CAT Quadratic Equations Bible Top 10 Must Solve Quadratic Equations Questions covering 90% of Syllabus – Page 4
All 38 Quadratic Equations questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 4 of 11.
A certain amount of water was poured into a 300 litre container and the remaining portion of the container was filled with milk. Then an amount of this solution was taken out from the container which was twice the volume of water that was earlier poured into it, and water was poured to refill the container again. If the resulting solution contains 72% milk, then the amount of water, in litres, that was initially poured into the container was
Answer: 30
Solution — first steps
Step 1. Let initial water be W litres; initial milk is 300−W.
Step 2. The withdrawal is 2W litres, leaving the milk multiplied by (300−2W)/300.
Step 2. The withdrawal is 2W litres, leaving the milk multiplied by (300−2W)/300.
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For any non-zero real number \(x\), let \(f(x)+2 f\left(\frac{1}{x}\right)=3 x\). Then, the sum of all possible values of \(x\) for which \(f(x)=3\), is
Answer: B. -3
Solution — first steps
Step 1. Write A=f(x), B=f(1/x). The original equation is A+2B=3x.
Step 2. Replacing x by 1/x gives 2A+B=3/x.
Step 2. Replacing x by 1/x gives 2A+B=3/x.
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The sum of all distinct real values of \(x\) that satisfy the equation \(10^x+\frac{4}{10^x}=\frac{17}{2}\), is
Answer: A. \(2 \log _{10} 2\)
Solution — first steps
Step 1. Let t=10^x>0. The equation becomes t+4/t=17/2.
Step 2. Multiply by 2t: 2t²−17t+8=0.
Step 2. Multiply by 2t: 2t²−17t+8=0.
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The number of integer solutions of equation \( 2|x|\left(x^2+1\right)=5 x^2 \) is
Answer: 3
Solution — first steps
Step 1. Because the left side contains |x|, set t=|x|≥0 and note x²=t².
Step 2. The equation becomes 2t(t²+1)=5t².
Step 2. The equation becomes 2t(t²+1)=5t².
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Arvind travels from town A to town B, and Surbhi from town B to town A, both starting at the same time along the same route. After meeting each other, Arvind takes 6 hours to reach town B while Surbhi takes 24 hours to reach town A. If Arvind traveled at a speed of 54 km/h, then the distance, in km, between town A and town B is:
Answer: D. 972 km
Solution — first steps
The total distance between town A and B, given the speeds and time taken, is 972 km.
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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.