NDA Solved Paper 2019 - I | NDA (National Defence Academy) Past Year Papers PDF Download

 Page 1


1.
What is thenth term of the sequence
25 125 625 3125 , , , , - - …?
(a) ( ) -
-
5
2 1 n
(b) ( ) -
+
1 5
2 1 n n
(c) ( ) -
- +
1 5
2 1 1 n n
(d) ( ) -
- +
1 5
1 1 n n
Ê
(d) Given, sequence 25,
- - 125 625 3125 , ,
Here,
T
T
T
T
2
1
3
2
= = ...........
So, this sequence in GP whose common
ratio is -5.
then a r = = - 25 5 ,
?nth term of sequence =
-
ar
n 1
= -
-
25 5
1
( )
n
= - ×
- -
( ) 1 5 5
1 2 1 n n
= -
- +
( ) 1 5
1 1 n n
2.
Suppose X = { , , , } 1 2 3 4 and R is a
relation on X . If
R = {( , ), ( , ), ( , ), ( , ), ( , ), 1 1 2 2 3 3 1 2 2 1
( , ), ( , )} 2 3 3 2 , then which one of the
following is correct?
(a) R is reflexive and symmetric, but not
transitive
(b) R is symmetric and transitive, but not
reflexive
(c) R is reflexive and transitive, but not
symmetric
(d) R is neither reflexive nor transitive, but
symmetric
Ê
(d) We have, X = { , , , } 1 2 3 4
R = { (1,1), (2, 2), (3, 3),
(1, 2), (2, 1), (2, 3), (3, 2)}
Since, ( , ) 4 4 ?R,
Hence, R is not reflexive.
Since, ( , ) , ( , ) 1 2 2 3 ? ? R R but
( , ) , 1 3 ?R R is not transitive.
(1, 2), (2, 3) ?R
and also (2, 1), (3, 2) ?R
?R is symmetric.
Hence,R is neither reflexive nor
transitive but symmetric.
3. A relationR is defined on the setN of
natural numbers as
xRy x xy y ? - + =
2 2
4 3 0. Then,
which one of the following is
correct?
(a) R is reflexive and symmetric, but not
transitive
(b) R is reflexive and transitive, but not
symmetric
(c) R is reflexive, symmetric and transitive
(d) R is reflexive, but neither symmetric
nor transitive
Ê
(d) Given, xRy ? x xy y
2 2
4 3 0 - + =
For reflexive
xRx ?x x x
2 2 2
4 3 0 - + =
So, ( , ) , x x R x N ? ? ?
Hence, R is reflexive.
For symmetric
xRy ? x xy y
2 2
4 3 0 - + =
? yRx ? y xy x
2 2
4 3 - +
It is not clear, that y xy x
2 2
4 3 - + is equal
to zero or not.
i.e. ( , ) x y R ? but ( , ) , y x R x y N ? ·? ?
Hence,R is not symmetric.
For transitive
xRy ?x xy y
2 2
4 3 0 - + =
yRz ? y yz z
2 2
4 3 - + = 0 (let)
xRz ?x xz z
2 2
4 3 - +
It is not clear, thatx xz z
2 2
4 3 - + is
equal to zero or not.
So, ( , ) ,( , ) x y R y z R ? ?
? ( , ) , , x z R x y z N ? ? ?
Hence,R is not transitive.
4.
If A x Z x = ? - = { : }
3
1 0 and
B x Z x x = ? + + = { : }
2
1 0 , where,Z
is set of complex numbers, then
what is A B n equal to?
(a) Null set
(b)
- + - - ?
?
?
?
?
?
1 3
2
1 3
2
i i
,
(c)
- + - - ?
?
?
?
?
?
1 3
4
1 3
4
i i
,
(d)
1 3
2
1 3
2
+ - ?
?
?
?
?
?
i i
,
Ê
(b) We have, A x Z x = ? - = { : }
3
1 0
and B x Z x x = ? + + = { : }
2
1 0
A
i i
=
- + - - ?
?
?
?
?
?
1
1 3
2
1 3
2
, ,
B
i i
=
- + - - ?
?
?
?
?
?
1 3
2
1 3
2
,
A B
i i
n =
- + - - ?
?
?
?
?
?
1 3
2
1 3
2
,
5.
Consider the following statements
for the two non-empty setsA andB.
1. ( ) ( ) ( ) A B A B A B n ? n ? n
= ? A B
2. ( ( )) A A B A B ? n = ?
Which of the above statements
is/are correct?
(a) Only 1 (b) Only 2
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(a) We have,
1. ( ) ( ) ( ) A B A B A B A B n ? n ? n = ?
LHS = n ? n ? n ( ) ( ) ( ) A B A B A B
= n ? ? n { ( )} ( ) A B B A B
[by distributive property]
PAPER : I Mathematics
Page 2


1.
What is thenth term of the sequence
25 125 625 3125 , , , , - - …?
(a) ( ) -
-
5
2 1 n
(b) ( ) -
+
1 5
2 1 n n
(c) ( ) -
- +
1 5
2 1 1 n n
(d) ( ) -
- +
1 5
1 1 n n
Ê
(d) Given, sequence 25,
- - 125 625 3125 , ,
Here,
T
T
T
T
2
1
3
2
= = ...........
So, this sequence in GP whose common
ratio is -5.
then a r = = - 25 5 ,
?nth term of sequence =
-
ar
n 1
= -
-
25 5
1
( )
n
= - ×
- -
( ) 1 5 5
1 2 1 n n
= -
- +
( ) 1 5
1 1 n n
2.
Suppose X = { , , , } 1 2 3 4 and R is a
relation on X . If
R = {( , ), ( , ), ( , ), ( , ), ( , ), 1 1 2 2 3 3 1 2 2 1
( , ), ( , )} 2 3 3 2 , then which one of the
following is correct?
(a) R is reflexive and symmetric, but not
transitive
(b) R is symmetric and transitive, but not
reflexive
(c) R is reflexive and transitive, but not
symmetric
(d) R is neither reflexive nor transitive, but
symmetric
Ê
(d) We have, X = { , , , } 1 2 3 4
R = { (1,1), (2, 2), (3, 3),
(1, 2), (2, 1), (2, 3), (3, 2)}
Since, ( , ) 4 4 ?R,
Hence, R is not reflexive.
Since, ( , ) , ( , ) 1 2 2 3 ? ? R R but
( , ) , 1 3 ?R R is not transitive.
(1, 2), (2, 3) ?R
and also (2, 1), (3, 2) ?R
?R is symmetric.
Hence,R is neither reflexive nor
transitive but symmetric.
3. A relationR is defined on the setN of
natural numbers as
xRy x xy y ? - + =
2 2
4 3 0. Then,
which one of the following is
correct?
(a) R is reflexive and symmetric, but not
transitive
(b) R is reflexive and transitive, but not
symmetric
(c) R is reflexive, symmetric and transitive
(d) R is reflexive, but neither symmetric
nor transitive
Ê
(d) Given, xRy ? x xy y
2 2
4 3 0 - + =
For reflexive
xRx ?x x x
2 2 2
4 3 0 - + =
So, ( , ) , x x R x N ? ? ?
Hence, R is reflexive.
For symmetric
xRy ? x xy y
2 2
4 3 0 - + =
? yRx ? y xy x
2 2
4 3 - +
It is not clear, that y xy x
2 2
4 3 - + is equal
to zero or not.
i.e. ( , ) x y R ? but ( , ) , y x R x y N ? ·? ?
Hence,R is not symmetric.
For transitive
xRy ?x xy y
2 2
4 3 0 - + =
yRz ? y yz z
2 2
4 3 - + = 0 (let)
xRz ?x xz z
2 2
4 3 - +
It is not clear, thatx xz z
2 2
4 3 - + is
equal to zero or not.
So, ( , ) ,( , ) x y R y z R ? ?
? ( , ) , , x z R x y z N ? ? ?
Hence,R is not transitive.
4.
If A x Z x = ? - = { : }
3
1 0 and
B x Z x x = ? + + = { : }
2
1 0 , where,Z
is set of complex numbers, then
what is A B n equal to?
(a) Null set
(b)
- + - - ?
?
?
?
?
?
1 3
2
1 3
2
i i
,
(c)
- + - - ?
?
?
?
?
?
1 3
4
1 3
4
i i
,
(d)
1 3
2
1 3
2
+ - ?
?
?
?
?
?
i i
,
Ê
(b) We have, A x Z x = ? - = { : }
3
1 0
and B x Z x x = ? + + = { : }
2
1 0
A
i i
=
- + - - ?
?
?
?
?
?
1
1 3
2
1 3
2
, ,
B
i i
=
- + - - ?
?
?
?
?
?
1 3
2
1 3
2
,
A B
i i
n =
- + - - ?
?
?
?
?
?
1 3
2
1 3
2
,
5.
Consider the following statements
for the two non-empty setsA andB.
1. ( ) ( ) ( ) A B A B A B n ? n ? n
= ? A B
2. ( ( )) A A B A B ? n = ?
Which of the above statements
is/are correct?
(a) Only 1 (b) Only 2
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(a) We have,
1. ( ) ( ) ( ) A B A B A B A B n ? n ? n = ?
LHS = n ? n ? n ( ) ( ) ( ) A B A B A B
= n ? ? n { ( )} ( ) A B B A B
[by distributive property]
PAPER : I Mathematics
= n ? n ( ) ( ) A U A B
[ ] Q B B U ? =
= ? n A A B ( )
= ? n ? ( ) ( ) A A A B
= n ? U A B ( ) = ? = A B RHS
Hence, 1 is correct.
2. A A B A B ? n = ? ( )
LHS = ? n A A B ( )
= ? n ? ( ) ( ) A A A B
= n ? U A B ( )
= ? ? ? A B A B
Hence, 2 is false.
?Only 1 is correct.
6.
Let X be a non-empty set and let
A B C , , be subsets of X . Consider the
following statements.
1. A C A B C B ? ? n ? n ( ) ( ),
( ) ( ) A B C B ? ? ?
2. ( ) ( ) A B C B n ? n for all sets
B A C ? ?
3. ( ) ( ) A B C B ? ? ? for all sets
B A C ? ?
Which of the above statements are
correct?
(a) Only 1 and 2 (b) Only 2 and 3
(c) Only 1 and 3 (d) 1, 2 and 3
Ê
(d) Let X = { , , , } 12 3 4
A B C = = = { , }, { , , }, { , , } 1 2 2 3 4 1 2 3
A ? C
A B C B n = n = { }, { , } 2 2 3
Clearly, ( ) ( ) A B C B n ? n
A B C B ? = ? = { , , , }, ( ) { , , , } 1 2 3 4 1 2 3 4
( ) ( ) A B C B ? ? n
Hence, Statement 1 is correct.
2.( ) ( ) A B C B n ? n for all setsB ?A ?C
Hence, Statement 2 is also correct.
3. ( ) ( ) A B C B ? ? ? for all sets
B ?A ? C
Hence, Statement 3 is also correct.
7.
IfB =
?
?
?
?
?
?
?
?
?
?
3 2 0
2 4 0
1 1 0
, then what is adjoint
of B equal to?
(a)
0 0 0
0 0 0
2 1 8 - -
?
?
?
?
?
?
?
?
?
?
(b)
0 0 2
0 0 1
0 0 8
-
-
?
?
?
?
?
?
?
?
?
?
(c)
0 0 2
0 0 1
0 0 0
?
?
?
?
?
?
?
?
?
?
(d) It does not exist
Ê
(a) We have,B =
?
?
?
?
?
?
?
?
?
?
3 2 0
2 4 0
1 1 0
Co-factor of B,
B B B
11 12 13
0 0 2 = = = - , ,
B B B
21 22 23
0 0 1 = = = - , ,
B B B
31 32 33
0 0 8 = = = , ,
adj B
B B B
B B B
B B B
=
?
?
?
?
?
?
?
?
?
?
'
11 12 13
21 22 23
31 32 33
=
-
-
?
?
?
?
?
?
?
?
?
?
'
0 0 2
0 0 1
0 0 8
=
- -
?
?
?
?
?
?
?
?
?
?
0 0 0
0 0 0
2 1 8
8.
What are the roots of the equation
| | ? x x x
2
6 2 - - = +
(a) -2 1 4 , , (b) 0 2 4 , ,
(c) 0 1 4 , , (d) -2 2 4 , ,
Ê
(d) We have,
| | x x x
2
6 2 - - = +
? | ( ) ( )| x x x - - = + 3 2 2
Case I x < 2
x x x
2
6 2 - - = +
x x
2
2 8 0 - - =
x x x
x x x
2
4 2 8 0
4 2 4 0
- + - =
- + - =
?
?
?
?
?
?
( ) ( )
( ) ( ) x x - + = 4 2 0
x = - 2 but x ? 4 [Qx < 2]
Case II 2 3 = < x
x x x
2
6 2 - - = - + ( )
x x x
2
6 2 0 - - + + =
x
2
4 0 - =
x = ± 2
x x = ? - 2 2 but [Qx ?( , ) 2 3 ]
Case III x = 3
x x x
2
6 2 - - = +
x x
2
2 8 0 - - =
( ) ( ) x x + - = 2 4 0
x x = ? - 4 2 but [Qx = 3]
? x = - 2 2 4 , ,
9.
If A =
?
?
?
?
?
?
0 1
1 0
, then the matrix A is
a/an
(a) singular matrix
(b) involutory matrix
(c) nilpotent matrix
(d) idempotent matrix
Ê
(b) We have, A =
?
?
?
?
?
?
0 1
1 0
| | A = - 1
Since,| | A ? 0
Hence, A is not singular.
A A A
2
0 1
1 0
0 1
1 0
= · =
?
?
?
?
?
?
·
?
?
?
?
?
?
=
?
?
?
?
?
?
1 0
0 1
A I
2
=
Hence, A is involutory matrix.
10.
If
x i
y i
i i
i
-
-
?
?
?
?
?
?
?
?
?
?
= +
3 1
1
0 2
6 11 , then what
are the values of x and y
respectively?
(a) -3 4 , (b) 3 4 ,
(c) 3 4 , - (d) -3, -4
Ê
(a) We have,
x i
y i
i i
i
-
-
= +
3 1
1
0 2
6 11
? x i y i i ( ) ( ) - + - - - = + 2 3 2 6 11
? 2 3 2 6 11 x y x y i i + + - + = + ( )
On equating real and imaginary parts, on
both sides,
we get 2 3 6 x y + = ...(i)
and - + = x y 2 11 ...(ii)
On solving Eqs. (i) and (ii), we get
x = - 3 and y = 4
11.
The common roots of the equations
z z z
3 2
2 2 1 0 + + + =
andz z
2017 2018
1 0 + + = are
(a) -1 , ? (b) 1
2
,?
(c) -1
2
, ? (d) ? ? ,
2
Ê
(d) We have, z z z
3 2
2 2 1 0 + + + =
( ) ( ) z z z + + + = 1 1 0
2
? z + = 1 0 or z z
2
1 0 + + =
z = -1
or z=
- ± - 1 1 4
2
=
- + - - 1 3
2
1 3
2
i i
, = ? ? ,
2
Now, z z
2017 2018
1 0 + + =
Put z = - 1 ,
LHS = - + - + ( ) ( ) 1 1 1
2017 2018
= - + + 1 1 1
= ? 1 0 (RHS)
? z = - 1is not a root of equation.
Put z =?,
LHS = + + ( ) ( ) ? ?
2017 2018
1
= + + ( ) . ( ) . ? ? ? ?
3 672 3 672 2
1
= + + ? ?
2
1 [ ] Q ?
3
1 =
[ ] Q1 0
2
+ + = ? ?
= = 0 RHS
? z =? is a root of equation.
put z = ?
2
,
LHS = + + ( ) ( ) ? ?
2 2017 2 2018
1
= + + ? ?
4034 4036
1
= + + ( ) . ( ) . ? ? ? ?
3 1344 2 3 1345
1
= + + = ? ?
2
1 0 RHS
? z = ?
2
is a root of equation.
Hence, ? ? ,
2
are the common roots of
these equations.
2
Page 3


1.
What is thenth term of the sequence
25 125 625 3125 , , , , - - …?
(a) ( ) -
-
5
2 1 n
(b) ( ) -
+
1 5
2 1 n n
(c) ( ) -
- +
1 5
2 1 1 n n
(d) ( ) -
- +
1 5
1 1 n n
Ê
(d) Given, sequence 25,
- - 125 625 3125 , ,
Here,
T
T
T
T
2
1
3
2
= = ...........
So, this sequence in GP whose common
ratio is -5.
then a r = = - 25 5 ,
?nth term of sequence =
-
ar
n 1
= -
-
25 5
1
( )
n
= - ×
- -
( ) 1 5 5
1 2 1 n n
= -
- +
( ) 1 5
1 1 n n
2.
Suppose X = { , , , } 1 2 3 4 and R is a
relation on X . If
R = {( , ), ( , ), ( , ), ( , ), ( , ), 1 1 2 2 3 3 1 2 2 1
( , ), ( , )} 2 3 3 2 , then which one of the
following is correct?
(a) R is reflexive and symmetric, but not
transitive
(b) R is symmetric and transitive, but not
reflexive
(c) R is reflexive and transitive, but not
symmetric
(d) R is neither reflexive nor transitive, but
symmetric
Ê
(d) We have, X = { , , , } 1 2 3 4
R = { (1,1), (2, 2), (3, 3),
(1, 2), (2, 1), (2, 3), (3, 2)}
Since, ( , ) 4 4 ?R,
Hence, R is not reflexive.
Since, ( , ) , ( , ) 1 2 2 3 ? ? R R but
( , ) , 1 3 ?R R is not transitive.
(1, 2), (2, 3) ?R
and also (2, 1), (3, 2) ?R
?R is symmetric.
Hence,R is neither reflexive nor
transitive but symmetric.
3. A relationR is defined on the setN of
natural numbers as
xRy x xy y ? - + =
2 2
4 3 0. Then,
which one of the following is
correct?
(a) R is reflexive and symmetric, but not
transitive
(b) R is reflexive and transitive, but not
symmetric
(c) R is reflexive, symmetric and transitive
(d) R is reflexive, but neither symmetric
nor transitive
Ê
(d) Given, xRy ? x xy y
2 2
4 3 0 - + =
For reflexive
xRx ?x x x
2 2 2
4 3 0 - + =
So, ( , ) , x x R x N ? ? ?
Hence, R is reflexive.
For symmetric
xRy ? x xy y
2 2
4 3 0 - + =
? yRx ? y xy x
2 2
4 3 - +
It is not clear, that y xy x
2 2
4 3 - + is equal
to zero or not.
i.e. ( , ) x y R ? but ( , ) , y x R x y N ? ·? ?
Hence,R is not symmetric.
For transitive
xRy ?x xy y
2 2
4 3 0 - + =
yRz ? y yz z
2 2
4 3 - + = 0 (let)
xRz ?x xz z
2 2
4 3 - +
It is not clear, thatx xz z
2 2
4 3 - + is
equal to zero or not.
So, ( , ) ,( , ) x y R y z R ? ?
? ( , ) , , x z R x y z N ? ? ?
Hence,R is not transitive.
4.
If A x Z x = ? - = { : }
3
1 0 and
B x Z x x = ? + + = { : }
2
1 0 , where,Z
is set of complex numbers, then
what is A B n equal to?
(a) Null set
(b)
- + - - ?
?
?
?
?
?
1 3
2
1 3
2
i i
,
(c)
- + - - ?
?
?
?
?
?
1 3
4
1 3
4
i i
,
(d)
1 3
2
1 3
2
+ - ?
?
?
?
?
?
i i
,
Ê
(b) We have, A x Z x = ? - = { : }
3
1 0
and B x Z x x = ? + + = { : }
2
1 0
A
i i
=
- + - - ?
?
?
?
?
?
1
1 3
2
1 3
2
, ,
B
i i
=
- + - - ?
?
?
?
?
?
1 3
2
1 3
2
,
A B
i i
n =
- + - - ?
?
?
?
?
?
1 3
2
1 3
2
,
5.
Consider the following statements
for the two non-empty setsA andB.
1. ( ) ( ) ( ) A B A B A B n ? n ? n
= ? A B
2. ( ( )) A A B A B ? n = ?
Which of the above statements
is/are correct?
(a) Only 1 (b) Only 2
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(a) We have,
1. ( ) ( ) ( ) A B A B A B A B n ? n ? n = ?
LHS = n ? n ? n ( ) ( ) ( ) A B A B A B
= n ? ? n { ( )} ( ) A B B A B
[by distributive property]
PAPER : I Mathematics
= n ? n ( ) ( ) A U A B
[ ] Q B B U ? =
= ? n A A B ( )
= ? n ? ( ) ( ) A A A B
= n ? U A B ( ) = ? = A B RHS
Hence, 1 is correct.
2. A A B A B ? n = ? ( )
LHS = ? n A A B ( )
= ? n ? ( ) ( ) A A A B
= n ? U A B ( )
= ? ? ? A B A B
Hence, 2 is false.
?Only 1 is correct.
6.
Let X be a non-empty set and let
A B C , , be subsets of X . Consider the
following statements.
1. A C A B C B ? ? n ? n ( ) ( ),
( ) ( ) A B C B ? ? ?
2. ( ) ( ) A B C B n ? n for all sets
B A C ? ?
3. ( ) ( ) A B C B ? ? ? for all sets
B A C ? ?
Which of the above statements are
correct?
(a) Only 1 and 2 (b) Only 2 and 3
(c) Only 1 and 3 (d) 1, 2 and 3
Ê
(d) Let X = { , , , } 12 3 4
A B C = = = { , }, { , , }, { , , } 1 2 2 3 4 1 2 3
A ? C
A B C B n = n = { }, { , } 2 2 3
Clearly, ( ) ( ) A B C B n ? n
A B C B ? = ? = { , , , }, ( ) { , , , } 1 2 3 4 1 2 3 4
( ) ( ) A B C B ? ? n
Hence, Statement 1 is correct.
2.( ) ( ) A B C B n ? n for all setsB ?A ?C
Hence, Statement 2 is also correct.
3. ( ) ( ) A B C B ? ? ? for all sets
B ?A ? C
Hence, Statement 3 is also correct.
7.
IfB =
?
?
?
?
?
?
?
?
?
?
3 2 0
2 4 0
1 1 0
, then what is adjoint
of B equal to?
(a)
0 0 0
0 0 0
2 1 8 - -
?
?
?
?
?
?
?
?
?
?
(b)
0 0 2
0 0 1
0 0 8
-
-
?
?
?
?
?
?
?
?
?
?
(c)
0 0 2
0 0 1
0 0 0
?
?
?
?
?
?
?
?
?
?
(d) It does not exist
Ê
(a) We have,B =
?
?
?
?
?
?
?
?
?
?
3 2 0
2 4 0
1 1 0
Co-factor of B,
B B B
11 12 13
0 0 2 = = = - , ,
B B B
21 22 23
0 0 1 = = = - , ,
B B B
31 32 33
0 0 8 = = = , ,
adj B
B B B
B B B
B B B
=
?
?
?
?
?
?
?
?
?
?
'
11 12 13
21 22 23
31 32 33
=
-
-
?
?
?
?
?
?
?
?
?
?
'
0 0 2
0 0 1
0 0 8
=
- -
?
?
?
?
?
?
?
?
?
?
0 0 0
0 0 0
2 1 8
8.
What are the roots of the equation
| | ? x x x
2
6 2 - - = +
(a) -2 1 4 , , (b) 0 2 4 , ,
(c) 0 1 4 , , (d) -2 2 4 , ,
Ê
(d) We have,
| | x x x
2
6 2 - - = +
? | ( ) ( )| x x x - - = + 3 2 2
Case I x < 2
x x x
2
6 2 - - = +
x x
2
2 8 0 - - =
x x x
x x x
2
4 2 8 0
4 2 4 0
- + - =
- + - =
?
?
?
?
?
?
( ) ( )
( ) ( ) x x - + = 4 2 0
x = - 2 but x ? 4 [Qx < 2]
Case II 2 3 = < x
x x x
2
6 2 - - = - + ( )
x x x
2
6 2 0 - - + + =
x
2
4 0 - =
x = ± 2
x x = ? - 2 2 but [Qx ?( , ) 2 3 ]
Case III x = 3
x x x
2
6 2 - - = +
x x
2
2 8 0 - - =
( ) ( ) x x + - = 2 4 0
x x = ? - 4 2 but [Qx = 3]
? x = - 2 2 4 , ,
9.
If A =
?
?
?
?
?
?
0 1
1 0
, then the matrix A is
a/an
(a) singular matrix
(b) involutory matrix
(c) nilpotent matrix
(d) idempotent matrix
Ê
(b) We have, A =
?
?
?
?
?
?
0 1
1 0
| | A = - 1
Since,| | A ? 0
Hence, A is not singular.
A A A
2
0 1
1 0
0 1
1 0
= · =
?
?
?
?
?
?
·
?
?
?
?
?
?
=
?
?
?
?
?
?
1 0
0 1
A I
2
=
Hence, A is involutory matrix.
10.
If
x i
y i
i i
i
-
-
?
?
?
?
?
?
?
?
?
?
= +
3 1
1
0 2
6 11 , then what
are the values of x and y
respectively?
(a) -3 4 , (b) 3 4 ,
(c) 3 4 , - (d) -3, -4
Ê
(a) We have,
x i
y i
i i
i
-
-
= +
3 1
1
0 2
6 11
? x i y i i ( ) ( ) - + - - - = + 2 3 2 6 11
? 2 3 2 6 11 x y x y i i + + - + = + ( )
On equating real and imaginary parts, on
both sides,
we get 2 3 6 x y + = ...(i)
and - + = x y 2 11 ...(ii)
On solving Eqs. (i) and (ii), we get
x = - 3 and y = 4
11.
The common roots of the equations
z z z
3 2
2 2 1 0 + + + =
andz z
2017 2018
1 0 + + = are
(a) -1 , ? (b) 1
2
,?
(c) -1
2
, ? (d) ? ? ,
2
Ê
(d) We have, z z z
3 2
2 2 1 0 + + + =
( ) ( ) z z z + + + = 1 1 0
2
? z + = 1 0 or z z
2
1 0 + + =
z = -1
or z=
- ± - 1 1 4
2
=
- + - - 1 3
2
1 3
2
i i
, = ? ? ,
2
Now, z z
2017 2018
1 0 + + =
Put z = - 1 ,
LHS = - + - + ( ) ( ) 1 1 1
2017 2018
= - + + 1 1 1
= ? 1 0 (RHS)
? z = - 1is not a root of equation.
Put z =?,
LHS = + + ( ) ( ) ? ?
2017 2018
1
= + + ( ) . ( ) . ? ? ? ?
3 672 3 672 2
1
= + + ? ?
2
1 [ ] Q ?
3
1 =
[ ] Q1 0
2
+ + = ? ?
= = 0 RHS
? z =? is a root of equation.
put z = ?
2
,
LHS = + + ( ) ( ) ? ?
2 2017 2 2018
1
= + + ? ?
4034 4036
1
= + + ( ) . ( ) . ? ? ? ?
3 1344 2 3 1345
1
= + + = ? ?
2
1 0 RHS
? z = ?
2
is a root of equation.
Hence, ? ? ,
2
are the common roots of
these equations.
2
12.
If C n C n ( , ) ( , ) 20 2 20 2 + = - , then
what isn equal to
(a) 8 (b) 10
(c) 12 (d) 16
Ê
(b) We have,C n C n ( , ) ( , ) 20 2 20 2 + = -
?
20
2
20
2
C C
n n + -
=
? n n + + - = 2 2 20
[Q
n
x
n
y
C C = ? x y n + = ]
? n = 10
13.
There are 10 points in a plane. No
three of these points are in a straight
line. What is the total number of
straight lines which can be formed
by joining the points?
(a) 90 (b) 45
(c) 40 (d) 30
Ê
(b) Given, 10 points in a plane where no
three of these points are in straight line.
Total number of straight line formed from
10 points is
10
2
10
2 8
10 9
2
45 C = =
×
=
!
! !
14. The equation px qx r
2
0 + + =
(where p q r , , , all are positive) has
distinct real rootsa andb. Which one
of the following is correct?
(a)a b > > 0 0 ,
(b)a b < < 0 0 ,
(c)a b > < 0 0 ,
(d)a b < > 0 0 ,
Ê
(b) Given, px qx r
2
0 + + = , where
p q r , , > 0anda andb are distinct roots.
? a b
q
p
+ =
-
and ab r =
Now, r > 0
? ab > 0
? a b > > 0 0 , ...(i)
or a b < < 0 0 , ...(ii)
Now,
-
<
q
p
0 q p , > 0
? a b + < 0
a b < < 0 0 , ...(iii)
From Eqs. (i), (ii) and (iii), we get
? a < 0 and b < 0
15.
If A = ? { , { , }} ? µ , then the power set
of A is
(a) { , { }, { }, { }} f f ? ?,µ
(b) {f, ? ?,µ ? ? µ { },{ },{ ,{ , }}}
(c) { { },{ },{ ,{ , }}} f, ? ?,µ ? ? µ
(d) {{ },{ },{ ,{ , }}} ? ?,µ ? ? µ
Ê
(b) We have, A = { , { , }} ? ? µ
{ }
P A ( ) { , { }, { , } , { , { , }}} = f ? ? µ ? ? µ
Directions (Q. Nos. 16 and 17) Read
the information carefully and answer
the given questions.
In a school, all the students play atleast one
of three indoor games– chess, carrom and
table tennis. 60 play chess, 50 play table
tennis, 48 play carrom, 12 play chess and
carrom, 15 play carrom and table tennis, 20
play table tennis and chess.
16.
What can be the minimum number of
students in the school?
(a) 123 (b) 111 (c) 95    (d) 63
Ê
(b) Let
A = Student play chess
B = Student play table tennis
C = Student play carrom
Given, n A n B nC ( ) , ( ) , ( ) = = = 60 50 48
n A B n B C ( ) , ( ) n = n = 20 15
n A C ( ) n = 12
For minimum number of students in
school
n A B C ( ) n n must be zero.
?n A B C n A n B nC ( ) ( ) ( ) ( ) ? ? = + +
- n - n n A B n B C ( ) ( )
- n + n n n A C n A B C ( ) ( )
= + + - - - + 60 50 48 20 15 12 0 = 111
17.
What can be the maximum number of
students in the school?
(a) 111 (b) 123
(c) 125 (d) 135
Ê
(b) For maximum number of students in
school n A B C ( ) n n must be 12.
? n A B C ( ) ? ?
= + + - - - + 60 50 48 20 15 12 12
= 123
18.
If A is an identity matrix of order 3,
then its inverse ( ) A
-1
(a) is equal to null matrix
(b) is equal to A
(c) is equal to 3A (d) does not exist
Ê
(b) Given, A =
?
?
?
?
?
?
?
?
?
?
1 0 0
0 1 0
0 0 1
? A A
-
=
?
?
?
?
?
?
?
?
?
?
=
1
1 0 0
0 1 0
0 0 1
19.
A is a square matrix of order 3 such
that its determinant is 4. What is the
determinant of its transpose?
(a) 64 (b) 36
(c) 32 (d) 4
Ê
(d) Given, | | A = 4
? | | A' = 4 [Q| | | |] A A = '
20.
From 6 programmers and 4 typists,
an office wants to recruit 5 people.
What is the number of ways this can
be done so as to recruit atleast one
typist?
(a) 209 (b) 210
(c) 246 (d) 242
Ê
(c) We have,
6 programmers and 4 typists
Number of ways of 5 recruit people such
that atleast one typist
= + +
4
1
6
4
4
2
6
3
4
3
6
2
C C C C C C
+
4
4
6
1
C C
= × + × + × + × 4 15 6 20 4 15 1 6
= + + + 60 120 60 6 = 246
21.
What is the number of terms in the
expansion of[( ) ( ) ] 2 3 2 3
2 2 2
x y x y - + ?
(a) 4 (b) 5
(c) 8 (d) 16
Ê
(b) Given, [( ) ( ) ] 2 3 2 3
2 2 2
x y x y - +
= - [ ] 4 9
2 2 4
x y
? Total number of terms = + = 4 1 5
22.
In the expansion of( ) 1 +ax
n
, the first
three terms are respectively 1 12 , x
and 64
2
x . What isn equal to?
(a) 6 (b) 9
(c) 10 (d) 12
Ê
(b) Given, first three terms of expansion
( ) , , 1 1 12 64
2
+ ax x x
n
is ,
Now,
( )
( )
1 1
1
2
2 2
+ = + +
-
+ ax nax
n n
a x
n
K
On equating first three terms, we get
na = 12 and
n n
a
( ) -
=
1
2
64
2
On putting the value ofa in
n n
a
( ) -
=
1
2
64
2
, we get
n n
n
( ) - ?
?
?
?
?
?
=
1
2
12
64
2
?
144 1
2
64
( ) n
n
-
=
? n = 9
23.
The numbers 1, 5 and 25 can be three
terms (not necessarily
consecutive)of
(a) only one AP
(b) more than one but finite numbers of
APs
(c) infinite number of APs
(d) finite number of GPs
Ê
(d) We have, 1, 5, 25 be three terms.
Clearly, 1, 5, 25 are finite number of GPs.
3
Page 4


1.
What is thenth term of the sequence
25 125 625 3125 , , , , - - …?
(a) ( ) -
-
5
2 1 n
(b) ( ) -
+
1 5
2 1 n n
(c) ( ) -
- +
1 5
2 1 1 n n
(d) ( ) -
- +
1 5
1 1 n n
Ê
(d) Given, sequence 25,
- - 125 625 3125 , ,
Here,
T
T
T
T
2
1
3
2
= = ...........
So, this sequence in GP whose common
ratio is -5.
then a r = = - 25 5 ,
?nth term of sequence =
-
ar
n 1
= -
-
25 5
1
( )
n
= - ×
- -
( ) 1 5 5
1 2 1 n n
= -
- +
( ) 1 5
1 1 n n
2.
Suppose X = { , , , } 1 2 3 4 and R is a
relation on X . If
R = {( , ), ( , ), ( , ), ( , ), ( , ), 1 1 2 2 3 3 1 2 2 1
( , ), ( , )} 2 3 3 2 , then which one of the
following is correct?
(a) R is reflexive and symmetric, but not
transitive
(b) R is symmetric and transitive, but not
reflexive
(c) R is reflexive and transitive, but not
symmetric
(d) R is neither reflexive nor transitive, but
symmetric
Ê
(d) We have, X = { , , , } 1 2 3 4
R = { (1,1), (2, 2), (3, 3),
(1, 2), (2, 1), (2, 3), (3, 2)}
Since, ( , ) 4 4 ?R,
Hence, R is not reflexive.
Since, ( , ) , ( , ) 1 2 2 3 ? ? R R but
( , ) , 1 3 ?R R is not transitive.
(1, 2), (2, 3) ?R
and also (2, 1), (3, 2) ?R
?R is symmetric.
Hence,R is neither reflexive nor
transitive but symmetric.
3. A relationR is defined on the setN of
natural numbers as
xRy x xy y ? - + =
2 2
4 3 0. Then,
which one of the following is
correct?
(a) R is reflexive and symmetric, but not
transitive
(b) R is reflexive and transitive, but not
symmetric
(c) R is reflexive, symmetric and transitive
(d) R is reflexive, but neither symmetric
nor transitive
Ê
(d) Given, xRy ? x xy y
2 2
4 3 0 - + =
For reflexive
xRx ?x x x
2 2 2
4 3 0 - + =
So, ( , ) , x x R x N ? ? ?
Hence, R is reflexive.
For symmetric
xRy ? x xy y
2 2
4 3 0 - + =
? yRx ? y xy x
2 2
4 3 - +
It is not clear, that y xy x
2 2
4 3 - + is equal
to zero or not.
i.e. ( , ) x y R ? but ( , ) , y x R x y N ? ·? ?
Hence,R is not symmetric.
For transitive
xRy ?x xy y
2 2
4 3 0 - + =
yRz ? y yz z
2 2
4 3 - + = 0 (let)
xRz ?x xz z
2 2
4 3 - +
It is not clear, thatx xz z
2 2
4 3 - + is
equal to zero or not.
So, ( , ) ,( , ) x y R y z R ? ?
? ( , ) , , x z R x y z N ? ? ?
Hence,R is not transitive.
4.
If A x Z x = ? - = { : }
3
1 0 and
B x Z x x = ? + + = { : }
2
1 0 , where,Z
is set of complex numbers, then
what is A B n equal to?
(a) Null set
(b)
- + - - ?
?
?
?
?
?
1 3
2
1 3
2
i i
,
(c)
- + - - ?
?
?
?
?
?
1 3
4
1 3
4
i i
,
(d)
1 3
2
1 3
2
+ - ?
?
?
?
?
?
i i
,
Ê
(b) We have, A x Z x = ? - = { : }
3
1 0
and B x Z x x = ? + + = { : }
2
1 0
A
i i
=
- + - - ?
?
?
?
?
?
1
1 3
2
1 3
2
, ,
B
i i
=
- + - - ?
?
?
?
?
?
1 3
2
1 3
2
,
A B
i i
n =
- + - - ?
?
?
?
?
?
1 3
2
1 3
2
,
5.
Consider the following statements
for the two non-empty setsA andB.
1. ( ) ( ) ( ) A B A B A B n ? n ? n
= ? A B
2. ( ( )) A A B A B ? n = ?
Which of the above statements
is/are correct?
(a) Only 1 (b) Only 2
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(a) We have,
1. ( ) ( ) ( ) A B A B A B A B n ? n ? n = ?
LHS = n ? n ? n ( ) ( ) ( ) A B A B A B
= n ? ? n { ( )} ( ) A B B A B
[by distributive property]
PAPER : I Mathematics
= n ? n ( ) ( ) A U A B
[ ] Q B B U ? =
= ? n A A B ( )
= ? n ? ( ) ( ) A A A B
= n ? U A B ( ) = ? = A B RHS
Hence, 1 is correct.
2. A A B A B ? n = ? ( )
LHS = ? n A A B ( )
= ? n ? ( ) ( ) A A A B
= n ? U A B ( )
= ? ? ? A B A B
Hence, 2 is false.
?Only 1 is correct.
6.
Let X be a non-empty set and let
A B C , , be subsets of X . Consider the
following statements.
1. A C A B C B ? ? n ? n ( ) ( ),
( ) ( ) A B C B ? ? ?
2. ( ) ( ) A B C B n ? n for all sets
B A C ? ?
3. ( ) ( ) A B C B ? ? ? for all sets
B A C ? ?
Which of the above statements are
correct?
(a) Only 1 and 2 (b) Only 2 and 3
(c) Only 1 and 3 (d) 1, 2 and 3
Ê
(d) Let X = { , , , } 12 3 4
A B C = = = { , }, { , , }, { , , } 1 2 2 3 4 1 2 3
A ? C
A B C B n = n = { }, { , } 2 2 3
Clearly, ( ) ( ) A B C B n ? n
A B C B ? = ? = { , , , }, ( ) { , , , } 1 2 3 4 1 2 3 4
( ) ( ) A B C B ? ? n
Hence, Statement 1 is correct.
2.( ) ( ) A B C B n ? n for all setsB ?A ?C
Hence, Statement 2 is also correct.
3. ( ) ( ) A B C B ? ? ? for all sets
B ?A ? C
Hence, Statement 3 is also correct.
7.
IfB =
?
?
?
?
?
?
?
?
?
?
3 2 0
2 4 0
1 1 0
, then what is adjoint
of B equal to?
(a)
0 0 0
0 0 0
2 1 8 - -
?
?
?
?
?
?
?
?
?
?
(b)
0 0 2
0 0 1
0 0 8
-
-
?
?
?
?
?
?
?
?
?
?
(c)
0 0 2
0 0 1
0 0 0
?
?
?
?
?
?
?
?
?
?
(d) It does not exist
Ê
(a) We have,B =
?
?
?
?
?
?
?
?
?
?
3 2 0
2 4 0
1 1 0
Co-factor of B,
B B B
11 12 13
0 0 2 = = = - , ,
B B B
21 22 23
0 0 1 = = = - , ,
B B B
31 32 33
0 0 8 = = = , ,
adj B
B B B
B B B
B B B
=
?
?
?
?
?
?
?
?
?
?
'
11 12 13
21 22 23
31 32 33
=
-
-
?
?
?
?
?
?
?
?
?
?
'
0 0 2
0 0 1
0 0 8
=
- -
?
?
?
?
?
?
?
?
?
?
0 0 0
0 0 0
2 1 8
8.
What are the roots of the equation
| | ? x x x
2
6 2 - - = +
(a) -2 1 4 , , (b) 0 2 4 , ,
(c) 0 1 4 , , (d) -2 2 4 , ,
Ê
(d) We have,
| | x x x
2
6 2 - - = +
? | ( ) ( )| x x x - - = + 3 2 2
Case I x < 2
x x x
2
6 2 - - = +
x x
2
2 8 0 - - =
x x x
x x x
2
4 2 8 0
4 2 4 0
- + - =
- + - =
?
?
?
?
?
?
( ) ( )
( ) ( ) x x - + = 4 2 0
x = - 2 but x ? 4 [Qx < 2]
Case II 2 3 = < x
x x x
2
6 2 - - = - + ( )
x x x
2
6 2 0 - - + + =
x
2
4 0 - =
x = ± 2
x x = ? - 2 2 but [Qx ?( , ) 2 3 ]
Case III x = 3
x x x
2
6 2 - - = +
x x
2
2 8 0 - - =
( ) ( ) x x + - = 2 4 0
x x = ? - 4 2 but [Qx = 3]
? x = - 2 2 4 , ,
9.
If A =
?
?
?
?
?
?
0 1
1 0
, then the matrix A is
a/an
(a) singular matrix
(b) involutory matrix
(c) nilpotent matrix
(d) idempotent matrix
Ê
(b) We have, A =
?
?
?
?
?
?
0 1
1 0
| | A = - 1
Since,| | A ? 0
Hence, A is not singular.
A A A
2
0 1
1 0
0 1
1 0
= · =
?
?
?
?
?
?
·
?
?
?
?
?
?
=
?
?
?
?
?
?
1 0
0 1
A I
2
=
Hence, A is involutory matrix.
10.
If
x i
y i
i i
i
-
-
?
?
?
?
?
?
?
?
?
?
= +
3 1
1
0 2
6 11 , then what
are the values of x and y
respectively?
(a) -3 4 , (b) 3 4 ,
(c) 3 4 , - (d) -3, -4
Ê
(a) We have,
x i
y i
i i
i
-
-
= +
3 1
1
0 2
6 11
? x i y i i ( ) ( ) - + - - - = + 2 3 2 6 11
? 2 3 2 6 11 x y x y i i + + - + = + ( )
On equating real and imaginary parts, on
both sides,
we get 2 3 6 x y + = ...(i)
and - + = x y 2 11 ...(ii)
On solving Eqs. (i) and (ii), we get
x = - 3 and y = 4
11.
The common roots of the equations
z z z
3 2
2 2 1 0 + + + =
andz z
2017 2018
1 0 + + = are
(a) -1 , ? (b) 1
2
,?
(c) -1
2
, ? (d) ? ? ,
2
Ê
(d) We have, z z z
3 2
2 2 1 0 + + + =
( ) ( ) z z z + + + = 1 1 0
2
? z + = 1 0 or z z
2
1 0 + + =
z = -1
or z=
- ± - 1 1 4
2
=
- + - - 1 3
2
1 3
2
i i
, = ? ? ,
2
Now, z z
2017 2018
1 0 + + =
Put z = - 1 ,
LHS = - + - + ( ) ( ) 1 1 1
2017 2018
= - + + 1 1 1
= ? 1 0 (RHS)
? z = - 1is not a root of equation.
Put z =?,
LHS = + + ( ) ( ) ? ?
2017 2018
1
= + + ( ) . ( ) . ? ? ? ?
3 672 3 672 2
1
= + + ? ?
2
1 [ ] Q ?
3
1 =
[ ] Q1 0
2
+ + = ? ?
= = 0 RHS
? z =? is a root of equation.
put z = ?
2
,
LHS = + + ( ) ( ) ? ?
2 2017 2 2018
1
= + + ? ?
4034 4036
1
= + + ( ) . ( ) . ? ? ? ?
3 1344 2 3 1345
1
= + + = ? ?
2
1 0 RHS
? z = ?
2
is a root of equation.
Hence, ? ? ,
2
are the common roots of
these equations.
2
12.
If C n C n ( , ) ( , ) 20 2 20 2 + = - , then
what isn equal to
(a) 8 (b) 10
(c) 12 (d) 16
Ê
(b) We have,C n C n ( , ) ( , ) 20 2 20 2 + = -
?
20
2
20
2
C C
n n + -
=
? n n + + - = 2 2 20
[Q
n
x
n
y
C C = ? x y n + = ]
? n = 10
13.
There are 10 points in a plane. No
three of these points are in a straight
line. What is the total number of
straight lines which can be formed
by joining the points?
(a) 90 (b) 45
(c) 40 (d) 30
Ê
(b) Given, 10 points in a plane where no
three of these points are in straight line.
Total number of straight line formed from
10 points is
10
2
10
2 8
10 9
2
45 C = =
×
=
!
! !
14. The equation px qx r
2
0 + + =
(where p q r , , , all are positive) has
distinct real rootsa andb. Which one
of the following is correct?
(a)a b > > 0 0 ,
(b)a b < < 0 0 ,
(c)a b > < 0 0 ,
(d)a b < > 0 0 ,
Ê
(b) Given, px qx r
2
0 + + = , where
p q r , , > 0anda andb are distinct roots.
? a b
q
p
+ =
-
and ab r =
Now, r > 0
? ab > 0
? a b > > 0 0 , ...(i)
or a b < < 0 0 , ...(ii)
Now,
-
<
q
p
0 q p , > 0
? a b + < 0
a b < < 0 0 , ...(iii)
From Eqs. (i), (ii) and (iii), we get
? a < 0 and b < 0
15.
If A = ? { , { , }} ? µ , then the power set
of A is
(a) { , { }, { }, { }} f f ? ?,µ
(b) {f, ? ?,µ ? ? µ { },{ },{ ,{ , }}}
(c) { { },{ },{ ,{ , }}} f, ? ?,µ ? ? µ
(d) {{ },{ },{ ,{ , }}} ? ?,µ ? ? µ
Ê
(b) We have, A = { , { , }} ? ? µ
{ }
P A ( ) { , { }, { , } , { , { , }}} = f ? ? µ ? ? µ
Directions (Q. Nos. 16 and 17) Read
the information carefully and answer
the given questions.
In a school, all the students play atleast one
of three indoor games– chess, carrom and
table tennis. 60 play chess, 50 play table
tennis, 48 play carrom, 12 play chess and
carrom, 15 play carrom and table tennis, 20
play table tennis and chess.
16.
What can be the minimum number of
students in the school?
(a) 123 (b) 111 (c) 95    (d) 63
Ê
(b) Let
A = Student play chess
B = Student play table tennis
C = Student play carrom
Given, n A n B nC ( ) , ( ) , ( ) = = = 60 50 48
n A B n B C ( ) , ( ) n = n = 20 15
n A C ( ) n = 12
For minimum number of students in
school
n A B C ( ) n n must be zero.
?n A B C n A n B nC ( ) ( ) ( ) ( ) ? ? = + +
- n - n n A B n B C ( ) ( )
- n + n n n A C n A B C ( ) ( )
= + + - - - + 60 50 48 20 15 12 0 = 111
17.
What can be the maximum number of
students in the school?
(a) 111 (b) 123
(c) 125 (d) 135
Ê
(b) For maximum number of students in
school n A B C ( ) n n must be 12.
? n A B C ( ) ? ?
= + + - - - + 60 50 48 20 15 12 12
= 123
18.
If A is an identity matrix of order 3,
then its inverse ( ) A
-1
(a) is equal to null matrix
(b) is equal to A
(c) is equal to 3A (d) does not exist
Ê
(b) Given, A =
?
?
?
?
?
?
?
?
?
?
1 0 0
0 1 0
0 0 1
? A A
-
=
?
?
?
?
?
?
?
?
?
?
=
1
1 0 0
0 1 0
0 0 1
19.
A is a square matrix of order 3 such
that its determinant is 4. What is the
determinant of its transpose?
(a) 64 (b) 36
(c) 32 (d) 4
Ê
(d) Given, | | A = 4
? | | A' = 4 [Q| | | |] A A = '
20.
From 6 programmers and 4 typists,
an office wants to recruit 5 people.
What is the number of ways this can
be done so as to recruit atleast one
typist?
(a) 209 (b) 210
(c) 246 (d) 242
Ê
(c) We have,
6 programmers and 4 typists
Number of ways of 5 recruit people such
that atleast one typist
= + +
4
1
6
4
4
2
6
3
4
3
6
2
C C C C C C
+
4
4
6
1
C C
= × + × + × + × 4 15 6 20 4 15 1 6
= + + + 60 120 60 6 = 246
21.
What is the number of terms in the
expansion of[( ) ( ) ] 2 3 2 3
2 2 2
x y x y - + ?
(a) 4 (b) 5
(c) 8 (d) 16
Ê
(b) Given, [( ) ( ) ] 2 3 2 3
2 2 2
x y x y - +
= - [ ] 4 9
2 2 4
x y
? Total number of terms = + = 4 1 5
22.
In the expansion of( ) 1 +ax
n
, the first
three terms are respectively 1 12 , x
and 64
2
x . What isn equal to?
(a) 6 (b) 9
(c) 10 (d) 12
Ê
(b) Given, first three terms of expansion
( ) , , 1 1 12 64
2
+ ax x x
n
is ,
Now,
( )
( )
1 1
1
2
2 2
+ = + +
-
+ ax nax
n n
a x
n
K
On equating first three terms, we get
na = 12 and
n n
a
( ) -
=
1
2
64
2
On putting the value ofa in
n n
a
( ) -
=
1
2
64
2
, we get
n n
n
( ) - ?
?
?
?
?
?
=
1
2
12
64
2
?
144 1
2
64
( ) n
n
-
=
? n = 9
23.
The numbers 1, 5 and 25 can be three
terms (not necessarily
consecutive)of
(a) only one AP
(b) more than one but finite numbers of
APs
(c) infinite number of APs
(d) finite number of GPs
Ê
(d) We have, 1, 5, 25 be three terms.
Clearly, 1, 5, 25 are finite number of GPs.
3
24.
The sum of ( ) p q + th and ( ) p q - th
terms of an AP is equal to
(a) ( ) 2p th term (b) ( ) 2q th term
(c) twice the p th term
(d) twice theq th term
Ê
(c) Let a is first term and d is common
difference of AP.
a a p q d
p q +
= + + - ( ) 1
and a a p q d
p q -
= + - - ( ) 1
Sum of ( ) p q + th and ( ) p q - th terms
= + = + -
+ -
a a a p d
p q p q
2 2 2 ( )
= + - 2 1 ( ( ) ) a p d = 2a
p
= twice of p th term
25. IfA is a square matrix of ordern > 1,
then which one of the following is
correct?
(a) det ( ) - = A det A
(b) det ( ) ( ) - = - A
n
1 det A
(c) det ( ) - = - A det A
(d) det ( ) - = A n det A
Ê
Sol. (b) We know that if A is a square
matrix of ordern > 1 , then
det( ) ( ) - = - A
n
1 det A
For example If A =
?
?
?
?
?
?
2 3
4 5
,
then - =
- -
- -
?
?
?
?
?
?
A
2 3
4 5
?det A =
?
?
?
?
?
?
2 3
4 5
= - 10 12 = - 2 …(i)
and det( ) - =
- -
- -
?
?
?
?
?
? A
2 3
4 5
= - 10 12 = - 2
= - - ( ) ( ) 1 2
2
[Q here n = 2]
= - ( ) 1
2
det A [from Eq. (i)]
if A =
-
?
?
?
?
?
?
?
?
?
?
1 2 3
3 1 0
4 3 2
Then, - =
- - -
- -
- -
?
?
?
?
?
?
?
?
?
?
A
1 2 3
3 1 0
4 3 2
? det A =
-
1 2 3
3 1 0
4 3 2
= - - - - - + - 1 2 0 2 6 0 3 9 4 ( ) ( ) ( )
= - + + 2 12 15 = 25
and det ( ) - =
- - -
- -
- -
?
?
?
?
?
?
?
?
?
A
1 2 3
3 1 0
4 3 2
= - - - + - - - - 1 2 0 2 6 0 3 9 4 ( ) ( ) ( )
= - - 2 12 15 = - 25
= - ( ) 1 25
3
[here n = 3]
= - ( ) 1
3
det A [from Eq. (i)]
26. What is the least value of
25 cosec 36sec
2 2
x x + ?
(a) 1
(b) 11
(c) 120
(d) 121
Ê
(d) Given, 25 36 cosec sec
2 2
x x +
= + + + 25 1 36 1
2 2
( cot ) ( tan ) x x
= + + + 25 25 36 36
2 2
cot tan x x
= + + + 25 36 25 36
2 2
cot tan x x
= + - + × × 61 5 6 2 5 6
2
( cot tan ) x x
= + = 61 60 121 [Qminimum value of
( cot tan ) 5 6 0
2
x x - = ]
? Minimum value of
25 cosec
2
x x + = 36 121
2
sec
Directions (Q. Nos. 27 and 28) Read
the information carefully and answer
the given questions.
Let A andB be3 3 × matrices with det A = 4
and det B = 3 .
27. What is det ( ) 2AB equal to?
(a) 96
(b) 72
(c) 48
(d) 36
Ê
(a) A and B be ( ) 3 3 × matrices with
det A = 4 and det B = 3
We know that,
det det det ( ) ( ) ( ) KAB K A B
n
= ×
where,n is the order or A andB,K is a real
number.
?det det det ( ) ( ) 2 2
3
AB A B = ×
[Qn = 3 and k = 2]
= × × 8 4 3
= 96
28. What is det ( ) 3
1
AB
-
equal to?
(a) 12 (b) 18
(c) 36 (d) 48
Ê
(c) A and B be ( ) 3 3 × matrices with
det A = 4 and det B = 3
We know that,
det det
det
( ) ( )
( )
KAB K A
B
n -
= ×
1
1
,
where n is the order of A and B, K is a
real number]
?det ( ) ( ) ( )
det
3 3
1
1 3
AB A
B
-
= × det
= × × 27 4
1
3
= 36
Directions (Q. Nos. 29 and 30) Read
the information carefully and answer
the given questions.
A complex number is given by
z
i
i
=
+
- -
1 2
1 1
2
( )
.
29.
What is the modulus ofz?
(a) 4 (b) 2 (c) 1 (d)
1
2
Ê
(c) We have, z
i
i
=
+
- -
1 2
1 1
2
( )
z
i
i
=
+
- - -
1 2
1 1 1 2 ( )
=
+
+
=
1 2
1 2
1
i
i
? | | z = 1
30.
What is the principal argument ofz?
(a) 0 (b)
p
4
(c)
p
2
(d) p
Ê
(a) arg ( ) tan
( )
( )
z
z
z
=
?
?
?
?
?
?
-1
lm
Re
=
?
?
?
?
?
?
= =
- -
tan tan
1 1
0
1
0 0
31.
What is the value of
sin cos sin sin
cos cos cos sin
34 236 56 124
28 88 178
° ° - ° °
° ° + ° 208°
?
(a) -2 (b) -1 (c) 2 (d) 1
Ê
(a) We have,
sin cos sin sin
cos cos cos sin
34 236 56 124
28 88 178
° ° - ° °
° ° + ° 208°
=
° ° + °
- ° ° + °
°
sin cos ( )
sin sin ( )
cos cos
34 180 56
56 90 34
28 88° + ° + °
° + °
cos ( )
sin ( )
90 88
180 28
=
- ° ° - ° °
° ° + °
sin cos sin cos
cos cos sin sin
34 56 56 34
28 88 88 28°
=
- ° + °
° - °
sin ( )
cos ( )
56 34
88 28
=
- °
°
sin
cos
90
60
=
-
= -
1
1
2
2
32.
tan54° can be expressed as
(a)
sin cos
sin cos
9 9
9 9
° + °
° - °
(b)
sin cos
sin cos
9 9
9 9
° - °
° + °
(c)
cos sin
cos sin
9 9
9 9
° + °
° - °
(d)
sin
cos
36
36
°
°
Ê
(c) We have, tan tan ( ) 54 45 9 ° = ° + °
=
° + °
- ° °
tan tan
tan tan
45 9
1 45 9
=
+ °
- °
1 9
1 9
tan
tan
=
° + °
° - °
cos sin
cos sin
9 9
9 9
4
Page 5


1.
What is thenth term of the sequence
25 125 625 3125 , , , , - - …?
(a) ( ) -
-
5
2 1 n
(b) ( ) -
+
1 5
2 1 n n
(c) ( ) -
- +
1 5
2 1 1 n n
(d) ( ) -
- +
1 5
1 1 n n
Ê
(d) Given, sequence 25,
- - 125 625 3125 , ,
Here,
T
T
T
T
2
1
3
2
= = ...........
So, this sequence in GP whose common
ratio is -5.
then a r = = - 25 5 ,
?nth term of sequence =
-
ar
n 1
= -
-
25 5
1
( )
n
= - ×
- -
( ) 1 5 5
1 2 1 n n
= -
- +
( ) 1 5
1 1 n n
2.
Suppose X = { , , , } 1 2 3 4 and R is a
relation on X . If
R = {( , ), ( , ), ( , ), ( , ), ( , ), 1 1 2 2 3 3 1 2 2 1
( , ), ( , )} 2 3 3 2 , then which one of the
following is correct?
(a) R is reflexive and symmetric, but not
transitive
(b) R is symmetric and transitive, but not
reflexive
(c) R is reflexive and transitive, but not
symmetric
(d) R is neither reflexive nor transitive, but
symmetric
Ê
(d) We have, X = { , , , } 1 2 3 4
R = { (1,1), (2, 2), (3, 3),
(1, 2), (2, 1), (2, 3), (3, 2)}
Since, ( , ) 4 4 ?R,
Hence, R is not reflexive.
Since, ( , ) , ( , ) 1 2 2 3 ? ? R R but
( , ) , 1 3 ?R R is not transitive.
(1, 2), (2, 3) ?R
and also (2, 1), (3, 2) ?R
?R is symmetric.
Hence,R is neither reflexive nor
transitive but symmetric.
3. A relationR is defined on the setN of
natural numbers as
xRy x xy y ? - + =
2 2
4 3 0. Then,
which one of the following is
correct?
(a) R is reflexive and symmetric, but not
transitive
(b) R is reflexive and transitive, but not
symmetric
(c) R is reflexive, symmetric and transitive
(d) R is reflexive, but neither symmetric
nor transitive
Ê
(d) Given, xRy ? x xy y
2 2
4 3 0 - + =
For reflexive
xRx ?x x x
2 2 2
4 3 0 - + =
So, ( , ) , x x R x N ? ? ?
Hence, R is reflexive.
For symmetric
xRy ? x xy y
2 2
4 3 0 - + =
? yRx ? y xy x
2 2
4 3 - +
It is not clear, that y xy x
2 2
4 3 - + is equal
to zero or not.
i.e. ( , ) x y R ? but ( , ) , y x R x y N ? ·? ?
Hence,R is not symmetric.
For transitive
xRy ?x xy y
2 2
4 3 0 - + =
yRz ? y yz z
2 2
4 3 - + = 0 (let)
xRz ?x xz z
2 2
4 3 - +
It is not clear, thatx xz z
2 2
4 3 - + is
equal to zero or not.
So, ( , ) ,( , ) x y R y z R ? ?
? ( , ) , , x z R x y z N ? ? ?
Hence,R is not transitive.
4.
If A x Z x = ? - = { : }
3
1 0 and
B x Z x x = ? + + = { : }
2
1 0 , where,Z
is set of complex numbers, then
what is A B n equal to?
(a) Null set
(b)
- + - - ?
?
?
?
?
?
1 3
2
1 3
2
i i
,
(c)
- + - - ?
?
?
?
?
?
1 3
4
1 3
4
i i
,
(d)
1 3
2
1 3
2
+ - ?
?
?
?
?
?
i i
,
Ê
(b) We have, A x Z x = ? - = { : }
3
1 0
and B x Z x x = ? + + = { : }
2
1 0
A
i i
=
- + - - ?
?
?
?
?
?
1
1 3
2
1 3
2
, ,
B
i i
=
- + - - ?
?
?
?
?
?
1 3
2
1 3
2
,
A B
i i
n =
- + - - ?
?
?
?
?
?
1 3
2
1 3
2
,
5.
Consider the following statements
for the two non-empty setsA andB.
1. ( ) ( ) ( ) A B A B A B n ? n ? n
= ? A B
2. ( ( )) A A B A B ? n = ?
Which of the above statements
is/are correct?
(a) Only 1 (b) Only 2
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(a) We have,
1. ( ) ( ) ( ) A B A B A B A B n ? n ? n = ?
LHS = n ? n ? n ( ) ( ) ( ) A B A B A B
= n ? ? n { ( )} ( ) A B B A B
[by distributive property]
PAPER : I Mathematics
= n ? n ( ) ( ) A U A B
[ ] Q B B U ? =
= ? n A A B ( )
= ? n ? ( ) ( ) A A A B
= n ? U A B ( ) = ? = A B RHS
Hence, 1 is correct.
2. A A B A B ? n = ? ( )
LHS = ? n A A B ( )
= ? n ? ( ) ( ) A A A B
= n ? U A B ( )
= ? ? ? A B A B
Hence, 2 is false.
?Only 1 is correct.
6.
Let X be a non-empty set and let
A B C , , be subsets of X . Consider the
following statements.
1. A C A B C B ? ? n ? n ( ) ( ),
( ) ( ) A B C B ? ? ?
2. ( ) ( ) A B C B n ? n for all sets
B A C ? ?
3. ( ) ( ) A B C B ? ? ? for all sets
B A C ? ?
Which of the above statements are
correct?
(a) Only 1 and 2 (b) Only 2 and 3
(c) Only 1 and 3 (d) 1, 2 and 3
Ê
(d) Let X = { , , , } 12 3 4
A B C = = = { , }, { , , }, { , , } 1 2 2 3 4 1 2 3
A ? C
A B C B n = n = { }, { , } 2 2 3
Clearly, ( ) ( ) A B C B n ? n
A B C B ? = ? = { , , , }, ( ) { , , , } 1 2 3 4 1 2 3 4
( ) ( ) A B C B ? ? n
Hence, Statement 1 is correct.
2.( ) ( ) A B C B n ? n for all setsB ?A ?C
Hence, Statement 2 is also correct.
3. ( ) ( ) A B C B ? ? ? for all sets
B ?A ? C
Hence, Statement 3 is also correct.
7.
IfB =
?
?
?
?
?
?
?
?
?
?
3 2 0
2 4 0
1 1 0
, then what is adjoint
of B equal to?
(a)
0 0 0
0 0 0
2 1 8 - -
?
?
?
?
?
?
?
?
?
?
(b)
0 0 2
0 0 1
0 0 8
-
-
?
?
?
?
?
?
?
?
?
?
(c)
0 0 2
0 0 1
0 0 0
?
?
?
?
?
?
?
?
?
?
(d) It does not exist
Ê
(a) We have,B =
?
?
?
?
?
?
?
?
?
?
3 2 0
2 4 0
1 1 0
Co-factor of B,
B B B
11 12 13
0 0 2 = = = - , ,
B B B
21 22 23
0 0 1 = = = - , ,
B B B
31 32 33
0 0 8 = = = , ,
adj B
B B B
B B B
B B B
=
?
?
?
?
?
?
?
?
?
?
'
11 12 13
21 22 23
31 32 33
=
-
-
?
?
?
?
?
?
?
?
?
?
'
0 0 2
0 0 1
0 0 8
=
- -
?
?
?
?
?
?
?
?
?
?
0 0 0
0 0 0
2 1 8
8.
What are the roots of the equation
| | ? x x x
2
6 2 - - = +
(a) -2 1 4 , , (b) 0 2 4 , ,
(c) 0 1 4 , , (d) -2 2 4 , ,
Ê
(d) We have,
| | x x x
2
6 2 - - = +
? | ( ) ( )| x x x - - = + 3 2 2
Case I x < 2
x x x
2
6 2 - - = +
x x
2
2 8 0 - - =
x x x
x x x
2
4 2 8 0
4 2 4 0
- + - =
- + - =
?
?
?
?
?
?
( ) ( )
( ) ( ) x x - + = 4 2 0
x = - 2 but x ? 4 [Qx < 2]
Case II 2 3 = < x
x x x
2
6 2 - - = - + ( )
x x x
2
6 2 0 - - + + =
x
2
4 0 - =
x = ± 2
x x = ? - 2 2 but [Qx ?( , ) 2 3 ]
Case III x = 3
x x x
2
6 2 - - = +
x x
2
2 8 0 - - =
( ) ( ) x x + - = 2 4 0
x x = ? - 4 2 but [Qx = 3]
? x = - 2 2 4 , ,
9.
If A =
?
?
?
?
?
?
0 1
1 0
, then the matrix A is
a/an
(a) singular matrix
(b) involutory matrix
(c) nilpotent matrix
(d) idempotent matrix
Ê
(b) We have, A =
?
?
?
?
?
?
0 1
1 0
| | A = - 1
Since,| | A ? 0
Hence, A is not singular.
A A A
2
0 1
1 0
0 1
1 0
= · =
?
?
?
?
?
?
·
?
?
?
?
?
?
=
?
?
?
?
?
?
1 0
0 1
A I
2
=
Hence, A is involutory matrix.
10.
If
x i
y i
i i
i
-
-
?
?
?
?
?
?
?
?
?
?
= +
3 1
1
0 2
6 11 , then what
are the values of x and y
respectively?
(a) -3 4 , (b) 3 4 ,
(c) 3 4 , - (d) -3, -4
Ê
(a) We have,
x i
y i
i i
i
-
-
= +
3 1
1
0 2
6 11
? x i y i i ( ) ( ) - + - - - = + 2 3 2 6 11
? 2 3 2 6 11 x y x y i i + + - + = + ( )
On equating real and imaginary parts, on
both sides,
we get 2 3 6 x y + = ...(i)
and - + = x y 2 11 ...(ii)
On solving Eqs. (i) and (ii), we get
x = - 3 and y = 4
11.
The common roots of the equations
z z z
3 2
2 2 1 0 + + + =
andz z
2017 2018
1 0 + + = are
(a) -1 , ? (b) 1
2
,?
(c) -1
2
, ? (d) ? ? ,
2
Ê
(d) We have, z z z
3 2
2 2 1 0 + + + =
( ) ( ) z z z + + + = 1 1 0
2
? z + = 1 0 or z z
2
1 0 + + =
z = -1
or z=
- ± - 1 1 4
2
=
- + - - 1 3
2
1 3
2
i i
, = ? ? ,
2
Now, z z
2017 2018
1 0 + + =
Put z = - 1 ,
LHS = - + - + ( ) ( ) 1 1 1
2017 2018
= - + + 1 1 1
= ? 1 0 (RHS)
? z = - 1is not a root of equation.
Put z =?,
LHS = + + ( ) ( ) ? ?
2017 2018
1
= + + ( ) . ( ) . ? ? ? ?
3 672 3 672 2
1
= + + ? ?
2
1 [ ] Q ?
3
1 =
[ ] Q1 0
2
+ + = ? ?
= = 0 RHS
? z =? is a root of equation.
put z = ?
2
,
LHS = + + ( ) ( ) ? ?
2 2017 2 2018
1
= + + ? ?
4034 4036
1
= + + ( ) . ( ) . ? ? ? ?
3 1344 2 3 1345
1
= + + = ? ?
2
1 0 RHS
? z = ?
2
is a root of equation.
Hence, ? ? ,
2
are the common roots of
these equations.
2
12.
If C n C n ( , ) ( , ) 20 2 20 2 + = - , then
what isn equal to
(a) 8 (b) 10
(c) 12 (d) 16
Ê
(b) We have,C n C n ( , ) ( , ) 20 2 20 2 + = -
?
20
2
20
2
C C
n n + -
=
? n n + + - = 2 2 20
[Q
n
x
n
y
C C = ? x y n + = ]
? n = 10
13.
There are 10 points in a plane. No
three of these points are in a straight
line. What is the total number of
straight lines which can be formed
by joining the points?
(a) 90 (b) 45
(c) 40 (d) 30
Ê
(b) Given, 10 points in a plane where no
three of these points are in straight line.
Total number of straight line formed from
10 points is
10
2
10
2 8
10 9
2
45 C = =
×
=
!
! !
14. The equation px qx r
2
0 + + =
(where p q r , , , all are positive) has
distinct real rootsa andb. Which one
of the following is correct?
(a)a b > > 0 0 ,
(b)a b < < 0 0 ,
(c)a b > < 0 0 ,
(d)a b < > 0 0 ,
Ê
(b) Given, px qx r
2
0 + + = , where
p q r , , > 0anda andb are distinct roots.
? a b
q
p
+ =
-
and ab r =
Now, r > 0
? ab > 0
? a b > > 0 0 , ...(i)
or a b < < 0 0 , ...(ii)
Now,
-
<
q
p
0 q p , > 0
? a b + < 0
a b < < 0 0 , ...(iii)
From Eqs. (i), (ii) and (iii), we get
? a < 0 and b < 0
15.
If A = ? { , { , }} ? µ , then the power set
of A is
(a) { , { }, { }, { }} f f ? ?,µ
(b) {f, ? ?,µ ? ? µ { },{ },{ ,{ , }}}
(c) { { },{ },{ ,{ , }}} f, ? ?,µ ? ? µ
(d) {{ },{ },{ ,{ , }}} ? ?,µ ? ? µ
Ê
(b) We have, A = { , { , }} ? ? µ
{ }
P A ( ) { , { }, { , } , { , { , }}} = f ? ? µ ? ? µ
Directions (Q. Nos. 16 and 17) Read
the information carefully and answer
the given questions.
In a school, all the students play atleast one
of three indoor games– chess, carrom and
table tennis. 60 play chess, 50 play table
tennis, 48 play carrom, 12 play chess and
carrom, 15 play carrom and table tennis, 20
play table tennis and chess.
16.
What can be the minimum number of
students in the school?
(a) 123 (b) 111 (c) 95    (d) 63
Ê
(b) Let
A = Student play chess
B = Student play table tennis
C = Student play carrom
Given, n A n B nC ( ) , ( ) , ( ) = = = 60 50 48
n A B n B C ( ) , ( ) n = n = 20 15
n A C ( ) n = 12
For minimum number of students in
school
n A B C ( ) n n must be zero.
?n A B C n A n B nC ( ) ( ) ( ) ( ) ? ? = + +
- n - n n A B n B C ( ) ( )
- n + n n n A C n A B C ( ) ( )
= + + - - - + 60 50 48 20 15 12 0 = 111
17.
What can be the maximum number of
students in the school?
(a) 111 (b) 123
(c) 125 (d) 135
Ê
(b) For maximum number of students in
school n A B C ( ) n n must be 12.
? n A B C ( ) ? ?
= + + - - - + 60 50 48 20 15 12 12
= 123
18.
If A is an identity matrix of order 3,
then its inverse ( ) A
-1
(a) is equal to null matrix
(b) is equal to A
(c) is equal to 3A (d) does not exist
Ê
(b) Given, A =
?
?
?
?
?
?
?
?
?
?
1 0 0
0 1 0
0 0 1
? A A
-
=
?
?
?
?
?
?
?
?
?
?
=
1
1 0 0
0 1 0
0 0 1
19.
A is a square matrix of order 3 such
that its determinant is 4. What is the
determinant of its transpose?
(a) 64 (b) 36
(c) 32 (d) 4
Ê
(d) Given, | | A = 4
? | | A' = 4 [Q| | | |] A A = '
20.
From 6 programmers and 4 typists,
an office wants to recruit 5 people.
What is the number of ways this can
be done so as to recruit atleast one
typist?
(a) 209 (b) 210
(c) 246 (d) 242
Ê
(c) We have,
6 programmers and 4 typists
Number of ways of 5 recruit people such
that atleast one typist
= + +
4
1
6
4
4
2
6
3
4
3
6
2
C C C C C C
+
4
4
6
1
C C
= × + × + × + × 4 15 6 20 4 15 1 6
= + + + 60 120 60 6 = 246
21.
What is the number of terms in the
expansion of[( ) ( ) ] 2 3 2 3
2 2 2
x y x y - + ?
(a) 4 (b) 5
(c) 8 (d) 16
Ê
(b) Given, [( ) ( ) ] 2 3 2 3
2 2 2
x y x y - +
= - [ ] 4 9
2 2 4
x y
? Total number of terms = + = 4 1 5
22.
In the expansion of( ) 1 +ax
n
, the first
three terms are respectively 1 12 , x
and 64
2
x . What isn equal to?
(a) 6 (b) 9
(c) 10 (d) 12
Ê
(b) Given, first three terms of expansion
( ) , , 1 1 12 64
2
+ ax x x
n
is ,
Now,
( )
( )
1 1
1
2
2 2
+ = + +
-
+ ax nax
n n
a x
n
K
On equating first three terms, we get
na = 12 and
n n
a
( ) -
=
1
2
64
2
On putting the value ofa in
n n
a
( ) -
=
1
2
64
2
, we get
n n
n
( ) - ?
?
?
?
?
?
=
1
2
12
64
2
?
144 1
2
64
( ) n
n
-
=
? n = 9
23.
The numbers 1, 5 and 25 can be three
terms (not necessarily
consecutive)of
(a) only one AP
(b) more than one but finite numbers of
APs
(c) infinite number of APs
(d) finite number of GPs
Ê
(d) We have, 1, 5, 25 be three terms.
Clearly, 1, 5, 25 are finite number of GPs.
3
24.
The sum of ( ) p q + th and ( ) p q - th
terms of an AP is equal to
(a) ( ) 2p th term (b) ( ) 2q th term
(c) twice the p th term
(d) twice theq th term
Ê
(c) Let a is first term and d is common
difference of AP.
a a p q d
p q +
= + + - ( ) 1
and a a p q d
p q -
= + - - ( ) 1
Sum of ( ) p q + th and ( ) p q - th terms
= + = + -
+ -
a a a p d
p q p q
2 2 2 ( )
= + - 2 1 ( ( ) ) a p d = 2a
p
= twice of p th term
25. IfA is a square matrix of ordern > 1,
then which one of the following is
correct?
(a) det ( ) - = A det A
(b) det ( ) ( ) - = - A
n
1 det A
(c) det ( ) - = - A det A
(d) det ( ) - = A n det A
Ê
Sol. (b) We know that if A is a square
matrix of ordern > 1 , then
det( ) ( ) - = - A
n
1 det A
For example If A =
?
?
?
?
?
?
2 3
4 5
,
then - =
- -
- -
?
?
?
?
?
?
A
2 3
4 5
?det A =
?
?
?
?
?
?
2 3
4 5
= - 10 12 = - 2 …(i)
and det( ) - =
- -
- -
?
?
?
?
?
? A
2 3
4 5
= - 10 12 = - 2
= - - ( ) ( ) 1 2
2
[Q here n = 2]
= - ( ) 1
2
det A [from Eq. (i)]
if A =
-
?
?
?
?
?
?
?
?
?
?
