Solution to Final (0607_1) of MA1505

Solution to Final (0607_1) of MA1505 - NATIONAL UNIVERSITY...

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Unformatted text preview: NATIONAL UNIVERSITY OF SINGAPORE FACULTY OF SCIENCE SEMESTER 1 EXAMINATION 2006—2007 MA1505 MATHEMATICS I November 2006 Time allowed. 2 hours Matriculation Number: ULHHTT INSTRUCTIONS TO CANDIDATES 1. 531 Write down your matriculation number neatly in the space provided above. This booklet (and only this booklet) will be collected at the end of the examination. Do not insert any loose pages in the booklet. . This examination paper consists of EIGHT (8) questions and comprises THIRTY THREE (33) printed pages. . Answer ALL questions. For each question? write your answer in the box and your working in the space provided inside the booklet following that question. . The marks for each question are indicated at the beginning of the question. Candidates may use calculators. However, they should lay out systematically the various steps in the calculations. ‘ For official use only. Do not write below this line. | Question Marks MA 1505 Examination Question 1 (a) [5 marks] Given that y = t + t2 + t5 and {I} = t3 w t2? find the value of 56% at the point corresponding to t = 1. (Show your working below and on the next page.) am _%: I+2X+5fi ‘— X’? WM Bil—7.x atxrl/ 09.. (+145: 09 ref—HF.- Obfi 3—2 :7: MA1505 Examination Question 1 (b) [5 marks] Let A be the point (0, a) and B be the point (0, a + b)? Where a and b are two positive constants. Let P denote a variable point (:15, 0)? Where as > 0. Find the value of x (in terms of a and b) that gives the largest angle [APB Answer 1 (b) W (Show your working below and on the nest page.) a! a+b _ "in 9: T K d9” , n+5) ! if 502': W x1 + (+33 X H [T x?" #02165) + Q. S ————--.—-"‘—" 1 {1+ CW)?" K 1% a. firm) } r ..(Q+b)0(1:f41) + M i ff 411+ mull 0‘ 11‘ 4'2) [){Wfixlf 014+!» “kl : /___’_'_______,_ ‘lfiéCMLJIlCXLMv alt? + - W W 0- M f' I - I W MW- 4 MA 1505 Examination Question 2 (a) [5 marks] The region R in the first quadrant of the my—plane is bounded by the curve y = :53, the x-axis and the tangent to y 2 $3 at the point (1:1). Find the area of R. Answer 2(a) I __..-—-—- {2 (Show your working below and on the next page.) 2 firm-9m: '(Zfluyfiflj o 5 I '3- 2. 3 % 1H”? "4‘" L “'6 3 4r“ I?- '1 MA1505 Examination Question 2 (b) [5 marks] A thin rod of 2 unit length is placed on the m—axis non: 3: = 0 to a: = 2. Its density varies across the length given by the function 6($)_ 6+3: 0§x<1 — 9—2313332. Find the m—coerdinate 0f the center of gravity of the rod. Answer ?,3 200) 3L5, (Show your working below and on the'nemt page.) 2 l 2 3?: U f." l: 500"“ fight/K) (ix + {liq—flu) dx { z 2(3 EFL—fix? -— @Xl'gxloftx 3 ! ,Wf/f [gm—$1]; + [fi’X-‘ij: /é_jf 3. : 3*§+"’°””§ 2+3 ¥3 f g+i+1¢P—<~‘i’+/ j: Z ...___.— MA1505 Examination Question 3 (a) [5 marks] Find the radius of eovergence of the power series 1"}. Z (—1) (5x + 2)”. n + 1 7120 Answer 3(3) 1 5 {Show you?" working below and on the next page.) M! H G") fl. (5 K102)“ k.“ : flfffiz} ———5{5X+ZI limbo)” m1 nfl o}. J— |§x+214|2=> (X‘C5)l<5 10 E % % MAlSO-fi Examination g Question. 3 (b) [5 marks} I Letf(x}=i.r—§ forallx-E(O,fi}.Let fl {M‘s Aaswer 3F“) 3-— f’irf— 2) mafif’fl | 1 | i . 5 H h % a‘ «=— 3- sziiifxm/zwffi i. £793.... 7 i .5. E 9 Ti- : 'r _."_ z w __ V é. I? {6’ 2)fi‘MHAP{X} i H E 0 2 E a Eff; '5? E r F 3%?2' (“aflmmflwfri (gamma ; D F E z E 17/?— 1E": J? in” i “ 2 "In -- — 5wde E “fifgx Z/Q’ML Lay:ng «pg-"3)5wflfij? 1*? r? J E z '2’ i g ' - .r 3 E i f 2’ I .i r _I . _ i i 2 EH!" fig” .2. flfl : :vfl‘ “ES-fl P4 dig/{Infl— il fif‘lli C )J V] ; 12 MA 1505 Examination Question 4 (a) [5 marks] Find the distance from the point (2, —1, 4) t0 the line r(t) =i+2j+7k+t(—3i+j —3k). Answer 4(a) rig-i- «J T? (Show your working below and on the next page.) LIL/t Z): CZ; "l/S‘)‘(’/2/ ?): (I; —3/ '3) .9 1-3 7’ "° —— '9 ’3 M: ‘3“‘0L'3£ :4170311‘3'31'1) ‘f -v a .1 i #1 , , 4 —.=’ '2’ ’< : -'— :_- 2A ’2 +CP£ & UL I .53 -3 mo + d .35 { “'3 14 MA1505 Examination Question 4 (b) [5 marks] Let f (x, y) be a differentiable function of two variables such that f(2, 1) = 1506 and g (2,1) = 4. It was found that if the point Q moved from [2, 1) a distance 0.1 unit towards (3, 0), the value of f became 1505. Estimate the value of g; (2, 1). Answer 40”) MW (Show you?“ working below and on the next page.) M %(21l)zq- (3/0)—(2,!) _-__________________,.__. izmeflrfim 0") “t 0’0) Haw-WI _.L. -I) I: «E (I) J. — — 4% v 9190,”:(4‘Xsltal‘ml‘ «E - HOS—(5‘06 x ESE-(0.1): 4.19; 1 fed? ~. —(0JE:¢ *Q 16 MA1505 Examination Question 5 (a) [5 marks] Find and classify all the critical points of f (3-3;) = 4333; — 2332 — 3J4 — 81. ("l/'OMUID (0,0) arm FM (Show your working below and on the next page.) £50 => 47 wxzo :> X=fl - 0 3 . 3 43,20 :7 m—trj :0 z) X: j --~@ Answer 5(a) l8 MA1505 Examination Question 5 (b) [5 marks] Let k be a positive constant. Evaluate ff mgexydwdy D where D is the plane region given by 1 D: 0£x32kand0£ygéfi (Show your working below and on the next page.) H flexjddj : fmgfi KieX’djfofx 3? 0 o 20 MA1505 Examination Question 6 (a) [5 marks] fl/gsmwy] gang-w) Evaluate Answer 6(a) (Show your “working below and on the next page.) .I! . 3+ , Jgggmfl—i—‘W , “if; [ 1?de 22 MA1505 Examination Question 6 (b) [5 marks] /f/D \rr|d$dydz where D is the spherical ball of radius 2 centered at the origin . Answer 6(b) 6? TT {Show your working below and on the next page.) Me Me SM X: mquccoQ) 7:. Yfmx¢fl-8/ ;: mfi grim 09/5 9. oscfisr 059$ 27:", / l we ddjd; : flux? dVohf 6((9_ Evaluate 24 MA1505 Examination Question 7 (a) [5 marks] A force given by the vector field F = (y + z)i + (:17 + 2yz)j + (I? + y2)k moves a particle from point PU), U? 0) to point Q (1, 2, 3). Find the work done by F. W (Show you?“ working below and on the next page.) . 31 +2 :2..(+ 2 ’wafg):§.£(x+273)l 5E§4flfi§flwfll 330 7}) W x 7) ‘F 217+} 3) ,F:)(7+>(}+?C7/}) {+27};ij :> 33:17; => 2?: fwm 73: Xj 443 +71} “10(3) 79}: X+7L:_.—) {who :5) flaw. 13:“‘7fl‘3W? é 0» WM~ . Um : 19mm» 73mm) :fi ‘ #5? YM% F5 Jaw 47 me) :(b 22b 350/ 0513] f .7 .: QM : j! FWm). vfmot O : (3612+r03t)05(' :13; H26 MA1505 Examination Question 7 (b) [5 marks] Evaluate the line integral [C(lnx/1+x2—y3)da:+ (51:3"f—H1 —sin3’y) dy where O is the boundary with positive orientation of the region between the circles 392 + y2 : 1 and 3::2 + y2 = 4. Answer 700) £1 2 (Show you?" working below and on the next page.) Lei inMmszajlzl fo‘f 3? :BmeJ Weir/0&9 0! ’2. _,_,._,‘f ,gIUY]! " 4 ‘ 1 28 MA 1 505 Examination Question 8 (a) [5 marks] Evaluate f f3 F o d8, where F = y2i + mgj + zk and S is the portion of the plane a: + y + z — 1 = 0 in the first octant. The orientation of S is given by the upward normal vector. Answer 8(a) I (Show your working below and on the next page.) x+Lj+S~l=o =3 3 : {ex-j -‘-&7WWU654 ?(‘4mf)= 01:? WV} + (IF-“"‘th‘ A -'-> —=> ‘L Yu:i*£ M $y=37—23 5 R 2]“va u1+(f~“—V);6{u 6H” K Mr I . 1.]. [WEuZ—Hva-Vflmdv 0 0 q:{"1/- ’1. zf' [wufiafl Hwéfl —1/'0t]“__0 41’— 0 1 :J’ffzvl_v3r'U’+-3L(I-V) {ff-"W “£04” I o \. “,1. — 1 i («-V) : [twirvfiéfi—tft‘” W w‘ o 30 :P' ’L‘ 3 MA 1 505 Examinatimi Question 8 (b) [5 marks] Using the method of separation of variables, solve the partial difi'erential equation mix —— guy = 0, Wherem > 0 and y > 0. (Show you?" working below and on the next page.) Let Vt: X Y / 2 'x x’y—— WO/ : 0 32 ...
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Solution to Final (0607_1) of MA1505 - NATIONAL UNIVERSITY...

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