MATHEMATICS I & II - VIDYADHAN COLLEGE
MATHEMATICS I & IIDIPLOMA COURSE IN ENGINEERINGFIRST SEMESTERA Publication underGovernment of TamilnaduDistribution of Free Textbook Programme( NOT FOR SALE )Untouchability is a sinUntouchability is a crimeUntouchability is inhumanDIRECTORATE OF TECHNICAL EDUCATIONGOVERNMENT OF TAMIL NADU
Government of TamilnaduFirst Edition – 2011ChairpersonThiru Kumar Jayanth I.A.SCommissioner of Technical EducationDirectorate of Technical Education, Chennai –25Co-ordinatorConvenerEr. S. GovindarajanP.L. SankarPrincipalDr. Dharmambal GovernmentPolytechnic CollegeTharamani, Chennai – 113Lecturer (Selection Grade)Rajagopal Polytechnic CollegeGudiyatham-632602ReviewerDr. S. Paul RajAssociate Professor, Dept of MathematicsAnna University, MIT Campus, Chennai - 42.AuthorsR. RamadossLecturer, (Selection Grade)TPEVR Govt. Polytechnic CollegeVellore-632002B.R. NarasimhanLecturer (Selection Grade)Arulmigu Palaniandavar PolytechnicCollegePalani-624601M. DevarajanLecturer (Selection Grade)Dr. Dharmambal GovernmentPolytechnic College for WomenTaramani, Chennai-600113Dr. L. RamuppillaiLecturer (Selection Grade)Thiagarajar Polytechnic CollegeSelam-636005K. ShanmugamLecturer (Selection Grade)Government Polytechnic CollegePurasawalkam, Chennai-600 012Dr.A. ShanmugasundaramLecturer (Selection Grade)Valivalam Desikar Polytechnic CollegeNagappattinam-611 001M. RamalingamLecturer (Selection Grade)Government Polytechnic CollegeTuticorin-628008R. SubramanianLecturer (Selection Grade)Arasan Ganesan Polytechnic CollegeSivakasi-626130Y. Antony LeoLecturerMothilal Nehru GovernmentPolytechnic CollegePondicherry-605008C. SaravananLecturer (Senior Scale)Annamalai Polytechnic CollegeChettinad-630102This book has been prepared by the Directorate of Technical EducationThis book has been printed on 60 G.S.M PaperThrough the Tamil Nadu Text Book Corporation
FOREWORDWe take great pleasure in presenting this book of mathematics tothe students of Polytechnic Colleges. This book is prepared inaccordance with the new syllabus framed by the Directorate ofTechnical Education, Chennai.This book has been prepared keeping in mind, the aptitude andattitude of the students and modern methods of education. The lucidmanner in which the concepts are explained, make the teachinglearning process more easy and effective. Each chapter in this book isprepared with strenuous efforts to present the principles of the subjectin the most easy-to-understand and the most easy-to-workoutmanner.Each chapter is presented with an introduction, definition,theorems, explanation, worked examples and exercises given are forbetter understanding of concepts and in the exercises, problems havebeen given in view of enough practice for mastering the concept.We hope that this book serves the purpose i.e., the curriculumwhich is revised by DTE, keeping in mind the changing needs of thesociety, to make it lively and vibrating. The language used is very clearand simple which is up to the level of comprehension of students.List of reference books provided will be of much helpful for furtherreference and enrichment of the various topics.We extend our deep sense of gratitude to Thiru.S.Govindarajan,Co-ordinator and Principal, Dr. Dharmambal Government polytechnicCollege for women, Chennai and Thiru. P.L. Sankar, convener,Rajagopal polytechnic College, Gudiyatham who took sincere effortsin preparing and reviewing this book.Valuable suggestions and constructive criticisms forimprovement of this book will be thankfully acknowledged.Wishing you all success.Authorsiii
SYLLABUSFIRST SEMESTER MATHEMATICS - IUNIT - IDETERMINANTS1.1 Definition and expansion of determinants of order 2 and 3 .Propertiesof determinants .Cramer's rule to solve simultaneous equations in 2and 3 unknowns-simple problems.1.2 Problems involving properties of determinants1.3 Matrices :Definition of matrix .Types of matrices .Algebra of matricessuch as equality, addition, subtraction, scalar multiplication andmultiplication of matrices. Transpose of a matrix, adjoint matrix andinverse matrix-simple problems.UNIT - IIBINOMIAL THEOREM2.1 Definition of factorial notation, definition of Permutation andCombinations with formula. Binomial theorem for positive integralindex (statement only), finding of general and middle terms. Simpleproblems.2.2 Problems finding co-efficient of xn, independent terms. Simpleproblems. Binomial Theorem for rational index, expansions, onlyupto – 3 for negative integers. Simple Expansions2.3 Partial Fractions :Definition of Polynomial fraction, proper andimproper fractions and definition of partial fractions.To resolve proper fraction into partial fraction with denominatorcontaining non repeated linear factors, repeated linear factors andirreducible non repeated quadratic factors. Simple problems.iv
UNIT - IIISTRAIGHT LINES3.1 Length of perpendicular distance from a point to the line andperpendicular distance between parallel lines. Simple problems.Angle between two straight lines and condition for parallel andperpendicular lines. Simple problems3.2 Pair of straight lines Through origin :Pair of lines passing through theorigin ax2 2hxy by2 0 expressed in the form (y-m1x)(y-m2x) 0.Derivation of angle between pair of straight lines. Condition forparallel and perpendicular lines. Simple problems3.3 Pair of straight lines not through origin: Condition for generalequation of the second degree ax2 2hxy by2 2gx 2fy c 0 torepresent pair of lines.(Statement only) Angle between them,condition for parallel and perpendicular lines. Simple problems.UNIT - IVTRIGONOMETRY4.1 Trigonometrical ratio of allied angles-Expansion of Sin(A B) andcos(A B)- problems using above expansion4.2 Expansion of tan(A B) and Problems using this expansion4.3 Trigonometrical ratios of multiple angles (2A only) and sub-multipleangles. Simple problems.UNIT - VTRIGONOMETRY5.1 Trigonometrical ratios of multiple angels (3A only) Simpleproblems.5.2 Sum and Product formulae-Simple Problems.5.3 Definition of inverse trigonometric ratios, relation between inversetrigonometric ratios-Simple Problemsv
FIRST SEMESTER MATHEMATICS IIUNIT - ICIRCLES1.1 Equation of circle – given centre and radius. General Equation ofcircle – finding center and radius. Simple problems.1.2 Equation of circle through three non collinear points – concyclicpoints. Equation of circle on the line joining the points (x1,y1) and(x2,y2) as diameter. Simple problems.1.3 Length of the tangent-Position of a point with respect to a circle.Equation of tangent (Derivation not required). Simple problems.UNIT-IIFAMILY OF CIRCLES:2.1 Concentric circles – contact of circles (internal and external circles) –orthogonal circles – condition for orthogonal circles.(Result only).Simple Problems2.2 Limits:Definition of limits x n - an na n-1x a x-aLtsin q 1,q 0 qLtLtq 0tan q 1q(q in radian)[Results only] – Problems using the above results.2.3 Differentiation:Definition – Differentiation of u xn, sinx, cosx, tanx,vcotx, secx, cosecx, logx, ex, u v, uv, uvw,(Results only).Simple problems using the above results.UNIT- III3.1 Differentiation of function of functions and Implicit functions. SimpleProblems.vi
3.2 Differentiation of inverse trigonometric functions and parametricfunctions. Simple problems.3.3 Successive differentiation up to second order (parametric form notincluded) Definition of differential equation, formation of differentialequation. Simple ProblemsUNIT- IVAPPLICATION OF DIFFERENTIATION–I4.1 Derivative as a rate measure-simple Problems.4.2 Velocity and Acceleration-simple Problems4.3 Tangents and Normals-simple ProblemsUNIT-VAPPLICATION OF DIFFERENTIATION –II5.1 Definition of Increasing function, Decreasing function and turningpoints. Maxima and Minima (for single variable only) – SimpleProblems.5.2 Partial Differentiation: Partial differentiation of two variable up tosecond order only. Simple problems.5.3 Definition of Homogeneous functions-Eulers theorem-SimpleProblems.vii
FIRST SEMESTERMATHEMATICS - IContentsPage NoUnit – 1DETERMINANTS .11.1 Introduction . 11.2 Problems Involving Properties of Determinants . 121.3 Matrices . 19Unit – 2BINOMIAL THEOREM. 442.1 Introduction . 452.2 Binomial Theorem . 472.3 Partial Fractions . 55Unit – 3STRAIGHT LINES . 693.1 Introduction . 693.2 Pair of straight lines through origin . 813.3 Pair of straight lines not through origin . 89Unit – 4TRIGONOMETRY . 1014.1 Trigonometrical Ratios ofRelated or Allied Angles . 1044.2 Compound Angles (Continued) . 1074.3 Multiple Angles of 2A Only andSub – Multiple Angles. 119Unit - 5TRIGONOMETRY . 1295.1 Trigonometrical Ratios of Multiple Angle of 3A . 1295.2 Sum and Product Formulae . 1355.3 Inverse Trigonometric Function . 142MODEL QUESTION PAPER . 161viii
MATHEMATICS – IIContentsPage NoUnit – 1CIRCLES . 1671.1 Circles . 1671.2 Concyclic Points. 1711.3 Length of the Tangent to a circlefrom a point(x1,y1) . 176Unit – 2FAMILY OF CIRCLES . 1852.1 Family of circles . 1852.2 Definition of limits . 1912.3 Differentiation. 196Unit – 3Differentiation Methods . 2063.1 Differentiation of function of functions . 2063.2 Differentiation of Inverse Trigonometric Functions .2143.3 successive differentiation . 223Unit – 4APPLICATION OF DIFFERENTIATION .2324.1 Derivative as a Rate Measure . 2324.2 Velocity and Acceleration . 2384.3 Tangents and Normals . 242Unit – 5APPLICATION OF DIFFERENTIATION-II . 2545.1 Introduction . 2545.2. Partial derivatives. 2715.3 Homogeneous Functions . 277MODEL QUESTION PAPER . 292ix
SEMESTER IMATHEMATICS – IUNIT – IDETERMINANTS1.1Definition and expansion of determinants of order 2 and 3Properties of determinants Cramer’s rule to solve simultaneousequations in 2 and 3 unknowns-simple problems.1.2Problems involving properties of determinants1.3MatricesDefinition of matrix. Types of matrices. Algebra of matrices suchas equality, addition, subtraction, scalar multiplication andmultiplication of matrices. Transpose of a matrix, adjoint matrixand inverse matrix-simple problems.1.1. DETERMINANTThe credit for the discovery of the subject of determinant goes tothe German mathematician, Gauss. After the introduction ofdeterminants, solving a system of simultaneous linear equationsbecomes much simpler.Definition:Determinant is a square arrangement of numbers (real orcomplex) within two vertical lines.Example :a1a2b1is a determinantb2Determinant of second order:The symbola bc dconsisting of 4 numbers a, b, c and arranged intwo rows and two columus is called a determinant of second order.The numbers a,b,c, and d are called elements of the determinantThe value of the determinant is Δ ad - bc1
Examples:1.2.2 35 1463 5 (2) (1) – (5) (3) 2 – 15 -13 (4) (-5) – (6) (3) -20 – 18 -38Determent of third order:a1The expression a 2a3b1b2b3c1c 2 consisting ofc3nine elements arranged in three rows and three columns is called adeterminant of third orderThe value of the determinant is obtained by expanding thedeterminant along the first rowΔ a1b2b3c2a- b1 2c3a3c2a c1 2c3a3b2b3 a1 (b 2 c 3 - b 3 c 2 ) - b1 (a 2 c 3 - a 3 c 2 ) c 1 (a 2b 3 - a 3 b 2 )Note: The determinant can be expanded along any row or column.Examples: 1(1 8) 2 (2 20 ) 3 ( 4 5)1 2 3 1( 7) 2 ( 18 ) 3 ( 1)2 1 4(1) 7 36 35 2 1 10 36 26 3(6 4) 1( 2 3)3 01 3(10 ) 1(1)2 3 4(2) 30 11 1 2 31Minor of an elementDefinition :Minor of an element is a determinant obtained by deleting theththrow and column in which that element occurs. The Minor of I row JColumn element is denoted by mij2
Example:1 1 30 4211 5 3Minor of 3 0 4 0-44 -4411 5Minor of 0 1 3 3-15 -125 3Cofactor of an elementDefinition :ththCo-factor of an element in i row,j column is the signed minor ofththI row J Column element and is denoted by Aij.i j(i.e) Aij (-1) mijThe sign is attached by the rule (-1) i jExample3 2 42 1 07 11 6Co-factor of -2 (-1)Co-factor of 7 (-1)1 23 12 07 63 (-1) (12) -12 2 44 (-1) (0-4) -41 0Properties of Determinants:Property 1:The value of a determinant is unaltered when the rows andcolumns are interchanged.a1 a 2(i.e) If Δ b1 b 2c1 c 2a3a1 b1Tb 3 and Δ a 2 b 2c3a3 b3then Δ ΔT3c1c2 ,c3
Property 2:If any two rows or columns of a determinant are interchanged thevalue of the determinant is changed in its sign.a1 a 2If Δ b1 b 2c1 c 2a3b1 b 2b 3 and Δ1 a1 a 2c3c1 c 2b3a3 ,c3then Δ1 -ΔNote: R1 and R2 are interchanged.Property 3:If any two rows or columns of a determinant are identical, thenthe value of the determinant is zero.a1 a 2(i.e) The value of a1 a 2c1 c 2a3a 3 is zeroc3Since R1 R2Property 4:If each element of a row or column of a determinant is multipliedby any number K / 0, then the value of the determinant is multiplied bythe same number K.a1 b1If Δ a 2 b 2a3 b3c1c2c3Ka1 Kb1 Kc 1c2 ,and Δ1 a 2 b 2a3b3c3then Δ1 KΔProperty 5:If each element of a row or column is expressed as the sum oftwo elements, then the determinant can be expressed as the sum oftwo determinants of the same order.4
