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

RBSE Solutions for Class 12 Maths