CAT 2023 Question Paper | Quant Slot 1

CAT Questions

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 22 : A lab experiment measures the number of organisms at 8 am every day. Starting with 2 organisms on the first day, the number of organisms on any day is equal to 3 more than twice the number on the previous day. If the number of organisms on the \(n^{\text {th }}\) day exceeds one million, then the lowest possible value of \(n\) is

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Video Explanation

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Explanatory Answer

1 Million = 10⁶
Population on…
Day 1: 2
Day 2: \( 2 ^ { 2 } + 3 \)
Day 3: \( 2 ^ { 3 } + 3 \cdot 2 + 3 \)
Day 4: \( 2 ^ { 4 } + 3 \cdot 2 ^ { 2 } + 3 \cdot 2 + 3 \)
Day n: \( 2 ^ { n } + 3 \left( 2 ^ { n - 2 } + 2 ^ { n - 3 } + \cdots + 1 \right) \)
= \( 2 ^ { n } + 3 \left( 2 ^ { n - 1 } - 1 \right) \)
= \( 2 ^ { n - 1 } ( 2 + 3 ) - 3 \)
= \( 2 ^ { n - 1 } ( 4 + 1 ) - 3 \)
= \( 2 ^ { n + 1 } + 2 ^ { n - 1 } - 3 \)
\( 10 ^ { 6 } = 2 ^ { 6 } \cdot 5 ^ { 6 } \)
\( 10 ^ { 6 } > 2 ^ { 6 } \cdot 4 ^ { 6 } \)
\( 10 ^ { 6 } > 2 ^ { 6 } \cdot 2 ^ { 12 } \)
\( 10 ^ { 6 } > 2 ^ { 18 } \)
Putting n = 18 in \( 2 ^ { n + 1 } + 2 ^ { n - 1 } - 3 \), we get 655357, which is less than 10⁶
On the 19^th day the population will be more than double the population on the 18^th day, which is more than 2 × 655357, which is definitely more than 10⁶.

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The answer is '19'