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

 Page 1


1.
If both p and q belong to the set
{ , , , } 1 2 3 4 , then how many equations
of the form
px qx
2
1 0 + + = will have real roots?
(a) 12 (b) 10
(c) 7 (d) 6
Ê
(d) Equation px qx
2
1 0 + + = , has real
roots, where p and q belong to the set
{ , , , } 1 2 3 4 .
? q p
2
4 0 - =
[Q for real roots of a quadratic  equation
b ac
2
4 0 - = ]
It is possible if value of
( , ) ( , ), ( , ), ( , ), ( , ), ( , ) p q = 1 2 1 3 1 4 2 3 2 4
and ( , ) 3 4
Hence, the number of equations are 6.
2.
What is the value of
1 2 3 4 5 101 - + - + - + ...... ?
(a) 51 (b) 55
(c) 110 (d) 111
Ê
(a) Given series,
= - + - + - + 1 2 3 4 5 101 ......
= + + + + ( ..... ) 1 3 5 101
- + + + + ( ..... ) 2 4 6 100
= + + + ( .... 1 3 5 51terms)
- + + + ( ...... 2 4 6 50 terms)
= + - ×
51
2
2 51 1 2 [ ( ) ]
- + - ×
50
2
4 50 1 2 [ ( ) ]
[Q both series are AP and
S
n
a n d
n
= + -
2
2 1 [ ( ) ]
= × - ×
51
2
102
50
2
102
= - = 2601 2550 51
3.
If A B , and C are subsets of a given
set, then which one of the following
relations is not correct?
(a) A A B A B ? n = ? ( )
(b) A A B A n ? = ( )
(c) ( ) ( ) ( ) A B C A C B C n ? = ? n ?
(d) ( ) ( ) ( ) A B C A C B C ? n = n ? n
Ê
(a) LetU be the set and A B , andC are the
subset ofU.
We know that, A A B A ? n = ( ) ,
So option (a) is not correct.
A A B A n ? = ( ) , so option (b) is correct.
( ) ( ) ( ) A B C A C B C n ? = ? n ? ,
so option (c) is correct.
and ( ) ( ) ( ) A B C A C B C ? n = n ? n
so option (d) is correct.
4.
If the sum of firstn terms of a series
is ( ), n + 12 then what is its third
term?
(a) 1 (b) 2
(c) 3 (d) 4
Ê
(a) Sum of firstn term of a series = + n 12
?a a a
1 2 3
+ + +.........+ = + a n
n
12
Put n = 1 , a
1
1 12 13 = + =
Putn = 2,a a
1 2
2 12 + = + ?a a
1 2
14 + =
?13 14
2
+ = a ?a
2
1 =
Put n = 3
a a a
1 2 3
3 12 + + = +
? 13 1 15
3
+ + = a
? a
3
15 14 1 = - =
5.
What is the value ofk for which the
sum of the squares of the roots of
2 2 2 1 0
2
x k x k - - - + = ( ) ( ) is
minimum?
(a) - 1 (b) 1
(c)
3
2
(d) 2
Ê
(c) Let a ß , be the roots of equation.
2 2 2 1 0
2
x k x k - - - + = ( ) ( )
Qa ß =
(
+
-
= -
2 2
2
2
k
k
)
,
aß =
- + ( ) k 1
2
We know that
a ß a ß aß
2 2 2
2 + = + - ( )
= - + ×
+
( ) k
k
2 2
1
2
2
= + - + + k k k
2
4 4 1
= - + k k
2
3 5
= - + - + k k
2
3
9
4
9
4
5
= -
?
?
?
?
?
?
+ k
3
2
11
4
2
a ß
2 2
+ is minimum, if k -
?
?
?
?
?
?
=
3
2
0
? k =
3
2
6.
If the roots of the equation
a b c x bc a x c a b ( ) ( ) ( ) - + - + - =
2
0
are equal, then which one of the
following is correct?
(a) a b , andc are in AP
(b) a b , andc are in GP
(c) a b , andc are in HP
(d) a b , and c do not follow any regular
pattern
Ê
(c) The roots of the equation
a b c x bc a x c a b ( ) ( ) ( ) - + - + - =
2
0
are equal.
?b c a a b c c a b
2 2
4 0 ( ) ( ). ( ) - - - - =
[Q ax bx c
2
0 + + = of roots are real if
b ac
2
4 0 - =
?b c a ca ac ab b
2 2 2 2
2 4 ( ) ( + - - -
- + = ac bc) 0
PAPER : I Mathematics
Page 2


1.
If both p and q belong to the set
{ , , , } 1 2 3 4 , then how many equations
of the form
px qx
2
1 0 + + = will have real roots?
(a) 12 (b) 10
(c) 7 (d) 6
Ê
(d) Equation px qx
2
1 0 + + = , has real
roots, where p and q belong to the set
{ , , , } 1 2 3 4 .
? q p
2
4 0 - =
[Q for real roots of a quadratic  equation
b ac
2
4 0 - = ]
It is possible if value of
( , ) ( , ), ( , ), ( , ), ( , ), ( , ) p q = 1 2 1 3 1 4 2 3 2 4
and ( , ) 3 4
Hence, the number of equations are 6.
2.
What is the value of
1 2 3 4 5 101 - + - + - + ...... ?
(a) 51 (b) 55
(c) 110 (d) 111
Ê
(a) Given series,
= - + - + - + 1 2 3 4 5 101 ......
= + + + + ( ..... ) 1 3 5 101
- + + + + ( ..... ) 2 4 6 100
= + + + ( .... 1 3 5 51terms)
- + + + ( ...... 2 4 6 50 terms)
= + - ×
51
2
2 51 1 2 [ ( ) ]
- + - ×
50
2
4 50 1 2 [ ( ) ]
[Q both series are AP and
S
n
a n d
n
= + -
2
2 1 [ ( ) ]
= × - ×
51
2
102
50
2
102
= - = 2601 2550 51
3.
If A B , and C are subsets of a given
set, then which one of the following
relations is not correct?
(a) A A B A B ? n = ? ( )
(b) A A B A n ? = ( )
(c) ( ) ( ) ( ) A B C A C B C n ? = ? n ?
(d) ( ) ( ) ( ) A B C A C B C ? n = n ? n
Ê
(a) LetU be the set and A B , andC are the
subset ofU.
We know that, A A B A ? n = ( ) ,
So option (a) is not correct.
A A B A n ? = ( ) , so option (b) is correct.
( ) ( ) ( ) A B C A C B C n ? = ? n ? ,
so option (c) is correct.
and ( ) ( ) ( ) A B C A C B C ? n = n ? n
so option (d) is correct.
4.
If the sum of firstn terms of a series
is ( ), n + 12 then what is its third
term?
(a) 1 (b) 2
(c) 3 (d) 4
Ê
(a) Sum of firstn term of a series = + n 12
?a a a
1 2 3
+ + +.........+ = + a n
n
12
Put n = 1 , a
1
1 12 13 = + =
Putn = 2,a a
1 2
2 12 + = + ?a a
1 2
14 + =
?13 14
2
+ = a ?a
2
1 =
Put n = 3
a a a
1 2 3
3 12 + + = +
? 13 1 15
3
+ + = a
? a
3
15 14 1 = - =
5.
What is the value ofk for which the
sum of the squares of the roots of
2 2 2 1 0
2
x k x k - - - + = ( ) ( ) is
minimum?
(a) - 1 (b) 1
(c)
3
2
(d) 2
Ê
(c) Let a ß , be the roots of equation.
2 2 2 1 0
2
x k x k - - - + = ( ) ( )
Qa ß =
(
+
-
= -
2 2
2
2
k
k
)
,
aß =
- + ( ) k 1
2
We know that
a ß a ß aß
2 2 2
2 + = + - ( )
= - + ×
+
( ) k
k
2 2
1
2
2
= + - + + k k k
2
4 4 1
= - + k k
2
3 5
= - + - + k k
2
3
9
4
9
4
5
= -
?
?
?
?
?
?
+ k
3
2
11
4
2
a ß
2 2
+ is minimum, if k -
?
?
?
?
?
?
=
3
2
0
? k =
3
2
6.
If the roots of the equation
a b c x bc a x c a b ( ) ( ) ( ) - + - + - =
2
0
are equal, then which one of the
following is correct?
(a) a b , andc are in AP
(b) a b , andc are in GP
(c) a b , andc are in HP
(d) a b , and c do not follow any regular
pattern
Ê
(c) The roots of the equation
a b c x bc a x c a b ( ) ( ) ( ) - + - + - =
2
0
are equal.
?b c a a b c c a b
2 2
4 0 ( ) ( ). ( ) - - - - =
[Q ax bx c
2
0 + + = of roots are real if
b ac
2
4 0 - =
?b c a ca ac ab b
2 2 2 2
2 4 ( ) ( + - - -
- + = ac bc) 0
PAPER : I Mathematics
?b c a b ab c a bc
2 2 2 2 2 2
2 4 + - –
+ + - = 4 4 4 0
2 2 2 2
ab c a c abc
?b c a b ab c
2 2 2 2 2
2 + +
- - + = 4 4 4 0
2 2 2 2
a bc abc a c
?b c a ac abc a c
2 2 2
2 4 ( ) ( ) + + - +
+ = 4 0
2 2
a c
? b c a abc a c
2 2
4 ( ) ( ) + - + + = ( ) 2 0
2
ac
?[ ( ) ] bc a ac + - = 2 0
2
?b c a ac ( ) + - = 2 0
?bc a ac ( ) + = 2 ?b
ac
c a
=
+
2
So, a b , andc are is HP.
7.
If | | x x x x
2 2
3 2 3 2 - + > - + , then
which one of the following is
correct?
(a) x = 1or x = 2 (b) 1 2 = = x
(c) 1 2 < < x
(d) x is any real value except 3 and 4
Ê
(c)| | x x x x
2 2
3 2 3 2 - + > - +
? - - + > - + ( ) x x x x
2 2
3 2 3 2
[if x x
2
3 2 0 - + < , and x x
2
3 2 0 - + >
not possible]
? - - + > 2 3 2 0
2
( ) x x
? x x
2
3 2 0 - + >
? x x x
2
2 2 0 - - + >
? ( )( ) x x - - > 2 1 0
? 1 2 < < x is correct.
8.
A geometric progression (GP)
consists of 200 terms. If the sum of
odd terms of the GP is m, and the
sum of even terms of the GP is n,
then what is its common ratio?
(a) m n / (b) n m /
(c) m n m + ( / ) (d) n m n + ( / )
Ê
(b) Let a ar ar , , ......
2
200 terms be a
geometric progression.
Where, a is the first terms and r be the
common ratio.
GP of odd termsa ar ar , , , .....
2 4
100 terms.
GP of even terms ar ar ar , , ,
3 5
…… 100
terms.
?Sum of odd terms of the GP = m
?
a r
r
m
{ }
200
1
1
-
-
= …(i)
Sum of even terms of the GP = n
?
ar r
r
n
( }
200
1
1
-
-
= …(ii)
Dividing of Eq. (i) by Eq. (ii),
?
1
r
m
n
= ?r
n
m
=
Hence, the common ratio of the GP is
n
m
.
9.
If a set A contains 3 elements and
another set B contains 6 elements,
then what is the minimum number
of elements that ( ) A B ? can have?
(a) 3 (b) 6
(c) 8 (d) 9
Ê
(b) n A n B ( ) , ( ) = = 3 6
?The minimum number of elements in
A B ? = 6
i.e n A B ( ) ? = 6
(because maxn A B ( ) n = 3
10.
What is the number of diagonals of
an octagon?
(a) 48
(b) 40
(c) 28
(d) 20
Ê
(d) The number of vertices of an octagon
= 8
?The number of points in a plane = 8
?Total number of straight line form by 8
points =
8
2
C
[Q 1 straight line form by 2
points]
= =
×
=
8
2 6
8 7
2
28
!
! !
? The number of diagonals of an octagon
= Total number
of straight line form by 8 points - number
of sides of octagon
= - = 28 8 20
11.
What is the value of the determinant
1 2 3
2 3 4
3 4 5
! ! !
! ! !
! ! !
?
(a) 0 (b) 12
(c) 24 (d) 36
Ê
(c) Given determinant
=
1 2 3
2 3 4
3 4 5
! ! !
! ! !
! ! !
=
1 2 6
2 6 24
6 24 120
=
1 0 0
2 2 6
6 12 48
[byC C C C C C
2 2 1 3 3 2
2 3 ? - ? - , ]
= - - + 1 96 72 0 0 ( )
[expression w.r.t. first row]
= 24
12.
What are the values of x that satisfy
the equation
x
x
x
x
0 2
2 2 1
1 1 1
3 0 2
2 1
0 1 1
0
2
+ = ?
(a) - ± 2 3
(b) - ± 1 3
(c) - ± 1 6
(d) - ± 2 6
Ê
(d) Given equation,
x
x
x
x
0 2
2 2 1
1 1 1
3 0 2
2 1
0 1 1
0
2
+ =
? x x x ( ) ( ) ( ) 2 1 0 2 2 2 3 2 1 - - + - + -
- + - = 0 2 0 0
2
( ) x
[expression w.r.t. first row]
? x x x x + - + + = 4 4 3 2 0
2
? 2 8 4 0
2
x x + - =
? x x
2
4 2 0 + - =
? x =
- ± - - 4 16 4 1 2
2
( ) ( )
=
- ±
=
- ± 4 24
2
4 2 6
2
= - ± 2 6
13.
If x a b c + + + = 0, then what is the
value of
x a b c
a x b c
a b x c
+
+
+
?
(a) 0 (b) ( ) a b c + +
2
(c) a b c
2 2 2
+ + (d) a b c + + - 2
Ê
(a) Given, x a b c + + + = 0
x a b c
a x b c
a b x c
+
+
+
=
+ + +
+ + + +
+ + + +
x a b c b c
x a b c x b c
x a b c b x c
[byC C C C
1 1 2 3
? + + ]
= + + + ( ) x a b c
1
1
1
b c
x b c
b x c
+
+
[x a b c + + + common fromC
1
] = 0
[ ] Q x a b c + + + = 0
14.
IfA =
-
-
?
?
?
?
?
?
1 1
1 1
, then the expression
A A
3 2
2 - is
(a) a null matrix (b) an identity matrix
(c) equal to A (d) equal to -A
Ê
(a) A =
-
-
?
?
?
?
?
?
1 1
1 1
?A A A
2
1 1
1 1
1 1
1 1
= · =
-
-
?
?
?
?
?
?
·
-
-
?
?
?
?
?
?
=
+ - -
- - +
?
?
?
?
?
?
=
-
-
?
?
?
?
?
?
1 1 1 1
1 1 1 1
2 2
2 2
and A A A
3 2
2 2
2 2
1 1
1 1
= · =
-
-
?
?
?
?
?
?
-
-
?
?
?
?
?
?
.
=
+ - -
- - +
?
?
?
?
?
?
=
-
-
?
?
?
?
?
?
2 2 2 2
2 2 2 2
4 4
4 4
2
Page 3


1.
If both p and q belong to the set
{ , , , } 1 2 3 4 , then how many equations
of the form
px qx
2
1 0 + + = will have real roots?
(a) 12 (b) 10
(c) 7 (d) 6
Ê
(d) Equation px qx
2
1 0 + + = , has real
roots, where p and q belong to the set
{ , , , } 1 2 3 4 .
? q p
2
4 0 - =
[Q for real roots of a quadratic  equation
b ac
2
4 0 - = ]
It is possible if value of
( , ) ( , ), ( , ), ( , ), ( , ), ( , ) p q = 1 2 1 3 1 4 2 3 2 4
and ( , ) 3 4
Hence, the number of equations are 6.
2.
What is the value of
1 2 3 4 5 101 - + - + - + ...... ?
(a) 51 (b) 55
(c) 110 (d) 111
Ê
(a) Given series,
= - + - + - + 1 2 3 4 5 101 ......
= + + + + ( ..... ) 1 3 5 101
- + + + + ( ..... ) 2 4 6 100
= + + + ( .... 1 3 5 51terms)
- + + + ( ...... 2 4 6 50 terms)
= + - ×
51
2
2 51 1 2 [ ( ) ]
- + - ×
50
2
4 50 1 2 [ ( ) ]
[Q both series are AP and
S
n
a n d
n
= + -
2
2 1 [ ( ) ]
= × - ×
51
2
102
50
2
102
= - = 2601 2550 51
3.
If A B , and C are subsets of a given
set, then which one of the following
relations is not correct?
(a) A A B A B ? n = ? ( )
(b) A A B A n ? = ( )
(c) ( ) ( ) ( ) A B C A C B C n ? = ? n ?
(d) ( ) ( ) ( ) A B C A C B C ? n = n ? n
Ê
(a) LetU be the set and A B , andC are the
subset ofU.
We know that, A A B A ? n = ( ) ,
So option (a) is not correct.
A A B A n ? = ( ) , so option (b) is correct.
( ) ( ) ( ) A B C A C B C n ? = ? n ? ,
so option (c) is correct.
and ( ) ( ) ( ) A B C A C B C ? n = n ? n
so option (d) is correct.
4.
If the sum of firstn terms of a series
is ( ), n + 12 then what is its third
term?
(a) 1 (b) 2
(c) 3 (d) 4
Ê
(a) Sum of firstn term of a series = + n 12
?a a a
1 2 3
+ + +.........+ = + a n
n
12
Put n = 1 , a
1
1 12 13 = + =
Putn = 2,a a
1 2
2 12 + = + ?a a
1 2
14 + =
?13 14
2
+ = a ?a
2
1 =
Put n = 3
a a a
1 2 3
3 12 + + = +
? 13 1 15
3
+ + = a
? a
3
15 14 1 = - =
5.
What is the value ofk for which the
sum of the squares of the roots of
2 2 2 1 0
2
x k x k - - - + = ( ) ( ) is
minimum?
(a) - 1 (b) 1
(c)
3
2
(d) 2
Ê
(c) Let a ß , be the roots of equation.
2 2 2 1 0
2
x k x k - - - + = ( ) ( )
Qa ß =
(
+
-
= -
2 2
2
2
k
k
)
,
aß =
- + ( ) k 1
2
We know that
a ß a ß aß
2 2 2
2 + = + - ( )
= - + ×
+
( ) k
k
2 2
1
2
2
= + - + + k k k
2
4 4 1
= - + k k
2
3 5
= - + - + k k
2
3
9
4
9
4
5
= -
?
?
?
?
?
?
+ k
3
2
11
4
2
a ß
2 2
+ is minimum, if k -
?
?
?
?
?
?
=
3
2
0
? k =
3
2
6.
If the roots of the equation
a b c x bc a x c a b ( ) ( ) ( ) - + - + - =
2
0
are equal, then which one of the
following is correct?
(a) a b , andc are in AP
(b) a b , andc are in GP
(c) a b , andc are in HP
(d) a b , and c do not follow any regular
pattern
Ê
(c) The roots of the equation
a b c x bc a x c a b ( ) ( ) ( ) - + - + - =
2
0
are equal.
?b c a a b c c a b
2 2
4 0 ( ) ( ). ( ) - - - - =
[Q ax bx c
2
0 + + = of roots are real if
b ac
2
4 0 - =
?b c a ca ac ab b
2 2 2 2
2 4 ( ) ( + - - -
- + = ac bc) 0
PAPER : I Mathematics
?b c a b ab c a bc
2 2 2 2 2 2
2 4 + - –
+ + - = 4 4 4 0
2 2 2 2
ab c a c abc
?b c a b ab c
2 2 2 2 2
2 + +
- - + = 4 4 4 0
2 2 2 2
a bc abc a c
?b c a ac abc a c
2 2 2
2 4 ( ) ( ) + + - +
+ = 4 0
2 2
a c
? b c a abc a c
2 2
4 ( ) ( ) + - + + = ( ) 2 0
2
ac
?[ ( ) ] bc a ac + - = 2 0
2
?b c a ac ( ) + - = 2 0
?bc a ac ( ) + = 2 ?b
ac
c a
=
+
2
So, a b , andc are is HP.
7.
If | | x x x x
2 2
3 2 3 2 - + > - + , then
which one of the following is
correct?
(a) x = 1or x = 2 (b) 1 2 = = x
(c) 1 2 < < x
(d) x is any real value except 3 and 4
Ê
(c)| | x x x x
2 2
3 2 3 2 - + > - +
? - - + > - + ( ) x x x x
2 2
3 2 3 2
[if x x
2
3 2 0 - + < , and x x
2
3 2 0 - + >
not possible]
? - - + > 2 3 2 0
2
( ) x x
? x x
2
3 2 0 - + >
? x x x
2
2 2 0 - - + >
? ( )( ) x x - - > 2 1 0
? 1 2 < < x is correct.
8.
A geometric progression (GP)
consists of 200 terms. If the sum of
odd terms of the GP is m, and the
sum of even terms of the GP is n,
then what is its common ratio?
(a) m n / (b) n m /
(c) m n m + ( / ) (d) n m n + ( / )
Ê
(b) Let a ar ar , , ......
2
200 terms be a
geometric progression.
Where, a is the first terms and r be the
common ratio.
GP of odd termsa ar ar , , , .....
2 4
100 terms.
GP of even terms ar ar ar , , ,
3 5
…… 100
terms.
?Sum of odd terms of the GP = m
?
a r
r
m
{ }
200
1
1
-
-
= …(i)
Sum of even terms of the GP = n
?
ar r
r
n
( }
200
1
1
-
-
= …(ii)
Dividing of Eq. (i) by Eq. (ii),
?
1
r
m
n
= ?r
n
m
=
Hence, the common ratio of the GP is
n
m
.
9.
If a set A contains 3 elements and
another set B contains 6 elements,
then what is the minimum number
of elements that ( ) A B ? can have?
(a) 3 (b) 6
(c) 8 (d) 9
Ê
(b) n A n B ( ) , ( ) = = 3 6
?The minimum number of elements in
A B ? = 6
i.e n A B ( ) ? = 6
(because maxn A B ( ) n = 3
10.
What is the number of diagonals of
an octagon?
(a) 48
(b) 40
(c) 28
(d) 20
Ê
(d) The number of vertices of an octagon
= 8
?The number of points in a plane = 8
?Total number of straight line form by 8
points =
8
2
C
[Q 1 straight line form by 2
points]
= =
×
=
8
2 6
8 7
2
28
!
! !
? The number of diagonals of an octagon
= Total number
of straight line form by 8 points - number
of sides of octagon
= - = 28 8 20
11.
What is the value of the determinant
1 2 3
2 3 4
3 4 5
! ! !
! ! !
! ! !
?
(a) 0 (b) 12
(c) 24 (d) 36
Ê
(c) Given determinant
=
1 2 3
2 3 4
3 4 5
! ! !
! ! !
! ! !
=
1 2 6
2 6 24
6 24 120
=
1 0 0
2 2 6
6 12 48
[byC C C C C C
2 2 1 3 3 2
2 3 ? - ? - , ]
= - - + 1 96 72 0 0 ( )
[expression w.r.t. first row]
= 24
12.
What are the values of x that satisfy
the equation
x
x
x
x
0 2
2 2 1
1 1 1
3 0 2
2 1
0 1 1
0
2
+ = ?
(a) - ± 2 3
(b) - ± 1 3
(c) - ± 1 6
(d) - ± 2 6
Ê
(d) Given equation,
x
x
x
x
0 2
2 2 1
1 1 1
3 0 2
2 1
0 1 1
0
2
+ =
? x x x ( ) ( ) ( ) 2 1 0 2 2 2 3 2 1 - - + - + -
- + - = 0 2 0 0
2
( ) x
[expression w.r.t. first row]
? x x x x + - + + = 4 4 3 2 0
2
? 2 8 4 0
2
x x + - =
? x x
2
4 2 0 + - =
? x =
- ± - - 4 16 4 1 2
2
( ) ( )
=
- ±
=
- ± 4 24
2
4 2 6
2
= - ± 2 6
13.
If x a b c + + + = 0, then what is the
value of
x a b c
a x b c
a b x c
+
+
+
?
(a) 0 (b) ( ) a b c + +
2
(c) a b c
2 2 2
+ + (d) a b c + + - 2
Ê
(a) Given, x a b c + + + = 0
x a b c
a x b c
a b x c
+
+
+
=
+ + +
+ + + +
+ + + +
x a b c b c
x a b c x b c
x a b c b x c
[byC C C C
1 1 2 3
? + + ]
= + + + ( ) x a b c
1
1
1
b c
x b c
b x c
+
+
[x a b c + + + common fromC
1
] = 0
[ ] Q x a b c + + + = 0
14.
IfA =
-
-
?
?
?
?
?
?
1 1
1 1
, then the expression
A A
3 2
2 - is
(a) a null matrix (b) an identity matrix
(c) equal to A (d) equal to -A
Ê
(a) A =
-
-
?
?
?
?
?
?
1 1
1 1
?A A A
2
1 1
1 1
1 1
1 1
= · =
-
-
?
?
?
?
?
?
·
-
-
?
?
?
?
?
?
=
+ - -
- - +
?
?
?
?
?
?
=
-
-
?
?
?
?
?
?
1 1 1 1
1 1 1 1
2 2
2 2
and A A A
3 2
2 2
2 2
1 1
1 1
= · =
-
-
?
?
?
?
?
?
-
-
?
?
?
?
?
?
.
=
+ - -
- - +
?
?
?
?
?
?
=
-
-
?
?
?
?
?
?
2 2 2 2
2 2 2 2
4 4
4 4
2
Now,
A A
3 2
2
4 4
4 4
2
2 2
2 2
- =
-
-
?
?
?
?
?
?
-
-
-
?
?
?
?
?
?
=
-
-
?
?
?
?
?
?
+
-
-
?
?
?
?
?
?
4 4
4 4
4 4
4 4
=
- - +
- + -
?
?
?
?
?
?
4 4 4 4
4 4 4 4
=
?
?
?
?
?
?
0 0
0 0
= a null matrix
15.
Letm andn m n ( ) < be the roots of the
equation x x
2
16 39 0 - + = . If four
terms p q r , , and s are inserted
betweenm andn to form an AP, then
what is the value of p q r s + + + ?
(a) 29 (b) 30
(c) 32 (d) 35
Ê
(c) m and n be the roots of the equation
x x m n
2
16 39 0 - + = < ( ).
? m n + = 16 …(i)
and mn = 39 …(ii)
We know that,n m m n mn - = + - ( )
2
4
( ) Q m n <
= - 256 156 = 100
n m - = 10 …(iii)
Solving the Eqs. (ii) and (iii),n m = = 13 3 ,
Four terms p q r , , and s are inserted
between m and n to form an AP.
? AP is 3 13 , , , , , p q r s
Here, a l n = = = 3 13 6 , ,
? l a n d = + - ( ) 1
13 3 6 1 = + - ( )d
? d =2
? p a d = + = + = 3 2 5,
q a d = + = + = 2 3 4 7
r a d = + = + = 3 3 6 9,
d a d = + = + = 4 3 8 11
Now, p q r s + + + = + + + 5 7 9 11
= 32
16.
Under which one of the following
conditions will the quadratic
equation
x mx
2
2 0 + + = always have real
roots?
(a) 2 3 8
2
= < m (b) 3 4
2
= < m
(c) m
2
8 = (d) m
2
3 =
Ê
(c) The quadratic equation
x mx
2
2 0 + + = ,
have real roots.
? m
2
4 1 2 0 - = ( )( )
[quadratic equation ax bx c
2
0 + + =
have real roots ifb ac
2
4 0 - = ]
? m
2
8 0 - =
? m
2
8 =
17.
What is the value of
i i +
?
?
?
?
?
?
+
-
?
?
?
?
?
?
3
2
3
2
2019 2019
?
(a) 1
(b) - 1
(c) 2i
(d) - 2i
Ê
(c)
i i + ?
?
?
?
?
?
+
- ?
?
?
?
?
?
3
2
3
2
2019 2019
= +
?
?
?
?
?
?
- -
?
?
?
?
?
?
3
2
1
2
3
2
1
2
2019 2019
i i
= +
?
?
?
?
?
?
cos sin
p p
6 6
2019
i
– cos sin
p p
6 6
2019
-
?
?
?
?
?
?
i
= + cos sin
2019
6
2019
6
p p
i
- + cos sin
2019
6
2019
6
p p
i
[De-moivre’s theorem
(cos sin ) cos sin ] ? ? ? ? ± = ± i n i n
n
= 2
2019
6
i sin
p
= × +
?
?
?
?
?
?
2 168 2
3
6
i sin p
p
= 2
3
6
i sin
p
[ sin ( ) sin , Q 2n n p ? ? + = is an integer]
= = 2
2
2 i i sin
p
18.
If a and ß are the roots of
x x
2
1 0 + + = , then what is
( ) a ß
j j
j
+
=
?
0
3
equal to?
(a) 8 (b) 6
(c) 4 (d) 2
Ê
(d) a and ß are the roots of the equation
x x
2
1 0 + + =
? a ß + = - 1
and aß = 1
Now, ( ) ( ) a ß a ß
j j
j
+ = +
=
?
0 0
0
3
+ + + + + + ( ) ( ) ( ) a ß a ß a ß
1 1 2 2 3 3
= + + - + + + - ( ) ( ) { } 1 1 1 2 2
2 2
a ß aß aß
+ + + - ( ) ( ) a ß a ß aß
2 2
= - + + - + - 2 1 2 1
2
{( ) } ( ) a ß aß
{ } a ß aß aß
2 2
2 3 + + -
= + - - - + - 1 1 2 1 3 1
2 2
{( ) ( )} {( ) ( )} a ß
= - - - - 1 1 1 3
2
{( ) }
= - - = ( ) 1 3 2
19. In a school, 50% students play cricket
and 40% play football. If 10% of
students play both the games, then
what per cent of students play
neither cricket nor football?
(a) 10% (b) 15% (c) 20% (d) 25%
Ê
(c) Students, who play cricket = 50%
Students, who play football = 40%
Students who play both games = 10%
Students who play only cricket
= - = 50 10 40%
Students who play only football
= - = 40 10 30%
?Total students who play any game
= + + = 40 30 10 80%
? Students who play neither cricket nor
football = - = 100 80 20%
20. If A x x = = = { : } 0 2 and B y y = { : is
a prime number}, then what is
A B n equal to?
(a) f (b) {1} (c) {2}      (d) {1, 2}
Ê
(c) A x x = = = { : } 0 2 = { , , } 0 1 2
and B y y = { : is a prime number}
= { , , , , , ..... } 2 3 5 7 11
?A B n = n { , , } { , , , , , ......} 0 1 2 2 3 5 7 11
= { } 2
21.
Ifx i = + 1 , then what is the value of
x x x
6 4 2
1 + + + ?
(a) 6 3 i - (b) - + 6 3 i
(c) - - 6 3 i (d) 6 3 i +
Ê
(c) Given, x i = + 1
= +
?
?
?
?
?
?
2
1
2 2
i
= +
?
?
?
?
?
?
2
4 4
cos sin
p p
i
Now, x x x
6 4 2
1 + + +
= + + + x x x
4 2 2
1 1 1 ( ) ( )
= + + ( ) ( ) x x
2 4
1 1
= +
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
( ) cos sin 2
4 4
1
2
2
p p
i
( ) cos sin 2
4 4
1
4
4
p p
+
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
i
= +
?
?
?
?
?
?
+
?
?
?
?
?
?
2
4
2
4
1 cos sin
2p p
i
4
4
4
4
4
1 cos sin
p p
+
?
?
?
?
?
?
+
?
?
?
?
?
?
i
[ (cos sin ) cos sin ] Q ? ? ? ? + = + i n i n
n
= +
?
