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# RD Sharma Solution CLass 12 Mathematics Chapter 27 Direction Cosines and Direction Ratios

## Chapter 27 – Direction Cosines and Direction Ratios Exercise Ex. 27.1

Question 1

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

Solution 1

Let l, m and n be the direction cosines of a line.

l = cos 90° = 0

Question 2

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

Solution 2

Question 3

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

Solution 3

Question 4

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

Solution 4

Question 5

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

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

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

2l + 2m – n = 0, mn + ln + lm = 0

Solution 19

## Chapter 27 – Direction Cosines and Direction Ratios Exercise Ex. 27VSAQ

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

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

Solution 17

Question 18

Write direction cosines of a line parallel to -axis.

Solution 18

Question 19

Solution 19

Question 20

Write the distance of a point P (a, b, c) from x-axis.

Solution 20

## Chapter 27 – Direction Cosines and Direction Ratios Exercise MCQ

Question 1

For every point P(x, y, z) on the xy-plane,

1. x = 0
2. y = 0
3. z = 0
4. x = y = z = 0
Solution 1

Correct option: (c)

Every point on xy-plane z co-ordinate is always zero.

Question 2

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

1. x = 0, y = 0, z ≠ 0
2. x = 0, z = 0, y ≠ 0
3. y = 0, z = 0, x ≠ 0
4. x = y = z = 0
Solution 2

Correct option: (c)

Point (x, y, z) is on the x-axis. Hence, y and z co-ordinate will be zero except the origin.

Question 3

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

1. 2
2. 3
3. 4
4. all of these
Solution 3

Correct option: (d)

Coordinates of the points given are diagonally opposite vertices of a parallelepiped. Hence, edges of parallelepiped can be 5-2, 7-3, 9-7 3,4,2.

Question 4

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

Solution 4

Correct option : (a)

Question 5

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

1. internally in the ratio 2 : 3
2. externally in the ratio 2 : 3
3. internally in the ratio 3 : 2
4. externally in the ratio 3 : 2

Solution 5

Correct option: (b)

Question 6

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

1. 2
2. 1
3. -1
4. -2
Solution 6

Correct option: (c)

Question 7

The distance of the point P(a, b, c) from the x-axis is

Solution 7

Correct option: (a)

Question 8

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

1. 3 : 4 internally
2. 3 : 1 externally
3. 1 : 2 internally
4. 2 : 1 externally

Solution 8

Correct option: (b)

Question 9

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

1. 3 : 2 internally
2. 3 : 2 externally
3. 2 : 1 internally
4. 2 : 1 externally

Solution 9

Correct option: (b)

Question 10

A(3, 2, 0), B(5, 3, 2) and C(-9, 6, -3) are the vertices of a triangle ABC. If the bisector of BAC meets BC at D, then coordinates of D are

1. (19/8, 57/16, 17/16)
2. (-19/8, 57/16, 17/16)
3. (19/8, -57/16, 17/16)
4. none of these
Solution 10

Correct option: (a)

Question 11

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

1. (-1, 2, -2)
2. (1, 2, 2)
3. (-1/9, 2/9, -2/9)
4. (3, 6, -9)
Solution 11

Correct option: (a)

Directions of OP from the origin

(-1,2,-2)=(x,y,z)

Question 12

The angle between the two diagonals of a cube is

Solution 12

Correct option: (d)

Question 13

If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos2α + cos2β + cos2γ + cos2δ is equal to

Solution 13

Correct option: (c)

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