Free Printable Precalculus Final Study

1.

Simplify √-4

a)

2i

b)

-2i

c)

2

d)

4i

2.

Simplify √-23

a)

-√23

b)

23i

c)

23

d)

i√23

3.

Simplify √-25

a)

i√25

b)

5i

c)

-5i

d)

25i

4.

Simplify 2√-32

a)

2i√32

b)

4i√2

c)

8i√2

d)

-16i√2

5.

$i^2=?$

a)

i

b)

-1

c)

-i

d)

1

6.

$i^{63}\ =?$

a)

i

b)

-1

c)

-i

d)

1

7.

Simplify:
-3i(4 + 2i)

a)

6 - 12i

b)

6 + 12i

c)

-12i - 6i²

d)

-12i + 6i²

8.

Simplify:
(2 - 3i)(5 + 4i)

a)

22 - 7i

b)

22 + 7i

c)

22 - 7i²

d)

22 + 7i²

9.

Simplify:
i⁷

a)

i

b)

-i

c)

1

d)

-1

10.

Simplify.

(a)

11.

What is the real portion of the number

3+5i

a)

3

b)

5

c)

-3

d)

-5

12.

What is the imaginary portion of the number $-7-12i$

a)

7

b)

12

c)

$-7$

d)

$-12$

13.

Two complex numbers are added. How do we do this?

a)

We add up all 4 numbers you see.

b)

We add up all the big numbers.

c)

We add only the real parts. Because we cannot see the imaginary parts.

d)

We add the real parts together, and then separately add the imaginary parts together. Your final answer is in the form. $a+bi$

14.

What complex number is represented by the graph?

a)

-3-i

b)

-3+i

c)

1+3i

d)

1-3i

15.

What is the radius of a unit circle?

a)

1

b)

2

c)

1/2

d)

0

16.

The hypotenuse of any right triangle from the center of the unit circle to an edge is always

a)

1

b)

sqt2/2

c)

sqt3/2

d)

1/2

17.

Based on your unit circle cos(0^o)=

a)

1

b)

0

c)

-1

d)

1/2

18.

Based on your unit circle sin(90^o)=

a)

1/2

b)

0

c)

-1

d)

1

19.

sin(7π/4)

a)

1/2

b)

-√2/2

c)

√2/2

d)

-1/2

20.

sin(3π/2)

a)

0

b)

1

c)

-1

d)

1/2

21.

Simplify the expression:
$\left(1\ +\ 5i\right)\ +\ \left(1\ -\ 5i\right)$

a)

$2\ +\ 10i$

b)

$2\ +\ 10i^2$

c)

$2$

d)

$-8$

22.

Simplify the expression:
$\left(8\ +\ 9i\right)\ +\ \left(4\ -\ 6i\right)$

a)

$17\ +\ 3i$

b)

$12\ +\ 3i^2$

c)

$17\ -2i$

d)

$12\ +\ 3i$

23.

Simplify the expression:
$\left(5\ +\ 14i\right)\ -\ \left(10\ +\ 2i\right)$

a)

$5\ -16i$

b)

$-5\ +\ 16i$

c)

$5\ -\ 12i$

d)

$-5\ +\ 12i$

24.

$2\left(3\ +\ 4i\right)-\left(4\ -\ 6i\right)$

a)

$2\ +\ 14i$

b)

$10\ +\ 2i$

c)

$-1\ -2i$

d)

$-2\ -2i$

25.

Simplify the expression:
$\left(5\ +\ 4i\right)\ -\ \left(-1\ -\ 2i\right)$

a)

$6\ +6i$

b)

$4\ +\ 2i$

c)

$6\ -\ 6i$

d)

$4\ -\ 2i$

26.

(-4i)(3i)(4i)

a)

24

b)

-48i

c)

48i

d)

60

27.

(i)(3i)

a)

-3i

b)

-3

c)

3

d)

2i

28.

(-3-i)(6-i)

a)

-21-7i

b)

-19-3i

c)

-15-5i

d)

-17-9i

29.

a)

A

b)

B

c)

C

d)

D

30.

Determine the modulus of : $\left|3-2i\right|$

a)

$\sqrt{-13}$

b)

$\sqrt{-1}$

c)

$\sqrt{13}$

d)

$5$

31.

What is the correct set-up to find the absolute value of -9+5i?

a)

$-9+5$

b)

$\sqrt{\left(-9+5i\right)^2}$

c)

$\sqrt{\left(-9\right)^2+\left(5\right)^2}$

d)

$\sqrt{-9+5}$

32.

Convert the polar coordinates to rectangular form

a)

(1, 1)

b)

(1, -1)

c)

(-1, 0)

d)

(-1, -1)

33.

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 )

34.

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)

35.

Which point represents (3, 3π/4)?

a)

F

b)

E

c)

B

d)

A

36.

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

37.

$\left(1-7i\right)^2$

(a)

38.

Which of the following is the complex conjugate of 3 + 4i?

a)

-3 + 4i

b)

-3 - 4i

c)

3 - 4i

d)

7i

39.

a)

A

b)

B

c)

C

d)

D

40.

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

41.

Add to Simplify:
(4x - 2x³) + (5x³ - 4x + 5)

a)

3x³ + 5

b)

3x³ + 8x + 5

c)

3x² + 5

d)

7x³ + 8x + 5

42.

Find the sum.
(3 - 2x + 2x²) + (4x - 5 + 3x²)

a)

7x - 7x + 5x²

b)

5x²+ 2x - 2

c)

5x²

d)

5x² + 6x + 8

43.

What is the type of polynomial 3x⁴ + 2x² + 5x ?

a)

monomial

b)

binomial

c)

trinomial

d)

polynomial

44.

Simplify the expression.
(4a³ - 8a - 4a²) + (7a³ - 7 - 6a)

a)

11a³ - 4a² - 14a - 7

b)

5a³ - 4a² - 14a - 7

c)

5a³ - 4a² - 20a - 7

d)

5a³ - 9a² - 20a - 7

45.

Add the polynomials (x-2) and (x²+3)

a)

x²+x+1

b)

x³+5

c)

x³+1

d)

x³-1

46.

What is the degree of the polynomial 3x⁴ + 2x² + 5x ?

a)

1

b)

2

c)

3

d)

4

47.

(3x²+xy-5y²) +
(2x²-xy+3y²)

a)

5x²-2xy+8y²

b)

5x²-2y²

c)

5x²+2y²

d)

5x²+2xy-8y²

48.

Add or Subtract to Simplify:
(4x - 2x³) + (5x³ - 4x + 5)

a)

3x³ + 5

b)

3x³ + 8x + 5

c)

3x² + 5

d)

7x³ + 8x + 5

49.

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

a)

7x + 7y

b)

7x + 5y

c)

7x + 6y

d)

3x - 5y

50.

What is the perimeter of the triangle shown?

a)

9r²+r-5

b)

9r⁴+r²-4

c)

9r²-r+4

d)

9r²+r-4

51.

Add or Subtract to Simplify:
(4x²+ x) - (x²+ 2x)

a)

3x²- x

b)

4x²+ 2x

c)

3x² + 3x

d)

3x² + 2x

52.

Add or Subtract to Simplify:
(4x - 2x³) + (5x³ - 4x + 5)

a)

3x³ + 5

b)

3x³ + 8x + 5

c)

3x² + 5

d)

7x³ + 8x + 5

53.

a)

b)

c)

d)

54.

Select the correct expansion of

(1 - x)⁵

a)

1 − 5x + 10x² − 10x³ + 5x⁴ − x⁵

b)

1 + 5x + 10x² + 10x³ + 5x⁴ + x⁵

c)

-1 + 5x - 10x² + 10x³ - 5x⁴ + x⁵

d)

1 − 5x + 10x² + 10x³ + 5x⁴ − x⁵

55.

Find the fourth term of the expansion (5+3y)⁵.

a)

2025y⁴

b)

750y³

c)

75y⁴

d)

6750y³

56.

What is the Binomial expansion of (x + 1)⁵ ?

a)

x⁵ + 5x⁴ + 10x³ + 10x² + 5x + 1

b)

x⁵ + 5x⁴ + 15x³ + 15x² + 5x + 1

c)

x⁵ + 6x⁴ + 15x³ + 15x² + 6x + 1

d)

x⁵ + 1

57.

Which row of Pascal's Triangle would you use to expand (x+y)³?

a)

1

b)

1 1

c)

1 2 1

d)

1 3 3 1

58.

How many terms are the in the expansion of (1+b)⁵

a)

5

b)

6

c)

25

d)

50

e)

3

59.

State the number of complex zeros for each function.

a)

3

b)

2

c)

5

d)

4

60.

State the number of complex zeros for each function.

a)

7, 5, 3, or 1

b)

5

c)

4

d)

7

61.

Is this division problem worked correctly?

a)

This is correct!

b)

This is incorrect!

62.

What should you add to this problem?

(2x³ + 5x² + 9) ÷ (x + 3)

a)

0x⁴

b)

0x³

c)

0x²

d)

0x

63.

a)

2x - 4

b)

2x + 4

c)

2x - 1 + -16/4x-5

d)

2x - 4 + -40/4x-5

64.

Divide
(4b² + b - 2) ÷ (4b + 5)

a)

b + 1 - 3/4b +5

b)

b - 1 + 3/4b - 5

c)

b + 1 + 3/4b + 5

d)

b - 1 + 3/4b + 5

65.

(3x³ + 5x - 1) ÷ (x + 1)

a)

3x³ - 3x² + 8x - 9

b)

3x² - 3x + 8 - 9/x+1

c)

3x² + 3x + 8 + 7/x+1

d)

3x³ + 3x² + 8x + 7

66.

Identify the zeros:

a)

x=-2, x=-1, x=0

b)

x=-2, x=0

c)

x=-1, x=0, x=2

d)

x=-2, x=0, x=1

67.

a)

A (-2,0) (2,0)

b)

B (-2,0) (1,-3)

c)

C (0,-2) (-3,1)

d)

D (0,-4) (0,-2)

68.

Determine the y-intercept
f(x) = x³ - 5x² - 25x + 125

a)

(0,0)

b)

(0,25)

c)

(0,125)

d)

(0,5)

69.

Which one is the same as fοg(x)

a)

g(f(x))

b)

f(g(x))

70.

Given f(x) = 3x + 10
and g(x) = x - 2
Find f(g(5))

a)

19

b)

23

c)

-10

d)

None of these.

71.

f(x) = 3x + 10
g(x) = x - 2
Find (f∘g)(0)

a)

16

b)

4

c)

-4

d)

None of these.

72.

f(x) = 3x + 10
g(x) = x - 2
Find g(f(-10))

a)

-42

b)

-26

c)

-18

d)

None of these.

73.

What is f(g(3))?

a)

1/8

b)

8

c)

1/4

d)

1/2

74.

Given that f(x) = x² + x and g(x) = 3x + 1, find the value of g(f(-2)).

a)

-14

b)

20

c)

6

d)

7

75.

a)

9 - √17

b)

4

c)

2

d)

√8

76.

f(x) = x² + 3
g(x) = 4x - 3
Find (fg)(-1)

a)

4

b)

-52

c)

-4

d)

52

77.

(L3) Given f(x)= x²+ 25 and g(x)= x - 5, find (f+g)(x).

a)

x²+ x - 20

b)

x²+ x + 20

c)

-x²+ x - 20

d)

x²- x - 20

78.

Which expression represents g(f(x)) if f(x) = 2x - 8
and
g(x) = 4x

a)

8x - 32

b)

8x² - 32x

c)

8x - 8

d)

6x - 8

79.

If f(x) = x² and g(x) = 3x - 1, find f(g(x))

a)

9x² - 6x + 1

b)

3x - 1

c)

3x² - 1

d)

9x² - 1

80.

Given f(x) = 2x and g(x) = x²+3 , find f(g(x)).

a)

x²+2x+3

b)

4x²+3

c)

2x²+3

d)

2x²+6

81.

Find the exact value of $\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)$

a)

$-\frac{\pi}{3}$

b)

$\frac{\pi}{3}$

c)

$-\frac{\pi}{6}$

d)

$\frac{\pi}{6}$

82.

Q1. The principal value of $\ Sin^{-1}\left(\frac{1}{\sqrt{2}}\right)\ is\$

a)

$\frac{\pi}{4}$

b)

$\frac{\pi}{6}$

c)

$\frac{\pi}{2}$

83.

Q3. The principal value of $\ Tan^{-1}\left(-1\right)\ is$

a)

$-\frac{\pi}{4}$

b)

$-\frac{\pi}{3}$

c)

$-\frac{\pi}{6}$

84.

Which equation is true?

a)

sin (x) = 6/8

b)

cos (x) = 8/6

c)

cos (x) = 6/8

d)

tan (x) = 6/8

85.

Which equation would you use to solve for the unknown angle?

a)

x = sin^–1(4/5)

b)

x = cos^–1(4/5)

c)

x = cos^–1(5/4)

d)

x = tan^–1(4/5)

86.

Evaluate: cos^-1(√3/2)

a)

π/3

b)

2π/3

c)

π/6

d)

5π/6

87.

Find all solutions to the equation $\sin\theta\ =\ .6$ . You answer should be in degrees. Consider n to be any integer

a)

$36.9\degree\ and\ 143.1\degree$

b)

$36.9\degree\ +\ 360\degree n$

c)

$36.9\degree\ +\ 2\pi n\ \ and\ 143.1\degree\ +2\pi n$

d)

$36.9\degree\ +\ 360\degree n\ \ and\ \ 143.1\degree\ +\ 360n$

88.

Find all solutions for 2sin(6x) + √3 = 0.

Your answer should be in radians.

a)

x = .175 + πn/3 and x = .873 + πn/3

b)

x = 1.920 + 2πn and x = 1.222 + 2πn

c)

x = .698 + πn/3 and x = .873 + πn/3

d)

x = 4.189 + 2πn and x = 5.236 + 2πn

89.

Use Sum or Difference Identities to find the exact value of each expression.

cos(75°)

a)

1/4

b)

c)

d)

90.

$\cos70^o\cos40^o-\sin70^o\sin40^o\$ is equivalent to

a)

$\cos30^o$

b)

$\cos70^{o\ }$

c)

$\cos110^o$

d)

$\sin70^{o\ }$

91.

$Find\ the\ exact\ value\ of\ \tan\ \frac{\pi}{12}$

a)

$2-\sqrt{3}$

b)

$-2+\sqrt{3}$

c)

$\sqrt{3}$

d)

$\frac{\sqrt{3}}{3}$

92.

Which of the following is NOT

a solution to

tan θ = 0 ?

a)

θ = 0

b)

θ = π /2

c)

θ = π

d)

θ = 2π

93.

A ladder is leaning against a house at an angle of 39 degrees. If the ladder is 12 ft. long, how far is the bottom of the ladder from the house?

a)

15.4 ft.

b)

9.3 ft.

c)

20.2 ft.

d)

5.3 ft.

94.

Which equation would you use to solve for the unknown angle?

a)

x = sin^–1(4/5)

b)

x = cos^–1(4/5)

c)

x = cos^–1(5/4)

d)

x = tan^–1(4/5)

95.

A vector is a ray with both ____________ and _____________.

a)

Wings and halos

b)

Space and time

c)

Direction and magnitude

d)

Radicals and exponents

96.

Which are equivalent vectors?

a)

u and -u

b)

u and a

c)

u and v

d)

u and b

97.

When the vector's initial point is at the origin, it is in ___.

a)

quadrant bearing

b)

true bearing

c)

resultant

d)

standard position.

98.

Two vectors that have the same magnitude and direction

a)

Parallel vectors

b)

Equivalent vectors

c)

Opposite vectors

99.

Two vectors that have the same magnitude but opposite direction

a)

Parallel vectors

b)

Equivalent vectors

c)

Opposite vectors

100.

If a vector has direction 40 degrees West of North, what quadrant would it lie in?

a)

Quadrant 1

b)

Quadrant 2

c)

Quadrant 3

101.

What is the x component of the vector in the diagram?

a)

104N

b)

60N

c)

-104N

d)

95N

102.

What is the y component of the vector in the diagram?

a)

104N

b)

60N

c)

-104N

d)

95N

103.

What is the x component of the vector?

a)

14cos(30)

b)

14sin(30)

c)

-14cos(30)

d)

-14sin(30)

104.

What is the y component of the vector?

a)

14sin(30)

b)

-14sin(30)

c)

14cos(30)

d)

-14cos(30)

105.

$p=<3,\ -9>$

Find the magnitude to the nearest tenth.

(a)

106.

Find the magnitude of a vector using ...

a)

Sine & Cosine

b)

Inverse Tangent

c)

Pythagorean Theorem

d)

Multiplication

107.

Find the magnitude of AB if the initial point is (-3, 1) and the terminal point is (3, 9).

a)

6

b)

8.2

c)

10

d)

10.4

108.

Find the magnitude of the vector <-4, -6>.

a)

7.211

b)

52

c)

3.162

d)

4.472

109.

Find the magnitude of the vector <-2, 1>.

a)

5

b)

2.236

c)

1

d)

2.646

110.

State vector x in terms of a.

a)

2a

b)

$\frac{1}{2}$ a

c)

-2a

d)

$-\frac{1}{2}$ a

111.

If vector a= (4, -5) and vector b= (-7, 4) Find a+b

a)

(3, 9)

b)

(-3, -1)

c)

(11, 1)

d)

(-3, -9)

112.

If vector a= (4, -5) and vector b= (-7, 4) Find a-b

a)

(-11, -9)

b)

(11, 9)

c)

(11, -9)

d)

(-3, -1)

113.

Let vector a= (5, -7) and vector b= (-12, 2) Find 2b-4a

a)

(44, -32)

b)

(-4, 24)

c)

(-44, 32)

d)

(4, -24)

114.

If vector a= (4, -5) and vector b= (-7, 4) Find 2a-3b

a)

(29, -22)

b)

(13, -2)

c)

(1, 7)

d)

(29, -2)

115.

If c=<-5, 4> and d=<8, 0>, calculate 2c+d

a)

<-2, 8>

b)

<-2, 0>

c)

<6, 8>

d)

<6, 0>

116.

If w = <2, -5> and y = <2, 0>, find 2w + y.

a)

<-10, 2>

b)

<4, -5>

c)

<-6, 10>

d)

<6, -10>

117.

An airplane flying due south (270°) 485 mph against a 25 mph wind blowing due west(180°). Determine the new speed and direction of the plane.

a)

Direction 267.0°
Magnitude 485.6mph

b)

Direction 93.0°
Magnitude 485.6mph

c)

Direction 237.0°
Magnitude 485.6mph

d)

Direction 87.0°
Magnitude 510.0mph

118.

George Washington throws a baseball with an initial velocity of 20.2 feet per second at an angle of 69° with the horizontal. Find the initial vertical and horizontal velocity of the ball.

a)

Horizontal 18.9
Vertical 7.2

b)

Horizontal 7.2
Vertical 18.9

c)

Horizontal 15.2
Vertical 9.6

d)

Horizontal 9.6
Vertical 15.2

119.

Which of these is not a conic section?

a)

Ellipse

b)

Hyperbola

c)

Rectangle

d)

Parabola

120.

Why are Conic Sections given this name?

a)

They are the cross-sections given when a cone is sliced.

b)

Because it's part of 'da rules'

c)

They are the cross-sections given when any 3D image is sliced.

121.

In the equation (x-3)²+(y+4)²=121, the radius of the circle is

a)

242

b)

11

c)

121

d)

22

122.

In the equation (x-3)²+(y+4)²=121, the radius of the circle is

a)

242

b)

11

c)

121

d)

22

123.

x + 3 = (y - 1)²

a)

Parabola Opens Up

b)

Parabola Opens Down

c)

Parabola Opens Right

d)

Parabola Opens Left

124.

What direction does the parabola open?
(y-9)²= -40(x+4)

a)

Up

b)

Down

c)

Left

d)

Right

125.

What are the vertices of the shifted ellipse?

a)

(1, 5) and (1, -5)

b)

(0, 5) and (0, -5)

c)

(1, 1) and (1, -1)

d)

(0, 1) and (0, -1)

126.

What is the center of the shifted ellipse?

a)

(0,0)

b)

(-3,1)

c)

(1,-3)

d)

(3,-1)

127.

What are the foci of the shifted ellipse?

a)

(-5,1) and (-1,1)

b)

(±√3,0)

c)

(0,1) and (-6,1)

d)

(√3 - 3,1) and (-√3 - 3,1)

128.

The foci of an ellipse are located at (-2, 3) & (2, 3), where is the center?

a)

(0, 3)

b)

(3, 0)

c)

(-2, 0)

d)

(2, 0)

129.

Write the equation of a circle with center (7, 0) with radius 3.

a)

(x - 7)² + y² = 9

b)

x² + (y -7)² = 9

c)

(x - 7)² + y² = 3

d)

x² + (y -7)² = 3

130.

Classify the conic section.

a)

Circle

b)

Ellipse

c)

Line

d)

None of these

131.

Classify this conic:
36x² – 5y² + 8x + 2y + 12 = 0

a)

Hyperbola

b)

Parabola

c)

Ellipse

d)

Circle

132.

Classify this conic:
6x² + 6y² – 3x + 4y = 0

a)

Circle

b)

Parabola

c)

Hyperbola

d)

Ellipse

133.

Classify this conic:
x²+5xy+8y²−10x+3y+10=0

a)

Ellipse

b)

Circle

c)

Hyperbola

d)

Parabola

134.

Identify the conic: 2x² + 6y – 9 = 0

a)

Parabola

b)

Hyperbola

c)

Ellipse

d)

Circle

135.

Classify this conic:

a)

Hyperbola

b)

Ellipse

c)

Parabola

d)

Circle

136.

Classify this conic:

a)

Ellipse

b)

Hyperbola

c)

Degenerate Conic Section

d)

Circle

137.

Which conic do you get from slicing a double cone on a slant (not through the base)?

a)

circle

b)

ellipse

c)

ellipse (except at the apex)

d)

hyperbola

138.

What equation is used to find the foci points on an ellipse?

a)

c² = a² + b²

b)

c² = a² - b²

c)

1/4d

d)

(x-h)² + (y-k)² = r²

139.

Identify the center and the length of a and b.

a)

C: (0,4) a= 25, b=4

b)

C: (0,0) a= 4, b=25

c)

C: (0,5) a= 2, b=5

d)

C: (0,0) a= 5, b=2

140.

For the ellipse with the following characteristics:
Vertices ( 4, 3), (4, - 9)
Length of minor axis is 8
What is the center of this ellipse?

a)

(4, 6)

b)

( - 4, 3)

c)

(6, 4)

d)

(4, -3)

141.

What is the equation of the circle?

a)

(x+1)² + (y +1)² = 3

b)

(x+1)² + (y -1)² = 3

c)

(x+1)² + (y +1)² = 9

d)

(x-1)² + (y -1)² = 9

142.

Given the equation of the ellipse and the C value, write the foci ordered pair

a)

(4 ± 7.42, -2)

b)

(4, - 2 ± 7.42)

c)

(- 4 ± 7.42, 2)

d)

( - 4, 2 ± 7.42)

143.

Which type of conic section is this equation?

a)

circle

b)

ellipse

c)

hyperbola

d)

parabola

144.

Check the box that shows a parabola.

a)

b)

c)

d)

e)

145.

What type of conic section is shown?

a)

Ellipse

b)

Hyperbola

c)

Parabola

d)

Circle

Precalculus Final Study

Total questions: 145

Worksheet time: 7hrs 11mins

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