The rank of the matrix $\begin{bmatrix} 2 & -3 & 4 & 0 \\ 5 & -4 & 2 & 1 \\ 1 & -3 & 5 & -4 \end{bmatrix}$ 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:

Suppose $\left| \begin{array}{cc} f'(x) & f(x) \\ f''(x) & f'(x) \end{array} \right| = 0$ where $f(x)$ is a continuously differentiable function with $f'(x) \ne 0$ and satisfies $f(0) = 1$ and $f'(0) = 2$. Then the number of solution$(s)$ of the equation $f(x) = x^2$ is equal to:

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 $\int_0^{\frac{\pi}{4}} (2|f(x)| + 5f'(x)) \, dx$.

For $\triangle ABC$,evaluate the determinant: $\left|\begin{array}{ccc}0 & \sin A & \tan B \\ -\sin ( B + C ) & 0 & \cos C \\ \tan ( A + C ) & -\cos C & 0\end{array}\right|=$ . . . . . . .

In a matrix $A$,if all the sub-matrices of order $k$ are singular and there is at least one non-singular sub-matrix of order $r$ $(r < k)$,then the rank $(\rho)$ of the matrix $A$: