**Sample Paper – 2015**

**Class – XII**

Subject –

Subject –

**Mathematics**

__SECTION A ( 1X10=1M)__
(Questions 1 to 10 carry 1 mark each

**)**
1. Let
* be the binary operation on N given by a *b = HCF of a and b. Find 20*16

2. What
is Sin

^{-1}(Sin 7Ï€/6) ?
3. Find
x and y if =

4. If A is a square matrix of
order 3 and = 64 then find .

6. Find the adj A of .

7.

8. Find the value of Î± so that = Î±i + 2j + k is perpendicular to = 4i – 9j +
2k

9. Find the unit vector in the
direction of if

10. Find k if the lines and
are
perpendicular.

__SECTION B(Q. 11 to 22 carry 4 marks each)__
11. Show
that the relation R on NXN defined by ( a,b) R (c,d) a+d=
b+c is an equivalence relation. (or)

Let f : R R be a function defined by f(x) =
4 + 3x . Show that f is invertible and find the inverse of f.

12. Prove that tan

^{-1}( - )/+) = Ï€/4 – ½ Cos^{-1}x .
13.
Using
properties of determinants Prove that = 4.

14.
Test
the continuity of the following function at x = 0 ,

If x = a ( t +
Sint ) , y = a ( 1 – Cost ) , show that y’’ = 1/a, at t= ( or ) If
x

^{p}y^{q}= ,Prove that y’ = y/x.
15. Find the intervals where the
function f (x) =2x

^{3}– 9x^{2}+ 12x + 30 is a) increasing b) decreasing.
16. Evaluate:

(or)

Evaluate as sum of limits

17. Solve the differential
equation x

^{2}y’ = x^{2}-2 +xy
( or)

Form the differential equation representing
the family of ellipses having foci on x-axis and centre at the origin.

Solve the differential
equation Cos

^{2}x y’ + y = tanx.
i.

18. Three vectors ,
satisfying
the condition ++
= 0 . Evaluate the quantity + + if = 1 , = 4 = 2.

19. Find the shortest
distance between the lines = I +
j + K ( 2i – j + k ) and = ( 2i + j
- k ) + p( 3i -5j+2k).

20. In a factory which
manufactures bolts, machine A, B and C respectively 25%, 35% and 40% of the
bolts, Of their output s 5,4,and 2 percent are respectively defective bolts. A
nolt is drawn random from the product and is found to be defective. What is the
probability that it is manufactured from machine A?

__SECTION C ( Each question carries 6 marks)__
21. Find
the inverse of using elementary transformation. ( or)
if A = find A

^{-1}and hence solve the equations 2x+3y+z= 11, -3x+2y+z=4, 5x-4y-2z = -9
24 .Find the maximum
area of the isosceles triangle inscribed in an ellipse x

^{2}/a^{2}+ y^{2}/b^{2}= 1, whose vertex lies along the major axis. (or) Show that the maximum value of the cylinder which can be inscribed in a sphere of radius 5 is 500Ï€ cm^{3}.
25.Prove that

26. Make a rough sketch of the
region given below and find its area using integration. { (x,y) : 0≤y≤2x+3,}.

27. Find the foot of the
perpendicular and the perpendicular distance of the point (3,2,1) from the
plane 2x-y+z +1=0. Find the image of
the point in the plane.

28. From a lot of 30 bulbs
which includes 6 defective, a sample of 4 bulbs is drawn at random with replacement.
Find the mean and variance of the number of defective bulbs.

29. A furniture firm
manufactures chairs and tables each requiring the use of three machines A,B and
C . Production of the chair requires 2 hrs on machine A, 1 hr on machine B,
and 1 hr on machine C.Each table
requires 1 hr on machine A, 1 hr on
machine B and 3 hrs on machine C. The
profit obtained by selling one chair is
Rs. 30 while by selling one table Rs. 60. The total time available per week on
machine A is 70 hrs, machine B 40 hrs,
and on machine C 90 hrs. How many chairs
and tables should be made per week so as to maximize profit? Formulate the
problem as LPP and solve it graphically.
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