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Friday, 9 October

JEE Main Maths · Complex Numbers and Quadratic Equations

Quadratic Roots in JEE Main

Quadratic Roots (Maths, Complex Numbers and Quadratic Equations) came 46 times in 34 of the 42 JEE Main shifts Lumi analysed, across 19 distinct ideas. 29 of them came in 2024–26; it was last asked in 2026.

Questions
46
in 34 shifts
Ideas asked
19
8 asked more than once
Numerical answer
11
24% of its questions
Pattern
Asked most years
last asked 2026
Questions each year
06121’175’203’214’224’2311’246’2512’26

In the shifts Lumi analysed for each year.

Difficulty and format
Easy 8Medium 31Hard 7

Format

  • Single correct34 Q
  • Numerical value11 Q
  • Two statements1 Q
Ideas inside the topic
19 ideas
Where it sits

Part of Complex Numbers and Quadratic Equations, which had 79 questions in all. Quadratic Roots accounts for 58% of them. In Lumi's JEE taxonomy it belongs to Algebra.

Some questions sit in other chapters too: Conic Sections, Integrals, Probability, Relations and Functions.

Every question

All 46 Quadratic Roots questions

Newest first, each with its options and official answer.

  1. 2 Apr 2026, Shift 1 · Q1Let α,α+2,α∈Z\alpha, \alpha+2, \alpha \in \mathbb{Z}, be the roots of the quadratic equation…HardSingle correct
  2. 2 Apr 2026, Shift 1 · Q23Let a,b,c∈{1,2,3,4}a, b, c \in \{1, 2, 3, 4\}. If the probability, that ax2+22 bx+c>0ax^2+2\sqrt2\,bx+c>0 for all x∈Rx \in \mathbf{R}, is mn\frac{m}{n},…MediumNumerical value
  3. 2 Apr 2026, Shift 2 · Q1Let α,β\alpha, \beta be the roots of the equation x2−3x+r=0x^2 - 3x + r = 0, and α2,2β\frac{\alpha}{2}, 2\beta be the roots of the equation…MediumSingle correct
  4. 2 Apr 2026, Shift 2 · Q12Let the parabola y=x2+px+qy = x^2 + px + q passing through the point (1,−1)(1, -1) be such that the distance between its vertex and the xx-axis is…MediumSingle correct
  5. 4 Apr 2026, Shift 2 · Q2Let S={z∈C:z2+4z+16=0}S=\{z\in\mathbb{C}:z^2+4z+16=0\}. Then ∑z∈S∣z+3 i∣2\sum_{z\in S}|z+\sqrt{3}\,i|^2 is equal to:EasySingle correct
  6. 4 Apr 2026, Shift 2 · Q4If α=1\alpha=1 and β=1+i2\beta=1+i\sqrt{2}, where i=−1i=\sqrt{-1} are two roots of the equation x3+ax2+bx+c=0x^3+ax^2+bx+c=0, a,b,c∈Ra,b,c\in\mathbb{R}, then…MediumSingle correct
  7. 5 Apr 2026, Shift 1 · Q1Let a,b∈Ca, b \in \mathbb{C}. Let α,β\alpha, \beta be the roots of the equation x2+ax+b=0x^2 + ax + b = 0. If β−α=11\beta - \alpha = \sqrt{11} and…MediumSingle correct
  8. 5 Apr 2026, Shift 2 · Q1Let α,β\alpha, \beta be the roots of the equation x2−x+p=0x^2-x+p=0 and γ,δ\gamma, \delta be the roots the equation x2−4x+q=0x^2-4x+q=0;…MediumSingle correct
  9. 5 Apr 2026, Shift 2 · Q2Let z1,z2∈Cz_1, z_2 \in \mathbb{C} be the distinct solutions of the equation z2+4z−(1+12i)=0z^2+4z-(1+12i)=0. Then ∣z1∣2+∣z2∣2|z_1|^2+|z_2|^2 is equal to :MediumSingle correct
  10. 6 Apr 2026, Shift 1 · Q2Let one root of the quadratic equation in xx: (k2−15k+27)x2+9(k−1)x+18=0(k^2-15k+27)x^2+9(k-1)x+18=0 be twice the other. Then the length of the latus rectum of…MediumSingle correct
  11. 6 Apr 2026, Shift 2 · Q2Consider the quadratic equation (n2−2n+2)x2−3x+(n2−2n+2)2=0(n^2-2n+2)x^2-3x+(n^2-2n+2)^2=0, n∈Rn\in\mathbf{R}. Let α\alpha be the minimum value of the product of…MediumSingle correct
