Straight Lines PYQs Set 3 (PDF)

Set 3 of 15 · Questions 61-90 of 442

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Straight Lines · Set 3 (Q61-90)
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61. $The roots of a cubic equation f (x) = 0 are diminished by \frac{- 3}{2} so as to remove the term containing x^{2} and the transformed equation is 8 x^{3} - 54 x - 78 = 0 . Then the equation f (x) = 0 is$ (1) $2 x^{3} - 9 x^{2} - 6 = 0$ (2) $2 x^{3} - 9 x^{2} - 27 x - 6 = 0$ (3) $8 x^{3} + 3 x^{2} - 6 = 0$ (4) $8 x^{3} - 9 x^{2} - 12 x - 15 = 0$ 62. $If a straight line L passing through the point (5, - 3) is inclined at an angle of 60^{\circ} to the line 3 x + y - 9 = 0 and L intersects X -axis then the equation of L is$ (1) $x - 3 y - 3 - 53 = 0$ (2) $3 x - y - 3 - 53 = 0$ (3) $3 x - y + 3 + 53 = 0$ (4) $x - 3 y + 3 + 53 = 0$ 63. $A point P (x, y) is such that the sum of squares of its distances from (a, 0) and (- a, 0) is 2 b^{2} . The equation representing the locus of P is$ (1) $x^{2} + y^{2} = b^{2} + a^{2}$ (2) $x^{2} + y^{2} = b^{2} - a^{2}$ (3) $x^{2} + y^{2} = b^{2} - 2 a^{2}$ (4) $x^{2} + y^{2} = b^{2} + 2 a^{2}$ 64. $The lines represented by a x^{2} + 2 hxy + b y^{2} = 0 are perpendicular to each other if$ (1) $h^{2} = a + b$ (2) $a + b = 0$ (3) $h^{2} = ab$ (4) $h = 0$ 65. $Let A be the point of intersection of the lines 3 x + 2 y = 14, 5 x - y = 6 and B be the point of intersection of the lines 4 x + 3 y = 8, 6 x + y = 5 . The distance of the point P (5, - 2) from the line AB is$ (1) $\frac{13}{2}$ (2) $8$ (3) $\frac{5}{2}$ (4) $6$ 66. $If the axes are rotated through an angle 45^{\circ}, the coordinates of the point (22, - 32) in the new system are$ (1) $(33, - 5)$ (2) $(- 1, - 5)$ (3) $(53, - 7)$ (4) $(7, - 3)$ 67. $The equations of the lines perpendicular to x^{2} - 5 xy + 4 y^{2} = 0 and passing through (2, 1) is$ (1) $4 x^{2} + 5 xy + y^{2} - 13 x - 1 = 0$ (2) $4 x^{2} + 5 xy + y^{2} - 5 x - 10 y - 7 = 0$ (3) $4 x^{2} + 5 xy + y^{2} - 4 x - 4 y - 15 = 0$ (4) $4 x^{2} + 5 xy + y^{2} - 21 x - 12 y + 27 = 0$ 68. $Let A = (2, 0) and B = (0, - 2) . Let P be any point such that the sum of the distances of P from A and B is 4 . Then the equation of the locus of the point P is$ (1) $3 x^{2} - 2 xy + 3 y^{2} - 4 x + 12 y + 16 = 0$ (2) $3 x^{2} - 2 xy + 3 y^{2} - 8 x + 8 y = 0$ (3) $3 x^{2} + 2 xy + 3 y^{2} + 8 x - 8 y = 0$ (4) $3 x^{2} + 2 xy + 3 y^{2} + 4 x - 12 y + 16 = 0$ 69. $The locus of the incentre of the triangle formed by the lines xy - 4 x - 4 y + 16 = 0 and x + y = 5 is$ (1) $x - y = 0$ (2) $x + y = 0$ (3) $x - 2 y = 0$ (4) $2 x - y = 0$ 70. $A pair of lines S = 0 together with the lines given by the equation 8 x^{2} - 14 xy + 3 y^{2} + 10 x + 10 y - 25 = 0 form a parallelogram. If its diagonals intersect at the point (3, 2), then the equation S = 0, is$ (1) $6 x^{2} - 9 xy + y^{2} - 25 x + 30 y + 25 = 0$ (2) $8 x^{2} - 14 xy + 3 y^{2} - 25 x + 30 y + 50 = 0$ (3) $8 x^{2} - 14 xy + 3 y^{2} - 50 x + 50 y + 75 = 0$ (4) $6 x^{2} + 14 xy - 3 y^{2} - 30 x + 40 y - 75 = 0$ 71. $Let A (1, 2) and C (- 3, - 6) be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to the line 7 x - y = 14 . If B (α, β) and D (γ, δ) are the other two vertices, then \| α + β + γ + δ \| is equal to$ (1) $6$ (2) $9$ (3) $3$ (4) $1$ 72. $The straight line passing through (- 1, 1) and remaining parallel to the line common to the pairs of lines provided bv 6 x^{2} - xy - 12 y^{2} = 0 and 15 x^{2} + 14 xy - 8 y^{2} = 0, is$ (1) $5 x - 2 y + 7 = 0$ (2) $3 x + 4 y - 1 = 0$ (3) $3 x - 4 y + 7 = 0$ (4) $2 x - 3 y + 5 = 0$ 73. $If the straight line passing through the point P (3, 4) makes an angle \frac{π}{6} with the positive direction of X -axis and meets the line 12 x + 5 y + 10 = 0 at Q, then the length of PQ is$ (1) $\frac{44}{12 2 + 9}$ (2) $\frac{66}{12 3 + 5}$ (3) $\frac{132}{12 3 + 5}$ (4) $\frac{148}{6 2 + 3}$ 74. $The orthocentre of the triangle with vertices A (0, 0), B (0, \frac{3}{2}), C (- 5, 0) is$ (1) $(\frac{5}{2}, \frac{3}{4})$ (2) $(\frac{- 5}{2}, \frac{3}{4})$ (3) $(- 5, \frac{3}{2})$ (4) $(0, 0)$ 75. $A (- 4, 0) and B (4, 0) are two fixed points. C and D are two points on Y -axis such that CD = 4 and C is a point below D . Then the locus of the point of intersection of the lines AC and BD is$ (1) $x^{2} - y^{2} - xy = 0$ (2) $x^{2} + 2 xy - 16 = 0$ (3) $(x + y)^{2} - 16 = 0$ (4) $2 xy = 16 + y^{2} + x^{2}$ 76. $The centroid of the triangle formed by the lines x + 3 y = 10 and 6 x^{2} + xy - y^{2} = 0 is$ (1) $(\frac{1}{3}, \frac{- 7}{3})$ (2) $(\frac{- 1}{3}, \frac{- 7}{3})$ (3) $(\frac{- 1}{3}, \frac{7}{3})$ (4) $(\frac{1}{3}, \frac{7}{3})$ 77. $If θ is the acute angle between the lines \frac{x}{a} + \frac{y}{b} = 1, \frac{x}{b} + \frac{y}{a} = 1 then sin θ =$ (1) $(\frac{2 ab}{a ^{2} + b ^{2}})$ (2) $(\frac{a - b}{a + b})$ (3) $(\frac{a ^{2} - b ^{2}}{2 ab})$ (4) $(\frac{a ^{2} - b ^{2}}{a ^{2} + b ^{2}})$ 78. $Let ABC be the triangle such that the equations of lines AB and AC be 3 y - x = 2 and x + y = 2, respectively, and the points B and C lie on x -axis. If P is the orthocentre of the triangle ABC, then the area of the triangle PBC is equal to$ (1) $8$ (2) $4$ (3) $10$ (4) $6$ 79. $The angle between the lines 2 x + 11 y - 7 = 0 and x + 3 y + 5 = 0 is\dots\dots..$ (1) $tan^{- 1} (\frac{17}{13})$ (2) $tan^{- 1} (\frac{4}{35})$ (3) $tan^{- 1} (\frac{1}{7})$ (4) $tan^{- 1} (7)$ 80. $Let L be the line joining the origin to the point of intersection of the lines represented by 2 x^{2} - 3 xy - 2 y^{2} + 10 x + 5 y = 0 . If L is perpendicular to the line kx + y + 3 = 0, then k is equal to$ (1) $\frac{1}{2}$ (2) $- \frac{1}{2}$ (3) $-1$ (4) $\frac{1}{3}$ 81. $The lines a x^{2} + 2 hxy + b y^{2} = 0 are at right angles if$ (1) $a + b = 0$ (2) $a + b = 1$ (3) $h^{2} - ab = 0$ (4) $a = b$ 82. $Let A (1, 0), B (2, - 1) and C (\frac{7}{3}, \frac{4}{3}) be three points. If the equation of the bisector of the angle ABC is αx + βy = 5, then the value of α^{2} + β^{2} is$ (1) $13$ (2) $10$ (3) $5$ (4) $8$ 83. $If L is a line passing through the point (- 1, 1) and parallel to the common line of the pairs of lines 6 x^{2} - xy - 12 y^{2} = 0 and 15 x^{2} + 14 xy - 8 y^{2} = 0, then the equation of pair of lines joining the origin to the points of intersection of the curve 2 x^{2} - xy - y^{2} + x - y = 0 and the line L is$ (1) $x^{2} - xy - y^{2} = 0$ (2) $x^{2} + xy - y^{2} = 0$ (3) $x^{2} - y^{2} = 0$ (4) $2 x^{2} - 3 xy - 6 y^{2} = 0$ 84. $Suppose 0 (0, 0) is the origin and the line L = x + y - λ = 0 meets the curve x^{2} + y^{2} - 2 x - 4 y + 2 = 0 at A and B . If ∠ AOB = 90^{\circ}, then the distance between such lines L = 0 is$ (1) $\frac{1}{2}$ (2) $\frac{3}{2}$ (3) $2$ (4) $2 + 1$ 85. $The acute angle between the lines x = - y, z = 0 and x = 0, z = 0 is$ (1) $π /6$ (2) $π /18$ (3) $π /3$ (4) $π /4$ 86. $The least distance from origin to a point on the line y = x +3 which lies at a distance of 2 units from (0, 3) is$ (1) $13 + 62$ (2) $10 + 62$ (3) $10 - 62$ (4) $13 - 62$ 87. $Two lines L_{1} and L_{2} passing through the point P (1, 2) cut the line x + y = 4 at a distance of \frac{6}{3} units from P . Then the angles made by L_{1}, L_{2} with positive X -axis are$ (1) $\frac{π}{3}, \frac{π}{6}$ (2) $\frac{π}{8}, \frac{3 π}{8}$ (3) $\frac{π}{12}, \frac{5 π}{12}$ (4) $\frac{π}{4}, \frac{π}{8}$ 88. $a x^{2} - 4 xy - 2 y^{2} = 0 represents a pair of lines. If 0 is the angle between these lines, cos θ = \frac{1}{5} and the possible values of ' a' are a_{1} and a_{2} (a_{1} < a_{2}) then a_{1} + 3 a_{2} =$ (1) $11$ (2) $10$ (3) $-5$ (4) $-6$ 89. $If the straight line 2 x - 3 y + 17 = 0 is perpendicular to the line passing through the points (7, 17) and (15, β), then β equals$ (1) $-5$ (2) $29$ (3) $5$ (4) $-29$ 90. $If the line, 2 x - y + 3 = 0 is at a distance \frac{1}{5} and \frac{2}{5} form the lines 4 x - 2 y + α = 0 and 6 x - 3 y + β = 0, respectively, then the sum of all possible values of α and β is :$