JEE Main Maths · Quadratic Equations PYQ
JEE Main Quadratic Equations PYQ (2002–2025)
Quadratic Equations is a P2 Algebra chapter — ~4% JEE Main weightage with 1 question per session. Foundation for Complex Numbers (quadratic with negative discriminant) and Calculus (optimization). Short chapter with formulaic problems.
Quadratic Equations PYQs from 2002 to 2025 covering roots, discriminant, common roots, quadratic inequalities and location of roots. Every solution uses KaTeX-rendered math.
Quadratic Equations & Inequalities at a Glance
Key Sub-Topics & What's Tested
Quadratic Equation Basics
Standard form ax² + bx + c = 0, roots α and β, sum α+β = -b/a, product αβ = c/a, discriminant D = b² - 4ac.
Nature of Roots
D > 0 → real distinct roots. D = 0 → equal real roots. D < 0 → complex conjugate roots. Rational/irrational roots condition.
Sum & Product of Roots
If α, β are roots: α+β = -b/a, αβ = c/a. Quadratic with roots α, β: x² - (α+β)x + αβ = 0.
Common Roots of Two Quadratics
One common root between ax² + bx + c = 0 and a'x² + b'x + c' = 0: (ca' - c'a)² = (ab' - a'b)(bc' - b'c). Both roots common: a/a' = b/b' = c/c'.
Quadratic Inequalities
ax² + bx + c > 0 or < 0 depending on sign of a and discriminant. Sign of parabola between/outside roots.
Location of Roots
Conditions for roots to lie in given interval (both positive, both negative, one in (a,b)), using signs of f(a), f(b), and discriminant.
Transformation of Roots
Forming new quadratic whose roots are related to original: reciprocals, squares, shifted by constant, etc.
Biquadratic & Rational Equations
Equations reducible to quadratic form (substitute t = x²), rational equations reducible to quadratic after cross-multiplication.
Question Type Distribution
| Question Type | Share (approx) | Example Pattern |
|---|---|---|
| Finding Roots / Discriminant | 25% | For x² + kx + 1 = 0 to have real roots, find range of k. |
| Common Roots | 20% | Find k such that x² - 3x + 2 = 0 and x² - kx + 8 = 0 have a common root. |
| Quadratic Inequalities | 20% | Solve x² - 5x + 6 < 0 for x. |
| Transformed Roots | 15% | If α, β are roots of x² + 2x + 5 = 0, find quadratic whose roots are 1/α and 1/β. |
| Location / Sign Conditions | 10% | Find k such that both roots of x² - (k-2)x + 1 = 0 lie in (1, 3). |
| Biquadratic Substitution | 10% | Solve x⁴ - 5x² + 4 = 0 using t = x² substitution. |
How to Solve Quadratic Equations & Inequalities PYQs
- 1Discriminant first for 'nature of roots' questions. D = b² - 4ac. Sign of D tells you whether roots are real, distinct, or complex. One-line check.
- 2Vieta's formulas: sum and product. α + β = -b/a. αβ = c/a. Don't find individual roots if the question asks for sums/products — use Vieta directly.
- 3For inequalities, sketch the parabola. Find roots, determine if parabola opens up (a > 0) or down (a < 0). Solution interval is intuitive from graph.
- 4Location of roots: evaluate f at interval endpoints. For both roots in (a,b), f(a)·f(b) > 0 (same sign) AND D ≥ 0 AND vertex x-coordinate lies in (a,b).
- 5Transformed roots: use Vieta on new roots. For roots 1/α, 1/β of new quadratic: sum = (1/α + 1/β) = (α+β)/(αβ), product = 1/(αβ). Build new quadratic.
Common Mistakes That Cost Marks
- Confusing sum and product of roots. α + β = -b/a (note negative sign). αβ = c/a (positive). Sign error on the sum is the #1 mistake.
- Forgetting complex roots come in conjugate pairs. If z is a root of a real-coefficient quadratic, z̄ is also a root. Basic theorem.
- Wrong inequality direction after multiplying by negative. Multiplying both sides of inequality by negative flips the inequality. Common sign error.
- Missing D ≥ 0 condition in location-of-roots. Location conditions assume real roots exist. Without D ≥ 0, there are no real roots to locate.
- Applying biquadratic formula without verifying range of t. After t = x², only t ≥ 0 gives real x. Negative t roots don't yield real x values.
Related JEE Main Practice
Frequently asked questions
How many Quadratic PYQs should I solve?
Target 40–60 PYQs across 2010–2025. Short chapter, formulaic problems — 30 hours of practice builds strong pattern recognition.
Is Quadratic Equations connected to other chapters?
Yes — Complex Numbers (quadratic with negative discriminant), Calculus (optimization via vertex), Polynomial Inequalities (higher-degree extensions). Foundational.
Do I need to memorise discriminant formula?
Yes — D = b² - 4ac. Memorise this plus Vieta's formulas. Takes 5 minutes, saves hours in PYQ practice.
What's the most-tested Quadratic pattern?
Finding range of k for specific root-nature condition (e.g., "for real distinct roots, k lies in..."). Appears in 2-3 PYQs per 10-year window.
Are quadratic inequalities computationally heavy?
No — sketch parabola, find roots, identify sign regions. Most inequalities solve in under 2 minutes once approach is clear.
Do I need to memorise the common-roots condition formula?
Yes — for single common root: (ca' - c'a)² = (ab' - a'b)(bc' - b'c). Memorise or know how to derive. Appears in PYQs occasionally.
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