$(a)$ Calculate the distance at which an object should be placed in front of a convex lens of focal length $10\, cm$ to obtain a virtual image of double its size.
$(b)$ In the above given case,find the magnification,if the image formed is real. Express it in terms of the relation between $v$ and $u$.

(A) Given: Focal length $f = +10\, cm$,Magnification $m = +2$ (for virtual image).
Using the magnification formula $m = \frac{v}{u}$,we get $v = mu = 2u$.
Using the lens formula $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$:
$\frac{1}{10} = \frac{1}{2u} - \frac{1}{u} = \frac{1-2}{2u} = -\frac{1}{2u}$.
Solving for $u$: $2u = -10$,so $u = -5\, cm$.
The object should be placed at a distance of $5\, cm$ in front of the lens.
$(b)$ If the image formed is real,then the magnification $m$ is negative. For a real image of double the size,$m = -2$. The relation between $v$ and $u$ is given by $m = \frac{v}{u}$,so $v = mu = -2u$.