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3D Geometry

36 questions · Mathematics · JEE Advanced

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PYQ Library
JEE Advanced
Mathematics
3D Geometry

3D Geometry

36 questions · Mathematics · JEE Advanced

Let $L_{1}$ be the line of intersection of the planes given by the equations $2 x + 3 y + z = 4$ and $x + 2 y + z = 5$ . Let $L_{2}$ be the line passing through the point $P (2, - 1, 3)$ and parallel to $L_{1}$ . Let $M$ denote the plane given by the…2025 · Shift 1 · Q21 · Multiple correct
Let $R^{3}$ denote the three-dimensional space. Take two points $P = (1, 2, 3)$ and $Q = (4, 2, 7)$ . Let $dist (X, Y)$ denote the distance between two points $X$ and $Y$ in $R^{3}$ . Let $S = {X \in R^{3} : (dist (X, P))^{2} - (dist (X, Q))^{2} = 50} and T = {Y \in R^{3} : (dist (Y, Q))^{2} - (dist (Y, P))^{2} = 50} .$ …2024 · Shift 1 · Q24 · Multiple correct
Let $γ \in R$ be such that the lines $L_{1} : \frac{x + 11}{1} = \frac{y + 21}{2} = \frac{z + 29}{3}$ and $L_{2} : \frac{x + 16}{3} = \frac{y + 11}{2} = \frac{z + 4}{γ}$ intersect. Let $R_{1}$ be the point of intersection of $L_{1}$ and $L_{2}$ . Let $O = (0, 0, 0)$ … Includes table2024 · Shift 1 · Q33 · MCQ
A straight line drawn from the point $P (1, 3, 2)$ , parallel to the line $\frac{x - 2}{1} = \frac{y - 4}{2} = \frac{z - 6}{1}$ , intersects the plane $L_{1} : x - y + 3 z = 6$ at the point $Q$ . Another straight line which passes through $Q$ and is perpendicular…2024 · Shift 2 · Q23 · Multiple correct
Let $ℓ_{1}$ and $ℓ_{2}$ be the lines $r_{1} = λ (\hat{i} + \hat{j} + \hat{k})$ and $r_{2} = (\hat{j} - \hat{k}) + μ (\hat{i} + \hat{k})$ , respectively. Let $X$ be the set of all the planes $H$ that contain the line $ℓ_{1}$ . For a plane $H$ … Includes table2023 · Shift 1 · Q33 · MCQ
Let $P_{1}$ and $P_{2}$ be two planes given by $P_{1} : 10 x + 15 y + 12 z - 60 = 0 P_{2} : - 2 x + 5 y + 4 z - 20 = 0$ Which of the following straight lines can be an edge of some tetrahedron whose two faces lie…2022 · Shift 1 · Q29 · Multiple correct
Let $S$ be the reflection of a point $Q$ with respect to the plane given by $r = - (t + p) \overset{}{^} + t \overset{}{^} + (1 + p) \hat{k}$ where $\overset{}{^}, \overset{}{^}, \hat{k}$ are real parameters and $\overset{}{^}, \overset{}{^}, \hat{k}$ are the unit vectors along…2022 · Shift 1 · Q30 · Multiple correct
Let L1 and L2 be the following straight lines. $L_{1} : \frac{x - 1}{1} = \frac{y}{- 1} = \frac{z - 1}{3}$ and $L_{2} : \frac{x - 1}{- 3} = \frac{y}{- 1} = \frac{z - 1}{1}$ . Suppose the straight line $L : \frac{x - α}{l} = \frac{y - 1}{m} = \frac{z - γ}{- 2}$ …2020 · Shift 1 · Q29 · Multiple correct
Let $α$ 2 + $β$ 2 + $γ$ 2 $e$ 0 and $α$ + $γ$ = 1. Suppose the point (3, 2, $-$ 1) is the mirror image of the point (1, 0, $-$ 1) with respect to the plane $α$ x + $β$ y + $γ$ z = $δ$ . Then which of…2020 · Shift 2 · Q28 · Multiple correct
Let L1 and L2 denote the lines $r = i + λ (- i + 2 j + 2 k)$ , $λ \in$ R and $r = μ (2 i - j + 2 k), μ \in R$ respectively. If L3 is a line which is…2019 · Shift 1 · Q27 · Multiple correct
Three lines are given by $r = λ i, λ \in R$ , $r = μ (i + j), μ \in R$ and $r = v (i + j + k), v \in R$ Let the lines cut the plane x + y + z = 1 at the points A, B…2019 · Shift 1 · Q35 · Numerical
Three lines $L_{1} : r = λ i$ , $λ \in$ R, $L_{2} : r = k + μ j$ , $μ \in$ R and $L_{3} : r = i + j + v k$ , v $\in$ R are given. For which point(s) Q on L2 can we find a point P…2019 · Shift 2 · Q22 · Multiple correct
Let P1 : 2x + y $-$ z = 3 and P2 : x + 2y + z = 2 be two planes. Then, which of the following statement(s) is(are) TRUE?2018 · Shift 1 · Q21 · Multiple correct
Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment PQ is perpendicular to the plane x + y = 3 and the mid-point of PQ lies in the plane x + y = 3) lies on the Z-axis. Let the distance of P…2018 · Shift 2 · Q30 · Numerical
Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the X-axis, Y-axis and Z-axis, respectively, where O(0, 0, 0) is the origin. Let $S (\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$ be the centre of…2018 · Shift 2 · Q31 · Numerical
The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes 2x + y $-$ 2z = 5 and 3x $-$ 6y $-$ 2z = 7 is2017 · Shift 2 · Q25 · MCQ
Consider a pyramid $O P QR S$ located in the first octant $(x \geq 0, y \geq 0, z \geq 0)$ with $O$ as origin, and $O P$ and $O R$ along the $x$ -axis and the $y$ -axis, respectively. The base $O P QR$ of the pyramid is a square with $O P = 3.$ …2016 · Shift 1 · Q20 · Multiple correct
Let $P$ be the image of the point $(3, 1, 7)$ with respect to the plane $x - y + z = 3.$ Then the equation of the plane passing through $P$ and containing the straight line $\frac{x}{1} = \frac{y}{2} = \frac{z}{1}$ is2016 · Shift 2 · Q22 · MCQ
In $R^{3},$ consider the planes $P_{1} : y = 0$ and $P_{2} : x + z = 1.$ Let $P_{3}$ be the plane, different from $P_{1}$ and $P_{2}$ , which passes through the intersection of $P_{1}$ and $P_{2} .$ If the distance of the point $(0, 1, 0)$ from…2015 · Shift 1 · Q33 · Multiple correct
In $R^{3},$ let $L$ be a straight lines passing through the origin. Suppose that all the points on $L$ are at a constant distance from the two planes $P_{1} : x + 2 y - z + 1 = 0$ and $P_{2} : 2 x - y + z - 1 = 0.$ Let $M$ be the locus of the…2015 · Shift 1 · Q34 · Multiple correct
From a point $P (λ, λ, λ),$ perpendicular $P Q$ and $P R$ are drawn respectively on the lines $y = x, z = 1$ and $y = - x, z = - 1.$ If $P$ is such that $∠ QP R$ is a right angle, then the possible value(s) of $λ$ …2014 · Shift 1 · Q29 · Multiple correct
Perpendiculars are drawn from points on the line $\frac{x + 2}{2} = \frac{y + 1}{- 1} = \frac{z}{3}$ to the plane $x + y + z = 3$ . The foot of perpendiculars lie on the line2013 · Shift 1 · Q28 · MCQ
A line $l$ passing through the origin is perpendicular to the lines $l_{1} : (3 + t) i + (- 1 + 2 t) j + (4 + 2 t) k, - \infty < t < \infty l_{2} : (3 + 2 s) i + (3 + 2 s) j + (2 + s) k, - \infty < s < \infty$ …2013 · Shift 1 · Q39 · Multiple correct
Consider the lines $L_{1} : \frac{x - 1}{2} = \frac{y}{- 1} = \frac{z + 3}{1}, L_{2} : \frac{x - 4}{1} = \frac{y + 3}{1} = \frac{z + 3}{2}$ and the planes $P_{1} : 7 x + y + 2 z = 3, P_{2} = 3 x + 5 y - 6 z = 4.$ Let $a x + b y + cz = d$ be…2013 · Shift 2 · Q23 · MCQ
Two lines $L_{1} : x = 5, \frac{y}{3 - α} = \frac{z}{- 2}$ and $L_{2} : x = α, \frac{y}{- 1} = \frac{z}{2 - α}$ are coplanar. Then $α$ can take value(s)2013 · Shift 2 · Q24 · Multiple correct
The point $P$ is the intersection of the straight line joining the points $Q (2, 3, 5)$ and $R (1, - 1, 4)$ with the plane $5 x - 4 y - z = 1.$ If $S$ is the foot of the perpendicular drawn from the point $T (2, 1, 4)$ to $QR,$ then the length of the…2012 · Shift 1 · Q35 · MCQ
If the straight lines $\frac{x - 1}{2} = \frac{y + 1}{k} = \frac{z}{2}$ and $\frac{x + 1}{5} = \frac{y + 1}{2} = \frac{z}{k}$ are coplanar, then the plane (s) containing these two lines is (are)2012 · Shift 2 · Q22 · Multiple correct
The equation of a plane passing through the line of intersection of the planes $x + 2 y + 3 z = 2$ and $x - y + z = 3$ and at a distance $\frac{2}{3}$ from the point $(3, 1, - 1)$ is2012 · Shift 2 · Q23 · MCQ
If the distance between the plane $A x - 2 y + z = d$ and the plane containing the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $\frac{x - 2}{3} = \frac{y - 3}{4} = \frac{z - 4}{5}$ is $6,$ then $∣ d ∣$ …2010 · Shift 1 · Q41 · Numerical
Equation of the plane containing the straight line $\frac{x}{2} = \frac{y}{3} = \frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3} = \frac{y}{4} = \frac{z}{2}$ and $\frac{x}{4} = \frac{y}{2} = \frac{z}{3}$ …2010 · Shift 1 · Q43 · MCQ
Match the statement in Column- $I$ with the values in Column- $I I$ Column- $I$ (A) A line from the origin meets the lines $\frac{x - 2}{1} = \frac{y - 1}{- 2} = \frac{z + 1}{1}$ and $\frac{x - \frac{8}{3}}{2} = \frac{y + 3}{- 1} = \frac{z - 1}{1}$ …2010 · Shift 2 · Q20 · MCQ
If the distance of the point $P (1, - 2, 1)$ from the plane $x + 2 y - 2 z = α,$ where $α > 0,$ is $5,$ then the foot of the perpendicular from $P$ to the planes is2010 · Shift 2 · Q23 · MCQ
Let $P (3, 2, 6)$ be a point in space and $Q$ be a point on the line $r = (i - j + 2 k) + μ (- 3 i + j + 5 k)$ Then the value of $μ$ for which the…2009 · Shift 1 · Q23 · MCQ
A line with positive direction cosines passes through the point P(2, $-$ 1, 2) and makes equal angles with the coordinate axes. The line meets the plane $2 x + y + z = 9$ at point Q. The length of the line segment PQ equals2009 · Shift 2 · Q24 · MCQ
Consider three planes $P_{1} : x - y + z = 1 P_{2} : x + y - z = 1 P_{3} : x - 3 y + 3 z = 2$ Let $L_{1}, L_{2}, L_{3}$ be the lines of intersection of the planes $P_{2}$ and $P_{3}, P_{3}$ and $P_{1}, P_{1}$ and $P_{2},$ respectively. STATEMENT - 1Z:…2008 · Shift 1 · Q26 · MCQ
Consider the lines, $L_{1} : \frac{x + 1}{3} = \frac{y + 2}{1} = \frac{z + 1}{2} L_{2} : \frac{x - 2}{1} = \frac{y - 2}{2} = \frac{z - 3}{3}$ The distance of the point $(1, 1, 1)$ from the plane passing through the point $(- 1, - 2, - 1)$ …2008 · Shift 2 · Q26 · MCQ