CAT 2020 Quant Question [Slot 1] with Solution 08

Question

If y is a negative number such that \({2^{{y^2}{{\log }_3}5}} = {5^{{{\log }_2}3}}\), then y equals

  1. \({\log _2}(1/3)\)
  2. \( - {\log _2}(1/3)\)
  3. \({\log _2}(1/5)\)
  4. \( - {\log _2}(1/5)\)
Option: 1
Solution:

\({2^{{y^2}{{\log }_3}5}} = {5^{{{\log }_2}3}}\)

\({\left( {{2^{{{\log }_3}5}}} \right)^{{y^2}}} = {5^{{{\log }_2}3}}\)

\({\left( {{5^{{{\log }_3}2}}} \right)^{{y^2}}} = {5^{{{\log }_2}3}}\)

\({5^{{y^2}{{\log }_3}2}} = {5^{{{\log }_2}3}}\)

\( \Rightarrow {y^2}{\log _3}2 = {\log _2}3\)

\({y^2} = \left( {{{\log }_2}3} \right)\left( {{{\log }_2}3} \right)\)

 is negative )

\(y = {\log _2}{3^{ - 1}} = {\log _2}\frac{1}{3}\)

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CAT 2020 Quant questions with Solutions