?
?
?
?
?
+
?
?
?
?
?
?
2
2 2
1 cos sin
p p
i
[ (cos sin ) ] 4 1 p p + + i
= + + - + + [ ( ) ] [ ( ) ] 2 0 1 4 1 0 1 i
= + - + ( ) ( ) 2 1 4 1 i = - - 6 3 i
3
Page 4


1.
If both p and q belong to the set
{ , , , } 1 2 3 4 , then how many equations
of the form
px qx
2
1 0 + + = will have real roots?
(a) 12 (b) 10
(c) 7 (d) 6
Ê
(d) Equation px qx
2
1 0 + + = , has real
roots, where p and q belong to the set
{ , , , } 1 2 3 4 .
? q p
2
4 0 - =
[Q for real roots of a quadratic  equation
b ac
2
4 0 - = ]
It is possible if value of
( , ) ( , ), ( , ), ( , ), ( , ), ( , ) p q = 1 2 1 3 1 4 2 3 2 4
and ( , ) 3 4
Hence, the number of equations are 6.
2.
What is the value of
1 2 3 4 5 101 - + - + - + ...... ?
(a) 51 (b) 55
(c) 110 (d) 111
Ê
(a) Given series,
= - + - + - + 1 2 3 4 5 101 ......
= + + + + ( ..... ) 1 3 5 101
- + + + + ( ..... ) 2 4 6 100
= + + + ( .... 1 3 5 51terms)
- + + + ( ...... 2 4 6 50 terms)
= + - ×
51
2
2 51 1 2 [ ( ) ]
- + - ×
50
2
4 50 1 2 [ ( ) ]
[Q both series are AP and
S
n
a n d
n
= + -
2
2 1 [ ( ) ]
= × - ×
51
2
102
50
2
102
= - = 2601 2550 51
3.
If A B , and C are subsets of a given
set, then which one of the following
relations is not correct?
(a) A A B A B ? n = ? ( )
(b) A A B A n ? = ( )
(c) ( ) ( ) ( ) A B C A C B C n ? = ? n ?
(d) ( ) ( ) ( ) A B C A C B C ? n = n ? n
Ê
(a) LetU be the set and A B , andC are the
subset ofU.
We know that, A A B A ? n = ( ) ,
So option (a) is not correct.
A A B A n ? = ( ) , so option (b) is correct.
( ) ( ) ( ) A B C A C B C n ? = ? n ? ,
so option (c) is correct.
and ( ) ( ) ( ) A B C A C B C ? n = n ? n
so option (d) is correct.
4.
If the sum of firstn terms of a series
is ( ), n + 12 then what is its third
term?
(a) 1 (b) 2
(c) 3 (d) 4
Ê
(a) Sum of firstn term of a series = + n 12
?a a a
1 2 3
+ + +.........+ = + a n
n
12
Put n = 1 , a
1
1 12 13 = + =
Putn = 2,a a
1 2
2 12 + = + ?a a
1 2
14 + =
?13 14
2
+ = a ?a
2
1 =
Put n = 3
a a a
1 2 3
3 12 + + = +
? 13 1 15
3
+ + = a
? a
3
15 14 1 = - =
5.
What is the value ofk for which the
sum of the squares of the roots of
2 2 2 1 0
2
x k x k - - - + = ( ) ( ) is
minimum?
(a) - 1 (b) 1
(c)
3
2
(d) 2
Ê
(c) Let a ß , be the roots of equation.
2 2 2 1 0
2
x k x k - - - + = ( ) ( )
Qa ß =
(
+
-
= -
2 2
2
2
k
k
)
,
aß =
- + ( ) k 1
2
We know that
a ß a ß aß
2 2 2
2 + = + - ( )
= - + ×
+
( ) k
k
2 2
1
2
2
= + - + + k k k
2
4 4 1
= - + k k
2
3 5
= - + - + k k
2
3
9
4
9
4
5
= -
?
?
?
?
?
?
+ k
3
2
11
4
2
a ß
2 2
+ is minimum, if k -
?
?
?
?
?
?
=
3
2
0
? k =
3
2
6.
If the roots of the equation
a b c x bc a x c a b ( ) ( ) ( ) - + - + - =
2
0
are equal, then which one of the
following is correct?
(a) a b , andc are in AP
(b) a b , andc are in GP
(c) a b , andc are in HP
(d) a b , and c do not follow any regular
pattern
Ê
(c) The roots of the equation
a b c x bc a x c a b ( ) ( ) ( ) - + - + - =
2
0
are equal.
?b c a a b c c a b
2 2
4 0 ( ) ( ). ( ) - - - - =
[Q ax bx c
2
0 + + = of roots are real if
b ac
2
4 0 - =
?b c a ca ac ab b
2 2 2 2
2 4 ( ) ( + - - -
- + = ac bc) 0
PAPER : I Mathematics
?b c a b ab c a bc
2 2 2 2 2 2
2 4 + - –
+ + - = 4 4 4 0
2 2 2 2
ab c a c abc
?b c a b ab c
2 2 2 2 2
2 + +
- - + = 4 4 4 0
2 2 2 2
a bc abc a c
?b c a ac abc a c
2 2 2
2 4 ( ) ( ) + + - +
+ = 4 0
2 2
a c
? b c a abc a c
2 2
4 ( ) ( ) + - + + = ( ) 2 0
2
ac
?[ ( ) ] bc a ac + - = 2 0
2
?b c a ac ( ) + - = 2 0
?bc a ac ( ) + = 2 ?b
ac
c a
=
+
2
So, a b , andc are is HP.
7.
If | | x x x x
2 2
3 2 3 2 - + > - + , then
which one of the following is
correct?
(a) x = 1or x = 2 (b) 1 2 = = x
(c) 1 2 < < x
(d) x is any real value except 3 and 4
Ê
(c)| | x x x x
2 2
3 2 3 2 - + > - +
? - - + > - + ( ) x x x x
2 2
3 2 3 2
[if x x
2
3 2 0 - + < , and x x
2
3 2 0 - + >
not possible]
? - - + > 2 3 2 0
2
( ) x x
? x x
2
3 2 0 - + >
? x x x
2
2 2 0 - - + >
? ( )( ) x x - - > 2 1 0
? 1 2 < < x is correct.
8.
A geometric progression (GP)
consists of 200 terms. If the sum of
odd terms of the GP is m, and the
sum of even terms of the GP is n,
then what is its common ratio?
(a) m n / (b) n m /
(c) m n m + ( / ) (d) n m n + ( / )
Ê
(b) Let a ar ar , , ......
2
200 terms be a
geometric progression.
Where, a is the first terms and r be the
common ratio.
GP of odd termsa ar ar , , , .....
2 4
100 terms.
GP of even terms ar ar ar , , ,
3 5
…… 100
terms.
?Sum of odd terms of the GP = m
?
a r
r
m
{ }
200
1
1
-
-
= …(i)
Sum of even terms of the GP = n
?
ar r
r
n
( }
200
1
1
-
-
= …(ii)
Dividing of Eq. (i) by Eq. (ii),
?
1
r
m
n
= ?r
n
m
=
Hence, the common ratio of the GP is
n
m
.
9.
If a set A contains 3 elements and
another set B contains 6 elements,
then what is the minimum number
of elements that ( ) A B ? can have?
(a) 3 (b) 6
(c) 8 (d) 9
Ê
(b) n A n B ( ) , ( ) = = 3 6
?The minimum number of elements in
A B ? = 6
i.e n A B ( ) ? = 6
(because maxn A B ( ) n = 3
10.
What is the number of diagonals of
an octagon?
(a) 48
(b) 40
(c) 28
(d) 20
Ê
(d) The number of vertices of an octagon
= 8
?The number of points in a plane = 8
?Total number of straight line form by 8
points =
8
2
C
[Q 1 straight line form by 2
points]
= =
×
=
8
2 6
8 7
2
28
!
! !
? The number of diagonals of an octagon
= Total number
of straight line form by 8 points - number
of sides of octagon
= - = 28 8 20
11.
What is the value of the determinant
1 2 3
2 3 4
3 4 5
! ! !
! ! !
! ! !
?
(a) 0 (b) 12
(c) 24 (d) 36
Ê
(c) Given determinant
=
1 2 3
2 3 4
3 4 5
! ! !
! ! !
! ! !
=
1 2 6
2 6 24
6 24 120
=
1 0 0
2 2 6
6 12 48
[byC C C C C C
2 2 1 3 3 2
2 3 ? - ? - , ]
= - - + 1 96 72 0 0 ( )
[expression w.r.t. first row]
= 24
12.
What are the values of x that satisfy
the equation
x
x
x
x
0 2
2 2 1
1 1 1
3 0 2
2 1
0 1 1
0
2
+ = ?
(a) - ± 2 3
(b) - ± 1 3
(c) - ± 1 6
(d) - ± 2 6
Ê
(d) Given equation,
x
x
x
x
0 2
2 2 1
1 1 1
3 0 2
2 1
0 1 1
0
2
+ =
? x x x ( ) ( ) ( ) 2 1 0 2 2 2 3 2 1 - - + - + -
- + - = 0 2 0 0
2
( ) x
[expression w.r.t. first row]
? x x x x + - + + = 4 4 3 2 0
2
? 2 8 4 0
2
x x + - =
? x x
2
4 2 0 + - =
? x =
- ± - - 4 16 4 1 2
2
( ) ( )
=
- ±
=
- ± 4 24
2
4 2 6
2
= - ± 2 6
13.
If x a b c + + + = 0, then what is the
value of
x a b c
a x b c
a b x c
+
+
+
?
(a) 0 (b) ( ) a b c + +
2
(c) a b c
2 2 2
+ + (d) a b c + + - 2
Ê
(a) Given, x a b c + + + = 0
x a b c
a x b c
a b x c
+
+
+
=
+ + +
+ + + +
+ + + +
x a b c b c
x a b c x b c
x a b c b x c
[byC C C C
1 1 2 3
? + + ]
= + + + ( ) x a b c
1
1
1
b c
x b c
b x c
+
+
[x a b c + + + common fromC
1
] = 0
[ ] Q x a b c + + + = 0
14.
IfA =
-
-
?
?
?
?
?
?
1 1
1 1
, then the expression
A A
3 2
2 - is
(a) a null matrix (b) an identity matrix
(c) equal to A (d) equal to -A
Ê
(a) A =
-
-
?
?
?
?
?
?
1 1
1 1
?A A A
2
1 1
1 1
1 1
1 1
= · =
-
-
?
?
?
?
?
?
·
-
-
?
?
?
?
?
?
=
+ - -
- - +
?
?
?
?
?
?
=
-
-
?
?
?
?
?
?
