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

If $y = \sin(px)$ and $y_n$ is the $n^{th}$ derivative of $y$,then $\left| \begin{array}{ccc} y & y_1 & y_2 \\ y_3 & y_4 & y_5 \\ y_6 & y_7 & y_8 \end{array} \right|$ is equal to:

Let $A = \begin{bmatrix} 1 & 0 & -1 & -3 \\ 0 & 1 & 1 & k-1 \\ 0 & 0 & k-1 & 1 \end{bmatrix}$ and $k \in R$. Then,the value of $k$,if it exists,for which the rank of $A$ is $2$,is

Which of the following matrices has rank $3$?

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

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: