2912math Max Final Test 2010

(1)

Sample Paper - 2010 Sample Paper - 2010 Subject – Mathematics Subject – Mathematics Class – XII Class – XII General Instructions General Instructions SECTION A: SECTION A: 1. If f: R 1. If f: R →→ R R be given be given by by f ( f ( x) =x) = 3 3 1 1 3 3 )) 3 3

(( −− xx then findthen find fof fof (( x x))

2. Simplify : 2. Simplify : ccosos

θ

_      θ θ θ θ − − θ θ θ θ c cooss si sinn s sinin c cooss + + sisinn

θ

_      θ θ θ θ θ θ − − θ θ si sinn c cooss c cooss si sinn

3. . Write the points where

3. . Write the points where the functionthe function f f x ( ( ) x ) = = x x xx+ + −−11is differentiable.is differentiable. 4. Find a matrix X such that B – 2A + X = O, where A =

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

     = =       − − 33 11 2 2 --0 0 B B a andnd 1 1 3 3 3 3 5 5 .. 5. If

5. If aa ==55;; bb ==1313 aanndd aa ××bb ==2525 ,, findfind aa ..bb

6. Evaluate 6. Evaluate dxdx x x 1 1 x x 2 2 sin sin 2 2 1 1 0 0 1 1













+

∫ ∫

−− 7. Evaluate 7. Evaluate

_∫ ∫

−

5 5 4 4 1 1 4 4 )) (( x x d dxx x x x x 8. Find

8. Find _a_a −−_b_b _if_i_f _tw_t_w_o_o _vectors_vectors _a_a _an_a_nd_d _b_b _a_ar_re_e _su_such_ch _that_that _a_a ==22,, _b_b ==33 _an_a_nd_d _a_a .. _b_b ==44

9. Find the value of

9. Find the value of ii..(( j j××k k ))++ j j..(( ii××k k )) ++ k k ..(( ii××jj))

10. Find a unit vector perpendicular to each of the vectors

10. Find a unit vector perpendicular to each of the vectors aa

+

bb aannd d aa

−

bb wherewhere

k k j j ii b b a annd d k k j j ii a a == ++ ++ == ++22 ++33 SECTION B: SECTION B:

(2)

11. Let f: N → R be a function defined as f ( x ) = 4x + 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 x sin 2 ) x 1 ( sin−1

−

−1

=

π

_{. OR} Prove that 1 2 2 2 2 2 1 cos 2 1 4 1 1 1 1 tan x x x x x − − ₌ ₊         − − + − + + π , 1 2 1 ≤ ≤ − x

13. If x , y, z are different and

0

1

0

1

3 2 3 2 3 2

=+

=

+

=∆

t h e n

s h o

t h a

x

z

y

y y

x

x x

14. If the function      ≥ < < + ≤ = 5 x ; 7 5 x 3 ; b ax 3 x : 1 ) x (

f _{is continuous at x = 3 and x = 5, then find the}

value of a & b. 15. ₍ ₎2 1 1 1 1 0 1 1 x dx dy that prove x for x y y x If + − = ≤ < − = + + + _OR 0 ) 1 ( 1 1 , ₂ 2 2 2 cos 1

−

≤

−

=

− a y dx dy x dx y d x that show x e y If a x

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

integer.

(3)

Prove that

_∫

4

₊

0 ) tan 1 log( π θ θ d = ₈ log 2 π

18. Solve

(

tan−1 y

−

x

)

dy

=

(1

+

y2)dx OR Solve: ( 1 + ex/y _{) dx + e}x/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 [0,2π ]. OR

Prove that the curves y 2 _{= 4ax and x y = c}2 _{cut at right angles if c}4_{= 32 a}4

20. being is them of one each and c b a vectors three be c b a Let , , =3, =4, =5

Perpendicular to the sum of the other two find a +b +c OR

If with reference to the right handed system of mutually perpendicular unit vectors i , j and k , α=3i − j, β=2i + j−3k then express β in the form of β = β +₁ β₂ , where

1

β is parallel to and β2 is perpendicular to . (hots)

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

k s j s i s r and k t j t i t r =( 1 − ) +( −2) +(3 −2 ) =( +1) +(2 −1) − (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?

(4)

23. Using elementary transformation find the inverse of the matrix













1

3

2

1

2

1

0

24. Prove that of all the triangles inscribed in a circle ,the equilateral triangle triangle has the maximum 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 = ₂ log 2 π

−

OR

Evaluate limit as a sum

4 2 2 2 3 4 3 2 x x dx − +

∫

1 3 4 1 9 y 16 x ellipse the by bounded region smaller the of area the Find 26. 2 2 = + =

+ _and _the _line x y

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.

(5)

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 takes one hour to prepare enough material to fill 1,000 bottles of B and there are 66 hours available 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]