NDA Solved Paper 2017 - II | NDA (National Defence Academy) Past Year Papers PDF Download

 Page 1


1. If x x
x
+ + = + log ( ) log log
10 10 10
1 2 5 6
then x is equal to
(a) 2, -3 (b) 2 only (c) 1 (d) 3
2. The remainder and the quotient of the
binary division ( ) ( ) 101110 110
2 2
¸ are
respectively
(a) ( ) 111
2
and ( ) 100
2
(b) ( ) 100
2
and ( ) 111
2
(c) ( ) 101
2
and ( ) 101
2
(d) ( ) 100
2
and ( ) 100
2
3. The matrix A has x rows and x + 5
columns. The matrix B has y rows and
11-y columns. Both AB and BA exist.
What are the values of x and y
respectively?
(a) 8 and 3 (b) 3 and 4
(c) 3 and 8 (d) 8 and 8
4. If S nP
n n Q
n
= +
- ( ) 1
2
, where S
n
denotes the sum of the firstn terms of an
AP, then the common difference is
(a) P Q + (b) 2 3 P Q +
(c) 2Q (d)Q
5. The roots of the equation
( ) ( ) ( ) q r x r p x p q - + - + - =
2
0 are
(a)
( )
( )
,
r p
q r
-
-
1
2
(b)
( )
( )
,
p q
q r
-
-
1
(c)
( )
( )
,
q r
p q
-
-
1
(d)
( )
( )
,
r p
p q
-
-
1
2
6. If E is the universal set and
A B C = È , then the set
E E E E E A - - - - - ( ( ( ( )))) is
same as the set
(a) B C ¢ È ¢ (b) B C È
(c) B C ¢ Ç ¢ (d) B C Ç
7. If A = {x x : is a multiple of 2},
B= {x x : is a multiple of 5} and
C = {x x : is a multiple of 10},
then A B C Ç Ç ( ) is equal to
(a) A (b) B
(c)C
(d) {x x : is a multiple of 100}
8. Ifa andb are the roots of the
equation 1 0
2
+ + = x x , then
the matrix product
1 b
a a
é
ë
ê
ù
û
ú
a b
b 1
é
ë
ê
ù
û
ú
is equal to
(a)
1 1
1 2
é
ë
ê
ù
û
ú
(b)
- -
-
é
ë
ê
ù
û
ú
1 1
1 2
(c)
1 1
1 2
-
-
é
ë
ê
ù
û
ú
(d)
- -
- -
é
ë
ê
ù
û
ú
1 1
1 2
9. If| | a denotes the absolute value
of an integer, then which of the
following are correct?
1. | | | || | ab a b =
2. | | | | | | a b a b + £ +
3. | | | | | | a b a b - ³ -
Select the correct answer using the
code given below.
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
10. How many different permutation
can be made out of the letters of the
word ‘PERMUTATION’?
(a) 19958400 (b) 19954800
(c) 19952400 (d) 39916800
11. If A
i i
i i
=
-
+
é
ë
ê
ù
û
ú
4 6 10
14 6 4
and k
i
=
1
2
,
wherei = -1, thenkA is equal to
(a)
2 3 5
7 2 3
+
-
é
ë
ê
ù
û
ú
i
i
(b)
2 3 5
7 2 3
-
+
é
ë
ê
ù
û
ú
i
i
(c)
2 3 7
5 2 3
-
+
é
ë
ê
ù
û
ú
i
i
(d)
2 3 5
7 2 3
+
+
é
ë
ê
ù
û
ú
i
i
12. The sum of all real roots of the
equation | | | | x x - + - - = 3 3 2 0
2
is
(a) 2 (b) 3
(c) 4 (d) 6
13. It is given that the roots of the
equation x x P
2
3
4 0 - - = log are
real. For this the minimum value of
P is
(a)
1
27
(b)
1
64
(c)
1
81
(d) 1
PAPER : I Mathematics
Page 2


1. If x x
x
+ + = + log ( ) log log
10 10 10
1 2 5 6
then x is equal to
(a) 2, -3 (b) 2 only (c) 1 (d) 3
2. The remainder and the quotient of the
binary division ( ) ( ) 101110 110
2 2
¸ are
respectively
(a) ( ) 111
2
and ( ) 100
2
(b) ( ) 100
2
and ( ) 111
2
(c) ( ) 101
2
and ( ) 101
2
(d) ( ) 100
2
and ( ) 100
2
3. The matrix A has x rows and x + 5
columns. The matrix B has y rows and
11-y columns. Both AB and BA exist.
What are the values of x and y
respectively?
(a) 8 and 3 (b) 3 and 4
(c) 3 and 8 (d) 8 and 8
4. If S nP
n n Q
n
= +
- ( ) 1
2
, where S
n
denotes the sum of the firstn terms of an
AP, then the common difference is
(a) P Q + (b) 2 3 P Q +
(c) 2Q (d)Q
5. The roots of the equation
( ) ( ) ( ) q r x r p x p q - + - + - =
2
0 are
(a)
( )
( )
,
r p
q r
-
-
1
2
(b)
( )
( )
,
p q
q r
-
-
1
(c)
( )
( )
,
q r
p q
-
-
1
(d)
( )
( )
,
r p
p q
-
-
1
2
6. If E is the universal set and
A B C = È , then the set
E E E E E A - - - - - ( ( ( ( )))) is
same as the set
(a) B C ¢ È ¢ (b) B C È
(c) B C ¢ Ç ¢ (d) B C Ç
7. If A = {x x : is a multiple of 2},
B= {x x : is a multiple of 5} and
C = {x x : is a multiple of 10},
then A B C Ç Ç ( ) is equal to
(a) A (b) B
(c)C
(d) {x x : is a multiple of 100}
8. Ifa andb are the roots of the
equation 1 0
2
+ + = x x , then
the matrix product
1 b
a a
é
ë
ê
ù
û
ú
a b
b 1
é
ë
ê
ù
û
ú
is equal to
(a)
1 1
1 2
é
ë
ê
ù
û
ú
(b)
- -
-
é
ë
ê
ù
û
ú
1 1
1 2
(c)
1 1
1 2
-
-
é
ë
ê
ù
û
ú
(d)
- -
- -
é
ë
ê
ù
û
ú
1 1
1 2
9. If| | a denotes the absolute value
of an integer, then which of the
following are correct?
1. | | | || | ab a b =
2. | | | | | | a b a b + £ +
3. | | | | | | a b a b - ³ -
Select the correct answer using the
code given below.
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
10. How many different permutation
can be made out of the letters of the
word ‘PERMUTATION’?
(a) 19958400 (b) 19954800
(c) 19952400 (d) 39916800
11. If A
i i
i i
=
-
+
é
ë
ê
ù
û
ú
4 6 10
14 6 4
and k
i
=
1
2
,
wherei = -1, thenkA is equal to
(a)
2 3 5
7 2 3
+
-
é
ë
ê
ù
û
ú
i
i
(b)
2 3 5
7 2 3
-
+
é
ë
ê
ù
û
ú
i
i
(c)
2 3 7
5 2 3
-
+
é
ë
ê
ù
û
ú
i
i
(d)
2 3 5
7 2 3
+
+
é
ë
ê
ù
û
ú
i
i
12. The sum of all real roots of the
equation | | | | x x - + - - = 3 3 2 0
2
is
(a) 2 (b) 3
(c) 4 (d) 6
13. It is given that the roots of the
equation x x P
2
3
4 0 - - = log are
real. For this the minimum value of
P is
(a)
1
27
(b)
1
64
(c)
1
81
(d) 1
PAPER : I Mathematics
14. If A is a square matrix, then the
value ofadj adj A A
T T
- ( ) is equal
to
(a) A
(b) 2 | | A I, where I is the identity matrix
(c) null matrix whose order is same as
that of A
(d) unit matrix whose order is same as
that of A
15. The value of the product
6 6 6 6
1
2
1
4
1
8
1
16
´ ´ ´ ´ … up to infinite
terms is
(a) 6 (b) 36 (c) 216   (d) 512
16. The value of the determinant
cos sin
sin cos
2 2
2 2
2
2 2
q q
2
q q
½
½
½
½
½
½
½
½
½
½
for all values ofq, is
(a)1 (b) cosq (c) sinq (d) cos2q
17. The number of terms in the
expansion of ( ) ( ) x a x a + + -
100 100
after simplification is
(a) 202 (b) 101 (c) 51 (d) 50
18. In the expansion of ( ) 1
50
+ x , the
sum of the coefficients of odd
powers of x is
(a) 2
26
(b) 2
49
(c) 2
50
(d) 2
51
19. Ifa b c , , are non-zero real numbers,
then the inverse of the matrix
A
a
b
c
=
é
ë
ê
ê
ê
ù
û
ú
ú
ú
0 0
0 0
0 0
is equal to
(a)
a
b
c
-
-
-
é
ë
ê
ê
ê
ù
û
ú
ú
ú
1
1
1
0 0
0 0
0 0
(b)
1
0 0
0 0
0 0
1
1
1
abc
a
b
c
-
-
-
é
ë
ê
ê
ê
ù
û
ú
ú
ú
(c)
1
1 0 0
0 1 0
0 0 1
abc
é
ë
ê
ê
ê
ù
û
ú
ú
ú
(d)
1
0 0
0 0
0 0
abc
a
b
c
é
ë
ê
ê
ê
ù
û
ú
ú
ú
20. A person is to count 4500 notes. Let
a
n
denote the number of notes he
counts in the n
th
minute. If
a a a
1 2 3
= = = …= = a
10
150, anda
10
,
a
11
, a
12
, … are in AP with the
common difference -2, then the
time taken by him to count all the
notes is
(a) 24 minutes (b) 34 minutes
(c) 125 minutes (d) 135 minutes
21. The smallest positive integer n for
which
1
1
1
+
-
æ
è
ç
ö
ø
÷
=
i
i
n
, is
(a) 1 (b) 4 (c) 8 (d) 16
22. If we define a relation R on the set
N N ´ as ( , ) ( , ) a b R c d Û
a d b c + = + for all ( , ), ( , ) a b c d
Î ´ N N, then the relation is
(a) symmetric only
(b) symmetric and transitive only
(c) equivalence relation
(d) reflexive only
23. If y x x x = + + +
2 3
… up to
infinite terms where x < 1, then
which one of the following is
correct?
(a) x
y
y
=
+ 1
(b) x
y
y
=
- 1
(c) x
y
y
=
+ 1
(d) x
y
y
=
- 1
24. If a and b are the roots of the
equation 3 2 1 0
2
x x + + = , then the
equation whose roots are a b +
-1
andb a +
-1
is
(a) 3 8 16 0
2
x x + + =
(b) 3 8 16 0
2
x x - - =
(c) 3 8 16 0
2
x x + - =
(d) x x
2
8 16 0 + + =
25. The value of
1 1 1
3 3
2
3
4
log log log e e e
+ + + …up
to infinite terms is
(a) log
e
9 (b) 0 (c) 1 (d) log
e
3
26. A tea party is arranged for 16
people along two sides of a long
table with eight chairs on each side.
Four particular men wish to sit on
one particular side and two
particular men on the other side.
The number of ways they can be
seated is
(a) 24 8 8 ´ ´ ! ! (b) ( !) 8
3
(c) 210 8 8 ´ ´ ! ! (d) 16!
27. The system of equations
kx y z + + = 1, x ky z k + + = and
x y kz k + + =
2
has no solution ifk
equals.
(a) 0 (b) 1
(c)-1 (d) -2
28. If 13 23 33 3
2 3
. . . . + + +¼+n
n
=
- + ( ) 2 1 3
4
n b
a
then a and b are
respectively
(a) n, 2 (b) n, 3
(c) n+ 1 2 , (d) n+ 1 3 ,
29. In DPQR, Ð = R
p
2
. If tan
P
2
æ
è
ç
ö
ø
÷
and
tan
Q
2
æ
è
ç
ö
ø
÷
are the roots of the
equation ax bx c
2
0 + + = , then
which one of the following is
correct?
(a) a b c = + (b) b c a = +
(c)c a b = + (d) b c =
30. If z
z
-
½
½
½
½
½
½
=
4
2, Then the maximum
value of | | z is equal to
(a) 1 3 + (b) 1 5 +
(c) 1 5 - (d) 5 1 -
31. The angle of elevation of a
stationary cloud from a point 25 m
above a lake is 15° and the angle of
depression of its image in the lake is
45°. The height of the cloud above
the lake level is
(a) 25 m (b) 25 3 m
(c) 50 m (d) 50 3 m
32. The value of
tan tan tan tan 9 27 63 81 ° - ° - ° + °
is equal to
(a)-1 (b) 0
(c) 1 (d) 4
33. The value of 3 20 20 cosec ° - ° sec
is equal to
(a) 4 (b) 2
(c) 1 (d) -4
2
Page 3


1. If x x
x
+ + = + log ( ) log log
10 10 10
1 2 5 6
then x is equal to
(a) 2, -3 (b) 2 only (c) 1 (d) 3
2. The remainder and the quotient of the
binary division ( ) ( ) 101110 110
2 2
¸ are
respectively
(a) ( ) 111
2
and ( ) 100
2
(b) ( ) 100
2
and ( ) 111
2
(c) ( ) 101
2
and ( ) 101
2
(d) ( ) 100
2
and ( ) 100
2
3. The matrix A has x rows and x + 5
columns. The matrix B has y rows and
11-y columns. Both AB and BA exist.
What are the values of x and y
respectively?
(a) 8 and 3 (b) 3 and 4
(c) 3 and 8 (d) 8 and 8
4. If S nP
n n Q
n
= +
- ( ) 1
2
, where S
n
denotes the sum of the firstn terms of an
AP, then the common difference is
(a) P Q + (b) 2 3 P Q +
(c) 2Q (d)Q
5. The roots of the equation
( ) ( ) ( ) q r x r p x p q - + - + - =
2
0 are
(a)
( )
( )
,
r p
q r
-
-
1
2
(b)
( )
( )
,
p q
q r
-
-
1
(c)
( )
( )
,
q r
p q
-
-
1
(d)
( )
( )
,
r p
p q
-
-
1
2
6. If E is the universal set and
A B C = È , then the set
E E E E E A - - - - - ( ( ( ( )))) is
same as the set
(a) B C ¢ È ¢ (b) B C È
(c) B C ¢ Ç ¢ (d) B C Ç
7. If A = {x x : is a multiple of 2},
B= {x x : is a multiple of 5} and
C = {x x : is a multiple of 10},
then A B C Ç Ç ( ) is equal to
(a) A (b) B
(c)C
(d) {x x : is a multiple of 100}
8. Ifa andb are the roots of the
equation 1 0
2
+ + = x x , then
the matrix product
1 b
a a
é
ë
ê
ù
û
ú
a b
b 1
é
ë
ê
ù
û
ú
is equal to
(a)
1 1
1 2
é
ë
ê
ù
û
ú
(b)
- -
-
é
ë
ê
ù
û
ú
1 1
1 2
(c)
1 1
1 2
-
-
é
ë
ê
ù
û
ú
(d)
- -
- -
é
ë
ê
ù
û
ú
1 1
1 2
9. If| | a denotes the absolute value
of an integer, then which of the
following are correct?
1. | | | || | ab a b =
2. | | | | | | a b a b + £ +
3. | | | | | | a b a b - ³ -
Select the correct answer using the
code given below.
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
10. How many different permutation
can be made out of the letters of the
word ‘PERMUTATION’?
(a) 19958400 (b) 19954800
(c) 19952400 (d) 39916800
11. If A
i i
i i
=
-
+
é
ë
ê
ù
û
ú
4 6 10
14 6 4
and k
i
=
1
2
,
wherei = -1, thenkA is equal to
(a)
2 3 5
7 2 3
+
-
é
ë
ê
ù
û
ú
i
i
(b)
2 3 5
7 2 3
-
+
é
ë
ê
ù
û
ú
i
i
(c)
2 3 7
5 2 3
-
+
é
ë
ê
ù
û
ú
i
i
(d)
2 3 5
7 2 3
+
+
é
ë
ê
ù
û
ú
i
i
12. The sum of all real roots of the
equation | | | | x x - + - - = 3 3 2 0
2
is
(a) 2 (b) 3
(c) 4 (d) 6
13. It is given that the roots of the
equation x x P
2
3
4 0 - - = log are
real. For this the minimum value of
P is
(a)
1
27
(b)
1
64
(c)
1
81
(d) 1
PAPER : I Mathematics
14. If A is a square matrix, then the
value ofadj adj A A
T T
- ( ) is equal
to
(a) A
(b) 2 | | A I, where I is the identity matrix
(c) null matrix whose order is same as
that of A
(d) unit matrix whose order is same as
that of A
15. The value of the product
6 6 6 6
1
2
1
4
1
8
1
16
´ ´ ´ ´ … up to infinite
terms is
(a) 6 (b) 36 (c) 216   (d) 512
16. The value of the determinant
cos sin
sin cos
2 2
2 2
2
2 2
q q
2
q q
½
½
½
½
½
½
½
½
½
½
for all values ofq, is
(a)1 (b) cosq (c) sinq (d) cos2q
17. The number of terms in the
expansion of ( ) ( ) x a x a + + -
100 100
after simplification is
(a) 202 (b) 101 (c) 51 (d) 50
18. In the expansion of ( ) 1
50
+ x , the
sum of the coefficients of odd
powers of x is
(a) 2
26
(b) 2
49
(c) 2
50
(d) 2
51
19. Ifa b c , , are non-zero real numbers,
then the inverse of the matrix
A
a
b
c
=
é
ë
ê
ê
ê
ù
û
ú
ú
ú
0 0
0 0
0 0
is equal to
(a)
a
b
c
-
-
-
é
ë
ê
ê
ê
ù
û
ú
ú
ú
1
1
1
0 0
0 0
0 0
(b)
1
0 0
0 0
0 0
1
1
1
abc
a
b
c
-
-
-
é
ë
ê
ê
ê
ù
û
ú
ú
ú
(c)
1
1 0 0
0 1 0
0 0 1
abc
é
ë
ê
ê
ê
ù
û
ú
ú
ú
(d)
1
0 0
0 0
0 0
abc
a
b
c
é
ë
ê
ê
ê
ù
û
ú
ú
ú
20. A person is to count 4500 notes. Let
a
n
denote the number of notes he
counts in the n
th
minute. If
a a a
1 2 3
= = = …= = a
10
150, anda
10
,
a
11
, a
12
, … are in AP with the
common difference -2, then the
time taken by him to count all the
notes is
(a) 24 minutes (b) 34 minutes
(c) 125 minutes (d) 135 minutes
21. The smallest positive integer n for
which
1
1
1
+
-
æ
è
ç
ö
ø
÷
=
i
i
n
, is
(a) 1 (b) 4 (c) 8 (d) 16
22. If we define a relation R on the set
N N ´ as ( , ) ( , ) a b R c d Û
a d b c + = + for all ( , ), ( , ) a b c d
Î ´ N N, then the relation is
(a) symmetric only
(b) symmetric and transitive only
(c) equivalence relation
(d) reflexive only
23. If y x x x = + + +
2 3
… up to
infinite terms where x < 1, then
which one of the following is
correct?
(a) x
y
y
=
+ 1
(b) x
y
y
=
- 1
(c) x
y
y
=
+ 1
(d) x
y
y
=
- 1
24. If a and b are the roots of the
equation 3 2 1 0
2
x x + + = , then the
equation whose roots are a b +
-1
andb a +
-1
is
(a) 3 8 16 0
2
x x + + =
(b) 3 8 16 0
2
x x - - =
(c) 3 8 16 0
2
x x + - =
(d) x x
2
8 16 0 + + =
25. The value of
1 1 1
3 3
2
3
4
log log log e e e
+ + + …up
to infinite terms is
(a) log
e
9 (b) 0 (c) 1 (d) log
e
3
26. A tea party is arranged for 16
people along two sides of a long
table with eight chairs on each side.
Four particular men wish to sit on
one particular side and two
particular men on the other side.
The number of ways they can be
seated is
(a) 24 8 8 ´ ´ ! ! (b) ( !) 8
3
(c) 210 8 8 ´ ´ ! ! (d) 16!
27. The system of equations
kx y z + + = 1, x ky z k + + = and
x y kz k + + =
2
has no solution ifk
equals.
(a) 0 (b) 1
(c)-1 (d) -2
28. If 13 23 33 3
2 3
. . . . + + +¼+n
n
=
- + ( ) 2 1 3
4
n b
a
then a and b are
respectively
(a) n, 2 (b) n, 3
(c) n+ 1 2 , (d) n+ 1 3 ,
29. In DPQR, Ð = R
p
2
. If tan
P
2
æ
è
ç
ö
ø
÷
and
tan
Q
2
æ
è
ç
ö
ø
÷
are the roots of the
equation ax bx c
2
0 + + = , then
which one of the following is
correct?
(a) a b c = + (b) b c a = +
(c)c a b = + (d) b c =
30. If z
z
-
½
½
½
½
½
½
=
4
2, Then the maximum
value of | | z is equal to
(a) 1 3 + (b) 1 5 +
(c) 1 5 - (d) 5 1 -
31. The angle of elevation of a
stationary cloud from a point 25 m
above a lake is 15° and the angle of
depression of its image in the lake is
45°. The height of the cloud above
the lake level is
(a) 25 m (b) 25 3 m
(c) 50 m (d) 50 3 m
32. The value of
tan tan tan tan 9 27 63 81 ° - ° - ° + °
is equal to
(a)-1 (b) 0
(c) 1 (d) 4
33. The value of 3 20 20 cosec ° - ° sec
is equal to
(a) 4 (b) 2
(c) 1 (d) -4
2
34. Anglea is divided into two parts A
and B such that A B x - = and
tan : tan : A B p q = . The value of
sinx is equal to
(a)
( )sin p q
p q
+
-
a
(b)
p
p q
sina
+
(c)
p
p q
sina
-
(d)
( )sin p q
p q
-
+
a
35. The value of
sin tan
- - æ
è
ç
ö
ø
÷
+
æ
è
ç
ö
ø
÷
1 1
3
5
1
7
is equal to
(a) 0 (b)
p
4
(c)
p
3
(d)
p
2
36. The angles of elevation of the top of
a tower from the top and foot of a
pole are respectively 30° and 45°. If
h
T
is the height of the tower andh
P
is the height of the pole, then which
of the following are correct?
1.
2
3 3
2
h h
h
P T
P
+
=
2.
h h h
T P P
-
+
=
3 1
2
3.
2
4 3
( ) h h
h
P T
P
+
= +
Select the correct answer using the
code given below.
(a) 1 and 3 only (b) 2 and 3 only
(c) 1 and 2 only (d) 1, 2 and 3
37. In a triangle ABC, a b c - + = 2 0.
The value of cot cot
A C
2 2
æ
è
ç
ö
ø
÷
æ
è
ç
ö
ø
÷
is
(a)
9
2
(b) 3 (c)
3
2
(d) 1
38. 1
2 2
+ = - +
æ
è
ç
ö
ø
÷
sin sin cos A
A A
is
true if
(a)
3
2
5
2
p p
< < A only  (b)
p p
2
3
2
< < A only
(c)
3
2
7
2
p p
< < A (d) 0
3
2
< < A
p
39. In triangle ABC, if
sin sin sin
cos cos cos
2 2 2
2 2 2
2
A B C
A B C
+ +
+ +
=
then the triangle is
(a) right-angled (b) equilateral
(c) isosceles (d) obtuse-angled
40. The principal value of sin
-1
x lies in
the interval
(a) -
æ
è
ç
ö
ø
÷
p p
2 2
, (b) -
é
ë
ê
ù
û
ú
p p
2 2
,
(c) 0
2
,
p é
ë
ê
ù
û
ú
(d) [ , ] 0 p
41. The points ( , ), ( , ) a b 0 0 , ( , ) - - a b and
( , ) ab b
2
are
(a) the vertices of a parallelogram
(b) the vertices of a rectangle
(c) the vertices of a square
(d) collinear
42. The length of the normal from
origin to the planex y z + - = 2 2 9 is
equal to
(a) 2 units (b) 3 units
(c) 4 units (d) 5 units
43. If a b , and g are the angles which
the vectorOP
¾®
(O being the origin)
makes with positive direction of the
coordinate axes, then which of the
following are correct?
1. cos cos sin
2 2 2
a b g + =
2. sin sin cos
2 2 2
a b g + =
3. sin sin sin
2 2 2
2 a b g + + =
Select the correct answer using the
code given below.
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
44. The angle between the lines
x y + - = 3 0 and x y - + = 3 0 is a
and the acute angle between the
lines x y - + = 3 2 3 0 and
3 1 0 x y - + = is b. Which one of
the following is correct?
(a)a b = (b) a b >
(c)a b < (d) a b = 2
45. Let a
®
= + -
$ $
$
i j k 2 , b
®
= - + 2 3
$ $
$
i j k
andg
®
= + + 2 6
$ $
$
i j k be three vectors.
Ifa
®
andb
®
are both perpendicular to
the vectord andd g × = 10, then what
is the magnitude ofd?
(a) 3 units (b) 2 3 units
(c)
3
2
unit (d)
1
3
unit
46. If $ a and
$
b are two unit vectors, then the
vector ($
$
) ($
$
) a b a b + ´ ´ is parallel to
(a) (
$
$
) a b - (b) (
$
$
) a b +
(c) (
$
$
) 2a b - (d) (
$
$
) 2a b +
47. A force F
®
= + +
$ $
$
i j k 3 2 acts on a
particle to displace it from the point
A i j k (
$ $
$
) + - 2 3 to the point
B i j k (
$ $
$
) 3 5 - + . The work done by
the force will be
(a) 5 units (b) 7 units
(c) 9 units (d) 10 units
48. For any vector a
®
|
$
| |
$
| |
$
| a a a
® ® ®
´ + ´ + ´ i j k
2 2 2
is equal to
(a)| | a
®
2
(b) 2
2
| | a
®
(c) 3
2
| | a
®
(d) 4
2
| | a
®
49. A man running round a racecourse
notes that the sum of the distances
of two flag-posts from him is
always 10 m and the distance
between the flag-posts is 8 m. The
area of the path he encloses is
(a) 18p square metres
(b) 15p square metres
(c) 12p square metres
(d) 8p square metres
50. The distance of the point (1, 3) from
the line 2 3 6 x y + = , measured
parallel to the line 4 4 x y + = , is
(a)
5
13
units (b)
3
17
units
(c) 17 units (d)
17
2
units
51. If the vectors ai j k
$ $
$
+ + ,
$ $
$
i bj k + +
and
$ $
$
i j ck + + ( , , ) a b c ¹ 1 are
coplanar, then the value of
1
1
1
1
1
1 -
+
-
+
- a b c
is equal to
(a) 0 (b) 1
(c) a b c + + (d) abc
52. The point of intersection of the line
joining the points ( , , ) - - 3 4 8 and
( , , ) 5 6 4 - with XY-plane is
(a)
7
3
8
3
0 , , -
æ
è
ç
ö
ø
÷
(b) - -
æ
è
ç
ö
ø
÷
7
3
8
3
0 , ,
(c) -
æ
è
ç
ö
ø
÷
7
3
8
3
0 , , (d)
7
3
8
3
0 , ,
æ
è
ç
ö
ø
÷
3
Page 4


1. If x x
x
+ + = + log ( ) log log
10 10 10
1 2 5 6
then x is equal to
(a) 2, -3 (b) 2 only (c) 1 (d) 3
2. The remainder and the quotient of the
binary division ( ) ( ) 101110 110
2 2
¸ are
respectively
(a) ( ) 111
2
and ( ) 100
2
(b) ( ) 100
2
and ( ) 111
2
(c) ( ) 101
2
and ( ) 101
2
(d) ( ) 100
2
and ( ) 100
2
3. The matrix A has x rows and x + 5
columns. The matrix B has y rows and
11-y columns. Both AB and BA exist.
