JEE Main 2025 Apr 3 Shift 2, Mathematics Q2: Lines in Space

Let $\alpha$ be the angle with the positive x-axis. Given $\beta =\gamma =\alpha /2$. Direction cosines satisfy:

${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$.

Substitute $\beta =\gamma =\alpha /2$:

${\cos }^2\alpha +2{\cos }^2(\alpha /2)=1$.

Using $2{\cos }^2(\alpha /2)=1+\cos \alpha$:

${\cos }^2\alpha +\cos \alpha =0\Rightarrow \cos \alpha (\cos \alpha +1)=0$.

So $\cos \alpha =0$ or $\cos \alpha =-1$, giving $\alpha =\pi /2$ or $\alpha =\pi$.

Correspondingly $\beta =\alpha /2=\pi /4$ or $\pi /2$.

Sum of possible values of $\beta =\frac{\pi }{4}+\frac{\pi }{2}=\frac{3\pi }{4}$.