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!