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A quadrilateral has 10 parts – 4 sides, 4 angles and 2 diagonals. Five measurements can determine a quadrilateral uniquely. (Scroll down to continue …)

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Curve: Curve is a figure formed on a plane surface by joining a number of non linear points without lifting a pencil.

Open Curve: An open curve is a curve which does not end at the same starting point or which does not intersect itself.

Closed Curve: Closed curve is a curve which intersects itself or which starts and ends at the same point.

Simple Closed Curve: A simple closed curve is a closed curve that does not intersect itself.

Polygon: Polygon is a closed figure bounded by three or more line segments such that each line segment intersects the other two line segements at exactly two other points (vertices) as shown in the following figures.

Polygons are classified into two types on the basis of interior angles: as (i) Convex polygon (ii) Concave polygon

(a) Convex Polygon: In this case, each angle is either acute or obtuse (angle < 180 ^o) as shown in the following figures.

Concave Polygon: In this case, any one angle is reflex (angle > 180°) and one diagonal is outside the polygon as shown in the following figures.

On the basis of sides, there are two types of polygons :

(a) Regular Polygon: A convex polygon is called a regular polygon, if all its sides and angles are equal as shown in the following figures.

Each angle of a regular polygon of n-sides =

Important results on polygon :

For a regular polygon of n sides (n > 2).

(b) Irregular Polygon: A polygon in which all the sides are unequal as shown in the following figures,

Triangle :

A simple closed figure bounded by three line segments is called a triangle, it has three sides as AB, BC and AC; three vertices as A, B and C and three interior angles A, ZB and ZCand the sum of all angles is 180°.

i.e., <A + <B + <C = 180°

Quadrilateral:

A simple closed figure bounded by four line segments is called a quadrilateral, it has four sides i.e.,

AB, BC, CD and AD and four vertices as A, B, Can d D and the sum of all angles of a quadrilateral is 360.

Practical Geometry

Five measurements can determine a quadrilateral uniquely.

A quadrilateral can be constructed uniquely if the lengths of its four sides and a diagonal is given.

A quadrilateral can be constructed uniquely if its two diagonals and three sides are

known.

A quadrilateral can be constructed uniquely if its two adjacent sides and three angles

are known.

A quadrilateral can be constructed uniquely if its three sides and two included angles

are given.

Quadrilateral

Quadrilateral is a closed figure with four sides.

Characteristics of a quadrilateral

Angle Sum Property of a Quadrilateral:

Qudrilateral is a four sided closed figure.

Sum of all angles of a quadrilateral is 360°.

Types Of Quadrilaterals

Quadrilaterals are broadly classified into three categories as:

(i) Kite

(ii) Trapezium

(ii) Parallelogram

Kite:

Kite

(i) Kite has no parallel sides

(ii) Kite has a pair of equal adjacent sides.

(ii) It is not a parallelogram

Characteristics Of Kite:

Perimeter Of Square

Area Of Kite

Trapezium:

Trapezium is a quadrilateral with the following characteristics:

(i) One pair of opposite sides is parallel to each other.

(ii) The other pair of opposite sides may not be parallel to each other.

Characteristics Of Trapezium

(i) Sum of all angles of a quadrilateral is 360°.

(ii) One pair of opposite sides is parallel to each other.

(iii) The other pair of opposite sides need not be parallel to each other.

Types Of Trapezium:

Quadrilaterals are broadly classified into two categories as:

(i) Isosceles Trapezium.

(ii) Scalene Trapezium.

(i) Right Trapezium.

Isosceles Trapezium:

Isosceles Trapezium is a quadrilateral with the following characteristics:

(i) One pair of opposite sides is parallel to each other.

(ii) The other pair of opposite sides are equal.

(iii) The other pair of opposite sides need not be parallel to each other.

Isosceles Trapezium is a trapezium with the following characteristics:

(i) One pair of opposite sides is parallel to each other.

(ii) The other pair of opposite sides are equal.

(iii) The other pair of opposite sides need not be parallel to each other.

Characteristics Of Isosceles Trapezium

(i) Sum of all angles of a quadrilateral is 360°.

(ii) One pair of opposite sides is parallel to each other.

(iii) The other pair of opposite sides are equal.

(iv) The other pair of opposite sides need not be parallel to each other.

Scalene Trapezium:

Scalene trapezium: Classified by the length of the legs or the measurement of their angles.

Characteristics Of Scalene Trapezium

Right Trapezium:

Right trapezium: Has one pair of parallel sides and one pair of right angles.

Characteristics Of Right Trapezium

Perimeter Of Trapezium

Area Of Trapezium

Parallelogram:

Parallelogram is a quadrilateral with the following characteristics:

(i) Two pairs of opposite sides are parallel to each other.

(ii) Two pairs of opposite sides are equal in length.

Characteristics of a parallelogram

(i) Sum of all angles of a Parallelogram is 360°.

(ii) Two pairs of opposite sides are parallel to each other.

(ii) Two pairs of opposite sides are equal in length.

(ii) Two pairs of opposite angles are equal.

(iii) Diagonals bisect each other.

(iv) Diagonals need not be equal to each other.

(v) Diagonals divide it into two congruent triangles.

Types Of Parallelogram

Parallelograms are broadly classified into three categories as:

(i) Rectangle

(ii) Rhombus

(iii) Square

Perimeter Of Parallelogram

Area Of Parallelogram

Rectangle:

Rectangle is a quadrilateral with the following characteristics:

(i) Two pairs of opposite sides are parallel to each other.

(ii) Two pairs of opposite sides are equal in length.

(iii) All four angles are right angles. (each angle is 90 ^o).

Characteristics Of Rectangle

(i) Sum of all angles of a quadrilateral is 360°.

(ii) Two pairs of opposite sides are parallel to each other.

(ii) Two pairs of opposite sides are equal in length.

(iii) All four angles are right angles. (each angle is 90 ^o).

(iii) Diagonals bisect each other.

(iv) Diagonals are equal to each other.

(v) Diagonals of a rectangle divide it into two congruent triangles.

Conclusions:

Every Rectangle is a Parallelogram. But Every Parallelogram need not to be a Rectangle.

Condition for a rhombus to be a square:

If all four angles of a parallelogram are right angles. (each angle is 90 ^o), the parallelogram becomes a Rectangle.

Perimeter Of Rectangle

Area Of Recatangle

Rhombus:

Rhombus is a quadrilateral with the following characteristics:

(i) Two pairs of opposite sides are parallel to each other.

(ii) All four sides are equal in length.

Characteristics Of Rhombus

(i) Sum of all angles of a quadrilateral is 360°.

(ii) Two pairs of opposite sides are parallel to each other.

(ii) All four sides are equal in length.

(ii) Two pairs of opposite angles are equal.

(iii) Diagonals bisect each other.

(iv) Diagonals need not be equal to each other.

(v) Diagonals divide a Rhombus into two congruent triangles.

Conclusions:

Every Rhombus is a Parallelogram. But Every Parallelogram need not to be a Rhombus.

Condition for a rhombus to be a square:

If all the sides of a parallelogram are equal, the parallelogram becomes a Rhombus.

Perimeter Of Rhombus

Area Of Rhombus

Square:

Square is a quadrilateral with the following characteristics:

(i) Two pairs of opposite sides are parallel to each other.

(ii) All four sides are equal in length.

(iii) All four angles are right angles. (each angle is 90 ^o).

Characteristics Of Square

(i) Sum of all angles of a quadrilateral is 360°.

(ii) Two pairs of opposite sides are parallel to each other.

(iii) All four sides are equal in length.

(iv) All four angles are right angles. (each angle is 90 ^o).

(v) Diagonals bisect each other.

(vi) Diagonals need not be equal to each other.

(vii) Diagonals divide a Rhombus into two congruent triangles.

Conclusions:

Every square is a Rhombus. But Every Rhombus need not to be a square.

Condition for a rhombus to be a square:

If all the angles of a rhombus are right angles (euqal to 90^o), the rhombus becomes a square.

2. Every Square is a prallelogram. But Every prallelogram need not to be a square.

Condition for a prallelogram to be a square:

(i) If all the angles of a parallelogram are right angles (euqal to 90^o), and all the sides of a parallelogram are equal in length, the parallelogram becomes a square.

3. Every Square is a rectangle. But Every Rectangle need not to be a square.

Condition for a Rectangle to be a square:

If all the sides of a Rectangle are equal in length, the Rectangle becomes a square.

If all the sides of a parallelogram are equal, the parallelogram becomes a Rhombus.

Perimeter Of Square

Area Of Square

Important Points To Remember

The diagonals of a parallelogram are equal if and only if it is a rectangle.
If a diagonal of a parallelogram bisects one of the angles of the parallelogram then it also bisects the opposite angle.
In a parallelogram, the bisectors of any two consecutive angles intersect at a right angle.
The angle bisectors of a parallelogram form a rectangle.

