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Saturday, 31 October

JEE Main 2021 · MathsMultiple choiceSingle correctHardMulti-step

JEE Main 31 August 2021, Shift 1, Maths Q72

Question 72 of 90 in this shift, Maths question 12 of 30, Section A.

Let ff be a non-negative function in [0,1][0, 1] and twice differentiable in (0,1)(0, 1). If ∫0x1−(f′(t))2 dt=∫0xf(t) dt\int_0^x\sqrt{1-(f'(\mathrm{t}))^2}\,\mathrm{dt}=\int_0^x f(\mathrm{t})\,\mathrm{dt}, 0≤x≤10\leq x\leq1 and f(0)=0f(0)=0, then lim⁡x→01x2∫0xf(t) dt\lim_{x\to0}\frac{1}{x^2}\int_0^x f(\mathrm{t})\,\mathrm{dt} :
  1. (1)equals 0
  2. (2)equals 12\frac{1}{2}Official answer
  3. (3)equals 1
  4. (4)does not exist

Official answer

Option 2

NTA final key.

Chapter
Integrals
Topic
Leibniz Rule
Idea tested
Constant Limit Leibniz

Same topic in other shifts

All Leibniz Rule questions
  1. 4 Apr 2026, Shift 2 · Q25Let ff be a twice differentiable function such that f(x)=∫0xtan⁡(t−x) dt−∫0xf(t)tan⁡t dtf(x)=\int_0^x\tan(t-x)\,dt-\int_0^x f(t)\tan t\,dt,…HardNumerical value
  2. 5 Apr 2026, Shift 2 · Q19Let f:[1,∞)→Rf:[1,\infty)\to\mathbf{R} be a differentiable function defined as f(x)=∫1xf(t) dt+(1−x)(log⁡ex−1)+ef(x)=\int_1^x f(t)\,dt+(1-x)(\log_e x-1)+e. Then the value of…HardSingle correct
  3. 2 Apr 2025, Shift 2 · Q20Let f:[1,∞)→[2,∞)f : [1, \infty) \to [2, \infty) be a differentiable function. If 10∫1xf(t) dt=5xf(x)−x5−910\int_1^x f(t)\,\mathrm{d}t = 5xf(x) - x^5 - 9 for all…MediumSingle correct
  4. 4 Apr 2024, Shift 2 · Q9Let f(x)=∫0x(t+sin⁡(1−et))dt, x∈Rf(x)=\int_0^x\left(t+\sin\left(1-e^t\right)\right)dt,\ x\in\mathbb{R}. Then, lim⁡x→0f(x)x3\lim_{x\to0}\frac{f(x)}{x^3} is equal toHardSingle correct
  5. 9 Apr 2024, Shift 2 · Q9…HardSingle correct
  6. 9 Apr 2024, Shift 2 · Q13Let ∫0x1−(y′(t))2 dt=∫0xy(t) dt, 0≤x≤3, y≥0, y(0)=0\int_0^x\sqrt{1-(y'(t))^2}\,dt=\int_0^x y(t)\,dt,\ 0\le x\le 3,\ y\ge0,\ y(0)=0. Then at x=2x=2, y′′+y+1y''+y+1 is equal toHardSingle correct

Question text from the official JEE Main paper published by NTA; answer from the final answer key. Chapter, topic and idea tags, difficulty and skill are Lumi’s. Lumi analyses 108 of the 174 JEE Main shifts held from 2017 to 2026.