HARD

JEE Advanced

IMPORTANT

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Let $L_{1}$ and $L_{2}$ be the following straight lines.
$L_{1} : \frac{x - 1}{1} = \frac{y}{- 1} = \frac{z - 1}{3}$ and $L_{2} : \frac{x - 1}{- 3} = \frac{y}{- 1} = \frac{z - 1}{1}$
Suppose the straight line $L : \frac{x - α}{l} = \frac{y - 1}{m} = \frac{z - γ}{- 2}$ 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?

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Important Questions on Three Dimensional Geometry

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Three Dimensional Geometry>JEE Advanced Paper 1 - 2019>Q 8

Let

L_{1}

and

L_{2}

denotes the lines,

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

and

\vec{r} = μ (2 \hat{i} - \hat{j} + 2 \hat{k}), μ \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}

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Three Dimensional Geometry>JEE Advanced Paper 1 - 2019>Q 9

Three lines are given by

$\vec{r} = λ \hat{i}, λ \in R$ ,

$\vec{r} = μ (\hat{i} + \hat{j}), μ \in R$ and

$\vec{r} = ν (\hat{i} + \hat{j} + \hat{k}), ν \in R .$

Let the lines cut the plane $x + y + z = 1$ at the points $A, B$ and $C$ respectively. If the area of the triangle $A B C$ is $Δ$ then the value of ${(6 Δ)}^{2}$ equals_________

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Three Dimensional Geometry>JEE Advanced Paper 1 - 2018>Q 10

Let

P_{1} : 2 x + y - z = 3 a n d P_{2} : x + 2 y + z = 2

be two planes. Then, which of the following statement(s) is (are) TRUE?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Three Dimensional Geometry>JEE Advanced Paper 1 - 2016>Q 11

Consider a pyramid

O P Q R S

located in the first octant

(x \geq 0, y \geq 0, z \geq 0)

with

O

, as origin, and

O P

and

O R

along the

x ‐ a x i s

and the

y ‐ a x i s

, respectively. The base

O P Q R

of the pyramid is a square with

O P = 3

. The point S is directly above the mid-point T of diagonal,

O Q

, such that,

T S = 3

. Then

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Three Dimensional Geometry>JEE Advanced Paper 1 - 2015>Q 12

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}

1

and the distance of a point

(α, β, γ)

, from

P_{3}

2

, then which of the following relations is(are) true ?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Three Dimensional Geometry>JEE Advanced Paper 1 - 2015>Q 13

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 + 2 y - z + 1 = 0

and

P_{2} : 2 x - y + z - 1 = 0 .

Let

M

be the locus of the foot of the perpendiculars drawn from the points on

L

to the plane

P_{1} .

Which of the following point(s) lie(s) on

M ?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Three Dimensional Geometry>JEE Advanced Paper 1 - 2014>Q 14

From a point

P (λ, λ, λ),

perpendiculars

P Q

and

P R

are drawn respectively on the lines

y = x, z = 1

and

y = - x, z = - 1

. If

P

is such that

∠ Q P R

is a right angle, then the possible value (s) of

λ

is (are)

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Three Dimensional Geometry>JEE Advanced Paper 1 - 2013>Q 15

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

Important Questions on Three Dimensional Geometry

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