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Algebra

Some Basic Rules

Rule of Addition: if the numbers are of same sign, add them and give the same

sign. e.g. Rule of Subtraction: if the numbers of opposite sign, subtract them and give the

sign of larger number. e.g.

Rules of Multiplication and DivisionPositive number Positive number positive number e.g.

Negative number

negative number

positive number e.g.

Positive numbernegative number positive number e.g. or

Rules of Multiplication and Division are same

When two signs come consecutively, first change the signs according to rules of

Multiplication/division and then use rules of Addition and Subtraction.e.g.

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Like and Unlike termsThe terms having same variables, with same index of variables, they are

referred to as Like terms. Otherwise they are called unlike terms.

e.g.

Like terms Unlike terms

Only like terms should be Added and Subtracted.

e.g. Unlike terms cant be Added or subtracted.

e.g. But Multiplication and Subtraction should be done on all the terms. Viz. like

and unlike.

e.g.

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1. Arithmetic and Geometric Progression

General Term :

Where, nth term, First term, total terms, common

difference between any two consecutive terms.

Sum of First terms of an A.P: [ ] [ ] where, last term

Specific terms in A.P.

o Three consecutive terms: o Four consecutive terms: o Five consecutive terms:

General term of G.P. :

Sum of First terms of G.P:

Specific terms in G.P.

o Three consecutive terms:

o Four consecutive terms:

o Five consecutive terms:

Arithmetic Mean

Geometric Mean

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2. Quadratic Equations

General form/ Standard form of Quadratic equation Methods of Solving Quadratic equations

1. Factorization method:

2.Formula method:

3. Completing Square Method:First, write the equation in the form of If the coefficient of if not 1, make it 1 by dividing both sides of the equationby the coefficient of. Then find the third termThird term * +

coefficient of

Nature of roots of a Quadratic equation

Discriminant

1. If , then quadratic equation has real and unequal roots2. If , the quadratic equation has real and equal roots3. If , the quadratic equation has not real roots.

The relation between Roots and Coefficients of quadratic equation

If and are the roots of the quadratic equation then,

To form a quadratic equation, from roots of the equation

If and are the given roots, then the quadratic equation can be formed byusing by formula,

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Equation which can be converted/reducible to quadratic equations

Some identities used in this chapter are,

OR

OR

OR

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3. Equations in two variables

Methods of solving simultaneous equations

1. Graphical method2. Determinant method (Cramers Rule)

Determinant method (Cramers Rule): if and are

the two equations, then and where,

| | | |

If Conditions of consistency of simultaneous equations

If and are the two simultaneous equations, then1. If

then equations have unique solution

2. If

then equations have no solutions

3. If

then equations have infinitely many solutions

Tips to form the simultaneous equations in the word problems

If the given words come in the word problems, use proper signs for given words to

prepare the equations.

is

less / smaller / younger than

greater than / exceeds

multiple of / times

Two consecutive natural numbers: and

Two even/odd consecutive numbers: and

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4. Probability

If one coin is tossed, { } If two coins are tossed simultaneously, {} If three coins are tossed simultaneously, or one coin is tossed three times,

{} If a Die is thrown, {} If a coin is tossed and die is thrown simultaneously,

{ }

If two dies are thrown,

{

}

Playing cards

Total cards = 52

26 Red cards 26 Black cards

13 heart 13 diamond 13 spade 13 club 1 king 1 king 1 king 1 king

1 queen 1 queen 1 queen 1 queen

1 jack 1 jack 1 jack 1 jack

1-10 1-10 1-10 1-10

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Probability of an event

Property of probability ()

Addition theorem of Probability

Tips:

a) If color is asked: 26 cards of each color

b) If sign is asked: 13 cards of each sign

c) If number is asked: 4 cards of each number

d) If king, queen or jack is asked: 4 card of king, 4 cards of queen and 4 cards of jack

e) Perfect square: 1, 4, 9, 16, 25, 36, and so on.

f) Primary numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, and so on.

g) At least 4 means: 4 and more than 4 are allowed

h) At most 4 means: 4 and less than 4 and zero(none) also allowed

i) Greater/more than 8 means: 9, 10, 11, 12, and so on. (but not 8)

j) Less than 8 means: 7, 6, 5, 4, and so on zero also. (but not 8)

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5. Statistics - I

Measures of Central Tendency

1. Mean for grouped frequency distribution

a)Direct method

4 columns

Class Intervals Class Mark frequency

Total

frequency Class mark; total frequency

b) Assumed mean method/shift of origin method5 columns

Class

Intervals

Class mark

Frequency

total

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3. Mode for grouped frequency distribution

If class intervals are of inclusive type, make 3 columns and if class intervals are of

exclusive type, 2 columns

2 or 3 columns

Class intervals Class boundaries Frequency

Mode * +

4. Inter-relation between measures of central tendencyMean Mode (Mean Median)

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5. Statistics - II

1. Pie Diagram

3 columns

Component No. of Component Measure of Central angle

Total

No. of component

Graphical Representation of Statistical Data

1. Histogram1. Make class intervals continuous if they are not continuous(extended

class intervals).

2. Take extended class intervals on X-axis and Frequencies on Y-axis

2. Frequency polygon and frequency curve1. Make two additional Classes of zero frequency.

2. Make three columns viz. Class intervals, Class marks, and Frequency.

3. Take Class marks on X-axis and frequency on Y-axis.

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3. Ogive curves or Cumulative frequency curves1. Less than cumulative frequency curve: 4 columns

Take class intervals on X-axis and frequency on Y-axis

ExtendedClasses

Frequencies

Upper Class

boundaries Less than cumulative

frequencies

Addition: from up to

down

Total

2. More than Cumulative frequency curvesTake class intervals on X-axis and frequency on Y-axis

Extended

Classes

Frequencies

Lower Class

boundaries More than cumulative

frequencies

Addition: from down

to up

Total