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# Mathematics2011 set 1

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Mathematics 2 011 question paper XII Mathematics

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### Mathematics2011 set 1

1. 1. ~~, 65/1 1 P.T 00 1 : 100 -""'", ~;~~~~:;"".; ~t~,~;., l1fUffi .0. ' .J! am I if ~ ,~,c.;t I ~ qj) I f(;r& tf{ ~-~ ~ I ~ 11 ~ not write ~ ~ ~ ~ 'qjT I ~ ~ ~-~~ fcfaUT "q:jT ~-~ I 't1 ~ ~ ~, ~ students will read the question paper only and will on the answer script during this period. qj) ~-~ ~ ~ "(fq:)" ~ ~ ~ ~ 29 ~ if the a.m., 10.30 to a.m. 10.15 From a.m. 10.15 question The paper. question this read to allotted been has time before question the of Number Serial the down be should paper question the of side hand right the on given ~~~I ~ - ~ m -~"q)T 3"(R" ~ Q5)s a:rrtif ~ answer-book. the of page title the I I I I I I I Candidates * I ~ ~ 10.30 ~ ~ at distributed be write I 1~(9.1 "~I ~ -q:;r if ~ ~-~ ~ fCfj ~ "qj"{ J Marks: tf{ ~ ~ ~ 3lT ~ ffl ~ I I .,;,.,.; fiR"?: 15 ~-~ ~ fCfj ~ "qj"{ . I q")1~ ~ 10.15 ~ ~ f<-I~-t1 , IIi I ~ ~ qj) 'qjT number Code I 3:rfi:rq;frif MATHEMATICS ~-:1I"I'111 ~ M attempting 4. ~ 4- Please I q GI~I'1 ~ will minutes 15 ';f I Maximum hours 3 if ~ paper . I f!TJi ~ ~ ~ . m ~-~ ~ . ~-~ ~ . . 3 "'~ ~ . . . ~ql~ 10.15 . No. : ~ . ~ Roll allowed: llme 65/1 No. Code I SOS Series 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
2. 2. / i? . / i? fit 1qj 1 e9": J j rf/ q, <{ :t q / ~ it fIJr;rif ~ 12 if -ar ~ / i? q;r 3iq; ~ ~ it fIJr;rif if ~ 10 if 3T ~ / ~. 31Rqp;f ~ I:/W : permitted. not is calculators of Use (v) questions. such all in alternatives the of one only attempt to have You each. marks six of questions 2 and each marks four of questions 4 in provided been has choice' internal However, choice. overall no is There (iv) question. the of requirement exact the per as or sentence one word, one in answered be to are A Section in questions All (iii) each. marks six of questions 7 of comprises C Section and each marks four of questions 12 of comprises B Section each, mark one of questions 10 of comprises A Section C. Band A, sections The question paper consists of 29 questions divided 3i' ;:.tt if' ~ 7 if tT ~ / i? q;r 3iq; :qn- ~ it fIJr;rif if ~ ff/4I/o<I All questions are compulsory. (ii) nPn" if m ~ 31P.fCfT qr;.p;r ~ ~ ~ a-frf iJi JI;'l;if / i? q;r 3iq; ~ if.. 31Jrrq;1 it 4 qJc? aiW ~ 7ft (fi"( ( if' ;r(ff ~ if I:/W if 3T ~ if iiit i:ft;r ?9U-.sT if ~ if JI;'l;if I:/W 'l!:« / i? ~ 31T;:rrfi:q; if JI;'l;if w ~~;1T~if/ w if 29 ~ ~ 3rj11frr ;:.tt Jr2177J 2 ~ 1[Uf W~ iJi qJc? aiW iii ) ( ii) 'q:jl;;rT ~ , (iv) ( <"1;~~ (v) (i) 2 65/1 (i) - --' . General Instructions: into three aT,-ar Z1" nPn" / -llllllr ~
3. 3. - 65/1 3 ',.. ) . sm sm + . cos 75° -1 I '~~: _c"::~~t , ~ 'Q 3 cos 2n) ~ I ~ "j:fR" ( ~ 3] A. of -1 ~ if ~ cos 0 ue va f ~ ~ ~ -1 terms . "q:jT 3 sm . 2n) ~ sin 75° m sin 15° A 3] I ~ A cos 15° i:I1T 15° sm- I m [ 2' 1( A-I [2 . ~, sin 75° wrIte , sin prrnCIpa .. m ~, 5 + 3) 2n ~ 5 e t IS at W . h 5 : ~ ( cos if i 15° cos 1 = cos- = ~ 3. A "j:fR" h 3W3lif; 5. If I I A ~ . 2 ~ * 4. ( . 2n) . . 3 I ~ arI;rqr ~ f~ ~ fqj ~ I ~ ~ ~ ~ B A~ 6)} (3, 5), (2, 4), {(I, = f ~ m 7} 6, 5, {4, = B 3}, 2, {I, = A trRT not. or one-one is f whether State B. to A from function a be 6)} (3, 5), (2, 4), {(I, = f let and 7} 6, 5, {4, = B 3}, 2, {I, = A Let 1. I -ff qj7 3iCli 1 ~ ~ n-q; 10 :if 1 ~ ~ each. mark 1 carry 10 to 1 numbers 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. ~ ~
4. 4. -.c~'- 1:fT";f~ . dx b)3 ~ f : (ax + b)3 dx Evaluate: In 7. + (ax Evaluate: f 6. ~ : In 1:fT";f ~ Write of the (1, 0, 0) "(f~ (0, 1, 1) ~ Write the points (1, ~ I vector 1 - j i 1 on the vector 4 7 I ~4-iIct>(UI~ ~~'1:&r'qjT~ 2 z-6 -. . y+4 = x-5 - b = . - 1. y 0 equatIon ~ = ~ = ~ I 65/1 7 1 i + j. ~~~I +j f . vector 1 3 3 0, 0) and 1 i 1 'qjT~ 1 e t W the firc;rR "qR;ft '1:&r ~ ~-~~ of the h . rIte . projection i-j ~ 1 10 joining 1). ~an 9. line gIven 1, direction-cosines me (0, the a 8. 2
5. 5. '0" ~~,!",",' B ~~ a+b, { each. function the Find 7. + JR' if : as defined is 5} 4, 3, 2, 1, {O, set the on * operation binary A OR = I lOx ii' = 3iq; fog = gof that marks 4 i?Ii f(x) JfR as Jffflq; 4 carry 22 11 '(fq:; R ~ 22 such R R f: ~ R Let : g it 11 defined mr JfR 11. numbers be Question to I i I SECTION a+b<6 6 and element 'a' ~ ~ 'a'. I t ~ of JR' = each inverse W~~ the ~ ~ b operation being : t ~ 'Sf"q)"R ~ 6 * < b + ~ a ~~ ~ ~ ~ b, 5} a { + 4, 3, 2, 1, {O, ~~ ~ = gof this a, 7 - + 6 lOx = ~ + a for with R, ~ if identity "[T'{T the invertible ~ R R ~ f: R : g is is fog zero set ~ the f(x) that of ~ Show - a + b 6, a*b= 12. Prove that: [ ,Ji-,;-x - .J:f+x -1 +.J~ .Jl-=-x ] x 1 7t = 4 - "2 -1 cos 'a' ~ ~ -~:S: x1 :S: 1 ' x, ~~~: tan q;r ~ 6 ~ ~ b + "(fm a I t t ~ a - 6 d~~~ 6, ~ q;r - b ~ + ~ ~ ~ t cYj~~un(.j ~ ~ ~ a a:*b= 65/1 ! ~ c' 5 1 :S: x ~ - x, "2 1 -1 :S: 1 7t cos ] - .Ji-=x 4 - .JI-,;-x +,Ji -=-x .JI-,;-x = [ -1 tan P.T.G.
