Science (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

\[\int\limits_0^1 \left\{ x \right\} dx,\] where {x} denotes the fractional part of x.  

If \[\left[ \cdot \right] and \left\{ \cdot \right\}\] denote respectively the greatest integer and fractional part functions respectively, evaluate the following integrals:

\[\int\limits_0^\pi \frac{1}{1 + \sin x} dx\] equals

The value of \[\int\limits_0^\pi \frac{x \tan x}{\sec x + \cos x} dx\] is __________ .

The value of \[\int\limits_0^{2\pi} \sqrt{1 + \sin\frac{x}{2}}dx\] is 

The value of the integral \[\int\limits_0^{\pi/2} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} dx\]  is 

\[\int\limits_0^\infty \frac{1}{1 + e^x} dx\]  equals

\[\int_0^\frac{\pi^2}{4} \frac{\sin\sqrt{x}}{\sqrt{x}} dx\] equals

What is the angle between vectors \[\vec{a} \text{ and } \vec{b}\] with magnitudes 2 and \[\sqrt{3}\] respectively? Given \[\vec{a} . \vec{b} = \sqrt{3} .\]

\[\int\limits_0^{\pi/2} \frac{1}{2 + \cos x} dx\] equals

\[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\vec{a} . \vec{b} = 6, \left| \vec{a} \right| = 3 \text{ and } \left| \vec{b} \right| = 4 .\] Write the projection of \[\vec{a} \text{ on } \vec{b}\] 

Find the cosine of the angle between the vectors \[4 \hat{i} - 3 \hat{j} + 3 \hat{k} \text{ and } 2 \hat{i} - \hat{j} - \hat{k} .\]