 3 miles against the stream. Find the rate of the stream. A. 1.5 miles per hour B. 1 mile per hour C. 0.8 mile per hour D. 0.6 mile per hour

98. EE Boa Board rd E Exam xam Oc October tober 1990 A man left his home at past 3 o’clock PM as indicated in his wall clock, between 2 to 3 hours after, he returns home and noticed the hands of the clock interchanged. At what time did the man leave his home? A. 3:31.47 B. 3:21.45 C. 3:46.10 D. 3:36.50 Board rd E Exam xam A April pril 19 1990 90 99. EE Boa A storage battery discharges at a rate which is proportional to the charge. If the charge is reduced by 50% of its original value at the end of 2 days,...