!:1:No. \ I I 5 I 5 I I g I 4 I

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1 \ Series SSO \!:1:No. \ I I 5 I 5 I I g I 4 I -:r. Code No. ISET-21 65/2/RU w cm qi) - - "@"&I Candidates must write the Code on the title page of the answer-book. fc!:i -l:b: -q 12 I Wo{-l:B: -ij m2j «t cfi1s q;) ID;l dtr- - fu"&1 fc!:i -l:b: -ij 26 I cnt'3'ti R-f@-11. cnlsn"mn I -l:b: q;) 15 cnt TT<TT t I -l:b: cnt fc«rur -q I 'B ID;T -l:b: q;) <) "3tR- 'CR "3tR 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 26 questions. Please write down the Serial Number of the question before attempting it. 15 minute time has been allotted to read this question paper. The question paper will be distributed at a.m. From a.m. to a.m., the students will read the question paper only and will not write any answer on the answer-book during this period.,fc),a MATHEMATICS a';me.allowed : 3 hours 1 3ffi: 100 Maximum Marks : 100 P.T.O.

2 Hllllrel f.t#1r: (i) 'Rfft'J!R31f.rcrrf I (ii) qi?: ff #Ii 'JIR-fl;f q 26 JlN [ I (iii) &U5 3l "JlR" 1-6 wii 3ffrr flg-yffl: qtc? JlN f 3/!( JlFqqi JIN "cfi #r'!. 1 3Tcfi f.'mfftr I (iv) &U5 w wii -Yffl: I JTcFiTr 'JIR f 3/k- JlFqqi JIN #fl! 4 3Tcfi f.'mfftr f I (v) &Us 'fr w wii -Yffl: II "cfi JrFI f 3/k- JlN "cfi #fl! 6 3Tcfi RWfftcr f I (vi) Yff< fmrt JlWJl m # JrFI c/it 3Tcf.'Pl I General Instructions : (i) (ii) (iii) (iv) All questions are compulsory. Please check that this question paper contains 26 questions: Questions 1-6 in Section A are very short-answer type questions carrying 1 mark each. Questions 7-19 in Section Bare long-answer I type questions carrying 4 marks each. (v) Questions in Section Care long-answer II type questions carrying 6 marks each. (ui) Please write down the serial number of the questwn before attempting it.

$I\ I\ I\ If a, b and c are mutually perpendicular unit vectors, then find the I\ I\ I\ value of 12 a$ $+ b + c I. 2 I\ I\ I\ Write a unit vector perpendicular to both the vectors a = i + j + k db=i+j.$

3 3' SECTION A Question numbers 1 to 6 carry 1 mark each. I\ I\ I\ If a, b and c are mutually perpendicular unit vectors, then find the I\ I\ I\ value of 12 a + b + c I. 2 I\ I\ I\ Write a unit vector perpendicular to both the vectors a = i + j + k db=i+j The equations of a line are 5x - 3 = 15y + 7 = 3 - loz. Write the direction cosines of the line. 4. x+y y+z z+x Write the value of == z X y

4 Write the sum of the order and degree of the following differential equation : ~ ~ wf)cfi{oi cf>t li~lcfi~h ~ kif Q, :./ (1 + y 2 ) + (2xy - cot y) dy = 0 dx Write the integrating factor of the following differential equation : (l +y 2 ) + (2xy - coty) dy = 0 dx ~<if SECTIONB Question numbers 7 to 19 carry 4 marks each am:~ fcfi A. (adj A) = IAII 3-1 Find the adjoint of the matrix A = : ( that A. (adj A)= I Al ] - : and hence show

5 @I'-'(~~ x eb ~ ~ ~ f(x) = lx-11 + Ix+ 11, ~all x =-1 <lll'r x = 1 ~ ~qci,#,{lq ~ t I I Show that the function f(x) = Ix Ix + 11, for all x E B, is not differentiable at the points x = - 1 and x = 1. ~ y=emsin-lx t m~'rf; {1-x2) d2y - xdy - m2y = 0.. ' dx2 dx d 2 y dy 2 If y = em sm x then show that (1 - x ) - - x - - m Y = 0. ' dx 2 dx ~ f(x) = Jx 2 + 1; g(x) = X + l ~ h(x) = 2x - 3 t ffi f' [h'{g'(x)}] ~ x cfiln!v_ I If f (x) = Jx 2 + 1; g(x) = x + 1 and h (x) = 2x - 3, then find f'[h'{g'(x)}]. x / l!r ~,j;)r,jq_ : J (3-2x) J2 + x - x 2 dx J x 2 + X + 1 dx (x 2 + 1) (x + 2) Evaluate: J (3-2x) J2 + x - x 2 dx OR Evaluate: J x 2 + X + 1 dx (x 2 + l)(x + 2)

6 ~ ~ 'ffti:1,~4 3q~$t1 ~ ~ ~m cf;) 511R11ffia ~ ~ ~ ~ ~ ~ (i)tr-tr~ (ii)'4mwu, om (iii) ~ cfil~~' m"cf{>ffcr~cxflf ~-srcmt: (i) ~ 50 (ii) ~ 20 (iii) ~ 40 (i) (ii) (iii) X y z ~~ffl~, ~~ ~-~TJTcIT-q~TRTT~Wo~ I ~~~~ ~ ~ -q~~cf@t~~~ I To promote the making of toilets for women, an organisation tr ied to generate awareness through (i) house calls (ii) letters, and (iii) announcements. The cost for each mode per attempt is given below : (i) ~ 50 (ii) ~ 20 (iii) ~ 40 The number of attempts made in three villages X, Y, and Z are given below: X y z (i) (ii) Find the total cost incurred by the organisation for the three villages separately, using matrices. (iii) Write one value generated by the organisation in the society.

7 Solve for x: cot-1(:xy + 1) + cot-i(yz + 1) + cot- 1(zx + 1) = 0 x - y y-z z - x ( 0 < xy, yx, zx < 1) Prove the following : OR eot- 1 (:xy + 1 ] + eot- 1 (yz + 1 ] + eot- 1 (zx + 1 ] = O x - y y-z z - x ( 0 < xy, yx, zx < 1) a 2 + ab ab be b2 b 2 + be ac + e 2 ae Using properties of determinants, prove the following: a2 a 2 +ab ab be b2 b 2 + be ac + e 2 ac

$/ -+ -+,.-,,,r.,r::i' ~ ~ ma cbl~v. ~ ~ ( t - b ) ~ ( t ~ b ) $TT ~ ~Mic«(~ I\ I\ -+ " " I\ -+ I\ I\ -+ ~ 4-5k. a=i+2j+k, b=2i+jomc =3 1 - J -+,.,. I\ -+ I\ I\ -+ If a = i + 2 j + k, b = 2 i + j and c = 3.$

8 / -+ -+,.-,,,r.,r::i' ~ ~ ma cbl~v. ~ ~ ( t - b ) ~ ( t ~ b ) $TT ~ ~Mic«(~ I\ I\ -+ " " I\ -+ I\ I\ -+ ~ 4-5k. a=i+2j+k, b=2i+jomc =3 1 - J -+,.,. I\ -+ I\ I\ -+ If a = i + 2 j + k, b = 2 i + j and c = 3. _ 4 j _ 5 k, then n a (-+ b ) and ( c - b ). unit vector perpendicular to both of the vectors a - I\ I\ I\ fid f=::.. ~ ~ --4 ~ -m 00 cf;t BJ.ftcf>{UI ~,.,.~ (1, 2, - 4) ~ QI~\ \!11"1 /\ /\ I\ I\ I\ 7k) <'f2tt ~'.~ 003TT i = (B i -19j + 10k) + A(3 i - l 6 J + -+ I\ "" "" " ~ <'f~i r =(15i +29j +5k) +µ (3 i +8j -5k )G1.,I ~ (1M!cf~~I ~ircn ~3TT (- 1, 2, 0) cf'm (2, 2, - 1) ~ ~ q@ ~ ~ cf;t BJ.ftcf>{OI ~ ~ ~ OO x - 1 = 2y + 1 = z + 1 ~ ~ i I Find the equation of a line passing through the point (1, 2, - 4) and -+ I\ I\ I\ I\.I\ I\ perpendicular to two lines r = (8 i - 19 j + 10 k ) + A(3_L - 16 j + 7 k) -+ I\ I\ /\ /\ Ji. /\ - and r = (15 i + 29 j + 5 k) + µ (3 i + 8 j - 5 k ). OR Find the equation of the plane passing through the poin ts (-1, 2, O), (2, 2, -1) and parallel to the line x - l = 2 Y + 1 = z + 1 / ~ om ~ 52 "CJm cf;r ~ ~~ ~ -il ~ 3 ~ 3t1Ufn: ~~ ~ m~ ~ ~ ~ I ~ ~ qff1 cfit cflt SIIWchctl ~ ~ ~ I ~: ~ cf)l ltt~ ~~ I ~ WM ~ 6 9itaJUT -~ ~ I m;n X ~ ~ ~ t ~ ~'Q 9P(X = 4) = P(X = 2) cfit ~ cf>«ft i I ~lf><."iitt ctt )llf¾<:flitt ~ ~ I...)-'hree cards are drawn successively '?.ti.b repl. acement from a well shuffled pack of 52 cards. Fmd the probability dist 'b. n ution of the nu b of spades. Hence find the mean of the distribution. m er OR. For 6 trials of an experiment, let X be a binomial. the relation 9P(X = 4) = P(X = 2). Find the probabil:anate Which satisfies ity of success. IIIIIRU 8

9 n/4 J 00s 3 x ~sin 2 0 Find: Tt/4 J cos 3 x~sin2 0,J.B. ~ ~ : Find: f f logx dx (x + 1) 2 logx dx (x + 1) 2 Question numbers 20 to 26 carry 6 marks each. ~~ lcf>(?t-l ~ -a fir.& ~ fcf> "qsf; y 2 = 4x ~ x 2 = 4y, OOJTI x = 0, x = 4, y = 4 ~ y = o-a M qtf ~~en) <'fi;r ~ mril-ij ~ cf>b ~ 1 Using integration, prove that the curves y 2 = 4x and x2 = 4y divide the area of the square bounded by x = 0, i = 4, y = 4, and y = O into three equal parts. 65/2/RU 9 P.T.O.

10 y2 2 xy - x ~I ~ ~ ffl:ftcfi{oi (tan- 1 y - x) dy = (1 + y 2 ) dx cnt ~ ~ m<f ~' ~ ~ ~~y=ot<flx=lt I 2 '-'Show that the differential equation dy = --=Y'---- is homogeneous and dx xy - x 2 also solve it. OR Find the particular solution of the differential equation (tan- 1 y- x) dy = (1 + y 2 ) dx, given that x = 1 when y = 0../' f.r'5, P(3, 4, 4),I;\ ou f.r'5, ~ <;!\ '1lo ~ ' afol f.r.s.311 A(3, - 4, - 5) ail{ B(2, -3, 1) ~ m ~ ~ cffffi ~ ~ 2x + y + z = 7 cflt Sl@'i;{ G cfl«fi ~ I Find the distance of the point P(3, 4, 4) from the point, where the line joining the points A(3, - 4, - 5) and B(2, - 3, 1) intersects the plane 2x + y + z = 7../ 11:x) = 5x 2 + 6x - 9 i;m 1ISTI '!ier f : R+ -> [- 9, ~I 'R f<l<ir ~ I ~ ~ flf; '!ier f, r-'(y) = (.J54 + :y -3 J ~ m>i "'.j{j!,qaft ~ I X * y = X + y + xy, 'r;/ x, y E X. ffl ~ ~ ~ ~ (*) ~ sb.lf4f.lil4, o?.tt ~1~=q4 t I ~ ~ c1)l dn4'4cifi ~ 1ft' ~ ~ <lwt X ~ ~ ~ c1)t 51@~1-1 1ft' ~ ~ I 10

11 ~erf:r~ [- 9 ].. ~c "" ' 00 given by f(x) = 5x 2 + 6x - 9. Prove that f is invertible with r-'(y) = (./54 + 5y - 3} 5 OR Ab" mary operation* is defined on the set x = R- { -1} by x * Y = x + Y + xy, V x, y E X. Check whether * is commutative and associative. Find its identity element and also find the inverse of each element ofx. P cf>t cfq ~ ~ ~ ~ ~ q5fi x 2 = 9p (9 - y) (f'lff x 2 = p (y + 1) ~-~ cn1 Bl.Jcf>lOI th cfitw ~ I Find the value of p for when the curves x 2 = 9p (9 - y) and x 2 = p (y + 1) cut each other at right angles. / 11!1' ~ ohf.l ii, q,1\'511~ ii ~ (>ia) A, B 31\( C FJ ~ ii, l!oro: 30%, 50% 3fu: 20% ~ ~ ~ I ~ ~ ~ ~ cnt sfi1ffi: 3, 4 3fu: 1 ~ 'qttt ~ (~) Wffi ~ I ~ ~ ~ ~ -ij ~ ~ ~ lll~~lll f-1<hlc'il ~ ~ 3fu: ~ ~ lti<tt ~ ~ I~ Sllf¾<hal ~ ~ ~ ~ ~ lffir B ~ ~ ~RT~Pl<Jl ~ I In a factory which manufactures bolts, machines A, B and C manufacture respectively 30%, 50% and 20% of the bolts. Of their outputs 3, 4 and 1. percent respectively are defective bolts. A bolts is drawn at random from the product and is found to ~e.. defecti~e. Find the pi:obability that this is --. not manufactured by machine B. 65/2/RU 11 P.T.O.

Operating C 1 C 1 C 2 and C 2 C 2 C 3, we get = 0, as R 1 and R 3 are identical. Ans: 0

Operating C 1 C 1 C 2 and C 2 C 2 C 3, we get = 0, as R 1 and R 3 are identical. Ans: 0 Q. Write the value of MATHEMATICS y y z z z y y y z z z y Operating R R + R, we get y z y z z y z y ( y z) z y Operating C C C and C C C, we get 0 0 0 0 ( y z) z y y ( y z)( ) z y y 0 0 0 0 = 0, as R and