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Quadratic Equations

JEE Mains Maths. Try each one first, then open its answer.

Top 10 JEE Mains PYQs to practise first

The questions and kinds of question asked most often come first. Each one says how often and in which papers.

  1. Question 1

    This type asked 10 times: 2021 Paper 2, 2021 Paper 15, 15 Apr 2023, 30 Jan 2024, 5 Apr 2024, 8 Apr 2024, 24 Jan 2025, 7 Apr 2025, 22 Jan 2026 S1, 24 Jan 2026 S1. This one: 24 Jan 2026 S1 See all 10 of this type

    70
    The number of the real solutions of the equation : x∣x+3∣+∣x−1∣−2=0x|x + 3| + |x - 1| - 2 = 0 is

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  2. Question 2

    This type asked 8 times: 2021 Paper 4, 2021 Paper 26, 29 Jul 2022, 27 Jan 2024, 6 Apr 2024, 9 Apr 2024, 2 Apr 2025, 3 Apr 2025. This one: 3 Apr 2025 See all 8 of this type

    53
    Let α\alpha and β\beta be the roots of x2+3x−16=0x^{2}+\sqrt{3}x- 16 = 0, and γ\gamma and δ\delta be the roots of x2+3x−1=0x^{2}+ 3x - 1 = 0. If Pn=αn+βnP_{n}=\alpha^{n}+\beta^{n} and Qn=γn+δnQ_{n}=\gamma^{n}+\delta^{n}, then P25+3P242P23+Q25−Q23Q24\frac{P_{25}+\sqrt{3}P_{24}}{2P_{23}}+\frac{Q_{25}-Q_{23}}{Q_{24}} is equal to

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  3. Question 3

    This type asked 7 times: 2021 Paper 1, 14 Jun 2022, 26 Jun 2022, 26 Jul 2022, 25 Jan 2023, 9 Apr 2024, 22 Jan 2025. This one: 22 Jan 2025 See all 7 of this type

    137
    Let αθ\alpha_{\theta} and βθ\beta_{\theta} be the distinct roots of 2x2+(cos⁡θ)x−1=0,θ∈(0,2π)2x^{2}+ (\cos\theta)x - 1 = 0,\theta \in (0,2\pi). If mm and MM are the minimum and the maximum values of αθ4+βθ4\alpha_{\theta}^{4}+\beta_{\theta}^{4}, then 16(M+m)16(M + m) equals :

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  4. Question 4

    This type asked 5 times: 2021 Paper 3, 2021 Paper 18, 2021 Paper 21, 31 Jan 2024, 29 Jan 2025. This one: 29 Jan 2025 See all 5 of this type

    126
    If the set of all a∈Ra \in R, for which the equation 2x2+(a−5)x+15=3a2x^{2}+ (a - 5)x + 15 = 3a has no real root, is the interval (α,β)(\alpha,\beta), and X={x∈Z:α<x<β}X = \{ x \in Z:\alpha < x < \beta\}, then ∑x∈Xx2\sum_{x \in X} x^{2} is equal to

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  5. Question 5

    This type asked 5 times: 2021 Paper 17, 28 Jun 2022, 24 Jan 2023, 31 Jan 2023, 11 Apr 2023. This one: 11 Apr 2023 See all 5 of this type

    171
    The number of points, where the curve f(x)=e8x−e6x−3e4x−e2x+1,x∈Rf(x) =e^{8x}-e^{6x}- 3e^{4x}-e^{2x}+ 1,x \in R cuts xx-axis, is equal to

  6. Question 6

    This type asked 5 times: 2021 Paper 19, 25 Jun 2022, 27 Jun 2022, 29 Jan 2023, 30 Jan 2023. This one: 30 Jan 2023 See all 5 of this type

    178
    If the value of real number a>0a > 0 for which x2−5ax+1=0x^{2}- 5ax + 1 = 0 and x2−ax−5=0x^{2}- ax - 5 = 0 have a common real root is 32β\frac{3}{\sqrt{2\beta}} then β\beta is equal to

  7. Question 7

    This type asked 4 times: 2021 Paper 23, 14 Jun 2022, 27 Jun 2022, 8 Apr 2024. This one: 8 Apr 2024 See all 4 of this type

    62
    The sum of all the solutions of the equation (8)2x−16⋅(8)x+48=0(8)^{2x}- 16 \cdot (8)^{x}+ 48 = 0 is :

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  8. Question 8

    This type asked 3 times: 2021 Paper 25, 23 Jan 2026 S2, 8 Apr 2026 S2. This one: 8 Apr 2026 S2 See all 3 of this type

    71
    The sum of squares of all real solution of the equation log⁡(x+1)(2x2+5x+3)=4−log⁡(2x+3)(x2+2x+1)\log_{(x + 1)}\left( 2x^{2} + 5x + 3 \right) = 4 - \log_{(2x + 3)}\left( x^{2} + 2x + 1 \right) is equal to ____\_\_\_\_ .

  9. Question 9

    This type asked 3 times: 1 Feb 2023, 21 Jan 2026 S2, 6 Apr 2026 S2. This one: 6 Apr 2026 S2 See all 3 of this type

    66
    Let lim⁡x→2(tan⁡(x−2))(rx2+(p−2)x−2p)(x−2)2=5\lim_{x \rightarrow 2} \frac{(\tan(x - 2))\left( rx^{2}+ (p - 2)x - 2p \right)}{(x - 2)^{2}}= 5 for some r,p∈Rr,p \in R. If the set of all possible values of q , such that the roots of the equation rx2−px+q=0rx^{2}- px + q = 0 lie in (0,2)(0,2), be the interval (α,β](\alpha,\beta\rbrack, then 4(α+β)4(\alpha+\beta) equals.

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  10. Question 10

    This type asked 3 times: 2 Apr 2025, 7 Apr 2025, 4 Apr 2026 S2. This one: 4 Apr 2026 S2 See all 3 of this type

    55
    If the quadratic equation (λ+2)x2−3λx+4λ=0(\lambda+ 2)x^{2}- 3\lambda x+ 4\lambda= 0, λ≠−2\lambda\neq - 2, has two positive roots, then the number of possible integral values of λ\lambda is :

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