JEE Main 2025 (Online) 3rd April Evening ShiftQuestion 1
Each of the angles $\beta$ and $\gamma$ 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 $\beta$ is
3D Geometry
Solution
$\alpha=2\beta$, $\gamma=\beta$ (given $\beta=\gamma$). Use $\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1$: $\cos^22\beta+2\cos^2\beta=1$. Let $t=\cos2\beta$: $t^2+(1+t)=1\Rightarrow t(t+1)=0$. $t=0$: $2\beta=\frac\pi2$, $\beta=\frac\pi4$. $t=-1$: $2\beta=\pi$, $\beta=\frac\pi2$ (direction $-\hat i$, valid). Other roots such as $\beta=\frac{3\pi}4$ give $\alpha=\frac{3\pi}2>\pi$, invalid. Sum $=\frac\pi4+\frac\pi2=\frac{3\pi}4$. Option (C).