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If θ is the angle, in degrees, between the longest diagonal of the cube and any one of the edges of the cube, then, cos θ =
1. \(\frac{1}{2}\)
2. \(\frac{\sqrt3}{2}\)
3. \(\frac{1}{\sqrt3}\)
4. \(\frac{1}{\sqrt2}\)

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Correct Answer - Option 3 : \(\frac{1}{\sqrt3}\)

Concept:

The longest diagonal would be from one corner vertex to the diagonally opposite corner vertex.

The diagonal of a square face of cube, a side of the cube and the longest diagonal will form a right angled triangle with longest diagonal as the hypotenuse.

Formula used:

 cos θ = Base/Hypotenues 

Calculation:

Length of diagonal of a side = √(a2 + a2) = a√2

Length of the longest diagonal = √[a2 + (a√2)2] = a√3

Then, cos θ = Base/Hypotenues = a/a√3 = 1/√3

∴ The required result will be 1/√3.

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