If $f(x) = \left| \begin{array}{ccc} 2\cos^2 2x & \sin 2x & -\sin x \\ \sin 2x & 2\sin^2 x & \cos x \\ \sin x & -\cos x & 0 \end{array} \right|$,then the value of $\int_{0}^{\frac{\pi}{2}} f'(x) \,dx$ is equal to

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:

The rank of the matrix $\begin{bmatrix} 1 & -1 & 1 \\ 1 & 1 & -1 \\ -1 & 1 & 1 \end{bmatrix}$ 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 $\alpha, \beta, \gamma$ are the roots of the equation $x^3+qx+r=0$ (where $r \neq 0$) and they are in $A$.$P$. Then the rank of the matrix $\begin{bmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{bmatrix}$ is

If $f(x) = \begin{vmatrix} 1 + \sin x + \sin 2x + \sin 3x & \frac{3 + \sin 2x}{2} & \frac{-2 + \sin 3x}{3} \\ 3 + 4 \sin x & \frac{3}{2} & \frac{4}{3} \sin x \\ 1 + \sin x & \frac{1}{2} \sin x & \frac{1}{3} \end{vmatrix}$, then $\int_0^{\pi / 2} (f(x) + f^{\prime}(x)) dx =$