CAT Inequalities Bible Top 10 Must Solve Inequalities Questions covering 80% of Syllabus – Page 4
All 38 Inequalities questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 4 of 11.
Aman invests Rs 4000 in a bank at a certain rate of interest, compounded annually. If the ratio of the value of the investment after 3 years to the value after 5 years is 25:36, then the minimum number of years required for the value to exceed Rs 20000 is _____
Answer: 9
Solution — first steps
16 → n=9
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In a group of 250 students, the percentage of girls was at least 44% and at most 60%. The rest of the students were boys. Each student opted for either swimming or running or both. If 50% of the boys and 80% of the girls opted for swimming while 70% of the boys and 60% of the girls opted for running, then the minimum and maximum possible number of students who opted for both swimming and running, are
Answer: 72 and 80
Solution — first steps
Step 1. Let G be girls and B=250−G be boys. The bounds give 110≤G≤150.
Step 2. Because every student chooses at least one activity, inclusion–exclusion fixes the overlap: both=0.2B+0.4G.
Step 2. Because every student chooses at least one activity, inclusion–exclusion fixes the overlap: both=0.2B+0.4G.
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The equation \(x^3+(2 r+1) x^2+(4 r-1) x+2=0\) has -2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of \(r\) is
Answer: 2
Solution — first steps
Step 1. Substitution x=−2 gives zero for every r, so it does not determine r.
Step 2. Factor the polynomial as (x+2)[x²+(2r−1)x+1].
Step 2. Factor the polynomial as (x+2)[x²+(2r−1)x+1].
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In an examination, the average marks of 4 girls and 6 boys is 24. Each of the girls has the same marks while each of the boys has the same marks. If the marks of any girl is at most double the marks of any boy, but not less than the marks of any boy, then the number of possible distinct integer values of the total marks of 2 girls and 6 boys is
Answer: B. 21
Solution — first steps
Step 1. Let each girl's mark be g and each boy's mark be b. Then 4g+6b=240.
Step 2. Impose b≤g≤2b. Using b=(240−4g)/6 gives 24≤g≤240/7.
Step 2. Impose b≤g≤2b. Using b=(240−4g)/6 gives 24≤g≤240/7.
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Any non-zero real numbers x, y such that y ≠ 3 and (x/y) < ((x+3)/(y-3)) will satisfy the condition
Answer: D. If y < 0, then -x < y
Solution — first steps
Solution: We are given that (x/y) 0, or -x < y. Thus, option D is correct: if y < 0, then -x < y.
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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.