CAT Maxima and Minima Bible Top 10 Must Solve Maxima and Minima Questions covering 90% of Syllabus – Page 5
All 40 Maxima and Minima questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 5 of 9.
If a certain amount of money is divided equally among \(n\) persons, each one receives Rs 352 . However, if two persons receive Rs 506 each and the remaining amount is divided equally among the other persons, each of them receive less than or equal to Rs 330 . Then, the maximum possible value of \(n\) is
Answer: 16
Solution — first steps
Step 1. Total money is 352n.
Step 2. After two people receive 506 each, the remaining n−2 people each get (352n−1012)/(n−2).
Step 2. After two people receive 506 each, the remaining n−2 people each get (352n−1012)/(n−2).
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Let \( n \) be any natural number such that \( 5^{n-1} < 3^{n+1} \).
The least integer value of \( m \) that satisfies \( 3^{n+1} < 2^{n+m} \) for all such \( n \) is?
The least integer value of \( m \) that satisfies \( 3^{n+1} < 2^{n+m} \) for all such \( n \) is?
Answer: C. 5
Solution — first steps
The least value of \( m \) is 5. For \( n = 5 \), we need \( 3^6 < 2^{5 + m} \).
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Let \( n \) and \( m \) be two positive integers such that there are exactly 41 integers greater than \( 8^m \) and less than \( 8^n \), which can be expressed as powers of 2. Then, the smallest possible value of \( n + m \) is:
Answer: C. 16
Solution — first steps
Answer: C. 16 Detailed Solution: Given that there are 41 integers between \( 8^m \) and \( 8^n \), which are powers of 2.
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The population of a town in 2020 was 100,000. The population decreased by \( y\% \) from 2020 to 2021, and increased by \( x\% \) from 2021 to 2022.
If the population in 2022 was greater than in 2020 and \( x - y = 10 \), what was the lowest possible population in 2021?
If the population in 2022 was greater than in 2020 and \( x - y = 10 \), what was the lowest possible population in 2021?
Answer: B. 73,000
Solution — first steps
The lowest population in 2021 is 73,000. Using the formula for successive percentage change, the population reduces by 27%, giving \( 73,000 \) as the minimum.
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Two collectors A and B collect coins weekly in the ratio 3:4. If A collects a multiple of 7 in 5 weeks and B collects a multiple of 24 in 3 weeks, what is the minimum number of coins collected by A in one week?
Answer: B. 42
Solution — first steps
The minimum number of coins collected by A in one week is 42. The least common multiple of the constraints gives \( x = 14 \), making \( 3x = 42 \).
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Last day solving checklist
- Name the unknown. Include its unit and state whether it is a count, amount, rate or parameter.
- Form the relationships. Translate each given condition into an equation or restriction.
- Choose the method. Use the formula, substitution, elimination, a sign chart or a bound appropriate to this concept.
- Check restrictions. Check integer counts, signs, nonzero denominators, domains and all original conditions.
- Verify the answer. Substitute into the original question; for an extreme, show a feasible example reaches it.
All 16 repeating CAT PYQs and how much each adds
| CAT PYQ | Repeating CAT PYQ | Questions | Representative | Cumulative questions | Cumulative share |
|---|---|---|---|---|---|
| 1 | Integer allocation from linear totals and bounds | 9 | CAT 2022 S1 Q8 | 9 | 16.67% |
| 2 | Divisibility and factor constraints with feasible candidates | 9 | CAT 2022 S1 Q4 | 18 | 33.33% |
| 3 | Ordered or balanced integer values with fixed totals | 6 | CAT 2021 S3 Q7 | 24 | 44.44% |
| 4 | Smallest integer crossing an exponential threshold | 6 | CAT 2021 S3 Q20 | 30 | 55.56% |
| 5 | AM GM minimum or maximum with equality | 4 | CAT 2021 S3 Q19 | 34 | 62.96% |
| 6 | Set overlap extrema with attainability | 3 | CAT 2022 S1 Q17 | 37 | 68.52% |
| 7 | Digit restrictions and positional extrema | 3 | CAT 2021 S2 Q12 | 40 | 74.07% |
| 8 | Compare finitely many algebraic parameter cases | 3 | CAT 2023 S2 Q21 | 43 | 79.63% |
| 9 | Quadratic vertex by completing the square | 2 | CAT 2021 S3 Q22 | 45 | 83.33% |
| 10 | Extreme integer parameter from a discriminant | 2 | CAT 2023 S1 Q7 | 47 | 87.04% |
| 11 | Endpoint extrema of piecewise or linear expressions | 2 | CAT 2022 S1 Q22 | 49 | 90.74% |
| 12 | Largest common term of APs | 1 | CAT 2023 S2 Q22 | 50 | 92.59% |
| 13 | Bound a logarithmic condition and construct equality | 1 | CAT 2024 S2 Q9 | 51 | 94.44% |
| 14 | Maximise an integer representation of a surd | 1 | CAT 2024 S1 Q2 | 52 | 96.30% |
| 15 | Minimum labour cost with integer paid days | 1 | CAT 2025 S1 Q18 | 53 | 98.15% |
| 16 | Maximum admissible percentage decline | 1 | CAT 2023 S3 Q10 | 54 | 100% |
Based on CAT Actual Quant Papers.