If a*b denotes the larger of 'a' and 'b' and if a o b = (a*b) + 3, then write the value of (5) o (10), where and o are binary operations.

If the matrix $\mathrm{A} = \left[\begin{array}{ccc}0& \mathrm{a}& -3\\ 2& 0& -1\\ \mathrm{b}& 1& 0\end{array}\right]$ is skew symmertic, find the values of 'a' and 'b'.

Find the value of ${\tan }^{-1} \sqrt{3} - co{t}^{-1} (-\sqrt{3})$

The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x³ - 0.02 x² + 30 x +5000. Find the marginal cost when 3 units are produced, whereby marginal cost we mean the instantaneous rate of total cost at any level of output.

5.

Differentiate ${\tan }^{-1} \left(\frac{1 + \cos \mathrm{x}}{\sin \mathrm{x}}\right)$ with respect x.

Given, $\mathrm{A} = \left[\begin{array}{cc}2& -3\\ -4& 7\end{array}\right]$ compute A^-1 and show that 2A^- = 9I-A

Prove that:$3 {\sin }^{-1} \mathrm{x} = {\sin }^{-1} (3\mathrm{x} - 4{\mathrm{x}}^3), \mathrm{X} \in \left[-\frac{1}{2}, \frac{1}{2}\right]$

Using properties of determinants, prove that

$\left|\begin{array}{ccc}1& 1& 1+3\mathrm{x}\\ 1+ 3\mathrm{y}& 1& 1\\ 1& 1+3\mathrm{z}& 1\end{array}\right| = 9 (3\mathrm{xyz} + \mathrm{xy} + \mathrm{yz} + \mathrm{zx})$

Find the equations of the tangent and the normal, to curve 16x² + 9y² = 145 at the point (x₁,y₁) where x₁ = 2 and y₁ > 0

Find the intervals in which the function f(x)  =  is

a) Strictly increasing

b) Strictly decreasing