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BASIC GEOMETRICAL IDEAS | Study
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Basic Geometrical Ideas | Speed Notes
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Geometry
Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a 🎉geometer.
Space
Space: space refers to a set of points that form a particular type of structure.
Plane
A plane is a flat surface that extends infinitely in all directions
Point
Point is an exact position or location in space with no dimensions.
Mathematically, a point is defined as a circle with zero radius.
Since it is not possible its is represented by a very small dot.
A point is usually represented by a capital letter.
In mathematical terms, pont is a cirlce with no radius. It does mean that a very very small circle.
Lines And Its 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 end points.
The line segment has two end points. Note: A line Segment has two endpoints. (both Initial and end points). (Scroll down to continue …).
Parallel Lines
Lines that are always the same distance apart from each other and that never meet are called Parallel lines.
Note: Parallel lines do not have any common point.
Intersecting Lines
Lines that meet at a point with each other are called intersecting lines.
Intersecting lines meet at a point.
Angle
An angle is made up of two rays starting from a common end point.
An angle leads to three divisions of a region:
On the angle, the interior of the angle and the exterior of the angle.
Curve
Curve is a drawing (straight or non-straight) made without lifting the pencil may be called a curve.
Note: In this sense, a line is also a curve.
Types of Curves
Cureves are two types. They are (i) Simple or Open curve (ii) Closed curve.
Simple Curve
Simple or open curve is a curve that does not cross (intersect) itself.
Closed Curve
Closed curve is a curve that crosses (intersects) itself.
Curves–Concave And Convex
In geometry, a curve is a line or shape that is drawn smoothly and continuously in a plane with bends or turns.
Mathematicians define a curve as any shape that can be drawn without lifting the pen.
In Mathematics, A curve is a continuous and smooth line that is defined by a mathematical function or parametric equations.
Concave And Convex curves:
Curves are of two types. They are concave curve and convex curve.
Concave Curve:
A curve is concave is a curve that curves inward, resembling a cave.
Examples:
   – The interior of a circle.
   – The graph of a concave function like y = -x².
Convex Curves:
A curve is convex if it curves outward.
Examples:
   – The exterior of a circle.
   – The graph of a convex function like y = x².
Polygon
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.
Part of Polygon
(i) Sides Of The Polygon
The line segments of a polygone are called sides of the polygon.
(ii) Adjacent Sides Of Polygon
Adjacent sides of a polygon are thesides of a polygon with a common end point.
(iii) Vertex Of Polygon
Vertext of a polygon is a point at which a pair of sides meet.
(iv) Adjacent Vertices
Adjacent vertices of polygon are the end points of the same side of the polygon.
(v) diagonal
Diagonal of a polygone is a line segment that joins the non-adjacent vertices of the 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°
Triangle
A triangle is a three-sided polygon.
In other terms triangle is a three sided closed figure.
Quadrilateral
A quadrilateral is a four-sided polygon. (It shouldbe named cyclically).
In any
similar relations exist for the other three angles.
Circle And Its Parts
A circle is the path of a point moving at the same distance from a fixed point.
Centre Of Circle
Centre Of Circle is a point that is equidistant from any point on the boundary of the circle.
In other words the centre of the circles is a centre point of the circle.
Radius Of Circle
Radius of circle is the distance between the centre of the circle and any point on the boundary of the circle.
Circumference Of Circle
Circumference of circle is the length of the boundary of the circle.
Chord Of Circle
A chord of a circle is a line segment joining any two points on the circle.
A diameter is a chord passing throughthe Centre of the circle.
Sector Of Circle
A sector is the region in the interior of a circle enclosed by an arc on one side and a pair of radii on the other two sides.
Segment Of Circle
A segment of a circle is a region in the interior of the circle enclosed by an arc and a chord.
The diameter of a circle divides it into two semi-circles.
The diameter of a circle divides it into two semi-circles.

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UNDERSTANDING ELEMENTARY SHAPES | Study
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Understanding Elementary Shapes |Speed Notes
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The distance betweenthe end pointsof a line segment is its length. A graduatedruler and the divider are useful to compare lengthsof line segments. When a hand of a clock moves from one position to another position we have an examplefor an angle. (Scroll down to continue …)
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One full turn of the hand is 1 revolution.
A right angle is ¼ revolution and a straight angle is ½ a revolution. We use a protractor to measure the size of an angle in degrees.
The measure of a right angle is 90° and hence that of a straight angle is 180°.
An angle is acute if its measure is smaller than that of a right angle and is obtuseif its measure is greaterthan that of a right angle and less than a straightangle.
A reflex angle is largerthan a straight angle.
Two intersecting lines are perpendicular if the anglebetween them is 90°.
The perpendicular bisector of a line segmentis a perpendicular to the line segmentthat divides it into two equal parts.
Triangles can be classified as follows based on their angles:
Triangles can be classified as follows based on the lengths of their sides:
Polygons are namedbased on theirsides.
Quadrilaterals are furtherclassified with reference to their properties.
Quadrilateral
Quadrilateral is a closed figure with four sides.
Characteristics of a quadrilateral
Angle Sum Property of a Quadrilateral:
Sum of all angles of a quadrilateral is 360°.
Types Of Quadrilaterals
Quadrilaterals are broadly classified into two categories as:
(i) Trapezium
(ii) 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
Kite:
(i) Kite has a pair of equal adjacent sides.
(ii) It is not a parallelogram
Characteristics Of Kite:
Perimeter Of Square
Area Of Kite
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.
·We see aroundus many three dimensional shapes.Cubes, cuboids, spheres,
cylinders, cones,prisms and pyramidsare some of them.
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INTEGERS | Study
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Integers | Speed Notes
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We haveseen that there are times whenwe need touse numbers witha negative sign. This is when we want to go below zero on the number line. These are called negative numbers. Some examples of their use can be in temperature scale, water level in lake or river, level of oil in tank etc. (Scroll down to continue …)
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They are also used to denote debit account or outstanding dues. The collection of numbers…, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, … is called integers. So, – 1,– 2, – 3, – 4, … called negative numbers are negative integers and 1, 2, 3, 4, … called positive numbers are the positive integers. We havealso seen howone more thangiven number givesa successor andone less than given number gives predecessor. We observe that (a) When we havethe same sign,add and putthe same sign. (i) When two positive integers are added, we get a positive integer [e.g.. (+3) + (+2) = + 5]. (ii) When two negative integers are added, we get a negative integer [e.g.. (–2) +(–1)= – 3]. (b) When one positive and one negative integers are added we subtract them as whole numbers by considering thenumbers without their sign and thenput the signof the bigger number with the subtraction obtained. The bigger integer is decided by ignoring thesigns of theintegers [e.g.. (+4)+ (–3) =+ 1 and(–4) + (+3)= – 1]. (c) The subtraction ofan integer isthe same asthe addition ofits additive inverse. We have shownhow addition andsubtraction of integers can also beshown on a number line.

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DECIMALS | Study
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Decimals | Speed Notes
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To understand the parts of one whole (i.e. a unit) we represent by a block divided into 10 eaual parts means (1/10) ^th of a unit. (Scroll down to continue …)
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Addition of Decimals: Decimalscan be added by writingthem with equal number of decimals places.Example: add 0.005,6.5 and 20.04.
Solution: Convert the given decimals as 0.005, 6.500 and 20.040. 0.005+ 6.500 + 20.040 = 26.545
Subtraction of Decimals: Decimalscan be subtracted by writingthem with equalnumber of decimalplaces.
Example: Subtract the given decimals as 5.674 and 12.500 12.500– 5.674 = 6.826
ComparingDecimals: Decimalsnumberscanbecompare The givendecimals have distinctwhole number part, so we compare wholenumber part only. The whole number part of 45.32 is greater than 35.69. Therefore, 45.32>35.69.
Using Decimals: Many dailylife problems can be solvedby converting different units of measurements such as money,length, weight, etc. in the decimal form.
Money:
100 paise = 1 Rupee
1 paise = 1/100 Rupee = 0.01 Rs. 5 paise = 5/100 Rs. = 0.05 Rs.
105 paise = 1 Rs. +5 paise = 1.05 Rs.
7 Rs. 8 paise= 7 Rs. + 0.08 Rs = 7.08 Rs.
7 Rs. 80 paise = 7 Rs. + 0.80 Rs. = 7.80 Rs.
Length:
10 mm = 1 cm
1mm = 1/10 cm = 0.1 cm 100 cm = 1 m
1 cm = 1/100 m = 0.01 m 1000 m = 1 km
1 m = 1/1000 km = 0.001km
Weight:
1000 g = 1 kg
1 g = 1/1000kg = 0.001 kg
25 g = 25/1000kg = 0.025 kg

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DATA HANDLING | Study
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Data Handling | Speed Notes
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Data: A collection of numbers gathered to give someinformation. Recording Data:Data can becollected from different sources. Pictograph: The representation of data through pictures of objects. It helps answer the questions onthe data ata glance. (Scroll down to continue …)
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Bar Graph: Pictorial representation of numerical datain the formof bars (ractangles) of equal width and varying heights. We have seen that data is a collection of numbers gathered to give some information.
To get a particular information from the givendata quickly, thedata can be arranged ina tabular formusing tally marks. We learnt how a pictograph represents data in the formof pictures, objects or parts ofobjects.
We have also seen how to interpret a pictograph and answer the related questions.
We havedrawn pictographs using symbols to represent a certain number of items orthings.
We havediscussed how torepresent data byusing a bardiagram or abar graph.
Ina bar graph, bars of uniform width are drawn horizontally or vertically with equal spacing between them.
Thelength of eachbar gives therequired information.
To do this we also discussed the process of choosing a scale for the graph. For example, 1unit = 100students.
We havealso practised reading a given bargraph.
We have seen howinterpretations from thesame can bemade.

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MENSURATION | Study
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Perimeter is the length of the boundary of the geometric shape.
In other words the distance covered along theboundary forming aclosed figure whenyou go round the figure once.
(a) Perimeter of arectangle = 2 × (length + breadth) (b) Perimeter of a square = 4 × length ofits side. (Scroll down to continue …)
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Mensuration
Perimeter is the length of the boundary of the geometric shape.
In other words the distance covered along theboundary forming aclosed figure whenyou go round the figure once.
(a) Perimeter of arectangle = 2 × (length + breadth)
(b) Perimeter of a square = 4 × length ofits side
(c) Perimeter of anequilateral triangle =3 × length of a side
(d) Perimeter of a regular pentagon has five equal sides = 5 × length of a sides Figures in which all sides and angles are equal are called regular closed figures.
The amount of surface enclosed by a closed figure is called its area. To calculate the area of a figure using a squared paper, the following conventions are adopted :
(a) Ignore portions ofthe area thatare less thanhalf a square.
(b) If more than half a square is in a region. Count it as one square
(c) If exactly half the square is counted, take its area as
Area of a rectangle = length × breadth Area of a square = side × side

