Three Dimensional Geometry Test

Three Dimensional Geometry

This is Three Dimensional Geometry Test-03 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
Find the shortest distance between the lines \(\frac{{x + 1}}{7} = \frac{{y + 1}}{{ - 6}} = \frac{{z + 1}}{1}\;and\;\frac{{x - 3}}{1} = \frac{{y - 5}}{{ - 2}} = \frac{{z - 7}}{1}\)
  • A
    \(2\sqrt {23} \)
  • B
    \(2\sqrt {27} \)
  • C
    \(2\sqrt {29} \)
    Correct
  • D
    \(2\sqrt {31} \)
2
Find the shortest distance between the lines : \(\vec r = \hat i + 2\hat j + 3\hat k\) +\(\lambda \left( {\hat i - 3\hat j + 2\hat k.} \right)\;and\;\vec r = 4\hat i + 5\hat j + 6\hat k\) +\(\mu \left( {2\hat i + 3\hat j + \hat k.} \right)\)
  • A
    \(\frac{3}{{\sqrt {27} }}\)
  • B
    \(\frac{3}{{\sqrt {17} }}\)
  • C
    \(\frac{3}{{\sqrt {23} }}\)
  • D
    \(\frac{3}{{\sqrt {19} }}\)
    Correct
3
Find the shortest distance between the lines \(\vec r = \left( {1 - t} \right)\hat i + \left( {t - 2} \right)\hat j + (3 - 2t)\hat k\)\(\;and\;\vec r = \left( {s + 1} \right)\hat i + \left( {2s - 1} \right)\hat j - (2s + 1)\hat k\)
  • A
    \(\frac{8}{{\sqrt {33} }}\)
  • B
    \(\frac{8}{{\sqrt {29} }}\)
    Correct
  • C
    \(\frac{8}{{\sqrt {35} }}\)
  • D
    \(\frac{8}{{\sqrt {31} }}\)
4
Find the angle between the following pairs of lines: \(\;\vec r = 3\hat i + \hat j - 2\hat k\) +\(\lambda \left( {\hat i - \hat j - 2\hat k.} \right)\;and\;\vec r = 2\hat i - \hat j - 56\hat k\) +\(\mu \left( {3\hat i - 5\hat j - 4\hat k.} \right)\)
  • A
    \(\theta = {\sin ^{ - 1}}\left( {\frac{8}{{5\sqrt 3 }}} \right)\)
  • B
    \(\theta = {\cos ^{ - 1}}\left( {\frac{8}{{5\sqrt 3 }}} \right)\)
    Correct
  • C
    \(\theta = {\tan ^{ - 1}}\left( {\frac{8}{{5\sqrt 3 }}} \right)\)
  • D
    \( \theta = {\cot ^{ - 1}}\left( {\frac{8}{{5\sqrt 3 }}} \right)\)
5
In the vector form, equation of a plane which is at a distance d from the origin, and \(\hat n\)is the unit vector normal to the plane through the origin is
  • A
    \(\vec r.\hat n = - d\)
  • B
    \(\vec r.\hat n = 2d\)
  • C
    \(\vec r.\hat n = - 2d\)
  • D
    \(\vec r.\hat n = d\)
    Correct
6
Equation of a plane which is at a distance d from the origin and the direction cosines of the normal to the plane are l, m, n is.
