3D Geometry · JEE Main previous year questions

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2024 20 questions

Q262024 · 9 Apr 2024 · Shift 2· Numerical answer
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 ____.
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Answer: 62
Q122024 · 8 Apr 2024 · Shift 1
Let $P (x, y, z)$ be a point in the first octant, whose projection in the $x y$ -plane is the point $Q$ . Let $O P = γ$ ; the angle between $O Q$ and the positive $x$ -axis be $θ$ ; and the angle between $O P$ and the positive $z$ -axis be $ϕ$ , where $O$ is the origin. Then the distance of $P$ from the $x$ -axis is
1. A. $γ 1 - sin^{2} ϕ cos^{2} θ$
2. B. $γ 1 + cos^{2} θ sin^{2} ϕ$
3. C. $γ 1 + cos^{2} ϕ sin^{2} θ$
4. D. $γ 1 - sin^{2} θ cos^{2} ϕ$
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Answer: A
Q192024 · 8 Apr 2024 · Shift 1
If the shortest distance between the lines $L_{1} : r = (2 + λ) \hat{i} + (1 - 3 λ) \hat{j} + (3 + 4 λ) \hat{k}, λ \in R$ $L_{2} : r = 2 (1 + μ) \hat{i} + 3 (1 + μ) \hat{j} + (5 + μ) \hat{k}, μ \in R$ is $\frac{m}{n}$ , where $g cd (m, n) = 1$ , then the value of $m + n$ equals
1. A. $384$
2. B. $387$
3. C. $390$
4. D. $377$
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Answer: B
Q22024 · 8 Apr 2024 · Shift 2
If the shortest distance between the lines $\frac{x - λ}{2} = \frac{y - 4}{3} = \frac{z - 3}{4}$ and $\frac{x - 2}{4} = \frac{y - 4}{6} = \frac{z - 7}{8}$ is $\frac{13}{29}$ , then a value of $λ$ is :
1. A. $\frac{13}{25}$
2. B.1
3. C. $- 1$
4. D. $- \frac{13}{25}$
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Answer: B
Q252024 · 8 Apr 2024 · Shift 2· Numerical answer
Let $P (α, β, γ)$ be the image of the point $Q (1, 6, 4)$ in the line $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$ . Then $2 α + β + γ$ is equal to ____
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Answer: 11
Q112024 · 6 Apr 2024 · Shift 2
Let $P (α, β, γ)$ be the image of the point $Q (3, - 3, 1)$ in the line $\frac{x - 0}{1} = \frac{y - 3}{1} = \frac{z - 1}{- 1}$ and $R$ be the point $(2, 5, - 1)$ . If the area of the triangle $PQR$ is $λ$ and $λ^{2} = 14 K$ , then $K$ is equal to :
1. A.18
2. B.81
3. C.72
4. D.36
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Answer: B
Q42024 · 5 Apr 2024 · Shift 2
Let $(α, β, γ)$ be the image of the point $(8, 5, 7)$ in the line $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 2}{5}$ . Then $α + β + γ$ is equal to :
1. A.16
2. B.20
3. C.18
4. D.14
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Answer: D
Q242024 · 5 Apr 2024 · Shift 2· Numerical answer
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 _________.
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Answer: 25
Q52024 · 4 Apr 2024 · Shift 2
Let P be the point of intersection of the lines $\frac{x - 2}{1} = \frac{y - 4}{5} = \frac{z - 2}{1}$ and $\frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z - 3}{2}$ . Then, the shortest distance of P from the line $4 x = 2 y = z$ is
1. A. $\frac{3 14}{7}$
2. B. $\frac{5 14}{7}$
3. C. $\frac{14}{7}$
4. D. $\frac{6 14}{7}$
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Answer: A
Q292024 · 4 Apr 2024 · Shift 2· Numerical answer
Consider a line $L$ passing through the points $P (1, 2, 1)$ and $Q (2, 1, - 1)$ . If the mirror image of the point $A (2, 2, 2)$ in the line $L$ is $(α, β, γ)$ , then $α + β + 6 γ$ is equal to _________.
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Answer: 6
Q202024 · 1 Feb 2024 · Shift 1
If the shortest distance between the lines $\frac{x - λ}{- 2} = \frac{y - 2}{1} = \frac{z - 1}{1}$ and $\frac{x - 3}{1} = \frac{y - 1}{- 2} = \frac{z - 2}{1}$ is 1 , then the sum of all possible values of $λ$ is :
1. A.0
2. B. $23$
3. C. $33$
4. D. $- 23$
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Answer: B
Q292024 · 1 Feb 2024 · Shift 1· Numerical answer
Let the line of the shortest distance between the lines $L_{1} : r = (\hat{i} + 2 \hat{j} + 3 \hat{k}) + λ (\hat{i} - \hat{j} + \hat{k})$ and $L_{2} : r = (4 \hat{i} + 5 \hat{j} + 6 \hat{k}) + μ (\hat{i} + \hat{j} - \hat{k})$ intersect $L_{1}$ and $L_{2}$ at $P$ and $Q$ respectively. If $(α, β, γ)$ is the mid point of the line segment $PQ$ , then $2 (α + β + γ)$ is equal to ____.
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Answer: 21
Q172024 · 1 Feb 2024 · Shift 2
If the mirror image of the point $P (3, 4, 9)$ in the line $\frac{x - 1}{3} = \frac{y + 1}{2} = \frac{z - 2}{1}$ is $(α, β, γ)$ , then $14 (α + β + γ)$ is :
1. A.102
2. B.138
3. C.132
4. D.108
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Answer: D
Q42024 · 31 Jan 2024 · Shift 2
Let $(α, β, γ)$ be the mirror image of the point $(2, 3, 5)$ in the line $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ . Then, $2 α + 3 β + 4 γ$ is equal to
1. A.32
2. B.31
3. C.33
4. D.34
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Answer: C
Q82024 · 31 Jan 2024 · Shift 2
The shortest distance, between lines $L_{1}$ and $L_{2}$ , where $L_{1} : \frac{x - 1}{2} = \frac{y + 1}{- 3} = \frac{z + 4}{2}$ and $L_{2}$ is the line, passing through the points $A (- 4, 4, 3), B (- 1, 6, 3)$ and perpendicular to the line $\frac{x - 3}{- 2} = \frac{y}{3} = \frac{z - 1}{1}$ , is
1. A. $\frac{141}{221}$
2. B. $\frac{24}{117}$
3. C. $\frac{42}{117}$
4. D. $\frac{121}{221}$
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Answer: A
Q272024 · 30 Jan 2024 · Shift 1· Numerical answer
If $d_{1}$ is the shortest distance between the lines $x + 1 = 2 y = - 12 z, x = y + 2 = 6 z - 6$ and $d_{2}$ is the shortest distance between the lines $\frac{x - 1}{2} = \frac{y + 8}{- 7} = \frac{z - 4}{5}, \frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 6}{- 3}$ , then the value of $\frac{32 3 d _{1}}{d _{2}}$ is :
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Answer: 16
Q242024 · 29 Jan 2024 · Shift 1· Numerical answer
A line with direction ratios $2, 1, 2$ meets the lines $x = y + 2 = z$ and $x + 2 = 2 y = 2 z$ respectively at the points P and Q. If the length of the perpendicular from the point $(1, 2, 12)$ to the line PQ is $l$ , then $l^{2}$ is _______.
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Answer: 65
Q212024 · 29 Jan 2024 · Shift 2· Numerical answer
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 $O M \cdot O N$ is equal to _________.
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Answer: 9
Q132024 · 27 Jan 2024 · Shift 1
If the shortest distance between the lines $\frac{x - 4}{1} = \frac{y + 1}{2} = \frac{z}{- 3}$ and $\frac{x - λ}{2} = \frac{y + 1}{4} = \frac{z - 2}{- 5}$ is $\frac{6}{5}$ , then the sum of all possible values of $λ$ is :
1. A.10
2. B.5
3. C.7
4. D.8
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Answer: D
Q52024 · 27 Jan 2024 · Shift 2
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. A. $(1, - 2, 1 + 2)$
2. B. $(3, - 4, 3 + 22)$
3. C. $(3, 4, 3 - 22)$
4. D. $(1, 2, 1 - 2)$
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Answer: C