What are the values of x and y
respectively?
(a) 8 and 3 (b) 3 and 4
(c) 3 and 8 (d) 8 and 8
4. If S nP
n n Q
n
= +
- ( ) 1
2
, where S
n
denotes the sum of the firstn terms of an
AP, then the common difference is
(a) P Q + (b) 2 3 P Q +
(c) 2Q (d)Q
5. The roots of the equation
( ) ( ) ( ) q r x r p x p q - + - + - =
2
0 are
(a)
( )
( )
,
r p
q r
-
-
1
2
(b)
( )
( )
,
p q
q r
-
-
1
(c)
( )
( )
,
q r
p q
-
-
1
(d)
( )
( )
,
r p
p q
-
-
1
2
6. If E is the universal set and
A B C = È , then the set
E E E E E A - - - - - ( ( ( ( )))) is
same as the set
(a) B C ¢ È ¢ (b) B C È
(c) B C ¢ Ç ¢ (d) B C Ç
7. If A = {x x : is a multiple of 2},
B= {x x : is a multiple of 5} and
C = {x x : is a multiple of 10},
then A B C Ç Ç ( ) is equal to
(a) A (b) B
(c)C
(d) {x x : is a multiple of 100}
8. Ifa andb are the roots of the
equation 1 0
2
+ + = x x , then
the matrix product
1 b
a a
é
ë
ê
ù
û
ú
a b
b 1
é
ë
ê
ù
û
ú
is equal to
(a)
1 1
1 2
é
ë
ê
ù
û
ú
(b)
- -
-
é
ë
ê
ù
û
ú
1 1
1 2
(c)
1 1
1 2
-
-
é
ë
ê
ù
û
ú
(d)
- -
- -
é
ë
ê
ù
û
ú
1 1
1 2
9. If| | a denotes the absolute value
of an integer, then which of the
following are correct?
1. | | | || | ab a b =
2. | | | | | | a b a b + £ +
3. | | | | | | a b a b - ³ -
Select the correct answer using the
code given below.
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
10. How many different permutation
can be made out of the letters of the
word ‘PERMUTATION’?
(a) 19958400 (b) 19954800
(c) 19952400 (d) 39916800
11. If A
i i
i i
=
-
+
é
ë
ê
ù
û
ú
4 6 10
14 6 4
and k
i
=
1
2
,
wherei = -1, thenkA is equal to
(a)
2 3 5
7 2 3
+
-
é
ë
ê
ù
û
ú
i
i
(b)
2 3 5
7 2 3
-
+
é
ë
ê
ù
û
ú
i
i
(c)
2 3 7
5 2 3
-
+
é
ë
ê
ù
û
ú
i
i
(d)
2 3 5
7 2 3
+
+
é
ë
ê
ù
û
ú
i
i
12. The sum of all real roots of the
equation | | | | x x - + - - = 3 3 2 0
2
is
(a) 2 (b) 3
(c) 4 (d) 6
13. It is given that the roots of the
equation x x P
2
3
4 0 - - = log are
real. For this the minimum value of
P is
(a)
1
27
(b)
1
64
(c)
1
81
(d) 1
PAPER : I Mathematics
14. If A is a square matrix, then the
value ofadj adj A A
T T
- ( ) is equal
to
(a) A
(b) 2 | | A I, where I is the identity matrix
(c) null matrix whose order is same as
that of A
(d) unit matrix whose order is same as
that of A
15. The value of the product
6 6 6 6
1
2
1
4
1
8
1
16
´ ´ ´ ´ … up to infinite
terms is
(a) 6 (b) 36 (c) 216   (d) 512
16. The value of the determinant
cos sin
sin cos
2 2
2 2
2
2 2
q q
2
q q
½
½
½
½
½
½
½
½
½
½
for all values ofq, is
(a)1 (b) cosq (c) sinq (d) cos2q
17. The number of terms in the
expansion of ( ) ( ) x a x a + + -
100 100
after simplification is
(a) 202 (b) 101 (c) 51 (d) 50
18. In the expansion of ( ) 1
50
+ x , the
sum of the coefficients of odd
powers of x is
(a) 2
26
(b) 2
49
(c) 2
50
(d) 2
51
19. Ifa b c , , are non-zero real numbers,
then the inverse of the matrix
A
a
b
c
=
é
ë
ê
ê
ê
ù
û
ú
ú
ú
0 0
0 0
0 0
is equal to
(a)
a
b
c
-
-
-
é
ë
ê
ê
ê
ù
û
ú
ú
ú
1
1
1
0 0
0 0
0 0
(b)
1
0 0
0 0
0 0
1
1
1
abc
a
b
c
-
-
-
é
ë
ê
ê
ê
ù
û
ú
ú
ú
(c)
1
1 0 0
0 1 0
0 0 1
abc
é
ë
ê
ê
ê
ù
û
ú
ú
ú
(d)
1
0 0
0 0
0 0
abc
a
b
c
é
ë
ê
ê
ê
ù
û
ú
ú
ú
20. A person is to count 4500 notes. Let
a
n
denote the number of notes he
counts in the n
th
minute. If
a a a
1 2 3
= = = …= = a
10
150, anda
10
,
a
11
, a
12
, … are in AP with the
common difference -2, then the
time taken by him to count all the
notes is
(a) 24 minutes (b) 34 minutes
(c) 125 minutes (d) 135 minutes
21. The smallest positive integer n for
which
1
1
1
+
-
æ
è
ç
ö
ø
÷
=
i
i
n
, is
(a) 1 (b) 4 (c) 8 (d) 16
22. If we define a relation R on the set
N N ´ as ( , ) ( , ) a b R c d Û
a d b c + = + for all ( , ), ( , ) a b c d
Î ´ N N, then the relation is
(a) symmetric only
(b) symmetric and transitive only
(c) equivalence relation
(d) reflexive only
23. If y x x x = + + +
2 3
… up to
infinite terms where x < 1, then
which one of the following is
correct?
(a) x
y
y
=
+ 1
(b) x
y
y
=
- 1
(c) x
y
y
=
+ 1
(d) x
y
y
=
- 1
24. If a and b are the roots of the
equation 3 2 1 0
2
x x + + = , then the
equation whose roots are a b +
-1
andb a +
-1
is
(a) 3 8 16 0
2
x x + + =
(b) 3 8 16 0
2
x x - - =
(c) 3 8 16 0
2
x x + - =
(d) x x
2
8 16 0 + + =
25. The value of
1 1 1
3 3
2
3
4
log log log e e e
+ + + …up
to infinite terms is
(a) log
e
9 (b) 0 (c) 1 (d) log
e
3
26. A tea party is arranged for 16
people along two sides of a long
table with eight chairs on each side.
Four particular men wish to sit on
one particular side and two
particular men on the other side.
The number of ways they can be
seated is
(a) 24 8 8 ´ ´ ! ! (b) ( !) 8
3
(c) 210 8 8 ´ ´ ! ! (d) 16!
27. The system of equations
kx y z + + = 1, x ky z k + + = and
x y kz k + + =
2
has no solution ifk
equals.
(a) 0 (b) 1
(c)-1 (d) -2
28. If 13 23 33 3
2 3
. . . . + + +¼+n
n
=
- + ( ) 2 1 3
4
n b
a
then a and b are
respectively
(a) n, 2 (b) n, 3
(c) n+ 1 2 , (d) n+ 1 3 ,
29. In DPQR, Ð = R
p
2
. If tan
P
2
æ
è
ç
ö
ø
÷
and
tan
Q
2
æ
è
ç
ö
ø
÷
are the roots of the
equation ax bx c
2
0 + + = , then
which one of the following is
correct?
(a) a b c = + (b) b c a = +
(c)c a b = + (d) b c =
30. If z
z
-
½
½
½
½
½
½
=
4
2, Then the maximum
value of | | z is equal to
(a) 1 3 + (b) 1 5 +
(c) 1 5 - (d) 5 1 -
31. The angle of elevation of a
stationary cloud from a point 25 m
above a lake is 15° and the angle of
depression of its image in the lake is
45°. The height of the cloud above
the lake level is
(a) 25 m (b) 25 3 m
(c) 50 m (d) 50 3 m
32. The value of
tan tan tan tan 9 27 63 81 ° - ° - ° + °
is equal to
(a)-1 (b) 0
(c) 1 (d) 4
33. The value of 3 20 20 cosec ° - ° sec
is equal to
(a) 4 (b) 2
(c) 1 (d) -4
2
34. Anglea is divided into two parts A
and B such that A B x - = and
tan : tan : A B p q = . The value of
sinx is equal to
(a)
( )sin p q
p q
+
-
a
(b)
p
p q
sina
+
(c)
p
p q
sina
-
(d)
( )sin p q
p q
-
+
a
35. The value of
sin tan
- - æ
è
ç
ö
ø
÷
+
æ
è
ç
ö
ø
÷
1 1
3
5
1
7
is equal to
(a) 0 (b)
p
4
(c)
p
3
(d)
p
2
36. The angles of elevation of the top of
a tower from the top and foot of a
pole are respectively 30° and 45°. If
h
T
is the height of the tower andh
P
is the height of the pole, then which
of the following are correct?
1.
2
3 3
2
h h
h
P T
P
+
=
2.
h h h
T P P
-
+
=
3 1
2
3.
2
4 3
( ) h h
h
P T
P
+
= +
Select the correct answer using the
code given below.
(a) 1 and 3 only (b) 2 and 3 only
(c) 1 and 2 only (d) 1, 2 and 3
37. In a triangle ABC, a b c - + = 2 0.
The value of cot cot
A C
2 2
æ
è
ç
ö
ø
÷
æ
è
ç
ö
ø
÷
is
(a)
9
2
(b) 3 (c)
3
2
(d) 1
38. 1
2 2
+ = - +
æ
è
ç
ö
ø
÷
sin sin cos A
A A
is
true if
(a)
3
2
5
2
p p
< < A only  (b)
p p
2
3
2
< < A only
(c)
3
2
7
2
p p
< < A (d) 0
3
2
< < A
p
39. In triangle ABC, if
sin sin sin
cos cos cos
2 2 2
2 2 2
2
A B C
A B C
+ +
+ +
=
then the triangle is
(a) right-angled (b) equilateral
(c) isosceles (d) obtuse-angled
40. The principal value of sin
-1
x lies in
the interval
(a) -
æ
è
ç
ö
ø
÷
p p
2 2
, (b) -
é
ë
ê
ù
û
ú
p p
2 2
,
(c) 0
2
,
p é
ë
ê
ù
û
ú
(d) [ , ] 0 p
41. The points ( , ), ( , ) a b 0 0 , ( , ) - - a b and
( , ) ab b
2
are
(a) the vertices of a parallelogram
(b) the vertices of a rectangle
(c) the vertices of a square
(d) collinear
42. The length of the normal from
origin to the planex y z + - = 2 2 9 is
equal to
(a) 2 units (b) 3 units
(c) 4 units (d) 5 units
43. If a b , and g are the angles which
the vectorOP
¾®
(O being the origin)
makes with positive direction of the
coordinate axes, then which of the
following are correct?
1. cos cos sin
2 2 2
a b g + =
2. sin sin cos
2 2 2
a b g + =
3. sin sin sin
2 2 2
2 a b g + + =
Select the correct answer using the
code given below.
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
44. The angle between the lines
x y + - = 3 0 and x y - + = 3 0 is a
and the acute angle between the
lines x y - + = 3 2 3 0 and
3 1 0 x y - + = is b. Which one of
the following is correct?
(a)a b = (b) a b >
(c)a b < (d) a b = 2
45. Let a
®
= + -
$ $
$
i j k 2 , b
®
= - + 2 3
$ $
$
i j k
andg
®
= + + 2 6
$ $
$
i j k be three vectors.
Ifa
®
andb
®
are both perpendicular to
the vectord andd g × = 10, then what
is the magnitude ofd?
(a) 3 units (b) 2 3 units
(c)
3
2
unit (d)
1
3
unit
46. If $ a and
$
b are two unit vectors, then the
vector ($
$
) ($
$
) a b a b + ´ ´ is parallel to
(a) (
$
$
) a b - (b) (
$
$
) a b +
(c) (
$
$
) 2a b - (d) (
$
$
) 2a b +
47. A force F
®
= + +
$ $
$
i j k 3 2 acts on a
particle to displace it from the point
A i j k (
$ $
$
) + - 2 3 to the point
B i j k (
$ $
$
) 3 5 - + . The work done by
the force will be
(a) 5 units (b) 7 units
(c) 9 units (d) 10 units
48. For any vector a
®
|
$
| |
$
| |
$
| a a a
® ® ®
´ + ´ + ´ i j k
2 2 2
is equal to
(a)| | a
®
2
(b) 2
2
| | a
®
(c) 3
2
| | a
®
(d) 4
2
| | a
®
49. A man running round a racecourse
notes that the sum of the distances
of two flag-posts from him is
always 10 m and the distance
between the flag-posts is 8 m. The
area of the path he encloses is
(a) 18p square metres
(b) 15p square metres
(c) 12p square metres
(d) 8p square metres
50. The distance of the point (1, 3) from
the line 2 3 6 x y + = , measured
parallel to the line 4 4 x y + = , is
(a)
5
13
units (b)
3
17
units
(c) 17 units (d)
17
2
units
51. If the vectors ai j k
$ $
$
+ + ,
$ $
$
i bj k + +
and
$ $
$
i j ck + + ( , , ) a b c ¹ 1 are
coplanar, then the value of
1
1
1
1
1
1 -
+
-
+
- a b c
is equal to
(a) 0 (b) 1
(c) a b c + + (d) abc
52. The point of intersection of the line
joining the points ( , , ) - - 3 4 8 and
( , , ) 5 6 4 - with XY-plane is
(a)
7
3
8
3
0 , , -
æ
è
ç
ö
ø
÷
(b) - -
æ
è
ç
ö
ø
÷
7
3
8
3
0 , ,
(c) -
æ
è
ç
ö
ø
÷
7
3
8
3
0 , , (d)
7
3
8
3
0 , ,
æ
è
ç
ö
ø
÷
3
53. If the angle between the lines
whose direction ratios are ( , , ) 2 1 2 -
and x, , 3 5 is
p
4
, then the smaller
value of x is
(a) 52 (b) 4 (c) 2 (d) 1
54. The position of the point ( , ) 1 2
relative to the ellipse
2 7 20
2 2
x y + = is
(a) outside the ellipse
(b) inside the ellipse but not at the focus
(c) on the ellipse
(d) at the focus
55. The equation of straight line which
cuts off an intercept of 5 units on
negative direction of Y-axis and
makes and angle 120° with positive
direction of X-axis is
(a) y x + + = 3 5 0
(b) y x - + = 3 5 0
(c) y x + - = 3 5 0
(d) y x - - = 3 5 0
56. The equation of the line passing
through the point (2, 3) and the
point of intersection of lines
2 3 7 0 x y - + = and 7 4 2 0 x y + + =
is
(a) 21 46 180 0 x y + - =
(b) 21 46 96 0 x y - + =
(c) 46 21 155 0 x y + - =
(d) 46 21 29 0 x y - - =
57. The equation of the ellipse whose
centre is at origin, major axis is
along X-axis with eccentricity
3
4
and latus rectum 4 units is
(a)
x y
2 2
1024
7
64
1 + = (b)
49
1024
7
64
1
2 2
x y
+ =
(c)
7
1024
49
64
1
2 2
x y
+ = (d)
x y
2 2
1024 64
1 + =
58. The equation of the circle which
passes through the points (1, 0), (0,
-6) and (3, 4) is
(a) 4 4 142 47 140 0
2 2
x y x y + + + + =
(b) 4 4 142 47 138 0
2 2
x y x y + - - + =
(c) 4 4 142 47 138 0
2 2
x y x y + - + + =
(d) 4 4 150 49 138 0
2 2
x y x y + + - + =
59. A variable plane passes through a
fixed point ( , , ) a b c and cuts the axes
in A B , and C respectively. The
locus of the centre of the sphere
OABC,O being the origin, is
(a)
x
a
y
b
z
c
+ + = 1 (b)
a
x
b
y
c
z
+ + = 1
(c)
a
x
b
y
c
z
+ + = 2 (d)
x
a
y
b
z
c
+ + = 2
60. The equation of the plane passing
through the line of intersection of
the planes x y z + + = 1,
2 3 4 7 x y z + + = , and perpendicular
to the plane x y z - + = 5 3 5 is given
by
(a) x y z + + - = 2 3 6 0
(b) x y z + + + = 2 3 6 0
(c) 3 4 5 8 0 x y z + + - =
(d) 3 4 5 8 0 x y z + + + =
61. The inverse of the functiony
x
= 5
ln
is
(a) x y y = >
1
5
0
ln
,
(b) x y y = >
ln
,
5
0
(c) x y y = <
1
5
0
ln
,
(d) x y y = > 5 0 ln ,
62. A function is defined as follows :
f x
x
x
x
x
( )
,
,
=
- ¹
=
ì
í
ï
î
ï
2
0
0 0
Which one of the following is
correct in respect of the above
function?
(a) f x ( ) is continuous at x = 0 but not
differentiable at x = 0
(b) f x ( ) is continuous as well as
differentiable at x = 0
(c) f x ( )is discontinuous at x = 0
(d) None of the above
63. If y x
x
x
=
¥
(cos )
(cos )
(cos )
, then
dy
dx
is
equal to
(a)-
-
y x
y x
2
1
tan
ln(cos )
(b)
y x
y x
2
1
tan
ln(cos ) +
(c)
y x
y x
2
1
tan
ln(sin ) -
(d)
y x
y x
2
1
sin
ln(sin ) +
64. Consider the following
1. x x +
2
is continuous at x = 0
2. x
x
+ cos
1
is discontinuous at
x = 0
3. x
x
2
1
+ cos is continuous atx = 0
Which of the above are correct?
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
65. Consider the following statements :
1.
dy
dx
at a point on the curve gives
slope of the tangent at that
point.
2. If a t ( ) denotes acceleration of a
particle, then a t dt c
ò
+ ( ) gives
velocity of the particle.
3. If s t ( ) gives displacement of a
particle at time t, then
ds
dt
gives
its acceleration at that instant.
Which of the above statements
is/are correct?
(a) 1 and 2 only (b) 2 only
(c) 1 only (d) 1, 2 and 3
66. If y
x
x
x
x
=
+
-
æ
è
ç
ö
ø
÷
+
-
+
æ
è
ç
ö
ø
÷
- -
sec sin
1 1
1
1
1
1
,
then
dy
dx
is equal to
(a) 0 (b) 1
(c)
x
x
-
+
1
1
(d)
x
x
+
-
1
1
67. What is tan (sec tan )
-
+
ò
1
x x dx
equal to?
(a)
px x
C
4 4
2
+ + (b)
px x
C
2 4
2
+ +
(c)
p p x x
C
4 4
2
+ + (d)
px x
C
4 4
2
- +
68. A function is defined in ( , ) 0 ¥ by
f x
x x
x x
x x
( )
for
ln for
ln . for
=
- < £
< £
- + < < ¥
æ
è
ç
ç
1 0 1
1 2
2 1 05 2
2
ç
Which one of the following is
correct in respect of the derivative
of the function, i.e., f x ¢ ( )?
(a) f x x ¢ = ( ) 2 for 0 1 < £ x
(b) f x x ¢ = - ( ) 2 for 0 1 < £ x
(c) f x x ¢ = - ( ) 2 for 0 1 < < x
(d) f x ¢ = ( ) 0 for 0< < ¥ x
4
Page 5


1. If x x
x
+ + = + log ( ) log log
10 10 10
1 2 5 6
then x is equal to
(a) 2, -3 (b) 2 only (c) 1 (d) 3
2. The remainder and the quotient of the
binary division ( ) ( ) 101110 110
2 2
¸ are
respectively
(a) ( ) 111
2
and ( ) 100
2
(b) ( ) 100
2
and ( ) 111
2
(c) ( ) 101
2
and ( ) 101
2
(d) ( ) 100
2
and ( ) 100
2
3. The matrix A has x rows and x + 5
columns. The matrix B has y rows and
11-y columns. Both AB and BA exist.
What are the values of x and y
respectively?
(a) 8 and 3 (b) 3 and 4
(c) 3 and 8 (d) 8 and 8
4. If S nP
n n Q
n
= +
- ( ) 1
2
, where S
n
denotes the sum of the firstn terms of an
AP, then the common difference is
(a) P Q + (b) 2 3 P Q +
(c) 2Q (d)Q
5. The roots of the equation
( ) ( ) ( ) q r x r p x p q - + - + - =
2
0 are
(a)
( )
( )
,
r p
q r
-
-
1
2
(b)
( )
( )
,
p q
q r
-
-
1
(c)
( )
( )
,
q r
p q
-
-
1
(d)
( )
( )
,
r p
p q
-
-
1
2
6. If E is the universal set and
A B C = È , then the set
E E E E E A - - - - - ( ( ( ( )))) is
same as the set
(a) B C ¢ È ¢ (b) B C È
(c) B C ¢ Ç ¢ (d) B C Ç
7. If A = {x x : is a multiple of 2},
B= {x x : is a multiple of 5} and
C = {x x : is a multiple of 10},
then A B C Ç Ç ( ) is equal to
(a) A (b) B
(c)C
(d) {x x : is a multiple of 100}
8. Ifa andb are the roots of the
equation 1 0
2
+ + = x x , then
the matrix product
1 b
a a
é
ë
ê
ù
û
ú
a b
b 1
é
ë
ê
ù
û
ú
is equal to
(a)
1 1
1 2
é
ë
ê
ù
û
ú
(b)
- -
-
é
ë
ê
ù
û
ú
1 1
1 2
(c)
1 1
1 2
-
-
é
ë
ê
ù
û
ú
(d)
- -
- -
é
ë
ê
ù
û
ú
1 1
1 2
9. If| | a denotes the absolute value
of an integer, then which of the
following are correct?
1. | | | || | ab a b =
2. | | | | | | a b a b + £ +
3. | | | | | | a b a b - ³ -
Select the correct answer using the
code given below.
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
10. How many different permutation
can be made out of the letters of the
word ‘PERMUTATION’?
(a) 19958400 (b) 19954800
(c) 19952400 (d) 39916800
11. If A
i i
i i
=
-
+
é
ë
ê
ù
û
ú
4 6 10
14 6 4
and k
i
=
1
2
,
wherei = -1, thenkA is equal to
(a)
2 3 5
7 2 3
+
-
é
ë
ê
ù
û
ú
i
i
(b)
2 3 5
7 2 3
-
+
é
ë
ê
ù
û
ú
i
i
(c)
2 3 7
5 2 3
-
+
é
ë
ê
ù
û
ú
i
i
(d)
2 3 5
7 2 3
+
+
é
ë
ê
ù
û
ú
i
i
12. The sum of all real roots of the
equation | | | | x x - + - - = 3 3 2 0
2
is
(a) 2 (b) 3
(c) 4 (d) 6
13. It is given that the roots of the
equation x x P
2
3
4 0 - - = log are
real. For this the minimum value of
P is
(a)
1
27
(b)
1
64
(c)
1
81
(d) 1
PAPER : I Mathematics
14. If A is a square matrix, then the
value ofadj adj A A
T T
- ( ) is equal
to
(a) A
(b) 2 | | A I, where I is the identity matrix
(c) null matrix whose order is same as
that of A
(d) unit matrix whose order is same as
that of A
15. The value of the product
6 6 6 6
1
2
1
4
1
8
1
16
´ ´ ´ ´ … up to infinite
terms is
(a) 6 (b) 36 (c) 216   (d) 512
16. The value of the determinant
cos sin
sin cos
2 2
2 2
2
2 2
q q
2
q q
½
½
½
½
½
½
½
½
½
½
for all values ofq, is
(a)1 (b) cosq (c) sinq (d) cos2q
17. The number of terms in the
expansion of ( ) ( ) x a x a + + -
100 100
after simplification is
(a) 202 (b) 101 (c) 51 (d) 50
18. In the expansion of ( ) 1
50
+ x , the
sum of the coefficients of odd
powers of x is
(a) 2
26
(b) 2
49
(c) 2
50
(d) 2
51
19. Ifa b c , , are non-zero real numbers,
then the inverse of the matrix
A
a
b
c
=
é
ë
ê
ê
ê
ù
û
ú
ú
ú
0 0
0 0
0 0
is equal to
(a)
a
b
c
-
-
-
é
ë
ê
ê
ê
ù
û
ú
ú
ú
1
1
1
0 0
0 0
0 0
(b)
1
0 0
0 0
0 0
1
1
1
abc
a
b
c
-
-
-
é
ë
ê
ê
ê
ù
û
ú
ú
ú
(c)
1
1 0 0
0 1 0
0 0 1
abc
é
ë
ê
ê
ê
ù
û
ú
ú
ú
(d)
1
0 0
0 0
0 0
abc
a
b
c
é
ë
ê
ê
ê
ù
û
ú
ú
ú
20. A person is to count 4500 notes. Let
a
n
denote the number of notes he
counts in the n
th
minute. If
a a a
1 2 3
= = = …= = a
10
150, anda
10
,
a
11
, a
12
, … are in AP with the
common difference -2, then the
time taken by him to count all the
notes is
(a) 24 minutes (b) 34 minutes
(c) 125 minutes (d) 135 minutes
21. The smallest positive integer n for
which
1
1
1
+
-
æ
è
ç
ö
ø
÷
=
i
i
n
, is
(a) 1 (b) 4 (c) 8 (d) 16
22. If we define a relation R on the set
N N ´ as ( , ) ( , ) a b R c d Û
a d b c + = + for all ( , ), ( , ) a b c d
Î ´ N N, then the relation is
(a) symmetric only
(b) symmetric and transitive only
(c) equivalence relation
(d) reflexive only
23. If y x x x = + + +
2 3
… up to
infinite terms where x < 1, then
which one of the following is
correct?
(a) x
y
y
=
+ 1
(b) x
y
y
=
- 1
(c) x
y
y
=
+ 1
(d) x
y
y
=
- 1
24. If a and b are the roots of the
equation 3 2 1 0
2
x x + + = , then the
equation whose roots are a b +
-1
andb a +
-1
is
(a) 3 8 16 0
2
x x + + =
(b) 3 8 16 0
2
x x - - =
(c) 3 8 16 0
2
x x + - =
(d) x x
2
8 16 0 + + =
25. The value of
1 1 1
3 3
2
3
4
log log log e e e
+ + + …up
to infinite terms is
(a) log
e
9 (b) 0 (c) 1 (d) log
e
3
26. A tea party is arranged for 16
people along two sides of a long
table with eight chairs on each side.
Four particular men wish to sit on
one particular side and two
particular men on the other side.
The number of ways they can be
seated is
(a) 24 8 8 ´ ´ ! ! (b) ( !) 8
3
(c) 210 8 8 ´ ´ ! ! (d) 16!
27. The system of equations
kx y z + + = 1, x ky z k + + = and
x y kz k + + =
2
has no solution ifk
equals.
(a) 0 (b) 1
(c)-1 (d) -2
28. If 13 23 33 3
2 3
. . . . + + +¼+n
n
=
- + ( ) 2 1 3
4
n b
a
then a and b are
respectively
(a) n, 2 (b) n, 3
(c) n+ 1 2 , (d) n+ 1 3 ,
29. In DPQR, Ð = R
p
2
. If tan
P
2
æ
è
ç
ö
ø
÷
and
tan
Q
2
æ
è
ç
ö
ø
÷
are the roots of the
equation ax bx c
2
0 + + = , then
which one of the following is
correct?
(a) a b c = + (b) b c a = +
(c)c a b = + (d) b c =
30. If z
z
-
½
½
½
½
½
½
=
4
2, Then the maximum
value of | | z is equal to
(a) 1 3 + (b) 1 5 +
(c) 1 5 - (d) 5 1 -
31. The angle of elevation of a
stationary cloud from a point 25 m
above a lake is 15° and the angle of
depression of its image in the lake is
45°. The height of the cloud above
the lake level is
(a) 25 m (b) 25 3 m
(c) 50 m (d) 50 3 m
32. The value of
tan tan tan tan 9 27 63 81 ° - ° - ° + °
is equal to
(a)-1 (b) 0
(c) 1 (d) 4
33. The value of 3 20 20 cosec ° - ° sec
is equal to
(a) 4 (b) 2
(c) 1 (d) -4
2
34. Anglea is divided into two parts A
and B such that A B x - = and
tan : tan : A B p q = . The value of
sinx is equal to
(a)
( )sin p q
p q
+
-
a
(b)
p
p q
sina
+
(c)
p
p q
sina
-
(d)
( )sin p q
p q
-
+
a
35. The value of
sin tan
- - æ
è
ç
ö
ø
÷
+
æ
è
ç
ö
ø
÷
1 1
3
5
1
7
is equal to
(a) 0 (b)
p
4
(c)
p
3
(d)
p
2
36. The angles of elevation of the top of
a tower from the top and foot of a
pole are respectively 30° and 45°. If
h
T
is the height of the tower andh
P
is the height of the pole, then which
of the following are correct?
1.
2
3 3
2
h h
h
P T
P
+
=
2.
h h h
T P P
-
+
=
3 1
2
3.
2
4 3
( ) h h
h
P T
P
+
= +
Select the correct answer using the
code given below.
(a) 1 and 3 only (b) 2 and 3 only
(c) 1 and 2 only (d) 1, 2 and 3
37. In a triangle ABC, a b c - + = 2 0.
The value of cot cot
A C
2 2
æ
è
ç
ö
ø
÷
æ
è
ç
ö
ø
÷
is
(a)
9
2
(b) 3 (c)
3
2
(d) 1
38. 1
2 2
+ = - +
æ
è
ç
ö
ø
÷
sin sin cos A
A A
is
true if
(a)
3
2
5
2
p p
< < A only  (b)
p p
2
3
2
< < A only
(c)
3
2
7
2
p p
< < A (d) 0
3
2
< < A
p
39. In triangle ABC, if
sin sin sin
cos cos cos
2 2 2
2 2 2
2
A B C
A B C
+ +
+ +
=
then the triangle is
(a) right-angled (b) equilateral
(c) isosceles (d) obtuse-angled
40. The principal value of sin
-1
x lies in
the interval
(a) -
æ
è
ç
ö
ø
÷
p p
2 2
, (b) -
é
ë
ê
ù
û
ú
p p
2 2
,
(c) 0
2
,
p é
ë
ê
ù
û
ú
(d) [ , ] 0 p
41. The points ( , ), ( , ) a b 0 0 , ( , ) - - a b and
( , ) ab b
2
are
(a) the vertices of a parallelogram
(b) the vertices of a rectangle
(c) the vertices of a square
(d) collinear
42. The length of the normal from
origin to the planex y z + - = 2 2 9 is
equal to
(a) 2 units (b) 3 units
(c) 4 units (d) 5 units
43. If a b , and g are the angles which
the vectorOP
¾®
(O being the origin)
makes with positive direction of the
coordinate axes, then which of the
following are correct?
1. cos cos sin
2 2 2
a b g + =
2. sin sin cos
2 2 2
a b g + =
3. sin sin sin
2 2 2
2 a b g + + =
Select the correct answer using the
code given below.
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
44. The angle between the lines
x y + - = 3 0 and x y - + = 3 0 is a
and the acute angle between the
lines x y - + = 3 2 3 0 and
3 1 0 x y - + = is b. Which one of
the following is correct?
(a)a b = (b) a b >
(c)a b < (d) a b = 2
45. Let a
®
= + -
$ $
$
i j k 2 , b
®
= - + 2 3
$ $
$
i j k
andg
®
= + + 2 6
$ $
$
i j k be three vectors.
Ifa
®
andb
®
are both perpendicular to
the vectord andd g × = 10, then what
is the magnitude ofd?
(a) 3 units (b) 2 3 units
(c)
3
2
unit (d)
1
3
unit
46. If $ a and
$
b are two unit vectors, then the
vector ($
$
) ($
$
) a b a b + ´ ´ is parallel to
(a) (
$
$
) a b - (b) (
$
$
) a b +
(c) (
$
$
) 2a b - (d) (
$
$
) 2a b +
47. A force F
®
= + +
$ $
$
i j k 3 2 acts on a
particle to displace it from the point
A i j k (
$ $
$
) + - 2 3 to the point
B i j k (
$ $
$
) 3 5 - + . The work done by
the force will be
(a) 5 units (b) 7 units
(c) 9 units (d) 10 units
48. For any vector a
®
|
$
| |
$
| |
$
| a a a
® ® ®
´ + ´ + ´ i j k
2 2 2
is equal to
(a)| | a
®
2
(b) 2
2
| | a
®
(c) 3
2
| | a
®
(d) 4
2
| | a
®
49. A man running round a racecourse
notes that the sum of the distances
of two flag-posts from him is
always 10 m and the distance
between the flag-posts is 8 m. The
area of the path he encloses is
(a) 18p square metres
(b) 15p square metres
(c) 12p square metres
(d) 8p square metres
50. The distance of the point (1, 3) from
the line 2 3 6 x y + = , measured
parallel to the line 4 4 x y + = , is
(a)
5
13
units (b)
3
17
units
(c) 17 units (d)
17
2
units
51. If the vectors ai j k
$ $
$
+ + ,
$ $
$
i bj k + +
and
$ $
$
i j ck + + ( , , ) a b c ¹ 1 are
coplanar, then the value of
1
1
1
1
1
1 -
+
-
+
- a b c
is equal to
(a) 0 (b) 1
(c) a b c + + (d) abc
52. The point of intersection of the line
joining the points ( , , ) - - 3 4 8 and
( , , ) 5 6 4 - with XY-plane is
(a)
7
3
8
3
0 , , -
æ
è
ç
ö
ø
÷
(b) - -
æ
è
ç
ö
ø
÷
7
3
8
3
0 , ,
(c) -
æ
è
ç
ö
ø
÷
7
3
8
3
0 , , (d)
7
3
8
3
0 , ,
æ
è
ç
ö
ø
÷
3
53. If the angle between the lines
whose direction ratios are ( , , ) 2 1 2 -
and x, , 3 5 is
p
4
, then the smaller
value of x is
(a) 52 (b) 4 (c) 2 (d) 1
54. The position of the point ( , ) 1 2
relative to the ellipse
2 7 20
2 2
x y + = is
(a) outside the ellipse
(b) inside the ellipse but not at the focus
(c) on the ellipse
(d) at the focus
55. The equation of straight line which
cuts off an intercept of 5 units on
negative direction of Y-axis and
makes and angle 120° with positive
direction of X-axis is
(a) y x + + = 3 5 0
(b) y x - + = 3 5 0
(c) y x + - = 3 5 0
(d) y x - - = 3 5 0
56. The equation of the line passing
through the point (2, 3) and the
point of intersection of lines
2 3 7 0 x y - + = and 7 4 2 0 x y + + =
is
(a) 21 46 180 0 x y + - =
(b) 21 46 96 0 x y - + =
(c) 46 21 155 0 x y + - =
(d) 46 21 29 0 x y - - =
57. The equation of the ellipse whose
centre is at origin, major axis is
along X-axis with eccentricity
3
4
and latus rectum 4 units is
(a)
x y
2 2
1024
7
64
1 + = (b)
49
1024
7
64
1
2 2
x y
+ =
(c)
7
1024
49
64
1
2 2
x y
+ = (d)
x y
2 2
1024 64
1 + =
58. The equation of the circle which
passes through the points (1, 0), (0,
-6) and (3, 4) is
(a) 4 4 142 47 140 0
2 2
x y x y + + + + =
(b) 4 4 142 47 138 0
2 2
x y x y + - - + =
(c) 4 4 142 47 138 0
2 2
x y x y + - + + =
(d) 4 4 150 49 138 0
2 2
x y x y + + - + =
59. A variable plane passes through a
fixed point ( , , ) a b c and cuts the axes
in A B , and C respectively. The
locus of the centre of the sphere
OABC,O being the origin, is
(a)
x
a
y
b
z
c
+ + = 1 (b)
a
x
b
y
c
z
+ + = 1
(c)
a
x
b
y
c
z
+ + = 2 (d)
x
a
y
b
z
c
+ + = 2
60. The equation of the plane passing
through the line of intersection of
the planes x y z + + = 1,
2 3 4 7 x y z + + = , and perpendicular
to the plane x y z - + = 5 3 5 is given
by
(a) x y z + + - = 2 3 6 0
(b) x y z + + + = 2 3 6 0
(c) 3 4 5 8 0 x y z + + - =
(d) 3 4 5 8 0 x y z + + + =
61. The inverse of the functiony
x
= 5
ln
is
(a) x y y = >
1
5
0
ln
,
(b) x y y = >
ln
,
5
0
(c) x y y = <
1
5
0
ln
,
(d) x y y = > 5 0 ln ,
62. A function is defined as follows :
f x
x
x
x
x
( )
,
,
=
- ¹
=
ì
í
ï
î
ï
2
0
0 0
Which one of the following is
correct in respect of the above
function?
(a) f x ( ) is continuous at x = 0 but not
differentiable at x = 0
(b) f x ( ) is continuous as well as
differentiable at x = 0
(c) f x ( )is discontinuous at x = 0
(d) None of the above
63. If y x
x
x
=
¥
(cos )
(cos )
(cos )
, then
dy
dx
is
equal to
(a)-
-
y x
y x
2
1
tan
ln(cos )
(b)
y x
y x
2
1
tan
ln(cos ) +
(c)
y x
y x
2
1
tan
ln(sin ) -
(d)
y x
y x
2
1
sin
ln(sin ) +
64. Consider the following
1. x x +
2
is continuous at x = 0
2. x
x
+ cos
1
is discontinuous at
x = 0
3. x
x
2
1
+ cos is continuous atx = 0
Which of the above are correct?
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
65. Consider the following statements :
1.
dy
dx
at a point on the curve gives
slope of the tangent at that
point.
2. If a t ( ) denotes acceleration of a
particle, then a t dt c
ò
+ ( ) gives
velocity of the particle.
3. If s t ( ) gives displacement of a
particle at time t, then
ds
dt
gives
its acceleration at that instant.
Which of the above statements
is/are correct?
(a) 1 and 2 only (b) 2 only
(c) 1 only (d) 1, 2 and 3
66. If y
x
x
x
x
=
+
-
æ
è
ç
ö
ø
÷
+
-
+
æ
è
ç
ö
ø
÷
- -
sec sin
1 1
1
1
1
1
,
then
dy
dx
is equal to
(a) 0 (b) 1
(c)
x
x
-
+
1
1
(d)
x
x
+
-
1
1
67. What is tan (sec tan )
-
+
ò
1
x x dx
equal to?
(a)
px x
C
4 4
2
+ + (b)
px x
C
2 4
2
+ +
(c)
p p x x
C
4 4
2
+ + (d)
px x
C
4 4
2
- +
68. A function is defined in ( , ) 0 ¥ by
f x
x x
x x
x x
( )
for
ln for
ln . for
=
- < £
< £
- + < < ¥
æ
è
ç
ç
1 0 1
1 2
2 1 05 2
2
ç
Which one of the following is
correct in respect of the derivative
of the function, i.e., f x ¢ ( )?
(a) f x x ¢ = ( ) 2 for 0 1 < £ x
(b) f x x ¢ = - ( ) 2 for 0 1 < £ x
(c) f x x ¢ = - ( ) 2 for 0 1 < < x
(d) f x ¢ = ( ) 0 for 0< < ¥ x
4
69. Which one of the following is
correct in respect of the function
f x x x x ( ) ( )( ) = - + 1 1 ?
(a) The local maximum value is larger
than local minimum value
(b) The local maximum value is smaller
than local minimum value
(c) The function has no local maximum
(d) The function has no local minimum
70. Consider the following statements :
1. Derivative of f x ( ) may not exist
at some point.
2. Derivative of f x ( ) may exist
finitely at some point.
3. Derivative of f x ( ) may be
infinite (geometrically) at some
point.
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
71. The maximum value of
lnx
x
is
(a)e (b)
1
e
(c)
2
e
(d) 1
72. The function f x x x ( ) | | = -
3
is
(a) odd
(b) even
(c) both even and odd
(d) neither even nor odd
73. If l
d
dx
e
x
1
= ( )
sin
l
e e
h
x
x h x
2
0
=
-
®
+
lim
sin( ) sin
l e xdx
x
3
=
ò
sin
cos
then which one of the following is
correct?
(a) l l
1 2
¹ (b)
d
dx
l l ( )
3 2
=
(c) l dx l
3 2
ò
= (d) l l
2 3
=
74. The general solution of
dy
dx
ax h
by k
=
+
+
represents a circle only when
(a) a b = = 0 (b) a b = - ¹ 0
(c) a b h k = ¹ = 0, (d) a b = ¹ 0
75. If lim
sin
x
x
x
l
®
=
p
2
and lim
cos
x
x
x
m
®¥
= ,
then which one of the following is
correct?
(a) l m = = 1 1 , (b) l m = = ¥
2
p
,
(c) l m = =
2
0
p
, (d) l m = = ¥ 1 ,
76. What is 1
2
0
2
+
ò
sin
x
dx
p
equal to?
(a) 8 (b) 4
(c) 2 (d) 0
77. The area bounded by the curve
| | | | x y + = 1
(a) 1 square unit
(b) 2 2 square units
(c) 2 square units
(d) 2 3 square units
78. Ifx is any real number, then
x
x
2
4
1+
belongs to which one of the
following intervals?
(a) (0, 1) (b) 0
1
2
,
æ
è
ç
ù
û
ú
(c) 0
1
2
,
æ
è
ç
ö
ø
÷
(d) [0, 1]
79. The left-hand derivative of
f x x x ( ) [ ]sin( ) = p at x k =
wherek is an integer and [ ] x is the
greatest integer function, is
(a) ( ) ( ) - - 1 1
k
k p (b) ( ) ( ) - -
-
1 1
1 k
k p
(c) ( ) -1
k
kp (d) ( ) -
-
1
1 k
kp
80. If f x
x
( )= -
2
1, then on the interval
[ , ] 0 p which one of the following is
correct?
(a) tan[ ( )] f x , where [ ] × is the greatest
integer function, and
1
f x ( )
are both
continuous
(b) tan[ ( )] f x , where [ ] × is the greatest
integer function, and f x
-1
( )are both
continuous
(c) tan[ ( )] f x , where [ ] × is the greatest
integer function, and
1
f x ( )
are both
discontinuous
(d) tan[ ( )] f x , where [ ] × is the greatest
integer function is discontinuous
but
1
f x ( )
is continuous
81. The order and degree of the
differential equation
1
2
3
2
2
2
2
+
æ
è
ç
ö
ø
÷
é
ë
ê
ê
ù
û
ú
ú
=r
é
ë
ê
ù
û
ú
dy
dx
d y
dx
are respectively
(a) 3 and 2 (b) 2 and 2
(c) 2 and 3 (d) 1 and 3
82. If y
x
x
=
+
æ
è
ç
ö
ø
÷
-
cos
1
2
2
1
, then
dy
dx
is
equal to
(a)-
+
2
1
2
x
for all| | x < 1
(b)-
+
2
1
2
x
for all| | x > 1
(c)
2
1
2
+ x
for all| | x < 1
(d) None of the above
83. The set of all points, where the
function f x e
x
( )= -
-
1
2
is
differentiable, is
(a) ( , ) 0 ¥ (b) ( , ) -¥ ¥
(c) ( , ) ( , ) -¥ È ¥ 0 0 (d) ( , ) - ¥ 1
84. Match List-I with List-II and select
the correct answer using the code
given below the lists :
List-I
(Function)
List-II
(Maximum value)
A. sin cos x x + 1. 10
B. 3 4 sin cos x x + 2. 2
C. 2sin cos x x + 3. 5
D. sin cos x x + 3 4. 5
Code
A B C D
(a) 2 3 1 4
(b) 2 3 4 1
(c) 3 2 1 4
(d) 3 2 4 1
85. If f x x x x ( ) ( ) = - + 1 , then f x ( )
is
(a) continuous but not differentiable at
x = 0
(b) differentiable at x = 0
(c) not continuous at x = 0
(d) None of the above
5