  • A
    lx + my + nz = – d
  • B
    lx – my + nz = d
  • C
    lx + my + nz = d
    Correct
  • D
    – lx + my + nz = d
7
The equation of a plane through a point whose position vector is \(\vec a\)and perpendicular to the vector \(\vec N\). is
  • A
    \(\left( {\vec r - \vec a} \right).\vec N = 0\)
    Correct
  • B
    \(\left( {\vec r + \vec a} \right).\vec N = 1\)
  • C
    \( - \left( {\vec r - \vec a} \right).\vec N = 0\)
  • D
    \(\left( {\vec r - \vec a} \right).\vec N = 1\)
8
Equation of a plane passing through three non collinear points (x1, y1, z1),(x2, y2, z2) and (x3, y3, z3) is
  • A
    \(\left| {\begin{array}{*{20}{c}} {x + {x_1}}&{y - {y_1}}&{z - {z_1}} \\ {{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}} \\ {{x_3} - {x_1}}&{{y_3} - {y_1}}&{{z_3} - {z_1}} \end{array}} \right|\)= 0
  • B
    \(\left| {\begin{array}{*{20}{c}} {x - {x_1}}&{y + {y_1}}&{z - {z_1}} \\ {{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}} \\ {{x_3} - {x_1}}&{{y_3} - {y_1}}&{{z_3} + {z_1}} \end{array}} \right|\)= 0
  • C
    \(\left| {\begin{array}{*{20}{c}} {x - {x_1}}&{y - {y_1}}&{z - {z_1}} \\ {{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}} \\ {{x_3} - {x_1}}&{{y_3} - {y_1}}&{{z_3} - {z_1}} \end{array}} \right|\)= 0
    Correct
  • D
    \(\left| {\begin{array}{*{20}{c}} {x - {x_1}}&{y - {y_1}}&{z - {z_1}} \\ {{x_2} - {x_1}}&{{y_2} + {y_1}}&{{z_2} - {z_1}} \\ {{x_3} + {x_1}}&{{y_3} - {y_1}}&{{z_3} - {z_1}} \end{array}} \right|\)= 0
9
Vector equation of a plane that contains three non collinear points having position vectors\(\vec a,\;\vec b\;and\;\vec c\;is\;\)
  • A
    \(\left( {\vec r + \vec a} \right).\left[ {\left( {\vec b - \vec a} \right) \times \left( {\vec c - \vec a} \right)} \right]\)
  • B
    \(\left( {\vec r - \vec a} \right).\left[ {\left( {\vec b - \vec a} \right) \times \left( {\vec c + \vec a} \right)} \right]\)
  • C
    \(\left( {\vec r - \vec a} \right).\left[ {\left( {\vec b + \vec a} \right) \times \left( {\vec c - \vec a} \right)} \right]\)
  • D
    \(\left( {\vec r - \vec a} \right).\left[ {\left( {\vec b - \vec a} \right) \times \left( {\vec c - \vec a} \right)} \right]\)
    Correct
10
Equation of a plane that cuts the coordinates axes at (a, 0, 0), (0, b, 0) and (0, 0, c) is
  • A
    \(\frac{x}{a} + \frac{y}{b} + \frac{z}{{2c}} = 1\)
  • B
    \(\frac{x}{a} + \frac{y}{{2b}} + \frac{z}{c} = 1\)
  • C
    \(\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\)
    Correct
  • D
    \(\frac{x}{{2a}} + \frac{y}{b} + \frac{z}{c} = 1\)
11
Vector equation of a plane that passes through the intersection of planes\(\vec r.{\overrightarrow n _1} = {d_1}\;and\;\vec r.{\overrightarrow n _2} = {d_2}\) expressed in terms of a non – zero constant \(\lambda \) is
  • A
    \(\vec r.\left( {\overrightarrow {{n_1}} - \lambda \overrightarrow {{n_2}} } \right) = - {d_1} + \lambda {d_2}\)
  • B
    \(\vec r.\left( { - \overrightarrow {{n_1}} - \lambda \overrightarrow {{n_2}} } \right) = {d_1} - \lambda {d_2}\)
  • C
    \(\vec r.\left( { - \overrightarrow {{n_1}} + \lambda \overrightarrow {{n_2}} } \right) = - {d_1} - \lambda {d_2}\)
  • D
    \(\vec r.\left( {\overrightarrow {{n_1}} + \lambda \overrightarrow {{n_2}} } \right) = {d_1} + \lambda {d_2}\)
    Correct
12
Cartesian equation of a plane that passes through the intersection of two given planes \({A_1}x{\text{ }} + {\text{ }}{B_1}y{\text{ }} + {\text{ }}{C_1}z{\text{ }} + {\text{ }}{D_1} \\ = {\text{ }}0{\text{ }}and{\text{ }}{A_2}x{\text{ }} + {\text{ }}{B_2}y{\text{ }} + {\text{ }}{C_2}z{\text{ }} + {\text{ }}{D_2} = {\text{ }}0\) is
  • A
