If $\mathrm{f}(x)=x^3-10 x^2+200 x-10$, then

The number of common tangents to the circles $x^2+y^2-x=0$ and $x^2+y^2+x=0$ is /are

Let $\bar{a}=2 \hat{i}+\hat{j}-2 \hat{k}$ and $\bar{b}=\hat{i}+\hat{j}$. If $\bar{c}$ is a vector such that $\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=|\overline{\mathrm{c}}|,|\overline{\mathrm{c}}-\overline{\mathrm{a}}|=2 \sqrt{2}$ and the angle between $(\overline{\mathrm{a}} \times \overline{\mathrm{b}})$ and $\overline{\mathrm{c}}$ is $30^{\circ}$, then $|(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times \overline{\mathrm{c}}|$ is equal to

The equation $\mathrm{e}^{\sin x}-\mathrm{e}^{-\sin x}=4$ has ̱_________ solutions.