Arts (English Medium) Class 12 - CBSE Question Bank Solutions

Find the distance between the parallel planes 2x − y + 3z − 4 = 0 and 6x − 3y + 9z + 13 = 0.

Find the distance between the planes \[\vec{r} \cdot \left( \hat{i}  + 2 \hat{j}  + 3 \hat{k}  \right) + 7 = 0 \text{ and } \vec{r} \cdot \left( 2 \hat{i}  + 4 \hat{j}  + 6 \hat{k}  \right) + 7 = 0 .\]

Evaluate : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .

Find a unit vector perpendicular to both the vectors \[\vec{a} + \vec{b} \text { and } \vec{a} - \vec{b}\] ,where \[\vec{a} = \hat{i}+ \hat{j} + \hat{k} , \vec{b} =\hat {i} + 2 \hat{j} + 3 \hat{k}\].

The distance between the planes 2x + 2y − z + 2 = 0 and 4x + 4y − 2z + 5 = 0 is 

The image of the point (1, 3, 4) in the plane 2x − y + z + 3 = 0 is

The distance of the line \[\vec{r} = 2 \hat{i} - 2 \hat{j} + 3 \hat{k} + \lambda\left( \hat{i} - \hat{j}+ 4 \hat{k}  \right)\]  from the plane \[\vec{r} \cdot \left( \hat{i} + 5 \hat{j} + \hat{k} \right) = 5\] is

If a plane passes through the point (1, 1, 1) and is perpendicular to the line \[\frac{x - 1}{3} = \frac{y - 1}{0} = \frac{z - 1}{4}\] then its perpendicular distance from the origin is ______.

\[\text{ If } \vec{a} = \hat { i }  + 3 \hat { j }  - 2 \hat { k } \text{ and }  \vec{b} = - \hat { i }  + 3 \hat { k }   , \text{ find }  \left| \vec{a} \times \vec{b} \right| .\]

If \[\vec{a} = 3 \hat { i } + 4 \hat { j } \text{ and }  \vec{b} = \hat { i  } + \hat{ j }  + \hat{ k } ,\]  find the value of \[\left| \vec{a} \times \vec{b} \right| .\]

If \[\vec{a} = 2 \hat{ i } + \hat{ k }  , \vec{b} = \hat { i }  + \hat{ j } + \hat{ k }  ,\]  find the magnitude of  \[\vec{a} \times \vec{b} .\]

 Find a unit vector perpendicular to both the vectors  \[4 \hat{ i } - \hat{ j }  + 3 \hat{ k } \text{ and }  - 2 \hat{ i  } + \hat{ j }  - 2 \hat{ k }  .\]

Find a unit vector perpendicular to the plane containing the vectors  \[\vec{a} = 2 \hat{ i } + \hat{ j }  + \hat{ k } \text{ and }  \vec{b} = \hat{ i } + 2 \hat{ j }  + \hat{ k } .\]

Find the magnitude of \[\vec{a} = \left( 3 \hat{ k }  + 4 \hat{ j } \right) \times \left( \hat{ i }  + \hat{ j }  - \hat{ k }  \right) .\]

Find a vector of magnitude 49, which is perpendicular to both the vectors  \[2 \hat{ i }   + 3 \hat{ j }  + 6 \hat{ k }  \text{ and } 3 \hat{ i }  - 6 \hat{ j }  + 2 \hat{ k }  .\]