Algebra
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This document contains 3 identities related to multiplication and addition of numbers: 1) To multiply the sum of two numbers by a third number, we multiply each number in the sum separately and add the results, following the identity (x+y)z = xz + yz. 2) To multiply the sum of two numbers by the sum of two other numbers, we multiply each number in the first sum by each number in the second sum and add the results, as expressed by the identity (x+y)(u+v) = xu + xv + yu + yv. 3) The document presents the identities (x-y)2 = x2 - 2xy +
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The document contains mathematical integrals expressed in equations. It includes:
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This document provides 15 examples of word problems involving numbers. Each example presents a multi-step word problem, shows the steps to define variables, write equations, and solve for the unknown values. The examples cover a range of problem types including finding missing numbers based on relationships between amounts, averages, sums, differences, products, and ratios.
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This document contains 13 mathematical expressions with their pronunciations. The expressions include: N squared plus N plus four; N equals two squared 2k plus one; P squared equals two q squared; eleven times u + eleven times t equals fifty five; X squared plus four thirds x plus four ninths equals two thirds plus four ninths; squared root of x plus ten plus five over squared root of x plus ten equals six; A equals pi r squared plus two pi r h; x cubed plus b x squared plus cx plus d equals zero; q equals minus d plus one over bc minus two over twenty seven b cubed; two x cubed equals seven x minus one; f of x equals four x squared minus three
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The document provides examples of evaluating expressions involving integers. It begins with the expression -3 - 5 + (-4) - (-2). It then introduces a flow chart for adding and subtracting integers based on rules for using a number line. The flow chart shows that when adding, the value increases, and when subtracting, the value decreases. Several practice problems with solutions are provided. It concludes with homework assignments and prompts for creating a practice problem with mistakes.
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2. The first problem involves finding the length of one side of a rectangular field given the total area. The second finds the height of a water tank given its base dimensions.
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2. Students are asked to solve expressions by adding numbers with the same sign, such as negative numbers, and putting the result in the same sign.
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2. One problem involves two APs where the ratio of the sum of the first n terms is 7n+1:4n+27. It is shown that the ratio of the 11th terms is 4:3.
3. Another problem proves that if an AP has (2n+1) terms, the ratio of the sum of odd terms to the sum of even terms is (n+1):n.
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1) An integral from 0 to infinity of (5x^2 + 9)^-1/2 dx which is evaluated to ln(u) from 14 to infinity and equals ln(infinity) - ln(14).
2) An integral from 0 to infinity of e^(x(ln(x))^2) dx which is evaluated by substituting u = ln(x) and equals -1.
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This document contains 13 mathematical expressions with their pronunciations. The expressions include: N squared plus N plus four; N equals two squared 2k plus one; P squared equals two q squared; eleven times u + eleven times t equals fifty five; X squared plus four thirds x plus four ninths equals two thirds plus four ninths; squared root of x plus ten plus five over squared root of x plus ten equals six; A equals pi r squared plus two pi r h; x cubed plus b x squared plus cx plus d equals zero; q equals minus d plus one over bc minus two over twenty seven b cubed; two x cubed equals seven x minus one; f of x equals four x squared minus three
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The document provides examples of evaluating expressions involving integers. It begins with the expression -3 - 5 + (-4) - (-2). It then introduces a flow chart for adding and subtracting integers based on rules for using a number line. The flow chart shows that when adding, the value increases, and when subtracting, the value decreases. Several practice problems with solutions are provided. It concludes with homework assignments and prompts for creating a practice problem with mistakes.
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2. The first problem involves finding the length of one side of a rectangular field given the total area. The second finds the height of a water tank given its base dimensions.
3. The third determines the radius of a cone given its volume.
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1. The document provides examples of adding integers with the same sign and different signs.
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3. Examples show adding integers visually using a number line and by writing addition sentences for combinations of positive and negative numbers.
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1. The document contains examples of arithmetic progressions and their properties. It includes problems involving finding terms, common differences, and sums of APs.
2. One problem involves two APs where the ratio of the sum of the first n terms is 7n+1:4n+27. It is shown that the ratio of the 11th terms is 4:3.
3. Another problem proves that if an AP has (2n+1) terms, the ratio of the sum of odd terms to the sum of even terms is (n+1):n.
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Algebra
- 1. WELCOME
- 2. ST.THOMAS TRAINING COLLEGE PREPARED BY DEEPU P KRISHNAN REG NO : 13386002 MATHEMATICS OPTIONAL
- 3. Identity
- 4. In order to multiply the sum of two numbers by a third number, we have to multiply each number in the sum separately and add using algebra, (x+y)z=xz+yz, for all numbers x,y,z.
- 6. 101×26 =(100+1)26 =2600+26 =2626
- 7. To find the product of a sum of two numbers by another sum of two numbers, we have to multiply each number of the first sum by each number of the second sum and add all these products together.
- 8. Using Algebra, (x+y) (u+v)= xu + xv + yu + yv, for all numbers x,y,u,v
- 10. (8+6)(10+5) = 8×10+8×5+6×10+6×5 =80+40+60+30 =210
- 12. (X+Y)(X-Y)=X²-Y²