# 2021

Sem -1 Solutions

## Polytechnic Engineering Mathematics – Sem -1 Solutions

Sem -1 Solutions

Education is the acquisition of knowledge, skill value, etc.
Education contributes to the development of society.
www.basicsinmaths.com website has been given material for mathematics Polytechnic Students.

Polytechnic Engineering Mathematics – Sem -1 Solutions solutions in PDF Files are designed by the ‘Basics In Maths” Team. These Pdf Files are very useful for students who are prepared for polytechnic examinations.

## TRANSFORMATIONS

Transformations  PDF File is designed by the ‘Basics In Maths” Team. This Pdf File is very useful for students who are prepared for polytechnic examinations.

### PART – 1

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PART – 2

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## INVERSE TRIGONOMETRIC FUNCTIONS

The inverse trigonometric functions solutions  PDF File is designed by the ‘Basics In Maths” Team. This Pdf File is very useful for students who are prepared for polytechnic examinations.

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## SOLUTIONS OF SIMULTANEOUS EQUATIONS

Solutions of simultaneous equations  PDF File is designed by the ‘Basics In Maths” Team. This Pdf File is very useful for students who are prepared for polytechnic examinations.

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## SOLUTIONS OF TRIANGLES

Solutions of triangles  PDF File is designed by the ‘Basics In Maths” Team. This Pdf File is very useful for students who are prepared for polytechnic examinations.

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## Sem – 2 Concept – Engineering Mathematics

Sem – 2 Concept

Education is the acquisition of knowledge, skill value, etc.
Education contributes to the development of society.
website has been given material for math Polytechnic Students.

Engineering Maths  Sem – 2 Concept solutions in PDF Files are designed by the ‘Basics In Maths” Team. These Pdf Files are very useful for students who are prepared for polytechnic examinations.

TS 6th Class Maths Concept

### Polytechnic-Engineering Mathematics -SEM-2-Concept.pdf

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TS 10th class maths concept (E/M)

## Sem -2 Solutions – Polytechnic Engineering Mathematics

Sem -2 Solutions

Education is the acquisition of knowledge, skill value, etc.
Education contributes to the development of society.
website has been given material for mathematics Polytechnic Students.

Polytechnic Engineering Mathematics – Sem -2 Solutions in PDF Files are designed by the ‘Basics In Maths” Team. These Pdf Files are very useful for students who are prepared for polytechnic examinations.

#### Tangents and Normals

Tangents and Normals solutions  PDF File is designed by the ‘Basics In Maths” Team. This Pdf File is very useful for students who are prepared for polytechnic examinations.

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### Rate Measure

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#### Maxima and Minima

The Maxima and Minim solutions  PDF File is designed by the ‘Basics In Maths” Team. This Pdf File is very useful for students who are prepared for polytechnic examinations.

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# TS Inter Maths 1A &1B Practice papers (Reduced Syllabus)

###### TS Inter Maths 1A and 1B Practice papers as per reduced syllabus were designed by the ‘Basics in Maths‘ team.

These Practice papers to do help the intermediate First-year Maths students.

# TS Inter Maths 1A and 1B Practice papers as per reduced syllabus are very useful in IPE examinations.

## MATHS 1A PRACTICE PAPER – 1

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## MATHS 1A PRACTICE PAPER – 2

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## MATHS 1A PRACTICE PAPER – 3

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## MATHS 1A PRACTICE PAPER – 4

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## MATHS 1B PRACTICE PAPER – 1

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## MATHS 1B PRACTICE PAPER – 2

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## MATHS 1B PRACTICE PAPER – 3

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## MATHS 1B PRACTICE PAPER – 4

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

### What is PDF?

PDF, or Portable Document Format, is an open file format used for exchanging electronic documents. Documents, forms, images, and web pages encoded in PDF can be correctly displayed on any device, including smartphones and tablets. If you distribute your reports in PDF, you can be sure that all of your pupils or friend will be able to open and read it on their PCs, Macs or Android smartphones.

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

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

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

IPE examinations.

### MATHEMATICS 1A

Mathematical Induction

Matrices

Trigonometric Equations

Inverse Trigonometric Functions

TS 10th class maths concept (E/M)

### MATHEMATICS 1B

DC’s and Dr’s

Tangents and Normals

Rate Measure

Maxima and Minima

Rolles and Langrange’s Theorem

TS 6th Class Maths Concept

## English Grammar 4 Competitive Exams & School Education

english grammar

This content is designed by the ‘Basics in Maths‘ team.

## English language:

The English Language is important to communicate and interact with other people around us. It keeps us in contact with other people.
An example of the importance of a language is the ‘English language’ because it is the international language and has become the most important language to people in many parts of the world.
British brought with them their language English to India.

ఇంగ్లీష్ భాష:

మన చుట్టూ ఉన్న ఇతర వ్యక్తులతో సందేశించాదానికికి  మరియు మన చుట్టూ ఉన్న ఇతర వ్యక్తులతో సంభాషించడానికి భాష ముఖ్యం. ఇది మనల్ని ఇతర వ్యక్తులతో సంప్రదించుటకు దోహదపడుతుంది.
ఒక భాష యొక్క ప్రాముఖ్యతకు ఒక ఉదాహరణ ‘ఆంగ్ల భాష’ ఎందుకంటే ఇది అంతర్జాతీయ భాష మరియు ప్రపంచంలోని అనేక ప్రాంతాల ప్రజలకు అత్యంత ముఖ్యమైన భాషగా మారింది.
బ్రిటిష్ వారు తమ భాష ఇంగ్లిష్ ను భారతదేశానికి తీసుకువచ్చారు.

### English Grammar:

Grammar is the way we arrange words to make proper sentences. Grammar rules about how to speak and write in a language. english grammar is the grammar of the English language. English grammar started out based on Old English,

ఇంగ్లిష్ వ్యాకరణం:

వ్యాకరణం అనేది సరైన వాక్యాలు చేయడానికి పదాలను ఏర్పాటు చేసే విధానం. వ్యాకరణం అనగా  ఒక భాషలో ఎలా మాట్లాడాలి మరియు ఎలా రాయాలి అనే నియమాలు. ఆంగ్ల వ్యాకరణం ఆంగ్ల భాష యొక్క వ్యాకరణం. ఓల్డ్ ఇంగ్లిష్ ఆధారంగా ఇంగ్లిష్ గ్రామర్ ప్రారంభమైంది,

## Introduction

There are 26 letters in the English Language. Those are called as ‘Alphabet’

There are two parts to Alphabet.

• Vowels (a, e, i, o, u) [ 5 letters]
• Consonants (Remaining 21 letters)

Without vowel (sound or structure) we cannot create even a single word in English.

## PARTS OF SPEECH (భాషాభాగాలు)

#### NOUN(నామవాచకం):

A noun is a naming word. The noun means the name of the person, things, places, or animals

(నామవాచకం ఒక వ్యక్తి యొక్క, ఒక వస్తువు యొక్క లేదా జంతువు యొక్క పేరును తెలుపుతుంది)

Ex: Ramu goes to college by car

Seetha went to school by bus        → underlined words are nouns

### Kinds of Nouns (According to their usage):

Proper Noun: A proper noun denotes one particular person, place, or thing.

(Proper Noun, ఒక ప్రత్యేక వ్యక్తి, వస్తువు లేదా జంతువు యొక్క పేరును తెలుపుతుంది)

Ex: Raju, Hyderabad, The Ganga, etc.

Common Noun: A common noun is a name given commonly to a person, place or thing.

(CommonNoun, ఒకే జాతికి చెందిన వ్యక్తి, వస్తువు లేదా జంతువు యొక్క పేరును తెలుపుతుంది)

Ex: boy, girl, animal, river, city, etc.

Collective noun:  A collective noun denotes a group or collection of persons or things taken as one.

(Collective Noun, వ్యక్తుల, వస్తువుల లేదా జంతువుల యొక్క గుంపును తెలుపుతుంది)

Ex: herd, army, committee, flock, etc.,

Material noun: A Material noun denote the name of a particular kind of metal, liquid, or substance.

(Material Noun, ఒక నిర్దిష్ట రకం లోహం, ద్రవం లేదా పదార్థం యొక్క పేరును తెలియజేస్తుంది)

Ex: salt, sand, gold, rice, paddy, etc.,

### Kinds of Nouns (According to their Meaning):

Concrete Noun:  A Concrete noun denotes something that can be tasted, something that can be touched or seen, something that exists physically.

(కాంక్రీట్ నామవాచకం దేనినైనా రుచి చూడవచ్చు, ఏదైనా తాకవచ్చు లేదా చూడవచ్చు, భౌతికంగా ఉన్నదాన్ని సూచిస్తుంది.)

Ex: Pencil, boy, girl, gold, silver, rice, etc.,

Note: Proper nouns and Material nouns are Concrete nouns.

Abstract Noun: An Abstract noun denotes something maybe an idea or emotion.

Ex: born, sad, joy, bravery, freedom, etc.,

### PRONOUN(సర్వనామం):

A pronoun is a word that is used instead of a noun.

సర్వనామం ను నామవాచకానికి బదులుగా వాడుతాము.

Ex:      Ramu went to the Ground, he played cricket.

పై వాక్యం లో రాము కు బదులుగా he వాడబడినది.

### Types of Pronouns:

Personal Pronouns: Personal pronoun refers to a particular person or thing. (దీనిని వ్యక్తి పేరు కి బదులుగా ఉపయోగిస్తారు)

These are three types

ఇవి మూడు రకాలు

I person: Talks about himself (తన గురించి చెప్పేది. ఉదా : నేను, నాకు, మేము , మాకు, మొ ||)

Ex:  I – we – my – us etc.,

II person: what it says about others (ఎదుటి వారి గురించి చెప్పేది. ఉదా : నీవు , మీరు , మీకు  మొ ||)

Ex: you, yours

III Person: Talks about the third person between the discussion of two people (ఇద్దరి వ్యక్తుల సంభాషణ మధ్య మూడో వ్యక్తి గురించి చెప్పే ది. ఉదా : అతను , ఆమె , అతనికి  , ఆమెకి , వారికి  మొ ||)

Ex: he, she, it, they. Etc.,

##### Reflexive Pronoun:

Reflexive Pronouns are used when the subject and the object of a sentence are the same. They can act as either objects or indirect objects. (ఒక వ్యక్తి చేసిన పని ఫలితాన్ని తానే పొందినప్పుడు వీటిని వాడుతారు)

Ex: myself, himself, themself, yourselves

##### Relative Pronouns:

A relative pronoun introduces a clause. It refers to some noun going before and also joins two sentences together.  (రెండు వాక్యములను కలుపుటకు వాడుతాము లేదా ఒక వాక్యములో అంతకుముందే చెప్పబడిన nouns ను refer చేస్తాయి)

Ex: who ……. Persons కు

Which ……. Places కు

That …… Things కు వాడుతారు

Demonstrative Pronoun: Demonstrative pronouns always identify nouns, whether those nouns are named specifically or not (ఇది, దేనినైనా లేక వేనినైన ఎత్తి చూపడానికి ఉపయోగపడుతుంది)

Ex: this, that, those, these, etc.,

Distributive Pronoun: Distributive pronouns refer to persons or things one at a time. (ఒకే సమయం లో ఎందరికో చెందేవి)

Ex; each, either, neither, etc.,

Indefinite Pronouns: Indefinite pronouns refer to people or things without saying exactly who or what they are. (ఫలానా వ్యక్తీ, ఫలానా వస్తువు గురించి కాకుండా ఎవరో ఒక వ్యక్తి, ఎదో ఒక వస్తువు గురించి Indefinite Pronouns తెలియజేస్తాయి)

Ex: somebody, none, all, nobody, etc.,

Interrogative Pronouns: These are used to ask questions (ప్రశ్నలు అడగడానికి వాడుతాము)

Ex: What, who, why

Subject (కర్త ): Subject means noun or pronoun or noun and pronoun.

An adjective is used with a noun to add something to its meaning (ఒక విశేషణం నామవాచకంతో దాని అర్థానికి ఏదైనా జోడించడానికి ఉపయోగించబడుతుంది)

Ex: large, big, small, honest, wise, etc.,

Qualitative Adjective: It indicates the characteristic of a person or an object (ఇది ఒక వ్యక్తి లేదా ఒక వస్తువు యొక్క లక్షణాన్ని తెలుపుతుంది)

Ex: honest, wise, small, big, etc.,

Quantitative adjective: It shows how much of a thing is (ఇవి ఎంత అనే అర్థంలో వాడుతాము)

Ex: some, much, little, enough, etc.,

Numeral Adjectives:  It shows how many things are meant (సంఖ్యాత్మకమైనవి. ఎన్ని అనే పదానికి సమాధానంగా వచ్చేవి)

Ex: few, many, most, five, three, etc.,

These, that, those, this వంటి నామవాచకం తో కలిపి వస్తే వాటిని Demonstrative Adjectives అంటారు.

Ex: This boy is tall

That girt is clever

### Verb (క్రియ)

Verb (క్రియ) లేకుండా ఇంగ్లీష్ లో వాక్యము లు ఉండవు. ఇంగ్లీష్ వాక్యానికి ‘క్రియ’ ప్రాణం ‍వంటిది.

A verb is a word that tells something about an action or state.

Verb (క్రియ) లేకుండా ఇంగ్లీష్ లో వాక్యము లు ఉండవు. ఇంగ్లీష్ వాక్యానికి ‘క్రియ’ ప్రాణం ‍వంటిది.

A verb is a word that tells something about an action or state. (పనులను తెలియజేయు పదాలను verbs అంటారు)

Ex: I go to school

I am a student

He plays Cricket

We sit at the table

There are two kinds of verbs:

1. Main verb
2. Helping verb

#### Main verbs:

A verb that has an individual meaning is called the Main verb (వ్యక్తిగత అర్థాన్ని కలిగి ఉన్న క్రియను ప్రధాన క్రియ అని అంటారు)

Ex: go, come, take, sing, play, etc.,

Helping verbs:

A verb that does not have any individual meaning is called Helping verb (వ్యక్తిగత అర్థం లేని క్రియను Helping verb అని అంటారు)

Helping verb is mainly used to identify the tense. (ప్రధానంగా టెన్స్ గుర్తించడానికి Helping verb ఉపయోగించబడుతుంది)

Ex: do, does, Is, am, are, have, has, had, will, shall, etc.,

#### Types of Verbs:

Transitive Verb: A transitive verb is a verb that denotes an action that passes over from the subject to an object. (Object ను కలిగి యుండే   verb ను transitive verb అంటారు)

Ex: He writes a letter

Raju sings a song

In Transitive Verb:  An intransitive verb is a verb that denotes an action that does not pass over to an object. (Object లేని verb ను Intransitive verb అంటారు)

Ex: the boy plays

The Bird sings

Ex: quickly, very, quietly, clearly etc.,

### Preposition (విభక్త్యర్థ పదం)

A preposition is a word that is placed before a noun or pronoun to show the relation between a person, place, or thing. (ఒక నామవాచకముకు లేదా సర్వనామమునకు ముందున్చబడి వాక్యంలోని ఇతర పదం లేక పదాలతో అ నామవాచకం లేదా సర్వనామం యొక్క సంబందాన్ని తెలిపే పదం)

#### Kinds of Prepositions

Simple Prepositions:  at, by, in, for, off, of, up, to, with, etc., are Simple Prepositions

Compound Prepositions: about, across, along, among, behind, before, below, besides, inside, within, without, etc., are called Compound Prepositions.

Phrase Prepositions: according to, Infront of, in favor of, because of, with regard to, etc., are known as Phrase Prepositions.

### Conjunction (సముచ్చయం):

Conjunction combines sentences or words together. (కొన్ని వాక్యాలను లేదా పదాలను కలిపే పదం)

Ex: and, or, but, so, if, as, since, when, etc.,

### Interjection (భావోద్వేగ ప్రకటన):

An Interjection is a word that expresses some sudden feelings or emotions.

(మానసిక భావాలను లేక ఉద్రేకాలను తెల్పుటకు వాడే మాటలను Interjections అని అంటారు).

Note: Interjections తరువాత ఆశ్చర్యార్ధకము (!) అనే గుర్తు ఉంచి దాని తరువాత వచ్చేమాట మొదటి అక్షరము Capital letters తో ప్రారంభించవలెను.

Ex: Hello! What are you doing here?

Alas! He  injured

Hurrah! I won the game

## Engineering Mathematics SEM – 1Concept

Mathematics Notes for Polytechnic SEM – 1 is Designed by the ” Basics in Maths” team. Here we can learn Concepts in Basic Engineering mathematics Polytechnic Sem – I.

This Material is very Useful for Basic Engineering Mathematics Polytechnic Sem – I Students.

By learning These Notes, Basic Engineering Mathematics Polytechnic Sem – I Students can Write their Exam successfully and fearlessly.

Engineering Mathematics SEM – 1Concept

#### LOGARITHMS

Logarithm: For ant two positive real numbers a, b, and a ≠ 1. If the real number x such then ax = b, then x is called logarithm of b to the base a. it is denoted by

#### Standard formulae of logarithms:

##### Logarithmic Function:

Let a be a positive real number and a ≠ 1. The function f: (o, ∞) → R Defined by f(x) =

PARTIAL FRACTIONS

Fractions:

If f(x) and g(x) are two polynomials, g(x) ≠ 0, then   is called rational fraction.

Ex:

etc.  are rational fractions.

##### Proper Fraction:

A rational fraction is said to be a Proper fraction if the degree of g(x) is greater than the degree of f(x).

Ex:

etc. are the proper fractions.

Improper Fraction:

A rational fraction is said to be an Improper fraction if the degree of g(x) is less than the degree of f(x).

Ex:

etc. are the Improper fractions.

##### Partial Fractions:

Expressing rational fractions as the sum of two or more simpler fractions is called resolving a given fraction into a partial fraction.

∎ If R(x) =  is proper fraction, then

Case(i): – For every factor of g(x) of the form (ax + b) n, there will be a sum of n partial fractions of the form:

Case(ii): – For every factor of g(x) of the form (ax2 + bx + c) n, there will be a sum of n partial fractions of the form:

∎ If R(x) = is improper fraction, then

Case (i): – If degree f(x) = degree of g(x),   where k is the quotient of the highest degree term of f(x) and g(x).

Case (ii): – If f(x) > g(x)

R(x) =

### MATRICES AND DETERMINANTS

Matrix: A set of numbers arranged in the form of a rectangular array having rows and columns is called 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 ‘m’ rows and ‘n’ columns is said to be of order m x n read as m by n.

Ex:

##### Types Of Matrices

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

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

Principal diagonal (diagonal) Matrix: If A = [a ij] is a square matrix of order ‘n’ the elements a11, a22, a33, ………. An n 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).

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 element of a square matrix is ‘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 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 are of the same order and the corresponding elements of A and B are equal.

If A and B are two matrices of the same order, then the matrix obtained by adding the corresponding elements of A and B is called the sum of A and B. It is denoted by A + B.

Subtraction of matrices:

If A and B are two matrices of the same order, then the matrix obtained by subtracting the corresponding elements of A and B is called the difference from A to B.

##### 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.

A m x n  . Bp x q = AB mx q; n = p

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 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.

Ex: – minor of a3 is     = b1c2 – b2c1

Minor of b2 is   = a1c3 – a3c1

Cofactor of an element: The cofactor of an element in i th row and j th column of A3×3 matrix is defined as its minor multiplied by (- 1) i+j.

###### Properties of determinants:

If each element of a row (column) f a square matrix is zero, then the determinant of that matrix is zero.

Ex: –    A =   ⇒ det(A) = 0

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.

Ex: –    A =     ⇒ det(A) = 0

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.

Ex: –

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 is zero, otherwise it is said to be non-singular matrix.

Ex: –   A =      det(A) = 4 – 4 = 0

∴ A is singular matrix

B =

Det(B) = 4 + 4 = 8≠ 0

∴ B is non-singular

Adjoint 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.

Let A = and     cofactor matrix 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 a unit matrix of the same order as A and B.

1. (A– I)– I = A
2. (AI)– I = (A-I)I
3. (AB)-I = B-I A-I
4. A-I =

#### Compound Angles

The algebraic sum of two or more angles is called a ‘compound angle’. Thus, angles A + B, A – B, A + B + C etc., are Compound Angles

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) =

tan ( + A) =

tan ( − A) =

cot (+ A) =

cot (− A) =

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

##### Multiple and Sub Multiple Angles

If A is an angle, then its integral multiples 2A, 3A, 4A, … are called ‘multiple angles ‘of A and the multiple of A by fraction like are called ‘submultiple angles.

