NDA Solved Paper 2020 - I & II | NDA (National Defence Academy) Past Year Papers PDF Download

 Page 1


1. If matrix A
i i
i i
=
-
- -
?
?
?
?
?
?
1
1
where
i = - 1, then which one of the
following is correct?
(a)A is hermitian
(b)A is skew-hermitian
(c) ( ) A A
T
+ is hermitian
(d)( ) A A
T
+ is skew-hermitian
Ê
(c) We have
A
i i
i i
=
-
- -
?
?
?
?
?
?
1
1
Now, A
i i
i i
=
+ -
+
?
?
?
?
?
?
1
1
( ) A
i i
i i
T
=
+
- +
?
?
?
?
?
?
1
1
Now, consider
X A A
i i
i i
T
= + =
+
- +
?
?
?
?
?
?
( )
1
1
+
-
- -
?
?
?
?
?
?
1
1
i i
i i
=
-
?
?
?
?
?
?
2 2
2 2
i
i
X
i
i
=
- ?
?
?
?
?
?
2 2
2 2
( ) X
i
i
T
=
-
?
?
?
?
?
?
2 2
2 2
= X
Hence, X A A
T
= + ( ) is a hermitian
matrix.
2. The term independent of x in the
binomial expansion of
2
2
10
x
x -
?
?
?
?
?
?
is equal to
(a) 180 (b) 120
(c) 90 (d) 72
Ê
(a) The general term in the binomial
expansion of
2
2
10
x
x -
?
?
?
?
?
?
is
T C
x
x
r r
r
r
+
-
=
?
?
?
?
?
?
-
1
10
2
10
2
( )
= -
-
- + +
10 10
20 2
2
2 1 C x
r
r
r
r
r
( ) ( )
For independent of x, put
- + + = 20 2
2
0 r
r
?
5
2
20
r
=
? r = 8
? T C
8 1
10
8
10 8 8
2 1
+
-
= - ( ) ( )
=
×
×
× ×
10 9
2 1
2 1
2
= 180
3. If ( ) 1 2
2 6
0 1 2
2
+ - = + + x x a a x a x
+ + K a x
12
12
, then what isa a
0 1
-
+ - + - + a a a a
2 3 4 12
K equal to?
(a) 32 (b) 64
(c) 2048 (d) 4096
Ê
(b) We have,
( ) 1 2
2 6
0 1 2
2
+ - = + + x x a a x a x
+ + .... a x
12
12
Put x = - 1both sides, we get
( ) .... 1 2 1
2 6
0 1 2 12
- - = - + - + a a a a
?a a a a
0 1 2 12
6
2 - + - + = - ..... ( ) = 64
4. IfC n C n ( , ) ( , ) 20 2 20 2 + = - , then
what is n equal to?
(a) 18 (b) 25
(c) 10 (d) 12
Ê
(c) Given,C(20,n + 2) = - C n ( , ) 20 2
? C ( , ( )) ( , ) 20 20 2 20 2 - + = - n C n
[ ( , ) ( , )] QC n r C n n r = -
? 20 2 2 - + = - ( ) n n
? 20 2 2 = + + - n n
? 2 20 n =
? n = 10
5. For how many values of k, is the
matrix
0 4
0 5
1
k
k
k k
- -
- -
?
?
?
?
?
?
?
?
?
?
singular?
(a) Only one (b) Only two
(c) Only four (d) Infinite
Ê
(d) The condition for singular matrix is
0 4
0 5
1
0
k
k
k k
- -
- -
=
Expanding alongR
1
, we get
0 5 4 0 0
2
- - + - + = k k k k ( ) ( )
? 4 4 0
2 2
k k - =
? 0 0 = , ? ? k R
Hence, for infinite values ofk, given
matrix is singular.
6. The number ( ) 1101101 1011011
2
+
can be written in decimal system as
(a) ( ) 198
10
(b) ( ) 199
10
(c) ( ) 200
10
(d) ( ) 201
10
Ê
(c) Now, ( ) 1101101 1 2 0 2
2
6
= × + ×
5
+ × + × + × + × 1 2 1 2 0 2 1 2
4 3 2 1
+ × 1 2
0
= + + + + + + 64 0 16 8 0 2 1
= ( ) 91
10
and
( ) 1011011 1 2 1 2 0 2
2
6 5 4
= × + × + ×
+ × + × + × + × 1 2 1 2 0 2 1 2
3 2 1 0
= + + + + + + 64 32 0 8 4 0 1= ( ) 109
10
NDA/NA Solved Paper 2020 (I & II) 1
PAPER : I Mathematics
Page 2


1. If matrix A
i i
i i
=
-
- -
?
?
?
?
?
?
1
1
where
i = - 1, then which one of the
following is correct?
(a)A is hermitian
(b)A is skew-hermitian
(c) ( ) A A
T
+ is hermitian
(d)( ) A A
T
+ is skew-hermitian
Ê
(c) We have
A
i i
i i
=
-
- -
?
?
?
?
?
?
1
1
Now, A
i i
i i
=
+ -
+
?
?
?
?
?
?
1
1
( ) A
i i
i i
T
=
+
- +
?
?
?
?
?
?
1
1
Now, consider
X A A
i i
i i
T
= + =
+
- +
?
?
?
?
?
?
( )
1
1
+
-
- -
?
?
?
?
?
?
1
1
i i
i i
=
-
?
?
?
?
?
?
2 2
2 2
i
i
X
i
i
=
- ?
?
?
?
?
?
2 2
2 2
( ) X
i
i
T
=
-
?
?
?
?
?
?
2 2
2 2
= X
Hence, X A A
T
= + ( ) is a hermitian
matrix.
2. The term independent of x in the
binomial expansion of
2
2
10
x
x -
?
?
?
?
?
?
is equal to
(a) 180 (b) 120
(c) 90 (d) 72
Ê
(a) The general term in the binomial
expansion of
2
2
10
x
x -
?
?
?
?
?
?
is
T C
x
x
r r
r
r
+
-
=
?
?
?
?
?
?
-
1
10
2
10
2
( )
= -
-
- + +
10 10
20 2
2
2 1 C x
r
r
r
r
r
( ) ( )
For independent of x, put
- + + = 20 2
2
0 r
r
?
5
2
20
r
=
? r = 8
? T C
8 1
10
8
10 8 8
2 1
+
-
= - ( ) ( )
=
×
×
× ×
10 9
2 1
2 1
2
= 180
3. If ( ) 1 2
2 6
0 1 2
2
+ - = + + x x a a x a x
+ + K a x
12
12
, then what isa a
0 1
-
+ - + - + a a a a
2 3 4 12
K equal to?
(a) 32 (b) 64
(c) 2048 (d) 4096
Ê
(b) We have,
( ) 1 2
2 6
0 1 2
2
+ - = + + x x a a x a x
+ + .... a x
12
12
Put x = - 1both sides, we get
( ) .... 1 2 1
2 6
0 1 2 12
- - = - + - + a a a a
?a a a a
0 1 2 12
6
2 - + - + = - ..... ( ) = 64
4. IfC n C n ( , ) ( , ) 20 2 20 2 + = - , then
what is n equal to?
(a) 18 (b) 25
(c) 10 (d) 12
Ê
(c) Given,C(20,n + 2) = - C n ( , ) 20 2
? C ( , ( )) ( , ) 20 20 2 20 2 - + = - n C n
[ ( , ) ( , )] QC n r C n n r = -
? 20 2 2 - + = - ( ) n n
? 20 2 2 = + + - n n
? 2 20 n =
? n = 10
5. For how many values of k, is the
matrix
0 4
0 5
1
k
k
k k
- -
- -
?
?
?
?
?
?
?
?
?
?
singular?
(a) Only one (b) Only two
(c) Only four (d) Infinite
Ê
(d) The condition for singular matrix is
0 4
0 5
1
0
k
k
k k
- -
- -
=
Expanding alongR
1
, we get
0 5 4 0 0
2
- - + - + = k k k k ( ) ( )
? 4 4 0
2 2
k k - =
? 0 0 = , ? ? k R
Hence, for infinite values ofk, given
matrix is singular.
6. The number ( ) 1101101 1011011
2
+
can be written in decimal system as
(a) ( ) 198
10
(b) ( ) 199
10
(c) ( ) 200
10
(d) ( ) 201
10
Ê
(c) Now, ( ) 1101101 1 2 0 2
2
6
= × + ×
5
+ × + × + × + × 1 2 1 2 0 2 1 2
4 3 2 1
+ × 1 2
0
= + + + + + + 64 0 16 8 0 2 1
= ( ) 91
10
and
( ) 1011011 1 2 1 2 0 2
2
6 5 4
= × + × + ×
+ × + × + × + × 1 2 1 2 0 2 1 2
3 2 1 0
= + + + + + + 64 32 0 8 4 0 1= ( ) 109
10
NDA/NA Solved Paper 2020 (I & II) 1
PAPER : I Mathematics
? ( ) 1101101 1011011
2
+
= + ( ) ( ) 1101101 1011011
2 2
= + ( ) ( ) 91 109
10 10
= ( ) 200
10
7. What is the value of
1
10
1024 10
1
5
5 5 5
log log log - + 3125?
(a) 0        (b) 1 (c) 2             (d) 3
Ê
(a)
1
10
1024 10
1
5
3125
5 5 5
log log log - +
= - × +
1
10
2 5 2
1
5
5
5
10
5 5
5
log log ( ) log
= - + +
10
10
2 5 2
5
5
5
5 5 5 5
log [log log ] log
[ ]
Q log log log mn m n = +
= - + + log [ log ]
5 5
2 1 2 1 [ log ] Q
m
m = 1
= 0
8. Ifx ab
c
= log ( ),y bc
a
= log ( ),
z ca
b
= log ( ), then which of the
following is correct?
