Let pk(x)=x^3-12x+k, where k is a real constant, and let N(k) denote the number of distinct real roots of pk(x)=0.

Differentiate: $p_k'(x)=3x^2-12=3(x-2)(x+2)$, so $p_k$ has a local maximum at $x=-2$ with value $p_k(-2)=16+k$ and a local minimum at $x=2$ with value $p_k(2)=k-16$. Three distinct real roots need $16+k>0$ and $k-16<0$, i.e. $|k|<16$; a repeated root occurs when $k=\pm16$ (two distinct roots); one real root when $|k|>16$. Thus $k=10$ gives $3$ roots (A true); $k=16$ gives a double root at $2$ and a simple root at $-4$, so $N=2$ (B true); $k=-20$ has $|k|>16$, hence a single real root (C false); $k=-16$ gives roots $4$ and a double root $-2$, so $N=2$, not $1$ (D false). Numerical roots confirm this.