NDA Solved Paper 2018 - II | NDA (National Defence Academy) Past Year Papers PDF Download

 Page 1


1. What is the value of log
7
log
7
7 7 7 equal to ?
(a) 3 7
2
log (b) 1 3 7
2
- log
(c) 1 3 2
7
- log (d)
7
8
Ê
(c) We have,
7 7 7 7 7 7 7
1
2
1
4
1
8
7
8
= · · =
Now, log log
7 7
7 7 7
= log log ( )
7 7
7
8
7
=
?
?
?
?
?
?
log
7
7
8
= - log log
7 7
7 8
[Qlog log log
m
n
m n = - ]
= - log log
7 7
3
7 2 = - 1 3 2
7
log
[Qlog log
b
n
b
a n a = ]
2. If an infinite GP has the first term x
and the sum 5, then which one of the
following is correct?
(a) x < -10
(b) - < < 10 0 x
(c) 0 10 < < x
(d) x > 10
Ê
(c) Given that first term of an infinity GP is
x and sum = 5
?
x
r 1
5
-
=
[Q sum of infinity GP =
-
a
r 1
]
?
x
r
5
1 = -
? r
x
= - 1
5
Where, | | r < 1
? - < - < 1
5
1
x
? - < - < 2
5
0
x
? - < - < 10 0 x
? 10 0 > > x
3. Consider the following expressions
1. x x
x
+ -
2
1
2. ax bx x c
d
x
e
x
2
2
+ + - + -
3. 3 5 x x ab
2
- +
4.
2
2 3
x ax b - +
5.
1 2
5 x x
-
+
Which of the above are rational
expressions?
(a) 1, 4 and 5 (b) 1, 3, 4 and 5
(c) 2, 4 and 5 (d) 1 and 2
Ê
(a) We know that, rational expressions
are those expression which can be write
in the form of
p x
q x
q x
( )
( )
, ( ) ? 0
So, 1, 4, 5 are rational expressions
4. A square matrix A is called
orthogonal if
(a) A A =
2
(b) ' =
-
A A
1
(c) A A =
-1
(d) A A = '
where A' is the transpose of A
Ê
(b) A square matrix is called an
orthogonal matrix if AA I ' =
multiply. by A
- 1
? A AA A I
- -
' =
1 1
( ) ?IA A ' =
- 1
? A A ' =
- 1
5. IfA, B and C are subsets of a
universal set, then which one of the
following is not correct?
(a) A B C A B A C ? n = ? n ? ( ) ( ) ( )
(b) A A B B A A '? ? = 'n ' ? ( ) ( )
(c) A B C C B A '? ? = 'n ' n ' ( ) ( )
(d) ( ) ( ) ( ) A B C A C B C n ? = ? n ?
where A' is the complement of A
Ê
(c) Let A B , and C are subsets of a
universal set.
Let A B C = = = { } , { } , { } 1 2 3
? = { , , } 1 2 3 , A' = { , } 2 3 ,B' = { , } 1 3 ,
C ' = { , } 1 2
by checking options, we get
LHS = A B C '? ? ( )
= ? { , } { } 2 3
= { , } 2 3
RHS = ( ) C B A ' n ' n '
= n n ( { , } { , } ) { , } 1 2 1 3 2 3
= ' n ( { } ) { , } 2 2 3
= n { , } { , } 1 3 2 3
= { } 3
LHS ? RHS
So, option (c) is wrong
6. Let x be the number of integers
lying between 2999 and 8001 which
have at least two digits equal.Thenx
is equal to
(a) 2480 (b) 2481
(c) 2482 (d) 2483
Ê
(b) We have, x be the number lying
between 2999 and 8001
if repetition allowed
total numbers = × × × = 5 10 10 10 5000
if repetition not allowed
?total numbers = × × × = 5 9 8 7 2520
PAPER : I Mathematics
Page 2


1. What is the value of log
7
log
7
7 7 7 equal to ?
(a) 3 7
2
log (b) 1 3 7
2
- log
(c) 1 3 2
7
- log (d)
7
8
Ê
(c) We have,
7 7 7 7 7 7 7
1
2
1
4
1
8
7
8
= · · =
Now, log log
7 7
7 7 7
= log log ( )
7 7
7
8
7
=
?
?
?
?
?
?
log
7
7
8
= - log log
7 7
7 8
[Qlog log log
m
n
m n = - ]
= - log log
7 7
3
7 2 = - 1 3 2
7
log
[Qlog log
b
n
b
a n a = ]
2. If an infinite GP has the first term x
and the sum 5, then which one of the
following is correct?
(a) x < -10
(b) - < < 10 0 x
(c) 0 10 < < x
(d) x > 10
Ê
(c) Given that first term of an infinity GP is
x and sum = 5
?
x
r 1
5
-
=
[Q sum of infinity GP =
-
a
r 1
]
?
x
r
5
1 = -
? r
x
= - 1
5
Where, | | r < 1
? - < - < 1
5
1
x
? - < - < 2
5
0
x
? - < - < 10 0 x
? 10 0 > > x
3. Consider the following expressions
1. x x
x
+ -
2
1
2. ax bx x c
d
x
e
x
2
2
+ + - + -
3. 3 5 x x ab
2
- +
4.
2
2 3
x ax b - +
5.
1 2
5 x x
-
+
Which of the above are rational
expressions?
(a) 1, 4 and 5 (b) 1, 3, 4 and 5
(c) 2, 4 and 5 (d) 1 and 2
Ê
(a) We know that, rational expressions
are those expression which can be write
in the form of
p x
q x
q x
( )
( )
, ( ) ? 0
So, 1, 4, 5 are rational expressions
4. A square matrix A is called
orthogonal if
(a) A A =
2
(b) ' =
-
A A
1
(c) A A =
-1
(d) A A = '
where A' is the transpose of A
Ê
(b) A square matrix is called an
orthogonal matrix if AA I ' =
multiply. by A
- 1
? A AA A I
- -
' =
1 1
( ) ?IA A ' =
- 1
? A A ' =
- 1
5. IfA, B and C are subsets of a
universal set, then which one of the
following is not correct?
(a) A B C A B A C ? n = ? n ? ( ) ( ) ( )
(b) A A B B A A '? ? = 'n ' ? ( ) ( )
(c) A B C C B A '? ? = 'n ' n ' ( ) ( )
(d) ( ) ( ) ( ) A B C A C B C n ? = ? n ?
where A' is the complement of A
Ê
(c) Let A B , and C are subsets of a
universal set.
Let A B C = = = { } , { } , { } 1 2 3
? = { , , } 1 2 3 , A' = { , } 2 3 ,B' = { , } 1 3 ,
C ' = { , } 1 2
by checking options, we get
LHS = A B C '? ? ( )
= ? { , } { } 2 3
= { , } 2 3
RHS = ( ) C B A ' n ' n '
= n n ( { , } { , } ) { , } 1 2 1 3 2 3
= ' n ( { } ) { , } 2 2 3
= n { , } { , } 1 3 2 3
= { } 3
LHS ? RHS
So, option (c) is wrong
6. Let x be the number of integers
lying between 2999 and 8001 which
have at least two digits equal.Thenx
is equal to
(a) 2480 (b) 2481
(c) 2482 (d) 2483
Ê
(b) We have, x be the number lying
between 2999 and 8001
if repetition allowed
total numbers = × × × = 5 10 10 10 5000
if repetition not allowed
?total numbers = × × × = 5 9 8 7 2520
PAPER : I Mathematics
So, x = atleast two digit repeated
= - + 5000 2520 1
= 2481
[Q add 1 because of number 8000]
7. The sum of the series
3 1
1
3
1
9
- + - +.... is equal to
(a)
20
9
(b)
9
20
(c)
9
4
(d)
4
9
Ê
(c) Given series
3 1
1
3
1
9
- + - + ..... are in GP
? r =
- 1
3
S
n
=
- -
?
?
?
?
?
?
3
1
1
3
QS
a
r
n
=
-
?
?
?
?
?
?
1
=
3
4
3
=
9
4
Directions (Q. Nos. 8 and 9)
Consider the information given below
and answer the two items that follow.
A survey was conducted among 300
students. It was found that 125 students
like to play cricket, 145 students like to
play football and 90 students like to play
tennis. 32 students like to play exactly two
games out of the three games.
8. How many stdudents like to play all
the three games ?
(a) 14 (b) 21
(c) 28 (d) 35
Ê
(a) Let,
A be the set of students like to play cricket
B be the set of students like to play
football.
C be the set of students like to play tennis.
We have,
n A B C ( ) ? ? = 300
n A ( ) = 125
n B ( ) = 145
n C ( ) = 90
n A B C n A n B n C ( ) ( ) ( ) ( ) ? ? = + +
- n + n + n [ ( ) ( ) ( )] n A B n B C n C A
+ n n n A B C ( )
? 300 125 145 90 = + +
- n + n + n [ ( ) ( ) ( )] n A B n B C n C A
+ n n n A B C ( )
?n A B n B C n C A ( ) ( ) ( ) n + n + n
= + n n 60 n A B C ( ) ...(i)
Again,
n A B n B C n C A ( ) ( ) ( ) n + n + n
- n n = 3 32 n A B C ( )
?n A B n B C n C A ( ) ( ) ( ) n + n + n
= + n n 32 3n A B C ( ) ...(ii)
From Eqs. (i) and (ii), we get
60 32 + n n = n A B C ( )
+ n n 3n A B C ( )
? 2 28 n A B C ( ) n n =
? n A B C ( ) n n = 14
9. How many students like to play
exactly only one game?
(a) 196 (b) 228
(c) 254 (d) 268
Ê
(c) Number of students like to play
exactly one
game = + + n A n B n C ( ) ( ) ( )
- n + n + n 2 [ ( ) ( ) ( )] n A B n B C n C A
+ n n 3n A B C ( )
= + + - + × 125 145 90 2 32 3 14 [ ]
+ × 3 14
= - 360 106
= 254
10. If a and ß ( 0) ? are the roots of the
quadratic equation x x
2
0 + - = a ß ,
then the quadratic expression
- + + x x
2
a ß, where x R ? has
(a) Least value -
1
4
(b) Least value -
9
4
(c) Greatest value
1
4
(d) Greatest value
9
4
Ê
(d) a and ß are the roots of quadratic
equation.
x x
2
0 + - = a ß
So, (aß ß = - ) ?aß ß + = 0
? ß a ( ) + = 1 0
a = - 1 [Qß ? 0]
a ß a + = -
? 2 0 a ß + =
? ß = 2
? - + + x x
2
a ß [Qa ß = - = 1 2 , ]
= - - + x x
2
2
Greatest value = -
-
-
?
?
?
?
?
?
2
1
4 1
2
( )
( )
[Q Greatest value = -
?
?
?
?
?
? c
b
a
2
2
4
]
= -
-
?
?
?
?
?
? 2
1
4
= + 2
1
4
=
9
4
11. What is the coefficient of the middle
term in the binomial expansion of
( ) 2 3
4
+ x ?
(a) 6 (b) 12
(c) 108 (d) 216
Ê
(d) We have, ( ) 2 3
4
+ x
Here,n = 4, so middle term is
4
2
1 +
?
?
?
?
?
?
th
= 3
rd
term
T C x
3
4
2
2 2
2 3 = × × ( )
[T C a b
r
n
r
r n r
+
-
=
1
]1
=
×
×
× ×
4 3
2 1
4 9
2
x
T x
3
2
216 =
Hence, coefficient of middle term is 216.
12. For a square matrix A, which of the
following properties hold?
1. ( ) A A
- -
=
1 1
2. det ( )
det
A
A
-
=
1
1
3. ( ) ? ? A A
- -
=
1 1
, where ? is a scalar
Select the correct answer using the
code given below.
(a) 1 and 2 (b) 2 and 3
(c) 1 and 3 (d) 1, 2 and 3
Ê
(d) For a square matrix A
Statement 1
( ) A A
- -
=
1 1
Statement 1 is true
Statement 2
det ( )
det
A
A
-
=
1
1
Statement 2 is correct
Statement 3
( ) ? ? A A
- -
=
1 1
, where ? is a scalar.
So, Statement 3 is correct.
13. Which one of the following factors
does the expansion of the
determinant
x
x
x
y
y
y
2
3
3
5
5
10
3
9
27
contain ?
(a) x - 3
(b) x y -
(c) y - 3
(d) x y - 3
Ê
(a) We have,
x y
x y
x y
3
5 9
10 27
2 3
3 5
?
?
?
?
?
?
?
?
2
Page 3


1. What is the value of log
7
log
7
7 7 7 equal to ?
(a) 3 7
2
log (b) 1 3 7
2
- log
(c) 1 3 2
7
- log (d)
7
8
Ê
(c) We have,
7 7 7 7 7 7 7
1
2
1
4
1
8
7
8
= · · =
Now, log log
7 7
7 7 7
= log log ( )
7 7
7
8
7
=
?
?
?
?
?
?
log
7
7
8
= - log log
7 7
7 8
[Qlog log log
m
n
m n = - ]
= - log log
7 7
3
7 2 = - 1 3 2
7
log
[Qlog log
b
n
b
a n a = ]
2. If an infinite GP has the first term x
and the sum 5, then which one of the
following is correct?
(a) x < -10
(b) - < < 10 0 x
(c) 0 10 < < x
(d) x > 10
Ê
(c) Given that first term of an infinity GP is
x and sum = 5
?
x
r 1
5
-
=
[Q sum of infinity GP =
-
a
r 1
]
?
x
r
5
1 = -
? r
x
= - 1
5
Where, | | r < 1
? - < - < 1
5
1
x
? - < - < 2
5
0
x
? - < - < 10 0 x
? 10 0 > > x
3. Consider the following expressions
1. x x
x
+ -
2
1
2. ax bx x c
d
x
e
x
2
2
+ + - + -
3. 3 5 x x ab
2
- +
4.
2
2 3
x ax b - +
5.
1 2
5 x x
-
+
Which of the above are rational
expressions?
(a) 1, 4 and 5 (b) 1, 3, 4 and 5
(c) 2, 4 and 5 (d) 1 and 2
Ê
(a) We know that, rational expressions
are those expression which can be write
in the form of
p x
q x
q x
( )
( )
, ( ) ? 0
So, 1, 4, 5 are rational expressions
4. A square matrix A is called
orthogonal if
(a) A A =
2
(b) ' =
-
A A
1
(c) A A =
-1
(d) A A = '
where A' is the transpose of A
Ê
(b) A square matrix is called an
orthogonal matrix if AA I ' =
multiply. by A
- 1
? A AA A I
- -
' =
1 1
( ) ?IA A ' =
- 1
? A A ' =
- 1
5. IfA, B and C are subsets of a
universal set, then which one of the
following is not correct?
(a) A B C A B A C ? n = ? n ? ( ) ( ) ( )
(b) A A B B A A '? ? = 'n ' ? ( ) ( )
(c) A B C C B A '? ? = 'n ' n ' ( ) ( )
(d) ( ) ( ) ( ) A B C A C B C n ? = ? n ?
where A' is the complement of A
Ê
(c) Let A B , and C are subsets of a
universal set.
Let A B C = = = { } , { } , { } 1 2 3
? = { , , } 1 2 3 , A' = { , } 2 3 ,B' = { , } 1 3 ,
C ' = { , } 1 2
by checking options, we get
LHS = A B C '? ? ( )
= ? { , } { } 2 3
= { , } 2 3
RHS = ( ) C B A ' n ' n '
= n n ( { , } { , } ) { , } 1 2 1 3 2 3
= ' n ( { } ) { , } 2 2 3
= n { , } { , } 1 3 2 3
= { } 3
LHS ? RHS
So, option (c) is wrong
6. Let x be the number of integers
lying between 2999 and 8001 which
have at least two digits equal.Thenx
is equal to
(a) 2480 (b) 2481
(c) 2482 (d) 2483
Ê
(b) We have, x be the number lying
between 2999 and 8001
if repetition allowed
total numbers = × × × = 5 10 10 10 5000
if repetition not allowed
?total numbers = × × × = 5 9 8 7 2520
PAPER : I Mathematics
So, x = atleast two digit repeated
= - + 5000 2520 1
= 2481
[Q add 1 because of number 8000]
7. The sum of the series
3 1
1
3
1
9
- + - +.... is equal to
(a)
20
9
(b)
9
20
(c)
9
4
(d)
4
9
Ê
(c) Given series
3 1
1
3
1
9
- + - + ..... are in GP
? r =
- 1
3
S
n
=
- -
?
?
?
?
?
?
3
1
1
3
QS
a
r
n
=
-
?
?
?
?
?
?
1
=
3
4
3
=
9
4
Directions (Q. Nos. 8 and 9)
Consider the information given below
and answer the two items that follow.
A survey was conducted among 300
students. It was found that 125 students
like to play cricket, 145 students like to
play football and 90 students like to play
tennis. 32 students like to play exactly two
games out of the three games.
8. How many stdudents like to play all
the three games ?
(a) 14 (b) 21
(c) 28 (d) 35
Ê
(a) Let,
A be the set of students like to play cricket
B be the set of students like to play
football.
C be the set of students like to play tennis.
We have,
n A B C ( ) ? ? = 300
n A ( ) = 125
n B ( ) = 145
n C ( ) = 90
n A B C n A n B n C ( ) ( ) ( ) ( ) ? ? = + +
- n + n + n [ ( ) ( ) ( )] n A B n B C n C A
+ n n n A B C ( )
? 300 125 145 90 = + +
- n + n + n [ ( ) ( ) ( )] n A B n B C n C A
+ n n n A B C ( )
?n A B n B C n C A ( ) ( ) ( ) n + n + n
= + n n 60 n A B C ( ) ...(i)
Again,
n A B n B C n C A ( ) ( ) ( ) n + n + n
- n n = 3 32 n A B C ( )
?n A B n B C n C A ( ) ( ) ( ) n + n + n
= + n n 32 3n A B C ( ) ...(ii)
From Eqs. (i) and (ii), we get
60 32 + n n = n A B C ( )
+ n n 3n A B C ( )
? 2 28 n A B C ( ) n n =
? n A B C ( ) n n = 14
9. How many students like to play
exactly only one game?
(a) 196 (b) 228
(c) 254 (d) 268
Ê
(c) Number of students like to play
exactly one
game = + + n A n B n C ( ) ( ) ( )
- n + n + n 2 [ ( ) ( ) ( )] n A B n B C n C A
+ n n 3n A B C ( )
= + + - + × 125 145 90 2 32 3 14 [ ]
+ × 3 14
= - 360 106
= 254
10. If a and ß ( 0) ? are the roots of the
quadratic equation x x
2
0 + - = a ß ,
then the quadratic expression
- + + x x
2
a ß, where x R ? has
(a) Least value -
1
4
(b) Least value -
9
4
(c) Greatest value
1
4
(d) Greatest value
9
4
Ê
(d) a and ß are the roots of quadratic
equation.
x x
2
0 + - = a ß
So, (aß ß = - ) ?aß ß + = 0
? ß a ( ) + = 1 0
a = - 1 [Qß ? 0]
a ß a + = -
? 2 0 a ß + =
? ß = 2
? - + + x x
2
a ß [Qa ß = - = 1 2 , ]
= - - + x x
2
2
Greatest value = -
-
-
?
?
?
?
?
?
2
1
4 1
2
( )
( )
[Q Greatest value = -
?
?
?
?
?
? c
b
a
2
2
4
]
= -
-
?
?
?
?
?
? 2
1
4
= + 2
1
4
=
9
4
11. What is the coefficient of the middle
term in the binomial expansion of
( ) 2 3
4
+ x ?
(a) 6 (b) 12
(c) 108 (d) 216
Ê
(d) We have, ( ) 2 3
4
+ x
Here,n = 4, so middle term is
4
2
1 +
?
?
?
?
?
?
th
= 3
rd
term
T C x
3
4
2
2 2
2 3 = × × ( )
[T C a b
r
n
r
r n r
+
-
=
1
]1
=
×
×
× ×
4 3
2 1
4 9
2
x
T x
3
2
216 =
Hence, coefficient of middle term is 216.
12. For a square matrix A, which of the
following properties hold?
1. ( ) A A
- -
=
1 1
2. det ( )
det
A
A
-
=
1
1
3. ( ) ? ? A A
- -
=
1 1
, where ? is a scalar
Select the correct answer using the
code given below.
(a) 1 and 2 (b) 2 and 3
(c) 1 and 3 (d) 1, 2 and 3
Ê
(d) For a square matrix A
Statement 1
( ) A A
- -
=
1 1
Statement 1 is true
Statement 2
det ( )
det
A
A
-
=
1
1
Statement 2 is correct
Statement 3
( ) ? ? A A
- -
=
1 1
, where ? is a scalar.
So, Statement 3 is correct.
13. Which one of the following factors
does the expansion of the
determinant
x
x
x
y
y
y
2
3
3
5
5
10
3
9
27
contain ?
(a) x - 3
(b) x y -
(c) y - 3
(d) x y - 3
Ê
(a) We have,
x y
x y
x y
3
5 9
10 27
2 3
3 5
?
?
?
?
?
?
?
?
2
[C C C
1 1 3
? - mes,]
=
-
-
-
?
?
?
?
?
?
?
?
x y
x y
x y
3 3
9 5 9
27 10 27
2 3
3 5
= - +
+ +
?
?
?
?
?
?
?
?
( ) x
y
x y
x x y
3
1 3
3 5 9
9 3 10 27
3
2 5
14. What is the adjoint of the matrix
cos ( )
sin ( )
sin ( )
cos( )
-
- -
- -
-
?
?
?
?
?
?
?
?
?
?
?
(a)
cos
sin
sin
cos
?
?
?
? -
- ?
?
?
?
?
?
(b)
cos
sin
sin
cos
?
?
?
?
?
?
?
?
?
?
(c)
cos
sin
sin
cos
?
?
?
? -
?
?
?
?
?
?
(d)
cos
sin
sin
cos
?
?
?
?
- ?
?
?
?
?
?
Ê
(a) We have,
A =
- - -
- - -
?
?
?
?
?
?
cos ( ) sin ( )
sin ( ) cos ( )
? ?
? ?
A =
?
?
?
?
?
?
cos sin
sin cos
? ?
? ?
Now, C
11
= cos ?
C
12
= - sin ?
C
21
= - sin ?
C
22
= cos ?
adj A
T
=
-
-
?
?
?
?
?
?
cos sin
sin cos
? ?
? ?
=
-
-
?
?
?
?
?
?
cos sin
sin cos
? ?
? ?
15. What is the value of
- +
?
?
?
?
?
? +
- -
?
?
?
?
?
?
1 3
2
1 3
2
3 3
i i
a n
wherei = -1 ?
(a) 3 (b) 2
(c) 1 (d) 0
Ê
(b) We have,
- + ?
?
?
?
?
? +
- - ?
?
?
?
?
?
1 3
2
1 3
2
3 3
i i
n n
= + ( ) ( ) ? ?
3 2 3 n n
Q ? ? =
- +
=
- - ?
?
?
?
?
?
?
?
?
?
?
?
?
?
1 3
2
1 3
2
2
i i
,
= + ( ) ( ) ? ?
3 3 2 n n
( ) ( ) 1 1
2 n n
+ [Q?
3
1 = ]
= + 1 1= 2
16. There are 17 cricket players, out of
which 5 players can bowl. In how
many ways can a team of 11 players
be selected so as to include 3
bowlers?
(a)C ( , ) 17 11
(b)C ( , ) 12 8
(c)C C ( , ) ( , ) 17 5 5 3 ×
(d)C C ( , ) ( , ) 5 3 12 8 ×
Ê
(d) There are 17 cricket players, out of
which 5 players can bowl.
required number of ways = ×
12
8
5
3
C C
= × C C ( , ) ( , ) 12 8 5 3
17. What is the value of
log log ?
9 8
27 32 +
(a)
7
2
(b)
19
6
(c) 4 (d) 7
Ê
(b) We have,
log log
9 8
27 32 +
= + log log
3
3
2
5
2
3 3 2
= +
3
2
3
5
3
2
3 2
log log
Q log log
a
n
a m
b
n
m
b =
?
?
?
?
?
?
= +
3
2
5
3
=
19
6
18. If A and B are two invertible square
matrices of same order, then what is
( ) AB
-1
equal to?
(a) B A
- - 1 1
(b) A B
- - 1 1
(c) B A
-1
(d) A B
-1
Ê
(a) If A and B are two invertible square
matrices of same order, then
( ) AB B A
- - -
=
1 1 1
19. If a b c + + = 0, then one of the
solutions of
a x
c
b
c
b x
a
b
a
c x
-
-
-
= 0 is
(a) x a =
(b) x
a b c
=
+ + 3
2
2 2 2
( )
(c) x
a b c
=
+ + 2
3
2 2 2
( )
(d) x = 0
Ê
(d) We have,
a x c b
c b x a
b a c x
-
-
-
?
?
?
?
?
?
?
?
= 0
R R R R
1 1 2 3
? + +
?
a b c x a b c x a b c x
c b x a
b a c x
+ + - + + - + + -
-
-
?
?
?
?
?
?
?
?
= 0
?
- - -
-
-
?
?
?
?
?
?
?
?
=
x x x
c b x a
b a c x
0
? ( ) -x
1 1 1
0 c b x a
b a c x
-
-
?
?
?
?
?
?
?
?
=
? x = 0
Hence, x = 0 is a solution
20. What should be the value of x, so
that the matrix
2
8
4
-
?
?
?
?
?
?
x
does not
have an inverse?
(a) 16 (b) -16
(c) 8 (d) -8
Ê
(b) Let, A
x
=
-
?
?
?
?
?
?
2 4
8
Matrix does not have any solution if
| | A = 0
2 32 0 x + =
2 32 x = -
x = -
32
2
x = - 16
21. The system of equations
2 3 5 x y z + - =
3 2 2 5 x y z - + =
and 5 3 16 x y z - - =
(a) is inconsistent
(b) is consistent, with a unique solution
(c) is consistent, with infinitely many
solutions
(d) has its solution lying along X-axis in
three-dimensional space
Ê
(b) The system of equations
2 3 5 x y z + - =
3 2 2 5 x y z - + =
and 5 3 0 x y z - - =
A =
-
-
- -
?
?
?
?
?
?
?
?
?
?
2 1 3
3 2 2
5 3 1
| | [ ( ) ( )] [ ( ) ( )] A = - - - - - - - 2 2 1 2 3 1 3 1 2 5
+ - - - - ( ) [ ( ) ( )( )] 3 3 3 2 5
= - - - 2 8 1 13 3 1 ( ) ( ) ( )
= + - 16 13 3 = ? 26 0
So, System is consistent with unique
solution.
22. Which one of the following is correct
in respect of the cube roots of unity?
(a) They are collinear
(b) They lie on a circle of radius 3
(c) They form an equilateral triangle
(d) None of the above
Ê
(c) We know that, cube roots of unity is
1
2
, , ? ? , where ? =
- + 1 3
2
i
and
?
2
1 3
2
=
- - i
3
Page 4


1. What is the value of log
7
log
7
7 7 7 equal to ?
(a) 3 7
2
log (b) 1 3 7
2
- log
(c) 1 3 2
7
- log (d)
7
8
Ê
(c) We have,
7 7 7 7 7 7 7
1
2
1
4
1
8
7
8
= · · =
Now, log log
7 7
7 7 7
= log log ( )
7 7
7
8
7
=
?
?
?
?
?
?
log
7
7
8
= - log log
7 7
7 8
[Qlog log log
m
n
m n = - ]
= - log log
7 7
3
7 2 = - 1 3 2
7
log
[Qlog log
b
n
b
a n a = ]
2. If an infinite GP has the first term x
and the sum 5, then which one of the
following is correct?
