UPSC CDS 1 2022 GK Paper

Concept:A convex lens forms a real, inverted image of the same size as the object when the object is placed at twice the focal length ($2F$).For a convex lens, the lens formula is $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$, where $f$ is focal length, $v$ is image distance, and $u$ is object distance (both measured from the optical centre).Explanation:For a real image of the same size, the magnification $m = \frac{v}{u} = -1$ (since the image is inverted).Thus, $v = -u$. (Note: $u$ is negative as per Cartesian sign convention, $v$ is positive for real image.)Substitute $v = -u$ into the lens formula: $\frac{1}{f} = \frac{1}{-u} - \frac{1}{u}$.Simplify: $\frac{1}{f} = -\frac{1}{u} - \frac{1}{u} = -\frac{2}{u}$.Given $f = 15$ cm, we get $\frac{1}{15} = -\frac{2}{u}$.Cross‑multiplying: $u = -30$ cm.The negative sign indicates the object is placed in front of the lens at a distance of $30$ cm.Answer:The object should be placed at a distance of $30$ cm from the lens. (Option C)