How many questions from Three Dimensional Geometry appear in JEE?

Three Dimensional Geometry has appeared in 176 of the last 186 JEE Main and JEE Advanced papers — about 95% of them — contributing 344 questions in total across those papers.

Is Three Dimensional Geometry an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 95% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Three Dimensional Geometry questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Three Dimensional Geometry JEE Previous Year Questions with Answers

Q1·MathematicsMultiple correctJEE Advanced 2026

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 ?

(A)The equation of the plane $P$ is $7x - 5y + z = -10$
(B)The distance between the planes $P$ and $P_1$ is 30
(C)The distance of the plane $P$ from the origin is $2\sqrt{3}$
(D)The acute angle between the plane $P$ and the plane $2x + 2y + z = 3$ is $\cos^{-1}\left( \frac{1}{3\sqrt{3}} \right)$

Q2·MathematicsMultiple correctJEE Advanced 2026

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 ?

(A)The orthocentre of the triangle $PRT$ is $\left(\dfrac{23}{5}, -4, \dfrac{31}{5}\right)$
(B)The orthocentre of the triangle $PRT$ is $(4, 3, 5)$
(C)The area of the triangle $PRT$ is $6\sqrt{5}$
(D)The area of the triangle $PRT$ is $18\sqrt{5}$

Q3·MathematicsSingle correctJEE Main 2026

Let the foot of perpendicular from the point

(\lambda, 2, 3)

on the line

\dfrac{x-4}{1} = \dfrac{y-9}{2} = \dfrac{z-5}{1}

be the point

(1, \mu, 2)

. Then the distance between the lines

\dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{z+4}{6}

and

\dfrac{x-\lambda}{2} = \dfrac{y-\mu}{3} = \dfrac{z+5}{6}

is equal to:

(A) $\dfrac{12}{7}$
(B) $\dfrac{\sqrt{145}}{7}$
(C) $\dfrac{\sqrt{146}}{7}$
(D) $\dfrac{\sqrt{143}}{7}$

Q4·MathematicsNumericalJEE Main 2026

Let a line

L_1

pass through the origin and be perpendicular to the lines

L_2 : \vec{r} = (3 + t)\hat{i} + (2t - 1)\hat{j} + (2t + 4)\hat{k}

and

L_3 : \vec{r} = (3 + 2s)\hat{i} + (3 + 2s)\hat{j} + (2 + s)\hat{k}

,

t, s \in \mathbb{R}

. If

(a, b, c)

,

a \in \mathbb{Z}

, is the point on

L_3

at a distance of

\sqrt{17}

from the point of intersection of

L_1

and

L_2

, then

(a + b + c)^2

is equal to ________.

Q5·MathematicsSingle correctJEE Main 2026

The shortest distance between the lines

\frac{x-4}{1} = \frac{y-3}{2} = \frac{z-2}{-3}

and

\frac{x+2}{2} = \frac{y-6}{4} = \frac{z-5}{-5}

is :

(A) $\frac{5\sqrt{6}}{6}$
(B) $2\sqrt{5}$
(C) $3\sqrt{5}$
(D) $4\sqrt{5}$

Q6·MathematicsNumericalJEE Main 2026

Let the image of the point

P(0, -5, 0)

in the line

\frac{x-1}{2} = \frac{y}{1} = \frac{z+1}{-2}

be the point

R

and the image of the point

Q\left(0, \frac{-1}{2}, 0\right)

in the line

\frac{x-1}{-1} = \frac{y+9}{4} = \frac{z+1}{1}

be the point

S

. Then the square of the area of the parallelogram

PQRS

is __________.

Q7·MathematicsSingle correctJEE Main 2026

Let the image of the point P(1, 6, a) in the line L :

\frac{x}{1} = \frac{y-1}{2} = \frac{z-a+1}{b}

, b > 0, be

\left(\frac{a}{3}, 0, a+c\right)

. If S(α, β, γ), α > 0, is the point on L such that the distance of S from the foot of perpendicular from the point P on L is

2\sqrt{14}

, then α + β + γ is equal to:

(A)19
(B)20
(C)21
(D)22

Q8·MathematicsSingle correctJEE Main 2026

Let a line L be perpendicular to both the lines L

_1

:

\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}

and L

_2

:

\frac{x-2}{1} = \frac{y-4}{4} = \frac{z-6}{7}

. If θ is the acute angle between the lines L and L

_3

:

\frac{x-\frac{8}{7}}{2} = \frac{y-\frac{4}{7}}{1} = \frac{z}{2}

, then tan θ is equal to:

(A) $\frac{3}{2}\sqrt{2}$
(B) $\frac{5}{2}\sqrt{2}$
(C) $\frac{5}{3}\sqrt{2}$
(D) $\frac{4}{3}\sqrt{2}$

Q9·MathematicsSingle correctJEE Main 2026

The square of the distance of the point

P(5,6,7)

from the line

\frac{x-2}{2} = \frac{y-5}{3} = \frac{z-2}{4}

is equal to:

(A)3
(B)5
(C)6
(D)8

Q10·MathematicsSingle correctJEE Main 2026

The square of the distance of the point of intersection of the lines

\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(a\hat{i} - \hat{j})

,

a \neq 0

and

\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + a\hat{k})

from the origin is:

(A)5
(B)10
(C)17
(D)26

Q11·MathematicsSingle correctJEE Main 2026

Let a triangle PQR be such that P and Q lie on the line

\frac{x+3}{8} = \frac{y-4}{2} = \frac{z+1}{2}

and are at a distance of 6 units from R(1, 2, 3). If (α, β, γ) is the centroid of △PQR, then α + β + γ is equal to :

(A)4
(B)5
(C)6
(D)8

Q12·MathematicsSingle correctJEE Main 2026

If the distance of the point (a, 2, 5) from the image of the point (1, 2, 7) in the line

\frac{x}{1} = \frac{y-1}{1} = \frac{z-2}{2}

is 4, then the sum of all possible values of a is equal to :

(A)11
(B)9
(C)6
(D)4

Q13·MathematicsSingle correctJEE Main 2026

A line with direction ratios

1, -1, 2

intersects the lines

\frac{x}{2} = \frac{y}{3} = \frac{z+1}{3}

and

\frac{x+1}{-1} = \frac{y-2}{1} = \frac{z}{4}

at the points

P

and

Q

, respectively. If the length of the line segment

PQ

is

\alpha

, then

225\alpha^2

is equal to:

(A) $1024$
(B) $1014$
(C) $1104$
(D) $1204$

Q14·MathematicsSingle correctJEE Main 2026

The shortest distance between the lines

\vec{r} = \left(\frac{1}{3}\hat{i} + 2\hat{j} + \frac{8}{3}\hat{k}\right) + \lambda(2\hat{i} - 5\hat{j} + 6\hat{k})

and

\vec{r} = \left(-\frac{2}{3}\hat{i} - \frac{1}{3}\hat{k}\right) + \mu(\hat{j} - \hat{k})

,

\lambda, \mu \in \mathbb{R}

, is:

(A) $\sqrt{5}$
(B)3
(C) $2\sqrt{3}$
(D) $\sqrt{15}$

Q15·MathematicsSingle correctJEE Main 2026

If

\left(2\alpha + 1, \alpha^{2} - 3\alpha, \frac{\alpha - 1}{2}\right)

is the image of

(\alpha, 2\alpha, 1)

in the line

\frac{x - 2}{3} = \frac{y - 1}{2} = \frac{z}{1}

, then the possible value(s) of α is (are)

(A)Only 3
(B)Only 3 and −1
(C)Only 3, $\frac{1}{4}$ and −1
(D)Only 3 and $\frac{1}{4}$

Q16·MathematicsSingle correctJEE Main 2026

The square of the distance of the point

(-2, -8, 6)

from the line

\frac{x-1}{1} = \frac{y-1}{2} = \frac{z}{-1}

along the line

\frac{x+5}{1} = \frac{y+5}{-1} = \frac{z}{2}

is equal to:

(A) $3$
(B) $6$
(C) $8$
(D) $12$

Q17·MathematicsSingle correctJEE Main 2026

Let a line

L

passing through the point

(1, 1, 1)

be perpendicular to both the vectors

2\hat{i} + 2\hat{j} + \hat{k}

and

\hat{i} + 2\hat{j} + 2\hat{k}

. If

P(a, b, c)

is the foot of perpendicular from the origin on the line

L

, then the value of

34(a + b + c)

is :

(A)50
(B)80
(C)100
(D)120

Q18·MathematicsSingle correctJEE Main 2026

Let the point A be the foot of perpendicular drawn from the point

P(a, b, 0)

on the line

\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-\alpha}{3}

. If the midpoint of the line segment PA is

\left(0, \frac{3}{4}, \frac{-1}{4}\right)

, then the value of

a^2 + b^2 + \alpha^2

is equal to:

(A) $1$
(B) $2$
(C) $6$
(D) $9$

Q19·MathematicsSingle correctJEE Main 2026

If the point of intersection of the lines

\frac{x+1}{3}=\frac{y+a}{5}=\frac{z+b+1}{7}

and

\frac{x-2}{1}=\frac{y-b}{4}=\frac{z-2a}{7}

lies on xy-plane, then the value of

a+b

is :

(A)2
(B)5
(C)7
(D)9

Q20·MathematicsNumericalJEE Main 2026

If the distance of the point P(43,

\alpha

,

\beta

),

\beta < 0

, from the line

\vec{r} = 4\hat{i} - \hat{k} + \mu(2\hat{i} + 3\hat{k})

,

\mu \in \mathbf{R}

along a line with direction ratios 3, –1, 0 is

13\sqrt{10}

, then

\alpha^2 + \beta^2

is equal to ________.

Q21·MathematicsSingle correctJEE Main 2026

Let Q(a, b, c) be the image of the point P(3, 2, 1) in the line

\frac{x-1}{1} = \frac{y}{2} = \frac{z-1}{1}

. Then the distance of Q from the line

\frac{x-9}{3} = \frac{y-9}{2} = \frac{z-5}{-2}

is

(A)6
(B)8
(C)7
(D)5

Q22·MathematicsSingle correctJEE Main 2026

If the distances of the point (1, 2, a) from the line

\dfrac{x-1}{1} = \dfrac{y}{2} = \dfrac{z-1}{1}

along the lines

L_1 : \dfrac{x-1}{3} = \dfrac{y-2}{4} = \dfrac{z-a}{b}

and

L_2 : \dfrac{x-1}{1} = \dfrac{y-2}{4} = \dfrac{z-a}{c}

are equal, then a + b + c is equal to

(A)7
(B)5
(C)6
(D)4

Q23·MathematicsNumericalJEE Main 2026

Let a line L passing through the point P(1, 1, 1) be perpendicular to the lines

\frac{\mathrm{x} - 4}{4} = \frac{\mathrm{y} - 1}{1} = \frac{\mathrm{z} - 1}{1}

and

\frac{\mathrm{x} - 17}{1} = \frac{\mathrm{y} - 71}{1} = \frac{\mathrm{z}}{0}

. Let the line L intersect the yz-plane at the point Q. Another line parallel to L and passing through the point S(1,0, –1) intersects the yz-plane at the point R. Then the square of the area of the parallelogram PQRS is equal to ______.

Q24·MathematicsSingle correctJEE Main 2026

Let the lines

L_{1}

:

\vec{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda\left(2\hat{i}+3\hat{j}+4\hat{k}\right)

, λ∈ℝ and

L_{2}

:

\vec{r}=\left(4\hat{i}+\hat{j}\right)+\mu\left(5\hat{i}+2\hat{j}+\hat{k}\right)

, μ∈ℝ, intersect at the point R. Let P and Q be the points lying on lines

L_{1}

and

L_{2}

, respectively, such that

\left|\overrightarrow{PR}\right|=\sqrt{29}

and

\left|\overrightarrow{PQ}\right|=\sqrt{\frac{47}{3}}

. If the point P lies in the first octant, then

27(QR)^{2}

is equal to

(A) $340$
(B) $360$
(C) $320$
(D) $348$

Q25·MathematicsSingle correctJEE Main 2026

The sum of all values of

\alpha

, for which the shortest distance between the lines

\frac{x+1}{\alpha} = \frac{y-2}{-1} = \frac{z-4}{-\alpha}

and

\frac{x}{\alpha} = \frac{y-1}{2} = \frac{z-1}{2\alpha}

is

\sqrt{2}

, is

(A)8
(B)−6
(C)6
(D)−8

Q26·MathematicsSingle correctJEE Main 2026

Let the direction cosines of two lines satisfy the equations :

4\ell + m - n = 0

and

2mn + 10n\ell + 3\ell m = 0

. Then the cosine of the acute angle between these lines is :

(A) $\frac{10}{\sqrt{38}}$
(B) $\frac{20}{3\sqrt{38}}$
(C) $\frac{10}{7\sqrt{38}}$
(D) $\frac{10}{3\sqrt{38}}$

Q27·MathematicsSingle correctJEE Main 2026

The vertices B and C of a triangle ABC lie on the line

\frac{x}{1} = \frac{1-y}{-2} = \frac{z-2}{3}

. The coordinates of A and B are (1, 6, 3) and (4, 9,

\alpha

) respectively and C is at a distance of 10 units from B. The area (in sq. units) of

\Delta ABC

is:

(A) $5\sqrt{13}$
(B) $15\sqrt{13}$
(C) $20\sqrt{13}$
(D) $10\sqrt{13}$

Q28·MathematicsNumericalJEE Main 2026

If the image of the point P(a, 2, a) in the line

\frac{x}{2} = \frac{y+a}{1} = \frac{z}{1}

is Q and the of image of Q in the line

\frac{x-2b}{2} = \frac{y-a}{1} = \frac{z+2b}{-5}

is P, then a + b is equal to………..

Q29·MathematicsSingle correctJEE Main 2026

Let P(α, β, γ) be the point on the line

\frac{x-1}{2} = \frac{y+1}{-3} = z

at a distance

4\sqrt{14}

from the point (1, −1, 0) and nearer to the origin. Then the shortest distance, between the lines

\frac{x-\alpha}{1} = \frac{y-\beta}{2} = \frac{z-\gamma}{3}

and

\frac{x+5}{2} = \frac{y-10}{1} = \frac{z-3}{1}

, is equal to

(A) $7\sqrt{\frac{5}{4}}$
(B) $4\sqrt{\frac{7}{5}}$
(C) $4\sqrt{\frac{5}{7}}$
(D) $2\sqrt{\frac{7}{4}}$

Q30·MathematicsSingle correctJEE Main 2026

Let L be the line

\frac{x+1}{2} = \frac{y+1}{3} = \frac{z+3}{6}

and let S be the set of all points (a, b, c) on L, whose distance from the line

\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-9}{0}

along the line L is 7. Then

\sum\limits_{(a,b,c) \in S} (a + b + c)

is equal to :

(A)34
(B)28
(C)40
(D)6

Q31·MathematicsSingle correctJEE Main 2026

If the image of the point P (1, 2 , a) in the line

\frac{x-6}{3} = \frac{y-7}{2} = \frac{7-z}{2}

is Q(5, b, c), then

a^2 + b^2 + c^2

is equal to

(A)293
(B)264
(C)298
(D)283

Q32·MathematicsSingle correctJEE Main 2026

Let the line L pass through the point (−3, 5, 2) and make equal angles with the positive coordinate axes. If the distance of L from the point (−2, r, 1) is

\sqrt{\frac{14}{3}}

, then the sum of all possible values of r is :

(A)12
(B)16
(C)6
(D)10

Q33·MathematicsSingle correctJEE Main 2026

Let (α, β, γ) be the co-ordinates of the foot of the perpendicular drawn from the point (5, 4, 2) on the line

\vec{r}=(-\hat{i}+3\hat{j}+\hat{k})+\lambda(2\hat{i}+3\hat{j}-\hat{k})

. Then the length of the projection of the vector

\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}

on the vector

6\hat{i}+2\hat{j}+3\hat{k}

is :

(A) $\frac{15}{7}$
(B)4
(C) $\frac{18}{7}$
(D)3

Q34·MathematicsSingle correctJEE Main 2026

Let the line

L_1

be parallel to the vector

-3\hat{i} + 2\hat{j} + 4\hat{k}

and pass through the point (2, 6, 7) and the line

L_2

be parallel to the vector

2\hat{i} + \hat{j} + 3\hat{k}

and pass through the point (4, 3, 5). If the line

L_3

is parallel to the vector

-3\hat{i} + 5\hat{j} + 16\hat{k}

and intersects the lines

L_1

and

L_2

at the points C and D, respectively, then

\left|\overrightarrow{CD}\right|^2

is equal to :

(A)171
(B)290
(C)312
(D)89

Q35·MathematicsMultiple correctJEE Advanced 2025

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. The which of the following statements is (are) TRUE?

(A)The length of the line segment PQ is $9\sqrt{3}$
(B)The length of the line segment QR is 15
(C)The area of ΔPQR is $\frac{3}{2}\sqrt{234}$
(D)The acute angle between the line segments PQ and PR is $\cos^{-1}\left(\frac{1}{2\sqrt{3}}\right)$

Q36·MathematicsIntegerJEE Main 2025

Let the area of the triangle formed by the lines

x+2=y-1=z

,

\dfrac{x-3}{5}=\dfrac{y}{-1}=\dfrac{z-1}{1}

and

\dfrac{x}{-3}=\dfrac{y-3}{3}=\dfrac{z-2}{1}

be

A

. Then

A^2

is equal to ___

Q37·MathematicsSingle correctJEE Main 2025

Let the values of

\lambda

for which the shortest distance between the lines

\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}

and

\dfrac{x-\lambda}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5}

is

\dfrac{1}{\sqrt6}

be

\lambda_1

and

\lambda_2

. Then the radius of the circle passing through the points

(0,0)

,

(\lambda_1,\lambda_2)

and

(\lambda_2,\lambda_1)

is:

(A) $\dfrac{5\sqrt2}{3}$
(B) $4$
(C) $\dfrac{\sqrt2}{3}$
(D) $3$

Q38·MathematicsSingle correctJEE Main 2025

Let the line L pass through

(1,1,1)

and intersect the lines

\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z-1}{4}

and

\dfrac{x-3}{1}=\dfrac{y-4}{2}=\dfrac{z-1}{1}

. Then, which of the following points lies on the line L?

(A) $(4,22,7)$
(B) $(5,4,3)$
(C) $(10,-29,-50)$
(D) $(7,15,13)$

Q39·MathematicsSingle correctJEE Main 2025

If the equation of the line passing through the point

\left(0,-\dfrac{1}{2},0\right)

and perpendicular to the lines

\vec{r}=\lambda(\hat{i}+a\hat{j}+b\hat{k})

and

\vec{r}=(\hat{i}-\hat{j}-6\hat{k})+\mu(-b\hat{i}+a\hat{j}+5\hat{k})

is

\dfrac{x-1}{-2}=\dfrac{y+4}{d}=\dfrac{z-c}{-4}

, then

a+b+c+d

is equal to:

(A)10
(B)14
(C)13
(D)12

Q40·MathematicsSingle correctJEE Main 2025

If the shortest distance between the lines

\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}

and

\dfrac{x}{1}=\dfrac{y}{\alpha}=\dfrac{z-5}{1}

is

\dfrac{5}{\sqrt{6}}

, then the sum of all possible values of

\alpha

is

(A) $\dfrac{3}{2}$
(B) $-\dfrac{3}{2}$
(C)3
(D) $-3$

Q41·MathematicsSingle correctJEE Main 2025

Consider the lines

L_1:x-1=y-2=z

and

L_2:x-2=y=z-1

. Let the feet of the perpendiculars from the point

P(5,1,-3)

on the lines

L_1

and

L_2

be

Q

and

R

respectively. If the area of the triangle

PQR

is

A

, then

4A^2

is equal to:

(A)139
(B)147
(C)151
(D)143

Q42·MathematicsSingle correctJEE Main 2025

Let

A

and

B

be two distinct points on the line

L:\dfrac{x-6}{3}=\dfrac{y-7}{2}=\dfrac{z-7}{-2}

. Both

A

and

B

are at a distance

2\sqrt{17}

from the foot of perpendicular drawn from the point

(1,2,3)

on the line

L

. If

O

is the origin, then

\overrightarrow{OA}\cdot\overrightarrow{OB}

is equal to:

(A)49
(B)47
(C)21
(D)62

Q43·MathematicsSingle correctJEE Main 2025

Let the shortest distance between the lines

\dfrac{x-3}{3}=\dfrac{y-\alpha}{-1}=\dfrac{z-3}{1}

and

\dfrac{x+3}{-3}=\dfrac{y+7}{2}=\dfrac{z-\beta}{4}

be

3\sqrt{30}

. Then the positive value of

5\alpha+\beta

is:

(A)42
(B)46
(C)48
(D)40

Q44·MathematicsSingle correctJEE Main 2025

Let

A

be the point of intersection of the lines

L_1:\dfrac{x-7}{1}=\dfrac{y-5}{0}=\dfrac{z-3}{-1}

and

L_2:\dfrac{x-1}{3}=\dfrac{y+3}{4}=\dfrac{z+7}{5}

. Let

B

and

C

be the points on the lines

L_1

and

L_2

respectively such that

AB=AC=\sqrt{15}

. Then the square of the area of the triangle

ABC

is:

(A)54
(B)63
(C)57
(D)60

Q45·MathematicsSingle correctJEE Main 2025

Let the values of

p

, for which the shortest distance between the lines

\dfrac{x+1}{3}=\dfrac{y}{4}=\dfrac{z}{5}

and

\vec{r}=(p\hat{i}+2\hat{j}+\hat{k})+\lambda(2\hat{i}+3\hat{j}+4\hat{k})

is

\dfrac{1}{\sqrt6}

, be

a, b

,

(a<b)

. Then the length of the latus rectum of the ellipse

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

is:

(A)9
(B) $\dfrac{3}{2}$
(C) $\dfrac{2}{3}$
(D)18

Q46·MathematicsSingle correctJEE Main 2025

The distance of the point

(7,10,11)

from the line

\dfrac{x-4}{1}=\dfrac{y-4}{0}=\dfrac{z-2}{3}

along the line

\dfrac{x-9}{2}=\dfrac{y-13}{3}=\dfrac{z-17}{6}

is:

(A)18
(B)14
(C)12
(D)16

Q47·MathematicsSingle correctJEE Main 2025

Let a line passing through the point

(4,1,0)

intersect the line

L_1:\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}

at the point

A(\alpha,\beta,\gamma)

and the line

L_2:x-6=y=-z+4

at the point

B(a,b,c)

. Then

\begin{vmatrix}1&0&1\\\alpha&\beta&\gamma\\a&b&c\end{vmatrix}

is equal to:

(A)8
(B)16
(C)12
(D)6

Q48·MathematicsSingle correctJEE Main 2025

Each of the angles

\beta

and

\gamma

that a given line makes with the positive

y

- and

z

-axes, respectively, is half of the angle that this line makes with the positive

x

-axis. Then the sum of all possible values of the angle

\beta

is:

(A) $\dfrac{3\pi}{4}$
(B) $\pi$
(C) $\dfrac{\pi}{2}$
(D) $\dfrac{3\pi}{2}$

Q49·MathematicsSingle correctJEE Main 2025

Line

L_1

passes through the point

(1,2,3)

and is parallel to the z-axis. Line

L_2

passes through the point

(\lambda,5,6)

and is parallel to the y-axis. Let for

\lambda=\lambda_1,\lambda_2

(\lambda_2<\lambda_1)

, the shortest distance between the two lines be 3. Then the square of the distance of the point

(\lambda_1,\lambda_2,2)

from the line

L_1

is:

(A)40
(B)32
(C)25
(D)37

Q50·MathematicsSingle correctJEE Main 2025

If the image of the point

P(1, 0, 3)

in the line joining the points

A(4, 7, 1)

and

B(3, 5, 3)

is

Q(\alpha, \beta, \gamma)

, then

\alpha + \beta + \gamma

is equal to:

(A) $\dfrac{47}{3}$
(B) $\dfrac{46}{3}$
(C)18
(D)13

Q51·MathematicsSingle correctJEE Main 2025

Let the vertices

Q

and

R

of the triangle

PQR

lie on the line

\dfrac{x+3}{5}=\dfrac{y-1}{2}=\dfrac{z+4}{3}

,

QR=5

and the coordinates of the point

P

be

(0,2,3)

. If the area of the triangle

PQR

is

\dfrac{m}{n}

then:

(A) $m-5\sqrt{21}\,n=0$
(B) $2m-5\sqrt{21}\,n=0$
(C) $5m-2\sqrt{21}\,n=0$
(D) $5m-21\sqrt{2}\,n=0$

Q52·MathematicsSingle correctJEE Main 2025

The line

L_1

is parallel to the vector

\vec{a}=-3\hat{i}+2\hat{j}+4\hat{k}

and passes through the point

(7,6,2)

and the line

L_2

is parallel to the vector

\vec{b}=2\hat{i}+\hat{j}+3\hat{k}

and passes through the point

(5,3,4)

. The shortest distance between the lines

L_1

and

L_2

is:

(A) $\dfrac{23}{\sqrt{38}}$
(B) $\dfrac{21}{\sqrt{57}}$
(C) $\dfrac{23}{\sqrt{57}}$
(D) $\dfrac{21}{\sqrt{38}}$

Q53·MathematicsSingle correctJEE Main 2025

Let a straight line L pass through the point

P(2,-1,3)

and be perpendicular to the lines

\dfrac{x-1}{2}=\dfrac{y+1}{1}=\dfrac{z-3}{-2}

and

\dfrac{x-3}{1}=\dfrac{y-2}{3}=\dfrac{z+2}{4}

. If the line L intersects the yz-plane at the point Q, then the distance between the points P and Q is:

(A)2
(B) $\sqrt{10}$
(C)3
(D) $2\sqrt{3}$

Q54·MathematicsSingle correctJEE Main 2025

Let P be the foot of the perpendicular from the point

(1, 2, 2)

on the line

L:\dfrac{x-1}{1}=\dfrac{y+1}{-1}=\dfrac{z-2}{2}

. Let the line

\vec{r}=\left(-\hat{i}-2\hat{k}\right)+\lambda\left(\hat{i}-\hat{j}+\hat{k}\right)

,

\lambda\in R

, intersect the line L at Q. Then

2(PQ)^2

is equal to:

(A)27
(B)25
(C)29
(D)19

Q55·MathematicsSingle correctJEE Main 2025

Let

L_1:\dfrac{x-1}{1}=\dfrac{y-2}{-1}=\dfrac{z-1}{2}

and

L_2:\dfrac{x+1}{-1}=\dfrac{y-2}{2}=\dfrac{z}{1}

be two lines. Let

L_3

be a line passing through the point

(\alpha,\beta,\gamma)

and be perpendicular to both

L_1

and

L_2

. If

L_3

intersects

L_1

, then

|5\alpha-11\beta-8\gamma|

equals:

(A)18
(B)16
(C)25
(D)20

Q56·MathematicsSingle correctJEE Main 2025

If the image of the point

(4, 4, 3)

in the line

\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-1}{3}

is

(\alpha, \beta, \gamma)

, then

\alpha+\beta+\gamma

is equal to

(A)9
(B)12
(C)8
(D)7

Q57·MathematicsSingle correctJEE Main 2025

Let

A(x, y, z)

be a point in xy-plane, which is equidistant from three points

(0, 3, 2)

,

(2, 0, 3)

and

(0, 0, 1)

. Let

B=(1, 4, -1)

and

C=(2, 0, -2)

. Then among the statements (S1):

\triangle ABC

is an isosceles right angled triangle and (S2): the area of

\triangle ABC

is

\frac{9\sqrt{2}}{2}

.

(A)both are true
(B)only (S1) is true
(C)only (S2) is true
(D)both are false

Q58·MathematicsSingle correctJEE Main 2025

The square of the distance of the point

\left(\frac{15}{7},\frac{32}{7},7\right)

from the line

\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}

in the direction of the vector

\hat{i}+4\hat{j}+7\hat{k}

is:

(A)54
(B)41
(C)66
(D)44

Q59·MathematicsIntegerJEE Main 2025

Let P be the image of the point

Q(7,-2,5)

in the line

L:\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z}{4}

and

R(5,p,q)

be a point on L. Then the square of the area of

\triangle PQR

is __________.

Q60·MathematicsSingle correctJEE Main 2025

Let the line passing through the points

(-1,2,1)

and parallel to the line

\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z}{4}

intersect the line

\dfrac{x+2}{3}=\dfrac{y-3}{2}=\dfrac{z-4}{1}

at the point P. Then the distance of P from the point

Q(4,-5,1)

is :

(A) $5$
(B) $10$
(C) $5\sqrt6$
(D) $5\sqrt5$

Q61·MathematicsSingle correctJEE Main 2025

Let in a

\triangle ABC

, the length of the side AC be 6, the vertex B be

(1,2,3)

and the vertices A, C lie on the line

\dfrac{x-6}{3}=\dfrac{y-7}{2}=\dfrac{z-7}{-2}

. Then the area (in sq. units) of

\triangle ABC

is

(A) $42$
(B) $21$
(C) $56$
(D) $17$

Q62·MathematicsSingle correctJEE Main 2025

If the square of the shortest distance between the lines

\dfrac{x-2}{1}=\dfrac{y-1}{2}=\dfrac{z+3}{-3}

and

\dfrac{x+1}{2}=\dfrac{y+3}{4}=\dfrac{z+5}{-5}

is

\dfrac{m}{n}

, where

m,n

are coprime numbers, then

m+n

is equal to :

(A)6
(B)9
(C)21
(D)14

Q63·MathematicsSingle correctJEE Main 2025

The distance of the line

\dfrac{x-2}{2}=\dfrac{y-6}{3}=\dfrac{z-3}{4}

from the point

(1,4,0)

along the line

\dfrac{x}{1}=\dfrac{y-2}{2}=\dfrac{z+3}{3}

is :

(A) $\sqrt{17}$
(B) $\sqrt{14}$
(C) $\sqrt{15}$
(D) $\sqrt{13}$

Q64·MathematicsSingle correctJEE Main 2025

Let

P

be the foot of the perpendicular from the point

Q(10,-3,-1)

on the line

\dfrac{x-3}{7}=\dfrac{y-2}{-1}=\dfrac{z+1}{-2}

. Then the area of the right angled triangle

PQR

, where

R

is the point

(3,-2,1)

, is

(A) $9\sqrt{15}$
(B) $\sqrt{30}$
(C) $8\sqrt{15}$
(D) $3\sqrt{30}$

Q65·MathematicsSingle correctJEE Main 2025

The perpendicular distance, of the line

\dfrac{x-1}{2}=\dfrac{y+2}{-1}=\dfrac{z+3}{2}

from the point

P(2,-10,1)

, is:

(A)6
(B) $5\sqrt{2}$
(C) $3\sqrt{5}$
(D) $4\sqrt{3}$

Q66·MathematicsSingle correctJEE Main 2025

Let

L_1:\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}

and

L_2:\dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5}

. Which point lies on the line of shortest distance between

L_1

and

L_2

?

(A) $\left(-\tfrac53,-7,1\right)$
(B) $\left(2,3,\tfrac13\right)$
(C) $\left(\tfrac83,-1,\tfrac13\right)$
(D) $\left(\tfrac{14}{3},-3,\tfrac{22}{3}\right)$

Q67·MathematicsIntegerJEE Main 2025

Let

L_1:\dfrac{x-1}{3}=\dfrac{y-1}{-1}=\dfrac{z+1}{0}

and

L_2:\dfrac{x-2}{2}=\dfrac{y+4}{0}=\dfrac{z+4}{\alpha}

meet at B. If P is the foot of perpendicular from

A(1,-1,1)

on

L_2

, find

26\,\alpha\,(PB)

(as integer).

Q68·MathematicsSingle correctJEE Main 2025

Let a line pass through two distinct points

P(-2,-1,3)

and

Q

, and be parallel to the vector

3\hat{i}+2\hat{j}+2\hat{k}

. If the distance of the point

Q

from the point

R(1,3,3)

is 5, then the square of the area of

\triangle PQR

is equal to:

(A)136
(B)140
(C)144
(D)148

Q69·MathematicsMultiple correctJEE Advanced 2024

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 + 3z = 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 : 2x - y + z = -4

at the point R. then which of the following statements is(are) TRUE?

(A)The length of the line segment PQ is $\sqrt{6}$
(B)The coordinates of R are (1, 6, 3)
(C)The centroid of the triangle PQR is $\left(\frac{4}{3}, \frac{14}{3}, \frac{5}{3}\right)$
(D)The perimeter of the triangle PQR is $\sqrt{2} + \sqrt{6} + \sqrt{11}$

Q70·MathematicsSingle correctJEE Advanced 2024

Let

\gamma \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}{\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 entries in List-II. The correct option is:

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. $\sqrt{\frac{2}{3}}$

(A)(P) $\to$ (3) (Q) $\to$ (4) (R) $\to$ (1) (S) $\to$ (2)
(B)(P) $\to$ (5) (Q) $\to$ (4) (R) $\to$ (1) (S) $\to$ (2)
(C)(P) $\to$ (3) (Q) $\to$ (4) (R) $\to$ (1) (S) $\to$ (5)
(D)(P) $\to$ (3) (Q) $\to$ (1) (R) $\to$ (4) (S) $\to$ (5)

Q71·MathematicsMultiple correctJEE Advanced 2024

Let

R^{3}

denote the three dimensional space. Take two points

P = (1, 2, 3)

and

Q = (4, 2, 7)

. Let

\mathrm{dist}(X, Y)

denote the distance between two points

X

and

Y

in

R^{3}

. Let

S = \{X \in R^{3} : (\mathrm{dist}(X, P))^{2} - (\mathrm{dist}(X, Q))^{2} = 50\}

and

T = \{Y \in R^{3} : (\mathrm{dist}(Y, Q))^{2} - (\mathrm{dist}(Y, P))^{2} = 50\}

. Then which of the following statements is(are) TRUE ?

(A)There is a triangle whose area is 1 and all of whose vertices are from $S$ .
(B)There are two distinct points $L$ and $M$ in $T$ such that each point on the line segments $LM$ is also in $T$ .
(C)There are infinitely many rectangles of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$ .
(D)There is a square of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$ .

Q72·MathematicsSingle correctJEE Main 2024

Consider the line L passing through the points

(1, 2, 3)

and

(2, 3, 5)

. The distance of the point

\left(\dfrac{11}{3}, \dfrac{11}{3}, \dfrac{19}{3}\right)

from the line L along the line

\dfrac{3x-11}{2}=\dfrac{3y-11}{1}=\dfrac{3z-19}{2}

is equal to:

(A) $3$
(B) $5$
(C) $4$
(D) $6$

Q73·MathematicsSingle correctJEE Main 2024

Let the line 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}{2}

. Then which of the following points lies on L?

(A) $\left(-\frac{1}{3}, 1, 1\right)$
(B) $\left(-\frac{1}{3}, 1, -1\right)$
(C) $\left(-\frac{1}{3}, -1, -1\right)$
(D) $\left(-\frac{1}{3}, -1, 1\right)$

Q74·MathematicsSingle correctJEE Main 2024

The shortest distance between the line

\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:

(A) $\frac{187}{\sqrt{563}}$
(B) $\frac{178}{\sqrt{563}}$
(C) $\frac{185}{\sqrt{563}}$
(D) $\frac{179}{\sqrt{563}}$

Q75·MathematicsNumericalJEE Main 2024

The square of the distance of the image of the point

(6, 1, 5)

in the line

\dfrac{x-1}{3}=\dfrac{y}{2}=\dfrac{z-2}{4}

, from the origin is _______.

