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.