Free Printable Complex Numbers 2

1.

Are the roots of the function f(x)=3x²+3x+3 real or imaginary? (hint: graph it)

a)

Real Roots

b)

Imaginary Roots

c)

Impossible to tell

d)

Both

2.

$Write\ \frac{7-3i}{5-2i}$ as a single complex number.

a)

$\frac{7}{5}+\frac{3}{2}i$

b)

$\frac{41}{29}-\frac{1}{29}i$

c)

$1-\frac{1}{29}i$

d)

$\frac{41}{29}-i$

3.

Which of the following would be the correct factored form with roots 2, 5i, and 6?

a)

(x-6)(x-2)(x+5i)(x-5i)

b)

(x+6)(x+2)(x+5i)

c)

(x-6)(x-2)(x-5i)

d)

(x+6)(x+2)(x-5i)

4.

If a+bi is a root of a polynomial, then _______ is also a root.

a)

i²

b)

a-bi

c)

a+bi

d)

-i²

5.

Use the Complex Conjugate Root Theorem to state another complex solution. Given that -2-5i is a root.

a)

-2-5i

b)

-2+5i

c)

2+5i

d)

5i

6.

Find a polynomial function of degree 3 with zeros 3i and -2.

a)

P(x) = x³+2x²+9x+11

b)

P(x) = x³-2x²+9x+18

c)

P(x) = x³+2x²+9x+18

d)

P(x) = x²-3ix+2x-6i

7.

Write a polynomial function of least degree with integral coefficients that has the given zeros.

5, -2i

a)

x³-5x²+4x-20

b)

x³-5x²+4x-23

c)

x³-5x²+4x-19

d)

x³-10x²+4x-20

8.

Use the Complex Conjugate Root Theorem to state another complex solution. Given that -7i is a root.

a)

7i

b)

-7i+1

c)

1-7i

d)

1+7i

9.

Find the modulus of the complex number. Also state what quadrant the point would land in.

-3 + i

a)

√10, Quadrant 2

b)

√10, Quadrant 3

c)

2, Quadrant 2

d)

3, Quadrant 3

10.

Express the given point in rectangular form.

4 ( cos π /3 + i sin π /3)

a)

2 + 2√3 i

b)

2 + √3 i

c)

2 - 2√3 i

d)

-2 + 2√3 i

11.

Write the complex number in polar form

5\ +\ 5i

a)

√50 (cos π/2 + i sin π/2)

b)

50 (cos π/4 + i sin π/4)

c)

√50 (cos 5π/4 + i sin 5π/4)

d)

√50 (cos π/4 + i sin π/4)

12.

Express the number given in trig form

3\ -\ 3i

a)

$3\sqrt{2}\left(\cos\ \left(-45\degree\right)+i\sin\ \left(-45\degree\right)\right)$

b)

$3\left(\cos\ \frac{7\pi}{7}+i\sin\ \frac{7\pi}{4}\right)$

c)

$3\sqrt{2}\left(\cos\ \frac{7\pi}{4}+i\sin\ \frac{7\pi}{4}\right)$

d)

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

13.

Find the product of $z_1=3\left(\cos\ \frac{2\pi}{3}+i\sin\ \frac{2\pi}{3}\right)$ and

$z_2=5\left(\cos\ \frac{\pi}{4}+i\sin\ \frac{\pi}{4}\right)$

a)

$15\left(\cos\ \frac{3\pi}{7}+i\sin\ \frac{3\pi}{7}\right)$

b)

$15\left(\cos\ \frac{\pi}{6}+i\sin\ \frac{\pi}{6}\right)$

c)

$15\left(\cos\ \frac{5\pi}{12}+i\sin\ \frac{5\pi}{12}\right)$

d)

$15\left(\cos\ \frac{11\pi}{12}+i\sin\ \frac{11\pi}{12}\right)$

14.

If a point is in Quadrant II, what do you need to add to the angle after calculating $Tan\ \theta$

a)

add nothing

b)

add $180\degree$

c)

add $270\degree$

d)

add $360\degree$

15.

Which complex point is graphed at point C in the complex plane shown?

a)

(-4, -1)

b)

-4 - i

c)

4 + i

d)

-4 + i

16.

Which complex point is graphed at point E in the complex plane shown?

a)

3i

b)

-3i

c)

3

d)

-3

17.

Use De Moivre's Theorem to find the indicated power of the complex number $\left[3\left(\cos\ \frac{\pi}{4}+i\sin\ \frac{\pi}{4}\right)\right]^{10}$ for $0\le\theta\le2\pi$

a)

$3^{10}\left(\cos\ \frac{5\pi}{2}+i\sin\ \frac{5\pi}{2}\right)$

b)

$3^{10}\left(\cos\ \frac{\pi}{2}+i\ \sin\ \frac{\pi}{2}\right)$

c)

$3^{10\ }\left(\cos\ \frac{\pi}{4}+\ i\ \sin\ \frac{\pi}{4}\ \right)$

d)

$3\ \left(\cos\ \frac{\pi}{2}+\ \sin\ \frac{\pi}{2}\right)$

18.

