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JEE Main 31-Jan-2024 Paper - Shift 1 — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEEJEE Main 31-Jan-2024 Paper - Shift 14 Marks
Question
Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$ and the angle between $\vec{b}$ and $\vec{c}$ is $\alpha$, then $192 \sin ^2 \alpha$ is equal to$........$
✓

Answer

$\overrightarrow{ b } \cdot \overrightarrow{ c }=(2 \overrightarrow{ a } \times \overrightarrow{ b }) \cdot \overrightarrow{ b }-3|b|^2$
$|b|| c | \cos \alpha=-3|b|^2$
$| c | \cos \alpha=-12, \text { as }| b |=4$
$\overrightarrow{ a } \cdot \overrightarrow{ b }=2$
$\cos \theta=\frac{1}{2} \Rightarrow \theta=\frac{\pi}{3}$
$| c |^2=|(2 \overrightarrow{ a } \times \overrightarrow{ b })-3 \overrightarrow{b}|^2$
$=64 \times \frac{3}{4}+144=192$
$|c|^2 \cos ^2 \alpha=144$
$192 \cos ^2 \alpha=144$
$192 \sin ^2 \alpha=48$

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