CAT 2018 Quant Questions

Question:
\(\frac{1}{{{{\log }_2}100}} - \frac{1}{{{{\log }_4}100}} + \frac{1}{{{{\log }_5}100}} - \frac{1}{{{{\log }_{10}}100}} + \frac{1}{{{{\log }_{20}}100}} - \frac{1}{{{{\log }_{25}}100}} + \frac{1}{{{{\log }_{50}}100}} = ?\)

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Correct Answer: 1

We know that \(\frac{1}{{lo{g_a}b}}\) = \(\frac{{lo{g_x}a}}{{lo{g_x}b}}\)

Therefore, we can say that \(\frac{1}{{lo{g_2}100}}\) = \(\frac{{lo{g_{10}}2}}{{lo{g_{10}}100}}\)

\( \Rightarrow \) \(\frac{1}{{lo{g_2}100}} - \frac{1}{{lo{g_4}100}} + \frac{1}{{lo{g_5}100}} - \frac{1}{{lo{g_{10}}100}} + \frac{1}{{lo{g_{20}}100}} - \frac{1}{{lo{g_{25}}100}} + \frac{1}{{lo{g_{50}}100}}\)

\( \Rightarrow \) \(\frac{{lo{g_{10}}2}}{{lo{g_{10}}100}}\)-\(\frac{{lo{g_{10}}4}}{{lo{g_{10}}100}}\)+\(\frac{{lo{g_{10}}5}}{{lo{g_{10}}100}}\)-\(\frac{{lo{g_{10}}10}}{{lo{g_{10}}100}}\)+\(\frac{{lo{g_{10}}20}}{{lo{g_{10}}100}}\)-\(\frac{{lo{g_{10}}25}}{{lo{g_{10}}100}}\)+\(\frac{{lo{g_{10}}50}}{{lo{g_{10}}100}}\)

We know that \(lo{g_{10}}100 = 2\)

\( \Rightarrow \) \(\frac{1}{2}*[lo{g_{10}}2 - lo{g_{10}}4 + lo{g_{10}}5 - lo{g_{10}}10 + lo{g_{10}}20 - lo{g_{10}}25 + lo{g_{10}}50]\)

\( \Rightarrow \) \(\frac{1}{2}*[lo{g_{10}}\frac{{2*5*20*50}}{{4*10*25}}]\)

\( \Rightarrow \) \(\frac{1}{2}*[lo{g_{10}}10]\)

\( \Rightarrow \) \(\frac{1}{2}\)

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CAT 2018 Quant Questions
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