Sample Paper – 2015
Class – XII
Subject – Mathematics
Class – XII
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 α so that
= αi + 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-1x .




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
xp y q =
,Prove
that y’ = y/x.


15. Find the intervals where the
function f (x) =2x3 – 9x2 + 12x + 30 is a) increasing b) decreasing.
16. Evaluate: 

(or)
Evaluate as sum of limits
17. Solve the differential
equation x2y’ = x2-2
+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 Cos2x 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 x2/a2 + y2/b2
= 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€ cm3.

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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