D.
Past Board Exam Problems
in Advanced Engineering Mathematics
1.
2.
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
determinant
4 - 1 2 3
x=
A.
B.
C.
D.
3.
4.
2
10
14
0
3
2
EE Board Exam 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 Exam 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 Exam 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 Exam 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
CE Board Exam November 1997
Given the matrix equation, solve for x
and y.
 1 1  x  2 
3 2   y  =  0 

   
A. -4, 6
B. -4, 2
C. -4, -2
D. -4, -6
CE Board Exam May 1996
1 2
Element of matrix B =
0 - 5
3 6
4 1
Find the elements of the product of
the two matrices, matrix BC.
11
8
A.
answer
- 20 - 5
Element of matrix C =
5.
6.
by
2 1
0 1
4 5
B.
- 11 8
19 5
C.
- 10 9
- 19 6
D.
- 11 9
- 20 - 4
EE Board Exam April 1997
Simplify: j29 + j21 + j
A. j3
B. 1 – j
C. 1 + j
j2
10. EE Board Exam October 1997
What is j4 cube times j2 square?
A. -j8
B. j8
C. -8
D. -j28
11. EE Board Exam April 1997
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 1997
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 1997
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 1997
If A = -2 – j3 and B = 3 + j4, what is
A/B?
18 - j
A.
25
-18 - j
B.
answer
25
-18 + j
C.
25
18 + j
D.
25
15. EE Board Exam October 1997
4 + j3
Rationalize
2− j
A. 1 + j2
11 + j10
B.
5
5 + j2
C.
5
D. 2 + j2
16. EE Board Exam October 1997
(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 Board Exam April 1996
What is the simplified expression of
6 + j2.5
?
the complex number
3 + j4
A. -0.32 + j0.66
B. 1.12 – j0.66
C. 0.32 - j0.66
D. -1.75 + j1.03
18. EE Board Exam April 1997
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
19. EE Board Exam June 1990
50 + j35
Find the quotient of
.
8 + j5
A. 6.47 cis (3°)
B. 4.47 cis (3°)
C. 7.47 cis (30°)
D. 2.47 cis (53°)
20. EE Board Exam March 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
21. EE Board Exam April 1999
 π
Evaluate cosh j 
 4
A. 0.707
B. 1.41 + j0.866
C. 0.5 + j0.707
D. j0.707
22. EE Board Exam April 1999
 π
Evaluate tanh j 
 3
A.
B.
C.
D.
0.5 + j1.732
j0.866
j1.732
0.5 + j0.866
23. EE Board Exam April 1999
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 Board Exam October 1997
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 Board Exam March 1998
Given the following series:
x3 x5
sin x = x +
+ ....
3! 5!
x2 x4
cos x = 1+
+ ....
2! 4!
x2 x3
e x = 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 Board Exam October 1997
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 Board Exam April 1997
Evaluate the determinant
1
2
3
5
- 1
3
2
- 3
-
C.
489
389
326
452
D.
29. EE Board Exam April 1997
Given the equations:
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
31. EE Board Exam October 1997
2 3 1
If A = - 1 2 4 , what is cofactor of
5 7
the second
element?
row,
third
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 Board Exam October 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
30. EE Board Exam April 1997
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)
0
C.
B.
3 - 4 - 3 - 4
A.
B.
C.
D.
-
cofactor with the first row, second
column element?
3 2 
A. − 

0 −1
28. EE Board Exam April 1997
Evaluate the determinant
2 14 3
1
1 - 2
0 5
32. EE Board Exam October 1997
3 1 2
If A =  −2 −1 0  , what is the
 0 2 −1
4
2
5
0
1
2 3
B.
D.
- 2 - 1 - 2
3
1
4
A.
B.
C.
D.
A.
column
B.
0 0 0 
0 0 0 


0 0 0 
C.
0 0 1
0 1 0 


 1 0 0 
D.
1 1 1
1 1 1


1 1 1
34. EE Board Exam April 1996
 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
35. EE Board Exam October 1997
Given:
4 5 0
1 0 0
B= 0 1 0 ,
0 0 1
A= 6 7 3
1 2 5
What is A times B equal to?
4 0 0
A. 0 7 0
0 0 5
0 0 0
B.
8 9 4
2 3 5
4 5 0
D.
6 7 3 answer
1 2 5
36. EE Board Exam April 1997
2 1
- 1 2
Matrix
+ 2 Matrix
=
- 1 3
1 1
A.
Matrix
B.
Matrix
C.
Matrix
D.
Matrix
- 2 4
2
3 1 2
 1 2 −1


 −2 −1 0 
D.
1 3 2
 −1 −2 0 


 2 2 −1
38. EE Board Exam April 1997
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.
2
- 1 2
1
1
2
1
- 1 3
0 5
1 5
answer
37. EE Board Exam October 1997
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
40. EE Board Exam April 1997
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 Board Exam March 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 Board Exam April 1997
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 Board Exam April 1999
Simplify the expression j1997 + j1999.
A. 0
B. -j
C. 1 + j
D. 1 – j
44. ECE Board Exam November 1998
Find the value of (1 + j)5
A. 1 – j
B. -4(1 + j)
1+j
4(1 + j)
45. ECE Board Exam November 1991
Evaluate the determinant
1 6 0
4 2 7
0 5 3
A.
B.
C.
D.
110
-101
101
-110
46. ME Board Exam April 1997
Evaluate the value of
−10
multiplied by −7 .
A. j
39. EE Board Exam April 1995, 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)
0 7 0
1 0 0
6 7 0
C.
C.
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 Board Exam Problems in Algebra
51. CE Board Exam May 1997
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
52. CE Board Exam May 1996
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
=
58. CE Board Exam November 1997
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 Board Exam November 1991
Solve for x in the given equation.
4 3
8 2 8x = 2
A.
B.
C.
D.
4
2
3
5
54. CE Board Exam November 1997
Find the remainder if we divide 4y3 +
18y2 + 8y – 4 by 2y + 3.
A. 10
B. 11
C. 15
D. 13
55. CE Board Exam November 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 Board Exam November 1996
Find the 6th term of the expansion of
æ1
ö16
çç - 3÷
÷
÷
èç2a
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
128a11
33669
answer
256a11
39396
128a11
57. CE Board Exam November 1993,
ECE November 1993
59. CE Board Exam November 1992,
May 1994
If loga 10 = 0.25, what is the value of
log10 a?
A. 2
B. 4
C. 6
D. 8
60. CE Board Exam November 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
61. CE Board Exam November 1994
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 Board Exam May 1998
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
63. CE Board Exam May 1995
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 Board Exam November 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
65. CE Board Exam May 1997
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 Board Exam May 1993, 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 Board Exam May 1993, May
1994, November 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 Board Exam May 1995
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 Board Exam May 1998
Determine
the
sum
of
the
progression if there are 7 arithmetic
means between 3 and 35.
A. 171
B. 182
C. 232
D. 216
70. CE Board Exam May 1991
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 Board Exam November 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
72. CE Board Exam May 1995
The numbers 28, x + 2, 112 form a
geometric progression. What is the
10th term?
A. 14336
B. 13463
C. 16433
D. 16344
73. CE Board Exam November 1994
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
74. CE Board Exam May 1998
Find the sum of 1, -1/5, 1/25, …
A. 5/6
B. 2/3
C. 0.84
D. 0.72
75. CE Board Exam May 1992
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.
B.
C.
D.
58 centavos
60 centavos
62 centavos
64 centavos
76. EE Board Exam October 1992
Find
the
value
of
x
x + 1 2x
+
= 47 - 2x
3
4
A. 16.47
B. 12.84
C. 18.27
D. 20.17
in
77. EE Board Exam October 1991
Find the value of x in the equations:
éA A ù
é3A 4A ù
ú= A
10 ê + ú= A 2 ê êx
ú
êx
y
y ú
ë
û
ë
û
A. 50/9
B. 80/9
C. 70/9
D. 60/9
78. EE Board Exam October 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
79. EE Board Exam October 1993
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
80. EE Board Exam April 1997
Multiply (2x + 5y)(5x – 2y)
A. 10x2 – 21xy + 10y2
B. -10x2 + 21xy + 10y2
C. 10x2 + 21xy – 10y2
D. -10x2 – 21xy – 10y2
81. EE Board Exam March 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 Board Exam October 1997
If f(x) = 2x2 + 2x + 4, what is f(2)?
A. 4x + 2
16
x2 + x + 2
8
83. EE Board Exam April 1997
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
84. EE Board Exam April 1996, March
1998
The polynomial x3 + 4x2 – 3x + 8 is
divided by x – 5, the remainder is
A. 175
B. 140
C. 218
D. 200
85. EE Board Exam October 1993
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 Board Exam October 1997
Find the values of x in the equation
24x2 + 5x – 1 = 0
A. (1/6, 1)
B. (1/6, 1/5)
C. (1/2, 1/5)
D. (1/8, -1/3)
87. EE Board Exam October 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 Board Exam October 1992
Given: log 6 + x log 4 = log 4 + log
(32 + 4x). Find x.
A. 2
B. 3
C. 4
D. 6
89. EE Board Exam April 1997
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 Board Exam April 1996
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 Board Exam April 1993
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
91. EE Board Exam October 1997
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%
97. EE Board Exam April 1997
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 Board Exam October 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 Board Exam October 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 Board Exam April 1997
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 Board Exam April 1997
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 Board Exam October 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
99. EE Board Exam April 1990
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,
how long will it take it reduce the
charge to 25% of its original charge?
A. 3
B. 4
C. 5
D. 6
100. EE Board Exam March 1998
The electric
power which a
transmission line can transmit is
proportional to the product of its
design voltage and current capacity,
and inversely to the transmission
distance. A 115-kilovolt line rated at
100 amperes can transmit 150
megawatts over 150 km. How much
power, in megawatts can a 230kilovolt line rated at 150 amperes
transmit over 100 km?
A. 785
B.
C.
D.
485
675
595
101. EE Board Exam March 1998
A bookstore purchased a best selling
price book at Php200.00 per copy. At
what price should this book be sold
so that, giving a 20% discount, the
profit is 30%?
A. Php450
B. Php500
C. Php357
D. Php400
102. EE Board Exam March 1998
In a certain community of 1,200
people, 60% are literate. Of the
males, 50% are literate and of the
females 70% are literate. What is the
female population?
A. 850
B. 500
C. 550
D. 600
103. EE Board Exam March 1998
Gravity causes a body to fall 16.1 ft in
the first second, 48.3 in the 2nd
second, 80.5 in the 3rd second. How
far did the body fall during the 10th
second?
A. 248.7 ft
B. 308.1 ft
C. 241.5 ft
D. 305.9 ft
104. EE Board Exam April 1997
A stack of bricks has 61 bricks in the
bottom layer, 58 bricks in the second
layer, 55 bricks in the third layer, and
so until there are 10 bricks in the last
layer. How many bricks are there all
together?
A. 638
B. 637
C. 639
D. 649
105. EE Board Exam April 1997
Once a month, a man puts some
money into the cookie jar. Each
month he puts 50 centavos more into
a jar than the month before. After 12
years, he counted his money, he had
Php5,436. How much money did he
put in the jar in the last month?
A. Php73.50
B. Php75.50
C. Php74.50
D. Php72.50
106. EE Board Exam April 1997
A girl on a bicycle coasts downhill
covers 4 feet the 1st second, 12 feet
the 2nd second, and in general, 8 feet
more each second than the previous
second. If she reaches the bottom at
the end of 14 seconds, how far did
she coasts?
A. 782 feet
B. 780 feet
C. 784 feet
D. 786 feet
112. EE Board Exam April 1997
If equal spheres are piled in the form
of a complete pyramid with an
equilateral triangle as base, find the
total number of spheres in the pile if
each side of the base contains 4
spheres.
A. 15
B. 20
C. 18
D. 21
107. EE Board Exam October 1991
The fourth term in the geometric
progression is 216 and the 6th term is
1944. Find the 8th term.
A. 17649
B. 17496
C. 16749
D. 17964
113. EE Board Exam October 1997
In the series 1, 1, 1/2, 1/6, 1/24, …,
determine the 6th term.
A. 1/80
B. 1/74
C. 1/100
D. 1/120
108. EE Board Exam April 1997
The seventh term is 56 and the
twelfth term is -1792 of a geometric
progression. Find the common ratio
and the first term. Assume the ratios
are equal.
A. -2, 5/8
B. -1, 5/8
C. -1, 7/8
D. -2, 7/8
109. EE Board Exam March 1998
Determine the sum of the infinite
series:
n
S=
A.
B.
C.
D.
1 1 1
 1
+ +
+L+  
3 9 27
3
4/5
3/4
2/3
1/2
110. EE Board Exam October 1994
A rubber ball is made to fall from a
height of 50 feet and is observed to
rebound 2/3 of the distance it falls.
How far will the ball travel before
coming to rest if the ball continues to
fall in this manner?
A. 200 feet
B. 225 feet
C. 250 feet
D. 275 feet
111. EE Board Exam April 1990
What is the fraction in lowest term
equivalent to 0.133133133133?
A. 133/666
B. 133/777
C. 133/888
D. 133/999
114. ECE Board Exam April 1999
If 16 is 4 more than 4x, find 5x – 1.
A. 14
B. 3
C. 12
D. 5
115. ECE Board Exam April 1991
Simplify
- 1
2
B.
5
C.
y2
x2
3
D.
y2
x2
118. ECE Board Exam April 1991
Simplify
7a+2 – 8(7)a+1 + 5(7)a + 49(7)a-2
A. -5a
B. -3a
C. -7a
D. -4a
119. ECE Board Exam April 1993
Solve
for
x
y
z
=
=
(b - c) (a - c) (a - b)
A. x – z
B. x + z
C. a + b
D. a – b
- 5
2
1/x2y7z5
1/x2y7z3
1/x2y5z7
1/x5y7z2
116. ECE Board Exam November 1993
Simplify the following equation
121. ECE Board Exam April 1993
Evaluate y =
5x
x+ 3
2x + 1
+
2x 2 + 7x + 3 2x 2 - 3x - 2 x 2 + x - 6
4
A.
answer
x+ 3
2
B.
x- 3
4
C.
x- 3
2
D.
x+ 3
117. ECE Board Exam April 1991
ïìï
ï 2
Simplify: ïí x 3
ïï
ïï
î
5
A.
é- 1 êx 3 y
ê
êë
y2
answer
x
1
2
1 ïü6
- 2 ù2 ïï
x 2 y- 2 3 ú ï
(
)
y:
120. ECE Board Exam April 1993
Solve for x in the following equations:
27x = 9y
81y3-x = 243
A. 1
B. 1.5
C. 2
D. 2.5
(x 2 y 3 z- 2 )(x- 3 yz- 3 )
(xyz- 3 )
A.
B.
C.
D.
3
y2
x
ú
úû
ý
ïï
ïï
þ
A.
B.
C.
D.
4(5 2n+ 1) - 10(52n- 1)
2(52n )
y = 5n
y=9
y = 52n
y = 18
122. ECE Board Exam April 1990
Given:
(an )(am ) = 100,000
anm = 1,000,000
an
am
Find a
A. 12
B. 9
C. 11
D. 10
= 10
123. ECE Board Exam November 1991
Give the factors of a2 – x2
A. 2a – 2x
B.
C.
D.
(a + x)(a – x)
(a + x)(a + x)
2x – 2a
124. ECE Board Exam November 1990
(a – b)3 = ?
A. a3 – 3a2b + 3ab2 + b3
B. a3 – 3a2b – 3ab2 – b3
C. a3 + 3a2b + 3ab2 – b3
D. a3 – 3a2b + 3ab2 – b3
125. ECE Board Exam April 1998
What is the least common factor of
10 and 32?
A. 320
B. 2
C. 180
D. 90
126. ECE Board Exam April 1999
Given: f(x) = (x + 3)(x – 4) + 4. When
f(x) is divided by (x – k), the
remainder is k. Find k.
A. 2
B. 4
C. 6
D. 8
127. ECE Board Exam April 1999
Find the mean proportional of 4 and
36.
A. 72
B. 24
C. 12
D. 20
128. ECE Board Exam April 1998
The arithmetic mean of 80 numbers
is 55. If two numbers namely 250 and
850 are removed, what is the
arithmetic mean of the remaining
numbers?
A. 42.31
B. 50
C. 38.62
D. 57.12
129. ECE Board Exam April 1998
The arithmetic mean of 6 numbers is
17. If two numbers are added to the
progression, the new set of numbers
will have an arithmetic mean of 19.
What are the two numbers of their
difference is 4?
A. 21, 25
B. 23, 27
C. 8, 12
D. 16, 20
130. ECE Board Exam March 1996
The equation of whose roots are the
reciprocal of the roots of 2x2 – 3x – 5 =
0 is
A.
B.
C.
D.
5x2 + 3x – 2 = 0
2x2 + 3x – 5 = 0
3x2 – 3x + 2 =
2x2 + 5x – 3 = 0
131. ECE Board Exam April 1990
Solve for the value of “a” in the
equation a8 – 17a4 + 16 = 0.
A. ±2
B. ±3
C. ±4
D. ±5
132. ECE Board Exam April 1998
In the expression of (x + 4y)12, the
numerical coefficient of the 5th term
is
A. 63,360
B. 126,720
C. 506,880
D. 253,440
133. ECE Board Exam November 1995
What is the sum of the coefficients of
the expansion (2x – 1)20?
A. 0
B. 1
C. 2
D. 3
134. ECE Board Exam April 1995
What is the sum of the coefficients of
the expansion (x + y – z)8?
A. 0
B. 1
C. 2
D. 3
135. ECE Board Exam November 1990
Log (MN) is equal to
A. Log M – N
B. Log M + N
C. N Log M
D. Log M + Log N
136. ECE Board Exam April 1998
What is the value of log2 5 + log3 5?
A. 7.39
B. 3.79
C. 3.97
D. 9.37
137. ECE Board Exam November 1995
Given: logb y = 2x + logb x. Which of
the following is true?
A. y = b2x
B. y = 2xb
C. y = 2x/b
D. y = xb2x
138. ECE Board Exam November 1991
Given: logb 1024 = 5/2. Find b.
A. 2560
B.
C.
D.
16
4
2
139. ECE Board Exam April 1993
Solve for the value of x in the
following equation: x3log x = 100x
A. 12
B. 8
C. 30
D. 10
140. ECE Board Exam November 1998
If log of 2 to the base 2 plus log of x
to the base 2 is equal to 2, then the
value of x is:
A. 4
B. -2
C. 2
D. -1
141. ECE Board Exam April 1995, April
1999
Mary is 24 years old. Mary is twice as
old as Ann was when Mary was as
old as Ann is now. How old is Ann
now?
A. 16
B. 18
C. 12
D. 15
142. ECE Board Exam November 1995
Pedro can paint a fence 50% faster
than Juan and 20% faster than Pilar,
and together they can paint a given
fence in 4 hours. How long will it take
Pedro to paint the same fence if he
had to work alone?
A. 6
B. 8
C. 10
D. 12
143. ECE Board Exam April 1999
Mike, Louie and Joy can mow the
lawn in 4, 6 and 7 hours respectively.
What fraction of the yard can they
mow in 1 hour if they work together?
A. 47/84
B. 45/84
C. 84/47
D. 36/60
144. ECE Board Exam November 1991
Crew No. 1 can finish installation of
an antenna tower in 200 man-hours
while Crew No. 2 can finish the same
job in 300 man-hours. How long will it
take both crews to finish the same
job, working together?
A. 100 man-hours
B. 120 man-hours
C.
D.
140 man-hours
160 man-hours
145. ECE Board Exam March 1996
Ten less than four times a certain
number is 14. Determine the number.
A. 6
B. 7
C. 8
D. 9
146. ECE Board Exam March 1996
The sum of two numbers is 21 and
one number is twice the other. Find
the numbers.
A. 6, 15
B. 7, 14
C. 8, 13
D. 9, 12
147. ECE Board Exam November 1997
The denominator of a certain fraction
is three more than twice the
numerator. If 7 is added to both terms
of the fraction, the resulting fraction is
3/5. Find the original fraction.
A. 8/5
B. 13/5
C. 5/13
D. 3/5
148. ECE Board Exam November 1998
A man rows downstream at the rate
of 8 mph and upstream at the rate of
2 mph. How far downstream should
he go if he is to return in 7/4 hours
after leaving?
A. 2.5 miles
B. 3.3 miles
C. 3.1 miles
D. 2.7 miles
149. ECE Board Exam April 1990
The resistance of a wire varies
directly with its length and inversely
with its area. If a certain piece of wire
10 mm long and 0.10 cm in diameter
has a resistance of 100 ohms, what
will be its resistance be if it is
uniformly stretched so that its length
becomes 12 m?
A. 80
B. 90
C. 144
D. 120
150. ECE Board Exam November 1993
If x varies directly as y and inversely
as z, and x = 14 when y = 7 and z =
2, find the value of x when y = 16 and
z = 4.
A. 14
B. 4
C.
D.
16
6
151. ECE Board Exam November 1993
Jojo bought a second hand DVD
player and then sold it to Rudy at a
profit of 40%. Rudy then sold the
player to Noel at a profit of 20%. If
Noel paid Php2,856 more that it cost
to Jojo, how much did Jojo paid for
the unit?
A. Php4,000
B. Php4,100
C. Php4,200
D. Php4,300
152. ECE Board Exam March 1996
A merchant has three items on sale,
namely a radio for P 50, a clock for P
30 and a flashlight for P 1. At the end
of the day, he sold a total of 100 of
the three items and has taken exactly
P 1000 on the total sales. How many
radios did he sell?
A. 16
B. 20
C. 18
D. 24
153. ECE Board Exam November 1998
Find the 30th term in the arithmetic
progression 4, 7, 10, …
A. 75
B. 88
C. 90
D. 91
154. ECE Board Exam April 1995
A besiege fortress is held by 5700
men who have provisions for 66
days. If the garrison looses 20 men
each day, for how many days can the
provision hold out?
A. 72
B. 74
C. 76
D. 78
155. ECE Board Exam November 1995
Find the fourth term of the
progression 1/2, 0.2, 0.125 …
A. 1/10
B. 1/11
C. 0.102
D. 0.099
156. ECE Board Exam April 1999
Determine x so that: x, 2x + 7, 10x – 7
will be a geometric progression.
A. 7, -7/12
B. 7, -5/6
C. 7, -14/5
D. 7, -7/6
157. ECE Board Exam November 2001,
April 1999
If one third of the air in a tank is
removed by each stroke of an air
jump, what fractional part of the total
air is removed in 6 strokes?
A. 0.7122
B. 0.9122
C. 0.6122
D. 0.8122
158. ECE Board Exam April 1998
The sum of the first 10 terms of a
geometric progression 2, 4, 8, … is
A. 1023
B. 2046
C. 225
D. 1596
159. ECE Board Exam April 1998
Find the sum of the
progression 6, -2, 2/3 …
A. 9/2
B. 5/2
C. 7/2
D. 11/2
infinite
160. ECE Board Exam November 1998
Find the ratio of an infinite geometric
progression if the sum is 2 and the
first term is 1/2.
A. 1/3
B. 1/2
C. 3/4
D. 1/4
161. ECE Board Exam April 1998
Find the 1987th digit in the decimal
equivalent to 1785/9999 starting from
the decimal point.
A. 8
B. 1
C. 7
D. 5
162. ECE March 1996
For a particular experiment you need
5 liters of a 10% solution. You find
7% and 12% solution on the shelf.
How much of the 7% solution should
you mix with appropriate amount of
the 12% solution to get 5 liters of a
10% solution?
A. 1.5 liter
B. 2.5 liters
C. 2 liters
D. 3 liters
163. ECE November 1999
The sum of the digits of a two-digit
number is 11. If the digits are
reversed, the resulting number is
seven more than twice the original
number. What is the original number?
