NIMCET 2012 Question Paper with solutions

Given that,
f (a + b) = f (a) × f (b) ... (i)
and f (5) = 2 , f ' (0) = 3
By definition :
f ' (5) = $\underset{\mathit{h}\to 0}{\lim }$ $\frac{\mathit{f}\left(5+\mathit{h}\right)-\mathit{f}\left(5\right)}{\mathit{h}}$
f ' (5) = $\underset{\mathit{h}\to 0}{\lim }$ $\frac{\mathit{f}\left(5\right)\times \mathit{f}\left(\mathit{h}\right)-\mathit{f}\left(5\right)}{\mathit{h}}$
f ' (5) = f (5) . $\underset{\mathit{h}\to 0}{\lim }$ $\frac{\mathit{f}\left(\mathit{h}\right)-1}{\mathit{h}}$
By ‘L'hospital rule,
f ' (5) = f (5) . f ' (0)
⇒ f ' (5) = 2 × 3 = 6