Three-Dimensional Geometry PYQs Set 15 (PDF)

Set 15 of 16 · Questions 421-450 of 471

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Three-Dimensional Geometry · Set 15 (Q421-450)
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421. $Let a plane P has the points i, j and i + j + k . Let L be the line through the point A and parallel to the vector i - j + k . If the plane P and line L intersect at a point B (0, 3, 2) and the distance from A to B is 33 units, then equations of the normal to the plane P through A are$ (1) $\frac{x - 3}{1} = \frac{y}{1} = \frac{z - 5}{- 1}$ (2) $\frac{x + 3}{1} = \frac{y - 6}{1} = \frac{z - 1}{- 1}$ (3) $\frac{x + 3}{1} = \frac{y}{1} = \frac{z - 5}{- 1}$ (4) $\frac{x + 3}{1} = \frac{y - 6}{1} = \frac{z + 1}{1}$ 422. $The equation of the plane passing through the line of intersection of planes π_{1} = 2 x + 6 y + 4 z - 7 = 0, π_{2} = x - y - 2 z - 2 = 0 and perpendicular to the plane x + y + 2 z - 5 = 0 is$ (1) $3 x + y - 2 z = 0$ (2) $6 x + 2 y - 4 z + 55 - 0$ (3) $6 x + 2 y - 4 z - 15 = 0$ (4) $3 x + y - 2 z - 15 = 0$ 423. $Let the plane P : 8 x + α_{1} y + α_{2} z + 12 = 0 be parallel to the line L : \frac{x + 2}{2} = \frac{y - 3}{3} = \frac{z + 4}{5} . If the intercept of P on the y -axis is 1, then the distance between L and P is$ (1) $14$ (2) $\frac{6}{14}$ (3) $\frac{2}{7}$ (4) $\frac{7}{2}$ 424. $A point on the plane passing through the points (2, 1, 4), (0, - 1, 0) and (0, 0, 1) is$ (1) $(- 2, 1, - 4)$ (2) $(2, 1, 4)$ (3) $(2, 1, 4)$ (4) $(- 2, - 1, - 4)$ 425. $Let the line L_{1} be parallel to the vector - 3 i + 2 j + 4 k and pass through the point (2, 6, 7), and the line L_{2} be parallel to the vector 2 i + j + 3 k and pass through the point (4, 3, 5) . If the line L_{3} is parallel to the vector - 3 i + 5 j + 16 k and intersects the lines L_{1} and L_{2} at the points C and D , respectively, then \| CD \|^{2} is equal to :$ (1) $89$ (2) $312$ (3) $171$ (4) $290$ 426. $Let the values of λ for which the shortest distance between the lines \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} and \frac{x - λ}{3} = \frac{y - 4}{4} = \frac{z - 5}{5} is \frac{1}{6} be λ_{1} and λ_{2} . Then the radius of the circle passing through the points (0, 0), (λ_{1}, λ_{2}) and (λ_{2}, λ_{1}) is$ (1) $3$ (2) $\frac{5 2}{3}$ (3) $\frac{2}{3}$ (4) $4$ 427. $If (x, y, z) be an arbitrary point lying on a plane P which passes through the point (42, 0, 0) (0, 42, 0) and (0, 0, 42), then the value ofexpression 3 + \frac{x - 11}{( y - 19 ) ^{2} ( z - 12 ) ^{2}} + \frac{y - 19}{( x - 11 ) ^{2} ( z - 12 ) ^{2}} + \frac{z - 12}{( x - 11 ) ^{2} ( y - 19 ) ^{2}} - \frac{x + y + z}{14 ( x - 11 ) ( y - 19 ) ( z - 12 )}$ (1) $3$ (2) $0$ (3) $39$ (4) $- 45$ 428. $E (1, 0, 0), F (0, 2, 0), G (0, 0, 3) are respectively the midpoints of the sides AB, BC, CA of △ ABC . If a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2}, are respectively the direction ratios of AF and BG , then \frac{a _{1}^{2} + b _{1}^{2} + c _{1}^{2}}{a _{2}^{2} + b _{2}^{2} + c _{2}^{2}} =$ (1) $\frac{26}{41}$ (2) $\frac{13}{26}$ (3) $\frac{17}{43}$ (4) $\frac{13}{43}$ 429. $Let A (x, y, z) be a point in xy -plane, which is equidistant from three points (0, 3, 2), (2, 0, 3) and (0, 0, 1). Let B = (1, 4, - 1) and C = (2, 0, - 2) . Then among the statements ( S 1 ) : △ ABC is an isosceles right angled triangle, and (S2) : the area of △ ABC is \frac{9 2}{2},$ (1) $both are true$ (2) $only (S2) is true$ (3) $only (S1) is true$ (4) $both are false$ 430. $Let the lines L_{1} : r = i + 2 j + 3 k + λ (2 i + 3 j + 4 k), λ \in R and L_{2} : r = (4 i + j) + μ (5 i + 2 j + k), μ \in R, intersect at the point R . Let P and Q be the points lying on lines L_{1} and L_{2}, respectively, such that \| PR \| = 29 and \| PQ \| = \frac{47}{3} . If the point P lies in the first octant, then 27 (QR)^{2} is equal to$ (1) $348$ (2) $340$ (3) $320$ (4) $360$ 431. $If the foot of the perpendicular drawn from (1, 9, 7) to the line passing through the point (3, 2, 1) and parallel to the planes x + 2 y + z = 0 and 3 y - z = 3 is (α, β, γ), then α + β + γ is equal to$ (1) $- 1$ (2) $3$ (3) $1$ (4) $5$ 432. $The line, that is coplanar to the \frac{x + 3}{- 3} = \frac{y - 1}{1} = \frac{z - 5}{5} is$ (1) $\frac{x + 1}{- 1} = \frac{y - 2}{2} = \frac{z - 5}{4}$ (2) $\frac{x + 1}{- 1} = \frac{y - 2}{2} = \frac{z - 5}{5}$ (3) $\frac{x - 1}{- 1} = \frac{y - 2}{2} = \frac{z - 5}{5}$ (4) $\frac{x + 1}{1} = \frac{y - 2}{2} = \frac{z - 5}{5}$ 433. $Let the lines L_{1} : \frac{x + 5}{3} = \frac{y + 4}{1} = \frac{z - α}{- 2} and L_{2} : 3 x + 2 y + z - 2 = 0 = x - 3 y + 2 z - 13 be coplanar. If the point P (a, b, c) on L_{1} is nearest to the point Q (- 4, - 3, 2), then ∣ a ∣ + ∣ b ∣ + ∣ c ∣ is equal to$ (1) $12$ (2) $14$ (3) $8$ (4) $10$ 434. $Let P be the point of intersection of the line \frac{x + 3}{3} = \frac{y + 2}{1} = \frac{1 - z}{2} and the plane x + y + z = 2 . If the distance of the point P from the plane 3 x - 4 y + 12 z = 32 is q, then q and 2 q are the roots of the equation$ (1) $x^{2} -18x - 72 = 0$ (2) $x^{2} -18x + 72 = 0$ (3) $x^{2} +18x+ 72 = 0$ (4) $x^{2} +18x - 72 = 0$ 435. $Let P be the plane passing through the line \frac{x - 1}{1} = \frac{y - 2}{- 3} = \frac{z + 5}{7} and the point (2, 4, -3). If the image of the point (-1, 3, 4) in the plane P is (α, β, γ), then α + β + γ is equal to$ (1) $10$ (2) $12$ (3) $9$ (4) $11$ 436. $The shortest distance between the lines \frac{x + 2}{1} = \frac{y}{- 2} = \frac{z - 5}{2} and \frac{x - 4}{1} = \frac{y - 1}{2} = \frac{z + 3}{0} is$ (1) $8$ (2) $6$ (3) $7$ (4) $9$ 437. $Let the position vectors of two points P and Q be 3 i - j + 2 k and i + 2 j - 4 k respectively. Let R and S be two points such that the direction ratios of lines PR and QS are (4, - 1, 2) and (- 2, 1, - 2), respectively. Let lines PR and QS intersect at T . If the vector TA is perpendicular to both PR and QS and thelength of vector TA is 5 units, then themodulus of a position vector of A is:$ (1) $482$ (2) $171$ (3) $5$ (4) $227$ 438. $Let the image of the point (1, 0, 7) in the line \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} be the point (α, β, γ) . Then which one of the following points lies on the line passing through (α, β, γ) and making angles \frac{2 π}{3} and \frac{3 π}{4} with y -axis and z -axis respectively and an acute angle with x -axis?$ (1) $(1, - 2, 1 + 2)$ (2) $(1, 2, 1 - 2)$ (3) $(3, 4, 3 - 22)$ (4) $(3, - 4, 3 + 22)$ 439. $Line L_{1} passes through the point (1, 2, 3) and is parallel to z -axis. Line L_{2} passes through the point (λ, 5, 6) and is parallel to y -axis. Let for λ = λ_{1}, λ_{2}, λ_{2} < λ_{1}, the shortest distance between the two lines be 3. Then the square of the distance of the point (λ_{1}, λ_{2}, 7) from the line L_{1} is$ (1) $25$ (2) $32$ (3) $40$ (4) $37$ 440. $Which of the following is false? 1.If (a, b, c) are direction ratios of a line, then a^{2} + b^{2} + c^{2} \neq = 1 . 2.The direction cosines of a line can be its direction ratios but not vice-versa. 3. If (l, m, n) is one set of direction cosines, then (- l, - m, - n) is also a valid set. 4. If (l_{1}, m_{1}, n_{1}) and (l_{2}, m_{2}, n_{2}) are direction cosines of perpendicular lines, then l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 1 .$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 441. $The distance of the point (1,-2, 3) from the plane x - y + z = 5 measured parallel to a line, whose direction ratios are 2, 3,-6 is :$ (1) $3$ (2) $5$ (3) $2$ (4) $1$ 442. $Let O be the origin and the position vector of A and B be 2 i + 2 j + k and 2 i + 4 j + 4 k respectively. If the internal bisector of ∠ AOB meets the line AB at C, then the length of OC is$ (1) $\frac{2}{3} 31$ (2) $\frac{2}{3} 34$ (3) $\frac{3}{4} 34$ (4) $\frac{3}{2} 31$ 443. $If (2, 3, - 1) is the foot of the perpendicular from (4, 2, 1) to a plane, then the equation of the plane is$ (1) $2 x - y + 2 z = 0$ (2) $2 x - y + 2 z + 1 = 0$ (3) $2 x + y + 2 z - 5 = 0$ (4) $2 x + y + 2 z - 1 = 0$ 444. $Let a line L passing through the point P (1, 1, 1) be perpendicular to the lines \frac{x - 4}{4} = \frac{y - 1}{1} = \frac{z - 1}{1} and \frac{x - 17}{1} = \frac{y - 71}{1} = \frac{z}{0} . Let the line L intersect the yz - plane at the point Q. Another line parallel to L and passing through the point S (1, 0, - 1) intersects the yz -plane at the point R. Then the square of the area of the parallelogram PQRS is equal to ________ .$ 445. $Let a line with direction ratios a, - 4 a, - 7 be perpendicular to the lines with direction ratios 3, - 1, 2 b and b, a, - 2 . If the point of intersection of the line \frac{x + 1}{a ^{2} + b ^{2}} = \frac{y - 2}{a ^{2} - b ^{2}} = \frac{z}{1} and the plane x - y + z = 0 is (α, β, γ), then α + β + γ is equal to ___________ .$ 446. $If the distance of the point (1, - 2, 3) from the plane x + 2 y - 3 z + 10 = 0 measured parallel to the line \frac{x - 1}{3} = \frac{2 - y}{m} = \frac{z + 3}{1} is \frac{7}{2}, then the value of ∣ m ∣ is equal to$ 447. $Let the image of the point P (1, 2, 3) in the plane 2 x - y + z = 9 be Q . If the coordinates of the point R are (6, 10, 7), then the square of the area of the triangle P QR is _________$ 448. $Let the line L : \frac{x - 1}{2} = \frac{y + 1}{- 1} = \frac{z - 3}{1} intersect the plane 2 x + y + 3 z = 16 at the point P . Let the point Q be the foot of perpendicular from the point R (1, - 1, - 3) on the line L . If α is the area of triangle P QR, then α^{2} is equal to ________$ 449. $Let d be the distance between the foot of perpendiculars of the points P (1, 2, - 1) and Q (2, - 1, 3) on the plane - x + y + z = 1 . Then d^{2} is equal to_______.$ 450. $If the lines \frac{x - k}{1} = \frac{y - 2}{2} = \frac{z - 3}{3} and \frac{x + 1}{3} = \frac{y + 2}{2} = \frac{z + 3}{1} are co-planar, then the value of k is$