A. 38
B. 53
C. 83
D. 44
164. ECE November 1999
Find the sum of the roots of 5x2 – 10x
+ 2 = 0.
A. -1/2
B. -2
C. 2
D. 1/2
165. EE November 1999, November 2000
The time required for the two
examinees to solve the same
problem differs by two minutes.
Together they can solve 32 problems
in one hour. How long will it take for
the slower problem solver to solve
the same problem?
A. 5 minutes
B. 2 minutes
C. 3 minutes
D. 4 minutes
166. ECE November 2000
Find the value of m that will make 4x2
– 4mx + 4m + 5 is a perfect square
trinomial.
A. 3
B. -2
C. 4
D. 5
167. ECE April 2001
Ana is 5 years older than Beth. In 5
years, the product of their ages will
be 1.5 times the product of their
present ages. How old is Beth now?
A. 27
B. 20
C. 25
D. 18
168. ECE April 2001
Find the coefficients of the term
involving b4 in the expansion of (a2 –
2b)10.
A. -3360
B. 10!
C. -960
D. 3360
169. ECE April 2001
One pipe can fill a tank in 6 hours
and another pipe can fill the same
tank in 3 hours. A drain pipe can
empty the tank in 24 hours. With all
tanks open, how long in hours will it
take to fill the tank?
A.
B.
C.
D.
2.228 hours
2.812 hours
2.322 hours
2.182 hours
170. ECE April 2001
The seating section in a Coliseum
has 30 seats in the first row, 32 seats
in the second row, 34 seats in the
third row, and so on, until the tenth
row is reached, after which there are
ten rows, each containing 50 seats.
Find the total number of seats in the
section.
A. 900
B. 910
C. 890
D. 1000
171. ECE November 2001
A piece of paper is 0.05 inches thick.
Each time the paper is folded into
half, the thickness is doubled. If the
paper was folded 12 times, how thick
in feet the folded paper be?
A. 10.24
B. 12.34
C. 17.10
D. 11.25
172. ECE November 2001
It takes an airplane one hour and
forty-five minutes to travel 500 miles
against the wind and covers the
same distance in one hour and fifteen
minutes with the wind. What is the
speed of the airplane?
A. 342.85 mph
B. 375.50 mph
C. 450.50 mph
D. 285.75 mph
173. ECE November 2002, April 2004
At exactly what time after 2 o’clock
will the hour hand and the minute
hand extend in opposite directions for
the first time?
A. 2:43 and 0.64 sec
B. 2:43 and 6.30 sec
C. 2:43 and 40.5 sec
D. 2:43 and 37.8 sec
174. ECE November 2002
The sum of the ages of Peter and
Paul is 21. Peter will be twice as old
as Paul 3 years form now. What is
the present age of Peter?
A. 8
B. 6
C. 18
D. 15
175. ECE November 2002
A multimillionaire left his entire estate
to his wife, daughter, son and
bodyguard. His daughter and son got
half the total value of the estate in the
ratio 3:2. His wife got twice value as
much as the share of the son. If the
bodyguard received half a million
pesos, what is the total value of the
estate?
A. 6.5 million
B. 5 million
C. 7 million
D. 6 million
176. ECE November 2002, April 2004
A speed boat can make a trip of 100
miles in one hour and 30 minutes if it
travels upstream. If it travels
downstream, it will take one hour and
fifteen minutes to travel the same
distance. What is the speed of the
boat in calm water?
A. 193.45 mph
B. 73.33 mph
C. 146.67 mph
D. 293.33 mph
177. ECE April 2003
Simplify the expression: the square
root of the cube root of 64x60.
A. 4x4
B. 8x2
C. 2x6
D. 2x10
178. ECE April 2003
A man can do a job three times as
fast as a boy. Working together it
would take them 6 hours to do the
same job. How long will it take the
man to do the job alone?
A. 9 hours
B. 8 hours
C. 7 hours
D. 10 hours
179. ECE April 2003
A company sells 80 units and makes
P 80 profit. Its sells 110 units and
makes P 140 profit. If the profit is a
linear function of the number of units
sold, what is the average profit per
unit if the company sells 250 units?
A. P 1.76
B. P 1.68
C. P 1.66
D. P 1.86
180. ECE November 2003
At approximately what time after 12
o’clock will the hour hand and the
minute hands of a clock form an
angle of 120
time?
A. 50 min
o’clock
B. 43 min
o’clock
C. 38 min
o’clock
D. 30 min
o’clock
degrees for the second
and 30 sec after 12
and 38 sec after 12
and 35 sec after 12
186. ECE April 2003
What are the first four terms of the
sequence whose general term is n2 +
1?
A. 1, 4, 9, 16
B. 2, 5, 10, 17
C. 5, 10, 17, 26
D. 2, 4, 6, 10
and 45 sec after 12
181. ECE November 2003
A company sells 80 units and makes
P 80 profit. It sells 110 units and
makes P 140 profit. If the profit is a
linear function of the number of units
sold, what is the average profit per
unit of the company sells 250 units?
A. P 1.76
B. P 1.68
C. P 1.66
D. P 1.86
182. ECE November 2003
What is the remainder when the
polynomial x4 – 5x3 + 5x2 + 7x + 6 id
divided by x + 2?
A. 16
B. 32
C. 48
D. 68
183. ECE November 2003
Harry is one-third as old as Ron and
8 years younger than Hermione. If
Harry is 8 years old, what is the sum
of their ages?
A. 40
B. 45
C. 48
D. 50
184. ECE November 2003
The sum of the three consecutive
even integers is 78. What is the
largest number?
A. 24
B. 28
C. 32
D. 30
185. ECE November 2003
An iron bar four meters long has a
300 pound weight hung on one end
and a 200 pound weight hung at the
opposite end. How far from the 300
pound weight should the fulcrum be
located to balance the bar?
A. 2.5 meters
B. 1.0 meter
C. 1.6 meters
D. 2.0 meters
187. ECE April 2004
Peter can do a job 50 percent faster
than Paul and 20 percent faster than
John. If they work together, they can
finish the job in 4 days. How many
days will it take Peter to finish the job
if he is to work alone?
A. 18
B. 10
C. 12
D. 16
188. ECE April 2004
Solve for x in the following equation:
x + 4x + 7x + 10x + … + 64x = 1430.
A. 4
B. 3
C. 2
D. 1
189. ECE April 2004
If kx3 – (k + 3)x2 + 13 is divided by x –
4, and the remainder is 157, then the
value of k is
A. 6
B. 4
C. 5
D. 3
190. ECE April 2004
If 16 is four more than 3x, then x2 + 5
= ____?
A. 16
B. 21
C. 3
D. 4
191. ECE April 2001, November 2002
Four positive integers form an
arithmetic progression. If the product
of the 1st and the last term is 70 and
the 2nd and the third term is 88, find
the 1st term.
A. 5
B. 3
C. 14
D. 8
192. ECE April 2004
A professional organization is
composed of x ECEs and 2x EEs. If 6
ECEs are replaced by 6 EEs, 1/6 of
the members will be ECEs. Solve for
x.
A.
B.
C.
D.
12
24
36
1
193. ECE April 2004
The average rate of production of
printed circuit board (PCB) is 1 unit
for every 2 hours work by two
workers. How many PCB’s can be
produced in one month by 60 workers
working 200 hours during the month?
A. 4000
B. 5000
C. 6000
D. 3000
194. ECE November 2004
What is the sum of all even integers
from 10 to 500?
A. 87,950
B. 124,950
C. 62,730
D. 65,955
195. ECE April 2005
From the equation 7x2 + (2k – 1)x – 3k
+ 2 = 0, determine the value of k so
that the sum and product of the roots
are equal.
A. 2
B. 4
C. 1
D. 3
196. ECE April 2005
What is the equation form of the
statement: “The amount by which
100 exceeds four times a given
number?”
A. 4x(100)
B. 100 + 4x
C. 100 – 4x
D. 4x – 100
197. ECE April 2005
Candle A and candle B of equal
length are lighted at the same time
and burning until candle A is twice as
long as candle B. Candle A is
designed to fully burn in 8 hours
while candle B for 4 hours. How long
will they be lighted?
A. 3 hours and 30 minutes
B. 2 hours and 40 minutes
C. 3 hours
D. 2 hours
198. ECE April 2005
Solve for x if 8^x = 2^(y + 2) and
16^(3x – y) = 4^y.
A. 2
B. 4
C.
D.
1
3
199. ECE April 2005
What is the sum of all odd integers
between 10 and 500?
A. 87,950
B. 124,950
C. 62,475
D. 65,955
200. ECE April 2005
How many terms of the progression
3, 5, 7, should there be so that their
sum was 2600.
A. 60
B. 50
C. 52
D. 55
201. ECE April 2005
If the 1st term of the geometric
progression is 27 and the 4th term is 1, the third term is
A. 3
B. 2
C. -3
D. -2
202. GE Board Exam February 1991
The product of 1/4 and 1/5 of a
number is 500. What is the number?
A. 50
B. 75
C. 100
D. 125
203. GE Board Exam February 1997
At what after 12:00 noon will the hour
hand and minute hand of the clock
first form an angle of 120 degrees?
A. 12:18.818
B. 12:21.818
C. 12:22.818
D. 12:24.818
204. GE Board Exam February 1994
From the time 6:15 pm to the time
7:45 pm of the same day, the minute
hand of a standard clock describes
an arc of
A. 60°
B. 90°
C. 180°
D. 540°
205. GE Board Exam July 1993
A Geodetic Engineering student got a
score of 30% on Test 1 of the five
number test in Surveying. On the last
number he got 90% in which a
constant difference more on each
number that he had on the
immediately preceding one. What
was his average score in Surveying?
A. 50%
B. 55%
C. 60%
D. 65%
The value of (3 to 2.5 power) square
is equal to
A. 729
B. 140
C. 243
D. 81
206. GE Board Exam February 1994
Robert is 15 years older than his
brother Stan. However “y” years ago,
Robert was twice as old as Stan. If
Stan is now “b” years old and b > y,
find the value of (b – y).
A. 15
B. 16
C. 17
D. 18
212. ME Board Exam April 1996
Factor the expression x2 + 6x + 8 as
completely as possible
A. (x + 4)(x + 2)
B. (x – 4)(x + 2)
C. (x – 4)(x – 2)
D. (x + 6)(x + 2)
207. ME Board Exam October 1995
Solve for the value of x and y.
4x + 2y = 5
13x – 3y = 2
A. y = 1/2, x = 3/2
B. y = 3/2, x = 1/2
C. y = 2, x = 1
D. y = 3, x = 1
208. ME Board Exam October 1996
Solve the simultaneous equations
2x2 – 3y2 = 6
3x2 + 2y2 = 35
A. x = 3 or -3; y = 2 or -2
B. x = 3 or -3; y = -2 or 1
C. x = 3 or -3; y = -2 or -1
D. x = 3 or -3; y = 2 or -3
209. ME Board Exam October 1996
Solve for the simultaneous equations:
x + y = -4
x+z–1=0
y+z+1=0
A. x = -1, y = -5, z = 3
B. x = 1, y = 2, z = -3
C. x = -1, y = -3, z = 2
D. x = -2, y = -3. z = -1
210. ME Board Exam October 1996
x- 2
Resolve
into partial
x 2 - 7x + 12
fraction.
6
2
A.
x- 4 x- 3
3
5
B.
x- 4 x- 3
6
5
C.
answer
x- 4 x- 3
7
5
D.
x- 4 x- 3
211. ME Board Exam October 1996
213. ME Board Exam April 1995
Factor the expression 3x3 – 3x2 – 18x
A. 3x(x – 3)(x + 2)
B. 3x(x + 3)(x + 2)
C. 3x(x + 3)(x – 2)
D. 3x(x – 3)(x – 2)
214. ME Board Exam April 1995
Simplify bm/n.
A.
bm
n
B.
bm+ n
C.
n m
D.
b
answer
m
b
n
215. ME Board Exam April 1998
Find the value of x which will satisfy
the following expression:
x- 2 = x + 2
A. 3/2
B. 9/4
C. 18/6
D. none of these
216. ME Board Exam April 1996
If x to the 3/4 power equals 8, x
equals
A. -9
B. 6
C. 9
D. 16
217. ME Board Exam October 1996
Solve for x that satisfies the equation
6x2 – 7x – 5 = 0
A. 5/3 or -1/2
B. 3/2 or 3/8
C. 7/5 or -7/15
D. 3/5 or 3/4
218. ME Board Exam April 1996
Solve for x: 10x2 + 10x + 1 = 0
A. -0.113, -0.887
B. -0.331, -0.788
C.
D.
-0.113, -0.788
-0.311, -0.887
219. ME Board Exam April 1997
What is the value of log to the base
10 of 10003.3?
A. 10.9
B. 99.9
C. 9.9
D. 9.5
220. ME Board Exam October 1996
Which value is equal to log to the
base e of e to the -7x power?
A. -7x
B. 10 to the -7x power
C. 7
D. -7 log to the base 10
221. ME Board Exam April 1996
Log of the nth root of x equals log of x
to 1/n power and also equal to
log x
A.
answer
n
B. n log x
log(x to the 1/n power)
C.
n
D. (n – 1)log x
222. ME Board Exam April 1997
What expression is equal to log x –
log(y + z)?
A. log x + log y + log z
B. log [x/(y + z)]
C. log x – log y – log z
D. log y + log (x + z)
223. ME Board Exam October 1997
Find the value of x if log12 x = 2.
A. 144
B. 414
C. 524
D. 425
224. ME Board Exam April 1998
A pump can pump out water from a
tank in 11 hours. Another pump can
pump out water from the same tank
in 20 hours. How long will it take both
pumps to pump out water in the tank?
A. 7 hours
B. 6 hours
C. 7 1/2 hours
D. 6 1/2 hours
225. ME Board Exam April 1995
If A can do the work in “x” days and B
in “y” days, how lone will they finish
the job working together?
x+y
A.
xy
B.
C.
D.
x+y
2
xy
answer
x+y
xy
226. ME Board Exam April 1995
A and B working together can finish
painting a house in 6 days. A working
alone can finish it in 54 days less
than B. How long will it take each of
them to finish the work alone?
A. 8, 13
B. 10, 15
C. 6, 11
D. 7, 12
227. ME Board Exam October 1994
On one job, two power shovels
excavate 20,000 cubic meters of
earth, the larger shovel working 40
hours and the smaller for 35 hours.
On another job, they removed 40,000
cubic meters with the larger shovel
working 70 hours and the smaller
working 90 hours. How much earth
can each remove in 1 hour working
alone?
A. 169.2, 287.3
B. 178.3, 294.1
C. 173.9, 347.8
D. 200.1, 312.4
228. ME Board Exam October 1992
A
Chemist
of
a
distillery
experimented
on
two
alcohol
solutions of different strength, 35%
alcohol
and
50%
alcohol,
respectively. How many cubic meters
of each strength must he use to
produce a mixture of 60 cubic meters
that contain 40% alcohol?
A. 20 m3 of solution with 35%
alcohol, 40 m3 of solution with
50% alcohol
B. 50 m3 of solution with 35%
alcohol, 20 m3 of solution with
50% alcohol
C. 20 m3 of solution with 35%
alcohol, 50 m3 of solution with
50% alcohol
D. 40 m3 of solution with 35%
alcohol, 20 m3 of solution with
50% alcohol
229. ME Board Exam October 1994
Two thousand (2000) kg of steel
containing 8% nickel is to be made
by mixing a steel containing 14%
nickel with another containing 6%
nickel. How much of each is needed?
A.
B.
C.
D.
1000 kg of steel with 14% nickel,
500 kg of steel with 6% nickel
750 kg of steel with 14% nickel,
1250 kg of steel with 6% nickel
500 kg of steel with 14% nickel,
1500 kg of steel with 6% nickel
1250 kg of steel with 14% nickel,
750 kg of steel with 6% nickel
230. ME Board Exam October 1991
The time required for an elevator to
lift a weight varies directly with the
weight and the distance through
which it is to be lifted and inversely
as the power of the motor. If it takes
30 seconds for a 10 hp motor to lift
100 lbs through 50 feet, what size of
motor is required to lift 800 lbs in 40
seconds through 40 feet?
A. 42
B. 44
C. 46
D. 48
231. ME Board Exam October 1996
The arithmetic mean of a and b is
a+b
A.
answer
2
B.
C.
D.
ab
ab
2
a−b
2
232. ME Board Exam April 1995
In a pile of logs, each layer contains
one more log than the layer above
and the top contains just one log. If
there are 105 logs in the pile, how
many layers are there?
A. 11
B. 12
C. 13
D. 14
233. ME Board Exam April 1999
If the sum is 220 and the first term is
10, find the common difference if the
last term is 30.
A. 2
B. 5
C. 3
D. 2/3
234. ME Board Exam October 1996
A product has a current selling price
of Php325.00. If its selling price is
expected to decline at the rate of
10% per annum because of
obsolescence, what will be its selling
price four years hence?
A.
B.
C.
D.
Php213.23
Php202.75
Php302.75
Php156.00
235.
A.
B.
C.
D.
236.
A.
B.
C.
D.
237.
A.
B.
C.
D.
Past Board Exam Problems
in Analytic Geometry
238. CE Board Exam November 1992
The two points on the lines 2x – 3y +
4 = 0 which are at a distance 2 from
the line 3x + 4y – 6 = 0 are
A. (-5,1) and (-5,2)
B. (64,-44) and (4,-4)
C. (8,8) and (12,12)
D. (44,-64) and (-4,4)
239. CE Board Exam November 1992
The distance from the point (2,1) to
the line 4x – 3y + 5 = 0 is
A. 1
B. 2
C. 3
D. 4
240. CE Board Exam November 1996
Determine the distance from (5,10) to
the line x – y = 0.
A. 3.33
B. 3.54
C. 4.23
D. 5.45
241. CE Board Exam May 1992
Find the distance between the given
lines 4x – 3y = 12 and 4x – 3y = -8.
A. 3
B. 4
C. 5
D. 6
242. CE Board Exam November 1995
What is the slope of the line 3x + 2y +
1 = 0?
A. 3/2
B. 2/3
C. -3/2
D. -2/3
243. CE Board Exam May 1996
What is the equation of the line that
passes thru (4,0) and is parallel to
the line x – y – 2 = 0?
A. x – y + 4 = 0
B. x + y + 4 = 0
C. x – y – 4 = 0
D. x – y = 0
244. CE Board Exam May 1997
Find the slope having a parametric
equation of x = 2 + t and y = 5 – 3t.
A. 2
B. 3
C. -2
D. -3
A.
B.
C.
D.
2.1
2.3
2.5
2.7
250. CE Board Exam May 1993, ECE
Board Exam November 1993, April
1994
The focus of the parabola y2 =16x is
at
A. (4,0)
B. (0,4)
C. (3,0)
D. (0,3)
251. CE Board Exam November 1994
What is the vertex of the parabola x2
= 4(y – 2)?
A. (2,0)
B. (0,2)
C. (3,0)
D. (0,3)
245. CE Board Exam May 1995
What is the radius of the circle x2 + y2
– 6y = 0?
A. 2
B. 3
C. 4
D. 5
252. CE Board Exam May 1995
What is the length of the latus rectum
of the curve x2 = 20y?
A. √20
B. 20
C. 5
D. √5
246. CE Board Exam November 1995
What are the coordinates of the
center of the curve x2 + y2 – 2x – 4y –
31 = 0?
A. (-1, 1)
B. (-2, 2)
C. (1, 2)
D. (2, 1)
253. CE Board Exam November 1994
What is the area enclosed by the
curve 9x2 + 25y2 – 225 = 0?
A. 47.1
B. 50.2
C. 63.8
D. 72.3
247. CE Board Exam May 1998
Find the slope of the line having a
parametric equation y = 4t + 6 and x
= t + 1.
A. 1
B. 2
C. 3
D. 4
248. CE Board Exam May 1996
How far from the y-axis is the center
of the curve 2x2 + 2y2 + 10x – 6y – 55
= 0?
A. -2.5
B. -3.0
C. -2.75
D. 3.25
249. CE Board Exam November 1993
The shortest distance form A(3,8) to
the circle x2 + y2 + 4x – 6y = 12 is
equal to
254. CE Board Exam May 1993
The length of the latus rectum for the
ellipse x2/64 + y2/16 = 1 is equal to
A. 2
B. 3
C. 4
D. 5
255. CE Board Exam November 1992
The earth’s orbit is an ellipse with the
sun at one of the foci. If the farthest
distance of the sun from the earth is
105.50 million km and the nearest
distance of the sun from the earth is
78.25 million km, find the eccentricity
of the ellipse.
A. 0.15
B. 0.25
C. 0.35
D. 0.45
256. CE Board Exam November 1995
How far from the x-axis is the focus F
of the hyperbola x2 – 2y2 + 4x + 4y + 4
= 0?
A. 4.5
B. 3.4
C. 2.7
D. 2.1
257. CE Board Exam May 1996
What is the equation of the
asymptote
of
the
hyperbola
x 2 y2
−
= 1?
9
4
A. 2x – 3y = 0
B. 3x -2y = 0
C. 2x – y = 0
D. 2x + y = 0
258. EE Board Exam April 1994
Find the distance between A (4, -3)
and B (-2, 5).
A. 11
B. 9
C. 10
D. 8
259. EE Board Exam April 1995
The line segment connecting (x, 6)
and (9, y) is bisected by the point (7,
3). Find the values of x and y.
A. 14, 6
B. 33, 12
C. 5, 0
D. 6, 9
260. EE Board Exam October 1997
Find the distance of the line 3x + 4y =
5 from the origin.
A. 4
B. 3
C. 2
D. 1
261. EE Board Exam April 1995
Find the distance between the line 3x
+ y – 12 = 0 and 3x + y– 4 = 0.
16
A.
10
12
B.
10
4
C.
10
8
D.
answer
10
262. EE Board Exam April 1994
Given three vertices of a triangle
whose coordinates are A (1, 1), B (3,
-3) and C (5, -3). Find the area of the
triangle
A.
B.
C.
D.
3
4
5
6
263. EE Board Exam April 1997
A line passes thru (1, -3) and (-4, 2).
Write the equation of the line in slope
intercept form.
A. y – 4 = x
B. y = -x – 2
C. y = x - 4
D. y – 2 = x
264. EE Board Exam October 1997
What is the x-intercept of the line
passing through (1,4) and (4,1)?
A. 4.5
B. 5
C. 4
D. 6
265. EE Board Exam October 1997
Find the location of the focus of the
parabola y2 + 4x – 4y – 8 = 0.
A. (2.5,-2)
B. (3,1)
C. (2,2)
D. (-2.5,-2)
266. EE Board Exam April 1997
The center of a circle is at (1, 1) and
one point on its circumference is (-1,
3). Find the other end of the diameter
through (-1, -3).
A. (2, 4)
B. (3, 5)
C. (3, 6)
D. (1, 3)
267. EE Board Exam October 1997
Find the major axis of the ellipse x2 +
4y2 – 2x – 8y + 1 = 0
A. 2
B. 10
C. 4
D. 6
268. EE Board Exam October 1993
4x2 – y2 = 16 is the equation of a/an
A. parabola
B. hyperbola
C. circle
D. ellipse
269. EE Board Exam October 1993
Find the eccentricity of the curve
9x2 – 4y2 – 36x + 8y = 4.
A. 1.80
B. 1.92
C. 1.86
D. 1.76
270. EE Board Exam October 1994
The semi-transverse axis of the
x 2 y2
−
= 1 is
hyperbola
9
4
A. 2
B. 3
C. 4
D. 5
271. EE Board Exam April 1994
Find the equation of the hyperbola
whose asymptotes are y = ±2x and
which passes through (5/2, 3).
