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

## 1. INTEGERS

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

1, 2, 3… Etc.

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

0, 1, 2, 3 …

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

….,-3, -2, -1, 0, 1, 2, 3,…..

Integers on a number line:

Operations on integers:

3 + 4 = 7

-2 + 4 = 2

Subtraction of integers on a number line:-

6 – 3 = 3

Multiplication of integers on a number line:-

2 × 3 ( 2 times of 3) = 6

3 × (- 4 ) ( 3 times of -4) = -12

Multiplication of two negative integers:

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

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

Multiplication of more than two negative integers:

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

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

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

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

Note:-

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

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

Division of integers:

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

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

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

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

Properties of integers:

1.Closure property:-

2.commutative property:-

3.associative property:-

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

•For any integer ‘a’, a + 0 = 0 + a

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

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

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

Multiplicative identity:-

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

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

•1 is the multiplicative identity.

multiplicative inverse:-

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

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

distributive property:-

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

3 × (2 + 4) = 18

(3 × 2) + (3 × 4) = 6 + 12 = 18

∴ 3 × (2 + 4) = (3 × 2) + (3 × 4).

## 2. FRACTIONS, DECIMALS AND RATIONAL NUMBERS

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

The proper and improper fractions:

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

Ex: – 1/5, 2/3, and so on.

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

Ex: – 5/2,11/5 and so on.

Comparing fractions:

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

Ex: ,    and so on

Unlike fractions: –

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

Ex: –     and so on.

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

∗ Like Fractions:

∗ Unlike fractions:

Subtraction of fractions:

∗ Like fractions:

Ex:

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

Ex:

Multiplication of fractions:

Multiplication of fraction by a whole number: –

Multiplication of numbers means adding repeatedly.

Ex: –

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

2.Multiplication of fraction with a fraction: –

multiplication of two fractions =

Division of fractions:

Ex: – 2 ÷

⇒ 6 one-thirds in two wholes

Reciprocal of fraction: reciprocal of a fraction is   .

Note:

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

Ex:

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

2.Division of a fraction by another fraction: –

Decimal number or fractional decimal:

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

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

Note: –

• while adding or subtracting decimal numbers, the digits in the same places must be added or subtracted.
• While writing the numbers one below the other, the decimal points must become one below the other. Decimal places 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 on a variable. It says that two expressions are equal.

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

Ex: – x + 2 = 5; 2 = x + 3

Balanced equation:

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

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

Ex: 8 + 3 = 11

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

RHS = 11 + 2 = 13

∴ LHS = RHS

8 +3 = 11 if subtract 2 on both sides

LHS = 8 + 3 – 2 = 9

RHS = 11 – 2 = 9

∴ LHS = RHS

Using algebraic equations in solving day to day problems:

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

## 4. LINES AND ANGLES

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

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

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

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

⇒ ∠AOC and ∠BOC adjacent angles.

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

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

Corresponding angles: –

Two angles which are lies 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 corresponding angles.

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

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

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

Transversal on parallel lines:

If pair of parallel lines are intersected by a transversal then the angles of each pair of corresponding angles are equal

⇒ ∠1, =∠5; ∠2= ∠6; ∠3= ∠7 and ∠4= ∠8

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

∠3= ∠5; ∠4= ∠6

•If pair of parallel lines are intersected by a transversal then the angles of each pair of exterior alternate angles are equal.

∠1= ∠7; ∠2= ∠8

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

∠3+∠6= 1800; ∠4+ ∠5 = 1800

Note:

1. If a transversal intersects two lines and the pair of corresponding angles are equal, then the lines are parallel.
2. If a transversal intersects two lines and the pair of alternate angles are equal, then the lines are parallel.
3. If a transversal intersects two lines and the interior angles on the same side of the transversal are supplementary, then the lines are parallel.

## 5. TRIANGLE AND ITS PROPERTIES

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

In ∆ABC,

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

Classification of triangles:

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

According to sides:

Based on sides triangles are three types:

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

According to angles:

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

Relationship between the sides of a triangle:

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

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

The altitude of a triangle:

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

Median of a triangle:

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

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

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

Angle-sum property of a triangle:
Some of the angles in a triangle is 1800

∠A + ∠B + ∠C = 1800

An exterior angle of a triangle:

When one side of the triangle is produced, the angle thus formed is called an exterior triangle.

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

x0+ y0 = z0

## 6.RATIO – APPLICATIONS

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

The ratio is represented by the symbol ‘:’

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

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

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

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

The product of means = the product of extremes

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

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

Percentages:

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

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

Profit and Loss:

Selling price = SP; Cost price = CP

If SP > CP, then we get profit

Profit = SP – CP

Profit percentage =

SP = CP + profit

If SP < CP, then we get a loss

Loss = CP – SP

Loss percentage =

SP = CP – Loss

Simple interest:

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

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

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

Amount = principle + interest

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

## 7.DATA HANDLING

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

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

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

Arithmetic Mean (A.M) =

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

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

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

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

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

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

Arrange given data in ascending or descending order.

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

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

Bar graph:

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

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

Ex:

Double bar graph:

It represents two observations side by side.

Ex:

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

The angle of each sector =

Ex:

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

## 8.CONGRUENCY OF TRIANGLES

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

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

Congruency of Triangles:

Two triangles are said to be congruent if (i) their corresponding angles are equal (ii) their corresponding sides are equal.
Ex: In ∆ ABC, ∆ DEF

∠A ≅ ∠D; ∠B≅ ∠E; ∠C ≅ ∠F

AB ≅ DE; BC≅ EF; AC ≅ DF

∴∆ABC ≅ ∆DEF

The criterion for congruency of Triangles:

1.Side -Side -Side congruency (SSS): –
If three side of a triangle is equal to the corresponding three sides of another triangle, then the triangles are congruent.

∴∆ABC ≅ ∆DEF

2.Side -Angle -Side congruency (SAS): –
If two sides and the angle included between the two sides of a triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

∴ ∆ABC ≅ ∆DEF

3.Angle – Side -Angle congruency (ASA): –
If two angles and included side of a triangle are equal to the corresponding two angles and included side of another triangle, then the triangles are congruent.

∴ ∆ABC ≅ ∆DEF

4.Right angle – Hypotenuse – Side congruence (RHS): –

If the hypotenuse and one side of a right-angled triangle are equal to the corresponding hypotenuse and side of the other right-angled triangle, then the triangles are Equal.

∴∆ABC ≅ ∆DEF

## 9.CONSTRUCTION OF TRIANGLES

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

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

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

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

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

Step of constructions:

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

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

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

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

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

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

Step of constructions:

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

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

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

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

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

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

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

Step of constructions:

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

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

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

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

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

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

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

Step of constructions:

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

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

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

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

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

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

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

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

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

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

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

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

## 10.ALGEBRAIC EXPRESSIONS

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

Ex: m, x, a, etc.

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

Ex: 1, 3,  etc.

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

Ex: 3x, –4x, x are like terms

3x, 4y, 4 are unlike terms

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

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

5 is a numerical coefficient

x is the literal coefficient

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

Ex: 2x – 3. 3x, 2 +3 – 4 etc.

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

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

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

Ex: x + y, xy, x – 3, 4x + 2 etc.

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

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

Types of Algebraic expressions:

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

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

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

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

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

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

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

x2 y +yz2 + xy2 – 4xy + 8yz etc.

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

Ex: Degree of 5xy3

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

Sum of exponents = 1 + 3 = 4

∴ degree of 5xy3 is 4

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

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

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

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

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

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

Subtraction of like terms:

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

Ex: 3x − 2x = (3 − 2) x = x

4x2y −2 x2y = (4 −2) x2y = 2x2y

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

The standard form of an Expression:

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

Ex: 5 – 2x2 + 4x +3x3

Standard form is 3x3 – 2x2 + 4x + 5

Finding the value of an expression:

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

Sol: given expression is x3 + y + 3

Substitute x = 1 and y = 2 in above expression

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

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

Column or Vertical method:

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

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

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

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

Sol:

Row or Horizontal method.

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

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

Step –3: Simplify the coefficients.

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

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

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

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

= (1 + 3) x2 + (3 – 2 + 3) x + 6

=4x2 + 4x + 6

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

Subtraction of algebraic expressions:

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

Column or Vertical method:

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

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

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

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

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

Sol:

Row or Horizontal method:

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

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

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

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

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

Sol:   3x2 + 4x – 3 – (x2 + 3x + 5)

= 3x2 + 4x – 3 – x2 – 3x – 5

= (3 – 1) x2 + (4 – 3) x + (– 3 – 5)

= 2x2 + x – 8

## 11.EXPONENTS

We know that,

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

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

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

am is in exponential form

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

Laws of exponents:

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

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

Ex: 1324 in standard form is 1.324 × 103.

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

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

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

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

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

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

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

Interior and exterior of a Quadrilateral:

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

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

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

Angle sum property of a quadrilateral:

The Sum of the angle in a Quadrilateral is 3600

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

1.Trapezium:

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

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

2.Kite:

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

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

3.Parallelogram:

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

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

Properties of parallelogram: –

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

4.Rhombus:

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

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

Properties of Rhombus: –

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

5.Rectangle:

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

Properties of Rectangle: –

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

6.Square:

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

Properties of Square: –

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

## 13.AREA AND PERIMETER

Area of a parallelogram:

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

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

Area of a Triangle:

Area of triangle = ½ b × h square units.

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

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

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

Area of rhombus = ½ d1 × d2 square units.

Circumference of the circle:

Circumference of circle = 2πr = πd

Area of the rectangular path:

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

## 14.UNDERSTANDING 2D AND 3D SHAPES

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

Nets of 3D shapes:

1.Cube:

2.Cylinder:

3.Pyramid:

Oblique Sketches:

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

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

Step-1: Draw the front face

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

Step-3: Join the corresponding corners.

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

Isometric Sketches:

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

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

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

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

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

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

## 15.SYMMETRY

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

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

Regular polygon:

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

Lines of symmetry for Regular polygons:

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

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

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

The angle of rotation of the equilateral triangle is 1200

The angle of rotation of a square is 900

Order of rotational symmetry:

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

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

The order of rotational symmetry for a square is 4.

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

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

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

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

Visit My Youtube Channel: Clock on Below Logo

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

## 1. KNOWING OUR NUMBERS

### Comparing numbers:

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

• Now Compare   5432 and 4678…

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

Order of numbers:

##### • Ascending Order: –

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

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

##### • Descending Order: –

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

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

Formations of numbers

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

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

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

• Greatest two-digit number is 99.

• Greatest three-digit number is 999.

• Greatest four-digit number is 9999.

### Place value

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

Ex:- 3458

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

Expanded form

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

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

= 3×1000 + 4×100 + 5×10 + 8×1

• Note:-

1 hundred = 10 tens

1 thousand = 10 hundreds

1 lakh = 100 thousands = 1000 hundreds

### Reading and Writing the numbers

Place value table for Indian system :

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

Place value table for International system :

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

### Use of commas:

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

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

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

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

Note:-10 millimetres = 1centimeter

100 centimetres = 1 meter

1000 meters = 1 kilometer

1000 milligrams = 1 gram

1000 grams = 1 kilo gram

## 2. WHOLE NUMBERS

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

1, 2, 3… Etc.

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

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

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

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

0, 1, 2, 3 …

Representation of whole number on the number line:

• Draw a line mark a point on it.

• Label it as ‘0’

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

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

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

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

Addition of the whole number can represent on the number line

Ex:-  3 + 2 = 5

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

### Subtraction on the number line:

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

Ex :-5 – 3 = 2

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

Multiplication on the number line:

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

Properties of whole numbers

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

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

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

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

Multiplication:- Whole numbers are closed under multiplication.

Ex:- 3 × 2 = 6, 6 is a whole number.

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

Ex:-  3 ÷ 2 is not a whole number.

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

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

Subtraction:- Whole numbers are not commutative under subtraction.

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

Multiplication:- Whole numbers are commutative under multiplication.

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

Division:- Whole numbers are not commutative under division.

Ex:-  3 ÷ 2  ≠ 2 ÷ 3.

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

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

Subtraction:- Whole numbers are not associative under subtraction.

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

Multiplication:- Whole numbers are associative under multiplication.

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

Division:- Whole numbers are not associative under division.

Ex:-  ( 3 ÷ 2) ÷ 5 ≠3 ÷ (2 ÷ 5).

Distributive property:

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

Note :Division by zero is not defining.

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

Thus, 0 is the additive identity.

2 ×1 = 2, 4 × 1 = 4 and so on.

Thus, 1 is a multiplicative identity.

Patterns:

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

The number 3 as shown as

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

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

• Some numbers can be shown as triangles.

3 can be shown as                         6 can be shown as

## 3. PLAYING WITH NUMBERS

Divisibility Rule:

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

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

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

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

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

∴ 231 is divisible by 3

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

∴ 436 is not divisible by 3.

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

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

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

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

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

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

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

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

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

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

∴ 4232 is divisible by 8.

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

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

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

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

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

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

Ex:- 6545

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

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

Now difference is 10 – 10 = 0

∴ 6545 is divisible by 11.

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

6 = 1×6

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

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

2) Every number is a factor of itself.

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

4) Factors of a given number are countable.

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

2, 3, 5, 7, …. Are prime numbers

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

4, 6,8,9….. are composite numbers.

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

2) 2 is the smallest prime number

3) 4 is the smallest composite number.

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

Ex:- (2, 3), (4,5) ……

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

Ex:- (2,3), (3,5), (17,19)…..

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

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

24 = 1 × 24 = 2 × 12 = 3 × 8 = 4 × 6

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

Ex:- 24 = 2 × 12

2 × 3 × 4

2 × 3 × 2 × 2

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

Methods of prime factorization:

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

fallow the procedure.

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

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

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

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

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

Ex:- 12, 9

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

Factors of 9 are:  1, 3 and 9

∴ Common factors of 12, 9 are 1,3

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

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

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

Factors of 9 = 1, 3, 9

Common factors of 12, 9 = 1,3

Highest common factor is 3

∴ H.C.F of 12, 9 is 3

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

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

9 = 3  × 3

12 =3 ×  2× 2

The common factor of 12, 9 is 3

∴ H.C.F of 12, 9 is 3

Continue division method:

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

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

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

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

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

Ex:- LCM of 3 and 4

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

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

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

∴ LCM of 3, 4 is 12.

Methods of finding LCM:

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

i)  Express each number as the product of prime factors

Prime factors of 6 =   2 × 3

Prime factors of 15 = 5× 3

ii)  Take the common factors both: 3

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

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

∴ LCM of 6 and 15 = (3) × 2 × 5 = 30.

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

i.    Arrange the given numbers in a row.

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

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

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

Ex:-

∴ LCM of 6, 15 = 3 × 2 × 5 = 30.

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

Ex:- 6 × 15 = 30 × 3 = 90.

## 4.BASIC GEOMETRICAL IDEAS

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

Geo means Earth and metron means measurement.

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

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

It is denoted by

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

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

It is denoted by

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

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

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

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

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

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

∗ A closed curve has three parts

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

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

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

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

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

Here two rays OA, OB are arms of the angle

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

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

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

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

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

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

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

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

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

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

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

## 5. MEASURES OF LINES AND ANGLES

Measure of line segment:

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

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

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

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

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

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

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

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

∗ This could be made accurate by using a Divider

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

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

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

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

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

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

The protractor:

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

∗To measure an angle 72° using protractor-

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

Hence, ∠ABC = 72°.

Types of angles:

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

Perpendicular Lines

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

## 6.INTEGERS

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

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

Z = {…, -3, -2, -1, 0, 1, 2, 3…}.

Representation of integers on a number line: –

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

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

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

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

Ex: – 3 + 2 =5, −3 – 2 = −5.

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

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

Addition of integers on a number line:

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

∴ 3 + 4 = 7

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

∴ − 3 − 4 = −7

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

Subtraction of integers on a number line:

Subtract 3 from 6

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

∴ 6 − 3 = 3.

Subtract −3 from 6

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

∴ 6 – (−3) = 9.

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

## 7. FRACTIONS AND DECIMALS

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

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

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

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

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

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

The numerator and the denominator:

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

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

Representing fractions pictorially:

Representing fractions on a number line:

Mark on a number line

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

Ex: –

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

Ex: –

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

Ex: –

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

A mixed fraction is in the form of

We can convert it into improper fraction by

Ex: –

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

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

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

Ex: –

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

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

Comparing fractions:

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

Ex: – and so on.

Unlike fractions: –

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

Ex: – and so on.

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

Ascending order and Descending order: –

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

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

Like fractions: –

Ex:

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

Ex: –

Subtraction of fractions

Like fractions: –

Ex: –

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

Ex: –

Decimal fractions:

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

Ex: –

## 8. DATA HANDLING

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

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

Organization of data: –

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

∗ Example for representing tally marks:

Pictograph:

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

Example for representing data by pictograph

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

Bar graph:

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

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

Construction of bar graph: –

Steps to construction: –

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

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

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

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

70 ÷ 10 = 7, 40 ÷ 10 = 4 and so on.

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

## 9. INTRODUCTION TO ALGEBRA

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

Patterns:

To make a triangle, 3 matchsticks are used

For making 2 triangles we have six matchsticks

For making 3 triangles we have nine matchsticks

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

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

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

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

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

Use of variables:

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

Perimeter square = 4s, s is the variable

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

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

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

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

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

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

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

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

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

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

Ex: 4y = 20

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

Solution of an equation (Root of the equation):

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

Ex: 3x = 15

If x = 5; LHS = 3×5 = 15

RHS = 15

∴ solution of above equation is 5

Trial and error method:

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

Ex: solve 2n = 10

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

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

## 10. PERIMETER & AREA

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

Length of the rectangle = l, breadth = b

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

= l + l +b + b

P = 2 (l + b) units.

Perimeter of a Square:

Length of the side  of a square   = a

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

= a + a + a + a

= 4a units.

Perimeter of an Equilateral Triangle:

Length of each side   = a

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

= a + a + a

= 3a units.

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

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

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

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

Area of a Rectangle:

Length of the rectangle = l, breadth = b

Area of the Rectangle = l × b square units.

Area of a Square:

Length of the side of a square   = a

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

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

## 11. RATIO AND PROPORTION

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

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

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

Simplest form of ratio:

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

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

Division of a given quantity in a given ratio:

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

Total parts = a + b

First part =and second part =

Ex: Divide 1200 in the ratio 2 : 3

Ans: Total parts = 2 + 3 = 5

First part = = 2 × 240 = 480

Second part =  = 3× 240 = 720.

Proportion:

Equalities of ratios is called proportion.

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

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

Unitary method:

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

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

Sol:  Given that cost of 5 pens = ₹20

Cost of one pen = 20 ÷ 5= 4

Cost of 12 pens = 4 × 12 = 48

∴ cost of 12 pens = ₹ 48.

## 12. SYMMETRY

Symmetry:

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

Symmetry is the mirror image of an object.

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

Line of symmetry:

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

Line of symmetry can be horizontal, vertical or diagonal.

Ex:

The English alphabet which have

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

## 13. PRACTICAL GEOMETRY

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

1.A Ruler (Scale)

2.The compasses

3.The divider

4.Protractor

∗ The ruler is used to measure lines.

∗ The compasses is used for constructing.

∗ The protractor is used for measuring angles.

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

Construction of a line segment of a given length:

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

1.By using Ruler: –

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

Steps of construction: –

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

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

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

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

1.By using the Compasses: –

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

Steps of construction: –

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

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

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

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

Construction of a circle:

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

Steps of construction: –

Step-1: Open the compasses for radius 3 cm

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

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

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

Construction of perpendicular bisector a line segment:

Steps of construction: –

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

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

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

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

Steps of construction: –

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

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

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

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

Construction of Angles using Protractor:

Let us suppose we want to construct ∠ABC = 500

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

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

Step-3: Mark a point A at 500

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

Constructing a copy an of Angle of un known measure:

Let ∠A is given, measure is not known

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

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

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

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

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

Construction to bisect a given angle:

Let an angle say ∠AOB be given

Steps of construction: –

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

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

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

Let two arcs intersects at S

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

Observe ∠AOS = ∠SOB

CONSTRUCTION ANGLES OF SPERCIAL MEASURES:

Construction of 600 angle: –

Steps of construction: –

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

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

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

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

Construction of 1200 angle: –

Steps of construction: –

Step-1: Draw a ray OA

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

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

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

Step-5: Join OR. Then ∠POR = 1200.

## 14. UNDERSDTANDING 3D AND 2D SHAPES

Cuboid:
3D- shapes or Solids:

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

Cuboid:

Objects like match boxes, erasers are the examples for cuboid

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

Cube:

A dice is an Example for cube.

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

Cylinder:

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

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

Cone:

joker cap is the example for cone

base of the cone is a circle.

Sphere:

Balls, laddoos are the examples for globe.

Triangular prism:

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

Pyreamid:

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

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

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

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

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

## 1. REAL NUMBERS

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

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

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

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

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

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

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

• In p/q, if prime factorisation of q is in form 2m 5n, then p/q is terminating decimal. Otherwise non-terminating repeating decimal.
• Decimal numbers with the finite no. of digits is called terminating Decimal numbers with the infinite no. of digits is called non-terminating decimal. In a decimal, a digit or a sequence of digits in the decimal part keeps repeating itself infinitely. Such decimals are called non-terminating repeating decimals.

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

• If ax = N then x = ${log_{a}}^{N}$

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

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

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

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

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

Ex: – 1,4,9, …etc.

Test for a number to be a perfect square:

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

Ex: – 36

Prime factors of 36 = 2× 2× 3× 3

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

∴ 36 is a perfect square.

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

Methods of Finding Square root of given Number

Prime factorization method: –

Steps:

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

Ex: – To find the square root of 16

Prim factors of 16 = 2 ×2× 2× 2

= 2 × 2 = 4

∴ square root of 16 = 4

Division method: –

Steps:

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

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

Ex: – To find the square root of 225

Properties of a perfect square:

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

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

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

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

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

Ex: –

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

Ex: – (0.3)2 = 0.09 < 0.03.

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

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

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

Ex: – 20, 120,1000 and so on.

The square root of a number in decimal form

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

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

Ex: – To find the square root of 79.21

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

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

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

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

Cube of a number:

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

Ex: – cube of 0.3 = 0.33 = 0.027

Cube of 2 = 23 = 8

Perfect cube:

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

Ex: – 1, 8, 27, and so on.

Cube root:

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

Cube root of x is denoted by

Methods of finding the cube root of the given Number

Prime factorization method: –

Steps:

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

Ex: – 27

Prim factors of 27 = 3×3×3

= 3

∴ cube root of 27 = 3

Test for a number to be perfect cube:

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

Ex: – 2744

Prime factors of 2744 = 2×2×2 × 7×7×7

∴ 2744 is a perfect cube.

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

## 1. RATIONAL AND IRRATIONAL NUMBERS

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

N = {1, 2, 3…}

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

W = {0, 1, 2, 3…}

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

Z = {…-3, -2, -1, 0, 1, 2, 3…}

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

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

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

Equal rational numbers:

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

The order of Rational numbers:

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

The absolute value of rational numbers:

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

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

To find rational number between given numbers:

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

Ex: – insert two rational number between 1 and 2

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

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

To rational numbers in a single step: –

Ex:- insert two rational number between 1 and 2

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

(∵ 1 + 2 = 3)

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

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

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

The decimal form of rational numbers

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

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

1.Terminating decimals: –

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

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

2.Non-Terminating repeating decimals: –

Irrational numbers:

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

Ex:- $\dpi{120}&space;\fn_cm&space;\sqrt{2},\,&space;\sqrt{5},\pi$ and so on.

• Calculation of square roots:
• There is a reference of irrationals in the calculation of square roots in Sulba Sutra.
• Procedure to finding $\dpi{120}&space;\fn_cm&space;\sqrt{2}$ value:

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

## 1. Goods and Service Tax

Two types of taxes in the Indian Government:

1.Direct taxes: –

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

2.Indirect taxes: –

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

Goods and Service Tax (GST):

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

Registration under GST:

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

Input and Output GST:

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

One currency one tax:

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

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

GST rate slabs:

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

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

• Simple tax system.

• Elimination of multiplicity of taxes.

• Development of a common market nation-wide.

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

Benefits of GST for Consumers:

• Single and transparent System.

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

• Increase in purchasing power and savings.

• Single tax system, simple and easy to administer.

• Higher revenue efficiency.

• Better control on leakage and tax evasion.

Types of GST in India

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

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

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

Reverse charge Mechanism:

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

Composition shame:

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

Input Tax Credit (ITC)

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

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

GST payable = Output GST – ITC

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

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

Utilization of ITC:

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

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

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

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

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

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

GST Returns:

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

A GST registered person has to submit the following returns:

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

## 2.Banking

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

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

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

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

Where p is the principle

n is no. of months

r is the rate of interest

Maturity value = (p × n) + I

## 3.Shares and Dividend

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

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

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

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

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

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

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

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

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

Some formulae:

Note:

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

## 4. Linear In equations

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

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

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

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

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

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

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

Inequations – Properties:

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

Ex: 3 < 5

3 + 2< 5 + 2

5 < 7 (no change in inequality)

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

Ex: 3 < 5

subtract 2 on both sides

3 – 2 < 5 – 2

1 < 3 (no change in inequality)

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

Ex: 3 < 5

Multiply 2 on both sides

3 × 2< 5 × 2

6 < 10 (no change in inequality)

6 < 8

Divide 2 on both sides

6 ÷ 2< 8 ÷ 2

3< 4 (no change in inequality)

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

Ex: 3 < 5

Multiply 2 on both sides

3 × –2< 5 × –2

–6 > –10 (change in inequality)

6 < 8

Divide 2 on both sides

6 ÷ –2< 8 ÷ –2

–3 > –4 (change in inequality)

Note:

• a < b iff b > a
• a > b iff b < a

Ex: x < 4 ⇔ 4 > x

x > 3 ⇔ 3 < x

Method of solving Liner Inequations:

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

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

Sol: given in equation is

3x – 2 < 2 + x

3x – 2 + 2< 2 + x + 2

3x < 4 + x

3x – x < 4

2x < 4

Dividing both sides by 2

x < 2

∴ Solution set = { 0, 1}

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

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

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

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

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

Solving Quadratic equation by using Factorization method:

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

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

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

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

Ex: Solve x2 – 3x – 4 = 0

x2 – 4x + x – 4 = 0

x (x – 4) + 1 (x – 4) = 0

(x – 4) (x + 1) = 0

x – 4 = 0 or x + 1 = 0

x = 4 or x =– 1

∴ Solution set = {– 1, 4}

Solving Quadratic equation by using Formula:

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

Ex: Solve x2 – 3x – 4 = 0

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

Compare with ax2 + bx + c = 0

a = 1, b = – 3, c = – 4

x = 4 or x = – 1

∴ Solution set = {– 1, 4}

Nature of the roots:

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

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

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

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

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

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

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

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

Method of Solving word problems in Quadratic equation:

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

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

Step – 3: Solve for x.

## 7.Ratio and Proportion

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

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

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

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

Lowest terms of a Ratio:

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

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

Comparison of Ratios:

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

Proportion:

An equality of ratios is called a proportion.

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

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

product of extremes = product of means

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

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

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

Note:

Results on Ratio and Proportion:

## 8.Remainder Theorem and Factor Theorem

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

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

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

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

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

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

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

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

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

Remainder theorem:

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

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

Sol: Given p(x) = 2x – 1

Remainder = p (3)

= 2(3) – 1

= 6 – 1 = 5

∴ remainder is 5

Note:

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

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

Note:

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

## 9. Matrices

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

The horizontal lines in a matrix are called its rows.

The vertical lines in a matrix are called its columns.

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

Ex:

Elements of a matrix:

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

A = [aij]m × n

A =

a11 means element in first row and first column

a12 means element in first row and second column

a22 means element in second row and second column

a32 means element in third row and second column

and so on.

Types of Matrices

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

Ex:

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

Ex:

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

Ex:

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

Ex:

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

Ex:

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

Ex:

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

Ex: If  ⟹ a=p; b = q; c = r; d = s

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

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

Ex:

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

Ex:

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

Ex:

Product of Matrices:

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

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

## 10. Arithmetic Progressions

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

Ex: 1, 2, 3, ……

2, 4, 6, 8, ….

2, 4, 8, 16, …

Arithmetic Progression (A.P.):

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

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

First term: a

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

nth term: Tn = a + (n – 1) d

Sum of the n terms of A.P.:

Sum of the n terms of A.P. is

Where a is first term and l is last term.

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

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

## 11. Geometric Progressions

Terms: a, a r, a r2…, a rn – 1

First term: a

Common ratio:

nth term: Tn = a rn – 1

Sum of the n terms of G.P.:

Sum of the n terms of G.P. is

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

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

## 12. Reflection

Coordinate Axes:

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

The horizontal line XOX’ is called X – axis.

The vertical line YOY’ is called Y – axis.

The point of intersection axes is called ‘origin’.

Coordinates of a point:

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

x is called x coordinate or abscissa of P

y is called y coordinate or ordinate of P

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

∴ Any point on the X – axis is (x, 0)

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

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

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

The equation of X – axis is y = 0.

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

The equation of Y – axis is x = 0.

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

Reflection

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

Image of a point in a line:

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

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

Reflection of a point in a line:

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

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

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

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

Combination of Reflection:

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

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

## 13. Section and Mid – Point Formula

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

P (x, y) =

Mid pint Formula:

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

Centroid of the triangle:

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

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

G =

## 14. Equation of a Straight line

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

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

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

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

Slope or Gradiant of a line:

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

m = tan θ

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

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

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

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

Types of equation of a straight line:

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

Slope intercept form:

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

Slope point form:

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

y – y1 = m (x – x1)

Two points form:

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

Intercept form:

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

Note: –

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

m1 = m2

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

m1 × m2 = – 1

slope of a line perpendicular to a line AB =

## 15. Similarity

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

Similarity as a Size transformation:

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

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

Properties of size transformation:

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

If k > 1, then the transformation is enlargement.

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

If k < 1, then the transformation is reduced.

Note:

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

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

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

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

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

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

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

## 16. Similarity of Triangles

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

If ∆ABC ~ ∆DEF, then

• ∠A = ∠D, ∠B = ∠E and ∠C = ∠F

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

Axioms of Similar Triangles

SAS – Axiom:

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

In ∆ABC and ∆DEF, ∠A = ∠D and

AA – Axiom : If two triangles have two pairs of corresponding angles equal, then the triangles are equal.

SSS – Axiom: If two triangles have their three sides of corresponding sides proportional, then the triangles are similar.

Basic proportionality Theorem:

In a triangle a line drawn parallel to one side divides the other two sides in the same ratio (proportional).

In ∆ ABC, DE ∥ BC ⇒

Converse of Basic proportionality Theorem:

If a line divides any two sides of a triangle proportionally, the line is parallel to the third side.

In ∆ ABC,  ⇒ DE ∥ BC.

∎ Ina triangle the internal bisector of an angle divides the opposite side in the ratio of the sides containing the angle.

In ∆ ABC, AD bisects ∠A then

∎ The areas of two similar triangles are proportional to the squares of their corresponding sides.

If ∆ABC ~ ∆DEF, then

∎ The areas of two similar triangles are proportional to the squares of their corresponding medians.

If ∆ABC ~ ∆DEF and AM, DN are medians of   ∆ABC and ∆DEF respectively then

∎ The areas of two similar triangles are proportional to the squares of their corresponding altitudes.

If ∆ABC ~ ∆DEF and AM, DN are altitudes of ∆ABC and ∆DEF respectively then

## 17. Loci

Locus: Locus is the path traced out by a moving point which moves according to some given geometrical conditions.

The plural form of Locus is ‘Loci’ read as ‘losai’

∎ The locus of the point which is equidistance from two given fixed points is the perpendicular bisector of the line segment joining the given fixed       points.

∎ Every point on the perpendicular bisector of AB is equidistance from A and B.

∎ The locus of the point which is equidistance from two intersecting lines is the pair of lines bisecting the angles formed by the given lines.

∎Every point on the angular bisector of two intersecting lines is equidistance from the lines.

## 18. Angle and Cyclic Properties of a Circle

∎ The angle subtended by an arc of a circle is double the angle subtended by it at any point on the circle.

∠AOB = 2 ∠ACB

∎ Angles in a same segment of a circle are equal.

∎ The angle in a semi-circle is 900

∎ If an arc of a circle subtends a right angle at any point on the remaining part of the circle, then the arc is semi-circle.

A quadrilateral is said to be cyclic if all the vertices passing through the circle.

The opposite angles of cyclic quadrilateral are supplementary.

∎ If pair of opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.

∎ The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

∎ Every cyclic parallelogram is a rectangle.

∎ An isosceles trapezium is always cyclic and its diagonals are equal.

∎ The mid-point of hypotenuse of a right-angled triangle is equidistance from its vertices.

## 19. Tangent Properties of Circles

Tangent: A line which intersect the circle at only one point is called Tangent to the circle.

∎ The tangent at any point of a circle and radius through the point are perpendicular to each other.

∎ If two tangents are drawn to a circle from an exterior point, then

• The tangents are equal in length.
• The tangents subtend equal angle at the centre.
• The tangents are equally inclined to the line joining the point and the centre if the circle.

Intersecting Chord and Tangents

Segment of a chord:

If P is a point on a chord AB of a circle, then we say that P divides AB internally into two segments PA and PB.

If AB is a chord of a circle and P is a point on AB produced, we say that P divides AB externally into two segments PA and PB.

∎ If two chords of a circle intersect internally or externally, then the product of the lengths of their segments are equal.

Alternate segments:

In the given figure APB is a tangent to the circle at point at a point P and PQ is a chord

The chord PQ divides the circle into two segments PSR and PSQ are called alternate segments.

The angle between a tangent and a chord through the point of contact is equal to an angle in the alternate segment.

