Design of Question Paper Mathematics Class XII

Design of Question Paper
Mathematics - Class XII

Time : 3 hours

Weightage of marks over different dimensions of the question paper shall be as follows :

Weightage to different topics/content units


Max. Marks : 100

Topics
Relations and functions
Algebra
Calculus
Vectors & three-dimensional Geometry
Linear programming
Probability

Marks
10
13
44
17
06
10
100

Weightage to different forms of questions

Forms of Questions

Very Short Answer questions (VSA)
Short answer questions (SA)
Long answer questions (LA)
Total

(1)

Class XII
MATHEMATICS
Blue-Print I

(2)

Sample Question Paper - I
MATHEMATICS
Class XII

Time : 3 Hours

General Instructions

All questions are compulsory.

SECTION - A

Which one of the following graphs represent the function of x ? Why ?

1.

y

y

x

x

(a)

(b)

What is the principal value of

2.

2π 
2π 

−1 
cos −1  cos
 + sin  sin
 ?
3 
3 



3.

A matrix A of order 3 × 3 has determinant 5. What is the value of | 3A | ?

4.

For what value of x, the following matrix is singular ?

5 − x
 2


x + 1
4 


Find the point on the curve y = x 2 − 2 x + 3 , where the tangent is parallel to x-axis.

5.

What is the angle between vectors a & b with magnitude √3 and 2 respectively ? Given a . b = 3 .

6.


Cartesian equations of a line AB are.

7.

2x −1 4 − y z + 1
=
=
2
7
2

Write the direction ratios of a line parallel to AB.

8.

Write a value of


(x )dx

3 log x


Write the position vector of a point dividing the line segment joining points A and B with position vectors a & b
externally in the ratio

9.

ˆ j
1 : 4, where a = 2 i + 3ˆ + 4k and b = − iˆ + ˆ + k
j ˆ

3 − 1
 2 1 4


If A = 
 and B = 2 2 
 4 1 5
1 3 



10.

Write the order of AB and BA.

Show that the function f : R → R defined by f (x ) =

11.

inverse of the function f.

OR
Examine which of the following is a binary operation

(i)

SECTION - B

2x −1
, x ∈ R is one-one and onto function. Also find the
3

a+b
, a, b ∈ N
2

a*b=

a+b
, a, b ∈ Q
2

a*b =

(ii)

for binary operation check the commutative and associative property.

Prove that

12.

 63 
5
3
tan −1   = sin −1   + cos −1  
 16 
 13 
5


13.

Using elementary transformations, find the inverse of

2 − 6
1 − 2 



OR

Using properties of determinants, prove that

− bc

b 2 + bc c 2 + bc

a 2 + ac

a 2 + ab b 2 + ab

c 2 + ac = (ab + bc + ca )3

− ac

− ab

Find all the points of discontinuity of the function f defined by

14.

x + 2,
x ≤1
f (x) = x − 2, 1 < x < 2
0,
x≥2

p q
If x y = (x + y )

15.

p +q

, prove that

dy y
=
dx x

OR

 1 + x2 + 1 − x2 
dy
−1
 , 0 < | x | <1
, if y = tan 
Find
dx
 1 + x2 − 1 − x2 



Evaluate

16.

(x + 1)(x
∫ (x + 3)(x

2

2

) dx
− 5)

+4

2

2

17.

A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lower most. Its semi-

−1
vertical angle is tan   . Water is poured into it at a constant rate of 5 cubic meter per minute. Find the rate at

which the level of the water is rising at the instant when the depth of water in the tank is 10m.

Evaluate the following integral as limit of sum

18.

Evaluate

19.

20.

Find the vector equation of the line parallel to the line

1
2

x −1 3 − y z + 1
=
=
and passing through (3, 0, –4). Also
5
2
4

(5)

find the distance between these two lines.

In a regular hexagon ABC DEF, if AB = a and BC = b , then express CD, DE, EF, FA, AC, AD, AE and CE

21.

in terms of a and b .

A football match may be either won, drawn or lost by the host country’s team. So there are three ways of
forecasting the result of any one match, one correct and two incorrect. Find the probability of forecasting at least
three correct results for four matches.
OR
A candidate has to reach the examination centre in time. Probability of him going by bus or scooter or by other

22.

1
1
3 1 3
, , respectively. The probability that he will be late is and respectively, if he
4
10 10 5
3
travels by bus or scooter. But he reaches in time if he uses any other mode of transport. He reached late at the
centre. Find the probability that he travelled by bus.

means of transport are

SECTION - C

23.

Find the matrix P satisfying the matrix equation

 2 1   − 3 2  1 2 
3 2 P  5 − 3 = 2 − 1


 
 

Find all the local maximum values and local minimum values of the function

24.

f (x ) = sin 2 x − x,

−

OR

π
π
<x<
2
2

A given quantity of metal is to be cast into a solid half circular cylinder (i.e., with rectangular base and semicircular
ends). Show that in order that the total surface area may be minimum, the ratio of the length of the cylinder to the

diameter of its circular ends is π : (π + 2 ).

Sketch the graph of

25.

| x − 2 | +2, x ≤ 2
f (x ) =  2
x>2
 x − 2,

What does the value of this integral represent on the graph ?

Evaluate

Solve the following differential equation 1 − x 2

26.

(

) dy − xy = x , given y = 2 when x = 0
dx

(6)

2

27.

Find the foot of the perpendicular from P(1, 2, 3) on the line

x−6 y−7 z−7
=
=
3
2
−2

Also obtain the equation of the plane containing the line and the point (1, 2, 3)

28.

Let X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability
of getting admission in x number of colleges. It is given that

if x = 0 or 1
kx

P(X = x ) = 2kx
if x = 2
,
k (5 − x ) if x = 3 or 4


k is +ve constant

(a) Find the value of k.
(b) What is the probability that you will get admission in exactly two colleges?
(c) Find the mean and variance of the probability distribution.

OR
Two bags A and B contain 4 white 3 black balls and 2 white and 2 black balls respectively. From bag A two balls
are transferred to bag B. Find the probability of drawing
(a) 2 white balls from bag B ?
(b) 2 black balls from bag B ?
(c) 1 white & 1 black ball from bag B ?

29.
A catering agency has two kitchens to prepare food at two places A and B. From these places ‘Mid-day Meal’ is
to be supplied to three different schools situated at P, Q, R. The monthly requirements of the schools are respec-
tively 40, 40 and 50 food packets. A packet contains lunch for 1000 students. Preparing capacity of kitchens A
and B are 60 and 70 packets per month respectively. The transportation cost per packet from the kitchens to
schools is given below :

How many packets from each kitchen should be transported to school so that the cost of transportation is mini-
mum ? Also find the minimum cost.

Posted by: http://educationexampoint.blogspot.in