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Engineering Mathematics Syllabus ( M-1, M-2 & M-3)

Engineering Mathematics Syllabus Eng. Maths – M1 Syllabus Engineering Mathematics Syllabus Unit – I: Ordinary Differential Equations: Basic concepts and definitions of 1st order differential equations Formation of differential equations solution of differential equations: variable separable homogeneous, equations reducible to homogeneous form exact differential equation equations reducible to exact form, linear differential equation equations reducible to linear form (Bernoulli’s equation) orthogonal trajectories applications of differential equations Unit – II: Linear Differential equations of 2nd and higher-order: Second-order linear homogeneous equations with constant coefficients Differential operators Solution of homogeneous equations Euler-Cauchy equation Linear dependence and independence Wronskian; Solution of nonhomogeneous equations General solution, complementary function Particular integral; solution by variation of parameters Undetermined coefficients Higher-order linear homogeneous equations applications. Unit – III: Differential Calculus (Two and Three variables) : Taylor’s Theorem Maxima and Minima Lagrange’s multipliers Unit – IV: Matrices, determinants, linear system of equations: Basic concepts of an algebra of matrices; Types of matrices; Vector Space, Sub-space, Basis, and dimension, linear system of equations; Consistency of linear systems The rank of a matrix. Gauss elimination; the inverse of a matrix by the Gauss-Jordan method; Linear dependence and independence Linear transformation, inverse transformation, and applications of matrices Determinants Cramer’s rule. Unit – V: Matrix-Eigen value problems: Eigenvalues, Eigenvectors, Cayley Hamilton theorem, basis, complex matrices; quadratic form; Hermitian, Skew Hermitian forms; Similar matrices; diagonalisation of matrices; Transformation of forms to the principal axis (conic section). Engineering Mathematics Syllabus – M2 Syllabus Unit I: Laplace Transforms: Laplace Transform, .Inverse Laplace Transform, Linearity, transform of derivatives and Integrals, Unit Step function, Dirac delta function, Second Shifting theorem, Differentiation and Integration of Transforms, Convolution, Integral Equation, Application to solve differential and integral equations, Systems of differential equations. Unit II: Series Solution of Differential Equations: Power series; the radius of convergence, power series method, Fresenius method; Special functions: Gamma function, Beta function; Legendre’s and Bessel’s equations; Legendre’s function, Bessel’s function, orthogonal functions, and generating functions. Unit III: Fourier series, Integrals and Transforms: Periodic functions, Even and Odd functions, Fourier series, Half Range Expansion, Fourier Integrals, Fourier sine and cosine transforms, Fourier Transform Unit IV: Vector Differential Calculus: Vector and Scalar functions and fields, Derivatives, Gradient of a scalar field, Directional derivative, Divergence of a vector field, Curl of a vector field. Unit V: Vector Integral Calculus: Line integral, Double Integral, Green’s theorem, Surface Integral, Triple Integral, Divergence Theorem for Gauss, Stroke’s Theorem Engineering Mathematics Syllabus M3 Syllabus UNIT I: Basic Probability Probability spaces, conditional probability, independent events, and Bayes theorem. Random variables: Discrete and continuous random variables, Expectation of random variables, Moments and Variance of Random Variables. UNIT II: Probability distributions: Binomial, Poisson, evaluation of statistical parameters for these distributions, Poisson approximation to the binomial distribution. Continuous random variables and their properties, distribution functions and density functions, Normal and Exponential, evaluation of statistical parameters for these distributions. UNIT III: Testing of hypothesis: Test of significance Basic testing of a hypothesis. The null and alternate hypothesis Types of errors Level of significance and critical region. Large sample test for a single proportion, The difference in proportions, A single mean, the difference of means, Small sample tests: Tests for single mean, The difference of means Test for the ratio of variances. UNIT IV: Complex variables (Differentiation): Limit Continuity and differentiation of complex functions, Analyticity, Cauchy – Riemann equations (without proof) Finding a harmonic conjugate, Elementary analytic functions and their properties. UNIT V: Complex variables (Integration): Line integral, Cauchy’s theorem, Cauchy’s integral formula, Zero of analytic functions, singularities Taylor’s series Laurent’s series Residues, Cauchy Residue Theorem conformal mappings Mobius transformations and their properties. Basics In Maths Visit my YouTube Channel: Click on the logo below

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Telugu Grammer( తెలుగు వ్యాకరణం )

Education / Nayini Satyanarayana Reddy

Telugu Grammer( తెలుగు వ్యాకరణం ) Telugu Grammer( తెలుగు వ్యాకరణం ): మన మనస్సు లోని భావాలను , అనుభూతులను పైకి చెప్పడానికి మాతృభాష ఎంతో ఉపయోగం , అందుకే మాతృ భాష తల్లి లాంటిది అంటారు. మనం మన మాతృ భాషని గౌరవించాలి . గ్రహణ సామర్థ్యం పెరగడానికి మాతృ భాష లో విద్యా బోధన ఎంతగానో ఉపయోగ పడుతుంది. మాతృ భాష లో బోధించడం వలన విద్యార్థుల్లో సృజనాత్మకత పెరుగుతుంది. భాషా భాగాలు : భాషకు ప్రాణం భావ ప్రసరణ . ఈ భావ ప్రసరణ ఒకరి నుండి మరొకరికి చేరాలి . ఇలా చేరడానికి కొన్ని పదాలు వాక్యాలు అవసరం. ఇలా వాక్యం లోని ఉపయోగాన్ని బట్టి భాషకు ఐదు ప్రధాన భాగాలుగా విభజించారు. అవి :− 1. నామవాచకం 2. సర్వనామం 3. విశేషణం 4. క్రియ 5. అవ్యయము 1.నామవాచకం : నామము అనగా పేరు.ఒక వ్యక్తిని గాని, వస్తువుని గాని ,గుణమును గాని, జాతిని గాని తెలుపును. ఉదా :- ధర్మరాజు , హైదరాబాద్ , బంతిపువ్వు , ఆవు మొ|| నవి . 2. సర్వనామము : నామ వాచకాలకు బదులుగా వాడే వాటిని “ సర్వనామాలు “ అంటారు. “సర్వ” అనగా సమస్తము. ఉదా :- అది, ఇది, అతడు , ఆమె ,అన్ని ,కొన్ని మొ|| నవి. 3. విశేషణం : నామవాచకము మరియు సర్వనామముల యొక్క గుణమును తెలియజేయునది . ఉదా :- మంచి , చెడు , లావు , పొట్టి , పొడుగు , ఎత్తు మొ|| నవి. 4. క్రియ : పనులను, స్తితిగతులను తెలియజేయునది . ఉదా :- రాస్తున్నాడు , వెళ్తున్నాడు , పాడుతున్నాడు మొ|| నవి. 5. అవ్యయము : వ్యయము అనగా నశించేది, అవయము అనగా నశించనిది . లింగ, వచన, విభక్తుల ప్రసక్తిగాని, వచన ఆకాంక్ష లేని వాటిని అవ్యయములు అంటారు . ఉదా :- అక్కడ, ఇక్కడ, ఆహా , భళా మొ|| నవి. సంధులు ∗ వ్యాకరణ భాషలో రెండు స్వరాల కలయికను సంధి అంటారు . ∗ రెండు అచ్చుల మధ్య జరిగే మార్పును సంధి కార్యం అంటారు. ∗ సంధి జరిగే మొదటి పదo చివరి అక్షరం లోని అచ్చును ‘పూర్వ పదం’ అంటారు . ∗ సంధి జరిగే రెండవ పదం మొదటి అక్షరం లోని అచ్చును ‘పర పదం ‘ అంటారు . ఉదా :- రామ + అయ్యా: ‘ రామ’ లోని ‘మ’ లో ‘అ’ పూర్వ పదం ‘అయ్యా’ లోని ‘అ ‘ పర పదం . అత్వ సంధి(అకార సంధి ): అత్తునకు సంధి బహుళంగా వస్తుంది . ఉదా :-1) మేఅల్లుడు = మేన + అల్లుడు 2) లేకేమి = లేక + ఏమి 3) రాకుంటే = రాక + ఉంటే 4) పోవుటెట్లు = పోవుట + ఎట్లు ఇత్వ సంధి ( ఇకార సంధి): ఏమ్యాదులకు ఇత్తునకు సంధి . ఉదా :- 1) ఏమంటివి = ఏమి + అంటివి 2) పైకెత్తినారు = పైకి + ఎత్తినారు 3) వచ్చిరిపుడు = వచ్చిరి + ఇపుడు 4) మనిషన్నవాడు = మనిషి + అన్నవాడు ఉత్వ సంధి ( ఉకార సంధి ): ఉత్తునకు అచ్చు పరమైనపుడు సంధి నిత్యంగా వస్తుంది . ఉదా :- 1) రాముడతడు = రాముడు + అతడు 2) మనమున్నాము = మనము + ఉన్నాము 3) అతడెక్కడ = అతడు + ఎక్కడ 4) మనసైన = మనసు + ఐన యదగామ సంధి : అంది లేని చోట అచ్చుల మద్య ‘య్’ వచ్చి చేరడాన్ని “యడాగమం” అంటారు . ఉదా :- 1) మాయమ్మ = మా + అమ్మ 2) హరియతడు = హరి + అతడు 3) మాయిల్లు = మా + ఇల్లు ఆమ్రేడిత సంధి : అచ్చునకు ఆమ్రేడితం పరమైతే సంధి తరచుగా వస్తుంది . ఉదా :- 1) ఆహాహా = ఆహా +ఆహా 2) ఔరౌర = ఔర + ఔర 3) అరెరే = అరె + అరె 4) ఏమిటేమిటి = ఏమిటి + ఏమిటి గసడదవాదేశ సంధి : ప్రథమ మీది పరుషాలకు గ, స ,డ , ద ,వ లు బహుళంగా వస్తాయి . ఉదా :- 1) కొలువుసేసి = కొలువు + చేసి 2) కూరగాయలు = కూర + కాయ 3) పాలువోయక = ఆలు + పోయక 4) తల్లిదండ్రులు = తల్లి + తండ్రి త్రిక సంధి : త్రికము మీది అసంయుక్త హల్లునకు దిత్వం బహుళంగా వస్తుంది . ఆ , ఈ , ఏ లు త్రికంఅనబడుతాయి ద్విరుక్తమైన హల్లు పరమైనపుడు, అచ్చికమైన దీర్ఘానికి హ్రస్వం వస్తుంది. ఉదా :- 1) ఇక్కాలము = ఈ + కాలము 2) అక్కోమరుండు = ఆ + కొమరుండు 3) ఎవ్వాడు = ఏ + వాడు 4) అచ్చోట = ఆ + చోట రుగాగమ సంధి : పేదాది శబ్దాలకు ‘ఆల‘ శబ్దo పరమైతే కర్మదారాయం లో రుగాగం వస్తుంది . ఉదా :- 1) మనుమరాలు = మనుమా + ఆలు 2) ధీరురాలు = దీరు + ఆలు. 3) పేదరాలు = పేద + ఆలు 4) బాలెంతరాలు = బాలెంత + ఆలు 5 ) ముద్దరాలు = ముద్ద +ఆలు 6) జవరాలు = జావా + ఆలు సవర్ణ దీర్ఘ సంధి: అ, ఇ , ఉ, ఋ లకు అవే అచ్చులు పరమైతే వాని దీర్గాలు ఎకాదేశంగా వస్తాయి . ఉదా :- 1) రామానుజుడు = రామ + అనుజుడు 2 ) రామాలయం = రామ + ఆలయం. 3) భానూఉదయం = భాను + ఉదయం 4) కవీంద్రుడు = కవి + ఇంద్రుడు. 5 ) పితౄణం = పితృ +ఋణం 6) వదూపేతుడు = వధు + ఉపేతుడు గుణసంధి : ఇ , ఉ , ఋ పరమైతే ఏ, ఓ , ఆర్ లు క్రమంగా ఎకాదేసంగా వస్తాయి . ఉదా :- 1) రాజేంద్రుడు = రాజ + ఇంద్రుడు 2 ) పరోపకారం = పర + ఉపకారం 3) రాజర్షి = రాజ + ఋషి 4)

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ICSE 8th Maths Concept| Basics In Maths

Education, ICSE / Nayini Satyanarayana Reddy

ICSE 8th Maths Concept ICSE 8th Maths Concept 3. SQUARES AND SQUARE ROOTS, CUBES AND CUBE ROOTS Square: A square number is a number raised to the power 2. The number obtained by multiplying the number by itself. Ex: – 1) square of 5 = 52 = 5 × 5 = 25, 2) square of 3 = 32 = 3× 3 = 9 ∗If a natural number p can be expressed as q2, where q is also natural, then p is called a square number. Ex: – 1,4,9, …etc. Test for a number to be a perfect square: If a number is expressed as the product of pairs of equal factors, then it is called a perfect square. Ex: – 36 Prime factors of 36 = 2× 2× 3× 3 36 can be expressed as the product of pairs of equal factors. ∴ 36 is a perfect square. Square Root: the square root of a number x is that number when multiplied by itself gives x as the product. The square root of x is denoted by Methods of Finding Square root of given Number Prime factorization method: – Steps: Resolve the given number into prime factors. Make pairs of similar factors. The product of prime factors, choosing one out of every pair gives the square root of the given number. Ex: – To find the square root of 16 Prim factors of 16 = 2 ×2× 2× 2 = 2 × 2 = 4 ∴ square root of 16 = 4 Division method: – Steps: Mark off the digits in pairs starting with the unit place. Each pair and remaining one digit are called a period. Think of the largest number whose square is equal to or just less than the first period. Take this number as the divisor as well as quotient. Subtract the product of divisor and quotient from the first period and bring down the next period to the right of the remainder. this becomes the new dividend. Now, the new divisor is obtained by taking twice the quotient and annexing with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of the new divisor and this digit is equal to or just less than the new dividend. Repeat steps 2, 3, and 4 till all the periods have been taken up. Thus, the obtained quotient is the required square root. Ex: – To find the square root of 225 Properties of a perfect square: 1. The square of an even number is always an even number. Ex: – 22 = 4 (4 is even), 62 = 36 (36 is even), here 2, 6 are an even number. 2. The square of an odd number is always an odd number. Ex: – 32 = 9 (9 is even), 152 = 225 (225 is even), here 3, 15 are an odd number. 3. The square of a proper fraction is a proper fraction less than the given fraction. Ex: – 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 ICSE 8th Maths Concept 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 ICSE 8th Maths Concept Methods of finding the cube root of the given Number Prime factorisation method: – Steps: Resolve the given number into prime factors. Make triplets of similar factors. The product of prime factors, choosing one out of every triplet gives the cube root of the given number. Ex: – 27 Prime factors of 27 = 3×3×3 = 3 ∴ cube root of 27 = 3 Test for a number to be a 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. Visit my YouTube Channel: Click on the logo below

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ICSE Class 9 Maths Made Easy: From Confusion to Confidence

Education, ICSE / Nayini Satyanarayana Reddy

ICSE Class 9 Maths Made Easy ICSE Class 9 Maths 9th Class Maths Concept: This note is prepared by the Basics in Maths team.These notes help ICSE 9th class Maths students develop interest in mathematics and remove fear. These notes cover all the topics covered in the ICSE 9th class Maths syllabus and include plenty of formulae and concepts to help you solve all the types of ICSE 9th Math problems appear in CBSE board and entrance examinations ICSE Class 9 Maths Made Easy 1. RATIONAL AND IRRATIONAL NUMBERS Natural numbers: counting numbers 1, 2, 3… called Natural numbers.The symbol N represents natural numbers. N = {1, 2, 3…} Whole numbers: Natural numbers with 0 form whole numbers. The symbol W represents whole numbers. W = {0, 1, 2, 3…} Integers: Integers include zero, whole numbers greater than zero, and whole numbers less than zero. They are represented by I or Z. Z = {…-3, -2, -1, 0, 1, 2, 3…} Rational number Any number expressed as p/q, with p and q as integers and q ≠ 0, is known as a rational number. The symbol used is Q. ∗ A rational number may have both its numerator and denominator positive or negative. For convenience, the denominator is assumed not to be negative. Ex: can be written as but our convenience, we can take Equal rational numbers: For any 4 integers a, b, c, and d (b, d ≠ 0), we have ⇒ ad = bc The order of Rational numbers: If are two rational numbers such that b> 0 and d > 0 then ⇒ ad > bc Absolute value of rational numbers: The absolute value of a rational number is always positive. The absolute value of is denoted by . Ex: – absolute value of To find a rational number between given numbers: Mean method: – A rational number between two numbers a and b is Ex: – insert two rational numbers between 1 and 2 1 < < 2 ⟹ 1 < < 2 1 < < 2 ⟹ 1 < 2 To rational numbers in a single step: – Ex:- insert two rational numbers between 1 and 2 To find two rational numbers, we can use 1 and 2 as rational numbers with the same denominator, 3 (∵ 1 + 2 = 3) 1 = and 2 = Note: – There are infinitely many rational numbers between two numbers. The decimal form of rational numbers ∗ A rational number is written either as a terminating decimal or as a non-terminating repeating decimal. Converting decimal form into the form: 1. Terminating decimals: – 1.2 = 1.35 = 2. Non-Terminating repeating decimals: – Irrational numbers: The numbers which are not written in the form of , where p, q are integers, and q ≠ 0 are called rational numbers. Rational numbers are denoted by QI or S. Every irrational number is expressed as a non-terminating, non-repeating decimal. Ex:- and so on. Calculation of square roots: There is a reference to irrationals in the calculation of square roots in the Sulba Sutra. Procedure for finding value: Visit my YouTube Channel: Click on Below Logo

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ICSE X Class Maths Concept

Education, ICSE / Nayini Satyanarayana Reddy

ICSE X Class Maths ICSE X Class Maths Concept designed by the ‘Basics in Maths’ team. These notes to do help the ICSE 10th class Maths students fall in love with mathematics and overcome their fear. These notes cover all the topics covered in the ICSE 10th class Maths syllabus and include plenty of formulae and concept to help you solve all the types of 10th class Mathematics problems asked in the ICSE board and entrance examinations. 1. Goods and Service Tax Two types of taxes in the Indian Government: 1.Direct taxes: – These are the taxes paid by an organisation or individual directly to the government. These include Income tax, Capital gain tax and Corporate tax. 2.Indirect taxes: – These are the taxes on goods and services paid by the customer, collected by an individual or an organisation and deposited with the Government. Earlier there were several indirect taxes levied by the central and state Governments. Goods and Service Tax (GST): GST is a comprehensive indirect tax for the whole nation. It makes India one unified common market. Registration under GST: Any individual or organisation that has an annual turnover of more than ₹ 20 lakh is to be registered under GST. Input and Output GST: For any individual or organisation, the GST paid on purchases is called the ‘Input GST’ and the GST collection on sale of goods is called the ‘Output GST’. The input GST is set off against the output GST and the difference between the two is payable in the Government account. One currency one tax: There is a uniform GST rate on any particular goods or services across all states and Union Territories of India. This is called ‘One currency one tax’. Note: Assam was the first state to implement GST and Jammu & Kashmir was the last. GST rate slabs: However, the tax on gold is kept at 3% and on rough precious and semi-precious is kept at 0.25%. The multitier GST tax rate system in India has been developed keeping in mind that essential commodities should be taxed less than luxury goods. Benefits of GST for Traders: • Simple tax system. • Elimination of multiplicity of taxes. • Development of a common market nation-wide. • Reduction of cascading effect. • Lower taxes result in the reduction of costs making in the domestic market. Benefits of GST for Consumers: • Single and transparent System. • Elimination of cascading effect has resulted in the reduction in the costs of goods and services. • Increase in purchasing power and savings. Benefits of GST for Traders: • Single tax system, simple and easy to administer. • Higher revenue efficiency. • Better control on leakage and tax evasion. Types of GST in India Central GST (CGST): For any intrastate supply half of the GST collected as the output GST is deposited with the Central Governments as CGST. State GST or Union Territory GST (SGST/UGST): For any local supply (supply with in the same state or Union Territory) half of the GST is deposited with the respective state or Union Territory Government as the beneficiary. This is called SGST/UGST. Integrated GST (IGST): The GST levied on the supply of goods or services in the case of interstate trade within India or in the case of exports/imports is known as IGST. Reverse charge Mechanism: There are cases where the chargeability gets reversed, that is the receiver becomes liable to pay the tax and deposit it to the Government Account. Composition shame: The composition is meant for small dealers and service providers with an annual turnover less than ₹ 1.5 crores and also for Restaurant service providers. Under this scheme the rates of GST are: Input Tax Credit (ITC) When a dealer sells his goods, he charges the output GST from his customer which he has to deposit in the government account, but in running his business he had paid input GST on the goods he had availed. This input GST, he utilizes as Input Tax credit and deposits the exes amount of output GST with the Government. Input Tax credit is a provision of reducing the GST already paid on inputs in order to avoid the cascading of taxes. GST payable = Output GST – ITC Claiming ITC: A dealer registered under GST can claim ITC only if: He possesses the tax invoice. He has received the said goods/services He has filed the returns. The tax paid by him has been paid to the government by his supplier. Utilization of ITC: The Amount of ITC available to any registered dealer shall be utilized to reduce the out put tax liability in the sequence shown in the table. E – ledgers under GST: An E – ledger is an electronic form of a pass book available to all GST registrants on the GST portal. These are of three types: (i) Electric cash ledger (ii) Electric credit ledger and (iii) Electric Liability Register (i) Electric cash ledger: It contains the amounts of GST deposited in each to the government. (ii) Electric credit ledger: It contains the balance of ITC available to the dealer. (iii) Electric credit ledger: It contains all the Tax liability of the dealer. GST Returns: These are the information provided from time to time by the dealer to the Government regarding the ITC, output Tax liability and the amounts of GST deposited. A GST registered person has to submit the following returns: E – Way bill: E – Way bill is an electronic way bill that can be generated on the E – Way bill portal. A registered person can not transport goods whose value exceeds ₹ 50,000 in a vehicle without an e – way bill. When an E – way bill is generated, a unique e – way bill number (EBN) is allocated and is available to the supplier, the transporter and recipient. A dealer must generate an E – way bill if he has to transport them for returning to the supplier.

