As given:

u = -2.4m = −240cm ;

v = 12cm

Accoring to the lens formula:

$$\frac{1}{f} = \frac{1}{v} – \frac{1}{u}$$

= $$\frac{1}{f} = \frac{1}{12} – \frac{1}{-240}$$

= $$\frac{1}{f} = \frac{1}{12} + \frac{1}{240}$$ $$\frac{1}{f} = \frac{21}{240}cm$$

When a glass plate is interposed between lens and film, so shift produced by it will be

Shift t = $$(1 – \frac{1}{\mu}) = 1$$

= $$(1 – \frac{1}{1.5}) =1$$

= $$(1 – \frac{2}{3})$$

= $$\frac{1}{3}cm$$

Inorder to get image at film, the lens should form image at a distance:

$${v}’ = 12 – \frac{1}{3}$$

= $$\frac{35}{3}cm$$

Now, by using the lens formula:

$$\Rightarrow \frac{21}{240}cm = \frac{3}{35}cm – \frac{1}{{u}’}$$ $$\Rightarrow \frac{1}{{u}’} = \frac{3}{35}cm – \frac{21}{240}cm$$ $$\Rightarrow \frac{1}{5} = [\frac{3}{7}cm – \frac{21}{48}cm]$$ $$\Rightarrow \frac{1}{{u}’} = \frac{1}{5} [\frac{144 – 147}{336}]$$ $$\Rightarrow {u}’ = -560 cm = -5.6m$$

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