Mathematics · 2022
JEE Main · 26 July 2022, Shift 1 · Q10
If a = lim_n → ∞ Σ_k =1^n (2 n)/(n^2+ k^2) and f(x)=√((1- cos x)/(1+ cos x)), x ∈(0,1), then:
If $\displaystyle \mathrm{a}=\lim _{\mathrm{n} \rightarrow \infty} \sum_{\mathrm{k}=1}^{\mathrm{n}} \frac{2 \mathrm{n}}{\mathrm{n}^2+\mathrm{k}^2}$ and $\displaystyle f(x)=\sqrt{\frac{1-\cos x}{1+\cos x}}, x \in(0,1)$, then :
Official answer
(3)
$\displaystyle \sqrt{2} f\left(\frac{\mathrm{a}}{2}\right)=f^{\prime}\left(\frac{\mathrm{a}}{2}\right)$
More from Definite Integration
- If the value of the integral ∫_0^5 (x+[x])/(e^x-[x]) d x=α e^-1+β, where α, β ∈ R, 5 α+6 β=0, and [x] denotes the greatest integer less than or equal…2021
- If for all real triplets (a, b, c), f(x)=a+b x+cx^2; then ∫_0^1 f(x) d x is equal to:2020
- If the value of the integral ∫_0^1 / 2 x^2/((1-x^2)^3 / 2) d x is k/6, then k is equal to:2020
- Let 0 ≤ x ≤ 2Max{(9-x^2)/(5-x)}=α and 0 ≤ x ≤ 2Min{(9-x^2)/(5-x)}=β. If ∫_β-8/3^2 α-1 Max{(9-x^2)/(5-x), x} d x=α_1+α_2 log_e(8/15) then α_1+α_2 is…2022
- Let the function f:[0,2] → R be defined as f(x)= {e^min {x^2, x-[x]}, x ∈[0,1); e^[x- log_e x], x ∈[1,2]} where [t] denotes the greatest integer less…2023
- If I_m, n=∫_0^1 x^m-1(1-x)^n-1 d x, for m, n ≥ 1, and ∫_0^1 (x^m-1+x^n-1)/((1+x)^m+n) d x=α I_m, n, α ∈ R, then α equals \ \ \ \.2021
- If [·] represents the greatest integer function, then the value of |∫_0^√((π)/2)[[x^2]- cos x] d x| is \ \ \ \.2021
- The value of k ∈ N for which the integral I_n=∫_0^1(1-x^k)^n d x, n ∈ N, satisfies 147 I_20=148 I_21 is2024
JEE Main 2022 Mathematics question, with the answer from NTA’s final answer key. Where our answers come from.