• Skip to main content
  • Skip to secondary menu
  • Skip to primary sidebar
  • Skip to footer
  • RBSE Model Papers
    • RBSE Class 12th Board Model Papers 2022
    • RBSE Class 10th Board Model Papers 2022
    • RBSE Class 8th Board Model Papers 2022
    • RBSE Class 5th Board Model Papers 2022
  • RBSE Books
  • RBSE Solutions for Class 10
    • RBSE Solutions for Class 10 Maths
    • RBSE Solutions for Class 10 Science
    • RBSE Solutions for Class 10 Social Science
    • RBSE Solutions for Class 10 English First Flight & Footprints without Feet
    • RBSE Solutions for Class 10 Hindi
    • RBSE Solutions for Class 10 Sanskrit
    • RBSE Solutions for Class 10 Rajasthan Adhyayan
    • RBSE Solutions for Class 10 Physical Education
  • RBSE Solutions for Class 9
    • RBSE Solutions for Class 9 Maths
    • RBSE Solutions for Class 9 Science
    • RBSE Solutions for Class 9 Social Science
    • RBSE Solutions for Class 9 English
    • RBSE Solutions for Class 9 Hindi
    • RBSE Solutions for Class 9 Sanskrit
    • RBSE Solutions for Class 9 Rajasthan Adhyayan
    • RBSE Solutions for Class 9 Physical Education
    • RBSE Solutions for Class 9 Information Technology
  • RBSE Solutions for Class 8
    • RBSE Solutions for Class 8 Maths
    • RBSE Solutions for Class 8 Science
    • RBSE Solutions for Class 8 Social Science
    • RBSE Solutions for Class 8 English
    • RBSE Solutions for Class 8 Hindi
    • RBSE Solutions for Class 8 Sanskrit
    • RBSE Solutions

RBSE Solutions

Rajasthan Board Textbook Solutions for Class 5, 6, 7, 8, 9, 10, 11 and 12

  • RBSE Solutions for Class 7
    • RBSE Solutions for Class 7 Maths
    • RBSE Solutions for Class 7 Science
    • RBSE Solutions for Class 7 Social Science
    • RBSE Solutions for Class 7 English
    • RBSE Solutions for Class 7 Hindi
    • RBSE Solutions for Class 7 Sanskrit
  • RBSE Solutions for Class 6
    • RBSE Solutions for Class 6 Maths
    • RBSE Solutions for Class 6 Science
    • RBSE Solutions for Class 6 Social Science
    • RBSE Solutions for Class 6 English
    • RBSE Solutions for Class 6 Hindi
    • RBSE Solutions for Class 6 Sanskrit
  • RBSE Solutions for Class 5
    • RBSE Solutions for Class 5 Maths
    • RBSE Solutions for Class 5 Environmental Studies
    • RBSE Solutions for Class 5 English
    • RBSE Solutions for Class 5 Hindi
  • RBSE Solutions Class 12
    • RBSE Solutions for Class 12 Maths
    • RBSE Solutions for Class 12 Physics
    • RBSE Solutions for Class 12 Chemistry
    • RBSE Solutions for Class 12 Biology
    • RBSE Solutions for Class 12 English
    • RBSE Solutions for Class 12 Hindi
    • RBSE Solutions for Class 12 Sanskrit
  • RBSE Class 11

RBSE Solutions for Class 12 Maths Chapter 1 Composite Functions Ex 1.3

May 22, 2019 by Fazal Leave a Comment

Rajasthan Board RBSE Class 12 Maths Chapter 1 Composite Functions Ex 1.3

Question 1.
Determine whether each of the following operation define a binary operation on the given set or not. Also, Justify your answer.
(i) a*b = a, on N
(ii) a*b = a + b – 3, on N
(iii) a*b = a + 3b, on N
(iv) a*b = a/b, on Q
(v) a*b = a – b, on R
Solution:
(i) a*b = a, on N
Here, * is a binary operation because
a, b ∈ N
= a*b = a ∈ N
Here, a*b ∈ N
Hence, * is a binary operation.

(ii) a*b = a + b – 3 ∈ N
Here, a*b is not a binary operation because,
1 ∈ N, 2 ∈ N
then 1 + 2 – 3 = 0 ∈ N

(iii) a*b= a + 3b ∈ N
Here, * is a binary operation because
1 ∈ N, 2 ∈ N
then 1 + 3 × 2 = 1 + 6 = 7 ∈N

(iv) a*b = \(\frac { a }{ b }\) ∈ Q
Here, a*b is not a binary operation because,
Let a = 22 ∈ Q and b= 7 ∈ Q
But
RBSE Solutions for Class 12 Maths Chapter 1 Composite Functions Ex 1.3

(v) a*b= a – b ∈ R
Here, * is a binary operation because,
a ∈ R, b ∈ R
⇒ a – b ∈ R, ∀ a, b ∈ R

Question 2.
Determine which of the following binary operation is commutative and which is associative :
(i) * on N defined as a*b = 2ab
(ii) * on N defined as a*b = a + b + aab
(iii) * on Z defined as a*b = a – b
(iv) * on Q defined as a*b = ab + 1
(v) * on R defined as a*b = a + b – 7
Solution:
(i) Given a*b = 2ab
Commutativity: Let a, b ∈ N
a*b = 2ab
= 2b.a
= b*a
So, a*b = b*a
∴ * is a commutative operation.

Associativity: Let a, b, c ∈ N
(a*b)*c = 2(ab)*2c = 2ab + c
= 2c*2(ab) = 2c +ab
a*(b*c) = 2a*2(bc) = 2a + bc
2ab+c ≠ 2a+bc
It is clear that (a*b)*c ≠ a*(b*c)
So, (a*b)*c is not an associative operation.
Hence, a*b = 2ab is commutative but not associative.

(ii) Given a*b= a + b + a2b
Commutativity: Let a, b ∈ N
a*b = a + b + a2b
b*a = b + a + b2a
a*b ≠ b*a .
So, * is not a commutative operation.

Associativity: Let a, b, c ∈ N
(a*b)*c = (a + b + a2b)*c
a*(b*c) = a*(b + c + b2c)
t is clear that (a*b)*c ≠ a*(b*c)
So * is not an associative operation.
Hence, a*b = a + b + a2b is neither commutative nor associative.

(iii) Given, a*b = a – b
Commutativity:
a*b = a – b, (a, b ∈ Z)
b*a = b – a, (a, b ∈ Z)
a*b* ≠ b*a
So * is not a commutative operation.

Associativity :
(a*b)*c = (a – b)*c
= a – b-c
a*(b*c) = a*(b-c)
= a – b + c
∵ (a*b)*c ≠ a*(b*c)
So, it is not associative operation.
It is clear that
a*b = a – b is neither commutative nor associative.

(iv) Given, a*b = ab + 1
Commutativity: Let a, b ∈ Q
a*b = ab + 1 and : b*a= ba + 1
⇒ a*b= b*a
∴ It is commutative.
∴ Addition and multiplication of rational number is commutative.

Associativity: Let a, b, c ∈ Q
(a*b)*c = (ab + 1)*c
= ab + 1 + c
(b*c)*a = (bc + 1) +a
= (a*b)*c ≠ (b*c)*a
So, * is not associative.
It is clear from above that a*b = ab + 1 is commutative but not associative.

(v) Given, a*b = a + b – 7
Commutativity: In R,
a*b = a + b – 7
= b + a – 7
= b*a’

Associativity :
(a*b)*c = (a + b – 7)*c
= (a + b – 7) + c – 7
= a + b + c – 14
a*(b*c) = a*(b + c – 7)
= a + (b + c – 7) – 7
= a + b + c – 14
So, (a*b)*c = a*(b*c)
Hence, it is clear that a*b = a + b – 7 are commutative and associative.

Question 3.
If * be an operation on Z, defined as a*b = a + b + 1, ∀ a, b ∈ Z then prove that * is commutative and associative, find its identity element. Also, find inverse element of any integer in Z.
Solution:
Given a*b = a + b + 1, ∀ a, b ∈ z

Commutativity :
a*b = a + b + 1
a*b = b + a + 1
= b*a
∴ a*b = b*a
∴ * is commutative operation.

Associativity :
(a*b) * c = (a + b + 1)*c
= a + b + 1 + c +1
a + b + c + 2
Again a*(b*c) = a*(b + c + 1)
= a + b + c + 1 + 1
= a + b + c + 2
a*(b*c)= (a*b)*c
∴ * is associative operation
Identity : If e is identity element, then
a*e = a
⇒ a + e + 1 = a
⇒ e = -1
So, – 1 ∈ Z is identity element.
Inverse : Let inverse of a is x, then by definition
a*x = -1 [∵ – 1 is identity]
⇒ a + x + 1 = -1
⇒ x = -(a + 2) ∈ Z
Inverse element a-1 = -(a + 2).

Question 4.
If * be a binary operation defined on R – {1}. as a*b = a + b – ab, ∀ a, b ∈ R – {1}
prove that * is commutative and associative. Find identity element and also find inverse of a.
Solution:
If a, b ∈ R – {1} by definition
a*b = a + b – ab
= b + a – ba
= b*a
∴ * is a binary operation.
Again, (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 – bc – ac + abc …(i)
and a*(b*c) = a*(b + c – bc)
= a + (b + c – bc) – a (b + c – bc)
= a + b + c – bc – ab – ac + abc
= a + b + c – ab – bc – ac + abc … (ii)
From eqs. (i) and (ii),
(a*b)*c= a*(b*c)
∴ * is associative operation.
Let e is identity of *, then for a ∈ R,
a*e = a (from definition of identity)
⇒ a + e – ae = a
⇒ e(1 – a)= 0
⇒ e = 0 ∈ R – {1}
∵ 1 – a ≠ 0
∴ 0 is identity of *.
Let b is inverse of a.
a*b = e
a + b – ab = 0.e
b – ab = -a
b(1 – a) = -a
RBSE Solutions for Class 12 Maths Chapter 1 Composite Functions Ex 1.3

Question 5.
Four functions are defined on set Ro, Such that,
f1(x) = x, f2(x) = -x, f3(x) = 1/x, f4(x) = – 1/x Construct the composition table for the composition of functions f1, f2, f3, f4. Also, find identity element and inverse of every element.
Solution:
Given,
f1(x) = x, f2(x) = – x,
RBSE Solutions for Class 12 Maths Chapter 1 Composite Functions Ex 1.3
RBSE Solutions for Class 12 Maths Chapter 1 Composite Functions Ex 1.3
RBSE Solutions for Class 12 Maths Chapter 1 Composite Functions Ex 1.3
∴ Table for above operations
RBSE Solutions for Class 12 Maths Chapter 1 Composite Functions Ex 1.3
Hence, it is clear that f1 is identity function of f1, f2, f3, f4.
Hence, inverse element is itself too.

RBSE Solutions for Class 12 Maths

Share this:

  • Click to share on WhatsApp (Opens in new window)
  • Click to share on Twitter (Opens in new window)
  • Click to share on Facebook (Opens in new window)

Related

Filed Under: Class 12

Reader Interactions

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Primary Sidebar

Recent Posts

  • RBSE Solutions for Class 7 Our Rajasthan in Hindi Medium & English Medium
  • RBSE Solutions for Class 6 Our Rajasthan in Hindi Medium & English Medium
  • RBSE Solutions for Class 7 Maths Chapter 15 Comparison of Quantities In Text Exercise
  • RBSE Solutions for Class 6 Maths Chapter 6 Decimal Numbers Additional Questions
  • RBSE Solutions for Class 11 Psychology in Hindi Medium & English Medium
  • RBSE Solutions for Class 11 Geography in Hindi Medium & English Medium
  • RBSE Solutions for Class 3 Hindi
  • RBSE Solutions for Class 3 English Let’s Learn English
  • RBSE Solutions for Class 3 EVS पर्यावरण अध्ययन अपना परिवेश in Hindi Medium & English Medium
  • RBSE Solutions for Class 3 Maths in Hindi Medium & English Medium
  • RBSE Solutions for Class 3 in Hindi Medium & English Medium

Footer

RBSE Solutions for Class 12
RBSE Solutions for Class 11
RBSE Solutions for Class 10
RBSE Solutions for Class 9
RBSE Solutions for Class 8
RBSE Solutions for Class 7
RBSE Solutions for Class 6
RBSE Solutions for Class 5
RBSE Solutions for Class 12 Maths
RBSE Solutions for Class 11 Maths
RBSE Solutions for Class 10 Maths
RBSE Solutions for Class 9 Maths
RBSE Solutions for Class 8 Maths
RBSE Solutions for Class 7 Maths
RBSE Solutions for Class 6 Maths
RBSE Solutions for Class 5 Maths
RBSE Class 11 Political Science Notes
RBSE Class 11 Geography Notes
RBSE Class 11 History Notes

Copyright © 2023 RBSE Solutions

 

Loading Comments...