JEE Main 2023 · MathsMultiple choiceSingle correctMediumMulti-step

JEE Main 10 April 2023, Shift 1, Maths Q9

Question 9 of 90 in this shift, Maths question 9 of 30, Section A.

Let

f

be a differentiable function such that

x^2f(x)-x=4\int_0^x tf(t)\,dt

f(1)=\frac{2}{3}

. Then

18f(3)

is equal to

(1) $150$
(2) $160$ Official answer
(3) $180$
(4) $210$

Official answer

Option 2

NTA final key.

Chapter: Differential Equations
Topic: Leibniz Rule
Idea tested: Solving Integral Equations

Same idea in other shifts

Asked 11× in all

4 Apr 2026, Shift 2 · Q25Let $f$ be a twice differentiable function such that $f(x)=\int_0^x\tan(t-x)\,dt-\int_0^x f(t)\tan t\,dt$ ,…HardNumerical value
5 Apr 2026, Shift 2 · Q19Let $f:[1,\infty)\to\mathbf{R}$ be a differentiable function defined as $f(x)=\int_1^x f(t)\,dt+(1-x)(\log_e x-1)+e$ . Then the value of…HardSingle correct
2 Apr 2025, Shift 2 · Q20Let $f : [1, \infty) \to [2, \infty)$ be a differentiable function. If $10\int_1^x f(t)\,\mathrm{d}t = 5xf(x) - x^5 - 9$ for all…MediumSingle correct
9 Apr 2024, Shift 2 · Q13Let $\int_0^x\sqrt{1-(y'(t))^2}\,dt=\int_0^x y(t)\,dt,\ 0\le x\le 3,\ y\ge0,\ y(0)=0$ . Then at $x=2$ , $y''+y+1$ is equal toHardSingle correct
1 Feb 2024, Shift 2 · Q24Let $f:(0,\infty)\to\mathbf{R}$ and $F(x)=\int_0^x t f(t)\,dt$ . If $F(x^2)=x^4+x^5$ , then $\sum_{r=1}^{12}f(r^2)$ is equal to ________.MediumNumerical value
10 Apr 2023, Shift 2 · Q11Let $f$ be a continuous function satisfying $\int_0^{t^2} (f(x) + x^2)\, dx = \frac{4}{3}t^3,\ \forall t > 0$ . Then…MediumSingle correct
25 Jul 2022, Shift 2 · Q26Let $f$ be a twice differentiable function on $\mathbb{R}$ . If $f'(0)=4$ and…HardNumerical value
27 Jul 2022, Shift 2 · Q28Let f be a differentiable function satisfying $f(x) = \frac{2}{\sqrt{3}} \int_0^{\sqrt{3}} f\left(\frac{\lambda^2 x}{3}\right) d\lambda$ ,…HardNumerical 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.