Three-Dimensional Geometry PYQs Set 8 (PDF)

Set 8 of 16 · Questions 211-240 of 471

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Three-Dimensional Geometry · Set 8 (Q211-240)
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211. $Angle between the lines of intersection of the planes x - y = 0, 2 x + y + z = 0 and 2 x - z = 0, x + y - 3 z = 0 is$ (1) $60^{\circ}$ (2) $45^{\circ}$ (3) $30^{\circ}$ (4) $90^{\circ}$ 212. $The vector equation of the plane which is at a distance \frac{3}{14} from the origin and the normal from the origin is 2 i - 3 j + k is$ (1) $r \cdot (2 i - 3 j + k) = 3$ (2) $r \cdot (i + j + k) = 9$ (3) $r \cdot (i + 2 j) = 3$ (4) $r \cdot (2 i + k) = 3$ 213. $If 2 i - j + 3 k, - 12 i - j - 3 k, - i + 2 j - 4 k and λi + 2 j - k are the position vectors of four coplanar points, then λ =$ (1) $9$ (2) $-2$ (3) $8$ (4) $6$ 214. $If the two lines \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4} and \frac{x - 3}{1} = \frac{y - k}{2} = \frac{z}{1} have a point in common, then k =$ (1) $\frac{2}{9}$ (2) $- \frac{2}{9}$ (3) $\frac{9}{2}$ (4) $0$ 215. $If P (2, 3, 4), Q (- 1, - 2, 1) and R (5, 8, 7) are three points, then$ (1) $direction ratios of PQ and QR are not in proportion$ (2) $P, Q, R lie on the same line$ (3) $P, Q, R do not lie on the same line$ (4) $None of these$ 216. $Let the direction cosines of two lines satisfy the equations: 4 l + m - n = 0 and 2 mn + 10 nl + 3 lm = 0 . Then the cosine of the acute angle between these lines is :$ (1) $\frac{20}{3 38}$ (2) $\frac{10}{3 38}$ (3) $\frac{10}{7 38}$ (4) $\frac{10}{38}$ 217. $The direction cosines of the line making angles \frac{π}{4}, \frac{π}{3} and θ (0 < θ < \frac{π}{2}) respectively with X, Y and Z -axes, are$ (1) $\frac{1}{2}, \frac{1}{2}, \frac{1}{2}$ (2) $\frac{1}{2}, \frac{1}{2}, \frac{3}{2}$ (3) $\frac{1}{2}, \frac{1}{2}, \frac{1}{2}$ (4) $\frac{1}{2}, \frac{3}{2}, \frac{1}{2}$ 218. $Let P (α, β, γ) be the point on the line \frac{x - 1}{2} = \frac{y + 1}{- 3} = z at a distance 414 from the point (1, - 1, 0) and nearer to the origin. Then the shortest distance, between the lines \frac{x - α}{1} = \frac{y - β}{2} = \frac{z - γ}{3} and \frac{x + 5}{2} = \frac{y - 10}{1} = \frac{z - 3}{1}, is equal to$ (1) $4 \frac{7}{5}$ (2) $2 \frac{7}{4}$ (3) $7 \frac{5}{4}$ (4) $4 \frac{5}{7}$ 219. $Let a straight line L pass through the point P (2, - 1, 3) and be perpendicular to the lines \frac{x - 1}{2} \frac{y + 1}{1} = \frac{z - 3}{- 2} and \frac{x - 3}{1} = \frac{y - 2}{3} = \frac{z + 2}{4} . If the line L intersects the yz -plane at the point Q, then the distance between the points P and Q is:$ (1) $10$ (2) $23$ (3) $2$ (4) $3$ 220. $The cartesian equation of a line is \frac{x + 2}{3} = \frac{y - 4}{2} = \frac{z - 5}{5}, then the vector equation of the line is$ (1) $r = (3 i + 2 j + 5 k) + λ (10 i + 25 j - 16 k)$ (2) $r = (- 2 i + 4 j + 5 k) + λ (10 i + 25 j - 16 k)$ (3) $r = (- 2 i + 4 j + 5 k) + λ (3 i + 2 j + 5 k)$ (4) $r = (2 i - 4 j - 5 k) + λ (- 3 i + 2 j - 5 k)$ 221. $Let S be the set of all real values of λ such that a plane passing through the points (- λ^{2}, 1, 1), (1, - λ^{2}, 1) and (1, 1, - λ^{2}) also passes through the point- (- 1, - 1, 1) . Then S is equal to :$ (1) ${3}$ (2) ${3, - 3}$ (3) ${1, - 1}$ (4) ${3, - 3}$ 222. $If the vertices of a triangle ABC are A (1, 2, 3) B (h, - 3, 0) and C (- 4, k, - 1) and the centroid of the triangle is (5, - 1, \frac{2}{3}) then triangle ABC is$ (1) $an obtuse angled triangle$ (2) $an acute angled triangle$ (3) $an isosceles triangle$ (4) $a right-angled triangle$ 223. $If the angle between the lines whose direction cosines are (- \frac{2}{21}, \frac{C}{21}, \frac{1}{21}) and (\frac{3}{54}, \frac{3}{54}, - \frac{6}{54}) is \frac{π}{2}, then the value of C is$ (1) $6$ (2) $4$ (3) $-4$ (4) $2$ 224. $Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6) . Then the image of R in the plane P is:$ (1) $(6, 5, - 2)$ (2) $(6,-5,2)$ (3) $(2, - 3, 4)$ (4) $(2, - 5, 6)$ 225. $Let the equation