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Given that, the ratio of Amal and Bimal scores is $11:14.$

- Let, the score of Amal $ = 11x$
- Then, the score of Bimal $ = 14x$
- Where $x$ is some constant.

Let their scores increases by $k.$

So, $\frac{11x+k}{14x+k} = \frac{47}{56} $

$ \Rightarrow 56(11x+k) = 47(14x+k) $

$ \Rightarrow 616x+56K = 658x+47k $

$ \Rightarrow 56k – 47k = 658x – 616x $

$ \Rightarrow 9k = 42x $

$ \Rightarrow \boxed{k = \frac{42}{9}x} $

Now, the score of Bimal:

- Bimal new score $ = 14x + k = 14x + \frac{42}{9}x $
- Bimal original score $ = 14x $

$\therefore$ The ratio of Bimal’s new score to that of his original score $ = \frac{14x+\frac{42x}{9}}{14x} = \frac{14x \left(1+\frac{3}{9} \right)}{14x}= \frac{9+3}{9} = \frac{12}{9} = \frac{4}{3} $

Therefore, the ratio of Bimal's new score to that of his original score is $4:3.$

Correct Answer $: \text{C}$