RD SHARMA Solutions for Class 12-science Maths Chapter 3 - Binary Operations

Chapter 3 - Binary Operations Exercise Ex. 3.1

Question 1(i)

Solution 1(i)

Question 1(ii)
Solution 1(ii)
Question 1(iii)
Solution 1(iii)

Question 1(iv)
Solution 1(iv)
Question 1(v)
Solution 1(v)
Question 1(vi)

Determine whether the following operation defines a binary operation on the

apostrophe circled dot apostrophe space o n space N space d e f i n e d space b y space a space circled dot space b space equals space a to the power of b space plus space b to the power of a space f o r space a l l space a comma space b space element of space N

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

Question 1(vii)
Solution 1(vii)
Question 2

Solution 2

Question 3

Let * be a binary opeartor o the set I of integers, defined by a*b = 2a + b - 3.

find the value of 3*4

Solution 3

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

now,

3*4 = 2 × 3 + 4 - 3

      = 10 - 3

      = 7

Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9

The binary operation & : R × R → R is defined as a * b = 2a + b.

Find (2*3)*4.

Solution 9

Question 10

Solution 10

Chapter 3 - Binary Operations Exercise Ex. 3.2

Question 1
Solution 1

Question 2
Solution 2
Question 3
Solution 3
Question 4(i)

Solution 4(i)

Question 4(ii)
Solution 4(ii)
Question 4(iii)
Solution 4(iii)
Question 4(iv)

Check the commutativity and associativity of each of the following binary operations:

 

 

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

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.

 

 

Question 4(v)
Solution 4(v)
Question 4(vi)
Solution 4(vi)
Question 4(vii)
Solution 4(vii)

Question 4(viii)

Check the commutativity and associativity of the following binary operation :

'*' on R defined by a*b = a + b - 7 for all a, b ε Q

Solution 4(viii)

Question 4(ix)

Check the commutativity and associativity of the following binary operation :

'*' on Q defined by a*b = (a - b)2 for all a, b ε Q.

Solution 4(ix)

Question 4(x)

Check the commutativity and associativity of the following binary operation :

'*' on Q defined by a*b = ab + 1 for all a, b ε Q

 

Solution 4(x)

Question 4(xi)

Check the commutativity and associativity of the following binary operation :

'*' on N, defined by a*b = ab for all a, b, ε N.

Solution 4(xi)

Question 4(xii)

Check the commutativity and associativity of the following binary operation :

'*' on Z, a*b = a - b for all a, b, ε Z.

Solution 4(xii)

Question 4(xiii)

Check the commutativity and associativity of the following binary operation :

apostrophe asterisk times apostrophe space on space straight Q comma space defined space by space straight a asterisk times straight b space equals space ab over 4 space for space all space straight a comma space straight b comma space element of space straight N.

Solution 4(xiii)

Question 4(xiv)

Check the commutativity and associativity of each of the following binary operations:

 

 

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

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.

 

Question 4(xv)

Check the commutativity and associativity of each of the following binary operations:

 

 

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

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.

 

Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8

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)

Question 9
Solution 9
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14

Let S be the set of all rational numbers except 1 and be defined on S by a b = a + b - ab, for all a, b s.

Prove that:

(i) is a binary operation on S

(ii) is commutative as well as associative.

Solution 14

Chapter 3 - Binary Operations Exercise Ex. 3.3

Question 1

Solution 1

Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4

Chapter 3 - Binary Operations Exercise Ex. 3.4

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4

Solution 4

Question 5

Solution 5

Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8

Chapter 3 - Binary Operations Exercise Ex. 3.5

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9

Question 10
Solution 10

Chapter 3 - Binary Operations Exercise MCQ

Question 1

If a*b = a2 + b2, then the value of (4*5) * 3 is

 

a. (42 + 52) + 32

b. (4 + 5)2 + 32

c. 412 + 32

d. (4 + 5 + 3)2

Solution 1

Question 2

If a*b denote the bigger among a and b and if a. b = (a * b) + 3, then 4.7 =

 

a. 14

b. 31

c. 10

d. 8

Solution 2

Question 3

On the power set p of a non-empty set A, we define an operation Δ by

Then which are the following statements is true about Δ

 

a. Commutative and associative without an identity

b. Commutative but not associative with an identity

c. Associative but not commutative without an identity

d. Associative and commutative with an identity

 

Solution 3

Question 4

If the binary operation* on Z is defined by a*b = a2 - b2 + ab + 4, then value of (2*3)  * 4 is

 

a. 233

b. 33

c. 55

d.  - 55

Solution 4

Question 5

For the binary operation * on Z is defined by a*b = a + b + 1 the identity element is

 

a. 0

b. -1

c. 1

d. 2

Solution 5

Question 6

If a binary operation * is defined on the set Z of integers as a * b = 3a - b, then the value of (2*3) * 4 is

 

a. 2

b. 3

c. 4

d. 5

Solution 6

 

Question 7

Q+ denote the set of all positive rational numbers. If the binary operation on Q+ is defined as   then the inverse of 3 is

 

a. 4/3

b. 2

c. 1/3

d. 2/3

Solution 7

Question 8

If G is the set of all matrices of the form   , where x R - {0}, then the identity element with respect to the multiplication of matrices as binary operation, is

 

Solution 8

 

Question 9

Q+ is the set of all positive rational numbers with the binary operation * defined by   for all a, b Q+. Then inverse of an element a Q+ is

 

a. A

b. 

c. 

d. 

Solution 9

Question 10

If the binary operation is defined on the set Q+ of all positive rational numbers by

  

Solution 10

Question 11

Let * be a binary operation defined on set Q - {1} by the rule

   

a*b = a + b - ab. Then, the identity element for * is

 

Solution 11

Question 12

Which of the following is true?

 

a. * defined by a * b =   is a binary operation on Z

b. * defined by a * b =   is a binary operation on Q

c. All binary commutative operations are associative

d. Subtraction is a binary operation on N.

Solution 12

Question 13

The binary operations * defined on n by a * b = a+ b + ab for all a, b N is.

 

a. Commutative only

b. Associative only

c. Commutative and associative both

d. None of these

Solution 13

Question 14

If a binary operation * is defined by a * b = a2 + b2 + ab + 1, then (2 * 3) * 2 is equal to

 

a. 20

b. 40

c. 400

d. 445

Solution 14

Question 15

Let * be a binary operation on R defined by a * b = ab + 1. Then, * is

 

a. Commutative but not associative

b. Associative but not commutative

c. Neither commutative nor associative

d. Both commutative and associative

Solution 15

Question 16

Subtraction of integers is

 

a. Commutative but not associative

b. Commutative and associative

c. Associative but not commutative

d. Neither commutative nor associative

Solution 16

Question 17

The law a + b = b + a is called

 

a. Closure law

b. Associative law

c. Commutative law

d. Neither commutative nor associative

Solution 17

Question 18

An operation * is defined on the set Z of non-zero integers by   for all a, b Z. Then the property satisfied is

 

a. Closure

b. Commutative

c. Associative

d. None of these

Solution 18

Question 19

On Z an operation * is defined by a * b = a2 + b2 for all a, b Z. Then operation * on Z is

 

a. Commutative and associative

b. Associative but not commutative

c. Not associative

d. Not a binary operation

Solution 19

Question 20

A binary operation * on Z defined by a * b = 3a + b for all a, b Z, is

 

a. Commutative

b. Associative

c. Not commutative

d. Commutative and associative

Solution 20

Question 21

Let * be a binary operation on Q+ defined by a * b =   for all a, b Q+. The inverse of 0.1 is

 

 

a. 105

b. 104

c. 106

d. None of these

 

Solution 21

Question 22

Let * be a binary operation on N defined by a * b = a + b + 10 for all a, b N. The identify element for * in N is

 

a. -10

b. 0

c. 10

d. Non-existent

Solution 22

Question 23

Consider the binary operation * defined on Q - {1} by the rule a * b = a + b - ab for all a, b Q - {1}. The identity element in Q - {1} is

 

a. 0

b. 1

c. 

d. -1

Solution 23

Question 24

For the binary operation * defined on R - {- 1} by the rule a * b = a + b + ab for all a, b R - {1}, the inverse of a is

 

a. -a

b. 

c. 

d. a2

Solution 24

Question 25

For the multiplication of matrices as a binary operation on the set of all matrices of the form   , a, b R the inverse    is

 

Solution 25

 

Question 26

On the set Q+ of all positive rational numbers a binary operation * is defined by a * b =   for all a, b Q+. The inverse of 8 is

 

a. 

b. 

c. 2

d. 4

Solution 26

Question 27

Let * be a binary option defined on Q+ by the rule a * b =   for all a, b, Q+. The inverse of 4 * 6 is

 

a. 

b. 

c. 

d. None of these

Solution 27

Question 28

The number of binary operations that can be defined on a set of 2 element is

 

a. 8

b. 4

c. 16

d. 64

Solution 28

Question 29

The number of commutative binary operations that can be defined on a set of 2 elements is

 

a. 8

b. 6

c. 4

d. 2

Solution 29

Chapter 3 - Binary Operations Exercise Ex.3VSAQ

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Write the inverse of 5 under multiplication modulo 11 on the set {1, 2, ..., 10}.

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

L e t space asterisk times space b e space a space b i n a r y space o p e r a t i o n space o n space s e t space o f space i n t e g e r s space I comma space d e f i n e d space b y space a asterisk times b equals 2 a plus b minus 3.
F i n d space t h e space v a l u e space o f space 3 asterisk times 4.

Solution 20

G i v e n space b i n a r y space o p e r a t i o n space i s space a asterisk times b equals 2 a plus b minus 3
T h u s comma space 3 asterisk times 4 equals 2 cross times 3 plus 4 minus 3 equals 6 plus 4 minus 3 equals 7