Science (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

Write a vector satisfying \[\vec{a} . \hat{i} = \vec{a} . \left( \hat{i} + \hat{j} \right) = \vec{a} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 1 .\]

If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, find the angle between \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} .\]

If \[\vec{a} \text{ and } \vec{b}\] are mutually perpendicular unit vectors, write the value of \[\left| \vec{a} + \vec{b} \right| .\] 

The value of \[\int\limits_0^\pi \frac{1}{5 + 3 \cos x} dx\] is

If \[\vec{a} , \vec{b} \text{ and } \vec{c}\] are mutually perpendicular unit vectors, write the value of \[\left| \vec{a} + \vec{b} + \vec{c} \right| .\] 

Find the angle between the vectors \[\vec{a} = \hat{i} - \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} + \hat{j} - \hat{k} .\]

\[\int\limits_0^{2a} f\left( x \right) dx\]  is equal to

If f (a + b − x) = f (x), then \[\int\limits_a^b\] x f (x) dx is equal to

The value of \[\int\limits_0^1 \tan^{- 1} \left( \frac{2x - 1}{1 + x - x^2} \right) dx,\] is

The value of \[\int\limits_0^{\pi/2} \log\left( \frac{4 + 3 \sin x}{4 + 3 \cos x} \right) dx\] is 

The value of \[\int\limits_{- \pi/2}^{\pi/2} \left( x^3 + x \cos x + \tan^5 x + 1 \right) dx, \] is 

For what value of λ are the vectors \[\vec{a} = 2 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}\] perpendicular to each other?

Find the projection of \[\vec{a} \text{ on } \vec{b} \text{ if } \vec{a} \cdot \vec{b} = 8 \text{ and } \vec{b} = 2 \hat{i} + 6 \hat{j} + 3 \hat{k} .\] 

Write the value of p for which \[\vec{a} = 3 \hat{i} + 2 \hat{j} + 9 \hat{k} \text{ and } \vec{b} = \hat{i} + p \hat{j} + 3 \hat{k}\]    are parallel vectors . 

Find the value of λ if the vectors \[2 \hat{i} + \lambda \hat{j} + 3 \hat{k} \text{ and } 3 \hat{i} + 2 \hat{j} - 4 \hat{k}\] are perpendicular to each other.