The method of dimensional analysis can be used to derive which of the following relations?
 a) \[\text{No}\ {{\text{e}}^{\lambda }}\] b) Asin$\left( w+kt \right)$
 c)\[\dfrac{\text{1}}{2}\text{m}{{\text{v}}^{\text{2}}}\ \text{+}\ \dfrac{\text{1}}{2}{{\text{w}}^{\text{2}}}\] d) None

HintIn this type of question where we are asked that the dimensional formula is used to derive which of the following relations, we need to observe all the options given in the question one-by-one. We know that dimensional formula for any quantity does not include the constant term in it's formula, and So, it's not easy to find the exact correct formula using dimensional formula.

Complete step-by-step solution
This problem deals with the dimensional analysis method.
Dimensional is the analysis of the relationships between different physical quantities by identifying their base quantities and units of measure and tracking these dimensions as calculations or comparisons are performed.
It is a problem solving method that uses the fact that any number or expression can be multiplied by one without changing its value. It is a useful technique.
We use dimensional analysis for three prominent reasons:
First:- Consistency of a dimensional equation.
Second:- Derive relation between physical quantities in physical phenomena.
Third:- To change units from one system to another.
Here, in question, we are asked whether any of the given formula among
\[\text{No}\ {{\text{e}}^{\lambda }}\], Asin$\left( w+kt \right)$, \[\dfrac{\text{1}}{2}\text{m}{{\text{v}}^{\text{2}}}\ \text{+}\ \dfrac{\text{1}}{2}{{\text{w}}^{\text{2}}}\]Can be derived through dimensional analysis.
None of these can be derived using dimensional analysis because dimensional analysis can only tell the dimensional quantities that constitute the formula and not the constant values, No A and $\dfrac{1}{2}$which can be determined by dimensional analysis.

So, answer is option D→none

Note:In this type of questions, we use dimensional analysis for three prominent reasons which are as follows:-
First:- Consistency of a dimensional equation.
Second:- Derive relation between physical quantities in physical phenomena.
Third:- To change units from one system to another.