(a)xyz = 1
(b)x y z + + = 1
(c) ( ) ( ) ( ) 1 1 1 1
1 1 1
+ + + + + =
- - -
x y z
(d)( ) ( ) ( ) 1 1 1 1
2 2 2
+ + + + =
- - -
x y z
Ê
(c) We have, x ab
c
= log ( )
y bc
a
= log ( )
z ca
b
= log ( )
Now, 1 + = + x c ab
c c
log log ( )
= log ( )
c
abc
1 + = y abc
a
log ( )
and 1 + = z abc
b
log ( )
Now, ( ) ( ) ( ) 1 1 1
1 1 1
+ + + + +
- - -
x y z
= +
- -
[log ( )] [log ( )]
c a
abc abc
1 1
+
-
[log ( )]
b
abc
1
= + +
1 1 1
log ( ) log ( ) log ( )
c a b
abc abc abc
= + +
log
log( )
log
log( )
log
log( )
c
abc
a
abc
b
abc
Q log
log
log
m
n
n
m
=
?
?
?
?
?
?
=
+ + log log log
log( )
c a b
abc
= =
log( )
log( )
abc
abc
1
9.
LetA
x y y
x x y
=
+
-
?
?
?
?
?
?
2
,B =
-
?
?
?
?
?
?
2
1
andC =
?
?
?
?
?
?
3
2
. If AB C = , then what is
the value of the determinant of the
matrixA?
(a) - 10 (b) - 14
(c) - 24 (d) - 34
Ê
(b) Given, A
x y y
x x y
=
+
-
?
?
?
?
?
?
2
B =
-
?
?
?
?
?
?
2
1
andC =
?
?
?
?
?
?
3
2
Also given, AB C =
?
x y y
x x y
+
-
?
?
?
?
?
?
-
?
?
?
?
?
?
=
?
?
?
?
?
?
2
2
1
3
2
?
2 2
4
3
2
x y y
x x y
+ -
- +
?
?
?
?
?
?
=
?
?
?
?
?
?
?
2
3
3
2
x y
x y
+
+
?
?
?
?
?
?
=
?
?
?
?
?
?
On equating the corresponding
elements, we get
2 3 x y + = and 3 2 x y + =
? x = - 1and y = 5
? A =
- +
× - - -
?
?
?
?
?
?
=
- -
?
?
?
?
?
?
1 5 5
2 1 1 5
4 5
2 6
?The determinant of matrix A is
A =
- -
4 5
2 6
= - + = - 24 10 14
10. If 15 4 5 . . = = x , then which one of
the following is correct?
(a) ( ) ( ) 2 3 2 9 0 x x - - >
(b)( ) ( ) 2 3 2 9 0 x x - - <
(c) ( ) ( ) 2 3 2 9 0 x x - - =
(d)( ) ( ) 2 3 2 9 0 x x - - =
Ê
(d) We have, 15 4 5 . . = = x
?
3
2
9
2
= = x ? 3 2 9 = = x
? ( ) ( ) 2 3 2 9 0 x x - - =
11. LetS = { , , , } 1 2 3K . A relation R on
S S × is defined by xRy if
log log
a a
x y > whena =
1
2
. Then
the relation is
(a) reflexive only
(b) symmetric only
(c) transitive only
(d) both symmetric and transitive
Ê
(c) We have,S = { , , , ....} 1 2 3
and log log
a a
x y >
Here, a = ?
1
2
0 1 ( , )
? log log
a a
x y > ? x y <
Now, x R x ? x x < which is not
possible. So it is not reflexive relation.
Now, x R y ? x y <
But y x < | , so it is not symmetric relation.
Now, x R y and y R z
? x y < and y z < ? x z < ? x R z
Hence, it is transitive relation only.
12. What is the value of the
determinant
i i i
i i i
i i i
2 3
4 6 8
9 12 15
?
?
?
?
?
?
?
?
?
?
where
i = - 1 ?
(a) 0 (b) - 2 (c) 4i (d) - 4i
Ê
(d) Let ? =
i i i
i i i
i i i
2 3
4 6 8
9 12 15
=
- -
-
-
i i
i i
1
1 1 1
1
[ , , ] Qi i i i
2 3 4
1 1 = - = - =
= - + - - - + i i i i i i ( ) ( ) ( ) 1 1 1
[Expanding alongR
1
]
= - - - - i i i i i
2 2
2 = - 4i
13. LetA
a h g
h b f
g f c
=
?
?
?
?
?
?
?
?
?
?
andB
x
y
z
=
?
?
?
?
?
?
?
?
?
?
,
then what is AB equal to?
(a)
ax hy gz
y
z
+ + ?
?
?
?
?
?
?
?
?
?
(b)
ax hy gz
hx by fz
z
+ +
+ +
?
?
?
?
?
?
?
?
?
?
(c)
ax hy gz
hx by fz
gx fy cz
+ +
+ +
+ +
?
?
?
?
?
?
?
?
?
?
(d) [ax hy gz hx by fz + + + +
gx fy cz + + ]
Ê
(c) Now,
AB
a h g
h b f
g f c
x
y
z
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
+ +
+ +
+ +
?
?
?
?
?
?
?
?
?
?
ax hy gz
hx by fz
gx fy cz
14. What is the number of ways in
which the letters of the word
‘ABLE’ can be arranged so that the
vowels occupy even places?
(a) 2 (b) 4 (c) 6 (d) 8
Ê
(b) In a given word ‘ABLE’ Vowels are
{ , } A E .
1 2 3 4
As, vowel occupy even places, so two
vowels occupy the places 2 and 4.
Therefore, the number of ways of
occupying the vowels in even places is
2!.
Now, we have two consonants and
these consonants occupy the odd
places 1 and 3. Therefore, the number of
ways of occupying the  consonants in
odd places is 2!.
?Total number of ways = × 2 2 ! !
= × 2 2 = 4
2
Page 3


1. If matrix A
i i
i i
=
-
- -
?
?
?
?
?
?
1
1
where
i = - 1, then which one of the
following is correct?
(a)A is hermitian
(b)A is skew-hermitian
(c) ( ) A A
T
+ is hermitian
(d)( ) A A
T
+ is skew-hermitian
Ê
(c) We have
A
i i
i i
=
-
- -
?
?
?
?
?
?
1
1
Now, A
i i
i i
=
+ -
+
?
?
?
?
?
?
1
1
( ) A
i i
i i
T
=
+
- +
?
?
?
?
?
?
1
1
Now, consider
X A A
i i
i i
T
= + =
+
- +
?
?
?
?
?
?
( )
1
1
+
-
- -
?
?
?
?
?
?
1
1
i i
i i
=
-
?
?
?
?
?
?
2 2
2 2
i
i
X
i
i
=
- ?
?
?
?
?
?
2 2
2 2
( ) X
i
i
T
=
-
?
?
?
?
?
?
2 2
2 2
= X
Hence, X A A
T
= + ( ) is a hermitian
matrix.
2. The term independent of x in the
binomial expansion of
2
2
10
x
x -
?
?
?
?
?
?
is equal to
(a) 180 (b) 120
(c) 90 (d) 72
Ê
(a) The general term in the binomial
expansion of
2
2
10
x
x -
?
?
?
?
?
?
is
T C
x
x
r r
r
r
+
-
=
?
?
?
?
?
?
-
1
10
2
10
2
( )
= -
-
- + +
10 10
20 2
2
2 1 C x
r
r
r
r
r
( ) ( )
For independent of x, put
- + + = 20 2
2
0 r
r
?
5
2
20
r
=
? r = 8
? T C
8 1
10
8
10 8 8
2 1
+
-
= - ( ) ( )
=
×
×
× ×
10 9
2 1
2 1
2
= 180
3. If ( ) 1 2
2 6
0 1 2
2
+ - = + + x x a a x a x
+ + K a x
12
12
, then what isa a
0 1
-
+ - + - + a a a a
2 3 4 12
K equal to?
(a) 32 (b) 64
(c) 2048 (d) 4096
Ê
(b) We have,
( ) 1 2
2 6
0 1 2
2
+ - = + + x x a a x a x
+ + .... a x
12
12
Put x = - 1both sides, we get
( ) .... 1 2 1
2 6
0 1 2 12
- - = - + - + a a a a
?a a a a
0 1 2 12
6
2 - + - + = - ..... ( ) = 64
4. IfC n C n ( , ) ( , ) 20 2 20 2 + = - , then
what is n equal to?
(a) 18 (b) 25
(c) 10 (d) 12
Ê
(c) Given,C(20,n + 2) = - C n ( , ) 20 2
? C ( , ( )) ( , ) 20 20 2 20 2 - + = - n C n
[ ( , ) ( , )] QC n r C n n r = -
? 20 2 2 - + = - ( ) n n
? 20 2 2 = + + - n n
? 2 20 n =
? n = 10
5. For how many values of k, is the
matrix
0 4
0 5
1
k
k
k k
- -
- -
?
?
?
?
?
?
?
?
?
?
singular?
(a) Only one (b) Only two
(c) Only four (d) Infinite
Ê
(d) The condition for singular matrix is
0 4
0 5
1
0
k
k
k k
- -
- -
=
Expanding alongR
1
, we get
0 5 4 0 0
2
- - + - + = k k k k ( ) ( )
? 4 4 0
2 2
k k - =
? 0 0 = , ? ? k R
Hence, for infinite values ofk, given
matrix is singular.
6. The number ( ) 1101101 1011011
2
+
can be written in decimal system as
(a) ( ) 198
10
(b) ( ) 199
10
(c) ( ) 200
10
(d) ( ) 201
10
Ê
(c) Now, ( ) 1101101 1 2 0 2
2
6
= × + ×
5
+ × + × + × + × 1 2 1 2 0 2 1 2
4 3 2 1
+ × 1 2
0
= + + + + + + 64 0 16 8 0 2 1
= ( ) 91
10
and
( ) 1011011 1 2 1 2 0 2
2
6 5 4
= × + × + ×
+ × + × + × + × 1 2 1 2 0 2 1 2
3 2 1 0
= + + + + + + 64 32 0 8 4 0 1= ( ) 109
10
NDA/NA Solved Paper 2020 (I & II) 1
PAPER : I Mathematics
? ( ) 1101101 1011011
2
+
= + ( ) ( ) 1101101 1011011
2 2
= + ( ) ( ) 91 109
10 10
= ( ) 200
10
7. What is the value of
1
10
1024 10
1
5
5 5 5
log log log - + 3125?
(a) 0        (b) 1 (c) 2             (d) 3
Ê
(a)
1
10
1024 10
1
5
3125
5 5 5
log log log - +
= - × +
1
10
2 5 2
1
5
5
5
10
5 5
5
log log ( ) log
= - + +
10
10
2 5 2
5
5
5
5 5 5 5
log [log log ] log
[ ]
Q log log log mn m n = +
= - + + log [ log ]
5 5
2 1 2 1 [ log ] Q
m
m = 1
= 0
8. Ifx ab
c
= log ( ),y bc
a
= log ( ),
z ca
b
= log ( ), then which of the
following is correct?
(a)xyz = 1
(b)x y z + + = 1
(c) ( ) ( ) ( ) 1 1 1 1
1 1 1
+ + + + + =
- - -
x y z
(d)( ) ( ) ( ) 1 1 1 1
2 2 2
+ + + + =
- - -
x y z
Ê
(c) We have, x ab
c
= log ( )
y bc
a
= log ( )
z ca
b
= log ( )
Now, 1 + = + x c ab
c c
log log ( )
= log ( )
c
abc
1 + = y abc
a
log ( )
and 1 + = z abc
b
log ( )
Now, ( ) ( ) ( ) 1 1 1
1 1 1
+ + + + +
- - -
x y z
= +
- -
[log ( )] [log ( )]
c a
abc abc
1 1
+
-
[log ( )]
b
abc
1
= + +
1 1 1
log ( ) log ( ) log ( )
c a b
abc abc abc
= + +
log
log( )
log
log( )
log
log( )
c
abc
a
abc
b
abc
Q log
log
log
m
n
n
m
=
?
?
?
?
?
?
=
+ + log log log
log( )
c a b
abc
= =
log( )
log( )
abc
abc
1
9.
LetA
x y y
x x y
=
+
-
?
?
?
?
?
?
2
,B =
-
?
?
?
?
?
?
2
1
andC =
?
?
?
?
?
?
3
2
. If AB C = , then what is
the value of the determinant of the
matrixA?
(a) - 10 (b) - 14
(c) - 24 (d) - 34
Ê
(b) Given, A
x y y
x x y
=
+
-
?
?
?
?
?
?
2
B =
-
?
?
?
?
?
?
2
1
andC =
?
?
?
?
?
?
3
2
Also given, AB C =
?
x y y
x x y
+
-
?
?
?
?
?
?
-
?
?
?
?
?
?
=
?
?
?
?
?
?
2
2
1
3
2
?
2 2
4
3
2
x y y
x x y
+ -
- +
?
?
?
?
?
?
=
?
?
?
?
?
?
?
2
3
3
2
x y
x y
+
+
?
?
?
?
?
?
=
?
?
?
?
?
?
On equating the corresponding
elements, we get
2 3 x y + = and 3 2 x y + =
? x = - 1and y = 5
? A =
- +
× - - -
?
?
?
?
?
?
=
- -
?
?
?
?
?
?
1 5 5
2 1 1 5
4 5
2 6
?The determinant of matrix A is
A =
- -
4 5
2 6
= - + = - 24 10 14
10. If 15 4 5 . . = = x , then which one of
the following is correct?
(a) ( ) ( ) 2 3 2 9 0 x x - - >
(b)( ) ( ) 2 3 2 9 0 x x - - <
(c) ( ) ( ) 2 3 2 9 0 x x - - =
(d)( ) ( ) 2 3 2 9 0 x x - - =
Ê
(d) We have, 15 4 5 . . = = x
?
3
2
9
2
= = x ? 3 2 9 = = x
? ( ) ( ) 2 3 2 9 0 x x - - =
11. LetS = { , , , } 1 2 3K . A relation R on
S S × is defined by xRy if
log log
a a
x y > whena =
1
2
. Then
the relation is
(a) reflexive only
(b) symmetric only
(c) transitive only
(d) both symmetric and transitive
Ê
(c) We have,S = { , , , ....} 1 2 3
and log log
a a
x y >
Here, a = ?
1
2
0 1 ( , )
? log log
a a
x y > ? x y <
Now, x R x ? x x < which is not
possible. So it is not reflexive relation.
Now, x R y ? x y <
But y x < | , so it is not symmetric relation.
Now, x R y and y R z
? x y < and y z < ? x z < ? x R z
Hence, it is transitive relation only.
12. What is the value of the
determinant
i i i
i i i
i i i
2 3
4 6 8
9 12 15
?
?
?
?
?
?
?
?
?
?
where
i = - 1 ?
(a) 0 (b) - 2 (c) 4i (d) - 4i
Ê
(d) Let ? =
i i i
i i i
i i i
2 3
4 6 8
9 12 15
=
- -
-
-
i i
i i
1
1 1 1
1
[ , , ] Qi i i i
2 3 4
1 1 = - = - =
= - + - - - + i i i i i i ( ) ( ) ( ) 1 1 1
[Expanding alongR
1
]
= - - - - i i i i i
2 2
2 = - 4i
13. LetA
a h g
h b f
g f c
=
?
?
?
?
?
?
?
?
?
?
andB
x
y
z
=
?
?
?
?
?
?
?
?
?
?
,
then what is AB equal to?
(a)
ax hy gz
y
z
+ + ?
?
?
?
?
?
?
?
?
?
(b)
ax hy gz
hx by fz
z
+ +
+ +
?
?
?
?
?
?
?
?
?
?
(c)
ax hy gz
hx by fz
gx fy cz
+ +
+ +
+ +
?
?
?
?
?
?
?
?
?
?
(d) [ax hy gz hx by fz + + + +
gx fy cz + + ]
Ê
(c) Now,
AB
a h g
h b f
g f c
x
y
z
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
+ +
+ +
+ +
?
?
?
?
?
?
?
?
?
?
ax hy gz
hx by fz
gx fy cz
14. What is the number of ways in
which the letters of the word
‘ABLE’ can be arranged so that the
vowels occupy even places?
(a) 2 (b) 4 (c) 6 (d) 8
Ê
(b) In a given word ‘ABLE’ Vowels are
{ , } A E .
1 2 3 4
As, vowel occupy even places, so two
vowels occupy the places 2 and 4.
Therefore, the number of ways of
occupying the vowels in even places is
2!.
Now, we have two consonants and
these consonants occupy the odd
places 1 and 3. Therefore, the number of
ways of occupying the  consonants in
odd places is 2!.
?Total number of ways = × 2 2 ! !
= × 2 2 = 4
2
15. What is the maximum number of
points of intersection of 5
non-overlapping circles?
(a) 10 (b) 15 (c) 20 (d) 25
Ê
(c) The maximum number of points of
intersection of 5 non-overlapping circles
= Selection of two circles × 2
[Q Two intersecting circles
cut at two points]
= ×
5
2
2 C =
×
×
× =
5 4
2 1
2 20
Directions (Q. Nos. 16-18)
Consider the following Venn diagram,
where X,Y andZ are three sets. Let the
number of elements inZ be denoted by
n(Z) which is equal to 90.
16. If the number of elements in Y and
Z are in the ratio 4 : 5, then what is
the value of b?
(a) 18 (b) 19 (c) 21 (d) 23
17. What is the value of
n X n Y n Z n X Y ( ) ( ) ( ) ( ) + + - n
- n - n n Y Z n X Z ( ) ( )
+ n n n X Y Z ( )?
(a)a b + + 43 (b)a b + + 63
(c)a b + + 96 (d)a b + + 106
18. If the number of elements
belonging to neither X, nor Y, nor Z
is equal to p, then what is the
number of elements in the
complement of X?
(a) p b + + 60 (b) p b + + 40
(c) p a + + 60 (d) p a + + 40
Solutions (16-18)
Givenn Z ( ) = 90
?12 18 17 90 + + + = c
? c = - 90 47 = 43
Ê
16. (c) Also given,
n Y
n Z
( )
( )
=
4
5
?
16 18 17
90
4
5
+ + +
=
b
? 51 72 + = b
? b = - 72 51
= 21
Ê
17.(d) Now,
n X n Y n Z n X Y ( ) ( ) ( ) ( ) + + - n
- n - n + n n n Y Z n X Z n X Y Z ( ) ( ) ( )
= ? ? n X Y Z ( )
= + + + + + + a b c 12 18 16 17
= + + + a b c 63
= + + + a b 43 63 [ ] Qc = 43
= + + a b 106
Ê
18. (a) Complement of X
= + + + p b c 17
= + + + p b 43 17 [ ] Qc = 43
= + + p b 60
Directions (Q. Nos. 19 and 20) Read
the following information and answer
the two items that follow.
Let
tan
tan
3A
A
K = , where tan A ? 0
and K ?
1
3
.
19. What is tan
2
A equal to?
(a)
K
K
+
-
3
3 1
(b)
K
K
-
-
3
3 1
(c)
3 3
3
K
K
-
-
(d)
K
K
+
+
3
3 1
Ê
(b) Given,
tan
tan
3A
A
K =
=
-
-
=
3
1 3
3
2
tan tan
( tan ) tan
A A
A A
K
?
3
1 3
2
2
-
-
=
tan
tan
A
A
K
? K K A A - = - 3 3
2 2
tan tan
? K A K - = - 3 3 1
2
tan ( )
? tan
2
3
3 1
A
K
K
=
-
-
20. For real values of tan A,K cannot
lie between
(a)
1
3
and 3 (b)
1
2
and 2
(c)
1
5
and 5 (d)
1
7
and 7
Ê
(a) For real values of tanA,K lies when
K
K
-
-
=
3
3 1
0 and 3 1 0 K - ?
?( ) ( ) K K - - = 3 3 1 0 andK ?
1
3
? K <
1
3
andK = 3
Hence, for real values of tanA,K cannot
lie between
1
3
3 ,
?
?
?
?
?
?
.
Directions (Q. Nos. 21 and 22) Read
the following information and answer
the two items that follow.
ABCD is a trapezium such that AB andCD
are parallel andBC is perpendicular to
them. Let ? = ? = ADB ABD ? a , , BC =p
andCD =q.
21. Consider the following
1. AD AB sin sin ? a =
2. BD AB sin sin ( ) ? ? a = +
Which of the above is/are correct?
(a) 1 Only (b) 2 Only
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(c) We have, ? = ADB ?, ? = ABD a,
BC p = andCD q =
1. In ?ABD, use Sine rule,
sin sin ? a
AB AD
=
? AD AB sin sin ? a = , which is correct.
2. In ?ABD, ? = - + A p ? a ( )
Use Sine rule in ?ABD,
sin sin A
BD AB
=
?
?
sin( ( )] sin p ? a ? - +
=
BD AB
…(i)
? AB BD sin( ) sin , ? a ? + =
which is correct.
Hence, both statements are
correct.
22. What is AB equal to?
(a)
( ) sin
cos sin
p q
p q
2 2
+
+
?
? ?
(b)
( ) cos
cos sin
p q
p q
2 2
-
+
?
? ?
(c)
( ) sin
cos sin
p q
q p
2 2
+
+
?
? ?
(d)
( ) cos
cos sin
p q
q p
2 2
-
+
?
? ?
Ê
(a) In right angle, ?BCD,
? = ° - B 90 a
BD p q = +
2 2
and sinB
CD
BD
=
? sin( ) 90
2 2
° - =
+
a
q
p q
[ ] Q ? = °- B 90 a
3
16
18
17 12
a b
Y X
Z
c
16
18
17 12
a b
Y X
Z
c
q D
C
B A
p
?
90°-a
a
Page 4


1. If matrix A
i i
i i
=
-
- -
?
?
?
?
?
?
1
1
where
i = - 1, then which one of the
following is correct?
(a)A is hermitian
(b)A is skew-hermitian
(c) ( ) A A
T
+ is hermitian
(d)( ) A A
T
+ is skew-hermitian
Ê
(c) We have
A
i i
i i
=
-
- -
?
?
?
?
?
?
1
1
Now, A
i i
i i
=
+ -
+
?
?
?
?
?
?
1
1
( ) A
i i
i i
T
=
+
- +
?
?
?
?
?
?
1
1
Now, consider
X A A
i i
i i
T
= + =
+
- +
?
?
?
?
?
?
( )
1
1
+
-
- -
?
?
?
?
?
?
1
1
i i
i i
=
-
?
?
?
?
?
?
2 2
2 2
i
i
X
i
i
=
- ?
?
?
?
?
?
2 2
2 2
( ) X
i
i
T
=
-
?
?
?
?
?
?
2 2
2 2
= X
Hence, X A A
T
= + ( ) is a hermitian
matrix.
2. The term independent of x in the
binomial expansion of
2
2
10
x
x -
?
?
?
?
?
?
is equal to
(a) 180 (b) 120
(c) 90 (d) 72
Ê
(a) The general term in the binomial
expansion of
2
2
10
x
x -
?
?
?
?
?
?
is
T C
x
x
r r
r
r
+
-
=
?
?
?
?
?
?
-
1
10
2
10
2
( )
= -
-
- + +
10 10
20 2
2
2 1 C x
r
r
r
r
r
( ) ( )
For independent of x, put
- + + = 20 2
2
0 r
r
?
5
2
20
r
=
? r = 8
? T C
8 1
10
8
10 8 8
2 1
+
-
= - ( ) ( )
=
×
×
× ×
10 9
2 1
2 1
2
= 180
3. If ( ) 1 2
2 6
0 1 2
2
+ - = + + x x a a x a x
+ + K a x
12
12
, then what isa a
0 1
-
+ - + - + a a a a
2 3 4 12
K equal to?
(a) 32 (b) 64
(c) 2048 (d) 4096
Ê
(b) We have,
( ) 1 2
2 6
0 1 2
2
+ - = + + x x a a x a x
+ + .... a x
12
12
Put x = - 1both sides, we get
( ) .... 1 2 1
2 6
0 1 2 12
- - = - + - + a a a a
?a a a a
0 1 2 12
6
2 - + - + = - ..... ( ) = 64
4. IfC n C n ( , ) ( , ) 20 2 20 2 + = - , then
what is n equal to?
(a) 18 (b) 25
(c) 10 (d) 12
Ê
(c) Given,C(20,n + 2) = - C n ( , ) 20 2
? C ( , ( )) ( , ) 20 20 2 20 2 - + = - n C n
[ ( , ) ( , )] QC n r C n n r = -
? 20 2 2 - + = - ( ) n n
? 20 2 2 = + + - n n
? 2 20 n =
? n = 10
5. For how many values of k, is the
matrix
0 4
0 5
1
k
k
k k
- -
- -
?
?
?
?
?
?
?
?
?
?
singular?
(a) Only one (b) Only two
(c) Only four (d) Infinite
Ê
(d) The condition for singular matrix is
0 4
0 5
1
0
k
k
k k
- -
- -
=
Expanding alongR
1
, we get
0 5 4 0 0
2
- - + - + = k k k k ( ) ( )
? 4 4 0
2 2
k k - =
? 0 0 = , ? ? k R
Hence, for infinite values ofk, given
matrix is singular.
6. The number ( ) 1101101 1011011
2
+
can be written in decimal system as
(a) ( ) 198
10
(b) ( ) 199
10
(c) ( ) 200
10
(d) ( ) 201
10
Ê
(c) Now, ( ) 1101101 1 2 0 2
2
6
= × + ×
5
+ × + × + × + × 1 2 1 2 0 2 1 2
4 3 2 1
+ × 1 2
0
= + + + + + + 64 0 16 8 0 2 1
= ( ) 91
10
and
( ) 1011011 1 2 1 2 0 2
2
6 5 4
= × + × + ×
+ × + × + × + × 1 2 1 2 0 2 1 2
3 2 1 0
= + + + + + + 64 32 0 8 4 0 1= ( ) 109
10
NDA/NA Solved Paper 2020 (I & II) 1
PAPER : I Mathematics
? ( ) 1101101 1011011
2
+
= + ( ) ( ) 1101101 1011011
2 2
= + ( ) ( ) 91 109
10 10
= ( ) 200
10
7. What is the value of
1
10
1024 10
1
5
5 5 5
log log log - + 3125?
(a) 0        (b) 1 (c) 2             (d) 3
Ê
(a)
1
10
1024 10
1
5
3125
5 5 5
log log log - +
= - × +
1
10
2 5 2
1
5
5
5
10
5 5
5
log log ( ) log
= - + +
10
10
2 5 2
5
5
5
5 5 5 5
log [log log ] log
[ ]
Q log log log mn m n = +
= - + + log [ log ]
5 5
2 1 2 1 [ log ] Q
m
m = 1
= 0
8. Ifx ab
c
= log ( ),y bc
a
= log ( ),
z ca
b
= log ( ), then which of the
following is correct?
(a)xyz = 1
(b)x y z + + = 1
(c) ( ) ( ) ( ) 1 1 1 1
1 1 1
+ + + + + =
- - -
x y z
(d)( ) ( ) ( ) 1 1 1 1
2 2 2
+ + + + =
- - -
x y z
Ê
(c) We have, x ab
c
= log ( )
y bc
a
= log ( )
z ca
b
= log ( )
Now, 1 + = + x c ab
c c
log log ( )
= log ( )
c
abc
1 + = y abc
a
log ( )
and 1 + = z abc
b
log ( )
Now, ( ) ( ) ( ) 1 1 1
1 1 1
+ + + + +
- - -
x y z
= +
- -
[log ( )] [log ( )]
c a
abc abc
1 1
+
-
[log ( )]
b
abc
1
= + +
1 1 1
log ( ) log ( ) log ( )
c a b
abc abc abc
= + +
log
log( )
log
log( )
log
log( )
c
abc
a
abc
b
abc
Q log
log
log
m
n
n
m
=
?
?
?
?
?
?
=
+ + log log log
log( )
c a b
abc
= =
log( )
log( )
abc
abc
1
9.
LetA
x y y
x x y
=
+
-
?
?
?
?
?
?
2
,B =
-
?
?
?
?
?
?
2
1
andC =
?
?
?
?
?
?
3
2
. If AB C = , then what is
the value of the determinant of the
matrixA?
(a) - 10 (b) - 14
(c) - 24 (d) - 34
Ê
(b) Given, A
x y y
x x y
=
+
-
?
?
?
?
?
?
2
B =
-
?
?
?
?
?
?
2
1
andC =
?
?
?
?
?
?
3
2
Also given, AB C =
?
x y y
x x y
+
-
?
?
?
?
?
?
-
?
?
?
?
?
?
=
?
?
?
?
?
?
2
2
1
3
2
?
2 2
4
3
2
x y y
x x y
+ -
- +
?
?
?
?
?
?
=
?
?
?
?
?
?
?
2
3
3
2
x y
x y
+
+
?
?
?
?
?
?
=
?
?
?
?
?
?
On equating the corresponding
elements, we get
2 3 x y + = and 3 2 x y + =
? x = - 1and y = 5
? A =
- +
× - - -
?
?
?
?
?
?
=
- -
?
?
?
?
?
?
1 5 5
2 1 1 5
4 5
2 6
?The determinant of matrix A is
A =
- -
4 5
2 6
= - + = - 24 10 14
10. If 15 4 5 . . = = x , then which one of
the following is correct?
(a) ( ) ( ) 2 3 2 9 0 x x - - >
(b)( ) ( ) 2 3 2 9 0 x x - - <
(c) ( ) ( ) 2 3 2 9 0 x x - - =
(d)( ) ( ) 2 3 2 9 0 x x - - =
Ê
(d) We have, 15 4 5 . . = = x
?
3
2
9
2
= = x ? 3 2 9 = = x
? ( ) ( ) 2 3 2 9 0 x x - - =
11. LetS = { , , , } 1 2 3K . A relation R on
S S × is defined by xRy if
log log
a a
x y > whena =
1
2
. Then
the relation is
(a) reflexive only
(b) symmetric only
(c) transitive only
(d) both symmetric and transitive
Ê
(c) We have,S = { , , , ....} 1 2 3
and log log
a a
x y >
Here, a = ?
1
2
0 1 ( , )
? log log
a a
x y > ? x y <
Now, x R x ? x x < which is not
possible. So it is not reflexive relation.
Now, x R y ? x y <
But y x < | , so it is not symmetric relation.
Now, x R y and y R z
? x y < and y z < ? x z < ? x R z
Hence, it is transitive relation only.
12. What is the value of the
determinant
i i i
i i i
i i i
2 3
4 6 8
9 12 15
?
?
?
?
?
?
?
?
?
?
where
i = - 1 ?
(a) 0 (b) - 2 (c) 4i (d) - 4i
Ê
(d) Let ? =
i i i
i i i
i i i
2 3
4 6 8
9 12 15
=
- -
-
-
i i
i i
1
1 1 1
1
[ , , ] Qi i i i
2 3 4
1 1 = - = - =
= - + - - - + i i i i i i ( ) ( ) ( ) 1 1 1
[Expanding alongR
1
]
= - - - - i i i i i
2 2
2 = - 4i
13. LetA
a h g
h b f
g f c
=
?
?
?
?
?
?
?
?
?
?
andB
x
y
z
=
?
?
?
?
?
?
?
?
?
?
,
then what is AB equal to?
(a)
ax hy gz
y
z
+ + ?
?
?
?
?
?
?
?
?
?
(b)
ax hy gz
hx by fz
z
+ +
+ +
?
?
?
?
?
?
?
?
?
?
(c)
ax hy gz
hx by fz
gx fy cz
+ +
+ +
+ +
?
?
?
?
?
?
?
?
?
?
(d) [ax hy gz hx by fz + + + +
gx fy cz + + ]
Ê
(c) Now,
AB
a h g
h b f
g f c
x
y
z
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
+ +
+ +
+ +
?
?
?
?
?
?
?
?
?
?
ax hy gz
hx by fz
gx fy cz
14. What is the number of ways in
which the letters of the word
‘ABLE’ can be arranged so that the
vowels occupy even places?
(a) 2 (b) 4 (c) 6 (d) 8
Ê
(b) In a given word ‘ABLE’ Vowels are
{ , } A E .
1 2 3 4
As, vowel occupy even places, so two
vowels occupy the places 2 and 4.
Therefore, the number of ways of
occupying the vowels in even places is
2!.
Now, we have two consonants and
these consonants occupy the odd
places 1 and 3. Therefore, the number of
ways of occupying the  consonants in
odd places is 2!.
?Total number of ways = × 2 2 ! !
= × 2 2 = 4
2
15. What is the maximum number of
points of intersection of 5
non-overlapping circles?
(a) 10 (b) 15 (c) 20 (d) 25
Ê
(c) The maximum number of points of
intersection of 5 non-overlapping circles
= Selection of two circles × 2
[Q Two intersecting circles
cut at two points]
= ×
5
2
2 C =
×
×
× =
5 4
2 1
2 20
Directions (Q. Nos. 16-18)
Consider the following Venn diagram,
where X,Y andZ are three sets. Let the
number of elements inZ be denoted by
n(Z) which is equal to 90.
16. If the number of elements in Y and
Z are in the ratio 4 : 5, then what is
the value of b?
(a) 18 (b) 19 (c) 21 (d) 23
17. What is the value of
n X n Y n Z n X Y ( ) ( ) ( ) ( ) + + - n
- n - n n Y Z n X Z ( ) ( )
+ n n n X Y Z ( )?
(a)a b + + 43 (b)a b + + 63
(c)a b + + 96 (d)a b + + 106
18. If the number of elements
belonging to neither X, nor Y, nor Z
is equal to p, then what is the
number of elements in the
complement of X?
(a) p b + + 60 (b) p b + + 40
(c) p a + + 60 (d) p a + + 40
Solutions (16-18)
Givenn Z ( ) = 90
?12 18 17 90 + + + = c
? c = - 90 47 = 43
Ê
16. (c) Also given,
n Y
n Z
( )
( )
=
4
5
?
16 18 17
90
4
5
+ + +
=
b
? 51 72 + = b
? b = - 72 51
= 21
Ê
17.(d) Now,
n X n Y n Z n X Y ( ) ( ) ( ) ( ) + + - n
- n - n + n n n Y Z n X Z n X Y Z ( ) ( ) ( )
= ? ? n X Y Z ( )
= + + + + + + a b c 12 18 16 17
= + + + a b c 63
= + + + a b 43 63 [ ] Qc = 43
= + + a b 106
Ê
18. (a) Complement of X
= + + + p b c 17
= + + + p b 43 17 [ ] Qc = 43
= + + p b 60
Directions (Q. Nos. 19 and 20) Read
the following information and answer
the two items that follow.
Let
tan
tan
3A
A
K = , where tan A ? 0
and K ?
1
3
.
19. What is tan
2
A equal to?
(a)
K
K
+
-
3
3 1
(b)
K
K
-
-
3
3 1
(c)
3 3
3
K
K
-
-
(d)
K
K
+
+
3
3 1
Ê
(b) Given,
tan
tan
3A
A
K =
=
-
-
=
3
1 3
3
2
tan tan
( tan ) tan
A A
A A
K
?
3
1 3
2
2
-
-
=
tan
tan
A
A
K
? K K A A - = - 3 3
2 2
tan tan
? K A K - = - 3 3 1
2
tan ( )
? tan
2
3
3 1
A
K
K
=
-
-
20. For real values of tan A,K cannot
lie between
(a)
1
3
and 3 (b)
1
2
and 2
(c)
1
5
and 5 (d)
1
7
and 7
Ê
(a) For real values of tanA,K lies when
K
K
-
-
=
3
3 1
0 and 3 1 0 K - ?
?( ) ( ) K K - - = 3 3 1 0 andK ?
1
3
? K <
1
3
andK = 3
Hence, for real values of tanA,K cannot
lie between
1
3
3 ,
?
?
?
?
?
?
.
Directions (Q. Nos. 21 and 22) Read
the following information and answer
the two items that follow.
ABCD is a trapezium such that AB andCD
are parallel andBC is perpendicular to
them. Let ? = ? = ADB ABD ? a , , BC =p
andCD =q.
21. Consider the following
1. AD AB sin sin ? a =
2. BD AB sin sin ( ) ? ? a = +
Which of the above is/are correct?
(a) 1 Only (b) 2 Only
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(c) We have, ? = ADB ?, ? = ABD a,
BC p = andCD q =
1. In ?ABD, use Sine rule,
sin sin ? a
AB AD
=
? AD AB sin sin ? a = , which is correct.
2. In ?ABD, ? = - + A p ? a ( )
Use Sine rule in ?ABD,
sin sin A
BD AB
=
?
?
sin( ( )] sin p ? a ? - +
=
BD AB
…(i)
? AB BD sin( ) sin , ? a ? + =
which is correct.
Hence, both statements are
correct.
22. What is AB equal to?
(a)
( ) sin
cos sin
p q
p q
2 2
+
+
?
? ?
(b)
( ) cos
cos sin
p q
p q
2 2
-
+
?
? ?
(c)
( ) sin
cos sin
p q
q p
2 2
+
+
?
? ?
(d)
( ) cos
cos sin
p q
q p
2 2
-
+
?
? ?
Ê
(a) In right angle, ?BCD,
? = ° - B 90 a
BD p q = +
2 2
and sinB
CD
BD
=
? sin( ) 90
2 2
° - =
+
a
q
p q
[ ] Q ? = °- B 90 a
3
16
18
17 12
a b
Y X
Z
c
16
18
17 12
a b
Y X
Z
c
q D
C
B A
p
?
90°-a
a
? cosa =
+
q
p q
2 2
and cosB
BC
BD
=
?cos( ) 90
2 2
° - =
+
a
p
p q
? sina =
+
p
p q
2 2
From eq. (i),
sin( ( )) sin p ? a ? - +
=
BD AB
? AB
BD
=
+
sin
sin( )
?
? a
=
+ p q
2 2
sin
sin cos cos sin
?
? a + ? a
[ ] QBD p q = +
2 2
=
+
+
+
+
p q
q
p q
p
p q
2 2
2 2 2 2
sin
sin cos
?
? ?
=
+
+
( )sin
sin cos
p q
q p
2 2
?
? ?
23. If tan
cos sin
cos sin
? =
° - °
° + °
17 17
17 17
, then
what is the value of ??
(a) 0°      (b) 28° (c) 38° (d) 52°
Ê
(b) We have,
tan
cos sin
cos sin
? =
° - °
° + °
17 17
17 17
=
- °
+ °
1 17
1 17
tan
tan
[Divide numerator and
denominator by cos17°]
? tan tan( ) ? = ° - ° 45 17
Q tan( )
tan tan
tan tan
45 17
45 17
1 45 17
° - ° =
° - °
+ ° °
?
?
?
?
?
?
? tan tan ? = ° 28
? ? = ° 28
24. A andB are positive acute angles
such that cos sin 2 3
2
B A = and
3 2 2 2 sin sin A B = . What is the
value of ( ) A B + 2 ?
(a)
p
6
(b)
p
4
(c)
p
3
(d)
p
2
Ê
(d) We have, cos sin 2 3
2
B A =
and 3 2 2 2 sin sin A B =
?
2 2
2
3 2
3
2
sin
cos
sin
sin
B
B
A
A
=
? 2
2
2
2
2
sin
cos
sin cos
sin
B
B
A A
A
=
×
? tan cot 2B A =
? tan tan 2
2
B A = -
?
?
?
?
?
?
p
? 2
2
B A = -
p
? A B + = 2
2
p
25. What is sin cos sin 3 3 4
3
x x x + +
- 3 sin x + - 3 4
3
cos cos x x equal
to?
(a) 0 (b) 1
(c) 2 2 sin x (d) 4 4 cos x
Ê
(a) sin cos ( sin sin ) 3 3 4 3
3
x x x x + + -
+ - ( cos cos ) 3 4
3
x x
= + - - sin cos sin cos 3 3 3 3 x x x x = 0
26.
The value of ordinate of the graph
ofy x = + 2 cos lies in the interval
(a) [0, 1    (b) [0, 3]   (c) [ , ] - 1 1 (d) [1, 3]
Ê
(d) We know that,
- = = 1 1 cosx
? - + = + = + 1 2 2 1 2 cosx
? 1 3 = = y
? y ?[ , ] 1 3
27. What is the value of
8 10 20 40 cos cos cos ° · ° · ° ?
(a) tan 10° (b) cot 10°
(c) cosec 10° (d) sec 10°
Ê
(b) 8 10 20 40 cos cos cos ° ° °
= ° ° ° ×
°
°
8 10 20 40
10
10
cos cos cos
sin
sin
=
° ° ° °
°
4 2 10 10 20 40
10
( sin cos ) cos cos
sin
=
° ° °
°
4 20 20 40
10
sin cos cos
sin
Q2 2 sin cos sin A A A =
=
° ° °
°
2 2 20 20 40
10
( sin cos ) cos
sin
=
× ° °
°
2 40 40
10
sin cos
sin
=
°
°
sin
sin
80
10
=
° - °
°
sin( )
sin
90 10
10
=
°
°
cos
sin
10
10
= ° cot10
28. What is the value of
cos cos 48 12 ° - °?
(a)
5 1
4
-
(b)
1 5
4
-
(c)
5 1
2
+
(d)
1 5
8
-
Ê
(b) cos cos 48 12 ° - °
= -
° + ° ?
?
?
?
?
?
° - ° ?
?
?
?
?
?
2
48 12
2
48 12
2
sin sin
Q cos cos sin
sin
C D
C D
C D
- = -
+ ?
?
?
?
?
?
- ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
2
2
2
?
= - ° ° 2 30 18 sin sin
= - × ×
-
2
1
2
5 1
4
=
- 1 5
4
29. Consider the following statements:
1. IfABC is a right-angled triangle,
right-angled at A and if
sin B =
1
3
, thencosecC = 3.
2. Ifb B c C cos cos = and if the
triangle ABC is not right-angled,
then ABC must be isosceles.
Which of the above statements
is/are correct?
(a) 1 Only (b) 2 Only
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(b) 1. We have, sinB =
1
3
?
AC
BC
=
1
3
? AC k = andBC k = 3
Use pythagoras theorem in ?ABC,
AB BC AC = - ( ) ( )
2 2
= - ( ) ( ) 3
2 2
k k
= - 9
2 2
k k
= 8
2
k
= 2 2k
Now, cosec C
BC
AB
=
= =
3
2 2
3
2 2
k
k
, which is not correct.
2. Suppose we consider ?ABC is an
isosceles triangle.
? ? = ? B C
Also we have,b B c C cos cos =
? b C c C cos cos =
[putB C = ]
? b c = , Which is correct.
4
B
C
A
A
b c
B C
Page 5


1. If matrix A
i i
i i
=
-
- -
?
?
?
?
?
?
1
1
where
i = - 1, then which one of the
following is correct?
(a)A is hermitian
(b)A is skew-hermitian
(c) ( ) A A
T
+ is hermitian
(d)( ) A A
T
+ is skew-hermitian
Ê
(c) We have
A
i i
i i
=
-
- -
?
?
?
?
?
?
1
1
Now, A
i i
i i
=
+ -
+
?
?
?
?
?
?
1
1
( ) A
i i
i i
T
=
+
- +
?
?
?
?
?
?
1
1
Now, consider
X A A
i i
i i
T
= + =
+
- +
?
?
?
?
?
?
( )
1
1
+
-
- -
?
?
?
?
?
?
1
1
i i
i i
=
-
?
?
?
?
?
?
2 2
2 2
i
i
X
i
i
=
- ?
?
?
?
?
?
2 2
2 2
( ) X
i
i
T
=
-
?
?
?
?
?
?
2 2
2 2
= X
Hence, X A A
T
= + ( ) is a hermitian
matrix.
2. The term independent of x in the
binomial expansion of
2
2
10
x
x -
?
?
?
?
?
?
is equal to
(a) 180 (b) 120
(c) 90 (d) 72
Ê
(a) The general term in the binomial
expansion of
2
2
10
x
x -
?
?
?
?
?
?
is
T C
x
x
r r
r
r
+
-
=
?
?
?
?
?
?
-
1
10
2
10
2
( )
= -
-
- + +
10 10
20 2
2
2 1 C x
r
r
r
r
r
( ) ( )
For independent of x, put
- + + = 20 2
2
0 r
r
?
5
2
20
r
=
? r = 8
? T C
8 1
10
8
10 8 8
2 1
+
-
= - ( ) ( )
=
×
×
× ×
10 9
2 1
2 1
2
= 180
3. If ( ) 1 2
2 6
0 1 2
2
+ - = + + x x a a x a x
+ + K a x
12
12
, then what isa a
0 1
-
+ - + - + a a a a
2 3 4 12
K equal to?
(a) 32 (b) 64
(c) 2048 (d) 4096
Ê
(b) We have,
( ) 1 2
2 6
0 1 2
2
+ - = + + x x a a x a x
+ + .... a x
12
12
Put x = - 1both sides, we get
( ) .... 1 2 1
2 6
0 1 2 12
- - = - + - + a a a a
?a a a a
0 1 2 12
6
2 - + - + = - ..... ( ) = 64
4. IfC n C n ( , ) ( , ) 20 2 20 2 + = - , then
what is n equal to?
(a) 18 (b) 25
(c) 10 (d) 12
Ê
(c) Given,C(20,n + 2) = - C n ( , ) 20 2
? C ( , ( )) ( , ) 20 20 2 20 2 - + = - n C n
[ ( , ) ( , )] QC n r C n n r = -
? 20 2 2 - + = - ( ) n n
? 20 2 2 = + + - n n
? 2 20 n =
? n = 10
5. For how many values of k, is the
matrix
0 4
0 5
1
k
k
k k
- -
- -
?
?
?
?
?
?
?
?
?
?
singular?
(a) Only one (b) Only two
(c) Only four (d) Infinite
Ê
(d) The condition for singular matrix is
0 4
0 5
1
0
k
k
k k
- -
- -
=
Expanding alongR
1
, we get
0 5 4 0 0
2
- - + - + = k k k k ( ) ( )
? 4 4 0
2 2
k k - =
? 0 0 = , ? ? k R
Hence, for infinite values ofk, given
matrix is singular.
6. The number ( ) 1101101 1011011
2
+
can be written in decimal system as
(a) ( ) 198
10
(b) ( ) 199
10
(c) ( ) 200
10
(d) ( ) 201
10
Ê
(c) Now, ( ) 1101101 1 2 0 2
2
6
= × + ×
5
+ × + × + × + × 1 2 1 2 0 2 1 2
4 3 2 1
+ × 1 2
0
= + + + + + + 64 0 16 8 0 2 1
= ( ) 91
10
and
( ) 1011011 1 2 1 2 0 2
2
6 5 4
= × + × + ×
+ × + × + × + × 1 2 1 2 0 2 1 2
3 2 1 0
= + + + + + + 64 32 0 8 4 0 1= ( ) 109
10
NDA/NA Solved Paper 2020 (I & II) 1
PAPER : I Mathematics
? ( ) 1101101 1011011
2
+
= + ( ) ( ) 1101101 1011011
2 2
= + ( ) ( ) 91 109
10 10
= ( ) 200
10
7. What is the value of
1
10
1024 10
1
5
5 5 5
log log log - + 3125?
(a) 0        (b) 1 (c) 2             (d) 3
Ê
(a)
1
10
1024 10
1
5
3125
5 5 5
log log log - +
= - × +
1
10
2 5 2
1
5
5
5
10
5 5
5
log log ( ) log
= - + +
10
10
2 5 2
5
5
5
5 5 5 5
log [log log ] log
[ ]
Q log log log mn m n = +
= - + + log [ log ]
5 5
2 1 2 1 [ log ] Q
m
m = 1
= 0
8. Ifx ab
c
= log ( ),y bc
a
= log ( ),
z ca
b
= log ( ), then which of the
following is correct?
(a)xyz = 1
(b)x y z + + = 1
(c) ( ) ( ) ( ) 1 1 1 1
1 1 1
+ + + + + =
- - -
x y z
(d)( ) ( ) ( ) 1 1 1 1
2 2 2
+ + + + =
- - -
x y z
Ê
(c) We have, x ab
c
= log ( )
y bc
a
= log ( )
z ca
b
= log ( )
Now, 1 + = + x c ab
c c
log log ( )
= log ( )
c
abc
1 + = y abc
a
log ( )
and 1 + = z abc
b
log ( )
Now, ( ) ( ) ( ) 1 1 1
1 1 1
+ + + + +
- - -
x y z
= +
- -
[log ( )] [log ( )]
c a
abc abc
1 1
+
-
[log ( )]
b
abc
1
= + +
1 1 1
log ( ) log ( ) log ( )
c a b
abc abc abc
= + +
log
log( )
log
log( )
log
log( )
c
abc
a
abc
b
abc
Q log
log
log
m
n
n
m
=
?
?
?
?
?
?
=
+ + log log log
log( )
c a b
abc
= =
log( )
log( )
abc
abc
1
9.
LetA
x y y
x x y
=
+
-
?
?
?
?
?
?
2
,B =
-
?
?
?
?
?
?
2
1
andC =
?
?
?
?
?
?
3
2
. If AB C = , then what is
the value of the determinant of the
matrixA?
(a) - 10 (b) - 14
(c) - 24 (d) - 34
Ê
(b) Given, A
x y y
x x y
=
+
-
?
?
?
?
?
?
2
B =
-
?
?
?
?
?
?
2
1
andC =
?
?
?
?
?
?
3
2
Also given, AB C =
?
x y y
x x y
+
-
?
?
?
?
?
?
-
?
?
?
?
?
?
=
?
?
?
?
?
?
2
2
1
3
2
?
2 2
4
3
2
x y y
x x y
+ -
- +
?
?
?
?
?
?
=
?
?
?
?
?
?
?
2
3
3
2
x y
x y
+
+
?
?
?
?
?
?
=
?
?
?
?
?
?
On equating the corresponding
elements, we get
2 3 x y + = and 3 2 x y + =
? x = - 1and y = 5
? A =
- +
× - - -
?
?
?
?
?
?
=
- -
?
?
?
?
?
?
1 5 5
2 1 1 5
4 5
2 6
?The determinant of matrix A is
A =
- -
4 5
2 6
= - + = - 24 10 14
10. If 15 4 5 . . = = x , then which one of
the following is correct?
(a) ( ) ( ) 2 3 2 9 0 x x - - >
(b)( ) ( ) 2 3 2 9 0 x x - - <
(c) ( ) ( ) 2 3 2 9 0 x x - - =
(d)( ) ( ) 2 3 2 9 0 x x - - =
Ê
(d) We have, 15 4 5 . . = = x
?
3
2
9
2
= = x ? 3 2 9 = = x
? ( ) ( ) 2 3 2 9 0 x x - - =
11. LetS = { , , , } 1 2 3K . A relation R on
S S × is defined by xRy if
log log
a a
x y > whena =
1
2
. Then
the relation is
(a) reflexive only
(b) symmetric only
(c) transitive only
(d) both symmetric and transitive
Ê
(c) We have,S = { , , , ....} 1 2 3
and log log
a a
x y >
Here, a = ?
1
2
0 1 ( , )
? log log
a a
x y > ? x y <
Now, x R x ? x x < which is not
possible. So it is not reflexive relation.
Now, x R y ? x y <
But y x < | , so it is not symmetric relation.
Now, x R y and y R z
? x y < and y z < ? x z < ? x R z
Hence, it is transitive relation only.
12. What is the value of the
determinant
i i i
i i i
i i i
2 3
4 6 8
9 12 15
?
?
?
?
?
?
?
?
?
?
where
i = - 1 ?
(a) 0 (b) - 2 (c) 4i (d) - 4i
Ê
(d) Let ? =
i i i
i i i
i i i
2 3
4 6 8
9 12 15
=
- -
-
-
i i
i i
1
1 1 1
1
[ , , ] Qi i i i
2 3 4
1 1 = - = - =
= - + - - - + i i i i i i ( ) ( ) ( ) 1 1 1
[Expanding alongR
1
]
= - - - - i i i i i
2 2
2 = - 4i
13. LetA
a h g
h b f
g f c
=
?
?
?
?
?
?
?
?
?
?
andB
x
y
z
=
?
?
?
?
?
?
?
?
?
?
,
then what is AB equal to?
(a)
ax hy gz
y
z
+ + ?
?
?
?
?
?
?
?
?
?
(b)
ax hy gz
hx by fz
z
+ +
+ +
?
?
?
?
?
?
?
?
?
?
(c)
ax hy gz
hx by fz
gx fy cz
+ +
+ +
+ +
?
?
?
?
?
?
?
?
?
?
(d) [ax hy gz hx by fz + + + +
gx fy cz + + ]
Ê
(c) Now,
AB
a h g
h b f
g f c
x
y
z
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
+ +
+ +
+ +
?
?
?
?
?
?
?
?
?
?
ax hy gz
hx by fz
gx fy cz
14. What is the number of ways in
which the letters of the word
‘ABLE’ can be arranged so that the
vowels occupy even places?
(a) 2 (b) 4 (c) 6 (d) 8
Ê
(b) In a given word ‘ABLE’ Vowels are
{ , } A E .
1 2 3 4
As, vowel occupy even places, so two
vowels occupy the places 2 and 4.
Therefore, the number of ways of
occupying the vowels in even places is
2!.
Now, we have two consonants and
these consonants occupy the odd
places 1 and 3. Therefore, the number of
ways of occupying the  consonants in
odd places is 2!.
?Total number of ways = × 2 2 ! !
= × 2 2 = 4
2
15. What is the maximum number of
points of intersection of 5
non-overlapping circles?
(a) 10 (b) 15 (c) 20 (d) 25
Ê
(c) The maximum number of points of
intersection of 5 non-overlapping circles
= Selection of two circles × 2
[Q Two intersecting circles
cut at two points]
= ×
5
2
2 C =
×
×
× =
5 4
2 1
2 20
Directions (Q. Nos. 16-18)
Consider the following Venn diagram,
where X,Y andZ are three sets. Let the
number of elements inZ be denoted by
n(Z) which is equal to 90.
16. If the number of elements in Y and
Z are in the ratio 4 : 5, then what is
the value of b?
(a) 18 (b) 19 (c) 21 (d) 23
17. What is the value of
n X n Y n Z n X Y ( ) ( ) ( ) ( ) + + - n
- n - n n Y Z n X Z ( ) ( )
+ n n n X Y Z ( )?
(a)a b + + 43 (b)a b + + 63
(c)a b + + 96 (d)a b + + 106
18. If the number of elements
belonging to neither X, nor Y, nor Z
is equal to p, then what is the
number of elements in the
complement of X?
(a) p b + + 60 (b) p b + + 40
(c) p a + + 60 (d) p a + + 40
Solutions (16-18)
Givenn Z ( ) = 90
?12 18 17 90 + + + = c
? c = - 90 47 = 43
Ê
16. (c) Also given,
n Y
n Z
( )
( )
=
4
5
?
16 18 17
90
4
5
+ + +
=
b
? 51 72 + = b
? b = - 72 51
= 21
Ê
17.(d) Now,
n X n Y n Z n X Y ( ) ( ) ( ) ( ) + + - n
- n - n + n n n Y Z n X Z n X Y Z ( ) ( ) ( )
= ? ? n X Y Z ( )
= + + + + + + a b c 12 18 16 17
= + + + a b c 63
= + + + a b 43 63 [ ] Qc = 43
= + + a b 106
Ê
18. (a) Complement of X
= + + + p b c 17
= + + + p b 43 17 [ ] Qc = 43
= + + p b 60
Directions (Q. Nos. 19 and 20) Read
the following information and answer
the two items that follow.
Let
tan
tan
3A
A
K = , where tan A ? 0
and K ?
1
3
.
19. What is tan
2
A equal to?
(a)
K
K
+
-
3
3 1
(b)
K
K
-
-
3
3 1
(c)
3 3
3
K
K
-
-
(d)
K
K
+
+
3
3 1
Ê
(b) Given,
tan
tan
3A
A
K =
=
-
-
=
3
1 3
3
2
tan tan
( tan ) tan
A A
A A
K
?
3
1 3
2
2
-
-
=
tan
tan
A
A
K
? K K A A - = - 3 3
2 2
tan tan
? K A K - = - 3 3 1
2
tan ( )
? tan
2
3
3 1
A
K
K
=
-
-
20. For real values of tan A,K cannot
lie between
(a)
1
3
and 3 (b)
1
2
and 2
(c)
1
5
and 5 (d)
1
7
and 7
Ê
(a) For real values of tanA,K lies when
K
K
-
-
=
3
3 1
0 and 3 1 0 K - ?
?( ) ( ) K K - - = 3 3 1 0 andK ?
1
3
? K <
1
3
andK = 3
Hence, for real values of tanA,K cannot
lie between
1
3
3 ,
?
?
?
?
?
?
.
Directions (Q. Nos. 21 and 22) Read
the following information and answer
the two items that follow.
ABCD is a trapezium such that AB andCD
are parallel andBC is perpendicular to
them. Let ? = ? = ADB ABD ? a , , BC =p
andCD =q.
21. Consider the following
1. AD AB sin sin ? a =
2. BD AB sin sin ( ) ? ? a = +
Which of the above is/are correct?
(a) 1 Only (b) 2 Only
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(c) We have, ? = ADB ?, ? = ABD a,
BC p = andCD q =
1. In ?ABD, use Sine rule,
sin sin ? a
AB AD
=
? AD AB sin sin ? a = , which is correct.
2. In ?ABD, ? = - + A p ? a ( )
Use Sine rule in ?ABD,
sin sin A
BD AB
=
?
?
sin( ( )] sin p ? a ? - +
=
BD AB
…(i)
? AB BD sin( ) sin , ? a ? + =
which is correct.
Hence, both statements are
correct.
22. What is AB equal to?
(a)
( ) sin
cos sin
p q
p q
2 2
+
+
?
? ?
(b)
( ) cos
cos sin
p q
p q
2 2
-
+
?
? ?
(c)
( ) sin
cos sin
p q
q p
2 2
+
+
?
? ?
(d)
( ) cos
cos sin
p q
q p
2 2
-
+
?
? ?
Ê
(a) In right angle, ?BCD,
? = ° - B 90 a
BD p q = +
2 2
and sinB
CD
BD
=
? sin( ) 90
2 2
° - =
+
a
q
p q
[ ] Q ? = °- B 90 a
3
16
18
17 12
a b
Y X
Z
c
16
18
17 12
a b
Y X
Z
c
q D
C
B A
p
?
90°-a
a
? cosa =
+
q
p q
2 2
and cosB
BC
BD
=
?cos( ) 90
2 2
° - =
+
a
p
p q
? sina =
+
p
p q
2 2
From eq. (i),
sin( ( )) sin p ? a ? - +
=
BD AB
? AB
BD
=
+
sin
sin( )
?
? a
=
+ p q
2 2
sin
sin cos cos sin
?
? a + ? a
[ ] QBD p q = +
2 2
=
+
+
+
+
p q
q
p q
p
p q
2 2
2 2 2 2
sin
sin cos
?
? ?
=
+
+
( )sin
sin cos
p q
q p
2 2
?
? ?
23. If tan
cos sin
cos sin
? =
° - °
° + °
17 17
17 17
, then
what is the value of ??
(a) 0°      (b) 28° (c) 38° (d) 52°
Ê
(b) We have,
tan
cos sin
cos sin
? =
° - °
° + °
17 17
17 17
=
- °
+ °
1 17
1 17
tan
tan
[Divide numerator and
denominator by cos17°]
? tan tan( ) ? = ° - ° 45 17
Q tan( )
tan tan
tan tan
45 17
45 17
1 45 17
° - ° =
° - °
+ ° °
?
?
?
?
?
?
? tan tan ? = ° 28
? ? = ° 28
24. A andB are positive acute angles
such that cos sin 2 3
2
B A = and
3 2 2 2 sin sin A B = . What is the
value of ( ) A B + 2 ?
