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Mathematics · 2022 · 2 marks
Three Dimensional GeometryDirection Cosines and Direction Ratios of a Line2 marksApplyvery short answer
CBSE 2022 · Region 5 · Set 1 · Q5
If a line makes an angle α, β, γ with the coordinate axes, then find the value of cos 2 α+ cos 2 β+ cos 2 γ.
If a line makes an angle $\displaystyle \alpha, \beta, \gamma$ with the coordinate axes, then find the value of $\displaystyle \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma$.
Marking-scheme solution
\[\begin{aligned}
& l=\cos \alpha, m=\cos \beta, n=\cos \gamma \Rightarrow \cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1 \\
& \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma \\
& \quad=2 \cos ^{2} \alpha-1+2 \cos ^{2} \beta-1+2 \cos ^{2} \gamma-1 \\
& \quad=2\left(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma\right)-3 \\
& \quad=2-3 \\
& \quad=-1
\end{aligned}
\]
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