NDA Solved Paper 2021 - I | NDA (National Defence Academy) Past Year Papers PDF Download

 Page 1


1. The smallest positive integern for
which
1
1
1
2
-
+
?
?
?
?
?
?
=
i
i
n
wherei = -1, is
(a) 2 (b) 4 (c) 6 (d) 8
Ê
(a)
1
1
1
2
-
+
?
?
?
?
?
? =
i
i
n
, wherei = - 1
1
1
1
1
1
2
-
+
×
-
-
?
?
?
?
?
? =
i
i
i
i
n
1 2
1
1
2
2
2
+ -
-
?
?
?
?
?
?
?
?
=
i i
i
n
1 1 2
1 1
1
2
- -
+
?
?
?
?
?
? =
i
n
? ( ) ( ) - = - i i
n
2
4
? n
2
4 =
n = 2
Hence, option (a) is correct.
2. The value of x, satisfying the
equation log sin
cosx
x = 1, where
0
2
< < x
p
, is
(a)
p
12
(b)
p
3
(c)
p
4
(d)
p
6
Ê
(c)log sin
cos x
x = 1 , where 0
2
< < x
p
?(cos ) sin x x
1
= ?cos sin x x =
?tan tan tan x x = ? = 1 p / 4
? x = p / 4
Hence, option (c) is correct.
3. If ? is the value of the determinant
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
then what is the value of the
following determinant?
pa b qc
pa b qc
pa b qc
1 1 1
2 2 2
3 3 3
(p ? 0 or 1,q ? 0 or 1)
(a) p? (b)q?
(c) ( ) p q + ? (d) pq?
Ê
(d) Given,
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
= ?
?
pa b qc
pa b qc
pa b qc
1 1 1
2 2 2
3 3 3
= · · p q
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
= pq?
Hence, option (d) is correct.
4. IfC C C C
n 0 1 2
, , , ..., are the
coefficients in the expansion of
( ) 1 + x
n
, then what is the value of
C C C C
n 1 2 3
+ + + + ... ?
(a) 2
n
(b) 2 1
n
-
(c) 2
1 n -
(d) 2 2
n
-
Ê
(b)Q( ) 1
0 1 2
+ = + + x C C x C
n n n n
x C x
n
n
n 2
+ … +
and we know that
n n n n
n
n
C C C C
0 1 2
2 + + + + = ...
?
n n n
n
n n
C C C C
1 2 0
2 + + … + = -
= - 2 1
n
Hence, option (b) is correct.
5. Ifa b c + + = 4 and
ab bc ca + + = 0, then what is the
value of the following
determinant?
a b c
b c a
c a b
(a) 32 (b) - 64
(c) - 128 (d) 64
Ê
(b) Let
? =
a b c
b c a
c a b
=
+ +
+ +
+ +
a b c b c
a b c c a
a b c a b
(byC C C C
1 1 2 3
? + + )
= + + ( ) a b c
b c
c a
a b
1
1
1
(To take commona b c + + fromC
1
)
= + +
- -
- - ( ) a b c
b c c a
c a a b
a b
0
0
1
(by R R R R R R
1 1 2 2 2 3
? - ? - , )
= + + - - - - ( )[( )( ) ( ) ] a b c b c a b c a
2
= + + - - + ( )( a b c ab b ca bc
2
PAPER : I Mathematics
Page 2


1. The smallest positive integern for
which
1
1
1
2
-
+
?
?
?
?
?
?
=
i
i
n
wherei = -1, is
(a) 2 (b) 4 (c) 6 (d) 8
Ê
(a)
1
1
1
2
-
+
?
?
?
?
?
? =
i
i
n
, wherei = - 1
1
1
1
1
1
2
-
+
×
-
-
?
?
?
?
?
? =
i
i
i
i
n
1 2
1
1
2
2
2
+ -
-
?
?
?
?
?
?
?
?
=
i i
i
n
1 1 2
1 1
1
2
- -
+
?
?
?
?
?
? =
i
n
? ( ) ( ) - = - i i
n
2
4
? n
2
4 =
n = 2
Hence, option (a) is correct.
2. The value of x, satisfying the
equation log sin
cosx
x = 1, where
0
2
< < x
p
, is
(a)
p
12
(b)
p
3
(c)
p
4
(d)
p
6
Ê
(c)log sin
cos x
x = 1 , where 0
2
< < x
p
?(cos ) sin x x
1
= ?cos sin x x =
?tan tan tan x x = ? = 1 p / 4
? x = p / 4
Hence, option (c) is correct.
3. If ? is the value of the determinant
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
then what is the value of the
following determinant?
pa b qc
pa b qc
pa b qc
1 1 1
2 2 2
3 3 3
(p ? 0 or 1,q ? 0 or 1)
(a) p? (b)q?
(c) ( ) p q + ? (d) pq?
Ê
(d) Given,
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
= ?
?
pa b qc
pa b qc
pa b qc
1 1 1
2 2 2
3 3 3
= · · p q
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
= pq?
Hence, option (d) is correct.
4. IfC C C C
n 0 1 2
, , , ..., are the
coefficients in the expansion of
( ) 1 + x
n
, then what is the value of
C C C C
n 1 2 3
+ + + + ... ?
(a) 2
n
(b) 2 1
n
-
(c) 2
1 n -
(d) 2 2
n
-
Ê
(b)Q( ) 1
0 1 2
+ = + + x C C x C
n n n n
x C x
n
n
n 2
+ … +
and we know that
n n n n
n
n
C C C C
0 1 2
2 + + + + = ...
?
n n n
n
n n
C C C C
1 2 0
2 + + … + = -
= - 2 1
n
Hence, option (b) is correct.
5. Ifa b c + + = 4 and
ab bc ca + + = 0, then what is the
value of the following
determinant?
a b c
b c a
c a b
(a) 32 (b) - 64
(c) - 128 (d) 64
Ê
(b) Let
? =
a b c
b c a
c a b
=
+ +
+ +
+ +
a b c b c
a b c c a
a b c a b
(byC C C C
1 1 2 3
? + + )
= + + ( ) a b c
b c
c a
a b
1
1
1
(To take commona b c + + fromC
1
)
= + +
- -
- - ( ) a b c
b c c a
c a a b
a b
0
0
1
(by R R R R R R
1 1 2 2 2 3
? - ? - , )
= + + - - - - ( )[( )( ) ( ) ] a b c b c a b c a
2
= + + - - + ( )( a b c ab b ca bc
2
PAPER : I Mathematics
- - + c a ca
2 2
2 )
= - + + ( ) a b c ( ) a b c ab bc ca
2 2 2
+ + - - -
= - + + + + ( )[( ) a b c a b c
2
- + + 3( )] ab bc ca
= - - = - ( )[ ] 4 16 0 64.
6. The number of integer values ofk, for which the equation
2 2 1 sinx k = + has a solution, is
(a) zero (b) one
(c) two (d) four
Ê
(c) Given,
2 2 1 sinx k = +
Q - = = 1 1 sinx ? - = = 2 2 2 sinx
- - = - = - 2 1 2 1 2 1 sinx
- = = 3 2 1 k
-
= =
3
2
1
2
k ? - · = = · 1 5 0 5 k
?Integer values ofk = - 1 0 ,
Hence, option (c) is correct.
7. Ifa a a a
1 2 3 9
, , , ..., are in GP, then what is the value of the
following determinant?
ln ln ln
ln ln ln
ln ln ln
a a a
a a a
a a a
1 2 3
4 5 6
7 8 9
(a) 0 (b) 1
(c) 2 (d) 4
Ê
(a) Let first term and common ratio of GP area and r respectively.
?
log log log
log log log
log log log
a a a
a a a
a a a
1 2 3
4 5 6
7 8 9
=
log log log
log log log
log log log
a ar ar
ar ar ar
ar ar a
2
3 4 5
6 7
r
8
=
+ +
+ + +
log log log log log
log log log log log
a a r a r
a r a r a
2
3 4 5log
log log log log log log
r
a r a r a r + + + 6 7 8
[Qlogmn = + log log m n]
= +
+
log log log
log log log log
log log log log
a r r
a r r r
a r r r
3
6
(byC C C
2 2 1
? - andC C C
3 3 2
? - )
= 0 [QC C
2 3
= ]
8. If the roots of the quadratic equation x x k
2
2 0 + + = are
real, then
(a)k < 0 (b)k = 0
(c)k < 1 (d)k = 1
Ê
(d) Given quadratic equation,
x x k
2
2 0 + + = … (i)
Since, roots are real
? D = 0 ?b ac
2
4 0 - =
( ) ( )( ) 2 41 0
2
- = k ?4 4 = k ? k = 1
Hence, option (d) is correct.
9. Ifn = 100!, then what is the value of the following?
1 1 1
2 3 4
log log log n n n
+ + + + ...
log
1
100
n
(a) 0 (b) 1 (c) 2 (d) 3
Ê
(b)
1 1 1
2 3 100
log log log n n n
+ + … +
= + + + + log log log ... log
n n n n
2 3 4 100
= · · · … log ( )
n
2 3 4 5 100
= log ( !)
! 100
100 [ !] Qn = 100
= 1 [ log ] Q
a
a
= 1
Hence, option (b) is correct.
10. Ifz i = + 1 , wherei = -1, then what is the modulus of
z
z
+
2
?
(a) 1 (b) 2 (c) 3 (d) 4
Ê
(b) z i = + 1 , wherei = - 1
z
z
i
i
+ = + +
+
2
1
2
1
( )
( )
= + +
+
×
-
-
( )
( )
( )
( )
1
2
1
1
1
i
i
i
i
= + +
-
( )
( )
1
2 1
2
i
i
= + + - | | 1 1 i i = = | | 2 2
Hence, option (b) is correct.
11. If A and B are two matrices such that AB is of ordern n × ,
then which one of the following is correct?
(a) A andB should be square matrices of same order.
(b) Either A orB should be a square matrix.
(c) Both A andB should be of same order.
(d) Orders of A andB need not be the same.
Ê
(d) Given that, order of matrix AB n n = ×
If we take A
n p ×
and B
p n ×
, then AB will be of ordern n × .
So, orders of A and B need not be the same, is correct.
Hence, option (d) is correct.
12. How many matrices of different orders are possible with
elements comprising all prime numbers less than 30?
(a) 2 (b) 3 (c) 4 (d) 6
Ê
(c)QPrime numbers less than30 = {2, 3, 5, 7, 11, 13, 17, 19, 23,
29}
? Number of elements = 10
?Possible order of matrices with
10 elements = × 10 1 ,1 10 × ,2 5 × , 5 2 ×
?Nubmer of matrices of different order = 4
Hence, option (c) is correct.
13. Let, A
p q
r s
=
where p q r , , ands are any four different prime numbers
less than 20. What is the maximum value of the
determinant?
(a) 215 (b) 311 (c) 317 (d) 323
Ê
(c) A
p q
r s
= , prime numbers less than 20
= {2, 3, 5, 7, 11, 13, 17, 19} ? A ps rq = -
For maximum values of A, p and s must be maximum andr andq
must be minimum.
Then, p = 17,s = 19,r = 2,q = 3
? A = × - × 17 19 2 3
= - = 323 6 317
Hence, option (c) is correct.
Page 3


1. The smallest positive integern for
which
1
1
1
2
-
+
?
?
?
?
?
?
=
i
i
n
wherei = -1, is
(a) 2 (b) 4 (c) 6 (d) 8
Ê
(a)
1
1
1
2
-
+
?
?
?
?
?
? =
i
i
n
, wherei = - 1
1
1
1
1
1
2
-
+
×
-
-
?
?
?
?
?
? =
i
i
i
i
n
1 2
1
1
2
2
2
+ -
-
?
?
?
?
?
?
?
?
=
i i
i
n
1 1 2
1 1
1
2
- -
+
?
?
?
?
?
? =
i
n
? ( ) ( ) - = - i i
n
2
4
? n
2
4 =
n = 2
Hence, option (a) is correct.
2. The value of x, satisfying the
equation log sin
cosx
x = 1, where
0
2
< < x
p
, is
(a)
p
12
(b)
p
3
(c)
p
4
(d)
p
6
Ê
(c)log sin
cos x
x = 1 , where 0
2
< < x
p
?(cos ) sin x x
1
= ?cos sin x x =
?tan tan tan x x = ? = 1 p / 4
? x = p / 4
Hence, option (c) is correct.
3. If ? is the value of the determinant
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
then what is the value of the
following determinant?
pa b qc
pa b qc
pa b qc
1 1 1
2 2 2
3 3 3
(p ? 0 or 1,q ? 0 or 1)
(a) p? (b)q?
(c) ( ) p q + ? (d) pq?
Ê
(d) Given,
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
= ?
?
pa b qc
pa b qc
pa b qc
1 1 1
2 2 2
3 3 3
= · · p q
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
= pq?
Hence, option (d) is correct.
4. IfC C C C
n 0 1 2
, , , ..., are the
coefficients in the expansion of
( ) 1 + x
n
, then what is the value of
C C C C
n 1 2 3
+ + + + ... ?
(a) 2
n
(b) 2 1
n
-
(c) 2
1 n -
(d) 2 2
n
-
Ê
(b)Q( ) 1
0 1 2
+ = + + x C C x C
n n n n
x C x
n
n
n 2
+ … +
and we know that
n n n n
n
n
C C C C
0 1 2
2 + + + + = ...
?
n n n
n
n n
C C C C
1 2 0
2 + + … + = -
= - 2 1
n
Hence, option (b) is correct.
5. Ifa b c + + = 4 and
ab bc ca + + = 0, then what is the
value of the following
determinant?
a b c
b c a
c a b
(a) 32 (b) - 64
(c) - 128 (d) 64
Ê
(b) Let
? =
a b c
b c a
c a b
=
+ +
+ +
+ +
a b c b c
a b c c a
a b c a b
(byC C C C
1 1 2 3
? + + )
= + + ( ) a b c
b c
c a
a b
1
1
1
(To take commona b c + + fromC
1
)
= + +
- -
- - ( ) a b c
b c c a
c a a b
a b
0
0
1
(by R R R R R R
1 1 2 2 2 3
? - ? - , )
= + + - - - - ( )[( )( ) ( ) ] a b c b c a b c a
2
= + + - - + ( )( a b c ab b ca bc
2
PAPER : I Mathematics
- - + c a ca
2 2
2 )
= - + + ( ) a b c ( ) a b c ab bc ca
2 2 2
+ + - - -
= - + + + + ( )[( ) a b c a b c
2
- + + 3( )] ab bc ca
= - - = - ( )[ ] 4 16 0 64.
6. The number of integer values ofk, for which the equation
2 2 1 sinx k = + has a solution, is
(a) zero (b) one
(c) two (d) four
Ê
(c) Given,
2 2 1 sinx k = +
Q - = = 1 1 sinx ? - = = 2 2 2 sinx
- - = - = - 2 1 2 1 2 1 sinx
- = = 3 2 1 k
-
= =
3
2
1
2
k ? - · = = · 1 5 0 5 k
?Integer values ofk = - 1 0 ,
Hence, option (c) is correct.
7. Ifa a a a
1 2 3 9
, , , ..., are in GP, then what is the value of the
following determinant?
ln ln ln
ln ln ln
ln ln ln
a a a
a a a
a a a
1 2 3
4 5 6
7 8 9
(a) 0 (b) 1
(c) 2 (d) 4
Ê
(a) Let first term and common ratio of GP area and r respectively.
?
log log log
log log log
log log log
a a a
a a a
a a a
1 2 3
4 5 6
7 8 9
=
log log log
log log log
log log log
a ar ar
ar ar ar
ar ar a
2
3 4 5
6 7
r
8
=
+ +
+ + +
log log log log log
log log log log log
a a r a r
a r a r a
2
3 4 5log
log log log log log log
r
a r a r a r + + + 6 7 8
[Qlogmn = + log log m n]
= +
+
log log log
log log log log
log log log log
a r r
a r r r
a r r r
3
6
(byC C C
2 2 1
? - andC C C
3 3 2
? - )
= 0 [QC C
2 3
= ]
8. If the roots of the quadratic equation x x k
2
2 0 + + = are
real, then
(a)k < 0 (b)k = 0
(c)k < 1 (d)k = 1
Ê
(d) Given quadratic equation,
x x k
2
2 0 + + = … (i)
Since, roots are real
? D = 0 ?b ac
2
4 0 - =
( ) ( )( ) 2 41 0
2
- = k ?4 4 = k ? k = 1
Hence, option (d) is correct.
9. Ifn = 100!, then what is the value of the following?
1 1 1
2 3 4
log log log n n n
+ + + + ...
log
1
100
n
(a) 0 (b) 1 (c) 2 (d) 3
Ê
(b)
1 1 1
2 3 100
log log log n n n
+ + … +
= + + + + log log log ... log
n n n n
2 3 4 100
= · · · … log ( )
n
2 3 4 5 100
= log ( !)
! 100
100 [ !] Qn = 100
= 1 [ log ] Q
a
a
= 1
Hence, option (b) is correct.
10. Ifz i = + 1 , wherei = -1, then what is the modulus of
z
z
+
2
?
(a) 1 (b) 2 (c) 3 (d) 4
Ê
(b) z i = + 1 , wherei = - 1
z
z
i
i
+ = + +
+
2
1
2
1
( )
( )
= + +
+
×
-
-
( )
( )
( )
( )
1
2
1
1
1
i
i
i
i
= + +
-
( )
( )
1
2 1
2
i
i
= + + - | | 1 1 i i = = | | 2 2
Hence, option (b) is correct.
11. If A and B are two matrices such that AB is of ordern n × ,
then which one of the following is correct?
(a) A andB should be square matrices of same order.
(b) Either A orB should be a square matrix.
(c) Both A andB should be of same order.
(d) Orders of A andB need not be the same.
Ê
(d) Given that, order of matrix AB n n = ×
If we take A
n p ×
and B
p n ×
, then AB will be of ordern n × .
So, orders of A and B need not be the same, is correct.
Hence, option (d) is correct.
12. How many matrices of different orders are possible with
elements comprising all prime numbers less than 30?
(a) 2 (b) 3 (c) 4 (d) 6
Ê
(c)QPrime numbers less than30 = {2, 3, 5, 7, 11, 13, 17, 19, 23,
29}
? Number of elements = 10
?Possible order of matrices with
10 elements = × 10 1 ,1 10 × ,2 5 × , 5 2 ×
?Nubmer of matrices of different order = 4
Hence, option (c) is correct.
13. Let, A
p q
r s
=
where p q r , , ands are any four different prime numbers
less than 20. What is the maximum value of the
determinant?
(a) 215 (b) 311 (c) 317 (d) 323
Ê
(c) A
p q
r s
= , prime numbers less than 20
= {2, 3, 5, 7, 11, 13, 17, 19} ? A ps rq = -
For maximum values of A, p and s must be maximum andr andq
must be minimum.
Then, p = 17,s = 19,r = 2,q = 3
? A = × - × 17 19 2 3
= - = 323 6 317
Hence, option (c) is correct.
14. IfA andB are square matrices of order 2 such that
det( ) det( ), AB BA = then which one of the following is correct?
(a) A must be a unit matrix
(b)B must be a unit matrix
(c) Both A andB must be unit matrices
(d) A andB need not be unit matrices
Ê
(d) A
2 2 ×
and B
2 2 ×
are two matrices
and| | | | AB BA = ?| || | | || | A B B A =
Let A =
?
?
?
?
?
?
1 2
3 4
,B =
-
-
?
?
?
?
?
?
2 1
3 2 1 2 / /
then,| | | | AB BA =
Hence, we can say A and B need not be the unit matrices.
Hence, option (d) is correct.
15. What is cot cot 2 4 x x - cot cot 4 6 x x - cot cot 6 2 x x equal
to?
(a) - 1 (b) 0 (c) 1 (d) 2
Ê
(c)Qcot cot( ) 6 2 4 x x x = +
cot
cot cot
cot cot
6
2 4 1
2 4
x
x x
x x
=
· -
+
Qcot ( )
cot cot
cot cot
A B
A B
A B
+ =
-
+
?
?
?
?
?
?
1
?cot cot cot cot 6 2 6 4 x x x x · + · = · - cot cot 2 4 1 x x
?cot cot cot cot 2 4 4 6 x x x x · - · - · = cot cot 6 2 1 x x
Hence, option (c) is correct.
16. If tanx = -
3
4
and x is in the second quadrant, then what
is the value of sin cos x x · ?
(a)
6
25
(b)
12
25
(c) -
6
25
(d) -
12
25
Ê
(d) Given,
tanx =
- 3
4
and x is in the 2nd quadrant.
Let perpendicular be3k and base be 4k, then
Hypotenuse = + = ( ) ( ) 3 4 5
2 2
k k k
sinx =
3
5
andcosx =
- 4
5
?sin cos x x · = ×
- ?
?
?
?
?
?
=
- 3
5
4
5
12
25
Hence, option (d) is correct.
17. What is the value of the following?
cosec
7
6
5
3
p p ?
?
?
?
?
?
?
?
?
?
?
?
sec
(a)
4
3
(b) 4 (c) - 4 (d) -
4
3
Ê
(c)cosec
7
6
5
3
p p ?
?
?
?
?
?
·
?
?
?
?
?
?
sec = +
?
?
?
?
?
?
· -
?
?
?
?
?
?
cosec p
p
p
p
6
2
3
sec
= - · cosec
p p
6 3
sec = - × 2 2 = - 4
Hence, option (c) is correct.
18. If the determinant
x
x
1 3
0 0 1
1 4
0 =
then what is x equal to?
(a) - 2 or 2 (b) - 3 or 3 (c) - 1 or 1 (d) 3 or 4
Ê
(c) Given,
x
x
1 3
0 0 1
1 4
0 = … (i)
- - = 1 1 0
2
( ) x ? x
2
1 0 - = ? x
2
1 =
? x = ± 1 ?1 0
2
- = x
? x
2
1 =
x = + - 1 1 ,
Hence, option (c) is correct.
19. What is the value of the following?
tan tan tan ...tan tan 31 33 35 57 59 ° ° ° ° °
(a) - 1 (b) 0 (c) 1 (d) 2
Ê
(c) tan º tan º tan º tan º tan º 31 33 35 57 59 · · … ·
= · °· … ° ° - ° tan º tan tan º tan tan( ) 31 33 35 45 90 33 x xK
· ° - tan( º) 90 31
= °· °· °… ° tan tan tan cot 31 33 35 35 · °· ° cot cot 33 31
= °· ° · °· ° (tan cot ) (tan cot ) 31 31 33 33 · °· ° … (tan cot ) 35 35
= · · 1 1 1… = 1
Hence, option (c) is correct.
20. If f x ( ) =
1
2
3 1
1
2 1 2
1
1
1 1
x
x x
x
x x
x x
x
x x
x x x ( – )
( – )
( – )( – )
( )
( )( – )
+
+
+
then what isf f f ( ) ( ) ( ) - + + 1 0 1 equal to?
(a) 0 (b) 1
(c) 100 (d) - 100
Ê
(a) f x
x x
x x x x x
x x x x x x x
( ) ( ) ( )
( ) ( )( ) ( )(
=
+
- +
- - - +
1 1
2 1 1
3 1 2 1 2 1 - 1)
f( ) - =
-
- = 1
1 1 0
2 2 0
6 12 0
0 ? f( ) 0
1 0 1
0 0 0
0 4 0
0 = =
f( ) 1
1 1 2
2 0 2
0 0 0
0 = =
?f f f ( ) ( ) ( ) - + + 1 0 1 = + + = 0 0 0 0
Hence, option (a) is correct.
21. The equation sin cos
- -
- =
1 1
6
x x
p
has
(a) no solution (b) unique solution
(c) two solutions (d) infinite number of solutions
Ê
(b)Qsin cos
- -
- =
1 1
6
x x
p
… (i)
and we know that sin cos
- -
+ =
1 1
2
x x
p
… (ii)
Adding Eqs. (i) and (ii), we get
2
6 2
1
sin
-
= + x
p p
? 2
2
3
1
sin
-
= x
p
? sin
-
=
1
3
x
p
x = = sin
p
3
3
2
Hence, the given equation has a unique solution.
Hence, option (b) is correct.
Page 4


1. The smallest positive integern for
which
1
1
1
2
-
+
?
?
?
?
?
?
=
i
i
n
wherei = -1, is
(a) 2 (b) 4 (c) 6 (d) 8
Ê
(a)
1
1
1
2
-
+
?
?
?
?
?
? =
i
i
n
, wherei = - 1
1
1
1
1
1
2
-
+
×
-
-
?
?
?
?
?
? =
i
i
i
i
n
1 2
1
1
2
2
2
+ -
-
?
?
?
?
?
?
?
?
=
i i
i
n
1 1 2
1 1
1
2
- -
+
?
?
?
?
?
? =
i
n
? ( ) ( ) - = - i i
n
2
4
? n
2
4 =
n = 2
Hence, option (a) is correct.
2. The value of x, satisfying the
equation log sin
cosx
x = 1, where
0
2
< < x
p
, is
(a)
p
12
(b)
p
3
(c)
p
4
(d)
p
6
Ê
(c)log sin
cos x
x = 1 , where 0
2
< < x
p
?(cos ) sin x x
1
= ?cos sin x x =
?tan tan tan x x = ? = 1 p / 4
? x = p / 4
Hence, option (c) is correct.
