The focus of the parabola \(y^2 + 4y - 8x + 20 = 0\) is at the point:
Let \( f(x) = \begin{cases} x^2 - \alpha, & \text{if } x < 1 \\ \beta x - 3, & \text{if } x \geq 1 \end{cases} \). If \( f \) is continuous at \( x = 1 \), then the value of \( \alpha + \beta \) is:
The value of \[ \left(\frac{10i}{(2-i)(3-i)}\right)^{2024} \] is equal to:
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:
Let \( f(x) = 2 - 7 \sin{\left( \frac{2x}{7} \right)} \). Then the maximum value of \( f(x) \) is:
Let \[ A = \begin{pmatrix} 3 & -2 & 1 \\ -1 & 3 & -1 \end{pmatrix} \] and \[ B = \begin{pmatrix} 1 \\ \alpha \\ -1 \end{pmatrix}. \] If \[ AB = \begin{pmatrix} -2 \\ 6 \end{pmatrix}, \] then the value of \( \alpha \) 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:
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:
The general solution of the differential equation \( \frac{dy}{dx} = xy - 2x - 2y + 4 \) 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:
Let \( S \) denote the set of all subsets of integers containing more than two numbers. A relation \( R \) on \( S \) is defined by:
\[ R = \{ (A, B) : \text{the sets } A \text{ and } B \text{ have at least two numbers in common} \}. \]
Then the relation \( R \) is:
The coefficient of \( x^{14}y \) in the expansion of \( (x^2 + \sqrt{y})^9 \) is:
The vectors \(\vec{a} = 4\mathbf{i} - 3\mathbf{j} - \mathbf{k}\) and \(\vec{b} = 3\mathbf{i} + 2\mathbf{j} + \lambda\mathbf{k}\) are perpendicular to each other. Then the value of \(\lambda\) is equal to:
The value of \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\cos^2 x}{1 + 2^{-x}} \, dx \] is equal to: