1. The problem states that the graph of $y = (mx + n)^2 - 1$ passes through the origin $(0,0)$ and we need to determine which statements about $m$ and $n$ are true. 2. Since the graph passes through the origin, substitute $x=0$ and $y=0$ into the equation: $$0 = (m \cdot 0 + n)^2 - 1 = n^2 - 1$$ 3. From this, we get: $$n^2 - 1 = 0 \implies n^2 = 1$$ 4. So statement II ($n^2 = 1$) must be true. 5. The vertex of the parabola given by $y = (mx + n)^2 - 1$ is at the point where the expression inside the square is zero: $$mx + n = 0 \implies x = -\frac{n}{m}$$ 6. Substituting back to find the vertex $y$-coordinate: $$y = (m(-\frac{n}{m}) + n)^2 - 1 = 0 - 1 = -1$$ 7. Therefore, the vertex coordinates are: $$\left(-\frac{n}{m}, -1\right)$$ 8. Statement III claims the vertex is at $(-n, -1)$, which is false unless $m=1$. 9. Regarding statement I ($m > 0$), the problem does not provide information to conclude the sign of $m$, so it is not necessarily true. 10. Hence, only statement II is always true. Final answer: A. II only