Mathematics 🧮

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01 Activity Questions 1.1

1. Below is a list of numbers: 22, -17, 47, -2000, 0, 1, 43, 1729, 6174, -63, 100, 32, -9. How many natural numbers are there in the given list? a) 6 b) 7 c) 8 d) 9 Solution Based on the sources, the set of natural numbers is denoted by N. This set includes 0, 1, 2, 3, 4, 5, and so on. The sources explicitly state that whenever they are talking about natural numbers, it always includes a 0, even though some books may not. Natural numbers are primarily used for counting.

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02 Comprehensive Course on Sets and Functions

1. Numbers and Basic Operations Definition: Natural Numbers (ℕ) The set of counting numbers starting from 0. ℕ = {0, 1, 2, 3, 4, ...} Definition: Integers (ℤ) All positive, negative whole numbers, and zero. ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...} 1.1 Arithmetic Operations Definition: Basic Arithmetic Operations Addition (+): Combining two or more numbers into a single number. Subtraction (-): Finding the difference between two numbers. Multiplication (×): Repeated addition of the same number. Division (÷): Repeated subtraction or splitting into equal parts. Modulo (mod): The remainder when one number is divided by another. Example: Arithmetic Operations 1. 5 + 2 = 7 2. 9 - 4 = 5 3. 3 × 4 = 12 (adding 3 four times) 4. 18 ÷ 3 = 6 (dividing 18 into 3 equal parts) 5. 10 mod 3 = 1 (when 10 is divided by 3, the remainder is 1) Practice Problems: Arithmetic Operations Try These Problems:

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03 natural numbers and the operations

Here’s an explanation of natural numbers and their operations in a notes format, designed for ease of understanding with emojis: Natural Numbers and Operations 🔢 Natural Numbers (N) Natural numbers are primarily used for counting objects (e.g., 7 balls ⚽️⚽️⚽️⚽️⚽️⚽️⚽️, 7 pencils ✏️✏️✏️✏️✏️✏️✏️). They are denoted by the symbol N. In this context, the set of natural numbers includes 0 (i.e., {0, 1, 2, 3, 4, …}). 💡 Historically, there can be some confusion in different books about whether 0 is included, but here, it always is. Arithmetic Operations

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04 Set theory

Here’s an explanation of set theory, presented in a notes format with emojis for easy understanding: Set Theory 📚 What is a Set? At its most basic level, a set is a collection of items. These items are called elements or members. Sets can be finite (e.g., days of the week 🗓️, months in a year) or infinite (e.g., the set of integers 🔢). Key Characteristics of Sets ✨

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05 Sets and Functions:- Detailed Course with Examples and Practice

1. Numbers and Basic Operations 1.1 Natural Numbers and Integers Natural Numbers ($\mathbb{N}$): Counting numbers starting from 0. $\mathbb{N} = {0, 1, 2, 3, 4, …}$ Integers ($\mathbb{Z}$): All positive, negative whole numbers, and zero. $\mathbb{Z} = {…, -3, -2, -1, 0, 1, 2, 3, …}$ 1.1.3 Arithmetic Operations Operation Description Example Addition (+) Combine numbers $5 + 2 = 7$ Subtraction (-) Find the difference $9 - 4 = 5$ Multiplication (×) Repeated addition $3 × 4 = 12$ Division (÷) Repeated subtraction $18 ÷ 3 = 6$ Modulo (mod) Remainder after division $10 \mod 3 = 1$ Practice:

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06 Sets and Functions:- Enhanced Course with Interactive Elements

1. Numbers and Basic Operations 1.1 Natural Numbers and Integers **Definition (Natural Numbers - $\mathbb{N}$):** The set of counting numbers starting from 0. $\mathbb{N} = \{0, 1, 2, 3, 4, ...\}$ **Definition (Integers - $\mathbb{Z}$):** All positive/negative whole numbers and zero. $\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$ 1.2 Rational and Real Numbers **Theorem (Rational Numbers - $\mathbb{Q}$):** Numbers of the form $\frac{p}{q}$ where $p, q \in \mathbb{Z}$ and $q \neq 0$. *Example:* $\frac{2}{5}$, $\frac{10}{20} = \frac{1}{2}$ (reduced form). **Theorem (Irrational Numbers):** Cannot be expressed as $\frac{p}{q}$. *Examples:* $\sqrt{2}$, $\pi$. **Definition (Real Numbers - $\mathbb{R}$):** Union of rational and irrational numbers. 2. Sets 2.1 Set Basics **Definition (Set):** A collection of distinct objects. *Notation:* $\{1, 2, 3\}$. **Definition (Cardinality):** Number of elements in a set. *Example:* $|\{1, 2, 3\}| = 3$. 2.2 Subsets and Set Comprehension **Theorem (Subset):** $X \subseteq Y$ if every element of $X$ is in $Y$. *Example:* $\{1, 2\} \subseteq \{1, 2, 3\}$. **Definition (Set Comprehension):** Constructs a subset using a rule. *Example:* $\{x^2 \mid x \in \mathbb{Z}, x \text{ even}\}$ (squares of even integers). 3. Relations 3.1 Cartesian Product and Binary Relations **Definition (Cartesian Product):** $X \times Y = \{(x, y) \mid x \in X, y \in Y\}$. *Example:* $A = \{a, b\}, B = \{1, 2\}$ $A \times B = \{(a, 1), (a, 2), (b, 1), (b, 2)\}$. **Definition (Binary Relation):** A subset of $X \times Y$. *Example:* $R = \{(a, 1), (b, 2)\}$. 3.2 Properties of Relations Property Definition Example Reflexive $(x, x) \in R$ for all $x$ in $S$. ${(1,1), (2,2)}$ Symmetric If $(x, y) \in R$, then $(y, x) \in R$. ${(1,2), (2,1)}$ Transitive If $(x, y), (y, z) \in R$, then $(x, z) \in R$. ${(1,2), (2,3), (1,3)}$ Equivalence Reflexive, symmetric, and transitive. ${(1,1), (2,2), (1,2)}$ 4. Functions 4.1 Function Basics **Definition (Function):** A relation where each input maps to exactly one output. *Notation:* $f: X \rightarrow Y$. **Theorem (Types of Functions):** - **Injective:** Each input maps to a unique output. - **Surjective:** Co-domain equals the range. - **Bijective:** Both injective and surjective. 4.2 Function Operations **Definition (Composition):** $(f \circ g)(x) = f(g(x))$. *Example:* $f(x) = 2x$, $g(x) = x + 1$ $(f \circ g)(x) = 2(x + 1) = 2x + 2$. **Definition (Inverse Function):** $f^{-1}$ exists if $f$ is bijective. *Example:* $f(x) = 3x + 2$, $f^{-1}(y) = \frac{y-2}{3}$. 5. Visual Aids 5.1 Venn Diagrams **Example (Set Operations):** ![Venn Diagram](https://byjus.com/maths/wp-content/uploads/2021/11/Venn-Diagram-1.png) - **Union (A ∪ B):** All elements in A or B. - **Intersection (A ∩ B):** Common elements in A and B. - **Complement (A'):** Elements not in A. 6. Interactive Quizzes **Quiz 1: Sets and Relations** List all subsets of ${a, b}$.

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07 Activity Questions 1.2

1. Which of the following option(s) is(are) true? Solution Based on the sources and our conversation history, we can determine which of the given inequalities between fractions are true by finding a common denominator and comparing the numerators. Rational numbers, which include fractions, can be written in the form p/q. To compare two fractions which have different denominators, there is no way to directly compare them. The only way is to convert them into equivalent fractions such that they have the same denominator. A number that is a multiple of both denominators can be used as the common denominator. Once the denominators are the same, you can add the numerators, or in this case, compare them.

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08 Rational numbers

Here’s an explanation of rational numbers, presented in a notes format with emojis for easy understanding: Rational Numbers (Q) 💭 What are they? Rational numbers are numbers that can be written as a ratio of two integers. They are expressed in the form p/q ➗, where ‘p’ and ‘q’ are both integers. The top number, ‘p’, is called the numerator ⬆️, and the bottom number, ‘q’, is called the denominator ⬇️. The set of rational numbers is denoted by the symbol Q. 💡 The ‘Q’ stands for quotient, linking to the idea of a ratio. Extending from other number sets

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09 Activity Questions 1.3

1. Which of the following statement(s) is(are) false? a) The sum of two natural numbers is always a natural number b) The difference between two integers is always an integer c) The product of two rational numbers is always a real number d) The product of two irrational numbers is always an irrational number Solution The statement that is false is:

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10 real and complex numbers

Here’s an explanation of real numbers and complex numbers in a notes format, designed for ease of understanding with emojis: Real Numbers (R) 🌍 Real numbers are an expansion of rational numbers and fill up the entire number line 📏, including all the “gaps” that rational numbers leave. They are denoted by the symbol R. What fills the gaps? Irrational Numbers 💫 Irrational numbers are those that cannot be written as a simple fraction p/q, where p and q are integers. They are simply numbers that are not rational. A classic example is the square root of 2 (√2). You can physically draw a line segment of length √2 (e.g., the hypotenuse of a square with sides of length 1). However, it cannot be precisely expressed as a ratio of two integers. This fact was known to ancient Greeks like Pythagoras, and its irrationality was reportedly proved by his follower Hippasus around 500 BCE, shocking the Pythagoreans who believed rational numbers formed the basis of all science. In general, the square root of any integer that is not a perfect square (e.g., √3, √5, √6) is an irrational number. Other well-known irrational numbers include pi (π) (the ratio of a circle’s circumference to its diameter) and e (used in natural logarithms). These numbers have infinite non-repeating decimal expansions. Density Property 🌊 Just like rational numbers, real numbers are dense: you can always find another real number between any two distinct real numbers (for example, by taking their average). This means there are no “gaps” in the real number line. Relationship to other Number Sets 🌳 Every natural number is an integer, every integer is a rational number, and every rational number is a real number. The set of natural numbers (N) is a subset of integers (Z). The set of integers (Z) is a subset of rational numbers (Q). The set of rational numbers (Q) is a proper subset of real numbers (R). This means that while all rational numbers are real numbers, there are real numbers (the irrationals) that are not rational. This hierarchical relationship can be visualized using Venn diagrams, where N is the innermost circle, followed by Z, then Q, and finally R as the largest encompassing circle. “Size” of Infinity ✨ Even though rational numbers are dense, the set of real numbers has a larger “size” or cardinality of infinity than the set of natural numbers, integers, or rational numbers. This implies there are vastly more irrational numbers than rational numbers. Complex Numbers (C) 🌌 The Need for Expansion 🚧 When dealing with operations like square roots of negative numbers, the existing real number system falls short. For instance, if you try to find the square root of -1 (√-1), you cannot find a real number that, when multiplied by itself, yields a negative result. This is because the rule for multiplication of signs states that if two numbers have the same sign (either both positive or both negative), their product is always positive. This limitation is also seen when solving quadratic equations: if the discriminant (b² - 4ac) is less than 0, it means you’d be taking the square root of a negative number, which implies no real solutions. Introducing Complex Numbers ✨ To allow for the square roots of negative numbers, a new class of numbers called complex numbers was created. Complex numbers extend the real number system. Symbol ℂ While the provided sources do not explicitly state a symbol for complex numbers, they follow a pattern of using single letters (N, Z, Q, R) for other number sets. Complex numbers are commonly denoted by C (or ℂ). This information is not directly from the provided sources and you may want to independently verify it. Beyond this Course 📚 The provided sources indicate that the study of complex numbers is generally beyond the scope of this particular course.

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11 Activity Questions 1.4

Q1. Which of the following sets are same? (i) {Ankitha, Keerthana, Raju, Suresh} (ii) {Raju, Ankitha, Keerthana, Raju, Ankitha, Suresh} (iii) {Keerthana, Suresh, Dheeraj, Raju, Ankitha} (iv) {Suresh, Raju, Ankitha, Keerthana} (v) {Dheeraj, Raju, Soumya, Keerthana} a) (i) and (ii) b) (iii),(iv) and (v) c) (i) and (iv) d) (i),(ii) and (iv) Solution Q2. Suppose X = {3, π, Tiger, Ball, -40, Dhoni}. Which of the following statement(s) is(are) true about X?

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12 Activity Questions 1.5

Question 1 Which of the following is a correct representation of set comprehension? ○ {x ; x ∈ ℕ, x is even} ○ {x | x ∈ ℕ, x is even} ○ {x is even | x ∈ ℕ | x} ○ {x is even ; x ∈ ℕ, x} Solution Question 2 Which of the following is the set of natural numbers that are multiples of 3 or 5?

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13 construction of subsets and set operation

Here’s an explanation of how subsets are constructed and common set operations, presented in a notes format for easy understanding with emojis: Construction of Subsets (Set Comprehension) 🛠️ What it is Set comprehension is a notation used to define subsets from existing sets. It’s especially useful for infinite sets where you can’t just list all the elements explicitly. It provides a precise and compact way to describe a set. Think of it like a recipe 🧑‍🍳 for building a new set by selecting items from an existing one based on certain rules. The Three Main Parts ✨ Set comprehension generally involves three components:

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14 Activity Questions 1.6

Q1. Which of the following sets is(are) infinite? Set of all Indian Nobel laureates Set of squares of all odd natural numbers Set of all countries in the world Set of all leap years Solution Q2. Which of the following set comprehension defines real numbers in interval [2, 0) ∪ (4, 8]?

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15 Sets - Examples

Based on the sources and our conversation history, a set is fundamentally understood as a collection of items. These items are called members or elements of the set. A crucial aspect of sets is that the order in which the members are listed does not matter, and duplicate members do not change the set. The cardinality of a set is the number of items or elements it contains. For finite sets, this is a straightforward count.

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16 Activity Questions 1.7

Q1. In a class of 40 students, 20 took Mathematics, 18 took Statistics, 16 took English, 7 took Mathematics and Statistics, 7 took Mathematics and English, 9 took Statistics and English, and 3 took all the three courses. How many students are not taking any of these courses? 3 4 5 6 Solution Q2. In a survey, it is found that 30 people like apples, 45 people like oranges, and 20 people like both apples and oranges. How many people like only one fruit? Solution Q3. In a class, 50 students play cricket, 35 play football, 14 play both, and 9 play neither. How many students are there in the class?

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17 natural numbers and the operations

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18 Activity Questions 1.8

Q1. Let A = {1, 4} and B = {2, 4, 6, 8}. Which of the following is the Cartesian product of A and B? ○ {(1, 2), (1, 4), (1, 6), (1, 8), (4, 1), (4, 4), (4, 6), (4, 8)} ○ {(1, 2), (1, 4), (1, 6), (1, 8), (4, 2), (4, 4), (4, 6), (4, 8)} ○ {(1, 2), (1, 4), (1, 6), (1, 8), (2, 4), (4, 2), (4, 6), (4, 8)}

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19 Relations

Based on the provided sources and our conversation history, a relation is fundamentally a way to describe connections or relationships between elements from one or more sets. Here’s a breakdown of what the sources say about relations: A relation is a collection of ordered pairs, or more generally, n-tuples of elements drawn from a Cartesian product of sets. They allow us to construct new sets from old sets. A relation is formally defined as a subset of a Cartesian product of sets. For example, a relation between two sets X and Y is a subset of X × Y. The order of elements is important in the ordered pairs (or n-tuples) that form a relation. For instance, the pair (a, b) is distinct from (b, a) unless a = b. Relations can be defined either by explicitly listing the pairs (for finite relations) or implicitly using set comprehension notation, which applies a filter or condition to elements from the Cartesian product. A relation describes special tuples within a Cartesian product that satisfy a certain condition or are of interest to us. Visualising Relations:

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20 Activity Questions 1.9

Q1. Suppose f : ℤ → ℤ is a function defined by f(k) = k³ + 4k - 10. The value of f(k) at k = 4 is ______ This is a fill-in-the-blank question asking for a numerical answer. Solution Q2. Let f(x) = |x| + 5 and Dom(f) = {c ∈ ℝ | f(x) ∈ ℝ}. Which of the following is(are) true?

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21 Functions

Based on the sources and our conversation history, functions are a fundamental concept in mathematics, closely related to sets and relations. At their core, functions are a special type of relation. While a relation is any collection of ordered pairs (or n-tuples) describing connections between elements of sets, a function is a rule that specifically tells us how to convert an input into an output. Formally, a function, denoted as f : X → Y, from a set X (the domain or input set) to a set Y (the codomain or set of possible output values), must satisfy two key conditions for every element in the domain:

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22 Activity Questions 1.10

Answer the questions 1-3, based on following information: Let A = {x|x ∈ ℕ, x < 10 and x is odd} B = {y|y ∈ ℕ, y is a perfect square and 15 < y < 40} Q1. Which of the following is a subset of B × A? ○ {(36, 3), (25, 5), (36, 6)} ○ {(1, 25), (6, 36), (7, 25), (3, 36)} ○ {(16, 5), (25, 9), (36, 3), (16, 1)}

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23 Relations - Examples

Based on the sources and our conversation history, a relation is defined as a collection of ordered pairs or n-tuples from a Cartesian product of sets. It is formally a subset of this Cartesian product. Relations are used to describe connections or relationships between elements. Here are several examples of relations discussed in the sources: Relations on Numbers: A relation between two sets of natural numbers, N, could be the set of pairs (n, m) where m is 1 plus n. This results in pairs like (0, 1), (2, 3), (17, 18), etc., which form a subset of N × N. The divisibility relation on natural numbers (N × N) or integers (Z × Z) is a relation where a pair (d, n) is included if d is a factor of n (meaning n mod d = 0). For example, (2, 82) and (14, 56) would be in this relation. This relation can be defined using set comprehension as the set of pairs (d, n) in N × N such that d divides n. A relation could be defined by pairs of integers (p, q) where their greatest common divisor is 1. This is related to the concept of reduced fractions. The “less than” relation (<) on numbers is a relation where a pair (a, b) is included if a is less than b. Similarly, the “strictly less than” relation is also an example. An equivalence relation example is connecting numbers that have the same remainder modulo 5. Numbers like 7 and 22 are related in this way because both have a remainder of 2 when divided by 5. A relation on the Cartesian product N × R could be the set of pairs (m, r) such that r is the square root of m. Pairs like (2, √2), (4, 2), (7, √7) would be in this relation. The relation associated with a function f(x) = x² is the set of pairs (x, y) such that y = x². When plotted, these points form the graph of the function, a parabola. Relations between Different Sets:

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24 Activity Questions 1.11

Q1. If Dom(f) = {x ∈ ℝ, f(x) ∈ ℝ} defined by f(x) = (x + 12)/(4x - 8), then the domain of the function f is ______ ○ ℝ ○ ℝ \ {1/4} ○ ℝ \ {-12} ○ ℝ \ {2} Solution Q2. The product of the minimum value of the function f(x) = 9|x| - 8 and the maximum value of the function g(x) = 11 - |x + 8| is ______

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25 FUnctions - Examples

Based on the sources and our conversation history, functions are described as a special type of relation where each element in the input set is mapped to exactly one element in the output set. Abstractly, a function can be thought of as a machine that produces an output for a given input. Here are several examples of functions found in the sources: Functions on Numbers: The Square Function: This is a frequently used example. Given an input x, it returns x². It can be written as f(x) = x². The domain and codomain are often considered the set of real numbers (R). The range is the set of non-negative real numbers, as the output of squaring any real number is always positive or zero. The graph of this function is a parabola. Linear Functions: These are functions of the form f(x) = ax + b or mx + c, where a (or m) and b (or c) are real numbers and a ≠ 0. They define a straight line when graphed. A specific example given is 3.5x + 5.7. Another linear function example is f(x) = 7x + 2 and f(x) = x. Quadratic Functions: These are functions of the form f(x) = ax² + bx + c, where a ≠ 0, and a, b, c are real numbers. The graph of any quadratic function is always a parabola. An example of a shifted parabola is 5x² + 3. Polynomial Functions: A general polynomial function of degree n is described as f(x) = anxⁿ + an-1xⁿ⁻¹ + ... + a₀x⁰, where an ≠ 0 and n is a natural number. They consist of mathematical terms added together, involving only addition, subtraction, multiplication, and natural exponents of variables. An example given is f(x) = x³ + 5. Exponential Functions: These are of the form f(x) = aˣ, where a > 0 and a ≠ 1. The natural exponential function, f(x) = eˣ, is a specific example where e > 1. Other examples include f(x) = 2ˣ and f(x) = (1/2)ˣ. Logarithmic Functions: These are of the form f(x) = logₐ(x), where a > 0 and a ≠ 1. They are the inverse of exponential functions. The natural logarithmic function is f(x) = loge x = ln x, and the common logarithmic function is f(x) = log₁₀ x = log x. The domain is the set of all positive real numbers. Square Root Function: The function f(x) = √x is discussed. By convention, this usually refers to the positive square root. The domain depends on the allowed codomain; if the codomain is restricted to real numbers, the domain is [0, ∞). If complex numbers are allowed as output, the domain can be all real numbers. Absolute Value Function: Denoted by f(x) = |x|, this function returns x if x ≥ 0 and -x if x < 0. It is used as an example to check for injectivity (it is not one-to-one) and continuity at x=0 (it is continuous). Step Functions: Examples include the Floor function, f(x) = ⌊x⌋ (greatest integer value less than or equal to x), and the Ceiling function, f(x) = ⌈x⌉ (smallest integer value greater than or equal to x). Trigonometric Functions: Examples mentioned include sin x, cos x, and tan x. f(x) = sin x is also used to check for differentiability. Constant Function: f(x) = c is used to illustrate differentiation. Rational Function: An example of a real-valued function given is f(x) = (5x+9)/(2x). Function Defined on an Interval: f(x) = 2x - 1 defined on the interval `` is used in the context of calculating area under a curve. Function used in SSE: f(x) = 2x - 2 is implicitly used in a sum squared error calculation example. Bounded Function Example: f(x) = 1/(x² + 1) is shown to be a bounded function with 0 ≤ f(x) ≤ 1. Functions on Other Sets:

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26 Prime NUmbers

Based on the sources and our conversation history, here’s a comprehensive overview of prime numbers: Definition: A prime number is a natural number that has no factors other than 1 and itself. It must have exactly two factors. Factors: The only factors of a prime number p are 1 and p. Why 1 is Not Prime: It is important that a prime number must have two separate factors. While 1 has 1 as a factor (because 1 times 1 is 1), it has only one factor, which is 1 itself. Therefore, 1 is technically not considered a prime number. Smallest Primes: The smallest prime number is 2 because it has exactly two factors: 1 and itself. The next prime numbers are 3, 5, and 7. Even Numbers: After the number 2, no even numbers can be prime because they are all multiples of 2, meaning 2 divides them in addition to 1 and themselves. For example, 4 is divisible by 2, and 6 is not prime because it’s a multiple of 3. Generating Primes (Sieve of Eratosthenes): There is a method called the sieve of Eratosthenes to generate prime numbers. You start by listing numbers (e.g., from 1 to 100). You know 1 is not prime. You take the first unmarked number, which is 2, declare it a prime, and then knock off all its multiples (all the even numbers) as non-primes. Then, you look for the next number that hasn’t been marked off, which is 3, declare it a prime, and mark off all its multiples (some of which might already be marked). You continue this process; the next unmarked number will be the next prime (e.g., 5 is found this way). This method is a good way to generate primes up to a certain number without missing any. Prime Factorization: A very important fact is that every number can be uniquely factorized into the prime numbers that form it. This is also called the prime factorization. For example, 12 can be written as 2 × 6 or 4 × 3, but its fundamental unique prime factorization is 2 × 2 × 3, or using exponentiation, 2² × 3. Similarly, 126 is 2 × 3² × 7. This unique decomposition property is used implicitly a lot. Infinitude of Primes: It is a known result that the set of prime numbers is an infinite set. There cannot be a largest prime number. Euclid provided a proof for this. The proof involves assuming there is a finite list of all primes (p₁, p₂, …, pk), constructing a new number N by multiplying all these primes together and adding 1 (N = p₁ × p₂ × … × pk + 1). This new number N must be larger than any prime in the list. If the list was exhaustive of all primes, N must be composite (not prime). If N is composite, it must have a prime factor, and this prime factor must be in the original list (say pⱼ). So, pⱼ divides N. However, pⱼ also divides the product p₁ × p₂ × … × pk (since pⱼ is one of the factors). A property of divisibility states that if a number divides a sum (a+b) and also divides one part (a), it must divide the other part (b). In this case, pⱼ divides N (the sum) and pⱼ divides the product (one part), so pⱼ must divide 1 (the other part). But pⱼ is a prime number, which is by definition greater than 1, and therefore cannot divide 1. This is a contradiction, meaning the initial assumption (that the list of primes is finite) must be false. Thus, the set of primes is infinite, and there is no largest prime. Distribution: Prime numbers have been extensively studied in an area called number theory. One topic is their distribution within the natural numbers. As numbers get larger, the gaps between primes tend to become larger. The function π(x) denotes the number of primes less than or equal to a given number x. For large x, π(x) is approximately x / log(x). Applications: Despite seemingly abstract, prime numbers are actually quite useful. One important application is in cryptography. Cryptography affects day-to-day life, such as protecting electronic commerce transactions. Much of this encryption relies on the existence of large prime numbers and the fact that it is difficult to factorize the product of two large primes. Computational Problems: There are two related computational problems: checking if a number is prime (primality testing) and finding the prime factors of a number (factorization). Primality testing can be done efficiently. However, there is no efficient way to factorize a large number. This paradox (being able to check if a number is prime efficiently, but not being able to factorize it quickly if it isn’t prime) is why primes are important in cryptography. Set Representation: The set of prime numbers can be defined as a subset of the natural numbers. Using set comprehension, the set of primes (P) can be defined as the set of natural numbers p such that the factors of p consist of exactly two elements {1, p}, and p is not 1. In summary, prime numbers are foundational in number theory, possess unique properties like the basis for unique prime factorization, are infinite in quantity, and have significant practical applications, particularly in securing digital communications.

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27 Why is a number irrational?

Based on the sources and our conversation history, a number is considered irrational primarily because it cannot be written in the form of p/q, where p and q are integers. In simpler terms, irrational numbers are defined as numbers that are not rational. Here are some key points and examples from the sources explaining why numbers are irrational: Definition: The set of rational numbers (Q) consists of numbers that can be expressed as a fraction p/q, where p and q are integers. Irrational numbers are those real numbers that fall outside this definition.

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28 set versus collection

Based on the sources and our conversation history, the key distinction between a set and a collection arises from foundational issues in set theory, particularly when dealing with infinite collections. Here’s a breakdown: Informally, a set is often described as a collection of items. Another source defines a set as a collection of well-defined items. For finite collections, the idea that any collection is a set seems reasonable. Examples of finite sets given include the days of the week, the factors of a number like 24, the set of natural numbers less than 10, and months in a year. However, the sources explicitly state that not every collection is in fact a set. This becomes particularly problematic when dealing with infinite collections. The reason for this caution stems from what is known as Russell’s Paradox. This paradox questioned the fundamental idea that any collection could be considered a set, especially in the context of trying to use set theory as a foundation for all of mathematics. Bertrand Russell’s work demonstrated that the collection of “all sets” leads to a problem if one assumes it is itself a set. The paradox is illustrated by considering a collection, say ‘S’, defined using set comprehension as the subset of all sets that do not contain themselves. If we assume that the collection of all sets exists as a set, then this subset ‘S’ is also a set because it is defined by applying a condition (set comprehension) to an existing set (the set of all sets). The paradox then asks whether this set ‘S’ belongs to itself. If ‘S’ belongs to itself, then by its own definition (sets in S do not contain themselves), it cannot belong to itself. This is a contradiction. If ‘S’ does not belong to itself, then by the condition used to define ‘S’ (sets in S are those that do not contain themselves), it must belong to itself. This is also a contradiction. The conclusion from this paradox is that the initial assumption that leads to the contradiction must be false. The assumption that goes wrong is the assumption that there is a set of all sets. Therefore, not every collection can be called a set; specifically, the collection of all sets does not exist as a set. In modern mathematics, when one is unsure if a collection satisfies the necessary properties to be a set, it is sometimes safer to refer to it as a class. A class is simply a collection of objects that may not have the implied properties expected from sets. The sources advise that when working with sets, especially in this course, it is safe to assume that known collections of numbers like natural numbers (N), integers (Z), rational numbers (Q), and real numbers (R) are sets. New sets should be constructed from these existing, well-defined sets using standard set operations like union, intersection, Cartesian product, and set comprehension. The caution is against “manufacturing sets out of thin air”, such as the problematic “set of all sets”. In summary, while “set” is often used informally to mean “collection,” the sources highlight that in formal mathematics, particularly concerning infinite collections and foundational issues, not every collection is a valid set due to paradoxes like Russell’s Paradox. The collection of all sets is a key example of a collection that is not a set. The term “class” is sometimes used for collections that might not be sets.

