Using properties of determinants, prove that:

$\left|\begin{array}{ccc}{\mathrm{a}}^2 +1& \mathrm{ab}& \mathrm{ac}\\ \mathrm{ba}& {\mathrm{b}}^{2 }+ 1& \mathrm{bc}\\ \mathrm{ca}& \mathrm{cb}& {\mathrm{c}}^2 + 1\end{array}\right|$ = a² + b^{2 +} c²+ 1

Using matrix method, solve the following system of equation:

x - 2y = 10, 2x + y + 3z = 8 and - 2y + z = 7.

If ${\cos }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\mathrm{y}\right) + {\cos }^{-1}\left(\mathrm{z}\right) = \mathrm{\pi }$, prove that: x² + y² + z² + 2xyz = 1

P, Q and R reprsent switches in on position and P', Q' and R' represent switches in off position. Construct a switching circuit representing the polynomial PR + Q(Q' + R)(P + QR). Using Boolean algebra, simplify the polynomial expression and construct the simplified circuit.

Verify Rolle's theorem for the function f(x) = ${\mathrm{e}}^{\mathrm{x}}\left(\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)\right) \mathrm{on} \left[\frac{\mathrm{\pi }}{4}, \frac{5\mathrm{\pi }}{4}\right]$

16.

Find the equation of the parbola with latus-rectum joining points (4, 6) and (4, - 2).

If y = $\frac{{\mathrm{xsin}}^{-1}\left(\mathrm{x}\right)}{\sqrt{1 - {\mathrm{x}}^2}}$, prove that: $\left(1 - {\mathrm{x}}^2\right)\frac{d\mathrm{y}}{d\mathrm{x}} = \mathrm{x} + \frac{\mathrm{y}}{\mathrm{x}}$

A wire of length 50m is cut into two pieces. One piece of the wire is bent in the shape of a square and the other in the shape of a circle. What should be the length of each piece so that the combined area of the two is minimum.

Evaluate: $\int \frac{\mathrm{x} + \sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}d\mathrm{x}$

Sketch the graphs of the curve y² = x and y² = 4 - 3x and find the area enclosed between them.