of the plane, that passes through the point (1, 4,-3) and contains the line of intersection of the planes 3x - 2y + 4z - 7 = 0 and x + 5y - 2z + 9 = 0, be αx + βy + γz+ 3 = 0, then α + β + γ is equal to :$ (1) $-23$ (2) $-15$ (3) $23$ (4) $15$ 226. $Three non-coplanar vectors a, b, c are the coterminous edges of a parallelepiped. If a and b determine the base of the parallelepiped then its height is$ (1) $\frac{\| [ ab c ] \|}{\| b \times c \|}$ (2) $\frac{\| [ ab c ] \|}{\| a \times b \|}$ (3) $\frac{\| [ ab c ] \|}{\| a \times c \|}$ (4) $\frac{\| [ ab c ] \|}{\| b + c \|}$ 227. $The equation of the plane containing the straight line \frac{x}{2} = \frac{y}{3} = \frac{z}{4} and perpendicular to the plane containing the straight lines \frac{x}{3} = \frac{y}{4} = \frac{z}{2} and \frac{x}{4} = \frac{y}{2} = \frac{z}{3} is:$ (1) $x - 2 y + z = 0$ (2) $3 x + 2 y - 3 z = 0$ (3) $x + 2 y - 2 z = 0$ (4) $5 x + 2 y - 4 z = 0$ 228. $The plane through the intersection of the planes x + y + z = 1 and 2 x + 3 y - z + 4 = 0 and parallel to y -axis also passes through the point:$ (1) $(- 3, 0, - 1)$ (2) $(- 3, 1, 1)$ (3) $(3, 3, - 1)$ (4) $(3, 2, 1)$ 229. $The combined equation for a pair of planes is S \equiv 2 x^{2} - 6 y^{2} - 12 z^{2} + 18 yz + 2 zx + xy = 0 . If one of the planes is parallel to x + 2 y - 2 z = 5, then the acute angle between the planes S = 0 is$ (1) $cos^{- 1} (\frac{16}{21})$ (2) $\frac{π}{2}$ (3) $\frac{2 π}{3}$ (4) $sin^{- 1} (\frac{7}{15})$ 230. $Let the line L pass through the point (- 3, 5, 2) and make equal angles with the positive coordinate axes. If the distance of L from the point (- 2, r, 1) is \frac{14}{3}, then the sum of all possible values of r is :$ (1) $10$ (2) $6$ (3) $12$ (4) $16$ 231. $The mirror image of the point (1, 2, 3) in a plane is (- \frac{7}{3}, - \frac{4}{3}, - \frac{1}{3}) Which of the following points lies on this plan?$ (1) $(-1,1,-1)$ (2) $(- 1, - 1, - 1)$ (3) $(- 1, - 1, 1)$ (4) $(1, - 1, 1)$ 232. $A (2, 3, - 4), B (- 3, 3, - 2), C (- 1, 4, 2) and D (3, 5, 1) are the vertices of a tetrahedron. If E, F, G are the centroids of its faces containing the point A, then the centroid of the triangle EFG is$ (1) $(\frac{1}{9}, \frac{15}{9}, \frac{- 3}{9})$ (2) $(\frac{1}{4}, \frac{15}{4}, \frac{- 3}{4})$ (3) $(\frac{4}{9}, \frac{11}{3}, \frac{- 10}{9})$ (4) $(\frac{- 1}{9}, \frac{12}{9}, \frac{1}{9})$ 233. $The lines \frac{x - 1}{2} = \frac{y + 1}{2} = \frac{z - 1}{4} and \frac{x - 3}{1} = \frac{y - k}{2} = \frac{z}{1} intersect each other at point$ (1) $(- 2, - 4, 5)$ (2) $(- 2, - 4, - 5)$ (3) $(2, 4, - 5)$ (4) $(2, - 4, - 5)$ 234. $If a line makes an angle of \frac{π}{3} with each of x and y -axis, then the acute angle made by z -axis is$ (1) $\frac{π}{6}$ (2) $\frac{π}{3}$ (3) $\frac{π}{2}$ (4) $\frac{π}{4}$ 235. $If the image of the point P (1, 2, a) in the line \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{7 - z}{2} is Q (5, b, c), then a^{2} + b^{2} + c^{2} is equal to$ (1) $298$ (2) $293$ (3) $264$ (4) $283$ 236. $The acute angle between the line r = (i + 2 j + k) + λ (i + j + k) and the plane r \cdot (2 i - j + k) = 5$ (1) $cos^{- 1} (\frac{2}{3})$ (2) $sin^{- 1} (\frac{2}{3})$ (3) $tan^{- 1} (\frac{2}{3})$ (4) $sin^{- 1} (\frac{2}{3})$ 237. $The equation of the planes parallel to the plane x - 2 y + 2 z - 3 = 0 which are at unit distance from the point (1, 2, 3) is + by + cz + d = 0 . If (b - d) = K (c - a), then the positive value of K is$ 238. $Let Q and R be the feet of perpendiculars from the point P (a, a, a) on the lines x = y, z = 1 and x = - y, z = - 1 respectively. If ∠ QPR is a right angle, then 12 a^{2} is equal to____$ 239. $Suppose the line \frac{x - 2}{α} = \frac{y - 2}{- 5} = \frac{z + 2}{2} lies on the plane x + 3y - 2z + β = 0. Then (α + β)is equal to.$ 240. $Let a line L pass through the point P (2, 3, 1) and be parallel to the line x + 3 y - 2 z - 2 = 0 = x - y + 2 z . If the distance of L from the point (5, 3, 8) is α, then 3 α^{2} is equal to ______$