Question

If ${a^b} = 4 - ab$ and ${b^a} = 1$, where a and are b positive integers, find a.
$
  {\text{A}}{\text{. 0}} \\
  {\text{B}}{\text{. 1}} \\
  {\text{C}}{\text{. 2}} \\
  {\text{D}}{\text{. 3}} \\
$

Answer 1

Hint- A number of the form ${b^a}$ can be equal to one only if either b = 1 or a = 0.
In this question we have been given ${a^b} = 4 - ab$ and ${b^a} = 1$
And we know that ${b^a}$ can be equal to one only if either $b = 1$ or $a = 0$.
But we are given that $a$ and are $b$ positive integers
So, $a \ne 0$ $ \Rightarrow b = 1$
Now if we put $b = 1$ in the equation ${a^b} = 4 - ab$
We get,
${a^1} = 4 - a\left( 1 \right)$
$ \Rightarrow 2a = 4$
So, $a = 2$
Here the correct answer is option (C).
Note- In these types of questions, the most important part is the domain of our variables, most of us miss this point and get struck as they give two answers. So, all the given constraints should be taken into consideration while solving the question.