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CBSE Class XII 2011 : MATHEMATICS

9 pages, 55 questions, 25 questions with responses, 39 total responses,    0    0
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~~, 65/1 1 P.T 0 0 1 : 1 00 - ""'", ~ ;~~~~:;"".; ~ t~,~;., l1fUffi .0. ' . J! a m I i f ~ , ~\,c.;t I ~ ~ ~ ' qjT I ~ q j) I f (;r& t f{ ~ -~ ~ I ~ 1 1 ~ not write ~ q f cfa\UT " q:jT ~ -~ I ' t1 ~ ~ ~ ~ ~ ~-~~ j) ~ -~ ~ ~ " (fq:)" ~ ~ ~ , ~ 2 9 students will read the question paper only and will on the answer script during this period. M ~ ~ ~ ~ ~ i f t he a .m., 1 0.30 t o a .m. 1 0.15 F rom a .m. 1 0.15 a t d istributed q uestion T he p aper. q uestion t his r ead t o a llotted b een h as t ime b efore q uestion t he o f N umber S erial t he d own b e s hould p aper q uestion t he o f s ide h and r ight t he o n g iven ~~~I ~ - ~ m - ~"q)T 3 "(R" ~ Q 5)s a :rrtif ~ a nswer-book. t he o f p age t itle t he I I I I I I I Candidates arks: I 1 ~(9.1 " ~I 1 0.30 - q:;r ~ ~ -~ ~ f Cfj b e w rite I * 3 tf{ ~ i f ~ 3 lT\ ~ f fl ~ " qj"{ I J ~ . ,;,.,.; ~ ~ ~ -~ ~ f Cfj ~ ~ . I q f iR"?: 1 5 ~ " qj"{ n umber C ode I ")1~ ~ 1 0.15 ~ ~ ~ q j) ' qjT f<-I~-t1 , I Ii I ~ M P lease I M MATHEMATICS ~ -:1I"I'111 4 . ~ 4 - w ill m inutes attempting q G I~I'1 ~ i f ~ 1 5 I aximum h ours 3 ~ paper ' ;f I :rfi:rq;frif f m ~ . I !TJi ~ ~ ~ -~ ~ . -~ ~ . 3 "'~ ~ . . ~ 10.15 . . . ql~ ~ . No. : a llowed: . ~ Roll ' llme 6 5/1 N o. C ode I S OS S eries I frirlr R~ ;' qj)-s ;{. must write the Code on 0. ";1. Please check that this question paper contains 11 printed pages. written on the title page of the answer-book by the candidate,. Please check that this question paper contains 29 questions. it. any answer ~j{ 0" ...",.-c' -.f / i ? . / i ? f it \ 1 e 9": n J j r f/ q , < { : t q / 3 i' ~ i t f IJr;rif i f' ~ 7 i f t T ~ / i ? q ;r 3 iq; : qn- ~ i t f IJr;rif if ~ 1 2 i f - ar ~ / i ? q ;r 3 iq; ~ ~ i t f IJr;rif i f ~ 1 0 i f 3 T ~ / ~ . 3 1Rqp;f ~ \ I:/W : p ermitted. n ot i s c alculators o f U se (v) q uestions. such a ll i n a lternatives t he o f o ne o nly a ttempt t o h ave Y ou e ach. marks s ix o f q uestions 2 a nd e ach m arks f our o f q uestions 4 i n provided b een h as c hoice' i nternal H owever, c hoice. o verall n o i s T here (iv) q uestion. t he o f r equirement e xact t he p er a s o r sentence o ne w ord, o ne i n a nswered b e t o a re A S ection i n q uestions A ll (iii) e ach. m arks s ix o f q uestions 7 o f c omprises C S ection a nd each m arks f our o f q uestions 1 2 o f c omprises B S ection e ach, mark o ne o f q uestions 1 0 o f c omprises A S ection C . B and A , sections The question paper consists of 29 questions divided Pn" i f ; :.tt ~ / i ? q ;r 3 iq; ~ ~ ff/4I/o<I All questions are compulsory. (ii) 1qj 3 m 4 3 1P.fCfT q r;.p;r ~ ~ ~ a -frf i Ji J I;'l;if \ I:/W i f 3 T ~ if.. 1Jrrq;1 i t i q Jc? a iW ~ 7 ft ( fi"( ( i f' ; r(ff ~ i f ~~\;1T~if/ if iiit i:ft;r 9U-.sT ? if ~ f J I;'l;if \ I:/W ' l!: / i ? ~ 3 1T;:rrfi:q; i f J I;'l;if w ~ 1 [Uf w if 29 ~ ~ 3 rj11frr ; :.tt J r2177J 2 q Jc? a iW W~ i ' q:jl;;rT ~ iii ) ( ii) Ji < , (iv) ( "1;~~ (v) (i) 2 65/1 (i) - --' . General Instructions: into three aT,-ar Pn" / n \Z1" - llllllr ~ - 65/1 3 ' ,.. ) . s m s m + . cos 75 -1 I ' ~~: _ c"::~~t , ~ ' Q ~ 3 c os 2n) ~ I A " j:fR" ( ~ 3] . o f c os -1 ~ i f ~ " 0 u e f ~ ~ ~ t erms -1 q:jT 3 . 2n) ~ m . A A s I ~ i :I1T 3] m v a p rrnCIpa I m sin 15 sin 75 A cos 15 -I 1 5 s m- 1( ~ [ 2' w [2 rIte , sin 75 . , 5 + e t .. m ~ , : ~ ( cos 3 ) 2n ~ s in 1 5 cos I S a t W .h 5 i 5 f i 3 1 = cos- = ~ "j:fR" 3. A I h W3lif; 5. f I I A ~ . 2 ~ * 4. ( . 2n) . . 3 I ~ a rI;rqr ~ f ~ ~ f qj ~ I ~ ~ ~ ~ B A~ 6 )} ( 3, 5 ), ( 2, 4 ), { (I, = f ~ m 7 } 6 , 5 , { 4, = B 3 }, 2 , { I, = A trRT n ot. o r o ne-one i s f w hether S tate B . t o A f rom f unction a b e 6 )} ( 3, 5 ), ( 2, 4 ), { (I, = f l et a nd 7 } 6 , 5 , { 4, = B 3 }, 2 , { I, = A L et 1. I - ff q j7 3 iCli 1 ~ ~ n -q; 1 0 : if 1 ~ ~ e ach. m ark 1 c arry 1 0 t o 1 n umbers Question ' ~ 3:r ~r SECTION : A ";4;!'ii1Y~& c/fi~1~f(:! ? Evaluate: cos 75 :'" ,. -2 . -2 If a matrix has 5 elements, write all possible orders it can have. P.T.G. ~ ~ -.c~'- 1:fT";f~ . d x b )3 ~ f : (ax + b)3 dx Evaluate: In 7. + ( ax Evaluate: f 6. ~ : In 1:fT";f ~ Write of the ~an (1, 0, 0) "(f~ (0, 1, 1) ~ Write the the points (1, ~ I 1\ -j i 1\ on the vector ~ 4 7 I ~ 4-iIct>(UI~ ~ ~'1:&r'qjT~ 2 - . z-6 = - . y+4 = x-5 - b g . y 1. Iven 0 e ~ = ~ = ~ I 65/1 7 1\ i + j. ~~I + j f quatIon v 1\ 3 3 0, 0) and 1\ i ' qjT~ i vector . e t r Ite . 1\ h . W 10 joining firc;rR "qR;ft '1:&r ~ ~-~~ of the 1\ -j ~ 1\ projection ector 9. line 1). m 1, direction-cosines e (0, the a 8. 2 '0" ~~,!",",' B ~ ~ e ach. m a+b, { if f unction t he F ind 7 . + l : a s d efined i s 5 } 4 , 3 , 2 , 1 , { O, s et t he o n * o peration b inary A OR J Ox I R' = ii' = f f (x) 3iq; og a s = g of t arks 4 c 2 2 t 1 1 arry 4 i?Ii d JfR efined Jffflq; hat R '(fq:; s ~ 22 uch R R f : ~ R L et : g it 11 b mr JfR 11. numbers e Question o I i I SECTION a+b<6 6 and each 'a' ~ ~ I t ~ element 'a'. J W of R' ~~ inverse : t ~ ' Sf"q)"R ~ 6 * < b + ~ a ~ ~ ~ ~ ~ b , 5 } a { + 4 , 3 , 2 , 1 , { O, ~~ ~ = " the ~ operation being [T'{T 7 ~ this a, f + = g of l - b + for 6 Ox = ~ f R a i identity with , ~ f 6 the invertible ~ R R ~ f : R : g is is og zero set ~ the (x) that of ~ Show - a+b , a*b= 12. Prove that: [ ,Ji-,;-x - .J:f+x -1 +.J~ .Jl-=-x ] x 1 7t = 4 - "2 -1 cos ' a' ~ ~ q -~:x1 S: :S: 1 ' x, ~~~: tan ;r ~ 6 ~ ~ b + " (fm a I t t ~ d a - 6 ~~~ 6 , q ~ ;r - b ~ + ~ ~ ~ t cYj~~un(.j ~ ~ ~ a a:*b= 65/1 ! ~ c' 5 1 : S: x : ~ - x , c " 2 1 -1 S: 1 7t os ] - .Ji-=x 4 - .JI-,;-x +,Ji -=-x .JI-,;-x = [ -1 tan P.T.G. -2 2x - 3 2x - 9 2x - 27 : ~ ~ 3x - 16 = 0 x- 8 ~ 3x - 4 x- 4 ~ x ~ R Jor1I(.1I~(1 ~ m q jT " "J]um ~ ~1\fTJl~1 3x - 64 x 14. Find the relationship between 'a' and 'b' so that the function 'f' defined by: { ax + 1, if x:S 3 f(x) = is continuous at x = 3. if x>3 ~ , " qft~ 3 > I x ~ ~ ~ 3 + ~ , ~ 3 x :S ~ 1 , ax q jT ~ ~ ' b' ~ ' a' m bx ~, {log (x e)} = f (x) ~ ' f', ~ ~ { + dx . 2 log x - - a o w s Y , - x dy - t th h e - x Y If - OR bx + 3, cosO) m I on 7t] 2 6 m 0 .03~~F~, I ~ ~ F ~ I"'~G . q: ~ R~~'1 ~ ~~~~~9~~~~, ~ 65/1 ~~ 3"{?AaJ c m, 0 I ~ ~ 2. ~ .03 o f e rror a ~ ~ its surface area. .q: ] ~ + cose) n w ith c m 9 a [ 0, e , - e s error in calculating in 4 - = y f Cfi ~ (2 s m easured i s s phere a o f r adius t he If ~ then find the approximate . [0 . - t" u f nc m . creasmg . . , (xe)} OR + a -e. 4 sin e (2 n = I Prove t h at y S . {log . 2 dx 15 X l og = ~ ~ I ~ ~ ...~~..' G ~II~~ ~ ( 11 ~ ~ , y - x e - Y X '-11~ -n& 3"{?AaJ that d a) -L dx - + (2x d (1 + x 2 ) - 2y ~ m ~ , y ) l og ( ; t an = - + (2x a) dx2 17. Evaluate: n/2 f dx : ~ f n/2 x + sin x dx 1+cosx C 0 x dy , ~ ,. ~; - y dx = .J;i-~7dx Solve the following d (y + 3x 2) - x "" : d "''-': "." . x ~ ~ - ~1.i1~,UI qj) R equation: -~7 = d differe~tial x ~ RI'--1I(1~d 19. y - following . d the y Solve x 18. i ~ d -y = 0 x + sin x dx 1 + cosx 0 1=fr;:r 0 == , x ~ dx f -{ (1 + x2) d2 show Cfi (; log y ), = tan 16. If x differential - equation: =x ' . : ~ ~ ~1.i1~,UI - RI'--1I(1I(9d ~ qj) dy =x (y + 3x 2dx ) dy 20. Using vectors, find the area of the triangle with vertices A(1, 1, 2), 5 ) 3 , B (2, 2 ), 1 , A (l, m ~ 7 - --~ P.T.D. ~ - 65/1 ~- ~ ~ ~ - q)1 ~ ~ I ' q}""{, ~ 5 ) m 5 , - q)1 C (l, ~ ~ B(2, 3, 5) and C(l, 5, 5). J "'-- b ~ n'l ~"l --~ : ~11~~ dx ~ ~ J"t;; J n/3 : --=~~+ and takes t he a nd 2 k) + 4 ] + ( 3i A + ~ 1 \ 2 k) + ' :' 4 J ' :' + ( 31 + 1 \ 2 k) + ' :' J " I n c oins. I t wo ~ ~ c ontaining ~ e ach ~ ~ I II a nd ~ ~ I I ~ I , out a coin. If the coin is of gold, what b ox a c hooses p erson A c oin. s ilver o ne a nd g old o ne i s t here I II, box box I, both coins are gold coins, in box II, both are silver coins and in at random is the " 10 ~ ~ ~ ~ ~ " G1";1) ~ ~ ~ ~ q:;r I ~, m ~ f u"iiiit f iij ~ ~ ~ I ~ t , t I II~, ~ } fll(.f~r:ff ~ ~ ~ ~ ~ 'i;fTm-q:;rt? ~ ~ ~ m t , - q:;r m ~ ~I<!~~J ~ ~ ~ ~ ~ ~ ~ I II ~ " ~ ~ ~ I G1";1) I I I , n ~ I t ~, ~ ~ ~ t ~ , ~ - q:;r ~ R~I(.'1ctl ~ m ~ probability that the other coin in the box is also of gold? 65/1 ~ 6 O . - = = x ~ ~ x -~ ~ I 3 + x 2 + - ( 21 = r ~ 5 = 1 k ) b oxes + \ ~ I = Y = (2i - ] ~ 1 \ 1 1 0) - 1 j \ 5 , 1 i dentical ( i . \ - 1 , ( - t hree r ~ ~ ~ G iven 28. to x = 5. j + k) \ 1 - \ ~ . (i =- 6 x (- 1, - 5, - 10), from the point of point k) of the ' distance r ~ ' (fP-1T ~ ~ ~ above x-axis and between intersection of the line "1 plane + 31 and evaluate the area under the :' the = Ix y A ) m q :jf I 3 - + Find q:;r x I ' (f'qj 0 Y = = x 27. 31 x+ I of l:fi = curve y dx -- (:;": Sketch the graph ~ 26. 7 -5 .J-c:x ~ J I ~ ~ 1:Jr;r 3:r~ nl6 , ~ rr Determine d esktQP 4 0,000 a c omputers R s. o a - the number f d emand nd c omputers 2 5,000 R m onthly s. p ersonal c ost o f w t otal ill t ypes t t will not exceed 250 units. he t hat hat t wo s ell m e stimates odel t o p lans p ortable a H e a nd m erchant respectively. model A - of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs. 70 lakhs and his profit on the desktop model is Rs. 4,500 and on the portable model is Rs. 5,000. ' ~ ~ , ~ I ~ ~ m "(f4T ~ . ~ ~ ~ 4,500~. W?J1R ~ ~ ~ ~ m ~ ~ 1ft ' - ;rf:r "!iA1m ~ ~ ~ ~ ~ ~ ~ m~ ~ ~ ~ ~ o ;f1i't ~ 0 5 ~ 2 ~ , m G-'fq ~ 4 1 :{lfuCfi ~ m ~ 7 0 5 ,000~.~, 0,000~. - q;~ ~ ~ 2 - q;~ f 3 1f~ C:ti 5,000~. " 5fqjT{ " GT ~ " qj~~~~~~~? ~ ~~ ~. ~ "(f4T i-~ifq ~ ~ ~ "(f4T ml:fi~~~1 ~ r ~~ ~ ~ W m'{T ~ W ?J1R ~ ~ ~ ~ ~ ~ Make an L.P.P. and solve it graphically. 65/1 . rll.li..iiiiii...iii_..ii~.;;;;;;;;;;;;; 11 ~ ~~~~ ' . t,-~ ,

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