Download Sample Exam B - PSU Math Home

MATH 26
NAME
FINAL EXAM
SECTION NUMBER
INSTRUCTOR
SAMPLE B
On your scantron, write and bubble your PSU ID, Section Number, and Test Version. Failure
to correctly code these items may result in a loss of 5 points on the exam.
On your scantron, bubble letters corresponding to your answers on indicated questions. It
is a good idea for future review to circle your answers in the test booklet.
Check that your exam contains 25 questions numbered sequentially.
Answer Questions 1-25 on your scantron.
Each multiple choice question is worth 6 points.
THE USE OF A CALCULATOR, CELL PHONE, OR ANY
OTHER ELECTRONIC DEVICE IS NOT PERMITTED IN THIS
EXAMINATION.
THE USE OF NOTES OF ANY KIND IS NOT PERMITTED
DURING THIS EXAMINATION.
MATH 26
FINAL EXAM, SAMPLE B
PAGE 2
1. Which of the following is the angle of least positive measure that is coterminal with
θ = −714◦ ?
a) 6◦
b) 3◦
c) 714◦
d) 14◦
e) 720◦
6
2. Which of the following is the area of the sector of a circle of radius
inches formed by a
π
◦
central angle of 60 ?
a) 2 square inches
b)
6
square inches
π
c)
3
square inches
π
d)
1080
square inches
π2
e)
24
square inches
π
3. A point at the edge of a circular fishing reel has a linear speed of 64π cm per second. Which
of the following is the radius of the reel if it makes 2 revolutions per second?
a) 32π cm
b) 16 cm
1
cm
16
π
d)
cm
16
c)
e) 32 cm
MATH 26
FINAL EXAM, SAMPLE B
PAGE 3
4. The triangles shown below are similar. Which of the following options lists the correct values
of x and y?
a) x = 6, y = 8
b) x = 6, y = 16
3
c) x = , y = 4
2
d) x = 4, y =
3
2
9
9
e) x = , y =
8
4
5. Evaluate sin2
π
π
+ tan .
3
4
a) 1
b)
5
4
c)
7
4
√
3
2
d) +
4
2
1 √
e) + 3
4
MATH 26
FINAL EXAM, SAMPLE B
PAGE 4
11π
.
6. Evaluate sec −
3
a) −2
2
b) √
3
2
c) − √
3
d)
1
2
e) 2
1
7. The point
, y in Quadrant IV lies on the unit circle and corresponds to an angle θ. Find
4
the value of sin θ.
a)
1
4
1
4
√
15
c)
4
√
15
d) −
4
b) −
e)
3
4
MATH 26
FINAL EXAM, SAMPLE B
PAGE 5
π
8. Which of the following sets consists only of x-intercepts for the function f (x) = cos 3x −
?
2
π 5π
a)
,
7 6
nπ π o
,
b)
2 3
π 3π
c)
,
3 4
π 2π
,
d)
3 3
π 3π
e)
,
5 5
9. Let f (x) = sin(3x − π) and g(x) = 2 sin(6x − 2π). Which of the following is TRUE?
a) The graphs of f and g have the same period.
b) The graphs of f and g have the same phase-shift.
c) The graphs of f and g have the same x-intercepts.
d) The graphs of f and g have the same amplitude.
e) The graphs of f and g have nothing in common.
10. Which of the following is a vertical asymptote for f (x) = − sec(6x − 3π)?
π
6
π
b) x =
2
a) x =
c) x =
5π
6
2π
3
π
e) x =
4
d) x =
MATH 26
FINAL EXAM, SAMPLE B
−1
11. Find the exact value of cos
1
− .
2
π
3
π
b) −
6
a) −
c)
2π
3
d)
5π
6
e)
5π
3
12. What is the exact value of cos(tan−1 (2))?
2
a) √
5
1
b) √
5
2
c) − √
5
1
d) − √
5
e) The value does not exist.
13. Which of the following expressions is equivalent to
a) cot(x)
b) − cot(x)
c) − cot(7x)
d) cot(7x)
e) − cot7 (x)
cos(7x)
?
sin(−7x)
PAGE 6
MATH 26
FINAL EXAM, SAMPLE B
PAGE 7
14. Simplify the following trigonometric expression.
1
1
−
1 − cos(2x) 1 + cos(2x)
a) 2 cot x csc x
b) −2 tan(2x) sec(2x)
c) 2 csc2 (2x)
d) 2 cot(2x) csc(2x)
e) 2 sec2 x
15. Suppose that the terminal side of angle α lies in Quadrant I and the terminal side of angle
3
2
β lies in Quadrant IV. If sin α = and cos β = , which of the following is the value of
3
5
sin(α − β)?
a)
b)
c)
d)
e)
√
6+4 5
15
√
6−4 5
15
√
8+3 5
15
√
8−3 5
15
√
3 5−8
15
MATH 26
FINAL EXAM, SAMPLE B
π
16. Use a half-angle formula to evaluate sin −
.