∎ If  is not an add multiple of

#### PROPERTIES OF TRIANGLES

In ∆ABC,

Lengths AB = c; BC = a; AC =b

The area of the triangle is denoted by ∆.

Perimeter of the triangle = 2s = a + b + c

A = ∠CAB; B = ∠ABC; C = ∠BCA.

Sine rule:

In ∆ABC,

⟹ a = 2R sin A; b = 2R sin B; c = 2R sin C

Where R is the circumradius and a, b, c, are lengths of the sides of ∆ABC.

Cosine rule:

In ∆ABC,

###### Projection rule:

In ∆ABC,

a = b cos C + c cos B

b = a cos C + c cos A

c = a cos B + b cos A

###### Tangent rule (Napier’s analogy):

Area of the triangle:

In ∆ABC, a, b, and c are sides

S = and area of the triangle

### HYPERBOLIC FUNCTIONS

The function f: R→R defined by f(x) = ∀ x ∈ R is called the ‘hyperbolic sin’ function. It is denoted by Sinh x.

∴Sinh x =

Similarly,

###### Identities:

cosh2x – sinh2 x = 1

cosh2x = 1 + sinh2 x

sinh2 x = cosh2 x – 1

sech2 x = 1 – tanh2 x

tanh2 x = 1 – sesh2 x

cosech2 x = coth2 x – 1

coth2 x = 1 + coth2 x

###### Addition formulas of hyperbolic functions:

Sinh (x + y) = Sinh x Cosh y + Cosh x Sinh y

Sinh (x − y) = Sinh x Cosh y − Cosh x Sinh y

Cosh (x + y) = Cosh x Cosh y + Sinh x Sinh y

Cosh (x − y) = Cosh x Cosh y − Sinh x Sinh y

tanh (x + y) =

tanh (x − y) =

coth (x + y) =

sinh 2x = 2 sinh x cosh 2x =

cosh 2x = cosh2x + sinh2 x = 2 cosh2x – 1 = 1 + 2 sinh2x =

tanh 2x =

###### Inverse hyperbolic functions:

Sinh−1x =  ∀ x ∈ R

Cosh−1x =   ∀ x ∈ (1, ∞)

Tanh−1x =    ∀ < 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 the imaginary part.

3 + 4i, 2 – 5i, – 3 + 2i are the examples for Complex numbers.

• z = x +i y 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:

The multiplicative inverse of the complex number z is 1/z.

z = x + i y then 1/z = x – i y/ x2 + y2

Conjugate complex numbers:

The complex numbers x + i y, x – i y are called conjugate complex numbers.

Conjugate of z is denoted by

The sum and product of two conjugate complex numbers are real.

If z1, z2 are two complex numbers then

###### Modulus and amplitude of complex numbers:

Modulus: – If z = x + i y, 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 cos θ, y = r sin θ

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}

Note:

∎ e = cos θ + i sin θ

∎ e−iθ = cos θ − i sin θ

##### De- Moiver’s theorem

For any integer n and real number θ, (cos θ + i sin θ) n = cos n θ + i sin n θ.

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)

(cos θ + i sin θ) -n = cos n θ – i sin n θ

(cos θ + i sin θ) (cos θ – i sin θ) = cos2θ – i2 sin2θ = cos2θ + sin2θ = 1.

cos θ + i sin θ = 1/ cos θ – i sin θ and cos θ – i sin θ = 1/ cos θ + i sin θ

(cos θ – i sin θ) n = (1/ (cos θ –+i sin θ)) n = (cos θ + i sin θ)-n = cos n θ – i sin n θ

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 (cos θ + 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 coincided 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

