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MTH082 Review Problems for Test 2
This is not a sample test. These problems are designed to get you started on your review for the test. Study the
homework from the textbook, your class notes, and your textbook for a more complete review.
Section 14.1
1. Simplify each problem and express it in the form a + bj .
a) j(3− 5 j)
b)
6 − −8
2
Section 14.2
2. Perform the indicated operation. Express answers in the form a + bj .
a) (4 + 2 j) + (7 − 3 j)
b) (3 + 5 j)(5 − 3 j)
c)
4+2j
6−2j
Section 14.3
3. Locate each point in the complex plane and express the number in rectangular form.
a) 2cis(115! )
b) 4 /240!
c) 5 cos(60! ) + j sin(60! )
(
)
4. Locate each point in the complex plane and express the number in polar form.
a) 3 + 3 j
b) −2 − 7 j
c) 6 − 3 j
Section 14.4
5. Express each number in exponential form.
⎛
⎛π⎞
⎛ π ⎞⎞
a) 3 ⎜ cos ⎜ ⎟ + j sin ⎜ ⎟ ⎟
b) −4 − 3 j
⎝ 4⎠
⎝ 4⎠⎠
⎝
c) 15 /150!
6. Perform each of the operations.
a) 5e2 j i 3e3 j
b) ( 5e7 j )
2
(
c) 27e6 j
)
1
3
d) 15e20 j ÷ 3e5 j
Section 14.5
7. Perform each of the operations and give your answers in polar form.
a) 3cis(48! ) i 5cis(23! )
b) 2 /30! i 5 /11!
c)
4 /174 !
2 /12!
8. Find the cube roots of −27 j .
Section 14.6
9. Use the formulas: Z = Z1 + Z 2 and Z =
Z1Z 2
to find the total impedence a) in series, and b) in parallel,
Z1 + Z 2
when Z1 = 4 − 7 j and Z 2 = −3+ 4 j
10. Use the RLC diagram, at right, to help determine the magnitude and
phase angle of Z, if: R = 35Ω , X L = 45Ω , and X C = 25Ω .
XL
imaginary
axis
Z = R + jX
X = XL - X C
φ
-X C
R
Real
Axis
Section 10.1
11. Find the period, amplitude, frequency, and sketch one cycle of each function. Clearly show the scale of
your graphs.
a) y = 7sin(π x)
b) y = 5sin(2π x)
c) y = 3sin(2x)
Section 10.3
12. Give the period, amplitude, frequency, and horizontal displacement, and sketch a graph of one cycle of
each function. Clearly show the scale of your graphs.
π⎞
π⎞
⎛
⎛
a) y = 5sin ⎜ 2x + ⎟
b) y = 3sin ⎜ π x − ⎟
⎝
⎝
2⎠
3⎠
Section 10.3
13. For the function y = 3sin(4x) − 2sin(2x) , determine the period and maximum amplitude, if they exist.
Determine a reasonable viewing window and graph two cycles on a graphing calculator.
Section 10.5
14. Sketch the sine waves represented by the given phasor diagrams.
4
Β
a)
b)
Α
4
30
8
−45
Α
3.5
Β
15. A weight hanging from a spring vibrates in simple harmonic motion. The amplitude is 5cm and the
frequency is 6Hz. Write an equation for the weight’s position y, at time t, if y = 0 when t = 0 sec.
16. In an ac circuit, the current, I, is given by: I = 5sin(60π t) . What is the amplitude, period, frequency, and
angular velocity?
SOLUTIONS:
1. a) 5 + 3 j
b) 3 − 1.414 j
3. a) −.845 + 1.813 j
b) −2 − 3.464 j
4. a) 4.243 cis(45! )
π
j
2. a) 11− j
c) 15e2.618 j
6. a) 15e5 j
b) 25e14 j
c) 3e2 j
7. a) 15 cis(71! )
b) 10 /41!
c) 2 /162!
8. 3j, −2.598 − 1.5 j , 2.598 − 1.5 j
Math 082/TM/W’15/02/11/2016
c)
1 1
+ j
2 2
c) 2.5 + 4.330 j
b) 7.280 cis(254.0! )
b) 5e3.786 j
5. a) 3e 4
b) 30 + 16 j
c) 6.708 cis(333.4 ! )
d) 5e15 j
9. a) 1− 3 j
b) −9.5 + 8.5 j
10. 40.31 /29.7!
2
11. a) period: 2
amplitude: 7
frequency: ½
c) period: π
amplitude: 3
1
frequency:
π
b) period: 1
amplitude: 5
frequency: 1
7
5
5
1
2
.25
.5
.75
1
-5
-7
3
−π
π/4
π/2
3π/4
π
−3
−5
12. a) period: π
amplitude: 5
1
frequency:
π
5
π
4
H Displacement: −
π
4
π
4
π
2
3π
4
-5
b) period: 2
amplitude: 3
1
frequency:
2
1
H Displacement:
3
3
1/3
4/3
7/3
−3
13. period: π , Maximum Amplitude: 5
14. a) yA = 8sin(x)
b) yA = 4 sin(x)
yB = 4 sin(x + 30! )
yB = 3.5sin(x − 45! )
15. y = 5sin(12π t)
16. amplitude: 5, period:
Math 082/TM/W’15/02/11/2016
1
, frequency: 30, ω : 60π
30
3