Sample Paper – 2014
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 
.
= 64 then find .
5. 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
= αi + 2j + k is perpendicular to = 4i – 9j + 2k9. Find the unit vector in the direction of 
if
if10. Find k if the lines 
and 
are perpendicular.
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)
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.
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 .
- )/+) = Ï€/4 – ½ Cos-1x . 13. Using properties of determinants Prove that 
= 4
.
= 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.
( 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
+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.
, 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).
= 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
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 = -924 .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.
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. 
