I Series SSO I Code N~. 65/1 ep)g ;:r. Roll No. ~-;:f. Candidates must write the Code on the title page of the answer-book. -cRTaw-TI ep)g 051 3m-~~cPl ~ ~-% ~ ~fffiY I • Please check that this question paper contains 12 printed pages. • Code number given on the right hand side of the question paper should be written on the title page of the answer-book by the candidate. • Please check that this question paper contains 29 questions. • Please write down the Serial Number of the question before attempting it. • 15 minutes time has been allotted to read this question paper. The question paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the student will read the question paper only and will not write any answer on the answer script during this period. • ~ ~ C/1{~ fcf; ~ ~-lBf # ~ ~ 12 ~ I • ~-lBf # ~ m2T ctr 3ffi fG:Q: ~ irn-~ cfit ~ ~-~ it; ~-~ lR ~ • ~ ~ C/1{ ~ fcf; ~ ~-lBf # 29 ~ ~ I • ~ ~ CfIT ~ k1"S-t1 ~ ~ it ~t ~ CfIT ~ ~ M • ~ ~-lBf cfit ~ it; ~ 15 f1:Rz Cf)f ~ ~ 1flfT ~ I ~ -lBf Cf)f fcrcwr ~ # 10.15 qit fcf;IN J1Rqrlf ~. I (ii) ff1 >IN q;r it 29 ~ ~ ~ rft;:r ~ it ~ ~ : 3{, 7i{ (f2lT "fT I "@V5 31 it 10 ~ ~. FiRit it ~ ~ 3fq; CfiT ~ I "(ffUg G[ it 12 >IN ~. fJRif iT ~ rm 3fq; CfiT ~ ( "(ffUg "fT it 7 >IN t FiRit it JTFitcn fJT 3fq; CfiT ~ I (iii) "(ffUg 31 it "fTsft mit ~ ~ ~ Tl,Cfi qrq:zr 31~ ~ ctt 3WH4C/?r1f 31¥ff( ~ \57T ~ t I (iv) q'" -uf >IN q;r it fc!q:;fq ~ ~. ( fiR W qr( 3fqif r:rrft 4 ~ it rr:qr E9? a:fqif r:rrft 2 ~ it 3TFfffi:q) fcrcfic;rr ~ ( #1 ~ m if it 3fJT1Ch1 f!!E tt fc!q:;fq CfiRT ~ I (v) ;}Wlq;c1c< it mctt 31J1fFrr ;rtt. ~ 65/1 2-3YJ = [ -51 23J. SECTION A ~31 Questions number 1 to 10 carry 1 mark each. JTR #?§ZlT 1 it 10 ffCfi ~ JTR 1 aicn CfiT ~ I 1. /Find the value of x, if l__?~ [32Xy+-Yx -Y3J = [ -51 23J. x Cf)f liR ~ ~ ?:1ft: [32Xy -+xY /2. /Let * be a binary operation on N given by a * b = RCF (a, b), a, b E N. ~ Write the value of 22 * 4. ! ) l1RT *, N en: ~ fu31TmU ~ ~ it a * b = RCF (a, b) &m ~~, ~ a, b E N ~ I 22 * 4 CfiT 11R ~ I \ ~ate: 65/1 1/12 Jo 3 p.T.a.5. ~rite the principal value of cos-1 (cos 7;). /cos-1 (cos 7;) <:fiTlj&i liB ~ I 6. /Write the value of the following determinant: ~ la-b b-c c-a b-c c-a c-a a-b a-b b-c Rq ~I\fU Ien <:fiTliB ft1furQ: : a-b b-c c-a b-c c-a a-b 2 2x Rq -B x <:fiTliB ~ ~: x 4 ::0 2 2x 8. /' Find the value of p if ,-/1\ 1\ 1\ 1\ 1\ ~ ~ (2i + 6j + 27 k) x (i + 3j + p k):: 0 . p<:fiTllR~~~ 1\ 1\ 1\ 1\ 1\ 1\ ~ (2 i + 6 j + 27 k) x (i + 3 j + p k) = 