  12. 6 Apr 2026, Shift 2 · Q3Let S={z∈C:z2+6 iz−3=0}S=\{z\in\mathbb{C}: z^2+\sqrt{6}\,iz-3=0\}. Then ∑z∈Sz8\sum_{z\in S} z^8 is equal to :MediumSingle correct
  13. 2 Apr 2025, Shift 1 · Q17Let Pn=αn+βnP_n=\alpha^n+\beta^n, n∈Nn\in\mathbf{N}. If P10=123P_{10}=123, P9=76P_9=76, P8=47P_8=47 and P1=1P_1=1, then the quadratic equation having roots…MediumSingle correct
  14. 3 Apr 2025, Shift 1 · Q3Let 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…MediumSingle correct
  15. 3 Apr 2025, Shift 1 · Q4Let z∈Cz \in \mathbb{C} be such that z2+3iz−2+i=2+3i\frac{z^2 + 3i}{z - 2 + i} = 2 + 3i. Then the sum of all possible values of z2z^2 isMediumSingle correct
  16. 3 Apr 2025, Shift 2 · Q4Let the equation x(x+2)(12−k)=2x(x + 2)(12 - k) = 2 have equal roots. Then the distance of the point (k,k2)\left(k, \frac{k}{2}\right) from the line…EasySingle correct
  17. 7 Apr 2025, Shift 2 · Q2If the range of the function f(x)=5−xx2−3x+2f(x)=\frac{5-x}{x^2-3x+2}, x≠1,2x\ne 1,2, is (−∞,α]∪[β,∞)(-\infty,\alpha]\cup[\beta,\infty), then α2+β2\alpha^2+\beta^2 is…MediumSingle correct
  18. 8 Apr 2025, Shift 2 · Q2The sum of the squares of the roots of ∣x−2∣2+∣x−2∣−2=0|x-2|^2 + |x-2| - 2 = 0 and the squares of the roots of x2−2∣x−3∣−5=0x^2 - 2|x-3| - 5 = 0, isMediumSingle correct
  19. 6 Apr 2024, Shift 2 · Q21Let α,β\alpha,\beta be roots of x2+2x−8=0x^2+\sqrt{2}x-8=0. If Un=αn+βnU_n=\alpha^n+\beta^n, then U10+2U92U8\frac{U_{10}+\sqrt{2}U_9}{2U_8} is equal to ________.EasyNumerical value
  20. 8 Apr 2024, Shift 1 · Q3The sum of all the solutions of the equation (8)2x−16⋅(8)x+48=0(8)^{2x}-16\cdot(8)^x+48=0 is :EasySingle correct
  21. 9 Apr 2024, Shift 1 · Q2Let α,β\alpha,\beta be the roots of the equation x2+22 x−1=0x^2+2\sqrt{2}\,x-1=0. The quadratic equation, whose roots are α4+β4\alpha^4+\beta^4 and…MediumSingle correct
  22. 9 Apr 2024, Shift 2 · Q3Let α,β; α>β\alpha, \beta;\ \alpha>\beta, be the roots of the equation x2−2x−3=0x^2-\sqrt{2}x-\sqrt{3}=0. Let Pn=αn−βn, n∈NP_n=\alpha^n-\beta^n,\ n\in\mathbb{N}.…MediumSingle correct
  23. 29 Jan 2024, Shift 1 · Q21Let α,β\alpha,\beta be the roots of the equation x2−x+2=0x^2-x+2=0 with Im(α)>Im(β)Im(\alpha)>Im(\beta). Then α6+α4+β4−5α2\alpha^6+\alpha^4+\beta^4-5\alpha^2 is…MediumNumerical value
  24. 30 Jan 2024, Shift 2 · Q22The number of real solutions of the equation x(x2+3∣x∣+5∣x−1∣+6∣x−2∣)=0x\left(x^2 + 3|x| + 5|x-1| + 6|x-2|\right) = 0 is ________.MediumNumerical value
  25. 31 Jan 2024, Shift 1 · Q2Let S be the set of postive integral values of aa for which ax2+2(a+1)x+9a+4x2−8x+32<0, ∀x∈R\frac{ax^2+2(a+1)x+9a+4}{x^2-8x+32}<0,\ \forall x\in\mathbb{R}. Then, the…EasySingle correct
  26. 31 Jan 2024, Shift 1 · Q7Let aa be the sum of all coefficients in the expansion of (1−2x+2x2)2023(3−4x2+2x3)2024(1-2x+2x^2)^{2023}(3-4x^2+2x^3)^{2024} and…HardSingle correct
  27. 31 Jan 2024, Shift 1 · Q8For 0<c<b<a0<c<b<a, let (a+b−2c)x2+(b+c−2a)x+(c+a−2b)=0(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0 and α≠1\alpha\neq1 be one of its root. Then, among the two statements (I) If…MediumTwo statements
  28. 31 Jan 2024, Shift 2 · Q3Let z1z_1 and z2z_2 be two complex numbers such that z1+z2=5z_1+z_2=5 and z13+z23=20+15iz_1^3+z_2^3=20+15i. Then, ∣z14+z24∣|z_1^4+z_2^4| equals -HardSingle correct
