PDF Files || Inter Maths 1A &1B || (New)

By [email protected] Reddy / October 11, 2021 / Education

Inter Maths 1A &1B|| PDF Files (New)

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Inter Mathematics 1A ands 1B Pdf Files| these Files were designed by the ‘Basics in Maths‘ team. These notes to do help the intermediate First-year Maths students.

Inter Maths – 1A & 1B PDF Files are very useful in IPE examinations.

These notes cover all the topics covered in the intermediate First-year Maths syllabus and include plenty of solutions to help you solve all the major types of Math problems asked in the

IPE examinations.

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MATHEMATICS 1A

Mathematical Induction

Addition Of Vectors SAQ’S

Matrices

Trigonometric Equations

Inverse Trigonometric Functions

TS 10th class maths concept (E/M)

MATHEMATICS 1B

DC’s and Dr’s

Tangents and Normals

Rate Measure

Maxima and Minima

Rolles and Langrange’s Theorem

TS 6th Class Maths Concept

Visit my YouTube Channel: Click on Below Logo

TS VII CLASS MATHS CONCEPT FEATURE IMAGE

TS 7th Class Maths Concept

By [email protected] Reddy / November 12, 2020 / Education

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.

These concepts were designed by the ‘Basic in Maths’ team. These notes to do help students fall in love with mathematics and overcome fear.

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:

integers

Operations on integers:

addition of integers:

3 + 4 = 7 integers addition

-2 + 4 = 2

addion of integers

Subtraction of integers on a number line:-

6 – 3 = 3 subtraction of integers

Multiplication of integers on a number line:-

2 × 3 ( 2 times of 3) = 6 multiplication on number line

3 × (- 4 ) ( 3 times of -4) = -12

multiplication of itegers

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.

multiplication rule

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:-

closure

2.commutative property:-

commutative property

3.associative property:-

associative property

Additive identity:-

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

•For any integer ‘a’, a + 0 = 0 + a

•0 is the additive identity.

Additive inverse:-

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.

Addition of fractions:

∗ Like Fractions:

$vii math addition of fracrtions$

∗ Unlike fractions:

$vii maths additon of unlike fractions$

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: –

$multiplying a fraction with whole number$

• 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 ÷ $ts vii math division of whole number by a fraction$

⇒ 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

ts vii maths tansposing rules

Using algebraic equations in solving day to day problems:

Read the problem carefully.
Denote the unknown or quantity to be found with some letters such as x, y, z …etc.
Write the problem in the form of an algebraic equation by making a relation among the quantities.
Solve the equation.
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:

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

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.

TS VII Maths Construction of Triangles 5 new

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 + 4x³is 3

degree of 3xy + 6x²y + 5x²y² is 4

Addition of like terms:

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

Addition of algebraic expressions:

The addition of expressions can be obtained by adding like terms.

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

Additive inverse of an expression:

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 = a^m

a^m is in exponential form

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

Laws of exponents:

a^m × aⁿ = a^{m + n}
(a^m)ⁿ = a^mn
a^m = aⁿ ⇒ m = n
(ab)^m = a^m.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.

12.QUADRILATERALS

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

In Quadrilateral ABCD

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

Diagonal of a Quadrilateral:

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.

Adjacent sides of a Quadrilateral:

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.

Adjacent angles of a Quadrilateral:

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.

Opposite sides of a Quadrilateral:

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.

Opposite angles of a Quadrilateral:

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.

Convex 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.

Concave 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

Types of Quadrilaterals:

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.

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TS VI CL;ASS MATHS CONCEPT FEATURE IMAGE

TS 6 th class Maths Concept || Basics In Maths

By [email protected] Reddy / November 11, 2020 / Education

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

Studying maths in the 6^th 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 :

place value table 1

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

placevalue table 2

Place value table for International system :

place value table 3 Ex:- represents the number in 635,218,924 in place value table

place value table 4

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:

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.

Addition:-Whole numbers are closed under addition.

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.

Addition:-Whole numbers are commutative under addition.

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 ÷ ].

Addition:-Whole numbers are associative under addition.

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 triangle number 3 6 can be shown as

triangle form 6

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.

DIVISION METHOD

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 PRIME FACTORISATION METHOD

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 DIVISION METHOD

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:-

LCM1

∴ 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

Zero is less than every positive integer.
Zero is greater than every negative integer.
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:

Add 3 and 4

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
Add −3 and −4

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.

$TS vi Math fractions and decimals 1$

• 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., $TS vi Math fractions and decimals 2$ 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., $TS vi Math fractions and decimals 3$ 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., $TS vi Math fractions and decimals 4$ of the circle.

The numerator and the denominator:

$TS vi Math fractions and decimals 5$

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:

$TS vi Math fractions and decimals 9$

Representing fractions on a number line:

Mark $TS vi Math fractions and decimals 7$ on a number line

$TS vi Math fractions and decimals 8$

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: – $TS vi Math fractions and decimals 10$

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

Ex: – $TS vi Math fractions and decimals 11$

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

Ex: – $TS vi Math fractions and decimals 12$

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

A mixed fraction is in the form of $TS vi Math fractions and decimals 13$

We can convert it into improper fraction by $TS vi Math fractions and decimals 14$

Ex: – $TS vi Math fractions and decimals 15$

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

$TS vi Math fractions and decimals 16$

Equivalent fractions are arising when we multiply both the numerator and denominator by the same number.
Equivalent fraction of $TS vi Math fractions and decimals 2$ are $TS vi Math fractions and decimals 17$ 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: – $TS vi Math fractions and decimals 18$

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: – $TS vi Math fractions and decimals 19$ are like fractions and $TS vi Math fractions and decimals 20$ 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: – $TS vi Math fractions and decimals 21$ 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: – $TS vi Math fractions and decimals 22$ 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: $TS vi Math fractions and decimals 23$ are in ascending order and $TS vi Math fractions and decimals 24$ are in descending order.

Addition of fractions:

Like fractions: –

$TS vi Math fractions and decimals 25$

Ex: $TS vi Math fractions and decimals 27$ $TS vi Math fractions and decimals 26$

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: – $TS vi Math fractions and decimals 28$ $TS vi Math fractions and decimals 29$

Subtraction of fractions

Like fractions: –

$TS vi Math fractions and decimals 30$

Ex: – $TS vi Math fractions and decimals 31$

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

Ex: – $TS vi Math fractions and decimals 32$

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: – $TS vi Math fractions and decimals 33$

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 n^th 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 n^th 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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TS 10th Class Maths Concept (T/M)

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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.

12. SYMMETRY

Symmetry:

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

Symmetry is the mirror image of an object.

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

Line of symmetry:

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

Line of symmetry can be horizontal, vertical or diagonal.

Ex:

The English alphabet which have

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

13. PRACTICAL GEOMETRY

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

1.A Ruler (Scale)

2.The compasses

3.The divider

4.Protractor

∗ The ruler is used to measure lines.

∗ The compasses is used for constructing.

∗ The protractor is used for measuring angles.

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

Construction of a line segment of a given length:

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

1.By using Ruler: –

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

Steps of construction: –

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

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

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

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

1.By using the Compasses: –

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

Steps of construction: –

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

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

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

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

Construction of a circle:

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

Steps of construction: –

Step-1: Open the compasses for radius 3 cm

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

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

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

Construction of perpendicular bisector a line segment:

Steps of construction: –

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

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

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

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

Steps of construction: –

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

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

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

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

Construction of Angles using Protractor:

Let us suppose we want to construct ∠ABC = 50⁰

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

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

Step-3: Mark a point A at 50⁰

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

Constructing a copy an of Angle of un known measure:

Let ∠A is given, measure is not known

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

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

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

Step-4: Set your compasses with BC as the radius.

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.

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TS 10th class maths concept (E/M)

By [email protected] Reddy / November 11, 2020 / Education

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 2^m 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 a^x = N then x = ${log_{a}}^{N}$

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

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ICSE 8th Maths Concept| Basics In Maths

By [email protected] Reddy / July 18, 2020 / Education

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:

Resolve the given number into prime factors.
Make pairs of similar factors.
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:

Mark off the digits in pairs starting with the unit place. Each pair and remaining one digit are called a period.
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.
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.
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: – $square of a proper fractin$

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:

Resolve the given number into prime factors.
Make triplets of similar factors.
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.

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ICSE IX Class Maths Concept

By [email protected] Reddy / July 18, 2020 / Education

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 9^thclass Maths students fall in love with mathematics and overcome their fear.

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

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: – $\inline \fn_jvn -\frac{2}{3}$ can be written as $\inline \fn_jvn \frac{-2}{3}=\frac{2}{-3}$ but our convenience we can take $\inline \fn_jvn \frac{-2}{3}$

Equal rational numbers:

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

The order of Rational numbers:

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

The absolute value of rational numbers:

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

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

To find rational number between given numbers:

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

Ex: – insert two rational number between 1 and 2

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

1 < $\inline \dpi{120} \fn_jvn \frac{3}{2}$ $\inline \dpi{120} \fn_jvn < \frac{\frac{3}2{+2}}{2}$ < 2 ⟹ 1 < $\inline \dpi{120} \fn_jvn \frac{3}{2}< \frac{7}{4}$ $\dpi{120} \fn_jvn <$ 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 = $\fn_jvn \frac{1\times 3}{3}$ and 2 = $\inline \dpi{120} \fn_jvn \frac{2\times 3}{3}$

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

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 $\dpi{120} \fn_jvn \frac{p}{q}$ the form:

1. Terminating decimals: –

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

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

2. Non-Terminating repeating decimals: –

Irrational numbers:

The numbers which are not written in the form of $\dpi{120} \fn_cm \frac{p}{q}$ , where p, q are integers, and q ≠ 0 are called rational numbers. Rational numbers are denoted by Q^Ior S.
Every irrational number can be expressed as a non-terminating decimal or non-repeating decimal.

Ex:- $\dpi{120} \fn_cm \sqrt{2},\, \sqrt{5},\pi$ 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 $\dpi{120} \fn_cm \sqrt{2}$ value:

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ICSE X Class Maths Concept

By [email protected] Reddy / July 18, 2020 / Education

ICSE X Class Maths

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

These notes cover all the topics covered in the ICSE 10^thclass Maths syllabus and include plenty of formulae and concept to help you solve all the types of 10^th 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.

Benefits of GST for Traders:

• Simple tax system.

• Elimination of multiplicity of taxes.

• Development of a common market nation-wide.

• Reduction of cascading effect.

• 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.

Benefits of GST for Traders:

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

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

Add 2 on both sides

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

Add 2 on both sides

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}

5.Quadratic Equations

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: x² – 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.

6.Problems on Quadratic equations

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 x^n-1 + a₂ x^n-2 + …+ a_n-1 x + a_n, where a₀, a₁, …, a_n 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 i^th row and j^th column is called the (i, j)^th element of the matrix and it is denoted by a_ij.

A = [a_ij]_{m × n}

A =

a₁₁ means element in first row and first column

a₁₂means element in first row and second column

a₂₂means element in second row and second column

a₃₂means 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 = [a_ij] 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 A^T.

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 = [a_ik]_mxn and B = [b_kj]_nxp be two matrices, then the matrix C = [c_ij]_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 = … = a_n – a_{n -1}

n^th term: T_n = 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 n^th 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 n^th term from the end is l – (n – 1) d .

11. Geometric Progressions

Terms: a, a r, a r²…, a r^{n – 1}

First term: a

Common ratio:

n^th term: T_n = a r^{n – 1}

Sum of the n terms of G.P.:

Sum of the n terms of G.P. is

To find the n^th 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 n^th 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) ⇒ R_x (x, y) = (x, –y)

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

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

Combination of Reflection:

R_x. R_y = R_y. R_x = R_o
R_x. R_o = R_o. R_x = R_y
R_y. R_o = R_o. R_y = R_x

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

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

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: –

If two lines are parallel then their slopes are equal

m₁ = m₂

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.

∠ACB = ∠ADB

∎ 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.

Cyclic Quadrilateral:

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:

External radius = R

Internal radius = r

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.

Radius of Sphere = r

Surface area = 4πr² sq. units

Volume = πr³ cubic. Units

Spherical Shell:

The solid enclosed between two concentric spheres is called spherical shell

External radius = R

Internal radius = r

Thickness of the cylinder = R – r

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

Hemi sphere:

Radius of hemisphere = r

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)

BC = Base = adjacent side of θ(adj)

AC is hypotenuse (hyp)

Trigonometric ratios:

Quotient relations:

Trigonometric Identities:
(i) sin²θ + 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:

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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, …, x_n be the class marks of the given frequency distribution and f₁, f₂, f₃, …, f_n be the corresponding frequencies, then plot the points (x₁, f₁), (x₂, f₂), …. (x_n, f_n) 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, …, x_n is

Mean =

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

Note:

If the mean of a data x₁, x_{2, …}x_nis m, then

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

If x₁, x₂, …. x_nare of n observations occurs f₁, f₂, …. f_n times respectively then mean is

Mean of grouped data:

Methods of finding mean:

Class mark (mid value) =

Direct method: ; x_i is class mark of i^th class, f_i is frequency of class.

Assumed mean method: ; d_i = x_i – a and a is assumed mean.

Step – deviation method: ; 𝛍_i = , 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 Q₁and 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

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11th Class Maths Concept

By [email protected] Reddy / July 16, 2020 / Education

11th Class Maths: This note is designed by ‘Basics in Maths’ team. These notes to do help the CBSE 11^thclass Maths students fall in love with mathematics and overcome their fear.

These notes cover all the topics covered in the CBSE 11^thclass Maths syllabus and include plenty of formulae and concept to help you solve all the types of11^thMath problems asked in the CBSE board and entrance examinations.

1. SETS

Well-defined objects:

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

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

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

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

∗ Set theory was developed by George canter.

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

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

Methods of representing sets:

Roster or table or listed form: –

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

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

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

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

Set builder form:

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

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

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

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

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

Ex: – A = {1, 2,3, 4} → finite set

B = {1, 2, 3, 4….}

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

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

⟹ A = B.

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

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

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

Ex: – A = {1,2,3}

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

Intervals:

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

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

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

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

Universal set: – A set that contains all the subsets of it under our consideration is called a universal set.

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

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

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

Ex: – A = {1, 2, 3}, B = {a, b, c}

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

∴ A = B.

Venn diagrams:

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

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

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TS Inter second year Maths 2B Concept

By [email protected] Reddy / July 14, 2020 / Education

TS Inter second year

TS Inter second year: This note is designed by the ‘Basics in Maths’ team. These notes to do help the TS intermediate second-year Maths students fall in love with mathematics and overcome the fear.

These notes cover all the topics covered in the TS I.P.E second year maths 2B syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations.

TS Inter second year

1. CIRCLES

Circle: In a plane, 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 (C) of the circle and the constant distance is called the radius(r) of the circle

Unit circle: If the radius of the circle is 1 unit, then that circle is called the unit circle.

Point Circle: A circle is said to be a point circle if its radius is zero. A point circle contains only one point in the centre of the circle. •

∗ The equation of the circle with centre (h, k) and radius r is

(x – h)² + (y – k)² = r²

∗ The equation of the circle with centre origin and radius r is

x² + y² = r²

⇒ x² + y² = r² is called standard form of the circle.

The general equation of the second degree ax² + 2hxy + by² + 2gx + 2fy + c = 0, where a, b, f, g, h and c are real numbers, represent a circle iff (i) a = b ≠ 0 (ii) h = 0 and (iii) g² + f² + c ≥ 0

∗ The general equation of the circle is x² + y² + 2gx + 2fy + c = 0

It’s centre c = (– g, – f) and radius

∗ The equation of the circle passing through origin is x² + y² + 2gx + 2fy = 0.

∗ The equation of the circle whose centre on the x-axis is x² + y² + 2gx + c = 0.

∗ The equation of the circle having centre on y-axis is x² + y² + 2fy + c = 0.

∗ The circles which have the same centre are called concentric circles.

∗ The equation of the circle concentric with the circle x² + y² + 2gx + 2fy + c = 0 is

x² + y² + 2gx + 2fy + k = 0.

∗ The length of the intercept made by a circle x² + y² + 2gx + 2fy + c = 0 on

x -axis is if g² – c > 0
y -axis is if f² – c > 0

Note: –

(a) if g² – c = 0, then A₁ A₂ = 0 ⇒ the circle touches the x- axis at only one point.

(b) if f² – c = 0, then B₁ B₂ = 0 ⇒ the circle touches the y- axis at only one point.

(d) if f² – c < 0, then the circle does not meet the y- axis.

∗ The equation of the circle having the line segment joining A (x₁, y₁) and B (x₂, y₂) as a diameter is

(x – x₁) (x – x₂) + (y – y₁) (y – y₂) = 0.

∗ Let A, B be any two points on a circle then,

The line is called the secant line of the circle.
The line segment is called the chard of the circle.
AB is called the length of the chord.

∗ A chord passing through the centre is called the diameter of the circle.

∗ The angle subtended by a chord on the circumference of at any point is equal.

The perpendicular bisector of a chord of a circle is asses through the centre of the circle.

∗ The angle in a semicircle is 90⁰.

∗ The equation of the circle passing through three non-collinear points A (x₁, y₁), B (x₂, y₂), C (x₃, y₃) is

Where c_i = − (x² + y²) and i = 1,2,3

∗ centre of the circle is

Parametric form:

If P (x, y) is a point on the circle with centre (h, k) and radius r, then

X = h + r cosθ, y = k + r sinθ 0 ≤ θ ≤ 2π.

⇒ A point n the circle x² + y² = r² is taken as (r cosθ, r sinθ) and simply denoted by θ.

Note:

If the centre of the circle is the origin, then the parametric equations are x = r cosθ, y = r, 0 ≤ θ ≤ 2π.
The point (h + rcosθ₁, k + r sin θ₁) is referred to as the point θ₁ on the circle having the centre (h, k) and radius r.

Notations:

S = x² + y² + 2gx + 2fy + c

S₁ = xx₁ + yy₁ + g(x +x₁) + f (y +y₁) + c

S₁₁= x₁² + y₁² + 2gx₁ +2fy₁ + c

S₁₂ = x₁x₂+ y₁y₂ + g(x_{1 +}x₂ ) + f (y_{1 +}y₂) + c

Position of a point with respect to the circle:

A circle divides the plane into three parts.
1. The interior of the circle

2. The circumference which is the circular curve.

3. The exterior of the circle.

Power of point:

Les S = 0 be a circle with radius ‘r’ and centre ‘C’ and P (x₁, y₁) be a point on the circle, then CP – r² is called the power of point ‘P’ concerning S = 0.

The power of point P (x₁, y₁) w.r.t. S = 0 is S₁₁.

•Let S = 0 be a circle in a plane and P (x₁, y₁) be any point in the same plane. then

P lies in the interior of the circle ⇔ S₁₁ < 0.
P lies on the circle ⇔ S₁₁ = 0.
Plies in the exterior of the circle ⇔ S₁₁ > 0.

Secant and tangent of a circle:

Let P be any point on the circle and Q be neighbourhood point of P lying on the circle. join P and Q, then the line PQ is the secant line.

The limiting position of the secant line PQ when Q is approached to the point P along the circle is called a tangent to the circle at P.

Length of the tangent:

If P is any point on the circle S = 0 and T is any exterior point of the circle, then PT is called the length of the tangent.

∗ If S = 0 is a circle and P (x₁, y₁) is an exterior point with respect o S = 0, then the length of the tangent from P (x₁, y₁) to S =0 is

Condition for a line to be a tangent:

A straight-line y = mx + c (i) meet the circle x² + y² = r²in two distinct points if
Touch the circle x² + y² = r²if
Does not touch the circle x² + y² = r²in two distinct points if

Note:

For all real values of m, the straight line is a tangent to the circle x² + y² = r² and the slope of the line is m.
A straight-line y = mx + c is a tangent to the circle x² + y² = r² if c = .
The equation of a tangent to the circle S ≡ x² + y² + 2gx + 2fy + c = 0 having the slope m is where r is the radius of the circle.
The circle S ≡ x² + y² + 2gx + 2fy + c = 0 touches (i) x – axis if g² = c (ii) y – axis if f²= c.

Chord joining two points on a circle:

If P (x₁, y₁) and Q (x₂, y₂) are two points on the circle S = 0 then the equation of secant line PQ is S₁ + S₂ = S₁₂.

Equation of tangent at a point on the circle:

The equation of the tangent at the point (x₁, y₁) to the circle S ≡ x² + y² + 2gx + 2fy + c = 0 is S₁ = 0.

The equation of the tangent at the point (x₁, y₁) to the circle x² + y² = r² is xx₁ + yy₁– r² = 0.

Point of contact:

If a straight-line lx + my + n = 0 touches the circle S ≡ x² + y² + 2gx + 2fy + c = 0 at P (x₁, y₁), then this line is the tangent to the circle S = 0. And the equation of the tangent is

(x₁ + g) x + (y₁ +f) y + (gx₁ + fy₁ + c) = 0.

∗ The equation of the chord joining two points θ₁, θ₂ on the circle x² + y² + 2gx + 2fy + c = 0 is

∗ The equation of the chord joining the points θ₁, θ₂ on the circle x² + y² = r² is

∗ The equation of the tangent at P(θ) on the circle x² + y² + 2gx + 2fy + c = 0 is

∗ The equation of the tangent at P(θ) on the circle x² + y² = r² is

x cosθ + y sinθ = r.

Normal:

The normal at any point P of the circle is the line which is passing through P and is perpendicular to the tangent at P.

The equation of the normal at P (x₁, y₁) of the circle S ≡ x² + y² + 2gx + 2fy + c = 0 is

(x – x₁) (y₁ + g) – (y – y₁) (x₁ + g) = 0.

The equation of the normal at P (x₁, y₁) of the circle x² + y² = r² is xy₁ – yx₁ = 0.

Chord of contact and Polar:

∗ If P (x₁, y₁) is an external point of the circle S ≡ x² + y² + 2gx + 2fy + c = 0, then there exists two tangents from P to the circle S = 0.

Chord of contact: –

If the tangents are drawn through P (x₁, y₁)

to a circle S = 0 touch the circle at points A and B then the secant line AB is called the chord of contact of P with respect to S = 0

∗ If P (x₁, y₁) is an external point of the circle S ≡ x² + y² + 2gx + 2fy + c = 0, then the equation of the chord of contact of P with respect to S =0 is S₁ = 0.

Note:

If the point P (x₁, y₁) is on the circle S = 0, then the tangent itself can be defined as the chord of contact.
If the point P (x₁, y₁) s an interior point of the circle S = 0, then the chord of contact does not exist.

Pole and Polar: –

Let S = 0 be a circle and P be any point if any line is drawn through the point Pin the plane other than the centre of S = 0. then the points of intersection meet the circle in two points A and B, of tangents drawn at A and B lie on a line called polar of P and P, is called Pole of polar.

∗ The equation of the polar of P (x₁, y₁) with respect to the circle S = 0 is S₁ = 0.

Note: –

If Plies outside the circle S = 0, then the polar of P meets the circle in two points and the polar becomes the chord f contact of P.
If P lies on the circle S = 0, then the polar P becomes the tangent at P o the circle.
If P lies inside the circle S = 0, then the polar of P does not meet the circle.
If P is the centre of the circle S = 0, then the polar of P does not exist.
The pole of the line lx + my + n = 0 with respect to the circle x² + y² = r² is
The pole of the line lx + my + n = 0 with respect to the circle x² + y² + 2gx + 2fy + c = 0 is
The polar of P (x₁, y₁) with respect to the circle S = 0 passes through Q (x₂, y₂) ⟺ polar of Q passes through P.

Conjugate points: Two P and Q are said to be conjugate points with respect to the circle S = 0, if the polar of P with respect to S = 0 passes through Q.

⇒ The condition for the points P (x₁, y₁), Q (x₂, y₂) to be conjugate with respect to the circle S = 0 is S₁₂= 0.

Conjugate lines: Two lines L₁ = 0 and L₂ = 0 are said to conjugate lines with respect to the circle S = 0 if the pole of L₁ = 0 is lies on L₂ = 0.

⇒ The condition for the lines l₁x + m₁y + n₁ = 0 and l₂x + m₂y + n₂ = 0 to be conjugate with respect to the circle S ≡ x² + y² + 2gx + 2fy + c = 0 is

r² (l₁l₂ + m₁m₂) = (l₁g + m₁f – n₁) (l₂g + m₂f – n₂)

⟹ The condition for the lines l₁x + m₁y + n₁ = 0 and l₂x + m₂y + n₂ = 0 to be conjugate with respect to the circle S ≡ x² + y² = r² is r² (l₁l₂ + m₁m₂) = n₁n₂

Inverse points: Let S = 0 be a circle with centre C and radius r. two points P and Q are said to be inverse points with respect to the circle S = 0 if

C, P, Q are collinear.
P, Q lies on the same side of C.
CQ = r².

⟹ If lies inside of the circle S = 0, then Q lies outside of the circle.

⟹ If P lies on the circle S = 0, then P =Q.

⟹ Let S = 0 be a circle with centre C and radius r. The polar of P meets the line CP in Q iff P, Q is inverse points.

⟹ f P, Q are inverse points with respect to S = 0, then P, Q are conjugate points with respect to the circle S = 0.

⟹ If P, Q are inverse points with respect to S = 0, then Q is the foot of the perpendicular from P on the polar of P with respect to the circle S = 0.

Equation of the chord with the given middle point:

The equation of the chord of the circle S = 0 having P (x₁, y₁) as its midpoint is S₁ = S₁₁.

Common tangents to the circle:

⟹ A straight line L = 0 is said to be a common tangent to the circle S = 0 and S^’= 0 if it is a tangent to both S = 0 and S’ = 0.

⟹ Two circles are said to touch each other if they have only one common tangent.

The relative position of two circles:

Let C₁, C₂ centres and r₁, r₂ be the radii of two circles S = 0 and S’ = 0respectively.

1.If C₁C₂ > r₁+ r₂, then two circles do not intersect.

⟹2 direct common tangents and

2 transverse common tangents

Total 4 common tangents

⟹P divides C₁C₂ in the ratio r₁: r₂internally

Here P is called the internal centre of similitude (I.C.S)

⟹ Q divides C₁C₂ in the ratio r₁: r₂externally

Here Q is called the external centre of similitude (E.C.S)

2.If C₁C₂ = r₁+ r₂, two circles touch each other

⟹ Q divides C₁C₂ in the ratio r₁: r₂externally

⟹ two direct common tangents and one common tangent. Total 3 tangents

Here Q is called the external centre of similitude (E.C.S)

⟹ two direct common tangents

Here Q is called the external centre of similitude (E.C.S)
⟹ Q divides C₁C₂ in the ratio r₁: r₂externally

⟹ internal centre of similitude does not exist.

⟹ only one common tangent

⟹ internal centre of similitude does not exist.

⟹ no. of common tangents zero.

Note: the combined equation of the pair of tangents drawn from an external point P (x₁, y₁) to the circle S = 0 is S S₁₁ = S₁².

2.SYSTEM OF CIRCLES

A set of circles is said to be a system of circles if it contains at least two circles.

The angle between two intersecting circles:

⟹ If two circles S = 0 and S’ = 0 intersect at P then the angle between the tangents of two circles at P is called angle between the circles at P.

⟹ If two circles S = 0 and S’ = 0 intersect at P and Q then the angle between the tangents of two circles at P and Q are equal.

⟹ If d is the distance between the centres of the two intersecting circles with radii r₁, r₂ and θ is the angle between the circles then.

⟹ If θ is the angle between the circles x² + y² + 2gx + 2fy + c = 0 and x² + y² + 2g’x + 2f’y + c’ = 0 then

⟹ Two intersecting lines are said to be Orthogonal if the angle between the circles is a right angle.

Condition for the orthogonality:

⟹ The condition that the two circles x² + y² + 2gx + 2fy + c = 0 and x² + y² + 2g’x + 2f’y + c’ = 0 cut each other orthogonally is 2gg’ + 2ff’ = c + c’.

⟹ If d is the distance between the centres of the two intersecting circles with radii r₁, r₂. Two circles cut orthogonally if d² = r₁² + r₂².

∎ If S = 0, S’ = 0 are two circles intersecting at two distinct points, then S – S’ = 0 represents a common chord of these two circles.

∎ If S = 0, S’ = 0 are two circles touch each other, then S – S’ = 0 represents a common tangent of these two circles.

∎ If S ≡ x² + y² + 2gx + 2fy + c = 0 and L ≡ lx + my + n = 0 are the equation of the circle and a line respectively intersecting each other, then S + λ L = 0 represent a circle passing through the point intersection of S = 0 and L = 0 ∀ λ ∈ ℛ.

Radical axes:

The radical axis of two circles s defined as the locus of the point which moves so that its powers with respect to the two circles are equal.

(OR)
The locus of a point, for which the powers with respect to given non-concentric circles are equal, is a straight line is called Radical axis of the given circles.

∎ The equation of Radical axis f the circles S = 0 and S’ = 0 is S – S’ = 0.

→ The radical axis of any two circles is perpendicular to the line joining their centres.

→ The lengths of tangents from a point on the radical axis of two circles are equal if exist.

→ Radical axis of two circles bisects all common tangents of the two circles.

∎ If the centres of any three circles are non-collinear then the radical axis of each pair of circles chosen from these three circles re concurrent.

Radical centre: The point of concurrence of radical axes of each pair of three circles is called radical centre (see above figure).

∎ If the circle S = 0 cuts the each of the two circle S’ = 0 and S’’ =0 orthogonally then the centre of S =0 lies on the radical axis of S’ = 0 and S’’ = 0.

∎ Radical axis of two circles is

The ’common chord’ if the two circles intersect at two distinct points.
The ‘common tangent’ at the point of contact if the two circles touch each other.

→ The radical axis of any two circles bisects the line joining the points of contact of common tangents to the circles.

→ Let S = 0, S’ = 0 and S’’ =0 be three circles whose centres are non- collinear and no two circles of these are intersecting then the circles having

Radical centre of these circles as the centre of the circle.
Length of the tangent from the radical centre to any one of these three circles cuts the given three circles orthogonally.

CONIC SECTIONS

Conic: The locus of a point moving on a plane such that its distance from a fixed point and a fixed straight line is in the constant ratio is called Conic.

The locus of a point moving on a plane such that its distance from a fixed point and a fixed line on the plane are in a constant ratio ‘e’, is called a Conic.

Focus: The fixed point is called focus and it is denoted by S.

Directrix: The fixed straight line is called the directrix.

Eccentricity: The constant ratio is called eccentricity and it is denoted by ‘e’.

⟹ Conic is the locus of a point P moving on a plane such that SP/PM = e, PM is the perpendicular distance from P to directrix at M.

⟹ If e = 1, then the conic is parabola.

⟹ if 0 < e < 1, then the conic is Ellipse.

⟹ if e > 1, then the conic is Hyperbola.

3.PARABOLA

If e = 1, then the conic is parabola.

The standard form of parabola is y² = 4ax.
Focus S = (a. 0).
Equation of directrix is x + a = 0.
Vertex A = (0, 0) and A is the mid-point of SZ.

• Equation of the parabola with focus (α, β) and directrix lx + my + n = 0 is

If the focus is situated on the left side of the directrix, the equation of the parabola with vertex as the origin and the axis is X-axis is y² = – 4ax.
The vertex being the origin, if the axis of the parabola is taken as Y – axis, equation of the parabola is x² = 4 ay or x² = – 4 ay according to the focus is above or below the X-axis.

Nature of the curve:

The nature of the parabola f the equation y² = 4 ax (a>0)

F y = 0, then 4 ax = 0 and x = 0

∴ the curve passes through the origin.

If x = 0, then y² = 0. Which gives y = 0. Y – axis is the tangent to the parabola at origin.
Let P(x, y) be any point on the parabola (a>0) and y² = 4 ax, we have x ≥ 0 and

∴ for any positive real value of x, we obtain two value of y of equal magnitude but opposite in sign. This shows that the curve is symmetric about X-axis and lies in the first and fourth quadrants.

The curve does not exist on the left side of the Y-axis since x ≥ 0 for any point (x, y) on the parabola.

Chord: The line segment joining two points on a parabola is called a chord.

Focal chord: A chord which is passing through focus is called Focal Chord.

Double ordinate: A chord through a point P on the parabola, which is perpendicular to the axis of the parabola is called Double ordinate.

Latus rectum: The double ordinate passing through the focus is called Latus rectum.

⟹ Length of Latus rectum = 4a.

Various forms of the parabola

1. y²= 4ax

focus: (a, 0)

equation of directrix: x + a = 0

axis of parabola: y = 0

vertex: (0. 0)

2. y²= −4ax

equation of directrix: x − a = 0

focus: (−a, 0)

axis of parabola: y = 0

vertex: (0. 0)

3. x²= 4ay

focus: (0, a)

equation of directrix: y + a = 0

axis of parabola: x = 0

vertex: (0. 0)

4. x²= −4ax

focus: (0, −a)

equation of directrix: y − a = 0

vertex: (0. 0) axis of parabola: x = 0

5. (y – k) ²= 4a (x – h)

focus: (h + a, k)

equation of directrix: x – h + a = 0

axis of parabola: y – k = 0

vertex: (h. k)

6. (y – k) ²= −4a (x – h)

focus: (h – a, k)
equation of directrix: x – h – a = 0

axis of parabola: y – k = 0

vertex: (h. k)

7. (x – h) ²= 4a (y – k)

focus: (h, k + a)

equation of directrix: y – k + a = 0

axis of parabola: x – h = 0

vertex: (h. k)

8. (x – h)²= −4a (y – k)

focus: (h, k – a)

equation of directrix: y – k – a = 0

vertex: (h. k)
axis of parabola: x – h =0

focus: (α, β)

equation of directrix: lx + my + n = 0

axis of parabola: m (x – α) – l (y – β) = 0

vertex: A

Note:

Equation of the parabola whose axis parallel to X – axis is x = ly² + my + n.
Equation of the parabola whose axis parallel to Y – axis is y = lx² + mx + n.

Focal distance: The distance of a point on the parabola from its focus is called Focal distance.

⟹ Focal distance of parabola s x₁ + a

Parametric equations of a parabola:

The point (at², 2at) satisfy the equation y² = 4ax, the parametric equations of parabola are x = at², y = 2at. The point P(at², 2at) is generally denoted by the point ‘t’ or P(t).

Notation:

S ≡ y² – 4 ax
S₁ ≡yy₁ – 2a (x + x₁)
S₁₂ ≡ y₁y₂ – 2a (x₁ + x₂)
S₁₁ ≡ y₁² – 4 ax₁

Equation of a tangent and normal at a point on the parabola:

∎ y = mx + c is a tangent to the parabola y² = 4ax, then c = a/m or a =cm, and the point of contact is (a/m², 2a/m).

∎ if m = 0, the line y = c is parallel to the axis of the parabola (i.e., x – axis)

y = c ⟹ c² = 4ax ⟹ x = c² /4a

∴ point of contact is (c²/4a, c).

∎ if m ≠ 0 and c = 0, then

Y = mx ⟹ x = 4a/m² and y = 4a/m

∴ point of contact is (4a/m², 4a/m).

∎ The equation of the chord joining the points (x₁, y₁) and (x₂, y₂) is S₁ + S₂ = S₁₂.

∎ The equation of the tangent at P (x₁, y₁) to the parabola S = 0 is S₁= 0.

∎ The equation of the normal at P (x₁, y₁) is (y – y₁) = – y₁/2a (x – x₁).

Parametric form:

∎ The equation of the tangent at a point ‘t’ on the parabola y² = 4ax is x – yt + at² = 0.

∎ Equation of the normal at a point ‘t’ on the parabola y² = 4ax is y + xt = 2at + at³.

∎ The condition for the straight-line lx + my + n = 0 to be a tangent to the parabola y² = 4 ax is

am² = nl and point of contact is (n/l, –2 am/l).

∎ common tangent to the parabolas y² = 4 ax and x² = 4 by is x a^1/3 + y b^1/3 + a^2/3 b^2/3 = 0.

∎ The equation of the chord of contact of the external point (x₁, y₁) w.r. t parabola S = 0 is S₁ = 0.

∎ The equation of the polar of the point (x₁, y₁) w.r. t parabola S = 0 is S₁ = 0.

∎ The pole of the line lx + my + n = 0 w.r.t. parabola y² = 4ax is (n/l, -2am/l).

∎ If two points P (x₁, y₁), Q (x₂, y₂) are conjugate points w.r.t. parabola S = 0, then S₁₂ = 0.

∎ The lines l₁x + m₁y + n₁ = 0 and l₂x + m₂y + n₂ = 0 are conjugate lines with respect to the parabola y² = 4 ax, then l₁n₂ + l₂n₁ = 2a m₁m₂.

4.ELLIPSE

Ellipse: A conic with eccentricity less than unity s called Ellipse.

∎ Equation of Ellipse in standard form is

⇒b² = a² (1 – e²) ⇒ e² – 1 = -b²/a²

Major and Minor axis:

⟹ The line segment AA’ and BB’ of length 2a and 2b respectively are axes of the Ellipse.

⟹ If a > b AA’ is called Major axis and BB’ is called Minor axis and vice-versa if a<b.

Various form of Ellipse:

Major axis: along the x-axis

Length of Major axis:2a

Minor axis: along y – axis

Length of Minor Axis:2b

Centre: (0, 0)

Foci: S = (ae, 0) and S’ = (–ae, 0)

Equation of directrices: x = a/e and x = –a/e

Eccentricty:

Major axis: along y – axis

Length of Major axis:2b

Minor axis: along x – axis

Length of Minor Axis:2a

Centre: (0, 0)

Foci: S = (0, be) and S’ = (0, –be)

Equation of directrices: x = b/e and x = –b/e

Eccentricty:

Centre not at the origin

Major axis: along with y = k

Length of Major axis:2a

Minor axis: along x = h

Length of Minor Axis:2b

Centre: (h, k)

Foci: S = (h +ae, k) and S’ = (h – ae, k)

Equation of directrices: x = h + a/e and x = h – a/e

Eccentricty:

Major axis: along x = h

Length of Major axis:2b

Minor axis: along with y = k

Length of Minor Axis:2a

Centre: (h, k)

Foci: S = (h, k + be) and S’ = (h, k – be)

Equation of directrices: xy = k + b/e and y = k – b/e

Eccentricty:

Chord: The line segment joining two points on a parabola is called a chord of Ellipse.

Focal chord: A chord which is passing through one of the foci is called Focal Chord.

Latus rectum: A focal chord perpendicular to the major axis of the Ellipse is called Latus Rectum. Ellipse has two latera recta.

Length of the Latus rectum:

1.The coordinates of the four ends of the latera recta of the ellipse

L = (ae, b²/a), L’ = (ae, -b²/a) and L₁ = (-ae, b²/a), L₁’= (-ae, -b²/a).

length of the Latus rectum = 2b²/a.

2.length of the Latus rectum of an ellipse is 2a²/b and the coordinates of the four ends of the latera recta are

L = (a²/b, be), L’ = (-a²/b, be) and L₁ = (a²/b, -be), L₁’ = (-a²/b, -be).

3. The equation of the Latus rectum of the Ellipse through S is x = ae and through S’ is x = -ae.

4. The equation of the Latus rectum of the Ellipse through S is y = be and through S’ is y = -be.

5. If P (x, y) is any point on the Ellipse whose foci are S and S’, then SP +S’P is constant.

Auxiliary circle: The circle described on the major axis of an Ellipse as the diameter is called the Auxiliary circle of the Ellipse. The Auxiliary circle of the Ellipse is x² + y² = a².

Parametric equations: The parametric equations of the Ellipse are x = a cosθ and y = b sinθ.

Notation:

Equation of Tangent and Normal

∎ The equation of any tangent to the Ellipse can be written as

∎ The condition for a straight-line y = mx + c to be a tangent to the Ellipse is c² = am² + b².

∎ The point of contact of two parallel tangents to the Ellipse are (-a²m/c, b²/c) and (a²m/c, -b²/c)

∎ The equation of the chord joining two points (x₁, y₁) and (x₂, y₂) on the Ellipse S = 0 is S₁ + S₂ = S₁₂.

∎ The equation of the Normal at P (x₁, y₁) to the Ellipse is

∎ Equation of the tangent at P(θ) on the Ellipse

∎ Equation of the normal at P(θ) on the Ellipse S = 0 is

∎ When θ = 0, π; equation of Normal is y =0.

∎ When θ = π/2, 3π/2; equation of Normal is x =0.

∎ The condition for the line lx + my + n = 0 to be a tangent the Ellipse S = 0 is a²l² + b²m² = n².

∎ The condition for the line x cosα + y sinα = p to be a tangent the Ellipse S = 0 is a² cos² α + b² sin² α = p².

∎ The pole of the line lx + my + n = 0 with respect to the Ellipse S = 0 is (-a²l/n, -b²m/n).

∎ The condition for the two lines l₁x + m₁y + n₁ = 0 and l₂x + m₂y + n₂ = 0 to be conjugate with respect to the Ellipse S = 0 is a²l₁l₂ + b²m₁m₂ = n₁n₂. a²l₁l₂.

5.HYPERBOLA

Hyperbola: Hyperbola is a conic in which the eccentricity is greater than the unity.
Standard form of Hyperbola:

Equation of Hyperbola in standard form is

Centre: C (0, 0)

Foci: (± ae, 0)

Directrix: x = ± a/e.

Ecdntricity:

Notation:

Rectangular Hyperbola:

If in a Hyperbola the length of the transverse axis (2a) is equal to the length of the conjugate axis(2b), then the hyperbola is called rectangular hyperbola.

Its equation is x² – y² = a² and eccentricity is .

Auxiliary circle: The circle described on the transverse axis of hyperbola as diameter is called the auxiliary circle of the hyperbola.

The equation of the auxiliary circle of S = 0 is x² + y² = a².

Parametric equations: The parametric equations of the Parabola S = 0 are x = a secθ and y = b tanθ.

Conjugate Hyperbola:

The hyperbola whose transverse and conjugate axis are respectively the conjugate and transverse axis of a given hyperbola is called a conjugate hyperbola.

The equation of hyperbola conjugate to S ≡ is S’ ≡

∎ For

The transverse axis lies on along X-axis and its length is 2a.

The conjugate axis lies on along Y-axis and its length is 2b.
∎ For

The transverse axis lies on along Y-axis and its length is 2b.

The conjugate axis lies on along X-axis and its length is 2a.

Various form of Hyperbola:

Let and

1.Hyperbola

The transverse axis along X-axis: y =0

Length of the transverse axis:2a

The conjugate axis along Y-axis: x = 0

Length of the conjugate axis: 2b

Centre: (0, 0)

Foci: (± ae, 0)

Equation of the directrices: x = ± a/e

Eccentricity:

2.Conjugate Hyperbola

The transverse axis along Y-axis: x = 0

Length of the transverse axis:2b

The conjugate axis along X-axis: y = 0

Length of the conjugate axis: 2a

Centre: (0, 0)

Foci: (0, ± be)

Equation of the directrices: y = ± b/e

Eccentricity:

Centre not at the origin:

3.Hyperbola

The transverse axis along X-axis: y = k

Length of the transverse axis:2a

The conjugate axis along Y-axis: x = h

Length of the conjugate axis:2b

Centre: (h, k)

Foci: (h± ae, k)

Equation of the directrices: x = h± a/e

Eccentricity:

4.Hyperbola

The transverse axis along Y-axis: x = h

Length of the transverse axis:2b

The conjugate axis along X-axis: y = k

Length of the conjugate axis: 2a

Centre: (h, k)

Foci: (h, k ± be)

Equation of the directrices: y = k± b/e

Eccentricity:

Equation of tangent and normal at a point on the Hyperbola:

∎ The equation of the tangent at P (x₁, y₁) to the hyperbola S = 0 is S₁ = 0.

∎ The equation of the tangent at P(θ) on the hyperbola S = 0 is

∎ The equation of the Normal at P (x₁, y₁) to the hyperbola S = 0 is

∎ Equation of the normal at P(θ) on the Hyperbola S = 0 is

∎ The condition for a straight-line y = mx + c to be a tangent to the hyperbola S = 0 is c² = am² − b².

Asymptotes of a hyperbola:

The equations of asymptotes of hyperbola S = 0 are and the joint equation of asymptotes is

6. INTEGRATION

∎ Let E be a subset of R such that E contains a right or left the neighbourhood of each of its points and let f: E → R be a function. If there is a function F on E such that F’(x) = f(x) ∀ x ∈ E, then we call F an anti-derivative of f or a primitive of f.

Indefinite integral: Let f: I→R. Suppose that f has an antiderivative F on I. Then we say that f has an integral on I and for any real constant c, we call F + c an indefinite integral of over I, denote it by ∫f(x) dx and read it as ‘integral’ f(x) dx.

∫f = ∫f(x) dx = F(x) + c. here c is called constant of integration.

In the indefinite integral ∫f(x)dx, f is called ‘integrand’ and x is called the variable of integration.

⟹

⟹ if f: I⟶R is differentiable on I, then ∫f’(x)dx = f(x) + c.

Standard forms:

Properties of integrals:

∎ ∫ (f ±g) (x) dx = ∫f(x) dx ± ∫g(x) dx + c

∎ ∫(af) (x) dx = a ∫f(x) dx + c

∎ ∫ (f₁ + f₂ + … + f_n) (x) dx = ∫f₁(x) dx + ∫f₂(x) dx + … +∫f_n(x) dx +c

∎ ∫f(g(x)) g’(x) dx = F(g(x)) + c

∎ ∫f(ax + b) dx = 1/a F(ax +b) + c

Some important formulae:

∫e^ax dx = 1/a e^ax + c
∫sin (ax + b) dx = -1/a cos (ax + b) + c
∫cos (ax + b) dx = 1/a sin (ax + b) + c
∫sec² (ax + b) dx = 1/a tan (ax + b) + c
∫ cosec² (ax + b) dx = 1/a cot (ax + b) + c
∫cosec (ax + b) cot (ax + b) dx = -1/a cosec (ax + b) + c
∫sec (ax + b) tan (ax + b) dx = 1/a sec (ax + b) + c

Integration by parts:

Let u, v real valued differentiable functions in I. Suppose that u,v has an integral on I, then uv’ has an integral on I and

∫(uv’) (x) dx = (uv) – ∫(u’v) (x) dx + c or ∫(uv) dx = u ∫v dx – ∫ [u’ ∫v dx] dx + c

Integration of exponential functions:

∫e^x dx = e^x + c; ∫x e^x dx = (x – 1) e^x + c

∫ e^x [f(x) +f’(x)] dx = e^x f(x) + c

Integration of logarithmic functions:

∫log x dx = x log x – x + c

Integration of inverse trigonometric functions:

Evaluation of integrals form :

Working rule: reduce ax² + bx + c to the form of a[(x + α)² + β] and then integrate using the substitution t = x + α.