Q76·MathematicsSingle correctJEE Main 2024

If the shortest distance between the lines

L_1:\vec r=(2+\lambda)\hat i+(1-3\lambda)\hat j+(3+4\lambda)\hat k

,

\lambda\in\mathbb{R}

and

L_2:\vec r=2(1+\mu)\hat i+3(1+\mu)\hat j+(5+\mu)\hat k

,

\mu\in\mathbb{R}

is

\dfrac{m}{\sqrt n}

, where

\gcd(m,n)=1

, then the value of

m+n

equals:

(A) $384$
(B) $387$
(C) $377$
(D) $390$

Q77·MathematicsSingle correctJEE Main 2024

If the shortest distance between the lines

\dfrac{x-\lambda}{2}=\dfrac{y-4}{3}=\dfrac{z-3}{4}

and

\dfrac{x-2}{4}=\dfrac{y-4}{6}=\dfrac{z-7}{8}

is

\dfrac{13}{\sqrt{29}}

, then a value of

\lambda

is

(A) $-\tfrac{13}{25}$
(B) $\tfrac{13}{25}$
(C) $1$
(D) $-1$

Q78·MathematicsSingle correctJEE Main 2024

Let

P(x,y,z)

be a point in the first octant, whose projection in the xy-plane is the point

Q

. Let

OP=\gamma

; the angle between

OQ

and the positive x-axis be

\theta

; and the angle between

OP

and the positive z-axis be

\phi

, where

O

is the origin. Then the distance of

P

from the x-axis is:

(A) $\gamma\sqrt{1-\sin^2\phi\cos^2\theta}$
(B) $\gamma\sqrt{1+\cos^2\theta\sin^2\phi}$
(C) $\gamma\sqrt{1-\sin^2\theta\cos^2\phi}$
(D) $\gamma\sqrt{1+\cos^2\phi\sin^2\theta}$

Q79·MathematicsNumericalJEE Main 2024

Let

P(\alpha,\beta,\gamma)

be the image of the point

Q(1,6,4)

in the line

\dfrac{x}{1}=\dfrac{y-1}{2}=\dfrac{z-2}{3}

. Then

2\alpha+\beta+\gamma

is equal to _______ .

Q80·MathematicsNumericalJEE Main 2024

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

\frac{44}{\sqrt{30}}

, then the largest possible value of

|\lambda|

is equal to ______.

Q81·MathematicsSingle correctJEE Main 2024

Let

P(\alpha, \beta, \gamma)

be the image of the point

Q(3, -3, 1)

in the line

\frac{x-0}{1} = \frac{y-3}{1} = \frac{z-1}{-1}

and R be the point

(2, 5, -1)

. If the area of the triangle PQR is

\lambda

and

\lambda^2 = 14K

, then K is equal to:

(A)36
(B)72
(C)18
(D)81

Q82·MathematicsSingle correctJEE Main 2024

The shortest distance between the lines

\dfrac{x-3}{2}=\dfrac{y+15}{-7}=\dfrac{z-9}{5}

and

\dfrac{x+1}{2}=\dfrac{y-1}{1}=\dfrac{z-9}{-3}

is

(A) $6\sqrt{5}$
(B) $4\sqrt{3}$
(C) $5\sqrt{3}$
(D) $8\sqrt{3}$

Q83·MathematicsNumericalJEE Main 2024

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 points

(2,-5,11)

and

(-6,7,-5)

. Then the length of the line segment

PQ

is equal to _______.

Q84·MathematicsSingle correctJEE Main 2024

If the line

\dfrac{2-x}{3} = \dfrac{3y-2}{4\lambda+1} = 4 - z

makes a right angle with the line

\dfrac{x+3}{3\mu} = \dfrac{1-2y}{6} = \dfrac{5-z}{7}

, then

4\lambda + 9\mu

is equal to:

(A) $13$
(B) $4$
(C) $5$
(D) $6$

Q85·MathematicsSingle correctJEE Main 2024

Let

d

be the distance of the point of intersection of the lines

\dfrac{x+6}{3}=\dfrac{y}{2}=\dfrac{z+1}{1}

and

\dfrac{x-7}{4}=\dfrac{y-9}{3}=\dfrac{z-4}{2}

from the point

(7, 8, 9)

. Then

d^2 + 6

is equal to:

(A) $72$
(B) $69$
(C) $75$
(D) $78$

Q86·MathematicsSingle correctJEE Main 2024

Let

(\alpha,\beta,\gamma)

be the image of the point

(8,5,7)

in the line

\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z-2}{5}

. Then

\alpha+\beta+\gamma

is equal to:

(A)16
(B)18
(C)14
(D)20

Q87·MathematicsNumericalJEE Main 2024

Let the point

(-1,\alpha,\beta)

lie on the line of the shortest distance between the lines

\dfrac{x+2}{-3}=\dfrac{y-2}{4}=\dfrac{z-5}{2}

and

\dfrac{x+2}{-1}=\dfrac{y+6}{2}=\dfrac{z-1}{0}

. Then

(\alpha-\beta)^2

is equal to __________.

Q88·MathematicsNumericalJEE Main 2024

Consider a line L passing through the points

P(1,2,1)

and

Q(2,1,-1)

. If the mirror image of the point

A(2,2,2)

in the line L is

(\alpha,\beta,\gamma)

, then

\alpha+\beta+6\gamma

is equal to

Q89·MathematicsSingle correctJEE Main 2024

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 point

P

, be

(\alpha,\beta,\gamma)

. Then

\alpha^2+\beta^2+\gamma^2

is equal to:

(A) $155$
(B) $150$
(C) $160$
(D) $165$

Q90·MathematicsNumericalJEE Main 2024

If the shortest distance between the lines

\dfrac{x+2}{2}=\dfrac{y+3}{3}=\dfrac{z-5}{4}

and

\dfrac{x-3}{1}=\dfrac{y-2}{-3}=\dfrac{z+4}{2}

is

\dfrac{38}{3\sqrt5}k

and

\displaystyle\int_0^k[x^2]\,dx=\alpha-\sqrt\alpha

, where

[x]

denotes the greatest integer function, then

6\alpha^3

is equal to ___

Q91·MathematicsSingle correctJEE Main 2024

Let P be the point of intersection of the lines

\dfrac{x-2}{1}=\dfrac{y-4}{5}=\dfrac{z-2}{2}

and

\dfrac{x-3}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{3}

. Then the shortest distance of P from the line

4x=2y=z

is

(A) $\tfrac{5\sqrt{14}}{7}$
(B) $\tfrac{\sqrt{14}}{7}$
(C) $\tfrac{3\sqrt{14}}{7}$
(D) $\tfrac{6\sqrt{14}}{7}$

Q92·MathematicsSingle correctJEE Main 2024

If the shortest distance between the lines

\dfrac{x-\lambda}{-2}=\dfrac{y-2}{1}=\dfrac{z-1}{1}

and

\dfrac{x-\sqrt3}{1}=\dfrac{y-1}{-2}=\dfrac{z-2}{2}

is

1

, then the sum of all possible values of

\lambda

is:

(A) $0$
(B) $2\sqrt3$
(C) $3\sqrt3$
(D) $-2\sqrt3$

Q93·MathematicsSingle correctJEE Main 2024

If the mirror image of the point

P(3,4,9)

in the line

\dfrac{x-1}{3}=\dfrac{y+1}{2}=\dfrac{z-2}{1}

is

(\alpha,\beta,\gamma)

, then

14(\alpha+\beta+\gamma)

is:

(A)102
(B)138
(C)108
(D)132

Q94·MathematicsNumericalJEE Main 2024

Let the line of the shortest distance between the lines

L_1:\vec r=(\hat i+2\hat j+3\hat k)+\lambda(\hat i-\hat j+\hat k)

and

L_2:\vec r=(4\hat i+5\hat j+6\hat k)+\mu(\hat i+\hat j-\hat k)

intersect

L_1

and

L_2

at

P

and

Q

respectively. If

(\alpha,\beta,\gamma)

is the midpoint of the line segment

PQ

, then

2(\alpha+\beta+\gamma)

is equal to ___

Q95·MathematicsSingle correctJEE Main 2024

Let P and Q be the points on the line

\dfrac{x+3}{8}=\dfrac{y-4}{2}=\dfrac{z+1}{2}

which are at a distance of 6 units from the point R(1, 2, 3). If the centroid of the triangle PQR is

(\alpha,\beta,\gamma)

, then

\alpha^2+\beta^2+\gamma^2

is:

(A)26
(B)36
(C)18
(D)24

Q96·MathematicsSingle correctJEE Main 2024

The distance of the point

Q(0,2,-2)

from the line passing through the point

P(5,-4,3)

and perpendicular to the lines

\vec{r}=\left(-3\hat{i}+2\hat{k}\right)+\lambda\left(2\hat{i}+3\hat{j}+5\hat{k}\right)

,

\lambda\in R

and

\vec{r}=\left(\hat{i}-2\hat{j}+\hat{k}\right)+\mu\left(-\hat{i}+3\hat{j}+2\hat{k}\right)

,

\mu\in R

is

(A) $\sqrt{86}$
(B) $\sqrt{20}$
(C) $\sqrt{54}$
(D) $\sqrt{74}$

Q97·MathematicsNumericalJEE Main 2024

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 AB lies at a distance of 21 units, from the point A. The distance between the points

P(a,b,c)

and

Q(4,-12,3)

is equal to ______.

Q98·MathematicsSingle correctJEE Main 2024

The shortest distance between lines

L_1

and

L_2

, where

L_1:\dfrac{x-1}{2}=\dfrac{y+1}{-3}=\dfrac{z+4}{2}

and

L_2

is the line passing through the points

A(-4,4,3)

,

B(-1,6,3)

and perpendicular to the line

\dfrac{x-3}{-2}=\dfrac{y}{3}=\dfrac{z-1}{1}

, is

(A) $\dfrac{121}{\sqrt{221}}$
(B) $\dfrac{24}{\sqrt{117}}$
(C) $\dfrac{141}{\sqrt{221}}$
(D) $\dfrac{42}{\sqrt{117}}$

Q99·MathematicsSingle correctJEE Main 2024

Let

(\alpha,\beta,\gamma)

be the mirror image of the point

(2,3,5)

in the line

\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}

. Then

2\alpha+3\beta+4\gamma

is equal to

(A)32
(B)33
(C)31
(D)34

Q100·MathematicsNumericalJEE Main 2024

Let Q and R be the feet of perpendiculars from the point

P(a,a,a)

on the lines

x=y

,

z=1

and

x=-y

,

z=-1

respectively. If

\angle QPR

is a right angle, then

12a^2

is equal to ______.

Q101·MathematicsSingle correctJEE Main 2024

Let

L_1:\vec{r}=(\hat{i}-\hat{j}+2\hat{k})+\lambda(\hat{i}-\hat{j}+2\hat{k})

,

\lambda\in R

;

L_2:\vec{r}=(\hat{j}-\hat{k})+\mu(3\hat{i}+\hat{j}+p\hat{k})

,

\mu\in R

and

L_3:\vec{r}=\delta(\ell\hat{i}+m\hat{j}+n\hat{k})

,

\delta\in R

be three lines such that

L_1

is perpendicular to

L_2

and

L_3

is perpendicular to both

L_1

and

L_2

. Then the point which lies on

L_3

is:

(A) $(-1,7,4)$
(B) $(-1,-7,4)$
(C) $(1,7,-4)$
(D) $(1,-7,4)$

Q102·MathematicsSingle correctJEE Main 2024

Let

(\alpha,\beta,\gamma)

be the foot of perpendicular from the point

(1,2,3)

on the line

\dfrac{x+3}{5}=\dfrac{y-1}{2}=\dfrac{z+4}{3}

. Then

19(\alpha+\beta+\gamma)

is equal to:

(A) $102$
(B) $101$
(C) $99$
(D) $100$

Q103·MathematicsNumericalJEE Main 2024

If

d_1

is the shortest distance between the lines

x+1=2y=-12z

,

x=y+2=6z-6

and

d_2

is the shortest distance between the lines

\dfrac{x-1}{2}=\dfrac{y+8}{-7}=\dfrac{z-4}{5}

,

\dfrac{x-1}{2}=\dfrac{y-2}{1}=\dfrac{z-6}{-3}

, then the value of

\dfrac{32\sqrt3\,d_1}{d_2}

is:

Q104·MathematicsNumericalJEE Main 2024

Let a line passing through the point

(-1,2,3)

intersect the lines

L_1:\dfrac{x-1}{3}=\dfrac{y-2}{2}=\dfrac{z+1}{-2}

at

M(\alpha,\beta,\gamma)

and

L_2:\dfrac{x+2}{-3}=\dfrac{y-2}{-2}=\dfrac{z-1}{4}

at

N(a,b,c)

. Then the value of

\dfrac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2}

equals ______.

Q105·MathematicsSingle correctJEE Main 2024

Let

P(3,2,3)

,

Q(4,6,2)

and

R(7,3,2)

be the vertices of

\triangle PQR

. Then, the angle

\angle QPR

is:

(A) $\dfrac{\pi}{6}$
(B) $\cos^{-1}\left(\dfrac{7}{18}\right)$
(C) $\cos^{-1}\left(\dfrac{1}{18}\right)$
(D) $\dfrac{\pi}{3}$

Q106·MathematicsNumericalJEE Main 2024

Let

O

be the origin, and

M

and

N

be the points on the lines

\dfrac{x-5}{4}=\dfrac{y-4}{1}=\dfrac{z-5}{3}

and

\dfrac{x+8}{12}=\dfrac{y+2}{5}=\dfrac{z+11}{9}

respectively such that

MN

is the shortest distance between the given lines. Then

\overrightarrow{OM}\cdot\overrightarrow{ON}

is equal to ___.

Q107·MathematicsNumericalJEE Main 2024

A line with direction ratios

2,1,2

meets the lines

x=y+2=z

and

x+2=2y=2z

respectively at the point P and Q. If the length of the perpendicular from the point

(1,2,12)

to the line PQ is

l

, then

l^2

is ______.

Q108·MathematicsSingle correctJEE Main 2024

Let PQR be a triangle with

R(-1,4,2)

. Suppose

M(2,1,2)

is the mid point of PQ. The distance of the centroid of

\triangle PQR

from the point of intersection of the line

\dfrac{x-2}{0}=\dfrac{y}{2}=\dfrac{z+3}{-1}

and

\dfrac{x-1}{1}=\dfrac{y+3}{-3}=\dfrac{z+1}{1}

is

(A) $69$
(B) $9$
(C) $\sqrt{69}$
(D) $\sqrt{99}$

Q109·MathematicsSingle correctJEE Main 2024

If the shortest distance between the lines

\dfrac{x-4}{1}=\dfrac{y+1}{2}=\dfrac{z}{-3}

and

\dfrac{x-\lambda}{2}=\dfrac{y+1}{4}=\dfrac{z-2}{-5}

is

\dfrac{6}{\sqrt5}

, then the sum of all possible values of

\lambda

is:

(A)5
(B)8
(C)7
(D)10

Q110·MathematicsNumericalJEE Main 2024

The lines

\dfrac{x-2}{1}=\dfrac{y}{-2}=\dfrac{z-7}{8}

and

\dfrac{x+3}{4}=\dfrac{y+2}{3}=\dfrac{z+2}{1}

intersect at the point P. If the distance of P from the line

\dfrac{x+1}{2}=\dfrac{y-1}{3}=\dfrac{z-1}{1}

is

l

, then

14l^2

is equal to __________.

Q111·MathematicsSingle correctJEE Main 2024

The distance of the point

(7,-2,11)

from the line

\dfrac{x-6}{1}=\dfrac{y-4}{0}=\dfrac{z-8}{3}

along the line

\dfrac{x-5}{2}=\dfrac{y-1}{-3}=\dfrac{z-5}{6}

, is:

(A)12
(B)14
(C)18
(D)21

Q112·MathematicsSingle correctJEE Main 2024

Let the image of the point

(1,0,7)

in the line

\dfrac{x}{1}=\dfrac{y-1}{2}=\dfrac{z-2}{3}

be the point

(\alpha,\beta,\gamma)

. Then which one of the following points lies on the line passing through

(\alpha,\beta,\gamma)

and making angles

\dfrac{2\pi}{3}

and

\dfrac{3\pi}{4}

with y-axis and z-axis respectively and an acute angle with x-axis ?

(A) $\left(1,-2,1+\sqrt2\right)$
(B) $\left(1,2,1-\sqrt2\right)$
(C) $\left(3,4,3-2\sqrt2\right)$
(D) $\left(3,-4,3+2\sqrt2\right)$

Q113·MathematicsSingle correctJEE Advanced 2023

Let Q be the cube with the set of vertices

\{(x_1, x_2, x_3) \in R^3 : x_1, x_2, x_3 \in \{0, 1\}\}

. Let F be the set of all twelve lines containing the diagonals of the six faces of the cube Q. Let S be the set of all four lines containing the main diagonals of the cube Q; for instance, the line passing through the vertices (0, 0, 0) and (1, 1, 1) is in S. For lines

\ell_1

and

\ell_2

, let

d(\ell_1, \ell_2)

denote the shortest distance between them. Then the maximum value of

d(\ell_1, \ell_2)

as

\ell_1

varies over F and

\ell_2

varies over S, is

(A) $\frac{1}{\sqrt{6}}$
(B) $\frac{1}{\sqrt{8}}$
(C) $\frac{1}{\sqrt{3}}$
(D) $\frac{1}{\sqrt{12}}$

Q114·MathematicsSingle correctJEE Advanced 2023

Let

\ell_1

and

\ell_2

be the lines

\vec{r_1} = \lambda\left(\hat{i} + \hat{j} + \hat{k}\right)

and

\vec{r_2} = \left(\hat{j} - \hat{k}\right) + \mu\left(\hat{i} + \hat{k}\right)

, 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(H_0)

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. The correct option is:

List – I	List – II
P.The value of $d(H_0)$ 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. $\frac{1}{\sqrt{2}}$

(A)(P) → (2) (Q) → (4) (R) → (5) (S) → (1)
(B)(P) → (5) (Q) → (4) (R) → (3) (S) → (1)
(C)(P) → (2) (Q) → (1) (R) → (3) (S) → (2)
(D)(P) → (5) (Q) → (1) (R) → (4) (S) → (2)

Q115·MathematicsSingle correctJEE Main 2023

Let

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}{1}

and

\frac{x+\lambda}{3}=\frac{y}{-4}=\frac{z-6}{0}

is

13

. Then

8\left|\sum_{\lambda\in S}\lambda\right|

is equal to

(A)304
(B)308
(C)306
(D)302

Q116·MathematicsNumericalJEE Main 2023

If the line

x = y = z

intersects the line

x\sin A + y\sin B + z\sin C - 18 = 0 = x\sin 2A + y\sin 2B + z\sin 2C - 9

, where

A, B, C

are the angles of a triangle

ABC

, then

80\left(\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}\right)

is equal to____

Q117·MathematicsNumericalJEE Main 2023

Let the plane

P

contain the line

2x + y - z - 3 = 0 = 5x - 3y + 4z + 9

and be parallel to the line

\frac{x + 2}{2} = \frac{3 - y}{-4} = \frac{z - 7}{5}

. Then the distance of the point

A(8, -1, -19)

from the plane

P

measured parallel to the line

\frac{x}{-3} = \frac{y - 5}{4} = \frac{2 - z}{-12}

is equal to____

Q118·MathematicsSingle correctJEE Main 2023

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, \gamma)

. Then the distance of

Q

from the origin is:

(A) $\sqrt{29}$
(B) $\sqrt{35}$
(C) $\sqrt{42}$
(D) $\sqrt{38}$

Q119·MathematicsSingle correctJEE Main 2023

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 passes through the point

(A) $(1, -2, 1)$
(B) $(0, 5, -2)$
(C) $(-2, 5, 0)$
(D) $(2, 0, 1)$

Q120·MathematicsSingle correctJEE Main 2023

The line, that is coplanar to the line

\dfrac{x+3}{-3} = \dfrac{y-1}{1} = \dfrac{z-5}{5}

, is

(A) $\dfrac{x+1}{1} = \dfrac{y-2}{2} = \dfrac{z-5}{5}$
(B) $\dfrac{x+1}{-1} = \dfrac{y-2}{2} = \dfrac{z-5}{5}$
(C) $\dfrac{x+1}{-1} = \dfrac{y-2}{2} = \dfrac{z-5}{4}$
(D) $\dfrac{x-1}{-1} = \dfrac{y-2}{2} = \dfrac{z-5}{5}$

Q121·MathematicsSingle correctJEE Main 2023

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 between the lines

\vec r=(\hat i-\hat j)+\lambda(2\hat i+\hat k)

and

\vec r=(2\hat i-\hat j)+\mu(\hat i-\hat j+\hat k)

is:

(A) $3\sqrt6$
(B) $3\sqrt5$
(C) $2\sqrt6$
(D) $2\sqrt5$

Q122·MathematicsSingle correctJEE Main 2023

Let the equation of the plane passing through the line of intersection of the planes

x+2y+az=2

and

x-y+z=3

be

5x-11y+bz=6a-1

. For

c\in\mathbb{Z}

, if the distance of this plane from the point

(a,-c,c)

is

\dfrac{2}{\sqrt a}

, then

\dfrac{a+b}{c}

is equal to:

(A) $-2$
(B) $2$
(C) $-4$
(D) $4$

Q123·MathematicsSingle correctJEE Main 2023

Let N be the foot of perpendicular from the point

P(1, -2, 3)

on the line passing through the points

(4, 5, 8)

and

(1, -7, 5)

. Then the distance of N from the plane

2x - 2y + z + 5 = 0

is

(A)6
(B)9
(C)7
(D)8

Q124·MathematicsNumericalJEE Main 2023

Let the image of the point

\left(\dfrac{5}{3},\dfrac{5}{3},\dfrac{8}{3}\right)

in the plane

x-2y+z-2=0

be

P

. If the distance of point

Q(6,-2,\alpha),\,\alpha>0

from

P

is

13

, then

\alpha

is equal to _____.

Q125·MathematicsNumericalJEE Main 2023

Let the plane

x+3y-2z+6=0

meet the co-ordinate axes at the points A, B, C. If the orthocentre of the triangle ABC is

\left(\alpha,\beta,\dfrac67\right)

, then

98(\alpha+\beta)^2

is equal to _________.

Q126·MathematicsSingle correctJEE Main 2023

Let the plane P:

4x-y+z=10

be rotated by an angle

\dfrac{\pi}{2}

about its line of intersection with the plane

x+y-z=4

. If

\alpha

is the distance of the point

(2,3,-4)

from the new position of the plane P, then

35\alpha

is equal to

(A)90
(B)85
(C)126
(D)170

Q127·MathematicsSingle correctJEE Main 2023

Let the lines

l_1:\dfrac{x+5}{3}=\dfrac{y+4}{1}=\dfrac{z-\alpha}{-2}

and

l_2:3x+2y+z-2=0=x-3y+2z-13

be coplanar. If the point P

(a,b,c)

on

l_1

is nearest to the point Q

(-4,-3,2)

, then

|a|+|b|+|c|

is equal to

(A)12
(B)14
(C)10
(D)8

Q128·MathematicsSingle correctJEE Main 2023

Let

\vec{a}

be a non-zero vector parallel to the line of intersection of the two planes described by

\hat{i} + \hat{j}, \hat{i} + \hat{k}

and

\hat{i} - \hat{j}, \hat{j} - \hat{k}

. If

\theta

is the angle between the vector

\vec{a}

and the vector

\vec{b} = 2\hat{i} - 2\hat{j} + \hat{k}

and

\vec{a} \cdot \vec{b} = 6

, then the ordered pair

\left(\theta, |\vec{a} \times \vec{b}|\right)

is equal to

(A) $\left(\frac{\pi}{4}, 3\sqrt{6}\right)$
(B) $\left(\frac{\pi}{3}, 3\sqrt{6}\right)$
(C) $\left(\frac{\pi}{3}, 6\right)$
(D) $\left(\frac{\pi}{4}, 6\right)$

Q129·MathematicsNumericalJEE Main 2023

Let the line

\ell:x=\dfrac{1-y}{-2}=\dfrac{z-3}{\lambda}

,

\lambda\in\mathbb{R}

meet the plane

P:x+2y+3z=4

at the point

(\alpha,\beta,\gamma)

. If the angle between the line

\ell

and the plane

P

is

\cos^{-1}\left(\sqrt{\dfrac{5}{14}}\right)

, then

\alpha+2\beta+6\gamma

is equal to

Q130·MathematicsSingle correctJEE Main 2023

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

A(3,4,\alpha)

and

B(2,\alpha,a)

from the plane

P

are

2

and

3

respectively, then the positive value of

a

is

(A) $6$
(B) $4$
(C) $3$
(D) $5$

Q131·MathematicsSingle correctJEE Main 2023

If equation of the plane that contains the point

(-2, 3, 5)

and is perpendicular to each of the planes

2x + 4y + 5z = 8

and

3x - 2y + 3z = 5

is

\alpha x + \beta y + \gamma z + 97 = 0

then

\alpha + \beta + \gamma =

(A)18
(B)17
(C)16
(D)15

Q132·MathematicsNumericalJEE Main 2023

Let a line

l

pass through the origin and be perpendicular to the lines

l_1 : \vec{r} = (\hat{i} - 11\hat{j} - 7\hat{k}) + \lambda(\hat{i} + 2\hat{j} + 3\hat{k}), \lambda \in \mathbb{R}

and

l_2 : \vec{r} = (-\hat{i} + \hat{k}) + \mu(2\hat{i} + 2\hat{j} + \hat{k}), \mu \in \mathbb{R}

. If

P

is the point of intersection of

l

and

l_1

, and

Q(\alpha, \beta, \gamma)

is the foot of perpendicular from

P

on

l_2

, then

9(\alpha + \beta + \gamma)

is equal to _______ .

Q133·MathematicsSingle correctJEE Main 2023

Let the line passing through the points

P(2,-1,2)

and

Q(5,3,4)

meet the plane

x-y+z=4

at the point

R

. Then the distance of the point

R

from the plane

x+2y+3z+2=0

measured parallel to the line

\dfrac{x-7}{2}=\dfrac{y+3}{2}=\dfrac{z-2}{1}

is equal to

(A) $\sqrt{31}$
(B) $\sqrt{189}$
(C) $\sqrt{61}$
(D) $3$

Q134·MathematicsSingle correctJEE Main 2023

Let

(\alpha, \beta, \gamma)

be the image of the point

P(2, 3, 5)

in the plane

2x + y - 3z = 6

. Then

\alpha + \beta + \gamma

is equal to

(A) $10$
(B) $5$
(C) $12$
(D) $9$

Q135·MathematicsSingle correctJEE Main 2023

Let the image of the point

P(1,2,6)

in the plane passing through the points

A(1,2,0)

,

B(1,4,1)

and

C(0,5,1)

be

Q(\alpha,\beta,\gamma)

. Then

(\alpha^2+\beta^2+\gamma^2)

is equal to:

(A) $65$
(B) $70$
(C) $76$
(D) $62$

Q136·MathematicsSingle correctJEE Main 2023

Let P be the point of intersection of the line

\dfrac{x+3}{3}=\dfrac{y+2}{1}=\dfrac{1-z}{2}

and the plane

x+y+z=2

. If the distance of the point P from the plane

3x-4y+12z=32

is q, then q and 2q are the roots of the equation

(A) $x^2-18x-72=0$
(B) $x^2+18x+72=0$
(C) $x^2-18x+72=0$
(D) $x^2+18x-72=0$

Q137·MathematicsSingle correctJEE Main 2023

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-y}{3}=\frac{z-6}{1}

at the points A and B respectively. Then the distance of the mid-point of the line segment AB from the plane

2x-2y+z=14

is:

(A) $4$
(B) $\frac{10}{3}$
(C) $3$
(D) $\frac{11}{3}$

Q138·MathematicsSingle correctJEE Main 2023

Let two vertices of triangle ABC be

(2,4,6)

and

(0,-2,-5)

, and its centroid be

(2,1,-1)

. If the image of third vertex in the plane

x+2y+4z=11

is

(\alpha,\beta,\gamma)

, then

\alpha\beta+\beta\gamma+\gamma\alpha

is equal to

(A) $72$
(B) $74$
(C) $76$
(D) $70$

Q139·MathematicsSingle correctJEE Main 2023

The shortest distance between the lines

\dfrac{x+2}{1}=\dfrac{y}{-2}=\dfrac{z-5}{2}

and

\dfrac{x-4}{1}=\dfrac{y-1}{2}=\dfrac{z+3}{0}

is

(A) $6$
(B) $9$
(C) $7$
(D) $8$

Q140·MathematicsNumericalJEE Main 2023

Let the foot of perpendicular from the point

A(4,3,1)

on the plane

P:x-y+2z+3=0

be N. If

B(5,\alpha,\beta),\ \alpha,\beta\in\mathbb{Z}

is a point on plane P such that the area of the triangle ABN is

3\sqrt{2}

, then

\alpha^2+\beta^2+\alpha\beta

is equal to _______ .

Q141·MathematicsNumericalJEE Main 2023

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 equal distance from the plane

2x+3y-6z+7=0

. If

\lambda_{1}>\lambda_{2}

, then the distance of the point

(\lambda_{1}-\lambda_{2},\lambda_{2},\lambda_{1})

from the line

\frac{x-5}{1}=\frac{y-1}{2}=\frac{z+7}{2}

is

Q142·MathematicsSingle correctJEE Main 2023

For

a,b\in\mathbb{Z}

and

|a-b|\leq10

, let the angle between the plane

P:ax+y-z=b

and the line

l:x-1=a-y=z+1

be

\cos^{-1}\left(\dfrac{1}{3}\right)

. If the distance of the point

(6,-6,4)

from the plane P is

3\sqrt{6}

, then

a^{4}+b^{2}

is equal to

(A)25
(B)85
(C)48
(D)32

Q143·MathematicsNumericalJEE Main 2023

Let

P_1

be the plane

3x-y-7z=11

and

P_2

be the plane passing through the points

(2,-1,0)

,

(2,0,-1)

and

(5,1,1)

. If the foot of the perpendicular drawn from the point

(7,4,-1)

on the line of intersection of the planes

P_1

and

P_2

is

(\alpha,\beta,\gamma)

, then

\alpha+\beta+\gamma

is equal to

Q144·MathematicsSingle correctJEE Main 2023

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

(A) $3\sqrt{6}$
(B) $6\sqrt{3}$
(C) $6\sqrt{2}$
(D) $2\sqrt{6}$

Q145·MathematicsSingle correctJEE Main 2023

Let P be the plane passing through the line

\dfrac{x-1}{1}=\dfrac{y-2}{-3}=\dfrac{z+5}{7}

and the point

(2,4,-3)

. If the image of the point

(-1,3,4)

in the plane P is

(\alpha,\beta,\gamma)

, then

\alpha+\beta+\gamma

is equal to

(A)12
(B)11
(C)9
(D)10

Q146·MathematicsSingle correctJEE Main 2023

If the equation of the plane containing the line

x+2y+3z-4=0=2x+y-z+5

and perpendicular to the plane

\vec{r}=(\hat{i}-\hat{j})+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}-2\hat{j}+3\hat{k})

is

ax+by+cz=4

, then

(a-b+c)

is equal to

(A)20
(B)24
(C)22
(D)18

Q147·MathematicsSingle correctJEE Main 2023

One vertex of a rectangular parallelopiped is at the origin

O

and the lengths of its edges along

x,\,y

and

z

axes are

3,\,4

and

5

units respectively. Let

P

be the vertex

(3,4,5)

. Then the shortest distance between the diagonal

OP

and an edge parallel to the

z

axis, not passing through

O

or

P

, is:

(A) $\dfrac{12}{\sqrt5}$
(B) $\dfrac{12}{5\sqrt5}$
(C) $12\sqrt5$
(D) $\dfrac{12}{5}$

Q148·MathematicsNumericalJEE Main 2023

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 the minimum value of

8\alpha\beta

is

Q149·MathematicsNumericalJEE Main 2023

Let

Q

be the foot of perpendicular from the point

P(0,2,3)

on the plane

2x-y+z=9

. If the coordinates of the point

R

are

(6,10,7)

, then the square of the area of the triangle

PQR

is _____.

Q150·MathematicsSingle correctJEE Main 2023

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})=-5

. If

P

passes through the point

(0,2,-2)

, then the square of distance of the point

(12,12,18)

from the plane

P

is

(A)620
(B)156
(C)310
(D)144

Q151·MathematicsSingle correctJEE Main 2023

Let the line

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 the plane

2x+y-3z=4

. Then the distance of the point

P(1,-9,2)

from the line

L

is

(A)9
(B) $\sqrt{54}$
(C) $\sqrt{69}$
(D) $\sqrt{74}$

Q152·MathematicsSingle correctJEE Main 2023

If the equation of the plane passing through the line of intersection of the planes

2x-y+z=3

and

4x-3y+5z+9=0

and parallel to the line

\dfrac{x+1}{-2}=\dfrac{y+3}{4}=\dfrac{z-2}{5}

is

ax+by+cz+6=0

, then

a+b+c

is equal to:

(A)14
(B)12
(C)13
(D)15

Q153·MathematicsNumericalJEE Main 2023

Let

\alpha x+\beta y+\gamma z=1

be the equation of a plane through the point

(3,-2,5)

and perpendicular to the line joining the points

(1,2,3)

and

(-2,3,5)

. Then the value of

\alpha\beta\gamma

is equal to

Q154·MathematicsSingle correctJEE Main 2023

Let the plane P pass through the intersection of the planes

x+3y-z=2

and

x+2y+3z=6

and be perpendicular to the plane

2x+y-z=0

. If d is the distance of P from the point

(-7,1,1)

, then

d^2

is equal to:

(A) $\dfrac{250}{83}$
(B) $\dfrac{250}{82}$
(C) $\dfrac{15}{53}$
(D) $\dfrac{25}{83}$

Q155·MathematicsSingle correctJEE Main 2023

The shortest distance between the lines

\frac{x-5}{1}=\frac{y-2}{2}=\frac{z-4}{-3}

and

\frac{x+3}{1}=\frac{y+5}{4}=\frac{z-1}{-5}

is

(A) $5\sqrt{3}$
(B) $7\sqrt{3}$
(C) $6\sqrt{3}$
(D) $4\sqrt{3}$

Q156·MathematicsNumericalJEE Main 2023

The point of intersection C of the plane

8x+y+2z=0

and the line joining the points

A(-3,-6,1)

and

B(2,-4,-3)

divides the line segment AB internally in the ratio k:1. If a, b, c (

[a],[b],[c]

are coprime) are the direction ratios of the perpendicular from the point C on the line

\dfrac{1-x}{1}=\dfrac{y+4}{2}=\dfrac{z+2}{3}

, then

|a+b+c|

is equal to

Q157·MathematicsSingle correctJEE Main 2023

Let the image of the point

P(2,-1,3)

in the plane

x+2y-z=0

be

Q

. Then the distance of the plane

3x+2y+z+29=0

from the point

Q

is

(A) $\frac{24\sqrt{2}}{7}$
(B) $2\sqrt{14}$
(C) $3\sqrt{14}$
(D) $\frac{22\sqrt{2}}{7}$

Q158·MathematicsSingle correctJEE Main 2023

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 distance of

P

from the point

(3,-2,2)

is

(A) $5$
(B) $4$
(C) $7$
(D) $6$

Q159·MathematicsSingle correctJEE Main 2023

The foot of perpendicular from the origin

O

to a plane

P

which meets the co-ordinate axes at the points

A,B,C

is

(2,a,4),\ a\in\mathbb{N}

. If the volume of the tetrahedron

OABC

is

144

unit

^3

, then which of the following points is NOT on

P

?

(A) $(0,6,3)$
(B) $(0,4,4)$
(C) $(2,2,4)$
(D) $(3,0,4)$

Q160·MathematicsSingle correctJEE Main 2023

Let the shortest distance between the lines

L:\frac{x-5}{-2}=\frac{y-\lambda}{0}=\frac{z+\lambda}{1},\ \lambda\ge 0

and

L_1:x+1=y-1=4-z

be

2\sqrt{6}

. If

(\alpha,\beta,\gamma)

lies on

L

, then which of the following is NOT possible ?

(A) $\alpha-2\gamma=19$
(B) $2\alpha+\gamma=7$
(C) $2\alpha-\gamma=9$
(D) $\alpha+2\gamma=24$

Q161·MathematicsSingle correctJEE Main 2023

Let the plane

P:8x+\alpha_1 y+\alpha_2 z+12=0

be parallel to the line

L:\dfrac{x+2}{2}=\dfrac{y-3}{3}=\dfrac{z+4}{5}

. If the intercept of

P

on the

y

-axis is 1, then the distance between

P

and

L

is:

(A) $\sqrt{\dfrac72}$
(B) $\sqrt{\dfrac27}$
(C) $\dfrac{6}{\sqrt{14}}$
(D) $\sqrt{14}$

Q162·MathematicsNumericalJEE Main 2023

Let the line

L: \frac{x-1}{2} = \frac{y+1}{-1} = \frac{z-3}{1}

intersect the plane

2x + y + 3z = 16

at the point

P

. Let the point

Q

be the foot of perpendicular from the point

R(1, -1, -3)

on the line

L

. If

\alpha

is the area of triangle

PQR

, then

\alpha^2

is equal to

Q163·MathematicsNumericalJEE Main 2023

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{k}) = 15

. Let

L

be the line that meets

P_2

at the point

(4, -2, 5)

and makes an angle

\theta

with the normal of

P_2

. If

\alpha

is the angle between

L

and

P_2

, then

(\tan^2 \theta)(\cot^2 \alpha)

is equal to

Q164·MathematicsNumericalJEE Main 2023

If

\lambda_1<\lambda_2

are two values of

\lambda

such that the angle between the planes

P_1:\vec r\cdot(3\hat i-5\hat j+\hat k)=7

and

P_2:\vec r\cdot(\lambda\hat i+\hat j-3\hat k)=9

is

\sin^{-1}\left(\dfrac{2\sqrt6}{5}\right)

, then the square of the length of perpendicular from the point

(38\lambda_1,10\lambda_2,2)

to the plane

P_1

is

Q165·MathematicsNumericalJEE Main 2023

If the equation of the plane passing through the point

(1,1,2)

and perpendicular to the line

x-3y+2z-1=0=4x-y+z

is

Ax+By+Cz=1

, then

140(C-B+A)

is equal to

Q166·MathematicsNumericalJEE Main 2023

Let a line

L

pass through the point

P(2, 3, 1)

and be parallel to the line

x + 3y - 2z - 2=0=x - y + 2z

. If the distance of

L

from the point

(5, 3, 8)

is

\alpha

, then

3\alpha^2

is equal to _______.

Q167·MathematicsSingle correctJEE Main 2023

Let a unit vector

\overrightarrow{OP}

make angles

\alpha,\beta,\gamma

with the positive directions of the co-ordinate axes OX, OY, OZ respectively, where

\beta\in\left(0,\dfrac{\pi}{2}\right)

. If

\overrightarrow{OP}

is perpendicular to the plane through points

(1,2,3),(2,3,4)

and

(1,5,7)

, then which one of the following is true?

(A) $\alpha\in\left(0,\dfrac{\pi}{2}\right)$ and $\gamma\in\left(0,\dfrac{\pi}{2}\right)$
(B) $\alpha\in\left(0,\dfrac{\pi}{2}\right)$ and $\gamma\in\left(\dfrac{\pi}{2},\pi\right)$
(C) $\alpha\in\left(\dfrac{\pi}{2},\pi\right)$ and $\gamma\in\left(\dfrac{\pi}{2},\pi\right)$
(D) $\alpha\in\left(\dfrac{\pi}{2},\pi\right)$ and $\gamma\in\left(0,\dfrac{\pi}{2}\right)$

Q168·MathematicsSingle correctJEE Main 2023

If a plane passes through the points

(-1, k, 0), (2, k, -1), (1, 1, 2)

and is parallel to the line

\dfrac{x - 1}{1}=\dfrac{2y + 1}{2}=\dfrac{z + 1}{-1}

, then the value of

\dfrac{k^2 + 1}{(k - 1)(k - 2)}

is:

(A) $\dfrac{17}{5}$
(B) $\dfrac{13}{6}$
(C) $\dfrac{6}{13}$
(D) $\dfrac{5}{17}$

Q169·MathematicsSingle correctJEE Main 2023

A vector

\vec{v}

in the first octant is inclined to the x-axis at

60^{\circ}

, to the y-axis at

45^{\circ}

and to the z-axis at an acute angle. If a plane passing through the points

(\sqrt{2}, -1, 1)

and

(a, b, c)

is normal to

\vec{v}

, then:

(A) $\sqrt{2}a + b + c = 1$
(B) $a + \sqrt{2}b + c = 1$
(C) $a + b + \sqrt{2}c = 1$
(D) $\sqrt{2}a - b + c = 1$

Q170·MathematicsSingle correctJEE Main 2023

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_1

and the line

\dfrac{x+1}{2}=\dfrac{y+4}{-3}=\dfrac{z}{2}

is:

(A) $\dfrac{11}{3}$
(B) $\dfrac{19}{3}$
(C) $7$
(D) $9$

Q171·MathematicsNumericalJEE Main 2023

Let the co-ordinates of one vertex of

\triangle ABC

be

A(0, 2, \alpha)

and the other two vertices lie on the line

\dfrac{x + \alpha}{5} = \dfrac{y - 1}{2} = \dfrac{z + 4}{3}

. For

\alpha \in \mathbb{Z}

, if the area of

\triangle ABC

is 21 sq. units and the line segment

BC

has length

2\sqrt{21}

units, then

\alpha^2

is equal to ________ .

Q172·MathematicsNumericalJEE Main 2023

Let the equation of the plane

P

containing the line

x + 10 = \dfrac{8 - y}{2} = z

be

ax + by + 3z = 2(a + b)

and the distance of the plane

P

from the point

(1, 27, 7)

be

c

. Then

a^2 + b^2 + c^2

is equal to ________ .

Q173·MathematicsSingle correctJEE Main 2023

The shortest distance between the lines

\dfrac{x-1}{2}=\dfrac{y-2}{1}=\dfrac{z-6}{-3}

and

\dfrac{x-1}{2}=\dfrac{y+8}{-7}=\dfrac{z-4}{5}

is:

(A) $5\sqrt3$
(B) $2\sqrt3$
(C) $3\sqrt3$
(D) $4\sqrt3$

Q174·MathematicsSingle correctJEE Main 2023

If the lines

\dfrac{x-1}{1}=\dfrac{y-2}{2}=\dfrac{z+3}{1}

and

\dfrac{x-a}{2}=\dfrac{y+2}{3}=\dfrac{z-3}{1}

intersect at the point

P

, then the distance of the point

P

from the plane

z=a

is:

(A) $28$
(B) $16$
(C) $10$
(D) $22$

Q175·MathematicsSingle correctJEE Main 2023

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 distance of the point

C

from the origin is

\sqrt5

. If

ab<0

and

P

is the point

(a-b,\,b,\,2b-a)

then

CP^2

is equal to:

(A) $\dfrac{97}{3}$
(B) $\dfrac{17}{3}$
(C) $\dfrac{16}{3}$
(D) $\dfrac{73}{3}$

Q176·MathematicsSingle correctJEE Main 2023

The foot of perpendicular of the point

(2,0,5)

on the line

\dfrac{x+1}{2} = \dfrac{y-1}{5} = \dfrac{z+1}{-1}

is

(\alpha, \beta, \gamma)

. Then, which of the following is NOT correct?

(A) $\dfrac{\beta}{\gamma} = -5$
(B) $\dfrac{\gamma}{\alpha} = \dfrac{5}{8}$
(C) $\dfrac{\alpha}{\beta} = -8$
(D) $\dfrac{\alpha\beta}{\gamma} = \dfrac{4}{15}$

Q177·MathematicsSingle correctJEE Main 2023

Consider the lines

L_1

and

L_2

given by

L_1:\dfrac{x-1}{2}=\dfrac{y-3}{1}=\dfrac{z-2}{2}

,

L_2:\dfrac{x-2}{1}=\dfrac{y-2}{2}=\dfrac{z-3}{3}

. A line

L_3

having direction ratios

1,-1,-2

, intersects

L_1

and

L_2

at the points

P

and

Q

respectively. Then the length of line segment

PQ

is:

(A) $3\sqrt{2}$
(B) $4\sqrt{3}$
(C) $4$
(D) $2\sqrt{6}$

Q178·MathematicsSingle correctJEE Main 2023

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,-1

is equal to:

(A) $\sqrt{14}$
(B) $3$
(C) $\sqrt{6}$
(D) $2\sqrt{3}$

Q179·MathematicsSingle correctJEE Main 2023

The shortest distance between the lines

x+1=2y=-12z

and

x=y+2=6z-6

is

(A) $\dfrac{3}{2}$
(B) $2$
(C) $\dfrac{5}{2}$
(D) $3$

Q180·MathematicsNumericalJEE Main 2023

If the shortest distance between the line joining the points

(1,2,3)

and

(2,3,4)

, and the line

\dfrac{x-1}{2}=\dfrac{y+1}{-1}=\dfrac{z-2}{0}

is

a

, then

28a^{2}

is equal to

Q181·MathematicsNumericalJEE Main 2023

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+z-2

be

ax+by+cz=65

. Then the distance of the point

(a,b,c)

from the plane

2x+2y-z+16=0

is _______.

Q182·MathematicsSingle correctJEE Main 2023

Let the plane containing the line of intersection of the planes

P_1:x+(\lambda+4)y+z=1

and

P_2:2x+y+z=2

pass through the points

(0,1,0)

and

(1,0,1)

. Then the distance of the point

(2\lambda,\lambda,-\lambda)

from the plane

P_2

is

(A) $4\sqrt6$
(B) $3\sqrt6$
(C) $5\sqrt6$
(D) $2\sqrt6$

Q183·MathematicsSingle correctJEE Main 2023

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-z=3

is

(\alpha,\beta,\gamma)

, then

\alpha+\beta+\gamma

is equal to

(A) $3$
(B) $1$
(C) $-1$
(D) $5$

Q184·MathematicsNumericalJEE Main 2023

The shortest distance between the lines

\dfrac{x-2}{3}=\dfrac{y+1}{2}=\dfrac{z-6}{2}

and

\dfrac{x-6}{3}=\dfrac{1-y}{2}=\dfrac{z+8}{0}

is equal to _______ .

Q185·MathematicsSingle correctJEE Main 2023

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:

(A) $5$
(B) $4$
(C) $5\sqrt{2}$
(D) $4\sqrt{2}$

Q186·MathematicsNumericalJEE Main 2023

If the shortest distance between the lines

\dfrac{x+\sqrt6}{2}=\dfrac{y-\sqrt6}{3}=\dfrac{z-\sqrt6}{4}

and

\dfrac{x-\lambda}{3}=\dfrac{y-2\sqrt6}{4}=\dfrac{z+2\sqrt6}{5}

is

6

, then the square of sum of all possible values of

\lambda

is

Q187·MathematicsSingle correctJEE Main 2023

The distance of the point

(-1,9,-16)

from the plane

2x+3y-z=5

measured parallel to the line

\dfrac{x+4}{3}=\dfrac{2-y}{4}=\dfrac{z-3}{12}

is

(A) $31$
(B) $13\sqrt{2}$
(C) $20\sqrt{2}$
(D) $26$

Q188·MathematicsMultiple correctJEE Advanced 2022

Let

S

be the reflection of a point

Q

with respect to the plane given by

\vec{r} = -\left(t + p\right)\hat{i} + t\hat{j} + \left(1 + p\right)\hat{k}

where

t

,

p

are real parameters and

\hat{i}

,

\hat{j}

,

\hat{k}

are the unit vectors along the three positive coordinate axes. If the position vectors of

Q

and

S

are

10\hat{i} + 15\hat{j} + 20\hat{k}

and

\alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}

respectively, then which of the following is/are TRUE?

(A) $3\left(\alpha + \beta\right) = -101$
(B) $3\left(\beta + \gamma\right) = -71$
(C) $3\left(\gamma + \alpha\right) = -86$
(D) $3\left(\alpha + \beta + \gamma\right) = -121$

Q189·MathematicsMultiple correctJEE Advanced 2022

Let

P_{1}

and

P_{2}

be two planes given by

P_{1} : 10x + 15y + 12z - 60 = 0

,

P_{2} : -2x + 5y + 4z - 20 = 0

. Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on

P_{1}

and

P_{2}

?

(A) $\frac{x-1}{0} = \frac{y-1}{0} = \frac{z-1}{5}$
(B) $\frac{x-6}{-5} = \frac{y}{2} = \frac{z}{3}$
(C) $\frac{x}{-2} = \frac{y-4}{5} = \frac{z}{4}$
(D) $\frac{x}{1} = \frac{y-4}{-2} = \frac{z}{3}$

Q190·MathematicsSingle correctJEE Main 2022

If the foot of the perpendicular from the point

A(-1, 4, 3)

on the plane

P : 2x + my + nz = 4

, is

\left(-2, \frac{7}{2}, \frac{3}{2}\right)

, then the distance of the point A from the plane P, measured parallel to a line with direction ratios

3, -1, -4

, is equal to :

(A)1
(B) $\sqrt{26}$
(C) $2\sqrt{2}$
(D) $\sqrt{14}$

Q191·MathematicsNumericalJEE Main 2022

Let a line with direction ratios a, –4a, –7 be perpendicular to the lines with direction ratios 3, –1, 2b and b, a, –2. If the point of intersection of the line

\frac{x+1}{a^{2}+b^{2}} = \frac{y-2}{a^{2}-b^{2}} = \frac{z}{1}

and the plane

x - y + z = 0

is

(\alpha,\,\beta,\,\gamma)

, then

\alpha + \beta + \gamma

is equal to __________.

Q192·MathematicsSingle correctJEE Main 2022

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:

(A) $\frac{21}{2}$
(B) $\frac{59}{8}$
(C) $\frac{57}{8}$
(D) $\frac{95}{8}$

Q193·MathematicsSingle correctJEE Main 2022

Let Q be the foot of perpendicular drawn from the point

P(1, 2, 3)

to the plane

x + 2y + z = 14

. If R is a point on the plane such that

\angle PRQ = 60^\circ

, then the area of

\triangle PQR

is equal to:

(A) $\frac{\sqrt{3}}{2}$
(B) $\sqrt{3}$
(C) $2\sqrt{3}$
(D) $3$

Q194·MathematicsSingle correctJEE Main 2022

A plane P is parallel to two lines whose direction ratios are

-2, 1, -3

, and

-1, 2, -2

and it contains the point

(2,2,-2)

. Let P intersect the co-ordinate axes at the points A, B, C making the intercepts

\alpha, \beta, \gamma

. If V is the volume of the tetrahedron OABC, where O is the origin and

p=\alpha+\beta+\gamma

, then the ordered pair

(V,p)

is equal to

(A) $(48,-13)$
(B) $(24,-13)$
(C) $(48,11)$
(D) $(24,-5)$

Q195·MathematicsSingle correctJEE Main 2022

The foot of the perpendicular from a point on the circle

x^{2} + y^{2} = 1

,

z = 0

to the plane

2x + 3y + z = 6

lies on which one of the following curves ?

(A) $(6x + 5y - 12)^{2} + 4(3x + 7y - 8)^{2} = 1$ , $z = 6 - 2x - 3y$
(B) $(5x + 6y - 12)^{2} + 4(3x + 5y - 9)^{2} = 1$ , $z = 6 - 2x - 3y$
(C) $(6x + 5y - 14)^{2} + 9(3x + 5y - 7)^{2} = 1$ , $z = 6 - 2x - 3y$
(D) $(5x + 6y - 14)^{2} + 9(3x + 7y - 8)^{2} = 1$ , $z = 6 - 2x - 3y$

Q196·MathematicsSingle correctJEE Main 2022

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 P be the plane containing these two lines. Then which of the following points does <b>NOT</b> lies on P?

(A) $(0,-2,-2)$
(B) $(-5,0,-1)$
(C) $(3,-1,0)$
(D) $(0,4,5)$

Q197·MathematicsNumericalJEE Main 2022

Let P(−2, −1, 1) and

Q\left( \frac{56}{17}, \frac{43}{17}, \frac{111}{17} \right)

be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are

\alpha

,

-1

,

\beta

, where both

\alpha

and

\beta

are integers of minimum absolute values, then

\alpha^{2} + \beta^{2}

is equal to __________.

Q198·MathematicsSingle correctJEE Main 2022

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\sqrt{6}

units and

Q(\alpha_{1}, \alpha_{2}, \alpha_{3})

is the image of the point P in this line, then

a + \sum_{i=1}^{3}\alpha_{i}

is equal to :

(A) $7$
(B) $8$
(C) $12$
(D) $14$

Q199·MathematicsNumericalJEE Main 2022

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

4ax-y+5z-7a=0=2x-5y-z-3

,

a\in\mathbb{R}

at the point

P\left(\alpha,\beta,\gamma\right)

. Then the value of

\alpha+\beta+\gamma

equals ______.

Q200·MathematicsSingle correctJEE Main 2022

If the line of intersection of the planes

ax + by = 3

and

ax + by + cz = 0

,

a > 0

makes an angle

30^{\circ}

with the plane

y - z + 2 = 0

, then the direction cosines of the line are :

(A) $\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0$
(B) $\frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}}, 0$
(C) $\frac{1}{\sqrt{5}}, -\frac{2}{\sqrt{5}}, 0$
(D) $\frac{1}{2}, -\frac{\sqrt{3}}{2}, 0$

Q201·MathematicsSingle correctJEE Main 2022

If the plane P passes through the intersection of two mutually perpendicular planes 2x + ky − 5z = 1 and 3kx − ky + z = 5, k < 3 and intercepts a unit length on positive x-axis, then the intercept made by the plane P on the y-axis is

(A) $\frac{1}{11}$
(B) $\frac{5}{11}$
(C)6
(D)7

Q202·MathematicsNumericalJEE Main 2022

The largest value of a, for which the perpendicular distance of the plane containing the lines

\vec{r}=\left(\hat{i}+\hat{j}\right)+\lambda\left(\hat{i}+a\hat{j}-\hat{k}\right)

and

\vec{r}=\left(\hat{i}+\hat{j}\right)+\mu\left(-\hat{i}+\hat{j}-a\hat{k}\right)

from the point

(2,1,4)

is

\sqrt{3}

, is____________.

Q203·MathematicsSingle correctJEE Main 2022

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 – 2y + 3z – 5 is :

(A) $\sqrt{\dfrac{21}{2}}$
(B) $\sqrt{\dfrac{9}{2}}$
(C) $\sqrt{\dfrac{73}{2}}$
(D)1

Q204·MathematicsNumericalJEE Main 2022

Let Q and 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 point P(4, 2, 7). Then the square of the area of the triangle PQR is______________.

Q205·MathematicsNumericalJEE Main 2022

The plane passing through the line L:

\ell x-y+3(1-\ell)z=1

,

x+2y-z=2

and perpendicular to the plane

3x+2y+z=6

is

3x-8y+7z=4

. If

\theta

is the acute angle between the line L and the y-axis, then

415\cos^{2}\theta

is equal to______.

Q206·MathematicsNumericalJEE Main 2022

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 angle of

\cos^{-1}\left( \sqrt{\frac{2}{27}} \right)

with the plane

P : ax - y - z = 0

,

(a > 0)

. If the image of the point

(1, 1, -5)

in the plane P is

(\alpha, \beta, \gamma)

, then

\alpha + \beta - \gamma

is equal to ________.

Q207·MathematicsSingle correctJEE Main 2022

Let 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 the straight lines

\frac{x}{2} = \frac{y}{3} = \frac{z}{5}

and

\frac{x}{3} = \frac{y}{7} = \frac{z}{8}

. If d is the distance of P from the point

(2, -5, 11)

, then

d^{2}

is equal to :

(A) $\frac{147}{2}$
(B) $96$
(C) $\frac{32}{3}$
(D) $54$

Q208·MathematicsNumericalJEE Main 2022

Let

P_1 : \vec{r} \cdot (2\hat{i} + \hat{j} - 3\hat{k}) = 4

be a plane. Let

P_2

be another plane which passes through the points (2, -3, 2) (2, − 2, − 3) and (1, −4, 2). If the direction ratios of the line of intersection of

P_1

and

P_2

be 16, α, β, then the value of α + β is equal to _____.

Q209·MathematicsSingle correctJEE Main 2022

Let

\frac{x - 2}{3} = \frac{y + 1}{-2} = \frac{z + 3}{-1}

lie on the plane

px - qy + z = 5

, for some

p, q \in \mathbb{R}

. The shortest distance of the plane from the origin is:

(A) $\sqrt{\frac{3}{109}}$
(B) $\sqrt{\frac{5}{142}}$
(C) $\sqrt{\frac{5}{71}}$
(D) $\sqrt{\frac{1}{142}}$

Q210·MathematicsSingle correctJEE Main 2022

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 contains the line

\vec{r} = -\hat{k} + \lambda\left(\hat{i} + \hat{j} + 2\hat{k}\right), \lambda \in \mathbb{R}

. Then, which of the following points lies on T?

(A) $(2, 1, 0)$
(B) $(1, 2, 1)$
(C) $(1, 2, 2)$
(D) $(1, 3, 2)$

Q211·MathematicsSingle correctJEE Main 2022

If the mirror image of the point (2, 4, 7) in the plane

3x - y + 4z = 2

is (a, b, c), the

2a + b + 2c

is equal to :

(A)54
(B)50
(C)−6
(D)−42

Q212·MathematicsSingle correctJEE Main 2022

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, c, d

are integers

d > 0

and

\gcd(|a|, |b|, |c|, d) = 1

, then the value of

a + 7b + c + 20d

is equal to

(A) $18$
(B) $20$
(C) $24$
(D) $22$

Q213·MathematicsSingle correctJEE Main 2022

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

\mathrm{PQ}=\mathrm{PR}=\sqrt{18}

where the point P is (1, −2, 3), then the area of the triangle PQR is equal to

(A) $\frac{2}{3}\sqrt{38}$
(B) $\frac{4}{3}\sqrt{38}$
(C) $\frac{8}{3}\sqrt{38}$
(D) $\sqrt{\frac{152}{3}}$

Q214·MathematicsNumericalJEE Main 2022

Let the image of the point

P(1, 2, 3)

in the line

L : \frac{x-6}{3} = \frac{y-1}{2} = \frac{z-2}{3}

be

Q

. let

R(\alpha, \beta, \gamma)

be a point that divides internally the line segment

PQ

in the ratio

1 : 3

. Then the value of

22(\alpha + \beta + \gamma)

is equal to

Q215·MathematicsSingle correctJEE Main 2022

Let the plane

P : \vec{r} \cdot \vec{a} = d

contain the line of intersection of two planes

\vec{r} \cdot \left(\hat{i} + 3\hat{j} - \hat{k}\right) = 6

and

\vec{r} \cdot \left(-6\hat{i} + 5\hat{j} - \hat{k}\right) = 7

. If the plane P passes through the point

\left(2,\, 3,\, \frac{1}{2}\right)

, then the value of

\frac{|13\vec{a}|^2}{d^2}

is equal to

(A)90
(B)93
(C)95
(D)97

Q216·MathematicsSingle correctJEE Main 2022

The acute angle between the planes

P_1

and

P_2

, when

P_1

and

P_2

are the planes passing through the intersection of the planes 5x + 8y + 13z − 29 = 0 and 8x − 7y + z − 20 = 0 and the points (2, 1, 3) and (0, 1, 2), respectively, is

(A) $\frac{\pi}{3}$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{6}$
(D) $\frac{\pi}{12}$

Q217·MathematicsSingle correctJEE Main 2022

The shortest distance between the lines

\dfrac{x-3}{2}=\dfrac{y-2}{3}=\dfrac{z-1}{-1}

and

\dfrac{x+3}{2}=\dfrac{y-6}{1}=\dfrac{z-5}{3}

is :

(A) $\dfrac{18}{\sqrt{5}}$
(B) $\dfrac{22}{3\sqrt{5}}$
(C) $\dfrac{46}{3\sqrt{5}}$
(D) $6\sqrt{3}$

Q218·MathematicsSingle correctJEE Main 2022

If two straight lines whose direction cosines are given by the relations

l + m - n = 0

,

3l^{2} + m^{2} + cnl = 0

are parallel, then the positive value of c is :

(A)6
(B)4
(C)3
(D)2

Q219·MathematicsNumericalJEE Main 2022

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 = 5, then 7

\beta

- 9

\gamma

is equal to _____________.

Q220·MathematicsSingle correctJEE Main 2022

Let the foot of the perpendicular from the point (1, 2, 4) on the line

\dfrac{x+2}{4}=\dfrac{y-1}{2}=\dfrac{z+1}{3}

be P. Then the distance of P from the plane

3x+4y+12z+23=0

(A) $5$
(B) $\dfrac{50}{13}$
(C) $4$
(D) $\dfrac{63}{13}$

Q221·MathematicsSingle correctJEE Main 2022

If the two lines

l_{1} : \frac{x-2}{3} = \frac{y+1}{-2}

,

z = 2

and

l_{2} : \frac{x-1}{1} = \frac{2y+3}{\alpha} = \frac{z+5}{2}

perpendicular, then an angle between the lines

l_{2}

and

l_{3} : \frac{1-x}{3} = \frac{2y-1}{-4} = \frac{z}{4}

is :

(A) $\cos^{-1}\left(\frac{29}{4}\right)$
(B) $\sec^{-1}\left(\frac{29}{4}\right)$
(C) $\cos^{-1}\left(\frac{2}{29}\right)$
(D) $\cos^{-1}\left(\frac{2}{\sqrt{29}}\right)$

Q222·MathematicsSingle correctJEE Main 2022

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 image of the point

\left(2, -\frac{1}{2}, 2\right)

in the rotated plane is B(a, b, c), then :

(A) $\frac{a}{8} = \frac{b}{5} = \frac{c}{-4}$
(B) $\frac{a}{4} = \frac{b}{5} = \frac{c}{-2}$
(C) $\frac{a}{8} = \frac{b}{-5} = \frac{c}{4}$
(D) $\frac{a}{4} = \frac{b}{5} = \frac{c}{2}$

Q223·MathematicsSingle correctJEE Main 2022

If the lines

\vec{r}=\left(\hat{i}-\hat{j}+\hat{k}\right)+\lambda\left(3\hat{j}-\hat{k}\right)

and

\vec{r}=\left(\alpha\hat{i}-\hat{j}\right)+\mu\left(2\hat{i}-3\hat{k}\right)

are co-planar, then distance of the plane containing these two lines from the point (α, 0, 0) is :

(A) $\frac{2}{9}$
(B) $\frac{2}{11}$
(C) $\frac{4}{11}$
(D)2

Q224·MathematicsSingle correctJEE Main 2022

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

\frac{\pi}{2}

, then the plane after the rotation passes through the point :

(A)(2, –2, 0)
(B)(–2, 2, 0)
(C)(1, 0, 2)
(D)(–1, 0, –2)

Q225·MathematicsSingle correctJEE Main 2022

Let Q be the mirror image of the point P(1, 0, 1) with respect to the plane S : x + y + z = 5. If a line L passing through (1, −1, −1), parallel to the line PQ meets the plane S at R, then

QR^2

is equal to:

(A)2
(B)5
(C)7
(D)11

Q226·MathematicsNumericalJEE Main 2022

Let

l_{1}

be the line in xy-plane with x and y intercepts

\frac{1}{8}

and

\frac{1}{4\sqrt{2}}

respectively, and

l_{2}

be the line in zx-plane with x and z intercepts

-\frac{1}{8}

and

-\frac{1}{6\sqrt{3}}

respectively. If d is the shortest distance between the line

l_{1}

and

l_{2}

, then

d^{-2}

is equal to

Q227·MathematicsNumericalJEE Main 2022

Let the lines

L_1 : \vec{r} = \lambda(\hat{i} + 2\hat{j} + 3\hat{k})

, λ∈R

L_2 : \vec{r} = (\hat{i} + 3\hat{j} + \hat{k}) + \mu\left(\hat{i} + \hat{j} + 5\hat{k}\right)

; μ∈R intersect at the point S. If a plane ax + by − z + d = 0 passes through S and is parallel to both the lines

L_1

and

L_2

, then the value of a + b + d is equal to ________

Q228·MathematicsSingle correctJEE Main 2022

Let P be the plane passing through the intersection of the planes

\vec{r} \cdot \left(\hat{i} + 3\hat{j} - \hat{k}\right) = 5

and

\vec{r} \cdot \left(2\hat{i} - \hat{j} + \hat{k}\right) = 3

, and the point

(2,1,-2)

. Let the position vectors of the points X and Y be

\hat{i} - 2\hat{j} + 4\hat{k}

and

5\hat{i} - \hat{j} + 2\hat{k}

respectively. Then the points

(A)X and X + Y are on the same side of P
(B)Y and Y $-$ X are on the opposite sides of P
(C)X and Y are on the opposite sides of P
(D)X + Y and X $-$ Y are on the same side of P

Q229·MathematicsSingle correctJEE Main 2022

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

2x + y + 3z = 1

is

(A) $\dfrac{\pi}{6}$
(B) $\dfrac{\pi}{4}$
(C) $\dfrac{\pi}{3}$
(D) $\dfrac{5\pi}{12}$

Q230·MathematicsSingle correctJEE Main 2022

If the shortest distance between the lines

\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{\lambda}

and

\dfrac{x-2}{1}=\dfrac{y-4}{4}=\dfrac{z-5}{5}

is

\dfrac{1}{\sqrt{3}}

, then the sum of all possible values of

\lambda

is :

(A)16
(B)6
(C)12
(D)15

Q231·MathematicsNumericalJEE Main 2022

If the shortest distance between the line

\vec{r} = \left(-\hat{i} + 3k\right) + \lambda\left(\hat{i} - a\hat{j}\right)

and

\vec{r} = \left(-\hat{j} + 2k\right) + \mu\left(\hat{i} - \hat{j} + k\right)

is

\sqrt{\dfrac{2}{3}}

, then the integral value of a is equal to

Q232·MathematicsNumericalJEE Main 2022

Let a line having direction ratios 1, -4, 2 intersect the lines

\dfrac{x-7}{3}=\dfrac{y-1}{-1}=\dfrac{z+2}{1}

and

\dfrac{x}{2}=\dfrac{y-7}{3}=\dfrac{z}{1}

at the point A and B. Then

(AB)^{2}

is equal to ____ .

Q233·MathematicsNumericalJEE Advanced 2021

Let

\alpha

,

\beta

and

\gamma

be real numbers such that the system of linear equations

x + 2y + 3z = \alpha

4x + 5y + 6z = \beta

7x + 8y + 9z = \gamma - 1

is consistent. Let

|M|

represent the determinant of the matrix

M = \begin{bmatrix} \alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}

Let P be the plane containing all those

(\alpha, \beta, \gamma)

for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P. The value of D is ________.

Q234·MathematicsSingle correctJEE Main 2021

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

(A) $\sqrt{26}$
(B) $2\sqrt{5}$
(C) $2\sqrt{6}$
(D) $4\sqrt{2}$

Q235·MathematicsSingle correctJEE Main 2021

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 following points lies on P?

(A) $\left(3, 1, -\frac{1}{2}\right)$
(B) $\left(-2, 0, -\frac{1}{2}\right)$
(C)(0, 2, –4)
(D)(4, 0, – 2)

Q236·MathematicsNumericalJEE Main 2021

The square of the distance of the point of intersection of the line

\frac{x-1}{2} = \frac{y-2}{3} = \frac{z+1}{6}

and the plane

2x - y + z = 6

from the point

(-1, -1, 2)

is ______.

Q237·MathematicsSingle correctJEE Main 2021

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 :

(A) $\frac{1}{\sqrt{2}}$
(B) $\frac{5}{2}$
(C) $\frac{\sqrt{42}}{2}$
(D) $\frac{\sqrt{34}}{2}$

Q238·MathematicsSingle correctJEE Main 2021

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 - 7 = 0

and

x + 5y - 2z + 9 = 0

, be

\alpha x + \beta y + \gamma z + 3 = 0

, then

\alpha + \beta + \gamma

is equal to :

(A) $-23$
(B) $-15$
(C)23
(D)15

Q239·MathematicsNumericalJEE Main 2021

Suppose the line

\frac{x - 2}{\alpha} = \frac{y - 2}{-5} = \frac{z + 2}{2}

lies on the plane

x + 3y - 2z + \beta = 0

. Then

(\alpha + \beta)

is equal to ______.

Q240·MathematicsSingle correctJEE Main 2021

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 :

(A) $3$
(B) $5$
(C) $2$
(D) $1$

Q241·MathematicsSingle correctJEE Main 2021

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

(A) $\frac{\pi}{2}$
(B) $\pi - \cos^{-1}\left(\frac{4}{9}\right)$
(C) $\cos^{-1}\left(\frac{8}{9}\right)$
(D) $\frac{\pi}{3}$

Q242·MathematicsSingle correctJEE Main 2021

Equation of a plane at a distance

\sqrt{\frac{2}{21}}

from the origin, which contains the line of intersection of the planes

x - y - z - 1 = 0

and

2x + y - 3z + 4 = 0

, is :

(A) $3x - y - 5z + 2 = 0$
(B) $3x - 4z + 3 = 0$
(C) $-x + 2y + 2z - 3 = 0$
(D) $4x - y - 5z + 2 = 0$

Q243·MathematicsNumericalJEE Main 2021

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, γ) be a point of this plane. Then the square of the length of the line segment SR is ___________ .

Q244·MathematicsSingle correctJEE Main 2021

The equation of the plane passing through the line of intersection of the planes

\vec{r}\cdot\left(\hat{i} + \hat{j} + \hat{k}\right) = 1

and

\vec{r}\cdot\left(2\hat{i} + 3\hat{j} - \hat{k}\right) + 4 = 0

and parallel to the x-axis is:

(A) $\vec{r}\cdot\left(\hat{j} - 3\hat{k}\right) + 6 = 0$
(B) $\vec{r}\cdot\left(\hat{i} + 3\hat{k}\right) + 6 = 0$
(C) $\vec{r}\cdot\left(\hat{i} - 3\hat{k}\right) + 6 = 0$
(D) $\vec{r}\cdot\left(\hat{j} - 3\hat{k}\right) - 6 = 0$

Q245·MathematicsSingle correctJEE Main 2021

A hall has a square floor of dimension 10m

\times

10m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is

\cos^{-1}\frac{1}{5}

, then the height of the hall (in meters) is :

(A) $5$
(B) $2\sqrt{10}$
(C) $5\sqrt{3}$
(D) $5\sqrt{2}$

Q246·MathematicsSingle correctJEE Main 2021

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 following points lies on P ?

(A) $(-1, 1, 2)$
(B) $(0, 1, 1)$
(C) $(1, 0, 1)$
(D) $(2, -1, 1)$

Q247·MathematicsNumericalJEE Main 2021

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 ________.

Q248·MathematicsNumericalJEE Main 2021

Let the line L be the projection of the line

\frac{x-1}{2} = \frac{y-3}{1} = \frac{z-4}{2}

in the plane

x - 2y - z = 3

. If d is the distance of the point

(0, 0, 6)

from L, then

d^{2}

is equal to ________.

Q249·MathematicsSingle correctJEE Main 2021

Let P be the plane passing through the point (1,2,3) and the line of intersection of the planes

\vec{r} \cdot \left(\hat{i} + \hat{j} + 4\hat{k}\right) = 16

and

\vec{r} \cdot \left(-\hat{i} + \hat{j} + \hat{k}\right) = 6

. Then which of the following points does NOT lie on P ?

(A) $(3, 3, 2)$
(B) $(6, -6, 2)$
(C) $(4, 2, 2)$
(D) $(-8, 8, 6)$

Q250·MathematicsSingle correctJEE Main 2021

For real numbers α and β ≠ 0, if the point of intersection of the straight lines

\frac{x - \alpha}{1} = \frac{y - 1}{2} = \frac{z - 1}{3}

and

\frac{x - 4}{\beta} = \frac{y - 6}{3} = \frac{z - 7}{3}

, lies on the plane x + 2y − z = 8, then α − β is equal to :

(A)3
(B)9
(C)7
(D)5

Q251·MathematicsSingle correctJEE Main 2021

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 + by + cz + 8 = 0

. Then the value of

a + b + c

is equal to :

(A)4
(B)8
(C)5
(D)3

Q252·MathematicsNumericalJEE Main 2021

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………….

Q253·MathematicsNumericalJEE Main 2021

Let a plane P pass through the point

\left(3,7,-7\right)

and contain the line,

\frac{x-2}{-3}=\frac{y-3}{2}=\frac{z+2}{1}

. If distance of the plane P from the origin is d, then

d^2

is equal to ____.

Q254·MathematicsNumericalJEE Main 2021

If the lines

\dfrac{x-k}{1} = \dfrac{y-2}{2} = \dfrac{z-3}{3}

and

\dfrac{x+1}{3} = \dfrac{y+2}{2} = \dfrac{z+3}{1}

are co-planar, then the value of k is......

Q255·MathematicsSingle correctJEE Main 2021

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 and point Q. If

\alpha

is the acute angle between the lines PN and PQ, then

\cos\alpha

is equal to……

(A) $\frac{1}{\sqrt{5}}$
(B) $\frac{\sqrt{3}}{2}$
(C) $\frac{1}{\sqrt{3}}$
(D) $\frac{1}{2\sqrt{3}}$

Q256·MathematicsSingle correctJEE Main 2021

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

\frac{1}{\sqrt{38}}

, then the integral value of

\lambda

is equal to :

(A)3
(B)2
(C)−1
(D)5

Q257·MathematicsSingle correctJEE Main 2021

Let L be the line of intersection of planes

\vec{r}.\left(\hat{i} - \hat{j} + 2\hat{k}\right) = 2

and

\vec{r}.\left(2\hat{i} + \hat{j} - \hat{k}\right) = 2

. If

P(\alpha, \beta, \gamma)

is the foot of perpendicular on L from the point

(1, 2, 0)

, then the value of

35(\alpha + \beta + \gamma)

is equal to :

(A) $101$
(B) $143$
(C) $134$
(D) $119$

Q258·MathematicsSingle correctJEE Main 2021

The lines

x=ay-1=z-2

and

x=3y-2=bz-2,\ (ab\neq 0)

are coplanar, if :

(A) $a=2,\ b=2$
(B) $b=1,\ a\in R-\{0\}$
(C) $a=2,\ b=3$
(D) $a=1,\ b\in R-\{0\}$

Q259·MathematicsSingle correctJEE Main 2021

Consider the line L given by the equation

\frac{x-3}{2}=\frac{y-1}{1}=\frac{z-2}{1}

. Let Q be the mirror image of the point

(2,3,-1)

with respect to L. Let a plane P be such that it passes through Q, and the line L is perpendicular to P. Then which of the following points is one the plane P ?

(A) $(1,2,2)$
(B) $(-1,1,2)$
(C) $(1,1,1)$
(D) $(1,1,2)$

Q260·MathematicsNumericalJEE Main 2021

If the shortest distance between the lines

\vec{r}_1 = \alpha\hat{i} + 2\hat{j} + 2\hat{k} + \lambda\left(\hat{i} - 2\hat{j} + 2\hat{k}\right), \lambda \in R

,

\alpha > 0

and

\vec{r}_2 = -4\hat{i} - \hat{k} + \mu\left(3\hat{i} - 2\hat{j} - 2\hat{k}\right), \mu \in R

is 9, then

\alpha

is equal to........

Q261·MathematicsNumericalJEE Main 2021

Let P be a plane passing through the points

\left(1, 0, 1\right)\left(1, -2, 1\right)

and

\left(0, 1, -2\right)

. Let a vector

\vec{a} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}

be such that

\vec{a}

is parallel to the plane P, perpendicular to

\left(\hat{i} + 2\hat{j} + 3\hat{k}\right)

and

\vec{a}.\left(\hat{i} + \hat{j} + 2\hat{k}\right) = 2

, then

\left(\alpha - \beta + \gamma\right)^2

equals........

Q262·MathematicsNumericalJEE Main 2021

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-3}{4} = \frac{y-2}{-3} = \frac{z+5}{7}

. If the point (1, −1, α) lies on the plane P, then the value of |5α| is equal to ________ .

Q263·MathematicsNumericalJEE Main 2021

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 = 0

. If

(b - d) = K(c - a)

, then the positive value of K is

Q264·MathematicsNumericalJEE Main 2021

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 integers, then the minimum value of

(a^{2} + b^{2} + c^{2} + d^{2})

is

Q265·MathematicsNumericalJEE Main 2021

Let the mirror image of the point (1, 3, a) with respect to the plane

\vec{r} \cdot \left(2\hat{i} - \hat{j} + \hat{k}\right) - b = 0

be (−3, 5, 2). Then the value of |a + b| is equal to ________ .

Q266·MathematicsNumericalJEE Main 2021

Let P be an arbitrary point having sum of the squares of the distance from the planes

x+y+z=0

,

lx-nz=0

and

x-2y+z=0

, equal to 9. If the locus of the point P is

x^{2}+y^{2}+z^{2}=9

, then the value of

l-n

is equal to _______.

Q267·MathematicsNumericalJEE Main 2021

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 point

(-2, 1, 3)

is

ax + by + cz - 7 = 0

, then the value of

2a + b + c - 7

is ______ .

Q268·MathematicsSingle correctJEE Main 2021

The equation of the plane which contains the y-axis and passes through the point

(1, 2, 3)

is :

(A) $x + 3z = 10$
(B) $x + 3z = 0$
(C) $3x + z = 6$
(D) $3x - z = 0$

Q269·MathematicsSingle correctJEE Main 2021

If the equation of plane passing through the mirror image of a point

(2, 3, 1)

with respect to line

\frac{x+1}{2}=\frac{y-3}{1}=\frac{z+2}{-1}

and containing the line

\frac{x-2}{3}=\frac{1-y}{2}=\frac{z+1}{1}

is

\alpha x + \beta y + \gamma z = 24

, then

\alpha + \beta + \gamma

is equal to :

(A) $20$
(B) $19$
(C) $18$
(D) $21$

Q270·MathematicsSingle correctJEE Main 2021

Let P be a plane lx + my + nz = 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 :

(A)1.5
(B)3
(C)2
(D)4

Q271·MathematicsSingle correctJEE Main 2021

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

3+\frac{x-11}{(y-19)^{2}(z-12)^{2}}+\frac{y-19}{(x-11)^{2}(z-12)^{2}}+\frac{z-12}{(x-11)^{2}(y-19)^{2}}-\frac{x+y+z}{14(x-11)(y-19)(z-12)}

(A) $0$
(B) $3$
(C) $39$
(D) $-45$

Q272·MathematicsSingle correctJEE Main 2021

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, –1) and D respectively, then the length of line segment CD is equal to :

(A) $\sqrt{31}$
(B) $\sqrt{41}$
(C) $\sqrt{55}$
(D) $\sqrt{66}$

Q273·MathematicsSingle correctJEE Main 2021

If the foot of the perpendicular from point (4, 3, 8) on the line

L_{1}:\frac{x-a}{l}=\frac{y-2}{3}=\frac{z-b}{4}

,

l \neq 0

is (3, 5, 7), then the shortest distance between the line

L_{1}

and line

L_{2}:\frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}

is equal to :

(A) $\frac{1}{2}$
(B) $\frac{1}{\sqrt{6}}$
(C) $\sqrt{\frac{2}{3}}$
(D) $\frac{1}{\sqrt{3}}$

Q274·MathematicsNumericalJEE Main 2021

If the distance of the point (1, -2, 3) from the plane

x + 2y - 3z + 10 = 0

measured parallel to the line,

\frac{x-1}{3}=\frac{2-y}{m}=\frac{z+3}{1}

is

\sqrt{\frac{7}{2}}

, then the value of

|m|

is equal to _______.

Q275·MathematicsSingle correctJEE Main 2021

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

(A) $-\frac{44}{5}$
(B) $\frac{39}{5}$
(C) $-\frac{39}{5}$
(D) $\frac{44}{5}$

Q276·MathematicsSingle correctJEE Main 2021

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 + \beta + \gamma)

equals:

(A)47
(B)39
(C)43
(D)41

Q277·MathematicsSingle correctJEE Main 2021

Let L be a line obtained from the intersection of two planes x + 2y + z = 6 and y + 2z = 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 :

(A)142
(B)68
(C)136
(D)102

Q278·MathematicsSingle correctJEE Main 2021

Consider the three planes

P_{1}

: 3x + 15y + 21z = 9,

P_{2}

: x − 3y − z= 5, and

P_{3}

: 2x + 10 y + 14z = 5 Then, which one of the following is true?

(A) $P_{1}$ and $P_{3}$ are parallel.
(B) $P_{2}$ and $P_{3}$ are parallel.
(C) $P_{1}$ and $P_{2}$ are parallel.
(D) $P_{1}$ , $P_{2}$ and $P_{3}$ all are parallel.

Q279·MathematicsNumericalJEE Main 2021

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 point (−2, −21, 29) and (−1, −16, 23), then

\left(\frac{\lambda}{11}\right)^{2}-\frac{4\lambda}{11}-4

is equal to _______.

Q280·MathematicsNumericalJEE Main 2021

A line ‘λ’ passing through origin is perpendicular to the lines

_{1}

:r=(3+t )ˆi+−+(12t )ˆj+(4+2t)kˆ

_{2}

:r=(3+2s )ˆi+(3+2s )ˆj+(2+s )kˆ If the co-ordinates of the point in the first octant on ‘λ2’ at the distance of 17 from the point of intersection of ‘λ’ and ‘λ1’ are (a, b, c), then 18(a+b+c) is equal to ______.

Q281·MathematicsSingle correctJEE Main 2021

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 OP on this plane is of length:

(A)2 5
(B)2 3
(C)2 11
(D)2 7

Q282·MathematicsSingle correctJEE Main 2021

The equation of the line through the point (0, 1, 2) and perpendicular to the line

\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{-2}

is :

(A) $\frac{x}{-3} = \frac{y-1}{4} = \frac{z-2}{3}$
(B) $\frac{x}{3} = \frac{y-1}{4} = \frac{z-2}{3}$
(C) $\frac{x}{3} = \frac{y-1}{-4} = \frac{z-2}{3}$
(D) $\frac{x}{3} = \frac{y-1}{4} = \frac{z-2}{-3}$

Q283·MathematicsSingle correctJEE Main 2021

Let

\alpha

be the angle between the lines whose direction cosines satisfy the equations

l + m - n = 0

and

l^{2} + m^{2} - n^{2} = 0

. Then the value of

\sin^{4}\alpha + \cos^{4}\alpha

is :

(A) $\frac{3}{4}$
(B) $\frac{1}{2}$
(C) $\frac{5}{8}$
(D) $\frac{3}{8}$

Q284·MathematicsNumericalJEE Main 2021

Let

\lambda

be an integer. If the shortest distance between the lines

x-\lambda=2y-1=-2z

and

x=y+2\lambda=z-\lambda

is

\frac{\sqrt{7}}{2\sqrt{2}}

, then the value of

|\lambda|

is

Q285·MathematicsSingle correctJEE Main 2021

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:

(A) $3x - 10y - 2z + 11 = 0$
(B) $6x - 5y - 2z - 2 = 0$
(C) $11x + y + 17z + 38 = 0$
(D) $6x - 5y + 2z + 10 = 0$

Q286·MathematicsSingle correctJEE Main 2021

The distance of the point

(1, 1, 9)

from the point of intersection of the line

\frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z - 5}{2}

and the plane

x + y + z = 17

is:

(A) $\sqrt{38}$
(B) $19\sqrt{2}$
(C) $2\sqrt{19}$
(D) $38$

Q287·MathematicsSingle correctJEE Main 2021

The vector equation of the plane passing through the intersection of the planes

\vec{r}\cdot(\hat{i}+\hat{j}+\hat{k})=1

and

\vec{r}\cdot(\hat{i}-2\hat{j})=-2,

and the point (1,0,2) is :

(A) $\vec{r}\cdot\left(\hat{i}-7\hat{j}+3\hat{k}\right)=\frac{7}{3}$
(B) $\vec{r}\cdot\left(\hat{i}+7\hat{j}+3\hat{k}\right)=7$
(C) $\vec{r}\cdot\left(3\hat{i}+7\hat{j}+3\hat{k}\right)=7$
(D) $\vec{r}\cdot\left(\hat{i}+7\hat{j}+3\hat{k}\right)=\frac{7}{3}$

Q288·MathematicsSingle correctJEE Main 2021

Let

a,b\in R

. If the mirror image of the point

P(a,6,9)

with respect to the line

\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}

is

(20,b,-a-9)

, then

|a+b|

is equal to :

(A)86
(B)88
(C)84
(D)90

Q289·MathematicsMultiple correctJEE Advanced 2020

Let

\alpha, \beta, \gamma, \delta

be real numbers such that

\alpha^{2} + \beta^{2} + \gamma^{2} \neq 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?

(A) $\alpha + \beta = 2$
(B) $\delta - \gamma = 3$
(C) $\delta + \beta = 4$
(D) $\alpha + \beta + \gamma = \delta$

Q290·MathematicsMultiple correctJEE Advanced 2020

Let

L_{1}

and

L_{2}

be the following straight line.

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 - \alpha}{l} = \frac{y - 1}{m} = \frac{z - \gamma}{-2}

lies in the plane containing

L_{1}

and

L_{2}

, and passes through the point of intersection of

L_{1}

and

L_{2}

. If the line L bisects the acute angle between the lines

L_{1}

and

L_{2}

, then which of the following statements is/are TRUE?

(A) $\alpha - \gamma = 3$
(B) $l + m = 2$
(C) $\alpha - \gamma = 1$
(D) $l + m = 0$

Q291·MathematicsSingle correctJEE Main 2020

A plane P meets the coordinate axes at A, B and C respectively. The centroid of

\triangle ABC

is given to be (1, 1, 2). Then the equation of the line through this centroid and perpendicular to the plane P is:

(A) $\frac{x-1}{2} = \frac{y-1}{1} = \frac{z-2}{1}$
(B) $\frac{x-1}{1} = \frac{y-1}{1} = \frac{z-2}{2}$
(C) $\frac{x-1}{2} = \frac{y-1}{2} = \frac{z-2}{1}$
(D) $\frac{x-1}{2} = \frac{y-1}{2} = \frac{z-2}{2}$

Q292·MathematicsSingle correctJEE Main 2020

The shortest distance between the lines

\dfrac{x-1}{0}=\dfrac{y+1}{-1}=\dfrac{z}{1}

and

x+y+z+1=0, 2x-y+z+3=0

is:

(A) $1$
(B) $\dfrac{1}{\sqrt{3}}$
(C) $\dfrac{1}{\sqrt{2}}$
(D) $\dfrac{1}{2}$

Q293·MathematicsSingle correctJEE Main 2020

If (a, b, c) is the image of the point (1, 2, –3) in the line,

\dfrac{x+1}{2} = \dfrac{y-3}{-2} = \dfrac{z}{-1}

, then a + b+ c is equal to

(A)3
(B)1
(C)2
(D)-1

Q294·MathematicsSingle correctJEE Main 2020

If for some

\alpha\in R

, the lines

L_1:\dfrac{x+1}{2}=\dfrac{y-2}{-1}=\dfrac{z-1}{1}

and

L_2:\dfrac{x+2}{\alpha}=\dfrac{y+1}{5-\alpha}=\dfrac{z+1}{1}

are coplanar, then the line

L_2

passes through the point:

(A) $(2,-10,-2)$
(B) $(10,-2,-2)$
(C) $(10,2,2)$
(D) $(-2,10,2)$

Q295·MathematicsSingle correctJEE Main 2020

The distance of the point (1, –2, 3) from the plane x − y + z = 5 measured parallel to the line

\frac{x}{2} = \frac{y}{3} = \frac{z}{-6}

is:

(A) $\frac{1}{7}$
(B)7
(C) $\frac{7}{5}$
(D)1

Q296·MathematicsNumericalJEE Main 2020

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, b \in R

, then the distance of the point

(3, 2, -1)

from the plane P is __________.

Q297·MathematicsSingle correctJEE Main 2020

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:

(A) $x-y-2z=1$
(B) $2x+y-z=1$
(C) $x-2y+z=1$
(D) $x+2y-z=1$

Q298·MathematicsSingle correctJEE Main 2020

The lines

\vec{r}=\left(\hat{i}-\hat{j}\right)+\ell\left(2\hat{i}+\hat{k}\right)

and

\vec{r}=\left(2\hat{i}-\hat{j}\right)+m\left(\hat{i}+\hat{j}-\hat{k}\right)

(A)intersect when $\ell=2$ and $m=\frac{1}{2}$
(B)intersect when $\ell=1$ and $m=2$
(C)do not intersect for any values of $\ell$ and $m$
(D)intersect for all values of $\ell$ and $m$

Q299·MathematicsSingle correctJEE Main 2020

The plane which bisects the line joining the points

(4, -2, 3)

and

(2, 4, -1)

at right angles also passes through the point:

(A) $(0, -1, 1)$
(B) $(4, 0, 1)$
(C) $(4, 0, -1)$
(D) $(0, 1, -1)$

Q300·MathematicsNumericalJEE Main 2020

Let a plane P contain two lines

\vec{r} = \hat{i} + \lambda \left( \hat{i} + \hat{j} \right), \lambda \in R

and

\vec{r} = -\hat{j} + \mu \left( \hat{j} - \hat{k} \right), \mu \in R

. If

Q\left( \alpha, \beta, \gamma \right)

is the foot of the perpendicular drawn from the point M

(1, 0, 1)

to P, then

3 \left( \alpha + \beta + \gamma \right)

equals ____________ .

Q301·MathematicsNumericalJEE Main 2020

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 ______

Q302·MathematicsSingle correctJEE Main 2020

The mirror image of the point (1, 2, 3) in a plane is

\left(-\dfrac{7}{3},-\dfrac{4}{3},-\dfrac{1}{3}\right)

. Which of the following points lies on this plane?