Mid Point Theorem

A line segment joining the mid points of any two sides of a triangle is parallel to the third side and length of the line segment is half of the parallel side.

Converse Of Mid Point Theorem

A line through the midpoint of a side of a triangle parallel to another side bisects the third side.

Intercept Theorem

If there are three parallel lines and the intercepts made by them on one transversal are equal then the intercepts on any other transversal are also equal.

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Angle Sum Property of a Quadrilateral

The sum of the four angles of a quadrilateral is 360°

If we draw a diagonal in the quadrilateral, it divides it into two triangles.

And we know the angle sum property of a triangle i.e. the sum of all the three angles of a triangle is 180°.

The sum of angles of ∆ADC = 180°.

The sum of angles of ∆ABC = 180°.

By adding both we get ∠A + ∠B + ∠C + ∠D = 360°

Hence, the sum of the four angles of a quadrilateral is 360°.

Example

Find ∠A and ∠D, if BC∥ AD and ∠B = 52° and ∠C = 60° in the quadrilateral ABCD.

Solution:

Given BC ∥ AD, so ∠A and ∠B are consecutive interior angles.

So ∠A + ∠B = 180° (Sum of consecutive interior angles is 180°).

∠B = 52°

∠A = 180°- 52° = 128°

∠A + ∠B + ∠C + ∠D = 360° (Sum of the four angles of a quadrilateral is 360°).

∠C = 60°

128° + 52° + 60° + ∠D = 360°

∠D = 120°

∴ ∠A = 128° and ∠D = 120 °.

Types of Quadrilaterals

Remark: A square, Rectangle and Rhombus are also a parallelogram.

Properties of a Parallelogram

Theorem 1: When we divide a parallelogram into two parts diagonally then it divides it into two congruent triangles.

∆ABD ≅ ∆CDB

Theorem 2: In a parallelogram, opposite sides will always be equal.

Theorem 3: A quadrilateral will be a parallelogram if each pair of its opposite sides will be equal.

Here, AD = BC and AB = DC

Then ABCD is a parallelogram.

Theorem 4: In a parallelogram, opposite angles are equal.

In ABCD, ∠A = ∠C and ∠B = ∠D

Theorem 5: In a quadrilateral, if each pair of opposite angles is equal, then it is said to be a parallelogram. This is the reverse of Theorem 4.

Theorem 6: The diagonals of a parallelogram bisect each other.

Here, AC and BD are the diagonals of the parallelogram ABCD.

So the bisect each other at the centre.

DE = EB and AE = EC

Theorem 7: When the diagonals of the given quadrilateral bisect each other, then it is a parallelogram.

This is the reverse of the theorem 6.

The Mid-point Theorem

1. If a line segment joins the midpoints of the two sides of the triangle then it will be parallel to the third side of the triangle.

If AB = BC and CD = DE then BD ∥ AE.

2. If a line starts from the midpoint of one line and that line is parallel to the third line then it will intersect the midpoint of the third line.

If D is the midpoint of AB and DE∥ BC then E is the midpoint of AC.

Example

Prove that C is the midpoint of BF if ABFE is a trapezium and AB ∥ EF.D is the midpoint of AE and EF∥ DC.

Solution:

Let BE cut DC at a point G.

Now in ∆AEB, D is the midpoint of AE and DG ∥ AB.

By midpoint theorem, G is the midpoint of EB.

Again in ∆BEF, G is the midpoint of BE and GC∥ EF.

So, by midpoint theorem C is the midpoint of BF.

Hence proved.

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November 5, 2023
Direct And Inverse Proportions | Study
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Variations: If the values of two quantities depend on each other in such a way that a change in one causes corresponding change in the other, then the two quantities are said to be in variation. (Scroll down to continue …)
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Direct Variation or Direct Proportion:
Extra:
Two quantities x and y are said to be in direct proportion if they increase (decrease) together in such a manner that the ratio of their corresponding values remains
constant. That is if
=k [k is a positive number, then x and y are said to vary directly.
In such a case if y₁, y₂are the values of y corresponding to the values x₁, x of x
respectively then = .
If the number of articles purchased increases, the total cost also increases. More than money deposited in a bank, more is the interest earned.
Quantities increasing or decreasing together need not always be in direct proportion, same in the case of inverse proportion.
When two quantities x and y are in direct proportion (or vary directly), they are
written as
. Symbol
stands for ‘is proportion to’.
Inverse Proportion: Two quantities x and y are said to be in inverse proportion if an increase in x causes a proportional decrease in y (and vice-versa) in such a manner that the product of their corresponding values remains constant. That is, if xy
= k, then x and y are said to vary inversely. In this case if y₁, y₂are the values of y
corresponding to the values x₁, x₂of x respectively then
^x1^{, Y}1 = ^x2^{, y}2 or
=
When two quantities x and y are in inverse proportion (or vary inversely), they are
written as x
. Example: If the number of workers increases, time taken to finish
the job decreases. Or If the speed will increase the time required to cover a given distance decreases.

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Factorisation: Representation of an algebraic expression as the product of two or more expressions is called factorization. Each such expression is called a factor of the given algebraic expression. (Scroll down to continue …)
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When we factorise an expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions.
An irreducible factor is a factor which cannot be expressed further as a product of factors.
A systematic way of factorising an expression is the common factor method. It consists of three steps:
Write each term of the expression as a product of irreducible factors
Look for and separate the common factors and
Combine the remaining factors in each term in accordance with the distributive law.
Sometimes, all the terms in a given expression do not have a common factor; but the terms can be grouped in such a way that all the terms in each group have a common factor. When we do this, there emerges a common factor across all the groups leading to the required factorisation of the expression. This is the method of regrouping.
In factorisation by regrouping, we should remember that any regrouping (i.e., rearrangement) of the terms in the given expression may not lead to factorisation. We must observe the expression and come out with the desired regrouping by trial and error.
A number of expressions to be factorised are of the form or can be put into the form: a²+ 2ab + b², a²– 2ab + b², a²– b²and x² + (a + b)x + ab. These expressions can be easily factorised using Identities I, II, III and IV
a²+ 2ab + b²= (a + b)²
a²– 2ab + b²= (a – b)²
a²– b²= (a + b) (a – b)
Factorisation
x² + (a + b)x + ab = (x + a)(x + b)
In expressions which have factors of the type (x + a) (x + b), remember the numerical term gives ab.
Its factors, a and b, should be so chosen that their sum, with signs taken care of, is the coefficient of x.
We know that in the case of numbers, division is the inverse of multiplication. This idea is applicable also to the division of algebraic expressions.
In the case of division of a polynomial by a monomial, we may carry out the division either by dividing each term of the polynomial by the monomial or by the common factor method.
In the case of division of a polynomial by a polynomial, we cannot proceed by dividing each term in the dividend polynomial by the divisor polynomial. Instead, we factorise both the polynomials and cancel their common factors.
In the case of divisions of algebraic expressions that we studied in this chapter, we have Dividend = Divisor × Quotient.
In general, however, the relation is Dividend = Divisor × Quotient + Remainder
Thus, we have considered in the present chapter only those divisions in which the remainder is zero.
There are many errors students commonly make when solving algebra exercises.
You should avoid making such errors.
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Graphical presentation of data is easier to understand.
A bar graph is used to show comparison among categories.
A pie graph is used to compare parts of a whole.
A Histogram is a bar graph that shows data in intervals. (Scrol down to continue …)
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Introduction to Graphs
A line graph displays data that changes continuously over periods of time. A line graph which is a whole unbroken line is called a linear graph.
For fixing a point on the graph sheet we need, x-coordinate and y-coordinate.
The relation between dependent variable and
through a graph.
independent variable is shown
A Bar Graph: A pictorial representation of numerical data in the form of bars (rectangles) of uniform width with equal spacing. The length (or height) of each bar
represents the given number.
A Pie Graph: A pie graph is used to compare parts of a whole. The various
observations or components are represented by the sectors of the circle.
A Histogram: Histogram is a type of bar diagram, where the class intervals are shown on the horizontal axis and the heights of the bars (rectangles) show the frequency of the class interval, but there is no gap between the bars as there is no gap between the
class intervals.
Linear Graph: A line graph in which all the line segments form a part of a single line. Coordinates: A point in Cartesian plane is represented by an ordered pair of numbers.
Ordered Pair: A pair of numbers written in specified order.
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Exponents And Powers | Study
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Numbers with exponents obey the following laws of exponents.
Very small numbers can be expressed in standard form using negative exponents. (Scroll down to continue …)
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Use of Exponents to Express Small Number in Standard form:
Very large and very small numbers can be expressed in standard form.
Standard form is also called scientific notation form.
(iii) A number written as number such that is said to be in standard form if m is a decimal and n is either a positive or a negative integer.
Examples: 150,000,000,000 = 1.5 x 10¹¹.
Exponential notation is a powerful way to express repeated multiplication of the same number.
For any non-zero rational number ‘a’ and a natural number n, the product a x a x a x x a(n times) = aⁿ.
It is known as the nth power of ‘a’ and is read as ‘a’ raised to the power n’.
The rational number a is called the base and n is called exponent.
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Comparing Quantities | Study
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Ratio: Comparing by division is called ratio. Quantities written in ratio have the sameunit. Ratio has no unit. (Scroll down to continue …)
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Equality of two ratios is called proportion. Product of extremes = Product of means
Percentage: Percentage means for every hundred. The result of any division in
whichthe divisor is 100 is a percentage. The divisor is denoted by a special
symbol %, read as percent. Profit and Loss:
(i) Cost Price (CP): The amount for which an article is bought. (ii) Selling Price (SP): The amount for which an article is sold. Additional expenses made after buying an article are included in the cost price
and are known as overhead expenses. These may include expenses like amount
spent onrepairs, labour charges, transportation, etc. Discount is a reduction given on marked price. Discount = Marked Price – Sale
Price. Discount can be calculated when discount percentage is given. Discount
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Algebraic Expressions And Identities | Study
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Expressions are formed from variables and constants.
Constant: A symbol having a fixed numerical value.
Example: 2,, 2.1, etc. (Scroll down to continue …)
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Variable: A symbol which takes various numerical values. Example: x, y, z, etc.
Algebric Expression: A combination of constants and variables connected by the sign
+, -, and is called algebraic expression.
Terms are added to form expressions.
Terms themselves are formed as product of factors.
Expressions that contain exactly one, two and three terms are called monomials, binomials and trinomials respectively.
In general, any expression containing one or more terms with non-zero coefficients (and with variables having non- negative exponents) is called a polynomial.
Like terms are formed from the same variables and the powers of these variables are the same, too.
Coefficients of like terms need not be the same.
While adding (or subtracting) polynomials, first look for like terms and add (or subtract) them; then handle the unlike terms.
There are number of situations in which we need to multiply algebraic expressions: for example, in finding area of a rectangle, the sides of which are given as expressions.
Monomial: An expression containing only one term. Example: -3, 4x, 3xy, etc.
Binomial: An expression containing two terms. Example: 2x-3, 4x+3y, xy-4, etc.,
Polynomial: In general, any expression containing one or more terms with non-zero coefficients (and with variables having non-negative exponents).
A polynomial may contain any number of terms, one or more than one.
A monomial multiplied by a monomial always gives a monomial.
Multiplication of a Polynomial and a monomial:
While multiplying a polynomial by a monomial, we multiply every term in the polynomial by the mononomial.
Trinomial: An expression containing three terms.
Example:
3x+2y+5z, etc.
In carrying out the multiplication of a polynomial by a binomial (or trinomial), we multiply term by term, i.e., every term of the polynomial is multiplied by every term in the binomial (or trinomial).
Note that in such multiplication, we may get terms in the product which are like and have to be combined.
An identity is an equality, which is true for all values of the variables in the equality.
On the other hand, an equation is true only for certain values of its variables.
An equation is not an identity.
The following are the standard identities:
(a + b)²= a²+ 2ab + b²
(a – b)²= a²– 2ab +b²
(a + b)(a – b) = a²– b²
(x + a) (x + b) = x² + (a + b) x + ab
The above four identities are useful in carrying out squares and products of algebraic expressions.
They also allow easy alternative methods to calculate products of numbers and so on.
Coefficients: In the term of an expression any of the factors with the sign of the term is called the coefficient of the product of the other factors.
Terms: Various parts of an algebraic expression which are separated by + and – signs. Example: The expression 4x + 5 has two terms 4x and 5.
Constant Term: A term of expression having no lateral factor.
Like term: The term having the same literal factors. Example 2xy and -4xy are like terms.
(iii) Unlike term: The terms having different literal factors.
Example:
are unlike terms.
and 3xy
Factors: Each term in an algebraic expression is a product of one or more number (s) and/or literals. These number (s) and/or literal (s) are known as the factor of that term. A constant factor is called numerical factor, while a variable factor is known as
a literal factor. The term 4x is the product of its factors 4 and x.

