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

The value of the integral

$$ \int_1^{100} \frac{[x]}{x} d x $$

where $[x]$ is the greatest integer less than or equal to $x$ for any real number $x$, is