Free Printable Q3 Exam Review

1.

What is the modulus?

a)

The absolute value

b)

can be denoted as r

c)

the distance from the origin to the point on the graph

d)

All of the above

2.

What is the argument?

a)

$\tan^{-1}\left(\frac{b}{a}\right)$

b)

The angle measured from the positive real axis to the plotted point "line"

c)

The absolute value of $\theta$

d)

All of the above

3.

What is the complex conjugate of a complex number?

a)

Changing the sign of the imaginary part

b)

Multiplying the complex number by scalar -1

c)

Can be denoted as $\left|z\right|$

d)

All of the above

4.

What is true about the nth root of a complex number?

a)

They are the answer(s) when you take the $\frac{1}{n}$ power of a complex number

b)

They are the answer(s) when you take the n power of a complex number

c)

Can only find it when the complex number is in polar form

d)

Can only find it when the complex number is in rectangular form

5.

Which of the following is a real number?

a)

$\pi$

b)

$\log102$

c)

$4i^{22}$

d)

All of the above

6.

Which of the following is an imaginary number?

a)

$\left(1+2i\right)^3$

b)

$\left(2i\right)^{\frac{1}{5}}$

c)

$i\sqrt{\left(-16\right)}$

d)

All of the above

7.

Which of the following is a complex number?

a)

$\left(4-2i\right)^2$

b)

$\left(-i\pi\right)^3$

c)

$-41i\sqrt{-20}$

d)

All of the above

8.

Which of the following complex numbers is in rectangular form?

a)

-41

b)

$2e^{i35}$

c)

$3\left(\cos15+i\sin15\right)$

d)

None of the above

9.

Which of the following complex numbers is in standard Euler form?

a)

$4ie$

b)

$3e^i$

c)

$-4e^{i120}$

d)

All of the above

10.

Which of the following complex numbers is in standard polar form?

a)

$-8\left(\cos\pi+i\sin\pi\right)$

b)

$3\left(\sin45+i\cos45\right)$

c)

$3i\left(\cos3\pi+i\sin3\pi\right)$

d)

None of the above

11.

What does z* represent?

a)

The absolute value of a complex number z

b)

The angle of a complex number z

c)

The complex conjugate of a complex number z

d)

None of the above

12.

What does $\left|z\right|$ represent?

a)

The modulus of a complex number z

b)

The argument of a complex number z

c)

The complex conjugate of a complex number z

d)

None of the above

13.

Find the argument of $\frac{x+yi}{x^3-y^2}$ if x and y are both positive real numbers where x>y

a)

45

b)

$\tan^{-1}\left(\frac{\left(x+y\right)}{\left(x^3-y^2\right)}\right)$

c)

$360-\tan^{-1}\left(\frac{\left(x+y\right)}{\left(x^3-y^2\right)}\right)$

d)

$\tan^{-1}\left(\frac{y}{x}\right)$

14.

Let z be a complex number - find the conjugate of z

$z^{ }+3i=\left(4-2i\right)^2$

a)

12+19i

b)

-217-456i

c)

4-5i

15.

Find x and y using system of equations:

$-2\sqrt{x^2+y^2}=5\left(2-i\right)+x-yi$

a)

y=-5, x=0 or $\frac{4}{3}$

b)

y=-5, x=0

c)

y=5, x=5 or $-\frac{3}{4}$

d)

y=5, x=-5

16.

Simplify $\left[4\left(\cos135+i\sin135\right)\right]^3$

a)

$4^{\frac{1}{3}}\left(\cos\left(\frac{135}{3}\right)+i\sin\left(\frac{135}{3}\right)\right),$ $\ 4^{\frac{1}{3}}\left(\cos\left(\frac{135+360}{3}\right)+i\sin\left(\frac{135+360}{3}\right)\right),\$ $4^{\frac{1}{3}}\left(\cos\left(\frac{135+720}{3}\right)+i\sin\left(\frac{135+720}{3}\right)\right)$

b)

$4^3\left(\cos\ 135+i\sin135\right)$

c)

$4^3\left(\cos45+i\sin45\right)$

d)

$4^{\frac{1}{3}}\left(\cos135+i\sin135\right),$ $4^{\frac{1}{3}}\left(\cos45+i\sin45\right)$

17.

Simplify: $\left(-2+3i\right)^3$

a)

46+9i

b)

-5-12i

c)

$\sqrt{13}^3\left(\cos56+i\sin56\right)$

d)

$13\sqrt{13}\left(\cos56^3+i\sin56^3\right)$

18.

Given the following zeros for a polynomial of degree 4, find the polynomial

2, -2, 4, -4

a)

$f\left(x\right)=x^4-20x^2+64$

b)

$x^4+8x^3-20x^2+64x$

c)

$f\left(x\right)=x^4-12x^{3.}+12x^2+96x+64$

d)

None of the above

19.

Find the argument of $-1+i\sqrt{3}$

a)

60 degrees

b)

56.85 degrees

c)

120 degrees

d)

178.95 degrees

20.

Given that you have a polynomial of degree 5 (i.e. $f\left(x\right)=ax^5+bx^4+cx^3+dx^2+ex+k$ ) how many solutions will there be for x?

a)

5

b)

4

c)

2

d)

Not enough information

21.

Given that you have a polynomial of degree 5 (i.e. $f\left(x\right)=ax^5+bx^4+cx^3+dx^2+ex+k$ ) with solutions 0, 3, -3 and 2+i - what is the final solution?

a)

1

b)

2-i

c)

3-i

d)

Not enough information

22.

Given that you have a polynomial of degree 5 (i.e. $f\left(x\right)=ax^5+bx^4+cx^3+dx^2+ex+k$ ) with solutions 0, 3, -3 and 2+i - how can you find the polynomial coefficients?

a)

$x\left(x+3\right)\left(x-3\right)\left(x-\left(2+i\right)\right)$ and simplify

b)

$x\left(x+3\right)\left(x-3\right)\left(x-\left(2+i\right)\right)\left(x-\left(2-i\right)\right)$ and simplify

c)

$f\left(x\right)=0x^5+3x^4-3x^3+\left(2+i\right)x^2$

d)

Not enough information

23.

Given $z=12-3i$ , find |z|

a)

$\left|z\right|=\tan^{-1}\left(-\frac{3}{12}\right)$

b)

$\left|z\right|=\sqrt{12^2+3^2}$

c)

$\left|z\right|=\sqrt{12^2-3^2}$

d)

$\left|z\right|=360-\tan^{-1}\left(\frac{3}{12}\right)$

24.

Given $z=8-4i$ , find the z*

a)

8+4i

b)

2+i

c)

2-i

d)

8-4i

25.

Simplify $\left[3\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)\right]^5$

a)

$3^5\left(\cos\frac{5\pi}{3}+i\sin\frac{5\pi}{3}\right)$

b)

$3^{\frac{1}{5}}\left(\cos\left(\frac{\left(\frac{\pi}{3}\right)}{5}\right)+i\sin\left(\frac{\left(\frac{\pi}{3}\right)}{5}\right)\right)$

c)

$3^5\left(\cos\left(\frac{\left(\frac{\pi}{3}+2\pi k\right)}{5}\right)+i\sin\left(\frac{\left(\frac{\pi}{3}+2\pi k\right)}{5}\right)\ \right)$ for k=0, 1, 2, 3, 4

d)

$3^{\frac{1}{5}}e^{i\left(\frac{5\pi}{3}\right)}$

26.

Simplify $\left(13+8i\right)^2$

a)

13-8i

b)

169+64i

c)

105+208i

d)

$169+208i+64i^2$

27.

How would we start to simplify the following? $\frac{\left(3-6i\right)}{4i}$

a)

multiply by 4i

b)

divide by -4i

c)

multiply by $\frac{i}{i}$

d)

Already simplified

28.

Simplify $\frac{\left(8+5i\right)}{\left(2-7i\right)}$

a)

$-\frac{\left(8+5i\right)}{45}$

b)

$\frac{\left(16+66i+35i^2\right)}{53}$

c)

$\frac{\left(-19+66i\right)}{53}$

d)

$-\frac{\left(-19+66i\right)}{45}$

29.

What complex number does the following represent?

a)

-2+3i

b)

2-3i

c)

3-2i

d)

-3+2i

30.

a)

$6\sqrt{3}\left(\cos135+i\sin135\right)$

b)

$6\sqrt{2}\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right)$

c)

$6\sqrt{2}\left(\cos225+i\sin225\right)$

d)

$6\sqrt{2}\left(\cos45+i\sin45\right)$

31.

a)

$e^{i59.04}$

b)

$\sqrt{34}e^{i1.03}$

c)

$4e^{i300.06}$

d)

$4e^{i2.11}$

32.

What quadrant it 13 degrees in?

a)

1

b)

2

c)

3

d)

4

33.

What quadrant it 131 degrees in?

a)

1

b)

2

c)

3

d)

4

34.

What quadrant it 241 degrees in?

a)

1

b)

2

c)

3

d)

4

35.

What quadrant it 280 degrees in?

a)

1

b)

2

c)

3

d)

4

36.

What quadrant is $\frac{3\pi}{10}$ radians in?

a)

1

b)

2

c)

3

d)

4

37.

What quadrant is $\frac{26\pi}{16}$ radians in?

a)

1

b)

2

c)

3

d)

4

38.

What quadrant is $\frac{105\pi}{20}$ radians in?

a)

1

b)

2

c)

3

d)

4

39.

Given $\left(27+i\sqrt{13}\right)+\left(a+bi\right)^{ }=\sqrt{5^3+b^2}$ , how would you find the value of a or b?

a)

Guess the value of a and find the value of b

b)

Seperate into a system of equations

c)

Simplify and a is the real part/b is the imaginary part

d)

Not enough information

40.

What quadrant it 815 degrees in?

a)

1

b)

2

c)

3

d)

4

41.

What quadrant is $\frac{4\pi}{7}$ radians in?

a)

1

b)

2

c)

3

d)

4

42.

What quadrant is $\frac{19\pi}{13}$ radians in?

a)

1

b)

2

c)

3

d)

4

43.

Find the argument of $\left(\ -\sqrt{6}-\sqrt{2}\right)+\left(\sqrt{6}-\sqrt{2}\right)i$

a)

$\frac{11\pi}{12}$

b)

$-\frac{\pi}{12}$

c)

135

d)

-15

44.

Find all zeros for the following equation $z^4-16=0$

a)

$\pm2$

b)

4

c)

$\pm\left(2+2i\right),\ \pm\left(2-2i\right)$

d)

$\pm2,\ \pm2i$

45.

What is the modulus of

$\left(-2\sqrt{6}+2\sqrt{2}\right)+\left(2\sqrt{6}+2\sqrt{2}\right)i$ ?

a)

$\frac{7\pi}{12}$

b)

8

c)

$\frac{\pi}{12}$

d)

1

Q3 Exam Review

Total questions: 45

Worksheet time: 2hrs 36mins

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