Let \( f(x) = \log_e(x) \) and let \( g(x) = \frac{x - 2}{x^2 + 1} \). Then the domain of the composite function \( f \circ g \) is:
\[ \int \frac{4x \cos \left( \sqrt{4x^2 + 7} \right)}{\sqrt{4x^2 + 7}} \, dx \]
The value of \[ \left(\frac{10i}{(2-i)(3-i)}\right)^{2024} \] is equal to:
Let \(f(x) = a^{3x}\) and \(a^5 = 8\). Then the value of \(f(5)\) is equal to:
For \(1 \leq x<\infty\), let \(f(x) = \sin^{-1}\left(\frac{1}{x}\right) + \cos^{-1}\left(\frac{1}{x}\right)\). Then \(f'(x) =\)
\[ \int_0^{\frac{\pi}{4}} (\tan^3 x + \tan^5 x) \, dx \]
Let \( I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{\tan^2 x}{1+5^x} \, dx \). Then:
If \(\sec \theta + \tan \theta = 2 + \sqrt{3}\), then \(\sec \theta - \tan \theta\) is:
An assignment of probabilities for outcomes of the sample space \( S = \{1, 2, 3, 4, 5, 6\} \) is as follows:
\[ \begin{array}{c|c c c c c c} 1 & 2 & 3 & 4 & 5 & 6 \\ \hline k & 3k & 5k & 7k & 9k & 11k \end{array} \]
If this assignment is valid, then the value of \( k \) is:
If \( a = \tan^{-1}\left(\frac{4}{3}\right) \) and \( b = \tan^{-1}\left(\frac{1}{3}\right) \), where \( 0<a, b<\frac{\pi}{2} \), then \( a - b \) is:
For a hyperbola, the vertices are at \( (6, 0) \) and \( (-6, 0) \). If the foci are at \( (2\sqrt{10}, 0) \) and \( -2\sqrt{10}, 0) \), then the equation of the hyperbola is:
The equation of the line passing through the point \((-9,5)\) and parallel to the line \(5x - 13y = 19\) is:
If a line makes angles \(\alpha\), \(\beta\), and \(\gamma\) with the positive directions of the x, y, and z-axis respectively, then \(\cos 2\alpha + \cos 2\beta + \cos 2\gamma\) equals:
The first term and the 6th term of a G.P. are 2 and \( \frac{64}{243} \) respectively. Then the sum of first 10 terms of the G.P. is: