Q41 / 75 · JEE Main 2025 (28 Jan Shift 1) · Mathematics

Let for some function $\mathrm{y}=f(x), \int_0^x t f(t) d t=x^2 f(x), x\gt0$ and $f(2)=3$. Then $f(6)$ is equal to

  • Correct answer
    A
    1
  • B
    3
  • C
    6
  • D
    2

Solution

$\begin{aligned} & \int_0^x t f(t) d t=x^2 f(x) \\ & \Rightarrow \quad x f(x)=2 x f(x)+x^2 f^{\prime}(x) \\ & \Rightarrow x^2 \cdot f^{\prime}(x)=-x f(x) \\ & \Rightarrow \quad \frac{f^{\prime}(x)}{f(x)}=-\frac{1}{x} \\ & \\ & \quad \int \frac{f^{\prime}(x)}{f(x) d x}=-\int \frac{1}{x} d x \\ & \Rightarrow \quad \ln f(x)=-\ln x+C \\ & \\ & \quad f(2)=3 \Rightarrow \ln 3=-\ln 2+C \\ & \quad C=\ln 6 \\ & \Rightarrow \quad f(x)=\frac{6}{x} \Rightarrow f(6)=1\end{aligned}$

Hence, the correct answer is option: 'A'.

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