Perfect Mathematics - II

Transcription

1 Std. I Sci. Perfect Mathematics - II Edition: July 2014 Mr. Vinodkumar J. Pandey.Sc. (Mathematics) G. N. Khalsa College, Mumbai Dr. Sidheshwar S. ellale M.Sc.,.Ed., PhD. (Maths) Department of Mathematics, Dayanand Science College, Latur. Mr. Vinod Singh M.Sc. (Mathematics) Mumbai University Published by Target PULICTIONS PVT. LTD. Shiv Mandir Sabhagriha, Mhatre Nagar, Near LIC Colony, Mithagar oad, Mulund (E), Mumbai Off.Tel: mail@targetpublications.org Website: Price : ` 260/- Printed at: India Printing Works 42, G.D. mbekar Marg, Wadala, Mumbai Target ll rights reserved Publications Pvt. Ltd. No part of this book may be reproduced or transmitted in any form or by any means, C.D. OM/udio Video Cassettes or electronic, mechanical including photocopying; recording or by any information storage and retrieval system without permission in writing from the Publisher.

2 Written according to the New Text book ( ) published by the Maharashtra State oard of Secondary and Higher Secondary Education, Pune. Std. I Sci. Perfect Mathematics - II Salient Features : Exhaustive coverage of entire syllabus. Covers answers to all textual and miscellaneous exercises. Precise theory for every topic. Neat, labelled and authentic diagrams. Written in systematic manner. Self evaluative in nature. Practice problems and multiple choice questions for effective preparation. TEID : 752

3 PEFCE In the case of good books, the point is not how many of them you can get through, but rather how many can get through to you. Std. I Sci. : PEFECT MTHEMTICS - II is a complete and thorough guide critically analysed and extensively drafted to boost the students confidence. The book is prepared as per the Maharashtra State board syllabus and provides answers to all textual questions. t the beginning of every chapter, topic wise distribution of all textual questions including practice problems has been provided for simpler understanding of different types of questions. Neatly labelled diagrams have been provided wherever required. and Multiple Choice Questions help the students to test their range of preparation and the amount of knowledge of each topic. Important theories and formulae are the highlights of this book. The steps are written in systematic manner for easy and effective understanding. 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 est of luck to all the aspirants! No. Topic Name Page No. 1 Sets, elations and Functions 1 2 Logarithms 39 3 Complex Numbers 67 4 Sequence and Series Permutations and Combinations Method of Induction and inomial Theorem Limits Differentiation Integration Statistics (Measures of Dispersion) Probability 389 Log Tables Logarithms 439 ntilogarithms 441

4 Chapter 01: Sets, elations and Functions 01 Sets, relations and functions Type of Problems Exercise Q. Nos. To describe sets in oster form 1.1 Q.1 (i.,, i) (ased on Exercise 1.1) Q.1 (i.,, i) 1.1 Q.2 (i.,, i), Q.12(i. to iv.) To describe sets in Set-uilder form (ased on Exercise 1.1) Miscellaneous (ased on Miscellaneous) Q.2 (i.,, i) Q.11 (i. to iv.) Q.1 (i.,, i) Q.1(i., ) Operations on Sets Ordered Pairs 1.1 Q.3 to Q.11 Q.13 (i. to iv.) (ased on Exercise 1.1) Q.3 to Q.10 Q.12 (i., ) Miscellaneous Q.2, 3, 4 (ased on Miscellaneous) Q.2, 3, Q.1, 2, 6, 11 (ased on Exercise 1.2) Q.1, 2, 5, Q.3, 4, 5 Cartesian product of two Sets (ased on Exercise 1.2) Miscellaneous (ased on Miscellaneous) Q.3, 4 Q.5 (i., ) Q Q.7, 8, 9, 10 To find domain and range of a given relation (ased on Exercise 1.2) Q.6, 7, 8 Miscellaneous Q.6, 7 (ased on Miscellaneous) Q.6, 7 1 1

5 Std. I Sci.: Perfect Maths - II 1.3 Q.9, 10 Types of Functions (ased on Exercise 1.3) Q.9, 10 Miscellaneous Q.9, 10, 22 (ased on Miscellaneous) Q.9, 10, Q.3, 5, 6, 7 To find values of the given function (ased on Exercise 1.3) Q.3, 4, 6, 7 Miscellaneous Q.13 to Q.17 (ased on Miscellaneous) Q.12 to Q.16 Operations on functions 1.3 Q.11 (i. to i) (ased on Exercise 1.3) Q.11 (i. to i) 1.3 Q.12 to Q.15 Composite function (ased on Exercise 1.3) Q.12 to Q.15 Miscellaneous Q.18 to Q.21 (ased on Miscellaneous) Q.17 to Q Q.16 Inverse function Q.16 (ased on Exercise 1.3) Miscellaneous Q.11, 12 (ased on Miscellaneous) Q Q.1, 2, 4, 8 To find domain and range of a given function (ased on Exercise 1.3) Q.1, 2, 5, 8 Miscellaneous Q.8, 23, 24 (ased on Miscellaneous) Q.8 2

6 Syllabus: Sets, Subset, Intervals, Types of sets, Power set, Ordered pair, Cartesian products of two sets, Definition of elation, Domain, Co-domain and ange of elation, Types of elation, Definition of function, Types of functions, Operations on functions, Composite function, Inverse function, inary operation, eal valued functions of the real variable. Introduction ll basic concepts of modern mathematics are based on set theory. The concepts involving logic can be explained more easily with the help of set theory. It plays a crucial role in the study of relations, functions, probability and is used extensively in various other branches of mathematics. We shall briefly revise and study some more concepts about sets. Sets set is a well-defined collection of objects. These objects may be actually listed or may be specified by a rule. set is usually denoted by the capital letters,, C, N,, etc. Each object in a set is called an element or a member of the set and is denoted by the small letters a, b, c, etc. If x is an element of set, then we write it as x and read it as x belongs to and if y is not an element of set, then we write it as y and read it as y does not belong to. If = {2, 4, 6, 8}, then 4, 7, 8, 10 The set of natural numbers, whole numbers, integers, rational numbers and real numbers are denoted by N, W, I, Q and respectively. Methods of epresentation of Sets There are two methods of representing a set which are as follows: i. oster method (Listing method): In this method all the elements are listed or tabulated. The elements are separated by commas and are enclosed within two braces(curly brackets). Chapter 01: Sets, elations and Functions The set of all positive even integers less than 9 can be written as = {2, 4, 6, 8}. 3 3 Set-uilder method: In this method, the set is described by the characteristic property of its elements. In general, if all the elements of set satisfy some property P, then write in set-builder notation as = {x/x has property P} and read it as is the set of all x such that x has the property P. Let = {3, 4, 5, 6, 7, 8} Using the set-builder method, can be written as = {x/x N, 3 x 8} Since = {3, 4, 5, 6, 7, 8} can also be stated as the set of natural numbers from 3 to 8 including 3 and 8. Some standard sets are as follows: N = set of all natural numbers = {1, 2, 3,..} Z or I = set of all integers = {. 3, 2, 1, 0, 1, 2, 3 } Q = set of all rational numbers p = /p,q Z,q 0 q Subset: Set is called a subset of set, if every element of set is also an element of set i.e., if x, then x. We denote this relation as and read it as is a subset for. It s clear that i. Every set is a subset of itself i.e.,. n empty set φ is a subset of every set. If = {2, 4, 6, 8} and = {2, 4, 6, 8, 10, 12}, then. If, then is called a superset of, denoted by. Proper subset: If every element of set is an element of set and contains at least one element which is not in, then is said to be a proper subset of and it is denoted as.

7 If = {2, 3, 5, 6}, = {1, 2, 3, 4, 5, 6, 7} Here every element of set i.e., 2, 3, 5, 6 is an element of set. ut contains elements 1, 4, 7 which are not in. Hence in the case we say that is a proper subset of and is denoted by. Intervals Open interval If p, q and p < q, then the set {x/x, p < x < q} is called open interval and is denoted by (p, q). Here all the numbers between p and q (p, q) except p and q. p x q (p, q) = {x/x, p < x < q} Closed interval If p, q and p < q, then the set {x/x, p x q} is called closed interval and is denoted by [p,q]. Here all the numbers between p and q [p, q] including p, q. [p, q] = {x/x, p x q} Semi-closed interval If p, q and p < q, then the set {x/x, p x < q} is called semi-closed interval and is denoted by [p, q). p x q [p, q) = {x/x, p x < q} [p, q) includes p but excludes q. Semi-open interval If p, q and p < q, then the set {x/x, p < x q} is called semi-open interval and is denoted by (p, q]. p x q (p, q] = {x/x, p < x q} (p, q] excludes p but includes q. emarks: i. Set of all real numbers > p i.e., (p, ) = [x/x, x > p} 4 p p x q Set of all real numbers p i.e., [p, ) = {x/x, x p} Set of real numbers < q i.e., (, q) {x/x, x < q} q Set of real numbers q i.e., (, q] = {x/x, x q} Std. I Sci.: Perfect Maths - II i Set of all real numbers is (, ) = (, ) = {x/x, < x < } Types of sets p Empty set: set which does not contain any element is called an empty set and it is denoted by φ or { }. It is also called null set or void set. = {x/x N, 3 < x < 4} = {x/x is a positive integer < 1} Note: The set {0} and {φ} are not empty sets as they contain one element, namely 0 and φ respectively. Singleton set: set which contains only one element is called a singleton set. = {5}, = {3}, = {x/x N, 1 < x < 3} The set = set of all integers which are neither positive nor negative is a singleton set since = {0} Finite set: set which contains countable number of elements is called a finite set. = {a, b, c} = {1, 2, 3, 4, 5} C = {a, e, i, o, u} q

8 Infinite set: set which contains uncountable number of elements is called an infinite set. N = {1, 2, 3, 4 } Z = { 3, 2, 1, 0, 1, 2, 3,..} Note: i. n empty set is a finite set. N, W, I, Q and are infinite sets. Equal sets: Two sets and are said to be equal if they have the same elements and we denote this as =. From this definition it follows that two sets and are equal if and only if and If = {1, 2, 3, 4}, = {2, 4, 1, 3}, then =. Equivalent sets: Two sets and are said to be equivalent, if they contain the same number of elements and we denote it as. If = {1, 2, 3, 4, 5}, = {a, b, c, d, e}, then n() = n() and are equivalent sets. Note: Equal sets are always equivalent but equivalent sets need not be equal. Universal set: non-empty set of which all the sets under consideration are subsets, is called a universal set. It is usually denoted by or U. If = {1, 2, 3, 4}, = {2, 8, 13, 15} and C = {1, 2, 3,, 50} are sets under consideration, then the set N of all natural numbers can be taken as the universal set. Venn diagram: set is represented by any closed figure such as circle, rectangle, triangle, etc. The diagrams representing sets are called venn diagrams. i. = {4, 6, 9} = {a, b, c, d, e, f} = {b, e, f} b.e.f.a.d.c Operations on sets Chapter 01: Sets, elations and Functions Complement of a set: Let be a subset of a universal set then the set of all those elements of which do not belong to is called the complement of set and it is denoted by or c. Thus, = {x/x, x } The shaded region in the above figure represents. Let = {1, 2, 3, 4, 5, 6, 7, 8, 9} be an universal set and = {1, 3, 5, 6, 8}. Then = {2, 4, 7, 9} Properties: If is the universal set and,, then i. ( ) = = φ i φ = iv. = φ v. = vi. If, then. Union of sets: If and are two sets, then the set of those elements which belong to or to or to both and is called the union of the sets and and is denoted by. i.e., = {x/x or x } The shaded portion in the below venn diagram represents. i. If = {1, 2, 3, 4}, = {2, 4, 6, 8}, then = {1, 2, 3, 4, 6, 8} If is the set of all odd integers and is the set of all even integers, then is the set of all integers. 5 5

9 Properties: If,, C are any three sets, then i. φ = = i = (Commutative law) iv. ( ) C = ( C) (ssociative law) v. = (Idempotent law) vi. If, then = v, Intersection of sets If and are two sets, then the set of those elements which belong to both and i.e., which are common to both and is called the intersection of the sets and and is denoted by. Thus, = { x/x and x } The shaded portion in the below venn diagram represents. If = {1, 2, 3, 4, 5}, = {1, 3, 5, 7, 9}, then = {1, 3, 5} Properties: If,, C are any three sets, then i. φ = φ = i = (Commutative law) iv. ( ) C = ( C) (ssociative law) v. = (Idempotent law) vi. If, then = v, Std. I Sci.: Perfect Maths - II = φ Difference of sets: If and are two sets then the set of all the elements of which are not in is called difference of sets and and is denoted by. Thus, = {x/x and x } Similarly, = {x/x and x } In the below venn diagrams shaded region represents and. If = {1, 2, 3, 4, 5, 6}, = {2, 4, 6, 8}, then = {1, 3, 5} and = {8} Distributive Properties of union and intersection If a, b, c, then a (b + c) = (a b) + (a c) This is known as distributive property of multiplication over addition. In set theory, the operation of union and intersection of sets are both distributive over each other i.e., If,, C are any three sets, then i. ( C) = ( ) ( C) ( C) = ( ) ( C) We verify these distributive laws using Venn diagrams shown below. The shaded portion in each figure shows the set obtained by performing the operation given below the figure. i. C = C Disjoint sets: Two sets and are said to be disjoint, if they have no element in common i.e., = φ. If = {2, 4, 6} and = {3, 5, 7}, then = φ and are disjoint sets. The venn diagram of the disjoint sets and is shown below: 6 ( C) ( ) ( C) C = C ( C) ( ) ( C)

10 De-Morgan s laws If and are two subsets of a universal set, then i. ( ) = ( ) = We verify these laws using Venn diagrams shown below. The shaded portion in each figure shows the set obtained by performing the operation below the figure: i. = ( ) = Chapter 01: Sets, elations and Functions Exercise Describe the following sets in oster form: i. {x/x is a letter of the word MIGE } 1 9 x/ xisanintegerand < x < 2 2 i {x/x = 2n, n N} i. Let = {x/x is a letter of the word MIGE } = {M,,, I, G, E} 1 9 Let = x/ xisan integer and < x < 2 2 = {0, 1, 2, 3, 4} i Let C = {x/x = 2n, n N} C = {2, 4, 6, 8,.} ( ) Number of elements in a set Let be a set. Then the total number of elements in it is denoted by n(). Let = {8, 9, 10, 11, 12} n () = 5 The number of elements in the empty set φ is zero. i.e., n (φ) = 0 esults: For given sets,, C i. n( ) = n() + n() n( ) When and are disjoint sets, then n( ) = n() + n() i n( ) + n( ) = n() iv. n( ) + n( ) = n() v. n( ) + n( ) + n( ) = n( ) vi. n( C) = n() + n() + n(c) n( ) n( C) n( C) + n( C) Power set The set of all subsets of set is called the power set of and it is denoted by P(). If = {a, b, c}, then P() ={φ, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}} Note: If contains n elements, then the power set of i.e., P() contains 2 n elements. 2. Describe the following sets in Set-uilder form: i. {0} {0, ± 1, ± 2, ± 3} i ,,,,,, i. Let = {0} 0 is a whole number but it is not a natural number = {x / x W, x N} Let = {0, ± 1, ± 2, ± 3} is the set of elements which belongs to Z from 3 to 3 = {x / x Z, 3 x 3} i Let C =,,,,,, n C = x/x =,n N,n 7 2 n If = {x / 6x 2 + x 15 = 0} = {x / 2x 2 5x 3 = 0} C = {x / 2x 2 x 3 = 0}, then find i. C C 2 x/6x + x 15= 0 = { } 6x 2 + x 15 = 0 6x x 9x 15 = 0 7 7

11 2x(3x + 5) 3(3x + 5) = 0 (3x + 5) (2x 3) = 0 5 x = 3 or x = =, 3 2 = {x/2x 2 5x 3 = 0} 2x 2 5x 3 = 0 2x 2 6x + x 3 = 0 2x(x 3) + 1(x 3) = 0 (x 3)(2x + 1) = 0 x = 3 or x = =,3 2 C = {x/2x 2 x 3 = 0} 2x 2 x 3 = 0 2x 2 3x + 2x 3 = 0 x(2x 3) + 1(2x 3) = 0 (2x 3) (x + 1) = 0 x = 3 or x = C = 1, 2 Thus, i. C =,,3 1, =, 1,,, C = { } 4. If,, C are the sets of the letters in the words college, marriage and luggage respectively, then verify that [ ( C)] = [( ) ( C)]. = {c, o, l, g, e} = {m, a, r, i, g, e,} C = {l, u, g, a, e} C = {m, a, r, i, g, e, l, u} ( C) = {c, o} = {c, o, l} C = {c, o} [( ) ( C)] = {c, o} = ( C) [ ( C)] = [( ) ( C)] 8 Std. I Sci.: Perfect Maths - II 5. If = {1, 2, 3, 4}, = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} and universal set = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then verify the following: i. ( C) = ( ) ( C) ( C) = ( ) ( C) i ( ) = iv. ( ) = v. = ( ) ( ) vi. = ( ) ( ) v n( ) = n() + n() n( ) = {1, 2, 3, 4}, = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} ={1, 2, 3, 4, 5, 6, 7, 8, 9, 10} i. ( C) = {4, 5, 6} ( C)= {1, 2, 3, 4, 5, 6} ( ) = {1, 2, 3, 4, 5, 6} ( C) = {1, 2, 3, 4, 5, 6, 7, 8} ( ) ( C) = {1, 2, 3, 4, 5, 6} ( C) = ( ) ( C) ( C) = {3, 4, 5, 6, 7, 8} ( C) = {3, 4} = {3, 4} C = {4} ( ) ( C) = {3, 4} ( C) = ( ) ( C) i = {1, 2, 3, 4, 5, 6} ( ) ={7, 8, 9, 10} = {5, 6, 7, 8, 9, 10}, = {1, 2, 7, 8, 9, 10} = {7, 8, 9, 10} ( ) = ( ) iv. = {3, 4} ( ) = {1, 2, 5, 6, 7, 8, 9, 10} = {5, 6, 7, 8, 9, 10} = {1, 2, 7, 8, 9, 10} = {1, 2, 5, 6, 7, 8, 9, 10} ( ) = v. = {3, 4} = {1, 2} ( ) ( ) = {1, 2, 3, 4} = ( ) ( ) vi. = {3, 4} = {5, 6} ( ) ( ) = {3, 4, 5, 6} = ( ) ( )

12 v = {1, 2, 3, 4}, = {3, 4, 5, 6}, = {3, 4}, = {1, 2, 3, 4, 5, 6} n() = 4, n() = 4, n( ) = 2, n( ) = 6 n() + n() n( ) = = 6 n( ) = n() + n() n( ) Chapter 01: Sets, elations and Functions No. of students who failed in IEEE or IIT entrance = n( C) = n() + n(c) n( C) = = If and are subsets of the universal set and n() = 50, n() = 35, n() = 20, n( ) = 5, find i. n( ) n( ) i n( ) iv. n( ). n() = 50, n() = 35, n() = 20, n( ) = 5 i. n( ) = n() [n( ) ] = n() n( ) = 50 5 = 45 n( ) = n() + n() n( ) = = 10 i n( ) = n() n( ) = = 10 iv. n( ) = n() n( ) = = In a class of 200 students who appeared certain examinations, 35 students failed in MHT-CET, 40 in IEEE and 40 in IIT entrance, 20 failed in MHT-CET and IEEE, 17 in IEEE and IIT entrance, 15 in MHT-CET and IIT entrance and 5 failed in all three examinations. Find how many students i. did not fail in any examination. failed in IEEE or IIT entrance. Let = set of students who failed in MHT-CET = set of students who failed in IEEE C = set of students who failed in IIT entrance = set of all students n() = 200, n() = 35, n() = 40, n(c) = 40, n( )= 20, n( C) = 17, n( C) = 15, n( C) = 5 i. n( C) = n() + n() + n(c) n( ) n( C) n( C) + n( C) = = 68 No. of students who did not fail in any exam = n() n( C) = = From amongst 2000 literate individuals of a town, 70% read Marathi newspapers, 50% read English newspapers and 32.5% read both Marathi and English newspapers. Find the number of individuals who read i. at least one of the newspapers. neither Marathi nor English newspaper. i only one of the newspapers. Let M = set of individuals who read Marathi newspapers E = set of individuals who read English newspapers = set of all literate individuals n() = 2000, n(m) = = n(e) = = n(m E) = = n(m E) = n(m) + n(e) n(m E) = = 1750 i. No. of individuals who read at least one of the newspapers = n(m E) = i No. of individuals who read neither Marathi nor English newspaper = n(m E ) = n(m E) = n() n(m E) = = 250 No. of individuals who read only one of the newspaper = n(m E ) + n(m E) = n(m E) n(m E) = = 1100