$\underline{\textbf{Answer:}\Rightarrow}\;\;\mathbf{A.}$
$$\begin {align}\textbf{Given:}\;\;\;\;\;\;\;\mathrm a &= \sum_{x=5}^{13}(x-9)^2\\ \text b &= \sum_{x=5}^{13}(x-10)^2 \\&=\sum_{x = 5}^{13}((x-9)-1)^2\\&=\sum_{x = 5}^{13}[(x-9)^2 + 1 -2(x-9)] \\&=\sum_{x = 5}^{13}(x-9)^2 + \sum_{x=5}^{13}1-2\sum _{x = 5}^{13}(x-9)\\&=a + \sum_{x = 5}^{13}1-2 \underbrace {\sum_{x=5}^{x = 13}(x-9)}_\text {= 0 }\\&=a + 9 -2(-4-3-2-1-0+1+2+3+4)\\&=a + 9 \end {align}$$
Now, we get:
$$\begin{align} &b = a + 9\\ &\Rightarrow b-a > 0\\ &\Rightarrow b\gt a \\ &\text{or,} ~a \lt b \end {align}$$
$\therefore \textbf A$ is the correct answer.