3. If ? is the value of the determinant
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
then what is the value of the
following determinant?
pa b qc
pa b qc
pa b qc
1 1 1
2 2 2
3 3 3
(p ? 0 or 1,q ? 0 or 1)
(a) p? (b)q?
(c) ( ) p q + ? (d) pq?
Ê
(d) Given,
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
= ?
?
pa b qc
pa b qc
pa b qc
1 1 1
2 2 2
3 3 3
= · · p q
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
= pq?
Hence, option (d) is correct.
4. IfC C C C
n 0 1 2
, , , ..., are the
coefficients in the expansion of
( ) 1 + x
n
, then what is the value of
C C C C
n 1 2 3
+ + + + ... ?
(a) 2
n
(b) 2 1
n
-
(c) 2
1 n -
(d) 2 2
n
-
Ê
(b)Q( ) 1
0 1 2
+ = + + x C C x C
n n n n
x C x
n
n
n 2
+ … +
and we know that
n n n n
n
n
C C C C
0 1 2
2 + + + + = ...
?
n n n
n
n n
C C C C
1 2 0
2 + + … + = -
= - 2 1
n
Hence, option (b) is correct.
5. Ifa b c + + = 4 and
ab bc ca + + = 0, then what is the
value of the following
determinant?
a b c
b c a
c a b
(a) 32 (b) - 64
(c) - 128 (d) 64
Ê
(b) Let
? =
a b c
b c a
c a b
=
+ +
+ +
+ +
a b c b c
a b c c a
a b c a b
(byC C C C
1 1 2 3
? + + )
= + + ( ) a b c
b c
c a
a b
1
1
1
(To take commona b c + + fromC
1
)
= + +
- -
- - ( ) a b c
b c c a
c a a b
a b
0
0
1
(by R R R R R R
1 1 2 2 2 3
? - ? - , )
= + + - - - - ( )[( )( ) ( ) ] a b c b c a b c a
2
= + + - - + ( )( a b c ab b ca bc
2
PAPER : I Mathematics
- - + c a ca
2 2
2 )
= - + + ( ) a b c ( ) a b c ab bc ca
2 2 2
+ + - - -
= - + + + + ( )[( ) a b c a b c
2
- + + 3( )] ab bc ca
= - - = - ( )[ ] 4 16 0 64.
6. The number of integer values ofk, for which the equation
2 2 1 sinx k = + has a solution, is
(a) zero (b) one
(c) two (d) four
Ê
(c) Given,
2 2 1 sinx k = +
Q - = = 1 1 sinx ? - = = 2 2 2 sinx
- - = - = - 2 1 2 1 2 1 sinx
- = = 3 2 1 k
-
= =
3
2
1
2
k ? - · = = · 1 5 0 5 k
?Integer values ofk = - 1 0 ,
Hence, option (c) is correct.
7. Ifa a a a
1 2 3 9
, , , ..., are in GP, then what is the value of the
following determinant?
ln ln ln
ln ln ln
ln ln ln
a a a
a a a
a a a
1 2 3
4 5 6
7 8 9
(a) 0 (b) 1
(c) 2 (d) 4
Ê
(a) Let first term and common ratio of GP area and r respectively.
?
log log log
log log log
log log log
a a a
a a a
a a a
1 2 3
4 5 6
7 8 9
=
log log log
log log log
log log log
a ar ar
ar ar ar
ar ar a
2
3 4 5
6 7
r
8
=
+ +
+ + +
log log log log log
log log log log log
a a r a r
a r a r a
2
3 4 5log
log log log log log log
r
a r a r a r + + + 6 7 8
[Qlogmn = + log log m n]
= +
+
log log log
log log log log
log log log log
a r r
a r r r
a r r r
3
6
(byC C C
2 2 1
? - andC C C
3 3 2
? - )
= 0 [QC C
2 3
= ]
8. If the roots of the quadratic equation x x k
2
2 0 + + = are
real, then
(a)k < 0 (b)k = 0
(c)k < 1 (d)k = 1
Ê
(d) Given quadratic equation,
x x k
2
2 0 + + = … (i)
Since, roots are real
? D = 0 ?b ac
2
4 0 - =
( ) ( )( ) 2 41 0
2
- = k ?4 4 = k ? k = 1
Hence, option (d) is correct.
9. Ifn = 100!, then what is the value of the following?
1 1 1
2 3 4
log log log n n n
+ + + + ...
log
1
100
n
(a) 0 (b) 1 (c) 2 (d) 3
Ê
(b)
1 1 1
2 3 100
log log log n n n
+ + … +
= + + + + log log log ... log
n n n n
2 3 4 100
= · · · … log ( )
n
2 3 4 5 100
= log ( !)
! 100
100 [ !] Qn = 100
= 1 [ log ] Q
a
a
= 1
Hence, option (b) is correct.
10. Ifz i = + 1 , wherei = -1, then what is the modulus of
z
z
+
2
?
(a) 1 (b) 2 (c) 3 (d) 4
Ê
(b) z i = + 1 , wherei = - 1
z
z
i
i
+ = + +
+
2
1
2
1
( )
( )
= + +
+
×
-
-
( )
( )
( )
( )
1
2
1
1
1
i
i
i
i
= + +
-
( )
( )
1
2 1
2
i
i
= + + - | | 1 1 i i = = | | 2 2
Hence, option (b) is correct.
11. If A and B are two matrices such that AB is of ordern n × ,
then which one of the following is correct?
(a) A andB should be square matrices of same order.
(b) Either A orB should be a square matrix.
(c) Both A andB should be of same order.
(d) Orders of A andB need not be the same.
Ê
(d) Given that, order of matrix AB n n = ×
If we take A
n p ×
and B
p n ×
, then AB will be of ordern n × .
So, orders of A and B need not be the same, is correct.
Hence, option (d) is correct.
12. How many matrices of different orders are possible with
elements comprising all prime numbers less than 30?
(a) 2 (b) 3 (c) 4 (d) 6
Ê
(c)QPrime numbers less than30 = {2, 3, 5, 7, 11, 13, 17, 19, 23,
29}
? Number of elements = 10
?Possible order of matrices with
10 elements = × 10 1 ,1 10 × ,2 5 × , 5 2 ×
?Nubmer of matrices of different order = 4
Hence, option (c) is correct.
13. Let, A
p q
r s
=
where p q r , , ands are any four different prime numbers
less than 20. What is the maximum value of the
determinant?
(a) 215 (b) 311 (c) 317 (d) 323
Ê
(c) A
p q
r s
= , prime numbers less than 20
= {2, 3, 5, 7, 11, 13, 17, 19} ? A ps rq = -
For maximum values of A, p and s must be maximum andr andq
must be minimum.
Then, p = 17,s = 19,r = 2,q = 3
? A = × - × 17 19 2 3
= - = 323 6 317
Hence, option (c) is correct.
14. IfA andB are square matrices of order 2 such that
det( ) det( ), AB BA = then which one of the following is correct?
(a) A must be a unit matrix
(b)B must be a unit matrix
(c) Both A andB must be unit matrices
(d) A andB need not be unit matrices
Ê
(d) A
2 2 ×
and B
2 2 ×
are two matrices
and| | | | AB BA = ?| || | | || | A B B A =
Let A =
?
?
?
?
?
?
1 2
3 4
,B =
-
-
?
?
?
?
?
?
2 1
3 2 1 2 / /
then,| | | | AB BA =
Hence, we can say A and B need not be the unit matrices.
Hence, option (d) is correct.
15. What is cot cot 2 4 x x - cot cot 4 6 x x - cot cot 6 2 x x equal
to?
(a) - 1 (b) 0 (c) 1 (d) 2
Ê
(c)Qcot cot( ) 6 2 4 x x x = +
cot
cot cot
cot cot
6
2 4 1
2 4
x
x x
x x
=
· -
+
Qcot ( )
cot cot
cot cot
A B
A B
A B
+ =
-
+
?
?
?
?
?
?
1
?cot cot cot cot 6 2 6 4 x x x x · + · = · - cot cot 2 4 1 x x
?cot cot cot cot 2 4 4 6 x x x x · - · - · = cot cot 6 2 1 x x
Hence, option (c) is correct.
16. If tanx = -
3
4
and x is in the second quadrant, then what
is the value of sin cos x x · ?
