Pre-Calculus
Midterm Exam Review
Name: _____________________________________
Sequences & Series
a a 
Sn  n  1 n 
 2 
1 rn 

S n  a1
1  r 
right term value  left term value
big n  small n
an  a1  n  1d
d
an = (a1)(r)(n – 1)
r  big n small n
right term value
left term value
S
a1
1 r
1. Which of the following is an arithmetic sequence?
a) 2, 4, 8, 14, 22, …
b) 1, 5, 6, 10, 11, …
c) 3, 9, 21, 39, 63, …
d) –3, 0, 6, 15, 27, …
e) 3, 8, 13, 18, 23, …
2. What is a rule for the nth term of the arithmetic sequence with a8 = 21 and a14 = 45?
a) a n  4n  11
b) a n  4n  8
d) a n  4n  7
c) a n  4n  1
e) a n  4n  11
3. What is the common ratio of an infinite geometric series whose sum is 125 and the first term is a1 = 625?
a) –4
c) 
b) –2
1
2
d) 2
e) 4
d) –187
e) –100
d) 10
e) does not converge
d) 57.6
e) 160
11
4. Find the sum of the series:  1  3k 
k 1
a) –396
b) –374
c) –198
5. What is the sum of the series: 3 + 1.8 + 1.08 + 0.648 + ….?
a) 5
b) 7.5
c) 8
6. What is S25 for the arithmetic series 4 + 4.2 + 4.4 + 4.6 + 4.8 + …?
a) 5
b) 10
c) 25
7. What is the twelfth term of the sequence of a geometric sequence; -6, 18, -54…..
a) 3,188,646
b) -3,188,646
c) 1,062,882
d) -1,062,882
e) - 3
8. How to do you write the series 4 + 6 + 8 + 10 using sigma notation?
4
4
a)  2k
4
b)  2k  2
k 1
k 1
4
4
c)  k  1
e)  2k  2
d)  k  3
k 1
k 1
k 1
9. How do you write the series 4 + 6 + 9 + 13.5 using sigma notation?
4
a)
1.5n
4
4
n 1
b)
1.54
4
n 1
c)
n 1
 2k
n 1
n 1
4
10. Which series is represented by
 41.5
2
4
d)
 n1.5
4
4
n 1
e)
1.5n  1
n 1

k ?
k 2
a) 3 + 10 + 21
b) 10 + 21 + 36
c) 3 + 7 + 11
d) 10 + 14 + 18
e) 10 + 21 + 32
4
11. What is the common ratio of the sequence
a)
1
2
b)
3
2
3 3 3 3
, , ,
,... ?
100 50 25 12.5
c) 2
d) 4
e) 6
d) 91
e) 100
d) 3
e) 4.5
12. What is S6 for the geometric series 0.25 – 0.75 + 2.25 – 6.75 + …?
a) –60.75
b) –45.5
c) 22
13. What is the common difference of the sequence 3, 4.5, 6, 7.5, …?
a) 1
b) 1.5
c) 2
14. An infinite geometric series has a sum of 200 and a common ratio
a) 40
b) 60
4
. Which is the first term of this series?
5
c) 80
d) 100
e) 180
Combinatorics
n

