List of practice Questions

Consider the system of linear equations
x + 2y + z = 5
2x + ay + 4z = 12
2x + 4y + 6z = b
The values of a and b such that there exists a non-trivial null space and the system admits infinite solutions are

Consider a matrix \( A = \begin{pmatrix} -5 & a \\ -2 & -2 \end{pmatrix} \), where \( a \) is a constant. If the eigenvalues of \( A \) are \(-1\) and \(-6\), then the value of \( a \), rounded off to the nearest integer, is ___

The digit in the unit’s place of the product $3^{999} \times 7^{1000}$ is __________.

Eigenvalues of the matrix $\begin{bmatrix}0 & 1 \\ -2 & 3 \end{bmatrix}$ are

The system of linear equations in real \( (x, y) \) given by \[ \begin{pmatrix} x \\ y \end{pmatrix} \begin{pmatrix} 2 & 5 - 2\alpha \\ \alpha & 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \] involves a real parameter \( \alpha \) and has infinitely many non-trivial solutions for special value(s) of \( \alpha \). Which one or more among the following options is/are non-trivial solution(s) of \( (x, y) \) for such special value(s) of \( \alpha \)?

The limit \[ p = \lim_{x \to \pi} \left( \frac{x^2 + \alpha x + 2\pi^2}{x - \pi + 2 \sin x} \right) \] has a finite value for a real \( \alpha \). The value of \( \alpha \) and the corresponding limit \( p \) are