3D Geometry · JEE Main previous year questions

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2021 21 questions

Q152021 · 1 Sept 2021 · Shift 2
Let the acute angle bisector of the two planes $x - 2 y - 2 z + 1 = 0$ and $2 x - 3 y - 6 z + 1 = 0$ be the plane P. Then which of the following points lies on P?
1. A. $(3, 1, - \frac{1}{2})$
2. B. $(- 2, 0, - \frac{1}{2})$
3. C. $(0, 2, - 4)$
4. D. $(4, 0, - 2)$
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Answer: B
Q192021 · 26 Aug 2021 · Shift 2
Let P be the plane passing through the point $(1, 2, 3)$ and the line of intersection of the planes $r \cdot (\hat{i} + \hat{j} + 4 \hat{k}) = 16$ and $r \cdot (- \hat{i} + \hat{j} + \hat{k}) = 6$ . Then which of the following points does NOT lie on P?
1. A. $(3, 3, 2)$
2. B. $(6, - 6, 2)$
3. C. $(4, 2, 2)$
4. D. $(- 8, 8, 6)$
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Answer: C
Q282021 · 26 Aug 2021 · Shift 2· Numerical answer
Let Q be the foot of the perpendicular from the point $P (7, - 2, 13)$ on the plane containing the lines $\frac{x + 1}{6} = \frac{y - 1}{7} = \frac{z - 3}{8}$ and $\frac{x - 1}{3} = \frac{y - 2}{5} = \frac{z - 3}{7}$ . Then $(PQ)^{2}$ , is equal to ________.
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Answer: 96
Q162021 · 27 Jul 2021 · Shift 2
For real numbers $α$ and $β \neq = 0$ , if the point of intersection of the straight lines $\frac{x - α}{1} = \frac{y - 1}{2} = \frac{z - 1}{3}$ and $\frac{x - 4}{β} = \frac{y - 6}{3} = \frac{z - 7}{3}$ , lies on the plane $x + 2 y - z = 8$ , then $α - β$ is equal to :
1. A.5
2. B.9
3. C.3
4. D.7
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Answer: D
Q292021 · 27 Jul 2021 · Shift 2· Numerical answer
The distance of the point P(3, 4, 4) from the point of intersection of the line joining the points Q(3, $- 4$ , $- 5$ ) and R(2, $- 3$ , 1) and the plane 2x + y + z = 7, is equal to ____.
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Answer: 7
Q202021 · 25 Jul 2021 · Shift 1
Let the foot of perpendicular from a point P(1, 2, $- 1$ ) to the straight line $L : \frac{x}{1} = \frac{y}{0} = \frac{z}{- 1}$ be N. Let a line be drawn from P parallel to the plane x + y + 2z = 0 which meets L at point Q. If $α$ is the acute angle between the lines PN and PQ, then $cos α$ is equal to ____.
1. A. $\frac{1}{5}$
2. B. $\frac{3}{2}$
3. C. $\frac{1}{3}$
4. D. $\frac{1}{2 3}$
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Answer: C
Q152021 · 22 Jul 2021 · Shift 2
Let L be the line of intersection of planes $r \cdot (\hat{i} - \hat{j} + 2 \hat{k}) = 2$ and $r \cdot (2 \hat{i} + \hat{j} - \hat{k}) = 2$ . If $P (α, β, γ)$ is the foot of perpendicular on L from the point $(1, 2, 0)$ , then the value of $35 (α + β + γ)$ is equal to :
1. A. $101$
2. B. $119$
3. C. $143$
4. D. $134$
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Answer: B
Q262021 · 22 Jul 2021 · Shift 2· Numerical answer
The area (in sq. units) of the region bounded by the curves $x^{2} + 2 y - 1 = 0$ , $y^{2} + 4 x - 4 = 0$ and $y^{2} - 4 x - 4 = 0$ , in the upper half plane is ________.
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Answer: 2
Q242021 · 20 Jul 2021 · Shift 1· Numerical answer
If the shortest distance between the lines $r_{1} = α \hat{i} + 2 \hat{j} + 2 \hat{k} + λ (\hat{i} - 2 \hat{j} + 2 \hat{k})$ , $λ \in R$ , $α > 0$ and $r_{2} = - 4 \hat{i} - \hat{k} + μ (3 \hat{i} - 2 \hat{j} - 2 \hat{k})$ , $μ \in R$ is 9, then $α$ is equal to ________.
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Answer: 6
Q282021 · 18 Mar 2021 · Shift 1· Numerical answer
Let the plane $a x + b y + cz + d = 0$ bisect the line joining the points $(4, - 3, 1)$ and $(2, 3, - 5)$ at the right angles. If a, b, c, d are integers, then the minimum value of $(a^{2} + b^{2} + c^{2} + d^{2})$ is _______.