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ALGEBRA | Study
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Algebra | Speed Notes
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Algebra: A generalization of arithmetic in which letters representing numbers are combined according to the rules of arithmetic.
We looked at patterns of making letters and other shapes using matchsticks.
We learnt how to write the general relation between the number of matchsticks required for repeating a given shape. (Scroll down to continue …)
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The number of times a given shape is repeated varies;
it takes on values 1,2,3, It is a variable, denoted by some letter like n.
A variable takes on different values, its value is not fixed.
The length of a square can have any value.
It is a variable.
But the number of angles of a triangle has a fixed value 3.
It is not a variable.
We may use any letter a, b, c … x, y, z etc., to show a variable.
A variable allows us to express relations in any practical situation.
Variables are numbers, although their valueis not fixed.
We can do the operations ofaddition, subtraction, multiplication and division on them just as in the case of fixed numbers.
Using different operations we can form expressions with variables like x –3, x +3, 2n, 5m, 3p, 2y + 3, 3l – 5, etc.
Variables allow us to express many common rules in both geometry and arithmetic in a general way.
For example, the rule that the sum of two numbers remains the same if the order in which thenumbers are taken is reversed canbe
expressed as a + b = b +a.
Here, the variables a and b stand for any number, 1, 32, 1000– 7, – 20, etc.
An equation is a condition on a variable.
It is expressed by saying that an expression with avariable is equal to a fixed number, e.g. x– 3 =10.
An equation has twosides, LHS and RHS, between them is the equal (=) sign.
Solution of an Equation: The value ofthe variable inan equation which satisfies the equation.
For getting thesolution of anequation, one method is the trial and error method.
In this method, we give somevalue to the variable and check whether it satisfies the equation.
We go on giving this way different values to the variable until we find the right which satisfies the equation.
The LHS of an equation is equal to its RHS only for a definite value of the variable in the equation.
We say that this definite value of the variable satisfies the equation.
This value itself is called the solution of the equation.
For getting the solution of an equation, one method is the trial and error method.
In this method, we give some value to the variable and check whether it satisfies the equation.
We goon giving this way different values to the variable` until we find the right value which satisfies the equation.
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RATIO AND PROPORTION | Study
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Ratio and Proportion Comparison By Taking Difference | Speed Notes
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CHAPTER 12 Ratio and Proportion Comparison by taking difference: For comparing quantities of thesame type, wecommonly use themethod of taking difference between thequantities. Some times thecomparison by difference does not makebetter sense thanthe comparison by division. (Scroll down to continue)
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Comparison by Division: In many situations, a more meaningful comparison between quantities is made byusing division, i.e.. by seeing how many times one quantity is to the other quantity. This method is known ascomparison by ratio. The comparison of two numbers or quantities bydivision is knownas the ratio. Symbol ‘:’is used todenote ratio. For comparison by ratio, thetwo quantities mustbe in thesame unit. Ifthey are not,they should beexpressed in thesame unit before the ratio istaken. For example, Isha’s weight is25 kg andher father’s weight is 75 kg.We say thatIsha’s father’s weight and Isha’s weight are in theratio 3 : 1

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Integers | Study
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Integers | Speed Notes
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Integers are a bigger collection of numbers which is formed by whole numbers and their negatives. You have studied inthe earlier class, about the representation of integers onthe number lineand their addition and subtraction. (Scroll down to continue …)
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We now study theproperties satisfied by addition andsubtraction.
(a) Integers are closed for addition and subtraction both. That is, a + b and a – b are again integers, where a andb are anyintegers.
(b) Addition is commutative forintegers, i.e., a + b = b + a for allintegers a andb.
(c) Addition is associative for integers, i.e., (a + b) + c = a + (b + c) for all integers a, b and c.
(d) Integer 0 is the identity under addition. That is, a + 0 = 0 + a = a for every integer a. We studied, how integers could be multiplied, andfound that product of a positive and a negative integer is a negative integer, whereas the product of two negative integers isa positive integer. For example, –2 × 7 = –14 and –3 × – 8 =24.
Product of even number of negative integers is positive, whereas the product of odd number of negative integers is negative. Integers showsome properties under multiplication.
(a) Integers are closed under multiplication. Thatis, a × b isan integer forany two integers a and b.
(b) Multiplication is commutative for integers. Thatis, a × b = b × a forany integers a and b.
(c) The integer 1 is theidentity under multiplication, i.e., 1 × a = a × 1 = a forany integer a.
(d) Multiplication is associative for integers, i.e.,(a × b) × c = a × (b × c) for anythree integers a,b and c.
Under addition and multiplication, integers show a property called distributive property.
That is, a× (b +c) = a × b+ a × c forany three integers a, b andc.
The properties of commutativity, associativity under addition and multiplication, and the distributive property help us to make our calculations easier. We alsolearn how to divide integers. We found that,
(a) When a positive integer is divided by a negative integer, the quotient obtained is a negative integer and vice-versa. (b) Division of a negative integer by another negative integer gives a positive integer as quotient. For any integer a,we have
1) The numbers. . . , —4,—3, —1, 0, 1, 2,3, 4, etc.are integers.
2) 1, 2, 3, 4, 5. . . . are positive integers and —1,-2, —3,.. are negative integers.
3) 0 isan integer which is neither positive nornegative.
4). On an integer number line, all numbers to the right of 0 arepositive integers andall numbers tothe left of0 are negative integers.
5) 0 is less than everypositive integer and greater than everynegative integer.
6) Every positive integer is greater than every negative integer.
7) Two integers thatare at thesame distance from 0, but onopposite sides of it are called opposite numbers.
8. The greater the number, the lesser is its opposite.
9. The sumof an integer and its opposite is zero.
10. The absolute valueof an integer is the numerical value of theinteger without regard to its sign.
The absolute value of an integer a isdenoted by |a| and is given by a,if a is positive or 0 a = -a,if a is negative
11. The sum oftwo integers of the same sign is an integer of the same sign whose absolute value is equal to the sum of the absolute values of the given integers.
12. The sum of two integers of opposite signs is an integer whose absolute value is the difference of the absolute values of addend and whose sign isthe sign ofthe addend having greater absolute value.
13. To subtract an integer b from another integer a, we change the sign ofb and addit to a. Thus, a − b = a + (−b)
14. All properties of operations onwhole numbers aresatisfied by theseoperations on integers.
15. If aand b are two integers, then(a − b) is alsoan integer.
16. −a and aare negative oradditive inverses of each other.
17. To find theproduct of twointegers, we multiply theirabsolute values andgive the result a plus signif both thenumbers have the same sign or a minussign otherwise.
18. To find thequotient of oneinteger divided by another non-zero integer, we divide their absolute values and give the result a plus sign if both the numbers have the same sign or a minus signotherwise.
19. All the properties applicable to wholenumbers are applicable to integers in addition, the subtraction operation has the closure property.
20. Any integer whenmultiplied or divided by 1 gives itself and whenmultiplied or divided by-1 gives its opposite.
21. When expression hasdifferent types ofoperations, some operations haveto be performed before the others. That is, each operation has its own precedence. The order in which operations are performed is division, multiplication, addition and finally subtraction (DMAS).
22. Brackets are usedin an expression when we wanta set of operations to be performed before the others.
23. While simplifying anexpression containing brackets, the operations within the innermost set of brackets are performed first and then those brackets are removed followed by the ones immediately after them tillall the brackets are removed.
24. While simplifying arithmetic expressions involving various brackets and operations, we use BODMAS rule.

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Fractions:
4. A fraction whose numerator is less than the denominator is called a proper fraction.
5. A fraction whose numerator is more than or equal to the denominator is called animproper fraction.
6. A combination of a whole number and a proper fraction is called a mixed fraction.
7. To get a fractionequivalent to a given fraction,we multiply (or divide) its numerator and denominator by the same non-zero number.
8. Fractions having the same denominators are called like fractions. Otherwise, they are calledunlike fractions.
9. A fraction is said to be in its lowest termsif its numerator and denominator have no commonfactor other than 1.
10. To compare fractions, we use the followingsteps:
Step I Find the LCMof the denominators of the given fractions.
Step II Converteach fraction to itsequivalent fraction with denominator equal to the LCM obtained in step I.
Step Ill Arrangethe fractions in ascending or descending order byarranging numerators in ascending or descending order.
11. To convert unlike fractions into like fractions, we use the following steps:Step I Find the LCM of the denominators of the given fractions.
Step II Convert each of the given fractions into an equivalent fraction having denominator equal to the LCM obtained in step I.
12. To add (or subtract)fractions, we may use the following steps:Step I Obtain the fractionsand their denominators.
Step II Find the LCMof the denominators.
Step III Convert each fraction into an equivalent fraction having its denominator equal to the LCM obtainedin step II.
Step IV Add (or subtract) like fractions obtained in Step Ill.
Step III Convert each fraction into an equivalent fraction having its denominator equal to the LCM obtainedin step II.
Step IV Add (or subtract) like fractions obtained in Step Ill.
14. Two fractions are said to be reciprocal of each other, if their product is 1. The reciprocal of a non zero fraction a/b is b/a.
15. The divisionof a fraction a/b by a non-zero fraction c/d is the product of a/b with the
reciprocal of c/d.
Decimals:
1. Decimals are an extension of our number system.
2. Decimals are fractionswhose denominators are 10, 100, 1000 etc.
3. A decimal has two parts, namely, the whole numberpart and decimal part.
4. The number of digits containedin the decimal part of a decimal number is known as the numberof decimal places.
5. Decimals having the same number of places are called like decimals, otherwise they are knownas unlike decimals.
6. We have, 0.1 = 0.10 = 0.100 etc, 0.5 = 0.50 = 0.500 etc and so on. That is by annexing zeros on the right side of the extreme right digit of the decimalpart of a number does not alterthe value of the number.
7. Unlike decimals may be converted into like decimals by annexing the requisite numberof zeros on the right side of the extreme right digit in the decimal part.
8. Decimal numbers may be convertedby using the following steps.Step I Obtain the decimalnumbers
Step II Compare the whole partsof the numbers. The number with greater whole part will be greater. If the whole parts are equal, go to next step.
Step Ill Compare the extreme left digits of the decimal parts of two numbers. The number with greater extreme left digit will be greater. If the extreme left digits of decimal parts are equal,then compare the next digits and so on.
9. A decimal can be converted into a fractionby using the following steps:Step I: Obtain the decimal.
Step II: Take the numerator as the number obtained by removing the decimal point from the given decimal.
Step III: Take the denominator as the number obtainedby inserting as many zeros with 1 (e.g.10, 100 or 1000 etc.)as there are number of places in the decimal part.
10. Fractions can be converted into decimals by using the following steps:
Step I: Obtain the fractionand convert it into an equivalent fraction with denominator 10 or 100 or 1000 if it is not so.
Step II: Write its numeratorand mark decimal point after one place or two places or threeplaces from right towards left if the denominator is 10 or 100 or 1000 respectively. If the numerator is short of digits, insert zeros at the left of the numerator.
11. Decimals can be added or subtracted by using the following steps:Step I: Convert the given decimals to like decimals.
Step II: Write the decimals in columns with their decimal pointsdirectly below each other so that tenthscome under tenths, hundredths come and hundredths and so on.
Step III: Addor subtract as we add or subtract whole numbers.
Step IV: Place the decimal point, in the answer, directly below the other decimal points.
12. In order to multiply a decimal by 10, 100, 1000 etc., we use the following rules:
Rule I: On multiplying a decimal by 10, the decimalpoint is shiftedto the right by one place.
Rule II: On multiplying a decimal by 100, the decimal point is shiftedto the right by two places.
Rule III: On multiplying a decimal by 1000, the decimal point is shiftedto the right by threeplaces, and so on.
13. A decimal can be multiplied by a whole number by using following steps:
Step I: Multiply the decimal without the decimalpoint by the given whole number.
Step II: Mark the decimal point in the product to have as many placesof decimal as are there in the given decimal.
14. To multiply a decimal by another decimal, we follow following steps:
Step I: Multiply the two decimalswithout decimal point just like whole numbers.
Step II: Insert the decimal point in the product by countingas many places from the right to left as the sum of the number of decimalplaces of the given decimals.
15. A decimal can be dividedby 10, 100, 1000 etc by using the followingrules:
Rule I When a decimal is divided by 10, the decimal point is shifted to the left by one place.
Rule II When a decimal is divided by 100, the decimal point is shifted to the left by two places.
Rule III When a decimal is divided by 1000, the decimal point is shiftedto the left by threeplaces.
16. A decimal can be divided by a whole number by using the following steps:Step I: Check the whole number part of the dividend.
Step II: If the wholenumber part of the dividend is less than the divisor,then place a 0 in the onesplace in the quotient. Otherwise, go to step Ill.
Step III: Divide the whole number part of the dividend.
Step IV: Place the decimal point to the right of ones place in the quotient obtained in step I.
Step V: Divide the decimal part of the dividend by the divisor. If the digits of the dividend are exhausted, then place zeros to the right of dividendand remainder each time and continue the process.
17. A decimal can be divided by a decimal by using the following steps:
Step 1 Multiple the dividend and divisor by 10 or 100 or 1000 etc. to convert the divisor into a whole number.
Step II Divide the new dividendby the whole number obtainedin step I.
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Data Handling | Study
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The collection, recording and presentation of data help us organiseour experiences and draw inferences from them.
Before collecting data we need to know what we would use it for.
The data that is collected needs to be organised in a propertable, so that it becomeseasy to understand and interpret. (Scroll down to continue …).
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Average is a numberthat represents or shows the central tendencyof a group of observations or data.
Arithmetic mean is one of the representative values of data.
Mean = sum of all observations/ Number of observations.
Mode is another form of central tendency or representative value.
The mode of a set of observations is the observation that occurs most often.
If each of the value in a data is occurring one time, then all are mode.
Sometimes we also say that this data has no mode since none of them is occurring frequently.
Median is also a form of representative value.
It refers to the value which lies in the middle of the data with half of the observations above it and the other half below it.
.
A bar graph is a representation of numbers using bars of uniform widths.
Double bar graphshelp to comparetwo collections of data at a glance.
Double bar graphshelp to comparetwo collections of data at a glance.
There are situations in our life, that are certain to happen, some that are impossible and some that may or may not happen.
The situation that may or may not happen has a chanceof happening.
Probability: A branch of mathematics that is capable of calculating the chance or likelihood of an event taking place (in percentage terms).
If you have 10 likelihoods and you want to calculate the probability of 1 event taking place,it is said that its probability is 1/10 or event has a 10% probability of taking place.
Events that have many possibilities can have probability between 0 and 1.
Important Formulae – Data Handling
1. A trial is anaction which results in one or several outcomes. 2. An experiment in whichthe result ofa trial cannot be predicted inadvance is called a random experiment.
3. An event associated to a random experiment is thecollection of someoutcomes of theexperiment.
4. An event associated witha random experiment is said tohappen if anyone of theoutcomes satisfying thedefinition of theevent is anoutcome of theexperiment when it is performed.
5. The Empirical probability ofhappening of an event E is defined as: P(E)= Number of trials in which the event happened/ Total number of trials.

