JEE Advanced
2026
MSQ
Let L be the straight line joining the points P(1, 2, –1) and Q(2, 3, 1). Let S be the foot of the perpendicular drawn from the point R(4, –1, 5) to the line L. Another line passing through R intersects L at a point T such that the point S divides the line segment PT internally in the ratio $|PS| : |ST| = 1 : 2$, where $|PS|$ and $|ST|$ are the lengths of the line segments PS and ST, respectively.
Then which of the following statements is (are) TRUE?
JEE Advanced
2026
MSQ
Let P be the plane such that it contains the straight line $\frac{x-1}{2}=\frac{y-3}{3}=\frac{z+2}{1}$ and is perpendicular to the plane $x+2y+3z=4$. Let $P_1$ be the plane which passes through the point $(4,2,2)$ and is parallel to P.
Then which of the following statements is (are) TRUE?
JEE Advanced
2025
MSQ
Let $L_1$ be the line of intersection of the planes given by the equations
$2x + 3y + z = 4$ and $x + 2y + z = 5$.
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 equation
$2x + y - 2z = 6$.
Suppose that the line $L_2$ meets the plane $M$ at the point $Q$. Let $R$ be the foot of the perpendicular drawn from $P$ to the plane $M$.
Then which of the following statements is (are) TRUE?
JEE Advanced
2024
MCQ
Let $\gamma \in \mathbb{R}$ be such that the lines $L_1: \frac{x+11}{1}=\frac{y+21}{2}=\frac{z+29}{3}$ and $L_2: \frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{\gamma}$ intersect. Let $R_1$ be the point of intersection of $L_1$ and $L_2$. Let $O=(0,0,0)$, and $\hat{n}$ denote a unit normal vector to the plane containing both the lines $L_1$ and $L_2$.
Match each entry in List-I to the correct entry in List-II.
| List-I |
List-II |
| (P) $\gamma$ equals |
(1) $-\hat{i} - \hat{j} + \hat{k}$ |
| (Q) A possible choice for $\hat{n}$ is |
(2) $\sqrt{\frac{3}{2}}$ |
| (R) $\overrightarrow{OR_1}$ equals |
(3) $1$ |
| (S) A possible value of $\overrightarrow{OR_1} \cdot \hat{n}$ is |
(4) $\frac{1}{\sqrt{6}} \hat{i} - \frac{2}{\sqrt{6}} \hat{j} + \frac{1}{\sqrt{6}} \hat{k}$ |
|
(5) $\sqrt{\frac{2}{3}}$ |
The correct option is :
JEE Advanced
2024
MSQ
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 to the plane $L_1$ intersects the plane $L_2: 2 x-y+z=-4$ at the point $R$. Then which of the following statements is (are) TRUE?
JEE Advanced
2024
MSQ
Let $\mathbb{R}^3$ denote the three-dimensional space. Take two points $P=(1,2,3)$ and $Q=(4,2,7)$. Let $\operatorname{dist}(X, Y)$ denote the distance between two points $X$ and $Y$ in $\mathbb{R}^3$. Let
$ \begin{gathered} S=\left\{X \in \mathbb{R}^3:(\operatorname{dist}(X, P))^2-(\operatorname{dist}(X, Q))^2=50\right\} \text { and } \\ T=\left\{Y \in \mathbb{R}^3:(\operatorname{dist}(Y, Q))^2-(\operatorname{dist}(Y, P))^2=50\right\} . \end{gathered} $
Then which of the following statements is (are) TRUE?
JEE Advanced
2023
MCQ
Let $\ell_1$ and $\ell_2$ be the lines $\vec{r}_1=\lambda(\hat{i}+\hat{j}+\hat{k})$ and $\vec{r}_2=(\hat{j}-\hat{k})+\mu(\hat{i}+\hat{k})$, respectively. Let $X$ be the set of all the planes $H$ that contain the line $\ell_1$. For a plane $H$, let $d(H)$ denote the smallest possible distance between the points of $\ell_2$ and $H$. Let $H_0$ be a plane in $X$ for which $d\left(H_0\right)$ is the maximum value of $d(H)$ as $H$ varies over all planes in $X$.
Match each entry in
List-I to the correct entries in
List-II.
| List - I |
List - II |
| (P) The value of $d\left(H_0\right)$ is |
(1) $\sqrt{3}$ |
| (Q) The distance of the point $(0,1,2)$ from $H_0$ is |
(2) $\frac{1}{\sqrt{3}}$ |
| (R) The distance of origin from $H_0$ is |
(3) 0 |
| (S) The distance of origin from the point of intersection of planes $y=z, x=1$ and $H_0$ is |
(4) $\sqrt{2}$ |
|
(5) $\frac{1}{\sqrt{2}}$ |
The correct option is:
JEE Advanced
2022
MSQ
Let $P_{1}$ and $P_{2}$ be two planes given by
$
\begin{aligned}
&P_{1}: 10 x+15 y+12 z-60=0 \\\\
&P_{2}:-2 x+5 y+4 z-20=0
\end{aligned}
$
Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P_{1}$ and $P_{2}$ ?
JEE Advanced
2022
MSQ
Let $S$ be the reflection of a point $Q$ with respect to the plane given by
$
\vec{r}=-(t+p) \hat{\imath}+t \hat{\jmath}+(1+p) \hat{k}
$
where $t, p$ are real parameters and $\hat{\imath}, \hat{\jmath}, \hat{k}$ are the unit vectors along the three positive coordinate axes. If the position vectors of $Q$ and $S$ are $10 \hat{\imath}+15 \hat{\jmath}+20 \hat{k}$ and $\alpha \hat{\imath}+\beta \hat{\jmath}+\gamma \hat{k}$ respectively, then which of the following is/are TRUE ?
JEE Advanced
2020
MSQ
Let $\alpha $2 + $\beta $2 + $\gamma $2 $ \ne $ 0 and $\alpha $ + $\gamma $ = 1. Suppose the point (3, 2, $-$1) is the mirror image of the point (1, 0, $-$1) with respect to the plane $\alpha $x + $\beta $y + $\gamma $z = $\delta $. Then which of the following statements is/are TRUE?
JEE Advanced
2020
MSQ
Let L1 and L2 be the following straight lines.
${L_1}:{{x - 1} \over 1} = {y \over { - 1}} = {{z - 1} \over 3}$ and ${L_2}:{{x - 1} \over { - 3}} = {y \over { - 1}} = {{z - 1} \over 1}$.
Suppose the straight line
$L:{{x - \alpha } \over l} = {{y - 1} \over m} = {{z - \gamma } \over { - 2}}$
lies in the plane containing L1 and L2 and passes through the point of intersection of L1 and L2. If the line L bisects the acute angle between the lines L1 and L2, then which of the following statements is/are TRUE?
JEE Advanced
2019
MSQ
Three lines ${L_1}:r = \lambda \widehat i$, $\lambda $ $ \in $ R,
${L_2}:r = \widehat k + \mu \widehat j$, $\mu $ $ \in $ R and
${L_3}:r = \widehat i + \widehat j + v\widehat k$, v $ \in $ R are given.
For which point(s) Q on L2 can we find a point P on L1 and a point R on L3 so that P, Q and R are collinear?
JEE Advanced
2019
MSQ
Let L1 and L2 denote the lines
$r = \widehat i + \lambda ( - \widehat i + 2\widehat j + 2\widehat k)$, $\lambda $$ \in $ R
and $r = \mu (2\widehat i - \widehat j + 2\widehat k),\,\mu \in R$
respectively. If L3 is a line which is perpendicular to both L1 and L2 and cuts both of them, then which of the following options describe(s) L3?
JEE Advanced
2018
MSQ
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?
JEE Advanced
2017
MCQ
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 is
JEE Advanced
2016
MCQ
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 ${x \over 1} = {y \over 2} = {z \over 1}$ is
JEE Advanced
2016
MSQ
Consider a pyramid $OPQRS$ located in the first octant $\left( {x \ge 0,y \ge 0,z \ge 0} \right)$ with $O$ as origin, and $OP$ and $OR$ along the $x$-axis and the $y$-axis, respectively. The base $OPQR$ of the pyramid is a square with $OP=3.$ The point $S$ is directly above the mid-point, $T$ of diagonal $OQ$ such that $TS=3.$ Then
JEE Advanced
2015
MSQ
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 + 2y - z + 1 = 0$ and ${P_2}:2x - y + z - 1 = 0.$ Let $M$ be the locus of the feet of the perpendiculars drawn from the points on $L$ to the plane ${P_1}.$ Which of the following points lie (s) on $M$?
