GOVERNMENT OF TAMIL NADU BUSINESS MATHEMATICS AND STATISTICS HIGHER SECONDARY FIRST YEAR. VOLUME - I

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1 GOVERNMENT OF TAMIL NADU BUSINESS MATHEMATICS AND STATISTICS HIGHER SECONDARY FIRST YEAR VOLUME - I Untouchability is Inhuman and a Crime A publication under Free Tetbook Programme of Government of Tamil Nadu Department of School Education Preliminary_XI_EM.indd :44:

2 Government of Tamil Nadu First Edition - 08 NOT FOR SALE Content Creation The wise possess all State Council of Educational Research and Training SCERT 08 Printing & Publishing Tamil NaduTetbook and Educational Services Corporation (ii) II Preliminary_XI_EM.indd :44:

3 HOW TO USE THE BOOK Career Options List of Further Studies & Professions. Learning Objectives: Learning objectives are brief statements that describe what students will be epected to learn by the end of school year, course, unit, lesson or class period. Additional information about the concept. Amazing facts, Rhetorical questions to lead students to Mathematical inquiry Eercise Assess students critical thinking and their understanding ICT To enhance digital skills among students Web links List of digital resources To motivate the students to further eplore the content digitally and take them in to virtual world Miscellaneous Problems Additional problems for the students Glossary Tamil translation of Mathematical terms References List of related books for further studies of the topic Lets use the QR code in the tet books! How? Download the QR code scanner from the Google PlayStore/ Apple App Store into your smartphone Open the QR code scanner application Once the scanner button in the application is clicked, camera opens and then bring it closer to the QR code in the tet book. Once the camera detects the QR code, a url appears in the screen.click the url and goto the content page. (iii) Preliminary_XI_EM.indd :44:

4 CAREER OPTIONS IN BUSINESS MATHEMATICS Higher Secondary students who have taken commerce with Business mathematics can take up careers in BCA, B.Com., and B.Sc. Statistics. Students who have taken up commerce stream, have a good future in banking and financial institutions. A lot of students choose to do B.Com with a specialization in computers. Higher Secondary Commerce students planning for further studies can take up careers in professional fields such as Company Secretary, Chartered Accountant (CA), ICAI and so on. Others can take up bachelor s degree in commerce (B.Com), followed by M.Com, Ph.D and M.Phil. There are wide range of career opportunities for B.Com graduates. After graduation in commerce, one can choose MBA, MA Economics, MA Operational and Research Statistics at Postgraduate level. Apart from these, there are several diploma, certificate and vocational courses which provide entry level jobs in the field of commerce. Career chart for Higher Secondary students who have taken commerce with Business Mathematics. Scope for further Courses Institutions studies B.Com., B.B.A., B.B.M., B.C.A., Government Arts & Science Colleges, C.A., I.C.W.A, C.S. Aided Colleges, Self financing B.Com (Computer), B.A. Colleges. Shri Ram College of Commerce (SRCC), Delhi Symbiosis Society s College of Arts & Commerce, Pune. St. Joseph s College, Bangalore B.Sc Statistics Presidency College, Chepauk, Chennai. Madras Christian College, Tambaram Loyola College, Chennai. D.R.B.C.C Hindu College, Pattabiram, Chennai. M.Sc., Statistics B.B.A., LLB, B.A., LLB, B.Com., LL.B. (Five years integrated Course) M.A. Economics (Integrated Five Year course) Admission based on All India Entrance Eamination Government Law College. School of ecellence, Affiliated to Dr.Ambethkar Law University Madras School of Economics, Kotturpuram, Chennai. M.L. Ph.D., B.S.W. School of Social studies, Egmore, Chennai M.S.W (iv) Preliminary_XI_EM.indd :44:

5 CONTENTS Matrices and Determinants -9. Determinants. Inverse of a Matri. Input Output Analysis 7 Algebra Partial fractions 40. Permutations 47. Combinations 58.4 Mathematical induction 64.5 Binomial Theorem 70 Analytical Geometry 8-. Locus 84. System of Straight Lines 86. Pair of Straight Lines 9.4 Circles 99.5 Conics 08 4 Trigonometry Trigonometric ratios 6 4. Trigonometric ratios of Compound angles 4. Transformation formulae Inverse Trigonometric functions 45 5 Differential Calculus Functions and their graphs Limits and Derivatives 7 5. Differentiation techniques 89 Answers 0-8 Glossary 9-0 Books for Reference E- Book Assessment DIGI Links (v) Preliminary_XI_EM.indd :44:

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7 SYLLABUS. MATRICES AND DETERMINANTS (5 periods) Determinants: Recall - Types of matrices - Determinants Minors Cofactors - Properties of determinants. Inverse of a matri: Singular matri Non singular matri - Adjoint of a matri Inverse of a matri - of a system of linear equation. Input - Output analysis: Hawkins Simon conditions.. ALGEBRA (0 periods) Partial fractions: Denominator contains non - repeated Linear factors - Denominator contains Linear factors, repeated for n times - Denominator contains quadratic factor, which cannot be factorized into linear factors. Permutations: Factorial - Fundamental principle of counting Permutation - Circular permutation. Combinations - Mathematical Induction - Binomial Theorem.. ANALYTICAL GEOMETRY (0 periods) Locus: Equation of a locus. System of straight lines: Recall Various forms of straight lines - Angle between two straight lines Distance of a point from a line - Concurrence of three lines. Pair of straight lines: Combined equation of pair of straight lines - Pair of straight lines passing through the origin - Angle between pair of straight lines passing through the origin - The condition for general second degree equation to represent the pair of straight lines. Circles: The equation of a circle when the centre and radius are given - Equation of a circle when the end points of a diameter are given - General equation of a circle - Parametric form of a circle Tangents Length of tangent to the circle. CONICS: Parabola. 4. TRIGONOMETRY (5 periods) Trigonometric ratios: Quadrants - Signs of the trigonometric ratios of an angle θ as θ varies from 0º to 60º - Trigonometric ratios of allied angles. Trigonometric ratios of compound angles: Compound angles - Sum and difference formulae of sine, cosine and tangent - Trigonometric ratios of multiple angles. Transformation formulae: Transformation of the products into sum or difference - Transformation of sum or difference into product. Inverse Trigonometric Functions: Properties of Inverse Trigonometric Functions. 5. DIFFERENTIAL CALCULUS (5 periods) Functions and their graphs: Some basic concepts Function - Various types of functions - Graph of a function. Limits and derivatives: Eistence of limit - Indeterminate forms and evaluation of limits Continuous function - Differentiability at a point - Differentiation from first principle. Differentiation techniques: Some standard results - General rules for differentiation - Successive differentiation. (vi) Preliminary_XI_EM.indd :44:

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9 Chapter MATRICES AND DETERMINANTS Learning Objectives After studying this chapter, the students will be able to understand z the definition of matrices and determinants z the properties of determinants z the concept of inverse matri z the concept of adjoint matri z the solving simultaneous linear equations z the input-output analysis. Determinants Introduction The idea of a determinant was believed to be originated from a Japanese Mathematician Seki Kowa (68) while systematizing the old Chinese method of solving simultaneous equations whose coefficients were represented by calculating bamboos or sticks. Later the German Mathematician Gottfried Wilhelm Von Leibnitz formally developed determinants. The present vertical notation was given in 84 by Arthur Cayley. Determinant was invented independently by Crammer whose well known rule for solving simultaneous equations was published in 750. Seki Kowa In class X, we have studied matrices and algebra of matrices. We have also learnt that a system of algebraic equations can be epressed in the form of matrices. We know that the area of a triangle with vertices (, y ) (, y ) and (, y ) is 6 ( y- y) + ( y- y) + ( y- Matrices and Determinants CH-.indd :4:4

10 To minimize the difficulty in remembering this type of epression, Mathematicians developed the idea of representing the epression in determinant form and the above epression can be represented in the form y y y Thus a determinant is a particular type of epression written in a special concise form. Note that the quantities are arranged in the form of a square between two vertical lines. This arrangement is called a determinant. In this chapter, we study determinants up to order three only with real entries. Also we shall study various properties of determinant (without proof), minors, cofactors, adjoint, inverse of a square matri and business applications of determinants. We have studied matrices in the previous class. Let us recall the basic concepts and operations on matrices... Recall Matri Definition. A matri is a rectangular arrangement of numbers in horizontal lines (rows) and vertical lines (columns). Numbers are enclosed in square brackets or a open brackets or pair of double bars. It is denoted by A, B, C, 4 For eample, A G Order of a matri If a matri A has m rows and n columns, then A is called a matri of order m n. For eample, If, A G then order of A is R V General form of a Matri a a a ggai gganw WWWW a a a ggai ggan Matri of order m n is represented as ggg ggg ggg ggg ggg ggg Sam am am ggami ggamnw T X It is shortly written as 6 aij@ mn i,,... m; j,,... n Here a ij is the element in the i th row and j th column of the matri. th Std. Business Mathematics CH-.indd :4:5

11 Types of matrices Row matri A matri having only one row is called a row matri. For eample A 6 a a a... ai... an@ n ; B 7 A X Column matri A matri having only one column is called a column matri R V SaW SaW S W. For eamples, A S W, B G S. W - 5 S. W Sam W m T X Zero matri (or) Null matri If all the elements of a matri are zero, then it is called a NOTE zero matri. It is represented by an English alphabet O Zero matri can be R V 0 0 of any order. For eamples, O 6@ 0, O 0 0 G, O S0 0 0W T X are all zero matrices. Square matri If the number of rows and number of columns of a matri are equal then it is called a square matri. 8 For eamples, A 4 G is a square matri of order R V S - 0 W B S 0 5 W is a square matri of order. S S 5 W -4W T X Triangular matri A square matri, whose elements above or below the main diagonal (leading diagonal) are all zero is called a triangular matri. 0 0 For eamples, A > H, B > H NOTE The sum of the diagonal elements of a square matri is called the trace of matri. Matrices and Determinants CH-.indd :4:6

12 Diagonal matri A square matri in which all the elements other than the main diagonal elements are zero is called the diagonal matri. 5 For eample, A > H is a diagonal matri of order 0 0 Scalar matri A diagonal matri with all diagonal elements are equal to K (a scalar) is called a scalar matri. 5 For eample, A > H is a scalar matri of order Unit matri (or) Identity matri A scalar matri having each diagonal element equal to one (unity) is called a Unit matri. For eamples, I 0 0 G is a unit matri of order. R V S 0 0W I S0 0W is a unit matri of order. S0 0 W T X Multiplication of a matri by a scalar If A 6 a is a matri of any order and if k is a scalar, then the scalar multiplication of A by the scalar k is defined as ka 6 kaij@ for all i, j. In other words to multiply a matri A by a scalar k, multiply every element of A by k. For eample, if A 4-8 G then, A G -4 6 Negative of a matri The negative of a matri A 6 aij@ m # n is defined as - A 6 - a m# n for all i, j and is obtained by changing the sign of every element For eample, if A G then - A - - G Equality of matrices Two matrices A and B are said to be equal if (i) they have the same order and (ii) their corresponding elements are equal. 4 th Std. Business Mathematics CH-.indd :4:7

13 Addition and subtraction of matrices Two matrices A and B can be added, provided both the matrices are of the same order. Their sum A+B is obtained by adding the corresponding entries of both the matrices A and B. Symbolically if A [a ij ] m n and B [b ij ] m n, then A + B [a ij + b ij ] m n Similarly A B A + ( B) [a ij ] m n + [ b ij ] m n [a ij b ij ] m n Multiplication of matrices Multiplication of two matrices is possible only when the number of columns of the first matri is equal to the number of rows of the second matri. The product of matrices A and B is obtained by multiplying every row of matri A with the corresponding elements of every column of matri B element-wise and add the results. Let A 6 a be an m # p matri and B 6 b be a p # n matri, then the product AB is a matri C 6 c of order m # n. Transpose of a matri Let A 6 a be a matri of order m # n. The transpose of A, denoted by A T of order n # m is obtained by interchanging either rows into columns or columns into rows of A. For eample, if NOTE A 4 5 G then 6 A T > It is believed that the students might be familiar with the above concepts and our present syllabus continues from the following. H Definition. To every square matri A of order n with entries as real or comple numbers, we can associate a number called determinant of matri A and it is denoted by A or det (A) or Δ. Thus determinant can be formed by the elements of the matri A. If A a a a a G then its A aa - aa a a a a Matrices and Determinants 5 CH-.indd :4:7

14 Eample. Evaluate: ( )( 4) -(- 4 ) Minors: To evaluate the determinant of order or more, we define minors and cofactors. Let A [a ij ] be a determinant of order n. The minor of an arbitrary element a ij is the determinant obtained by deleting the i th row and j th column in which the element a ij stands. The minor of a ij is denoted by M ij... Cofactors: The cofactor is a signed minor. The cofactor of aij is denoted by A ij and is defined as i+ j Aij (- ) Mij a The minors and cofactors of a, a, a in a third order determinant a a a a a are as follows a a a a a (i) Minor of a is M aa - aa a a a a Cofactor of a is A ( ) + - M a a - a a a a (ii) Minor of a is M a a a a - a a a a + a a Cofactor of a is A (- ) M - -( aa - aa) a a (iii) Minor of a is M a a aa - aa a a + a a Cofactor of is A (- ) M a a - a a a a Eample. Find the minor and cofactor of all the elements in the determinant 4-6 th Std. Business Mathematics CH-.indd :4:8

15 Minor of M Minor of M 4 Minor of 4 M Minor of M + Cofactor of A (- ) M + Cofactor of - A (- ) M -4 + Cofactor of 4 A (- ) M + Cofactor of A (- ) M Eample. Find the minor and cofactor of each element of the determinant Minor of is M Minor of is M Minor of is M 4 Minor of is M Minor of is M 4 Minor of 5 is M 4 Minor of 4 is M Minor of is M Minor of 0 is M Cofactor of is A (- ) M M -5 + Cofactor of is A (- ) M - M 0 Matrices and Determinants 7 CH-.indd :4:9

16 + Cofactor of is A (- ) M M -6 + Cofactor of is A (- ) M - M + Cofactor of is A (- ) M M -8 + Cofactor of 5 is A (- ) M - M + Cofactor of 4 is A (- ) M M Cofactor of is A ( ) + - M - M - + Cofactor of 0 is A (- ) M M 4 For eample If D a a a a a a a a a, then NOTE Value of a determinant can be obtained by using any row or column Δ a A + a A + a A (or) a M a M + a M (epanding along R ) Δ a A + a A + a A (or) a M a M + a M (epanding along C ) Eample.4 4 Evaluate: (Minor of ) (Minor of ) + 4 (Minor of 4) Properties of determinants (without proof). The value of a determinant is unaltered when its rows and columns are interchanged. If any two rows (columns) of a determinant are interchanged then the value of the determinant changes only in sign.. If the determinant has two identical rows (columns) then the value of the determinant is zero. 8 th Std. Business Mathematics CH-.indd :4:9

17 4. If all the elements in a row (column) of a determinant are multiplied by constant k, then the value of the determinant is multiplied by k. 5. If any two rows (columns) of a determinant are proportional then the value of the determinant is zero. 6. If each element in a row (column) of a determinant is epressed as the sum of two or more terms, then the determinant can be epressed as the sum of two or more determinants of the same order. 7. The value of the determinant is unaltered when a constant multiple of the elements of any row (column) is added to the corresponding elements of a different row (column) in a determinant. Eample.5 Show that + a y y+ b z z+ c 0 a b c y z y z y z + a y+ b z+ c y z + a b c a b c a b c a b c Eample.6 - Evaluate ( )(+) ( ) e o+ e o e o ! 4 8 Eample Solve NOTE The value of the determinant of triangular matri is equal to the product of the main diagonal elements. Matrices and Determinants 9 CH-.indd :4:0

18 Eample & ( )( )( ) 0,, Evaluate (since R / R ) Eample.9 Evaluate a b c a b c (a b) (b c) (c a) a a 0 a- b a - b R " R- R b b 0 b- c b - c R R R " - c c c c 0 a- b ^a- bh^a+ bh 0 b- c ^b- ch^b+ ch c c ^a- bh^b- ch 0 0 c a+ b b+ c c ^a- bh^b- ch " b+ c- ^a+ ^a- bh^b- ch^c- ah. Eercise.. Find the minors and cofactors of all the elements of the following determinants (i) (ii) Evaluate th Std. Business Mathematics CH-.indd :4:0

19 . Solve: Find AB if A G and B 0 G - 5. Solve: Evaluate: 7. Prove that a b c a b c a b c bc ca ab - bc - ca - ab b+ c c+ a a+ b 0 - a ab ac 8. Prove that ab - b bc 4a bc ac bc - c. Inverse of a matri.. Singular matri : A Square matri A is said to be singular, if A 0.. Non singular matri: A square matri A is said to be non singular, if A! 0 Eample.0 Show that G is a singular matri. 4 Let A G 4 A ` A is a singular matri. Matrices and Determinants CH-.indd :4:

20 Eample. Show that 8 G is non singular. 4 Let A 8 G 4 A ! 0 ` A is a non-singular matri.. Adjoint of a Matri If A and B are non singular matrices of the same order then AB and BA are also non singular matrices of the same order. The adjoint of a square matri A is defined as the transpose of a cofactor matri. Adjoint of the matri A is denoted by adj A i.e., adj A 6Aij@ T where 6Aij@ is the cofactor matri of A. Eample. Find adj A for A G 4 A G 4 T adj A 6 A 4 - G - Eample. RS SSSS Find adjoint of A 0 S-4 T A ij ( ) i+j M ij th Std. Business Mathematics V W X A ( ) + M NOTE (i) adja A n - n is the order of the matri A (ii) ka n k A n is the order of the matri A n (iii) adj( ka) k - adja n is the order of the matri A (iv) A( adj A) ( adj AA ) AlI (v) AdjI II, is the unit matri. (vi) adj (AB) (adj B)(adj A) (vii) AB A B CH-.indd :4:

21 A ( ) + 0 M A ( ) + M ( 4) 4 A ( ) + M ^ 0 - ]- gh - A ( ) + M A ( ) + M - - -] - 8 g 7-4 A ( ) M 0 - ( - ) 0 A ( ) + M A ( ) + - M [A ij ] > - - 7H 0 T Adj A 6 A Inverse of a matri > H Let A be any non-singular matri of order n. If there eists a square matri B of order n such that AB BA I then, B is called the inverse of A and is denoted by A - We have A (adj A) (adj A)A ( A c adj A A A I m c adj A A ma I ( AB BA I where B A - A adja A adja Matrices and Determinants CH-.indd :4:

22 NOTE (i) If B is the inverse of A then, A is the inverse of B i.e., B A & A B (ii) AA A A I (iii) The inverse of a matri, if it eists, is unique. (iv) Order of inverse of A will be the same as that of order of A. (v) I - (vi) ( AB) I, where I is the identity matri. B A (vii) A - A Let A be a non-singular matri, then A I + A A - Eample.4 4 If A G then, find A A G - A Since A is a nonsingular matri, A - eists Now adj A -4 G A - A adja -4 G 6 Eample.5-6 If A G then, find A th Std. Business Mathematics CH-.indd :4:4

23 A Since A is a singular matri, A - does not eist. Eample.6 If A > H then find A - A ( - 0) - 4( 6-8) + 40 ( - 0) ( - 5) - 4( - ) + 40 ( ) ! 0 Since A is a nonsingular matri, A - eists. A - 5; A 8; A - 4 A ; A - ; A 0 A 0; A - ; A 6 A > H adj A A > H - adja A R 5 - S 0 T -4 - V 0 - W X > H Matrices and Determinants 5 CH-.indd :4:5

24 Eample.7 If A G satisfies the equation A - ka + I O then, find k and also A A G A A.A -. G G 7 G 4 7 Given A - ka+ I O & G- k G+ G G k + - k + 0 & G 0 0 G 4- k k k k 0 0 & - - G G 4 - k 8- k 0 0 & 4- k 0 & k 4. Also A 4! 0 adj A A - - G - A adja - G - - G - 6 th Std. Business Mathematics CH-.indd :4:5

25 Eample.8 If A G, B 0 - G then, show that ( AB) B A ` A - eists. A - -! 0 B 0-0 +! 0 B - also eists. AB 0 - G G G AB -! 0 ( AB) - eists. adj(ab) - G - ( AB) - adj( AB) AB - G G - - adj A G - () A - A adja - - G - - G - Matrices and Determinants 7 CH-.indd :4:6

26 adj B G - 0 B - B adjb G - 0 B - A - G G - 0 G - - G () - From () and (), ( AB) B A Hence proved. Eample.9 7 R Show that the matricesa > 4 H and 5 4 B V - - are inverses of each other S W R V S W -4-5 R V S W T X S 7W S W 6 5 AB S W S W S W S W S 6-0 W T X S W T X R V R V S 7W S- 4-5W S4 W 5 S- -6 5W S W S 6-0W T X T X > 0 H > H > H I 8 th Std. Business Mathematics CH-.indd :4:7

27 R V S-4-5 W S W 6 5 BA S- - W S W > 6-0 S W T X > - - H > > H 7 H 7 H > H Thus AB BA I > H I Therefore A and B are inverses of each other. Eercise.. Find the adjoint of the matri A G 4. If A > 4 H then verify that A ( adj A) A I and also find A 4. Find the inverse of each of the following matrices (i) G (ii) G (iii) > 0 4H (iv) > - -H If A G and B -6 - G, then verify adj (AB) (adj B)(adj A) - 5. If A > 0H then, show that (adj A)A O If A > 4-4H then, show that the inverse of A is A itself If A > 0 -H then, find A. - Matrices and Determinants 9 CH-.indd :4:8

28 R V S W S W 8. Show that the matrices A > H and B S - - W S W other S W T X If A G and 5 B G then, verify that ( AB ) B A 0. Find m if the matri 9 m 7 4 > H has no inverse. are inverses of each If X > -5 Hand Y th Std. Business Mathematics p - q..5 of a system of linear equations > H then, find p, q if Y X - Consider a system of n linear non homogeneous equations with n unknowns,,... n as a+ a ann b a+ a ann b a + a a b m m mn n m This can be written in matri form as R V R V R V Saa... an W SW Sb W Sa a... a W S W S n b W S W S W S W S W S. W S. W S W S. W S. W Sam am... a W S mn W S n b W m T X T X T X Thus we get the matri equation AX B () R V R a a... a V R S n W S W b V S W Sa a... a W S n W Sb W S W S W S W where A S W ; X S. W and B S. W S W S. W S W Sam am... a W S. W mn S T X W T n Sb X m W T X From () solution is X A B CH-.indd :4:40

29 Eample.0 Solve by using matri inversion method: + 5y + y 7 The given system can be written as 5 G G y 7 G i.e., AX B X A B where A 5 G, X y G and B 7 G A 5 ] 0 A eists. adj A A - 5 G - A adja -5 G - - X A B -5 G G G G - G G y - and y. Matrices and Determinants CH-.indd :4:40

30 Eample. Solve by using matri inversion method: - y+ z 8; + y- z ; 4- y+ z 4 The given system can be written as y z 8 4 > H > H > H i.e., AX B X A B - Here A > -H, X 4 - y z 8 4 > H and B > H - A ] 0 A eists A A 8 A 0 A 5 A 6 A A A 9 A 7 [A ij ] adj A [A ij ] T th Std. Business Mathematics A -0 7 > H A adja > H CH-.indd :4:40

31 - 7 X A B > yh > H z, y and z. Eample > H > H > H > H > H The cost of 4 kg onion, kg wheat and kg rice is `0. The cost of kg onion, 4 kg wheat and 6 kg rice is `560. The cost of 6 kg onion, kg wheat and kg rice is `80. Find the cost of each item per kg by matri inversion method. Let, y and z be the cost of onion, wheat and rice per kg respectively. Then, 4 + y + z 0 It can be written as y z + 4y + 6z y + z > H > H > H AX B 4 where A > 4 6H, X 6 A ] 0 y z > H and B > H Matrices and Determinants CH-.indd :4:4

32 A eist A 0 A 0 A 0 A 5 A 0 A 0 A 0 A 0 A 0 [A ij ] > H adj A [A ij ] T > H A A adja > - H X A B y z > H > H > H > H > H > > H > H , y 40 and z 60. H Cost of kg onion is `0, cost of kg wheat is `40 and cost of kg rice is ` th Std. Business Mathematics CH-.indd :4:4

33 Step ICT Corner Epected final outcomes Open the Browser and type the URL given (or) Scan the QR Code. GeoGebra Work book called th BUSINESS MATHS will appear. In this several work sheets for Business Maths are given, Open the worksheet named Simultaneous Equations Step - Type the parameters for three simultaneous equations in the respective boes. You will get the solutions for the equations. Also, you can see the planes for the equations and the intersection point() on the right-hand side. Solve yourself and check the answer. If the planes do not intersect then there is no solution. Step Step Step 4 Browse in the link th Business Maths: (or) scan the QR Code Matrices and Determinants 5 CH-.indd :4:4

34 Eercise.. Solve by matri inversion method : + y 5 0 ; y + 0. Solve by matri inversion method : (i) y + z ; + y z ; + y 5z 8 (ii) y + z ; + z ; + y + z 4. (iii) z 0 ; 5 + y 4 ; y + z 5. A sales person Ravi has the following record of sales for the month of January, February and March 009 for three products A, B and C He has been paid a commission at fied rate per unit but at varying rates for products A, B and C Months Sales in Units Commission A B C January February March Find the rate of commission payable on A, B and C per unit sold using matri inversion method. 4. The prices of three commodities A, B and C are `, `y and `z per unit respectively. P purchases 4 units of C and sells units of A and 5 units of B. Q purchases units of B and sells units of A and unit of C. R purchases unit of A and sells 4 units of B and 6 units of C. In the process P, Q and R earn `6,000, `5,000 and `,000 respectively. By using matri inversion method, find the prices per unit of A, B and C. 5. The sum of three numbers is 0. If we multiply the first by and add the second number and subtract the third we get. If we multiply the first by and add second and third to it, we get 46. By using matri inversion method find the numbers. 6. Weekly ependiture in an office for three weeks is given as follows. Assuming that the salary in all the three weeks of different categories of staff did not vary, calculate the salary for each type of staff, using matri inversion method. week Number of employees A B C Total weekly Salary (in rupees) st week nd week 4500 rd week th Std. Business Mathematics CH-.indd :4:4

35 . Input output Analysis Input Output analysis is a technique which was invented by Prof. Wassily W.Leontief. Input Output analysis is a form of economic analysis based on the interdependencies between economic sectors. The method is most commonly used for estimating the impacts of positive or negative economic shocks and analyzing the ripple effects throughout an economy. The foundation of Input - Output analysis involves input output tables. Such tables include a series of rows and columns of data that quantify the supply chain for sectors of the economy. Industries are listed in the heads of each row and each column. The data in each column corresponds to the level of inputs used in that industry s production function. For eample the column for auto manufacturing shows the resources required for building automobiles (ie., requirement of steel, aluminum, plastic, electronic etc.,). Input Output models typically includes separate tables showing the amount of labour required per rupee unit of investment or production. Consider a simple economic model consisting of two industries A and A where each produces only one type of product. Assume that each industry consumes part of its own output and rest from the other industry for its operation. The industries are thus interdependent. Further assume that whatever is produced that is consumed. That is the total output of each industry must be such as to meet its own demand, the demand of the other industry and the eternal demand (final demand). Our aim is to determine the output levels of each of the two industries in order to meet a change in final demand, based on knowledge of the current outputs of the two industries, of course under the assumption that the structure of the economy does not change. Let a ij be the rupee value of the output of A i consumed by A j, i, j, Let and be the rupee value of the current outputs of A and A respectively. Let d and d be the rupee value of the final demands for the outputs of A and A respectively. The assumptions lead us to frame the two equations a + a + d ; a + a + d... () Let b ij a ij j i, j, a a a a b ; b ; b ; b Matrices and Determinants 7 CH-.indd :4:4

36 The equations () take the form b + b + d b + b + d The above equations can be rearranged as ^ - bh - b d - b+ ^ - bh d The matri form of the above equations is - b -b G G d G -b - b d 0 b b ) e o- e o G d G 0 b b d ( B)X D b b where B G, I b b 0 0 G X and D d G G d By solving we get X(I B) - D The matri B is known as the Technology matri... The Hawkins Simon conditions Hawkins Simon conditions ensure the viability of the system. If B is the technology matri then Hawkins Simon conditions are Eample. (i) the main diagonal elements in I B must be positive and (ii) I- B must be positive. The technology matri of an economic system of two industries is G. Test whether the system is viable as per Hawkins Simon conditions th Std. Business Mathematics CH-.indd :4:4

37 B G I B G G G I- B (0.)(0.) ( 0.)( 0.9) < 0 Since I- B is negative, Hawkins Simon conditions are not satisfied. Therefore, the given system is not viable. Eample.4 The following inter industry transactions table was constructed for an economy of the year 06. Industry Final consumption Total output 500, ,500,750,600 4,650 8,000 Labours 50 4, Construct technology co-efficient matri showing direct requirements. Does a solution eist for this system. a 500 a a 750 a a b 500 a 0.0; b Matrices and Determinants 9 CH-.indd :4:4

38 a b 750 a 0. 70; b The technology matri is G I B 0 0 G G G, elements of main diagonal are positive Now, I- B (0.8)(0.8) ( 0.7)( 0.) >0 Since diagonal elements of are positive and I- B is positive, Hawkins Simon conditions are satisfied Hence this system has a solution. Eample.5 In an economy there are two industries P and P and the following table gives the supply and the demand position in crores of rupees. Production sector Consumption sector P P Final demand Gross output P P Determine the outputs when the final demand changes to 5 for P and 4 for P. a 0 a 5 50 a 0 a 0 60 b b a ; b a ; b 0 th Std. Business Mathematics a a 0 60 CH-.indd :4:4

39 R V S 5 The technology matri is B 5 W S W S 5 W T XR 0 V S 5 I B 0 G 5 W S W S 5 W R V S 4 T X - 5 W S 5 W -, elements of main diagonal are positive S 5 W T X 4-5 Now, I- B Main diagonal elements of I- B are positive and I- B is positive The problem has a solution R V S 5 adj (I B) W S W 4 S 5 5 W T X (I B) ( )( Since 0,( ) eist) I B adj I B I B I B - -! adj( I B) R V S W S W 5 4 S 7 > W H 4 T X X (I B) D, where D 5 G > H G G 4 04 The output of industry P should be `50 crores and P should be `04 crores. Eample.6 An economy produces only coal and steel. These two commodities serve as intermediate inputs in each other s production. 0.4 tonne of steel and 0.7 tonne of coal are needed to produce a tonne of steel. Similarly 0. tonne of steel and 0.6 tonne of coal are required to produce a tonne of coal. No capital inputs are needed. Do you think that Matrices and Determinants CH-.indd :4:44

40 the system is viable? and 5 labour days are required to produce a tonne s of coal and steel respectively. If economy needs 00 tonnes of coal and 50 tonnes of steel, calculate the gross output of the two commodities and the total labour days required. Here the technology matri is given under Steel Coal Final demand Steel Coal Labour days The technology matri is B G I B 0 0 G G G I- B (0.6)(0.4) ( 0.7)( 0.) Since the diagonal elements of I B are positive and value of I- B is positive, the system is viable adj (I B) G (I B) ( ) I- B adj I - B G X (I B) D, where D 50 G G G G G th Std. Business Mathematics CH-.indd :4:44

41 Steel output 76.5 tonnes Coal output tonnes Total labour days required 5(steel output) + ( coal output) 5(76.5) + (558.8) labour days. Eercise.4. The technology matri of an economic system of two industries is Test whether the system is viable as per Hawkins Simon conditions G The technology matri of an economic system of two industries is G. Test whether the system is viable as per Hawkins-Simon conditions The technology matri of an economic system of two industries is G. Test whether the system is viable as per Hawkins-Simon conditions. 4. Two commodities A and B are produced such that 0.4 tonne of A and 0.7 tonne of B are required to produce a tonne of A. Similarly 0. tonne of A and 0.7 tonne of B are needed to produce a tonne of B. Write down the technology matri. If 6.8 tonnes of A and 0. tonnes of B are required, find the gross production of both of them. 5. Suppose the inter-industry flow of the product of two industries are given as under. Production sector X Consumption sector Y Domestic demand Total output X Y Determine the technology matri and test Hawkin s -Simon conditions for the viability of the system. If the domestic demand changes to 80 and 40 units respectively, what should be the gross output of each sector in order to meet the new demands. Matrices and Determinants CH-.indd :4:45

42 6. You are given the following transaction matri for a two sector economy. Sector Sales Final demand Gross output (i) Write the technology matri (ii) Determine the output when the final demand for the output sector alone increases to units. 7. Suppose the inter-industry flow of the product of two sectors X and Y are given as under. Production sector X Consumption Sector Y Domestic demand Gross output X Y Find the gross output when the domestic demand changes to for X and 8 for Y. Choose the correct answer Eercise The value of if 0 is (a) 0, (b) 0, (c), (d),. The value of + y y+ z z+ y z y z is (a) y z (b) + y + z (c) + y + z (d) 0. The cofactor of 7 in the determinant is (a) 8 (b) 8 (c) 7 (d) 7 4 th Std. Business Mathematics CH-.indd :4:45

43 4. If D then is (a) D (b) D (c) D (d) D 5. The value of the determinant a b 0 (a) abc (b) 0 (c) a b c (d) abc 6. If A is square matri of order then ka is 0 0 c is (a) k A (b) k A (c) k A 7. adj] ABg is equal to (d) k A T T (a) adja adjb (b) adja adjb (c) adjb adja T T (d) adjb adja The inverse matri of f- p is (a) 0 f p 7-5 (b) 0 f - 5 p 0 5 (c) 7 f p 0-5 (d) 7 f p a b 9. If A e c d o such that ad - bc! 0 then A - is (a) d b e o d b ad - bc (b) e o - c a ad - bc c a d -b (c) e o d -b ad - bc (d) e o - c a ad - bc c a 0. The number of Hawkins -Simon conditions for the viability of an input-output analysis is (a) (b) (c) 4 (d). The inventor of input - output analysis is (a) Sir Francis Galton (c) Prof. Wassily W. Leontief (b) Fisher (d) Arthur Caylay Matrices and Determinants 5 CH-.indd :4:48

44 . Which of the following matri has no inverse (a) (c) - e o (b) -4 cos a sin a e o (d) - sin a cos a e -4 e - - o sin a cos a sin a o cos a. The inverse matri of e o is 5 - (a) e o (b) -5 - (c) e o (d) e - e 5 o - 5 o - 4. If A e - -4 o then A^adjAh is (a) -4 - e o (b) e o (c) - e 0 0 o (d) 0 e o 0 5. If A and B non-singular matri then, which of the following is incorrect? (a) A I implies A - A (b) I - I (c) If AX B then X B - A (d) If A is square matri of order then adj A A 6. The value of y 5 4z is - -y -z (a) 5 (b) 4 (c) 0 (d) 7. If A is an invertible matri of order then det^a h be equal to - (a) det(a) (b) det ] Ag (c) (d) 0 8. If A is matri and A 4 then A - is equal to (a) 4 (b) 6 (c) (d) 4 9. If A is a square matri of order and A then adja is equal to (a) 8 (b) 7 (c) (d) 9 6 th Std. Business Mathematics CH-.indd :4:5

45 0. The value of y y z - yz - z - y is (a) (b) 0 (c) (d) -yz. If A cos i sin i G, then A is equal to - sin i cos i (a). If. If 4cos i (b) 4 (c) (d) a a a D a a a and A ij is cofactor of a ij, then value of D is given by a a a (a) aa + aa + aa (b) a A + a A + a A (c) aa + aa + aa (d) a A + a A + a A then the value of is (a) (b) 6 (c) (d) If 5 then the value of is (a) 5 (b) 5 (c) 5 (d) 0 5. If any three rows or columns of a determinant are identical then the value of the determinant is (a) 0 (b) (c) (d) Miscellaneous Problems. Solve: Evaluate Without actual epansion show that the value of the determinant 0 4. Show that ab ac ab 0 bc ac bc 0 abc is zero. Matrices and Determinants 7 CH-.indd :4:54

46 R V 5. If A 4 7 verify that A^adjAh ^ adjah] Ag A I. S - W T X R V - 6. If A S W then, find the Inverse of A. S 5 - W T X 7. If A - > H show that A - 4A+ 5I O and also find A - 8. Solve by using matri inversion method : - y+ z, - y 0, y- z. 9. The cost of Kg of Wheat and Kg of Sugar is `70. The cost of Kg of Wheat and Kg of Rice is `70. The cost of Kg of Wheat, Kg of Sugar and Kg of rice is `70. Find the cost of per kg each item using matri inversion method. 0. The data are about an economy of two industries A and B. The values are in crores of rupees. User Final Total Producer A B demand output A B Find the output when the final demand changes to 00 for A and 600 for B Summary z Determinant is a number associated to a square matri. a a z If A a a then its A a a - a a z Minor of an arbitrary element a ij of the determinant of the matri is the determinant obtained by deleting the i th row and j th column in which the element a ij stands. The minor of a ij is denoted by M ij. z The cofactor is a signed minor. The cofactor of aij is denoted by A ij and is defined as i+ j A (- ) M. ij z Let a a a a a a ij a a a, then Δ a A + a A + a A or a M a M + a M 8 th Std. Business Mathematics CH-.indd :4:56

47 T z adj A 7A ij A where [A ij ] is the cofactor matri of the given matri. z adja A n -, where n is the order of the matri A. z A^adjAh ^ adjah] Ag A I. z adji I, I is the unit Matri. z adj(ab) (adj B)(adj A). z A square matri A is said to be singular, if A 0. z A square matri A is said to be non-singular if A! 0 z Let A be any square matri of order n and I be the identity matri of order n. If there eists a square matri B of order n such that AB BA I then, B is called the inverse of A and is denoted by A z Inverse of A ( if it eists ) is A - A adja z Hawkins - Simon conditions ensure the viability of the system. If B is the technology matri then Hawkins-Simon conditions are (i) the main diagonal elements in I- B must be positive and (ii) I- B must be positive.. Adjoint Matri Analysis Cofactor Determinant Digonal Matri Input Inverse Matri Minors Non-Singular Matri Output Scalar Singular Matri Transpose of a Matri Triangular Matri GLOSSARY ச ர ப ப அண பக ப பப ய வ இண க கப ரண அண க சகப ண ம ல வ ட ட அண உள ள ட ந ர ம ற அண ச ற றண க சகப ண கள ப ச யமற றக சகப ண அண வ ள ய ட த ச ய ல ப ச யக சகப ண அண ந ர ந ரல ம ற ற அண ம க சகப ண அண Matrices and Determinants 9 CH-.indd :4:57

48 Chapter ALGEBRA Learning Objectives After studying this chapter, the students will be able to understand z definition of partial fractions z the various techniques to resolve into partial fraction z different cases of permutation such as word formation z different cases of combination such as word formation z difference between permutation and combination z the concept and principle of mathematical induction z Binomial epansion techniques Introduction Algebra is a major component of mathematics that is used to unify mathematics concepts. Algebra is built on eperiences with numbers and operations along with geometry and data analysis. The word algebra is derived from the Arabic word al-jabr. The Arabic mathematician Al-Khwarizmi has traditionally been known as the Father of algebra. Algebra is used to find perimeter, area, volume of any plane figure and solid. Al-Khwarizmi. Partial Fractions Rational Epression p] g An epression of the form q ] g rational algebraic epression. n n- a 0 + a + g + a m m- b + b + g + b 0 n m,[q() 0] is called a If the degree of the numerator p () is less than that of the denominator q () then p] g q] g is called a proper fraction. If the degree of p () is greater than or equal to that of p] g q () then q ] g is called an improper fraction. 40 th Std. Business Mathematics Ch-.indd :6:4

49 An improper fraction can be epressed as the sum of an integral function and a proper fraction. That is, For eample, Partial Fractions p] g q] g f r] g ] g + q] g + + We can epress the sum or difference of two fractions - and - fraction. i.e., ] -g] -g 5-8 ] -g] -g as a single Process of writing a single fraction as a sum or difference of two or more simpler fractions is called splitting up into partial fractions. Generally if p () and q () are two rational integral algebraic functions of and p] g the fraction q ] g be epressed as the algebraic sum (or difference) of simpler fractions p] g according to certain specified rules, then the fraction q ] g is said to be resolved into partial fractions... Denominator contains non-repeated Linear factors p] g In the rational fraction q ] g, if q ] g is the product of non-repeated linear factors p] g of the form ] a + bg] c+ dg, then q ] g can be resolved into partial fraction of the form A B a + b + c + d, where A and B are constants to be determined. + 7 For eample ] -g] -g ] - A g + ] - B g, the values of A and B are to be determined. Eample. Find the values of A and B if Let ^ - h A B ^ h + A B Multiplying both sides by ] - g] + g, we get Algebra 4 Ch-.indd :6:6

50 Put in () we get, A] g A] + g+ B] - g... () ` A Put - in () we get, A(0)+B( ) ` B Eample. Resolve into partial fractions : Write the denominator into the product of linear factors. Here ] -g] -g 7 - Let ] - A g + ] - B g () Multiplying both the sides of () by (-)(-), we get 7 - A] - g+ B] - g..() Put in (),we get B () & B 0 Put in (),we get 4 - A] -g & A - Substituting the values of A and B in (), we get ] - g + ] - g 4 th Std. Business Mathematics Ch-.indd :6:8

51 ] 7 - g ] 7 - g < F ] < F g ] g ] - g ] - g - 0 ] - g + ] - g Eample. Resolve into partial fraction : + 4 ^ - 4h] + g Write the denominator into the product of linear factors Here ^ - 4h] + g ] - g] + g] + g + 4 ` ^ - 4h] + g ] - A g + ] + B g + ] C + g...() Multiplying both the sides by ( )(+)(+), we get + 4 A] + g] + g+ B] - g] + g+ C] - g] + g... () Put in (), we get A] 0g + B] -4g]- g+ C] 0g ` B Put in (), we get + 4 A] 4g] g+ B] 0g+ C] 0g ` A Put in (), we get A] 0g+ B] 0g+ C] -g] g Algebra 4 Ch-.indd :6:

52 ` C - Substituting the values of A, B and C in (), we get + 4 ^ - 4h] + g ] - g + ( + ) Denominator contains Linear factors, repeated n times p () In the rational fraction q (), if q () is of the form ] a + bg n [the linear factor is p () the product of (a+ b), n times], then q () can be resolved into partial fraction of the form. A A A An n a + b g + ] g ] a + bg ] a + bg ] a + bg A For eample, ] + 4g] + g B C ] + 4g + ] + g + ] + g Eample.4 A Find the values of A, B and C if ] - g] + g B C ] + g A ] - g] + g B C ] + g A] + g + B] - g] + g+ C] - g... () Put in () A] + g + B] 0g+ C] 0g ` A 4 Put in () - A] 0g+ B] 0g+ C] --g ` C Equating the constant term on both sides of (),we get A-B- C 0 & B A- C ` B 4 44 th Std. Business Mathematics Ch-.indd :6:4

53 Eample.5 + Resolve into partial fraction ] - g ] + g + A B Let ] - g ] + g ] - g + C + ] - g ] + g Multiplying both the sides by ] - g ] + g, we get... () + A] - g] + g+ B] + g+ C] - g () Put in (), we have + A] 0g+ B] 5g+ C] 0g ` B 5 Put in (), we have + A] 0g+ B] 0g+ C] --g - ` C 5 Equating the coefficient of on both the sides of (), we get 0 A+ C A -C ` A 5 Eample.6 Substituting the values of A, B and C in (), + ] - g ] + g Resolve into partial fraction 9 ] - g] + g 5] - g + - 5] - g 5] + g. 9 ] - g] + g. Multiplying both sides by ] - g] + g A B C ] - g + ] + g + ] + g () ` 9 A] + g + B] - g] + g+ C] - g () Put in () Algebra 45 Ch-.indd :6:7

54 9 A] 0g+ B] 0g+ C] --g ` C Put in () 9 A] g + B] 0g+ C] 0g ` A Equating the coefficient of on both the sides of (), we get 0 A+ B B -A ` B Substituting the values of A, B and C in (), we get 9 ] - g] + g ] + g.. Denominator contains quadratic factor, which cannot be factorized into linear factors p] g In the rational fraction q] g, if q ] g is the quadratic factor a + b + c and cannot p] g be factorized into linear factors, then q ] g can be resolved into partial fraction of the A + B form a. + b + c Eample.7 Resolve into partial fractions: + ] - g^ + h Here + cannot be factorized into linear factors. Let + ] - g^ + h A B c Multiplying both sides by ] - g^ + h... () + A^ + h + ] B + Cg] -g... () Put in () 46 th Std. Business Mathematics Ch-.indd :6:9

55 + A] + g+ 0 ` A Put 0 in () 0+ A] 0+ g + ] 0+ Cg] -g A C ` C Equating the coefficient of on both the sides of (), we get A + B 0 B -A - ` B Substituting the values of A, B and C in (), we get - + ] - g^ + h ] - g - ^ + h Eercise. Resolve into partial fractions for the following: ] + g 0. ^ + 4h] + g. Permutations.. Factorial ] - g] + g - ] + g] -g - ] + g^ + h ] - g] + g -5-7 ] - g + ] - g] + g For any natural number n, n factorial is the product of the first n natural numbers and is denoted by n! or n. For any natural number n, n! n] n-g] n-g # # Algebra 47 Ch-.indd :6:4

56 For eamples, 5! 5#4### 0 4! 4###4! ##6! #! NOTE 0! For any natural number n n! n] n-g] n-g # # n] n - g! n] n-g] n-g! In particular, 8! 8 ( ) 8 7! Eample.8 Evaluate the following: (i) 7! 6! (ii) 5 8!! (iii)! 6 9!! (i) 7! 7# 6 6! 6!! 7. (ii) 5 8!! 8# 7# 6# 5! 5! 6. (iii)! 6 9!! 9# 8# 7# 6! 6! #! 9# 8# 7 # # 84. Eample.9 Rewrite 7! in terms of 5! 7! 7 6! 7 6 5! 4 5! 48 th Std. Business Mathematics Ch-.indd :6:4

