A concave mirror forms a real image three times larger than the object on a screen. Object and screen are moved until the image becomes twice the size of object. If the shift of object is 6 cm, the shift of the screen & focal length of mirror are

$\frac{4}{3x}=\frac{\text{1}}{\text{f}}\to \left(1\right)$

$\frac{1}{2\left(6+x\right)}+\frac{1}{6+x}=\frac{1}{\text{f}}$

$\frac{3}{2\left(6+x\right)}=\frac{\text{1}}{\text{f}}\to \left(2\right)$

From (1) and (2)

$\frac{4}{3x}=\frac{3}{2\left(6+x\right)}$

$\therefore ax=8\left(6+x\right)$

$ax=48+8x$

$x=48\text{cm}$

Shift the screen $=\text{3}x-\text{2}\left(6+x\right)$

$=3\times 48-2\left(6+48\right)$

$=36\text{cm}$

$\frac{3}{2\left(6+x\right)}-\frac{\text{1}}{\text{f}}$

$\frac{3}{2\left(6+48\right)}=\frac{\text{1}}{\text{f}}$

$\frac{3}{108}=\frac{\text{1}}{\text{f}}$

$\frac{\text{1}}{\text{f}}=\frac{1}{36}$

$\text{f}=36\text{cm}$