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

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

Question 3.
Angle between two diagonals of a cube is :
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
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 :
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise

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

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
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
whose direction ratio’s are 3, 0, 0.
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise

Question 5.
vector form of line
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
(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:
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
∴ Position vector of point A
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
∴ Direction ratio of line are -2,-5, 13
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
∴ Vector equation of line
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
Hence, (a) is the correct option.

Question 6.
If lines
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous ExerciseRBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
are perpendicular to each other than value of ? is :
(a) 0
(b) 1
(c) -1
(d) 2
Solution:
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise

Question 7.
Shortest distance between lines
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
(a) 10 unit
(b) 12 unit
(c) 14 unit
(d) None of these
Solution:
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise

Question 8.
Angle between line
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
Solution:
We know that angle between two lines
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise

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 :
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
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 (\(\frac { a }{ 3 } \),\(\frac { b }{ 3 } \),\(\frac { c }{ 3 } \))
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise

Question 11.
Position vectors of two points are
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
Equation of plane passing through Q and perependicular of PQ is
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
Solution:
Let position vector of point P.
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
and position vector of point Q.
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
then \(\overrightarrow { PQ } \) = position vector of Q- position of vector of P
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
∴ Equation of plane passing through point Q (\(\overrightarrow { b } \)) perpendicular to PQ is
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise

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

Question 12.
Relation between direction cosines of two lines are l – 5m + 3n = 0 and 7l2 + 5m2 – 3n2 = 0
Find these lines.
Solution:
Given
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise

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

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

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

Question 15.
Find the equation of plane, passes through P (-2,1,2) and is parallel to the two vectors
RBSE Solutions for Class 12 Maths Chapter 14 Three Dimensional Geometry Miscellaneous Exercise
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 Chapter 14 Three Dimensional Geometry Miscellaneous Exercise

RBSE Solutions for Class 12 Maths