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# CAT 2019 Quant Question with Solution 36

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

1. $1 \leq x \leq 2$
2. $-3 \leq x \leq 3$
3. $1 \leq x \leq 3$
4. $-1 \leq x \leq 3$

The expression will be real only if ${{\log }_{e}}\frac{4x-{{x}^{2}}}{3}\ge 0$

Or $\frac{4x-{{x}^{2}}}{3}\ge {{e}^{0}}$

$\Rightarrow \frac{4x-{{x}^{2}}}{3}\ge 1$

$\Rightarrow 4x-{{x}^{2}}\ge 3$

$\Rightarrow {{x}^{2}}-4x+3\le 0$

$\Rightarrow \left( x-1 \right)\left( x-3 \right)\le 0$

$1 \leq x \leq 3$

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