Lesson 3.1 factorisation
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The document discusses different methods for factorizing quadratic expressions. It explains that the traditional method relies on multi-step processes, while the Vedic method makes factorization simpler using two sub-sutras called "Anurupyena" and "Adyamadyenantyamantyena", which mean "proportionately" and "the first by the first and the last by the last". This Vedic method splits the middle term of a quadratic expression into two parts proportionately so that the factors can be determined directly from the coefficients and terms of the expression.
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This module introduces quadratic functions. It discusses identifying quadratic functions as those with the highest exponent of 2, rewriting quadratic functions in general form f(x) = ax^2 + bx + c and standard form f(x) = a(x-h)^2 + k, and the key properties of quadratic graphs including the vertex and axis of symmetry. The module provides examples of identifying quadratic functions from equations and ordered pairs/tables and rewriting quadratic functions between general and standard form using completing the square.Calculus 08 techniques_of_integration
Calculus 08 techniques_of_integrationtutulk
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Trigonometric substitutions can help evaluate integrals involving trigonometric functions. The document provides examples of using trigonometric substitutions to evaluate integrals of powers of sine and cosine functions. Specifically:
1) Integrals of powers of sine and cosine can be evaluated by rewriting the functions using trigonometric identities and then substituting variables. For example, sin5x can be rewritten as sinx(sin2x)2 and evaluated using the substitution u=cosx.
2) More complex integrals may require multiple steps and identities. For example, evaluating sin6x requires rewriting sin2x using an identity and then integrating four resulting terms.
3) Trigonometric substitutions allow rewriting integrals involvingMaths Notes - Differential Equations
Maths Notes - Differential EquationsJames McMurray
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This document summarizes methods for solving ordinary differential equations (ODEs). It discusses:
1) Types of ODEs including order, degree, linear/nonlinear.
2) Four methods for solving 1st order ODEs: separable variables, homogeneous equations, exact equations, and integrating factors.
3) Solutions to higher order linear ODEs using complementary functions and particular integrals.
4) Finding complementary functions and particular integrals for ODEs with constant coefficients.
Em05 pe
Em05 penahomyitbarek
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The document provides examples and explanations for solving various types of polynomial equations, beginning with linear equations and then simultaneous linear equations. It then introduces quadratic equations.
Key steps for solving linear equations include isolating the variable, usually by subtracting or dividing both sides by a constant. For simultaneous linear equations, common techniques are elimination (subtracting equations to remove a variable) and substitution.
Several examples demonstrate solving systems of two and three linear equations graphically or algebraically. The document emphasizes setting up equivalent equations to efficiently eliminate variables. Finally, quadratic equations are introduced but no examples are provided.More from A V Prakasam (6)
Basics of Vedic Mathematics - Multiplication (1 of 2)
Basics of Vedic Mathematics - Multiplication (1 of 2)A V Prakasam
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An e-learning module based on ‘Vedic Mathematics’ by Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaja.
Note: The PowerPoint version of this presentation has animation and transitions. So some slides may appear cluttered or odd since these features may not be supported in ºÝºÝߣShare.Lesson 2.2 division
Lesson 2.2 divisionA V Prakasam
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The document describes the Paravartya method for division of algebraic expressions. It begins by explaining the method using an example of dividing 12x^2 - 8x - 32 by x - 2. It shows setting up the problem in two columns, with the divisor in column 1 and decreasing powers of x in column 2. The dividend coefficients are placed below, and the quotient terms are calculated by multiplying the prior term by the divisor coefficient and adding. Several more examples are provided to illustrate dividing polynomials. The method is then explained for divisors with more than one term, and examples are given. Finally, applications to dividing numbers are demonstrated.Lesson 2.1 division
Lesson 2.1 divisionA V Prakasam
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The document discusses the Nikhilam method for division. It provides examples and explanations for dividing numbers with divisors of 9, 8, 7 and 6. There are four cases covered: 1) divisor of 9 with 2-digit or more dividends, 2) divisor of 9 with 3+ digit dividends, 3) divisor of 9 with remainders greater than 9, and 4) divisors of 8, 7, 6. The method involves placing digits below a line, adding adjacent digits, and using remainders to get the quotient and remainder. Worked examples with step-by-step explanations are provided for each case.Lesson 1.2 multiplication
Lesson 1.2 multiplicationA V Prakasam
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The Urdhva-Tirayk Sutra describes a method for multiplying multi-digit numbers by representing each number as the sum of its digits multiplied by powers of ten, then distributing and combining terms. It provides rules for determining each digit of the product by multiplying corresponding digits of the factors and adding carry values. Examples show applying the method to multiply numbers, calculate areas, volumes, and solve word problems involving currency conversions. The use of vinculums to represent negative digits is also introduced.Lesson 1.1 multiplication
Lesson 1.1 multiplicationA V Prakasam
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This document introduces the Nikhilam Sutra, a method of Vedic mathematics for multiplication. It explains the principles and provides examples of multiplying numbers near and away from multiples of 10 using appropriate bases. The key steps are to make the numbers equal in digits, choose a base, find the differences from the base, add the numbers and differences, and multiply the differences. It also covers cases where the numbers are slightly above or below multiples and proportional methods for numbers with rational relationships. Practice problems are provided to demonstrate applying the sutra.Lesson 2.3 division
Lesson 2.3 divisionA V Prakasam
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The document describes the Urdhva Tiryak method of division from Vedic mathematics. It provides examples to illustrate the method. For algebraic divisions, the divisor terms are divided in descending order to obtain the quotient terms. The examples show finding the quotient and remainder when dividing polynomials using this method. In the recap, it notes this method is well-suited for algebraic expressions but more difficult for arithmetic calculations, and each of the Vedic division methods only apply to certain cases rather than providing a single general division algorithm.Lesson 3.1 factorisation
- 1. Factorisation of Simple Quadratics Factorisation of Simple Quadratcis Example 1: Factorise x2+7x+10 When the coefficient of x2 is 1, factorisation is easy even according to the current system. Split the middle coefficient into two parts so that their sum is the middle coefficient and their product is the absolute term. In this example split 7 into 5 and 2 sothat 5+2=7 and 5X2=10. Then the facotrs can be written as (x+2)(X+5) But in the case of quadratics whose coefficent of x2 is not unit, the students depend upon a multi-step method. For example: Example 2: Factorise 2x2 +5x+2 2x2 +5x+2 = 2x2+4x+x+2 = 2x(x+2)+1(x+2) = (2x+1)(x+2) The vedic system, however, makes the factorisation symple through two small sub-sutras – Anurupyena and Adyamadyenantyamantyena, which mean ‘proportionately’ and ‘the first by the first and the last by the last’. This will be explained in the next slide.
- 2. Factorisation of Simple Quadratics Consider the previous example – 2x2+5x+2 (i) Split the middle quotient, 5x, into two parts so that the coefficient of the first part is the same as the ratio of the second part to the last quotient. In other words split 5 into 4 and 1 so that the ration2:4 is the same as the ratio1:2. now this ratio 1:2 is one factor which is x+s (ii) The second factor is obtained by dividing the first quotient of the quadratic, i.e. 2x 2, by the first quotient of the factor already found i.e. x, giving 2x and the last quotient of the quadratic i.e. 2 by the last quotient of the factor already found i.e. by 2 giving 1. The second factory is then 2x+1 The following additional examples would be found useful; Examples: 1. 2x2-5x-3 = (x+3)(2x-1) 2. 2x2+7x+5 = (x+1)(2x+5) 3. 2x2+9x+10 = (x+2)(2x+5) 4. 2x2-5x-3 = (x-3)(2x+1) 5. 9x2-15x+4 = (3x-1)(3x-4)