M2 Exam Preparation: Key Concepts and Strategies

Cristian emphasizes mastering M1 content for M2 success through structured video lessons and practice exercises.

### Complete Summary of M2: Study Strategies and Key Contents
* 📚 Cristian welcomes you to the M2 summary, where you will learn all the specific contents of this test, highlighting that **more than half of the material comes from M1**.
* 🧠 Although M2 questions are generally **less difficult**, they will be addressed effectively so that you can **answer them smoothly**.
* 📊 The structure of the video includes a **theoretical introduction** followed by DEMRE exercises, covering questions from the **PCU, PDT, and PAES**, with **cumulative guides** at the end of each thematic axis.
* 📖 A free book is offered that includes **content summaries**, **solved exercises and cumulative guides**, as well as a special section on **data sufficiency**.
* 🎯 The goal is that, by the end of the M2 overview, you will be able to **solve all the specific test questions**, with a focus on the **advanced difficulty of the M1 questions**.
* 🚀 You are invited to join the M30M Ultra Intensive in November, where you will learn how to **solve questions from both tests**, ensuring a **high score in your desired race**.
* 💰 Financial mathematics is introduced as an **essential tool for analyzing financial problems**, covering topics such as **simple and compound interest**, **mortgage loans**, and **salary deductions**.
* 📈 In the financial mathematics overview, key concepts such as **capital**, **interest rate**, and **amortization** are explained, providing practical examples to facilitate the understanding of these topics.

### Understanding Compound Interest and Financial Credits
* 📈 Compound interest is calculated not only on the **initial capital**, but also on the **accumulated interest**, which allows for a **faster growth of the capital** compared to simple interest.
* 💰 For example, if Valentina invests $1,000 at a rate of **10% annual compound interest**, her final principal after 2 years will be **$1,210**.
* 📊 In a comparison, after 5 years, simple interest would generate a principal of **$1,500**, while compound interest would reach approximately **$1,610**.
* 📉 The nominal rate is the percentage announced by the financial entity, but it does not consider the **frequency of capitalization**, while the effective rate does, resulting in **10.47%** in the case of Valentina.
* 🏠 In the area of mortgage loans, the loan amount is usually a **percentage of the value of the property** and the interest rate can be **fixed or variable**, with the CAE being a **crucial indicator for comparing costs between entities**.
* 💳 Consumer loans are loans for personal expenses, with **higher interest rates and shorter terms**, often between **6 months and 5 years**.
* 👷 In the pension system, workers must contribute **10% of their taxable salary** to their AFP, where returns vary according to the fund chosen, and it is recommended to **invest in riskier funds when younger**.
* 📊 Finally, it highlights the importance of understanding how **compound interest and other financial concepts** affect capital growth and lending, emphasizing the need for **good financial management**.

### Finance and Logarithms: Interest Calculations and Properties of Logarithms
* 💰 Erik must pay a total interest of **36,000 pesos per year** for a loan of **180,000 pesos**, which translates into **3,000 pesos per month** in 12 installments.
* 📈 A person who made a **30-day renewable term deposit** at a rate of **0.3%** and renewed with a **100% increase in the rate** will see his principal grow at the end of the two periods.
* The profit is calculated by subtracting the initial capital of **K** from the final capital, which is obtained by multiplying **K by 1.003 and then by 1.006**, resulting in an expression that represents the profit generated.
* 💵 The net salary is determined by subtracting the contributions from the taxable salary, where the total deductions are **17%** and the non-taxable salary is **150,000 pesos**.
* 🏦 A bank offers a quarterly rate of **2.5%** for time deposits, and upon depositing **2,500 pesos**, the final principal after **2 years (or 8 quarters)** is calculated using the compound interest formula.
* 📊 In the field of logarithms, it is established that the **logarithm of a positive argument and a positive base other than one** can take real values, and properties such as **addition and subtraction of logarithms** can be used to simplify calculations.
* 🔍 The relationship between logarithms, powers, and roots is explored, highlighting that the **logarithm of an argument in base B** is related to the **exponent necessary for B raised to that exponent** to be equal to the argument.
* 📉 The properties of logarithms as a function of the base are analyzed, establishing that when the base is **greater than one**, the logarithm **increases with the argument**, while if the base is **less than one**, the logarithm **decreases as the argument increases**.

### Solving Equations and Properties of Logarithms: Strategies and Examples
* 📊 By subtracting the equations, we obtain that $25 - 19 = 6$ is equivalent to $1.5 M1 - 1.5 m2$, which can be factored as **$1.5(M1 - m2)$**.
* ➗ Dividing both sides by (1.5), it is found that **(6 / 1.5 = 4)**, confirming that the difference sought is **4**, resulting in option **B** as the correct answer.
* 🔍 In the case of logarithms, it is established that $log_8(2) = x$, which implies that $8^x = 2$. By equating bases, it is concluded that **$x = \frac{1}{3}$**.
* Using properties of logarithms, it is shown that the **logarithm of a root** can be expressed as a **multiplication of logarithms**, allowing to **simplify the expression**.
* 📝 In an exercise on logarithms, it is stated that if **log(log_M(P)) = 3**, the logarithm can be expressed in terms of **log(M)** and **log(P)** for ease of calculation.
* 📈 For the logarithm of periodic (0.1) in base 3, it is transformed to a fraction, allowing to calculate that the result is **(-2)**.
* 📊 A problem on bacterial growth is presented, where it is stated that the **initial population of bacteria doubles every hour**, leading to the conclusion that **B = 4**.
* 📝 Finally, the compatibility of systems of equations is discussed, highlighting the cases of **unique, infinite or unsolvable solutions**, and a summary of the **power function and its graphical representation** is given.

### Homothety and Potential Functions: Graphs and Exercises
* 📊 In this video potential functions are evaluated, starting with the evaluation of f(-1) and f(1), where both results are **-1** when solving **-1 raised to -6**.
* The functions are plotted and the asymptotes at the evaluated points are identified, emphasizing that **both points are relevant to the graph**.
* 📈 Three functions are introduced: f(x) = x³, g(x) = x², and h(x) = x⁴, and their intersections at the points **(0,0)** and **(1,1)** are examined.
* 🔍 It is observed that for values of x between 0 and 1, the function **x² is greater than x³**, which in turn is **greater than x⁴**, while for values greater than 1, the relationship is **reversed**.
* ✏ Exercises on function inequalities are solved, confirming that for **x > 1**, **f(x) < g(x) < h(x)** is true.
* 📏 When graphing functions with different homothety ratios, it is explained how **larger or smaller figures** are obtained depending on the value of the ratio.
* 🔄 The properties of homothety are discussed, including how to calculate areas and perimeters of homothetic figures, highlighting that the **area is multiplied by the ratio squared**.
* 📐 Finally, practical exercises are carried out to apply the concepts of **homothety** and it is concluded with examples of how to **determine the homothety ratio** and the **properties of the resulting figures**.

### Homothety and Trigonometric Ratios: Concepts and Exercises
* 📐 In the homothety analysis, it is determined that the **vertex of the triangle is the center and the right angle**, which allows us to **rule out certain transformation options**.
* 🔄 It is identified that the homothetic figure is generated from an **inverse homothety**, where the **negative ratio** indicates that the small triangle is located on the **opposite side** of the large triangle.
* 📏 When calculating the homothetic ratio between two rectangles, it is established that the **homothetic side of 3 units** is related to the **original side of 7 units**, resulting in a **positive ratio**.
* 📊 In the case of a triangle ABC with homothety applied, it is concluded that the side **AB is parallel to the segment PQ**, confirming the relationship between the figures.
* 📐 When applying homothety to a square in the first quadrant, it is established that if the ratio is **positive**, the resulting figure will **remain on the same side** with respect to the center of homothety.
* 🔍 A lens that produces an inverted image is examined and it is determined that the **homotopy ratio must be greater than one**, discarding options that do not meet this condition.
* 📏 In the study of variance and standard deviation, it is explained that the variance is calculated by **subtracting each data from the average**, **squaring it**, and **averaging the result**.
* 📈 It is concluded that the **standard deviation is the square root of the variance**, providing a measure of **dispersion of the data in the same units as the original set**.

### Probability, Standard Deviation and Combinatorics in Statistical Data
* 📊 In solving statistical problems, one begins by establishing equations that relate the data and their respective variables, as in the case of the sum of **"a" and "b" that resulted in 11**.
* 📈 The variance is calculated from the difference between each data and the average, and is expressed as the **sum of the squares of these differences** divided by the total data.
* 📉 When calculating the standard deviation, it is important to remember that it is **not altered by adding a constant value to all the data**, which simplifies the calculation process.
* 💡 In the course analysis, it was observed that although the averages are the same, the **standard deviations may differ**, indicating that the **dispersion of the grades is not the same**.
* 🔍 In conditional probability problems, the formula that relates the **joint probability of two events** to the probability of **one given the other** is used, as in the case of Cristian and his followers.
* 📋 The organization of data in double-entry tables facilitates the calculation of probabilities by allowing the **relationships between different groups to be clearly visualized**.
* In combinatorics, **additive and multiplicative principles** are applied to calculate the **total number of options available** in situations such as the choice of **juice flavors** and **glass sizes**.
* 🔗 Finally, when dealing with complex problems, it is essential to **break them down into simpler parts** and use **known formulas and properties** to arrive at an **effective solution**.

