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Saturday, 31 October

JEE Main · Maths · Integrals

King's Property

JEE Main asked King's Property 18 times in 18 of the 108 shifts Lumi analysed, in 2020, 2021, 2022, 2023, 2024 and 2025. It belongs to Definite Integral Properties, in Integrals (Maths). The longest wait between two years asking it was 1 year. In 2020, 2021, 2023, 2024 and 2025 it came in more than one shift of the same year.

Updated 2026-10-24

Times asked
18
in 18 shifts
Years
6 of 8
First asked
2020
last in 2025
Pattern
Asked most years
across the analysed shifts

Questions on this idea a shift, by year

2017–2026. Asked in 2020, 2021, 2022, 2023, 2024 and 2025. Years differ in how many shifts Lumi analysed, so each year is scaled to one shift of 25 Maths questions.

00.20.40’170.17’200.14’210.04’220.14’230.33’240.25’250’26

Raw count: 2017: 0 in 1 shift · 2020: 2 in 12 shifts · 2021: 4 in 24 shifts · 2022: 1 in 24 shifts · 2023: 2 in 12 shifts · 2024: 7 in 18 shifts · 2025: 2 in 8 shifts · 2026: 0 in 9 shifts.

Shift by shift

Each square is one analysed shift, in date order. Open a filled one for that paper.

  • 20260 of 9
  • 20252 of 8
  • 20247 of 18
  • 20232 of 12
  • 20221 of 24
  • 20214 of 24
  • 20202 of 12
  • 20170 of 1

Asked in that shift More than once Not asked

How JEE Main framed it

Single correct is the usual form. Skill tested: multi-step.

  • Definite integral using king's property
  • Ratio of integrals using substitution and symmetry

Difficulty

Easy 1Medium 9Hard 8

4 of the 18 questions had a numerical answer.

Every question

All 18 questions

From the official JEE Main shift papers, newest first. Open one for the full question, the options and the official answer.

