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

 Page 1


1. If x x
2
1 0 + + = , then what is the
value of x x x
199 200 201
+ + ?
(a) - 1 (b) 0
(c) 1 (d) 3
Ê
(b) Given that,
x x
2
1 0 + + = ...(i)
? x x x
199 200 201
+ + = + + x x x
199 2
1 ( )
= × x
199
0
= 0
2. If x y z , , are in GP, then which of
the following is/are correct?
1. ln( ), ln( ), ln( ) 3 3 3 x y z are in AP.
2. xyz x + ln( ), xyz y + ln( ),
xyz z + ln( ) are in HP.
Select the correct answer using the
code given below.
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
Ê
(a) Given thatx, y, z are in GP.
? y xz
2
= ...(i)
(1) If log( ) 3x , log( ) 3y , log( ) 3z are in AP
Then, 2 3 3 3 log( ) log( ) log( ) y x z = +
9 9
2
y xz = ( )
9 9
2
y xz = ( )
y xz
2
=
Hence, statement (1) is correct.
Hence, we can say ifx, y, z are in GP.
?log x, logy, log z are in AP.
? xyz x + log ,xyz y + log ,xyz z + log
are in AP.
Hence, Statement (2) is wrong.
?Option (a) is correct.
3. If log
10
2, log ( )
10
2 1
x
- , log ( )
10
2 3
x
+
are in AP, then what is x equal to?
(a) 0 (b) 1
(c) log
2
5 (d) log
5
2
Ê
(c) Given that, log
10
2, log ( )
10
2 1
x
- ,
log ( )
10
2 3
x
+ are in AP.
?2 2 1 2 2 3
10 10 10
log ( ) log log ( )
x x
- = + +
log ( ) log ( )
10
2
10
2 1 2 2 3
x x
- = +
? 2 1 2 2 2 2 6
2x x x
+ - · = · +
? ( ) ( ) 2 4 2 5 0
2 x x
- - =
Let 2
x
y =
? y y
2
4 5 0 - - =
( ) ( ) y y - + = 5 1 0
? y = 5 or y = - 1
(Ignore because 2
x
cannot be negative)
? y = 5 ? 2 5
x
=
x = log
2
5
Hence, option (c) is correct.
4. LetS = {2, 3, 4, 5, 6, 7, 9}. How many
different 3-digit numbers (with all
digits different) from S can be made
which are less than 500?
(a) 30 (b) 49
(c) 90 (d) 147
Ê
(c) Let S = {2, 3, 4, 5, 6, 7, 9}
? nS ( ) = 7
Three digit number less than
= × × = 3 6 5 90
?Option (c) is correct.
Note Hundreds digit can be filled with
3 choices that are 2, 3, 4.
Similarly, tens digit can be filled
with 6 ways and unit digit can be
filled with 5 ways.
5. If p = (1111 … up ton digits), then
what is the value of 9
2
p p + ?
(a) 10
n
p (b) 2 10 p
n
·
(c) 10 1
n
p - (d) 10 1
n
p +
Ê
(a) Given that,
p = (1111... upton digits)
= + + + +
-
1 10 10 10
2 1
K
n
=
-
-
1 10 1
10 1
( )
n
Q K a ar ar ar
a r
r
n
n
+ + + + =
-
-
?
?
?
?
?
?
- 2 1
1
1
( )
? p
n
=
- 10 1
9
? 9 10 1 p
n
= -
? 9 1 10 p
n
+ =
? 9 10
2
p p p
n
+ = ·
? Hence, option (a) is correct.
6. The quadratic equation
3 5 3 5 0
2 2 2
x k k x k k - + + - = ( ) has
real roots of equal magnitude and
opposite sign. Which one of the
following is correct?
(a) 0
5
3
< < k
(b) 0
3
5
< < k only
(c)
3
5
5
3
< < k
(d) No such value of k exists.
Ê
(d) Since, we know that if a quadratic
equationax bx c
2
0 + + = has real
roots of equal magnitude and opposite
sign.
Then, b = 0 ...(i)
and product of roots < 0 ...(ii)
PAPER : I Mathematics
500 =
3 6 5
Page 2


1. If x x
2
1 0 + + = , then what is the
value of x x x
199 200 201
+ + ?
(a) - 1 (b) 0
(c) 1 (d) 3
Ê
(b) Given that,
x x
2
1 0 + + = ...(i)
? x x x
199 200 201
+ + = + + x x x
199 2
1 ( )
= × x
199
0
= 0
2. If x y z , , are in GP, then which of
the following is/are correct?
1. ln( ), ln( ), ln( ) 3 3 3 x y z are in AP.
2. xyz x + ln( ), xyz y + ln( ),
xyz z + ln( ) are in HP.
Select the correct answer using the
code given below.
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
Ê
(a) Given thatx, y, z are in GP.
? y xz
2
= ...(i)
(1) If log( ) 3x , log( ) 3y , log( ) 3z are in AP
Then, 2 3 3 3 log( ) log( ) log( ) y x z = +
9 9
2
y xz = ( )
9 9
2
y xz = ( )
y xz
2
=
Hence, statement (1) is correct.
Hence, we can say ifx, y, z are in GP.
?log x, logy, log z are in AP.
? xyz x + log ,xyz y + log ,xyz z + log
are in AP.
Hence, Statement (2) is wrong.
?Option (a) is correct.
3. If log
10
2, log ( )
10
2 1
x
- , log ( )
10
2 3
x
+
are in AP, then what is x equal to?
(a) 0 (b) 1
(c) log
2
5 (d) log
5
2
Ê
(c) Given that, log
10
2, log ( )
10
2 1
x
- ,
log ( )
10
2 3
x
+ are in AP.
?2 2 1 2 2 3
10 10 10
log ( ) log log ( )
x x
- = + +
log ( ) log ( )
10
2
10
2 1 2 2 3
x x
- = +
? 2 1 2 2 2 2 6
2x x x
+ - · = · +
? ( ) ( ) 2 4 2 5 0
2 x x
- - =
Let 2
x
y =
? y y
2
4 5 0 - - =
( ) ( ) y y - + = 5 1 0
? y = 5 or y = - 1
(Ignore because 2
x
cannot be negative)
? y = 5 ? 2 5
x
=
x = log
2
5
Hence, option (c) is correct.
4. LetS = {2, 3, 4, 5, 6, 7, 9}. How many
different 3-digit numbers (with all
digits different) from S can be made
which are less than 500?
(a) 30 (b) 49
(c) 90 (d) 147
Ê
(c) Let S = {2, 3, 4, 5, 6, 7, 9}
? nS ( ) = 7
Three digit number less than
= × × = 3 6 5 90
?Option (c) is correct.
Note Hundreds digit can be filled with
3 choices that are 2, 3, 4.
Similarly, tens digit can be filled
with 6 ways and unit digit can be
filled with 5 ways.
5. If p = (1111 … up ton digits), then
what is the value of 9
2
p p + ?
(a) 10
n
p (b) 2 10 p
n
·
(c) 10 1
n
p - (d) 10 1
n
p +
Ê
(a) Given that,
p = (1111... upton digits)
= + + + +
-
1 10 10 10
2 1
K
n
=
-
-
1 10 1
10 1
( )
n
Q K a ar ar ar
a r
r
n
n
+ + + + =
-
-
?
?
?
?
?
?
- 2 1
1
1
( )
? p
n
=
- 10 1
9
? 9 10 1 p
n
= -
? 9 1 10 p
n
+ =
? 9 10
2
p p p
n
+ = ·
? Hence, option (a) is correct.
6. The quadratic equation
3 5 3 5 0
2 2 2
x k k x k k - + + - = ( ) has
real roots of equal magnitude and
opposite sign. Which one of the
following is correct?
(a) 0
5
3
< < k
(b) 0
3
5
< < k only
(c)
3
5
5
3
< < k
(d) No such value of k exists.
Ê
(d) Since, we know that if a quadratic
equationax bx c
2
0 + + = has real
roots of equal magnitude and opposite
sign.
Then, b = 0 ...(i)
and product of roots < 0 ...(ii)
PAPER : I Mathematics
500 =
3 6 5
In the given quadratic equation,
3 5 3 5 0
2 2 2
x k k x k k - - + - = ( )
a = 3,b k k = - + ( )
2
5 ,c k k = - 3 5
2
By Eq. (i), b = 0
? - + = ( ) k k
2
5 0
? k k ( ) + = 5 0
? k = - 0 5 ,
By Eq. (ii), Product of roots < 0
c
a
< 0
?
3 5
3
0
2
k k -
<
? k k ( ) 3 5 0 - <
? 0
5
3
< < k
From (i) and (ii) no such values ofk exists.
Hence, option (d) is correct.
7. Ifa n n
n
= ( !), then what is
a a a a
1 2 3 10
+ + + … + equal to?
(a) 10 1 ! - (b) 11 1 ! +
(c) 10 1 ! + (d) 11 1 ! -
Ê
(d) Given,a n n
n
= ( !)
= + - ( ) ( !) n n 1 1
= + - ( ) ! ! n n n 1
= + - ( )! ! n n 1
? a
1
2 1 = - ! !
a
2
3 2 = - ! !
... ... ... ... ... ...
a
10
11 10 = - ! !
?a a a a
1 2 3 10
+ + + + ...
= + + - + - + + - 2 1 3 2 4 3 11 10 ! ! ! ! ! ! ! ! K
= - 11 1 ! !
= - 11 1 !
?Option (d) is correct.
8. If p andq are the non-zero roots of
the equation x px q
2
0 + + = , then
how many possible values canq
have?
(a) Nil (b) One
(c) Two (d) Three
Ê
(b) Given quadratic equation
x px q
2
0 + + =
and roots are p andq (non zero)
? Sum of roots =
- coefficient of
coefficient of
2
x
x
p q p + = - ...(i)
? Product of roots =
constant term
coefficient of
2
x
pq q = ...(ii)
? pq q - = 0
q p ( ) - = 1 0
Q q ? 0 ? p - = 1 0
p = 1
From Eq. (i)
p q p + = -
q p = - = - 2 2 1 ( )
q = - 2
?Option (b) is correct.
