Using properties of determinnats, prove that:

$\left|\begin{array}{ccc}\mathrm{x}& \mathrm{y}& \mathrm{z}\\ {\mathrm{x}}^2& {\mathrm{y}}^2& {\mathrm{z}}^2\\ \mathrm{y} + \mathrm{z}& \mathrm{z} + \mathrm{x}& \mathrm{x} + \mathrm{y}\end{array}\right| = (\mathrm{x} - \mathrm{y})(\mathrm{y} - \mathrm{z})(\mathrm{z} - \mathrm{x})(\mathrm{x} +\hspace{0.17em}\mathrm{y} +\hspace{0.17em}\mathrm{z})$

Solve for x: ${\sin }^{-1}\left(\mathrm{x}\right) + {\sin }^{-1}\left(1 - \mathrm{x}\right) = {\cos }^{-1}\left(\mathrm{x}\right)$

Construct a circuit diagram for the following Boolean Function:

(BC + A)(A'B' + C') + A'B'C'

Using laws of Boolean Algebra, simplify the function and draw the simplified circuit.

Verify Lagrange's Mean value theorem for the function f(x) = $\sqrt{{\mathrm{x}}^2 - \mathrm{x}}$ in the interval [1, 4].

From the following information, find the equation of the Hyperbola and the equation of its Transverse Axis:

Focus: (- 2, 1), Directrix: 2x - 3y + 1 = 0, e = $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{3}$}\right.$

If y = ${\left({\cot }^{-1}\left(\mathrm{x}\right)\right)}^2$, show that $\left(1 + {\mathrm{x}}^2\right)\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + 2\mathrm{x}(1 + {\mathrm{x}}^2)\frac{d\mathrm{y}}{d\mathrm{x}} = 2$

Find the maximum value of the cylinder which can be inscribed in a sphere of radius $3\sqrt{3}$ cm. (Leave the answer in terms of $\mathrm{\pi }$)

Evaluate: $\int \frac{{\cos }^{-1}\left(\mathrm{x}\right)}{{\mathrm{x}}^2}d\mathrm{x}$

Find the area bounded by the curve y = 2x - x², and the line y = x

20.

Find the Karl Pearson's co-efficient of correlation between x and y for the following data:

x	16	18	21	20	22	26	27	15
y	22	25	24	26	25	30	33	14