JEE Mathematics 2025 Question with Solution

Given: $f: [0,3] \to A$ with $f(x)=2x^3-15x^2+36x+7$ and $g:[0,\infty)\to B$ with $g(x)=\frac{x}{x^{2025}+1}$. Both functions are onto.

Find: $n(S)$ where $S=\{x\in\mathbb{Z}: x\in A \text{ or } x\in B\}$.

For $f(x)$, find the range on $[0,3]$.

$$f'(x)=6x^2-30x+36=6(x-2)(x-3)$$

So the critical points in the interval are $x=2$ and $x=3$. Now evaluate:

$$f(0)=7$$ $$f(2)=2(8)-15(4)+72+7=16-60+72+7=35$$ $$f(3)=2(27)-15(9)+108+7=54-135+108+7=34$$

Hence the minimum value is $7$ and the maximum value is $35$, so

$$A=[7,35]$$

Therefore the integers in $A$ are $7,8,9,\ldots,35$, whose count is

$$35-7+1=29$$

Now consider $g(x)=\frac{x}{x^{2025}+1}$ for $x\ge 0$.

$$g(0)=0$$

and as $x\to\infty$,

$$g(x)\to 0$$

Also, the function attains a positive maximum value less than $1$, so the range includes $0$ and all other values lie below $1$. Thus the only integer belonging to $B$ is $0$.

Therefore,

$$S=\{0,7,8,9,\ldots,35\}$$

and hence

$$n(S)=29+1=30$$

Therefore, the correct option is A.](streamdown:incomplete-link)

Given: $f: [0,3] \to A$ and $g:[0,\infty)\to B$ are onto.

Find: The number of integers in $A\cup B$.

For $f$, use derivative to locate extrema:

$$f'(x)=6x^2-30x+36=6(x-2)(x-3)$$

Checking endpoint and critical-point values:

$$f(0)=7, \qquad f(2)=35, \qquad f(3)=34$$

Since $f$ is continuous on the closed interval $[0,3]$, its range is the full interval from minimum to maximum:

$$A=[7,35]$$

So the integers in $A$ are exactly $7$ through $35$, giving

$$29$$

integers.

For $g$, differentiate:

$$g'(x)=\frac{(x^{2025}+1)-x(2025x^{2024})}{(x^{2025}+1)^2}$$ $$=\frac{1-2024x^{2025}}{(x^{2025}+1)^2}$$

Thus the maximum occurs when

$$1-2024x^{2025}=0$$

that is,

$$x^{2025}=\frac{1}{2024}$$

At that point the maximum value is positive but less than $1$. Also $g(0)=0$. Hence

$$B\subset [0,1)$$

and the only integer in $B$ is $0$.

So,

$$S=\{x\in\mathbb{Z}:x\in A \text{ or } x\in B\}=\{0,7,8,9,\ldots,35\}$$

Therefore,

$$n(S)=29+1=30$$

So the correct option is A.](streamdown:incomplete-link)