1 1 1 1
1 1 1 1
2 2
2 2
and A A A
3 2
2 2
2 2
1 1
1 1
= · =
-
-
?
?
?
?
?
?
-
-
?
?
?
?
?
?
.
=
+ - -
- - +
?
?
?
?
?
?
=
-
-
?
?
?
?
?
?
2 2 2 2
2 2 2 2
4 4
4 4
2
Now,
A A
3 2
2
4 4
4 4
2
2 2
2 2
- =
-
-
?
?
?
?
?
?
-
-
-
?
?
?
?
?
?
=
-
-
?
?
?
?
?
?
+
-
-
?
?
?
?
?
?
4 4
4 4
4 4
4 4
=
- - +
- + -
?
?
?
?
?
?
4 4 4 4
4 4 4 4
=
?
?
?
?
?
?
0 0
0 0
= a null matrix
15.
Letm andn m n ( ) < be the roots of the
equation x x
2
16 39 0 - + = . If four
terms p q r , , and s are inserted
betweenm andn to form an AP, then
what is the value of p q r s + + + ?
(a) 29 (b) 30
(c) 32 (d) 35
Ê
(c) m and n be the roots of the equation
x x m n
2
16 39 0 - + = < ( ).
? m n + = 16 …(i)
and mn = 39 …(ii)
We know that,n m m n mn - = + - ( )
2
4
( ) Q m n <
= - 256 156 = 100
n m - = 10 …(iii)
Solving the Eqs. (ii) and (iii),n m = = 13 3 ,
Four terms p q r , , and s are inserted
between m and n to form an AP.
? AP is 3 13 , , , , , p q r s
Here, a l n = = = 3 13 6 , ,
? l a n d = + - ( ) 1
13 3 6 1 = + - ( )d
? d =2
? p a d = + = + = 3 2 5,
q a d = + = + = 2 3 4 7
r a d = + = + = 3 3 6 9,
d a d = + = + = 4 3 8 11
Now, p q r s + + + = + + + 5 7 9 11
= 32
16.
Under which one of the following
conditions will the quadratic
equation
x mx
2
2 0 + + = always have real
roots?
(a) 2 3 8
2
= < m (b) 3 4
2
= < m
(c) m
2
8 = (d) m
2
3 =
Ê
(c) The quadratic equation
x mx
2
2 0 + + = ,
have real roots.
? m
2
4 1 2 0 - = ( )( )
[quadratic equation ax bx c
2
0 + + =
have real roots ifb ac
2
4 0 - = ]
? m
2
8 0 - =
? m
2
8 =
17.
What is the value of
i i +
?
?
?
?
?
?
+
-
?
?
?
?
?
?
3
2
3
2
2019 2019
?
(a) 1
(b) - 1
(c) 2i
(d) - 2i
Ê
(c)
i i + ?
?
?
?
?
?
+
- ?
?
?
?
?
?
3
2
3
2
2019 2019
= +
?
?
?
?
?
?
- -
?
?
?
?
?
?
3
2
1
2
3
2
1
2
2019 2019
i i
= +
?
?
?
?
?
?
cos sin
p p
6 6
2019
i
– cos sin
p p
6 6
2019
-
?
?
?
?
?
?
i
= + cos sin
2019
6
2019
6
p p
i
- + cos sin
2019
6
2019
6
p p
i
[De-moivre’s theorem
(cos sin ) cos sin ] ? ? ? ? ± = ± i n i n
n
= 2
2019
6
i sin
p
= × +
?
?
?
?
?
?
2 168 2
3
6
i sin p
p
= 2
3
6
i sin
p
[ sin ( ) sin , Q 2n n p ? ? + = is an integer]
= = 2
2
2 i i sin
p
18.
If a and ß are the roots of
x x
2
1 0 + + = , then what is
( ) a ß
j j
j
+
=
?
0
3
equal to?
(a) 8 (b) 6
(c) 4 (d) 2
Ê
(d) a and ß are the roots of the equation
x x
2
1 0 + + =
? a ß + = - 1
and aß = 1
Now, ( ) ( ) a ß a ß
j j
j
+ = +
=
?
0 0
0
3
+ + + + + + ( ) ( ) ( ) a ß a ß a ß
1 1 2 2 3 3
= + + - + + + - ( ) ( ) { } 1 1 1 2 2
2 2
a ß aß aß
+ + + - ( ) ( ) a ß a ß aß
2 2
= - + + - + - 2 1 2 1
2
{( ) } ( ) a ß aß
{ } a ß aß aß
2 2
2 3 + + -
= + - - - + - 1 1 2 1 3 1
2 2
{( ) ( )} {( ) ( )} a ß
= - - - - 1 1 1 3
2
{( ) }
= - - = ( ) 1 3 2
19. In a school, 50% students play cricket
and 40% play football. If 10% of
students play both the games, then
what per cent of students play
neither cricket nor football?
(a) 10% (b) 15% (c) 20% (d) 25%
Ê
(c) Students, who play cricket = 50%
Students, who play football = 40%
Students who play both games = 10%
Students who play only cricket
= - = 50 10 40%
Students who play only football
= - = 40 10 30%
?Total students who play any game
= + + = 40 30 10 80%
? Students who play neither cricket nor
football = - = 100 80 20%
20. If A x x = = = { : } 0 2 and B y y = { : is
a prime number}, then what is
A B n equal to?
(a) f (b) {1} (c) {2}      (d) {1, 2}
Ê
(c) A x x = = = { : } 0 2 = { , , } 0 1 2
and B y y = { : is a prime number}
= { , , , , , ..... } 2 3 5 7 11
?A B n = n { , , } { , , , , , ......} 0 1 2 2 3 5 7 11
= { } 2
21.
Ifx i = + 1 , then what is the value of
x x x
6 4 2
1 + + + ?
(a) 6 3 i - (b) - + 6 3 i
(c) - - 6 3 i (d) 6 3 i +
Ê
(c) Given, x i = + 1
= +
?
?
?
?
?
?
2
1
2 2
i
= +
?
?
?
?
?
?
2
4 4
cos sin
p p
i
Now, x x x
6 4 2
1 + + +
= + + + x x x
4 2 2
1 1 1 ( ) ( )
= + + ( ) ( ) x x
2 4
1 1
= +
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
( ) cos sin 2
4 4
1
2
2
p p
i
( ) cos sin 2
4 4
1
4
4
p p
+
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
i
= +
?
?
?
?
?
?
+
?
?
?
?
?
?
2
4
2
4
1 cos sin
2p p
i
4
4
4
4
4
1 cos sin
p p
+
?
?
?
?
?
?
+
?
?
?
?
?
?
i
[ (cos sin ) cos sin ] Q ? ? ? ? + = + i n i n
n
= +
?
?
?
?
?
?
+
?
?
?
?
?
?
2
2 2
1 cos sin
p p
i
[ (cos sin ) ] 4 1 p p + + i
= + + - + + [ ( ) ] [ ( ) ] 2 0 1 4 1 0 1 i
= + - + ( ) ( ) 2 1 4 1 i = - - 6 3 i
3
22. What is the value of
2
1
2
1
2
1
2
+
+
+
+ 8 ...
?
(a) 2 1 - (b) 2 1 + (c) 3   (d) 4
Ê
(b) Let, x = +
+
+
+ 8
2
1
2
1
2
1
2 ...
? x
x
= + 2
1
? x x
2
2 1 = +
? x x
2
2 1 0 - - =
? x =
± - - - 2 2 4 1 1
2
2
( ) ( ) ( )
=
± 2 8
2
=
± 2 2 2
2
= ± 1 2
= + 2 1 ( ) Q x > 2
23.
If P n r ( , ) = 2520 and C n r ( , ) = 21,
then what is the value of
C n r ( , ) + + 1 1 ?
(a) 7 (b) 14
(c) 28 (d) 56
Ê
(c) IfP n r ( , ) = 2520 andC n r ( , ) = 21 ,
?
n
r
p = 2520
?
n
n r
!
( )! -
= 2520 …(i)
and
n
r
C = 21
?
n
r n r
!
! ( )! -
= 21 …(ii)
From Eqs. (i) and (ii), we get
2520
21
r!
=
? r! = =
2520
21
120
? r! ! = 5
? r = 5
Putting the value ofr in Eq. (i),
n
n
!
( )! -
=
5
2520
? n n n n n ( ) ( ) ( )( ) - - - - 1 2 3 4
= × × × × 7 6 5 4 3
? n = 7
Now,C n r ( , ) + + 1 1 =
+
+
n
r
C
1
1
= =
+
+
7 1
5 1
8
6
C C
= =
× 8
6 2
8 7
2
!
! !
= 28
24.
How many terms are there in the
expansion of
( ) ( ) ? 1 2 1 4 4
2 5 2 5
+ + + + + x x y y
(a) 12 (b) 20
(c) 21 (d) 22
Ê
(d) Given expansion,
( ) ( ) 1 2 1 4 4
2 5 2 5
+ + + + + x x y y
= + + + [( ) ] [( ) ] 1 1 2
2 5 2 5
x y
= + + + ( ) ( ) 1 1 2
10 10
x y
? Total number of terms in given
expansion.
= + + + ( ) ( ) 10 1 10 1 = 22
[Q total number of terms in expansion of
( ) 1 1 + = + x n
n
]
25.
If the middle term in the expansion
of x
x
n
2
2
1
+
?
?
?
?
?
?
is 184756
10
x , then
what is the value ofn?
(a) 10 (b) 8
(c) 5 (d) 4
Ê
(a) The middle term in the expansion of
x
x
n
2
2
1
+
?
?
?
?
?
?
= +
?
?
?
?
?
?
2
2
1
n
th term [Q 2nis even]
= + ( ) n 1th term.
According to the question,
Value of middle term = 184756
10
x
?
2 2 2
1
n
n
n n
n
C x
x
( )
- ?
?
?
?
?
?
= 184756
10
x
[QT C x a
r
n
r
n r r
+
-
=
1
in expansion
of ( ) x a
n
+ ]
?
2 4 2 10
184756
n
n
n n n
C x x ( )
- -
=
?
2 10
184756
n
n
n
C x x ( ) =
Comparing the power of x both sides
n = 10
26.
If A =
?
?
?
?
?
?
?
?
?
?
1
2
3
2
3
4
and B =
?
?
?
?
?
?
1 2
2 1
, then
which one of the following is
correct?
(a) BothAB and BA exist
(b) NeitherAB nor BA exists
(c) AB exists butBA does not exist
(d) AB does not exist butBA exists
Ê
(c)We have, A =
?
?
?
?
?
?
?
?
?
?
1
2
3
2
3
4
andB =
?
?
?
?
?
?
1 2
2 1
order of A = × 3 2 and order ofB = × 2 2
Q Number of column of A = Number of
row of B
?AB exists.
and number of column of B ? Number of
raw of A
?BA does not exist.
Hence, AB exists butBA does not exist.
27.
If n! has 17 zeros, then what is the
value ofn?
(a) 95 (b) 85
(c) 80
(d) No such value ofn exists
Ê
(b) We know that each interval of 5!is one
zero.
i.e. 5! has one zero.
10! has two zeros.
?85! has 17 zeros.
Hence, the value ofn is 85.
28. Let A B ? = { |( )( ) x x a x b - - > 0,
wherea b < }. What areA andB equal
to?
(a) A x x a = > { | } and B x x b = > { | }
(b) A x x a = < { | } and B x x b = > { | }
(c) A x x a = < { | } and B x x b = < { | }
(d) A x x a = > { | } and B x x b = < { | }
Ê
(c) Let A B ? = { :( )( ) x x a x b - - > 0,
where a b < }.
It is possible if x a - < 0 and x b - < 0
or x a < and x b <
?A x x a = < { : } and B x x b = < { : }
29.
If the constant term in the expansion
of x
k
x
-
?
?
?
?
?
?
2
10
is 405, then what can
be the values ofk?
(a) ±2 (b) ±3
(c) ±5 (d) ±9
Ê
(b) Let ( ) r + 1 th term in the expansion of
x
k
x
-
?
?
?
?
?
?
2
10
is constant.
? T C x
k
x
r r
r
r
+
-
=
- ?
?
?
?
?
? 1
10 10
2
( )
[QT C x a
r
n
r
n r r
+
-
=
1
in expansion
of ( ) x a
n
+ ]
? 405
10
10
2
2
= · -
-
-
C x k
r
r
r
r
( ) ( )
? 405
10
10 5
2
= · -
-
C x k
r
r
r
( ) ( ) …(i)
For constant term
10 5
2
0
-
=
r
?10 5 0 - = r
? r = 2
Putting the value ofr, in Eq. (i),
405
10
2
2
= - C k .( )
? 405
10
2 8
2
= ×
!
! !
k
? 405
10 9
2
2
=
×
.k
? k
2
405
45
=
? k
2
9 =
? k = ± 3
4
Page 5


1.
If both p and q belong to the set
{ , , , } 1 2 3 4 , then how many equations
of the form
px qx
2
1 0 + + = will have real roots?
(a) 12 (b) 10
(c) 7 (d) 6
Ê
(d) Equation px qx
2
1 0 + + = , has real
roots, where p and q belong to the set
{ , , , } 1 2 3 4 .
? q p
2
4 0 - =
[Q for real roots of a quadratic  equation
b ac
2
4 0 - = ]
It is possible if value of
( , ) ( , ), ( , ), ( , ), ( , ), ( , ) p q = 1 2 1 3 1 4 2 3 2 4
and ( , ) 3 4
Hence, the number of equations are 6.
2.
What is the value of
1 2 3 4 5 101 - + - + - + ...... ?
(a) 51 (b) 55
(c) 110 (d) 111
Ê
(a) Given series,
= - + - + - + 1 2 3 4 5 101 ......
= + + + + ( ..... ) 1 3 5 101
- + + + + ( ..... ) 2 4 6 100
= + + + ( .... 1 3 5 51terms)
- + + + ( ...... 2 4 6 50 terms)
= + - ×
51
2
2 51 1 2 [ ( ) ]
- + - ×
50
2
4 50 1 2 [ ( ) ]
[Q both series are AP and
S
n
a n d
n
= + -
2
2 1 [ ( ) ]
= × - ×
51
2
102
50
2
102
= - = 2601 2550 51
3.
If A B , and C are subsets of a given
set, then which one of the following
relations is not correct?
(a) A A B A B ? n = ? ( )
(b) A A B A n ? = ( )
(c) ( ) ( ) ( ) A B C A C B C n ? = ? n ?
(d) ( ) ( ) ( ) A B C A C B C ? n = n ? n
Ê
(a) LetU be the set and A B , andC are the
subset ofU.
We know that, A A B A ? n = ( ) ,
So option (a) is not correct.
A A B A n ? = ( ) , so option (b) is correct.
( ) ( ) ( ) A B C A C B C n ? = ? n ? ,
so option (c) is correct.
and ( ) ( ) ( ) A B C A C B C ? n = n ? n
so option (d) is correct.
4.
If the sum of firstn terms of a series
is ( ), n + 12 then what is its third
term?
(a) 1 (b) 2
(c) 3 (d) 4
Ê
(a) Sum of firstn term of a series = + n 12
?a a a
1 2 3
+ + +.........+ = + a n
n
12
Put n = 1 , a
1
1 12 13 = + =
Putn = 2,a a
1 2
2 12 + = + ?a a
1 2
14 + =
?13 14
2
+ = a ?a
2
1 =
Put n = 3
a a a
1 2 3
3 12 + + = +
? 13 1 15
3
+ + = a
? a
3
15 14 1 = - =
5.
What is the value ofk for which the
sum of the squares of the roots of
2 2 2 1 0
2
x k x k - - - + = ( ) ( ) is
minimum?
(a) - 1 (b) 1
(c)
3
2
(d) 2
Ê
(c) Let a ß , be the roots of equation.
2 2 2 1 0
2
x k x k - - - + = ( ) ( )
Qa ß =
(
+
-
= -
2 2
2
2
k
k
)
,
aß =
- + ( ) k 1
2
We know that
a ß a ß aß
2 2 2
2 + = + - ( )
= - + ×
+
( ) k
k
2 2
1
2
2
= + - + + k k k
2
4 4 1
= - + k k
2
3 5
= - + - + k k
2
3
9
4
9
4
5
= -
?
?
?
?
?
?
+ k
3
2
11
4
2
a ß
2 2
+ is minimum, if k -
?
?
?
?
?
?
=
3
2
0
? k =
3
2
6.
If the roots of the equation
a b c x bc a x c a b ( ) ( ) ( ) - + - + - =
2
0
are equal, then which one of the
following is correct?
(a) a b , andc are in AP
(b) a b , andc are in GP
(c) a b , andc are in HP
(d) a b , and c do not follow any regular
pattern
Ê
(c) The roots of the equation
a b c x bc a x c a b ( ) ( ) ( ) - + - + - =
2
0
are equal.
?b c a a b c c a b
2 2
4 0 ( ) ( ). ( ) - - - - =
[Q ax bx c
2
0 + + = of roots are real if
b ac
2
4 0 - =
?b c a ca ac ab b
2 2 2 2
2 4 ( ) ( + - - -
- + = ac bc) 0
PAPER : I Mathematics
?b c a b ab c a bc
2 2 2 2 2 2
2 4 + - –
+ + - = 4 4 4 0
2 2 2 2
ab c a c abc
?b c a b ab c
2 2 2 2 2
2 + +
- - + = 4 4 4 0
2 2 2 2
a bc abc a c
?b c a ac abc a c
2 2 2
2 4 ( ) ( ) + + - +
+ = 4 0
2 2
a c
? b c a abc a c
2 2
4 ( ) ( ) + - + + = ( ) 2 0
2
ac
?[ ( ) ] bc a ac + - = 2 0
2
?b c a ac ( ) + - = 2 0
?bc a ac ( ) + = 2 ?b
ac
c a
=
+
2
So, a b , andc are is HP.
7.
If | | x x x x
2 2
3 2 3 2 - + > - + , then
which one of the following is
correct?
(a) x = 1or x = 2 (b) 1 2 = = x
(c) 1 2 < < x
(d) x is any real value except 3 and 4
Ê
(c)| | x x x x
2 2
3 2 3 2 - + > - +
? - - + > - + ( ) x x x x
2 2
3 2 3 2
[if x x
2
3 2 0 - + < , and x x
2
3 2 0 - + >
not possible]
? - - + > 2 3 2 0
2
( ) x x
? x x
2
3 2 0 - + >
? x x x
2
2 2 0 - - + >
? ( )( ) x x - - > 2 1 0
? 1 2 < < x is correct.
8.
A geometric progression (GP)
consists of 200 terms. If the sum of
odd terms of the GP is m, and the
sum of even terms of the GP is n,
then what is its common ratio?
(a) m n / (b) n m /
(c) m n m + ( / ) (d) n m n + ( / )
Ê
(b) Let a ar ar , , ......
2
200 terms be a
geometric progression.
Where, a is the first terms and r be the
common ratio.
GP of odd termsa ar ar , , , .....
2 4
100 terms.
GP of even terms ar ar ar , , ,
3 5
…… 100
terms.
?Sum of odd terms of the GP = m
?
a r
r
m
{ }
200
1
1
-
-
= …(i)
Sum of even terms of the GP = n
?
ar r
r
n
( }
200
1
1
-
-
= …(ii)
Dividing of Eq. (i) by Eq. (ii),
?
1
r
m
n
= ?r
n
m
=
Hence, the common ratio of the GP is
n
m
.
9.
If a set A contains 3 elements and
another set B contains 6 elements,
then what is the minimum number
of elements that ( ) A B ? can have?
(a) 3 (b) 6
(c) 8 (d) 9
Ê
(b) n A n B ( ) , ( ) = = 3 6
?The minimum number of elements in
A B ? = 6
i.e n A B ( ) ? = 6
(because maxn A B ( ) n = 3
10.
What is the number of diagonals of
an octagon?
(a) 48
(b) 40
(c) 28
(d) 20
Ê
(d) The number of vertices of an octagon
= 8
?The number of points in a plane = 8
?Total number of straight line form by 8
points =
8
2
C
[Q 1 straight line form by 2
points]
= =
×
=
8
2 6
8 7
2
28
!
! !
? The number of diagonals of an octagon
= Total number
of straight line form by 8 points - number
of sides of octagon
= - = 28 8 20
11.
What is the value of the determinant
1 2 3
2 3 4
3 4 5
! ! !
! ! !
! ! !
?
(a) 0 (b) 12
(c) 24 (d) 36
Ê
(c) Given determinant
=
1 2 3
2 3 4
3 4 5
! ! !
! ! !
! ! !
=
1 2 6
2 6 24
6 24 120
=
1 0 0
2 2 6
6 12 48
[byC C C C C C
2 2 1 3 3 2
2 3 ? - ? - , ]
= - - + 1 96 72 0 0 ( )
[expression w.r.t. first row]
= 24
12.
What are the values of x that satisfy
the equation
x
x
x
x
0 2
2 2 1
1 1 1
3 0 2
2 1
0 1 1
0
2
+ = ?