9. If ? =
a b c
d e f
g h i
then what is
3 5 4 7 6
3 5 4 7 6
3 5 4 7 6
d g a g g
e h b h h
f i c i i
+ +
+ +
+ +
equal to?
(a) ? (b) 7?
(c) 72? (d) - 72?
Ê
(d) Given, ? =
a b c
d e f
g h i
=
+ +
+ +
+ +
3 5 4 7 6
3 5 4 7 6
3 5 4 7 6
d g a g g
e h b h h
f i c i i
=
+ +
+ +
+ +
6
3 5 4 7
3 5 4 7
3 5 4 7
d g a g g
e h b h h
f i c i i
ByC C C
1 1 3
5 ? - ,C C C
2 2 3
7 ? -
= 6
3 4
3 4
3 4
d a g
e b h
f c i
= × × 6 3 4
d a g
e b h
f c i
ByC C
2 1
?
= - 72
a d g
b e h
c f i
ByR C ?
= - = - 72 72
a b c
d e f
g h i
?
Hence, option (d) is correct.
10. If
1 1 1
b c c a a b + + +
, , are in HP, then
which of the following is/are correct?
1. a b c , , are in AP
2. ( ) b c +
2
, ( ) c a +
2
, ( ) a b +
2
are in GP.
Select the correct answer using the
code given below.
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(a) Given that,
1
b c +
,
1
c a +
,
1
a b +
are in HP.
?b c c a a b + + + , , are in AP.
?( ) ( ), ( ) a b c b c a b c + + - + + +
- + + + - + ( ), ( ) ( ) c a a b c a b are in AP.
? a b c , , are in AP.
2. From 1;a b c , , are in AP.
? b a d c a d = + = + , 2
where,d is common difference.
? ( ) ( ) b c a d a d + = + + +
2 2
2
= + ( ) 2 3
2
a d
( ) ( ) ( ) c a a d a a d + = + + = +
4 4 4
2 2 2
( ) ( ) ( ) a b a a d a d + = + + = +
2 2 2
2
Here, ( ) ( ) .( ) c a b c a b + = + +
4 2 2
So,( ) , ( ) ,( ) b c c a a b + + +
2 2 2
are not in G.P.
Hence, option (a) is correct.
11. If A
a
=
?
?
?
?
?
?
1
0 1
,
wherea ?ù, then what is
A A A
100 50 25
2 - - equal to?
(a) - 2I (b) - I
(c) 2 I (d) I
where I is the identity matrix.
Ê
(a) A
a
a =
?
?
?
?
?
?
?
1
0 1
ù
The sequence for given matrix A is
A
na
n
=
?
?
?
?
?
?
1
0 1
?A A A
100 50 25
2 - - =
?
?
?
?
?
?
1 100
0 1
a
-
?
?
?
?
?
?
1 50
0 1
a
-
?
?
?
?
?
?
2
1 25
0 1
a
=
- - - -
- - - -
?
?
?
?
?
?
1 1 2 100 50 50
0 0 0 1 1 2
a a a
=
-
-
?
?
?
?
?
?
= -
2 0
0 2
2I
Hence, option (a) is correct.
12. If
a b a b c
a b a b c
a b a b c
- - -
- - + -
- - - - +
- = kabc 0
( , , ) a b c ? ? ? 0 0 0
then what is the value ofk?
(a) - 4 (b) - 2
(c) 2 (d) 4
Ê
(a) Given that,
a b a b c
a b a b c
a b a b c
kabc
- - -
- - + -
- - - - +
- = 0
(a ? 0,b ? 0,c ? 0)
R R R
1 1 2
? +
0 0 2
0
-
- - + -
- - - - +
- =
c
a b a b c
a b a b c
kabc
- - - - - 2c a b a b [( ) ( ) ( ) ]- = kabc 0
- - = 2 2 0 c ab kabc ( )
- = kabc abc 4
? k = - 4
Hence, option (a) is correct.
2
Page 3


1. If x x
2
1 0 + + = , then what is the
value of x x x
199 200 201
+ + ?
(a) - 1 (b) 0
(c) 1 (d) 3
Ê
(b) Given that,
x x
2
1 0 + + = ...(i)
? x x x
199 200 201
+ + = + + x x x
199 2
1 ( )
= × x
199
0
= 0
2. If x y z , , are in GP, then which of
the following is/are correct?
1. ln( ), ln( ), ln( ) 3 3 3 x y z are in AP.
2. xyz x + ln( ), xyz y + ln( ),
xyz z + ln( ) are in HP.
Select the correct answer using the
code given below.
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
Ê
(a) Given thatx, y, z are in GP.
? y xz
2
= ...(i)
(1) If log( ) 3x , log( ) 3y , log( ) 3z are in AP
Then, 2 3 3 3 log( ) log( ) log( ) y x z = +
9 9
2
y xz = ( )
9 9
2
y xz = ( )
y xz
2
=
Hence, statement (1) is correct.
Hence, we can say ifx, y, z are in GP.
?log x, logy, log z are in AP.
? xyz x + log ,xyz y + log ,xyz z + log
are in AP.
Hence, Statement (2) is wrong.
?Option (a) is correct.
3. If log
10
2, log ( )
10
2 1
x
- , log ( )
10
2 3
x
+
are in AP, then what is x equal to?
(a) 0 (b) 1
(c) log
2
5 (d) log
5
2
Ê
(c) Given that, log
10
2, log ( )
10
2 1
x
- ,
log ( )
10
2 3
x
+ are in AP.
?2 2 1 2 2 3
10 10 10
log ( ) log log ( )
x x
- = + +
log ( ) log ( )
10
2
10
2 1 2 2 3
x x
- = +
? 2 1 2 2 2 2 6
2x x x
+ - · = · +
? ( ) ( ) 2 4 2 5 0
2 x x
- - =
Let 2
x
y =
? y y
2
4 5 0 - - =
( ) ( ) y y - + = 5 1 0
? y = 5 or y = - 1
(Ignore because 2
x
cannot be negative)
? y = 5 ? 2 5
x
=
x = log
2
5
Hence, option (c) is correct.
4. LetS = {2, 3, 4, 5, 6, 7, 9}. How many
different 3-digit numbers (with all
digits different) from S can be made
which are less than 500?
(a) 30 (b) 49
(c) 90 (d) 147
Ê
(c) Let S = {2, 3, 4, 5, 6, 7, 9}
? nS ( ) = 7
Three digit number less than
= × × = 3 6 5 90
?Option (c) is correct.
Note Hundreds digit can be filled with
3 choices that are 2, 3, 4.
Similarly, tens digit can be filled
with 6 ways and unit digit can be
filled with 5 ways.
5. If p = (1111 … up ton digits), then
what is the value of 9
2
p p + ?
(a) 10
n
p (b) 2 10 p
n
·
(c) 10 1
n
p - (d) 10 1
n
p +
Ê
(a) Given that,
p = (1111... upton digits)
= + + + +
-
1 10 10 10
2 1
K
n
=
-
-
1 10 1
10 1
( )
n
Q K a ar ar ar
a r
r
n
n
+ + + + =
-
-
?
?
?
?
?
?
- 2 1
1
1
( )
? p
n
=
- 10 1
9
? 9 10 1 p
n
= -
? 9 1 10 p
n
+ =
? 9 10
2
p p p
n
+ = ·
? Hence, option (a) is correct.
6. The quadratic equation
3 5 3 5 0
2 2 2
x k k x k k - + + - = ( ) has
real roots of equal magnitude and
opposite sign. Which one of the
following is correct?
(a) 0
5
3
< < k
(b) 0
3
5
< < k only
(c)
3
5
5
3
< < k
(d) No such value of k exists.
Ê
(d) Since, we know that if a quadratic
equationax bx c
2
0 + + = has real
roots of equal magnitude and opposite
sign.
Then, b = 0 ...(i)
and product of roots < 0 ...(ii)
PAPER : I Mathematics
500 =
3 6 5
In the given quadratic equation,
3 5 3 5 0
2 2 2
x k k x k k - - + - = ( )
a = 3,b k k = - + ( )
2
5 ,c k k = - 3 5
2
By Eq. (i), b = 0
? - + = ( ) k k
2
5 0
? k k ( ) + = 5 0
? k = - 0 5 ,
By Eq. (ii), Product of roots < 0
c
a
< 0
?
3 5
3
0
2
k k -
<
? k k ( ) 3 5 0 - <
? 0
5
3
< < k
From (i) and (ii) no such values ofk exists.
Hence, option (d) is correct.
7. Ifa n n
n
= ( !), then what is
a a a a
1 2 3 10
+ + + … + equal to?
(a) 10 1 ! - (b) 11 1 ! +
(c) 10 1 ! + (d) 11 1 ! -
Ê
(d) Given,a n n
n
= ( !)
= + - ( ) ( !) n n 1 1
= + - ( ) ! ! n n n 1
= + - ( )! ! n n 1
? a
1
2 1 = - ! !
a
2
3 2 = - ! !
... ... ... ... ... ...
a
10
11 10 = - ! !
?a a a a
1 2 3 10
+ + + + ...
= + + - + - + + - 2 1 3 2 4 3 11 10 ! ! ! ! ! ! ! ! K
= - 11 1 ! !
= - 11 1 !
?Option (d) is correct.
8. If p andq are the non-zero roots of
the equation x px q
2
0 + + = , then
how many possible values canq
have?