(a) x < -10
(b) - < < 10 0 x
(c) 0 10 < < x
(d) x > 10
Ê
(c) Given that first term of an infinity GP is
x and sum = 5
?
x
r 1
5
-
=
[Q sum of infinity GP =
-
a
r 1
]
?
x
r
5
1 = -
? r
x
= - 1
5
Where, | | r < 1
? - < - < 1
5
1
x
? - < - < 2
5
0
x
? - < - < 10 0 x
? 10 0 > > x
3. Consider the following expressions
1. x x
x
+ -
2
1
2. ax bx x c
d
x
e
x
2
2
+ + - + -
3. 3 5 x x ab
2
- +
4.
2
2 3
x ax b - +
5.
1 2
5 x x
-
+
Which of the above are rational
expressions?
(a) 1, 4 and 5 (b) 1, 3, 4 and 5
(c) 2, 4 and 5 (d) 1 and 2
Ê
(a) We know that, rational expressions
are those expression which can be write
in the form of
p x
q x
q x
( )
( )
, ( ) ? 0
So, 1, 4, 5 are rational expressions
4. A square matrix A is called
orthogonal if
(a) A A =
2
(b) ' =
-
A A
1
(c) A A =
-1
(d) A A = '
where A' is the transpose of A
Ê
(b) A square matrix is called an
orthogonal matrix if AA I ' =
multiply. by A
- 1
? A AA A I
- -
' =
1 1
( ) ?IA A ' =
- 1
? A A ' =
- 1
5. IfA, B and C are subsets of a
universal set, then which one of the
following is not correct?
(a) A B C A B A C ? n = ? n ? ( ) ( ) ( )
(b) A A B B A A '? ? = 'n ' ? ( ) ( )
(c) A B C C B A '? ? = 'n ' n ' ( ) ( )
(d) ( ) ( ) ( ) A B C A C B C n ? = ? n ?
where A' is the complement of A
Ê
(c) Let A B , and C are subsets of a
universal set.
Let A B C = = = { } , { } , { } 1 2 3
? = { , , } 1 2 3 , A' = { , } 2 3 ,B' = { , } 1 3 ,
C ' = { , } 1 2
by checking options, we get
LHS = A B C '? ? ( )
= ? { , } { } 2 3
= { , } 2 3
RHS = ( ) C B A ' n ' n '
= n n ( { , } { , } ) { , } 1 2 1 3 2 3
= ' n ( { } ) { , } 2 2 3
= n { , } { , } 1 3 2 3
= { } 3
LHS ? RHS
So, option (c) is wrong
6. Let x be the number of integers
lying between 2999 and 8001 which
have at least two digits equal.Thenx
is equal to
(a) 2480 (b) 2481
(c) 2482 (d) 2483
Ê
(b) We have, x be the number lying
between 2999 and 8001
if repetition allowed
total numbers = × × × = 5 10 10 10 5000
if repetition not allowed
?total numbers = × × × = 5 9 8 7 2520
PAPER : I Mathematics
So, x = atleast two digit repeated
= - + 5000 2520 1
= 2481
[Q add 1 because of number 8000]
7. The sum of the series
3 1
1
3
1
9
- + - +.... is equal to
(a)
20
9
(b)
9
20
(c)
9
4
(d)
4
9
Ê
(c) Given series
3 1
1
3
1
9
- + - + ..... are in GP
? r =
- 1
3
S
n
=
- -
?
?
?
?
?
?
3
1
1
3
QS
a
r
n
=
-
?
?
?
?
?
?
1
=
3
4
3
=
9
4
Directions (Q. Nos. 8 and 9)
Consider the information given below
and answer the two items that follow.
A survey was conducted among 300
students. It was found that 125 students
like to play cricket, 145 students like to
play football and 90 students like to play
tennis. 32 students like to play exactly two
games out of the three games.
8. How many stdudents like to play all
the three games ?
(a) 14 (b) 21
(c) 28 (d) 35
Ê
(a) Let,
A be the set of students like to play cricket
B be the set of students like to play
football.
C be the set of students like to play tennis.
We have,
n A B C ( ) ? ? = 300
n A ( ) = 125
n B ( ) = 145
n C ( ) = 90
n A B C n A n B n C ( ) ( ) ( ) ( ) ? ? = + +
- n + n + n [ ( ) ( ) ( )] n A B n B C n C A
+ n n n A B C ( )
? 300 125 145 90 = + +
- n + n + n [ ( ) ( ) ( )] n A B n B C n C A
+ n n n A B C ( )
?n A B n B C n C A ( ) ( ) ( ) n + n + n
= + n n 60 n A B C ( ) ...(i)
Again,
n A B n B C n C A ( ) ( ) ( ) n + n + n
- n n = 3 32 n A B C ( )
?n A B n B C n C A ( ) ( ) ( ) n + n + n
= + n n 32 3n A B C ( ) ...(ii)
From Eqs. (i) and (ii), we get
60 32 + n n = n A B C ( )
+ n n 3n A B C ( )
? 2 28 n A B C ( ) n n =
? n A B C ( ) n n = 14
9. How many students like to play
exactly only one game?
(a) 196 (b) 228
(c) 254 (d) 268
Ê
(c) Number of students like to play
exactly one
game = + + n A n B n C ( ) ( ) ( )
- n + n + n 2 [ ( ) ( ) ( )] n A B n B C n C A
+ n n 3n A B C ( )
= + + - + × 125 145 90 2 32 3 14 [ ]
+ × 3 14
= - 360 106
= 254
10. If a and ß ( 0) ? are the roots of the
quadratic equation x x
2
0 + - = a ß ,
then the quadratic expression
- + + x x
2
a ß, where x R ? has
(a) Least value -
1
4
(b) Least value -
9
4
(c) Greatest value
1
4
(d) Greatest value
9
4
Ê
(d) a and ß are the roots of quadratic
equation.
x x
2
0 + - = a ß
So, (aß ß = - ) ?aß ß + = 0
? ß a ( ) + = 1 0
a = - 1 [Qß ? 0]
a ß a + = -
? 2 0 a ß + =
? ß = 2
? - + + x x
2
a ß [Qa ß = - = 1 2 , ]
= - - + x x
2
2
Greatest value = -
-
-
?
?
?
?
?
?
2
1
4 1
2
( )
( )
[Q Greatest value = -
?
?
?
?
?
? c
b
a
2
2
4
]
= -
-
?
?
?
?
?
? 2
1
4
= + 2
1
4
=
9
4
11. What is the coefficient of the middle
term in the binomial expansion of
( ) 2 3
4
+ x ?
(a) 6 (b) 12
(c) 108 (d) 216
Ê
(d) We have, ( ) 2 3
4
+ x
Here,n = 4, so middle term is
4
2
1 +
?
?
?
?
?
?
th
= 3
rd
term
T C x
3
4
2
2 2
2 3 = × × ( )
[T C a b
r
n
r
r n r
+
-
=
1
]1
=
×
×
× ×
4 3
2 1
4 9
2
x
T x
3
2
216 =
Hence, coefficient of middle term is 216.
12. For a square matrix A, which of the
following properties hold?
1. ( ) A A
- -
=
1 1
2. det ( )
det
A
A
-
=
1
1
3. ( ) ? ? A A
- -
=
1 1
, where ? is a scalar
Select the correct answer using the
code given below.
(a) 1 and 2 (b) 2 and 3
(c) 1 and 3 (d) 1, 2 and 3
Ê
(d) For a square matrix A
Statement 1
( ) A A
- -
=
1 1
Statement 1 is true
Statement 2
det ( )
det
A
A
-
=
1
1
Statement 2 is correct
Statement 3
( ) ? ? A A
- -
=
1 1
, where ? is a scalar.
So, Statement 3 is correct.
13. Which one of the following factors
does the expansion of the
determinant
x
x
x
y
y
y
2
3
3
5
5
10
3
9
27
contain ?
(a) x - 3
(b) x y -
(c) y - 3
(d) x y - 3
Ê
(a) We have,
x y
x y
x y
3
5 9
10 27
2 3
3 5
?
?
?
?
?
?
?
?
2
[C C C
1 1 3
? - mes,]
=
-
-
-
?
?
?
?
?
?
?
?
x y
x y
x y
3 3
9 5 9
27 10 27
2 3
3 5
= - +
+ +
?
?
?
?
?
?
?
?
( ) x
y
x y
x x y
3
1 3
3 5 9
9 3 10 27
3
2 5
14. What is the adjoint of the matrix
cos ( )
sin ( )
sin ( )
cos( )
-
- -
- -
-
?
?
?
?
?
?
?
?
?
?
?
(a)
cos
sin
sin
cos
?
?
?
? -
- ?
?
?
?
?
?
(b)
cos
sin
sin
cos
?
?
?
?
?
?
?
?
?
?
(c)
cos
sin
sin
cos
?
?
?
? -
?
?
?
?
?
?
(d)
cos
sin
sin
cos
?
?
?
?
- ?
?
?
?
?
?
Ê
(a) We have,
A =
- - -
- - -
?
?
?
?
?
?
cos ( ) sin ( )
sin ( ) cos ( )
? ?
? ?
A =
?
?
?
?
?
?
cos sin
sin cos
? ?
? ?
Now, C
11
= cos ?
C
12
= - sin ?
C
21
= - sin ?
C
22
= cos ?
adj A
T
=
-
-
?
?
?
?
?
?
cos sin
sin cos
? ?
? ?
=
-
-
?
?
?
?
?
?
cos sin
sin cos
? ?
? ?
15. What is the value of
- +
?
?
?
?
?
? +
- -
?
?
?
?
?
?
1 3
2
1 3
2
3 3
i i
a n
wherei = -1 ?
(a) 3 (b) 2
(c) 1 (d) 0
Ê
(b) We have,
- + ?
?
?
?
?
? +
- - ?
?
?
?
?
?
1 3
2
1 3
2
3 3
i i
n n
= + ( ) ( ) ? ?
3 2 3 n n
Q ? ? =
- +
=
- - ?
?
?
?
?
?
?
?
?
?
?
?
?
?
1 3
2
1 3
2
2
i i
,
= + ( ) ( ) ? ?
3 3 2 n n
( ) ( ) 1 1
2 n n
+ [Q?
3
1 = ]
= + 1 1= 2
16. There are 17 cricket players, out of
which 5 players can bowl. In how
many ways can a team of 11 players
be selected so as to include 3
bowlers?
(a)C ( , ) 17 11
(b)C ( , ) 12 8
(c)C C ( , ) ( , ) 17 5 5 3 ×
(d)C C ( , ) ( , ) 5 3 12 8 ×
Ê
(d) There are 17 cricket players, out of
which 5 players can bowl.
required number of ways = ×
12
8
5
3
C C
= × C C ( , ) ( , ) 12 8 5 3
17. What is the value of
log log ?
9 8
27 32 +
(a)
7
2
(b)
19
6
(c) 4 (d) 7
Ê
(b) We have,
log log
9 8
27 32 +
= + log log
3
3
2
5
2
3 3 2
= +
3
2
3
5
3
2
3 2
log log
Q log log
a
n
a m
b
n
m
b =
?
?
?
?
?
?
= +
3
2
5
3
=
19
6
18. If A and B are two invertible square
matrices of same order, then what is
( ) AB
-1
equal to?
(a) B A
- - 1 1
(b) A B
- - 1 1
(c) B A
-1
(d) A B
-1
Ê
(a) If A and B are two invertible square
matrices of same order, then
( ) AB B A
- - -
=
1 1 1
19. If a b c + + = 0, then one of the
solutions of
a x
c
b
c
b x
a
b
a
c x
-
-
-
= 0 is
(a) x a =
(b) x
a b c
=
+ + 3
2
2 2 2
( )
(c) x
a b c
=
+ + 2
3
2 2 2
( )
(d) x = 0
Ê
(d) We have,
a x c b
c b x a
b a c x
-
-
-
?
?
?
?
?
?
?
?
= 0
R R R R
1 1 2 3
? + +
?
a b c x a b c x a b c x
c b x a
b a c x
+ + - + + - + + -
-
-
?
?
?
?
?
?
?
?
= 0
?
- - -
-
-
?
?
?
?
?
?
?
?
=
x x x
c b x a
b a c x
0
? ( ) -x
1 1 1
0 c b x a
b a c x
-
-
?
?
?
?
?
?
?
?
=
? x = 0
Hence, x = 0 is a solution
20. What should be the value of x, so
that the matrix
2
8
4
-
?
?
?
?
?
?
x
does not
have an inverse?
(a) 16 (b) -16
(c) 8 (d) -8
Ê
(b) Let, A
x
=
-
?
?
?
?
?
?
2 4
8
Matrix does not have any solution if
| | A = 0
2 32 0 x + =
2 32 x = -
x = -
32
2
x = - 16
21. The system of equations
2 3 5 x y z + - =
3 2 2 5 x y z - + =
and 5 3 16 x y z - - =
(a) is inconsistent
(b) is consistent, with a unique solution
(c) is consistent, with infinitely many
solutions
(d) has its solution lying along X-axis in
three-dimensional space
Ê
(b) The system of equations
2 3 5 x y z + - =
3 2 2 5 x y z - + =
and 5 3 0 x y z - - =
A =
-
-
- -
?
?
?
?
?
?
?
?
?
?
2 1 3
3 2 2
5 3 1
| | [ ( ) ( )] [ ( ) ( )] A = - - - - - - - 2 2 1 2 3 1 3 1 2 5
+ - - - - ( ) [ ( ) ( )( )] 3 3 3 2 5
= - - - 2 8 1 13 3 1 ( ) ( ) ( )
= + - 16 13 3 = ? 26 0
So, System is consistent with unique
solution.
22. Which one of the following is correct
in respect of the cube roots of unity?
(a) They are collinear
(b) They lie on a circle of radius 3
(c) They form an equilateral triangle
(d) None of the above
Ê
(c) We know that, cube roots of unity is
1
2
, , ? ? , where ? =
- + 1 3
2
i
and
?
2
1 3
2
=
- - i
3
They form an equilateral triangle.
23. Ifu v , andw (all positive) are thep
th
,
q
th
and r
th
terms of a GP, then the
determinant of the matrix
ln
ln
ln
u
v
w
p
q
r
1
1
1
?
?
?
?
?
?
?
?
?
?
is
(a) 0
(b) 1
(c) ( )( ) ( ) p q q r r p - - -
(d) ln ln ln u v w × ×
Ê
(a) Given that u v , and w are the p
th
,q
th
andr
th
term of GP
?u aR v aR
p q
= =
- - 1 1
, [Qa aR
n
n
=
-1
]
and w aR
r
=
- 1
We have,
ln
ln
ln
u p
v q
w r
1
1
1
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
-
-
-
ln
ln
ln
a R p
a R q
a R r
p
q
r
1
1
1
1
1
1
=
+ -
+ -
+ -
?
?
?
?
?
?
?
?
ln ln
ln ln
ln ln
a p R p
a q R q
a r R r
1 1
1 1
1 1
=
?
?
?
?
?
?
?
?
+
-
-
-
ln
ln
ln
( ) ln
( ) ln
( )
a p
a q
a r
p R p
q R q
r
1
1
1
1 1
1 1
1 ln R r 1
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
+
-
-
-
?
?
?
?
?
?
?
?
ln ln a
p
q
r
R
p p
q q
r r
1 1
1 1
1 1
1 1
1 1
1 1
C C C
2 2 3
? -
= +
- -
- -
- -
?
?
?
?
?
?
?
?
0
1 1 1
1 1 1
1 1 1
ln R
p p
q q
r r
= 0
24. Let the coefficient of the middle term
of the binomial expansion of
( ) 1
2
+x
n
be a and those of two
middle terms of the binomial
expansion of ( ) 1
2 1
+
-
x
n
be ß and ?.
Which one of the following relations
is correct?
(a) a ß ? > + (b) a ß ? < +
(c) a ß ? = + (d) a ß? =
Ê
(c) We have,( ) 1
2
+ x
n
Middle term = +
?
?
?
?
?
?
2
2
1
n
th
term
= + ( ) n 1
th
term
Coefficient of ( ) n + 1
th
term =
2n
n
C
a =
2n
n
C
Again, we have binomial expansion of
( ) 1
2 1
+
-
x
n
coefficient of middle terms
are,
? ß =
- 2 1 n
n
C
and ? =
-
-
2 1
1
n
n
C
Now, ß ? + = +
- -
-
2 1 2 1
1
n
n
n
n
C C
[ ] Q
n
r
n
r
n
r
C C C + =
-
+
1
1
=
2n
n
C
25. Let A x x = ? - = = [ : ] R 1 1 ,
B y y = ? - = = [ : ] R 1 1 andS be the
subset of A B × , defined by
S x y A B x y = ? × + = [( , ) : ]
2 2
1 .
Which one of the following is
correct ?
(a)S is a one-one function from A into B
(b)S is a many-one function from A intoB
(c)S is a bijective mapping from A intoB
(d)S is not a function
Ê
(d) Given that,
A x R x = ? - = = { : } 1 1 ,
B y R y = ? - = < { : } 1 1
andS x y A B x y = ? × + = { ( , ) : }
2 2
1
By vertical line test. when we draw a
vertical line, then line cuts the circle in two
points. Hence,S is not a function.
26. Let T
r
be the r
th
term of an AP for
r = 1 2 3 , , , .... If for some distinct
positive integers m and n we have
T n
m
= 1 / and T
n
= 1 / m, then what
isT
mn
equal to ?
(a) ( ) mn
-1
(b) m n
- -
+
1 1
(c) 1
(d) 0
Ê
(c) Let first term of an AP is a and
common difference isd
Given that,
T
n
m
=
1
a m d
n
+ - = ( ) 1
1
...(i)
and T
m
n
=
1
a n d
m
+ - = ( ) 1
1
...(ii)
Subtracting Eq. (ii) from Eq. (i), we get
( ) ( ) m d n d
n m
- - - = - 1 1
1 1
? ( ) m n d
m n
mn
- =
-
? d
mn
=
1
Put in Eq. (i),
a m
mn n
+ - = ( ) 1
1 1
? a
n mn n
+ - =
1 1 1
? a
mn
=
1
Now, T a mn d
mn
= + - ( ) 1
= + -
1
1
1
mn
mn
mn
( )
= + -
1
1
1
mn mn
T
mn
= 1
27. Suppose f x ( ) is such a quadratic
expression that it is positive for all
real x.
Ifg x f x f x f x ( ) ( ) ( ) ( ) = + ' + '' ,
then for any real x
(a) g x ( ) < 0 (b) g x ( ) > 0
(c) g x ( ) = 0 (d) g x ( ) = 0
Ê
(b) Given that f x ( ) is a quadratic
expression
Letf x ax bx c a ( ) , = + + >
2
0
? b ac
2
4 0 - < [ ( ) ] ? > f x 0
? b ac
2
4 <
Now, f x ax b ' = + ( ) 2
and f x a ' ' = ( ) 2
We have,
g x f x f x f x ( ) ( ) ( ) ( ) = + ' + ' '
= + + + + + ax bx c ax b a
2
2 2
= + + + + + ax b a x a b c
2
2 2 ( )
Now, ( ) ( ) b a a a b c + - + + 2 4 2
2
= + + - - - b ab a a ab ac
2 2 2
4 4 8 4 4
= - - b ac a
2 2
4 8 < 0
[ ] Qb ac
2
4 0 - <
? g x ( ) > 0
4
Y
X X'
Y'
-1
-1
1
1
(–½, 3 v
?
Y
A (1,2)
X
120º
120º
120º
B
C
2
)
(–½, 3 v
?
2
)
Y´
X´
1
Page 5


1. What is the value of log
7
log
7
7 7 7 equal to ?
(a) 3 7
2
log (b) 1 3 7
2
- log
(c) 1 3 2
7
- log (d)
7
8
Ê
(c) We have,
7 7 7 7 7 7 7
1
2
1
4
1
8
7
8
= · · =
Now, log log
7 7
7 7 7
= log log ( )
7 7
7
8
7
=
?
?
?
?
?
?
log
7
7
8
= - log log
7 7
7 8
[Qlog log log
m
n
m n = - ]
= - log log
7 7
3
7 2 = - 1 3 2
7
log
[Qlog log
b
n
b
a n a = ]
2. If an infinite GP has the first term x
and the sum 5, then which one of the
following is correct?
(a) x < -10
(b) - < < 10 0 x
(c) 0 10 < < x
(d) x > 10
Ê
(c) Given that first term of an infinity GP is
x and sum = 5
?
x
r 1
5
-
=
[Q sum of infinity GP =
-
a
r 1
]
?
x
r
5
1 = -
? r
x
= - 1
5
Where, | | r < 1
? - < - < 1
5
1
x
? - < - < 2
5
0
x
? - < - < 10 0 x
? 10 0 > > x
3. Consider the following expressions
1. x x
x
+ -
2
1
2. ax bx x c
d
x
e
x
2
2
+ + - + -
3. 3 5 x x ab
2
- +
4.
2
2 3
x ax b - +
5.
1 2
5 x x
-
+
Which of the above are rational
expressions?
(a) 1, 4 and 5 (b) 1, 3, 4 and 5
(c) 2, 4 and 5 (d) 1 and 2
Ê
(a) We know that, rational expressions
are those expression which can be write
in the form of
p x
q x
q x
( )
( )
, ( ) ? 0
So, 1, 4, 5 are rational expressions
4. A square matrix A is called
orthogonal if
(a) A A =
2
(b) ' =
-
A A
1
(c) A A =
-1
(d) A A = '
where A' is the transpose of A
Ê
(b) A square matrix is called an
orthogonal matrix if AA I ' =
multiply. by A
- 1
? A AA A I
- -
' =
1 1
( ) ?IA A ' =
- 1
? A A ' =
- 1
5. IfA, B and C are subsets of a
universal set, then which one of the
following is not correct?
(a) A B C A B A C ? n = ? n ? ( ) ( ) ( )
(b) A A B B A A '? ? = 'n ' ? ( ) ( )
(c) A B C C B A '? ? = 'n ' n ' ( ) ( )
(d) ( ) ( ) ( ) A B C A C B C n ? = ? n ?
where A' is the complement of A
Ê
(c) Let A B , and C are subsets of a
universal set.
Let A B C = = = { } , { } , { } 1 2 3
? = { , , } 1 2 3 , A' = { , } 2 3 ,B' = { , } 1 3 ,
C ' = { , } 1 2
by checking options, we get
LHS = A B C '? ? ( )
= ? { , } { } 2 3
= { , } 2 3
RHS = ( ) C B A ' n ' n '
= n n ( { , } { , } ) { , } 1 2 1 3 2 3
= ' n ( { } ) { , } 2 2 3
= n { , } { , } 1 3 2 3
= { } 3
LHS ? RHS
So, option (c) is wrong
6. Let x be the number of integers
lying between 2999 and 8001 which
have at least two digits equal.Thenx
is equal to
(a) 2480 (b) 2481
(c) 2482 (d) 2483
Ê
(b) We have, x be the number lying
between 2999 and 8001
if repetition allowed
total numbers = × × × = 5 10 10 10 5000
if repetition not allowed
?total numbers = × × × = 5 9 8 7 2520
PAPER : I Mathematics
So, x = atleast two digit repeated
= - + 5000 2520 1
= 2481
[Q add 1 because of number 8000]
7. The sum of the series
3 1
1
3
1
9
- + - +.... is equal to
(a)
20
9
(b)
9
20
(c)
9
4
(d)
4
9
Ê
(c) Given series
3 1
1
3
1
9
- + - + ..... are in GP
? r =
- 1
3
S
n
=
- -
?
?
?
?
?
?
3
1
1
3
QS
a
r
n
=
-
?
?
?
?
?
?
1
=
3
4
3
=
9
4
Directions (Q. Nos. 8 and 9)
Consider the information given below
and answer the two items that follow.
A survey was conducted among 300
students. It was found that 125 students
like to play cricket, 145 students like to
play football and 90 students like to play
tennis. 32 students like to play exactly two
games out of the three games.
8. How many stdudents like to play all
the three games ?
(a) 14 (b) 21
(c) 28 (d) 35
Ê
(a) Let,
A be the set of students like to play cricket
B be the set of students like to play
football.
C be the set of students like to play tennis.
We have,
n A B C ( ) ? ? = 300
n A ( ) = 125
n B ( ) = 145
n C ( ) = 90
n A B C n A n B n C ( ) ( ) ( ) ( ) ? ? = + +
- n + n + n [ ( ) ( ) ( )] n A B n B C n C A
+ n n n A B C ( )
? 300 125 145 90 = + +
- n + n + n [ ( ) ( ) ( )] n A B n B C n C A
+ n n n A B C ( )
?n A B n B C n C A ( ) ( ) ( ) n + n + n
= + n n 60 n A B C ( ) ...(i)
Again,
n A B n B C n C A ( ) ( ) ( ) n + n + n
- n n = 3 32 n A B C ( )
?n A B n B C n C A ( ) ( ) ( ) n + n + n
= + n n 32 3n A B C ( ) ...(ii)
From Eqs. (i) and (ii), we get
60 32 + n n = n A B C ( )
+ n n 3n A B C ( )
? 2 28 n A B C ( ) n n =
? n A B C ( ) n n = 14
9. How many students like to play
exactly only one game?
(a) 196 (b) 228
(c) 254 (d) 268
Ê
(c) Number of students like to play
exactly one
game = + + n A n B n C ( ) ( ) ( )
- n + n + n 2 [ ( ) ( ) ( )] n A B n B C n C A
+ n n 3n A B C ( )
= + + - + × 125 145 90 2 32 3 14 [ ]
+ × 3 14
= - 360 106
= 254
10. If a and ß ( 0) ? are the roots of the
quadratic equation x x
2
0 + - = a ß ,
then the quadratic expression
- + + x x
2
a ß, where x R ? has
(a) Least value -
1
4
(b) Least value -
9
4
(c) Greatest value
1
4
(d) Greatest value
9
4
Ê
(d) a and ß are the roots of quadratic
equation.
x x
2
0 + - = a ß
So, (aß ß = - ) ?aß ß + = 0
? ß a ( ) + = 1 0
a = - 1 [Qß ? 0]
a ß a + = -
? 2 0 a ß + =
? ß = 2
? - + + x x
2
a ß [Qa ß = - = 1 2 , ]
= - - + x x
2
2
Greatest value = -
-
-
?
?
?
?
?
?
2
1
4 1
2
( )
( )
[Q Greatest value = -
?
?
?
?
?
? c
b
a
2
2
4
]
= -
-
?
?
?
?
?
? 2
1
4
= + 2
1
4
=
9
4
11. What is the coefficient of the middle
term in the binomial expansion of
( ) 2 3
4
+ x ?
(a) 6 (b) 12
(c) 108 (d) 216
Ê
(d) We have, ( ) 2 3
4
+ x
Here,n = 4, so middle term is
4
2
1 +
?
?
?
?
?
?
th
= 3
rd
term
T C x
3
4
2
2 2
2 3 = × × ( )
[T C a b
r
n
r
r n r
+
-
=
1
]1
=
×
×
× ×
4 3
2 1
4 9
2
x
T x
3
2
216 =
Hence, coefficient of middle term is 216.
12. For a square matrix A, which of the
following properties hold?
1. ( ) A A
- -
=
1 1
2. det ( )
det
A
A
-
=
1
1
3. ( ) ? ? A A
- -
=
1 1
, where ? is a scalar
Select the correct answer using the
code given below.
