JEE MAIN - Mathematics (2025 - 28th January Evening Shift - No. 19)
Let $f:[0,3] \rightarrow$ A be defined by $f(x)=2 x^3-15 x^2+36 x+7$ and $g:[0, \infty) \rightarrow B$ be defined by $g(x)=\frac{x^{2025}}{x^{2025}+1}$, If both the functions are onto and $S=\{ x \in Z ; x \in A$ or $x \in B \}$, then $n(S)$ is equal to :
29
31
30
36
Explanation
as $f(x)$ is onto hence $A$ is range of $f(x)$
$$\text { now } \begin{aligned} f^{\prime}(x) & =6 x^2-30 x+36 \\ & =6(x-2)(x-3) \end{aligned}$$
$$f(2)=16-60+72+7=35$$
$$\begin{aligned} & \mathrm{f}(3)=54-135+108+7=34 \\ & \mathrm{f}(0)=7 \end{aligned}$$
hence range $\in[7,35]=\mathrm{A}$
also for range of $g(x)$
$$\begin{aligned} & g(x)=1-\frac{1}{x^{2025}+1} \in[0,1)=B \\ & s=\{0,7,8, \ldots . .35\} \text { hence } n(s)=30 \end{aligned}$$
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