Verify Lagrange's Mean Value Theorem for the following function:

f(x) = $2\sin \left(\mathrm{x}\right) + \sin \left(2\mathrm{x}\right) \mathrm{on} \left[0, \mathrm{\pi }\right]$

Find the equation of hyperbola whose foci are (0, $\pm \sqrt{10}$) and passing through the point (2, 3).

Show that the rectangle of maximum perimeter which can be inscribed in a circle o radius 10 cm is a square of side 10$\sqrt{2}$ cm.

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

Solve the differential equation:

${\mathrm{e}}^{\frac{\mathrm{x}}{\mathrm{y}}}\left(1 - \frac{\mathrm{x}}{\mathrm{y}}\right) + \left(1 +\hspace{0.17em}{\mathrm{e}}^{\frac{\mathrm{x}}{\mathrm{y}}}\right)\frac{d\mathrm{x}}{d\mathrm{y}} = 0$, when x = 0, y = 1

36.

Using vectors, prove that angle in a semicircle is a right angle.

Box I contains two white and three black balls. Box II contains four white and one black balls and box III contains three white and four black balls. A dice having three red, two yellow, and one green face, is thrown to select the box. If red face turns up, we pick up box I, if yellow face turns up, we pick up box II, otherwise we pick up box III. Then, we draw a ball from the selected box. If the ball drawn is white, what is the probability that the dice had turned up with a red face?

Five dice are thrown simulteneously. If the occurence of an odd number in a single dice is considered a success, find the probability of maximum three successses.