∠QPB = ∠PSQ and ∠APQ = ∠PRQ

## 21. Volume and Surface Area of Solids

Cylinder:

Solids like circular pillars, circular pipes, circular pencils etc. are said to be in cylindrical shape.

Radius of the base = r

Height of the cylinder = h

Curved surface area = 2πrh sq. units

Total surface area = 2πr (r + h) sq. units

Volume = πr2h cubic. units

Hollow Cylinder:

Height = h

Thickness of the cylinder = R – r

Area of cross section = π (R2 – r2) sq. units

Volume of material = πh (R2 – r2) cubic. units

Curved surface area = 2πh (R+ r) sq. units

Total surface area = 2π (Rh + rh + R2 – r2) sq. units

Cone:

Radius of the base = r

Height of the cylinder = h

Slant height = l

l2 = r2 + h2 ⇒ l =

Curved surface area = πrl sq. units

Total surface area = πr (r + l) sq. units

Volume = πr2h cubic. Units

Sphere:

Objectives like football, throwball, etc. are said to be the shape of sphere.

Surface area = 4πr2 sq. units

Volume = πr3 cubic. Units

Spherical Shell:

The solid enclosed between two concentric spheres is called spherical shell

Thickness of the cylinder = R – r

Volume of the material =  π (R3 – r3) cubic. Units

Hemi sphere:

Curved Surface area = 2πr2 sq. units

Surface area = 3πr2 sq. units

Volume = πr3 cubic. Units

## 22.Trigonometrical Identities

The word Trigonometry derived from Greek word, tri three, gonia angle and metron to measure.

Angle: – The figure formed by two rays meeting at a common end point is an angle.

Naming the sides in a right-angled triangle:

AB = Perpendicular =opposite side of θ (opp)

AC is hypotenuse (hyp)

Trigonometric ratios:

Quotient relations:

Trigonometric Identities:
(i) sin
2θ + cos2θ = 1          (ii) sec2θ − tan2θ = 1        (iii) cosec2θ − cot2θ = 1

∎ sin2θ = 1 − cos2θ; cos2θ = 1 – sin2θ

∎ sec2θ = tan2θ + 1; sec2θ – 1 =tan2θ

∎ Cosec2θ = Cot2θ + 1; Cosec2θ – 1 = Cot2θ

Trigonometric tables:

A trigonometric Table Consist of three parts:

• A column on the extreme left containing degree from 00 to 890.
• The columns headed by 0’, 6’, 12’, 18’, 24’, 30’, 36’, 42’, 48’ and 54’.
• Five columns of mean differences, headed by 1’, 2’, 3’, 4’ and 5’

Note:

• The mean difference is added in case of ‘sines’, ‘tangents’ and ‘secants’
• The mean difference is subtracted in case of ‘cosines’, ‘cotangents’ and ‘cosecants’

Relation between degrees and minutes:

10 = 60’⇒ 1’ =

Trigonometric Tables Charts:

By clicking below you get Sin, Cosine and Tangent Tables

Taking too long?

| Open in new tab

## 23. Heights and Distances

Horizontal line: A line which is parallel to earth from observation point to object is horizontal line
Line of sight: The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer.

Angle of elevation: The line of sight is above the horizontal line and angle between the line of sight and horizontal line is called angle of elevation.

Angle of depression: The line of sight is below the horizontal line and angle between the line of sight and horizontal line is called angle of depression.

Solving procedure:

∎All the objects such as tower, trees, buildings, ships, mountains etc. shall be consider as linear for mathematical convenience.

∎The angle of elevation or angle of depression is considered with reference to the horizontal line.

∎The height of observer neglected, if it is not given in the problem.

∎To find heights and distances we need to draw figures and with the help of these figures we can solve the problems.

## 24. Graphical Representation of Statistical Data

Data: A set of given facts in numerical figures is called data.

Frequency: The number of times an observation occurs is called its frequency.

Frequency Distribution: The tabular arrangement of data showing the frequency of each observation is called its frequency distribution.

Class interval: Each group into which the raw data is condensed is called a class interval.

Class limits: Each class interval is bounded by two figures is called Class limits.

Left side part of class limit is called ‘Lower limit’

Right side part of class limit is called ‘Upper limit’

Inclusive form:  In each class, the data related to both the lower and upper limits are included in the same class, is called Inclusive form.

Ex: 1 – 10, 11 – 20, 21 – 30 etc.

Exclusive form: In each class, the data related to the upper limits are excluded is called Exclusive form.

Ex: 0 – 10, 10 – 20, 20 – 30 etc.

Class size = upper limit – lower limit

Class mark =  [lower limit + upper limit]

Note:

In an inclusive form, Adjustment factor =  [lower limit of one class – upper limit of previous class]

Histogram:  A histogram is a graphical representation of a frequency distribution in an exclusive form, in the form of rectangles with class interval as bases and the corresponding frequencies as heights

Method of drawing a Histogram:

Step-1:  If the given frequency distribution is in inclusive, then convert them into the exclusive form

Step-2: Choose a suitable scale on the X – axis and mark the class intervals on it.

Step-3: Choose a suitable scale on the Y – axis and mark the frequencies on it.

Step-4: Draw rectangle with class intervals as bases and the corresponding frequencies as the corresponding heights.

Example:

Frequency polygon:

Let x1, x2, x3, …, xn be the class marks of the given frequency distribution and f1, f2, f3, …, fn be the corresponding frequencies, then plot the points (x1, f1), (x2, f2), …. (xn, fn) on a graph paper and join these points by a line segment. complete the diagram in the form of polygon by taking two or more classes.

Example:

Cumulative Frequency curve or Ogive:

In order to represent a frequency distribution by an Ogive, we mark the upper class along X– axis and the corresponding cumulative frequencies along Y – axis and join these points by free hand curve, called Ogive.

Example:

## 25. Measures of Central Tendency

Average of a Data:

For a given data a single value of the variable representing the entire data which describes the characteristics of the data is called average of the data.

An average tends to lie centrally with the values of the variable arranged in ascending order of magnitude. So, we call an average a measure of central tendency of the data.

Three measures of central tendency are:   (i) Mean   (ii) Median  and (iii) Mode

Average of a Data:

For a given data a single value of the variable representing the entire data which describes the characteristics of the data is called average of the data.

An average tends to lie centrally with the values of the variable arranged in ascending order of magnitude. So, we call an average a measure of central tendency of the data.

Three measures of central tendency are:   (i) Mean   (ii) Median  and (iii) Mode

Mean

Mean for Un Grouped data:

The mean of ‘n’ observations x1, x2, x3, …, xn is

Mean =

The Symbol Σ is called ‘sigma’ stands for summation of the data.

Note:

If the mean of a data x1, x2, … xn is m, then

• Mean of (x1+k), (x2 + k), …. (xn + k) = m + k
• Mean of (x1−k), (x2 − k), …. (xn − k) = m −k
• Mean of (k x1), (k x2), …. (k xn) = k m

If x1, x2, …. xn are of n observations occurs f1, f2, …. fn times respectively then mean is

Mean of grouped data:

Methods of finding mean:

Class mark (mid value) =

Direct method: ;  xi is class mark of ith class, fi is frequency of class.

Assumed mean method: ;  di = xi – a and a is assumed mean.

Step – deviation method:  ; 𝛍i =  , h is class size.

## 26. Median, Quartile and Mode

Median

Median is the middle most observation of given data.

For un grouped data:

First, we arrange given observations into ascending or descending order.

If n is odd median =  observation.

If n is even median =

For grouped data:

Median = , where

l is the lower boundary of median class

f is the frequency of median class

c.f is the preceding cumulative frequency of the median class

h is the class size

Quartiles

The observations which divides the whole set of observations into 4 equal parts are known as Quartiles.

Lower Quartile (First Quartile): If the variates are arranged in ascending order, then the observations lying midway between the lower extreme and the median is called the Lower Quartile. It is denoted by Q1.

If n is Even Q1 observation

If n is Odd Q1 =  observation

Middle Quartile: The middle Quartile is the median, denoted by Q2.

Upper Quartile (Third Quartile): If the variates are arranged in ascending order, then the observations lying midway between and the median the upper extreme is called the Upper Quartile. It is denoted by Q3.

If n is Even Q =  observation

If n is Odd Q1 = observation

Range: The difference between the biggest and the smallest observations is called the Range.

Interquartile Range: The difference between the upper quartile and Lower quartile is called the inter quartile.

Range = Q3 – Q2

Semi – interquartile range:

Semi – interquartile Range = ½ [ Q3 – Q2]

Estimating median:

Step 1: If the given frequency distribution is not continuous, convert into the continuous form.

Step 2: Prepare the cumulative frequency table.

Step 3: Draw Ogive for the cumulative frequency distribution given above

Step 4: Let sum of the frequencies = N.

Step 5: Mark a point A on Y- axis corresponding to

Step 6: From A draw Horizontal line to meet Ogive curve at P. From P draw a vertical line PM to meet X – axis at M. Then the abscissa of M gives the Median.

Estimating Q1 and Q2:

To locate the value of Q1 on Ogive curve, we mark the point along

Y – axis, corresponding to  and proceed similarly.

To locate the value of Q3 on Ogive curve, we mark the point along

Y – axis, corresponding to  and proceed similarly.

Mode

The value of a data which is occurred most frequently is called Mode.

Modal class: The class with maximum frequency is called the Modal class.

Estimation of Mode from Histogram:

Step 1: If the given frequency distribution is not continuous, convert it into a continuous form.

Step 2: Draw a histogram to represent the above data.

Step 3: from the upper corner of the highest rectangle, draw line segments

To meet the opposite corners of adjacent rectangles, diagonally

Let these line segments intersect at P.

Step 4: Draw PM perpendicular to X-axis at M, Then the abscissa of M is The Mode

## 27. Probability

J Cordon Italian mathematician wrote the first book on probability named “the book of games and chance”.

Probability:

It is the concept which numerically measures the degree of certainty of the occurrence of an event.

Some words in probability:

Experiment: A repeatable procedure with a set of possible results.

Trial: By a trial, we mean experimenting.

Outcome: a possible result of an experiment.

Sample space: All the possible outcomes of an experiment.

Sample point: Just one of the possible outcomes.

Event: One or more outcomes of an experiment.

Probability of occurrence of an Event (Classical definition):

In a random experiment Let S be the sample space and E be the event, then E ⊆ S. The probability of occurrence of E is defined as:

P(E) =

Deck of cards: A deck of playing cards consists of 52 cards which are divided into 4 suits of 13 cards each. They are black spade     , black clubs, red heart     and red diamond       . The cards in each suit are: 2, 3, 4, 5, 6, 7, 8, 9 ,10, Ace, Jack, Queen and King. Jack, Queen and King are called face (picture) cards.

Impossible event: If there is no probability of an event to occur then it is impossible event. Its probability is zero.

Sure or certain event: If the probability of an event is 1 then it is sure or certain event.

Complimentary event: Let E denote the event, ‘not E’ is called complimentary event of E. It is denoted by  . P ( ) = 1 – P(E) ⟹ P ( ) +P(E) = 1.

0 ≤ P(E) ≤ 1

# These notes cover all the topics covered in the CBSE 11th class Maths syllabus and include plenty of formulae and concept to help you solve all the types of11thMath problems asked in the CBSE board and entrance examinations.

## 1. SETS

Well defined objects:

1. All objects in a set must have the same general similarity or property.
2. Must be able to confirm whether something belongs to the set or not.

Set: – A collection of well-defined objects is called a set.

∗ Sets are usually denoted by capital English alphabets like A, B, C and so on.

∗ The elements in set are taken as small English alphabets like a, b, c and so on.

∗ Set theory was developed by George canter.

• If any object belongs to a set, then it is called objects / elements. We denote by ∈ to indicate that it belongs to. If it does not belong to the set then it is denoted by ∉.

Ex: – 1 ∈ N, 0 ∈ W, −1 ∈ Z , 0 ∉ N etc.

Methods of representing sets:

Roster or table or listed form: –

In this form all the elements of set are listed, the elements are separated by commas and enclosed within braces { }.

Ex: – set of vowels in English alphabet = {a, e, I, o, u},

set of even natural numbers less than 10 = {2, 4, 6, 8} etc.

Note: – In roster form, an element is not repeated.  We can list the elements in any order.

Set builder form:

Pointing an element in a set to x (or any symbols such as y, z, etc.) followed by a colon(:), next to :  write the properties or properties of the elements in that set and placing in a flower brackets  is called the set builder  form.: Or / symbols read as ‘such that’

Ex: – {2, 4, 6, 8} = {x / x is an even and x ∈N, x< 10},

{a, e, i, o, u} = {x : x is a vowel in English alphabet}.

Null set: – (empty set or void set) the set which has no elements is called as null set. It is denoted by ∅ or { }.

Finite and infinite sets: – If a set which contains finite no. of elements then it is called finite set. If a set contains infinite no. of elements then it is called infinite set.

Ex: – A = {1, 2,3, 4} → finite set

B = {1, 2, 3, 4….}

Equal sets: – two set A and B are said to be equal sets if they have same elements., and write as A = B

Ex: – A = {1, 2, 3, 4}, B = {3, 1, 4, 2}

⟹ A = B.

Sub set: – for any two sets A and B, if every element of set A is in set B, then we can say that A is subset of B. It is denoted by A ⊂ B.

Ex: – If A = {1, 2, 3, 4, 5, 6, 7, 8}, subsets of A are {1}, {1, 3, 5}, {1,2,3,4} and so on.

Power set: – set of all the subsets of a set A is called power set of A. It is denoted by p(A).

Ex: – A = {1,2,3}

P(A) = {{1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1, 2, 3}, ∅}.

Intervals:

∗ Open interval: – (a, b) = {x: a< x <b} → set of rational numbers lies between a and b.

∗ Closed interval: – [a, b] = {x: a≤ x ≤b} → set of rational numbers lies between a and b, including a and b.

∗ Open – closed: – (a, b] = {x: a< x ≤b} → set of rational numbers lies between a and b, excluding a and including b.

∗ Closed-open: -[a, b) = {x: a≤ x <b} → set of rational numbers lies between a and b, including a and excluding b.

Universal set: – A set which contains all the subset of it under our consideration is called universal set.

Cardinal number of a set: – Number of elements in a set A is called cardinal number of that set A. It is denoted by n(A).

• If a set has n elements, then no. of elements of that set has 2n

Equivalent sets: – two set A and B are said to be equivalent sets if n(A) = n(B) (they have same cardinal number).

Ex: – A = {1, 2, 3}, B = {a, b, c}

n(A) = 3 and n(B) = 3

∴ A = B.

Venn diagrams:

U = {1, 2, 3, 4, 5, 6}
the relationship between sets are usually represented by means of diagrams which are known as ‘Venn diagrams. These diagrams consist of rectangle and circles. Universal set is represented by rectangles and subsets by circles.

U = {1, 2, 3, 4, 5, 6} A = {1, 2, 3} B = {1, 2}

## TS inter 1st year Maths Blueprint

These blue prints designed by ‘Basics in Maths’ team. These to do help the TS intermediate first year Maths students fall in love with mathematics and overcome the fear.

These blue prints cover all the topics of  the TS I.P.E  first  year maths syllabus and  help inI.P.E exams.

# These notes cover all the topics covered in the TS I.P.E second year maths 2A 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.

## 1. COMPLEX NUMBERS

•  The equation x2 + 1 = 0 has no roots in real number system.

∴ scientists imagined a number ‘i’ such that i2 = − 1.

Complex number:  if x, y are any two real numbers then the general form of the complex number is

z = x + i y;  where x real part and y is imaginary part.

∗ z =  + iy can be written as (x, y)

∗If z1 = x1 + i y1, z2 = x2 + i y2, then

∗ z1 + z2 = (x1 + x2, y1 + y2) = (x1 + x2) + i (y1 + y2)

∗ z1 − z2 = (x1 − x2, y1 − y2) = (x1 − x2) + i (y1 − y2)

∗ z1∙   z2 = (x1 x2 −y1 y2, x1y2 + x2y1) = (x1x2 −y1 y2) + i (x1y2 +x2 y1)

∗ z1/ z2 = (x1x2 + y1 y2/x22 +y22, x2 y1 – x1y2/ x22 +y22)

= (x1x2 + y1 y2/x22 +y22) + i (x2 y1 – x1y2/ x22 +y22)

Multiplicative inverse of complex number:

Multiplicative inverse of complex number z is 1/z.

z = x + i y then 1/z = x – i y/ x2 + y2

Conjugate complex number:

• The complex numbers x + iy, x – iy are called conjugate complex numbers.
• The sum and product of two conjugate complex numbers are real.
• If z1, z2 are two complex numbers then

Modulus and amplitude of complex number:

Modulus: – If z = x + iy, then the non-negative real number is called modulus of z and it is denoted by or ‘r’.

Amplitude: – The complex number z = x + i y represented by the point P (x, y) on the XOY plane. ∠XOP = θ is called amplitude of z or argument of z.

∗ x = r cosθ, y = r sinθ

⇒ x2 + y2 = r2 cos2θ + r2 sin2θ = r2 (cos2θ + sin2θ) = r2(1)

⇒ x2 + y2 = r2

⇒ r =   and  = r.

∗ Arg (z) = tan−1(y/x)

∗ Arg (z1.z2) = Arg (z1) + Arg (z2) + nπ for some n ∈ { −1, 0, 1}

∗ Arg(z1/z2) = Arg (z1) − Arg (z2) + nπ for some n ∈ { −1, 0, 1}

Argand plane: The plane contains all complex numbers is called Argand plane. This was introduced by the mathematician Gauss (1777-1855), who first thought that complex numbers can be represented as a two-dimensional plane.

The square root of a complex number:

## 2.DE- MOIVER’S THEOREM

De- Moiver’s theorem: For any integer n and real number θ, (cosθ + i sinθ) n = cos nθ + i sin nθ.

cos α + i sin α can be written as cis α

cis α.cis β= cis (α + β)

1/cisα = cis(-α)

cisα/cisβ = cis (α – β)

(cosθ + i sinθ) -n = cos nθ – i sin nθ

(cosθ + i sin θ) (cosθ – i sin θ) = cos2θ – i2 sin2θ = cos2θ + sin2θ = 1.

cosθ + i sin θ = 1/ cosθ – i sin θ and cosθ – i sin θ = 1/ cosθ + i sin θ

(cosθ – i sin θ) n = (1/ (cosθ –+i sin θ)) n = (cosθ + i sin θ)-n = cos nθ – i sin nθ

nth root of a complex number: let n be a positive integer and z0 ≠ 0 be a given complex number. Any complex number z satisfying z n = z0 is called an nth root of z0. It is denoted by z01/n or

let z = r (cosθ + i sin θ) ≠ 0 and n be a positive integer. For k∈ {0, 1, 2, 3…, (n – 1)}   let . Then a0, a1, a2, …, an-1 are all n distinct nth roots of z and any nth root of z is coincide with one of them.

nth root of unity:  Let n be a positive integer greater than 1 and

Note:

• The sum of the nth roots of unity is zero.
• The product of nth roots of unity is (– 1) n – 1.
• The nth roots of unity 1, ω, ω2, …, ωn-1 are in geometric progression with common ratio ω.

Cube root of unity:

x3 – 1 = 0 ⇒ x3 = 1

x =11/3

Quadratic Expression: If a, b, c are real or complex numbers and a ≠ 0, then the expression ax2 + bx + c is called a quadratic expression in variable ‘x’.

∎ A complex number α is said to be a zero of the quadratic expression ax2 + bx + c if aα2 + bα + c = 0.

Quadratic Equation:  If a, b, c are real or complex numbers and a ≠ 0, then ax2 + bx + c = 0 is called a quadratic equation in variable ‘x’.

∎ A complex number α is said to be root or solution of the quadratic equation ax2 + bx + c = o if aα2 + bα + c = 0.

The roots of a quadratic equation:

∎ The zeroes of the quadratic expression ax2 + bx + c are same as the roots of quadratic equation ax2 + bx + c = o.

∎ The roots of the quadratic equation are

If α, β are the roots of the quadratic equation ax2 bx +c= 0, then α + β = -b/a and αβ = c/a.

Discriminate:

If ax2 + bx + c = 0 is a quadratic equation, then b2 – 4ac is called the discriminant of quadratic equation. b2 – 4ac is also the discriminant of quadratic expression ax2 + bx + c. It is denoted by ∆

∴ ∆ = b2 – 4 ac.

Nature of the roots:

The nature of the roots of the quadratic equation as follows:

1. If ∆ > 0, then roots are real and distinct.
2. If ∆ = 0, then roots are real and equal.
3. If < 0, then roots are imaginary.

Note:

• If ∆ > 0 and b2 – 4 ac is a perfect square, then the roots are rational and distinct.
• If ∆ < 0 and b2 – 4 ac is not a perfect square, then the roots are irrational and distinct. Further, the roots are conjugate surds.

If α, β are the roots of the quadratic equation ax2 + bx +c= 0, then ax2 + bx + c = a (x – α) (x – β)

The quadratic equation whose roots are α, β is (x – α) (x – β) = 0 ⇒ x2 – (α + β) x + αβ = 0.

∎ The necessary and sufficient condition for the quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 to have common root is (c1a2 – c2a1)2 = (a1b2 – a2b1) (b1c2 – b2c1)

the common root is (c1a2 – c2a1)/ (a1b2 – a2b1).

If f(x) = ax2 + bx +c= 0 is a quadratic equation then

1. The quadratic equation whose roots are the reciprocals of the roots of f(x) = 0 is f(1/x) = 0.
2. Whose roots are greater than by ‘k’ then those of f(x) = 0 is f (x – k) = 0.
• Whose roots are smaller by ‘k’ than those of f(x) = 0 is f (x + k) = 0.
1. Whose roots are multiplied by ‘k’ of hose f(x) = 0 is f (x/k) = 0.

Sign of quadratic Expressions – change in signs:

∎ If the roots of quadratic equation ax2 + bx + c = 0 are complex roots, then for x ∈ R, ax2 + bx + c and ‘a’ have the same sign.

∎ If the roots of quadratic equation ax2 + bx + c = 0 are real and equal then for x ∈ R – {-b/2a}, ax2 + bx + c and ‘a’ have the same sign.

∎ Let α, β are the roots of the quadratic equation ax2 + bx +c= 0and α < β, then

• x ∈ R, α < x < β ⇒ ax2 + bx + c and ‘a’ have the opposite sign.
• x ∈ R, x < α or x> β ⇒ ax2 + bx + c and ‘a’ have the same sign.

Maximum and Minimum values:

Let f(x) = ax2 + bx +c is a quadratic expression then

• if a < 0, then f(x) has maximum value at x = -2b/a and maximum value is 4ac – b2/4a.
• if a > 0, then f(x) has minimum value at x = -2b/a and minimum value is 4ac – b2/4a.

A quadratic in equation in one variable is of the form ax2 + bx +c > 0 or ax2 + bx +c ≥ 0 or  ax2 + bx +c< 0 or ax2 + bx +c ≤ 0 where a, b, c are real numbers and a ≠ 0. The values of x which satisfy given inequation are called the solution of the in equations.

⟹ Quadratic inequations are solved by two methods (i) Algebraic method (ii) Graphical method.

## 4.THEORY OF EQUATIONS

Polynomial: If n is non- negative integer and a0, a1, a2. …, an are real or complex numbers and a0 ≠ 0, then an expression

f(x) = a0 xn + a1 xn – 1+a2xn – 2 + … + an is called polynomial in x of degree n.

a0, a1, a2. …, an are called the coefficients of the polynomial f(x), a0 is called leading coefficient and an is called constant term.

Monic polynomial: A polynomial with leading coefficient is 1 is called a monic polynomial.

Remainder theorem: let f(x) be a polynomial of degree n> 0. Let a ∈C. Then there exist a polynomial q(x) of degree n – 1 such that

f(x) = (x – a) q(x) + f(a).

Factor theorem: let f(x) be a polynomial of degree n> 0. Let a ∈ C. We say that (x – a) is factor of f(x) if there exist a polynomial q(x) such that f(x) = (x – a) q(x).

⟹ let f(x) be a polynomial of degree n> 0, then (x – a) is factor of f(x) iff f(a) = 0.

The fundamental theorem of algebra: Every non-constant polynomial equation has at least one root.

⟹ The set of all roots of a polynomial f(x) = 0 of degree n> 0 is non-empty and has at most n elements. Also there exist α1, α 2. …, α n in C such that f(x) = a (x – α1) (x – α2) (x – α3) … (x – αn) where a is the leading coefficient of f(x).

The relations between the roots and the coefficients:

Let xn + p1 xn – 1+ p2xn – 2 + … + pn = 0 be a polynomial equation of degree n

Let α1, α 2. …, α n be roots.

xn + p1 xn – 1+ p2xn – 2 + … + pn = (x – α1) (x – α2) (x – α3) … (x – αn)

= xn – (α1+ α 2+ …+ α n) xn – 1+( α1 α 2+ α2 α 3 +…+ α n-1 α n) xn-2 – … (-1)n α1. α 2…. α n

These equalities the relation between the roots and coefficient of the polynomial equation whose leading coefficient is 1.

Note:

1.For the quadratic equation: Let α, β be the roots of the quadratic equation ax2 + bx + c = 0, then

Sum of the roots = α + β = -b/a

Product of the roots = αβ = c/a

2.For the cubic equation: Let α, β and γ be the roots of the cubic equation ax3+ bx2 + cx + d = 0, then

Sum of the roots = α + β + γ = -b/a

Product of the roots taken two at a time= αβ + β γ + γ α = c/a

Product of the roots = αβ γ = -d/a

Notation: Let α, β and γ be the roots of a cubic polynomial, then

α + β + γ is denoted by ∑α,  αβ + β γ + γ α is denoted by ∑ αβ , 1/α + 1/β + 1/γ is denoted by ∑1/α and  α2 β + β2 α  + α2 γ + γ2 β + β2 γ + γ2 α = ∑ α2 β + ∑ α β2

synthetic division: This method has two types

∗ Finding the quotient and remainder, when

a0 xn + a1 xn – 1+a2xn – 2 + … + an (a0 ≠ 0) is divided by (x – a).

∗ Finding the quotient and remainder, when

a0 xn + a1 xn – 1+a2xn – 2 + … + an (a0 ≠ 0) is divided by x2 – px – q.

Method of finding the quotient and remainder, when a0 xn + a1 xn – 1+a2xn – 2 + … + an (a0 ≠ 0) is divided by (x – a)- (Horner’s Method):

The procedure of the above method:

• First write down the coefficients of xn, xn-1, …x, x0. If any term with xk (0 ≤ k < 1) is missing, take the coefficient of it as zero.
• Draw a vertical line to the left of ‘a0’ and write ‘a’ to the left of the vertical line on the same horizontal level as that of ‘a0’.
• Under a0 write 0 and draw the horizontal line below it. Below the horizontal line and below 0, write the sum a + 0 as the first term of the 3rd row, which is equal to b0 with a and write this product below a1 in the second row. The sum a1 + ab0 is b­1. Write this in the 3rd row next to b0. Continue this process until the terms of the second and the third rows are filled.
• From the table, the quotient is b0 xn-1 + b1 xn – 2+ … + bn-1 and the remainder is R = an + abn-1.

Note:

1. If the divisor is (x + a) then the above method can be used by replacing a with – a.
2. If the divisor is ax – b, then replace a by b/a.

Method of finding the quotient and remainder, when  a0 xn + a1 xn – 1+a2xn – 2 + … + an (a0 ≠ 0) is divided by x2 – px – q:

The procedure of the above method:

• First write down the coefficients of xn, xn-1, …x, x0. If any term with xk (0 ≤ k < 1) is missing, take the coefficient of it as zero.
• Draw a vertical line to the left of ‘a0’ and write p, q as column figures to the left of the vertical line in the second and third rows respectively. These are the negatives of the coefficient of x and the constant term in the divisor. Draw a horizontal line below the third row.
• Put 0 in two rows underneath a0 write this sum a + 0 + 0 as the first term of the 4th row, which is equal to b0. Next, multiply b0 with p and write this product below a1 and write the next column entry as 0. The sum a1 + pb0 + 0 is b­1. Write this in the 4th row underneath a1 multiply b1 with p and b0 with q and write this product underneath a2. let the sum of a2, pb1 and qb0 be b2. continue this process until the terms an-1 are obtained. Name the 4th row under an-1 R1. below an put 0 and qbn-2 in the second and third rows respectively. Let the sum as an, 0 qbn-2 be R2. write it in the 4th row below an.

Trial and Error method: To find a root of f(x) = 0, we have to find out a value of x, for which f(x) = 0. Some times we can do this by inspection. This method is called trial and error method.

Multiple roots or repeated roots: let f(x) be a polynomial of degree n > 0. Let α1, α 2. …, α n be the roots of f(x) = 0 so that f(x) = a0 (x – α1) (x – α2) (x – α3) … (x – αn). A complex number α is said to be a root of f(x) = 0 of multiplicity m, if α = αk for exactly m values of k among 1,2, 3…, n. Roots of multiplicity m>1 are called multiple roots or repeated roots.

Roots of multiplicity 1 are called simple roots.

∎ Let f(x) be a polynomial of degree n> 0. Let α be a root of f(x) = 0 of multiplicity m. If m>1, then α is a root of the equation f’(x) = 0 of multiplicity m – 1. If m = 1, then f’(α) ≠ 0

∎ Let f(x) be a polynomial of degree n > 0. Let α be a root of f(x) = 0 of multiplicity m, then α is a root of the equation f(k)(x) = 0 of multiplicity m –k (k = 1, 2, 3, …, m – 1).

∎ Let f(x) be a polynomial of degree n> 0. Let α be a root of f(x) = 0 of multiplicity m Iff f(α) = f’(α) = … = f (m – 1) (α) and f(m)(α) ≠ 0.

Procedure to find multiple roots:  Let f(x) be a polynomial. First, we find f’(x) and then find the HCF of f(x) and f’(x). Now we note that, if α is a root of the HCF of multiplicity k, then α is a multiple order of (k + 1) of f(x) =0.

∎ let f(x) be a polynomial with real coefficients. Let α ∈ C, then

∎ Let f(x) be a polynomial of degree n > 0, with real coefficients. Let a0 be the leading coefficient of f(x).

1. If the equation f(x) = 0 has no real roots, then n is even and f(α), a0 have the same sign for all real values of α
2. If n is odd, then the equation f(x) = 0 has at least one real root.

∎ Let f(x) be a polynomial of degree n > 0, with real coefficients. Let a and b be rational numbers, b > 0 and irrational. Then   is a root of f(x) = 0 if and only if another root is a .

Roots with the change of sign:

If α1, α 2. …, α n are the roots of f(x) = 0, then -α1, – α 2. …, – α n are the roots of f(-x) = 0.

Roots multiplied by a given number:

If α1, α 2. …, α n are the roots of f(x) = 0, then for any non-zero complex number k, the roots of f(x/k) = 0 are kα1, k α 2. …, kα n.

Roots subtracted by a given number:

If α1, α 2. …, α n are the roots of f(x) = 0, then α1-h, α 2-h. …, – α n-h are the roots of f (x +h) = 0.

Roots added by a given number:

If α1, α 2. …, α n are the roots of f(x) = 0, then α1+h, α 2+h. …, – α n+h are the roots of f (x -h) = 0.

Reciprocal roots:

Let α1, α 2. …, α n are the roots of f(x) = 0. Suppose none of them non- zero, then 1/α1,1/ α 2. …,1/ α n are the roots of xn f (1/x) = 0.

∎ if α is a root of f(x) = 0, then α2 is a root of

Reciprocal equation:

Let f(x) be a polynomial of degree n > 0 is said to be reciprocal if f(0) ≠ 0 and ∀x ∈C-{0}, where a0 is the leading coefficient of f(x).

If f(x) is a reciprocal polynomial, then the equation f(x) = 0 is reciprocal equation.

∎ If f(x) = a0 xn + a1 xn – 1+a2xn – 2 + … + an  be a polynomial of degree n > 0, then f(x) is reciprocal iff an – k = ak for k = 0, 1, 2,… , n or  an – k = – ak for k = 0, 1, 2,… , n .

The reciprocal polynomial of class one and class two:

A reciprocal polynomial f(x) of degree n with leading coefficient a0 is said to be class one or class two according to as f(0) = a0 or  –a0.

If f(x) is a reciprocal polynomial then the equation f(x) = 0 is said to be the reciprocal equation of class one or class two according to as f(x) is a reciprocal polynomial of class one or class two.

Note:

∎ For an odd degree, the reciprocal equation of class one -1 is the root and for an odd degree reciprocal equation of class two, 1 is root.

∎ For an even degree, the reciprocal equation of class two, – 1 and 1 roots.

∎ To solve the reciprocal equation of order 2m, divide the equation by xm and put x + 1/x = y or x – 1/x = y according to the equation of class one or class two.  The degree of the transformed equation is m.

∎ For an odd degree reciprocal equation. To find the roots of it, divide f(x) by (x + 1) or (x – 1) according as the equation of class one or class two. Let Q(x) be the quotient obtained, then f(x) = (x+1) Q(x) or f(x) = (x – 1) Q(x) according as the equation of class one or class two and Q(x) is even degree reciprocal polynomial. The roots of Q(x) = 0 can be obtained by above procedure.

## 5.PERMUTATIONS AND COMBINATIONS

The fundamental principle of counting: if a work w1 can be performed in ‘m’ different ways and a second work w2 can be performed in ‘n ‘different ways, then the two works can be performed in ‘mn’ ways.

Permutation: From a given finite set of elements selecting some or all of them and arranging them in a line is called a ‘linear permutation’ or ‘permutation’.

Circular permutation

Permutations of ‘n’ dissimilar thing taken ‘r’ at a time:

∎ If n, r are positive integers and r ≤ n, then the no. of permutations of n dissimilar things taken as ‘r’ at a time is n (n – 1) (n – 2) (n – 3) … (n – r + 1).

Notation:

The number of permutations of n dissimilar things taken as r at a time is denoted by   nPr or P (n, r) (1≤r≤n).

nPr  =  n (n – 1) (n – 2) (n – 3) … (n – r + 1)

∎ If n ≥1 and 0≤r≤n, then

nPn = n! and nP0 = 1.

∎ For 1≤r≤n , nPr = n. (n – 1) P (r – 1)

∎ If n, r are positive integers and 1 ≤r < n, then nPr = (n – 1) Pr + r. (n – 1) P (r – 1).

∎ The sum of all r-digit numbers that can be formed using the given ‘n’ non-zero digits (1 ≤r ≤n≤9) is

(n – 1) P (r – 1) × [ sum of the given digits × 1111… 1(r times)]

∎If ‘0’ is one digit among the given ‘digits, then we get that the sum of all r-digit numbers that can be formed using the given ‘n’ digits including ‘0’ is

{ (n – 1) P (r – 1) × [ sum of the given digits × 1111… 1(r times)]} – { (n – 2) P (r – 2) × [ sum of the given digits × 1111… 1((r-1) times)]}.

Note: If a set A has m elements and the set B has n elements, then the no. of injections into A to B is nPm if m ≤n and 0 if m> n.

Permutations when repetitions are allowed:

∎ Let n and r be positive integers. If the repetition of things is allowed, then the no. of permutations of ‘n’ dissimilar things taken ‘r’ at a time is nr.

Palindrome: A number or a word which reads the same either from left to right or right to left is called a palindrome.

Ex:  121, 1331, ATTA, AMMA etc.

Note: The no. of palindromes with r distinct letters that can be formed using given n distinct letters is

(i) nr/2 if r is even (ii) nr+1/2 if r is odd.

Circular permutation: From a given finite set of elements selecting some or all of them and arranging them around a circle is called a ‘circular permutation’.

The no. of circular permutations of ‘n’ dissimilar things (taken all at a time) is (n – 1)!

∎ In case of the garlands of flowers, chains of beads etc, no. of circular permutations = ½ (n – 1)!

Permutations with constraint repetitions:

∎ The no. of linear permutations of ‘n’ things n which ‘p’ things are alike and the rest are different is

∎ The no. of linear permutations of ‘n’ things n which ‘p’ like things of one kind, q like things of the second kind, r like things of the third kind and the rest are different is

Combinations:

A combination is only a selection. There is no importance to the order or arrangement of things in a combination.

∎ The no. of combinations of ‘n’ dissimilar things taken ‘r’ at a time is denoted by nCr or C (n, r)

∎ For 0≤r≤n, nCr = n C n – r

∎ If m, n are distinct positive integers, then the no, of ways of dividing (m + n) things into two groups containing m things and ‘n’ things is

∎ If m, n, p are distinct positive integers, then the no, of ways of dividing (m + n + p) things into three groups containing m things, ‘n’ things and ‘p’ things is

∎ The no. of ways of dividing 2n dissimilar things into two equal groups containing ‘n’ things in each case is

∎ The no. of ways of dividing ‘mn’ dissimilar things into m equal groups containing ‘n’ things in each case is

∎ The no. of ways of distributing ‘mn’ dissimilar things equally among  m  persons is

• For 0 ≤ r, s ≤ n, if nCr = nCs then r =s or n = r + s.

• If 1 ≤ r ≤ n, then nCr-1 + nCr = (n+1) Cr.

• If 2 ≤ r ≤ n, then nCr-2 +2 nCr-1 = (n+2) Cr.

• If p things are alike of one kind, q things are alike of the second kind and r things are alike of the third kind, then the number of ways of selecting any no, of things out of these (p + q +r) things is (p + 1) (q+1)(r+1) – 1.

• The number of ways of selecting one or more things out of ‘n’ dissimilar things is 2n – 1.

• If p1, p2,…, pn are distinct primes and α1, α 2,…, α n are positive integers, then the number of positive divisors of is (α1+1)( α2+1) … (αk + 1).

Exponents of a prime in n! (n ∈ z+): Exponents of a prime number ‘p’ in n! is the largest integer ‘k’ such that pk divides n!

## 6.BINOMIAL THEOREM

Binomial: Binomial means two terms connected by either ‘+’ or ‘– ‘.

Binomial expansions:

(x + y)1 = x + y

(x + y)2 = x2 + 2xy + y2

(x + y)3 = x3 + 3x2 y + 3xy2 + y3 and so on, are called binomial expansions.

Binomial coefficients:

Coefficients of expansion (x + y) are 1, 1.

Coefficients of expansion (x + y)2 are 1, 2, 1

Coefficients of expansion (x + y) are 1, 3, 3, 1

And so on, are called binomial coefficients.

Pascal triangle:

Binomial theorem: Let n be a positive integer and x, a be real numbers, then

(x + a) n = nC0 xn a0 + nC1xn – 1 a1 + nC2 x n – 2 a2 +… + nCr xn – r ar + … + nCn x0 an

Note: –

Let n be a positive integer and x, a be real numbers, then

(i) (x + a) n = ∑ nCr xn – r ar

(ii) The expansion of (x + a) n has (n + 1) terms.

(iii) The rth term in the expansion of (x + a) n, which is denoted by Tr, is given by Tr = nCr-1 xn – r +1 ar-1 for 1≤ r≤ n + 1.

The general term of the binomial expansion:

In the expansion of (x + a)n, the (r + 1)th term is called the general term of the binomial expansion and it is given by Tr+1 = nCr xn –r  ar for 0≤ r≤ n.

(x – a) n = nC0 xn (-a)0 + nC1xn – 1 (-a)1 + nC2 x n – 2 (-a)2 +… + nCr xn – r (-a) r + … + nCn x0 (-a) n

= nC0 xn a0nC1xn – 1 a1 + nC2 x n – 2 a2 –… +(–1) r nCr xn – r ar + … + (–1) n nCn x0 an

And the general term is Tr+1 = (–1) r nCr xn –r ar for 0≤ r≤ n.

Trinomial Expansion: Let n ∈ N and a, b, c ∈ R, then (a + b + c) n can be expand using the binomial theorem taking a as the first term and (b + c) as the second term

(a + b + c) n = (a + (b+ c)) n = ∑ nC0 an-r (b + c) r (0≤ r≤ n)

⟹ no. of terms in the expansion of (a + b + c) n =

Middle terms in (x + a) n:

∎ if n is even then  term is the middle term.

∎ if n is odd then terms are the middle terms.

Binomial coefficients: The coefficients in the binomial expansion (x + a) n are nC0, nC1, …, nCr, …, nCn these coefficients are called binomial coefficients. When n is fixed these coefficients are denoted by C0, C1, …, Cr, …, Cn. respectively.

Note:

• The binomial expansion of (1 + x) n = nC0 + nC1x + nC2 x 2 +… + nCn xn. This expansion is called the standard binomial expansion.

With the standard notation, if n is a positive integer, then

• C0 + C1 + C2 …+ Cn = 2n
• C0 + C2 + C4 …+ Cn = 2n-1 if n is even
• C0 + C2 + C4 …+ Cn-1 = 2n-1 if n is odd
• C1 + C3 + C5 …+ Cn-1 = 2n-1 if n is even
• C1 + C3 + C5 …+ Cn = 2n-1 if n is odd

Integral part and Fractional part: If x is any real number, then there exist an integer n such that n ≤ x < n+ 1. This integer n is called an integral part of the real number x and it is denoted by [x]. The real number x – [x] is called fractional part of x and it is denoted by {x}.

Numerically greatest term:  In the binomial expansion of (1 + x) n, the rth term Tr is called numerically greatest term if,

⟹ if  = p, where p is a positive integer then, pth and (p + 1)th are the numerically greatest terms.

⟹ if = p + F, where p is a positive integer and 0 < F < 1 then, (p + 1)th  is the numerically greatest term.

⟹ To find the numerically greatest term(s) in the binomial expansion of (a + x)n we write (a + x)n = an(1 + x/a)n and then find the numerically greatest term(s) by using above rules.

Largest Binomial coefficient:

The largest binomial coefficient(s) among nC0, nC1, …, nCr, …, nCn is (are)

(i) if n is even integer.

(ii) if n is an odd integer

Binomial theorem for Rational Index:

If m is a rational number and x is a real number such that – 1 < x < 1, then

Rational Index: –

## 7. PARTIAL FRACTIONS

Rational fraction:  If f(x) and g(x) are two polynomials and g(x) is a non-zero polynomial, then is called a rational fraction or polynomial fraction or simply a fraction.

Ex:

Proper and Improper Fractions: A rational fraction is called a Proper fraction if the degree of f(x) is less than the degree of g(x). Otherwise, it is called an improper fraction.

Ex: is a proper fraction and is an Improper fraction.

Irreducible Polynomial:  A polynomial f(x) is said to be irreducible if it can not be express as a product of two polynomials g(x) and h(x) such that the degree of each polynomial is less than the degree of f(x). If f(x)is not irreducible then we say that f(x) is reducible.

Ex: 3x – 1, x2 + x + 1 are irreducible polynomials.

Division Algorithm for Polynomials: If f(x) and g(x) are two polynomials with g(x) ≠ 0, then there exist unique polynomials q(x) and r(x) such that f(x) = q(x) g(x) + r(x) , where either r(x) = 0 or the degree of r(x) is less than the degree of g(x).

Partial Fraction:  If a proper fraction is expressed as the sum of two or more proper fractions, wherein the power of the denominator of irreducible polynomials, then each proper fraction in the sum is called a partial fraction of the given fraction.

Partial Fraction of  when g(x) contains linear factors:

Rule – 1:   Let  be a proper fraction. To each non- repeated factor of g(x), there will be a partial fraction of the form where A is a non-zero real number, to be determined.

Rule – 2:   Let  be a proper fraction. To each factor (ax + b)n, a ≠ 0 where ‘n’ is a positive integer, of g(x) there will be a partial fraction of the form where A1, A2, …, An are to be determined constants. Note that An ≠ 0 and Rule –1 is a particular case of Rule-2 for n = 1.

Partial Fraction of  when g(x) contains irreducible factors:

Rule – 3: Let  be a proper fraction. To each non- repeated quadratic factor (ax2 + bx + c), a ≠ 0 of g(x) there will be a partial fraction of the form where A, B are real numbers, to be determined.

Rule – 4: Let  be a proper fraction. If n (>1)∈ N is the largest exponent so that (ax2 + bx + c)n, a ≠ 0) factor of g(x) there will be a partial fraction of the form where A1, A2, …, An and B1, B2, …, Bn are real numbers, to be determined.

Partial Fraction of when  is an Improper fraction:

Case (1): If degree f(x) = degree of g(x) then by                                                                                                division algorithm there exist a unique constant k and r(x) such that f(x) = k g(x) + r(x), where either r(x) = 0 or the degree of r(x) is less than the degree of g(x) and the constant k is the quotient of the coefficient of the highest degree terms of f(x) and g(x).

can be expressed as k +   where is a proper fraction which can be resolved into a partial fraction using the above rules.

Case (2): If degree f(x) > degree of g(x) then by division algorithm can be expressed as q(x) +   where q(x) is a non-zero polynomial and is a proper fraction which can be resolved into a partial fraction using above rules.

## 8. MEASURES OF DISPERSION

The measure of dispersion: In a measure of central tendency, we have to know a measure to describe the variability. This method is called a measure of dispersion.

Measuring dispersion of a data is significant because it determines the reliability of an average by pointing out as to how far an average is representative of the entire data.

Some measures of dispersion are: (i) Range (ii) Mean deviation (iii) Standard deviation

Range:

For ungrouped data, the range is the difference between the maximum and minimum value of the series of observations.

For grouped data range is approximated as the difference between the upper limit of the largest class and the lower limit of the smallest class.

Mean deviation:

To find the dispersion of values of x from a central value ‘a’ we find the deviation about ‘a’. They are   (x – a)’s. To find the mean deviation we have to sum up all such deviations.

Mean deviation from the mean for ungrouped data:

Let x1, x2, …, xn be n observations of discrete data.

Steps for finding the Mean deviation from the mean for ungrouped data:

1. First, we have to find the mean ( ) of the n observations. Let it be ‘a’
2. Find the deviations of each xi from ‘a’, i.e., x1 – a, x2 – a, …, xn – a.
3. Find the absolute values of i.e., of these deviations by ignoring the negative sign, if any, in the deviation computed in step 2.
4. Find the arithmetic mean of the absolute values of the deviations.

M.D from the mean =

Mean deviation from the median for ungrouped data:

Let x1, x2, …, xn be n observations of discrete data.

Steps for finding the Mean deviation from the median for ungrouped data:

1. First, we have to find the median of the n observations. Let it be ‘a’
2. Find the deviations of each xi from ‘a’, i.e., x1 – a, x2 – a, …, xn – a.
3. Find the absolute values of i.e., of these deviations by ignoring the negative sign, if any, in the deviation computed in step 2.
4. Find their arithmetic mean as M.D from median =

Mean deviation for a grouped data:

A data can be arranged or grouped as a frequency distribution in two ways: (i) Discrete frequency distribution and (ii) Continuous frequency distribution.

(i) Discrete frequency distribution:

If x1, x2, …, xn are of ‘n’ observations occurring with frequencies f1, f2, …, fn Then we can represent this data in the following manner:

 xi x1 x2 x3 … xn fi f1 f2 f3 … fn

This form is called the discrete frequency distribution

Mean deviation about the mean and median:

Mean =

M.D(mean) =

M.D(median) =

Where N is total frequency.

(ii) Continuous frequency distribution: –

Continuous distribution is a series in which the data is classified into different class-intervals along with their respective frequencies.

Mean deviation about the means and median:

Mean =

M.D(mean) =

Median =

M.D(median) =

Where N is total frequency.

Step- Deviation method: If the midpoints of the class intervals xi as well as their associated frequencies are very large then we use this method.

Arithmetic mean =

Where

Variance and Standard Deviation of un grouped data:

If x1, x2, …, xn are n observations and is their mean, then

We have the following cases:

Case(i): if = 0, then each= 0 which implies all observations are equal to the mean  and there is no dispersion.

Case(ii): if  is small, then it indicates that each observation xi is very close to the mean  and hence the degree of dispersion is low.

Case(iii): if  is larger, then it indicates the higher degree of dispersion of the observations from the mean .

Variance = σ2=

Standard deviation

∎ The coefficient of variation of a distribution (C.V.) =

## 9. PROBABILITY

Random Experiment: If the result of an experiment is not certain and is any one of the several possible outcomes, then the experiment is called ‘random experiment.

Sample space: The set of all possible outcomes of an experiment is called ample space when ever the experiment conducted and is denoted by ‘S’.

Event: Any subset of the sample space is called an event.

Complimentary of an event: The complementary of an event E , is denoted by Ec , is the event given by Ec = S – E which is called the complimentary event of E.

Equally likely events: two events are said to be equally likely events when chance of occurrence of one event is equal to that of other.

Exhaustive events: A set of events is said to be exhaustive if the performance of the experiment always result in the occurrence of the at least one of them.

The events E1, E2, …, En are said to be exhaustive if E1∪ E2∪…, ∪ En = S.

Mutually Exclusive events: A set of events is said to be mutually exclusive if happening of one of them prevents the happening of any one of remaining events.

The events E1, E2, …, En are said to be exhaustive if Ei ∩ Ej =∅ for i ≠ j, 1 ≤i, j≤ n.

Classical definition of Probability: In a random experiment, let there be n mutually exclusive, exhaustive and equally likely events.

E be the event of the experiment. ‘m’ elementary events are favourable to an event E, then the probability of E is defined as P (E) =

For any event E, 0 ≤ P(E) ≤1.

∎ If Ec is the non-occurrence of E, then the probability of on-occurrence of E is P (Ec)

P (Ec) = 1 – P (E) ⇒ P (Ec) + P (E) = 1

Limitations of the Classical definition of the probability:

1. If the out comes of the random experiment are not equally likely, then the probability of an event in such experiment is not defined.
2. If the random experiment contains infinitely many out comes, then his definition cannot be applied to find the probability of an event in such an experiment.

Relative frequency (Statistical or Emperical) definition probability:

Suppose a random experiment is repeated n times, out of which an event E occurs m(n) times, then the ratio  is called the nth relative frequency of the event E.

Let r1, r2, …, rn be the sequence. If rn tends to a definite limit, , l is defined to be the probability of the event E and we write P (E) =

Deficiencies of the relative frequency definition of probability:

1. Repeating a random experiment infinitely many times is practically impossible.
2. The sequence of relative frequencies is assumed to tend to a definite limit, which may not exist.
3. The values r1, r2, …, rn are not real variables. Therefore, it is not possible to prove the existence and the uniqueness of the limit of rn as n → ∞, by applying methods used in calculus.

Probability Function:

Let S be the sample space of a random experiment, which is finite. Then a function P: S → R satisfying the following axioms is called a Probability function.

(i) P (E) ≥ 0 ∀ E ∈ S (axiom of non-negativity)

(ii) P (S) = 1 (axiom of certainty).

(iii) If E1, E2 ∈ S and E1 ∩ E2 =∅, then P (E1 ∪ E2) = P (E1) + P (E2) (axiom of additivity).

For each E ∈ S, the real number P (E) s called the probability of the event E. If E = {a}, then we write p(a) instead of P ({a}).

Note:

1. P (∅) = 0 for any sample space S, S ∅ = S and S∩ ∅ = ∅. P (S) = (S ∅) = P (S) + P (∅) = P (S) (∵ P (∅) = 0).
1. If S is countably infinite, then axiom (iii) of the above definition is to be replaced by (iii)*: if is a sequence of pairwise mutually exclusive events, then
2. Suppose S be a sample space of a random experiment. Let P be a probability function. If E1, E2, …, En are finitely many pairwise mutually exclusive events, then P (E1∪ E2∪…, ∪ En) = P (E1) + P (E2) + … + P (En)

⟹ If E1, E2 are any two events in a sample space S, then
If E1, E2 are any two events in a sample space S and P is a probability function, then P (E1∪ E2) = P (E1) + P (E2) – P (E1∩ E2)

P (E1 – E2) = P (E1) – P (E1∩ E2)

P (E2 – E1) = P (E2) – P (E1∩ E2)

Set – theoretic descriptions:

 Event Set-theoretic description Event A or Event B to occur A∪B Both event A and B occur A∩B Neither A nor B occur (A∪B) c = Ac ∩ Bc A occurs but B does not occur A ∩ Bc or A\B Exactly one of the event A, B to occur (A∩B) c ∪ (Ac ∩ B) or (A – B) ∪ (B – A)                 or (A∪B) – (A ∩ B) Not more than one of the events A, B occurs (A∩B) c ∪ (Ac ∩ B) ∪ (Ac ∩ Bc) Event B occurs whenever event A occurs A ⊆ B

Conditional event: If A, B are two events of random experiment, then the event of happening (occurring) B after the event A happens(occurs) is called conditional event. It is denoted by B\A.

Conditional probability:  If A, B are two events and P (A) ≠ 0, then the probability of B after the event A has occurred is called conditional probability. It is denoted by P (B/A) and is defined by                  P(B/A) =

Multiplication theorem of probability: If A, B are two events of random experiment with P (A) > 0 and P (B) > 0, then P (A∩B) = P (A) P (B/A) = P (B) P (A/B).

Two events A and B said to be independent if P (B/A) = P(B) or P (A/B) = P (A)

Two events A and B said to be independent if P (A∩B) = P (A). P (B)

The events A1, A2, …, An are   said to be independent if P (A1∩ A2∩ …∩ An) = P (A1). PA2). …. P(An).

Bayes Theorem:

If A1, A2, …, An are mutually exclusive and exhaustive events in a sample space S such that P (Ai) > 0 for i = 1, 2, 3, …, n and E is an event with P (E) > 0, then

## 10. RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Random variable: Let S be a sample space of a random experiment. A real valued function X: S → R is called random variable.

∎ A set A is said to be countable if there exist a bijection from A into a subset of N.

Probability distribution Function:  If X: S → R is a random variable connected with a random experiment and P is a probability function associate with it. The unction F: R → R defined by F(x) = P (X ≤ x) is called probability distribution function of the random variable X.

Discrete or discontinuous Random variable:   Let S be a sample space, a random variable X: S → R is said to be Discrete or discontinuous if the range of X is countable.

I.POBABILITY DISTRIBUTION:

If X: S → R is a discrete random variable with range {x1, x2, x3………}, then {P(X = xr; r = 1, 2, 3, …, n} is called probability distribution of X.

The table for the probability distribution of the discrete random variable X is:

 X = xi x1 x2 x3 … xn P (X = xi) P (x1) P (x2) P (x3) … P (xn)

Mean (μ) =∑xi P (X = xi)

Variance = σ2 = ∑ (xi – μ)2 P (X = xi) = ∑ xi2 P (X = xi) – μ2

Standard deviation is σ.

II.BINOMIAL or BERNOULLI DISTRIBUTION:

Let n be a positive integer and p be the random number such that 0 < p < 1. A random variable X with range {0, 1, 2, 3, …, n} is said to have a Binomial distribution with parameters n and p, if

P (X = x) = nCx px qn – x for x = 0, 1, 2, …, n and q = 1 – p.

Mean = np

Variance = npq.

III. POISSON DISTRIBUTION:

Let λ > 0 be a real number. A random variable X with range {1, 2, 3, …, n} is said to be Poisson distribution with parameter λ if,

Mean = λ

Variance = λ

Poisson distribution as a limiting form of Binomial distribution:

Poisson distribution can be derived as the limiting case of binomial distribution in the following case.

If λ > 0 for each positive integer n > λ, let Xn be the Binomial variable B (n, λ/n). using the fact that we can prove that for every non-negative integer k,

# These notes cover all the topics covered in the TS I.P.E  first year maths 1B 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.

## 0.COORDINATE GEOMETRY( BASICS)

• Distance between two points A(x1, y1), B(x2, y2) is

• distance between a point A(x1, y1) to the origin is

• The midpoint of two points A(x1, y1), B(x2, y2) is

•     If P divides the line segment joining the points A(x1, y1), B(x2, y2) in the ratio m:n then the coordinates of P are

• Area of the triangle formed by the vertices A (x1, y1), B (x2, y2) and C (x3, y3) is

## 1. LOCUS

Locus: The set of points that are satisfying a given condition or property is called the locus of the point.

Ex:- If a point P is equidistant from the points A and B, then AP =BP

Ex 2: – set of points that are at a constant distance from a fixed point.

here the locus of a point is a circle.

• In a right-angled triangle PAB, the right angle at P and P is the locus of the point, then

AB2 = PA2 + PB2

•Area of the triangle formed by the vertices A (x1, y1), B (x2, y2), and C (x3, y3) is

## 2.CHANGE OF AXES

Transformation of axes:

When  the origin is shifted to  (h, k), without changing the direction of axes then

•To remove the first degree terms of the equation ax2  + 2hxy + by2 +2gx +2fy+ c = 0, origin should be shifted to the point

•If the equation ax2 + by2 +2gx +2fy+ c = 0, origin should be shifted to the point

Rotation of axes:

When the  axes are rotated through an angle θ then

•To remove the xy term of the equation ax2 + 2hxy + by2  = 0, axes should be rotated through an angle θ is given by

## 3.STRAIGHT LINES

Slope:-  A-line makes an angle θ with the positive direction of the X-axis, then tan θ is called the slope of the line.

It is denoted by “m”.

m= tan θ

• The slope of the x-axis is zero.

• Slope of any line parallel to the x-axis is zero.

• The y-axis slope is undefined.

• The slope of any line parallel to the y-axis is also undefined.

• The slope of the line joining the points A (x1, y1) and B (x2, y2) is

Slope of the line ax + by + c = 0 is

### Types of the equation of a straight line:

• Equation of x- axis is y = 0.
• Equation of any line parallel to the x-axis is y = k, where k is the 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 the distance from the 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

• The equation of the line ∥ el    to ax +by + c = 0 is ax +by + k = 0.

• The equation of the line ⊥ler   to ax +by + c = 0 is bx −ay + k = 0.

Note: –

1. If two lines are parallel then their slopes are equal

m1 = m2

1. If two lines are perpendicular then product of their slopes is – 1

m1 × m2 = – 1

1. The area of the triangle formed by the line ax + by + c = 0 with the coordinate axes is
2. The area of the triangle formed by the line   with the coordinate axes is

Perpendicular distance (Length of the perpendicular):

The perpendicular distance from a point P (x1, y1) to the line ax + by + c = 0 is

• The perpendicular distance from origin to the line ax + by + c = 0 is

Distance between two parallel lines:

•The distance between the parallel lines ax1 + by1 + c1 = 0 and ax2 + by2 + c2 = 0 is

Perpendicular form or Normal form:

The equation of the line which is at a distance of ‘p’ from the origin and α (0≤ α ≤ 3600) is the angle made by the perpendicular with the positive direction of the x-axis is x cosα + y sinα = p.

Symmetric form:

The equation of the line passing through point P (x1, y1) and having inclination θ is

Parametric form:

if P (x, y) is any point on the line passing through A (x1, y1) and

making inclination θ, then

x = x1 + r cos θ, y = y1 + r sin θ

where ‘r’  is the distance from P to A.

• The ratio in which the line L ≡ ax + by + c = 0 divide the line segment joining the points A (x1, y1), B (x2, y2) is – L11: L22.

Where L11 = ax1 + by1 + c and L22 = ax2 + by2 + c.

Note: – the points A (x1, y1), B (x2, y2) lie on the same side or opposite side of line L = 0 according to L11 and L22 have the same sign or opposite sign.

∗  x-axis divides the line segment joining the points A (x1, y1), B (x2, y2) in the ratio – y1: y2.

∗  y-axis divides the line segment joining the points A (x1, y1), B (x2, y2) in the ratio – x1: x2.

Point of intersection of two lines:

the point of intersection of two lines a1x + b1y + c = 0 and a2x + b2y + c = 0 is

#### Concurrent Lines:

Three or more lines are said to be concurrent lines if they have a point in common.

The common point is called the point of concurrence.

∗  The condition that the lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 and a3x + b3y + c3 = 0 to be concurrent is

a3(b1c2 – b2c1) + b3(c1a2 – c2a1) + c3(a1b2 – a2b1).

∗ The condition that the lines ax + hy +g = 0, hx + by + f = 0 and gx +fy + c = 0 is

abc + 2fgh – af2 – bg2 – ch2 = o.

Note: – if two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 said to be identical (same) if

#### Family of a straight line:

Family of straight lines: – A set of straight lines having a common property is called a family of straight lines.

Let L1 ≡ a1x + b1y + c1 = 0 and L2 ≡ a2x + b2y + c2 =0 represent two intersecting lines, theThe equation λ1 L1 + λ2 L2 = 0 represent a family of straight lines passing through the point of intersection of the lines L1 = 0 and L2 = 0.

∗  The equation of the straight line passing through the point of intersection of the lines L1 = 0 and L2 = 0 is L1 + λL2 = 0.

The angle between two lines:

If θ is the angle between the lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 then

∗  If θ is an acute angle then

∗ If θ is the angle between two lines, then (π – θ) is another angle between two lines.

∗ If θ≠π/2 is angle between the lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, then

∗ If m1, m2 are the slopes of two lines then

Note: – The lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are

∗ Parallel iff

∗ Perpendicular iff a1a2 + b1b2 = 0

The foot of the perpendicular:

If Q (h. k) is the foot of the perpendicular from a point P (x1, y1)to the line ax + by +c = 0 then

Image of the point:

If Q (h. k) is the image of point P (x1, y1)   with respect to the line ax + by +c = 0 then

Collinear Points:

If three points are said to be collinear, then they lie on the same line.

∗ If A, B, and C are collinear, then

Slope of AB = Slope of BC (or) Slope of BC = Slope of AC (or) Slope of AB = Slope of AC

## 4. PAIR OF STRAIGHT LINES

∎ ax2 + 2hxy + by2 = 0 is called the second-degree homogeneous equation in two variable x and y.

This equation always represents a pair of straight lines which are passing through the origin.

∎ If l1x + m1y = 0 and l2x + m2y = 0 are two lines represented by the equation ax2 + 2hxy + by2 = 0, then ax2 + 2hxy + by2 = (l1x + m1y) (l2x + m2y)

⇒ a = l1l2; 2h = l1m2 + l2m1; b = m1m2

∎ If m1, m2 are the slopes of the lines represented by the equation ax2 + 2hxy + by2 = 0, then

m1+ m2 = – 2h/b and m1 m2 = a/b

∎ The lines represented by the equation ax2 + 2hxy + by2 = 0 are

∎ If h2 = ab, then the lines represented by the equation ax2 + 2hxy + by2 = 0 are coincident.

∎ If two lines represented by the equation ax2 + 2hxy + by2 = 0 are equally inclined to the coordinate axes then h = 0 and ab < 0.

∎ The equation of the pair of lines passing through the point (h, k) and

(i) Parallel to the lines represented by the equation ax2 + 2hxy + by2 = 0 is

a (x – h)2 + 2h (x – h) (y – k) + b (y – k)2 = 0

(ii) Perpendicular to the lines represented by the equation ax2 + 2hxy + by2 = 0 is

b (x – h)2 – 2h (x – h) (y – k) + a (y – k)2 = 0

Angle between the lines:

If θ is the angle between the lines represented by the equation ax2 + 2hxy + by2 = 0, then

∎ If a + b = 0, then two lines are perpendicular.

Area of the triangle:

The area of the triangle formed by the lines ax2 + 2hxy + by2 = 0 and the line lx + my + n = 0 is

Angular Bisectors:

⇒ the angle between angular bisectors is always 900
L1 = o, L2 = o are two non-parallel lines the locus of the point P such that the perpendicular distance from P to the first lie is equal to the perpendicular distance from P to second line is called the angular bisector of two lines.

⇒ If two lines are a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, then the angular bisectors are

∎ The equation of the pair of angular bisectors of ax2 + 2hxy + by2 = 0 is h (x2 – y2) = (a – b =) xy.

∎ If ax2 + 2hxy + by2 + 2gx 2fy + c= 0 represents a pair of straight lines then

(i) abc + 2fgh – af2 – bg2 – ch2 = 0

(ii) h2 ≥ ab, g2 ≥ ac and f2 ≥ bc

If two lines represented by ax2 + 2hxy + by2 + 2gx 2fy + c= 0 are l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0, then

ax2 + 2hxy + by2 + 2gx 2fy + c = (l1x + m1y + n1) (l2x + m2y + n2)

a = l1l2; 2h = l1m2 + l2m1; b = m1m2 ; 2g = l1n2 + l2n1;  2f =  m1n2 + m2n1 and c = n1n2

The point of intersection of the lines represented by ax2 + 2hxy + by2 + 2gx 2fy + c= 0 is

∎ If two line have same homogeneous path then the lines represented by the first pair is parallel to the lines represented by the second pair.

ax2 + 2hxy + by2 + 2gx 2fy + c= 0 …………… (2)

ax2 + 2hxy + by2 = 0 …………… (1)

equation (1) and equation (2) form a parallelogram, one of the diagonals of parallelogram which is not passing through origin is 2gx + 2fy + c = 0.

∎ If two lines represented by ax2 + 2hxy + by2 + 2gx 2fy + c= 0 are parallel then

• h2 = ab (ii) af2 = bg2 (iii) hf = bg, gh = ab

Distance between parallel lines is

## 5. THREE DIMENSIONAL COORDINATES

• Let X’OX, Y’OY be two mutually perpendicular lines passing through a fixed point ‘O’. These two lines determine the XOY – plane (XY- plane). Draw the line Z’OZ perpendicular to XY – plane and passing through ‘O’.

The fixed point ‘O’ is called origin and three mutually perpendicular lines X’OX, Y’OY, Z’OZ are called Rectangular coordinate axes.

Three coordinate axes taken two at a time determine three planes namely XOY- plane, YOZ-plane, ZOY-plane or XY-plane, YZ-plane, ZX-plane respectively.

For every point P in space, we can associate an ordered triad (x, y, z) of real numbers formed by its coordinates.

The set of points in space is referred to as ‘Three-Dimensional Space’ or R3 – Space.

∗ If P (x, y, z) is a point in a space, then

x is called x-coordinate of P

y is called y-coordinate of P

z is called z-coordinate of P

Distance between two points in space:

∗ Distance between the points A (x1, y1, z1) and B (x2, y2, z2) is

∗ Distance between the point P (x, y, z) to the origin is

Translation of axes:

When the origin is shifted to the point (h, k, l), then

X = x – h; Y = y – k; Z = z – l and x = X + h; y = Y + k; z = Z + l

∗ The foot of the perpendicular from P (x, y, z) to X-axis is A (x, 0, 0).

The perpendicular distance of P from X-axis is

Similarly,

The perpendicular distance of P from Y-axis is

The perpendicular distance of P from Z-axis is

Collinear points: If three or more points lie on the same line are called collinear points.

Section formula:

The point dividing the line segment joining the points A (x1, y1, z1) and B (x2, y2, z2) in the ratio m : n is given by

The mid-point of the line segment joining the points A (x1, y1, z1) and B (x2, y2, z2) is

The centroid of the triangle whose vertices are A (x1, y1, z1), B (x2, y2, z2) and C (x3, y3, z3) is

Tetrahedron:

→ It has 4 vertices and 6 edges.
→ A Tetrahedron is a closed figure formed by four planes not all passing through the same point.

→ Each edge arises as the line of intersection of two of the four planes.

→ The line segment joining the vertices to the centroid of opposite face. The point of concurrence is        called centroid of Tetrahedron.

→ Centroid divides the line segment in the ratio 3:1.

