Which of the following expression can be deduced on the basis of dimensional analysis? (All symbols have their usual meanings)

Option (1):
\(x=A \cos (\omega t)\)
- \(A\) : displacement \(\rightarrow[L]\)
- \(\cos (\omega t)\) : dimensionless
E) Dimensionally correct. However, dimensional analysis cannot deduce the presence of a trigonometric function like cosine.
Cannot be deduced via dimensional analysis.
Option (2):
\(N=N_0 e^{-\lambda t}\)
- \(\quad N_0\) : number of nuclei \(\rightarrow\) dimensionless
- \(\lambda\) : decay constant \(\rightarrow\left[T^{-1}\right]\)
- \(\lambda t\) : dimensionless \(\Rightarrow\) exponential argument is fine
E. Again, though dimensionally correct, the exponential form can't be derived using dimensional analysis.
Cannot be deduced via dimensional analysis.
Option (3):
\(F=6 \pi \eta r \nu\)
- \(\eta\) : dynamic viscosity \(\rightarrow\left[M L^{-1} T^{-1}\right]\)
- \(r\) : length \(\rightarrow[L]\)
- \(\nu\) : velocity \(\rightarrow\left[L T^{-1}\right]\)
\(\Rightarrow \text { RHS: }\left[M L^{-1} T^{-1}\right] \cdot[L] \cdot\left[L T^{-1}\right]=\left[M L T^{-2}\right]=\text { Force }\)
Dimensionally consistent
The form (ignoring the constant \(6 \pi\)) can be derived using dimensional analysis.
\(\checkmark\) Correct
Option (4):
None of these
Since option (3) can be deduced dimensionally, this is incorrect.
Incorrect