CAT Quadratic Equations Bible Top 10 Must Solve Quadratic Equations Questions covering 90% of Syllabus – Page 2
All 38 Quadratic Equations questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 2 of 11.
The sum of all possible real values of x for which
log_(x-3)(x² - 9) = log_(x-3)(x + 1) + 2 is :
log_(x-3)(x² - 9) = log_(x-3)(x + 1) + 2 is :
Answer: C. (3+√33)/2
Solution — first steps
log_(x-3)((x²−9)/(x+1)) = 2 By definition of log : (x²−9)/(x+1) = (x−3)² Now equating : (x²−9)/(x+1) = x²−6x+9 x²−9 = (x²−6x+9)(x+1) Solve the cubic equation: x³−6x²+3x+18 = 0 Factor the cubic equation: (x−3)(x²−3x−6) = 0 Solve x²−3x−6 = 0 using the quadratic formula: x = (3 ± √33)/2 Sum of solutions: Sum = (3 + √33)/2 Option 3 is the answer
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For real values of x, the range of the function f(x) = (2x - 3)/(2x² + 4x - 6)
Answer: C. (-oo , 1/8] U [1/2 , oo)
Solution — first steps
Correct Answer: 3 We can factor the denominator: 2x² + 4x - 6 = 2(x + 3)(x - 1) So, the function becomes: f(x) = (2x - 3) / (2(x + 3)(x - 1)) , thus x ≠ -3 , 1 Range of f(x) f(0) = 1/2 f(1) = not possible f(2) = 1/10 f(3) = 1/8 Only option 3 includes f(0) and f(3) Thus option 3 is the answer
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f(x) = (x²+ 3x)(x² + 3x + 2), then the sum of all real roots of the equation √f(x) + 1 = 9701, is
Answer: C. -3
Solution — first steps
(a + 2) = a² + 2a √fx + 1 = √a² + 2a + 1 = 9701 √(a+1)² = 9701 , a = 9700 Thus x² + 3x = 9700 , x² + 3x - 9700 = 0 Sum of roots = -b/a = -3/1 = -3 Thus option 3 is the answer
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ABCD is a trapezium in which AB is parallel to DC and AD is perpendicular to AB and AB = 3DC. If a circle inscribed in a trapezium touching all the sides has a radius 3cm , then the area in sq cm , of the trapezium is :
Answer: A. 48
Solution — first steps
Given: Let the trapezium be ABCD. AB || DC (The parallel sides are the bases).
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Two places A and B are 45 kms apart and connected by a straight road. Anil goes from A to B while Sunil goes from B to A. Starting at the same time, they cross each other in exactly 1 hour 30 minutes. If Anil reaches B exactly 1 hour 15 minutes after Sunil reaches A, the speed of Anil, in km per hour, is
Answer: A. 12
Solution — first steps
5s = 45 \Rightarrow a + s = 30 \quad \text{(Equation 1)} \) It's also given: Anil takes 1 hour 15 minutes more than Sunil to reach destination. 25as \) Multiply entire equation by 4: \( 6s^2 = 6a^2 + 5as \) Substitute from Equation 1: \( s = 30 - a \) \( 6(30 - a)^2 = 6a^2 + 5a(30 - a) \) Left: \( 6(900 - 60a + a^2) = 5400 - 360a + 6a^2 \) Right: \( 6a^2 + 150a - 5a^2 = a^2 + 150a \) So: \( 5400 - 360a = 150a - 5a^2 \) Rearranging: \( 5400 - 510a + 5a^2 = 0 \) Divide the equation by 5: \( a^2 - 102a + 1080 = 0 \) Solve using quadratic formula: \( a = \frac{102 \pm \sqrt{102^2 - 4 \cdot 1080}}{2} = \frac{102 \pm \sqrt{6084}}{2} = \frac{102 \pm 78}{2} \) So, \( a = \frac{102 + 78}{2} = 90 \) or \( a = \frac{102 - 78}{2} = 12 \) Final Answer: \( \boxed{12 \text{ km/h}} \)
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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.