# 3D Geometry · Mathematics · JEE Main

Start Practice## MCQ (Single Correct Answer)

JEE Main 2024 (Online) 9th April Evening Shift

Consider the line $$\mathrm{L}$$ passing through the points $$(1,2,3)$$ and $$(2,3,5)$$. The distance of the point $$\left(\frac{11}{3}, \frac{11}{3},...

JEE Main 2024 (Online) 9th April Morning Shift

The shortest distance between the lines $$\frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5}$$ and $$\frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1}$$ is:

JEE Main 2024 (Online) 9th April Morning Shift

Let the line $$\mathrm{L}$$ intersect the lines $$x-2=-y=z-1,2(x+1)=2(y-1)=z+1$$ and be parallel to the line $$\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{...

JEE Main 2024 (Online) 8th April Evening Shift

If the shortest distance between the lines $$\frac{x-\lambda}{2}=\frac{y-4}{3}=\frac{z-3}{4}$$ and $$\frac{x-2}{4}=\frac{y-4}{6}=\frac{z-7}{8}$$ is $$...

JEE Main 2024 (Online) 8th April Morning Shift

Let $$P(x, y, z)$$ be a point in the first octant, whose projection in the $$x y$$-plane is the point $$Q$$. Let $$O P=\gamma$$; the angle between $$O...

JEE Main 2024 (Online) 8th April Morning Shift

If the shortest distance between the lines
$$\begin{array}{ll}
L_1: \vec{r}=(2+\lambda) \hat{i}+(1-3 \lambda) \hat{j}+(3+4 \lambda) \hat{k}, & \lambda...

JEE Main 2024 (Online) 6th April Evening Shift

Let $$\mathrm{P}(\alpha, \beta, \gamma)$$ be the image of the point $$\mathrm{Q}(3,-3,1)$$ in the line $$\frac{x-0}{1}=\frac{y-3}{1}=\frac{z-1}{-1}$$ ...

JEE Main 2024 (Online) 6th April Morning Shift

If $$A(3,1,-1), B\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right), C(2,2,1)$$ and $$D\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right)$$ are the ...

JEE Main 2024 (Online) 6th April Morning Shift

The shortest distance between the lines $$\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$$ and $$\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$$ is

JEE Main 2024 (Online) 5th April Evening Shift

Let $$(\alpha, \beta, \gamma)$$ be the image of the point $$(8,5,7)$$ in the line $$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{5}$$. Then $$\alpha+\beta+\...

JEE Main 2024 (Online) 5th April Morning Shift

If the line $$\frac{2-x}{3}=\frac{3 y-2}{4 \lambda+1}=4-z$$ makes a right angle with the line $$\frac{x+3}{3 \mu}=\frac{1-2 y}{6}=\frac{5-z}{7}$$, the...

JEE Main 2024 (Online) 5th April Morning Shift

Let $$\mathrm{d}$$ be the distance of the point of intersection of the lines $$\frac{x+6}{3}=\frac{y}{2}=\frac{z+1}{1}$$ and $$\frac{x-7}{4}=\frac{y-9...

JEE Main 2024 (Online) 4th April Evening Shift

Let $$\mathrm{P}$$ be the point of intersection of the lines $$\frac{x-2}{1}=\frac{y-4}{5}=\frac{z-2}{1}$$ and $$\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-3...

JEE Main 2024 (Online) 4th April Morning Shift

Let the point, on the line passing through the points $$P(1,-2,3)$$ and $$Q(5,-4,7)$$, farther from the origin and at a distance of 9 units from the p...

JEE Main 2024 (Online) 1st February Evening Shift

Consider a $\triangle A B C$ where $A(1,3,2), B(-2,8,0)$ and $C(3,6,7)$. If the angle bisector of $\angle B A C$ meets
the line $B C$ at $D$, then the...

JEE Main 2024 (Online) 1st February Evening Shift

Let $\mathrm{P}$ and $\mathrm{Q}$ be the points on the line $\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2}$ which are at a distance of 6 units from the po...

JEE Main 2024 (Online) 1st February Evening Shift

If the mirror image of the point $P(3,4,9)$ in the line
$\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-2}{1}$ is $(\alpha, \beta, \gamma)$, then 14 $(\alpha+\b...

JEE Main 2024 (Online) 1st February Morning Shift

If the shortest distance between the lines $\frac{x-\lambda}{-2}=\frac{y-2}{1}=\frac{z-1}{1}$ and $\frac{x-\sqrt{3}}{1}=\frac{y-1}{-2}=\frac{z-2}{1}$ ...

JEE Main 2024 (Online) 31st January Evening Shift

Let $$(\alpha, \beta, \gamma)$$ be the mirror image of the point $$(2,3,5)$$ in the line $$\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$$. Then, $$2 \alp...

JEE Main 2024 (Online) 31st January Evening Shift

The shortest distance, between lines $$L_1$$ and $$L_2$$, where $$L_1: \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+4}{2}$$ and $$L_2$$ is the line, passing t...

JEE Main 2024 (Online) 30th January Evening Shift

Let $$L_1: \vec{r}=(\hat{i}-\hat{j}+2 \hat{k})+\lambda(\hat{i}-\hat{j}+2 \hat{k}), \lambda \in \mathbb{R}$$,
$$L_2: \vec{r}=(\hat{j}-\hat{k})+\mu(3 \h...

JEE Main 2024 (Online) 30th January Morning Shift

Let $$(\alpha, \beta, \gamma)$$ be the foot of perpendicular from the point $$(1,2,3)$$ on the line $$\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$$. The...

JEE Main 2024 (Online) 30th January Morning Shift

Let $$A(2,3,5)$$ and $$C(-3,4,-2)$$ be opposite vertices of a parallelogram $$A B C D$$. If the diagonal $$\overrightarrow{\mathrm{BD}}=\hat{i}+2 \hat...

JEE Main 2024 (Online) 29th January Evening Shift

Let $$\mathrm{P}(3,2,3), \mathrm{Q}(4,6,2)$$ and $$\mathrm{R}(7,3,2)$$ be the vertices of $$\triangle \mathrm{PQR}$$. Then, the angle $$\angle \mathrm...

JEE Main 2024 (Online) 29th January Morning Shift

Let $$O$$ be the origin and the position vectors of $$A$$ and $$B$$ be $$2 \hat{i}+2 \hat{j}+\hat{k}$$ and $$2 \hat{i}+4 \hat{j}+4 \hat{k}$$ respectiv...

JEE Main 2024 (Online) 29th January Morning Shift

Let $$P Q R$$ be a triangle with $$R(-1,4,2)$$. Suppose $$M(2,1,2)$$ is the mid point of $$\mathrm{PQ}$$. The distance of the centroid of $$\triangle ...

JEE Main 2024 (Online) 27th January Evening Shift

Let the image of the point $$(1,0,7)$$ in the line $$\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$$ be the point $$(\alpha, \beta, \gamma)$$. Then which on...

JEE Main 2024 (Online) 27th January Morning Shift

The distance, of the point $(7,-2,11)$ from the line $\frac{x-6}{1}=\frac{y-4}{0}=\frac{z-8}{3}$ along the line $\frac{x-5}{2}=\frac{y-1}{-3}=\frac{z-...

JEE Main 2024 (Online) 27th January Morning Shift

If the shortest distance between the lines $\frac{x-4}{1}=\frac{y+1}{2}=\frac{z}{-3}$ and $\frac{x-\lambda}{2}=\frac{y+1}{4}=\frac{z-2}{-5}$ is $\frac...

JEE Main 2023 (Online) 15th April Morning Shift

Let the foot of perpendicular of the point $P(3,-2,-9)$ on the plane passing through the points $(-1,-2,-3),(9,3,4),(9,-2,1)$ be $Q(\alpha, \beta, \ga...

JEE Main 2023 (Online) 15th April Morning Shift

Let the system of linear equations
$-x+2 y-9 z=7$
$-x+3 y+7 z=9$
$-2 x+y+5 z=8$
$-3 x+y+13 z=\lambda$
has a unique solution $x=\alpha, y=\beta, z=\ga...

JEE Main 2023 (Online) 15th April Morning Shift

Let $\mathrm{S}$ be the set of all values of $\lambda$, for which the shortest distance between the lines $\frac{x-\lambda}{0}=\frac{y-3}{4}=\frac{z+6...

JEE Main 2023 (Online) 13th April Evening Shift

The line, that is coplanar to the line $$\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$$, is :

JEE Main 2023 (Online) 13th April Evening Shift

The plane, passing through the points $$(0,-1,2)$$ and $$(-1,2,1)$$ and parallel to the line passing through $$(5,1,-7)$$ and $$(1,-1,-1)$$, also pass...

JEE Main 2023 (Online) 13th April Evening Shift

Let $$\mathrm{N}$$ be the foot of perpendicular from the point $$\mathrm{P}(1,-2,3)$$ on the line passing through the points $$(4,5,8)$$ and $$(1,-7,5...

JEE Main 2023 (Online) 13th April Morning Shift

Let the equation of plane passing through the line of intersection of the planes $$x+2 y+a z=2$$ and $$x-y+z=3$$ be $$5 x-11 y+b z=6 a-1$$. For $$c \i...

JEE Main 2023 (Online) 13th April Morning Shift

The distance of the point $$(-1,2,3)$$ from the plane $$\vec{r} \cdot(\hat{i}-2 \hat{j}+3 \hat{k})=10$$ parallel to the line of the shortest distance ...

JEE Main 2023 (Online) 12th April Morning Shift

Let the lines $$l_{1}: \frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha}{-2}$$ and $$l_{2}: 3 x+2 y+z-2=0=x-3 y+2 z-13$$ be coplanar. If the point $$\mathrm...

JEE Main 2023 (Online) 12th April Morning Shift

Let the plane P: $$4 x-y+z=10$$ be rotated by an angle $$\frac{\pi}{2}$$ about its line of intersection with the plane $$x+y-z=4$$. If $$\alpha$$ is t...

JEE Main 2023 (Online) 11th April Evening Shift

Let the line passing through the points $$\mathrm{P}(2,-1,2)$$ and $$\mathrm{Q}(5,3,4)$$ meet the plane $$x-y+z=4$$ at the point $$\mathrm{R}$$. Then ...

JEE Main 2023 (Online) 11th April Evening Shift

Let P be the plane passing through the points $$(5,3,0),(13,3,-2)$$ and $$(1,6,2)$$.
For $$\alpha \in \mathbb{N}$$, if the distances of the points $$...

JEE Main 2023 (Online) 11th April Morning Shift

Let $$(\alpha, \beta, \gamma)$$ be the image of the point $$\mathrm{P}(2,3,5)$$ in the plane $$2 x+y-3 z=6$$. Then $$\alpha+\beta+\gamma$$ is equal to...

JEE Main 2023 (Online) 11th April Morning Shift

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 $$...

JEE Main 2023 (Online) 10th April Evening Shift

Let the image of the point $$\mathrm{P}(1,2,6)$$ in the plane passing through the points $$\mathrm{A}(1,2,0), \mathrm{B}(1,4,1)$$ and $$\mathrm{C}(0,5...

JEE Main 2023 (Online) 10th April Evening Shift

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...

JEE Main 2023 (Online) 10th April Morning Shift

The shortest distance between the lines $${{x + 2} \over 1} = {y \over { - 2}} = {{z - 5} \over 2}$$ and $${{x - 4} \over 1} = {{y - 1} \over 2} = {{z...

JEE Main 2023 (Online) 10th April Morning Shift

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 p...

JEE Main 2023 (Online) 10th April Morning Shift

Let P be the point of intersection of the line $${{x + 3} \over 3} = {{y + 2} \over 1} = {{1 - z} \over 2}$$ and the plane $$x+y+z=2$$. If the distanc...

JEE Main 2023 (Online) 8th April Evening Shift

For $$\mathrm{a}, \mathrm{b} \in \mathbb{Z}$$ and $$|\mathrm{a}-\mathrm{b}| \leq 10$$, let the angle between the plane $$\mathrm{P}: \mathrm{ax}+y-\ma...

JEE Main 2023 (Online) 8th April Evening Shift

Let $$\mathrm{P}$$ 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 th...

JEE Main 2023 (Online) 8th April Morning Shift

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 :

JEE Main 2023 (Online) 8th April Morning Shift

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 $\vec{r}=(\hat{i}-\hat{j})+\lambda(\hat{i}...

JEE Main 2023 (Online) 6th April Evening Shift

A plane P contains the line of intersection of the plane $$\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=6$$ and $$\vec{r} \cdot(2 \hat{i}+3 \hat{j}+4 \hat{k...

JEE Main 2023 (Online) 6th April Evening Shift

Let the line $$\mathrm{L}$$ 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 th...

JEE Main 2023 (Online) 6th April Morning Shift

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...

JEE Main 2023 (Online) 6th April Morning Shift

One vertex of a rectangular parallelopiped is at the origin $$\mathrm{O}$$ and the lengths of its edges along $$x, y$$ and $$z$$ axes are $$3,4$$ and ...

JEE Main 2023 (Online) 1st February Evening Shift

Let the plane P pass through the intersection of the planes $$2x+3y-z=2$$ and $$x+2y+3z=6$$, and be perpendicular to the plane $$2x+y-z+1=0$$. If d is...

JEE Main 2023 (Online) 1st February Morning Shift

The shortest distance between the lines $${{x - 5} \over 1} = {{y - 2} \over 2} = {{z - 4} \over { - 3}}$$ and $${{x + 3} \over 1} = {{y + 5} \over 4}...

JEE Main 2023 (Online) 1st February Morning Shift

Let the image of the point $$P(2,-1,3)$$ in the plane $$x+2 y-z=0$$ be $$Q$$. Then the distance of the plane $$3 x+2 y+z+29=0$$ from the point $$Q$$ i...

JEE Main 2023 (Online) 31st January Evening Shift

Let the plane $\mathrm{P}: 8 x+\alpha_{1} y+\alpha_{2} z+12=0$ be parallel to the line $\mathrm{L}: \frac{x+2}{2}=\frac{y-3}{3}=\frac{z+4}{5}$. If the...

JEE Main 2023 (Online) 31st January Evening Shift

The foot of perpendicular from the origin $\mathrm{O}$ to a plane $\mathrm{P}$ which meets the co-ordinate axes at the points $\mathrm{A}, \mathrm{B},...

JEE Main 2023 (Online) 31st January Evening Shift

Let $P$ be the plane, passing through the point $(1,-1,-5)$ and perpendicular to the line joining the points $(4,1,-3)$ and $(2,4,3)$. Then the distan...

JEE Main 2023 (Online) 31st January Evening Shift

If a point $\mathrm{P}(\alpha, \beta, \gamma)$ satisfying
$$\left( {\matrix{
\alpha & \beta & \gamma \cr
} } \right)\left( {\matrix{
2 & ...

JEE Main 2023 (Online) 31st January Morning Shift

Let the shortest distance between the lines
$$L: \frac{x-5}{-2}=\frac{y-\lambda}{0}=\frac{z+\lambda}{1}, \lambda \geq 0$$ and
$$L_{1}: x+1=y-1=4-z$$...

JEE Main 2023 (Online) 30th January Evening Shift

A vector $\vec{v}$ in the first octant is inclined to the $x$-axis at $60^{\circ}$, to the $y$-axis at 45 and to the $z$-axis at an acute angle. If a ...

JEE Main 2023 (Online) 30th January Evening Shift

If a plane passes through the points $(-1, k, 0),(2, k,-1),(1,1,2)$ and is parallel to the line $\frac{x-1}{1}=\frac{2 y+1}{2}=\frac{z+1}{-1}$, then t...

JEE Main 2023 (Online) 30th January Morning Shift

The line $$l_1$$ passes through the point (2, 6, 2) and is perpendicular to the plane $$2x+y-2z=10$$. Then the shortest distance between the line $$l_...

JEE Main 2023 (Online) 29th January Evening Shift

The plane $$2x-y+z=4$$ intersects the line segment joining the points A ($$a,-2,4)$$ and B ($$2,b,-3)$$ at the point C in the ratio 2 : 1 and the dist...

JEE Main 2023 (Online) 29th January Evening Shift

If the lines $${{x - 1} \over 1} = {{y - 2} \over 2} = {{z + 3} \over 1}$$ and $${{x - a} \over 2} = {{y + 2} \over 3} = {{z - 3} \over 1}$$ intersect...

JEE Main 2023 (Online) 29th January Evening Shift

The shortest distance between the lines $${{x - 1} \over 2} = {{y + 8} \over -7} = {{z - 4} \over 5}$$ and $${{x - 1} \over 2} = {{y - 2} \over 1} = {...

JEE Main 2023 (Online) 25th January Evening Shift

The foot of perpendicular of the point (2, 0, 5) on the line $${{x + 1} \over 2} = {{y - 1} \over 5} = {{z + 1} \over { - 1}}$$ is ($$\alpha,\beta,\ga...

JEE Main 2023 (Online) 25th January Evening Shift

The shortest distance between the lines $$x+1=2y=-12z$$ and $$x=y+2=6z-6$$ is :

JEE Main 2023 (Online) 25th January Morning Shift

The distance of the point P(4, 6, $$-$$2) from the line passing through the point ($$-$$3, 2, 3) and parallel to a line with direction ratios 3, 3, $$...

JEE Main 2023 (Online) 25th January Morning Shift

Consider the lines $$L_1$$ and $$L_2$$ given by
$${L_1}:{{x - 1} \over 2} = {{y - 3} \over 1} = {{z - 2} \over 2}$$
$${L_2}:{{x - 2} \over 1} = {{y - ...

JEE Main 2023 (Online) 24th January Evening Shift

If the foot of the perpendicular drawn from (1, 9, 7) to the line passing through the point (3, 2, 1) and parallel to the planes $$x+2y+z=0$$ and $$3y...

JEE Main 2023 (Online) 24th January Evening Shift

Let the plane containing the line of intersection of the planes P1 : $$x+(\lambda+4)y+z=1$$ and P2 : $$2x+y+z=2$$ pass through the points (0, 1, 0) an...

JEE Main 2023 (Online) 24th January Morning Shift

The distance of the point (7, $$-$$3, $$-$$4) from the plane passing through the points (2, $$-$$3, 1), ($$-$$1, 1, $$-$$2) and (3, $$-$$4, 2) is :...

JEE Main 2023 (Online) 24th January Morning Shift

The distance of the point ($$-1,9,-16$$) from the plane $$2x+3y-z=5$$ measured parallel to the line $${{x + 4} \over 3} = {{2 - y} \over 4} = {{z - 3}...

JEE Main 2022 (Online) 29th July Evening Shift

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 $$\an...

JEE Main 2022 (Online) 29th July Evening Shift

If $$(2,3,9),(5,2,1),(1, \lambda, 8)$$ and $$(\lambda, 2,3)$$ are coplanar, then the product of all possible values of $$\lambda$$ is:

JEE Main 2022 (Online) 29th July Morning Shift

If the foot of the perpendicular from the point $$\mathrm{A}(-1,4,3)$$ on the plane $$\mathrm{P}: 2 x+\mathrm{m} y+\mathrm{n} z=4$$, is $$\left(-2, \f...

JEE Main 2022 (Online) 28th July Evening Shift

Let the lines $$\frac{x-1}{\lambda}=\frac{y-2}{1}=\frac{z-3}{2}$$ and $$\frac{x+26}{-2}=\frac{y+18}{3}=\frac{z+28}{\lambda}$$ be coplanar and $$\mathr...

JEE Main 2022 (Online) 28th July Evening Shift

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 ...

JEE Main 2022 (Online) 28th July Morning Shift

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...

JEE Main 2022 (Online) 27th July Evening Shift

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 $$2 \sqr...

JEE Main 2022 (Online) 27th July Evening Shift

If the line of intersection of the planes $$a x+b y=3$$ and $$a x+b y+c z=0$$, a $$>0$$ makes an angle $$30^{\circ}$$ with the plane $$y-z+2=0$$, then...

JEE Main 2022 (Online) 27th July Morning Shift

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...

JEE Main 2022 (Online) 26th July Evening Shift

A vector $$\vec{a}$$ is parallel to the line of intersection of the plane determined by the vectors $$\hat{i}, \hat{i}+\hat{j}$$ and the plane determi...

JEE Main 2022 (Online) 26th July Morning Shift

The length of the perpendicular from the point $$(1,-2,5)$$ on the line passing through $$(1,2,4)$$ and parallel to the line $$x+y-z=0=x-2 y+3 z-5$$ i...

JEE Main 2022 (Online) 25th July Evening Shift

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...

JEE Main 2022 (Online) 25th July Evening Shift

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 :

JEE Main 2022 (Online) 25th July Morning Shift

Let $$\mathrm{P}$$ be the plane containing the straight line $$\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5}$$ and perpendicular to the plane containing...

JEE Main 2022 (Online) 30th June Morning Shift

The distance of the point (3, 2, $$-$$1) from the plane $$3x - y + 4z + 1 = 0$$ along the line $${{2 - x} \over 2} = {{y - 3} \over 2} = {{z + 1} \ove...

JEE Main 2022 (Online) 29th June Evening Shift

Let $${{x - 2} \over 3} = {{y + 1} \over { - 2}} = {{z + 3} \over { - 1}}$$ lie on the plane $$px - qy + z = 5$$, for some p, q $$\in$$ R. The shortes...

JEE Main 2022 (Online) 29th June Evening Shift

Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2y + 2z = 16. Let T be a plane passing through the point Q and contain...

JEE Main 2022 (Online) 29th June Morning Shift

If the mirror image of the point (2, 4, 7) in the plane 3x $$-$$ y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal to :

JEE Main 2022 (Online) 28th June Evening Shift

Let the plane ax + by + cz = d pass through (2, 3, $$-$$5) and is perpendicular to the planes 2x + y $$-$$ 5z = 10 and 3x + 5y $$-$$ 7z = 12. If a, b,...

JEE Main 2022 (Online) 28th June Morning Shift

If two distinct point Q, R lie on the line of intersection of the planes $$ - x + 2y - z = 0$$ and $$3x - 5y + 2z = 0$$ and $$PQ = PR = \sqrt {18} $$ ...

JEE Main 2022 (Online) 28th June Morning Shift

The acute angle between the planes P1 and P2, when P1 and P2 are the planes passing through the intersection of the planes $$5x + 8y + 13z - 29 = 0$$ ...

JEE Main 2022 (Online) 28th June Morning Shift

Let the plane $$P:\overrightarrow r \,.\,\overrightarrow a = d$$ contain the line of intersection of two planes $$\overrightarrow r \,.\,\left( {\wid...

JEE Main 2022 (Online) 27th June Evening Shift

Let the foot of the perpendicular from the point (1, 2, 4) on the line $${{x + 2} \over 4} = {{y - 1} \over 2} = {{z + 1} \over 3}$$ be P. Then the di...

JEE Main 2022 (Online) 27th June Evening Shift

The shortest distance between the lines $${{x - 3} \over 2} = {{y - 2} \over 3} = {{z - 1} \over { - 1}}$$ and $${{x + 3} \over 2} = {{y - 6} \over 1}...

JEE Main 2022 (Online) 27th June Morning Shift

If two straight lines whose direction cosines are given by the relations $$l + m - n = 0$$, $$3{l^2} + {m^2} + cnl = 0$$ are parallel, then the positi...

JEE Main 2022 (Online) 26th June Evening Shift

If the plane $$2x + y - 5z = 0$$ is rotated about its line of intersection with the plane $$3x - y + 4z - 7 = 0$$ by an angle of $${\pi \over 2}$$, t...

JEE Main 2022 (Online) 26th June Evening Shift

If the lines $$\overrightarrow r = \left( {\widehat i - \widehat j + \widehat k} \right) + \lambda \left( {3\widehat j - \widehat k} \right)$$ and $$...

JEE Main 2022 (Online) 26th June Evening Shift

Let $$\overrightarrow a = \widehat i + \widehat j + 2\widehat k$$, $$\overrightarrow b = 2\widehat i - 3\widehat j + \widehat k$$ and $$\overrightar...

JEE Main 2022 (Online) 26th June Morning Shift

If the two lines $${l_1}:{{x - 2} \over 3} = {{y + 1} \over {-2}},\,z = 2$$ and $${l_2}:{{x - 1} \over 1} = {{2y + 3} \over \alpha } = {{z + 5} \over ...

JEE Main 2022 (Online) 26th June Morning Shift

Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x $$-$$ 3y + 5z = 8. If the mirror i...

JEE Main 2022 (Online) 25th June Evening Shift

Let p be the plane passing through the intersection of the planes $$\overrightarrow r \,.\,\left( {\widehat i + 3\widehat j - \widehat k} \right) = 5$...

JEE Main 2022 (Online) 25th June Morning Shift

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), parall...

JEE Main 2022 (Online) 24th June Evening Shift

If the shortest distance between the lines $${{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over \lambda }$$ and $${{x - 2} \over 1} = {{y - 4} \ov...

JEE Main 2022 (Online) 24th June Evening Shift

Let the points on the plane P be equidistant from the points ($$-$$4, 2, 1) and (2, $$-$$2, 3). Then the acute angle between the plane P and the plane...

JEE Main 2021 (Online) 1st September Evening Shift

Let the acute angle bisector of the two planes x $$-$$ 2y $$-$$ 2z + 1 = 0 and 2x $$-$$ 3y $$-$$ 6z + 1 = 0 be the plane P. Then which of the followin...

JEE Main 2021 (Online) 1st September Evening Shift

The distance of line $$3y - 2z - 1 = 0 = 3x - z + 4$$ from the point (2, $$-$$1, 6) is :

JEE Main 2021 (Online) 31st August Evening Shift

The distance of the point ($$-$$1, 2, $$-$$2) from the line of intersection of the planes 2x + 3y + 2z = 0 and x $$-$$ 2y + z = 0 is :

JEE Main 2021 (Online) 31st August Morning Shift

Let the equation of the plane, that passes through the point (1, 4, $$-$$3) and contains the line of intersection of the planes 3x $$-$$ 2y + 4z $$-$$...

JEE Main 2021 (Online) 27th August Evening Shift

The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m $$-$$ n = 0 and mn + nl + lm = 0, is :

JEE Main 2021 (Online) 27th August Evening Shift

The equation of the plane passing through the line of intersection of the planes $$\overrightarrow r .\left( {\widehat i + \widehat j + \widehat k} \r...

JEE Main 2021 (Online) 27th August Morning Shift

The distance of the point (1, $$-$$2, 3) from the plane x $$-$$ y + z = 5 measured parallel to a line, whose direction ratios are 2, 3, $$-$$6 is :

JEE Main 2021 (Online) 27th August Morning Shift

Equation of a plane at a distance $$\sqrt {{2 \over {21}}} $$ from the origin, which contains the line of intersection of the planes x $$-$$ y $$-$$ z...

JEE Main 2021 (Online) 26th August Evening Shift

Let P be the plane passing through the point (1, 2, 3) and the line of intersection of the planes $$\overrightarrow r \,.\,\left( {\widehat i + \wideh...

JEE Main 2021 (Online) 26th August Morning Shift

A plane P contains the line $$x + 2y + 3z + 1 = 0 = x - y - z - 6$$, and is perpendicular to the plane $$ - 2x + y + z + 8 = 0$$. Then which of the f...

JEE Main 2021 (Online) 27th July Evening Shift

For real numbers $$\alpha$$ and $$\beta$$ $$\ne$$ 0, if the point of intersection of the straight lines$${{x - \alpha } \over 1} = {{y - 1} \over 2} =...

JEE Main 2021 (Online) 27th July Morning Shift

Let the plane passing through the point ($$-$$1, 0, $$-$$2) and perpendicular to each of the planes 2x + y $$-$$ z = 2 and x $$-$$ y $$-$$ z = 3 be ax...

JEE Main 2021 (Online) 25th July Morning Shift

Let the foot of perpendicular from a point P(1, 2, $$-$$1) to the straight line $$L:{x \over 1} = {y \over 0} = {z \over { - 1}}$$ be N. Let a line be...

JEE Main 2021 (Online) 22th July Evening Shift

Let L be the line of intersection of planes $$\overrightarrow r .(\widehat i - \widehat j + 2\widehat k) = 2$$ and $$\overrightarrow r .(2\widehat i +...

JEE Main 2021 (Online) 22th July Evening Shift

If the shortest distance between the straight lines $$3(x - 1) = 6(y - 2) = 2(z - 1)$$ and $$4(x - 2) = 2(y - \lambda ) = (z - 3),\lambda \in R$$ is ...

JEE Main 2021 (Online) 20th July Evening Shift

The lines x = ay $$-$$ 1 = z $$-$$ 2 and x = 3y $$-$$ 2 = bz $$-$$ 2, (ab $$\ne$$ 0) are coplanar, if :

JEE Main 2021 (Online) 20th July Evening Shift

Consider the line L given by the equation $${{x - 3} \over 2} = {{y - 1} \over 1} = {{z - 2} \over 1}$$. Let Q be the mirror image of the point (2, 3,...

JEE Main 2021 (Online) 17th March Evening Shift

If the equation of plane passing through the mirror image of a point (2, 3, 1) with respect to line $${{x + 1} \over 2} = {{y - 3} \over 1} = {{z + 2}...

JEE Main 2021 (Online) 17th March Morning Shift

The equation of the plane which contains the y-axis and passes through the point (1, 2, 3) is :

JEE Main 2021 (Online) 16th March Evening Shift

If the foot of the perpendicular from point (4, 3, 8) on the line $${L_1}:{{x - a} \over l} = {{y - 2} \over 3} = {{z - b} \over 4}$$, l $$\ne$$ 0 is ...

JEE Main 2021 (Online) 16th March Evening Shift

If (x, y, z) be an arbitrary point lying on a plane P which passes through the points (42, 0, 0), (0, 42, 0) and (0, 0, 42), then the value of the exp...

JEE Main 2021 (Online) 16th March Morning Shift

Let the position vectors of two points P and Q be 3$$\widehat i$$ $$-$$ $$\widehat j$$ + 2$$\widehat k$$ and $$\widehat i$$ + 2$$\widehat j$$ $$-$$ 4$...

JEE Main 2021 (Online) 16th March Morning Shift

Let P be a plane lx + my + nz = 0 containing the line, $${{1 - x} \over 1} = {{y + 4} \over 2} = {{z + 2} \over 3}$$. If plane P divides the line segm...

JEE Main 2021 (Online) 16th March Morning Shift

If for a > 0, the feet of perpendiculars from the points A(a, $$-$$2a, 3) and B(0, 4, 5) on the plane lx + my + nz = 0 are points C(0, $$-$$a, $$-$...

JEE Main 2021 (Online) 26th February Evening Shift

If the mirror image of the point (1, 3, 5) with respect to the plane 4x $$-$$ 5y + 2z = 8 is ($$\alpha$$, $$\beta$$, $$\gamma$$), then 5($$\alpha$$ + ...

JEE Main 2021 (Online) 26th February Evening Shift

Let L be a line obtained from the intersection of two planes x + 2y + z = 6 and y + 2z = 4. If point P($$\alpha$$, $$\beta$$, $$\gamma$$) is the foot ...

JEE Main 2021 (Online) 26th February Morning Shift

Consider the three planesP1 : 3x + 15y + 21z = 9,P2 : x $$-$$ 3y $$-$$ z = 5, and P3 : 2x + 10y + 14z = 5Then, which one of the following is true?...

JEE Main 2021 (Online) 26th February Morning Shift

If (1, 5, 35), (7, 5, 5), (1, $$\lambda$$, 7) and (2$$\lambda$$, 1, 2) are coplanar, then the sum of all possible values of $$\lambda$$ is :

JEE Main 2021 (Online) 25th February Evening Shift

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 $$\overrig...

JEE Main 2021 (Online) 25th February Morning Shift

The equation of the line through the point (0, 1, 2) and perpendicular to the line $${{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over { - 2}}$$ ...

JEE Main 2021 (Online) 25th February Morning Shift

Let $$\alpha$$ be the angle between the lines whose direction cosines satisfy the equations l + m $$-$$ n = 0 and l2 + m2 $$-$$ n2 = 0. Then the value...

JEE Main 2021 (Online) 24th February Evening Shift

Let a, b$$ \in $$R. If the mirror image of the point P(a, 6, 9) with respect to the line $${{x - 3} \over 7} = {{y - 2} \over 5} = {{z - 1} \over { - ...

JEE Main 2021 (Online) 24th February Evening Shift

The vector equation of the plane passing through the intersection of the planes $$\overrightarrow r .\left( {\widehat i + \widehat j + \widehat k} \ri...

JEE Main 2021 (Online) 24th February Morning Shift

The equation of the plane passing through the point (1, 2, -3) and perpendicular to the
planes 3x + y - 2z = 5 and 2x - 5y - z = 7, is :

JEE Main 2021 (Online) 24th February Morning Shift

The distance of the point (1, 1, 9) from the point of intersection of the line
$${{x - 3} \over 1} = {{y - 4} \over 2} = {{z - 5} \over 2}$$
and the p...

JEE Main 2020 (Online) 6th September Evening Slot

A plane P meets the coordinate axes at A, B
and C respectively. The centroid of $$\Delta $$ABC is
given to be (1, 1, 2). Then the equation of the
line...

JEE Main 2020 (Online) 6th September Morning Slot

The shortest distance between the lines
$${{x - 1} \over 0} = {{y + 1} \over { - 1}} = {z \over 1}$$ and x + y + z + 1 = 0, 2x – y + z
+ 3 = 0 is :...

JEE Main 2020 (Online) 5th September Evening Slot

If for some $$\alpha $$ $$ \in $$ R, the lines
L1 : $${{x + 1} \over 2} = {{y - 2} \over { - 1}} = {{z - 1} \over 1}$$ and
L2 : $${{x + 2} \over \alph...

JEE Main 2020 (Online) 5th September Morning Slot

If (a, b, c) is the image of the point (1, 2, -3) in the line $${{x + 1} \over 2} = {{y - 3} \over { - 2}} = {z \over { - 1}}$$, then a + b + c is :...

JEE Main 2020 (Online) 4th September Evening Slot

The distance of the point (1, –2, 3) from the plane x – y + z = 5 measured parallel to the line $${x \over 2} = {y \over 3} = {z \over { - 6}}$$ is :...

JEE Main 2020 (Online) 3rd September Evening Slot

The plane which bisects the line joining, the
points (4, –2, 3) and (2, 4, –1) at right angles
also passes through the point :

JEE Main 2020 (Online) 3rd September Morning Slot

The foot of the perpendicular drawn from the
point (4, 2, 3) to the line joining the points
(1, –2, 3) and (1, 1, 0) lies on the plane :

JEE Main 2020 (Online) 2nd September Evening Slot

A plane passing through the point (3, 1, 1)
contains two lines whose direction ratios are 1,
–2, 2 and 2, 3, –1 respectively. If this plane also
passe...

JEE Main 2020 (Online) 2nd September Morning Slot

The plane passing through the points (1, 2, 1),
(2, 1, 2) and parallel to the line, 2x = 3y, z = 1
also passes through the point :

JEE Main 2020 (Online) 8th January Evening Slot

The mirror image of the point (1, 2, 3) in a plane
is $$\left( { - {7 \over 3}, - {4 \over 3}, - {1 \over 3}} \right)$$. Which of the following
points...

JEE Main 2020 (Online) 8th January Morning Slot

The shortest distance between the lines
$${{x - 3} \over 3} = {{y - 8} \over { - 1}} = {{z - 3} \over 1}$$ and
$${{x + 3} \over { - 3}} = {{y + 7} \ov...

JEE Main 2020 (Online) 7th January Morning Slot

Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point
(2, 1, 6). Then the image of R in the plane P is :

JEE Main 2019 (Online) 12th April Evening Slot

A plane which bisects the angle between the two given planes 2x – y + 2z – 4 = 0 and x + 2y + 2z – 2 = 0,
passes through the point :

JEE Main 2019 (Online) 12th April Evening Slot

The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines
$$\overrightarrow r = \left( {\widehat i + \widehat...

JEE Main 2019 (Online) 12th April Morning Slot

If the line $${{x - 2} \over 3} = {{y + 1} \over 2} = {{z - 1} \over { - 1}}$$
intersects the plane 2x + 3y – z + 13 = 0 at a point P and the plane
3...

JEE Main 2019 (Online) 10th April Evening Slot

If the plane 2x – y + 2z + 3 = 0 has the distances
$${1 \over 3}$$
and
$${2 \over 3}$$
units from the planes 4x – 2y + 4z + $$\lambda $$ = 0 and
2x ...

JEE Main 2019 (Online) 10th April Evening Slot

A perpendicular is drawn from a point on the line $${{x - 1} \over 2} = {{y + 1} \over { - 1}} = {z \over 1}$$ to the plane x + y + z = 3 such that t...