0 . ~, 9 •• ~ coo'lrideinatthee adxieresc. tion cosines of a line equally inclined to the three ~ ~ ~ ~ ~ ~ it (fRT rrj~~licj) ~ 'R ~ cfiTuT ~ m I 10. If P is a unit vector and (~ -p) . (~ + p) :: 80, then find I~ I. ~ Ii ~ lIDfCfi ~ ~ w:rr (~ -p) . (~ + p) :: 80 ~, m I}( I <:fiTliB ~~I 65/1 4SECTION B "{§US. Cif Questions number 11 to 22 carry 4 marks each. >IN til§lrT 11 "# 22 (fCJi ~ J:r:'R"~ 4 3iq; !I ~:gth x of a rectangle is decreasing at the rate of 5 em/minute ~ .... and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rate of change of (a) the perimeter, (b) the area of the rectangle. OR Find the intervals in which the function f given by f(x) = sin x + cos x, o:s; x :s; 2n, is strictly increasing or strictly decreasing. ~ :jWffi cn1 ~ x, 5 -wIT/fir;rG cn1 ~ -B ere Wt w ~ ~ y, 4 -wIT/fiRz cn1 G:\ -B ~ wr * I \ifiilx = 8 WTI ~ y = 6 W:ft *, ~ ~ ~ (31)~, (q) ~ ~ YKqJrj cn1 ~ ~ ~ I ~ ~ ~ ~ ~ f(x) = sin x + cos x, O:s; x :s; 2n mT ~ T.:fim f, f.1tR qm ~ f.1tR f-,1+lYlrj * I 12. If sin y = x sin (a + y), prove that dy dx OR _ sin2 (a+y) sIn a (cos x)Y = (sin y)X, find dy dx 7.1fc:: sin y = x sin (a + y) W, 'ill ft:r.& ~ fcn dy dx = sin2 (a + y) SIn a 65/1 7.1fc:: (cos x)Y= (sin y)X *, 'ill dy dx 5 p.T.a. 1----------_ n2' if n is even Find whether the function f is bijective. n+l -2 ' fen) == n2 ' &FJl@:~~~f:N~N~ \ ~ ~ % cp.:rr ~ f ~ ~ (bijective) ~ OR Evaluate: r//J x sin-1 x dx 1iH~~: J dx = J5 -4x -2x2 3l~ 1iH~~: J x sin-1 x dx 6 65/1:V/'sin -1 x , soh w that 15. If y = /1 _ x2 2 d2y dy (1 -x ) --3x --y = 0 dx2 dx ..,..,& sin -1 x '111." Y = -=== ~1-x2 2 d2y dy (1 -x ) --3x --y = 0 . dx2 dx L///16. On a multiple choice examination with three possible answers (out of which only one is correct) for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing ? ~ ~-fqctl01l~ ~ if -qfq ~ ~ ~ ~ it cfR ~~ ~ ~ (ftRif it ~ ~ ~ ~) I ~ CflTT J.l!f~ctlc11~ fen ~ ~ ~ ~ ~ 'iffi "lIT 31f~ ~it~~~? ('I-7.'.'>USin~ of determinants, prove the foUowing: 11+ p 1 + p + q 2 3 + 2p 1 + 3p + 2q I = 1 3 6 + 3p 1 + 6p + 3q r ". it ~ q)f m~,Rkf ~ ~ : 11+ p 1 + p + q 23 3 + 2p 6 + 3p 1 + 3p + 2q I = 1 1 + 6p + 3q 18. Solve the following differential equation : x ddyx = y _ x tan (yx)~. Rkf ~ ~41ctl~UI cit ~ ~ : x ddyx = y _ x tan ( yx ) 65/1 7 p.T.a.19. Solve the following differential equation : 2 dy cos x -+ y = tan x dx f.:r8 ~ o/;-il1lch-lUI ~ QC1~ : 2 dy cos x -dx + y = tan x ~Find the shortest distance between the following two lines : -7 !\ !\ !\ r = (1 + A) i + (2 -A) j + (A + 1) k; -7 !\!\!\ !\!\!\ r = (2 i -j -k) + !l (2 i + j + 2 k). -7 !\ !\ !\ r = (1 + A) i + (2 -A) j + (A + 1) k ; -7 !\!\!\ !\!\!\ r = (2 i -j -k) + !l (2 i + j + 2 k). ~:" Prove the following: -ll/~1 + sin x /"" cot ~=== ,~ OR -+ -~J11--ssiinn xx J = x2 , Solve for x : 2 tan-1 (cos x) = tan-1 (2 cosec x) f.:r8~~~:cot-1 (_~~1=l=++=ssi=ninx=x=--+~_.1~-=-l=s-=ins=xinJ=x= x2,. X E (0, n4) 3l$!fCfT x ~ fu"Q: QC1~ : 2 tan-1 (cos x) = tan-1 (2 cosec x) 65/1 8A (3, -1, 2), of the point 1\ 1\ 1\ 22. The scalar product of the vector i + j + k with the unit vector along 1\ 1\ 1\ 1\ 1\ 1\ the sum of vectors 2 i + 4 j -5 k and Ai + 2 j + 3 k is equal to one. Find the value of A. 1\ 1\ 1\ 1\ 1\ 1\ ~ 2i + 4j -5k ~ Ai + 2j + 3k ~ <.jljl\.h~ cn1 ~ if ~ ~ 1\ 1\ 1\ it ~ i + j + k q;r ~ ~UI"\.h~ 1 ~ I A q;r +rR ~ q,~ll.--t/~ SECTION C ~ ll' Questions number 23 to 29 carry six marks each. ,If-J1H if&rT 23 it 29 rfcfi JRitq; J1H it 6 ~ ~ I 2~ the equation of the plane determined by the points B (5, 2, 4) and C (-1, -1, 6). Also find the distance P (6, 5, 9) from the plane. ~an A (3, -1, 2), B (5, 2, 4) ~ C (-1, -1, 6) ID\TRmft; w:rm1 q;r ...:1 ..... ~ ~ J-~ P (6, 5, 9) ctr ~ w:rm1 it ~ ~ ~ ~ I ~~--a/rea of the region included between the parabola; = x and the line x + y = 2. 25~~aluate : i\ 65/1 9 p.T.a.~g matrices, solve the following system of equations: x+y+z=6 x + 2z = 7 3x + y + z = 12 OR Obtain thfIeonlveleoluteorhwssmfemmeinegagnttraixry 3 0-1 A = 12 3 0 0 4 1 ~ cnr m'8:, R8 ~~Iq,{UI~ it ~ ~ '" x+y+z=6 x + 2z = 7 3x + y + z = 12 3l~ ~ ~3TI ihmIDU R8 ~ cnr ~ ~ ~ 3 0 -1 A = 12o 34 o1 ~oloured balls are distributed in three bags as shown in the following table: . Colour of the ball Bag Black WRheitde 1 23 412 534 A bag is selected at random and then two balls are randomly drawn from the selected bag. They happen to be black and red. What is the probability that they came from bag I ? 6511 10/OR \/;/A manufacturer can sell x items at a ~ ~CfiTtrT 'Cfil"ffi ~m 1 23 412 534 ~ ~ <11~~<11 ~ '1flIT (f~ ~ it ?J 7R ~I~~~I ~ ~ ~ ~ ~ am m 'IT{ ~ I 541fl1ctldl q:rcfiR f.:rffi ~ fcfi "3it 3if~ ~m? ~OOcn~W1BIT~~w:FiIDU~~ I 29. If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is ~3 pnce of Rs. (5 -~1)00 each. The cost price of x items is Rs. (\x5 + 500). Find the number of items he should sell to earn maximum profit. 65/1 11 p.T.a.65/1 ~ ~ (~ + 500)~. ~ I ~ cn1 ~ ~ ~ ~ ~ dB 31f~ (1N ~~itft;m:~~ I 12