If $f: N \to Z$ is defined by $f(n) = \det \begin{vmatrix} n & -1 & -5 \\ -2n^2 & 3(2k+1) & 2k+1 \\ -3n^3 & 3(2k+1) & 3(k+2)+1 \end{vmatrix}$, where $k \in N$ and $\sum_{n=1}^k f(n) = 98$, then $k$ is equal to:

Let $a, b \in R-\{0\}$,and $I_2$ be the identity matrix of order $2$. Then the rank of the block matrix $\begin{bmatrix} a I_2 & b I_2 \\ a I_2 & b I_2 \end{bmatrix}$ is

If $A = \begin{bmatrix} 2 & 4 & 5 \\ 4 & 8 & 10 \\ -6 & -12 & -15 \end{bmatrix}$,then the rank of $A$ is equal to

$f(x) = \left| \begin{array}{ccc} \sin^2 x & -2 + \cos^2 x & \cos 2x \\ 2 + \sin^2 x & \cos^2 x & \cos 2x \\ \sin^2 x & \cos^2 x & 1 + \cos 2x \end{array} \right|, x \in [0, \pi]$. The maximum value of $f(x)$ is equal to $.....$

Let $f(x) = \left| \begin{array}{ccc} \cos x & x & 1 \\ 2 \sin x & x & 2x \\ \sin x & x & x \end{array} \right|$. Then,$\lim_{x \rightarrow 0} \frac{f(x)}{x^2}$ is

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