r 0
n
Cr a n  r b r
15. How many distinguishable permutations of the letters in CABANA are there?
a) 60
b) 120
c) 180
d) 360
e) 720
d) 120
e) 125
16. How many distinguishable permutations of the letters in STORE are there?
a) 15
b) 24
c) 60
17. In an activity club with 24 students, the offices of president, vice president, secretary, and treasurer will be filled. In how many ways
can the offices be filled?
a) 24
b) 72
c) 96
d) 10,626
e) 255,024
18. In how many ways can a 9 person committee be chosen from a group of 12 people?
a) 9
b) 12
c) 220
d) 60,480
e) 79,833,600
d) 945
e) 2835
d) 720
e) 2160
19. What is the coefficient of x4 in the expansion of (3x + 1)7?
a) 35
b) 105
c) 315
20. What is the coefficient of x3 in the expansion of (2x – 3)5?
a) 36
b) 72
c) 108
21. A movie theater sells 3 sizes of popcorn (small, medium, and large) with 3 choices of toppings (no butter, butter, extra butter). How
many possible ways can a bag of popcorn be purchased?
a) 1
b) 3
c) 9
d) 27
e) 81
22. In how many ways can 10 runners finish a race first, second, or third?
a) 3
b) 10
c) 300
d) 720
e) 1000
23. How many possible license plates can be issued with two letters followed by three numbers? (Letters and digits may be repeated)
a) 25,000
b) 67,600
c) 250,000
d) 676,000
e) None of these
d) 22,100
e) 132,600
24. In how many ways can 3 cards be chosen from a standard deck of 52 cards?
a) 156
b) 2210
c) 12,600
25. A musician has 5 songs she can play at an open mic night. How many different ways can she play 2 of them?
a) 5
b) 10
c) 20
d) 60
e) 120
26. A printer has 8 colors of ink, but George can only pick 3 to use on a pamphlet that he is printing. How many different color
combinations can he choose?
a) 24
b) 56
c) 120
d) 336
e) 40,320
27. A math class is made up of 11 boys and 10 girls. How many ways can the teacher choose one boy and one girl to solve a problem for
the class?
a) 21
b) 22
c) 110
d) 121
e) 144
28. Which of the following sets represents the shaded region in the Venn Diagram?
A
a) A B
c) A '  B
b) A B
B
d) A  B '
e) B
29. You are forming a 7-member committee from 10 men and 13 women. The committee must consist of 4 men and 3 women. How
many different 7-member committees are possible?
a) 60060
b) 86486400
c) 245157
d) 1235591280
e) None of these
Probability
P(A or B) = P(A) + P(B) – P(A & B)
P(A & B) = P(A)  P(B)
P(B | A) =
𝑃(𝐵)∙𝑃(𝐴)
𝑃(𝐴)
30. A spinner has equal regions numbered 1 through 15. What is the probability that the spinner will stop on an even number or a
multiple of 3?
2
7
4
1
a)
b)
c)
d)
e) 12
3
9
5
3
31. If P(A) = 0.6, P(B) = 0.3, and P(A or B) = 0.8, what is P(A and B)?
a) 0.1
b) 0.2
c) 0.3
d) 0.4
e) 0.5
32. A card is randomly selected from a standard deck of 52 cards. What is the probability that it is a 5 or an ace?
a) 0.25
b) 0.208
c) 0.154
d) 0.106
e) 0.082
33. What is the probability of 3 or fewer successes for a binomial experiment consisting of 8 trials with the probability of 0.47 of
success on each trial?
a) 0.243
b) 0.315
c) 0.431
d) 0.569
e) 0.757
34. A fair coin is flipped three times. What is the probability of obtaining three tails?
a) 0.125
b) 0.167
c) 0.25
d) 0.50
e) 0.75
35. A fair coin is flipped 6 times. What is the probability of obtaining at least 4 heads?
a) 0.234
b) 0.891
c) 0.344
d) 0.109
e) none of these
36. What is the probability that in a family of seven children exactly two are girls? Assume a boy and a girl are equally likely.
a)
7
128
b)
7
64
c)
21
128
d)
2
7
e)
5
7
37. A fair coin is tossed 10 times. What is the probability of obtaining at least six heads?
a) 0.0013
b) 0.6230
c) 0.8281
d) 0.1719
e) 0.3770
d) 0.8
e) 1.2
38. Events A and B are independent, P(A) = 0.2, P(B) = 0.6, what is P(A and B)?
a) 0.12
b) 0.4
c) 0.55
39. Events A and B are dependent, P(A) = 60%, and P(B | A) = 30%. What is P(A and B)?
a) 10%
b) 18%
c) 24%
d) 30%
40. Five cards are dealt from a well-shuffled deck of playing cards. The expression
a) one red king or two red kings
d) at least two black queens
2
e) 90%
C1  50 C 4  2 C 2  50 C3
represents the probability of:
52 C 5
b) one jack or two black tens
e) two red kings and two black jacks
c) one black jack and two black tens
41. A bag contains 6 red marbles, 3 blue marbles, and 7 green marbles. If 3 marbles are randomly selected (without replacement) from
the bag, what is the probability that they are all blue?
a) .0018
b) .0015
c) .0066
d) 0.1
e) .0017
42. High school students were surveyed about lunch. 55% of all students bring a lunch from home. The remaining students buy a
school lunch. The school lunch offers pizza, grilled cheese, and tacos. Of the students who buy lunch, 50% like pizza, 42% like tacos,
and the remaining 8% like grilled cheese. If a student was picked at random what is the probability that a student buys a grilled cheese at
school?
a) 1%
b) 4%
c) 8%
d) 12%
e) 100%
Conics – reference your yellow cards. You will not get to use these cards on the midterm, but something very similar will be provided.
2
2
 x  1  y  2 