(a)
6
25
(b)
12
25
(c) -
6
25
(d) -
12
25
Ê
(d) Given,
tanx =
- 3
4
and x is in the 2nd quadrant.
Let perpendicular be3k and base be 4k, then
Hypotenuse = + = ( ) ( ) 3 4 5
2 2
k k k
sinx =
3
5
andcosx =
- 4
5
?sin cos x x · = ×
- ?
?
?
?
?
?
=
- 3
5
4
5
12
25
Hence, option (d) is correct.
17. What is the value of the following?
cosec
7
6
5
3
p p ?
?
?
?
?
?
?
?
?
?
?
?
sec
(a)
4
3
(b) 4 (c) - 4 (d) -
4
3
Ê
(c)cosec
7
6
5
3
p p ?
?
?
?
?
?
·
?
?
?
?
?
?
sec = +
?
?
?
?
?
?
· -
?
?
?
?
?
?
cosec p
p
p
p
6
2
3
sec
= - · cosec
p p
6 3
sec = - × 2 2 = - 4
Hence, option (c) is correct.
18. If the determinant
x
x
1 3
0 0 1
1 4
0 =
then what is x equal to?
(a) - 2 or 2 (b) - 3 or 3 (c) - 1 or 1 (d) 3 or 4
Ê
(c) Given,
x
x
1 3
0 0 1
1 4
0 = … (i)
- - = 1 1 0
2
( ) x ? x
2
1 0 - = ? x
2
1 =
? x = ± 1 ?1 0
2
- = x
? x
2
1 =
x = + - 1 1 ,
Hence, option (c) is correct.
19. What is the value of the following?
tan tan tan ...tan tan 31 33 35 57 59 ° ° ° ° °
(a) - 1 (b) 0 (c) 1 (d) 2
Ê
(c) tan º tan º tan º tan º tan º 31 33 35 57 59 · · … ·
= · °· … ° ° - ° tan º tan tan º tan tan( ) 31 33 35 45 90 33 x xK
· ° - tan( º) 90 31
= °· °· °… ° tan tan tan cot 31 33 35 35 · °· ° cot cot 33 31
= °· ° · °· ° (tan cot ) (tan cot ) 31 31 33 33 · °· ° … (tan cot ) 35 35
= · · 1 1 1… = 1
Hence, option (c) is correct.
20. If f x ( ) =
1
2
3 1
1
2 1 2
1
1
1 1
x
x x
x
x x
x x
x
x x
x x x ( – )
( – )
( – )( – )
( )
( )( – )
+
+
+
then what isf f f ( ) ( ) ( ) - + + 1 0 1 equal to?
(a) 0 (b) 1
(c) 100 (d) - 100
Ê
(a) f x
x x
x x x x x
x x x x x x x
( ) ( ) ( )
( ) ( )( ) ( )(
=
+
- +
- - - +
1 1
2 1 1
3 1 2 1 2 1 - 1)
f( ) - =
-
- = 1
1 1 0
2 2 0
6 12 0
0 ? f( ) 0
1 0 1
0 0 0
0 4 0
0 = =
f( ) 1
1 1 2
2 0 2
0 0 0
0 = =
?f f f ( ) ( ) ( ) - + + 1 0 1 = + + = 0 0 0 0
Hence, option (a) is correct.
21. The equation sin cos
- -
- =
1 1
6
x x
p
has
(a) no solution (b) unique solution
(c) two solutions (d) infinite number of solutions
Ê
(b)Qsin cos
- -
- =
1 1
6
x x
p
… (i)
and we know that sin cos
- -
+ =
1 1
2
x x
p
… (ii)
Adding Eqs. (i) and (ii), we get
2
6 2
1
sin
-
= + x
p p
? 2
2
3
1
sin
-
= x
p
? sin
-
=
1
3
x
p
x = = sin
p
3
3
2
Hence, the given equation has a unique solution.
Hence, option (b) is correct.
22. What is the value of the following?
(sin cos ) (sin cos ) 24 66 24 66 ° + ° ° - °
(a) - 1 (b) 0 (c) 1 (d) 2
Ê
(b)(sin º cos º)(sin º cos º) 24 66 24 66 + -
= + (sin º cos º) 24 66
{sin( º) cos º} 90 66 66 ° - -
[Qsin( ) cos 90° - = ? ?]
= + - (sin º cos º)(cos º cos º) 24 66 66 66
= + (sin º cos º)( ) 24 66 0 = 0
Hence, option (b) is correct.
23. A chord subtends an angle 120° at
the centre of a unit circle. What is
the length of the chord?
(a) 2 1 - units (b) 3 1 - units
(c) 2 units (d) 3 units
Ê
(d) Given, radius of the circle = 1unit
? = AOB 120º
By using cosine rule,
cos º 120
2
2 2 2
=
+ -
· ·
OA OB AB
OA OB
… (i)
Let AB x = unit,OA = 1unit,OB = 1unit
From Eq. (i),
-
=
+ -
· ·
1
2
1 1
2 1 1
2
x
? - = - 1 2
2
x
? x
2
3 = ? x = 3 unit
Hence, option (d) is correct.
24. What is ( cot 1 + - ? ?) cosec
( tan sec ) 1 + + ? ? equal to?
(a) 1 (b) 2 (c) 3 (d) 4
Ê
(b)( cot ) 1 + - ? ? cosec
( tan sec ) 1 + + ? ?
= + -
?
?
?
?
?
?
1
1 cos
sin sin
?
? ?
1
1
+ +
?
?
?
?
?
?
sin
cos cos
?
? ?
=
+ - ?
?
?
?
?
?
+ + ?
?
?
?
?
?
sin cos
sin
sin cos
cos
? ?
?
? ?
?
1 1
=
+ -
·
(sin cos
sin cos
? ?)
? ?
2 2
1
=
+ + sin cos sin cos –
sin cos
2 2
2 1 ? ? ? ?
? ?
=
+ · -
·
1 2 1 sin cos
sin cos
? ?
? ?
= 2
Hence, option (b) is correct.
25. What is
1
1
1
1
2
2
2
+
+
-
-
-
?
?
?
?
?
?
tan
cot
tan
cot
?
?
?
?
equal to?
(a) 0 (b) 1
(c) 2 tan? (d) 2cot?
Ê
(a)
1
1
1
1
2
2
2
+
+
-
-
-
?
?
?
?
?
?
tan
cot
tan
cot
?
?
?
?
=
+
+
-
-
-
?
?
?
?
?
?
?
?
?
?
1
1
1
1
1
1
2
2
2
tan
tan
tan
tan
?
?
?
?
=
+
+
?
?
?
?
?
? tan
tan
tan
2
2
2
1
1
?
?
?
-
-
-
?
?
?
?
?
?
tan ( tan )
tan
? ?
?
1
1
2
= - = tan tan
2 2
0 ? ?
Hence, option (a) is correct.
26. What is the interior angle of a
regular octagon of side length 2 cm?
(a)
p
2
(b)
3
4
p
(c)
3
5
p
(d)
3
8
p
Ê
(b) Given, length of side of regular
octagon = 2 cm
QSum of interior angles of octagon
= - × ( ) º 8 2 180
= × ° 6 180
[Qsum of interior angles of polygon
= - × ( ) º n 2 180 ]
?Interior angle =
× 6 180
8
º
= = 135
3
4
º
p
Hence, option (b) is correct.
27. If 7 24 25 sin cos ? ? + = , then what is
the value of (sin cos ? ?) + ?
(a) 1 (b)
26
25
(c)
6
5
(d)
31
25
Ê
(d) Given,7 24 25 sin cos ? ? + =
Since, we know that if
a b c sin cos ? ? + =
then b a a b c sin cos ? ? - = + -
2 2 2
Q 7 24 25 sin cos ? ? + = … (i)
?24 7 7 24 25
2 2 2
sin cos ? ? - = + -
24 7 0 sin cos ? ? - = … (ii)
Eq. (i) × 7 + Eq. (ii) × 24
49 168 175 sin cos ? ? + =
576 168 0 sin cos ? ? - =
625 175 sin? =
sin? = =
175
625
7
25
?cos? = -
?
?
?
?
?
?
= 1
7
25
24
25
2
?sin cos ? ? + = +
7
25
24
25
=
31
25
Hence, option (d) is correct.
28. A ladder 6 m long reaches a point
6 m below the top of a vertical
flagstaff. From the foot of the
ladder, the elevation of the top of
the flagstaff is 75°. What is the
height of the flagstaff ?
(a) 12 m (b) 9 m
(c) ( ) 6 3 + m (d) ( ) 6 3 3 + m
Ê
(d) Let AC be a vertical flagstaff.
? CD = 6 m, BD = 6 m
? = ° CBD 75
Let AD h = meter
In ?ABC
90 75 180 + + ? = ° C [Q sum of interior
angle of triangle is 180°]
? = ° C 15
In ?BCD,
BD CD = ? ? = ? = BCD CBD 15º
? ? = - = ABD 75 15 60 º º º
In ?ABD, sin º 60
6
=
h
?
3
2 6
=
h
h = 3 3 m
?Height of the flagstaff = + ( ) h 6 m
= + ( ) 3 3 6 m
Hence, option (d) is correct.
29. The shadow of a tower is found to
be x metre longer, when the angle
of elevation of the sun changes
from 60° to 45°. If the height of the
tower is 5 3 3 ( ) + m, then what is x
equal to?
(a) 8 m (b) 10 m
(c) 12 m (d) 15 m
Ê
(b) In the given diagram,
AB represents the position of tower,
whereh = + 5 3 3 ( )m
CD x = m
In ?ABC,
tan º
( )
60
5 3 3
=
+
BC
? 3
5 3 3
=
+ ( )
BC
D
A
C
6m
60°
75°
15°
6m
h
B
120°
1 1
O
A B
D C B
A
x y
5(3+ 3) v m
45° 60°
Page 5


1. The smallest positive integern for
which
1
1
1
2
-
+
?
?
?
?
?
?
=
i
i
n
wherei = -1, is
(a) 2 (b) 4 (c) 6 (d) 8
Ê
(a)
1
1
1
2
-
+
?
?
?
?
?
? =
i
i
n
, wherei = - 1
1
1
1
1
1
2
-
+
×
-
-
?
?
?
?
?
? =
i
i
i
i
n
1 2
1
1
2
2
2
+ -
-
?
?
?
?
?
?
?
?
=
i i
i
n
1 1 2
1 1
1
2
- -
+
?
?
?
?
?
? =
i
n
? ( ) ( ) - = - i i
n
2
4
? n
2
4 =
n = 2
Hence, option (a) is correct.
2. The value of x, satisfying the
equation log sin
cosx
x = 1, where
0
2
< < x
p
, is
(a)
p
12
(b)
p
3
(c)
p
4
(d)
p
6
Ê
(c)log sin
cos x
x = 1 , where 0
2
< < x
p
?(cos ) sin x x
1
= ?cos sin x x =
?tan tan tan x x = ? = 1 p / 4
? x = p / 4
Hence, option (c) is correct.
3. If ? is the value of the determinant
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
then what is the value of the
following determinant?
pa b qc
pa b qc
pa b qc
1 1 1
2 2 2
3 3 3
(p ? 0 or 1,q ? 0 or 1)
(a) p? (b)q?
(c) ( ) p q + ? (d) pq?
Ê
(d) Given,
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
= ?
?
pa b qc
pa b qc
pa b qc
1 1 1
2 2 2
3 3 3
= · · p q
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
= pq?
Hence, option (d) is correct.
4. IfC C C C
n 0 1 2
, , , ..., are the
coefficients in the expansion of
( ) 1 + x
n
, then what is the value of
C C C C
n 1 2 3
+ + + + ... ?
(a) 2
n
(b) 2 1
n
-
(c) 2
1 n -
(d) 2 2
n
-
Ê
(b)Q( ) 1
0 1 2
+ = + + x C C x C
n n n n
x C x
n
n
n 2
+ … +
and we know that
n n n n
n
n
C C C C
0 1 2
2 + + + + = ...
?
n n n
n
n n
C C C C
1 2 0
2 + + … + = -
= - 2 1
n
Hence, option (b) is correct.
5. Ifa b c + + = 4 and
ab bc ca + + = 0, then what is the
value of the following
determinant?
a b c
b c a
c a b
(a) 32 (b) - 64
(c) - 128 (d) 64
Ê
(b) Let
? =
a b c
b c a
c a b
=
+ +
+ +
+ +
a b c b c
a b c c a
a b c a b
(byC C C C
1 1 2 3
? + + )
= + + ( ) a b c
b c
c a
a b
1
1
1
(To take commona b c + + fromC
1
)
= + +
- -
- - ( ) a b c
b c c a
c a a b
a b
0
0
1
(by R R R R R R
1 1 2 2 2 3
? - ? - , )
= + + - - - - ( )[( )( ) ( ) ] a b c b c a b c a
2
= + + - - + ( )( a b c ab b ca bc
2
PAPER : I Mathematics
- - + c a ca
2 2
2 )
= - + + ( ) a b c ( ) a b c ab bc ca
2 2 2
+ + - - -
= - + + + + ( )[( ) a b c a b c
2
- + + 3( )] ab bc ca
= - - = - ( )[ ] 4 16 0 64.
6. The number of integer values ofk, for which the equation
2 2 1 sinx k = + has a solution, is
(a) zero (b) one
(c) two (d) four
Ê
(c) Given,
2 2 1 sinx k = +
Q - = = 1 1 sinx ? - = = 2 2 2 sinx
- - = - = - 2 1 2 1 2 1 sinx
- = = 3 2 1 k
-
= =
3
2
1
2
k ? - · = = · 1 5 0 5 k
?Integer values ofk = - 1 0 ,
Hence, option (c) is correct.
7. Ifa a a a
1 2 3 9
, , , ..., are in GP, then what is the value of the
following determinant?
ln ln ln
ln ln ln
ln ln ln
a a a
a a a
a a a
1 2 3
4 5 6
7 8 9
(a) 0 (b) 1
(c) 2 (d) 4
Ê
(a) Let first term and common ratio of GP area and r respectively.
?
log log log
log log log
log log log
a a a
a a a
a a a
1 2 3
4 5 6
7 8 9
=
log log log
log log log
log log log
a ar ar
ar ar ar
ar ar a
2
3 4 5
6 7
r
8
=
+ +
+ + +
log log log log log
log log log log log
a a r a r
a r a r a
2
3 4 5log
log log log log log log
r
a r a r a r + + + 6 7 8
[Qlogmn = + log log m n]
= +
+
log log log
log log log log
log log log log
a r r
a r r r
a r r r
3
6
(byC C C
2 2 1
? - andC C C
3 3 2
? - )
= 0 [QC C
2 3
= ]
8. If the roots of the quadratic equation x x k
2
2 0 + + = are
real, then
(a)k < 0 (b)k = 0
(c)k < 1 (d)k = 1
Ê
(d) Given quadratic equation,
x x k
2
2 0 + + = … (i)
Since, roots are real
? D = 0 ?b ac
2
4 0 - =
( ) ( )( ) 2 41 0
2
- = k ?4 4 = k ? k = 1
Hence, option (d) is correct.
9. Ifn = 100!, then what is the value of the following?
1 1 1
2 3 4
log log log n n n
+ + + + ...
log
1
100
n
(a) 0 (b) 1 (c) 2 (d) 3
Ê
(b)
1 1 1
2 3 100
log log log n n n
+ + … +
= + + + + log log log ... log
n n n n
2 3 4 100
= · · · … log ( )
n
2 3 4 5 100
= log ( !)
! 100
100 [ !] Qn = 100
= 1 [ log ] Q
a
a
= 1
Hence, option (b) is correct.
10. Ifz i = + 1 , wherei = -1, then what is the modulus of
z
z
+
2
?
(a) 1 (b) 2 (c) 3 (d) 4
Ê
(b) z i = + 1 , wherei = - 1
z
z
i
i
+ = + +
+
2
1
2
1
( )
( )
= + +
+
×
-
-
( )
( )
( )
( )
1
2
1
1
1
i
i
i
i
= + +
-
( )
( )
1
2 1
2
i
i
= + + - | | 1 1 i i = = | | 2 2
Hence, option (b) is correct.
11. If A and B are two matrices such that AB is of ordern n × ,
then which one of the following is correct?
(a) A andB should be square matrices of same order.
(b) Either A orB should be a square matrix.
(c) Both A andB should be of same order.
(d) Orders of A andB need not be the same.
Ê
(d) Given that, order of matrix AB n n = ×
If we take A
n p ×
and B
p n ×
, then AB will be of ordern n × .
So, orders of A and B need not be the same, is correct.
Hence, option (d) is correct.
12. How many matrices of different orders are possible with
elements comprising all prime numbers less than 30?
(a) 2 (b) 3 (c) 4 (d) 6
Ê
(c)QPrime numbers less than30 = {2, 3, 5, 7, 11, 13, 17, 19, 23,
29}
? Number of elements = 10
?Possible order of matrices with
10 elements = × 10 1 ,1 10 × ,2 5 × , 5 2 ×
?Nubmer of matrices of different order = 4
Hence, option (c) is correct.
13. Let, A
p q
r s
=
where p q r , , ands are any four different prime numbers
less than 20. What is the maximum value of the
determinant?
(a) 215 (b) 311 (c) 317 (d) 323
Ê
(c) A
p q
r s
= , prime numbers less than 20
= {2, 3, 5, 7, 11, 13, 17, 19} ? A ps rq = -
For maximum values of A, p and s must be maximum andr andq
must be minimum.
Then, p = 17,s = 19,r = 2,q = 3
? A = × - × 17 19 2 3
= - = 323 6 317
Hence, option (c) is correct.
14. IfA andB are square matrices of order 2 such that
det( ) det( ), AB BA = then which one of the following is correct?
(a) A must be a unit matrix
(b)B must be a unit matrix
(c) Both A andB must be unit matrices
(d) A andB need not be unit matrices
Ê
(d) A
2 2 ×
and B
2 2 ×
are two matrices
and| | | | AB BA = ?| || | | || | A B B A =
Let A =
?
?
?
?
?
?
1 2
3 4
,B =
-
-
?
?
?
?
?
?
2 1
3 2 1 2 / /
then,| | | | AB BA =
Hence, we can say A and B need not be the unit matrices.
Hence, option (d) is correct.
15. What is cot cot 2 4 x x - cot cot 4 6 x x - cot cot 6 2 x x equal
to?
(a) - 1 (b) 0 (c) 1 (d) 2
Ê
(c)Qcot cot( ) 6 2 4 x x x = +
cot
cot cot
cot cot
6
2 4 1
2 4
x
x x
x x
=
· -
+
Qcot ( )
cot cot
cot cot
A B
A B
A B
+ =
-
+
?
?
?
?
?
?
1
?cot cot cot cot 6 2 6 4 x x x x · + · = · - cot cot 2 4 1 x x
?cot cot cot cot 2 4 4 6 x x x x · - · - · = cot cot 6 2 1 x x
Hence, option (c) is correct.
16. If tanx = -
3
4
and x is in the second quadrant, then what
is the value of sin cos x x · ?
(a)
6
25
(b)
12
25
(c) -
6
25
(d) -
12
25
Ê
(d) Given,
tanx =
- 3
4
and x is in the 2nd quadrant.
Let perpendicular be3k and base be 4k, then
Hypotenuse = + = ( ) ( ) 3 4 5
2 2
k k k
sinx =
3
5
andcosx =
- 4
5
?sin cos x x · = ×
- ?
?
?
?
?
?
=
- 3
5
4
5
12
25
Hence, option (d) is correct.
17. What is the value of the following?
cosec
7
6
5
3
p p ?
?
?
?
?
?
?
?
?
?
?
?
sec
(a)
4
3
(b) 4 (c) - 4 (d) -
4
3
Ê
(c)cosec
7
6
5
3
p p ?
?
?
?
?
?
·
?
?
?
?
?
?
sec = +
?
?
?
?
?
?
· -
?
?
?
?
?
?
cosec p
p
p
p
6
2
3
sec
= - · cosec
p p
6 3
sec = - × 2 2 = - 4
Hence, option (c) is correct.
18. If the determinant
x
x
1 3
0 0 1
1 4
0 =
then what is x equal to?
(a) - 2 or 2 (b) - 3 or 3 (c) - 1 or 1 (d) 3 or 4
Ê
(c) Given,
x
x
1 3
0 0 1
1 4
0 = … (i)
- - = 1 1 0
2
( ) x ? x
2
1 0 - = ? x
2
1 =
? x = ± 1 ?1 0
2
- = x
? x
2
1 =
x = + - 1 1 ,
Hence, option (c) is correct.
19. What is the value of the following?
tan tan tan ...tan tan 31 33 35 57 59 ° ° ° ° °
(a) - 1 (b) 0 (c) 1 (d) 2
Ê
(c) tan º tan º tan º tan º tan º 31 33 35 57 59 · · … ·
= · °· … ° ° - ° tan º tan tan º tan tan( ) 31 33 35 45 90 33 x xK
· ° - tan( º) 90 31
= °· °· °… ° tan tan tan cot 31 33 35 35 · °· ° cot cot 33 31
= °· ° · °· ° (tan cot ) (tan cot ) 31 31 33 33 · °· ° … (tan cot ) 35 35
= · · 1 1 1… = 1
Hence, option (c) is correct.
20. If f x ( ) =
1
2
3 1
1
2 1 2
1
1
1 1
x
x x
x
x x
x x
x
x x
x x x ( – )
( – )
( – )( – )
( )
( )( – )
+
+
+
then what isf f f ( ) ( ) ( ) - + + 1 0 1 equal to?
(a) 0 (b) 1
(c) 100 (d) - 100
Ê
(a) f x
x x
x x x x x
x x x x x x x
( ) ( ) ( )
( ) ( )( ) ( )(
=
+
- +
- - - +
1 1
2 1 1
3 1 2 1 2 1 - 1)
f( ) - =
-
- = 1
1 1 0
2 2 0
6 12 0
0 ? f( ) 0
1 0 1
0 0 0
0 4 0
0 = =
f( ) 1
1 1 2
2 0 2
0 0 0
0 = =
?f f f ( ) ( ) ( ) - + + 1 0 1 = + + = 0 0 0 0
Hence, option (a) is correct.
21. The equation sin cos
- -
- =
1 1
6
x x
p
has
(a) no solution (b) unique solution
(c) two solutions (d) infinite number of solutions
Ê
(b)Qsin cos
- -
- =
1 1
6
x x
p
… (i)
and we know that sin cos
- -
+ =
1 1
2
x x
p
… (ii)
Adding Eqs. (i) and (ii), we get
2
6 2
1
sin
-
= + x
p p
? 2
2
3
1
sin
-
= x
p
? sin
-
=
1
3
x
p
x = = sin
p
3
3
2
Hence, the given equation has a unique solution.
Hence, option (b) is correct.
22. What is the value of the following?
(sin cos ) (sin cos ) 24 66 24 66 ° + ° ° - °
(a) - 1 (b) 0 (c) 1 (d) 2
Ê
(b)(sin º cos º)(sin º cos º) 24 66 24 66 + -
= + (sin º cos º) 24 66
{sin( º) cos º} 90 66 66 ° - -
[Qsin( ) cos 90° - = ? ?]
= + - (sin º cos º)(cos º cos º) 24 66 66 66
= + (sin º cos º)( ) 24 66 0 = 0
Hence, option (b) is correct.
23. A chord subtends an angle 120° at
the centre of a unit circle. What is
the length of the chord?
(a) 2 1 - units (b) 3 1 - units
(c) 2 units (d) 3 units
Ê
(d) Given, radius of the circle = 1unit
? = AOB 120º
By using cosine rule,
cos º 120
2
2 2 2
=
+ -
· ·
OA OB AB
OA OB
… (i)
Let AB x = unit,OA = 1unit,OB = 1unit
From Eq. (i),
-
=
+ -
· ·
1
2
1 1
2 1 1
2
x
? - = - 1 2
2
x
? x
2
3 = ? x = 3 unit
Hence, option (d) is correct.
24. What is ( cot 1 + - ? ?) cosec
( tan sec ) 1 + + ? ? equal to?
(a) 1 (b) 2 (c) 3 (d) 4
Ê
(b)( cot ) 1 + - ? ? cosec
( tan sec ) 1 + + ? ?
= + -
?
?
?
?
?
?
1
1 cos
sin sin
?
? ?
1
1
+ +
?
?
?
?
?
?
sin
cos cos
?
? ?
=
+ - ?
?
?
?
?
?
+ + ?
?
?
?
?
?
sin cos
sin
sin cos
cos
? ?
?
? ?
?
1 1
=
+ -
·
(sin cos
sin cos
? ?)
? ?
2 2
1
=
+ + sin cos sin cos –
sin cos
2 2
2 1 ? ? ? ?
? ?
=
+ · -
·
1 2 1 sin cos
sin cos
? ?
? ?
= 2
Hence, option (b) is correct.
25. What is
1
1
1
1
2
2
2
+
+
-
-
-
?
?
?
?
?
?
tan
cot
tan
cot
?
?
?
?
equal to?
(a) 0 (b) 1
(c) 2 tan? (d) 2cot?
Ê
(a)
1
1
1
1
2
2
2
+
+
-
-
-
?
?
?
?
?
?
tan
cot
tan
cot
?
?
?
?
=
+
+
-
-
-
?
?
?
?
?
?
?
?
?
?
1
1
1
1
1
1
2
2
2
tan
tan
tan
tan
?
?
?
?
=
+
+
?
?
?
?
?
? tan
tan
tan
2
2
2
1
1
?
?
?
-
-
-
?
?
?
?
?
?
tan ( tan )
tan
? ?
?
1
1
2
= - = tan tan
2 2
0 ? ?
Hence, option (a) is correct.
26. What is the interior angle of a
regular octagon of side length 2 cm?
(a)
p
2
(b)
3
4
p
(c)
3
5
p
(d)
3
8
p
Ê
(b) Given, length of side of regular
octagon = 2 cm
QSum of interior angles of octagon
= - × ( ) º 8 2 180
= × ° 6 180
[Qsum of interior angles of polygon
= - × ( ) º n 2 180 ]
?Interior angle =
× 6 180
8
º
= = 135
3
4
º
p
Hence, option (b) is correct.
27. If 7 24 25 sin cos ? ? + = , then what is
the value of (sin cos ? ?) + ?
(a) 1 (b)
26
25
(c)
6
5
(d)
31
25
Ê
(d) Given,7 24 25 sin cos ? ? + =
Since, we know that if
a b c sin cos ? ? + =
then b a a b c sin cos ? ? - = + -
2 2 2
Q 7 24 25 sin cos ? ? + = … (i)
?24 7 7 24 25
2 2 2
sin cos ? ? - = + -
24 7 0 sin cos ? ? - = … (ii)
Eq. (i) × 7 + Eq. (ii) × 24
49 168 175 sin cos ? ? + =
576 168 0 sin cos ? ? - =
625 175 sin? =
sin? = =
175
625
7
25
?cos? = -
?
?
?
?
?
?
= 1
7
25
24
25
2
?sin cos ? ? + = +
7
25
24
25
=
31
25
Hence, option (d) is correct.
28. A ladder 6 m long reaches a point
6 m below the top of a vertical
flagstaff. From the foot of the
ladder, the elevation of the top of
the flagstaff is 75°. What is the
height of the flagstaff ?
(a) 12 m (b) 9 m
(c) ( ) 6 3 + m (d) ( ) 6 3 3 + m
Ê
(d) Let AC be a vertical flagstaff.
? CD = 6 m, BD = 6 m
? = ° CBD 75
Let AD h = meter
In ?ABC
90 75 180 + + ? = ° C [Q sum of interior
angle of triangle is 180°]
? = ° C 15
In ?BCD,
BD CD = ? ? = ? = BCD CBD 15º
? ? = - = ABD 75 15 60 º º º
In ?ABD, sin º 60
6
=
h
?
3
2 6
=
h
h = 3 3 m
?Height of the flagstaff = + ( ) h 6 m
= + ( ) 3 3 6 m
Hence, option (d) is correct.
29. The shadow of a tower is found to
be x metre longer, when the angle
of elevation of the sun changes
from 60° to 45°. If the height of the
tower is 5 3 3 ( ) + m, then what is x
equal to?
(a) 8 m (b) 10 m
(c) 12 m (d) 15 m
Ê
(b) In the given diagram,
AB represents the position of tower,
whereh = + 5 3 3 ( )m
CD x = m
In ?ABC,
tan º
( )
60
5 3 3
=
+
BC
? 3
5 3 3
=
+ ( )
BC
D
A
C
6m
60°
75°
15°
6m
h
B
120°
1 1
O
A B
D C B
A
x y
5(3+ 3) v m
45° 60°
? BC = + 5 3 1 ( ) m
In ?ABD,
tan º
( )
45
5 3 3
=
+
BD
? 1
5 3 3
=
+ ( )
BD
? BD = + 5 3 3 ( ) m
Since, x BD BC = -
x = + - + 5 3 3 5 3 1 ( ) ( )
x = + - - 5 3 3 3 1 ( )
x = 10 m
Hence, option (b) is correct.
30. If 3 4 cos sin ? ? = , then what is the
value of tan( ) 45° + ? ?
(a) 10 (b) 7 (c)
7
2
(d)
7
4
Ê
(b) If 3 4 cos sin ? ? =
?
3
4
=
sin
cos
?
?
? tan? =
3
4
?tan(
tan º tan
tan tan
45
45
1 45
° + =
+
- °·
?)
?
?
=
+
- ×
1
3
4
1 1
3
4
=
+
-
=
4 3
4 3
7
Hence, option (b) is correct.
31. tan cot
- -
+ =
1 1
2
x x
p
holds, when
(a)x R ?
(b)x R ? - - ( , ) 1 1 only
(c)x R ? - { } 0 only
(d)x R ? - - [ , ] 1 1 only
Ê
(a) Since, tan cot
- -
+ =
1 1
2
x x
p
for all x R ? .
Hence, option (a) is correct.
32. If tanA =
1
7
, then what is cos2A
equal to?
(a)
24
25
(b)
18
25
(c)
12
25
(d)
6
25
Ê
(a) tanA =
1
7
?cos
tan
tan
( / )
( / )
2
1
1
1 1 7
1 1 7
2
2
2
2
A
A
A
=
-
+
=
-
+
=
-
+
=
49 1
49 1
48
50
cos2
24
25
A =
Hence, option (a) is correct.
33. The sides of a triangle arem n , and
m n mn
2 2
+ + . What is the sum
of the acute angles of the triangle?
(a) 45° (b) 60° (c) 75° (d) 90°
Ê
(b) Let AB m = , AC n =
BC m n mn = + +
2 2
By using cosine rule,
cosA
AB AC BC
AB AC
=
+ -
·
2 2 2
2
?cosA
m n m n mn
mn
=
+ - - -
2 2 2 2
2
?cosA =
- 1
2
? A = 120º
?? + ? = - ? B C A 180
[Q sum of interior angle is 180°]
= - 180 120 º º
? + ? = B C 60º
Hence, option (b) is correct.
34. What is the area of the triangle
ABC with sidesa = 10cm,c = 4cm
and angle B = ° 30 ?
(a) 16 cm
2
(b) 12 cm
2
(c) 10 cm
2
(d) 8 cm
2
Ê
(c) Given, a = 10 cm
c = 4 cm
? = B 30º
QArea of triangle = ?
1
2
ac B sin( )
= × × ×
1
2
10 4 30 sin º = × ×
1
2
40
1
2
= 10 sq cm
Hence, option (c) is correct.
35. Consider the following statements
1. A = { , , } 1 3 5 and B = { , , } 2 4 7 are
equivalent sets.
2. A = { , , } 1 5 9 and B = { , , , , } 1 5 5 9 9
are equal sets
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
(c) A = { , , } 1 3 5 and B = { , , } 2 4 7
Since, number of elements are same in
both the sets.
? A and B are equivalent sets.
If A = { , , } 1 5 9 ,B = { , , , , } 1 5 5 9 9
Which is nothing butB = { , , } 1 5 9
Since, elements are same in A andB
? A and B are equal sets
Hence, option (c) is correct.
36. Consider the following statements
1. The null set is a subset of every
set.
2. Every set is a subset of itself.
3. If a set has 10 elements, then its
power set will have 1024
elements.
Which of the above statements are
correct?
(a) 1 and 2 only (b) 2 and 3 only
(c) 1 and 3 only (d) 1, 2 and 3
Ê
(d) Since we know that null set is a
subset of every set and every set is a
subset of itself.
If n A ( ) = 10
?n P A ( ( )) = = 2 1024
10
?all the given statements are true.
Hence, option (d) is correct.
37. Let R be a relation defined as xRy if
and only if 2 3 20 x y + = , where
x y N , ? . How many elements of
the form ( , ) x y are there in R ?
(a) 2 (b) 3
(c) 4 (d) 6
Ê
(b)QxRy x y ? + = 2 3 20
where, x y , ?¥
Q y
x
=
- 20 2
3
All ordered pair which satisfies the given
relations are (1, 6), (4, 4), (7, 2).
? R = {( , ),( , ),( , )} 1 6 4 4 7 2
? n R ( ) = 3
Hence, option (b) is correct.
38. Consider the following statements
1. A function f :¢ ¢ ? , defined by
f x x ( ) = + 1, is one-one as well
as onto.
2. A function f :¥ ¥ ? , defined by
f x x ( ) = + 1, is one-one but not
onto.
Which of the above statement(s)
is/are correct?
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
A
B C
m n
vm n mn
2 2
+ +
A
B C
30°
c b
a