A convex lens forms a real image of the same size as an object placed…

The correct answer is Option (3) → \( -\frac{1}{2} \)

A convex lens gives a real image as large as the object only when the object is at a distance \( 2f \). The image then lies at \( 2f \) on the other side. So the first position gives the focal length. The lens formula with the new object distance then gives the magnification \( m=v/u \).

From the same-size image:

    \( 2f=40\,\mathrm{cm} \)

    \( \Rightarrow\; f=+20\,\mathrm{cm} \)

The object is now \( 40+20=60\,\mathrm{cm} \) from the lens, so \( u=-60\,\mathrm{cm} \). The image distance is:

    \( \begin{aligned} \frac{1}{v} &= \frac{1}{f}+\frac{1}{u} \\ &= \frac{1}{20\,\mathrm{cm}}-\frac{1}{60\,\mathrm{cm}} \\ &= \frac{1}{30\,\mathrm{cm}} \end{aligned} \)

    \( \Rightarrow\; v=+30\,\mathrm{cm} \)

The magnification of the new image is:

    \( \begin{aligned} m &= \frac{v}{u} \\ &= \frac{+30\,\mathrm{cm}}{-60\,\mathrm{cm}} \\ &= -\frac{1}{2} \end{aligned} \)

The negative sign shows that the image is real and inverted. Moving the object away from \( 2f \) has made the image half the size of the object.