All questions of Chapter 11 - Three Dimensional Geometry for Commerce Exam

• a)
  (- , + , +)
• b)
  (-, – , +)
• c)
  ( + , – , -)
• d)
  ( + , + , -)

• a)
  
• b)
  
• c)
  
• d)
  

• A:

  1 , 0 , -2/3

• B:

  3/√13, 0, -2/√13

• C:

  3/4, 0, -2/4

• D:

  -3/√12, 0, -2/√13

The answer is b.

• a)
  0°
• b)
  cos^-1(1/√3)
• c)
  90°
• d)
  180°

• a)
  -2, 9, -6
• b)
  -4, 18, -12
• c)
  2/11, 9/11, 6/11
• d)
  -2/11, 9/11, -6/11

• a)
  The planes are perpendicular
• b)
  The planes are parallel
• c)
  The planes are at 45^∘
• d)
  The planes are at 55^∘

If the direction cosines of a line from the positive X-axis and Y-axis areThe angle of the line through Z-axis is:​

• a)
  30°
• b)
  45°
• c)
  135°
• d)
  60°

• a)
  1, 1, 1
• b)
  ± 1/√3, 1/√3, 1/√3
• c)
  1, 0, 0
• d)
  1/3, 1/3, 1/3

• a)
  1, 0, 0
• b)
  0, 0, 0
• c)
  0, 1, 0
• d)
  0, 0, 1

• a)
  
• b)
  
• c)
  
• d)
  

The angle between the pair of lines given byand  is:

• a)
  
• b)
  
• c)
  
• d)
  

The angle between the lines  and  is:

• a)
  
• b)
  
• c)
  
• d)
  

For which value of a lines  and  are perpendicular?

• a)
  11/70
• b)
  5
• c)
  1
• d)
  70/11

• a)
  
• b)
  
• c)
  
• d)
  

• a)
  x + z + 2 = 0
• b)
  – x – z + 2 = 0
• c)
  x – z + 2 = 0
• d)
  x + z + 5 = 0

• a)
  lx – my + nz = d
• b)
  – lx + my + nz = d
• c)
  lx + my + nz = d
• d)
  lx + my + nz = – d

• a)
  Proportional to the direction cosine l of the line
• b)
  Inversely Proportional to the direction cosines of the line
• c)
  Inversely Proportional to the direction cosine l of the line
• d)
  Proportional to the direction cosines of the line.

The length of the shortest distance between the lines  and  is:

• a)
  Zero
• b)
  6√3
• c)
  144
• d)
  12√3

• a)
  
• b)
  
• c)
  
• d)
  

• a)
  Both A and R are true and R is the correct explanation of A
• b)
  Both A and R are true but R is NOT the correct explanation of A
• c)
  A is true but R is false
• d)
  A is false but R is True

• a)
  
• b)
  
• c)
  
• d)
  

• a)
  1, 1, 1,
• b)
  0, 0, 0
• c)
  1/3, 1/3, 1/3
• d)
  1/√3, 1/√3, 1/√3

• a)
  9/13
• b)
  7/13
• c)
  5/13
• d)
  3/13

• a)
  
• b)
  
• c)
  
• d)
  

The equation of a plane through a point whose position vector is  perpendicular to the vector  . is

• a)
  
• b)
  
• c)
  
• d)
  

• a)
  
• b)
  
• c)
  
• d)
  

Vector equation of a plane that passes through the intersection of planes  expressed in terms of a non – zero constant λ is

• a)
  
• b)
  
• c)
  
• d)
  

The vector and cartesian equations of the planes that passes through the point (1, 0, – 2) and the normal to the plane is

• a)
  
• b)
  
• c)
  
• d)
  

In the vector form, equation of a plane which is at a distance d from the origin, and  is the unit vector normal to the plane through the origin is

• a)
  
• b)
  
• c)
  
• d)
  

• a)
  The tangents of the angles made by the line with the negative directions of the coordinate axes.
• b)
  The sines of the angles made by the line with the positive directions of the coordinate axes.
• c)
  The cotangents of the angles made by the line with the negative directions of the coordinate axes.
• d)
  The cosines of the angles made by the line with the positive directions of the coordinate axes.

• a)
  0, 1, 0; 2
• b)
  1, 0, 0; 3
• c)
  1, 0, 1; 3
• d)
  0, 0, 1; 2

is a vector joining two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂). If  Direction cosines of  are

• a)
  
• b)
  
• c)
  
• d)
  

• a)
  17/3
• b)
  19/3
• c)
  13/3
• d)
  15/3

• a)
  2x + 3y – 7z = 5
• b)
  2x + 5y – 3z = 5
• c)
  3x + 3y – 3z = 5
• d)
  2x + 3y – 3z = 5

Vector equation of a line that passes through the given point whose position vector is  and parallel to a given vector  is

• a)
  
• b)
  
• c)
  
• d)
  

Find the values of p so that the lines   are at right angles.

• a)
  p =72/15
• b)
  p =70/11
• c)
  p =70/12
• d)
  p =71/13

• a)
  numbers which are same as direction angles of a line
• b)
  numbers which are proportional to the direction angles of a line
• c)
  numbers which are proportional to the direction cosines of a line
• d)
  none

• a)
  ± 1/√2, ± 1/√2, ± 1/√2
• b)
  1/√3, 1/√3, 1/√3
• c)
  1/√2, 1/√2, 1/√2
• d)
  ± 1/√3, ± 1/√3, ± 1/√3

• a)
  
• b)
  
• c)
  
• d)
  

Find the shortest distance between the lines :   

• a)
  
• b)
  
• c)
  
• d)
  

• a)
  
• b)
  
• c)
  
• d)
  

Shortest distance between 

• a)
  
• b)
  
• c)
  
• d)
  

Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as

Assertion (A): The angle between the straight lines

Reason (R): Skew lines are lines in different planes which are parallel and intersecting.

• a)
  Both A and R are true and R is the correct explanation of A
• b)
  Both A and R are true but R is NOT the correct explanation of A
• c)
  A is true but R is false
• d)
  A is false but R is True

• a)
  They form the angles α, β & γ with each other
• b)
  Any two must be perpendicular to each other.
• c)
  All three are mutually perpendicular
• d)
  no relation between these three planes

• a)
  180° - α, 180° - β, 180° - γ
• b)
  cos - α, cos - β, cos - γ
• c)
  90° - α, 90° - β, 90° - γ
• d)
  sin α, sin β, sin γ

• a)
  
• b)
  
• c)
  
• d)
  

Distance between 

• a)
  
• b)
  
• c)
  
• d)
  

• a)
  
• b)
  
• c)
  
• d)
  

Shortest distance between the lines 

• a)
  
• b)
  
• c)
  
• d)