# 2020

## 7th Class Maths Concept

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

### 1. INTEGERS

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 including zero.

0, 1, 2, 3 …

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

….,-3, -2, -1, 0, 1, 2, 3,…..

Integers on a number line:

Operations on integers:

3 + 4 = 7

-2 + 4 = 2

Subtraction of integers on a number line:-

6 – 3 = 3

Multiplication of integers on a number line:-

2 × 3 ( 2 times of 3) = 6

3 × (- 4 ) ( 3 times of -4) = -12

Multiplication of two negative integers:

• To multiply two negative integers, first, we multiply them as whole numbers and put plus sign before the result.
• The multiplication of two negative integers is always negative.

Ex:- -3 × -2 = 6,  -10 × -2 = 20 and so on.

Multiplication of more than two negative integers:

• If we multiply three negative integers, then the result will be a negative integer.

Ex:- -3 ×   -4 ×   -5 = -60,  -1× -7 × -4 = -28 and so on.

• If we multiply four negative integers, then the result will be a positive integer.

Ex:- -3 ×   -4 ×  -5 × -2  = 120,  -1× -7 × -4  × -2 = 56 and so on.

Note:-

1. If the no. of negative integers is even, then the result will be positive.

2. If the no. of negative integers is odd, then the result will be negative.

Division of integers:

• The division is the inverse of multiplication.
• When we divide a negative integer by a positive integer or a positive integer by a negative integer, we divide them as whole numbers then put negative signs for the quotient.

Ex:- -3 ÷ 1 = 3, 4 ÷ -2 = -2 and so on.

When we divide a negative integer by a negative integer, we get a positive number as the quotient.

Ex:- -3 ÷ -1 = 3, -4 ÷ -2 = 2 and so on.

#### Properties of integers:

1.Closure property:-

2.commutative property:-

3.associative property:-

1 + 0 = 0 + 1 = 1,   10 + 0 = 0 + 10 = 10

•For any integer ‘a’, a + 0 = 0 + a

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

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

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

Multiplicative identity:-

2 × 1 = 1 × 2 = 2,    5 × 1 = 1 × 5 = 5

•For any integer ‘a’, a × 1 = 1 × a = a

•1 is the multiplicative identity.

multiplicative inverse:-

For any integer ‘a’, 1/a × a = a × 1/a = 1

• multiplicative inverse of a = 1/a
• Multiplicative inverse of  1/a = a.

distributive property:-

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

3 × (2 + 4) = 18

(3 × 2) + (3 × 4) = 6 + 12 = 18

∴ 3 × (2 + 4) = (3 × 2) + (3 × 4).

### 2. FRACTIONS, DECIMALS AND RATIONAL NUMBERS

Fraction: A fraction is a number that represents a part of the whole. A group of objects is divided into equal parts, then each part is called a fraction.

The proper and improper fractions:

In a proper fraction, the numerator is less than the denominator.

Ex: – 1/5, 2/3, and so on.

In an improper fraction, the numerator is greater than the denominator.

Ex: – 5/2,11/5 and so on.

Comparing fractions:

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

Ex: ,    and so on

Unlike fractions: –

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

∗ Like Fractions:

∗ Unlike fractions:

Subtraction of fractions:

∗ Like fractions:

Ex:

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

Ex:

Multiplication of fractions:

Multiplication of fraction by a whole number: –

Multiplication of numbers means adding repeatedly.

Ex: –

• To multiply a whole number with a proper or improper fraction, we multiply the whole number with the numerator of the fraction, keeping the denominator the same.

2.Multiplication of fraction with a fraction: –

multiplication of two fractions =

Division of fractions:

Ex: – 2 ÷

⇒ 6 one-thirds in two wholes

Reciprocal of fraction: reciprocal of a fraction is   .

Note:

• dividing by a fraction is equal to multiplying the number by its reciprocal.
• For dividing a number by mixed fraction, first, convert the mixed fraction into an improper fraction and then solve it.

Ex:

1.Division of a whole number by a fraction: –

2.Division of a fraction by another fraction: –

Decimal number or fractional decimal:

In a decimal number, a dot(.) or a decimal point separates the whole part of the number from the fractional part.

The part right side of the decimal point is called the decimal part of the number as it represents a part of 1. The part left to the decimal point is called the integral part of the number.

Note: –

• while adding or subtracting decimal numbers, the digits in the same places must be added or subtracted.
• While writing the numbers one below the other, the decimal points must become one below the other. Decimal places are made equal by placing zeroes on the right side of the decimal numbers.
###### Comparison of decimal numbers:

while comparing decimal numbers, first we compare the integral parts. If the integral parts are the same, then compare the decimal part.

Ex: – which is bigger: 13.5 or 14.5

Ans: 14.5

Which is bigger: 13.53 or 13. 25

Ans: 13.53

Multiplication of decimal numbers:

For example, we multiply 0.1 × 0.1

Multiplication of decimal numbers by 10, 100, and 1000: –

Here, we notice that the decimal point in the product shifts to the right side by as many zeroes as in 10, 100, and 1000.

Division of decimal number:

Division of decimal number by 10,100 and 1000: –

Here, we notice that the decimal point in the product shifts to the left side by as many zeroes as in 10, 100, and 1000.

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 a bigger collection of integers, negative fractional numbers, positive fractional numbers.

Ex: – 1, 2, -1/2, 0 etc.

### 3. SIMPLE EQUATIONS

Equation: Equation is the condition of a variable. It says that two expressions are equal.

• An equation has two sides LHS and RHS, on both sides of the equality of sign.
• One of the expressions of the equation is must have a variable.
• If we interchange the expressions from LHS to RHS, the equation remains the same

Ex: – x + 2 = 5; 2 = x + 3

Balanced equation:

In an equation, if LHS =RHS, then that equation is balanced.

If the same number is added or subtracted on both sides of the balanced equation, the equation remains will the same.

Ex: 8 + 3 = 11

If add 2 on both sides ⇒ LHS = 8 + 3 + 2 = 13

RHS = 11 + 2 = 13

∴ LHS = RHS

8 +3 = 11 if subtract 2 on both sides

LHS = 8 + 3 – 2 = 9

RHS = 11 – 2 = 9

∴ LHS = RHS

###### Using algebraic equations in solving day to day problems:
2. Denote the unknown or quantity to be found with some letters such as x, y, z …etc.
3. Write the problem in the form of an algebraic equation by making a relation among the quantities.
4. Solve the equation.
5. Check the solution

### 4. LINES AND ANGLES

Complimentary angles: When the sum of the angles is 900, the angles are called complementary angles.

Ex: 300, 600; 200, 700 and soon.Supplementary angles: When the sum of the angles is 1800, the angles are called  Supplementary angles.

Ex: 1200, 600; 1100, 700 and soon.

Adjacent angles: The angle having a common Arm and a common vertex are called Adjacent angles.

⇒ ∠AOC and ∠BOC adjacent angles.

Vertically opposite angle: If two lines are intersecting at a point, then the angles that are formed opposite to each other at that point are called vertically opposite angles.

Transversal: A line that intersects two or more lines at distinct points is called a transversal.

#### Angles made by a transversal:

Corresponding angles:

Two angles that lie on 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 side of the transversal and both interior or exterior are called Alternate angles.

∠1, ∠7; ∠2, ∠8 are exterior alternate angles

∠3, ∠5; ∠4, ∠6 are interior alternate angles.

∠3, ∠6; ∠4, ∠5 interior angles same side of the transversal.

Transversal on parallel lines:

If pair of parallel lines are intersected by a transversal then the angles of 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 of 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 of 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 of each pair of interior angles on 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.

### 5. TRIANGLE AND ITS PROPERTIES

Triangle:
A closed figure formed by three-line segments is called a triangle.

In ∆ABC,

• Three sides are
• Three angles are ∠ABC, ∠BCA, ∠ACB
• Three vertices are A, B, C.

Classification of triangles:

Triangles can be classified according to the properties of their sides and angles.

According to sides:

Based on sides triangles are three types:

• Scanlan triangle (ii) Isosceles Triangle (iii) equilateral triangle

According to angles:

• Acute-angled triangle (ii) Right-angled triangle (iii) Obtuse-angled triangle

Relationship between the sides of a triangle:

1. The sum of the lengths of any two sides of a triangle is greater than the third side.

1. The difference between the lengths of any two sides of a triangle is less than the third side.

The altitude of a triangle:

We can draw three altitudes in a triangle.
A perpendicular line drawn from a vertex to its opposite side of a triangle is called the Altitude of the triangle.

Median of a triangle:

In a triangle, a line drawn from the vertex to the mid-point of its opposite side is called the median of the triangle.

Medians of a triangle are concurrent. We can draw three medians in a triangle.

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

Angle-sum property of a triangle:
Some 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 an exterior triangle.

Exterior angle property:- The exterior angle of a triangle is equal to the sum of two interior opposite angles.

x0+ y0 = z0

## 6.RATIO – APPLICATIONS

Ratio: Comparison of two quantities of the same kind is called ‘Ratio.

The ratio is represented by the symbol ‘:’

If the ratio of two quantities ‘a’ and ‘b’ is a : b, then we read this as ‘a is to b’

The quantities ‘a’ and ‘b’ are called terms of the ratio.

Proportion: if two ratios are equal, then they are said to be proportional.
‘a’ is called as first term or antecedent and ‘b’ is called a second term or consequent.

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

The product of means = the product of extremes

Unitary method:  The method in which we first find the value of one unit and then the value of the required no. of units is known as the unitary method.

Direct proportion: In two quantities, when one quantity increase(decreases) the other quantity also increases(decreases) then two quantities are in direct proportion.

Percentages:

‘per cent’ means for a hundred or per every hundred. The symbol % is used to denote the percentage.

1% means 1 out of 100, 17% means 17 out of 100.

Profit and Loss:

Selling price = SP; Cost price = CP

If SP > CP, then we get profit

Profit = SP – CP

Profit percentage =

SP = CP + profit

If SP < CP, then we get a loss

Loss = CP – SP

Loss percentage =

SP = CP – Loss

Simple interest:

Principle: – The money borrowed or lent out for a certain period is called the Principle.

Interest: – The extra money, for keeping the principle paid by the borrower is called interest.

Amount: – The amount that is paid back is equal to the sum of the borrowed principal and the interest.

Amount = principle + interest

Interest (I) =   where R is the rate of interest.

## 7.DATA HANDLING

Data: The information which is in the form of numbers or words and helps in taking decisions or drawing conclusions is called data.

Observations: The numerical entries in the data are called observations.

Arithmetic Mean: The average data is also called an Arithmetic mean.

