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This image contains multiple math problems from various areas such as geometry, calculus, linear algebra, and analysis. Here's a breakdown of the problems presented:

1. **Triangle Area (3D Coordinates)**: Finding the area of the triangle given by three vertices \(A(1,2,1)\), \(B(-3,4,6)\), and \(C(1,8,3)\).

2. **Volume and Surface Area of Parallelepiped**: Using vectors from four points \(A(1,2,5)\), \(B(4,8,1)\), \(C(-3,2,3)\), and \(D(0,3,9)\) to find the volume and surface area.

3. **Angle between Two Vectors**: Proving the relationship between the angle \(\theta\) between vectors \(\mathbf{u}\) and \(\mathbf{v}\), i.e., \(\tan \theta = \frac{\|\mathbf{u} \times \mathbf{v}\|}{\mathbf{u} \cdot \mathbf{v}} \).

4. **Systems of Linear Equations**: Solving a system using:
   - Cramer's rule
   - Gaussian elimination

5. **Limit Evaluation**: Evaluating the following limits:
   - \(\lim_{x \to 1} \frac{x^6 - 1}{x^4 - 1}\)
   - \(\lim_{x \to 3} \frac{x^2 - 9}{3 - x}\)
   - \(\lim_{x \to 0} \frac{1 - \cos x}{\sin x}\)

6. **Mean Value Theorem**: Finding values \(a\), \(b\), \(c\), and \(d\) such that the given piecewise function satisfies the Mean Value Theorem on \([-1, 2]\).

7. **Derivatives**: Finding the derivatives of the following functions:
   - \(f(x) = \ln \left( \frac{x(x-1)}{x-2} \right)\)
   - \(y = \sqrt{\cos(\sin^2 x)}\)

Let me know which problem you'd like to start solving, or if you'd like solutions for multiple!

Math problems involving triangle area, parallelepiped volume, vector angles, systems of equations, limits, Mean Value Theorem, and derivatives.

Explore solutions to complex math problems, including finding the area of a triangle in 3D, calculating the volume of a parallelepiped, evaluating limits, and using the Mean Value Theorem. These problems cover topics in calculus, linear algebra, and vector analysis. Suitable for advanced high school or undergraduate students.

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