3D Geometry · JEE Main previous year questions

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2023 57 questions

Q182023 · 13 Apr 2023 · Shift 2
Let $N$ be the foot of perpendicular from the point $P (1, - 2, 3)$ on the line passing through the points $(4, 5, 8)$ and $(1, - 7, 5)$ . Then the distance of $N$ from the plane $2 x - 2 y + z + 5 = 0$ is :
1. A. $7$
2. B. $6$
3. C. $9$
4. D. $8$
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Answer: A
Q192023 · 12 Apr 2023 · Shift 1
Let the plane $P : 4 x - y + z = 10$ be rotated by an angle $\frac{π}{2}$ about its line of intersection with the plane $x + y - z = 4$ . If $α$ is the distance of the point $(2, 3, - 4)$ from the new position of the plane $P$ , then $35 α$ is equal to :
1. A. $126$
2. B. $105$
3. C. $85$
4. D. $90$
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Answer: A
Q302023 · 12 Apr 2023 · Shift 1· Numerical answer
Let the plane $x + 3 y - 2 z + 6 = 0$ meet the co-ordinate axes at the points A, B, C. If the orthocenter of the triangle ABC is $(α, β, \frac{6}{7})$ , then $98 (α + β)^{2}$ is equal to _________.
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Answer: 288
Q172023 · 11 Apr 2023 · Shift 1
Let $(α, β, γ)$ be the image of the point $P (2, 3, 5)$ in the plane $2 x + y - 3 z = 6$ . Then $α + β + γ$ is equal to :
1. A.10
2. B.9
3. C.5
4. D.12
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Answer: A
Q182023 · 11 Apr 2023 · Shift 1
If equation of the plane that contains the point $(- 2, 3, 5)$ and is perpendicular to each of the planes $2 x + 4 y + 5 z = 8$ and $3 x - 2 y + 3 z = 5$ is $α x + β y + γ z + 97 = 0$ then $α + β + γ =$
1. A.15
2. B.16
3. C.17
4. D.18
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Answer: A
Q182023 · 11 Apr 2023 · Shift 2
Let the line passing through the points $P (2, - 1, 2)$ and $Q (5, 3, 4)$ meet the plane $x - y + z = 4$ at the point $R$ . Then the distance of the point $R$ from the plane $x + 2 y + 3 z + 2 = 0$ measured parallel to the line $\frac{x - 7}{2} = \frac{y + 3}{2} = \frac{z - 2}{1}$ is equal to :
1. A. $31$
2. B. $189$
3. C. $61$
4. D. $3$
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Answer: D
Q192023 · 11 Apr 2023 · Shift 2
Let $P$ be the plane passing through the points $(5, 3, 0)$ , $(13, 3, - 2)$ and $(1, 6, 2)$ . For $α \in N$ , if the distances of the points $A (3, 4, α)$ and $B (2, α, a)$ from the plane $P$ are $2$ and $3$ respectively, then the positive value of $a$ is :
1. A. $6$
2. B. $4$
3. C. $5$
4. D. $3$
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Answer: B
Q272023 · 11 Apr 2023 · Shift 2· Numerical answer
Let the line $l : x = \frac{1 - y}{- 2} = \frac{z - 3}{λ}$ , $λ \in R$ meet the plane $P : x + 2 y + 3 z = 4$ at the point $(α, β, γ)$ . If the angle between the line $l$ and the plane $P$ is $cos^{- 1} (\frac{5}{14})$ , then $α + 2 β + 6 γ$ is equal to _________.
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Answer: 11
Q22023 · 10 Apr 2023 · Shift 1
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. A.8
2. B.7
3. C.6
4. D.9
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Answer: D
Q162023 · 10 Apr 2023 · Shift 1
Let two vertices of a triangle ABC be $(2, 4, 6)$ and $(0, - 2, - 5)$ , and its centroid be $(2, 1, - 1)$ . If the image of the third vertex in the plane $x + 2 y + 4 z = 11$ is $(α, β, γ)$ , then $α β + β γ + γ α$ is equal to :
1. A.72
2. B.74
3. C.76
4. D.70
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Answer: B
Q182023 · 10 Apr 2023 · Shift 1
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 2q are the roots of the equation :
1. A. $x^{2} + 18 x - 72 = 0$
2. B. $x^{2} - 18 x - 72 = 0$
3. C. $x^{2} + 18 x + 72 = 0$
4. D. $x^{2} - 18 x + 72 = 0$
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Answer: D
Q162023 · 10 Apr 2023 · Shift 2
Let the image of the point $P (1, 2, 6)$ in the plane passing through the points $A (1, 2, 0)$ , $B (1, 4, 1)$ and $C (0, 5, 1)$ be $Q (α, β, γ)$ . Then $(α^{2} + β^{2} + γ^{2})$ is equal to :
1. A.76
2. B.62
3. C.70
4. D.65
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Answer: D
Q172023 · 10 Apr 2023 · Shift 2
Let the line $\frac{x}{1} = \frac{6 - y}{2} = \frac{z + 8}{5}$ intersect the lines $\frac{x - 5}{4} = \frac{y - 7}{3} = \frac{z + 2}{1}$ and $\frac{x + 3}{6} = \frac{3 - y}{3} = \frac{z - 6}{1}$ at the points A and B respectively. Then the distance of the mid-point of the line segment AB from the plane $2 x - 2 y + z = 14$ is :
1. A.3
2. B. $\frac{10}{3}$
3. C.4
4. D. $\frac{11}{3}$
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Answer: C
Q272023 · 10 Apr 2023 · Shift 2· Numerical answer
Let the foot of perpendicular from the point $A (4, 3, 1)$ on the plane $P : x - y + 2 z + 3 = 0$ be N. If $B (5, α, β), α, β \in Z$ is a point on plane P such that the area of the triangle ABN is $32$ , then $α^{2} + β^{2} + α β$ is equal to _________.
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Answer: 7
Q72023 · 8 Apr 2023 · Shift 1
The shortest distance between the lines $\frac{x - 4}{4} = \frac{y + 2}{5} = \frac{z + 3}{3}$ and $\frac{x - 1}{3} = \frac{y - 3}{4} = \frac{z - 4}{2}$ is :
1. A. $36$
2. B. $62$
3. C. $63$
4. D. $26$
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Answer: A
Q192023 · 8 Apr 2023 · Shift 1
If the equation of the plane containing the line $x + 2 y + 3 z - 4 = 0 = 2 x + y - z + 5$ and perpendicular to the plane $r = (\hat{i} - \hat{j}) + λ (\hat{i} + \hat{j} + \hat{k}) + μ (\hat{i} - 2 \hat{j} + 3 \hat{k})$ is $a x + b y + cz = 4$ , then $(a - b + c)$ is equal to :
1. A. $18$
2. B. $22$
3. C. $20$
4. D. $24$
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Answer: B
Q302023 · 8 Apr 2023 · Shift 1· Numerical answer
Let $λ_{1}, λ_{2}$ be the values of $λ$ for which the points $(\frac{5}{2}, 1, λ)$ and $(- 2, 0, 1)$ are at equal distance from the plane $2 x + 3 y - 6 z + 7 = 0$ . 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 ___________.
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Answer: 9
Q172023 · 8 Apr 2023 · Shift 2
For $a, b \in Z$ and $∣ a - b ∣ \leq 10$ , let the angle between the plane $P : a x + y - z = b$ and the line $l : x - 1 = a - y = z + 1$ be $cos^{- 1} (\frac{1}{3})$ . If the distance of the point $(6, - 6, 4)$ from the plane $P$ is $36$ , then $a^{4} + b^{2}$ is equal to :
1. A. $48$
2. B. $85$
3. C. $32$
4. D. $25$
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Answer: C
Q192023 · 8 Apr 2023 · Shift 2
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. A. $10$
2. B. $12$
3. C. $9$
4. D. $11$
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Answer: A
Q282023 · 8 Apr 2023 · Shift 2· Numerical answer
Let $P_{1}$ be the plane $3 x - y - 7 z = 11$ and $P_{2}$ be the plane passing through the points $(2, - 1, 0), (2, 0, - 1)$ , and $(5, 1, 1)$ . If the foot of the perpendicular drawn from the point $(7, 4, - 1)$ on the line of intersection of the planes $P_{1}$ and $P_{2}$ is $(α, β, γ)$ , then $α + β + γ$ is equal to ___________.
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Answer: 11
Q42023 · 6 Apr 2023 · Shift 1
One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along $x, y$ and $z$ axes are $3, 4$ and $5$ units respectively. Let P be the vertex $(3, 4, 5)$ . Then the shortest distance between the diagonal OP and an edge parallel to $z$ axis, not passing through O or P is :
1. A. $\frac{12}{5}$
2. B. $125$
3. C. $\frac{12}{5}$
4. D. $\frac{12}{5 5}$
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Answer: C
Q162023 · 6 Apr 2023 · Shift 1
If the equation of the plane passing through the line of intersection of the planes $2 x - y + z = 3, 4 x - 3 y + 5 z + 9 = 0$ and parallel to the line $\frac{x + 1}{- 2} = \frac{y + 3}{4} = \frac{z - 2}{5}$ is $a x + b y + cz + 6 = 0$ , then $a + b + c$ is equal to :
1. A.13
2. B.15
3. C.14
4. D.12
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Answer: C
Q302023 · 6 Apr 2023 · Shift 1· Numerical answer
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 PQR is ________.
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Answer: 594
Q142023 · 6 Apr 2023 · Shift 2
A plane P contains the line of intersection of the plane $r \cdot (\hat{i} + \hat{j} + \hat{k}) = 6$ and $r \cdot (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) = - 5$ . If P passes through the point $(0, 2, - 2)$ , then the square of distance of the point $(12, 12, 18)$ from the plane P is :
1. A.310
2. B.620
3. C.1240
4. D.155
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Answer: B
Q192023 · 6 Apr 2023 · Shift 2
Let the line L pass through the point $(0, 1, 2)$ , intersect the line $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and be parallel to the plane $2 x + y - 3 z = 4$ . Then the distance of the point $P (1, - 9, 2)$ from the line L is :
1. A.9
2. B. $74$
3. C. $69$
4. D. $54$
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Answer: B