COMPETITION TEST - 2

Q.1 The equation kx2 + 4xy + 5y2 = 0 represents two lines inclined at an angle π if k is :

(A) 54

(B) 45

(C) 45−

(D) 165

Q.2 The combined equation of the pair of lines through the point (1, 0) and parallel to the linesrepresented by 2x2 – xy – y2 = 0 is :(A) 2x2 – xy – 2y2 + 4x – y = 6 (B) 2x2 – xy – y2 – 4x – y + 2 = 0(C) 2x2 – xy – y2 – 4x + y + 2 = 0 (D) 2x2 – xy – y2 – 4x + y + 1 = 0

Q.3 If one end of the diameter of the circle x2 + y2 – 4x – 2y – 5 = 0 is (3, 4) the co-ordinatesof the other end are :(A) (1, –2) (B) (–1, 2) (C) (–1, –2) (D) (1, 2)

Q.4 If the equation 2x2 + hxy + 2y2 + 4x – 6y + k = 0 represents a circle of radius 4 units thenthe values of h and k are :(A) 0, 3 (B) 0, –3 (C) –3, 0 (D) 2, –3

Q.5 The equation of circle touching both axis in the first quadrant and radius 10 is :(A) x2 + y2 – 20x – 20y + 100 = 0 (B) x2 + y2 + 20x + 20y + 100 = 0(C) x2 + y2 – 20x + 20y + 100 = 0 (D) x2 + y2 + 20x – 20y + 100 = 0

Q.6 The equation of the parabola with vertex at (0, 0) and the axis as y-axis and passing throughthe point (6, –3) is :(A) x2 + 4y = 0 (B) x2 + 8y = 0 (C) x2 + 12y = 0 (D) x2 – 12y = 0

Q.7 The eccentricity of the parabola y2 = –8x is :(A) 2 (B) –2 (C) –1 (D) 1

Q.8 If pair of lines represented by ax2 + 2hxy + by2 = 0 are such that the sum of the slopes ofthe lines is three times the product, of their slopes then(A) 3b + 2h = 0 (B) 3a + 2h = 0 (C) 3h + 2a = 0 (D) –3a + 2h = 0

Q.9 The equation 2x2 – 3xy – py2 + x + qy – 1 = 0 represents two mutually perpendicular linesif :(A) p = 3, q = 2 (B) p = 2, q = 3 (C) p = –2, q = 3 (D) p = 2, q = –3

Q.10 The condition that the lines ax2 + 2hxy + by2 = 0 are equally inclined to the x-axis is :(A) a + b = 0 (B) h2 – ab = 0 (C) h = 0 (D) a = b

Q.11 If the acute angle between the lines ax2 + 2hxy + by2 = 0 is π/3 then (a + 3b) (3a + b)is :(A) h2 (B) 2h2 (C) 3h2 (D) 4h2

Q.12 If one of the lines ax2 + 2hxy + by2 = 0 bisects the angle between the co-ordinate axes, then(A) (a – b)2 = 4h2 (B) (a + b)2 = 4h2 (C) 4ab = h2 (D) (a + b)2 = 2h2

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COMPETITION TEST - 2

Q.13 If the latus rectum is equal to half of its major axis then the eccentricity of the ellipse is

(A) 12

(B) 1

2 2(C)

1

2(D)

13

Q.14 The distance between foci and the distance between directrices of the ellipse x2 + 4y2 = 4is :

(A) 2 3 , 8 3 (B) 8

3, 2 3 (C) 2 3 ,

8

3(D) 3 ,

4

3

Q.15 The equation of the circle passing through the point (4, 5) and having centre at (2, 2) is(A) x2 + y2 + 4x + 4y – 5 = 0 (B) x2 + y2 – 4x – 4y – 5 = 0(C) x2 + y2 – 4x = 13 (D) x2 + y2 – 4x + 4y + 5 = 0

Q.16 The lines 2x – 3y = 5 and 3x – 4y = 7 are the diameters of a circle of area 154 sq. units.The equation of the circle is :(A) x2 + y2 + 2x – 2y = 62 (B) x2 + y2 – 2x + 2y = 47(C) x2 + y2 + 2x – 2y = 47 (D) x2 + y2 – 2x + 2y = 62

Q.17 If the line x – 1 = 0 is the directrix of the parabola y2 – kx + 8 = 0 then one of the valuesof k is :(A) 1/6 (B) 8 (C) 4 (D) 1/4

