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The document you uploaded is a 2020 entrance exam for the University of Tsukuba's Graduate School of Systems and Information Engineering. It contains questions related to Mathematics, Microeconomics, and Urban and Regional Planning.

Let's work on the mathematics section of the exam. Here’s a breakdown of the questions from the Mathematics part:

### Problem I
1. **Relationship between \( \beta \in \mathbb{R} \) and the dimension of the linear space \( L \)**: You are asked to explain how the value of \( \beta \) affects the dimension of the space spanned by the given vectors.
2. **Basis for the linear space \( L \) when its dimension is 2**: Determine a specific \( \beta \) that makes the dimension of \( L \) equal to 2 and find a basis for this space.
3. **Expression of \( L \) and Euclidean distance function**: Using the basis found in (2), express \( L \) and find the Euclidean distance function between a vector depending on \( \gamma \) and the points on \( L \).
4. **Minimization of the Euclidean distance**: Find the vector \( \hat{r}(\gamma) \) that minimizes the Euclidean distance for any \( \gamma \in \mathbb{R} \).
5. **Eigenvector and eigenvalue problem**: When \( \hat{r}(\gamma) \) is an eigenvector of a given matrix, find the corresponding \( \gamma \) and its eigenvalue.

### Problem II
1. **Limit and continuity**: Determine the right-hand limits of two functions as \( x \) approaches 0, and the conditions for the functions to be continuous at \( x = 0 \).
2. **Definition of derivative and application**: Provide the definition of a derivative, its interpretation, and an application to social problems. Then, express a given limit as a derivative.
3. **Graphing and integration**: Draw the graph of a given function, compute an improper integral, and generalize this result for an arbitrary natural number.

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Explore and solve advanced mathematics problems from the University of Tsukuba's 2020 entrance exam for Graduate School. Includes questions on linear algebra, vector spaces, limits, derivatives, and integration. Suitable for graduate-level students.

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