Let $\lim_{n\to\infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2n}{(n^2+1)\sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8n}{(n^2+4)\sqrt{n^4+16}}+\ldots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2n\cdot n^2}{(n^2+n^2)\sqrt{n^4+n^4}}\right)$ be $\frac{\pi}{k}$, using only the principal values of the inverse trigonometric functions. Then $k^2$ is equal to ________.

JEE Main 2024 · MathsNumerical answerNumerical valueHardMulti-step

JEE Main 9 April 2024, Shift 1, Maths Q28

Question 28 of 90 in this shift, Maths question 28 of 30, Section B.

Let

\lim_{n\to\infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2n}{(n^2+1)\sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8n}{(n^2+4)\sqrt{n^4+16}}+\ldots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2n\cdot n^2}{(n^2+n^2)\sqrt{n^4+n^4}}\right)

\frac{\pi}{k}

, using only the principal values of the inverse trigonometric functions. Then

k^2

is equal to ________.

Official answer

NTA final key.

Chapter: Integrals
Topic: Integral as Limit of Sum
Idea tested: Summation to Riemann Form

Same idea in other shifts

Asked 9× in all

30 Jan 2024, Shift 1 · Q10The value of $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n^3}{(n^2+k^2)(n^2+3k^2)}$ is :MediumSingle correct
13 Apr 2023, Shift 1 · Q7Among (S1) : $\lim_{n\to\infty}\frac{1}{n^2}(2+4+6+\ldots\ldots+2n)=1$ (S2) :…MediumTwo statements
25 Jul 2022, Shift 1 · Q27If…MediumNumerical value
25 Jul 2022, Shift 2 · Q7…MediumSingle correct
26 Jul 2022, Shift 1 · Q10If $\mathrm{a} = \lim_{n \to \infty} \sum_{k=1}^{n} \dfrac{2n}{n^2 + k^2}$ and $f(x) = \sqrt{\dfrac{1 - \cos x}{1 + \cos x}}$ ,…MediumSingle correct
24 Jun 2022, Shift 2 · Q7…HardSingle correct
30 Jun 2022, Shift 1 · Q11 $\lim_{n\to\infty} \sum_{r=1}^{n} \frac{r}{2r^2 - 7rn + 6n^2}$ is equal to :MediumSingle correct
25 Feb 2021, Shift 2 · Q74 $\lim_{n\to\infty}\left[\frac{1}{n} + \frac{n}{(n+1)^2} + \frac{n}{(n+2)^2} + \ldots\ldots\ldots + \frac{n}{(2n-1)^2}\right]$ is equal to :MediumSingle 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.