Three-Dimensional Geometry PYQs Set 10 (PDF)

Set 10 of 16 · Questions 271-300 of 471

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Three-Dimensional Geometry · Set 10 (Q271-300)
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271. $If p = 4 i - j + k is a point and q = 9 i - 2 j + 6 k is a vector, then the perpendicular distance of origin from the plane passing through p and perpendicular to q is$ (1) $4$ (2) $32$ (3) $9$ (4) $11$ 272. $If the angle between the planes r \cdot (11 i - 2 j + α k) = 7 and r \cdot (2 i + 4 j - 2 k) = 5 is \frac{π}{2}, then α =$ (1) $2$ (2) $3$ (3) $5$ (4) $7$ 273. $The shortest distance between the lines x + 1 = 2 y = - 12 z and x = y + 2 = 6 z - 6 is$ (1) $2$ (2) $3$ (3) $\frac{5}{2}$ (4) $\frac{3}{2}$ 274. $If (\frac{9}{4}, \frac{5}{4}, \frac{15}{4}) is the centroid of a tetrahedron whose vertices are (a, 2, 1), (1, b, 4), (4, 0, c) and (1, 1, 7), then$ (1) $a = b = c$ (2) $a = b = c + 1$ (3) $b = c = a + 1$ (4) $a = c = b + 1$ 275. $The point of intersection of the line x + 1 = \frac{y + 3}{3} = \frac{- z + 2}{2} with the plane 3 x + 4 y + 5 z = 10 is$ (1) $(2, 6, - 4)$ (2) $(- 2, 6, - 4)$ (3) $(2, 6, 4)$ (4) $(2, - 6, - 4)$ 276. $Two particles P and Q located at the points P (t, t^{3} - 16 t - 3), Q (t + 1, t^{3} - 6 t - 6) are moving in a plane, the minimum distance between the points in their motion is$ (1) $1$ (2) $5$ (3) $169$ (4) $49$ 277. $If a non-zero vector a is parallel to the line of intersection of the plane determined by the vectors j - k, 3 j - 2 k and the plane determined by the vectors 2 i + 3 j i - 3 j, then the angle between the vectors a and i + j + k is$ (1) $sin^{- 1} (\frac{2}{3})$ (2) $cos^{- 1} (\pm \frac{2}{3})$ (3) $tan^{- 1} 3$ (4) $cos^{- 1} (\pm \frac{1}{3})$ 278. $Let a = 2 i - 3 j - 5 k and b = 3 i + 2 j - 5 k be two vectors and r be a vector in the plane of a and b . If r is orthogonal to the vector 5 i - 2 j + 3 k and the magnitude of r is 94, then \| r \cdot b \| =$ (1) $36$ (2) $38$ (3) $42$ (4) $46$ 279. $If (2, 3, c) are the direction ratios of a ray passing through the point C (5, q, 1) and also the mid point of the line segment joining the points A (p, - 4, 2) and B (3, 2, - 4) then c . (p + 7 q) =$ (1) $17$ (2) $34$ (3) $21$ (4) $28$ 280. $The plane which bisects the line joining, the points (4, - 2, 3) and (2, 4, - 1) at right angles also passes through the point:$ (1) $(4, 0, 1)$ (2) $(0, - 1, 1)$ (3) $(0, 1, - 1)$ (4) $(4, 0, - 1)$ 281. $Consider the lines L_{1} : \frac{x + 1}{3} = \frac{y + 2}{1} = \frac{z + 1}{2} L_{2} : \frac{x - 2}{1} = \frac{y + 2}{2} = \frac{z - 3}{3}, then the unit vector perpendicular to both L_{1} and L_{2} is$ (1) $\frac{- i + 7 j + 5 k}{5 3}$ (2) $\frac{- i - 7 j + 5 k}{5 3}$ (3) $\frac{i - 7 j + 5 k}{5 3}$ (4) $\frac{i + 7 j + 5 k}{5 3}$ 282. $The distance of the line \frac{x - 2}{2} = \frac{y - 6}{3} = \frac{z - 3}{4} from the point (1, 4, 0) along the line \frac{x}{1} = \frac{y - 2}{2} = \frac{z + 3}{3} is:$ (1) $17$ (2) $15$ (3) $14$ (4) $13$ 283. $The lines r = (i^- j^) + 1 (2 i^+ k^) and r = (2 i^- j^) + m (i^+ j^- k^)$ (1) $do not intersect for any values of 1 and m$ (2) $intersect for all values of 1 and m$ (3) $intersect when 1 = 2 and m = \frac{1}{2}$ (4) $intersect when 1 = 1 and m = 2$ 284. $The distance of the point P (4, 6, - 2) from the line passing through the point (- 3, 2, 3) and parallel to a line with direction ratios 3, 3, - 1 is equal to$ (1) $3$ (2) $6$ (3) $23$ (4) $14$ 285. $If △ ABC is right angled at A, where A \equiv (4, 2, x), B \equiv (3, 1, 8) and C \equiv (2, - 1, 2), then the value of x is$ (1) $4$ (2) $2$ (3) $3$ (4) $1$ 286. $A plane P contains the line x + 2y + 3z + 1 = 0 = x - y - z - 6, and is perpendicular to the plane -2x + y + z + 8 = 0. Then which of the following points lies on P?