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.