CAT Quadratic Equations Bible Top 10 Must Solve Quadratic Equations Questions covering 90% of Syllabus – Page 9
All 38 Quadratic Equations questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 9 of 11.
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 ↑
If \((3 + 2\sqrt{2})\) is a root of the equation \(ax^{2} + bx + c = 0\) and \((4 + 2\sqrt{3})\) is a root of the equation \(ay^{2} + my + n = 0\) where \(a, b, c, m\) and \(n\) are integers, then the value of \(\left(\dfrac{b}{m} + \dfrac{c - 2b}{n}\right)\) is
Answer: D. 4
Solution — first steps
Step 1. Integer coefficients imply conjugate roots 3±2√2 in the first quadratic.
Step 2. Their sum 6 and product 1 give b=−6a, c=a.
Step 2. Their sum 6 and product 1 give b=−6a, c=a.
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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 15 repeating CAT PYQs and how much each adds
| CAT PYQ | Repeating CAT PYQ | Questions | Representative | Cumulative questions | Cumulative share |
|---|---|---|---|---|---|
| 1 | Quadratic modelling of counts journeys and work rates | 9 | CAT 2021 S1 Q16 | 9 | 17.65% |
| 2 | Root sums products and conjugate roots | 7 | CAT 2021 S2 Q19 | 16 | 31.37% |
| 3 | Successive percentages replacement and price changes | 6 | CAT 2021 S2 Q4 | 22 | 43.14% |
| 4 | Discriminant conditions and integer parameters | 4 | CAT 2022 S1 Q5 | 26 | 50.98% |
| 5 | Modulus or square root branches producing quadratics | 4 | CAT 2021 S1 Q8 | 30 | 58.82% |
| 6 | Exponential substitutions and quadratic exponents | 4 | CAT 2023 S2 Q2 | 34 | 66.67% |
| 7 | Vertex repeated root and perfect square forms | 3 | CAT 2021 S3 Q22 | 37 | 72.55% |
| 8 | Logarithms converted to quadratic equations | 3 | CAT 2023 S1 Q1 | 40 | 78.43% |
| 9 | Shared roots eliminated between equations | 2 | CAT 2024 S1 Q20 | 42 | 82.35% |
| 10 | Rational expression range through a discriminant | 2 | CAT 2022 S3 Q15 | 44 | 86.27% |
| 11 | Grouped identities reduce coupled equations | 2 | CAT 2021 S2 Q3 | 46 | 90.20% |
| 12 | Geometric length constraints producing quadratics | 2 | CAT 2021 S3 Q11 | 48 | 94.12% |
| 13 | Balanced AP totals producing a quadratic index | 1 | CAT 2025 S1 Q4 | 49 | 96.08% |
| 14 | GP terms related by a quadratic product | 1 | CAT 2025 S2 Q8 | 50 | 98.04% |
| 15 | Consistency of a linear system gives a quadratic parameter | 1 | CAT 2023 S3 Q3 | 51 | 100% |
Based on CAT Actual Quant Papers.