JEE (Main) · Mathematics, Class 11
Quadratic Equations: JEE Main PYQ, 2020–2025
Quadratic Equations (Mathematics, Class 11) has been asked 112 times in the 111 JEE (Main) shift papers from 2020–2025: about 1 questions a paper, 4% of all Mathematics questions. It is the 16th most-asked of 19 Mathematics chapters. The 2025 shifts had 15 from it in all. Last checked 28 October 2026.
Questions per year, all shifts
Year by year.
| Year | Questions |
|---|---|
| 2020 | 15in 14 papers |
| 2021 | 29in 26 papers |
| 2022 | 21in 24 papers |
| 2023 | 12in 12 papers |
| 2024 | 20in 18 papers |
| 2025 | 15in 17 papers |
JEE (Main) is sat in many shifts a year, each with its own paper; the counts add up every shift. 2021–2024 papers had 30 questions a subject, the others 25.
Inside the chapter
Topics asked most.
- Roots in real and complex systems45
- Relations between roots and coefficients42
- Nature of roots19
- Forming equations from given roots6
Try it
10 Quadratic Equations questions to practise.
Our own questions, written in the JEE Main pattern and checked, each with an explanation. Pick an option to see the answer.
Q1 · Mathematics · Quadratic Equations
The sum of the roots of a quadratic equation is −3/2 and their product is 1/2. Which of the following is an equation with integer coefficients having these roots?
Answer: (A)
The equation is x² − (sum)x + product = 0, which is x² + (3/2)x + 1/2 = 0. Multiplying by 2 clears the fractions and gives 2x² + 3x + 1 = 0.
Q2 · Mathematics · Quadratic Equations
The real numbers x satisfying |x − 1|² + |x − 1| − 6 = 0 are the real solutions of this equation. What is the sum of the squares of all its real solutions?
Answer: (A)
Put t = |x − 1|, so t ≥ 0. The equation becomes t² + t − 6 = 0, which factorises as (t + 3)(t − 2) = 0, so t = 2 or t = −3. Since a modulus cannot be negative, only t = 2 is valid. Then |x − 1| = 2 gives x − 1 = 2 or x − 1 = −2, so x = 3 or x = −1. The sum of squares is 3² + (−1)² = 10. Do not accept t = −3 (that adds a false solution, giving 13). Do not drop the negative branch of the modulus (that gives 9). Do not report the sum of the solutions, 2.
Q3 · Mathematics · Quadratic Equations
If α and β are the roots of x² − 4x + 2 = 0, the quadratic equation with integer coefficients (and positive leading coefficient) whose roots are α + 1/β and β + 1/α is:
Answer: (A)
For roots α + 1/β and β + 1/α, find the sum and the product in terms of S = α+β = 4 and P = αβ = 2. Sum = S + (1/α + 1/β) = S + S/P = 4 + 2 = 6. Product = (α + 1/β)(β + 1/α) = αβ + 1 + 1 + 1/(αβ) = P + 2 + 1/P = 2 + 2 + 1/2 = 9/2. The equation is x² − 6x + 9/2 = 0, which becomes 2x² − 12x + 9 = 0.
Q4 · Mathematics · Quadratic Equations
If α and β are the roots of 2x² − 5x + 3 = 0, which equation has roots 1/α and 1/β?
Answer: (A)
From 2x² − 5x + 3 = 0, α + β = 5/2 and αβ = 3/2. For the reciprocals, the sum is (α + β)/(αβ) = (5/2)/(3/2) = 5/3 and the product is 1/(αβ) = 2/3. The equation is x² − (5/3)x + 2/3 = 0, or 3x² − 5x + 2 = 0. Reversing the order of the coefficients gives the same result.
Q5 · Mathematics · Quadratic Equations
Let z₁ and z₂ be the roots of the equation z² − (2 + i)z + (−1 + 7i) = 0 over the complex numbers (i² = −1). What is |z₁|² + |z₂|²?
Answer: (A)
The discriminant is (2 + i)² − 4(−1 + 7i) = 3 + 4i + 4 − 28i = 7 − 24i. Since (4 − 3i)² = 16 − 9 − 24i = 7 − 24i, the square root is ±(4 − 3i). The roots are z = [(2 + i) ± (4 − 3i)]/2, which gives 3 − i and −1 + 2i. Check: the sum is 2 + i and the product is −1 + 7i. Then |3 − i|² = 10 and |−1 + 2i|² = 5, so the total is 15.
