2912math Max Final Test 2010

8/8/2019 2912math Max Final Test 2010

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Sample Paper - 2010

Subject – Mathematics

Class – XII

General Instructions

SECTION A:

1. If f: R R→ be given by f ( x) = 3

1

3 )3( x− then find )( x fof

2. Simplify : θcos

θθ−

θθ

cossin

sincos

+ θsin

θθ

θ−θ

sincos

cossin

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

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

=

− 13

2-0 Band

13

35.

5. If b.afind ,25baand 13b ;5a =×==

6. Evaluate dxx1x2sin

2

1

0

1

+∫ −

7. Evaluate ∫ −5

4

1

4 )(

x

dx x x

8. Find 4.3,2 ===− baand bathat suchareband avectorstwoif ba

9. Find the value of ).().().( jik k i jk ji ×+×+×

10. Find a unit vector perpendicular to each of the vectors wherebaand ba −+

k jiband k jia 32 ++=++=

SECTION B:

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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 :2

xsin2)x1(sin11 π

=−− −− . OR

Prove that21

22

22

1 cos2

1

411

11tan x

x x

x x −− +=

−−+

−++ π

, 12

1≤≤

− x

13. If x , y, z are different and 010

1

1

1

32

32

32

=+=

+

+

+

=∆ xt h a s h ot h e n

z z z

y y y

x x x

14. If the function

≥

<<+

≤

=

5x;7

5x3;bax

3x:1

)x(f is continuous at x = 3 and x = 5, then find the

value of a & b.

15.( ) 21

111011

xdx

dythat prove x for x y y x If

+

−=≤<−=+++ OR

0)1(11,2

2

22cos

1

=−−−≤≤−=−

yadx

dy x

dx

yd xthat show xe y If

xa

16. Evaluate: dx and hence find : dx , where n is positive

integer.

17.Evaluate:( )( )∫ +− x x x x

dx

cossin2cos2sinOR

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Prove that

∫ +

4

0)tan1log(

π

θ θ d = 2log8

π

18. Solve ( ) dx ydy x y )1(tan21 +=−− OR

Solve: ( 1 + ex/y ) dx + ex/y ( 1 – x/y ) dy = 0

19. Find the intervals in which the function f(x) = sin x - cos x is increasing or decreasing on

[0, π 2 ]. OR

Prove that the curves y 2 = 4ax and x y = c2 cut at right angles if c4 = 32 a4

20.

being isthemof oneeachand cbavectorsthreebecbaLet 5,4,3,, ===

Perpendicular to the sum of the other two find cba ++ OR

If with reference to the right handed system of mutually perpendicular unit vectors i

, j

and

k

, k3 ji2 , ji3

−+=β−=α then express β in the form of β = 1β + 2β , where

1β is parallel to and 2β is perpendicular to . (hots)

21. Find the shortest distance between the lines whose vector equations are

k s j si sr and k t jt it r )12()12()1()23()2()1( +−−++=−+−+−= OR

Find the image of the point P( 6,5,9) on the plane determined by the points

A ( 3,-1,2 ) , B ( 5,2,4 ) and C ( -1,-1,6 )

22. The probability of a shooter hitting a target is 3/4. How many minimum number of times

must he/ she fire so that the probability of hitting the target at least once is more than

0.99?

SECTION C:

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23. Using elementary transformation find the inverse of the matrix

113

321

210

24. Prove that of all the triangles inscribed in a circle ,the equilateral triangle triangle has themaximum area. OR

A water tank has the shape of an inverted right circular cone with its axis vertical and

vertex lowermost .Its semi vertical angle is tan-1 ( 0.5) .Water is poured in to it at a constant

rate of 5 cubic meter per hour .Find the rate at which the level of the water is rising at an

instant when the depth of the water tank is 4m.

25. Prove that ∫ 2

0

)log(sin

π

dx x = ∫ 2

0

)log(cos

π

dx x = 2log2

π − OR

Evaluate limit as a sum4

2

2

2 34

3 2

x x dx− +∫

1

341

9

y

16

x

ellipsethe by boundedregionsmaller theof areatheFind26.

22

=+=+y x

linetheand

27. Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the

number of heads. If she gets 1, 2,3 or 4 she tosses a coin once and notes whether a head or tail

is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2,3 or 4

with the die.

28. A line makes angles δ γ β α and ,, with the diagonals of a cube , prove that

3

4coscoscoscos

2222

=+++ δ γ β α

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29. A manufacturer of patent medicines is preparing a production plan on medicines A and B. There is sufficient raw material available to make 20,000 bottles of A and 40,000

bottles of B, but there are only 45,000 bottles into which either of medicines can be put.Further, it takes 3 hours to prepare enough material to fill 1,000 bottles of A and it takesone hour to prepare enough material to fill 1,000 bottles of B and there are 66 hoursavailable for this operation. The profit is Rs.8 per bottle for A and Rs. 7 per Bottle for B.How should the manufacturer schedule his production in order to maximize his profit.

Paper Submitted By:- AJAY KAKKAR MATHGURU

Email :- [email protected]

Phone No. 9811400749

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2912math Max Final Test 2010