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

If \[\int\limits_0^a \frac{1}{1 + 4 x^2} dx = \frac{\pi}{8},\] then a equals

If \[\hat{a} , \hat{b}\] are unit vectors such that \[\hat{a} + \hat{b}\]  is a unit vector, write the value of \[\left| \hat{a} - \hat{b} \right| .\] 

If \[\int\limits_0^1 f\left( x \right) dx = 1, \int\limits_0^1 xf\left( x \right) dx = a, \int\limits_0^1 x^2 f\left( x \right) dx = a^2 , then \int\limits_0^1 \left( a - x \right)^2 f\left( x \right) dx\] equals

The value of \[\int\limits_{- \pi}^\pi \sin^3 x \cos^2 x\ dx\] is 

If \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and } \vec{a} . \vec{b} = 2, \text{ find } \left| \vec{a} - \vec{b} \right| .\]

If \[\vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = - \hat{j} + \hat{k} ,\]  find the projection of \[\vec{a} \text{ on } \vec{b}\] 

For any two non-zero vectors, write the value of \[\frac{\left| \vec{a} + \vec{b} \right|^2 + \left| \vec{a} - \vec{b} \right|^2}{\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2} .\] 

The derivative of \[f\left( x \right) = \int\limits_{x^2}^{x^3} \frac{1}{\log_e t} dt, \left( x > 0 \right),\] is

Write the projections of \[\vec{r} = 3 \hat{i} - 4 \hat{j} + 12 \hat{k}\] on the coordinate axes. 

If \[I_{10} = \int\limits_0^{\pi/2} x^{10} \sin x\ dx,\]  then the value of I₁₀ + 90I₈ is

Write the component of \[\vec{b}\] along \[\vec{a}\] 

Write the value of \[\left( \vec{a} . \hat{i} \right) \hat{i} + \left( \vec{a} . \hat{j} \right) \hat{j} + \left( \vec{a} . \hat{k} \right) \hat{k} ,\]  where \[\vec{a}\] is any vector. 

Find the value of θ ∈(0, π/2) for which vectors \[\vec{a} = \left( \sin \theta \right) \hat{i} + \left( \cos \theta \right) \hat{j} \text{ and } \vec{b} = \hat{i} - \sqrt{3} \hat{j} + 2 \hat{k}\] are perpendicular.

Write the projection of \[\hat{i} + \hat{j} + \hat{k}\] along the vector \[\hat{j}\] 

The value of the integral \[\int\limits_{- 2}^2 \left| 1 - x^2 \right| dx\] is ________ .