Fundamentals of Engineering Exam Sample Math Questions
Directions: Select the best answer.
y
of y  x 2 z  3z 2 x  6( x  z ) is:
x
a. 2 xz  3 z 2  6
b. x 2 z  6 zx  6 z
c. 2x  9
d. 2x  6z  6
1. The partial derivative
2. If the functional form of a curve is known, differentiation can be used to determine all of
the following EXCEPT the
a. concavity of the curve.
b. location of the inflection points on the curve.
c. number of inflection points on the curve.
d. area under the curve between certain bounds.
3. Which of the following choices is the general solution to this differential equation:
dy
 5 y  0; y (0)  1 ?
dt
a. e 5t
b. e 5 t
c. e
5 t
d. 5e 5t
4. If D is the differential operator, then the general solution to D  2  y  0
2
C1 e 4 x
b. C1 e 2 x
c. e 4 x C1  C 2 x 
d. e 2 x C1  C 2 x 
a.
5. A particle traveled in a straight line in such a way that its distance S from a given point
on that line after time t was S  20t 3  t 4 . The rate of change of acceleration at time t=2
is:
a. 72
b. 144
c. 192
d. 208
6. Which of the following is a unit vector perpendicular to the plane determined by the
vectors A=2i + 4j and B=i + j - k?
a. -2i + j - k
b.
c.
d.
1
5
1
6
1
6
(i + 2j)
(-2i + j - k)
(-2i - j - k)
7. If sin( y )  x  y , then y  using implicit differentiation would be
a.
y
cos( y )  x
b.
 sin( y )
x2
c.
cos( y)
x
d.
x
cos( x)  x
(Questions 8-10) Under certain conditions, the motion of an oscillating spring and mass is
described by the differential equation
d 2x
 16 x  0, where x is displacement in meters and t is
dt 2
time in seconds. At t=0, the displacement is .08 m and the velocity is 0 m per second; that is
x(0)  .08 and x (0)  0.
8. The solution that fits the initial conditions is:
a. x   sin( 4t )
b. x  .02 sin( 4t )  .08 cos( 4t )
c. x  4 cos( 4t )
d. x  .08 cos( 4t )
9. The maximum amplitude of the motions is:
a. 0.02 m
b. 0.08 m
c. 0.16 m
10. The period of motion is
a.  / 2 sec
b.  sec
c. 2 sec
d. 0.32 m
d. 3 sec
11. The equation of the line normal to the curve defined by the function y ( x)  8  2 x 2 at
the point (1,6) is:
a.
b.
c.
d.
y6
4
x 1
y6 1

x 1 4
y6
 4
x 1
y6
 4
x 1
12. The Laplace transform of the step function of magnitude a is:
a.
1
sa
b.
a
s
c.
a
sa
d.
a
s2
e. s + a
13. The only point of inflection on the curve representing the equation y  x 3  x 2  3 is at x
equal to:
a.
2
3
b.  1
3
c. 0
d. 1
3
e. 2
3
14. The indefinite integral of x 3  x  1 is
a.
b.
c.
d.
x4
4
x3
3
x3
3
x4
4
x2
2
x2

2
x2

2
x2

2

x
x
 xC
 xC
15. Consider a function of x equal to the determinant shown here: f ( x) 
x x2
. The first
x4 x3
derivative f (x ) of this function with respect to x is equal to
a.
b.
c.
d.
3x 2  8 x 3
3x 2  7 x 6
4x3  6x5
x4  x6
Questions 16-18, relate to the three vectors A, B, and C in Cartesian coordinates; the unit
vectors i, j, and k are parallel to the x, y, and z axes, respectively.
A = 2i+3j+k
B = 4i-2j-2k
C = i-k
16. The angle between the vectors A and B is:
a.  180 
b. 0 
c. 45
d. 90 
e. 180 
17. The scalar projection of A on C is:
a.

2
2
b. 
1
14
c. 0
d.
1
14
e.
2
2
18. The area of the parallelogram formed by vectors B and C is:
a.
12
b. 12
c.
17
d. 17
e. 24
19. A paraboloid 4 units high is formed by rotating x 2  y about the y-axis. Compute the
volume for this paraboloid.
a. 4
b. 6
c. 8
d. 10
Questions 20-21. The position x in kilometers of a train traveling on a straight track is given as a
function of time t in hours by the following equation, x 
1 4
t  4t 3  16t 2 . The train moves
4
from point P to point Q and back to point P according to the equation above. The direction from
P to Q is positive; P is the position at time t = 0 hours.
20. What is the train’s velocity at time t = 4hours?
a. -16 km/h
b. -8 km/h
c. 0 km/h
d. 32 km/h
e. 64 km/h
21. What is the train’s acceleration at time t = 4 hours?
a. -16 km/h 2
b. 0 km/h 2
c. 12 km/h 2
d. 16 km/h 2
e. 32 km/h 2
22. Let y  f (x ) be a continuous function on the interval from x=a to x=b. The average
value of the curve between a and b is:
b
b
 xdx
2
 x dx
a.
a
c.
ba
b

b.
a
ba
b
 f ( x)dx
f 2 ( x) dx
a
d.
ba
a
ba
5 1 
1 2 3


23. Let A  
, and B  6 0 . The (2,1) entry of AB is:



1 2 9
4 7
a. 29
b. 53
c. 33
d. 64
24. The general equation of second degree is Ax 2  Bxy  Cy 2  Dx  Ey  F  0.
The values of the coefficients in the general equation for the parabola shown in the diagram
could be
a. B, C, D, and F each equal to 0; A and E both positive.
b. B, C, D, and F each equal to 0; A negative and E positive.
c. A, B, E, and F each equal to 0; C and D positive.
d. A, B, E, and F each equal to 0; D negative and C positive.
e. A, D, E, and F each equal to 0; B negative and C positive.
25. The diagram shown below shows two intersecting curves: y = x, and y 
1 2
x .
3
Which of the following expressions gives the distance from the y-axis to the centroid of
the area bounded by the two curves shown in the diagram?
3
x
a.
2
dx
0
3

