Advanced Engineering Math Exam Problems

Past Board Exam Problems D. j2
in Advanced Engineering Mathematics 14. EE Board Exam October 1997
6. EE Board Exam April 1997 If A = -2 – j3 and B = 3 + j4, what is
Write in the form a + jb the A/B?
1. CE Board Exam May 1994 expression j3217 – j427 + j18 18 - j
The expression 3 + j4 is a complex A. 1 + j2 A.
25
number. Compute its absolute value. B. 1 – j
-18 - j
A. 4 C. -1 + j2 B. answer
B. 5 D. 1 + j 25
C. 6 -18 + j
D. 7 7. EE Board Exam October 1993 C.
25
Write the polar form of the vector 3 + 18 + j
2. CE Board Exam November 1996 j4. D.
Compute the value of x by A. 6 cis 53.1 deg 25
determinant B. 10 cis 53.1 deg
C. 5 cis 53.1 deg 15. EE Board Exam October 1997
4 - 1 2 3
D. 8 cis 53.1 deg 4 + j3
2 0 2 1 Rationalize
x= 2− j
10 3 0 1 8. EE Board Exam April 1995 A. 1 + j2
14 2 4 5 Simplify (3 – j)2 – 7(3 – j) + 10. 11 + j10
A. -(3 + j) B.
A. -32 5
B. 3 + j
B. -28 5 + j2
C. 3 – j C.
C. 16
D. -(3 – j) 5
D. 52
D. 2 + j2
9. EE Board Exam April 1996
3. CE Board Exam November 1997
If A = 40ej120°, B = 20 cis(-40), C = 16. EE Board Exam October 1997
Given the matrix equation, solve for x
26.46 + j0, solve for A + B + C. (2 + j3)(5 − j)
and y. Simplify
A. 27.7 cis(45°)
 1 1  x  2  B. 35.1 cis(45°) (3 − j2)2
3 2   y  =  0  C. 30.8 cis(45°) A. (221 – j91)/169
    
D. 33.4 cis(45°) B. (21 + j52)/13
A. -4, 6
C. (-7 + j17)/13
B. -4, 2
10. EE Board Exam October 1997 D. (-90 + j220)/169
C. -4, -2
D. -4, -6 What is j4 cube times j2 square?
A. -j8 17. EE Board Exam April 1996
B. j8 What is the simplified expression of
4. CE Board Exam May 1996
C. -8 6 + j2.5
1 2 D. -j28 the complex number ?
Element of matrix B = 3 + j4
0 - 5
A. -0.32 + j0.66
11. EE Board Exam April 1997 B. 1.12 – j0.66
3 6
Element of matrix C = What is the simplified complex
4 1 C. 0.32 - j0.66
expression of (4.33 + j2.5) square?
D. -1.75 + j1.03
Find the elements of the product of A. 12.5 + j21.65
the two matrices, matrix BC. B. 20 + j20
18. EE Board Exam April 1997
11 8 C. 15 + j20
Perform the operation: 4(cos 60° + j
A. answer D. 21.65 + j12.5
- 20 - 5 sin 60°) divided by 2(cos 30° + j sin
30°)] in rectangular coordinates.
- 11 8 12. EE Board Exam November 1997
B. A. square root of 3 – j2
Find the principal 5th root of [50(cos
19 5 B. square root of 3 – j
150° + j sin 150°)].
C. square root of 3 + j
- 10 9 A. 1.9 + j1.1
C. D. square root of 3 + j2
B. 3.26 – j2.1
- 19 6
C. 2.87 + j2.1
19. EE Board Exam June 1990
- 11 9 D. 2.25 – j1.2
D. 50 + j35
- 20 - 4 Find the quotient of .
13. EE Board Exam October 1997 8 + j5
What is the quotient when 4 + j8 is A. 6.47 cis (3°)
5. EE Board Exam April 1997 divided by j3? B. 4.47 cis (3°)
Simplify: j29 + j21 + j A. 8 – j4 C. 7.47 cis (30°)
A. j3 B. 8 + j4 D. 2.47 cis (53°)
B. 1 – j C. -8 + j4
C. 1 + j D. -8 – j4

, One term of a Fourier series in 2 3
20. EE Board Exam March 1998 cosine form is 10cos 40πt. Write it in A.
Three vectors A, B and C are related 0 5
exponential form.
as follows: A/B = 2 at180°, A + C = -5 A. 5ej40πt 2 3
+ j15, C = conjugate of B. Find A. B. 5ej40πt + 5e-j40πt B. - answer
0 5
A. 5 – j5 C. 10e-j40πt
B. -10 + j10 D. 10ej40πt 1 7
C. 10 – j10 C. -
2 0
D. 15 + j15 27. EE Board Exam April 1997
Evaluate the determinant 3 1
D.
21. EE Board Exam April 1999 1 2 3 5 7
 π - 2 - 1 - 2
Evaluate cosh j 
 4 3 1 4 32. EE Board Exam October 1997
A. 0.707 3 1 2
A. 4
B. 1.41 + j0.866
B. 2 If A =  −2 −1 0  , what is the
C. 0.5 + j0.707
C. 5  0 2 −1
D. j0.707
D. 0 cofactor with the first row, second
22. EE Board Exam April 1999 column element?
28. EE Board Exam April 1997
 π Evaluate the determinant 3 2 
Evaluate tanh j  A. −  
 3 2 14 3 1 0 −1
1 5 - 1 3  −2 −1
A. 0.5 + j1.732 B. 0 2
B. j0.866 1 - 2 2 - 3  
C. j1.732 3 - 4 - 3 - 4 3 2 
D. 0.5 + j0.866 C. 0 −1
A. 489  
B. 389
23. EE Board Exam April 1999 C. 326  −2 0 
Evaluate ln (2 + j3). D. −  answer
D. 452  0 −1
A. 1.34 + j0.32
B. 2.54 + j0.866 29. EE Board Exam April 1997
C. 2.23 + j0.21 33. EE Board Exam October 1997
Given the equations: If a 3 x 3 matrix and its inverse are
D. 1.28 + j0.98 x+y+z=2 multiplied, write the product.
3x – y – 2z = 4
24. EE Board Exam October 1997 5x – 2y + 3z = -7 1 0 0
Evaluate the terms at t = 1 of the Solve for y by determinants A. 0 1 0  answer
Fourier series 2ej10πt + 2e-j10πt A. 1 0 0 1
A. 2 + j B. -2
B. 2 C. 3 0 0 0 
C. 4 D. 0 0 0 0 
B.  
D. 2 + j2
0 0 0 
30. EE Board Exam April 1997
25. EE Board Exam March 1998 Solve the equations by Cramer’s 0 0 1
Given the following series: Rule 0 1 0 
C.  
x3 x5 2x – y + 3z = -3
sin x = x - + + ....  1 0 0 
3! 5! 3x + 3y – z = 10
x2 x4 -x – y + z = -4 1 1 1
cos x = 1- + + .... A. (2, 1, -1) 1 1 1
2! 4! D.  
B. (2, -1, 1)
x2 x3 C. (1, 2, -1) 1 1 1
e x = 1+ x + + + ....
2! 3! D. (-1, -2, 1)
What relation can you draw from 34. EE Board Exam April 1996
these series? 31. EE Board Exam October 1997
 1 −1 2 
A. ex = cos x + sin x 2 3 1
If matrix  2 1 3  is multiplied by
B. ejx = cos x + jsin x If A = - 1 2 4 , what is cofactor of
C. ejx = jcos x + sin x 0 −1 1
D. jex = icos x + jsin x 0 5 7
x x
the second row, third column  y  is equal to zero, then matrix  y 
26. EE Board Exam October 1997 element?    
 z   z 
is

, A. 3 3 1 2 C. 1+j
B. 1  1 2 −1 D. 4(1 + j)
C.  
C. 0
D. -2  −2 −1 0 
45. ECE Board Exam November 1991
1 3 2
 −1 −2 0  Evaluate the determinant
35. EE Board Exam October 1997 D.   1 6 0
Given:  2 2 −1
4 2 7
4 5 0 1 0 0
0 5 3
A= 6 7 3 B= 0 1 0 , 38. EE Board Exam April 1997
What is the inverse Laplace A. 110
1 2 5 0 0 1 B. -101
transform of k divided by [(s square)
What is A times B equal to? + (k square)]? C. 101
4 0 0 A. cos kt D. -110
A. 0 7 0 B. sin kt
C. (e exponent kt) 46. ME Board Exam April 1997
0 0 5 D. 1.00 Evaluate the value of −10
0 0 0 multiplied by −7 .
39. EE Board Exam April 1995, April
B. 0 7 0 A. j
1997
1 0 0 The Laplace transform of cos wt is B. 70 answer
6 7 0 A. s/[(s square) + (w square]
C. - 70
B. w/[(s square) + (w square]
C. 8 9 4 C. w/(s + w) D. 17
2 3 5 D. s/(s + w)
4 5 0 47.
40. EE Board Exam April 1997
D. 6 7 3 answer Find the Laplace transform of 2/(s + A.
1 2 5 1) – 4/(s + 3). B.
A. 2 e(exp -t) – 4 e(exp -3t) C.
B. e(exp -2t) + e(exp -3t) D.
36. EE Board Exam April 1997 C. e(exp -2t) – e(exp -3t)
2 1 - 1 2 D. [2 e(exp -t)][1 – 2 e(exp -3t)] 48.
Matrix + 2 Matrix =
- 1 3 1 1
41. EE Board Exam March 1998 A.
- 2 4 Determine the inverse Laplace
A. Matrix B.
2 2 200 C.
transform of I(s) = 2
- 1 2 s − 50s + 10625 D.
B. Matrix A. i(t) = 2e-25t sin 100t
1 1 49.
B. i(t) = 2te-25t sin 100t
2 1 C. i(t) = 2e-25t cos 100t
C. Matrix D. i(t) = 2te-25t cos 100t A.
- 1 3 B.
0 5 42. EE Board Exam April 1997 C.
D. Matrix answer D.
1 5 The inverse Laplace transform of
s/[(s square) + (w square)] is
A. sin wt 50.
B. w
37. EE Board Exam October 1997 A.
C. e exponent wt
3 1 2 D. cos wt B.
Transpose the matrix  −2 −1 0  C.
43. ECE Board Exam April 1999 D.
 0 2 −1
Simplify the expression j1997 + j1999.
 −1 2 0  A. 0 Past Board Exam Problems in Algebra
A.  0 −1 −2  B. -j
  C. 1 + j
 2 1 3  51. CE Board Exam May 1997
D. 1 – j
Find the value of w in the following
 3 −2 0 
 1 −1 2  answer 44. ECE Board Exam November 1998 equations
B.   3x – 2y + w = 11
Find the value of (1 + j)5
2 0 −1 A. 1 – j x + 5y – 2w = -9
B. -4(1 + j) 2x + y – 3w = -6
A. 3

, B. 2 Find the value of log8 48. 64. CE Board Exam November 1995
C. 4 A. 1.86 In how many minutes after 7 o’clock
D. -2 B. 1.68 will the hands be directly opposite
C. 1.78 each other for the first time?
52. CE Board Exam May 1996 D. 1.98 A. 5.22 minutes
Find the value of A in the equation: B. 5.33 minutes
x 2 + 4x + 10 A B(2x + 2) C 58. CE Board Exam November 1997 C. 5.46 minutes
3 2
= + 2 + 2 Evaluate the log6 845 = x. D. 5.54 minutes
x + 2x + 5x x x + 2x + 5 x + 2x + 5 A. 3.76
B. 5.84 65. CE Board Exam May 1997
A. -2 C. 4.48 What time after 3 o’clock will the
B. 1/2 D. 2.98 hands of the clock be together for the
C. -1/2 first time?
D. 2 59. CE Board Exam November 1992, A. 3:02.30
May 1994 B. 3:17.37
53. CE Board Exam November 1991 If loga 10 = 0.25, what is the value of C. 3:14.32
Solve for x in the given equation. log10 a? D. 3:16.36
4 3 A. 2
8 2 8x = 2 B. 4 66. CE Board Exam May 1993, April
A. 4 C. 6 2004
B. 2 D. 8 Given that “w” varies directly as the
C. 3 product of “x” and “y” and inversely
D. 5 60. CE Board Exam November 1993 as the square of “z” and that w = 4
It takes Butch twice as long as it when x = 2, y = 6 and z = 3. Find the
54. CE Board Exam November 1997 takes Dan to do a certain piece of value of “w” when x = 1, y = 4 and z =
Find the remainder if we divide 4y3 + work. Working together they can do 2.
18y2 + 8y – 4 by 2y + 3. the work in 6 days. How long would it A. 3
A. 10 take Dan to do it alone? B. 4
B. 11 A. 9 days C. 5
C. 15 B. 10 days D. 6
D. 13 C. 11 days
D. 12 days 67. CE Board Exam May 1993, May
55. CE Board Exam November 1993 1994, November 1994
A 400-mm φ pipe can fill a tank alone 61. CE Board Exam November 1994 How many terms of the progression
in 5 hours and another 600-mm φ An airplane flying with the wind, took 3, 5, 7, … must be taken in order that
pipe can fill the tank alone in 4 hours. 2 hrs to travel 1000 km and 2.5 hrs in their sum will be 2600?
A drain pipe 300-mm φ can empty flying back. What was the wind A. 48
the tank in 20 hours. With all the velocity in kph? B. 49
three pipes open, how long will it take A. 50 kph C. 50
to fill the tank? B. 60 kph D. 51
A. 2.00 hours C. 70 kph
B. 2.50 hours D. 40 kph 68. CE Board Exam May 1995
C. 2.25 hours What is the sum of the progression 4,
D. 2.75 hours 62. CE Board Exam May 1998 9, 14, 19… up to the 20th term?
A boat travels downstream in 2/3 of A. 1030
56. CE Board Exam November 1996 the time as it goes going upstream. If B. 1035
Find the 6th term of the expansion of the velocity of the river’s current is 8 C. 1040
æ1 ö16 kph, determine the velocity of the D. 1045
çç - 3÷
÷ boat in still water.
èç2a ÷
ø A. 40 kph 69. CE Board Exam May 1998
66939 B. 50 kph Determine the sum of the
A. -
C. 30 kph progression if there are 7 arithmetic
256a11
D. 60 kph means between 3 and 35.
66339 A. 171
B. - answer
128a11 63. CE Board Exam May 1995 B. 182
33669 In how many minutes after 2 o’clock C. 232
C. - will the hands of the clock extend in D. 216
256a11
opposite directions for the first time?
39396 A. 42.4 minutes 70. CE Board Exam May 1991
D. -
128a11 B. 42.8 minutes In the “Gulf War” in the Middle East,
C. 43.2 minutes the allied forces captures 6400 of
57. CE Board Exam November 1993, D. 43.6 minutes Saddam’s soldiers and with
ECE November 1993 provisions on hand it will last for 216