CAT 2022 Slot 3 QA Question & Solution

AlgebraMedium

If $c=\frac{16x}{y}+\frac{49y}{x}$ for some non-zero real numbers x and y, then c cannot take the value

$60$

$-50$

$-70$

$-60$

Let $\frac{x}{y}\ be\ t$

Therefore, $c=16t\ +\ \frac{49}{t}$

Applying AM>= GM

$\frac{\left(16t\ +\ \frac{49}{t}\right)}{2}\ge\ \left(16t\times\frac{49}{t}\right)^{\frac{1}{2}}$

$16t\ +\ \frac{49}{t}\ge56$

When t is positive then c is greater than equal to 56.

When t is negative then c is less than equal to -56.

Therefore $c\ \in\ \left(-\infty,\ -56\right]\ ∪\ \left[56,\infty\ \right]$

As -50 is not in the range of c so it is the answer