Differentiate the following function w.r.t. x:

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

Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a - b| is even}, is an equivalence relation.

Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ if (x² + y²)² = xy.

If y =3 cos ( log x ) + 4 sin ( log x ), then show that ${\mathrm{x}}^2 \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = 0$

Find the equation of the tangent to the curve $\mathrm{y} = \sqrt{3\mathrm{x} - 2}$ which is parallel to the line 4x – 2y + 5 = 0

Find the intervals in which the function f given by $\mathrm{f} ( \mathrm{x} ) = {\mathrm{x}}^3 + \frac{1}{{\mathrm{x}}^3}, \mathrm{x} \ne 0$ is

(i) increasing

(ii) decreasing.

Prove that: ${\sin }^{-1} \left(\frac{4}{5}\right) + {\sin }^{-1} \left(\frac{5}{13}\right) + {\sin }^{-1} \left(\frac{16}{65}\right) = \frac{\mathrm{\pi }}{2}$

Solve for x: ${\tan }^{-1} 3\mathrm{x} + {\tan }^{-1} 2\mathrm{x} = \frac{\mathrm{\pi }}{4}$

19.

Using properties of determinants prove the following:

$\left|\begin{array}{ccc}\mathrm{a} & \mathrm{b}& \mathrm{c}\\ \mathrm{a} - \mathrm{b} & \mathrm{b} - \mathrm{c} & \mathrm{c} - \mathrm{a}\\ \mathrm{b} + \mathrm{c} & \mathrm{c} + \mathrm{a} & \mathrm{a} + \mathrm{b}\end{array}\right| = {\mathrm{a}}^3 + {\mathrm{b}}^3 + {\mathrm{c}}^3 - 3\mathrm{abc}$

Using matrices, solve the following system of equations:
2x – 3y + 5 = 11
3x + 2y – 4z = -5
x + y – 2z= -3