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.
If \( x = a(0 - \sin \theta) \), \( y = a(1 + \cos \theta) \), find \[ \frac{dy}{dx}. \]
Integrate \[ \int \frac{\sin(\tan^{-1} x)}{1 + x^2} \, dx. \]
Solve: \[ \frac{dy}{dx} = \frac{1 + y^2}{1 + x^2}. \]
If \( e^y(x + 1) = 1 \), show that \[ \frac{d^2y}{dx^2} = \left( \frac{dy}{dx} \right)^2. \]
A relation \( R = \{(a, b) : a = b - 2, b \geq 6 \} \) is defined on the set \( \mathbb{N} \). Then the correct answer will be:
Derive an expression for energy stored in a charged capacitor. A spherical metal ball of radius 15 cm carries a charge of 2μC. Calculate the electric field at a distance of 20 cm from the center of the sphere.
Draw a neat labelled diagram of Ferry's perfectly black body. Compare the rms speed of hydrogen molecules at 227°C with rms speed of oxygen molecules at 127°C. Given that molecular masses of hydrogen and oxygen are 2 and 32, respectively.
Distinguish between an ammeter and a voltmeter. (Two points each).The displacement of a particle performing simple harmonic motion is \( \frac{1}{3} \) of its amplitude. What fraction of total energy is its kinetic energy?