(a) Nil (b) One
(c) Two (d) Three
Ê
(b) Given quadratic equation
x px q
2
0 + + =
and roots are p andq (non zero)
? Sum of roots =
- coefficient of
coefficient of
2
x
x
p q p + = - ...(i)
? Product of roots =
constant term
coefficient of
2
x
pq q = ...(ii)
? pq q - = 0
q p ( ) - = 1 0
Q q ? 0 ? p - = 1 0
p = 1
From Eq. (i)
p q p + = -
q p = - = - 2 2 1 ( )
q = - 2
?Option (b) is correct.
9. If ? =
a b c
d e f
g h i
then what is
3 5 4 7 6
3 5 4 7 6
3 5 4 7 6
d g a g g
e h b h h
f i c i i
+ +
+ +
+ +
equal to?
(a) ? (b) 7?
(c) 72? (d) - 72?
Ê
(d) Given, ? =
a b c
d e f
g h i
=
+ +
+ +
+ +
3 5 4 7 6
3 5 4 7 6
3 5 4 7 6
d g a g g
e h b h h
f i c i i
=
+ +
+ +
+ +
6
3 5 4 7
3 5 4 7
3 5 4 7
d g a g g
e h b h h
f i c i i
ByC C C
1 1 3
5 ? - ,C C C
2 2 3
7 ? -
= 6
3 4
3 4
3 4
d a g
e b h
f c i
= × × 6 3 4
d a g
e b h
f c i
ByC C
2 1
?
= - 72
a d g
b e h
c f i
ByR C ?
= - = - 72 72
a b c
d e f
g h i
?
Hence, option (d) is correct.
10. If
1 1 1
b c c a a b + + +
, , are in HP, then
which of the following is/are correct?
1. a b c , , are in AP
2. ( ) b c +
2
, ( ) c a +
2
, ( ) a b +
2
are in GP.
Select the correct answer using the
code given below.
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(a) Given that,
1
b c +
,
1
c a +
,
1
a b +
are in HP.
?b c c a a b + + + , , are in AP.
?( ) ( ), ( ) a b c b c a b c + + - + + +
- + + + - + ( ), ( ) ( ) c a a b c a b are in AP.
? a b c , , are in AP.
2. From 1;a b c , , are in AP.
? b a d c a d = + = + , 2
where,d is common difference.
? ( ) ( ) b c a d a d + = + + +
2 2
2
= + ( ) 2 3
2
a d
( ) ( ) ( ) c a a d a a d + = + + = +
4 4 4
2 2 2
( ) ( ) ( ) a b a a d a d + = + + = +
2 2 2
2
Here, ( ) ( ) .( ) c a b c a b + = + +
4 2 2
So,( ) , ( ) ,( ) b c c a a b + + +
2 2 2
are not in G.P.
Hence, option (a) is correct.
11. If A
a
=
?
?
?
?
?
?
1
0 1
,
wherea ?ù, then what is
A A A
100 50 25
2 - - equal to?
(a) - 2I (b) - I
(c) 2 I (d) I
where I is the identity matrix.
Ê
(a) A
a
a =
?
?
?
?
?
?
?
1
0 1
ù
The sequence for given matrix A is
A
na
n
=
?
?
?
?
?
?
1
0 1
?A A A
100 50 25
2 - - =
?
?
?
?
?
?
1 100
0 1
a
-
?
?
?
?
?
?
1 50
0 1
a
-
?
?
?
?
?
?
2
1 25
0 1
a
=
- - - -
- - - -
?
?
?
?
?
?
1 1 2 100 50 50
0 0 0 1 1 2
a a a
=
-
-
?
?
?
?
?
?
= -
2 0
0 2
2I
Hence, option (a) is correct.
12. If
a b a b c
a b a b c
a b a b c
- - -
- - + -
- - - - +
- = kabc 0
( , , ) a b c ? ? ? 0 0 0
then what is the value ofk?
(a) - 4 (b) - 2
(c) 2 (d) 4
Ê
(a) Given that,
a b a b c
a b a b c
a b a b c
kabc
- - -
- - + -
- - - - +
- = 0
(a ? 0,b ? 0,c ? 0)
R R R
1 1 2
? +
0 0 2
0
-
- - + -
- - - - +
- =
c
a b a b c
a b a b c
kabc
- - - - - 2c a b a b [( ) ( ) ( ) ]- = kabc 0
- - = 2 2 0 c ab kabc ( )
- = kabc abc 4
? k = - 4
Hence, option (a) is correct.
2
13. What is i
n
n
n
=
+
?
1
8 7
equal to,
wherei = - 1?
(a) - 1 (b) 1
(c) i (d) - i
Ê
(a) LetS i
n
n
n
=
=
+
?
1
8 7
S i i i i
n
= + + + +
+ 2 3 8 7
...
=
-
-
?
?
?
?
?
?
=
-
-
?
?
?
?
?
?
+ + +
i
i
i
i
i
i
n n
( )
( ) 8 7 4 2 1 3
1
1
1
1
=
-
-
?
?
?
?
?
? i
i
i
3
1
1
[Q i i
n r r 4 +
= ]
=
- -
-
?
?
?
?
?
?
=
- -
-
i
i
i
i i
i
1
1 1
2
=
-
-
= -
1
1
1
i
i
14. Ifz x iy = + , wherei = - 1, then
what does the equation
zz z z z + + + - = | | ( )
2
4 48 0
represent?
(a) Straight line
(b) Parabola
(c) Circle
(d) Pair of straight lines
Ê
(c) Given, z x iy = +
? z x iy = -
? z z x + = 2
and z x y
2
2 2
= +
? zz z z z + + + - = | | ( )
2
4 48 0
( )( ) ( ) x iy x iy x y x + - + + + - =
2 2
4 2 48 0
x y x y x
2 2 2 2
8 48 0 + + + + - =
2 2 8 48 0
2 2
x y x + + - =
x y x
2 2
4 24 0 + + - =
which represents circle.
Hence, option (c) is correct.
15. Which one of the following is a
square root of 2 2
2 2
a a b + + ,
wherea b , ?ú ?
(a) a ib a ib + + -
(b) a ib a ib + - -
(c) 2a ib +
(d) 2a ib - , where i = - 1
Ê
(a) 2 2
2 2
a a b + +
= + - + - 2 2
2 2 2
a ib ib a i b
= + + - ( ) ( ) a ib a ib
+ + - 2 ( ) ( ) a ib a ib
= + + - ( ) a ib a ib
2
Hence, square root of
2 2
2 2
a a b + + = + + - a ib a ib
16. If sin? and cos? are the roots of the
equationax bx c
2
0 + + = , then
which one of the following is
correct?
(a) a b ac
2 2
2 0 + - =
(b) - + + = a b ac
2 2
2 0
(c) a b ac
2 2
2 0 - + =
(d) a b ac
2 2
2 0 + + =
Ê
(c) Given, equationax bx c
2
0 + + = …(i)
QRoots are sin ? and cos ?
? sin cos ? ? + = -
b
a
and sin cos ? ? · =
c
d
On squaring both sides, we get
sin cos sin cos
2 2
2
2
2 ? ? ? ? + + =
b
a
1 2
2
2
+ =
c
a
b
a
a a c b ( ) + = 2
2
? a b ac
2 2
2 0 - + =
?Option (c) is correct.
17. IfC n ( , ) 4 ,C n ( , ) 5 andC n ( , ) 6 are in
AP, then what is the value ofn?
(a) 7 (b) 8 (c) 9 (d) 10
Ê
(a)
n n
C C
4 5
· and
n
C
6
are in AP.
? 2
5 4 6
· = +
n n n
C C C
2
5 5 4 4
n
n
n
n
!
!( )!
!
!( )! -
=
-
+
-
n
n
!
! ( )! 6 6
2
5 4 5 6
( !)
( !)( ) ( )!
n
n n - -
=
- - -
+
×
?
?
?
?
?
?
n
n n n
!
!( )! ( ) ( ) 4 6
1
4 5
1
6 5
2
5 5
1
4 5
1
30 ( ) ( ) ( ) n n n -
=
- -
+
2 8 5
5 9 20
1
30
2
n
n n
- -
- +
=
( )
30 2 13 5 45 100
2
( ) n n n - = - +
5 105 490 0
2
n n - + =
n n
2
21 98 0 - + =
( ) ( ) n n - - = 14 7 0
n = 14 or n = 7
? Option (a) is correct.
18. How many 4-letter words (with or
without meaning) containing two
vowels can be constructed using
only the letters (without repetition)
of the word ‘LUCKNOW’?
(a) 240 (b) 200
(c) 150 (d) 120
Ê
(a) In LUCKNOW, there are 2 vowels
and 5 consonants.
?4 letter words = · ·
5
2
2
2
4 C C !
= × × = 10 1 24 240
?Option (a) is correct.
19. Suppose 20 distinct points are
placed randomly on a circle. Which
of the following statements is/are
correct?
1. The number of straight lines that
can be drawn by joining any two
of these points is 380.
2. The number of triangles that can
be drawn by joining any three of
these points is 1140.
Select the correct answer using the
code given below.
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(b) Given, that there are 20 distinct
points on a circle and we have to draw a
straight line by joining any two of these
points.
Hence, number of straight lines
= =
×
=
20
2
20 19
2
190 C
? Statement (1) is wrong.
and Number of triangle
= =
× ×
× ×
=
20
3
20 19 18
1 2 3
1140 C
?Statement (2) is correct.
Hence, option (b) is correct.
20. How many terms are there in the
expansion of
a
b
b
a
2
2
2
2
21
2 + +
?
?
?
?
?
?
wherea ? 0,b ? 0 ?
(a) 21 (b) 22
(c) 42 (d) 43
Ê
(d)
a
b
b
a
2
2
2
2
21
2 + +
?
?
?
?
?
?
?
a
b
b
a
+
?
?
?
?
?
?
?
?
?
?
?
?
2
21
= +
?
?
?
?
?
?
a
b
b
a
42
Since, we know that number of terms in
the expansion of ( ) a b
n
+ = + n 1
Hence, total number of terms
= + = 42 1 43
?Option (d) is correct.
21. For what values ofk is the system
of equations 2 3 1 0
2
k x y + - = ,
7 2 3 0 x y - + = , 6 1 0 kx y + + =
consistent?
(a)
3 11
10
±
(b)
21 161
10
±
(c)
3 7
10
±
(d)
4 11
10
±
Ê
(b) Given equations,
2 3 1 0
2
k x y + - =
7 2 3 0 x y - + =
6 1 0 kx y + + =
3
Page 4


1. If x x
2
1 0 + + = , then what is the
value of x x x
199 200 201
+ + ?
(a) - 1 (b) 0
(c) 1 (d) 3
Ê
(b) Given that,
x x
2
1 0 + + = ...(i)
? x x x
199 200 201
+ + = + + x x x
199 2
1 ( )
= × x
199
0
= 0
2. If x y z , , are in GP, then which of
the following is/are correct?
1. ln( ), ln( ), ln( ) 3 3 3 x y z are in AP.
2. xyz x + ln( ), xyz y + ln( ),
xyz z + ln( ) are in HP.
Select the correct answer using the
code given below.
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
Ê
(a) Given thatx, y, z are in GP.
? y xz
2
= ...(i)
(1) If log( ) 3x , log( ) 3y , log( ) 3z are in AP
Then, 2 3 3 3 log( ) log( ) log( ) y x z = +
9 9
2
y xz = ( )
9 9
2
y xz = ( )
y xz
2
=
Hence, statement (1) is correct.
Hence, we can say ifx, y, z are in GP.
?log x, logy, log z are in AP.
? xyz x + log ,xyz y + log ,xyz z + log
are in AP.
Hence, Statement (2) is wrong.
?Option (a) is correct.
3. If log
10
2, log ( )
10
2 1
x
- , log ( )
10
2 3
x
+
are in AP, then what is x equal to?
(a) 0 (b) 1
(c) log
2
5 (d) log
5
2
Ê
(c) Given that, log
10
2, log ( )
10
2 1
x
- ,
log ( )
10
2 3
x
+ are in AP.
?2 2 1 2 2 3
10 10 10
log ( ) log log ( )
x x
- = + +
log ( ) log ( )
10
2
10
2 1 2 2 3
x x
- = +
? 2 1 2 2 2 2 6
2x x x
+ - · = · +
? ( ) ( ) 2 4 2 5 0
2 x x
- - =
Let 2
x
y =
? y y
2
4 5 0 - - =
( ) ( ) y y - + = 5 1 0
? y = 5 or y = - 1
(Ignore because 2
x
cannot be negative)
? y = 5 ? 2 5
x
=
x = log
2
5
Hence, option (c) is correct.
4. LetS = {2, 3, 4, 5, 6, 7, 9}. How many
different 3-digit numbers (with all
digits different) from S can be made
which are less than 500?
(a) 30 (b) 49
(c) 90 (d) 147
Ê
(c) Let S = {2, 3, 4, 5, 6, 7, 9}
? nS ( ) = 7
Three digit number less than
= × × = 3 6 5 90
?Option (c) is correct.
Note Hundreds digit can be filled with
3 choices that are 2, 3, 4.
Similarly, tens digit can be filled
with 6 ways and unit digit can be
filled with 5 ways.
5. If p = (1111 … up ton digits), then
what is the value of 9
2
p p + ?
(a) 10
n
p (b) 2 10 p
n
·
(c) 10 1
n
p - (d) 10 1
n
p +
Ê
(a) Given that,
p = (1111... upton digits)
= + + + +
-
1 10 10 10
2 1
K
n
=
-
-
1 10 1
10 1
( )
n
Q K a ar ar ar
a r
r
n
n
+ + + + =
-
-
?
?
?
?
?
?
- 2 1
1
1
( )
? p
n
=
- 10 1
9
? 9 10 1 p
n
= -
? 9 1 10 p
n
+ =
? 9 10
2
p p p
n
+ = ·
? Hence, option (a) is correct.
6. The quadratic equation
3 5 3 5 0
2 2 2
x k k x k k - + + - = ( ) has
real roots of equal magnitude and
opposite sign. Which one of the
following is correct?
(a) 0
5
3
< < k
(b) 0
3
5
< < k only
(c)
3
5
5
3
< < k
(d) No such value of k exists.
Ê
(d) Since, we know that if a quadratic
equationax bx c
2
0 + + = has real
roots of equal magnitude and opposite
sign.
Then, b = 0 ...(i)
and product of roots < 0 ...(ii)
PAPER : I Mathematics
500 =
3 6 5
In the given quadratic equation,
3 5 3 5 0
2 2 2
x k k x k k - - + - = ( )
a = 3,b k k = - + ( )
2
5 ,c k k = - 3 5
2
By Eq. (i), b = 0
? - + = ( ) k k
2
5 0
? k k ( ) + = 5 0
? k = - 0 5 ,
By Eq. (ii), Product of roots < 0
c
a
< 0
?
3 5
3
0
2
k k -
<
? k k ( ) 3 5 0 - <
? 0
5
3
< < k
From (i) and (ii) no such values ofk exists.
Hence, option (d) is correct.
7. Ifa n n
n
= ( !), then what is
a a a a
1 2 3 10
+ + + … + equal to?
(a) 10 1 ! - (b) 11 1 ! +
(c) 10 1 ! + (d) 11 1 ! -
Ê
(d) Given,a n n
n
= ( !)
= + - ( ) ( !) n n 1 1
= + - ( ) ! ! n n n 1
= + - ( )! ! n n 1
? a
1
2 1 = - ! !
a
2
3 2 = - ! !
... ... ... ... ... ...
a
10
11 10 = - ! !
?a a a a
1 2 3 10
+ + + + ...
= + + - + - + + - 2 1 3 2 4 3 11 10 ! ! ! ! ! ! ! ! K
= - 11 1 ! !
= - 11 1 !
?Option (d) is correct.
8. If p andq are the non-zero roots of
the equation x px q
2
0 + + = , then
how many possible values canq
have?
(a) Nil (b) One
(c) Two (d) Three
Ê
(b) Given quadratic equation
x px q
2
0 + + =
and roots are p andq (non zero)
? Sum of roots =
- coefficient of
coefficient of
2
x
x
p q p + = - ...(i)
? Product of roots =
constant term
coefficient of
2
x
pq q = ...(ii)
? pq q - = 0
q p ( ) - = 1 0
Q q ? 0 ? p - = 1 0
p = 1
From Eq. (i)
p q p + = -
q p = - = - 2 2 1 ( )
q = - 2
?Option (b) is correct.
9. If ? =
a b c
d e f
g h i
then what is
3 5 4 7 6
3 5 4 7 6
3 5 4 7 6
d g a g g
e h b h h
f i c i i
+ +
+ +
+ +
equal to?
(a) ? (b) 7?
(c) 72? (d) - 72?
Ê
(d) Given, ? =
a b c
d e f
g h i
=
+ +
+ +
+ +
3 5 4 7 6
3 5 4 7 6
3 5 4 7 6
d g a g g
e h b h h
f i c i i
=
+ +
+ +
+ +
6
3 5 4 7
3 5 4 7
3 5 4 7
d g a g g
e h b h h
f i c i i
ByC C C
1 1 3
5 ? - ,C C C
2 2 3
7 ? -
= 6
3 4
3 4
3 4
d a g
e b h
f c i
= × × 6 3 4
d a g
e b h
f c i
ByC C
2 1
?
= - 72
a d g
b e h
c f i
ByR C ?
= - = - 72 72
a b c
d e f
g h i
?
Hence, option (d) is correct.
10. If
1 1 1
b c c a a b + + +
, , are in HP, then
which of the following is/are correct?
1. a b c , , are in AP
2. ( ) b c +
2
, ( ) c a +
2
, ( ) a b +
2
are in GP.
Select the correct answer using the
code given below.
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(a) Given that,
1
b c +
,
1
c a +
,
1
a b +
are in HP.
?b c c a a b + + + , , are in AP.
?( ) ( ), ( ) a b c b c a b c + + - + + +
- + + + - + ( ), ( ) ( ) c a a b c a b are in AP.
? a b c , , are in AP.
2. From 1;a b c , , are in AP.
? b a d c a d = + = + , 2
where,d is common difference.
? ( ) ( ) b c a d a d + = + + +
2 2
2
= + ( ) 2 3
2
a d
( ) ( ) ( ) c a a d a a d + = + + = +
4 4 4
2 2 2
( ) ( ) ( ) a b a a d a d + = + + = +
2 2 2
2
Here, ( ) ( ) .( ) c a b c a b + = + +
4 2 2
So,( ) , ( ) ,( ) b c c a a b + + +
2 2 2
are not in G.P.
Hence, option (a) is correct.
11. If A
a
=
?
?
?
?
?
?
1
0 1
,
wherea ?ù, then what is
A A A
100 50 25
2 - - equal to?
(a) - 2I (b) - I
(c) 2 I (d) I
where I is the identity matrix.
Ê
(a) A
a
a =
?
?
?
?
?
?
?
1
0 1
ù
The sequence for given matrix A is
A
na
n
=
?
?
?
?
?
?
1
0 1
?A A A
100 50 25
2 - - =
?
?
?
?
?
?
1 100
0 1
a
-
?
?
?
?
?
?
1 50
0 1
a
-
?
?
?
?
?
?
2
1 25
0 1
a
=
- - - -
- - - -
?
?
?
?
?
?
1 1 2 100 50 50
0 0 0 1 1 2
a a a
=
-
-
?
?
?
?
?
?
= -
2 0
0 2
2I
Hence, option (a) is correct.
12. If
a b a b c
a b a b c
a b a b c
- - -
- - + -
- - - - +
- = kabc 0
( , , ) a b c ? ? ? 0 0 0
then what is the value ofk?
(a) - 4 (b) - 2
(c) 2 (d) 4
Ê
(a) Given that,
a b a b c
a b a b c
a b a b c
kabc
- - -
- - + -
- - - - +
- = 0
(a ? 0,b ? 0,c ? 0)
R R R
1 1 2
? +
0 0 2
0
-
- - + -
- - - - +
- =
c
a b a b c
a b a b c
kabc
- - - - - 2c a b a b [( ) ( ) ( ) ]- = kabc 0
- - = 2 2 0 c ab kabc ( )
- = kabc abc 4
? k = - 4
Hence, option (a) is correct.
2
13. What is i
n
n
n
=
+
?
1
8 7
equal to,
wherei = - 1?
(a) - 1 (b) 1
(c) i (d) - i
Ê
(a) LetS i
n
n
n
=
=
+
?
1
8 7
S i i i i
n
= + + + +
+ 2 3 8 7
...
=
-
-
?
?
?
?
?
?
=
-
-
?
?
?
?
?
?
+ + +
i
i
i
i
i
i
n n
( )
( ) 8 7 4 2 1 3
1
1
1
1
=
-
-
?
?
?
?
?
? i
i
i
3
1
1
[Q i i
n r r 4 +
= ]
=
- -
-
?
?
?
?
?
?
=
- -
-
i
i
i
i i
i
1
1 1
2
=
-
-
= -
1
1
1
i
i
14. Ifz x iy = + , wherei = - 1, then
what does the equation
zz z z z + + + - = | | ( )
2
4 48 0
represent?
(a) Straight line
(b) Parabola
(c) Circle
(d) Pair of straight lines
Ê
(c) Given, z x iy = +
? z x iy = -
? z z x + = 2
and z x y
2
2 2
= +
? zz z z z + + + - = | | ( )
2
4 48 0
( )( ) ( ) x iy x iy x y x + - + + + - =
2 2
4 2 48 0
x y x y x
2 2 2 2
8 48 0 + + + + - =
2 2 8 48 0
2 2
x y x + + - =
x y x
2 2
4 24 0 + + - =
which represents circle.
Hence, option (c) is correct.
15. Which one of the following is a
square root of 2 2
2 2
a a b + + ,
wherea b , ?ú ?
(a) a ib a ib + + -
(b) a ib a ib + - -
(c) 2a ib +
(d) 2a ib - , where i = - 1
Ê
(a) 2 2
2 2
a a b + +
= + - + - 2 2
2 2 2
a ib ib a i b
= + + - ( ) ( ) a ib a ib
+ + - 2 ( ) ( ) a ib a ib
= + + - ( ) a ib a ib
2
Hence, square root of
2 2
2 2
a a b + + = + + - a ib a ib
16. If sin? and cos? are the roots of the
equationax bx c
2
0 + + = , then
which one of the following is
correct?
(a) a b ac
2 2
2 0 + - =
(b) - + + = a b ac
2 2
2 0
(c) a b ac
2 2
2 0 - + =
(d) a b ac
2 2
2 0 + + =
Ê
(c) Given, equationax bx c
2
0 + + = …(i)
QRoots are sin ? and cos ?
? sin cos ? ? + = -
b
a
and sin cos ? ? · =
c
d
On squaring both sides, we get
sin cos sin cos
2 2
2
2
2 ? ? ? ? + + =
b
a
1 2
2
2
+ =
c
a
b
a
a a c b ( ) + = 2
2
? a b ac
2 2
2 0 - + =
?Option (c) is correct.
17. IfC n ( , ) 4 ,C n ( , ) 5 andC n ( , ) 6 are in
AP, then what is the value ofn?
(a) 7 (b) 8 (c) 9 (d) 10
Ê
(a)
n n
C C
4 5
· and
n
C
6
are in AP.
? 2
5 4 6
· = +
n n n
C C C
2
5 5 4 4
n
n
n
n
!
!( )!
!
!( )! -
=
-
+
-
n
n
!
! ( )! 6 6
2
5 4 5 6
( !)
( !)( ) ( )!
n
n n - -
=
- - -
+
×
?
?
?
?
?
?
n
n n n
!
!( )! ( ) ( ) 4 6
1
4 5
1
6 5
2
5 5
1
4 5
1
30 ( ) ( ) ( ) n n n -
=
- -
+
2 8 5
5 9 20
1
30
2
n
n n
- -
- +
=
( )
30 2 13 5 45 100
2
( ) n n n - = - +
5 105 490 0
2
n n - + =
n n
2
21 98 0 - + =
( ) ( ) n n - - = 14 7 0
n = 14 or n = 7
? Option (a) is correct.
18. How many 4-letter words (with or
without meaning) containing two
vowels can be constructed using
only the letters (without repetition)
of the word ‘LUCKNOW’?
(a) 240 (b) 200
(c) 150 (d) 120
Ê
(a) In LUCKNOW, there are 2 vowels
and 5 consonants.
?4 letter words = · ·
5
2
2
2
4 C C !
= × × = 10 1 24 240
?Option (a) is correct.
19. Suppose 20 distinct points are
placed randomly on a circle. Which
of the following statements is/are
correct?
1. The number of straight lines that
can be drawn by joining any two
of these points is 380.
2. The number of triangles that can
be drawn by joining any three of
these points is 1140.
Select the correct answer using the
code given below.
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(b) Given, that there are 20 distinct
points on a circle and we have to draw a
straight line by joining any two of these
points.
Hence, number of straight lines
= =
×
=
20
2
20 19
2
190 C
? Statement (1) is wrong.
and Number of triangle
= =
× ×
× ×
=
20
3
20 19 18
1 2 3
1140 C
?Statement (2) is correct.
Hence, option (b) is correct.
20. How many terms are there in the
expansion of
a
b
b
a
2
2
2
2
21
2 + +
?
?
?
?
?
?
wherea ? 0,b ? 0 ?
(a) 21 (b) 22
(c) 42 (d) 43
Ê
(d)
a
b
b
a
2
2
2
2
21
2 + +
?
?
?
?
?
?
?
a
b
b
a
+
?
?
?
?
?
?
?
?
?
?
?
?
2
21
= +
?
?
?
?
?
?
a
b
b
a
42
Since, we know that number of terms in
the expansion of ( ) a b
n
+ = + n 1
Hence, total number of terms
= + = 42 1 43
?Option (d) is correct.
21. For what values ofk is the system
of equations 2 3 1 0
2
k x y + - = ,
7 2 3 0 x y - + = , 6 1 0 kx y + + =
consistent?
(a)
3 11
10
±
(b)
21 161
10
±
(c)
3 7
10
±
(d)
4 11
10
±
Ê
(b) Given equations,
2 3 1 0
2
k x y + - =
7 2 3 0 x y - + =
6 1 0 kx y + + =
3
For consistency, determinant formed by
the equations
2 3 1
7 2 3
6 1 1
0
2
k
k
-
- =
2 2 3 3 7 18
2
k k ( ) ( ) - - - -
- + = 1 7 12 0 ( ) k
- - + - - = 10 21 54 7 12 0
2
k k k
- - - = 10 42 28 0
2
k k
5 21 14 0
2
k k - + =
k =
± - 21 441 280
10
k =
± 21 161
10
Hence, option (b) is correct.
22. The inverse of a matrix A is given
by
-
-
?
?
?
?
?
?
?
?
2 1
3
2
1
2
What is A equal to?
(a)
1 2
3 4
?
?
?
?
?
?
(b)
1 2
3 4
-
-
?
?
?
?
?
?
(c)
1 2
3 4 -
?
?
?
?
?
?
(d)
- ?
?
?
?
?
?
1 2
3 4
Ê
(a) A =
-
-
?
?
?
?
?
?
?
?
2 1
3
2
1
2
? | | ( ) A = - -
?
?
?
?
?
?
- = - ? 2
1
2
3
2
1
2
0
A
11
1
2
= - , A
12
3
2
= -
A
21
1 = - , A
22
2 = -
?adj A =
- -
- -
?
?
?
?
?
?
?
?
?
?
1
2
1
3
2
2
?A
A
A
-
= =
-
- -
- -
?
?
?
?
?
?
?
?
?
?
1
1 1
1
2
1
2
1
3
2
2
| |
adj
=
?
?
?
?
?
?
?
?
?
?
2
1
2
1
3
2
2
=
?
?
?
?
?
?
1 2
3 4
Hence, option (a) is correct.
23. What is the period of the function
f x x ( ) ln( sin ) = + 2
2
?
(a)
p
2
(b) p (c) 2 p (d) 3p
Ê
(b) f x x ( ) ln( sin ) = + 2
2
Q Period of sin
2
x is p.
and f x x ( ) log { sin ( )} p p + = + + 2
2
= + log { sin } 2
2
x
= f x ( )
Hence, period of ln( sin ) 2
2
+ = x p
?Option (b) is correct.
24. If sin( ) A B + = 1 and
2 1 sin( ) A B - = , where 0 < A, B <
p
2
,
then what is tan : tan A B equal to?
(a) 1 : 2 (b) 2 : 1
(c) 1 : 3 (d) 3 : 1
Ê
(d) Given, sin( ) A B + = 1
and 2 1 sin( ) A B - =
0 < A,B<
p
2
Q sin( ) sin A B + = = 1
2
p
? A B + =
p
2
...(i)
2 1 sin( ) A B - =
? sin( ) sin A B - = =
1
2 6
p
? A B - =
p
6
...(ii)
Now, adding Eq. (i) and Eq. (ii), we get
2
2 6
A = +
p p
=
4
6
p
A =
p
3
,B =
p
6
? tan : tan tan : tan A B =
p p
3 6
= 3
1
3
: = 3 1 :
?Option (d) is correct.
25. Consider a regular polygon with
10 sides. What is the number of
triangles that can be formed by
joining the vertices which have no
common side with any of the sides
of the polygon?
(a) 25 (b) 50
(c) 75 (d) 100
Ê
(b) Given number of sides ( ) n = 10
Number of triangles which have no
common side with any of the sides of the
polygon =
- - n n n ( ) ( )
!
