The lines $$\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5} \quad$$ and $$\frac{x+2}{4}=\frac{y-1}{3}=\frac{z+1}{2}$$

A curve passes through the point $$\left(1, \frac{\pi}{6}\right)$$. Let the slope of the curve at each point $$(x, y)$$ be $$\frac{y}{x}+\sec \left(\frac{y}{x}\right), x>0$$, then, the equation of the curve is

For the following shaded area, the linear constraints except $$x,y \ge 0$$ are

Let $$\omega \neq 1$$ be a cube root of unity and $$S$$ be the set of all non-singular matrices of the form $$\left[\begin{array}{ccc}1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1\end{array}\right]$$ where each of $$a, b$$ and $$c$$ is either $$\omega$$ or $$\omega^2$$, then the number of distinct matrices in the set $$\mathrm{S}$$ is