1
  x  3 x

dx

2
0
1
x
b.
2
dx
0
1

1
  x  3 x
0
3
c.

  x
0
3
2
1 
 x 3 dx
3 

1

1
3

1
2
  x  3 x
0
3
d.
  x  3 x
0
3
  x  3 x
0

dx

2
2

dx


dx


dx

26. Find the radius for the circle with the equation x 2  6( x  y)  y 2  2 .
a. 1
b. 2
c. 3
d. 4
sin x
x 0
x
27. Find the value of the limit: lim
a. 0
b. ½
c. 1
d. 2
28. Find the area bounded by x  y 2 and y  x 2 .
a. 1/6
b. 1/3
c. ½
d. 2
29. Find the solution as t   , for y   4 y   4 y  20.
a. y = 0
b. y  e 2t
c. y  te 2t
d. y = 5
30. It takes a pigeon X, 2 hours to fly straight home averaging 20 km/h. It takes pigeon Y, 2
hours and 20 minutes to fly straight home averaging 30 km/h. If both pigeons start from
the same spot but fly at an angle 120  to each other, about how far apart do they live?
a. 96 km
b. 87 km
c. 80 km
d. 73 km
31. A matrix B has eigenvalues (characteristic values)  found from which of the following
equations?
a. B  I  0

BI  x
B 

d. Bx  I  0
b.
c.



32. The divergence of the vector function u  xyi  2 y 2 j  yzk at the point (0, 1, 1) is
a. -4
b. -1
c. 0
d. 4
33. A right circular cone cut parallel with the axis of symmetry could reveal a
a. circle
b. hyperbola
c. ellipse
d. parabola
34. If y  cos(x), then
dy

dx
a.
sin( x )
b.
 tan( x) cos( x)
1
sec( x )
d. sec( x) sin( x)
c.
35. Write the compex product 3  4i 1  2i  in polar form.
a.
5.39;158 
b. 11.18;79.7 
c. 11.18;10.3 
d. 5.39;21.8 
  
Questions 36-38: Given the set of equations represented by a i , j x j  ri  , where
2 x1  3x 2  12
2 x1  x 2  2
 
36. Determine the adjoint matrix a i , j
a.
1  3 
  2 2



1 1 

  3 2
1 1 

3 2
b. 
1 3

1 2
c. 
d. 
 
37. Determine the inverse matrix a i , j
a.
 1  3 
  2  2


1
1  1 3 
4 2  2
b.
c.
1 1 1 
2  3 2
d.
1 1  3
5  1 2
 
38. Determine the eigenvalues (characteristic values) associated with the matrix a i , j .
a. 2, 6
b. 5, 0
c. 0, 4
d. 4, -1
39. If log a 10  0.25 , then log 10 a 
a. 4
b. .5
c. .25
d. 2
40. Find the distance between the points P ( 1, -3, 5 ) and Q ( -3, 4, -2 ).
a.
10
b.
14
c. 8
d.
114
41. The rectangular coordinates of a point are (-3, -5.2 ), find the polar coordinates for the
same point.
a. (6,120  ) b. (6,120  ) c. (6,120 ) d. (6,150  )
42. Find the x-value of the absolute minimum for the equation y  6 x  x 3 in the interval
3  x  3.
a. -3
b.  2
c. 0
d. 3
Problems 43 & 44 use the following information: The population of a bacteria colony doubles
every 3 days and has a present population of 50000.
43. What is the equation describing the population growth?
a. 50,000 1t / 3
b.
50,000 2t / 3
c.
50,000  (2)
d.
50,000  (2) t
t
3
44. How long will it take for the population to triple?
a. 1.6 days
b. 4.5 days
c. 4.8 days
d. 7.5 days
Problems 45-46 are based on the following equation: r cos 2 a  sin a
45. What conic section is described by this equation?
a. Circle
b. Ellipse
c. Parabola
d. Hyperbola
46. What are the coordinates of the focus of the conic section?
a. (1,-1) b. (0, -1/4) c. (1/4, 0) d. ( 0, 1/4)
47. The area enclosed by the curve r  2(sin   cos ) is
a. 
b. /2
c. 2
d. 4
48. The derivative of cos 3 5 x is
a. 3 sin 2 5 x
b. 15 sin 2 5 x
c. cos 2 5 x sin x
d.  15 cos 2 (5 x) sin 5 x
2
49.
x
2
1  x 3 dx 
0
a. 52/9
50. If sin  
a.
b.
c.
d.
b. 0
a
c. 52/3
d. 26/3
, which of the following statements must be true?
a2  b2
b 
tan 1   
a 2
b
tan 1  
a
b

cos 1
 
a2  b2 2
a
cos 1

a2  b2