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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Three Dimensional Geometry

NDA 2017 · Mathematics · Three Dimensional Geometry

If α, β and γ are the angles which the vector OP (O being the...

If α,β\alpha, \beta and γ\gamma are the angles which the vector OP→\overrightarrow{\mathrm{OP}} (O being the origin) makes with positive direction of the coordinate axes, then which of the following are correct? 1. cos⁡2α+cos⁡2β=sin⁡2γ2.sin⁡2α+sin⁡2β=cos⁡2γ3.sin⁡2α+sin⁡2β+sin⁡2γ=2\cos ^2 \alpha+\cos ^2 \beta=\sin ^2 \gamma 2. \sin ^2 \alpha+\sin ^2 \beta=\cos ^2 \gamma 3. \sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma=2 Select the correct answer using the code given below.

Options

  1. A11 and 22 only
  2. B22 and 33 only
  3. C11 and 33 only
  4. D1,21,2 and 33

Correct answer

C. 11 and 33 only

Step-by-step solution

We know, cos⁡2α+cos⁡2β+cos⁡2γ=1⇒cos⁡2α+cos⁡2β=1−cos⁡2γ=sin⁡2γ∴\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1\Rightarrow \cos ^2 \alpha+\cos ^2 \beta=1-\cos ^2 \gamma=\sin ^2 \gamma\therefore Statement 1 is correct. Now, cos⁡2α+cos⁡2β+cos⁡2γ=1\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1⇒1−sin⁡2α+1−sin⁡2β+1−sin⁡2γ=1&⇒3−(sin⁡2α+sin⁡2β+sin⁡2γ)=1&⇒sin⁡2α+sin⁡2β+sin⁡2γ=2\begin{aligned} & \Rightarrow 1-\sin ^2 \alpha+1-\sin ^2 \beta+1-\sin ^2 \gamma=1 \& \Rightarrow 3-\left(\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma\right)=1 \& \Rightarrow \sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma=2\end{aligned}∴\therefore Statement 3 is correct.
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