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The document provides examples and techniques for changing the subject of a formula. It demonstrates flipping both sides of an equation, multiplying or dividing both sides by the same term, and isolating the term to be made the subject by collecting like terms on one side of the equation and leaving the term by itself on the other side. Common mistakes discussed include incorrectly flipping terms individually rather than both sides of the equation and prematurely making denominators the same.

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Sec 1 Maths Notes Equations

1) The document provides steps for solving simple linear equations with no fractions and fractional equations.
2) For linear equations with no fractions, the steps are to expand if there is a bracket, group like terms to one side, and then solve for the variable.
3) For fractional equations, the steps are to multiply both sides by the common denominator to clear the fractions, then group like terms and solve.

Sec 3 A Maths Notes Indices

1. The document discusses solving exponential equations with one, two, or three terms using properties of exponents such as changing bases to the same term and equating powers.
2. Examples are provided for solving two-term exponential equations by making the bases equal and equations with three terms by substituting variables, changing bases to the same term, and equating powers.
3. Solving exponential equations as products using properties such as treating exponents as multipliers is also demonstrated through examples.

Sec 4 A Maths Notes Maxima Minima

The document contains two examples of maximum and minimum problems involving differentiation.
Example 1 asks the reader to find the minimum volume of a cone given that a sphere must fit inside it. It is found that the minimum volume occurs when the radius of the cone is 28.577 cm.
Example 2 involves finding the maximum volume of a cylinder inscribed in a sphere. The maximum volume is calculated to be 104,000 cm3, occurring when the height of the cylinder is 28.5 cm.
The document provides guidance on solving maximum and minimum problems using differentiation, illustrated through these two examples involving geometric shapes.

Sec 3 E Maths Notes Coordinate Geometry

This document provides examples and explanations of using the distance formula and equations of lines in coordinate geometry. It defines the distance formula and shows how to calculate the distance between two points with given coordinates. It also demonstrates how to determine the gradient and y-intercept of a line given its equation, find the equation of a line given the gradient and a point or two points, and find values related to lines parallel or intersecting given lines.

Tam 2nd

The document discusses techniques for finding the slope of a tangent line to a function at a given point using derivatives. It provides examples of finding the slope for various functions, including polynomials, trigonometric functions, and implicitly defined functions, using the definition of derivative and rules like the product rule. Approximations of slope using finite differences are also covered. Guidelines for performing implicit differentiation are outlined.

solar system

The document discusses three special factoring formulas:
1) Perfect square trinomials can be factored using the difference of two squares formula.
2) The difference of two squares formula factors expressions of the form (a-b)^2 into (a+b)(a-b).
3) The sum or difference of two cubes can be factored into a binomial times a trinomial using specific formulas.

Difference of Two Squares

This document provides instructions on factoring polynomials using the difference of two squares formula. It begins with the objectives and a review of perfect squares. It then presents the difference of two squares formula a2 - b2 = (a - b)(a + b) and provides examples of factoring expressions like x2 - 25 and 3x2 - 75 using this formula. The document stresses that for an expression to be factorable as a difference of two squares, it must be a binomial with two terms that are perfect squares separated by a subtraction sign. It provides a table for students to identify whether additional expressions can be factorized this way. In the conclusion, it summarizes the key things to remember about factoring the difference of

FACTORING

L1. The document provides information on different factoring techniques:
L1.1 Common monomial factoring, difference of two squares, sum and difference of two cubes, and perfect square trinomials.
L1.2 Steps are outlined for each technique with examples provided.
L1.3 Key aspects of each technique are highlighted such as requiring the first and last term to be perfect squares for difference of two squares.

Sec 1 Maths Notes Equations

1) The document provides steps for solving simple linear equations with no fractions and fractional equations.
2) For linear equations with no fractions, the steps are to expand if there is a bracket, group like terms to one side, and then solve for the variable.
3) For fractional equations, the steps are to multiply both sides by the common denominator to clear the fractions, then group like terms and solve.

Sec 3 A Maths Notes Indices

1. The document discusses solving exponential equations with one, two, or three terms using properties of exponents such as changing bases to the same term and equating powers.
2. Examples are provided for solving two-term exponential equations by making the bases equal and equations with three terms by substituting variables, changing bases to the same term, and equating powers.
3. Solving exponential equations as products using properties such as treating exponents as multipliers is also demonstrated through examples.

Sec 4 A Maths Notes Maxima Minima

The document contains two examples of maximum and minimum problems involving differentiation.
Example 1 asks the reader to find the minimum volume of a cone given that a sphere must fit inside it. It is found that the minimum volume occurs when the radius of the cone is 28.577 cm.
Example 2 involves finding the maximum volume of a cylinder inscribed in a sphere. The maximum volume is calculated to be 104,000 cm3, occurring when the height of the cylinder is 28.5 cm.
The document provides guidance on solving maximum and minimum problems using differentiation, illustrated through these two examples involving geometric shapes.

Sec 3 E Maths Notes Coordinate Geometry

This document provides examples and explanations of using the distance formula and equations of lines in coordinate geometry. It defines the distance formula and shows how to calculate the distance between two points with given coordinates. It also demonstrates how to determine the gradient and y-intercept of a line given its equation, find the equation of a line given the gradient and a point or two points, and find values related to lines parallel or intersecting given lines.

Tam 2nd

The document discusses techniques for finding the slope of a tangent line to a function at a given point using derivatives. It provides examples of finding the slope for various functions, including polynomials, trigonometric functions, and implicitly defined functions, using the definition of derivative and rules like the product rule. Approximations of slope using finite differences are also covered. Guidelines for performing implicit differentiation are outlined.

solar system

The document discusses three special factoring formulas:
1) Perfect square trinomials can be factored using the difference of two squares formula.
2) The difference of two squares formula factors expressions of the form (a-b)^2 into (a+b)(a-b).
3) The sum or difference of two cubes can be factored into a binomial times a trinomial using specific formulas.

Difference of Two Squares

This document provides instructions on factoring polynomials using the difference of two squares formula. It begins with the objectives and a review of perfect squares. It then presents the difference of two squares formula a2 - b2 = (a - b)(a + b) and provides examples of factoring expressions like x2 - 25 and 3x2 - 75 using this formula. The document stresses that for an expression to be factorable as a difference of two squares, it must be a binomial with two terms that are perfect squares separated by a subtraction sign. It provides a table for students to identify whether additional expressions can be factorized this way. In the conclusion, it summarizes the key things to remember about factoring the difference of

FACTORING

L1. The document provides information on different factoring techniques:
L1.1 Common monomial factoring, difference of two squares, sum and difference of two cubes, and perfect square trinomials.
L1.2 Steps are outlined for each technique with examples provided.
L1.3 Key aspects of each technique are highlighted such as requiring the first and last term to be perfect squares for difference of two squares.

factoring perfect square trinomial

The document discusses factoring perfect square trinomials. It defines a perfect square trinomial as having the first and last terms as perfect squares, and the middle term twice the product of the first and last terms. The document provides examples of factoring perfect square trinomials using the multiplication breaker map (MBM) method. This involves taking the square root of the first and last terms, checking if the middle term satisfies the definition, and then factoring the expression as the sum or difference of the two terms squared. The document concludes with an activity having students work in groups to factor various perfect square trinomials using MBM.

1.1 ss factoring the difference of two squares

This document discusses factoring polynomials that are the difference of two squares using the formula a2 - b2 = (a + b)(a - b). It provides examples of factoring polynomials like x2 - 16, 9x2 - 100, and 36m2 - 49n4. It also lists 10 practice problems for factoring polynomials that are differences of two squares and references where readers can learn more.

Factoring difference of two squares

1. Factoring the difference of two squares involves writing an expression in the form a2 - b2 as the product of the sum and difference of two binomials.
2. To factor an expression using this method, take the square root of the first and last terms and write them as the sum and difference of two binomials that have the same first and last terms.
3. The key characteristics for an expression to be factorable as the difference of two squares are: it has two terms, the first term is a perfect square, it uses subtraction, and the last term is a perfect square.

Factoring the difference of two squares

The document discusses factoring the difference of two squares. It states that the factors of the difference between two squares are the product of the sum and difference of the two numbers. It then provides examples of factoring different expressions involving the difference of two squares in three steps: 1) take the square root of each term, 2) use the results to form the sum and difference of the numbers, and 3) write the factored expression as the product of the sum and difference.

