JEE Main 2022 · MathsMultiple choiceSingle correctMediumMulti-step

JEE Main 29 July 2022, Shift 1, Maths Q5

Question 5 of 90 in this shift, Maths question 5 of 30, Section A.

\lim_{x \to 0} \frac{\alpha e^x + \beta e^{-x} + \gamma \sin x}{x \sin^2 x} = \frac{2}{3}

, where

\alpha, \beta, \gamma \in \mathbf{R}

, then which of the following is NOT correct ?

(1) $\alpha^2 + \beta^2 + \gamma^2 = 6$
(2) $\alpha\beta + \beta\gamma + \gamma\alpha + 1 = 0$
(3) $\alpha\beta^2 + \beta\gamma^2 + \gamma\alpha^2 + 3 = 0$ Official answer
(4) $\alpha^2 - \beta^2 + \gamma^2 = 4$

Official answer

Option 3

NTA final key.

Chapter: Limits and Derivatives
Topic: L'Hopital's Rule
Idea tested: Determining Unknown Constants from Finite Limits

Same idea in other shifts

Asked 6× in all

2 Apr 2026, Shift 1 · Q16If $\lim_{x\to2}\frac{\sin\left(x^3-5x^2+ax+b\right)}{\left(\sqrt{x-1}-1\right)\log_e(x-1)}=m$ , then $a+b+m$ is equal to :HardSingle correct
2 Apr 2025, Shift 1 · Q7For $\alpha, \beta, \gamma\in\mathbf{R}$ , if $\lim_{x\to0}\frac{x^2\sin\alpha x+(\gamma-1)e^{x^2}}{\sin2x-\beta x}=3$ , then…MediumSingle correct
2 Apr 2025, Shift 2 · Q17If $\lim_{x\to0} \frac{\cos(2x) + a\cos(4x) - b}{x^4}$ is finite, then $(a+b)$ is equal to :MediumSingle correct
25 Jul 2022, Shift 1 · Q5If $\lim_{n\to\infty}\left(\sqrt{n^2-n-1}+n\alpha+\beta\right)=0$ , then $8(\alpha+\beta)$ is equal to :MediumSingle correct
25 Feb 2021, Shift 2 · Q84If $\lim_{x\to 0} \frac{ax - (e^{4x} - 1)}{ax(e^{4x} - 1)}$ exists and is equal to b, then the value of $a - 2b$ is ________.MediumNumerical value

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.