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Install Now**NCERT Solutions class 12 Maths Exercise 1.2** solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

**Download NCERT solutions for Relations and Functions as PDF.**

## NCERT Solutions class 12 Maths Relations and Functions

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

**Ans.**

**Part I**: and

If then

is one-one.

is onto.

**Part II**: If

where N

is one-one.

But every real number belonging to co-domain may not have a pre-image in N.

e.g. N is not onto.

**2. Check the injectivity and surjectivity of the following functions:**

**(i) **** given by **** **

**(ii) **** given by **

**(iii) **** given by **

**(iv) **** given by **

**(v) **** given by **

**Ans. (i)** given by

If then

is injective.

There are such numbers of co-domain which have no image in domain N.

e.g. 3 co-domain N, but there is no pre-image in domain of

therefore is not onto. is not surjective.

**(ii) ** given by

Since, Z = therefore,

and 1 have same image. is not injective.

There are such numbers of co-domain which have no image in domain Z.

e.g. 3 co-domain, but domain of is not surjective.

**(iii)** given by

As

and 1 have same image. is not injective.

e.g. co-domain, but domain R of is not surjective.

**(iv) ** given by

If then

i.e., for every N, has a unique image in its co-domain. is injective.

There are many such members of co-domain of which do not have pre-image in its domain e.g., 2, 3, etc.

Therefore is not onto. is not surjective.

**(v)** given by

If then

i.e., for every Z, has a unique image in its co-domain. is injective.

There are many such members of co-domain of which do not have pre-image in its domain.

Therefore is not onto.

**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 **** **

**Ans. **Function , given by

and

is not one-one.

All the images of R belong to its domain have integers as the images in co-domain. But no fraction proper or improper belonging to co-domain of has any pre-image in its domain.

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

**Ans. **Modulus Function , given by

Now

contains

Thus negative integers are not images of any element. is not one-one.

Also second set R contains some negative numbers which are not images of any real number.

is not onto.

**5. Show that the Signum Function ****, given by **** is neither one-one nor onto.**

**Ans. **Signum Function , given by

for

is not one-one.

Except no other members of co-domain of has any pre-image its domain.

is not onto.

#### NCERT Solutions class 12 Maths Exercise 1.2

**6. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let **** = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that **** is one-one.**

**Ans. **A = {1, 2, 3}, B = {4, 5, 6, 7} and = {(1, 4), (2, 5), (3, 6)}

Here, and

Here, also distinct elements of A have distinct images in B.

Therefore, is not one-one and is not bijective.

#### NCERT Solutions class 12 Maths Exercise 1.2

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

**(i) **** defined by **** **

**(ii) **** defined by **** **

**Ans. (i) ** defined by

Now, if R, then and

And if , then is one-one.

Again, if every element of Y (– R) is image of some element of X (R) under i.e., for every Y, there exists an element in X such that

Now

is onto or bijective function.

**(ii) ** defined by

Now, if R, then and

And if , then

is not one-one.

Again, if every element of Y (– R) is image of some element of X (R) under i.e., for every Y, there exists an element in X such that

Now,

is not onto.

Therefore, is not bijective.

#### NCERT Solutions class 12 Maths Exercise 1.2

**8. Let A and B be sets. Show that **** : A x B **** B x A such that **** is bijective function.**

**Ans. **Injectivity: Let and A x B such that

and

=

= for all A x B

So, is injective.

Surjectivity: Let be an arbitrary element of B x A. Then B and A.

A x B

Thus, for all B x A, their exists A x B such that

So, A x B B x A is an onto function, therefore is bijective.

**9. Let **** be defined by **** for all **** N.**

**State whether the function **** is bijective. Justify your answer.**

**Ans. ** be defined by

(a) and

The elements 1, 2, belonging to domain of have the same image 1 in its co-domain.

So, is not one-one, therefore, is not injective.

(b) Every number of co-domain has pre-image in its domain e.g., 1 has two pre-images 1 and 2.

So, is onto, therefore, is not bijective.

#### NCERT Solutions class 12 Maths Exercise 1.2

**10. Let A = R – {3} and B = R – {1}. Consider the function **** : A **** B defined by **** Is **** one-one and onto? Justify your answer.**

**Ans. **A = R – {3} and B = R – {1} and

Let A, then and

Now, for

is one-one function.

Now

=

Therefore, is an onto function.

#### NCERT Solutions class 12 Maths Exercise 1.2

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

**(A) **** is one-one onto**

**(B) **** is many-one onto**

**(C) **** is one-one but not onto**

**(D) **** is neither one-one nor onto**

**Ans. ** and

Let , then and

Therefore, is not one-one function.

Now,

and

Therefore, is not onto function.

Therefore, option (D) is correct.

#### NCERT Solutions class 12 Maths Exercise 1.2

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

**(A) **** is one-one onto**

**(B) **** is many-one onto**

**(C) **** is one-one but not onto**

**(D) **** is neither one-one nor onto**

**Ans. **Let R such that

Therefore, is one-one function.

Now, consider R (co-domain of ) certainly R (domain of )

Thus for all R (co-domain of ) there exists R (domain of ) such that

Therefore, is onto function.

Therefore, option (A) is correct.

## NCERT Solutions class 12 Maths Exercise 1.2

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

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