Power point for theorem

The document discusses the Pythagorean theorem and distance formula. It asks questions about what the variables represent in each, what type of triangle the Pythagorean theorem applies to, and the steps to use the distance formula. It then provides an example of using the distance formula to find the distance between the points (9,8) and (7,5), which equals the square root of 13 or about 3.6.

Math academy-partial-fractions-notes

This document provides an introduction to partial fractions. It defines key terms like polynomials, rational functions, and proper and improper fractions. It then outlines the three main cases for splitting a fraction into partial fractions: (1) a linear factor (ax+b), (2) a repeated linear factor, and (3) a quadratic factor (ax^2+bx+c). For each case, it provides an example of how to write the fraction as a sum of partial fractions. It concludes by emphasizing two important checks: (1) the fraction must be proper, and (2) the denominator must be completely factorized before attempting to write it as partial fractions.

05 perfect square, difference of two squares

This document discusses perfect squares and the difference of two squares. It defines a perfect square as a number that can be expressed as a square, such as 9, 16, or 81. Any expression of the form a^2, (a+b)^2, or (k-h)^2 is also a perfect square. Perfect squares are helpful for expanding and factorizing expressions. For example, (c+d)^2 = c^2 + 2cd + d^2. The document also discusses how to find the area of a square when one side is increased by some amount b, using the formula (a+b)^2 = a^2 + 2ab + b^2. It concludes by explaining that

Factoring the difference of two squares

A powerpoint presentation of factoring the difference of two squares with Math Quiz Show as the motivation

Section 1.5 distributive property (algebra)

The document discusses the distributive property and how it is used to simplify algebraic expressions. It provides examples of distributing terms over addition and subtraction, such as 5(x + 7) = 5x + 35. It also discusses like terms and how they can be combined when simplifying expressions.

Algebra 7 Point 6

1) This document discusses different techniques for factoring polynomials completely into their prime factors.
2) It provides examples of factoring polynomials using the greatest common factor, recognizing special cases like the difference of squares, and factoring trinomials that are perfect squares or can be written as the product of two binomials.
3) The reader is given examples of completely factoring various polynomials and is instructed to complete additional practice problems at the end.

5.3 Solving Quadratics by Finding Square Roots

This document discusses solving quadratic equations by taking square roots. It covers the definition and properties of square roots, including the product and quotient properties. It then demonstrates how to simplify radical expressions using these properties. Finally, it shows how to solve quadratic equations by taking the square root of both sides and using sign notation. Examples are provided for each concept along with practice problems for the reader.

common monomial factor

The document discusses factoring polynomials by finding the greatest common factor (GCF). It provides examples of factoring polynomials by finding the GCF of the numerical coefficients and variable terms. Students are then given practice problems to factor polynomials by finding the GCF and writing the factored form.

Factoring Perfect Square Trinomial

The document discusses perfect square trinomials and how to factor them. It provides examples of factoring various square trinomials using the properties that the first and last terms must be perfect squares, and the middle term must be twice the product of the square roots of the first and last terms. It then has students practice factoring several square trinomial examples on their own.

Lecture 10

Parametric equations define a curve where x and y are defined in terms of a third variable, called the parameter. The document provides examples of parametric equations and how to eliminate the parameter to obtain a single equation relating x and y, whose graph is the same as that defined by the original parametric equations. Examples shown include linear, circular and parabolic curves. Exercises are provided to eliminate parameters from additional parametric equations and sketch the resulting curves.

Factoring difference of squares

The document discusses factoring the difference of two squares. It explains that factoring the difference of two squares is the reverse of multiplying the sum and difference of the same two terms. An example of factoring x^2 - 9 into (x + 3)(x - 3) is provided to illustrate this process. The document then provides another example, factoring 4x^2 - 25 into (2x + 5)(2x - 5).

Polynomials Test Answers

This document contains a polynomials test with multiple questions:
1) Find the products of several polynomials, including (x-1)(x+3) and (x^2 - x - 1)(x + 7).
2) For a triangle with base x + 1 and height 3x, express the area in terms of x using the area formula A = 1/2bh.
3) Divide the polynomial (x^2 + 10x + 21) by (x + 3).

Markov number

Markov numbers are positive integers that satisfy the Markov Diophantine equation: x2 + y2 + z2 = 3xyz. Only a few Markov numbers exist below 100, and they have been extensively studied. Key results include:
- Markov numbers satisfy several number theoretic properties like being mutually co-prime.
- The distribution of Markov numbers below a given value x has been bounded.
- The Markov and Euclid trees describe the relationships between Markov triples.
- Several theorems prove properties like uniqueness of Markov numbers under certain conditions.

Project in Mathematics. Functions

This document is a student's math project submitted to their teacher. It covers several topics in math including:
1) Evaluating functions by substituting values into equations. Examples are worked out for different functions.
2) Inverse functions and how to think of a function transforming one value into another.
3) Synthetic division, working through an example problem.
4) Exponential functions, working through examples of exponential equations.
5) Logarithm functions, explaining how to arrange logarithm equations before solving them and working through examples.
6) Converting between degrees and radians, working through examples of both degree to radian and radian to degree conversions.

Determinants

The document provides instructions on calculating the determinant of matrices. It begins with an overview of determinants and their properties. It then provides examples of calculating the determinant of 2x2 and 3x3 matrices using expansion by minors and diagonals. The key steps shown are expanding the determinants using the minor of each element, where the sign of each term depends on its position in the matrix.

Distributive property

The students will learn to use the distributive property to simplify expressions by distributing terms being multiplied to terms inside parentheses. The distributive property distributes the number outside the parentheses to each term inside. Examples are provided to demonstrate distributing terms and combining like terms to simplify expressions.

Sec 3 A Maths Notes Indices

1) The document provides examples of solving exponential equations with various methods depending on whether the equation has two terms, three or more terms, or involves indices as products or quotients.
2) Key steps include splitting equations, letting one term equal a variable, raising both sides to the same power, and changing all terms to have the same base before equating exponents.
3) Examples range from simple equations like 82=x to more complex ones involving subtraction, addition, and multiplication of terms with different bases and exponents like (23)3=x+2-x.

JC Vectors summary

1. The document provides revision on various topics in vectors including ratio theorem, scalar and vector products, lines, planes, perpendiculars, reflections, angles, distances, direction cosines, and geometric meanings.
2. Key concepts covered include using scalar and vector products to find angles between lines, planes, and determining if lines or planes are parallel/perpendicular.
3. Methods for finding the foot of a perpendicular from a point to a line or plane, reflecting lines and planes, and determining relationships between lines and planes are summarized.

factoring perfect square trinomial

The document discusses factoring perfect square trinomials. It defines a perfect square trinomial as having the first and last terms as perfect squares, and the middle term twice the product of the first and last terms. The document provides examples of factoring perfect square trinomials using the multiplication breaker map (MBM) method. This involves taking the square root of the first and last terms, checking if the middle term satisfies the definition, and then factoring the expression as the sum or difference of the two terms squared. The document concludes with an activity having students work in groups to factor various perfect square trinomials using MBM.

1.1 ss factoring the difference of two squares

This document discusses factoring polynomials that are the difference of two squares using the formula a2 - b2 = (a + b)(a - b). It provides examples of factoring polynomials like x2 - 16, 9x2 - 100, and 36m2 - 49n4. It also lists 10 practice problems for factoring polynomials that are differences of two squares and references where readers can learn more.

Factoring difference of two squares

1. Factoring the difference of two squares involves writing an expression in the form a2 - b2 as the product of the sum and difference of two binomials.
2. To factor an expression using this method, take the square root of the first and last terms and write them as the sum and difference of two binomials that have the same first and last terms.
3. The key characteristics for an expression to be factorable as the difference of two squares are: it has two terms, the first term is a perfect square, it uses subtraction, and the last term is a perfect square.

Factoring the difference of two squares

The document discusses factoring the difference of two squares. It states that the factors of the difference between two squares are the product of the sum and difference of the two numbers. It then provides examples of factoring different expressions involving the difference of two squares in three steps: 1) take the square root of each term, 2) use the results to form the sum and difference of the numbers, and 3) write the factored expression as the product of the sum and difference.

Power point for theorem

The document discusses the Pythagorean theorem and distance formula. It asks questions about what the variables represent in each, what type of triangle the Pythagorean theorem applies to, and the steps to use the distance formula. It then provides an example of using the distance formula to find the distance between the points (9,8) and (7,5), which equals the square root of 13 or about 3.6.

Math academy-partial-fractions-notes

This document provides an introduction to partial fractions. It defines key terms like polynomials, rational functions, and proper and improper fractions. It then outlines the three main cases for splitting a fraction into partial fractions: (1) a linear factor (ax+b), (2) a repeated linear factor, and (3) a quadratic factor (ax^2+bx+c). For each case, it provides an example of how to write the fraction as a sum of partial fractions. It concludes by emphasizing two important checks: (1) the fraction must be proper, and (2) the denominator must be completely factorized before attempting to write it as partial fractions.

05 perfect square, difference of two squares

This document discusses perfect squares and the difference of two squares. It defines a perfect square as a number that can be expressed as a square, such as 9, 16, or 81. Any expression of the form a^2, (a+b)^2, or (k-h)^2 is also a perfect square. Perfect squares are helpful for expanding and factorizing expressions. For example, (c+d)^2 = c^2 + 2cd + d^2. The document also discusses how to find the area of a square when one side is increased by some amount b, using the formula (a+b)^2 = a^2 + 2ab + b^2. It concludes by explaining that

Factoring the difference of two squares

A powerpoint presentation of factoring the difference of two squares with Math Quiz Show as the motivation

Section 1.5 distributive property (algebra)

The document discusses the distributive property and how it is used to simplify algebraic expressions. It provides examples of distributing terms over addition and subtraction, such as 5(x + 7) = 5x + 35. It also discusses like terms and how they can be combined when simplifying expressions.

Algebra 7 Point 6

1) This document discusses different techniques for factoring polynomials completely into their prime factors.
2) It provides examples of factoring polynomials using the greatest common factor, recognizing special cases like the difference of squares, and factoring trinomials that are perfect squares or can be written as the product of two binomials.
3) The reader is given examples of completely factoring various polynomials and is instructed to complete additional practice problems at the end.

5.3 Solving Quadratics by Finding Square Roots

This document discusses solving quadratic equations by taking square roots. It covers the definition and properties of square roots, including the product and quotient properties. It then demonstrates how to simplify radical expressions using these properties. Finally, it shows how to solve quadratic equations by taking the square root of both sides and using sign notation. Examples are provided for each concept along with practice problems for the reader.

common monomial factor

The document discusses factoring polynomials by finding the greatest common factor (GCF). It provides examples of factoring polynomials by finding the GCF of the numerical coefficients and variable terms. Students are then given practice problems to factor polynomials by finding the GCF and writing the factored form.

Factoring Perfect Square Trinomial

The document discusses perfect square trinomials and how to factor them. It provides examples of factoring various square trinomials using the properties that the first and last terms must be perfect squares, and the middle term must be twice the product of the square roots of the first and last terms. It then has students practice factoring several square trinomial examples on their own.

Lecture 10

Parametric equations define a curve where x and y are defined in terms of a third variable, called the parameter. The document provides examples of parametric equations and how to eliminate the parameter to obtain a single equation relating x and y, whose graph is the same as that defined by the original parametric equations. Examples shown include linear, circular and parabolic curves. Exercises are provided to eliminate parameters from additional parametric equations and sketch the resulting curves.

Factoring difference of squares

The document discusses factoring the difference of two squares. It explains that factoring the difference of two squares is the reverse of multiplying the sum and difference of the same two terms. An example of factoring x^2 - 9 into (x + 3)(x - 3) is provided to illustrate this process. The document then provides another example, factoring 4x^2 - 25 into (2x + 5)(2x - 5).

Polynomials Test Answers

This document contains a polynomials test with multiple questions:
1) Find the products of several polynomials, including (x-1)(x+3) and (x^2 - x - 1)(x + 7).
2) For a triangle with base x + 1 and height 3x, express the area in terms of x using the area formula A = 1/2bh.
3) Divide the polynomial (x^2 + 10x + 21) by (x + 3).

Markov number

Markov numbers are positive integers that satisfy the Markov Diophantine equation: x2 + y2 + z2 = 3xyz. Only a few Markov numbers exist below 100, and they have been extensively studied. Key results include:
- Markov numbers satisfy several number theoretic properties like being mutually co-prime.
- The distribution of Markov numbers below a given value x has been bounded.
- The Markov and Euclid trees describe the relationships between Markov triples.
- Several theorems prove properties like uniqueness of Markov numbers under certain conditions.

