Std. XII Commerce Mathematics and Statistics I

` Written according to the New Tet book (-4) published by the Maharashtra State Board of Secondary and Higher Secondary Education, Pune. Std. XII Commerce Mathematics and Statistics I Third Edition: April 6 Salient Features : Precise Theory for every Topic. Ehaustive coverage of entire syllabus. Topic-wise distribution of all tetual questions and practice problems at the beginning of every chapter. Relevant and important formulae wherever required. Covers answers to all Tetual Questions. Practice problems based on Tetual Eercises and Board Questions (March 8 March 6) included for better preparation and self evaluation. Multiple Choice Questions at the end of every chapter. Two Model Question papers based on the latest paper pattern. Includes Board Question Papers of March and October 4, 5 and March 6. Printed at: Repro India Ltd., Mumbai No part of this book may be reproduced or transmitted in any form or by any means, C.D. ROM/Audio Video Cassettes or electronic, mechanical including photocopying; recording or by any information storage and retrieval system without permission in writing from the Publisher. 4_57_JUP P.O. No. 659

Preface Mathematics is not just a subject that is restricted to the four walls of a classroom. Its philosophy and applications are to be looked for in the daily course of our life. The knowledge of mathematics is essential for us, to eplore and practice in a variety of fields like business administration, banking, stock echange and in science and engineering. With the same thought in mind, we present to you "Std. XII Commerce: Mathematics and Statistics-I" a complete and thorough book with a revolutionary fresh approach towards content and thus laying a platform for an in depth understanding of the subject. This book has been written according to the revised syllabus and includes two model question papers based on the latest paper pattern. At the beginning of every chapter, topic-wise distribution of all tetual questions including practice problems have been provided for simpler understanding of various types of questions. Every topic included in the book is divided into sub-topics, each of which are precisely eplained with the associated theories. We have provided answer keys for all the tetual questions and miscellaneous eercises. In addition to this, we have included practice problems based upon solved eercises which not only aid students in self evaluation but also provide them with plenty of practice. We've also ensured that each chapter ends with a set of Multiple Choice Questions so as to prepare students for competitive eaminations. We are sure this study material will turn out to be a powerful resource for students and facilitate them in understanding the concepts of Mathematics in the most simple way. The journey to create a complete book is strewn with triumphs, failures and near misses. If you think we ve nearly missed something or want to applaud us for our triumphs, we d love to hear from you. Please write to us on: mail@targetpublications.org Yours faithfully Publisher Best of luck to all the aspirants! BOARD PAPER PATTERN Time: Hours Total Marks: 8. One theory question paper of 8 marks and duration for this paper will be hours.. For Mathematics and Statistics, (Commerce) there will be only one question paper and two answer papers. Question paper will contain two sections viz. Section I and Section II. Students should solve each section on separate answer books.

Section I Q.. This Question will have 8 sub-questions, each carring two marks. Students will have to attempt any 6 out of the given 8 sub-questions. Q.. This Question carries 4 marks and consists of two sub parts (A) and (B) as follows: (A) It contains sub-questions of marks each. Students will have to attempt any out of the given sub-questions. (B) It contains sub-questions of 4 marks each. Students will have to attempt any out of the given sub-questions. Q.. This Question carries 4 marks and consists of two sub parts (A) and (B) as follows: (A) It contains sub-questions of marks each. Students will have to attempt any out of the given sub-questions (B) It contains sub-questions of 4 marks each. Students will have to attempt any out of the given sub-questions. Section II Q.4. This Question will have 8 sub-questions, each carring two marks. Students will have to attempt any 6 out of the given 8 sub-questions. Q.5. This Question carries 4 marks and consists of two sub parts (A) and (B) as follows: (A) It contains sub-questions of marks each. Students will have to attempt any out of the given sub-questions. (B) It contains sub-questions of 4 marks each. Students will have to attempt any out of the given sub-questions. Q.6. This Question carries 4 marks and consists of two sub parts (A) and (B) as follows: (A) It contains sub-questions of marks each. Students will have to attempt any out of the given sub-questions (B) It contains sub-questions of 4 marks each. Students will have to attempt any out of the given sub-questions. Evaluation Scheme for Practical i. Duration for practical eamination for each batch will be one hour. ii. Total marks : MARKWISE DISTRIBUTION [ Marks] [4 Marks] [4 Marks] [ Marks] [4 Marks] [4 Marks] Unitwise Distribution of Marks Section - I Sr.No. Units Marks with Option 4 5 6 7 Mathematical Logic Matrices Continuity Differentiation Application of Derivative Integration Definite Integrals 8 8 8 8 8 8 Total 58

Unitwise Distribution of Marks Section - II Sr. No. Units Marks with Option. Commercial Arithmetic: Ratio, Proportion, Partnership Commission, Brokerage, Discount Insurance, Annuity. Demography 8. Bivariate Data Correlation 8 4. Regression Analysis 7 5. Random Variable and Probability Distribution 8 6. Management Mathematics 4 Total 58 Weightage of Objectives Sr. No. Objectives Marks Marks with Option Percentage 4 Knowledge Understanding Application Skill 8 8 45 6. 7.5 4..5 Total 8 6. Weightage of Types of Questions Sr. No. Types of Questions Marks Marks with Option Percentage Objective Type Short Answer Long Answer 4 4 6 48 4 Total 8 6. No. Topic Name Page No. Mathematical Logic Matrices 4 Continuity 4 Differentiation 5 5 Applications of Derivative 88 6 Integration 8 7 Definite Integrals 8 Model Question Paper - I Model Question Paper - II 5 Board Questions Paper March 4 7 Board Questions Paper October 4 9 Board Questions Paper March 5 Board Questions Paper October 5 Board Questions Paper March 6 5