(a) - ± 2 3
(b) - ± 1 3
(c) - ± 1 6
(d) - ± 2 6
Ê
(d) Given equation,
x
x
x
x
0 2
2 2 1
1 1 1
3 0 2
2 1
0 1 1
0
2
+ =
? x x x ( ) ( ) ( ) 2 1 0 2 2 2 3 2 1 - - + - + -
- + - = 0 2 0 0
2
( ) x
[expression w.r.t. first row]
? x x x x + - + + = 4 4 3 2 0
2
? 2 8 4 0
2
x x + - =
? x x
2
4 2 0 + - =
? x =
- ± - - 4 16 4 1 2
2
( ) ( )
=
- ±
=
- ± 4 24
2
4 2 6
2
= - ± 2 6
13.
If x a b c + + + = 0, then what is the
value of
x a b c
a x b c
a b x c
+
+
+
?
(a) 0 (b) ( ) a b c + +
2
(c) a b c
2 2 2
+ + (d) a b c + + - 2
Ê
(a) Given, x a b c + + + = 0
x a b c
a x b c
a b x c
+
+
+
=
+ + +
+ + + +
+ + + +
x a b c b c
x a b c x b c
x a b c b x c
[byC C C C
1 1 2 3
? + + ]
= + + + ( ) x a b c
1
1
1
b c
x b c
b x c
+
+
[x a b c + + + common fromC
1
] = 0
[ ] Q x a b c + + + = 0
14.
IfA =
-
-
?
?
?
?
?
?
1 1
1 1
, then the expression
A A
3 2
2 - is
(a) a null matrix (b) an identity matrix
(c) equal to A (d) equal to -A
Ê
(a) A =
-
-
?
?
?
?
?
?
1 1
1 1
?A A A
2
1 1
1 1
1 1
1 1
= · =
-
-
?
?
?
?
?
?
·
-
-
?
?
?
?
?
?
=
+ - -
- - +
?
?
?
?
?
?
=
-
-
?
?
?
?
?
?
1 1 1 1
1 1 1 1
2 2
2 2
and A A A
3 2
2 2
2 2
1 1
1 1
= · =
-
-
?
?
?
?
?
?
-
-
?
?
?
?
?
?
.
=
+ - -
- - +
?
?
?
?
?
?
=
-
-
?
?
?
?
?
?
2 2 2 2
2 2 2 2
4 4
4 4
2
Now,
A A
3 2
2
4 4
4 4
2
2 2
2 2
- =
-
-
?
?
?
?
?
?
-
-
-
?
?
?
?
?
?
=
-
-
?
?
?
?
?
?
+
-
-
?
?
?
?
?
?
4 4
4 4
4 4
4 4
=
- - +
- + -
?
?
?
?
?
?
4 4 4 4
4 4 4 4
=
?
?
?
?
?
?
0 0
0 0
= a null matrix
15.
Letm andn m n ( ) < be the roots of the
equation x x
2
16 39 0 - + = . If four
terms p q r , , and s are inserted
betweenm andn to form an AP, then
what is the value of p q r s + + + ?
(a) 29 (b) 30
(c) 32 (d) 35
Ê
(c) m and n be the roots of the equation
x x m n
2
16 39 0 - + = < ( ).
? m n + = 16 …(i)
and mn = 39 …(ii)
We know that,n m m n mn - = + - ( )
2
4
( ) Q m n <
= - 256 156 = 100
n m - = 10 …(iii)
Solving the Eqs. (ii) and (iii),n m = = 13 3 ,
Four terms p q r , , and s are inserted
between m and n to form an AP.
? AP is 3 13 , , , , , p q r s
Here, a l n = = = 3 13 6 , ,
? l a n d = + - ( ) 1
13 3 6 1 = + - ( )d
? d =2
? p a d = + = + = 3 2 5,
q a d = + = + = 2 3 4 7
r a d = + = + = 3 3 6 9,
d a d = + = + = 4 3 8 11
Now, p q r s + + + = + + + 5 7 9 11
= 32
16.
Under which one of the following
conditions will the quadratic
equation
x mx
2
2 0 + + = always have real
roots?
(a) 2 3 8
2
= < m (b) 3 4
2
= < m
(c) m
2
8 = (d) m
2
3 =
Ê
(c) The quadratic equation
x mx
2
2 0 + + = ,
have real roots.
? m
2
4 1 2 0 - = ( )( )
[quadratic equation ax bx c
2
0 + + =
have real roots ifb ac
2
4 0 - = ]
? m
2
8 0 - =
? m
2
8 =
17.
What is the value of
i i +
?
?
?
?
?
?
+
-
?
?
?
?
?
?
3
2
3
2
2019 2019
?
(a) 1
(b) - 1
(c) 2i
(d) - 2i
Ê
(c)
i i + ?
?
?
?
?
?
+
- ?
?
?
?
?
?
3
2
3
2
2019 2019
= +
?
?
?
?
?
?
- -
?
?
?
?
?
?
3
2
1
2
3
2
1
2
2019 2019
i i
= +
?
?
?
?
?
?
cos sin
p p
6 6
2019
i
– cos sin
p p
6 6
2019
-
?
?
?
?
?
?
i
= + cos sin
2019
6
2019
6
p p
i
- + cos sin
2019
6
2019
6
p p
i
[De-moivre’s theorem
(cos sin ) cos sin ] ? ? ? ? ± = ± i n i n
n
= 2
2019
6
i sin
p
= × +
?
?
?
?
?
?
2 168 2
3
6
i sin p
p
= 2
3
6
i sin
p
[ sin ( ) sin , Q 2n n p ? ? + = is an integer]
= = 2
2
2 i i sin
p
18.
If a and ß are the roots of
x x
2
1 0 + + = , then what is
( ) a ß
j j
j
+
=
?
0
3
equal to?
(a) 8 (b) 6
(c) 4 (d) 2
Ê
(d) a and ß are the roots of the equation
x x
2
1 0 + + =
? a ß + = - 1
and aß = 1
Now, ( ) ( ) a ß a ß
j j
j
+ = +
=
?
0 0
0
3
+ + + + + + ( ) ( ) ( ) a ß a ß a ß
1 1 2 2 3 3
= + + - + + + - ( ) ( ) { } 1 1 1 2 2
2 2
a ß aß aß
+ + + - ( ) ( ) a ß a ß aß
2 2
= - + + - + - 2 1 2 1
2
{( ) } ( ) a ß aß
{ } a ß aß aß
2 2
2 3 + + -
= + - - - + - 1 1 2 1 3 1
2 2
{( ) ( )} {( ) ( )} a ß
= - - - - 1 1 1 3
2
{( ) }
= - - = ( ) 1 3 2
19. In a school, 50% students play cricket
and 40% play football. If 10% of
students play both the games, then
what per cent of students play
neither cricket nor football?
(a) 10% (b) 15% (c) 20% (d) 25%
Ê
(c) Students, who play cricket = 50%
Students, who play football = 40%
Students who play both games = 10%
Students who play only cricket
= - = 50 10 40%
Students who play only football
= - = 40 10 30%
?Total students who play any game
= + + = 40 30 10 80%
? Students who play neither cricket nor
football = - = 100 80 20%
20. If A x x = = = { : } 0 2 and B y y = { : is
a prime number}, then what is
A B n equal to?
(a) f (b) {1} (c) {2}      (d) {1, 2}
Ê
(c) A x x = = = { : } 0 2 = { , , } 0 1 2
and B y y = { : is a prime number}
= { , , , , , ..... } 2 3 5 7 11
?A B n = n { , , } { , , , , , ......} 0 1 2 2 3 5 7 11
= { } 2
21.
Ifx i = + 1 , then what is the value of
x x x
6 4 2
1 + + + ?
(a) 6 3 i - (b) - + 6 3 i
(c) - - 6 3 i (d) 6 3 i +
Ê
(c) Given, x i = + 1
= +
?
?
?
?
?
?
2
1
2 2
i
= +
?
?
?
?
?
?
2
4 4
cos sin
p p
i
Now, x x x
6 4 2
1 + + +
= + + + x x x
4 2 2
1 1 1 ( ) ( )
= + + ( ) ( ) x x
2 4
1 1
= +
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
( ) cos sin 2
4 4
1
2
2
p p
i
( ) cos sin 2
4 4
1
4
4
p p
+
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
i
= +
?
?
?
?
?
?
+
?
?
?
?
?
?
2
4
2
4
1 cos sin
2p p
i
4
4
4
4
4
1 cos sin
p p
+
?
?
?
?
?
?
+
?
?
?
?
?
?
i
[ (cos sin ) cos sin ] Q ? ? ? ? + = + i n i n
n
= +
?
?
?
?
?
?
+
?
?
?
?
?
?
2
2 2
1 cos sin
p p
i
[ (cos sin ) ] 4 1 p p + + i
= + + - + + [ ( ) ] [ ( ) ] 2 0 1 4 1 0 1 i
= + - + ( ) ( ) 2 1 4 1 i = - - 6 3 i
3
22. What is the value of
2
1
2
1
2
1
2
+
+
+
+ 8 ...
?
(a) 2 1 - (b) 2 1 + (c) 3   (d) 4
Ê
(b) Let, x = +
+
+
+ 8
2
1
2
1
2
1
2 ...
? x
x
= + 2
1
? x x
2
2 1 = +
? x x
2
2 1 0 - - =
? x =
± - - - 2 2 4 1 1
2
2
( ) ( ) ( )
=
± 2 8
2
=
± 2 2 2
2
= ± 1 2
= + 2 1 ( ) Q x > 2
23.
If P n r ( , ) = 2520 and C n r ( , ) = 21,
then what is the value of
C n r ( , ) + + 1 1 ?
(a) 7 (b) 14
(c) 28 (d) 56
Ê
(c) IfP n r ( , ) = 2520 andC n r ( , ) = 21 ,
?
n
r
p = 2520
?
n
n r
!
( )! -
= 2520 …(i)
and
n
r
C = 21
?
n
r n r
!
! ( )! -
= 21 …(ii)
From Eqs. (i) and (ii), we get
2520
21
r!
=
? r! = =
2520
21
120
? r! ! = 5
? r = 5
Putting the value ofr in Eq. (i),
n
n
!
( )! -
=
5
2520
? n n n n n ( ) ( ) ( )( ) - - - - 1 2 3 4
= × × × × 7 6 5 4 3
? n = 7
Now,C n r ( , ) + + 1 1 =
+
+
n
r
C
1
1
= =
+
+
7 1
5 1
8
6
C C
= =
× 8
6 2
8 7
2
!
! !
= 28
24.
How many terms are there in the
expansion of
( ) ( ) ? 1 2 1 4 4
2 5 2 5
+ + + + + x x y y
(a) 12 (b) 20
(c) 21 (d) 22
Ê
(d) Given expansion,
( ) ( ) 1 2 1 4 4
2 5 2 5
+ + + + + x x y y
= + + + [( ) ] [( ) ] 1 1 2
2 5 2 5
x y
= + + + ( ) ( ) 1 1 2
10 10
x y
? Total number of terms in given
expansion.