Q6 · Mathematics · Quadratic Equations
For real p, consider E1: x² + 2px + 3p + 4 = 0 and E2: x² − 2px + 5p − 6 = 0. Treat equal roots as real roots. For how many integer values of p does exactly one of E1 and E2 have real roots?
Answer: (A)
For E1: D/4 = p² − 3p − 4 = (p − 4)(p + 1) ≥ 0, so p ≤ −1 or p ≥ 4. For E2: D/4 = p² − 5p + 6 = (p − 2)(p − 3) ≥ 0, so p ≤ 2 or p ≥ 3. The E1 set lies entirely inside the E2 set, so exactly one equation has real roots when E2 is real and E1 is not, i.e. −1 < p < 4 with p ≤ 2 or p ≥ 3. That gives −1 < p ≤ 2 or 3 ≤ p < 4, with integers 0, 1, 2, 3. The count is 4.
Q7 · Mathematics · Quadratic Equations
Let p and q be real numbers with q ≠ 0. What is the nature of the roots of x² + px − q² = 0?
Answer: (C)
D = p² − 4(1)(−q²) = p² + 4q². Since p² ≥ 0 and 4q² > 0 (as q ≠ 0), we get D > 0 for all real p. So the roots are always real and distinct. This also follows from the product of roots being −q² < 0, which gives opposite signs.
Q8 · Mathematics · Quadratic Equations
For which real value of k does the equation x² − (k² − 5k + 6)x + (k² − 9) = 0 have two distinct real roots that are negatives of each other?
Answer: (A)
Roots that are negatives of each other have sum zero, so k² − 5k + 6 = 0, which gives k = 2 or k = 3. The roots must also be distinct, so the product must be negative. Then the roots are ±√(−product). For k = 2, the product is k² − 9 = −5 < 0 and the roots are ±√5. For k = 3, the product is 0 and the roots are 0, 0, which are not distinct. So only k = 2 works.
Q9 · Mathematics · Quadratic Equations
Let α and β be the roots of x² − 6x − 2 = 0, and let aₙ = αⁿ − βⁿ for n ≥ 1. Then the value of (a₁₀ − 2a₈)/(2a₉) is:
Answer: (A)
Each root satisfies t² = 6t + 2. Multiply by t⁸ to get t¹⁰ = 6t⁹ + 2t⁸. Apply this to t = α and t = β and subtract: α¹⁰ − β¹⁰ = 6(α⁹ − β⁹) + 2(α⁸ − β⁸), so a₁₀ = 6a₉ + 2a₈. Then a₁₀ − 2a₈ = 6a₉, and (a₁₀ − 2a₈)/(2a₉) = 6a₉/(2a₉) = 3.
Q10 · Mathematics · Quadratic Equations
A quadratic equation with leading coefficient 1 has roots 0 and 7. The equation is:
Answer: (A)
Sum of roots = 0 + 7 = 7 and product = 0 × 7 = 0. So x² − 7x + 0 = 0, or x² − 7x = 0. Check: x(x − 7) = 0 gives x = 0 or x = 7.
Free with an account
All 112 real JEE Main questions from Quadratic Equations.
The questions from the actual papers, with the official answer and an explanation for each — chapter by chapter, or as full timed papers.
Solve them freeQuestions
About Quadratic Equations in JEE Main.
How many questions come from Quadratic Equations in JEE Main?
Quadratic Equations (Mathematics, Class 11) has been asked 112 times in the 111 JEE (Main) shift papers from 2020–2025: about 1 questions a paper, 4% of all Mathematics questions. It is the 16th most-asked of 19 Mathematics chapters. The 2025 shifts had 15 from it in all.
Which topics of Quadratic Equations are asked most in JEE Main?
Roots in real and complex systems (45), Relations between roots and coefficients (42), Nature of roots (19), Forming equations from given roots (6), counted over the papers from 2020–2025.
Is Quadratic Equations important for JEE Main 2027?
It is in the syllabus in force, so it can be asked. By past papers it ranks 16 of 19 in Mathematics; use that to decide how much time it gets, not whether to study it.
Where can I solve Quadratic Equations JEE Main previous year questions?
On Nikaalle, free: sign up and practise every past JEE Main question from Quadratic Equations with an explanation for each, or sit the full papers timed. The ten questions on this page are our own, written in the same pattern.
Your JEE Main 2027 plan, chapter by chapter.
A week-wise plan that gives the heavy chapters their time. Free, from a non-profit trust.