Let $l, m, n \in R$ and $A = \begin{bmatrix} 1 & r & r^2 & l \\ r & r^2 & 1 & m \\ r^2 & 1 & r & n \end{bmatrix}$. Then the set of all real values of $r$ for which the rank of $A$ is $3$, is

Find the rank of the matrix $\begin{bmatrix} 1 & 4 & -1 \\ 2 & 3 & 0 \\ 0 & 1 & 2 \end{bmatrix}$.

If $A = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix}$ where $a = 7^x$,$b = 7^{7^x}$,$c = 7^{7^{7^x}}$,then $\int |A| \, dx$ (where $|A|$ is the determinant of the matrix $A$) is equal to:

If $a_1, a_2, a_3, \dots, a_n$ form a geometric progression,find the value of the determinant: $\left| \begin{array}{ccc} \log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{array} \right|$.

If $f(x) = \left| \begin{array}{ccc} \sin(x + \alpha) & \sin(x + \beta) & \sin(x + \gamma) \\ \cos(x + \alpha) & \cos(x + \beta) & \cos(x + \gamma) \\ \sin(\alpha + \beta) & \sin(\beta + \gamma) & \sin(\gamma + \alpha) \end{array} \right|$ and $f(10) = 10$,then $f(\pi)$ is equal to

If the rank of the matrix $\begin{bmatrix} x & x & x \\ x & x^2 & x \\ x & x & x+1 \end{bmatrix}$ is $1$, then: