CAT 2022 Slot 3 · Quant · Question 15

Both terms depend only on the ratio of x to y, so write t=\frac{x}{y}.

Then c=16t+\frac{49}{t}, with t any non-zero real.

Take t>0 first, where both terms are positive and AM-GM applies.

That gives c\ge 2\sqrt{16t\times \frac{49}{t}}=2\sqrt{784}=2\times 28=56.

So for positive t, the value of c is at least 56.

Now take t<0, and write t=-s with s>0.

Then c=-16s-\frac{49}{s}=-\left( 16s+\frac{49}{s}\right).

The bracket is again at least 56, so c\le -56.

Putting the two together, c is either at least 56 or at most -56.

Nothing strictly between -56 and 56 is attainable.

Checking the options: -70 and -60 are below -56, and 60 is above 56 — all reachable.

But -50 lies inside the forbidden band, so c cannot equal -50 — the third option.

Shortcut. The expression is the classic kt+\frac{K}{t}, whose values avoid the open interval between -2\sqrt{kK} and 2\sqrt{kK}. Here 2\sqrt{16\times 49}=56, and only one option sits inside that gap.