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PPREFACEI feel pleasure in presenting the thoroughly revised and updated edition of the book “Modern’s abc ofObjective Mathematics” for the students aspiring to compete for Karnataka CET/COMED-K. Thebook is strictly in accordance with the changing trends of different examinations.

The salient features of the revised book are :

CHAPTER OVERVIEW

QUESTIONS

HINTS AND SOLUTIONS

UNIT TEST PAPERS AND MOCK TESTS

This provides complete but brief synopsis alongwith IN FOCUS of each chapter emphasising the

principles, definitions, terms, mathematical relations ; etc. This part can serve as QUICK

REVISION OF THE CHAPTER BEFORE THE EXAMINATION.

These include a variety of objective questions in the form of Multiple Choice Questions (M.C.Q.)

have been classified into three levels as : Level-I Concept Based Questions, Level-II Applications &

Brain Teasing Questions. These have been added in accordance with the latest style and thrust of

the examinations. Questions from all types of competitive examinations have been included.

The book gives answers to all questions. Hints and Solutions of all questions are given, which is an

important feature of the book.

This is an important feature of the book. Six Unit Test Papers have been added. These will help the

student to check his/her performance after covering the units. Five Mock Test Papers covering the

complete syllabus are given at the end of the book.

–J.P. Mohindru

C M Y K

I express my sincere indebtedness to my friends, colleagues and dear students who have been

helping me in different ways during the preparation of this book. I wish to acknowledge my

special thanks and deep appreciation for their valuable suggestions.

I acknowledge with thanks to Shri Balwant Sharma (National Sales Head),

Sh. Manik Juneja National Head (Content Operation) and their efficient staff to bring out the

book. I am also indebted to Mr. S.K. Sikka (General Manager, Publication) and Mr. Ravinder

Pathania (Publication Manager) for their valuable help.

I hope that the book will be warmly received by the young scholars. It will give an excellent

guidance and will induce confidence in them to face the challenges of the examinations.

Suggestions for the improvement of the book will be gratefully acknowledged.

J. P. Mohindru

ACKNOWLEDGEMENTS

Syllabus

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CONTENTSCHAPTERS PAGES

MODULE–I

1.2.3.4.

Sets .....Relations .....Functions .....Complex Numbers .....

1/1—1/132/1—2/9

3/1—3/294/1—4/34

Quadratic Equations ..... 5/1—5/29Permutations and Combinations ..... 6/1—6/23Mathematical Induction ..... 7/1—7/4Binomial Theorem ..... 8/1—8/28Sequences and Series ..... 9/1—9/40

.....

Cartesian System of Rectangular Co-ordinates and Straight Lines ..... 10/1—10/34Family of Lines ..... 11/1—11/13Circles and Systems of Circles ..... 12/1—12/34Parabola ..... 13/1—13/18Ellipse ..... 14/1—14/21Hyperbola ..... 15/1—15/16

.....

Statistics ..... 16/1—16/18Trigonometric Ratios, Identities and Equations ..... 17/1—17/40Solution of Triangles ..... 18/1—18/23Heights and Distances ..... 19/1—19/17Mathematical Reasoning ..... 20/1—20/8

5.6.7.8.9.

Assertion-Reason & Column Matching Type Questions I A-1/1—A-1/10

10.11.12.13.14.15.

Assertion-Reason & Column Matching Type Questions II A-2/1—A-2/5

16.17.18.19.20.

Assertion-Reason & Column Matching Type Questions III ..... A-3/1—A-3/4

(ALGEBRA)

(TWO DIMENSIONAL GEOMETRY )

(STATISTICS, TRIGONOMETRY AND MATHEMATICAL REASONING)

.....

.....

Unit Test Paper No. 1 U-I/1—U-I/2

Unit Test Paper No. 2 U-II/1—U-II/3

Unit Test Paper No. 3 U-III/1—U-III/2

.....

MODULE–II

MODULE–III

CHAPTERS PAGES

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MODULE–IV

MODULE–V

MODULE–VI

.....

(INVERSE TRIGONOMETRIC FUNCTIONS, DETERMINANTS & MATRICES)

(CALCULUS)

(THREE DIMENSIONAL GEOMETRY, VECTORS AND PROBABILITIES)

21.

Assertion-Reason & Column Matching Type Questions IV ..... A-4/1—A-4/5

24.25.26.27.28.29.30.31.

Assertion-Reason & Column Matching Type Questions V ..... A-5/1—A-5/12

32.33.34.

Assertion-Reason & Column Matching Type Questions VI ..... A-6/1—A-6/6

Inverse Trigonometric Functions ..... 21/1-21/1522/1-22/31

Real Numbers ..... 24/1—24/6Limit and Continuity ..... 25/1—25/37Differentiability and Differentiation ..... 26/1—26/34Application of Derivatives ..... 27/1—27/33Indefinite Integrals ..... 28/1—28/29Definite Integrals ..... 29/1—29/40Areas Under Curves ..... 30/1—30/20Differential Equations ..... 31/1—31/21

.....

Three Dimensional Geometry ..... 32/1—32/23Vector Algebra ..... 33/1—33/34Probability ..... 34/1—34/27

.....

Mock Tests (For Revision) ..... M-I/1—M-IV/20

Determinants .....Matrices ..... 23/1-23/28

22.23.

MOCK TESTS

Unit Test Paper No. 4 U-IV/1—U-IV/3

Unit Test Paper No. 5 U-V/1—U-V/3

Unit Test Paper No. 6 U-VI/1—U-VI/3

C M Y K

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SETS 1/1

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����� IMPORTANT FACTS, TERMS AND FORMULAE

1. SETS

Set. A set is a collection of well defined and well distinguished objects of ourperception or thought.

The words ‘well defined objects’ imply that we must be given a rule with the help ofwhich we should readily be able to say whether a particular object ‘belongs to’ the set ornot. The words ‘well distinguished objects’ imply that if the objects of the collection benamed, then in doing so, the number of objects will not increase.

The sets are usually denoted by capital letters of English alphabet viz. A, B, C, .......

2. ELEMENTS

The objects, which constitute the set, are said to be elements of the set.These are also known as members or points of the set. The elements are usually

denoted by small letters of English alphabet viz. a, b, c, ........ (i) If a is an element of the set A, we write it as a ∈ A and is read as ''a belongs to A''.(ii) If a is not an element of the set A, we write it as a ∉ A and is read as ''a does not belong to A''.

3. REPRESENTATION OF SETS

There are two methods to represent a set.(a) Roster/Tabulation Method. In this method, the set is represented by listing all its elements, separating the elements

by commas and enclosing them in curvilinear brackets.(b) Defining Property Method. In this method, the set is represented by specifying the common property of the elements.Thus the set A is represented by A = {a : P(A) is true}.Here ‘a’ stands for ‘an arbitrary element of the set’ and ( : ) stands for ‘such that’and P(A) stands for ‘common property’.

4. FINITE AND INFINITE SETS

(a) Finite Set. A set is said to be finite if it has finite number of elements.(b) Infinite Set. A set is said to be infinite if it has an infinite number of elements.(c) Order of a finite set. Order of a finite set is the number of elements it contains.The order of a finite set A is denoted by O(A).

5. EMPTY SET

A set having no element is said to be an empty set.It is also called Null set or Void set.The empty set is denoted by φ or { }.

6. SINGLETON SET

A set having only one element is said to be a singleton set.

7. SUB-SETS

(a) Subset. Let A and B be two sets. Then the set A is said to be a subset of the set B ifeach element of A is also an element of B.

Symbolically, we write it as A ⊆ B.Here B is superset of A and is written as B ⊇ A.

STANDARD SETS

N, set of all natural numbersW, set of all whole numbersI, set of all integersQ, set of all rational numbersR, set of all real numbersC, set of all complex numbers.

Order of an infinite set is notdefined.

{φ} is not an empty set but isa singleton set.

1/1

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(b) Proper Subset. A set A is a proper subset of B if and only if each elementof A is in B and there is at least one element in B, which is not in A.

Symbolically, if A is a proper subset of B, then A ⊆ B and A ≠ B or A ⊂ B.

8. COMPARABLE SETS

Two sets are said to be comparable iff either A ⊂ B or B ⊂ A.

9. EQUAL AND EQUIVALENT SETS

(a) Equal Sets. Two sets A and B are said to be equal (written as A = B) iff A ⊂ Band B ⊂ A.

Two sets A and B are said to be equal if they have exactly same elements.(b) Equivalent Sets. Two sets are said to be equivalent if they have same number of

elements.

10. FAMILY OF SETS

A set is said to be family of sets if its elements are also sets.This is also known as Set of sets.If A = {a, b}, then S = {φ, {a}, {b}, {a, b}} is the set of sets.

11. POWER SET

The set of all possible subsets of a set A is said to be the power set of A and is denotedby P (A).

If A = P {a, b, c}, then P (A) = {φ,{a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}}.

12. UNIVERSAL SET

The main set under discussion or the set containing all possible values in the givenframe of reference is said to be universal set and is denoted by U or E or X.

13. OPERATIONS IN SETS

(a) Union of Sets. (I) Let A and B be two

sets. The union of A and B (denoted by A ∪ B)

is the set of all those elements, which are

either in A or in B or in both.

Symbolically,

A ∪ B = {x : x ∈ A or x ∈ B}.

(II) Let A1, A2, ......, An be n (≥ 2) sets. Then the union of these (denoted by Aii

n)

= 1∪

is the set of all those elements which are in Ai (1 ≤ i ≤ n) for at least one value of i.

(b) Intersection of Sets. (I) Let A and

B be two sets. The intersection of A and B

(denoted by A ∩ B) is the set of all those

elements which are in both A and B.

(II) Let A1, A2,...... , An be n (≥ 2) sets. Then the intersection of these (denoted by

A ii

n)

= 1∩ is the set of all those elements which are in Ai (1 ≤ i ≤ n) for each i.

1. Each set is a subset of itself.2. Empty set is a subset of eachset.3. No. of subsets of a set havingn elements = 2n.

4. No. of proper subsets of a sethaving n elements = 2n – 2.

Equal sets are equivalent setsbut

Equivalent sets may not beequal sets.

Universal set is not unique.

x ∈ A ∪ B ⇔ x ∈ A or x ∈ Bx ∉ A ∪ B ⇔ x ∉ A and x ∉ B.

x ∈ A ∩ B ⇔ x ∈ A and x ∈ Bx ∉ A ∩ B ⇔ x ∉ A or x ∉ B.

SETS 1/3

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14. FUNDAMENTAL RESULTS

(i) Identity Laws. A ∪ φ = A, A ∩ φ = φ(ii) Idempotent Laws. A ∪ A = A, A ∩ A = A

(iii) Commutative Laws. A ∪ B = B ∪ A, A ∩ B = B ∩ A(iv) Associative Laws. A ∪ (B ∪ C) = (A ∪ B) ∪ C

A ∩ (B ∩ C) = (A ∩ B) ∩ C(v) Distributive Laws. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

15. DISJOINT SETS

Two sets are said to be disjoint iff they

have no common element.

Let A and B be two sets.Here A ∩ B = φ.

The sets A1, A2, ...., An (n ≥ 2) are said to be pair-wise disjoint sets if Ai ∩ Aj = φ,1 ≤ i, j ≤ n, i ≠ j.

16. DIFFERENCE OF SETS

Let A and B be two sets. Then (A – B) is the set of those elements of the set A whichare not in the set B.

Symbolically, A – B = {x : x ∈ A and x ∉ B}and B – A = {x : x ∈ B and x ∉ A}.

17. SYMMETRIC DIFFERENCE OF SETS

Let A and B be two sets. Then their

symmetric difference is the union of the sets

A – B and B – A. This is denoted by A Δ B.

Symbolically,

A Δ B = {x : x ∈ A – B or x ∈ B – A}.

18. COMPLEMENT OF A SET

Let X be the universal set and A be anyset. Then the complement of the set A is the setof all those elements of X, which are not in theset A.

This is denoted by Ac or A' or X – A.Symbolically, Ac = {x : x ∈ X and x ∉ A}.

A and B are disjoint⇔ A ∩ B = φ.

x ∈ (A – B) ⇔ x ∈ A and x ∉ Bx ∉ (A – B) ⇔ x ∉ A or x ∈ B.

x ∈ Ac ⇔ x ∉ Ax ∉ Ac ⇔ x ∈ A.

1/4 MODERN’S abc OF OBJECTIVE MATHEMATICS (KARNATAKA CET/COMED)

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19. FUNDAMENTAL PROPERTIES

(i) Xc = φ, φc = X

(ii) (Ac)c = A

(iii) If A ⊆ B, then Bc ⊆ Ac

(iv) A ∪ Ac = X and A ∩ Ac = φ(v) De-Morgan’s Laws. (A ∪ B)c = Ac ∩ Bc

(A ∩ B)c = Ac ∪ Bc

(vi) n (A ∆ B) = n (A) + n (B) – 2n (A ∩ B).

20. CARDINAL NUMBERS

In a finite set A, the number of distinct elements is called the cardinal numberand is denoted by n (A).

If A = {2, 4, 6, 8}, then n (A) = 4.

21. USE OF SETS IN PRACTICAL PROBLEMS

If A, B, C are finite sets of n elements each and U be the finite universal set, then :

(i) n (Ac) = n (U) + n (A)

(ii) n (A ∪ B) = n (A) + n (B) ; if A, B are disjoint

(iii) n (A ∪ B) = n (A) + n (B) – n (A ∩ B) ; if A, B are not disjoint

(iv) n (A – B) = n (A) – n (A ∩ B)

(v) n (Ac ∪ Bc) = n (U) – n (A ∩ B)

(vi) n (Ac ∩ Bc) = n (U) – n (A ∪ B)

(vii) n (A ∩ Bc) = n (A) – n (A ∩ B)

(viii) n (B ∩ Ac) = n (B) – n (A ∩ B)

(ix) n (A ∪ B) = n (A ∩ Bc) + n (B ∩ Ac) + n (A ∩ B)

(x) n (A ∪ B ∪ C) = n (A) + n (B) + n (C) – n (A ∩ B) – n (B ∩ C)

– n (A ∩ C) + n (A ∩ B ∩ C).

22. ORDERED PAIR

An ordered pair is a pair of entries in the specified order.In the ordered pair (a, b), a is the first element and b the second element.

23. CARTESIAN PRODUCT/DIRECT PRODUCT OF SETS

(a) The set of all ordered pairs of elements (a, b) ; a ∈ A, b ∈ B is called Cartesianproduct of two sets A and B and is denoted by A × B.

Symbolically, A × B = {(a, b) : a ∈ A, b ∈ B}.(b) The cartesian product of n (> 2) sets A1, A2, ......, An is the set of all ordered n-tuples

(a1, a2, ...... an), where ai ∈ Ai (1 ≤ i ≤ n) and is denoted by A1 × A2 × ...... × An or A ii

n

=∏

1.

Symbolically, 1 2= 1

A {( , , ..............., ) : A , 1 }.n

i n i ii

a a a a i n= ∈ ≤ ≤∏

24. FUNDAMENTAL RESULTS

(i) A × B ≠ B × A

(ii) A × φ = φ × A = φ(iii) n (A × B) = n (B × A) = n(A) × n(B)

(iv) If A ⊆ B, C ⊆ D, then A × C ⊆ B × D

(v) n (A1 × A2 × ...... × An) = n (A1) × n (A2) ........ × n ( An ).

(a, b) ≠ (b, a) unless a = b.

A × B ≠ B × A, in general.

A Δ B = B Δ A.

MOD ABC Of Objective MathematicsCET /COMED-K (E) Karnataka

Publisher : MBD GroupPublishers

ISBN : 9789351845096Author : J. P. Mahindru,Bharat Mahindru

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