📘 MATHEMATICS

1. The problem is to understand and compare quantities, a key concept in Class 8 NCERT math. 2. Comparing quantities involves finding the ratio or difference between two or more qu

1. The problem is to verify whether a given function is a metric on $\mathbb{R}^n$. 2. A function $d: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ is a metric if it satisfies t

1. **Problem:** Calculate the average of the numbers 4, 8, and 12. 2. **Formula:** The average (mean) of $n$ numbers $x_1, x_2, ..., x_n$ is given by:

1. **Problem Statement:** Define a convex function. 2. **Definition:** A function $f: \mathbb{R}^n \to \mathbb{R}$ is called convex if for all $x, y \in \mathbb{R}^n$ and for all $

1. Let's start by understanding what a function is. A function is a rule that assigns each input exactly one output. For example, if you have a function $f(x) = x^2$, it means for

1. The user asks for all formulas used in math. 2. There are many formulas across different branches of math such as algebra, geometry, calculus, and physics.

1. Тодорхойлолт: f(x) функц нь (0; \pi) завсарт өгөгдсөн байна. \n\n2. Зорилго: f(x)-г тэгш болон сондгой функц байхаар үргэлжлүүлж, Фурьегийн цуваанд задлах.\n\n3. Фурьегийн цувра

1. Дано множество A = \{1, 2, 3, 4, 5\} и множество B = \{4, 5, 6, 7\}. Найдите \(A \cup B\) (объединение множеств). 2. Для тех же множеств A и B найдите \(A \cap B\) (пересечение

1. **حل المعادلة 8x^3 - 27 = 0** - المعادلة من الشكل $$a^3 - b^3 = 0$$ حيث $$a = 2x$$ و $$b = 3$$.

1. Problem: Find $n(B)$ if $n(A \times B) = 6$ and $A = \{1, 3\}$. Formula: $n(A \times B) = n(A) \times n(B)$.

1. **Problem Statement:** Define the following terms with examples: (a) ε - δ definition of a limit of a function μ(x)

1. Let's start by understanding what it means to apply a formula in math appropriately. 2. A formula is a mathematical rule or relationship expressed using symbols and variables.

1. **Énoncé du problème :** Déterminer la valeur de vérité et la négation des propositions données.

1. **Problem Statement:** Understand the mathematical constant $e$ and its value. 2. **Definition:** The number $e$ is a fundamental mathematical constant known as Euler's number.

1. **Problem Statement:** Calculate the factorial of a very large number, such as $1000!$. 2. **Definition:** The factorial of a positive integer $n$, denoted $n!$, is the product

1. The problem is to understand what "pure maths" refers to and its scope. 2. Pure mathematics is the branch of mathematics that deals with abstract concepts and theoretical framew

1. The problem is to understand the concept of consistency in a mathematical or logical context. 2. Consistency means that a set of statements or equations does not contain any con

1. The problem asks to identify the figure represented by the number $2 \times 10^{48}$. 2. This number is written in scientific notation, where $2$ is the coefficient and $10^{48}

1. The problem is to find the general solution of a differential equation or algebraic problem (please specify the exact problem for precise solution). 2. Generally, for a differen

1. **Problème 1 : Montrer que $\frac{(4^{n+1} + 4^n)^2}{(2^{2n+1} - 2^{2n})^2} \in \mathbb{N}$ pour $n \in \mathbb{N}$.** 2. **Comparer : $2 - \sqrt{5}$ et $\frac{-1}{2 + \sqrt{5}}

1. The problem is to understand the concept of "notation" in mathematics and how to use it effectively. 2. Notation is a system of symbols and signs used to represent numbers, oper