\[ \begin{array}{c|c} X = x & P(X = x) \\ \hline 1 & 3K^2 \\ 3 & K \\ 5 & K^2 \\ 2 & 2K \end{array} \]
\[ f(x) = \begin{cases} \frac{(4^x - 1)^4 \cot(x \log 4)}{\sin(x \log 4) \log(1 + x^2 \log 4)}, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases} \]
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 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: