Three-Dimensional Geometry PYQs Set 16 (PDF)

Set 16 of 16 · Questions 451-471 of 471

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Three-Dimensional Geometry · Set 16 (Q451-471)
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451. $The octant in which the point (2, - 4, - 7) lies is$ (1) $Eighth$ (2) $Fourth$ (3) $Third$ (4) $Fifth$ 452. $The point (1, - 3, 4) lies in the octant$ (1) $Third$ (2) $Fourth$ (3) $Eighth$ (4) $Second$ 453. $A parallelogram has vertices A (4, 4, - 1), B (5, 6, - 1), C (6, 5, 1) and D (x, y, z) . Then the vertex D is$ (1) $(5, 1, 0)$ (2) $(- 5, 0, 1)$ (3) $(5, 3, 1)$ (4) $(5, 1, 3)$ 454. $The distance of the plane 3 x + 4 y + 5 z + 19 = 0 from the point (1, - 1, 1) measured along a line parallel to the line with direction ratios 2, 3, 1 is$ (1) $\frac{23}{5 2}$ (2) $\frac{71}{5 2}$ (3) $14$ (4) $23$ 455. $A plane meets the X, Y, Z - axes in A, B, C respectively. If the centroid of the triangle ABC is (2, - 3, 5) then the perpendicular distance from origin to the given plane is$ (1) $\frac{7}{40}$ (2) $\frac{6}{7}$ (3) $\frac{8}{50}$ (4) $\frac{90}{19}$ 456. $The distance of the point P (a, b, c) from the x -axis is$ (1) $a^{2} + b^{2}$ (2) $b^{2} + c^{2}$ (3) $a$ (4) $a^{2} + c^{2}$ 457. $Equation of line passing through the point (2, 3, 1) and parallel to the line of intersection of the plane x - 2 y - z + 5 = 0 and x + y + 3 z = 6 is$ (1) $\frac{x - 2}{- 5} = \frac{y - 3}{- 4} = \frac{z - 1}{3}$ (2) $\frac{x - 2}{4} = \frac{y - 3}{3} = \frac{z - 1}{2}$ (3) $\frac{x - 2}{5} = \frac{y - 3}{- 4} = \frac{z - 1}{3}$ (4) $\frac{x - 2}{5} = \frac{y - 3}{4} = \frac{z - 1}{3}$ 458. $The length of perpendicular drawn from the point (3, - 1, 11) to the line \frac{x}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} is$ (1) $33$ (2) $66$ (3) $53$ (4) $29$ 459. $Let D be the foot of the perpendicular drawn from the point A (2, 0, 3) to the line joining the points B (0, 4, 1) and C (- 2, 0, 4) . Then the ratio in which D divides BC is$ (1) $3 : 2$ (2) $26 : 17$ (3) $18 : 11$ (4) $16 : 9$ 460. $The x -intercept of a plane π passing through the point (1, 1, 1) is \frac{5}{2} and the perpendicular distance from the origin to the plane π is \frac{5}{7} . If the y -intercept of the plane π is negative and the z -intercept is positive then its y -intercept is$ (1) $\frac{- 5}{3}$ (2) $\frac{- 5}{6}$ (3) $\frac{- 3}{2}$ (4) $\frac{- 5}{2}$ 461. $The angle between the lines, whose direction cosines l, m, n satisfy the equations l + m + n = 0 and 2 l^{2} + 2 m^{2} - n^{2} = 0, as$ (1) $60^{\circ}$ (2) $180^{\circ}$ (3) $90^{\circ}$ (4) $30^{\circ}$ 462. $A (2, 1, 2), B (1, 0, 0), C (1 + 3, 3, - 6) are vertices of a triangle. If the length of the median drawn through A is λ 9 - 23 + 26 then λ =$ (1) $4$ (2) $3$ (3) $2$ (4) $1$ 463. $If - 2, \frac{4}{3}, \frac{- 4}{5} are the intercepts made by a plane on X, Y, Z -axes respectively then the direction cosines of a normal to this plane are$ (1) $(\frac{- 1}{3}, \frac{2}{3}, \frac{- 2}{3})$ (2) $(\frac{2}{3 5}, \frac{- 4}{3 5}, \frac{5}{3 5})$ (3) $(\frac{- 4}{57}, \frac{4}{57}, \frac{- 5}{57})$ (4) $(\frac{2}{38}, \frac{- 3}{38}, \frac{5}{38})$ 464. $The image of the point (1, 6, 3) in the line \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} is$ (1) $(1, 0, 7)$ (2) $(7, 0, 1)$ (3) $(2, 7, 0)$ (4) $(- 1, - 6, - 3)$ 465. $The angle between two diagonals of a cube is$ (1) $cos^{- 1} (\frac{1}{3})$ (2) $30^{\circ}$ (3) $cos^{- 1} (\frac{1}{3})$ (4) $45^{\circ}$ 466. $If A (1, 2, 3), B (2, - 3, 1), C (3, 2, - 1) are three vertices of a tetrahedron ABCD and G (\frac{5}{2}, \frac{3}{2}, \frac{9}{4}) is its centroid then the point which divides GD in the ratio 1 : 2 is$ (1) $(6, 1, 3) 1$ (2) $(3, \frac{8}{3}, 3)$ (3) $(\frac{1}{3}, \frac{2}{3}, 1)$ (4) $(3, \frac{8}{3}, \frac{7}{2})$ 467. $If [x] denotes the greatest integer \leq x, then lim_{n \to \infty} \frac{1}{n ^{3}} ([1^{2} x] + [2^{2} x] + [3^{2} x] + \dots + [n^{2} x]) =$ (1) $\frac{x}{2}$ (2) $\frac{x}{3}$ (3) $\frac{x}{6}$ (4) $0$ 468. $If a, b and c are non-coplanar vectors and the four points with position vectors 2 a + 3 b - c, a - 2 b + 3 c, 3 a + 4 b - 2 c and k a - 6 b + 6 c are coplanar, then k =$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 469. $P_{1} = r . (i + j + 2 k) = 9 and P_{2} = r . (2 i - j + k) = 15 . Let L be the line that meets P_{2} at the point (4, - 2, 5) and makes an angle θ with the normal of P_{2} . If α is the angle between L and P_{2} then (tan^{2} θ) (cot^{2} α) is equal to ________$ 470. $If the lines \frac{x - 1}{2} = \frac{2 - y}{- 3} = \frac{z - 3}{α} and \frac{x - 4}{5} = \frac{y - 1}{2} = \frac{z}{β} intersect, then the magnitude of the minimum value of 8 α β is ____$ 471. $Let λ_{1}, λ_{2} be the values of A for which the points (\frac{5}{2}, 1, λ) and (- 2, 0, 1) are at equal distance from the plane 2 x + 3 y - 6 z + 7 If λ_{1} > λ_{2} then the distance of the point λ_{1}, - λ_{2}, λ_{2}, λ_{1} from the line \frac{x - 5}{1} = \frac{y - 1}{2} = \frac{z + 7}{2} is ______$