Cbse 2009 Maths Board Paper Class Xii

I

Series SSO

65/1

Code N~.

I

ep)g ;:r.

Roll No.

Candidates

must

~-;:f.

the title page of the answer-book. 3m-~~cPl

-cRTaw-TI ep)g 051

~ ~fffiY

write the Code on

~

~-%

I

•

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

•

~

•

~-lBf

•

~

~

•

~

~

•

~

~

C/1{ ~

#~

~-lBf

10.15 qit ~

fcf;

m2T ctr 3ffi

C/1{ ~ CfIT

fcf;

it; ~

cfit ~ fcf;
~

31cff~ it; GRR

fG:Q:

~-lBf

~

k1"S-t1

~

#~

~-lBf

~

I

if ~-~

~

~

12 ~

irn- ~

~

# 29 ~ ~

it

cfit ~

~

~

CfIT

~

1flfT

10.15 qit ~ 10.30 qit ~ lR

cnT{

~

it; ~-~

~-~

lR ~

I

~t

15 f1:Rz Cf)f ~

I

~

~

~

M

~

I ~

~

~

~

#

-lBf Cf)f fcrcwr ~ ~-lBf

cfit ~

3:fk

Marks:

100

I

MATHEMATICS

Time allowed : 3 hours

Maximum

a:rr~

6511

1

3:fcn :

100

p.T.a.

General Instructions: (i)

All questions are compulsory.

(ii)

The question paper consists of 29 questions divided into three sections A, Band C. Section A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four marks each and Section C comprises of 7 questions of six marks each.

(iii)

All questions

(iv)

There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions.

(v)

Use of calculators is not permitted.

'R7TT'Pl

in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.

f.IW :

(i)

~

(ii)

ff1 >IN 31

>IN J1Rqrlf ~. I

it

q;r

10

iT ~

it 29 ~

~ ~ rft;:r ~

FiRit it ~

~

~.

rm

3fq;

CfiT

~

(

~ "(ffUg

it ~ 3fq;

"fT

CfiT

~ : ~

it 7 >IN

I

3{,

"(ffUg G[

t FiRit it

7i{ (f2lT

"fT

I

"@V5

it 12 >IN ~. fJRif JTFitcn fJT 3fq;

CfiT

~ I

(iii)

"(ffUg 31¥ff(

(i v)

31

it

~

it

q;r

a:fqif

2 ~

fc!q:;fq CfiRT ~

(v)

65/1

;}Wlq;c1c<

t

\57T ~

q-uf >IN '" r:rrft

m it ~

"fTsft

it

fc!q:;fq

~ ~

31~

ctt 3WH4C/?r1f

~

I

~

~. (

fiR

it

3TFfffi:q)

fcrcfic;rr

ctt

31J1fFrr

;rtt. ~

I

m

Tl,Cfi qrq:zr

2

~

W qr( 3fqif (

#1 ~

4

~

m if

it

r:rrft

it 3fJT1Ch1

rr:qr

f!!E

E9?

tt

SECTION A ~31

Questions number JTR

1 it 10

#?§ZlT

1 to ffCfi

10 carry

~

1 mark

JTR 1 aicn

CfiT

each. ~

I

1. / Find the value of x, if

l__

2y - x [3X+Y

?~

x Cf)f liR ~

-x +Y

2y [3X

/2. ~ ! )

~

51

3 2J.

- 3YJ = [ - 5 1

3 2J.

3 -YJ

= [-

?:1ft:

/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

a, b

E

en: ~

fu31TmU ~

N ~ I 22 * 4

CfiT 11R

~ ~

it

a * b = RCF (a, b) &m ~~,

~

I

\ ~ate: 1/12

J o

65/1

3

p.T.a.

5. ~rite /

the principal

cos-1 (cos 7;)

6. /Write

I

<:fiT lj&i liB ~

the value of the following determinant: la-b b-c c-a

~

b-c

c-a

a-b

c-a

a-b

b-c

Rq ~I\fU Ien

<:fiT liB

-B

ft1furQ:

:

a-b

b-c

c-a

b-c

c-a

a-b

2

Rq

value of cos-1 (cos 7;).

2x

x <:fiT liB ~ x 4

~:

::0 2

8 ,-

~

.

/' Find

/

2x

the value of p if

(2

i

1\

1\

1\

1\

1\

~

1\

1\

1\

1\

1\

~

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

0.

p<:fiTllR~~~ 1\

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

~,

9 ••

~

coordinate 'lie the axes. direction

~ ~ ~~ ~ 10.

If

P

cosines

~

it

of a line equally

is a unit vector and (~

~ Ii ~ lIDfCfi~

~

w:rr (~

p) . (~

-

+

p) . (~

~~I

65/1

'R ~

(fRT rrj~~licj) ~

4

inclined cfiTuT

p) :: 80, +

p) :: 80

to the three ~

m

then find

I

I~ I.

~, m I}( I <:fiT liB

SECTION B "{§US.

Cif

Questions number 11 to 22 carry 4 marks each. >IN til§lrT 11

~:gth ~ ....

"#

22

(fCJi

J:r:'R"~ 4 3iq;

~

!

I

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 by f(x) = sin x + cos x, o:s; x :s; 2n, is strictly increasing or strictly decreasing.

~ G:\

:jWffi cn1 ~

wr

-B ~

(q) ~

5 -wIT/fir;rG cn1 ~

x,

* I \ifiilx

~

YKqJrj

=8

cn1 ~

WTI

~

-B ere Wt w ~

y =6

~

W:ft

y,

~

*, ~ ~

4 -wIT/fiRz cn1

~

(31)~,

I

~

~

~

~

~

qm ~ f.1tR 12.

~ f-,1+lYlrj

f(x) = sin x + cos x,

*

O:s;

x

:s; 2n

mT ~

T.:fim

f, f.1tR

I

If sin y = x sin (a + y), prove that

_ sin2 (a+y) sIn a dx

dy

OR dy dx

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

7.1fc::

7.1fc::

65/1

sin

y

= x sin (a + y)

(cos x)Y = (sin y)X

*,

W,

'ill

'ill

ft:r.& ~

fcn

dy = sin2 (a + y) dx SIn a

dy dx

5

p.T.a.

1

-----

-----

n 2'

if n is even

Find whether the function f is bijective.

fen)

==

n+l 2 ' n 2'

&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

6 65/1

_

V :

15.

If

,

/'sin -1 x

y =

/1 _ x2

d2y dx2

2

(1 - x ) -

..,..,& '111."

Y

=

soh w that dy - 3x - y = 0 dx

sin -1 x -=== ~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 the probability that a candidate would get four or more correct answers just by guessing ? ~

~-fqctl01l~

~

~

if -qfq ~

~

~) I ~

CflTT

~ ~

it

~

J.l!f~ctlc11~ fen ~

~

cfR ~~ ~

~

~ ~

~ (ftRif 'iffi

it ~ "lIT

31f~

~it~~~?

'-.'

(I7'.>USin~

of determinants, prove the foUowing: 11+

p

2

3 + 2p

1 + 3p + 2q

3

6 + 3p

1 + 6p + 3q

r ". it ~

q)f

11+

18.

m~,

Rkf ~

p

1 + 3p + 2q

3

6 + 3p

1 + 6p + 3q

Solve the following differential

1

:

I

equation

1

=

:

= y _ x tan (y) x~. ~41ctl~UI cit ~

dx x dy

=

1 + p + q

3 + 2p

Rkf ~

I

~

2

x dy dx

65/1

1 + p + q

=

~

:

y _ x tan ( yx )

7

p.T.a.

Solve the following differential equation :

19.

dy

2

cos x -

dx

f.:r8 ~

+ y = tan x

~

o/;-il1lch-lUI

2

QC1~

:

dy

cos x - + y = tan x dx ~Find

the shortest distance between the following two lines : -7 !\ !\ !\ r = (1 + A) i + (2 - A) j + (A + 1) k;

!\!\!\

-7

= (2 i -

r

j -

!\!\!\

k) + !l (2 i +

j

+ 2 k).

-7 !\ !\ !\ r = (1 + A) i + (2 - A) j + (A + 1) k ; -7 !\!\!\ r = (2 j

i-

~:"

-

k) + !l (2

!\!\!\

i

+

j

+ 2 k).

Prove the following: ,~

/"" cot

~=== ~1 - sin x J -ll/ ~1 + sin x -+ -J1OR

2, = x

Solve for x : 2 tan-1 (cos x) = tan-1 (2 cosec x) f.:r8~~~:

2. ~1 + sin x - ~1 - sin x J = x, cot-1 (_~=l=+=s=in=x=-+_.~-=l=-=s=in=x 3l$!fCfT

x~

fu"Q:

QC1~

:

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

65/1

8

X E

4 (0, n)

22.

1\

1\

i +

j

+ k with the unit vector along

5 k and

Ai + 2 j + 3 k is equal to one.

The scalar product of the vector 1\

1\

the sum of vectors 2 i + 4 j Find the value of A. 1\

~

1\

1\

2i + 4j - 5k 1\

1\

i +j

it ~

1\

1\

~

1\ 1\

1\

1\

1\

1\

Ai + 2j + 3k

~

<.jljl\.h~

if ~

cn1 ~

~

1\

+ k

q;r ~

1 ~ I

~UI"\.h~

A q;r +rR ~ ~

q,~ll.--t/

SECTION C

~ ,If-

ll'

Questions number 23 to 29 carry six marks each. J1H if&rT 23 it 29

2~

rfcfi

JRitq;

J1H it 6 ~

~ I

A (3, -1, 2),

the equation of the plane determined by the points B (5, 2, 4)

and

C (-1, -1, 6). Also

find

the

distance

of the point

P (6, 5, 9) from the plane.

~an ~

A (3, -1, 2), B (5, 2, 4) ~

~

J-~

P (6, 5, 9) ctr ~

C (-1, -1, 6) ID\TRmft; w:rm1 q;r ...:1 w:rm1 it ~

~ ~

~

.....

i

I

\

~~area --/ the line

25~~aluate

65/1

of the region included between the parabola;

= x and

x + y = 2.

:

9

p.T.a.

~g

matrices, solve the following system of equations: x+y+z=6 x + 2z = 7 3x + y + z = 12

ofmatrix OR elementary 4-01 01 Inverse the the usmg following 33

AObtain = 12 0

~

'"

cnr

m '8:, R8 ~~Iq,{UI~

it ~ ~

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

3x + y + z = 12 3l~

~

~3TI

ih

3 A =

~oloured table:

12

o

m

IDU

R8 ~

0

-1

3

o

4

1

cnr

~

~

~

balls are distributed in three bags as shown in the following . 24531 124RedColour of the ball White Black

Bag

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

I

6511

10

m

53124 24 ~

~

~CfiTtrT

'Cfil"ffi

~

~

m

<11~~<11 ~

'IT{

I

~

541fl1ctldl

it

(f~ ~

'1flIT

q:<TI ~

fcl)

~.

?J 7R

~I~~~I

(D ~

~

if

~

~

~

-B ~

~

?

~

am

~

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 expectation is 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 can buy, how should he invest his money in order to )llaximise the profit? Formulate this as a linear programming problem and solve it graphically. ~

~

"¥9

W~ ~

5,760 ~. ~ (f~ ~ if (f~ ~

~

22~;

C1fl1 1R

~

~

~m? 29.

~ '"

~

~ fc;rQ:

{9~~III) ~

~ ~)

~

if ~

~

I ~

~

20 ~

3if~

240~.

(f~ ~

{9~C\rjl

~

fc;rQ:

"3it

I

1R ~

urn ~

>rcfiR

311Rr

m

W

31RTT ~

mfR
Lfrn f.:rffi ~ ~

WIT

I ~ fcfi ~ I ~

f.:rffi ~

~OOcn~W1BIT~~w:FiIDU~~

ch9r ~ ~

fcfi

~

360~. W fcl)

c.nT o;Q

"3it 3if~

I

If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is ~

3

/

\/ ;/

A manufacturer

OR

can sell x items at a 100 pnce of Rs. (5 - ~)

The cost price of x items is Rs. (x \5 + 500). items he should sell to earn maximum profit. 65/1

11

Find

the

each.

number

of

p.T.a.

~

~

(~

~~itft;m:~~

65/1

+

500)~. ~ I ~

cn1 ~

I

12

~

~

~

~

dB 31f~

(1N