JEE Main 2022 · MathsMultiple choiceSingle correctHardMulti-step

JEE Main 24 June 2022, Shift 2, Maths Q7

Question 7 of 90 in this shift, Maths question 7 of 30, Section A.

\lim_{n\to\infty}\left[\frac{n^2}{(n^2+1)(n+1)} + \frac{n^2}{(n^2+4)(n+2)} + \frac{n^2}{(n^2+9)(n+3)} + \ldots + \frac{n^2}{(n^2+n^2)(n+n)}\right]

is equal to

(1) $\frac{\pi}{8} + \frac{1}{4}\log_e 2$ Official answer
(2) $\frac{\pi}{4} + \frac{1}{8}\log_e 2$
(3) $\frac{\pi}{4} - \frac{1}{8}\log_e 2$
(4) $\frac{\pi}{8} + \log_e \sqrt{2}$

Official answer

Option 1

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

9 Apr 2024, Shift 1 · Q28Let…HardNumerical value
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
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.