Three-Dimensional Geometry PYQs Set 13 (PDF)

Set 13 of 16 · Questions 361-390 of 471

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Three-Dimensional Geometry · Set 13 (Q361-390)
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361. $If M is the foot of the perpendicular drawn from P (1, 2, - 1) to the plane passing through the point A (3, - 2, 1) and perpendicular to the vector 4 i + 7 j - 4 k, then the length of PM is$ (1) $\frac{16}{3}$ (2) $\frac{18}{5}$ (3) $\frac{22}{9}$ (4) $\frac{28}{9}$ 362. $The square of the distance of the point (\frac{15}{7}, \frac{32}{7}, 7) from the line \frac{x + 1}{3} = \frac{y + 3}{5} = \frac{z + 5}{7} in the direction of the vector i + 4 j + 7 k is:$ (1) $54$ (2) $44$ (3) $41$ (4) $66$ 363. $If the foot of the perpendicular drawn from the origin to a plane is (1, 2, 3), then a point on that plane is$ (1) $(3, 2, 1)$ (2) $(7, 2, 1)$ (3) $(7, 3, - 1)$ (4) $(6, - 3, 4)$ 364. $Let P be the plane containing the straight line \frac{x - 3}{9} = \frac{y + 4}{- 1} = \frac{z - 7}{- 5} and perpendicular to the plane containing the straight lines \frac{x}{2} = \frac{y}{3} = \frac{z}{5} and \frac{x}{3} = \frac{y}{7} = \frac{z}{8} . If d is the distance of P from the point (2,-5, 11), then d^{2} is equal to:$ (1) $\frac{147}{2}$ (2) $96$ (3) $\frac{32}{3}$ (4) $54$ 365. $The distance of the point (-1, 2,-2) from the line of intersection of the planes 2x + 3y + 2z = 0 and x - 2y + z = 0 is :$ (1) $\frac{1}{2}$ (2) $\frac{5}{2}$ (3) $\frac{42}{2}$ (4) $\frac{34}{2}$ 366. $The vertices B and C of a triangle ABC lie on the line \frac{x}{1} = \frac{1 - y}{- 2} = \frac{z - 2}{3} . The coordinates of A and B are (1, 6, 3) and (4, 9, α) respectively and C is at a distance of 10 units from B. The area (in sq. units) of △ ABC is:$ (1) $2013$ (2) $513$ (3) $1513$ (4) $1013$ 367. $A (3, 2, - 1), B (4, 1, 1), C (6, 2, 5) are three points. If D, E, F are three points which divide BC, CA, AB respectively in the same ratio 2 : 1, then the centroid of △ DEF is$ (1) $(\frac{13}{3}, \frac{5}{3}, \frac{5}{3})$ (2) $(13, 5, 5)$ (3) $(4, 2, 1)$ (4) $(\frac{11}{3}, \frac{4}{3}, \frac{1}{3})$ 368. $Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles ABC, ACD and ADB be 5,6 and 7 square units respectively. Then the area (in square units) of the △ BCD is equal to:$ (1) $110$ (2) $12$ (3) $340$ (4) $73$ 369. $A vector a is parallel to the line of intersection of the plane determined by the vectors i^, i^+ j^and the plane determined by the vectors i^- j, i^+ k^. The obtuse angle between a and the vector b = i^- 2 j^+ 2 k^is$ (1) $\frac{3 π}{4}$ (2) $\frac{2 π}{3}$ (3) $\frac{4 π}{5}$ (4) $\frac{5 π}{6}$ 370. $Let the lines \frac{x - 1}{λ} = \frac{y - 2}{1} = \frac{z - 3}{2} and \frac{x + 26}{- 2} = \frac{y + 18}{3} = \frac{z + 28}{λ} be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lies on P?$ (1) $(0,-2,-2)$ (2) $(-5,0,-l)$ (3) $(3,-l,0)$ (4) $(0,4,5)$ 371. $Let P (α, β, γ) be the image of the point Q (3, - 3, 1) in the line \frac{x - 0}{1} = \frac{y - 3}{1} = \frac{z - 1}{- 1} and R be the point (2, 5, - 1) . If the area of the triangle PQR is λ and λ^{2} = 14 K, then K is equal to :$ (1) $36$ (2) $81$ (3) $72$ (4) $18$ 372. $The angle between the straight lines, whose direction cosines are given by the equations 2l + 2 m - n = 0 and mn + nl+ l m = 0, is :$ (1) $\frac{π}{2}$ (2) $π - cos^{- 1} (\frac{4}{9})$ (3) $cos^{- 1} (\frac{8}{9})$ (4) $\frac{π}{3}$ 373. $Let the line L intersect the lines x - 2 = - y = z - 1, 2 (x + 1) = 2 (y - 1) = z + 1 and be parallel to the line \frac{x - 2}{3} = \frac{y - 1}{1} = \frac{z - 2}{2} . Then which of the following points lies on L ?