57 Eample.0 Find n, if n 9! + 0!! 9! + 0! n! 9! + 0 # 9! n! 9! : + 0 D n! 9! # 0 n!! n 9! # # 0! # 0! # 0! # 0!... Fundamental principle of counting Multiplication principle of counting: Consider the following situation in an auditorium which has three entrance doors and two eit doors. Our objective is to find the number of ways a person can enter the hall and then come out. Assume that, P, P and P are the three entrance doors and S and S are the two eit doors. A person can enter the hall through any one of the doors P, P or P in ways. After entering the hall, the person can come out through any of the two eit doors S or S in ways. P P S S P S S S P S P S P S Hence the total number of ways of entering the hall and coming out is P S P S # 6 ways. S P S These possibilities are eplained in the given flow chart (Fig..). Fig.. The above problem can be solved by applying multiplication principle of counting. Algebra 49 Ch-.indd :6:44

58 Eample. Definition. There are two jobs. Of which one job can be completed in m ways, when it has completed in any one of these m ways, second job can be completed in n ways, then the two jobs in succession can be completed in m # n ways. This is called multiplication principle of counting. Find the number of 4 letter words, with or without meaning, which can be formed out of the letters of the word NOTE, where the repetition of the letters is not allowed. Let us allot four vacant places (boes) to represent the four letters. N,O,T,E of the given word NOTE. 4 Fig :. The first place can be filled by any one of the 4 letters in 4 ways. Since repetition is not allowed, the number of ways of filling the second vacant place by any of the remaining letters in ways. In the similar way the third bo can be filled in ways and that of last bo can be filled in way. Hence by multiplication principle of counting, total number of words formed is 4# # # 4 words. Eample. If each objective type questions having 4 choices, then find the total number of ways of answering the 4 questions. Since each question can be answered in 4 ways, the total number of ways of answering 4 questions 4# 4# 4# 456 ways. Eample. How many digits numbers can be formed if the repetition of digits is not allowed? Let us allot boes to represent the digits of the digit number 50 th Std. Business Mathematics Ch-.indd :6:44

59 00 s place 0 s place unit s place Fig :. Three digit number never begins with 0. ` 00 s place can be filled with any one of the digits from to 9 in 9 ways. Since repetition is not allowed, 0 s place can be filled by the remaining 9 numbers (including 0) in 9 ways and the unit place can be filled by the remaining 8 numbers in 8 ways. Then by multiplication principle of counting, number of digit numbers 9#9# Addition principle of counting Let us consider the following situation in a class, there are 0 boys and 8 girls. The class teacher wants to select either a boy or a girl to represent the class in a function. Our objective is to find the number of ways that the teacher can select the student. Here the teacher has to perform either of the following two jobs. One student can be selected in 0 ways among 0 boys. One student can be selected in 8 ways among 8 girls Hence the jobs can be performed in ways This problem can be solved by applying Addition principle of counting. Eample.4 Definition. If there are two jobs, each of which can be performed independently in m and n ways respectively, then either of the two jobs can be performed in (m+n) ways. This is called addition principle of counting. There are 6 books on commerce and 5 books on accountancy in a book shop. In how many ways can a student purchase either a book on commerce or a book on accountancy? Out of 6 commerce books, a book can be purchased in 6 ways. Algebra 5 Ch-.indd :6:44

60 Out of 5 accountancy books, a book can be purchased in 5 ways. Then by addition principle counting the total number of ways of purchasing any one book is 5+6 ways. Eercise.. Find if. Evaluate 6! + 7! 8!. n! r!( n - r)! when n 5 and r.. If (n+)! 60[(n )!], find n 4. How many five digits telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 with no digit appears more than once? 5. How many numbers lesser than 000 can be formed using the digits 5, 6, 7, 8 and 9 if no digit is repeated?..4 Permutation Suppose we have a fruit salad with combination of APPLES, GRAPES & BANANAS. We don t care what order the fruits are in. They could also be bananas, grapes and apples or grapes, apples and bananas. Here the order of miing is not important. Any how, we will have the same fruit salad. But, consider a number lock with the number code 95. If we make more than three attempts wrongly, then it locked permanently. In that situation we do care about the order 95. The lock will not work if we input -5-9 or The number lock will work if it is eactly We have so many such situations in our practical life. So we should know the order of arrangement, called Permutation. Definition. The number of arrangements that can be made out of n things taking r at a time is called the number of permutation of n things taking r at a time. For eample, the number of three digit numbers formed using the digits,, taking all at a time is 6 6 three digit numbers are,,,,, 5 th Std. Business Mathematics Ch-.indd :6:45

61 Notations: If n and r are two positive integers such that 0 < r < n, then the number of all permutations of n distinct things taken r at a time is denoted by P(n,r ) (or) npr We have the following theorem without proof. Theorem : npr Eample.5 Evaluate : 5P and P(8, 5) n! ( n- r)! 5P 5! ] 5- g! 5!! 60. P(8, 5) 8P 5 8! ] 8-5g! 8!! !! 670 Results (i) 0! (ii) np 0! ] n - n 0g! n n!! (iii) np!! n n n] n - g ] - g! ] n - g! (iv) np n n! n ] n- ng! 0!! n! n (v) np r n(n ) (n )... [n (r )] Eample.6 Evaluate: (i) 8P (ii) 5P 4 (i) 8P 8# 7# 6 6. (ii) 5P 4 5# 4# # 0. Algebra 5 Ch-.indd :6:46

62 Eample.7 In how many ways 7 pictures can be hung from 5 picture nails on a wall? The number of ways in which 7 pictures can be hung from 5 picture nails on a wall 7! is nothing but the number of permutation of 7 things taken 5 at a time 7P 5 ] 7-5g! 50 ways. Eample.8 Find how many four letter words can be formed from the letters of the word LOGARITHMS ( words are with or without meanings) There are 0 letters in the word LOGARITHMS Therefore n 0 Since we have to find four letter words, r 4 Hence the required number of four letter words np r 0 p Eample.9 If np r 60, find n and r. np r Therefore n 6 and r 4. Permutation of repeated things: P 4 The number of permutation of n different things taken r at a time, when repetition is allowed is n r. 54 th Std. Business Mathematics Ch-.indd :6:46

63 Eample.0 Using 9 digits from,,, 9, taking digits at a time, how many digits numbers can be formed when repetition is allowed? Here, n 9 and r ` Number of digit numbers n r 9 79 numbers Permutations when all the objects are not distinct: The number of permutations of n things taken all at a time, of which p things are of n! one kind and q things are of another kind, such that p + q n is pq!!. In general, the number of permutation of n objects taken all at a time, p are of one kind, p are of another kind, p are of third kind, p k are of k th kind such that p n! + p + +p k n is. p! p!... pk! Eample. How many distinct words can be formed using all the letters of the following words. (i) MISSISSIPPI (ii) MATHEMATICS. (i) There are letters in the word MISSISSIPPI In this word M occurs once I S occurs 4 times occurs 4 times P occurs twice. Therefore required number of permutation (ii) In the word MATHEMATICS! 44!!! There are letters Here M occurs twice Algebra 55 Ch-.indd :6:47

64 T occurs twice A occurs twice and the rest are all different. Therefore number of permutations!!!!..5 Circular permutation In the last section, we have studied permutation of n different things taken all together is n!. Each permutation is a different arrangement of n things in a row or on a straight line. These are called Linear permutation. Now we consider the permutation of n things along a circle, called circular permutation. Consider the four letters A, B, C, D and the number of row arrangement of these 4 letters can be done in 4! Ways. Of those 4! arrangements, the arrangements ABCD, BCDA, CDAB, DABC are one and the same when they are arranged along a circle. C D A B D B A C B D C A C( g, f) C( g, f) C( g, f) C( g, f) A B C D Fig..4 4! So, the number of permutation of 4 things along a circle is 4! In general, circular permutation of n different things taken all at the time (n )! NOTE If clock wise and anti-clockwise circular permutations are considered to be same (identical), the number of circular permutation of n objects taken all at a time ] n - g! is Eample. In how many ways 8 students can be arranged (i) in a line (ii) along a circle 56 th Std. Business Mathematics Ch-.indd :6:47

65 (i) Number of permutations of 8 students along a line 8P 8 8! (ii) When the students are arranged along a circle, then the number of permutation is (8 )! 7! Eample. In how many ways 0 identical keys can be arranged in a ring? Since keys are identical, both clock wise and anti-clockwise circular permutation are same. The number of permutation is ] n - g! ] 0 - g! 9! Eample.4 Find the rank of the word RANK in dictionary. s Here maimum number of the word in dictionary formed by using the letters of the word RANK is 4! The letters of the word RANK in alphabetical order are A, K, N, R Number of words starting with A! 6 Number of words starting with K! 6 Number of words starting with N! 6 Number of words starting with RAK! Number of words starting with RANK 0! ` Rank of the word RANK is Rank of word in dictionary The rank of a word in a dictionary is to arrange the words in alphabetical order. In this dictionary the word may or may not be meaningful. Algebra 57 Ch-.indd :6:47

66 Eercise.. If np 4 (np ), find n.. In how many ways 5 boys and girls can be seated in a row, so that no two girls are together?. How many 6- digit telephone numbers can be constructed with the digits 0,,,,4,5,6,7,8,9 if each numbers starts with 5 and no digit appear more than once? 4. Find the number of arrangements that can be made out of the letters of the word ASSASSINATION 5. (a) In how many ways can 8 identical beads be strung on a necklace? (b) In how many ways can 8 boys form a ring? 6. Find the rank of the word CHAT in dictionary.. Combinations A combination is a selection of items from a collection such that the order of selection does not matter. That is the act of combining the elements irrespective of their order. Let us suppose that in an interview two office assistants are to be selected out of 4 persons namely A, B, C and D. The selection may be of AB, AC, AD, BC, BD, CD (we cannot write BA, since AB and BA are of the same selection). Number of ways of selecting persons out of 4 persons is 6.It is represented as 4C 4 6. The process of different selection without repetition is called Combination. Definition.4 Combination is the selection of n things taken r at a time without repetition. Out of n things, r things can be selected in nc r ways. nc r n! r! ] n - r g, 0 < r < n! Here n 0 but r may be 0. For eample, out of 5 balls balls can be selected in 5C ways. 5C 5!! ] 5- g!! 5 5 4!! b l 0 ways. 58 th Std. Business Mathematics Ch-.indd :6:48

67 Eample.5 Find 8C 8 7 8C 8 Permutations are for lists (order matters) and combinations are for groups (order doesn t matter) Properties (i) nc 0 nc n. (ii) nc n. (iii) nc n] n - g! (iv) nc nc y, then either y or + y n. (v) nc r nc. n r (vi) nc r + nc r (n + )C r. np r (vii) nc r r! Eample.6 If nc 4 nc 6, find C n. If nc ncy, then + y n. Here nc 4 nc 6 ` n C n C 0 Eample.7 C 66 If np r 70 ; nc r 0, find r Algebra 59 Ch-.indd :6:48

68 We know that nc r Eample.8 0 ( r. If 5C r 5C r+, find r 5C r 5C r+ npr r! 70 r! 70 r! 0 6! Then by the property, nc nc y ( + y n, we have 60 th Std. Business Mathematics r + r + 5 ( r. Eample.9 From a class of students, 4 students are to be chosen for competition. In how many ways can this be done? The number of combination C 4 Eample.0! 4!] - 4 g!! 4!] 8 g.! A question paper has two parts namely Part A and Part B. Each part contains 0 questions. If the student has to choose 8 from part A and 5 from part B, in how many ways can he choose the questions? In part A, out of 0 questions 8 can be selected in 0C 8 ways. In part B out of 0 questions 5 can be selected in 0C 5. Therefore by multiplication principle the total number of selection is Ch-.indd :6:49

69 0C 8 0C 5 0C 0C Eample. A Cricket team of players is to be formed from 6 players including 4 bowlers and wicket-keepers. In how many different ways can a team be formed so that the team contains at least bowlers and at least one wicket-keeper? A Cricket team of players can be formed in the following ways: (i) bowlers, wicket keeper and 7 other players can be selected in 4C C 0C7 ways 4C C 0C7 4C C 0C 960 ways (ii) bowlers, wicket keepers and 6 other players can be selected in 4C C 0C6 ways 4C C 0C6 4C C 0C ways (iii) 4 bowlers and wicket keeper and 6 other players 4C # C # 0C 4C # C # 0C 40 ways bowlers, wicket keepers and 5 other players can be selected in 4C # C # 0C ways 4 5 4C # C # 0C 5 ways 4 5 By addition principle of counting. Total number of ways Algebra 6 Ch-.indd :6:50

70 Eample. If 4] ncg ] n + g C, find n 4( nc ) ] n + g C nn ( - ) 4 # ] n+ g] n+ g] ng # # ] n - g (n+) (n+) ] n - g ^n + n+ h n - 9n+ 4 0 ] n-g] n-7g 0 & n, n 7 Eample. If ] n + g Cn 45, find n ] n + g Cn 45 + n+ -n ] n g C 45 ] n + g C 45 ( n + ) ] n + g 45 n + n ] n+ g] n-8g 0 n - is not possible n, 8 ` n 8 6 th Std. Business Mathematics Ch-.indd :6:5

71 Step ICT Corner Epected final outcomes Open the Browser and type the URL given (or) Scan the QR Code. GeoGebra Work book called th BUSINESS MATHS will appear. In this several work sheets for Business Maths are given, Open the worksheet named Combination Eercise Step - Combination practice eercise will open. You can generate as many questions you like by clicking New question Solve the problem yourself and check your answer by clicking on the Show solution bo. Also click Short method and follow this method. Step Step Step 4 Browse in the link th Business Maths: (or) scan the QR Code Algebra 6 Ch-.indd :6:5

72 Eercise.4. If np r 680 and nc r 70, find n and r.. Verify that 8C 4 + 8C 9C 4.. How many chords can be drawn through points on a circle? 4. How many triangles can be formed by joining the vertices of a heagon? 5. Out of 7 consonants and 4 vowels, how many words of consonants and vowels can be formed? 6. If four dice are rolled, find the number of possible outcomes in which atleast one die shows. 7. There are 8 guests at a dinner party. They have to sit 9 guests on either side of a long table, three particular persons decide to sit on one particular side and two others on the other side. In how many ways can the guests to be seated? 8. If a polygon has 44 diagonals, find the number of its sides. 9. In how many ways can a cricket team of players be chosen out of a batch of 5 players? (i) There is no restriction on the selection. (ii) A particular player is always chosen. (iii) A particular player is never chosen. 0. A Committee of 5 is to be formed out of 6 gents and 4 ladies. In how many ways this can be done when (i) atleast two ladies are included (ii) atmost two ladies are included.4 Mathematical induction Mathematical induction is one of the techniques which can be used to prove variety of mathematical statements which are formulated in terms of n, where n is a positive integer. 64 th Std. Business Mathematics Ch-.indd :6:5

73 Mathematical Induction is used in the branches of Algebra, Geometry and Analysis where it turns out to be necessary to prove some truths of propositions. The principle of mathematical induction: Eample.4 Let P(n) be a given statement for n! N. (i) Initial step: Let the statement is true for n i.e., P() is true and (ii) Inductive step: If the statement is true for n k (where k is a particular but arbitrary natural number) then the statement is true for n k +. i.e., truth of P(k) implies the truth of P(k+). Then P(n) is true for all natural numbers n. Using mathematical induction method, Prove that n n ] n + g, n! N. Let the given statement P(n) be defined as +++.n n ] n + g for n! N. Step : Put n LHS P() ] + g RHS P() LHS RHS for n ` P() is true Step : Let us assume that the statement is true for n k. i.e., P(k) is true k] k + g k is true Step : To prove that P(k + ) is true P(k + ) k + (k +) P(k) + k + Algebra 65 Ch-.indd :6:54

74 ` P(k +) is true k ] k + g + k + k ] k+ g+ ] k+ g ] k+ g] k+ g Thus if P(k) is true, then P(k +) is also true. ` P(n) is true for all n! N n] n + g Hence n, n! N Eample.5 for all n By the principle of Mathematical Induction, prove that (n ) n,! N. Let P(n) denote the statement (n ) n Put n LHS ` P() is true. Assume that P(k) is true. RHS LHS RHS i.e., (k ) k To prove: P(k + ) is true. ` (k ) + (k+) P] kg + ] k + g k + k+ ] k + g 66 th Std. Business Mathematics Ch-.indd :6:55

75 P(k + ) is true whenever P(k) is true. ` P(n) is true for all n Eample.6! N. By Mathematical Induction, prove that n n ] n+ g] n+ g 6, for all n! N. Let P(n) denote the statement: n n ] n+ g] n+ g 6 Put n ` LHS ] + g] + g RHS 6 LHS RHS ` P() is true. Assume that P(k) is true. P(k) k Now, k] k+ g] k+ g k + (k+) p] kg + ] k + g ` P(k + ) is true whenever P(k) is true ` P(n) is true for alln! N. k ] k+ g] k+ g k 6 + ] + g k ] k+ g] k+ g+ 6] k + g 6 ] k + g6 k ] k+ g+ 6] k+ g@ 6 k k ] + g^ + 7 k + 6h 6 ] k + g] k + g] k + g 6 Algebra 67 Ch-.indd :6:57

76 Eample.7 Show by the principle of mathematical induction that n is a divisible by 7, for all n! N. Let the given statement P(n) be defined as P(n) n Step : Put n ` P() 7 is divisible by 7 i.e., P() is true. Step : Let us assume that the statement is true for n k i.e., P(k) is true. We assume k is divisible by 7 k & - 7m Step : To prove that P( + ) is true P(k +) ] k + g ] k + g - k $ - Eample.8 which is divisible by 7 68 th Std. Business Mathematics k $ 8- k $ ] 7+ g- k k $ 7+ - k k $ 7+ 7m 7( + m) P(k +) is true whenever p(k) is true ` P(n) is true for all n Hence the proof.! N. By the principle of mathematical induction prove that n + n is an even number, for all n! N. Ch-.indd :6:59

77 Let P(n) denote the statement that, n + n is an even number. Put n Let us assume that P(k) is true. ` + +, an even number. ` k + k is an even number is true ` take k + k m () To prove P(k + ) is true ` ] k+ g + ] k+ g k + k+ + k + k + k+ k + m + (k + ) by () (m + k + ) ( a multiple of ) ` (k + ) + (k + ) is an even number ` P(k + ) is true whenever P(k) is true. P(n) is true for n! N. Eercise.5 By the principle of mathematical induction, prove the following n n ( n+ ) 4 for all n! N n(n + ) n ] n+ g] n+ g, for all n! N n n(n+), for all n! N (n ) n ] n - g, for all n! N. 5. n is a divisible by 8, for all n! N. n n 6. a - b is divisible by a b, for all n! N n is divisible by 4, for all n! N. Algebra 69 Ch-.indd :7:00

78 8. n(n + ) (n+) is divisible by 6, for all n! N. 9. n > n, for all n! N..5 Binomial theorem An algebraic epression of sum or the difference of two terms is called a binomial. For eample y, 5a b 5 7 ^ + h ] - g, c + y m, c p + p m, d 4 + n y etc., are binomials. The Binomial theorem or Binomial Epression is a result of epanding the powers of binomials. It is possible to epand ( + y ) n into a sum involving terms of the form a b y c, eponents b and c are non-negative integers with b + c n, the coefficient a of each term is a positive integer called binomial coefficient. Epansion of ( + a) was given by Greek mathematician Euclid on 4th century and there is an evidence that the binomial theorem for cubes i.e., epansion of (+a) was known by 6th century in India. The term Binomial coefficient was first introduced by Michael Stifle in 544.Blaise Pascal (9 June 6 to 9 August 66) a French Mathematician, Physicist, inventor, writer and catholic theologian. In his Treatise on Arithmetical triangle of 65 described the convenient tabular presentation for Binomial coefficient now called Pascal s triangle. Sir Issac Newton generalized the Binomial theorem and made it valid for any rational eponent. Now we study the Binomial theorem for ( + a) n Theorem(without proof) If and a are real numbers, then for all n! N ] + ag n n 0 n- n - n- r r nc a+ nc a + nc a nc a nc - a + nc a nc a n n n n - n 0 / r 0 r r n- r r NOTE When n 0, (+a) 0 When n, ] + ag C+ Ca + a 70 th Std. Business Mathematics 0 a When n, ] + ag C + C + Ca + a+ a 0 When n, ] + ag C + Ca+ C a + Ca 0 + a+ a + a and so on. Ch-.indd :7:0

79 Given below is the Pascal s Triangle showing the co- efficient of various terms in Binomial epansion of ] + ag n n 0 n n n n n Fig..5 NOTE (i) Number of terms in the epansion of ( + a) n is n+ (ii) Sum of the indices of and a in each term in the epansion is n (iii) nc0, nc, nc, nc,... ncr,... ncnare also represented as C0, C, C, C,..., Cr,..., Cn, are called Binomial co-efficients. (iv) Since ncr ncn- r,, for r 0,,,, n in the epansion of ( + a) n, binomial co-efficients of terms equidistant from the beginning and from the end of the epansion are equal. (v) Sum of the co-efficients in the epansion of (+) n is equal of n (vi) In the epansion of (+) n, the sum of the co-efficients of odd terms the sum of the co-efficients of the even terms n (vii) General term in the epansion of (+a) n is t r+ nc Middle term of ] + ag n r n- r r a Case (i) If n is even, then the number of terms n + is odd, then there is only one middle term, given by t n + Case (ii) If n is odd, then the number of terms (n+) is even.. Therefore, we have two middle terms given by t n + and t n + Algebra 7 Ch-.indd :7:04

80 Sometimes we need a particular term in the epansion of ( + a) n. For this we first write the general term tr +. The value of r can be obtained by taking the term tr + as the required term. To find the term independent of (term without ), equate the power of in tr + to zero, we get the value of r. By substituting the value of r in tr +, we get the term independent of. NOTE ] + ag n n 0 n- n - - nc a+ nc a + nc a nc a a (i) To find ( a) n, replace by a 0 r n r r n.() ] - ag n n 0 n- n nc a+ nc - - a + nc - - ] g ] ag nc ]- ag ]-ag 0 r n r r n n 0 n- nc a nc a n- nc a r n- r... ( ) ncr a r n... ( ) a n 0 - ] g+ ] g ] g ] g...() Note that we have the signs alternatively. (ii) If we put a in (), we get n r ] + g + nc+ nc ncr nc n n...() (iii) If we replace by in (), we get n r r n n ] - g - nc+ nc ncr ]- g ncn ( ) Eample.9 Epand ( + y) 5 using binomial theorem. ] + ag n n 0 n- n - n- r r nc a+ nc a + nc a nc a nc - a + nc a n r n - n 0 ^+ yh 5 5C 5 5C 4 y 5C ] g + ] g ^ h + ] g ^ yh + 5C ] g ^yh + 0 5C ] g^yh + 5C ^yh y y ] g ^ h + ^ h_ i+ _ y i+ 5] g8y + 4y y+ 70 y y + 80 y + 4y n 7 th Std. Business Mathematics Ch-.indd :7:06

81 Eample.40 Using binomial theorem, epand c Eample.4 + m 4 c + m C 4 ^ h + 4 ^ h + 4C ^ h c m + 4C ^ hc m + 4C4 c m Using binomial theorem, evaluate (0) 5 (0) 5 (00 + ) 5 (00) 5 + 5C (00) 4 + 5C (00) + 5C (00) + 5C4 (00) + 5C ( ) + 0(000000) + 0(0000) + 5(00) Eample.4 Find the 5 th term in the epansion of c - General term in the epansion of ] + ag n - is tr + ncr a To find the 5 th term of For this take r 4 c - m 0 4 m 0 n r r 4 Then, t4+ t5 0C4 6 ] g c- m (Here n 0,, a ) 4 0C 6 4 ] g 8 Eample Find the middle term in the epansion of b - l () Algebra 7 Ch-.indd :7:09

82 0 Compare b - l with ] + ag n n 0,, a - Since n 0, we have terms (odd) ` 6 th term is the middle term. n- r r The general term is tr + ncr a () To get t6, put r 5 t + t 0C ^ C h b l 5 ]-g Eample.44 Find the middle term in the epansion of b + 9yl 9 Compare b + 9yl 9 with ( + a) n Since n 9,we have 0 terms(even) t + t + ` There are two middle terms namely, General term in the epansion of ( + a) n is n n t i.e., 9 t +, 9 + s tr + nc a r n- r r () Here t5 and t6 are middle terms. put r 4 in (), t4+ t5 9C 4 b l $ ^9yh 74 th Std. Business Mathematics 9C 4 5 $ 9 y y put r 5 in (), t5+ t6 9C 5 b l $ ^9yh C 5 4 $ 9 y 9854 y $ 5 $ 9 y Ch-.indd :7:

83 Eample.45 Find the Coefficient of 0 in the binomial epansion of b - l n- r r General term of ( + a) n is tr + ncr a Compare b - l with ( + a) n - r - tr + Cr ^ h b l r () - r ] r C - g r r ]- g b l C ]-g $ r -r r r -r To find the co-efficient of 0, take r 0 r 4 Put r 4 in (), t5 C C $ 4 4 r ` Co-efficient of 0 is C Eample.46 9 Find the term independent of in the epansion of c + m. n- r r General term in the epansion of ( + a) n is tr + ncr a Compare c + ` tr 9 m with ( + a) n + 9C 9 - r r ] g c tr r r 9Cr r + 9Cr r $ m r 9 - r 9- r To get the term independent of, equating the power of as 0 9 r 0 ` r Put r in () t+ 9C 5 $ 9C 5 79 r... () Algebra 75 Ch-.indd :7:6

84 Step ICT Corner Epected final outcomes Open the Browser and type the URL given (or) Scan the QR Code. GeoGebra Work book called th BUSINESS MATHS will appear. In this several work sheets for Business Maths are given, Open the worksheet named Pascal s Triangle Step - Pascal s triangle page will open. Click on the check boes View Pascal Triangle and View Pascal s Numbers to see the values. Now you can move the sliders n and r to move the red point to respective n C r. Now click on view Combination to see the n C r calculation. Step Step Step 4 Browse in the link th Business Maths: (or) scan the QR Code 76 th Std. Business Mathematics Ch-.indd :7:7

85 Eercise.6. Epand the following by using binomial theorem 7 (i) ] a- bg 4 (ii) c + y m (iii) c + m 6. Evaluate the following using binomial theorem: (i) (0) 4 (ii) (999) 5. Find the 5 th term in the epansion of ( y). 4. Find the middle terms in the epansion of (i) b + l (ii) b + l (iii) c 8 - m 0 5. Find the term independent of in the epansion of (i) 9 5 b - l (ii) - c m (iii) b + l 6. Prove that the term independent of in the epansion of b + n l is $ $ 5..., ] n - g n n!. n 7. Show that the middle term in the epansion of ] + g is 5.. ( n- ) n n n! Eercise.7 Choose the correct answer. If nc nc then the value of nc4 is (a) (b) (c) 4 (d) 5. The value of n, when np 0 is (a) (b) 6 (c) 5 (d) 4. The number of ways selecting 4 players out of 5 is (a) 4! (b) 0 (c) 5 (d) 5 Algebra 77 Ch-.indd :7:9

86 4. If npr 70] ncrg, then r is equal to (a) 4 (b) 5 (c) 6 (d) 7 5. The possible out comes when a coin is tossed five times (a) 5 (b) 5 (c) 0 (d) 5 6. The number of diagonals in a polygon of n sides is equal to (a) nc (b) nc - (c) nc - n (d) nc - 7. The greatest positive integer which divide n] n+ g] n+ g] n+ g for all n! N is (a) (b) 6 (c) 0 (d) 4 8. If n is a positive integer, then the number of terms in the epansion of ] + ag n is (a) n (b) n + (c) n (d) n 9. For all n > 0, nc + nc + nc + + nc n is equal to (a) n (b) n (c) n (d) n 0. The term containing in the epansion of ^- yh 7 is (a) rd (b) 4 th (c) 5 th (d) 6 th. The middle term in the epansion of b + l is (a) 0C4 b l (b) 0C5 (c) 0C6 (d) C 78 th Std. Business Mathematics 0 6. The constant term in the epansion of b + l is (a) 56 (b) 65 (c) 6 (d) 60. The last term in the epansion of ^+ h 8 is 4. If (a) 8 (b) 6 (c) 8 (d) 7 k 4 ] + 4g] -g then k is equal to (a) 9 (b) (c) 5 (d) 7 5. The number of letter words that can be formed from the letters of the word number when the repetition is allowed are (a) 06 (b) (c) 6 (d) 00 Ch-.indd :7:

87 6. The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is (a) 8 (b) (c) 9 (d) 6 7. There are 0 true or false questions in an eamination. Then these questions can be answered in (a) 40 ways (b) 0 ways (c) 04 ways (d) 00 ways 8. The value of ] 5C0+ 5Cg + ] 5C+ 5Cg + ] 5C+ 5Cg+ ] 5C+ 5C4g+ ] 5C4+ 5C5g is (a) 6 (b) 5 (c) 8 (d) 7 9. The total number of 9 digit number which have all different digit is (a) 0! (b) 9! (c) 9 9! (d) 0 0! 0. The number of ways to arrange the letters of the word CHEESE (a) 0 (b) 40 (c) 70 (d) 6. Thirteen guests has participated in a dinner. The number of handshakes happened in the dinner is (a) 75 (b) 78 (c) 86 (d). Number of words with or without meaning that can be formed using letters of the word EQUATION, with no repetition of letters is (a) 7! (b)! (c) 8! (d) 5!. Sum of Binomial co-efficient in a particular epansion is 56, then number of terms in the epansion is (a) 8 (b) 7 (c) 6 (d) 9 4. The number of permutation of n different things taken r at a time, when the repetition is allowed is (a) r n (b) n r n! n! (c) ] n- rg (d)! ] n+ rg! 5. Sum of the binomial coefficients is (a) n (b) n (c) n (d) n+7 Algebra 79 Ch-.indd :7:

88 Miscellaneous Problems. Resolve into Partial Fractions : ] - g] + g. Resolve into Partial Fractions: Decompose into Partial Fractions: 4. Decompose into Partial Fractions: ] + g] -g ] - g^ - + h 5. Evaluate the following epression. (i) 7! 6! (ii) 5 8!! (iii)! 6 9!! 6. How many code symbols can be formed using 5 out of 6 letters A, B, C, D, E, F so that the letters a) cannot be repeated b) can be repeated c) cannot be repeated but must begin with E d) cannot be repeated but end with CAB. 7. From 0 raffle tickets in a hat, four tickets are to be selected in order. The holder of the first ticket wins a car, the second a motor cycle, the third a bicycle and the fourth a skateboard. In how many different ways can these prizes be awarded? 8. In how many different ways, Mathematics, Economics and History books can be selected from 9 Mathematics, 8 Economics and 7 History books? 9. Let there be red, yellow and green signal flags. How many different signals are possible if we wish to make signals by arranging all of them vertically on a staff? 0. Find the Co-efficient of in the epansion of c + 80 th Std. Business Mathematics Summary z For any natural number n, n factorial is the product of the first n natural numbers and is denoted by n! or n. z For any natural number n n! n(n ) (n ) z 0! z If there are two jobs, each of which can be performed independently in m and n ways respectively, then either of the two jobs can be performed in (m+n) ways. z The number of arrangements that can be made out of n things taking r at a time is called the number of permutation of n things taking r at a time. m 7 Ch-.indd :7:4

89 z z z z np np np np n! ] n- rg!!! ] n - n 0g! n n!!! n n nn ^ - h ] - g! ] n - g! n! n ] n- ng! 0!! n! r q n n z npr nn ( -)( n-) [ n-( r - )] z The number of permutation of n different things taken r at a time, when repetition is allowed is n r. z The number of permutations of n things taken all at a time, of which p things are n! of one kind and q things are of another kind, such that p+q n is pq!! z Circular permutation of n different things taken all at the time (n )! z The number of circular permutation of n identical objects taken all at a time is ] n - g! z Combination is the selection of n things taken r at a time without repetition. z Out of n things, r things can be selected in ncr ways. n! z ncr, 0 # r # n r! ^n- rh! z ncn. z nc n. z nc nn ^ - h! z nc ncy, then either y or + y n. z ncr ncn- r. z nc + nc - ^n+ hc r r r n z ( a) nco n a o nc n - a nc n - n- r r a + + ncr a + ncn - a + nc a nc a n 0 n n / r 0 r n- r r z Number of terms in the epansion of (+a) n is n+ n - z Sum of the indices of and a in each term in the epansion of (+a) n is n Algebra 8 Ch-.indd :7:7

90 z nc0, nc, nc, nc ncr ncnare also represented as C, C, C, C Cr C called Binomial co-efficients. 0 n, z SincenCr ncn- r, for r 0,,, n in the epansion of (+a) n, Binomial co -efficients which are equidistant from either end are equal z nc nc, nc nc, nc nc 0 n n- n - z Sum of the co-efficients is equal of n z Sum of the co-efficients of odd terms sum of the co-efficients of even terms n- z General term in the epansion of ^+ ah n - istr+ nc r a n r r GLOSSARY Binomial ஈர ற ப ப Circular Permutation வட ட வர ச ம ற றம Coefficient க ழ Combination ச ர வ Factorial க ரண யப ப ர க ம Independent term ச ர உற ப ப Linear factor ஒர பட க ரரண Mathematical Induction கண தத கதரக த த ற தல Middle term ம ய உற ப ப Multiplication Principle of counting எண ண தல ன ப ர க ல க ரளச Partial fraction பக த ப ப ன னங ள Pascal s Triangle ப ஸ ல ன ம க க ணம Permutations வர ச ம ற றங ள Principle of counting எண ண தல ன க ரளச Rational Epression வ க தம ற ச ரசவ 8 th Std. Business Mathematics Ch-.indd :7:8

91 Chapter ANALYTICAL GEOMETRY Learning Objectives After studying this chapter, the students will be able to understand z the locus z the angle between two lines. z the concept of concurrent lines. z the pair of straight lines z the general equation and parametric equation of a circle. z the centre and radius of the general equation of a circle. z the equation of a circle when the etremities of a diameter are given. z the equation of a tangent to the circle at a given point. z identification of conics. z the standard equation of a parabola, its focus, directri, latus rectum and commercial applications Introduction The word Geometry is derived from the word geo meaning earth and metron meaning measuring. The need of measuring land is the origin of geometry. Geometry is the study of points, lines, curves, surface etc., and their properties. The importance of analytical geometry is that it establishes a correspondence between geometric curves and algebraic equations. A systematic study of geometry by the use of algebra was first carried out by celebrated French Philosopher and Rene Descartes ( ) mathematician Rene Descartes ( ), in his book La Geometry, published in 67. The resulting combination of analysis and geometry is referred now as analytical geometry. He is known as the father of Analytical geometry Analytical geometry is etremely useful in the aircraft industry, especially when dealing with the shape of an airplane fuselage. Analytical Geometry 8 Ch-.indd :7:54

92 . Locus Definition. The path traced by a moving point under some specified geometrical condition is called its locus... Equation of a locus Any relation in and y which is satisfied by every point on the locus is called the equation of the locus. For eample, (i) The locus of a point P^, yh whose distance from a fied point C(h, k) is constant, is a circle. The fied point C is called the centre. a C a Fig.. a P P (ii) The locus of a point whose distances from two points A and B are equal is the perpendicular bisector of the line segment AB. A B Straight line is the locus of a point which moves in the same direction. Eample. Fig.. A point in the plane moves so that its distance from the origin is thrice its distance from the y- ais. Find its locus. Let P^, yh be any point on the locus and A be the foot of the perpendicular from P^, y h to the y- ais. Given that OP AP OP 9AP 84 th Std. Business Mathematics Ch-.indd :7:54

93 ^ - 0h + ^y - 0h 9 + y y 0 ` The locus of P^, y h is 8 - y 0 Eample. Find the locus of the point which is equidistant from (, ) and (, 4). Let A(, ) and B (, 4) be the given points Let P^, y h be any point on the locus. Given that PA PB. PA PB ^ - h + ^y + h ^ - h + ^y + 4h y + 6y y + 8y + 6 i.e., -y - 0 Eample. i.e., -y- 6 0 The locus of P^, y h is -y Find the locus of a point, so that the join of ( 5, ) and (, ) subtends a right angle at the moving point. Let A( 5, ) and B(, ) be the given points Let P^, y h be any point on the locus. Given that + APB 90 o. Triangle APB is a right angle triangle. BA PA +PB Analytical Geometry 85 Ch-.indd :7:57

94 (+5) + ( ) ^ + 5h + ^y - h + ^ - h + ^y - h y - y y - 4y + 4 i.e., + y + 4-6y i.e., + y + -y - 0 The locus of P^, y h is + y --y- 0 Eercise.. Find the locus of a point which is equidistant from (, ) and ais.. A point moves so that it is always at a distance of 4 units from the point (, ). If the distance of a point from the points (,) and (,) are in the ratio :, then find the locus of the point. 4. Find a point on ais which is equidistant from the points (7, 6) and (, 4). 5. If A(, ) and B(, ) are two fied points, then find the locus of a point P so that the area of triangle APB 8 sq.units.. System of straight lines.. Recall In lower classes, we studied the basic concept of coordinate geometry, like distance formula, section - formula, area of triangle and slope of a straight lines etc. We also studied various form of equations of lines in X std. Let us recall the equations of straight lines. Which will help us for better understanding the new concept and definitions of XI std co-ordinate geometry. Various forms of straight lines: (i) Slope-intercept form Equation of straight line having slope m and y-intercept c is y m+c (ii) Point- slope form Equation of Straight line passing through the given point P^, yh and having a slope m is y- y m^- h 86 th Std. Business Mathematics Ch-.indd :7:59

95 (iii) Two-Point form Equation of a straight line joining the given points A^, yh, B^, yh is y- y - y - y -. In determinant form, equation of straight line joining two given points A^, yh and B^, y h is y y y 0 (iv) Intercept form Equation of a straight line whose and y intercepts are a and b, is a y + b. (v) General form Equation of straight line in general form is a + by + c 0 where a, b and c are constants and a, b are not simultaneously zero... Angle between two straight lines Let l and l be two straight lines represented by the equations l: y m + c and l : y m+ c intersecting at P. m If i and i are two angles made by l and l with - ais then slope of the lines are tani and m tan i. y From fig., if i is angle between the lines l l & l then l i i-i ` tan i tan^i - i h tani- tan i + tani tan i O P i i i tanθ m- m + mm Fig.. θ tan - m- m + mm Analytical Geometry 87 Ch-.indd :8:0

96 NOTE m- m (i) If + mm is positive, then i, the angle between l and l is acute and if it is negative, then, i is the obtuse. (ii) We know that two straight lines are parallel if and only if their slopes are equal. (iii) We know that two lines are perpendicular if and only if the product of their slopes is. (Here the slopes m and m are finite.) The straight lines -ais and y-ais are perpendicular to each other. But, the condition mm - is not true because the slope of the -ais is zero and the slope of the y-ais is not defined. Eample.4 Find the acute angle between the lines - y+ 0 and + y+ 0 Let m and m be the slopes of - y+ 0 and + y+ 0 Now m, m Let i be the angle between the given lines m- m tan i + mm 88 th Std. Business Mathematics + + ^- h i tan - ^h.. Distance of a point from a line (i) The length of the perpendicular from a point P(l, m) to the line a + by + c 0 al + bm + c is d a + b (ii) The length of the perpendicular form the origin (0,0) to the line a + by + c 0 c is d a + b Eample.5 Show that perpendicular distances of the line - y+ 5 0from origin and from the point P(, ) are equal. Ch-.indd :8:04

97 Given line is - y+ 5 0 Perpendicular distance of the given line from P(, ) is Distance of (0,0) from the given line The given line is equidistance from origin and (, ) Eample.6 If the angle between the two lines is 4 r and slope of one of the lines is, then find the slope of the other line. by We know that the acute angle i between two lines with slopes mand m is given m- m tan i + mm r Given m andi 4 r - m ` tan 4 + m - m d + m n + m - m & m Hence the slope of the other line is...4 Concurrence of three lines If two lines l and l meet at a common point P, then that P is called point of intersection of l and l. This point of intersection is obtained by solving the equations of l and l. If three or more straight lines will have a point in common then they are said to be concurrent. The lines passing through the common point are called concurrent lines and the common point is called concurrent point. Analytical Geometry 89 Ch-.indd :8:06

98 Conditions for three given straight lines to be concurrent Let a + by + c 0 " () a + by + c 0 " () a + by + c 0 " () be the equations of three straight lines, then the condition that these lines to be concurrent is Eample.7 a a a b b b c c c 0 Show that the given lines -4y- 08, - y and -y- 7 0 are concurrent and find the concurrent point. Given lines -4y () 8- y... () -y () Conditon for concurrent lines is i.e., a a a b b b c c c 0 (77 99) + 4( 56+66) ( 4+) & Given lines are concurrent. To get the point of concurrency solve the equations () and () Equation () # & 6 8y 6 Equation () # & 6 9y y 5 90 th Std. Business Mathematics Ch-.indd :8:07

99 When y5 from () 8 88 Point of concurrency is (, 5) Eample.8 If the lines -5y- 05, + y- 7 0and + ky 0 are concurrent, find the value of k. Given the lines are concurrent. Therefore a a a b b b c c c k 0 (5+) k( +55) 0 & 4k 68. ` k. Eample.9 A private company appointed a clerk in the year 0, his salary was fied as `0,000. In 07 his salary raised to `5,000. (i) Epress the above information as a linear function in and y where y represent the salary of the clerk and -represent the year (ii) What will be his salary in 00? Let y represent the salary (in Rs) and represent the year year () salary (y) 0( ) 0,000(y ) 07( ) 5,000(y ) 00? The equation of straight line epressing the given information as a linear equation in and y is Analytical Geometry 9 Ch-.indd :8:08

100 y- y y - y y - 0, 000 5, 000-0, y - 0, In 00 his salary will be y ,000 y 000 9,9,000 y 000(00) 9,9,000 y `8000 Eercise.. Find the angle between the lines whose slopes are and. Find the distance of the point (4,) from the line- 4y+ 0. Show that the straight lines + y- 4 0, + 0and - y are concurrent. 4. Find the value of a for which the straight lines + 4y ; - 7y - and a -y are concurrent. 5. A manufacturer produces 80 TV sets at a cost of `,0,000 and 5 TV sets at a cost of `,87,500. Assuming the cost curve to be linear, find the linear epression of the given information. Also estimate the cost of 95 TV sets.. Pair of straight lines.. Combined equation of the pair of straight lines Let us consider the two individual equations of straight lines l + my + n 0 and l+ my + n 0 Then their combined equation is ^l + my + nh ^l + my + nh 0 9 th Std. Business Mathematics Ch-.indd :8:0

101 ll + ^lm + lmhy + mmy + ^ln + lnh+ ^mn + mnhy+ n n 0 Hence the general equation of pair of straight lines can be taken as a + hy+ by + g+ fy+ c 0 where abc,,, f, g and h are all constants... Pair of straight lines passing through the origin The homogeneous equation a + hy+ by 0... () of second degree in and y represents a pair of straight lines passing through the origin. Let y m and y m be two straight lines passing through the origin. Then their combined equation is ^y-mh^y-m h 0 & mm m m - ^ + h y + y 0... () () and () represent the same pair of straight lines a ` mm h b - m + ^ mh a & mm b and m m b h + -. a i.e., product of the slopes b and sum of the slopes -.. Angle between pair of straight lines passing through the origin The equation of the pair of straight lines passing through the origin is a + hy+ by 0 Let m and m be the slopes of above lines. Here m + m - b h a and mm b. Let i be the angle between the pair of straight lines. b h Then m- m tan i + mm Analytical Geometry 9 Ch-.indd :8:

102 ± h - ab a+ b Let us take i as acute angle ` i tan - h - ab < F a+ b NOTE (i) If i is the angle between the pair of straight lines a + hy+ by + g+ fy+ c 0, then i tan - h - ab < F a+ b (ii) If the straight lines are parallel, then h ab. (iii) If the straight lines are perpendicular, then a+ b 0 i.e., cqefficient qf + cqefficient qf y 0..4 The condition for general second degree equation to represent the pair of straight lines NOTE The condition for a general second degree equation in, y namely The condition in determinant form is a + hy+ by + g+ fy+ c 0 to represent a pair of straight lines is abc+ fgh af bg ch 0. Eample.0 Find the combined equation of the given straight lines whose separate equations are + y- 0 and + y The combined equation of the given straight lines is ^+ y-h^+ y- 5h 0 i.e., + y- + 4y + y -y-0-5y+ 5 0 i.e., + 5y+ y -- 7y+ 5 0 a h g h b f g f c 0 94 th Std. Business Mathematics Ch-.indd :8:7

103 Eample. Show that the equation + 5y+ y y+ 4 0 represents a pair of straight lines. Also find the angle between them. Compare the equation + 5y+ y y+ 4 0 with a + hy+ by + g+ fy+ c 0, we get 5 7 a, b, h, g, f and c 4 Condition for the given equation to represent a pair of straight lines is abc+ fgh af bg ch abc+ fgh af bg ch Hence the given equation represents a pair of straight lines. Let θ be the angle between the lines. h - ab 5 Then i tan - < F a+ tan 4-6 b - > H 5 ` i tan - b 5 l Eample. The slope of one of the straight lines a + hy+ by 0 is twice that of the other, show that 8h 9ab Let m and m be the slopes of the pair of straight lines a + hy+ by 0 ` m+ m - b h a and mm b It is given that one slope is twice the other, so let m ` m + m - b h and m$ m b a m Analytical Geometry 95 Ch-.indd :8:9

104 ` h m - b and m & b- h b l a b & & 8h 9b b a 8h 9ab b a Eample. Show that the equation + 7y+ y y+ 0 represent two straight lines and find their separate equations. 96 th Std. Business Mathematics Compare the equation + 7y+ y y+ 0 with a + hy+ by + g+ fy+ c 0, we get, a, b, h, g, f, c 7 5 a h g a 7 5 Now h b f 5 5 g f c 5 b6-7 4 l- b l+ 5 b l b- 4 l- 7 $ $ Hence the given equation represents a pair of straight lines. Now consider, + 7y+ y + 6y+ y+ y ^+ yh+ y^+ yh ^+ yh^+ yh Let + 7y+ y y+ ^+ y+ lh^+ y+ mh Comparing the coefficient of, l + m 5 () Comparing the coefficient of y, l + m 5 () Solving () and (), we get m and l ` The separate equations are + y+ 0 and + y+ 0. Ch-.indd :8:

105 Eample.4 Show that the pair of straight lines 4 - y+ 9y + 8-7y+ 8 0 represents a pair of parallel straight lines and find their separate equations. The given equation is 4 - y+ 9y + 8-7y+ 8 0 Here a 4, b 9 and h 6 h - ab Hence the given equation represents a pair of parallel straight lines. Now 4 - y+ 9y ^- yh Consider 4 - y+ 9y + 8-7y+ 8 0 & ^- yh + 9 ^ - yh Put y z z + 9z (z+)(z+8) 0 z+ 0 z+8 0 y+ 0 y+8 0 Hence the separate equations are Eample.5 y+ 0 and y+8 0 Find the angle between the straight lines + 4y+ y 0 The given equation is + 4y+ y 0 Here a, b and h. Analytical Geometry 97 Ch-.indd :8:5

106 If i is the angle between the given straight lines, then Eample.6 θ tan - h - ab < F a+ b tan - 4 < F - tan - ^ h i r For what value of k does + 5y+ y y+ k 0 represent a pair of straight lines. 5 Here a, b, h 5, g, f 9, c k. The given line represents a pair of straight lines if, abc+ fgh -af -bg - ch i.e., 4k k 0 & 6k k 0 & 9k 5 ` k 8 Eercise.. If the equation a + 5y - 6y + + 5y+ c 0 represents a pair of perpendicular straight lines, find a and c.. Show that the equation - 0y+ y + 4-5y+ 0 represents a pair of straight lines and also find the separate equations of the straight lines.. Show that the pair of straight lines 4 + y+ 9y -6-9y+ 0 represents two parallel straight lines and also find the separate equations of the straight lines. 4. Find the angle between the pair of straight lines -5y- y + 7+ y th Std. Business Mathematics Ch-.indd :8:7

107 .4 Circles Definition. A circle is the locus of a point which moves in such a way that its distance from a fied point is always constant. The fied point is called the centre of the circle and the constant distance is the radius of the circle..4. The equation of a circle when the centre and radius are given. Let C(h, k) be the centre and r be the radius of the circle Let P(, y) be any point on the circle y C r R P(, y) CP r CP r O L M ( h) + (y k) r is the equation of the Fig..4 circle. In particular, if the centre is at the origin, the equation of circle is + y r Eample.7 Find the equation of the circle with centre at (, ) and radius is 4 units. Equation of circle is ^- hh + ^y- kh r Here (h, k) (, ) and r 4 Equation of circle is ^- h + ^y+ h y + y+ 6 + y - 6+ y- 6 0 Eample.8 Find the equation of the circle with centre at origin and radius is units. Analytical Geometry 99 Ch-.indd :8:8

108 Equation of circle is + y r Here r i.e equation of circle is + y 9.4. Equation of a circle when the end points of a diameter are given Let A^, yh and B^, yh be the end points of a diameter of a circle and P(, y) be any point on the circle. P(, y) We know that angle in the semi circle is 90 o ` APB 90c ` (Slope of AP) (Slope of BP) A(, y ) C B(, y ) y- y y- y d n # d n - - Fig..5 ^y-yh^y-yh -^-h^-h & ^-h^- h+ ^y-yh^y-yh 0 is the required equation of circle. Eample.9 Find the equation of the circle when the end points of the diameter are (, 4) and (, ). Equation of a circle when the end points of the diameter are given is ^-h^- h+ ^y-yh^y-yh 0 Here ^, yh (, 4) and ^, yh (, ) ` ] -g] - g+ _ y- 4i_ y+ i 0 + y -5-y General equation of a circle The general equation of circle is + y + g + fy + c 0 where g, f and c are constants. 00 th Std. Business Mathematics Ch-.indd :8:

109 i.e + y + g + fy c + g+ g - g + y + fy + f - f c ^+ gh - g + ^y+ fh - f c ^+ gh + ^y+ f h g + f -c 7 - ^- gha + 7y - ^-f ha 7 g + f -ca Comparing this with the circle^- hh + ^y- kh r We get, centre is ^-g, -fh and radius is g + f -c NOTE The general second degree equation a + by + hy+ g+ fy+ c 0 represents a circle if (i) a b i.e., co-efficient of co-efficient of y. (ii) h 0 i.e., no y term. Eample.0 Find the centre and radius of the circle + y y- 4 0 Equation of circle is + y y- 4 0 Here g 4, f and c 4 Centre C^-g, - fh C^4, - h and Radius: r g + f - c Eample unit. For what values of a and b does the equation a ] - g + by + ] b - g y y- 0 represents a circle? Write down the resulting equation of the circle. Analytical Geometry 0 Ch-.indd :8:

110 The given equation is a ] - g + by + ] b - g y y- 0 As per conditions noted above, (i) coefficient of y 0 & b - 0 ` b (ii) coefficient of Coefficient of y & a b a & a 4 ` Resulting equation of circle is + y y - 0 Eample. If the equation of a circle + y + a + by 0 passing through the points (, ) and (, ), find the values of a and b The circle + y + a + by 0 passing through (, ) and (, ) ` We have a + b 0 and + + a + b 0 & a+ b 5 () and a + b () solving () and (), we get a, b,. Eample. If the centre of the circle + y + - 6y+ 0 lies on a straight line a + y + 0, then find the value of a Centre C(, ) It lies on a + y + 0 a a 8 0 th Std. Business Mathematics Ch-.indd :8:5

111 Eample.4 Show that the point (7, 5) lies on the circle + y y 0 and find the coordinates of the other end of the diameter through this point. Let A(7, 5) Equation of circle is + y y- 0 Substitute (7, 5) for (, y), we get + y y ]- g - 6] 7g + 4] -5g - ` (7, 5) lies on the circle Here g and f ` Centre C(, ) Let the other end of the diameter be B (, y) Midpoint of AB + y, C, 7-5 c m ^ - h + 7 y y Other end of the diameter is (, ). Eample.5 Find the equation of the circle passing through the points (0,0), (, ) and (,0). Let the equation of the circle be + y + g + fy + c 0 The circle passes through the point (0, 0) c 0... () The circle passes through the point (, ) Analytical Geometry 0 Ch-.indd :8:6

112 + + g() + f() + c 0 g + 4f + c 5... () The circle passes through the point (, 0) g() + f() 0 + c 0 Solving (), () and (), we get g -, f - 4 and c 0 ` The equation of the circle is - + y + ( - ) + b 4 l y+ 0 0 ie., + y -4- y Parametric form of a circle 4g + c 4... () Consider a circle with radius r and centre at the origin. Let Py (, ) be any point on the circle. Assume that OP makes an angle i with the positive direction of -ais. Draw PM perpendicular to -ais. Y P(, y) From the figure, cosi r & r cos i r sini i r & y rsin i O M X The equations r cos i, y r sin i are called the parametric equations of the circle + y r. Here i is called the parameter and 0 # i # r. Fig..6 Eample.6 Find the parametric equations of the circle Here r 5 & r 5 Parametric equations are r cos i, y r sin i & 5cosi, y 5sin i, 0 # i # r + y 5 04 th Std. Business Mathematics Ch-.indd :8:9

113 Eercise.4. Find the equation of the following circles having (i) the centre (,5) and radius 5 units (ii) the centre (0,0) and radius units. Find the centre and radius of the circle (i) + y 6 (ii) + y -- 4y+ 5 0 (iii) 5 + 5y + 4-8y- 6 0 (iv) ] + g] - 5g+ ^y-h^y- h 0. Find the equation of the circle whose centre is (, ) and having circumference 6r 4. Find the equation of the circle whose centre is (,) and which passes through (,4) 5. Find the equation of the circle passing through the points (0, ),(4, ) and (, ). 6. Find the equation of the circle on the line joining the points (,0), (0,) and having its centre on the line + y 7. If the lines + y 6 and + y 4 are diameters of the circle, and the circle passes through the point (, 6) then find its equation. 8. Find the equation of the circle having (4, 7) and (, 5) as the etremities of a diameter. 9. Find the Cartesian equation of the circle whose parametric equations are cos i, y sin i, 0 # i # r..4.5 Tangents The equation of the tangent to the circle + y + g + fy + c 0 at ^, yh is + yy + g ( + ) + f( y+ y ) + c 0. Corollary: The equation of the tangent at ^, yh to the circle + y a is + yy a C( g, f) P(, y ) Fig..7 Analytical Geometry T 05 Ch-.indd :8:4

114 NOTE To get the equation of the tangent at ^, yh to the circle Eample.7 replace by, y + y + g + fy + c 0... () by yy, by + and y by y + y in equation (). Find the equation of tangent at the point (, 5) on the circle + y + - 8y The equation of the tangent at, y ^ h to the given circle + y + - 8y+ 7 0 is yy + + # y y ^ + h - 8 # + ^ h Here ^, yh (, 5) - + 5y+ ( -) - 4( y+ 5) y+ --4y y+ -6-8y y+ 0 is the required equation. Length of the tangent to the circle Length of the tangent to the circle + y + g + fy + c 0 from a point P ^, yh is PT + y + g + fy + c T C( g, f) P(, y ) 06 th Std. Business Mathematics Fig..8 Ch-.indd :8:4

115 Eample.8 NOTE (i) If the point P is on the circle then PT 0 (ii) If the point P is outside the circle then PT > 0 (iii) If the point P is inside the circle then PT < 0 Find the length of the tangent from the point (,) to the circle + y y+ 8 0 The length of the tangent to the circle + y + g + fy + c 0 from a point ^, yh is + y + g + fy + c Length of the tangent + y y () + 4() + 8 [Here ^, y h (, )] 49 Eample.9 Length of the tangent 7 units Determine whether the points P(0,), Q(5,9), R(, ) and S(, ) lie outside the circle, on the circle or inside the circle + y y- 8 0 The equation of the circle is + y y- 8 0 PT + y y -8 At P(0, ) PT < 0 At Q(5, 9) QT > 0 At R(, ) RT > 0 At S(, ) ST ` The point P lies inside the circle. The points Q and R lie outside the circle and the point S lies on the circle. Analytical Geometry 07 Ch-.indd :8:45

116 Result Condition for the straight line y m + c to be a tangent to the circle + y a is c a ( + m ) Eample.0 Find the value of k so that the line + 4y- k 0 is a tangent to + y The given equations are + y and + 4y- k 0 The condition for the tangency is c a ( + m ) Here a 64, m - 4 and c k 4 k c a ( + m ) & 6 64b l k 64 5 k ± th Std. Business Mathematics Eercise.5. Find the equation of the tangent to the circle + y y- 8 0 at (, ). Determine whether the points P(, 0), Q(, ) and R(, ) lie outside the circle, on the circle or inside the circle + y -4-6y Find the length of the tangent from (, ) to the circle + y - + 4y Find the value of P if the line + 4y- P 0 is a tangent to the circle + y 6.5 Conics Definition. Fied line (directri) l M P(Moving point) If a point moves in a plane such that its distance from a fied point bears a constant ratio to its perpendicular distance from a fied straight line, then the path described by the moving point is called a conic. F(Fied point) In figure, the fied point F is called focus, the fied straight line l is called directri and P is the Fig..9 moving point such that PM FP e, a constant. Here the locus of P is called a conic and the constant e is called the eccentricity of the conic. Ch-.indd :8:48

117 Based on the value of eccentricity we can classify the conics namely, a) If e, then, the conic is called a parabola b) If e <, then, the conic is called an ellipse c) If e >, then, the conic is called a hyperbola. The general second degree equation a + hy+ by + g+ fy+ c 0 represents, (i) a pair of straight lines if abc+ fgh -af -bg - ch 0 (ii) a circle if a b and h 0 If the above two conditions are not satisfied, then a + hy+ by + g+ fy+ c 0 represents, (i) a parabola if h ab 0 (ii) an ellipse if h ab < 0 (iii) a hyperbola if h ab > 0 In this chapter, we study about parabola only..5. Parabola Definition.4 The locus of a point whose distance from a fied point is equal to its distance from a fied line is called a parabola. y 4a is the standard equation of the parabola. It is open rightward..5. Definitions regarding a parabola: y 4a M( a, y) y P(, y) Z( a, 0) V F(a, 0) Fig..0 Analytical Geometry 09 Ch-.indd :8:49

118 Focus The fied point used to draw the parabola is called the focus (F). Here, the focus is F(a, 0). Directri The fied line used to draw a parabola is called the directri of the parabola. Here, the equation of the directri is a. Ais The ais of the parabola is the ais of symmetry. The curve y 4a is symmetrical about -ais and hence -ais or y 0 is the ais of the parabola y 4a. Note that the ais of the parabola passes through the focus and perpendicular to the directri. Verte The point of intersection of the parabola with its ais is called its verte. Here, the verte is V(0, 0). Focal distance The distance between a point on the parabola and its focus is called a focal distance Focal chord A chord which passes through the focus of the parabola is called the focal chord of the parabola. Latus rectum It is a focal chord perpendicular to the ais of the parabola. Here, the equation of the latus rectum is a. Length of the latus rectrum is 4a..5. Other standard parabolas :. Open leftward : y - 4aa 6 0@ If > 0, then y become imaginary. i.e., the curve eist for 0. y 4a y a F( a, 0) V Fig... Open upward : 4aya 6 0@ 0 th Std. Business Mathematics Ch-.indd :8:50

119 If y < 0, then becomes imaginary. i.e., the curve eist for y 0. y 4ay F(0, a) V y a Fig... Open downward : - 4aya 6 0@ If y > 0, then becomes imaginary. i.e., the curve eist for y 0. y y a V 4ay F(0, a) Fig.. Equations y 4a y 4a 4ay 4ay Ais y 0 y Verte V(0, 0) V(0, 0) V(0, 0) V(0, 0) Focus F(a, 0) F( a, 0) F(0, a) F(0, a) Equation of directri a a y a y a Length of Latus rectum 4a 4a 4a 4a Equation of Latus rectum a a y a y a The process of shifting the origin or translation of aes. Consider the oy system. Draw a line parallel to -ais (say X ais) and draw a line parallel to y ais (say Y ais). Let P^, yhbe a point with respect to oy system and P^X, Yh be the same point with respect to XOlY system. Analytical Geometry Ch-.indd :8:5

120 y Y (X, Y) P(, y) O Y O h (0, 0) k M L X Fig..4 Let the co-ordinates of Ol with respect to oy system be (h, k) The co-ordinate of P with respect to oy system: OL OM + ML h + X i.e) X + h similarly y Y + k ` The new co-ordinates of P with respect to XOl Y system. X h and Y y k.5.4 General form of the standard equation of a parabola, which is open rightward (i.e., the verte other than origin) : Consider a parabola with verte V whose co-ordinates with respect to XOY system is (0, 0) and with respect to oy system is (h, k). Since it is open rightward, the equation of the parabola w.r.t. XOY system is Y 4aX. By shifting the origin X -h and Y y- k, the equation of the parabola with respect to old oy system is^y- kh 4a^- hh. This is the general form of the standard equation of the parabola, which is open rightward. Similarly the other general forms are ^y- kh -4a^-hh (open leftwards) th Std. Business Mathematics Ch-.indd :8:5

121 ^- hh 4a^y- kh (open upwards) ^- hh -4a^y-kh (open downwards) Eample. Find the equation of the parabola whose focus is (,) and whose directri is - y+ 0. F is (, ) and directri is - y+ 0 - y+ 0 M P(, y) let P^, yh be any point on the parabola. FP For parabola PM FP PM F(, ) FP ^- h + ^y- h y - 6y+ 9 Fig..5 + y -- 6y+ 0 y + 0 ^ PM + - y + h + ^ + - y +h ^- y+ h + y + 4-y- 4y+ 4 PM FP PM + y -4- y+ 0 + y -y- 4y The required equation of the parabola is & + y + y -8-8y+ 6 0 Eample. Find the focus, the verte, the equation of the directri, the ais and the length of the latus rectum of the parabola y - Analytical Geometry Ch-.indd :8:56

122 The given equation is of the form y - 4a where 4a, a. Y The parabola is open left, its focus is F^-a, qh, F^-0, h Its verte is Vhk ^, h V^00, h F(, 0) V(0,0) X The equation of the directri is a i.e., Fig..6 Its ais is ais whose equation is y 0. Length of the latus rectum 4a Eample. Show that the demand function 0p-0 - p is a parabola and price is maimum at its verte. 0p-0 -p 4 th Std. Business Mathematics -p p p + 0p- 5 -_ p - 0p+ 5i -^p -5h Put X -5 and P p-5 X - P i.e P - X It is a parabola open downward. ` At the verte p 5 when 5. i.e., price is maimum at the verte. Eample.4 Find the ais, verte, focus, equation of directri and length of latus rectum for the parabola + 6-4y + 0 Ch-.indd :8:58

123 X +, Y y- 4y y ^ h + 4y ^ + h ^ + h 4(y ) X- and y Y+ X 4Y Comparing with X 4aY, 4a 4 a Referred to (, y) Referred to (X, Y) X-, y Y+ Ais y 0 Y 0 y Verte V(0,0) V(0,0) V(,) Focus F(0, a) F(0, ) F(, 4) Equation of directri (y a) Y y Length of Latus rectum (4a) 4() 4 4 Eample.5 The supply of a commodity is related to the price by the relation 5P- 5. Show that the supply curve is a parabola. The supply price relation is given by 5p 5 & 4aP Where X and P p- 5(p ) ` the supply curve is a parabola whose verte is ( X 0, P 0) i.e., The supply curve is a parabola whose verte is (0, ) Analytical Geometry 5 Ch-.indd :9:0

124 Step ICT Corner Epected final outcomes Open the Browser and type the URL given (or) Scan the QR Code. GeoGebra Work book called th BUSINESS MATHS will appear. In this several work sheets for Business Maths are given, Open the worksheet named Locus-Parabola Step - Locus-Parabola page will open. Click on the check boes on the Left-hand side to see the respective parabola types with Directri and Focus. On right-hand side Locus for parabola is given. You can play/pause for the path of the locus and observe the condition for the locus. Step Step Step 4 Browse in the link th Business Maths: (or) scan the QR Code 6 th Std. Business Mathematics Ch-.indd :9:0

125 Eercise.6. Find the equation of the parabola whose focus is the point F^-, -hand the directri is the line 4 y+ 0.. The parabola y k passes through the point (4, ). Find its latus rectum and focus.. Find the verte, focus, ais, directri and the length of latus rectum of the parabola y 8y Find the co-ordinates of the focus, verte, equation of the directri, ais and the length of latus rectum of the parabola (a) y 0 (b) 8y (c) - 6y 5. The average variable cost of a monthly output of tonnes of a firm producing a valuable metal is ` Show that the average variable cost curve is a parabola. Also find the output and the average cost at the verte of the parabola. 6. The profit ` y accumulated in thousand in months is given by y Find the best time to end the project. Eercise.7 Choose the correct answer. If m and m are the slopes of the pair of lines given by a + hy+ by 0, then the value of m+ m is (a) b h (b) - b h (c) a h h (d) a. The angle between the pair of straight lines 7y+ 4y 0 is (a) tan - b l (b) tan - b l (c) tan - c m 5 (d) tan - 5 d n. If the lines -y- 5 0 and -4y are the diameters of a circle, then its centre is (a) (, ) (b) (,) (c) (, ) (d) (, ) 4. The intercept of the straight line + y- 0 is (a) (b) (c) (d) Analytical Geometry 7 Ch-.indd :9:05

126 5. The slope of the line 7+ 5y- 8 0 is (a) 5 7 (b) 5 7 (c) 7 5 (d) The locus of the point P which moves such that P is at equidistance from their coordinate aes is (a) y (b) y - (c) y (d) y - 7. The locus of the point P which moves such that P is always at equidistance from the line + y+ 7 0 is (a) + y+ 0 (b) - y+ 0 (c) - y+ 0 (d) + y If k + y - y 0 represent a pair of lines which are perpendicular then k is equal to (a) (b) 8 th Std. Business Mathematics (c) (d) 9. (, ) is the centre of the circle + y + a + by - 4 0, then its radius (a) (b) (c) 4 (d) 0. The length of the tangent from (4,5) to the circle + y 6 is (a) 4 (b) 5 (c) 6 (d) 5. The focus of the parabola 6y is (a) (4,0) (b) ( 4, 0) (c) (0, 4) (d) (0, 4). Length of the latus rectum of the parabola y - 5. (a) 5 (b) 5 (c) 5 (d) 5. The centre of the circle + y - + y- 9 0 is (a) (,) (b) (, ) (c) (,) (d) (, ) 4. The equation of the circle with centre on the ais and passing through the origin is (a) - a+ y 0 (b) y - ay+ 0 (c) + y a (d) - ay+ y 0 Ch-.indd :9:09

127 5. If the centre of the circle is ( a, b) and radius is a - b, then the equation of circle is (a) + y + a + by + b 0 (b) + y + a + by - b 0 (c) + y -a -by - b 0 (d) + y -a - by + b 0 6. Combined equation of co-ordinate aes is (a) - y 0 (b) + y 0 (c) y c (d) y 0 7. a + 4y + y 0 represents a pair of parallel lines then a is (a) (b) (c) 4 (d) 4 8. In the equation of the circle + y 6 then y intercept is (are) (a) 4 (b)6 (c) ±4 (d) ±6 9. If the perimeter of the circle is 8π units and centre is (,) then the equation of the circle is (a) - + y- ^ h ^ h 4 (b) ^- h + ^y- h 6 (c) y- 4 ^ h ^ h (d) + y 4 0. The equation of the circle with centre (, 4) and touches the ais is (a) ^- h + ^y- 4h 4 (b) ^- h + ^y+ 4h 6 (c) ^- h + ^y- 4h 6 (d) + y 6. If the circle touches ais, y ais and the line 6 then the length of the diameter of the circle is (a) 6 (b) (c) (d) 4. The eccentricity of the parabola is (a) (b) (c)0 (d). The double ordinate passing through the focus is (a) focal chord (b) latus rectum (c) directri (d) ais Analytical Geometry 9 Ch-.indd :9:

128 4. The distance between directri and focus of a parabola y 4a is (a) a (b)a (c) 4a (d) a 5. The equation of directri of the parabola y - is (a) (b) 4-0 (c) (d) Miscellaneous Problems. A point P moves so that P and the points (, ) and (, 5) are always collinear. Find the locus of P.. As the number of units produced increases from 500 to 000 and the total cost of production increases from ` 6000 to `9000. Find the relationship between the cost (y) and the number of units produced () if the relationship is linear.. Prove that the lines 4+ y 0, 4y 5 and 5+ y 7are concurrent. 4. Find the value of p for which the straight lines 8p + ( - py ) + 0 and p + py are perpendicular to each other. 5. If the slope of one of the straight lines a + hy+ by 0 is thrice that of the other, then show that h 4ab. 6. Find the values of a and b if the equation a ] - g + by + ] b - 8g y y- 0 represents a circle. 7. Find whether the points ^-, -h, ^0, h and ^-, -4h lie above, below or on the line + y If ^4, h is one etremity of a diameter of the circle + y - + 6y- 5 0, find the other etremity. 9. Find the equation of the parabola which is symmetrical about -ais and passing through (, ). 0. Find the ais, verte, focus, equation of directri length of latus rectum of the parabola^y- h 4^- h 0 th Std. Business Mathematics Ch-.indd :9:6

129 Summary z Angle between the two intersecting lines y m + c and - m- m y m+ c is i tan + mm z Condition for the three straight lines a + by + c 0, a + by + c 0, and a b c a + by + c 0 to be concurrent is a b c 0 a b c z The condition for a general second degree equation a + hy+ by + g+ fy+ c 0 to represent a pair of straight lines is abc+ fgh af bg ch 0. z Pair of straight lines passing through the origin is a + hy+ by 0. z Let a + hy+ by 0 be the straight lines passing through the origin then the a product of the slopes is - b and sum of the slopes is b h. z If i is the angle between the pair of straight lines! h - ab z a + hy+ by + g+ fy+ c 0, then i tan - < a+ F. b z The equation of a circle with centre (h, k) and the radius r is^- hh + ^y- kh r z The equation of a circle with centre at origin and the radius r is + y r. z The equation of the circle with ^, yh and ^, yh as end points of a diameter is ^-h^- h+ ^y-yh^y-yh 0. z The parametric equations of the circle + y r are r cos i, y r sin i 0 # i # r z The equation of the tangent to the circle + y + g + fy + c 0 at ^, yh is + yy + g ( + ) + f( y+ y ) + c 0 z The equation of the tangent at, y ^ h to the circle + y a is + yy a. z Length of the tangent to the circle + y + g + fy + c 0 from a point ^, yh is + y + g + fy + c. z Condition for the straight line y m+ c to be a tangent to the circle + y a is c a ^ + m h. z The standard equation of the parabola is y 4a Analytical Geometry Ch-.indd :9:9

130 GLOSSARY Centre ம யம Chord ந ண Circle வட டம Concurrent Line ஒர ப ள ள வழள க க ட Conics க ம ப வவடட ள Diameter வள ட டம Directi இயக க வர Equation சமன ட Focal distance க வள யத த ரம Focus க வள யம Latus rectum ச வவ லம Length of the tangent த ட க டட ன ந ளம Locus நள யமப ம அலலத இயங க வர Origin ஆ ள Pair of straight line இரட ட ந ர க ட Parabola பரவள யம Parallel line இண க ட Parameter த ண யலக Perpendicular line ச ங க த த க ட Point of concurrency ஒர ங ள ம வ ப ப ள ள Point of intersection வவடட ம ப ள ள Radius ஆரம Straight line ந ர க ட Tangent த ட க ட Verte ம ன th Std. Business Mathematics Ch-.indd :9:9

131 Chapter 4 TRIGONOMETRY Learning Objectives After studying this chapter, the students will be able to understand z trigonometric ratio of angles z addition formulae, multiple and sub-multiple angles z transformation of sums into products and vice versa z basic concepts of inverse trigonometric functions z properties of inverse trigonometric functions Introduction The word trigonometry is derived from the Greek word tri (meaning three), gon (meaning sides) and metron (meaning measure). In fact, trigonometry is the study of relationships between the sides and angles of a triangle. Around second century A.D. George Rheticus was the first to define the trigonometric functions in terms of right angles. The study of trigonometry was first started in India. The ancient Indian Mathematician, Aryabhatta, Brahmagupta, Bhaskara I and Bhaskara II obtained important results. Brahmagupta Bhaskara I gave formulae to find the values of sine functions for angles more than 90 degrees.the earliest applications of trigonometry were in the fields of navigation, surveying and astronomy. Currently, trigonometry is used in many areas such as electric circuits, describing the state of an atom, predicting the heights of tides in the ocean, analyzing a musical tone. Trigonometry Ch-4.indd :9:59

132 Recall. sinθ side opposite to angle i Hypotenuse side opposite to angle i. tanθ Adjacent side to angle i Hypotenuse 5. secθ Adjacent side to angle i. cosθ Adjacent side toangle i Hypotenuse Adjacent side to angle i 4. cotθ side opposite to angel i Hypotenuse 6. cosecθ side opposite to angle i Relations between trigonometric ratios. sinθ cosec i. cosθ sec i. tanθ sin i cos i 4. tanθ cot i or cosecθ sin i or secθ cos i cos i or cotθ sin i or cotθ tan i Trigonometric Identities. sin i+ cos i. + tan i sec θ. + cot i cosec θ. Angle Angle is a measure of rotation of a given ray about its initial point. The ray OA is called the initial side and the final position of the ray OB after rotation is called the terminal side of the angle. The point O of rotation is called the verte. O Initial side Angle AOB If the direction of rotation is anticlockwise, then angle is said to be positive and if the Fig. 4. direction of rotation is clockwise, then angle is said to be negative. Terminal side B A Measurement of an angle Two types of units of measurement of an angle which are most commonly used namely degree measure and radian measure. 4 th Std. Business Mathematics Ch-4.indd :40:0

133 Degree measure th If a rotation from the initial position to the terminal position b 60 l of the revolution, the angle is said to have a measure of one degree and written as o. A degree is divided into minutes and minute is divided into seconds. One degree 60 minutes 60l One minute 60 seconds (60m ) Radian measure r B r The angle subtended at the centre of the circle by an arc equal to the length of the radius of the circle is called a radian, and it is denoted by c O c r A Fig. 4. NOTE The number of radians in an angle subtended by an arc of a circle at the centre of a circle is length of the arc radius i.e., i r s, where θ is the angle subtended at the centre of a circle of radius r by an arc of length s. Relation between degrees and radians We know that the circumference of a circle of radius unit is π. One complete revolution of the radius of unit circle subtends π radians. A circle subtends at the centre an angle whose degree measure is60c. Eample 4. π radians 60c π radians 80c radian 80c r Convert (i) 60c into radians (ii) degree 4r 5 radians into degree (iii) 4 radians into Trigonometry 5 Ch-4.indd :40:0

134 i) 60c 60 # 80 r 8 radians 9 r 4r 4 80 ii) 5 radians 5 # c r r 44c 80 iii) 4 radian 4 r 4 # 80 # 7 4c9lll 5 4. Trigonometric ratios 4.. Quadrants: Y Let XOX l and Yl OY be two lines at right angles to each other as in the figure. We call XOX l and Yl OY as X ais and Y ais respectively. Clearly these aes divided the entire plane into four equal parts called Quadrants. X II III O I IV X The parts XOY, YOX ', XOYl ' and YlOX are known as the first, second, third and the Y fourth quadrant respectively. Fig. 4. Eample 4. Find the quadrants in which the terminal sides of the following angles lie. (i) (i) - 70c (ii) - 0c (iii) 5c. (i) The terminal side of - 70c lies in IV quadrant. Y X O X P 6 th Std. Business Mathematics Y Fig. 4.4 Ch-4.indd :40:05

135 (ii) The terminal side -0cof lies in I quadrant. Y P X O X Y Fig. 4.5 (iii) 5c ] # 60g + 80c+ 65c the terminal side lies in III quadrant. Y X X P Y Fig Signs of the trigonometric ratios of an angle θ as it varies from 0º to 60º In the first quadrant both and y are positive. So all trigonometric ratios are positive. In the second quadrant ] 90c i 80cg is negative and y is positive. So trigonometric ratios sin θ and cosec θ are positive. In the third quadrant] 80c i 70cg both and y are negative. So trigonometric ratios tan θ and cot θ are positive. In the fourth quadrant ] 70c < i < 60cg is positive and y is negative. So trigonometric ratios cos θ and sec θ are positive. A function f () is said to be odd function if f( - ) f. () sini, tan i, cotiand cosec i are all odd function. A function f () is said to be even function if f( - ) f. () cosiand sec i are even function. Trigonometry 7 Ch-4.indd :40:06

136 Y II Quadrant sin & cosec I Quadrant All X III Quadrant tan & cot O IV Quadrant cos & sec X Y ASTC : All Sin Tan Cos Fig Trigonometric ratios of allied angles: Two angles are said to be allied angles when their sum or difference is either zero or a multiple of 90c. The angles - i, 90c! i, 80c! i, 60c! i etc.,are angles allied to the angle θ. Using trigonometric ratios of the allied angles we can find the trigonometric ratios of any angle. 90c - i 90c + i 80c - i 80c + i 70c - i 70c + i 60c - i 60c + i Angle/ - i or or or or or or or or Function r r r r - i + i r- i r+ i - i + i r- i r+ i Sine - sin i cos i cos i sin i - sin i - cos i - cos i - sin i sin i Cosine cos i sin i - sin i - cos i - cos i - sin i sin i cos i cos i tangent cotangent - tan i cot i - cot i - tan i tan i cot i - cot i - tan i tan i - cot i tan i - tan i - cot i cot i tan i - tan i - cot i cot i secant sec i cosec i - cosec i - sec i - sec i - cosec i cosec i sec i sec i cosecant - cosec i sec i sec i cosec i - cosec i - sec i - sec i - cosec i cosec i Eample 4. Table : 4. Find the values of each of the following trigonometric ratios. (i) sin 50c (ii) cos( - 0c) (iii) cosec 90c (iv) tan( - 5c) (v) sec485c 8 th Std. Business Mathematics Ch-4.indd :40:0

137 (i) sin50c sin( 90c+ 60c) Since 50c lies in the second quadrant, we have sin50c sin( 90c+ 60c) cos60c (ii) we have cos( 0 o ) cos0 o Since 0 lies in the third quadrant, we have cos0 o cos(80 o + 0 o ) cos0 o (iii) cosec90 o cosec(60 o + 0 o ) cosec0 o (iv) tan( 5 o ) tan(5 o ) tan( 60 o + 5 o ) tan5 o tan(90 o + 45 o ) ( cot45 o ) (v) sec485 o sec(4 60 o + 45 o ) sec45 o Eample 4.4 Prove that sin600 o cos 90 o + cos 480 o sin 50 o - sin600 o sin(60 o +40 o ) sin40 o sin(80 o + 60 o ) sin 60 o - cos90 o cos(60 o + 0 o ) cos0 o cos480 o cos(60 o + 0) cos0 o cos(80 o 60 o ) cos60 o sin50 o sin(80 o 0 o ) sin0 o - Now sin 600 o cos90 o + cos480 o sin50 o b - lb l+ b- lb l Trigonometry 9 Ch-4.indd :40:

138 Eample 4.5 Prove that sin] -igtan] 90c-igsec] 80c-ig sin( 80 + i) cot( 60 -i) cosec] 90c - ig L.H.S sin] -igtan] 90c-igsec] 80c-ig sin( 80 + i) cot( 60 -i) cosec] 90c - ig ]-sin igcot i] -sec ig ]-sinig]-cot igsec i 0 th Std. Business Mathematics Eercise 4.. Convert the following degree measure into radian measure (i) 60 o (ii) 50 o (iii) 40 o (iv) 0 o. Find the degree measure corresponding to the following radian measure. r 9r r (i) 8 (ii) 5 (iii) (iv) 8. Determine the quadrants in which the following degree lie. (i) 80 o (ii) 40 o (iii) 95 o 4. Find the values of each of the following trigonometric ratios. (i) sin00 o (ii) cos( 0 o ) (iii) sec90 o (iv) tan( 855 o ) (v) cosec5 o 5. Prove that: (i) tan( 5 o ) cot( 405 o ) tan( 765 o ) cot(675 o ) 0 (ii) r sin 6 cosec r r + 6 cos (iii) r 5r 5r 5r secb - i lsecb i - l+ tanb + i ltanb i - l- 6. If A, B, C, D are angles of a cyclic quadrilateral, prove that : cosa + cosb + cosc + cosd 0 7. Prove that : (i) sin] 80c- igcos] 90c+ igtan] 70 -igcot] 60 -ig sin] 60 - igcos] 60 + igsin] 70 - igcosec] -ig (ii) sini$ cos i ' sin b r r - il $ cosec i + cos b - il $ sec i Ch-4.indd :40:

139 8. Prove that : cos50 o cos0 o + sin90 o cos0 o 9. Prove that: (i) tan] r+ gcot] -rg- cos] r- gcos] r+ g sin (ii) sin] 80c+ Agcos] 90c-Agtan] 70c - Ag sin cos sin 540c- A cos 60c+ A cosec 70c + A - A A ] g ] g ] g 0. If sin 5, tan r r i { and < i < r < { < 8tani - 5sec { 4. Trigonometric ratios of compound angles 4.. Compound angles, then find the value of When we add or subtract angles, the result is called a compound angle. i.e.,the algebraic sum of two or more angles are called compound angles For eample, If A, B, C are three angles then A ± B, A + B + C, A B + C etc., are compound angles. 4.. Sum and difference formulae of sine, cosine and tangent (i) sin] A+ Bg sina cosb + cosa sinb (ii) sin] A- Bg sina cosb cosa sinb (iii) cos] A+ Bg cosa cosb sina sinb (iv) cos] A- Bg cosa cosb + sina sinb Eample 4.6 (v) tan(a+b) (vi) tan(a B) tana+ tan B - tanatan B tana- tan B + tanatan B 4 If cosa r 5 and cosb, < A, B < r, find the value of (i) sin(a B) (ii) cos(a+b). Since r < A, B < r, both A and B lie in the fourth quadrant, ` sina and sinb are negative. 4 Given cosa 5 and cosb, Trigonometry Ch-4.indd :40:5

140 Therefore, sina (i) (ii) Eample cos A Sin B - -cos B cos(a + B) cosa cosb sina sinb 4 5 # - b - 5 l# b - 5 l sin(a B) sina cosb cosa sinb b lb l- b lb 5 l Find the values of each of the following trigonometric ratios. (i) sin5 o (ii) cos( 05 o ) (iii) tan75 o (iv) sec65 o (i) sin5 o sin(45 o 0 o ) sin45ccos 0c- cos45csin 0c # - # - (ii) cos( 05 o ) cos05 o cos(60 o + 45 o ) cos60 o cos45 o sin60 o sin45 o # - # - (iii) tan75 o tan(45 o + 0 o ) th Std. Business Mathematics Ch-4.indd :40:7

141 tan45c+ tan 0c - tan45ctan 0c (iv) cos 65º cos(80º 5º) cos 5º cos(60º 45º) (cos60º cos45º + sin60º sin 45º) - e # + # - e + o ` sec65º + - # + - ^ - h Eample 4.8 sin] A If tana m tanb, prove that sin ] A + - o Bg B g m m + - Given tana m tanb sin A cos A sin B m cos B sinacos B cosasin B m Componendo and dividendo rule: a If b d c a+ b c+ d, then a - b c - d Applying componendo and dividendo rule, we get sinacos B+ cosasin B m + sinacos B - cosasin B m - sin] A+ Bg sin] A - B g m m + -, which completes the proof. Trigonometry Ch-4.indd :40:9

142 4.. Trigonometric ratios of multiple angles Identities involving sina, cosa, tana, sina etc., are called multiple angle identities. (i) sin A sin^a+ Ah sinacos A+ cos Asin A sin Acos A. (ii) cos A cos( A+ A) cos AcosA- sin AsinA cos A sin A (iii) sin A sin(a+a) sin AcosA+ cos Asin A ^sin AcosAhcos A+ ^- sin Ahsin A sin Acos A+ sin A-sin A sin A^- sin Ah+ sin A - sin A sina 4sin A Thus we have the following multiple angles formulae. (i) sina sina cosa tan A (ii) sina + tan A. (i) cosa cos A sin A (ii) cosa cos A (iii) cosa sin A - tan A (iv) cosa + tan A tan A. tana - tan A 4. sina sina- 4sin A 5. cosa 4cos A- cos A 6. tana tana- tan A - tan A NOTE (i) cos A + cos A ] g (or) cos A! (ii) sin A - cos A ] g (or) sin A! + cos A - cos A 4 th Std. Business Mathematics Ch-4.indd :40:

143 Eample 4.9 Show that Eample 4.0 sin i + cos i sin i + cos i tan i sinicos i sin i tan i cos i cos i Using multiple angle identity, find tan60 o tan A tan A - tan A Put A 0 o in the above identity, we get tan60 o tan 0c - tan 0c - # Eample 4. If tana 7 and tanb, show that cosa sin4b Now cosa - + tan A tan A sin4b sinb cosb # () + tan B tan B # - + tan B + tan B 4 5 () From() and () we get, cosa sin4b. Eample 4. If tana - sin cos B B then prove that tana tanb Trigonometry 5 Ch-4.indd :40:

144 Consider - cos B sin B tana B sin B B sin cos - sin cos B B B sin B B tan cos tan B A B A B ` tana tanb Eample 4. If tan a and tan b 7 then prove that ^a+ bh r 4. tana+ tan b tan^a+ bh - tana$ tan b tan a # tana tan a tan^a+ bh # a r + b 4 tan 4 r 6 th Std. Business Mathematics Eercise 4.. Find the values of the following : (i) cosec5º (ii) sin(-05º) (iii) cot75º. Find the values of the following r r r r (i) sin76º cos6º cos76º sin6º (ii) sin 4 cos + cos 4 sin (iii) cos70º cos0º sin70º sin0º (iv) cos 5º sin 5º. If sin A 5, 0 < A< r and cos B, A r - r < < find the values of the following : (i) cos(a+b) (ii) sin(a B) (iii) tan(a B) Ch-4.indd :40:5

145 4. If cosa r 4 and cosb 7 where A, B are acute angles prove that A B 5. Prove that tan80º tan85º tan 5º 6. If cot α -, sec β 5, where π < α < tan(α + β). State the quadrant in which α + β terminates. r r and < b < r, find the value of 7. If A+B 45º, prove that (+tana)(+tanb) and hence deduce the value of tan c 8. Prove that (i) sin] A+ 60cg+ sin] A- 60cg sin A (ii) tan4atan AtanA+ tan A+ tana- tan 4A 0 9. (i) If tanθ find tanθ (ii) If sin A, find sina 0. If sina 5, find the values of cosa and tana sin] B- Cg sin] C- Ag sin] A- Bg. Prove that cosbcos C + cosc cos A + cosacos B 0.. If tana- tan B and cotb- cot A y prove that cot] A- Bg + y.. If sina+ sin b a and cosa+ cos b b, then prove that cos(α β) 4. Find the value of tan 8 r. a + b If tan a 7, sin b Prove that a+ b r 0 4 where 0 < a < r 0 < b <. 4. Transformation formulae The two sets of transformation formulae are described below. r and 4.. Transformation of the products into sum or difference sinacos B+ cos Asin B sin] A+ Bg () Trigonometry 7 Ch-4.indd :40:8

146 sinacos B - cosasin B sin] A- Bg () cosacos B- sinasin B cos] A+ Bg () cosacos B+ sinasin B cos] A- Bg (4) Adding () and (), we get sinacos B sin] A+ Bg+ sin] A- Bg Subtracting () from (), we get cosa sinb sin] A+ Bg- sin] A- Bg Adding () and (4), we get cosacos B cos] A+ Bg+ cos] A- Bg Subtracting () from (4), we get sina sinb cos] A- Bg- cos] A+ Bg Thus, we have the following formulae; sina cosb sin] A+ Bg+ sin] A- Bg cosa sinb sin] A+ Bg- sin] A- Bg cosacos B cos] A+ Bg+ cos] A- Bg sinasin B cos] A- Bg- cos] A+ Bg Also sin A sin B sin] A+ Bgsin] A-Bg cos A- sin B cos] A+ Bgcos] A-Bg 4.. Transformation of sum or difference into product C D C D sinc + sind sinb + lcosb - l C D C D sinc sind cosb + lsinb - l C D C D cosc + cosd cosb + lcosb - l C D C D cosc cosd - sinb + lsinb - l 8 th Std. Business Mathematics Ch-4.indd :40:4

147 Eample 4.4 Epress the following as sum or difference (i) sinicos i (ii) cosicos i (iii) sin4isin i A 5A (iv) cos7i cos 5i (v) cos cos (vii) cosasin 5A (i) sinicos i sin] i+ ig+ sin] i- ig sini+ sin i (ii) cosicos i cos] i+ ig+ cos] i- ig cos4i+ cos i (iii) sin4isin i cos] 4i- ig- cos] 4i+ ig (vi) cos9isin 6i cosi- cos 6i (iv) cos7i cos 5i 6 cos] 7 i+ 5 ig+ cos] 7 i- 5 ig@ cosi+ cos i 5? A 5A (v) cos cos A 5A A 5A cosb + l+ cosb - l 8A - A ; cos + cosb le cos A cos A ]- g@ cos4 A+ cos A 5? (vi) cos9isin 6i 6 sin] 9 i+ 6 ig- sin] 9 i- 6 ig@ 6 sin] 5ig- sin i@ (vii) cosasin 5 A sin] A+ 5Ag- sin] A- 5Ag sin8a+ sin A Eample 4.5 Show that sin0csin 40csin 80c 8 Trigonometry 9 Ch-4.indd :40:6

148 Eample th Std. Business Mathematics LHS sin0csin 40csin 80c sin 0csin^60c- 0chsin^60c+ 0ch sin 0c6 sin 60c- sin sin 0c8 4 - sin 0cB - 4sin 0 sin 0c: c D 4 sin0c- 4sin 0c 4 sin 4 60 c 4 8 Show that sin0csin 40csin60csin 80c 6 L.H.S sin0csin 40csin60csin 80c sin60csin 0csin] 60c- 0cgsin] 60c+ 0cg sin 0c^sin 60c- sin 0ch sin 0 cb 4 - sin 0cl Eample 4.7 b lb 4 l^sin0c- 4sin 0ch. b lb 4 lsin 60c b lb 4 lb l 6 R.H.S. Prove that cos A + cos ] A+ 0cg+ cos ] A- 0cg cos A + cos ] A+ 0cg+ cos ] A- 0cg cos A + cos ] A + 0cg + ^ - sin ] A - 0c gh (Since cos A - sin A) cos A cos A ^ ] + cgh - sin ] A - 0cg cos A + + cos[ ^ A+ 0 c ) + ( A- 0 c h] cos[ ] A+ 0 c g- ] A- 0 c g] Ch-4.indd :40:9

149 cos A + + cosa cos40 o (Since cos A- sin B cos] A+ Bgcos] A-Bg ) cos A + + ^ cos A - hcos] 80 c + 60 c g (since cos A + cos A) cos A + + ^ cos A - h] - cos 60 c g cos A + + ^cos A - hb- l cos A+ - cos A+ Eample 4.8 Convert the following into the product of trigonometric functions. (i) sin9a+ sin 7A (ii) sin7i- sin 4i (iii) cos8a+ cos A (iv) cos4a- cos 8a (v) cos0c- cos 0c (vi) cos75c+ cos 45c (vii) 9A+ 7A 9A-7A (i) sin9a+ sin 7A sinb lcosb l 6A A sinb lcosb l sin8acos A 7i+ 4i 7i-4i (ii) sin7i- sin 4i cosb lsinb l i i cosb lsinb l 8A+ A 8A-A (iii) cos8a+ cos A cosb lcosb l 0A A cosb lcosb - 4 l cos0a cos] -Ag cos0acos A 4a+ 8a 4a-8a (iv) cos4a- cos 8a - sinb lsinb l a 4a sinb lsinb l sin6asin a 0c+ 0c 0c-0c (v) cos0c- cos 0c - sinb lsinb l sinb 50 c lsinb 0 c l sin5csin 5c Trigonometry 4 Ch-4.indd :40:4

