2018/19 AHSMC Part 1 Problem 9

The given inequality may be written as follows: $$kx^2 \le 10 \iff x^2 \le \frac{10}{k} \iff x \in \left[ -\sqrt{\frac{10}{k}},\ \sqrt{\frac{10}{k}} \right].$$ The inequality $kx^2 \le 10$ has exactly five positive integer solutions if and only if $$\sqrt{\frac{10}{k}} \in [5,6) \iff \frac{10}{k} \in [25,36) \iff \frac{1}{k} \in [2.5, 3.6).$$ Since the inequality $kx \le 1$ is equivalent to $x \le \dfrac{1}{k}$, the number of positive integer solutions is two if $\dfrac{1}{k} \in [2.5, 3)$ and three if $\dfrac{1}{k} \in [3, 3.6)$. The answer is (B).