A. 4x2 + y2 + 16 = 0
B. 4x2 + y2 – 16 = 0
C. x2 – 4y2 – 16 = 0
D. 4x2 – y2 = 16
272. EE Board Exam April 1997
Find the polar equation of the circle, if
its center is at (4,0) and the radius 4.
A. r – 8 cos θ = 0
B. r – 6 cos θ = 0
C. r – 12 cos θ = 0
D. r – 4 cos θ = 0
273. EE Board Exam October 1997
Given the polar equation r = 5 sin θ.
Determine
the
rectangular
coordinates (x, y) of a point in the
curve when θ = 30°.
A. (2.17, 1.25)
B. (3.08, 1.5)
C. (2.51, 4.12)
D. (6, 3)
274. ECE Board Exam April 1999
The linear distance between -4 and
17 on the number line is
A. 13
B. 21
C. -17
D. -13
275. ECE Board Exam November 1998
Determine the coordinates of the
point which is three-fifths of the way
from the point (2,-5) to the point (3,5).
A. (-1,1)
B. (-2,-1)
C. (-1,-2)
D. (1,-1)
276. ECE Board Exam November 1998
The segment from (-1, 4) to (2, -2) is
extended three times its own length.
Find the terminal point.
A. (11, -24)
B. (-11, -20)
C. (11, -18)
D. (11, -20)
277. ECE Board Exam April 1999
Find the inclination of the line passing
through (-5, 3) and (10, 7).
A. 14.73 degrees
B. 14.93 degrees
C. 14.83 degrees
D. 14.63 degrees
278. ECE Board Exam November 1990
In Cartesian coordinates, the vertices
of a triangle are defined by the
following points: (-2, 0), (4, 0) and (3,
3). What is its area?
A. 8 sq. units
B. 9 sq. units
C. 10 sq. units
D. 11 sq. units
279. ECE Board Exam November 1990
In Cartesian coordinates, the vertices
of a square are: (1, 1), (0, 8), (4, 5)
and (-3, 4). What is its area?
A. 20 sq. units
B. 30 sq. units
C. 25 sq. units
D. 35 sq. units
280. ECE Board Exam April 1999
If the points (-2, 3), (x, y) and (-3,
5) lie on a straight line, then the
equation of the line is ____.
A. x – 2y – 1 = 0
B. 2x + y – 1 = 0
C. x + 2y – 1 = 0
D. 2x + y + 1 = 0
C.
D.
2x – y + 2 = 0
2x + y + 2 = 0
284. ECE Board Exam April 1998
Determine B such that 3x + 2y – 7 = 0
is perpendicular to 2x – By + 2 = 0.
A. 5
B. 4
C. 3
D. 2
285. ECE Board Exam April 1998
The diameter of a circle described by
9x2 + 9y2 =16 is
A. 4/3
B. 16/9
C. 8/3
D. 4
286. ECE Board Exam April 1998
Find the value of k for which the
equation x2+ y2 + 4x – 2y – k = 0
represents a point circle.
A. 5
B. 6
C. -6
D. -5
287. ECE Board Exam April 1999
Given the equation of the curve, 3x2
+ 2x – 5y + 7 = 0. Determine the
curve.
A. parabola
B. ellipse
C. circle
D. parabola
281. ECE Board Exam April 1999
Two vertices of a triangle are (2, 4)
and (-2, 3) and the area is 2 square
units, the locus of the third vertex is
____.
A. 4x – y = 14
B. 4x + 4y = 14
C. x + 4y = 12
D. x – 4y = -10
288. ECE Board Exam April 1998
Find the equation of the axis of
symmetry of the function y = 2x2 – 7x
+ 5.
A. 7x + 4 = 0
B. 4x + 7 = 0
C. 4x – 7 = 0
D. x – 2 = 0
282. ECE Board Exam April 1998
Find the area of the triangle which
the line 2x – 3y + 6 = 0 forms with the
coordinate axis.
A. 3
B. 4
C. 5
D. 2
289. ECE Board Exam November 1997
Compute the focal length and the
length of the latus rectum of parabola
y2 + 8x – 6y + 25 = 0.
A. 2, 8
B. 4, 16
C. 16, 64
D. 1, 4
283. ECE Board Exam November 1998
A line passes through point (2, 2).
Find the equation of the line if the
length of the line segment intercepted
by the coordinate axes is the square
root of 5.
A. 2x + y – 2 = 0
B. 2x – y – 2 = 0
290. ECE Board Exam April 1998
Point P(x,y) moves with a distance
from point (0,1) one half of its
distance from the line y = 4. The
equation of its locus is
A. 2x2 – 4y2 = 5
B. 4x2 + 3y2 = 12
C. 2x2 + 5y2 = 3
D.
x2 + 2y2 = 4
291. ECE Board Exam April 1998
The major axis of the elliptical path in
which the earth moves around the
sun is approximately 186,000,000
miles and the eccentricity of the
ellipse is 1/60. Determine the apogee
of the earth.
A. 93,000,000 miles
B. 91,450,000 miles
C. 94,335,100 miles
D. 94,550,000 miles
292. ECE Board Exam April 1994, April
1999
Find the equation of the directrix of
the parabola y2 = 16x.
A. x = 2
B. x = -2
C. x = 4
D. x = -4
293. ECE November 1999
The semi-major axis of an ellipse is 4
and its semi-minor axis is 3. The
distance from the center to the
directrix is
A. 6.047
B. 6.532
C. 0.6614
D. 6.222
294. ECE November 1997
The midpoint of the line segment
between P1(x, y) and P2(-2, 4) is
Pm(2, -1). Find the coordinates of P1.
A. (6, -6)
B. (6, -5)
C. (5, -6)
D. (-6, 6)
295. ECE November 1997
Given the ellipse (x2/36) + (y2/32) = 1,
Determine the distance between the
foci.
A. 8
B. 4
C. 2
D. 3
296. ECE November 1997
Find the coordinates of the point (2,
4) with respect to the translated axis
with origin at (1, 3).
A. (1, -1)
B. (-1, -1)
C. (1, 1)
D. (-1, -1)
297. ECE November 1998
A line passes through point (2, 2).
Find the equation of the line if the
length of the segment intercepted by
the coordinate axes is the square
root of 5.
A. 2x + y – 2 = 0
B. 2x – y – 2 = 0
C. 2x – y + 2 = 0
D. 2x + y + 2 = 0
298. ECE November 199
A point moves so that its distance
from the point (2, -1) is equal to the
distance from the x-axis. The
equation of the locus is
A. x2 – 4x + 2y + 5 = 0
B. x2 – 4x - 2y + 5 = 0
C. x2 + 4x + 2y + 5 = 0
D. x2 + 4x - 2y - 5 = 0
299. ECE November 1999
The point of intersection of the planes
x + 5y – 2z = 9, 3x – 2y + z = -3 and x
+ y + z = 2 is at
A. (1, 2, 1)
B. (2, 1, -1)
C. (1, -1, 2)
D. (-1, -1, 2)
300. ECE November 1999, April 2005
Given the points (3, 7) and (-4, 7).
Solve for the distance between them.
A. 15.65
B. 17.65
C. 16.65
D. 14.65
301. ECE November 1999
Find the distance of the directrix from
the center of an ellipse if its major
axis is 10 and its minor axis is 8.
A. 8.5
B. 8.1
C. 8.3
D. 8.7
302. ECE April 2000
Find the coordinates of the vertex of
the parabolas y = x2 – 4x + 1 by
making use of the fact that at the
vertex, the slope of the tangent is
zero.
A. (2, -3)
B. (-2, -3)
C. (-1, -3)
D. (3, -2)
303. ECE April 2000
Find the area of the hexagon
ABCDEF formed by joining the points
A(1,4), B(0,-3), C(2,3), D(-1,2), E(-2,1) and F(3,0).
A. 24
B. 20
C. 22
D.
15
D.
x–y–1=0
304. ECE April 2000
The parabolic antenna has an
equation y2 + 8x = 0. Determine the
length of the latus rectum.
A. 8
B. 10
C. 12
D. 9
311. ECE April 2002
Find the value of k if the distance
from the point (2,1) to the line 5x +
12y + k = 0 is 2.
A. 5
B. 2
C. 4
D. 3
305. ECE November 2000
A line 4x + 2y – 2 = 0 is coincident
with the line
A. 4x + 4y - 2 = 0
B. 4x + 3y + 3 = 0
C. 8x + 4y – 2 = 0
D. 8x + 4y – 4 = 0
312. ECE November 2002
Determine the farthest distance from
the point (3, 7) to the circle x2 + y2 +
4x – 6y – 12 = 0.
A. 6.40
B. 1.40
C. 11.40
D. 4.60
306. ECE April 2001
Find the equation of the parabola
whose axis is parallel to the x-axis
and passes through the points (3,1),
(0,0) and (8,-4).
A. x2 – 2x – y = 0
B. x2 + 2x + y = 0
C. y2 + 2y + x = 0
D. y2 + 2y – x = 0
307. ECE April 2001, November 2002
The directrix of a parabola is the line
y = 5 and its focus is at the point (4,3). What is the length of the latus
rectum?
A. 18
B. 14
C. 16
D. 12
308. ECE November 2001
A point P(x, 2) is equidistant from the
points (-2, 9) and (4, -7). The value of
x is
A. 11/3
B. 20/3
C. 19/3
D. 6
309. ECE November 2001
Find the angle between the planes 3x
– y + z – 5 = 0 and x + 2y + 2z + 2 = 0.
A. 62.45°
B. 52.45°
C. 82.45°
D. 72.45°
310. ECE November 2001
Find the equation of the line where
the x-intercept is 2 and the yintercept is -2.
A. 2x + 2y + 2 = 0
B. x – y – 2 = 0
C. 2y – 2x + 2 = 0
313. ECE November 2002
Find
the
equation
of
the
perpendicular bisector of the line
joining (4, 0) and (6, 3).
A. 4x + 6y – 29 = 0
B. 4x + 6y + 29 = 0
C. 4x – 6y + 29 = 0
D. 4x – 6y – 29 = 0
314. ECE April 2003
Given the points (3,7) and (-4,-7).
Solve for the distance between them
A. 14.65
B. 15.65
C. 17.65
D. 16.65
315. ECE April 2003
Determine the vertex of the parabola
y = x2 + 8x + 2.
A. (18, 4)
B. (-4, -18)
C. (4, 18)
D. (-4, 18)
316. ECE April 2003
What is the equation of the circle with
its center at the origin and if the point
(1, 1) lies on the circumference of the
circle?
A. (x + 1)2 + (y + 1)2 = 2
B. (x + 1)2 + (y + 1)2 = 4
C. x2 + y2 = 2
D. x2 + y2 = 4
317. ECE April 2003
What is the distance of the line 4x –
3y + 5 = 0 from the point (4, 2)?
A. 5
B. 4
C. 2
D. 3
318. ECE April 2003
If the lines 4x – y + 2 = 0 and x + 2ky
+ 1 = 0 are perpendicular to each
other, determine the value of k.
A. 3
B. 4
C. 1
D. 2
319. ECE April 2003
A triangle is drawn with vertices at (1, -1), (1, 3) and (4, 1). What is the
median from vertex (4, 1)?
A. 10 units
B. 4 units
C. 5 units
D. 6 units
320. ECE November 2003
What is the equation of the circle at
the origin and a radius of 5?
A. x2 + y2 = 1
B. x2 + y2 = 25
C. x2 + y2 = 10
D. x2 + y2 = 5
321. ECE November 2003
What is the equation of the line
through (-3, 5) which makes an angle
of 45 degrees with the line 2x + y =
12?
A. x + 3y – 12 = 0
B. x + 3y + 18 = 0
C. x + 2y – 7 = 0
D. x – 3y – 18 = 0
322. ECE November 2003
Determine the acute angle between
the lines y – 3x = 2 and y – 4x = 9.
A. 4.39 deg
B. 3.75 deg
C. 5.35 deg
D. 2.53 deg
323. ECE November 2003
Determine the equation of the
perpendicular
bisector
of
the
segment PQ if P(-2, 3) and Q(4, -5).
A. 3y – 3x + 7 = 0
B. 4x – 3y + 7 = 0
C. 6x – 8y – 14 = 0
D. 3x – 4y – 7 = 0
324. ECE April 2004
Find the volume of the pyramid
formed in the first octant by the plane
6x + 10y + 5z – 30 = 0 and the
coordinate axes.
A. 13
B. 12
C. 14
D. 15
325. ECE April 2004
A circle with its center in the first
quadrant is tangent to both x and y
axes. If the radius is 4, what is the
equation of the circle?
A. (x + 4)2 + (y + 4)2 = 16
B. (x – 8)2 + (y – 8)2 = 16
C. (x – 4)2 + (y – 4)2 = 16
D. (x + 4)2 + (y – 4)2 = 16
326. ECE April 2004
A circle is described by the equation
x2 + y2 – 16x = 0. What is the length of
the chord which is 4 units from the
center of the circle?
A. 6.93 units
B. 13.86 units
C. 11.55 units
D. 9.85 units
327. ECE April 2004
What is the equation of the line that
passes through (-3, 5) and is parallel
to the line 4x – 2y + 2 = 0?
A. 4x – 2y + 22 = 0
B. 2x + y + 10 = 0
C. 4x + 2y – 11 = 0
D. 2x – y + 11 = 0
328. ECE April 2004
What is the distance between the line
x + 2y + 8 = 0 and the point (5,-2)?
A. 4.20
B. 4.44
C. 4.02
D. 4.22
329. ECE November 1995
If the product of the slopes of any two
straight lines is negative 1, one of
these lines is said to be
A. parallel
B. skew
C. perpendicular
D. non-intersecting
330. ECE March 1996
The line passing through the focus
and perpendicular to the directrix of a
parabola is called
A. latus rectum
B. axis of parabola
C. tangent line
D. secant line
331. ECE November 1996
It represents the distance of a point
from the y-axis.
A. ordinate
B. coordinate
C. abscissa
D. polar distance
332. ECE April 1994
In a conic section, if the eccentricity >
1, the locus is
A. an ellipse
B. a hyperbola
C. a parabola
D. a circle
333. ECE April 1998
It can be defined as the set of all
points in the plane the sum of whose
distances from two fixed points is
constant.
A. circle
B. ellipse
C. hyperbola
D. parabola
334. ECE April 1998
If the equation is unchanged by the
substitution of –x for x, its curve is
symmetric with respect to the
A. y-axis
B. x-axis
C. origin
D. line 45 degrees with the axis
335. ECE November 1997
A line which is perpendicular to the xaxis has a slope equal to
A. zero
B. 2
C. one
D. infinity
336. ECE November 1997
In an ellipse, a chord which contains
a focus and is in a line perpendicular
to the major axis is a
A. latus rectum
B. minor
C. focal width
D. conjugate axis
337. ME Board Exam October 1996
What is the length of the line with a
slope of 4/3 from a point (6, 4) to the
y-axis?
A. 10
B. 25
C. 50
D. 75
338. ME Board Exam April 1998
Find the slope of the line defined by y
– x = 5.
A. 1
B. 1/4
C. -1/2
D. 5 + x
339. ME Board Exam April 1997
Find the equation of a straight line
with a slope of 3 and a y-intercept of
1.
A. 3x + y – 1 =0
B. 3x – y + 1 = 0
C. x + 3y + 1 = 0
D. x – 3y – 1 =0
340. ME Board Exam April 1998
The equation of the line that
intercepts the x-axis at x = 4 and the
y-axis at y = -6 is,
A. 3x + 2y = 12
B. 2x – 3y = 12
C. 3x – 2y = 12
D. 2x + 3y = 12
341. ME Board Exam April 1998
What is the radius of a circle with the
following equation: x2 – 6x + y2 – 4y –
12 = 0?
A. 3.46
B. 7
C. 5
D. 6
342. ME Board Exam October 1996
The equation x2 + y2 – 4x + 2y – 20 = 0
describes
A. a circle of radius 5 centered at
the origin
B. an ellipse centered at (2,-1)
C. a sphere centered at the origin
D. a circle of radius 5 centered at
(2,-1)
343. ME Board Exam April 1997
In the equation y = -x2 + x + 1, where
is the curve facing?
A. upward
B. facing left
C. facing right
D. downward
344. ME Board Exam October 1997
The general equation of a conic
section is given by the following
equation: Ax2 + Bxy + Cy2 + Dx + Ey
+ F = 0. A curve maybe identified as
an ellipse by which of the following
conditions?
A. B2 - 4AC < 0
B. B2 – 4AC = 0
C. B2 – 4AC > 0
D. B2 – 4AC = 1
345. ME Board Exam April 1997
What is the radius of the sphere
center at the origin that passes the
point 8, 1, 6?
A. 10
B. 9
C. √101
D.
10.5
346. ME Board Exam October 1996
What are the x and y coordinates of
the focus of the conic section
described by the following equation?
(Angle θ corresponds to a right
triangle with adjacent side x, opposite
side y and the hypotenuse r)
r – sin2 θ = cos θ
A. (1/4, 0)
B. (0, π/2)
C. (0, 0)
D. (-1/2, 0)
347. ME April 1997
What is the graph of the equation Ax2
+ Bx + Cy2 + Dy + E = 0?
A. circle
B. ellipse
C. parabola
D. hyperbola
348. ME October 1997
What type of curve is generated by a
point which moves in uniform circular
motion about an axis, while travelling
at a constant speed, v, parallel to the
x-axis?
A. helix
B. spiral of Archimedes
C. hypocycloid
D. cycloid
349. ME April 1998
Points that lie in the same plane
A. coplanar
B. oblique
C. collinear
D. parallel
Past Board Exam Problems
in Differential Calculus
350. CE Board Exam November 1997
Evaluate Lim
x→ 1 x 2
A.
B.
C.
D.
x2 -1
+ 3x - 4
1/5
2/5
3/5
4/5
351. CE Board Exam November 1994
What is the derivative with respect to
x of (x + 1)3 – x3?
A. 3x + 6
B. 3x – 3
C. 6x – 3
D. 6x + 3
352. CE Board Exam May 1997
Find the derivative of arcos 4x.
-4
A.
answer
1 - 16x 2
4
B.
1 - 16x 2
-4
C.
1 - 4x 2
4
D.
1 - 4x 2
353. CE Board Exam November 1996
Find the derivative of
A.
)2
(x + 1
x
-
(x + 1)3
x
)3
(x + 1
x
2
B.
4(x + 1) 2(x + 1)3
x
x
C.
2(x + 1)2 (x + 1)3
x
x3
D.
3(x + 1)2 (x + 1)3
answer
x
x2
354. CE Board Exam November 1995
The derivative with respect to x of
2cos2 (x2 + 2)
A. 2sin (x2 + 2) cos (x2 + 2)
B. -2sin (x2 + 2) cos (x2 + 2)
C. 8xsin (x2 + 2) cos (x2 + 2)
D. -8xsin (x2 + 2) cos (x2 + 2)
355. CE November 1997
What is the first derivative of y =
arcsin 3x?
3
A. −
1 + 9x 2
3
B.
1 + 9x 2
3
C. −
1 + 9x 2
3
D.
answer
1 + 9x 2
356. CE May 1999
Find the second derivative of y = x-2
at x = 2.
A. 96
B. 0.375
C. -0.25
D. -0.875
357. CE Board Exam November 1993
Find the second derivative of y by
implicit differentiation from the
equation 4x2 + 8y2 = 36.
A. 64x2
9
B. - y 3 answer
4
C. 32xy
16 3
D. y
9
358. CE Board Exam May 1998
Find the slope of the curve x2 + y2 –
6x + 10y + 5 = 0 at point (1, 0).
A. 1/5
B. 2/5
C. 1/4
D. 2
359. CE Board Exam May 1996
Find the slope of the tangent to the
curve y = 2x – x2 + x3 at (0, 2).
A. 1
B. 2
C. 3
D. 4
360. CE Board Exam May 1996
Find the slope of the ellipse x2 + 4y2 –
10x – 16y + 5 = 0 at the point where y
= 2 + 80.5 and x = 7.
A. -0.1463
B. -0.1538
C. -0.1654
D. -0.1768
361. CE Board Exam May 1995
Find the equation of the line normal
to x2 + y2 = 25 at the point (4, 3).
A. 5x + 3y = 0
B. 3x – 4y = 0
C. 3x + 4y = 0
D. 5x – 3y = 0
362. CE November 1998
Determine the slope of the curve x2 +
y2 – 6x – 4y – 21 = 0 at (0, 7).
A. 3/5
B. -2/5
C. -3/5
D. 2/5
363. CE May 1998
Find the slope of the line whose
parametric equations are x = 4t + 6
and y = t – 1.
A. -4
B. 1/4
C. 4
D. -1/4
364. CE November 1999
Find the slope of the curve y = 64(4 +
x)1/2 at (0, 12).
A. 0.67
B. 1.5
C. 1.33
D. 0.75
365. CE May 1999
Find the angle that the line 2y – 9x –
18 = 0 makes with the x-axis.
A. 74.77°
B. 4.5°
C. 47.77°
D. 77.47°
366. CE November 1998
What is the radius of curvature at
point (1, 2) of the curve 4x – y2 = 0?
A. 6.21
B. 5.21
C. 5.66
D. 6.66
367. CE November 1997
Find the radius of curvature of the
curve y2 - 4x = 0 at point (4, 4).
A. 24.4
B. 25.4
C. 23.4
D. 22.4
368. CE November 1999
Find the radius of curvature of the
curve x = y3 at (1, 1).
A. 4.72
B. 3.28
C. 4.67
D. 5.27
369. CE November
The chords of the ellipse 64x2 + 25y2
= 1600 having equal slopes of 1/5 are
bisected by its diameter. Determine
the equation of the diameter of the
ellipse?
A. 5x – 64y = 0
B. 64x – 5y = 0
C. 5x + 64y = 0
D. 64x + 5y = 0
370. CE Board Exam May 1995
A wall “h” meters high is 2 m away
from the building. The shortest ladder
that can reach the building with one
end resting on the ground outside the
wall is 6 m. How high is the wall in
meters?
A. 2.34
B. 2.24
C. 2.44
D. 2.14
371. CE Board Exam May 1997
Find the minimum amount of tin
sheet that can be made into a closed
cylinder having a volume of 108 cu.
inches in square inches.
A. 125.50
B. 127.50
C. 129.50
D. 123.50
372. CE Board Exam November 1996
A Norman window is in the shape of
a rectangle surmounted by a semicircle. What is the ratio of the width of
the rectangle to the total height so
that it will yield a window admitting
the most light for a given perimeter?
A. 1
B. 1/2
C. 2
D. 2/3
373. CE Board Exam May 1998
Determine the diameter of a closed
cylindrical tank having a volume of
11.3 cu. m to obtain the minimum
surface area.
A. 1.22
B. 1.64
C. 2.44
D. 2.68
374. CE Board Exam November 1998
Water is pouring into a conical vessel
15 cm deep and having a radius of
3.75 cm across the top if the rate at
which the water rises is 2 cm/sec,
how fast is the water flowing into the
conical vessel when the water is 4
cm deep?
A. 2.37 m3/sec
B. 5.73 m3/sec
C. 6.28 m3/sec
D. 4.57 m3/sec
375. CE Board Exam May 1997
Car A moves due east at 30 kph at
the same instant car B is moving S
30° E, with a speed of 60 kph. If the
distance from A to B is 30 km, find
how fast is the distance between
them separating after one hour.
A. 36 kph
B. 38 kph
C. 40 kph
D. 45 kph
376. CE Board Exam November 1996
A car starting at 12:00 noon travels
west at a speed of 30 kph. Another
car starting from rest starting from the
same point at 2:00 pm travels north
at 45 kph. Find how fast (in kph) the
two are separating at 4:00 pm?
