Discussion

Explanation:

Let us calculate time interval for hands of a normal clock to coincide.

The two hands coincide at 12 O'clock. Next they will coincide between 1 and 2 O'clock.

⇒ $\left|30\mathrm{h}-\frac{11}{2}\mathrm{m}\right|$ = 0
⇒ $\left|30\times 1-\frac{11}{2}\mathrm{m}\right|$ = 0
⇒ 30 - $\frac{11}{2}$m = 0
⇒ m = $\frac{60}{11}$ = 5$\frac{5}{11}$ minutes.

∴ The two hands coincide again at 1 : 5$\frac{5}{11}$.

⇒ From 12 O'clok till 1 : 5$\frac{5}{11}$ it takes 65$\frac{5}{11}$ minutes for the two hands to coincide again.

For a normal clock, hands coincide every $65\frac{5}{11}$ minutes, but for the faulty clock hands coincide every 64 minutes.

[If you look at the faulty clock you would get a sense that $65\frac{5}{11}$ minutes have passed when actually only 64 minutes have passed.]

∴ The faulty clock gains $1\frac{5}{11}$ minutes every 64 minutes.

∴ Time gained in a day i.e., 24 hours = $\frac{1\frac{5}{11}}{64}\times 24\times 60$ = $\frac{360}{11}$ = $32\frac{8}{11}$ minutes.

Hence, option (d).

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