CAT 2019 Slot 2 QA Question & Solution

AlgebraMedium

Question

If x is a real number, then $\sqrt{\log_{e}{\frac{4x - x^2}{3}}}$ is a real number if and only if

Options

$1 \leq x \leq 3$

$1 \leq x \leq 2$

$-1 \leq x \leq 3$

$-3 \leq x \leq 3$

Solution

$\sqrt{\log_{e}{\frac{4x - x^2}{3}}}$ will be real if $\log_e\ \frac{\ 4x-x^2}{3}\ \ge\ 0$

$\frac{\ 4x-x^2}{3}\ >=\ 1$

$\ 4x-x^2-3\ >=\ 0$

$\ x^2-4x+3\ =<\ 0$

1=< x=< 3