JEE Main 2019 (Online) 10th April Morning Slot

If the length of the perpendicular from the point ($$\beta $$, 0, $$\beta $$) ($$\beta $$ $$ \ne $$ 0) to the line,
$${x \over 1} = {{y - 1} \over 0} ...

JEE Main 2019 (Online) 10th April Morning Slot

If Q(0, –1, –3) is the image of the point P in the plane 3x – y + 4z = 2 and R is the point (3, –1, –2), then the
area (in sq. units) of $$\Delta $$PQ...

JEE Main 2019 (Online) 9th April Evening Slot

The vertices B and C of a $$\Delta $$ABC lie on the line,
$${{x + 2} \over 3} = {{y - 1} \over 0} = {z \over 4}$$ such that BC = 5 units. Then the
ar...

JEE Main 2019 (Online) 9th April Evening Slot

Let P be the plane, which contains the line of
intersection of the planes, x + y + z – 6 = 0 and
2x + 3y + z + 5 = 0 and it is perpendicular to the
xy...

JEE Main 2019 (Online) 9th April Morning Slot

A plane passing through the points (0, –1, 0)
and (0, 0, 1) and making an angle $${\pi \over 4}$$ with the
plane y – z + 5 = 0, also passes through t...

JEE Main 2019 (Online) 9th April Morning Slot

If the line, $${{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 2} \over 4}$$ meets the plane,
x + 2y + 3z = 15 at a point P, then the distance of P from...

JEE Main 2019 (Online) 8th April Evening Slot

The vector equation of the plane through the line
of intersection of the planes x + y + z = 1 and 2x
+ 3y+ 4z = 5 which is perpendicular to the plane
...

JEE Main 2019 (Online) 8th April Evening Slot

If a point R(4, y, z) lies on the line segment joining
the points P(2, –3, 4) and Q(8, 0, 10), then the
distance of R from the origin is :

JEE Main 2019 (Online) 8th April Morning Slot

The length of the perpendicular from the point
(2, –1, 4) on the straight line,
$${{x + 3} \over {10}}$$= $${{y - 2} \over {-7}}$$ = $${{z} \over {1}}...

JEE Main 2019 (Online) 8th April Morning Slot

The magnitude of the projection of the vector
$$\mathop {2i}\limits^ \wedge + \mathop {3j}\limits^ \wedge + \mathop k\limits^ \wedge $$ on the ve...

JEE Main 2019 (Online) 8th April Morning Slot

The equation of a plane containing the line of
intersection of the planes 2x – y – 4 = 0 and
y + 2z – 4 = 0 and passing through the point
(1, 1, 0) is...

JEE Main 2019 (Online) 12th January Evening Slot

Let S be the set of all real values of $$\lambda $$ such that a plane passing through the points (–$$\lambda $$2, 1, 1), (1, –$$\lambda $$2, 1) and (1...

JEE Main 2019 (Online) 12th January Evening Slot

If an angle between the line, $${{x + 1} \over 2} = {{y - 2} \over 1} = {{z - 3} \over { - 2}}$$ and the plane, $$x - 2y - kz = 3$$ is $${\cos ^{ - 1}...

JEE Main 2019 (Online) 12th January Morning Slot

The perpendicular distance from the origin to the plane containing the two lines, $${{x + 2} \over 3} = {{y - 2} \over 5} = {{z + 5} \over 7}$$ and $$...

JEE Main 2019 (Online) 12th January Morning Slot

A tetrahedron has vertices P(1, 2, 1), Q(2, 1, 3), R(–1, 1, 2) and O(0, 0, 0). The angle between the faces OPQ and PQR is :

JEE Main 2019 (Online) 11th January Evening Slot

Two lines $${{x - 3} \over 1} = {{y + 1} \over 3} = {{z - 6} \over { - 1}}$$ and $${{x + 5} \over 7} = {{y - 2} \over { - 6}} = {{z - 3} \over 4}$$ i...

JEE Main 2019 (Online) 11th January Evening Slot

If the point (2, $$\alpha $$, $$\beta $$) lies on the plane which passes through the points (3, 4, 2) and (7, 0, 6) and is perpendicular to the plane ...

JEE Main 2019 (Online) 11th January Morning Slot

The plane containing the line $${{x - 3} \over 2} = {{y + 2} \over { - 1}} = {{z - 1} \over 3}$$ and also containing its projection on the plane 2x +...

JEE Main 2019 (Online) 11th January Morning Slot

The direction ratios of normal to the plane through the points (0, –1, 0) and (0, 0, 1) and making an angle $${\pi \over 4}$$ with the plane y $$-$$ ...

JEE Main 2019 (Online) 10th January Evening Slot

On which of the following lines lies the point of intersection of the line, $${{x - 4} \over 2} = {{y - 5} \over 2} = {{z - 3} \over 1}$$&...

JEE Main 2019 (Online) 10th January Evening Slot

The plane which bisects the line segment joining the points (–3, –3, 4) and (3, 7, 6) at right angles, passes through which one of the following point...

JEE Main 2019 (Online) 10th January Morning Slot

The plane passing through the point (4, –1, 2) and parallel to the lines $${{x + 2} \over 3} = {{y - 2} \over { - 1}} = {{z + 1} \over 2}$$...

JEE Main 2019 (Online) 10th January Morning Slot

Let A be a point on the line $$\overrightarrow r = \left( {1 - 3\mu } \right)\widehat i + \left( {\mu - 1} \right)\widehat j + \left( {2 + 5\mu } \r...

JEE Main 2019 (Online) 9th January Evening Slot

The equation of the plane containing the straight line $${x \over 2} = {y \over 3} = {z \over 4}$$ and perpendicular to the plane containing the strai...

JEE Main 2019 (Online) 9th January Evening Slot

If the lines x = ay + b, z = cy + d and x = a'z + b', y = c'z + d' are perpendicular, then :

JEE Main 2019 (Online) 9th January Morning Slot

The equation of the line passing through (–4, 3, 1), parallel
to the plane x + 2y – z – 5 = 0 and intersecting
the line $${{x + 1} \over { - 3}} = {{...

JEE Main 2019 (Online) 9th January Morning Slot

The plane through the intersection of the planes x + y + z = 1 and 2x + 3y – z + 4 = 0 and parallel to y-axis
also passes through the point :

JEE Main 2018 (Online) 16th April Morning Slot

If the angle between the lines, $${x \over 2} = {y \over 2} = {z \over 1}$$
and $${{5 - x} \over { - 2}} = {{7y - 14} \over p} = {{z - 3} \over 4}\,\...

JEE Main 2018 (Online) 16th April Morning Slot

The sum of the intercepts on the coordinate axes of the plane passing through the point ($$-$$2, $$-2,$$ 2) and containing the line joining the points...

JEE Main 2018 (Offline)

If L1 is the line of intersection of the planes 2x - 2y + 3z - 2 = 0, x - y + z + 1 = 0 and L2 is the line of
intersection of the planes x + 2y - z - ...

JEE Main 2018 (Offline)

The length of the projection of the line segment joining the points (5, -1, 4) and (4, -1, 3) on the plane,
x + y + z = 7 is :

JEE Main 2018 (Online) 15th April Evening Slot

An angle between the lines whose direction cosines are gien by the equations,
$$l$$ + 3m + 5n = 0 and 5$$l$$m $$-$$ 2mn + 6n$$l$$ = 0, is :

JEE Main 2018 (Online) 15th April Evening Slot

A plane bisects the line segment joining the points (1, 2, 3) and ($$-$$ 3, 4, 5) at rigt angles. Then this plane also passes through the point :

JEE Main 2018 (Online) 15th April Morning Slot

A variable plane passes through a fixed point (3,2,1) and meets x, y and z axes at A, B and C respectively. A plane is drawn parallel to yz -plane thr...

JEE Main 2018 (Online) 15th April Morning Slot

An angle between the plane, x + y + z = 5 and the line of intersection of the planes, 3x + 4y + z $$-$$ 1 = 0 and 5x + 8y + 2z + 14 =0, is :

JEE Main 2017 (Online) 9th April Morning Slot

If x = a, y = b, z = c is a solution of the system of linear equations
x + 8y + 7z = 0
9x + 2y + 3z = 0
x + y + z = 0
such that the point (a, b, c) li...

JEE Main 2017 (Online) 9th April Morning Slot

If a variable plane, at a distance of 3 units from the origin, intersects the coordinate
axes at A, B and C, then the locus of the centroid of $$\Delt...

JEE Main 2017 (Online) 9th April Morning Slot

If the line, $${{x - 3} \over 1} = {{y + 2} \over { - 1}} = {{z + \lambda } \over { - 2}}$$ lies in the plane, 2x−4y+3z=2, then the shortest distance ...

JEE Main 2017 (Online) 8th April Morning Slot

The coordinates of the foot of the perpendicular from the point (1, $$-$$2, 1) on the plane containing the lines, $${{x + 1} \over 6} = {{y - 1} \over...

JEE Main 2017 (Online) 8th April Morning Slot

The line of intersection of the planes $$\overrightarrow r .\left( {3\widehat i - \widehat j + \widehat k} \right) = 1\,\,$$ and
$$\overrightarrow r ...

JEE Main 2017 (Offline)

If the image of the point P(1, –2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to the line,
$${x \over 1} = {y \over 4} = {z \over 5}$$ ...

JEE Main 2017 (Offline)

The distance of the point (1, 3, – 7) from the plane passing through the point (1, –1, – 1), having normal
perpendicular to both the lines
$${{x - 1} ...

JEE Main 2016 (Online) 10th April Morning Slot

ABC is a triangle in a plane with vertices
A(2, 3, 5), B(−1, 3, 2) and C($$\lambda $$, 5, $$\mu $$).
If the median through A is equally inclined to ...

JEE Main 2016 (Online) 10th April Morning Slot

The number of distinct real values of $$\lambda $$ for which the lines
$${{x - 1} \over 1} = {{y - 2} \over 2} = {{z + 3} \over {{\lambda ^2}}}$$ an...

JEE Main 2016 (Online) 9th April Morning Slot

The shortest distance between the lines $${x \over 2} = {y \over 2} = {z \over 1}$$ and
$${{x + 2} \over { - 1}} = {{y - 4} \over 8} = {{z - 5} \over...

JEE Main 2016 (Online) 9th April Morning Slot

The distance of the point (1, − 2, 4) from the plane passing through the point
(1, 2, 2) and perpendicular to the planes x − y + 2z = 3 and 2x − 2y + ...

JEE Main 2016 (Offline)

If the line, $${{x - 3} \over 2} = {{y + 2} \over { - 1}} = {{z + 4} \over 3}\,$$ lies in the planes, $$lx+my-z=9,$$ then $${l^2} + {m^2}$$ is equal ...

JEE Main 2016 (Offline)

The distance of the point $$(1,-5,9)$$ from the plane $$x-y+z=5$$ measured along the line $$x=y=z$$ is :

JEE Main 2015 (Offline)

The equation of the plane containing the line $$2x-5y+z=3; x+y+4z=5,$$ and parallel to the plane, $$x+3y+6z=1,$$ is :

JEE Main 2015 (Offline)

The distance of the point $$(1, 0, 2)$$ from the point of intersection of the line $${{x - 2} \over 3} = {{y + 1} \over 4} = {{z - 2} \over {12}}$$ an...

JEE Main 2014 (Offline)

The image of the line $${{x - 1} \over 3} = {{y - 3} \over 1} = {{z - 4} \over { - 5}}\,$$ in the plane $$2x-y+z+3=0$$ is the line :

JEE Main 2014 (Offline)

The angle between the lines whose direction cosines satisfy the equations $$l+m+n=0$$ and $${l^2} = {m^2} + {n^2}$$ is :

JEE Main 2013 (Offline)

Distance between two parallel planes $$2x+y+2z=8$$ and $$4x+2y+4z+5=0$$ is :

JEE Main 2013 (Offline)

If the lines $${{x - 2} \over 1} = {{y - 3} \over 1} = {{z - 4} \over { - k}}$$ and $${{x - 1} \over k} = {{y - 4} \over 2} = {{z - 5} \over 1}$$ are ...

AIEEE 2012

A equation of a plane parallel to the plane $$x-2y+2z-5=0$$ and at a unit distance from the origin is :

AIEEE 2012

If the line $${{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over 4}$$ and $${{x - 3} \over 1} = {{y - k} \over 2} = {z \over 1}$$ intersect, then ...

AIEEE 2011

If the angle between the line $$x = {{y - 1} \over 2} = {{z - 3} \over \lambda }$$ and the plane
$$x+2y+3z=4$$ is $${\cos ^{ - 1}}\left( {\sqrt {{5 \...

AIEEE 2011

Statement - 1 : The point $$A(1,0,7)$$ is the mirror image of the point
$$B(1,6,3)$$ in the line : $${x \over 1} = {{y - 1} \over 2} = {{z - 2} \over...

AIEEE 2010

Statement-1 : The point $$A(3, 1, 6)$$ is the mirror image of the point $$B(1, 3, 4)$$ in the plane $$x-y+z=5.$$
Statement-2 : The plane $$x-y+z=5$$...

AIEEE 2010

A line $$AB$$ in three-dimensional space makes angles $${45^ \circ }$$ and $${120^ \circ }$$ with the positive $$x$$-axis and the positive $$y$$-axis ...

AIEEE 2009

Let the line $$\,\,\,\,\,$$ $${{x - 2} \over 3} = {{y - 1} \over { - 5}} = {{z + 2} \over 2}$$ lie in the plane $$\,\,\,\,\,$$ $$x + 3y - \alpha z + \...

AIEEE 2009

The projections of a vector on the three coordinate axis are $$6,-3,2$$ respectively. The direction cosines of the vector are :

AIEEE 2008

The line passing through the points $$(5,1,a)$$ and $$(3, b, 1)$$ crosses the $$yz$$-plane at the point $$\left( {0,{{17} \over 2}, - {{ - 13} \over ...

AIEEE 2008

If the straight lines $$\,\,\,\,\,$$ $$\,\,\,\,\,$$ $${{x - 1} \over k} = {{y - 2} \over 2} = {{z - 3} \over 3}$$ $$\,\,\,\,\,$$ and$$\,\,\,\,\,$$ $$...

AIEEE 2007

If a line makes an angle of $$\pi /4$$ with the positive directions of each of $$x$$-axis and $$y$$-axis, then the angle that the line makes with the ...

AIEEE 2007

If $$(2,3,5)$$ is one end of a diameter of the sphere $${x^2} + {y^2} + {z^2} - 6x - 12y - 2z + 20 = 0,$$ then the coordinates of the other end of the...

AIEEE 2007

Let $$L$$ be the line of intersection of the planes $$2x+3y+z=1$$ and $$x+3y+2z=2.$$ If $$L$$ makes an angle $$\alpha $$ with the positive $$x$$-axis,...

AIEEE 2006

The image of the point $$(-1, 3,4)$$ in the plane $$x-2y=0$$ is :

AIEEE 2006

The two lines $$x=ay+b, z=cy+d;$$ and $$x=a'y+b' ,$$ $$z=c'y+d'$$ are perpendicular to each other if :

AIEEE 2005

If the angel $$\theta $$ between the line $${{x + 1} \over 1} = {{y - 1} \over 2} = {{z - 2} \over 2}$$ and
the plane $$2x - y + \sqrt \lambda \,\,z...

AIEEE 2005

The distance between the line
$$\overrightarrow r = 2\widehat i - 2\widehat j + 3\widehat k + \lambda \left( {i - j + 4k} \right),$$ and the plane
...

AIEEE 2005

The angle between the lines $$2x=3y=-z$$ and $$6x=-y=-4z$$ is :

AIEEE 2005

If the plane $$2ax-3ay+4az+6=0$$ passes through the midpoint of the line joining the centres of the spheres
$${x^2} + {y^2} + {z^2} + 6x - 8y - 2z = ...

AIEEE 2005

The plane $$x+2y-z=4$$ cuts the sphere $${x^2} + {y^2} + {z^2} - x + z - 2 = 0$$ in a circle of radius

AIEEE 2004

A line makes the same angle $$\theta $$, with each of the $$x$$ and $$z$$ axis.
If the angle $$\beta \,$$, which it makes with y-axis, is such that $...

AIEEE 2004

Distance between two parallel planes
$$\,2x + y + 2z = 8$$ and $$4x + 2y + 4z + 5 = 0$$ is :

AIEEE 2004

The intersection of the spheres
$${x^2} + {y^2} + {z^2} + 7x - 2y - z = 13$$ and
$${x^2} + {y^2} + {z^2} - 3x + 3y + 4z = 8$$
is the same as the int...

AIEEE 2004

A line with direction cosines proportional to $$2,1,2$$ meets each of the lines $$x=y+a=z$$ and $$x+a=2y=2z$$ . The co-ordinates of each of the point...

AIEEE 2004

If the straight lines
$$x=1+s,y=-3$$$$ - \lambda s,$$ $$z = 1 + \lambda s$$ and $$x = {t \over 2},y = 1 + t,z = 2 - t,$$ with parameters $$s$$ and $$...

AIEEE 2003

The shortest distance from the plane $$12x+4y+3z=327$$ to the sphere $${x^2} + {y^2} + {z^2} + 4x - 2y - 6z = 155$$ is

AIEEE 2003

The two lines $$x=ay+b,z=cy+d$$ and $$x = a'y + b',z = c'y + d'$$ will be perpendicular, if and only if :

AIEEE 2003

The lines $${{x - 2} \over 1} = {{y - 3} \over 1} = {{z - 4} \over { - k}}$$ and $${{x - 1} \over k} = {{y - 4} \over 2} = {{z - 5} \over 1}$$ are cop...

AIEEE 2003

The radius of the circle in which the sphere
$${x^2} + {y^2} + {z^2} + 2x - 2y - 4z - 19 = 0$$ is cut by the plane
$$x+2y+2z+7=0$$ is ...

AIEEE 2003

Two systems of rectangular axes have the same origin. If a plane cuts then at distances $$a,b,c$$ and $$a', b', c'$$ from the origin then

AIEEE 2002

A plane which passes through the point $$(3,2,0)$$ and the line
$${{x - 4} \over 1} = {{y - 7} \over 5} = {{z - 4} \over 4}$$ is :

AIEEE 2002

The $$d.r.$$ of normal to the plane through $$(1, 0, 0), (0, 1, 0)$$ which makes an angle $$\pi /4$$ with plane $$x+y=3$$ are :

## Numerical

JEE Main 2024 (Online) 9th April Evening Shift

The square of the distance of the image of the point $$(6,1,5)$$ in the line $$\frac{x-1}{3}=\frac{y}{2}=\frac{z-2}{4}$$, from the origin is _________...

