CAT 2025 Question Paper | Quant Slot 3

CAT Questions

CAT 2025 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 21 : Let p, q and r be three natural numbers such that their sum is 900, and r is a perfect square whose value lies between 150 and 500. If p is not less than 0.3q and not more than 0.7q, then the sum of the maximum and minimum possible values of p is

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Video Explanation

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Explanatory Answer

Given: \[ p+q+r=900 \]
and \(r\) is a perfect square between 150 and 500.
Also: \[ 0.3q\leq p\leq0.7q \]

Since: \[ q=900-r-p \]

To maximize \(p\), take the minimum possible \(r\).
The smallest perfect square greater than 150 is: \[ r=169 \]

Using the upper bound \(p\leq0.7q\):
\[ p\leq0.7(900-169-p) \]
\[ 1.7p\leq511.7 \]
\[ p\leq301 \]
Taking \(p=301\):
\[ q=900-169-301=430 \]
and \(301=0.7(430)\), so it is valid.

To minimize \(p\), take the maximum possible \(r\).
The largest perfect square below 500 is: \[ r=484 \]

Using the lower bound \(p\geq0.3q\):
\[ p\geq0.3(900-484-p) \]
\[ 1.3p\geq124.8 \]
\[ p\geq96 \]
Taking \(p=96\):
\[ q=900-484-96=320 \]
and \(96=0.3(320)\), so it is valid.

Sum of maximum and minimum possible values:
\[ 301+96=397 \]

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The answer is '397'

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