Continuity Chapter : Continuity Type of Problems Eercise Q. Nos. Continuity of Standard Function Eamine the Continuity of a Function at a given point Types of Discontinuity (Removable Discontinuity/ Irremovable Discontinuity) Find the value of Function if it is Continuous at given point Find the value of k/a/b if the Function is Continuous at a given point/points. Find the points of Discontinuity for the given Functions. Q. Practice Problems (Based on Eercise.) Q.. Q.,, Practice Problems (Based on Eercise.) Miscellaneous Q. Practice Problems (Based on Miscellaneous). Q.4 Practice Problems (Based on Eercise.) Q.,, 9,,, 4, 5, 6, 7 Q.,, 8 Q.4 Miscellaneous Q., Practice Problems (Based on Miscellaneous) Q.. Q.5, 7 Practice Problems (Based on Eercise.) Miscellaneous Q., 4 Practice Problems (Based on Miscellaneous) Q.5, 7,, 4 Q.4. Q.6, 8, 9 Practice Problems (Based on Eercise.) Miscellaneous Q.5, 6, 7, 8 Practice Problems (Based on Miscellaneous) Miscellaneous Q.9 Q.6, 8,, 8, 9,,,,, 5, 6 Q.5, 6, 7

Std. XII : Commerce (Maths I) Syllabus:. Continuity of a function at a point. Algebra of continuous functions. Types of discontinuity.4 Continuity of some standard functions Introduction Continuity is the state of being continuous and continuous means without any interruption or disturbance. For eample, the price of a commodity and its demand are inversely proportional. The graph of demand curve of a commodity is a continuous curve without any breaks or gaps. Y X X O Demand Y Note: A graph consisting of jumps is not a graph of continuous function.. Continuity of a function at a point Definition: A function f is said to be continuous at a point a in the domain of f, if a i.e. if f() eists and a f() a a f() f (a). f() f (a) If any of the above conditions is not satisfied by the function, then it is discontinuous at that point. The point is known as point of discontinuity. eg., Consider the function, f() + 7, 4 5 5, 4 Since, f() has different epressions for the value of left hand and right hand its have to be found out. 4 Price f() 4 Also, f (4) 5 (4) 5 5 and f() 4 (5 5) 5 4 5 5 ( + 7) 4 + 7 5 4 f() 4 4 f() f (4) f() is continuous at 4. Continuity of a function on its domain Definition: A real valued function f : D R where D R is said to be a continuous function on D, if it is continuous at every point in the domain D. eg., Consider the functions, i. f() 4 + + ii. f() sin These two functions are continuous on every domain D, where D R.. Algebra of continuous functions If f and g are two real valued functions defined on the same domain, which are continuous at a, then. the function kf is continuous at a, for any constant k R.. the function f g is continuous at a. the function f. g is continuous at a 4. the function f is continuous at a, when g g (a) 5. composite functions, f[g()] and g[f()], if well defined are continuous functions at a.. Types of discontinuity. Removable discontinuity: A real valued function f is said to have removable discontinuity at a in its domain, if f() eists but f() f (a) a a i.e. f() f() f(a) a a This type of discontinuity can be removed by redefining the function f at a as f (a) f(). a eg., Consider the function, 5 6 f(),, Here, f() 56... [,, ] f() eists

Chapter : Continuity Also, f () f() f (). (given) function f is discontinuous at, This discontinuity can be removed by redefining f as follows: 56 f(),, is a point of removable discontinuity.. Irremovable discontinuity: A real valued function f is said to have irremovable discontinuity at a, if f() does not eist i.e. a f() a a f() or one of the its does not eist. Such a function can not be redefined as continuous function. eg., Consider the function, f() + +,, Here, and f() f() + + () + () + 8 f() () 8 f() it of the function does not eist. f has irremovable discontinuity at.4 Continuity of some standard functions. Constant function: The constant function f() k (where k R is a constant). The function is continuous for all belonging to its domain. eg., f(), f() log, f() e 7. Polynomial function: The function f() a + a + a +. + a n n, where n N, a, a. a n R is continuous for all belonging to domain of. eg., f() + 5 + 9, f() 5 + 9, f() 4 6, R.. Rational function: If f and g are two polynomial functions having same domain then the rational function f is continuous in its g domain at points where g(). eg., 56 Consider the function, 9 Here, f() + 5 + 6 and g() 9 Given function is continuous on its domain, where 9 i.e., ( + ) ( ) i.e., +, i.e.,, The function is continuous on its domain ecept at,. 4. Trigonometric function: sin (a + b) and cos (a + b), where a, b R are continuous functions for all R. eg., sin (5 + ), cos (7 ) R. Note: Tangent, cotangent, secant and cosecant functions are continuous on their respective domains. 5. Eponential function: f() a, a >, a, R is an eponential function, which is continuous for all R. eg., f(), f(), f() e R, where a >, a. 6. Logarithmic function: f() log a where a >, a is a logarithmic function which is continuous for every positive real number i.e. for all R + eg., f() log a 7, f() log a 9 R, where a >, a. Some Important Formulae Algebra of its: If f() and g() are any two functions,. [f() + g()] f() + g().. 4. 5. a [f() g()] a [f()g()] a f( ) a g( ) a a a f() a a f() g() a a f ( ) a g( ), where a g() g() a [k.f()] k f(), where k is a constant. a

