Let a unit vector O P → make angle α , β , γ with the positive directions of the co-ordinate axes OX , OY , OZ respectively, where β ∈ 0 , π 2 . O P → 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 ?

α ∈ π 2 , π and γ ∈ π 2 , π

α ∈ 0 , π 2 and γ ∈ 0 , π 2

α ∈ π 2 , π and γ ∈ 0 , π 2

α ∈ 0 , π 2 and γ ∈ π 2 , π

Let a unit vector O P → make angle α , β , γ with the positive directions of the co-ordinate axes OX , OY , OZ respectively, where β ∈ 0 , π 2 . O P → 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 ?

α ∈ π 2 , π and γ ∈ π 2 , π

MEDIUM

JEE Main

IMPORTANT

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Let a unit vector $\vec{O P}$ make angle $α, β, γ$ with the positive directions of the co-ordinate axes $OX, OY$ , $OZ$ respectively, where $β \in (0, \frac{π}{2})$ . $\vec{O P}$ 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 ?

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Important Questions on Three Dimensional Geometry

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 20 - Three Dimensional Geometry>JEE Main - 30 January 2023 Shift 1>Q 189

λ_{1} < λ_{2}

are two values of

λ

such that the angle between the planes

P_{1} : \vec{r} (3 \hat{i} - 5 \hat{j} + \hat{k}) = 7

and

P_{2} : \vec{r} \cdot (λ \hat{i} + \hat{j} - 3 \hat{k}) = 9

\sin^{- 1} (\frac{2 \sqrt{6}}{5})

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

(38 λ_{1}, 10 λ_{2}, 2)

to the plane

P_{1}

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 20 - Three Dimensional Geometry>JEE Main - 30 January 2023 Shift 1>Q 190

If the equation of the plane passing through the point

(1, 1, 2)

and perpendicular to the line

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

A x + B y + C z = 1

, then

140 (C - B + A)

is equal to

EASY

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 20 - Three Dimensional Geometry>JEE Main - 30 January 2023 Shift 2>Q 191

A vector

\vec{v}

in the first octant is inclined to the

x

axis at

60 °

, to the

y

-axis at

45 °

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

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 20 - Three Dimensional Geometry>JEE Main - 30 January 2023 Shift 2>Q 192

If a plane passes through the points

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

(1, 1, 2)

and is parallel to the line

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

= \frac{z + 1}{- 1}

, then the value of

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

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 20 - Three Dimensional Geometry>JEE Main - 30 January 2023 Shift 2>Q 193

Let a line

L

pass through the point

P (2, 3, 1)

and be parallel to the line

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

. If the distance of

L

from the point

(5, 3, 8)

α

, then

3 α^{2}

is equal to ________

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 20 - Three Dimensional Geometry>JEE Main - 31 January 2023 Shift 1>Q 194

Let the shortest distance between the lines

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

and

L_{1} : x + 1 = y - 1 = 4 - z

2 \sqrt{6}

. If

(α, β, γ)

lies on

L

, then which of the following is NOT possible?

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 20 - Three Dimensional Geometry>JEE Main - 31 January 2023 Shift 1>Q 195

Let

θ

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

θ

with the normal of

P_{2}

. If

α

is the angle between

L

and

P_{2}

then

(\tan^{2} θ) (\cot^{2} α)

is equal to _____ .

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 20 - Three Dimensional Geometry>JEE Main - 31 January 2023 Shift 1>Q 196

Let the line

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

intersect the plane

2 x + y + 3 z = 16

at the point

P

. Let the point

Q

be the foot of perpendicular from the point

R (1, - 1, - 3)

on the line

L

. If

α

is the area of triangle

P Q R

. then

α^{2}

is equal to _____ .

Let a unit vector OP→ make angle α,β,γ with the positive directions of the co-ordinate axes OX,OY, OZ respectively, where β∈0,π2. 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 ?

Important Questions on Three Dimensional Geometry

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