Verified Solution • 2025–26 Session moderate

QJEEM24Mafd28: JEE Main 2024 Mathematics - Class 12 JEEM24Mafd28 Solution: Let be a differentiable function such that . If the lim x → 0 x …

Let f:[−π2,π2]→Rf:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathbf{R} be a differentiable function such that f(0)=12f(0)=\frac{1}{2} . If the limx→0x∫0xf(t)dtex2−1=α, then 8α28 \alpha^2 is equal to :

Multiple Choice Options

  • A 4
  • B 2
  • C 1
  • D 16

Step-by-Step Verified Solution

amp;lim⁡x→0x∫0xf(t)dt(ex2−1x2)×x2amp;lim⁡x→0∫0xf(t)dtx(lim⁡x→0ex2−1x2=1)amp;=lim⁡x→0f(x)1 (using L Hospital) amp;f(0)=12amp;α=12amp;8α2=2\begin{aligned}& \lim _{x \rightarrow 0} \frac{x \int_0^x f(t) d t}{\left(\frac{e^{x^2}-1}{x^2}\right) \times x^2} \\\\& \lim _{x \rightarrow 0} \frac{\int_0^x f(t) d t}{x} \quad\left(\lim _{x \rightarrow 0} \frac{e^{x^2}-1}{x^2}=1\right) \\\\& =\lim _{x \rightarrow 0} \frac{f(x)}{1} \text { (using L Hospital) } \\\\& f(0)=\frac{1}{2} \\\\& \alpha=\frac{1}{2} \\\\& 8 \alpha^2=2\end{aligned}

Related Learning Resources & Exam Preparation

JEE Main PYQ Question Bank
Browse year-wise authentic previous years' questions with solutions
Authentic CBT Mock Tests
Simulate official exam environment with live timer and percentile scoring