McL 151 MT 3 VB KEY - SCORE 50 NAME K4” STUDENT NUMBER...

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Unformatted text preview: SCORE: 50 NAME: K4” / STUDENT NUMBER: University of Victoria Department of Mathematics & Statistics Math 151 A03 & A04 Mid-Term Exam #3 Version B F riday, March 26‘”, 2010 PLEASE ANSWER EACH OF THE 17 QUESTIONS INCLUDED IN THIS BOOKLET GOOD LUCK!! TIME ALLOWED: 45 MINUTES For multiple choice questions where a numerical answer is required, if your exact answer is not listed amongst the choices offered then choose the value closest to your unrounded answer; ' Show ALL your work in the space provided. ° Do not allow yourself to get stuck on any one Question keep movingll SECTION ONE: MULTIPLE CHOICE Questions 1 — 11 (22 MARKS IN TOTAL) EACH QUESTION IS WORTH TWO MARKS. PLEASE RECORD YOUR ANSWER BY PLACING THE CORRECT BLOCK CAPITAL ALONGSIDE THE CORRESPONDING QUESTION NUMBER ON THE SEPARATE ANSWER SHEET ON THE FRONT PAGE. ANSWER CHOICES MUST BE CLEAR IN ORDER TO RECEIVE CREDIT. ANSWERS MUST BE TRAN SF ERRED TO THE SEPARATE ANSWER SHEET DURING THE EXAM — YOU WILL NOT RECEIVE EXTRA TIME AT THE END TO DO THIS. Show all your work in the Space provided. For verification purposes Show all calculations. Unverified answers may be disallowed. SECTION TWO: WRITTEN QUESTIONS Questions 12 - 17 (28 MARKS IN TOTAL) MULTIPLE CHOICE ANSWER SHEET VERSION B ** PLEASE DO NOT DETACH FROM THE UESTION PAPER NAME: STUDENT NUMBER: LEGAL SIGNATURE: MULTIPLE CHOICE ANSWER SHEET Indicate your answer by entering clearly the letter associated with the solution which best answers each individual question in the space to the right of the question number. BLOCK CAPITALS should be used. If your answer is not clear it will be marked wrong and if your answer is not supported by workings on your question paper you may receive no credit for your answer. PLEASE ENTER YOUR FINAL CHOICES INTO THE BLANKS BELOW DURING THE EXAM' YOU WILL NOT BE GIVEN EXTRA TIME TO DO SO AT THE END. TOTAL CORRECT: ____________________________________________________________________________________________________________________________ -— SECTION ONE: MULTIPLE CHOICE (VERSION B) EACH QUESTION IS WORTH TWO MARKS. IF YOUR EXACT ANSWER IS NOT LISTED THEN PICK THE ANSWER CLOSEST TO YOUR ANSWER. PLEASE INDICATE YOUR FINAL ANSWER BY PLACING THE CORRECT BLOCK CAPITAL IN THE BOX PROVIDED TO THE RIGHT OF EACH QUESTION NUMBER ON THE SEPARATE ANSWER SHEET ON THE FRONT PAGE. THE SEPARATE ANSWER SHEET SHOULD BE COMPLETED DURING THE EXAM; YOU WILL NOT BE GIVEN EXTRA TIME TO COMPLETE IT AT THE END! 1. GivenA= 0 0 0 0 1 5 ,find (125 15 3/2 —7 12 —1/2 5 1 3 —5 19 2 8 7 6 4 —3/2 0 .12 22 A) 0 B) 3/2 C) -12 D) 3 E) 2 F)7 G) None of the given Options. —1 0 3 2. Given A: 4 m2 9 ,find AT. 5 8 10 T I A —/ (ATM. (!‘~r n ( u- —1 0 5 A) AT: 4 —2 9 3 8 10 —1 4 5 B) AT: 0 —2 8 3 9 10 -1 0 3 C) AT: 4 M2 9 5 8 10 D) None of the above. —I 0 3 1 0 2 Given A: 4 —2 9 and B: 4 —3 12 , find A+B. 5 8 10 1 0 5 0 4 4 A) A+B= 4 —5 21 7 20 15 . B) A+B= 8 —5 21 _V 8 15 - ‘ C) 21 815 h: + t2! El @000 | Ln 0 0 5 D) A+B= 8 ~1 21 6 8 15 E) None of the above. 1 2 1 Given = “y ,find (x,y). 4 8 3x+y 8 A)(—1,3) B)(—1,2) C)(2,—1) D)(2,4) E) (1, 1) F) (4,2) G)(3,—1)H)Noneofthese. , 1 ;;,.. 1 ,{_ r 1 -. 5' f 1.2.1 f Luk. _/ ,7 : — '8 ,27 I 4" ' \ :3 i :. /, | l x " n W. "i: c: x , 1 A. . A clothing store sells men’s shirts for $25, silk ties for $8, and wool suits for $300. Last month, the store had sales consisting of 100 shirts, 200 ties, and 50 suits. Write a row vector R to represent the prices of each item and a column vector C to represent the corresponding number of items sold. What is the total revenue? [$ signs are omitted in the matrices] 25 A) C = [100 200 50], R = 8 and the total revenue is RC = $19,100 300 100 B) R = [25 8 300], C = 200 and the total revenue is RC = $19,100 50 100 C) R = [25 8 300], C = 200 and the total revenue is RC 2 $10,000 50 D) None of the above. [5' . r ’ if :75 6 _ 1 3 6. Given A I [—2 —6 1 , find its inverse if it exists. _ 1 —1 —2 —6 —3 -1 —6 —3 —l —2 A) Inverse does not exrst. B) —E[ 3 6] C) [ ] D)—[ ] E) [ ] 1N" '\.:~’:-‘L ti'R‘zfif 75% 7a A”? r Suppose a system of equations contains 3 equations With 4 variables. Which of the following statements is/are guaranteed to be true? i) The system will either have one solution, or it will have infinitely many solutions. ii) If the system is written as the matrix equation AX=B, the matrix A is guaranteed to have an inverse. Ni) x" _ _ @l iii) It IS impossible for the system to have exactly one solution. 7/3in - \" A) None B) Only (iii) C) Only (ii) D) Only (i) E) (i) and (ii) F) (i) and (iii) G) (ii) and (iii) H) All . . , _ 2x + 5 y = 10 Determine the value of k for Wthh th1s system has no solut10n. ky 1 x _ = _ A) 0 r g. . g. ‘ c) -1 i ' .a D) .4 a w L E) ".4 {I:7M_I.: ‘, O f+ 1H: F) -2.5 ‘—~ W G) i c;- > W ‘V um"!- H) Impossible for the system to have no solution. 5“ y i . ~, , s 2 5 6 0 2 2 15 —1 lfD= , E: and DE: ,findED. 3 7 -—2 3 —1 4 21 —1 ADE—ED—le '1 B ED*12 6 CED‘ dfd ) ~ —— 4 21 _1 ) _6 14 ) 1s un e me . BED—4152 EN fthb ) — _1 21 4 ) oneo ea ove. ix} 11. a—Zb—c=53 2a—3b+c=8 4a~5b+5€=23 10. The system of linear equations written in Matrix Form is: 1—2 —1 1—2 —153 a 1—2 —1 A)2—31 B)2—3 1 8 C)b = 2—3 1 [53823] 4 —5 5 4 —5 5 23 c 4 45 5 1 —2 —1 a 1 42 —1 53 D)[a b c]: 2 —3 l [53 8 23] E) b = 2 —3 1 s 4 —5 5 c 4 —5 5 23 1 —2 —1 a 1 _2 —1 a 53 F)2—31b=[53823] G)2—31b=8 r4 —5 57 VC 4 ~5 5 C) 23 +115» Err-f“! T fi {.471 '5 7"- H)Noneofthese. 1 A u 3 ;l i 3' B: 3 if: 5 (J 1 Hank is participating in a bowling contest. He takes a little While to warm up so his initial bali generally ends up in the gutter on 8 out of 10 occasions. If his first ball (i.e. the initial throw) did not end up in the gutter there is a 60% chance that on his next turn it will end up in the gutter. If his first ball did end up in the gutter there is a 90% chance that his next ball will strike at least one pin. This pattern continues indefinitely. What is the probability that his third ball will end up striking at least one pin? ' A) 0.00 '4” B) 0.10 C) 0.20 D) 0.30 ,.;.-. a. E) 0.40 34".: c - 5,? i . . i t F) 0.50 '* G) 0.60 :7 , : H) 0.70 I) 0.80 J) 0.90 K) 1.00 L) None of the above. X. . z SECTION TWO: WRITTEN QUESTIONS ALL WORKINGS MUST BE SHOWN IN ORDER TO RECEIVE ANY CREDIT ALL ANSWERS SHOULD BE GIVEN CORRECT TO TWO DECIMAL PLACES UNLESS STATED OTHERWISE. [4t] 12. Solve the given system of linear equations by using the inverse of the coefficient matrix, 3x — 2y = 6 —x + y = 4 I Gauss-Jordan elimination to arrive at your inverse. ' ALL workings must be clearly shown in order to receive any credit. You must use A I A I E For" X mi» WE r“ I I if 7 mt :. n ‘f .