The general solution of the differential equation \( \frac{dy}{dx} = xy - 2x - 2y + 4 \) is:
If \( f'(x) = 4x\cos^2(x) \sin\left(\frac{x}{4}\right) \), then \( \lim_{x \to 0} \frac{f(\pi + x) - f(\pi)}{x} \) is equal to:
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:
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:
The value of the limit \(\lim_{x \to 0} \frac{(2 + \cos 3x) \sin^2 x}{x \tan(2x)}\) is equal to:
The coefficient of \( x^{14}y \) in the expansion of \( (x^2 + \sqrt{y})^9 \) is:
If \(\sec \theta + \tan \theta = 2 + \sqrt{3}\), then \(\sec \theta - \tan \theta\) is:
The focus of the parabola \(y^2 + 4y - 8x + 20 = 0\) is at the point:
A particle is moving along the curve \( y = 8x + \cos y \), where \( 0 \leq y \leq \pi \). If at a point the ordinate is changing 4 times as fast as the abscissa, then the coordinates of the point are:
The integral \(\int e^x \sqrt{e^x} \, dx\) equals:
If \( a = \frac{1 + \tan \theta + \sec \theta}{2 \sec \theta} \) and \( b = \frac{\sin \theta}{1 - \sec \theta + \tan \theta} \), then \( \frac{a}{b} \) is equal to:
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:
\[ f(x) = \begin{cases} x\left( \frac{\pi}{2} + x \right), & \text{if } x \geq 0 \\ x\left( \frac{\pi}{2} - x \right), & \text{if } x < 0 \end{cases} \]
The area bounded by the parabola \(y = x^2 + 2\) and the lines \(y = x\), \(x = 1\) and \(x = 2\) (in square units) 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 value of \[ \left(\frac{10i}{(2-i)(3-i)}\right)^{2024} \] is equal to:
The line \(y = 5x + 7\) is perpendicular to the line joining the points \((2, 12)\) and \((12, k)\). Then the value of \(k\) is equal to: