Three-Dimensional Geometry PYQs Set 4 (PDF)

Set 4 of 16 · Questions 91-120 of 471

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Three-Dimensional Geometry · Set 4 (Q91-120)
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91. $A (27, - 243, 81) is a point in space. B, C, D are images of A with respect to XY, YZ and ZX planes respectively. If the centroid of the triangle BCD is (α, β, γ), then α + β + γ =$ (1) $-25$ (2) $45$ (3) $25$ (4) $-45$ 92. $The vector equation of the line passing through the point having position vector 2 i + j - 3 k and perpendicular to vectors i + j + k and i + 2 j - k is$ (1) $r = (2 i + j - 3 k) + λ (- 3 i - 2 j + k)$ (2) $r = (2 i + j - 3 k) + λ (- 3 i + 2 j - k)$ (3) $r = (2 i + j - 3 k) + λ (- 3 i + 2 j + k)$ (4) $r = (2 i + j - 3 k) + λ (i + 2 j - k)$ 93. $If the shortest distance between the lines \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} and \frac{x}{1} = \frac{y}{α} = \frac{z - 5}{1} is \frac{5}{6}, then the sum of all possible values of α is$ (1) $\frac{3}{2}$ (2) $3$ (3) $-3$ (4) $- \frac{3}{2}$ 94. $If the line joining the points A (7, p, 2) and B (q, - 2, 5) is parallel to the line joining the points C (2, - 3, 5) and D (- 6, - 15, 11), then the value of p^{2} + q^{2} is equal to$ (1) $25$ (2) $16$ (3) $9$ (4) $7$ 95. $The point of intersection of the lines r = 2 b + t (6 c - a) and r = a + s (b - 3 c) is$ (1) $a + b + c$ (2) $b - c - 6 a$ (3) $2 a - b + c$ (4) $a + 2 b - 6 c$ 96. $Equation of the plane passing through (1, - 1, 2) and perpendicular to the planes x + 2 y - 2 z = 4 and 3 x + 2 y + z = 6 is$ (1) $6 x - 7 y - 4 z - 5 = 0$ (2) $6 x + 7 y - 4 z + 5 = 0$ (3) $6 x - 7 y + 4 z + 5 = 0$ (4) $6 x + 7 y + 4 z - 5 = 0$ 97. $The shortest distance between the lines \frac{x - 3}{2} = \frac{y + 15}{- 7} = \frac{z - 9}{5} and \frac{x + 1}{2} = \frac{y - 1}{1} = \frac{z - 9}{- 3} is$ (1) $83$ (2) $43$ (3) $53 `$ (4) $63$ 98. $The transformed equation of 2 x^{2} + 3 y^{2} - z^{2} - 8 x + 18 y + 2 z + 9 = 0 when the axes are translated to the point (2, - 3, 1) is$ (1) $2 x^{2} + 3 y^{2} - z^{2} = 25$ (2) $2 x^{2} + 3 y^{2} + z^{2} = 25$ (3) $2 x^{2} - 3 y^{2} - z^{2} = 25$ (4) $2 x^{2} + 3 y^{2} - z^{2} = 50$ 99. $If the equation of the plane that contains the point (-2, 3, 5) and is perpendicular to each of the planes 2x + 4y + 5z =8 and 3x- 2y + 3z = 5 is α x + β y + γ z + 97 = 0 then α + β + γ$ (1) $15$ (2) $18$ (3) $16$ (4) $17$ 100. $Let the plane P pass through the intersection of the planes 2 x + 3 y - z = 2 and x + 2 y + 3 z = 6, and be perpendicular to the plane 2 x + y - z + 1 = 0 . If d is the distance of P from the point (- 7, 1, 1), then d is equal to$ (1) $\frac{250}{83}$ (2) $\frac{15}{53}$ (3) $\frac{25}{83}$ (4) $\frac{250}{82}$ 101. $If the lines \frac{x - k}{2} = \frac{y + 1}{3} = \frac{z - 1}{4} and \frac{x - 3}{1} = \frac{y - \frac{9}{2}}{2} = \frac{z}{1} intersect, then the value of k is$ (1) $\frac{1}{2}$ (2) $-1$ (3) $1$ (4) $\frac{3}{2}$ 102. $If the lines x + 2 ay + a = 0, x + 3 by + b = 0, x + 4 cy + c = 0 are concurrent, then a, b and c are in$ (1) $arithmetic progression$ (2) $geometric progression$ (3) $harmonic progression$ (4) $arithmetic-geometric progression$ 103. $If the line, \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 2}{4} meets the plane, x + 2 y + 3 z = 15 at a point P, then the distance of P from the origin is:$ (1) $5 /2$ (2) $25$ (3) $9/2$ (4) $7/2$ 104. $The vector equation of any plane passing through the line of intersection of the planes r \cdot m_{1} = q_{1} and r \cdot m_{2} = q_{2} is given by r (m_{1} + λ m_{2}) = q_{1} + λ q_{2} for λ \in R . The vector equation of a plane passing through the point 2 i - 3 j + k and the line of intersection of the planes r \cdot (i - 2 j + 3 k) = 5 and r \cdot (3 i + j - 2 k) = 7$ (1) $r \cdot (- 2 i - 3 j + 5 k) = - 2$ (2) $r \cdot (7 i - k) = 19$ (3) $r \cdot (4 i - j + k) = 12$ (4) $r \cdot (8 i + 5 j - 9 k) = 16$ 105. $A plane containing two lines whose direction ratios are (- 1, 2, 1) and (1, 3, 2) passes through the point (2, 1, k) . If this plane also passes through the point (3, - 1, 4), then k =$ (1) $5$ (2) $3$ (3) $6$ (4) $-3$ 106. $If lines \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4} and x - 3 = \frac{y - k}{2} = z intersect then the value of k is$ (1) $\frac{9}{2}$ (2) $\frac{1}{2}$ (3) $\frac{5}{2}$ (4) $\frac{7}{2}$ 107. $If planes x - cy - bz = 0, cx - y + az = 0 and bx + ay - z = 0 pass through straight line, then a^{2} + b^{2} + c^{2} =$ (1) $1 - abc$ (2) $abc - 1$ (3) $1 - 2 abc$ (4) $2 abc - 1$ 108. $The angle between the line of intersection of the two planes r \cdot (2 i + 2 j - 3 k) = 5, r \cdot (3 i + 3 j - 5 k) = 3 and the line r = 3 i + 2 j + k + t (5 i + 5 j - 7 k) is$ (1) $cos^{- 1} (\frac{- 1}{28})$ (2) $cos^{- 1} (\frac{41}{17 99})$ (3) $\frac{π}{2}$ (4) $\frac{π}{3}$ 109. $The equation of the plane through (4, 4, 0) and perpendicular to the planes 2 x + y + 2 z + 3 = 0 and 3 x + 3 y + 2 z - 8 = 0 is$ (1) $4 x + 3 y + 3 z = 28$ (2) $4 x - 2 y - 3 z = 8$ (3) $4 x + 2 y + 3 z = 24$ (4) $4 x + 2 y - 3 z = 24$ 110. $A tetrahedron has vertices O (0, 0, 0), A (1, 2, 1), B (2, 1, 3), C (- 1, 1, 2) . If θ is the angle between the faces OAB and ABC. then cos θ =$ (1) $\frac{1}{2}$ (2) $\frac{19}{35}$ (3) $\frac{3}{2}$ (4) $\frac{17}{31}$ 111. $The cartesian equation of the plane passing through the point (3, - 2, - 1) and parallel to the vectors b = i - 2 j + 4 k and c = 3 i + 2 j - 5 k is$ (1) $2 x - 17 y - 8 z + 63 = 0$ (2) $3 x + 17 y + 8 z - 36 = 0$ (3) $2 x + 17 y + 8 z + 36 = 0$ (4) $3 x - 16 y + 8 z - 63 = 0$ 112. $Let i - j + 2 k and i + 2 j - 2 k be the position vectors of points A and B respectively. If C is a point on the line joining A and B such that BC = 10, then the position vector of C can be$ (1) $i + 8 j - 10 k$ (2) $i + 4 j - 6 k$ (3) $i - 8 j + 10 k$ (4) $i - 4 j - 6 k$ 113. $The shortest distance between the line r = 2 i - 2 j + 3 k + λ (i - j + 4 k) and the plane r \cdot (i + 5 j + k) = 5 is$ (1) $\frac{1}{3 3}$ (2) $\frac{5}{3 3}$ (3) $\frac{10}{3 3}$ (4) $\frac{11}{3 3}$ 114. $If planes r \cdot (p i - j + 2 k) + 3 = 0 and r \cdot (2 i - p j - k) - 5 = 0 include angle \frac{π}{3}, then the value of p is$ (1) $1, - 3$ (2) $- 1, 3$ (3) $-3$ (4) $3$ 115. $If the direction ratios a, b, c of a line L satisfy the relations ab + bc + ca = 0 and 6 ab + 9 bc + 8 ca = 0, then the direction cosines of the line L are$ (1) $\frac{1}{3}, \frac{1}{3}, \frac{1}{3}$ (2) $\frac{2}{7}, \frac{1}{7}, \frac{- 2}{7}$ (3) $\frac{- 1}{6}, \frac{3}{6}, \frac{2}{6}$ (4) $\frac{- 3}{7}, \frac{2}{7}, \frac{- 6}{7}$ 116. $The plane 2 x - 3 y + 6 z - 11 = 0 makes an angle sin^{- 1} (α) with X -axis. The value of α is equal to$ (1) $\frac{3}{2}$ (2) $\frac{2}{7}$ (3) $\frac{2}{3}$ (4) $\frac{3}{7}$ 117. $If the distance of the point P (43, α, β), β < 0, from the line r = 4 i - k + μ (2 i + 3 k), μ \in R along a line with direction ratios 3, - 1, 0 is 1310, then α^{2} + β^{2} is equal to____$ 118. $If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), ant the line \frac{x - 1}{2} = \frac{y + 1}{- 1} = \frac{z - 2}{0} is α, then 28 α^{2} is equal to _______$ 119. $Let O be the origin, and M and N be the points on the lines \frac{x - 5}{4} = \frac{y - 4}{1} = \frac{z - 5}{3} and \frac{x + 8}{12} = \frac{y + 2}{5} = \frac{z + 11}{9} respectively such that MN is the shortest distance between the given lines. Then OM \cdot ON is equal to______$ 120. $A line passes through A (4, - 6, - 2) and B (16, - 2, 4) . The point P (a, b, c) where a, b, c are non-negative integers, on the line AB lies at a distance of 21 units, from the point A . The distance between the points P (a, b, c) and Q (4, - 12, 3) is equal to____$