Three-Dimensional Geometry PYQs Set 14 (PDF)

Set 14 of 16 · Questions 391-420 of 471

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Three-Dimensional Geometry · Set 14 (Q391-420)
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391. $If the image of the point P (1, 0, 3) in the line joining the points A (4, 7, 1) and B (3, 5, 3) is Q (α, β, γ), then α + β + γ is equal to:$ (1) $\frac{46}{3}$ (2) $18$ (3) $13$ (4) $\frac{47}{3}$ 392. $If P = (3, 12, 4) and Q is a point on the line OP such that OQ = 3 then the sum of all the coordinates of Q is$ (1) $\pm \frac{10}{13}$ (2) $\pm \frac{28}{13}$ (3) $\pm \frac{19}{13}$ (4) $\pm \frac{57}{13}$ 393. $A plane P is parallel to two lines whose direction ratios are -2, 1, -3, and -1, 2, -2 and it contains the point (2, 2, -2). Let P intersect the co-ordinate axes at the points A, B, C making the intercepts α, β, γ . If V is the volume of the tetrahedron OABC, where O is the origin and p = α + β + γ, then the ordered pair (V, p) is equal to$ (1) $(48,-13)$ (2) $(24,-13)$ (3) $(48, 11)$ (4) $(24,-5)$ 394. $Let A (3, 0, - 1), B (2, 10, 6) and C (1, 2, 1) be the vertices of a triangle and M be the midpoint of AC . If G divides BM in the ratio, 2 : 1, then cos (∠ GOA) (O being the origin ) is equal to:$ (1) $\frac{1}{2 15}$ (2) $\frac{1}{15}$ (3) $\frac{1}{6 10}$ (4) $\frac{1}{30}$ 395. $The foot of the perpendicular from a point on the circle x^{2} + y^{2} = 1, z = 0 to the plane 2x + 3y + z = 6 lies on which one of the following curves ?$ (1) $(6 x + 5 y - 12)^{2} + 4 (3 x + 7 y - 8)^{2} = 1, z = 6 - 2x - 3y$ (2) $(5 x + 6 y - 12)^{2} + 4 (3 x + 5 y - 9)^{2} = 1, z = 6 - 2x - 3y$ (3) $(6 x + 5 y - 14)^{2} + 9 (3 x + 5 y - 7)^{2} = 1, z = 6 - 2x - 3y$ (4) $(5 x + 6 y - 14)^{2} + 9 (3 x + 7 y - 8)^{2} = 1, z = 6 - 2x - 3y$ 396. $The distance of the point Q (0, 2, - 2) form the line passing through the point P (5, - 4, 3) and perpendicular to the lines r = (- 3 i + 2 k) + λ (2 i + 3 j + 5 k), λ \in R and r = (i - 2 j + k) + μ (- i + 3 j + 2 k), μ \in R is$ (1) $86$ (2) $20$ (3) $54$ (4) $74$ 397. $The acute angle between the planes P, and P_{2}, when P_{1} and P_{2} are the planes passing through the intersection of the planes 5x + 8y + 13z - 29 = 0 and 8x - 7y + z - 20 = 0 and the points (2, 1, 3) and (0,1, 2), respectively, is$ (1) $\frac{π}{3}$ (2) $\frac{π}{4}$ (3) $\frac{π}{6}$ (4) $\frac{π}{12}$ 398. $Let the line \frac{x}{1} = \frac{6 - y}{2} = \frac{z + 8}{5} intersect the lines \frac{x - 5}{4} = \frac{y - 7}{3} = \frac{z + 2}{1} and \frac{x + 3}{6} = \frac{3 - y}{3} = \frac{z - 6}{1} at the points A and B respectively. Then the distance of the mid-point of the line segment AB from the plane 2x - 2y + z = 14 is$ (1) $3$ (2) $\frac{11}{3}$ (3) $4$ (4) $\frac{10}{3}$ 399. $The position vectors of the vertices A, B and C of a triangle are 2 i - 3 j + 3 k, 2 i + 2 j + 3 k and - i + j + 3 k respectively. Let l denotes the length of the angle bisector AD of ∠ BAC where D is on the line segment BC, then 2 l^{2} equals:$ (1) $49$ (2) $42$ (3) $50$ (4) $45$ 400. $Let the point A divide the line segment joining the points P (- 1, - 1, 2) and Q (5, 5, 10) internally in the ratio r : 1 (r > 0) . If O is the origin and (OQ \cdot OA) - \frac{1}{5} \| OP \times OA \|^{2} = 10, then the value of r is:$ (1) $7$ (2) $14$ (3) $3$ (4) $7$ 401. $Let a line pass through two distinct points P (- 2, - 1, 3) and Q, and be parallel to the vector 3 i + 2 j + 2 k . If the distance of the point Q from the point R (1, 3, 3) is 5 , then the square of the area of △ PQR is equal to:$ (1) $148$ (2) $136$ (3) $144$ (4) $140$ 402. $Let the image of the point P (2, - 1, 3) in the plane x + 2 y - z = 0 be Q . Then the distance of the plane 3 x + 2 y + z + 29 = 0 from the point Q$ (1) $\frac{22 2}{7}$ (2) $\frac{24 2}{7}$ (3) $214$ (4) $314$ 403. $If the point (a, 8, - 2) divides the line segment joining the points (1, 4, 6) and (5, 2, 10) in the ratio m : n then \frac{2 m}{n} - \frac{a}{3} =$ (1) $-7$ (2) $1$ (3) $-2$ (4) $3$ 404. $If the angle between the planes r \cdot (mi - j + 2 k) + 3 = 0 and r \cdot (2 i - mj - k) - 5 = 0 is \frac{π}{3} then m =$ (1) $2$ (2) $+3$ (3) $3$ (4) $-2$ 405. $If a plane passes through the points (- 1, k, 0), (2, k, - 1), (1, 1, 2) and is parallel to the line \frac{x - 1}{1} = \frac{2 y + 1}{2} = \frac{z + 1}{- 1}, then the value of \frac{k ^{2} + 1}{( k - 1 ) ( k - 2 )} is$ (1) $\frac{17}{5}$ (2) $\frac{5}{17}$ (3) $\frac{6}{13}$ (4) $\frac{13}{6}$ 406. $If the line joining the points (k, 3, 4), (4, 7, 8) is parallel to the line joining the points (- 1, - 2, 1), (1, 2, l), then k + l =$ (1) $2$ (2) $5$ (3) $7$ (4) $-3$ 407. $If the lines \frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z - 1}{λ} and \frac{x - 2}{3} = \frac{y - 3}{2} = \frac{z - 2}{3} are coplanar, then sin^{- 1} (sin λ) + cos^{- 1} (cos λ) is equal to$ (1) $8 - 2 π$ (2) $6 - π$ (3) $3 π - 8$ (4) $4 π - 8$ 408. $If an angle between the line, \frac{x + 1}{2} = \frac{y - 2}{1} = \frac{z - 3}{- 2} and the plane, x - 2 y - k z = 3 is cos^{- 1} (\frac{2 2}{3}), then a value of k is$ (1) $\frac{5}{3}$ (2) $\frac{3}{5}$ (3) $- \frac{3}{5}$ (4) $- \frac{5}{3}$ 409. $Let P be the plane, passing through the point (1, - 1, - 5) and perpendicular to the line joining the points (4, 1, - 3) and (2, 4, 3) . then the distance of P from the point (3, - 2, 2) is$ (1) $6$ (2) $4$ (3) $5$ (4) $7$ 410. $Let P (x, y, z) be a point in the first octant, whose projection in the xy -plane is the point Q . Let OP = γ; the angle between OQ and the positive x -axis be θ; and the angle between OP and the positive z -axis be φ, where O is the origin. Then the distance of P from the x -axis is$ (1) $γ 1 - sin^{2} ϕ cos^{2} θ$ (2) $γ 1 - sin^{2} θ cos^{2} ϕ$ (3) $γ 1 + cos^{2} ϕ sin^{2} θ$ (4) $γ 1 + cos^{2} θ sin^{2} ϕ$ 411. $If a point P (α, β, γ) satisfying (α β γ) 2981034888 = (000) lies on the plane 2 x + 4 y + 3 z = 5, then 6 α + 9 β + 7 γ is equal to$ (1) $- 1$ (2) $\frac{11}{5}$ (3) $\frac{5}{4}$ (4) $11$ 412. $Let the mirror image of the point (1, 3, a) with respect to the plane r \cdot (2 i - j + k) - b = 0 be (- 3, 5, 2) .Then the value of la + bl is equal to$ 413. $If the image of the point P (a, 2, a) in the line \frac{x}{2} = \frac{y + a}{1} = \frac{z}{1} is Q and the image of Q in the line \frac{x - 2 b}{2} = \frac{y - a}{1} = \frac{z + 2 b}{- 5} is P , then a + b is equal to____.$ 414. $The lines \frac{x - 2}{2} = \frac{y}{- 2} = \frac{z - 7}{16} and \frac{x + 3}{4} = \frac{y + 2}{3} = \frac{z + 2}{1} intersect at the point P . If the distance of P from the line \frac{x + 1}{2} = \frac{y - 1}{3} = \frac{z - 1}{1} is l, then 14 l^{2} is equal to$ 415. $If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through (α, 7, 1) is (\frac{5}{3}, \frac{7}{3}, \frac{17}{3}) then α is equal to \dots \dots \dots \dots \dots$ 416. $Let the point (- 1, α, β) lie on the line of the shortest distance between the lines \frac{x + 2}{- 3} = \frac{y - 2}{4} = \frac{z - 5}{2} and \frac{x + 2}{- 1} = \frac{y + 6}{2} = \frac{z - 1}{0} . Then (α - β)^{2} is equal to______$ 417. $The square of the distance of the image of the point (6, 1, 5) in the line \frac{x - 1}{3} = \frac{y}{2} = \frac{z - 2}{4}, from the origin is_____$ 418. $Let P(-2, -1, 1) and Q (\frac{56}{17}, \frac{43}{17}, \frac{111}{17}) be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are α, - 1, β, where both α and β are integers of minimum absolute values, then α^{2} + β^{2} is equal to_________.$ 419. $If the line x = y = z intersects the line x sin A + y sin B + z sin C - 18 = 0 = x sin 2A + y sin 2B + z sin 2C -9, where A, B, C are the angles of a triangle ABC, then 80 (sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2}) is equal to _______$ 420. $Let α x + β y + γ z = 1 be the equation of a plane passing through the point (3, - 2, 5) and perpendicular to the line joining the points (1, 2, 3) and (- 2, 3, 5) . Then the value of α β γ is equal to _______$