MATHEMATICS Max Marks : 80

SECTION – I
( SINGLE CORRECT CHOICE TYPE )
This section contains 7 multiple choice questions. Each question has 4 choices (A), (B), (C) and
(D) for its answer, out of which ONLY ONE is correct
47. The equation of the line belonging to the family of the lines given by

3x + 4y + 6 +  (x + y + 2) = 0 (where  being a parameter) and situated at the

greatest distance from the point (2, 3) is

A) 4x – 3y + 8 = 0 B) 4x + 3y + 8 = 0

C) 4x + 3y – 8 = 0 D) 3x - 4y + 6 = 0

48. The sides of a triangle are 11, 15 and k, k ∈ I+ . Then the number of values of k such

that the triangle is obtuse is

A) 7 B) 12 C) 13 D) 14
π
49. Two lines L1 and L2 are drawn from (α, α )(α > 0) making an angle with lines
4
L3 ≡ x + y = f (α ) and L4 ≡ x + y + f (α) = 0 . L1 intersects L3 and L4 at A and B, and L2

intersects L3 and L4 at C and D respectively. ( f (α) < 2α ) . If area of trapezium

ABDC is independent of α , then f (α ) can be (k is an integer )

k k
A) k α B) C) D) k α 2
α α2
50. Radius of the circle which is drawn on a normal chord of y 2  8 x as diameter and it

passes through the vertex of the parabola is

A) 2 3 B) 3 3 C) 6 3 D) none of these

x2 y2
51. There are exactly two points on the ellipse 2 + 2 = 1 whose distance from the center
a b

a 2  2b 2
of the ellipse are equal to . Eccentricity of this ellipse is equal to
2

3 1 1 2
A) B) C) D)
2 3 2 3
52. If a parabola touches the x-axis at (6,0) and y-axis at (0,3) and lies in the first
quadrant, its axis of symmetry is px  qy  r  0  p  q  and H.C.F of (p,q,r) =1, then
pqr 

A) 10 B) 12 C) 13 D) 15
2
15
 15 
53. The Value of 
r 0
15
Cr  r   is
 2

A) 210.15 B) 212.15 C) 213.15 D) 215.15

SECTION – II
( MULTIPLE CORRECT CHOICE TYPE )
This section contains 4 multiple choice questions. Each question has 4 choices (A), (B), (C) and
(D) for its answer, out of which ONE OR MORE is/ are correct
54. Let P (x1, y1) and Q(x2, y2), y1 < 0, y2 < 0, be the end points of the latus rectum of the
ellipse x2 + 4y2 = 4. The equations of parabolas with latus rectum PQ are

(A) x2 + 2 3 y = 3 + 3 (B) x2 – 2 3 y = 3 + 3

(C) x2 + 2 3 y = 3 – 3 (D) x2 – 2 3 y = 3 – 3
55. Let A and B be two distinct points on the parabola y2 = 4x. If the axis of the parabola
touches a circle of radius r having AB as its diameter, then the slope of the line joining
A and B can be
1 1 2 2
(A)  (B) (C) (D) 
r r r r

56. If the circle x2 + y2 = a2 cuts a rectangular hyperbola xy = c2 in

 c  c  c   c
A  ct1 ,  , B  ct 2 ,  , C  ct 3 ,  and D  ct 4 ,  , then
 t 1 t  t 2   t 3   4 

(A) t1t2t3t4 = 1 (B) t1 + t2 + t3 + t4 = 0

1 1 1 1 1
(C)    =0 (D)  0
t1 t 2 t 3 t 4 t1 t 2
57. Tangents are drawn to the circle x 2  y 2  32 from a point A lying on the x – axis. The

tangents cut the y – axis at points B and C, then the possible coordinates of A such that

the area of ABC is minimum, are


A) 8 2, 0  B) 8, 0  C)  8,0  
D) 8 2, 0 
SECTION – III
 (COMPREHENSION TYPE)
This section contains 2 groups of questions. Each group has 2&3 multiple choice questions
based on a paragraph. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of
which ONLY ONE is correct.
Paragraph-1
Each of the circles S1  x 2  y 2  4y  1  0;S2  x 2  y 2  6x  y  8  0 and
S3  x 2  y 2  4x  4y  37  0 touches the other two. Let P1 , P2 , P3 be the point of contact to

the circles S1  0 , S2  0 ; S2  0 , S3  0 ; S3  0 , S1  0 respectively and C1 , C2 , C3 are the

centres of S1 ,S2 ,S3 respectively . And T be the point of concurrence of tangets at
P1 , P2 , P3 to the circle.