  29. 31 Jan 2024, Shift 2 · Q24Let a,b,ca, b, c be the lengths of three sides of a triangle satisfying the condition (a2+b2)x2−2b(a+c)x+(b2+c2)=0(a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0. If the set of all…HardNumerical value
  30. 6 Apr 2023, Shift 1 · Q2The sum of all the roots of the equation ∣x2−8x+15∣−2x+7=0|x^2-8x+15|-2x+7=0 is:MediumSingle correct
  31. 8 Apr 2023, Shift 2 · Q22Let mm and nn be the numbers of real roots of the quadratic equations x2−12x+[x]+31=0x^2 - 12x + [x] + 31 = 0 and x2−5∣x+2∣−4=0x^2 - 5|x + 2| - 4 = 0…HardNumerical value
  32. 11 Apr 2023, Shift 1 · Q24If aa and bb are the roots of the equation x2−7x−1=0x^2 - 7x - 1 = 0, then the value of…MediumNumerical value
  33. 13 Apr 2023, Shift 2 · Q3Let α,β\alpha, \beta be the roots of the equation x2−2x+2=0x^2-\sqrt{2}x+2=0. Then α14+β14\alpha^{14}+\beta^{14} is equal toMediumSingle correct
  34. 28 Jul 2022, Shift 1 · Q26For p,q∈Rp,q\in\mathbf{R}, consider the real valued function f(x)=(x−p)2−q, x∈Rf(x)=(x-p)^2-q,\ x\in\mathbf{R} and q>0q>0. Let a1,a2,a3a_1,a_2,a_3 and a4a_4 be in…MediumNumerical value
  35. 28 Jul 2022, Shift 1 · Q30The sum of all real values of xx for which 3x2−9x+17x2+3x+10=5x2−7x+193x2+5x+12\frac{3x^2-9x+17}{x^2+3x+10}=\frac{5x^2-7x+19}{3x^2+5x+12} is equal to __________.MediumNumerical value
  36. 24 Jun 2022, Shift 1 · Q7If the sum of the squares of the reciprocals of the roots α\alpha and β\beta of the equation 3x2+λx−1=03x^2+\lambda x-1=0 is 15, then…MediumSingle correct
  37. 29 Jun 2022, Shift 1 · Q8Let α\alpha and β\beta be the roots of the equation x2+(2i−1)=0x^2 + (2i - 1) = 0. Then, the value of ∣α8+β8∣|\alpha^8 + \beta^8| is equal to :MediumSingle correct
  38. 3 Aug 2021, Shift 2 · Q62If the equations 2x2+kx−5=02x^2 + kx - 5 = 0 and x2−3x−4=0x^2 - 3x - 4 = 0 have one root in common, then a value of 'kk' isEasySingle correct
  39. 25 Jul 2021, Shift 1 · Q72The number of real roots of the equation e6x−e4x−2e3x−12e2x+ex+1=0e^{6x}-e^{4x}-2e^{3x}-12e^{2x}+e^x+1=0 is :HardSingle correct
  40. 25 Jul 2021, Shift 1 · Q90If α,β\alpha,\beta are roots of the equation x2+5(2)x+10=0x^2+5(\sqrt2)x+10=0, α>β\alpha>\beta and Pn=αn−βnP_n=\alpha^n-\beta^n for each positive integer n,…MediumNumerical value
  41. 8 Jan 2020, Shift 1 · Q52If the equation, x2+bx+45=0x^2+bx+45=0 (b∈Rb\in R) has conjugate complex roots and they satisfy ∣z+1∣=210|z+1|=2\sqrt{10}, then :MediumSingle correct
  42. 8 Jan 2020, Shift 1 · Q71The least positive value of 'a' for which the equation, 2x2+(a−10)x+332=2a2x^2+(a-10)x+\frac{33}{2}=2a has real roots is ______.EasyNumerical value
  43. 9 Jan 2020, Shift 2 · Q52Let a,b∈Ra,b\in\mathbf{R}, a≠0a\ne0 be such that the equation, ax2−2bx+5=0ax^2-2bx+5=0 has a repeated root α\alpha, which is also a root of the…MediumSingle correct
  44. 2 Sep 2020, Shift 1 · Q52Let α\alpha and β\beta be the roots of the equation, 5x2+6x−2=05x^{2}+6x-2=0. If Sn=αn+βnS_n=\alpha^{n}+\beta^{n}, n=1,2,3,…n=1, 2, 3, \ldots, then :EasySingle correct
  45. 6 Sep 2020, Shift 2 · Q52If α\alpha and β\beta are the roots of the equation 2x(2x+1)=12x(2x+1)=1, then β\beta is equal to :MediumSingle correct
  46. 2 Apr 2017 (offline) · Q62If, for a positive integer n, the quadratic equation, x(x+1)+(x+1)(x+2)+…+(x+n−1‾)(x+n)=10nx(x+1)+(x+1)(x+2)+\ldots+(x+\overline{n-1})(x+n)=10n has two consecutive integral…HardSingle correct