QJEEM25Ma8711: JEE Main 2025 Mathematics - Class 11 JEEM25Ma8711 Solution: Let A be defined by and be defined by , If both the functions…

Question

Verified Solution • 2025–26 Session moderate

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 :

Accepted Answer

as $f(x)$ is onto hence $A$ is range of $f(x)$

$now \begin{aligned} f^{'} (x) & = 6 x^{2} - 30 x + 36 \\ = 6 (x - 2) (x - 3) \end{aligned}$

$f (2) = 16 - 60 + 72 + 7 = 35$

$\begin{aligned} f (3) = 54 - 135 + 108 + 7 = 34 \\ f (0) = 7 \end{aligned}$

hence range $\in[7,35]=\mathrm{A}$

also for range of $g(x)$

$\begin{aligned} g (x) = 1 - \frac{1}{x^{2025} + 1} \in [0, 1) = B \\ s = {0, 7, 8, \dots . .35} hence n (s) = 30 \end{aligned}$

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