Project in Mathematics. Functions

This document is a student's math project submitted to their teacher. It covers several topics in math including:
1) Evaluating functions by substituting values into equations. Examples are worked out for different functions.
2) Inverse functions and how to think of a function transforming one value into another.
3) Synthetic division, working through an example problem.
4) Exponential functions, working through examples of exponential equations.
5) Logarithm functions, explaining how to arrange logarithm equations before solving them and working through examples.
6) Converting between degrees and radians, working through examples of both degree to radian and radian to degree conversions.

Determinants

The document provides instructions on calculating the determinant of matrices. It begins with an overview of determinants and their properties. It then provides examples of calculating the determinant of 2x2 and 3x3 matrices using expansion by minors and diagonals. The key steps shown are expanding the determinants using the minor of each element, where the sign of each term depends on its position in the matrix.

Distributive property

The students will learn to use the distributive property to simplify expressions by distributing terms being multiplied to terms inside parentheses. The distributive property distributes the number outside the parentheses to each term inside. Examples are provided to demonstrate distributing terms and combining like terms to simplify expressions.

factoring perfect square trinomial

factoring perfect square trinomial

1.1 ss factoring the difference of two squares

1.1 ss factoring the difference of two squares

Factoring difference of two squares

Factoring difference of two squares

Factoring the difference of two squares

Factoring the difference of two squares

Power point for theorem

Power point for theorem

Math academy-partial-fractions-notes

Math academy-partial-fractions-notes

05 perfect square, difference of two squares

05 perfect square, difference of two squares

Factoring the difference of two squares

Factoring the difference of two squares

Section 1.5 distributive property (algebra)

Section 1.5 distributive property (algebra)

Algebra 7 Point 6

Algebra 7 Point 6

5.3 Solving Quadratics by Finding Square Roots

5.3 Solving Quadratics by Finding Square Roots

common monomial factor

common monomial factor

Factoring Perfect Square Trinomial

Factoring Perfect Square Trinomial

Lecture 10

Lecture 10

Factoring difference of squares

Factoring difference of squares

Polynomials Test Answers

Polynomials Test Answers

Markov number

Markov number

Project in Mathematics. Functions

Project in Mathematics. Functions

Determinants

Determinants

Distributive property

Distributive property

Sec 3 A Maths Notes Indices

1) The document provides examples of solving exponential equations with various methods depending on whether the equation has two terms, three or more terms, or involves indices as products or quotients.
2) Key steps include splitting equations, letting one term equal a variable, raising both sides to the same power, and changing all terms to have the same base before equating exponents.
3) Examples range from simple equations like 82=x to more complex ones involving subtraction, addition, and multiplication of terms with different bases and exponents like (23)3=x+2-x.

JC Vectors summary

1. The document provides revision on various topics in vectors including ratio theorem, scalar and vector products, lines, planes, perpendiculars, reflections, angles, distances, direction cosines, and geometric meanings.
2. Key concepts covered include using scalar and vector products to find angles between lines, planes, and determining if lines or planes are parallel/perpendicular.
3. Methods for finding the foot of a perpendicular from a point to a line or plane, reflecting lines and planes, and determining relationships between lines and planes are summarized.

Functions JC H2 Maths

This document defines functions and their key properties such as domain, range, and inverse functions. It provides examples of determining whether a function is one-to-one and finding its inverse. Composite functions are discussed, including ensuring the range of the inner function is contained within the domain of the outer function. Self-inverse functions are introduced, where the inverse of a function is equal to the function itself.

Vectors2

The document discusses various topics related to vectors and planes:
1. It explains the vector product in three forms - mathematical calculation, 3D picture, and in terms of sine. It provides an example to calculate the vector product of two vectors.
2. It discusses the different forms of the equation of a plane - parametric, scalar product, and Cartesian forms. It provides examples to write the equation of a plane in these different forms.
3. It explains how to find the foot of the perpendicular from a point to a plane. It provides examples to find the foot and shortest distance.
4. It discusses how to find acute angles between lines, planes, and a line and plane. Examples

Complex Numbers 1 - Math Academy - JC H2 maths A levels

The document provides lessons on complex numbers. It defines a complex number as being of the form z = x + iy, where x and y are real numbers. It discusses operations like addition, subtraction, multiplication and division of complex numbers. It also defines the complex conjugate and gives some examples of performing operations on complex numbers.

