CAT Maxima and Minima Bible Top 10 Must Solve Maxima and Minima Questions covering 90% of Syllabus – Page 6
All 40 Maxima and Minima questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 6 of 9.
For some real numbers \(a\) and \(b\), the system of equations \(x+y=4\) and \((a+5) x+\left(b^2-15\right) y=8 b\) has infinitely many solutions for \(x\) and \(y\). Then, the maximum possible value of \(a b\) is
Answer: B. 33
Solution — first steps
Step 1. For infinitely many solutions, the second equation must be a scalar multiple of x+y=4.
Step 2. Since its right side is 8b, the scalar is 2b.
Step 2. Since its right side is 8b, the scalar is 2b.
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A quadratic equation \(x^2+b x+c=0\) has two real roots. If the difference between the reciprocals of the roots is \(\frac{1}{3}\), and the sum of the reciprocals of the squares of the roots is \(\frac{5}{9}\), then the largest possible value of \((b+c)\) is
Answer: 9
Solution — first steps
Step 1. Let u=1/r and v=1/s for the two real roots r,s.
Step 2. Given (u−v)²=1/9 and u²+v²=5/9, so 2uv=4/9 and uv=2/9.
Step 2. Given (u−v)²=1/9 and u²+v²=5/9, so 2uv=4/9 and uv=2/9.
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The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is
Answer: 468
Solution — first steps
Step 1. A number has15 positive divisors only with prime-exponent pattern14 or(4,2), since15=15 or5×3.
Step 2. Thus the forms are p^14 or p^4q² with different primes.
Step 2. Thus the forms are p^14 or p^4q² with different primes.
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Let \(n\) be any natural number such that \(5^{n-1} \lt 3^{n+1}\). Then, the least integer value of \(m\) that satisfies \(3^{n+1} \lt 2^{n+m}\) for each such \(n\), is
Answer: 5
Solution — first steps
Step 1. Rewrite the first inequality as (5/3)^(n−1)9. Since the base exceeds 1, allowed n are 1 through 5.
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A fruit seller has a stock of mangoes, bananas and apples with at least one fruit of each type. At the beginning of a day, the number of mangoes make up 40% of his stock. That day, he sells half of the mangoes, 96 bananas and 40% of the apples. At the end of the day, he ends up selling 50% of the fruits. The smallest possible total number of fruits in the stock at the beginning of the day is
Answer: 340
Solution — first steps
Step 1. Let total fruit count be T; mangoes=2T/5. Write apples A and bananas B, so A+B=3T/5.
Step 2. Total sold is T/5+96+2A/5=T/2. Therefore A=3T/4−240 and B=240−3T/20.
Step 2. Total sold is T/5+96+2A/5=T/2. Therefore A=3T/4−240 and B=240−3T/20.
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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.