Three-Dimensional Geometry PYQs Set 12 (PDF)

Set 12 of 16 · Questions 331-360 of 471

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Three-Dimensional Geometry · Set 12 (Q331-360)
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331. $Let the acute angle bisector of the two planes x - 2y - 2z + 1 = 0 and 2x - 3y - 6z + 1 = 0 be the plane P. Then which of the following points lies on P?$ (1) $(3, 1, - \frac{1}{2})$ (2) $(- 2, 0, - \frac{1}{2})$ (3) $(0, 2,-4)$ (4) $(4, 0,-2)$ 332. $The length of the perpendicular from the point (1, -2, 5) on the line passing through (1, 2, 4) and parallel to the line x + y-z = 0 = x -2y + 3z-5 is :$ (1) $\frac{21}{2}$ (2) $\frac{9}{2}$ (3) $\frac{73}{2}$ (4) $1$ 333. $lines x = ay - 1 = z - 2 and x = 3y - 2 = bz - 2, (ab \neq 0) are coplanar, if :$ (1) $b = 1, a \in R - {0}$ (2) $a = 1, b \in R - {0}$ (3) $a = 2, b = 2$ (4) $a = 2, b = 3$ 334. $Let P be the foot of the perpendicular from the point (1, 2, 2) on the line L : \frac{x - 1}{1} = \frac{y + 1}{- 1} = \frac{z - 2}{2} . Let the line r = (- i + j - 2 k) + λ (i - j + k), λ \in R, intersect the line L at Q. Then 2 (PQ)^{2} is equal to:$ (1) $25$ (2) $19$ (3) $29$ (4) $27$ 335. $If the line of intersection of the planes ax+by=3 and ax+by+cz=0, a > 0 makes an angle 30° with the plane y-z+2=0, then the direction cosines of the line are :$ (1) $\frac{1}{2}, \frac{1}{2}, 0$ (2) $\frac{1}{2}, \frac{- 1}{2}, 0$ (3) $\frac{1}{5}, - \frac{2}{5}, 0$ (4) $\frac{1}{2}, - \frac{3}{2}, 0$ 336. $Let Q be the mirror image of the point P(l, 0, 1) with respect to the plane S :x + y + z = 5. If a line L passing through (1, -1, -1), parallel to the line PQ meets the plane S at R, then Q R^{2} is equal to:$ (1) $2$ (2) $5$ (3) $7$ (4) $11$ 337. $Let the foot of the perpendicular drawn from the point (1, 2, 3) to a plane be (- 1, 3, - 2) . Then the perpendicular distance from the origin to the plane is$ (1) $\frac{5}{30}$ (2) $\frac{15}{2}$ (3) $\frac{2}{15}$ (4) $\frac{1}{3}$ 338. $Let Q be the mirror image of the point P(l, 2, 1) with respect to the plane x + 2y + 2z = 16. Let T be a plane passing through the point Q and contains the line r = - k^+ λ (i^+ j^+ 2 k^), λ \in R .Then, which of the following points lies on T?$ (1) $(2, 1,0)$ (2) $(1,2,1)$ (3) $(l,2,2)$ (4) $(l,3,2)$ 339. $Let \frac{x - 2}{3} = \frac{y + 1}{- 2} = \frac{z + 3}{- 1} lie on the plane px - qy +z=5, for some p, q \in R. The shortest distance of the plane from the origin is:$ (1) $\frac{3}{109}$ (2) $\frac{5}{142}$ (3) $\frac{5}{71}$ (4) $\frac{1}{142}$ 340. $The perpendicular distance, of the line \frac{x - 1}{2} = \frac{y + 2}{- 1} = \frac{z + 3}{2} from the point P (2, - 10, 1), is:$ (1) $6$ (2) $52$ (3) $43$ (4) $35$ 341. $Let P be a plane l x + my + nz = 0 containing the line, \frac{1 - x}{1} = \frac{y + 4}{2} = \frac{z + 2}{3} . If plane P divides the linesegment AB joining points A (- 3, - 6, 1) and B (2, 4, - 3) in ratio k : 1 then the value of k is equal to :$ (1) $1.5$ (2) $3$ (3) $2$ (4) $4$ 342. $If x -coordinate of a point P on the line joining the points Q (2, 2, 1) and R (5, 2, - 2) is 4 , then the y -coordinate of P =$ (1) $- \frac{1}{2} (x -coordinate of P)$ (2) $-2 (z-coordinate of P )$ (3) $2(z-coordinate of P )$ (4) $Sum of x and z coordinates of P$ 343. $If the extremities of a diagonal of a square are (1, 2, 3) and (2, - 3, 5), then its side is of length$ (1) $6$ (2) $15$ (3) $15$ (4) $3$ 344. $The foot of perpendicular of the point (2, 0, 5) the line \frac{x + 1}{2} = \frac{y - 1}{5} = \frac{z + 1}{- 1} is (α, β, γ) . Then, which of following is NOT correct?