$ (1) $(-1, 1, 2)$ (2) $(0, 1, 1)$ (3) $(1, 0, 1)$ (4) $(2,-1, 1)$ 287. $Let in a △ ABC, the length of the side AC be 6 , the vertex B be (1, 2, 3) and the vertices A, C lie on the line \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{- 2} . Then the area (in sq. units) of △ ABC is:$ (1) $17$ (2) $21$ (3) $56$ (4) $42$ 288. $If the two lines l_{1} : \frac{x - 2}{3} = \frac{y + 1}{- 2}, z = 2 and l_{2} : \frac{x - 1}{1} = \frac{2 y + 3}{α} = \frac{z + 5}{2} perpendicular, then an angle between the lines l_{2} and l_{3} : \frac{1 - x}{3} = \frac{2 y - 1}{- 4} = \frac{z}{4} is :$ (1) $cos^{- 1} (\frac{29}{4})$ (2) $sec^{- 1} (\frac{29}{4})$ (3) $cos^{- 1} (\frac{2}{29})$ (4) $cos^{- 1} (\frac{2}{29})$ 289. $If the plane P passes through the intersection of two mutually perpendicular planes 2x + ky - 5z = 1 and 3kx -ky + z = 5, k<3 and intercepts a unit length on positive x-axis, then the intercept made by the plane P on the y-axis is$ (1) $\frac{1}{11}$ (2) $\frac{5}{11}$ (3) $6$ (4) $7$ 290. $The line l_{1} passes through the point 2, 6, 2) and is perpendicular to the plane 2 x + y - 2 z = 10 . then the shortest distance between the line l_{1} and the line \frac{x + 1}{2} = \frac{y + 4}{- 3} = \frac{z}{2} is$ (1) $7$ (2) $\frac{19}{3}$ (3) $203$ (4) $9$ 291. $The distance of the point (- 2, 4, - 5) from the line \frac{x + 3}{3} = \frac{y - 4}{5} = \frac{z + 8}{6} is$ (1) $\frac{37}{10}$ (2) $\frac{37}{10}$ (3) $\frac{37}{10}$ (4) $\frac{37}{10}$ 292. $The ratio in which B (\frac{33}{5}, \frac{28}{5}, \frac{38}{5}) divides the line segment joining A (3, 2, 4) and C (9, 8, 10) is$ (1) $2 : 1$ (2) $1 : 3$ (3) $3 : 2$ (4) $5 : 3$ 293. $Let S be the set of all values of λ, for which the shortest distance between the lines \frac{x - λ}{0} = \frac{y - 3}{4} = \frac{z + 6}{1} and \frac{x + λ}{3} = \frac{y}{- 4} = \frac{z - 6}{0} is 13. Then 8 \sum_{λ \in s} λ is equal to$ (1) $306$ (2) $304$ (3) $308$ (4) $302$ 294. $Let P (3, 2, 3), Q (4, 6, 2) and R (7, 3, 2) be the vertices of △ PQR . Then, the angle ∠ QPR is$ (1) $\frac{π}{6}$ (2) $cos^{- 1} (\frac{7}{18})$ (3) $cos^{- 1} (\frac{1}{18})$ (4) $\frac{π}{3}$ 295. $The foot of the perpendicular drawn from the point (4, 2, 3) to the line joining the points (1, - 2, 3) and (1, 1, 0) lies on the plane:$ (1) $x - 2 y + z = 1$ (2) $x + 2 y - z = 1$ (3) $x - y - 2 y = 1$ (4) $2 x + y - z = 1$ 296. $The vertices B and C of a Δ ABC lie on the line, \frac{x + 2}{3} = \frac{y - 1}{0} = \frac{z}{4} such that BC = 5 units. Then the area (in sq. units) of this triangle, given that the point A (1, - 1, 2), is:$ (1) $517$ (2) $234$ (3) $6$ (4) $34$ 297. $If the equation of the plane passing through the line of intersection of the planes 2 x - 7 y + 4 z - 3 = 0, 3 x - 5 y + 4 z + 11 = 0 and the point (- 2, 1, 3) is ax + by + cz - 7 = 0, then the value of 2 a + b + c - 7 is:$ 298. $If the shortest distance between the lines \frac{x + 2}{2} = \frac{y + 3}{3} = \frac{z - 5}{4} and \frac{x - 3}{1} = \frac{y - 2}{- 3} = \frac{z + 4}{2} is \frac{38}{3 5} k, and \int_{0}^{k} (x^{2}) d x = α - α, where [x] denotes the greatest integer function, then 6 α^{3} is equal to ______$ 299. $Let the image of the point P(l, 2, 3) in the line L : \frac{x - 6}{3} = \frac{y - 1}{2} = \frac{z - 2}{3} be Q. let R_{(} α, β, γ_{)} be a point that divides internally the line segment PQ in the ratio 1 : 3. Then the value of 2 2_{(} α + β + γ_{)} is equal to$ 300. $Let the mirror image of the point (a, b, c) with respect to the plane 3x - 4y + 12z + 19 = 0 be (a- 6, β, γ). If a + b + c = 5, then 7 β - 9 γ is equal to __________ .$