CAT Inequalities Bible Top 10 Must Solve Inequalities Questions covering 80% of Syllabus – Page 9
All 38 Inequalities questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 9 of 11.
Refer to the passage above ↑
If \(c = \dfrac{16x}{y} + \dfrac{49y}{x}\) for some non-zero real numbers \(x\) and \(y\), then \(c\) cannot take the value
Answer: B. -50
Solution — first steps
Step 1. Let t=x/y≠0; c=16t+49/t.
Step 2. If t>0, AM–GM gives c≥2√(16×49)=56.
Step 2. If t>0, AM–GM gives c≥2√(16×49)=56.
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Refer to the passage above ↑
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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Refer to the passage above ↑
Suppose \(k\) is any integer such that the equation \(2x^{2} + kx + 5 = 0\) has no real roots and the equation \(x^{2} + (k - 5)x + 1 = 0\) has two distinct real roots for \(x\). Then, the number of possible values of \(k\) is
Answer: A. 9
Solution — first steps
Step 1. No real roots in first quadratic requires k²−400.
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Refer to the passage above ↑
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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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 21 repeating CAT PYQs and how much each adds
| CAT PYQ | Repeating CAT PYQ | Questions | Representative | Cumulative questions | Cumulative share |
|---|---|---|---|---|---|
| 1 | Exponential bounds and smallest integer periods | 7 | CAT 2021 S2 Q15 | 7 | 12.73% |
| 2 | Discriminant bounds for real roots or ranges | 6 | CAT 2022 S1 Q5 | 13 | 23.64% |
| 3 | Linear allocation bounds with nonnegative counts | 6 | CAT 2022 S1 Q8 | 19 | 34.55% |
| 4 | Absolute value intervals and sign cases | 5 | CAT 2021 S1 Q2 | 24 | 43.64% |
| 5 | Ordered values and average bounds | 5 | CAT 2022 S2 Q8 | 29 | 52.73% |
| 6 | Logarithmic bounds and monotonicity | 4 | CAT 2024 S2 Q9 | 33 | 60.00% |
| 7 | Rational sign charts and excluded poles | 3 | CAT 2021 S1 Q1 | 36 | 65.45% |
| 8 | Set overlap bounds and feasible endpoints | 3 | CAT 2022 S1 Q17 | 39 | 70.91% |
| 9 | Finite integer enumeration within bounds | 3 | CAT 2025 S1 Q17 | 42 | 76.36% |
| 10 | Floor intervals and endpoint inclusion | 2 | CAT 2022 S1 Q12 | 44 | 80.00% |
| 11 | Factorial size bounds | 1 | CAT 2022 S2 Q13 | 45 | 81.82% |
| 12 | Bounds on eligible AP terms | 1 | CAT 2022 S3 Q17 | 46 | 83.64% |
| 13 | Remainders with an upper population bound | 1 | CAT 2022 S3 Q19 | 47 | 85.45% |
| 14 | Polygon side inequality | 1 | CAT 2022 S3 Q22 | 48 | 87.27% |
| 15 | Speed bound with a palindrome restriction | 1 | CAT 2023 S1 Q8 | 49 | 89.09% |
| 16 | Translate time bounds to work rate bounds | 1 | CAT 2023 S3 Q8 | 50 | 90.91% |
| 17 | Quadratic percentage growth inequality | 1 | CAT 2023 S3 Q10 | 51 | 92.73% |
| 18 | Geometric regions defined by inequalities | 1 | CAT 2024 S1 Q4 | 52 | 94.55% |
| 19 | Square bound with an equality condition | 1 | CAT 2024 S1 Q8 | 53 | 96.36% |
| 20 | Compare quadratic extrema | 1 | CAT 2025 S1 Q8 | 54 | 98.18% |
| 21 | Range by completing the square and reciprocal bounds | 1 | CAT 2021 S2 Q13 | 55 | 100% |
Based on CAT Actual Quant Papers.