4 5
3
?Number of triangles
=
- - 10 10 4 10 5
6
( ) ( )
=
× × 10 6 5
6
= 50
Hence, option (b) is correct.
26. Consider all the real roots of the
equation x x
4 2
10 9 0 - + = .
What is the sum of the absolute
values of the roots?
(a) 4 (b) 6
(c) 8 (d) 10
Ê
(c) Given equation,
x x
4 2
10 9 0 - + =
Let y x =
2
? y y
2
10 9 0 - + =
( ) ( ) y y - - = 9 1 0
? y = 9 or y = 1
x
2
9 = or y = 1
x
2
9 = or x
2
1 =
x = ± 3,x = ± 1
? Sum = + - + + - | | | | | | | | 3 3 1 1
= 8
Hence, option (c) is correct.
27. Consider the expansion of ( ) 1 + x
n
.
Let p q r , , ands be the coefficients
of first, second,nth and ( ) n + 1 th
terms respectively. What is
( ) ps qr + equal to?
(a) 1 2 + n (b) 1 2
2
+ n
(c) 1
2
+ n (d) 1 4 + n
Ê
(c) Given, ( ) 1 + x
n
In the above expansion, ( ) r + 1 th term
T C x
r
n
r
r
+
=
1
? T C x T C x
n n
1 0 2 1
= ° = ' ,
p = 1 ...(i)
?
n
C q
1
=
n q = ...(ii)
T C x
n
n
n
n
=
-
-
1
1
,
T C x
n
n
n
n
+
=
1
? r n = ...(iii)
? s = 1 ...(iv)
? ( ) ps qr + = · + · 1 1 n n = + 1
2
n
= + ( ) 1
2
n
Hence, option (c) is correct.
28. Let sin sin sin
- - -
+ + =
1 1 1
3
2
x y z
p
for 0 = x ,y z , = 1. What is the value
of x y z
1000 1001 1002
+ + ?
(a) 0 (b) 1
(c) 3 (d) 6
Ê
(c) Let sin sin sin
- - -
+ + =
1 1 1
3
2
x y z
p
Which is only possible when,
sin
-
=
1
2
x
p
, sin
-
=
1
2
y
p
,
sin
-
=
1
2
z
p
? x = 1 , y = 1 , z = 1
? x y z
1000 1001 1002
+ +
= + + 1 1 1= 3
Hence, option (c) is correct.
4
Page 5


1. If x x
2
1 0 + + = , then what is the
value of x x x
199 200 201
+ + ?
(a) - 1 (b) 0
(c) 1 (d) 3
Ê
(b) Given that,
x x
2
1 0 + + = ...(i)
? x x x
199 200 201
+ + = + + x x x
199 2
1 ( )
= × x
199
0
= 0
2. If x y z , , are in GP, then which of
the following is/are correct?
1. ln( ), ln( ), ln( ) 3 3 3 x y z are in AP.
2. xyz x + ln( ), xyz y + ln( ),
xyz z + ln( ) are in HP.
Select the correct answer using the
code given below.
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
Ê
(a) Given thatx, y, z are in GP.
? y xz
2
= ...(i)
(1) If log( ) 3x , log( ) 3y , log( ) 3z are in AP
Then, 2 3 3 3 log( ) log( ) log( ) y x z = +
9 9
2
y xz = ( )
9 9
2
y xz = ( )
y xz
2
=
Hence, statement (1) is correct.
Hence, we can say ifx, y, z are in GP.
?log x, logy, log z are in AP.
? xyz x + log ,xyz y + log ,xyz z + log
are in AP.
Hence, Statement (2) is wrong.
?Option (a) is correct.
3. If log
10
2, log ( )
10
2 1
x
- , log ( )
10
2 3
x
+
are in AP, then what is x equal to?
(a) 0 (b) 1
(c) log
2
5 (d) log
5
2
Ê
(c) Given that, log
10
2, log ( )
10
2 1
x
- ,
log ( )
10
2 3
x
+ are in AP.
?2 2 1 2 2 3
10 10 10
log ( ) log log ( )
x x
- = + +
log ( ) log ( )
10
2
10
2 1 2 2 3
x x
- = +
? 2 1 2 2 2 2 6
2x x x
+ - · = · +
? ( ) ( ) 2 4 2 5 0
2 x x
- - =
Let 2
x
y =
? y y
2
4 5 0 - - =
( ) ( ) y y - + = 5 1 0
? y = 5 or y = - 1
(Ignore because 2
x
cannot be negative)
? y = 5 ? 2 5
x
=
x = log
2
5
Hence, option (c) is correct.
4. LetS = {2, 3, 4, 5, 6, 7, 9}. How many
different 3-digit numbers (with all
digits different) from S can be made
which are less than 500?
(a) 30 (b) 49
(c) 90 (d) 147
Ê
(c) Let S = {2, 3, 4, 5, 6, 7, 9}
? nS ( ) = 7
Three digit number less than
= × × = 3 6 5 90
?Option (c) is correct.
Note Hundreds digit can be filled with
3 choices that are 2, 3, 4.
Similarly, tens digit can be filled
with 6 ways and unit digit can be
filled with 5 ways.
5. If p = (1111 … up ton digits), then
what is the value of 9
2
p p + ?
(a) 10
n
p (b) 2 10 p
n
·
(c) 10 1
n
p - (d) 10 1
n
p +
Ê
(a) Given that,
p = (1111... upton digits)
= + + + +
-
1 10 10 10
2 1
K
n
=
-
-
1 10 1
10 1
( )
n
Q K a ar ar ar
a r
r
n
n
+ + + + =
-
-
?
?
?
?
?
?
- 2 1
1
1
( )
? p
n
=
- 10 1
9
? 9 10 1 p
n
= -
? 9 1 10 p
n
+ =
? 9 10
2
p p p
n
+ = ·
? Hence, option (a) is correct.
6. The quadratic equation
3 5 3 5 0
2 2 2
x k k x k k - + + - = ( ) has
real roots of equal magnitude and
opposite sign. Which one of the
following is correct?
(a) 0
5
3
< < k
(b) 0
3
5
< < k only
(c)
3
5
5
3
< < k
(d) No such value of k exists.
Ê
(d) Since, we know that if a quadratic
equationax bx c
2
0 + + = has real
roots of equal magnitude and opposite
sign.
Then, b = 0 ...(i)
and product of roots < 0 ...(ii)
PAPER : I Mathematics
500 =
3 6 5
In the given quadratic equation,
3 5 3 5 0
2 2 2
x k k x k k - - + - = ( )
a = 3,b k k = - + ( )
2
5 ,c k k = - 3 5
2
By Eq. (i), b = 0
? - + = ( ) k k
2
5 0
? k k ( ) + = 5 0
? k = - 0 5 ,
By Eq. (ii), Product of roots < 0
c
a
< 0
?
3 5
3
0
2
k k -
<
? k k ( ) 3 5 0 - <
? 0
5
3
< < k
From (i) and (ii) no such values ofk exists.
Hence, option (d) is correct.
7. Ifa n n
n
= ( !), then what is
a a a a
1 2 3 10
+ + + … + equal to?
(a) 10 1 ! - (b) 11 1 ! +
(c) 10 1 ! + (d) 11 1 ! -
Ê
(d) Given,a n n
n
= ( !)
= + - ( ) ( !) n n 1 1
= + - ( ) ! ! n n n 1
= + - ( )! ! n n 1
? a
1
2 1 = - ! !
a
2
3 2 = - ! !
... ... ... ... ... ...
a
10
11 10 = - ! !
?a a a a
1 2 3 10
+ + + + ...
= + + - + - + + - 2 1 3 2 4 3 11 10 ! ! ! ! ! ! ! ! K
= - 11 1 ! !
= - 11 1 !
?Option (d) is correct.
8. If p andq are the non-zero roots of
the equation x px q
2
0 + + = , then
how many possible values canq
have?
(a) Nil (b) One
(c) Two (d) Three
Ê
(b) Given quadratic equation
x px q
2
0 + + =
and roots are p andq (non zero)
? Sum of roots =
- coefficient of
coefficient of
2
x
x
p q p + = - ...(i)
? Product of roots =
constant term
coefficient of
2
x
pq q = ...(ii)
? pq q - = 0
q p ( ) - = 1 0
Q q ? 0 ? p - = 1 0
p = 1
From Eq. (i)
p q p + = -
q p = - = - 2 2 1 ( )
q = - 2
?Option (b) is correct.
9. If ? =
a b c
d e f
g h i
then what is
3 5 4 7 6
3 5 4 7 6
3 5 4 7 6
d g a g g
e h b h h
f i c i i
+ +
+ +
+ +
equal to?
(a) ? (b) 7?
(c) 72? (d) - 72?
Ê
(d) Given, ? =
a b c
d e f
g h i
=
+ +
+ +
+ +
3 5 4 7 6
3 5 4 7 6
3 5 4 7 6
d g a g g
e h b h h
f i c i i
=
+ +
+ +
+ +
6
3 5 4 7
3 5 4 7
3 5 4 7
d g a g g
e h b h h
f i c i i
ByC C C
1 1 3
5 ? - ,C C C
2 2 3
7 ? -
= 6
3 4
3 4
3 4
d a g
e b h
f c i
= × × 6 3 4
d a g
e b h
f c i
ByC C
2 1
?
= - 72
a d g
b e h
c f i
ByR C ?
= - = - 72 72
a b c
d e f
g h i
?
Hence, option (d) is correct.
10. If
1 1 1
b c c a a b + + +
, , are in HP, then
which of the following is/are correct?
1. a b c , , are in AP
2. ( ) b c +
2
, ( ) c a +
2
, ( ) a b +
2
are in GP.
Select the correct answer using the
code given below.
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(a) Given that,
1
b c +
,
1
c a +
,
1
a b +
are in HP.
?b c c a a b + + + , , are in AP.
?( ) ( ), ( ) a b c b c a b c + + - + + +
- + + + - + ( ), ( ) ( ) c a a b c a b are in AP.
? a b c , , are in AP.
2. From 1;a b c , , are in AP.
? b a d c a d = + = + , 2
where,d is common difference.
? ( ) ( ) b c a d a d + = + + +
2 2
2
= + ( ) 2 3
2
a d
( ) ( ) ( ) c a a d a a d + = + + = +
4 4 4
2 2 2
( ) ( ) ( ) a b a a d a d + = + + = +
2 2 2
2
Here, ( ) ( ) .( ) c a b c a b + = + +
4 2 2
So,( ) , ( ) ,( ) b c c a a b + + +
2 2 2
are not in G.P.
Hence, option (a) is correct.
11. If A
a
=
?
?
?
?
?
?
1
0 1
,
wherea ?ù, then what is
A A A
100 50 25
2 - - equal to?
(a) - 2I (b) - I
(c) 2 I (d) I
where I is the identity matrix.
Ê
(a) A
a
a =
?
?
?
?
?
?
?
1
0 1
ù
The sequence for given matrix A is
A
na
n
=
?
?
?
?
?
?
1
0 1
?A A A
100 50 25
2 - - =
?
?
?
?
?
?
1 100
0 1
a
-
?
?
?
?
?
?
1 50
0 1
a
-
?
?
?
?
?
?
2
1 25
0 1
a
=
- - - -
- - - -
?
?
?
?
?
?
1 1 2 100 50 50
0 0 0 1 1 2
a a a
=
-
-
?
?
?
?
?
?
= -
2 0
0 2
2I
Hence, option (a) is correct.
12. If
a b a b c
a b a b c
a b a b c
- - -
- - + -
- - - - +
- = kabc 0
( , , ) a b c ? ? ? 0 0 0
then what is the value ofk?
(a) - 4 (b) - 2
(c) 2 (d) 4
Ê
(a) Given that,
a b a b c
a b a b c
a b a b c
kabc
- - -
- - + -
- - - - +
- = 0
(a ? 0,b ? 0,c ? 0)
R R R
1 1 2
? +
0 0 2
0
-
- - + -
- - - - +
- =
c
a b a b c
a b a b c
kabc
- - - - - 2c a b a b [( ) ( ) ( ) ]- = kabc 0
- - = 2 2 0 c ab kabc ( )
- = kabc abc 4
? k = - 4
Hence, option (a) is correct.
2
13. What is i
n
n
n
=
+
?
1
8 7
equal to,
wherei = - 1?
(a) - 1 (b) 1
(c) i (d) - i
Ê
(a) LetS i
n
n
n
=
=
+
?
1
8 7
S i i i i
n
= + + + +
+ 2 3 8 7
...
=
-
-
?
?
?
?
?
?
=
-
-
?
?
?
?
?
?
+ + +
i
i
i
i
i
i
n n
( )
( ) 8 7 4 2 1 3
1
1
1
1
=
-
-
?
?
?
?
?
? i
i
i
3
1
1
[Q i i
n r r 4 +
= ]
=
- -
-
?
?
?
?
?
?
=
- -
-
i
i
i
i i
i
1
1 1
2
=
-
-
= -
1
1
1
i
i
14. Ifz x iy = + , wherei = - 1, then
what does the equation
zz z z z + + + - = | | ( )
2
4 48 0
represent?
(a) Straight line
(b) Parabola
(c) Circle
(d) Pair of straight lines
Ê
(c) Given, z x iy = +
? z x iy = -
? z z x + = 2
and z x y
2
2 2
= +
? zz z z z + + + - = | | ( )
2
4 48 0
( )( ) ( ) x iy x iy x y x + - + + + - =
2 2
4 2 48 0
x y x y x
2 2 2 2
8 48 0 + + + + - =
2 2 8 48 0
2 2
x y x + + - =
x y x
2 2
4 24 0 + + - =
which represents circle.
Hence, option (c) is correct.
15. Which one of the following is a
square root of 2 2
2 2
a a b + + ,
wherea b , ?ú ?
(a) a ib a ib + + -
(b) a ib a ib + - -
(c) 2a ib +
(d) 2a ib - , where i = - 1
Ê
(a) 2 2
2 2
a a b + +
= + - + - 2 2
2 2 2
a ib ib a i b
= + + - ( ) ( ) a ib a ib
+ + - 2 ( ) ( ) a ib a ib
= + + - ( ) a ib a ib
2
Hence, square root of
2 2
2 2
a a b + + = + + - a ib a ib
16. If sin? and cos? are the roots of the
equationax bx c
2
0 + + = , then
which one of the following is
correct?
(a) a b ac
2 2
2 0 + - =
(b) - + + = a b ac
2 2
2 0
(c) a b ac
2 2
2 0 - + =
(d) a b ac
2 2
2 0 + + =
Ê
(c) Given, equationax bx c
2
0 + + = …(i)
QRoots are sin ? and cos ?
? sin cos ? ? + = -
b
a
and sin cos ? ? · =
c
d
On squaring both sides, we get
sin cos sin cos
2 2
2
2
2 ? ? ? ? + + =
b
a
1 2
2
2
+ =
c
a
b
a
a a c b ( ) + = 2
2
? a b ac
2 2
2 0 - + =
?Option (c) is correct.
17. IfC n ( , ) 4 ,C n ( , ) 5 andC n ( , ) 6 are in
AP, then what is the value ofn?
(a) 7 (b) 8 (c) 9 (d) 10
Ê
(a)
n n
C C
4 5
· and
n
C
6
are in AP.
? 2
5 4 6
· = +
n n n
C C C
2
5 5 4 4
n
n
n
n
!
!( )!
!
!( )! -
=
-
+
-
n
n
!
! ( )! 6 6
2
5 4 5 6
( !)
( !)( ) ( )!
n
n n - -
=
- - -
+
×
?
?
?
?
?
?
n
n n n
!
!( )! ( ) ( ) 4 6
1
4 5
1
6 5
2
5 5
1
4 5
1
30 ( ) ( ) ( ) n n n -
=
- -
+
2 8 5
5 9 20
1
30
2
n
n n
- -
- +
=
( )
30 2 13 5 45 100
2
( ) n n n - = - +
5 105 490 0
2
n n - + =
n n
2
21 98 0 - + =
( ) ( ) n n - - = 14 7 0
n = 14 or n = 7
? Option (a) is correct.
18. How many 4-letter words (with or
without meaning) containing two
vowels can be constructed using
only the letters (without repetition)
of the word ‘LUCKNOW’?
(a) 240 (b) 200
(c) 150 (d) 120
Ê
(a) In LUCKNOW, there are 2 vowels
and 5 consonants.
?4 letter words = · ·
5
2
2
2
4 C C !
= × × = 10 1 24 240
?Option (a) is correct.
19. Suppose 20 distinct points are
placed randomly on a circle. Which
of the following statements is/are
correct?
1. The number of straight lines that
can be drawn by joining any two
of these points is 380.
2. The number of triangles that can
be drawn by joining any three of
these points is 1140.
Select the correct answer using the
code given below.
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
Ê
(b) Given, that there are 20 distinct
points on a circle and we have to draw a
straight line by joining any two of these
points.
Hence, number of straight lines
= =
×
=
20
2
20 19
2
190 C
? Statement (1) is wrong.
and Number of triangle
= =
× ×
× ×
=
20
3
20 19 18
1 2 3
1140 C
?Statement (2) is correct.
Hence, option (b) is correct.
20. How many terms are there in the
expansion of
a
b
b
a
2
2
2
2
21
2 + +
?
?
?
?
?
?
wherea ? 0,b ? 0 ?
(a) 21 (b) 22
(c) 42 (d) 43
Ê
(d)
a
b
b
a
2
2
2
2
21
2 + +
?
?
?
?
?
?
?
a
b
b
a
+
?
?
?
?
?
?
?
?
?
?
?
?
2
21
= +
?
?
?
?
?
?
a
b
b
a
42
Since, we know that number of terms in
the expansion of ( ) a b
n
+ = + n 1
Hence, total number of terms
= + = 42 1 43
?Option (d) is correct.
21. For what values ofk is the system
of equations 2 3 1 0
2
k x y + - = ,
7 2 3 0 x y - + = , 6 1 0 kx y + + =
consistent?
(a)
3 11
10
±
(b)
21 161
10
±
(c)
3 7
10
±
(d)
4 11
10
±
Ê
(b) Given equations,
2 3 1 0
2
k x y + - =
7 2 3 0 x y - + =
6 1 0 kx y + + =
3
For consistency, determinant formed by
the equations
2 3 1
7 2 3
6 1 1
0
2
k
k
-
- =
2 2 3 3 7 18
2
k k ( ) ( ) - - - -
- + = 1 7 12 0 ( ) k
- - + - - = 10 21 54 7 12 0
2
k k k
- - - = 10 42 28 0
2
k k
5 21 14 0
2
k k - + =
k =
± - 21 441 280
10
k =
± 21 161
10
Hence, option (b) is correct.
22. The inverse of a matrix A is given
by
-
-
?
?
?
?
?
?
?
?
2 1
3
2
1
2
What is A equal to?
(a)
1 2
3 4
?
?
?
?
?
?
(b)
1 2
3 4
-
-
?
?
?
?
?
?
(c)
1 2
3 4 -
?
?
?
?
?
?
(d)
- ?
?
?
?
?
?
1 2
3 4
Ê
(a) A =
-
-
?
?
?
?
?
?
?
?
2 1
3
2
1
2
? | | ( ) A = - -
?
?
?
?
?
?
- = - ? 2
1
2
3
2
1
2
0
A
11
1
2
= - , A
12
3
2
= -
A
21
1 = - , A
22
2 = -
?adj A =
- -
- -
?
?
?
?
?
?
?
?
?
?
1
2
1
3
2
2
?A
A
A
-
= =
-
- -
- -
?
?
?
?
?
?
?
?
?
?
1
1 1
1
2
1
2
1
3
2
2
| |
adj
=
?
?
?
?
?
?
?
?
?
?
2
1
2
1
3
2
2
=
?
?
?
?
?
?
1 2
3 4
Hence, option (a) is correct.
23. What is the period of the function
f x x ( ) ln( sin ) = + 2
2
?
(a)
p
2
(b) p (c) 2 p (d) 3p
Ê
(b) f x x ( ) ln( sin ) = + 2
2
Q Period of sin
2
x is p.
and f x x ( ) log { sin ( )} p p + = + + 2
2
= + log { sin } 2
2
x
= f x ( )
Hence, period of ln( sin ) 2
2
+ = x p
?Option (b) is correct.
24. If sin( ) A B + = 1 and
2 1 sin( ) A B - = , where 0 < A, B <
p
2
,
then what is tan : tan A B equal to?
(a) 1 : 2 (b) 2 : 1
(c) 1 : 3 (d) 3 : 1
Ê
(d) Given, sin( ) A B + = 1
and 2 1 sin( ) A B - =
0 < A,B<
p
2
Q sin( ) sin A B + = = 1
2
p
? A B + =
p
2
...(i)
2 1 sin( ) A B - =
? sin( ) sin A B - = =
1
2 6
p
? A B - =
p
6
...(ii)
Now, adding Eq. (i) and Eq. (ii), we get
2
2 6
A = +
p p
=
4
6
p
A =
p
3
,B =
p
6
? tan : tan tan : tan A B =
p p
3 6
= 3
1
3
: = 3 1 :
?Option (d) is correct.
25. Consider a regular polygon with
10 sides. What is the number of
triangles that can be formed by
joining the vertices which have no
common side with any of the sides
of the polygon?
(a) 25 (b) 50
(c) 75 (d) 100
Ê
(b) Given number of sides ( ) n = 10
Number of triangles which have no
common side with any of the sides of the
polygon =
- - n n n ( ) ( )
!
4 5
3
?Number of triangles
=
- - 10 10 4 10 5
6
( ) ( )
=
× × 10 6 5
6
= 50
Hence, option (b) is correct.
26. Consider all the real roots of the
equation x x
4 2
10 9 0 - + = .
What is the sum of the absolute
values of the roots?
(a) 4 (b) 6
(c) 8 (d) 10
Ê
(c) Given equation,
x x
4 2
10 9 0 - + =
Let y x =
2
? y y
2
10 9 0 - + =
( ) ( ) y y - - = 9 1 0
? y = 9 or y = 1
x
2
9 = or y = 1
x
2
9 = or x
2
1 =
x = ± 3,x = ± 1
? Sum = + - + + - | | | | | | | | 3 3 1 1
= 8
Hence, option (c) is correct.
27. Consider the expansion of ( ) 1 + x
n
.
Let p q r , , ands be the coefficients
of first, second,nth and ( ) n + 1 th
terms respectively. What is
( ) ps qr + equal to?
(a) 1 2 + n (b) 1 2
2
+ n
(c) 1
2
+ n (d) 1 4 + n
Ê
(c) Given, ( ) 1 + x
n
In the above expansion, ( ) r + 1 th term
T C x
r
n
r
r
+
=
1
? T C x T C x
n n
1 0 2 1
= ° = ' ,
p = 1 ...(i)
?
n
C q
1
=
n q = ...(ii)
T C x
n
n
n
n
=
-
-
1
1
,
T C x
n
n
n
n
+
=
1
? r n = ...(iii)
? s = 1 ...(iv)
? ( ) ps qr + = · + · 1 1 n n = + 1
2
n
= + ( ) 1
2
n
Hence, option (c) is correct.
28. Let sin sin sin
- - -
+ + =
1 1 1
3
2
x y z
p
for 0 = x ,y z , = 1. What is the value
of x y z
1000 1001 1002
+ + ?
(a) 0 (b) 1
(c) 3 (d) 6
Ê
(c) Let sin sin sin
- - -
+ + =
1 1 1
3
2
x y z
p
Which is only possible when,
sin
-
=
1
2
x
p
, sin
-
=
1
2
y
p
,
sin
-
=
1
2
z
p
? x = 1 , y = 1 , z = 1
? x y z
1000 1001 1002
+ +
= + + 1 1 1= 3
Hence, option (c) is correct.
4
29. Let sin sin cos cos x y x y + = + for
all x y , ?ú. What is tan
x y
2 2
+
?
?
?
?
?
?
equal to?
(a) 1 (b) 2
(c) 2 (d) 2 2
Ê
(a) Given that,
sin sin cos cos x y x y + = + ? ? x ú
2
2 2
sin cos
x y x y + ?
?
?
?
?
?
- ?
?
?
?
?
?
=
+ ?
?
?
?
?
?
·
- ?
?
?
?
?
?
2
2 2
cos cos
x y x y
?
2
2 2
2
2 2
sin cos
cos cos
x y x y
x y x y
+ ?
?
?
?
?
?
·
- ?
?
?
?
?
?
+ ?
?
?
?
?
?
·
- ?
?
?
?
?
?
= 1
tan
x y + ?
?
?
?
?
?
=
2
1 or tan
x y
2 2
1 +
?
?
?
?
?
?
=
Hence, option (a) is correct.
30. Let A =
-
?
?
?
?
?
?
0 2
2 0
and ( ) , mI nA A + =
2
wherem n , are
positive real numbers and I is the
identity matrix. What is ( ) m n +
equal to?
(a) 0 (b)
1
2
(c) 1 (d)
3
2
Ê
(d) Let A =
-
?
?
?
?
?
?
0 2
2 0
and ( ) mI nA +
2
= A
where,I is Identify matrix
QmI nA m n + =
?
?
?
?
?
?
+
-
?
?
?
?
?
?
1 0
0 1
0 2
2 0
=
?
?
?
?
?
?
+
-
?
?
?
?
?
?
m
m
n
n
0
0
0 2
2 0
=
-
?
?
?
?
?
?
m n
n m
2
2
?( ) mI nA A + =
2
m n
n m
m n
n m
2
2
2
2 -
?
?
?
?
?
?
-
?
?
?
?
?
?
=
-
?
?
?
?
?
?
0 2
2 0
m n mn
mn m n
2 2
2 2
4 4
4 4
-
- -
?
?
?
?
?
?
=
-
?
?
?
?
?
?
0 2
2 0
? 4 2 mn = and m n
2 2
4 0 - =
mn =
1
2
and m n = ± 2
When, m n = 2
( ) ( ) 2
1
2
n n =
n = ±
1
2
?m = ± 1
? m n + = + = 1
1
2
3
2
Hence, option (d) is correct.
31. What is the value of the following?
cot sin cot
- - ?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
1 1
3
5
3
2
(a)
6
17
(b)
7
16
(c)
16
7
(d)
17
6
Ê
(a) cot sin cot
- -
+
?
?
?
?
?
?
1 1
3
5
3
2
?cot cot cot
- -
-
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
1
2
1
1
3
5
3
5
3
2
?
?
?
?
?
?
?
[Q sin cot
- -
=
-
?
?
?
?
?
?
?
?
1 1
2
1
x
x
x
and cot cot cot x y
xy
x y
+ =
-
+
?
?
?
?
?
?
?
?
?
- - 1 1
1
?cot cot cot
- - ?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
1 1
4
3
3
2
?cot cot
-
× -
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
1
4
3
3
2
1
4
3
3
2
?
1
17
6
6
17
=
Hence, option (a) is correct.
32. Let 4 3
2
sin x = , where 0 = = x p.
What is tan3x equal to?
(a) - 2 (b) - 1
(c) 0 (d) 1
Ê
(c) Given that 4 3
2
sin x = , 0 = = x p
? sin
2
3
4
x =
? sin sin x = =
3
2 3
p
or sin
2
3
p
? x =
p
3
,
2
3
p
? tan tan 3
3
3
x =
p
= tan p = 0
Also tan tan 3 3
2
3
x =
?
?
?
?
?
?
p
= tan 2 p = 0
?Option (c) is correct.
33. Let p q , and 3 be respectively the
first, third and fifth terms of an AP.
Letd be the common difference. If
the product ( ) pq is minimum, then
what is the value ofd?
(a) 1 (b)
3
8
(c)
9
8
(d)
9
4
Ê
(c) Given that first term of AP = p
? a p = …(i)
Where,a denotes first term.
and a q
3
= ,a
5
3 =
? a d q + = 2 ...(ii)
a d + = 4 3 ...(iii)
? pq a a d = + ( ) 2
= - - + ( ) ( ) 3 4 3 4 2 d d d
= - - ( ) ( ) 3 4 3 2 d d
= - + 9 18 8
2
d d
Let f d d = - + 9 18 8
2
f d ' = - + 0 18 16
= - + 18 16d
For maxima and minima
f' = 0
? - + = 18 16 0 d
? d = =
18
16
9
8
.
Now, '' = f 16 (Positive)
So,f will be maximum atd =
9
8
.
Hence, option (c) is correct.
34. Consider the following statements
in respect of the roots of the
equation x
3
8 0 - =
1. The roots are non-collinear.
2. The roots lie on a circle of unit
radius.
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
(a) x
3
8 0 - =
?( ) ( ) x x x - + + = 2 2 4 0
2
x = 2, 2?, 2
2
?
Where, ? =
- + 1 3
2
i
Hence, roots are non-collinear and will
lie on a circle of 2 unit radius.
Hence, option (a) is correct.
35. Let the equation secx x p · = cosec
have a solution, where p is a
positive real number. What should
be the smallest value of p?
(a)
1
2
(b) 1
(c) 2
(d) Minimum does not exist
Ê
(c) sec x x p · = cosec
?
1
sinx x
p
·
=
cos
?
2
2 sin cos x x
p =
2
2 sin x
p =
5