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29 Degree of infinity

Based on the sources and our conversation history, the concept of “Degrees of Infinity” refers to the idea that different infinite sets can have different “sizes” or cardinalities. While the term “Degree of Affinity” was not found in the sources, “Degrees of Infinity” was discussed as a way to understand the magnitude of infinite sets. Here’s what the sources explain about this concept: Cardinality is the term used to denote the number of elements in a set. For finite sets, determining cardinality is straightforward – you simply count the elements. This count results in a natural number. The challenge arises when dealing with infinite sets. Examples of infinite sets discussed include the natural numbers (N), the integers (Z), the rational numbers (Q), and the real numbers (R). Some of these sets, like natural numbers and integers, are described as discrete, while rational numbers are described as dense. Despite these structural differences, the question is whether they all have the same “size” or if there are more elements in one infinite set than another. This leads to the core question: are there degrees of infinity?. To compare the sizes of infinite sets, the concept of a bijection is used. If a bijection (a one-to-one and onto function) exists between the set of natural numbers (N) and another set X, it means you can effectively pair up the elements of N with the elements of X. This process allows you to enumerate the elements of X, essentially listing them out in an ordered sequence. Sets that can be enumerated in this way are considered to have the same cardinality as the natural numbers. Examples in the sources include the set of integers and even pairs of integers (Z cross Z), suggesting they can be enumerated. However, the sources demonstrate that the set of real numbers (R) has a higher degree of infinity than the natural numbers. This is shown using a method similar to Cantor’s diagonal argument. By considering infinite sequences of 0s and 1s (which can represent real numbers, specifically those between 0 and 1), it’s proven that no matter how you try to list or enumerate all such sequences, you can always construct a new sequence that is not on your list. This argument shows that the set of infinite 0,1 sequences is not countable. Since these sequences can be mapped to real numbers in the interval, this proves that even this small fraction of the real numbers, and therefore the entire set of real numbers (R), cannot be enumerated. This distinction—being able to enumerate (like N, Z, Q) versus not being able to enumerate (like R)—means that the set of real numbers is “larger” in terms of cardinality than the set of natural numbers, integers, or rational numbers. This demonstrates that there are indeed different degrees of infinity. The sources also briefly mention the continuum hypothesis, a significant open question in set theory concerning whether there exist infinite sets with cardinality strictly between that of the natural numbers (countable infinity) and the real numbers (uncountable infinity). It is noted that this question was shown to be independent of the standard axioms of set theory. In summary, the concept of “degrees of infinity” highlights that not all infinite sets are the same size; sets like the real numbers are proven to be “larger” (uncountable) than sets like the natural numbers (countable), establishing different levels of infinity.

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30 Activity Questions 2.1

Question 1 (Multiple Choice) 1. Choose the correct option with respect to the points P(5, -3), Q(-3, 3), R(0, -100), and S(-2.5, 0) on the rectangular coordinate system. Options: Point R does not lie in any quadrant Points P and R lie in Quadrant III Points S and Q lie in Quadrant II Points R and S cannot be represented on the rectangular coordinate system Solution Question 2 (Multiple Select Questions - MSQ)

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31 area of a triangle

Based on the sources and our previous conversation, the area of a triangle formed by three points in a rectangular coordinate system is a concept discussed, particularly when the three points are non-collinear. Here’s what the sources say about calculating the area of a triangle: Using the Formula: The area ($\Delta$) of a triangle formed by three points with coordinates (x₁, y₁), (x₂, y₂), and (x₃, y₃) in the XY plane is given by the formula: $\Delta = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$. The formula includes a modulus (absolute value) because the area of any region is always positive.

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32 distance formula

Based on the sources, the distance formula is a key concept within the rectangular coordinate system used to calculate the length between points or between a point and a line. Different formulas are used depending on what distance you need to find. Here are the main distance formulas discussed in the sources: Distance between any two points The distance between two points (x1, y1) and (x2, y2) in the Cartesian plane (XY plane) is given by the formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. This formula is fundamental for finding the distance between any two points on a coordinate plane. The derivation of this formula relies on constructing a right-angle triangle using the two points and a third auxiliary point that shares one coordinate with each of the original points. The lengths of the two sides of the right-angle triangle parallel to the axes are the absolute differences of the respective coordinates, specifically $|x_2 - x_1|$ and $|y_2 - y_1|$. Applying the Pythagorean theorem, where the distance between the two points is the hypotenuse, results in the formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. The squares in the formula mean that the order of subtraction (x1-x2 or x2-x1) does not affect the result. A specific case is finding the distance of a point (x, y) from the origin (0, 0). Using the same logic, the distance from the origin is $\sqrt{x^2 + y^2}$. For example, the distance between points (2,4) and (-4,12) is calculated as $\sqrt{((-4)-2)^2 + (12-4)^2} = \sqrt{(-6)^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. Distance of a line from a given point

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33 Distance of a line from a given point

Drawing upon the information from the sources and our conversation history, we can discuss the distance of a point from a given line, particularly when the line is expressed in its General Form. The problem is to determine the distance of a point P, with coordinates (x₁, y₁), from a line l, given by the equation Ax + By + C = 0. This form, Ax + By + C = 0, is known as the general form of the equation of a line, and it is a versatile representation as it can describe any straight line [Conversation history].

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34 equation of a perpendicular line passing through a point

Drawing upon the provided sources and our conversation history, we can determine the equation of a line perpendicular to a given line and passing through a specific point, often starting with the General Form of the given line. The general equation of a line is expressed as Ax + By + C = 0. This form is particularly useful as it can represent any straight line, including vertical lines, which have an undefined slope. For this to represent a line, A and B cannot be zero simultaneously.

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35 equation of parallel and perpendicular lines in general form

Based on the sources and our conversation, the General Form of the equation of a straight line is a powerful representation because it can represent any straight line. This includes vertical lines, which some other forms (like the standard slope-intercept form y = mx + c) cannot represent because their slope is undefined. The general equation of a line is given by: Ax + By + C = 0 For this equation to represent a line, the coefficients A and B cannot be simultaneously equal to 0.

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36 General Equation of line

Based on the sources and our conversation history, the General Form is presented as a comprehensive algebraic representation for any straight line in the rectangular coordinate system. Here’s a breakdown of the key information about the general equation of a line: The Equation The general form of the equation of a straight line is given by Ax + By + C = 0. Universality This form is powerful because it can represent any straight line, including those that are vertical. Unlike some other forms (like slope-intercept y = mx + c), the general form can handle vertical lines which have an undefined slope. Condition for a Line For the equation Ax + By + C = 0 to represent a line, the coefficients A and B cannot be simultaneously equal to 0. Individually, A can be 0 (resulting in a horizontal line) or B can be 0 (resulting in a vertical line), but they cannot both be zero at the same time. Relationship to Other Forms All other forms of linear equations, such as the slope-point form, slope-intercept form, two-point form, and intercept form, can be rearranged into this general form. Extracting Geometric Properties The general form allows for the determination of geometric properties of the line. Slope (m): For a non-vertical line (where B ≠ 0), the slope can be found by rearranging the equation into the slope-intercept form (y = mx + c). By solving Ax + By + C = 0 for y, we get By = -Ax - C, which gives y = (-A/B)x - C/B. Thus, the slope is m = -A/B. This was shown using the example 3x - 4y + 12 = 0, where A=3 and B=-4, giving a slope of -3/(-4) = 3/4. Y-intercept: For a non-vertical line (where B ≠ 0), the y-intercept occurs where x = 0. Substituting x = 0 into Ax + By + C = 0 gives By + C = 0, so y = -C/B. The y-intercept is (0, -C/B). In the example 3x - 4y + 12 = 0, C=12 and B=-4, so the y-intercept is -12/(-4) = 3 [Conversation history]. X-intercept: For a non-horizontal line (where A ≠ 0), the x-intercept occurs where y = 0. Substituting y = 0 into Ax + By + C = 0 gives Ax + C = 0, so x = -C/A. The x-intercept is (-C/A, 0). In the example 3x - 4y + 12 = 0, C=12 and A=3, so the x-intercept is -12/3 = -4 [Conversation history]. Special Cases (Vertical and Horizontal Lines): If B = 0 (and A ≠ 0), the equation becomes Ax + C = 0, which simplifies to x = -C/A. This represents a vertical line. Vertical lines have an undefined slope. If A = 0 (and B ≠ 0), the equation becomes By + C = 0, which simplifies to y = -C/B. This represents a horizontal line. Horizontal lines have a slope of 0. Usefulness The general form is particularly useful for certain calculations, such as finding the distance of a point from a line or the distance between two parallel lines, where the equations are typically given in this form. In essence, the general equation Ax + By + C = 0 provides a unified algebraic framework to represent all straight lines, offering flexibility in deriving or converting to other forms and facilitating the calculation of key geometric properties.

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37 Parallel Lines

Based on the sources and our conversation, the concepts of parallel and perpendicular lines are extensively discussed, particularly in relation to the slope of a line. The slope, denoted by ’m’, characterises the direction and steepness of a line. It is defined as the ratio of the change in y to the change in x between any two points on the line, or as the tangent of the angle of inclination (θ) the line makes with the positive x-axis (m = tan(θ)).

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38 Rectangular Coordinate System

Based on the sources, a Rectangular Coordinate System, also known as a Cartesian coordinate system, is a system used to specify each point in a plane by a set of numerical coordinates. It is important for studying algebraic properties of geometric objects such as points, lines, and planes. The system allows for the unique identification of a point on a plane. Here are the key components and features of the rectangular coordinate system:

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39 Representation of a Line - 2

Certainly. Building on our previous discussion about representing lines algebraically, the sources present several different forms of equations used for this purpose. While the slope characterises a non-vertical line’s direction and steepness, it alone is not sufficient to uniquely define a line. To uniquely represent a line, additional information, such as a point it passes through or its intercepts, is needed. We have already touched upon several representations, including: Horizontal lines (y = a) Vertical lines (x = b) The Point-Slope Form (y - y₀) = m(x - x₀), which requires a known point (x₀, y₀) and the slope (m). The Two-Point Form (y - y₁) = (y₂ - y₁)/(x₂ - x₁) * (x - x₁), requiring two distinct points (x₁, y₁) and (x₂, y₂). This form essentially calculates the slope from the two points and uses the point-slope form. The Slope-Intercept Form (y-intercept) y = mx + c, using the slope (m) and the y-intercept (c), where the line crosses the y-axis at (0, c). The Slope-Intercept Form (x-intercept) y = m(x - d), using the slope (m) and the x-intercept (d), where the line crosses the x-axis at (d, 0). The Intercept Form x/a + y/b = 1, which uses both the x-intercept (a) and the y-intercept (b). The sources note that this form is easy to remember: “below x you put x-intercept below y you put y-intercept and equate it with 1”. All these forms provide an algebraic condition on the coordinates (x, y) that any point on the line must satisfy. If a point’s coordinates satisfy the equation, it lies on the line.

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40 Representation of a Line 1

Based on the sources and our conversation, representing a line algebraically in the rectangular coordinate system is a key concept, allowing us to describe a geometric object – a straight line – using equations. We’ve discussed the slope of a line, which describes its direction and steepness. However, the slope alone cannot uniquely determine a line; many parallel lines can have the same slope. To represent a non-vertical line uniquely, you need more than just the slope. The sources explain different forms of linear equations that provide this unique representation, often requiring a point the line passes through in addition to, or allowing for the calculation of, the slope.

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41 Section Formula

The Section Formula is a concept used within the rectangular coordinate system to determine the coordinates of a point that divides a line segment connecting two given points. This point lies on the line containing the two given points. The formula depends on whether the point divides the line segment internally or externally. Internal Division: If a point P(x, y) divides the line segment connecting points A(x₁, y₁) and B(x₂, y₂) internally in the ratio m : n, its coordinates are given by:

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42 slope of a line

Okay, let’s discuss the slope of a line based on the information from the sources and our conversation. The slope of a straight line, often denoted by the letter ’m’, is a fundamental concept in the rectangular coordinate system that describes both the direction and steepness of the line. Here’s a breakdown of what the sources tell us about the slope of a line: Definition and Calculation: In simple terms, the slope is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate between any two distinct points on the line. This is also referred to as the “rise by run” ratio, where “rise” is the change in the vertical direction (parallel to the y-axis) and “run” is the change in the horizontal direction (parallel to the x-axis). Given two points on a line, (x₁, y₁) and (x₂, y₂), the numerical value of the slope (m) can be calculated using the formula: m = (y₂ - y₁) / (x₂ - x₁). It’s important to take the difference in y-coordinates and the corresponding difference in x-coordinates in the same order. For instance, if you subtract y₁ from y₂, you must subtract x₁ from x₂ in the denominator. The sources also relate the slope to the inclination of the line. The inclination (theta, θ) is the angle the line makes with the positive x-axis, measured in an anti-clockwise direction. The slope is equal to the tangent of this angle: m = tan(θ). The typical range for the inclination angle is from 0 to 180 degrees. Special Cases:

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43 Straight line fit

Based on the sources and our conversation history, “Straight Line Fit” is a concept that deals with finding a straight line that best represents or fits a given set of points. This is particularly relevant when dealing with real-world data or observations where points may not all lie perfectly on a single line. The goal of straight line fitting is to find the equation of a line, typically in the form of y = mx + c (for non-vertical lines), that minimises the “distance” between the line and the given data points.

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44 Activity Questions 2.2

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45 Activity Questions 2.3

Question 1 (Fill in the blank) The coordinates of the midpoint of points P(4, -2) and Q(-1, -1) are _______ Options: (0.5, -0.5) (1.5, -1.5) (-0.5, -0.5) (-0.5, 0.5) Solution Question 2 (Fill in the blank)

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46 Activity Questions 2.4

Multiple Choice Questions (MCQ): Question 1: Choose the correct statement based on the three points P(0, 10), Q(-20, -30) and R(10, 30) Options: The given points form a triangle of area 5 square units The given points form a triangle of area 15 square units The given points do not form a triangle None of the above Solution Question 2: The area of the triangle formed by the midpoints of line segments PQ, QR, and RP where the coordinates of P, Q, and R are (0, 0), (3, 0), and (3, 4) respectively, is ______

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47 Activity Questions 2.5

Multiple Choice Questions (MCQ): Question 1: Find the slope of a line passing through the origin and the point (-3, -2) Options: 1/6 -2/3 -1/3 3/2 Solution Question 2: If the slope of a line passing through P(1, 0) and Q(-2, k) is 1, then the value of k is _____

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48 Activity Questions 2.6

Multiple Choice Questions (MCQ): Question 1: If a line is perpendicular to the X-axis, then the slope of such line is _______ Options: 0 Not defined 1 -1 Solution Question 2: If a line is parallel to a line which is perpendicular to the Y-axis, then the slope of the first line is _______

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49 Activity Questions 2.7

Multiple Choice Questions (MCQ): Question 1: Which of the following represents a equation of the horizontal line? Options: y = 0 x = 5 x = -2 x = 0 Solution Question 2: The equation of a line parallel to the X-axis and passing through the point (-2, 0) is _____

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50 Activity Questions 2.8

Multiple Choice Questions (MCQ): Question 1: The equation of a line passing through (-1, -1) with value of slope 1 is _____ Options: y = -x y = x y = -x - 1 y = -x + 1 Solution Question 2: The equation of a line which cuts the X-axis at (5, 0) and Y-axis at (0, 5) is _____

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51 Activity Questions 2.9

Multiple Choice Questions (MCQ): Question 1: If the x-intercept and the y-intercept of a straight line are -6 and 7 respectively, then choose the correct equation of the line. Options: 7x - 6y + 42 = 0 -6x + 7y - 1 = 0 7x - 6y - 1 = 0 -6x + 7y - 2 = 0 Solution Question 2: The slope of the line 6x - 2y + 8 = 0 is _____

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52 Activity Questions 2.10

Multiple Select Question Question 1: Which of the following statements are true? Options: Lines 2x + 3y - 8 = 0 and 3x - y - 2 = 0 are parallel lines Lines 3x + 5y - 10 = 0 and 6x + 10y - 26 = 0 are parallel lines Lines 6x + 8y - 20 = 0 and 4x - 3y = 0 are perpendicular to each other

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53 Activity Questions 2.11

Multiple Choice Question Question 1: The equation of a line passing through the point (3, 4) and perpendicular to the line 3x + 4y - 8 = 0 is Options: 8x - 6y = 0 2x + 8y = 38 8x + 4y = 5 x + y = 1 Solution Numerical Answer Type

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54 Activity Questions 2.12

Multiple Select Questions Question 1: If the general form of a line is 3x + 2y - 5 = 0, then choose the correct set of options. Options: The slope of the given line is -3/2 The x-intercept is 3 The point where the given line cuts the X-axis is (5/3, 0) The y-intercept is 2 The point where the given line cuts the Y-axis is (0, 5/2) Solution Question 2: Given the point (-2, 1) and the line -3x + 4y - 7 = 0, choose the correct set of options.

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55 Activity Questions 2.13

Numerical Answer Type Question 1: If a line fit y = x + 1 is given for the data as shown in Table AQ-3.1, then compute the Sum Squares Error (SSE). Table AQ-3.1: x 1 2 3 4 6 y 1 1 2 5 7 Solution Multiple Choice Question Question 2: If the relation between x and y is as shown in Table AQ-3.2, then which among the following lines is the best fit?

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56 Activity Questions 3.1

Question 1: Which of the graphs in Figure 1 represents the following function: $y = x^2 - x + 1$? This is a multiple choice question with four options (A, B, C, and D), each showing different graphs. The question asks students to identify which graph correctly represents the given quadratic function.

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57 Quadratic functions

A quadratic function is a type of function described by an equation in the form f(x) = ax² + bx + c, where a is not equal to 0. The condition that ‘a’ must not be 0 is crucial, because if a were 0, the equation would reduce to f(x) = bx + c, which is a linear function. The name “quadratic” is related to the term “square”. The graph of any quadratic function is always a parabola.

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58 Activity Questions 3.2

Question 1 The curve on the surface of the banana as shown in Figure 2 can be described using the equation $y = x^2 + 2x + 4$. An ant (shown in blue color in Figure 2) is walking from one end of the banana to the other end. What will be the x-coordinate of the ant’s location once it reaches the vertex of its path?

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59 Examples of Quadratic functions

Based on the provided sources, a quadratic function is described by an equation of the form f(x) = ax² + bx + c, where ‘a’ is not equal to 0. The name “quadratic” is related to the term “square”. The graph of any quadratic function is always a parabola. Here are some examples of quadratic functions and how they are discussed in the sources: y = x²: This is given as a standard prototype example. It is the form where b=0 and c=0, and a=1. Points like (0,0), (1,1), (-1,1), (2,4), and (-2,4) can be plotted to graph it. The graph forms an upward parabola shape. For this function, the slope at any point x is 2x. Setting the slope to 0 (2x = 0) gives x=0, which is the x-coordinate of the vertex where the minimum value is attained. The y-coordinate of the vertex (at x=0) is 0, which is the minimum value. It shows symmetry about the y-axis because, for instance, 2² is the same as -2². f(x) = x² + 2x + 1: This function can be graphed by generating a table of ordered pairs and plotting them. Examples of points given are (-2,1), (-1,0), (0,1), and (1,4). The axis of symmetry for this function is x = -1. This is found using the formula x = -b/(2a), where a=1 and b=2. The point where the axis of symmetry meets the parabola is the vertex. For this function, the vertex is at x = -1. Substituting x=-1 into the function gives f(-1) = (-1)² + 2(-1) + 1 = 1 - 2 + 1 = 0. The y-intercept is 1 (when x=0). The minimum value attained is 0, which is the y-coordinate of the vertex. This minimum occurs at the vertex where the slope is 0. f(x) = x² + 8x + 9: For this function, the y-intercept is 9 (when x=0). The axis of symmetry is x = -b/(2a) = -8/(2*1) = -4. The vertex is at x = -4. The y-coordinate of the vertex is f(-4) = (-4)² + 8(-4) + 9 = 16 - 32 + 9 = -7. This value (-7) represents the minimum since a > 0. (Calculation outside of sources, but based on source concepts). f(x) = -x² + 1: In this function, a = -1, b = 0, and c = 1. The y-intercept is 1. The axis of symmetry is x = -b/(2a) = -0/(2*(-1)) = 0, which is the y-axis. The vertex is at x = 0. The y-coordinate is f(0) = -(0)² + 1 = 1. The vertex is (0,1). Since a is negative (a < 0), the curve opens downwards. The y-coordinate of the vertex (1) represents the maximum value attained by the function. The graph of this function never intersects the x-axis. f(x) = 5x² + 3: This is an example of a parabola that has been shifted upwards. Similar to y=x², this function can only take positive values if the +3 term were absent. f(x) = x² + 6x + 8: For this function, a=1, b=6, and c=8. The y-intercept is 8. The axis of symmetry is x = -b/(2a) = -6/(2*1) = -3. The roots (or x-intercepts) are -4 and -2. The value of the function at these points is 0. The vertex is at x = -3. The y-coordinate is f(-3) = (-3)² + 6(-3) + 8 = 9 - 18 + 8 = -1. This is the minimum value since a > 0. Since a > 0 and the vertex value (-1) is negative, the curve opens up and crosses the x-axis at two points, resulting in two real roots. x² + 1: This is mentioned in the context of solving the quadratic equation x² + 1 = 0. It’s noted that b=0, so the graph is symmetric about the y-axis. The graph of this function never intersects the x-axis. The discriminant (b² - 4ac) for x² + 1 = 0 is 0² - 4(1)(1) = -4, which is less than 0, indicating no real roots. 3x² + 10x - 8: This is given as an example of a quadratic equation in standard form (ax² + bx + c = 0). It was derived from the intercept form with roots 2/3 and -4. x² - 4x + 4: This is an example used to demonstrate factoring. The product of the last terms is 4 and the sum of the cross products is -4. It factors into (x-2)(x-2). Setting this equal to 0 (x² - 4x + 4 = 0) shows it has one real root, which is repeated (x=2). x² - 25: This example is used to show factoring of a difference of squares. It factors into (x+5)(x-5). Setting this equal to 0 (x² - 25 = 0) gives the roots -5 and 5. 9x² - 12x + 4: This is an example used to calculate the discriminant. Here, a=9, b=-12, c=4. The discriminant is b² - 4ac = (-12)² - 4(9)(4) = 144 - 144 = 0. Since the discriminant is 0, it has only one real rational root (repeated). 2x² + 16x + 33: This is another example used to calculate the discriminant. Here, a=2, b=16, c=33. The discriminant is b² - 4ac = (16)² - 4(2)(33) = 256 - 264 = -8. Since the discriminant is less than 0, it has no real roots. These examples illustrate various properties of quadratic functions, such as how the coefficient ‘a’ affects the direction the parabola opens, how to find the y-intercept (c), the importance of the axis of symmetry x = -b/(2a) and the vertex for graphing, how the vertex determines the minimum or maximum value, and the relationship between the function’s zeros and the roots of the corresponding quadratic equation. The slope of a quadratic function is variable and given by 2ax + b, reaching zero at the vertex. Quadratic equations can be solved using methods like graphing (finding x-intercepts), factoring, completing the square, or the quadratic formula. The discriminant (b² - 4ac) within the quadratic formula indicates the number and type of real roots.