A.
B.
C.
D.
49
51
53
55
377. CE Board Exam May 1996
Two railroad tracks are perpendicular
to each other. At 12:00 pm there is a
train at each track approaching the
crossing at 50 kph, one being 100 km
and the other 150 km away from the
crossing. How fast in kph is the
distance between the two trains
changing at 4:00 pm?
A. 67.08
B. 68.08
C. 69.08
D. 70.08
378. CE Board Exam May 1998
Water is running into a hemispherical
bowl having a radius of 10 cm at a
constant rate of 3 cm3/min. When the
water is x cm deep, the water level is
rising at the rate of 0.0149 cm/min.
What is the value of x?
A. 3
B. 2
C. 4
D. 5
379. CE May 1999
The number of newspaper copies
distributed is given by C = 50t2 – 200t
+ 10000, where t is in years. Find the
minimum
number
of
copies
distributed from 1995 to 2002
A. 9850
B. 9800
C. 10200
D. 7500
380. CE May 1999
A rectangular box having a square
base and open at the top is to have a
capacity of 16823 cc. Find the height
of the box to use the least amount of
material.
A. 16.14 cm
B. 32.28 cm
C. 18.41 cm
D. 28.74 cm
381. CE May 1998, November 1999
The volume of a closed cylindrical
tank is 11.3 cubic meters. If the total
surface is a minimum, what is its
base radius, in m?
A. 1.44
B. 1.88
C. 1.22
D. 1.66
382. CE May 1998
An object moves along a straight line
such that, after t minutes, its distance
from its starting point is D = 20t + 5/(t
+ 1) meters. At what speed, in
m/minute will it be moving at the end
of 4 minutes?
A. 39.8
B. 49.8
C. 29.8
D. 19.8
383. CE May 1996
The speed of the traffic flowing past a
certain downtown exit between the
hours of 1:00 pm and 6:00 pm is
approximately V = t3 – 10.5t2 + 30t +
20 miles per hour, where t = number
of hours past noon. What is the
fastest speed of the traffic between
1:00 pm and 6:00 pm in mph?
A. 50
B. 46
C. 32.5
D. 52
384. CE May 1997
A car drives east from point A at 30
kph. Another car B starting from B at
the same time drives S 30° W toward
A at 60 kph. B is 30 km away from A.
How fast in kph is the distance
between the two cars changing after
one hour?
A. 76.94 kph
B. 78.94 kph
C. 75.94 kph
D. 77.94 kph
385. CE November 1998
There is a constant inflow of a liquid
into a conical vessel 15 ft deep and
7.5 feet in diameter at the top. Water
is rising at the rate of 2 feet per
minute when the water is 4 feet deep.
What is the rate of inflow in cu. Ft per
minute?
A. 8.14
B. 7.46
C. 9.33
D. 6.28
386. CE May 2003
What is the radius of curvature of the
curve y2 = 16x = 0 at the point (4, 8)?
A. -0.044
B. -0.088
C. -0.066
D. -0.033
Suppose that x years after founding
in 1975, a certain employee
association has a membership of f(x)
= 100(2x3 – 45x2 + 264x), at what time
between 1975 and 1989 was the
membership smallest?
A. 1983
B. 1985
C. 1984
D. 1986
388. CE November 2002
A 3 meter long steel pipe has its
upper end leaning against a vertical
wall and lower end on a level ground.
The lower end moves away at a
constant rate of 2 cm/s. How fast is
the upper end moving down, in cm/s,
when the lower end is 2 m from the
wall?
A. 1.81
B. 1.66
C. 1.79
D. 1.98
389. CE May 2002
A particle moves according to the
parametric equations:
y = 2t2 and x = t3
where x and y are displacement (in
meters) in x and y direction,
respectively and t is time in seconds.
Determine the acceleration of the
body after t = 3 seconds.
A. 12.85 m/s2
B. 18.44 m/s2
C. 21.47 m/s2
D. 5.21 m/s2
390. CE May 2002
Determine the shortest distance from
point (4, 2) to the parabola y2 = 8x.
A. 2.83
B. 3.54
C. 2.41
D. 6.32
391. CE November 2001
Water flows into a tank having the
form of a frustum of a right circular
cone. The tank is 4 m tall with upper
radius of 1.5 m and the lower radius
of 1 m. When the water in the tank is
1.2 m deep, the surface rises at the
rate of 0.012 m/s. Calculate the
discharge of water flowing into the
tank in m3/s.
A. 0.02
B. 0.05
C. 0.08
D. 0.12
387. CE November 2002
392. CE November 2003
The motion of a particle is defined by
the parametric equation x = t3 and y =
2t3. Determine the velocity when t =
2.
A. 14.42
B. 16.25
C. 12.74
D. 18.63
393. CE November 2003
The sum of two numbers is K. The
product of one by the cube of the
other is to be a minimum. Determine
one the numbers.
A. 3K/4
B. 3K/8
C. 3K/2
D. 3K/7
394. EE Board Exam April 1993
1 − cos x
Evaluate Lim
x →0
x2
A. 0
B. 1/2
C. 2
D. -1/2
395. EE Board Exam October 1994
3x 4 − 2x 2 + 7
Evaluate Lim
x →∞ 5x 3 + x − 3
A. Undefined
B. 3/5
C. infinity
D. 0
396. EE Board Exam October 1997
Differentiate y = ex cos x2
A. -ex sin x2
B. ex (cos x2 – 2x sin x2)
C. ex cos x2 – 2x sin x2
D. -2xex sin x
397. EE Board Exam October 1997
Differentiate y = sec (x2 + 2)
A. 2x cos (x2 + 2)
B. -cos (x2 + 2) cot (x2 + 2)
C. 2x sec (x2 + 2) tan (x2 + 2)
D. cos (x2 + 2)
398. EE Board Exam October 1997
Differentiate y = log(x2+1)2
A. 4x(x2 + 1)
4 x log10 e
B.
answer
( x 2 + 1)
C. log e(x)(x2 + 1)
D. 2x(x2 + 1)
399. EE Board Exam October 1997
Differentiate (x2 + 2)1/2
A.
( x 2 + 1)1/ 2
2
B.
C.
D.
x
( x 2 + 2)1/ 2
answer
2x
( x 2 + 2)1/ 2
(x2 + 2)3/2
400. EE Board Exam October 1997
If y = (t2 + 2)2 and t = x1/2, determine
dy/dx.
A. 3/2
B.
C.
D.
2x 2 + 2x
3
2(x + 2)
x5/2 + x1/2
401. EE Board Exam April 1995
Find y’ if y = arcsin (cos x).
A. -1
B. -2
C. 1
D. 2
402. EE Board Exam October 1997
If y = 4cos x + sin 2x, what is the
slope of the curve when x = 2?
A. -2.21
B. -4.94
C. -3.25
D. -2.21
403. EE Board Exam April 1997
Locate the points of inflection of the
curve y = f(x) = x2ex.
A. −2 ± 3
B.
2± 2
C.
−2 ± 2 answer
D.
2± 3
404. EE Board Exam April 1990
The sum of two positive numbers is
50. What are the numbers if their
product is to be the largest possible.
A. 24 and 26
B. 28 and 22
C. 25 and 25
D. 20 and 30
405. EE Board Exam March 1998
A triangle has variable sides x, y, z
subject to the constraint such that the
perimeter P is fixed to 18 cm. What is
the maximum possible area for the
triangle?
A. 15.59 cm2
B. 18.71 cm2
C. 17.15 cm2
D. 14.03 cm2
406. EE Board Exam October 1997
A farmer has enough money to build
only 100 meters of fence. What are
the dimensions of the field he can
enclose this maximum area?
A. 25 m by 25 m
B. 15 m by 35 m
C. 20 m by 30 m
D. 22.5 m by 27.5 m
407. EE Board Exam April 1997
The cost of fuel in running a
locomotive is proportional to the
square of the speed and is $25 per
hour for a speed of 25 miles per hour.
Other costs amount to $ 100 per
hour, regardless of the speed. What
is the speed which will make the cost
per mile a minimum?
A. 40
B. 55
C. 50
D. 45
408. EE Board Exam April 1997
A poster is to contain 300 m2 of
printed matter with margins of 10 cm
at the top and bottom and 5 cm at
each side. Find the over-all
dimensions, if the total area of the
poster is a minimum.
A. 27.76 cm, 47.8 cm
B. 20.45 cm, 35.6 cm
C. 22.24 cm, 44.5 cm
D. 25.55 cm, 46.7 cm
409. EE Board Exam March 1998
A fencing is limited to 20 ft in length.
What is the maximum rectangular
area that can be fenced in using two
perpendicular corner sides of an
existing wall?
A. 120
B. 100
C. 140
D. 190
410. EE Board Exam October 1992
The cost per hour of running a motor
boat is proportional to the cube of the
speed. At what speed will the boat
run against a current of 8 kph in order
to go a distance most economically?
A. 10 kph
B. 13 kph
C. 11 kph
D. 12 kph
411. EE Board Exam October 1993
A boatman is at A which is 4.5 km
from the nearest point B on a straight
shore BM. He wishes to reach in
minimum time a point C situated on
the shore 9 km from B. How far from
C should he land if he can row at the
rate of 6 kph and can walk at the rate
of 7.5 kph?
A. 4.15 km
B. 3.0 km
C. 3.25 km
D. 4.0 km
412. EE Board Exam October 1993
At any distance x from the source of
light, the intensity of illumination
varies directly as the intensity of the
source and inversely as the square of
x. Suppose that there is a light at A
and another at B having an intensity
8 times that of A. The distance AB is
4 m. At what point from A on the line
AB will the intensity of illumination be
least?
A. 2.15 m
B. 1.33 m
C. 1.50 m
D. 1.92 m
413. EE Board Exam April 1997
The coordinates (x, y) in feet of a
moving particle P is given by x = cos t
– 1 and y = 2sin t + 1, where t is the
time in seconds. At what extreme
rates in fps is P moving along the
curve?
A. 3 and 2
B. 3 and 1
C. 2 and 0.5
D. 2 and 1
414. EE Board Exam October 1993
Water is flowing into a conical cistern
at the rate of 8 m3/min. If the height of
the inverted cone is 12 m and the
radius of its circular opening is 6 m.
How fast is the water level rising
when the water is 4 m deep?
A. 0.64 m/min
B. 0.56 m/min
C. 0.75 m/min
D. 0.45 m/min
415. EE Board Exam October 1993
A standard cell has an emf “E” of 1.2
volts. If the resistance “R” of the
circuit is increasing at a rate of 0.03
ohm/sec, at what rate is the current
“I” changing at the instant when the
resistance is 6 ohms? Assume
Ohm’s law E = IR.
A. -0.002 amp/sec
B. 0.004 amp/sec
C. -0.001 amp/sec
D. 0.003 amp/sec
416. ECE Board Exam November 1991
(
Evaluate the limit Lim x 2 + 3x - 4
x →4
A.
B.
C.
D.
)
24
26
28
30
417. ECE Board Exam November 1994
Evaluate Lim(2 - x ) tan
x →1
A.
B.
C.
D.
πx
2
e2π
e2/π
0
∞
426. ECE Board Exam November 1991
Give the slope of the curve at the
point (1, 1): y = x3/4 – 2x + 1.
A. 1/4
B. -1/4
C. 1 1/4
D. -1 1/4
419. ECE Board Exam April 1993
x2 - 4
x® 2 x - 2
Evaluate lim
0
2
4
6
420. ECE November 1997
Evaluate the limit (ln x)/x as x
approaches positive infinity.
A. 1
B. 0
C. e
D. infinity
421. ECE Board Exam November 1991
Differentiate the equation y =
A.
B.
C.
D.
x 2 + 2x
(x + 1)2
x2
x +1
answer
x
x +1
2x
2x 2
x +1
422. ECE November1997
Given the equation y = (elnx)2, find y’
A. lnx
B. 2( ln x) /x
C. 2x
D. 2eln x
423. ECE March 1996
424. ECE November 1997
If y = x(ln x), find d2y/dx2.
A. 1/x2
B. -1/x
C. 1/x
D. -1/x2
425. ECE Board Exam November 1991
Find the slope of the line tangent to
the curve y = x3 – 2x + 1 at x = 1.
A. 1
B. ½ asss
C. 1/3
D. 1/4
418. ECE Board Exam April 1998
x- 4
Evaluate lim 2
x ® 4 x - x - 12
A. undefined
B. 0
C. infinity
D. 1/7
A.
B.
C.
D.
The derivative of ln (cos x) is
A. sec x
B. -sec x
C. -tan x
D. tan x
427. ECE November 1998
Find the slope of x2y = 8 at the point
(2, 2).
A. 2
B. -1
C. -1/2
D. -2
428. ECE Board Exam April 1999
Find the coordinates of the vertex of
the parabola y = x2 – 4x + 1 by
making use of the fact that at the
vertex, the slope of the tangent line is
zero.
A. (2, -3)
B. (3, -2)
C. (-1, -3)
D. (-2, 3)
429. ECE Board Exam April 1999
Find the equation of the line normal
to x2 + y2 = 5 at the point (2, 1).
A. y = 2x
B. x = 2y
C. 2x+ 3y = 3
D. x + y = 1
430. ECE Board Exam November 1991
In the curve y = 2 + 12x – x3, find the
critical points.
A. (2, 18) and (-2, -14)
B. (-2, 18) and (2, -14)
C. (2, 18) and (2, -14)
D. (-2, 18) and (-2, 14)
431. ECE Board Exam November 1996
Find the radius of curvature of a
parabola y2 – 4x = 0 at point (4, 4).
A. 22.36 units
B. 25.78 units
C. 20.33 units
D. 15.42 units
432. ECE Board Exam November 1996
Find the radius of curvature at any
point in the curve y + ln cos x = 0.
A. cos x
B. 1.5707
C. sec x
D. 1
433. ECE Board Exam April 1999
Find the minimum distance from the
point (4, 2) to the parabola y2 = 8x.
A.
4 3
B.
2 2 answer
C.
3
D.
2 3
434. ECE April 1998
Find the point in the parabola y2 = 4x
at which the rate of change of the
ordinate and abscissa are equal.
A. (1, 2)
B. (2, 1)
C. (4, 4)
D. (-1, 4)
435. ECE Board Exam November 1996
Given a cone of diameter x and
altitude of h. What percent is the
volume of the largest cylinder which
can be inscribed in the cone to the
volume of the cone?
A. 44%
B. 46%
C. 56%
D. 65%
436. ECE Board Exam April 1998
A statue 3 m high is standing on a
base of 4 m high. If an observer’s eye
is 1.5 m above the ground, how far
should he stand from the base in
order that the angle subtended by the
statue is a maximum?
A. 3.41 m
B. 3.51 m
C. 3.71 m
D. 4.41 m
437. ECE Board Exam November 1991
A balloon is released from the ground
100 meters from an observer. The
balloon rises directly upward at the
rate of 4 meters per second. How fast
is the distance between them
changing after 1 second?
A. 1.68 m/sec
B. 1.36 m/sec
C. 1.55 m/sec
D. 1.49 m/sec
438. ECE Board Exam April 1998
A balloon is rising vertically over a
point A on the ground at the rate of
15 ft/sec. A point B on the ground is
level with and 30 ft from A. When the
balloon is 40 ft from A, at what rate is
its distance from B changing?
A. 13 fps
B. 15 fps
C. 12 fps
D. 10 fps
439. ECE Board Exam November 1998
What is the allowable error in
measuring the edge of the cube that
is intended to hold 8 cu. m, if the
error of the computed volume is not
to exceed 0.03 cu. m?
A. 0.002
B. 0.003
C. 0.0025
D. 0.001
440. ECE Board Exam November 1997,
November 1999
2
If y = x ln x, find
A.
B.
C.
D.
d y
dx 2
-1/x
-1/x2
1/x2
1/x
441. ECE Board Exam April 1999
The depth of water in a cylindrical
tank 4 m in diameter is increasing at
the rate of 0.7 m/min. Find the rate at
which the water is flowing into the
tank.
A. 2.5 m3/min
B. 1.5 m3/min
C. 6.4 m3/min
D. 8.8 m3/min
442. ECE Board Exam November 1999
Two posts, one 8 m high and the
other 12 m high are 15 m apart. If the
posts are supported by a cable
running from the top if the first post to
a stake on the ground and then back
to the top of the second post, find the
distance to the lower post to the
stake to use minimum amount of
wire.
A. 6 m
B.
C.
D.
8m
9m
4m
443. ECE Board Exam April 2000
Find the approximate increase by the
use of differentials, in the volume of
the sphere if the radius increases
from 2 to 2.05 in one second.
A. 2.12
B. 2.51
C. 2.86
D. 2.25
444. ECE Board Exam April 2000, April
1999
What is the area of the largest
rectangle that can be inscribed in a
semi-circle of radius 10?
A. 2 50
B. 100
C. 1000
D.
50
445. ECE Board Exam April 2000
If ln(ln y) + ln y = ln x, find y’.
A. x/(x + y)
B. x/(x – y)
C. y/(x + y)
D. y/(x - y)
446. ECE Board Exam April 2000
The volume of the sphere is
increasing at the rate of 6 cm3/hr. At
what rate is its surface area
increasing (in cm2/hr) when the
radius is 50 cm?
A. 0.50
B. 0.30
C. 0.40
D. 0.24
447. ECE Board Exam April 2000,
November 2001
Water is running out a conical funnel
at the rate of 1 cu. in per second. If
the radius of the base of the funnel is
4 in and the altitude is 8 in., find the
rate at which the water level is
dropping when it is 2 in. from the top.
A. -3/2 pi in/s
B. 2/3 pi in/s
C. -4/9 pi in/s
D. -1/9 pi in/s
448. ECE Board Exam November 2000
If y = 2x + sin 2x, find x if y’= 0.
A. π/2
B. π/4
C. 2π/3
D. 3π/2
449. ECE Board Exam November 2000
The equation of the line tangent to
the curve y = x + 5/x at point P(1, 3)
is
A. 4x – y + 7 = 0
B. x + 4y – 7 = 0
C. 4x + y – 7 = 0
D. x – 4y + 7 = 0
450. ECE Board Exam November 2000
If y = arctan (ln x), find dy/dx at x =
1/e.
A. e
B. e/2
C. e/3
D. e^2
451. ECE Board Exam November 2000
Find the change in y = 2x – 3 if x
changes from 3.3 to 3.5.
A. 0.4
B. 0.2
C. 0.5
D. 0.3
452. ECE Board Exam April 2001
The radius of a sphere is r inches at
time t seconds. Find the radius when
the rates of increase of the surface
area and the radius are numerically
equal.
A. 1/8π in.
B. 1/4π in.
C. 2π in.
D. π2 in.
453. ECE Board Exam November 2001
The distance of a body travels as a
function of time and is defined by x(t)
= 18t + 9t2. What is the velocity at t =
3?
A. 36
B. 18
C. 72
D. 54
454. ECE Board Exam November 2001
Find the height of a right circular
cylinder of maximum volume, which
can be inscribed in a sphere of radius
10 cm.
A. 14.12 cm
B. 15.11 cm
C. 12.81 cm
D. 11.55 cm
455. ECE Board Exam November 2001
What is the second derivative of a
function y = 5x3 + 2x + 1?
A. 25x
B. 30x
C. 18
D. 30
456. ECE Board Exam April 1999, April
2002
Find the minimum distance from the
point (4,2) to the parabola y2 = 8x.
A. 4 3
B.
2 2 answer
C.
3
D.
2 3
457. ECE Board Exam November 2002,
November 2004
A statue 3.2 m high stands on a
pedestal such that its foot is 0.4 m
above an observer’s eye level. How
far from the statue must the observer
stand in order that the angle
subtended by the statue will be a
maximum?
A. 1.1 m
B. 1.5 m
C. 1.2 m
D. 1.4 m
458. ECE Board Exam November 2002
A person in a rowboat is 3 km from a
point P on a straight shore while his
destination is 5 km directly east of
point P. If he is able to row 4 km per
hour and walk 5 km per hour, how far
from the destination must he land on
the shore in order to reach his
destination in the shortest possible
time?
A. 1 km
B. 2.5 km
C. 3 km
D. 2 km
459. ECE Board Exam November 2002
What is the slope of the curve y = 1 +
x2 at the point where y = 10?
A. 8
B. 3
C. 9
D. 6
460. ECE Board Exam November 2002
Given the equation: 2y3 = 3x2 – 5.
Determine the slope of the line
tangent at (4, 1).
A. 4
B. 3
C. 1/4
D. 1
461. ECE Board Exam April 2002
What is the maximum area of a
rectangle that can be inscribed in a
right triangle with base of 8 cm and a
height of 6 cm?
A. 12 sq cm
B.
C.
D.
48 sq cm
24 sq cm
50 sq cm
462. ECE Board Exam April 2003
Solve for the radius of a right circular
cone of maximum volume which can
be inscribed in a sphere of radius 12
cm.
A. 2 2
B.
3 2
C.
8 2 answer
D.
5 2
463. ECE Board Exam April 2003
Determine the slope of the tangents
to the parabola y = -x2 + 5x – 6 at the
points of intersection with the x-axis.
A. 2
B. -4
C. 1
D. -2
464. ECE Board Exam April 2003
A drop of ink is placed on a piece of
paper and causes a circular blot that
increases in area at the rate of 1 sq
mm/sec. At what rate does the radius
of the bolt increase when its area is 1
sq mm?
A. 1/ π
B. π2/2
C.
π / 2 π answer
D.
π/ π
465. ECE Board Exam April 2003
Solve for dy/dx if x = 2 + t and y = 1 +
t 2.
A. 2x
B. t
C. 0
D. 2t
466. ECE Board Exam November 2003
Determine the equation of the line
tangent to the parabola y = x2 at the
point (1, 1).
A. y = 2x + 1
B. y = 2 – x
C. y = 2x – 1
D. y = 2 + x
467. ECE Board Exam November 2003
A fisherman on a wharf 1.2 meters
above the level of still water is pulling
a rope tied to a boat at the rate of 2
meters per minute. How fast is the
boat approaching the wharf when
there are two meters of rope out?
A. 2.5 m/min
B.
C.
D.
1.25 m/min
2.0 m/min
3.0 m/min
468. ECE Board Exam November 2003
What is the second derivative of y
with respect to w in the following
equation: y = (3w2 – 4)(3w2 + 4)?
A. 36w3
B. 9w4
C. 9w16
D. 108w2
469. ECE Board Exam November 2003
A stone is thrown into still water and
causes concentric circular ripples.
The radius of the ripples increases at
the rate of 12 inches/sec. At what
rate does the area of the ripples
increases (in sq inch per sec) when
its radius is 3 inches?
A. 402.55
B. 275.60
C. 226.19
D. 390.50
470. ECE Board Exam November 2004
In how many equal parts can a wire,
50 cm long be cut so that the product
of its parts is a maximum?
A. 15
B. 19
C. 13
D. 20
471. ECE Board Exam November 2003
æsin x 3 ÷
ö
ç
÷
Evaluate lim çç
÷
2÷
x® 0 ç
èsin x ÷
ø
A. -1
B. 2pi
C. 0
D. -2
472. ECE Board Exam November 2004
A conical vessel 1 cm deep and with
a radius of 6 cm at the top, is being
filled with water. If the rate at which
the water rises is 2 cm/sec, how fast
is the volume increasing when the
water is 4 cm deep?
A. 3 pi
B. 4 pi
C. 8 pi
D. 16 pi
473. ECE Board Exam November 2004
A customer is using a straw to drink
iced tea from a right circular glass at
the rate of 6 cubic cm per minute. If
the height of the glass is 12 cm and
the diameter is 6 cm, how fast is the
level of the iced tea decreasing at a
constant rate in cm per min?
A. 0.212
B. 1.570
C. 0.318
D. 0.747
474. ECE Board Exam November 2003
A condominium is to be constructed
in a rectangular lot with a perimeter
of 800 m. What is the largest area
that can be enclosed by fencing the
perimeter?
A. 5 hectares
B. 4 hectares
C. 6 hectares
D. 3.5 hectares
475. ECE Board Exam April 2005
The cost of a product is a function of
the quantity q of the product: c(q) = q2
– 2000q + 100. What should be the
quantity for which the cost is a
minimum?
A. 2500
B. 1000
C. 2000
D. 1500
476. ECE November 1996
A rectangular field is to be fenced
into four equal parts. What is the size
of the largest field that can be fenced
this way with a fencing length of 1500
feet if the division is to be paralleled
to one side?
A. 65,200
B. 62,500
C. 64,500
D. 63,500
477. ECE November 1997
Three sides of a trapezoid are each 8
cm long. How long is the 4th side,
when the area of the trapezoid has
the greatest value?
A. 16 cm
B. 15 cm
C. 12 cm
D. 10 cm
478. ECE March 1996, November 1996
The altitude of a cylinder of maximum
volume that can be inscribed in a
right circular cone of radius r and
height h is
A. h/3
B. 2h/3
C. 3h/2
D. h/4
479. ECE November 1995
The height of a right circular cylinder
is 50 inches and decreases at the
rate of 4 inches per second, while the
radius of the base is 20 inches and
increases at the rate of one inch per
second. At what rate is the volume
changing?
A. 11310 cu. m/sec
B. 1275 cu. m/sec
C. 11130 cu. m/sec
D. 1257 cu. m/sec
480. ECE November 1995, March 1996
A point on the curve where the
second derivative of a function is
equal to zero is called
A. maxima
B. minima
C. point of inflection
D. point of intersection
481. ECE November 1995
The point on the curve where the first
derivate of a function is zero and the
second derivative is positive is called
A. maxima
B. minima
C. point of inflection
D. point of intersection
482. ECE November 1996
At the minimum point, the slope of
the tangent line is
A. negative
B. infinity
C. positive
D. zero
483. ECE November 1996
At the inflection point of y = f(x)
where x = a,
A. f”(a) = is not equal to zero
B. f”(a) = 0
C. f”(a) > 0
D. f”(a) < 0
484. ECE April 1998
Point of the derivatives, which do not
exist (and so equals zero) are called
A. stationary points
B. maximum points
C. maximum and minimum points
D. minimum point
485. ECE November 1997
If the second derivative of the
equation of a curve is equal to the
negative of the equation of the same
curve, the curve is:
A. a cissoid
B. a paraboloid
C. a sinusoid
D. an exponential
D.
ax
486. ME Board Exam April 1998
492. ME April 1996
Find the derivative of the function 2x2
+ 8x + 9 with respect to x.
A. DF(x) = 4x - 8
B. DF(x) = 2x + 9
C. DF(x) = 2x + 8
D. DF(x) = 4x + 8
x 2 - 16
4 x-4
Evaluate Lim
x→
A.
B.
C.
D.
0
1
8
16
487. ME Board Exam October 1997
Compute
the
following
x+ 4
lim
x® ¥ x - 4
A. 1
B. 0
C. 2
D. infinite
limit:
488. ME Board Exam April 1997
What is the first derivative of the
expression (xy)x = e?
A. 0
x
B.
y
C.
D.
-y
(1 + ln xy )
x
(1 - ln xy)
-y
x
answer
489. ME Board Exam April 1998
Find the derivative with respect to x
the function
A.
B.
C.
D.
2 - 3x
2
- 2x 2
2 - 3x 2
-3x
2 - 3x 2
answer
- 3x 2
2 - 3x
3x
2
2 - 3x 2
490. ME April 1996
Find the derivative of (x + 5)/(x2 – 1)
with respect to x.
A. DF(x) = (-x2 – 10x – 1)/(x2 - 1)2
B. DF(x) = (-x2 + 10x – 1)/(x2 - 1)2
C. DF(x) = (x2 – 10x – 1)/(x2 - 1)2
D. DF(x) = (-x2 – 10x + 1)/(x2 + 1)2
491. ME April 1996
If a is a simple constant, what is the
derivative of y = xa?
A. axa-1
B. (a – 1)x
C. xa-1
493. ME April 1998
Differentiate ax2 + b to the 1/2 power.
A. -2ax
B. 2ax
C. 2ax+ b
D. ax + 2b
494. ME April 1997
If y = cos x, what is dy/dx?
A. sec x
B. -sec x
C. sin x
D. -sin x
495. ME October 1997
Given the function f(x) = x3 – 5x + 2,
find the value of the first derivative at
x = 2, f’(2).
A. 7
B. 3x2 - 5
C. 2
D. 8
496. ME Board Exam April 1998
Find the partial derivative with
respect to x of the function xy2 – 5y +
6.
A. y2 – 5
B. y2
C. xy – 5y
D. 2xy
497. ME Board Exam October 1997
Find the second derivative of x3 – 5x2
+x=0
A. 10x - 5
B. 6x - 10
C. 3x + 10
D. 3x2 – 5x
498. ME Board Exam April 1998
Given the function f(x) = x to the 3rd
power -6x + 2. Find the first derivative
at x = 2.
A. 6
B. 7
C. 3x2 - 5
D. 8
499. ME April 1996
Find the slope of the line tangent to
the curve y = x3 – 2x + 1 at x = 1.
A. 1
B.
C.
D.
1/2
1/3
1/4
500. ME April 1996
Find the slope of the tangent to a
parabola y = x2 at a point on a curve
where x = 1/2.
A. 0
B. 1
C. 1/4
D. -1/2
501. ME Board Exam April 1998
A box is to be constructed from a
piece of zinc 20 sq. in. by cutting
equal squares from each corner and
turning up the zinc to form the side.
What is the volume of the largest box
that can be so constructed?
A. 599.95 cu. inches
B. 592.58 cu. inches
C. 579.50 cu. inches
D. 622.49 cu. inches
502. ME Board Exam April 1996
The cost C of a product is a function
of the quantity x of the product: C(x)
= x2 – 4000x + 50. Find the quantity
for which the cost is a minimum.
A. 1000
B. 1500
C. 2000
D. 3000
503. ME Board Exam October 1996
What is the maximum profit when the
profit versus production function is as
given below? P is profit and x is unit
of production.
P = 200,000 – x – [1.1/(x + 1)]8
A. 285,000
B. 200,000
C. 250,000
D. 305,000
504. ME Board Exam October 1996
Water is pouring into swimming pool.
After t hours, there are t + t1/2 gallons
in the pool. At what rate is the water
pouring into the pool when t = 9
hours?
A. 7/6 gph
B. 8/7 gph
C. 6/5 gph
D. 5/4 gph
505. ME October 1997
A function is given below, what x
value maximizes y?
y2 + y + x2 – 2x = 5
A. 2.23
B. -1
C.
D.
5
1
506. ME April 1998
If y = x to the 3rd power – 3x. Find the
maximum value of y.
A. 0
B. -1
C. 1
D. 2
507. ME April 1998
As x increases uniformly at a rate of
0.002 feet per second, at what rate is
the expression (1 + x) to the 3rd
power increasing when x becomes 8
feet?
A. 430 cfs
B. 0.300 cfs
C. 0.486 cfs
D. 0.346 cfs
508. ME April 1998
The distance a body travels is a
function of time and is given by x(t) =
16t + 8t2. Find its velocity at t = 3.
A. 64
B. 56
C. 54
D. 44
509.
A.
B.
C.
D.
Past Board Exam Problems
in Differential Equations
510. CE Board Exam May 1997
Find the differential equation of the
family of lines passing through the
origin.
A. ydx – xdy = 0
B. xdy – ydx = 0
C. xdx + ydy = 0
D. ydx + xdy = 0
511. CE Board Exam May 1996
What is the differential equation of
the family of parabolas having their
vertices at the origin and their foci on
the x-axis?
A. 2xdy – ydx = 0
B. xdy + ydx = 0
C. 2ydx – xdy = 0
dy
D.
-x=0
dx
512. CE Board Exam November 1995
Determine the differential equation of
the family of line passing through (h,
k).
A. (y – k)dx – (x – h)dy = 0
B. (y – h) + (y – k) = dy/dx
C. (x – h)dx – (y – k)dy = 0
D. (x + h)dx – (y – k)dy = 0
513. EE Board Exam March 1998
Solve the differential equation: x (y –
1) dx + (x + 1) dy = 0. If y = 2 when x
= 1, determine y when x = 2.
A. 1.80
B. 1.48
C. 1.55
D. 1.63
514. EE Board Exam October 1997
If dy = x2 dx; what is the equation of y
in terms of x if the curve passes
through (1, 1)?
A. x2 – 3y + 3 = 0
B. x3 – 3y + 2 = 0
C. x3 + 3y2 + 2 = 0
D. x3 + 2y + 2 = 0
515. EE Board Exam October 1997
Solve the differential equation dy – x
dx = 0, if the curve passes through
(1, 0).
A. 3x2 + 2y – 3 = 0
B. 2y + x2 – 1 = 0
C. x2 – 2y – 1 = 0
D. 2x2 + 2y – 2 = 0
516. EE Board Exam October 1995
Find the general solution y’ = y sec x.
A. y = C (sec x + tan x)
B. y = C (sec x – tan x)
C. y = C sec x tan x
D. y = C (sec2x + tan x)
517. EE Board Exam April 1998
(
)
Solve y- x 2 + y 2 dx-xdy = 0
A.
x 2 + y 2 + y = C answer
B.
x2 + y2 + y = C
C.
x+y +y=C
D.
x2 - y + y = C
518. EE Board Exam April 1996
Solve xy’ (2y – 1) = y (1 – x)
A. ln (xy) = 2 (x – y) + C
B. ln (xy) = x – 2y + C
C. ln (xy) = 2y – x + C
D. ln (xy) = x + 2y + C
519. EE Board Exam April 1996
Solve (x + y) dy = (x – y) dx.
A. x2 + y2 = C
B.
C.
D.
x2 + 2xy + y2 = C
x2 - 2xy – y2 = C
x2 – 2xy + y2 = C
520. EE Board Exam April 1997
Radium decomposes at a rate
proportional to the amount at any
instant. In 100 years, 100 mg of
radium decomposes to 96 mg. How
many mg will be left after 100 years?
A. 88.60
B. 95.32
C. 92.16
D. 90.72
521. ECE Board Exam April 1998
The equation y2 = cx is the general
solution of
2y
A. y' =
x
2x
B. y' =
y
C.
D.
y
answer
2x
x
y' =
2y
y' =
522. ECE Board Exam November 1998
Find the equation of the curve at
every point of which the tangent line
has a slope of 2x.
A. x = -y2 + C
B. y = -x2 + C
C. y = x2 + C
D. x = y2 + C
523. ECE Board Exam April 1998
Solve (cos x cos y - cot x) dx – sin x
sin y dy = 0
A. sin x cos y = ln (c cos x)
B. sin x cos y = ln (c sin x)
C. sin x cos y = -ln (c sin x)
D. sin x cos y = -ln (c cos x)
524. ECE Board Exam November 1994
Find the differential equation whose
general solution is y = C1x + C2ex
A. (x – 1) y” – xy’ + y = 0
B. (x + 1) y” – xy’ + y = 0
C. (x – 1) y” + xy’ + y = 0
D. (x + 1) y” + xy’ + y = 0
525. ECE Board Exam November 1998
Find the equation of the family of
orthogonal trajectories of the system
of parabolas y2 = 2x + C.
A. y = Ce-x
B. y = Ce2x
C. y = Cex
D. y = Ce-2x
526. ME Board Exam April 1996
What is the solution of the first order
differential equation y(k + 1) = y(k) +
5?
5
A. y(k ) = 4 k
B. y(k) = 20 + 5k
C. y(k ) = C - k , C is a constant
D. The solution is non-existent for
real values of y
527. ME October 1997
Given the following simultaneous
differential equations:
dx
dy
2
−3
+x−y =k
dt
dt
dx
dy
3
+2
− x + cos t = 0
dt
dt
Solve for dy/dt.
2
3
5
3 
cos t + x − y − k 
A.

9
2
2
2 
B.
C.
D.
1
1
3 
−  sin t + x + y 2 − k 
6
9
2 
1
[5x − 3y − 3k − 2 cos t ] answer
13
2 
5
3
3 
cos t + x − y − k 
13 
2
2
2 
528. ME Board Exam April 1998
If the nominal interest rate is 3%, how
much is Php5,000 worth in 10 years
in a continuously compounded
account?
A. Php5,750
B. Php6.750
C. Php7,500
D. Php6,350
529. ME Board Exam October 1997
A
nominal
interest
of
3%
compounded continuously is given on
the account. What is the accumulated
amount of Php10,000 after 10 years?
A. Php13,620.10
B. Php13,500.10
C. Php13,650.20
D. Php13,498.60
A.
B.
C.
D.
D.
540.
A.
B.
C.
D.
532.
A.
B.
C.
D.
541.
A.
B.
C.
D.
533.
A.
B.
C.
D.
542.
A.
B.
C.
D.
534.
A.
B.
C.
D.
543.
A.
B.
C.
D.
535.
A.
B.
C.
D.
544.
A.
B.
C.
D.
536.
A.
B.
C.
D.
545.
A.
B.
C.
D.
537.
A.
B.
C.
D.
546.
A.
B.
C.
D.
538.
A.
B.
C.
D.
530.
A.
B.
C.
D.
547.
A.
B.
C.
D.
539.
531.
A.
B.
C.
548.
B.
C.
D.
A.
B.
C.
D.
0.022
0.033
0.044
C.
D.
555. CE Board Exam May 1995
What is the integral of cos 2x esin2x
dx?
A. esin 2x/2 + C
B. -esin 2x/2 + C
C. -esin 2x + C
D. esin 2x + C
549.
A.
B.
C.
D.
562. CE May 1997
Evaluate the integral of x(x – 5)12 dx
with limits from 5 to 6
A. 81/182
B. 82/182
C. 83/182
D. 84/182
563. CE November 1996
Past Board Exam Problems
in Integral Calculus
556. CE Board Exam November 1996
Evaluate the integral of (3x2 + 9y2) dx
dy if the interior limit has an upper
limit of y and a lower limit of 0, and
whose outer limit has an upper limit
of 2 and lower limit of 0.
A. 10
B. 20
C. 30
D. 40
550. CE Board Exam November 1995
What is the integral of sin6 x cos3 x dx
if the lower limit is zero and the upper
limit is π/2?
A. 0.0203
B. 0.0307
C. 0.0417
D. 0.0543
557. CE May 1999
4dx
Evaluate
3x + 2
A. 4 ln (3x + 2) + C
B. 4/3 ln (3x + 2) + C
C. 1/3 ln (3x + 2) + C
D. 2 ln (3x + 2) + C
551. CE Board Exam November 1994
What is the integral of sin5 x dx if the
lower limit is 0 and the upper limit is
π/2?
A. 0.233
B. 0.333
C. 0.433
D. 0.533
558. CE May 1994
∫
Evaluate the integral of e x
2
+1
2xdx .
x +1
B.
e
+C
ln 2
e2x + C
C.
D.
e x +1 + C answer
2xex + C
2
559. CE May 1995
What is the integral of cos 2x esin
dx?
A. -esin 2x + C
B. esin 2x/2 + C
C. esin 2x + C
D. -esin 2x/2 + C
553. CE Board Exam May 1997
560. CE May 1992
6
Evaluate
x(x − 5)12 dx
5
∫
0.456
0.556
0.656
0.756
554. CE Board Exam November 1996
1 xdx
Evaluate
0 ( x + 1)8
A. 0.011
∫
Evaluate
A.
B.
C.
D.
∫
xdx
if it
( x + 1) 8
has an upper limit of 1 and a lower
limit of 0.
A. 0.022
B. 0.056
C. 0.043
D. 0.031
565. CE November 1998
Evaluate the integral of 3 (sin x)^3 dx
using lower limit of 0 and upper limit
= π/2.
A. 2.0
B. 1.7
C. 1.4
D. 2.3
2
A.
Evaluate the integral of
564. CE November 1997, November 1994
Using lower limit = 0 and upper limit =
π/2, what is the integral of 15 sin7 x
dx?
A. 6.783
B. 6.857
C. 6.648
D. 6.539
552. CE Board Exam May 1996
Find the integral of 12sin5 x cos5 x dx
if the lower limit = 0 the upper limit =
π/2.
A. 0.2
B. 0.3
C. 0.4
D. 0.5
A.
B.
C.
D.
0.114
0.186
2x
tan θ ln sec θdθ
2 (ln sec θ)2 + C
(ln sec θ)2 + C
1/2 (ln sec θ) + C
1/2 (ln sec θ)2 + C
561. CE November 1999
Evaluate the integral of x cos 2x dx
with limits from 0 to π/4
A. 0.143
B. 0.258
566. CE May 1998, May 1996 November
1995
Evaluate the integral of 5 cos6 x sin2
x dx using lower limit = 0 and upper
limit = π/2
A. 0.5046
B. 0.3068
C. 0.6107
D. 0.4105
567. ECE April 1998
Evaluate the integral cos8 3A dA from
0 to π/6.
A. 35π/768
B. 23π/765
C. 27π/363
D. 12π/81
568. CE May 1999
2 2y
Evaluate ∫1 ∫0 ( x 2 + y 2 )dxdy
A.
B.
35/2
19/2
C.
D.
17/2
37/2
569. CE November 2002
Determine the value of the integral of
sin5 3x dx with limits from 0 to π/6.
A. 0.324
B. 0.178
C. 0.275
D. 0.458
570. CE November 2001
Evaluate the integral of x cos (4x) dx
with lower limit of 0 and upper limit of
π/4.
A. 1/8
B. -1/8
C. 1/16
D. -1/16
571. CE Board Exam November 1994
What is the area bounded by x2 = -9y
and the line y + 1 = 0?
A. 3 sq units
B. 4 sq units
C. 5 sq units
D. 6 sq units
572. CE Board Exam May 1995
What is the area (in square units)
bounded by the curve y2 = x and the
line x – 4 = 0?
A. 30/3
B. 31/3
C. 32/3
D. 29/3
573. CE Board Exam May 1996
What is the area (in square units)
bounded by the curve y2 = 4x and x2
= 4y?
A. 5.33
B. 6.67
C. 7.33
D. 8.67
574. CE Board Exam May 1997
Find the area enclosed by the curve
x2 + 8y + 16 = 0, the y-axis and the
line x – 4 = 0.
A. 7.67 sq units
B. 8.67 sq units
C. 9.67 sq units
D. 10.67 sq units
575. CE Board Exam November 1996,
November 1998
Find the area enclosed by r2 = a2 cos
2θ.
A. a
B. 2a
C. a2
D. a3
576. CE Board Exam May 1997
The area enclosed by the ellipse x2/9
+ y2/4 = 1 is revolved about the line x
= 3. What is the volume generated?
A. 355.3
B. 360.1
C. 370.3
D. 365.1
577. CE Board Exam May 1996
The area in the second quadrant of
the circle x2 + y2 = 36 is revolved
about the line y + 10 = 0. What is the
volume generated?
A. 2218.33
B. 2228.83
C. 2233.43
D. 2208.53
578. CE Board Exam November 1995
The area bounded by the curve y2 =
12x and the line x = 3 is revolved
about the line x = 3. What is the
volume generated?
A. 179
B. 181
C. 184
D. 186
579. CE Board Exam November 1994
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 line y – 2 = 0?
A. 28.41
B. 26.81
C. 25.83
D. 27.32
580. CE Board Exam May 1995
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. 53.26
B. 52.26
C. 51.26
D. 50.26
581. CE Board Exam November 1995
Find the moment of inertia, with
respect to x-axis of the area bounded
by the parabola y2 = 4x and the line x
= 1.
A. 2.03
B. 2.13
C. 2.33
D. 2.53
582. CE November 1997
What is the area within the curve r2 =
16 cos θ?
A. 26
B. 28
C. 30
D. 32
583. CE May 1999
Find the area enclosed by r2 = 2a2
cos θ.
A. 2a2
B. a2
C. 4a2
D. 3a2
584. CE May 1998
Find the length of the arc of x3 + y2 =
64 from x = -1 to x = -3 in the second
quadrant.
A. 2.24
B. 2.61
C. 2.75
D. 2.07
585. CE May 1998
The area in the first quadrant,
bounded by the curve y = 2x1/2, the yaxis and the line y – 6 = 0 is revolved
about the line y = 6. Find the centroid
of the solid formed.
A. (2.2, 6)
B. (1.6, 6)
C. (1.8, 6)
D. (2.0, 6)
586. CE November 1998
A solid is formed by revolving about
the y-axis, the area bounded by the
curve x3 = y, the y-axis and the line x
= 8. Find its centroid.
A. (0, 4.75)
B. (0, 4.5)
C. (0, 5.25)
D. (0, 5)
587. CE November 1995
Find the moment of inertia of the area
bounded by the parabola y2 = 4x, xaxis and the line x =1, with respect to
the x-axis.
A. 1.067
B. 1.244
C. 0.968
D. 0.867
588. CE November 2002
Find the length of one arc of the
curve whose parametric equations
are x = 2θ – 2sin θ and y = 2 – 2cos θ.
A. 16
B.
C.
D.
18
14
12
589. CE May 2002
A conical tank 12 ft high and 120 ft
across the top is filled with a liquid
that weighs 62.4 pcf. How much work
is done in pumping all the liquid at
the top of the tank?
A. 58,811 ft-lb
B. 63,421 ft-lb
C. 59,475 ft-lb
D. 47,453 ft-lb
590. CE November 2001
Determine the moment of inertia of
the area bounded by the curve x2 =
4y, the line x – 4 = 0 and the x-axis
with respect to the y-axis.
A. 51.2
B. 25.1
C. 52.1
D. 21.5
591. CE November 2001
Determine the area bounded by the
curves x = 1/y, 2x – y = 0, x = 6 and
the x-axis.
A. 2.138
B. 2.328
C. 2.324
D. 2.638
592. CE May 2001
Find the area bounded by the curve y
= sin x and the x-axis from x = π/3 to
x = π.
A. 9 square units
B. 12 square units
C. 8 square units
D. 6 square units
593. CE May 2001
A body moves such that its
acceleration as a function of time is a
= 2 + 12t, where “t” is in minutes and
“a” is in m/min2. Its velocity after 1
min is 11 m/min. Find its velocity after
2 minutes.
A. 31 m/min
B. 23 m/min
C. 45 m/min
D. 18 m/min
594. CE November 2003
What is the perimeter of the curve r =
4(1 – sin θ)?
A. 32.47
B. 30.12
C. 25.13
D. 28.54
595. CE November 2003
Find the surface area generated by
rotating the first quadrant portion of
the curve x2 = 6 – 8y about the y-axis.
A. 58.41
B. 64.25
C. 61.27
D. 66.38
596. EE Board Exam March 1998
1
Integrate
with respect to x
3x + 4
and evaluate the result from x = 0
and x = 2.
A. 0.278
B. 0.336
C. 0.305
D. 0.252
597. EE Board Exam April 1997
Evaluate the integral of ln x dx, the
limits are 1 and e.
A. 0
B. 1
C. 2
D. 3
598. EE Board Exam October 1997
10 2log10 edx
Evaluate ∫
1
x
A. 2.0
B. 49.7
C. 3.0
D. 5.12
599. EE Board Exam April 1997
Find the integral of [(e exp x – 1)]
divided by [e exp x + 1] dx
A. ln (e exp x – 1) square + x + C
B. ln (e exp x + 1) – x + C
C. ln (e exp x – 1) + x + C
D. ln (e exp x + 1) square – x + C
600. EE Board Exam April 1997
Evaluate the double integral of r sin u
dr du, the limits of r are 0 and cos u
and the limits of u are 0 and pi.
A. 1
B. 1/2
C. 0
D. 1/3
601. EE Board Exam April 1996
Evaluate
A.
B.
C.
D.
π/2 1 2
∫ ∫∫
0
0 0
zdzr 2 dr sin udu
2/3
4/3
1/3
5/3
602. EE Board Exam April 1993
Find the area of the region bounded
by y2 = 8x and y = 2x.
A. 1.22 sq. units
B. 1.33 sq. units
C. 1.44 sq. units
D. 1.55 sq. units
603. EE Board Exam October 1997
Find the area bounded by the curve y
= x2 + 2, and the lines x = 0 and y = 0
and x = 4.
A. 88/3
B. 64/3
C. 54/3
D. 64/5
604. EE Board Exam April 1997
Find the area bounded by the
parabolas y = 6x – x2 and y = x2 – 2x.
A. 44/3 sq. units
B. 64/3 sq. units
C. 74/3 sq. units
D. 54/3 sq. units
605. EE Board Exam October 1997
Find the area bounded by the line x –
2y + 10 = 0, the x-axis, the y-axis and
x = 10.
A. 75
B. 50
C. 100
D. 25
606. ECE Board Exam April 1999
What is the integral of (3t-1)3dt?
1
A.
(3t − 1) 4 + C answer
12
1
B.
(3t − 4) 4 + C
12
1
C.
(3t − 1) 4 + C
4
1
D.
(3t − 1)3 + C
4
607. ECE Board Exam November 1998
Evaluate the integral of dx/(x + 2)
from -6 to -10.
A. 21/2
B. 1/2
C. ln 3
D. ln 2
608. ECE Board Exam November 1998,
ME April 1998
Integrate x cos (2x2 + 7)dx
1
A.
sin(2x 2 + 7) + C answer
4
1
B.
cos(2x 2 + 7) + C
4
C.
D.
sin x
2
+C
4(x + 7)
sin (2x2 + 7) + C
609. ECE Board Exam April 1997
Evaluate the integral of sin6 xdx from
0 to π/2
π
A.
32
2π
B.
17
3π
C.
32
5π
answer
D.
32
610. ECE Board Exam April 1998
Evaluate
A.
B.
C.
D.
π /6
∫0
(cos3A)8 dA
27 π
363
35 π
answer
768
23 π
765
12 π
81
611. ECE Board Exam November 1991
Evaluate the integral of cos2 y dy
y sin2y
A.
+
+ C answer
2
4
B. y + 2 cos y + C
y sin2y
C.
+
+C
4
4
D. y + sin 2y + C
612. ECE Board Exam November 1998
Integrate the square root of (1 – cos
x)dx.
1
A. −2 2 cos x + C answer
2
B.
C.
D.
−2 2 cos x + C
1
2 2 cos x + C
2
2 2 cos x + C
613. ECE Board Exam April 1998
Find the area (in square units)
bounded by the parabolas x2 – 2y = 0
and x2 + 2y – 8 = 0.
A. 11.7
B. 4.7
C. 9.7
D. 10.7
614. ECE April 2002, April 2005
The integral of 34xdx is equal to
A. 44x/ln 3 + C
B. 34x/ln 27 + C
C. 34x/ln 81 + C
D. 44x/ln 12 + C
615. ECE November 2002
Determine the area bounded by the
lines x = 1, x = 3, the x-axis and the
graph f(x) = x2 – 3x.
A. 3.33 square units
B. 2.75 square units
C. 5.67 square units
D. 4.50 square units
616. ECE April 2003
Solve for the area bounded by y = 2x
– x2 and the x-axis.
A. 2/3
B. 4/3
C. 5/3
D. 7/2
617. ECE April 2003
Determine the slope of the tangents
to the parabola y = -x2 + 5x – 6 at the
points of intersection with the x-axis.
A. 2
B. -4
C. 1
D. -2
618. ECE April 2003, November 2003
What is the integral of (2sec2 x – sin
x) dx?
A. 2cosx + tan x + C
B. 2tan x + sin x + C
C. 2sin x + cos x + C
D. 2tan x + cos x + C
619. ECE April 2003
Determine the area bounded by the
curve y = 1/(x^2), the y-axis and the
lines y = 1 and y = 5.
A. 2.0
B. 3.3
C. 2.6
D. 2.47
620. ECE November 2003
Evaluate the integral sin^(4) theta (d)
from 0 to pi/2.
A. 2π/3
B. 3π/5
C. 3π/16
D. π/4
621. ECE November 2003
What is the integral of cos xdx from
pi/3 to pi/2?
A. 0.134
B. 0.500
C.
D.
0.707
0.293
622. ECE November 2004
Determine the coordinates of the
centroid of the area bounded by the
curve x2 = -(y – 4), the x-axis and the
y-axis on the first quadrant.
A. (3/9, 9/5)
B. (3/4, 8/5)
C. (6/5, 4/7)
D. (4/3, 5/8)
623. ECE April 2005
What is the area bounded by the
curve defined by the equation x2 – 8y
= 0 and its latus rectum?
A. 5.33
B. 10.67
C. 7.33
D. 3.66
624. ECE April 2005
The area in the first quadrant
bounded by the parabola 12y = x2,
the y-axis, and the line y = 3,
revolves about the line y = 3. What is
the generated volume?
A. 72.75
B. 80.75
C. 90.48
D. 85.25
625. ECE April 2005
The area bounded by the graphs of y
= 2x + 3 and y = x2 revolves about
the x-axis. Determine the volume
generated.
A. 228
B. 329
C. 255
D. 375
626. ECE November 1995
Find the area bounded by y = (11 –
x)1/x, the lines 3x = 2 and x = 10 and
the x-axis.
A. 19.456 sq units
B. 20.567 sq units
C. 22.567 sq units
D. 21.478 sq units
627. ECE November 1996
Find the area bounded by the y-axis
and x = 4 – y2/3.
A. 25.6
B. 28.1
C. 12.8
D. 56.2
628. ECE November 1995
The velocity of a body is given by f(t)
= sin (πt), where the velocity is given
in meters per second and t is given in
seconds. The distance covered in
meters between t = 0.25 and 0.5
seconds is close to
A. 0.5221 m
B. -0.2251 m
C. 0.2251 m
D. -0.5221 m
629. ECE November 1995
The rate of change of a function of y
with respect to x equals 2 – y, and y =
8 when x = 0. Find y when x = ln (2).
A. 2
B. -5
C. 5
D. -2
630. ME Board Exam April 1995, April
1997
Integrate (7x3 + 4x2)dx
A.
7x3 4x 2
+
+C
3
2
B.
7x 4 4x 2
+
+C
4
5
C.
D.
7x 4 4x 3
+
+ C answer
4
3
4x
7x 4 −
+C
2
D.
13.23
635. ME Board Exam October 1997
What is the area bounded by the
curve y = x3, the x-axis and the line x
= -2 and x = 1?
A. 4.25
B. 2.45
C. 5.24
D. 5.42
636. ME Board Exam April 1999
Find the area in the first quadrant
bounded by the parabola y2 = 4x, x =
1 and x = 3.
A. 9.555
B. 9.955
C. 5.955
D. 5.595
637. ME Board Exam April 1998
What is the area between y = 0, y =
3x2, x = 0 and x = 2?
A. 8
B. 24
C. 12
D. 6
638.
631. ME Board Exam October 1997
Evaluate the integral of cos x dx
limits from π/4 to π/2.
A. 0.423
B. 0.293
C. 0.923
D. 0.329
632. ME Board Exam April 1995, October
1997
The integral of cos x with respect to x
is
A. sin x + C
B. sec x + C
C. -sin x + C
D. csc x + C
A.
B.
C.
D.
Past Board Exam Problems
in Plane Geometry
639. CE Board Exam May 1997
How many sides have a polygon if
the sum of the interior angles is
1080°?
A. 5
B. 6
C. 7
D. 8
633. ME April 1998
Integrate x cos (2x2 + 7)dx
A. 1/4 sin (2x2 + 7) + C
B. sin (2x2 + 7) + C
C. 1/4 cos (2x2 + 7) + C
D. 1/4 (sin(-θ))(2x2 + 7) + C
640. CE Board Exam November 1994
In a triangle ABC, angle A = 45° and
C = 70°. The side opposite angle C is
40 m long. What is the length of the
side opposite angle A?
A. 26.1 m
B. 27.1 m
C. 29.1 m
D. 30.1 m
634. ME Board Exam April 1999
Find the area bounded
parabola x2 = 4y and y = 4.
A. 21.33
B. 33.21
C. 31.32
641. CE Board Exam May 1995
In triangle ABC, angle C = 70°, A =
45°, AB = 40 m. What is the length of
the median drawn from vertex A to
side BC?
A. 36.3 m
by
the
B.
C.
D.
36.6 m
36.9 m
37.2 m
642. CE Board Exam May 1996
What is the radius of the circle
circumscribing an isosceles right
triangle having an area of 162 sq.
cm?
A. 12.73 m
B. 13.52 m
C. 14.18 m
D. 15.55 m
643. CE Board Exam May 1996
Two sides of a triangle are 50 m and
60 m long. The angle included
between these sides is 30°. What is
the interior angle opposite the longest
side?
A. 93.74°
B. 92.74°
C. 90.74°
D. 86.38°
644. CE Board Exam November 1995
The area of a circle circumscribing
about an equilateral triangle is 254.47
sq. m. What is the area of the triangle
in sq. m?
A. 100.25
B. 102.25
C. 104.25
D. 105.25
645. CE Board Exam May 1995
What is the area in sq. m of the circle
circumscribed about an equilateral
triangle with a side 10 cm long?
A. 104.7
B. 105.7
C. 106.7
D. 107.7
646. CE Board Exam November 1992
The area of a triangle inscribed in a
circle is 39.19 cm2 and the radius of
the circumscribing circle is 7.14 cm. If
the two sides of the inscribed triangle
are 8 cm and 10 cm, respectively,
find the third side.
A. 11 cm
B. 12 cm
C. 13 cm
D. 14 cm
647. CE Board Exam November 1994
The area of a triangle is 8346 sq m
and two of its interior angles are
37°25’ and 56°17’. What is the length
of the longest side?
A. 171.5 m
B. 181.5 m
C.
D.
191.5 m
200.5 m
648. CE Board Exam May 1998
A circle having an area of 452 sq. m
is cut into two segments by a chord
which is 6 cm from the center of the
circle. Compute the area of the
bigger segment.
A. 354.89 sq. m
B. 363.56 sq. m
C. 378.42 sq. m
D. 383.64 sq. m
649. CE Board Exam November 1996
Find the area of a quadrilateral
having sides AB = 10 cm, BC = 5 cm,
CD = 14.14 cm and DA = 15 cm, if
the sum of the opposite angles is
equal to 225°.
A. 96 sq m
B. 100 sq m
C. 94 sq m
D. 98 sq m
650. CE Board Exam October 1997
A quadrilateral has sides equal to 12
m, 20 m, 8 m and 16.97 m
respectively. If the sum of the two
opposite angles is equal to 225°, find
the area of the quadrilateral.
A. 100 m2
B. 124 m2
C. 168 m2
D. 158 m2
651. EE Board Exam April 1992
The angle subtended by an arc is
24°. If the radius of the circle is 45
cm, find the length of arc.
A. 16.85 cm
B. 17.85 cm
C. 18.85 cm
D. 19.85 cm
652. EE Board Exam April 1994
Given a triangle, C = 100°, a = 15 m,
b = 20 m. Find c.
A. 26 m
B. 27 m
C. 28 m
D. 29 m
653. EE Board Exam April 1991
From a point outside of an equilateral
triangle, the distances to the vertices
are 10 m, 18 m and 10 m,
respectively. What is the length of
one side of a triangle?
A. 17.75 m
B. 18.50 m
C. 19.95 m
D. 20.50 m
654. EE Board Exam April 1991
The sides of a triangle are 8 cm, 10
cm and 14 cm. Determine the radius
of the inscribed circle.
A. 2.25 cm
B. 2.35 cm
C. 2.45 cm
D. 2.55 cm
655. EE Board Exam April 1991
The sides of a triangle are 8 cm, 10
cm and 14 cm. Determine the radius
of the circumscribing circle.
A. 7.14 cm
B. 7.34 cm
C. 7.54 cm
D. 7.74 cm
656. EE Board Exam April 1992
Two perpendicular chords both 5 cm
from the center of a circle divide the
circle into four parts. If the radius of
the circle is 13 cm, find the area of
the smallest part.
A. 30 cm2
B. 31 cm2
C. 32 cm2
D. 33 cm2
657. EE Board Exam March 1998
A rhombus has diagonals of 32 and
26 inches. Determine its area.
A. 360 in2
B. 280 in2
C. 320 in2
D. 400 in2
658. EE Board Exam October 1992
Determine
the
area
of
the
quadrilateral shown, OB = 80 cm, AO
= 120 cm, OD = 150 cm and ϕ = 25°.
Figure here
A.
B.
C.
D.
2721.66 cm2
2271.66 cm2
2172.66 cm2
2217.66 cm2
659. EE Board Exam April 1990
Find the area (in cm2) of a regular
octagon inscribed in a circle of radius
10 cm.
A. 283
B. 289
C. 298
D. 238
660. EE A Board Exam pril 1990
In a circle with diameter of 10 m, a
regular five pointed star touching the
circumference is inscribed. What is
the area of the part not covered by
the star in m2?
A. 40.5 m2
B. 45.5 m2
C. 50.5 m2
D. 55.5 m2
661. EE Board Exam April 1993
Find the area of a regular pentagon
whose side is 25 m and the apothem
is 17.2 m.
A. 1075 m2
B. 1085 m2
C. 1080 m2
D. 1095 m2
662. EE Board Exam March 1998
A regular pentagon has sides of 20
cm. An inner pentagon with sides of
10 cm is inside and concentric to the
larger pentagon. Determine the area
inside and concentric to the larger
pentagon but outside of the smaller
pentagon.
A. 430.70 cm2
B. 573. 26 cm2
C. 473.77 cm2
D. 516.14 cm2
663. EE Board Exam March 1999
Determine the area of a regular 6star polygon if the inner regular
hexagon has 10 cm sides.
A. 441.68 cm2
B. 467.64 cm2
C. 519.60 cm2
D. 493.62 cm2
664. ECE Board Exam November 1998
Find the angle in mils subtended by a
line 10 yards long at a distance of
5000 yards.
A. 1
B. 2
C. 2.5
D. 4
665. ECE Board Exam April 1999
Assuming that the earth is a sphere
whose radius is 6400 km. Find the
distance along a 3 degree arc at the
equator of the earth’s surface.
A. 335.10 km
B. 533.10 km
C. 353.10 km
D. 353.01 km
666. ECE Board Exam November 1992
Given a circle whose diameter AB
equals 2 m. If two points C and D lie
on the circle and angles ABC and
BAD are 18° and 36°, respectively,
find the length of the major arc CD.
A.
B.
C.
D.
1.26 m
1.36 m
1.63 m
1.45 m
667. ECE Board Exam November 1998
Each angle of the regular dodecagon
is equal to ____ degrees.
A. 135
B. 150
C. 125
D. 105
668. ECE Board Exam March 1996
The sum of the interior angles of a
polygon is 540°. Find the number of
sides.
A. 3
B. 4
C. 5
D. 6
669. ECE Board Exam April 1991
Find the sum of the interior angles of
the vertices of a five pointed
inscribed in a circle.
A. 150°
B. 160°
C. 170°
D. 180°
670. ECE Board Exam November 2004,
March 1996
A circle of radius 6 has half of its area
removed by cutting off a border of
uniform width. Find the width of the
border.
A. 1.76 cm
B. 1.35 cm
C. 1.98 cm
D. 2.03 cm
671. ECE Board Exam April 1991
A square section ABCD has one of its
sides equal to x. Point E is inside the
square forming an equilateral triangle
BEC having one side equal in length
to the side of the square. Find the
angle AED.
A. 130°
B. 140°
C. 150°
D. 160°
672. ECE Board Exam April 1998
The angle of a sector is 30 degrees
and the radius is 15 cm. What is the
area of a sector?
A. 59.8 cm2
B. 89.5 cm2
C. 58.9 cm2
D. 85.9 cm2
673. ECE Board Exam April 1998
The distance between the centers of
the three circles which are mutually
tangent to each other externally are
10, 12 and 14 units. The area of the
largest circle is
A. 72π
B. 23π
C. 64π
D. 16π
674. ECE Board Exam November 1993
The arc of sector is 9 units and its
radius 3 units. What is the area of the
sector in square units?
A. 12.5
B. 13.5
C. 14.5
D. 15.5
675. ECE Board Exam April 1992
A swimming pool is constructed in
the shape of two partially overlapping
identical circles. Each of the circles
has a radius of 9 m and each circle
passes through the center of the
other. Find the area of the swimming
pool.
A. 380 m2
B. 390 m2
C. 400 m2
D. 410 m2
676. ECE Board Exam November 1995
A rectangle ABCD which measures
18 cm by 24 cm is folded once,
perpendicular to diagonal AC, so that
the opposite vertices A and C
coincide. Find the length of the fold.
A. 20.5 cm
B. 21.5 cm
C. 22.5 cm
D. 23.5 cm
677. ECE Board Exam April 1998
A trapezoid has an area of 360 m2
and an altitude of 2 m. Its two bases
have ratio of 4:5. What are the
lengths of the bases?
A. 12, 15
B. 7, 11
C. 8, 10
D. 16, 20
678. ECE Board Exam April 1998
If the sides of a parallelogram and an
included angle are 6, 10 and 80°
respectively, find the length of the
shorter diagonal.
A. 10.63
B. 10.37
C. 10.73
D. 10.23
679. ECE November 1997, November
1999
A regular octagon is inscribed in a
circle of radius 10. Find the area of
the octagon.
A. 228.2
B. 288.2
C. 238.2
D. 282.6
680. ECE November 1997
One side of a regular octagon is 2.
Find the area of the octagon.
A. 31.0
B. 21.4
C. 19.3
D. 13.9
681. ECE November 1997
A piece of wire is shaped to enclose
the square whose area is 169 sq. cm.
It is then reshape to enclose the
rectangle whose length is 15 cm. The
area of the rectangle is
A. 156 sq. cm
B. 165 sq. cm
C. 175 sq. cm
D. 170 sq. cm
682. ECE November 1998
Two triangles have equal bases. The
altitude of one triangle is 3 units more
than its base and the altitude of the
other is 3 units less than its base.
Find the altitudes, if the areas of the
triangle differ by 21 sq. units.
A. 4 and 10
B. 3 and 9
C. 6 and 12
D. 5 and 11
683. ECE November 1998
If
an
equilateral
triangle
is
circumscribed about a circle of 10
cm, determine the side of the
triangle.
A. 64.21 cm
B. 36.44 cm
C. 32.10 cm
D. 34.64 cm
684. ECE April 2000
One leg of right triangle s 20 cm and
the hypotenuse is 2 cm longer than
the other leg. Find the length of the
hypotenuse.
A. 10
B. 15
C. 25
D. 20
685. ECE April 2000
The area of a rhombus is 132 sq. m.
If its shorter diagonal is 12 m, find the
longer diagonal.
A. 20 m
B. 24 m
C. 38 m
D. 22 m
686. ECE April 2000
One of the diagonals of a rhombus is
25 units and the area is 75 square
units. Determine the length of the
sides.
A. 15.47
B. 12.85
C. 12.58
D. 18.25
687. ECE April 2000
You are given one coin, 5 cm
diameter and a large supply of coins
with diameter 2 cm. What is the
maximum number of the smaller
coins that may be arranged
tangentially around the larger without
any overlap?
A. 8
B. 7
C. 10
D. 12
688. ECE November 2000
A piece of wire of length 50 cm is cut
into two parts. Each part is then bent
to form a square. It is found that the
total area of the square is 100 m sq.
m. Find the difference in length of the
sides of the two squares.
A. 6.62
B. 5.32
C. 5.44
D. 6.61
689. ECE November 2002
A right triangle is inscribed in a circle
such that one side of the triangle is
the diameter of a circle. If one of the
acute angles of the triangle measures
60° and the side opposite that angle
has length 15, what is the area of the
circle?
A. 175.15
B. 223.73
C. 235.62
D. 228.61
690. ECE November 2002
What is the ratio of the area of a
square inscribed in a circle to the
area of the square circumscribing the
circle?
A. 1/3
B. 2/5
C.
D.
2/3
1/2
691. ECE November 2003
If the radius of Quezon Memorial
Circle is decreased by 28%, then its
area is decreased by _____.
A. 46.81%
B. 41.86%
C. 41.68%
D. 48.16%
692. ECE April 2003
What is the apothem of a regular
polygon having an area of 235 and a
perimeter of 60?
A. 6.5
B. 8.5
C. 5.5
D. 7.5
between the two planes is expressed
by measuring the
A. dihedral angle
B. plane angle
C. polyhedral angle
D. reflex angle
698. ECE April 1995
A five pointed star is also known as
A. pentagon
B. pentagrom
C. pentagram
D. quintagon
699. ECE April 1995
The area bounded by two concentric
circles is called
A. ring
B. disk
C. annulus
D. sector
693. ECE April 2004, April 1998
The sides of a triangle are 8, 15 and
17 units. If each side is doubled, by
how many square units will the area
of the triangle increase?
A. 120
B. 180
C. 320
D. 240
700. ECE November 1996
In plane geometry, two circular arcs
that together make up a full circle are
called
A. coterminal arcs
B. conjugate arcs
C. half arcs
D. congruent arcs
694. ECE April 2004
What is the area of a rhombus whose
diagonals are 12 and 24 cm,
respectively?
A. 122 cm2
B. 96 cm2
C. 27 cm2
D. 144 cm2
701. ECE April 1998
The area of the region bounded by
two concentric circles is called
A. washer
B. ring
C. annulus
D. circular disk
695. ECE April 2004
A wire with a length of 52 inches is
cut into two unequal parts. Each part
is bent to form a square. If the sum of
the area for the two squares is 97
square inch, what is the area of the
smaller square?
A. 75
B. 25
C. 16
D. 81
696. ECE November 2005
If the sides of a triangle are 3, 4, 5 m,
the area of the inscribed circle is
A. pi square m
B. 2pi square m
C. 3/4 pi square m
D. 3 pi/2 square m
697. ECE April 1995, March 1996
When two planes intersect with each
other, the amount of divergence
702. GE Board Exam February 1992
A regular hexagon is inscribed in a
circle whose diameter is 20 m. Find
the area of the 6 segments of the
circle formed by the sides of the
hexagon.
A. 36.45 sq. m
B. 63.54 sq. m
C. 45.63 sq. m
D. 54.36 sq. m
703. ME Board Exam April 1990
A rat fell on a bucket of a water wheel
with diameter of 600 cm which
traveled an angle of 190° before it
dropped from the bucket. Calculate
for the linear cm that the rat was
carried by the bucket before it fell.
A. 950
B. 965
C. 985
D. 995
It is a plane closed curve, all points of
which are the same distance from a
point within, called the center.
A. arc
B. circle
C. radius
D. chord
704. ME Board Exam April 1999
How many sides are there in a
polygon if each interior angle is 165
degrees?
A. 12
B. 24
C. 20
D. 48
705. ME Board Exam April 1999
Find each interior angle
hexagon.
A. 90°
B. 120°
C. 150°
D. 180°
of
a
706. ME Board Exam April 1996
The area of a circle is 89.42 sq
inches. What is its circumference?
A. 32.25 in
B. 33.52 in
C. 35.33 in
D. 35.55 in
707. ME Board Exam April 1991
Find the difference of the area of the
square inscribed in a semi-circle
having a radius of 15 m. The base of
the square lies on the diameter of the
semi-circle.
