NCERT Solutions For Class 12 Maths Chapter 1 Exercise 1.4 (Ex 1.4)

NCERT refers to the National Council of Education and Research Training, which is a government organisation set up to improve the quality of school education in India. Intending to develop an independent, self-sufficient and growth-oriented education system, NCERT has the responsibility of developing and publishing textbooks. It also develops educational kits and digital material to aid students in their studies. CBSE and many other state boards follow NCERT textbooks. In fact, it is recommended to refer to NCERT books for various competitive exams like NEET, JEE, UPSC etc. NCERT books are designed in such a manner that they are easy to comprehend by the students and help in their overall development. Each textbook has numerous examples and exercises alongside tables for better learning of the concepts. One can start learning the basics from these books and gradually move toward a higher level.

Mathematics can be an intimidating subject for many. It may be not everyone’s best subject, but if prepared well, it can fetch one a good percentage. Mathematics is a subject that requires daily practice and spaced repetition. NCERT Mathematics is considered easy by a lot of students. One can easily clear the basics of Mathematics by referring to NCERT books. Students can easily attempt higher-level problems and twisted questions if they have a strong and deep understanding of the concepts, which can be achieved through NCERT textbooks.

Class 12 Mathematics is relatively easier than class 11 Mathematics. Those who have a good grasp of Class 11 Mathematics, will not face any trouble solving NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4. However, scoring good in Class 12 Mathematics is crucial for everyone as it determines their overall percentage in the Board Examination. Board Examination percentage is determinant of a student’s future as it is the criteria for admission in many reputed colleges like DU, JNU etc. NCERT Class 12 Mathematics covers numerous basic topics like Relation and Functions, Inverse Trigonometric Functions, Continuity and Differentiability, Vector Algebra and Three Dimensional Geometry etc. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 is a part of the first chapter that is taught in Mathematics for Class 12.  These topics also comprise the syllabus of undergraduate-level engineering exams like JEE Mains, BITSAT, VITEEE, etc. Apart from Class 12, Extramarks also provides NCERT Solutions Class 11, NCERT Solutions Class 10, NCERT Solutions Class 9, etc. These solutions are foundational, and they act as stepping stones for more complex concepts.

NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 provided by Extramarks is in a simplified language that could be understood easily by students of various academic backgrounds. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 identifies all the critical problems around relations and functions and gives a detailed step-by-step explanation of the problems given in the textbook. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 must be analysed by students to clear their doubts and confusion related to this exercise and chapter as a whole. Overall, NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 is to make students’ practice easy and efficient, and help them gain confidence. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 can also come in handy for further preparation for competitive examinations.

NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.4 (Ex 1.4)

Students can access NCERT Solutions for Class 12 Chapter 1 – Relations and Functions on Extramarks. These solutions are devised keeping the marking scheme in mind. These solutions are also prepared by teachers with years of experience. Accuracy and efficiency in Mathematics matter a lot which can be achieved through daily practice and regular revision. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 helps students attain consistency in their practice.  Additionally, Extramarks solutions are devised in such a manner that they can help students learn how to frame their answers in the board examination to score high marks. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 contains 13 practice questions on binary operations. These questions are centred around determining commutative and associative binary operations and invertible binary operations. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 are also a reliable source of study material. Extramarks NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 are in comprehensible language and with highlighted formulas. Extramarks NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 can be used by students to recheck their methods.  Extramarks NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4, Exercise 1.3, Exercise 1.2, and Exercise 1.1, can all prove to be instrumental in students’ preparation process.

NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4, on Extramarks are easily accessible, and these solutions are designed to help students in their preparation journey and clear their doubts. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 solutions will help students tally their answers. The benefit of NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 is that one can save time by simply referring to them when stuck at a problem.

PDFs for downloads are available. By downloading the PDF for NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 the students do not have to worry about their internet connection accessibility. Students can download the PDF for NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 and practise the questions offline at their own pace.

Access NCERT Solutions for Class 12 Maths Chapter 1 – Relations and Functions

Extramarks provides easy access to NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4. Solutions are designed by experts to make students’ preparation more effective. Solutions are authentic and without errors. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 are constructed in a way that all the fundamentals of the chapter are covered. Extramarks, also, has a similarly good quality NCERT Solutions Class 8, NCERT Solutions Class 7, and NCERT Solutions Class 6 etc.

NCERT Solutions for Class 12 Maths

Students can find NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 on the Extramarks website. The solutions are available in PDF format. It helps students prepare better for their upcoming board examinations. Not only that, but NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 PDF Download also helps students study for competitive exams that follow soon after the end of the board exams. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 PDF Download is also simply accessible with a single click of a button. Solutions are cross-checked to avoid errors that may lead the students to get confused. Highly trained teachers with years of experience have curated these solutions. These solutions have been made available on the website of Extramarks.

NCERT Solution Class 12 Maths of Chapter 1 All Exercises 

Apart from NCERT Solutions Class 12 Chapter 1 Maths Exercise 1.4, Extramarks also provides NCERT Solution Class 12 Maths of Chapter 1. All exercises are available on the Extramarks website. There are a total of 4 exercises in Chapter 1 Relations and Functions. Exercise 1.1, Exercise 1.2, and Exercise 1.3 are essential for solving Exercise 1.4. Extramarks has NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4, Exercise 1.3, Exercise 1.2, and Exercise 1.1.

Exercise 1.1 is centred around determining whether the type of relation is reflexive, symmetric or transitive. It also has questions asking whether two variables are related or not. Exercise 1.2 has questions which deal with functions and identify whether a function is a one-to-one function or an onto function. Exercise 1.3 has questions on the concept of the composition of two functions and the inverse of a function. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 quizzes students on the concept of binary operations and whether binary operations are associative and commutative. All these exercises are conceptually interrelated, one can simply not skip any exercise as it will lead to confusion.

This set of exercises clears the basics for Relations and Functions that will further aid students in their preparation for competitive exams. NCERT Solution Class 12 Maths of Chapter 1 All Exercises are linked below.

Chapter 1 Relations And Functions Other Exercises
Exercise 1.1
16 Questions And Solutions (3 Short Answer, 13 Long Answers)
Exercise 1.2
12 Questions And Solutions (5 Short Answer, 7 Long Answers)
Exercise 1.3
14 Questions And Solutions (4 Short Answer, 10 Long Answers)

Overview of Class 12 Maths Chapter 1

This chapter is related to Relations and Functions, which is generally not a hard concept to grasp since it has been already introduced in class 11. If students wish to review and test their previous knowledge of the topic, they can refer to NCERT solutions of Class 11 that are available on the Extramarks website alongside other study material and animated videos. Functions and Binary Operations are new additions. With proper guidance and enough practice, one can easily master this topic. This is where the Extramarks NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 comes in handy. Since the solutions are precisely written in comprehensible language, students can tally their solutions with the Extramarks NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 and can also learn how to approach the problems.

NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 revolves around Relations and Functions as a fundamental part of Mathematics, which helps define different concepts and various specified valued functions alongside graphs. Relations and Functions are essential topics in Algebra.  Relations and Functions have two different meanings Mathematically. Students usually get confused about their differences. The difference between a relation and a function is that a Relation can have many outputs for a single input, but a function has a single input for a single output.

To understand NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4, students have to remember the notion of relations and functions, domain, co-domain and range which have been already introduced in Class 11. This was introduced along with different types of specific real-valued functions and their graphs. The concept of the term ‘relation’ in Mathematics is derived from the meaning of relation in the English language, according to which two objects or quantities are related if there is an identifiable connection or link between the two objects or quantities. In Mathematical terms, relations are defined as a recognizable connection or link between two objects or quantities. It is a subset of the Cartesian product. There are various types of relations like empty, universal, reflexive, transitive, symmetric, and equivalence relations. However, Class 12 Maths NCERT Solutions Chapter 1 Exercise 1.4 is primarily focused on binary operations.

These are the key relations – 

Empty relation is the relation R in X given by R = φ ⊂ X × X.

Universal relation is the relation R in X given by R = X × X.

Reflexive relation R in X is a relation with (a, a) ∈ R ∀ a ∈ X.

Symmetric relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R.

Transitive relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R.

Equivalence relation R in X is a relation which is reflexive, symmetric, and transitive.

The idea of a function along with some special functions like identity function, constant function, polynomial function, rational function, modulus and signum function etc. have been covered in class 11. Graphs have been already introduced in Class 11. Students have also been acquainted with addition, subtraction, multiplication and division of the two functions. A function in Mathematics is envisioned as a rule, which gives a distinctive output for every input x. A function is a law that defines the relationship between one variable ( independent variable) and another variable ( dependent variable). It also covers types of functions like one-to-one and onto functions, composite functions, and inverse functions. Class 12 Maths Chapter 1 Exercise 1.4 Solutions is an extension of the topic and can come in handy in preparation.

There 3 new types of functions that have been introduced – 

A function f : X → Y is one-one (or injective) if f(x 1 ) = f(x 2 ) ⇒ x 1 = x 2 ∀ x 1 , x 2 ∈ X.

A function f : X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that f(x) = y.

A function f : X → Y is one-one and onto (or bijective), if f is both one-one and onto.

The composition of functions and invertible functions have also been discussed in the chapter.

In NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4, students have already been introduced to four fundamental operations, namely addition, subtraction, multiplication and division. When three numbers are added, initially the first two numbers are added and the result is then added to the third number. Thus, addition, multiplication, subtraction and division are examples of a binary operation, as ‘binary’ stands for two. Even in Mathematical terms, binary operations refer to a set of operations such as addition, subtraction, division, and multiplication that are carried out by an arbitrary set called ‘X’. Availability of  NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 focusing on this topic is a single click away on Extramarks. Extramarks also provides NCERT Solutions Class 5, NCERT Solutions Class 4, NCERT Solutions Class 4, NCERT Solutions Class 3, NCERT Solutions Class 2 and NCERT Solutions Class 1 which are related to these four fundamental operations.

Important Topics covered in Exercise 1.4 of NCERT Solutions for Class 12 Maths Chapter 1

Concepts of paramount importance have been covered in Class 12 Chapter 1 Relations and Functions. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4, is primarily based on binary operations of sets. Extramarks NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4, are a great assistance tool for students in their preparation for both their Class 12 Board Examination and Competitive exams like JEE, UPSC and NDA exams.

NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 answers medium to hard-level questions which are based on binary operations. Binary operations on sets is an operation whose two domains and the codomain are the same set. The fundamental operations introduced in primary classes, Addition (+), Subtraction (-), Division (/) and Multiplication (*) are the four principle binary operations used on sets. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 has answers related to these four binary operations and ways to calculate them using numerous functions and properties. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 has solutions to the questions that ask to determine whether a binary operation is commutative or associative. Additionally, NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 contains answers to questions asking about the existence of identity elements.

Key learnings from Exercise 1.4 of NCERT Solutions for Class 12 Maths Chapter 1

In Chapter 1 Relations and Functions, the first three exercises are based on types of relations and functions that are an extension of previously acquainted topics. However, NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 is related to an extended and higher level version of an old concept i.e. Binary Operations. Below are some vital learnings from NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 :

A binary operation ∗ on a set A is a function ∗ from A × A to A.

An element e ∈ X is the identity element for binary operation ∗ : X × X → X, if a ∗ e = a = e ∗ a ∀ a ∈ X.

An element a ∈ X is invertible for binary operation ∗ : X × X → X, if there exists b ∈ X such that a ∗ b = e = b ∗ a where e is the identity for the binary operation ∗. The element b is called the inverse of a and is denoted by a –1.

An operation ∗ on X is commutative if a ∗ b = b ∗ a ∀ a, b in X.

An operation ∗ on X is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c)∀ a, b, c in X.

After concluding this exercise with the help of NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4, students can determine whether given functions are binary or not, using commutative and associative properties mainly.

How NCERT Exercise 1.4 ch 1 Class 12 Maths Can Help You Reach Your Goals?  

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Q.1 Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.

(i) On Z+, define * by a * b = a − b

(ii) On Z+, define * by a * b = ab

(iii) On R, define * by a * b = ab2

(iv) On Z+, define * by a * b = |a − b|

(v) On Z+, define * by a * b = a

Ans

( i ) On Z + , * is defined by a * b = a−b.      Here, ( 1,2 )∈ Z +     so, 1*2= 1−2=−1∉ Z +     Thus,  * is not a binary operation. ( ii )  On Z + , * is defined by a*b = ab. Here, a,b∈ Z + and ab∈ Z +     This means that * carries each pair (a, b) to a unique element       a * b = ab in Z + .      Therefore, *  is a binary operation. ( iii )  On R, * is defined by a * b = ab 2 .     It is seen that for each a, b∈R, there is a unique element ab 2 in R.  This means that * carries each pair (a, b) to a unique element a * b = ab 2 in R.  Therefore, * is a binary operation. ( iv )  On Z + , * is defined by a*b = |a−b|.        It is seen that for each a, b∈ Z + , there is a unique element       |a− b| in Z + .       This means that * carries each pair (a, b) to a unique element       a*b = |a− b| in Z + .       Therefore, * is a binary operation. ( v )  On Z + , * is defined by a*b = a.       * carries each pair (a, b) to a unique element a * b = a in Z + .       Therefore, * is a binary operation. MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8IrFz0xbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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Q.2

For each binary operation*defined below, determinewhether *is commutative or associative.   (i) On Z, definea*b=a−b (ii) On Q, definea*b=ab+1(iii) OnQ, definea*b=ab2 (iv) OnZ+, definea*b=2ab(v) OnZ+, definea*b=ab(vi) OnR−{−1}, definea*b=ab+1

Ans

(i) On Z, * is defined by a * b = a−b.      It can be observed that  1 * 2 = 1−2 = 1 and 2 * 1 = 2−1 = 1.∴    1*2≠2 * 1; where 1, 2∈ZHence, the operation * is not commutative.Also we have:(1 * 2) * 3 = (1−2)*3 =−1 * 3 =−1−3 =−41 * (2 * 3) = 1 * (2−3) = 1 * −1 = 1 − (−1) = 2∴(1 * 2) * 3≠1 * (2 * 3) ; where 1, 2, 3∈ZHence, the operation * is not associative.(ii) On Q, * is defined by a * b = ab + 1. It is known that:               ab = ba ∀a, b∈Q⇒ab + 1 = ba + 1                   ∀a, b∈Q⇒     a * b = a * b∀a, b∈QTherefore, the operation * is commutative.It can be observed that:   (1 * 2) * 3 = (1 × 2 + 1) * 3 = 3 * 3 = 3 × 3 + 1 = 10   1 * (2 * 3) = 1 * (2 × 3 + 1) = 1 * 7 = 1 × 7 + 1 = 8∴(1 * 2) * 3≠1 * (2 * 3) ; where 1, 2, 3∈QTherefore, the operation * is not associative.(iii)On Q, * is defined by a*b=ab2      It is known that:ab = ba∀a, b∈Q⇒     ab2=ba2∀a, b∈Q⇒ a*b=b*a∀a, b∈QTherefore, the operation * is commutative.For all a, b, c∈Q, we have:(a*b)*c=(ab2)*c=(ab2)c2=abc4a*(b*c)=a*(bc2)=a(bc2)2=abc4So,  (a*b)*c=a*(b*c)∀a, b, c∈QTherefore, the operation * is associative.(iv) On Z+, * is defined by a*b = ab.It can be observed that:     1*2=12=1 and 2*1=22=4∴  1*2≠2*1;    where  1,2∈Z+Therefore, the operation * is not commutative.For Associative Law:(3*4)*5=(32)*5=(32)2=813*(4*5)=3*(42)=32=9∴(3*4)*5≠3*(4*5)  where 2, 3, 4∈Z+Therefore, the operation * is not associative.(vi) On R, *−{−1} is defined by  a*b=ab+1It can be observed that  2*3=23+1=24=12and           3*2=32+1=33=1∴    2*3≠3*2 ; where 2, 3∈R−{−1}Therefore, the operation * is not commutative.For​ Associative Law:(2*3)*4=(23+1)*4=(24)*4=(12)*4=(12)4+1=1102*(3*4)=2*(34+1)=2*(35)=2(35)+1=285=54∴(2*3)*4≠2*(3*4);   where 2,3,4∈R−{−1}Therefore, the operation * is not associative.

Q.3

Consider the binary operation ∧ on theset {1, 2, 3, 4, 5}definedby  a∧b=min{a,b}.Write the operation tableof the operation ∧

Ans

The binary operation∧on the set {1, 2, 3, 4, 5} is defined as a∧b= min{a,b} ∀a,b∈{1, 2, 3, 4, 5}.Thus, the operation table for the given operation∧can be given as:

  ^ 1 2 3 4 5
1 1 1 1 1 1
2 1 2 2 2 2
3 1 2 3 3 3
4 1 2 3 4 4
5 1 2 3 4 5

 

 

 

 

 

Conside ra binary operation*on the set {1,2,3,4,5} given      by the following multiplication table  (Table 1.2)(i) Compute(2*3)*4and2*(3*4)(ii) Is*commutative?(iii) Compute(2*3)*(4*5)

Q.4

* 1 2 3 4 5
1 1 1 1 1 1
2 1 2 2 2 2
3 1 2 3 3 3
4 1 2 3 4 4
5 1 2 3 4 5

 

 

 

 

 

Ans

(i) (2*3)*4=1*4=1     2*(3*4)=2*1=1(ii) For every a, b∈{1, 2, 3, 4, 5}, we have a * b = b * a.      Therefore, the operation * is commutative.(iii)(2 * 3) = 1 and (4 * 5) = 1∴(2 * 3) * (4 * 5) = 1 * 1 = 1

Q.5 Let *′ be the binary operation on the set {1, 2, 3, 4, 5} defined by a *′ b = H.C.F. of a and b. Is the operation *′ same as the operation * defined in Exercise 4 above? Justify your answer.

Ans

The binary operation *’R on the set {1, 2, 3 4, 5} is defined as a *’ b = H.C.F of a and b. The operation table for the operation *’ can be given as:

*’ 1 2 3 4 5
1 1 1 1 1 1
2 1 2 1 2 1
3 1 1 1 1 1
4 1 2 1 4 1
5 1 1 1 1 1

 

 

 

 

We see that the operation tables for the operations * and *’ are the same. Thus, the operation *’ is same as the operation*

Q.6

Let * be the binary operation on Ngiven by a*b=L.C.M. of aand b. Find(i) 5*7,20*16(ii) Is*commutative?(iii) Is*associative?(iv) Find the identity of*in N(v) Which element so f Narein vertible f or the operation*?

Ans

The binary operation * on N is defined as a * b = L.C.M. of a and b.(i) 5 * 7 = L.C.M. of 5 and 7     = 35 20 * 16 = L.C.M of 20 and 16 = 80(ii) It is known that:L.C.M of a and b = L.C.M of b and a       ∀a, b∈N.∴                      a * b = b * aThus, the operation * is commutative.(iii) For a, b, c∈N, we have:      (a * b) * c = (L.C.M of a and b) * c           = LCM of a, b, and c       a * (b * c)= a * (LCM of b and c)           = L.C.M of a, b, and c∴   (a * b) * c = a * (b * c)Thus, the operation * is associative.(iv) It is known that:L.C.M. of a and 1 = a = L.C.M. 1 and a ∀ a∈ N⇒                     a * 1 = a = 1 * a ∀ a∈ NThus, 1 is the identity of * in N.(v) An element a in N is invertible with respect to the operation *  if there exists an element b in  N, such that a * b = e = b * a.Here,       e = 1This means that:L.C.M of a and b = 1 = L.C.M of b and aThis case is possible only when a and b are equal to 1.Thus, 1 is the only invertible element of N with respect to the operation *.

Q.7

Is*defined on the set {1, 2, 3, 4, 5} by a*b= L.C.M. of aand b a binary operation? Justify your  answer.

Ans

The operation * on the set A = {1, 2, 3, 4, 5} is defined asa * b = L.C.M. of a and b.Then, the operation table for the given operation * can be givenas:

* 1 2 3 4 5
1 1 2 3 4 5
2 2 2 6 4 10
3 3 6 3 12 15
4 4 4 12 4 20
5 5 10 15 20 5

It can be observed from the obtained table that:

3 * 2 = 2 * 3 = 6∈A, 5 * 2 = 2 * 5 = 10∈A,

3 * 4 = 4 * 3 = 12∈A

3 * 5 = 5 * 3 = 15∈A, 4 * 5 = 5 * 4 = 20∉A

Hence, the given operation * is not a binary operation.

Q.8 Let * be the binary operation on N defined by
a * b = H.C.F. of a and b. Is * commutative? Is * associative? Does there exist identity for this binary operation on N?

Ans

The binary operation * on N is defined as: a * b = H.C.F. of a and b Since, H.C.F. of a and b= H.C.F. of b and a    ∀a, b∈N. ∴                   a*b=b*a Thus, the operation * is commutative. For a, b, c∈N, we have: (a * b)* c=(H.C.F. of a and b) * c    =H.C.F. of a, b, and c a *(b * c)=a *(H.C.F. of b and c)    =H.C.F. of a, b, and c ∴(a*b)*c=a*(b*c) Thus, the operation * is associative. Let an element e∈N will be the identity for the operation   * if a*e=a=e* a ∀a∈N. But this relation is not true for all a∈N. Thus, the operation * does not have any identity in N. MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8IrFz0xbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@2014@

Q.9

Let * be a binary operation on the set  Q   of   rationalnumbers as follows:(i) a*b=a–b (ii) a*b=a2+b2(iii )a*b=a+ab (iv)a*b=a–b2(v)   a*b=ab4   vi a*b=ab2Find which of the binary operation s are commutativeand which are as sociative.

Ans

(i)  Since,  a*b=a−b ∀a, b∈Q=−(b−a) ∀a, b∈Q=−b*a ∀a, b∈QSo, the operation * is not cummutative.Now, let a, b, c∈Q, then  (a*b)*c=(a−b)*c=a−b−cand          a*(b*c)=a*(b−c)=a−(b−c)=a−b+cSo,   (a*b)*c≠a*(b*c)Thus, the operation * is not associative.(ii)Since,  a*b=a2+b2 ∀a,b∈Q=b2+a2 ∀a,b∈Q=b*a ∀a,b∈QSo, the operation * is cummutative.Now, let a,b,c∈Q, then  (a*b)*c=(a2+b2)*c=(a2+b2)2+c2and          a*(b*c)=a*(b2+c2)=a2+(b2+c2)2So,   (a*b)*c≠a*(b*c)Thus, the operation * is not associative.(iii)Since,  a*b=a+ab ∀a,b∈Qand   b*a=b+ba ∀a,b∈Q∴ a*b≠b*a ∀a,b∈QSo, the operation * is not cummutative.Now, let a,b,c∈Q, then  (a*b)*c=(a+ab)*c=(a+ab)+(a+ab)c=a+ab+ac+abcand          a*(b*c)=a*(b+bc)=a+a(b+bc)=a+ab+abcSo,   (a*b)*c≠a*(b*c)Thus, the operation * is not associative.(iv)Since,  a*b=(a−b)2 ∀a,b∈Q   ={−(b−a)}2                      =(b−a)2 ∀a,b∈Q   =b*a∴ a*b≠b*a           ∀a,b∈QSo, the operation * is not cummutative.Now, let a,b,c∈Q, then  (a*b)*c={(a−b)2}*c={(a−b)2−c}2=(a2−2ab+b2−c)2and         a*(b*c)=a*(b−c)2={a−(b−c)2}2=(a−b2+2bc−c2)2So,   (a*b)*c≠a*(b*c)Thus, the operation * is not associative.(v)Since,  a*b=ab4∀a,b∈Q   =ba4∀a,b∈Q   =b*a∴ a*b=b*a          ∀a,b∈QSo, the operation * is cummutative.Now, let a,b,c∈Q, then  (a*b)*c=(ab4)*c=(ab4)c4=abc16and          a*(b*c)=a*(bc4)=a(bc4)4=abc16So,   (a*b)*c=a*(b*c)Thus, the operation * is associative.(vi)Since,  a*b=ab2∀a,b∈Q   b*a =ba2∀a,b∈Q∴   a*b≠b*a ∀a,b∈QSo, the operation * is not cummutative.Now, let a,b,c∈Q, then  (a*b)*c=(ab2)*c=(ab2)c2=ab2c2and          a*(b*c)=a*(bc2)=a(bc2)2=ab2c4So,   (a*b)*c≠a*(b*c)Thus, the operation * is not associative.Hence, the operations defined in (ii), (iv), (v) are commutative and the operation defined in (v) is associative.

Q.10

Let * be a binary operation on the set  Q  of   rationalnumbersas follows:(i) a*b=a–b (ii)a*b=a2+b2(iii) a*b=a+ab (iv)a*b=a–b2(v)   a*b=ab4    vi a*b=ab2

Find which of the operations given above has identity.

Ans

An element e∈Q will be the identity element for the operation * if    a * e = a = e * a, ∀a∈Q.However, there is no such element e∈Q with respect to each of the six operations satisfying the above condition.Thus, none of the six operations in above question has identity.

Q.11

Let A=N×Nand*be the binary operation on A defined by(a, b)*(c, d)=(a+c, b+d)Show that *is commutative and associative. Find the identity element for*onA, if any.

Ans

A = N × N* is a binary operation on A and is defined by:(a, b) * (c, d) = (a + c, b + d)Let (a, b), (c, d)∈AThen,    a, b, c, d∈NSince,(a, b) * (c, d) = (a + c, b + d)and  (c, d) * (a, b) = (c + a, d + b)     = (a + c, b + d)[Addition is commutative in the set of natural numbers]∴(a, b)*(c, d) = (c, d)*(a, b)Therefore, the operation * is commutative.For associativity:Now, let (a, b), (c, d), (e, f)∈AThen, a, b, c, d, e, f∈NSo,    {(a, b) * (c, d)}*(e,f)={(a + c, b + d)}*(e,f)             =(a + c+ e, b + d+f )    (a, b) * {(c, d)*(e,f)}=(a, b) * {(c + e, d+ f)}             =(a + c + e, b + d + f)∴{(a, b) * (c, d)}*(e,f)=(a, b) * {(c, d)*(e,f)}Therefore, the operation * is associative.An element e=(e1+e2)∈A will be an identity element for the operation * if a*e=a=e*a∀a=(a1,a2)∈Ai.e.,   (a1+e1,a2+e2)=(a1,a2)=(e1+a1,e2+a2)which is not true for any element in A.Therefore, the operation * does not have any identity element.

Q.12

State whether the following statements are true or false.Justify.(i)For an arbitrary binary operation*on a set N,a*a=a“a*N.(ii) If*is acommutative binary operationon N, thena*(b*c)=(c*b)*a

Ans

(i) Define an operation * on N as:      a * b = a + b ∀a, b∈NThen, in particular, for b = a = 5, we have:      5 * 5 = 5 + 5=10≠5Therefore, statement (i) is false.(ii) R.H.S. = (c * b) * a    = (b * c) * a [* is commutative]    = a * (b * c) [Again, as * is commutative]    = L.H.S.∴ a * (b * c) = (c * b) * aTherefore, statement (ii) is true.

Q.13

Consider a binary operation*on N  defined as a*b=a3+b3.Choose the correct answer.(A )Is*both a ssociative and commutative?(B) Is*commutative but not associative?(C)Is*associative but not commutative?(D)Is*neither commutative nor associative?

Ans

On N, the operation * is defined as a*b = a 3 + b 3 . For, a, b∈N, we have: a*b= a 3 + b 3 = b 3 + a 3 =b*a   [Addition is commutative in N] Therefore, the operation * is commutative. For associative: Since, 2,3,4∈N So,             ( 2*3 )*4=( 2 3 + 3 3 )*4 =35*4 = 35 3 + 4 3 =42939 and           2*( 3*4 )=2*( 3 3 + 4 3 ) =2*( 91 ) = 2 3 + 91 3 =8+753571 =753579 ∴           ( 2*3 )*4≠2*( 3*4 ); where 2,3,4∈N Therefore, the operation * is not associative. Hence, the operation * is commutative, but not associative. Thus, the correct answer is B. MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8IrFz0xbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@00E8@

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FAQs (Frequently Asked Questions)

1. What does Chapter 1 of Class 12 Maths entail?

Chapter 1 of Class 12 NCERT Maths textbook is titled Relations and Functions. The chapter forms an important part of the Class 12 Maths syllabus. This chapter is also essential from an examination point of view. This chapter helps students understand the types of relations and functions and binary operations. Hence, the chapter is significant to clear one’s basics. Additionally, this chapter helps students understand the succeeding chapters. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 is primarily helpful in the topic of binary operations. Extramarks provides NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4, Exercise 1.3, Exercise 1.2, and Exercise 1.1 on its website.

2. What is the best way to approach Chapter 1 of Class 12 Mathematics?

Students can start by recalling their already acquired knowledge of the topic Relations and Functions from Class 11. After revising, students must go through all the formulas and definitions. It should then be followed by an in-depth analysis of the textbook examples to understand the methods used to solve the problems. Then, the students can start practising the textbook exercises. Students can also use NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4, Exercise 1.3, Exercise 1.2 and Exercise 1.1 in case they get stuck on a problem. 

3. Why is it important to solve all questions of Class 12 Maths Chapter 1?

Mathematics is a subject that can be mastered through daily practice without putting in extra effort to memorise stuff. If students do not practise regularly they are bound to forget the formulas and their usage. NCERT textbooks come packed with exercises that quiz students on their understanding of the concepts. Therefore, it is necessary to practise all the questions of Chapter 1 Class 12 Maths. These questions are also essential for preparing for competitive exams like JEE, BITSAT, UPSC, NDA etc. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 and other exercises can come in handy while solving these questions.

4. Where can students find the easiest solutions to NCERT Chapter 1 Class 12 Maths Solutions?

The correct answers to the questions can be found at the back of the NCERT textbook. Students can verify whether the method used by them is correct or not. However, the textbook does not provide step-by-step methods for all the questions. Extramarks is a prominent website known for its easily accessible and comprehensive study material. Students can access NCERT solutions on the Extramarks website that are written in easy-to-understand language. Extramarks NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4, Exercise 1.3, Exercise 1.2, and Exercise 1.1 make the preparation process less complicated and more straightforward.  

5. What is Exercise 1.4 of Chapter 1 Class 12 Maths all about?

Exercise 1.4 is about binary operations, commutative binary operations and associative binary operations. The concept is an extension of an old concept. The students must be aware of four basic Mathematical operations i.e addition, subtraction, multiplication and division. Extramarks NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 can be used by students to practice this exercise.  

6. Is Chapter 1 of Class 12 Mathematics easy?

Chapter 1 of Class 12 Mathematics titled Relations and Functions is relatively easy as its basics have already been introduced in Class 11. However, without proper rehearsal of the topic, one is bound to find it difficult. Students must practise the topic thoroughly and actively recall it to maintain proficiency in it. NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4, Exercise 1.3, Exercise 1.2 and Exercise 1.1 can make the preparation easier.

7. How many types of Functions and Relations are there?

NCERT Textbook of class 12 covers six types of relations – Transitive Relation, Equivalence Relation, Empty Relation, Universal Relation, Symmetric Relation and Reflexive Relation. It also covers four types of functions – One-to-one function, Onto function, One-to-one and Onto function and Invertible function. 

8. Is Chapter 1 of Class 12 NCERT Mathematics essential for competitive exams?

Chapter 1 of Class 12 NCERT Mathematics is related to Relations and Functions. The topic Relations and Functions is one of the most important ones. NCERT textbooks are important for conceptual learning. To ace competitive exams, one must have an in-depth understanding and clarity of the basic concepts. NCERT textbooks are concise and cover the basics. Hence, Chapter 1 Class 12 NCERT Mathematics is essential for competitive exams. Extramarks NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4, Exercise 1.3, Exercise 1.2, Exercise 1.1 can help in making the preparation process easy for students. 

This was a complete explanation of the NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4. The exercise can be challenging but it is based on the topics and concepts covered previously in Class 11. However, the NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 do help students in practising important questions from the textbook while understanding the steps and process of solving the questions. The NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4 are curated by experts which makes it a reliable source of learning and understanding. Check out the Extramarks website for NCERT Solutions Class 12 Maths Chapter 1 Exercise 1.4.