RD SHARMA Solutions for Class 12-science Maths Chapter 27 - Direction Cosines and Direction Ratios

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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

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios


Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 2

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

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 3

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

Solution 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 16(i)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 16(i)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 16(ii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 16(ii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 16(iii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 16(iii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 16(iv)

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

2l + 2m - n = 0, mn + ln + lm = 0

Solution 16(iv)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios 

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

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

 

Solution 4

Correct option : (a)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios 

 

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)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios 

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)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios 

Question 7

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

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

 

Solution 7

Correct option: (a)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios 

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)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios 

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)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios 

 

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)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios 

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

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 12

Correct option: (d)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios 

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

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 13

Correct option: (c)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios 

 

Chapter 27 - Direction Cosines and Direction Ratios Exercise Ex. 27VSAQ

Question 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios
Question 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 17

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

Solution 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 18

Write direction cosines of a line parallel to -axis.

Solution 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Solution 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios

Question 20

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

Solution 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Direction Cosines And Direction Ratios