Fundamentals of Engineering Exam Sample Math Questions Directions: Select the best answer.

1. The partial derivative xy∂

∂ of )(63 22 zxxzzxy +++= is:

a. 632 2 ++ zxz b. zzxzx 662 ++ c. 92 +x d. 662 ++ zx

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:

;05 =+ ydtdy

1)0( =y ?

a. te 5 b. te 5− c. te 5− d. te 55 −

4. If D is the differential operator, then the general solution to ( ) 02 2 =+ yD

a. xeC 41

−

b. xeC 21

−

c. ( )xCCe x21

4 +−

d. ( )xCCe x21

2 +−

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 4320 ttS −= . 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. 51

(i + 2j)

c. 61

(-2i + j - k)

d. 61

(-2i - j - k)

7. If yxy ⋅=)sin( , then yʹ′ using implicit differentiation would be

a. xy

y−)cos(

b. 2

)sin(xy−

c. xy)cos(

d. xx

x+)cos(

(Questions 8-10) Under certain conditions, the motion of an oscillating spring and mass is

described by the differential equation ,0162

2

=+ xdtxd

where x is displacement in meters and t is

time in seconds. At t=0, the displacement is .08 m and the velocity is 0 m per second; that is 08.)0( =x and .0)0( =ʹ′x

8. The solution that fits the initial conditions is:

a. )4sin( tx −= b. )4cos(08.)4sin(02. ttx += c. )4cos(4 tx = d. )4cos(08. tx =

9. The maximum amplitude of the motions is: a. 0.02 m b. 0.08 m c. 0.16 m d. 0.32 m

10. The period of motion is

a. 2/π sec b. π sec c. π2 sec d. π3 sec

11. The equation of the line normal to the curve defined by the function 228)( xxy −= at the point (1,6) is:

a. 416=

−−xy

b. 41

16=

−−xy

c. 416

−=−−xy

d. 416

−=++xy

12. The Laplace transform of the step function of magnitude a is:

a. as +1

b. sa

c. asa+

d. 2sa

e. s + a

13. The only point of inflection on the curve representing the equation 323 −+= xxy is at x

equal to: a. 3

2− b. 31− c. 0 d. 3

1 e. 32

14. The indefinite integral of 13 +− xx is

a. xxx+−

24

24

b. xxx+−

23

23

c. Cxxx++−

23

23

d. Cxxx++−

24

24

15. Consider a function of x equal to the determinant shown here: 34

2

)(xxxx

xf = . The first

derivative )(xf ʹ′ of this function with respect to x is equal to

a. 32 83 xx − b. 62 73 xx + c. 53 64 xx − d. 64 xx −

Questions 16-17, 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. 22

− b. 141

− c. 0 d. 141

e. 22

18. The polar coordinates for the Cartesian coordinate pair ( -3, -5.2) is:

a. !150,6 b. !120,6 − c. !60,6 d. !150,6 −

19. A paraboloid 4 units high is formed by rotating yx =2 about the y-axis. Compute the volume for this paraboloid.

a. 4π b. 6π c. 8π d. 10π

20. Let )(xfy = be a continuous function on the interval from x=a to x=b. The average

value of the curve between a and b is:

a. ab

dxxb

a

−

∫ 2

c. ab

xdxb

a

−

∫

b. ab

dxxfb

a

−

∫ )(2

d. ab

dxxfb

a

−

∫ )(

21. Let ⎥⎦

⎤⎢⎣

⎡=

921321

A , and

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

701

465

B . The (2,1) entry of AB is:

a. 29 b. 53 c. 33 d. 64

22. The general equation of second degree is .022 =+++++ FEyDxCyBxyAx

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.

23. Use Euler’s method to compute x(0.15) from the differential equation 13 += xdtdx

, with

initial condition 2)0( −=x , where the step size h = 0.15. a. -3.05 b. -2.7 c. 2.7 d. 3.05

24. A graph of 0523 =−− xx indicates there is a root close to x = 2. Use one iteration of Newton’s method (aka Newton-Raphson method), to find the approximation, 2x given that our guess is .21 =x

a. 1.9 b. 2.05 c. 2.1 d. 2.09

25. The diagram shown below shows two intersecting curves: y = x, and 2

31 xy = .

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?

a.

∫

∫

⎟⎠

⎞⎜⎝

⎛+

3

0

2

3

0

2

31 dxxx

dxx

b.

∫

∫

⎟⎠

⎞⎜⎝

⎛+

1

0

2

1

0

2

31 dxxx

dxx

c.

∫

∫

⎟⎠

⎞⎜⎝

⎛−

⎟⎠

⎞⎜⎝

⎛−

3

0

2

3

0

32

3131

dxxx

dxxx

d.

∫

∫

⎟⎠

⎞⎜⎝

⎛−

⎟⎠

⎞⎜⎝

⎛−

3

0

2

3

0

3

3131

dxxx

dxxx

26. Find the value of the limit: xx

x

sinlim0→

a. 0 b. ½ c. 1 d. 2

27. *The following equation describes a 2nd order system: )(452

2

txydxdy

dxyd

=++ . The

system may be described as: a. non-linear b. over-damped c. critically damped d. under-damped

28. Find the solution as ∞→t , for .2044 =+ʹ′+ʹ′ʹ′ yyy

a. y = 0 b. tey 2−= c. ttey 2−= d. y = 5

29. 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

30. A matrix B has Eigen-values (characteristic values) λ found from which of the following equations?

a. 0=− IB λ

b. xBI !λ= c. λ=B

d. 0=− IxB λ!

31. The divergence of the vector function kyzjyixyu!!!

−+= 22 at the point (0, 1, 1) is a. -4 b. -1 c. 0 d. 4

32. A right circular cone cut parallel with the axis of symmetry could reveal a a. circle b. hyperbola c. ellipse d. parabola

33. If ),cos(xy = then =dxdy

a. )sin(x c. )sec(

1x

b. )cos()tan( xx− d. )sin()sec( xx

34. Write the complex product ( )( )ii 2143 −+ in polar form.

a. !158;39.5 − b. !7.79;18.11 c. !3.10;18.11 − d. !8.21;39.5 −

Questions 35-37: Given the set of equations represented by [ ][ ] [ ]ijji rxa =, , where

221232

21

21

=+

=+

xxxx

35. Determine the adjoint matrix [ ]+jia ,

a. ⎥⎦

⎤⎢⎣

⎡

−

−

2231

b. ⎥⎦

⎤⎢⎣

⎡

−

−

2311

c. ⎥⎦

⎤⎢⎣

⎡

2311

d. ⎥⎦

⎤⎢⎣

⎡

2131

36. Determine the inverse matrix [ ] 1,

−jia

a. ⎥⎦

⎤⎢⎣

⎡

−−

−−

2231

b. ⎥⎦

⎤⎢⎣

⎡

−

−

2231

41

c. ⎥⎦

⎤⎢⎣

⎡

−

−

2311

21

d. ⎥⎦

⎤⎢⎣

⎡

−

−

2131

51

37. Determine the Eigen-values (characteristic values) associated with the matrix [ ]jia , .

a. 2, 6 b. 5, 0 c. 0, 4 d. 4, -1

38. If 25.010log =a , then =a10log a. 4 b. 5. c. .25 d. 2

39. Find the distance between the points P ( 1, -3, 5 ) and Q ( -3, 4, -2 ).

a. 10 b. 14 c. 8 d. 114

40. The roots of 1

6116 23

++++

=x

xxxF , are:

a. -1, -2, -3 b. 2, -3 c. -2, -3 d. 2, 3

41. Find the x-value of the absolute minimum for the equation 36 xxy −= in the interval 33 ≤≤− x .

a. -3 b. 2− c. 0 d. 3 Problems 42 & 43 use the following information: The population of a bacteria colony doubles every 3 days and has a present population of 50000.

42. What is the equation describing the population growth? a. ( )3/1000,50 t+

b. 3/2000,50 t

c. 3)2(000,50t

⋅

d. t)2(000,50 ⋅

43. 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

44. *The area enclosed by the curve )cos(sin2 θθ +=r is

a. π b. π/2 c. 2π d. 4π

45. The derivative of x5cos3 is a. x5sin3 2 b. x5sin15 2 c. xxsin5cos2 d. xx 5sin)5(cos15 2−

46. =+∫ dxxx2

0

32 1

a. 52/9 b. 0 c. 52/3 d. 26/3

47. If 22

sinba

a+

=α , which of the following statements must be true?

a. απ−=−

2tan 1

ab

b. α−=−

ab1tan

c. απ−=

+

−

2cos

22

1

bab

d. α=+

−

22

1cosba

a

48. Suppose 2)( ttf = . The area under the curve for 20 ≤≤ t , estimated by using the

trapezoidal rule with 5.0=Δt , is most nearly: a. 4.0 b. 2.75 c. 2.67 d. 1.33

49. The expression 3 )sin()cos( θθ ⋅+ i is exactly equal to:

a. ( ) ( )θθ 3sin3cos ⋅+ i c. ( ) ( )3sin3cos θθ ⋅+ i

b. ( ) ( )33 sincos θθ ⋅− i d. ( ) ( )θθ 3sin3cos ⋅− i

50. The equation of a sphere with center at ( 0, 1, -2 ) and a radius of 9 is: a. 81)2()1( 222 =++−+ zyx

b. 81)2()1( 222 =−+++ zyx

c. 9)2()1( 222 =−+++ zyx

d. 9)2()1( 222 =++−+ zyx