11.

(i) Is the binary operation *, defined on set N, given by  a * b = $\frac{\mathrm{a} + \mathrm{b}}{2}$  for all a,b $\in$N, commutative?

(ii) Is the above binary operation * associative?

Prove the following:

${\tan }^{-1}\frac{1}{3} + {\tan }^{-1}\frac{1}{5} + {\tan }^{-1}\frac{1}{7} + {\tan }^{-1}\frac{1}{8}$

Let $\mathrm{A}= \left|\begin{array}{ccc}3& 2& 5\\ 4& 1& 3\\ 0& 6& 7\end{array}\right|.$ Express A as sum of two matrices such that one is symmetric and the other is skew symmetric.

If A = $\left[\begin{array}{ccc}1 & 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$, verify that A² – 4A – 5I = 0.

For what value of k is the following function continuous at x = 2?

$\mathrm{f} ( \mathrm{x} ) =\begin{array}{c}2\mathrm{x} + 1 ; \mathrm{x}<2\\ \mathrm{k} ; \mathrm{x} = 2\\ 3\mathrm{x} - 1 ; \mathrm{x}>1 \end{array}$

Find the equation of tangent to the curve x = sin 3t, y = cos 2t, at t = $\frac{\mathrm{\pi }}{4}$

Using properties of determinants, prove the following:

$\left|\begin{array}{ccc}\mathrm{\alpha }& \mathrm{\beta }& \mathrm{\gamma }\\ {\mathrm{\alpha }}^2& {\mathrm{\beta }}^2& {\mathrm{\gamma }}^2\\ \mathrm{\beta } + \mathrm{\gamma } & \mathrm{\gamma } + \mathrm{\alpha } & \mathrm{\alpha } + \mathrm{\beta }\end{array}\right| = \left(\mathrm{\alpha } - \mathrm{\beta }\right) \left(\mathrm{\beta } - \mathrm{\gamma } \right) \left(\mathrm{\gamma } - \mathrm{\alpha }\right) \left(\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma }\right)$

Show that the rectangle of maximum area that can be inscribed in a circle is a square.

Show that the height of the cylinder of maximum volume that can be inscribed in a cone of height h is $\frac{1}{3}$h.

Differentiate the following with respect of x:

$\mathrm{y} = {\tan }^{-1} \left(\frac{\sqrt{1 + \mathrm{x}} - \sqrt{1 - \mathrm{x}}}{\sqrt{1 + \mathrm{x}} + \sqrt{1 - \mathrm{x}}} \right)$