— [.T‘E i!” ;‘ . x t f 1 r r I m i: i 3%; ‘ l u“ I f Aflm u I J l 7 | h {/2 [r J “Aw __fly 7 ‘ I 2!" I" j " !' .,- ,0 X A; E “Mug.” 1 1,3 : K I «J r r 2‘ A I t z J r W X ATE ("V 11;?” ” (1‘17 :7 l: ‘ {* 7 Li ' ' 1" g L d W H‘ - i i ,4 i EI / ‘5 f? C:1 i’kc’t "‘l 4 a J 5 /r f xJI-h- -——JIL»J [5] 13. Consider a two—state Markov Chain with transition matrix T = Ulla» U'IIN The states are in the order 1, 2. 17 . . . . , . it t 4/11: 54‘. a"? [U a) What kind of Markov Chain IS thls‘? (“1‘1 i” f“ i “ “I t ’V ' [1 Mark] b) In the long run, what is the probability that the process is in state 2? [4 Marks] I 4 . . ‘ wk ’1‘ WC - 36 F 4 :;l: 3 T i glint}. K _ t)!‘ f , I. M ’ ‘ in. H2}? [ FD: '7 i. T- - ‘3' i i 1:}: :1“. i - I i \ f“ "I at ' 'V ,1 L 3.1») x j “ j ’ T 3/ t) w: ii;g.-,fiif 9/735; ‘7 7’ 1“ :> Jr; 3., x b / ..\ n E ,. a?“ H i 2 y C «if a» n . t. l T r ‘1 “7‘43 t r i\ i r k "s i . e" 2‘ '_ r L N 1 O 0 0 _ _ _ _ .2 .3 .2 .3 [6] 14. A 4-state Markov Cham has transmon matrix T : 0 a 0 0 . .7 .1 0 .2 The current states as presented are in the order 1, 3, 2, 4. =12 5/4 15/32 0.8 ~03 0 5/4 0 .8 ] We also know that [ a) What kind of Markov Chain is this? 4 “"‘e-ri‘s“ '~ ' i1 Mark] b) If the process begins in state 4, what is the probability that the process will eventually enter state 1? [5 MarksLFinal answer must be given as a fraction in simplest form, 1‘ . r __‘ _. 3' 5 C; m ~_ , , l t c 4 L. x a _ r" . - f _ \g-ur r ("Eff 2' r‘E‘ E e fl '7 S k ‘- ‘ H .« ,i : r“ 3 r. 5 ‘\ “I ‘T ‘ r T } 1 E: l t 7 J 5: " d _ l , ‘ J ‘i [ H\ i" L \ w <4 a __ ' I. ERIK A a 1' O \ V j I Sill-j“ j "‘k. l 1' (_ A I 2 u‘ u- , r. g < I iii; E A J i [LL/(fl: ~ za r , . v I u * If ‘\ ' C 9' i' 7 f x'; 7 r I) 1‘ I f) J/if [ i J _ {/3 r i“ E f / i 3‘ r 5 W( m“ I x: / iLtu prslaictsiéii'af/ \ T ‘ L‘Q‘ “ 3 v 4 - ,, ' e [2] 15. Let A, B, C, D, E all be square matrices of size two, each with an inverse. Solve the following matrix equation for D: — : O é D C r 413% if; " " - {i fix}; (é ‘ i 3:: 13-5 1 (ii. (I V { m__ ; ___ > 5] 16' Formmate’ but do mt solve! a linear programming problem for the following scenario. A boat manufacturer makes fishing boats that are sold for a profit of $400 each, and canoes that are sold for a profit of $500 each. Each canoe requires 100 assembly hours and 25 finishing hours and each fishing boat requires 75 assembly hours and 50 finishing hours. The manufacturer has a total of 8000 assembly hours and 3000 finishing hours available. How many fishing boats and how many canoes should be made to maximize the manufacturer’s profit? (Let x = the number of fishing boats and y = the number of canoes.) .| g) H 3 r: i ‘ r r: L—\. ['2 it 5: "E U u ‘l '75 3. $003 7 ,- 575; a Fri-:7 % 23: ~ »-~=* 35' 3,, .a 7 [c2] 17. Solve the following linear programming problem graphically using the method of corners. hfimmfiemmbhmmmeC=x+3y&anm 2x + y a 8 x+y26 x20, 3120 "5v “.37: ‘ff« ’ Q— 7 x k - 2 .3: F .n f" , Gt r 'rk k- z‘ 1 It] — g 77 , I r f S- l .4. ,J, U 6: A (of 8) a 2A}. M1KlM$ZELsz (Uh an (w. (-5- o a 1k " “ "'\ i ML «In ' "4 w v: f b l J- : 7* A 3 - , ‘ I ' (J E: a I l \ ’ LLWKL"! z'xt-ai .4 _ batik If fir: Q {LQJ .i’\'\r LFJ / s I - h \ i E . E z' j , -' A ‘w "\Nerur’mv : ¢ i V ‘ (w. H , ML!) r . 10 ...
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McL 151 MT 3 VB KEY - SCORE 50 NAME K4” STUDENT NUMBER...

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