Arithmetic Mean (A.M) =

The arithmetic mean always lies between the highest and lowest observations of the data.

When all the values of the data set are increased or decreased by a certain number, the mean also increases or decreases by the same number.

Mode: The most frequently occurring observation in data is called Mode.

If data has two modes, then it is called bimodal data.

Note: If each observation in a data is repeated an equal no. of times, then the data has no mode.

Median: The middlemost observation in data is called the Median.

Arrange given data in ascending or descending order.

If a data has an odd no. of observations, then the middle observation is the median.

If a data has even no. of observations, then the median is the average of middle observations.

Bar graph:

Bar graphs are made up of uniform width which can be drawn horizontally or vertically with equal spacing between them.

The length of each bar tells us the frequency of the particular item.

Ex:

Double bar graph:

It represents two observations side by side.

Ex:

Pie chart: A circle can be divided into sectors to represent the given data

The angle of each sector =

Ex:

 Budget Amount in rupees Food 1200 Education 800 Others 2000 Savings 5000 Total income 9000

## 8.CONGRUENCY OF TRIANGLES

Congruent figures: Two figures are said to be congruent if they have the same shape and size.

Congruency of line segments:  If two-line segments have the same length, then they are congruent. Conversely, if two-line segments are congruent, then they have the same length.

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

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 ≅ ∆DEF

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

## 9.CONSTRUCTION OF TRIANGLES

The no. of measurements required to construct a triangle = 3

A triangle can be drawn in any of the situations given below:

• Three sides of a triangle
• Two sides and the angle included between them.
• Two angles and the side included between them.
• The hypotenuse and one adjacent side of the right-angled triangle.

Construction of a triangle when measurements of the three sides are given:

Ex: construct a triangle ABC with sides AB = 4cm, BC = 7cm and AC = 5cm

Step of constructions:

Step -1: Draw a rough sketch of the triangle and label it with the given measurements.

Step -2: Draw a line segment of BC of length 7cm.

Step -3: with centre B, draw an arc of radius 4cm, draw another arc from C with radius 5cm such that it intersects first at A.

Step -4: join A, B and A, C. The required triangle ABC is constructed.

Construction of a triangle when two sides and the included angle given:

EX: construct a triangle ABC with sides AB = 4cm, BC = 6cm and ∠B=600

Step of constructions:

Step -1: Draw a rough sketch of the triangle and label it with the given measurements.

Step -2: Draw a line segment of AB of length 4cm.

Step -3: draw a ray BX making an angle 600 with AB.

Step -4: draw an arc of radius 5cm from B, which cuts ray BX at C.

Step -5: join C and A, we get the required ∆ABC.

Construction of a triangle when two angles and the side between the angles given:

Ex: construct a triangle PQR with sides QR = 4cm, ∠Q= 1200 and ∠R= 400

Step of constructions:

Step -1: Draw a rough sketch of a triangle and label it with the given measurements.

Step -2: Draw a line segment QR of length 4 cm.

Step -3: Draw a ray RX, making an angle 400 with QR.

Step -4: Draw a ray QY, making an angle 1000 with QR, which intersects ray RX.

Step -5: Mark the intersecting point of the two rays as P. Required triangle PQR is constructed.

Construction of a triangle when two sides and the non-included angles are given:

Ex: construct a triangle MAN with sides MN = 4cm, AM = 3cm and ∠A= 400

Step of constructions:

Step -1: Draw a rough sketch of a triangle and label it with the given measurements.

Step -2: Draw a line segment MA of length 0f 5cm.

Step -3: Draw a ray AX making an angle 400 with the line segment MA.

Step -4: With M as the centre and radius 3 cm draw an arc to cut ray AX. Mark the intersecting point as N.

Step -5: join M, N, then we get the required triangle MAN.

Construction of a right-angled triangle when hypotenuse and sides are given:

Ex: construct a triangle ABC, right angle at B and AB = 4cm, Ac = 5cm

Step of constructions:
Step -1: Draw a rough sketch of a triangle and label it with the given measurements.

Step -2: Draw a line segment BC of length 0f 4cm.

Step -3: Draw a ray BX perpendicular to BC at B

Step -4: Draw an arc from C with a radius of 5cm to intersect ray BX at A.

Step -5: Join A, C, then we get the required triangle ABC.

## 10.ALGEBRAIC EXPRESSIONS

Variable: It is a dependent term. It takes different value.

Ex: m, x, a, etc.

Constant:  It is an independent term. It has a fixed value.

Ex: 1, 3,  etc.

Like terms and Unlike terms: If the terms contain the same variable with the same exponents, then they are like terms otherwise, unlike terms.

Ex: 3x, –4x, x are like terms

3x, 4y, 4 are unlike terms

Coefficient: Coefficient is a term which the multiple of another term (s)

EX: In 5x. 5 is the coefficient of x and x is the coefficient of 5

5 is a numerical coefficient

x is the literal coefficient

Expression: An expression is a single term or a combination of terms connected by the symbols ‘+’ (plus) or ‘−’ (minus).

Ex: 2x – 3. 3x, 2 +3 – 4 etc.

Numerical Expressions:  If every term of an expression is constant, then the expression is called numerical expression.

Ex: 2 + 3 + 5, 2 – 4 – 7, 1 + 5 – 4 etc.

Algebraic expression:  If an expression at least one algebraic term, then the expression is called an algebraic expression.

Ex: x + y, xy, x – 3, 4x + 2 etc.

Note: Plus (+) and Minus (−) separate the terms

Multiplication (×) and Division (÷) do not separate the terms.

Types of Algebraic expressions:

Monomial: – If an expression has only one term, then it is called a monomial.

Ex: 2x2, 3y, x, y, xyz etc.

Binomial: If an expression has two unlike terms, then it is called binomial.

Ex: 2x+ 3y, x2+ y, x +yz2 etc.

Trinomial: If an expression has three unlike terms, then it is called trinomial.  Ex: 2x+ 3y + 4xy, x2+ y + z, x2 y +yz2 + xy2 etc.

Multinomial: If an expression has more than three unlike terms, then it is called multinomial.

Ex: 2x+ 3y + 4xy +5, x2+ y + z – 4y + 6 ,

x2 y +yz2 + xy2 – 4xy + 8yz etc.

Degree of a monomial: The sum of all exponents of the variables present in a monomial is called the degree of the monomial.

Ex: Degree of 5xy3

An exponent of x is 1 and an exponent of y is 3

Sum of exponents = 1 + 3 = 4

∴ degree of 5xy3 is 4

Degree of an Algebraic Expression: The highest exponent of all the terms of an expression is called the degree of an Algebraic expression.

Ex: degree of x2 + 3x + 4x3 is 3

degree of 3xy + 6x2y + 5x2y2 is 4

The sum of two or more like terms is a like term with a numerical coefficient that is equal to the sum of the numerical coefficients of all the like terms in addition.

Ex: 3x + 2x = (3 + 2) x = 5x

4x2y + x2y = (4 + 1) x2y = 5x2y

Subtraction of like terms:

The difference of two like terms is a like term with a numerical coefficient is equal to the difference between the numerical coefficients of the two like terms.

Ex: 3x − 2x = (3 − 2) x = x

4x2y −2 x2y = (4 −2) x2y = 2x2y

Note: (i) addition and subtraction are not done for unlike terms. (ii) If no terms of an expression are alike then it is said to be in the simplified form.

The standard form of an Expression:

In an expression, if the terms are in such a way that the degree of the terms is in descending order, then the expression is said to be in standard form.

Ex: 5 – 2x2 + 4x +3x3

Standard form is 3x3 – 2x2 + 4x + 5

Finding the value of an expression:

Example: find the value of expression x3 + y + 3, when x = 1 and y = 2

Sol: given expression is x3 + y + 3

Substitute x = 1 and y = 2 in above expression

(1)3 + 2 + 3 = 1 + 2 + 3 = 6

This is in two ways: (i) Column or Vertical method (ii) Row or Horizontal method.

Column or Vertical method:

Step –1: Write the expression in standard form if necessary.

Step –2: write one expression below the other such that the like terms come in the same column.

Step –3: Add the like terms column-wise and write the result just below the concerned column.

Ex: Add x2 + 3x + 5, 3 – 2x + 3x2 and 3x – 2

Sol:

Row or Horizontal method.

Step –1: Write the expression in standard form if necessary.

Step –2:  Re-arrange them term by grouping the like terms.

Step –3: Simplify the coefficients.

Step –4: Write the resultant expression in standard form.

Ex: Add x2 + 3x + 5, 3 – 2x + 3x2 and 3x –2

Sol: (x2 + 3x + 5) + (3 – 2x + 3x2) + (3x –2)

= (x2 + 3x2) + (3x – 2x + 3x) + (5 + 3 – 2)

= (1 + 3) x2 + (3 – 2 + 3) x + 6

=4x2 + 4x + 6

For every algebraic expression there exist another algebraic expression such that their sum is zero. These two expressions are called the additive inverse of each other.

Subtraction of algebraic expressions:

This is in two ways: (i) Column or Vertical method (ii) Row or Horizontal method.

Column or Vertical method:

Step –1: Write the expression in standard form if necessary.

Step –2: write one expression below the other such that the expression to be subtracted comes in the second row and the like terms come one below the other.

Step –3: Change the sign of every term of the expression in the second row to get the additive inverse of the expression.

Step –4: Add the like terms column-wise and write the result just below the concerned column.

Ex: Subtract: x2 + 3x + 5 from 3x2 + 4x – 3

Sol:

Row or Horizontal method:

Step –1: Write the expressions in one row with the expression to be subtracted in a bracket with assigning a negative sign to it.

Step –2:  Add the additive inverse of the second expression to the first expression.

Step –3: Group the like terms and add or subtract.

Step –4: Write the resultant expression in standard form.

Ex: Subtract: x2 + 3x + 5 from 3x2 + 4x – 3

Sol:   3x2 + 4x – 3 – (x2 + 3x + 5)

= 3x2 + 4x – 3 – x2 – 3x – 5

= (3 – 1) x2 + (4 – 3) x + (– 3 – 5)

= 2x2 + x – 8

## 11.EXPONENTS

We know that,

a × a = a2 (a raised to the power of 2)

a × a × a = a3 (a raised to the power of 3)

a × a × a × a × a × a ×…. m times = am

am is in exponential form

a is called base, m is called exponent or index.

Laws of exponents:

• am × an = am + n
• (am)n = amn
• am = an ⇒ m = n
• (ab)m = am.an
• a0 = 1

Standard form:  A number that is expressed as the product of the largest integer exponent of 10 and a decimal number between 1 and 10 is said to be in standard form.

