If the rank of the matrix $A=\begin{bmatrix} 1 & 2 & 1 & -1 \\ -1 & 2 & 3 & 5 \\ 0 & 1 & k & k \end{bmatrix}$ is $2$ and $k$ is a real number,then $k$ is a root of the following quadratic equation:

Let $f$ be a twice differentiable function defined on $R$ such that $f(0)=1$,$f^{\prime}(0)=2$ and $f^{\prime}(x) \neq 0$ for all $x \in R$. If $\left|\begin{array}{ll}f(x) & f^{\prime}(x) \\ f^{\prime}(x) & f^{\prime \prime}(x)\end{array}\right|=0$ for all $x \in R$,then the value of $f(1)$ lies in the interval:

If $f(x) = \left| \begin{array}{ccc} \cos(2x) & \cos(2x) & \sin(2x) \\ -\cos x & \cos x & -\sin x \\ \sin x & \sin x & \cos x \end{array} \right|$,then:
$A$. $f'(x) = 0$ at exactly three points in $(-\pi, \pi)$
$B$. $f'(x) = 0$ at more than three points in $(-\pi, \pi)$
$C$. $f(x)$ attains its maximum at $x = 0$
$D$. $f(x)$ attains its minimum at $x = 0$

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:

If $\left|\begin{array}{ccc}x^2+3x & x+1 & x-3 \\ x-1 & 2-x & x+4 \\ x-3 & x-3 & 3x\end{array}\right|=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4$,then find the value of $(a_1+a_3)+2(a_0+a_2+a_4)$.

The determinant $\left| \begin{array}{ccc} ^x{C_1} & ^x{C_2} & ^x{C_3} \\ ^y{C_1} & ^y{C_2} & ^y{C_3} \\ ^z{C_1} & ^z{C_2} & ^z{C_3} \end{array} \right|$ equals: