# Studying maths in the 6th  class successfully meaning that children take responsibility for their own learning and learn to apply the concepts to solve problems.

## 1. KNOWING OUR NUMBERS

### Comparing numbers:

##### • We can compare the numbers by counting the digits in the numbers.

• Now Compare   5432 and 4678…

5432 is greater as the digits at the ten thousand place in 5432 is greater than that in  4678.

Order of numbers:

##### • Ascending Order: –

arrange the numbers from smallest to the greatest; this order is called Ascending order.

Ex:- 23, 44, 65, 79, 100

##### • Descending Order: –

arrange the numbers from greatest to the smallest, this order is called Ascending order.

Ex:- 100,79, 65, 33, 23

Formations of numbers

• Form the largest and smallest possible numbers using the digits 3, 2, 4, 1 without repetition

• Largest number formed by arranging the given digits in descending order _ 4321.

• Smallest number formed by arranging the given digits in ascending order _ 1234.

• Greatest two-digit number is 99.

• Greatest three-digit number is 999.

• Greatest four-digit number is 9999.

### Place value

• Place value is the positional notation, which defines the position of a digit.

Ex:- 3458

8 is one place, 5 is tens place, 4 is hundreds place and 3 is thousands place.

Expanded form

• It refers to expand the numbers to see the value of each digit.

Ex :- 3458 = 3000 + 400 + 50 + 8

= 3×1000 + 4×100 + 5×10 + 8×1

• Note:-

1 hundred = 10 tens

1 thousand = 10 hundreds

1 lakh = 100 thousands = 1000 hundreds

### Reading and Writing the numbers

Place value table for Indian system :

Example: Represents the number in 6,35,21,892 in place value table

Place value table for International system :

Ex:- represents the number in 635,218,924 in place value table

### Use of commas:

• Indian system of numeration:- in the Indian system of numeration we use ones, tens, hundreds, thousands, lakhs and crores. The first comma comes after three digits from the right, the second comma comes two digits latter and the third comma comes after another two digits.E

Ex:-  “three crores thirty-five lakh seventeen thousand four hundred thirty” can be written as.3,35,17,430

• International system of numeration:- in the International system of numeration we use ones, tens, hundreds, thousands, millions and billions.

Ex:- “ six hundred thirty-five million two hundred eighteen thousand nine hundred twenty-four” can be written as 635,218,924.

Note:-10 millimetres = 1centimeter

100 centimetres = 1 meter

1000 meters = 1 kilometer

1000 milligrams = 1 gram

1000 grams = 1 kilo gram

## 2. WHOLE NUMBERS

Natural numbers: All the counting numbers starting from 1 are called Natural numbers.

1, 2, 3… Etc.

Successor and Predecessor: If we add 1 to any natural number, we get the next number, which is called the Successor. If we subtract 1 from any natural number, we get the previous number, which is called Predecessor.

Ex: – successor of 23 is 24 and predecessor of 32 is 31.

Note:- There is no predecessor of 1 in natural numbers.

Whole numbers: Whole numbers are the collection of natural numbers.

0, 1, 2, 3 …

Representation of whole number on the number line:

• Draw a line mark a point on it.

• Label it as ‘0’

• Mark as many points at equal distance to the right of 0.

• Label the points as 1, 2, 3, 4, … respectively.

• The distance between any two consecutive points is the unit distance.

•  The distance between 2 and 4 is 2 units, like as the distance between 2 and 6 is 4 units
• The number on the write is always greater than the number on the left
• The number on the left of any number is always smaller than that number

Addition of the whole number can represent on the number line

Ex:-  3 + 2 = 5

Start from three, we add 3 to 2. We make two jumps to the right of the number line as shown above. We reach at 5.

### Subtraction on the number line:

Subtraction of the whole number can be represented on the number line

Ex :-5 – 3 = 2

Start from 5, we subtract 3 from 5. We make three jumps to the left of the number line shown as above. We reach at 2.

Multiplication on the number line:

For multiplying 2 and 3, start from 0, make 2 jumps using 3 units at a time to the right, as you reach to 6. Thus, 2 × 3 =6.

Properties of whole numbers

Closer property: Two whole numbers are said to be closed if their operation (+, -, ×,÷) is always closed.

Ex: 3, 2 are whole numbers ⟹ 3 + 2 = 5 ( 5 is whole number)

Subtraction:- Whole numbers are not closed under subtraction as their difference not always a whole number.

Ex:- 2 – 3 = −1 ( −1 is not a whole number)

Multiplication:- Whole numbers are closed under multiplication.

Ex:- 3 × 2 = 6, 6 is a whole number.

Division:- Whole numbers are not closed under division, as their division is not always a whole number.

Ex:-  3 ÷ 2 is not a whole number.

Commutative property: Two whole numbers are said to be commutative if the result is the same when we change their position.

Ex: 3, 2 are whole numbers ⟹ 3 + 2 = 5 and 2 + 3 = 5 ( 3 + 2 = 2 + 3).

Subtraction:- Whole numbers are not commutative under subtraction.

Ex:- 2 – 3 = −1 and 3 – 2 = 1( 2 −3 ≠ 3 – 2 ).

Multiplication:- Whole numbers are commutative under multiplication.

Ex:- 3 × 2 = 6 and 2 ×3 = 6 (3 × 2 = 2 ×3)

Division:- Whole numbers are not commutative under division.

Ex:-  3 ÷ 2  ≠ 2 ÷ 3.

Associative property: For any three whole numbers a, b and c if (a ⨀ b)⨀ c = a ⨀ (b ⨀ c), then whole numbers are associative under operation ⨀. [ ⨀ = +, –, × and ÷ ].

Ex: ( 3 + 2) + 5 = 10 and  3 + (2 + 5) = 10 ⟹ ( 3 + 2) + 5 =   3 + (2 + 5)

Subtraction:- Whole numbers are not associative under subtraction.

Ex:- : ( 3 − 2) − 5 = −4  and  3 − (2 − 5) = 6 ⟹ ( 3 + 2) + 5 ≠3 + (2 + 5)

Multiplication:- Whole numbers are associative under multiplication.

Ex:- (3 × 2) ×5 = 30 and 3 ×(2 × 5) = 30  ⟹  (3 × 2) ×5 =  3 ×(2 × 5)

Division:- Whole numbers are not associative under division.

Ex:-  ( 3 ÷ 2) ÷ 5 ≠3 ÷ (2 ÷ 5).

Distributive property:

For any three whole numbers a, b and c, a×(b + c) = (a × b) +( a × c).

Note :Division by zero is not defining.

2 +0 = 2, 5 + 0 = 5 and so on.

Thus, 0 is the additive identity.

2 ×1 = 2, 4 × 1 = 4 and so on.

Thus, 1 is a multiplicative identity.

Patterns:

• Every number can be arranged as a line. The number 2 is shown as

The number 3 as shown as

• Some numbers can be shown as rectangles. 8 can be shown as

• Some numbers can be arranged as squares. 9 can be shown as

• Some numbers can be shown as triangles.

3 can be shown as                         6 can be shown as

## 3. PLAYING WITH NUMBERS

Divisibility Rule:

The process of checking whether a number is divisible by a given number or not without actual division is called divisibility rule for that number.

Divisibility by 2:- a number is divisible by 2 if its once place is either 0, 2, 4, 6 or 8.

Ex:- 26 is divisible by 2. 35 not divisible by 2.

Divisibility by 3:- if the sum of the digits of a number is divisible by 3, then that number is divisible by 3.

Ex:- 231 → 2 + 3 +1 =6, 6 is divisible by 3

∴ 231 is divisible by 3

436 → 4 + 3 + 6 = 13, 13 is not divisible by 3

∴ 436 is not divisible by 3.

Divisibility by 4:- if the last two digits of a number is divisible by 4, then that number is divisible by 4.

Ex:- 436, 36 is divisible by 4            ∴ 436 is divisible by 4

623, 23 is not divisible by 4      ∴ 623 is not divisible by 4.

Divisibility by 5:- a number is divisible by 5, if its once place is either 0 or 5.

Ex:- 20, 25 are divisible by 5. 22, 46 are not divisible by 5.

Divisibility by 6:- a number is divisible by 6, if it is divisible by both 3 and 2.

Ex:-  242 is divisible by both 2 and 3     ∴ 242 is divisible by 6

232 is divisible by 3 but not 2        ∴ 232 is not divisible by 6

Divisibility by 8:- if the last three digits of a number is divisible by 8, then that number is divisible by 8.

Ex:- 4232, last three digits 232 are divisible by 8

∴ 4232 is divisible by 8.

Divisibility by 9:- if the sum of the digits of a number is divisible by 9, then that number is divisible by 9.

Ex:-  459, 4 + 5 + 9 = 18 → 18 is divisible by 9       ∴ 459 is divisible by 9

532, 5 + 3 + 2 = 10 → 10 is not divisible by 9       ∴ 532 is not divisible by 9.

Divisibility by 10:- a number is divisible by 10 if its once place is 0.

Ex:- 20 is divisible by 10. 22, 45 are not divisible by 10.

Divisibility by 11:- A number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of the digits at even places is either 0 or 11.

Ex:- 6545

Sum of the digits at odd places = 5 + 5 = 10

Sum of the digits at even places = 4 + 6 = 10

Now difference is 10 – 10 = 0

∴ 6545 is divisible by 11.

Factors: a number which divides the other number exactly is called a factor of that number.

6 = 1×6

= 2×3      ⟹  factors of 6 are: 1, 2, 3 and 6

Note- 1)1 is a factor of every number.

2) Every number is a factor of itself.

3) Every factor is less than are equal to the given number.

4) Factors of a given number are countable.

Prime numbers: The numbers, which have only two factors 1, and itself are called prime numbers.

2, 3, 5, 7, …. Are prime numbers

Composite numbers: The number, which has more than two factors are called composite numbers.

4, 6,8,9….. are composite numbers.

• Note: – 1) 1 is neither prime nor composite

2) 2 is the smallest prime number

3) 4 is the smallest composite number.

Co – prime number: The number which has no common factor except 1 is called co-prime number.

Ex:- (2, 3), (4,5) ……

Twin – primes: If the difference of two prime numbers is 2, then those numbers are called twin prime numbers.

Ex:- (2,3), (3,5), (17,19)…..

Factorization: When a number is expressed as the product of its factors, we say that the number has been factorized. The process of finding the factors is called Factorisation.

Ex:-  factors of 24 are: 1, 2, 3, 4, 6, 8, 12 and 24

24 = 1 × 24 = 2 × 12 = 3 × 8 = 4 × 6

Prime factorisation: The process of finding the prime factors is called prime factorisation.

Ex:- 24 = 2 × 12

2 × 3 × 4

2 × 3 × 2 × 2

∴ Prime factorisation of 24 is 2 × 2 × 2 × 3.

Methods of prime factorization:

Division method:- Prime factorization of 12 using the division method,

fallow the procedure.

Start dividing by the least prime factor. Continue division till the resulting number to be divided is 1.

The prime factorization of 12 is 2 × 2 × 3.

Factor tree method:- To find the prime factorization of 24, using the factor tree method we proceed as follows:

• Express 24 as a product of two numbers.
• Factorise 4 and 6 further, since they are composite numbers.
• Continue till all factors are prime numbers.
•  The prime factorization of 24 is 2 × 2 × 2 × 3.

Common factors: Common factors are those numbers, which are factors of all the given numbers.

Ex:- 12, 9

Factors of 12 are:  1, 2, 3, 4, 6 and 12

Factors of 9 are:  1, 3 and 9

∴ Common factors of 12, 9 are 1,3

Highest Common Factor (H.C.F):- The highest common factor of two or more numbers is the highest of their common factors. It is also called ad Greatest Common Divisor(G.C.D).

Ex:- H.C.F of 12, 9

Factors of 12 = 1, 2, 3,4, 6, 12

Factors of 9 = 1, 3, 9

Common factors of 12, 9 = 1,3

Highest common factor is 3

∴ H.C.F of 12, 9 is 3

### Method of finding HCF: Prime factorization method:

The HCF of 9 , 12 can be found by the prime factorization method as follows.

9 = 3  × 3

12 =3 ×  2× 2

The common factor of 12, 9 is 3

∴ H.C.F of 12, 9 is 3

Continue division method:

Euclid invented this method. Divide the larger number by smaller and then divide the previous divisor by the remainder until the remainder zero. The last divisor is the HCF of given numbers.

Common multiple multiples of 3 are 3, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39,42…

Multiples of  4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52….

Common multiples of 3 and 4 are  12, 24, 36….

Least common multiple (LCM):- The least common multiple of two or more given numbers is the lowest of their common multiple.

Ex:- LCM of 3 and 4

Multiples of 3 =    3, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39,42…

Multiples of 4 =   4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52….

Common multiples of 3 and 4 =  12, 24, 36….

∴ LCM of 3, 4 is 12.

Methods of finding LCM:

1.    Prime factorization method:-  the LCM of  6, 15 by using prime factorization method is as follows:

i)  Express each number as the product of prime factors

Prime factors of 6 =   2 × 3

Prime factors of 15 = 5× 3

ii)  Take the common factors both: 3

iii)  Take the extra factors of both 6 and 15 i.e., 2 and 5

iv)  The product of all common factors of two numbers and extra common factors of both finds LCM.

∴ LCM of 6 and 15 = (3) × 2 × 5 = 30.

2.    Division method:- To find LCM of 6 and 15:

i.    Arrange the given numbers in a row.

ii.  Then divide the least prime number, which divides at least two of the given numbers, and carry forward the numbers, which are not divisible by that number if any.

iii.Repeat the process till no numbers have a common factor other than 1.

iv. LCM is the product of the divisors and the remaining numbers.

Ex:-

∴ LCM of 6, 15 = 3 × 2 × 5 = 30.

Note:-  the product of LCM and HCF of given numbers = the product of given numbers.

Ex:- 6 × 15 = 30 × 3 = 90.

## 4.BASIC GEOMETRICAL IDEAS

The term ‘geometry’ is derived from the Greek word ‘geometron’.

Geo means Earth and metron means measurement.

Point: Point is a location or position on the surface of the plane. It is denoted by capital letters of the English alphabet.

Line: It is made up of infinitely many points with infinity length.

It is denoted by

Ray: Ray is a part of a line. It begins at a point and goes on endlessly n a specific direction.

It is denoted by
Line segment: It is a part of the line with the finite length.

It is denoted by

Intersecting lines: If two lines are meeting at the same point, then those lines are called intersecting lines. That pint is called the point of intersection.

Parallel lines: The lines, which are never meet at any point, are called parallel lines.

Curve: Anything, which is not straight, is called Curve.

Simple curve: – A curve that does not cross itself.

Open curve: – A curve in which its endpoints do not meet.

Closed curve: – A curve that has no endpoint is called a closed curve.

∗ A closed curve has three parts

The Interior of the curve: – It refers to the inside area of the curve. (B)

The exterior of the curve: – It refers to the outside area of the curve. (A)

On the curve: – It refers to the inside area of the curve. (C)

Polygon: – polygon is a simple closed figure made by line segments.

Angle: the figure formed by two rays having a common end is called an angle.

Here two rays OA, OB are arms of the angle

O is the Vertex. It is denoted by ∠AOB or ∠ BOA.

Triangle: A simple closed figure formed by the three line segments is a triangle. The line segments are called sides of the triangle.

• AB, BC and AC are sides of a triangle.
• A,  Band C are vertices of a triangle.
• ∠ABC, ∠BAC and ∠ACB are angles of the triangle.
• This triangle is denoted by ∆ABC.

Quadrilateral: A simple closed figure formed by the four-line segments is a Quadrilateral.

• AB, BC, CD and DA are the sides of the quadrilateral.
• A, B, C and D are the vertices of the quadrilateral.
• ∠A, ∠B, ∠C and ∠D are the angles of quadrilateral.
• AB, DC and BC, AD are opposite sides of the quadrilateral.
• AB, BC; AD, DC; DC, BC and AD, AB are adjacent sides( the sides which have common vertex are called adjacent sides)
• A, C and B, D are opposite vertices and also opposite angles.
• AC and BD diagonals of a rectangle (A line segment joining opposite vertices is called diagonal).

Circle: The set of points that are at a constant distance from a fixed point is called a circle. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle.

• O is the center of the circle.
• OA, OB, and OC radii of the circle
• AB is the diameter of the circle.
• PQ is a chord.

Circumference of the circle: – the length of the boundary of the circle is called the circumference of the circle.

Arc: – The part of the circumference is called Arc. From the above fig.  is an arc.APisarc.

Sector: – Region enclosed by an arc and two radii is called a sector.

Segment: – The region enclosed by arc and chord is called a segment of the circle.

## 5. MEASURES OF LINES AND ANGLES

Measure of line segment:

• A line segment is a part of the line with two endpoints.
• This makes it possible to measure a line segment.
• This measure of each line segment is its ‘length’.
• We use length to compare line segments.
• We can compare the length of two line segments by: (i) simple observation (ii) tracing on a paper and (iii) using instruments.

Simple observation: – We can tell which line segment is greater than other just by observing the two-line segments but it is not sure.

Here we can clearly say that CD > AB but sometimes it is difficult to tell which one is greater.

1. Tracing on a paper: – In this method we have to trace one line on paper then put the traced line segment on the other line to check which one is greater.

But this is a difficult method because every time to measure the different size of line segments we have to make a separate line segment.

Comparing by instruments: – To compare any two-line segments accurately, we use ruler(scale) and divider.

∗ We can use a ruler to measure the length of a line segment.

Put the zero mark at point A and then move toward l to measure the length of the line segment, but it may have some errors based on the thickness of the ruler.

∗ This could be made accurate by using a Divider

• Put the one end of the divider on point A and open it to put another end on point B.
• Now pick up the divider without disturbing the opening and place it on the ruler so that one end lies on “0”.
• Read the marking on the other end and we can compare the two line.

Measure of an angle: Angle is formed two rays or two-line segments.

• We can understand the concept of right and straight angles by directions.
• There are four directions-North, South, East and West.
• When we move from North to East then it forms an angle of 90°, which is called Right Angle.

• When we move from North to South then it forms an angle of 180°, which is called Straight Angle.
• When we move four right angles in the same direction then we reach to the same position again i.e. if we make a clockwise turn from North to reach to North again then it forms an angle of 360°, which is called a Complete Angle. This is called one revolution.

∗ In a clock, there are two hands i.e. minute hand and hour hand, which moves clockwise in every minute. When the clock hand moves from one position to another then turns through an angle.

• When a hand starts from 12 and reaches to 12 again then it is said to be completed a
• s were the ray moves in the opposite direction of the hands of a clock are called anti – clockwise angles. These are denoted by positive measure.
• Angles were the ray moves in the direction of the hands of a clock are called clockwise angles. These are denoted by negative measure.

The protractor:

• By observing an angle we can only get the type of angle but to compare it properly we need to measure it.
• An angle is measured in the “degree”. One complete revolution is divided into 360 equal parts so each part is one degree. We write it as 360° and read as “three hundred sixty degrees”.
• We can measure the angle using a ready to use device called Protractor.
• It has a curved edge, which is divided into 180 equal parts. It starts from 0° to 180° from right to left and vice versa.

∗To measure an angle 72° using protractor-

• Place the protractor on the angle in such a way that the midpoint of protractor comes on the vertex B of the angle.
• Adjust it so that line BC comes on the straight line of the protractor.
• Read the scale, which starts from 0° coinciding with the line BC.
• The point where the line AB comes on the protractor is the degree measure of the angle.

Hence, ∠ABC = 72°.

Types of angles:

 Type of angle Measure Zero angle 0° Right angle 90° Straight angle 180° Complete angle 360° Acute angle Between 0° to 90° Obtuse angle Between 90° to 180° Reflex angle Between 180° to 360°

Perpendicular Lines

If two lines intersect with each other and form an angle of 90° then they must be perpendicular to

## 6.INTEGERS

There several situations in our daily life, where we use these numbers to represent loss or profit; past or future; low or high temperature. The numbers on the left side of zero are called negative numbers.

Integers: The numbers which are positive, zero and negative numbers together are called as integers and they are denoted by I or Z.

Z = {…, -3, -2, -1, 0, 1, 2, 3…}.

Representation of integers on a number line: –

• The numbers which are on the right side of zero are positive numbers and which are on the left side of zero are negative numbers.
• 0 is neither positive nor negative.
• On a number line, the number increases as we move to right and decrease as we move to the left.

∴ -3 < -2 <   -1 <   0   < 1 <   2   < 3 <   4   <  5  so on.

• Note: – 1. Any positive integer is always greater than any negative integer
1. Zero is less than every positive integer.
2. Zero is greater than every negative integer.
3.   Zero doesn’t come in any of the negative and positive integers.

1. If two integers have same sign, then add the integers and put that sign before the result.

Ex: – 3 + 2 =5, −3 – 2 = −5.

1. If two integers have different sign, then subtract smaller one from bigger and put the bigger one sign before the result.

Ex: – 3 − 2 =1, −3 + 2 = −1, −10 + 5 = −5.

Addition of integers on a number line:

• On the number line, we first move three steps to the right of 0 to reach 3, then we move 4 steps to the right of 3 and to reach 7

∴ 3 + 4 = 7

• On the number line, we first move three steps to the left of 0 to reach −3, then we move 4 steps to the left of −3 and to reach −7.

∴ − 3 − 4 = −7

∗ Any two distinct numbers that give zero when added to each other are additive inverse each other.

Subtraction of integers on a number line:

Subtract 3 from 6

• On the number line, we first move 6 steps to the right of 0 to reach 6, then we move 3 steps to the left of 6 and to reach 3.

∴ 6 − 3 = 3.

Subtract −3 from 6

On the number line, we first move 6 steps to the right of 0 to reach 6. For – 3 we have to move left but for – ( −3) we move in the opposite direction. Thus, we move 3 steps to the left of 6 and to reach 9.

∴ 6 – (−3) = 9.

• Subtraction of integers is the same as the addition of their additive inverse.

## 7. FRACTIONS AND DECIMALS

A fraction means a part of a group of a whole.

The ‘whole’ here could be an object or the group of objects. But all the parts of the whole must be equal. The ‘whole’ here could be an object or the group of objects. However, all the parts of the whole must be equal.

• Fig(i) is the whole. The complete circle.

• In Fig (ii), we divide the circle into two equal parts, then the shaded portion is the half ie., of the circle.

• In Fig (iii), we divide the circle into three equal parts, then the shaded portion is the one third of the circle i.e., of the circle.

• In Fig (iv), we divide the circle into four equal parts, then the shaded portion is the one fourth of the circle i.e., of the circle.

The numerator and the denominator:

The upper part of the fraction is called ‘numerator’. It tells the no. of parts we have.

The lower part of the fraction is called ‘denominator’. It tells the total parts in whole.

Representing fractions pictorially:

Representing fractions on a number line:

Mark on a number line

Proper fractions: In a fraction if the numerator is less than denominator then, then it is called proper fraction. If we represent a proper fraction on a number line then it is always lies between 0 and 1.

Ex: –

Improper fractions: In a fraction if the numerator is greater than denominator then, then it is called improper fraction.

Ex: –

Mixed fractions: – The fraction made by the combination of whole number and a part is called mixed fraction.

Ex: –

Note: Only improper fractions can be represented as mixed fractions.

A mixed fraction is in the form of

We can convert it into improper fraction by

Ex: –

Equivalent fractions: – Equivalent fractions those fractions which represent the same part of whole.

• Equivalent fractions are arising when we multiply both the numerator and denominator by the same number.
• Equivalent fraction of are    and so on

Standard form of a fraction (simplest or lowest form):- A fraction is said to be in standard form if both the numerator and denominator of that fraction have no common factor except 1.

Ex: –

Like and Unlike fractions: The fractional numbers that have the same denominators are called fractional numbers and have not the same denominator are called unlike fractions.

Ex: –   are like fractions and  are un like fractions.

Comparing fractions:

Like fractions: – We have to compare the like fractions with the numerator only, because the like fractions have same denominator. The fraction with greater numerator is greater and the fraction with smaller numerator is smaller.

Ex: – and so on.

Unlike fractions: –

With same numerator: For comparing unlike fractions, we have to compare denominators when the numerator is same. The fraction with greater denominator is smaller and the fraction with smaller denominator is smaller.

Ex: – and so on.

Note: – To find the equivalent fractions of both the fractions with the same denominator, we have to take the LCM of their denominators.

Ascending order and Descending order: –

When we write numbers in a form that they increase from the left to right then they are in the Ascending order. When we write numbers in a form that they decrease from the left to right then they are in the Descending order.

Ex: – For fractions: are in ascending order and are in descending order.

Like fractions: –

Ex:

Un like fractions: – For adding unlike fractions, first we have to find the equivalent fraction of given fractions and then add them as like fractions.

Ex: –

Subtraction of fractions

Like fractions: –

Ex: –

Un like fractions: – First we have to find the equivalent fraction of given fractions and then subtract them as like fractions.

Ex: –

Decimal fractions:

A fraction where the denominator is a power of ten is called decimal fraction. We can write decimal fraction with a decimal point (.). it makes easier to do addition, subtraction and multiplication on fractions.

Ex: –

## 8. DATA HANDLING

Data: collection of information in the form of numbers or words is called data.

Recording data: Recording of data depends on the requirement of the data. We can record data in different ways.

Organization of data: –

• Data is difficult to read.
• We have to organize it.
• Data can be organized in a tabular form.
• Data is represented in tabular form using frequency distribution and the tally marks.
• Frequency tells the no. of times the observations is happened.
• Tally marks show the frequency of the data.

∗ Example for representing tally marks:

Pictograph:

If the data is represented by the picture of objects instead if numbers, then it is called pictograph. Pictures make it is easy to understand the data and answer the questions to related it by observing the pictures.

Example for representing data by pictograph

• Drawing a pictograph is difficult to draw some difficult pictures.
• For understanding every one, e must use proper symbols.

Bar graph:

• Bar graphs are used to represent the independent observations with frequencies.

• In a bar graph, bars of uniform width are drawn horizontally or vertically with equal spacing between them.

Construction of bar graph: –

Steps to construction: –

1.Draw two perpendicular lines one horizontal (x – axis) and one vertical (y – axis).

2.Along the x- axis mark ‘items’ and along the x – axis mark ‘cost of items’.

3.Select a suitable scale 1cm = 10(rupees).

4.Calculate the heights of the bars by dividing the frequencies with the scale

70 ÷ 10 = 7, 40 ÷ 10 = 4 and so on.

5.Draw rectangular vertical bars of same width on the x- axis with heights calculated above.

## 9. INTRODUCTION TO ALGEBRA

Algebra is the use of letters or symbols to represent number. It helps us to study about un known quantities.

Patterns:

To make a triangle, 3 matchsticks are used

For making 2 triangles we have six matchsticks

For making 3 triangles we have nine matchsticks

• of matchsticks required for making 1 triangle = 3 = 3 × 1
• of matchsticks required for making 2 triangles = 6 = 3 × 2
• of matchsticks required for making 3 triangles = 9 = 3 × 3

Thus the no. of matchsticks for making ‘n’ triangles = 3 × n = 3n.

Variable: Variable is a unknown quantity that may change. It is a dependent term.

In the above pattern, the rule is 3n, here ‘n’ is the variable.

• We can use lower case alphabets are used as variable.
• Numbers cannot use as variables, since they have fixed value.
• Variables help us to solve other problems also.
• Variables can take different values; they have no fixed value.
• Mathematical operations addition, subtraction, multiplication and division can be done on the variables.

Use of variables:

perimeter of a polygon is the sum of the lengths of all its sides.

Perimeter square = 4s, s is the variable

Perimeter of rectangle = 2 (l + b); l, b are variables.

To find the nth term from the given pattern: 3, 6, 9…

 Number 3 6 9 12 15 … Pattern 3×1 3×2 3×3 3×4 3×5 …

From the table we observe that, the first number is 3×1, the second number is 3×2, the third number is 3×3 and so on.

∴ the nth term of pattern 3, 6, 9, 12, = 3n, here n is variable.

Simple equation: simple equation is a condition to be satisfied by the variables. Equation has equality sign between its two sides.

Ex: 5m = 10, 2x + 1 = 0 etc.

L.H.S and R.H.S of an equation:

The expression which is at the left of equal sign of an equation is called Left Hand Side (L.H.S)

The expression which is at the right side of equal sign of an equation is called Right Hand Side (R.H.S)

Ex: 4y = 20

L.H.S = 4y and R.H.S = 20

Solution of an equation (Root of the equation):

Solution or Root of an equation is the values of variable for which L.H.S and R.H.S are equal.

Ex: 3x = 15

If x = 5; LHS = 3×5 = 15

RHS = 15

∴ solution of above equation is 5

Trial and error method:

By using this method, we get the solution of given equation.

Ex: solve 2n = 10

 Substituting value of n Value of L. H. S Value of R. H. S Whether LHS and RHS are equal 1 2×1 = 2 10 Not equal 2 2×2 = 4 10 Not equal 3 2×3 = 6 10 Not equal 4 2×4 = 8 10 Not equal 5 2×5 = 10 10 Equal

When n = 5, LHS = RHS ∴ solution of equation is 5.

## 10. PERIMETER & AREA

Perimeter: Perimeter is the distance covered along the boundary forming a closed figure when you go around the figure once.
Perimeter of a Rectangle:

Length of the rectangle = l, breadth = b

Perimeter of rectangle = sum of the lengths of its sides.

= l + l +b + b

P = 2 (l + b) units.

Perimeter of a Square:

Length of the side  of a square   = a

Perimeter of rectangle = sum of the lengths of its sides.

= a + a + a + a

= 4a units.

Perimeter of an Equilateral Triangle:

Length of each side   = a

Perimeter of rectangle = sum of the lengths of its sides.

= a + a + a

= 3a units.

Polygon: A polygon is a simple closed figure bounded by line segments.

Regular polygon: A polygon which has equal side and equal angles, is called Regular polygon.

The perimeter of regular polygon of ‘n’ sides whose length ‘a’ = na.

Area: The amount of surface enclosed by a closed figure is called its area.

Area of a Rectangle:

Length of the rectangle = l, breadth = b

Area of the Rectangle = l × b square units.

Area of a Square:

Length of the side of a square   = a

Area of a Square = a × a = a2 square units.

Note:  The area of the square is more than the area of any other rectangle having the same perimeter.

## 11. RATIO AND PROPORTION

Ratio:  Ratio is the comparison of two quantities of same kind.

The ratio of two quantities a and b is written as a: b and read as ‘a is to b’.

‘a’ is called first term or antecedent and ‘b’ is called second term or consequent.

Simplest form of ratio:

If a ratio is written in terms of whole numbers with no common factors other than 1, then the ratio is said to be in the ‘simplest form’ or in the ‘lowest terms’.

Ex: the simplest form of 5 : 15 is 1 : 3.

Division of a given quantity in a given ratio:

Let us suppose that, if a quantity ‘c’ divided into two parts in the ratio a: b, then

Total parts = a + b

First part =and second part =

Ex: Divide 1200 in the ratio 2 : 3

Ans: Total parts = 2 + 3 = 5

First part = = 2 × 240 = 480

Second part =  = 3× 240 = 720.

Proportion:

Equalities of ratios is called proportion.

If a : b = c : d, then a ,b ,c and dare in proportion. This is represented as a : b ∷ c : d.

If a, b, c and d are in proportion, then ad = bc.

Unitary method:

In this method, first we find the value of one unit and then the value of the required number of units.

Ex: If the cost of 5 pens is ₹ 20, then find the cost of 12 pens.

Sol:  Given that cost of 5 pens = ₹20

Cost of one pen = 20 ÷ 5= 4

Cost of 12 pens = 4 × 12 = 48

∴ cost of 12 pens = ₹ 48.

## 12. SYMMETRY

Symmetry:

The word symmetry comes from Greek word. It means ‘to measure together’.

Symmetry is the mirror image of an object.

Symmetry means that one object becomes exactly like another when we move it in some way: turn, flip or slide.
Ex:

Line of symmetry:

A line along which you can fold a figure so that two parts of it coincide exactly is called a ‘line of symmetry’.

Line of symmetry can be horizontal, vertical or diagonal.

Ex:

The English alphabet which have

• Vertical line of symmetry: A, H, I, M, O, U, V, W and X
• Horizontal line of symmetry: B, C, D, E, H, I, K, O and X
• No line of symmetry: D, G, J, L, N, P, Q, R, S, Y AND Z.

## 13. PRACTICAL GEOMETRY

The following instruments from a geometry box are used to construct figures:

1.A Ruler (Scale)

2.The compasses

3.The divider

4.Protractor

∗ The ruler is used to measure lines.

∗ The compasses is used for constructing.

∗ The protractor is used for measuring angles.

∗ Divider is used to make equal line segments or mark point on a line.

Construction of a line segment of a given length:

We can construct a line segment in two ways: 1) By using Ruler 2) By using the Compasses

1.By using Ruler: –

Let us suppose we want to draw a line segment AB of length 3.5 cm

Steps of construction: –

Step-1: Place the ruler on a paper and hold it firmly.

Step-2: Mark a point with sharp edged pencil against ‘0cm’ mark of the ruler.

Step-3: Name the point as A. Mark another point against 5 small divisions just after the 3cm mark. Name this point as B

Step-4: Join A and B along the edge of the ruler. AB is the required line segment of length 3.5cm.

1.By using the Compasses: –

Let us suppose we want to draw a line segment AB of length 3.5 cm

Steps of construction: –

Step-1: draw a line l. Mark a point on the line l.

Step-2:
place the metal pointer of the compasses on the zero mark of the ruler. open the compasses so that the pencil point touches the 3.5cm mark on the ruler.

Step-4: on the line l, we got the line segment AB of length 3.5cm.

Step-3:
place the pointer on A on the line l and draw arc to cut the line. Mark the point where the arc cuts the line as B.

Construction of a circle:

Let us suppose we want to draw a circle of radius 3 cm

Steps of construction: –

Step-1: Open the compasses for radius 3 cm

Step-2: Mark a point with sharp edged pencil. This is the centre.

Step-3: Place the pointer of the compasses firmly at the centre.

Step-4: Without moving its metal point, slowly rotate the pencil and till it come back to the straight point.

Construction of perpendicular bisector a line segment:

Steps of construction: –

Step-1: Draw a line segment AB.
Step-2:
Set the compasses as radius more than half of the length of line segment AB.
Step-3: With A as centre, draw arcs below and above the line segment

Step-4: With same radius and B as the centre draw two arcs above and below the line segment to cut the previous arcs. Name the intersecting points of arcs as M and N.

Step-5: Join the points M and N. then, the line MN is the required perpendicular bisector of the line segment AB.

Construction of perpendicular to a line, through a point which is not on it:

Steps of construction: –

Step-1: Draw a line l and a point A not on it

Step-2: With A as centre draw an arc which intersects the given line at two points M and N.

Step-3: Using the same radius and with M and N as centres construct two arcs that intersect at a point B on the other side of the line.

Step-4: Join A and B. AB is the perpendicular of the given line l.

Construction of Angles using Protractor:

Let us suppose we want to construct ∠ABC = 500

Steps of construction: –
Step-1:
Draw a ray BC of any length.

Step-2: Place the centre point of the protractor at B and the line aligned with the

Step-3: Mark a point A at 500

Step-4: join AB. ∠ABC is the required angle.

Constructing a copy an of Angle of un known measure:

Let ∠A is given, measure is not known

Steps of construction: –
Step-1:
Draw a line and choose a point A on it.

Step-2: Now place the compasses at A and draw an arc to cut the rats AC and AB.

Step-3:  Use the same compasses setting to draw an arc with P as centre, cutting l at Q.

Step-5: Place the compasses pointer at Q and draw an arc to cut the existing arc at R.

Step-6: Join PR. It has the same measure as ∠BAC.

Construction to bisect a given angle:

Let an angle say ∠AOB be given

Steps of construction: –

Step-1: With O as the centre and ray convenient radius, draw an arc PQ cutting OA and OB at P and Q respectively.

Step-2: With P as the centre and any radius slightly more than half of the length of PQ, draw an arc in the interior of the given angle.

Step-3: With Q as the centre and without alternating radius draw another arc in the interior of ∠AOB.

Let two arcs intersects at S

Step-4: Draw ray , then is the bisector of ∠AOB

Observe ∠AOS = ∠SOB

CONSTRUCTION ANGLES OF SPERCIAL MEASURES:

Construction of 600 angle: –

Steps of construction: –

Step-1: Draw a line l and mark a point O on it.

Step-2: Place the pointer at O and draw an arc of convenient radius which cuts the line at P (say).

Step-3:  With the centre P and the same radius as in the step-2. Now draw an arc that passes through O.

Step-4:  Let the two arcs intersects at Q. Join OQ. We get ∠POQ = 600.

Construction of 1200 angle: –

Steps of construction: –

Step-1: Draw a ray OA

Step-2: Place the pointer of the compasses at O. With O as the centre and any convenient radius draw an arc cutting OA at P.

Step-3: With P as the centre and the same radius as in the step-2 draw an arc which cuts the first arc at Q.

Step-4: With Q as the centre and the same radius as in the step-2 draw an arc which cuts the first arc at R.

Step-5: Join OR. Then ∠POR = 1200.

## 14. UNDERSDTANDING 3D AND 2D SHAPES

Cuboid:
3D- shapes or Solids:

The object which have a length, breadth and height (or depth) are called ‘three dimensional’ or ‘3D- shapes’ or ‘Solids’.

Cuboid:

Objects like match boxes, erasers are the examples for cuboid

A cuboid has 6- Faces, 8- Vertices and 12 – Edges

Cube:

A dice is an Example for cube.

A cube has 6- Faces, 8- Vertices and 12 – Edges

Cylinder:

Objects like wooden log, a piece of pipe are the examples for cylinder

the top and base of the cylinder are circular in shape.

Cone:

joker cap is the example for cone

base of the cone is a circle.

Sphere:

Balls, laddoos are the examples for globe.

Triangular prism:

If the base of a prism is triangle, then it is called triangular prism.

Pyreamid:

A pyramid is a solid shape with a base and point vertex.

If a pyramid has triangular base, then it is called triangular pyramid

If a pyramid has  square base, then it is called square pyramid.

Polygon: A polygon is a closed figure made with linesegments.

RegularPolygon: A polygon with all equal sides and all equal angles is called a regular polygon.

## 3.SQUARES AND SQUARE ROOTS, CUBES AND CUBE ROOTS

Square:  Square number is the number raised to the power 2. The number obtained by the number multiplied by itself.

Ex: – 1) square of 5 = 52 = 5 × 5 = 25, 2) square of 3 = 32 = 3× 3 = 9

∗If a natural number p can be expressed as q2, where q is also natural, then p is called a square number.

Ex: – 1,4,9, …etc.

Test for a number to be a perfect square:

If a number is expressed as the product of pairs of equal factors, then it is called a perfect square.

Ex: – 36

Prime factors of 36 = 2× 2× 3× 3

36 can be expressed as the product of pairs of equal factors.

∴ 36 is a perfect square.

Square Root: the square root of a number x is that number when multiplied by itself gives x as the product. The square root of x is denoted by

Methods of Finding Square root of given Number

Prime factorization method: –

Steps:

1. Resolve the given number into prime factors.
2. Make pairs of similar factors.
3. The product of prime factors, choosing one out of every pair gives the square root of the given number.

Ex: – To find the square root of 16

Prim factors of 16 = 2 ×2× 2× 2

= 2 × 2 = 4

∴ square root of 16 = 4

Division method: –

Steps:

1. Mark off the digits in pairs starting with the unit place. Each pair and remaining one digit are called a period.
2. Think of the largest number whose square is equal to or just less than the first period. Take this number as the divisor as well as quotient.
3. Subtract the product of divisor and quotient from the first period and bring down the next period to the right of the remainder. this becomes the new dividend.
4. Now, the new divisor is obtained by taking twice the quotient and annexing with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of the new divisor and this digit is equal to or just less than the new dividend.

Repeat steps 2, 3, and 4 till all the periods have been taken up. Thus, the obtained quotient is the required square root.

Ex: – To find the square root of 225

Properties of a perfect square:

1. The square of an even number is always an even number.

Ex: – 22 = 4 (4 is even), 62 = 36 (36 is even), here 2, 6 are an even number.

2. The square of an odd number is always an odd number.

Ex: – 32 = 9 (9 is even), 152 = 225 (225 is even), here 3, 15 are an odd number.

3. The square of a proper fraction is a proper fraction less than the given fraction.

Ex: –

4. The square of decimal fraction less than 1 is smaller than the given decimal.

Ex: – (0.3)2 = 0.09 < 0.03.

5. A number ending with 2, 3, 7, or 8 is never a perfect square.

Ex: – 72, 58, 23 are not perfect squares.

6. A number ending with an odd no. of zeros is never a perfect square

Ex: – 20, 120,1000 and so on.

The square root of a number in decimal form

Make the no. of decimal places even, by affixing a zero, if necessary. Now periods and find out the square root by the long division method.

Put the decimal point in the square root as soon as the integral part is exhausted.

Ex: – To find the square root of 79.21

The square root of a decimal number which is not perfect square:

if the square root is required to correct up to two places of decimal, we shall find it up 3 places of decimal and then round it off up to two decimal places.

if the square root is required to correct up to three places of decimal, we shall find it up 4 places of decimal and then round it off up to three decimal places.

Ex: – To find the square root of 0.8 up to two decimal places

Cube of a number:

The cube of a number is that number raised to the power 3.

Ex: – cube of 0.3 = 0.33 = 0.027

Cube of 2 = 23 = 8

Perfect cube:

If a number is a perfect cube, then it can be written as the cube of some natural numbers.

Ex: – 1, 8, 27, and so on.

Cube root:

The cube root of a number x is that number which when multiplied by itself three times gives x as the product.

Cube root of x is denoted by

Methods of finding the cube root of the given Number

Prime factorization method: –

Steps:

1. Resolve the given number into prime factors.
2. Make triplets of similar factors.
3. The product of prime factors, choosing one out of every triplet gives the cube root of the given number.

Ex: – 27

Prim factors of 27 = 3×3×3

= 3

∴ cube root of 27 = 3

Test for a number to be perfect cube:

A given number is a perfect cube if it can be expressed as the product of triplets of equal factors.

Ex: – 2744

Prime factors of 2744 = 2×2×2 × 7×7×7

∴ 2744 is a perfect cube.

# Math problems asked in the CBSE board and entrance examinations.

## 1. RATIONAL AND IRRATIONAL NUMBERS

Natural numbers: counting numbers 1, 2, 3… called Natural numbers. The set of natural numbers is denoted by N.

N = {1, 2, 3…}

Whole numbers: Natural numbers including 0 are called whole numbers. The set of whole numbers denoted by W.

W = {0, 1, 2, 3…}

Integers: All positive numbers and negative numbers including 0 are called integers. The set of integers is denoted by I or Z.

Z = {…-3, -2, -1, 0, 1, 2, 3…}

Rational number: The number, which is written in the form of, where p, q are integers and q ≠ o is called a rational number. It is denoted by Q.

∗ In a rational number, the numerator and the denominator both can be positive or negative, but our convenience can take a positive denominator.

Ex: – can be written as   but our convenience we can take

Equal rational numbers:

For any 4 integers a, b, c, and d (b, d ≠ 0), we have  ⇒ ad = bc

The order of Rational numbers:

If  are two rational numbers such that b> 0 and d > 0 then  ⇒ ad > bc

The absolute value of rational numbers:

The absolute value of a rational number is always positive. The absolute value of   is denoted by .

Ex: – absolute value of

To find rational number between given numbers:

• Mean method: – A rational number between two numbers a and b is

Ex: – insert two rational number between 1 and 2

1 <   < 2   ⟹     1 <    < 2

1 <  < 2   ⟹   1 <   2

To rational numbers in a single step: –

Ex:- insert two rational number between 1 and 2

To find two rational numbers, we 1 and 2 as rational numbers with same denominator 3

(∵ 1 + 2 = 3)

1 =     and 2 =

Note: – there are infinitely many rational numbers between two numbers.

The decimal form of rational numbers

∗ Every rational number can be expressed as a terminating decimal or non-terminating repeating decimal.

Converting decimal form into   form:

1.Terminating decimals: –

1.2 =

1.35 =

2.Non-Terminating repeating decimals: –

Irrational numbers:

• The numbers which are not written in the form of  , where p, q are integers, and q ≠ 0 are called rational numbers. Rational numbers are denoted by QI or S.
• Every irrational number can be expressed as a non-terminating decimal or non-repeating decimal.

Ex:- and so on.

• Calculation of square roots:
• There is a reference of irrationals in the calculation of square roots in Sulba Sutra.
• Procedure to finding value:

# These notes cover all the topics covered in the ICSE 10th class Maths syllabus and include plenty of formulae and concept to help you solve all the types of 10th class Mathematics problems asked in the ICSE board and entrance examinations.

## 1. Goods and Service Tax

Two types of taxes in the Indian Government:

1.Direct taxes: –

These are the taxes paid by an organisation or individual directly to the government. These include Income tax, Capital gain tax and Corporate tax.

2.Indirect taxes: –

These are the taxes on goods and services paid by the customer, collected by an individual or an organisation and deposited with the Government. Earlier there were several indirect taxes levied by the central and state Governments.

Goods and Service Tax (GST):

GST is a comprehensive indirect tax for the whole nation. It makes India one unified common market.

Registration under GST:

Any individual or organisation that has an annual turnover of more than ₹ 20 lakh is to be registered under GST.

Input and Output GST:

For any individual or organisation, the GST paid on purchases is called the ‘Input GST’ and the GST collection on sale of goods is called the ‘Output GST’. The input GST is set off against the output GST and the difference between the two is payable in the Government account.

One currency one tax:

There is a uniform GST rate on any particular goods or services across all states and Union Territories of India. This is called ‘One currency one tax’.

Note: Assam was the first state to implement GST and Jammu & Kashmir was the last.

GST rate slabs:

However, the tax on gold is kept at 3% and on rough precious and semi-precious is kept at 0.25%.

The multitier GST tax rate system in India has been developed keeping in mind that essential commodities should be taxed less than luxury goods.

• Simple tax system.

• Elimination of multiplicity of taxes.

• Development of a common market nation-wide.

• Lower taxes result in the reduction of costs making in the domestic market.

Benefits of GST for Consumers:

• Single and transparent System.

• Elimination of cascading effect has resulted in the reduction in the costs of goods and services.

• Increase in purchasing power and savings.

• Single tax system, simple and easy to administer.

• Higher revenue efficiency.

• Better control on leakage and tax evasion.

Types of GST in India

Central GST (CGST): For any intrastate supply half of the GST collected as the output GST is deposited with the Central Governments as CGST.

State GST or Union Territory GST (SGST/UGST): For any local supply (supply with in the same state or Union Territory) half of the GST is deposited with the respective state or Union Territory Government as the beneficiary. This is called SGST/UGST.

Integrated GST (IGST): The GST levied on the supply of goods or services in the case of interstate trade within India or in the case of exports/imports is known as IGST.

Reverse charge Mechanism:

There are cases where the chargeability gets reversed, that is the receiver becomes liable to pay the tax and deposit it to the Government Account.

Composition shame:

The composition is meant for small dealers and service providers with an annual turnover less than ₹ 1.5 crores and also for Restaurant service providers. Under this scheme the rates of GST are:

Input Tax Credit (ITC)

When a dealer sells his goods, he charges the output GST from his customer which he has to deposit in the government account, but in running his business he had paid input GST on the goods he had availed. This input GST, he utilizes as Input Tax credit and deposits the exes amount of output GST with the Government.

Input Tax credit is a provision of reducing the GST already paid on inputs in order to avoid the cascading of taxes.

GST payable = Output GST – ITC

Claiming ITC: A dealer registered under GST can claim ITC only if:

• He possesses the tax invoice.
• He has received the said goods/services
• He has filed the returns.
• The tax paid by him has been paid to the government by his supplier.

Utilization of ITC:

The Amount of ITC available to any registered dealer shall be utilized to reduce the out put tax liability in the sequence shown in the table.

E – ledgers under GST:
An E – ledger is an electronic form of a pass book available to all GST registrants on the GST portal. These are of three types:

(i) Electric cash ledger (ii) Electric credit ledger and (iii) Electric Liability Register

(i) Electric cash ledger: It contains the amounts of GST deposited in each to the government.

(ii) Electric credit ledger: It contains the balance of ITC available to the dealer.

(iii) Electric credit ledger: It contains all the Tax liability of the dealer.

GST Returns:

These are the information provided from time to time by the dealer to the Government regarding the ITC, output Tax liability and the amounts of GST deposited.

A GST registered person has to submit the following returns:

E – Way bill:
E – Way bill is an electronic way bill that can be generated on the E – Way bill portal. A registered person can not transport goods whose value exceeds ₹ 50,000 in a vehicle without an e – way bill. When an E – way bill is generated, a unique e – way bill number (EBN) is allocated and is available to the supplier, the transporter and recipient. A dealer must generate an E – way bill if he has to transport them for returning to the supplier.

## 2.Banking

To encourage the habit of saving income groups, banks and post offices provide recurring deposit schemes.

Maturity period: An investor deposits a fixed amount every month for a fixed time period is called the maturity period,

Maturity value:  At the end of the maturity period, the investor gets the amount deposited with the interest. The total amount received by the investor is called Maturity value.

Interest =

Where p is the principle

n is no. of months

r is the rate of interest

Maturity value = (p × n) + I

## 3.Shares and Dividend

Capital: The total amount of money needed to run the company is called Capital.

Nominal value (N.V): – The original value of a share is called the nominal value. It is also called as face value (F.V), printed value (P.V) or registered Value (R.V).

Market value: – The price of a share at a particular time is called market value (M.V). This value changes from time to time.

Shares: The whole capital is divided in to small units is called shares.

Share at par: – If the market value of a share is equal to face value of a share, then that share is called a share at par.

Share at a premium or Above par: – If the market value of a share is greater than the face value of the share then, the share is called share at a premium or above par.

Share at discount: – If the market value of a share is lesser than the face value of the share then, the share is called share at discount.

Dividend: – The profit distributed to the shareholders from a company at the end of the year is called a dividend.

The dividend is always calculated as the percentage of face value of the share.

Some formulae:

Note:

• The face value of a share always remains the same
• The market value of a share changes from time to time.
• Dividend is always paid on the face value of a share

## 4. Linear In equations

Linear inequations: A statement of inequality between two expressions involving a single variable x with highest power one is called linear inequation.

Ex: 3x – 3 < 3x + 5; 2x + 10 ≥ x – 2 etc.

General forms of Inequations: The general forms of the linear inequations are: (i) ax + b < c   (ii) ax + by ≤ c    (iii)  ax + by ≥ c    (iv) ax + by > c, where a, b and c are real numbers and a ≠ 0.

Domain of the variable or Replacement Set: The set form which the value of the variable x is replaced in an inequation is called the Domain of the variable.

Solution set: The set of all whole values of x from the replacement set which satisfy the given inequation is called the solution set.

Ex: Solution set of x < 6, x ∈ N is {1, 2, 3, 4, 5}

Solution set of x ≤ 6, x ∈ W is {0, 1, 2, 3, 4, 5, 6}

Inequations – Properties:

• Adding the same number or expression to each side of an inequation does not change the inequality.

Ex: 3 < 5

3 + 2< 5 + 2

5 < 7 (no change in inequality)

• Subtracting the same number or expression to each side of an inequation does not change the inequality.

Ex: 3 < 5

subtract 2 on both sides

3 – 2 < 5 – 2

1 < 3 (no change in inequality)

• Multiplying or Dividing the same positive number or expression to each side of an inequation does not change the inequality.

Ex: 3 < 5

Multiply 2 on both sides

3 × 2< 5 × 2

6 < 10 (no change in inequality)

6 < 8

Divide 2 on both sides

6 ÷ 2< 8 ÷ 2

3< 4 (no change in inequality)

•Multiplying or Dividing the same negative number or expression to each side of an inequation can change(reverse) the inequality.

Ex: 3 < 5

Multiply 2 on both sides

3 × –2< 5 × –2

–6 > –10 (change in inequality)

6 < 8

Divide 2 on both sides

6 ÷ –2< 8 ÷ –2

–3 > –4 (change in inequality)

Note:

• a < b iff b > a
• a > b iff b < a

Ex: x < 4 ⇔ 4 > x

x > 3 ⇔ 3 < x

Method of solving Liner Inequations:

• Simplify both sides by removing group symbols and collecting like terms.
• Remove fractions by multiplying both sides by an appropriate factor.
• Collect all variable terms on one side and all constants on the other side of the inequality sign.
• Make the coefficient of the variable 1.
• Choose the solution set from the replacement set.

Ex: Solve the inequation 3x – 2 < 2 + x, x ∈ W

Sol: given in equation is

3x – 2 < 2 + x

3x – 2 + 2< 2 + x + 2

3x < 4 + x

3x – x < 4

2x < 4

Dividing both sides by 2

x < 2

∴ Solution set = { 0, 1}

Quadratic Equation: An equation of the form ax2 + bx + c = 0, where a, b, and c are real and a ≠ 0 is called a Quadratic equation in a variable ‘x’.

Ex: x 2 – 3x + 4 = 0 is a quadratic equation in a variable ‘x’

t2 + 5t = 6 is a quadratic equation in a variable ’t’

Roots of a quadratic equation: A number α is called a root of the quadratic equation ax2 + bx + c = 0, if aα2 + bα + c = 0.

Solution set:  The set of elements representing the roots of a quadratic equation is called solution set of the give quadratic equation.

Solving Quadratic equation by using Factorization method:

Step – 1: Make the given equation into the standard form of ax2 + bx + c = 0.

Step – 2: Factorise ax2 + bx + c into two linear factors.

Step – 3: Put each linear factor equal to zero.

Step – 4: Solve these linear equations and get two roots of the given quadratic equation.

Ex: Solve x2 – 3x – 4 = 0

x2 – 4x + x – 4 = 0

x (x – 4) + 1 (x – 4) = 0

(x – 4) (x + 1) = 0

x – 4 = 0 or x + 1 = 0

x = 4 or x =– 1

∴ Solution set = {– 1, 4}

Solving Quadratic equation by using Formula:

The roots of the quadratic equation ax2 + bx + c = 0 are:

Ex: Solve x2 – 3x – 4 = 0

Sol: Given equation is x2 – 3x – 4 = 0

Compare with ax2 + bx + c = 0

a = 1, b = – 3, c = – 4

x = 4 or x = – 1

∴ Solution set = {– 1, 4}

Nature of the roots:

Discriminant: – For a quadratic equation ax2 + bx + c = 0, b2 – 4ac is called discriminant.

(i) If b2 – 4ac > 0, then roots are real and un equal.

Case – 1: b2 – 4ac > 0 and it is a perfect square, then roots are rational and unequal.

Case – 2: b2 – 4ac > 0 and it is not a perfect square, then roots are irrational and unequal.

(ii) If b2 – 4ac = 0, then roots are equal and real.

(iii) b2 – 4ac < 0, then roots are imaginary and un equal.

To solve word problems and determine unknown values, by forming quadratic equations from the information given and solving them by using methods of solving Quadratic equation.

The problems may be based on numbers, ages, time and work, time and distances, mensuration etc.

Method of Solving word problems in Quadratic equation:

Step – 1: Read the given problem carefully and assume the unknown be x.

Step – 2: Translate the given statement and form a quadratic equation in x.

Step – 3: Solve for x.

## 7.Ratio and Proportion

Ratio: Comparing two quantities of same kind by using division is called a ratio.

The ratio between two quantities ‘a’ and ‘b’ is written as a : b and read as ‘a is to b’

In the ratio a : b, ‘a’  is called ‘first term’ or ‘antecedent’ and ‘b’ is called ‘second term’ or ‘consequent’.

Note:  The value of a ratio remains un changed if both of its terms are multiplied or divided by the same number, which is not a zero.

Lowest terms of a Ratio:

In the ratio a : b, if a, b have no common factor except 1, then we say that a : b is in lowest terms.

Ex: 4 : 12 = 1 : 3 ( lowest terms)

Comparison of Ratios:

• (a : b) > (c : d) ⇔ ad > bc
• (a : b) = (c : d) ⇔ ad = bc
• (a : b) < (c : d) ⇔ ad < bc

Proportion:

An equality of ratios is called a proportion.

a, b, c and d are said to be in proportion if a : b = c : d and we write as a : b : : c : d.

a and d are ‘extremes’, b and c are ‘means’

product of extremes = product of means

Continued proportion: If a, b, c, d, e and f are in continued proportion, then

Mean proportion:  If then b2 = ac or b =  , b is called mean proportion between a and b.

Third proportional: If a : b = b : c, then c is called third proportional to a and b.

Note:

Results on Ratio and Proportion:

## 8.Remainder Theorem and Factor Theorem

Polynomial: An expression of the form p(x) = a0 xn + a1 xn-1 + a2 xn-2 + …+ an-1 x + an, where a0, a1, …, an are real numbers and a0 ≠ 0. Is called a polynomial of degree n.

Value of a polynomial: The value of a polynomial p(x) at x = a is obtained by substituting x = a in the given polynomial and is denoted by p(a).

Ex: If p(x) = 2x + 3, then find the value of p (1), p (0).

Sol: given p(x) = 2x + 3

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

p (0) = 2 (0) + 3 = 0 + 3 = 3

Division algorithm: On dividing a polynomial p(x) by a polynomial g(x), there exist quotient polynomial q(x) and remainder polynomial r(x) then

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

p(x) is dividend; g(x) is divisor; q(x) is quotient; r(x) is remainder.

Remainder theorem:

If a polynomial p(x) is divided by (x – a), then the remainder is p(a).

Ex: If p(x) = 2x – 1 is divided by (x – 3), then find reminder.

Sol: Given p(x) = 2x – 1

Remainder = p (3)

= 2(3) – 1

= 6 – 1 = 5

∴ remainder is 5

Note:

• If p(x) is divided by (x + a), then the remainder is p (– a).
• If p(x) is divided by (ax + b), then remainder is .
• If p(x) is divided by (ax – b), then remainder is .

Factor theorem: Let p(x) be a polynomial and ‘a’ be given real number, then (x – a) is a factor of p(x) ⇔ p(a) = 0.

Note:

• If (x + a) is the factor of p(x), then p (– a) = 0.
• If (ax + b) is the factor of p(x), then  = 0.
•  If (ax – b) is the factor of p(x),    = 0

## 9. Matrices

Matrix: A rectangular arrangement of numbers in the form of horizontal and vertical lines and enclosed by the brackets [ ] or parenthesis ( ), is called a matrix.

The horizontal lines in a matrix are called its rows.

The vertical lines in a matrix are called its columns.

Oder of Matrix: A matrix having ‘m’ rows and ‘n’ columns is said to be of order m x n read as m by n.

Ex:

Elements of a matrix:

An element of a matrix appearing in the ith row and jth column is called the (i, j)th element of the matrix and it is denoted by aij.

A = [aij]m × n

A =

a11 means element in first row and first column

a12 means element in first row and second column

a22 means element in second row and second column

a32 means element in third row and second column

and so on.

Types of Matrices

Row matrix & column Matrix: A matrix with only one row s called a row matrix and a matrix with only one column is called column matrix.

Ex:

Rectangular Matrix: A matrix in which the no. of rows is not equal to no. of columns is called Rectangular matrix.

Ex:

Square Matrix: A matrix in which the no. of rows is equal to no. of columns is called square matrix.

Ex:

Diagonal Matrix: If each non-diagonal elements of a square matrix is ‘zero’ then the matrix is called diagonal matrix.

Ex:

Identity Matrix or Unit Matrix: If each of non-diagonal elements of a square matrix is ‘zero’ and all diagonal elements are equal to ‘1’, then that matrix is called unit matrix

Ex:

Null Matrix or Zero Matrix: If each element of a matrix is zero, then it is called null matrix.

Ex:

Equality of matrices: matrices A and B are said to be equal if A and B of the same order and the corresponding elements of A and B are equal.

Ex: If  ⟹ a=p; b = q; c = r; d = s

Comparing Matrices: Comparison of two matrices is possible, if they have same order.

Transpose of Matrix: If A = [aij] is an m x n matrix, then the matrix obtained by interchanging the rows and columns is called the transpose of A. It is denoted by   AT.

Ex:

Addition of Matrices: If A and B are two matrices of the same order, then their sum A + B is the matrix obtained by adding the corresponding elements of A and B.

Ex:

Subtraction of Matrices: If A and B are two matrices of the same order, then their difference A + B is the matrix obtained by subtracting the elements of B from the corresponding elements of A.

Ex:

Product of Matrices:

Let A = [aik]mxn and B = [bkj]nxp be two matrices, then the matrix C = [cij]mxp   where

Note: Matrix multiplication of two matrices is possible when no. of columns of first matrix is equal to no. of rows of second matrix.

## 10. Arithmetic Progressions

Sequence:  The numbers which are arranged in a different order to some definite rule are said to form a sequence.

Ex: 1, 2, 3, ……

2, 4, 6, 8, ….

2, 4, 8, 16, …

Arithmetic Progression (A.P.):

A sequence in which each term differ from its preceding term by a constant is called an Arithmetic Progression (A.P.). The constant difference is called the common difference.

Terms: a, a + d, a + 2d…, a + (n – 1) d

First term: a

Common Difference: d = a2 – a1 = a3 – a2 = … = an – an -1

nth term: Tn = a + (n – 1) d

Sum of the n terms of A.P.:

Sum of the n terms of A.P. is

Where a is first term and l is last term.

To find the nth term from the end of an A.P.:

Let a be first term, d be the common difference and ‘l’ be the last term of a given A.P. then its nth term from the end is l – (n – 1) d .

## 11. Geometric Progressions

Terms: a, a r, a r2…, a rn – 1

First term: a

Common ratio:

nth term: Tn = a rn – 1

Sum of the n terms of G.P.:

Sum of the n terms of G.P. is

To find the nth term from the end of an G.P.:

Let a be first term, r be the common ratio and ‘l’ be the last term of a given G.P. then its nth term from the end is

## 12. Reflection

Coordinate Axes:

The position of the point in a plane is determined by two fixed mutually perpendicular lines XOX’ and YOY’ intersecting each other at ‘O’. These lines are called coordinate axes.

The horizontal line XOX’ is called X – axis.

The vertical line YOY’ is called Y – axis.

The point of intersection axes is called ‘origin’.

Coordinates of a point:

Let P be any point on the plane, the distance of P from X – axis is ‘x’ units and the distance of P from Y – axis is ‘y’ units, then we say that coordinates of P are (x, y).

x is called x coordinate or abscissa of P

y is called y coordinate or ordinate of P

The distance of any point on X – axis from X – axis is 0

∴ Any point on the X – axis is (x, 0)

The distance of any point on Y – axis from Y – axis is 0

∴ Any point on the Y – axis is (0, y).

The coordinates of the origin O are (0, 0).

The equation of X – axis is y = 0.

The equation of any line parallel to X – axis is y = k, where k is the distance from X – axis.

The equation of Y – axis is x = 0.

The equation of any line parallel to Y – axis is x = k, where k is the distance from Y – axis.

Reflection

Image of an object in a mirror: When an object is placed in front of a plane mirror, then its image is formed at the same distance behind the mirror as the distance of the object from the mirror.

Image of a point in a line:

Let P be a point and AB is a given line. Draw PM perpendicular to AB and produce PM

to Q such that PM = QM, then Q is called image of P with respect to the line AB.

Reflection of a point in a line:

Assume the given line as a mirror, the image of a given point is called the reflection of that point in the given line.

Reflection of P (x, y) in X – axis is P (x, –y) ⇒ Rx (x, y) = (x, –y)

Reflection of P (x, y) in Y – axis is P (–x, y) ⇒ Rx (x, y) = (–x, y)

Reflection of P (x, y) in the origin is P (–x, –y) ⇒ Rx (x, y) = (–x, –y)

Combination of Reflection:

• Rx. Ry = Ry. Rx = Ro
• Rx. Ro = Ro. Rx = Ry
• Ry. Ro = Ro. Ry = Rx

Invariant Points: A point P is said to be invariant in a given line if the image of P (x, y) in that line is P (x, y).

## 13. Section and Mid – Point Formula

Section formula: If P (x, y) divides the line segment joining the points A (x1, y1) and B (x2, y2) in the ratio m : n, then

P (x, y) =

Mid pint Formula:

The mid-point of the line segment joining the points A (x1, y1) and B (x2, y2) is

Centroid of the triangle:

The point of concurrence of medians of a triangle is called centroid of the triangle. It is denoted by G.

The centroid of the triangle formed by the vertices A (x1, x2), B (x2, y2) and C (x3, y3) is

G =

## 14. Equation of a Straight line

Inclination of a line: The angle of inclination of a line is the angle θ which is the part of the line above the X – axis makes with the positive direction of X – axis and measured in anticlockwise direction.

Horizontal line: A line which is parallel to X – axis is called horizontal line.

Vertical line: A line which is parallel to Y – axis is called vertical line.

Oblique line: A line which is neither parallel to X – axis nor parallel to Y – axis is called an oblique line.

Slope or Gradiant of a line:

A line makes an angle θ with the positive direction of x – axis then tan θ is called the slope of the line, it is denoted by ‘m’

m = tan θ

1. Slope of the x- axis is zero.
2. Slope of any line parallel to x- axis is zero.
3. Slope of y- axis is undefined.
4. Slope of any line parallel to y- axis is also undefined.
5. Slope of the line joining the points A (x1, y1) and B (x2, y2) is

1. Slope of the line ax + by + c = 0 is  =

Condition for collinearity: If three points A, B and C are lies on the same line then they are collinear points.

Condition for the collinearity is slope of AB = Slope of BC = slope of AC

Types of equation of a straight line:

• Equation of x- axis is y = 0.
• Equation of any line parallel to x – axis is y = k, where k is distance from above or below the x- axis.
• Equation of y- axis is x = 0.
• Equation of any line parallel to y – axis is x = k, where k is distance from left or right side of the y- axis.

Slope intercept form:

The equation of the line with slope m and y- intercept ‘c’ is y = mx + c.

Slope point form:

The equation of the line passing through the point (x1, y1) with slope ‘m’ is

y – y1 = m (x – x1)

Two points form:

The equation of the line passing through the points (x1, y1) and (x2, y2) is

Intercept form:

The equation of the line with x- intercept a, y – intercept b is

Note: –

1. If two lines are parallel then their slopes are equal

m1 = m2

1. If two lines are perpendicular then product of their slopes is – 1

m1 × m2 = – 1

slope of a line perpendicular to a line AB =

## 15. Similarity

Similar figures: If two figures have same shape but not in size, then they are similar.

Similarity as a Size transformation:

It is the process in which a given figure is enlarged or reduced by a scale factor ‘k’, such that the resulting figure is similar to the given figure.

The given figure is called an ‘object’ and the resulting figure is called its ‘image’

Properties of size transformation:

Let ‘k’ be the scale factor of a given size transformation, then

If k > 1, then the transformation is enlargement.

If k = 1, then the transformation is identity transformation.

If k < 1, then the transformation is reduced.

Note:

• Each side if the resulting figure = k times the corresponding side of the given figure.
• Area of the resulting figure = k2 × (Area of the given figure).
• Volume of the resulting figure = k3 × (Volume of given figure).

Model: The model of a plane figure and the actual figure are similar to one another.

Let the model of the plane figure drawn to the scale 1 : n, then scale factor k =  .

• Length of the model = k × (length of the actual figure)
• Area of the model = k2 × (Area of the actual figure).
• Volume of the model = k3 × (Volume of actual figure).

Map: The model of a plane figure and the actual figure are similar to one another.

Let the Map of the plane drawn to the scale 1 : n, then scale factor k =  .

• Length of the map = k × (Actual length)
• Area of the model = k2 × (Actual area).

## 16. Similarity of Triangles

Similar triangles: Two triangles are said to be similar If: (i) their corresponding angles are equal and (ii) their corresponding sides are in proportional (in same ratio).

If ∆ABC ~ ∆DEF, then

• ∠A = ∠D, ∠B = ∠E and ∠C = ∠F

Here k is scale factor, (i) if k > 1, then we get enlarged figures (ii) if k = 1, then we get congruent figures (iii) if k < 1, then we get reduced figures.

Axioms of Similar Triangles

SAS – Axiom:

If two triangles have a pair of corresponding angles equal and the corresponding sides including them proportional, then the triangle is similar.

In ∆ABC and ∆DEF, ∠A = ∠D and

AA – Axiom : If two triangles have two pairs of corresponding angles equal, then the triangles are equal.

SSS – Axiom: If two triangles have their three sides of corresponding sides proportional, then the triangles are similar.

Basic proportionality Theorem:

In a triangle a line drawn parallel to one side divides the other two sides in the same ratio (proportional).

In ∆ ABC, DE ∥ BC ⇒

Converse of Basic proportionality Theorem:

If a line divides any two sides of a triangle proportionally, the line is parallel to the third side.

In ∆ ABC,  ⇒ DE ∥ BC.

∎ Ina triangle the internal bisector of an angle divides the opposite side in the ratio of the sides containing the angle.

In ∆ ABC, AD bisects ∠A then

∎ The areas of two similar triangles are proportional to the squares of their corresponding sides.

If ∆ABC ~ ∆DEF, then

∎ The areas of two similar triangles are proportional to the squares of their corresponding medians.

If ∆ABC ~ ∆DEF and AM, DN are medians of   ∆ABC and ∆DEF respectively then

∎ The areas of two similar triangles are proportional to the squares of their corresponding altitudes.

If ∆ABC ~ ∆DEF and AM, DN are altitudes of ∆ABC and ∆DEF respectively then

## 17. Loci

Locus: Locus is the path traced out by a moving point which moves according to some given geometrical conditions.

The plural form of Locus is ‘Loci’ read as ‘losai’

∎ The locus of the point which is equidistance from two given fixed points is the perpendicular bisector of the line segment joining the given fixed       points.

∎ Every point on the perpendicular bisector of AB is equidistance from A and B.

∎ The locus of the point which is equidistance from two intersecting lines is the pair of lines bisecting the angles formed by the given lines.

∎Every point on the angular bisector of two intersecting lines is equidistance from the lines.

## 18. Angle and Cyclic Properties of a Circle

∎ The angle subtended by an arc of a circle is double the angle subtended by it at any point on the circle.

∠AOB = 2 ∠ACB

∎ Angles in a same segment of a circle are equal.

∎ The angle in a semi-circle is 900

∎ If an arc of a circle subtends a right angle at any point on the remaining part of the circle, then the arc is semi-circle.

A quadrilateral is said to be cyclic if all the vertices passing through the circle.

The opposite angles of cyclic quadrilateral are supplementary.

∎ If pair of opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.

∎ The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

∎ Every cyclic parallelogram is a rectangle.

∎ An isosceles trapezium is always cyclic and its diagonals are equal.

∎ The mid-point of hypotenuse of a right-angled triangle is equidistance from its vertices.

## 19. Tangent Properties of Circles

Tangent: A line which intersect the circle at only one point is called Tangent to the circle.

∎ The tangent at any point of a circle and radius through the point are perpendicular to each other.

∎ If two tangents are drawn to a circle from an exterior point, then

• The tangents are equal in length.
• The tangents subtend equal angle at the centre.
• The tangents are equally inclined to the line joining the point and the centre if the circle.

Intersecting Chord and Tangents

Segment of a chord:

If P is a point on a chord AB of a circle, then we say that P divides AB internally into two segments PA and PB.

If AB is a chord of a circle and P is a point on AB produced, we say that P divides AB externally into two segments PA and PB.

∎ If two chords of a circle intersect internally or externally, then the product of the lengths of their segments are equal.

Alternate segments:

In the given figure APB is a tangent to the circle at point at a point P and PQ is a chord

The chord PQ divides the circle into two segments PSR and PSQ are called alternate segments.

The angle between a tangent and a chord through the point of contact is equal to an angle in the alternate segment.

∠QPB = ∠PSQ and ∠APQ = ∠PRQ

## 21. Volume and Surface Area of Solids

Cylinder:

Solids like circular pillars, circular pipes, circular pencils etc. are said to be in cylindrical shape.

Radius of the base = r

Height of the cylinder = h

Curved surface area = 2πrh sq. units

Total surface area = 2πr (r + h) sq. units

Volume = πr2h cubic. units

Hollow Cylinder:

Height = h

Thickness of the cylinder = R – r

Area of cross section = π (R2 – r2) sq. units

Volume of material = πh (R2 – r2) cubic. units

Curved surface area = 2πh (R+ r) sq. units

Total surface area = 2π (Rh + rh + R2 – r2) sq. units

Cone:

Radius of the base = r

Height of the cylinder = h

Slant height = l

l2 = r2 + h2 ⇒ l =

Curved surface area = πrl sq. units

Total surface area = πr (r + l) sq. units

Volume = πr2h cubic. Units

Sphere:

Objectives like football, throwball, etc. are said to be the shape of sphere.

Surface area = 4πr2 sq. units

Volume = πr3 cubic. Units

Spherical Shell:

The solid enclosed between two concentric spheres is called spherical shell

Thickness of the cylinder = R – r

Volume of the material =  π (R3 – r3) cubic. Units

Hemi sphere:

Curved Surface area = 2πr2 sq. units

Surface area = 3πr2 sq. units

Volume = πr3 cubic. Units

## 22.Trigonometrical Identities

The word Trigonometry derived from Greek word, tri three, gonia angle and metron to measure.

Angle: – The figure formed by two rays meeting at a common end point is an angle.

Naming the sides in a right-angled triangle:

AB = Perpendicular =opposite side of θ (opp)

AC is hypotenuse (hyp)

Trigonometric ratios:

Quotient relations:

Trigonometric Identities:
(i) sin
2θ + cos2θ = 1          (ii) sec2θ − tan2θ = 1        (iii) cosec2θ − cot2θ = 1

∎ sin2θ = 1 − cos2θ; cos2θ = 1 – sin2θ

∎ sec2θ = tan2θ + 1; sec2θ – 1 =tan2θ

∎ Cosec2θ = Cot2θ + 1; Cosec2θ – 1 = Cot2θ

Trigonometric tables:

A trigonometric Table Consist of three parts:

• A column on the extreme left containing degree from 00 to 890.
• The columns headed by 0’, 6’, 12’, 18’, 24’, 30’, 36’, 42’, 48’ and 54’.
• Five columns of mean differences, headed by 1’, 2’, 3’, 4’ and 5’

Note:

• The mean difference is added in case of ‘sines’, ‘tangents’ and ‘secants’
• The mean difference is subtracted in case of ‘cosines’, ‘cotangents’ and ‘cosecants’

Relation between degrees and minutes:

10 = 60’⇒ 1’ =

Trigonometric Tables Charts:

By clicking below you get Sin, Cosine and Tangent Tables

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## 23. Heights and Distances

Horizontal line: A line which is parallel to earth from observation point to object is horizontal line
Line of sight: The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer.

Angle of elevation: The line of sight is above the horizontal line and angle between the line of sight and horizontal line is called angle of elevation.

Angle of depression: The line of sight is below the horizontal line and angle between the line of sight and horizontal line is called angle of depression.

Solving procedure:

∎All the objects such as tower, trees, buildings, ships, mountains etc. shall be consider as linear for mathematical convenience.

∎The angle of elevation or angle of depression is considered with reference to the horizontal line.

∎The height of observer neglected, if it is not given in the problem.

∎To find heights and distances we need to draw figures and with the help of these figures we can solve the problems.

## 24. Graphical Representation of Statistical Data

Data: A set of given facts in numerical figures is called data.

Frequency: The number of times an observation occurs is called its frequency.

Frequency Distribution: The tabular arrangement of data showing the frequency of each observation is called its frequency distribution.

Class interval: Each group into which the raw data is condensed is called a class interval.

Class limits: Each class interval is bounded by two figures is called Class limits.

Left side part of class limit is called ‘Lower limit’

Right side part of class limit is called ‘Upper limit’

Inclusive form:  In each class, the data related to both the lower and upper limits are included in the same class, is called Inclusive form.

Ex: 1 – 10, 11 – 20, 21 – 30 etc.

Exclusive form: In each class, the data related to the upper limits are excluded is called Exclusive form.

Ex: 0 – 10, 10 – 20, 20 – 30 etc.

Class size = upper limit – lower limit

Class mark =  [lower limit + upper limit]

Note:

In an inclusive form, Adjustment factor =  [lower limit of one class – upper limit of previous class]

Histogram:  A histogram is a graphical representation of a frequency distribution in an exclusive form, in the form of rectangles with class interval as bases and the corresponding frequencies as heights

Method of drawing a Histogram:

Step-1:  If the given frequency distribution is in inclusive, then convert them into the exclusive form

Step-2: Choose a suitable scale on the X – axis and mark the class intervals on it.

Step-3: Choose a suitable scale on the Y – axis and mark the frequencies on it.

Step-4: Draw rectangle with class intervals as bases and the corresponding frequencies as the corresponding heights.

Example:

Frequency polygon:

Let x1, x2, x3, …, xn be the class marks of the given frequency distribution and f1, f2, f3, …, fn be the corresponding frequencies, then plot the points (x1, f1), (x2, f2), …. (xn, fn) on a graph paper and join these points by a line segment. complete the diagram in the form of polygon by taking two or more classes.

Example:

Cumulative Frequency curve or Ogive:

In order to represent a frequency distribution by an Ogive, we mark the upper class along X– axis and the corresponding cumulative frequencies along Y – axis and join these points by free hand curve, called Ogive.

Example:

## 25. Measures of Central Tendency

Average of a Data:

For a given data a single value of the variable representing the entire data which describes the characteristics of the data is called average of the data.

An average tends to lie centrally with the values of the variable arranged in ascending order of magnitude. So, we call an average a measure of central tendency of the data.

Three measures of central tendency are:   (i) Mean   (ii) Median  and (iii) Mode

Average of a Data:

For a given data a single value of the variable representing the entire data which describes the characteristics of the data is called average of the data.

An average tends to lie centrally with the values of the variable arranged in ascending order of magnitude. So, we call an average a measure of central tendency of the data.

Three measures of central tendency are:   (i) Mean   (ii) Median  and (iii) Mode

Mean

Mean for Un Grouped data:

The mean of ‘n’ observations x1, x2, x3, …, xn is

Mean =

The Symbol Σ is called ‘sigma’ stands for summation of the data.

Note:

If the mean of a data x1, x2, … xn is m, then

• Mean of (x1+k), (x2 + k), …. (xn + k) = m + k
• Mean of (x1−k), (x2 − k), …. (xn − k) = m −k
• Mean of (k x1), (k x2), …. (k xn) = k m

If x1, x2, …. xn are of n observations occurs f1, f2, …. fn times respectively then mean is

Mean of grouped data:

Methods of finding mean:

Class mark (mid value) =

Direct method: ;  xi is class mark of ith class, fi is frequency of class.

Assumed mean method: ;  di = xi – a and a is assumed mean.

Step – deviation method:  ; 𝛍i =  , h is class size.

## 26. Median, Quartile and Mode

Median

Median is the middle most observation of given data.

For un grouped data:

First, we arrange given observations into ascending or descending order.

If n is odd median =  observation.

If n is even median =

For grouped data:

Median = , where

l is the lower boundary of median class

f is the frequency of median class

c.f is the preceding cumulative frequency of the median class

h is the class size

Quartiles

The observations which divides the whole set of observations into 4 equal parts are known as Quartiles.

Lower Quartile (First Quartile): If the variates are arranged in ascending order, then the observations lying midway between the lower extreme and the median is called the Lower Quartile. It is denoted by Q1.

If n is Even Q1 observation

If n is Odd Q1 =  observation

Middle Quartile: The middle Quartile is the median, denoted by Q2.

Upper Quartile (Third Quartile): If the variates are arranged in ascending order, then the observations lying midway between and the median the upper extreme is called the Upper Quartile. It is denoted by Q3.

If n is Even Q =  observation

If n is Odd Q1 = observation

Range: The difference between the biggest and the smallest observations is called the Range.

Interquartile Range: The difference between the upper quartile and Lower quartile is called the inter quartile.

Range = Q3 – Q2

Semi – interquartile range:

Semi – interquartile Range = ½ [ Q3 – Q2]

Estimating median:

Step 1: If the given frequency distribution is not continuous, convert into the continuous form.

Step 2: Prepare the cumulative frequency table.

Step 3: Draw Ogive for the cumulative frequency distribution given above

Step 4: Let sum of the frequencies = N.

Step 5: Mark a point A on Y- axis corresponding to

Step 6: From A draw Horizontal line to meet Ogive curve at P. From P draw a vertical line PM to meet X – axis at M. Then the abscissa of M gives the Median.

Estimating Q1 and Q2:

To locate the value of Q1 on Ogive curve, we mark the point along

Y – axis, corresponding to  and proceed similarly.

To locate the value of Q3 on Ogive curve, we mark the point along

Y – axis, corresponding to  and proceed similarly.

Mode

The value of a data which is occurred most frequently is called Mode.

Modal class: The class with maximum frequency is called the Modal class.

Estimation of Mode from Histogram:

Step 1: If the given frequency distribution is not continuous, convert it into a continuous form.

Step 2: Draw a histogram to represent the above data.

Step 3: from the upper corner of the highest rectangle, draw line segments

To meet the opposite corners of adjacent rectangles, diagonally

Let these line segments intersect at P.

Step 4: Draw PM perpendicular to X-axis at M, Then the abscissa of M is The Mode

## 27. Probability

J Cordon Italian mathematician wrote the first book on probability named “the book of games and chance”.

Probability:

It is the concept which numerically measures the degree of certainty of the occurrence of an event.

Some words in probability:

Experiment: A repeatable procedure with a set of possible results.

Trial: By a trial, we mean experimenting.

Outcome: a possible result of an experiment.

Sample space: All the possible outcomes of an experiment.

Sample point: Just one of the possible outcomes.

Event: One or more outcomes of an experiment.

Probability of occurrence of an Event (Classical definition):

In a random experiment Let S be the sample space and E be the event, then E ⊆ S. The probability of occurrence of E is defined as:

P(E) =

Deck of cards: A deck of playing cards consists of 52 cards which are divided into 4 suits of 13 cards each. They are black spade     , black clubs, red heart     and red diamond       . The cards in each suit are: 2, 3, 4, 5, 6, 7, 8, 9 ,10, Ace, Jack, Queen and King. Jack, Queen and King are called face (picture) cards.

Impossible event: If there is no probability of an event to occur then it is impossible event. Its probability is zero.

Sure or certain event: If the probability of an event is 1 then it is sure or certain event.

Complimentary event: Let E denote the event, ‘not E’ is called complimentary event of E. It is denoted by  . P ( ) = 1 – P(E) ⟹ P ( ) +P(E) = 1.

0 ≤ P(E) ≤ 1

# These notes cover all the topics covered in the CBSE 11th class Maths syllabus and include plenty of formulae and concept to help you solve all the types of11thMath problems asked in the CBSE board and entrance examinations.

## 1. SETS

Well defined objects:

1. All objects in a set must have the same general similarity or property.
2. Must be able to confirm whether something belongs to the set or not.

Set: – A collection of well-defined objects is called a set.

∗ Sets are usually denoted by capital English alphabets like A, B, C and so on.

∗ The elements in set are taken as small English alphabets like a, b, c and so on.

∗ Set theory was developed by George canter.

• If any object belongs to a set, then it is called objects / elements. We denote by ∈ to indicate that it belongs to. If it does not belong to the set then it is denoted by ∉.

Ex: – 1 ∈ N, 0 ∈ W, −1 ∈ Z , 0 ∉ N etc.

Methods of representing sets:

Roster or table or listed form: –

In this form all the elements of set are listed, the elements are separated by commas and enclosed within braces { }.

Ex: – set of vowels in English alphabet = {a, e, I, o, u},

set of even natural numbers less than 10 = {2, 4, 6, 8} etc.

Note: – In roster form, an element is not repeated.  We can list the elements in any order.

Set builder form:

Pointing an element in a set to x (or any symbols such as y, z, etc.) followed by a colon(:), next to :  write the properties or properties of the elements in that set and placing in a flower brackets  is called the set builder  form.: Or / symbols read as ‘such that’

Ex: – {2, 4, 6, 8} = {x / x is an even and x ∈N, x< 10},

{a, e, i, o, u} = {x : x is a vowel in English alphabet}.

Null set: – (empty set or void set) the set which has no elements is called as null set. It is denoted by ∅ or { }.

Finite and infinite sets: – If a set which contains finite no. of elements then it is called finite set. If a set contains infinite no. of elements then it is called infinite set.

Ex: – A = {1, 2,3, 4} → finite set

B = {1, 2, 3, 4….}

Equal sets: – two set A and B are said to be equal sets if they have same elements., and write as A = B

Ex: – A = {1, 2, 3, 4}, B = {3, 1, 4, 2}

⟹ A = B.

Sub set: – for any two sets A and B, if every element of set A is in set B, then we can say that A is subset of B. It is denoted by A ⊂ B.

Ex: – If A = {1, 2, 3, 4, 5, 6, 7, 8}, subsets of A are {1}, {1, 3, 5}, {1,2,3,4} and so on.

Power set: – set of all the subsets of a set A is called power set of A. It is denoted by p(A).

Ex: – A = {1,2,3}

P(A) = {{1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1, 2, 3}, ∅}.

Intervals:

∗ Open interval: – (a, b) = {x: a< x <b} → set of rational numbers lies between a and b.

∗ Closed interval: – [a, b] = {x: a≤ x ≤b} → set of rational numbers lies between a and b, including a and b.

∗ Open – closed: – (a, b] = {x: a< x ≤b} → set of rational numbers lies between a and b, excluding a and including b.

∗ Closed-open: -[a, b) = {x: a≤ x <b} → set of rational numbers lies between a and b, including a and excluding b.

Universal set: – A set which contains all the subset of it under our consideration is called universal set.

Cardinal number of a set: – Number of elements in a set A is called cardinal number of that set A. It is denoted by n(A).

• If a set has n elements, then no. of elements of that set has 2n

Equivalent sets: – two set A and B are said to be equivalent sets if n(A) = n(B) (they have same cardinal number).

Ex: – A = {1, 2, 3}, B = {a, b, c}

n(A) = 3 and n(B) = 3

∴ A = B.

Venn diagrams:

U = {1, 2, 3, 4, 5, 6}
the relationship between sets are usually represented by means of diagrams which are known as ‘Venn diagrams. These diagrams consist of rectangle and circles. Universal set is represented by rectangles and subsets by circles.

U = {1, 2, 3, 4, 5, 6} A = {1, 2, 3} B = {1, 2}

## TS 8th Class Maths Concept

Studying maths in VIII class successfully meaning that children take responsibility for their own learning and learn to apply the concepts to solve problems.

This notes is designed by the ‘Basics in Maths team’. These notes to do help students fall in love with mathematics and overcome fear.

## 1. RATIONAL NUMBERS

• Natural numbers: All the counting numbers starting from 1 are called Natural numbers.

1, 2, 3… Etc.

• Whole numbers: Whole numbers are the collection of natural numbers.

0, 1, 2, 3 …

• Integers: integers are the collection of whole numbers and negative numbers.

….., -3, -2, -1, 0, 1, 2, 3….

• Rational numbers: The numbers which are written in the form of p/q, where p, q are integers and q ≠ 0 are called rational numbers. Rational numbers are denoted by Q.

Properties of Rational numbers

• Natural numbers:

1.Closure property:-

### • Rational numbers:

#### 3. Associative  property:-

1 + 0 = 0 + 1 = 1,   3/2 + 0 = 0 + 3/2 = 3/2

• For any rational number ‘a’, a + 0 = 0 + a

• 0 is the additive identity.

2 + (-2) = (-2) + 2 = 0,  5 + (-5) = (-5) + 5 = 0

• For any rational number ‘a’, a+ (-a) = (-a) + a = 0

• Additive inverse of a = -a and additive inverse of (-a) = a

Multiplicative identity:-

2 × 1 = 1 × 2 = 2,    6 ×1/6 = 1 × 1/6 = 1/6

• For any rational number ‘a’, a × 1 = 1 × a = a

• 1 is the multiplicative identity.

Multiplicative inverse:-

2 × 1/2 = 1/2 × 2 =1

For any rational number ‘a’,a × 1/a = 1/a × a = 1

• multiplicative inverse of a =1/a

• Multiplicative inverse of  1/a= a.

Distributive property:-

For any three rational numbers  a, b and c, a × (b + c) = (a × b) + (a × c).

3/2×(5/3+1/5)=(3/2×5/3)+(3/2×1/5)

Representing rational numbers on a number line:

Ex: represent 29/6 on a number line

this lies between 4 and 5

Divide the number line between 4 and 5 into 6 equal parts. Mark 5th part counting from 4.

The role of zero:

• If 0 is added to any rational number, then the rational number remains the same.

For any rational number ‘a’ a + 0 = a = 0 + a

• 0 is the additive identity.

• Natural numbers does not have additive identity.

for any rational number ‘a’ a + (-a) = 0 = (-a) + a

3 + (-3) = 0,   10 + (-10) = 0

• additive inverse of ‘a’ is ‘-a’ and additive inverse of  ‘-a’ is ‘a’

The role of 1:

• If 1 is multiplied to any rational number, then the rational number remains same.

For any rational number ‘a’ a × 1 = a = 1 × a

• 1 is the multiplicative identity.

3 × 1/3 = 1 = 1/3 × 3

for any natural number ‘a’ a × 1/a = 1 = 1/a ×a

• Multiplicative inverse of ‘a’ is ‘1/a’ and multiplicative inverse of ‘1/a’ is ‘a’

Distributive property:

For any 3 rational numbers a, b and c, a (b + c) = ab + ac

Ex:-  1/3 (2/5 + 1/5) = 1/3(3/5) = 3/15

1/3× 2/5 + 1/3 × 1/5 = 2/5 + 1/5 = 3/5

Inserting rational numbers between given two numbers:

• There infinitely many rational numbers between given two numbers.

• We have two methods to find rational numbers between two numbers.

First method: – First we have to convert given rational numbers as the same denominator and write the rational numbers which come between given numbers.

Second method: – if a and b any given rational numbers then   is a rational number between a and b.

Decimal representation of rational numbers

The decimal expansion of rational is either terminating or non-terminating repeating decimal.

Note:-Decimal numbers with the finite no. of digits is called terminating Decimal numbers with the infinite no. of digits is called non- terminating decimal. In a decimal, a digit or a sequence of digits in the decimal part keeps repeating itself infinitely. Such decimals are called non- terminating repeating decimals.

Terminating decimals:

Consider a rational number

Non-terminating decimals:

Consider a rational number

## 2. LINEAR EQUATIONS IN ONE VARIABLE

Equations: An algebraic equation is the equality of algebraic expressions involving variables and constants.

• It has an equality sign.
• The expression on the left of the equality sign is called the LHS (Left Hand Side) and right of the equality is called RHS (Right Hand Side) of the equation.
• In an equation, the value of RHS and LHS are equal. This happens to be true only for certain values of the variable. This value is called the solution of the equation.

Linear equations: If the degree of the equation is 1, then it is called a linear equation.

Ex:  2x – 3 = 5, x = 3y, 5x + 3y = 3 and so on.

Simple equations or linear equations in one variable: An equation of the form ax + b = 0 or ax = b where a, b are constants and a≠0is called a linear equation in one variable or simple equation.

Ex: 2x + 3 = 7, x = 3, 2 – 3x = – 1 and so on.

Note:  if we transpose terms from LHS to RHS or RHS to LHS

‘+’ quantity becomes ‘– ‘quantity

‘–’ quantity becomes ‘+ ‘quantity

‘×’ quantity becomes ‘÷ ‘quantity

‘÷’ quantity becomes ‘× ‘quantity

Solving simple equation having the variable on one side:

Ex: solve the equation 2x + 32 = 2

Sol:                     2x = 2 – 32 (transpose 32 to RHS)

2x = – 30

x = – 30/2 (transposing to RHS)

∴ solution of 2x + 32 = 2 is – 10.

Solving simple equation having the variables on both sides:

Ex:  solve the equation 9y + 5 = 15y – 1

Sol: given equation is 9y + 5 = 15y – 1

9y – 15y = – 1 – 5 (Transposing 5 to RHS and 15y to LHS)

–6y = – 6

y = –6/–6 = 1

y = 1

∴ solution of the equation 9y + 5 = 15y – 1 is 1.

Method of cross Multiplication:

Multiply the numerator of the LHS by the denominator of the RHS and multiply the numerator of the RHS by the denominator of LHS. This method is called the cross multiplication method.

Reducing Equations to simpler form – Equations Reducible to Linear form:

Ex: solve the equation

Sol: given equation is

⇒ 7(5x + 2) = 12(2x + 3) (∵ by cross multiplication method)

⇒ 35x + 14 = 24x + 36

⇒ 35x – 24x = 36 – 14 (by transposing terms)

⇒ 11x = 22

∴ x = 2  is the solution of given equation.

It has 4 sides, 4 vertices, 4 angles and two diagonals.

1.Trapezium: A quadrilateral with at least one pair of parallel sides is called a trapezium.

Opposite sides are not equal and diagonals are not equal

2.Parallelogram: A quadrilateral with two pairs of opposite sides are parallel is called a parallelogram.

Opposite sides are parallel and equal

Opposite angles are equal

Diagonals are not equal.

Diagonals bisect each other.

3.Rectangle: A parallelogram with one of the angles 900 is called a rectangle.

Opposite sides are parallel and equal

Opposite angles are equal

Diagonals are equal.

Diagonals bisects each other.

4.Rhombus: A parallelogram with adjacent sides are equal is called rhombus.

All sides are equal

Opposite angles are equal

Diagonals are not equal.

Diagonals bisect each other and angle between diagonals is 900

5.Square: A rhombus with four right angles is called a square.

All sides are equal

Opposite angles are equal

Diagonals are equal.

Diagonals bisect each other and angle between diagonals is 900

6.Kite: A quadrilateral with two pairs of adjacent sides is called a kite.

We can draw quadrilaterals when the following measurements are given.

1. When 4 sides and one angle are given
2. When 4 sides and one diagonal are given
• When three sides and two diagonals are given
1. When two adjacent sides are given and three angles are given
2. When three side and two included angles are given

Type of Quadrilateral – No, of individual measurements:

 Type of quadrilateral No. of individual measurements Quadrilateral 5 Trapezium 4 Parallelogram 3 Rectangle 3 Rhombus 2 Square 1

Example 1:

Construct the quadrilateral PQRS with the measurements: PQ = 5.5cm, QR = 3.5 cm, RS= 4 cm, PS = 5 cm and ∠P = 450.

Steps of construction:

1. Construct a line segment PQ with a radius 5.5cm.
2. With the center, P draw a ray and an arc that are equal to 450 and 5 cm.
3. These intersecting points are kept as S.
4. With centers S, Q draws two arcs equal to Radius 4 cm, 3.5 cm respectively.
1. The intersecting point of these two arcs is kept as R.
2. Join Q, R, and S, R
3. Therefore, the required quadrilateral PQRS formed.

Example 2

Construct the parallelogram PQRS with the measurements: PQ = 4.5cm, QR = 3 cm and ∠PQR = 600

In parallelogram PQRS with the measurements: PQ = 4.5cm, QR = 3 cm and ∠PQR = 600

⇒ RS = 4.5cm, PS = 3 cm (in a parallelogram opposite sides are equal)

Steps of construction

1. Construct a line segment PQ with a radius 4.5cm.
2. With the center, Q draws a ray and an arc that are equal to 450 and 3 cm.
3. These intersecting points are kept as R.
4. With centers R, P draws two arcs equal to radius 4 cm, 3.5 cm respectively.
1. The intersecting point of these two arcs is kept as S.
2. Join P, S and R, S
3. Therefore, the required parallelogram PQRS formed.

## 4. EXPONENTS  AND POWERS

We know that a2 = a × a (two times)

a3 = a × a × a (three times)

⇒ a × a × a × a × a … m times = am

Here, am is called the exponent form.

• In exponent, form am, ‘a’ is base, ‘m’ is exponent, power, or index.
• We read am as a raised to the power of m.

Laws of exponents:

Express small numbers in Standard form by using exponents:

• If a number is expressed in the form of m ×10n where 1≤m<10, n is any integer, then that number is in standard form.
• Very small numbers can be expressed in standard form using negative exponents.

Ex:    express 0.0000456 in standard form

Sol : 0.0000456 = 456/10000000 = 456/107 = 456 × 10-7.

## 5. COMPARING QUANTITIES USING PROPORTION

Ratio: comparing two quantities of same kind by using division is called ratio.

Ratio of two quantities a and b is denoted by a: b.

Per cent: per cent means ‘per hundred’ or out of hundred’. The symbol % stands for percent.

Discount:

Marked price (M.P): – The price printed on an article by manufacturer is called marked price. It is also called as list price or usual price or catalogue price.

Discount: – Discount is the reduced marked price. It is generally given as percent of the marked price. Discount is always depending on the marked price.

Net price or selling price: – The difference between the M.P and discount is called net price or selling price.

Example:  A T.V is marked at ₹ 18000 and discount allowed on it is 10%. What is the amount of discount and its sale price?

Ans:  Given marked price = ₹18000, discount percentage = 10%

Now, discount = 10% 0f 18000 = = 1800

Selling price = marked price – discount = 18000 – 1800 = 16200.

∴ selling price = ₹ 16,200.

Profit and loss:

Cost price (C.P.): – Cost price is the price for which an article is bought or the price paid by a customer to by an article.

Selling price (S.P.): – Selling price is the price for which an article is sold.

Profit: – If the Selling price is greater than the cost price, then we get the profit.

Profit = S.P – C.P.

Loss:If the Selling price is less than the Cost price then, we get loss.

Loss = C.P – S.P.

Some formulae in profit and loss:

For-Profit:

For Loss:

Government collects taxes on every sale. This is called VAT. Shop keeper collect this from the customers and pay it to the Govt.

VAT is changed on the Selling price of an item and will be included in the bill. VAT is an increase percent of selling price.

Example:

The cost of an article is ₹ 500. The sales tax is 5%. Find the bill amount.

Ans: cost price of an article = ₹500

% of Sales tax = 5

Sales tax paid = ₹

Bill amount = cost price + sales tax paid

= 500 + 25

= ₹ 525.

Simple interest:

Principal: – The money which is borrowed is called ‘principal’.

Rate of interest: – percentage of interest per year is called rate of interest.

Time: – The period for which money is called time.

Interest: – The money which is paid for the use of the principal is called interest.

Amount: – The total money which is paid after the expiry of the time is called amount.

Compound Interest: Compound interest allows us to earn interest on interest.

• The time period after which interest is added to principal is called conversion period. When interest is compounded h yearly, there are two conversion periods in a year. In such case, alf year rate will be half of the annual rate.

## 6. SQUARE ROOTS AND CUBE ROOTS

Square:  Square number is the number raised to the power 2. The number obtained by the number multiplied by itself.

• If a natural number p can be expressed as q2, where q is also natural, then p is called a square number.

Ex: – 1) square of 9 = 92 = 9× 9 = 18, 2) square of 4 = 42 = 4× 4

Perfect Square:  A natural number is called a perfect square if it is the square of some natural number.

Ex: – 1,4,9, …etc.

Properties of perfect square:

1. The square of an even numbers is always an even number.

Ex: – 22 = 4 (4 is even), 62 = 36 (36 is even), here 2, 6 are an even number.

1. The square of an odd number is always an add number.

Ex: – 32 = 9 (9 is even), 152 = 225 (225 is even), here 3, 15 are an odd number.

• The square of a proper fraction is as proper fraction less than the given fraction.

Ex: –

1. The square of decimal fraction less than 1 is smaller than the given decimal.

Ex: – (0.3)2 = 0.09 < 0.03.

1. A number ending with 2, 3, 7 or 8 is never a perfect square.

Ex: – 72, 58, 23 are not perfect squares.

1. A number ending with an odd no. of zeros is never a perfect square

Ex: – 20, 120,1000 and so on.

Patterns in square numbers:

1. 1 + 3 = 4 = 22

1 + 3+ 5 = 9 = 32

1 + 3 + 5 +7 = 16 = 42

…………………………….

⇒ sum of n odd natural numbers = n2

1. Difference between two consecutive square numbers:

22 − 11 =4 −1 = 3 = 2 + 1

32 − 21 =9 −4 = 5 = 3 + 2

42 − 31 =16 −9 = 7 = 3 + 4

⇒ for any natural number ‘m’, (m + 1)2 – m2 = (m+1) + m

1. Pythagorean triplet:

Three natural numbers a, b and c are said to form a Pythagorean triplet if, c2 = a2 + b2

For every natural number a > 1, (2a, a2 – 1, a2 + 1).

Ex: – if we put a = 3 in (2a, a2 – 1, a2 + 1), then we get Pythagorean triplet (6, 8, 10).

1. Between two consecutive square numbers m2 and (m + 1)2, there are 2m non-perfect square numbers.

Ex: – 22, 32 are two consecutive square numbers

Non-perfect square numbers between 22 and 32 are:5, 6, 7, and 8

⇒ 2(2) = 4 Non-perfect square numbers are there in between 22 and 32

1. Using the identities (a + b)2 = a2 + 2ab + b2, (a – b)2 = a2 – 2ab + b2 to evaluate square numbers.

Ex: – 122 = (10 + 2) 2 = 102 + 2 (10) (2) + 22 = 100 + 400 + 4 = 144

92 = (10 – 1)2 = 102 – 2 (10) (1) + 12 = 100 – 20 + 1 = 81

1. Using the identity (a – b) (a + b) = a2 – b2 to find the product of two consecutive odd or two consecutive even numbers.

Ex: – 9 × 11 = (10 – 1) (10 + 1) = 102 – 1 = 99

20 × 22 = (21 – 1) (21 + 1) = 212 – 1 = 441 – 1 = 440.

Square Root: the square root of a number x is that number when multiplied by itself gives x as the product. The square root of x is denoted by .

Ex: –

Methods of Finding Square root of given Number

Prime factorization method: –

Steps:

1. Resolve the given number into prime factors.
2. Make pairs of similar factors.
3. The product of prime factors, choosing one out of every pair gives the square root of the given number.

Ex: – 16

Prim factors of 16 = 2 ×2× 2× 2

= 2 × 2 = 4

∴ square root of 16 = 4

Division method: –

Steps:

1. Mark off the digits in pairs starting with the unit place. Each pair and remaining one digit are called a period.
2. Think of the largest number whose square is equal to or just less than the first period. Take this number as the divisor as well as quotient.
3. Subtract the product of divisor and quotient from the first period and bring down the next period to the right of the remainder. this becomes the new dividend.
4. Now, a new divisor is obtained by taking twice the quotient and annexing with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of the new divisor and this digit is equal to or just less than the new dividend.
5. Repeat steps 2, 3, and 4 till all the periods have been taken up. Thus, the obtained quotient is the required square root.

Finding the square root through subtraction of successive odd numbers:

• Subtract first odd number (1) from a given number
• Subtract the second odd number (3) from the above result.
• Continue this process until the result will be zero (0).
• Count the steps involved above the process. No. of steps is the required answer.

Ex: find square root of 16

16 – 1 = 15; 15 – 3 = 12; 12 – 5 = 7; 7 – 7 = 0

After 4 steps we got 0.

∴ square root of 16 = 4.

The square root of a number in decimal form

Make the no. of decimal places even, by affixing a zero, if necessary. Now periods and find out the square root by the long division method.

Put the decimal point in the square root as soon as the integral part is exhausted.

Ex: – To find the square root of 79.21

The square root of a decimal number which is not perfect square:

if the square root is required to correct up to two places of decimal, we shall find it up 3 places of decimal and then round it off up to two decimal places.

if the square root is required to correct up to three places of decimal, we shall find it up 4 places of decimal and then round it off up to three decimal places.

Ex: – To find the square root of 0.8 up to 2 decimal places

Cube of a number:

The cube of a number is that number raised to the power 3.

Ex: – cube of 0.3 = 0.33 = 0.027

Cube of 2 = 23 = 8

Perfect cube:

If a number is a perfect cube, then it can be written as the cube of some natural numbers.

Ex: – 1, 8, 27, and so on.

Cube root:

The cube root of a number x is that number which when multiplied by itself three times gives x as the product.

Cube root of x is denoted by

Methods of finding the cube root of given Number

Prime factorization method: –

Steps:

1. Resolve the given number into prime factors.
2. Make triplets of similar factors.
3. The product of prime factors, choosing one out of every triplet gives the cube root of the given number.

Ex: – 27

Prim factors of 27 = 3 ×3×3

= 3

∴ cube root of 27 = 3

Estimating the cube root of a number:

Ex:  estimate the cube root of 2744

Start making groups of through estimation. The first group is 744 and the second group is 2

2      744

The first group i.e., 744will give us the units digit of the cube root. As 744 ends with 4, its cube root also ends with 4. So, the unit place of cube root will be 4.

In second group number is 2

We know that 13 < 2 < 23

As the smallest number is 1, t becomes the tens place of the required cube root.

∴  cube root of 2744 = 14.

## 7. FREQUENCY DISTRIBUTION TABLES AND GRAPHS

Data: An information available in the numerical form or verbal form or graphical form that helps in taking decisions or drawing conclusions is called data.

Measures of central tendency:

The measures of central tendency are 3 types. They are: 1. Arithmetic mean 2.  Median and 3.  Mode.

1.Arithmetic Mean:

Arithmetic mean of x1, x2, x3, …. x n is   ⇒

Where ∑xi represents the sum of all xi ’s; ‘i’ takes the values from 1 to n.

Arithmetic mean by deviation Method: –

A is assumed mean.

∎Sum of the deviations of all observations from the estimated mean is zero.

∎Arithmetic mean is a representative value of the entire data.

∎Arithmetic mean depends on both no. of observations and value of each observation in a data.

∎Arithmetic mean is unique value of data.

∎When all the observations of the data are increased or decreased by a certain number, the mean also increase or decrease by the same number.

∎When all the observations of the data are multiplied or divided by a certain number, the mean also multiplied or divided by the same number.

2.Median:

Median is the middle most value of the given data.

First, we arrange given data in ascending or descending order.

If n is the no. of observation in a data, then

Median =  observation, when n is odd.

Median =   when n is even.

∎Median is the middle most observation of the data.

∎It depends on no. of observations and middle observations of the ordered data

∎ It not effected by any change in extreme values.

3.Mode:

Mode is the most frequently occurring observation of given data.

∎Mode depends neither on no. of observations nor value of all observations.

∎It is used to analyse both numerical and verbal data.

∎ There may be two or three or many modes for the same data.

Grouped data:

If we organize the data by dividing it into convenient groups, then it is called Grouped data.

Frequency distribution or Frequency table:

Representation of classified distinct observations of the data with frequencies is called frequency distribution.

Class intervals: Small groups in a data are called Class intervals

Ex: 0 – 5, 5- 10, …

Limits and boundaries:

In the class interval 5 – 10, 5 is called lower limit and 10 is called upper limit.

Boundaries: Average of upper limit of first class and lower limit of second class is becomes the upper boundary first class and lower boundary of second class.

These boundaries are also called ‘true class limits’

Length of the class: Difference between upper and lower boundaries of a class is called Length of the class.

From the above table length of the class 0.5 – 10.5 is 10.5 – 0.5 = 10

Range: The difference between highest and least value of given data is called range of the data.

Construction of grouped frequency Distribution:

Ex: 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7

Highest value = 7; least value = 1

Range = 7 – 1 = 6.

Class interval =

=  = 0.8 approximately.

Characteristics of grouped frequency distribution:

1. It divide the data onto convenient and small groups called class intervals.1.
2. Class intervals like 1 – 10, 11 – 20 … are called inclusive class intervals. Both lower and upper limits of a particular class belong to that particular class.
3. Class intervals like 0 – 10, 10 – 20 … are called exclusive class intervals. Only lower limit of particular class belongs to that class but not its upper limit.
4. In exclusive class intervals, both limits and boundaries are equal.
5. In inclusive class intervals, limits and boundaries are not equal.
6. Individual values of all observations can not be identified from the frequency table, but the value of each observation of a particular class assumed to be the average of upper and lower boundaries of that class. This value is called ‘class mark’ or ‘mid value’.

Less than and greater than Cumulative frequencies:

The distribution that represents upper boundaries of the classes and their respective less than cumulative frequencies is called ‘less than cumulative frequency distribution’

The distribution that represents lower boundaries of the classes and their respective greater than cumulative frequencies is called ‘greater than cumulative frequency distribution’

 Class intervals frequency LB Greater than cumulative frequency UB Less than cumulative frequency 0 – 5 7 0 36 + 7 = 43 5 7 5 – 10 10 5 26 + 10 = 36 10 7+10 = 17 10 – 15 15 10 11 + 15 = 26 15 17 + 15 = 32 15 – 20 8 15 3 + 8 = 11 20 32 + 8 = 40 20 – 25 3 20 3 25 40 + 3 = 43

Graphical representation of the Data:

Bar graph:

A display of information using vertical or horizontal bars of uniform width and different lengths being proportional to the respective values is called bar graph.

Ex:

Histogram:

A graphical representation of frequency distribution of exclusive class intervals is called histogram.

Ex:

Steps on Construction:

Step-1:  If class intervals are inclusive, then convert them into the exclusive form

Step-2: Choose a suitable scale on the X – axis and mark the class intervals on it.

Step-3: Choose a suitable scale on the Y – axis and mark the frequencies on it.

Scale: On X – axis 1cm = 10 units

On Y – axis 1 cm = 10 units

Step-4: Draw rectangle with class intervals as bases and the corresponding frequencies as the corresponding heights.

Histogram with varying Base widths:

Ex:

Steps on Construction:

Step-1:  If class intervals are inclusive, then convert them into the exclusive form

Step-2: Choose a suitable scale on the X – axis and mark the class intervals on it.

Step-3: Choose a suitable scale on the Y – axis and mark the frequencies on it.

Scale: On X – axis 1cm = 10 units

On Y – axis 1 cm = 10 units

Step-4: Draw rectangle with class intervals as bases and the corresponding frequencies as the corresponding heights.

Frequency polygon:

Ex:

Steps on Construction:

Step-1:  Calculate the mid points of every class interval given in the data.

Step-2: Choose a suitable scale on the X – axis and mark the class intervals on it.

Step-3: Choose a suitable scale on the Y – axis and mark the frequencies on it.

Scale: On X – axis 1cm = 10 units

On Y – axis 1 cm = 10 units

Step-4: Draw the histogram for this data and mark the midpoints of the tope

Step-5: Join the mid points successfully.

Frequency curve:

Ex:

Steps on Construction:

Step-1:  find the class mark of the class intervals.

Step-2: Choose a suitable scale on the X – axis and mark the class intervals on it.

Step-3: Choose a suitable scale on the Y – axis and mark the frequencies on it.

Scale: On X – axis 1cm = 10 units

On Y – axis 1 cm = 2 units

Step-4:  Plot the points (which are in the above table) on graph

Step-5: Join the consecutive points by a free hand curve.

Less than cumulative frequency curve:

Ex:

 Class Frequency UB L. C. F 0 – 10 2 10 2 10 – 20 5 20 7 20 – 30 3 30 10 30 – 40 1 40 11 40 – 50 4 50 15 50 – 60 2 60 17

Steps on Construction:

Step-1:  If class intervals are inclusive, then convert them into the exclusive form

Step-2: Construct the less than cumulative frequency table.

Step-3: Choose a suitable scale on the X – axis and mark the upper boundaries class intervals on it.

Choose a suitable scale on the Y – axis and mark the cumulative frequencies on it.

Scale: On X – axis 1cm = 10 units

On Y – axis 1 cm = 2 units

Step-4:  If ‘x’ denotes the upper boundary of class interval and ‘y’ denotes the corresponding cumulative frequency of particular class, then plot (x, y) on the graph.

## 8.EXPLORING GEOMETRICAL FIGURES

Congruent figures: The figures which have same shape and size are called congruent figures.

Flip: Flip is a transformation in which a plane figure is reflected across a line, creating a mirror image of the original figure.

Rotation: Turning round centre is called Rotation. The distance from the centre to any point on the shape stays the same. Every point makes a circular round the centre.

Similar figures: The figures which have same shape but not in size are called similar figures.

Dilation: The method of drawing enlarged or reduced similar figures is called Dilation.

Constructing Dilation:

Ex: Construct a dilation with scale factor 3, of a triangle

Steps on Construction:

Step-1:  Draw a ∆ABC and choose the centre of dilation O which is not on the triangle. Join every vertex of the triangle from O and produce

Step-2: By using compasses, mark three points A’, B’, C’ on the projection so that OA’ = 3OA; OB’ = 3OB and OC’ = 3OC.

Step-3:  Join A’B’, B’C’, C’A’. We observe that ∆ABC~∆A’B’C’

Symmetry:

In symmetry there are 3 types: 1. Line of symmetry, 2. Rotational symmetry and 3. Point symmetry.

1.Line of symmetry: The lines which cuts the figures exactly halves is called line of symmetry.

2.Rotational symmetry:

When an object is rotated about its centre, it comes same position after some rotation, then it is called rotational symmetry.

No. of rotations to get initial position of an object is called ‘order of position.

Ex: When a rectangle is rotated about its centre its shape resembles the initial position two times.

Order of rotation of rectangle is 2.

3.Point symmetry:

The figure looks the same either we see it from upside or it from down side is called ‘point of symmetry.

Ex: H, S, I have point of symmetry.

Tessellations: The patterns formed by repeating figures to fill a plane without gaps or overlaps are called ‘Tessellations’.

## 9.AREA OF PLANE FIGURES

Area of triangle:

The area of triangle with base ‘b’ and height ‘h’ is  square units

Area of Rectangle:

The area of rectangle with breadth ‘b’ and length ‘l’ is l × b square units.

Area of Square:

The area of triangle with side ‘a’ is a × a = a2 square units.
Area of parallelogram:

The area of parallelogram with base ‘b’ and height ‘h’ is b × h square units.

Area of Rhombus:

The area of Rhombus with lengths of diagonal d1, d2 is square units.

Area of Trapezium:

The area of Trapezium whose lengths of parallel sides a, b and distance between the parallel side’s ‘h’ is    square units.

The area of Quadrilateral whose lengths of perpendiculars drawn from vertices to diagonal are h1, h2 and length of the diagonal ‘a’ is     square units.

.Area of circle:

The area of circle with radius ‘r’ is π r
2 square units.

Area of circular path or area of Ring:

Area of ring = area of outer circle – area of inner circle

= πR 2 – π r2

= π(R 2 –  r2) square units.

Length of the arc:

Length of the arc of a sector (l) is

Area of sector:

area of sector
=

## 10.DIRECT AND INVERSE PROPORTIONS

Proportion: If a : b = c : d, then a, b, c and d are in proportion.

Direct proportion:

x and y are any two quantities are said to be in direct proportion, if x is increase (decrease), then y is increase (decrease).

where k is any constant

If x1 and x2 are the values of x corresponding to the values of y1 and y2 of y respectively, then

Inverse proportion:

x and y are any two quantities are said to be in inverse proportion, if x is increase (decrease), then y is decrease (increase).

xy = k where k is any constant

If x1 and x2 are the values of x corresponding to the values of y1 and y2 of y respectively, then

⇒ x1y1 = x2y2.

Compound proportion:

Change in one quantity depends upon the change in two or more quantities in some proportion, then we equate the ratio of the first quantity to the compound ratio of the other two quantities.

• One quantity may be in direct proportion with the other two quantities.
• One quantity may be in inverse proportion with the other two quantities.
• One quantity may be in direct proportion with one of the two quantities and inverse proportion to with the remaining quantity.

## 11.ALGEBRAIC EXPRESSIONS

Term:  Term is the product of constant and one or more variables.

Ex: 2x, 3xy. 5x2yz etc.

Algebraic Expression: Terms are added or subtracted to form an Algebraic Expression.

Ex; 2x + 3, 2y – 3x, 4xyz – 3x3y etc.

Monomial: If an expression contains only one term then it is called monomial.

Ex; x, 3x, – 5yz

Binomial: If an expression contains two terms, then it is called Binomial.

Ex: x + 3, x – y, 3xy – 2zx etc.

Trinomial: If an expression contains three terms, then it is called Trinomial.

Ex: x + 3 – y, x + 3xy – y, 3xy – 2z + x etc.

Like and Unlike terms: If the terms have same variable with same exponent then they are called Like terms, other wise they are called Unlike terms.

Ex: 2xy, 5yx, – 4xy are like terms

2xy, 5yz, 6zx are Unlike terms.

Ex:  Add 4x2 – 3xy + 2y2 and x2 + xy – 6y2

Subtraction of algebraic expressions:
Ex:
Subtract x2 – 2xy + 3y2 from 5x2 + 6xy – y2

Multiplication of Algebraic expressions:

For finding the product of algebraic terms we add the power of same base variables.

1.Multiplying two monomials: –

Ex: 3 × x = x + x + x = 3x.

5x × 3y = (5 × 3) × (x × y) = 15 × xy = 15xy

5x × 3x = (5 × 3) × (x × x) = 15 × x2 = 15x2(5 × 3) × (x × y) = 15 × xy = 15xy

2.Multiplying three or more monomials: –

Ex: 3 × x × y= 3xy.

5x × 3y × 4z = (5 × 3 × 4) × (x × y × z) = 60 × xyz = 60xyz

3x2 × (– 4x) × 2x3 × 2 = (3 × – 4 × 2 × 2) × (x2 × x × x3) = – 48 x6

3.Multiplying a binomial by a monomial: –

Ex: 5x (3x – 4y) = (5x × 3x) + (5x × – 4y) = 15x2 – 20xy

4.Multiplying a Trinomial by a monomial: –

Ex: 5x (3x – 4y + 4z) = (5x × 3x) + (5x × – 4y) + (5x × 4z) = 15x2 – 20xy + 20 xz

5.Multiplying a Binomial by a Binomial: –

Ex: (x + y) (2x – 3y) = x (2x – 3y) + y (2x – 3y) = 2x2 – 6 xy + 2xy – 3 y2 = 2x2 – 4xy – 3y2

6.Multiplying a Binomial by a Trinomial: –

Ex: (x + y) (2x – 3y + z) = x (2x – 3y + z) + y (2x – 3y + z)

= 2x2 – 6 xy + xz + 2xy – 3 y2 + yz

= 2x2 – 4xy – 3y2 + xz + yz

Identity:  An equation is called an identity if it is satisfied by any value that replaces its variables. An equation is true for certain values for the variable in it, where as an identity is true for all its variables. There fore it is known as universally true equation.

Symbol for identity is denoted by ‘≡’ (read as identically equal to)

Some important identities:

• (a +b)2 ≡ a2 + 2ab + b2
• (a – b)2 ≡ a2 – 2ab + b2
• (a + b) (a – b) ≡ a2 – b2
• (a + b + c)2 ≡ a2 + b2 + c2 + 2ab + 2bc + 2ca
• (x + a) (x + b) ≡ x2 + (a + b) x + ab.

Geometrical verification of (a +b)2 ≡ a2 + 2ab + b2

Consider a square with side a + b

Area of square = (a + b)2

Procedure:

•Divide the square into four regions as shown in the figure.

•It consists of two squares with side ‘a’ and side ‘b’ respectively and two rectangles with length and breadth as ‘a’ and ‘b’ respectively.

•The area of given square is equal to sum of the areas of four regions.

⇒ Area of square = area of square with side a + area of square with side b + area of rectangle with sides a and b + area of the rectangle with sides and b

⇒ (a + b)2 = a2 + b2 + ab + ba

(a + b) 2 = a2 + 2ab + b2

∴ (a +b)2 ≡ a2 + 2ab + b2

Geometrical verification of (a – b)2 ≡ a2 – 2ab + b2

Consider the square with side ‘a’

The square is divided into four regions I, II, III and IV

Area of square = area of region I + area of region II + area of region III + area of region IV

a2 = b (a – b) + b2 + b (a – b) + (a – b)2

a2 = ab – b2 + b2 + ab – b2 + (a – b)2

a2 = ab + ab – b2 + (a – b)2

⇒ (a – b)2 = a2 – ab – ab + b2

(a – b)2 = a2 – 2ab + b2

Geometrical verification of (a + b) (a – b) ≡ a2 – b2

Consider the square with side ‘a’

Remove the square from this whose side is ‘b’ units, we get

a2 – b2 = area of region I + area of region II

= a (a – b) + b (a – b)

= (a – b) (a + b)

∴ (a + b) (a – b) ≡ a2 – b2

## 12.FACFTORISATION

Factorisation:

The process of writing given expression as a product of its factors is called ‘Factorisation’.

It is helped to write the algebraic expressions in simpler form.

Irreducible factor:

A factor which can not be further expressed as product of factors is an irreducible factor.

Factorisation by Method of common factors:

Ex: Factorise 3x + 15

3x + 15 = (3 × x) + (3 ×5) (writing each term as the product of irreducible factors)

3 is the common factor of both terms

Take 3 as the common

3x + 15 = 3 × (x + 5) = 3 (x + 5)

Factorisation by grouping the terms:

Ex: Factorise ax + by + ay + bx

Firs group the like terms

ax + by + ay + bx = (ax + bx) + (ay + by)

= x (a + b) + y (a + b) (by taking out common factor from each term)

= (a + b) (x + y) (by taking out common factor from each term)

Factorisation by using identities:

• (a +b)2 ≡ a2 + 2ab + b2
• (a – b)2 ≡ a2 – 2ab + b2
• (a + b) (a – b) ≡ a2 – b2 are the algebraic identities.

Example 1:

Factorise x2 + 4x + 4

Sol: x2 + 4x + 4 = x2 + 2 (2)(x) + (2)2

It is in the form of identity (a + b)2 = a2 + 2ab + b2

x2 + 4x + 4 = (x + 2)2 = (x + 2) (x + 2).

Example 2:

Factorise x2 – 4x + 4

Sol: x2 – 4x + 4= x2 –2 (2)(x) + (2)2

It is in the form of identity (a – b)2 = a2 – 2ab + b2

x2 + 4x + 4 = = x2 –2 (2)(x) + (2)2 =(x – 2)2 = (x – 2) (x –2).

Example 3:

Factorise 4x2 – 9y2

Sol: 4x2 – 9y2 = (2x)2 – (3y)2

It is in the form of identity (a – b) (a + b) = a2 – b2

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

Factors of the form (x + a) (x + b) = x2 + (a + b) x + ab:

Ex: x2 + 12x + 35

Here we have to find out factors 35 whose sum is 12

35 = 1 × 35               1 + 35 = 36

–1 × –35           –1 –35 = –36

7 × 5                 7 + 5 = 12

–7 × –5              –7 –5 = – 12

Now x2 + 12x + 35 = x2 + (7 + 5) x + 35

= x2 + 7x + 5x + 35

= x (x + 7) + 5 (x + 7)

= (x + 7) (x + 5)

Division of algebraic Expression:

1.Division of a monomial by another monomial:

Ex: 12x5 ÷ 3x

12x5 ÷ 3x =  =

= 4x4

2.Division of an expression by a monomial:

Ex: 4x3 + 10 x2 + 8x ÷ 2x

4x3 + 10 x2 + 8x = 2 × 2 × x × x × x + 2 × 5 × x × x + 2 × 2× 2 × x

= (2x) (2x2) + (2x) (5x) + (2x) (4)

= (2x) (2x3 + 5x + 4)

4x3 + 10 x2 + 8x ÷ 2x =

=

= 2x2 + 5x + 4

3.Division of an Expression by Expression:

Ex: (5x2 + 15x) ÷ (x + 3)

5x2 + 15x = 5x (x + 3)

(5x2 + 15x) ÷ (x + 3) =

=

= 5x

## 13.VISUALIZING 3-D IN 2-D

Various Geometrical Solids:

Some solids (3 – D objects) have flat faces and some solids have curved faces.

Polyhedron: 3 – D objects which have flat surfaces are called polyhedron.

Ex: book, dice, cube etc.

Non – Polyhedron: 3 – D objects which have curved faces are called Non – polyhedron.

Ex: ball, pipe etc.

Faces, Edges, and Vertices of 3D – objects:

Regular polyhedron:

The polyhedron, which has congruent faces, equal edges and vertices are formed by equal no. of edges is called regular polyhedron.

Ex: Cube, Tetrahedron.

Prism: The soiled object with two parallel and congruent polygonal faces and lateral faces as rectangles or parallelograms is called a prism.

If the base of the prism is triangle, then it is called triangular prism.

If the base of the prism is square, then it is called square prism.

If the base of the prism is pentagon, then it is called pentagonal prism.

Euler’s Relation (Formula):

E + 2 = F + V

Where E = No. of edges;

F = No. of faces and

V = No. of vertices

Net diagrams:

A net is a short of skeleton – outline in 2 – D, which, when folded the net results in 3 – D shape.

Ex:

Tetrahedron

Cube

## 14.SURFACE AREAS AND VOLUMES

Cuboid:

Lateral surface area (L.S.A) = 2h (l + b) square units.

Total surface area (T.S.A) = 2 (lb + bh + hl) square units.

Volume = lbh cubic units.

Cube:

Lateral surface area (L.S.A) = 4 a2 square units.

Total surface area (T.S.A) = 6 a2 square units.

Volume = a3 cubic units.

We measure volume of liquids in millilitres(ml) or litres(l)

1cm3 = 1 ml.

1000 cm3 = 1l.

1m3 = 1000000cm3 = 1000 l. = 1 kl. (kilo litre).

## 15.PLAYING WITH NUMBERS

Divisibility:

If a number ‘a’ divides another number ‘b’ completely, then ‘b’ is divisible by ‘a’.

Place value of digit:

Place value of 7 is 7 000000.

Place value of 6 is 6000

Place value of 3 is 30

Divisibility Rules:

Divisibility rule by 2: –

If the unit place of a given number is 0, 2, 4, 6, 8 then that number is divisible by 2.

Ex: 10, 12, 526 etc.

Divisibility rule by 3: –

If the sum of the digits of a given number is divisible by 3, then that number is divisible by 3.

Ex: 234

Sum of the digits = 2 + 3 + 4 = 9

9 is divisible 3

∴ 234 is divisible by 3

Divisibility rule by 4: –

If the last two digits of a given number is divisible by 4, then that number is divisible by 4.

Ex: 324

24 is divisible by 4

∴ 324 is divisible by 4

∴ 324 is divisible by 4

Divisibility rule by 5: –

If the units place of given number is 0 or 5, then it is divisible by 5.

Ex: 10, 15, 235, 480 etc.

Divisibility rule by 6: –

If a number is divisible by both 3 and 2 then that number is divisible by 6.

Ex: 324

324 is divisible by both 3 and 2

∴ 324 is divisible by 6

Divisibility rule by 7: –

Fist multiply the last digit of given number by 2,

subtract this result from the number formed by remaining digits of given number.

If that result is divisible by 7, then the given number is divisible by 7.

Ex: 112

Last digit is 2 ⇒ 2 × 2 = 4

Now 11 – 4 = 7

7 is divisible by 7

∴ 112 is divisible by 7.

Divisibility by 8: –

if the last three digits of a number is divisible by 8, then that number is divisible by 8.

Ex: – 4232, last three digits 232 are divisible by 8

∴ 4232 is divisible by 8.

Divisibility by 9: –

if the sum of the digits of a number is divisible by 9, then that number is divisible by 9.

Ex: – 459, 4 + 5 + 9 = 18 → 18 is divisible by 9       ∴ 459 is divisible by 9

532, 5 + 3 + 2 = 10 → 10 is not divisible by 9       ∴ 532 is not divisible by 9.

Divisibility by 10: –

a number is divisible by 10, if its once place is 0.

Ex: – 20 is divisible by 10. 22, 45 are not divisible by 10.

Divisibility by 11: –

A number is divisible by 11, if the difference between the sum of the digits at odd places and the sum of the digits at even places is either 0 or 11.

Ex: – 6545

Sum of the digits at odd places = 5 + 5 = 10

Sum of the digits at even places = 4 + 6 = 10

Now difference is 10 – 10 = 0

∴ 6545 is divisible by 11.

# This note is designed by the ‘Basics in Maths’ team. These notes to do help students fall in love with mathematics and overcome fear.

## 1.REAL NUMBERS

### Rational numbers:-

• The numbers which are written in the form of, where p, q are integers and q≠ 0 are called rational numbers. Rational numbers are denoted by Q.

ex:-  3/2, 3/5, 2, 1 and so on

• Natural numbers, Whole numbers, and Integers are rational numbers.
• The rational numbers do not have a unique representation.

Representation of rational number:

Represent

To find a rational number between given numbers:

Mean method:- A rational number between two numbers a and b is

Ex:- insert two rational number between 1 and 2

To find a rational number in a single step:-

Ex:- insert two rational number between 1 and 2

To find two rational numbers, we 1 and 2 as rational numbers with same denominator 3     (∵ 1 + 2 = 3)

The decimal form of rational numbers:

• Note:- Every rational number can be expressed as a terminating decimal or non-terminating repeating decimal.
• Converting decimal form into a fraction:
1. Terminating decimals:-  (i) 1.2 = 12/10 = 6/5

(ii) 1.35 =135/100 = 135/100 = 27/20

1. Non-Terminating repeating decimals:-

### Irrational numbers:

• The numbers which are not written in the form of, where p, q are integers and q ≠ 0 are called rational numbers. Rational numbers are denoted by QI or S.
• Every irrational number can be expressed as a non-terminating decimal or non-repeating decimal.

Ex:-

Calculation of square roots:

• There is a reference of irrationals in the calculation of square roots in Sulbha Sutra.
• Procedure to find   value:

#### Representing irrational numbers on a number line:

Ex:- Locate    on a number line

• At ‘O’ draw a unit square OABC on a number line with each side 1 unit in length.
• By Pythagoras theorem                          OB2 = OA2 + AB2

=  12 + 12

OB2 = 2

OB =

• Using a compass with centre O and radius OB, draw an arc on the right side to O intersecting the number line at the point
• The location of is now at k.

• Note:-  If a and b are two positive rational numbers such that ab is not a perfect square, this an irrational number lies between a and b.

### Real numbers

• The collection of all rational and irrational numbers is called real numbers.
• Real numbers cover all the points on the number line.
• Every real number is represented by a unique point on the number line.
• Ex:-   are some examples of real numbers.

#### Representing real numbers on the number line through successive magnifications:-

locating 2. 775 on a number line

### Operation on real numbers

• The sum, difference, product and quotient of irrational numbers need not be an irrational number.
• Irrational numbers are not closed under addition, subtraction, multiplication, and division.
• For any two real numbers a and b

#### Rationalizing the denominator:

• Rationalizing factor(R.F):-If the product of two irrational numbers is rational, then each of the two is the rationalizing factor to others.
• The rationalizing factor of a given irrational number is not unique. It is convenient to use the simplest of all R.F.s of given irrational number.
• Note:-

## 2. POLYNOMIALS AND FACTORIZATION

Polynomial: An algebraic expression in which the variables involved have only whole number powers is called a polynomial.

Ex: x2 , x3 + 1, x2 + xy + y2  and so on.

Polynomials in one variable: The polynomials which are in the form of (a constant) × (some power of variable) are called polynomials in one variable.

Ex: 2x, 4x, 3x2 + 1 and so on.

Degree of the polynomial: The degree of a term is the sum of the exponent of its variable factors. The degree of the polynomial is the highest power of its variable term.

Ex:  degree of 3x2 + 2x3 + 1 is 3

degree of 5x2y3 + 2xy + 3x3 is 5

a polynomial in one variable x of degree n is anxn + an-1xn-1 + …+a1x + a0. Where a0, a1…an are constants and an ≠ 0.

Types of polynomials:

1. According to no. of terms: –
 No. of non-zero terms Name of the polynomial Examples Terms 1 Monomial 3x 3x 2 Binomial -2 x + 7 -2x, 7 3 Trinomial 5x2 + 4x + 2 5x2, 4x, 2 More than 3 Multinomial 6x3 – 5x2 + 7x – 3 6x3, -5x2, x, -3

According to a degree: –

 Degree of the polynomial Name of the polynomial Example Not defined Zero polynomial 0 0 Constant polynomial -12, 4, 7 etc. 1 Linear polynomial 2x+3, x – 3 etc. 2 Quadratic polynomial 2x2 + 3x + 1, x2 – 4 etc. 3 Cubic polynomial 3x3 – 4x2 + 2x + 6 4 Bi quadratic polynomial 4x4 + 2x3 + 45x2 +9x + 7

Zero of the polynomial: Let p(x) be a polynomial, if p(x) = 0 then, x is the zero of the polynomial p(x).

Ex: p(x) = 2x – 2

P(1) = 2(1) – 2 = 2 – 2 = 0

∴ 1 is the zero of the polynomial.

Zero of the linear polynomial in one variable:

 Linear polynomial Zero of the polynomial x+ a – a x – a a ax + b -b/a ax – b b/a

Dividing polynomials:

If p(x) is divided by g(x), then there exists quotient polynomial q(x) and remainder r(x) such that

p(x) = q(x) × g(x) + r(x)

this is called division algorithm for polynomials.

Remainder theorem:

Let p(x) be a polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial (x – a), then the remainder is p(a).

Ex: if p(x) = 3x2 – 4 x + 2 is divided by the polynomial (x – 1), then find remainder.

Ans: Given p(x) = 3x2 – 4 x + 2

Remainder is p (1)

⇒ p (1) = 3(1)2 – 4 (1) + 2 = 3 – 4 + 2 = 5 – 4 = 1

∴ remainder is 1.

Factor theorem:

If p(x) s a polynomial of degree greater than or equal to one and a is any real number, then x – a is a factor of p(x), if p(a) = 0  and its converse if (x – a) is a factor of p(x), then p(a) = 0.

Ex:  if p(x) = x2 – 2x + 1, then show that (x – 1) is a factor of p(x)

Ans: given   polynomial is p(x) = x2 – 2x + 1

P (1) = (1)2 – 2(1) + 1 = 1 – 2 + 1 = 2 – 2 = 0

∴ x – 1is the factor of x2 – 2x + 1.

Algebraic identities:

(i ) (x + y)2 ≡ x2 + 2xy + y2         (ii) (x − y)2≡x2 − 2xy + y2        (iii) (x + y)(x – y)≡x2 – y2

(iv) (x + a) (x + b) ≡ x2 + (a + b) x + ab   (v) (x + y + z)2 ≡ x2 + y2 + z2 + 2xy + 2yz + 2zx

(v) (x +y)3 ≡ x3 + 3x2y + 3xy2 + y3 ≡ x3 + y3 + 3xy (x + y)

(vi) (x − y)3 ≡ x3 − 3x2y + 3xy2 + y3 ≡ x3 − y3 + 3xy (x − y)

(vii)  (x + y + z)(x2+ y2 + z2 – xy – yz – zx) ≡ x3 + y3 + z3 – 3xyz.

## 3.THE ELEMENTS OF GEOMETRY

Geometry: The word geometry derived from the Greek word ‘geo’ means earth and ‘metron’ means measure.

Euclid’s Elements:  Euclid wrote 13 books called ‘The Elements’. Euclid creates the first system of thought based on fundamental definitions, axioms, propositions rules of inference or logic.

Some definitions of Euclid’s 1st book of Elements are: (i) A ‘point’ is that that which has no part. (ii) A ‘line’ is the breathless length. (iii) The ends of a line are points. (iv) A straight line is a line which lies evenly with the points itself.  (v) A ‘surface’ is that which has length and breadth only. (vi) The edge of the surface are lines. (vii) A plane surface is a surface which lies evenly with the straight lines on itself.

Note:  In geometry, a point, a line and a plane are undefined terms.

Axioms: Axioms are statements that are self-evident or assumed to be true within the context of a particular mathematical system. Axioms are elf evident facts and do not require any proof.

Some of Euclid’s Axioms are:

1. Things which are equal to the same things are equal to another.
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders are also equal.
4. Things which coincide with one another are equal.
5. Things which are double of the same things are equal.
6. Things that are halves of the same things are equal.

Postulates: Postulates are used for the assumptions made in the geometry.

Euclid’s five postulates:

Postulate – 1: There is a unique line that passes through the given two distinct points.

Postulate – 2: A-line segment can be extended on either side to form a straight line.

Postulate –3: We can escribe a circle with any centre and any radius.

Postulate – 4: All right angles are equal.

Postulate – 5: If a straight line falling on two straight line makes the interior angles on the same side of it taken together is less than two right angles, then two straight lines, if produced infinitely, meet on that side on which the sum of the angles is less than two right angles.

Equivalent versions of Euclid’s fifth postulate:

From the fig. sum of the angles, x and y is less than 1800

1. Through a point not on a given line, exactly one parallel line may be drawn o the given line (John Playfair).
2. The sum of the angles of any triangle is constant and is equal to two right angles (Legendre).
3. If a straight line intersects any one of two parallel lines, then it will intersect others also (Proclus).
4. Straight lines parallel to the same straight line are parallel to one another (Proclus).

Conjecture or Hypothesis: The statements which are neither proved nor disproved are called conjectures.

## 4. LINES AND ANGLES Line: Line can be extended in both directions endlessly.

Ray:  It is a part of the line. It begins at a point and goes on endlessly in a specific direction.

Line segment: A part of the line with two endpoints is called a line segment.

Collinear points: If three or more points lie on the same line, then they are called collinear points.
A B and C are collinear points

Note: if ‘n’ points lie on a line, then no. of line segments =

Angle:  An angle is formed when two rays originate from the same point. The rays making an angle are called ‘Arms’ of the angle. The common point is called ‘vertex’.

Intersecting and Non-intersecting lines:  If two lines meet at any point, then the lines are intersecting lines. If two lines never meet at any point are called non-intersecting lines or parallel lines.

Concurrent lines: If two or more lines meet at the point, then that lines are called concurrent lines.
Complementary angles:  Two angles are said to be complementary angles if their um is 900.

The complementary angle of x0 is 900 – x0.

Supplementary angles:  Two angles are said to be supplementary angles if their um is 1800.

the supplementary angle of x0 is 1800 – x0.

Linear pair of angles: If a ray stands on a straight line, then the sum of the two adjacent angles is so formed is 1800.

Note: If the sum of two adjacent angles is 1800, then they are called linear pairs of angles.

If the sum of two adjacent angles is 1800, then they are called linear pairs of angles.

Note: – if the sum of two adjacent angles is 1800, then non-common arms of the angles form a line. This is the converse of a linear pair of angle axiom.

Angles at a point: We know that the sum of all the angles around a point is always 3600.

From the figure ∠ 1 + ∠2 ∠ 3 + ∠4 ∠5 = 3600

Vertically opposite angles:  When two lines intersect at a point, the angles with the same vertex and have no common arm are called vertically opposite angles.

Note:
If two lines intersect each other, then the pairs of vertically opposite angles are equal.
∠AOD, ∠BOC; ∠AOC, ∠BOD are vertically opposite angles.

⟹ ∠AOD = ∠BOC; ∠AOC = ∠BOD.

Lines and a transversal:

Transversal: A-line that intersects two or more lines at distinct points is called a transversal.

∠1, ∠2, ∠7 and ∠8 are exterior angles

∠3, ∠4, ∠5 and ∠6 are interior angles

Corresponding angles: –

Two angles which are lies the same side of the transversal and one interior and another one exterior are called corresponding angles.

∠1, ∠5; ∠2, ∠6; ∠3, ∠7 and ∠4, ∠8

Alternate angles: –

Two angles which are the lies opposite sides of the transversal and both interior or exterior are called corresponding angles.

∠1, ∠7; ∠2, ∠8 are exterior alternate angles

∠3, ∠5; ∠4, ∠6 are interior alternate angles.

∠3, ∠6; ∠4, ∠5 interior angles the same side of the transversal.

Transversal on parallel lines:

∗ If pair of parallel lines are intersected by a transversal then the angles each pair of corresponding angles are equal

⇒ ∠1, =∠5; ∠2= ∠6; ∠3= ∠7 and ∠4= ∠8

∗ If pair of parallel lines are intersected by a transversal then the angles each pair of interior alternate angles are equal.

∠3= ∠5; ∠4= ∠6

∗ If pair of parallel lines are intersected by a transversal then the angles each pair of exterior alternate angles are equal.

∠1= ∠7; ∠2= ∠8

∗ If pair of parallel lines are intersected by a transversal then the angles each pair of interior angles in the same side of the transversal are supplementary.

∠3+∠6= 1800; ∠4+ ∠5 = 1800

Note:

1. If a transversal intersects two lines and the pair of corresponding angles are equal, then the lines are parallel.
2. If a transversal intersects two lines and the pair of alternate angles are equal, then the lines are parallel.
3. If a transversal intersects two lines and the interior angles on the same side of the transversal are supplementary, then the lines are parallel.

Note: Lines that are parallel to the same line are parallel to each other.

Angle-sum property of a triangle:

The sum of the angles in a triangle is 1800

∠A + ∠B + ∠C = 1800

An exterior angle of a triangle:

When one side of the triangle is produced, the angle thus formed is called the exterior angle. Exterior angle property:- The exterior angle of a triangle is equal to the sum of two interior opposite angles.

x0+ y0 = z0

## 5.CO-ORDINATE GEOMETRY

The representation of a point on a plane with the idea of two references led to the development of a new branch of mathematics is known as Co-ordinate geometry.

Rene Descartes (1595 – 1650) developed the study of coordinate geometry. He found an association between algebraic equations and geometric curves and figures.

Cartesian system:

We use number line by making points on the line at equal distance.

Take two number line, perpendicular to each other in the plane. We locate the position of points concerning these lines

The point of intersection of these number lines is Origin it is denoted by ‘O’. The horizontal number line is XXI is called X-axis and the vertical number line YYI s called as Y-axis.

OX is a positive X-axis, OXΙ is a negative X-axis.

OY is a positive Y-axis, OYI is a negative Y-axis.

• The distance of a point from Y-axis is called x co-ordinate or
• The distance of a point from X-axis is called y co-ordinate or
• The co-ordinates of origin are (0,0).
• The x coordinate of a point on y-axis is 0.
• They c-ordinate of a point on y-axis is 0.
• Equation of x- axis is y = 0.
• Equation of y- axis is x = 0.

## 6.LINEAR EQUATIONS IN TWO VARIABLES

Equation: The expression which is connected by equality symbol (=) is called an equation.

Ex: x = 5, x + y = 2, x2 = 2y etc.

Linear equation: If the degree of the equation is 1, then it is called a linear equation.

Ex: x + 2y = 4, 3x – 5 = 0, 4y = 6 etc.

Linear equation in one variable: If a linear equation has only one variable, then it is called ‘linear equation in one variable’.

Ex: x + 3 = 4, 4x – 4 = 0, y – 3 = 0 etc.

Linear equation in two variables: If a linear equation has two variables, then it is called ‘linear equation in two variables.

Ex: x + 3y = 4, 4x – 4y = 0, y – 3x = 0 etc.

General form:

The general form of linear equation in two variables ‘x’ and ‘y’ is ax + by c = 0, where a, b, c are real numbers and a ≠ 0, b ≠ 0.

Solution: Any pair of values of ‘x’ and ‘y’ which satisfies the linear equations in two variables is called its solution.

Ex: x + y = 3

x = 2, y =1; x =1, y =2, x =0, y =3; x =3, y = 0 are some solutions of above equation.

∴ For a linear equation in two variables, we can find many solutions.

Note: An easy way of getting two solutions is put x = 0 and get the corresponding value of y. Similarly, we can put y= 0 and get the corresponding value of x.

Graph of linear equation in two variables:

Ex: x + y = 3

Sol: Given equation is x + y = 3

y = 3 – x

 x 0 1 2 3 y = 3 – x 3 2 1 0 (x, y) (0, 3) (1, 2) (2, 1) (3, 0)

Steps to draw the graph of the linear equations in two variables:

1. Write the linear equation.
2. Put x = 0 in the given equation and find the corresponding value of y.
3. Put y = 0 in the given equation and find the corresponding value of x.
4. Write the values of ‘x’ and its corresponding value of ‘y’ as coordinates of x and y respectively as (x, y) form.
5. Plot the points on graph paper.
6. Join these points.

Observations from the graph:

• Every solution of the linear equation in two variables represents a point on the line of the equation.
• Every point on this line is a solution to the linear equation in two variables.
• Any point that does not lie on this line not a solution to the equation.
• The collection points that give the solution of the linear equation in two variables is the graph of the linear equation.

Equation of the line parallel to X-axis:

Equation of X – axis is y = 0.

Equation of any line parallel to X-axis is y = k, where k is the distance from above or below of the X-axis.

Equation of the line parallel to Y-axis:

Equation of Y – axis is x = 0.

Equation of any line parallel to Y-axis is x = k, where k is the distance from left or right of the Y-axis.

## 7.TRIANGLES

Triangle: A figure made up of three-line segments is called a triangle.

Congruent figures: The figures which have the same shape and same size are called congruent figures.

If two-line segments have equal in length, then they are congruent.

Congruency of Triangles:

Two triangles are said to be congruent if (i) their corresponding angles are equal (ii) their corresponding sides are equal.

Ex: In ∆ ABC, ∆ DEF

∠A ≅ ∠D; ∠B≅ ∠E; ∠C ≅ ∠F

AB ≅ DE; BC≅ EF; AC ≅ DF

∴∆ABC ≅ ∆DEF

If two triangles are congruent, then corresponding parts of congruent triangles are equal (CPCT)

The criterion for congruency of Triangles:

1.Side -Side -Side congruency (SSS):

If three side of a triangle is equal to the corresponding three sides of another triangle, then the triangles are congruent.

∴ ∆ABC ≅ ∆DEF

2. Side -Angle -Side congruency (SAS):

If two sides and the angle included between the two sides of a triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

∴ ∆ABC ≅ ∆DE

3.Angle – Side -Angle congruency (ASA):

If two angles and included side of a triangle are equal to the corresponding two angles and included side of another triangle, then the triangles are congruent.

∴ ∆ABC ≅ ∆DEF

4.Right angle – Hypotenuse – Side congruence (RHS):

If the hypotenuse and one side of a right-angled triangle are equal to the corresponding hypotenuse and side of the other right-angled triangle, then the triangles are Equal.

∴∆ABC ≅ ∆DEF

5.A.S congruence: If two triangles are congruent if any two pairs of angles and one pair of corresponding angles are equal.

Some properties of the triangle:

1.Angles opposite to the equal side of an Isosceles triangle is equal.

2.The sides opposite to equal angles of a triangle are equal.

In equalities in a triangle:

1.If two sides of a triangle are equal, the angles opposite to the longer side is larger.

2.In any triangle, the side opposite to larger angle is longer.

3.The sum of any two sides of the triangle is greater than the third side.

4.The difference of any two sides of a triangle is less than the third side.

Diagonal: The line segment joining any two opposite sides of a quadrilateral is called diagonal of the quadrilateral.

In a quadrilateral, there are 4 interior angles. The sum of these 4 angles is 3600.

⇒ the sum of the angles in a quadrilateral ABCD = 3600

i.e., ∠A + ∠B+ ∠C + ∠D = 3600

Trapezium: In a quadrilateral one pair of opposite sides are parallel, then it is called Trapezium.

Parallelogram:  In a quadrilateral two pair of opposite sides are parallel, then it is called a parallelogram.

Properties of parallelogram:

1.Opposite sides and opposite angles are equal.

2.Diagonals are bisected each other and not equal in length.

3.A diagonal of a parallelogram divides it into two congruent triangles.

Rectangle: In a parallelogram one of the angles is a right angle, then it is called Rectangle.

Properties of Rectangle:

1.Opposite sides are equal.

2.Each angle is 900

3.Diagonals are bisected each other and equal in length.

A diagonal of a parallelogram divides it into two congruent triangles.

Rhombus: In a parallelogram two adjacent sides are equal, then it is called Rhombus.

Properties of Rhombus:

1.All sides are equal.

2.Opposite angles are equal.

3.Diagonals are bisected each other perpendicularly and not equal in length.

Square: In a Rhombus one of the angles is 900, then it is called a square.

Properties of Square:

1.all sides are equal.

2.Each angle is 900

3.Diagonals are bisected each other and equal in length.

Mid-point theorem:

The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half of it.

In ∆ABC, D, E are midpoints of side AB and AC ⇒DE ∥ BC and DE = ½ BC.

The converse of mid-point theorem:

The line drawn through the mid-point of one of the sides of a triangle and parallel to another side will bisect the third side.

## 9.STATISTICS

Data: The facts or figures which are numerical or collected with a definite purpose are called data.

Statistics: Extraction of meaning from the data is studied in a branch of mathematics is called statistics.

Primary data: The information was collected by an investigator with a defined objective, the data obtained is called Primary Data.

Ex: Heights of the students in a class, the population in a country etc.

Secondary data: The information collected from a source, which had already been recorded is called secondary data.

Ex: No. of obscenities in the last month (School attendance register).

Presentation of data: Once the data is collected, the investigator has to find out ways to present in the form which is meaningful, easy to understand and shows its main features at a glance.

Range: The difference between maximum and minimum observations in data is called Range of the data.

Un grouped Frequency distribution table:

The actual observations of the data are shown in the table with their frequencies is called Un frequency distribution table.

The data is tabulated by using the tally marks

Grouped Frequency distribution table:

Presenting the data in groups with their frequencies is called frequency distribution table.

MEASURES OF CENTRAL TENDENCY

Mean: Mean is the sum of the observations divided by the no. of observations.

Mean =

Mean for ungrouped data:

If x1, x2, … in are n observations then

Mean =

If x1, x2, … xn are n observations occurs f1, f2, … fn times respectively then

Mean =

Mean by deviation method:

Mean =

where a = assumed mean.

di = xi – a (deviation)

Median:

Median is the middlemost observation of a given data.

First, we arrange the given data into ascending order.

If n is odd then median = observation

If n is even then median =

Mode:

The mode is the value of the observation which occurs most frequently.

Observation with the maximum frequency is called mode.

## 10. SURFACE AREA AND VOLUMES

2 – D objects: The objects which have length and breadth are called two-dimensional figures (2 – D objects).

3 – D objects or Solids: The objects which have length, breadth and also height are called three-dimensional figures (3 – D objects).

Cuboid:

Lateral surface area (L.S.A) = 2h (l + b) square units.

Total surface area (T.S.A) = 2 (lb + bh + hl) square units.

Volume = lbh cubic units.

Cube:

Lateral surface area (L.S.A) = 4 a2 square units.

Total surface area (T.S.A) = 6 a2 square units.

Volume = a3 cubic units.

Cylinder:

Curved surface area (C.S.A) = 2πrh square units.

Total surface area (T.S.A) = 2πr (r +h) square units.

Volume = πr2 h cubic units.

Cone:

Curved surface area (C.S.A) = πrl square units.

Total surface area (T.S.A) = πr (r +l) square units.

Volume =  πr2 h cubic units.

Sphere:

Total surface area (T.S.A) = 4πr2 square units.

Volume =  πr3 cubic units.

Hemisphere:

Curved surface area (C.S.A) = 2πr2 square units.

Total surface area (T.S.A) = 3πr2 square units.

Volume =  πr3 cubic units.

## 11.AREAS

Planar region:

The part of the plane enclosed by a simple closed figure is called a planar region corresponding to that figure.

Area:

The magnitude or measure of the planer region corresponding to the figure is called its area. The area of a figure is a number associated with the part of the plane enclosed by the figure.

The unit area is the area of a square of a side of unit length.

This is defined as 1 square unit

∎ The area of the rectangle whose length ‘l’ and breadth ‘b’ is l × b square units.

The areas of two congruent figures are equal. If two figures have the same area, they need not be congruent.

parallelograms on the same base and between the same parallel are equal in area.

Parallelograms ABCD and ABQP have same base AB and lie between the parallel sides AB and QD

∴ Area of parallelogram ABCD = Area of parallelogram ABQP.

If a triangle and parallelogram have the same base and lies between the parallel lines, then the area of the triangle is equal to half of the area of the parallelogram.

Area of the triangle APB = ½ (Area of the parallelogram ABCD)

If two triangles same base and lies between the parallel lines, then areas of triangles are equal.

Area of ∆ABC = Area of ∆ABD.

The median of the triangle divides it into two triangles of equal area.

Area of ∆ABD = Area of ∆ADC.

## 12.CIRCLES

Circle:

A circle is a collection of all points in a plane which are at a constant distance from a fixed point on the plane.

The fixed point is called the centre of the circle and constant distance is called the radius of the circle.

A circle divides the plane into three parts: (i) inside the circle (interior of the circle) (ii) on the circle (iii) outside the circle (exterior of the circle).

Points A, B are the interior of the circle

Points G, C are on the circle

Points D, E and F are exterior of the circle

Chord: A line segment joining any two points on the circle is called a chord of the circle.

AB is the chord

Arc: The part of the circle between ant two points on it is called Arc of the circle.

Arc of a circle is denoted by

If the arc is smaller than the semicircle, then it is called ‘minor arc’

If the arc is greater than the semicircle, then it is called ‘major arc’

Segment:

The region between the chord and the minor arc is called the minor segment.

The region between the chord and the major segment is called the major segment.

Sector:

The area enclosed by an arc and the two radii joining the centre to the endpoints of an arc is called a sector.

Equal chords of a circle subtend equal angles at the centre.

If the angle subtended by the chords of a circle at the centre are equal, then the chords are equal.

The perpendicular from the centre of the circle to a chord bisects the chord.

If a line drawn from the centre of the circle bisects the chord then the line is perpendicular to that chord.

There is one and only one circle that passes through three non-collinear points.

Arcs of Equal length subtend equal angles at the centre.

The angle subtended by an arc at the centre is twice the angle subtended by it on the remaining arc of the circle.

The angle subtended by an arc in the same segment is equal.

If a line segment joining two points, subtends an equal angle at two other points lying on the same side of the line then the four points lie on a circle.

Concyclic points:

The points which lie on the same circle are called concyclic points.

If the vertices of a quadrilateral lie on the same circle, then it is called a cyclic quadrilateral.

The pairs of opposite angles of an acyclic quadrilateral are supplementary.

If the sum of any pair of opposite angles in a quadrilateral is 1800, then the quadrilateral is cyclic.

## 13.GEOMETRICAL CONSTRUCTIONS

To construct the perpendicular bisector of a given line segment:

Steps of construction:

Step – 1: Draw a line segment of AB.

Step – 2: Taking A as the centre with a radius more than half of AB, draw an arc on either side of the line segment AB.

Step – 3: Taking B as the centre, with the same radius as above, draw arcs so that they intersect the previously drawn arcs.

Step – 4: Marks the points of intersection as X and Y.

Step – 5: Join X and Y, then XY is the perpendicular bisector of AB.

To construct the bisector of a given angle:

Steps of construction:

Step – 1: Draw the given ∠BAC.

Step – 2: Taking A as the centre with any radius, draw an arc to intersect the rays AB and AC at D and E respectively.

Step – 3: Taking D and E as the centres, draw two arcs with equal radii to intersects each other at F.

Step – 4: Draw a ray AF. It the bisector of ∠BAC.

To construct a triangle, given its base, a base angle and sum of the other two sides:

construct a triangle ABC, BC = 4cm, AB + BC = 6cm and ∠B = 600

Steps of construction:

Step – 1: Draw a rough sketch of ∆ABC.

Step – 2: Draw the base BC = 4cm and construct ∠CBX = 600 at B.

Step – 3: With centre B and radius 6cm (AB + BC = 6cm) draw an arc on BX to meet at D.

Step – 4: Join CD and Draw the perpendicular bisector of CD to BD at A.

Step – 5: Join A to C get the required triangle ABC.

To construct a triangle, given its base, a base angle and difference of the other two sides:

construct a triangle ABC, BC = 4.5cm, AB – AC = 1.5cm and ∠B = 300

Steps of construction:

Step – 1: Draw a rough sketch of ∆ABC.

Step – 2: Draw the base BC = 4.5cm and construct ∠CBX = 300 at B.

Step – 3: With centre B and radius 1.5cm (AB – AC = 1.5cm) draw an arc on BX to meet at D.

Step – 4: Join CD and Draw the perpendicular bisector of CD to BD at A.

Step – 5: Join A to C get the required triangle ABC.

To construct a triangle, given its perimeter and its two base angles:

Construct triangle ABC, in which ∠A = 600, ∠B = 300 and AB + BC + CA = 10cm.

Steps of construction:

Step – 1: Draw a rough sketch of ∆ABC.

Step – 2: Draw a line segment XY = 10 cm.

Step – 3: Construct ∠YXC = 150 and ∠XYC = 300 which are angular bisectors of ∠B and ∠A respectively and they meet at C.

Step – 4: Draw the perpendicular bisectors of CX and CY to intersect XY At B and A respectively.

Step – 5: Join BC and AC, ∆ABC is the required triangle.

To construct a circle segment given a chord and given an angle:

Construct a segment of a circle on a chord of length 8 cm and containing the angle of 600

Step – 1: Draw a rough sketch of ∆ABC.

Step – 2: Draw a line segment AB = 8 cm.

Step – 3: Draw a ray AX such that ∠BAX = 300 and draw ray BY such that ∠ABY = 300

Step – 4: With centre ‘O’ and radius OA or OB, draw the circle.

Step – 5: Mark a point C on the arc of the circle. Join AC and BC. We get ∠ACB = 600

## 14. PROBABILITY

Random Experiment: If in an experiment all the possible outcomes are known in advance and none of the outcomes can be predicted with certainty, then such an experiment is called a random experiment.

Ex: Throwing a die, Tossing A coin.

Events: The possible outcomes of a trial are called events.

To measure the chance of its happening numerically we classify them as follows:

Certain: Something that must happen

Equally likely: something that has the same chance of occurring.

Less likely: Something that would occur with less chance.

More likely: Something that would occur with more chance.

Impossible: Something that cannot happen.

Probability of an event =

## 15.PROOFS IN MATHEMATICS

Statements:  The sentences that can be judged to be true or false but not both are called statements. The mathematical statements are cannot be ambiguous.

Hypothesis: A statement of an idea which explains a sense of observation is called Hypothesis.

Mathematical proof: A process which can establish the truth of a mathematical statement based purely on logical arguments is called a mathematical proof.

Conjecture:  The statements which are neither proved nor disproved are called Conjecture.

Axiom: The statements which are assumed to be without proof are called axioms.

Theorem: A mathematical statement whose truth has been established or proved is called Theorem.

Proof: A proof is made up of a successive sequence of mathematical statements.

Deductive reasoning: The logical reasoning which is used in the establishment of the truth of an unambiguous statement is called Deductive reasoning.

Inductive reasoning: The reasoning which is based on examining of variety of cases discovering pattern and forming conclusion is called inductive reasoning.

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