If the system of equations $a_1 x + b_1 y + c_1 z = 0$,$a_2 x + b_2 y + c_2 z = 0$,and $a_3 x + b_3 y + c_3 z = 0$ has only the trivial solution,then the rank of the matrix $A = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}$ is:

Let $A=\begin{bmatrix} -1 & -2 & -3 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{bmatrix}$, $B=\begin{bmatrix} 1 & -2 \\ -1 & 2 \end{bmatrix}$ and $C=\begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}$. If $a, b$ and $c$ respectively denote the ranks of $A, B$ and $C$, then the correct order of these numbers is:

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

Let $M$ and $m$ respectively be the maximum and the minimum values of $f(x) = \left| \begin{array}{ccc} 1+\sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1+\cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1+4\sin 4x \end{array} \right|$,$x \in R$. Then $M^4 - m^4$ is equal to:

$A$ is an $m \times n$ matrix of rank $4$. If $A$ contains an $m$-th order non-singular submatrix and $A^T A$ is a $7 \times 7$ matrix, then the number of rows of $A$ is:

If $f(x) = \left| \begin{array}{ccc} 2 \cos^2 x & \sin 2x & \sin x \\ \sin 2x & 2 \sin^2 x & -\cos x \\ \sin x & -\cos x & 0 \end{array} \right|$, then find the value of $\int_0^{\frac{\pi}{4}} (2|f(x)| + 5f'(x)) \, dx$.