If $f:[0, \pi / 2) \rightarrow R$ is defined as $f(\theta)=\left|\begin{array}{ccc}1 & \tan \theta & 1 \\ -\tan \theta & 1 & \tan \theta \\ -1 & -\tan \theta & 1\end{array}\right|$,then the range of $f$ is:

Let $A$ be a $3 \times 3$ real matrix such that $A^2(A-2I) - 4(A-I) = O$,where $I$ and $O$ are the identity and null matrices,respectively. If $A^5 = \alpha A^2 + \beta A + \gamma I$,where $\alpha, \beta$ and $\gamma$ are real constants,then $\alpha + \beta + \gamma$ is equal to:

In $\triangle ABC$,if $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=0$,then $\cos A \cos B+\cos B \cos C+\cos C \cos A=$

If $ P=\left|\begin{array}{ll}x & 1 \\ 1 & x\end{array}\right| $ and $ Q=\left|\begin{array}{lll}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{array}\right| $,then $ \frac{d Q}{d x}= $

If $A$ and $B$ are two square matrices of the same order and $(AB+BA)^{T}+(AB-BA)^{T}=2BA$,then:

Matrix $A$ satisfies $A^2 = 2A - I$,where $I$ is the identity matrix. Then for $n \ge 2$,$A^n$ is equal to $(n \in N)$: