1
JEE Main 2025 (Online) 28th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\int\limits_0^x t f(t) d t$. If $g\left(x^3\right)=x^6+x^7$, then value of $\sum\limits_{r=1}^{15} f\left(r^3\right)$ is :

A

270

B

340

C

310

D

320

2
JEE Main 2025 (Online) 28th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
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 :
A

29

B

31

C

30

D

36

3
JEE Main 2025 (Online) 28th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
For positive integers $n$, if $4 a_n=\left(n^2+5 n+6\right)$ and $S_n=\sum\limits_{k=1}^n\left(\frac{1}{a_k}\right)$, then the value of $507 S_{2025}$ is :
A

540

B

675

C

1350

D

135

4
JEE Main 2025 (Online) 28th January Evening Shift
Numerical
+4
-1
Change Language

Let $f(x)=\lim \limits_{n \rightarrow \infty} \sum\limits_{r=0}^n\left(\frac{\tan \left(x / 2^{r+1}\right)+\tan ^3\left(x / 2^{r+1}\right)}{1-\tan ^2\left(x / 2^{r+1}\right)}\right)$ Then $\lim\limits_{x \rightarrow 0} \frac{e^x-e^{f(x)}}{(x-f(x))}$ is equal to ___________.

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