2912math Max Final Test 2010

http://www.cbseguess.com / Sample Paper - 2010 Subject – Mathematics Class – XII

General Instructions SECTION A: 1. If f: R  → R be given by f ( x) = ( 3 − x 3 )

 cos θ 2. Simplify : cos θ  sin θ − sin

1 3

then find   fof   ( x)

sin θ 

sin θ − cos θ sin sin θ  + sin   cos θ sin θ  cos θ sin

3. . Write the points where the function f ( x) = x + x − 1 is differentiable.

5 − 3

3

4. Find a matrix X such that B – 2A + X = O, where A =  5.

If a

=5 ;

b =13

1

6. Evaluate

∫ 

sin

and

a ×b

=25 , find

a .b

a

are

 and B 1

0 = 3

b

= 3

- 2 1

. 

−1   2x

 dx     1 + x 2  

0

1

7. Evaluate

8. Find

a

( x

∫ 

4

−  x )  x

b

−

4

dx

5

if   tw o vectors

and  b

such

that 

a

= 2,

and 

a . b

9. Find the value of  i.(   j ×k  ) +  j.( i ×k  ) + k .( i × j ) 10. Find a unit vector perpendicular to each of the vectors

a

+b

and  a

−b

a = i +  j + k  and  b = i + 2  j +3k 

SECTION B:

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where

= 4

http://www.cbseguess.com / 11. Let f: N

 R

→

be a function defined as f ( x ) = 4x2 + 12x + 15 .Show that f: N

→S 

 where S is the range of f, is invertible .Find the inverse of f . 12. Solve for x : sin −1 (1 − x) − 2 sin −1 x =

Prove that

tan

−1

 1 + x 2 + 1 − x 2  2 2   1 + x − 1 − x

13. If x , y, z are different and

2

 x x 1 x+

3

2

3

2

3

π 2

.

OR 

 π   1  = + cos 4 2  

−1

x2 ,

−1 2

≤  x ≤ 1

∆  y=  y 1 y+ = 0 t  hs t he 1h+ x on= a0

 z  z 1 z+

1: x ≤ 3   14. If the function f (x ) = ax + b;3 < x < 5 is continuous at x = 3 and x = 5, then find the  7; x ≥ 5 

value of a & b. 15.  If    x 1 + y + y 1 + x

 If    y

=

16. Evaluate:

e

a cos

−1

x

= 0   for  −1