Using principal value, evaluate the following: ${\sin }^{-1} \left(\sin \frac{3\mathrm{\pi }}{5}\right)$ 

If the binary operation * on the set of integers Z, is defined by a * b = a + 3b² , then find the value of 2 * 4.

If A is an invertible matrix of order 3 and |A| = 5, then find |adj. A|.

If matrix A = (1, 2, 3), write AA’, where A’ is the transpose of matrix A.

Write the value of the determinant $\left|\begin{array}{ccc} 2 & 3 & 4 \\ 5 & 6 & 8\\ 6\mathrm{x} & 9\mathrm{x} & 12\mathrm{x}\end{array}\right|$

Evaluate: $\int {\sec }^2 \left( 7 - \mathrm{x} \right) \mathrm{dx}$

If ${\int }_0^1\left( 3{\mathrm{x}}^2 + 2\mathrm{x} + \mathrm{k} \right) \mathrm{dx} = 0$ , find the value of k.

Find the projection of $\overrightarrow{\mathrm{a}}$ on $\overrightarrow{\mathrm{b}}$ if $\overrightarrow{\mathrm{a}}. \overrightarrow{\mathrm{b}} =8 \mathrm{and} \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + 6\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$

Write a unit vector in the direction of $\overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}.$

Write the value of p for which $\overrightarrow{\mathrm{a}} = 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 9\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}}+\mathrm{p}\hat{\mathrm{j}} +3\hat{\mathrm{k}}$ are parallel vectors.