  1. 8 Apr 2025, Shift 2 · Q18Let f(x)f(x) be a positive function and I1=∫−1212x f(2x(1−2x)) dxI_1 = \int_{-\frac{1}{2}}^{1} 2x\, f\big(2x(1-2x)\big)\,dx and…HardSingle correct
  2. 3 Apr 2025, Shift 2 · Q18The integral ∫0π8x dx4cos⁡2x+sin⁡2x\int_0^{\pi} \frac{8x\,dx}{4\cos^2 x + \sin^2 x} is equal toMediumSingle correct
  3. 31 Jan 2024, Shift 1 · Q26Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a function defined by f(x)=4x4x+2f(x)=\frac{4^x}{4^x+2} and…HardNumerical value
  4. 27 Jan 2024, Shift 1 · Q12If (a,b)(a, b) be the orthocentre of the triangle whose vertices are (1,2)(1, 2), (2,3)(2, 3) and (3,1)(3, 1), and I1=∫abxsin⁡(4x−x2) dxI_1=\int_a^b x\sin(4x-x^2)\,dx,…HardSingle correct
  5. 1 Feb 2024, Shift 2 · Q11The value of ∫−01(2x3−3x2−x+1)13 dx\int_{-0}^{1}(2x^3-3x^2-x+1)^{\frac{1}{3}}\,dx is equal to :MediumSingle correct
  6. 1 Feb 2024, Shift 1 · Q10The value of the intergral ∫0π/4x dxsin⁡4(2x)+cos⁡4(2x)\int_0^{\pi/4}\frac{x\,dx}{\sin^4(2x)+\cos^4(2x)} equals :HardSingle correct
  7. 5 Apr 2024, Shift 2 · Q25If f(t)=∫0π2x dx1−cos⁡2tsin⁡2xf(t) = \int_0^{\pi} \frac{2x\,dx}{1-\cos^2 t \sin^2 x}, 0<t<π0 < t < \pi, then the value of…HardNumerical value
  8. 5 Apr 2024, Shift 1 · Q10The value of ∫−ππ2y(1+sin⁡y)1+cos⁡2y dy\int_{-\pi}^{\pi}\frac{2y(1+\sin y)}{1+\cos^2 y}\,dy is :MediumSingle correct
  9. 4 Apr 2024, Shift 1 · Q25If ∫0π/4sin⁡2x1+sin⁡xcos⁡xdx=1alog⁡e(a3)+πb3\int_0^{\pi/4}\frac{\sin^2x}{1+\sin x\cos x}dx=\frac{1}{a}\log_e\left(\frac{a}{3}\right)+\frac{\pi}{b\sqrt{3}}, where…HardNumerical value
  10. 15 Apr 2023, Shift 1 · Q10If ∫011(5+2x−2x2)(1+e(2−4x))dx=1αlog⁡e(α+1β)\int_0^1\frac{1}{\left(5+2x-2x^2\right)\left(1+e^{(2-4x)}\right)}dx=\frac{1}{\alpha}\log_e\left(\frac{\alpha+1}{\beta}\right),…HardSingle correct
  11. 6 Apr 2023, Shift 2 · Q11Let f(x)f(x) be a function satisfying f(x)+f(π−x)=π2, ∀x∈Rf(x)+f(\pi-x)=\pi^2,\ \forall x\in\mathbb{R}. Then ∫0πf(x)sin⁡x dx\int_0^{\pi}f(x)\sin x\,dx is equal toMediumSingle correct
  12. 26 Jun 2022, Shift 1 · Q28The value of the integral 48π4∫0π(3πx22−x3)sin⁡x1+cos⁡2x dx\frac{48}{\pi^4}\int_0^{\pi}\left(\frac{3\pi x^2}{2} - x^3\right)\frac{\sin x}{1 + \cos^2 x}\,dx is equal to…HardNumerical value
  13. 24 Feb 2021, Shift 2 · Q68Let f(x)f(x) be a differentiable function defined on [0, 2] such that f′(x)=f′(2−x)f'(x)=f'(2-x) for all x∈(0,2)x\in(0, 2), f(0)=1f(0)=1 and f(2)=e2f(2)=e^2. Then…MediumSingle correct
  14. 27 Aug 2021, Shift 1 · Q77∫616log⁡ex2log⁡ex2+log⁡e(x2−44x+484)dx\int_{6}^{16}\frac{\log_e x^2}{\log_e x^2+\log_e(x^2-44x+484)}dx is equal to :MediumSingle correct
  15. 4 Aug 2021, Shift 2 · Q70The value of the integral ∫16log⁡ex dxlog⁡ex2+log⁡e(x2−14x+49)\int_1^6\frac{\log_e x\,dx}{\log_e x^2+\log_e(x^2-14x+49)} isMediumSingle correct
  16. 4 Aug 2021, Shift 1 · Q69If I1=∫0πxecosec xdxI_1=\int_0^{\pi}xe^{\mathrm{cosec}\,x}dx and I2=∫0πecosec xdxI_2=\int_0^{\pi}e^{\mathrm{cosec}\,x}dx, then the ratio I1:I2I_1:I_2 is equal toMediumSingle correct
  17. 5 Sep 2020, Shift 1 · Q62The value of ∫−π/2π/211+esin⁡xdx\int_{-\pi/2}^{\pi/2}\frac{1}{1+e^{\sin x}}dx is :EasySingle correct
  18. 9 Jan 2020, Shift 1 · Q63The value of ∫02πxsin⁡8xsin⁡8x+cos⁡8x dx\int_0^{2\pi}\frac{x\sin^8 x}{\sin^8 x+\cos^8 x}\,\mathrm{d}x is equal to :MediumSingle correct

Shifts: 9 Jan 2020, Shift 1, 5 Sep 2020, Shift 1, 24 Feb 2021, Shift 2, 4 Aug 2021, Shift 2, 4 Aug 2021, Shift 1, 27 Aug 2021, Shift 1, 26 Jun 2022, Shift 1, 6 Apr 2023, Shift 2, 15 Apr 2023, Shift 1, 27 Jan 2024, Shift 1, 31 Jan 2024, Shift 1, 1 Feb 2024, Shift 2, 1 Feb 2024, Shift 1, 4 Apr 2024, Shift 1, 5 Apr 2024, Shift 2, 5 Apr 2024, Shift 1, 3 Apr 2025, Shift 2, 8 Apr 2025, Shift 2.

Same topic

Other ideas in Definite Integral Properties

Definite Integral Properties has 19 ideas JEE Main has asked; 13 of them came up in more than one question. The 12 most asked are below.

Topic analysisChapter analysisStudy this chapterAll repeated ideas