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60 Activity Questions 3.3

Question 1 Calculate the slope of the parabola at a point (x, y) obtained by plotting the following function: $y = x^2 + 2x + 4$ Multiple choice options: $2x + 2$ $+2x$ $-1.5x$ $0.5$ Solution Question 2 Calculate the slope of the parabola at a point (x, y) obtained by plotting the following function: $y = -5x^2 + 10x + 10$

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61 Slope of quadratic function

Drawing upon the provided sources and our conversation history, the slope of a quadratic function is a key concept discussed, contrasting it with the constant slope of a linear function. Here’s what the sources say about the slope of a quadratic function: Contrast with Linear Functions: For a linear function y = mx + c (or f(x) = mx + c), the quantity ’m’ represents the slope, and it is constant. This slope represents the sensitivity of the expression to changes in the x-direction, indicating how a small change in x affects the change in y. It is calculated as the ratio of change in y to change in x (rise by run) and can be related to the tangent of the angle of inclination with the positive x-axis. Unlike linear functions, the slope of a quadratic function is variable. Definition and Formula:

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62 Activity Questions 3.4

Question 1 A stone is thrown with an initial speed u (m/s) as shown in figure 3. The height of the stone’s trajectory above the ground is $H(t) = -5t^2 + \frac{1}{2}ut$ (where t is the time of flight). If the highest point in air that the stone can reach is 5m above the ground, then calculate the initial speed u. The question includes a diagram showing the parabolic trajectory of the stone reaching a maximum height of 5 meters.

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63 Solution of quadratic equation using graph

Based on the sources and our conversation history, we can explain how to solve a quadratic equation using the graphing method. First, let’s understand the connection between quadratic equations and quadratic functions. A quadratic equation is formed when a quadratic function, defined as f(x) = ax² + bx + c where a ≠ 0, is set equal to a specific value. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are integers and a is not equal to 0. The solutions to this equation are called the roots of the equation.

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64 Activity Questions 3.5

Question 1 The slope of a line which passes through the vertex and the y-intercept of the quadratic equation $x^2 + 10x - 5$ is Multiple choice options: $\sqrt{1230}$ $-5$ $5$ $\sqrt{650}$ Solution Question 2 Identify the point at which the slope of the equation $x^2 + 2x - 5$ is 10

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65 Slope Line & Parabola

Drawing on the sources and our conversation history, we can discuss the slope of a line and compare it to the slope of a parabola (which is the graph of a quadratic function). Slope of a Straight Line The concept of the slope of a line is typically studied before moving on to quadratic functions. The slope of a line describes both its direction and steepness. In a general sense, the slope tells you how a small change in the x-direction affects the change in the y-direction. It is considered a measure of the rate of change.

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66 Activity Questions 3.6 - Summary Lecture

Summary Lecture

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67 Summary Lecture (Quadratic Functions)

Based on the “Summary lecture”, this video summarises the topics covered regarding quadratic equations and functions. The lecture begins by positioning quadratic functions as a generalisation of the concept of a straight line or a linear function, which is typically in the form of mx + b. A quadratic function is defined in the form f(x) = ax² + bx + c, with the crucial condition that a is not equal to 0. If ‘a’ were equal to 0, the term with x² would disappear, and it would simply reduce to a linear function. The name “quadratic” is related to the term “square”.

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68 Activity Questions 3.7

Question 1 Choose the correct standard form of a quadratic equation with roots $\frac{2}{3}$ and $\frac{10}{3}$. Multiple choice options: $6x^2 - x - 70 = 0$ $\frac{1}{3}(6x^2 - x - 70) = 0$ $\frac{1}{6}(6x^2 - x - 7) = 0$ $(x - \frac{2}{3})(x + \frac{10}{3}) = 0$ Solution Question 2 Choose the correct option about a with the help of Figure AQ-5.1.

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69 Solution of quadratic equation using Factorization

Based on the sources and our conversation history, solving a quadratic equation using the factoring method is a common technique for finding its roots (solutions). A quadratic equation is obtained when a quadratic function, of the form ax² + bx + c where a is not equal to 0, is set equal to a value, often 0 for the standard form. The standard form is ax² + bx + c = 0, where a, b, and c are typically considered integers.

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70 Activity Questions 3.8

Multiple Choice Questions (MCQ): Question 1 What should be added in p(x) to make it perfect square, if p(x) = x² - 12x + 34? Multiple choice options: 1 2 3 4 Solution Question 2 Choose the correct option regarding equation x² - 12x + 37 = 0.

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71 Solution of quadratic equation using Square method

Solving a quadratic equation using the completing the square method is another technique for finding the roots of the equation, as discussed in the sources. This method involves transforming the equation into a specific form that allows you to take the square root easily. It also has a direct connection to the well-known Quadratic Formula. The general idea is to manipulate the quadratic equation so that one side of the equation becomes a perfect square trinomial (like (x + a)² or (x - a)²), and the other side is a constant.

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72 Activity Questions 3.9

Multiple Choice Questions (MCQ): Question 1 The quadratic equation $9x^2 + 6x + 1 = 0$ has Multiple choice options: Two distinct roots Equal roots No roots Inadequate information Solution Question 2 If two different quadratic equations have same discriminant then choose the correct option.

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73 Quadratic formula

Based on the sources and our conversation, the quadratic formula is a powerful tool used to find the roots (or solutions) of a quadratic equation. A quadratic equation is formed when a quadratic function, which is in the form ax² + bx + c where a is not equal to 0, is set equal to a value, often 0 for the standard form: ax² + bx + c = 0. The roots of this equation are the x-values for which the equation holds true. These roots correspond to the x-intercepts or zeros of the associated quadratic function f(x) = ax² + bx + c.

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74 Summary lecture

Based on the sources and our conversation, a “Summary lecture” appears to consolidate key concepts, particularly relating to quadratic functions and equations. These lectures summarise topics such as the definition and representation of quadratic functions, the relationship to quadratic equations, and various methods for finding the solutions or roots of these equations. Here’s a summary of the key points discussed in these summary lectures and related sources: Quadratic Function Definition and Forms: