State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive.

Write the value of  sin $\left[ \frac{\mathrm{\pi }}{3} - {\sin }^{-1} \left( - \frac{1}{2}\right) \right]$

For a 2 x 2 matrix, A = [ a_ij ] whose elements are given by  a_ij = $\frac{\mathrm{i}}{\mathrm{j}}$,  write the value of _a12 .

For what value of x, the matrix $\left[ \begin{array}{cc}5 - \mathrm{x} & \mathrm{x} + 1\\ 2 & 4\end{array} \right]$ is singular?

Write  A^-1  for A = $\left[ \begin{array}{cc}2 & 5\\ 1 & 3\end{array} \right]$

6.

Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. Write the operation table of the operation *.

Write the value of $\int \sec \mathrm{x} \left( \sec \mathrm{x} + \tan \mathrm{x} \right) \mathrm{dx}$

Write the value of $\int \frac{\mathrm{dx}}{{\mathrm{x}}^2 + 16}$

For what value of ‘a’ the vectors   $2 \hat{\mathrm{i}} - 3 \hat{\mathrm{j}} + 4 \mathrm{k} \mathrm{and} \mathrm{a} \hat{\mathrm{i}} + 6 \hat{\mathrm{j}} - 8 \mathrm{k}$  are collinear?

Write the direction cosines of the vector  $- 2 \hat{\mathrm{i}} + \hat{\mathrm{j}} - 5 \mathrm{k}.$