3D Geometry · JEE Main previous year questions

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2022 28 questions

Q172022 · 29 Jul 2022 · Shift 1
If the foot of the perpendicular from the point $A (- 1, 4, 3)$ on the plane $P : 2 x + m y + n z = 4$ , is $(- 2, \frac{7}{2}, \frac{3}{2})$ , then the distance of the point A from the plane P, measured parallel to a line with direction ratios $3, - 1, - 4$ , is equal to :
1. A.1
2. B. $26$
3. C. $22$
4. D. $14$
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Answer: B
Q292022 · 29 Jul 2022 · Shift 1· Numerical answer
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 ____.
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Answer: 10
Q182022 · 29 Jul 2022 · Shift 2
Let $Q$ be the foot of perpendicular drawn from the point $P (1, 2, 3)$ to the plane $x + 2 y + z = 14$ . If $R$ is a point on the plane such that $∠ P R Q = 6 0^{\circ}$ , then the area of $△ P QR$ is equal to :
1. A. $\frac{3}{2}$
2. B. $3$
3. C. $23$
4. D.3
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Answer: B
Q192022 · 28 Jul 2022 · Shift 1
The foot of the perpendicular from a point on the circle $x^{2} + y^{2} = 1, z = 0$ to the plane $2 x + 3 y + z = 6$ lies on which one of the following curves?
1. A. $(6 x + 5 y - 12)^{2} + 4 (3 x + 7 y - 8)^{2} = 1, z = 6 - 2 x - 3 y$
2. B. $(5 x + 6 y - 12)^{2} + 4 (3 x + 5 y - 9)^{2} = 1, z = 6 - 2 x - 3 y$
3. C. $(6 x + 5 y - 14)^{2} + 9 (3 x + 5 y - 7)^{2} = 1, z = 6 - 2 x - 3 y$
4. D. $(5 x + 6 y - 14)^{2} + 9 (3 x + 7 y - 8)^{2} = 1, z = 6 - 2 x - 3 y$
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Answer: B
Q222022 · 28 Jul 2022 · Shift 1· Numerical answer
Let $P (- 2, - 1, 1)$ and $Q (\frac{56}{17}, \frac{43}{17}, \frac{111}{17})$ be the vertices of the rhombus $PRQS$ . If the direction ratios of the diagonal $RS$ are $α, - 1, β$ , where both $α$ and $β$ are integers of minimum absolute values, then $α^{2} + β^{2}$ is equal to ____.
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Answer: 450
Q172022 · 28 Jul 2022 · Shift 2
Let the lines $\frac{x - 1}{λ} = \frac{y - 2}{1} = \frac{z - 3}{2}$ and $\frac{x + 26}{- 2} = \frac{y + 18}{3} = \frac{z + 28}{λ}$ be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lie on P?
1. A. $(0, - 2, - 2)$
2. B. $(- 5, 0, - 1)$
3. C. $(3, - 1, 0)$
4. D. $(0, 4, 5)$
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Answer: D
Q182022 · 28 Jul 2022 · Shift 2
A plane P is parallel to two lines whose direction ratios are $- 2, 1, - 3$ and $- 1, 2, - 2$ and it contains the point $(2, 2, - 2)$ . Let P intersect the co-ordinate axes at the points A, B, C making the intercepts $α, β, γ$ . If V is the volume of the tetrahedron OABC, where O is the origin, and $p = α + β + γ$ , then the ordered pair $(V, p)$ is equal to :
1. A. $(48, - 13)$
2. B. $(24, - 13)$
3. C. $(48, 11)$
4. D. $(24, - 5)$
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Answer: B
Q202022 · 27 Jul 2022 · Shift 1
If the plane $P$ passes through the intersection of two mutually perpendicular planes $2 x + k y - 5 z = 1$ and $3 k x - k y + z = 5, k < 3$ and intercepts a unit length on positive $x$ -axis, then the intercept made by the plane $P$ on the $y$ -axis is :
1. A. $\frac{1}{11}$
2. B. $\frac{5}{11}$
3. C.6
4. D.7
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Answer: D
Q292022 · 27 Jul 2022 · Shift 1· Numerical answer
Let the line $\frac{x - 3}{7} = \frac{y - 2}{- 1} = \frac{z - 3}{- 4}$ intersect the plane containing the lines $\frac{x - 4}{1} = \frac{y + 1}{- 2} = \frac{z}{1}$ and $4 a x - y + 5 z - 7 a = 0 = 2 x - 5 y - z - 3, a \in R$ at the point $P (α, β, γ)$ . Then the value of $α + β + γ$ equals _________.
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Answer: 12
Q112022 · 27 Jul 2022 · Shift 2
If the length of the perpendicular drawn from the point $P (a, 4, 2)$ , $a > 0$ on the line $\frac{x + 1}{2} = \frac{y - 3}{3} = \frac{z - 1}{- 1}$ is $26$ units and $Q (α_{1}, α_{2}, α_{3})$ is the image of the point $P$ in this line, then $a + \sum_{i = 1}^{3} α_{i}$ is equal to :
1. A. $7$
2. B. $8$
3. C. $12$
4. D. $14$
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Answer: B
Q162022 · 27 Jul 2022 · Shift 2
If the line of intersection of the planes $a x + b y = 3$ and $a x + b y + cz = 0$ , $a > 0$ makes an angle $3 0^{\circ}$ with the plane $y - z + 2 = 0$ , then the direction cosines of the line are :
1. A. $\frac{1}{2}, \frac{1}{2}, 0$
2. B. $\frac{1}{2}, \pm \frac{1}{2}, 0$
3. C. $\frac{1}{5}, - \frac{2}{5}, 0$
4. D. $\frac{1}{2}, - \frac{3}{2}, 0$
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Answer: B
Q32022 · 26 Jul 2022 · Shift 1
If the system of linear equations. $8 x + y + 4 z = - 2$ $x + y + z = 0$ $λ x - 3 y = μ$ has infinitely many solutions, then the distance of the point $(λ, μ, - \frac{1}{2})$ from the plane $8 x + y + 4 z + 2 = 0$ is :
1. A. $35$
2. B.4
3. C. $\frac{26}{9}$
4. D. $\frac{10}{3}$
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Answer: D
Q112022 · 25 Jul 2022 · Shift 2
The shortest distance between the lines $\frac{x + 7}{- 6} = \frac{y - 6}{7} = z$ and $\frac{7 - x}{2} = y - 2 = z - 6$ is :
1. A. $229$
2. B.1
3. C. $\frac{37}{29}$
4. D. $\frac{29}{2}$
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Answer: A
Q192022 · 25 Jul 2022 · Shift 2
A plane $E$ is perpendicular to the two planes $2 x - 2 y + z = 0$ and $x - y + 2 z = 4$ , and passes through the point $P (1, - 1, 1)$ . If the distance of the plane $E$ from the point $Q (a, a, 2)$ is $32$ , then $(P Q)^{2}$ is equal to :
1. A.9
2. B.12
3. C.21
4. D.33
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Answer: C
Q172022 · 30 Jun 2022 · Shift 1
The distance of the point $(3, 2, - 1)$ from the plane $3 x - y + 4 z + 1 = 0$ along the line $\frac{2 - x}{2} = \frac{y - 3}{2} = \frac{z + 1}{1}$ is equal to :
1. A.9
2. B.6
3. C.3
4. D.2
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Answer: C
Q202022 · 28 Jun 2022 · Shift 2
Let the plane $a x + b y + cz = d$ pass through $(2, 3, - 5)$ and is perpendicular to the planes $2 x + y - 5 z = 10$ and $3 x + 5 y - 7 z = 12$ . If $a, b, c, d$ are integers $d > 0$ and $g cd (∣ a ∣, ∣ b ∣, ∣ c ∣, d) = 1$ , then the value of $a + 7 b + c + 20 d$ is equal to :
1. A. $18$
2. B. $20$
3. C. $24$
4. D. $22$
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Answer: D
Q212022 · 28 Jun 2022 · Shift 2· Numerical answer
Let the image of the point $P (1, 2, 3)$ in the line $L : \frac{x - 6}{3} = \frac{y - 1}{2} = \frac{z - 2}{3}$ be Q. Let $R (α, β, γ)$ be a point that divides internally the line segment PQ in the ratio $1 : 3$ . Then the value of $22 (α + β + γ)$ is equal to ________.
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Answer: 125
Q302022 · 27 Jun 2022 · Shift 1· Numerical answer
Let the mirror image of the point $(a, b, c)$ with respect to the plane $3 x - 4 y + 12 z + 19 = 0$ be $(a - 6, β, γ)$ . If $a + b + c = 5$ , then $7 β - 9 γ$ is equal to ____.
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Answer: 137
Q92022 · 27 Jun 2022 · Shift 2
The shortest distance between the lines $\frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z - 1}{- 1}$ and $\frac{x + 3}{2} = \frac{y - 6}{1} = \frac{z - 5}{3}$ , is :
1. A. $\frac{18}{5}$
2. B. $\frac{22}{3 5}$
3. C. $\frac{46}{3 5}$
4. D. $63$
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Answer: A
Q192022 · 27 Jun 2022 · Shift 2
Let the foot of the perpendicular from the point $(1, 2, 4)$ on the line $\frac{x + 2}{4} = \frac{y - 1}{2} = \frac{z + 1}{3}$ be P. Then the distance of P from the plane $3 x + 4 y + 12 z + 23 = 0$ is :
1. A. $5$
2. B. $\frac{50}{13}$
3. C. $4$
4. D. $\frac{63}{13}$
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Answer: A
Q172022 · 26 Jun 2022 · Shift 1
Let the plane $2 x + 3 y + z + 20 = 0$ be rotated through a right angle about its line of intersection with the plane $x - 3 y + 5 z = 8$ . If the mirror image of the point $(2, - \frac{1}{2}, 2)$ in the rotated plane is $B (a, b, c)$ , then :
1. A. $\frac{a}{8} = \frac{b}{5} = \frac{c}{- 4}$
2. B. $\frac{a}{4} = \frac{b}{5} = \frac{c}{- 2}$
3. C. $\frac{a}{8} = - \frac{b}{5} = \frac{c}{4}$
4. D. $\frac{a}{4} = \frac{b}{5} = \frac{c}{2}$
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Answer: A
Q182022 · 26 Jun 2022 · Shift 2
If the plane $2 x + y - 5 z = 0$ is rotated about its line of intersection with the plane $3 x - y + 4 z - 7 = 0$ by an angle of $\frac{π}{2}$ , then the plane after the rotation passes through the point :
1. A. $(2, - 2, 0)$
2. B. $(- 2, 2, 0)$
3. C. $(1, 0, 2)$
4. D. $(- 1, 0, - 2)$
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Answer: C
Q192022 · 26 Jun 2022 · Shift 2
If the lines $r = (\hat{i} - \hat{j} + \hat{k}) + λ (3 \hat{j} - \hat{k})$ and $r = (α \hat{i} - \hat{j}) + μ (2 \hat{i} - 3 \hat{k})$ are co-planar, then the distance of the plane containing these two lines from the point $(α, 0, 0)$ is :
1. A. $\frac{2}{9}$
2. B. $\frac{2}{11}$
3. C. $\frac{4}{11}$
4. D.2
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Answer: B
Q202022 · 25 Jun 2022 · Shift 1
Let $Q$ be the mirror image of the point $P (1, 0, 1)$ with respect to the plane $S : x + y + z = 5$ . If a line $L$ passing through $(1, - 1, - 1)$ , parallel to the line $PQ$ meets the plane $S$ at $R$ , then $QR^{2}$ is equal to :
1. A. $2$
2. B. $5$
3. C. $7$
4. D. $11$
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Answer: B
Q292022 · 25 Jun 2022 · Shift 1· Numerical answer
Let the lines $L_{1} : r = λ (\hat{i} + 2 \hat{j} + 3 \hat{k})$ , $λ \in R$ $L_{2} : r = (\hat{i} + 3 \hat{j} + \hat{k}) + μ (\hat{i} + \hat{j} + 5 \hat{k})$ ; $μ \in R$ , intersect at the point $S$ . If a plane $a x + b y - z + d = 0$ passes through $S$ and is parallel to both the lines $L_{1}$ and $L_{2}$ , then the value of $a + b + d$ is equal to .
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Answer: 5