Free Printable Precalculus Semester 2 Skills Practice

1.

a)

sinθ

b)

cot²θ

c)

tan²θ

d)

cos²θ

2.

a)

secθ

b)

cos²θ

c)

sin²θ

d)

sin²θ/cos²θ

3.

a)

sin x

b)

cos x

c)

tan x

d)

1

4.

Factor: sin²x +3sinx + 2

a)

(sinx +2)(sinx+1)

b)

(sinx +3)(sinx-1)

c)

sinx(sinx+3)

d)

(sinx +3)(sinx+1)

5.

Factor: csc³x - csc²x - cscx + 1

a)

cscx+1

b)

cot²x(cscx-1)

c)

cos²x(sinx-1)

d)

1

6.

Simplify the trig expression.

$\frac{\cos\left(90\degree-A\right)}{\sin\left(90\degree-A\right)}$

Submission (no fractions)

Type the trig function.

Place a space between the abbreviation and angle A.

(a)

7.

Find the solution over the interval [0, 2π)

a)

π/6; 5π/6; 7π/6; 11π/6

b)

π/2; 3π/2

c)

π/2; 3π/2; π/6; 5π/6; 7π/6; 11π/6

d)

π/2

8.

Solve equation for $0\le\theta<2\pi$ .
$1+\tan^2\theta=4\tan^2\theta$

a)

$\theta=\frac{\pi}{6},\frac{5\pi}{6},\frac{7\pi}{6},\frac{11\pi}{6}$

b)

$\theta=\frac{\pi}{3},\frac{2\pi}{3},\frac{4\pi}{3},\frac{5\pi}{3}$

c)

$\theta=\frac{\pi}{6},\frac{7\pi}{6}$

d)

$\theta=\frac{\pi}{6},\frac{5\pi}{6},\frac{7\pi}{6}$

9.

Solve $3\sin^2x=7\sin^2x+4\sin x+1$ over $\left[0,2\pi\right)\$

a)

$x=\left\{\frac{7\pi}{6}\right\}$

b)

$x=\left\{0,\ \frac{\pi}{3},\ \frac{5\pi}{3}\right\}$

c)

$x=\left\{\frac{7\pi}{6},\ \frac{11\pi}{6}\right\}$

d)

$x=\left\{\frac{3\pi}{4},\ \frac{7\pi}{4}\right\}$

10.

Find all solutions.

2sin(6x) + √3 = 0

a)

x = π/18 + πn/3 and x = 5π/18 + πn/3

b)

x = 11π/18 + 2πn and x = 7π/18 + 2πn

c)

x = 4π/18 + πn/3 and x = 5π/18 + πn/3

d)

x = 4π/3 + 2πn and x = 5π/3 + 2πn

11.

Solve on the domain [0, 2π)
2sin x cos x = √2 cos x

a)

π/2, 3π/2

b)

0, π/4, 3π/4

c)

π/2, 3π/2, π/4, 3π/4

d)

No solution

12.

Solve on the domain [0, 2π)
cos² x + sin x + 1 = 0

a)

π

b)

3π/2

c)

π/6, 5π/6, 3π/2

d)

No solution

13.

What type of conic section is $2x^2+2y^2-4x-16y-9=0$ ?

a)

Circle

b)

Parabola

c)

Ellipse

d)

Hyperbola

14.

Find the foci of this ellipse?

a)

(8, -1), (0, -1)

b)

(4±√7, -1)

c)

(4, 3), (4, -5)

d)

(4, -1)

15.

What is the equation used to find the focal points on an ELLIPSE?

a)

c² = a² + b²

b)

c² = a² - b²

c)

1/4d

d)

(x-h)² + (y-k)² = r²

16.

What are the vertices of the hyperbola?

a)

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

b)

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

c)

(0, 25) and (-25, 0)

d)

(0, 0) and (0, 5)

17.

Write in rectangular form and idenitify the conic section.

x = 3sec θ - 2

y = 2tan θ + 3

a)

$9\left(x+2\right)^2-4\left(y-3\right)^2=1$ horizontal hyperbola

b)

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

horizontal hyperbola

c)

$\frac{y^2}{4}-\frac{x^2}{9}=1$ vertical hyperbola

d)

$\frac{\left(x-2\right)^2}{9}-\frac{\left(y+3\right)^2}{4}=1$ vertical hyperbola

18.

What is the value of a and b for an ellipse with the following vertices and foci
Vertices ( -7, -3), (13, - 3)
foci ( - 5, -3 ) , (11 , -3)

a)

a = 10 , b = 6

b)

a = 6 , b = 10

c)

a = 36 , b = 100

d)

a = 100 , b = 36

19.

Write an equation for the ellipse with the given characteristics:
Vertices ( 4, 3), (4, - 9)
Length of minor axis is 8

a)

(x-4)²/36 + (y+3)²/4 = 1

b)

(x-4)²/16 + (y+3)²/36=1

c)

(x-4)² / 4 + (y+3)²∕36 =1

d)

(y+3)²/36 - (x-4)²/8 = 1

20.

Find the directrix of the parabola:
(x - 4)² = 8(y + 3)

a)

y = - 3

b)

x= 5

c)

y= - 5

d)

y= 2

21.

What is the focus of the following parabola?
(y + 5)²= -12 (x - 2)

a)

(2, - 5)

b)

( -5, 2)

c)

(1 , -5 )

d)

(-1, -5 )

22.

What number would go in the blank?

a)

-2

b)

-4

c)

-8

d)

-16

23.

Put the ellipse in standard form

a)

A

b)

B

c)

C

d)

D

24.

A hyperbola has vertices (±5, 0) and one focus at (6, 0). What is the equation of the hyperbola in standard form?

What are the equations for the asymptotes?

__________________________________________________

a)

x²/25 + y²/11 = 1

b)

x²/5 - y²/11 = 1

c)

x²/11- y²/25 = 1

d)

x²/25 - y²/11= 1

25.

What are the foci of the hyperbola with the equation y²/12 - x²/5 = 1

a)

(0, ±√17)

b)

(±√17, 0)

c)

(0, ±√7)

d)

(±√7, 0)

26.

What is the equation describing the relationship between dimensions and foci of a hyperbola?

a)

c² = a² + b²

b)

c² = a² - b²

c)

4p

d)

(x-h)² + (y-k)² = r²

27.

same question, now solve it and answer the question:

Farmer Jo has 32 square feet of land in which to make an enclosure for bunnies, chicks, and penguins. There will be fencing put in around the entire enclosure and in the middle (as pictured) to create 3 sections, what dimensions should the overall enclosure be in order to use the least amount of fence?

a)

The overall enclosure should be 8 feet by 8 feet

b)

The overall enclosure should be 8 feet by 4 feet

c)

The overall enclosure should be 32 square feet

d)

The overall enclosure should be $\sqrt{128}=8\sqrt{2}$ feet on each dimension

28.

same question, but more ...
There will be fencing put in around the entire enclosure and in the middle (as pictured) to create 3 sections, what dimensions should the overall enclosure be in order to use the least amount of fence?

Which equation represents the fence amount (F) that you want to minimize?

a)

$F=2x+4y$

b)

$F=2x+2y$

c)

$F=xy$

d)

$F=3\left(x+y\right)$

29.

A closed rectangular shipping box with square base is to be made from 120 square inches of cardboard. What dimensions should the box be for maximum volume?

Choose the constraint and optimization equations that represent the problem.

a)

$x^2=120$ and $V=x^3$

b)

$2x+y=120$ and $V=xy$

c)

$x^2y=120$ and $V=2x^2+4xy$

d)

$2x^2+4xy=120$ and $V=x^2y$

30.

Which equation would be used to find the volume of the open box? Don't forget to write down the letter of your answer before clicking!

a)

$\ \ M.\ \ \ V=\ \left(50\right)\left(20\right)\left(x\right)$

b)

$E.\ \ \ \ V=\ \left(50-2x\right)\left(20-2x\right)\left(x\right)$

c)

$P.\ \ \ V=\ \left(50-x\right)\left(20-x\right)\ x$

31.

A farmer wants to construct a rectangular pigpen using 400 ft of fencing. The pen will be built next to an existing stone wall, so only three sides of fencing need to be constructed to enclose the pen. What dimensions should the farmer use to construct the pen with the largest possible area?

a)

100ft x 200ft

b)

102ft x 196 ft

c)

50 ft x 300 ft

d)

50 ft x 175 ft

32.

A rectangular page is to contain 144 sq. inches of print. The margins on each side are 1 inch. Find the dimensions of the page that would use the least paper.

Choose the equations that represent the problem.

a)

xy=144 and A=(x+1)(y+1)

b)

(x+1)(y+1)=144 and A=xy

c)

xy=144 and A=(x+2)(y+2)

d)

(x+2)(y+2)=144 and A=xy

33.

Multiply

a)

1 2
6 -8

b)

-3 -5
-5 -11

c)

-2 8
-6 -12

d)

2 -3
5 -2

34.

Given matrices A & B:
A is a (4x33) matrix
B is a (33x1) matrix
What are the dimensions of the product of A times B?

a)

4x1

b)

4x33

c)

33x1

d)

Matrix does not exist

35.

What must be true in order to ADD two matrices?

a)

They must be square.

b)

The dimensions must be equal.

c)

The determinant can't equal 0.

d)

The column of the 1st must equal the row of the 2nd.

36.

What is the entry at f₄₂?

a)

23

b)

-81

c)

-44

d)

99

e)

-56

37.

Calculator Allowed

a)

b)

c)

d)

38.

Select the correct matrix

a)

A

b)

B

c)

C

39.

Solve.

a)

(-2, 6, -2)

b)

$\left(\frac{6}{7},\ 2,\ \frac{22}{7}\right)$

40.

Nancy went to the grocery story. On Monday she purchased 4 apples and 6 bananas for a total of $13. On Wednesday she purchased 3 apples and 7 bananas for a total of $13.50. Which system of equations represents the situation?

a)

4x + 6y = 3
13.5x - 13y = 6

b)

x + y = 4
x - y = 6

c)

4x + 6y = 13
3x + 7y = 13.5

d)

4x - 6y = 13
3x - 7y = 13.5

41.

The school that Laura goes to is selling tickets to the annual talent show. On the first day of ticket sales the school sold 4 senior citizen tickets, 2 adult tickets and 5 child tickets for a total of $55. The school took in $67 on the second day by selling 7 senior citizen tickets, 2 adult tickets and 5 child tickets. On the third day the show earned $46 when they sold 2 senior citizen tickets, 4 adult tickets and 2 child tickets. What is the price each of one senior citizen ticket, one adult ticket and one child ticket?

a)

Adult: $12

Child: $8

Senior: $8

b)

Adult: $6

Child: $4

Senior: $5

c)

Adult: $7

Child: $5

Senior: $4

d)

None of the above.

42.

Find the measure of ∠ADB. (Round to a whole answer.)

a)

25°

b)

10°

c)

58°

d)

122°

43.

A = 60° , a = 170 , b = 180

a)

h < a < b

B = 66.49° , C = 7.51° , c = 157.82

B = 92.51° , C = 2.49° , c = 15.26

b)

h < a < b

B = 66.49° , C = 53.51° , c = 157.82

B = 113.51° , C = 6.49° , c = 22.18

c)

a = h

B = 22.02° , C = 127.98° , c = 252.25

d)

No possible triangles

44.

What is
the Bearing
of B
from A?

a)

45

b)

225

c)

90

d)

135

45.

What are the heading of the picture above?

a)

S26⁰E

b)

E26⁰S

c)

N26⁰W

d)

E26⁰N

46.

What is the magnitude of the resulting force?

a)

18.343 Newtons

b)

8.918 Newtons

c)

20 Newtons

d)

4 Newtons

47.

The president’s airplane, Air Force One, flies at 250 m/s to the east with respect to the air. The air is moving at 35 m/s to the north with respect to the ground. Find the velocity of Air Force One with respect to the ground.

a)

252 m/s

b)

222 m/s

c)

435 m/s

d)

125 m/s

48.

Convert true bearing 095° into quadrant bearing

a)

S5°E

b)

S85°E

c)

N95°E

d)

E5°S

49.

A plane is traveling 350 mph at a true bearing of $160\degree$ . A wind is blowing to the east at a speed of 20 mph. What is the resultant speed and direction of the airplane (as a true bearing)?

a)

Speed: 357.33

Direction: 158.1

b)

Speed: 357.33

Direction: 23.1

c)

Speed: 357.33

Direction: 66.9

d)

Speed: 357.33 mph

Direction: 156.9

50.

Use sum or difference angles identity to find the exact value for $\cos105^o$

a)

$\frac{\sqrt{6}+\sqrt{2}}{4}$

b)

$\frac{\sqrt{6}-\sqrt{2}}{4}$

c)

$\frac{-\sqrt{6}-\sqrt{2}}{4}$

d)

$\frac{\sqrt{2}-\sqrt{6}}{4}$

51.

a)

A

b)

B

c)

C

d)

D

52.

a)

A

b)

B

c)

C

d)

D

53.

Given $\sin x=\frac{3}{5}and\ \sin y=\frac{2}{3},$ where $x\ and\ y$ are both in first quadrant. $Evaluate\ \sin\left(x+y\right)$

a)

$\frac{3\sqrt{5}+8}{15}$

b)

$\frac{4\sqrt{5}+6}{15}$

c)

$\frac{2\sqrt{5}+12}{15}$

d)

$\frac{4\sqrt{5}+2}{5}$

54.

Find AB

a)

10

b)

8

c)

6.1

d)

4.9

55.

What is the measure of angle A?

a)

58 degrees

b)

61 degrees

c)

78 degrees

d)

74 degrees

56.

Which law would you use?

a)

Law of Sines

b)

Law of Cosines

57.

$m\angle H=$

a)

$44.1\degree$

b)

$44.6\degree$

c)

$45.2$

d)

$45.6\degree$

58.

$m=$

a)

12.7

b)

13.1

c)

13.6

d)

14.2

59.

The focus is at (-8,-39/4) and the vertex is at (-8,-10). What is the equation of the parabola?

a)

y=(x+8)²-10

b)

y=(x+11)²+7

c)

y=-(x+8)²-10

d)

y=(-x+11)²-10

60.

Given the following, write the equation of the ellipse.

a)

A

b)

B

c)

C

d)

D

61.

What is the standard equation of the parabola with vertex $focus\left(11,7\right)\ and\ directrix\ y=4?$

a)

$\left(y-5.5\right)^2=-6\left(x-11\right)$

b)

$\left(y-5.5\right)^2=6\left(x-11\right)$

c)

$\left(x-11\right)^2=-6\left(y-5.5\right)$

d)

$\left(x-11\right)^2=6\left(y-5.5\right)$

62.

Based on the graph, what is the standard equation of the parabola?

a)

$\left(y+4\right)^2=12\left(x-2\right)$

b)

$\left(y+4\right)^2=-12\left(x-2\right)$

c)

$\left(x-2\right)^2=12\left(y+4\right)$

d)

$\left(x-2\right)^2=-12\left(y+4\right)$

63.

Identify the vertex, focus, axis of symmetry, directrix, direction of opening, and graph of each parabola.
$x^2=-\frac{1}{2}y$

a)

A

b)

B

c)

C

d)

D

64.

Graph the HYPERBOLA with the given equation:

$\frac{\left(x-2\right)^2}{4}-\frac{\left(y+1\right)^2}{16}=1$

a)

b)

c)

d)

65.

Find an equation in standard form for the HYPERBOLA that satisfies the given conditions:

center: (3,-5)

focus: (10,-5)

vertex: (7, -5)

a)

$\frac{\left(x-3\right)^2}{16}+\frac{\left(y+5\right)^2}{49}=1$

b)

$\frac{\left(x+3\right)^2}{16}+\frac{\left(y+5\right)^2}{33}=1$

c)

$\frac{\left(x-3\right)^2}{33}-\frac{\left(y+5\right)^2}{16}=1$

d)

$\frac{\left(x-3\right)^2}{16}-\frac{\left(y+5\right)^2}{33}=1$

66.

Find the equation for the graph shown

a)

$\frac{\left(x+1\right)^2}{9}+\frac{\left(y+2\right)^2}{4}=1$

b)

$\left(x+1\right)^2+\left(y+2\right)^2=9$

c)

$\frac{\left(x+1\right)^2}{6}+\frac{\left(y+2\right)^2}{4}=1$

d)

$\frac{\left(y+2\right)^2}{9}+\frac{\left(x+1\right)^2}{4}=1$

67.

An NFL punter at the 20 yard line kicks a football downfield with an initial velocity of 75 ft/sec at an angle of 66⁰. The ball leaves his foot at a height of 3 feet.

Model the scenario using parametric equations.

a)

x = (20cos(66⁰))t; y=-16t² + (20sin(66⁰)t + 75

b)

x = (20cos(66⁰))t; y=-16t² + (20sin(66⁰)t + 3

c)

x = (75cos(66⁰))t; y=-16t² + (75sin(66⁰)t + 20

d)

x = (75cos(66⁰))t; y=-16t² + (75sin(66⁰)t + 3

68.

An NFL punter at the 20 yard line kicks a football downfield with an initial velocity of 75 ft/sec at an angle of 66⁰. The ball leaves his foot at a height of 3 feet.

How high will the ball be when it reaches the crossbar of the goalposts, which are 25 yards (75 feet) away from the kicker?

a)

-81.516

b)

74.734

c)

28.236

d)

25

69.

Find the center and the radius of a CIRCLE with the given equation:

$x^2+y^2-12x+16y+36=0$

a)

center: (-6,8)

radius: 6

b)

center: (36,64)

radius: 8

c)

center: (6,-8)

radius: 8

d)

center: (6,-8)

radius: 64

70.

Convert the equation x² + 4y² + 2x + 24y + 21 = 0 into standard form.

a)

b)

c)

d)

71.

For the given ellipse, find the center, vertices, foci and eccentricity.

a)

b)

c)

d)

72.

For the given hyperbola, find the center, vertices, foci, eccentricity and equation of the asymptotes.

a)

b)

c)

73.

Write the equation of the hyperbola in standard form.

9x²- 54x - 4y² + 16y +101 = 0

a)

b)

c)

d)

74.

An elliptical pool is 5 meters across and 8 meters long with decorative fountains located at the foci. How far from the center should the fountains be located? (Rounded to the nearest tenth) How far apart are the fountains?

a)

9.8 meters, 19.6 meters

b)

3.1 meters, 9.8 meters

c)

3.1 meters, 6.2 meters

d)

6.2 meters, 38.4 meters

75.

a)

13

b)

7

c)

6

d)

-3

76.

a)

0

b)

DNE

c)

3

d)

1

77.

Find $\lim_{x\rightarrow2}\frac{\text{}\frac{x}{x+2}-2}{x+4}$ .

a)

4

b)

-4

c)

$\frac{1}{4}$

d)

$-\frac{1}{4}$

78.

a)

0

b)

1

c)

2

d)

DNE

Precalculus Semester 2 Skills Practice

Total soal: 78

Worksheet time: 7hrs 30mins

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