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NDA 2013
Exam
NDA
Year
2013
Subject
Mathematics
Chapter
Three Dimensional Geometry

NDA 2013 · Mathematics · Three Dimensional Geometry

If θ is the acute angle between the diagonals of a cube, then which...

If θ\theta is the acute angle between the diagonals of a cube, then which one of the following is correct?

Options

  1. Aθ=30∘\theta=30^{{}^{\circ}}
  2. Bθ=45∘\theta=45^{{}^{\circ}}
  3. C2cos⁡θ=12 \cos \theta=1
  4. D3cos⁡θ=13 \cos \theta=1

Correct answer

D. 3cos⁡θ=13 \cos \theta=1

Step-by-step solution

If ' θ\theta ' be the angle between the diagonals of a cube, then cos⁡(θ)=1(−1)+(−1)(1)+1(1)1+1+1⋅1+1+1=−13⋅3=−13θ=cos⁡−1(−13)=cos⁡−1(13)\begin{aligned} \cos (\theta) & =\frac{1(-1)+(-1)(1)+1(1)}{\sqrt{1+1+1} \cdot \sqrt{1+1+1}} \\ & =\frac{-1}{\sqrt{3} \cdot \sqrt{3}}=-\frac{1}{3} \\ \theta & =\cos ^{-1}\left(-\frac{1}{3}\right)=\cos ^{-1}\left(\frac{1}{3}\right) \end{aligned} or, 3cos⁡(θ)=13 \cos (\theta)=1.
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