A. 171.5 cm2
B. 172.5 cm2
C. 173.5 cm2
D. 174.5 cm2
712. ME April 1998
One fourth of a great circle
A. cone
B. quadrant
C. circle
D. sphere
713.
A.
B.
C.
D.
714.
A.
B.
C.
D.
Past Board Exam Problems
in Probability and Statistics
708. ME Board Exam October 1996, April
1997
The area of a regular hexagon
inscribed in a circle of radius 1 is
A. 1.316
B. 2.945
C. 2.598
D. 3.816
715. CE Board Exam November 1996
How many 4 digit numbers can be
formed without repeating any digit
from the following digits: 1, 2, 3, 4
and 6?
A. 120
B. 130
C. 140
D. 150
709. ME Board Exam October 1996
The area of a circle is 89.42 sq.
inches. What is the length of the side
of a regular hexagon inscribed in a
circle?
A. 5.533 in
B. 5.335 in
C. 6.335 in
D. 7.335 in
716. CE Board Exam November 1998
A coin is tossed 3 times. What is the
probability of getting 3 tails up?
A. 1/8
B. 1/16
C. 1/4
D. 7/8
710. ME April 1998
The sum of the sides of a polygon
A. perimeter
B. square
C. hexagon
D. circumference
711. ME April 1998
717. CE Board Exam May 1996
A bag contains 3 white and 5 black
balls. If two balls are drawn in
succession without replacement,
what is the probability that both balls
are black?
A. 5/16
B. 5/28
C. 5/32
D.
5/14
718. CHE Board Exam November 1996
In how many ways can a committee
of three consisting of two chemical
engineers and one mechanical
engineer can be formed from four
chemical engineers and three
mechanical engineers?
A. 18
B. 64
C. 32
D. none of these
719. EE Board Exam October 1993
In a class of 40 students, 27 like
Calculus and 25 like Chemistry. How
many like both Calculus and
Chemistry?
A. 10
B. 11
C. 12
D. 13
720. EE Board Exam March 1998
In a commercial survey involving
1000 persons, 120 were found to
prefer brand x only, 200 prefer brand
y only, 150 prefer brand z only, 370
prefer either brand x or brand y but
not z, 450 prefer brand y or z but not
x and 370 prefer either brand z or y
but not y. How many persons have
no brand reference, satisfied with any
of the three brands?
A. 280
B. 230
C. 180
D. 130
721. EE Board Exam April 1997
A toothpaste firm claims that in a
survey of 54 people, they were using
either Colgate, Hapee or Close-Up
brands. The following statistics were
found: 6 people used all three
brands, 5 used only Hapee and
Close-Up, 18 used Hapee or CloseUp, 2 used Hapee, 2 used only
Hapee and Colgate, 1 used Close-Up
and Colgate, and 20 used only
Colgate. Is the survey worth paying
for?
A. neither yes or no
B. yes
C. no
D. either yes or no
722. EE Board Exam June 1990
How many permutations are there in
the letters PNRSCE are taken six at a
time?
A. 1440
B.
C.
D.
480
720
360
723. EE Board Exam April 1996
In how many ways can 6 distinct
books be arranged in a bookshelf?
A. 720
B. 120
C. 360
D. 180
724. EE Board Exam April 1997
What is the number of permutations
of the letters in the word BANANA?
A. 36
B. 60
C. 52
D. 42
725. EE Board Exam October 1997
Four different colored flags can be
hung in a row to make coded signal.
How many signals can be made if a
signal consists of the display of one
or more flags?
A. 64
B. 65
C. 68
D. 62
726. EE Board Exam June 1990, April
1993, CHE May 1994
In how many ways can 4 boys and 4
girls be seated alternately in a row of
8 seats?
A. 1152
B. 2304
C. 576
D. 2204
727. EE Board Exam October 1997
There are four balls of different
colors. Two balls are taken at a time
and arranged in a definite order. For
example, if a white and a red balls
are taken, one definite arrangement
is white first, red second, and another
arrangement is red first, white
second.
How
many
such
arrangements are possible?
A. 24
B. 6
C. 12
D. 36
728. EE Board Exam October 1997
There are four balls of different
colors. Two balls at a time are taken
and arranged any way. How many
such combinations are possible?
A. 36
B. 3
C.
D.
6
12
729. EE Board Exam March 1998
How many 6-number combinations
can be generated from the numbers
from 1 to 42 inclusive, without
repetition and with no regards to the
order of the numbers?
A. 850,668
B. 5,245,786
C. 188,848,296
D. 31,474,716
730. EE Board Exam April 1995
In mathematics examination, a
student may select 7 problems from a
set of 10 problems. In how many
ways can he make his choice?
A. 120
B. 530
C. 720
D. 320
731. EE Board Exam April 1997
How many committees can be
formed by choosing 4 men from an
organization of a membership of 15
men?
A. 1390
B. 1240
C. 1435
D. 1365
732. EE Board Exam October 1996
There are five main roads between
the cities A and B, and four between
B and C. In how many ways can a
person drive from A to C and return,
going through B on both trips without
driving on the same road twice?
A. 260
B. 240
C. 120
D. 160
733. EE Board Exam April 1991
There are 50 tickets in a lottery in
which there is a first prize and
second prize. What is the probability
of a man drawing a price if he owns 5
tickets?
A. 50%
B. 25%
C. 20%
D. 40%
734. EE Board Exam April 1996
The probability of getting at least 2
heads when a coin is tossed four
times is
A. 11/16
B. 13/16
C.
D.
1/4
3/8
735. EE Board Exam April 1991
In the ECE board examinations, the
probability that an examinee will pass
each subject is 0.8. What is the
probability that an examinee will pass
at least two subjects out of the three
board subjects?
A. 70.9%
B. 80.9%
C. 85.9%
D. 89.6%
736. ECE Board Exam November 1998
In a club of 40 executives, 33 like to
smoke Marlboro and 20 like to smoke
Philip Morris. How many like both?
A. 10
B. 11
C. 12
D. 13
737. EE Board Exam October 1990
From a bag containing 4 black balls
and 5 white balls, two balls are drawn
one at a time. Find the probability
that both balls are white. Assume that
the first ball is returned before the
second ball is drawn.
A. 25/81
B. 16/81
C. 5/18
D. 40/81
738. EE Board Exam October 1997
A group of 3 people enter a theater
after the lights had dimmed. They are
shown to the correct group of 3 seats
by the usher. Each person holds a
number stub. What is the probability
that each is in the correct seat
according to the numbers on seat
and stub?
A. 1/6
B. 1/4
C. 1/2
D. 1/8
739. ECE Board Exam November 1992
The probability for the ECE board
examinees from a certain school to
pass the subject Mathematics is 3/7
and for the subject Communications
is 5/7. If none of the examinees fails
both subject and there are 4
examinees
that
passed
both
subjects, find the number of
examinees from that school who took
the examinations.
A. 20
B. 25
C.
D.
30
28
C.
D.
740. ECE Board Exam November 1998
If 15 people won prizes in the state
lottery (assuming there are no ties),
how many ways can these 15 people
win first, second, third, fourth and fifth
prizes?
A. 4,845
B. 116,260
C. 360,360
D. 3,003
741. ECE March 1998
A person draws 3 balls in succession
from a box containing 5 red balls, 6
yellow balls and 7 green balls. Find
the probability of drawing the balls in
the order red, yellow and green.
A. 0.3894
B. 0.03489
C. 0.0894
D. 0.04289
742. ECE April 1998
The arithmetic mean of 6 numbers is
17. If two numbers are added to the
progression, the new set of numbers
will have an arithmetic mean of 19.
What are the two numbers if their
difference is 4?
A. 21, 25
B. 23, 27
C. 8, 12
D. 16, 20
743. ECE April 1998
The arithmetic mean
is 55. If two numbers
850 are removed,
arithmetic mean of
numbers?
A. 42.31
B. 50
C. 38.62
D. 57.12
of 80 numbers
namely 250and
what is the
the remaining
744. ECE November 1999
Find the probability of getting exactly
12 out of 30 questions on a true or
false question.
A. 0.12
B. 0.15
C. 0.08
D. 0.04
745. ECE April 2000
How many triangles are formed by 10
distinct points no three of which are
collinear?
A. 120
B. 56
320
720
746. ECE November 2001
In how many ways can 9 books be
arranged on a shelf so that 5 of the
books are always together?
A. 30,240
B. 14,400
C. 15,170
D. 14,200
747. ECE November 2001
Find the probability of getting a prime
number thrice by tossing a die 5
times.
A. 0.4225
B. 0.3125
C. 0.3750
D. 0.1625
748. ECE November 2002
How many 4-digit zip codes are there
if no digit is repeated?
A. 17,280
B. 720
C. 151,200
D. 5,040
749. ECE November 2002
During a board meeting, each
member shakes hands with all the
other members. If there were a total
of 91 handshakes, how many
members were in the group?
A. 12
B. 14
C. 13
D. 15
750. ECE November 2002
How many 3-digit area codes are
there for a telephone company if the
first digit may not be 0 or 1, and the
second digit must be 0 or 1?
A. 360
B. 160
C. 1000
D. 720
751. ECE November 2002
An urn contains white and black
balls. If the probability to pick a white
ball is equal to log x and the
probability that it will be black is equal
to log 2x, what is the value of x?
A. 1.515
B. 2.236
C. 1.732
D. 1.414
752. ECE April 2003
There are 2 copies each of 4 different
books. In how many ways can they
be arranged on a shelf?
A. 5040
B. 1260
C. 2520
D. 1680
753. ECE April 2003
During the board examinations, there
were 350 examinees from Luzon,
250 from Visayas and 400 from
Mindanao. The results of the exams
revealed that the flunkers from
Luzon, Visayas and Mindanao are
3%, 5% and 7% respectively. If a
name of a flunker is picked at
random, what is the probability that
he is from Mindanao?
A. 0.330
B. 0.549
C. 0.42
D. 0.375
754. ECE April 2003
If the odds against event E are 2:7,
find the probability of success.
A. 0.275
B. 0.375
C. 0.368
D. none of these
755. ECE April 2003
If the probability that a basketball
player sinks the basket at 3-point
range is 2/5, determine the
probability of shooting 5 out of 8
attempts.
A. 31.1%
B. 21.3%
C. 28.4%
D. 12.4%
756. ECE November 2003
A statistics of a machine factory
indicates that for every 1000 it
produces there is one reject unit. If a
customer buys 200 units, what is the
probability that the delivery will have
at least one reject unit?
A. 0.8186
B. 0.1814
C. 0.1918
D. 0.1655
757. ECE November 2004
From the digits 0, 1, 2, 3, 4, 5, 6, 7, 8,
9, find the number of six digit
combinations.
A. 84
B. 210
C. 510
D. 126
758. ECE April 2005
A bag contains 3 white balls and 5
red balls. If two balls are drawn in
succession without returning the first
ball drawn, what is the probability that
the balls drawn are both red?
A. 0.357
B. 0.107
C. 0.237
D. 0.299
759. ECE April 2005
A janitor with bunch of 9 keys is to
open a door but only one key can
open. What is the probability that he
will succeed in 3 trials?
A. 0.333
B. 0.255
C. 0.425
D. 0.375
760. ECE April 2005
If there are nine distinct items 3 at a
time, how many permutations will
there be?
A. 252
B. 720
C. 504
D. 336
761. ECE November 2005
Compute the standard deviation of
the following sets of numbers: 2, 4, 6,
8, 10 and 12
A. 3.416
B. 4.206
C. 3.742
D. 5.136
762. ECE Board Exam April 1998
A semiconductor company will hire 7
men and 4 women. In how many
ways can the company choose from
9 men and 6 women who qualified for
the position?
A. 680
B. 540
C. 480
D. 840
763. ECE Board Exam April 1994
There are 13 teams in a tournament.
Each team is to play with each other
only once. What is the minimum
number of days can they all play
without any team playing more than
one game in any day?
A. 11
B. 12
C. 13
D. 14
764. ECE Board Exam March 1996
The probability of getting a credit of
an examination is 1/3. If three
students are selected in random,
what is the probability that at least
one of them got a credit?
A. 19/27
B. 8/27
C. 2/3
D. 1/3
765. GE Board Exam February 1994
A survey of 100 persons revealed
that 72 of them had eaten at
restaurant P and that 52 of them had
eaten at restaurant Q. Which of the
following could not be the number of
persons in the surveyed group who
had eaten at both P and Q?
A. 20
B. 22
C. 24
D. 26
766. ME Board Exam April 1994
A PSME unit has 10 ME’s, 8 PME’s
and 6 CPM’s. If a committee of 3
members, one from each group is to
be formed, how many such
committees can be formed?
A. 2,024
B. 12,144
C. 480
D. 360
767. ME Board Exam October 1992
In how many ways can a PSME
Chapter with 15 directors choose a
President, a Vice President, a
Secretary, a Treasurer and an
Auditor, if no member can hold more
than one position?
A. 360,360
B. 32,720
C. 3,003
D. 3,603,600
768. ME Board Exam October 1997
In how many ways can you invite one
or more of your five friends in a
party?
A. 15
B. 31
C. 36
D. 25
769. ME Board Exam April 1994
From a box containing 6 red balls, 8
white balls and 10 blue balls, one ball
is drawn at random. Determine the
probability that it is red or white.
A. 1/3
B. 7/12
C.
D.
5/12
1/4
770. ME Board Exam April 1996
An urn contains 4 black balls and 6
white balls. What is the probability of
getting 1 black and 1 white ball in two
consecutive draws from the urn?
A. 0.24
B. 0.27
C. 0.53
D. 0.04
771.
A.
B.
C.
D.
772.
A.
B.
C.
D.
773.
A.
B.
C.
D.
774.
A.
B.
C.
D.
775.
A.
B.
C.
D.
776.
A.
B.
C.
D.
777.
A.
B.
C.
D.
778.
Past Board Exam Problems
in Solid Geometry
784. CE Board Exam May 1998
Find the volume of a cone to be
constructed from a sector having a
diameter of 72 cm and a central
angle of 150°.
A. 5533.32 cm3
B. 6622.44 cm3
C. 7710.82 cm3
D. 8866.44 cm3
779. CE Board Exam November 1994
What is the area in sq m of the zone
of a spherical segment having a
volume of 1470.265 cm m if the
diameter of the sphere is 30 m?
A. 465.5 m2
B. 565.5 m2
C. 665.5 m2
D. 656.5 m2
785. CE Board Exam November 1996
A conical vessel has a height of 24
cm and a base diameter of 12 cm. It
holds water to a depth of 18 cm
above its vertex. Find the volume (in
cm3) of its content.
A. 188.40
B. 298.40
C. 381.70
D. 412.60
A.
B.
C.
D.
780. CE Board Exam May 1995
A sphere having a diameter of 30 cm
is cut into 2 segments. The altitude of
the first segment is 6 cm. What is the
ratio of the area of the second to that
of the first?
A. 4:1
B. 3:1
C. 2:1
D. 3:2
781. CE Board Exam November 1996
If the edge of a cube is increased by
30%, by how much is the surface
area increased?
A. 30%
B. 33%
C. 60%
D. 69%
782. CE Board Exam May 1997
A circular cone having an altitude of 9
m is divided into 2 segments having
the same vertex. If the smaller
altitude is 6, find the ratio of the
volume of the small cone to the big
cone.
A. 0.186
B. 0.296
C. 0.386
D. 0.486
783. CE Board Exam November 1997
Find the volume of a cone to be
constructed from a sector having a
diameter of 72 cm and central angle
of 210°.
A. 12367.2 cm2
B. 13232.6 cm2
C. 13503.4 cm2
D. 14682.5 cm2
786. CE Board Exam May 1995
What is the height of a right circular
cone having a slant height of 10x
and a base diameter of 2x?
A. 2x
B. 3x
C. 3.317x
D. 3.162x
787. CE Board Exam November 1995
The ratio of the volume to the lateral
area of a right circular cone is 2:1. If
the altitude is 15 cm, what is the ratio
of the slant height to the radius?
A. 5:6
B. 5:4
C. 5:3
D. 5:2
788. CE Board Exam November 1994
A regular triangular pyramid has an
altitude of 9 m and a volume if 187.06
cu. m. What is the base edge in
meters?
A. 12
B. 13
C. 14
D. 15
789. CE Board Exam November 1995
The volume of the frustum of a
regular pyramid is 135 cu. m. The
lower base is an equilateral triangle
with an edge of 9 m. The upper base
is 8 m above the lower base. What is
the upper base edge in meters?
A. 2
B. 3
C. 4
D. 5
790. CE Board Exam November 1995
A circular cylinder with a volume of
6.5 cu. m us circumscribed about a
right prism whose base is an
equilateral triangle if side 1.25 m.
What is the altitude of the cylinder in
meters?
A. 3.50
B. 3.75
C. 4.00
D. 4.25
791. CE Board Exam May 1996
A circular cylinder is circumscribed
about a right prism having a square
base one meter on an edge. The
volume of the cylinder is 6.283 cu. m.
Find its altitude in meters.
A. 4.00
B. 3.75
C. 3.50
D. 3.25
792. CE Board Exam November 1997
The bases of a right prism are a
hexagon with one of each side equal
to 6 cm. The bases are 12 cm apart.
What is the volume of the right
prism?
A. 1211.6 cm3
B. 2211.7 cm3
C. 1212.5 cm3
D. 1122.4 cm3
793. CE Board Exam May 1996
A mixture compound of equal parts of
two liquids, one white and the other
black, was placed in a hemispherical
bowl. The total depth of the two
liquids is 6 inches. After standing for
a short time, the mixtures separated,
the white liquid settling below the
black. If the thickness of the segment
of the black liquid is 2 inches, find the
radius of the bowl in inches.
A. 7.33
B. 7.53
C. 7.73
D. 7.93
794. CE Board Exam November 1996
The volume of water in a spherical
tank having a diameter of 4 m is
5.236 m3. Determine the depth of the
water in the tank.
A. 1.0
B. 1.2
C. 1.4
D. 1.8
795. CE Board Exam May 1997
The corners of a cubical block
touched the closed spherical shell
that encloses it. If the volume of the
box is 2744 cm3, what volume in cm3
inside the shell is not occupied by the
block?
A. 2714.56
B. 3714.65
C. 4713.53
D. 4613.74
796. EE Board Exam October 1991
How many times do the volume of a
sphere increases if the radius is
doubled?
A. 4 times
B. 2 times
C. 6 times
D. 8 times
797. EE Board Exam April 1992
What is the volume of a frustum of a
cone whose upper base is 15 cm in
diameter and lower base 10 cm in
diameter with an altitude of 25 cm?
A. 3018.87 cm3
B. 3180.87 cm3
C. 3108.87 cm3
D. 3081.87 cm3
798. EE Board Exam April 1993
In a portion of an electrical railway
cutting, the areas of cross section
taken every 50 m are 2556, 2619,
2700, 2610 and 2484 sq m. Find its
volume.
A. 522,600 m3
B. 520,500 m3
C. 540,600 m3
D. 534,200 m3
799. EE Board Exam April 1996
Two vertical conical tanks are joined
at the vertices by a pipe. Initially the
bigger tank is full of water. The pipe
valve is open to allow the water to
flow to the smaller tank until it is full.
At this moment, how deep is the
water in the bigger tank? The bigger
tank has a diameter of 6 ft and a
height of 10 ft, the smaller tank has a
diameter of 6 ft and a height of 8 ft.
Neglect the volume of the water in
the pipeline.
A. 3 200 answer
B.
3
50
C.
3
25
D.
4
50
800. ECE Board Exam April 1995
Each side of a cube is increased by
1%. By what percent is the volume of
the cube increased?
A. 1.21%
B. 2.8%
C.
D.
3.03%
3.5%
801. ECE Board Exam November 1992
Given a sphere of diameter d, what is
the percentage increase in its
diameter when the surface area
increases by 21%?
A. 5%
B. 10%
C. 21%
D. 33%
802. ECE Board Exam November 1992
Given a sphere of diameter d, what is
the percentage increase in its volume
when the surface area increases by
21%?
A. 5%
B. 10%
C. 21%
D. 33%
803. ECE November 1995
Each side of a cube is increased by
10%. By what percent is the volume
of the cube increased?
A. 33.1%
B. 3.31%
C. 0.031%
D. 13.31%
804. ECE November 1996
A reservoir is shaped like a square
prism. If the area of its base is 225
square centimeters, how many liters
will it hold?
A. 337.5
B. 3.375
C. 3375
D. 33.75
805. ECE November 1999
A metal washer 1–inch in diameter is
pierced by a ½-inch hole. What is the
volume of the washer if it is 1/8-inch
thick?
A. 0.074
B. 0.047
C. 0.028
D. 0.082
806. ECE November 1999
Find the approximate change in the
volume of a cube of side “x” inches
caused by increasing its side by 1%.
A. 0.30 x2 in2
B. 0.02x2 in2
C. 0.010x2 in2
D. 0.03x2 in2
807. ECE November 1999
What is the distance between two
vertices of a cube which are farthest
from each other, if an edge measures
8 cm?
A. 13.86
B. 11.32
C. 16.93
D. 14.33
808. ECE April 2000, November 1999
A regular hexagon pyramid has a
slant height of 4 cm and the length of
each side of the base is 6 cm. Find
the lateral area.
A. 62 cm2
B. 52 cm2
C. 72 cm2
D. 82 cm2
809. ECE November 2000
The lateral area of the right circular
water tank is 92 cm2 and its volume is
342 cm3. Determine the radius.
A. 5.56 cm
B. 6.05 cm
C. 7.28 cm
D. 7.43 cm
810. ECE November 2000
A cone and cylinder have the same
height and the same volume Find the
ratio of the radius of the cone to the
radius of the cylinder?
A. 0.577
B. 1.732
C. 0.866
D. 1.414
811. ECE April 2001
It is desired that the volume of the
sphere be tripled. By how many times
will the radius increased?
A. 21/2
B. 31/3
C. 31/2
D. 33
812. ECE November 2001
If the lateral area of a right circular
cylinder is 68 and is volume is 220,
find its radius.
A. 4
B. 3
C. 5
D. 2
813. ECE November 2001
Find the increase in volume of a
spherical balloon when the radius is
increased from 2 to 3 inches.
A. 74.12 cu. in
B. 74.59 cu. In
C. 75.99 cu. in
D.
79.59 cu. in
814. ECE November 2001
A pyramid whose altitude is 5 ft
weight 800 lbs. At what distance from
its vertex must it be cut by a plane
parallel to its base so that the two
solids of equal weight will be formed?
A. 2.52 ft
B. 2.96 ft
C. 3.97 ft
D. 4.96 ft
815. ECE November 2001
The circumference of a great circle if
a sphere is 18π. Find the volume of a
sphere.
A. 3023.6
B. 3043.6
C. 3033.6
D. 3053.6
816. ECE November 2003, November
1999
The volume of two spheres is in the
ratio 27:343 and the sum of their radii
is 10. Find the radius of the smaller
sphere.
A. 5
B. 4
C. 3
D. 6
817. ECE April 2003
The area of one of the faces of an
icosahedron is 5 sq. m. What is the
total surface area of the said solid?
A. 89.3 sq. m
B. 100 sq. m
C. 97.3 sq. m
D. 78.2 sq. m
818. ECE November 2003
A cube of ice is 64 cu. ft. The ice
melts until it becomes a cube, which
is one-half of its original volume.
What is the length of the edge of the
new cube?
A. 7.31 ft
B. 3.17 ft
C. 1.73 ft
D. 3.71 ft
819. ECE April 2004
By how many percent will the volume
of a cube increase if its edge is
increase by 20%?