= + + + ( ) ( ) 10 1 10 1 = 22
[Q total number of terms in expansion of
( ) 1 1 + = + x n
n
]
25.
If the middle term in the expansion
of x
x
n
2
2
1
+
?
?
?
?
?
?
is 184756
10
x , then
what is the value ofn?
(a) 10 (b) 8
(c) 5 (d) 4
Ê
(a) The middle term in the expansion of
x
x
n
2
2
1
+
?
?
?
?
?
?
= +
?
?
?
?
?
?
2
2
1
n
th term [Q 2nis even]
= + ( ) n 1th term.
According to the question,
Value of middle term = 184756
10
x
?
2 2 2
1
n
n
n n
n
C x
x
( )
- ?
?
?
?
?
?
= 184756
10
x
[QT C x a
r
n
r
n r r
+
-
=
1
in expansion
of ( ) x a
n
+ ]
?
2 4 2 10
184756
n
n
n n n
C x x ( )
- -
=
?
2 10
184756
n
n
n
C x x ( ) =
Comparing the power of x both sides
n = 10
26.
If A =
?
?
?
?
?
?
?
?
?
?
1
2
3
2
3
4
and B =
?
?
?
?
?
?
1 2
2 1
, then
which one of the following is
correct?
(a) BothAB and BA exist
(b) NeitherAB nor BA exists
(c) AB exists butBA does not exist
(d) AB does not exist butBA exists
Ê
(c)We have, A =
?
?
?
?
?
?
?
?
?
?
1
2
3
2
3
4
andB =
?
?
?
?
?
?
1 2
2 1
order of A = × 3 2 and order ofB = × 2 2
Q Number of column of A = Number of
row of B
?AB exists.
and number of column of B ? Number of
raw of A
?BA does not exist.
Hence, AB exists butBA does not exist.
27.
If n! has 17 zeros, then what is the
value ofn?
(a) 95 (b) 85
(c) 80
(d) No such value ofn exists
Ê
(b) We know that each interval of 5!is one
zero.
i.e. 5! has one zero.
10! has two zeros.
?85! has 17 zeros.
Hence, the value ofn is 85.
28. Let A B ? = { |( )( ) x x a x b - - > 0,
wherea b < }. What areA andB equal
to?
(a) A x x a = > { | } and B x x b = > { | }
(b) A x x a = < { | } and B x x b = > { | }
(c) A x x a = < { | } and B x x b = < { | }
(d) A x x a = > { | } and B x x b = < { | }
Ê
(c) Let A B ? = { :( )( ) x x a x b - - > 0,
where a b < }.
It is possible if x a - < 0 and x b - < 0
or x a < and x b <
?A x x a = < { : } and B x x b = < { : }
29.
If the constant term in the expansion
of x
k
x
-
?
?
?
?
?
?
2
10
is 405, then what can
be the values ofk?
(a) ±2 (b) ±3
(c) ±5 (d) ±9
Ê
(b) Let ( ) r + 1 th term in the expansion of
x
k
x
-
?
?
?
?
?
?
2
10
is constant.
? T C x
k
x
r r
r
r
+
-
=
- ?
?
?
?
?
? 1
10 10
2
( )
[QT C x a
r
n
r
n r r
+
-
=
1
in expansion
of ( ) x a
n
+ ]
? 405
10
10
2
2
= · -
-
-
C x k
r
r
r
r
( ) ( )
? 405
10
10 5
2
= · -
-
C x k
r
r
r
( ) ( ) …(i)
For constant term
10 5
2
0
-
=
r
?10 5 0 - = r
? r = 2
Putting the value ofr, in Eq. (i),
405
10
2
2
= - C k .( )
? 405
10
2 8
2
= ×
!
! !
k
? 405
10 9
2
2
=
×
.k
? k
2
405
45
=
? k
2
9 =
? k = ± 3
4
30.
What isC C C ( , ) ( , ) ( , ) 47 4 51 3 50 3 + +
+ + + C C C ( , ) ( , ) ( , ) 49 3 48 3 47 3 equal
to?
(a)C( , ) 47 4 (b)C( , ) 52 5
(c)C( , ) 52 4 (d)C( , ) 47 5
Ê
(c)C C C ( , ) ( , ) ( , ) 47 4 51 3 50 3 + +
+ + + C C C ( , ) ( , ) ( , ) 49 3 48 3 47 3
= + + +
47
4
51
3
50
3
49
3
C C C C
+ +
48
3
47
3
C C
= + + +
51
3
50
3
49
3
48
3
C C C C
+ +
47
3
47
4
C C
= + + + +
51
3
50
3
49
3
48
3
48
4
C C C C C
[ ] Q
n
r
n
r
n
r
C C C + =
-
+
1
1
= + + +
51
3
50
3
49
3
49
4
C C C C
= + +
51
3
50
3
50
4
C C C
= +
51
3
51
4
C C
= =
52
4
52 4 C C( , )
31.
Leta b c , , be in AP andk ? 0 be a real
number. Which of the following are
correct?
1. ka kb kc , , are in AP
2. k a k b k c - - - , , are in AP
3.
a
k
b
k
c
k
, , are in AP
Select the correct answer using the
code given below.
(a) 1 and 2 only (b) 2 and 3 only
(c) 1 and 3 only (d) 1, 2 and 3
Ê
(d) a bc , , are in AP.
We know that equal number addition,
subtraction and multiply, divide, by equal
number of each term of an AP, the
resultent, series be an AP.
?ka kb kc , , are in AP (multiplying byk).
k a k b k c - - - , , are in AP (subtraction
fromk) and
a
k
b
k
c
k
, , are in AP (divide byk)
Hence, option (d) is correct answer.
32.
How many two-digit numbers are
divisible by 4?
(a) 21 (b) 22
(c) 24 (d) 25
Ê
(b) Series of two-digit number that
divisible by 4 is
12, 16, 20, ........., 96
This series is an AP
Here, A = 12,d = 4,l = 96
Let total number of terms ben.
? l a n d = + - ( ) 1
? 96 12 1 4 = + - ( ) n
? 84 1 4 = - ( ) n
? n - = 1 21
? n = + = 21 1 22
33.
LetS
n
be the sum of the firstn terms
of an AP. If S n n
n 2
2
3 14 = + , then
what is the common difference?
(a) 5 (b) 6
(c) 7 (d) 9
Ê
(c)S n n
n 2
2
3 14 = + (S
n
be the sum of first
n terms of an AP)
? S n n
n 2
2
3
2
2
7
2
2 = + .( ) ( )
Put 2n n =
we get,S
n n
n
= +
3
2
7
2
2
? T S S
n n n
= -
-1
= + - - - -
3
2
7
2
3
2
1
7
2
1
2
2
n
n
n n ( ) ( )
= + - + -
3
2
7
2
3
2
3
2
7
2
2 2
n n n n - +
7
2
7
2
2 . n
T n
n
= - 7 2
Put n = 1 2 , , ....
T
1
7 1 2 5 = - = ( )
T
2
7 2 2 12 = - = ( )
? d T T = - = - =
2 1
12 5 7
34.
If 3rd, 8th and 13th terms of a GP are
p q , and r respectively, then which
one of the following is correct?
(a)q pr
2
= (b) r pq
2
=
(c) pqr =1 (d) 2q p r = +
Ê
(a) Let first term and common ratio of a
GP be a and R.
? T aR p
3
2
= = …(i)
T aR q
8
7
= = …(ii)
T aR r
13
12
= = …(iii)
Multiplying of Eqs. (i) and (iii)
( ) ( ) aR aR pr
2 12
=
? a R pr
2 14
=
? ( ) aR pr
7 2
=
? q pr
2
= [from Eq. (ii)]
35.
What is the solution of x y = = 4 0 ,
and x y = - = 4 0 , ?
(a) x y = - = 4 0 , (b) x y = = 4 0 ,
(c) x y = - = 4 0 , (d) x y = - = 4 0 ,
Ê
(c) Given inequalities
x y = = 4 0 , …(i)
and x y = - = 4 0 , …(ii)
Possible value of x and y.
x = - - - - - { , , , , , , , , , , ...} 4 3 2 1 0 1 2 3 4 5
y = { , , , , .....} 0 1 2 3 4 …(i)
and x = - - - - { , , , , ...} 4 5 6 7 ,
y = - - - - { , , , , ...} 0 1 2 3 4 …(ii)
Take combine (i) and (ii),
x = - - - - { , , , ... } 4 5 6 7 , y = 0
or x y = - = 4 0 , .
36.
Ifx
x log
7
7 > wherex > 0, then which
one of the following is correct?
(a) x ? 8 ( , ) 0 (b) x ?
?
?
?
?
?
?
1
7
7 ,
(c) x ?
?
?
?
?
?
?
? 8 0
1
7
7 , ( , )
(d) x ? 8
?
?
?
?
?
?
1
7
,
Ê
(b) x
x log
7
7 > where x > 0.
Taking log on base 7 both sides
log . log log
7 7 7
7 x x >
[ log log ] Q
a
n
a
m n m =
? (log )
7
2
1 x > [ log ] Q
a
a = 1
? log ( )
7
1 x > ±
? x > ? 7
1
x > 7
and x x < ? <
-
7
1
7
1
Hence, x ?
?
?
?
?
?
?
1
7
7 ,
37.
How many real roots does the
equation x x
2
3 2 0 + + = | | have?
(a) Zero (b) One
(c) Two (d) Four
Ê
(a) Given equation, x x
2
3 2 0 + + = | |
Case I x x
2
3 2 0 + + = (when x > 0)
? x x x
2
2 2 0 + + + =
? x x x ( ) ( ) + + + = 1 2 1 0
? ( ) ( ) x x + + = 1 2 0
? x = - - 1 2 ,
Hence, no real roots because x > 0.
Case II x x
2
3 2 0 - + = (when x < 0)
? x x x
2
2 2 0 - - + =
? x x x ( ) ( ) - - - = 2 1 2 0
? ( )( ) x x - - = 2 1 0
? x = 1 2 ,
Hence, no real roots because x < 0.
? The number of real roots of given
equation is zero.
38.
Consider the following statements in
respect of the quadratic equation
4 0
2
( )( ) x p x q r - - - = ,
where p q , andr are real numbers.
1. The roots are real.
2. The roots are equal, if p q = and
r = 0.
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
(c) Given quadratic equation,
4 0
2
( ) ( ) x p x q r - - - =
? 4 4 4 4 0
2 2
x q p x pq r - + + - = ( )
Comparing it Eq. byax bx c
2
0 + + =
5