$A$ student focused the image of a candle flame on a white screen by placing the flame at various distances from a convex lens. He noted his observations:

Observation	Distance of flame from the lens $(cm)$	Distance of screen from the lens $(cm)$
$(i)$	$60$	$20$
$(ii)$	$40$	$24$
$(iii)$	$30$	$30$
$(iv)$	$24$	$40$
$(v)$	$15$	$70$

$(a)$ From the above table, find the focal length of the lens without using the lens formula.
$(b)$ Which set of observations is incorrect and why?
$(c)$ In which case will the size of the object and the image be the same? Give a reason for your answer.

(N/A) From the table of observations, it is clear that for observation $(iii)$, $u = 30 \, cm$ and $v = 30 \, cm$. This means that both $u$ and $v$ are equal to twice the focal length $(2f)$ of the convex lens.
Therefore, $2f = 30 \, cm$, which gives the focal length $f = 15 \, cm$.
$(b)$ The observation $(v)$ is incorrect. For this observation, $u = 15 \, cm$, which is equal to the focal length $(f = 15 \, cm)$. When an object is placed at the focus of a convex lens, the image is formed at infinity, not at $70 \, cm$.
$(c)$ The size of the image is equal to the size of the object when the magnification $m = 1$. Since $m = \frac{v}{u}$, the size of the image will be equal to the size of the object if $v = u$. This occurs in observation $(iii)$, where $u = 30 \, cm$ and $v = 30 \, cm$.