Mathematics Recruitment Test: Riemann Sums, Linear Algebra, and Integration

Math Problem Statement

Using Riemann sums, calculate limits, determine basis and matrix transformations in linear algebra, and evaluate integrals using substitution techniques.

Solution

The image contains three exercises from a mathematics recruitment test. Here's a breakdown of the tasks in each exercise:

Exercise 1: Riemann Sums and Limits

Using Riemann sums, calculate the following limits:

$\lim_{n \to \infty} \left( \sum_{k=1}^n \frac{1}{n^2 + k^2} \right)$
$\lim_{n \to \infty} \left( \sum_{k=1}^n \frac{k}{n^2 + k^2} \right)$
$\lim_{n \to \infty} \left( \sum_{k=1}^{n-1} \frac{k}{\sqrt{n^2 - k^2}} \right)$

Exercise 2: Linear Algebra - Endomorphism in $\mathbb{R}^3$

The endomorphism $f$ in $\mathbb{R}^3$ is represented by the matrix:

1 & 2 & -2 \\ 2 & 1 & -2 \\ 2 & 2 & -3 \end{pmatrix}$$ Tasks: 1. Prove that $$E_1 = \{ (x, y, z) \in \mathbb{R}^3 \ | \ f(x, y, z) = 0 \}$$ is a subspace of $$\mathbb{R}^3$$ and determine its basis. 2. Given $$b = (0, 1, 1)$$ and $$c = (1, 4, 4)$$, verify that $$b$$ and $$c$$ are linearly independent and form a basis of $$E_1$$. 3. With $$a = (1, 1, 1)$$, prove that $$B' = \{ a, b, c \}$$ is a basis of $$\mathbb{R}^3$$. 4. Find the transition matrix $$P$$ from basis $$B$$ to basis $$B'$$. 5. Calculate $$P^{-1}$$, the inverse of the transition matrix. --- ### **Exercise 3**: Integration with Variable Substitution Using substitution techniques, compute the following integrals: 1. $$I = \int_0^{\ln 2} \sqrt{1 + e^x} \, dx, \quad \text{(substitute } t = \sqrt{1 + e^x})$$ 2. $$J = \int_0^1 \sqrt{\frac{1 - x^2}{1 + x^2}} \, dx, \quad \text{(substitute } t = \cos x)$$ 3. $$K = \int_0^1 \frac{1}{1 + \sqrt{1 + x}} \, dx, \quad \text{(substitute } t = 1 + \sqrt{1 + x})$$ --- Would you like me to solve these problems step by step, or should I start with specific parts? Let me know how you'd like to proceed!

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Math Problem Analysis

Mathematical Concepts

Riemann Sums
Limits
Linear Algebra
Subspaces
Basis
Matrix Transformation
Variable Substitution
Integration

Formulas

Riemann sum definition
Limit formula for sequences and sums
Matrix operations and transformations
Change of basis matrix
Integration by substitution formula

Theorems

Basis and dimension theorem
Linear independence theorem
Inverse of a matrix

Suitable Grade Level

Undergraduate Level

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