CAT 2021

Slot 1

QA

Question & Solution

$5 - \log_{10}\sqrt{1 + x} + 4 \log_{10} \sqrt{1 - x} = \log_{10} \frac{1}{\sqrt{1 - x^2}}$

We can re-write the equation as: $5-\log_{10}\sqrt{1+x}+4\log_{10}\sqrt{1-x}=\log_{10}\left(\sqrt{1+x} \times \sqrt{1-x}\right)^{-1}$

$5-\log_{10}\sqrt{1+x}+4\log_{10}\sqrt{1-x}=(-1)\log_{10}(\sqrt{1+x}) + (-1)\log_{10}(\sqrt{1-x})$

$5=-\log_{10}\sqrt{1+x}+\log_{10}\sqrt{1+x}-\log_{10}\sqrt{1-x}-4\log_{10}\sqrt{1-x}$

$5=-5\log_{10}\sqrt{1-x}$

$\sqrt{1-x}=\frac{1}{10}$

Squaring both sides: $(\sqrt{1-x})^2=\frac{1}{100}$

$\therefore x = 1 - \frac{1}{100} = \frac{99}{100}$

Hence, $100x = 100 \times \frac{99}{100} = 99$