When a complex number is drawn on the complex plane. The distance from the origin is the value of

a)

the argument

b)

The modulus

c)

The real co-efficient

d)

The imaginary co-efficient

19.

The argument measures the angle

a)

clockwise from the y-axis

b)

anti clockwise from the y-axis

c)

anti clockwise from the x-axis

d)

clockwise from the x-axis

20.

For the number -1+5i the argument measured in degrees would be

a)

between 0 and 90°

b)

between 0 and -90°

c)

between 90° and 180°

d)

between -90° and -180°

21.

For the number 2-3i the argument measured in degrees would be

a)

between 0 and 90°

b)

between 0 and -90°

c)

between 90° and 180°

d)

between -90° and -180°

22.

Express the complex number in polar form.
-2 + 5i

a)

5.39 ( cos 1.95 + i sin 1.95)

b)

5.39 ( cos 65.06 + i sin 65.06)

c)

2.65 ( cos 1.95 - i sin 1.95 )

d)

2.65 ( cos 65. 06 - i sin 65.06 )

23.

Express the complex number in polar form.
6 + 2i

a)

6.32 ( cos 0.32 + i sin 0.32)

b)

6.32 ( cos 0.45 + i sin 0.45)

c)

1.73 ( cos 0.32 + i sin 0.32)

d)

1.73 ( cos 0.45 - i sin 0.45)

24.

What quadrant would the point in polar form land in?
4 ( cos π /3 + i sin π /3)

a)

Quadrant 4

b)

Quadrant 1

c)

Quadrant 2

d)

Quadrant 3

25.

Express the point in polar form in rectangular form.
4 ( cos π /3 + i sin π /3)

a)

2 + 2√3 i

b)

2 + √3 i

c)

2 - 2√3 i

d)

-2 + 2√3 i

26.

The distance between z=4-3i and w=-2-4i on the complex plane is...

a)

√35

b)

√35 i

c)

√37

d)

5i

27.

Let $w=-\frac{\sqrt{3}}{2}+\frac{1}{2}i$ . Calculate the modulus and argument.

a)

3, $\frac{2}{3}\pi$

b)

5, $\frac{3}{2}\pi$

c)

1, $\frac{5}{6}\pi$

d)

Not Possible

28.

If z=√2+√2 i then arg(z²)=

a)

0

b)

$\frac{\pi}{3}$

c)

$\frac{\pi}{2}$

d)

Not Possible

29.

Find the absolute value of the complex number. Also state what quadrant the point would land in.
2 + 3i

a)

√13, Quadrant 1

b)

√5, Quadrant 1

c)

√11, Quadrant 2

d)

√11, Quadrant 3

30.

for an imaginary number z=a+bi, the modulus squared $\left|z\right|^2$ is given by

a)

$\sqrt{a^2+b^2}$

b)

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

c)

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

d)

$a^2+b^2$

31.

i⁴²=

a)

i

b)

-i

c)

-1

d)

1

32.

The value of

\left(1+i\sqrt{3}\right)^6

is

a)

64

b)

-64

c)

1

d)

2

33.

The value of

\frac{1}{\left(\cos\theta+i\sin\theta\right)^6}

is

a)

$-\cos6\theta-i\sin6\theta$

b)

$\cos6\theta+i\sin6\theta$

c)

$\cos6\theta-i\sin6\theta$

d)

$-\cos6\theta+i\sin6\theta$

34.

$\left(\cos\ \frac{\pi}{6}\ -\ i\sin\ \frac{\pi}{6}\right)^{12}\ is\ equal\ to$

a)

-i

b)

-1

c)

i

d)

1

35.

$\cos4\theta$ is eqauivalent to

a)

$8\cos^4\theta+8\cos^2\theta+1$

b)

$8\cos^4\theta-8\cos^2\theta+1$

c)

$8\cos^4\theta-8\cos^2\theta-1$

d)

$8\cos^4\theta+8\cos^2\theta-1$

36.

$\sin4\theta$ is equivalent to

a)

$8\cos^3\theta\sin\theta+8\cos\theta\sin^3\theta$

b)

$4\cos^3\theta\sin\theta+4\cos\theta\sin^3\theta$

c)

$4\cos^3\theta\sin\theta-4\cos\theta\sin^3\theta$

d)

$8\cos^3\theta\sin\theta-4\cos\theta\sin^3\theta$

37.

$If\ z\ =\ \cos\theta\ +i\sin\theta,\ the\ value\ of\ z^3+\frac{1}{z^3}\ is$

a)

$2\cos3\theta$

b)

$\cos3\theta$

c)

$i\sin3\theta$

d)

$2i\sin3\theta$

Complex Numbers 2

Total questions: 37

Worksheet time: 3hrs 42mins

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