If
\[ \cosh^{-1}\left(\frac{5}{3}\right) + \sinh^{-1}\left(\frac{3}{4}\right) = k, \]
then \( e^k \) is:
\[ \sin^{-1} x - \cos^{-1} 2x = \sin^{-1} \left(\frac{\sqrt{3}}{2}\right) - \cos^{-1} \left(\frac{\sqrt{3}}{2}\right) \]
Then, \[ \tan^{-1} x + \tan^{-1} \left(\frac{x}{x+1}\right) = ? \]
\[ \text{sech}^{-1}\left(\frac{3}{5}\right) - \text{tanh}^{-1}\left(\frac{3}{5}\right) = ? \]
In a triangle ABC, if \( a = 5 \), \( b = 3 \), and \( c = 7 \), then the ratio:
\[ \sqrt{\frac{\sin(A - B)}{\sin(A + B)}} \]
The system of simultaneous linear equations :
\[ \begin{array}{rcl} x - 2y + 3z &=& 4 \\ 2x + 3y + z &=& 6 \\ 3x + y - 2z &=& 7 \end{array} \]
Evaluate the integral: \[ \int \frac{dx}{4 + 3\cot x} \]
Evaluate the integral: \[ \int \frac{dx}{(x+1)\sqrt{x^2 + 4}} \]
If \[ \int e^x (x^3 + x^2 - x + 4) \, dx = e^x f(x) + C, \] then \( f(1) \) is:
Evaluate the integral: \[ \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{dx}{\sec^2 x + (\tan^{2022} x - 1)(\sec^2 x - 1)} \]
The product of all the values of \(\bigl(\sqrt{3} - i\bigr)^{25}\) is ?
The number of common roots among the 12th and 30th roots of unity is ?
∫ √(2x2 - 5x + 2) dx = ∫ (41/60) dx,
and
-1/2 > α > 0, then α = ?
If \( x \) and \( y \) are two positive real numbers such that \( x + iy = \frac{13\sqrt{5} + 12i}{(2 - 3i)(3 + 2i)} \), then \( 13y - 26x = \):