$A$ convex lens has a focal length of $25\, cm$. Calculate the distance of the object from the lens if the image is to be formed on the opposite side of the lens at a distance of $75\, cm$ from the lens. What will be the nature of the image?

(D) Given: Focal length $f = +25\, cm$,Image distance $v = +75\, cm$ (since it is on the opposite side).
Using the lens formula: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$.
Substituting the values: $\frac{1}{75} - \frac{1}{u} = \frac{1}{25}$.
Rearranging for $u$: $\frac{1}{u} = \frac{1}{75} - \frac{1}{25} = \frac{1 - 3}{75} = \frac{-2}{75}$.
Thus,$u = -37.5\, cm$.
The negative sign indicates the object is placed $37.5\, cm$ in front of the lens.
Since $v$ is positive and the image is formed on the opposite side,the image is real and inverted. Also,since $|v| > |u|$ $(75 > 37.5)$,the image is magnified.