Integral Calculus
Integral Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Suppose $\begin{vmatrix} f'(x) & f(x) \\ f''(x) & f'(x) \end{vmatrix} = 0$ where $f(x)$ is differentiable function with $f'(x) \neq 0$ & satisfies $f(0) = 1, f'(0) = 2$. If $f(x) = e^{\lambda x} + \mu$ then $\lambda + \mu$ is ____.
Step-by-Step Solution
Key Concept: Riemann sums with careful parameter identification convert discrete geometric approximations into definite integrals.
The area of triangular sectors is approximated using Riemann sums. With $OB_k = k$ and $\angle AOB_k = \frac{k\pi}{2n}$, each sector has area $S_k = \frac{1}{2}k\sin\frac{k\pi}{2n}$. The total area is $L = \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n S_k = \frac{k}{2n^2}\sum_{n=1}^\infty \sin\frac{k\pi}{2n} = \frac{2}{\pi^2}$, obtained by converting the sum to a Riemann integral.
Correct Answer: 2