22. Point P (2, 4) is translated parallel to the line y – x – 1 = 0, through a distance 3 √2 so that its ordinate is decreased and it reaches at Q. If R is the mirror image of Q in the line y – x – 1 = 0, its coordinate are
(A) (– 1, 1) (B) (0, 0)
(C) (6, 6) (D) none of these
23. If the line y = √3 x cuts the curve x3 + y2 + 3x2 + 9 = 0 at the points A, B, C, then OA.OB.OC ( O being origin) equals
(A) 36 (B) 72
(C) 108 (D) none of these
24. Let O be the origin, and let A(1, 0), B(0, 1) be two points. If P(x, y) is a point such that xy > 0 and x + y < 1, then
(A) P lies either inside the ΔOAB or in the third quadrant
(B) P cannot be inside the ΔOAB
(C) P lies inside the ΔOAB
(D) none of these
25. Let ABC be a triangle with equation of sides AB, BC, CA respectively x – 2 =0, y – 5 = 0 and 5x + 2y – 10 = 0, then the orthocentre of triangle lies on the line
(A) x – y = 0 (B) 3x + y =1
(C) 4x + y = 13 (D) x – 2y =1
26. The foot of the perpendicular on the line 3x + y = λ drawn from the origin is C if the line cuts the x–axis and y–axis at A and B respectively then BC : CA is
(A) 1: 3 (B) 3 : 1
(C) 1: 9 (D) 9 : 1
28. In what ratio does the point (3, –2) divide the line segment joining the points (1, 4) and (–3, 16)?
(A) 1 : 3 (externally) (B) 3 : 1 (externally) (C) 1 : 3 (internally) (D) 3 : 1 (internally)
29. For what value of x will the points (x, –1), (2, 1) and (4, 5) lie on a line?
(A) 1 (B) 0 (C) 2 (D) none of these
30. The angle between straight lines x2 –y2 –2y –1 = 0 is
(A) 90° (B) 60° (C) 75° (D) 36°
31. The distance between the lines 4x + 3y = 11 and 8x + 6y = 15 is
(A) 7/2 (B) 7/10 (C) 4 (D) none of these
32. Find the length of the perpendicular from origin to the straight line 3x –y + 2 = 0
(A) 2 (B) –2/ √10 (C) 2/ √10 (D) none of these
33. If the sum of the slopes of the lines given by 4x2 + 2kxy –7y2 = 0 is equal to the product of the slopes then k =
(A) –4 (B) 4 (C) –2 (D) 2
34. Find the value of k, so that the equation –2x2 + xy + y2 –5x + y + k = 0 may represent a pairof straight lines
53. The straight lines of the family x(a+b) + y (a-b) = 2a (a and b being parameters) are
(A) not concurrent (B) Concurrent at (1, -1)
(C) Concurrent at (1, 1) (D) None of these
54. If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is
(A) square (B) a circle
(C) straight line (D) two intersecting lines
55. If the line y = mx meets the lines x + 2y – 1= 0 and 2x – y + 3 = 0 at the same point, then m is equal to
(A) 1 (B) -1
(C) 2 (D) -2
56. The area inclosed by 3|x| + 4|y| ≤ 12 is
(A) 6 squar units (B) 12 sq. units
(C) 24 square units (D) 36 square units
57. If a, b, c are in A.P. then line 2ax + 3by + 3c = 0 always passes through fixed point
(A) (2, –2) (B) (3/2, 2)
(C) (3/2, –2) (D) none of these
58. Equation (3a – 2b)x2 + (c –2a)y2 + 2hxy = 0 represents pair of straight lines which are perpendicular to each other then (a – b) is equal to
(A) b + c (B) b – c
(C) c – b (D) 2c
59. ax + by + c = 0 represents a line parallel to x–axis if
(A) a = 0, b = 0 (B) a = 0, b≠0
(C) a≠0, b = 0 (D) c = 0
60. If the angle between the two straight lines represented by 2x2+ 5xy+ 3y2+7y+4= 0 is tan–1m then m equals to
(A) 1/5 (B) 1
(C) 7/5 (D) 7
61. The diagonals of a parallelogram PQRS are along the straight lines ax + 2by = 50 and 4bx – 2ay =100. Then PQRS must be a
(A) rhombus (B) rectangle
(C) square (D) none of these
62. The area enclosed by |x| + |y| = 1 is
(A) 1 (B) 2
(C) 3 (D) 4
68. The distance of the line 2x –3y = 4 from the point (1, 1) in the direction of the line x + y = 1 is …………………………………………….
69. If the point (2, a) lies between the lines x + y = 1 and 2(x + y) = 5, then a lies between ……………………….. and …………………………………..
70. If mn = 1, then the lines mx + y = 1 and y – nx = 2 will be ………………………………………
71. If the point (2a –3, a2 –1) is on the same side of the line x + y –4 = 0 as that of the origin,then the set of values of a is ……………………………
72. The set of lines ax + by + c = 0 where 3a + 2b + 4c = 0 is concurrent at the point ………………………………………………..
73. If the image of the point (–2, 1) by a line mirror be (2, –1) then the equation of the line mirror is ……………………………………..…………
13. Equation of the bisector of angle B of the triangle ABC is y = x. If A is (2, 6) and B is (1, 1); equation of side BC is
(A) 2x + y – 3 = 0 (B) x – 5y + 4 = 0
(C) x – 6y + 5 = 0 (D) none of these
14. Vertex opposite to the side x + y – 2 = 0 of the equilateral triangle, with centroid at the origin; is
(A) (– 1, 1) (B) (2, 2)
(C) (– 2, – 2) (D) none of these
19. The number of lines that can be drawn from the point (2, 3), so that its distance from (-1, 6) is equal to 6, is
(A) 1 (B) 2
(C) 0 (D) infinite
20. If ΔOAB is an equilateral triangle (O is the origin and A is a point on the x-axis), then centroid of the triangle will be
(A) always rational (B) rational if B is rational
(C) rational if A is rational (D) never rational
(a point P(x, y) is said to be rational if both x and y are rational)
21. Equation of a straight line passing through the point (4, 5) and equally inclined to the lines
3x = 4y + 7 and 5y = 12x + 6 is
(A) 9x –7y = 1 (B) 9x + 7y = 71
(C) 7x – 9y = 73 (D) 7x – 9y + 17 = 0
22. Two vertices of a triangle are (5, -1) and (-2, 3). If the orthocentre of the triangle is the origin, then the third vertex is
(A) (-4, 7) (B) (-4, -7) (C) (4, -7) (D) (4, 7)
23. Drawn from the origin are two mutually perpendicular lines forming an isosceles triangle with the straight line 2x + y = a. Then the area of this triangle is ………………………….
24. Two particles start form the same point (2, –1), one moving 2 units along the line x + y = 1 and the other 5 units along the line x –2y = 4. If the particles move towards increasing y, then their new positions are ……………….., ……………………………
ANSWERS
19. If 3 a + 4 b + 2 c = 0, then the point of concurrent of the family of lines a x + b y + c = 0 and (1, 2) are
(A) on the same sides of the line 4 x - y + 1 = 0
(B) on the opposite side of the line 4 x - y + 1 = 0
(C) are at equal distances from the origin.
(D) None of these
20. If a, b, c are three consecutive integers, then the family of lines a x + b y + c = 0 are concurrent at the point,
(A) (1, 2) (B) (-2, 1)
(C) (1, -2) (D) None of these
ANSWERS
LEVEL −III
1. C
2. A
3. A
4. C
5. B
6. D
7. D
8. A
9. A
10. B
11. D
12. B
13. D
14. C
15. A
16. A
17. D
18. B
19. A
20. C
