Three Dimensional Geometry Test

Three Dimensional Geometry

This is Three Dimensional Geometry Test-01 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
Direction cosines of a line are
  • A
    The cotangents of the angles made by the line with the negative directions of the coordinate axes.
  • B
    The cosines of the angles made by the line with the positive directions of the coordinate axes.
    Correct
  • C
    The sines of the angles made by the line with the positive directions of the coordinate axes.
  • D
    The tangents of the angles made by the line with the negative directions of the coordinate axes.
2
If l, m, n are the direction cosines of a line, then
  • A
    \({l^2} + {\text{ 2}}{m^2} + {\text{ }}{n^2} = {\text{ }}1.\)
  • B
    \({l^2} + {\text{ }}{m^2} + {\text{ }}{n^2} = {\text{ }}1.\)
    Correct
  • C
    \(2{l^2} + {\text{ }}{m^2} + {\text{ }}{n^2} = {\text{ }}1.\)
  • D
    \({l^2} + {\text{ }}{m^2} + {\text{ 2}}{n^2} = {\text{ }}1.\)l2
3
\(\overrightarrow {PQ} \)is a vector joining two points P(x1, y1, z1) and Q(x2, y2, z2).If \(\left| {\overline {PQ} } \right| = d\),Direction cosines of \(\overrightarrow {PQ} \;\)are
  • A
    \(\frac{{{x_2} + {x_1}}}{d},\;\frac{{{y_2} - {y_1}}}{d},\;\frac{{{z_2} - {z_1}}}{d}\)
  • B
    \(\frac{{{x_2} - {x_1}}}{d},\;\frac{{{y_2} + {y_1}}}{d},\;\frac{{{z_2} - {z_1}}}{d}\)
  • C
    \(\frac{{{x_2} - {x_1}}}{d},\;\frac{{{y_2} - {y_1}}}{d},\;\frac{{{z_2} + {z_1}}}{d}\)
  • D
    \(\frac{{{x_2} - {x_1}}}{d},\;\frac{{{y_2} - {y_1}}}{d},\;\frac{{{z_2} - {z_1}}}{d}\)
    Correct
4
If l, m and n are the direction cosines of a line, Direction ratios of the line are the numbers which are
  • A
    Inversely Proportional to the direction cosines of the line
  • B
    Proportional to the direction cosines of the line.
    Correct
  • C
    Inversely Proportional to the direction cosine l of the line
  • D
    Proportional to the direction cosine l of the line
5
If l, m, n are the direction cosines and a, b, c are the direction ratios of a line then
  • A
    Option A
  • B
    Option B
    Correct
  • C
    Option C
  • D
    Option D
6
Skew lines are lines in different planes which are
  • A
    intersecting
  • B
    parallel and intersecting
  • C
    parallel
  • D
    neither parallel nor intersecting
    Correct
7
Angle between skew lines is
  • A
    the angle between two non intersecting lines drawn from any point parallel to each of the skew lines
  • B
    the angle between two intersecting lines drawn from any point perpendicular to each of the skew lines
  • C
    the angle between two intersecting lines drawn from any point parallel to each of the skew lines
    Correct
  • D
    the angle between two non intersecting lines drawn from any point anti – parallel to each of the skew lines
8
If \({l_1},{\text{ }}{m_1},{\text{ }}{n_1}\) and \({l_2},{\text{ }}{m_2},{\text{ }}{n_2}\) are the direction cosines of two lines; and \(\theta \) is the acute angle between the two lines; then
  • A
    \(\cos \theta = \left| {{l_1}{m_2} + {m_1}{m_2} + {n_1}{n_2}} \right|\)
  • B
    \(\cos \theta = \left| {{l_1}{l_2} + {l_1}{m_2} + {n_1}{n_2}} \right|\)
  • C
    \(\cos \theta = \left| {{l_1}{l_2} + {m_1}{m_2} + {l_1}{n_2}} \right|\)
  • D
    \(\cos \theta = \left| {{l_1}{l_2} + {m_1}{m_2} + {n_1}{n_2}} \right|\)
    Correct
9
If \({a_1},{\text{ }}{{\text{b}}_1},{\text{ }}{{\text{c}}_1}\) and \({a_2},{\text{ }}{{\text{b}}_2},{\text{ }}{{\text{c}}_2}\) are the direction ratios of two lines and \(\theta \) is the acute angle between the two lines; then
  • A
    \(\cos \theta = \left| {\frac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{a_2}}}{{\sqrt {a_1^2 + b_1^2 + c_1^2} \sqrt {a_2^2 + b_2^2 + c_2^2} }}} \right|\)
  • B
    \(\cos \theta = \left| {\frac{{{a_1}{b_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {a_1^2 + b_1^2 + c_1^2} \sqrt {a_2^2 + b_2^2 + c_2^2} }}} \right|\)
  • C
    \(\cos \theta = \left| {\frac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {a_1^2 + b_1^2 + c_1^2} \sqrt {a_2^2 + b_2^2 + c_2^2} }}} \right|\)
    Correct
  • D
    \(\cos \theta = \left| {\frac{{{a_1}{a_2} + {b_1}{c_2} + {c_1}{c_2}}}{{\sqrt {a_1^2 + b_1^2 + c_1^2} \sqrt {a_2^2 + b_2^2 + c_2^2} }}} \right|\)
10
Vector equation of a line that passes through the given point whose position vector is\(\vec a\) and parallel to a given vector \(\vec b\) is
  • A
    \(\vec r = \vec a + \lambda \vec b\)
    Correct
  • B
    \(\vec r = \vec a - \lambda \vec b\)
  • C
    \(\vec r = - \vec a - \lambda \vec b\)
  • D
    \(\vec r = - \vec a + \lambda \vec b\)
11
Equation of a line through a point \(\left( {{x_1},{\text{ }}{y_1},{\text{ }}{z_1}} \right)\) and having direction cosines l, m, n is
  • A
    \(\frac{{x - {x_1}}}{l} = \frac{{y - {y_1}}}{m} = \frac{{z - {z_1}}}{n}\)
    Correct
  • B
    \(\frac{{x - {x_1}}}{l} = \frac{{y - {z_1}}}{m} = \frac{{z - {z_1}}}{n}\)
  • C
    \(\frac{{x - {x_1}}}{l} = \frac{{y - {y_1}}}{m} = \frac{{z - {x_1}}}{n}\)
  • D
    \(\frac{{x - {y_1}}}{l} = \frac{{y - {y_1}}}{m} = \frac{{z - {z_1}}}{n}\)
12
Vector equation of a line that passes through two points whose position vectors are \(\vec a\;and\;\vec b\) is
  • A
    \(\vec r = - \vec a + \lambda \left( {\vec b + \vec a} \right)\)
  • B
    \(\vec r = - \vec a + \lambda \left( {\vec b - \vec a} \right)\)
  • C
    \(\vec r = \vec a + \lambda \left( {\vec b - \vec a} \right)\)
    Correct
  • D
    \(\vec r = \vec a + \lambda \left( {\vec b + \vec a} \right)\)
13
Cartesian equation of a line that passes through two points\(\left( {{x_1},{\text{ }}{y_1},{\text{ }}{z_1}} \right)\) and \(\left( {{x_2},{\text{ }}{y_2},{\text{ }}{z_2}} \right)\) is
  • A
    \(\frac{{x - {x_1}}}{{{x_2} - {x_1}}} = \frac{{y - {y_1}}}{{{y_2} - {y_1}}} = \frac{{z + {z_1}}}{{{z_2} - {z_1}}}\)
  • B
    \(\frac{{x - {x_1}}}{{{x_2} - {x_1}}} = \frac{{y - {y_1}}}{{{y_2} + {y_1}}} = \frac{{z - {z_1}}}{{{z_2} - {z_1}}}\)
  • C
    \(\frac{{x + {x_1}}}{{{x_2} - {x_1}}} = \frac{{y - {y_1}}}{{{y_2} + {y_1}}} = \frac{{z - {z_1}}}{{{z_2} - {z_1}}}\)
  • D
    \(\frac{{x - {x_1}}}{{{x_2} - {x_1}}} = \frac{{y - {y_1}}}{{{y_2} - {y_1}}} = \frac{{z - {z_1}}}{{{z_2} - {z_1}}}\)
    Correct
14
If \(\theta \) is the acute angle between \(\vec r = \overrightarrow {{a_1}} + \lambda \overrightarrow {{b_1}} \;and\;\vec r = \overrightarrow {{a_2}} + \lambda \overrightarrow {{b_2}} \;\) , then
  • A
    \(\cos \theta = \left| {\frac{{\overrightarrow {{b_1}} .\overrightarrow {{b_2}} }}{{\left| {\overrightarrow {{b_1}} } \right|\left| {\overrightarrow {{b_2}} } \right|}}} \right|\)
    Correct
  • B
    \(\tan \theta = \left| {\frac{{\overrightarrow {{b_1}} .\overrightarrow {{b_2}} }}{{\left| {\overrightarrow {{b_1}} } \right|\left| {\overrightarrow {{b_2}} } \right|}}} \right|\)
  • C
    \(\cot {\text{\theta }} = \left| {\frac{{\overrightarrow {{b_1}} .\overrightarrow {{b_2}} }}{{\left| {\overrightarrow {{b_1}} } \right|\left| {\overrightarrow {{b_2}} } \right|}}} \right|\)
  • D
    \(\sin \theta = \left| {\frac{{\overrightarrow {{b_1}} .\overrightarrow {{b_2}} }}{{\left| {\overrightarrow {{b_1}} } \right|\left| {\overrightarrow {{b_2}} } \right|}}} \right|\)
15
If \(\frac{{x - {x_1}}}{{{l_1}}} = \frac{{y - {y_1}}}{{{m_1}}} = \frac{{z - {z_1}}}{{{n_1}}}\;and\;\frac{{x - {x_1}}}{{{l_2}}} = \frac{{y - {y_1}}}{{{m_2}}} = \frac{{z - {z_1}}}{{{n_2}}}\)are the equations of the two lines, then the acute angle between the two lines is given by
  • A
    \(\cos \theta = \left| {{l_1}{l_2} + {m_1}{m_2} + {n_1}{n_2}} \right|\)
    Correct
  • B
    \(\tan \theta = \left| {{l_1}{l_2} + {m_1}{m_2} + {n_1}{n_2}} \right|\)
  • C
    \(\cot \theta = \left| {{l_1}{l_2} + {m_1}{m_2} + {n_1}{n_2}} \right|\)
  • D
    \(\sin \theta = \left| {{l_1}{l_2} + {m_1}{m_2} + {n_1}{n_2}} \right|\)