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STD 12 - 1. relation and function — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 1. relation and function4 Marks
MCQ
If $h\left( x \right) = \left[ {\ln \frac{x}{e}} \right] + \left[ {\ln \frac{e}{x}} \right]$ ,where [.] denotes greatest integer function, then which of the following is false ?
  • A
    Range of $h(x)$ is $\{-1, 0\}$
  • B
    $h(x)$ is a periodic function
  • C
    If $h(x) = -1$ , then $x$ can be rational as well as irrational
  • ✓
    If $h(x) = 0$ , then $x$ must be irrational
✓

Answer

Correct option: D.
If $h(x) = 0$ , then $x$ must be irrational
d
$h(x)=\left[\ln \frac{x}{e}\right]+\left[\ln \frac{e}{x}\right]=[t]+[-t], t=\ln \left(\frac{x}{e}\right)$

$=\left\{\begin{array}{cc}{0} & {\ln \frac{x}{e} \in I} \\ {-1} & {\ln \frac{x}{e} \notin I}\end{array}\right.$

Range is $\{-1,0\}$

If $\mathrm{h}(\mathrm{x})=0 \Rightarrow \frac{\mathrm{x}}{\mathrm{e}} \in \mathrm{k}, \mathrm{k} \in \mathrm{I}$

$\Rightarrow x$ can be rational or irrational

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