JEE Main 2021 · MathsMultiple choiceSingle correctEasyApplication

JEE Main 4 August 2021, Shift 2, Maths Q67

Question 67 of 90 in this shift, Maths question 7 of 30, Section A.

Let

f:\mathbb{R}\to\mathbb{R}

be twice differentiable and satisfy

f(x+y)=f(x)+f(y)

. If

f'(0)=1

, then which of the following is true?

(1) $f(0)=1$
(2) $f'(2)=1$ Official answer
(3) $f''(0)=1$
(4) $f'(1)=2$

Official answer

Option 2

NTA final key.

Chapter: Continuity and Differentiability
Topic: Continuity and Differentiability
Idea tested: Differentiability via Functional Equations

Same idea in other shifts

Asked 10× in all

5 Apr 2026, Shift 1 · Q19Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f\left(\frac{x+y}{3}\right) = \frac{f(x)+f(y)}{3}$ for all…MediumSingle correct
2 Apr 2025, Shift 1 · Q14Let $f:\mathbf{R}\to\mathbf{R}$ be a twice differentiable function such that…HardSingle correct
30 Jan 2024, Shift 2 · Q7Let $f: \mathbb{R} - \{0\} \to \mathbb{R}$ be a function satisfying $f\left(\frac{x}{y}\right) = \frac{f(x)}{f(y)}$ for all…MediumSingle correct
26 Jul 2022, Shift 1 · Q1Let $f : \mathbf{R} \rightarrow \mathbf{R}$ be a continuous function such that $f\,(3x) - f\,(x) = x$ . If $f\,(8) = 7$ , then $f\,(14)$ is…MediumSingle correct
26 Jun 2022, Shift 2 · Q21Let $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfy $f(x + y) = 2^x f(y) + 4^y f(x)$ , $\forall x, y \in \mathbb{R}$ . If $f(2) = 3$ , then…HardNumerical value
26 Feb 2021, Shift 1 · Q74Let $f$ be any function defined on $\mathbf{R}$ and let it satisfy the condition :…MediumSingle correct
18 Mar 2021, Shift 2 · Q90Let $f:\mathbb{R}\to\mathbb{R}$ satisfy the equation $f(x+y)=f(x)\cdot f(y)$ for all $x, y\in\mathbb{R}$ and $f(x)\neq0$ for any…MediumNumerical value
4 Sep 2020, Shift 1 · Q73Suppose a differentiable function $f(x)$ satisfies the identity $f(x+y) = f(x) + f(y) + xy^2 + x^2y$ , for all real $x$ and $y$ . If…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.