1. (a + b)(a – b) = a2 – b
2. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
3. (a ± b)2 = a2 + b2± 2ab
4. (a + b + c + d)2 = a2 + b2 + c2 + d2 + 2(ab + ac + ad + bc + bd + cd) 5. (a ± b)3 = a3 ± b3 ± 3ab(a ± b)
6. (a ± b)(a2 + b2 m ab) = a3 ± b3
7. (a + b + c)(a2 + b2 + c2 ab – bc – ca) = a3 + b3 + c3 – 3abc = 1/2 (a + b + c)[(a  b)2 + (b  c)2 + (c  a)2] 8. when a + b + c = 0, a3 + b3 + c3 = 3abc
9. (x + a)(x + b) (x + c) = x3 + (a + b + c) x2 + (ab + bc + ac)x + abc 10. (x – a)(x – b) (x – c) = x3 – (a + b + c) x2 + (ab + bc + ac)x – abc 11. a4 + a2b2 + b4 = (a2 + ab + b2)( a2 – ab + b2)
12. a4 + b4 = (a2 – √2ab + b2)( a2 + √2ab + b2)
13. an + bn = (a + b) (a n1 – a n2 b + a n3 b2 – a n4 b3 +…….. + b n1) (valid only if n is odd)
14. an – bn = (a – b) (a n1 + a n2 b + a n3 b2 + a n4 b3 +……… + b n1) {where n ϵ N)
15. (a ± b)2n is always positive while (a ± b)2n is always negative, for any real values of a and b 16. (a – b)2n = (b – a)2” and (a – b)2n+1 = – (b – a)2n+1 17. if α and β are the roots of equation ax2 + bx + c = 0, roots of cx” + bx + a = 0 are 1/α and 1/β. if α and β are the roots of equation ax2 + bx + c = 0, roots of ax2 – bx + c = 0 are α and β. 18.
* n(n + l)(2n + 1) is always divisible by 6.
* 32n leaves remainder = 1 when divided by 8
* n3 + (n + 1 )3 + (n + 2 )3 is always divisible by 9
* 102n + 1 + 1 is always divisible by 11
* n(n2 1) is always divisible by 6
* n2+ n is always even
* 23n1 is always divisible by 7
* 152n1 +l is always divisible by 16
* n3 + 2n is always divisible by 3
* 34n – 4 3n is always divisible by 17
* n! + 1 is not divisible by any number between 2 and n (where n! = n (n – l)(n – 2)(n – 3)…….3.2.1)
for eg 5! = 5.4.3.2.1 = 120 and similarly 10! = 10.9.8…….2.1= 3628800 19. Product of n consecutive numbers is always divisible by...
...The Euler’s Formula
Euler’s formula and Identity: eix = cos(x) + i(sin(x))
The world of math today is one with endless possibilities. It expands into many different and interesting topics, often being incorporated into our everyday lives. Today, I will talk about one of these topics; the most mindblowing and fascinating formula invented, called the “Euler’s formula”. This formula was created and introduced by mathematician Leonhard Euler. In essence, the formula establishes the deep relationship between trigonometric functions and the complex exponential function.
Euler’s formula: eix=cos(x)+isin(x); x being any real number
Wow  we're relating an imaginary exponent to sine and cosine! What is even more interesting is that the formula has a special case: when π is substituted for x in the above equation, the result is an amazing identity called the Euler’s identity:
eix=cos(x)+isin(x)
eiπ=cos(π)+isin(π)
eiπ= 1+i(0)
eiπ= 1
Euler’s identity: eiπ= 1
This formula is known to be a “perfect mathematical beauty”. The physicist Richard Feynman called it "one of the most remarkable, almost astounding, formulas in all of mathematics." This is because these three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental...
...
Beauty of Math!1 x 8 + 1 = 912 x 8 + 2 = 98123 x 8 + 3 = 9871234 x 8 + 4 = 987612345 x 8 + 5 = 98765123456 x 8 + 6 = 9876541234567 x 8 + 7 = 987654312345678 x 8 + 8 = 98765432123456789 x 8 + 9 = 9876543211 x 9 + 2 = 1112 x 9 + 3 = 111123 x 9 + 4 = 11111234 x 9 + 5 = 1111112345 x 9 + 6 = 111111123456 x 9 + 7 = 11111111234567 x 9 + 8 = 1111111112345678 x 9 + 9 = 111111111123456789 x 9 +10= 11111111119 x 9 + 7 = 8898 x 9 + 6 = 888987 x 9 + 5 = 88889876 x 9 + 4 = 8888898765 x 9 + 3 = 888888987654 x 9 + 2 = 88888889876543 x 9 + 1 = 8888888898765432 x 9 + 0 = 888888888Brilliant, isn't it?
And look at this symmetry:1 x 1 = 111 x 11 = 121111 x 111 = 123211111 x 1111 = 123432111111 x 11111 = 123454321111111 x 111111 = 123456543211111111 x 1111111 = 123456765432111111111 x 11111111 = 123456787654321111111111 x 111111111=123456789 87654321Now, take a look at this...101%From a strictly mathematical viewpoint:What Equals 100%? What does it mean to give MORE than 100%?Ever wonder about those people who say they are giving more than 100%?We have all been in situations where someone wants you to GIVE OVER 100%.How about ACHIEVING 101%?What equals 100% in life?Here's a little mathematical formula that might help answer these questions:
If:A B C D E F G H I J K L M N O P Q R S T U V W X Y ZIs represented as:1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26. If:HARDWO R K8+1+18+4+23+ 15+18+11 = 98%And:KNOWLE DGE11+14+15+23+...
...a+b≥x ORa+b≤x a+b>x OR a+b<x
e.g.
∣2x1∣≥3=
2x1≥3 OR 2x1≤3
2x≥3+1 2x≤3+1
2x≥4 2x≤2
x≥2 x≤1
Rationalizing
Any fraction involving surds needs to have the denominator rationalised.
Rationalise 23
23 = 23 ×1
=23 ×33 note that33 =1
=233×3
=233
The denominator is rationalised now
Beta
Trigonometry
Basic rules
Hypotenuse
Adjacent
Sin = Opposite/hypotenuse
Cos = Adjacent/hypotenuse
Tan = Opposite/Adjacent
Opposite
Cot = 1/tan
Sec= 1/sin
Csc = 1cos
Special rules
 300  450  600 
Sin  12  12  32 
Cos  32  12  12 
tan  13  1  2
2
3
300
600
1
1
3 
2
450
1
1
Area of triangle
There is an basic formula for finding the area of a triangle A=12BH but I’m going to look at how to find the area of a triangle WITHOUT the base OR the height.
X
Y
Z
As long were are given x,y AND z we can find the area of this triangle without using the base or height. B
A=12×X×Y×sinZ
A
C
c
a
b
AREA=12absinB You can’t find the area if you don’t meet these conditions
AREA=12bcsinA
AREA=12acsinB...
... Politeness formulas in Arabic:
When talking about politeness formulas in Arabic and in English and how they are different, it is crucial to take into account the distinction between propositional content of a formula and its illocutionary force potential. A good example showing the relationship between semantic content or propositional content and illocutionary force illustrates in using congratulations in English and “shukran” in Arabic which is equivalent to “thanks”. Sometimes illocutionary force is not completely predictable, but simply can be learnt by what people agree upon. For instance, there are three expressions in Arabic performing different forces according to the matter of conventions; \barakaallahufik\ “God bless you” is used to perform the act of thanking, whereas \barakafik\ “blessing in yourself” has a different force, addressing the family members and the relatives of the deceased. \mabruuk\ “blesses” is another formula used for congratulations of marriage or success in examinations.
There are many other expressions used by Arabs in one of the earliest means of demonstrating politeness in a second language which are” greetings”. Greetings actually are the first task one should know when learning a new language. In Arabic greetings; for example, one would not restrict themselves to say \ahlan wa sahlan\ “hello” rather they would precede it by \marhaba\ “welcome” and they may...
...CYPOP25.3 Evaluate the benefits of different types of formula that are commonly available
Breast milk is a natural form of baby food which is perfect for a baby. But there are some people who for different reasons can’t breastfeed, or have chosen not to, formula milk is the next best thing.
Scientists and medical experts have spent years developing highquality formula milks that will provide babies with the specific nutrition that they need.
There are many different types of formula feed and there are many different factors to take into account before choosing the right formula feed.
* Health
* Dietary needs
* Age
* Cost and preparation time of different formula milks.
There are three different types of formula:
* Cow’s milkbased formula
* Hydrolysed protein formula
* Soyabased formula
Most babies can have cow’s milkbased formula, however there are some who have a health or dietary reason why they can’t.
The different range of formula milks are described below.
(information obtained from the baby centre website)
Cow’s milkbased formula
Most baby formula milks are based on cow's milk, which is modified to resemble breastmilk as closely as possible. Manufacturers modify cow's milk for babies by adjusting carbohydrate,...
...Marketing Research Project
Pricing strategy for ‘Formula One India Racing Event’ for Retail Customers
Table of Contents
1. Marketing Research Objective: 3
2. Introduction 3
3. Mode of survey 5
4. Survey Questionnaire 6
5. References 8
1. Marketing Research Objective
What should be optimal price for different classes of tickets at ‘Formula One India Racing Event’ for Retail Customers?
1. Introduction
Formula One is the highest class of single seated auto racing authorized by the Fédération Internationale de l'Automobile (FIA). The F1 season consists of a series of races, known as Grand’s Prix held on purposebuilt circuits and public roads. The results of each race are combined to determine two annual World Championships, one for the drivers and one for the constructors. The sport is a massive television event and each race is watched by over 600 million people around world.
Europe is Formula One's traditional centre, where all of the teams are based, and where around half of the races take place. However, the sport's scope has expanded significantly in recent years and Formula one event are being organized in several Asian countries including China, Turkey, Singapore, South Korea. More recently, Indian Grand prix has been announced. The first Indian Grand Prix’s race will be held at the Jaypee International Race Circuit in Greater Noida on October 30, 2011. The 5.14 km...
...
In Mathematics, a periodic function is a function that repeats its values in regular intervals or periods. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity.
Periodic functions are those that repeat on a set interval. All trigonometric functions are periodic. They are useful because one can determine the value of the function anywhere in the domain. If a function is periodic, then there is some value n for which over the entire domain of the function. The smallest nvalue that fits the function is called the period of f(x). (see fig. 1)
This report will be outlining the importance of the periodic function and how to use it in everyday life.
Fig 1.
This photo above provides a clear definition of what a Translational symmetry is and how it is involved with Periodic functions.
In periodic functions amplitude is term used to explain some periodic values. “The maximum absolute value of a periodic curve measured along its vertical axis. It also measures the angle made with the positive horizontal axis by the vector representation of a complex number”. (http://www.thefreedictionary.com/amplitude, 2013) see fig. 1.1.
Fig. 1.1
A period is known for the cycle length of a curve, in a periodic function. The period is distance required for the function to complete once full cycle. (see fig 1.2)
Fig 1.2
A phase...
...Innovating new packaging by determining the best suitable level of efficiency with the use of calculus for Trini Chocolate Delights.
JABARI BOWMAN
ADD MATH
SBA
CLASS: 56
SCHOOL: ST. MARY’S COLLEGE
TEACHER: MR. YIPYOUNG
Problem Statement
The project’s purpose is to work out the best level of production to suit the new packaging for ‘Trini Chocolate Delights ltd.’
The company’s objective is to maximize produce of their new product but minimize manufacturing cost. To fulfill this task the use of calculus is needed along with other mathematical methods to help design a best suitable and cheap packaging that will be used to carry a grand amount of smaller products while cost remain in a stead/safe amount.
Mathematical Formulation
Below is a list of formulas applied to each question
A. Tan =

B. Substitution
Area of triangle + Area of rectangle= Area of pentagon
C. Differentiation
D. The quadratic equation
The quadratic formula
OR
Factorization of the quadratic formula
E.
Problem solution
A
Problem diagram
E B
ycm
D 8xcm C
i) F is the midpoint of line EB and it’s also the perpendicular of triangle EAB. The line AF cuts triangle EAF in half resulting in two isosceles triangles.
Since AFB= AFE and DC=EB, then FB and EF...