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Page 1
SECTION-A
1. A line passing through the point A(9, 0) makes an angle
of 30º with the positive direction of x-axis. If this line is
rotated about A through an angle of 15º in the clockwise
direction, then its equation in the new position is
(1) 9
32
??
?
y
x (2) 9
32
??
?
x
y
(3) 9
32
??
?
x
y (4) 9
32
??
?
y
x
Ans. (1)
Sol.
Eq
n
: y – 0 = tan15° (x – 9) ? y = (2 ? 3 ??(x ? 9)
2. Let S
a
denote the sum of first n terms an arithmetic
progression. If S
20
= 790 and S
10
= 145, then S
15
–
S
5
is :
(1) 395
(2) 390
(3) 405
(4) 410
Ans. (1)
Sol.
? ?
20
20
S 2a 19d 790
2
? ? ?
2a + 19d = 79 …..(1)
? ?
10
10
S 2a 9d 145
2
? ? ?
2a + 9d = 29 …..(2)
From (1) and (2) a = -8, d = 5
? ? ? ?
15 5
15 5
S S 2a 14d 2a 4d
22
? ? ? ? ?
? ? ? ?
15 5
16 70 16 20
22
? ? ? ? ? ?
= 405-10
= 395
3. If z = x + iy, xy ? 0 , satisfies the equation
2
z i z 0 ?? , then |z
2
| is equal to :
(1) 9
(2) 1
(3) 4
(4)
1
4
Ans. (2)
Sol.
2
2
z iz
z iz
??
?
|z
2
| = |z|
|z|
2
– |z| = 0
|z|(|z| – 1) = 0
|z| = 0 (not acceptable)
? |z| = 1
? |z|
2
= 1
4. Let
i 2 3
ˆ ˆ ˆ
a a i a j a k ? ? ? and
1 2 3
ˆ ˆ ˆ
b b i b j b k ? ? ? be
two vectors such that a 1; ? a.b 2 ? and b 4. ? If
? ?
c 2 a b 3b ? ? ? , then the angle between b and c
is equal to :
(1)
1
2
cos
3
?
??
??
??
(2)
1
1
cos
3
?
??
?
??
??
(3)
1
3
cos
2
?
??
?
??
??
??
(4)
1
2
cos
3
?
??
??
??
Ans. (3)
Page 2
SECTION-A
1. A line passing through the point A(9, 0) makes an angle
of 30º with the positive direction of x-axis. If this line is
rotated about A through an angle of 15º in the clockwise
direction, then its equation in the new position is
(1) 9
32
??
?
y
x (2) 9
32
??
?
x
y
(3) 9
32
??
?
x
y (4) 9
32
??
?
y
x
Ans. (1)
Sol.
Eq
n
: y – 0 = tan15° (x – 9) ? y = (2 ? 3 ??(x ? 9)
2. Let S
a
denote the sum of first n terms an arithmetic
progression. If S
20
= 790 and S
10
= 145, then S
15
–
S
5
is :
(1) 395
(2) 390
(3) 405
(4) 410
Ans. (1)
Sol.
? ?
20
20
S 2a 19d 790
2
? ? ?
2a + 19d = 79 …..(1)
? ?
10
10
S 2a 9d 145
2
? ? ?
2a + 9d = 29 …..(2)
From (1) and (2) a = -8, d = 5
? ? ? ?
15 5
15 5
S S 2a 14d 2a 4d
22
? ? ? ? ?
? ? ? ?
15 5
16 70 16 20
22
? ? ? ? ? ?
= 405-10
= 395
3. If z = x + iy, xy ? 0 , satisfies the equation
2
z i z 0 ?? , then |z
2
| is equal to :
(1) 9
(2) 1
(3) 4
(4)
1
4
Ans. (2)
Sol.
2
2
z iz
z iz
??
?
|z
2
| = |z|
|z|
2
– |z| = 0
|z|(|z| – 1) = 0
|z| = 0 (not acceptable)
? |z| = 1
? |z|
2
= 1
4. Let
i 2 3
ˆ ˆ ˆ
a a i a j a k ? ? ? and
1 2 3
ˆ ˆ ˆ
b b i b j b k ? ? ? be
two vectors such that a 1; ? a.b 2 ? and b 4. ? If
? ?
c 2 a b 3b ? ? ? , then the angle between b and c
is equal to :
(1)
1
2
cos
3
?
??
??
??
(2)
1
1
cos
3
?
??
?
??
??
(3)
1
3
cos
2
?
??
?
??
??
??
(4)
1
2
cos
3
?
??
??
??
Ans. (3)
Sol. Given a 1, b 4, a.b 2 ? ? ?
? ?
c 2 a b 3b ? ? ?
Dot product with a on both sides
c.a 6 ?? …..(1)
Dot product with b on both sides
b.c 48 ?? …..(2)
22
c.c 4 a b 9 b ? ? ?
? ?
2 2
2 2 2
c 4 a b a.b 9 b
??
? ? ?
??
??
? ? ? ? ? ? ? ?
2 2
c 4 1 4 4 9 16
??
? ? ?
??
??
? ?
2
c 4 12 144 ??
2
c 48 144 ??
2
c 192 ?
b.c
cos
bc
? ? ?
48
cos
192.4
?
? ? ?
48
cos
8 3.4
?
? ? ?
3
cos
23
?
? ? ?
1
33
cos cos
22
?
??
??
? ? ? ? ? ?
??
??
??
5. The maximum area of a triangle whose one vertex
is at (0, 0) and the other two vertices lie on the
curve y = -2x
2
+ 54 at points (x, y) and (-x, y)
where y > 0 is :
(1) 88
(2) 122
(3) 92
(4) 108
Ans. (4)
Sol.
(x, y)
(-x, y)
(0, 0)
Area of ? ?
?
0 0 1
1
x y 1
2
x y 1
?
?
?
? ? ?
1
xy xy xy
2
? ? ? ?
? ? ?
? ?
2
Area xy x 2x 54 ? ? ? ? ? ?
?
? ?
? ?
2
d
d
6x 54 0
dx dx
?
?
? ? ? ? ? ?at x = 3
Area = 3 (-2 × 9 + 54) = 108
6. The value of
? ? ? ?
3 n
2 2 2
n
k1
n
lim
n k n 3k
?
??
?
??
?
is :
(1)
? ?
2 3 3
24
??
(2)
? ?
13
8 4 3 3
?
?
(3)
? ?
13 2 3 3
8
??
(4)
? ?
8 2 3 3
?
?
Ans. (2)
Sol.
3 n
22
n
k1 4
22
n
lim
k 3k
n 1 1
nn
??
?
? ? ? ?
?? ? ? ? ?
? ? ? ?
? ? ? ?
?
3 n
22
n
k1
22
1n
lim
n
k 3k
11
nn
??
?
?
? ? ? ?
?? ? ? ? ?
? ? ? ?
? ? ? ?
?
? ?
1
22
0
dx
1
3 1 x x
3
?
??
??
??
??
?
Page 3
SECTION-A
1. A line passing through the point A(9, 0) makes an angle
of 30º with the positive direction of x-axis. If this line is
rotated about A through an angle of 15º in the clockwise
direction, then its equation in the new position is
(1) 9
32
??
?
y
x (2) 9
32
??
?
x
y
(3) 9
32
??
?
x
y (4) 9
32
??
?
y
x
Ans. (1)
Sol.
Eq
n
: y – 0 = tan15° (x – 9) ? y = (2 ? 3 ??(x ? 9)
2. Let S
a
denote the sum of first n terms an arithmetic
progression. If S
20
= 790 and S
10
= 145, then S
15
–
S
5
is :
(1) 395
(2) 390
(3) 405
(4) 410
Ans. (1)
Sol.
? ?
20
20
S 2a 19d 790
2
? ? ?
2a + 19d = 79 …..(1)
? ?
10
10
S 2a 9d 145
2
? ? ?
2a + 9d = 29 …..(2)
From (1) and (2) a = -8, d = 5
? ? ? ?
15 5
15 5
S S 2a 14d 2a 4d
22
? ? ? ? ?
? ? ? ?
15 5
16 70 16 20
22
? ? ? ? ? ?
= 405-10
= 395
3. If z = x + iy, xy ? 0 , satisfies the equation
2
z i z 0 ?? , then |z
2
| is equal to :
(1) 9
(2) 1
(3) 4
(4)
1
4
Ans. (2)
Sol.
2
2
z iz
z iz
??
?
|z
2
| = |z|
|z|
2
– |z| = 0
|z|(|z| – 1) = 0
|z| = 0 (not acceptable)
? |z| = 1
? |z|
2
= 1
4. Let
i 2 3
ˆ ˆ ˆ
a a i a j a k ? ? ? and
1 2 3
ˆ ˆ ˆ
b b i b j b k ? ? ? be
two vectors such that a 1; ? a.b 2 ? and b 4. ? If
? ?
c 2 a b 3b ? ? ? , then the angle between b and c
is equal to :
(1)
1
2
cos
3
?
??
??
??
(2)
1
1
cos
3
?
??
?
??
??
(3)
1
3
cos
2
?
??
?
??
??
??
(4)
1
2
cos
3
?
??
??
??
Ans. (3)
Sol. Given a 1, b 4, a.b 2 ? ? ?
? ?
c 2 a b 3b ? ? ?
Dot product with a on both sides
c.a 6 ?? …..(1)
Dot product with b on both sides
b.c 48 ?? …..(2)
22
c.c 4 a b 9 b ? ? ?
? ?
2 2
2 2 2
c 4 a b a.b 9 b
??
? ? ?
??
??
? ? ? ? ? ? ? ?
2 2
c 4 1 4 4 9 16
??
? ? ?
??
??
? ?
2
c 4 12 144 ??
2
c 48 144 ??
2
c 192 ?
b.c
cos
bc
? ? ?
48
cos
192.4
?
? ? ?
48
cos
8 3.4
?
? ? ?
3
cos
23
?
? ? ?
1
33
cos cos
22
?
??
??
? ? ? ? ? ?
??
??
??
5. The maximum area of a triangle whose one vertex
is at (0, 0) and the other two vertices lie on the
curve y = -2x
2
+ 54 at points (x, y) and (-x, y)
where y > 0 is :
(1) 88
(2) 122
(3) 92
(4) 108
Ans. (4)
Sol.
(x, y)
(-x, y)
(0, 0)
Area of ? ?
?
0 0 1
1
x y 1
2
x y 1
?
?
?
? ? ?
1
xy xy xy
2
? ? ? ?
? ? ?
? ?
2
Area xy x 2x 54 ? ? ? ? ? ?
?
? ?
? ?
2
d
d
6x 54 0
dx dx
?
?
? ? ? ? ? ?at x = 3
Area = 3 (-2 × 9 + 54) = 108
6. The value of
? ? ? ?
3 n
2 2 2
n
k1
n
lim
n k n 3k
?
??
?
??
?
is :
(1)
? ?
2 3 3
24
??
(2)
? ?
13
8 4 3 3
?
?
(3)
? ?
13 2 3 3
8
??
(4)
? ?
8 2 3 3
?
?
Ans. (2)
Sol.
3 n
22
n
k1 4
22
n
lim
k 3k
n 1 1
nn
??
?
? ? ? ?
?? ? ? ? ?
? ? ? ?
? ? ? ?
?
3 n
22
n
k1
22
1n
lim
n
k 3k
11
nn
??
?
?
? ? ? ?
?? ? ? ? ?
? ? ? ?
? ? ? ?
?
? ?
1
22
0
dx
1
3 1 x x
3
?
??
??
??
??
?
? ?
? ?
22
1
22
0
1
x 1 x
13 3
dx
1 32
1 x x
3
??
? ? ?
??
??
??
??
??
??
??
?
1
22
2 0
1 1 1
dx
2
1x
1
x
3
??
??
??
??
??
?
??
??
?
??
??
?? ??
?
? ? ? ?
1 1
11
0 0
11
3 tan 3x tan x
22
??
??
??
??
31
2 3 2 4 8 23
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ?
13
8. 4 3 3
?
?
?
7. Let g : RR ? be a non constant twice
differentiable such that
13
g' g'
22
? ? ? ?
?
? ? ? ?
? ? ? ?
. If a real
valued function f is defined as
? ? ? ? ? ?
1
f x g x g 2 x
2
? ? ? ? ?
??
, then
(1) f”(x) = 0 for atleast two x in (0, 2)
(2) f”(x) = 0 for exactly one x in (0, 1)
(3) f”(x) = 0 for no x in (0, 1)
(4)
31
f ' f ' 1
22
? ? ? ?
??
? ? ? ?
? ? ? ?
Ans. (1)
Sol. ? ?
? ? ? ?
31
g' g'
g' x g' 2 x 3 22
f ' x ,f ' 0
2 2 2
? ? ? ?
?
? ? ? ?
??
??
? ? ? ?
? ? ?
??
??
Also
13
g' g'
11 22
f ' 0, f ' 0
2 2 2
? ? ? ?
?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
31
f ' f ' 0
22
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
13
rootsin ,1 and 1,
22
? ? ? ?
?
? ? ? ?
? ? ? ?
? ? f " x ? is zero at least twice in
13
,
22
??
??
??
8. The area (in square units) of the region bounded by
the parabola y
2
= 4(x – 2) and the line y = 2x - 8
(1) 8
(2) 9
(3) 6
(4) 7
Ans. (2)
Sol. Let X = x – 2
y
2
= 4x, y = 2 (x + 2) – 8
y
2
= 4x, y = 2x – 4
4
2
2
y y 4
A
42
?
?
??
?
-2
4
= 9
9. Let y = y (x) be the solution of the differential
equation sec x dy + {2(1 – x) tan x + x(2 – x)}
dx = 0 such that y(0) = 2.Then y(2) is equal to :
(1) 2
(2) 2{1 – sin (2)}
(3) 2{sin (2) + 1}
(4) 1
Ans. (1)
Sol. ? ?
? ?
2
dy
2 x 1 sin x x 2x cos x
dx
? ? ? ?
Now both side integrate
? ? ? ?
? ?
? ? ? ?
2
y x 2 x 1 sin xdx x 2x sin x 2x 2 sin x dx
??
? ? ? ? ? ?
??
??
??
? ?
? ?
2
y x x 2x sin x ? ? ? ?
? ? y 0 0 2 ? ? ? ? ? ?
? ?
? ?
2
y x x 2x sin x 2 ? ? ?
? ? y 2 2 ?
Page 4
SECTION-A
1. A line passing through the point A(9, 0) makes an angle
of 30º with the positive direction of x-axis. If this line is
rotated about A through an angle of 15º in the clockwise
direction, then its equation in the new position is
(1) 9
32
??
?
y
x (2) 9
32
??
?
x
y
(3) 9
32
??
?
x
y (4) 9
32
??
?
y
x
Ans. (1)
Sol.
Eq
n
: y – 0 = tan15° (x – 9) ? y = (2 ? 3 ??(x ? 9)
2. Let S
a
denote the sum of first n terms an arithmetic
progression. If S
20
= 790 and S
10
= 145, then S
15
–
S
5
is :
(1) 395
(2) 390
(3) 405
(4) 410
Ans. (1)
Sol.
? ?
20
20
S 2a 19d 790
2
? ? ?
2a + 19d = 79 …..(1)
? ?
10
10
S 2a 9d 145
2
? ? ?
2a + 9d = 29 …..(2)
From (1) and (2) a = -8, d = 5
? ? ? ?
15 5
15 5
S S 2a 14d 2a 4d
22
? ? ? ? ?
? ? ? ?
15 5
16 70 16 20
22
? ? ? ? ? ?
= 405-10
= 395
3. If z = x + iy, xy ? 0 , satisfies the equation
2
z i z 0 ?? , then |z
2
| is equal to :
(1) 9
(2) 1
(3) 4
(4)
1
4
Ans. (2)
Sol.
2
2
z iz
z iz
??
?
|z
2
| = |z|
|z|
2
– |z| = 0
|z|(|z| – 1) = 0
|z| = 0 (not acceptable)
? |z| = 1
? |z|
2
= 1
4. Let
i 2 3
ˆ ˆ ˆ
a a i a j a k ? ? ? and
1 2 3
ˆ ˆ ˆ
b b i b j b k ? ? ? be
two vectors such that a 1; ? a.b 2 ? and b 4. ? If
? ?
c 2 a b 3b ? ? ? , then the angle between b and c
is equal to :
(1)
1
2
cos
3
?
??
??
??
(2)
1
1
cos
3
?
??
?
??
??
(3)
1
3
cos
2
?
??
?
??
??
??
(4)
1
2
cos
3
?
??
??
??
Ans. (3)
Sol. Given a 1, b 4, a.b 2 ? ? ?
? ?
c 2 a b 3b ? ? ?
Dot product with a on both sides
c.a 6 ?? …..(1)
Dot product with b on both sides
b.c 48 ?? …..(2)
22
c.c 4 a b 9 b ? ? ?
? ?
2 2
2 2 2
c 4 a b a.b 9 b
??
? ? ?
??
??
? ? ? ? ? ? ? ?
2 2
c 4 1 4 4 9 16
??
? ? ?
??
??
? ?
2
c 4 12 144 ??
2
c 48 144 ??
2
c 192 ?
b.c
cos
bc
? ? ?
48
cos
192.4
?
? ? ?
48
cos
8 3.4
?
? ? ?
3
cos
23
?
? ? ?
1
33
cos cos
22
?
??
??
? ? ? ? ? ?
??
??
??
5. The maximum area of a triangle whose one vertex
is at (0, 0) and the other two vertices lie on the
curve y = -2x
2
+ 54 at points (x, y) and (-x, y)
where y > 0 is :
(1) 88
(2) 122
(3) 92
(4) 108
Ans. (4)
Sol.
(x, y)
(-x, y)
(0, 0)
Area of ? ?
?
0 0 1
1
x y 1
2
x y 1
?
?
?
? ? ?
1
xy xy xy
2
? ? ? ?
? ? ?
? ?
2
Area xy x 2x 54 ? ? ? ? ? ?
?
? ?
? ?
2
d
d
6x 54 0
dx dx
?
?
? ? ? ? ? ?at x = 3
Area = 3 (-2 × 9 + 54) = 108
6. The value of
? ? ? ?
3 n
2 2 2
n
k1
n
lim
n k n 3k
?
