3D Geometry · JEE Main previous year questions

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2023 57 questions

Q42023 · 1 Feb 2023 · Shift 1
The shortest distance between the lines $\frac{x - 5}{1} = \frac{y - 2}{2} = \frac{z - 4}{- 3}$ and $\frac{x + 3}{1} = \frac{y + 5}{4} = \frac{z - 1}{- 5}$ is :
1. A. $73$
2. B. $53$
3. C. $43$
4. D. $63$
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Answer: D
Q202023 · 1 Feb 2023 · Shift 1
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 $Q$ is :
1. A. $214$
2. B. $\frac{22 2}{7}$
3. C. $\frac{24 2}{7}$
4. D. $314$
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Answer: D
Q102023 · 31 Jan 2023 · Shift 1
Let the shortest distance between the lines $L : \frac{x - 5}{- 2} = \frac{y - λ}{0} = \frac{z + λ}{1}, λ \geq 0$ and $L_{1} : x + 1 = y - 1 = 4 - z$ be $26$ . If $(α, β, γ)$ lies on $L$ , then which of the following is NOT possible?
1. A. $α + 2 γ = 24$
2. B. $2 α + γ = 7$
3. C. $α - 2 γ = 19$
4. D. $2 α - γ = 9$
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Answer: A
Q292023 · 31 Jan 2023 · Shift 1· Numerical answer
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 ___________.
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Answer: 180
Q302023 · 31 Jan 2023 · Shift 1· Numerical answer
Let $θ$ be the angle between the planes $P_{1} : r \cdot (\hat{i} + \hat{j} + 2 \hat{k}) = 9$ and $P_{2} : r \cdot (2 \hat{i} - \hat{j} + \hat{k}) = 15$ . 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 ___________.
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Answer: 9
Q162023 · 31 Jan 2023 · Shift 2
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 $P$ and $L$ is :
1. A. $\frac{6}{14}$
2. B. $14$
3. C. $\frac{2}{7}$
4. D. $\frac{7}{2}$
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Answer: B
Q182023 · 31 Jan 2023 · Shift 2
The foot of perpendicular from the origin $O$ to a plane $P$ which meets the co-ordinate axes at the points $A, B, C$ is $(2, a, 4), a \in N$ . If the volume of the tetrahedron $OABC$ is 144 $unit^{3}$ , then which of the following points is NOT on $P$ ?
1. A. $(3, 0, 4)$
2. B. $(0, 6, 3)$
3. C. $(0, 4, 4)$
4. D. $(2, 2, 4)$
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Answer: A
Q192023 · 31 Jan 2023 · Shift 2
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. A.5
2. B.4
3. C.6
4. D.7
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Answer: A
Q202023 · 31 Jan 2023 · Shift 2
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. A. $\frac{11}{5}$
2. B.11
3. C. $- 1$
4. D. $\frac{5}{4}$
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Answer: B
Q92023 · 30 Jan 2023 · Shift 1
Let a unit vector $O P$ make angles $α, β, γ$ with the positive directions of the co-ordinate axes OX, OY, OZ respectively, where $β \in (0, \frac{π}{2})$ . If $O P$ is perpendicular to the plane through points $(1, 2, 3), (2, 3, 4)$ and $(1, 5, 7)$ , then which one of the following is true?
1. A. $α \in (\frac{π}{2}, π)$ and $γ \in (\frac{π}{2}, π)$
2. B. $α \in (0, \frac{π}{2})$ and $γ \in (\frac{π}{2}, π)$
3. C. $α \in (\frac{π}{2}, π)$ and $γ \in (0, \frac{π}{2})$
4. D. $α \in (0, \frac{π}{2})$ and $γ \in (0, \frac{π}{2})$
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Answer: A
Q182023 · 30 Jan 2023 · Shift 1
The line $l_{1}$ passes through the point $(2, 6, 2)$ and is perpendicular to the plane $2 x + y - 2 z = 10$ . Then the shortest distance between the line $l_{1}$ and the line $\frac{x + 1}{2} = \frac{y + 4}{- 3} = \frac{z}{2}$ is :
1. A. $9$
2. B. $7$
3. C. $\frac{19}{3}$
4. D. $\frac{13}{3}$
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Answer: A
Q292023 · 30 Jan 2023 · Shift 1· Numerical answer
If the equation of the plane passing through the point $(1, 1, 2)$ and perpendicular to the line $x - 3 y + 2 z - 1 = 0 = 4 x - y + z$ is $A x + B y + C z = 1$ , then $140 (C - B + A)$ is equal to ____.
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Answer: 15
Q302023 · 30 Jan 2023 · Shift 1· Numerical answer
If $λ_{1} < λ_{2}$ are two values of $λ$ such that the angle between the planes $P_{1} : r \cdot (3 \hat{i} - 5 \hat{j} + \hat{k}) = 7$ and $P_{2} : r \cdot (λ \hat{i} + \hat{j} - 3 \hat{k}) = 9$ is $sin^{- 1} (\frac{2 6}{5})$ , then the square of the length of perpendicular from the point $(38 λ_{1}, 10 λ_{2}, 2)$ to the plane $P_{1}$ is ____.
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Answer: 315
Q292023 · 29 Jan 2023 · Shift 1· Numerical answer
Let the equation of the plane P containing the line $x + 10 = \frac{8 - y}{2} = z$ be $a x + b y + 3 z = 2 (a + b)$ and the distance of the plane $P$ from the point (1, 27, 7) be $c$ . Then $a^{2} + b^{2} + c^{2}$ is equal to ____.
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Answer: 355
Q142023 · 29 Jan 2023 · Shift 2
The shortest distance between the lines $\frac{x - 1}{2} = \frac{y + 8}{- 7} = \frac{z - 4}{5}$ and $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 6}{- 3}$ is :
1. A. $23$
2. B. $33$
3. C. $43$
4. D. $53$
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Answer: C
Q172023 · 29 Jan 2023 · Shift 2
The plane $2 x - y + z = 4$ intersects the line segment joining the points A $(a, - 2, 4)$ and B $(2, b, - 3)$ at the point C in the ratio 2 : 1 and the distance of the point C from the origin is $5$ . If $ab < 0$ and P is the point $(a - b, b, 2 b - a)$ then $C P^{2}$ is equal to :
1. A. $\frac{17}{3}$
2. B. $\frac{97}{3}$
3. C. $\frac{16}{3}$
4. D. $\frac{73}{3}$
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Answer: A
Q192023 · 29 Jan 2023 · Shift 2
If the lines $\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z + 3}{1}$ and $\frac{x - a}{2} = \frac{y + 2}{3} = \frac{z - 3}{1}$ intersect at the point P, then the distance of the point P from the plane $z = a$ is :
1. A.28
2. B.22
3. C.10
4. D.16
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Answer: A
Q82023 · 25 Jan 2023 · Shift 1
The distance of the point $P (4, 6, - 2)$ from the line passing through the point $(- 3, 2, 3)$ and parallel to a line with direction ratios $3, 3, - 1$ is equal to :
1. A. $3$
2. B. $14$
3. C. $6$
4. D. $23$
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Answer: B
Q102023 · 25 Jan 2023 · Shift 1
Consider the lines $L_{1}$ and $L_{2}$ given by $L_{1} : \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z - 2}{2}$ $L_{2} : \frac{x - 2}{1} = \frac{y - 2}{2} = \frac{z - 3}{3}$ . A line $L_{3}$ having direction ratios $1, - 1, - 2$ , intersects $L_{1}$ and $L_{2}$ at the points $P$ and $Q$ respectively. Then the length of line segment $P Q$ is
1. A. $43$
2. B. $26$
3. C. $4$
4. D. $32$
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Answer: B
Q302023 · 25 Jan 2023 · Shift 1· Numerical answer
Let the equation of the plane passing through the line $x - 2 y - z - 5 = 0 = x + y + 3 z - 5$ and parallel to the line $x + y + 2 z - 7 = 0 = 2 x + 3 y + z - 2$ be $a x + b y + cz = 65$ . Then the distance of the point (a, b, c) from the plane $2 x + 2 y - z + 16 = 0$ is ____.
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Answer: 9
Q72023 · 25 Jan 2023 · Shift 2
The shortest distance between the lines $x + 1 = 2 y = - 12 z$ and $x = y + 2 = 6 z - 6$ is :
1. A.3
2. B. $\frac{5}{2}$
3. C. $\frac{3}{2}$
4. D.2
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Answer: D
Q272023 · 25 Jan 2023 · Shift 2· Numerical answer
If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line $\frac{x - 1}{2} = \frac{y + 1}{- 1} = \frac{z - 2}{0}$ is $α$ , then $28 α^{2}$ is equal to ____.
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Answer: 18
Q162023 · 24 Jan 2023 · Shift 1
The distance of the point $(7, - 3, - 4)$ from the plane passing through the points $(2, - 3, 1), (- 1, 1, - 2)$ and $(3, - 4, 2)$ is :
1. A. $42$
2. B.4
3. C.5
4. D. $52$
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Answer: D
Q182023 · 24 Jan 2023 · Shift 1
The distance of the point $(- 1, 9, - 16)$ from the plane $2 x + 3 y - z = 5$ measured parallel to the line $\frac{x + 4}{3} = \frac{2 - y}{4} = \frac{z - 3}{12}$ is :
1. A. $132$
2. B.26
3. C. $202$
4. D.31
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Answer: B
Q222023 · 24 Jan 2023 · Shift 1· Numerical answer
The shortest distance between the lines $\frac{x - 2}{3} = \frac{y + 1}{2} = \frac{z - 6}{2}$ and $\frac{x - 6}{3} = \frac{1 - y}{2} = \frac{z + 8}{0}$ is equal to ____
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Answer: 14