The binary operation *: $\mathrm{R} \times \mathrm{R} \to \mathrm{R}$ is defined as $\mathrm{a} * \mathrm{b} = 2\mathrm{a} + \mathrm{b}$ Find (2*3)*4.

If $\mathrm{A} = \left(\begin{array}{cc}5& \mathrm{a}\\ \mathrm{b}& 0\end{array}\right)$and A is symmetric matrix, show that a = b.

Solve: $3{\tan }^{-1}\left(\mathrm{x}\right) + {\cot }^{-1}\left(\mathrm{x}\right) = \mathrm{\pi }$

Without expanding at any stage, find the value of:

$\left|\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{a} + 2\mathrm{x}& \mathrm{b} +\hspace{0.17em}2\mathrm{y}& \mathrm{c} + 2\mathrm{z}\\ \mathrm{x}& \mathrm{y}& \mathrm{z}\end{array}\right|$

Find the value of constant ‘k’ so that the function f(x) defined as:

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{{\mathrm{x}}^2 - 2\mathrm{x} - 3}{\mathrm{x} +\hspace{0.17em}1}, \mathrm{x} \ne - 1 \\ \mathrm{k}, \mathrm{x} = - 1\right\} \end{gathered}$$

is continuous at x = - 1.

6.

Find the approximate change in the volume ‘V’ of a cube of side x metres caused by decreasing the side by 1%.

Evaluate: $\int \frac{{\mathrm{x}}^3 +\hspace{0.17em}5{\mathrm{x}}^2 + 4\mathrm{x} +\hspace{0.17em}1}{{\mathrm{x}}^2}\mathrm{dx}$.

Find the differential equation of the family of concentric circle x² + y² = a².

If A and B are events such that P(A) = $\frac{1}{2}$, P(B) = $\frac{1}{3}$ and $\mathrm{P}(\mathrm{A}\cap \mathrm{B}) = \frac{1}{4}$ then find:

(a) P(A/B)

(b) P(B/A)

In a race, the probabilities of A and B winning the race are $\frac{1}{3}$ and $\frac{1}{6}$ respectively. Find the probability of neither of them winning the race.