→ The centroid of the Tetrahedron whose vertices are A (x1, y1, z1), B (x2, y2, z2), C (x3, y3, z3) and C (x4, y4, z4) is

The line segment joining the points (x1, y1, z1), (x2, y2, z2) is divided by

XY – plane in the ratio – z1: z2

YZ – plane in the ratio – x1: x2

XZ – plane in the ratio – y1: y2

## 6. DIRECTION COSINES AND RATIOS

Consider a ray OP passing through origin ‘O’ and making angles α, β, γ respectively with the positive direction of X, Y, Z axes.

Cos α, Cos β, Cos γ are called Direction Cosines (dc’s) of the ray OP.

Dc’s are denoted by (l, m, n), where l = Cos α, m = Cos β, n = Cos γ

• A line in a space has two directions, it has two sets of dc’s, one for each direction. If (l, m, n) is one set of dc’s, then (-l, -m, -n) is the other set.

• Suppose P (x, y, z) is any point in space such that OP = r. If (l, m, n) are dc’s of a ray OP then x = lr, y= mr, z = nr.

• If OP = r and dc’s of OP are (l, m, n) then the coordinates of P are (lr, mr, nr).

• If P (x, y, z) is a point in the space, then dc’s of OP are

• If (l, m, n) are dc’s of a line then l2 + m2 + n2 = 1.

⇒ cos2α + cos2β + cos2γ = 1.

Direction Ratios:

Any three real numbers which are proportional to the dc’s of a line are called direction ratios (dr’s) of that line.

• Let (a, b, c) be dr’s of a line whose dc’s are (l, m, n). Then (a, b, c) are proportional to (l, m, n)

and a2 + b2 + c2 ≠ 1.

• Dr’s of the line joining the points (x1, y1, z1), (x2, y2, z2) are (x2 – x1, y2 – y1, z2 – z1)

• If (a, b, c) are dr’s of a line then its dc’s are

• If (l1, m1, n1), (l2, m2, n2) are dc’s of two lines and θ is angle between them then Cos θ = l1l2 + m1m2 + n1n2

If two lines perpendicular then l1l2 + m1m2 + n1n2 = 0.

• If (a1, b1, c1), (a2, b2, c2) are dr’s of two lines and θ is the angle between them then

If two line are perpendicular then a1a2 + b1b2 + c1c2 = 0.

## 7. THE PLANE

Plane: A plane is a proper subset of R3 which has at least three non-collinear points and any two points in it.

∎ Equation of the plane passing through a given point A (x1, y1, z1), and perpendicular to the line whose dr’s (a, b, c) is a(x – x1) + a(y – y1)  + a(z – z1) = 0.

∎ The equation of the plane hose dc’s of the normal to the plane (l, m, n) and perpendicular distance from the origin to the pane p is lx + my + nz = p

∎ The equation of the plane passing through three non-collinear points A (x1, y1, z1), B (x2, y2, z2) and C (x3, y3, z3) is

∎ The general equation of the plane is ax + by + cz + d = 0, where (a, b, c) are Dr’s of the normal to the plane.

Normal form:

The equation of the plane ax + by + cz + d = 0 in the normal form is

Perpendicular distance:

The perpendicular distance from (x1, y1, z1) to the plane ax + by + cz + d = 0 is

The perpendicular distance from the origin to the plane ax + by + cz + d = 0 is

Intercepts:

X- intercept = aIf a plane cuts X –axis at (a, 0, 0), Y-axis at (0, b, 0) and Z-axis at (0, 0, c) then

Y-intercept = b

Z-intercept = c

The equation of the plane in the intercept form is

The intercepts of the plane ax + by + cz + d =0 is -d/a, -d/b, -d/c

∎ The equation of the plane parallel to ax + by + cz + d = 0 is ax + by + cz + k = 0.

∎ The equation of XY – plane is z = 0.

∎ The equation of YZ – plane is x = 0.

∎ The equation of XZ – plane is y = 0.

∎ Distance between the two parallel planes ax + by + cz + d1 =0 and ax + by + cz + d2 =0 is

The angle between two planes:

The angle between the normal to two planes is called the angle between the planes.

If θ is the angle between the planes a1 x + b1 y + c1 z + d1 =0 and a2 x + b2 y + c2 z + d2 =0 then

If two line are perpendicular then a1a2 + b1b2 + c1c2 = 0.

∎ The distance of the point P (x, y, z) from

## 8. LIMITS AND CONTINUITY

Intervals:

Let (a, b) ∈ R such that a ≤ b, then the set

• {x ∈ R: a ≤ x ≤ b}, is denoted by [a, b] and it is called as closed interval
• {x ∈ R: a < x < b}, is denoted by (a, b) and it is called as open interval
• {x ∈ R: a < x ≤ b}, is denoted by (a, b] and it is called as open closed interval
• {x ∈ R: a ≤ x < b}, is denoted by [a, b) and it is called as closed open interval
• {x ∈ R: x ≥ a}, is denoted by [a, ∞)
• {x ∈ R: x > a}, is denoted by (a, ∞)
• {x ∈ R: x ≤ a}, is denoted by (- ∞, a]
• x ∈ R: x < a}, is denoted by (- ∞, a)

Neighbourhood:
Let a ∈ R. If δ > 0, then the open interval (a – δ, a + δ) is called the δ – neighbourhood of ‘a’

Limit:

If f(x) is a function of x such that if x approaches to a constant value ‘a’, then the value of f(x) also approaches to ‘l’. Then the constant ‘I’ is called a limit of f(x) at x = a

Or

A real number l is called the limit of the function f, if for all ϵ> 0 there exist δ > 0 such that    whenever  ⟹

Properties of Limits:

Sand witch theorem:( Squeez Principle):

f, g, and h are functions such that f(x) ≤ g(x) ≤ h(x), then  and if

Left- hand and Right-hand Limits:

If x < a, then  is called left-hand limit

If x > a, then  is called right-hand limit

Note:

In Determinate forms:

if a function f(x) any of the following forms at x = a:

Then f(x) is said to be indeterminate at x = a.

### Continuity:

Condition 1:  If the condition is like x = a and x ≠ a, then we use following property.

If then f(x) is continuous at x = a, otherwise f(x) is not continuous.

Condition 2: If the condition is like x ≤ a and x >a, or x < a and x ≥a then we use following property.

If     then f(x) is continuous at x = a, otherwise f(x) is not continuous.

## 9.DIFFERENTIATION

Let f be a function defined on a neighbourhood of a real number ‘a’ if exist then we say that f is differentiable at x a and it is denoted by f'(a).

∴ f’(a) =

∎ If right hand derivative = left hand derivative, then f is differentiable at ‘a’.

i.e.,

First principle in derivative:

The first principle of the derivative of f at any real number ‘x’ is f’(x) =

∎ The differentiation of f(x) is denoted by

means differentiation of ‘y’ with respect to ‘x’

The derivative of constant function is zero i.e., f’(a) = 0 where ‘a’ is any constant.

∎ Let I be an interval in R u and v are real valued functions on I and x ∈ I. Suppose that u and v are differentiable at ‘x’, then

• (u ± v) is also differentiable at ‘x’ and (u ± v)’(x) = u’ (x) ± v’(x).
• ‘uv’ is also differentiable at ‘x’ and (uv)’(x) = u(x) v’(x) + v(x) u’(x).
• αu + βv is also differentiable at ‘x’ and (αu + βv)’(x) = αu’(x) + βv’(x), α, β are constants.
• is also differentiable at ‘x’ and

∎ (f o g)’ (x) = f’(g(x)). g’(x).

Formulae:

Derivative of Trigonometric & Inverse trigonometric functions:

Derivative of Hyperbolic & Inverse Hyperbolic functions:

Parametric Differentiation:

If x = f(t) and y = g(t) then the procedure of finding  in terms of the parameter ‘t’ is called parametric equations.

Implicitly differentiation:

An equation involving two or more variables is called an implicit equation.

ax2 + 2hxy + b y2 = 0 is an implicit equation in terms of x and y.

The process of finding    from an implicit equation is called implicitly differentiation.

Derivative of one function w.r.t.  another function:

The derivative of f(x) w.r.t g(x) is

Second order derivative:

Let y = f(x) be a function, if y is differentiable then the derivative of f is f’(x). If ‘(x) is again differentiable then the derivative of f’(x) is called second order derivative. And it is denoted by f” (x) or

## 10. ERRORS AND APPROXIMATIONS

Approximations:

Let y f(x) be a function defined an interval I and x ∈ I. If ∆x is any change in x, then ∆y be the corresponding change in y thus ∆y = f (x + ∆x) – f (x).

Let

where ϵ is very small

For ‘ϵ.∆x’ is very small and hence,

Approximate value is f (x + ∆x) = f(x) + f’(x). ∆x

Differential:

Let y f(x) be a function defined an interval I and x ∈ I. If ∆x is any change in x, then called differential of y = f(x) and it is denoted by df.

∴ dy = f’(x). ∆x

Errors:

Let y f(x) be a function defined an interval I and x ∈ I. If ∆x is any change in x, then ∆y be the corresponding change in y.

### The Following formulae will be used in Solving problems

CIRCLE:

If ‘r’ is radius, ‘d’ is diameter ‘P’ is the perimeter or circumference and A is area of the circle then

d= 2r, P = 2πr = πd and A = πr2sq.u

SECTOR:

If ‘r’ is the radius, ‘l’ is the length of arc and θ is of the sector then

Area = ½ l r = ½ r2θsq.u.

Perimeter = l + 2r = r (θ + 2) u.

CYLINDER:

Length of the Arc ‘l’ = rθ (θ must be in radians).

If ‘r is the radius of the base of cylinder and ‘h’ is the height of the cylinder, then

Area of base = πr2 sq.units.

Lateral surface area = 2πrh units.

Total surface area = 2πr (h + r) units.

Volume = πr2 h cubic units.

CONE:

If ‘r’ is the radius of base, ‘h’ is the height of cone and ‘l’ is slant height then

l 2 + r2 = h2

Lateral surface area = πrl units.

Total surface area = πr (l + r) sq. units.

Volume =  cubic units.

SPHERE:

If ‘r’ is the radius of the Sphere then

Surface area = πr2 sq. units.

Volume =  πr3 cubic units.

## 11. TANGENTS AND NORMALS

Tangent of a Curve:
If the secant line PQ approaches to the same position as Q moves along the curve and approaches to either side then limiting position is called a ‘Tangent line’ to the curve at P. The point P is called point of contact

Let y = f(x) be a curve, P a point on the curve. If Q(≠P) is another point on the curve then the line PD is called secant line.

Let y = f(x) be a curve and P (x, y) be a point on the curve. The slope of the tangent to the curve y = f(x) at P is called gradient of the curve.

Slope of the tangent to the curve y = f(x) at P (x, y) is m =

∎ The equation of the tangent at P (x1, y1) to the curve is y – y1 = m (x – x1) where m =

Normal of a curve:

Let y = f(x) be a curve and P (x, y) be a point on the curve. The line passing through P and perpendicular to the tangent of the curve y = f(x) at P is called Normal of the curve.

∎ The equation of the tangent at P (x1, y1) to the curve is y – y1 = -1/m (x – x1).
Slope of the normal is -1/m. where m =

Lengths of tangent, normal, subtangent and subnormal:

PT → Normal; QN → subnormal
PN → Tangent; QT → subtangent

∎ if m =  then

Angle between two curves:

If two curves intersect at a point P., then the angle between the tangents of the curves at P is called the angle between the curves at P.

∎ If m1, m2 are the slopes of two tangents of the two curves and θ is the angle between the curves then

Tanθ =

Note:

• If m1= m2, then two corves are touch each other.
• if m1× m2 = –1, then two curves intersect orthogonally.

## 12. RATE MEASURE

Average rate of change:

if y = f(x) then the average rate of change in y between x = x1 and x = x2 is defined as

Instantaneous rate of change:

if y = f(x), then the instantaneous rate of change of a unction f at x = x0 is defined as

Rectilinear Motion:

A motion of a particle in a line is called Rectilinear motion. The rectilinear motion is denoted by s = f(t) where f(t) is the rule connecting ‘s’ and ‘t’.

Velocity, Acceleration:

A particle starts from a fixed point and moves a distance ‘S’ along a straight-line during time ‘t’ then

Velocity =

Acceleration =

Note:

(i) If v> 0, then the particle s moving away from the straight point.

(ii) If v < 0, then particle s moving away towards the straight point.

(iii) If v = 0, then the particle comes rest.

## 13.ROLLE’S & LANGRANGEE’S THEOREM

Rolle’s Theorem:

Suppose a, b (a < b) are two real numbers. Let f: [a, b] → R be a function satisfying the following conditions:

(i) f is continuous on [a, b]

(ii) f is differentiable on (a, b) and

(iii) f(a) = f(b)

then there exists at least one c ∈ (a, b) such that f’(c)= 0.

Lagrange’s Theorem:

Suppose a, b (a < b) are two real numbers. Let f: [a, b] → R be a function satisfying the following conditions:

(i) f is continuous on [a, b]

(ii) f is differentiable on (a, b) and

then there exists at least one c ∈ (a, b) such that f’(c)=

## 14.INCREASING & DECREASING FUNCTIONS

Let f be a real function on an interval I then f is said to be

(i) an increasing function on I if

x1 < x2 ⇒ f (x1) ≤ f (x2) ∀ x1, x2 ∈ I

(ii) decreasing function on I if

x1 < x2 ⇒ f (x1) ≥ f (x2) ∀ x1, x2 ∈ I

Let f be a real function on an interval I then f is said to be

(i) strictly increasing function on I if

x1 < x2 ⇒ f (x1) < f (x2) ∀ x1, x2 ∈ I

(ii) strictly decreasing function on I if

x1 < x2 ⇒ f (x1) > f (x2) ∀ x1, x2 ∈ I

Let f(x) be a real valued function defined on I = (a, b) or [a, b) or (a, b] or [a, b]. Suppose f is continuous on I and differentiable in (a, b). If

(i) f’ (c) > 0 ∀ c ∈ (a, b), then f is strictly increasing on I

(ii) f’ (c) < 0 ∀ c ∈ (a, b), then f is strictly decreasing on I

(iii) f’ (c) ≥ 0 ∀ c ∈ (a, b), then f is increasing on I

(iv) f’ (c) ≤ 0 ∀ c ∈ (a, b), then f is decreasing on I

Critical point:

A point x = c in the domain of the function said to be ‘critical point’ of the function f if either f’ (c) = 0 or f’ (c) does not exists.

Stationary point:

A point x = c in the domain of the function said to be ‘stationary point’ of the function f if  f’ (c) = 0.

MAXIMA & MINIMA

Global maxima – Global minima:

Let D be an interval in R and f: D → R be a real function and c ∈ D. Then f is said to be

(i) a global maximum on D if f(c) ≥ f(x)

(ii) a global minimum on D if f(c) ≤ f(x)

Relative maximum:

Let D be an interval in R and f: D → R be a real function and c ∈ D. Then f is said to be relative maximum at c if there exist δ > 0 such that f(c) ≥ f(x) ∀ x ∈ (c – δ, c + δ).

Here, f (c) is called relative maximum value of f(x) at x = c and the point x = c is called point of relative maximum.

Relative minimum:

Let D be an interval in R and f: D → R be a real function and c ∈ D. Then f is said to be relative maximum at c if there exist δ > 0 such that f(c) ≤ f(x) ∀ x ∈ (c – δ, c + δ).

Here, f (c) is called relative maximum value of f(x) at x = c and the point x = c is called point of relative minimum.

The relative maximum and minimum value of f are called extreme values.

If f is either minima or maxima f’ (α) = 0.

Let f be a continuous function om [a, b] and α ∈ (a, b)

(i) If f’ (α) = 0 and f’’ (α) >0, then f(α) is relative minimum.

(ii) if f’ (α) = 0 and f’’ (α) <0, then f(α) is relative maximum

## 1.Functions

Set: A collection of well-defined objects is called a set.

Ordered pair: Two elements a and b listed in a specific order form. An ordered pair denoted by (a, b).

Cartesian product: Let A and B are two non-empty sets. The Cartesian product of A and B is denoted by A × B and is defined as a set of all ordered pairs (a, b) where a ϵ A and b ϵB

Relation: Let A and B are two non-empty sets the relation R from A to B is a subset of A×B.

⇒ R: A→B is a relation if  R⊂ A × B

#### Function:

A relation f: A → B is said to be a function if ∀ aϵ A there exists a unique element b such that (a, b) ϵ f.                                            (Or)

A relation f: A → B is said to be a function if

(i) x ϵ A ⇒ f(x) ϵ B

(ii)  x1 , x2 ϵ A , x1 = x2 in A  ⇒ f(x1) = f(x2) in B.

Note:   If A, B are two finite sets then the no. of   functions that can be defined from A to B is  n(B)n(A)

VARIOUS TYPES OF FUNCTIONS

One– one Function (Injective):- A function f: A→ B is said to be a one-one function or injective if different elements in A have different images in B.

(Or)

A function f: A→ B is said to be one-one function if f(x1) = f(x2) in B ⇒ x1 = x2 in A.

