Three-Dimensional Geometry PYQs Set 2 (PDF)

Set 2 of 16 · Questions 31-60 of 471

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Three-Dimensional Geometry · Set 2 (Q31-60)
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31. $If the equation of the plane which is at a distance of \frac{1}{3} units from the origin and perpendicular to a line whose directional ratios are (1, 2, 2) is x + py + qz + r = 0 then p^{2} + q^{2} + r^{2} =$ (1) $3$ (2) $5$ (3) $13$ (4) $2$ 32. $The plane passing through the point (4, - 1, 2) and parallel to the lines \frac{x + 2}{3} = \frac{y - 2}{- 1} = \frac{z + 1}{2} and \frac{x - 2}{1} = \frac{y - 3}{2} = \frac{z - 4}{3} also passes through the point:$ (1) $(1, 1, - 1)$ (2) $(1, 1, 1)$ (3) $(- 1, - 1, - 1)$ (4) $(- 1, - 1, 1)$ 33. $The equation of the line passing through (- 4, 3, 1), parallel to the plane x + 2 y - z - 5 = 0 and intersecting the line \frac{x + 1}{- 3} = \frac{y - 3}{2} = \frac{z - 2}{- 1} is :$ (1) $\frac{x - 4}{2} = \frac{y + 3}{1} = \frac{z + 1}{4}$ (2) $\frac{x + 4}{1} = \frac{y - 3}{1} = \frac{z - 1}{3}$ (3) $\frac{x + 4}{3} = \frac{y - 3}{- 1} = \frac{z - 1}{1}$ (4) $\frac{x + 4}{- 1} = \frac{y - 3}{1} = \frac{z - 1}{1}$ 34. $l, m, n are three-unit vectors in a right-handed system and L is a line through the points A, B, C whose position vectors are pl + 7 m - 6 n, 2 l + 5 m - 4 n and l + 4 m - 3 n respectively. If the equation of the plane containing L and the points (- p, p, p + 1) is ax + by + cz = 1, then p (a + b + c) =$ (1) $0$ (2) $\frac{- 40}{19}$ (3) $\frac{40}{19}$ (4) $-6$ 35. $The angle between the lines whose direction cosines are (\frac{3}{4}, \frac{1}{4}, \frac{3}{2}) and (\frac{3}{4}, \frac{1}{4}, \frac{- 3}{2}) is$ (1) $π$ (2) $\frac{π}{2}$ (3) $\frac{π}{3}$ (4) $\frac{π}{4}$ 36. $Let L_{1} : r = (i - j + 2 k) + λ (i - j + 2 k), λ \in R L_{2} : r = (j - k) + μ (3 i + j + p k), μ \in R and L_{3} : r = δ (l i + mj + nk) δ \in R Be three lines such that L_{1} is perpendicular to L_{2} and L_{3} is perpendicular to both L_{1} and L_{2} . Then the point which lies on L_{3} is$ (1) $(- 1, 7, 4)$ (2) $(- 1, - 7, 4)$ (3) $(1, 7, - 4)$ (4) $(1, - 7, 4)$ 37. $The line L_{1} is parallel to the vector a = - 3 i + 2 j + 4 k and passes through the point (7, 6, 2) and the line L_{2} is parallel to the vector b = 2 i + j + 3 k and passes through the point (5, 3, 4) . The shortest distance between the lines L_{1} and L_{2} is:$ (1) $\frac{23}{38}$ (2) $\frac{21}{38}$ (3) $\frac{23}{57}$ (4) $\frac{21}{57}$ 38. $The equation of the plane passing through the point (- 1, 2, 1) and perpendicular to the line joining the points (- 3, 1, 2) and (2, 3, 4) is$ (1) $r \cdot (5 i + 2 j + 2 k) = 1$ (2) $r \cdot (5 i + 2 j + 2 k) = - 1$ (3) $r \cdot (5 i - 2 j + 2 k) = - 5$ (4) $r \cdot (5 i - 2 j - 2 k) = 1$ 39. $The distance of the point (- 1, 9, - 16) from the plane 2 x + 3 y - z = 5 measured parallel to the line \frac{x + 4}{3} = \frac{2 - y}{4} = \frac{z - 3}{12} is$ (1) $132$ (2) $31$ (3) $26$ (4) $202$ 40. $Let the plane containing the line of intersection of the planes P 1 : x + (λ + 4) y + z = 1 and P_{2} : 2 x + y + z = 2 passes through the points (0, 1, 0) and (1, 0, 1) . Then the distance of the point (2 λ, λ, - λ) from the plane P_{2} is$ (1) $56$ (2) $46$ (3) $26$ (4) $36$ 41. $The line passing through (1, 1, - 1) and parallel to the vector i + 2 j - k meets the line \frac{x - 3}{- 1} = \frac{y + 2}{5} = \frac{z - 2}{- 4} at A and the plane 2 x - y + 2 z + 7 = 0 at B . Then AB is equal to$ (1) $6$ (2) $26$ (3) $36$ (4) $46$ 42. $The equation of the plane through the intersection of the planes x + y + z = 1 and 2 x + 3 y - z + 4 = 0 and parallel to X -axis is$ (1) $3 y - 2 z + 6 = 0$ (2) $3 y - z + 6 = 0$ (3) $y - 3 z + 6 = 0$ (4) $y + 3 z + 6 = 0$ 43. $The perpendicular distance from the origin to the planecontaining the two lines, \frac{x + 2}{3} = \frac{y - 2}{5} = \frac{z + 5}{7} and \frac{x - 1}{1} = \frac{y - 4}{4} = \frac{z + 4}{7} is :$ (1) $116$ (2) $\frac{11}{6}$ (3) $11$ (4) $611$ 44. $The equation of the plane through (- 1, 1, 2), whose normal makes equal acute angles with co-ordinate axes is$ (1) $r \cdot (i + j + k) = 2$ (2) $r \cdot (i + j + k) = 6$ (3) $r \cdot (3 i - 3 j + 3 k) = 2$ (4) $r \cdot (i - j + k) = 3$ 45. $Let the shortest distance between the lines \frac{x - 3}{3} = \frac{y - α}{- 1} = \frac{z - 3}{1} and \frac{x + 3}{- 3} = \frac{y + 7}{2} = \frac{z - β}{4} be 330 . Then the positive value of 5 α + β is$ (1) $42$ (2) $40$ (3) $48$ (4) $46$ 46. $The plane 3 x + 4 y + 6 z + 7 = 0 is rotated about the line r = (i + 2 j - 3 k) + t (2 i - 3 j + k) until the plane passes through origin. The equation of the plane in the new position is$ (1) $x + y + z = 0$ (2) $6 x + 3 y - 4 z = 0$ (3) $4 x - 5 y - 2 z = 0$ (4) $x + 2 y + 4 z = 0$ 47. $Let d be the distance of the point of intersection of the lines \frac{x + 6}{3} = \frac{y}{2} = \frac{z + 1}{1} and \frac{x - 7}{4} = \frac{y - 9}{3} = \frac{z - 4}{2} from the point (7, 8, 9) . Then d^{2} + 6 is equal to :$ (1) $69$ (2) $78$ (3) $72$ (4) $75$ 48. $The plane 2 x - y + z = 4 intersects the line segment joining the points A (a, - 2, 4) and B (2, b, - 3) at the point C in the ratio 2 : 1 and the distance of the point C from the origin is 5 . If ab < 0 and P is the point (a - b, b, 2 b - a) then C P^{2} is equal to$ (1) $\frac{17}{3}$ (2) $\frac{16}{3}$ (3) $\frac{73}{3}$ (4) $97/3$ 49. $Let A = (3, 4, 0), B = (4, 4, 4), C = (- 6, 2, 3), and D = (1, 1, 2) . If θ is the acute angle between the lines AB and CD then cos θ =$ (1) $\frac{4}{17 3}$ (2) $\frac{3}{17 3}$ (3) $\frac{12}{17 3}$ (4) $\frac{11}{17 3}$ 50. $If the plane 2 x - y + 2 z + 3 = 0 has the distances \frac{1}{2} and \frac{2}{3} units from the planes 4 x - 2 y + 4 z + λ = 0 and 2 x - y + 2 z + μ = 0 respectively, then the maximum value of λ + μ is equal to:$ (1) $9$ (2) $15$ (3) $5$ (4) $13$ 51. $The sine of the angle between the straight line \frac{x - 2}{3} = \frac{3 - y}{- 4} = \frac{z - 4}{5} and the plane 2 x - 2 y + z = 5 is$ (1) $\frac{3}{50}$ (2) $\frac{4}{5 2}$ (3) $\frac{2}{10}$ (4) $\frac{3}{50}$ 52. $The direction cosines l, m, n of two lines are satisfying 3 l + m + 5 n = 0 and 6 mn - 2 nl + 5 lm = 0 . If θ is the angle between those lines then \| cos θ \| =$ (1) $\frac{1}{6}$ (2) $\frac{1}{2}$ (3) $\frac{1}{6}$ (4) $\frac{1}{3}$ 53. $If the planes x + 3 y + kz = 0 and 3 x + y - 2 z = 0 are at right angles, then the value of k is$ (1) $-2$ (2) $2$ (3) $-3$ (4) $3$ 54. $The distance of a point a from the plane r \cdot m = q is given by \frac{\| a \cdot m ⇀ - q \|}{\| m \|} . If the distance of the point i + 2 j + 3 k from the plane r \cdot (2 i + 6 j - 9 k) = - 1 is p and the distance of the origin from this plane is q, then p - q =$ (1) $6$ (2) $5$ (3) $2$ (4) $1$ 55. $The distance of the point (1, 2, - 4) from the line \frac{x - 3}{2} = \frac{y - 3}{3} = \frac{z + 5}{6} is$ (1) $\frac{293}{7}$ (2) $\frac{293}{49}$ (3) $\frac{293}{49}$ (4) $\frac{293}{7}$ 56. $The angle between the lines 2 x = 3 y = - z and 6 x = - y = - 4 z is$ (1) $0^{\circ}$ (2) $45^{\circ}$ (3) $90^{\circ}$ (4) $30^{\circ}$ 57. $The image of the point (5, 2, 6) with respect to the plane x + y + z = 9 is$ (1) $(3, - 5, 2)$ (2) $(\frac{7}{2}, - 1, 5)$ (3) $(\frac{7}{3}, - \frac{2}{3}, \frac{10}{3})$ (4) $(\frac{7}{3}, \frac{2}{3}, - \frac{5}{3})$ 58. $If the shortest between the lines \frac{x + 6}{2} = \frac{y - 6}{3} = \frac{z - 6}{4} and \frac{x - λ}{3} = \frac{y - 2 6}{4} = \frac{z + 2 6}{5} is 6, then the square of sum of all possible values of λ is$ 59. $If the distance between the plane, 23 x - 10 y - 2 z + 48 = 0 and the plane containing the lines \frac{x + 1}{2} = \frac{y - 3}{4} = \frac{z + 1}{3} and \frac{x + 3}{2} = \frac{y + 2}{6} = \frac{z - 1}{λ} (λ \in R) is equal to$ 60. $If d_{1} is the shortest distance between the lines x + 1 = 2 y = - 12 z, x = y + 2 = 6 z - 6 and d_{2} is the shortest distance between the lines \frac{x - 1}{2} = \frac{y + 8}{- 7} = \frac{z - 4}{5}, \frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 6}{- 3}, then the value of \frac{32 3 d _{1}}{d _{2}} is :$