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Page 1
MATHEMATICS
SECTION - A
Multiple Choice Questions: This section contains 20
multiple choice questions. Each question has 4 choices
(1), (2), (3) and (4), out of which ONLY ONE is correct.
Choose the correct answer :
1. The probability that a randomly chosen 2 × 2 matrix
with all the entries from the set of first 10 primes, is
singular, is equal to :
(A)
4
133
10
(B)
3
18
10
(C)
3
19
10
(D)
4
271
10
Answer (C)
Sol. Let A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
Let E be the event that matrix of order 2 × 2 is singular
Case-I
All entries are same example
22
22
??
??
??
=
10
C
1
Case-II
Matrix with two prime numbers only example
35
35
??
??
??
=
10
C
2
× 2! × 2!
10 10
12
4 4 3
2! 2! 190 19
()
10 10 10
CC
PE
+ ? ?
= = =
2. Let the solution curve of the differential equation
22
16 ,
dy
x y y x
dx
- = + y(1) = 3 be y = y(x). Then
y(2) is equal to :
(A) 15 (B) 11
(C) 13 (D) 17
Answer (A)
Sol.
22
16
dy
x y y x
dx
- = +
y = 4x tan ?
2
4tan 4 sec
dy d
x
dx dx
?
= ? + ?
22
4 tan 4 sec 4 tan 4 sec
d
x x x x
dx
?
? + ? - ? = ?
sec
dx
d
x
? ? =
??
log |sec? + tan?| = log |x| + C
y(1) = 3 ? 3 = 4 tan?
35
tan sec
44
= ? = ? ? =
8
ln ln 1
4
C =+
? C = ln 2
? |sec? + tan?| = 2|x|
To find y(2) put x = 2
? tan
8
y
?=
(sec? + tan?)
2
= 16
sec tan 4
1
sec tan
4
15
2tan 2
48
y
? + ? = ?
? - ? = ?
? = = ?
? 15 y =
3. If the mirror image of the point (2, 4, 7) in the plane
3x – y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal
to :
(A) 54 (B) 50
(C) –6 (D) –42
Answer (C)
Sol. Mirror image of (2, 4, 7) in 3x – y + 4z = 2 is
(a, b, c) then
2 2 2
2 4 7 2(6 4 28 2)
3 1 4
3 ( 1) 4
a b c - - - - - + -
= = =
-
+ - +
2 4 7 28
3 1 4 13
a b c - - - -
= = =
-
58 80 21
13 13 13
a b c
--
= = =
116 80 42
22
13
a b c
- + -
+ + =
= –6
Page 2
MATHEMATICS
SECTION - A
Multiple Choice Questions: This section contains 20
multiple choice questions. Each question has 4 choices
(1), (2), (3) and (4), out of which ONLY ONE is correct.
Choose the correct answer :
1. The probability that a randomly chosen 2 × 2 matrix
with all the entries from the set of first 10 primes, is
singular, is equal to :
(A)
4
133
10
(B)
3
18
10
(C)
3
19
10
(D)
4
271
10
Answer (C)
Sol. Let A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
Let E be the event that matrix of order 2 × 2 is singular
Case-I
All entries are same example
22
22
??
??
??
=
10
C
1
Case-II
Matrix with two prime numbers only example
35
35
??
??
??
=
10
C
2
× 2! × 2!
10 10
12
4 4 3
2! 2! 190 19
()
10 10 10
CC
PE
+ ? ?
= = =
2. Let the solution curve of the differential equation
22
16 ,
dy
x y y x
dx
- = + y(1) = 3 be y = y(x). Then
y(2) is equal to :
(A) 15 (B) 11
(C) 13 (D) 17
Answer (A)
Sol.
22
16
dy
x y y x
dx
- = +
y = 4x tan ?
2
4tan 4 sec
dy d
x
dx dx
?
= ? + ?
22
4 tan 4 sec 4 tan 4 sec
d
x x x x
dx
?
? + ? - ? = ?
sec
dx
d
x
? ? =
??
log |sec? + tan?| = log |x| + C
y(1) = 3 ? 3 = 4 tan?
35
tan sec
44
= ? = ? ? =
8
ln ln 1
4
C =+
? C = ln 2
? |sec? + tan?| = 2|x|
To find y(2) put x = 2
? tan
8
y
?=
(sec? + tan?)
2
= 16
sec tan 4
1
sec tan
4
15
2tan 2
48
y
? + ? = ?
? - ? = ?
? = = ?
? 15 y =
3. If the mirror image of the point (2, 4, 7) in the plane
3x – y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal
to :
(A) 54 (B) 50
(C) –6 (D) –42
Answer (C)
Sol. Mirror image of (2, 4, 7) in 3x – y + 4z = 2 is
(a, b, c) then
2 2 2
2 4 7 2(6 4 28 2)
3 1 4
3 ( 1) 4
a b c - - - - - + -
= = =
-
+ - +
2 4 7 28
3 1 4 13
a b c - - - -
= = =
-
58 80 21
13 13 13
a b c
--
= = =
116 80 42
22
13
a b c
- + -
+ + =
= –6
4. Let ƒ : R ? R be a function defined by :
3
2
max{ 3 } ; 2
2 6 ; 2 3 ƒ( )
[ 3] 9 ; 3 5
2 1 ; 5
tx
t t x
x x x x
xx
xx
?
?
-?
?
?
?
+ - ? ? =
?
?
- + ? ?
?
+? ?
?
where [t] is the greatest integer less than or equal
to t. Let m be the number of points where ƒ is not
differentiable and
2
2
ƒ( ) . I x dx
-
=
?
Then the ordered
pair (m, I) is equal to :
(A)
27
3,
4
??
??
??
(B)
23
3,
4
??
??
??
(C)
27
4,
4
??
??
??
(D)
23
4,
4
??
??
??
Answer (C)
Sol.
3
max{ 3 } ; 2
tx
t t x
?
-?
g(t) = t
3
–3t ? g? (t) = 3t
2
–3 = 3(t – 1)(t + 1)
3
2
31
2 1 2
2 6 2 3
ƒ( )
9 3 4
10 4 5
11 5
2 1 5
x x x
x
x x x
x
x
x
x
xx
?
- ? -
?
- ? ?
?
?
+ - ? ?
?
?
=
?
??
?
??
?
?
=
?
+? ?
?
Points of non-differentiability = {2, 3, 4, 5}
? m = 4
2 1 2
3
2 2 1
ƒ( ) ( 3 ) 2 I x dx x x dx dx
-
- - -
= = - +
? ? ?
1
42
2
3 1 3
2(2 1) (4 6) 6
4 2 4 2
xx
-
-
??
??
= - + + = - - - + ??
??
??
??
??
=
27
4
5. Let
ˆ ˆ ˆ ˆ ˆ ˆ
3 , 3 4 a i j k b i j k = ? + - = - ? + and
ˆ ˆ ˆ
22 c i j k = + - where ?, ? ? R, be three vectors. If
the projection of a on c is
10
3
and
ˆ ˆ ˆ
6 10 7 , b c i j k ? = - + + then the value of ? + ? is
equal to :
(A) 3 (B) 4
(C) 5 (D) 6
Answer (A)
Sol.
ˆ ˆ ˆ
3 a i j k = ? + -
ˆ ˆ ˆ
34 b i j k = - ? +
ˆ ˆ ˆ
22 c i j k = + -
Projection of a on c is
10
3
ac
b
?
=
2 2 2
6 2 8 10
33
1 2 ( 2)
? + + ? +
==
+ + -
? 2 ?=
ˆ ˆ ˆ
6 10 7 b c i j k ? = - + +
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ
3 4 (2 8) 10 (6 ) 6 10 7
1 2 2
i j k
i j k i j k -? = ? - + + + ? = - + +
-
2 8 6 & 6 7 ? - = - + ? =
? 1 ?=
? + ? = 2 + 1 = 3
Page 3
MATHEMATICS
SECTION - A
Multiple Choice Questions: This section contains 20
multiple choice questions. Each question has 4 choices
(1), (2), (3) and (4), out of which ONLY ONE is correct.
Choose the correct answer :
1. The probability that a randomly chosen 2 × 2 matrix
with all the entries from the set of first 10 primes, is
singular, is equal to :
(A)
4
133
10
(B)
3
18
10
(C)
3
19
10
(D)
4
271
10
Answer (C)
Sol. Let A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
Let E be the event that matrix of order 2 × 2 is singular
Case-I
All entries are same example
22
22
??
??
??
=
10
C
1
Case-II
Matrix with two prime numbers only example
35
35
??
??
??
=
10
C
2
× 2! × 2!
10 10
12
4 4 3
2! 2! 190 19
()
10 10 10
CC
PE
+ ? ?
= = =
2. Let the solution curve of the differential equation
22
16 ,
dy
x y y x
dx
- = + y(1) = 3 be y = y(x). Then
y(2) is equal to :
(A) 15 (B) 11
(C) 13 (D) 17
Answer (A)
Sol.
22
16
dy
x y y x
dx
- = +
y = 4x tan ?
2
4tan 4 sec
dy d
x
dx dx
?
= ? + ?
22
4 tan 4 sec 4 tan 4 sec
d
x x x x
dx
?
? + ? - ? = ?
sec
dx
d
x
? ? =
??
log |sec? + tan?| = log |x| + C
y(1) = 3 ? 3 = 4 tan?
35
tan sec
44
= ? = ? ? =
8
ln ln 1
4
C =+
? C = ln 2
? |sec? + tan?| = 2|x|
To find y(2) put x = 2
? tan
8
y
?=
(sec? + tan?)
2
= 16
sec tan 4
1
sec tan
4
15
2tan 2
48
y
? + ? = ?
? - ? = ?
? = = ?
? 15 y =
3. If the mirror image of the point (2, 4, 7) in the plane
3x – y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal
to :
(A) 54 (B) 50
(C) –6 (D) –42
Answer (C)
Sol. Mirror image of (2, 4, 7) in 3x – y + 4z = 2 is
(a, b, c) then
2 2 2
2 4 7 2(6 4 28 2)
3 1 4
3 ( 1) 4
a b c - - - - - + -
= = =
-
+ - +
2 4 7 28
3 1 4 13
a b c - - - -
= = =
-
58 80 21
13 13 13
a b c
--
= = =
116 80 42
22
13
a b c
- + -
+ + =
= –6
4. Let ƒ : R ? R be a function defined by :
3
2
max{ 3 } ; 2
2 6 ; 2 3 ƒ( )
[ 3] 9 ; 3 5
2 1 ; 5
tx
t t x
x x x x
xx
xx
?
?
-?
?
?
?
+ - ? ? =
?
?
- + ? ?
?
+? ?
?
where [t] is the greatest integer less than or equal
to t. Let m be the number of points where ƒ is not
differentiable and
2
2
ƒ( ) . I x dx
-
=
?
Then the ordered
pair (m, I) is equal to :
(A)
27
3,
4
??
??
??
(B)
23
3,
4
??
??
??
(C)
27
4,
4
??
??
??
(D)
23
4,
4
??
??
??
Answer (C)
Sol.
3
max{ 3 } ; 2
tx
t t x
?
-?
g(t) = t
3
–3t ? g? (t) = 3t
2
–3 = 3(t – 1)(t + 1)
3
2
31
2 1 2
2 6 2 3
ƒ( )
9 3 4
10 4 5
11 5
2 1 5
x x x
x
x x x
x
x
x
x
xx
?
- ? -
?
- ? ?
?
?
+ - ? ?
?
?
=
?
??
?
??
?
?
=
?
+? ?
?
Points of non-differentiability = {2, 3, 4, 5}
? m = 4
2 1 2
3
2 2 1
ƒ( ) ( 3 ) 2 I x dx x x dx dx
-
- - -
= = - +
? ? ?
1
42
2
3 1 3
2(2 1) (4 6) 6
4 2 4 2
xx
-
-
??
??
= - + + = - - - + ??
??
??
??
??
=
27
4
5. Let
ˆ ˆ ˆ ˆ ˆ ˆ
3 , 3 4 a i j k b i j k = ? + - = - ? + and
ˆ ˆ ˆ
22 c i j k = + - where ?, ? ? R, be three vectors. If
the projection of a on c is
10
3
and
ˆ ˆ ˆ
6 10 7 , b c i j k ? = - + + then the value of ? + ? is
equal to :
(A) 3 (B) 4
(C) 5 (D) 6
Answer (A)
Sol.
ˆ ˆ ˆ
3 a i j k = ? + -
ˆ ˆ ˆ
34 b i j k = - ? +
ˆ ˆ ˆ
22 c i j k = + -
Projection of a on c is
10
3
ac
b
?
=
2 2 2
6 2 8 10
33
1 2 ( 2)
? + + ? +
==
+ + -
? 2 ?=
ˆ ˆ ˆ
6 10 7 b c i j k ? = - + +
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ
3 4 (2 8) 10 (6 ) 6 10 7
1 2 2
i j k
i j k i j k -? = ? - + + + ? = - + +
-
2 8 6 & 6 7 ? - = - + ? =
? 1 ?=
? + ? = 2 + 1 = 3
6. The area enclosed by y
2
= 8x and 2 yx = that lies
outside the triangle formed by
2 , 1 , 2 2, y x x y = = = is equal to :
(A)
16 2
6
(B)
11 2
6
(C)
13 2
6
(C)
52
6
Answer (C)
Sol.
( ) ( ) ( )
2, 2 2 , 1 , 2 2 , 1 , 2 A B C
Area = ( )
42
2
0
area
8
2
yy
dy BAC
??
- - ? ??
??
??
?
42
23
0
1
24 2
22
yy
AB BC
??
= - - ? ? ??
??
??
32 4 2 1
8 2 1 2
24 2
?
= - - ? ?
16 2 2
82
32
= - -
( )
2 13 2
48 32 3
66
= - - =
7. If the system of linear equations
2x + y – z = 7
x – 3y + 2z = 1
x + 4y + ?z = k, where ?, k ? R
has infinitely many solutions, then ? + k is equal to:
(A) –3 (B) 3
(C) 6 (D) 9
Answer (B)
Sol. 2x + y – z = 7
x – 3y + 2z = 1
x + 4y + ?z = k
2 1 1
1 3 2 7 21 0
14
-
? = - = - ? - =
?
3 ? = -
1
7 1 1
1 3 2
43 k
-
? = -
-
? 6 – k = 0 ? k = 6
3 6 3 k ? + = - + =
8. Let ? and ? be the roots of the equation x
2
+
(2i – 1) = 0. Then, the value of
88
? + ? is equal to:
(A) 50 (B) 250
(C) 1250 (D) 1500
Answer (A)
Sol. x
2
+ 2i – 1 = 0
?
2
= ?
2
= 1 – 2i
?
4
= (1 – 2i)
2
= 1 + (2i)
2
– 4i = –3 – 4i
?
8
= (–3 – 4i)
2
= 9 – 16 + 24i = –7 + 24i
( ) ( )
22
88
2 7 24 2 7 24 50 i ? + ? = - + = - + =
9. Let { , , , } ? ? ? ? ? ? be such that
( ) ( ) ( )
p q p q q ? ? ? ? is a tautology. Then ? is
equal to :
(A) ? (B) ?
(C) ? (D) ?
Answer (C)
Sol. ( ) p q q ??
( ) ~ p q q ??
( ) ~~ p q q = ? ?
( ) ( ) ~~ = ? ? ? p q q q
( ) ~ p q T = ? ?
~pq =?
Now ( ) ( ) ~ p q p q ? ? ?
( ) ( ) ~ ~ ~ p q p p q p q p q p q
T T F T T T
T F F F F T
F T T F T T
F F T F T T
? ? ? ? ?
? ? = ?
Page 4
MATHEMATICS
SECTION - A
Multiple Choice Questions: This section contains 20
multiple choice questions. Each question has 4 choices
(1), (2), (3) and (4), out of which ONLY ONE is correct.
Choose the correct answer :
1. The probability that a randomly chosen 2 × 2 matrix
with all the entries from the set of first 10 primes, is
singular, is equal to :
(A)
4
133
10
(B)
3
18
10
(C)
3
19
10
(D)
4
271
10
Answer (C)
Sol. Let A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
Let E be the event that matrix of order 2 × 2 is singular
Case-I
All entries are same example
22
22
??
??
??
=
10
C
1
Case-II
Matrix with two prime numbers only example
35
35
??
??
??
=
10
C
2
× 2! × 2!
10 10
12
4 4 3
2! 2! 190 19
()
10 10 10
CC
PE
+ ? ?
= = =
2. Let the solution curve of the differential equation
22
16 ,
dy
x y y x
dx
- = + y(1) = 3 be y = y(x). Then
y(2) is equal to :
(A) 15 (B) 11
(C) 13 (D) 17
Answer (A)
Sol.
22
16
dy
x y y x
dx
- = +
y = 4x tan ?
2
4tan 4 sec
dy d
x
dx dx
?
= ? + ?
22
4 tan 4 sec 4 tan 4 sec
d
x x x x
dx
?
? + ? - ? = ?
sec
dx
d
x
? ? =
??
log |sec? + tan?| = log |x| + C
y(1) = 3 ? 3 = 4 tan?
35
tan sec
44
= ? = ? ? =
8
ln ln 1
4
C =+
? C = ln 2
? |sec? + tan?| = 2|x|
To find y(2) put x = 2
? tan
8
y
?=
(sec? + tan?)
2
= 16
sec tan 4
1
sec tan
4
15
2tan 2
48
y
? + ? = ?
? - ? = ?
? = = ?
? 15 y =
3. If the mirror image of the point (2, 4, 7) in the plane
3x – y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal
to :
(A) 54 (B) 50
(C) –6 (D) –42
Answer (C)
Sol. Mirror image of (2, 4, 7) in 3x – y + 4z = 2 is
(a, b, c) then
2 2 2
2 4 7 2(6 4 28 2)
3 1 4
3 ( 1) 4
a b c - - - - - + -
= = =
-
+ - +
2 4 7 28
3 1 4 13
a b c - - - -
= = =
-
58 80 21
13 13 13
a b c
--
= = =
116 80 42
22
13
a b c
- + -
+ + =
= –6
4. Let ƒ : R ? R be a function defined by :
3
2
max{ 3 } ; 2
2 6 ; 2 3 ƒ( )
[ 3] 9 ; 3 5
2 1 ; 5
tx
t t x
x x x x
xx
xx
?
?
-?
?
?
?
+ - ? ? =
?
?
- + ? ?
?
+? ?
?
where [t] is the greatest integer less than or equal
to t. Let m be the number of points where ƒ is not
differentiable and
2
2
ƒ( ) . I x dx
-
=
?
Then the ordered
pair (m, I) is equal to :
(A)
27
3,
4
??
??
??
(B)
23
3,
4
??
??
??
(C)
27
4,
4
??
??
??
(D)
23
4,
4
??
??
??
Answer (C)
Sol.
3
max{ 3 } ; 2
tx
t t x
?
-?
g(t) = t
3
–3t ? g? (t) = 3t
2
–3 = 3(t – 1)(t + 1)
3
2
31
2 1 2
2 6 2 3
ƒ( )
9 3 4
10 4 5
11 5
2 1 5
x x x
x
x x x
x
x
x
x
xx
?
- ? -
?
- ? ?
?
?
+ - ? ?
?
?
=
?
??
?
??
?
?
=
?
+? ?
?
Points of non-differentiability = {2, 3, 4, 5}
? m = 4
2 1 2
3
2 2 1
ƒ( ) ( 3 ) 2 I x dx x x dx dx
-
- - -
= = - +
? ? ?
1
42
2
3 1 3
2(2 1) (4 6) 6
4 2 4 2
xx
-
-
??
??
= - + + = - - - + ??
??
??
??
??
=
27
4
5. Let
ˆ ˆ ˆ ˆ ˆ ˆ
3 , 3 4 a i j k b i j k = ? + - = - ? + and
ˆ ˆ ˆ
22 c i j k = + - where ?, ? ? R, be three vectors. If
the projection of a on c is
10
3
and
ˆ ˆ ˆ
6 10 7 , b c i j k ? = - + + then the value of ? + ? is
equal to :
(A) 3 (B) 4
(C) 5 (D) 6
Answer (A)
Sol.
ˆ ˆ ˆ
3 a i j k = ? + -
ˆ ˆ ˆ
34 b i j k = - ? +
ˆ ˆ ˆ
22 c i j k = + -
Projection of a on c is
10
3
ac
b
?
=
2 2 2
6 2 8 10
33
1 2 ( 2)
? + + ? +
==
+ + -
? 2 ?=
ˆ ˆ ˆ
6 10 7 b c i j k ? = - + +
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ
3 4 (2 8) 10 (6 ) 6 10 7
1 2 2
i j k
i j k i j k -? = ? - + + + ? = - + +
-
2 8 6 & 6 7 ? - = - + ? =
? 1 ?=
? + ? = 2 + 1 = 3
6. The area enclosed by y
2
= 8x and 2 yx = that lies
outside the triangle formed by
2 , 1 , 2 2, y x x y = = = is equal to :
(A)
16 2
6
(B)
11 2
6
(C)
13 2
6
(C)
52
6
Answer (C)
Sol.
( ) ( ) ( )
2, 2 2 , 1 , 2 2 , 1 , 2 A B C
Area = ( )
42
2
0
area
8
2
yy
dy BAC
??
- - ? ??
??
??
?
42
23
0
1
24 2
22
yy
AB BC
??
= - - ? ? ??
??
??
32 4 2 1
8 2 1 2
24 2
?
= - - ? ?
16 2 2
82
32
= - -
( )
2 13 2
48 32 3
66
= - - =
7. If the system of linear equations
2x + y – z = 7
x – 3y + 2z = 1
x + 4y + ?z = k, where ?, k ? R
has infinitely many solutions, then ? + k is equal to:
(A) –3 (B) 3
(C) 6 (D) 9
Answer (B)
Sol. 2x + y – z = 7
x – 3y + 2z = 1
x + 4y + ?z = k
2 1 1
1 3 2 7 21 0
14
-
? = - = - ? - =
?
3 ? = -
1
7 1 1
1 3 2
43 k
-
? = -
-
? 6 – k = 0 ? k = 6
3 6 3 k ? + = - + =
8. Let ? and ? be the roots of the equation x
2
+
(2i – 1) = 0. Then, the value of
88
? + ? is equal to:
(A) 50 (B) 250
(C) 1250 (D) 1500
Answer (A)
Sol. x
2
+ 2i – 1 = 0
?
2
= ?
2
= 1 – 2i
?
4
= (1 – 2i)
2
= 1 + (2i)
2
– 4i = –3 – 4i
?
8
= (–3 – 4i)
2
= 9 – 16 + 24i = –7 + 24i
( ) ( )
22
88
2 7 24 2 7 24 50 i ? + ? = - + = - + =
9. Let { , , , } ? ? ? ? ? ? be such that
( ) ( ) ( )
p q p q q ? ? ? ? is a tautology. Then ? is
equal to :
(A) ? (B) ?
(C) ? (D) ?
Answer (C)
Sol. ( ) p q q ??
( ) ~ p q q ??
( ) ~~ p q q = ? ?
( ) ( ) ~~ = ? ? ? p q q q
( ) ~ p q T = ? ?
~pq =?
Now ( ) ( ) ~ p q p q ? ? ?
( ) ( ) ~ ~ ~ p q p p q p q p q p q
T T F T T T
T F F F F T
F T T F T T
F F T F T T
? ? ? ? ?
? ? = ?
10. Let A = [a ij] be a square matrix of order 3 such that
a ij = 2
j–i
, for all i, j = 1, 2, 3. Then, the matrix A
2
+ A
3
+ … + A
10
is equal to :
(A)
10
33
2
A
??
-
??
??
??
(B)
10
31
2
A
??
-
??
??
??
(C)
10
31
2
A
??
+
??
??
??
(D)
10
33
2
A
??
+
??
??
??
Answer (A)
Sol.
0 1 2
11 12 13
1 0 1
21 22 23
2 1 0
31 32 33
2 2 2
2 2 2
2 2 2
a a a
A a a a
a a a
-
--
??
??
??
??
??
==
??
??
??
?? ??
??
2
1 2 4 1 2 4 3 6 12
1 1 3
1 2 1 2 3 6 3
2 2 2
1 1 1 1 3 3
1 1 3
4 2 4 2 4 2
AA
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
= = =
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
A
2
= 3A
A
3
= A.A
2
= A(3A) = 3A
2
= 3
2
A
A
4
= 3
3
A
Now
A
2
+ A
3
+ … + A
10
A[3
1
+ 3
2
+ 3
3
+ … + 3
9
]
9
3 3 1
31
A
??
-
??
=
-
( )
10
33
2
A
-
=
11. Let a set A = A1 ?A2 ?… ?A k, where A i ? A j = ?
for i ? j, 1 ? i, j ? k. Define the relation R from A to
A by R = {(x, y) : y ?A i if and only if x?A i, 1 ? i ?
k}. Then, R is :
(A) reflexive, symmetric but not transitive
(B) reflexive, transitive but not symmetric
(C) reflexive but not symmetric and transitive
(D) an equivalence relation
Answer (D)
Sol. R = {(x, y) : y?A i, iff x?Ai, 1 ? i ? k}
(1) Reflexive
(a, a) ? a?Ai iff a?Ai
(2) Symmetric
(a, b) ? a?Ai iff b?Ai
(b, a) ?R as b?Ai iff a?Ai
(3) Transitive
(a, b)?R & (b, c) ?R.
? a?Ai iff b?A i & b?A i iff c?A i
? a?Ai iff c?A i
? (a, c)?R.
? Relation is equivalence
12. Let ? ?
0
n
n
a
?
=
be a sequence such that a0 = a1 = 0
and a n + 2 = 2a n + 1 – a n + 1 for all n ? 0.
Then
2
7
n
n
n
a
?
=
?
is equal to :
(A)
6
343
(B)
7
216
(C)
8
343
(D)
49
216
Answer (B)
Sol.
2 1 0 1
2 1 & 0
n n n
a a a a a
++
= - + = =
2 1 0
2 1 1 a a a = - + =
3 2 1
2 1 3 a a a = - + =
4 3 2
2 1 6 a a a = - + =
5 4 3
2 1 10 a a a = - + =
3 24
2 3 4
2
7 7 7 7
n
n
n
a a a a
?
=
= + + +?
?
2 3 4 5
3 4 5
2 3 4
34
2 3 4
1 3 6 10
7 7 7 7
1 1 3 6
7
7 7 7
6 1 2 3
7
777
6 1 2
49
77
36 1 1 1
49
777
s
s
s
s
s
= + + + + ?
= + + + ?
= + + + ?
= + + ?
= + + + ?
Page 5
MATHEMATICS
SECTION - A
Multiple Choice Questions: This section contains 20
multiple choice questions. Each question has 4 choices
(1), (2), (3) and (4), out of which ONLY ONE is correct.
Choose the correct answer :
1. The probability that a randomly chosen 2 × 2 matrix
with all the entries from the set of first 10 primes, is
singular, is equal to :
(A)
4
133
10
(B)
3
18
10
(C)
3
19
10
(D)
4
271
10
Answer (C)
Sol. Let A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
Let E be the event that matrix of order 2 × 2 is singular
Case-I
All entries are same example
22
22
??
??
??
=
10
C
1
Case-II
Matrix with two prime numbers only example
35
35
??
??
??
=
10
C
2
× 2! × 2!
10 10
12
4 4 3
2! 2! 190 19
()
10 10 10
CC
PE
+ ? ?
= = =
2. Let the solution curve of the differential equation
22
16 ,
dy
x y y x
dx
- = + y(1) = 3 be y = y(x). Then
y(2) is equal to :
(A) 15 (B) 11
(C) 13 (D) 17
Answer (A)
Sol.
22
16
dy
x y y x
dx
- = +
y = 4x tan ?
2
4tan 4 sec
dy d
x
dx dx
?
= ? + ?
22
4 tan 4 sec 4 tan 4 sec
d
x x x x
dx
?
? + ? - ? = ?
sec
dx
d
x
? ? =
??
log |sec? + tan?| = log |x| + C
y(1) = 3 ? 3 = 4 tan?
35
tan sec
44
= ? = ? ? =
8
ln ln 1
4
C =+
? C = ln 2
? |sec? + tan?| = 2|x|
To find y(2) put x = 2
? tan
8
y
?=
(sec? + tan?)
2
= 16
sec tan 4
1
sec tan
4
15
2tan 2
48
y
? + ? = ?
? - ? = ?
? = = ?
? 15 y =
3. If the mirror image of the point (2, 4, 7) in the plane
3x – y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal
to :
(A) 54 (B) 50
(C) –6 (D) –42
Answer (C)
Sol. Mirror image of (2, 4, 7) in 3x – y + 4z = 2 is
(a, b, c) then
2 2 2
2 4 7 2(6 4 28 2)
3 1 4
3 ( 1) 4
a b c - - - - - + -
= = =
-
+ - +
2 4 7 28
3 1 4 13
a b c - - - -
= = =
-
58 80 21
13 13 13
a b c
--
= = =
116 80 42
22
13
a b c
- + -
+ + =
= –6
4. Let ƒ : R ? R be a function defined by :
3
2
max{ 3 } ; 2
2 6 ; 2 3 ƒ( )
[ 3] 9 ; 3 5
2 1 ; 5
tx
t t x
x x x x
xx
xx
?
?
-?
?
?
?
+ - ? ? =
?
?
- + ? ?
?
+? ?
?
where [t] is the greatest integer less than or equal
to t. Let m be the number of points where ƒ is not
differentiable and
2
2
ƒ( ) . I x dx
-
=
?
Then the ordered
pair (m, I) is equal to :
(A)
27
3,
4
??
??
??
(B)
23
3,
4
??
??
??
(C)
27
4,
4
??
??
??
(D)
23
4,
4
??
??
??
Answer (C)
Sol.
3
max{ 3 } ; 2
tx
t t x
?
-?
g(t) = t
3
–3t ? g? (t) = 3t
2
–3 = 3(t – 1)(t + 1)
3
2
31
2 1 2
2 6 2 3
ƒ( )
9 3 4
10 4 5
11 5
2 1 5
x x x
x
x x x
x
x
x
x
xx
?
- ? -
?
- ? ?
?
?
+ - ? ?
?
?
=
?
??
?
??
?
?
=
?
+? ?
?
Points of non-differentiability = {2, 3, 4, 5}
? m = 4
2 1 2
3
2 2 1
ƒ( ) ( 3 ) 2 I x dx x x dx dx
-
- - -
= = - +
? ? ?
1
42
2
3 1 3
2(2 1) (4 6) 6
4 2 4 2
xx
-
-
??
??
= - + + = - - - + ??
??
??
??
??
=
27
4
5. Let
ˆ ˆ ˆ ˆ ˆ ˆ
3 , 3 4 a i j k b i j k = ? + - = - ? + and
ˆ ˆ ˆ
22 c i j k = + - where ?, ? ? R, be three vectors. If
the projection of a on c is
10
3
and
ˆ ˆ ˆ
6 10 7 , b c i j k ? = - + + then the value of ? + ? is
equal to :
(A) 3 (B) 4
(C) 5 (D) 6
Answer (A)
Sol.
ˆ ˆ ˆ
3 a i j k = ? + -
ˆ ˆ ˆ
34 b i j k = - ? +
ˆ ˆ ˆ
22 c i j k = + -
Projection of a on c is
10
3
ac
b
?
=
2 2 2
6 2 8 10
33
1 2 ( 2)
? + + ? +
==
+ + -
? 2 ?=
ˆ ˆ ˆ
6 10 7 b c i j k ? = - + +
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ
3 4 (2 8) 10 (6 ) 6 10 7
1 2 2
i j k
i j k i j k -? = ? - + + + ? = - + +
-
2 8 6 & 6 7 ? - = - + ? =
? 1 ?=
? + ? = 2 + 1 = 3
6. The area enclosed by y
2
= 8x and 2 yx = that lies
outside the triangle formed by
2 , 1 , 2 2, y x x y = = = is equal to :
(A)
16 2
6
(B)
11 2
6
(C)
13 2
6
(C)
52
6
Answer (C)
Sol.
( ) ( ) ( )
2, 2 2 , 1 , 2 2 , 1 , 2 A B C
Area = ( )
42
2
0
area
8
2
yy
dy BAC
??
- - ? ??
??
??
?
42
23
0
1
24 2
22
yy
AB BC
??
= - - ? ? ??
??
??
32 4 2 1
8 2 1 2
24 2
?
= - - ? ?
16 2 2
82
32
= - -
( )
2 13 2
48 32 3
66
= - - =
7. If the system of linear equations
2x + y – z = 7
x – 3y + 2z = 1
x + 4y + ?z = k, where ?, k ? R
has infinitely many solutions, then ? + k is equal to:
(A) –3 (B) 3
(C) 6 (D) 9
Answer (B)
Sol. 2x + y – z = 7
x – 3y + 2z = 1
x + 4y + ?z = k
2 1 1
1 3 2 7 21 0
14
-
? = - = - ? - =
?
3 ? = -
1
7 1 1
1 3 2
43 k
-
? = -
-
? 6 – k = 0 ? k = 6
3 6 3 k ? + = - + =
8. Let ? and ? be the roots of the equation x
2
+
(2i – 1) = 0. Then, the value of
88
? + ? is equal to:
(A) 50 (B) 250
(C) 1250 (D) 1500
Answer (A)
Sol. x
2
+ 2i – 1 = 0
?
2
= ?
2
= 1 – 2i
?
4
= (1 – 2i)
2
= 1 + (2i)
2
– 4i = –3 – 4i
?
8
= (–3 – 4i)
2
= 9 – 16 + 24i = –7 + 24i
( ) ( )
22
88
2 7 24 2 7 24 50 i ? + ? = - + = - + =
9. Let { , , , } ? ? ? ? ? ? be such that
( ) ( ) ( )
p q p q q ? ? ? ? is a tautology. Then ? is
equal to :
(A) ? (B) ?
(C) ? (D) ?
Answer (C)
Sol. ( ) p q q ??
( ) ~ p q q ??
( ) ~~ p q q = ? ?
( ) ( ) ~~ = ? ? ? p q q q
( ) ~ p q T = ? ?
~pq =?
Now ( ) ( ) ~ p q p q ? ? ?
( ) ( ) ~ ~ ~ p q p p q p q p q p q
T T F T T T
T F F F F T
F T T F T T
F F T F T T
? ? ? ? ?
? ? = ?
10. Let A = [a ij] be a square matrix of order 3 such that
a ij = 2
j–i
, for all i, j = 1, 2, 3. Then, the matrix A
2
+ A
3
+ … + A
10
is equal to :
(A)
10
33
2
A
??
-
??
??
??
(B)
10
31
2
A
??
-
??
??
??
(C)
10
31
2
A
??
+
??
??
??
(D)
10
33
2
A
??
+
??
??
??
Answer (A)
Sol.
0 1 2
11 12 13
1 0 1
21 22 23
2 1 0
31 32 33
2 2 2
2 2 2
2 2 2
a a a
A a a a
a a a
-
--
??
??
??
??
??
==
??
??
??
?? ??
??
2
1 2 4 1 2 4 3 6 12
1 1 3
1 2 1 2 3 6 3
2 2 2
1 1 1 1 3 3
1 1 3
4 2 4 2 4 2
AA
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
= = =
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
A
2
= 3A
A
3
= A.A
2
= A(3A) = 3A
2
= 3
2
A
A
4
= 3
3
A
Now
A
2
+ A
3
+ … + A
10
A[3
1
+ 3
2
+ 3
3
+ … + 3
9
]
9
3 3 1
31
A
??
-
??
=
-
( )
10
33
2
A
-
=
11. Let a set A = A1 ?A2 ?… ?A k, where A i ? A j = ?
for i ? j, 1 ? i, j ? k. Define the relation R from A to
A by R = {(x, y) : y ?A i if and only if x?A i, 1 ? i ?
k}. Then, R is :
(A) reflexive, symmetric but not transitive
(B) reflexive, transitive but not symmetric
(C) reflexive but not symmetric and transitive
(D) an equivalence relation
Answer (D)
Sol. R = {(x, y) : y?A i, iff x?Ai, 1 ? i ? k}
(1) Reflexive
(a, a) ? a?Ai iff a?Ai
(2) Symmetric
(a, b) ? a?Ai iff b?Ai
(b, a) ?R as b?Ai iff a?Ai
(3) Transitive
(a, b)?R & (b, c) ?R.
? a?Ai iff b?A i & b?A i iff c?A i
? a?Ai iff c?A i
? (a, c)?R.
? Relation is equivalence
12. Let ? ?
0
n
n
a
?
=
be a sequence such that a0 = a1 = 0
and a n + 2 = 2a n + 1 – a n + 1 for all n ? 0.
Then
2
7
n
n
n
a
?
=
?
is equal to :
(A)
6
343
(B)
7
216
(C)
8
343
(D)
49
216
Answer (B)
Sol.
2 1 0 1
2 1 & 0
n n n
a a a a a
++
= - + = =
2 1 0
2 1 1 a a a = - + =
3 2 1
2 1 3 a a a = - + =
4 3 2
2 1 6 a a a = - + =
5 4 3
2 1 10 a a a = - + =
3 24
2 3 4
2
7 7 7 7
n
n
n
a a a a
?
=
= + + +?
?
2 3 4 5
3 4 5
2 3 4
34
2 3 4
1 3 6 10
7 7 7 7
1 1 3 6
7
7 7 7
6 1 2 3
7
777
6 1 2
49
77
36 1 1 1
49
777
s
s
s
s
s
= + + + + ?
= + + + ?
= + + + ?
= + + ?
= + + + ?
2
1
36
7
1
49
1
7
s
=
-
36 7
49 49 6
s
=
?
7
216
s =
13. The distance between the two points A and A?
which lie on y = 2 such that both the line segments
AB and A?B (where B is the point (2, 3)) subtend
angle
4
?
at the origin, is equal to
(A) 10 (B)
48
5
(C)
52
5
(D) 3
Answer (C)
Sol. Let A(?, 2) Given B(2, 3)
23
&
2
OA OB
mm ==
?
23
43
2
tan 1
23
4 2 6
1.
2
-
? - ?
?
= ? = ?
?+
+
?
4 – 3? = 2? + 6 & 4 – 3? = -2? - 6
2
& 10
5
-
? = ? =
2
, 2 & (10,2) and (2, 3)
5
A A B
??
-?
??
??
2 52
10
55
AA? = + =
14. A wire of length 22 m is to be cut into two pieces.
One of the pieces is to be made into a square and
the other into an equilateral triangle. Then, the
length of the side of the equilateral triangle, so that
the combined area of the square and the equilateral
triangle is minimum, is
(A)
22
9 4 3 +
(B)
66
9 4 3 +
(C)
22
4 9 3 +
(D)
66
4 9 3 +
Answer (B)
Sol.
4a + 3b = 22
Total area =
22
3
4
A a b =+
2
2
22 3 3
44
b
Ab
- ??
=+
??
??
22 3 3 3
2 .2 0
4 4 4
dA b
b
dB
-- ? ?? ?
= + =
? ?? ?
? ?? ?
?
33
(22 3 )
28
b
b =-
4 3 66 9 bb =-
66
9 4 3
b =
+
15. The domain of the function
1
2
1
1
2sin
41
cos
x
-
-
?? ??
?? ??
? - ?
??
??
?
??
??
is :
(A)
11
,
22
??
--
??
??
R
(B) ( ? ? ) ? ? , 1 1 , 0 -? - ? ? ?
(C) ? ?
11
, , 0
22
- ? ? ? ?
-? ? ? ?
? ? ? ?
? ? ? ?
(D) ? ?
11
, , 0
22
- ? ? ? ?
-? ? ? ?
?? ??
? ? ? ?
Answer (D)
Sol.
1
2
21
1 sin 1
41 x
-
??
- ? ?
??
?
? - ?
1
2
1
sin
22
41 x
-
??
- ? ?
-
2
1
11
41 x
- ? ?
-
2
2
14
10
(2 1)(2 1)
41
x
nx
x
? - ? ?
+-
-
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