$ (1) $\frac{αβ}{γ} = \frac{4}{15}$ (2) $\frac{α}{β} = - 8$ (3) $\frac{β}{γ} = - 5$ (4) $\frac{γ}{α} = \frac{5}{8}$ 345. $Let (α, β, γ) be the foot of perpendicular from the point (1, 2, 3) on the line \frac{x + 3}{5} = \frac{y - 1}{2} = \frac{z + 4}{3} . then 19 (α + β + γ) is equal to :$ (1) $102$ (2) $101$ (3) $99$ (4) $100$ 346. $If the equation of the plane passing through the line of intersection of the planes 2x-y + z = 3, 4x-3y+ 5z+9 = 0 and parallel to the \frac{x + 1}{- 2} = \frac{y + 3}{4} = \frac{z - 2}{5} ax + by + cz + 6 = 0, then a + b + c is equal to$ (1) $12$ (2) $14$ (3) $16$ (4) $13$ 347. $Let P = (- 1, 0), Q = (0, 0) and R = (3, 33) be three points. Then the equation of the bisector of the ∠ PQR is$ (1) $3 x + y = 0$ (2) $x + \frac{3}{2} y = 0$ (3) $\frac{- 3}{2} x + y = 0$ (4) $x + 3 y = 0$ 348. $Let the image of the point P (l, 2, 6) in the plane passing through the points A (l, 2, 0) and B (l, 4, 1) C (0, 5, 1) be Q (α, β, γ) . Then (α^{2} + β^{2} + γ^{2}) equal to$ (1) $65$ (2) $62$ (3) $76$ (4) $70$ 349. $Let the plane ax + by + cz = d pass through (2, 3, -5) and is perpendicular to the planes 2x + y - 5z = 10 and 3x + 5y - 7z = 12. If a, b, c, d are integers d > 0 and gcd (\|a\|, \|b\|, \|c\|, d) = 1, then the value of a + 7b + c + 20d is equal to$ (1) $18$ (2) $20$ (3) $24$ (4) $22$ 350. $If the foot of the perpendicular drawn from the point (1, 0, - 2) to the plane π is (2, 0, - 1) and the equation of the plane π is ax + by + cz = 2 then a^{2} + b^{2} + c^{2} =$ (1) $2$ (2) $8$ (3) $4$ (4) $9$ 351. $Let N be the foot of perpendicular from the point P (l,-2, 3) on the line passing through the points (4, 5, 8) and (1, -7, 5). Then the distance of N from the plane 2x -2y + z + 5 = 0 is$ (1) $8$ (2) $6$ (3) $9$ (4) $7$ 352. $Let P (α, β, γ) be the image of the point Q (1, 6, 4) in the line \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} . Then 2 α + β + γ is equal to______$ 353. $Let S be the mirror image of the point Q(1, 3, 4) with respect to the plane 2x - y + z + 3 = 0 and let R(3, 5, γ) be a point of this plane. Then the square of the length of the line segment SR is .$ 354. $Let a plane P pass through the point (3, 7,-7 and contain the line, \frac{x - 2}{- 3} = \frac{y - 3}{2} = \frac{z + 2}{1} . If distance of the plane P from the origin is d, then d^{2} is equal to$ 355. $Let the plane P contain the line 2x + y - z - 3 = 0 5x - 3y + 4z + 9 and be parallel to the line \frac{x + 2}{2} = \frac{3 - y}{- 4} = \frac{z - 7}{5} . Then the distance of the point A (8, -1, -19) from the plane P measured parallel to the line \frac{x}{- 3} = \frac{y - 5}{4} = \frac{2 - z}{- 12} is equal to ________$ 356. $A line with direction ratios 2, 1, 2 meets the lines x = y + 2 = z and x + 2 = 2 y = 2 z respectively at the point P and Q . if the length of the perpendicular from the point (1, 2, 12) to the line PQ is l, then l^{2} is$ 357. $Let the area of the triangle formed by the lines x + 2 = y - 1 = z, \frac{x - 3}{5} = \frac{y}{- 1} = \frac{z - 1}{1} and \frac{x}{- 3} = \frac{y - 3}{3} = \frac{z - 2}{1} be A . Then A^{2} is equal to______.$ 358. $Let a line passing through the point (- 1, 2, 3) intersect the lines L_{1} : \frac{x - 1}{3} = \frac{y - 2}{2} = \frac{z + 1}{- 2} at M (α, β, γ) and L_{2} : \frac{x + 2}{- 3} = \frac{y - 2}{- 2} = \frac{z - 1}{4} at N (a, b, c) . Then the value of \frac{( α + β + γ ) ^{2}}{( a + b + c ) ^{2}} equals____$ 359. $Let the image of the point (\frac{5}{3}, \frac{5}{3}, \frac{8}{3}) in the plane x - 2y + z- 2 = O be P. If the distance of the point Q (6,-2, α), α > 0, from P is 13, then α is equal to$ 360. $Let P_{1} be the plane 3x - y - 7z = 11 and P_{2} be the plane passing through the points (2,-1, 0), (2, 0,-1), and (5, 1, 1). If the foot of the perpendicular drawn from the point (7, 4,-1) on the line of intersection of the planes P_{1} and P_{2} is (α, β, γ), then α + β + γ is equal to$