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Showing posts with label class11-maths. Show all posts
Showing posts with label class11-maths. Show all posts

# Class 11 Maths - Cartesian Products of Two sets (Important Points)

Chapter: Relations and Functions

Let A and B be two nonempty sets. Then, the Cartesian product of A and B is the set denoted by (A×B), consisting of all ordered pairs (a, b) such that a ∈ A and b ∈ B.

∴ A × B = {(a, b): a ∈ A and b ∈ B }.

② If A = ϕ or B = ϕ (empty sets), we define A × B = ϕ

③ B × A = {(b, a) : b ∈ B and a ∈ A} and A × A = {(a,b):a,b ∈ A}.

# CBSE Class 11 Maths Chapter 2: Relations and Functions (1 Mark Based Questions with Answers) Part-1

Q1: What is an ordered pair?

Answer: An ordered pair is a pair of entries in the specified order.

Q2: What is ordered 2-tuple?

Answer: Another name of an ordered pair.

Q3: If A and B are any two sets, write to represent an ordered pair of elements of A and B?

Answer: (a, b) : a ∈ A, b ∈ B

Q4: Is (a, b) = (b, a)?

Answer: No. a, b) ≠ (b, a) unless a = b

# Class 11 Maths Standard CBSE Preboard Sample Question Paper (Set-1) 2020-21

Learning is a continual process. To improve scores, the best way is to learn and practice. Solving question papers help students to evaluate their knowledge and familiarise with different types of question patterns being asked in the examination.

Eduvictors provides CBSE Previous Year Papers, Study Notes and Sample Question Papers for Class 11 Maths, Physics, Accounts, English, Hindi, Business Studies and Physical Education with/without solutions to help students in their board exam preparation.

Here attached the Maths Preboard Sample Question Paper (2020-2021) for your practice.

## Sets - NCERT Exercise 1.6 Answers

Class 11 Maths

Q1: If X and Y are two sets such that n( X ) = 17, n( Y ) = 23 and n( X ∪ Y ) = 38, find n ( X ∩ Y ).

n( X ) = 17,
n( Y ) = 23 and
n( X ∪ Y ) = 38

We know that, n(X ∪ Y) = n(X)  + n(Y) - n (X ∩ Y)
∴  38 = 17 + 23  - n (X ∩ Y)
n (X ∩ Y) = 17 + 23 - 38 = 40 - 38 = 2
n (X ∩ Y) = 2.

## Sets - NCERT Exercise 1.5 Answers

CBSE Class 11 Maths

Q1: Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 1, 2, 3, 4}, B = { 2, 4, 6, 8 } and C = { 3, 4, 5, 6 }.
Find
(i) A'
(ii) B'
(iii) (A ∪ C)'
(iv) (A ∪ B)'
(v) (A')'
(vi) (B – C)'

(i) A' =  U - A = {5,6,7,8,9}

(ii) B'  = U - B = {1,3,5,7,9}

## Class 11  Maths Chapter Sets - NCERT Exercise 1.4 (Q 9 - Q 12)

NCERT Solutions

Q9: If A = {3, 6, 9, 12, 15, 18, 21},
B = { 4, 8, 12, 16, 20 },
C = { 2, 4, 6, 8, 10, 12, 14, 16 },
D = {5, 10, 15, 20 };
find
(i) A – B
(ii) A – C
(iii) A – D
(iv) B – A
(v) C – A
(vi) D – A
(vii) B – C
(viii) B – D
(ix) C – B
(x) D – B
(xi) C – D
(xii) D – C

(i) A – B    = {3, 6, 9, 15, 18, 21}

(ii) A – C  = {3, 9, 15, 18, 21}

(iii) A – D = {3, 6, 9, 12, 18, 21}

(iv) B – A  = {4, 8, 16, 20}

(v) C – A  = {2,4,8,10,14,16}

(vi) D – A = {5,10,20}

(vii) B – C = {20}

(viii) B – D = {4,8,12,16}

(ix) C – B = {2,6,10,14}

(x) D – B  = {5,10,15}

(xi) C – D = {2,4,6,8,12,14,16}

(xii) D – C = {5,15,20}

## Laws of Set Operations Class 11 Maths - Sets

Sets under the operations of union, intersection and complement satisfy various laws. These laws collectively called Algebra of sets.

Here follows few important Laws of Set Operations

1. Idempotent Laws:
(a) A ∪ A = A
(b) A ∩ A = A

## Class 11  Maths Chapter Sets - NCERT Exercise 1.4 (Q 6 - Q 8)

NCERT Solutions

Q6: If A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find
(i) A ∩ B
(ii) B ∩ C
(iii) A ∩ C ∩ D
(iv) A ∩ C
(v) B ∩ D
(vi) A ∩ (B ∪ C)
(vii) A ∩ D
(viii) A ∩ (B ∪ D)
(ix) ( A ∩ B ) ∩ ( B ∪ C )
(x) ( A ∪ D) ∩ ( B ∪ C)

(i) A ∩ B  = {7,9,11}

(ii) B ∩ C = {11, 13}

(iii) A ∩ C ∩ D = {A ∩ C} ∩ D  = {11} ∩ {15, 17} = φ

(iv) A ∩ C = {11}

(v) B ∩ D = {7, 9, 11, 13}  ∩ {15, 17} = φ

(vi) A ∩ (B ∪ C)
= {3, 5, 7, 9, 11}  ∩ ({7, 9, 11, 13} ∪ {11, 13, 15})
= {3, 5, 7, 9, 11}  ∩ {7, 9, 11, 13, 15}
= {7, 9, 11}

(vii) A ∩ D = φ

(viii) A ∩ (B ∪ D)
= {3, 5, 7, 9, 11}  ∩ ({7, 9, 11, 13} ∪ {15, 17})
= {3, 5, 7, 9, 11}  ∩ {7, 9, 11, 13, 15, 17}
= {7, 9, 11}

## CBSE Class 11 Maths Sets NCERT Exercise 1.4 Answers

(Q1 - Q5)

Q1: Find the union of each of the following pairs of sets:
(i) X = {1,3,5} Y = {1,2,3}

(ii) A = {a,e,i,o,u}, B = {a,b,c}

(iii) A = {x : x is a natural number and multiple of 3}
B = {x : x is a natural number less than 6}

(iv) A = {x : x is a natural number and 1 < x ≤ 6 }
B = {x : x is a natural number and 6 < x < 10 }

(v) A = {1, 2, 3}, B = φ

(i) X = {1,3,5} Y = {1,2,3}
X ∪ Y = {1,2,3,5}

## Sets - Operations Of Sets

Class 11 - Maths

COMPLEMENT OF A SET

If U  be the universal set and A is the subset of U, then the complement of A with respect to U, denoted by A',
A' is defined as
A' = {x : x ∈ U and x  ∉ A }

## Venn Diagrams

Chapter: Sets
Class 11 - Maths

Most of the relationship between sets can be represented by means of diagrams.

Figures representing sets in the form of an enclosed region in the plane are called Venn Digrams. In these diagrams, the universal set is usually represented by a rectangular region and its subsets by bounded regions inside the rectangular region.

VENN DIAGRAMS IN DIFFERENT SITUATIONS

## CBSE Class 11 NCERT Exercise 1.3

Q1: Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces :

(i) { 2, 3, 4 } . . . { 1, 2, 3, 4,5 }

(ii) { a, b, c } . . . { b, c, d }

(iii) {x : x is a student of Class XI of your school}. . .{x : x student of your school}

(iv) {x : x is a circle in the plane} . . .{x : x is a circle in the same plane with radius 1 unit}

## NCERT Exercise 1.2

Q1. Which of the following are examples of the null set
(i) Set of odd natural numbers divisible by 2

(ii) Set of even prime numbers

(iii) { x : x is a natural numbers, x < 5 and x > 7 }

(iv) { y : y is a point common to any two parallel lines}

## Different Types of Sets

Class 11 - Mathematics

Empty Set:
A set is said to be empty set if it does not have any element. It is denoted by Φ or {}. Empty set is also called a null set or void set.
For example, The set of all odd natural numbers divisible by 2 is an empty set because odd natural numbers are not divisible by 2.

Finite Set:
A set which contains finite number of elements is called a finite set.
For example, The set {x: x ∈  N and x ≤ 10} is a ﬁnite set because number of elements in the set are countable.

## CBSE Class 11 - Mathematics Annual Sample Question Paper (2018-19)

Getting good marks is very important for students of class 11. For this one must practice and attempt sample question papers within time limits.

Here follows a sample question paper on Mathematics subject to assess your knowledge about the subject.

For studying you may refer following books:

## Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace

Must Read Book For Mathematics Students and Teachers
 Euclid's Window - Buy Ebook or Hard Copy

Through Euclid's Window Leonard Mlodinow brilliantly and delightfully leads us on a journey through five revolutions in geometry, from the Greek concept of parallel lines to the latest notions of hyperspace. Here is an altogether new, refreshing, alternative history of math revealing how simple questions anyone might ask about space in the living room or in some other galaxy have been the hidden engine of the highest achievements in science and technology.

Mlodinow reveals how geometry's first revolution began with a "little" scheme hatched by Pythagoras: the invention of a system of abstract rules that could model the universe. That modest idea was the basis of scientific civilization. But further advance was halted when the Western mind nodded off into the Dark Ages. Finally in the fourteenth century an obscure bishop in France invented the graph and heralded the next revolution: the marriage of geometry and number. Then, while intrepid mariners were sailing back and forth across the Atlantic to the New World, a fifteen-year-old genius realized that, like the earth's surface, space could be curved. Could parallel lines really meet? Could the angles of a triangle really add up to more or less than 180 degrees? The curved-space revolution reinvented both mathematics and physics; it also set the stage for a patent office clerk named Einstein to add time to the dimensions of space. His great geometric revolution ushered in the modern era of physics.

## Divisibility Rules of Various Numbers

(2-25)

Divisibility by 2:
A number is divisible by 2 if its unit’s digit is even or 0.

Divisibility by 3:
A number is divisible by 3 if the sum of its digits are divisible by 3.

Divisibility by 4:
A number is divisible by 4 if the last 2 digits are divisible by 4, or if the last two digits are 0’s.

Divisibility by 5:
A number is divisible by 5 if its unit’s digit is 5 or 0.

Divisibility by 6:
A number is divisible by 6 if it is simultaneously divisible by 2 and 3.

## Mathematics -Syllabus (2017-18)

### CBSE Class 11 Mathematics

Here presents the prescribed syllabus for Class 11 Mathematics for the curriculum 2017-18.

## Sets

(Unit Test Paper)
CBSE Class 11 - Mathematics

1. Write the set of all vowels in the English alphabet which precede k. [1]

2. Is pair of sets equal? Give reasons. [1]
A = {2, 3} B = x : x is solution of x 2 + 5x + 6 = 0}

3. Write the following intervals in set builder form: [1]
(-3, 0) and [6, 12]

## SETS

Class 11 - Mathematics

#### NCERT Exercise 1.1

(i) The collection of all months of a year beginning with the letter J.
(ii) The collection of ten most talented writers of India.
(iii) A team of eleven best-cricket batsmen of the world.
(iv) The collection of all boys in your class.
(v) The collection of all natural numbers less than 100.
(vi) A collection of novels written by the writer Munshi Prem Chand.
(vii) The collection of all even integers.
(viii) The collection of questions in this Chapter.
(ix) A collection of most dangerous animals of the world.

(i) Yes
The collection of all months of a year beginning with the letter J is a well-defined collection of objects since one can definitely identify a month that belongs to this collection. Hence, this collection is a set.

(ii) No
The collection of ten most talented writers of India is not a well-defined collection because the criteria for determining a writer’s talent may vary from person to person. Hence, this collection is not a set.

(iii) No
A team of eleven best cricket batsmen of the world is not a well-defined collection because the criteria for determining a batsman’s talent may vary from person to person. Hence, this collection is not a set.

(iv) Yes
The collection of all boys in your class is a well-defined collection because you can definitely identify a boy who belongs to this collection. Hence, this collection is a set.

(v) Yes
The collection of all natural numbers less than 100 is a well-defined collection because one can definitely identify a number that belongs to this collection. Hence, this collection is a set.

(vi) Yes
A collection of novels written by the writer Munshi Prem Chand is a well-defined collection because one can definitely identify a book that belongs to this collection. Hence, this collection is a set.

(vii) Yes
The collection of all even integers is a well-defined collection because one can definitely identify an even integer that belongs to this collection. Hence, this collection is a set.

(viii) Yes
The collection of questions in this chapter is a well-defined collection because one can definitely identify a question that belongs to this chapter. Hence, this collection is a set.

(ix) No
The collection of most dangerous animals of the world is not a well-defined collection because the criteria for determining the generousness of an animal can vary from person to person. Hence, this collection is not a set.
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