Practice Final Exam
Name:________________________________
Hour:______
Pre-Calculus
1. Find the value of x in the figure below.
A. 418.8
B. 990.9
x
C. 730.1
35º
D. 975.9
25º
300
2. Use the Law of Sines or Law of Cosines to find side a in ABC, given A = 50º, C = 33.5º, and b = 76.
A. 105.5
B. No solution
C. 42.2
D. 58.6
3. Use the Law of Sines or Law of Cosines to find the largest angle in a triangle with sides 4, 6, 9.
A. 53.8
B. 32º
C. 127.2º
D. No Solution
√2
4. Find cos−1 ( 2 ).
𝜋
4
A.
B. −
𝜋
4
3𝜋
4
C.
7𝜋
4
D.
7
5. Evaluate sec (tan−1 24).
A.
25
7
B.
24
25
C.
25
24
D.
24
7
𝜋
6. Write the equation of the sine function with amplitude 3 and period 2 .
𝜋
2
A. 𝑦 = 3 sin ( ) 𝑥
B. 𝑦 = 3 sin 2𝑥
C. 𝑦 = 3 sin 4𝑥
7. State a sine equation that represents the graph below.
1
A. 𝑦 = 3 sin (3) 𝑥 + 1
B. 𝑦 = 2 sin 3𝑥 − 1
1
0
1
C. 𝑦 = 2 sin (3) 𝑥 + 1
3𝜋
6𝜋
D. 𝑦 = 2 sin 6𝜋𝑥 + 1
D. 𝑦 = 3 sin 4𝜋𝑥
8. Suppose a wheel has a diameter of 16 inches and it spins at a rate of 50 revolutions per minute.
Determine the linear velocity of the wheel in mi/h.
A. 2.4 mi/h
B. 4.8 mi/hr
C. 0.38 mi/hr
D. 28.6 mi/h
C. 1
D. csc 𝜃
C. csc 𝜃
D. sin2 𝜃
C. csc 𝜃
D. 1
9. Simplify: csc 𝜃 cos 𝜃 tan 𝜃 sec 𝜃
A. cos 𝜃
B. sec 𝜃
10. Simplify: csc 𝜃 − cot 𝜃 cos 𝜃
A. sin 𝜃
B. cot 𝜃
11. Simplify: (1 − sin2 𝜃)(1 + cot 2 𝜃)
A. cos 2 𝜃
B. cot 2 𝜃
12. Use a sum/difference formula to find the exact value of sin 105º.
A.
√2−√6
4
B.
√6+√2
2
C.
√6−√2
4
13. Use a double-angle formula to find sin 2x if cos 𝑥 =
A. −
√3
2
B.
√3
2
D.
√3
2
24
7
B.
3𝜋
2
1
C. − 2
336
527
C.
< 𝑥 < 2𝜋.
D. √3
14. Use a double-angle formula to find tan 2x if sec 𝑥 = −
A. −
and
√6+√2
4
25
7
and 180º < x < 270º.
24
7
336
D. − 527
15. Solve: cot 𝜃 = 1 for all x-values in the interval [0,2𝜋).
A.
𝜋
4
and
3𝜋
4
B.
3𝜋
4
and
7𝜋
4
C.
𝜋
4
and
5𝜋
4
D.
3𝜋
4
D.
𝜋
6
D.
𝜋 3𝜋 4𝜋 5𝜋
, , ,
2 2 3 3
and
5𝜋
4
16. Solve: sec 𝜃 + 2 = 0 for all x-values in the interval [0,2𝜋).
A.
2𝜋
3
and
4𝜋
3
B.
𝜋
3
and
5𝜋
3
C.
5𝜋
6
and
7𝜋
6
and
11𝜋
6
17. Solve: 2 sin2 𝑥 + sin 𝑥 = 0 for all x-values in the interval [0,2𝜋).
A. 0, 𝜋,
7𝜋 11𝜋
, 6
6
B.
𝜋 3𝜋 7𝜋 11𝜋
, , , 6
2 2 6
C. 0, 𝜋,
4𝜋 5𝜋
,
3 3
18. Solve: cos 2𝑥 = sin 𝑥 for all x-values in the interval [0,2𝜋).
𝜋 2𝜋
3
A. 𝜋, 3 ,
B.
3𝜋 𝜋 5𝜋
, ,
2 6 6
𝜋 5𝜋
6
C. 𝜋, 6 ,
19. Which graph represents the polar coordinate (−3,
A.
B.
D.
3𝜋 𝜋 2𝜋
, ,
2 3 3
2𝜋
)?
3
C.
D.
20. Convert the rectangular coordinate (5, 7) to a polar coordinate.
A. (√74, 54.50 )
B. (√74, 35.50 )
C. (√24, 54.50 )
D. (√24, 35.50 )
21. Convert the polar coordinate (-3, 240º) to a rectangular coordinate.
A. (−
3√3
3
, − 2)
2
3
B. (− 2 , −
3√3
)
2
3√3 3
, )
2 2
3 3√3
)
2
C. (
D. (2 ,
𝜋
22. Convert the polar coordinate (2, 2 ) to a rectangular coordinate.
A. (0, -2)
B. (0, 2)
C. (2, 0)
D. (-2, 0)
23. Find another polar representation of the point (−2, 𝜋) where r > 0.
A. (2, 3𝜋)
B. (2, 2𝜋)
24. Express 4 (cos
5𝜋
3
+ 𝑖 sin
A. 2 − 2𝑖√3
5𝜋
)
3
C. (−2, 𝜋 )
D. (−2, 2𝜋)
in rectangular form.
B. 2√3 − 2𝑖
C. −2 + 2𝑖√3
D. −2 − 2𝑖√3
25. Express −√3 + 𝑖 in polar form.
A. 2 (cos
5𝜋
6
+ 𝑖 sin
5𝜋
)
6
B. 2 (cos
2𝜋
3
+ 𝑖 sin
2𝜋
)
3
C. 2 (cos
11𝜋
6
+ 𝑖 sin
11𝜋
)
6
𝜋
6
𝜋
6
D. 2 (cos + 𝑖 sin )
5
26. Use DeMoivre’s Theorem to find (1 − √3𝑖) in polar form.
A. 32 (cos
55𝜋
6
+ 𝑖 sin
55𝜋
)
6
B. 32 (cos
5𝜋
3
+ 𝑖 sin
5𝜋
)
3
C. 32 (cos
25𝜋
3
+ 𝑖 sin
25𝜋
)
3
D. 2 (cos
5𝜋
3
+ 𝑖 sin
5𝜋
)
3
27. Find the vector in component form having magnitude |𝐯| = 4 and direction 𝜃=150º.
A. ⟨2, −2√3⟩
B. ⟨−2, 2√3⟩
C. ⟨2√3, −2⟩
D. ⟨−2√3, 2⟩
28. Find the direction of the vector 𝐯 = −3𝐢 + 3√3𝐣.
A. 300º
B. 60º
C. 120º
D. 240º
C. -30
D. 30
29. Find 𝐮 ∙ 𝐯 if 𝐮 = ⟨2, 6⟩ and 𝐯 = ⟨−3, 4⟩.
A. 18
B. 8
30. Find the angle between u and v if 𝐮 = ⟨4, 7⟩ and 𝐯 = ⟨−3, 5⟩.
A. -89º
B. 61º
C. 179º
D. 29º
31. Find the work done by the force F = 50i – 40j in moving an object from (-1, 5) to (60, 3).
A. 3130 ft/lb
B. 2630 ft/lb
C. 2970 ft/lb
D. 3030 ft/lb
32. For the function f whose graph is given, state the value of lim 𝑓(𝑥) , if it exists.
𝑥→1
A. Does not exist
B. 1
C. -1
D. 3
33. Evaluate the limit, if it exists: lim (𝑥 2 + 𝑥 + 1)
𝑥→−2
A. 3
B. -5
lim (
34. Evaluate the limit, if it exists: 𝑥→3
A. 1
C. Does not exist
D. 5
𝑥 2 − 2𝑥 − 3
)
𝑥 2 − 10𝑥 + 21
B. -1
C. Does not exist
D. 0
35. Find an equation of the tangent line to 𝑦 = 𝑥 2 − 3𝑥 + 5 at (2, 3).
A. y = x – 1
B. y = -x + 5
C. y = -x + 3
D. y = x + 1
Answers to Practice Exam
Pre-Calculus
1. A
2. D
3. C
4. A
5. C
6. C
7. C
8. A
9. B 10. A
11. B
12. D
13. A
14. D
15. C
16. A
17. A
18. B
19. C
20. A
21. D
22. B
23. B
24. A
25. A
26. C
27. D
28. C
29. A
30. B
31. A
32. A
33. A
34. B
35. D
Answers to Practice Exam
Pre-Calculus
1. A
2. D
3. C
4. A
5. C
6. C
7. C
8. A
9. B 10. A
11. B
12. D
13. A
14. D
15. C
16. A
17. A
18. B
19. C
20. A
21. D
22. B
23. B
24. A
25. A
26. C
27. D
28. C
29. A
30. B
31. A
32. A
33. A
34. B
35. D