# RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise

## 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) l2 + m2 + n2 = 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 7l2 + 5m2 – 3n2 = 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