\[ f(x) = \begin{cases} \frac{(4^x - 1)^4 \cot(x \log 4)}{\sin(x \log 4) \log(1 + x^2 \log 4)}, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases} \]
Find \( e^k \) if \( f(x) \) is continuous at \( x = 0 \).
\[ y = \sqrt{\sin(\log(2x)) + \sqrt{\sin(\log(2x)) + \sqrt{\sin(\log(2x))} + \dots \infty}} \]
Find \( \frac{dy}{dx} \) for the given function:
\[ y = \tan^{-1} \left( \frac{\sin^3(2x) - 3x^2 \sin(2x)}{3x \sin(2x) - x^3} \right). \]
The function \( y = 2x^3 - 8x^2 + 10x - 4 \) is defined on \([1,2]\). If the tangent drawn at a point \( (a,b) \) on the graph of this function is parallel to the X-axis and \( a \in (1,2) \), then \( a = \) ?
If \( m \) and \( M \) are respectively the absolute minimum and absolute maximum values of a function \( f(x) = 2x^3 + 9x^2 + 12x + 1 \) defined on \([-3,0]\), then \( m + M \) is:
Evaluate the integral: \[ \int \frac{dx}{4 + 3\cot x} \]
Evaluate the integral: \[ \int \frac{dx}{(x+1)\sqrt{x^2 + 4}} \]
Evaluate the integral: \[ I = \int_{-\frac{\pi}{15}}^{\frac{\pi}{15}} \frac{\cos 5x}{1 + e^{5x}} \, dx \]
The area of the region (in square units) enclosed by the curves \( y = 8x^3 - 1 \), \( y = 0 \), \( x = -1 \), and \( x = 1 \) is:
The range of gravitational forces is:
In a simple pendulum experiment for the determination of acceleration due to gravity, the error in the measurement of the length of the pendulum is 1% and the error in the measurement of the time period is 2%. The error in the estimation of acceleration due to gravity is:
A constant force of \[ \mathbf{F} = (8\hat{i} - 2\hat{j} + 6\hat{k}) \text{ N} \] acts on a body of mass 2 kg, displacing it from \[ \mathbf{r_1} = (2\hat{i} + 3\hat{j} - 4\hat{k}) \text{ m to } \mathbf{r_2} = (4\hat{i} - 3\hat{j} + 6\hat{k}) \text{ m}. \] The work done in the process is:
A ball 'A' of mass 1.2 kg moving with a velocity of 8.4 m/s makes a one-dimensional elastic collision with a ball 'B' of mass 3.6 kg at rest. The percentage of kinetic energy transferred by ball 'A' to ball 'B' is:
A body of mass \( m \) and radius \( r \) rolling horizontally with velocity \( V \), rolls up an inclined plane to a vertical height \( \frac{V^2}{g} \). The body is:
A simple pendulum is made of a metal wire of length \( L \), area of cross-section \( A \), material of Young's modulus \( Y \), and a bob of mass \( m \). This pendulum is hung in a bus moving with a uniform speed \( V \) on a horizontal circular road of radius \( R \). The elongation in the wire is:
If the excess pressures inside two soap bubbles are in the ratio \( 2:3 \), then the ratio of the volumes of the soap bubbles is:
The velocities of air above and below the surfaces of a flying aeroplane wing are 50 m/s and 40 m/s respectively. If the area of the wing is 10 m² and the mass of the aeroplane is 500 kg, then as time passes by (density of air = 1.3 kg/m³), the aeroplane will:
A liquid cools from a temperature of 368 K to 358 K in 22 minutes. In the same room, the same liquid takes 12.5 minutes to cool from 358 K to 353 K. The room temperature is:
At a pressure \( P \) and temperature \( 127^\circ C \), a vessel contains 21 g of a gas. A small hole is made into the vessel so that the gas leaks out. At a pressure of \( \frac{2P}{3} \) and a temperature of \( t^\circ C \), the mass of the gas leaked out is 5 g. Then \( t \) is:
Two closed pipes when sounded simultaneously in their fundamental modes produce 6 beats per second. If the length of the shorter pipe is 150 cm, then the length of the longer pipe is: \[ \text{(Speed of sound in air = 336 ms}^{-1}) \]