Note: No. of one-one functions that can be defined from A into B is n(B) p n(A)   if  n(A) ≤ n(B)

On to Function (Surjection): – A function f: A→ B is said to be onto function or surjection if for each yϵ B ∃ x ϵ A such that f(x) =y

Note: if n(A) = m and n(B) = 2 then no. of onto functions = 2m – 2

Bijection: – A function f: A→ B is said to be Bijection if it is both ‘one-one and ‘onto’.

Constant function:  A function f: A→ B is said to be constant function if f(x) = k ∀ xϵA

Identity function:  Let A be a non-empty set, then the function defined by I: A → A, I(x)=x is called identity function on A.

Equal function:  Two functions f and g are said to be equal if

(i)   They have same domain (D)

(ii)  f(x) = g(x) ∀ xϵ D

Even function:  A function f: A→ B is said to be even function if f (- x) = f(x) ∀ xϵ A

Odd function:   A function f: A→ B is said to be odd function if f (- x) = – f(x) ∀ xϵ A

Composite function:  If f: A→B, g: B→C are two functions then the composite relation is a function from A to C.

gof: A→C is a composite function and is defined by gof(x) = g(f(x)).

Step function:  A number x = I + F

I → integral part    = [x]

F → fractional part = {x}

∴ x = [x] + {x}

If y = [x] then domain = R and

Range = Z

0 ≤ x ≤ 1, [x] = 0

1≤ x ≤ 2, [x] = 1

-1 ≤ x ≤ 0, [x] = -1

If k is any integer [ x + k] = k + [x]

The value of [x] is lies in x – 1 < [x] ≤ 1.

Inverse function: If f: A → B is bijection then f -1  is exists

f-1: B → A is an inverse function of f.

### SOME IMPORTANT POINTS

of subsets of a set of n elements is 2n

of proper subsets of a set of n elements is 2n – 1

Let A and B are two non-empty finite sets and f: A → B is a function. This function will

One-one if n(A) ≤ n(B)

On to if n(A) ≥ n(B)

Bijection   if n(A) = n(B).

## 3. MATRICES

Matrix: An ordered rectangular array of elements is called a matrix

• Matrices are generally enclosed by brackets like
• Matrices are denoted by capital letters A, B, C and so on
• Elements in a matrix are real or complex numbers; real or complex real-valued functions.

Oder of Matrix: A matrix having rows and ‘n’ columns is said to be of order m x n. Read as m by n.

### Square Matrix: A matrix in which the no. of rows is equal to the no. of columns is called a square matrix.

Principal diagonal ( diagonal)  Matrix: If A  = [aij] is a square matrix of order ‘n’ the elements  a11 , a22 , a33 , ………. ann is said to constitute its principal diagonal.

Trace Matrix: The sum of the elements of the principal diagonal of a square matrix A is called the trace of the matrix. It is denoted by Tr (A).

Ex:-

Diagonal Matrix: If each non-diagonal element of a square matrix is ‘zero’ then the matrix is called a diagonal matrix.

Scalar Matrix: If each non-diagonal elements of a square matrix are ‘zero’ and all diagonal elements are equal to each other, then it is called a scalar matrix.

Identity Matrix or Unit Matrix: If each of the non-diagonal elements of a square matrix is ‘zero’ and all diagonal elements are equal to ‘1’, then that matrix is called a unit matrix.

Null Matrix or Zero Matrix: If each element of a matrix is zero, then it is called a null matrix.

Row matrix & column Matrix: A matrix with only one row s called a row matrix and a matrix with only one column is called a column matrix.

Triangular matrices:

A square matrix A = [aij] is said to be upper triangular if aij = 0   ∀ i > j

A square matrix A = [aij] is said to be lower triangular matrix aij = 0  ∀ i < j

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.

### Product of Matrices:

Let A = [aik]mxn and B = [bkj]nxp be two matrices ,then the matrix C = [cij]mxp  where

Note: Matrix multiplication of two matrices is possible when no. of columns of the first matrix is equal to no. of rows of the second matrix.

Transpose of Matrix: If A = [aij] is an m x n matrix, then the matrix obtained by interchanging the rows and columns is called the transpose of A. It is denoted by AI or AT.

Note: (i) (AI)I = A (ii) (k AI) = k . AI    (iii)  (A + B )T = AT + BT  (iv)  (AB)T = BTAT

Symmetric Matrix: A square matrix A is said to be symmetric if AT =A

If A is a symmetric matrix, then A + AT is symmetric.

Skew-Symmetric Matrix: A square matrix A is said to be skew-symmetric if AT = -A

If A is a skew-symmetric matrix, then A – AT is skew-symmetric

Minor of an element: Consider a square matrix

the minor an element in this matrix is defined as the determinant of the 2×2 matrix obtained after deleting the rows and the columns in which the element is present.

Cofactor of an element: The cofactor of an element in i th row and j th column of A3×3 matrix is defined as it’s minor multiplied by (- 1 ) i+j .

### Properties of determinants:

• If each element of a row (column) of a square matrix is zero, then the determinant of that matrix is zero.

• If A is a square matrix of order 3 and k is scalar then.
• If two rows (columns) of a square matrix are identical (same), then Det. Of that matrix is zero.

• If each element in a row (column) of a square matrix is the sum of two numbers then its determinant can be expressed as the sum of the determinants.

• If each element of a square matrix are polynomials in x and its determinant is zero when x = a, then (x-a) is a factor of that matrix.
• For any square matrix A  Det(A) =  Det (AI).
• Det(AB) = Det(A) . Det(B).
• For any positive integer n Det(An) = (DetA)n.

Singular and non-singular matrices: A Square matrix is said to be singular if its determinant is zero, otherwise it is said to be the non-singular matrix.

Ad joint of a matrix: The transpose of the matrix formed by replacing the elements of a square matrix A with the corresponding cofactors is called the adjoint of A.

Invertible matrix: Let A be a square matrix, we say that A is invertible if there exists a matrix B such that AB =BA = I, where I is the unit matrix of the same order as A and B.

Augmented matrix: The coefficient matrix (A) augmented with the constant column matrix (D) is called the augmented matrix. It is denoted by [AD].

Sub matrix: A matrix obtained by deleting some rows and columns (or both) of a matrix is called the submatrix of the given matrix.

Let A be a non-zero matrix. The rank of A is defined as the maximum of the order of the non-singular submatrices of A.

• Note: If A is a non-zero matrix of order 3 then the rank of A is:
• 1, if every 2×2 submatrix is singular
• 2, if A is singular and at least one of its 2×2 sub-matrices is non-singular

(iii)  3, if A is non – singular.

Consistent and Inconsistent: The system of linear equations is consistent if it has a solution, in-consistent if it has no solution.

• Note: The system of three equations in three unknowns AX = D has
• A unique solution if rank(A) = rank ([AD]) = 3
• Infinitely many solutions if rank (A) = ([AD]) < 3
• No solution if rank (A) ≠ rank ([AD])

### Solutions of a homogeneous system of linear equations:

The system of equations AX = 0 has

• The trivial solution only if rank(A) = 3
• An infinite no. of solutions if rank(A) < 3

Directed line: If A and B are two distinct points in the space, the ordered pair (A, B) denoted by AB is called a directed line segment with initial point A and terminal point B.

⇒ A directed line passes through three characteristics: (i) length (ii) support (iii) direction

Scalar: A quantity having magnitude only is called a scalar. We identify real numbers as a scalar.

Ex: – mass, length, temperature, etc.

Vector: A quantity having length and direction is called a vector.

Ex: – velocity, acceleration, force, etc.

⇒ If is a vector then its length is denoted by

Position of vector: If P (x, y, z) is any point in the space, then is called the position vector of the point P with respect to origin (O). This is denoted by

Like and unlike vectors:  If two vectors are parallel and having the same direction then they are called like vectors.

If two vectors are parallel and having opposite direction then they are called, unlike vectors.

Coplanar vectors:
Vectors whose supports are in the same plane or parallel to the same plane are called coplanar vectors.

Triangle law: If are two vectors, there exist three points A, B, and C in a space such that   defined by

Parallelogram law: If two vectors and represented by two adjacent sides of a parallelogram in magnitude and direction then their sum is represented in magnitude and direction by the diagonal of the parallelogram through their common point.

Scalar multiplication: Let be a vector and λ be a scalar then we define vector λ  to be the vector if either is zero vector or λ is the scalar zero; otherwise λ is the vector in the direction of with the magnitude if λ>0 and λ  = (−λ)(− ) if λ<0.

The angle between two non-zero vectors:   Let be two non-zero vectors, let  then ∠AOB has two values. The value of ∠AOB, which does not exceed 1800 is called the angle between the vectors and , it is denoted by ( ).

Section formula: Let be two position vectors of the points A and B with respect to the origin if a point P divides the line segment AB in the ratio m:n then

Linear combination of vectors:  let  be vectors x1, x2, x3…. xn be scalars, then the vector is called the linear combination of vectors.

Components: Consider the ordered triad (a, b, c) of non-coplanar vectors If r is any vector then there exist a unique triad (x, y, z) of scalars such that  . These scalars x, y, z are called the components of with respect to the ordered triad   (a, b, c).

• i, j, k are unit vectors along the X, Y and Z axes respectively and P(x, y, z) is any point in the space then = r = x i + y j +z k   and

Regular polygon: A polygon is said to be regular if all the sides, as well as all the interior angles, are equal.

• If a polygon has sides then the no. of diagonals of a polygon is
• The unit vector bisecting the angle between  is

### Vector equation of a line and plane

⇒The vector equation of the line passing through point A () and ∥el to the vector  is

Proof:-

Then AP,  are collinear vector proof: let P ( ) be any point on the line a

the equation of the line passing through origin and parallel to the vectoris

• the  vector equation of the line passing through the points A( )  and B(  )  is
• Cartesian equation of the line passing through A ( x1, y1, z1) and  B ( x2, y2, z2) is
• The vector equation of the plane passing through point A( ) and parallel to the vectors and is
• The vector equation of the plane passing through the point A( ), B( ) and parallel to the vector is
• The vector equation of the plane passing through the points A( ), B( ) and C( ) is

$large&space;bar{r}=&space;(1-t)bar{a}&space;+&space;t&space;bar{b}$

## 5.PRODUCT OF VECTORS

Dot product (Scalar product): Let are two vectors. The dot product or direct product of and  is denoted byand is defined as

• If = 0, = 0 ⟹  = 0.
• If ≠0, ≠ 0 then
• The dot product of two vectors is a scalar
• If are two vectors, then

• If θ is the angle between the vectors then.

⟹

⟹ If   > 0, then θ is an acute angle

⟹ If    < 0, then θ is obtuse angle 0

⟹ If    = 0, then  is perpendicular to

• If is any vector then

Component and Orthogonal Projection:

Let=,=  be two non-zero vectors. Let the plane passing through B ( ) and perpendicular to intersects

In M, then is called the component of on

• The component (projection) vector of  on is
• Length of the projection (component) =
• Component of perpendicular to =

If ,,    form a right-handed system of an orthonormal triad, then

• If then = a1b1 + a2b2 + a3b3
• If  then

Parallelogram law:

In a parallelogram, the sum of the squares of the lengths of the diagonals is equal to the sum of the squares of the lengths of its sides.

In ∆ABC, the length of the median through vertex A is

Vector equation of a plane:

The vector equation of the plane whose perpendicular distance from the origin is p and unit normal drawn from the origin towards the plane is,

•The vector equation of the plane passing through point A ( ) and perpendicular to the is

•If θ is the angle between the planes then

Cross product (vector product): Let and be two non-zero collinear vectors. The cross product of   and  is denoted by ×  (read as a cross ) and is defined as

• The vector × is perpendicular to both  and and also perpendicular to the plane containing them

• The unit vector perpendicular to both and  is

• Let then

• If and  are two sides of a triangle then the area of the triangle =

• If A ( ), B ()and C ( )are the vertices of a ∆ABC, then its area

• The area of the parallelogram whose adjacent sides and    is

• The area of the parallelogram whose diagonals  and    is

• If A ( ), B ( )and C ( )are three points then the perpendicular distance from A to the line passing through B, C is

Let,and be three vectors, then () . is called the scalar triple product of,andand it is denoted by

Ifthen

•In determinant rows(columns) are equal then the det. Value is zero.
•In a determinant, if we interchange any two rows or columns, then the sign of det. Is change.
•Four distinct points A, B, C, and D are said to be coplanar iff
The volume of parallelepiped:
If ,andare edges of a parallelepiped then its volume is
The volume of parallelepiped:
The volume of Tetrahedron with, and are coterminous edges is
The volume of Tetrahedron whose vertices are A, B, C and D is
Vector equation of a plane:
The vector equation of the plane passing through point A () and parallel to the vectors and is
The vector equation of the plane passing through the points A ( ) and B( ) and parallel to the vector is
The vector equation of the plane passing through the points A (), B( ) and C( ) is
Skew lines:
The lines which are neither intersecting nor parallel are called Skew lines

The shortest distance between the Skew lines:
If are two skew lines, then the shortest distance between them is

If A, B, C and D are four points, then the shortest distance between the line joining the points AB and CD is

•The plane passing through the intersection of the planes is
the perpendicular distance from point A (a ̅) to the plane is

Let ,and be three vectors, then is called the vector triple product of, and.

Scalar product of four vectors:

Vector product of four vectors:

## 6. TRIGONOMETRY UPTO TRANSFORMATIONS

The word ’trigonometry’ derived from the Greek words ‘trigonon’ and ‘metron’. The word ‘trigonon’ means a triangle and the word ‘metron’ means a measure.

Angle: An angle is a union of two rays having a common endpoint in a plane.

There are three systems of measurement of the angles.

• Sexagesimal system (British system)
• Centesimal system (French system)

Sexagesimal system: – In this system, a circle can be divided into 360 equal parts. Each part is called one degree (0). One circle = 3600

Further, each degree can be divided into 60 equal parts. Each part is called one minute (‘).

and each minute can be divided into 60 equal parts. Each part is called one second (“)

Sexagesimal system: – In this system, a circle can be divided into 400 equal parts. Each part is called one grade (g). One circle = 400g

Further, each grade can be divided into 100 equal parts. Each part is called one minute (‘).

and each minute can be divided into 100 equal parts. Each part is called one second (“)

Circular measure: Radian is defined as the amount of the angle subtended by an arc of length ’r’ of a circle of radius ‘r’.

One radian is denoted by 1c. One circle = 2πc

Relation between the three measures:

3600 = 400g = 2 πc

1800 = 200g = πc

Trigonometric Ratios:

Trigonometric identities: –

∗ sin2θ + cos2θ = 1

1 – cos2θ = sin2θ

1 – sin2θ = cos2θ

∗ sec2θ − tan2θ = 1

sec2θ = 1 + tan2θ

tan2θ = sec2θ – 1

(secθ − tanθ) (secθ + tanθ) = 1

∗  cosec2θ − cot2θ = 1

co sec2θ = 1 + cot2θ

cot2θ = cosec2θ – 1

(cosec θ – cot θ) (cosec θ + cot θ) = 1

• sin θ. cosec θ = 1

sec θ. cos θ = 1

tan θ. cot θ = 1

All Silver Tea Cups Rule:

Note: If 900 ±θ or 2700 ±θ then

‘sin’ changes to ‘cos’; ‘tan’ changes to ‘cot’; ‘sec’ changes to ‘cosec’

‘cos’ changes to ‘sin’; ‘cot’ changes to ‘tan’; ‘cosec’ changes to ‘sec’.

If 1800 ±θ or 3600 ±θ then, no change in ratios.

Values of Trigonometric Ratios:

Complementary angles: Two angles A and B are said to be complementary angles, if A + B = 900.

supplementary angles: Two angles A and B are said to be supplementary angles, if A + B = 1800.

Let E ⊆ R and f: E → R be a function, then f is called periodic function if there exists a positive real number ‘p’ such that

• (x + p) ∈ E ∀ x∈ E
• F (x+ p) = f(x) ∀ x∈ E

If such a positive real number ‘p’ exists, then it is called a period of f.

The algebraic sum of two or more angles is called a ‘compound angle’.

For any two real numbers A and B

sin (A + B) = sin A cos B + cos A Cos B

sin (A − B) = sin A cos B − cos A Cos B

cos (A + B) = cos A cos B − sin A sin B

cos (A − B) = cos A cos B + sin A sin B

tan (A + B) =

tan (A − B) =

cot (A + B) =

⋇ cot (A − B) =

sin (A + B + C) = ∑sin A cos B cos C − sin A sin B sin C

cos (A + B + C) = cos A cos B cos C− ∑cos A sin B sin C

tan (A + B + C) =

⋇ cot (A + B + C) =

⋇ sin (A + B) sin (A – B) = sin2 A – sin2 B = cos2 B – cos2 A

⋇ cos (A + B) cos (A – B) = cos2 A – sin2 B = cos2 B – sin2 A

Extreme values of trigonometric functions:

If a, b, c ∈ R such that a2 + b2 ≠ 0, then