Maths Q14 – JEE-MAIN 2025

Each of the angles

β

and

γ

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

-axis. Then the sum of all possible values of the angle

β

Clue 1 of 3: Direction Cosines Relationship

Recall the fundamental relationship between the direction cosines of a line in 3D space, which states that the sum of the squares of the direction cosines is 1.

🥷

Ninja Strategy•Range Analysis

By understanding the valid range for the angle $β$ (which is $[0, π / 2]$ ), one can eliminate options that exceed the maximum possible sum of $π$ .

🧠 Full Solution Path

Step 1: Formulate the Direction Cosine Equation✦ Active

Let the angles a line makes with the positive $x$ -, $y$ -, and $z$ -axes be $α$ , $β$ , and $γ$ respectively. The fundamental relation between their cosines (direction cosines) is:

\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1

Given that $β = \frac{α}{2}$ and $γ = \frac{α}{2}$ , substitute these into the equation:

\cos^{2} α + \cos^{2} (\frac{α}{2}) + \cos^{2} (\frac{α}{2}) = 1

\cos^{2} α + 2 \cos^{2} (\frac{α}{2}) = 1

Step 2: Simplify and Solve for

\cos α

○ Expand

Use the trigonometric identity $2 \cos^{2} x = 1 + \cos (2 x)$ . For $x = \frac{α}{2}$ , this becomes $2 \cos^{2} (\frac{α}{2}) = 1 + \cos α$ . Substitute this into the equation from Step 1:

\cos^{2} α + (1 + \cos α) = 1

Simplify the equation:

\cos^{2} α + \cos α = 0

\cos α (\cos α + 1) = 0

This yields two possible values for $\cos α$ :

\cos α = 0 or \cos α = - 1

💡 Teacher's Secret Hint

Remember that angles a line makes with axes are typically in the range $[0, π]$ .

Step 3: Find Possible Values of

β

and Their Sum○ Expand

For $\cos α = 0$ , the angle $α = \frac{π}{2}$ (since $0 \leq α \leq π$ ). Then, $β_{1} = \frac{α}{2} = \frac{π / 2}{2} = \frac{π}{4}$ .

For $\cos α = - 1$ , the angle $α = π$ (since $0 \leq α \leq π$ ). Then, $β_{2} = \frac{α}{2} = \frac{π}{2}$ .

The possible values for $β$ are $\frac{π}{4}$ and $\frac{π}{2}$ . The sum of all possible values of $β$ is:

Sum = \frac{π}{4} + \frac{π}{2} = \frac{π}{4} + \frac{2 π}{4} = \frac{3 π}{4}

💡 Teacher's Secret Hint

Ensure that the calculated angles $β$ are within the valid range $[0, π / 2]$ based on the initial conditions.

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Maths Question 14 – JEE-MAIN 2025

Clue 1 of 3: Direction Cosines Relationship

🧠 Full Solution Path