Std. XII : Commerce (Maths I) Limits of Algebraic functions:. a.. 4 a a a k k, where k is a constant. n a a n na n Limits of Trigonometric functions: sin. tan.. cos Limits of Eponential functions:.. a log a, where (a >, a ) ( + ) e Limits of Logarithmic functions: log. Eercise.. Are the following functions continuous on the set of real numbers? Justify your answers. i. f() 7 Given, f() 7 It is a constant function. f() is continuous on the set of real numbers i.e., R ii. f() e Given, f () e It is a constant function. [ e.788] f() is continuous on the set of real numbers i.e., R. iii. f () log 9 Given, f() log 9 Here, log 9 is a constant f() is a constant function f() is continuous on the set of real numbers i.e., R. iv. f() 7 4 5 + Given, f() 7 4 5 + It is a polynomial function f() is continuous on the set of real numbers i.e., R v. g() sin (4 ) Given, g() sin (4 ) It is a sine function f() is continuous on the set of real numbers i.e., R 5 +7 + vi. h() + + + Given, h() 5+ 7 + It is a rational function and is discontinuous if + + + But, R, + + + h() is continuous on the set of real numbers, ecept when + + + 69 vii. g() 69 Given, g() It is a rational function and is discontinuous, if + But R, + g () is continuous on the set of real numbers i.e., R viii. f() 5 Given, f() 5 It is an eponential function It is continuous on the set of real numbers i.e., R i. f() 5 Given, f() 5 It is the difference of two eponential functions It is continuous on the set of real numbers i.e., R

Chapter : Continuity (5 + 7). f() e Given, f() e (5 + 7) It is an eponential function It is continuous on the set of real numbers i.e., R. Eamine the continuity of the following functions at the given point. (All functions are defined on R R) i. f() + 9, for 4 +, for > ; at. [Mar 5] f() ( + 9) () + 9 5. (i) and f() (4 + ) 4() + 5. (ii) Also, f () () + 9 5. (iii) f() f() f(). [From (i), (ii) and (iii)] f is continuous at. ii. f() 6, for 4 4 8, for 4; at 4. [Oct 5] 6 f() 4 4 4 4 4 4 ( 4)( 4) 4 ( 4) 4 ( + 4).[ 4, 4, 4 ] 4 f() 4 + 4 8. (i) Also, f (4) 8. (ii)(given) f() f (4). [From (i) and (ii)] 4 f is continuous at 4. iii. f() 4, for 79, for ; at Consider, + 7 9 By synthetic division, we get 7 9 9 9 + 7 9 ( ) ( + + 9) f() 4 9 9 9 9 [,, ] ( ) 9 9 f.(i) Also, f( ).(ii)(given) f() f ( ).[From (i) and (ii)] f is discontinuous at. sin 5 iv. f(), for, for ; at. sin5 sin5 f() 5 5 sin5 5 5 5 () sin. [, 5, ] f() 5 Also, f(). (given) f() f () f is discontinuous at. 5

Std. XII : Commerce (Maths I) 7 v. f() f(), for, for ; at. [Oct 4] 7 7 7 7 () 7 7 9 7 7 7 7 7. [,, ] 7 6 f(). (i) Also, f(). (ii)(given) f() f().[from (i) and (ii)] f is continuous at. vi. f() 5 5, for 4 8, for 4; at 4. 5 f() 4 4 5 5 4 5 5 5 5 5 5 5 4 5 5 4 95 5 5 5 4 5 4 4 5 4 5 4 4 5 5 4 5...[ 4, 4, 4 ] 5 4 4 5 4 f() Also, f(4) 8 4 f() f(4). (i). (ii)(given) 6. [From (i) and (ii)] f is discontinuous at 4. vii. f() cos, for, for ; at [Oct 5] f() cos cos [, ] for all R, also, when, cos eists. Let cos finite number k (say) cos where k [, ] f(). (i) k 6

Also, f(). (ii)(given) f() f(). [From (i) and (ii)] f is continuous at. tan sin viii. f() sin sin, for < Consider, f() sin sin, for ; at tan sin sin sin tan sin sin 4sin sin [sin sin 4sin ] tan sin 4sin sin sin cos 4sin sin sin cos 4sin sin cos 4sin sin sin 4sin [ cos sin ] Also, Chapter : Continuity sin sin 4 sin 4 8 [,, f() 8 f() sin sin sin sin sin sin sin ]. (i) sin sin () (). [ sin ] sin.sin sin sin sin 4 4 sin f() f() f() does not eist f is discontinuous at. (ii) f().[from (i) and (ii)] 7

Std. XII : Commerce (Maths I) 9 i. f(), for < 6 4, for > 4 for ; at Consider, 9 f() 6 + 9 6 8 f() does not eist. But f() 4 f() f() f is discontinuous at. 6. f(), for < 4 4, for >, for, at 6 f() 6 6 6 6 6 log 6 + log log a [ log a ] log 6 f() log 9. (i) 4 4 f() 4 4 4 4 4 4 4 4 4 (log 4). 4 a [ log a ] f() (log 4)... (ii) f() f()... [From (i) and (ii)] f() does not eist f is discontinuous at. Discuss the continuity of the following functions. a i. f(),, a & a log a, ; at. a f() a a f() log a. (i).[,, a log a] Also, f() log a. (ii)(given) f() f().[from (i) and (ii)] f is continuous at 5 ii. f() 4, log 5, ; at. 4 5 f() 4 8

5 4 5 4 5 4 5 4 5 4 log5 log a...[ log a] log 4 log 5 log f() 4 log. (i) Also, f () log 5. (ii) (given) 4 f() f() f is discontinuous at iii. g() 5, e 5/, ; at. g() 5 5 5 g() e 5. (i)...[, 5 5 e Also g() g() g() g is discontinuous at 5. (ii)(given) e].[from (i) and (ii)] Chapter : Continuity log + iv. h(),, ; at. log h() log log () log...[,, ] h(). (i) Also, h(). (ii)(given) h() h().[from (i) and (ii)] h is continuous at v. f() tan, <, > e e, at. f() tan tan log...[,, a f() log f() e e e e tan tan log a, ]. (i) 9

Std. XII : Commerce (Maths I) f() f() e e e e e e e loge e e...[... (ii) f() does not eist f is discontinuous at log e] f().[from (i) and (ii)] 4. Discuss the continuity of the following functions at the points given against them. If the function is discontinuous, determine whether the discontinuity is removable. In that case, redefine the function, so that it becomes continuous. i. f(), for <, for ; at. f() ( ) () () 4 4 and f() f() 5 f() f() ( ) () 6. (i). (ii) f().[from (i) and (ii)] it of the function does not eist. f has irremovable discontinuity at 7 ii. f(), for < 8, for ; at. 7 f() and 9...[a b (a b)(a + ab + b )] ( + + 9) [, ] () + () + 9 f() 7.(i) f() f() 4 8 8().(ii) f() f().[from (i) and (ii)] it of the function does not eist f has irremovable discontinuity at iii. f() sin9, for, for ; at. sin9 f() sin9 9 9 9 sin9 9 9 [, 9, f() 9. (i) sin ] Also, f(). (ii) (given) f() f().[from (i) and (ii)]

Chapter : Continuity f has removable discontinuity at This discontinuity can be removed by redefining the function as: f() sin9, for 9, for ; at e iv. f(), for 5, for ; at e f() 5 e 5 e 5 log e 5 e [,, log e ] f() 5 Also, f() f() f(). (i). (ii)(given).[from (i) and (ii)] f has removable discontinuity at This discontinuity can be removed by redefining the function as: e f() 5, for, for ; at 5 v. f(), for 5, for ; at f() 4...[, ] f(). (i) Also, f() 5. (ii)(given) f() f().[from (i) and (ii)] f has removable discontinuity at This discontinuity can be removed by redefining the function as: f(), for, for ; at 5. If f is continuous at, then find f() 5 5 i. f(), Given, f is continuous at f() f() 5 5 5 5 5 5 5 log 5 f() (log 5) 5...[ 5 5 5 5 a log a ]

Std. XII : Commerce (Maths I) sin ii. f() log Given, f is continuous at f() f() sin... [, sin,, [Mar ] log sin log( ) sin sin sin sin sin log + sin sin sin log( + ) log sin, a loga] iii. f() 5 5, [Mar 5] tan Given, f is continuous at f() f() 5 5 tan 5 5 tan 5 5 tan 5 5 tan 5 tan 5.[,, tan 5 tan 5 tan log5 log f() (log 5) (log ) a log a, tan ] cos cos iv. f(), [Mar 6] Given, f is continuous at f() f() f () 4 cos cos 4cos coscos.[cos 4cos cos] 4cos 4cos 4coscos 4cos cos 4cossin 4 cos. 4. cos(). () sin.[ sin ]

Chapter : Continuity 6. Find the value of k, if the function i. g(), for k, for is continuous at Given, g is continuous at and g() k g() g() k ()... [ a k n a a n na ii. h() +, for < 5 + k, for is continuous at Given, h is continuous at h() h() h ().(i) Now, h() + () + and h() h() 5 k 5 k h() 5 + k 5 + k...[from (i)] k 4 n iii. f() tan 7, for k, for is continuous at [Mar 5] Given, f is continuous at and f () k f() f() k 7 tan7 tan 7 7 7 tan7 7 ] 7 () k 7.[,7, tan iv. h() + k, for 7, for 7 is continuous at 7 Given h is continuous at 7, h(7) h(7) h() 7 7 + k ( + k) 7 h(7) (7 + k) 7 + k or (7 + k) k 7 or 7 k k or k 7 k or k 7. ] sin 7. If f(), for, is continuous at, then find f. [Oct 5] Given, f is continuous at f f sin Put h h as,,h sin h f h h h cosh h... [sin cos ]