JEE Advanced
2015
MSQ
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 ${P_3}$ is $1$ and the distance of a point $\left( {\alpha ,\beta ,\gamma } \right)$ from ${P_3}$ is $2,$ then which of the following relations is (are) true?
JEE Advanced
2014
MCQ
From a point $P\left( {\lambda ,\lambda ,\lambda } \right),$ perpendicular $PQ$ and $PR$ are drawn respectively on the lines $y=x, z=1$ and $y=-x, z=-1.$ If $P$ is such that $\angle QPR$ is a right angle, then the possible value(s) of $\lambda $ is/(are)
JEE Advanced
2013
MCQ
Consider the lines
${L_1}:{{x - 1} \over 2} = {y \over { - 1}} = {{z + 3} \over 1},{L_2} : {{x - 4} \over 1} = {{y + 3} \over 1} = {{z + 3} \over 2}$
and the planes ${P_1}:7x + y + 2z = 3,{P_2} = 3x + 5y - 6z = 4.$ Let $ax+by+cz=d$ be the equation of the plane passing through the point of intersection of lines ${L_1}$ and ${L_2},$ and perpendicular to planes ${P_1}$ and ${P_2}.$
Match List $I$ with List $II$ and select the correct answer using the code given below the lists:
List $I$
(P.) $a=$
(Q.) $b=$
(R.) $c=$
(S.) $d=$
List $II$
(1.) $13$
(2.) $-3$
(3.) $1$
(4.) $-2$
JEE Advanced
2013
MCQ
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 line
JEE Advanced
2013
MSQ
Two lines ${L_1}:x = 5,{y \over {3 - \alpha }} = {z \over { - 2}}$ and ${L_2}:x = \alpha ,{y \over { - 1}} = {z \over {2 - \alpha }}$ are coplanar. Then $\alpha $ can take value(s)
JEE Advanced
2013
MSQ
A line $l$ passing through the origin is perpendicular to the lines
$$\,{l_1}:\left( {3 + t} \right)\widehat i + \left( { - 1 + 2t} \right)\widehat j + \left( {4 + 2t} \right)\widehat k,\,\,\,\,\, - \infty < t < \infty $$
$${l_2}:\left( {3 + 2s} \right)\widehat i + \left( {3 + 2s} \right)\widehat j + \left( {2 + s} \right)\widehat k,\,\,\,\,\, - \infty < s < \infty $$
Then, the coordinate(s) of the points(s) on ${l_2}$ at a distance of $\sqrt {17} $ from the point of intersection of $l$ and ${l_1}$ is (are)
JEE Advanced
2012
MCQ
The equation of a plane passing through the line of intersection of the planes $x+2y+3z=2$ and $x-y+z=3$ and at a distance ${2 \over {\sqrt 3 }}$ from the point $(3, 1, -1)$ is
JEE Advanced
2012
MCQ
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 $5x-4y-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 line segment $PS$ is
JEE Advanced
2012
MSQ
If the straight lines $\,{{x - 1} \over 2} = {{y + 1} \over k} = {z \over 2}$ and ${{x + 1} \over 5} = {{y + 1} \over 2} = {z \over k}$ are coplanar, then the plane (s) containing these two lines is (are)
JEE Advanced
2010
MCQ
Equation of the plane containing the straight line ${x \over 2} = {y \over 3} = {z \over 4}$ and perpendicular to the plane containing the straight lines ${x \over 3} = {y \over 4} = {z \over 2}$ and ${x \over 4} = {y \over 2} = {z \over 3}$ is
JEE Advanced
2010
MCQ
Match the statement in
Column-$I$ with the values in
Column-$II$
$\,\,\,\,$ $\,\,\,\,$ $\,\,\,\,$ Column-$I$
(A)$\,\,\,\,$ A line from the origin meets the lines $\,{{x - 2} \over 1} = {{y - 1} \over { - 2}} = {{z + 1} \over 1}$
and ${{x - {8 \over 3}} \over 2} = {{y + 3} \over { - 1}} = {{z - 1} \over 1}$ at $P$ and $Q$ respectively. If length $PQ=d,$ then ${d^2}$ is
(B)$\,\,\,\,$ The values of $x$ satisfying ${\tan ^{ - 1}}\left( {x + 3} \right) - {\tan ^{ - 1}}\left( {x - 3} \right) = {\sin ^{ - 1}}\left( {{3 \over 5}} \right)$ are
(C)$\,\,\,\,$ Non-zero vectors $\overrightarrow a ,\overrightarrow b $ and $\overrightarrow c \,\,$ satisfy $\overrightarrow a \,.\,\overrightarrow b \, = 0.$
$\left( {\overrightarrow b - \overrightarrow a } \right).\left( {\overrightarrow b + \overrightarrow c } \right) = 0$ and $2\left| {\overrightarrow b + \overrightarrow c } \right| = \left| {\overrightarrow b - \overrightarrow a } \right|.$
If $\overrightarrow a = \mu \overrightarrow b + 4\overrightarrow c \,\,,$ then the possible values of $\mu $ are
(D)$\,\,\,\,$ Let $f$ be the function on $\left[ { - \pi ,\pi } \right]$ given by $f(0)=9$
and $f\left( x \right) = \sin \left( {{{9x} \over 2}} \right)/\sin \left( {{x \over 2}} \right)$ for $x \ne 0$
The value of ${2 \over \pi }\int_{ - \pi }^\pi {f\left( x \right)dx} $ is
$\,\,\,\,$ $\,\,\,\,$ $\,\,\,\,$Column-$II$
(p)$\,\,\,\,$ $-4$
(q)$\,\,\,\,$ $0$
(r)$\,\,\,\,$ $4$
(s)$\,\,\,\,$ $5$
(t)$\,\,\,\,$ $6$
JEE Advanced
2010
MCQ
If the distance of the point $P(1, -2, 1)$ from the plane $x+2y-2z$$\, = \alpha ,$ where $\alpha > 0,$ is $5,$ then the foot of the perpendicular from $P$ to the planes is
JEE Advanced
2009
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 $2x + y + z = 9$ at point Q. The length of the line segment PQ equals
JEE Advanced
2009
MCQ
Let $P(3,2,6)$ be a point in space and $Q$ be a point on the line
$$\widehat r = \left( {\widehat i - \widehat j + 2\widehat k} \right) + \mu \left( { - 3\widehat i + \widehat j + 5\widehat k} \right)$$
Then the value of $\mu $ for which the vector ${\overrightarrow {PQ} }$ is parallel to the plane $x - 4y + 3z = 1$ is :
JEE Advanced
2008
MCQ
The distance of the point $(1, 1, 1)$ from the plane passing through the point $(-1, -2, -1)$ and whose normal is perpendicular to both the lines ${L_1}$ and ${L_2}$ is :
JEE Advanced
2008
MCQ
Consider three planes
$${P_1}:x - y + z = 1$$
$${P_2}:x + y - z = 1$$
$${P_3}:x - 3y + 3z = 2$$
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: At least two of the lines ${L_1},$ ${L_2}$ and ${L_3}$ are non-parallel and
STATEMENT - 2: The three planes doe not have a common point.
JEE Advanced
2007
MCQ
Consider the planes $3x-6y-2z=15$ and $2x+y-2z=5.$
STATEMENT-1: The parametric equations of the line of intersection of the given planes are $x=3+14t,y=1+2t,z=15t.$ because
STATEMENT-2: The vector ${14\widehat i + 2\widehat j + 15\widehat k}$ is parallel to the line of intersection of given planes.
JEE Advanced
2007
MCQ
Consider the planes $3 x-6 y-2 z=15$ and $2 x+y-2 z=5$.
STATEMENT - 1 : The parametric equations of the line of intersection of the given planes are $x=3+14 t, y=1+2 t, z=15 t$
STATEMENT - 2 : The vectors $14 \hat{i}+2 \hat{j}+15 \hat{k}$ is parallel to the line of intersection of the given planes.
JEE Advanced
2006
MCQ
Match the following:
| (i) |
$\sum\limits_{i = 1}^\infty {{{\tan }^{ - 1}}\left( {{1 \over {2{i^2}}}} \right) = t} $ then $\tan t=$ |
(A) |
0 |
| (ii) |
Sides $a,b,c$ of a triangle ABC are in AP and $\cos {\theta _1} = {a \over {b + c}},\cos {\theta _2} = {b \over {a + c}},\cos {\theta _3} = {c \over {a + b}}$, then ${\tan ^2}\left( {{{{\theta _1}} \over 2}} \right) + {\tan ^2}\left( {{{{\theta _3}} \over 2}} \right) = $ |
(B) |
1 |
| (iii) |
A line is perpendicular to $x + 2y + 2z = 0$ and passes through (0, 1, 0). The perpendicular distance of this line from the origin is |
(C) |
${{\sqrt 5 } \over 3}$ |
|
|
(D) |
2/3 |
JEE Advanced
2006
MCQ
A plane passes through $(1,-2,1)$ and is perpendicular to two planes $2 x-2 y+z=0$ and $x-y+2 z=4$. The distance of the plane from the point $(1,2,2)$ is:
JEE Advanced
2006
MSQ
Let ${\overrightarrow A }$ be vector parallel to line of intersection of planes ${P_1}$ and ${P_2}.$ Planes ${P_1}$ is parallel to the vectors $2\widehat j + 3\widehat k$ and $4\widehat j - 3\widehat k$ and that ${P_2}$ is parallel to $\widehat j - \widehat k$ and $3\widehat i + 3\widehat j,$ then the angle between vector ${\overrightarrow A }$ and a given vector $2\widehat i + \widehat j - 2\widehat k$ is
JEE Advanced
2005
MCQ
A variable plane at a distance of the one unit from the origin cuts the coordinates axes at $A,$ $B$ and $C.$ If the centroid $D$ $(x, y, z)$ of triangle $ABC$ satisfies the relation ${1 \over {{x^2}}} + {1 \over {{y^2}}} + {1 \over {{z^2}}} = k,$ then the value $k$ is
JEE Advanced
2005
MCQ
Find the equation of the plane containing the line $2 x-y+z-3=0,3 x+y+z=5$ and at a distance of $\frac{1}{\sqrt{6}}$ from the point $(2,1,-1)$.
JEE Advanced
2004
MCQ
If the lines ${{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over 4}$ and $\,{{x - 3} \over 1} = {{y - k} \over 2} = {z \over 1}$ intersect, then the value of $k$ is
JEE Advanced
2003
MCQ
The value of $k$ such that ${{x - 4} \over 1} = {{y - 2} \over 1} = {{z - k} \over 2}$ lies in the plane $2x -4y +z = 7,$ is
JEE Advanced
1994
MCQ
Let $\overrightarrow p $ and $\overrightarrow q $ be the position vectors of $P$ and $Q$ respectively, with respect to $O$ and $\left| {\overrightarrow p } \right| = p,\left| {\overrightarrow q } \right| = q.$ The points $R$ and $S$ divide $PQ$ internally and externally in the ratio $2:3$ respectively. If $OR$ and $OS$ are perpendicular then
JEE Advanced
1994
MCQ
Let $\alpha ,\beta ,\gamma $ be distinct real numbers. The points with position
vectors $\alpha \widehat i + \beta \widehat j + \gamma \widehat k,\,\,\beta \widehat i + \gamma \widehat j + \alpha \widehat k,\,\,\gamma \widehat i + \alpha \widehat j + \beta \widehat k$
JEE Advanced
1983
MCQ
The points with position vectors $60i+3j,$ $40i-8j,$ $ai-52j$ are collinear if
JEE Advanced
1983
MCQ
The volume of the parallelopiped whose sides are given by
$\overrightarrow {OA} = 2i - 2j,\,\overrightarrow {OB} = i + j - k,\,\overrightarrow {OC} = 3i - k,$ is