maths previous year question paper 2009

8/2/2019 Maths previous year question paper 2009

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Series ssa I Code No. 65/1Q5)s~.Ro~Xo.

I 1 - - I- . Candidates must write the Code onthe title pag;e of the answer-book .. tR Im a J T g m T m - g f f i 1 d 5 1 I v ~ " ~tR ~ fc;r I

I 1

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., thestudent will read the question paper only and will not write any answer onthe answer script during this period.

_ ~ ~ e m : - R Nl ~ ~-lf;J if ~ T5 12 % ' I ~-lf;J i t ~ ~ ~ a n t ~ T J T J : ~ ~ ~ rn;r 3m-~ ~ ~-'TO ~ -~ ~ ~ C fi"{ - R Nl ~ ~-lf;J i t 29 ~ ~ ~ ~ C f i T '3"ff{ fei&41 ~ ~ ~ ~, ~ qn ~ ~ ft11j ~ ~-lf;J it ~ ~ f c : : m : 15 fiRz c t i T W lG f ~- T f I 1 T ~ 1 ~ - -q;f q ;r f c R R u T ~ - q

10.15 ~ M~ 1 1 D ..1 5 ~ tI 10.30 .~ -rF5 ~ ~ ~-~ q : ; ) ~ mw 3 1 < I T c r ~ ~ ~ ~ - ~ - 1R ~ ~ ~ ~MATHEM.ATICS

lliUkl~e c:?lJw ed : 3 hours Maximum Marks: 100

~ a ic n : 100

65 1 1 P.T.O.


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General Instructions:(i) All questions ar_ecompulsory.(ii) The question paper consists of 29 questions divided into three

sections A, Band C . Section A comprises of 10 questions of onemark each, Section B comprises of 12 questions of four markseach and Section C comprises of 7 questions of six marks each.

(iii) All questions in Section A are to be answered in one word, onesentence or as per the exact requirement of the question.

(iu) There is no overall choice. However, internal choice has beenprovided in 4 questions of four marks each and 2 questions of sixmarks each. You have to attempt only one of the alternatiues in allsuch questions.(v) Use of calculators is not permitted.

(i)(ii)

(iii)

(iu)

(v)

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t r 4 t JfN .3 1 R q r l f ~ . IW JfN W it 29 JfN.~. : i f r f t ; : r &Vi f it f c r > : r r N m ~ : 31, .~. 0 2 T T tr I ?IfU53{ it 10 JfN ~ fir;rif i t ~ ~ 3ir t ; cg .~ I ?IfU5 ~ T i 12 JfN W fir;rifif ~ Tjfff 3fq; C f iT ~ I ?TJS tr if 7 JfN W f;r;rif it ~ fH .W E C f i T" f f f

7U5 31 T i f f ' 4 t m-~:rR ~ ~ .~ qrq


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SECTION A19U5 31

Questions number 1to 10 carry 1mark each.JfN i f&:lr 1 it 10 f(cfi ~ ffi" 1 _ ,3fq; 'liT " f f I

1. Find the value of x, if

[ 3 X + Y - Y ) [ 1 2 )2 y - x 3 = . - 5 3 .x e m ll'R ~ ~ < : r f u :G ; : : - : ) ~ [ - ~ : ) .

2. Let * ' be a binary operation on N given by a * b = ReF (a, b), a, bEN ..Write the value of 22 * 4.ttRT *, N 0/ ~ r n : 3 W : T f U ~ ~ c r f t a * ' b= HCF (a, b) ID :u ~ t ~a, bEN ~ I 2 2 *4 C f i T ll'R f C ; : r f u r Q : r

3. Evaluate:1 1 . J 2Jo1 / . J 2Jo

I. . . . Evaluate:J cosJX dxFx

- = - = - : m e r ~ :cos . J X-----,=--- dxJ X

3 P.T.D.


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5. Write the principal value of cos-1 ( C O S 7;).cos-l ( C O S 7 ; ) C O T ~ 1 I T - 1 ~ I

6. Write the value of the following determinant:a= -b b-c c-ab-c c-a a-b

b-c-a a-b

a-b b-c c-ab-c c-a , a-bc-a a-b b-c

7. Find the value of x from the following :x 4 ;;:;:02 2x

Rq it x C O T l 1 R m~ :x 4 = 02 2x

8. Find the value of p ifA A A A A A ~(2 i + 6 j + 27 k ) x (i + 3 j + p k);;:;: 0 .

p C f i T l1R m~ ! J f u :A A A A A A -7(2 i + 6 j + 27 k) x (i + 3 j + p k) = 0 .

9. Write the direction cosines of a line equally inclined to the threecoordinate axes.3\1 W r ~ ~ ~. ~ ~ ~ A ~ ~ li< b ~ lV WlR c i t u T ~ m l

10. If P is a unit vector and (~- I i ) . (;t + I t ) ; ; : ; :80, then find I : ; I * Ii~~ ~ ~ . r r ( ~ - I i ) . ( : ; + p) = 80 ~! ill I : ; 1 C f i T l1R~~

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SECTION B~~

Questions number 11 to 22 carry 4 marks each.JfR t k9w 11 it 22 r r q : ; ~ JfR it 4 Jiq) ? ' I

11. The length x of a rectangle isvdecreasing at the rate of 5 em/minuteand the width y is increasing at the rate of 4 em/minute. Whenx ::::8 em and y :; 6 em, find the rate of change of (a) the perimeter,(b) the area of the rectangle.

ORFind the intervals in which the function f given by

f(x) :::::sin x + cos x, 0 S; x < 2n,is strictly increasing or strictly decreasing.

~ 3lrl e E l ~ X,5 WTI/ f tRz c t t ~ ~ W W t ~ 3 fh : ~ y , 4 i t t ITlf tRz c t tG \ B ~ - ~ 1 ~ x = 8 i t t IT 3 f h : y = 6 W : f r t , (f


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13. Let f: N --7 N be defined by

fen) "'"n+l2 ' if n is odd for all n E N.n2' if n is even

Find whether the function .f is bijective.

ftn)"",n+l2n2'

"[fU 1:fR~ ~ ~ f: N --7 N~ 1~ . ~ . f c p q : < : f f ~ f ~ ~ (bijective) ~

14. Evaluate:J elxJ5- 4x-2x2

OR.Evaluate .:J x sin" x dx1=fR ~ ~.:J . . dxJS - 4x- 2x2

3l~1=fR~~: J x sin-1x dx

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sin-1x15. If y ';:::--==J i - x2 '.'. 2 d2y . dy(1 - x ) ~ - 3x ~ - y = 0dx dx

show that

~ y = sin-Ix. ' J l- x2

,2 d2y dy(1- x ) _' - - 3x _'" - y = 0dx2 dx16. On a multiple choice examination with three possible answers (out of

which only one is correct) for each of the five questions , what is theprobability that a candidate would get four or more correct answersjust by guessing? .~ ~ - R lC h ( ; q l l J if ~ ~ ~ ~ ~ ~ f f i " ; r ~ s w . : % ( ~ it ~" Q % ' ~ ~ ) I ~ C F IT 5 1 , 1? : l C h ( l I, ~ f c f i 1 t C f i ~ m~ ~ ~< n 3if~m& i m i t s w .: ~ ?

17. Using properties of determinants, prove the following:11+ P 1 + P + q2 3 + 2p 1 + 3p + 2q = 13 6 + 3p 1 '+ 6p + 3q .

t m f U l c h l & i ~ 9 iT m~}.p:;f, m ~ :1 1 + p 1 + p + q2 3 + 2p 1 + 3p + 2q - 13 6 + 3p 1 + 6p + 3q

18. Solve the following differential equation :x dy = y - xtan (.y)dx ,x

-:. - 3"{Q & ;,d fl1 4 1E h {U I ~' Q " C 1 ~ :

x ~ - y - x tan (~)

7 P.T.O.


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19. Solve the following differential equation :2 .. dy.cos x ~_ + y = tan xdx

~ ~fl41Cfl{OI c n l ~ ~ :2 dycos x _"_ + y = tan xdx

20. Find the shortest distance between the following two lines :~ " " ":::;1 + A) i + (2 - A) j + (A + 1) k ;~ " " " " A "r = (2 i-j - k) + Il (2i + j + 2 k).

~ A " Ar =(1+A)i +(2-A)j +(A+1)k;~ A A A "A"r = (2 i-j - k) + Il (2 i + j + 2 k ).

21. Prove the following:-1(. J 1 + sin xcot ..~1 + sin x

OR

+ J 1 - sin x) x- - J l . - sin x = 2'Solve for x:

2 tan-1 (cos x) = tan-1(2 cosec x)~ < h l f t : R ; ; " ~ :

cot-1( i~ : : : : : j~ = : : : J = ~ , XE (0, :)3:r2lmx&!ftnt~~:.

2tan -1 (cos x) = tan-1 (2 cosec x)


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A A A22. The scalar product of the vector i + J + k with the unit vector alongA A A A A Athe sum of vectors 2 i + 4j - 5 k and Ai + 2j + 3 k is equal to one.

Find the value of 'A .A A A A A A~ 2i + 4j - 5k d"~ A i + 2j + 3k ~ljllllfl(1 r t t Rm " ij ~ ~A A A~~ i+ j + k C f i f ~ 20l'1lf1(1 1 ~ I 'A 'fiT liR ~ ~

SECTION C~lJ

Questions number 23 to 29 carry six marks each.>rR #w;rr 23 # 29 'ffCfj ! T r W Jr.R i i i 6 afrn. # . I

23. Find the equation of the plane determined by the points A (3, -1, 2),B (5, 2, 4) and C (-1, -1, 6). Also find the distance of the pointP (6, 5, 9) from the plane.~a# A (3, -1, 2), B (5, .2, 4) d"~ C (-1, -1,6) '[fU ~ ~ C!if+ l 1 0 4 " I i $ { U I~ ~ I ~. P (6, 5, 9) r t t ~ ~ it ~ '4 t ~ ~ I

24. Find the area of the region included between the parabola y2 = x andthe line -x + y = 2.~y2=x d"2ITbsrrx+y=2t~$rC!if~~~

25-. Evaluate:

i!J x dx'. a2 cos2x +b2 sin2 xo65/1 9 P.T.D.


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26. Using matrices, solve the following system of equations:x+y+z:;;;:6x + 2z :;;;:73x + y + z ;:; 12

ORObtain the Inverse of the following matrix using elementaryoperations :

o34

~1o1

~ q)f w i t r T Cf l{ , f;p:;f fl4 1 Cfl'(U I ~ C f l T 'Q fr ~x+y+z:;;;:6x + 2z = 73x + y + z = 12

3l~

3A= 2

o

o34

~1

o1

27. Coloured balls are distributed in three bags as shown in the followingtable:

Colour of the ballBag IBlack White RedI 1 2 3

II 2 4 1III 4 5 3

A bag is selected at random and then two balls are randomly drawnfrom the selected bag. They happen to be black and red. What is theprobability that they came from bag I?

65 10


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~ ~Cf)f \'T

C f l I T 1 T ~ ~I 1 2 3II 2 4 1III 4 5 3

~ ~ 6 1 1 ~ 'O i j 6 l 1 'f1J 1 ' J 1 l T " 'M1 ~ ~ ? J iR 4 1 ~ ' l ' i j 6 l 1 ~ ~ a m : c r w ~ a m :" M " K 1 ~ ~ I S U ft lC f i d l C F I T W fc p it, ~ (I) ~ It ~ ~ ~ ?

28. A dealer wishes to purchase a number of fans and sewing machines.He has only Rs. 5,760 to invest and has a space for at most 20 items.A fan costs him Rs. 360 and a sewing machine Rs. 240. His expectationis that he can sell a fan at a profit of Rs. 22 and a sewing machine ata profit of Rs. 18. Assuming that he can sell all the items that he canbuy, how should he invest his money in order to maximise the profit?Formulate this as a linear programming problem and solve itgraphically,~ ~ ~ ~ ~ ~ ~ @ (j


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~ ~ ( . 5 - _ _ ! _ _ ) ~ . 5 l fu ~ c t t ?J it x ~ ~ ~ ~ 1 'x ~CfiT~" . 100 . . . .'~ ~ (~ +500).~. ~ I . ~ c t t % ~WCl ~ ~ ~ ~ ffi'q~cnG.~~~~ I

maths previous year question paper 2009

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