6. 6. - 2 2x - 3 2x - 9 2x - 27 : ~ ~ 3x - 16 = 0 x- 8 ~ 3x - 4 x- 4 ~ x ~ RJor1I(.1I~(1 ~ m qjT ""J]um ~ ~1fTJl~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 qjT ~ ~ 'b' ~ 'a' m bx ~, {log (x e)} = f(x) ~ 'f', ~ ~ { + dx . 2 log x - - a ow s Y , - x dy - t th h e - x Y If - OR bx + 3, cosO) 7t] 2 + cose) 0.03~~F~, I ~ ~ F ~I"'~G .q: ~R~~'1 ~ ~~~~~9~~~~, ~ 65/1 6 m~~ 3"{?AaJ cm, I ~ ~ ~ ~ .q: its surface area. 2. ~ 0.03 of error an with cm 9 ] ~ [ 0, e, - e error in calculating sin 4 - = y fCfi ~ (2 as measured is sphere a of radius the If ~ then find the approximate . [0 . - t" , f m . Ion . . OR + unc - e . 4 sin e (2 (xe)} mcreasmg = an Prove t h at y IS . {log . 2 dx 15 X log = ~ ~ I~ ~...~~..' G~II~~ ~ (11 ~ ~, y - x e - Y X '-11~ -n& 3"{?AaJ
7. 7. -{ (1 + x2) d2 show that d a) -L dx - + (2x ~ m ~, d2y - x2) (1 + y) log (; tan = - + (2x a) dx2 17. Evaluate: n/2 f 0 dx : ~ f n/2 x + sin x dx 1+cosx C 0 x dy - y dx = .J;i-~7dx Solve the following dx (y + 3x 2) - "" ,. : ~ ~ -qj) ~1.i1~,UI , ~~; "'-': "." '. dx equation: R-~7 = differe~tial dx ~ RI'--1I(1~d 19. y following . - the dy Solve x 18. i ~ dy - = x + sin x dx 1 + cosx 0 1=fr;:r 0 == , x ~ dx fCfi (; 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).
8. 8. J"'-- b ~n'l ~"l --~ : ~11~~ dx ~ ~J"t;; J n/3 : --=~~+ the and 2k) + 4] + (3i A + ~ 1 2k) + ':' 4J ':' + (31 A + 1 2k) + ':' In coins. I two ~ ~ containing ~ each ~ ~ III and ~~ II ~ I, 5 = and takes out a coin. If the coin is of gold, what box a chooses person A coin. silver one and gold one is there III, 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 ~ fu"iiiit fiij ~ ~ ~ I ~ t, t III~, ~ }fll(.f~r:ff ~ ~ ~ ~ ~ 'i;fTm-q:;rt? ~ ~ ~ m t, -q:;r m ~ ~I<!~~J ~ ~ ~ ~ ~ ~ ~ III ~ ~ ~ ~ I "G1";1) II I, ~ t n~, ~ ~ I 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 I = - (21 = r ~ ~ 1 ~ k) boxes + 1 10) - j 1 5, identical (i . 1 - 1, (- three r ~ ~ ~ Given 28. to x = 5. j + k) 1 Y = (2i - ] J" - 1 ~ . (i =- 6 x (- 1, - 5, - 10), from the point of point 2k) 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 ~ q:jf I 3 + Find -q:;r x I '(f'qj 0 Y = = x 27. 31 x + I of ml:fi = curve y dx -- )(:;": Sketch the graph ~ 26. 7 -5 .J-c:x ~ J I ~ ~ 1:Jr;r 3:r~ nl6 ", ~ rr
9. 9. Determine desktQP 40,000 a computers Rs. - the number of demand and computers 25,000 monthly Rs. personal cost of total will types will not exceed 250 units. the that that two sell estimates model to plans portable a He and merchant 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 ~ "(f4T . ~ ~ ~ 4,500~. ~W?J1R ~ ~ m ~ ~ ~ ~ m1ft -;rf:r '"!iA1m ~ ~ ~ ~ ~~ ~ ~m~ ~ ~ ~ ~ o;f1i't ~ 0 5 2 ~ ~, mG-'fq ~ 1:{lfuCfi ~ m ~ 70 5,000~.~, 40,000~. -q;~ ~ ~ -q;~ 31f~ fC:ti 25,000~. "5fqjT{ "GT ~ ~~ ~. ~ "(f4T i-~ifq ~ "qj~~~~~~~? ~~ "(f4T ~ml:fi~~~1 ~ r ~~ ~ ~ Wm'{T ~ W?J1R ~ ~ ~ ~ ~ ~ Make an L.P.P. and solve it graphically. 65/1 . rll.li..iiiiii...iii_..ii~.;;;;;;;;;;;;; 11 ~~~~~ '. t,-~ ,