(A)(1, −1, 1)
(B)(1, 1, 1)
(C)(−1, −1, −1)
(D)(−1, −1, 1)

Q303·MathematicsSingle correctJEE Main 2020

The shortest distance between the lines

\dfrac{x-3}{3}=\dfrac{y-8}{-1}=\dfrac{z-3}{1}

and

\dfrac{x+3}{-3}=\dfrac{y+7}{2}=\dfrac{z-6}{4}

is:

(A) $\dfrac{7}{2}\sqrt{30}$
(B) $3\sqrt{30}$
(C)3
(D) $2\sqrt{30}$

Q304·MathematicsNumericalJEE Main 2020

If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through

(\alpha,7,1)

is

\left(\dfrac{5}{3},\dfrac{7}{3},\dfrac{17}{3}\right)

, then

\alpha

is equal to __________.

Q305·MathematicsSingle correctJEE Main 2020

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:

(A)(6, 5, 2)
(B)(4, 3, 2)
(C)(6, 5, −2)
(D)(3, 4, −2)

Q306·MathematicsMultiple correctJEE Advanced 2019

Let

L_1

and

L_2

denote the lines

\vec{r} = \hat{i} + \lambda\left(-\hat{i} + 2\hat{j} + 2\hat{k}\right), \lambda \in R

and

\vec{r} = \mu\left(2\hat{i} - \hat{j} + 2\hat{k}\right), \mu \in R

respectively. If

L_3

is a line which is perpendicular to both

L_1

and

L_2

and cuts both of them, then which of the following options describe (s)

L_3

?

(A) $\vec{r} = \frac{2}{9}\left(4\hat{i} + \hat{j} + \hat{k}\right) + t\left(2\hat{i} + 2\hat{j} - \hat{k}\right), t \in R$
(B) $\vec{r} = \frac{1}{3}\left(2\hat{i} + \hat{k}\right) + t\left(2\hat{i} + 2\hat{j} - \hat{k}\right), t \in R$
(C) $\vec{r} = \frac{2}{9}\left(2\hat{i} - \hat{j} + 2\hat{k}\right) + t\left(2\hat{i} + 2\hat{j} - \hat{k}\right), t \in R$
(D) $\vec{r} = t\left(2\hat{i} + 2\hat{j} - \hat{k}\right), t \in R$

Q307·MathematicsNumericalJEE Advanced 2019

Three lines are given by

\vec{r} = \lambda\hat{i}, \lambda \in R

,

\vec{r} = \mu\left(\hat{i} + \hat{j}\right), \mu \in R

and

\vec{r} = v\left(\hat{i} + \hat{j} + \hat{k}\right), v \in R

. Let the lines cut the plane

x + y + z = 1

at the points A, B and C respectively. If the area of the triangle ABC is

\Delta

then the value of

(6\Delta)^2

equals

Q308·MathematicsMultiple correctJEE Advanced 2019

Three lines

L_1 : \vec{r} = \lambda\hat{i},\ \lambda \in \mathbb{R}

L_2 : \vec{r} = \hat{k} + \mu\hat{j},\ \mu \in \mathbb{R}

and

L_3 : \vec{r} = \hat{i} + \hat{j} + \nu\hat{k},\ \nu \in \mathbb{R}

are given. For which point(s) Q on

L_2

can we find a point P on

L_1

and a point R on

L_3

so that P, Q and R are collinear?

(A) $\hat{k} - \dfrac{1}{2}\hat{j}$
(B) $\hat{k} + \dfrac{1}{2}\hat{j}$
(C) $\hat{k}$
(D) $\hat{k} + \hat{j}$

Q309·MathematicsSingle correctJEE Main 2019

The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines

\vec{r} = (\hat{i}+\hat{j}) + \lambda(\hat{i}+2\hat{j}-\hat{k})

and

\vec{r} = (\hat{i}+\hat{j}) + \mu(-\hat{i}+\hat{j}-2\hat{k})

is:

(A) $\dfrac{1}{3}$
(B) $\sqrt{3}$
(C) $\dfrac{1}{\sqrt{3}}$
(D)3

Q310·MathematicsSingle correctJEE Main 2019

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:

(A)(1, 4, −1)
(B)(2, −4, 1)
(C)(2, 4, 1)
(D)(1, −4, 1)

Q311·MathematicsSingle correctJEE Main 2019

The distance of the point having position vector

-\hat{i} + 2\hat{j} + 6\hat{k}

from the straight line passing through the point (2, 3, -4) and parallel to the vector

6\hat{i} + 3\hat{j} - 4\hat{k}

is

(A)7
(B) $4\sqrt{3}$
(C) $2\sqrt{13}$
(D)6

Q312·MathematicsSingle correctJEE Main 2019

If the plane

2x - y + 2z + 3 = 0

has the distances

\dfrac{1}{3}

and

\dfrac{2}{3}

units from the planes

4x - 2y + 4z + \lambda = 0

and

2x - y + 2z + \mu = 0

, respectively, then the maximum value of

\lambda + \mu

us equal to

(A)15
(B)13
(C)5
(D)9

Q313·MathematicsSingle correctJEE Main 2019

A perpendicular is drawn from a point on the line

\dfrac{x-1}{2} = \dfrac{y+1}{-1} = \dfrac{z}{1}

to the plane

x + y + z = 3

such that the foot of the perpendicular Q also lies on the plane

x - y + z = 3

. Then the co-ordinates of Q are

(A) $(2, 0, 1)$
(B) $(-1, 0, 4)$
(C) $(1, 0, 2)$
(D) $(4, 0, -1)$

Q314·MathematicsSingle correctJEE Main 2019

The vertices B and C of a

\triangle ABC

lie on the line,

\dfrac{x+2}{3}=\dfrac{y-1}{0}=\dfrac{z}{4}

such that BC = 5 units. Then the area (in sq. units) of this triangle, given that the point A

(1,-1,2)

is:

(A) $2\sqrt{34}$
(B) $\sqrt{34}$
(C) $6$
(D) $5\sqrt{17}$

Q315·MathematicsSingle correctJEE Main 2019

A plane passing through the points

(0,-1,0)

and

(0,0,1)

and making an angle

\dfrac{\pi}{4}

with plane

y-z+5=0

, also passes through the point:

(A) $(\sqrt{2},1,4)$
(B) $(-\sqrt{2},-1,-4)$
(C) $(-\sqrt{2},1,-4)$
(D) $(\sqrt{2},-1,4)$

Q316·MathematicsSingle correctJEE Main 2019

If the line,

\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z-2}{4}

meets the plane,

x+2y+3z=15

at a point P, then the distance of P from the origin is:

(A) $\dfrac{\sqrt{5}}{2}$
(B) $2\sqrt{5}$
(C) $\dfrac{9}{2}$
(D) $\dfrac{7}{2}$

Q317·MathematicsSingle correctJEE Main 2019

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 - plane. Then the distance of the point

(0,0,256)

from P is equal to:

(A) $63\sqrt{5}$
(B) $205\sqrt{5}$
(C) $\dfrac{17}{\sqrt{5}}$
(D) $\dfrac{11}{\sqrt{5}}$

Q318·MathematicsSingle correctJEE Main 2019

The length of the perpendicular from the point (2, −1, 4) on the straight line,

\dfrac{x+3}{10}=\dfrac{y-2}{-7}=\dfrac{z}{1}

is:

(A)greater than 2 but less than 3
(B)less than 2
(C)greater than 4
(D)greater than 3 but less than 4

Q319·MathematicsSingle correctJEE Main 2019

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:

(A) $\sqrt{53}$
(B)6
(C) $2\sqrt{14}$
(D) $2\sqrt{21}$

Q320·MathematicsSingle correctJEE Main 2019

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:

(A) $x+3y+z=4$
(B) $2x-z=2$
(C) $x-3y-2z=-2$
(D) $x-y-z=0$

Q321·MathematicsSingle correctJEE Main 2019

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

x-y+z=0

is:

(A) $\vec{r}\times(\hat{i}-\hat{k})+2=0$
(B) $\vec{r}.(\hat{i}-\hat{k})-2=0$
(C) $\vec{r}.(\hat{i}-\hat{k})+2=0$
(D) $\vec{x}\times(\hat{i}-\hat{k})+2=0$

Q322·MathematicsSingle correctJEE Main 2019

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:

(A) $\cos^{-1}\left(\dfrac{17}{31}\right)$
(B) $\cos^{-1}\left(\dfrac{19}{35}\right)$
(C) $\cos^{-1}\left(\dfrac{9}{35}\right)$
(D) $\cos^{-1}\left(\dfrac{7}{31}\right)$

Q323·MathematicsSingle correctJEE Main 2019

Let S be the set of all real values of λ such that a plane passing through the points

(-\lambda^{2}, 1, 1)

,

(1, -\lambda^{2}, 1)

and

(1, 1, -\lambda^{2})

also passes through the point (−1, −1, 1). Then S is equal to:

(A) $\{\sqrt{3}\}$
(B) $\{\sqrt{3}, -\sqrt{3}\}$
(C) $\{1, -1\}$
(D) $\{3, -3\}$

Q324·MathematicsSingle correctJEE Main 2019

If an angle between the line,

\dfrac{x+1}{2} = \dfrac{y-2}{1} = \dfrac{z-3}{-2}

and the plane, x − 2y − kz = 3 is

\cos^{-1}\left(\dfrac{2\sqrt{2}}{3}\right)

, then a value of k is :

(A) $\sqrt{\dfrac{5}{3}}$
(B) $\sqrt{\dfrac{3}{5}}$
(C) $-\dfrac{3}{5}$
(D) $-\dfrac{5}{3}$

Q325·MathematicsSingle correctJEE Main 2019

The perpendicular distance from the origin to the plane containing the two lines,

\frac{x+2}{3}=\frac{y-2}{5}=\frac{z+5}{7}

and

\frac{x-1}{1}=\frac{y-4}{4}=\frac{z+4}{7}

, is:

(A) $11\sqrt{6}$
(B) $11/\sqrt{6}$
(C)11
(D) $6\sqrt{11}$

Q326·MathematicsSingle correctJEE Main 2019

The plane containing the line

\dfrac{x-3}{2}=\dfrac{y+2}{-1}=\dfrac{z-1}{3}

and also containing its projection on the plane

2x+3y-z=5

, contains which one of the following points?

(A) $(2,2,0)$
(B) $(-2,2,2)$
(C) $(0,-2,2)$
(D) $(2,0,-2)$

Q327·MathematicsSingle correctJEE Main 2019

If the point (2, α, β) lies on the plane which passes through the points (3, 4, 2) and (7, 0, 6) and is perpendicular to the plane

2x-5y=15

, then

2\alpha-3\beta

is equal to:

(A)12
(B)7
(C)5
(D)17

Q328·MathematicsSingle correctJEE Main 2019

Two lines

\dfrac{x-3}{1}=\dfrac{y+1}{3}=\dfrac{z-6}{-1}

and

\dfrac{x+5}{7}=\dfrac{y-2}{-6}=\dfrac{z-3}{4}

intersect at the point R. The reflection f R in the xy – plane has coordinates:

(A)(2, –4, –7)
(B)(2, 4, 7)
(C)(2, –4, 7)
(D)(–2, 4, 7)

Q329·MathematicsSingle correctJEE Main 2019

Let A be a point on the line

\vec{r} = (1 - 3\mu)\hat{i} + (\mu - 1)\hat{j} + (2 + 5\mu)\hat{k}

and B(3, 2, 6) be a point in the space. Then the value of

\mu

for which the vector

\overrightarrow{AB}

is parallel to the plane

x - 4y + 3z = 1

is:

(A) $\dfrac{1}{4}$
(B) $\dfrac{1}{8}$
(C) $\dfrac{1}{2}$
(D) $-\dfrac{1}{4}$

Q330·MathematicsSingle correctJEE Main 2019

If the lines

x = ay + b, z = cy + d

and

x = a'z + b', y = c'z + d'

are perpendicular, then:

(A) $cc' + a + a' = 0$
(B) $aa' + c + c' = 0$
(C) $ab' + bc' + 1 = 0$
(D) $bb' + cc' + 1 = 0$

Q331·MathematicsSingle correctJEE Main 2019

The plane through the intersection of the plane

x+y+z=1

and

2x+3y-z+4=0

and parallel to y – axis also pass through the point:

(A) $(-3,\ 0,\ -1)$
(B) $(-3,\ 1,\ 1)$
(C) $(3,\ 3,\ -1)$
(D) $(3,\ 2,\ 1)$

Q332·MathematicsSingle correctJEE Main 2019

The 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}

is:

(A) $x + 2y - 2z = 0$
(B) $x - 2y + z = 0$
(C) $5x + 2y - 4z = 0$
(D) $3x + 2y - 3z = 0$

Q333·MathematicsNumericalJEE Advanced 2018

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 from the x-axis be 5. If R is the image of P in the xy-plane, then the length of PR is ______ .

Q334·MathematicsMultiple correctJEE Advanced 2018

Let

P_{1} : 2x + y - z = 3

and

P_{2} : x + 2y + z = 2

be two planes. Then, which of the following statement(s) is (are) TRUE ?

(A)The line of intersection of $P_{1}$ and $P_{2}$ has direction ratios 1, 2, $-1$
(B)The line $\frac{3x-4}{9} = \frac{1-3y}{9} = \frac{z}{3}$ is perpendicular to the line of intersection of $P_{1}$ and $P_{2}$
(C)The acute angle between $P_{1}$ and $P_{2}$ is $60^{\circ}$
(D)If $P_{3}$ is the plane passing through the point $(4, 2, -2)$ and perpendicular to the line of intersection of $P_{1}$ and $P_{2}$ , then the distance of the point $(2, 1, 1)$ from the plane $P_{3}$ is $\frac{2}{\sqrt{3}}$

Q335·MathematicsSingle correctJEE Advanced 2017

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

(A) $14x + 2y - 15z = 1$
(B) $14x - 2y + 15z = 27$
(C) $14x + 2y + 15z = 31$
(D) $-14x + 2y + 15z = 3$

Q336·MathematicsMultiple correctJEE Advanced 2016

Consider a pyramid OPQRS located in the first octant (

x \geq 0

,

y \geq 0

,

z \geq 0

) with O as origin, and OP and OR along the x-axis and the y-axis, respectively. The bases 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

(A)the acute angle between OQ and OS is $\frac{\pi}{3}$
(B)the equation of the plane containing the triangle OQS is $x - y = 0$
(C)the length of the perpendicular from P to the plane containing the triangle OQS is $\frac{3}{\sqrt{2}}$
(D)the perpendicular distance from O to the straight line containing RS is $\sqrt{\frac{15}{2}}$

Q337·MathematicsSingle correctJEE Advanced 2016

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}

is

(A) $x + y - 3z = 0$
(B) $3x + z = 0$
(C) $x - 4y + 7z = 0$
(D) $2x - y = 0$

Q338·MathematicsMultiple correctJEE Advanced 2015

In

\mathbb{R}^{3}

, let

L

be a straight line 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

?

(A) $\left(0, -\dfrac{5}{6}, -\dfrac{2}{3}\right)$
(B) $\left(-\dfrac{1}{6}, -\dfrac{1}{3}, \dfrac{1}{6}\right)$
(C) $\left(-\dfrac{5}{6}, 0, \dfrac{1}{6}\right)$
(D) $\left(-\dfrac{1}{3}, 0, \dfrac{2}{3}\right)$

Q339·MathematicsMultiple correctJEE Advanced 2015

In

\mathbb{R}^{3}

, consider the planes

P_{1} : y = 0

and

P_{2} : x + z = 1

. Let

P_{3}

be a 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

(\alpha, \beta, \gamma)

from

P_{3}

is 2, then which of the following relations is (are) true ?

(A) $2\alpha + \beta + 2\gamma + 2 = 0$
(B) $2\alpha - \beta + 2\gamma + 4 = 0$
(C) $2\alpha + \beta - 2\gamma - 10 = 0$
(D) $2\alpha - \beta + 2\gamma - 8 = 0$

Q340·MathematicsMultiple correctJEE Advanced 2014

From a point

P(\lambda, \lambda, \lambda)

, perpendiculars

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)

(A) $\sqrt{2}$
(B) $1$
(C) $-1$
(D) $-\sqrt{2}$

Q341·MathematicsSingle correctJEE Advanced 2013

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} : 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	List-II
P. $a =$	1. $13$
Q. $b =$	2. $-3$
R. $c =$	3. $1$
S. $d =$	4. $-2$

(A)P-3, Q-2, R-4, S-1
(B)P-1, Q-3, R-4, S-2
(C)P-3, Q-2, R-1, S-4
(D)P-2, Q-4, R-1, S-3

Q342·MathematicsMultiple correctJEE Advanced 2013

Two lines

L_{1} : x = 5, \frac{y}{3 - \alpha} = \frac{z}{-2}

and

L_{2} : x = \alpha, \frac{y}{-1} = \frac{z}{2 - \alpha}

are coplanar. Then

\alpha

can take value(s)

(A) $1$
(B) $2$
(C) $3$
(D) $4$

Q343·MathematicsSingle correctJEE Advanced 2013

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 feet of perpendiculars lie on the line

(A) $\frac{x}{5}=\frac{y-1}{8}=\frac{z-2}{-13}$
(B) $\frac{x}{2}=\frac{y-1}{3}=\frac{z-2}{-5}$
(C) $\frac{x}{4}=\frac{y-1}{3}=\frac{z-2}{-7}$
(D) $\frac{x}{2}=\frac{y-1}{-7}=\frac{z-2}{5}$

Q344·MathematicsMultiple correctJEE Advanced 2013

A line

l

passing through the origin is perpendicular to the lines

l_1:(3+t)\hat{i}+(-1+2t)\hat{j}+(4+2t)\hat{k}

,

-\infty<t<\infty

,

l_2:(3+2s)\hat{i}+(3+2s)\hat{j}+(2+s)\hat{k}

,

-\infty<s<\infty

. Then, the coordinate(s) of the point(s) on

l_2

at a distance of

\sqrt{17}

from the point of intersection of

l

and

l_1

is (are)

(A) $(\frac{7}{3},\frac{7}{3},\frac{5}{3})$
(B) $(-1,-1,0)$
(C) $(1,1,1)$
(D) $(\frac{7}{9},\frac{7}{9},\frac{8}{9})$

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