MATHS QUESTION
XII – (CBSE)
RELATIONS AND FUNCTIONS
Each question carries 1/2 marks
6.
If f(x) = x + 7 and g(x) = x – 7, x R, find (fog) (7).
Let * be a binary operation, defined by a * b = 3a + 4b – 2, find 4 * 5.
If the binary operation * on the set of integers Z, is defined by a * b = a + 3b2 , the find the value of 2 * 4.
Let * be a binary operation on N given by a * b = HCF (a, b), a, b N. Write the value of 22 * 4.
If the binary operation *, defined on Q is defined as a * b = 2a + b – ab, for all a, b Q, find the value of 3 * 4.
If f : R
R be defined by f(x) = (3 – x3 )1/3 , then find fof (x).
7.
If f is an invertible function, defined as f (x)
8.
If f : R
R and g : R
R are given by f (x) = sin x and g(x) = 5x2 , find gof (x).
If f(x) = 27x3 and g (x) = 1/3 , find gof (x).
State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive.
Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is
one-one or not.
If f : R
R is defined by f (x) = 3x + 2, define f (f(x)).
Write fog, if f : R
R and g : R
R are given by f (x) = | x | and g(x) = |5x – 2|.
Write fog, if f : R
R and g : R
R are given by f (x) = 8x3 and g(x) = x1/3 .
Let * be a binary operation on N given by a * b = lcm (a, b) for all a, b N. Find 5 * 7.
The binary operation * : R × R
R is defined as a * b = 2a + b. Find (2 * 3) * 4.
If the bhinary operation * on the set Z of integers is defined by a * b = a + b – 5, the write the identify element for
the operation * in Z.
1.
2.
3.
4.
5.
9.
10.
11.
12.
13.
14.
15.
16.
17.
3x 4
, write f 1 (x).
5
Each question carries 4/6 marks
1.
2.
3.
4.
5.
Show that the relation R defined by (a, b) R (c, d)
a + d = b + c on the set N × N is an equivalence relation.
Show that the relation R defined by R = {(a, b) : a – b is divisible by 3; a, b N} is an equivalence relation.
Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is even}, is an equivalence
relation.
n 1
, if n is odd
2
Let f : N N be defined by f (n)
for all n N. Find whether the function f is bijective.
n
, if n iseven
2
Show that the relation R in the set of real numbers, defined as R = {(a, b) : a b2 } is neither reflexive, nor
6.
symmetric, nor transitive.
Let Z be the set of all integers and R be the relation on Z defined as R = {(a, b) : a, b
by 5}. Prove that R is an equivalence relation.
7.
Let * be a binary operation on Q, defined by a *b
Z and (a – b) is divisible
3ab
. Show that * is commutative as well as associative. Also
5
find its identify, if it exists.
8.
Show that the relation S in the set R of real numbers, defined as S = {(a, b) : a, b
R and a b3} is neither
reflexive, nor symmetric nor transitive.
Page 1
9.
Show that the relation S in the set A {x Z:0 x 12} given by S {(a,b):a, b Z,| a b| is divisible by 4}
is an equivalence relation. Find the set of all elements related to 1.
y 6 1
.
3
[ 5, ) given by f (x) 9x 2 6x 5. Show that f is invertible with f 1 (y)
10.
Consider f : R
11.
13.
Let A = N × N and * be a binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Show that * is
commutative and associative. Also, find the identify element for * on A, if any.
Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. Write the operation table
of the operation*.
Let f : R R be defined as f(x) = 10x + 7. Find the function g:R R such that gof = fog = IR.
14.
A binary operation * on the set {0, 1, 2, 3, 4, 5} is defined as : a * b =
12.
a b, if a b 6
Show that zero
a b 6, if a b 6
is the identify for this operation and each element a of the set is invertible with 6 – a, being the inverse of a.
15.
f 1 (y)
16.
[4, ) given by f (x) x 2 4. Show that f is invertible with the inverse f
Consider f : R
1
of f given by
y 4, where R+ is the set of all non-negative real numbers.
Let A = R – {3} and B = R – {1}. Consider the function f : A
B defined by f (x)
x 2
. Show that f is onex 3
one and onto and hence find f 1.
x 1, if x is odd
, is bijective (both one-one and onto).
x 1, if x is even
17.
Show that f : N
N, given by f (x)
18.
Consider the binary operation * : R R R and o: R R R defined as a * b = |a – b| and a o b = a for all a,
b R. Show that * is commutative but not associative, o is associative but not commutative.
19.
If f (x)
4x 3
2
,x
, show that f o f (x) = x for all x
6x 4
3
2
. What is the inverse of f?
3
INVERSE TRIGONOMETRIC FUNCTIONS
Each question carries 1/2 marks
1.
Show that sin 1(2x 1 x2 ) 2sin 1 x.
2.
Solve for x: tan 1 (x 1) tan 1 (x 1) tan
3.
Prove that 2tan
4.
Prove that sin
5.
Find the principal value of sin
6.
Write the principal value of sin
1
7.
Write the principal value of cos
1
8.
Write the principal value of sin
1
1
11
5
12
13
tan
11
cos
8
1
tan
4
5
1
sin
8
.
31
4
.
7
tan
1
1
63
16
1
.
3
.
5
3
.
2
3
.
2
1
2
cos
1
1
.
2
Page 2
9.
Write the principal value of sec 1 ( 2).
10.
Write the principal value of cot 1 (
11.
Write the principal value of sin
12.
What is the domain of the function sin 1 x?
13.
Evaluate sin
14.
Write the principal value of cos
1
cos
7
.
6
15.
Write the principal value of tan
1
tan
3
.
4
16.
Find the principal value of tan 1( 1).
17.
Using principal value evaluate the following : cos
18.
Solve for x : tan
19.
sin
3
1
1
3).
sin
4
.
5
1
.
2
1
tan 1 x, x 0.
2
1
2sin
Write the principal value of cos 1
2
1
cos
2
3
sin
1
sin
2
.
3
11
x
1 x
20.
Find the principal value of tan
21.
Find the value of cot (tan
1
1
1
.
2
1
3 sec 1 ( 2).
cot
1
).
Each question carries 4/6 marks
1
4
5
sin
1
5
13
1.
Prove that sin
2.
Solve for x : tan 1 3x tan 1 2x
sin
1
16
65
3.
4
Solve for x : 2tan 1 (cos x) tan 1 (2 cosec x)
4.
Prove the following : tan
5.
Solve the following for x : cos
6.
Solve for x : tan
7.
1
1
4
tan
1
2
9
1
x2 1
x2 1
2
1
3
cos 1
2
5
tan
2x
1
x
2
1
2
3
x 1
x 1
tan 1
.
x 2
x 2 4
1
1 x
cos 1
, x (0,1)
Prove the following : tan 1 x
2
1 x
1
1
12
13
Prove the following : cos
9.
Prove that tan 1 (1) tan 1 (2) tan 1 (3)
10.
Prove that tan
1
sin
1
8.
3
5
1
1
1
1
tan 1
tan 1
tan 1
3
5
7
8
sin
4
1
56
65
.
Page 3
2x
1 x2
11.
Prove the following : tan 1 x tan
12.
Prove the following : cos [tan 1 {sin (cot 1 x)}]
13.
Prove that tan
14.
Prove the following : cot
15.
Find the value of tan
16.
Prove that tan
17.
Prove the following : 2tan
18.
Prove that tan
19.
Prove the following :
20.
Prove that sin
21.
Prove the following : cos sin
22.
Prove that : sin
23.
Solve for x : 2tan 1 (sin x) tan 1 (2secx),x
1 sin x
1 sin x
1
x
y
1
tan
1 x
1 x
1
1
2
1
8
17
1
1
9
8
63
65
1
5
tan
sin
1
1
1
7
1
1
8
1
1
3
1
3
5
1
x
,x
2
1
cos 1 x,
2
4
tanh
9
sin
4
sin
1 sin x
1 sin x
1
2
1
1
3x x3
1 3x 2
1
0,
4
x y
.
x y
1
1 x
1 x
tan
tan
1 x2
2 x2
1
a
2b
tan
cos 1
.
4 2
b
a
1
a
cos 1
2
b
4
1
1
x 1
31
.
17
.
1
2 2
3
36
.
85
cos 1
3
3
cot 1
5
2
5
13
tan
4
9
sin
4
1
2
cos
6
5 13
3
5
1
2
MATRICES
Each question carries 1/2 marks
x 3
4
y 4 x y
5 4
, find x and y.
3 9
1.
If
2.
If matrix A = [1 2 3], write AA’, where A’ is the transpose of matrix A.
3.
Find the value of x, if
4.
If
5.
y 2x 5
x
3
If A = (aij )
3x y
2y x
y
3
1 2
.
5 3
7 5
, find the value of y.
2 3
2 3
1 4
0 7
5
9 and B (bij )
2
2 1
3 4
1 5
1
4 , then find a22 b21.
2
Page 4
1 2 3 1
3 4 2 5
7 11
, then write the value of k.
k 23
6.
If
7.
If A
8.
Write a square matrix of order 2, which is both symmetric and skew symmetric.
9.
From the matrix equation, find the value of x :
10.
If
11.
If a matrix has 5 elements, write all the possible orders it can have.
12.
x y z
Write the values of x – y + z from the following equations :
x z
y z
13.
1
Write the order of the product matrix 2 [2 3 4]
3
14.
If
15.
Simplify : cos
16.
Find the value of x + y from the following equations : 2
cos
sin
sin
cos
3 4 x
2 x 1
2 3
5 7
T
then for what value of
1
2
3
4
3 4
.
5 6
cos
sin
sin
cos
sin
sin
cos
cos
sin
.
x
5
7 y 3
3
1
4
2
7 6
15 14
1 2 1
, then find AT BT .
1 2 3
If A
18.
If A is a square matrix such that A2
19.
If x
1
1
9
5
7
4 6
, write the value of x.
9 x
3 4
1 4 and B
0 1
y
x y 4
5 3y
19
, find the value of x.
15
17.
2
3
is A an identify matrix?
A, then write the value of (I A)2 3A.
10
, write the value of x.
5
Each question carries 4/6 marks
1.
Let A
3 2 5
4 1 3 . Express A as a sum of two matrices such that one is symmetric and the other is skew0 6 7
symmetric.
2.
3.
If A
1 2 2
2 1 2 , verify that A2 4A 5I 0.
2 2 1
Using elementary transformations, find the inverse of the following matrix
1
2
2
2
5
4
3
7 .
5
Page 5
3 0
2 3
0 4
1
0
1
4.
Obtain the inverse of the following matrix, using elementary operations : A
5.
Using elementary row operations, find the inverse of the following matrix :
6.
Express the following matrix as the sum of a symmetric and skew symmetric matrix, and verify your result:
3
3
1
2
2
1
2 5
1 3
4
5
2
1
4 ,B [ 1 2 1]
3
7.
For the following matrices A and B, verify that (AB)' B'A'. A
8.
Using elementary transformations, find the inverse of the following matrix : A
9.
10.
Using elementary transformations, find the inverse of the matrix
1 3
3 0
2 1
Using elementary transformations, find the inverse of the matrix
1 1 2
1 2 3 .
3 1 1
6 5
.
5 4
2
1
0
DETERMINANTS
Each question carries 1/2 marks
a ib c id
.
c id a ib
1.
Evaluate
2.
2
Find the cofactor of a12 in the following : 6
1
3.
Evaluate
4.
If
5.
2 3
4
8 .
Write the value of the determinant 5 6
6x 9x 12x
6.
Find the value of x, from the following :
7.
If A
sin30
sin60
2x 5 3
5x 2 9
3
0
5
5
4 .
7
cos30
.
cos60
0, find x.
x 4
2 2x
0
1 2
, then find the value of k if | 2A | = k | A |.
4 2
Page 6
8.
0 2 0
What is the value of the determinant 2 3 4 ?
4 5 6
9.
2
Find the minor of the element of second row and third column (a 23 ) in the following determinant : 6
1
10.
If A is a square matrix of order 3 and | 3A | = k | A |, then write the value of k.
11.
What positive value of x makes the following pair of determinants equal?
12.
Write the adjoint of the following matrix :
2
4
13.
What is the value of the following determinant?
14.
For what value of x, the matrix
15.
Write A 1 for A
16.
Evaluate
17.
If A
18.
If
2
5
x x
1 x
cos15
sin75
3 5
0 4
5 7
2x 3 16 3
,
5 x 5 2
1
.
3
4 a b c
4 b c a
4 c a b
5 x x 1
is singular?
2
4
2 5
.
1 3
sin15
.
cos75
3
, write A 1 in terms of A.
2
3 4
, write the positive value of x.
1 2
5 3 8
2 0 1 , write the minor of the element a 23.
1 2 3
19.
If
20.
Let A be a square matrix of order 3 × 3. Write the value of |2A|, where |A| = 4.
21.
102 18 36
3 4
Write the value of the determinant 1
17 3 6
Each question carries 4/6 marks
1.
a b c
2a
2a
2b
b c a
2b
Using properties of determinants, prove that :
2c
2c
c a b
2.
b c a b
Using properties of determinants, show that : c a c a
a b b c
(a b c)3 .
(a b c) (a c)2
Page 7
3.
x
y
z
2
2
Prove the following, using properties of determinants : x
y
z2
y z z x x y
(x y)(y z)(z x)(x y z).
a2
4.
bc
c2 ac
Using properties of determinants, prove the following : a 2 ab
b2
ca
2
ab
b bc
c2
5.
Using matrices, solve the following system of equations : x + 2y + z = 7; x + 3z = 11; 2x – 3y = 1.
6.
1 1 1
Using properties of determinants, prove that : a b c
a3 b3 c3
7.
Show that the matrix A
8.
a b b c c a
Prove using the properties of determinants : b c c a a b
c a a b b c
9.
a 2 2a 2a 1 1
Using properties of determinants, prove that : 2a 1 a 2 1 (a 1)3.
3
3
1
10.
a
b
c
2
2
Using properties of determinants, prove that : a
b
c2
b c c a a b
11.
x x 2 1 x3
If x y z and y y2 1 y3
z z2 1 z3
12.
3x 8
3
3
3x 8
3
Solve for x 3
3
3
3x 8
13.
1 a 2 bc a3
Using properties of determinants, prove the following : 1 b2 ca b3
1 c2 ab c3
14.
1 a 2 b2
2ab
2b
2
2
2ab
1 a b
2a
Using properties of determinants, prove the following :
2b
2a
1 a 2 b2
15.
Using matrices, solve the following system of equations : 2x 3y 5z 11; 3x 2y 4z
(a b)(b c)(c a)(a b c)
3 1
satisfies the equation A2 5A 7I O. Hence find A 1.
1 2
a b c
2b c a.
c a b
(a b c)(a b)(b c)(c a)
0, then show that xyz = 1.
0.
a
16.
4a 2b2c2
b
(a b)(b c)(c a)(a 2 b2 c2 ).
(1 a 2 b2 )3.
5; x y 2z
3
c
Using properties of determinants, prove the following : a b b c c a
b c c a a b
a 3 b3 c3 3abc
Page 8
17.
a 2 1 ab
ac
2
Using properties of determinants, show that ab b 1 bc
1 a 2 b2 c2
ca
cb c2 1
18.
1 1 p
1 p q
Using properties of determinants, prove the following : 2 3 2p 4 3p 2q 1.
3 6 3p 10 6p 3q
19.
x y
x x
Using properties of determinants, prove that 5x 4y 4x 2x
10x 8y 8x 3x
20.
1 x x2
Using properties of determinants, prove that x2 1 x
x x2 1
21.
a bx c dx p qx
Using properties of determinants, prove that ax b cx d px q
u
v
w
22.
(b c)2
ab
ca
2
Using properties of determinants, prove that
ab
(a c)
bc
ac
bc
(a b)2
23.
x x2 1 px3
y y2 1 py3
z z2 1 px3
x3.
(1 x3 )2 .
a c p
(1 x ) b d q .
u v w
2
2 abc (a b c)3.
(1 pxyz)(x y)(y z)(z x), where p is any scalar.
24.
Using matrices, solve the following system of equations : x 2y 3z
25.
a b c
If a, b, c are positive and unequal, show that the value of determinant b c a is negative.
c a b
26.
If A
2
3
1
3
2
1
3x 2y 4z
4; 2x 3y 2z 2; 3x 3y 4z 11
5
4 , find A 1. Using A 1 solve the following system of equations : 2x 3y 5z 16;
2
4; x y 2z
3
27.
a bx2 c dx2 p qx2
Prove the following, using properties of determinants : ax2 b cx2 d px2 q
u
v
w
28.
x 4 2x
2x
Prove that 2x x 4 2x
2x
2x x 4
(x
4
b d q
1) a c p
u v w
(5x 4)(4 x)2 .
2
29.
Show that
2
4
2 2 2
2
Page 9
2
x
3 10
4
4;
y z
x
6 5
6
1;
y z
x
9 20
2.
y z
30.
Solve for x, y, z
31.
x 2 2x 3 3x 4
Using properties of determinants, solve the following for x : x 4 2x 9 3x 16
x 8 2x 27 3x 64
32.
x a
x
x
Using properties of determinants, solve the following for x : x
x a
x
x
x
x a
33.
a b 2c
a
b
Using properties of determinants, prove that
c
b c 2a
b
c
a
c a 2a
34.
y k
y
y
Prove using properties of determinants y
y k
y
y
y
y k
35.
Use product
1
0
3
1
2
2
2
3
4
2 0
9 2
6 1
1
3
2
0
0
2(a b c)3.
k 2 (3y k)
to solve the system of equations x y 2z 1;
2y 3z 1;
3x 2y 4z 2
36.
1 a 1
1
Prove without expanding that 1 1 b 1
1
1 1 c
abc ab bc ca abc 1
1
a
1 1
1
or 1
b c
a
1 1
is a
b c
factor of determinant.
37.
1 1 1
Using properties of determinants, prove that : a b c
a3 b3 c3
38.
b c c a a b
Prove that q r r p p q
y z z x x y
39.
Using matrices solve the following system of linear equations :
(a b)(b c)(c a)(a b c)
a b c
2p q r.
x y z
x y 2z 7;
3x 4y 5z
5,
2x y 3z 12
40.
b c a
a
c a
b
Using properties of determinants prove that b
c
c a b
41.
a
a b
a b c
Prove that : 2a 3a 2b 4a 3b 2c
3a 6a 3b 10a 6b 3c
42.
a
b
c
2
2
b
c2
Using properties of determinants prove that a
b c c a a b
4abc.
a3.
(a b c)(a b)(b c)(c a)
Page 10
43.
If A
3
15
5
1
1
6
2
1
5 and B
2
1
1
0
2
3
2
2
0 , find (AB) 1.
1
CONTINUITY AND DIFFERENTIABILITY
Each question carries 4/6 marks
kx2 , x 1
is continuous at x = 1.
4, x 1
1.
Find the value of k, if the function f (x)
2.
Find the derivative of the following functions w.r.t. x : (sin x)tan x
3.
If y = sin (log x), prove that x2
d2 y
dy
x
y 0
2
dx
dx
x2 25
, if x 5
is continuous at x = 5
x 5
k,
if x 5
4.
Find k, so that the function f (x)
5.
If y Aemx
6.
If y log
7.
For what value of k is the following function continuous at x = 2?
f (x)
Benx , prove that
d2 y
dy
(m n)
mny 0
2
dx
dx
1 cos2x
dy
, show that
2 cosec 2x.
1 cos x
dx
2x 1 ; if x 2
k
; if x 2
3x 1 ; if x 2
8.
Find the derivative of the following function w.r.t. x : y tan
9.
Find the derivative of the following function w.r.t. x : y
10.
Find the derivative of the following function w.r.t. x : y tan
11.
(cos x)secx .
1
1 x
1 x
1 x
.
1 x
1
x
x2 1 log
1
Determine the values of a, b and c for which the function f (x)
1 sin x
1 sin x
1
1
.
x2
1 sin x
.
1 sin x
sin(a 1)x sin x
, if x 0
x
c
, if x 0 may be
x bx2
b x3
x
, if x 0
continuous at x = 0.
12.
Find derivative of the following function w.r.t. x : y sin
13.
If x = a (cos
1
5x 12 1 x 2
13
dy
logtan ) and y = a sin θ, find the value of
at
2
dx
4
.
Page 11
14.
1 sin3 x
3cos2 x
, if
x
a
, if
x
b(1 sin x)
, if
( 2x)2
x
Let f (x)
2
2
. If f(x) be a continuous function at x
2
, find a and b.
2
15.
Find the derivative of the following function w.r.t. x : y x x
16.
If y [log (x
x2 1)]2 , show that (1 x 2 )
(sin x)x .
d2 y
dy
x
2 0.
2
dx
dx
x4 2x3 x2
tan 1 x
0
17.
Discuss the continuity of the following function at x = 0: f (x)
18.
Find the derivative of following function w.r.t. x : y
19.
Verify Lagrange’s mean value theorem for the function functions : f (x) x 2
20.
Find the derivative of the following function w.r.t. x : (x 2
, x 0
.
, x 0
secx 1
.
secx 1
y2 ) 2
2x 3 in [4, 6]
xy.
APPLICATION OF DERIVATIVES
Each question carries 4/6 marks
1.
2.
3.
Prove that curve
x
a
n
y
b
n
2 touches the straight line
x
a
y
2 at (a, b) for all values of n N at the
b
point (a, b).
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 the least when the depth of the tank is half of its
width.
Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and semi-vertical angle
30 is
4 3
h.
81
4.
Find the intervals in which function f(x) is increasing or descreasing : f (x) x3 12x 2 36x 17.
5.
A wire of length 28 metres is to be cut into two pieces. One of the pieces is to be made into a circle and the other
into a square. What should be the length of the two pieces so that the combined area of the square and the circle is
minimum?
Of all the rectangle each of which has perimeter 40 metres, find one which has maximum area. Find the area also.
Show that the right circular cyclinder of given volume open at the top has minimum total surface area, provided
its height is equal to radius of its base.
6.
7.
8.
Find the equation of tangent to the curve x = sin 3t, y = cos 2t, at t
9.
Show that the height of the right circular cylinder of maximum volume that can be inscribed in a given right
circular cone of height h is
10.
.
h
.
3
Show that the semi-vertical angle of the right circular cone of maximum volume and given slant height, is
tan
11.
4
1
2.
Find the interval(s) in which following function f(x) is increasing or decreasing : f (x) 2x3 9x 2 12x 15.
Page 12
12.
At what points will the tangent to the curve y 2x3 15x2 36x 21 be parallel to the x-axis? Also find the
13.
equations of the tangents to the curve at these points.
A point on the hypotenuse of a right angled triangle is at distance a and b from the sides. Show that the length of
the hypotenuse is at least (a 2/3 b2/3 )3/2 .
14.
Show that the curves x y2 and xy = k cut at right angles, if 8k2 1.
15.
Find the greatest area of isosceles triangle that can be inscribed in a given ellipse
16.
coinciding with one extremity of the major axis.
Show that the semi-vertical angle of a right circular cone of given surface area and maximum volume is
sin
1
x2
a2
y2
1 with its vertex
b2
1
.
3
1
,x 0
x3
17.
Find the interval(s) in which function f(x) is increasing or decreasing : f (x) x3
18.
Find the equation of the tangent to the curve y
19.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r.
A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2m and
20.
3x 2 which is parallel to the line 4x 2y 5 0.
volume is 8m3. If building of tank costs Rs. 70 per sq metre for the base and Rs. 45 per sq metre for sides, what
is the cost of least expensive tank?
INTEGRALS
Each question carries 1/2 marks
1.
Evaluate
x2
dx.
1 x3
2.
Evaluate
2cosx
dx.
3sin2 x
3.
Evaluate
2x dx
.
0 1 x2
4.
Evaluate
x 2 4x
dx.
x3 6x 2 5
5.
Evaluate
x cos6x
dx.
3x2 sin6x
6.
Evaluate
sec2 x
dx.
3 tan x
7.
Evaluate
sec2 (7 x)dx
8.
If
9.
Evaluate
sin x
dx
x
10.
Evaluate
sec2 x
dx
x
11.
Evaluate
dx.
12.
Evaluate
1
dx
x xlog x
13.
Evaluate
eax e
eax e
dx
14.
Evaluate
15.
Evaluate
log x
dx
x
16.
Evaluate
17.
Evaluate
sec2 (7 4x)dx.
18.
If (ax b)2 dx f (x) c, find f(x).
19.
Evaluate
20.
Evaluate sec x (sec x tan x) dx.
1
1
2
0
1
0
1
1 x
2
ax
ax
dx
.
1 x2
1
0
(3x 2 2x k)dx 0, find the value of k.
1
dx
2x 3
1
0
/2
/2
sin5 x dx.
Page 13
Each question carries 4/6 marks
1.
Evaluate
3.
Evaluate
5.
Evaluate
7.
Evaluate
9.
Prove that
10.
Prove that
12.
Evaluate
14.
Evaluate
16.
Prove that
18.
Evaluate
20.
Evaluate
3
(2x 2 3x 5) dx.
2.
Evaluate
sin x
dx
(1 cosx)(2 cosx)
4.
Evaluate
cos(x a)
dx
sin(x b)
6.
Evaluate
8.
Evaluate
0
1/ 2
0
a
0
(sin 1 x)
dx
(1 x2 )3/2
f (x)dx
/2
0
a
0
f (a x)dx. Using it, evaluate
sin x
dx
sin x cosx
4
.
x2 cot 1 x dx
1
0
cot 1 (1 x x 2 )dx.
a
0
/2
0
sin
1
x
a x
dx
a
(
2
2).
log (sin x) dx.
2
0
1
0
(3x 2 2x 1)dx.
x
4
x
dx
x2 1
x2 4
dx.
x 4 16
0
x tan x
dx.
secxcosecx
x 2 xdx.
11.
Evaluate
13.
Evaluate
15.
Prove that
17.
Evaluate
19.
Evaluate
1 sin x
dx.
1 sin x
1
dx
2
2
a sin x b2 cos2 x
tan
a
a
1
0
1
a x
dx a .
a x
log(1 x)
dx.
1 x2
ex
5 4ex e2x
dx.
(x 4)ex
dx.
(x 2)3
APPLICATION OF INTEGRALS
Each question carries 4/6 marks
1.
Find the area of the region enclosed between the two circles : x 2
y2 1, (x 1)2 y2 1.
2.
Using integration, find the area of the circle x 2
3.
Find the area of the region included between the parabola y
4.
Using integration, find the area of the region bounded by the parabola y2
5.
Find the area bounded by the lines x 2y 2, y x 1 and 2x y 7.
6.
Prove that the curves y2
y2 16 which is exterior to the parabola y2 6x.
3 2
x and the line 3x 2y 12 0.
4
4x and the circle 4x2 4y2 9.
4x and x 2 4y divide the area of the square bounded by x = 0, x = 4, y = 4 and y = 0
into three equal parts.
7.
Find the area of the region lying between the parabolas y2
4ax and x 2
8.
Find the area of the region included between the parabola y2
9.
Using integration, find the area of the region : {(x, y) :9x 2
10.
Using integration find the area of the region : {(x, y) : 25x 2 9y2
11.
Using integration find the area of the following region : (x, y) :
4ay, where a > 0.
x and the line x + y = 2.
y2 36 and 3x y 6}.
x2
9
225 and 5x 3y 15}.
y2
x
1
4
3
y
2
Page 14
5 x2
12.
Using integration find the area of the following region : (x, y):| x 1| y
13.
Compute, using integration, the area of the region bounded by the line y 4x 5, y 5 x and 4y x 5.
14.
Find the area of the circle 4x 2
15.
Using integration, find the area of the following region : {(x,y):| x 2| y
16.
Sketch the graph of y = |x + 3| and evaluate the area under the curve y = |x + 3| above x-axis and between x = - 6
to x = 0.
Using the method of integration find the area of the region bounded by the lines 2x y 4; 3x 2y 6 and
17.
4y2 9 which is interior of the parabola x 2
4y.
20 x2 }.
x 3y 5 0
18.
Using the method of integration, find the area of the region bounded by the lines 3x 2y 1 0, 2x 3y 21 0
and x 5y 9 0.
19.
Find the area of the region {(x, y) : x 2
20.
Using integration find the area of the region in the first quadrant enclosed by the x-axis, line x
circle x 2
21.
y2
y2 4,x y 2}.
3y and the
4.
Using the method of integration, find the area of the
ABC, coordinates of whose vertices are A(2, 0), B(4, 5)
and C(6, 3).
DIFFERENTIAL EQUATIONS
Each question carries 1/2 marks
1.
What is the degree of the following differential equation?
5x
dy
dx
2
d2 y
6y log x
dx2
Each question carries 4/6 marks
dy
dx
x . ex logx ex
.
xcos y
1.
Solve the differential equation
2.
Form the differential equation representing the family of curves y = A cos (x + B), where A and B are constants.
3.
Solve the differential equation x2
4.
5.
dy
2xy y2 .
dx
dy
Solve the differential equation cosx
y sin x, given that y = 2 when x = 0.
dx
Solve the differential equation (3xy y2 )dx (x 2 xy)dy 0.
6.
Form the differential equation representing the family of parabolas having vertex at origin and axis along positive
direction of x-axis.
7.
Solve the differential equation (1 x2 )
8.
Solve the differential equation
9.
dy
y tan 1 x.
dx
dy
y cosx sin x
dx
dy y
y
cosec
0; y 0 when x = 1.
Solve the differential equation
dx x
x
dy
dx
y x tan
y
.
x
10.
Solve the differential equation x
11.
Form the differential equation of the family of circles touching the y-axis at origin.
Page 15
12.
Form the differential equation representing the family of curves given by (x a)2
2y2 a 2 , where a is an
arbitrary constant.
14.
dy
(x 2) (y 2), find the solution curve passing through the point (1, -1).
dx
Solve the differential equation (x3 y3 )dy x 2 y dx 0.
15.
Solve the differential equation
16.
Solve the differential equation xlogx
13.
17.
18.
19.
20.
For the differential equation xy
dy
dx
ytan x, given that y = 1 when x = 0.
dy
2
y
logx.
dx
x
dy
2
Solve the differential equation (x2 1)
2xy 2 .
dx
x 1
dy
Show that the differential equation (x y)
x 2y, is homogeneous and solve it.
dx
dx
Solve the differential equation (3x2 y)
x, x 0, when x = 1, y = 1.
dy
Show that the following differential equation is homogeneous, and then solve it : y dx xlog
y
dy 2x dy 0.
x
VECTOR ALGEBRA
Each question carries 1/2 marks
1.
Find a unit vector in the direction of a 3i 2j 6k.
2.
Find the angle between the vectors a i j k and b i
3.
For what value of λ are the vectors a 2i
4.
If P (1, 5, 4) and Q (4, 1, - 2), find the direction ratios of PQ .
5.
If a i 2j k and b 3i
6.
If | a |
7.
If | a | 2, | b|
8.
If a i 2j 3k and b 2i 4j 9k, find a unit vector parallel to a b.
9.
If | a |
10.
Find a vector in the direction of a i 2j whose magnitude is 7.
11.
Find the projection of a on b if a . b 8 and b 2i 6j 3k.
12.
Write a unit vector in the direction of a 2i 6j 3k.
13.
Write the value of p for which a 3i 2j 9k and b i pj 3k are parallel vectors.
14.
Find the angle between two vectors a and b with magnitudes 1 and 2 respectively and when | a b |
15.
Find the value of p if (2i 6j 27k) (i 3j pk) 0.
16.
Write the direction cosines of a line equally inclined to the three coordinate axes.
17.
If p is a unit vector and (x p) (x p) 80, then find | x |.
18.
Write a vector of magnitude 15 units in the direction of vector i 2j 2k .
j k.
j k and b i 2j 3k perpendicular to each other?
j 5k, find a unit vector in the direction of a b.
3, | b| 2 and a . b 3, find the angle between a and b.
3 and a.b
3, find the angle between a and b.
3, | b| 2 and angle between a and b is 60º, find a.b.
3.
Page 16
19.
What is the cosine of the angle which the vector
2i j k makes with y-axis?
20.
Write a vector of magnitude 9 units in the direction of vector 2i j 2k .
ANSWERS
1.
6.
3 2 6
i
j
k
7 7 7
7.
6
1
3
2.
3.
i 2j 2k
3 3 3
8.
3
12.
2 6 3
i
j
k
7 7 7
13.
17.
6i 3j 6k
18.
5
2
2
3
14.
3
19.
9.
3
4
4.
3, 4, 6
5.
3
10.
7(i 2j)
5
15.
5i 10j 10k
20.
2
i
21
1
j
21
11.
8
7
16.
cos
4
k
21
1
2
3
Each question carries 4/6 marks
1.
Find the projection of b c on a, where a 2i 2j k, b i 2j 2k and c 2i j 4k.
2.
If a = i + j + k and b = j k, find a vector c such that a c b and a. b 3.
3.
If a b c 0 and | a | 3, | b| 5 and | c | 7, show that the angle between a and b is 60º.
4.
Show that the area of the parallelogram having diagonals (3i
5.
If i j k, 2i 5j, 3i 2j 3k and i 6j k are the position vectors of the points A, B, C and D, find the angle
j 2k) and (i 3j 4k) is 5 3 sq units.
between AB and CD. Deduce that AB and CD are collinear.
6.
The scalar product of the vector i j k with the unit vector along the sum of vectors 2i 4j 5k and
i 2j 3k is equal to one. Find the value of λ.
7.
Define the scalar and vector product of two vectors a and b. If for three non-zero vectors a , b and c ; a b =
8.
a c and a b a c, then show that b c.
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are
(2a b) and (a 3b) respectively, externally in the ratio 1:2. Also, show that P is the mid-point of the line
segment RQ.
9.
If a i
j k, b 4i 2j 3k and c i 2j k, find a vector of magnitude 6 units which is parallel to the
vector 2a b 3c.
10.
Let a i 4j 2k, b 3i 2j 7k and c 2i j 4k. Find a vector d which is perpendicular to both a and b
and c. d 18.
11.
The scalar product of the vector i 2j 3k and i 4j 5k is equal to one. Find the value of λ.
12.
Find a unit vector perpendicular to each of the vectors a b and a b, where a 3i 2j 2k and
b i 2j 2k.
13.
If the vectors a and b are such that | a | = 2, | b | = 1 and a b 1, then find the value of (3a 5b) . (2a 7b).
14.
Using vectors, find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).
15.
If Vectors a 2i 2j 3k, b
i 2j k and c 3i j are such that a
b is perpendicular to c , then find the
value of λ.
Page 17
16.
If a, b, c are three vectors such that | a | 5, | b | 12 and | c| 13, and a b c 0, find the value of
a b b c c a.
17.
Let a i 4j 2k, b 3i 2j 7k and c 2i j 4k. Find a vector p which is perpendicular to both a and b
and p c = 18.
18.
If
3i 4j 5k and
is perpendicular to
19.
If
20.
2i j 3k then express
sin
1
|a b|
2
2
where
1
2,
2
where
1
is parallel to
and
2
in the form
1
1
is parallel to
and
2
is
.
If a & b are unit vectors inclined at an angle
(a)
in the form
.
3i j and
perpendicular to
2i j 4k, then express
(b) cos
2
then prove that.
1
|a b|
2
(c) tan
| a b|
2 | a b|
21.
| a b| 60 | a b|
40 | b| 46 then show that | a | 22
22.
Show that the points A, B, C with position vectors 2i j j, i 3j 5k and 3i 4j 4k are the vertices of a right
angled triangle.
23.
Dot product of a vector with (i j 3k) , (i 3j 2k), (2i j 4k)
are 0, 5, 8 respectively then show that this
vector is (i 2j k) .
24.
Show that the area of triangle whose vertices are A (3, -1, 2), B(1, -1, -3), C(4, -3, 1) is
25.
Show that the area of parallelogram having diagonals (3i
165
2
j 2k) and (i 3.j 4k) is 5 3
ANSWERS
2
9.
2i 4j 4k
13.
0
18.
2i j 4k
2.
14.
5 2 2
i
j
k
3 3 3
c
1.
10.
3i
3
j 2k
2
3 4
i
j k
5 5
3.
60
5 3
4.
d 64i 2j 28k
15.
8
5.
11.
16.
13 9
i
j 3k
5 5
169
6.
2
7.
12.
2a b
8.
2 2 1
i
j k
3 3 3
(64i 2j 28k)
17.
19.
1
2i j 3k
3 1
i
j
2 2
1 3
i
j 3k
2 2
THREE DIMENSIONAL GEOMETRY
Each question carries 1/2 marks
1.
Find the direction cosines of the line passing through the following points : (-2, 4, -5), (1, 2, 3).
2.
Write the vector equation of the following line :
3.
Write the position vector of the midpoint of the vector joining the points P(2, 3, 4) and Q(4, 1, -2).
4.
Write the Cartesian equation of the following line given in vector form : r 2i j 4k
5.
Write the intercept cut off by the plane 2x y z 5 on x-axis.
6.
What are the direction cosines of a line which makes equal angles with the coordinate axes?
x 5
3
y 4
7
6 z
2
(i j k).
Page 18
7.
8.
9.
10.
If a line has direction ratios 2, -1, -2, then what are its direction cosines?
Find the distance of the plane 3x 4y 12z 3 from the origin.
Write the direction cosines of a line parallel to z-axis.
The x-coordinate of a point on the line joining the points Q(2, 2, 1) and R(5, 1, -2) is 4. Find its z-coordinate.
ANSWERS
1.
4.
3
2 8
,
,
77 77 77
x 2 y 1 z 4
1
1
1
2.
r (5i 4j 6k)
5.
5
2
1
,
3
6.
(3i 7j 2k)
1
,
3
7.
3
3i 2j k
3.
2 1 2
, ,
3 3 3
3
13
8.
001
9.
10.
-1
Each question carries 4/6 marks
1.
Find the equation of the plane which is perpendicular to the plane 5x 3y 6z 8 0 and which contains the
line of intersection of the planes x 2y 3z 4 0 and 2x y z 5 0.
x 3
1
y 5
2
z 7
x 1 y 1 z 1
and
1
7
6
1
2.
Find the shortest distance between the following lines :
3.
Find the point on the line
4.
Find the equation of the plane passing through the point (-1, -1, 2) and perpendicular to each of the planes
2x 3y 3z 2 and 5x 4y z 6.
5.
Find the length and the foot of the perpendicular drawn from the point (2, -1, 5) to the line
6.
From the point P(1, 2, 4) a perpendicular is drawn on the plane 2x y 2z 3 0. Find the equation, the length
x 2
3
y 1 z 3
at a distance 3 2 from the point (1, 2, 3).
2
2
x 11 y 2
10
4
z 8
.
11
and the coordinates of the foot of the perpendicular.
7.
Find the shortest distance between the line r i 2j 3k
8.
Find the distance of the point (1, -2, 3) from the plane x – y + z = 5 measured parallel to the line
9.
Find the value of λ so that the lines
1 x
3
7y 14
2
(2i 3j 4k) and r 2i 4j 5k
5z 10
7 7x
and
11
3
(3i 4j 5k).
x
2
y
3
z
.
6
y 5 6 z
are perpendicular to
1
5
each other.
10.
Find
the
shortest
r (2i j k)
distance
between
the
following
lines
:
r (1
)i (2
)j (
1)k
and
(2i j 2k).
x 1 3y 5 3 z
and the plane 10x 2y 11z 3.
2
9
6
y 1 z 5
x 1 y 2 z 5
and
are coplanar. Also find the equation of the
1
5
1
2
5
11.
Find the angle between the line
12.
Show that the lines
13.
plane containing the lines.
Find the equation of the perpendicular drawn from the ppoint (1, -2, 3) to the plane 2x 3y 4z 9 0. Also
14.
find the coordinates of the foot of the perpendicular.
Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and B(3, -1, 2) and parallel to the
line
x 4
1
y 3
4
x 3
3
z 1
.
7
Page 19
15.
Write the vector equations of the following lines and hence determine the distance between them :
x 1 y 2
2
3
16.
z 4 x 3 y 3 z 5
;
4
6
12
6
x 2 y 1 z 3
Find the points on the line
at a distance of 5 units from the point P(1, 3, 3).
3
2
2
17.
Find the coordinates of the foot of the perpendicular and the perpendicular distance of the point P(3, 2, 1) from
the plane 2x y z 1 0. Find also the image of the point in the plane.
18.
Find the equation of the plane passing through the point P(1, 1, 1) and containing the line
r ( 3i j 5k)
19.
(i 2j 5k).
Find the shortest distance between the following pair of lines and hence write whether the lines are intersecting or
not :
20.
(3i j 5k). Also, show that the plane contains the line r ( i 2j 5k)
x 1 y 1
x 1 y 2
z;
;z 2
2
3
5
1
Find the angle between the following pair of lines :
x 2
2
y 1 z 3
x 2
and
7
3
1
2y 8
4
z 5
and check
4
whether the lines are parallel or perpendicular.
21.
ANSWERS
1.
5.
11.
15.
19.
51x 15y 50z 173 0
14
sin
1
293
7
9
195
6.
8
21
x 1 y 2
2
1
2.
2 29
3.
z 4 11 19 34 1
,
, ,
,
2
9 9 9 3
12.
x 2y z 0
16.
( 2, 1,3) or (4,3,7)
20.
90 lines are perpendicular
13.
56 43 111
, ,
17 17 17
7.
1
6
x 1 y 2
2
3
17.
8.
1
4.
9x 17y 23z 20 0
9.
7
z 3
, ( 1,1, 1)
4
(1,3,0), ( 1,4, 1)
18.
3 2
10.
14.
x 19y 11z 0
r. (i 2j k) 0
LINEAR PROGRAMMING
Each question carries 4/6 marks
1.
2.
3.
If a young man rides his motorcycle at the speed of 25 km/hour, he had to spend Rs. 2 per km on petrol. If he
rides it at a faster speed of 40 km/hour, the petrol cost increases to Rs. 5 per km. He has at most Rs. 100 to spend
on petrol and wishes to find the maximum distance that he can travel in one hour. Express this as a LPP and solve
it graphically.
A factory owner purchases two types of machines A and B for his factory. The requirements and the limitations
for the machines are as follows :
Machine
Area occupied
Labour force
Daily output (in units)
2
A
1000 m
12 men
60
B
1200 m2
8 men
40
2
He has maximum area of 9000 m available and 72 skilled labourers who can operate both the machines. How
many machines of each type should he buy to maximize the daily output?
A farmer has a supply of chemical fertilizer of type A which contains 10% nitrogen and 5% phosphoric acid, and
type B which contains 6% nitrogen and 10% phosphoric acid. After testing the soil conditions of the field, the
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4.
farmer finds that he needs at least 14 kg of nitrogen and 14 kg of phosphoric acid is required for producing a good
crop. The fertilizer of type A costs Rs. 5 per kg and the type B costs Rs. 3 per kg. How many kg of each type of
the fertilizer should be used to meet the requirement at the minimum possible cost? Using LPP solve the above
problem graphically.
A diet is the contain at least 80 units of Vitamin A and 100 units of minerals. Two foods F1 and F2 are available.
Food F1 costs 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 mninerals. One unit of food F2 contains 6 units of Vitamin A and 3 units of minerals. Formulate this as a
5.
6.
7.
8.
9.
10.
11.
linear programming problem. Find graphically the minimum cost for diet that consists of mixture of these two
foods and also meets the minimal nutritional requirements.
A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5,760 to invest and has
space for at the most 20 items. A fan costs him Rs. 360 and a sewing machine Rs. 240. He expects to sell a fan at
a profit of Rs. 22 and a sewing machine for a profit of Rs. 18. Assuming that he can sell all the items he buy, how
should he invest money to maximize his profit? Solve it graphically.
One kind of cake requires 200 g of flour and 25 g of fat, and another kind of cake requires 100 g of flour and 50 g
of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that
there is no shortage of the other ingredients used in making the cakes. Formulate this problem as a linear
programming problem and solve graphically.
A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per da y is at
most 24. It takes 1 hour to make a ring and 30 minutes to make chain. The maximum number of hours available
per day is 16. If the profit on a ring is Rs. 300 and that on a chain is Rs. 190, find the number of rings and chains
that should be manufactured per day, so as to earn the maximum profit. Make it as a LPP and solve it graphically.
A factory makes two types of items A and B made of plywood. One piece of item A requires 5 minutes for
cutting and 10 minutes for assembling. One piece of item B requires 8 minutes for cutting and 8 minutes for
assembling. There are 3 hours and 20 minutes available for cutting and 4 hours for assembling. The profit on one
piece of item A is Rs. 5 and that on item B is Rs. 6. How many pieces of each type should the factory make so as
to maximize profit? Make it as LPP and solve it graphically.
A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of
craftman’s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman’s time. In
a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman’s time.
If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find the number of tennis rackets and
cricket bats that the factory must manufacture to earn the maximum profit. Make it as a LPP and solve
graphically.
A merchant plans to sell two types of personal computers – a desktop model and a portable model that will cost
Rs. 25,000 and Rs. 40,000 respectively. He estimates that the total monthly demand of computers will not exceed
250 units. Determine the number of units of each type of computers which the merchant should stock to get
maximum profit if he does not want to invest more than Rs. 70 lakh and his profit on the desktop model is Rs.
4,500 and on the portable model is Rs. 5,000. Make a LPP and solve it graphically.
A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of grindling/cutting
machine and a sprayer. It takes 2 hours on the grinding/cutting machine and 3 hours on the sprayer to
manufacture a pedestal lamp. It takes one hour on the grindling/cutting machine and 2 hours on the sprayer to
manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting
machine for at the most 12 hours. The proft from the sale of a lamp is Rs. 5 and that from a shade is Rs. 3.
Assuming that the manfucaturer can sell all the lamps and shades that he produces, how should he schedule his
daily production in order to maximize his profit? Make a LPP and solve it graphically.
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12.
13.
14.
A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to
produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts.
He earns a profit of Rs. 17.50 per package on nuts and Rs. 7 per package on bolts. How many packages of each
should be produced each day so as to maximize his profit, if he operates his machines for at the most 12 hours a
day? Formulate this problem as a linear programming problem and solve it graphically.
A dietician wishes to mix two types of food in such a way that vitamin contents of the mixture contain at least 8
units of vitamin A and 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C.
Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. If costs Rs. 5 per kg to purchase Food I and
Rs. 7 per kg to purchase Food II. Determine the minimum cost of such a mixture. Formulate this problem as a
LPP and solve it graphically.
A company produces soft drinks that has a contract which requires a minimum of 80 units of chemical A and 60
units of chemical B go into each bottle of the drink. The chemical are available in prepared mix packets from two
different suppliers. Supplier S has a packet of mix of 4 units of A and 2 units of B that costs Rs. 10 and the
supplier T has a packet of mix of 1 unit of A and 1 unit of B that costs Rs. 4. How many packets of mixes from S
and T should the company purchase to honour the contract requirement and yet minimize cost? Make a LPP and
solve graphically.
PROBABILITY
Each question carries 4/6 marks
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Find mean μ, variance
2
, for the following probability distribution :
X
0
1
2
3
P(X)
1/8
3/8
3/8
1/8
Find the binomial distribution for which mean is 4 and variance 3.
Bag A contains 6 red and 5 blue balls and another bag B contains 5 red and 8 blue balls. A ball is drawn from bag
A without seeing its colour and it is put into the bag B. Then a ball is drawn from bag B at random. Find the
probability that the ball drawn is blue in colour.
Out of 9 outstanding students of a school, there are 4 boys and 5 girls. A team of 4 students is to be selected for a
quiz competition. Find the probability that 2 boys and 2 girls are selected.
A bag X contains 2 white and 3 red balls and a bag Y contains 4 white and 5 red balls. One ball is drawn at
random from one of the bag and is found to be red. Find the probability that it was drawn from bag Y.
An urn contains 7 red and 4 blue balls. Two balls are drawn at random with replacement. Find the probability of
getting (i) 2 red balls (ii) 2 blue balls (iii) 1 red and 1 blue ball.
There are 2000 scooter drivers, 4000 car drivers and 6000 truck drivers all insured. The probabilities of an
accident involving a scooter, a car, a truck are 0.01, 0.03, 0.15 respectively. One of the insured drivers meets with
an accident. What is the probability that he is a scooter driver?
12 cards, numbered 1 to 12, are placed in a box, mixed up thoroughly and then one card is drawn at randomly
from the box. If it is known that the number on the drawn card is more than 3, find the probability that it is an
even number.
In a bulb factory machines A, B and C manufacture 60%, 30% and 10% bulbs respectively. 1%, 2% and 3% of
the bulbs produced respectively by A, B and C are found to be defective. A bulb is picked up at random from the
total production and found to be defective. Find the probability that this bulb was produced by the machine A.
Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Find the mean and standard deviation
of the number of kings.
In a factory which manufactures bolts, machines A, B and C manufacture respectively 25%, 35% and 40% of the
bolts. Of their outputs 5, 4 and 2 per cent are respectively defective bolts. A bolt is drawn at random from the
production and is found to e defective. Find the probability that it is manufactured by the machine B.
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12.
13.
14.
15.
16.
17.
18.
19.
20.
A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the
sixth throw of the die.
Three bags contain balls as shown in the table below :
Bag
Number of white balls Number of black balls Number of red balls
I
1
2
3
II
2
1
1
III
4
3
2
A bag is chosen at random and two balls are drawn from it. They happen to be white and red. What is the
probability that they came from the III bag?
Two groups are competing for the positions on Board of directors of a corporation. The probabilities that the first
and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of
introducing a new product is 0.7 and corresponding probability is 0.3 if second group wins. Find the probability
that the new product introduced was by second group.
There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes
up tails 25% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it
shows head, what is the probability that it was from the two headed coin?
A man is known to speak the truth 3 out of 5 times. He throws a die and reports that it is a number greater than 4.
Find the probability that it is actually a number greater than 4.
From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find
the mean and variance of the number of defective bulbs.
On a multiple choice examination with three possible answers (out of which only one is correct) for each of the
five questions, what is the probability that a candidate would get four or more correct answer just by guessing?
A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn at random and
are found to both clubs. Find the probability of the lost card being of club.
A family has 2 children. Find the probability that both are boys, if it is known that
(i)
at least one of the children is a boy.
(ii) the elder child is a boy
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1.
Consider the point A(0, 1) and B (2, 0) and P be a point on the line 4x 3 y 9 0 . Coordinates of P such
that | PA PB | is maximum, are
84 13
6 17
(c)
(d) (0, –3)
,
,
5 5
5 5
The vertices of a triangle are A( x1, x1 tan ) , B( x2 , x2 tan ) and C( x3 , x3 tan ) . If the circumecentre of
a
triangle ABC coincides with the origin and H(a, b) be orthocentre, then
b
cos cos cos
sin sin sin
sin sin sin
tan
tan tan
(a)
(b)
(c)
(d)
cos .cos .cos
sin .sin .sin
cos cos cos
tan .tan .tan
12 17
,
4 5
(a)
2.
3.
(b)
The joint equation of two altitudes of an equilateral triangle is ( 3x
y 8 4 3) (
3x y 12 4 3) 0 .
The third altitude has the equation
3x 2 4 3
3x 2 4 3
(d) y 10 0
If the area of the rhombus enclosed by the lines lx my n 0 be 2 square units, then
(a)
4.
(a) l, m, n are in G.P.
5.
(b) y 10 0
(c)
(b) l, n, m are in G.P.
(c) lm
(d) l n
n
m
If the point (1 cos ,sin ) lies between the region corresponding to acute angle between the lines
x y 0 and 6 y x, then
R n ,n I
(a)
6.
2 xy by
that the equation ax
(a) a
K2 b
2
(d) None of these
K2 b
(b) a
can be found such
2K (x y 1) 0 represents a pair of straight lines is
(c) K 2
The line lx my 1 intersects the circle x2
then a2 (l 2
a or K 2 a
(d) K
2a or K 2b
y2 a2 at points A, B, If AB subtends 45º at the origin
m2 )
(a) 4 2 2
8.
n ,n I
(c)
If a and b are positive numbers (a < b), then the range of value of K for which a real
2
7.
(2n 1) , n I
2
(b)
(b) 4 2 6
3
The equation x
2
6x y 11xy
2
(c) 2 6
3
6y
(d) 4
6
0 represent three straight lines passing through the origin, the
slopes of which form a/an
(a) A.P.
9.
(c) H.P.
2
If the slope of one of the lines represented by ax
then
2hxy by
(d) None of these
2
0 be the square of the other,
a b 8h2
h
ab
(a) 4
10.
(b) G.P.
(b) 6
(c) 8
2
The locus of the centre of a circle touching the circle x
(d) None of these
y2 4 y 2x 2 3 1 internally and tangents on
which from (1, 2) is making a 60º angle with each other, is
(a) ( x 1)2
( y 2)2 3 (b) ( x 2)2 ( y 1)2 1 2 3
Page 24
(c) x2
11.
y2 1
(d) None of these
The least distance between two points P and Q on the circles
x2 y2 8x 10 y 37 0 and
x2 y2 16x 55 0
(a) 5 units
12.
(c) 5 5 units
(b) 8 units
Tangents are drawn from O (origin) to touch the circle x2
(d) None of these
y2 2gx 2 fy c 0 at points P and Q. The
equation of the circle circumscribing triangle OPQ is
13.
(a) 2x2
2 y2 gx fy 0
(c) x2
y2 2gx 2 fy 0
(b) x2
y2 gx fy 0
(d) None of these
The line 4x 3 y 4 0 divides the circumference of the circle centered at (5, 3), in the ratio 1 : 2. Then the
equation of the circle is
14.
(a) x2
y2 10x 6 y 66 0
(b) x2
y2 10x 6 y 100 0
(c) x2
y2 10x 6 y 66 0
(d) x2
y2 10x 6 y 100 0
If p and q be the longest and the shortest distance respectively of the point (–7, 2) from any point ( , ) on
the curve whose equation is x2
(a) 2 11
15.
(b) 2 5
(c) 13
PQ is any focal chord of the parabola y2
(a) 40
16.
y2 10x 14 y 51 0 then G.M. of p and q is
(d) 11
32x . The length of PQ can never be less than
(b) 45
(c) 32
If the tangent at the point P( x1, y1) to the parabola y2
(d) 48
4ax meets the parabola y2 4a(x b) at Q and R,
then the mid-point of QR is
(a) ( x1 b, y1 b)
17.
(b) ( x1 b, y1 b)
(c) ( x1, y1 )
(d) ( x1 b, y1 b)
If perpendiculars be drawn from any two fixed points on the axis of a parabola equidistant from the focus on
any tangent to it, then the difference of their squares is [ l is the length of latus rectum and 2d is the distance
between two points]
(b) 2ld
(a) ld
18.
A focal chord of parabola y2
(c) 4ld
(d)
4x is inclined at an angle of
4
l2 d2
with positive x-direction, then the slope of
normal drawn at the ends of chord will satisfy the equation
(a) m2
19.
The parabola y2
(a) r
20.
2m 1 0
20
(b) m2
2m 1 0
(c) m2 1 0
(d) m2
2m 2 0
4x and the circle ( x 6)2 y2 r 2 will have no common tangent if ‘r’ is equal to
(b) r
20
(c) r
18
(d) R ( 20, 28)
The ends of line segement are P(1, 3) and Q(1, 1), R is a point on the line segement PQ such
that PR : RQ 1: . If R is an interior point of parabola y2
4x , then
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(0,1)
(a)
21.
3
,1
5
(b)
(1, )
(d)
Coordinates of the vertices B and C of a triangle ABC are (2, 0) and (8, 0) respectively. The vertex A is
varying in such a way that 4 tan
(a)
22.
( 1,0)
(c)
( x 5)2
25
y2
1
16
(b)
B C
tan
1. Then locus of A is
C
2
( x 5)2
16
y2
1
25
(c)
( x 5)2
25
y2
1
9
(d)
( x 5)2
9
y2
1
25
In an ellipse, if the lines joining a focus to the extremities of the minor axis make an equilateral triangle with
the minor axis, then the eccentricity of the ellipse is
(a)
23.
3
2
Tangents are drawn to the ellipse
(a) 27
24.
3
4
(b)
(b)
(d)
(c)
27
4
(d)
x2
a
27
55
y2
1 from two points on
b2
a2 b2 from the centre is
(b) 2b2
(c) a2
b2
(d) a2
The locus of the foot of perpendicular from the centre on any tangent to the ellipse
(a) A circle
1
2
y2
1at ends of laterarecta. The area of quadrilateral so formed is
5
The sum of the squares of the perpendiculars on any tangent to the ellipse
(a) 2a 2
26.
x2
9
27
2
the minor axis each at a distance
25.
1
2
(c)
(b) A pair of straight lines (c) Another ellipse
If the normal to the rectangular hyperbola xy
c2 at the point ct,
x2
a2
b2
y2
1 is
b2
(d) None of these
c
c
meets the curve again at ct ',
,
t
t'
then
3
(a) t t ' 1
27.
If | z i | 2 and z0
(a) 2
28.
3
(b) t t '
31
31 2
(d) tt '
(c) 7
The complex number z1, z2 and z3 satisfying
1
(d) 5
1 i 3
are the vertices of a triangle, which is
2
z1 z3
z2 z3
(b) Right angled isosceles
(c) Equilateral
(d) Obtuse angled isosceles
For all complex numbers z1, z2 satisfying | z1 | 12 and | z2 3 4i | 5 , the minimum value of | z1
(a) 0
30.
(c) tt ' 1
5 3i , then the maximum value of | iz z0 | is
(b)
(a) Of area zero
29.
1
If | z
(b) 2
2
2
1| | z |
(c) 7
(d) 17
(c) Circle
(d) None of these
z2 | is
1, then locus of z is
(a) Real axis
(b) Imaginary axis
TOP IC :
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
B
D
B
B
D
D
A
C
B
D
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11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B
B
A
A
C
C
A
B
B
A
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
A
A
A
A
D
B
C
C
B
B
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