Mock Test 2 — Determinants

If $\Delta(x) = \begin{vmatrix} x & 1 & 2 \\ x^2 & 2x & 3 \\ x^3 & 3x^2 & 4 \end{vmatrix}$, then the derivative value $\Delta'(0)$ is equal to:

If $A$ is an invertible matrix of order 3, the determinant of its adjoint matrix satisfies $|\text{adj}(A)| = |A|^2$. The value of $|\text{adj}(2A)|$ is:

The system of linear equations $x + y + z = 2$, $2x + 3y + 2z = 5$, $2x + 3y + (a^2-1)z = a+1$ has infinitely many solutions if:

If $A$ is a square matrix satisfying $A^2 - A + I = O$, then its inverse matrix $A^{-1}$ is identically equal to:

The value of the determinant $\begin{vmatrix} 1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \end{vmatrix}$, where $\omega$ is a non-real cube root of unity, is:

If the area of a triangle with vertices $(k, 0)$, $(4, 0)$, and $(0, 2)$ is 4 square units, the possible values of the integer parameter $k$ are:

If $\lambda_1, \lambda_2, \lambda_3$ are the eigenvalues of a $3 \times 3$ matrix $A$, then the eigenvalues of the squared matrix $A^2$ are:

If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, the matrix expression $A^2 - 5A - 2I$ evaluates to:

The number of distinct real roots of the equation $\begin{vmatrix} x & 0 & 1 \\ 0 & x & 0 \\ 1 & 0 & x \end{vmatrix} = 0$ is exactly:

If a system of 3 linear equations with 3 variables satisfies $\Delta = 0$ and $\Delta_x = 2, \Delta_y = 0, \Delta_z = 0$, the planes:

Find the positive integer eigenvalue $\lambda$ for the matrix $A = \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix}$.

If a matrix $A$ of order 3 satisfies $A \cdot \text{adj}(A) = 5I$, find the value of its determinant $|A|$.

Evaluate the value of the determinant $\begin{vmatrix} 1^2 & 2^2 & 3^2 \\ 2^2 & 3^2 & 4^2 \\ 3^2 & 4^2 & 5^2 \end{vmatrix} + 9$.

Assertion (A): If a matrix is orthogonal ($A A^T = I$), the value of its determinant must be either $1$ or $-1$.
Reason (R): The determinant of a product satisfies $|AA^T| = |A||A^T| = |A|^2$, and the determinant of the identity matrix is $|I| = 1$, so $|A|^2 = 1 \implies |A| = \pm 1$.

Assertion (A): The homogeneous system of linear equations $AX = 0$ has infinitely many solutions if its coefficient matrix satisfies $|A| = 0$.
Reason (R): For a homogeneous system, all modified determinants ($\Delta_x, \Delta_y, \Delta_z$) are always zero, so if $\Delta = 0$, the system meets the criteria for infinitely many solutions.