CAT 2023 Question Paper | Quant Slot 2

CAT 2023 Quant was dominated by Algebra followed by Arithmetic. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 5 : For some positive real number $x$, if $\log _{\sqrt{3}}(x)+\frac{\log _x(25)}{\log _x(0.008)}=\frac{16}{3}$, then the value of $\log _3\left(3 x^2\right)$ is

Video Explanation

Explanatory Answer

$( x ) + \frac { 5 ^ { 2 } } { ( 0.008 ) } = \frac { 16 } { 3 }$
$0.008 = \frac { 8 } { 1000 } = \frac { 2 ^ { 3 } } { 10 ^ { 3 } } = 5 ^ { - 3 }$
$( x ) + \frac { 5 ^ { 2 } } { 5 ^ { - 3 } } = \frac { 16 } { 3 }$
$( x ) - \frac { 2 } { 3 } \frac { \log \log x } { 5 } = \frac { 16 } { 3 }$
$( x ) = \frac { 16 } { 3 } + \frac { 2 } { 3 } = \frac { 18 } { 3 } = 6$
$x = ( \sqrt { 3 } ) ^ { 6 } = 3 ^ { \frac { 6 } { 2 } } = 3 ^ { 3 }$
$3 x ^ { 2 } = 3 \times 3 ^ { 6 } = 3 ^ { 7 }$
$\therefore \left( 3 x ^ { 2 } \right) = \left( 3 ^ { 7 } \right) = 7$

The question is " For some positive real number $x$, if $\log _{\sqrt{3}}(x)+\frac{\log _x(25)}{\log _x(0.008)}=\frac{16}{3}$, then the value of $\log _3\left(3 x^2\right)$ is "

Hence, the answer is '7'

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