(a)
p
6
(b)
p
4
(c)
p
3
(d)
p
2
Ê
(d) We have, cos sin 2 3
2
B A =
and 3 2 2 2 sin sin A B =
?
2 2
2
3 2
3
2
sin
cos
sin
sin
B
B
A
A
=
? 2
2
2
2
2
sin
cos
sin cos
sin
B
B
A A
A
=
×
? tan cot 2B A =
? tan tan 2
2
B A = -
?
?
?
?
?
?
p
? 2
2
B A = -
p
? A B + = 2
2
p
25. What is sin cos sin 3 3 4
3
x x x + +
- 3 sin x + - 3 4
3
cos cos x x equal
to?
(a) 0 (b) 1
(c) 2 2 sin x (d) 4 4 cos x
Ê
(a) sin cos ( sin sin ) 3 3 4 3
3
x x x x + + -
+ - ( cos cos ) 3 4
3
x x
= + - - sin cos sin cos 3 3 3 3 x x x x = 0
26.
The value of ordinate of the graph
ofy x = + 2 cos lies in the interval
(a) [0, 1    (b) [0, 3]   (c) [ , ] - 1 1 (d) [1, 3]
Ê
(d) We know that,
- = = 1 1 cosx
? - + = + = + 1 2 2 1 2 cosx
? 1 3 = = y
? y ?[ , ] 1 3
27. What is the value of
8 10 20 40 cos cos cos ° · ° · ° ?
(a) tan 10° (b) cot 10°
(c) cosec 10° (d) sec 10°
Ê
(b) 8 10 20 40 cos cos cos ° ° °
= ° ° ° ×
°
°
8 10 20 40
10
10
cos cos cos
sin
sin
=
° ° ° °
°
4 2 10 10 20 40
10
( sin cos ) cos cos
sin
=
° ° °
°
4 20 20 40
10
sin cos cos
sin
Q2 2 sin cos sin A A A =
=
° ° °
°
2 2 20 20 40
10
( sin cos ) cos
sin
=
× ° °
°
2 40 40
10
sin cos
sin
=
°
°
sin
sin
80
10
=
° - °
°
sin( )
sin
90 10
10
=
°
°
cos
sin
10
10
= ° cot10
28. What is the value of
cos cos 48 12 ° - °?
(a)
5 1
4
-
(b)
1 5
4
-
(c)
5 1
2
+
(d)
1 5
8
-
Ê
(b) cos cos 48 12 ° - °
= -
° + ° ?
?
?
?
?
?
° - ° ?
?
?
?
?
?
2
48 12
2
48 12
2
sin sin
Q cos cos sin
sin
C D
C D
C D
- = -
+ ?
?
?
?
?
?
- ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
2
2
2
?
= - ° ° 2 30 18 sin sin
= - × ×
-
2
1
2
5 1
4
=
- 1 5
4
29. Consider the following statements:
1. IfABC is a right-angled triangle,
right-angled at A and if
sin B =
1
3
, thencosecC = 3.
2. Ifb B c C cos cos = and if the
triangle ABC is not right-angled,
then ABC must be isosceles.
Which of the above statements
is/are correct?
(a) 1 Only (b) 2 Only
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(b) 1. We have, sinB =
1
3
?
AC
BC
=
1
3
? AC k = andBC k = 3
Use pythagoras theorem in ?ABC,
AB BC AC = - ( ) ( )
2 2
= - ( ) ( ) 3
2 2
k k
= - 9
2 2
k k
= 8
2
k
= 2 2k
Now, cosec C
BC
AB
=
= =
3
2 2
3
2 2
k
k
, which is not correct.
2. Suppose we consider ?ABC is an
isosceles triangle.
? ? = ? B C
Also we have,b B c C cos cos =
? b C c C cos cos =
[putB C = ]
? b c = , Which is correct.
4
B
C
A
A
b c
B C
30. Consider the following statements
1. If in a triangle ABC,A B = 2 and
b c = , then it must be an
obtuse-angled triangle.
2. There exists no triangle ABC
withA B = ° = ° 40 65 , and
a
c
= ° ° sin 40 15 cosec .
Which of the above statements
is/are correct?
(a) 1 Only (b) 2 Only
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(d) 1. We have, in ?ABC,
A B = 2 andb c =
? Angles opposite to equal sides are
equal.
? ? = ? C B
Also, A B C = = 2 2
In ?ABC, ? + ? + ? = ° A B C 180
? 2 180 C C C + + = °
? C =
°
= °
180
4
45
? B = ° 45 and A = ° 90
Thus, it shows that ?ABC is not an
obtuse angle triangle.
Hence, statement 1 is incorrect.
2. We have A = ° 40 ,B = ° 65
In ?ABC,
? + ? + ? = ° A B C 180
? 40 65 180 ° + ° + ? = ° C
? ? = ° C 75
Use sine rule in ?ABC,
a c
sin sin 40 75 °
=
°
a
c
= ° ° sin cos 40 75 ec ,
Hence, Statement 2 is incorrect.
Directions (Q. Nos. 31-33) Read the
following information and answer the
three items that follow.
Let a x b x c sin cos
2 2
+ = ,
b y a y d sin cos
2 2
+ =
and p x q y tan tan =
31. What is tan
2
x equal to?
(a)
c b
a c
-
-
(b)
a c
c b
-
-
(c)
c a
c b
-
-
(d)
c b
c a
-
-
Ê
(a) We have,
a x b x c sin cos
2 2
+ =
On dividing both sides by cos
2
x, we get
a x b c x tan ( ) sec
2 2
1 + = ×
? a x b c x tan ( tan )
2 2
1 + = +
[ sec tan ] Q
2 2
1 ? ? - =
? tan ( )
2
x a c c b - = -
? tan
2
x
c b
a c
=
-
-
…(i)
32. What is
d a
b d
-
-
equal to?
(a) sin
2
y (b) cos
2
y
(c) tan
2
y (d) cot
2
y
Ê
(c) We have,b y a y d sin cos
2 2
+ =
On dividing both sides by cos ,
2
y we get
b y a d y tan ( ) (sec )
2 2
1 + =
? b y a d y tan ( tan )
2 2
1 + = +
[ sec tan ] Q
2 2
1 ? ? - =
? tan ( )
2
y b d d a - = -
?
d a
b d
y
-
-
= tan
2
…(ii)
33. What is
p
q
2
2
equal to?
(a)
( ) ( )
( ) ( )
b c b d
a d a c
- -
- -
(b)
( ) ( )
( ) ( )
a d c a
b c d b
- -
- -
(c)
( ) ( )
( ) ( )
d a c a
b c d b
- -
- -
(d)
( ) ( )
( ) ( )
b c b d
c a a d
- -
- -
Ê
(b) We have,
p x q y tan tan =
On squaring both sides, we get.
p
q
y
x
2
2
2
2
=
tan
tan
=
- -
- -
( ) / ( )
( ) / ( )
d a b d
c b a c
[Q from eq. (i) and (ii)]
=
- -
- -
( ) ( )
( ) ( )
d a a c
b d c b
=
- -
- -
( ) ( )
( ) ( )
a d c a
d b b c
Directions (Q. Nos. 34-36) Read the
following information and answer the
three items that follow.
Let t sin cos
n
n n
= + ? ?
34. What is
t t
t t
3 5
5 7
-
-
equal to?
(a)
t
t
1
3
(b)
t
t
3
5
(c)
t
t
5
7
(d)
t
t
1
7
Ê
(a) We have,
t
n
n n
= sin cos ? + ?
Now,
t t
t t
3 5
5 7
3 3
5 5
5 5
-
-
=
+
- +
+
(sin cos )
(sin cos )
(sin cos
? ?
? ?
? ?)
(sin cos ) - +
7 7
? ?
=
- + -
- + -
(sin sin ) (cos cos )
(sin sin ) (cos c
3 5 3 5
5 7 5
? ? ? ?
? ? ? os )
7
?
=
- + -
- +
sin ( sin ) cos ( cos )
sin ( sin ) cos
3 2 3 2
5 2 5
1 1
1
? ? ? ?
? ? ? ( cos ) 1
2
- ?
=
+
+
sin cos cos sin
sin cos cos sin
3 2 3 2
5 2 5 2
? ? ? ?
? ? ? ?
=
+ sin cos (sin cos )
sin cos (sin cos )
2 2
2 2 3 3
? ? ? ?
? ? ? + ?
=
+
=
sin cos
sin cos
? ?
? + ?
3 3
1
3
t
t
35. What ist t
1
2
2
- equal to?
(a) cos 2? (b) sin 2?
(c) 2 cos ? (d) 2 sin ?
Ê
(b)t t
1
2
2
2
- = + (sin cos ) ? ?
- + (sin cos )
2 2
? ?
= + + sin cos sin cos
2 2
2 ? ? ? ?
- + (sin cos )
2 2
? ?
= = 2 2 sin cos sin ? ? ?
36. What is the value oft
10
where
? = ° 45 ?
(a) 1         (b)
1
4
(c)
1
16
(d)
1
32
Ê
(c) Now,t
10
10 10
= + sin cos ? ?
= ° + ° (sin ) (cos ) 45 45
10 10
[Put ? = ° 45 ]
=
?
?
?
?
?
?
+
?
?
?
?
?
?
1
2
1
2
10 10
=
?
?
?
?
?
?
2
1
2
5
= =
1
2
1
16
4
Directions (Q. Nos. 37-39) Read the
following information and answer the
three items that follow.
Let a ß = = ° 15 .
37. What is the value of sin cos a ß + ?
(a)
1
2
(b)
1
2 2
(c)
3
2 2
(d)
3
2
Ê
(d) sin cos a ß +
= +
?
?
?
?
?
?
1
2
1
2
sin cos a ß × 2
= ° + ° 2 45 45 (sin cos sin cos ) a ß
= ° ° + ° ° 2 15 45 45 15 (sin cos sin cos )
[ ] Q a ß = = ° 15
= ° + ° 2 15 45 sin( )
= ° = × 2 60 2
3
2
sin =
3
2
5
C
a b
A B
c
75°
40° 65°