Ex: 1324 in standard form is 1.324 × 103.

Quadrilateral: A Quadrilateral is a closed figure with four sides, four angles and four vertices.

• AB, BC, CD, and AD are sides.
• A, B, C and D are the vertices.
• ∠ABC, ∠BCD, ∠CDA and ∠DAC are the angles.

The line segment joining the opposite vertices of a quadrilateral are called the diagonals of the Quadrilateral. In the above figure AC, BD is the diagonals.

The two sides of a Quadrilateral that have a common vertex are called the adjacent sides of the Quadrilateral. From the above figure, AB, BC; BC, CD; CD, DA and DA, AB are the adjacent sides.

The two angles of a Quadrilateral that have a common side are called the adjacent angles of the Quadrilateral. From the above figure, ∠A, ∠B; ∠B, ∠C; ∠C, ∠D and ∠D, ∠A is the adjacent angles.

The two sides of a quadrilateral, which do not have a common vertex are called opposite sides of a quadrilateral. From the above figure, AB, CD; BC, DA are the opposite sides.

The two angles of a quadrilateral, which do not have a common side are called opposite angles of a quadrilateral. From the above figure, ∠A, ∠C; ∠B, ∠D are the opposite angles.

Interior and exterior of a Quadrilateral:

In a Quadrilateral ABCD, S, N are interior points, M, P are exterior points and A, B, C, D and Q are lies on the Quadrilateral.

A Quadrilateral is said to be a convex Quadrilateral if all line segments joining points in the interior of the Quadrilateral also lie in the interior of the Quadrilateral.

A Quadrilateral is said to be a concave Quadrilateral if all line segments joining points in the interior of the Quadrilateral not lie in the interior of the Quadrilateral.

Angle sum property of a quadrilateral:

The Sum of the angle in a Quadrilateral is 3600

In a Quadrilateral ABCD, ∠A + ∠B + ∠C + ∠D = 3600

1.Trapezium:

In a Quadrilateral, one pair of opposite sides are parallel then it is Trapezium.

In a Trapezium ABCD, AB∥ DC; AC, BD are diagonals.

2.Kite:

In a Quadrilateral two distinct consecutive pairs of sides are equal in length then it is called a Kite.

In a Kite ABCD, AB = BC; AD = DC AC, BD are diagonals.

3.Parallelogram:

In a Quadrilateral, two pairs of opposite sides are parallel then it is Parallelogram.

In a Parallelogram ABCD, AB∥ DC, AD∥ BC; AD, BD are diagonals.

Properties of parallelogram: –

• The opposite sides of a parallelogram are equal in length.
• The opposite angles are equal in measure.
• The sum of the adjacent angles is 1800
• Diagonals are bisected to each other and not equal in length.

4.Rhombus:

In a parallelogram in which two adjacent sides are equal, then it is a Rhombus.

In a Parallelogram ABCD, AB∥ DC, AD∥ BC; AD, BD are diagonals.

Properties of Rhombus: –

• All sides of a Rhombus are equal in length.
• The opposite angles are equal in measure.
• The sum of the adjacent angles is 1800
• Diagonals are bisected to each other perpendicularly and not equal in length.

5.Rectangle:

In a parallelogram all angles are equal, then it is a Rectangle.

Properties of Rectangle: –

• The opposite sides are equal in length.
• Each angle is 900.
• The sum of the adjacent angles is 1800
• Diagonals are bisected to each other and not equal in length.
• Each diagonal divides the rectangle into two congruent triangles.

6.Square:

In a rectangle adjacent sides are equal, then it is a Square.

Properties of Square: –

• All sides of a square are equal in length.
• Each angle is 900.
• The sum of the adjacent angles is 1800
• Diagonals are bisected to each other and equal in length.
• Each diagonal divides the square into two congruent triangles.

## 13.AREA AND PERIMETER

Area of a parallelogram:

Area of parallelogram (A) = b × h square units.

The area of the parallelogram is equal to the product of its base (b) and the height(h)

Area of a Triangle:

Area of triangle = ½ b × h square units.

The area of the triangle is equal to half the product of its base (b) and height (h).

In a Right-angled triangle, two of its sides can be the height.
Area of a Rhombus:

The area of the Rhombus is equal to half the product of its diagonals

Area of rhombus = ½ d1 × d2 square units.

Circumference of the circle:

Circumference of circle = 2πr = πd

Area of the rectangular path:

Area of Rectangular path = area of the outer rectangle – are of the inner rectangle

## 14. UNDERSTANDING 2D AND 3D SHAPES

Net: Net is a short of skeleton-outline in 2d, which when folded the result in 3d shape.

Nets of 3D shapes:

1.Cube:

2.Cylinder:

3.Pyramid:

Oblique Sketches:

Oblique sketches are drawn on a grid paper to visualise 3D shapes.

Ex: Draw an oblique sketch of a 3×3×3 cube

Step-1: Draw the front face

Step-2: Draw the opposite face, which is the same as the front face. The sketch is somewhat offset from Step-1

Step-3: Join the corresponding corners.

Step-4: Redraw using dotted lines for hidden edges.

Isometric Sketches:

Isometric sketches are drawn on a dot isometric paper to visualise 3D shapes.

Ex: Draw an oblique sketch of a 2×3×4 cuboid

Step-1: Draw a rectangle to show the front face.

Step-2:  Draw four parallel line segments of length 3cm.

Step-3: Connect the corresponding corners with appropriate line segments

Step-4: This is an isometric sketch of a cuboid

## 15.SYMMETRY

Line of symmetry: The line which divides a figure into two identical parts is called the line of symmetry or axis of symmetry.

An object can have one or more than one lines of symmetry.

Regular polygon:

If a polygon has equal sides and equal angles, then the polygon is called a Regular polygon.

Lines of symmetry for Regular polygons:

 Regular polygon No. of sides No. of axes of symmetry Triangle 3 3 Square 4 4 Pentagon 5 5 Polygon n n

Rotational symmetry: If we rotate a figure, about a fixed point by a certain angle and the figure looks the same as before, then the figure has rotational symmetry.

The angle of rotational symmetry:  The minimum angle of rotation of a figure to get the same figure as the original is called the angle of rotational symmetry or angle of rotation.

The angle of rotation of the equilateral triangle is 1200

The angle of rotation of a square is 900

Order of rotational symmetry:

The no. of times a figure, rotated through its angle of rotational symmetry before it comes to the original position is called the order of rotational symmetry.

The order of rotational symmetry for an equilateral triangle is 3.

The order of rotational symmetry for a square is 4.

Note: All figures have rotational symmetry of order 1, as can be rotated completely through 3600 to come back to its original position.

An object has rotational symmetry, only when the order of symmetry is more than 1.

• Some shapes have a line of symmetry and some have rotational symmetry and some have both.

Square, Equilateral triangle and Circle have both line and rotational symmetry.

## 6th maths notes|| TS 6 th class Maths Concept

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.

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. KNOWING OUR NUMBERS

•  Number: A number is a mathematical object used to count and measure.1,2,3…….etc.

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.

##### Addition on the number line:
•  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.

##### Identity under addition and multiplication:

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.

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 re 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.

#### Addition and subtraction of 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.

2.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.

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### 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.

### 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.

## TS 10th class maths concept

Studying mathematics successfully meaning that, children take responsibility for their own learning and learn to apply the concepts to solve problems.

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

## 1. REAL NUMBERS

• Rational number: The number, which is written in the form of p/q where p,q are integers q not equal to zero is called a rational number. It is denoted by Q.

• Irrational numbers:- the number, which is not rational is called an irrational number. It is denoted by Q’ or S.

• Euclid division lemma:- For any positive integers a and b, then q, r are integers exists uniquely satisfying the rules a = bq + r, 0 ≤ r < b.

• Prime number:- The number which has only two factors 1 and itself is called a prime number. (2, 3, 5, 7 …. Etc.)

• Composite number:- the number which has more than two factors is called a composite number. (4, 6, 8, 9, 10,… etc.)

• Co-prime numbers:- Two numbers  said to be co-prime numbers if they have no common factor except 1. [Ex: (1, 2), (3, 4), (4, 7)…etc.]

• To find HCF, LCM by using prime factorization method:  H. C.F= product of the smallest power of each common prime factors of given numbers. L.C.M = product of the greatest power of each prime factor of given numbers.

• In p/q, if prime factorisation of q is in form 2m 5n, then p/q is terminating decimal. Otherwise non-terminating repeating decimal.
• 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.

• ‘p’ is a prime number and ‘a’ is a positive integer, if p divides a2, then p divides a.

• If ax = N then x = ${log_{a}}^{N}$

(i) log (xy) = log(x) + log(y)  (ii) log (x/y) = log( x) – log( y) (iii) log (xm ) = m log (x)

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## Quantitative Aptitude For Competitive Exams

Quantitative Aptitude by the ‘Basics in Maths’ team. These notes to do help the students with any Competitive Exam.

These notes cover all the topics covered in any Competitive Exam syllabus and include plenty of formulae and concept to help you solve all the  Competitive examinations.

## Number System

Number: A number is a mathematical object used to count and measure.1,2,3……. etc.

the ten thousand places in 5432 are greater than that in 4978.

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:

The largest number formed by arranging the given digits in descending order _ 4321.

The smallest number formed by arranging the given digits in ascending order _ 1234.

The greatest two-digit number is 99.

The greatest three-digit number is 999.

The greatest four-digit number is 9999.

Place value

Place value is the positional notation, which defines a digit’s position.

Ex: – 1234 ⟶   4 is one’s place, 3 is tens place, 2 in the hundreds place and 1 is thousands place.

Face value

The face value of a digit in a numeral is its own value.

Ex: – 1234

Face value of r is 200

Face value of 3 is 30

Place value table for Indian system:

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 later and the third comma comes after another two digits.

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

The international system of numeration: – In an 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.

##### Types of Numbers:

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

1, 2, 3…

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

0, 1, 2, 3 …

Note:  All natural numbers are whole numbers but all whole numbers need not be natural numbers.

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, where p, q are integers and q ≠ 0 are called rational numbers. Rational numbers are denoted by Q.

Ex: – 2, 3, 0.3, and so on

Even numbers: The numbers which are divisible by 2 successfully are called even numbers.

EX: 2, 4, 6, 8, 10, …

Odd numbers:  The numbers which are not divisible by 2 are called Odd numbers.

Ex: 1, 3, 5,7, 9, 11, …

Note:

• The Sum of two even numbers is an even number.
• The Sum of two odd numbers is an odd number.
• The Sum of even and odd numbers is an odd number.
• The difference between the two even numbers is an even number.
• The difference between two odd numbers is an even number.
• The difference between even and odd numbers is an odd number.
• The product of two even numbers is an even number.
• The product of two odd numbers is an odd number.
• The product of even and odd numbers is an even number.

Prime numbers: The numbers, which have only two factors 1, and themselves 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 a co-prime number.

Ex: – (2, 3), (4,5) ……

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

Ex: – (2,3), (3,5), (17,19) ….

Special products:

(a + b)2 = a2 + 2ab + b2

(a − b)2 = a2 − 2ab + b2

(a + b) (a – b) = a2 − b2

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

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

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

a3 + b3 = (a + b) (a2 – ab + b2)

a3 – b3 = (a – b) (a2 + ab + b2)

if a + b + c = 0, then a3 + b3 + c3 = 3abc.

## Divisibility Rules

The process of checking whether a number is divisible by a given number or not without actual division is called the 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 are 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 it’s one 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 rule by 7: –

First, multiply the last digit of a given number by 2,

subtract this result from the number formed by the remaining digits of a 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 are 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.

Factorial of a number: For positive integer ‘n’, the continued production of first natural numbers is called factorial n and is denoted by n!

Ex: 4! = 4 × 3 × 2 × 1 = 24

7! = 7 × 6 × 5× 4 × 3 × 2 × 1 = 5040

Absolute value or Modulus of a number:   Absolute value of a number always gives a positive number.

= x if x ≥ 0

= −x if x < 0

Ex:

Greatest integral value: The greatest integral value of a number ‘n’ is denoted by [n]

n <[n]< n + 1

Ex: [3.2] = 3; [5.6] = 5; [– 5. 4] = – 6

## HCF and LCM

Factors: a number that 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

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

Perfect number: If the sum of the factors of a number (except the number itself) is equal to that number, then it is called a Perfect number.

Ex: 6

The factors of 6 are: 1, 2, 3, 6

1 + 2 + 3 = 6

Therefore 6 is a perfect number.

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

The 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.

Ex: – HCF of 9, 12

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.

∴ HCF of 9, 12 is 3

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:

Prime factorization method: –

The LCM of 6, 15 by using the prime factorization method is as follows:

Prime factors of 6 =   2 × 3

Prime factors of 15 = 5× 3

1.Express each number as a product of prime factors

2.Take the common factors both: 3

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

4.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.

Division method: – To find LCM of 6 and 15:

Arrange the given numbers in a row.

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.

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

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

Ex: –    Find the LCM of 6, 15

∴ 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.

∎ HCF of co-prime numbers is 1 and LCM is their product

∎ HCF of fractions =

∎ LCM of fractions =

## Fractions

Fraction: Fraction is a part of the whole. The whole may be a single object or group of objects.

In, 3 is the numerator and 4 is the denominator.

Proper fraction: if the numerator of a fraction is less than the denominator, then the fraction is called a proper fraction.

Ex:   etc.

Improper fraction: if the numerator of a fraction is greater than the denominator, then the fraction is called an improper fraction.

Ex: etc.

Mixed fraction: An improper fraction can be written as a combination of a whole and part. Such a fraction is called a mixed fraction.

Ex:

Decimal fractions: If the denominators of fractions are the powers of 10, then those fractions are called decimal fractions. We can write a decimal fraction with a decimal point (.). it makes it easier to do addition, subtraction, and multiplication on fractions.

Ex:  etc.

Parts in a decimal fraction:

In a decimal fraction, a dot(.) or a decimal point separates the whole part of the number from the fractional part.

The part right side of the decimal point is called the decimal part of the number as it represents a part of 1. The part left to the decimal point is called the integral part of the number.

In 2.35, 2 is an integral part and 35 is the decimal part

It is read as two-point three five

Addition and Subtracting of Decimal Fractions:

while adding or subtracting decimal numbers, the digits in the same places must be added or subtracted.

While writing the numbers one below the other, the decimal points must become one below the other. Decimal places are made equal by placing zeroes on the right side of the decimal numbers.

Subtract 3.12 from 5.327

Comparison of decimal numbers:

while comparing decimal numbers, first we compare the integral parts. If the integral parts are the same, then compare the decimal part.

Ex: – which is bigger: 13.5 or 14.5

Ans: 14.5

Which is bigger: 13.53 or 13. 25

Ans: 13.53

Multiplication of decimal fractions:

For example, we multiply 0.1 × 0.1

Multiplication of decimal fractions by 10, 100, and 1000: –

Here, we notice that the decimal point in the product shifts to the right side by as many zeroes as in 10, 100, and 1000.

Division of decimal fractions:

Division of decimal number by 10,100 and 1000:

Here, we notice that the decimal point in the product shifts to the left side by as many zeroes as in 10, 100, and 1000.

## BODMAS Rule

B ⟶ Bracket

O ⟶ Off (Product)

D ⟶ Division

S ⟶ Subtraction

Ex:  Solve:  7 – 12 ÷ 3 of 12

Sol: 7 – 12 ÷ 3 of 12 = 72 – 12 ÷ 3 of 2    ( 3 of 2 = 3× 2 = 6)

= 72 – 12 ÷ 6

= 72 – 2

= 70

## Squares and Square roots, Cubes and Cube roots

Square:  Square number is the number raised to 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

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

Ex: 1,4,9, …etc.

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:

###### 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 to 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 to 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

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).

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

Ex:  and so on.

## Surds

If ‘a’ is a positive rational number that can be expressed as the nth power of some rational number, then the irrational number  is called a Surd

n is called the order of surd, a is called ‘radicand’

Mixed surd:  A surd which has a rational factor other than 1, the other factor being irrational is called mixed surd.

Ex:

Pure surd: A surd which unity has its rational factor, other factor being a rational factor is called Pure surd.

Ex:  etc.

Rationalizing Factor (R.F):  If the product of two surds is a rational number, then each of the two surds is a rationalizing factor of the other. It is not unique.

## Logarithms

For any two positive integers a and x if aN = x then

Laws of logarithms:

## Ratio

The comparison of two quantities of the same kind by using division is called ‘Ratio’

The ratio of two quantities ‘a’ and ‘b’ is denoted by a: b, read as a is to b

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

Compound Ratio:  The compound ratio of a: b and c : d is denoted by a: b:: c : d

a : b : : c : d = a × c : b × d

the compound ratio of ( a : b), (c : d) and( x : y) is acx : bdy

The duplicate ratio of a: b is a2: b2

Sub Duplicate ratio of a: b is

Triplicate ratio of a: b is a3: b3

Sub Triplicate ratio of a: b is

## Allegation or Mixture

Allegation: Alligation is a rule, to find the ratio in which two or more quantities at the given price must be mixed to produce a mixture of the desired price.

Mixture: Mixture means mixing two or more quantities. It can be expressed in the form of percentage or ratio.

Mean price: The cost price of a unity quantity of the mixture is called the mean price.

Allegation Rule: If two quantities are mixed, then

Suppose a container contains ‘x’ units of liquid from which units are taken out and replaced by water. After n operations the quantity of pure liquid = units.

## Proportion

If two ratios are equal, then they are in proportion.

If a: b = c : d, then a, b, c and d are in proportion

a and d are extremes

b and c are means

product of means = product of extremes

Componendo and Dividendo: If, then

Unitary method:  The method in which we first find the value of one unit and then the value of the required no. of units is known as the unitary method.

Direct proportion: In two quantities, when one quantity increase(decreases) the other quantity also increases(decreases) then two quantities are in direct proportion.

Inverse proportion: In two quantities, when one quantity increase(decreases) the other quantity also decreases (increases) then two quantities are in direct proportion.

## Percentage

Per cent means per 100, ‘x’ per cent means ‘x’ hundredth.

x% =

Ex: 30% =

Converting a fraction into a percentage:

Ex: Convert into the percentage

Sol:  = %  =  20 %

∎ If the price of commodity decreased by R%, then the increase in consumption so as not to

decrease the expenditure is

∎ If the price of commodity increased by R%, then the increase in consumption so as not to

increase the expenditure is

Results on machines: If the present value of a machine is P and it deprecates at the rate of R% per year, then

• Value of the Machine After ‘n’ years =
• Value of the Machine ‘n’ years ago =

Results on Population: If the present Population is P and it increases at the rate of R% per year, then

• Population after ‘n’ years =
• Population ‘n’ years ago =

∎ If A is R% more than B, then B is less than A by

∎ If A is R% less than B, then B is more than A by

## 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 buy 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 lost.

Loss = C.P – S.P.

###### Some formulae in profit and loss:

For-Profit:

For Loss:

∎ When a person sells two similar items, one at the gain of x%, and another one at a loss of x%, then the loss percentage

∎If the selling price is equal to cost price but uses false weight then

Gain% =  %

∎ if the selling price of an item is at a profit of x% but uses false weight which is y% less than the original weight then

Gain % = %

∎ if the selling price of an item is at a loss of x% but uses false weight which is y% less than the original weight then

Loss or Gain % = %

## Simple interest

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

Rate of interest: – percentage of interest per year is called the rate of interest(R).

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

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

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

Amount = Principal + S.I

## Compound Interest

Compound interest allows us to earn interest on interest.

Amount = ( interest compounded annually)

P = principal, R = rate of interest, and n = no. of terms or time period.

Amount = ( interest compounded half-yearly)

∎When the rates are different for different years, like X%, Y%, and Z% for first, second and third year respectively, then

Amount =

∎ the present worth of ₹ p due n years hence is given then present worth =

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

## Partnership

A business which is undertaken by two or more persons jointly is called ‘Partnership. Those persons are called Partners.

Types of Partnership:

The partnership is of two types (i) General or Simple partnership  (ii) Compound partnership

(i) Simple partnership or General partnership:

In this type, the period of the investment is the same. Partners divide either profit and loss in the ratio of their investments.

If X and Y invests ₹x and ₹y respectively in a business for one year, then the end of the year

X’s share of profit: Share of s profit  = x: y

(ii) Compound Partnership:

In this type, the investment and period of time are different. Their investments have to be reduced to invests per month or year. The profit or loss will be divided in the ratio of these converted investments.

If X invests ₹x perp months and Y invests ₹y per q months, then

X’s share of profit: Share of s profit  = x: y

## Time and work

∎ If A can do a piece of work in ‘x’ days, then A’s one day’s work =

∎ If A’s one-day work =1/x, then A can finish the work in ‘x’ days.

