Maths portfolio

MATHS PORTFOLIO

Life is a math Equation

In order to gain the most

You have to know how to convert

Negatives into Positives

NOTE-

INDEX

• Click on the Chapter you want to see .

Coordinate Geometry

Number Systems

Polynomials

Introdution to Euclids geometry

Linear Equations in two Variables

Lines and Angles

Area of Parallelograms and Triangles

Quadrilaterals

Triangles

10

12

11

Heron's Formula

Constructions

Circles

NOTE-

INDEX

• Click on the Chapter you want to see .

15

13

14

Surface Area's and Volumes

Probability

Statistics

Chapter -1 Number Systems

Natural Numbers

1.

All Counting numbers are called Natural numbers. EG- 12, 98,76,56,34,67.....ETC

Whole Numbers

All Natural numbers with 0 are called Whole numbers. EG-0, 34, 67, 34, 95, 34,....ETC

2.

Integers

3.

Integers are set of numbers that include all the natural numbers (1, 2, 3, 4, 5, ....ETC) and their negatives and zero. EG-0, -2 , 3, -15, 55, -100....ETC

Real Numbers

Rational Numbers

Irrational Numbers

The numbers which cant be expressed as p/q form, where p and q are integers and q is not equal to 0 are called Irrational Numbers. EG- √2, π, √3....etc

The numbers of the form p/qwhere p and q are integers, co-prime and q is not equal to 0 are called rational numbers. EG- 2 ,-8,4/7...etc

VS

Terminating Decimals

Non - Teminating Decimals

If the Decimal expression of a rational number terminates, then the decimal is called Terminating Decimals.

A decimal in which a digit or a set of digits repeats periodically is called a non - Terminating or Recurring number.

ex: 3/4 = 4)300(0.75 -28 ------ 20 20. ------ 0 ------ MORE EX :7/5 , 2/5 ... ETC;

ex:20/3 = 3)2000... (6.6..... -18 ------- 20 -18 ------- 2 -------

VS

Operations on Real Numbers

Like Irrational numbers can be added and subtracted. To add and subtract keep their irrational factor the same, add or subtract their coefficients. EG- 3√2+2√2=5√2..etc

Identities

1. (√a+√b) (√a-√b) = a-b
2. (a+√b) (a-√b) = a2 -b
3. (√a+√b) (√c+√d) = √ac +√ad+√bc+√bd
4. (√a-√b)2 = a2+a√ab+b2
5. (√a-√b)2 = a2-a√ab+b2

Real Numbers

Rationalization

"The process of converting into an equivalent expression whose denominator is a rational number is called rationalizing the denominator or Rationalization"

Monomial

Binomial

"Conjugate irrational number - The binomial irrational number which differs only in sign (+ or -) between the terms is called a conjugate irrational number. EG- ( √a - √b ) is conjugate of (√a +√b)"

"Rationalize a monomial square root. 1.Multiply a same factor in the numerator and denominator. Rationalize, Multiply in both numerator and denominator. 2. Rationalize the expression in the denominator contains one or more square roots"

Laws of Exponents

1. a^m . a^n = a^m+n
2. (a^m)^n = a^m.n
3. a^m / a^n = a^m-n , m>n
4. a^m . b^n = (ab)^m
5. a^ -m = 1 / a^m
6. a^0 = 1
7. m√a = a^ 1/m

Represntation of Irrational Numbers on Number line

• If ABC is a right angle triangle with AB ,BC and AC as the perpendicular base and hypotenuse of the triangle respectively with AB = x units and BC = y units.
• Then , the hypotenuse of triangle AC is given by √x2 + y2

END OF CHAPTER 1

By Tejal

Click here to the Index page

Chapter - 2 Polynomials

Algebraic Expression

Terms

1.

An algebraic expression is an expression that is made up of variables and constants along with algebraic operations (addition,subtraction ,etc)

Polynomial Expression

Coefficient

Variable

Constant

In general, an expression with one and more than one term with non-negative integral exponents of a variable is known as a polynomial. EG- x7 +2x4 - 5 + 3x

2.

Types of Algebraic Expressions

1.

Monomial Expression

2.

Binomial Expression

3.

Trinomial Expression

4.

Polynomial Expression

Standard form of Polynomial

Let a, a, a,..... be real numbers, and let n be a non-negative integer. A Polynomial in x is an expression of the form an -1 xn-1 +....a1 x + a0 Where a n is not equal to 0

Degree of a Polynomial

The highest power of the variable in a polynomial is called the degree of the Polynomial.

Zero Polynomial

• In a Polynomial if an = an-1 = an - z -a2 = a = 0 = 0, then the polynomial is called zero polynomial .

EG - '0' is the zero Polynomial.

• Degree of a zero Polynomial is undefined.

Example

Name

Degree

Constant

2x + 3

Linear

x2 - x + 5

Quadratic

3x3 + 23

Cubic

Quartic

x4 + 2x3 - 3

x5 + 2x3 = 5x * 3

Quinti

Introduction of zero of a Polynomial

• Now , if p(x) = ax + b , a is not equal to 0 , is a linear polynomial, how can we find a zero of p(x) ?

EG- 4 may have given you some idea p(x) = 0 ax + b =0 , a is not equal to 0 ax = -b x= -b/a So, x = -b/a is the only zero of p(x) i.e, a linear polynomial has one and only one zero.

Representation of Polynomial

• If the variable in polynomial is x, we may denote the polynomial by p (x) , q(x) or r(x)
• So, for example , we may write -

p(x) = 2x2 + 5x -3 q(x) = x3 -1 r(y) = y3 + y + 1 s(u) = 2-u - u2 + 6u5

Remainder Theorem

• If a polynomial f(x) is divided by (x-a), the remainder is the constant f(a), and f(x) = q(x) . x(-a) + f(a)
• Where q(x) is a polynomial with degree one less than the degree of f(x).
• If a polynomial p(x) is divided by the binomial x-a , the remainder abtained is p(a).

Division Algorithm-

Divided = (Divisor x Quotient) + Remainder

Factor Theorem

• x-a is a factor of the polynomial p(x), if p(a) =0, where is any real number.
• This is an extension to remainder theorem where remainder is 0 , i.e p(a)=0
• CONVERSE - p(x) is divided by x-a if a is a real number, if p(a)=0 , then (x-a) is the factor of p(x).

Identities

1. (a + b)2 = a2 + b2 + 2ab
2. (a – b)2 = a2 + b2 – 2ab
3. a2 – b2 = (a + b) (a – b)
4. (x + a) (x + b) = x2 + (a + b)x + ab
5. (a + b)3 = a3 + b3 + 3ab(a + b)
6. (a – b)3 = a3 – b3 – 3ab (a – b)
7. (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
8. a3 + b3 + c3 – 3abc = (a + b + c) (a2 + b2 + c2 – ab – bc – ca)
9. a3 - b3 = (a - b ) (a2 + ab +b2)
10. a3 + b3 = (a + b) (a2 - ab + b2)

Derivation of Identities

(x + y + z)2

(x - y + z)2

VS

(x + y+ z)2= { x + y + (-z)}2 =x2 +y2 + (-z)2 + 2.x.y + 2.y.(-z) + 2.(-z).x =x2 + y2 + z2 +2xy -2yz -2zx

(x - y +z)2 ={x - (-y) +(-z)2 =x2 + (-y)2 + (-z)2 +2.x.-y + 2.-y * (-z) + 2.(-z) .x = x2 - y2 -z2 -2xy -2yz + 2zx

Factorization

Factorization is a method of breaking the arithmetic algebraic expressions into a product of their factors. If we multiply the factors again, then they will result in the original expression.

END OF CHAPTER 2

By Tejal

Click here to the Index page

Introduction

The French mathematician Rene Descartes in 1637 introduced the Cartesian system of coordinates for describing the position of a point in a plane. This idea has given rise to an important branch of mathematics, known as Coordinate Geometry.

Chapter -3 Coordinate Geometry

Origin

1.

The coordinate axes intersect each other at right angles, the point of intersection of these two axes is called Origin.

Quadrants

2.

The cartesian plane is divided into four equal parts, called quadrants. These are named in the order as I, II, III and IV starting with the upper right and going around in anticlockwise direction.

Chapter - 3 Co-ordinate Geometry

Cartesian Plane

A Cartesian plane is a graph with two axes, one is called the x-axis and the other one is the y-axis. These two axes are perpendicular to each other.

THANK YOU!!

By - Tejal