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Govt. of Tamilnadu Department of School Education Bridge Course 2011-2012 Class X- maths

Govt. of Tamilnadu Department of School Education Bridge Course 2011-2012 Class X- maths. Mathematicians. Pythagoras. 569 B.C. – 475 B.C. Greece First pure mathematician 5 beliefs Secret society Pythagorean theorem. Aristotle. 384 B.C. – 322 B.C. Greece Philosopher

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Govt. of Tamilnadu Department of School Education Bridge Course 2011-2012 Class X- maths

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  1. Govt. of TamilnaduDepartment of School EducationBridge Course 2011-2012Class X-maths

  2. Mathematicians

  3. Pythagoras • 569 B.C. – 475 B.C. • Greece • First pure mathematician • 5 beliefs • Secret society • Pythagorean theorem

  4. Aristotle • 384 B.C. – 322 B.C. • Greece • Philosopher • Studied mathematics in relation to science

  5. Euclid • 325 B.C. – 265 B.C. • Greece • Wrote The Elements • Geometry today

  6. Al-Khwarizmi • 780 A.D.-850 A.D. • Baghdad (in Iraq) • 1st book on Algebra • Algebra • Natural Number • Equation

  7. Introduction to Set Theory

  8. Sets and Their Elements A set A is a collection well defined elements. If x is an element of A, we write xA; if not: xA. Say: “x is a member of A”, or “x is in A”, or “x belongs to A” We use lowercase letters for elements, and capitals for sets. Notation: use set braces “{“,”}” around the elements. For example: A = {0,1,2,3,4} Hence 2  A = {0,1,2,3,4} and 6  A = {0,1,2,3,4} Another way to write it: A = {x | x is an integer between 0 and 4}.

  9. Some Sets The set A = {a, b, c, d} has 4 elements. The set A = {2, 4, 6, 8, …, 40} has 20 elements. The ellipsis, “ … ”, is used to mean we fill in the missing elements in the obvious manner or pattern, as there are too many to actually list out on paper. The set of natural numbers: N = {0,1,2,…} The set of integers: Z = {…,–2,–1,0,1,2,…} The set of positive integers: Z+ = {1,2,3,…}

  10. (Proper) Subsets A set A is a subset of B, if and only if all elementsof A are also elements of B: Notation: AB or BA. If A is not a subset of B, we write AB or BA. If AB and B contains an element that is not in A,then we say “A is a proper subset of B”: AB or BA. For all sets: AA.

  11. Special Sets, Cardinality The sets A and B are equal (A=B) if and only if each element of A is an element of B and vice versa. The empty set, denoted by or { }, is the set without elements. The universal set, denoted by Ω, is the set of all elements currently under consideration. The size or cardinality of a finite set A, denoted with |A|, is the number of (distinct) elements.Example: ||=0 |{2,4,8,16}| = 4

  12. How to Think of Sets The elements of a set do not have an ordering,hence {a,b,c} = {b,c,a} The elements of a set do not have multitudes,hence {a,a,a} = {a,a} = {a} The size of A is thus the number of different elements

  13. Intersection, Union The intersection of two sets A and B, is the set ABof elements x such that both xA and xB.Notation: AB = { x | xA and xB} The union of two sets A and B, is the set ABof elements x such that xA or xB.Notation: AB = { x | xA or xB} Sets A and B are (mutually) disjoint if AB=

  14. Complements The complement of a set A,denoted with Ā(or A' or Ac), are the elements that are not elements of A. Therefore, A Ā = 

  15. Set Manipulations

  16. AB _ A A B _ B Venn Diagrams Venn diagrams are used to depict the unions, subsets, complements, intersections etc. of sets: C

  17. Set Difference • A – B • The set difference “A minus B” is the set of elements that are in A, with those that are in B subtracted out. Another way of putting it is, it is the set of elements that are in A, and not in B. _ • Therefore, A – B = A  B

  18. Examples

  19. Cylinders

  20. Try It V = r2h The radius of the cylinder is 5 m, and the height is 4.2 m V = 3.14 · 52 · 4.2 Substitute the values you know. V =329.7

  21. Practice 13 cm - radius 7 cm - height V = r2h Start with the formula V = 3.14 x 132 x 7 substitute what you know = 3.14 x 169 x 7 Solve using order of Ops. = 3714.62 cm3

  22. Area of Triangle L=length H=height What is the area of this triangle? W=width

  23. TRIANGLE L=length H=height What is the area of this triangle? W=width COPY THIS!

  24. TRIANGLE L=length H=height What is the area of this triangle? W=width PASTE HERE!

  25. TRIANGLE L=length H=height What is the area of this triangle? W=width FLIP VERTICALLY!

  26. TRIANGLE L=length H=height What is the area of this triangle? W=width FLIP HORIZONTALLY!

  27. TRIANGLE L=length H=height What is the area of this triangle? W=width TRANSLATE TO HERE!

  28. TRIANGLE The area of the triangle is ½ the area of this parallelogram? L=length H=height W=width

  29. TRIANGLE H=height Area = ½ (Width x Height) (perpendicular height H) W=width

  30. TRIANGLE H=height Area = ½ (Width x Height) (perpendicular height H) W=width

  31. Polynomials What does each prefix mean? Mono One Bi Two Tri Three

  32. What about poly? one or more A polynomial is a monomial or a sum/difference of monomials. Important Note!! An expression is not a polynomial if there is a variable in the denominator.

  33. State whether each expression is a polynomial. If it is, identify it. 1) 7y - 3x + 4 trinomial 2) 10x3yz2 monomial 3) not a polynomial

  34. The degree of a monomial is the sum of the exponents of the variables.Find the degree of each monomial. 5x2 4a4b3c 3) 7x3Y5z

  35. To find the degree of a polynomial, find the largest degree of the terms. 1) 8x2 - 2x + 7 Degrees: 2 1 0 Which is biggest? 2 is the degree! 2) y7 + 6y4 + 3x4m4 Degrees: 7 4 8 8 is the degree!

  36. Put in descending order: • 8x - 3x2 + x4 - 4 x4 - 3x2 + 8x - 4 2) Put in descending order in terms of x: 12x2y3 - 6x3y2 + 3y - 2x -6x3y2 + 12x2y3 - 2x + 3y

  37. 3) Put in ascending order in terms of y: 12x2y3 - 6x3y2 + 3y - 2x -2x + 3y - 6x3y2 + 12x2y3 • Put in ascending order: 5a3 - 3 + 2a - a2 -3 + 2a - a2 + 5a3

  38. Write in ascending order in terms of y:x4 – x3y2 + 4xy–2x2y3 • x4 + 4xy– x3y2–2x2y3 • –2x2y3 – x3y2 + 4xy + x4 • x4 – x3y2–2x2y3 + 4xy • 4xy –2x2y3 – x3y2 + x4

  39. Addition And subtraction on polynomials1. Add the following polynomials:(9y - 7x + 15a) + (-3y + 8x - 8a) Group your like terms. 9y - 3y - 7x + 8x + 15a - 8a 6y + x + 7a

  40. 2. Add the following polynomials:(3a2 + 3ab - b2) + (4ab + 6b2) Combine your like terms. 3a2 + 3ab + 4ab - b2 + 6b2 3a2 + 7ab + 5b2

  41. Add the polynomials.+ Y X X2 Y X XY Y X Y 1 1 Y Y • x2 + 3x + 7y + xy + 8 • x2 + 4y + 2x + 3 • 3x + 7y + 8 • x2 + 11xy + 8 1 1 1 1 1 1 Y

  42. 3. Add the following polynomials using column form:(4x2 - 2xy + 3y2) + (-3x2 - xy + 2y2) Line up your like terms. 4x2 - 2xy + 3y2 + -3x2 - xy + 2y2 _________________________ x2 - 3xy + 5y2

  43. Rewrite subtraction as adding the opposite. (9y - 7x + 15a) + (+ 3y - 8x + 8a) Group the like terms. 9y + 3y - 7x - 8x + 15a + 8a 12y - 15x + 23a 4. Subtract the following polynomials:(9y - 7x + 15a) - (-3y + 8x - 8a)

  44. 6. Subtract the following polynomials using column form:(4x2 - 2xy + 3y2) - (-3x2 - xy + 2y2) Line up your like terms and add the opposite. 4x2 - 2xy + 3y2 + (+ 3x2+ xy - 2y2) -------------------------------------- 7x2 - xy + y2

  45. Find the sum or difference.(5a – 3b) + (2a + 6b) • 3a – 9b • 3a + 3b • 7a + 3b • 7a – 3b

  46. Find the sum or difference.(5a – 3b) – (2a + 6b) • 3a – 9b • 3a + 3b • 7a + 3b • 7a – 9b

  47. (6 3)(y3 y5)   (3 9)(m m2)(n2 n)   Multiplying Monomials Multiply. A. (6y3)(3y5) (6y3)(3y5) Group factors with like bases together. Multiply. 18y8 B. (3mn2) (9m2n) Group factors with like bases together. (3mn2)(9m2n) 27m3n3 Multiply.

  48. Remember! When multiplying powers with the same base, keep the base and add the exponents. x2x3= x2+3 = x5

  49. (3 6)(x3 x2)   (2 5)(r2)(t3 t)   Example 1 Multiply. a. (3x3)(6x2) Group factors with like bases together. (3x3)(6x2) Multiply. 18x5 b. (2r2t)(5t3) Group factors with like bases together. (2r2t)(5t3) Multiply. 10r2t4

  50. Multiplying a Polynomial by a Monomial Multiply. 4(3x2 + 4x – 8) 4(3x2 + 4x – 8) Distribute 4. (4)3x2 +(4)4x – (4)8 Multiply. 12x2 + 16x – 32

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