Practice Final Exam 2014
Name:________________________________
Hour:______
Pre-Calculus
1. Simplify: csc 𝜃 cos 𝜃 tan 𝜃 sec 𝜃
A. cos 𝜃
B. sec 𝜃
C. 1
D. csc 𝜃
C. csc 𝜃
D. sin2 𝜃
C. csc 𝜃
D. 1
2. Simplify: csc 𝜃 − cot 𝜃 cos 𝜃
A. sin 𝜃
B. cot 𝜃
3. Simplify: (1 − sin2 𝜃)(1 + cot 2 𝜃)
A. cos 2 𝜃
B. cot 2 𝜃
4. 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
5. Use a double-angle formula to find sin 2x if cos 𝑥 =
A. −
√3
2
B.
√3
2
√3
2
24
7
B.
and
1
C. − 2
6. Use a double-angle formula to find tan 2x if sec 𝑥 = −
A. −
D.
336
527
C.
24
7
3𝜋
2
√6+√2
4
< 𝑥 < 2𝜋.
D. √3
25
7
and 180º < x < 270º.
D. −
336
527
D.
7𝜋
4
√2
7. Find cos−1 ( 2 ).
A.
𝜋
4
B. −
𝜋
4
C.
3𝜋
4
7
8. Evaluate sec (tan−1 24).
A.
25
7
B.
24
25
C.
25
24
D.
24
7
D.
3𝜋
4
D.
𝜋
6
9. 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
and
5𝜋
4
10. 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
11. 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
D.
𝜋 3𝜋 4𝜋 5𝜋
, , ,
2 2 3 3
12. Solve: cos 2𝑥 = sin 𝑥 for all x-values.
𝜋
3
2𝜋
3
+ 2𝑘𝜋
𝜋
5𝜋
6
+ 2𝑘𝜋
𝜋
5𝜋
6
+ 2𝑘𝜋
2𝜋
3
+ 2𝑘𝜋
A. 𝜋 + 2𝑘𝜋, + 2𝑘𝜋,
B.
3𝜋
2
+ 2𝑘𝜋, 6 + 2𝑘𝜋,
C. 𝜋 + 2𝑘𝜋, 6 + 2𝑘𝜋,
D.
3𝜋
2
𝜋
3
+ 2𝑘𝜋, + 2𝑘𝜋,
13. Which graph represents the polar coordinate (−3,
A.
B.
2𝜋
)?
3
C.
D.
14. 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 )
15. 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
16. Convert the polar coordinate (2, 2 ) to a rectangular coordinate.
A. (0, -2)
B. (0, 2)
C. (2, 0)
D. (-2, 0)
17. Find another polar representation of the point (−2, 𝜋) where r > 0.
A. (2, 3𝜋)
B. (2, 2𝜋)
18. 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
19. 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
𝜋
D. 2 (cos 6 + 𝑖 sin 6 )
5
20. 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
21. 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〉
5𝜋
3
+ 𝑖 sin
5𝜋
)
3
22. Find the direction of the vector 𝐯 = −3𝐢 + 3√3𝐣.
A. 300º
B. 60º
C. 120º
D. 240º
C. -30
D. 30
23. Find 𝐮 ∙ 𝐯 if 𝐮 = 〈2, 6〉 and 𝐯 = 〈−3, 4〉.
A. 18
B. 8
24. Find the angle between u and v if 𝐮 = 〈4, 7〉 and 𝐯 = 〈−3, 5〉.
A. -89º
B. 61º
C. 179º
D. 29º
25. 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
26. Which is a vertex of the ellipse
A. (-1, 3)
(𝑥+1)2
25
C. 2970 ft/lb
+
(𝑦+2)2
4
D. 3030 ft/lb
= 1?
B. (-1, -2)
C. (4, -2)
D. (5, 2)
27. The endpoints of the major axis of an ellipse are (6, 3) and (-2, 3) and the endpoints of the minor
axis are (2, 0) and (2, 6). Find the equation of the ellipse.
A.
(𝑥−2)2
16
+
(𝑦−3)2
9
=1
B.
(𝑥−2)2
9
+
(𝑦−3)2
16
C.
