Q36 / 75 · JEE Main 2026 (24 January Shift 2) · Mathematics
Let $f$ be a function such that $3 f(x)+2 f\left(\frac{m}{19 x}\right)=5 x, x \neq 0$, where $m=\sum_{i=1}^{9}(i)^{2}$. Then $f(5)-f(2)$ is equal to
- Correct answerA18
- B36
- C9
- D-9
Solution
$m = \sum_{i=1}^{9} i^2 = \frac{9 \cdot 10 \cdot 19}{6} = 285$, so $\frac{m}{19} = 15$.
The equation becomes $3f(x) + 2f\left(\frac{15}{x}\right) = 5x$ ...(i)
Replace $x$ by $\frac{15}{x}$: $3f\left(\frac{15}{x}\right) + 2f(x) = \frac{75}{x}$ ...(ii)
From $3 \times (i) - 2 \times (ii)$: $5f(x) = 15x - \frac{150}{x}$, so $f(x) = 3x - \frac{30}{x}$.
$f(5) = 15 - 6 = 9$, $f(2) = 6 - 15 = -9$.
$f(5) - f(2) = 9 - (-9) = 18$.
The equation becomes $3f(x) + 2f\left(\frac{15}{x}\right) = 5x$ ...(i)
Replace $x$ by $\frac{15}{x}$: $3f\left(\frac{15}{x}\right) + 2f(x) = \frac{75}{x}$ ...(ii)
From $3 \times (i) - 2 \times (ii)$: $5f(x) = 15x - \frac{150}{x}$, so $f(x) = 3x - \frac{30}{x}$.
$f(5) = 15 - 6 = 9$, $f(2) = 6 - 15 = -9$.
$f(5) - f(2) = 9 - (-9) = 18$.
Hence, the correct answer is option: 'A'.