Let $f(x) = \left|\begin{array}{ccc} a & -1 & 0 \\ ax & a & -1 \\ ax^2 & ax & a \end{array}\right|$,where $a \in R$. Then the sum of the squares of all the values of $a$ for which $2f'(10) - f'(5) + 100 = 0$ is:

If $y = \left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c\end{array}\right|$,then $\frac{dy}{dx}$ 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:

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:

Let for $i = 1, 2, 3$,$p_i(x)$ be a polynomial of degree $2$ in $x$,$p'_i(x)$ and $p''_i(x)$ be the first and second order derivatives of $p_i(x)$ respectively. Let $A(x) = \begin{bmatrix} p_1(x) & p'_1(x) & p''_1(x) \\ p_2(x) & p'_2(x) & p''_2(x) \\ p_3(x) & p'_3(x) & p''_3(x) \end{bmatrix}$ and $B(x) = [A(x)]^T A(x)$. Then the determinant of $B(x)$

$A$ value of $b$ for which the rank of the matrix $A = \begin{bmatrix} 1 & 1 & -1 & 0 \\ 4 & 4 & -3 & 1 \\ b & 2 & 2 & 2 \\ 9 & 9 & b & 3 \end{bmatrix}$ is $3$,is