Elements of Mathematics Class 12>Chapter 13 - Three Dimensional Geometry>ADDITIONAL EXERCISES>Q 6

HARD

12th Odisha Board

IMPORTANT

Earn 100

Prove that the acute angle between the lines whose direction cosines are given by the relations $l + m + n = 0$ and $l^{2} + m^{2} - n^{2} = 0$ is $\frac{π}{3}$

Important Questions on Three Dimensional Geometry

HARD

12th Odisha Board

IMPORTANT

Elements of Mathematics Class 12>Chapter 13 - Three Dimensional Geometry>ADDITIONAL EXERCISES>Q 7

Prove that the three lines drawn from origin with direction cosines $l_{1}, m_{1}, n_{1}; l_{2}, m_{2}, n_{2}; l_{3}, m_{3}, n_{3}$ , are coplanar if $|\begin{matrix} l_{1} & m_{1} & n_{1} \\ l_{2} & m_{2} & n_{2} \\ l_{3} & m_{3} & n_{3} \end{matrix}| = 0$ .

HARD

12th Odisha Board

IMPORTANT

Elements of Mathematics Class 12>Chapter 13 - Three Dimensional Geometry>ADDITIONAL EXERCISES>Q 8

Prove that three lines drawn from origin with direction cosines proportional to

(1, - 1, 1), (2 - 3, 0), (1, 0, 3)

lie on one plane.

HARD

12th Odisha Board

IMPORTANT

Elements of Mathematics Class 12>Chapter 13 - Three Dimensional Geometry>ADDITIONAL EXERCISES>Q 9

Determine

k

so that the lines joining the points

P_{1} (k, 1, - 1)

and

P_{2} (2 k, 0, 2)

shall be perpendicular to the line from

P_{2}

to

P_{3} (2 + 2 k, k, 1)

.

HARD

12th Odisha Board

IMPORTANT

Elements of Mathematics Class 12>Chapter 13 - Three Dimensional Geometry>ADDITIONAL EXERCISES>Q 10

The angle between the lines whose direction ratios are proportional to

a, b, c

and

b - c, c - a, a - b

is of the form of

\frac{π}{a}

, then

a =

HARD

12th Odisha Board

IMPORTANT

Elements of Mathematics Class 12>Chapter 13 - Three Dimensional Geometry>ADDITIONAL EXERCISES>Q 11

O

is the origin and

A

is the point

(a, b, c)

. Find the equation of the plane through

A

at right angles to

OA

.

HARD

12th Odisha Board

IMPORTANT

Elements of Mathematics Class 12>Chapter 13 - Three Dimensional Geometry>ADDITIONAL EXERCISES>Q 12

Find the equation of the plane through $(6, 3, 1)$ and $(8, - 5, 3)$ parallel to $x -$ axis.