??
?
??
?
is :
(1)
? ?
2 3 3
24
??
(2)
? ?
13
8 4 3 3
?
?
(3)
? ?
13 2 3 3
8
??
(4)
? ?
8 2 3 3
?
?
Ans. (2)
Sol.
3 n
22
n
k1 4
22
n
lim
k 3k
n 1 1
nn
??
?
? ? ? ?
?? ? ? ? ?
? ? ? ?
? ? ? ?
?
3 n
22
n
k1
22
1n
lim
n
k 3k
11
nn
??
?
?
? ? ? ?
?? ? ? ? ?
? ? ? ?
? ? ? ?
?
? ?
1
22
0
dx
1
3 1 x x
3
?
??
??
??
??
?
? ?
? ?
22
1
22
0
1
x 1 x
13 3
dx
1 32
1 x x
3
??
? ? ?
??
??
??
??
??
??
??
?
1
22
2 0
1 1 1
dx
2
1x
1
x
3
??
??
??
??
??
?
??
??
?
??
??
?? ??
?
? ? ? ?
1 1
11
0 0
11
3 tan 3x tan x
22
??
??
??
??
31
2 3 2 4 8 23
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ?
13
8. 4 3 3
?
?
?
7. Let g : RR ? be a non constant twice
differentiable such that
13
g' g'
22
? ? ? ?
?
? ? ? ?
? ? ? ?
. If a real
valued function f is defined as
? ? ? ? ? ?
1
f x g x g 2 x
2
? ? ? ? ?
??
, then
(1) f”(x) = 0 for atleast two x in (0, 2)
(2) f”(x) = 0 for exactly one x in (0, 1)
(3) f”(x) = 0 for no x in (0, 1)
(4)
31
f ' f ' 1
22
? ? ? ?
??
? ? ? ?
? ? ? ?
Ans. (1)
Sol. ? ?
? ? ? ?
31
g' g'
g' x g' 2 x 3 22
f ' x ,f ' 0
2 2 2
? ? ? ?
?
? ? ? ?
??
??
? ? ? ?
? ? ?
??
??
Also
13
g' g'
11 22
f ' 0, f ' 0
2 2 2
? ? ? ?
?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
31
f ' f ' 0
22
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
13
rootsin ,1 and 1,
22
? ? ? ?
?
? ? ? ?
? ? ? ?
? ? f " x ? is zero at least twice in
13
,
22
??
??
??
8. The area (in square units) of the region bounded by
the parabola y
2
= 4(x – 2) and the line y = 2x - 8
(1) 8
(2) 9
(3) 6
(4) 7
Ans. (2)
Sol. Let X = x – 2
y
2
= 4x, y = 2 (x + 2) – 8
y
2
= 4x, y = 2x – 4
4
2
2
y y 4
A
42
?
?
??
?
-2
4
= 9
9. Let y = y (x) be the solution of the differential
equation sec x dy + {2(1 – x) tan x + x(2 – x)}
dx = 0 such that y(0) = 2.Then y(2) is equal to :
(1) 2
(2) 2{1 – sin (2)}
(3) 2{sin (2) + 1}
(4) 1
Ans. (1)
Sol. ? ?
? ?
2
dy
2 x 1 sin x x 2x cos x
dx
? ? ? ?
Now both side integrate
? ? ? ?
? ?
? ? ? ?
2
y x 2 x 1 sin xdx x 2x sin x 2x 2 sin x dx
??
? ? ? ? ? ?
??
??
??
? ?
? ?
2
y x x 2x sin x ? ? ? ?
? ? y 0 0 2 ? ? ? ? ? ?
? ?
? ?
2
y x x 2x sin x 2 ? ? ?
? ? y 2 2 ?
? ? be the foot of perpendicular from the
point (1, 2, 3) on the line
x 3 y 1 z 4
5 2 3
? ? ?
?? .
then ? ? 19 ? ? ? ? ? is equal to :
(1) 102
(2) 101
(3) 99
(4) 100
Ans. (2)
Sol.
(1, 2, 3)
P( ) ? ??? ???
Let foot P (5k – 3, 2k + 1, 3k – 4) ?
DR's AP: 5k 4, 2k 1 , 3k 7 ? ? ? ? ?
DR's Line: 5, 2, 3 ? ?
Condition of perpendicular lines (25k-20) + (4k-2) + (9k – 21)=0
Then
43
k
38
? ?
Then ? ? 19 ? ? ? ? ? =101
11. Two integers x and y are chosen with replacement
from the set {0, 1, 2, 3, ….., 10}. Then the
probability that | x y| 5 ?? is :
(1)
30
121
(2)
62
121
(3)
60
121
(4)
31
121
Ans. (1)
Sol. If x = 0, y = 6, 7, 8, 9, 10
If x = 1, y = 7, 8, 9, 10
If x = 2, y = 8, 9, 10
If x = 3, y = 9, 10
If x = 4, y = 10
If x = 5, y = no possible value
Total possible ways = (5 + 4 + 3 + 2 + 1) × 2
= 30
Required probability ?
30 30
11 11 121
?
?
?
12. If the domain of the function
? ? ? ? ? ?
1
1
e
2x
f x cos log 3 x
4
?
?
?? ?
? ? ?
??
??
is
? ? [ , ) y ?? ? ? , then ? ? ? ? ? is equal to :
(1) 12
(2) 9
(3) 11
(4) 8
Ans. (3)
Sol.
2x
11
4
?
? ? ?
2x
1
4
?
??
–4 ? 2 – |x| ? 4
–6 ? – |x| ? 2
–2 ? |x| ? 6
|x| ? 6
? x ? [–6, 6] …(1)
Now, 3 – x ? 1
And x ? 2 …(2)
and 3 – x > 0
x < 3 …(3)
From (1), (2) and (3)
? x ? [–6, 3) – {2}
? = 6
? = 3
? = 2
? + ? + ? = 11
13. Consider the system of linear equation x + y + z =
4 ?, x + 2y + 2z ? = 10 ? , x + 3y + 4 ?
2
z = ?
2
+15,
where ? , R ?? . Which one of the following
statements is NOT correct ?
(1) The system has unique solution if
1
2
?? and
1 ?? , 15
Page 5
SECTION-A
1. A line passing through the point A(9, 0) makes an angle
of 30º with the positive direction of x-axis. If this line is
rotated about A through an angle of 15º in the clockwise
direction, then its equation in the new position is
(1) 9
32
??
?
y
x (2) 9
32
??
?
x
y
(3) 9
32
??
?
x
y (4) 9
32
??
?
y
x
Ans. (1)
Sol.
Eq
n
: y – 0 = tan15° (x – 9) ? y = (2 ? 3 ??(x ? 9)
2. Let S
a
denote the sum of first n terms an arithmetic
progression. If S
20
= 790 and S
10
= 145, then S
15
–
S
5
is :
(1) 395
(2) 390
(3) 405
(4) 410
Ans. (1)
Sol.
? ?
20
20
S 2a 19d 790
2
? ? ?
2a + 19d = 79 …..(1)
? ?
10
10
S 2a 9d 145
2
? ? ?
2a + 9d = 29 …..(2)
From (1) and (2) a = -8, d = 5
? ? ? ?
15 5
15 5
S S 2a 14d 2a 4d
22
? ? ? ? ?
? ? ? ?
15 5
16 70 16 20
22
? ? ? ? ? ?
= 405-10
= 395
3. If z = x + iy, xy ? 0 , satisfies the equation
2
z i z 0 ?? , then |z
2
| is equal to :
(1) 9
(2) 1
(3) 4
(4)
1
4
Ans. (2)
Sol.
2
2
z iz
z iz
??
?
|z
2
| = |z|
|z|
2
– |z| = 0
|z|(|z| – 1) = 0
|z| = 0 (not acceptable)
? |z| = 1
? |z|
2
= 1
4. Let
i 2 3
ˆ ˆ ˆ
a a i a j a k ? ? ? and
1 2 3
ˆ ˆ ˆ
b b i b j b k ? ? ? be
two vectors such that a 1; ? a.b 2 ? and b 4. ? If
? ?
c 2 a b 3b ? ? ? , then the angle between b and c
is equal to :
(1)
1
2
cos
3
?
??
??
??
(2)
1
1
cos
3
?
??
?
??
??
(3)
1
3
cos
2
?
??
?
??
??
??
(4)
1
2
cos
3
?
??
??
??
Ans. (3)
Sol. Given a 1, b 4, a.b 2 ? ? ?
? ?
c 2 a b 3b ? ? ?
Dot product with a on both sides
c.a 6 ?? …..(1)
Dot product with b on both sides
b.c 48 ?? …..(2)
22
c.c 4 a b 9 b ? ? ?
? ?
2 2
2 2 2
c 4 a b a.b 9 b
??
? ? ?
??
??
? ? ? ? ? ? ? ?
2 2
c 4 1 4 4 9 16
??
? ? ?
??
??
? ?
2
c 4 12 144 ??
2
c 48 144 ??
2
c 192 ?
b.c
cos
bc
? ? ?
48
cos
192.4
?
? ? ?
48
cos
8 3.4
?
? ? ?
3
cos
23
?
? ? ?
1
33
cos cos
22
?
??
??
? ? ? ? ? ?
??
??
??
5. The maximum area of a triangle whose one vertex
is at (0, 0) and the other two vertices lie on the
curve y = -2x
2
+ 54 at points (x, y) and (-x, y)
where y > 0 is :
(1) 88
(2) 122
(3) 92
(4) 108
Ans. (4)
Sol.
(x, y)
(-x, y)
(0, 0)
Area of ? ?
?
0 0 1
1
x y 1
2
x y 1
?
?
?
? ? ?
1
xy xy xy
2
? ? ? ?
? ? ?
? ?
2
Area xy x 2x 54 ? ? ? ? ? ?
?
? ?
? ?
2
d
d
6x 54 0
dx dx
?
?
? ? ? ? ? ?at x = 3
Area = 3 (-2 × 9 + 54) = 108
6. The value of
? ? ? ?
3 n
2 2 2
n
k1
n
lim
n k n 3k
?
??
?
??
?
is :
(1)
? ?
2 3 3
24
??
(2)
? ?
13
8 4 3 3
?
?
(3)
? ?
13 2 3 3
8
??
(4)
? ?
8 2 3 3
?
?
Ans. (2)
Sol.
3 n
22
n
k1 4
22
n
lim
k 3k
n 1 1
nn
??
?
? ? ? ?
?? ? ? ? ?
? ? ? ?
? ? ? ?
?
3 n
22
n
k1
22
1n
lim
n
k 3k
11
nn
??
?
?
? ? ? ?
?? ? ? ? ?
? ? ? ?
? ? ? ?
?
? ?
1
22
0
dx
1
3 1 x x
3
?
??
??
??
??
?
? ?
? ?
22
1
22
0
1
x 1 x
13 3
dx
1 32
1 x x
3
??
? ? ?
??
??
??
??
??
??
??
?
1
22
2 0
1 1 1
dx
2
1x
1
x
3
??
??
??
??
??
?
??
??
?
??
??
?? ??
?
? ? ? ?
1 1
11
0 0
11
3 tan 3x tan x
22
??
??
??
??
31
2 3 2 4 8 23
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ?
13
8. 4 3 3
?
?
?
7. Let g : RR ? be a non constant twice
differentiable such that
13
g' g'
22
? ? ? ?
?
? ? ? ?
? ? ? ?
. If a real
valued function f is defined as
? ? ? ? ? ?
1
f x g x g 2 x
2
? ? ? ? ?
??
, then
(1) f”(x) = 0 for atleast two x in (0, 2)
(2) f”(x) = 0 for exactly one x in (0, 1)
(3) f”(x) = 0 for no x in (0, 1)
(4)
31
f ' f ' 1
22
? ? ? ?
??
? ? ? ?
? ? ? ?
Ans. (1)
Sol. ? ?
? ? ? ?
31
g' g'
g' x g' 2 x 3 22
f ' x ,f ' 0
2 2 2
? ? ? ?
?
? ? ? ?
??
??
? ? ? ?
? ? ?
??
??
Also
13
g' g'
11 22
f ' 0, f ' 0
2 2 2
? ? ? ?
?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
31
f ' f ' 0
22
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
13
rootsin ,1 and 1,
22
? ? ? ?
?
? ? ? ?
? ? ? ?
? ? f " x ? is zero at least twice in
13
,
22
??
??
??
8. The area (in square units) of the region bounded by
the parabola y
2
= 4(x – 2) and the line y = 2x - 8
(1) 8
(2) 9
(3) 6
(4) 7
Ans. (2)
Sol. Let X = x – 2
y
2
= 4x, y = 2 (x + 2) – 8
y
2
= 4x, y = 2x – 4
4
2
2
y y 4
A
42
?
?
??
?
-2
4
= 9
9. Let y = y (x) be the solution of the differential
equation sec x dy + {2(1 – x) tan x + x(2 – x)}
dx = 0 such that y(0) = 2.Then y(2) is equal to :
(1) 2
(2) 2{1 – sin (2)}
(3) 2{sin (2) + 1}
(4) 1
Ans. (1)
Sol. ? ?
? ?
2
dy
2 x 1 sin x x 2x cos x
dx
? ? ? ?
Now both side integrate
? ? ? ?
? ?
? ? ? ?
2
y x 2 x 1 sin xdx x 2x sin x 2x 2 sin x dx
??
? ? ? ? ? ?
??
??
??
? ?
? ?
2
y x x 2x sin x ? ? ? ?
? ? y 0 0 2 ? ? ? ? ? ?
? ?
? ?
2
y x x 2x sin x 2 ? ? ?
? ? y 2 2 ?
? ? be the foot of perpendicular from the
point (1, 2, 3) on the line
x 3 y 1 z 4
5 2 3
? ? ?
?? .
then ? ? 19 ? ? ? ? ? is equal to :
(1) 102
(2) 101
(3) 99
(4) 100
Ans. (2)
Sol.
(1, 2, 3)
P( ) ? ??? ???
Let foot P (5k – 3, 2k + 1, 3k – 4) ?
DR's AP: 5k 4, 2k 1 , 3k 7 ? ? ? ? ?
DR's Line: 5, 2, 3 ? ?
Condition of perpendicular lines (25k-20) + (4k-2) + (9k – 21)=0
Then
43
k
38
? ?
Then ? ? 19 ? ? ? ? ? =101
11. Two integers x and y are chosen with replacement
from the set {0, 1, 2, 3, ….., 10}. Then the
probability that | x y| 5 ?? is :
(1)
30
121
(2)
62
121
(3)
60
121
(4)
31
121
Ans. (1)
Sol. If x = 0, y = 6, 7, 8, 9, 10
If x = 1, y = 7, 8, 9, 10
If x = 2, y = 8, 9, 10
If x = 3, y = 9, 10
If x = 4, y = 10
If x = 5, y = no possible value
Total possible ways = (5 + 4 + 3 + 2 + 1) × 2
= 30
Required probability ?
30 30
11 11 121
?
?
?
12. If the domain of the function
? ? ? ? ? ?
1
1
e
2x
f x cos log 3 x
4
?
?
?? ?
? ? ?
??
??
is
? ? [ , ) y ?? ? ? , then ? ? ? ? ? is equal to :
(1) 12
(2) 9
(3) 11
(4) 8
Ans. (3)
Sol.
2x
11
4
?
? ? ?
2x
1
4
?
??
–4 ? 2 – |x| ? 4
–6 ? – |x| ? 2
–2 ? |x| ? 6
|x| ? 6
? x ? [–6, 6] …(1)
Now, 3 – x ? 1
And x ? 2 …(2)
and 3 – x > 0
x < 3 …(3)
From (1), (2) and (3)
? x ? [–6, 3) – {2}
? = 6
? = 3
? = 2
? + ? + ? = 11
13. Consider the system of linear equation x + y + z =
4 ?, x + 2y + 2z ? = 10 ? , x + 3y + 4 ?
2
z = ?
2
+15,
where ? , R ?? . Which one of the following
statements is NOT correct ?
(1) The system has unique solution if
1
2
?? and
1 ?? , 15
(2) The system is inconsistent if
1
2
?? and 1 ??
(3) The system has infinite number of solutions if
1
2
?? and 15 ??
(4) The system is consistent if
1
2
??
Ans. (2)
Sol. x + y + z = 4 ?, x + 2y + 2z ? = 10 ? , x + 3y + 4 ?
2
z = ?
2
+15,
2
1 1 1
1 2 2
1 3 4
? ? ?
?
= ? ?
2
21 ?? ?
? For unique solution
1
0, 2 1 0,
2
??
? ? ? ? ? ? ?
??
??
Let
1
0,
2
? ? ? ?
y x z
2
4 1 1
0, 10 2 1
15 3 1
?
? ? ? ? ? ? ?
??
? ? ? ? 15 1 ? ? ? ? ?
For infinite solution
1
, 1 or 15
2
? ? ? ?
14. If the circles ? ? ? ?
22 2
x 1 y 2 r ? ? ? ? and
22
x y 4x 4y 4 0 ? ? ? ? ? intersect at exactly two
distinct points, then
(1) 5 < r < 9
(2) 0 < r < 7
(3) 3 < r < 7
(4)
1
r7
2
??
Ans. (3)
Sol. If two circles intersect at two distinct points
1 2 1 2 1 2
r r C C r r ? ? ? ? ?
r 2 9 16 r 2 ? ? ? ? ?
|r – 2 | < 5 and r + 2 > 5
-5 < r – 2 < 5 r > 3 ……….(2)
-3 < r < 7 ………(1)
From (1) and (2)
3 < r < 7
15. If the length of the minor axis of ellipse is equal to
half of the distance between the foci, then the
eccentricity of the ellipse is :
(1)
5
3
(2)
3
2
(3)
1
3
(4)
2
5
Ans. (4)
Sol. 2b = ae
be
a2
?
2
e
e1
4
??
2
e
5
?
16. Let M denote the median of the following
frequency distribution.
Class 0-4 4-8 8-12 12-16 16-20
Frequency 3 9 10 8 6
Then 20 M is equal to :
(1) 416
(2) 104
(3) 52
(4) 208
Ans. (4)
Sol.
Class Frequency Cumulative
frequency
0-4 3 3
4-8 9 12
8-12 10 22
12-16 8 30
16-20 6 36
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