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Simple Equations | Study
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Simple Equations | Speed Notes
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An equation isa condition ona variable suchthat two expressions in the variable should have equalvalue.
Thevalue of thevariable for whichthe equation issatisfied is called the solution ofthe equation.
An equation remains the same if the LHSand the RHSare interchanged. (Scroll down to continue …)
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In case ofthe balanced equation, if we add the same number to both thesides, or subtract the same number from both the sides,
or
multiply both sidesby the same number, or divide both sidesby the samenumber, the balance remains un disturbed,
i.e.,the value of the LHS remains equal to the value of the RHS The above property gives a systematic method of solving an equation.
We carry out a series of identical mathematical operations on the two sides of the equation in such a waythat on oneof the sides we get justthe variable. Thelast step isthe solution of the equation.
Transposing means moving to the other side.
Transposition of a number has the same effect as adding same number to (or subtracting the same number from) both sides of the equation.
Whenyou transpose a number fromone side ofthe equation tothe other side, you change itssign.
For example, transposing +3 fromthe LHS tothe RHS in equation x + 3 = 8 gives x = 8 – 3 (= 5).
We can carry out the transposition of an expression in thesame way as the transposition of a number.
We havelearnt how to construct simple algebraic expressions corresponding to practical situations.
Wealso learnt how,using the technique of doing thesame mathematical operation (for example adding the samenumber) on bothsides, we could build an equation starting fromits solution.
Further, we also learnt that we could relate a given equation tosome appropriate problem/puzzlefrom the equation. practical situation and build a practical word.

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Lines and Angles | Study
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Basics Of Geometry

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Point: Point is an exact position in space with no dimensions, represented by a dot.
Ray: Ray is a part of a line that starts at an endpoint and extends infinitely away from the end point in single direction.
Line or Straight line: A line or straight line is perfectly straight and extends forever in both direction.
Line segment: A line segment is the part of a line between two points.
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Intersecting lines: Two or more lines that have one and only one point in common.
Point of intersection: Point of intersection is a common point at which the intersecting lines meet.
Transversal: Transversal is a line that intersects two or more lines which lie in the same plane at distinct points.
Parallel lines: Parallel lines are the lines on a plane which never meet. They are at a distance apart.
Complementary Angles: Complementary angles are the angles whose total is equal to 90^o .
Supplementary Angles: Suplementary angles are the angles whose total is equal to 180^o
Adjacent Angles: Adjacent Angles are the angles which have a common vertex and a common interior points.
Linear Pair of Angles: Linear pair of angles is a pair of adjacent angles whose non-common sides are opposite rays.
Vertically Opposite Angles: Vertically opposite angles are the angles formed by two intersecting lines which have the have common arms.
Angles made by Transversal:
When two lines are intersecting by a transversal, eight angles are formed.
Transversal of Parallel Lines: If two parallel lines are intersected by a transversal, each pair of:
Corresponding angles are congruent.
Alternateinterior angles are congruent.
Alternate exterior angles are congruent.
If the transversal is perpendicular to the parallellines, all of the angles formed are congruent to 90^o angles.
A line which intersects two or more given lines at distinct points is called a transversal to the given lines.
Lines in a plane are parallel if they do not intersect when produced indefinitely in either direction.
The distance between two intersecting lines is zero.
The distance between two parallel lines is the same everywhere and is equal to the perpendicular distance between them.
If two parallel lines are intersected by a transversal then:
pairs of alternate (interior orexterior) angles are equal.
pairs of corresponding angles are equal.
interior angles onthe same sideof the transversal are supplementary.
6. If two non-parallel lines are intersected by transversal then none of (i), (ii) and (iii) hold true in 5. 7.
If two lines are intersected by a transversal, then they are parallel if any one of the following is true:
The angles of a pair of corresponding angles are equal.
The angles of a pairof alternate interior angles are equal.
The angles of a pairof interior angles on the sameside of the transversal are supplementary.

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The Triangle And Its Properties | Study
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Basics Of Geometry

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A closed plane figure bounded by three linesegments. The six elements of a triangle are its three angles and thethree sides. The line segment joining a vertex of a triangle to the mid point of its opposite side is called a medianof the triangle. (Scroll down to continue …)
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Triangle:
A closed plane figure bounded by three line segments is called Triangle.
The six elements of a triangle are its three angles and the three sides. The line segment joining a vertex of a triangle to the midpoint of its
Median:
The opposite side is called the median of the triangle.
A triangle has three medians.
Altitude of the triangle:
The perpendicular line segment from vertex of a triangle to its opposite sides is called an altitude of the triangle.
A triangle has3 altitudes.
Type of triangle based onSides:
Equilateral Triangle:
A triangle is said to be equilateral, if each one of its sides has the same length. In An equilateral triangle, each angle measures 60°.
Isosceles Triangle:
A triangle is said to be isosceles, if atleast any two of its sides are of same length. The non-equal side of an isosceles triangle is called its base; the base angles of an isosceles triangle have equal measure.
Scalene Triangle:
A triangle having all sides of different lengths. It has no two angles equal.
Property of the lengths of sides of a triangle:
The sum of the lengths of any two sides of a triangle is greater than the length of the third side. The difference between the lengths of any two sides is smaller than the length of the third side. This property is useful to know if it is possible to draw a triangle when the lengths of the three sides are known.
Types of Triangle based on Angles:
(i) Right Angled Triangle:
A triangle one of whose angles measures
(ii) Obtuse Angled Triangle:
A triangle one of whose angles measures more than
(iii) Acute Angled Triangle:
A triangle each of whose angles measures less than In a right angled triangle, the side opposite to the right angle is called the hypotenuse and the other two sides are called its legs.
Pythagoras Property:
In a right-angled triangle, the square on the hypotenuse = the sum of the squares on its legs.If a triangle is not right-angled, this property does not hold good. Thisproperty is useful to decide whether a given triangle is right-angled
or not.
Exterior angle of a triangle:
An exterior angle of a triangle is formed, when a side of a triangle is produced. At each vertex, you have two ways of forming an exterior angle.
A property of exterior angles:
The measure of any exterior angle of a triangle is equal to the sum of the measures of its interior opposite angles.
The angle sum property of a triangle:
The total measure of the three angles of a triangle is 180°.
Property of the Lengths of Sides of a Triangle:
The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. The difference of the lengths of any two sides of a triangle is always smaller than the length of the third side.
Important Formulas – TheTriangles and its Properties
1. A triangle is a figure made up by three line segments joining, in pairs, three non-collinear points. That is, if A, B, C are three non-collinear points, the figure formed by three line segments AB,BC and CA is called a triangle with vertices A, B, C.
2. The three line segments forming a triangle are called the sides of the triangle.
3. The three sides and three angles of a triangle are together called the six parts or elements of the triangle.
4. A triangle whose two sides are equal, is called an isosceles triangle.
5. A triangle whose all sides are equal, is called an equilateral triangle.
6. A triangle whose no two sides are equal, is called a scalene triangle.
7. A triangle whose all the angles are acute is called an acute triangle.
8. A triangle whose one of the angles is a right angle is called a right triangle.
9. A triangle whose one of the angles is an obtuse angle is called an obtuse triangle.
10. The interior of a triangle is made up of all such points P of the plane, as are enclosed by the triangle.
11. The exterior of a triangle is that part of the plane which consists of those points Q, which are neither on the triangle nor in its interior.
12. The interior of a triangle together with the triangle itself is called the triangular region.
13. The sum of the angles of a triangle is two right angles or 180°.
14. If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the interior opposite angles.
15. In any triangle, an exterior angle is greater than either of the interior opposite angles.
16. The sum of any two sides of a triangle is greater than the third side.
17. In a right triangle, if a, b are the lengths of the sides and c that of the hypotenuse, then
18. If the sides of a triangle are of lengths a, b and c such that
then the triangle is right-angled and the side of length c is the hypotenuse.
19. Three positive numbers a, b, c in this order are said to form a Pythagorean triplet, if
Triplets (3, 4, 5) (5, 12,13), (8, 15, 17), (7,24, 25) and (12, 35,37) are somePythagorean triples.

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Comparing Quantities: Weare often requiredto compare two quantities, in our dailylife. They may be heights, weights, salaries, marks etc. To compare two quantities, their units must be the same.
We are often required to compare two quantities in our daily life. They may be heights, weights,salaries, marks etc. (Scroll down to continue …)
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While comparing heights of two persons with heights150 cm and 75 cm, we write it as the ratio 150 : 75 or 2 : 1.
Ratio: A ratio compares two quantities using a particular operation.
Percentage: Percentage are numerators of fractions with denominator 100. Percent is represent by the symbol% and means hundredth too.
Two ratios can be compared by converting them to like fractions. If the two fractions are equal,we say the two given ratios are equivalent.
If two ratios are equivalent then the four quantities are said to be in proportion. For example, the ratios 8 : 2 and 16 : 4 are equivalent therefore 8, 2, 16 and 4 are in proportion.
A way of comparing quantities is percentage. Percentages are numerators of fractions with denominator 100. Per cent means per hundred. For example 82% marks means
82 marks out of hundred.
Percentages are widely used in our daily life,
(a) We have learnt to find exact number when a certain per cent of the total quantity is given.
(b) When parts of a quantityare given to us as ratios, we have seen how to convert
them to percentages.
(c) The increase or decrease in a certainquantity can also be expressed as percentage.
(d) The profit or loss incurredin a certain transaction can be expressedin terms of percentages.
(e) While computing intereston an amount borrowed, the rate of interest is given in terms of per cents. For example, ` 800 borrowed for 3 years at 12% per annum. Simple Interest:Principal means the borrowed money.
The extra money paid by borrower for using borrowedmoney for given time is called interest(I).
The period for which the money is borrowed is called ‘TimePeriod’ (T).
Rate of interestis generally given in percentper year.
Interest, I = PTR/100
Total money paid by the borrower to the lenderis called the amount.

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Rational Number: A number that can be expressed in the form (Scroll down to continue …)
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14. Every positive rational number is greater than zero.
15. Every negative rational number is less than zero.
16. The rational numbers can be represented on the number line.
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Perimeter is the distance around a closed figure whereasarea is the part of plane occupied by the closedfigure.
Area is the measure of the part of plane or regionenclosed by it. (Scroll down to continue …)
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We have learnt how to find perimeter and area of a squareand rectangle earlierclass.
They are:
(a) Perimeter of a square = 4 × side
(b) Perimeter of a rectangle = 2 × (length + breadth)
(c) Area of a square = side × side
(d) Area of a rectangle = length × breadth Areaof a parallelogram = base × height

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Exponents are used to express large numbers in shorter form to make them easy to read, understand, compare and operate upon. (Scroll down to continue …)
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Expressing Large Numbers in the Standard Form: Any number can be expressed as a decimal number between 1.0 and 10.0 (including 1.0) multiplied by a power of 10. Such form of a number is called its standard form or scientific motion. Very large numbers are difficult to read, understand, compare and operate upon. To make all these easier, we use exponents, converting many of the large numbers in a shorter form. The following are exponential forms of some numbers?
Here, 10, 3 and 2 are the bases, whereas 4, 5 and 7 are their respective exponents. We also say, 10,000 is the 4th power of 10, 243 is the 5th power of 3, etc. Numbers in exponential form obey certain laws, which are: For any non-zero integers a and b and whole numbers m and n,
(g) (–1) even number = 1 (–1) odd number = – 1

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Congruence: The relation of two objects being congruent is called congruence. (Scroll down to continue …)
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Chapter – 7
Congruence of Triangles
SSS Congruence of two triangles: Under a given correspondence, two triangles are congruent if the three sides of the one are equal to the three corresponding sides of the other.
SAS Congruenceof two triangles: Under a given correspondence, two triangles are
congruent if two sides and the angleincluded between them in one of the triangles are equal to the corresponding sides and the angle included between them of the other triangle.
ASA Congruence of two triangles: Under a given correspondence, two triangles are congruent if two anglesand the side included betweenthem in one of the triangles are equal to the corresponding angles and the side included between them of the other triangle.
RHS Congruence of two right-angled triangles: Under a given correspondence, two right-angled triangles are congruent if the hypotenuse and a leg of one of the triangles are equal to the hypotenuse and the corresponding leg of the other triangle.
There is no such thing as AAA Congruence of two triangles: Two triangles with equal corresponding angles need not be congruent. In such a correspondence, one of them can be an enlarged copy of the other.
(They would be congruent only if they are exact copies of one another).

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Visualising Solid Shapes | Study
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The circle, thesquare, the rectangle, the quadrilateral and the triangle are examples of plane figures; the cube, the cuboid, the sphere, the cylinder, the cone and the pyramid areexamples of solid shapes. (Scroll down to continue …)
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Plane figures areof two-dimensions (2-D) and the solid shapes are of three- dimensions (3-D). The corners of a solid shape are called its vertices; theline segments ofits skeleton areits edges; and itsflat surfaces areits faces. A net is a skeleton-outline of a solid that can be folded to make it. The same solid can haveseveral types ofnets. Solid shapes can be drawn on a flat surface (like paper) realistically. We call this 2-D representation of a 3-Dsolid. Two types ofsketches of asolid are possible: (a) An oblique sketch does nothave proportional lengths. Still it conveys all important aspects of the appearance of the solid. (b) An isometric sketch is drawn on an isometric dot paper, a sample of which isgiven at theend of thisbook. In an isometric sketch of the solidthe measurements kept proportional. Visualising solidshapesis a veryuseful skill. Youshould be ableto see ‘hidden’ parts of thesolid shape. Different sections of a solid can be viewed in many ways: (a) One way is to viewby cutting or slicing the shape, whichwould result in the cross- section of thesolid. (b) Another way isby observing a 2-D shadow of a 3-Dshape. (c) A third wayis to lookat the shapefrom different angles; the front-view, theside- view and thetop view canprovide a lotof information aboutthe shape observed.
19. When a grouping symbol preceded by ‘ sign is removed or inserted, thenthe sign of eachterm of thecorresponding expression ischanged (from ‘ + ‘ to ‘−’ and from‘− ‘ to + ‘).

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Real Numbers | Study
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Introduction:
Euclid’s Division Lemma/Euclid’s Division Algorithm :
Given positive integers a and b, there exist unique integers q and r satisfying a=bq+r, 0 r<b.
This statement is nothing but a restatement of the long division process in which q is called the quotient and r is called the remainder. (Scroll down to continue …).
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Introduction:
Euclid’s Division Lemma/Euclid’s Division Algorithm:
Given positive integers a and b, there exist unique integers q and r satisfying a=bq+r, 0 r<b.
This statement is nothing but a restatement of the long division process in which q is called the quotient and r is called the remainder.
NOTE:
1. Lemma is a proven statement used for proving another statement.
2. Euclid’s Division Algorithm can be extended for all integers, except zero i.e., b 0.
HCF of two positive integers :
HCF of two positive integers a and b is the largest integer (say d ) that divides both a and b(a>b) and is obtained by the following method :
Step 1. Obtain two integers r and q, such that a=bq+r, 0r<b.
Step 2. If r=0, then b is the required HCF.
Step 3. If r0, then again obtain two integers using Euclid’s Division Lemma and continue till the remainder becomes zero. The divisor when remainder becomes zero, is the required HCF.
The Fundamental Theorem of Arithmetic :
Every composite number can be factorised as a product of primes and this factorisation is unique, apart from the order in which the prime factors occur.
Irrational Number :
A number is an irrational if and only if, its decimal representation is non-terminating and non-repeating (non-recurring).
OR
A number which cannot be expressed in the form of pq , q 0 and p, qI, will be an irrational number. The set of irrational numbers is generally denoted by Q.
NOTE:
1. The rational number pq will have a terminating decimal representation only, if in standard form, the prime factorisation of q, the denominator is of the form 2ⁿ5^m, where n, m are some non-negative integers.

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Polynomials | Study
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Introduction :
Polynomial :
Any expression of the form a₀xⁿ+a₁x^n-1+a₂x^n-2+….an is called a polynomial of degree n in variable x ; a₀≠0, where n is a non-negative integer and a₀, a₁, a₂, ….., and are real numbers, called the coefficients of the terms of the polynomial. (Scroll down to continue …)
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A polynomial in x can be denoted by the symbols p(x), q(x), f(x), g(x), etc.
Degree Of Polynomial: The highest power of x in p(x) is called the degree of the polynomial p(x).
Linear Polynomial : A polynomial of degree one is called a linear polynomial.
Quadratic Polynomial :
A polynomial of degree two is called a Quadratic Polynomial.
Generally, any quadratic polynomial in x is of the form ax²+bx+c, a ≠ 0 and a, b, c are real numbers.
Cubic Polynomial :
A polynomial of degree three is called a Cubic Polynomial.
Generally, any cubic polynomial in x is of the form ax³+bx²+cx+d, a≠0 and a, b, c, d are real numbers.
Value of a Polynomial :
If we replace x by ‘ -2’ in the polynomial p(x) = 3x³-2x²+x-1
we have p(-2) =3(-2)³-2(-2)²+(-2)-1
= -24-8-2-1 =-35
Thus, on replacing x by ‘ -2 ‘ in the polynomial p(x), we get -35, which is called the value of the polynomial.
Hence, if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of the polynomial p(x) at x=k, and generally, denoted by p(k).
Zeros of a Polynomial :
A real constant, k is said to be a zero of a polynomial p(x) in x, if p(k)=0
For example, the polynomial p(x) = x²+x-12 gives p(3)=3²+3-12=0 and p(-4)=(-4)²+(-4)-12=0.
Thus, 3 and -4 are two zeroes of the polynomial p(x).
A linear polynomial (degree one) has one and only one zero, given by;
Zero of the linear polynomial = -(constant term )coefficient of x
Geometrical Representation of the Zeroes of a Polynomial :
Let us consider a linear polynomial p(x)=3x-6.
We know that, graph of a linear polynomial is a straight line.
Therefore, graph of p(x)=3x-6 is a straight line passing through the points (1,-3),(3,3),(2,0).
Table for p(x)=3x – 6
From the graph of p(x)=3x-6, we observe that it intersects the x-axis at the point (2,0).
Zero of the polynomial [p(x)=3x-6] = -(-6)3 = 63 = 2.
Thus, we conclude that the zero of the polynomial p(x) = 3x – 6 is the x-coordinate of the point where the graph of p(x) = 3x – 6 intersects the x-axis.
Similarly, the zeroes of a quadratic polynomial, p(x) = ax²+bx+c, a≠0, are the x-coordinates of the points where the graph (parabola) of p(x)=ax²+bx+c, a≠0, intersects the x-axis.
Graph of p(x) = ax²+bx+c, a≠0 intersects the x-axis at the most in two points and hence the quadratic polynomial can have at most two distinct real zeros.
A cubic polynomial can have at most three distinct real zeros.
Relation between Zeroes and Coefficients of a Polynomial :
Let the quadratic polynomial be p(x) = ax²+bx+c, a≠0 and having zeroes as α and β, then
Sum of the zeroes = α + β
= -(coefficient of x) /(coefficient of x²) = -b/a
Product of the zeroes = αβ
Let the cubic polynomial be p(x) = ax³+bx²+cx+d, a≠0 and having zeroes as α , β and γ, then Sum of the zeroes = α + β + γ
α + β + γ = -(coefficient of x² )/(coefficient of x³)= -b/a
αβ = (constant term) /(coefficient of x²) = c/a
Sum of the products of zeroes taken two at a time αβ+βγ+γα
αβ+βγ+γα = (coefficient of x) /(coefficient of x³)= c/a
and
Product of the zeroes = αβγ
αβγ = (constant term) /(coefficient of x³)= -d/a
Division Algorithm for Polynomials :
For any two polynomials p(x) and g(x) ; g(x) ≠0, we can find two polynomials q(x) and r(x), such that p(x)=g(x) × q(x)+r(x).
Where r(x)=0 or degree of r(x) is less than the degree of g(x). Here, q(x) is called quotient, r(x) is called remainder, p(x) is called dividend and g(x) is called divisor. This result is known as a division algorithm for polynomials.

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