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Answer: 28
Q292021 · 18 Mar 2021 · Shift 1· Numerical answer
The equation of the planes parallel to the plane $x - 2 y + 2 z - 3 = 0$ which are at unit distance from the point $(1, 2, 3)$ is $a x + b y + cz + d = 0$ . If $(b - d) = k (c - a)$ , then the positive value of k is :
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Answer: 4
Q282021 · 18 Mar 2021 · Shift 2· Numerical answer
Let the mirror image of the point (1, 3, a) with respect to the plane $r \cdot (2 \hat{i} - \hat{j} + \hat{k}) - b = 0$ be ( $- 3$ , 5, 2). Then, the value of | a + b | is equal to ____.
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Answer: 1
Q292021 · 18 Mar 2021 · Shift 2· Numerical answer
Let P be a plane containing the line $\frac{x - 1}{3} = \frac{y + 6}{4} = \frac{z + 5}{2}$ and parallel to the line $\frac{x - 1}{4} = \frac{y - 2}{- 3} = \frac{z + 5}{7}$ . If the point $(1, - 1, α)$ lies on the plane P, then the value of $∣5 α ∣$ is equal to ____.
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Answer: 38
Q172021 · 16 Mar 2021 · Shift 1
Let P be a plane $l x + m y + n z = 0$ containing the line, $\frac{1 - x}{1} = \frac{y + 4}{2} = \frac{z + 2}{3}$ . If plane P divides the line segment AB joining points $A (- 3, - 6, 1)$ and $B (2, 4, - 3)$ in ratio $k : 1$ then the value of k is equal to :
1. A.2
2. B.3
3. C.1.5
4. D.4
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Answer: A
Q192021 · 16 Mar 2021 · Shift 1
If for $a > 0$ , the feet of perpendiculars from the points $A (a, - 2 a, 3)$ and $B (0, 4, 5)$ on the plane $l x + m y + n z = 0$ are points $C (0, - a, - 1)$ and D respectively, then the length of line segment CD is equal to :
1. A. $41$
2. B. $55$
3. C. $31$
4. D. $66$
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Answer: D
Q182021 · 26 Feb 2021 · Shift 1
Consider the three planes $P_{1} : 3 x + 15 y + 21 z = 9$ , $P_{2} : x - 3 y - z = 5$ , and $P_{3} : 2 x + 10 y + 14 z = 5$ Then, which one of the following is true?
1. A. $P_{1}$ and $P_{2}$ are parallel.
2. B. $P_{1}, P_{2}$ and $P_{3}$ all are parallel.
3. C. $P_{1}$ and $P_{3}$ are parallel.
4. D. $P_{2}$ and $P_{3}$ are parallel.
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Answer: C
Q282021 · 26 Feb 2021 · Shift 1· Numerical answer
Let $(λ, 2, 1)$ be a point on the plane which passes through the point $(4, - 2, 2)$ . If the plane is perpendicular to the line joining the points $(- 2, - 21, 29)$ and $(- 1, - 16, 23)$ , then $(\frac{λ}{11})^{2} - \frac{4 λ}{11} - 4$ is equal to _________.
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Answer: 8
Q162021 · 26 Feb 2021 · Shift 2
If the mirror image of the point $(1, 3, 5)$ with respect to the plane $4 x - 5 y + 2 z = 8$ is $(α, β, γ)$ , then $5 (α + β + γ)$ equals :
1. A.39
2. B.41
3. C.47
4. D.43
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Answer: C
Q202021 · 26 Feb 2021 · Shift 2
Let L be a line obtained from the intersection of two planes $x + 2 y + z = 6$ and $y + 2 z = 4$ . If point $P (α, β, γ)$ is the foot of perpendicular from $(3, 2, 1)$ on L, then the value of $21 (α + β + γ)$ equals :
1. A.102
2. B.142
3. C.136
4. D.68
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Answer: A
Q12021 · 25 Feb 2021 · Shift 2
The shortest distance between the line $x - y = 1$ and the curve $x^{2} = 2 y$ is :
1. A.0
2. B. $\frac{1}{2 2}$
3. C. $\frac{1}{2}$
4. D. $\frac{1}{2}$
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Answer: B
Q182021 · 25 Feb 2021 · Shift 2
A plane passes through the points A(1, 2, 3), B(2, 3, 1) and C(2, 4, 2). If O is the origin and P is (2, $- 1$ , 1), then the projection of $O P$ on this plane is of length :
1. A. $\frac{2}{7}$
2. B. $\frac{2}{5}$
3. C. $\frac{2}{3}$
4. D. $\frac{2}{11}$
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Answer: D