I Series SSO I Code N~. 65/1ep)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 bewritten 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 beforeattempting it.

• 15 minutes time has been allotted to read this question paper. The questionpaper 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.

• ~ ~ 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;<rr ~ I 10.15 qit ~ 10.30 qit ~ ~ ~ ~-lBf cfit ~ 3:fk

~ 31cff~ it; GRR if ~-~ lR cnT{ ~ ~ ~ I

MATHEMATICS

Time allowed : 3 hours

6511 1

Maximum Marks: 100

a:rr~ 3:fcn : 100

p.T.a.

General Instructions:

(i) All questions are compulsory.

(ii) The question paper consists of 29 questions divided into threesections 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.

(iv) 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 alternatives in allsuch questions.

(v) Use of calculators is not permitted.

'R7TT'Pl f.IW :

(i) ~ >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 m it ~ ~ ~ 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 ttfc!q:;fq CfiRT ~ I

(v) ;}Wlq;c1c< it m ctt 31J1fFrr ;rtt. ~

65/1 2

- YJ = [ 1 2J.3 - 5 3

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__?~ [3X+Y -YJ = [ 1 2J.2y - x 3 - 5 3

x Cf)f liR ~ ~ ?:1ft:

[3X + Y2y -x

/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 + 6 j + 27 k) x (i + 3 j + p k):: 0 .

p<:fiTllR~~~1\ 1\ 1\ 1\ 1\ 1\ ~

(2 i + 6 j + 27 k) x (i + 3 j + p k) = 0 .

, 99 •• ~ 'lie the direction cosines of a line equally inclined to the three~ coordinate axes.

~ ~ ~ ~ ~ ~ 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 4

SECTION 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 byf(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 dydx

OR

_ sin2 (a+y)sIn a

(cos x)Y = (sin y)X, finddydx

7.1fc:: sin y = x sin (a + y) W, 'ill ft:r.& ~ fcndydx

= sin2 (a + y)SIn a

65/1

7.1fc:: (cos x)Y = (sin y)X *, 'illdydx

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

665/1

: /' -1 x h w thatV sin so,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 theprobability that a candidate would get four or more correct answersjust by guessing ?

~ ~-fqctl01l~ ~ if -qfq ~ ~ ~ ~ it cfR ~~ ~ ~ (ftRif it ~~ ~ ~) I ~ CflTT J.l!f~ctlc11~ fen ~ ~ ~ ~ ~ 'iffi "lIT 31f~~it~~~?

(I7'.>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

2

3

3 + 2p

6 + 3p

1 + 3p + 2q I = 1

1 + 6p + 3q

18. Solve the following differential equation :

x dy = y _ x tan (y)dx x~.

Rkf ~ ~41ctl~UI cit ~ ~ :

x dy = y _ x tan ( y )dx x

65/1 7 p.T.a.

19. Solve the following differential equation :2 dycos x - + y = tan x

dx

f.:r8 ~ o/;-il1lch-lUI ~ QC1~ :

2 dycos x - + y = tan xdx

~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 ~===

,~ ~1 + sin x

OR+ -J1- sin x J = x ,- ~1 - sin x 2

Solve for x :

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

f.:r8~~~:

cot-1 (_~=l=+=s=in=x=-+_.~-=l=-=s=in=xJ = x, X E (0, n)~1 + sin x - ~1 - sin x 2. 4

3l$!fCfT

x ~ fu"Q: QC1~ :

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

65/1 8

A (3, -1, 2),

of the point

1\ 1\ 1\

22. The scalar product of the vector i + j + k with the unit vector along1\ 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

~~area 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=6x + 2z = 7

3x + y + z = 12

OR

Obtain

theInverseofthefollowingmatrixusmgelementary

operations 3

0-1A = 12

30

0

4 1

~ cnr m '8:, R8 ~~Iq,{UI~ it ~ ~'"

x+y+z=6x + 2z = 7

3x + y + z = 12

3l~

~ ~3TI ih m IDU R8 ~ cnr ~ ~ ~

3 0 -1

A = 12o

3

4

o

1

~oloured balls are distributed in three bags as shown in the followingtable: .

Colour of the ballBag Black

WhiteRed

I123

II241

III453

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 ?

6511 10

/ OR;/ A manufacturer can sell x items at a\/

~

~CfiTtrT

'Cfil"ffi

~mI

123

II

241

III

453

~ ~ <11~~<11 ~ '1flIT (f~ ~ it ?J 7R ~I~~~I ~ ~ ~ ~ ~ amm 'IT{ ~ I 541fl1ctldl q:<TI ~ fcl) ~. ~ (D ~ if -B ~ ~ ?

2~aler wishes to purchase a number of fans and s.ewi~K machines.He has only ~ 5,760 to invest and has a space' for at most 20 itl:l_t,Il.8.A fan costs hi!!?-Rs. 3QO and a se'Ying_mac.hine_Rs.240. His expectationis that he can sell a fan at a prQf1t {)J_~._~2 and a sewing machine at

a profit_ of Rs. 18. Assuming that he can sell all the items that he canbuy, how should he invest his money in order to )llaximise the profit?Formulate this as a linear programming problem and solve itgraphically.

~ ~ "¥9 W ~ ~ ~ {9~C\rjl ~ ~ I ~ Lfrn f.:rffi ~ ~ ~

5,760 ~. ~ (f~ ~ ~ fc;rQ: 3if~ 20 ~ ~ fc;rQ: m W I ~ ch9r 360~.if (f~ ~ ~ ~ 240~. if ~ ~ I "3it 31RTT ~ fcfi ~ ~ W c.nT

22~; C1fl1 1R (f~ ~ ~ mfR <it 18 ~. ~ 1R ~ WIT I ~ ~ fcl) o;Q

~ ~ {9~~III) ~ ~) ~ 311Rr urn ~ >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-angledtriangle is given, show that the area of the triangle is maximum when

the angle between them is ~3

pnce of Rs. (5 - ~) each.100

The cost price of x items is Rs. (x + 500). Find the number of\5items 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