The lengths of three unequal edges of a rectangular solid block are in G.P.

Since lengths of three unequal sides of a rectangular solid block are in G.P.

Let the length, breadth and height of rectangular solid block be $\frac{a}{r}, a$ and ar respectively.

∴ Volume of rectangular block is

$\frac{a}{r} \times a \times a r \quad[\because$ volume $=$ length $\times$ breadth $\times$ height $]$

i.e a3 = 216    (given)

i.e a = 6

Since surface area of rectangular block

$=2\left(\frac{a}{r} \cdot a+a \cdot a r+\frac{a}{r} a r\right)=252 \quad$ (given)

$(\because$ surface area $=2(l b+b h+h l))$

$\Rightarrow 2\left(\frac{a^{2}}{r}+a^{2} r+a^{2}\right)=252$

$\Rightarrow 2 a^{2}\left(\frac{1}{r}+r+1\right)=252$

$\Rightarrow 2 \times 36\left(\frac{1+r^{2}+r}{r}\right)=252$

$\Rightarrow 2\left(1+r^{2}+r\right)=7 r$

i. e $2 r^{2}+2+2 r=7 r$

i. e $2 r^{2}-5 r+2=0$

i.e $(2 r-1)(r-2)=0$

$\Rightarrow r=\frac{1}{2}, r=2$

$\therefore$ for $r=\frac{1}{2}$, length $=\frac{a}{r}=\frac{6}{\frac{1}{2}}=12$

Breadth = a = 6

Height $=\operatorname{ar}=6 \times \frac{1}{2}=3$ and for $r=2$,

length $=\frac{a}{r}=\frac{6}{2}=3$

breadth = a = 6

right = ar = 12

∴ length of largest edge is 12

Hence, the correct answer is option A.