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# RD Sharma solutions for Class 12 Mathematics chapter 27 - Direction Cosines and Direction Ratios

## Chapter 27 : Direction Cosines and Direction Ratios

#### Pages 22 - 23

Q 1 | Page 22

If a line makes angles of 90°, 60° and 30° with the positive direction of xy, and z-axis respectively, find its direction cosines

Q 2 | Page 23

If a line has direction ratios 2, −1, −2, determine its direction cosines.

Q 3 | Page 23

Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3) .

Q 4 | Page 23

Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.

Q 5 | Page 23

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, −4), (−1, 1, 2) and (−5, −5, −2).

Q 6 | Page 23

Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.

Q 7 | Page 23

Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.

Q 8 | Page 23

Find the acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.

Q 9 | Page 23

Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.

Q 10 | Page 23

Show that the line through points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and (1, 2, 5).

Q 11 | Page 23

Show that the line through the points (1, −1, 2) and (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Q 12 | Page 23

Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1) and (4, 3, −1).

Q 13 | Page 23

Find the angle between the lines whose direction ratios are proportional to abc and b − cc − aa− b.

Q 14 | Page 23

If the coordinates of the points ABCD are (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2), then find the angle between AB and CD.

Q 15 | Page 23

Find the direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2ln − mn= 0.

Q 16.1 | Page 23

Find the angle between the lines whose direction cosines are given by the equations
(i) m + n = 0 and l2 + m2 − n2 = 0

Q 16.2 | Page 23

Find the angle between the lines whose direction cosines are given by the equations

2l − m + 2n = 0 and mn + nl + lm = 0

Q 16.3 | Page 23

Find the angle between the lines whose direction cosines are given by the equations

l + 2m + 3n = 0 and 3lm − 4ln + mn = 0

Q 16.4 | Page 23

Find the angle between the lines whose direction cosines are given by the equations

2l + 2m − n = 0, mn + ln + lm = 0

#### Pages 24 - 25

Q 1 | Page 24

Define direction cosines of a directed line.

Q 2 | Page 24

What are the direction cosines of X-axis?

Q 3 | Page 24

What are the direction cosines of Y-axis?

Q 4 | Page 24

What are the direction cosines of Z-axis?

Q 5 | Page 24

Write the distances of the point (7, −2, 3) from XYYZ and XZ-planes.

Q 6 | Page 24

Write the distance of the point (3, −5, 12) from X-axis?

Q 7 | Page 24

Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).

Q 8 | Page 24

A line makes an angle of 60° with each of X-axis and Y-axis. Find the acute angle made by the line with Z-axis.

Q 9 | Page 25

If a line makes angles α, β and γ with the coordinate axes, find the value of cos 2α + cos 2β + cos 2γ.

Q 10 | Page 25

Write the ratio in which the line segment joining (abc) and (−a, −c, −b) is divided by the xy-plane.

Q 11 | Page 25

Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.

Q 12 | Page 25

Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.

Q 13 | Page 25

Write the distance of the point P (xyz) from XOY plane.

Q 14 | Page 25

Write the coordinates of the projection of point P (xyz) on XOZ-plane.

Q 15 | Page 25

Write the coordinates of the projection of the point P (2, −3, 5) on Y-axis.

Q 16 | Page 25

Find the distance of the point (2, 3, 4) from the x-axis.

Q 17 | Page 25

If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?

Q 18 | Page 25

Write direction cosines of a line parallel to z-axis.

Q 19 | Page 25

If a unit vector  vec a makes an angle $\frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text{ with } \hat{j}$ and an acute angle θ with $\hat{ k}$ ,then find the value of θ.

Q 20 | Page 25

Answer each of the following questions in one word or one sentence or as per exact requirement of the question:
Write the distance of a point P(abc) from x-axis.

Q 21 | Page 25

If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.

#### Pages 25 - 26

Q 1 | Page 25

For every point P (xyz) on the xy-plane,

x = 0

y = 0

z = 0

x = y = z = 0

Q 2 | Page 25

For every point P (xyz) on the x-axis (except the origin),

x = 0, y = 0, z ≠ 0

x = 0, z = 0, y ≠ 0

y = 0, z = 0, x ≠ 0

x = y = z = 0

Q 3 | Page 25

A rectangular parallelopiped is formed by planes drawn through the points (5, 7, 9) and (2, 3, 7) parallel to the coordinate planes. The length of an edge of this rectangular parallelopiped is

2

3

4

all of these

Q 4 | Page 25

A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7), parallel to the coordinate planes. The length of a diagonal of the parallelopiped is

7

sqrt(38)

sqrt(155)

none of these

Q 5 | Page 25

The xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)

internally in the ratio 2 : 3

externally in the ratio 2 : 3

internally in the ratio 3 : 2

externally in the ratio 3 : 2

Q 6 | Page 25

If the x-coordinate of a point P on the join of Q (2, 2, 1) and R (5, 1, −2) is 4, then its z-coordinate is

2

1

-1

-2

Q 7 | Page 25

The distance of the point P (abc) from the x-axis is

$\sqrt{b^2 + c^2}$

$\sqrt{a^2 + c^2}$

$\sqrt{a^2 + b^2}$

none of these

Q 8 | Page 26

Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is

3 : 1 internally

3 : 1 externally

1 : 2 internally

2 : 1 externally

Q 9 | Page 26

If P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio

3 : 2 externally

3 : 2 internally

2 : 1 internally

2 : 1 externally

Q 11 | Page 26

If O is the origin, OP = 3 with direction ratios proportional to −1, 2, −2 then the coordinates of P are

(−1, 2, −2)

(1, 2, 2)

(−1/9, 2/9, −2/9)

(3, 6, −9)

Q 12 | Page 26

The angle between the two diagonals of a cube is

(a) 30°

(b) 45°

(c) $\cos^{- 1} \left( \frac{1}{\sqrt{3}} \right)$

(d) $\cos^{- 1} \left( \frac{1}{3} \right)$

Q 13 | Page 26

If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos2 α + cos2 β + cos2γ + cos2 δ is equal to

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{4}{3}$

$\frac{8}{3}$

## RD Sharma solutions for Class 12 Mathematics chapter 27 - Direction Cosines and Direction Ratios

RD Sharma solutions for Class 12 Maths chapter 27 (Direction Cosines and Direction Ratios) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 27 Direction Cosines and Direction Ratios are Plane Passing Through the Intersection of Two Given Planes, Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point, Equation of a Plane in Normal Form, Three - Dimensional Geometry Examples and Solutions, Introduction of Three Dimensional Geometry, Equation of a Plane Passing Through Three Non Collinear Points, Relation Between Direction Ratio and Direction Cosines, Intercept Form of the Equation of a Plane, Coplanarity of Two Lines, Distance of a Point from a Plane, Angle Between Line and a Plane, Angle Between Two Planes, Angle Between Two Lines, Vector and Cartesian Equation of a Plane, Shortest Distance Between Two Lines, Equation of a Line in Space, Direction Cosines and Direction Ratios of a Line.

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