JAMIA 2023 — Mathematics PYQ

1. Understanding the Exponent

The term in the exponent is $x-[x]$, which is the fractional part of $x$, denoted as $\{x\}$. This function behaves very differently from a standard linear $x$.

• On the interval $[0,1)$, $\{x\}=x$.

• On the interval $[1,2)$, $\{x\}=x-1$.

• This pattern repeats every integer.

2. The Property of Periodic Functions

The function $e^{\{x\}}$ is periodic with period $T=1$. A fundamental property of definite integrals for periodic functions is:

Applying this to your problem where $n=1000$ and $T=1$:

3. Calculating the Single Unit

Between $0$ and $1$, the fractional part $\{x\}$ is simply $x$.

4. Conclusion

Multiplying the result of one period by the total number of periods (1000):