## Rajasthan Board RBSE Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise

Question 1.

Which of the following group is not direction cosines of a line :

(a) 1,1,1

(b) 0,0, -1

(c)-1,0,0

(d)0,-1,0

Solution:

Direction cosines of a line are proportional to direction ratio’s.

Let a, b and c are direction ratio’s, then according to question

Question 2.

Consider a point P such that OP = 6 and \(\bar { OP } \) makes angle 45° and 60° with OX and OY – axis respectively, then position vector of P will be :

Solution:

Question 3.

Angle between two diagonals of a cube is :

Solution:

Let the adjacent cores of cube of side ‘a’ are OA, OB, OR to be taken as coordinate axis.

Then the coordinates of the vertices of cube are following :

Question 4.

Direction cosines of 3i be

(a) 3,0,0

(b) 1,0,0

(c)-1, 0,0

(d)-3,0,0

Solution:

Given vector

whose direction ratio’s are 3, 0, 0.

Question 5.

vector form of line

(a) (3i + 4j – 7k) + ?(-2i – 5j + 13k)

(b) (- 2j – 5j + 13k) + ?(3i + 4j – 7k)

(c) (- 3i – 4j + 7k) + ?(- 2i – 5j + 13k)

(d) None of these

Solution:

∴ Position vector of point A

∴ Direction ratio of line are -2,-5, 13

∴ Vector equation of line

Hence, (a) is the correct option.

Question 6.

If lines

are perpendicular to each other than value of ? is :

(a) 0

(b) 1

(c) -1

(d) 2

Solution:

Question 7.

Shortest distance between lines

(a) 10 unit

(b) 12 unit

(c) 14 unit

(d) None of these

Solution:

Question 8.

Angle between line

Solution:

We know that angle between two lines

Question 9.

If equation lx + my + nz = p is normal form of a plane, then which of the following is not true :

(a) l, m, n are direction cosines of normal to the plane

(b) p is perpendicular distance from origin to plane

(c) for every value of p, plane passes through origin

(d) l^{2} + m^{2} + n^{2} = 1

Solution:

∵ P is distance of the plane from origin.

So, plane can pass through origin only if p = 0 otherwise not for other values.

Hence, (c) is correct option.

Question 10.

A plane meets axis in A, B and C such that centroid of ? ABC is (1, 2, 3) then equation of plane is :

Solution:

Let equation of plane \(\frac { x }{ a } \) + \(\frac { y }{ b } \) + \(\frac { z }{ c } \) = 1 which meets the coordinate axis on points A (a,0,0), B(0,b,0) and C (0,0,c), then centroid of ∆ABC will be (\(\frac { a }{ 3 } \),\(\frac { b }{ 3 } \),\(\frac { c }{ 3 } \))

Question 11.

Position vectors of two points are

Equation of plane passing through Q and perependicular of PQ is

Solution:

Let position vector of point P.

and position vector of point Q.

then \(\overrightarrow { PQ } \) = position vector of Q- position of vector of P

∴ Equation of plane passing through point Q (\(\overrightarrow { b } \)) perpendicular to PQ is

Question 12.

Relation between direction cosines of two lines are l – 5m + 3n = 0 and 7l^{2} + 5m^{2} – 3n^{2} = 0

Find these lines.

Solution:

Given

Question 13.

Projection of a line on axis are – 3, 4, – 12. Find length of line segment and direction cosines.

Solution:

Projection of a line coordinate axis are the direction ratios of a line.

If direction cosines are l, m, n then

Question 14.

Prove that the line joining the points (a, b, c) and (a’ b’, c’) passes through origin, if aa’+ bb’+ cc’ = pp’ where p and p’ are distance of points from origin.

Solution:

According to question, distance of points (a, b, c) and (a’, b’, c’) from origin.

Question 15.

Find the equation of plane, passes through P (-2,1,2) and is parallel to the two vectors

Solution:

∵ Plane passes through point P(- 2, 1, 2).

∴ Equation of plane is

a(x + 2) + b(y – 1) + c(z – 2) = 0

But plane is travelled to the vector