$ (1) $(- \frac{1}{3}, 1, - 1)$ (2) $(- \frac{1}{3}, - 1, 1)$ (3) $(- \frac{1}{3}, 1, 1)$ (4) $(- \frac{1}{3}, - 1, - 1)$ 374. $If A (4, 3, 5), B (0, - 2, 2) and C (3, 2, 1) are three points. Then, the coordinates of the point in which the bisector of ∠ BAC meets the side BC, is$ (1) $(\frac{15}{8}, \frac{4}{8}, \frac{11}{8})$ (2) $(\frac{12}{7}, \frac{2}{7}, \frac{10}{7})$ (3) $(\frac{9}{5}, \frac{2}{5}, \frac{7}{5})$ (4) $(\frac{3}{2}, 0, \frac{3}{2})$ 375. $The equation of the plane which contains the y -axis and passes through the point (1, 2, 3) is:$ (1) $x + 3 z = 10$ (2) $x + 3 z = 0$ (3) $3 x + z = 6$ (4) $3 x - z = 0$ 376. $If the shortest distance between the lines \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{λ} and \frac{x - 2}{1} = \frac{y - 4}{4} = \frac{z - 5}{5} is \frac{1}{3}, then the sum of all possible values of λ, is :$ (1) $16$ (2) $6$ (3) $12$ (4) $15$ 377. $Let the position vectors of the vertices A, B and C of a triangle be 2 i + 2 j + k, i + 2 j + 2 k and 2 i + j + 2 k respectively. Let l_{1}, l_{2} and l_{3} be the lengths of perpendiculars drawn from the ortho center of the triangle on the sides AB, BC and CA respectively, then l_{1}^{2} + l_{2}^{2} + l_{3}^{2} equals:$ (1) $\frac{1}{5}$ (2) $\frac{1}{2}$ (3) $\frac{1}{4}$ (4) $\frac{1}{3}$ 378. $Let P and Q be the points on the line \frac{x + 3}{8} = \frac{y - 4}{2} = \frac{z + 1}{2} which are at a distance of 6 units from the point R (1, 2, 3) . If the centroid of the triangle PQR is (α, β, γ), then α^{2} + β^{2} + γ^{2} is;$ (1) $26$ (2) $36$ (3) $18$ (4) $24$ 379. $Let L_{1} : \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} and L_{2} : \frac{x - 2}{3} = \frac{y - 4}{4} = \frac{z - 5}{5} be two lines. Then which of the following points lies on the line of the shortest distance between L_{1} and L_{2} ?$ (1) $(\frac{14}{3}, - 3, \frac{22}{3})$ (2) $(- \frac{5}{3}, - 7, 1)$ (3) $(2, 3, \frac{1}{3})$ (4) $(\frac{8}{3}, - 1, \frac{1}{3})$ 380. $If the image of the point (4, 4, 3) in the line \frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 1}{3} is (α, β, γ), then α + β + γ is equal to$ (1) $9$ (2) $12$ (3) $7$ (4) $8$ 381. $If for a > 0, the feet of perpendiculars from the points A (a, - 2 a, 3) and B (0, 4, 5) on the plane l x + my + nz = 0 are points C (0, - a, - 1) and D respectively, then the length of line segment CD is equal to$ (1) $31$ (2) $41$ (3) $55$ (4) $66$ 382. $The plane which bisects the line segment joining the points (- 3, - 3, 4) and (3, 7, 6) at right angles, passes through which one of the following points?$ (1) $(- 2, 3, 5)$ (2) $(4, - 1, 7)$ (3) $(2, 1, 3)$ (4) $(4, 1, - 2)$ 383. $Let the line L pass through (1, 1, 1) and intersect the lines \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4} and \frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z}{1} . Then, which of the following points lies on the line L ?$ (1) $(7, 15, 13)$ (2) $(4, 22, 7)$ (3) $(10, - 29, - 50)$ (4) $(5, 4, 3)$ 384. $Let the line L be the projection of the line \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z - 4}{2} in the plane x - 2y - z = 3. If is the distance of the point (0, 0, 6) from L, then d^{2} is equal to$ 385. $Let P be the point (10, - 2, - 1) and Q be the foot of the perpendicular drawn from the point R (1, 7, 6) on the line passing through the points (2, - 5, 11) and (- 6, 7, - 5) . Then the length of the line segment PQ is equal to________$ 386. $Consider a line L passing through the points P (1, 2, 1) and Q (2, 1, - 1) . If the mirror image of the point A (2, 2, 2) in the line L is (α, β, γ), then α + β + 6 γ is equal to______$ 387. $Let the line of the shortest distance between the lines L_{1} : r = (i + 2 j + 3 k) + λ (i - j + k) and L_{2} : r = (4 i + 5 j + 6 k) + μ (i + j - k) intersect L_{1} and L_{2} at P and Q respectively. If (α, β, γ) is the midpoint of the line segment PQ, then 2 (α + β + γ) is equal to$ 388. $Let a line having direction ratios 1, -4, 2 intersect the lines \frac{x - 7}{3} = \frac{y - 1}{- 1} = \frac{z + 2}{1} and \frac{x}{2} = \frac{y - 7}{3} = \frac{z}{1} at the point A and B. Then (A B)^{2} is equal to____.$ 389. $Let the plane ax + by + cz + d = 0 bisect the line joining the points (4, - 3, 1) and (2, 3, - 5) at the right angles. If a, b, c, d are integers, then the minimum value of (a^{2} + b^{2} + c^{2} + d^{2}) is$ 390. $Let Q be the foot of the perpendicular from the point P(7,-2,13) on the plane containing the lines \frac{x + 1}{6} = \frac{y - 1}{7} = \frac{z - 3}{8} and \frac{x - 1}{3} = \frac{y - 2}{5} = \frac{z - 3}{7} . Then (P Q)^{2}, is equal to .$