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Consider the following statements :

$S_{1}:x=\sqrt{log_{11}7}$ and $y=\sqrt{log_{7}11}$ , then $e^{y\ln7-x\ln7}$ is equal to $1$ .

$S_{2}: \log_{x}3>\log_{x}2$ is true for all value of $x\epsilon (0,1)\cup (1,\infty )$

$S_{3}:\left | x-2 \right |=[-\pi]$ , then $x$ is $6,-2$

$S_{4}:\log_{25}(2+\tan^{2}\theta)=0.5$ , then $\theta$ may be $\frac{4\pi }{3}\:\:or\:\:\frac{2\pi }{3}$

State, in order, whether $S_{1}, S_{2}, S_{3}, S_{4}$ are true or false
Option: 1
$T\:F\:F\:T$

Option: 2
$F\:F\:T\:T$

Option: 3
$F\:T\:F\:T$

Option: 4
$T\:T\:F\:T$

Answers (1)

best_answer

Greatest Integer Function -

$\left [ x \right ]=$ Greatest integer less than or equal to x

$\left ( for\: x\: \in\: R \right )$

- wherein

Range = Integers

$S_{1}:e^{y\:ln7-x\:ln11}=e^{ln\frac{7^{y}}{11^{x}}}=\frac{7^{y}}{11^{x}}=\frac{7^{\sqrt{log_{7}11}}}{11^{\sqrt{log_{11}7}}}$

$=\frac{7^{\frac{log_{7}11}{\sqrt{log_{7}11}}}}{11^{\sqrt{log_{11}7}}}=\frac{11^{\sqrt{log_{11}7}}}{11^{\sqrt{log_{11}7}}}=1$

$S_{2}:log_{x}3>log_{x}2\Rightarrow \:\:\:\:x>1$

$S_{3}:\left | x-2 \right |=[-\pi]$

$\left | x-2 \right |=-4\:\:\:no\:\:solution$

$S_{4}:log_{25}(2+\tan^{2}\theta)=\frac{1}{2}\Rightarrow 2+\tan^{2}\theta=5$

$\Rightarrow log_{5}(2+\tan^{2}\theta)=\frac{1}{2}\Rightarrow 2+\tan^{2}\theta=5$

$\Rightarrow \tan^{2}\theta=3\Rightarrow \tan\theta=\pm \sqrt{3}\\\\\\$

$\Rightarrow \theta\:may\:take\:value\:\frac{2\pi}{3}\:or\:\frac{4\pi}{3}$

Posted by

Devendra Khairwa

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