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Data Handling | Study
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Data Handling: Deals with the process of collecting data, presenting it and getting result.
Data mostly available to us in an unorganised form is called raw data. (Scroll down to continue …)
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Grouped data can be presented using histogram. Histogram is a type of bar diagram, where the class intervals are shown on the horizontal axis and the heights of the bars show the frequency of the class interval. Also, there is no gap between the bars as there is no gap between the class intervals.
In order to draw meaningful inferences from any data, we need to organise the data systematically.
Frequency gives the number of times that a particular entry occurs.
Raw data can be ‘grouped’ and presented systematically through ‘grouped frequency distribution’.
Statistics: The science which deals with the collection, presentation, analysis and interpretation of numerical data.
Observation: Each entry (number) in raw data.
Range: The difference between the lowest and the highest observation in a given data.
Array: Arranging raw data in ascending or descending order of magnitude. Data can also presented using circle graph or pie chart. A circle graph shows the relationship between a whole and its part.
There are certain experiments whose outcomes have an equal chance of occurring. A random experiment is one whose outcome cannot be predicted exactly in advance. Outcomes of an experiment are equally likely if each has the same chance of occurring.
Frequency: The number of times a particular observation occurs in the given data.
Class Interval: A group in which the raw data is condensed.
(i) Continuous: The upper limit of a class interval coincides with the lower limit of the next class.

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Squares and Square Roots | Speed Notes
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Square: Number obtained when a number is multiplied by itself. It is the number raised to the power 2. 2²= 2 x 2=4(square of 2 is 4).
If a natural number m can be expressed as n², where n is also a natural number, then m is a square number. (Scroll down to continue …)
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All square numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place. Square numbers can only have even number of zeros at the end. Square root is the inverse operation of square.
There are two integral square roots of a perfect square number.
Positive square root of a number is denoted by the symbol For example, 3²=9 gives
Perfect Square or Square number: It is the square of some natural number. If m=n², then m is a perfect square number where m and n are natural numbers. Example: 1=1 x 1=12, 4=2 x 2=2².
Properties of Square number:
A number ending in 2, 3, 7 or 8 is never a perfect square. Example: 152, 1028, 6593 etc.
A number ending in 0, 1, 4, 5, 6 or 9 may not necessarily be a square number. Example: 20, 31, 24, etc.
Square of even numbers are even. Example: 2²= 4, 4²=16 etc.
Square of odd numbers are odd. Example: 5²= 25, 9²= 81, etc.
A number ending in an odd number of zeroes cannot be a perferct square. Example: 10, 1000, 900000, etc.
The difference of squares of two consecutive natural number is equal to their sum. (n + 1)²– n²= n+1+n. Example: 4²– 3²=4 + 3=7. 12²– 11²=12+11 =23, etc.
A triplet (m, n, p) of three natural numbers m, n and p is called Pythagorean
triplet, if m²+ n²= p²: 3²+ 4²= 25 = 5²
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November 5, 2023
Cubes And Cube Roots | Study
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Cube number: Number obtained when a number is multiplied by itself three times. 2³= 2 x 2 x 2 = 8, 3³= 3 x 3 x 3=27, etc.
Numbers like 1729, 4104, 13832, are known as Hardy – Ramanujan Numbers. They
can be expressed as sum of two cubes in two different ways.
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Numbers obtained when a number is multiplied by itself three times are known as cube numbers. For example 1, 8, 27, … etc.
If in the prime factorisation of any number each factor appears three times, then the
number is a perfect cube.
The symbol
denotes cube root. For example
Perfect Cube: A natural number is said to be a perfect cube if it is the cube of some natural number. Example: 8 is perfect cube, because there is a natural number 2 such that 8 = 2³, but 18 is not a perfect cube, because there is no natural number whose cube is 18.
The cube of a negative number is always negative.
Properties of Cube of Number:
Cubes of even number are even.
Cubes of odd numbers are odd.
The sum of the cubes of first n natural numbers is equal to the square of their sum.
Cubes of the numbers ending with the digits 0, 1, 4, 5, 6 and 9 end with digits 0, 1, 4, 5, 6 and 9 respectively.
Cube of the number ending in 2 ends in 8 and cube of the number ending in 8 ends in 2.
Cube of the number ending in 3 ends in 7 and cube of the number ending in 7
ends in 3.
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Rational Numbers | Study
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Rational numbers are closed under the operations of addition, subtraction and multiplication, But not in division. (Scroll down to continue …)
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The operations addition and multiplication are
(i) commutative for rational numbers.
(ii) associative for rational numbers.
The rational number 0 is the additive identity for rational numbers.
The additive inverse of the rational number a/b is -a/b and vice- versa.
The reciprocal or multiplicative inverse of the rational number
is if a/b is c/d if (a/b)(c/d) =1
Distributive property of rational numbers:
For all rational numbers a, b and c, a(b + c) = ab + ac
and a(b – c) = ab – ac.
Rational numbers can be represented on a number line.
Between any two given rational numbers there are countless rational numbers.
The idea of mean helps us to find rational numbers between two rational numbers.
Positive Rationals: Numerator and Denominator both are either positive or negative.
Example: 2/3, -4/-5
Positive Rationals: Numerator and Denominator both are either positive or negative.
Example: -2/3, 4/-5
.

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Linear Equations In One Variable | Study
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A statement of equality of two algebraic expressions involving one or more variables. Example: x + 2 = 3
Linear Equation in One variable: The expressions which form the equation that contain single variable and the highest power of the variable in the equation is one. (Scroll down to continue …)
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Linear Equations in One Variable
An algebraic equation is an equality involving variables. It says that the value of the expression on one side of the equality sign is equal to the value of the expression on the other side.
The equations we study in Classes VI, VII and VIII are linear equations in one variable. In such equations, the expressions which form the equation contain only one variable. Further, the equations are linear, i.e., the highest power of the variable appearing in the equation is 1.
A linear equation may have for its solution any rational number.
An equation may have linear expressions on both sides. Equations that we studied in Classes VI and VII had just a number on one side of the equation.
Just as numbers, variables can, also, be transposed from one side of the equation to the other.
Occasionally, the expressions forming equations have to be simplified before we can solve them by usual methods. Some equations may not even be linear to begin with, but they can be brought to a linear form by multiplying both sides of the equation by a suitable expression.
The utility of linear equations is in their diverse applications; different problems on numbers, ages, perimeters, combination of currency notes, and so on can be solved
using linear equations.

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CBSE 9 | Mathematics – Study – Premium
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November 5, 2023

LINES AND ANGLES | Study

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Basic Terms And Definitions

1. Point – A Point is that which has no component. It is represented by a dot.

A point is shown with a capital letter.

Examples: A, B, C …..

Colinear And Non-colinear Points

Collinear and Non-collinear points

5. Collinear points: Points lying on the same line are called Collinear Points.

Non-collinear points: Points which do not lie on the same line are called Non-Collinear Points.

Line – When we join two distinct points then we get a line. A line has no endpoints; it can be extended on both sides infinitely.

Line Segment Line – Segment is the part of the line which has two endpoints.

Ray – Ray is also a part of the line that has only one endpoint and has no end on the other side. (Scroll to continue …)

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Lines And Their Types

Ray

A Ray is a straight path that stars at a point and extends infinitely in one direction.

Note: A ray is a portion of line starting at a point and extends in one direction endlessly. A ray has only one endpoint (Initial point).

Line or Straight Line

A line is a straight path that extends infinitely in two opposite directions. It can be treated as a combination of two rays starting from the same point but extending in the opposite directions.

Note: A line has no end points.

Line Segment:

A line segment is the part of a line between two points. (Segment means part).

The length of a line segment is the shortest length between two endpoints.

The line segment has two endpoints. Note: A line Segment has two endpoints. (both Initial and end points).

Intersecting Lines and Non-intersecting Lines

Intersecting Lines

Lines that meet or cross at a point with each other are called intersecting lines Or Non-parallel lines.

Intersecting lines meet at a point.

Parallel Lines

Lines that are always the same distance apart from each other and that never meet are called Parallel lines or Non-intersecting lines.

Note: Parallel lines do not have any common point.

Angles

When two rays begin from the same endpoint then they form an Angle. The two rays are the arms of the angle and the endpoint is the vertex of the angle.

Types of Angles By Measure

Complementary and Supplementary Angles

Complementary Angles are the different angles whose sum is 90°.

Suplementary Angles are the different angles whose sum is 180°

The sum of two adjacent angles formed by that ray is 180°

Angles Based On Position

Adjacent Angles: Two angles that share a common sideare called Adjacent Angles.

Linear Pair: A pair of adjacent angles whose non-common sides form a straight line (i.e., they are supplementary and add up to 180°).

Vertical (Opposite) Angles:

Angles opposite each other when two lines intersect. They are always equal.

Angles:

An angle is formed by two rays(called the sides or arms of the angle) with a common endpoint called the vertex.

Angles are measured in degrees(°) or radians, with a full rotation being 360°.

Special Angle Pairs:

Complementary Angles:

Two angles whose measures add up to 90°.

Supplementary Angles: Two angles whose measures add up to 180°

Angles in Polygons:

Interior Angles: Angles on the inside of a polygon. The sum of the interior angles of an n-sided polygon is \((n-2) \times 180°\).

Exterior Angles: Angles on the outside of a polygon. The sum of the exterior angles of any polygon is always 360°.

Angles Formed by Parallel Lines and Transversals:

When a transversal intersects two parallel lines, several angle pairs are formed:

Corresponding Angles: Angles in the same position relative to the two lines and the transversal. They are equal.

Alternate Interior Angles: Angles on opposite sides of the transversal and inside the parallel lines. They are equal.

Alternate Exterior Angles: Angles on opposite sides of the transversal and outside the parallel lines. They are equal.

– **Consecutive (Same-Side) Interior Angles:** Angles on the same side of the transversal and inside the parallel lines. They are supplementary.

Measuring Angles:

Protractor: A tool used to measure angles in degrees.

Radians: Another unit of angle measure. One full revolution (360°) is equal to \(2\pi\) radians.

Angle Relationships in Circles:

Central Angle: An angle whose vertex is the centre of the circle. The measure of a central angle is equal to the measure of the arc it intercepts.

Inscribed Angle: An angle whose vertex is on the circle and whose sides contain chords of the circle. The measure of an inscribed angle is half the measure of the intercepted arc.

Angles Formed by Tangents and Chords: The measure of the angle formed by a tangent and a chord through the point of contact is half the measure of the intercepted arc.

Angles Inside the Circle (but not at the centre): The measure of an angle formed by two intersecting chords is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Trigonometric Angles:

Standard Position: An angle with its vertex at the origin and one side on the positive x-axis.

Reference Angle: The acute angle formed by the terminal side of an angle and the x-axis.

Quadrantal Angles: Angles that are multiples of 90° (0°, 90°, 180°, 270°, 360°).

Angle Conversions:

Degrees to Radians: Multiply the number of degrees by \(\frac{\pi}{180}\).

Radians to Degrees: Multiply the number of radians by \(\frac{180}{\pi}\).

Key Properties and Theorems:

Angle Sum Property of a Triangle:

The sum of the interior angles of a triangle is always 180°.

Exterior Angle Theorem:

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

Polygon Interior Angles Theorem:

The sum of the interior angles of an n-sided polygon is \((n-2) \times 180°\).

Understanding these fundamental aspects of angles will enhance your comprehension of geometric principles and their applications.

Pairs of Angles Axioms

1. If a ray stands on a line, then the sum of two adjacent angles formed by that ray is 180°.

This shows that the common arm of the two angles is the ray which is standing on a line and the two adjacent angles are the linear pair of the angles. As the sum of two angles is 180° so these are supplementary angles too.

2. If the sum of two adjacent angles is 180°, then the arms which are not common of the angles form a line.

This is the reverse of the first axiom which says that the opposite is also true.

Vertically opposite Angles Theorem

When two lines intersect each other, then the vertically opposite angles so formed will be equal.

AC and BD are intersecting each other so ∠AOD = ∠BOC and ∠AOB = DOC.

Parallel Lines and a Transversal

If a line passes through two distinct lines and intersects them at distant points then this line is called Transversal Line.

Here line “l” is transversal of line m and n.

Exterior Angles – ∠1, ∠2, ∠7 and ∠8

Interior Angles – ∠3, ∠4, ∠5 and ∠6

Pairs of angles formed when a transversal intersects two lines-

1. Corresponding Angles:

∠ 1 and ∠ 5
∠ 2 and ∠ 6
∠ 4 and ∠ 8
∠ 3 and ∠ 7

2. Alternate Interior Angles:

∠ 4 and ∠ 6
∠ 3 and ∠ 5

3. Alternate Exterior Angles:

∠ 1 and ∠ 7
∠ 2 and ∠ 8

4. Interior Angles on the same side of the transversal:

∠ 4 and ∠ 5
∠ 3 and ∠ 6

Transversal Axioms

1. If a transversal intersects two parallel lines, then

Each pair of corresponding angles will be equal.
Each pair of alternate interior angles will be equal.
Each pair of interior angles on the same side of the transversal will be supplementary.

2. If a transversal intersects two lines in such a way that

Corresponding angles are equal then these two lines will be parallel to each other.
Alternate interior angles are equal then the two lines will be parallel.
Interior angles on the same side of the transversal are supplementary then the two lines will be parallel.

Lines Parallel To The Same Line

If two lines are parallel with a common line then these two lines will also be parallel to each other.

As in the above figure if AB ∥ CD and EF ∥ CD then AB ∥ EF.

Angle Sum Property of a Triangle

1. The sum of the angles of a triangle is 180º.

∠A + ∠B + ∠C = 180°

2. If we produce any side of a triangle, then the exterior angle formed is equal to the sum of the two interior opposite angles.

∠BCD = ∠BAC + ∠ABC.

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QUADRILATERALS | Study

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Quadrilateral

Any closed polygon with four sides, four angles and four vertices are called Quadrilateral. It could be regular or irregular. (Sroll down to continute …)

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Revision Notes – CBSE 09 Math – Quadrilaterals

Angle Sum Property of a Quadrilateral

The sum of the four angles of a quadrilateral is 360°

If we draw a diagonal in the quadrilateral, it divides it into two triangles.

And we know the angle sum property of a triangle i.e. the sum of all the three angles of a triangle is 180°.

The sum of angles of ∆ADC = 180°.

The sum of angles of ∆ABC = 180°.

By adding both we get ∠A + ∠B + ∠C + ∠D = 360°

Hence, the sum of the four angles of a quadrilateral is 360°.

Example

Find ∠A and ∠D, if BC∥ AD and ∠B = 52° and ∠C = 60° in the quadrilateral ABCD.

Solution:

Given BC ∥ AD, so ∠A and ∠B are consecutive interior angles.

So ∠A + ∠B = 180° (Sum of consecutive interior angles is 180°).

∠B = 52°

∠A = 180°- 52° = 128°

∠A + ∠B + ∠C + ∠D = 360° (Sum of the four angles of a quadrilateral is 360°).

∠C = 60°

128° + 52° + 60° + ∠D = 360°

∠D = 120°

∴ ∠A = 128° and ∠D = 120 °.

Types of Quadrilaterals

Remark: A square, Rectangle and Rhombus are also a parallelogram.

Properties of a Parallelogram

Theorem 1: When we divide a parallelogram into two parts diagonally then it divides it into two congruent triangles.

∆ABD ≅ ∆CDB

Theorem 2: In a parallelogram, opposite sides will always be equal.

Theorem 3: A quadrilateral will be a parallelogram if each pair of its opposite sides will be equal.

Here, AD = BC and AB = DC

Then ABCD is a parallelogram.

Theorem 4: In a parallelogram, opposite angles are equal.

In ABCD, ∠A = ∠C and ∠B = ∠D

Theorem 5: In a quadrilateral, if each pair of opposite angles is equal, then it is said to be a parallelogram. This is the reverse of Theorem 4.

Theorem 6: The diagonals of a parallelogram bisect each other.

Here, AC and BD are the diagonals of the parallelogram ABCD.

So the bisect each other at the centre.

DE = EB and AE = EC

Theorem 7: When the diagonals of the given quadrilateral bisect each other, then it is a parallelogram.

This is the reverse of the theorem 6.

The Mid-point Theorem

1. If a line segment joins the midpoints of the two sides of the triangle then it will be parallel to the third side of the triangle.

If AB = BC and CD = DE then BD ∥ AE.

2. If a line starts from the midpoint of one line and that line is parallel to the third line then it will intersect the midpoint of the third line.

If D is the midpoint of AB and DE∥ BC then E is the midpoint of AC.

Example

Prove that C is the midpoint of BF if ABFE is a trapezium and AB ∥ EF.D is the midpoint of AE and EF∥ DC.

Solution:

Let BE cut DC at a point G.

Now in ∆AEB, D is the midpoint of AE and DG ∥ AB.

By midpoint theorem, G is the midpoint of EB.

Again in ∆BEF, G is the midpoint of BE and GC∥ EF.

So, by midpoint theorem C is the midpoint of BF.

Hence proved.

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HERON’S FORMULA | Study
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Perimeter
Perimeter is defined as the outside boundary of any closed shape.
To calculate the perimeter of a given shape we need to add all the sides of the shape.
Example: The perimeter of a rectangle is the sum of its all four sides. The unit of the perimeter is the same as its length.
Perimeter of the Given rectangle = 3 + 7 + 3 + 7 cm
Perimeter of rectangle = 20 cm. (Scroll down to continue …)
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Area
Area of any closed figure is the surface enclosed by the perimeter. Unit of Area is the square of the unit of length.
Area of a triangle
The general formula to find the area of a triangle, if the height is given, is
Area of a Right Angled Triangle
To find the area of a right-angled triangle, we use the formula:
right-angled triangle, we take the two sides having the right angle, one as the base and one as height.
Example: Calculate area of a triangle of the Figure.
Data: base = 3 cm and height = 4 cm
Formula: Area of triangle = 1/2 × 3 × 4= 6 cm²
Result: Area of a triangle of the Figure is 6 cm² .
Remark: If you take base as 4 cm and height as 3 cm then also the area of the triangle will remain the same.
Area of Equilateral Triangle
Equilateral Triangle: Equilateral Triangle is defined as a triangle having three equal sides.
To calculate the area of the Equilateral Triangle ABC,
We calculate the height (altitude), AD by making the median of the triangle.
In the given example, the Height (altitude), AD touches Base of the equilateral triangle at the midpoint of BC, Say point, D.
Here the equilateral triangle ABC has three equal sides, such as:
AB = BC = AC = 10 cm.
Since, midpoint of BC divides the triangle into two right angle triangles.
The height, AD, is calculated using Pythagoras theorem.
According to Pythagoras theorem, AB² = AD² + BD²
On substituting the values we get,
(10)² = AD² + (5)²
AD² = (10)² – (5)²
AD² = 100 – 25 = 75
AD = 5√3
Now we can find the area of the triangle using the formula:
Area of triangle = 1/2 × base × height
On substituting the values we get,
Area of triangle = 1/2 × 10 × 5√3
25√3 cm²
Area of Isosceles Triangle
In the isosceles triangle also we need to find the height of the triangle then calculate the area of the triangle.
Here,
Area of a Triangle — by Heron’s Formula
The formula of the area of a triangle given by herons is called Heron’s Formula.
Heron’s Formula:
where a, b and c are the sides of the triangle and s is the semiperimeter
Generally, this formula is used when the height of the triangle is not possible to find or you can say if the triangle is a scalene triangle.
Here the sides of triangle are
AB = 12 cm
BC = 14 cm
AC = 6 cm
Application of Heron’s Formula in Finding Areas of Quadrilaterals
If we know the sides and one diagonal of the quadrilateral then we can find its area by using the Heron’s formula.
Find the area of the quadrilateral if its sides and the diagonal are given as follows.
Given, the sides of the quadrilateral
AB = 9 cm
BC = 40 cm
DC = 28 cm
AD = 15 cm
Diagonal is AC = 41 cm
Here, ∆ABC is a right angle triangle, so its area will be
Area of Quadrilateral ABCD = Area of ∆ABC + Area of ∆ADC
= 180 cm² + 126 cm²
= 306 cm²

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November 5, 2023
INTRODUCTION TO EUCLID’S GEOMETRY | Study
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Introduction to Euclid Geometry The necessity of geometry had been felt from ancient times in different parts of the world.
The practical problems faced by people of ancient civilization had developed this branch of mathematics.
Let us cite few examples.
With floods in the river, the demarcations of land owners on the river-side land were used to wipe out. (Scroll down to continue …)
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In order to redraw the boundaries, the idea of area was introduced, the idea of area was introduced.
The volumes of granaries could be measured by using geometry.
The existence of Egyptian pyramids indicates the use of geometry from olden times.
In Vedic period, there was a manual of geometrical construction, known as Sulbasutra’s.
Different geometrical shapes were constructed as altars to perform various Vedic rites.
The word Geometry originates from the green word ‘Geo’ (earth) and metrein (to measure) Through Geometry was developed and applied from ancient time in various part the world, it was not presented in a systematic manner.
Later in 300 BC, the Egyptian mathematician Euclid, collected all the known work and arranged it in a systematic manner.
‘Elements’ is a classic treatise in geometry which was written by Euclid.
This was the most influential book. The ‘element’ was used as a text book for several years in western Europe.
The ‘elements’ started with 28 definitions, five postulates and five common notions and systematically built the rest of plane and solid geometry.
The geometrical approach given by Euclid is known as Euclid method.
The Euclid method consists of making a small set of assumptions and then proving many other proposition from these assumptions.
The assumptions, made were obvious universal truth. The two types of assumption, made were ‘axioms’ and ‘postulates’.
Euclid’s Definitions Euclid listed 23 definitions in book 1 of the ‘elements’.
We list a few of them: 1) 2) 3) 4) 5) 6) 7) A point is that which has no part A line is a breadth less length The ends of a line are points A straight line is a line which lies evenly with the points on itself. A surface is that which has length and breadth only.
The edges of a surface are lines A plane surface is surface which lies evenly with straight lines on its self. Starting with these definitions, Euclid assumed certain assumptions, known as axioms and postulates.
Euclid’s Axioms Axioms were assumptions which were used throughout mathematics and are not specifically linked to geometry.
Few of Euclid’s axioms are:
1) Things which are equal to the same thing are equal to one another.
2) It equals are added to equals; the wholes are equal.
3) 4) 5) 6) 7) If equals are subtracted from equals, the remainders are equal.
Things which coincide one another are equal to one another.
The whole is greater than the part Things which are double of the same thing are equal to one another.
Things which are half of the same things are equal to one another.
All these axioms refer to magnitude of same kind.
Axiom – 1 can be written as follows: If x = Z and y = Z, then x = y
Axiom – 2 explains the following: If x = y, then x + Z = y + Z According to axiom – 3, If x = y, then x – Z = y – Z Axiom – 4 justifies the principle of superposition that every thing equals itself.
Axiom – 5, gives us the concept of comparison. If x is a part of y, then there is a quantity Z such that x = y + Z or x > y Note that magnitudes of the same kind can be added, subtracted or compared.
Euclid’s Postulates Euclid used the term postulate for the assumptions that were specific to geometry. Euclid’s five postulates are as follows: Postulate 1: A straight line may be drawn from any one point to any other point. Same may be stated as axiom 5.1 Given two distinct points, there is a unique line that passes through them.
Postulate 2: A terminated line can be produced indefinitely. Postulate 3: A circle can be drawn with any centre and any radius. Postulate 4: All right angles are equal to one another. Postulate 5: If a straight line falling on two straight lines makes the interior angle on the same side of it taken together less than two right angles, then two straight lines, if produced indefinitely, meet on that side on which the sum of the angles is less than two right angles. Postulates 1 to Postulates 4 are very simple and obvious and therefore they are taken a ‘self evident truths’. Postulates 5 is complex and it needs to be discussed. Suppose the line XY falls on two lines AB and CD such that ∠1 + ∠2 < 180°, then the lines AB and CD will intersect at a point. In the given figure, they intersect on left side of PQ, if both are produced. Note: In mathematics the words axiom and postulate may be used interchangeably, though they have distinct meaning according to Euclid. System of Consistent Axioms A system of axioms is said to be consistent, if it is impossible to deduce a statement from these axioms, which contradicts any of the given axioms or proposition. Proposition or Theorem The statement or results which were proved by using Euclid’s axioms and postulates are called propositions or Theorems. Theorem: Two distinct lines cannot have more than one point in common. Proof: Given: AB and CD are two lines. To prove: They intersect at one point or they do not intersect. Proof: Suppose the lines AB and CD intersect at two points P and Q. This implies the line AB passes through the points P and Q. Also the line CD passes through the points P and Q. This implies there are two lines which pass through two distinct point P and Q. But we know that one and only one line can pass through two distinct points. This axiom contradicts out assumption that two distinct lines can have more than one point in common. The lines AB and CD cannot pass through two distinct point P and Q. Equivalent Versions of Euclid’s Fifth Postulate The two different version of fifth postulate a) For every line l and for every point P not lying on l, there exist a unique line m passing through P and parallel to l. b) Two distinct intersecting lines cannot be parallel to the same line.

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NUMBER SYSTEMS | Study
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Introduction to Natural Numbers
Non-negative counting numbers excluding zero are called Natural Numbers.
N = 1, 2, 3, 4, 5, ……….
Whole Numbers
All natural numbers including zero are called Whole Numbers.
W = 0, 1, 2, 3, 4, 5, ……………. (Scroll down to continue …)
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Integers
All natural numbers, negative numbers and 0, together are called Integers.
Z = – 3, – 2, – 1, 0, 1, 2, 3, 4, …………..
Rational Numbers
The number ‘a’ is called Rational if it can be written in the form of r/s where ‘r’ and ‘s’ are integers and s ≠ 0,
Q = 2/3, 3/5, etc. all are rational numbers.
How to find a rational number between two given numbers?
To find the rational number between two given numbers ‘a’ and ‘b’.
Example:
Find 2 rational numbers between 4 and 5.
Solution:
To find the rational number between 4 and 5
To find another number we will follow the same process again.
Hence the two rational numbers between 4 and 5 are 9/2 and 17/4.
Remark: There could be unlimited rational numbers between any two rational numbers.
Irrational Numbers
The number ‘a’ which can’t be written in the form of p/q is called irrational. Here, p and q are integers and q ≠ 0. You can say that the numbers which are not rational are called Irrational Numbers.
Example – √7, √11 etc.
Real Numbers
All numbers including both rational and irrational numbers are called Real Numbers.
R = – 2, – (2/3), 0, 3 and √2
Real Numbers And Their Decimal Expansions
1. Rational Numbers
If the rational number is in the form of a/b, then we can get two situations by dividing a by b.
a. If the remainder becomes zero
While dividing if we get zero as the remainder after some steps then the decimal expansion of such a number is called terminating.
Example:
7/8 = 0.875
b. If the remainder does not become zero
While dividing if the decimal expansion continues and not becomes zero then it is called non-terminating or repeating expansion.
Example:
1/3 = 0.3333….
Hence, the decimal expansion of rational numbers could be terminating or non-terminating recurring and vice-versa.
2. Irrational Numbers
If we do the decimal expansion of an irrational number then it would be non –terminating non-recurring and vice-versa. i. e. the remainder does not become zero and also not repeated.
Example:
π = 3.141592653589793238……
Representing Real Numbers on the Number Line
To represent the real numbers on the number line, we use the process of successive magnification. We visualise the numbers through a magnifying glass on the number line.
Example:
Step 1: The number lies between 4 and 5, so we divide it into 10 equal parts. Now for the first decimal place, we will mark the number between 4.2 and 4.3.
Step 2: Now we will divide it into 10 equal parts again. The second decimal place will be between 4.26 and 4.27.
Step 3: Now we will again divide it into 10 equal parts. The third decimal place will be between 4.262 and 4.263.
Step 4: By doing the same process again we will mark the point at 4.2626.
Operations on Real Numbers
1. The sum, difference, product and quotient of two rational numbers will be rational.
Example:
2. If we add or subtract a rational number with an irrational number then the outcome will be irrational.
Example:
If 5 is a rational number. √7 is an irrational number. Then, 5 + √7 and 5 – √7 are irrational numbers.
3. If we multiply a non-zero rational number with an irrational number, the outcome will be irrational. If we divide a non-zero rational number with an irrational number, the outcome will also be irrational.
Example:
If 7 is a rational number and √5 is an irrational number then 7√7 and 7/√5 are irrational numbers.
4. The sum, difference, product and quotient of two irrational numbers could be rational or irrational.
Example:
Finding Roots of a Positive Real Number ‘x’ geometrically and mark it on the Number Line
To find √x geometrically
1. First, mark the distance x unit from point A on the line. This ensures that AB equals x unit.
2. From B mark a point C with the distance of 1 unit, so that BC = 1 unit.
3. Take the midpoint of AC and mark it as O. Then take OC as the radius and draw a semicircle.
4. From the point B draw a perpendicular BD which intersects the semicircle at point D.
The length of BD = √x.
To mark the position of √x on the number line, we will take AC as the number line. B will be zero. So C is point 1 on the number line.
Now we will take B as the centre and BD as the radius. We will draw the arc on the number line at point E.
Now E is √x on the number line.
Identities Related to Square Roots
If p and q are two positive real numbers
Examples:
1. Simplify
We will use the identity
2. Simplify
We will use the identity
Rationalising the Denominator
Rationalising the denominator means to convert the denominator containing a square root term into a rational number. This is done by finding the equivalent fraction of the given fraction.
For which we can use the identities of the real numbers.
Example:
Rationalise the denominator of 7/(7- √3).
Solution:
We will use the identityhere.
Laws of Exponents for Real Numbers
If we have a and b as the base and m and n as the exponents, then
1. a^m × aⁿ =a^m+n
2. (a^m)ⁿ = a^mn
4. a^mb^m = (ab)^m
5.a⁰ = 1
6. a¹ = a
7. 1/aⁿ = a^-n
Let a > 0 be a real number and n a positive integer.
Let a > 0 be a real number. Let m and n be integers. They have no common factors other than 1. Also, n > 0. Then,
Example:
Simplify the expression (2x³y⁴) (3xy⁵)².
Solution:
Here we will use the law of exponents
a^m × aⁿ =a^m+n and (a^m)ⁿ = a^mn
(2x³y⁴)(3xy⁵)²
(2x³y⁴)(3 ² x ² y¹⁰)
18. x³. x². y⁴. y¹⁰
18. x³⁺². y⁴⁺¹⁰
18x⁵y¹⁴
Here’s a simple outline for an eBook on Real Numbers:
Title: Understanding Real Numbers: A Comprehensive Guide
Table of Contents
Introduction to Real Numbers
What are Numbers?
Introduction to Real Numbers
Why Are Real Numbers Important?
Classification of Numbers
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
Properties of Real Numbers
Closure Property
Commutative Property
Associative Property
Distributive Property
Identity and Inverse Elements
The Real Number Line
Concept of Number Line
Plotting Real Numbers on the Number Line
Understanding Density of Real Numbers
Rational and Irrational Numbers
Definition of Rational Numbers
Properties of Rational Numbers
Definition of Irrational Numbers
Examples of Irrational Numbers (like √2, π, e)
Proving √2 is Irrational
Decimals and Real Numbers
Finite and Infinite Decimals
Terminating and Non-Terminating Decimals
Relationship between Decimals and Fractions
Operations on Real Numbers
Addition and Subtraction
Multiplication and Division
Operations with Decimals
Operations with Irrational Numbers
Absolute Value and Real Numbers
Definition of Absolute Value
Geometric Representation on the Number Line
Properties of Absolute Value
The Concept of Infinity
Understanding Infinite Sets
Limits and Real Numbers
Approaching Infinity on the Number Line
Applications of Real Numbers
In Geometry (Pythagorean Theorem)
In Calculus (Limits, Derivatives, and Integrals)
In Daily Life (Measurements, Finance, etc.)
Advanced Topics on Real Numbers
Real Numbers in Algebra
Real Numbers and Functions
Real Numbers and Continuity
Conclusion
Summary of Key Concepts
Importance of Mastering Real Numbers
How Real Numbers Apply to Higher Mathematics
Chapter 1: Introduction to Real Numbers
What Are Numbers?
Numbers are abstract symbols used to represent quantities. Throughout history, different types of numbers have been developed to address various mathematical problems.
Introduction to Real Numbers
Real numbers are all the numbers that can be found on the number line. This includes rational numbers (such as 5, -3, and 0.75) and irrational numbers (such as √2 and π). Together, they form the building blocks of modern mathematics.
Real numbers are used to measure continuous quantities like distance, time, and weight. They are integral to the concepts of calculus, physics, engineering, and many other fields.
Why Are Real Numbers Important?
Real numbers play a critical role in mathematics. They allow us to describe the size of objects, calculate areas and volumes, and express very large or very small values. Without real numbers, much of modern science and technology would not exist.
Chapter 2: Classification of Numbers
Natural Numbers
The set of natural numbers consists of counting numbers, such as 1, 2, 3, and so on. These are the simplest type of numbers and do not include zero.
Whole Numbers
Whole numbers are like natural numbers but also include zero. Thus, the set is {0, 1, 2, 3,…}.
Integers
Integers expand on whole numbers by including negative numbers. The set of integers is {…, -3, -2, -1, 0, 1, 2, 3,…}.
Rational Numbers
Rational numbers are numbers that can be expressed as a fraction of two integers (a/b), where b ≠ 0. Examples of rational numbers include 1/2, -4, and 0.75.
Irrational Numbers
Irrational numbers cannot be expressed as a fraction of two integers. Examples include √2, π, and e. These numbers have non-repeating, non-terminating decimal expansions.
Chapter 3: Properties of Real Numbers
Closure Property
The set of real numbers is closed under addition, subtraction, multiplication, and division (except division by zero). This means that the result of any of these operations on two real numbers will always yield another real number.
Commutative Property
For any two real numbers a and b:
Addition: a + b = b + a
Multiplication: a × b = b × a
Associative Property
For any three real numbers a, b, and c:
Addition: (a + b) + c = a + (b + c)
Multiplication: (a × b) × c = a × (b × c)
Distributive Property
The distributive property connects addition and multiplication:
a × (b + c) = (a × b) + (a × c)
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LINEAR EQUATIONS IN TWO VARIABLES | Study
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Linear Equations
The equation of a straight line is the linear equation. It could be in one variable or two variables.
Linear Equation in One Variable
The equation with one variable in it is known as a Linear Equation in One Variable. (Scroll down to continue …)
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The general form for Linear Equation in One Variable is px + q = s, where p, q and s are real numbers and p ≠ 0.
Example:
x + 5 = 10
y – 3 = 19
These are called Linear Equations in One Variable because the highest degree of the variable is one.
Graph of the Linear Equation in One Variable
We can mark the point of the linear equation in one variable on the number line.
x = 2 can be marked on the number line as follows –
Graph of the Linear Equation in One Variable
Linear Equation in Two Variables
An equation with two variables is known as a Linear Equation in Two Variables. The general form of the linear equation in two variables is
ax + by + c = 0
where a and b are coefficients and c is the constant. a ≠ 0 and b ≠ 0.
Example
6x + 2y + 5 = 0, etc.
Slope Intercept form
Generally, the linear equation in two variables is written in the slope-intercept form as this is the easiest way to find the slope of the straight line while drawing the graph of it.
The slope-intercept form is y = mx+c
Where m represents the slope of the line.
and c tells the point of intersection of the line with the y-axis.
Remark: If b = 0 i.e. if the equation is y = mx then the line will pass through the origin as the y-intercept is zero.
Solution of a Linear Equation
There is only one solution in the linear equation in one variable but there are infinitely many solutions in the linear equation in two variables.
As there are two variables, the solution will be in the form of an ordered pair, i.e. (x, y).
The pair which satisfies the equation is the solution to that particular equation.
Example:
Find the solution for the equation 2x + y = 7.
Solution:
To calculate the solution of the given equation we will take x = 0
2(0) + y = 7
y = 7
Hence, one solution is (0, 7).
To find another solution we will take y = 0
2x + 0 = 7
x = 3.5
So another solution is (3.5, 0).
Graph of a Linear Equation in Two Variables
To draw the graph of a linear equation in two variables, we need to draw a table to write the solutions of the given equation, and then plot them on the Cartesian plane.
By joining these coordinates, we get the line of that equation.
The coordinates which satisfy the given Equation lie on the line of the equation.
Every point (x, y) on the line is the solution x = a, y = b of the given Equation.
Any point, which does not lie on the line AB, is not a solution of Equation.
Example:
Draw the graph of the equation 3x + 4y = 12.
Solution:
To draw the graph of the equation 3x + 4y = 12, we need to find the solutions of the equation.
Let x = 0
3(0) + 4y = 12
y = 3
Let y = 0
3x + 4(0) = 12
x = 4
Now draw a table to write the solutions.
x 0 4
y 3 0
Now we can draw the graph easily by plotting these points on the Cartesian plane.
Linear Equation in Two Variables
Equations of Lines Parallel to the x-axis and y-axis
When we draw the graph of the linear equation in one variable then it will be a point on the number line.
x – 5 = 0
x = 5
This shows that it has only one solution i.e. x = 5, so it can be plotted on the number line.
But if we treat this equation as the linear equation in two variables then it will have infinitely many solutions and the graph will be a straight line.
x – 5 = 0 or x + (0) y – 5 = 0
This shows that this is the linear equation in two variables where the value of y is always zero. So the line will not touch the y-axis at any point.
x = 5, x = number, then the graph will be the vertical line parallel to the y-axis.
All the points on the line will be the solution of the given equation.
Equations of Lines Parallel to the x-axis and y-axis
Similarly if y = – 3, y = number then the graph will be the horizontal line parallel to the x-axis.

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COORDINATE GEOMETRY | Study
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Coordinate Geometry: Cartesian Plane | Speed Notes
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Cartesian System
A plane formed by two number lines, one horizontal
and the other vertical, such that they intersect each
other at their zeroes, and then they form a Cartesian
Plane.
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● The horizontal line is known as the x-axis and the vertical line is known
as the y-axis.
● The point where these two lines intersect each other is called the origin.
It is represented as ‘O’.
● OX and OY are the positive directions as the positive numbers lie in the
right and upward direction.
● Similarly, the left and the downward directions are the negative directions
as all the negative numbers lie there.
Quadrants of the Cartesian Plane The Cartesian plane is divided into four quadrants namely Quadrant I, Quadrant II, Quadrant III, and Quadrant IV anticlockwise from OX.
Coordinates of a Point To write the coordinates of a point we need to follow the following rules. ● Thex – coordinate of a point is marked by drawing perpendicular from the y-axis measured a length of the x-axis .It is also called the Abscissa.
They – coordinate of a point is marked by drawing a perpendicular from the x-axis measured a length of the y-axis .It is also called the Ordinate. ● While writing the coordinates of a point in the coordinate plane, the x – coordinate comes first, and then the y – coordinate. We write the coordinates in brackets. In figure, OB = CA = x coordinate (Abscissa), and CO = AB = y coordinate (Ordinate). We write the coordinate as (x, y).
Remarks:
As the origin, O has zero distance from the x-axis and the y-axis so its abscissa and ordinate are zero. Hence the coordinate of the origin is (0, 0). The relationship between the signs of the coordinates of a point and the quadrant of a point in which it lies.
Plotting a Point in the Plane if its Coordinates are Given
Steps to plot the point (2, 3) on the Cartesian plane:
● First of all, we need to draw the Cartesian plane by drawing the coordinate axes with 1 unit = 1 cm.
● To mark the x coordinates, starting from 0 moves towards the positive x-axis and counts to 2.
● To mark the y coordinate, starting from 2 moves upwards in the positive direction and counts to 3.
● Now this point is the coordinate (2, 3). Likewise, we can plot all the other points, like (-3, 1) and (-1.5,-2.5) in the figure.
Question: Are the coordinates (x, y) = (y, x)? Let x = (-4) and y = (-2) So (x, y) = (- 4, – 2) (y, x) = (- 2, – 4)
Let’s mark these coordinates on the Cartesian plane. You can see that the positions of both the points are different in the Cartesian plane. So, If x ≠ y, then (x, y) ≠ (y, x), and (x, y) = (y, x), if x = y.
Example: Plot the points (6, 4), (- 6,- 4), (- 6, 4) and (6,- 4) on the Cartesian plane.
Solution: Since both numbers 6, 4 are positive the point (6, 4) lies in the first quadrant. For x coordinate, we will move towards the right and count to 6. Then from that point go upward and count to 4. Mark that point as the coordinate (6, 4). Similarly, we can plot all the other three points.

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POLYNOMIALS | Study

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Polynomial

A polynomial is an algebraic expression that includes constants, variables, and exponents. It is the expression in which the variables have only positive integral powers.

Example

1. 4x³ + 3x² + x +3 is a polynomial in variable x.

2. 4x² + 3x^-1 – 4 is not a polynomial as it has negative power.

3. 3x^3/2+ 2x – 3 is not a polynomial. Polynomials

Polynomials are denoted by p (x), q (x), etc.
In the above polynomials, 2x², 3y, and 2 are the terms of the polynomial.
2 and 3 are the coefficients of x² and y, respectively.
x and y are the variables.
2 is the constant term, which has no variable.

Polynomials in One Variable

If there is only one variable in the expression, then this is called the polynomial in one variable.

Example

x³ + x – 4 is polynomial in variable x and is denoted by p(x).
r² + 2 is polynomial in variable r and is denoted by p(r).

Types of polynomials on the basis of the number of terms

Types of polynomials on the basis of the number of degrees

The highest value of the power of the variable in the polynomial is the degree of the polynomial.

Zeros of a Polynomial

If p(x) is a polynomial, then the number ‘a’ will be the zero of the polynomial with p(a) = 0. We can find the zero of the polynomial by equating it to zero.

Example: 1

The given polynomial is p(x) = x – 4

To find the zero of the polynomial, we will equate it to zero.

x – 4 = 0

x = 4

p(4) = x – 4 = 4 – 4 = 0

This shows that if we place 4 in place of x, we get the value of the polynomial as zero. So 4 is the zero of this polynomial. And also, we are getting the value 4 by equating the polynomial with 0.

So 4 is the zero of the polynomial or the root of the polynomial.

The root of the polynomial is basically the x-intercept of the polynomial.

If the polynomial has one root, it will intersect the x-axis at one point only, and if it has two roots, it will intersect at two points, and so on.

Example: 2

Find p (1) for the polynomial p (t) = t² – t + 1

p (1) = (1)² – 1 + 1

= 1 – 1 + 1

= 1

Remainder Theorem

We know the property of division which follows in the basic division, i.e.

Dividend = (Divisor × Quotient) + Remainder

This follows the division of polynomials.

If p(x) and g(x) are two polynomials in which the degrees of p(x) ≥ degree of g(x) and g(x) ≠ 0 are given, then we can get the q(x) and r(x) so that:

P(x) = g(x) q(x) + r(x),

where r(x) = 0 or degree of r(x) < degree of g(x).

It says that p(x) divided by g(x), gives q(x) as a quotient and r(x) as a remainder.

Let’s understand it with an example

Division of a Polynomial with a Monomial

We can see that ‘x’ is common in the above polynomial, so we can write it as

Hence, 3x²+ x + 1 and x the factors of 3x³ + x² + x.

Steps of the Division of a Polynomial with a Non –Zero Polynomial

Divide x² – 3x -10 by 2 + x

Step 1: Write the dividend and divisor in descending order, i.e., in the standard form. x² – 3x -10 and x + 2

Divide the first term of the dividend with the first term of the divisor.

x²/x = x this will be the first term of the quotient.

Step 2: Now multiply the divisor by this term of the quotient and subtract it from the dividend.

Step 3: Now the remainder is our new dividend, so we will repeat the process again by dividing the dividend by the divisor.

Step 4: – (5x/x) = – 5

Step 5:

The remainder is zero.

Hence x² – 3x – 10 = (x + 2)(x – 5) + 0

Dividend = (Divisor × Quotient) + Remainder

The Remainder Theorem says that if p(x) is any polynomial of degree greater than or equal to one and let ‘t’ be any real number and p (x) is divided by the linear polynomial x – t, then the remainder is p(t).

As we know,

P(x) = g(x) q(x) + r(x)

If p(x) is divided by (x-t) then

If x = t

P (t) = (t – t). q (t) + r = 0

To find the remainder or to check the multiple of the polynomial, we can use the remainder theorem.

Example:

What is the remainder if a⁴ + a³ – 2a² + a + 1 is divided by a – 1.

Solution:

P(x) = a⁴ + a³ – 2a² + a + 1

To find the zero of (a – 1), we need to equate it to zero.

a -1 = 0

a = 1

p (1) = (1)⁴ + (1)³ – 2(1)² + (1) + 1

= 1 + 1 – 2 + 1 + 1

= 2

So by using the remainder theorem, we can easily find the remainder after the division of the polynomial.

Factor Theorem

The factor theorem says that if p(y) is a polynomial with degree n≥1 and t is a real number, then

(y – t) is a factor of p(y), if p(t) = 0, and
P (t) = 0 if (y – t) is a factor of p (y).

Example: 1

Check whether g(x) = x – 3 is the factor of p(x) = x³– 4x²+ x + 6 using the factor theorem.

Solution:

According to the factor theorem, if x – 3 is the factor of p(x), then p(3) = 0, as the root of x – 3 is 3.

P (3) = (3)³– 4(3)² + (3) + 6

= 27 – 36 + 3 + 6 = 0

Hence, g (x) is the factor of p (x).

Example: 2

Find the value of k, if x – 1 is a factor of p(x) = kx²– √2x + 1

Solution:

As x -1 is the factor, p(1) = 0

Factorization of Polynomials

Factorization can be done by three methods

1. By taking out the common factor

If we have to factorise x² –x then we can do it by taking x common.

x(x – 1) so that x and x-1 are the factors of x² – x.

2. By grouping

ab + bc + ax + cx = (ab + bc) + (ax + cx)

= b(a + c) + x(a + c)

= (a + c)(b + x)

3. By splitting the middle term

x² + bx + c = x² + (p + q) + pq

= (x + p)(x + q)

This shows that we have to split the middle term in such a way that the sum of the two terms is equal to ‘b’ and the product is equal to ‘c’.

Example: 1

Factorize 6x² + 17x + 5 by splitting the middle term.

Solution:

If we can find two numbers p and q such that p + q = 17 and pq = 6 × 5 = 30, then we can get the factors.

Some of the factors of 30 are 1 and 30, 2 and 15, 3 and 10, 5 and 6, out of which 2 and 15 is the pair which gives p + q = 17.

6x² + 17x + 5 =6 x² + (2 + 15) x + 5

= 6 x² + 2x + 15x + 5

= 2 x (3x + 1) + 5(3x + 1)

= (3x + 1) (2x + 5)

Algebraic Identities

1. (x + y)²= x²+ 2xy + y²

2. (x – y)² = x²– 2xy + y²

3. (x + y) (x – y) = x² – y²

4. (x + a) (x + b) = x² + (a + b)x + ab

5. (x + y + z)²= x²+ y²+ z²+ 2xy + 2yz + 2zx

6. (x + y)³ = x³ + y³ + 3xy(x + y) = x³+ y³ + 3x²y + 3xy²

7. (x – y)³ = x³– y³ – 3xy(x – y) = x³– y³ – 3x²y + 3xy²

8. x³+ y³ = (x + y)(x² – xy + y²)

9. x³ – y³= (x – y)(x² + xy + y²)

10. x³+ y³ + z³– 3xyz = (x + y + z)(x² + y²+ z² – xy – yz – zx) x³ + y³ + z³ = 3xyz if x + y + z = 0

Example: 2

Factorize 8x³ + 27y³ + 36x²y + 54xy²

Solution:

The given expression can be written as

= (2x)³+ (3y)³ + 3(4x²) (3y) + 3(2x) (9y²)

= (2x)³ + (3y)³ + 3(2x)²(3y) + 3(2x)(3y)²

= (2x + 3y)³ (Using Identity VI)

= (2x + 3y) (2x + 3y) (2x + 3y) are the factors.

Example: 3

Factorize 4x² + y² + z² – 4xy – 2yz + 4xz.

Solution:

4x² + y² + z² – 4xy – 2yz + 4xz = (2x)²+ (–y)² + (z)² + 2(2x) (-y)+ 2(–y)(z) + 2(2x)(z)

= [2x + (- y) + z]² (Using Identity V)

= (2x – y + z)² = (2x – y + z) (2x – y + z)

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