CAT Maxima and Minima Bible Top 10 Must Solve Maxima and Minima Questions covering 90% of Syllabus – Page 9
All 40 Maxima and Minima questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 9 of 9.
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A donation box can receive only cheques of ₹100, ₹250, and ₹500. On one good day, the donation box was found to contain exactly 100 cheques amounting to a total sum of ₹15250. Then, the maximum possible number of cheques of ₹500 that the donation box may have contained, is
Answer: 12
Solution — first steps
Step 1. Let counts of 100, 250, 500 cheques be a, b, c, summing 100.
Step 2. Subtract 100 times the count from total value: 150b+400c=5250, or 3b+8c105.
Step 2. Subtract 100 times the count from total value: 150b+400c=5250, or 3b+8c105.
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In an examination, the average marks of students in sections A and B are 32 and 60, respectively. The number of students in section A is 10 less than that in section B. If the average marks of all the students across both the sections combined is an integer, then the difference between the maximum and minimum possible number of students in section A is
Answer: 63
Solution — first steps
Step 1. Let section A count a>0, section B count a+10.
Step 2. Combined average=(32a+60(a+10))/(2a+10)=46+70/(a+5).
Step 2. Combined average=(32a+60(a+10))/(2a+10)=46+70/(a+5).
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Consider six distinct natural numbers such that the average of the two smallest numbers is 14, and the average of the two largest numbers is 28. Then, the maximum possible value of the average of these six numbers is
Answer: D. 22.5
Solution — first steps
Step 1. Order numbers a<b<c<d<e<f. Given a+b = 28 and e+f(56).
Step 2. As e<f and integers, e≤27.
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Refer to the passage above ↑
The minimum possible value of \(\dfrac{x^{2} - 6x + 10}{3 - x}\), for \(x < 3\), is
Answer: B. 2
Solution — first steps
Step 1. Put t=3−x>0.
Step 2. Numerator x²−6x+10=(x−3)²+1=t²+1.
Step 2. Numerator x²−6x+10=(x−3)²+1=t²+1.
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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.