JEE Main 2024 (Online) 8th April Evening Shift

Let $$\mathrm{P}(\alpha, \beta, \gamma)$$ be the image of the point $$\mathrm{Q}(1,6,4)$$ in the line $$\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$$. The...

JEE Main 2024 (Online) 6th April Evening Shift

If the shortest distance between the lines $$\frac{x-\lambda}{3}=\frac{y-2}{-1}=\frac{z-1}{1}$$ and $$\frac{x+2}{-3}=\frac{y+5}{2}=\frac{z-4}{4}$$ is ...

JEE Main 2024 (Online) 6th April Morning Shift

Let $$P$$ be the point $$(10,-2,-1)$$ and $$Q$$ be the foot of the perpendicular drawn from the point $$R(1,7,6)$$ on the line passing through the poi...

JEE Main 2024 (Online) 5th April Evening Shift

Let the point $$(-1, \alpha, \beta)$$ lie on the line of the shortest distance between the lines $$\frac{x+2}{-3}=\frac{y-2}{4}=\frac{z-5}{2}$$ and $$...

JEE Main 2024 (Online) 4th April Evening Shift

Consider a line $$\mathrm{L}$$ passing through the points $$\mathrm{P}(1,2,1)$$ and $$\mathrm{Q}(2,1,-1)$$. If the mirror image of the point $$\mathrm...

JEE Main 2024 (Online) 1st February Morning Shift

Let the line of the shortest distance between the lines
$$
\begin{aligned}
& \mathrm{L}_1: \overrightarrow{\mathrm{r}}=(\hat{i}+2 \hat{j}+3 \hat{k})+\...

JEE Main 2024 (Online) 31st January Evening Shift

A line passes through $$A(4,-6,-2)$$ and $$B(16,-2,4)$$. The point $$P(a, b, c)$$, where $$a, b, c$$ are non-negative integers, on the line $$A B$$ li...

JEE Main 2024 (Online) 31st January Morning Shift

Let $$\mathrm{Q}$$ and $$\mathrm{R}$$ be the feet of perpendiculars from the point $$\mathrm{P}(a, a, a)$$ on the lines $$x=y, z=1$$ and $$x=-y, z=-1$...

JEE Main 2024 (Online) 30th January Evening Shift

Let a line passing through the point $$(-1,2,3)$$ intersect the lines $$L_1: \frac{x-1}{3}=\frac{y-2}{2}=\frac{z+1}{-2}$$ at $$M(\alpha, \beta, \gamma...

JEE Main 2024 (Online) 30th January Morning Shift

If $$\mathrm{d}_1$$ is the shortest distance between the lines $$x+1=2 y=-12 z, x=y+2=6 z-6$$ and $$\mathrm{d}_2$$ is the shortest distance between th...

JEE Main 2024 (Online) 29th January Evening Shift

Let O be the origin, and M and $$\mathrm{N}$$ be the points on the lines $$\frac{x-5}{4}=\frac{y-4}{1}=\frac{z-5}{3}$$ and $$\frac{x+8}{12}=\frac{y+2}...

JEE Main 2024 (Online) 29th January Morning Shift

A line with direction ratios $$2,1,2$$ meets the lines $$x=y+2=z$$ and $$x+2=2 y=2 z$$ respectively at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$. I...

JEE Main 2024 (Online) 27th January Evening Shift

The lines $$\frac{x-2}{2}=\frac{y}{-2}=\frac{z-7}{16}$$ and $$\frac{x+3}{4}=\frac{y+2}{3}=\frac{z+2}{1}$$ intersect at the point $$P$$. If the distanc...

JEE Main 2023 (Online) 15th April Morning Shift

Let the plane $P$ contain the line $2 x+y-z-3=0=5 x-3 y+4 z+9$ and be parallel to the line $\frac{x+2}{2}=\frac{3-y}{-4}=\frac{z-7}{5}$. Then the dist...

JEE Main 2023 (Online) 13th April Morning Shift

Let the image of the point $$\left(\frac{5}{3}, \frac{5}{3}, \frac{8}{3}\right)$$ in the plane $$x-2 y+z-2=0$$ be P. If the distance of the point $$Q(...

JEE Main 2023 (Online) 12th April Morning Shift

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 $$\mathrm{ABC}$$ is $$\left(\alpha...

JEE Main 2023 (Online) 11th April Evening Shift

Let the line $$l: x=\frac{1-y}{-2}=\frac{z-3}{\lambda}, \lambda \in \mathbb{R}$$ meet the plane $$P: x+2 y+3 z=4$$ at the point $$(\alpha, \beta, \gam...

JEE Main 2023 (Online) 11th April Morning Shift

Let a line $$l$$ pass through the origin and be perpendicular to the lines
$$l_{1}: \vec{r}=(\hat{\imath}-11 \hat{\jmath}-7 \hat{k})+\lambda(\hat{i}+2...

JEE Main 2023 (Online) 10th April Evening Shift

Let the foot of perpendicular from the point $$\mathrm{A}(4,3,1)$$ on the plane $$\mathrm{P}: x-y+2 z+3=0$$ be N. If B$$(5, \alpha, \beta), \alpha, \b...

JEE Main 2023 (Online) 8th April Evening Shift

Let $$\mathrm{P}_{1}$$ be the plane $$3 x-y-7 z=11$$ and $$\mathrm{P}_{2}$$ be the plane passing through the points $$(2,-1,0),(2,0,-1)$$, and $$(5,1,...

JEE Main 2023 (Online) 8th April Morning Shift

Let $$\lambda_{1}, \lambda_{2}$$ be the values of $$\lambda$$ for which the points $$\left(\frac{5}{2}, 1, \lambda\right)$$ and $$(-2,0,1)$$ are at eq...

JEE Main 2023 (Online) 6th April Evening Shift

If the lines $$\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha}$$ and $$\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta}$$ intersect, then the magnitude of ...

JEE Main 2023 (Online) 6th April Morning Shift

Let the image of the point $$\mathrm{P}(1,2,3)$$ in the plane $$2 x-y+z=9$$ be $$\mathrm{Q}$$. If the coordinates of the point $$\mathrm{R}$$ are $$(6...

JEE Main 2023 (Online) 1st February Evening Shift

The point of intersection $$\mathrm{C}$$ of the plane $$8 x+y+2 z=0$$ and the line joining the points $$\mathrm{A}(-3,-6,1)$$ and $$\mathrm{B}(2,4,-3)...

JEE Main 2023 (Online) 1st February Evening Shift

Let $$\alpha x+\beta y+\gamma z=1$$ be the equation of a plane passing through the point $$(3,-2,5)$$ and perpendicular to the line joining the points...

JEE Main 2023 (Online) 31st January Morning Shift

Let the line $$L: \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{1}$$ intersect the plane $$2 x+y+3 z=16$$ at the point
$$P$$. Let the point $$Q$$ be the fo...

JEE Main 2023 (Online) 31st January Morning Shift

Let $$\theta$$ be the angle between the planes $$P_{1}: \vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=9$$ and $$P_{2}: \vec{r} \cdot(2 \hat{i}-\hat{j}+\hat...

JEE Main 2023 (Online) 30th January Evening Shift

Let a line $L$ pass through the point $P(2,3,1)$ and be parallel to the line $x+3 y-2 z-2=0=x-y+2 z$. If the distance of $L$ from the point $(5,3,8)$ ...

JEE Main 2023 (Online) 30th January Morning Shift

If the equation of the plane passing through the point $$(1,1,2)$$ and perpendicular to the line $$x-3 y+ 2 z-1=0=4 x-y+z$$ is $$\mathrm{A} x+\mathrm{...

JEE Main 2023 (Online) 30th January Morning Shift

If $$\lambda_{1} ...

JEE Main 2023 (Online) 29th January Morning Shift

Let the equation of the plane P containing the line $$x+10=\frac{8-y}{2}=z$$ be $$ax+by+3z=2(a+b)$$ and the distance of the plane $$P$$ from the point...

JEE Main 2023 (Online) 29th January Morning Shift

Let the co-ordinates of one vertex of $$\Delta ABC$$ be $$A(0,2,\alpha)$$ and the other two vertices lie on the line $${{x + \alpha } \over 5} = {{y -...

JEE Main 2023 (Online) 25th January Evening Shift

If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line $${{x - 1} \over 2} = {{y + 1} \over { - 1}} = {{z ...

JEE Main 2023 (Online) 25th January Morning Shift

Let the equation of the plane passing through the line $$x - 2y - z - 5 = 0 = x + y + 3z - 5$$ and parallel to the line $$x + y + 2z - 7 = 0 = 2x + 3y...

JEE Main 2023 (Online) 24th January Evening Shift

If the shortest between the lines $${{x + \sqrt 6 } \over 2} = {{y - \sqrt 6 } \over 3} = {{z - \sqrt 6 } \over 4}$$ and $${{x - \lambda } \over 3} = ...

JEE Main 2023 (Online) 24th January Morning Shift

The shortest distance between the lines $${{x - 2} \over 3} = {{y + 1} \over 2} = {{z - 6} \over 2}$$ and $${{x - 6} \over 3} = {{1 - y} \over 2} = {{...

JEE Main 2022 (Online) 29th July Morning Shift

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 inter...

JEE Main 2022 (Online) 28th July Morning Shift

Let $$\mathrm{P}(-2,-1,1)$$ and $$\mathrm{Q}\left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)$$ be the vertices of the rhombus PRQS. If the di...

JEE Main 2022 (Online) 27th July Morning Shift

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...

JEE Main 2022 (Online) 26th July Evening Shift

The largest value of $$a$$, for which the perpendicular distance of the plane containing the lines $$
\vec{r}=(\hat{i}+\hat{j})+\lambda(\hat{i}+a \hat...

JEE Main 2022 (Online) 26th July Evening Shift

The plane passing through the line $$L: l x-y+3(1-l) z=1, x+2 y-z=2$$ and perpendicular to the plane $$3 x+2 y+z=6$$ is $$3 x-8 y+7 z=4$$. If $$\theta...

JEE Main 2022 (Online) 26th July Morning Shift

Let $$\mathrm{Q}$$ and $$\mathrm{R}$$ be two points on the line $$\frac{x+1}{2}=\frac{y+2}{3}=\frac{z-1}{2}$$ at a distance $$\sqrt{26}$$ from the poi...

JEE Main 2022 (Online) 25th July Morning Shift

The line of shortest distance between the lines $$\frac{x-2}{0}=\frac{y-1}{1}=\frac{z}{1}$$ and $$\frac{x-3}{2}=\frac{y-5}{2}=\frac{z-1}{1}$$ makes an...

JEE Main 2022 (Online) 30th June Morning Shift

Consider a triangle ABC whose vertices are A(0, $$\alpha$$, $$\alpha$$), B($$\alpha$$, 0, $$\alpha$$) and C($$\alpha$$, $$\alpha$$, 0), $$\alpha$$ > 0...

JEE Main 2022 (Online) 29th June Morning Shift

Let d be the distance between the foot of perpendiculars of the points P(1, 2, $$-$$1) and Q(2, $$-$$1, 3) on the plane $$-$$x + y + z = 1. Then d2 is...

JEE Main 2022 (Online) 29th June Morning Shift

Let $${P_1}:\overrightarrow r \,.\,\left( {2\widehat i + \widehat j - 3\widehat k} \right) = 4$$ be a plane. Let P2 be another plane which passes thro...

JEE Main 2022 (Online) 28th June Evening Shift

Let the image of the point P(1, 2, 3) in the line $$L:{{x - 6} \over 3} = {{y - 1} \over 2} = {{z - 2} \over 3}$$ be Q. Let R ($$\alpha$$, $$\beta$$, ...

JEE Main 2022 (Online) 27th June Morning Shift

Let the mirror image of the point (a, b, c) with respect to the plane 3x $$-$$ 4y + 12z + 19 = 0 be (a $$-$$ 6, $$\beta$$, $$\gamma$$). If a + b + c =...

JEE Main 2022 (Online) 25th June Evening Shift

Let l1 be the line in xy-plane with x and y intercepts $${1 \over 8}$$ and $${1 \over {4\sqrt 2 }}$$ respectively, and l2 be the line in zx-plane with...

JEE Main 2022 (Online) 25th June Morning Shift

Let the lines
$${L_1}:\overrightarrow r = \lambda \left( {\widehat i + 2\widehat j + 3\widehat k} \right),\,\lambda \in R$$
$${L_2}:\overrightarrow ...

JEE Main 2022 (Online) 24th June Morning Shift

Let a line having direction ratios, 1, $$-$$4, 2 intersect the lines $${{x - 7} \over 3} = {{y - 1} \over { - 1}} = {{z + 2} \over 1}$$ and $${x \over...

JEE Main 2022 (Online) 24th June Morning Shift

If the shortest distance between the lines $$\overrightarrow r = \left( { - \widehat i + 3\widehat k} \right) + \lambda \left( {\widehat i - a\wideha...

JEE Main 2021 (Online) 31st August Evening Shift

Suppose, the line $${{x - 2} \over \alpha } = {{y - 2} \over { - 5}} = {{z + 2} \over 2}$$ lies on the plane $$x + 3y - 2z + \beta = 0$$. Then $$(\al...

JEE Main 2021 (Online) 31st August Morning Shift

The square of the distance of the point of intersection of the line $${{x - 1} \over 2} = {{y - 2} \over 3} = {{z + 1} \over 6}$$ and the plane $$2x -...

JEE Main 2021 (Online) 27th August Evening Shift

Let S be the mirror image of the point Q(1, 3, 4) with respect to the plane 2x $$-$$ y + z + 3 = 0 and let R(3, 5, $$\gamma$$) be a point of this plan...

JEE Main 2021 (Online) 26th August Evening Shift

Let Q be the foot of the perpendicular from the point P(7, $$-$$2, 13) on the plane containing the lines $${{x + 1} \over 6} = {{y - 1} \over 7} = {{z...

JEE Main 2021 (Online) 26th August Morning Shift

Let the line L be the projection of the line $${{x - 1} \over 2} = {{y - 3} \over 1} = {{z - 4} \over 2}$$ in the plane x $$-$$ 2y $$-$$ z = 3. If d i...

JEE Main 2021 (Online) 27th July Evening Shift

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 p...

JEE Main 2021 (Online) 27th July Morning Shift

Let a plane P pass through the point (3, 7, $$-$$7) and contain the line, $${{x - 2} \over { - 3}} = {{y - 3} \over 2} = {{z + 2} \over 1}$$. If dista...

JEE Main 2021 (Online) 25th July Evening Shift

If the lines $${{x - k} \over 1} = {{y - 2} \over 2} = {{z - 3} \over 3}$$ and $${{x + 1} \over 3} = {{y + 2} \over 2} = {{z + 3} \over 1}$$ are co-pl...

JEE Main 2021 (Online) 20th July Morning Shift

Let P be a plane passing through the points (1, 0, 1), (1, $$-$$2, 1) and (0, 1, $$-$$2). Let a vector $$\overrightarrow a = \alpha \widehat i + \bet...

JEE Main 2021 (Online) 18th March Evening Shift

Let the mirror image of the point (1, 3, a) with respect to the plane $$\overrightarrow r .\left( {2\widehat i - \widehat j + \widehat k} \right) - b ...

JEE Main 2021 (Online) 18th March Evening Shift

Let P be a plane containing the line $${{x - 1} \over 3} = {{y + 6} \over 4} = {{z + 5} \over 2}$$ and parallel to the line $${{x - 1} \over 4} = {{y ...

JEE Main 2021 (Online) 18th March Morning Shift

Let the plane ax + by + 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 integer...

JEE Main 2021 (Online) 18th March Morning Shift

The equation of the planes parallel to the plane x $$-$$ 2y + 2z $$-$$ 3 = 0 which are at unit distance from the point (1, 2, 3) is ax + by + cz + d =...

JEE Main 2021 (Online) 17th March Evening Shift

Let P be an arbitrary point having sum of the squares of the distances from the planes x + y + z = 0, lx $$-$$ nz = 0 and x $$-$$ 2y + z = 0, equal to...

JEE Main 2021 (Online) 17th March Morning Shift

If the equation of the plane passing through the line of intersection of the planes 2x $$-$$ 7y + 4z $$-$$ 3 = 0, 3x $$-$$ 5y + 4z + 11 = 0 and the po...

JEE Main 2021 (Online) 16th March Evening Shift

If the distance of the point (1, $$-$$2, 3) from the plane x + 2y $$-$$ 3z + 10 = 0 measured parallel to the line, $${{x - 1} \over 3} = {{2 - y} \ove...

JEE Main 2021 (Online) 26th February Morning Shift

Let ($$\lambda$$, 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 p...

JEE Main 2021 (Online) 25th February Evening Shift

A line 'l' passing through origin is perpendicular to the lines$${l_1}:\overrightarrow r = (3 + t)\widehat i + ( - 1 + 2t)\widehat j + (4 + 2t)\wideh...

JEE Main 2021 (Online) 24th February Evening Shift

Let $$\lambda$$ be an integer. If the shortest distance between the lines x $$-$$ $$\lambda$$ = 2y $$-$$ 1 = $$-$$2z and x = y + 2$$\lambda$$ = z $$-$...

JEE Main 2020 (Online) 4th September Morning Slot

If the equation of a plane P, passing through the intersection of the planes, x + 4y - z + 7 = 0
and 3x + y + 5z = 8 is ax + by + 6z = 15 for some a, ...

JEE Main 2020 (Online) 3rd September Evening Slot

Let a plane P contain two lines
$$\overrightarrow r = \widehat i + \lambda \left( {\widehat i + \widehat j} \right)$$, $$\lambda \in R$$ and
$$\over...

JEE Main 2020 (Online) 9th January Evening Slot

If the distance between the plane,
23x – 10y – 2z + 48 = 0 and the plane
containing the lines
$${{x + 1} \over 2} = {{y - 3} \over 4} = {{z + 1} \over...

JEE Main 2020 (Online) 9th January Morning Slot

The projection of the line segment joining the
points (1, –1, 3) and (2, –4, 11) on the line
joining the points (–1, 2, 3) and (3, –2, 10)
is ________...

JEE Main 2020 (Online) 7th January Evening Slot

If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through ($$\alpha $$, 7, 1)
is
$$\left( {{5 \over 3},{7 \over 3},{{1...