Let $A$ be the matrix $\left[\begin{array}{ccc}\cos \theta & 0 & -\sin \theta \\ 1 & 1 & 1 \\ \sin \theta & 0 & \cos \theta\end{array}\right]$. For any natural number $k$, the determinant of $A^k$ is

Consider the vectors $\vec{a}=\hat{\imath}+x \hat{\jmath}+2 \hat{k}, \vec{b}=\hat{\imath}+2 \hat{\jmath}+x \hat{k}, \vec{c}=2 \hat{\imath}+\hat{\jmath}+3 \hat{k}$. The values of $x$ for Which there is at least one nonzero vector perpendicular to the vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are

Consider the tangent lines to the circle $x^2+y^2=1$ at points $P=(1,0)$ and $Q=\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$. If $R$ is the point of intersection of these two tangent lines, then $\angle P R Q$ is:

The function given by $f(x)=2 x^3-15 x^2+36 x-5$ is