(a) 1 and 2 (b) 2 and 3
(c) 1 and 3 (d) 1, 2 and 3
Ê
(d) For a square matrix A
Statement 1
( ) A A
- -
=
1 1
Statement 1 is true
Statement 2
det ( )
det
A
A
-
=
1
1
Statement 2 is correct
Statement 3
( ) ? ? A A
- -
=
1 1
, where ? is a scalar.
So, Statement 3 is correct.
13. Which one of the following factors
does the expansion of the
determinant
x
x
x
y
y
y
2
3
3
5
5
10
3
9
27
contain ?
(a) x - 3
(b) x y -
(c) y - 3
(d) x y - 3
Ê
(a) We have,
x y
x y
x y
3
5 9
10 27
2 3
3 5
?
?
?
?
?
?
?
?
2
[C C C
1 1 3
? - mes,]
=
-
-
-
?
?
?
?
?
?
?
?
x y
x y
x y
3 3
9 5 9
27 10 27
2 3
3 5
= - +
+ +
?
?
?
?
?
?
?
?
( ) x
y
x y
x x y
3
1 3
3 5 9
9 3 10 27
3
2 5
14. What is the adjoint of the matrix
cos ( )
sin ( )
sin ( )
cos( )
-
- -
- -
-
?
?
?
?
?
?
?
?
?
?
?
(a)
cos
sin
sin
cos
?
?
?
? -
- ?
?
?
?
?
?
(b)
cos
sin
sin
cos
?
?
?
?
?
?
?
?
?
?
(c)
cos
sin
sin
cos
?
?
?
? -
?
?
?
?
?
?
(d)
cos
sin
sin
cos
?
?
?
?
- ?
?
?
?
?
?
Ê
(a) We have,
A =
- - -
- - -
?
?
?
?
?
?
cos ( ) sin ( )
sin ( ) cos ( )
? ?
? ?
A =
?
?
?
?
?
?
cos sin
sin cos
? ?
? ?
Now, C
11
= cos ?
C
12
= - sin ?
C
21
= - sin ?
C
22
= cos ?
adj A
T
=
-
-
?
?
?
?
?
?
cos sin
sin cos
? ?
? ?
=
-
-
?
?
?
?
?
?
cos sin
sin cos
? ?
? ?
15. What is the value of
- +
?
?
?
?
?
? +
- -
?
?
?
?
?
?
1 3
2
1 3
2
3 3
i i
a n
wherei = -1 ?
(a) 3 (b) 2
(c) 1 (d) 0
Ê
(b) We have,
- + ?
?
?
?
?
? +
- - ?
?
?
?
?
?
1 3
2
1 3
2
3 3
i i
n n
= + ( ) ( ) ? ?
3 2 3 n n
Q ? ? =
- +
=
- - ?
?
?
?
?
?
?
?
?
?
?
?
?
?
1 3
2
1 3
2
2
i i
,
= + ( ) ( ) ? ?
3 3 2 n n
( ) ( ) 1 1
2 n n
+ [Q?
3
1 = ]
= + 1 1= 2
16. There are 17 cricket players, out of
which 5 players can bowl. In how
many ways can a team of 11 players
be selected so as to include 3
bowlers?
(a)C ( , ) 17 11
(b)C ( , ) 12 8
(c)C C ( , ) ( , ) 17 5 5 3 ×
(d)C C ( , ) ( , ) 5 3 12 8 ×
Ê
(d) There are 17 cricket players, out of
which 5 players can bowl.
required number of ways = ×
12
8
5
3
C C
= × C C ( , ) ( , ) 12 8 5 3
17. What is the value of
log log ?
9 8
27 32 +
(a)
7
2
(b)
19
6
(c) 4 (d) 7
Ê
(b) We have,
log log
9 8
27 32 +
= + log log
3
3
2
5
2
3 3 2
= +
3
2
3
5
3
2
3 2
log log
Q log log
a
n
a m
b
n
m
b =
?
?
?
?
?
?
= +
3
2
5
3
=
19
6
18. If A and B are two invertible square
matrices of same order, then what is
( ) AB
-1
equal to?
(a) B A
- - 1 1
(b) A B
- - 1 1
(c) B A
-1
(d) A B
-1
Ê
(a) If A and B are two invertible square
matrices of same order, then
( ) AB B A
- - -
=
1 1 1
19. If a b c + + = 0, then one of the
solutions of
a x
c
b
c
b x
a
b
a
c x
-
-
-
= 0 is
(a) x a =
(b) x
a b c
=
+ + 3
2
2 2 2
( )
(c) x
a b c
=
+ + 2
3
2 2 2
( )
(d) x = 0
Ê
(d) We have,
a x c b
c b x a
b a c x
-
-
-
?
?
?
?
?
?
?
?
= 0
R R R R
1 1 2 3
? + +
?
a b c x a b c x a b c x
c b x a
b a c x
+ + - + + - + + -
-
-
?
?
?
?
?
?
?
?
= 0
?
- - -
-
-
?
?
?
?
?
?
?
?
=
x x x
c b x a
b a c x
0
? ( ) -x
1 1 1
0 c b x a
b a c x
-
-
?
?
?
?
?
?
?
?
=
? x = 0
Hence, x = 0 is a solution
20. What should be the value of x, so
that the matrix
2
8
4
-
?
?
?
?
?
?
x
does not
have an inverse?
(a) 16 (b) -16
(c) 8 (d) -8
Ê
(b) Let, A
x
=
-
?
?
?
?
?
?
2 4
8
Matrix does not have any solution if
| | A = 0
2 32 0 x + =
2 32 x = -
x = -
32
2
x = - 16
21. The system of equations
2 3 5 x y z + - =
3 2 2 5 x y z - + =
and 5 3 16 x y z - - =
(a) is inconsistent
(b) is consistent, with a unique solution
(c) is consistent, with infinitely many
solutions
(d) has its solution lying along X-axis in
three-dimensional space
Ê
(b) The system of equations
2 3 5 x y z + - =
3 2 2 5 x y z - + =
and 5 3 0 x y z - - =
A =
-
-
- -
?
?
?
?
?
?
?
?
?
?
2 1 3
3 2 2
5 3 1
| | [ ( ) ( )] [ ( ) ( )] A = - - - - - - - 2 2 1 2 3 1 3 1 2 5
+ - - - - ( ) [ ( ) ( )( )] 3 3 3 2 5
= - - - 2 8 1 13 3 1 ( ) ( ) ( )
= + - 16 13 3 = ? 26 0
So, System is consistent with unique
solution.
22. Which one of the following is correct
in respect of the cube roots of unity?
(a) They are collinear
(b) They lie on a circle of radius 3
(c) They form an equilateral triangle
(d) None of the above
Ê
(c) We know that, cube roots of unity is
1
2
, , ? ? , where ? =
- + 1 3
2
i
and
?
2
1 3
2
=
- - i
3
They form an equilateral triangle.
23. Ifu v , andw (all positive) are thep
th
,
q
th
and r
th
terms of a GP, then the
determinant of the matrix
ln
ln
ln
u
v
w
p
q
r
1
1
1
?
?
?
?
?
?
?
?
?
?
is
(a) 0
(b) 1
(c) ( )( ) ( ) p q q r r p - - -
(d) ln ln ln u v w × ×
Ê
(a) Given that u v , and w are the p
th
,q
th
andr
th
term of GP
?u aR v aR
p q
= =
- - 1 1
, [Qa aR
n
n
=
-1
]
and w aR
r
=
- 1
We have,
ln
ln
ln
u p
v q
w r
1
1
1
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
-
-
-
ln
ln
ln
a R p
a R q
a R r
p
q
r
1
1
1
1
1
1
=
+ -
+ -
+ -
?
?
?
?
?
?
?
?
ln ln
ln ln
ln ln
a p R p
a q R q
a r R r
1 1
1 1
1 1
=
?
?
?
?
?
?
?
?
+
-
-
-
ln
ln
ln
( ) ln
( ) ln
( )
a p
a q
a r
p R p
q R q
r
1
1
1
1 1
1 1
1 ln R r 1
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
+
-
-
-
?
?
?
?
?
?
?
?
ln ln a
p
q
r
R
p p
q q
r r
1 1
1 1
1 1
1 1
1 1
1 1
C C C
2 2 3
? -
= +
- -
- -
- -
?
?
?
?
?
?
?
?
0
1 1 1
1 1 1
1 1 1
ln R
p p
q q
r r
= 0
24. Let the coefficient of the middle term
of the binomial expansion of
( ) 1
2
+x
n
be a and those of two
middle terms of the binomial
expansion of ( ) 1
2 1
+
-
x
n
be ß and ?.
Which one of the following relations
is correct?
(a) a ß ? > + (b) a ß ? < +
(c) a ß ? = + (d) a ß? =
Ê
(c) We have,( ) 1
2
+ x
n
Middle term = +
?
?
?
?
?
?
2
2
1
n
th
term
= + ( ) n 1
th
term
Coefficient of ( ) n + 1
th
term =
2n
n
C
a =
2n
n
C
Again, we have binomial expansion of
( ) 1
2 1
+
-
x
n
coefficient of middle terms
are,
? ß =
- 2 1 n
n
C
and ? =
-
-
2 1
1
n
n
C
Now, ß ? + = +
- -
-
2 1 2 1
1
n
n
n
n
C C
[ ] Q
n
r
n
r
n
r
C C C + =
-
+
1
1
=
2n
n
C
25. Let A x x = ? - = = [ : ] R 1 1 ,
B y y = ? - = = [ : ] R 1 1 andS be the
subset of A B × , defined by
S x y A B x y = ? × + = [( , ) : ]
2 2
1 .
Which one of the following is
correct ?
(a)S is a one-one function from A into B
(b)S is a many-one function from A intoB
(c)S is a bijective mapping from A intoB
(d)S is not a function
Ê
(d) Given that,
A x R x = ? - = = { : } 1 1 ,
B y R y = ? - = < { : } 1 1
andS x y A B x y = ? × + = { ( , ) : }
2 2
1
By vertical line test. when we draw a
vertical line, then line cuts the circle in two
points. Hence,S is not a function.
26. Let T
r
be the r
th
term of an AP for
r = 1 2 3 , , , .... If for some distinct
positive integers m and n we have
T n
m
= 1 / and T
n
= 1 / m, then what
isT
mn
equal to ?
(a) ( ) mn
-1
(b) m n
- -
+
1 1
(c) 1
(d) 0
Ê
(c) Let first term of an AP is a and
common difference isd
Given that,
T
n
m
=
1
a m d
n
+ - = ( ) 1
1
...(i)
and T
m
n
=
1
a n d
m
+ - = ( ) 1
1
...(ii)
Subtracting Eq. (ii) from Eq. (i), we get
( ) ( ) m d n d
n m
- - - = - 1 1
1 1
? ( ) m n d
m n
mn
- =
-
? d
mn
=
1
Put in Eq. (i),
a m
mn n
+ - = ( ) 1
1 1
? a
n mn n
+ - =
1 1 1
? a
mn
=
1
Now, T a mn d
mn
= + - ( ) 1
= + -
1
1
1
mn
mn
mn
( )
= + -
1
1
1
mn mn
T
mn
= 1
27. Suppose f x ( ) is such a quadratic
expression that it is positive for all
real x.
Ifg x f x f x f x ( ) ( ) ( ) ( ) = + ' + '' ,
then for any real x
(a) g x ( ) < 0 (b) g x ( ) > 0
(c) g x ( ) = 0 (d) g x ( ) = 0
Ê
(b) Given that f x ( ) is a quadratic
expression
Letf x ax bx c a ( ) , = + + >
2
0
? b ac
2
4 0 - < [ ( ) ] ? > f x 0
? b ac
2
4 <
Now, f x ax b ' = + ( ) 2
and f x a ' ' = ( ) 2
We have,
g x f x f x f x ( ) ( ) ( ) ( ) = + ' + ' '
= + + + + + ax bx c ax b a
2
2 2
= + + + + + ax b a x a b c
2
2 2 ( )
Now, ( ) ( ) b a a a b c + - + + 2 4 2
2
= + + - - - b ab a a ab ac
2 2 2
4 4 8 4 4
= - - b ac a
2 2
4 8 < 0
[ ] Qb ac
2
4 0 - <
? g x ( ) > 0
4
Y
X X'
Y'
-1
-1
1
1
(–½, 3 v
?
Y
A (1,2)
X
120º
120º
120º
B
C
2
)
(–½, 3 v
?
2
)
Y´
X´
1
28. Consider the following in respect of
matrices A,B andC of same order.
1. ( ) A B C A B C + + ' = ' + ' + '
2. ( ) AB A B ' = ' '
3. ( ) ABC C B A ' = ' ' '
Where A' is the transpose of the
matrix A. Which of the above are
correct?
(a) 1 and 2 (b) 2 and 3
(c) 1 and 3 (d) 1, 2 and 3
Ê
(c) Given that A B , andC are matrices of
same order
Statement 1
( ) A B C A B C ' + + ' = ' + ' + '
[ ( ) ] Q A B A B + ' = ' + '
So, Statement 1 is correct
Statement 2
We know that,
( ) AB B A ' = ' '
Hence, Statement 2 is incorrect
Statement 3
( ) ABC C B A ' = ' ' ' [ ( ) ] Q AB B A ' = ' '
Hence,  Statement 3 is correct.
29. The sum of the binary numbers
( ) 11011
2
, ( ) 10110110
2
and
( ) 10011 0
2
x y is the binary numbers
( ) 101101101
2
. What are the values of
x andy ?
(a) x y = = 1 1 , (b) x y = = 1 0 ,
(c) x y = = 0 1 , (d) x y = = 0 0 ,
Ê
(b) Sum of the binary number
( ) , ( ) 11011 10110110
2 2
and ( ) 10011
2
xoy is
( ) 101101101
2
So, (101101101 )
- 10110110
10110111
- 11011
10011100
Compare with ( ) 10011
2
x y o
We get, x = 1and y = 0
30. Let matrix B be the adjoint of a
square matrix A, I be the identity
matrix of same order asA. Ifk( ) ? 0 is
the determinant of the matrix A,
then what is AB equal to ?
(a) l (b) kl (c)k l
2
(d) ( / ) 1 k l
Ê
(b) Given,
B adjA l = = , identity Matrix
( A) = K
? AB A A = ( ) adj = ( ) A l = kl
31. If ( . ) 02 2
x
= and log .
10
2 03010 = , then
what is the value of x to the nearest
tenth?
(a) -10 0 . (b) -0 5 .
(c) -0 4 . (d) -0 2 .
Ê
(c) We have,
( . ) 0 2 2
x
=
taking log
10
both side
x log . log
10 10
02 2 =
? x log log
10 10
2
10
2
?
?
?
?
?
?
=
? x [log log ] log
10 10 10
2 10 2 - =
?x [ . ] . 0 3010 1 0 3010 - = [ log ] Q
a
a = 1
? x = - ˜ -
0 3010
0 6990
0 43
.
.
.
32. The total number of 5-digit numbers
that can be composed of distinct
digits from 0 to 9 is
(a) 45360 (b) 30240
(c) 27216 (d) 15120
Ê
(c) 5-digit number that can be
composed by distinct digits from 0 to 9 is
given as
required number = × × × × 9 9 8 7 6
= 27216
33. What is the determinant of the
matrix
x
z
y
y
x
z
y z
z x
x y
+
+
+
?
?
?
?
?
?
?
?
?
?
?
(a) ( ) ( ) ( ) x y y z z x - - -
(b) ( ) ( ) x y y z - -
(c) ( ) ( ) y z z x - -
(d) ( ) ( ) z x x y z - + +
2
Ê
(d) We have,
x y y z
z x z x
y z x y
+
+
+
?
?
?
?
?
?
?
?
R R R R
1 1 2 3
? + +
x y z x y z x y z
z x z x
y z x y
+ + + + + +
+
+
?
?
?
?
?
?
?
?
2 ( )
= + + +
+
?
?
?
?
?
?
?
?
( ) x y z z x z x
y z x y
1 1 2
C C C C C C
2 2 1 3 3 1
2 ? - ? - and
= + + - -
- -
?
?
?
?
?
?
?
?
( ) x y z z x z x z
y z y x y
1 0 0
= + + - - ( ) [ ( ) ( ) x y z x z x y 1
- - - ( ) ( )] x z z y
= + + - - - + ( ) [( ) ( )] x y z x z x y z y
= + + - - ( ) ( ) ( ) x y z x z x z
= + + - ( ) ( ) x y z z x
2
34. If A, B and C are the angles of a
triangle and
1 1
1
1
1
2 2 2
I A
A A
B
B B
C
C
+
+
+
+
+
+
sin
sin
sin
sin sin
sin
sin sin Sin C
= 0,
then which one of the following is
correct?
(a) The triangle ABC is isosceles
(b) The triangle ABC is equilateral
(c) The triangle ABC is scalene
(d)No conclusion can be drawn with
regard to the nature of the triangle
Ê
(a) We have,
1 1
1 1
1
1
2 2 2
+ +
+ +
+
+
sin sin
sin sin sin sin
sin A B
A A B B
C
C Sin Sin C
?
?
?
?
?
?
?
?
?
?
= 0
R R R R R R
1 1 2 3 3 2
? - ? - ,
- - -
+ + +
- -
sin sin sin
sin sin sin
sin sin sin
A B C
A B C
A B
1 1 1
1 1
2 2 2
1
0
C -
?
?
?
?
?
?
?
?
=
R R R
2 2 1
? +
- - -
- - -
?
?
?
?
?
?
?
?
=
sin sin sin
cos cos cos
A B C
A B C
1 1 1 0
2 2 2
R R R
3 3 2
? +
?
sin sin sin
cos cos cos
A B C
A B C
1 1 1
1 1 1
0
2 2 2
- - -
?
?
?
?
?
?
?
?
=
?
sin sin sin
sin sin sin
A B C
A B C
1 1 1 0
2 2 2
?
?
?
?
?
?
?
?
=
[C C C
1 1 2
? - C C C
2 2 3
? - ]
?
sin sin sin sin sin
sin sin sin sin sin
A B B C C
A B B C
- -
- -
0 0 1
2 2 2 2 2
C
?
?
?
?
?
?
?
?
= 0
?(sin sin ) (sin sin ) A B B C - -
1 1
0 0 0
2
sin
sin sin sin sin sin
C
A C B C C + +
?
?
?
?
?
?
?
?
?
?
? Sin sin A B - = 0
or sin sin B C - = 0
? sin sin A B =
and sin sin B C =
? A B =
and B C =
So, ABC is an isosceles triangle
5
9 9 8 7 6