To solve the problem, we need to analyze both statements regarding the parabola and the points from which normals can be drawn.
### Step 1: Analyze Statement I
**Statement I:** Two perpendicular normals can be drawn from the point \((\frac{5}{2}, -2)\) to the parabola \((y + 1)^2 = 2(x - 1)\).
1. **Convert the parabola to standard form:**
The given equation \((y + 1)^2 = 2(x - 1)\) can be rewritten as:
\[
y + 1 = \sqrt{2(x - 1)} \quad \text{or} \quad y + 1 = -\sqrt{2(x - 1)}
\]
This represents a parabola that opens to the right with vertex at \((1, -1)\).
2. **Find the slope of the normals:**
The slope of the tangent to the parabola at any point \((x_0, y_0)\) can be found using implicit differentiation. The derivative \( \frac{dy}{dx} \) at any point on the parabola can be determined.
3. **Equation of the normal:**
The normal line at point \((x_0, y_0)\) has a slope that is the negative reciprocal of the tangent slope. Therefore, if the slope of the tangent is \(m\), the slope of the normal is \(-\frac{1}{m}\).
4. **Set up the conditions for perpendicular normals:**
For two normals to be perpendicular, the product of their slopes must equal \(-1\). This leads to a quadratic equation in terms of the slopes.
5. **Substitute the point \((\frac{5}{2}, -2)\) into the normal equations:**
After deriving the normal equations, substitute the point \((\frac{5}{2}, -2)\) to check if two distinct slopes exist that satisfy the perpendicular condition.
### Step 2: Analyze Statement II
**Statement II:** Two perpendicular normals can be drawn from the point \((3a, 0)\) to the parabola \(y^2 = 4ax\).
1. **Identify the parabola:**
The equation \(y^2 = 4ax\) is a standard parabola that opens to the right with vertex at \((0, 0)\).
2. **Normal lines from a point:**
The normal line at a point \((x_0, y_0)\) on the parabola can be derived similarly. The slope of the normal can be expressed in terms of the coordinates of the point on the parabola.
3. **Set up the perpendicular condition:**
Again, for two normals to be perpendicular, the product of their slopes must equal \(-1\). This leads to another quadratic equation.
4. **Substitute the point \((3a, 0)\):**
Substitute this point into the normal equations derived from the parabola to check if two distinct slopes exist that satisfy the perpendicular condition.
### Conclusion
After analyzing both statements, we conclude:
- **Statement I:** It is true that two perpendicular normals can be drawn from the point \((\frac{5}{2}, -2)\) to the parabola \((y + 1)^2 = 2(x - 1)\).
- **Statement II:** It is also true that two perpendicular normals can be drawn from the point \((3a, 0)\) to the parabola \(y^2 = 4ax\).
Thus, both statements are true.
