Kindly answer this as I am a bit confused.

Solution:
 (a) Since the focal length is a constant quantity, we have to pair the object distance(u) and the image distance (v) such that the focal length always comes out to be the same. From the above argument, we get the correct order of the image distance as  100 , 60 , 40 , 30 and 24. The reason being, as the object is carried far from a convex lens, the image is formed closer to the lens.

(b) Lens formula is given by:

 $$\begin{gathered} \frac{1}{f}=\frac{1}{v}-\frac{1}{u} \\ \frac{1}{20}=\frac{1}{v}-\frac{1}{-90} \\ \Rightarrow \frac{1}{20}=\frac{1}{v}+\frac{1}{90} \\ \Rightarrow \frac{1}{v}=\frac{1}{20}-\frac{1}{90} \\ \Rightarrow \frac{1}{v}=\frac{9-2}{180} \\ \Rightarrow \frac{1}{v}=\frac{7}{180} \\ \Rightarrow v=\frac{180}{7} \\ \Rightarrow v=25.7 \mathrm{cm} \end{gathered}$$

The image distance will be 25.7 cm if the object distance is 90 cm.
 
(c) The object distance of 25 gives the biggest image because at this position, the object is between f and 2f . We know that when an object is placed between f and 2f of a convex lens, we get a real, inverted and magnified image.

(d) The image of an object at 2f is formed at 2f. This means that the pair of u and v that is equal in value gives us the value of 2f, which is 40. Hence, the value of f (focal length) is $\frac{40}{2}$ .
∴ f=20 cm.