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# RD Sharma Solution for Class 12 Science Mathematics Chapter 3 - Binary Operations

Our RD Sharma Textbook Solutions are considered extremely helpful for solving the tough questions which are asked in the CBSE Class 12 exam. TopperLearning Textbook Solutions are compiled by subject experts. Herein, you can find all the answers to the textbook questions for Chapter 3 - Binary Operations.

All our solutions are created in accordance with the latest CBSE syllabus, and they are amended from time to time to provide the most relevant answers. Our free RD Sharma Solutions for CBSE Class 12 Mathematics will strengthen your fundamentals in Mathematics and will help you in your attempts to score more marks in the final examination. CBSE Class 12 students can refer to our solutions any time — while doing their homework and while preparing for the exam.

Exercise/Page

## RD Sharma Solution for Class 12 Science Mathematics Chapter 3 - Binary Operations Page/Excercise 3.1

Solution 1(i)

Solution 1(ii)

Solution 1(iii)

Solution 1(iv)

Solution 1(v)

Solution 1(vi)

We have,

A b = ab + ba for all a, b, ϵ N

Let a ϵ N and b ϵ N

ab ϵ N and ba ϵ N

ab + ba ϵ N

a b ϵ N

Thus, the operation ‘’ defines a binary relation on N

Solution 1(vii)

Solution 2

Solution 3

It is given that, a*b = 2a + b - 3

now,

3*4 = 2 × 3 + 4 - 3

= 10 - 3

= 7

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

## RD Sharma Solution for Class 12 Science Mathematics Chapter 3 - Binary Operations Page/Excercise 3.2

Solution 1

Solution 2

Solution 3

Solution 4(i)

Solution 4(ii)

Solution 4(iii)

Solution 4(iv)

'' on Q defined by ab = a2 + b2 for all a, b ϵ Q

Commutativity:

For a, b ϵ Q

ab = a2 + b2 = b2 + a2 = ba

So, '' is commutative on Q.

Associativity:

For a, b, c ϵ Q

(ab) ⊙c = (a2 + b2) ⊙c = (a2 + b2)2 + c2

a⊙(b ⊙c) =a ⊙( b2 +c2)= a2 +(b2 + c2 )2

(ab) ⊙c ≠ a⊙(b ⊙c)

So, '' is not associative on Q.

Solution 4(v)

Solution 4(vi)

Solution 4(vii)

Solution 4(viii)

Solution 4(ix)

Solution 4(X)

Solution 4(xi)

Solution 4(xii)

Solution 4(xiii)

Solution 4(xiv)

'*' on Q defined by a*b = a + b - ab for all a, b ϵ Z

Commutativity:

For a, b ϵ Z

a*b = a + b - ab = b + a - ba = b*a

So, '*' is commutative on Z.

Associativity:

For a, b, c ϵ Z

(a*b) *c = (a + b - ab) *c

= a + b - ab + c + ac + bc - abc

a*(b*c )= a*( b + c - bc)

= a + b +c - bc + ab + ac + - abc

(a*b) *c ≠ a*(b*c )

So, '*' is not associative on Z.

Solution 4(xv)

'*' on Q defined by a*b = gcd (a, b) for all a, b ϵ N

Commutativity:

For a, b ϵ Q

a*b = gcd (a, b) = gcd (b, a) = b*a

So, '*' is commutative on N.

Associativity:

For a, b, c ϵ N

(a*b) *c = (gcd (a, b)) *c

= gcd (a, b, c)

=a*( gcd (b, c))

=a*(b*c)

(a*b) *c = a*(b*c )

So, '*' is associative on N.

Solution 5

Solution 6

Solution 7

Solution 8

Now consider (a * b) * c.

Thus, we have, (a * b) * c = (a + b + ab) * c

= a + b + ab + c +(a + b + ab)c

= a + b + ab + c + ac + bc + abc

= a + b + c + ab + ac + bc + abc   ---(i)

Solution 9

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 1

Solution 2

Solution 3

Solution 4

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

## Browse Study Material

TopperLearning provides step-by-step solutions for each question in each chapter in the RD Sharma textbook for class 9. Access the CBSE Class 9 Mathematics Chapter 3 - Binary Operations for free. The textbook questions have been solved by our subject matter experts to help you understand how to answer them. Our RD Sharma Textbook Solutions will help you to study and revise, and you can easily clear your fundamentals of Chapter 3 - Binary Operations.

# Text Book Solutions

CBSE XII Science - Mathematics

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