cube roots of unity are: 1

ω =  , ω2 =

ω2 +ω + 1 = 0 and ω3 = 1

### TRANSFORMATIONS

For A, B∈ R

⋇ sin (A + B) + sin (A – B) = 2sin A cos B

⋇ sin (A + B) −sin (A – B) = 2cos A sin B

⋇ cos (A + B) + cos (A – B) = 2 cos A cos B

⋇ cos (A + B) − cos (A – B) = − 2sin A sin B

For any two real numbers C and D

⋇ sin C + sin D = 2sin cos

⋇ sin C −sin D= 2cos  sin

⋇ cos C + cos D = 2 cos   cos

⋇ cos C − cos D = − 2sin   sin

If A + B + C = π or 1800, then

⋇ sin (A + B) = sin C; sin (B + C) = sin A; sin (A + C) = sin B

⋇ cos (A + B) = − cos C; cos (B + C) = −cos A; cos (A + C) = − cos B

If A + B + C = 900 or then

⋇ sin   = cos;   sin   = cos ;  sin   = cos

⋇ cos  = sin; cos   = sin; cos   = sin

If    then

⋇ sin (A + B) = cos C; sin (B + C) = cos A; sin (A + C) = cos B

⋇ cos (A + B) = sin C; cos (B + C) = sin A; cos (A + C) = sin B

### INVERSE TRIGONOMETRIC RATIOS

If A, B are two sets and f: A→ B is a bijection, then f-1 is existing and f-1: B → A is an inverse function.

### Solutions of Simultaneous Equations

#### Matrix Inversion Method:

Let a system of simultaneous equations be

a1 x + b1 y + c1z = d1

a2 x + b2 y + c2z = d2

a3 x + b3 y + c3z = d3

The matrix form of the above equations is

Therefore, the matrix equation is AX = B

If Det A ≠ 0, A-1 is exists

X = A-1 B

By using above Condition, we get the values of x, y and z

This Method is called as Matrix Inversion Method

#### Cramer’s Method:

Let system of simultaneous equations be

a1 x + b1 y + c1z = d1

a2 x + b2 y + c2z = d2

a3 x + b3 y + c3z = d3

1 is obtained by replacing the coefficients of x (1st column elements of ∆) by constant values

2 is obtained by replacing the coefficients of y (2nd column elements of ∆) by constant values

3 is obtained by replacing the coefficients of z (3rd column elements of ∆) by constant values

Now

This method is called Cramer’s Method

#### Gauss-Jordan Method:

Let a system of simultaneous equations be

a1 x + b1 y + c1z = d1

a2 x + b2 y + c2z = d2

a3 x + b3 y + c3z = d3

Augmented matrix: The coefficient matrix (A) augmented with the constant column matrix (B) is called the augmented matrix. It is denoted by [AD].

This Matrix is reduced to the standard form ofby using row operations

1. Interchanging any two rows
2. Multiplying the elements of any two elements by a constant.
3. Adding to the elements of one row with the corresponding elements of another row multiplied by a constant.

∴ The solution of a given system of simultaneous equations is x = α, y = β, and z = γ.

##### Procedure to get the standard form:
1. Take the coefficient of x as the unity as a first equation.
2. If 1 is there in the first-row first column, then make the remaining two elements in the first column zero.
3. After that, if one element in R2 or R3 is 1, then make the remaining two elements in that column C2 or C3 as zeroes.
4. If any row contains two elements as zeros and only non-zero divide that row elements with the non-zero element to get unity and make the remaining two elements in that column as zeros.

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TS TET 2022, Paper 2, 10 th Maths, chapter 14, Statistics, Practicew bits: https://youtu.be/RA6jDQQaRXw

Matrices Basic Concepts: https://youtu.be/_Gqg07ZquN0
Trace of Matrix, Addition of Matrices, Equality of Matrices: https://youtu.be/Hycj-UITfD8
Multiplication of two matrices P – 1:https://youtu.be/rMp96I6ag8s
Multiplication of two matrices P – 2:https://youtu.be/Axu5BbGKkbs
Multiplication of two matrices P – 3:https://youtu.be/1aSVztz9TII
transose of matrices: https://youtu.be/D8nhjYQJqm0
Smmetric and Skew Symmetric Matrices:https://youtu.be/JUv2RT35DaA

Adjoint and Inverse of Matrices P – 1:https://youtu.be/mPnkSkxKaxo

Adjoint and Inverse of Matrices P – 2:https://youtu.be/4fpvFTNpXC0

Adjoint and Inverse of Matrices P – 3:https://youtu.be/Ge-jHQRF4SM

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## Reduced Syllabus 2021 Telangana BIE Maths

Reduced Syllabus 2021

This content is designed by the ‘Basics in Maths‘ team.

Telangana BIE Maths Reduced Syllabus(2021) very useful  I.P.E  exam.

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