12
s
√
2+ 3
a)
4
s
√
2− 3
b)
4
s
√
2+ 3
c) −
4
s
√
2− 3
d) −
4
e) −
1
2
1
−1
17. Find the value of cos 2 sin
.
4
a) −
b)
1
2
7
8
7
8
√
3
d)
2
√
c) −
e) −
3
2
PAGE 8
MATH 26
FINAL EXAM, SAMPLE B
PAGE 9
18. Suppose α and β are two angles satisfying cos(α) = sin(β). Which of the following is equal
to cos(α − β) − cos(α + β)?
a) sin(2α)
b) sin(2β)
c) cos(2α)
d) cos(2β)
e) tan(2αβ)
19. Find all solutions to the equation 2 sin x cos x = sin x on the interval [0, 2π).
a)
π 5π
,
3 3
b)
π 11π
,
6 6
c)
π 2π
,
3 3
π 11π
d) 0, π, ,
6 6
π 5π
e) 0, π, ,
3 3
20. Find the lengths of the missing side of 4ABC (if possible) given that
6
a = 3, b = √ , A = 60◦ .
3
(a, b, c represent the lengths of the sides opposite to the angles A, B, C respectively.)
a) c =
√
3
b) c = 3
c) c = 6
d) c = 2
e) There is no such triangle.
MATH 26
21. In a triangle ∆ABC, a =
FINAL EXAM, SAMPLE B
√
PAGE 10
7, b = 2 and c = 1, find the angle A.
a) A = 30◦
b) A = 60◦
c) A = 90◦
d) A = 120◦
e) A = 150◦
22. Determine the length of the chord intercepted by a central angle of 30◦ in a circle of radius
of 2 cm.
√
2 cm
√
b) 3 cm
a)
c) 4 cm
q
√
d)
8 − 2 3 cm
q
√
e) 8 − 4 3 cm
23. What is the area of a triangle with sides 3, 5, 6?
√
56
√
b) 8
√
c) 792
√
d) 28
a)
e) 9
MATH 26
FINAL EXAM, SAMPLE B
24. Write the polar equation r = 3 cos θ in rectangular coordinates.
a) x2 + y 2 = 3x
b) x2 + y 2 = 3y
c) y 2 = 3x
p
d)
x2 + y 2 = 3x
e) x + y = 3x2
√ √
25. Convert the point (− 3, 3) from rectangular coordinates to polar coordinates.
√
π
3
√
3, −1
b)
a)
6,
π
3, −
4
√ 3π
d)
6,
4
√ 7π
e) 2 3,
4
c)
PAGE 11
MATH 26
FINAL EXAM, SAMPLE B
sin(α + β) = sin α cos β + cos α sin β
sin(α − β) = sin α cos β − cos α sin β
cos(α + β) = cos α cos β − sin α sin β
cos(α − β) = cos α cos β + sin α sin β
tan α + tan β
tan(α + β) =
1 − tan α tan β
tan α − tan β
tan(α − β) =
1 + tan α tan β
sin(2α) = 2 sin α cos α
cos(2α) = cos2 α − sin2 α = 1 − 2 sin2 α = 2 cos2 α − 1
r
1 − cos α
α
sin = ±
2
2
r
α
1 + cos α
cos = ±
2
2
α
1 − cos α
sin α
tan =
=
2
sin α
1 + cos α
1
sin α cos β = [sin(α + β) + sin(α − β)]
2
1
cos α sin β = [sin(α + β) − sin(α − β)]
2
1
cos α cos β = [cos(α − β) + cos(α + β)]
2
1
sin α sin β = [cos(α − β) − cos(α + β)]
2
α+β
α−β
cos
2
2
α−β
α+β
sin α − sin β = 2 sin
cos
2
2
α+β
α−β
cos α + cos β = 2 cos
cos
2
2
α+β
α−β
cos α − cos β = −2 sin
sin
2
2
sin α + sin β = 2 sin
PAGE 12
MATH 26
1. A
2. B
3. B
4. C
5. C
6. E
7. D
8. D
9. B
10. E
11. C
12. B
13. C
14. D
15. A
16. D
17. B
18. A
19. E
20. A
21. D
22. E
23. A
24. A
25. D
FINAL EXAM, SAMPLE B
PAGE 13