Three Dimensional Geometry Test

Three Dimensional Geometry

This is Three Dimensional Geometry Test-02 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
Shortest distance between two skew lines is
  • A
    The line segment from origin to both the lines
  • B
    The line segment at minimum angle to both the lines
  • C
    The line segment perpendicular to both the lines.
    Correct
  • D
    The line segment parallel to both the lines
2
Shortest distance between\(\vec r = \overrightarrow {{a_1}} + \lambda \overrightarrow {{b_2}} \;and\;\vec r = \overrightarrow {{a_1}} + \mu \overrightarrow {{b_2}} \) is
  • A
    \(S.D = \left| {\frac{{\left( {\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right).\left( {\overrightarrow {{a_2}} - \overrightarrow {{a_1}} } \right)}}{{\left| {\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right|}}} \right|\)
    Correct
  • B
    \(S.D = \left| {\frac{{\left( { - \overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right).\left( {\overrightarrow {{a_2}} - \overrightarrow {{a_1}} } \right)}}{{\left| {\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right|}}} \right|\)
  • C
    \(S.D = \left| {\frac{{\left( {\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right).\left( {\overrightarrow {{a_2}} + \overrightarrow {{a_1}} } \right)}}{{\left| {\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right|}}} \right|\)
  • D
    \(S.D = \left| {\frac{{\left( {\overrightarrow {{b_1}} \times - \overrightarrow {{b_2}} } \right).\left( {\overrightarrow {{a_2}} - \overrightarrow {{a_1}} } \right)}}{{\left| {\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right|}}} \right|\)
3
Shortest distance between the lines \(\frac{{x - {x_1}}}{{{a_1}}} = \frac{{y - {y_1}}}{{{b_1}}} = \frac{{z - {z_1}}}{{{c_1}}}\)and \(\frac{{x - {x_1}}}{{{a_2}}} = \frac{{y - {y_1}}}{{{b_2}}} = \frac{{z - {z_1}}}{{{c_2}}}\) is
  • A
    \(S.D. = \frac{{\left| {\begin{array}{*{20}{c}} {{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}} \\ {{a_1}}&{{b_1}}&{{c_1}} \\ {{a_2}}&{{b_2}}&{{c_2}} \end{array}} \right|}}{{\sqrt {{{\left( {{b_1}{c_2} - {b_2}{c_1}} \right)}^2} + {{\left( {{c_1}{a_2} - {c_2}{a_1}} \right)}^2} + {{\left( {{a_1}{b_2} - {a_2}{b_1}} \right)}^2}} }}\)
    Correct
  • B
    \(S.D. = \frac{{\left| {\begin{array}{*{20}{c}} {{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} + {z_1}} \\ {{a_1}}&{{b_1}}&{{c_1}} \\ {{a_2}}&{{b_2}}&{{c_2}} \end{array}} \right|}}{{\sqrt {{{\left( {{b_1}{c_2} - {b_2}{c_1}} \right)}^2} + {{\left( {{c_1}{a_2} - {c_2}{a_1}} \right)}^2} + {{\left( {{a_1}{b_2} - {a_2}{b_1}} \right)}^2}} }}\) .
  • C
    \(S.D. = \frac{{\left| {\begin{array}{*{20}{c}} {{x_2} + {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}} \\ {{a_1}}&{{b_1}}&{{c_1}} \\ {{a_2}}&{{b_2}}&{{c_2}} \end{array}} \right|}}{{\sqrt {{{\left( {{b_1}{c_2} - {b_2}{c_1}} \right)}^2} + {{\left( {{c_1}{a_2} - {c_2}{a_1}} \right)}^2} + {{\left( {{a_1}{b_2} - {a_2}{b_1}} \right)}^2}} }}\)
  • D
    \(S.D. = \frac{{\left| {\begin{array}{*{20}{c}} {{x_2} - {x_1}}&{{y_2} + {y_1}}&{{z_2} - {z_1}} \\ {{a_1}}&{{b_1}}&{{c_1}} \\ {{a_2}}&{{b_2}}&{{c_2}} \end{array}} \right|}}{{\sqrt {{{\left( {{b_1}{c_2} - {b_2}{c_1}} \right)}^2} + {{\left( {{c_1}{a_2} - {c_2}{a_1}} \right)}^2} + {{\left( {{a_1}{b_2} - {a_2}{b_1}} \right)}^2}} }}\)
4
Distance between\(\vec r = \overrightarrow {{a_1}} + \lambda \vec b\;and\;\vec r = \overrightarrow {{a_2}} + \mu \vec b\) is
  • A
    \(S.D. = \left| {\frac{{\vec b \times \left( {\overrightarrow {{a_2}} + \overrightarrow {{a_1}} } \right)}}{{\left| {\vec b} \right|}}} \right|\).
  • B
    \(S.D. = \left| {\frac{{\vec b \times \left( { - \overrightarrow {{a_2}} - \overrightarrow {{a_1}} } \right)}}{{\left| {\vec b} \right|}}} \right|\).
  • C
    \(S.D. = \left| {\frac{{ - \vec b \times \left( {\overrightarrow {{a_2}} - \overrightarrow {{a_1}} } \right)}}{{\left| {\vec b} \right|}}} \right|\)
  • D
    \(S.D. = \left| {\frac{{\vec b \times \left( {\overrightarrow {{a_2}} - \overrightarrow {{a_1}} } \right)}}{{\left| {\vec b} \right|}}} \right|\).
    Correct
5
If a line makes angles \(90^\circ ,{\text{ }}135^\circ ,{\text{ }}45^\circ \) with the x, y and z – axes respectively, find its direction cosines.
  • A
    \(1,\;\frac{{ - 1}}{{\sqrt 3 }},\;\frac{1}{{\sqrt 2 }}\)
  • B
    \(0,\;\frac{{ - 1}}{{\sqrt 2 }},\;\frac{1}{{\sqrt 3 }}\)
  • C
    \(0,\;\frac{{ - 1}}{{\sqrt 2 }},\;\frac{1}{{\sqrt 5 }}\)
  • D
    \(0,\;\frac{{ - 1}}{{\sqrt 2 }},\;\frac{1}{{\sqrt 2 }}\)
    Correct
6
If a line has the direction ratios – 18, 12, – 4, then what are its direction cosines ?
  • A
    \(\frac{{ - 7}}{{11}},\;\frac{6}{{11}},\frac{{ - 3}}{{11}}\)
  • B
    \(\frac{9}{{11}},\;\frac{6}{{11}},\frac{{ - 2}}{{11}}\)
  • C
    \(\frac{{ - 9}}{{11}},\;\frac{6}{{11}},\frac{{ - 2}}{{11}}\)
    Correct
  • D
    \(\frac{{ - 9}}{{11}},\;\frac{6}{{11}},\frac{2}{{11}}\)
7
Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector\(3\hat i + 2\hat j - 2\hat k\).
  • A
    \(\vec r = \widehat {2i} + 2\hat j + 3\hat k\) +\(\lambda \left( {3\hat i + 2\hat j - 2\hat k.} \right)\)
  • B
    \(\vec r = 3\hat i + 2\hat j + 3\hat k\) +\(\lambda \left( {3\hat i + 2\hat j - 2\hat k.} \right)\)
  • C
    \(\vec r = \hat i + 2\hat j + 3\hat k\) +\(\lambda \left( {3\hat i + 2\hat j - 2\hat k.} \right)\)
    Correct
  • D
    \(\vec r = 4\hat i + 2\hat j + 3\hat k\) +\(\lambda \left( {3\hat i + 2\hat j - 2\hat k.} \right)\)
8
Find the equation of the line in cartesian form that passes through the point with position vector\(2\hat i - \hat j + 4\hat k\) and is in the direction \(\hat i + 2\hat j - \hat k\).
  • A
    \(\frac{{x - 3}}{1} = \frac{{y + 1}}{2} = \frac{{z - 3}}{{ - 1}}\)
  • B
    \(\frac{{x - 5}}{1} = \frac{{y + 1}}{2} = \frac{{z - 5}}{{ - 1}}\)
  • C
    \(\frac{{x - 1}}{1} = \frac{{y + 1}}{2} = \frac{{z - 2}}{{ - 1}}\)
  • D
    \(\frac{{x - 2}}{1} = \frac{{y + 1}}{2} = \frac{{z - 4}}{{ - 1}}\)
    Correct
9
Find the equation of the line in cartesian form that passes through the point (– 2, 4, – 5) and parallel to the line given by\(\frac{{x + 3}}{3} = \frac{{y - 4}}{5} = \frac{{z + 8}}{6}\)
  • A
    \(\frac{{x + 3}}{3} = \frac{{y - 4}}{5} = \frac{{z + 5}}{6}\)
  • B
    \(\frac{{x + 2}}{3} = \frac{{y - 4}}{5} = \frac{{z + 5}}{6}\)
    Correct
  • C
    \(\frac{{x + 5}}{3} = \frac{{y - 4}}{5} = \frac{{z + 5}}{6}\)
  • D
    \(\frac{{x + 4}}{3} = \frac{{y - 4}}{5} = \frac{{z + 5}}{6}\)
10
The cartesian equation of a line is \(\frac{{x - 5}}{3} = \frac{{y + 4}}{7} = \frac{{z - 6}}{2}\). Write its vector form.
  • A
    \(\vec r = 5\hat i - 4\hat j + 2\hat k\) +\(\lambda \left( {3\hat i + 7\hat j + 2\hat k.} \right)\)
  • B
    \(\vec r = 6\hat i - 4\hat j + 6\hat k\) +\(\lambda \left( {3\hat i + 7\hat j + 2\hat k.} \right)\)
  • C
    \(\vec r = 5\hat i - 4\hat j + 6\hat k\) +\(\lambda \left( {3\hat i + 7\hat j + 2\hat k.} \right)\)
    Correct
  • D
    \(\vec r = 5\hat i - 5\hat j + 6\hat k\) +\(\lambda \left( {3\hat i + 7\hat j + 2\hat k.} \right)\)
11
Find the cartesian equations of the lines that passes through the origin and (5, – 2, 3).
  • A
    \(\frac{x}{5} = \frac{y}{{ - 2}} = \frac{z}{4}\)
  • B
    \(\frac{x}{5} = \frac{y}{{ - 1}} = \frac{z}{3}\)
  • C
    \(\frac{x}{6} = \frac{y}{{ - 2}} = \frac{z}{3}\)
  • D
    \(\frac{x}{5} = \frac{y}{{ - 2}} = \frac{z}{3}\)
    Correct
12
Find the vector equations of the line that passes through the points (3, – 2, – 5), (3, – 2, 6).
  • A
    \(\vec r = - 3\hat i - 2\hat j - 5\hat k\) +\(\lambda \left( {11\hat k.} \right)\)
  • B
    \(\vec r = - 3\hat i - 2\hat j + 5\hat k\) +\(\lambda \left( {11\hat k.} \right)\)
  • C
    \(\vec r = - 3\hat i + 2\hat j - 5\hat k\) +\(\lambda \left( {11\hat k.} \right)\)
  • D
    \(\vec r = 3\hat i - 2\hat j - 5\hat k\) +\(\lambda \left( {11\hat k.} \right)\)
    Correct
13
Find the angle between the following pairs of lines: \(\;\vec r = 2\hat i - 5\hat j + \hat k\) +\(\lambda \left( {3\hat i + 2\hat j + 6\hat k.} \right)\;and\;\vec r = 7\hat i - 6\hat k\) +\(\mu \left( {\hat i + 2\hat j + 2\hat k.} \right)\)
  • A
    \( \theta = {\cos ^{ - 1}}( {\frac{{19}}{{21}}}) \)
    Correct
  • B
    \(\theta = {\tan ^{ - 1}}\left( {\frac{{19}}{{21}}} \right)\)
  • C
    \(\theta = {\sin ^{ - 1}}\left( {\frac{{19}}{{21}}} \right)\)
  • D
    \(\theta = {\cot ^{ - 1}}\left( {\frac{{19}}{{21}}} \right)\)
14
Find the values of p so that the lines\(\frac{{1 - x}}{3} = \frac{{7y - 14}}{{2p}} = \frac{{z - 3}}{2}\;and\;\frac{{7 - 7x}}{{3p}} = \frac{{y - 5}}{1} = \frac{{6 - z}}{5}\)are at right angles.
  • A
    \(p = \frac{{70}}{{12}}\)
  • B
    \(p = \frac{{70}}{{11}}\)
    Correct
  • C
    \(p = \frac{{71}}{{13}}\)
  • D
    \(p = \frac{{72}}{{15}}\)
15
Find the shortest distance between the lines \(\vec r = \hat i + 2\hat j + \hat k\) +\(\lambda \left( {\hat i - \hat j + \hat k.} \right)\;and\;\vec r = 2\hat i - \hat j - \hat k\) +\(\mu \left( {2\hat i + \hat j + \widehat {2k}.} \right)\)
  • A
    \(\frac{{4\sqrt 2 }}{2}\)
  • B
    \(\frac{{3\sqrt 2 }}{2}\)
    Correct
  • C
    \(\frac{{5\sqrt 2 }}{2}\)
  • D
    \(\frac{{3\sqrt 2 }}{5}\)