22M:015 – FINAL EXAM
1. If a couple has 3 children, what is the probability of their having at least one son and at least one daughter (assuming equally likely outcomes for both sexes)?
a.
b.
c.
d.
e. None of the above.
2. In a game, a die is rolled. If an odd number shows, one wins twice the number; otherwise one loses three times the number. The expected win is:
a. $–4
b. $2
c. $–2
d. $1
e. None of the above.
3. The distance between the points (1,–2) and (2,–1) is
a. 2
b.
c.
d.
e. None of the above.
4. The minimum value of x subject to the constraints x + y ≤ 2, x – y ≥ 1, x ≥ 0, y ≥ 0 is:
a. 1
b. 2
c.
d.
e. None of the above.
5. The number of 3-digit numbers, with no repetition of digits, having 1 as one of the digits and not having 0 as the first digit, is:
a. 21
b. 18
c. 17
d. 16
e. None of the above.
6. The letters of the word BANANA can be arranged in
a. 150 ways
b. 140 ways
c. 120 ways
d. 90 ways
e. None of the above.
7. A, B are independent events in a probability space such that P(B – A) = .2, P(A) = .6. Then P(B) =
a. .4
b. .5
c. .2
d. .3
e. None of the above.
8. A club consists of 4 men and 6 women. The number of ways a committee of 4 can be formed if it must have at least one woman is:
a. 220
b. 210
c. 180
d. 150
e. None of the above.
9. A group, consisting of 3 men and 2 women, are to be seated on 5 chairs in a row. If the middle chair is to be occupied by a woman and end seats are to be taken by men, the number of ways it can be done is:
a. 120
b. 60
c. 48
d. 24
e. None of the above.
10. Two cards are drawn, in succession, from a deck of 52 cards without replacement. If the second card is black then the probability that the first card is also black is:
a.
b.
c.
d.
e. None of the above.
11. The line through (0,–1) and perpendicular to 2x – 3y = 0 also passes through:
a. (0,0)
b. (1,1)
c. (1,2)
d. (–2,2)
e. None of the above.
12. The value of the determinant 
is:
a. 0
b. 12
c. –15
d. –12
e. None of the above.
13. Assume x > 1 and xln x – x2 = 0. Then
a. 1 < x < 3
b. 3 < x < 5
c. 5 < x < 7
d. 7 < x < 9
e. None of the above.
14. The equation 
= 9 has:
a. 1 solution
b. 2 solutions
c. 3 solutions
d. 4 solutions
e. None of the above.
15. If r is distance of the center of the circle x2 + y2 + x – 2y = 1 from the origin then
a. 1 < r < 2
b. 2 < r < 3
c. 3 < r < 4
d. 4 < r < 5
e. None of the above.
16. An amount $6,000 is invested at the rate of r% compounded continuously. After 10 years the value of the investment is $10,000. Then
a. 2 < r < 4
b. 4 < r < 6
c. 6 < r < 7
d. 7 < r < 9
e. None of the above.
17. In a triangle a = 7, b = 3, c = 3. Then angle A satisfies:
a. 0 < A < 45
b. 45 < A < 90
c. 90 < A < 135
d. 135 < A < 180
e. None of the above.
18. If A = 
, then A2 =
a.
b.
c.
d.
e. None of the above.
19. If sec x = 
and x is in the third quadrant then cos 2x =
a.
b.
c.
d.
e. None of the above.
20. A coin is tossed 6 times. The probability of getting three heads, given that in the first two tossing turned up tails, is:
a. 0.5
b. 0.05
c. 0.25
d. 0.2
e. None of the above.
21. Two dice are rolled five times. The probability of getting sum = 6 exactly 3 times is:
a.
b.
c.
d.
e. None of the above.
22. sin(A + B) sin(A – B) =
a. sin2A – sin2B
b. sin2A + sin2B
c. cos2A – cos2B
d. cos2A + cos2B
e. None of the above.
23. A and B are finite sets such that n(A – B) = 10, n(B – A) = 9 and n(A 
B) = 23. Then n(A) =
a. 13
b. 14
c. 15
d. 18
e. None of the above.
24. The arc, of length 3, of a circle of radius r makes an angle of 110, in degrees, at the center. Then:
a. 0 < r < 5
b. 5 < r < 10
c. 10 < r < 15
d. 15 < r < 20
e. None of the above.
25.The system of equations x – y = 2, 3x – 3y – z = 1, x + y + z = –1, has
a. one solution
b. no solution
c. two solutions
d. infinity of solutions
e. None of the above.