.MATH 115 - FINAL EXAMDecember 16, 2005

~ame ~------------------------

Student no. (from ID)_______________________________

Flec. Day ~ time ~-----------------------------

Teaching Assistant ___________________________________

This is a Multiple choice, closed book, no calculator exam. You may use a 5" x 8" card.Show all your work.

PUT YOUR ANSWERS ON THE ANSWER SHEET (page 19). ~ext to thenumber for each problem circle your letter answer (for example if your answer to problem11 is C circle the letter C next to 11 on the answer sheet). Make sure that you give ananswer for each question and that you have circled the letter that you intended to!

There is a table for the area under the standard normal distribution at the end of theexam that you may need.

NAME (print) 1

1. X is a continuous random variable on the interval [0,1] whose density function isof the form kx2 for some constant k. What is Var(X)?A. 1/80B. 1/40C. 3/80D. 1/20E. 5/80F. 3/40G. fio

H. {io

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NAME (print) 2

2. A Geiger counter clicks, on the average, every 15 seconds. (The numher of clicksis a Poisson process.) Let X be the total number of clicks of three such geigercounters in a minute. How many clicks per minute is one standard deviation of X.A. 12B. 45C. J45D. 8E. JI2F. 6G. V6H. J8

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NAME (print) 3

3. Find

A. In(y) + xY In(x)B. -~ + xY In(x)y

C. -y\ +xY-1(yln(x)+1)D. x In(y) + xY-1ln(x)E. ~ + xY-1(yln(x) + 1) + 2YF. -~ + xy-1yG. 22+ xY In(x)yH. 0

fJ2

8x8y (~ + y2 + xY)

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NAME (print) 4

4. If the measurements of a and b to the nearest 1/10 of an inch are a = 5 inches andb = 6 inches then the maximum percentage error in calculating the area A = 1mb

of the ellipse ~~ + fz- = 1 is closest to:A. 307r%B. 1.17r%C. 187r%D. 117r%E. 37r%F. 61/9600%G. 10/3%H. 11/3%

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NAME (print) 5

5. The waiting time for an elevator is an exponentially distributed random variablewith mean 3 minutes. When your guest arrives at your floor you ask if he had towait longer than 6 minutes for the elevator. He says no. What is the probabilitythat he had to wait at least 3 minutes?

A. 1/2B. (e - l)/(e - 2)C. (e-1)/(e-2)D. (e-2)/(e-1)E. e-2F. e-1 - e-2G. (e-1 - e-2)/(e-2 - 1)H. (e-1 - e-2)/(1 - e-2)

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NAME (print) 6

6. Suppose that a random variable X is uniformly distributed on the interval [1,6].The expected value of 1/(X + 1) is:

A. 1/7B. 1/5C. In(7/2)/5D. In(7/2)/6E. In(7)/5F. 5/2G. 7/2H. l/ln(7/2)

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NAME (print) 7

7. Let w = In(l + X22)- arctan(x) and x = 3eUcos(v) + v. Find ~~ at 11= V = O.A. 147/110B. 196/110C. 0

D. 49/110E. 3/10F. 3

G. 4/10H. undefined

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NAME (print) 8

8. The tangent plane to the surface x2 - 2y2 + Z2 + yz = 2 at the point (2,1, -1)intersects the z-axis at the point:

A. (0,0, -4)B. (0,0,4)C. (0,0,-1)D. (0,0,3)E. (0,0, -3)F. (0,0,2)G. (0,0, -2)H. (0,0, V3)

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NAME (print)

9. Given that

A2 = (~1 ~) and A3 = (~ =~)

and compute the product P = ab. Then P =

A. 2B. -2C. 1D. -1E. 0F. 3G. -3H. 4

9

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NAME (print) 10

10. Find the shortest distance from the point (1,0, -2) to the plane x + 2y + z = 4.

A. J6B. 2J6C. 2D. 5J6/6E. 1F. J6/6G. J6/3H. 2)3

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NAME (print)

11.

11

A. O.B. 1.C.D.E.F.G.H.

e.

(e9 - 1)/2.eI6 - 1.(e16 - e)/2.e9 - 1.eI6 - e.

f4

13 ey2dydx =1 x-I

NAME (print) 12

12. Which of the following statements regarding the system of equations

{

x-y+z=13x - 3y + 3z = 32x + 4y + 6z = k

is true ?

A. The system has a unique solution for any value of k.B. The system has a unique solution only when k = 8.

C. The system has a unique solution only when k = O.

D. The system only has infinitely many solutions when k = 8.E. The system infinitely many solutions for every value of k.F. The system has a unique solution for k = 0 and infinitely many when k = 8.G. The system never has a solution.H. The system has infinitely many solutions when k > 0, a unique solution

when k = 0, and no solutions for k < O.

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NAME (print) 13

[

0 3 -2

]

13. A = 2 -1 13 2 -1

The sum of the entries in the third row of A-I equals:A. -2B. 0C. 2D. 4E. 8F. 10G. -4H. -8

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NAME (print) 14

14. A chest of drawers has three drawers: the top, the middle, and the bottom. Thetop drawer contains 3 black and 2 white pairs of socks, the middle drawer has 2black and 4 white pairs of socks, and the bottom drawer has 4 black and 2 whitepairs of socks. A drawer is selected at random (any drawer is equally likely to bechosen) and a pair of socks from the drawer is chosen at random. Given that thepair is white, what is the probability that it came from the middle drawer?A. 1/8B. 2/9C. 7/15D. 1/2E. 2/15F. 10/21G. 15/32H. 5/9

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NAME (print) 15

15. A bridge hand consists of 13 card from a standard 52-card deck. Find the proba-bility that a bridge hand contains all four aces?A. 113B .i.

C' H! 13!. 9! 52!D. 0

4!E. 52!

13!F. 1 - 52!G. ~H 48!

. 52!

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NAME (print) 16

16. Consider three events E, F and G. Assume that they have the following proba-bilities: Pr(E) = ~, Pr(F) = io' and Pr(G) = ~. Assumethat we also knowthefollowingconditionalprobabilitiesPr(EIF) = ~, Pr(EIG) = ~, Pr(FIG) = ~, andPr((E n G)IF) = ~. Then Pr(F U (E n G)) is (There may be more informationthan is needed to solve the problem):A. 1/4B. 1/10C. 16/25D. 3/5E. 2/9F. 7/10G. 3/4H. 19/20

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NAME (print) 17

17. The SAT score of students in a high school has a normal distribution with mean1000 and standard deviation 100. In order to be admitted to Prestigious University,the SAT score has to be at least 1200. If the high school has 1200 students, andthey all applied to PU, the number of students expected to be admitted lies in therange

A. 171- 200B. 91 - 101C. 80 - 91D. 61 - 80E. 31 - 60F. 151-171G. 102 - 150H. 0 - 30

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NAME (print) 18

18. An economy depends on two basic products: coal and gas. In order to produce 1unit of coal, one needs 0.4 units of coal and 0.5 units of gas. In order to produce1 unit of gas, one needs 0.2 units of coal and 0.6 units of gas. What Hhould theproduction be in order to satisfy a final demand of 2 million units of coal and 4million units of gas?

A. 7.6 million units of coal and 3.9 million units of gas.B. 15.2 million units of coal and 8.33 million units of gas.C. 11.43 million units of coal and 24.29 million units of gas.D. 13.9 million units of coal and 24.29 million units of gas.E. 17.1 million units of coal and 4.33 million units of gas.F. 13.2 million units of coal and 24.29 million units of gas.G. 3.3 million units of coal and 8.33 million units of gas.H. 11.3 million units of coal and 8.33 million units of gas.

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NAME (print) 19

ANSWER SHEET

Problem Answer

1) A B C D E F G H

2) A B C D E F G H

3) A B C D E F G H

4) A B C D E F G H

5) A B C D E F G H

6) A B C D E F G H

7) A B C D E F G H

8) A B C D E F G H

9) A B C D E F G H

10) A B C D E F G H

11) A B C D E F G H

12) A B C D E F G H

13) ABC D E F G H

14) A B C D E F G H

15) A B C D E F G H

16) ABC D E F G H

17) A B C D E F G H

18) A B C D E F G H

Nonnal Distribution Table http://www.math.upcnn.edu/-ccrokclnonnal.html

P(O<z< a)

r;-I0.00 1 0.01 1 0.02 1 0.03 1 0.04 I 0.05 1 0.06 1 0.07 1 0.08 1 0.09 I

ro:oIO.OOOO10.004010.008010.012010.016010.019910.023910.027910.0319[0.0359

ro:I10.0398 10.0438 10.0478 10.051710.055710.059610.063610. 067510.071410.0753

[f).2lo.0793 10.083210.0871 10.0910 10.094810.098710.102610.106410.110310.1141

ro:J10.1179 10.1217 10.1255 10.1293 10.1331 10.136810.140610.144310 .1480 [0.1517

roA10.1554 10.1591 10.162810.166410.1700 10.173610.177210.180810.1844[0.1879

[0:510.191510.1950 10.198510.201910.205410.208810.212310.215710.2190 [0.2224

~10.2257Io.2291Io.2324Io.2357Io.2389Io.242210.2454Io.248610.251710.2549

ro:7lo.2580 [0.261110.264210.267310.270410.273410.276410.2794 [(>.282310.2852"

f().810.2881 10.2910 10.293910.296710.299510.302310.305110.307810.3106 [0.3133

~10.3159Io.318610.3212Io.3238Io.3264Io.328910.331510.334010.3365[0.3389

fl.OI0.3413 10.3438/0.3461 10.348510.350810.3531 10.355410.35771 0.3599 [0.3621

fil10.3643 10.366510.368610.370810.372910.374910.3770 10.3790 10.3810 [0.3830

fi2lo.3849 10.386910.3888 10.390710.3925 10.394410.396210.39 8010.3997 [0.4015

rt310.4032 10.4049 10.4066 10.4082 10.409910.4115 10.4131 10.414710.416 2 [0.4177

ru10.4192 10.4207 10.4222 10.423610.4251 10.426510.427910.429210.43 0610.4319

fLS10.4332 10.4345 10.4357 10.4370 10.438210.439410.440610.441810.442910.4441

fL610.4452 10.4463 10.447410.448410.449510.450510.4515 10.452510.45 35 [0.4545

rL710.4554 10.4564 10.4573 10.458210.4591 10.459910.460810.461610.46 25 [0.4633

fLS10.4641 10.464910.465610.466410.467110.467810.468610.469310.46 99 [0.4706

fL910.4713 10.471910.472610.473210.473810.474410.4750 10.475610.4761 [0.4767

[2:010.477210.477810.478310.478810.479310.479810.480310.480810.481210.4817

fU10.4821 10.482610.4830 10.483410.483810.484210.484610.4850 10.4854 [0.4857

ri210.4861 10.486410.486810.4871 10.487510.487810.4881 10.488410.48 87 [0.4890

rD10.4893 10.489610.4898 10.4901 10.490410.490610.490910.491110.49 13 [0.4916

[2:410.491810.4920 10.4922/0.492510.492710.492910.493110.493210.493410.4936

[2:510.493810.4940 10.494110.494310.494510.494610.494810.494910.495110.4952

fi610.4953 10.495510.495610.4957/0.495910.4960 10.496110.496210.4963 [0.4964

ri710.4965 10.496610.4967 10.4968 10.496910.4970 10.4971 10.497210.4973 [0.4974

~10.497410.497510.497610.497710.497710.497810.497910.497910.4980 [0.4981

~10.498110.498210.498210.498310.498410.498410.498510.498510.4986 [0.4986

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