Algebra

Algebra formulae:

• (a+b)²=a²+2ab+b²

• (a-b)²=a²-2ab+b²

• (a+b)²+(a-b)²=2(a²+b²)

• a²+b²=(a-b)²+2ab

• (a+b)²-(a-b)²=4ab

• (a+b)²+(a-b)²=2(a²+b²)

• (a+b+c)²=a²+b²+c²+2ab+2bc+2ca

• (a+b)³=a³+3a²b+3ab²+b³=a³+b³+3ab(a+b)

• (a-b)³=a³-3a²b+3ab²-b³=a³-b³-3ab(a-b)

• a³+b³=(a+b)(a²-ab+b²)

• a³-b³=(a-b)(a²+ab+b²)

• (a+b+c)(a²+b²+c²-ab-bc-ca)=a³+b³+c³-3abc

• If a+b+c=0 then a³+b³+c³=3abc

• If a²+b²+c²=ab+bc+ca then a=b=c

• If A and B are non empty sets then n(A∪B)=n(A)+n(B)-n(A∩B)

• If A and B are disjoint sets then n(A∪B)=n(A)+n(B)

• If A and Bare any two non-empty sets then AΔB is called symmetric difference between A and B where AΔB=(A-B)∪(B-A)=(A∪B)-(A∩B)

• If A and B are any two non-empty sets and A⊂B then 1)A∪B=B 2)A∩B=A 3)A-B=∅

Probability :

Probability:

Random Experiment: An experiment that can be repeated any number of times under identical conditions in

which : i) All possible outcomes of the experiment are known in advance, ii) The actual outcome in a

particular case is not known in advance, is called a random experiment.

Example 1 : In an experiment of tossing an unbiased coin is a random experiment.

we have only two possible outcomes: Head (H) and Tail (T).

Example 2 : Rolling a fair die is also a random experiment.

we have only six possible outcomes: 1, 2, 3, 4, 5, 6.

Example 3 : Drawing a card from well shuffled pack of cards is also a random experiment.

3, we have Fifty two possible outcomes.

Simple Event : Any possible outcome of a random experiment is called an elementary or simple event.

Event: An event associated to a random experiment is a compound event if it is obtained by combining two or

more elementary events associated to the random experiment.

Example : Tossing an unbiased coin experiment.

we have only two possible outcomes:

possible outcomes are Head (H) , Tail (T) are simple events

In single throw of a die, the event “getting an even number” i.e., {2,4,6} is an event.

Note : Every simple event is also an event but every event is not simple event.