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JEE Main 20253 AprilEvening Shift
Exam
JEE Main
Year
2025
Date
3 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Three Dimensional Geometry

JEE Main 2025 · 3 April · Evening Shift · Mathematics · Three Dimensional Geometry

Each of the angles β and γ that a given line makes with the...

Each of the angles β\beta and γ\gamma that a given line makes with the positive y - and z -axes, respectively, is half of the angle that this line makes with the positive x -axes. Then the sum of all possible values of the angle β\beta is

Options

  1. A3π4\frac{3 \pi}{4}
  2. Bπ\pi
  3. Cπ2\frac{\pi}{2}
  4. D3π2\frac{3 \pi}{2}

Correct answer

A. 3π4\frac{3 \pi}{4}

Step-by-step solution

<br/>β=α2,γ=α2cos⁡2α+cos⁡2β+cos⁡2γ=1cos⁡2α+2cos⁡2α2=1cos⁡2α+cos⁡α=0cos⁡α(cos⁡α+1)=0cos⁡α=0,−1α=π2,π<br/>\begin{aligned}<br/>& \beta=\frac{\alpha}{2}, \gamma=\frac{\alpha}{2} \\ & \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1 \\ & \cos ^2 \alpha+2 \cos ^2 \frac{\alpha}{2}=1 \\ & \cos ^2 \alpha+\cos \alpha=0 \\ & \cos \alpha(\cos \alpha+1)=0 \\ & \cos \alpha=0,-1 \\ & \alpha=\frac{\pi}{2}, \pi<br/>\end{aligned}
Now β=α2⇒π4,π2\beta=\frac{\alpha}{2} \Rightarrow \frac{\pi}{4}, \frac{\pi}{2}
so sum is 3π4\frac{3 \pi}{4}
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