=1
(𝑥−3)2
16
+
(𝑦−1)2
9
28. Identify the equation of the asymptotes for the graph of the hyperbola
9
A. 𝑦 = ± 16 𝑥
4
B. 𝑦 = ± 3 𝑥
29. Find the foci of the hyperbola
A. (−2 ± √13, 1)
(𝑦−1)2
4
C. 𝑦 = ±
−
(𝑥+2)2
B. (−2, 1 ± √13)
9
16
𝑥
9
D.
=1
𝑥2
16
−
𝑦2
9
(𝑥−2)2
64
B. up
(𝑦−3)2
36
= 1.
3
D. 𝑦 = ± 4 𝑥
= 1.
C. (0, ±√13)
D. (−2, 1 ± √5)
30. In which direction does the parabola 𝑦 2 = −5𝑥 open?
A. right
+
C. down
D. left
=1
31. Given that the vertex of a parabola is (4, -2) and the focus is (4, 3), find the equation of the
parabola.
A. 𝑥 2 = 20𝑦
B. (𝑦 + 2)2 = 20(𝑥 − 4)
C. (𝑥 − 4)2 = 20(𝑦 + 2)
D.(𝑥 − 4)2 = 5(𝑦 + 2)
C. 1
D. 3
2
2
32. Find the sum of the sequence: ∑(𝑘 − 2)
𝑘=0
A. -1
B. 0
33. Find the nth term of the sequence: 53, 46, 39, 32, …
A. 𝑎𝑛 = 𝑛 − 7
B. 𝑎𝑛 = −7𝑛 + 46
C. 𝑎𝑛 = −7𝑛 + 60
D. 𝑎𝑛 = 53(−7)𝑛−1
34. Find the next 3 terms of the sequence: 𝑎𝑛 = 2𝑎𝑛−1 + 1 when 𝑎1 = −5.
A. -9, -17, -33
B. -10, -20, -40
C. 3, 5, 7
D. -4, -3, -2
35. If the 7th term of an arithmetic sequence is 27 and the 16th term is 63, find the 23rd term.
A. 223
B. 91
C. 4
D. 3
36. Which term of the sequence -290, -280, -270, -260, is 30?
A. 31st term
B. 33rd term
C. 32nd term
1
D. 30th term
2
37. The common ratio in a geometric sequence is 3 and the fifth term is 27. Find the third term.
A.
1
81
B. 6
C.
2
3
D.
2
19683
38. Find the 21st term in the expansion of (𝑥 + 𝑦)32 .
A. 225792840𝑥 12 𝑦 20
B. 129024480𝑥11 𝑦 21
C. 225792840𝑥 20 𝑦12
D. 129024480𝑥 20 𝑦12
lim 𝑓(𝑥) , if it exists.
39. For the function f whose graph is given, state the value of 𝑥→1
A. Does not exist
B. 1
C. -1
D. 3
40. Evaluate the limit, if it exists: lim (𝑥 2 + 𝑥 + 1)
𝑥→−2
A. 3
B. -5
lim (
41. Evaluate the limit, if it exists: 𝑥→3
A. 1
C. Does not exist
D. 5
C. Does not exist
D. 0
𝑥 2 − 2𝑥 − 3
)
𝑥 2 − 10𝑥 + 21
B. -1
42. 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
43. Find the derivative of the function 𝑓(𝑥) = 5𝑥 7 − 3𝑥 4 − 2𝑥 + 6.
A. 12𝑥 6 + 𝑥 3 − 1
B. 35𝑥 6 − 12𝑥 3 − 2
C. 35𝑥 6 − 12𝑥 3 − 2𝑥 + 1
D. 23𝑥 − 2
Answers
1. B
2. A
3. B
4. D
5. A
6. D
7. A
8. C
9. C
10. A
11. A
12. B
13. C
14. A
15. D
16. B
17. B
18. A
19. A
20. C
21. D
22. C
23. A
24. B
25. A
26. C
27. A
28. D
29. B
30. D
31. C
32. A
33. C
34. A
35. B
36. B
37. C
38. A
39. A
40. A
41. B
42. D
43. B