Three Dimensional Geometry Test-04

This is Three Dimensional Geometry Test-04 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

The angle \(\theta \) between the planes A1x + B1y + C1z + D1 = 0 and A2 x + B2 y + C2 z + D2 = 0 is given by

A
\(\tan \theta = \left| {\frac{{{A_1}{A_2} + {B_1}{B_2} + {C_1}{C_2}}}{{\sqrt {A_1^2 + B_1^2 + C_1^2} \sqrt {A_2^2 + B_2^2 + C_2^2} }}} \right|\)
B
\(\sin \theta = \left| {\frac{{{A_1}{A_2} + {B_1}{B_2} + {C_1}{C_2}}}{{\sqrt {A_1^2 + B_1^2 + C_1^2} \sqrt {A_2^2 + B_2^2 + C_2^2} }}} \right|\)
C
\(\cot \theta = \left| {\frac{{{A_1}{A_2} + {B_1}{B_2} + {C_1}{C_2}}}{{\sqrt {A_1^2 + B_1^2 + C_1^2} \sqrt {A_2^2 + B_2^2 + C_2^2} }}} \right|\)
D
\(\cos \theta = \left| {\frac{{{A_1}{A_2} + {B_1}{B_2} + {C_1}{C_2}}}{{\sqrt {A_1^2 + B_1^2 + C_1^2} \sqrt {A_2^2 + B_2^2 + C_2^2} }}} \right|\).
Correct

The distance of a point whose position vector is \(\vec a\) from the plane \(\vec r.\hat n = d\) is

A
\(\left| {d - \vec a.\hat n} \right|\)
Correct
B
\(2\left| {d - \vec a.\hat n} \right|\)
C
\(\left| {d + \vec a.\hat n} \right|\)
D
\(\left| {d +2 \vec a.\hat n} \right|\)

The distance d from a point P(x1, y1, z1) to the plane Ax + By + Cz + D = 0 is

A
\(d = \left| {\frac{{A{x_1} + B{y_1} + C{z_1} + 2D}}{{\sqrt {{A^2} + {B^2} + {C^2}} }}} \right|\)
B
\(d = \left| {\frac{{A{x_1} + 2B{y_1} + C{z_1} + D}}{{\sqrt {{A^2} + {B^2} + {C^2}} }}} \right|\)
C
\(d = \left| {\frac{{A{x_1} + B{y_1} + C{z_1} + D}}{{\sqrt {{A^2} + {B^2} + {C^2}} }}} \right|\)
Correct
D
\(d = \left| {\frac{{A{x_1} + B{y_1} + 2C{z_1} + D}}{{\sqrt {{A^2} + {B^2} + {C^2}} }}} \right|\)

Determine the direction cosines of the normal to the plane and the distance from the origin. Plane z = 2

A
0, 1, 0; 2
B
0, 0, 1; 2
Correct
C
1, 0, 0; 3
D
1, 0, 1; 3

Determine the direction cosines of the normal to the plane and the distance from the origin. Plane x + y + z = 1

A
\(\frac{1}{{\sqrt 3 }},\;\frac{1}{{\sqrt 3 }},\frac{1}{{\sqrt 3 }};\;\frac{1}{{\sqrt 3 }}\)
Correct
B
\(\frac{1}{{\sqrt 3 }},\;\frac{2}{{\sqrt 3 }},\frac{1}{{\sqrt 3 }};\;\frac{1}{{\sqrt 3 }}\)
C
\(\frac{2}{{\sqrt 3 }},\;\frac{1}{{\sqrt 3 }},\frac{1}{{\sqrt 3 }};\;\frac{1}{{\sqrt 3 }}\)
D
\(\frac{1}{{\sqrt 3 }},\;\frac{1}{{\sqrt 3 }},\frac{2}{{\sqrt 3 }};\;\frac{1}{{\sqrt 3 }}\)

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector\(3\hat i + 5\hat j - 6\hat k\)

A
\(\vec r.\left( {\frac{{3\hat i + 5\hat j - 6\hat k}}{{\sqrt {70} }}} \right) = 5\)
B
\(\vec r.\left( {\frac{{3\hat i + 5\hat j - 6\hat k}}{{\sqrt {70} }}} \right) = 9\)
C
\(\vec r.\left( {\frac{{3\hat i + 5\hat j - 6\hat k}}{{\sqrt {70} }}} \right) = 7\)
Correct
D
\(\vec r.\left( {\frac{{3\hat i + 5\hat j - 6\hat k}}{{\sqrt {70} }}} \right) = 11\)

Find the Cartesian equation of the plane\(\vec r.\left( {\hat i + \hat j - \hat k} \right) = 2\).

A
x + y – z = 5
B
x + y – z = 2
Correct
C
x + y – z = 4
D
x + y – z = 3

Find the Cartesian equation of the plane\(\vec r.\left( {2\hat i + 3\hat j - 4\hat k} \right) = 1\)

A
2x + 3y – 4 z = 4
B
2x + 3y – 4 z = 3
C
2x + 3y – 4 z = 1
Correct
D
2x + 3y – 4 z = 2

the vector and cartesian equations of the planes that passes through the point (1, 0, – 2) and the normal to the plane is\(\hat i + \hat j - \hat k\)

A
\(\left[ {\vec r - \left( {\hat i - 5\hat k} \right)} \right].\left( {\hat i + \hat j - \hat k} \right) = 0;\)\(x - y - z =3\)
B
\(\left[ {\vec r - \left( {\hat i - 2\hat k} \right)} \right].\left( {\hat i + \hat j - \hat k} \right) = 0\)\(x + y - z = 3\)
Correct
C
\(\left[ {\vec r + \left( {\hat i - 2\hat k} \right)} \right].\left( {\hat i + \hat j - \hat k} \right) = 0;\)\(x + y - z = 5\)
D
\(\left[ {\vec r - \left( {\hat i + 2\hat k} \right)} \right].\left( {\hat i + \hat j - \hat k} \right) = 0;\)\(x + y - z = 7\)

Find the vector and cartesian equations of the planes that passes through the point (1 ,4 ,6) and the normal to the plane is\(\hat i - 2\hat j + \hat k\)

A
\(\left[ {\vec r - \left( {\hat i + 5\hat j + 6\hat k} \right)} \right].\left( {\hat i - 2\hat j + \hat k} \right) = 0;x - 2y + 2z + 1 = 0\)
B
\(\left[ {\vec r - \left( {\hat i + 4\hat j + 6\hat k} \right)} \right].\left( {\hat i - 2\hat j + \hat k} \right) = 0;x - 2y + z + 1 = 0\)
Correct
C
\(\left[ {\vec r - \left( {\hat i + 4\hat j + 7\hat k} \right)} \right].\left( {\hat i - 2\hat j + \hat k} \right) = 0;x - 2y + z + 5 = 0\)
D
\(\left[ {\vec r - \left( {2\hat i + 4\hat j + 6\hat k} \right)} \right].\left( {\hat i - 2\hat j + \hat k} \right) = 0;x - 3y + z + 1 = 0\)

Find the equations of the planes that passes through three points (1, 1, 0), (1, 2, 1), (– 2, 2, – 1)

A
2x + 5y – 3z = 5
B
2x + 3y – 3z = 5
Correct
C
3x + 3y – 3z = 5
D
2x + 3y – 7z = 5

Find the intercepts cut off by the plane 2x + y – z = 5.

A
\(\frac{5}{2},\;5,\; - 5\)
Correct
B
\( - \frac{5}{2},\;5,\; - 5\)
C
\( - \frac{5}{2},\;5,\;5\)
D
\(\frac{5}{2},\; - 5,\; - 5\)

Find the equation of the plane with intercept 3 on the y – axis and parallel to ZOX plane.

A
y = 2
B
y = 3
Correct
C
y = 4
D
y = 5

Find the equation of the plane through the intersection of the planes 3x – y + 2z – 4 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).

A
7x – 5y + 4z – 10 = 0
B
7x – 5y + 4z – 8 = 0
Correct
C
7x – 5y + 4z – 9 = 0
D
7x – 5y + 4z – 11 = 0

Find the vector equation of the plane passing through the intersection of the planes\(\vec r.\left( {2\hat i + 2\hat j - 3\hat k} \right) = 7, \) \(\vec r.\left( {2\hat i + 5\hat j + 3\hat k} \right) = 9\) and through the point (2, 1, 3).

A
\(\vec r.\left( {7\hat i + 27\hat j + 21\hat k} \right) = 59\)
Correct
B
\(\vec r.\left( {38\hat i + 68\hat j + 3\hat k} \right) = 1\)
C
\(\vec r.\left( {7\hat i + 27\hat j + 3\hat k} \right) = 15\)
D
\(\vec r.\left( {38\hat i + 8\hat j + 3\hat k} \right) = 155\)