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You've uploaded a mathematical exam page, including multiple-choice and longer problem-solving questions. Let me break down the key problems and analyze them.

### Problem 7:
You are given the quadratic curve \(x^2 + axy + 4y^2 - 7x + y + 3 = 0\). The curve is a hyperbola when \(a = \pm 4\).

This problem likely tests knowledge of conic sections and their classifications based on their discriminant or structure.

### Multiple Choice Questions:
1. The trajectory of a point determined by a fixed plane and a line perpendicular to it appears to be a **B: Cylinder**.
   
2. Given three vectors \(\vec{a}, \vec{b}, \vec{c}\) such that \(\vec{a} + \vec{b} + \vec{c} = 0\), the correct choice for the vector cross-product is **C: \(\vec{b} \times \vec{c}\)**.

3. A line passing through a point \((2, -3, 5)\) with equal direction cosines relative to the coordinate axes, the distance from point \((1, 2, 3)\) is **A: \(\frac{\sqrt{42}}{3}\)**.

4. The value of \(t\) for the line equation \(\frac{x + 1}{t} = \frac{y - 3}{1} = \frac{z + 1}{2}\) and the plane \(2x + 4y - 5z + 1 = 0\) is **D: 3**.

5. The surface described by \(xy + yz + zx + z^2 - 4 = 0\) is **A: Cylinder**.

6. For the quadratic curve \(2xy + y^2 - 2x - 1 = 0\), the correct description is **B: Center curve with a unique singular point**.

### Problem Solving:
The next part involves vector calculations, specifically the volume of a parallelepiped. The cross-product and determinant methods are applied here. I can go through the steps if you'd like further details.

Would you like more detailed explanations on any particular problem, or should I break down the vector algebra in the last part?

### Follow-up Questions:
1. How do we classify conic sections based on their discriminant?
2. Can you explain the geometry of a cylinder in three-dimensional space?
3. How do direction cosines relate to the distance between a point and a line?
4. Why is the cross-product useful in vector calculations for volumes?
5. What is the geometric interpretation of a hyperbola when \(a = \pm 4\) in the equation?

### Tip:
For vector problems, always remember that the scalar triple product of three vectors gives the volume of the parallelepiped formed by those vectors.

You are given a quadratic curve equation x^2 + axy + 4y^2 - 7x + y + 3 = 0, and the curve is a hyperbola when a = ±4. The question involves solving for key properties of the curve and analyzing conic sections.

Explore the properties of quadratic curves and conic sections through the equation x^2 + axy + 4y^2 - 7x + y + 3 = 0. Learn about hyperbolas, vector algebra, and solving for the cross product. This advanced math problem is suitable for students at the undergraduate level or advanced high school grades (11-12).

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