A man born in the first half of the nineteenth century was x years old in the year x2. He was born in ?

Answer: Option A

Clearly, the man was born between 1800 and 1850 is 1849.
$$\text{And, }1849={\left(43\right)}^2$$
So, the man was 43 years old in 1849
$$\begin{array}{rl}& \text{Thus, he was born in }\\ & =\left(1849-43\right)\\ & =1806\end{array}$$