∎ If the work done by A is twice the work done by B, then

The ratio of work done by A and B is 2: 1

The ratio of time taken by A and B to finish the work is  1: 2

∎ If the work done by A is thrice as the work done by B, then

The ratio of work done by A and B is 3: 1

The ratio of time taken by A and B to finish the work is  1 : 3

## Time and Distance

If the distance travelled = d, Speed = s and Time taken to travel = t, then

Distance = Speed × Time

⇒ d = s × t

Speed = ; Time =

∎ Relation between the speed in kilo metre per hour and speed in meter per seconds is

18 km/hr = 5 m/sec

x km/ hr =  m/sec

x m/sec = km/sec

Relative Speed:

∎ In the same direction relative speed = Difference of the Speeds

∎ In the opposite direction relative speed = sum of the Speeds

∎ If the two-object moving in the same direction with speeds x km/hr and y km/hr respectively, then their relative speed = (x – y) km/hr.

∎ If the two-object move in the opposite direction with speeds x km/hr and y km/hr respectively, then their relative speed = (x + y) km/hr.

∎ Time is taken to meet the object =

## Pipes and Cisterns

Inlet: A pipe connected with a tank or cistern, that fills it is called Inlet.

Outlet:  A pipe connected with a tank or cistern, that emptying is called an outlet.

∎If a pipe can fill a tank in x hrs. then in one hour, it can fill l part of the tank.

∎If a pipe can empty a tank in y hrs. then in one hour, it can empty part of the tank.

∎ If one pipe can fill a tank in x hrs. and another pipe can empty the full tank in y hrs. then on opening, both the pipes, the tank can fill in 1           hour =

∎ If one pipe can fill a tank in x hrs. and another pipe can empty the full tank in y hrs. then on opening, both the pipes, the tank can empty in      1 hour =

## Boats and Streams

Downstream:  In water, the direction along the water flow (stream) is called Downstream

Upstream:  In water, the direction against the stream is called Upstream.

∎If the speed of an object moving in water is x km/hr and the speed of the water flow (stream) is y km/hr then

For upstream relative speed = (x – y) km/hr.

For downstream relative speed = (x + y) km/hr.

Speed in still water = ½ (x + y) km/hr

Rate of stream = ½ (x – y) km/hr

## For Train Problems

∎ Time is taken by a train of length x meters to cross (pass) a pole (or a standing man) is equal to the time taken by the train to cover x meters.

∎ Time is taken by a train of length x meters to cross a stationary object (like a bridge) of lengthy meters is the time taken by train to cover (x + y) meters.

∎ If the two trains moving in the same direction with speeds x km/hr and y km/hr respectively, then their relative speed = (x – y) km/hr.

∎ If the two trains moving in the opposite direction with speeds x km/hr and y km/hr respectively, then their relative speed = (x + y) km/hr.

∎ If two trains of length meters and b meters are moving in opposite directions at x km/hr and y km/hr then the time taken by the trains to cross each other =   hrs.

∎ If two trains of length meters and b meters are moving in the same direction at x km/hr and y km/hr then the time taken by the trains to cross each other =   hrs.

## Triangle

A closed figure with three-line segments is called a triangle.

A triangle has 3 sides, 3 angles, and 3 vertices.

Types of the triangle:

According to the sides:

Scalene triangle: – If no two sides of the triangle are equal then it is called a scalene triangle.

Isosceles triangle- If any two sides of a triangle are equal, then it is called an isosceles triangle.

Equilateral triangle: – If all sides of a triangle are equal, then it is called an equilateral triangle

According to the angles:

Acute angled triangle: – If all angles of a triangle are acute, then it is called an acute-angled triangle.

Right-angled triangle:– If one of the angles of a triangle is a right angle(900), then it is called a right-angled triangle.

Obtuse angled triangle: If one of the angles of a triangle is obtuse, then it is called an obtuse-angled triangle.

Relationship between the sides of a triangle:

∎In a triangle, the sum of the lengths of two sides is greater than the third side.

∎ In a triangle, the difference between the lengths of the two sides is less than the third side.

Altitude: In a triangle, a perpendicular line drawn from the vertex to its opposite side is called Altitude.

∎We can draw three altitudes in a triangle.

∎ In an Isosceles triangle, the altitude from the vertex bisects the base.

Median: In a triangle, a line drawn from the vertex to the midpoint of its opposite side is called the median.

∎We can draw three medians in a triangle.

∎ The median of a triangle divides it into two triangles of equal area.

Circumcentre:

The point of concurrency of medians in a triangle is called circumcentre.

∎ Circumcentre divides median in the ratio 2: 1.

Angle sum property of triangles:

Some of the angles in a triangle is 1800

An Exterior angle: The angle formed in the exterior of a triangle when one of the sides is produced is called an exterior of the triangle.

∎ An exterior angle of a triangle is equal to the sum of its interior opposite angles.

Vertical angle bisector theorem: The bisector of the vertical angle of a triangle divides the base in the ratio of the other two sides.

Similar triangles:

Two triangles are said to be similar if (i) their corresponding angles are equal (ii) their corresponding side are in proportion.

∎ The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

∎ The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding Altitudes.

Pythagorean theorem:

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In ∆ABC, ∠B = 900, then AC2 = AB2 + BC2

FORMULAE:

∎ Area of the triangle with base ‘b’ and height ‘h’ = b × h sq. Units

∎ Area of the triangle whose lengths of the sides a, b and c =  sq. Units

Where, s =

∎ If a and b are the lengths of any two sides of a triangle and 𝛉 is the angle between them, then area of the triangle =   ab sin 𝛉 sq. Units

∎ Area of an equilateral triangle with side a =    a2 sq. Units

∎The radius of the circumcircle of a triangle =

∎ The radius of the circumcircle of an equilateral triangle of side a =

∎ The radius of the incircle of a triangle of the area ∆ and semi perimeter s =

∎ The radius of the incircle of an equilateral triangle of side a =

Quadrilateral: A simple closed figure with four-line segments is called a quadrilateral.

Diagonal: The line joining opposite vertices of a quadrilateral is called diagonal.

Trapezium: In a quadrilateral one pair of opposite sides are parallel, then it is called Trapezium.

∎ The line joining the midpoints of non-parallel sides of a trapezium is parallel to each of the parallel sides and equal to half of their sum.

Parallelogram: In a quadrilateral two pairs of opposite sides are parallel, then it is called Parallelogram.

∎ The diagonals of a parallelogram bisect each other and not equal.

∎ The diagonal of a parallelogram divides it into two triangles of equal area.

∎ In a parallelogram opposite sides are equal and opposite angles are equal.

Rectangle: In a parallelogram one of the angles is 900, then it is called a rectangle.

∎ In a rectangle, diagonal is bisect each other and equal.

Rhombus: In a parallel gram adjacent sides are equal, then it is called Rhombus.

∎ In a Rhombus diagonal are bisect each other perpendicularly and unequal.

A parallelogram and rectangle on the same base and lie between the same parallel lines are equal in area.

FORMULAE:

∎ Area of the Trapezium = (sum of the parallel sides) × (distance between them)

∎ Area of the parallelogram = base × height

∎ Perimeter of rectangle = 2 (length + breadth)

Area if the rectangle = length × breadth

∎ Area of rhombus = × product of lengths of diagonals

∎ Perimeter of Square = 4 × side

Area of square = (side)2

∎Area of four walls of a room = 2 (length + breadth) × height

Circles:

Circle: A 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.

∎ Circumference of the circle = 2πr, where r is the radius of the circle.

Area of the circle = πr2

∎Length of an Arc = l = rθ, where θ is the angle subtended by an arc at the centre.

∎Area of sector =  πr2

∎Area of a regular polygon of side a = 1.72 a2

## Mensuration

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.

Diagonal =

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.

Diagonal =

Cylinder:

Curved surface area (C.S.A) = 2πr h square units.

Total surface area (T.S.A) = 2πr (r +h) square units.

Volume = πr2 h cubic units.

Cone:

Slant height =

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.

Volume =  (R2 + r2 + Rr) πh cubic units.

Frustum of a cone:

When a cone is cut by a plane parallel to the base of the cone then the portion between the plane and the base is called the frustum of the cone.

Slant height =  units

L.S.A = πl (r + R) square units.

T.S.A = πl [r2 + R2 + l (R + r)] square 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.

Pyramid:

Surface area = Area of the base + Area of each of the lateral faces

Volume =  × area of the base × height cubic units.

Average:

Average =

Average of n natural numbers = ½ (n + 1).

suppose A person covers a certain distance at x km/hr and an equal distance at y km/hr, then the average speed during the whole journey is  km/hr

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 it 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 =

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

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 a 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.

## Calendar

Ordinary year: A year having 365 days is called an ordinary year.  In ordinary years we have 52 complete weeks and 1 extra day.

7 months ⟶ 31 days.

4 months ⟶ 30 days.

1 month (February) ⟶ 28 days.

Leap year: A year having 366 days is called a leap year.  In leap years we have 52 complete weeks and 2 extra days.

7 months ⟶ 31 days.

4 months ⟶ 30 days.

1 month (February) ⟶ 29 days.

∎ A leap year is divisible by 4.

∎ A century to be a leap year if it must be divided by 400.

∎ In a century 76 ordinary years and 24 leap years.

Odd day: Extra days apart from the complete weeks in a given period is called an odd day.

∎ An ordinary year has 1 odd day.

∎ A leap year has two odd days.

∎ A century has 5 odd days. (76 × 1 + 24 × 2 = 124 odd days = 17 weeks = 5 odd days).

∎In 200 years ⟶ 3 odd days.

In 300 years ⟶ 1 odd day.

In 400 years ⟶ 0 odd day.

In 800 years ⟶ 0 odd day.

In 1200 years ⟶ 0 odd day.

Century − Code:

Month − odd day − Code:

Day – Code:

Formula:

## Clocks

A clock has two hands (i) Shorter hand or hour hand (ii) Longer hand or Minute hand

∎For every hour, both hands coincide once.

Hour hand:

Angle made by Hour hand in 12 hours = 3600

In one hour, an hour hand can make the angle = 300.

In one minute, the hour hand can make the angle =

Minute hand:

Angle made by Minute hand in 60 minutes = 3600

In one minute, Minute hand can make the angle = 60

Relative speed:

The relative speed of the Minute hand and Hour hand = =

∎ When two hands are rotated in the clockwise direction

1. They may be opposite to each other.
2. They may coincide with each other.
3. They may be perpendicular to each other.

Two hands – Coincide:

The angle between two hands = 00

In the span of 1 hour, they coincide only 1 time.

12 hours, they coincide 11 times.

1 day, they coincide 22 times.

Two hands – opposite direction:

The angle between two hands =1800

In the span of 1 hour, they coincide only 1 time.

12 hours, they coincide 11 times.

1 day, they coincide 22 times.

Two hands – perpendicular:

The angle between two hands = 900

In the span of 1 hour, they coincide 2 times.

12 hours, they coincide 22 times.

1 day, they coincide 44 times.

∎ When two hands are opposite to each other than those two hands are 30 minutes spaces apart.

∎ When two hands are perpendicular to each other than those two hands are 15 minutes spaces apart.

Formula:

M = (H × 300± θ)

Where M = Minutes

H = Hours (starting hour)

θ = angle between two hands

If starting hour is at 12’O clock, then take H = 0

## Engineering Mathematics Syllabus

### Eng. Maths – M1 Syllabus

Engineering Mathematics Syllabus

Unit – I: Ordinary Differential Equations:

Basic concepts and definitions of 1st order differential equations; Formation of differential equations; solution of

differential equations: variable separable, homogeneous, equations reducible to homogeneous form, exact differential equation, equations reducible to exact form, linear differential equation, equations reducible to linear form (Bernoulli’s equation); orthogonal trajectories, applications of differential equations.

Unit – II: Linear Differential equations of 2nd and higher-order:

Second-order linear homogeneous equations with constant coefficients; differential operators; solution of homogeneous equations; Euler-Cauchy equation; linear dependence and independence; Wronskian; Solution of nonhomogeneous equations: general solution, complementary function, particular integral; solution by variation of parameters; undetermined coefficients; higher order linear homogeneous equations; applications.

Unit – III: Differential Calculus (Two and Three variables) :

Taylor’s Theorem, Maxima, and Minima, Lagrange’s multipliers

Unit – IV: Matrices, determinants, linear system of equations:

Basic concepts of an algebra of matrices; types of matrices; Vector Space, Sub-space, Basis, and dimension, linear the system of equations; consistency of linear systems; the rank of a matrix; Gauss elimination; the inverse of a matrix by Gauss Jordan method; linear dependence and independence, linear transformation; inverse transformation; applications of matrices; determinants; Cramer’s rule.

Unit – V: Matrix-Eigen value problems:

Eigenvalues, Eigenvectors, Cayley Hamilton theorem, basis, complex matrices; quadratic form; Hermitian, Skew Hermitian forms; similar matrices; diagonalization of matrices; transformation of forms to principal axis (conic section).

### Engineering Mathematics Syllabus – M2 Syllabus

Unit I: Laplace Transforms:

Laplace Transform, Inverse Laplace Transform, Linearity, transform of derivatives and Integrals, Unit Step function, Dirac delta function, Second Shifting theorem, Differentiation and Integration of Transforms, Convolution, Integral Equation, Application to solve differential and integral equations, Systems of differential equations.

Unit II: Series Solution of Differential Equations:

Power series; the radius of convergence, power series method, Fresenius method; Special functions: Gamma function, Beta function; Legendre’s and Bessel’s equations; Legendre’s function, Bessel’s function, orthogonal functions; generating functions.

Unit III: Fourier series, Integrals and Transforms:

Periodic functions, Even and Odd functions, Fourier series, Half Range Expansion, Fourier Integrals, Fourier sine, and cosine transforms, Fourier Transform

Unit IV: Vector Differential Calculus:

Vector and Scalar functions and fields, Derivatives, Gradient of a scalar field, Directional derivative, Divergence of a vector field, Curl of a vector field.

Unit V: Vector Integral Calculus:

Line integral, Double Integral, Green’s theorem, Surface Integral, Triple Integral, Divergence Theorem for Gauss, Stroke’s Theorem

#### Engineering Mathematics Syllabus M3 Syllabus

UNIT I: Basic Probability

Probability spaces, conditional probability, independent events, and Bayes theorem.  Random variables: Discrete and continuous random variables, Expectation of random variables, Moments, and Variance of random variables.

UNIT II: Probability distributions:

Binomial, Poisson, evaluation of statistical parameters for these distributions, Poisson approximation to the binomial distribution.

Continuous random variables and their properties, distribution functions and density functions, Normal and Exponential, evaluation of statistical parameters for these distributions.

UNIT III: Testing of hypothesis:

Test of significance: Basic testing of hypothesis. The null and alternate hypothesis, types of errors, level of significance, critical region.

Large sample test for a single proportion, the difference of proportions, a single mean, the difference of means, small sample tests: Tests for single mean, the difference of means and test for ratio of variances.

UNIT IV: Complex variables (Differentiation):

Limit, Continuity and differentiation of complex functions, Analyticity, Cauchy – Riemann equations (without proof), finding harmonic conjugate, elementary analytic functions and their properties.

UNIT V: Complex variables (Integration):

Line integral, Cauchy’s theorem, Cauchy’s integral formula, Zero of analytic functions, singularities, Taylor’s series, Laurent’s series, Residues, Cauchy Residue theorem, conformal mappings, Mobius transformations and their properties.

## Telugu Grammer( తెలుగు వ్యాకరణం )

Telugu Grammer( తెలుగు వ్యాకరణం ): మన మనస్సు లోని భావాలను , అనుభూతులను పైకి చెప్పడానికి  మాతృభాష ఎంతో ఉపయోగం , అందుకే మాతృ భాష తల్లి లాంటిది అంటారు. మనం మన మాతృ భాషని గౌరవించాలి .

గ్రహణ సామర్థ్యం పెరగడానికి మాతృ భాష  లో విద్యా బోధన ఎంతగానో ఉపయోగ పడుతుంది.  మాతృ భాష లో బోధించడం వలన విద్యార్థుల్లో సృజనాత్మకత పెరుగుతుంది.

### భాషా భాగాలు :

భాషకు ప్రాణం భావ ప్రసరణ . ఈ భావ ప్రసరణ ఒకరి నుండి మరొకరికి చేరాలి . ఇలా చేరడానికి కొన్ని పదాలు వాక్యాలు అవసరం. ఇలా వాక్యం లోని ఉపయోగాన్ని బట్టి భాషకు ఐదు ప్రధాన భాగాలుగా విభజించారు.

అవి :− 1. నామవాచకం   2. సర్వనామం  3. విశేషణం  4. క్రియ  5. అవ్యయము

### 1.నామవాచకం :

నామము అనగా పేరు.ఒక వ్యక్తిని గాని, వస్తువుని గాని ,గుణమును గాని, జాతిని గాని తెలుపును.

ఉదా :- ధర్మరాజు , హైదరాబాద్ , బంతిపువ్వు , ఆవు మొ|| నవి .

#### 2. సర్వనామము :

నామ వాచకాలకు బదులుగా వాడే వాటిని “ సర్వనామాలు “ అంటారు. “సర్వ” అనగా సమస్తము.

ఉదా :- అది, ఇది, అతడు , ఆమె ,అన్ని ,కొన్ని మొ|| నవి.

#### 3.  విశేషణం  :

నామవాచకము మరియు సర్వనామముల యొక్క గుణమును తెలియజేయునది .

ఉదా :- మంచి , చెడు , లావు , పొట్టి , పొడుగు  , ఎత్తు  మొ|| నవి.

#### 4. క్రియ :

పనులను, స్తితిగతులను తెలియజేయునది .

ఉదా :- రాస్తున్నాడు , వెళ్తున్నాడు , పాడుతున్నాడు  మొ|| నవి.

#### 5. అవ్యయము :

వ్యయము అనగా నశించేది, అవయము అనగా నశించనిది .

లింగ, వచన, విభక్తుల ప్రసక్తిగాని, వచన ఆకాంక్ష లేని వాటిని అవ్యయములు అంటారు .

ఉదా :-  అక్కడ, ఇక్కడ, ఆహా , భళా  మొ|| నవి.

## సంధులు

వ్యాకరణ భాషలో రెండు  స్వరాల కలయికను  సంధి  అంటారు .

రెండు అచ్చుల మధ్య  జరిగే మార్పును  సంధి కార్యం  అంటారు.

సంధి జరిగే మొదటి పదo చివరి అక్షరం లోని అచ్చును ‘పూర్వ పదం’ అంటారు .

సంధి జరిగే రెండవ పదం మొదటి అక్షరం లోని అచ్చును ‘పర పదం ‘ అంటారు .

ఉదా :- రామ + అయ్యా:  ‘ రామ’  లోని     ‘మ’  లో  ‘అ’ పూర్వ పదం  ‘అయ్యా’  లోని  ‘అ ‘ పర పదం .

##### అత్వ సంధి(అకార సంధి ):

అత్తునకు సంధి  బహుళంగా వస్తుంది .

ఉదా :-1) మేఅల్లుడు = మేన + అల్లుడు                     3) లేకేమి  = లేక + ఏమి

2) రాకుంటే  = రాక + ఉంటే                          4) పోవుటెట్లు =  పోవుట  + ఎట్లు

##### ఇత్వ సంధి ( ఇకార సంధి):

ఏమ్యాదులకు ఇత్తునకు సంధి .

ఉదా :- 1) ఏమంటివి = ఏమి + అంటివి                       3) పైకెత్తినారు  = పైకి + ఎత్తినారు

2) వచ్చిరిపుడు  = వచ్చిరి + ఇపుడు                4) మనిషన్నవాడు  = మనిషి + అన్నవాడు

#### ఉత్వ సంధి ( ఉకార సంధి ):

ఉత్తునకు అచ్చు పరమైనపుడు సంధి నిత్యంగా వస్తుంది .

ఉదా :- 1) రాముడతడు = రాముడు + అతడు            3) మనమున్నాము = మనము + ఉన్నాము

2) అతడెక్కడ = అతడు + ఎక్కడ

4) మనసైన = మనసు + ఐన

#### యదగామ సంధి :

అంది లేని చోట అచ్చుల మద్య ‘య్’ వచ్చి  చేరడాన్ని “యడాగమం” అంటారు .

ఉదా :- 1) మాయమ్మ = మా + అమ్మ                             3) హరియతడు = హరి + అతడు

2) మాయిల్లు = మా + ఇల్లు

#### ఆమ్రేడిత సంధి :

అచ్చునకు ఆమ్రేడితం పరమైతే సంధి తరచుగా వస్తుంది .

ఉదా :- 1)  ఆహాహా = ఆహా + ఆహా                                  3)  ఔరౌర = ఔర  + ఔర

2) అరెరే  = అరె + అరె                                           4)  ఏమిటేమిటి = ఏమిటి + ఏమిటి

##### గసడదవాదేశ సంధి :

ప్రథమ మీది పరుషాలకు గ, స ,డ , ద ,వ   లు  బహుళంగా వస్తాయి .

ఉదా :-  1)  కొలువుసేసి = కొలువు + చేసి                   3) కూరగాయలు = కూర + కాయ

2) పాలువోయక = ఆలు + పోయక                4) తల్లిదండ్రులు  = తల్లి + తండ్రి

#### త్రిక సంధి :

త్రికము మీది అసంయుక్త హల్లునకు దిత్వం బహుళంగా వస్తుంది .

 ఆ , ఈ , ఏ  లు త్రికంఅనబడుతాయి

ద్విరుక్తమైన హల్లు పరమైనపుడు, అచ్చికమైన దీర్ఘానికి  హ్రస్వం వస్తుంది.

ఉదా :-  1)  ఇక్కాలము  = ఈ + కాలము                       3)  అక్కోమరుండు =  ఆ + కొమరుండు

2) ఎవ్వాడు = ఏ + వాడు                                    4) అచ్చోట = ఆ + చోట

###### రుగాగమ సంధి :

పేదాది శబ్దాలకు  ‘ఆల‘ శబ్దo పరమైతే కర్మదారాయం లో రుగాగం వస్తుంది .

ఉదా :- 1) మనుమరాలు = మనుమా + ఆలు                3) ధీరురాలు  = దీరు + ఆలు

2) పేదరాలు = పేద + ఆలు                                     4)  బాలెంతరాలు = బాలెంత + ఆలు

5 ) ముద్దరాలు = ముద్ద +ఆలు                               6) జవరాలు = జావా + ఆలు

###### సవర్ణ దీర్ఘ  సంధి:

అ, ఇ , ఉ,  ఋ లకు అవే అచ్చులు పరమైతే వాని దీర్గాలు ఎకాదేశంగా వస్తాయి .

ఉదా :-    1)  రామానుజుడు = రామ + అనుజుడు    2 )  రామాలయం = రామ + ఆలయం

3)  భానూఉదయం  = భాను + ఉదయం                  4)  కవీంద్రుడు = కవి + ఇంద్రుడు

5 ) పితౄణం = పితృ +ఋణం        6) వదూపేతుడు  = వధు + ఉపేతుడు

#### గుణసంధి :

ఇ , ఉ , ఋ  పరమైతే ఏ, ఓ , ఆర్  లు క్రమంగా ఎకాదేసంగా వస్తాయి .

ఉదా :-    1)  రాజేంద్రుడు  = రాజ  + ఇంద్రుడు          2 )  పరోపకారం  = పర  + ఉపకారం

3)  రాజర్షి   = రాజ  + ఋషి               4)   మహోన్నతి = మహా  + ఉన్నతి

#### యణాదేశ సంధి:

ఇ , ఉ, ఋ లకు అసవర్ణ అచ్చులు పరమైతే య , వ ,ర  లు వస్తాయి .

ఉదా :-    1)  అత్యవసరం  = అతి  + అవసరం    2 )  ప్రత్యేకం  = ప్రతి  + ఏకం

3)  అణ్వస్త్రం   = అణు  + అస్త్రం      4)  పిత్రార్జితం  = పితృ  + ఆర్జితం

5 ) పితౄణం = పితృ +ఋణం

#### వృద్ధి సంధి :

అకారినికి ఏ , ఐ  లు పరమైతే ‘ఐ ‘ కారము , ఓ , ఔ లు పరమైతే  ‘ఔ’ కారము వస్తాయి.

ఉదా :-    1)  వసుధైక  = వసుధ   + ఏక           2 )  సమైక్యం   = సమ   + ఐక్యం

3)  వనౌసది    = వన   + ఔసది         4)  పిత్రార్జితం  = పితృ  + ఆర్జితం

##### అనునాసిక సంధి :

వర్గ ప్రతమాక్షరాలకు ‘న’ గాని , ‘మ’ గాని ప్రమైతే అనునాసికాలు.

ఉదా :-    1)  వాజ్మయం   = వాక్   + మయం           2 )  జగన్నాథుడు   = జగత్  + నాథుడు

3)  అణ్వస్త్రం   = అణు  + అస్త్రం               4)  పిత్రార్జితం  = పితృ  + ఆర్జితం

5 ) తన్మయం  = తత్  + మయం

## సమాసాలు

సమాసం : వేరు వేరు అర్థాలు కల రెండు పదాలు కలిసి , ఏకంగా ఏర్పడితే దాన్ని ‘సమాసం’ అంటారు .

###### దంద్వ సమాసం:

రెండు కాని,  అంతకంటే ఎక్కువ కాని నామవాచకాల మద్య ఏర్పడే సమాసాన్ని ‘దంద్వ సమాసం ‘ అంటారు.

ఉదా : – 1) అన్నదమ్ములు  – అన్న, తమ్ముడు    2) తల్లిదడ్రులు – తల్లి, తండ్రి

3) మంచిచెడులు – మంచి, చెడు  4 ) కష్టసుఖాలు – కష్ట , సుఖము

###### ద్విగు సామాసం :

సమాసంలో మొదటి పదంలో సంఖ్య గల సమాసాన్ని ‘ద్విగు’ సమాసం అంటారు.

ఉదా : – 1) నవరసాలు   – నవ సంఖ్య  గల రసాలు    2) రెండుజడలు  – రెండు సంఖ్య  గల జడలు

3) నాలుగు వేదాలు – సంఖ్య గల వేదాలు

#### తత్పురుష సమాసం :

విభక్తి ప్రత్యాలు విగ్రహ వాక్యంలో ఉపయోగించే సమాసాలు ‘ తత్పురుష  సమాసాలు .

## విభక్తులు

• ఒక వాక్యం లోని వేరు వేరు పదాలకు అన్వయం కలిగించు పదాలను “ విభక్తులు” అంటారు
 విభక్తులు ప్రత్యయాలు ప్రథమా విభక్తి డు – ము – వు – లు ద్వితీయ విభక్తి నిన్ – నన్ – లన్ – కూర్చి – గురించి తృతీయ విభక్తి చేతన్ – చెన్ –  తోడన్ – తోన్ చతుర్థి విభక్తి కొరకున్  – కై పంచమ విభక్తి వలనన్ – కంటెన్ – పట్టి షష్ఠివిభక్తి యొక్క – లోన్ – లోపలన్ సప్తమి విభక్తి అందున్ – నన్ సంబోధన ప్రథమా  విభక్తి ఓరి – ఓయి – ఓసి
 సమాస పదం విగ్రహ వాక్యం సమాసం పేరు మద్యాహ్నము అన్నము యొక్క మద్య ప్రథమా తత్పురుష జటాధారి జడలను ధరించినవాడు ద్వితీయ  తత్పురుష రాజ పూజితుడు రాజు చే పూజితుడు తృతీయ  తత్పురుష వంట కట్టెలు వంట కొరకు కట్టెలు చతుర్థి  తత్పురుష అగ్నిభయం అగ్ని వల్ల భయం పంచమ  తత్పురుష భుజభలం భుజాల యొక్క భలం షష్ఠి తత్పురుష పుర జనులు పురమునందు  జనులు సప్తమి తత్పురుష

విశేషణ పూర్వపద కర్మధారయ సమాసం : విశేషణం పూర్వపదంగా (మొదటి) ఉండే సమాసం .

ఉదా : – తెల్ల గుర్రం – తెల్లదైన  గుర్రం,  ఇస్టార్థములు – ఇష్టమైన అర్థములు

సంభావన  పూర్వపద కర్మధారయ సమాసం :  సమాసం లోని పూర్వపడం సంజ్ఞావాచాకంగా , ఉత్తరపదం జాతి వాచకంగా ఉంటుంది .

ఉదా :- కాశిక పట్టణం – కాశిక అను పేరు గల పట్టణం, తెలంగాణా రాష్ట్రము – తెలంగాణ అను పేరు గల రాష్ట్రం

నైతత్పురుష సమాసం :  వ్యతిరేఖ పదాన్ని ఇచ్చే పదం .

ఉదా :- అసత్యం – సత్యం కానిది , నిరాదారం – ఆదారం కానిది , అనుచితం – ఉచితం కానుది .

#### అలంకారాలు

అలంకారం : చెప్పదలచిన విషయాన్ని అందంగా మలిచేది.

అంత్యాను ప్రాస అలంకారం:  ఒకే అక్షరం లేదా రెండు , మూడు అక్షరాలు వాక్యం చివర మాటి మాటికి వస్తే దాన్ని  అంత్యాను ప్రాస అలంకారం అంటారు .

ఉదా : – భాగవతమున భక్తి – భారతమున యుక్తి – రామ కథయే రక్తి  ఓ కూనలమ్మ .

వృత్యానుప్రాస అలంకారం: ఒకటి గాని అంతకంటే ఎక్కువ గాని హల్లులు పలుమార్లు వచ్చునట్లు చెబితే  వృత్యానుప్రాస అవితుంది.

ఉదా : – వీరు పొమ్మను వారు  వారు పోగబెట్టు వారు

కాకి కోకికాదు దా

చేకానుప్రాస అలంకారం: అర్థ భేదం తో కూడిన హల్లుల జంట వెంట వెంటనే వస్తే చేక్కనుప్రాస అనబడుతుంది .

ఉదా : – అ నాథ నాథ  నంద నంద న నీకు వందనం

నీకు వంద వందనాలు.

లాటాను ప్రాస అలంకారం : అర్థంలో భేధం లేకపోయినా , తాత్పర్యంలో భేదం ఉండేటట్లు , ఒక పదం రెండు సార్లు ప్రయోగించబడితే లాతానుప్రాస అనబడుతుంది.

ఉదా : – కమలాక్షునకు అర్పించు కరములు కరములు

యమకం : అచ్చులలో హల్లులలో మార్పు లేనట్టి అక్షరాల సమూహం అర్థ భేదంతో మళ్ళీ  ప్రయోగిన్చినట్లయితే యమకం అనబడుతుంది.

ఉదా : –  పురము నందు నంతిపురము

ముక్త పద గ్రస్తo : విడిచి పెట్టబడ్డ పద భాగాలను వ్యవదానం లేకుండా వెంటనే ప్రయోగించి చెబితే ముక్త ప్రదగ్రస్తం .

ఉదా :- సుదతీ సూదన మదనా

మదనా గ తురంగ పూర్ణ మణి మాయ సదనా

సదనా మయ గజ రాదనా .

ఉపమాలంకారం : ఉపమేయానికి ఉపమానం తో చక్కని పోలిక వర్ణించబడిన యెడల ఉపమాన అలంకారం అనబడుతుంది .

ఉదా : – 1) చేనేత కార్మికులు ఎలుకల్ల మాడి పోతున్నారు    2) నీ కీర్తి హంష  లాగ ఆకాశ గంగలో మునుగుతుంది

రూపకాలoకారం : ఉపమానానినికి , ఉపమేయానినికి భేదం లేనట్లు వర్ణించి చెబితే రూపకాలoకారం అంటారు.

ఉదా : – 1) సంసార సాగరాన్ని తరించడం మిక్కిలి కష్టం   2 ) మౌనికి తేనె పలుకులు అందరికి ఇష్టమే

ఉత్ప్రేక్ష అలంకారం : ఉపమానానినికి ఉన్న ధర్మాలు  ఉపమేయంలో ఉండడం చేత , ఉపమేయాన్ని  ఉపమానo గా ఊహించి చెబితే  ఉత్ప్రేక్ష  అలంకారం అంటారు.

ఉదా : –  1) ఆ మేడలు ఆకాశాన్ని  ముద్దడుతున్నాయా అన్నట్లు ఉన్నాయి    2) ఈ వెన్నెల పాలవెళ్లి యో  అన్నట్లుంది .

అతి శయోక్తి అలంకారం : ఒక వస్తువు గురించి కాని సందర్భాన్ని గురించి కాని ఉన్నదాని కంటే ఎక్కువ చేసి చెబితే అతి శయోక్తి అంటారు.

ఉదా : – మా నగరం లోని మేడలు ఆకాశాన్ని అంటుతున్నాయి

శ్లేషాలo కారం : అనేకమైన అర్థాలు కల శబ్దాలను ఉపయోగించి చెబితే శ్లేష అనబడుతుంది

ఉదా : – రాజు కువలయానంద  కరుడు

రాజు = ప్రభువు , చంద్రుడు       కువలయం = భూమి , కలువ పూలు

స్వభావోక్తి అలంకారం :  జాతి గుణం క్రియాదు లలో ఉన్నది ఉన్నట్లు చెప్పడం

ఉదా : – చెట్ల ఆకులు గాలికి కదులుతున్నాయి

## ఛందస్సు

పద్య లక్షణాన్ని తెలిపే శాస్త్రాన్ని ఛందస్ శాస్త్రం అంటారు .

ఒక మాత్ర కాలం లో ఉచ్చరించబడేది లఘువు (I )

రెండు మాత్రల  కాలం లో ఉచ్చరించబడేది  గురువు ( U  )

య  గణం   IUU                                            జ  గణం    IUI

మ  గణం  UUU                                           భ  గణం    UII

త   గణం   UUI                                             న  గణం III

ర   గణం    UIU                                              స  గణం  IIU

 పద్యం పేరు గణాలు యతి స్తానం అక్షరాల సంఖ్య ఉత్పల మాల భ, ర, న, భ, భ, ర, వ 10 20 చంపక మాల న, జ, భ, జ, జ, జ, ర 11 21 శార్దూలం మ, స, జ, స, త, త, గ 13 19 మత్తేభం స, భ, ర, న, మ, య, వ 14 20

## వాఖ్య నిర్మాణము – రకాలు

వాక్యాలు మూడు రకాలు : 1 ) సామాన్య వాక్యము 2) సంశ్లిష్ట వాక్యము       3) సంయుక్త వాక్యము .

• సామాన్య వాక్యము :- క్రియ ఉన్నా  లేకున్నా ఒకే ఒక్క భావాన్ని ప్రకటించే వాక్యాలను సామాన్య వాక్యాలు అంటారు .

ఉదా : (i ) సీత బజారుకు వెళ్ళింది .    (ii) పాము కాటేసింది   (iii ) మురళి మంచి బాలుడు

• సంశ్లిష్ట వాక్యము:- ఒక సమాపక క్రియ , ఒకటి గాని అంతకన్నా ఎక్కువ గాని అసామాపక క్రియలు ఉంటే  ఆ వాక్యాన్ని    సంశ్లిష్ట వాక్యము అంటారు .

ఉదా :- (i )  రాము అన్నము తిని , పడుకున్నాడు         (ii )  సీత బజారుకు వెళ్లి , సరుకులు కొన్నది

• సంయుక్త వాక్యము:- సమ ప్రాధాన్యం కల వాక్యాలను కలపడం వల్ల ఏర్పడే వాక్యాo ను సంయుక్త వాక్యము అంటారు.

ఉదా :-  (i ) అతడు నటుడు, రచయిత    (ii ) రాము మరియు సిత హైదరాబాద్ వెళ్లారు    (iii ) సీత చదువుతుంది ,

## కర్తరి – కర్మణి వాఖ్యాలు

• కర్తరి వాక్యము :- ఒక వాక్యంలో కర్తకు ప్రాధాన్యం ఇచ్చి , కర్మకు ద్వితీయ విభక్తి (నిన్ , నున్ , లన్ , కూర్చి , గురించి ) చేరితే ఆ వాక్యాని కర్తరి వాక్యం అంటారు.
• కర్మణి వాక్యము :-  ఒక వాక్యంలో క్రియకు ధాతువు చేరి  , కర్మకు తృతీయ  విభక్తి  ( చేతస్ , చేన్ , తోన్ , తోడన్ ) చేరితే ఆ వాక్యాని కర్తరి వాక్యం అంటారు.

ఉదా :-

 కర్తరి వాక్యము కర్మణి  వాక్యము 1 ప్రజలు శాంతిని కోరుతున్నారు ప్రజలచే శాంతి కోరబడుతుంది 2 మేం పెద్దలను గౌరవిస్తాము మాచే పెద్దలు గౌరవించ బడతారు 3 రాజు రైలును నడిపాడు రైలు రాజు చే నడపబడింది 4 భీముడు కొండలను పిండి చేసాడు కొండలు భీముని చే పిండి చేయబడెను 5 నా మీద రాళ్ళు విసురుతారు నా మీద రాళ్ళు విసరబడతాయి

ప్రత్యక్ష – పరోక్ష కథనాలు
ప్రతక్ష కథనం : ఒకరు చెప్పిన విషయాన్ని ఉన్నది ఉన్నట్లుగా చెప్పడం . ఒకరు చెప్పిన విషయం  “     “  చిహ్నాల మద్య ఉందును.

ఉదా:-   i )  “ నేను రస జీవిని “  అని చాసో అన్నాడు    (ii ) అంబేత్కర్  “  నేను ఎవరిని యాచిన్చను  “ అని అన్నాడు

• పరోక్ష కథనం :  ఒకరు చెప్పిన విషయాన్ని మన మాటల్లో చెప్పడం .    ఇందులో   “    “  చిహ్నాలు ఉండవు .

ఉదా:-  (i )  తాను రస  జీవినని చాసో అన్నాడు    (ii ) అబ్మేత్కర్ తాను ఎవరినీ  యాచిన్చనని  అన్నాడు

గమనిక : ప్రత్యక్ష కథనం  నుండి పరోక్షం లోకి మార్చునప్పుడు  జరుగు మార్పులు :

 ప్రత్యక్షం పరోక్షం నేను తాను ఆయన అతను, వాడు అది ఇది నాకు తనకు నా తన నన్ను తనను మేము తాము మాకు తమకు ఇది అది

## 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.

## ICSE IX Class Maths Concept

ICSE IX Class Maths Concept: This note is designed by the ‘Basics in Maths’ team. These notes to do help the ICSE 9th class Maths students fall in love with mathematics and overcome their fear.

These notes cover all the topics covered in the ICSE 9th class Maths syllabus and include plenty of formulae and concept to help you solve all the types of ICSE 9th

Math problems are 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: – $\inline&space;\fn_jvn&space;-\frac{2}{3}$ can be written as $\inline&space;\fn_jvn&space;\frac{-2}{3}=\frac{2}{-3}$  but our convenience we can take $\inline&space;\fn_jvn&space;\frac{-2}{3}$

Equal rational numbers:

For any 4 integers a, b, c, and d (b, d ≠ 0), we have $\inline&space;\fn_jvn&space;\frac{a}{b}=\frac{c}{d}$ ⇒ ad = bc

The order of Rational numbers:

If  are two rational numbers such that b> 0 and d > 0 then $\inline&space;\fn_jvn&space;\frac{a}{b}>&space;\frac{c}{d}$ ⇒ ad > bc

The absolute value of rational numbers:

The absolute value of a rational number is always positive. The absolute value of  $\inline&space;\fn_jvn&space;\frac{a}{b}$ is denoted by $\inline&space;\fn_jvn&space;\left&space;|&space;\frac{a}{b}&space;\right&space;|$ .

Ex: – absolute value of $\inline&space;\fn_jvn&space;-\frac{2}{3}=\frac{2}{3}$

To find rational number between given numbers:

• Mean method: – A rational number between two numbers a and b is $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{a&space;+&space;b}{2}$

Ex: – insert two rational number between 1 and 2

1 <  $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{1&space;+&space;2}{2}$ < 2   ⟹     1 <  $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{3}{2}$  < 2

1 <  $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{3}{2}$ $\inline&space;\dpi{120}&space;\fn_jvn&space;<&space;\frac{\frac{3}2{+2}}{2}$< 2   ⟹   1 < $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{3}{2}<&space;\frac{7}{4}$ $\dpi{120}&space;\fn_jvn&space;<$  2

To rational numbers in a single step: –

Ex:- insert two rational numbers between 1 and 2

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

(∵ 1 + 2 = 3)

1 =   $\fn_jvn&space;\frac{1\times&space;3}{3}$  and 2 = $\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{2\times&space;3}{3}$

$\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{3}{3}\left&space;(&space;1&space;\right&space;)<&space;\frac{4}{3}<&space;\frac{5}{3}<&space;\frac{6}{3}\left&space;(&space;2&space;\right&space;)$

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 a non-terminating repeating decimal.

Converting decimal form into $\dpi{120}&space;\fn_jvn&space;\frac{p}{q}$  the form:

1. Terminating decimals: –

1.2 =$\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{12}{10}=\frac{6}{5}$

1.35 =$\inline&space;\dpi{120}&space;\fn_jvn&space;\frac{135}{100}=\frac{27}{20}$

2. Non-Terminating repeating decimals: –

Irrational numbers:

• The numbers which are not written in the form of  $\dpi{120}&space;\fn_cm&space;\frac{p}{q}$, 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:- $\dpi{120}&space;\fn_cm&space;\sqrt{2},\,&space;\sqrt{5},\pi$ 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 $\dpi{120}&space;\fn_cm&space;\sqrt{2}$ value:

## ICSE X Class Maths

ICSE X Class Maths Concept designed by the ‘Basics in Maths’ team. These notes to do help the ICSE 10th class Maths students fall in love with mathematics and overcome their fear.

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 =    $p&space;times&space;frac{n(n+1))}{2times&space;12}times&space;frac{r}{100}$

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 met

hod:

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

20. Constructions

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