A. 72.80
B. 17.28
C. 80.00
D. 1.728
820. ECE November 2005
What is the volume of a hexagonal
prism 15 cm high and with one of its
sides equal to 6 cm?
A. 955 cm3
B. 1403 cm3
C. 810 cm3
D. 1205 cm3
sector off and joining the edges to
form a cone. Determine the angle
subtended by the sector removed.
A. 144°
B. 148°
C. 152°
D. 154°
821. ECE November 2005
The total volume of two spheres is
100pi cubic units. The ratio of their
areas is 4:9. What is the volume of
the smaller sphere in cubic units?
A. 75.85
B. 314.16
C. 71.79
D. 242.36
827. ME Board Exam April 1997
A cubical container that measures 2
inches on a side is tightly packed with
6 marbles and is filled with water. All
8 marbles are in contact with the
walls of the container and the
adjacent marbles. All of the marbles
are of the same size. What is the
volume of the water in the container?
A. 0.38 in3
B. 2.5 in3
C. 3.8 in3
D. 4.2 in3
822. ECE November 1996
Prisms are classified according to
their
A. diagonals
B. sides
C. vertices
D. bases
823. ECE November 1996
It is a polyhedron of which two faces
are equal polygons in parallel planes
and
the
other
faces
are
parallelograms
A. tetrahedron
B. prism
C. frustum
D. prismatoid
824. ECE November 1996
Polygons are classified according to
the number of
A. vertices
B. sides
C. diagonals
D. angles
825. ME Board Exam April 1996
Determine the volume of a right
truncated prism with the following
definitions. Let the corners of the
triangular base be defined by A, B
and C. The length of AB = 10 ft., BC
= 9 ft., and CA = 12 ft. The sides A, B
and C are perpendicular to the
triangular base and have a height of
8.6 ft., 7.1 ft., and 5.5 ft., respectively.
A. 413 ft3
B. 311 ft3
C. 313 ft3
D. 391 ft3
826. ME Board Exam October 1991
A circular of cardboard with a
diameter of 1 m be made into a
conical hat 40 cm high by cutting a
828. ME April 1998
The study of the properties of figures
of three dimensions
A. physics
B. plane geometry
C. solid geometry
D. trigonometry
829. ME April 1998
The volume of a circular cylinder is
equal to the product of its base and
altitude
A. postulate
B. theorem
C. corollary
D. axiom
830.
A.
B.
C.
D.
831.
A.
B.
C.
D.
832.
A.
B.
C.
D.
Past Board Exam Problems
in Trigonometry
833. CE Board Exam November 1993
If sin 3A = cos 6B then
A. A + B = 90°
B. A + 2B = 30°
C. A + B = 180°
D. none of these
834. CE Board Exam November 1993
If cos 65° + cos 55° = cos θ, find θ in
radians.
A. 0.765
B. 0.087
C. 1.213
D. 1.421
835. CE Board Exam November 1992
15 

Find the value of sin  arccos 
17 

A. 8/11
B. 8/19
C. 8/15
D. 8/17
836. CE Board Exam November 1992
If tan x = 1/2, tan y = 1/3. What is the
value of tan (x + y)?
A. 1/2
B. 1/6
C. 2
D. 1
837. CE Board Exam November 1993
Find the value of y in the given
equation: y = (1 + cos 2θ)tan θ
A. sin θ
B. cos θ
C. sin 2θ
D. cos 2θ
838. CE Board Exam May 1992
sin θ + cos θ tan θ
Find the value of
cos θ
A. 2sin θ
B. 2cos θ
C. 2tan θ
D. 2cot θ
839. CE Board Exam May 1994
If coversed sin θ = 0.134, find the
value of θ.
A. 30°
B. 45°
C. 60°
D. 90°
840. CE Board Exam November 1997
The angle of elevation of the top of
tower B from the top of the tower A is
28° and the angle of elevation of the
top of tower A from the base of the
tower is B is 46°. The two towers lie
in the same horizontal plane. If the
height of the tower B is 120 m, find
the height of tower A.
A. 66.3 m
B. 79.3 m
C. 87. 2 m
D. 90.7 m
841. CE Board Exam November 1997
Points A and B are 100 m apart and
are of the same elevation as the foot
of a building. The angles of elevation
of the top of the building from points
A and B are 21° and 32° respectively.
How far is A from the building?
A. 259.28 m
B. 265.42 m
C. 271.64 m
D. 277.29 m
842. EE Board Exam October 1996
Solve for x if tan 3x = 5 tan x.
A. 20.705°
B. 30.705°
C. 15.705°
D. 35.705°
843. EE Board Exam October 1997
If sin x cos x + sin 2x = 1, what are
the values of x?
A. 32.2°, 69.3°
B. -20.67°, 69.3°
C. 20.90°, 69.1°
D. -32.2°, 69.3°
844. EE Board Exam April 1997
Solve for G if csc (11G – 16 degrees)
= sec (5G + 26 degrees)/
A. 7 degrees
B. 5 degrees
C. 6 degrees
D. 4 degrees
845. EE Board Exam April 1992
What is the value of A between 270°
and 360° if 2sin2 A – sin A = 1?
A. 300°
B. 320°
C. 310°
D. 330°
846. EE Board Exam October 1991
The sine of a certain angle is 0.6.
Calculate the cotangent of the angle.
A. 4/3
B. 5/4
C. 4/5
D. 3/4
847. EE Board Exam March 1998
1
,
sin13 A
angle A in degrees.
A. 5 degrees
B. 6 degrees
C. 3 degrees
D. 7 degrees
If sec 2 A =
determine
the
848. EE Board Exam October 1992
Evaluate arc cot [2cos (arc sin 0.5)]
A. 30°
B. 45°
C. 60°
D. 90°
849. EE Board Exam March 1998
Solve for x in the equation: arctan (x
+ 1) + arctan (x – 1) = arctan (12).
A. 1.50
B. 1.34
C. 1.20
D. 1.25
850. EE Board Exam October 1997
The sides of a triangular lot are 138
m, 180 m and 190 m. The lot is to be
divided by a line bisecting the longest
side and down from the opposite
vertex. Find the length of the line.
A. 120 m
B. 130 m
C. 125 m
D. 128 m
851. EE Board Exam October 1997
The sides of a triangle are 195, 157
and 210 respectively. What is the
area of the triangle?
A. 73,250 sq. units
B. 10,250 sq. units
C. 14,586 sq. units
D. 11,260 sq. units
852. ECE Board Exam November 1998
Solve for A in the given equation cos2
A = 1 – cos2 A.
A. 45, 125, 225, 335 degrees
B. 45, 125, 225, 315 degrees
C. 45, 135. 225, 315 degrees
D. 45, 150, 220, 315 degrees
853. ECE Board Exam April 1991
Evaluate the following:
sin0° + sin1° + sin2° + L + sin89° + sin90°
cos0° + cos1° + cos 2° + L + cos89° + cos90°
A.
B.
C.
D.
1
0
45.5
10
854. ECE Board Exam April 1991
Simplify the following:
cos A + cosB sin A + sinB
+
sin A − sinB cos A − cosB
A. 0
B. sin A
C. 1
D. cos A
855. ECE Board Exam April 1991
2 sin θ cos θ − cos θ
Evaluate:
1 − sin θ + sin2 θ − cos 2 θ
A. sin θ
B. cos θ
C. tan θ
D. cot θ
856. ECE Board Exam April 1994
Solve for the value of “A” when sin A
= 3.5x and cos A = 5.5x.
A. 32.47°
B. 33.68°
C. 34.12°
D. 35.21°
857. ECE Board Exam November 1996
If sin A = 2.511x, cos A = 3.06x and
sin 2A = 3.939x, find the value of x.
A. 0.265
B. 0.256
C. 0.562
D. 0.626
858. ECE Board Exam April 1998
Points A and B 1000 m apart are
plotted on a plotted on a straight
highway running east and west. From
A, the bearing of a tower C is 32
degrees W of N and from B the
bearing of C is 26 degrees N of E.
Approximate the shortest distance of
tower C to the highway.
A. 364 m
B. 374 m
C. 394 m
D. 384 m
859. ECE Board Exam November 1998
Two triangles have equal bases. The
altitude of one triangle is 3 units more
than its base and the altitude of the
other is 3 units less than its base.
Find the altitude, if the areas of the
triangles differ by 21 square units.
A. 6 and 12
B. 3 and 9
C. 5 and 11
D. 4 and 10
860. ECE Board Exam April 1999
Sin (B – A) is equal to ____, when B =
270° and A is an acute angle.
A. -cos A
B.
C.
D.
cos A
-sin A
sin A
861. ECE Board Exam April 1999
If sec2 A is 5/2, the quantity 1 – sin2 A
is equivalent to
A. 2.5
B. 1.5
C. 0.40
D. 0.60
862. ECE Board Exam April
November 2000
(cos A)4 – (sin A)4 is equal to
A. cos 4A
B. cos 2A
C. sin 2A
D. sin 4A
1999,
863. ECE Board Exam April 1999
Of what quadrant is A, if sec A is
positive and csc A is negative?
A. IV
B. II
C. III
D. I
864. ECE Board Exam November 1998
Csc 520° is equal to
A. cos 20°
B. csc 20°
C. tan 45°
D. sin 20°
865. ECE Board Exam April 1993
Solve for θ in the following equation:
sin 2θ = cos θ
A. 30°
B. 45°
C. 60°
D. 15°
866. ECE Board Exam March 1996
Solve for x in the given equation:
π
arctan(2x) + arctan(x) =
4
A. 0.149
B. 0.281
C. 0.421
D. 0.316
867. ECE Board Exam April 1998
A man finds the angle of elevation of
the top of a tower to be 30 degrees.
He walks 85 m nearer the tower and
finds its angle of elevation to be 60
degrees. What is the height of the
tower?
A. 76.31 m
B. 73.31 m
C. 73.16 m
D. 73.61 m
868. ECE Board Exam April 1994
A pole casts a shadow 15 m long
when the angle of elevation of the
sun is 61°. If the pole is leaned 15°
from the vertical directly towards the
sun, determine the length of the pole.
A. 54.23 m
B. 48.23 m
C. 42.44 m
D. 46.21 m
869. ECE Board Exam November 1991
The captain of a ship views the top of
the lighthouse at ant angle of
elevation of 60° with the horizontal at
an elevation of 6 meters above sea
level. Five minutes later, the same
captain of the ship views the top of
the same lighthouse at an angle of
30° with the horizontal. Determine the
speed of the ship if the telescope is
known to be 50 meters above sea
level.
A. 0.265 m/sec
B. 0.155 m/sec
C. 0.169 m/sec
D. 0.210 m/sec
870. ECE Board Exam November 1998
If
an
equilateral
triangle
is
circumscribed about a circle of 10
cm, determine the side of the
triangle.
A. 34.64 cm
B. 64.12 cm
C. 36.44 cm
D. 32.10 cm
871. ECE Board Exam November 1998
The two legs of a triangle are 300
and 150 m each, respectively. The
angle opposite the 150 m side is 26°.
What is the third side?
A. 197.49 m
B. 218.61 m
C. 341.78 m
D. 282.15 m
872. ECE Board Exam April 1997
The sides of a triangle are 8, 15 and
17 units. If each side is doubled, how
many square units will the area of the
new triangle be?
A. 240
B. 420
C. 320
D. 200
873. ECE March 1996
The hypotenuse of a right triangle is
34 cm. Find the lengths of then two
legs if one leg is 14 cm longer than
the other.
A. 17 cm, 31 cm
B. 16 cm, 30 cm
C. 18 cm, 32 cm
D. 15 cm, 29 cm
874. ECE March 1996
If sin A = 4/5, in quadrant II, sin B =
7/25, B is quadrant I. Find sin (A + B).
A. 3/5
B. 4/5
C. 3/4
D. 2/5
875. ECE March 1998
If 77° + 0.40x = arctan (cot 0.25x),
solve for x.
A. 10°
B. 30°
C. 20°
D. 40°
876. ECE November 1997
Find the value of x in the equation
csc x + cot x = 3.
A. π/4
B. π/2
C. π/3
D. π/5
877. ECE April 1998
Find the angle in mils subtended by a
line 10 yards long at a distance of
5000 yards.
A. 1 mil
B. 2.04 mils
C. 4 mils
D. 2.5 mils
878. ECE April 1999
How many degrees is 4800 mils?
A. 135 deg
B. 270 deg
C. 235 deg
D. 142 deg
879. ECE November 1999
A railroad is to be laid-off in a circular
path. What should be the radius if the
track is to change direction by 30
degrees at a distance of 157.08 m?
A. 300 m
B. 200 m
C. 150 m
D. 250 m
880. ECE November 1999
If (2 log4 x) – (log4 9) = 2, find x.
A. 10
B. 13
C. 12
D. 11
881. ECE November 1999, November
2001
If arctan (x) + arctan (1/3) = π/4, the
value of x is
A. 1/2
B. 1/4
C. 1/3
D. 1/5
882. ECE November 1999
If tan 4A = cot 6A, then what is the
value of angle A?
A. 9°
B. 12°
C. 10°
D. 14°
883. ECE November 1999, November
2001
A central angle of 45° subtends an
arc of 12 cm. What is the radius of
the circle?
A. 15.28 cm
B. 12.82 cm
C. 12.58 cm
D. 15.82 cm
884. ECE November 1999
Given: y = 4cos 2x. Determine its
amplitude.
A. square root of 2
B. 8
C. 2
D. 4
885. ECE April 2000
If A + B + C = 180° and tan A + tan B
+ tan C = 5.67, find the value of (tan
A)(tan B)(tan C).
A. 1.89
B. 5.67
C. 1.78
D. 6.75
886. ECE April 2000
Three times the sine of a certain
angle is twice of the square of the
cosine of the same angle. Find the
angle.
A. 30°
B. 10°
C. 60°
D. 45°
887. ECE April 2001
Solve A of an oblique triangle ABC, if
a = 25, b = 16 and C = 94.1°.
A. 52 degrees and 40 minutes
B. 50 degrees and 30 minutes
C. 54 degrees and 30 minutes
D. 49 degrees and 32 minutes
888. ECE April 2001
If sin A = 2.5x and cos A = 5.5x, find
the value of A in degrees.
A. 24.44°
B. 54.34°
C. 42.47°
D. 35.74°
889. ECE April 2001
Triangle ABC is a right triangle with
right angle at C. If BC = 4 and the
altitude to the hypotenuse is 1, find
the area of the triangle ABC.
A. 2.43
B. 2.07
C. 2.70
D. 2.11
890. ECE April 2001
The measure of 2.25 revolutions
counterclockwise is
A. -810°
B. 810°
C. 805°
D. 825°
891. ECE November 2001
If cot 2A cot 68 = 1, then tan A is
equal to ____.
A. 0.194
B. 0.914
C. 0.419
D. 491
892. ECE April 2002, April 1999
Assuming that the earth is a sphere
whose radius is 6400 km. Find the
distance along a 3 degree arc at the
equator of the earth’s surface.
A. 335.10 km
B. 533.10 km
C. 353.10 km
D. 353.01 km
893. ECE November 2002
A certain angle has an explement 5
times the supplement. Find the angle.
A. 67.5 degrees
B. 108 degrees
C. 135 degrees
D. 58.5 degrees
894. ECE November 2002
Find the height of a tree if the angle
of elevation of its top changes from
20° to 40° as the observer advances
23 meters toward the base.
A. 13.78 m
B. 16.78 m
C. 14.78 m
D. 15.78 m
895. ECE November 2002
A wheel, 3 ft in diameter, rolls down
an inclined plane 30 degrees with the
horizontal. How high is the center of
the wheel when it is 5 ft from the
base of the plane?
A. 4 ft
B. 2.5 ft
C. 3 ft
D. 5 ft
896. ECE November 2002
If the complement of an angle A is
2/5 of its supplement, then A is ____.
A. 45°
B. 75°
C. 60°
D. 30°
897. ECE April 2003
One side of a right triangle is 15 cm
long and the hypotenuse is 10 cm
longer than the other side. What is
the length of the hypotenuse?
A. 13.5 cm
B. 6.5 cm
C. 12.5 cm
D. 16.25 cm
898. ECE April 2003
If tan A = 1/3 and cot B = 2, tan (A –
B) is equal to ____.
A. 11/7
B. -1/7
C. -11/7
D. 1/7
899. ECE April 2003
Three circle of radii 3, 4 and 5 inches,
respectively are tangent each other
externally. Find the largest angle of a
triangle formed by joining the centers.
A. 72.6 degrees
B. 75.1 degrees
C. 73.4 degrees
D. 73.3 degrees
900. ECE April 2003
Find the value of (sec A + tan A)/(sec
A – tan A), if csc A = 2.
A. 4
B. 2
C. 3
D. 1
901. ECE November 2003
If log 2 = x and log 3 = y, what is log
2.4 in terms of x and y?
A. 3x + 2y – 1
B. 3x + y - 1
C. 3x + y + 1
D. 3x – y + 1
902. ECE November 2003
Simplify the expression 4 cos y sin y
(1 – 2sin2 y).
A. sec 2y
B. cos 2y
C. tan 4y
D. sin 4y
903. ECE November 2003
What is the base B of the logarithm
function log 4 = 2/3?
A. 8
B. 2
C. 3
D. 5
904. ECE November 2003
If y = arcsec (negative square root of
2), what is the value of y in degrees?
A. 75°
B. 60°
C. 45°
D. 135°
905. ECE November 2003
If the tangent of an angle of a right
triangle is 0.75, what is the csc of the
angle?
A. 1.732
B. 1.333
C. 1.667
D. 1.414
906. ECE November 2003
If arctan 2x + arctan 3x = 45 degrees,
what is the value of x?
A. 1/6
B. 1/3
C. 1/5
D. 1/4
907. ECE November 2003
If 2log 3 (base x) + log 2 (base x) = 2
+ log 6 (base x), then x equals ____.
A. square root of 3
B. 3
C. 2
D. square root of 2
908. ECE April 2004
Given: log (2x – 3) = 1/2. Solve for the
x if the base is 9.
A. 3
B. 12
C. 4
D. 5
909. ECE November 2004
What is the value of x if log (base x)
1296 = 4?
A. 5
B. 3
C. 6
D. 4
910. ECE April 2004
If sin A = 4/5, sin B = 7/25, what sin
(A + B) if A is in the 3rd quadrant and
B is the 2nd quadrant.
A. -3/5
B. 4/5
C. 3/5
D. 2/5
911. ECE April 2005
A railroad is to be laid-off in a circular
path. What should be the radius if the
track is to change direction by 30
degrees at a distance of 300 m?
A. 300 m
B. 573 m
C. 275 m
D. 325 m
912. ECE Board Exam April 1995
The angle which the line of sight to
the object makes with the horizontal
which is above the eye of the
observer is called
A. angle of depression
B. angle of elevation
C. acute angle
D. bearing
913. ECE Board Exam April 1995
The median of a triangle is the line
connecting the vertex and the
midpoint of the opposite side. For a
given
triangle,
these
medians
intersects at a point which is called
the
A. Orthocenter
B. Circumcenter
C. centroid
D. incenter
914. ECE Board Exam March 1996, April
1996
The altitudes of the sides of a triangle
intersect at the point known as
A. orthocenter
B. circumcenter
C. centroid
D. incenter
915. ECE Board Exam April 1995
The arc length equal to the radius of
the circle is called
A. 1 radian
B. 1 quarter circle
C. π radian
D. 1 grad
916. GE Board Exam August 1994
A ship A started sailing S 42°35’ W at
the rate of 2 kph. After 2 hours, ship
B started at the same point going N
46°20’ W at the rate of 7 kph. After
how many hours will the second ship
be exactly north of ship A?
A. 3.68
B. 4.03
C. 5.12
D. 4.83
917. ME Board Exam April 1991
A man standing in a 48.5 meter
building high, has an eyesight height
of 1.5 m from the top of the building,
took a depression reading from the
top of another nearby building and
nearest wall, which are 50° and 80°
respectively. Find the height of the
nearby building in meters. The man is
standing at the edge of the building
and both buildings lie on the same
horizontal plane.
A. 39.49
B. 35.50
C. 30.74
D. 42.55
918. ME Board Exam October 1996
Angles are measured from the
positive horizontal axis, and the
positive direction is counterclockwise.
What are the values of sin B and cos
B in the 4th quadrant?
A. sin B > 0 and cos B < 0
B. sin B < 0 and cos B < 0
C. sin B > 0 and cos B > 0
D. sin B < 0 and cos B > 0
919. ME Board Exam April 1996
Simplify the equation sin2 θ(1 + cot2
θ)
A. 1
B. sin2 θ
C. sin2 θ sec2 θ
D. sec2 θ
920. ME Board Exam October 1995
Simplify the expression
sec A – (sec A) sin2 A.
A. cos2 A
B. cos A
C. sin2 A
D. sin A
921. ME Board Exam April 1998


3 

Evaluate Arc tan2 cos arcsin

2 



A. π/3
B. π/4
C. π/16
D. π/2
922. ME Board Exam April 1993
An aerolift airplane can fly at an
airspeed of 300 mph. If there is no
wind blowing towards the cast at 50
mph, what should be the plane’s
compass heading in order for its
course to be 30°? What will be the
plane’s ground speed if it flies in this
course?
A. 19.7°, 307.4 mph
B. 20.1°, 309.4 mph
C. 21.7°, 321.8 mph
D. 22.3°, 319.2 mph
923. ME Board Exam November 1994
A wire supporting a pole is fastened
to it 20 feet from the ground and to
the ground 15 feet from the pole.
Determine the length of the wire and
the angle it makes with the pole.
A. 24 ft, 53.13°
B. 24 ft, 36.87°
C. 25 ft, 53.13°
D. 25 ft, 36.87°
924. ME Board Exam April 1997
An observer wishes to determine the
height of a tower. He takes sight at
the top of the tower from A to B,
which are 50 ft apart at the same
elevation on a direct line with the
tower. The vertical angle at point A is
30° and at point B is 40°. What is the
height of the tower?
A. 85.60 ft
B. 92.54 ft
C. 110.29 ft
D. 143.97 ft
925. ME Board Exam April 1993
A PLDT tower and a monument stand
on a level plane. The angles of
depression of the top and bottom of
the monument viewed from the top of
the PLDT tower are 13° and 35°
respectively. The height of the tower
is 50 m. Find the height of the
monument.
A. 29.13 m
B. 30.11 m
C. 32.12 m
D. 33.51 m
926. ME October 1997
In general triangles the expression
(sinA)/a = (sin B)/b = (sin C)/c is
called
A. Euler’s formula
B. Law of cosines
C. Law of sines
D. Pythagorean theorem
927. ME October 1997
An angle more than pi radian but less
than 2*pi radians is
A. straight angle
B. obtuse angle
C. related angle
D. reflex angle
928.
A.
B.
C.
D.
929.
A.
B.
C.
D.
SPHERICAL TRIGONOMETRY
930. CE Board Exam May 1997
A spherical triangle ABC has an
angle C = 90° and sides a = 50° and
c = 80°. Find the value of “b” in
degrees.
A. 73.22
B. 74.33
C. 75.44
D. 76.55
931. EE Board Exam April 1997
A ship on a certain day is at latitude
20° N and longitude 140° E. After
sailing for 150 hours at a uniform
speed along a great circle route, it
reaches a point at latitude 10° S and
longitude 170° E. If the radius of the
earth is 3959 miles, find the speed in
miles per hour.
A. 17.4
B. 15.4
C. 16.4
D. 19.4
932. ECE Board Exam April 1997
The area of a spherical triangle
whose parts are A = 93°40’, B =
64°12’ C = 116°51’ and the radius of
the sphere is 100 m is
A. 15613 sq. m
B. 16531 sq. m
C. 18645 sq. m
D. 25612 sq. m
A.
B.
C.
D.