1. M. SRINIVASAN, PGT (MATHS)
SAMPLE PAPER
SOLVED
CLASS XII
MATHEMATICS
PRESENTED BY
M. SRINIVASAN, PGT (MATHS)
ZIET, KVS, MUMBAI

2. General Instructions
All questions are compulsory
The question paper consists of 29 questions divided into three sections A, B and C
Section A contains 10 questions of 1 mark each, Section B is of 12 questions of four marks each and Section C comprises of 7 questions of six marks each
There is no overall choice. However, internal choice has been provided in 4 questions of four marks and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions.
Use of calculators is not permitted

7. (AB – BA)’ = (AB)’ – (BA)’ = B’A’ – A’B’ = BA – AB
5. If A and B are symmetric matrices of the same order, then show that AB – BA is a skew symmetric matrix.
A’ = A & B’ = B
(AB – BA)’ = (AB)’ – (BA)’
= B’A’ – A’B’
= BA – AB
(AB – BA)’ = - (AB – BA)
AB – BA is a skew symmetric matrix.

8. 6.
Put t = x + log (sin x)
= log t
= log [ x + log (sinx)] + C

9. f(-x) = - f (x) f(x) is a odd function
7. Evaluate:
f(x) =
f(-x) = -
f(-x) = - f (x)
f(x) is a odd function

10. Parallel unit vector =

11. For T1  A, T1 is similar to T1. (T1 , T1) R  R is Reflexive
11. Show that the relation R, defined on the set A of all triangles as: R = {(T1 , T2): T1 is similar to T2} is an equivalence relation.
For T1  A, T1 is similar to T1.
(T1 , T1) R  R is Reflexive
For T1 , T2 A, Let T1 is similar to T2.
Then T2 is also similar to T2. Hence (T1 , T2) R
R is symmetric
For T1 , T2 , T3 A, Let T1 is similar to T2 and T2 similar to T3.
Then T1 similar to T3
Hence (T1 , T3) R  R is transitive
As R is reflexive, symmetric and transitive,
R is an equivalence relation

12. Hence f is injective (one-to-one)
11. Let f: N {0}  N {0} defined by
Show that f(x) is a bijective function
Let m and n are even number.
For f(m) = f(n), we have m – 1 = n – 1  m = n
Let m and n are odd numbers
For f(m) = f (n), we have m+1 = n+ 1  m = n
If m is even and n is odd then m  n
Then f(m) is odd and f(n) is even  f(m)  f(n)
Hence f is injective (one-to-one)

13. Hence f is surjective. (ONTO)
If n is odd natural number then there exists an even natural number n+1 such that
f(n+1) = n
If n is even natural number then there exists an even natural number n-1 such that
f(n-1) = n
Hence f is surjective. (ONTO)
Hence f is bijective

14. 12. Evaluate =

15. EXPAND WITH RESPECT TO FIRST COLUMN
13. Using properties of determinants prove that:
= (1 – x3)2
(1 – x3)2 = [(1 – x) (1 + x + x2)]2
EXPAND WITH RESPECT TO FIRST COLUMN
= [(1 – x) (1 + x + x2)]2

16. As f(x) is continuous at x = 1 Using the above equations,
14.If the function f(x) given by:
is continuous at x = 1, find the value of a and b.
As f(x) is continuous at x = 1
LHL = RHL = f(1)
Using LHL, 5a – 2 b = f(1) = 11
using RHL, 3a + b = f(1) = 11
Using the above equations,
a = 3 and b = 2

17. 15. Find
for y = = = = =

18. 15. If y = prove that (x2 + a2) y2 + x y1 = 0 (x2 + a2) y2 + x y1 = 0
Find the first derivative and cross multiplying
Find the second derivate
(x2 + a2) y2 + x y1 = 0

19. h = 3r V = Volume of water in the cone tan  =
16. Water is running in to a conical vessel, 15 cm deep and 5 cm in radius, at the rate of 0.1 cm3/sec. When the water is 6 cm deep, find at what rate is the level of water increasing.
V = Volume of water in the cone
tan  =
h = 3r

20. 17. EVALUATE =
PUT y = x2

21. Put cos x = t and sin x dx = - dt
17. EVALUATE
Multiply and divide by sinx =
Put cos x = t and sin x dx = - dt = =

22. I.F = The given DE is The General Solution is:
18. Solve the differential equation: (tan-1y – x)dy = (1 + y2) dx
The given DE is
I.F =
The General Solution is:

23. x y y’’ + x (y’)2 – y y’ = 0 Differentiating once
19. Form the differential equation of the family of hyperbolas having foci on x-axis and center at origin.
= 1
The equation of family of hyperbolas
Differentiating once
Differentiating again
x y y’’ + x (y’)2 – y y’ = 0

24. P(E) = P(AE or BE) = P(AE) + P(BE)
20. A bag A contains 8 white and 7 black balls while the other bag B contains 5 white and 4 black balls. One ball is randomly picked up from bag A and mixed up with the balls in the bag B. Then a ball is randomly drawn from it. Find the probability the ball drawn is white.
A = event of transferring a white ball
P(A) =
B = event of transferring a black ball
P(B) =
E = event of selecting white ball from II bag
P(E) = P(AE or BE) = P(AE) + P(BE) =

25. 20. Find the mean and variance of the number of heads in a two tosses of a coin.
Let X = Number of heads in two tosses of a coin
X takes the values: 0,1,2 X 1 2
P(X)