 1.
For numbers 43 – 45, refer to the ellipse represented by
16
9
43. Find the coordinates of the center.
a) (1, 2)
b) (–1, –2)
c) (–1, 2)
d) (–2, 1)
e) (1, –2)
44. Find the coordinates of the foci.

a) 1  7,  2

b) (5, –2) , (–3, –2)

c) 1,  2  7

d) (1, 4), (1, –8)
e) (–4, –2) , (6, –2)
45. Find the coordinates of the vertices and co-vertices.
a) (1, 2), (1, –6), (4, –2), (–2, –2)
b) (5, –2), (–3, –2), (1, 1), (1, –5)
c) (4, 2), (–2, 2), (1, 1), (1, –5)
d) (5, –2), (–3, –2), (1, 2), (1, –6)
e) Ellipses don’t have co-vertices
46. Find the coordinates of the foci for the hyperbola

a) 0,  2


b) 0,  6

y2 x2

 1.
4
2

c)  2, 0


d)  6, 0

e) None of these
47. Write the standard form of the equation of the hyperbola for which the transverse axis is 4 units long and vertical and the conjugate
axis is 3 units long.
a)
c)
e)
 x  1
2

2.25
 y  4
2

2.25
 y  4
 y  4
2
4

2
1
4
 x  1
2
1
4
 x  1
2.25
b)
d)
 y  4
2
2.25
 x  1

2
4

 x  1
2
1
4
 y  4
2.25
2
1
2
1
48. What is the directrix of the parabola with equation x2 = –28y?
a) x = 7
c) y = –7
b) x = 28
d) y = 7
e) y = 28
d) right
e) none
49. Determine the orientation of the parabola: focus (0, 4), directrix y = 1
a) up
b) down
Trigonometry
c) left
A = ½ r2
s = r
50. Which of the following is equivalent to
a) 285
s=
𝒅𝒆𝒈𝒓𝒆𝒆
𝟑𝟔𝟎
𝟐𝝅𝒓
A=
𝒅𝒆𝒈𝒓𝒆𝒆
𝟑𝟔𝟎
𝝅𝒓𝟐
7
?
4
b) 305
c) 315
d) 330
e) 375
51. Which of the following is equivalent to 240?
a)
5
6
b)
7
6
c)
5
4
d)
4
3
e)
8
3
52. What is the arc length of a sector with a radius of 5 cm and a central angle of 20?
a) 1.75 cm
b) 2.8 cm
c) 4.35 cm
d) 4 cm
e) 100 cm
53. What is the area of a sector with a radius of 10 cm and a central angle of
a) 40.08 cm2
b) 43.63 cm2
c) 48.72cm2
5
?
18
d) 51.43 cm2
e) 58.18 cm2
54. If (6, 2) is a point on the terminal side of an angle θ in standard position, what is the value of sin θ?
1
a)
b)
10
1
3
2
c)
10
d)
e) 3
10
55. The number of degrees in one revolution is:
a)

2
b) π
c) 2π
d) 180°
e) 360°
d) IV
e) none
56. The tangent and cosine functions are both negative in which quadrant?
a) I
b) II
c) III
57. What is the value of tan  when cos  = 
a) 
5
3
b) 
4
3
3
in quadrant 2?
5
c) 
5
4
d) 
4
5
e) 
3
5
58. Using a reference angle, sin 117° = ?
a) –sin 17°
b) –sin 63°
c) –sin 117°
d) sin 63°
e) sin 17°
3

59. Find the exact value of csc tan 1  .
4

a) 
5
3
60. If cot x = 
a) 
b) 
3
5
c)
3
5
d)
5
4
e)
5
3
c)
5
13
d)
12
13
e)
12
5
5
where 0 ≤ x ≤ π find cos x.
12
12
5
b) 
5
13
61. An airplane is at an elevation of 15,000 ft and approaches the airport with an angle of descent of 10°. What is the distance between
the airport and the point on the ground directly below the plane?
a) 15,231.399 ft
b) 85,069.227 ft
c) 86,381.557 ft
4
3
, which of the following is true?
62. If cos   and    
5
2
3
4
5
a) sin   
b) tan  
c) tan   
4
4
3
d) 230,114.595 ft
d) cot   
3
4
e) 342,901.569 ft
e) sin  
3
5