The determinant $\left|\begin{array}{lll}b^2-a b & b-c & b c-a c \\ a b-a^2 & a-b & b^2-a b \\ b c-a c & c-a & a b-a^2\end{array}\right|$ equals to

The number of distinct real roots of $\left|\begin{array}{lll}\cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is

If $A, B$ and $C$ are angles of a triangle, then the determinant $\left|\begin{array}{ccc}-1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1\end{array}\right|$ is equal to

If $f(t)=\left[\begin{array}{ccc}\cos t & t & 1 \\ 2 \sin t & t & 2 t \\ \sin t & t & t\end{array}\right]$, then $\lim _\limits{t \rightarrow 0} \frac{f(t)}{t^2}$ is equal to

The maximum value of $\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+\sin \theta & 1 \\ 1+\cos \theta & 1 & 1\end{array}\right|$ is (where, $\theta$ is real number)