Q.18 If the parameter of the point P on the parabola y2 = 20x is 2, then focal distance of P is(A) 20 (B) 16 (C) 5 (D) 25

Q.19 The parametric equations of the parabola are x = t2 + 1, y = 2t + 1. The Cartesian equationof its directrix is :(A) x = 0 (B) y = 0 (C) x +1 = 0 (D) y + 1 = 0

Q.20 If the slopes of one of the lines given by 4x2 + kxy + y2 = 0 is four times the other, thenvalue of k is :(A) 1 (B) –1 (C) 5 (D) ±5

Q.21 The lines 3x – 4y + 4 = 0 and 6x – 8y – 7 = 0 are tangents to the same circle. The radiusof this circle is :

(A) 45

(B) 710

(C) 34

(D) 32

Q.22 If 3x + y = 0 is a tangent to the circle which has its centre at the point (2, –1) then theequation of the other tangent to the circle from the origin is :(A) x – 3y = 0 (B) x + 3y = 0 (C) 3x – y = 0 (D) x + 2y = 0

Q.23 The value of k, if the line y = x + k touches the ellipse 2x2 + 3y2 = 1 is :

(A) ±65

(B) ±56

(C) ±32

(D) ±23

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COMPETITION TEST - 2

Q.24 The slopes of the tangent to the ellipse 2 2x y

7 4+ = 1 from the point (3, –2) are

(A) 1, 2 (B) –1, 3 (C) 0, 5 (D) 0, –6

Q.25 Separate equations of lines for a pair of lines whose equation is x2 + xy – 12y2 = 0 are :(A) x + 4y = 0 and x + 3y = 0 (B) 2x – 3y = 0 and x – 4y = 0(C) x + 4y = 0 and x – 3y = 0 (D) x – 4y = 0 and x + 3y = 0

Q.26 If the equation hxy + gx + fy + c = 0 represents pair of straight lines, then :(A) fg = ch (B) gh = fc (C) fh = gc (D) fh = –fc

Q.27 The line y = mx + c touches the parabola x2 = 4ay if

(A) c = –am (B) c = a

m−

(C) c = –am2 (D) c = 2

am

Q.28 Equation of the tangent at (–4, –4) on x2 = –4y is :(A) 2x – y + 4 = 0 (B) 2x + y – 4 = 0 (C) 2x – y – 12 = 0(D) 2x + y + 4 = 0

Q.29 Tangents drawn at the end points of focal chord of a parabola are(A) parallel (B) perpendicular (C) oblique (D) coincident

Q.30 If a2 + 14ab + b2 = 12h2, then the angle between the l ines represented byax2 + 2hxy + by2 = 0 is :(A) 30° (B) 60° (C) 45° (D) 90°

Q.31 If the acute angles between the pair of lines 3x2 – 7xy + 4y2 = 0 and 6x2 – 5xy + y2 = 0be θ1 and θ2 respectively, then :(A) θ1 = θ2 (B) θ1 = 2θ2 (C) 2θ1 = θ2 (D) θ1 = π – θ2

Q.32 The equation of the parabola is y2 = 12x. If m1, m2 are slopes of tangents through (–4, –1)to the parabola, then m1

2 + m22 =

(A) 19

(B) 2516

(C) 3616

(D) 1625

Q.33 If the line y = mx + 5 be tangent to the ellipse 7x2 + 9y2 = 63 then m =

(A) ±2 (B) ± 2 (C) ±1 (D) 0

Q.34 The product of the lengths of the perpendicular segments from the foci on any tangent tothe ellipse b2x2 + a2y2 = a2b2 is :(A) b2 (B) a2 (C) 2b (D) 2a

Q.35 Length of the latus rectum of a hyperbola 2 2

2 2

x y1

a b− = is :

(A) 22a

b(B)

2

2

a2b

(C) 22b

a(D)

2b2a

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COMPETITION TEST - 2

Q.36 The asymptotes of hyperbola passes through the :(A) foci of the hyperbola (B) centre of the hyperbola(C) vertices of the hyperbola (D) perpendicular to the directrices

Q.37 If the lines represented by the equation ax2 + bxy + cy2 = 0 make angles α and β with thex-axis then tan (a + b) =

(A) b

a c+(B)

ab c+

(C) b

a c−(D)

ca b

−+

Q.38 The combined equation of the lines through the origin and perpendicular to the lines given by3(x – 2)2 + (x – 2)(y + 2) – (y + 2)2 = 0 is :(A) x2 + xy – 3y2 = 0 (B) x2 + xy + 3y2 = 0(C) 3x2 – xy + y2 = 0 (D) 3x2 – xy – y2 = 0

Q.39 The combined equation of the lines through the origin and making an equilateral triangle withthe line x + y = 5 is :(A) x2 + xy + y2 = 0 (B) x2 – xy + y2 = 0(C) x2 + 4xy + y2 = 0 (D) x2 – 4xy + y2 = 0

Q.40 The circles x2 + y2 – 2x + 6y + 6 = 0 and x2 + y2 – 5x + 6y + 15 = 0 touch each otherthen the equation of their common tangent is :(A) x = 3 (B) y = 6(C) 7x + 2y – 21 = 0 (D) 7x + 12y + 21 = 0

Q.41 If the two circles 2x2 + 2y2 – 3x + 6y + k = 0 and x2 + y2 – 4x + 10y + 16 = 0 cut orthogonally,then the value of k is :(A) 41 (B) 14 (C) 4 (D) 0

Q.42 The parametric equations x = 2

2

a(1 t )1 t

−+

and y = 2

2at1 t+ represents a circle of radius =

(A) a (B) a2 (C) 2a (D) 4a2

Q.43 If the line x – y = 1 touches the hyperbola whose foci are (±3, 0), the equation of thehyperbola is :

(A) 2 2x y

14 5

+ = (B) 2 2y x

15 4

− = (C) 2 2x y

15 4

+ = (D) 2 2x y

15 4

− =

Q.44 The eccentricity of the conjugate hyperbola of the hyperbola x2 – 3y2 = 1 is :

(A) 2

3(B) 2 (C) 4 (D)

43

Q.45 The equations of the common tangents to the circle 5x2 + 5y2 = 16 and the hyperbola2 2x y

16 48− = 1 is :

(A) y = ±2(x ± 2) (B) y = (x ± 2) (C) y = 2(x – 2) (D) y = –2(x – 2)

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COMPETITION TEST - 2

Q.46 The line 3x – 4y = λ touches the circle x2 + y2 – 4x – 8y – 5 = 0 if the value of λ is :(A) 20 (B) –15 (C) –35 (D) 35

Q.47 If length of tangent segment from the point P(–2, 3) to the circle 2x2 + 2y2 = r2 is 232

then

r =

(A) 192

(B) 19 (C) 32

(D) 3

Q.48 The number of tangents which can be drawn from the point (1, 2) to the circlex2 + y2 = 5 are(A) 1 (B) 2 (C) 3 (D) 0

Q.49 Equation of the director circle of the circle x2 + y2 = 25 is :(A) x2 + y2 = 100 (B) x2 + y2 = 25 (C) x2 + y2 = 50 (D) x2 + y2 = 10

Q.50 The equation of the locus of the point that the tangents from which to the circle x2 + y

2 = 25

have slopes whose sum is 2 is :(A) x2 + xy + 25 = 0 (B) x2 + xy = 25(C) x2 = xy + 25 (D) x2 = xy – 25

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Q.1 - 50 carry 4 marks each.

(b) There is Negative Marking.

COMPETITION TEST #

MATHEMATICS

INSTRUCTIONS TO CANDIDATES

1. This paper contains 50 Qs. in all.

2. Each Questions has four options, only one option is correct.

3.

4. Answer sheet is provided at the end of the question paper booklet. You must write your name, Father's Name,Address and Roll No. in the upper portion of Answer Sheet and indicate your answers on Answer Sheet only.

Note : (a) There are 50 Questions in this paper.

(c) Read the following instructions very carefully before attempting the question paper.

SYLLABUSPair of Straight Lines, Circle, Conic

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MATHEMATICS

FOR OFFICE USE ONLY

No. of Correct Ans. No. of Wrong Ans. Marks

Total

Total Marks

Rank

A B C D

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

A B C D

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

A B C D

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

Name : ............................................................................................. Roll No. : ......................

Father's Name : ....................................................... Phone No : .................... Date : ......................

Address : .............................................................................................

ANSWER SHEET

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ANSWER KEYQue. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ans. B C A B A C D B B C D B C C B

Que. 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Ans. B C D A D C A B D C A C A B A

Que. 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

Ans. A B B A C C C A D A C A D B A

Que. 46 47 48 49 50

Ans. D D A A C

ANSWER KEYQue. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ans. B C A B A C D B B C D B C C B

Que. 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Ans. B C D A D C A B D C A C A B A

Que. 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

Ans. A B B A C C C A D A C A D B A

Que. 46 47 48 49 50

Ans. D D A A C

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