Recurrence

This document provides examples of recurrence relations and their solutions. It begins by defining convergence of sequences and limits. It then provides examples of recurrence relations, solving them using algebraic and graphical methods. One example finds the 6th term of a sequence defined by a recurrence relation to be 2.3009. Another example solves a recurrence relation algebraically to express the general term un in terms of n. The document emphasizes using graphical methods like sketching graphs to prove properties of sequences defined by recurrence relations.

Functions 1 - Math Academy - JC H2 maths A levels

Slides for Functions lecture 1.
JC H2 Maths
A levels Singapore
www.mathacademy.sg
Copyright 2015 Math Academy

Probability 2 - Math Academy - JC H2 maths A levels

The document provides information on probability concepts including Venn diagrams, union and intersection of events, useful probability formulas, mutually exclusive vs independent events, and examples testing these concepts. Specifically, it defines union as "taking everything in A and B", intersection as "taking common parts in A and B", provides formulas for probability of unions and intersections, and shows how to determine if events are mutually exclusive or independent using the probability of their intersection. It also includes worked examples testing concepts like mutually exclusive events and independence.

Probability 1 - Math Academy - JC H2 maths A levels

The document discusses conditional probability and provides examples. It defines conditional probability P(A|B) as the probability of event A occurring given that event B has already occurred. An example calculates probabilities for drawing marbles from a bag. Another example finds probabilities for selecting chocolates with different flavors from a box containing chocolates of various flavors. Formulas and step-by-step workings are provided for calculating conditional probabilities.

Sec 3 A Maths Notes Indices

Sec 3 A Maths Notes Indices

JC Vectors summary

JC Vectors summary

Functions JC H2 Maths

Functions JC H2 Maths

Vectors2

Vectors2

Complex Numbers 1 - Math Academy - JC H2 maths A levels

Complex Numbers 1 - Math Academy - JC H2 maths A levels

Recurrence

Recurrence

Functions 1 - Math Academy - JC H2 maths A levels

Functions 1 - Math Academy - JC H2 maths A levels

Probability 2 - Math Academy - JC H2 maths A levels

Probability 2 - Math Academy - JC H2 maths A levels

Probability 1 - Math Academy - JC H2 maths A levels

Probability 1 - Math Academy - JC H2 maths A levels

Natural birth techniques - Mrs.Akanksha Trivedi Rama University

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Natural birth techniques are various type such as/ water birth , alexender method, hypnosis, bradley method, lamaze method etcChapter 4 - Islamic Financial Institutions in Malaysia.pptx

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Wound healing PPT

This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
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- 1. 4048 2EM Algebra – Change Subject (1) Math Academy® All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. Math Academy® 1 Knowing is not enough; we must apply. Willing is not enough; we must do. Johann Wolfgang von Goethe Notes: Changing Subject of a Formula Example 1: Make the letter in the brackets the subject of the given formula. Example 2: Example 3: (3 ) 2 w x y c+ = ( )x z yx m = - ( )y 1x y m z - = m x y z - = m x y z - = 2 a b c x = + ( )x 2 1 a b c x + = 2 1x a b c = + 2 a x b c = + 2( ) a x b c = + Common mistake m m z x y - = FLIP EACH side Multiply on both sidesm ALONEy LHS and RHS one term each FLIP EACH side (not flip individually) Multiply on both sidesa Common mistake 1 1 2 x a b c = + LHS: ALONEx Technique: Flip
- 2. 4048 2EM Algebra – Change Subject (1) Math Academy® All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. Math Academy® 2 Example 4: Example 5: Example 6: 3 2x y k k y - = ( )x 2 R r T r + = - ( )r 1 11 =+ ba ( )b 1 1 1 b a = - 1 1a b a a = - 1 1a b a - = 1 1 b a a = - 1 a b a = - appears oncex Do not Cross-multiply appears twicer Cross-multiply Group terms with r Cross- Multiply Factorise r Make the term containing on one side.b Make Same Denominator Flip Do not make same denominator at this stage (else appear everywhere --- Tedious !!) b