150 75c+ 45c 75c-45c (vi) cos75c+ cos 45c cosb lcosb l 0c 0c cosb lcosb l cos60ccos 5c (vii) cos55c+ sin 55c cos55c + cos] 90c- 55cg Eample 4.9 cos55c+ cos 5c 55c+ 5c 55c-5c cosb lcosb l cos45ccos 0c d n$ cos 0c cos 0c If three angles A, B and C are in arithmetic progression, Prove that sina cotb cos C - - sin C cos A A- C A+ C sina- sin C sinb lcosb l cosc - cos A A+ C A-C sinb lsinb l A+ C cosb l A+ C A+ C cotb l sinb l Eample 4.0 Prove that cotb (since A, B, C are in A.P., B A + C ) sin5- sin + sin cos5 - cos tan sin5- sin + sin cos5 - cos sin5+ sin -sin cos5- cos sin cos sin cos5 - - cos sin ] cos - g sin sin sin cos - - sin sin cos tan 4 th Std. Business Mathematics Ch-4.indd :40:45

151 Eample 4. Find sin05 o + cos05 o sin05 o + cos05 o Eample 4. sin(90 o + 5 o )+cos05 o cos5 o + cos05 o cos05 o + cos5 o 05c+ 5c 05c-5c cosb lcosb l 0c 90c cosb lcosb l cos60 o cos45 o a- b Prove that ^cosa+ cos bh + ^sina+ sin bh 4 cos b l a+ b a-b cosa+ cos b cosb l$ cosb l.() a+ b a-b sina+ sin b sinb l$ cosb l.() Squaring and adding () and () ^cosa+ cos bh + ^sina+ sin bh 4 cos a+ b cos a-b 4 sin a+ b cos a-b b l b l+ b l b l a- b a+ b a+ b 4 cos b l; cos b l+ sin b le 4 cos a - b b l Eercise 4.. Epress each of the following as the sum or difference of sine or cosine: A A (i) sin 8 sin 8 (ii) cos] 60c+ Agsin] 0c+ Ag 7A 5A (iii) cos sin (iv) cos7i sin i Trigonometry 4 Ch-4.indd :40:47

152 . Epress each of the following as the product of sine and cosine (i) sina + sina (ii) cosa + cos4a (iii) sin6i - sin i (iv) cosi - cos i. Prove that (i) cos0ccos 40ccos 80c 8 (ii) tan0ctan 40ctan 80c 4. Prove that a- b (i) ^cosa- cos bh + ^sina- sin bh 4 sin b l (ii) sinasin( 60c+ A) sin( 60c- A) 4 sin A 5. Prove that (i) sin] A- Bgsin C + sin] B- Cgsin A + sin] C- Agsin B 0 r 9r r 5r (ii) cos cos + cos + cos 0 6. Prove that cosa cos A A (i) sina- - sin A tan cos7a cos 5A (ii) sin7a- + sin 5A cot A 7. Prove that cos0ccos 40ccos60ccos 80c 6 8. Evaluate (i) cos0 + cos 00 + cos 40 (ii) sin50 - sin 70c + sin 0 9. If cosa+ cos B and sina+ sin B 4, prove that tanb A+ B l 0. If sin^y+ z- h, sin^z+ - yh, sin^+ y-zh are in A.P, then prove that tan, tan y and tan z are in A.P A+ B. If coseca+ sec A cosecb+ sec B prove that cotb l tan Atan B 44 th Std. Business Mathematics Ch-4.indd :40:49

153 4.4 Inverse Trigonometric Functions 4.4. Inverse Trigonometric Functions The quantities such as sin, cos, tan functions. etc., are known as inverse trigonometric - i.e., if sin i, then i sin. The domains and ranges (Principal value branches) of trigonometric functions are given below Function Domain Range y sin [, ] -r r :, D cos - [, ] [0, r ] tan -r r R ^-h b, l -, cosec - -r r R (, ) :, D, y! 0 sec - -r r R (, ) :, D r, y! cot - R (0, r ) 4.4. Properties of Inverse Trigonometric Functions Property () (i) sin sin (ii) cos ] cos g (iii) tan tan (iv) cot ] cot g (v) sec sec (vi) cosec ] cosec g Property () (i) - - sin b l cosec () (ii) - - cos b l sec () (iii) - - tan b l cot () (iv) - - cosec b l sin () (v) - - sec b l cos () (vi) - - cot b l tan () Trigonometry 45 Ch-4.indd :40:5

154 Property () - - (i) sin (- ) - sin () - - (iii) tan (- ) - tan () - - (v) sec (- ) r - sec () Property (4) - - r (i) sin () + cos () - - r (iii) sec () + cosec () - - (ii) cos (- ) r - cos () - - (iv) cot (- ) - cot () - - (vi) cosec (- ) - cosec () - - r (ii) tan () + cot () Property (5) If y<, then - - y (i) tan () + tan () y tan - + d n - y - - y (ii) tan () - tan () y tan - - d n + y Property (6) sin () sin y + ^ h sin _ - y + y - i Eample 4. - Find the principal value of (i) sin - _ i ( ii) tan ^ - h (i) Let sin - _ i r y where - # y # ` sin y _ i sin b r 6 l 46 th Std. Business Mathematics y 6 r The principal value of sin - _ i r is 6 r (ii) Let tan - ^- h r y where - # y # ` tan y tanb- r The principal value of tan - ^- h is y r - r - r l Ch-4.indd :40:56

155 Eample Evaluate the following (i) cosbsin - 5 (i) Let bsin l i... () sin i 5 cos i - sin - i () l (ii) tanbcos l From () and (), we get - 5 Now cosbsin l cos i (ii) Let bcos l i 8 cos i 7 sin i - cos i () From () and (), we get Now tanbcos l sin i tan i cos i Eample Prove that (i) tan 7 tan - b l+ b l tan b 9 l (i) tan - (ii) cos tan 5 - b l+ b l tan b 7 l b 7 l+ tan - - b l tan - 7 > + - _ i_ i H 7 Trigonometry 47 Ch-4.indd :40:58

156 tan - b 90 0 l tan - b 9 l (ii) Let cos - b 5 4 l i. Then cos i ` tan i 4 ` cos - b 5 4 l tan - b 4 l 5 4 & i tan - b 4 l & sin i 5 Now cos Eample 4.6 b 5 4 l+ tan b 5 l - - tan b 4 l + tan tan - + e o - # 4 tan - b 7 l 5 Find the value of tan; r 4 tan - - b 8 le Let tan - b 8 l i Eample 4.7 Then tan i 8 b 5 l ` tan; r 4 tan - - b 8 le tan b r 4 - il Solve tan tan r tan 4 - tan i r + tan tan i b - l+ tan b + + l r tan b l b l 4 tan > tan H b _ i_ i l - - Given that tan - b - l+ tan b + + l r th Std. Business Mathematics Ch-4.indd :4:00

157 ` tan - b l r r tan 4-4 & 0 &! Eample 4.8 Simplify: sin ` j+ sin - - ` j We know that sin ] g + sin () y sin y + y - A ` sin b l+ sin b l sin - : D sin - : D sin - c m 9 Eample 4.9 Solve tan ] + g+ tan ] - g tan - b l - - & tan + h + ^- h < F tan - ^+ h^ - - ` h j - ^ tan - c 4 - ^4 - h m tan - ` j & 6 & 9 ` + Eample 4.0 If tan( + y) 4 and tan (), then find y Trigonometry 49 Ch-4.indd :4:0

158 tan(+y) 4 + y tan (4) tan ()+y tan (4) y tan (4) tan () tan ; + ] 4 # g E tan - b l y tan - : 7 8 D. Find the pricipal value of the following Eercise 4.4 (i) sin - b- l (ii) tan - ]-g (iii) cosec - ^h (iv) sec - ^- h. Prove that - - (i) tan ] g sin c m + (ii) tan b 4 l+ tan b 7 l r - -. Show that tan tan - b l+ b l tan b 4 l - - r 4. Solve tan + tan 4-5. Solve tan tan - - ] g ] g tan b 7 4 l 6. Evaluate (i) cos tan - 8 ` 4 jb (ii) sin8 cos - ` 5 4 jb - 7. Evaluate: cos sin ` ` j+ sin ` jj - m - m n 8. Prove that tan b n l r - tan b m+ - n l 4 9. Show that sin - 5 sin 7 8 b- l b- l cos Epress tan - cos : r sin, < < r - D -, in the simplest form. 50 th Std. Business Mathematics Ch-4.indd :4:05

159 Step ICT Corner Epected final outcomes Open the Browser and type the URL given (or) Scan the QR Code. GeoGebra Work book called th BUSINESS MATHS will appear. In this several work sheets for Business Maths are given, Open the worksheet named Inverse Trigonometric Functions Step - Inverse Trigonometric Functions page will open. Click on the check boes on the Left-hand side to see the respective Inverse trigonometric functions. You can move and observe the point on the curve to see the -value and y-value along the aes. Sine and cosine inverse are up to the ma limit. But Tangent inverse is having customised limit. Step Step Step 4 Browse in the link th Business Maths: (or) scan the QR Code Trigonometry 5 Ch-4.indd :4:05

160 Eercise 4.5 Choose the correct answer. The degree measure of r 8 is (a) 0c60l (b) c0l (c) c60l (d) 0c0l. The radian measure of 7c0l is 5r (a) 4 r (b) 4 (c) 7r 4. If tan i and i lies in the first quadrant then cos i is 5 (a) 6 (b) - 6 (c) 5 6 9r (d) 4 (d) The value of sin 5 o is + - (a) (b) (c) (d) 5. The value of sin] -40cg is (a) (b) - (c) (d) - 6. The value of cos] -480cgis (a) (b) - (c) (d) - 7. The value of sin8 o cos7 o + cos8 o sin7 o is (a) (b) (c) - (d) 0 8. The value of sin5º cos5º is (a) (b) (c) (d) 4 9. The value of seca sin(70 o + A) is (a) (b) cos A (c) sec A (d) 0. If sina + cosa then sina is equal to (a) (b) (c) 0 (d) 5 th Std. Business Mathematics Ch-4.indd :4:09

161 . The value of cos 45º sin 45º is (a) (b) (c) 0 (d). The value of sin 45º is (a) (b) (c) 4 (d) 0. The value of 4cos 40º cos40º is (a) 4. The value of (a) (b) tan 0c + tan 0c (b) is (c) (c) (d) (d) 5. If sin A then 4cos A - cos A is (a) (b) 0 (c) tan0c- tan 0c 6. The value of is - tan 0c (a) (b) (c) (d) 7. The value of cosec - d n is r r r r (a) 4 (b) (c) (d) 6-8. sec - + cosec (a) - r r (b) 9. If a and b be between 0 and then sin a is (a) If tan A and tan B (c) r (d) (d) r r and if cos^a+ bh and sin^a- bh 5 56 (b) 0 (c) 65 then tan^ A+ B h is equal to 64 (d) 65 (a) (b) (c) (d) 4. tan b r 4 - l is (a) b + tan - tan l (b) - tan tan b + l (c) - tan (d) + tan Trigonometry 5 Ch-4.indd :4:4

162 . sinbcos (a) 5-5 lis. The value of (a) - (b) 5 cosec] -45cg is (b) (c) 5 4 (d) 5 4 (c) (d) - 4. If p sec50c tan 50c then p is (a) cos50c (b) sin50c (c) tan50c (d) sec50c cos 5. b cosec l - -sin - cos is (a) cos - sin (b) sin - cos (c) (d) 0 Miscellaneous Problems. Prove that cos4+ cos + cos sin4+ sin + sin cot. Prove that cosec0c- sin 0c 4. Prove that cot 4] sin5+ sin g cot ] sin5- sin g. 4. If r tan 4, r < < then find the value of sin and cos 5. Prove that r r r sin 4 + cos 4 + sec Find the value of (i) sin75c (ii) tan5c 7. If sina, sin B 4 then find the value of sin] A+ Bg, where A and B are acute angles Show that cos sin 5-56 b l+ b l sin b 65 l 9. If cos( a+ b) 5 4 and sin( a- b) 5 where a+ b and a- bare acute, then find tana 0. Epress tan - cos- sin b cos+ sin l, 0 < < r in the simplest form. 54 th Std. Business Mathematics Ch-4.indd :4:8

163 Summary z If in a circle of radius r, an arc of length l subtends an angle of θ radians, then l r θ z sin + cos z tan + sec z cot + cosec z sin(α + β) sin α cos β + cos α sin β z sin(α β) sin α cos β cos α sin β z cos(α + β) cos α cos β sin α sin β z cos^a- bh cos a$ cosb+ sin asin b tana+ tan b z tan^a+ bh - tanatan b tana- tan b z tan^a- bh + tanatan b z sin a sin acos a z cosa cos a sin a - sin a cos a - tan a z tan a - tan a z sin a sina-4 sin a z cosa 4cos a- cos a tana - tan a z tan a - tan a + y -y z sin + sin y sin cos + y -y z sin - sin y cos sin + y -y z cos+ cos y cos cos + y -y z cos- cos y - sin sin z sin cos y 6 sin^+ yh+ sin^-yh@ z cos sin y 6 sin^+ yh-sin^-yh@ z coscos y 6 cos^+ yh+ cos^-yh@ z sin sin y 6 cos^-yh- cos^+ yh@ Trigonometry 55 Ch-4.indd :4:

164 GLOSSARY Allied angles Angle Componendo and dividendo. Compound angle Degree measure Inverse function Length of the arc. Multiple angle த ண க க ணங ள க ணம க ட டல மற ற ம கழ த தல வ க த சமம க டட க க ணம ப க அளவ ந ரம ற ச ரப வ ல ல ன ந ளம மடங க க ணம Quadrants க ல பக த கள Radian measure ர ட யன அளவ / ஆர யன அளவ Transformation formulae உர ம ற ற ச த த ரங ள Trignometric identities Trignometry Ratios த ர க ணம த ம ற ற ர ம கள த ர க ணம த வ க தங ள 56 th Std. Business Mathematics Ch-4.indd :4:

165 Chapter 5 DIFFERENTIAL CALCULUS Learning Objectives After studying this chapter, the students will be able to understand z the concept of limit and continuity z formula of limit and definition of continuity z basic concept of derivative z how to find derivative of a function by first principle z derivative of a function by certain direct formula and its respective application z how to find higher order derivatives Introduction Calculus is a Latin word which means that Pebble or a small stone used for calculation. The word calculation is also derived from the same Latin word. Calculus is primary mathematical tool for dealing with change. The concept of derivative is the basic tool in science of calculus. Calculus is essentially concerned with the rate of change of dependent variable with respect to an independent variable. Sir Issac Newton (64 77 CE) and the German mathematician G.W. Leibnitz ( CE) invented and developed the Isaac Newton subject independently and almost simultaneously. In this chapter we will study about functions and their graphs, limits, derivatives and differentiation technique. 5. Functions and their graphs Some basic concepts 5.. Quantity Anything which can be performed on basic mathematical operations like addition, subtraction, multiplication and division is called a quantity. G.W. Leibnitz Differential Calculus 57 Ch-5.indd :4:57

166 5.. Constant A quantity which retains the same value throughout a mathematical investigation is called a constant. Basically constant quantities are of two types (i) Absolute constants are those which do not change their values in any mathematical investigation. In other words, they are fied for ever. Eamples:,, p,... (ii) Arbitrary constants are those which retain the same value throughout a problem, but we may assign different values to get different solutions. The arbitrary constants are usually denoted by the letters a, b, c, Variable Eample: In an equation ym+4, m is called arbitrary constant. A variable is a quantity which can assume different values in a particular problem. Variables are generally denoted by the letters, y, z,... Eample: In an equation of the straight line y +, a b and y are variables because they assumes the co-ordinates of a moving point in a straight line and thus changes its value from point to point. a and b are intercept values on the aes which are arbitrary. There are two kinds of variables (i) A variable is said to be an independent variable when it can have any arbitrary value. (ii) A variable is said to be a dependent variable when its value depend on the value assumed by some other variable. Eample: In the equation y 5 - +, is the independent variable, y is the dependent variable and is the constant Intervals The real numbers can be represented geometrically as points on a number line called real line. The symbol R denotes either the real number system or the real line. A subset of the real line is an interval. It contains atleast two numbers and all the real numbers lying between them. A a Fig. 5. B b 58 th Std. Business Mathematics Ch-5.indd :4:57

167 (i) Open interval The set { : a < < b} is an open interval, denoted by (a, b). In this interval, the boundary points a and b are not included. For eample, in an open interval (4, 7), 4 and 7 are not elements of this interval. A B ( ) a Fig. 5. b But, 4.00 and 6.99 are elements of this interval. (ii) Closed interval The set { : a < < b} is a closed interval and is denoted by [a, b]. In the interval [a, b], the boundary points a and b are included. a b For eample, in an interval [4, 7], 4 and 7 are also Fig. 5. elements of this interval. Also we can make a mention about semi closed or semi open intervals. (a, b] { : a < < b} is called left open interval A B [ ] and [a, b) { : a < < b} is called right open interval. In all the above cases, b - a h is called the length of the interval Neighbourhood of a point Let a be any real number. Let e > 0 be arbitrarily small real number. Then ( a - e, a + e ) is called an e - neighbourhood of the point a and denoted by N a, e For eamples, 5 4 b5 -, l N, N, 9 & : < < , + 5 b l & : < < Differential Calculus 59 Ch-5.indd :4:58

168 5..6 Function Let X and Y be two non-empty sets of real numbers. If there eists a rule f which associates to every element! X, a unique element y! Y, then such a rule f is called a function (or mapping) from the set X to the set Y. We write f : X $ Y. The function notation y f() was first used by Leonhard Euler in The set X is called the domain of f, Y is called the co-domain of f and the range of f is defined as f(x) {f() /! X}. Clearly f(x) Y. A function of is generally denoted by the symbol f(), and read as function of or f of Classification of functions Functions can be classified into two groups. (i) Algebraic functions Algebraic functions are algebraic epressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division and raising to a fractional power. Eamples : y , y, y are algebraic + + functions. (a) y is a polynomial or rational integral function. + 7 (b) y is a rational function. + + (c) y + 6 is an irrational function. (ii) Transcendental functions A function which is not algebraic is called transcendental function. - Eamples: sin, sin, e, log a are transcendental functions. (a) sin, tan,... are trigonometric functions. (b) sin -, cos -,... are inverse trigonometric functions. (c) e,,,... are eponential functions. (d) log a, log e (sin ),... are logarithmic functions. 60 th Std. Business Mathematics Ch-5.indd :4:59

169 5..8 Even and odd functions A function f() is said to be an even function of, if f( - ) f( ). A function f() is said to be an odd function of, if f( - ) - f( ). Eamples: f() and f() cos are even functions. f() and f() sin are odd functions. NOTE f() + 5 is neither even nor odd function 5..9 Eplicit and implicit functions A function in which the dependent variable is epressed eplicitly in terms of some independent variables is known as eplicit function. Eamples: y + and y e + e - are eplicit functions of. If two variables and y are connected by the relation or function f(, y) 0 and none of the variable is directly epressed in terms of the other, then the function is called an implicit function. Eample: + y -y 0 is an implicit function Constant function If k is a fied real number then the function f() given by f() k for all! R is called a constant function. Eamples: y, f() 5 are constant functions. Graph of a constant function y f() is a straight line parallel to -ais 5.. Identity function A function that associates each real number to itself is called the identity function and is denoted by I. i.e. A function defined on R by f() for all R is an identity function. Eample: The set of ordered pairs {(, ), (, ), (, )} defined by f : A " A where A{,, } is an identity function. Graph of an identity function on R is a straight line passing through the origin and makes an angle of 45cwith positive direction of -ais Differential Calculus 6 Ch-5.indd :4:59

170 5.. Modulus function if $ 0 A function f() defined by f(), where ) is called the modulus - if 0 function. NOTE It is also called an absolute value function. 5 5, 5 Remark Domain is R and range set is [0, ) ( 5) Signum function A function f() is defined by f() if! 0 * is called the Signum function. 0 if 0 Remark Domain is R and range set is {, 0, } 5..4 Step function (i) Greatest integer function The function whose value at any real number is the NOTE greatest integer less than or equal to is called the greatest.5,., integer function. It is denoted by , 0., i.e. f : R$ R defined by f() is called the 4 4. greatest integer function. (ii) Least integer function The function whose value at any real number is the smallest integer greater than or equal to is called the least integer function. It is denoted by. i.e. f : R$ R defined by f() is called the least integer function NOTE 4.7 5, 7. 7, 5 5, Remark Function s domain is R and range is Z [set of integers] 5..5 Rational function p () A function f() is defined by the rule f (), q ( )! 0 is called rational q () function. + Eample: f () -,! is a rational function. 6 th Std. Business Mathematics Ch-5.indd :4:00

171 5..6 Polynomial function For the real numbers a0, a, a,..., an ; a0! 0 and n is a non-negative integer, a function f() given by f n n- n - ] g a 0 + a + a an is called as a polynomial function of degree n. Eample: f () is a polynomial function of degree Linear function For the real numbers a and b with a! 0, a function f() a + b is called a linear function. Eample: y + is a linear function Quadratic function For the real numbers a, b and c with a! 0, a function f() a + b + c is called a quadratic function. Eample: f () is a quadratic function Eponential function A function f() a, a! and a > 0, for all R is called an eponential function Remark Domain is R and range is (0, ) and (0, ) is a point on the graph Eamples: e, e + and are eponential functions Logarithmic function For > 0, a > 0 and a!, a function f() defined by f() log a is called the logarithmic function. Remark Domain is (0, ), range is R and (, 0) is a point on the graph Eample: f() log e (+), f() log e (sin) are logarithmic functions. 5.. Sum, difference, product and quotient of two functions If f() and g() are two functions having same domain and co-domain, then (i) (f + g) () f() + g() (ii) (fg)() f() g() Differential Calculus 6 Ch-5.indd :4:0

172 f (iii) c m() g f (), g ( )! 0 g () (iv) (cf)() cf(), c is a constant. 5.. Graph of a function The graph of a function is the set of points _, f] gi where belongs to the domain of the function and f() is the value of the function at. To draw the graph of a function, we find a sufficient number of ordered pairs _, f] gi belonging to the function and join them by a smooth curve. NOTE It is enough to draw a diagram for graph of a function in white sheet itself. Eample 5. Find the domain for which the functions f() and g() are equal. Given that f() g() & + 0 Eample 5. (+) ( ) 0 `, Domain is $ -,. f {(, ), (, )} be a function described by the formula f() a+b. Determine a and b? Given that f() a + b, f() and f() ` a + b and a + b & a and b 64 th Std. Business Mathematics Ch-5.indd :4:0

173 Eample 5. Eample 5.4 If f() +, then show that [f()] f( ) + f ` j f() +, f( ) + and f` j f() Now, LHS [f()] 8 + B + + ` + j f ( ) + f () f ( ) + f` j RHS Let f be defined by f() 5 and g be defined by - 5 g() + 5 if! - 5 * Find l such that f() g() for all R m if - 5 Given that ` f() g() for all R f( 5) g( 5) 5 5 l ` l 0 Eample 5.5 If f(), show that f ^ h$ fy ^ h f ^ + yh f() (Given) ` f( + y) +y. y f^h$ fy ^ h Differential Calculus 65 Ch-5.indd :4:0

174 Eample 5.6 If f( ), 0,,, then show that + f( ) (Given) + f] g - ` f 7f] ga f ] g + Eample 5.7 If + f( ) log, f[ f( )] show that f f( ). + + f( ) log (Given) + ` f c m + log log + + ( + ) log + ( ) Eample log f( ). If f() and g(), then find (i) (f+g)() (ii) (f g)() (iii) (fg)() (i) (ii) (f + g) () f() + g() + + if $ 0 ) ) - if < 0 0 (f g) () f() g() if $ 0 if < 0 - if $ 0 0 ) ) - (- ) if < 0 if $ 0 if < 0 66 th Std. Business Mathematics Ch-5.indd :4:05

175 (iii) (f g) () f() g() ) - if $ 0 if < 0 Eample 5.9 A group of students wish to charter a bus for an educational tour which can accommodate atmost 50 students. The bus company requires at least 5 students for that trip. The bus company charges ` 00 per student up to the strength of 45. For more than 45 students, company charges each student ` 00 less 5 times the number more than 45. Consider the number of students who participates the tour as a function, find the total cost and its domain. Let be the number of students who participate the tour. Then 5 < < 50 and is a positive integer. Formula : Total cost (cost per student) (number of students) (i) If the number of students are between 5 and 45, then the cost per student is ` 00 ` The total cost is y 00. (ii) If the number of students are between 46 and 50, then the cost per student is ` $ 00 - ( - 45) ` The total cost is y ` 09 - j ; 5 # # 45 ` y * 09 - and ; 46 # # 50 5 the domain is {5, 6, 7,..., 50}. Eample 5.0 Draw the graph of the function f() y f() ) - if $ 0 if < 0 Differential Calculus 67 Ch-5.indd :4:06

176 Choose suitable values for and determine y. Thus we get the following table y 0 Table : 5. Plot the points (-, ), (-, ), (0, 0), (, ), (, ) and draw a smooth curve. y (-, ) (, ) (-, ) (, ) y, < 0 y, > 0 The graph is as shown in the figure Eample 5. l (0, 0) yl Fig. 5.4 Draw the graph of the function f() - 5 Let y f() - 5 Choose suitable values for and determine y. Thus we get the following table y Table : 5. y Plot the points (-, 4), (-, -), (-, -4), (0, -5), (, -4), (, -), (, 4) and draw a smooth curve. (-, 4) (, 4) y 5 The graph is as shown in the figure Eample 5. Draw the graph of f() a, a! and a > 0 l (-, -) (-, -4) (0, -5) yl Fig 5.5 (, -) (, -4) We know that, domain set is R, range set is (0, ) and the curve passing through the point (0, ) 68 th Std. Business Mathematics Ch-5.indd :4:06

177 Case (i) when a > < if < 0 y f() a * if 0 > if > 0 We noticed that as increases, y is also increases and y > 0. ` The graph is as shown in the figure. y ya, a> l (0, l) o yl Fig 5.6 Case (ii) when 0 < a < > if < 0 y f() a * if 0 < if > 0 We noticed that as increases, y decreases and y > 0. ` The graph is as shown in the figure. ya, 0<a< y NOTE l (0, l) O yl Fig 5.7 (i) The value of e lies between and. i.e. < e < (ii) The graph of f() e is identical to that of f() a if a > and the graph of f() e is identical to that of f() a if 0 < a < Eample 5. Draw the graph of f() log a ; > 0, a > 0 and a!. Differential Calculus 69 Ch-5.indd :4:07

178 We know that, domain set is (0, ), range set is R and the curve passing the point (, 0) Case (i) when a > and > 0 < 0 if 0 < < f () loga * 0 if > 0 if > We noticed that as increases, f() is also increases. ` The graph is as shown in the figure. y f() log a, a> l O (, 0) yl Fig 5.8 Case (ii) when 0 < a < and > 0 > 0 if 0 < < f () loga * 0 if < 0 if > We noticed that increases, the value of f() decreases. ` The graph is as shown in the figure. y f() log a, 0<a< l O (, 0) 70 th Std. Business Mathematics yl Fig 5.9 Ch-5.indd :4:07

179 Step ICT Corner Epected final outcomes Open the Browser and type the URL given (or) Scan the QR Code. GeoGebra Work book called th BUSINESS MATHS will appear. In this several work sheets for Business Maths are given, Open the worksheet named Types of Functions Step - Types of Functions page will open. Click on the check boes of the functions on the Right-hand side to see the respective functions. You can move the slider and change the function. Observe each function. Step Step Step 4 Browse in the link th Business Maths: (or) scan the QR Code Differential Calculus 7 Ch-5.indd :4:07

180 Eercise 5.. Determine whether the following functions are odd or even? a (i) f() (ii) f ] g log_ + + i a + (iii) f() sin + cos (iv) f() - (v) f() +. Let f be defined by f() - k +, R. Find k, if f is an odd function.. If f () 4. If f () 5. For f () 6. If f () -, then show that f () + f` j 0 + -, then prove that f^f() h - +, write the epressions of f ` j and e and g () log e, then find f () (i) (f+g)() (ii) (fg)() (iii) (f)() (iv) (5g)() 7. Draw the graph of the following functions : (i) f () 6 - (ii) f () - (iii) f () (iv) f () e (v) f () e - (vi) f () 5. Limits and derivatives In this section, we will discuss about limits, continuity of a function, differentiability and differentiation from first principle. Limits are used when we have to find the value of a function near to some value. Definition 5. Let f be a real valued function of. Let l and a be two fied numbers. If f() approaches the value l as approaches to a, then we say that l is the limit of the function f() as tends to a, and written as lim f() l " a Notations: (i) (ii) L[f()] a lim f() is known as left hand limit of f() at a. " a - R[f()] a lim f() is known as right hand limit of f() at a. " a + 7 th Std. Business Mathematics Ch-5.indd :4:

181 5.. Eistence of limit lim f() eist if and only if both L[f()] " a a and R[f()] a eist and are equal. i.e., lim f() eist if and only if L[f()] " a a R[f()] a. NOTE L[f()] a and R[f()] a are shortly written as L[f(a)] and R[f(a)] 5.. Algorithm of left hand limit : L[f()] a (i) Write lim f(). " a - (ii) Put a h, h > 0 and replace "a by h"0. (iii) Obtain lim f(a h). h " 0 (iv) The value obtained in step (iii) is called as the value of left hand limit of the function f() at a. 5.. Algorithm of right hand limit : R[f()] a (i) Write lim f(). " a + (ii) Put a+h, h>0 and replace "a + by h"0. (iii) Obtain lim f(a+h). h " 0 (iv) The value obtained in step (iii) is called as the value of right hand limit of the function f() at a. Eample 5.4 Evaluate the left hand and right hand limits of the function - if! f() * - at. 0 if L[f()] lim f() " - lim f( h), h h " 0 lim ( - h) - h " 0 ( - h ) - Differential Calculus 7 Ch-5.indd :4:5

182 - h lim h " 0 - h lim h 0 h - " h lim h " 0 R[f()] lim f() " + lim f( + h) h " 0 lim ( + h) - h " 0 ( + h ) - lim h h " 0 h lim h " 0 NOTE Here, L[f()]! R[f()] ` lim f() does not eist. " Eample if 0 # Verify the eistence of the function f() ) at. 4 - if L[f()] lim f() " - lim f( h), h h " 0 lim [5( h) 4] h " 0 lim ( 5h) h " 0 R[f()] lim f() " + lim f( + h), + h h " 0 lim [4( + h) ( + h)] h " 0 4() () Clearly, L[f()] R[f()] ` lim f() eists and equal to. " 74 th Std. Business Mathematics Ch-5.indd :4:8

183 NOTE Let a be a point and f() be a function, then the following stages may happen (i) lim f() eists but f(a) does not eist. " a (ii) The value of f(a) eists but lim f() does not eist. " a (iii) Both lim f() and f(a) eist but are unequal. " a (iv) Both lim f() and f(a) eist and are equal. " a 5..4 Some results of limits If lim f() and lim g() eists, then " a " a (i) lim (f+g)() lim f() + lim g() " a " a " a (ii) lim (fg)() lim f() lim g() " a " a " a f lim f () (iii) limc g m " a ^h " a lim g (), whenever lim g ( )! 0 " a " a (iv) lim k f() k lim f(), where k is a constant. " a " a 5..5 Indeterminate forms and evaluation of limits Let f() and g() are two functions in which the limits eist. fa ^ h If lim f() lim g() 0, then takes " a " a ga ^ h f_ i imply that lim is meaningless. " a g_ i 0 0 form which is meaningless but does not In many cases, this limit eists and have a finite value. Finding the solution of such limits are called evaluation of the indeterminate form. 0 The indeterminate forms are 0,, 0#, -, 0 0, 0 and. Among all these indeterminate forms, 0 0 is the fundamental one Methods of evaluation of algebraic limits (i) Direct substitution (ii) Factorization (iii) Rationalisation (iv) By using some standard limits Differential Calculus 75 Ch-5.indd :4:9

184 5..7 Some standard limits - a (i) lim a - " a (ii) lim sin i 76 th Std. Business Mathematics n i " 0 i a - (iii) lim " 0 ( ) (iv) lim log + " 0 n - nan if n! Q lim tan i i " 0 i log e a if a>0 (v) lim + " 0 e (vi) lim ` + j e " e - (vii) lim Eample 5.6 " 0 Evaluate: lim( + 4-5). ", q is in radian. lim( + 4-5) " () + 4() 5 Eample Evaluate: lim +. " " lim + Eample 5.8 lim^ " h lim^ + h " ^h - 4^h sin- cos Evaluate : lim cos + sin. " r 4 5sin- cos lim cos + sin " r 4 lim^5sin - cos h r " 4 lim^cos + sin h r " 4 Ch-5.indd :4:

185 Eample 5.9 Evaluate: lim - -. " r 5 sin - r cos r r cos + sin 5 ^ h - 0 ^ h 0 ^ h + ^ h lim is of the type " 0 lim - - ^- h^ + + h lim ^ - h " Eample 5.0 " lim^ + + h " () + + Evaluate: lim + -. " 0 lim + - ^ + - h^ + + h lim " 0 " 0 ^ + + h lim " 0 ^ + + h lim " 0 ^ + + h.. + Eample 5. Evaluate: lim " lim " - a a a a a a 5 5 lim. - a - a - a - a 5 5 " a a 5 ^ h - 5 ^ah a a Differential Calculus 77 Ch-5.indd :4:4

186 Eample 5. sin Evaluate: lim sin 5. " 0 sin lim sin 5 " 0 Eample 5. sin # lim 0 sin 5 # 5 5 lim sin 5 " 0 lim sin 5 5 " * 4 " 0 5 ` j 6-5 Evaluate: lim " lim 4 5 lim 4 " + " Eample 5.4 Evaluate: lim tan` j. " Let y and y " 0 as " lim tan` j lim tan y y " y " 0. Eample 5.5 / n Evaluate: lim n n. " n lim n n 78 th Std. Business Mathematics nn ^ + h^n + h " / lim n " 6n Ch-5.indd :4:6

187 lim n n n n 6 $ + + ` j` n j` n j. n " 6 lim $ ` + nj` + nj. n " 6 ^ h^ h. Eample 5.6 log^ + h Show that lim 0 sin. " log^ + h lim 0 sin " lim log ^ + h ' # " 0 sin lim log ^ + h # 0 lim sin " ` j " 0 # Eercise 5.. Evaluate the following (i) lim + + " (iv) lim " (ii) lim " (v) lim " 5 8 a - - a a 5 8 (iii) lim n n " / (vi) lim sin " a. If lim + a lim^ + 6h, find the values of a. " a ". If lim " - - n 7 n 448, then find the least positive integer n If f^h 5, then find lim f^h - " a + b 5. Let f^h +, If lim f^h and lim f^h, then show that f( ) 0. " 0 " Differential Calculus 79 Ch-5.indd :4:9

188 Derivative Before going into the topic, let us first discuss about the continuity of a function. A function f() is continues at a if its graph has no break at a If there is any break at the point a then we say that the function is not continuous at the point a. If a function is continuous at all the point in an interval, then it is said to be a continuous in that interval Continuous function A function f() is continuous at a if (i) f(a) eists (ii) lim f() eists " a (iii) lim f() f(a) " a Observation In the above statement, if atleast any one condition is not satisfied at a point a by the function f(), then it is said to be a discontinuous function at a 5..9 Some properties of continuous functions Let f() and g() are two real valued continuous functions at a, then (i) f() + g() is continuous at a. (ii) f() g() is continuous at a. (iii) (iv) (v) (vi) kf() is continuous at a, k be a real number. f^h is continuous at a, if f(a) ] 0. f^h g^h is continuous at a, if g(a) ] 0. f^h is continuous at a. 80 th Std. Business Mathematics Ch-5.indd :4:0

189 Eample 5.7 Show that f() 5-4, if 0 # ) is continuous at 4 -, if < L7f] ga lim f] g " - lim f] - hg, h h " 0 lim 65] - hg- 4@ h " 0 5() 4 R7f] ga lim f] g " + lim f] + hg, + h h " 0 lim 74] + hg - ] - hga h " 0 4() () 4 Now, f() 5() ` lim f] g lim f] g f(), " - " + f() is continuous at. Eample 5.8 Verify the continuity of the function f() given by - if< f] g ) at. + if$ L7f] ga lim f] g " - lim f] - hg, h h " 0 lim " - ] - hg, h $ 0 lim h 0 h " 0 Differential Calculus 8 Ch-5.indd :4:

190 R7f] ga lim f] g + + lim f] + hg, + h h " 0 lim" + ] + hg, h " 0 lim] 4 + hg 4 h " 0 f] g Now, f() + 4. Here lim f] g! lim $ $ Hence f() is not continuous at. Z a() if < a ] If f() [ k if a then ] b() if > a \ L[f()] a lim f () lim () a - " a " a R[f()] a lim f () lim () b + " a " a It is applicable only when the function has different definition on both sides of the given point a. Observations (i) A constant function is everywhere continuous. (ii) The identity function is everywhere continuous. (iii) A polynomial function is everywhere continuous. (iv) The modulus function is everywhere continuous. (v) The eponential function a, a>0 is everywhere continuous. (vi) The logarithmic function is continuous in its domain. (vii) Every rational function is continuous at every point in its domain. Eercise 5.. Eamine the following functions for continuity at indicated points (a) f] g * - - 4, if! at 0, if (b) f] g * - - 9, if! at 6, if. Show that f() is continuous at Differentiability at a point Let f() be a real valued function defined on an open interval (a, b) and let 8 th Std. Business Mathematics Ch-5.indd :4:6

191 f] g- f] cg c! (a, b). f() is said to be differentiable or derivable at c if and only if lim c eists finitely. - " c This limit is called the derivative (or) differential co-efficient of the function f() at c and is denoted by (or) D f(c) (or) : d d _ f] gid. Thus. 5.. Left hand derivative and right hand derivative f] g- f] cg f] c- hg - f] cg (i) lim c - " c or lim h - h " 0 - is called the left hand derivative of f() at c and it denoted by f l^c - h or L[ f l (c)]. f] g- f] cg f] c+ hg - f] cg (ii) lim c - " c or lim h is called the right hand derivative of + h " 0 f() at c and it denoted by f l^c + h or R[ f l (c)]. Result f() is differentiable at c, L7fl] cga R7fl] cga Remarks: (i) If L7fl] cga! R7fl] cga, then we say A function may be that f() is not differentiable at c. continuous at a point but (ii) f() is differentiable at c f() is continuous at c. may not be differentiable at that point. Eample 5.9 c Show that the function f() is not differentiable at 0. f() ) - if if $ 0 < 0 L[ f l (0)] f] g- f] 0g lim, " f] 0 - hg - f] 0g lim h " 0 0-h -0, 0 h Differential Calculus 8 Ch-5.indd :4:9

192 lim 84 th Std. Business Mathematics f] -hg - f] 0g h h " 0 - lim -h - 0 h h " 0 - -h lim h h " 0 - h lim h h " 0 - lim] -g h " 0 f] g- f] 0g R[ f l (0)] lim - 0 " 0 + f] 0 + hg - f] 0g lim h " 0 0+ h -0, 0+h f] hg- f] 0g lim h " 0 h h - 0 lim h h " 0 h lim h " 0 h h lim h " 0 h lim h " 0 Here, L[ f l (0)]! R[ f l (0)] ` f() is not differentiable at 0 Eample 5.0 Show that f() differentiable at and find f l (). f() Ch-5.indd :4:4

193 f] g - f] g L[ f l ()] lim " - - f] - hg - f] g lim h " 0 -h - ] - hg - lim -h h " 0 + h -h- lim -h h " 0 lim] - h + g h " 0 f] g - f] g R[ f l ()] lim " + - f] + hg - f] g lim h " 0 + h - ] + hg - lim h " 0 h + h + h- lim h h " 0 lim] h + g h " 0 Here, L[ f l ()] R[ f l ()] ` f() is differentiable at and f l () 5.. Differentiation from first principle The process of finding the derivative of a function by using the definition that f] + hg - f] g f l] g lim h $ 0 h is called the differentiation from first principle. It is dy convenient to write f l() as d. Differential Calculus 85 Ch-5.indd :4:4

194 5.. Derivatives of some standard functions using first principle d. For! R, ^ d h n Proof: n n - Let then f() n f(+h) (+h) n d d _ f] gi lim h " 0 f] + hg - f] g h n ] + hg -] g lim h h " 0 n ] + hg -] g lim h " 0 ] + h g - n n z lim z - - z" n n, where z + h and z " as h " 0 n n n n - a n - < a lim a na a - F " i.e., d d ^ n h n n. d d ^e h e proof: Let f] g e then f(+h) e + h d d _ f] gi lim h " 0 f] + hg - f] g h e lim h " 0 + h - e h e e- e lim h h " 0 lim e h " 0 h h ; e - E h 86 th Std. Business Mathematics Ch-5.indd :4:46

195 e e lim; - E h h " 0 h e # < e a lim; - E h F e h " 0 h i.e., d d ^ h e. e. log d d ^ h Proof: e Let Then, f() log e f(+h) log e (+h) d d f] + hg - f] g _ f] gi lim h " 0 h loge] + hg- loge lim h " 0 h h loge b + l lim h Eample 5. h " 0 log lim h " 0 e log lim h " 0 b + h e h h b + l h l # [ lim log e ] + g a ] d Find d ^ h from first principle. " 0 Differential Calculus 87 Ch-5.indd :4:48

196 Let f() then f] + hg ] + hg d d _ f] gi lim h " 0 f] + hg - f] g h ] + hg - lim h h " 0 h+ h + h lim h h " 0 h " 0 lim^ + h+ h h + () 0 + () 0 d d ^ h Eample 5. d Find d Let f() e ^ e h from first principle. then f( + h) e (+h) d d _ f] gi lim h " 0 f] + hg - f] g h e lim h " 0 ] + hg - e h e $ e - e lim h h " 0 h e ^e h lim h h " 0 h - 88 th Std. Business Mathematics Ch-5.indd :4:5

197 e e lim h " 0 h - h e # e - [ a lim ] e " 0 Eercise 5.4. Find the derivative of the following functions from first principle. (i) (ii) e (iii) log(+) 5. Differentiation techniques In this section we will discuss about different techniques to obtain the derivatives of the given functions. 5.. Some standard results [formulae]. d d n n n - ^ h. d d () d d (k) 0, k is a constant d d (k) k, k is a constant d d b l - d d ^ h d d (e ) e d d (e a ) ae a d d (e a+b ) ae a+b d d _ e i- e. d d ^a h a logea Differential Calculus 89 Ch-5.indd :4:5

198 d. ^log d h e d. log a d e ] + g + a d 4. _ loge ] a + bgi d d 5. ^logah d log a 6. sin d d ] g cos e a ] a + bg d d (cos) sin d d (tan) sec d d (cot) cosec d d (sec) sec tan. d d (cosec ) cosec cot 5.. General rules for differentiation (i) Addition rule d d 7f] g+ g] ga d d 7f] ga+ d d 7g] ga (ii) Subtraction rule (or) Difference rule d d 7f] g- g] ga d d 7f] ga- d d 7g] ga (iii) Product rule d d 7f] g$ g] ga d d f g f d 7 ] ga ] g+ ] g 7 g ] ga d d (or) If u f() and v g() then, ] uv g d u d v d v d ] g+ ] u g d (iv) Quotient rule d d f g d f d f d ] g ] g 7 ] ga- ] g 7 g ] ga G d g ] g 7g] ga d u (or) If u f() and v g() then, b d v l v d ] u g d u d - ] v g d v 90 th Std. Business Mathematics Ch-5.indd :4:55

199 (v) Scalar product d d 7 cf ] ga c d 7 f ] ga, d where c is a constant. (vi) Chain rule: d d 7 f _ g ] gia f l_ g ] gi $ g l] g (or) If y f(t) and t g() then, Here we discuss about the differentiation for the eplicit functions using standard results and general rules for differentiation. Eample 5. dy d dy dt Differentiate the following functions with respect to. (i) (ii) 7e (iii) $ d dt - + (iv) sin (v) sin (vi) + + (i) d d ` j - (ii) d d ^7e h 7 d d (iii) Differentiating y dy d ^e h 7e - + with respect to ] + g d ] -g d d - ] - g ] + g d + ] g ] + g] -g- ] - g] g + ] g (iv) Differentiating y sin with respect to dy d d ] sin g d d + sin ^ d h cos+ sin ] cos+ sin g -6 ] + g Differential Calculus 9 Ch-5.indd :4:57

200 (v) y sin (or) (sin ) Let u sin then y u Differentiating u sin with respect to We get, du d cos Differentiating y u with respect to u dy We get, du u sin dy dy du d du $ d sin $ cos (vi) y + + Let u + + then y u Differentiating u + + with respect to du We get, d + Differentiating y u with respect to u. dy We get, du u dy dy d du + + du $ d $ ] + g sin log sin log log d d d d ^ ^ hh ^ ^ hh d log d ^ h ^ h d d 5 ^ + + h d^ d ^ + + h + + h d d ^ + + h 5 d d ^e h d ^ h e d d d ^ h ^ + ^ + h h h ^ th Std. Business Mathematics Ch-5.indd :4:00

201 Eample 5.4 If f() n and f l] g 5, then find the value of n. f() n ` f l] g n n Eample 5.5 f l] g n() n f l] g n f l] g 5 (given) ( n 5 If y and u u dy 9, find d. y u u dy du - u dy du - ^ - 9h ( a u 9) u 9 du d Now, dy dy d du du $ d - ^ - 9h $ 4 - ^ - 9h. Differential Calculus 9 Ch-5.indd :4:0

202 ICT Corner Epected final outcomes Step Open the Browser and type the URL given (or) Scan the QR Code. GeoGebra Work book called th BUSINESS MATHS will appear. In this several work sheets for Business Maths are given, Open the worksheet named Elementary Differentiation Step - Elementary Differentiation and Functions page will open. Click on the function check boes on the Right-hand side to see the respective graph. You can move the sliders a, b, c and d to change the co-efficient and work out the differentiation. Then click the Derivatives check bo to see the answer and respective graphs Step Step Step 4 Browse in the link th Business Maths: (or) scan the QR Code 94 th Std. Business Mathematics Ch-5.indd :4:0