Std. XII : Commerce (Maths I) 4 h h cosh cosh h cosh cos h 4h cosh sin h 4h cosh h sin h 4 h h h cosh sin 4 ().[ cos ] 4 4 f 8 e 8. If f() for <, a a for log 7 for >, b b is continuous at, then find a and b. [ Mar 6] Given, f is continuous at and f() f() f() f() f() f().(i) Now, e f() a e a e a loge a a.[,, log a ] f()...(ii) a log Also, f() 7 b log 7 7 b 7 f() 7 b Now, a a and 7 b b 7 a, b 7 7 log 7 b 7 7 b log... [,7, ]...(iii) 9. If f is continuous at and f() + a, for < + a + b, for and f(), then find a, b. Given, f is continuous at f() f() Now, f() + a.[from (i) and (ii)].[from (i) and (iii)]. (i) a f() + a and f() + a + b + a + b f() a + b + a a + b.[from (i)] b Also, f() + a + b, for and f() f() () + a + b + a + b a + b. (ii) Substituting b in (ii) we get a + a a, b

. Is the function f() + 5 cos + continuous at? Justify. Given, f() + 5 cos + f() ( + + ) 5 cos Let + + p() and 5 cos q() f() p() q()...(i) Now, p() + + p() is a polynomial function It is continuous at each value of and q() 5 cos Here, 5 is constant function and cos is cosine function which is continuous. q() is continuous at each value of f() is a difference of two continuous function which is always continuous f() is continuous at Miscellaneous Eercise. Discuss the continuity of the function. f() 64, for 4 9 5, for 4; at 4 f() 64 4 4 9 5 4 9 5 4 9 5 9 5 4 46 9 5 4 9 5 4 46 9 5 4 9 5 4 46 9 5 4 6 4 46 95 4 44 Chapter : Continuity 46 9 5 4 4 [ 4, 4, 4 ] 4 446 4 9 5 4 4 48 8 f() 6 4. (i) Also, f(4). (ii)(given) f() f(4) 4.[From (i) and (ii)] f is discontinuous at 4. Eamine the continuity of the function e f() log, for, for ; at. If discontinuous, then state whether the discontinuity is removable. If so, redefine and make it continuous. e e f() log 9 log 9 e 9 log 9 loge e log( ) e...,, loge, log () f(). (i) Also, f(). (ii)(given) 5

Std. XII : Commerce (Maths I) f() f().[from (i) and (ii)] f has removable discontinuity at This discontinuity can be removed by redefining the function as: f() e, for log( ), for ; at. The function f is defined as 7 8 f(), for 5 /5 /5, for > / / f() Eamine, if f is continuous at. 7 8 f() 5 7 7 5 5 7 7 7 7 5 5 5 5 7 n n 7 a... 5 na a 5 a 6 7 7 4 5 5 f() 8 5 and f() 5 5 5 5 5 5 n 5 5 n a...[ a a 4 5 4 5 5 5 f() 5 () f() f() f() does not eist f is discontinuous at n na 4. The function f defined as 8 8 f(), for 5 7 is continuous at. Find f() Given, function is continuous at 8 8 f() f() 5 7 8 8 5 7 5 7 5 7 n 8 8 8 8 8 8 5 7 5 7 8 8 88 5 7 5 7 8 8 77 5 7 88 8 8 7 5 7 8 8 8 ] 6

7 5 7 8 8 8...[, ] 7 5 7 7 8 8 8 8 8 48 f () 7 5. Find k if the function given below is continuous at cos sin f(), for k, for Given, f is continuous at f f() cos sin k cos sin cos [sin sin cos] cos sin Put h h as,, h cos h sin h h h h sinhcosh h h [cos sin,sin cos] h sinhsin h 8h h sin sin h h h h 4 4 h sin sinh. 8 h h h h 8 Chapter : Continuity h sinh. h,, h h 6. If the function given below is continuous at as well as at 4, then find the values of a and b. f() + a + b, +, 4 a + 5b, 4 [Oct 4] Given, f is continuous at f() f().(i) Now, f() + a + b () + a() + b f() 4 + a + b and f() ( + ) () + f() 8 4 + a + b 8.[From (i)] a + b 4.(ii) Also, f is continuous at 4 f() f().(iii) 4 Now, 4 and 4 4 f() 4 f() 4 f() 4 f() 8a + 5b + (4) + 4 4 a + 5b a(4) + 5b 4 8a + 5b.[From(iii)] 8a + 5b 4.(iv) 7

Std. XII : Commerce (Maths I) By eq. (iv) 5 eq. (iii), we get 8a 5b 4 a 5b a 6 a Substituting a in eq. (ii), we get + b 4 b 4 6 b a or b 7. Find a and b if f is continuous at, where sin f() + a,, cos + b, Given, f is continuous at and f () f() f() f () 8 f() Now, f() f() Put + h h as,, h f() h.(i) sin a sin h a h sin h a h h sin h h a h.[sin(+) sin] sin h h a h sin h a h h h () + a sin h.[h, h, ] h h f() + a Also f().(ii) cos b Put + h h as,, h f() cosh b h h cosh b h h h h cosh b h.[cos(+) cos] h sin b h.[ cos sin ] h sin h b h 4 4 h sin h b 4 h 4 h sin b h h h b h sin... [ h,, ] f() b.(iii) + a.[from (i) and (ii)] a + a and b.[from (i) and (iii)]

Chapter : Continuity b b a, b 8. Find k, if the function f is continuous at, where e sin i. f(), for k, for Given, f is continuous at and f() k f() f() k k e sin e sin e. log e. ().[ ii. f() sin e log e, sin ] 7, for k, for Given, f is continuous at and f() f() f() 7 9 k k 9 9 k k 9 k 9. k log9 log k log k log9... [ a log a ] log k log 9... [n log a log a n ] log k log k log iii. f(), for 5 k, for Given, f is continuous at and f() k f() f() log k 5 5 5 k 5 log 5 log log...[,, ] 9. Find the points of discontinuity, if any, for the following: i. f() cos cos Given, f() Let, + cos p() and + q() Consider, p() + cos Here, is always continuous for all real values of and cosine is a continuous function p() is a continuous function and q() + It is a polynonimal function It is continuous for all real values of f() is a continuous function. 9

