Verified Solution • 2025–26 Session moderate

QJEEM25Ma1fe1: JEE Main 2025 Mathematics - Class 11 JEEM25Ma1fe1 Solution: If , then is equal to :

If ∑r=113{1sin⁡(π4+(r−1)π6)sin⁡(π4+rπ6)}=a3+b,a,b∈Z\sum\limits_{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi}{4}+(r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4}+\frac{r \pi}{6}\right)}\right\}=a \sqrt{3}+b, a, b \in Z , then a2+b2a^2+b^2 is equal to :

Multiple Choice Options

  • A 10
  • B 4
  • C 8
  • D 2

Step-by-Step Verified Solution

amp;amp;1sin⁡π6∑r=113sin⁡[(π4+rπ6)−(π4)−(r−1)π6]sin⁡(π4+(r−1)π6)sin⁡(π4+rπ6)amp;1sin⁡π6∑r=113(cot⁡(π4+(r−1)π6)−cot⁡(π4+rπ6))amp;=23−2=α3+bamp; So a2+b2=8\begin{aligned}&\begin{aligned}& \frac{1}{\sin \frac{\pi}{6}} \sum_{r=1}^{13} \frac{\sin \left[\left(\frac{\pi}{4}+\frac{\mathrm{r} \pi}{6}\right)-\left(\frac{\pi}{4}\right)-(\mathrm{r}-1) \frac{\pi}{6}\right]}{\sin \left(\frac{\pi}{4}+(\mathrm{r}-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4}+\frac{\mathrm{r} \pi}{6}\right)} \\& \frac{1}{\sin \frac{\pi}{6}} \sum_{\mathrm{r}=1}^{13}\left(\cot \left(\frac{\pi}{4}+(\mathrm{r}-1) \frac{\pi}{6}\right)-\cot \left(\frac{\pi}{4}+\frac{\mathrm{r} \pi}{6}\right)\right) \\& =2 \sqrt{3}-2=\alpha \sqrt{3}+\mathrm{b}\end{aligned}\\&\text { So } \mathrm{a}^2+\mathrm{b}^2=8\end{aligned}

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