CAT Quadratic Equations Bible Top 10 Must Solve Quadratic Equations Questions covering 90% of Syllabus – Page 3
All 38 Quadratic Equations questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 3 of 11.
The sum of all real values of \(k\) for which \(\left(\frac{1}{8}\right)^k \times\left(\frac{1}{32768}\right)^{\frac{1}{3}}=\frac{1}{8} \times\left(\frac{1}{32768}\right)^{\frac{1}{k}}\), is
Answer: B. \(-\frac{2}{3}\)
Solution — first steps
Step 1. Convert every number to base 2: 1/8=2^-3 and 1/32768=2^-15.
Step 2. The left exponent of 2 is -3k-5; the right exponent is -3-15/k.
Step 2. The left exponent of 2 is -3k-5; the right exponent is -3-15/k.
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The roots \( \alpha, \beta \) of the equation \( 3x^2 + \lambda x - 1 = 0 \)
satisfy
\[
\frac{1}{\alpha^2} + \frac{1}{\beta^2} = 15.
\]
The value of \( (\alpha^3 + \beta^3)^2 \) is:
satisfy
\[
\frac{1}{\alpha^2} + \frac{1}{\beta^2} = 15.
\]
The value of \( (\alpha^3 + \beta^3)^2 \) is:
Answer: 4
Solution — first steps
Try Each Option - Algebra Checking Each Option 🔹 Option 1: \( (\alpha^3 + \beta^3)^2 = 9 \) Let \( \alpha + \beta = s \), \( \alpha\beta = -\frac{1}{3} \) Then \( \alpha^3 + \beta^3 = s^3 - 3\alpha\beta s = s^3 + s \) Try \( s = 2 \Rightarrow \alpha^3 + \beta^3 = 8 + 2 = 10 \Rightarrow (10)^2 = 100 \neq 9 \) Try \( s = 1 \Rightarrow 1 + 1 = 2 \Rightarrow (2)^2 = 4 \neq 9 \) ❌ Option 1 is incorrect 🔹 Option 2: \( (\alpha^3 + \beta^3)^2 = 4 \) Try \( s = 1 \Rightarrow \alpha^3 + \beta^3 = 1 + 1 = 2 \Rightarrow (2)^2 = 4 \) Now check the condition: \[ \alpha^2 + \beta^2 = s^2 - 2\alpha\beta = 1 + \frac{2}{3} = \frac{5}{3}, \quad (\alpha\beta)^2 = \frac{1}{9} \] \[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{5/3}{1/9} = 15 \quad \text{✅ Satisfied} \] ✅ Option 2 is correct 🔹 Option 3: \( (\alpha^3 + \beta^3)^2 = 16 \) Try \( s = 2 \Rightarrow \alpha^3 + \beta^3 = 8 + 2 = 10 \Rightarrow (10)^2 = 100 \) Try \( s = \sqrt{16} = 4 \Rightarrow Not possible, would make cubic sum too large ❌ Option 3 is incorrect 🔹 Option 4: \( (\alpha^3 + \beta^3)^2 = 1 \) Try \( s = 0 \Rightarrow \alpha^3 + \beta^3 = 0 - 0 = 0 \Rightarrow (0)^2 = 0 \) Try \( s = 1 \Rightarrow (1 + 1)^2 = 4 \) ❌ Option 4 is incorrect ✅ Final Answer: Option 2 — \(\boxed{4}\)
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After two successive increments, Gopal’s salary became 187.5% of his initial salary. If the percentage of salary increase in the second increment was twice of that in the first increment, then the percentage of salary increase in the first increment was ?
Answer: A. 25
Solution — first steps
08 xxx
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For any non-zero real x, let f(x) + 2 f(1/x) = 3x. Find the sum of all possible values of x for which f(x) = 3.
Answer: -3
Solution — first steps
f(x) + 2 f(1/x) = 3x f(1/x) + 2 f(x) = 3 / x Multiply 2nd by 2 and subtract: 3 f(x) = (6/x) - 3x → f(x) = (2/x) - x Set f(x)=3: (2/x)-x=3 → 2 - x² = 3x → x² + 3x -2 =0 x = (-3 ± √17)/2 Sum = -3
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A 300 litre container. Water x litres, rest milk. Remove 2x litres solution, refill water. Final solution 72% milk. Find x.
Answer: 30
Solution — first steps
Set up quadratic: x² - 450x +12600=0 x=30
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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.