# NCERT Solutions for Exercise 1.2 Class 12 Maths Chapter 1 - Relations and Functions

NCERT Solutions for Class 12 Maths chapter 1 exercise 1.2 is the most important exercise of chapter Relations and Functions and it includes topics like types of relations, functions, binary operations etc. **Exercise 1.2 Class 12 Maths **exposes students to questions like proving one to one functions etc. Such questions are generally asked in Board examinations. Solving NCERT syllabus for Class 12 Maths chapter 1 exercise 1.2 is recommended to students to score well in CBSE Class 12 board exam. The contribution of chapter Relations and Functions is high in competitive exams also like JEE main and NEET. There are many questions asked in previous years which are based on concepts of Class 12 Maths chapter 1 exercise 1.2. The NCERT chapter Relations and Functions also has the following exercise for practice.

Relations and Functions Exercise 1.1

Relations and Functions Exercise 1.3

Relations and Functions Exercise 1.4

Relations and Functions Miscellaneous Exercise

## ** NCERT Solutions For Class 12 Maths Chapter 1 Relations And Functions: Exercise 1.2 **

** Question:1 ** Show that the function defined by is one-one and onto,where R _{ ∗ } is the set of all non-zero real numbers. Is the result true, if the domain R _{ ∗ } is replaced by N with co-domain being same as R _{ ∗ } ?

** Answer: **

Given, is defined by .

One - One :

f is one-one.

Onto:

We have , then there exists ( Here ) such that

.

Hence, the function is one-one and onto.

If the domain R _{ ∗ } is replaced by N with co-domain being same as R _{ ∗ i.e. } defined by

g is one-one.

For ,

but there does not exists any x in N.

Hence, function g is one-one but not onto.

** Question:2(i) ** Check the injectivity and surjectivity of the following functions:

(i) given by

** Answer: **

One- one:

then

f is one- one i.e. injective.

For there is no x in N such that

f is not onto i.e. not surjective.

Hence, f is injective but not surjective.

** Question:2(ii) ** Check the injectivity and surjectivity of the following functions:

(ii) given by

Answer:

One- one:

For then

but

f is not one- one i.e. not injective.

For there is no x in Z such that

f is not onto i.e. not surjective.

Hence, f is neither injective nor surjective.

** Question:2(iii) ** Check the injectivity and surjectivity of the following functions:

(iii) given by

Answer:

One- one:

For then

but

f is not one- one i.e. not injective.

For there is no x in R such that

f is not onto i.e. not surjective.

Hence, f is not injective and not surjective.

** Question:2(iv) ** Check the injectivity and surjectivity of the following functions:

(iv) given by

** Answer: **

One- one:

then

f is one- one i.e. injective.

For there is no x in N such that

f is not onto i.e. not surjective.

Hence, f is injective but not surjective.

** Question:2(v) ** Check the injectivity and surjectivity of the following functions:

(v) given by

** Answer: **

One- one:

For then

f is one- one i.e. injective.

For there is no x in Z such that

f is not onto i.e. not surjective.

Hence, f is injective but not surjective.

** Question:3 ** Prove that the Greatest Integer Function , given by , is neither one-one nor onto, where denotes the greatest integer less than or equal to .

** Answer: **

One- one:

For then and

but

f is not one- one i.e. not injective.

For there is no x in R such that

f is not onto i.e. not surjective.

Hence, f is not injective but not surjective.

** Question:4 ** Show that the Modulus Function f : R → R, given by , is neither one-one nor onto, where is if is positive or 0 and is , if is negative.

** Answer: **

One- one:

For then

f is not one- one i.e. not injective.

For ,

We know is always positive there is no x in R such that

f is not onto i.e. not surjective.

Hence, , is neither one-one nor onto.

** Question:5 ** Show that the Signum Function , given by

is neither one-one nor onto.

** Answer: **

is given by

As we can see , but

So it is not one-one.

Now, f(x) takes only 3 values (1,0,-1) for the element -3 in codomain ,there does not exists x in domain such that .

So it is not onto.

Hence, signum function is neither one-one nor onto.

** Question:6 ** Let , and let be a function from A to B. Show that f is one-one.

** Answer: **

Every element of A has a distant value in f.

Hence, it is one-one.

** Question:7(i) ** In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i) defined by

** Answer: **

Let there be such that

f is one-one.

Let there be ,

Puting value of x,

f is onto.

f is both one-one and onto hence, f is bijective.

** Question:7(ii) ** In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(ii) defined by

** Answer: **

Let there be such that

For and

f is not one-one.

Let there be (-2 in codomain of R)

There does not exists any x in domain R such that

f is not onto.

Hence, f is neither one-one nor onto.

** Question:8 ** Let A and B be sets. Show that such that is

bijective function.

** Answer: **

Let

such that

and

f is one- one

Let,

then there exists such that

f is onto.

Hence, it is bijective.

** Question:9 ** Let be defined by for all . State whether the function f is bijective. Justify your answer.

** Answer: **

,

Here we can observe,

and

As we can see but

f is not one-one.

Let, (N=co-domain)

case1 n be even

For ,

then there is such that

case2 n be odd

For ,

then there is such that

f is onto.

f is not one-one but onto

hence, the function f is not bijective.

** Question:10 ** Let and . Consider the function defined by . Is f one-one and onto? Justify your answer.

** Answer: **

Let such that

f is one-one.

Let, then

such that

For any there exists such that

f is onto

Hence, the function is one-one and onto.

** Question:11 ** Let be defined as . Choose the correct answer.

(A) f is one-one onto

(B) f is many-one onto

(C) f is one-one but not onto

(D) f is neither one-one nor onto.

** Answer: **

One- one:

For then

does not imply that

example: and

f is not one- one

For there is no x in R such that

f is not onto.

Hence, f is neither one-one nor onto.

Option D is correct.

** Question:12 ** Let be defined as . Choose the correct answer.

(A) f is one-one onto

(B) f is many-one onto

(C) f is one-one but not onto

(D) f is neither one-one nor onto.

** Answer: **

One - One :

Let

f is one-one.

Onto:

We have , then there exists such that

.

Hence, the function is one-one and onto.

The correct answer is A .

**More About NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.2**

The NCERT Class 12 Maths chapter Relations and Functions has a total of 5 exercises including miscellaneous. Exercise 1.2 Class 12 Maths covers solutions to 12 main questions and their sub-questions. Most of the questions are related to proving a function one to one. Hence NCERT Solutions for Class 12 Maths chapter 1 exercise 1.2 can be referred for learning the concepts related to proof etc.

Also Read| NCERT Notes For Class 12 Mathematics Chapter 1

**Benefits of NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.2**

The Class 12th Maths chapter 1 exercise provided here is solved by subject matter experts having rich experience in the domain of competitive exam preparation.

Students can practice Exercise 1.2 Class 12 Maths to prepare various concepts like signus functions, one to one functions etc, many direct questions are asked in Board exams from this chapter.

These Class 12 Maths chapter 1 exercise 1.2 solutions can be referred by students to revise before the exam and clarify any doubt regarding solution of exercise questions.

NCERT Solutions for

**Class 12 Maths**chapter 1 exercise 1.2 provided here are most recommended solutions for students aspiring to score well in examinations.

### Also see-

NCERT exemplar solutions class 12 maths chapter 1

NCERT solutions for class 12 maths chapter 1

**NCERT Solutions Subject Wise**

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**Subject wise NCERT Exemplar solutions**

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