Mathematics · 3D Geometry

JEE Main 2025 — 3 April, Evening Shift — Question 28

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

Option A:
$\pi$
Option B:
$\frac{3 \pi}{4}$
Correct
Option C:
$\frac{\pi}{2}$
Option D:
$\frac{3 \pi}{2}$

Answer: B

Step-by-step solution

$\beta=\gamma=\frac{\alpha}{2}$

$\cos ^{2} \alpha+\cos ^{2} \beta \pm \cos ^{2} y=1$

$\cos ^{2} 2 \beta+\cos ^{2} \beta+\cos ^{2} \beta=1$

$\left(2 \cos ^{2} \beta-1\right)^{2}+\left(2 \cos ^{2} \beta-1\right)=0$

$\left(2 \cos ^{2} \beta-1\right)\left[\left(2 \cos ^{2} \beta-1+1\right]=0\right.$

$\cos ^{2} \beta=\frac{1}{2}$ or $\cos ^{2} \beta=0$

$\therefore \quad \beta=\frac{\pi}{4}, \frac{\pi}{2}$

$\therefore$ sum $=\frac{\pi}{4}+\frac{\pi}{2}=\frac{3 \pi}{4}$

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 3 April, Evening Shift
Subject: Mathematics
Chapter: 3D Geometry
Topic: Direction Cosines and Direction Ratios

The distance of the point (7,10,11) from the line x-4/1=y-4/0=z-2/3 along the line x-9/2=y-13/3=z-17/6 is

Answer: B

Step-by-step solution

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