If \[ y = 500 e^{7x} + 600 e^{-7x}, \quad \text{then show that} \quad \frac{d^2 y}{dx^2} = 49y. \]
Find the area of the region enclosed by the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \]
Prove that \[ \int_0^{\pi} \frac{x \tan x}{\sec x + \tan x} \, dx = \frac{\pi}{2} (\pi - 2). \]
Prove that the height of the cylinder of maximum volume inscribed in a sphere of radius \( R \) is \( \frac{2R}{\sqrt{3}} \).
Solve the differential equation \[ (x + y) \, dy + (x - y) \, dx = 0, \quad \text{if} \quad y = 1 \text{ when } x = 1. \]
Show that the vectors \( 2\hat{i} - \hat{j} + \hat{k}, \hat{i} - 3\hat{j} - 5\hat{k}, 3\hat{i} - 4\hat{j} - 4\hat{k} \) form the vertices of a right-angled triangle.
Find the angle between the lines \[ \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z-5}{-5} \quad \text{and} \quad \frac{x+3}{-3} = \frac{y-1}{2} = \frac{z-5}{5}. \]
Solve the differential equation \[ \frac{dy}{dx} + y \cot x = 2x + x^2 \cot x \quad \text{where} \quad x \neq 0. \]
Find the shortest distance between the lines \[ \mathbf{r}_1 = (i + 2j + k) + \lambda(i - j + k) \quad \text{and} \quad \mathbf{r}_2 = (2i - j - k) + \mu(2i + j + 2k). \]
If \( x\sqrt{1 + y} + y\sqrt{1 + x} = 0 \) for \( -1<x<1 \), then prove that \[ \frac{dy}{dx} = -\frac{1}{(1 + x^2)^2}. \]
Prove that the function \( f(x) = |x| \) is continuous at \( x = 0 \) but not differentiable.
A die is thrown twice. Let us represent the event 'obtaining an odd number on the first throw' by A and the event 'obtaining an odd number on the second throw' by B. Test the independency of the events A and B.
Solve the differential equation \[ (x - y) \frac{dy}{dx} = x + 2y. \]
Find the intervals in which the function \( f(x) = x^2 - 4x + 6 \) is
Find the value of \[ \int e^x \left( \tan^{-1} x + \frac{1}{1 + x^2} \right) dx. \]
Prove that the number of equivalence relations in the set \( \{1, 2, 3\} \) including \( \{(1, 2)\} \) and \( \{(2, 1)\} \) is 2.