26. 21. Find the area of the triangle with vertices A(1 , 1, 2), B(2, 3, 5) and C(1, 5, 3)
AREA OF TRIANGLE ABC =
Area of the triangle =

27. Use perpendicular distance from (0,0,0) = 1 The required planes are
22. Find the equation of the plane passing through the line of intersection of the planes 2x + 6y + 12 = 0 and x – y + 4z = 0 which are at a unit distance from the origin
Required plane : (2x + 6y + 12) +  (3x – y + 4z) = 0
Use perpendicular distance from (0,0,0) = 1
 = 2
The required planes are
2x + y + 2z + 3 = 0 and x – 2y +2 z – 3 = 0

28. 23. Using Matrices, solve the following system of equations:
2x – 3y + 5z = 11; 3x + 2y – 4z = -5 ; x + y – 2z = -3
AX = B
x = 1 ; y = 2 ; z = 3

29. Let x and y are length and breadth of the rectangle
24. A window is in the form of a rectangle surrounded by a semi-circular opening. The total perimeter of window is 10 meters. Find the dimensions of the window so as to admit maximum light through the whole opening.
Let x and y are length and breadth of the rectangle
Area is maximum for

30. SURFACE AREA MINIMUM WHEN x = 2y
24. An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of the material will be least when depth of the tank is half of its width.
For the tank:
length = x ; breadth = x and height = y
x = 2y
SURFACE AREA MINIMUM WHEN x = 2y
DEPTH OF THE TANK IS
HALF OF ITS WIDTH.

31. Put t = tan x MULTIPLY THE NUMERATOR & DENOMINATOR BY 25. Evaluate: dx
USE PROPERTIES OF INTEGRALS
USE PROPERTIES OF INTEGRALS
MULTIPLY THE NUMERATOR & DENOMINATOR BY
Put t = tan x

32. Point of intersection (1,0)
26. Find the area of the region: {(x , y) : x2 + y2 ≤ 1 ≤ x + y}
THE CURVES ARE
x2 + y2 ≤ 1 & x + y  1
CIRCLE & STRAIGHT LINE
Point of intersection (1,0)

33. 26. Compute the area bounded by the lines
x +2y = 2 ; y – x = 1 and 2x + y = 7
(2 , 3)
(4 , -1)
RANGE OF X : 0 TO 4

34. x +2y = 2
y – x = 1
2x + y = 7
AREA =
Area = 6 square units

35. The equation of the line through A and parallel to given line:
27. Find the distance of the point (1, 2, 3) from the plane x – y + z = 5 measured parallel to the line
The equation of the line through A and parallel to given line:
COORDINATES OF P [2K + 1 , 3K – 2 , -6K +3]
P LIES ON x – y + z = 5
The point P is
Distance of AP = 1

36. OBJECTIVE FUNCTION Min Z = 4x + 6y
28. A diet is to contain at least 80 units of vitamin A and units of minerals. Two foods F1 and F2 are available. Food F1 cost Rs.4/- per unit and F2 costs Rs.6/- per unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost of the diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.
x = Number of units of Food F1
y = Number of units of Food F2
Total cost = 4x + 6y
OBJECTIVE FUNCTION Min Z = 4x + 6y

37. CONSTRAINTS ARE ON VITAMINS AND VITAMINS MINUMUM MINERALS: 100 UNITS
TOTAL VITAMINS: 3x + 6y
MINUMUM VITAMINS: 80 UNITS
3x + 6y  80
TOTAL MINERALS: 4x + 3y
MINUMUM MINERALS: 100 UNITS
4x + 3y  100
THE LPP IS
Min Z = 4x + 6y SUBJECT TO
3x + 6y  80
4x + 3y  100
x, y  0

39. MINIMUM COST
Rs.104

40. A = EVENT OF BOLT MANUFACTURED BY MACHINE A
29. In a bolt factory, machines A, B and C manufacture 25%, 35% and 40% of the total. Of their output 5%, 4%, 2% are defective. A bolt is drawn at random from the product. a) What is the probability that the bolt drawn is defective? b) If the bolt drawn is found to be defective, Find the probability that it is a product of machine B? 
I EXPERIMENT: BOLT MANUFACTURED BY THE FACTORIES
A = EVENT OF BOLT MANUFACTURED BY MACHINE A
B = EVENT OF BOLT MANUFACTURED BY MACHINE B
C = EVENT OF BOLT MANUFACTURED BY MACHINE C

41. a) Probability of bolt drawn is defective = P (E)
E = EVENT OF DEFECTIVE DETECTED
a) Probability of bolt drawn is defective = P (E)

42. b) probability that defective is a product of machine B

44. f o f(x) = f[f(x)] = f(3x2-2)
2. If f: RR defined by f (x) = 3x2-2, find f o f
f o f(x) = f[f(x)] = f(3x2-2)
f o f (x) = 27x4 - 36x2 + 10

45. 3. If A and B are two matrices of the same order ,under what conditions
-AB + BA = 0
AB = BA

47. 5k – 6 = 0

48. 6. Evaluate:

49. 7. Evaluate:

51. 9. EVALUATE

53. (a – b) (b – c) (c – a) (a + b + c)
Using Properties of
determinants prove that
(a – b) (b – c) (c – a) (a + b + c)

54. Using Properties of
determinants prove that