Using prpoperties of determinants, prove that $\left|\begin{array}{ccc}\mathrm{b}+ \mathrm{c}& \mathrm{a}& \mathrm{a}\\ \mathrm{b}& \mathrm{a} + \mathrm{c}& \mathrm{b}\\ \mathrm{c}& \mathrm{c}& \mathrm{a} +\hspace{0.17em}\mathrm{b}\end{array}\right| = 4\mathrm{abc}$

Solve the following system of linear equation using matrix method.

3x + y + z = 1, 2x + 2z = 0, 5x + y + 2z = 2.

If ${\sin }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{x}\right) = \frac{\mathrm{\pi }}{2}$, prove that 2x² + 1 = $\sqrt{5}$

Write the Boolean function corrosponding to the switching circuit given below.

Verify the conditions of Rolle's Theorem for the following function.

$\mathrm{f}\left(\mathrm{x}\right) = \log ({\mathrm{x}}^2 +\hspace{0.17em}2) - \log \left(3\right) \mathrm{on} [- 1, 1]$

Find a point in the interval, where the tangent to the curve is parallel to x - axis.

16.

Find the equation of the standard ellipse, taking its axes as the coordinate axes, whose minor axis is equal to the distance between the foci and whose legth of latus rectum is 10. Also, find its eccentricity. 

If $\log \left(\mathrm{y}\right) = {\tan }^{-1}\left(\mathrm{x}\right)$, prove that:

$(1 + {\mathrm{x}}^2)\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} +\hspace{0.17em}(2\mathrm{x} - 1)\frac{d\mathrm{y}}{d\mathrm{x}}$ = 0

A rectangle is inscribed in semicircle of ardius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get maximum area. Also, find the maximum area.

Evaluate: $\int \frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{\sqrt{9 + 16\sin \left(2\mathrm{x}\right)}}d\mathrm{x}$

Find the area of the region bound by the curves y = 6x - x² and y - x² - 2x.