(i.e) If Δ a1 d1 b1 d2a2b2a3b3a1 b1then Δ a 2 b 2a3 b3c 1 d3c2 ,c3c1ª d1 d2«c 2 «a 2 b 2« a 3 b 3c3d3 º»c2 »c 3 »¼Property 6:If each element of a row or column of a determinant is multipliedby a constant K / 0 and then added to or subtracted from thecorresponding elements of any other row or column then the value ofthe determinant is unaltered.a1 b1Let Δ a 2 b 2a3 b3c1c2c3a1 ma 2 na 3Δ1 a2a3a1 b1 a2 b2a3 b3a2 Δ m a2a3b1 mb 2 nb 3b2b3c1ma 2c2 a2c3a3b2b2b3mb 2b2b3c2a3c 2 n a 2c3a3c1 mc 2 nc 3c2c3mc 2na 3c 2 a2c3a3b3b2b3nb 3b2b3nc 3c2c3c3c2c3Δ1 Δ m (0) n (0) ΔProperty 7:In a given determinant if two rows or columns are identical forx a, then (x-a) is a factor of the determinant.5
Let Δ 1aa31bb31For a b, Δ bb31cc31bb31cc3 0[ C1, and C2 are identical] (a-b) is a factor of ΔNotation :Usually the three rows of the determinant first row, second rowand third row are denoted by R1, R2 and R3 respectively and thecolumns by C1, C2 and C3If we have to interchange two rows say R1 and R2 the symboldouble sided arrow will be used. We will write like this R2 R2 itshould be read as “is interchanged with” similarly for columnsC2 C2.If the elements of R2 are subtracted from the correspondingelements of R1 , then we write R1 - R2 similarly for columns also.If the elements of one column say C1 ,‘m’ times the element ofC2 and n times that of C3 are added, we write like this C1 C1 m C2 n C3 . Here one sided arrow is to be read as “is changed to”Solution of simultaneous equations using Cramer’s rule:Consider the linear equations.a1x b1y c1a2 x b2 y c 2let Δ a1 b1a2 b26
Δx c 1 b1c 2 b2Δy a1a2c1c2ΔyΔxand y , provided Δ 0ΔΔx, y are unique solutions of the given equations. This method ofsolving the line equations is called Cramer’s rule.Then x Similarly for a set of three simultaneous equations in x, y and za 1 x b 1 y c 1 z d 1a 2 x b 2 y c 2 z d 2 anda 3 x b 3 y c 3 z d3, the solution of the system of equations,ΔyΔΔby cramer’s rule is given by,x x , y and z z ,ΔΔΔprovided Δ 0wherea1Δ a2a3a1Δ y a2a3b1b2b3d1d2d3c1c2c3d1Δ x d2d3b1b2b3c1c2c3c1a1 b1c 2 and Δ z a 2 b 2c3a3 b37d1d2d3
1.1 WORKED EXAMPLESPART – A1. Solvex 2 0x 3xSolution:x 2 0x 3x3 x 2 2x 0x(3 x 2) 0x 0 or x 2. Solve23x 8 02 xSolution:x 8 02 xx 2 16 0x 2 16x 4m 2 13. Find the value of ‘m’ when 3 4 2 0 7 3 0Solution:m 2 1Given 3 4 2 0 7 3 0Expanding the determinant along, R1 we havem(0-6)-2 (0 14) 1 (9 28) 0m(-6) -2 (14) 1 (37) 08
-6m -28 37 0-6m 9 0-6m -9m 93 2612 04. Find the Co-factor of element 3 in the determinant 1 3 456 7Solution:Cofactor of 3 A 22 (-1)2 21 05 74 (-1) (7-0) 7PART – B1. Using cramer’s rule, solve the following simultaneous equationsx y z 22x-y–2z -1x – 2y – z 1Solution:111Δ 2 1 21 2 1 1 (1-4) -1 (-2 2) 1 (-4 1) 1 (-3) -1 (0) 1 (-3) -3 -3 -6 0211Δx 1 1 21 2 19
2 (1-4) -1 (1 2) 1 (2 1) 2 (-3) -1 (3) 1 (3) -6 -3 3 -61 21Δy 2 1 21 1 1 1 (1 2) -2 (-2 2) 1 (2 1) 1 (3) -2 (0) 1 (3) 3 3 6112Δz 2 1 11 2 1 1(-1-2) -1 (2 1) 2 (-4 1) -3-3-6 -12 By Cramer’s rule,x Δx 6 1Δ 6z Δ z 12 2Δ 6y ΔyΔ 6 -1 62. Using Cramer’s rule solve: -2y 3z-2x 1 0-x y-z 5 0 -2z -4x y 4Solution:Rearrange the given equations in order-2x-2y 3z -1; -x y-z -5; -4x y-2z 4 2 2 3Δ 1 1 1 4 1 210
-2(-2 1) 2 (2-4) 3(-1 4) -2 (-1) 2 (-2) 3 (3) 2-4 9 73 1 2Δx 5 1 141 2 -1(-2 1) 2 (10 4) 3 (-5-4) 1 28-27 2 2 1 3Δy 1 5 1 4 4 2 -2 (10 4) 1 (2-4) 3 (-4-20) -2(14) 1 (-2) 3 (-24) -28-2-72 -102Δz 2 2 11 5 4 14 1 -2 (4 5) 2 (-4-20) -1 (-1 4) -18-48-3 -69x Δ y 102Δx 2Δ 69 ,y , and z z 77ΔΔ7Δ3. Using Cramer’s rule solve2x-3y 5x-8 4y11
Solution:2x-3y 5x-4y 8Δ 2 3 (2) (-4) – (-3) (1)1 4 -8 3 -5Δx 5 3 (5) (-4) – (-3) (8)8 4 -20 24 4Δy 2 5 16 – 5 111 8By Cramer’s ruleΔx44 Δ 55Δy1111y Δ 55x 1.2 PROBLEMS INVOLVING PROPERTIES OF DETERMINANTSPART-A1) Evaluate201111 71931411 30Solution:20 11 31Δ 11 7 419 11 3031 11 430Δ 031 7 411 30C1 C1 C2since C1 C312
2) Without expanding, find the value of1 23 1 23 6 91Solution:1 23Let Δ 1 1 23 6 93 2 12 13(1) 3( 2) 3(3)11 2 3 3 1 1 21 2 3 3 (0) 0, since R 1 R 33) Evaluate1 a b c1 b c a1 c a bSolution:1 a b cLet Δ 1 b c a1 c a b1 a b c b c 1 a b c c aC2 C 2 C31 a b c a b1 1 b c (a b c) 1 1 c a1 1 a b (a b c) (0) 0, since C1 C213
x yy z z x4) Prove that y z z xz xx yx y 0y zSolution:x yy z z xL.H.S y z z xz xx yx yy zx y y z z xy z z xy z z x x y z xx yz x x y y z x yy zC1 C1 C2 C30 y z z x 0 z x0 x yx y 0 R.H.Sy zPART – B1 x x21) Prove that 1 y y 2 (x-y) (y-z) (z-x)1 z z2Solution:1 x x2L.H.S 1 y y 21 z z20 x y x2 y0 y z y2 z21zz22R1 R1 R2, R2 R2 R314
0 1 x y (x-y) (y-z) 0 1 y z1 z z21 x y(expanded along the first column)1 y z (x-y) (y-z) (x-y) (y-z) [1(y z) –1(x y)] (x-y) (y-z) (z-x)L.H.S R.H.S12) Prove that aa31bb31c (a b c) (a-b) (b-c) (c-a)c3Solution:1L.H.S aa31bb31cc3oo1Δ
MATHEMATICS – I UNIT – I DETERMINANTS 1.1 Definition and expansion of determinants of order 2 and 3 Properties of determinants Cramer’s rule to solve simultaneous equations in 2 and 3 unknowns-simple problems. 1.2 Problems involving properties of determinants 1.3 Matrices Definition of matrix. Types of matrices. Algebra of matrices such
Texts of Wow Rosh Hashana II 5780 - Congregation Shearith Israel, Atlanta Georgia Wow ׳ג ׳א:׳א תישארב (א) ׃ץרֶָֽאָּהָּ תאֵֵ֥וְּ םִימִַׁ֖שַָּה תאֵֵ֥ םיקִִ֑לֹאֱ ארָָּ֣ Îָּ תישִִׁ֖ארֵ Îְּ(ב) חַורְָּ֣ו ם
IBDP MATHEMATICS: ANALYSIS AND APPROACHES SYLLABUS SL 1.1 11 General SL 1.2 11 Mathematics SL 1.3 11 Mathematics SL 1.4 11 General 11 Mathematics 12 General SL 1.5 11 Mathematics SL 1.6 11 Mathematic12 Specialist SL 1.7 11 Mathematic* Not change of base SL 1.8 11 Mathematics SL 1.9 11 Mathematics AHL 1.10 11 Mathematic* only partially AHL 1.11 Not covered AHL 1.12 11 Mathematics AHL 1.13 12 .
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English/Language Arts 4. Grade Level i. Ninth grade 5. Length of Class Time i. 90 minute class 6. Length of Time to Complete Unit Plan th i. The unit on Non-fiction began on March 8 and was competed on March th 30 . The class had instruction on the topic everyday of the week. Student population Contextual/Environmental Factors Source Implications for Instruction and Assessment Rural School .