1 2 3
3 1 0
4 3 2
Then, - =
- - -
- -
- -
?
?
?
?
?
?
?
?
?
?
A
1 2 3
3 1 0
4 3 2
? det A =
-
1 2 3
3 1 0
4 3 2
= - - - - - + - 1 2 0 2 6 0 3 9 4 ( ) ( ) ( )
= - + + 2 12 15 = 25
and det ( ) - =
- - -
- -
- -
?
?
?
?
?
?
?
?
?
A
1 2 3
3 1 0
4 3 2
= - - - + - - - - 1 2 0 2 6 0 3 9 4 ( ) ( ) ( )
= - - 2 12 15 = - 25
= - ( ) 1 25
3
[here n = 3]
= - ( ) 1
3
det A [from Eq. (i)]
26. What is the least value of
25 cosec 36sec
2 2
x x + ?
(a) 1
(b) 11
(c) 120
(d) 121
Ê
(d) Given, 25 36 cosec sec
2 2
x x +
= + + + 25 1 36 1
2 2
( cot ) ( tan ) x x
= + + + 25 25 36 36
2 2
cot tan x x
= + + + 25 36 25 36
2 2
cot tan x x
= + - + × × 61 5 6 2 5 6
2
( cot tan ) x x
= + = 61 60 121 [Qminimum value of
( cot tan ) 5 6 0
2
x x - = ]
? Minimum value of
25 cosec
2
x x + = 36 121
2
sec
Directions (Q. Nos. 27 and 28) Read
the information carefully and answer
the given questions.
Let A andB be3 3 × matrices with det A = 4
and det B = 3 .
27. What is det ( ) 2AB equal to?
(a) 96
(b) 72
(c) 48
(d) 36
Ê
(a) A and B be ( ) 3 3 × matrices with
det A = 4 and det B = 3
We know that,
det det det ( ) ( ) ( ) KAB K A B
n
= ×
where,n is the order or A andB,K is a real
number.
?det det det ( ) ( ) 2 2
3
AB A B = ×
[Qn = 3 and k = 2]
= × × 8 4 3
= 96
28. What is det ( ) 3
1
AB
-
equal to?
(a) 12 (b) 18
(c) 36 (d) 48
Ê
(c) A and B be ( ) 3 3 × matrices with
det A = 4 and det B = 3
We know that,
det det
det
( ) ( )
( )
KAB K A
B
n -
= ×
1
1
,
where n is the order of A and B, K is a
real number]
?det ( ) ( ) ( )
det
3 3
1
1 3
AB A
B
-
= × det
= × × 27 4
1
3
= 36
Directions (Q. Nos. 29 and 30) Read
the information carefully and answer
the given questions.
A complex number is given by
z
i
i
=
+
- -
1 2
1 1
2
( )
.
29.
What is the modulus ofz?
(a) 4 (b) 2 (c) 1 (d)
1
2
Ê
(c) We have, z
i
i
=
+
- -
1 2
1 1
2
( )
z
i
i
=
+
- - -
1 2
1 1 1 2 ( )
=
+
+
=
1 2
1 2
1
i
i
? | | z = 1
30.
What is the principal argument ofz?
(a) 0 (b)
p
4
(c)
p
2
(d) p
Ê
(a) arg ( ) tan
( )
( )
z
z
z
=
?
?
?
?
?
?
-1
lm
Re
=
?
?
?
?
?
?
= =
- -
tan tan
1 1
0
1
0 0
31.
What is the value of
sin cos sin sin
cos cos cos sin
34 236 56 124
28 88 178
° ° - ° °
° ° + ° 208°
?
(a) -2 (b) -1 (c) 2 (d) 1
Ê
(a) We have,
sin cos sin sin
cos cos cos sin
34 236 56 124
28 88 178
° ° - ° °
° ° + ° 208°
=
° ° + °
- ° ° + °
°
sin cos ( )
sin sin ( )
cos cos
34 180 56
56 90 34
28 88° + ° + °
° + °
cos ( )
sin ( )
90 88
180 28
=
- ° ° - ° °
° ° + °
sin cos sin cos
cos cos sin sin
34 56 56 34
28 88 88 28°
=
- ° + °
° - °
sin ( )
cos ( )
56 34
88 28
=
- °
°
sin
cos
90
60
=
-
= -
1
1
2
2
32.
tan54° can be expressed as
(a)
sin cos
sin cos
9 9
9 9
° + °
° - °
(b)
sin cos
sin cos
9 9
9 9
° - °
° + °
(c)
cos sin
cos sin
9 9
9 9
° + °
° - °
(d)
sin
cos
36
36
°
°
Ê
(c) We have, tan tan ( ) 54 45 9 ° = ° + °
=
° + °
- ° °
tan tan
tan tan
45 9
1 45 9
=
+ °
- °
1 9
1 9
tan
tan
=
° + °
° - °
cos sin
cos sin
9 9
9 9
4
Directions (Q. Nos. 33-35) Read the
given information carefully and answer
the given questions.
If p X Y = - cos sin ? ?,
q X Y = + sin cos ? ? and
p pq q AX BY
2 2 2 2
4 + + = + ,
0
2
= =
p
? .
33.
What is the value of ??
(a)
p
2
(b)
p
3
(c)
p
4
(d)
p
6
Ê
(c) We have,
p X Y = - cos sin ? ? ...(i)
q X Y = + sin cos ? ? ...(ii)
and p pq q AX BY
2 2 2 2
4 + + = + ...(iii)
From Eqs. (i) and (ii), we get
p q X Y
2 2 2
+ = - ( cos sin ) ? ?
+ + ( sin cos ) X Y ? ?
2
? p q X Y
2 2 2 2
+ = +
and pq X Y = - ( cos sin ) ? ?
( sin cos ) X Y ? ? +
? pq X Y = - ( )sin cos
2 2
? ?
+ XY cos 2?
? p pq q X Y
2 2 2 2
4 2 + + = + +
( ) X Y
2 2
-
sin cos 2 4 2 ? ? + XY
Given, p pq q AX BY
2 2 2 2
4 + + = +
? X Y X Y
2 2 2 2
2 + + - ( )
sin cos 2 4 2 ? ? + XY = + AX BY
2 2
2
Coefficient of XY = 0
? cos 2 0 ? =
? 2
2
?
p
=
? ?
p
=
4
34.
What is the value of A?
(a) 4
(b) 3
(c) 2
(d) 1
Ê
(b) X Y X Y
2 2 2 2
2
2
+ + - ( ) sin
p
= + AX BY
2 2
? X Y X Y
2 2 2 2
2 2 + + - = + AX BY
2 2
? 3
2 2 2 2
X Y AX BY - = +
? A B = = - 3 1 ,
35.
What is the value of B ?
(a) -1 (b) 0
(c) 1 (d) 2
Ê
(a) B = - 1
Directions (Q. Nos. 36 and 37) Read
the given information carefully and
answer the given questions.
It is given that cos(? - a) = a,
cos( ) ? - ß = b.
36.
What is cos( ) a - ß equal to ?
(a)ab a b + - - 1 1
2 2
(b)ab a b - - - 1 1
2 2
(c)a a b a 1 1
2 2
- - -
(d) a b b a 1 1
2 2
- + -
Ê
(a) Given cos ( ) ? a - = a
cos ( ) ? ß - = b
cos ( ) cos {( ) ( )} a ß ? ß ? a - = - - -
= - - + - cos ( ) cos ( ) sin ( ) ? ß ? a ? ß
sin ( ) ? a -
= + - - ab a b 1 1
2 2
37. What is sin ( ) cos
2
2 a - ß + (a - ß) ab
equal to?
(a)a b
2 2
+ (b)a b
2 2
-
(c)b a
2 2
- (d) - + ( ) a b
2 2
Ê
(a) sin ( ) cos ( )
2
2 a ß a ß - + - ab
= - - + - 1 2
2
cos ( ) cos ( ) a ß a ß ab
= - + - - + 1 1 1 2
2 2 2
( ) ab a b ab
( ) ab a b + - - 1 1
2 2
= - + - - + 1 1 1 2
2 2 2 2
[ ( ) ( ) a b a b ab
1 1 2 2
2 2 2 2
- - + + a b a b ab ]
( ) ( ( ) 1 1
2 2
- - a b
= - - + + - 1 1
2 2 2 2 2 2
a b a b a b
- - - + + 2 1 1 2 2
2 2 2 2
ab a b a b ab
1 1
2 2
- - a b
= + a b
2 2
38.
If sin cos a + a = p, then what is
cos ( )
2
2a equal to?
(a) p
2
(b) p
2
1 -
(c) p p
2 2
2 ( ) - (d) p
2
1 +
Ê
(c) We have, sin cos a a + = p
sin cos sin cos
2 2 2
2 a a a a + + = p
? 1 2
2
+ = sin a p
? sin2 1
2
a = - p
? sin ( )
2 2 2
2 1 a = - p
? 1 2 2 1
2 4 2
- = - + cos a p p
? cos
2 2 4
2 2 a = - p p
? cos ( )
2 2 2
2 2 a = - p p
39.
What is the value of
sin sec
- -
+ -
p
1 1
4
5
5
4 2
?
(a)
p
4
(b)
p
2
(c) p (d) 0
Ê
(d) We have,
sin
- -
+ -
1 1
4
5
5
4 2
sec
p
= + -
- -
sin cos
1 1
4
5
4
5 2
p
Q sec
- -
=
?
?
?
?
?
?
?
?
?
?
?
?
1 1
1
x
x
cos
= - =
p p
2 2
0 Q sin cos
- -
+ =
?
?
?
?
?
?
1 1
2
x x
p
40.
If sin cos
- -
+
-
-
+
1
2
1
2
2
2
1
1
1
p
p
q
q
=
-
-
tan
1
2
2
1
x
x
, then what is x equal
to?
(a)
p q
pq
+
+ 1
(b)
p q
pq
-
+ 1
(c)
pq
pq 1 +
(d)
p q
pq
+
- 1
Ê
(b) Given,
sin cos
- -
+
-
-
+
?
?
?
?
?
?
?
?
1
2
1
2
2
2
1
1
1
p
p
q
q
=
-
-
tan
1
2
2
1
x
x
? 2 2 2
1 1 1
tan tan tan
- - -
- = p q x
? tan tan
- -
-
+
?
?
?
?
?
? =
1 1
1
p q
pq
x
? x
p q
pq
=
-
+ 1
41. If tan? =
1
2
and tan f =
1
3
, then what
is the value of ( ) ? f + ?
(a) 0 (b)
p
6
(c)
p
4
(d)
p
2
Ê
(c) Given, tan , tan ? = f =
1
2
1
3
tan ( )
tan tan
tan tan
?
?
?
+ f =
+ f
- f 1
? tan (? + f) =
+
- ×
1
2
1
3
1
1
2
1
3
? tan (? + f) =
+
-
= =
3 2
6 1
5
5
1
? tan ( ) ? + f = 1
? ?
p
+ f = =
-
tan
1
1
4
5