Each of the angles and that a given line makes with the positive … - Vidyadip

MCQ

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

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

✓

Answer

A. $\frac{3 \pi}{4}$
$\beta=\frac{\alpha}{2}, \gamma=\frac{\alpha}{2}$
$\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$
$\cos ^{2} \alpha+2 \cos ^{2} \frac{\alpha}{2}=1$
$\cos ^{2} \alpha+\cos \alpha=0$
$\cos \alpha(\cos \alpha+1)=0$
$\cos \alpha=0,-1$
$\alpha=\frac{\pi}{2}, \pi$
Now $\beta=\frac{\alpha}{2} \Rightarrow \frac{\pi}{4}, \frac{\pi}{2}$
so sum is $\frac{3 \pi}{4}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "SECTION - A [MATHS - MCQ]" from JEE Main 3-April-2025 Paper - Shift 2→JEE Main 3-April-2025 Paper - Shift 2 — all question types→JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $ a, b, c$ are non-zero vectors such that $a\,\,.\,\,b = a\,\,.\,\,c,$ then which statement is true

Let $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ be a point in the first octant, whose projection in the xy-plane is the point $\mathrm{Q}$. Let $\mathrm{OP}=\gamma$; the angle between $OQ$ and the positive $\mathrm{x}$-axis be $\theta$; and the angle between $\mathrm{OP}$ and the positive $\mathrm{z}$-axis be $\phi$, where $\mathrm{O}$ is the origin. Then the distance of $\mathrm{P}$ from the $\mathrm{x}$-axis is :

Two dice are thrown. If first shows $5$, then the probability that the sum of the numbers appears on both is $8$ or more than $8$, is

If $|{z_1}|\, = \,|{z_2}|$ and $arg\,\,\left( {\frac{{{z_1}}}{{{z_2}}}} \right) = \pi $, then ${z_1} + {z_2}$ is equal to

$8^{th}$ term of the series $2\sqrt 2 + \sqrt 2 + 0 + .....$ will be

If $y = {1 \over {a - z}},$ then ${{dz} \over {dy}} = $

If ${\tan ^{ - 1}}x + {\tan ^{ - 1}}y + {\tan ^{ - 1}}z = \pi ,$ then $\frac{1}{{xy}} + \frac{1}{{yz}} + \frac{1}{{zx}} = $

Let $f :\left[\frac{1}{2}, 1\right] \rightarrow R$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f^{\prime}(x)<2 f(x)$ and $f\left(\frac{1}{2}\right)=1$. Then the value of $\int_{1 / 2}^1 f(x) d x$ lies in the interval

If $\alpha ,\beta $ and $\gamma $ are the roots of ${x^3} + px + q = 0$, then the value of ${\alpha ^3} + {\beta ^3} + {\gamma ^3}$ is equal to

If $a+b+c=1, a b+b c+c a=2$ and $a b c=3$, then the value of $a^{4}+b^{4}+c^{4}$ is equal to $....$