Match the entries of List-I with those of List-II. List-I P. Number of complex numbers z satisfying z^3=bar z Q. Minimum value of |z-3-4i| over all z…

P: taking moduli, $|z|^3=|z|$ gives $|z|=0$ or $1$. For $|z|=1$, $z^3=\bar z=1/z$ means $z^4=1$, giving $4$ solutions; with $z=0$ the total is $5$, i.e. (3). Q: the point $3+4i$ is at distance $5$ from the origin, so the least distance to the unit circle is $5-1=4$, i.e. (1). R: for $r=|z|$ we have $r-\tfrac1r\le\left|z+\tfrac1z\right|=2$, so $r^2-2r-1\le0$ and $r\le1+\sqrt2$; at $r=1+\sqrt2$ we have $r^2+r^{-2}=6$ and the condition $r^2+r^{-2}+2\cos2\theta=4$ is met with $\cos2\theta=-1$, so the bound is attained: (4). S: the roots $2,2\omega,2\omega^2$ form an equilateral triangle of circumradius $2$ and side $2\sqrt3$, with area $\tfrac{\sqrt3}{4}\cdot12=3\sqrt3$, i.e. (2). So P→3, Q→1, R→4, S→2.