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CBSE 10th Class Maths Concept

CBSE, Education / Nayini Satyanarayana Reddy

CBSE 10th Class Maths Concept CBSE 10th Class Maths: This concept note is designed for CBSE 10th class maths students, This concept notes is to help students for the CBSE board examination and other competitive exams also… CBSE 10th Class Maths 1. REAL NUMBERS Rational number: The number, which is written in the form of is called a rational number. It is denoted by Q. Irrational number: – the number, which is not rational is called an irrational number. It is denoted by Q’ or S. 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 are said to be co-prime numbers, if they have no common factor except 1. [Ex: (1, 2), (3, 4), (4, 7) …etc.] Euclid division lemma: – For any positive integers a and b, then q, rare integers exist uniquely satisfying the rules a = bq + r, 0 ≤ r < b. To find H.C.F by using Euclid’s division lemma: For any two integers a and b (a > b). Apply Euclid division lemma, to a and b, we find whole numbers q and r such that a = bq + r, 0≤r<b. If r = 0, b is the H.C.F of a and b. If r≠ 0, apply Euclid division lemma, to b and r. Continue the process till the remainder is zero. The divisor at this stage is the required H.C.F. Note: – Euclid division lemma also called a division algorithm. Euclid division lemma is stated for only positive integers, it can be extended for all integers except 0. The fundamental theorem of arithmetic: Every composite number can be expressed as a product of primes, and this factorization is unique, apart from the order in which prime factors occur. Ex: – 12 = 2 ×2× 3, 15 = 3× 5 and so on. CBSE 10th Class Maths Concept To find LCM and HCF by using the prime factorization method: H.C.F = product of the smallest power of each common prime factor of given numbers. L.C.M = product of the greatest power of each prime factor of given numbers. ‘p’ is a prime number and ‘a’ is a positive integer, if p divides a2, then p divides a. 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. Decimal expansion of rational numbers: Decimal expansion of rational numbers is either terminating or non-terminating repeating (recurring)decimals. Ex: – 1.34, 2.345, 1.2222… and so on. In p/q, if the prime factorization of q is in form 2m 5n, then p/q is a terminating decimal. Otherwise non-terminating repeating decimal. Decimal expansion of irrational numbers: Decimal expansion of irrational numbers is non-terminating decimals. Ex: – 1.414…., 1.314…. TS 10th class maths concept (E/M) Ts Inter Maths IA Concept PDF Files || Inter Mathematics 1A and 1B Visit My Youtube Channel: Click on below logo

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12 th Class Maths Concept CBSE

CBSE, Education / Nayini Satyanarayana Reddy

12 th Class Maths Concept 12 th Class Maths: This note is designed by the ‘Basics in Maths’ team. These notes are to help the CBSE 12th class Maths students fall in love with mathematics and overcome their fear. These notes cover all the topics covered in the CBSE 12th class Maths syllabus and include plenty of formulae and concepts to help you solve all the types of 12thMath problems asked in the CBSE board and entrance examinations. 12 th Class Maths Concept 1. RELATIONS AND FUNCTIONS Ordered pair: Two elements a and b listed in a specific order form. An ordered pair is 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 set of all ordered pairs (a, b) where a ϵ A and b ϵ B. Relation: Let A and B are two non-empty sets the relation R from A to B is subset of A×B. ⇒ R: A→B is a relation if R⊂ A × B Types of relations: Empty relation: – A relation in a set A is said to be an empty relation if no element of A is related to any element of A. R = ∅ ⊂ A × A Universal Relation: – A relation in a set A is said to be a universal relation if each element of A is related to every element of A R = A × A Both empty relations and universal relations are sometimes called trivial relations. Reflexive relation: A relation R in a set A is said to be reflexive if ∀a ∈ A ⇒ (a, a) ∈ R. Symmetric relation: A relation R in a set A is said to be symmetric, if ∀a, b ∈ A ⇒, (a, b) ∈ R ⇒ (b, a) ∈ R. Anti-Symmetric relation: A relation R in a set A is said to be Anti-symmetric, if ∀a, b ∈ A; (a, b) ∈ R (b, a) ∈ R ⇒ a = b. Transitive relation: A relation R in a set A is said to be Transitive, if ∀a, b, c∈ A; (a, b) ∈ R (b, c) ∈ R ⇒ a = c. Equivalence relation: A relation R in a set A is said to equivalence relation if it is reflexive, symmetric and transitive. Function: A relation f: X → Y is said to be a function if ∀ xϵ X, there exists a unique element y in Y such that (x, y) ϵ f. (Or) A relation f: A → B is said to be a function if (i) x ϵ X ⇒ f(x) ϵ Y (ii) x1, x2 ϵ X, x1 = x2 in X ⇒ f(x1) = f(x2) in Y. TYPES OF FUNCTIONS One– one Function (Injective): A function f: X→ Y is said to be a one-to-one function or injective if different elements in X have different images in Y. (Or) A function f: X→ Y is said to be one-one function if f(x1) = f(x2) in Y ⇒x1 = x2 in X. On to Function (Surjection): – A function f: X→ Y is said to be onto function or surjection if for each yϵ Y there exists x ϵ X such that f(x) = y. Bijection: – A function f: X→ Y is said to be Bijection if it is both ‘one-one’ and ‘onto’. Composite function: If f: A→B, g: B→C are two functions then the composite relation gof is a function from A to C. gof: A→C is a composite function and is defined by gof(x) = g(f(x)). Visit my Youtube Channel: Click on Below Logo

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11th Class Maths Concept

CBSE, Education / Nayini Satyanarayana Reddy

11th Class Maths Concept 11th Class Maths: This note is designed by ‘Basics in Maths’ team. These notes to do help the CBSE 11th class Maths students fall in love with mathematics and overcome their fear. 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: All objects in a set must have the same general similarity or property. Must be able to confirm whether something belongs to the set or not. Set: – A collection of well-defined objects is called a set. ∗ Sets are usually denoted by capital English alphabets like A, B, C, and so on. ∗ The elements in set are taken as small English alphabets like a, b, c, and so on. ∗ Set theory was developed by George canter. • If any object belongs to a set, then it is called an object/element. We denote by ∈ to indicate that it belongs to. If it does not belong to the set then it is denoted by ∉. Ex: – 1 ∈ N, 0 ∈ W, −1 ∈ Z, 0 ∉ N, etc. Methods of representing sets: Roster or table or listed form: – In this form all the elements of the set are listed, and the elements are separated by commas and enclosed within braces { }. Ex: – set of vowels in English alphabet = {a, e, I, o, u}, set of even natural numbers less than 10 = {2, 4, 6, 8} etc. Note: – In roster form, an element is not repeated. We can list the elements in any order. Set builder form: Pointing an element in a set to x (or any symbols such as y, z, etc.) followed by a colon(:), next to write the properties or properties of the elements in that set and placed in flower brackets is called the set builder form.: Or / symbols read as ‘such that’ Ex: – {2, 4, 6, 8} = {x / x is an even and x ∈N, x< 10}, {a, e, i, o, u} = {x : x is a vowel in English alphabet}. Null set: – (empty set or void set) the set which has no elements is called as a null set. It is denoted by ∅ or { }. Finite and infinite sets: – If a set contains a finite no. of elements then it is called a finite set. If a set contains an infinite no. of elements then it is called an infinite set. Ex: – A = {1, 2,3, 4} → finite set B = {1, 2, 3, 4….} Equal sets: – two sets A and B are said to be equal sets if they have the same elements., and write as A = B Ex: – A = {1, 2, 3, 4}, B = {3, 1, 4, 2} ⟹ A = B. Subset: – for any two sets A and B, if every element of set A is in set B, then we can say that A is a subset of B. It is denoted by A ⊂ B. Ex: – If A = {1, 2, 3, 4, 5, 6, 7, 8}, subsets of A are {1}, {1, 3, 5}, {1,2,3,4}, and so on. Power set: – set of all the subsets of a set A is called the power set of A. It is denoted by p(A). Ex: – A = {1,2,3} P(A) = {{1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1, 2, 3}, ∅}. Intervals: ∗ Open interval: – (a, b) = {x: a< x <b} → set of rational numbers lies between a and b. ∗ Closed interval: – [a, b] = {x: a≤ x ≤b} → set of rational numbers lies between a and b, including a and b. ∗ Open – closed: – (a, b] = {x: a< x ≤b} → set of rational numbers lies between a and b, excluding a and including b. ∗ Closed-open: -[a, b) = {x: a≤ x <b} → set of rational numbers lies between a and b, including a and excluding b. Universal set: – A set that contains all the subsets of it under our consideration is called a universal set. Cardinal number of a set: – Number of elements in a set A is called the cardinal number of that set A. It is denoted by n(A). • If a set has n elements, then no. of elements of that set has 2n Equivalent sets: – two set A and B are said to be equivalent sets if n(A) = n(B) (they have the same cardinal number). Ex: – A = {1, 2, 3}, B = {a, b, c} n(A) = 3 and n(B) = 3 ∴ A = B. Venn diagrams: U = {1, 2, 3, 4, 5, 6} The relationship between sets is usually represented by means of diagrams, which are known as ‘Venn diagrams. These diagrams consist of rectangles and circles. A universal set is represented by rectangles and subsets by circles. U = {1, 2, 3, 4, 5, 6} A = {1, 2, 3} B = {1, 2} Visit my YouTube Channel: Click on the logo below

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TS inter 1st year Maths Blueprint

Education, TS Inter Maths / Nayini Satyanarayana Reddy

TS inter1st year Maths Blueprint TS Inter 1st year Maths Blueprint: : These blueprints were designed by the ‘Basics in Maths’ team. These to-dos help the TS intermediate first-year Maths students fall in love with mathematics and overcome their fear. These blueprints cover all the topics of the TS I.P.E first-year maths syllabus and help in I.P.E exams. Maths IA Two-Mark Questions & Solutions Maths IB Two Marks Questions & Solutions Maths – IA Concept Maths – IB Concept TS inter 1st year Maths Blueprin Maths – IIA Concept Maths – IIB Concept YouTube

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TS Inter second year Maths 2B Concept

Education, TS Inter Maths, TS Inter Maths 2B / Nayini Satyanarayana Reddy

TS Inter second year TS Inter second year: This note is designed by the ‘Basics in Maths’ team. These notes to do help the TS intermediate second-year Maths students fall in love with mathematics and overcome the fear. These notes cover all the topics covered in the TS I.P.E second year maths 2B syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations. TS Inter second year 1. CIRCLES Circle: In a plane, the set of points that are at a constant distance from a fixed point is called a circle. ∗ The fixed point is called the centre (C) of the circle and the constant distance is called the radius(r) of the circle Unit circle: If the radius of the circle is 1 unit, then that circle is called the unit circle. Point Circle: A circle is said to be a point circle if its radius is zero. A point circle contains only one point in the centre of the circle. • ∗ The equation of the circle with centre (h, k) and radius r is (x – h)2 + (y – k)2 = r2 ∗ The equation of the circle with centre origin and radius r is x2 + y2 = r2 ⇒ x2 + y2 = r2 is called standard form of the circle. The general equation of the second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, where a, b, f, g, h and c are real numbers, represent a circle iff (i) a = b ≠ 0 (ii) h = 0 and (iii) g2 + f2 + c ≥ 0 ∗ The general equation of the circle is x2 + y2 + 2gx + 2fy + c = 0 It’s centre c = (– g, – f) and radius ∗ The equation of the circle passing through origin is x2 + y2 + 2gx + 2fy = 0. ∗ The equation of the circle whose centre on the x-axis is x2 + y2 + 2gx + c = 0. ∗ The equation of the circle having centre on y-axis is x2 + y2 + 2fy + c = 0. ∗ The circles which have the same centre are called concentric circles. ∗ The equation of the circle concentric with the circle x2 + y2 + 2gx + 2fy + c = 0 is x2 + y2 + 2gx + 2fy + k = 0. ∗ The length of the intercept made by a circle x2 + y2 + 2gx + 2fy + c = 0 on x -axis is if g2 – c > 0 y -axis is if f2 – c > 0 Note: – (a) if g2 – c = 0, then A1 A2 = 0 ⇒ the circle touches the x- axis at only one point. (b) if f2 – c = 0, then B1 B2 = 0 ⇒ the circle touches the y- axis at only one point. (c) if g2 – c < 0, then the circle does not meet the x- axis. (d) if f2 – c < 0, then the circle does not meet the y- axis. ∗ The equation of the circle having the line segment joining A (x1, y1) and B (x2, y2) as a diameter is (x – x1) (x – x2) + (y – y1) (y – y2) = 0. ∗ Let A, B be any two points on a circle then, The line is called the secant line of the circle. The line segment is called the chard of the circle. AB is called the length of the chord. ∗ A chord passing through the centre is called the diameter of the circle. ∗ The angle subtended by a chord on the circumference of at any point is equal. The perpendicular bisector of a chord of a circle is asses through the centre of the circle. ∗ The angle in a semicircle is 900. ∗ The equation of the circle passing through three non-collinear points A (x1, y1), B (x2, y2), C (x3, y3) is Where ci = − (x2 + y2) and i = 1,2,3 ∗ centre of the circle is Parametric form: If P (x, y) is a point on the circle with centre (h, k) and radius r, then X = h + r cosθ, y = k + r sinθ 0 ≤ θ ≤ 2π. ⇒ A point n the circle x2 + y2 = r2 is taken as (r cosθ, r sinθ) and simply denoted by θ. Note: If the centre of the circle is the origin, then the parametric equations are x = r cosθ, y = r, 0 ≤ θ ≤ 2π. The point (h + rcosθ1, k + r sin θ1) is referred to as the point θ1 on the circle having the centre (h, k) and radius r. Notations: S = x2 + y2 + 2gx + 2fy + c S1 = xx1 + yy1 + g(x +x1) + f (y +y1) + c S11 = x12 + y12 + 2gx1 +2fy1 + c S12 = x1x2 + y1y2 + g(x1 + x2 ) + f (y1 + y2) + c Position of a point with respect to the circle: A circle divides the plane into three parts. 1. The interior of the circle 2. The circumference which is the circular curve. 3. The exterior of the circle. Power of point: Les S = 0 be a circle with radius ‘r’ and centre ‘C’ and P (x1, y1) be a point on the circle, then CP – r2 is called the power of point ‘P’ concerning S = 0. The power of point P (x1, y1) w.r.t. S = 0 is S11. •Let S = 0 be a circle in a plane and P

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