Std. XII : Commerce (Maths I) 5 4 ii. f() 4 5 4 Given, f() 4 5 4 f() f() is a rational function f() will be discontinuous if ( + ) ( ) i.e., + or i.e., or f() is discontinuous at and 4 9 iii. f() 6 49 Given, f() 6 f() is a rational function f() will be discontinuous if 6 + 6 6 i.e., 4 6 64 6 4 6 i Value of is a comple number f() is continuous for all real values of 9 iv. f() sin 9 9 Given, f() sin 9 Let 9 p() and sin 9 q() Consider, p() 9 It is a polynomial function It is continuous function and q() sin 9 Here, sine is a continuous function and 9 is a constant function q() is continuous as sin f() is continuous function.. If possible, redefine the function to make it continuous. i. f(), for e, for ; at. f Put + h h as,, h f() h h h h h h f() e...[ e ]...(i) Also, f() e. (ii) (given) f() f().[from (i) and (ii)] f has removable discontinuity at This discontinuity can be removed by redefining the function as: f(), for e, for ; at 6 ii. f() tan, for sin log 5, for ; at. tan 6 f() sin tan 6 sin cos cos tan 6 tan cos tan 6. tan cos log 6 cos... [, tan, f() log 6 Also, f() log 5 f() f() a log a ].(given) f has removable discontinuity at 4

Chapter : Continuity This discontinuity can be removed by redefining the function as: tan 6 f() sin, for log 6, for ; at sin 5 iii. f(), for 5, for ; at. [Oct 4] sin 5 f() sin 5 5 5 5 5() f() 5 Also, f() 5 f() f() sin5 5 [, 5, sin ]. (given) f has removable discontinuity at This discontinuity can be removed by redefining the function as: sin 5 f(), for 5, for ; at cos iv. f() sin, for < sin, for > cos, for ; at. f() cos sin sin sin sin sin sin sin sin...[a b (a b) (a + ab + b )] sin sin sin.[ sinsin sin, sin] sin sin sin f ( ) and f() sin cos sin sin cos sin sin cos sin sin sin sin sin sin sin sin sin sin.[ sinsin sin, sin] sin sin f ( ) f ( ) 4 f() it of the function does not eist f has irremovable discontinuity at 4

Std. XII : Commerce (Maths I) v. f(), for, for >, for ; at. f() and +...[, ] + + + () 4 f() 5 4 f() 4.[, ] 4 f() f() f() it of the function does not eist. f has irremovable discontinuity at 4 4 vi. f(), for. 5, for ; at. 4 4 f() 4 4 + + + () () + () () + () () + (4) () 4 + + + 4.[ a n a a n na 4 4 n f(). (i) Also, f() 5. (ii)(given) f() f() f has removable discontinuity at ] 4

Chapter : Continuity This discontinuity can be removed by redefining the function as: 4 4 f(), for, for ; at Additional Problems for Practice Based on Eercise.. Are the following functions continuous on the set of real numbers? Justify your answer. (All functions are defined on R R) i. f() 8 + 9 ii. f() e 5 iii. f() log 9 iv. f() 8 5 4 + 5 + + + 9 v. h() cos(9 + 5) vi. 5 g() 4 9 vii. g() + 7. Eamine the continuity of the following funtions at the given point: i. f() sin + cos, for, for ; at ii. f() sin, for, for ; at iii. f() ( + ) /, for e, for ; at 6 iv. f(), for 7, for ; at v. f() + 6 +, for 4 + 8, for > 4; at 4 y e.sin y vi. f(y), for y y 4, for y ; at y 4 vii. f(), for 5 4 5 e, for at viii. f() sin ( + ), for tan sin, for > ; at 5 i. f(), for < 4, for ; at. f(), for, for ; at. Discuss the continuity of the following functions: 5 a a i. f(), for log a, for ; at ii. f() a, for a e, for ; at 5 log iii. g(), for 5, for ; at iv. f() 5 e sin for (log 5 + ), for ; at v. sin a f(), for, for ; at vi. f() sin, for, for ; at vii. f() cos, for, for ; at 4 viii. f(), for, for ; at [Mar 6] 4. Discuss the continuity of the following functions at the points given against them. If the function is discontinuous, determine whether the discontinuity is removable. In that case, redefine the function, so that it becomes continuous. 4

Std. XII : Commerce (Maths I) i. f() cos tan, for 9, for ; at ii. 5 f(), for < 6, for ; at iii. f() sin 5 for 5, for ; at iv. f() 4, for sin 8, for ; at v. sin( ) f(), for, for ; at 5. If f is continuous at, then find f(). sin i. f() 4 log ( ), log( a ) log( b ) ii. f() log( ) log( ) iii. f() tan cos sin iv. f() 6. Find the value of k, if the function i. g(), for k, for is continuous at ii. f() 8, k for, for is continuous at iii. f() log( k ), for sin 5, for is continuous iv. f() + k, for k, for < is continuous at v. k f(), for ( ) 5 4, for is continuous at 7. If f() cos[7( )], for is 5( ) continuous at, find f() 8. If the function f() kcos, for, for be continuous at, then find k 9. Is the function f() + + cos + sin 5 + continuous at 4? Justify. If the function f is continuous at, then find f(). Where f() for. [Mar 4]. If the function f is continuous at, then 5 find k where f(), for < k +, for > [Mar 4]. Discuss the continuity of the function f defined as 8 f (),, ; at [Mar 8]. Discuss the continuity of the function f defined as: 5 f (), if 5 e, if ; at [Oct 8] 4. Discuss the continuity of f () at, where 4 f () for 4, for [Mar 9] 5. Discuss the continuity of the function f defined as, f() +, if 6, if < ; at [Oct ] 44

Chapter : Continuity 4 6. If f(), for 4, for Discuss the continuity of f() at [Mar, Oct ] 7. Discuss the continuity of the following function: f, for e, for ; at [Oct ] 8. If f is continuous at, where f () + a, for b, for < Find a, b given that f (). [Mar 8] 9. Find k, if the function f defined as: cosk f (),, is continuous at [Oct 8]. Find k, if the function 64 f (), for 4 4 k, for 4 is continuous at 4 [Oct 9]. If f() is continuous at and it is defined as a a f, k, find k. [Mar ]. The function f defined as sin p f(), if > q + 5 6, if is continuous at. Find the values of p and q, given that f (). [Oct ]. If f() tan + a, for <, for + 4 b, for > is continuous at, then find the values of a and b. [Mar ] 9 4. Find f() if f(), is contiuous at. [Oct ] 5. If f is continuous at where e f(), a, then find a. [Mar ] 6. If f is continuous at where f() + a, b, <, find a and b. Given that f() [Oct ] Based on Miscellaneous Eercise. Eamine the continuity of the following funtions at the given point: 7 4 5 i. f(), for cos ii., 7 for ; at f() sin tan for <, for log( ), for > e 4 iii. f() ( 6), for 6, for 6; at 6 5 iv. f(), for < +, for ; at 6 v. f() ( ), for 7, for ; at. Discuss the continuity of the following functions: i. f() 5 4 for log, for ; at ii. ( ) f() tan.log( ) log 4, for 45

Std. XII : Commerce (Maths I). Discuss the continuity of the following functions at the points given against them. If the function is discontinuous, determine whether the discontinuity is removable. In that case, redefine the function, so that it becomes continuous: i. f() 4 e, 6 for log, for ; at ii. f(), for log, for ; at iii. f() iv. f() 6 64, for 8, for ; at (8 ), for sin log 4 8 log 8, for ; at 4. If f is continuous at, then find f(). i. f() ii. f() cos 4 e e sin 5, 5. Find the value of k, if the function sin f(), for k, for is continuous at 6. If f() sin 4 + a, for > 5 + 4 b, for <, for is continuous at, find a and b. [Mar 9,, Oct 9] 7. cos4 If f(), for < a, for, for > 6 4 is continuous at, then find the value of a. 8. Discuss the continuity of the function f at 5 5, where f(), cos cos6 for log5, for 8 [Mar 4] Multiple Choice Questions., If f() c, is continuous at, then c (A) (B) 4 (C) (D)., if If f() a b, if 5 is continuous, 7, if 5 then the value of a and b is (A), 8 (B), 8 (C), 8 (D), 8. The sum of two discontinuous functions (A) is always discontinuous. (B) may be continuous. (C) is always continuous. (D) may be discontinuous. 4. For what value of k the function 5 44,if f() is k,if continuous at? (A) (B) (C) (D) 4 4 log ( a ) log ( b ) 5. The function f() is not defined at. The value which should be assigned to f at so that it is continuous at, is (A) a b (B) a + b (C) log a + log b (D) log a log b 46

Chapter : Continuity 6. In order that the function f() ( + ) cot is continuos at, f() must be defined as (A) f() e (B) f() (C) f() e (D) None of these 7. If f() sin, sin is k, a continuous function, then k (A) (B) (C) (D) 8. A function f is continuous at a point a in the domain of f if (A) f() eists (B) a a f() f(a) (C) f() f(a) a (D) both (A) and (B). 9. Which of the following function is discontinuous? (A) f() (B) g() tan (C) h() (D) none of these. If the function f() continuous at, then k kcos,when, when (A) (B) 6 (C) (D) None of these. The points at which the function f() is discontinuous, are (A),4 (B), 4 (C),,4 (D),,4. Which of the following statement is true for graph f() log (A) Graph shows that function is continuous (B) Graph shows that function is discontinuous (C) Graph finds for negative and positive values of (D) Graph is symmetric along -ais is,when. If f(), then, when (A) f() (B) f() (C) f() is continuous at (D) All the above are correct 4. a,when a If f() a, then, when a (A) f() is continuous at a (B) f() is discontinuous at a (C) f() (D) None of these 5. cos4, when < If f() a when,,when 6 4 is continuous at, then the value of a will be (A) 8 (B) 8 (C) 4 (D) None of these 6. 4 6,when If f(), then 6, when (A) f() is continuous at (B) f() is discountinuous at (C) f() 6 (D) None of these 7. The values of A and B such that the function f() sin, AsinB,, is continuous cos, everywhere are (A) A, B (B) A, B (C) A, B (D) A, B 47

Std. XII : Commerce (Maths I) 8. If f() k k,for <, is,for continuous at, then k (A) 4 (B) (C) (D) 9. The function f() sin is (A) Continuous for all (B) Continuous only at certain points (C) Differentiable at all points (D) None of these sin cos. The function f() sin cos is not defined at. The value of f(), so that f() is continuous at, is (A) (B) (C) (D). 7 The function f() is discontinuous for (A) only (B) and only (C),, only (D),, and other values of. The function ' f is defined by f(), if >, f() k if and, if < is continuous, then the value of k is equal to (A) (B) (C) 4 (D). cos4 Function f(), where and 8 f() k, where is a continous function at then the value of k will be? (A) k (B) k (C) k (D) None of these 4., when/ If f(), when /, then,when/ (A) /f() (B) /f() (C) f() is continuous at 5. If f() 5 for 5 and f is 7 continuous at 5, then f(5) (A) (B) 5 (C) (D) 5 Answers to Additional Practice Problems Based on Eercise.. i. Polynomial function continuous ii. Constant function continuous iii Constant function continuous iv. Polynomial function continuous v. Cosine function continuous vi. Rational function continuous for all R, ecept when 4 + 9 + vii. Addition of eponential functions continuous. i. Continuous ii. Continuous iii. Continuous iv. Discontinuous v. Continuous vi. Discontinuous vii. Continuous viii. Continuous i. Discontinuous. Continuous. i. Discontinuous ii. Continuous iii. Continuous iv. Discontinuous v. Discontinuous vi. Continuous vii. Discontinuous viii. Discontinuous 4. i. Discontinuous, removable ii. Discontinuous, irremovable iii. Discontinuous, removable iv. Discontinuous, removable v. Discontinuous, removable 5. i. (log 4) ii. a + b iii. iv. 4 6. i. ii. iii. 5 iv. v. 4 48 (D) f() is discontinuous at 7. 49

Chapter : Continuity 8. 6 9. Addition of continuous functions. f() is continuous.. f(). k 4. Discontinuous. Discontinuous 4. Continuous 5. Discontinuous 6. Continuous 7. Discontinuous 8. a, b 9. k ± 4. k 48. log a. p, q Answers to Multiple Choice Question. (B). (C). (B) 4. (C) 5. (B) 6. (C) 7. (B) 8. (D) 9. (B). (B). (B). (A). (D) 4. (B) 5. (A) 6. (B) 7. (C) 8. (C) 9. (A). (C). (C). (B). (B) 4. (D) 5. (A). a, b 4. 6 5. a 6. a, b 4 Based on Miscellaneous Eercise. i. Discontinuous ii. Continuous iii. Discontinuous iv. Continuous v. Continuous. i. Discontinuous ii. Discontinuous. i. Discontinuous, removable ii. Discontinuous, removable iii. Discontinuous, removable iv. Discontinuous, removable 4. i. (log ) ii. 5. 9 4 6. a 5, b 7. 8 8. Discontinuous 49

Board Question Paper : March 6 BOARD QUESTION PAPER : MARCH 6 Notes: i. All questions are compulsory. ii. Figures to the right indicate full marks. iii. Answer to every question must be written on a new page. iv. L.P.P. problem should be solved on graph paper. v. Log table will be provided on request. vi. Write answers of Section I and Section II in one answer book. Section I Q.. Attempt any SIX of the following: [] i. If y (sin ), find d y d () ii. If A show that A A is a scalar matri. () iii. Write the negation of the following statements: (a) y N, y + 7 (b) If the lines are parallel then their slopes are equal. () iv. The total revenue R 7 where is number of items sold. Find for which total revenue R is increasing. () v. sec Evaluate: d tan 4 () vi. Find d y d (sin 5) () vii. Discuss the continuity of function f at Where f () 4, for, for () viii. State which of the following sentences are statements. In case of statement, write down the truth value: (a) Every quadratic equation has only real roots. (b) 4 is a rational number. () Q.. (A) Attempt any TWO of the following: [6][4] i. Solve the following equations by the inversion method: + y 5 and + y () ii. iii. Find and y, if 5 4 4 y Evaluate: tan d. () () 5

Std. XII : Commerce (Maths I) (B) Attempt any TWO of the following: [8] i. (a) Epress the truth of each of the following statements using Venn diagram. () All teachers are scholars and scholars are teachers. () If a quadrilateral is a rhombus then it is a parallelogram. (b) Write converse and inverse of the following statement: If Ravi is good in logic then Ravi is good in Mathematics. (4) ii. Find the area of the region bounded by the lines y + 8, and 4. (4) iii. 9 Evaluate: d (4) Q.. (A) Attempt any TWO of the following: [6][4] i. e If f () a, for <, a, for log ( 7 ), for >, b b Is continuous at then find a and b. () ii. If the function f is continuous at, then find f() cos cos where f (), () iii. If f() 4 + + k and f(), f() 4, find f(). () (B) Attempt any TWO of the following: [8] i. Find MPC (Marginal Propensity to Consume) and APC (Average Propensity to Consume) if the ependiture E c of a person with income I is given as E c (.) I + (.75) I when I. (4) ii. Cost of assembling wallclocks is 4 and labour charges are 5. Find the number of wallclocks to be manufactured for which marginal cost is minimum. (4) iii. If cos y k, y show that y d y d k. (4) 6