58. P2 and P3 are reflections of each other in the line

A) y = x B) y = x + 1 C) 2x  y  3  0 D) 2x  y  a  0
59. The point P2 divides C2C3 in the ratio
A) 1: 6 B) 1: 6 C) 6 :1 D) 6 :1
60. The area of quadrilateral TP2C3P3

A) 11 sq units B) 25 sq units C) 15 sq units D) 9 sq units

Paragraph-2

Let C : y  x 2  3 , D: y  kx 2 , L1 : x  a , L2 : x  1 ,  a  0

61. If the parabolas C and D intersect at a point A on the line L1, then the tangent line L at

A to the parabola D is

A) 2  a 2  3 x  ay  a3  3a  0 B) 2  a 2  3 x  ay  a3  3a  0

C)  a 2  3 x  2ay  2a3  6a  0 D) 2  a 2  3 x  ay  a 3  3a  0

62. If the tangent line L in above question meets the parabola C at a point B on the line :

L2, other than A then ‘a’ is equal to

A) – 3 B) – 2 C) 2 D) 1
 SECTION –IV
(INTEGER ANSWER TYPE)
This section contains 7 questions . The answer to each of the questions is a single digit integer,
ranging from 0 to 9. The appropriate bubbles below the respective question numbers in the
ORS have to be darkened.
63. If point P(x, y) is such that it moves on a hyperbola

(x  3) 2  (y  4) 2  x 2  y 2 = k2 + 1, then number of possible integral values of k is

x2
64. If P is any point on ellipse  4y 2  1 whose foci are at A and B then, max. value of
4
PA.PB is

65. If foot of perpendicular from focus upon any tangent of parabola y2 – 4y + 4x + 8 = 0

lies on the line x + k = 0 then k is
66. The parabola y = x2 – 5x + 4 cuts the x-axis at P and Q. A circle is drawn through P
and Q so that the origin lies outside it. The length of tangent to the circle from the
origin is equal to
67. Let the line x – 8y + k = 0, kI meets the rectangular hyperbola xy = 1 at the points
whose abscissae are integers then the number of such lines is
68. If k>0, p,q,r,s are non-zero constants and the graphs of f  x  kx  p  r and
g  x    kx  q  s are intersecting at exactly 2 points (1,4), (3,1), then
pq
 _________
k
69. Let S1  0, S2  0 be two circles intersect at P(6,4) and both are tangents to x-axis and
52
the line y=mx. If the product of radii of the two circles S1  0, S2  0 is , then the
3
value of m 2 is ____

MATHS
47 B 48 C 49 B 50 C 51 C 52 C
53 C 54 BC 55 CD 56 ABC 57 BC 58 A
59 B 60 C 61 D 62 B 63 3 64 4
65 1 66 2 67 4 68 4 69 3
MATHS
SINGLE ANSWERS
47. The family of lines 3x + 4y + 6 +   x  y  2  0 are concurrent at the

intersection of 3x + 4y + 6 = 0 and x + y + 2 = 0 i.e., at (–2, 0)

  the line through (–2, 0) lying at the greatest distance from A (2, 3) is the line through

B (–2, 0) perpendicular to AB.

Hence its equation is

4
y0    x  2 i.e., 4 x  3 y  8  0
3
48.4 < k < 26

for triangle to be obtuse k2 > 112 + 152 or

152 > 112 + k2

⇒ k ∈ {5,6,7,8,9,10,19,20,21,22,23,24,25}
49. Distance b/w AC and BD
= 2 f (α )
AC = 2(OP - OE)
A Y
B P(,)

E
X
O
F C
L3
D
L4

 f (α ) 
 
= 2  2α −
 2 

 f (α ) 
 
BD = 2  2α +
 2 

⇒ AC + BD = 4 2α

⇒ ar (□ ABCD) = 4α f (α )
50. If PQ where P(f), Q(f1) is a normal chord at P to the parabola | Eq.of the circle
having PQ is a diameter is  x  at 2    x  at12    y  2at   y  2at1   0 But , it passes
4 2
through the origin  t1t  4 | t1   t  | t 2  2 | PQ  12 3
t t
51. The given distance is clearly the length of semi major axis.
 a 2  2b 2
Thus, a
2

 2b 2  a 2

 2a 2 1  e 2  a 2 
1 1
 e2  e
2 2
52. Equation of the circle described in the segment joining (6,0),(0,3) as a diameter is

x 2  y 2  6 x  3 y  0 , if touches the directrix of the parabola at (0,0) |  of dirctrix of

 6 12 
the parabola is 2 x  y  0 , projection drawn from (0,3) to 2 x  y  0 is  , 
 5 5
 6 12 
Hence focus of the parabola is  ,  is equation of the axis of the parabola is
5 5 

5 x  10 y  18  0 .

15 15
225 15
53. 
r 0
15
Cr .r 2  15 r.15 Cr 
r 0 4
2

15 15

r . 2 15
Cr 15  r  1  1 14 Cr 1  15.214  15.14.213  215.60
r 0 r 1

225 15 15
 Required Sum   2  2 .60  225.214
4
ONE OR MORE THAN ONE ANSWERS
x 2 y2 1 3
54. Given ellipse is  1 e= 1 
4 1 4 2

 
(-ae,0) (ae,0)

   1
 1 Q P
 3,  
  3,    2
 2
 1 1
P  (ae, –b2/a) =  3,   , Q  (–ae, –b2/a) = (– 3, – )
 2 2

length of PQ = 2 3

 3 1 
V 1/2 V  0, 
Q  2 
P
S
 1 3 
V   0,  
V
 2 

PQ 3
VS = SV = =
4 2

 Equations of parabolas

are

 3 1 
x2 = –2 3y
   x2 + 2 3 y=3– 3
 2 

 1 3 
and x2 = 2 3 y
   x2 – 2 3y =3+ 3
 2 
55. Slope of line AB

2
B (t2 , 2t2)
(t12, 2t1) A 
 t 12  t 2 2 
 , t1  t 2 
 2 


y2 = 4x

(t 2  t1 )  2  2
M=   
(t 2  t1 ) (t 2  t1 )  t1  t 2  r

As |t1 + t2| = r
Any point on xy = c2 is  ct,  . As it lies on the given circle, we get
c
56.
t  

c2
c2t2 +  a 2  c2t4 - a2 t2 + c2 = 0
t2

a2
Thus t1t2t3t4 = 1, t1 +t2 + t3 + t4 = 0, t1t2 = -  , t1t2t3 = 0
c2

Thus, (a), (b), (c) are true.

57. Conceptual
PASSAGES
58. P2 , P3 are reflection to each other w.r.t y = x
59.  r2 : r3  1: 6
60. T  3, 3 P2  4, 1 C3  2, 2 C4  1, 4
 
61. A  a, a 2  3 Equation of tangent L is S1 = 0 is 2(a2-3) x –ay-a3 +3a = 0
62. The line L meets the parabola C: y= x2-3 at the Points for which

x2-3 =

2 a2  3  xa 2
 3 (x-a) (ax+6-a2) = 0 But x = 1 and x  a
a
a2  6
x  1  a  2,3
a
INTEGERS ANSWERS
63. |PS1 – PS2| = const (A)
for hyperbola
k2 + 1 < 5 , k2 < 4
-2 < k < 2 No. of integral values is 3.
64. PA + PB = 4

PA  PB
 (PA.PB)1/ 2  PA.PB  4.
2
65. (y – 2)2 = -4 (x + 1)

foot of  will lie on tangent at vertex. x + 1 = 0

 k = 1.
66. P  1,0  , Q   4,0 

(x – 1)(x – 4) + y2 + y = 0

THE LENGTH OF TANGENT FROM (0, 0) IS 4  2

8
67. x − + k = 0 ⇒ x 2 + kx − 8 = 0
x
−k ± k 2 + 32
⇒x=
2
2 2
Let k + 32 = λ (λ ∈ I )
⇒ λ 2 − k 2 = 32 ⇒ (λ − k )(λ + k ) = 32
The number of values of k is 4

68.

y  q 
 , s
k 

 3,1
1,4  p 
 , r
 x 
  p  q O
x
4
k

pq
 4
k

69. Let m  tan  , C1  r1 , r , tan  / 2 , C2  r2 , r2 tan  / 2 equation of the circle having centre at

 
( r , r tan  / 2 ),radius r is x 2  y 2  2rx  2r tan y  r 2 tan 2  0 , but , it passes through (6,4)
2 2

 r 2 tan 2  / 2  2r  6  4 tan  / 2  52  0

Here , r1 , r2 are the roots of the Q.E

52 52  2
 r1r2    tan 2    3     m2  3
 3  2 3
tan 2  
 2