If ${\left( \cos \mathrm{x} \right)}^{\mathrm{y}} = {\left( \cos \mathrm{y} \right)}^{\mathrm{x}}, \mathrm{find} \frac{\mathrm{dy}}{\mathrm{dx}}.$

If sin y = x sin (a + y), prove that $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{\displaystyle {\sin }^2}{\displaystyle }{\displaystyle \mathrm{a}}{\displaystyle }{\displaystyle +}{\displaystyle }{\displaystyle \mathrm{y}}}{\sin {\displaystyle }{\displaystyle \mathrm{a}}}$.

Let A = R – {3} and B = R – {1}. Consider the function f : A $\to$B  defined by $\mathrm{f} ( \mathrm{x} ) = \left( \frac{\mathrm{x} - 2}{\mathrm{x} - 3} \right)$. Show that f is one-one and onto and hence find f ^{- 1}.

Prove that  ${\tan }^{- 1} \left( \frac{ \cos \mathrm{x}}{ 1 + \sin \mathrm{x}} \right) = \frac{\mathrm{\pi }}{4} - \frac{\mathrm{\pi }}{2}, \mathrm{x} \in \left( - \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2} \right)$

Prove that   ${\sin }^{- 1} \left( \frac{8}{17} \right) + {\sin }^{- 1} \left( \frac{3}{5} \right) = {\cos }^{- 1} \left( \frac{36}{85} \right).$

Find the point on the curve  y = x³ – 11x + 5  at which the equation of tangent is  y = x – 11.

Using differentials, find the approximate value of  $\sqrt{ 49.5}.$

18.

Using properties of determinants prove the following:

$\left| \begin{array}{ccc}1 & 1& 1\\ \mathrm{a} & \mathrm{b}& \mathrm{c}\\ {\mathrm{a}}^3 & {\mathrm{b}}^3& {\mathrm{c}}^3\end{array} \right| = \left( \mathrm{a} - \mathrm{b} \right) \left( \mathrm{b} - \mathrm{c} \right) \left( \mathrm{c} - \mathrm{a} \right) \left( \mathrm{a} + \mathrm{b} +\mathrm{c} \right)$

If  y = 3 cos ( log x ) + 4 sin ( log x ), show that

${\mathrm{x}}^2 \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = 0$

Using matrices solve the following system of linear equations:

x - y + 2 z = 7

3 x + 4 y - 5 z = - 5

2 x - y + 3 z = 12