1.

If f(x) = x + 7 and g(x) = x – 7, x $\in$ R, find (fog) (7)

Evaluate:   $\sin \left[\frac{\mathrm{\pi }}{3} - {\sin }^{-1} \left(\frac{1}{2}\right)\right]$

Find the value of x and y if: $2\left[\begin{array}{cc}1& 3\\ 0& \mathrm{x}\end{array}\right] + \left[\begin{array}{cc}\mathrm{y}& 0\\ 1& 2\end{array}\right] = \left[\begin{array}{cc}5& 6\\ 1& 8\end{array}\right]$

Evaluate: $\left[\begin{array}{cc}\mathrm{a} + \mathrm{ib}& \mathrm{c} + \mathrm{id}\\ -\mathrm{c} + \mathrm{id}& \mathrm{a} - \mathrm{ib}\end{array}\right]$

Find the co-factor of a12 in the following:

$\left|\begin{array}{ccc}2& -3& 5\\ 6& 0& 4\\ 1& 5& -7\end{array}\right|$

Evaluate: $\int \frac{{\mathrm{x}}^2}{1 + {\mathrm{x}}^3}\mathrm{dx}$

Evaluate: ${\int }_0^4\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2} \mathrm{dx}$

Find a unit vector in the direction of  $\overrightarrow{\mathrm{a} }= 3\widehat{\mathrm{i} } - 2\hat{\mathrm{j}} + 6\mathrm{k}$

Find the angle between the vectors $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = \hat{1} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$

For what value of $\mathrm{\lambda }$ are the vectors  $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + \mathrm{\lambda } \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} - 2\hat{\mathrm{j}} -3\hat{\mathrm{k}}$ are perpendicular to each other?