Three-Dimensional Geometry PYQs Set 7 (PDF)

Set 7 of 16 · Questions 181-210 of 471

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Three-Dimensional Geometry · Set 7 (Q181-210)
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181. $If the distances of the point (1, 2, a) from the line \frac{x - 1}{1} = \frac{y}{2} = \frac{z - 1}{1} along the lines L_{1} : \frac{x - 1}{3} = \frac{y - 2}{4} = \frac{z - a}{b} and L_{2} : \frac{x - 1}{1} = \frac{y - 2}{4} = \frac{z - a}{c} are equal, then a + b + c is equal to$ (1) $5$ (2) $7$ (3) $4$ (4) $6$ 182. $The locus represented by xy + yz = 0 is$ (1) $a pair of perpendicular lines$ (2) $a pair of parallel lines$ (3) $a pair of parallel planes$ (4) $a pair of perpendicular planes$ 183. $A plane meets the coordinate axes at P, Q, R respectively. If the centroid of △ PQR is (1, \frac{1}{2}, \frac{1}{3}), then the equation of the plane is$ (1) $2 x + 4 y + 3 z = 5$ (2) $x + 2 y + 3 z = 3$ (3) $x + 4 y + 6 z = 5$ (4) $2 x - 2 y + 6 z = 3$ 184. $The shortest distance between lines L_{1} and L_{2}, where L_{1} : \frac{x - 1}{2} = \frac{y + 1}{- 3} = \frac{z + 4}{2} and L_{2} is the line passing through the points A (- 4, 4, 3) \cdot B (- 1, 6, 3) and perpendicular to the line \frac{x - 3}{- 2} = \frac{y}{3} = \frac{z - 1}{1}, is$ (1) $\frac{121}{221}$ (2) $\frac{24}{117}$ (3) $\frac{141}{221}$ (4) $\frac{42}{117}$ 185. $If Q (0, - 1, - 3) is the image of the point P in the plane 3 x - y + 4 z = 2 and R is the point (3, - 1, - 2), then the area (in sq. units) of Δ PQR is:$ (1) $213$ (2) $\frac{91}{4}$ (3) $\frac{91}{2}$ (4) $\frac{65}{2}$ 186. $If the direction cosines of the two lines satisfy the equation l + m + n = 0, 2 lm + 2 ln - mn = 0, then the acute angle between these lines is$ (1) $cos^{- 1} (\frac{1}{3})$ (2) $30^{\circ}$ (3) $cos^{- 1} (\frac{2}{3})$ (4) $60^{\circ}$ 187. $The equation of the plane passing through the points (1, 2, 3), (- 1, 4, 2) and (3, 1, 1) is$ (1) $5 x + 6 y + 2 z - 23 = 0$ (2) $5 x + y + 2 z - 13 = 0$ (3) $5 x + y + 12 z - 43 = 0$ (4) $5 x + y + z - 10 = 0$ 188. $If A = (1, 8, 4), B = (2, - 3, 1), then the direction cosines of a normal to the plane AOB is$ (1) $\frac{2}{78}, \frac{5}{78}, \frac{- 7}{78}$ (2) $\frac{2 10}{9}, \frac{7 10}{90}, \frac{- 19 10}{90}$ (3) $\frac{4}{218}, \frac{9}{218}, \frac{- 11}{218}$ (4) $\frac{2}{11}, \frac{6}{11}, \frac{- 9}{11}$ 189. $If the point of intersection of the lines r = i - 6 j + (p sec α) k + t (i + 2 j + k) and r = 4 j + k + λ (2 i + (p tan α) j + 2 k) is 8 i + 8 j + 9 k, (where 0 < α < \frac{π}{2}), then p =$ (1) $5$ (2) $3$ (3) $2$ (4) $0$ 190. $If the shortest distance between the lines \frac{x - λ}{2} = \frac{y - 4}{3} = \frac{z - 3}{4} and \frac{x - 2}{4} = \frac{y - 4}{6} = \frac{z - 7}{8} is \frac{13}{29}, then a value of λ is :$ (1) $-1$ (2) $- \frac{13}{25}$ (3) $\frac{13}{25}$ (4) $1$ 191. $If 2 i + 3 j - 4 k and - i + 2 j + k are the two diagonals of a parallelogram, then the area of the parallelogram in square units is$ (1) $\frac{1}{2} 170$ (2) $174$ (3) $\frac{87}{2}$ (4) $\frac{1}{4} 174$ 192. $The point which lies on the plane passing through the points i - 2 j - 3 k, 3 i - j + 4 k, - 3 i + 2 j - 5 k is$ (1) $- i + 3 j - 2 k$ (2) $7 i - 5 j - 6 k$ (3) $- i + 9 j + 14 k$ (4) $3 i - 7 j + 8 k$ 193. $The harmonic conjugate of (2, 3, 4) with respect to the points (3, - 2, 2), (6, - 17, - 4) is$ (1) $(\frac{1}{2}, \frac{1}{3}, \frac{1}{4})$ (2) $(\frac{18}{5}, - 5, \frac{4}{5})$ (3) $(\frac{- 18}{5}, \frac{5}{4}, \frac{4}{5})$ (4) $(\frac{18}{5}, - 5, \frac{- 4}{5})$ 194. $If the line \frac{2 - x}{3} = \frac{3 y - 2}{4 λ + 1} = 4 - z makes a right angle with the line \frac{x + 3}{3 μ} = \frac{1 - 2 y}{6} = \frac{5 - z}{7}, then 4 λ + 9 μ is equal to :$ (1) $4$ (2) $13$ (3) $5$ (4) $6$ 195. $If a line makes angles α, β, γ with the positive direction of X, Y and Z -axes respectively, then the value of sin^{2} α + sin^{2} β + sin^{2} γ =$ (1) $1$ (2) $2$ (3) $3$ (4) $-1$ 196. $The coordinates of the point which divides the line joining the points (2, 3, 4) and (3, - 4, 7) in the ratio 2 : 4 externally is$ (1) $(10, 1, 1)$ (2) $(1, 10, 1)$ (3) $(10, - 10, 10)$ (4) $(1, 1, 10)$ 197. $If the shortest distance between the lines \frac{x - λ}{- 2} = \frac{y - 2}{1} = \frac{z - 1}{1} and \frac{x - 3}{1} = \frac{y - 1}{- 2} = \frac{z - 2}{1} is 1 , then the sum of all possible values of λ is :$ (1) $0$ (2) $23$ (3) $33$ (4) $- 23$ 198. $Let a = i + 2 j + k, b = i - j + k and c = i + j - k, a vector in the plane of a and b whose projection on c is \frac{1}{3} is$ (1) $4 i + j - 4 k$ (2) $4 i - j + 4 k$ (3) $3 i + j - 3 k$ (4) $i + j - 2 k$ 199. $If plane x + 4 y - 2 z = 1 x + 7 y - 5 z = β x + 5 y + αz = 5 intersects in a line (R \times R \times R) then α + β is equal to$ (1) $0$ (2) $10$ (3) $- 10$ (4) $2$ 200. $If the equation of plane passing through themirror image of a point (2, 3, 1) with respect to line \frac{x + 1}{2} = \frac{y - 3}{1} = \frac{z + 2}{- 1} and containing the line \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z + 1}{1} is αx + βy + γz = 24 then α + β + γ is equal to :$ (1) $20$ (2) $19$ (3) $18$ (4) $21$ 201. $The equation of line passing through (3, - 1, 2) and perpendiculartothelines r = (i + j - k) + 1 (2 i - 2 j + k) and r = (2 i + j - 3 k) + m (i - 2 j + 2 k) is$ (1) $\frac{x + 3}{2} = \frac{y + 1}{3} = \frac{z - 2}{2}$ (2) $\frac{x - 3}{3} = \frac{y + 1}{2} = \frac{z - 2}{2}$ (3) $\frac{x - 3}{2} = \frac{y + 1}{3} = \frac{z - 2}{2}$ (4) $\frac{x - 3}{2} = \frac{y + 1}{2} = \frac{z - 2}{3}$ 202. $If a = i + 2 j + 3 k, b = 2 i + 3 j + k, c = 8 i + 13 j + 9 k and x a + y b + z c = 0, then \frac{xy}{z ^{2}} =$ (1) $-1$ (2) $-6$ (3) $6$ (4) $1$ 203. $If a plane x + y + z - 5 = 0 intersects the line joining A (1, 1, 1) and B (2, 2, 2) at P then AP : PB =$ (1) $1 : 2$ (2) $2 : 3$ (3) $3 : 2$ (4) $2 : 1$ 204. $The angle between the pair of lines \frac{x + 3}{3} = \frac{y - 1}{5} = \frac{z + 3}{4} and \frac{x + 1}{1} = \frac{y - 4}{4} = \frac{z - 5}{2} is$ (1) $0 = cos^{- 1} (\frac{27}{5})$ (2) $θ = cos^{- 1} (\frac{19}{21})$ (3) $θ = cos^{- 1} (\frac{8 3}{15})$ (4) $θ = cos^{- 1} (\frac{31}{5 42})$ 205. $If lines \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4} and \frac{x - 3}{1} = \frac{y - λ}{2} = \frac{z}{1} intersect each other, then λ =$ (1) $7/2$ (2) $3/2$ (3) $9/2$ (4) $5/2$ 206. $The cartesian equation of the plane passing through the point (1, - 2, 3) and perpendicular to the vector - i + 2 j - 3 k, is$ (1) $- x + 2 y - 3 z = 14$ (2) $x - 2 y + 3 z = 14$ (3) $x + 2 y - 3 z = 4$ (4) $- x + 2 y + 3 z = 14$ 207. $The distance between the lines \frac{x}{2} = \frac{y}{- 1} = \frac{z}{2} and \frac{x - 1}{2} = \frac{y - 1}{- 1} = \frac{z - 1}{2} is$ (1) $1 unit$ (2) $2 units$ (3) $2 units$ (4) $3 units$ 208. $If the points A (1, 3, 5), B (2, 4, 6), C (4, 5, k) form a right angled triangle then the number of possible values of k is$ (1) $2$ (2) $3$ (3) $0$ (4) $1$ 209. $Let A be a point on the line r = (1 - 3 μ) i + (μ - 1) j + (2 + 5 μ) k and B (3, 2, 6) be a point in the space. Then the value of μ for which the vector AB is parallel to the plane x - 4 y + 3 z = 1 is:$ (1) $\frac{1}{4}$ (2) $\frac{1}{8}$ (3) $\frac{1}{2}$ (4) $- \frac{1}{4}$ 210. $Let a plane P contain two lines r \to= i^+ λ (i^+ j^), λ \in R and r \to= - j^+ μ (j^- k^), μ \in R If Q (α, β, γ) is the foot of the perpendicular drawn from the point M (1, 0, 1) to P, then 3 (α + β + γ) equals$