### Combinations and Permutations: Counting Methods in Mathematics
* 🔢 To form four-digit numbers that are even using repeated digits, the thousand, hundred, ten, and unit positions must be considered, where the **unit must be even**.
* 📊 There are four options for the thousand, hundred and ten positions, but only **two options (2 and 4) for the unit**, resulting in a total of **128 possible combinations**.
* 🔄 The simple permutation allows the ordering of distinct elements and is calculated as a **factorial n**; for example, **4! equals 24**.
* 👥 The permutation with repetition is used when some elements are repeated, calculated as **n! divided by the factorial of the repetitions**.
* 🔄 In circular permutation, the number of ways to order n elements is **(n-1)!**, as in the case of four people at a round table giving **6**.
* 📈 Combinations are used when the **order does not matter**; for example, in a soccer tournament with 8 teams, **28 combinations of 2 teams** can be formed.
* 🍹 In combinations with repetition, the formula is **(n + k - 1) over k**, as in the case of choosing **2 drinks of 4 types**, resulting in **10 forms**.
* 🏆 Finally, the variation considers the order and is calculated as **n!/(n-k)!**, and in the case of an 8-team tournament for the first 2 places, there are **56 ways to do it**.

### Combinatorial and Binomial Models: Strategies and Exercises
* 📊 It starts with the solution of a factorial problem, where $ \frac{6!}{4!} $ is calculated, simplifying to **30** and confirming that the correct answer is **option C**.
* 🔤 Then, an exercise on the formation of words with **9 different letters** is presented, where combinations of **2, 3, and 4 letters** must be considered, using variations **without repetition**.
* 🔄 The importance of the **order in the combinations** is explained and it is emphasized that the **number of words formed** is equal to the sum of the variations of **9 over 2, 3 and 4**, using the factorial formula.
* 🚪 A new problem is presented on the organization of people crossing a door, where the simple permutation must be calculated after **removing one tenth of the participants**.
* 🚗 An exercise on the choice of automobiles is addressed, applying the **multiplicative principle** to calculate the combinations of makes, models, and colors, resulting in a total of **30 options**.
* 🎾 In a tennis tournament, we seek to determine how many matches are arranged between n players, using combinations and the formula **$ \frac{n(n-1)}{2} $** to obtain the correct answer.
* 📚 A problem on the selection of books from different disciplines is presented, where combinations are applied to determine the ways to **choose books from each category**, resulting in **option B**.
* 🔑 Finally, binomial models are analyzed, explaining their application in situations of **success and failure**, and exercises on dice throwing and book sales are solved, using the **binomial distribution formula** to calculate probabilities.

Resumen completo PAES M2 Regular 2024

Matemática en 30 minutos

Introducción al resumen de la M2 y su estructura

Presentación del video y su objetivo: aprender los contenidos específicos de la prueba M2.

Énfasis en que la prueba M2 incluye más de la mitad de contenido de la M1, pero con mayor dificultad.

Estructura del video: contenido teórico y ejercicios DEMRE (PCU, PDT, PAES).

Guías acumulativas al finalizar cada eje temático.

Libro gratuito descargable en PDF con resumen, ejercicios y guías acumulativas.

Apartado especial sobre suficiencia de datos en la descripción.

Repaso de los números reales (naturales, enteros, racionales, irracionales).

Énfasis en la suma de un racional con un irracional, resultando siempre un irracional.

Ejercicios de números reales con ejemplos del DEMRE.

Importancia de descartar alternativas para ahorrar tiempo.

Gráfico mostrando la proporción de preguntas de M1 y M2 en la prueba.

Invitación al Ultra Intensivo M30M (noviembre).

Incluye test diagnóstico, 50 horas de matemática en vivo, y alrededor de 1000 ejercicios.

Precio y beneficios de la inscripción temprana.

Contenido del Ultra Intensivo: M1 y M2, dificultad avanzada, mini ensayos.

Matemática financiera: conceptos y ejercicios

Introducción a la matemática financiera y su utilidad.

Contenidos: interés simple y compuesto, créditos hipotecarios y de consumo, AFP, sueldos y descuentos.

Explicación e ejemplos de interés simple y compuesto.

Explicación de créditos hipotecarios y sus conceptos.

Explicación de conceptos importantes relacionados con las AFP.

Explicación de sueldos, renta imponible y descuentos para trabajadores dependientes e independientes.