CAT 2023 Question Paper | Quant Slot 3

CAT 2023 Quant was dominated by Arithmetic followed by Algebra. 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 1 : For a real number $x$, if $\frac{1}{2}, \frac{\log _3\left(2^x-9\right)}{\log _3 4}$, and $\frac{\log _5\left(2^x+\frac{17}{2}\right)}{\log _5 4}$ are in an arithmetic progression, then the common difference is

Video Explanation

Explanatory Answer

$\frac { 1 } { 2 } , \frac { \log _ { 3 } \left( 2 ^ { x } - 9 \right) } { \log _ { 3 } ^ { 4 } } , \frac { \log _ { 5 } \left( 2 ^ { x } + \frac { 17 } { 2 } \right) } { \log _ { 5 } ^ { 4 } }$are in AP.
$\because \quad \frac { A } { B } = A$
$\frac { 1 } { 2 } , \left( 2 ^ { x } - 9 \right) , \left( 2 ^ { x } + \frac { 17 } { 2 } \right)$are in A.P
$2 , \left( 2 ^ { x } - 9 \right) , \left( 2 ^ { x } + \frac { 17 } { 2 } \right)$are in A.P
If $\log \log A , \log \log B , \log \log C$ all in $A . P$, then $A , B , C$ will be in $G . P$
$\therefore 2 , \left( 2 ^ { x } - 9 \right) , \left( 2 ^ { x } + \frac { 17 } { 2 } \right)$are in G.P
$\left( 2 ^ { x } - 9 \right) ^ { 2 } = 2 \times \left( 2 ^ { x } + \frac { 17 } { 2 } \right)$
$2 ^ { 2 x } + 81 - 9 \times 2 ^ { x + 1 } = 2 ^ { x + 1 } + 17$
$2 ^ { 2 x } - 10 \times 2 ^ { x + 1 } + 64 = 0$
$\left( 2 ^ { x } \right) ^ { 2 } - 20 \left( 2 ^ { x } \right) + 64 = 0$
Let $y = 2 ^ { x }$
$y ^ { 2 } - 20 y + 64 = 0$
$y ^ { 2 } - 16 y - 4 y + 64 = 0$
$y = 16$ or $y = 4$
$2 ^ { x } = 16$ or $2 ^ { x } = 4$
$x = 4$ or $x = 2$
if $x = 2 ; \left( 2 ^ { 2 } - 9 \right) = ( - 5 )$
But, since the argument of the log can't be negative, $x = 4$
∴ The common ratio of the G.P is $\frac { 2 ^ { x } - 9 } { 2 } = \frac { 2 ^ { 4 } - 9 } { 2 } = \frac { 16 - 9 } { 2 } = \frac { 7 } { 2 }$
$\therefore$ The common difference of the $A P = \left( \frac { 7 } { 2 } \right)$

The question is " For a real number $x$, if $\frac{1}{2}, \frac{\log _3\left(2^x-9\right)}{\log _3 4}$, and $\frac{\log _5\left(2^x+\frac{17}{2}\right)}{\log _5 4}$ are in an arithmetic progression, then the common difference is "

The answer is '$\log _4\left(\frac{7}{2}\right)$'

Choice C is the correct answer.

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