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Complex Numbers Worksheet

Complex Numbers Worksheet
  • Page 1
 1.  
Find the product of 5 (cos 30° + i sin 30°) and 4 (cos 90° + i sin 90°).
a.
10 (1 + i 3)
b.
10 (- 1 + i 3)
c.
10 (1 - i 3)
d.
- 10 (1 + i 3)


Solution:

5 (cos 30° + i sin 30°) × 4 (cos 90° + i sin 90°).

= 20 (cos 120° + i sin 120°)
[(cosθ1+i sinθ1)×(cosθ2+i sinθ2) = cos (θ1+θ2) + i sin (θ1+θ2).]

= 20 ( cos(180° - 60°) + i sin (180° - 60°))
[Rewrite 120° as (180° - 60°).]

= 20 (- cos 60° + i sin 60°)

= 20 (-1 / 2+ i 32)
[Use cos 60° = 1 / 2, sin 60° = 32.]

= 10 (- 1 + i 3)
[Simplify.]

So, the product of 5 (cos 30° + i sin 30°) and 4 (cos 90° + i sin 90°) = 10(- 1 + i 3).


Correct answer : (2)
 2.  
What is standard form of the complex number?
(5 + i)3
a.
110i + 74
b.
110 + 74i
c.
110i - 74
d.
110 - 74i


Solution:

(5 + i)3 = 125 + 75i + 15i2 + i3
[Use (a+b)3 = a3 + 3a2b + 3ab2 + b3.]

= 125 + 75i - 15 - i
[Use i2 = - 1, i3 = - i.]

= 110 + 74i
[Simplify.]

So, the standard form of the given complex number is 110 + 74i.


Correct answer : (2)
 3.  
Simplify:
- 49
a.
1 7
b.
- 7i
c.
7i
d.
7


Answer: (c)


Correct answer : (3)
 4.  
Simplify:
-16
a.
- 4i
b.
1 4
c.
4i
d.
4


Answer: (c)


Correct answer : (3)
 5.  
Simplify:
-13
a.
- i13
b.
i13
c.
13
d.
113


Answer: (b)


Correct answer : (2)
 6.  
Simplify:
-11
a.
11
b.
111
c.
i11
d.
- i11


Answer: (c)


Correct answer : (3)
 7.  
Simplify:
-7
a.
i7
b.
7
c.
- i7
d.
17


Answer: (a)


Correct answer : (1)
 8.  
Simplify:
(4 - 7i) + (- 3 + 2i)
a.
1 - 5i
b.
- 1 - 5i
c.
- 1 + 5i
d.
1 + 5i


Answer: (a)


Correct answer : (1)
 9.  
Simplify:
(- 6 + 5i) - (3 - 2i)
a.
- 6 + 7i
b.
7 - 9i
c.
- 9 + 7i
d.
- 6


Answer: (c)


Correct answer : (3)
 10.  
Simplify:
(7 - 3i)(7 + 3i)
a.
58
b.
- 49
c.
7
d.
49


Answer: (a)


Correct answer : (1)

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