JEE Advanced 2024 Paper 2 · Q08 · Sequences & Series

Question

Let $f : \mathbb{R} 	o \mathbb{R}$ be a function such that $f(x + y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$, and $g : \mathbb{R} 	o (0, \infty)$ be a function such that $g(x + y) = g(x)\,g(y)$ for all $x, y \in \mathbb{R}$. If $f\!\left(\dfrac{-3}{5}ight) = 12$ and $g\!\left(\dfrac{-1}{3}ight) = 2$, then the value of $f\!\left(\dfrac{1}{4}ight) + g(-2) - 8\,g(0)$ is ______.

SolveKar AI · Accepted Answer

The functional equations give $f(x) = kx$ and $g(x) = a^{x}$ for some constants. From $f(-3/5) = 12$: $-3k/5 = 12 \Rightarrow k = -20$. From $g(-1/3) = 2$: $a^{-1/3} = 2 \Rightarrow a = 1/8$. Hence $f(1/4) = -5$, $g(-2) = (1/8)^{-2} = 64$, and $g(0) = 1$. Required value $= -5 + 64 - 8(1) = 51$.

JEE Advanced 2024 Paper 2 · Q08 · Sequences & Series

Solution