Correct option is (2) [L-1T1]
Given equation is, \(v = \omega A \cos (\omega t - kx)\)
As we know, all trigonometric functions are dimensionless. Therefore, the term \(\cos (\omega t - kx)\) should be dimensionless.
⇒ \([\omega t - kx] = [M^0L^0T^0]\)
Two quantities can be subtracted if they have same dimensions.
Therefore,
\([\omega t] = [kx] = [M^0L^0T^0]\)
Dimension of time \(t = [T]\)
⇒ \([\omega ] = \frac{[M^0L^0 T^0]}{[T]} = [T^{-1}]\)
Also, dimension of position \(x = [L]\)
⇒ \([k] = \frac{[M^0L^0T^0]}{[L]} = [L^{-1}]\)
Therefore, dimension of \(\frac kw\) is given by
\(\left[\frac k\omega\right] = \frac{[L^{-1}]}{[T^{-1}]} = [L^{-1}T^1]\)