JEE Mathematics 2025 Question with Solution

Given: Let the angle with the positive $x$-axis be $\alpha$. Then $\beta = \frac{\alpha}{2}$ and $\gamma = \frac{\alpha}{2}$.

Find: The sum of all possible values of $\beta$.

For a line in three dimensions, if $l = \cos \alpha$, $m = \cos \beta$, and $n = \cos \gamma$ are the direction cosines, then

Substitute $\beta = \frac{\alpha}{2}$ and $\gamma = \frac{\alpha}{2}$:

So,

Using

we get

which simplifies to

Hence,

that is,

Therefore, either

or

Case 1:

Then

Case 2:

Then

So the possible values of $\beta$ are $\frac{\pi}{4}$ and $\frac{\pi}{2}$. Their sum is

Therefore, the correct option is A.

Given: $\beta = \frac{\alpha}{2}$ and $\gamma = \frac{\alpha}{2}$.

Find: All possible values of $\beta$ and their sum.

Start from the direction cosine identity:

Substituting the given relations,

Now use

So,

which gives

Hence,

Thus,

1. $\cos \alpha = 0$
2. $\cos \alpha = -1$

From the range of angles with positive axes, take $0 \le \alpha, \beta, \gamma \le \pi$.

If

then

and hence

If

then

and hence

Therefore the possible values are $\frac{\pi}{4}$ and $\frac{\pi}{2}$, and the required sum is

So the answer is $\frac{3\pi}{4}$.