Three Dimensional Geometry Test

Three Dimensional Geometry

This is Three Dimensional Geometry Test-05 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 equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0.
  • A
    – x – z + 2 = 0
  • B
    x + z + 2 = 0
  • C
    x – z + 2 = 0
    Correct
  • D
    x + z + 5 = 0
2
Find the angle between the planes whose vector equations are\(\vec r.\left( {2\hat i + 2\hat j - 3\hat k} \right) = 5,\;\;and\;\vec r.\left( {3\hat i - 3\hat j + 5\hat k} \right) = 3\)
  • A
    \({\sin ^{ - 1}}\left( {\frac{{15}}{{\sqrt {731} }}} \right)\)
  • B
    \({\tan ^{ - 1}}\left( {\frac{{15}}{{\sqrt {731} }}} \right)\)
  • C
    \({\cos ^{ - 1}}\left( {\frac{{15}}{{\sqrt {731} }}} \right)\)
    Correct
  • D
    \({\cot ^{ - 1}}\left( {\frac{{15}}{{\sqrt {731} }}} \right)\)
3
Determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0
  • A
    \({\cos ^{ - 1}}\left( {\frac{2}{5}} \right)\)
    Correct
  • B
    \({\cot ^{ - 1}}\left( {\frac{2}{5}} \right)\)
  • C
    \({\tan ^{ - 1}}\left( {\frac{2}{5}} \right)\)
  • D
    \({\sin ^{ - 1}}\left( {\frac{2}{5}} \right)\)
4
In the following case, determine whether the given planes are parallel orperpendicular, and in case they are neither, find the angles between them. 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0
  • A
    The planes are parallel
  • B
    The planes are at \(45^\circ \)
  • C
    The planes are at \(55^\circ \)
  • D
    The planes are perpendicular
    Correct
5
In the following case, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0
  • A
    The planes are perpendicular
  • B
    The planes are at \(55^\circ \)
  • C
    The planes are parallel
    Correct
  • D
    The planes are at \(45^\circ \)
6
In the following case, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. 2x – y + 3z – 1 = 0 and 2x – y + 3z + 3 = 0
  • A
    The planes are parallel
    Correct
  • B
    The planes are at {tex}45^\circ {/tex
  • C
    The planes are perpendicular
  • D
    The planes are at \(55^\circ \)
7
In the following case, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. 4x + 8y + z – 8 = 0 and y + z – 4 = 0
  • A
    \(43^\circ \)
  • B
    \(49^\circ \)
  • C
    \(45^\circ \)
    Correct
  • D
    \(47^\circ \)
8
Find the distance of the point (0, 0, 0) from the plane 3x – 4y + 12 z = 3
  • A
    \(\frac{9}{{13}}\)
  • B
    \(\frac{3}{{13}}\)
    Correct
  • C
    \(\frac{7}{{13}}\)
  • D
    \(\frac{5}{{13}}\)
9
Find the distance of the point (3, – 2, 1) from the plane 2x – y + 2z + 3 = 0
  • A
    \(\frac{{17}}{3}\)
  • B
    \(\frac{{19}}{3}\)
  • C
    \(\frac{{13}}{3}\)
    Correct
  • D
    \(\frac{{15}}{3}\)
10
Find the distance of the point (2, 3, – 5) from the plane x + 2y – 2z = 9
  • A
    3
    Correct
  • B
    5
  • C
    2
  • D
    4
11
Find the distance of the point (– 6, 0, 0) from the plane 2x – 3y + 6z – 2 = 0
  • A
    2
    Correct
  • B
    4
  • C
    3
  • D
    5
12
Equation of a plane perpendicular to a given line with direction ratios A, B, C and passing through a given point \(\left( {{x_1},{\text{ }}{y_1},{\text{ }}{z_1}} \right)\) is
  • A
    A (x + x1) + B (y – y1) + C (z – z1) = 0
  • B
    A (x – x1) + B (y – y1) + C (z – z1) = 1
  • C
    A (x – x1) + B (y + y1) + C (z – z1) = 1
  • D
    A (x – x1) + B (y – y1) + C (z – z1) = 0
    Correct
13
The angle \(\phi \) between the line \(\vec r = \vec a + \lambda \vec b\) and the plane \(\vec r.\hat n = d\) is
  • A
    \(\cot \phi = \;\left| {\frac{{\vec b.\hat n}}{{\left| {\vec b} \right|\left| {\hat n} \right|}}} \right|\)
  • B
    \(\cos \phi = \;\left| {\frac{{\vec b.\hat n}}{{\left| {\vec b} \right|\left| {\hat n} \right|}}} \right|\)
  • C
    \(\sin \phi = \;\left| {\frac{{\vec b.\hat n}}{{\left| {\vec b} \right|\left| {\hat n} \right|}}} \right|\)
    Correct
  • D
    \(\tan \phi = \;\left| {\frac{{\vec b.\hat n}}{{\left| {\vec b} \right|\left| {\hat n} \right|}}} \right|\)
14
Find the coordinates of the foot of the perpendicular drawn from the origin to 2x + 3y + 4z – 12 = 0
  • A
    \(\left( {\frac{{24}}{{29}},\frac{{36}}{{29}},\frac{{49}}{{29}}} \right)\)
  • B
    \(\left( {\frac{{24}}{{29}},\frac{{39}}{{29}},\frac{{48}}{{29}}} \right)\)
  • C
    \(\left( {\frac{{24}}{{29}},\frac{{36}}{{29}},\frac{{48}}{{29}}} \right)\)
    Correct
  • D
    \(\left( {\frac{{27}}{{29}},\frac{{36}}{{29}},\frac{{48}}{{29}}} \right)\)
15
Find the coordinates of the foot of the perpendicular drawn from the origin to 3y + 4z – 6 = 0
  • A
    \(\left( {0,\frac{{19}}{{25}},\frac{{24}}{{25}}} \right)\)
  • B
    \(\left( {1,\frac{{18}}{{25}},\frac{{24}}{{25}}} \right)\)
  • C
    \(\left( {0,\frac{{18}}{{25}},\frac{{24}}{{25}}} \right)\)
    Correct
  • D
    \(\left( {0,\frac{{18}}{{25}},\frac{{27}}{{25}}} \right)\)