    \(\left( {{A_1}x{\text{ }} + {\text{ }}{B_1}y\;--\;{C_1}z{\text{ }} + {\text{ }}{D_1}} \right){\text{ }} \\ + \lambda \;\left( {{A_2}x{\text{ }} + {\text{ }}{B_2}y{\text{ }} + {\text{ }}{C_2}z{\text{ }} + {\text{ }}{D_2}} \right){\text{ }} = {\text{ }}0.\)
  • B
    \(\left( {{\text{ }}--{\text{ }}{A_1}x{\text{ }} + {\text{ }}{B_1}y{\text{ }} + {\text{ }}{C_1}z{\text{ }} + {\text{ }}{D_1}} \right){\text{ }} \\ + \;\left( {{A_2}x{\text{ }} + {\text{ }}{B_2}y{\text{ }} + {\text{ }}{C_2}z{\text{ }} + {\text{ }}{D_2}} \right){\text{ }} = {\text{ }}0.\)
  • C
    \(\left( {{A_1}x\;--\;{B_1}y{\text{ }} + {\text{ }}{C_1}z{\text{ }} + {\text{ }}{D_1}} \right){\text{ }} \\ + \lambda \;\left( {{A_2}x{\text{ }} + {\text{ }}{B_2}y{\text{ }} + {\text{ }}{C_2}z{\text{ }} + {\text{ }}{D_2}} \right){\text{ }} = {\text{ }}0.\)
  • D
    \(\left( {{A_1}x{\text{ }} + {\text{ }}{B_1}y{\text{ }} + {\text{ }}{C_1}z{\text{ }} + {\text{ }}{D_1}} \right){\text{ }} \\ + \;\left( {{A_2}x{\text{ }} + {\text{ }}{B_2}y{\text{ }} + {\text{ }}{C_2}z{\text{ }} + {\text{ }}{D_2}} \right){\text{ }} = {\text{ }}0.\)
    Correct
13
Two lines \(\vec r = \overrightarrow {{a_1}} + \lambda \overrightarrow {{b_1}} \;and\;\vec r = \overrightarrow {{a_2}} + \mu \overrightarrow {{b_2}} \;\) are coplanar if
  • A
    \(\left( {\overrightarrow {{a_2}} - \overrightarrow {{a_1}} } \right).\left( {\overrightarrow {{b_1}} \times - \overrightarrow {{b_2}} } \right) = 0\)
  • B
    \(\left( {\overrightarrow {{a_2}} - \overrightarrow {{a_1}} } \right).\left( {\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right) = 0\)
    Correct
  • C
    \(\left( {\overrightarrow {{a_2}} - \overrightarrow {{a_1}} } \right).\left( { - \overrightarrow {{b_1}} \times \overrightarrow { - {b_2}} } \right) = 0\)
  • D
    \(\left( {\overrightarrow {{a_2}} - \overrightarrow {{a_1}} } \right).\left( { - \overrightarrow {{b_1}} \times \overrightarrow {{b_2}} } \right) = 0\)
14
In the Cartesian form two lines \(\frac{{x - {x_1}}}{{{a_1}}} = \frac{{y - {y_1}}}{{{b_1}}} = \frac{{z - {z_1}}}{{{c_1}}}\)and \(\frac{{x - {x_2}}}{{{a_2}}} = \frac{{y - {y_2}}}{{{b_2}}} = \frac{{z - {z_2}}}{{{c_2}}}\)are coplanar if
  • A
    \(\left| {\begin{array}{*{20}{c}} {{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}} \\ {{a_1}}&{{b_1}}&{{c_1}} \\ {{a_2}}&{{b_2}}&{ - {c_2}} \end{array}} \right| = 0\)
  • B
    \(\left| {\begin{array}{*{20}{c}} {{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}} \\ {{a_1}}&{{b_1}}&{{c_1}} \\ { - {a_2}}&{{b_2}}&{{c_2}} \end{array}} \right| = 0\)\(\)
  • C
    \(\left| {\begin{array}{*{20}{c}} {{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}} \\ {{a_1}}&{{b_1}}&{{c_1}} \\ {{a_2}}&{{b_2}}&{{c_2}} \end{array}} \right| = 0\)
    Correct
  • D
    \(\left| {\begin{array}{*{20}{c}} {{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}} \\ {{a_1}}&{{b_1}}&{{c_1}} \\ {{a_2}}&{ - {b_2}}&{{c_2}} \end{array}} \right| = 0\)
15
In vector form, if \(\theta \) is the angle between the two planes\(\vec r.\overrightarrow {{n_1}} = {d_1}\;and\;\vec r.\overrightarrow {{n_2}} = {d_2}\), then
  • A
    \(\theta = \;{\cos ^{ - 1}}\frac{{\left| {\overrightarrow {{n_1}} .\overrightarrow {{n_2}} } \right|}}{{\overrightarrow {{n_1}} \left| {\overrightarrow {{n_2}} } \right|}}\)
    Correct
  • B
    \(\theta = \;{\tan ^{ - 1}}\frac{{\left| {\overrightarrow {{n_1}} .\overrightarrow {{n_2}} } \right|}}{{\overrightarrow {{n_1}} \left| {\overrightarrow {{n_2}} } \right|}}\)
  • C
    \(\theta = \;{\sin ^{ - 1}}\frac{{\left| {\overrightarrow {{n_1}} .\overrightarrow {{n_2}} } \right|}}{{\overrightarrow {{n_1}} \left| {\overrightarrow {{n_2}} } \right|}}\)
  • D
    \(\theta = \;{\cot ^{ - 1}}\frac{{\left| {\overrightarrow {{n_1}} .\overrightarrow {{n_2}} } \right|}}{{\overrightarrow {{n_1}} \left| {\overrightarrow {{n_2}} } \right|}}\)