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Install Now**NCERT Solutions class 12 Maths Exercise 1.3 Class 12** Maths book 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. Let **** : {1, 3, 4} **** {1, 2, 5} and **** : {1, 2, 5} **** {1, 3} be given by **** = {(1, 2), (3, 5), (4, 1)} and **** = {(1, 3), (2, 3), (5, 1)}. Write down **** **

**Ans. ** = {(1, 2), (3, 5), (4, 1)} and = {(1, 3), (2, 3), (5, 1)}

Now, and

and

Hence, {(1, 3), (3, 1), (4, 3)}

**2. Let **** and **** be functions from R to R. Show that:**

**Ans. **(a) To prove:

L. H. S. = = = = R. H. S.

(b) To prove:

L. H. S. = = = = R. H. S.

**3. Find **** and ****, if:**

**(i) **** and **** **

**(ii) **** and **** **

**Ans. **To find: and

(i) and

and = =

(ii) and

and =

#### NCERT Solutions class 12 Maths Exercise 1.3

**4. If **** show that **** for all **** What is the inverse of **

**Ans. **Given:

L.H.S. = = = =

= = R.H.S.

Now,

Hence inverse of

**5. State with reason whether following functions have inverse:**

**(i) **** : {1, 2, 3, 4} **** {10} with **** = {(1, 10), (2, 10), (3, 10), (4, 10)}**

**(ii) **** : {5, 6, 7, 8} **** {1, 2, 3, 4} with **** = {(5, 4), (6, 3), (7, 4), (8, 2)}**

**(iii) **** : {2, 3, 4, 5} **** {7, 9, 11, 13} with **** = {(2, 7), (3, 9), (4, 11), (5, 13)}**

**Ans. **(i) = {(1, 10), (2, 10), (3, 10), (4, 10)}

It is many-one function, therefore has no inverse.

(ii) = {(5, 4), (6, 3), (7, 4), (8, 2)}

It is many-one function, therefore has no inverse.

(iii) = {(2, 7), (3, 9), (4, 11), (5, 13)}

is one-one onto function, therefore, has an inverse.

**6. Show that **** R given by **** is one-one. Find the inverse of the function **** Range **** **

**Ans. Part I:** R given by

Let , then and

When then

is one-one.

**Part II**: Let Range of

for some in

As

is onto.

Therefore,

**7. Consider **** : R **** R given by **** Show that **** is invertible. Find the inverse of **** **

**Ans. **Consider : R R given by

Let R, then and

Now, for , then is one-one.

Let Range of

is onto.

Therefore, is invertible and hence, .

**8. Consider **** given by **** Show that **** is invertible with the inverse **** of **** given by **** where **** is the set of all non-negative real numbers.**

**Ans. **Consider and

Let R , then and

is one-one.

Now

as

is onto.

Therefore, is invertible and .

**9. Consider **** given by **** Show that **** is invertible with **** **

**Ans. **Consider and

Let R , then and

Now, then

is one-one.

Now, again

= = =

=

= is onto.

Therefore, is invertible and .

**10. Let **** be an invertible function. Show that **** has unique inverse.**

**(Hint: Suppose **** and **** are two inverses of **** Then for all **** Use one-one ness of ****).**

**Ans. **Given: be an invertible function.

Thus is 1 – 1 and onto and therefore exists.

Let and be two inverses of Then for all Y,

The inverse is unique and hence has a unique inverse.

**11. Consider ****: {1, 2, 3} **** **** given by **** and **** Find **** and show that **** **

**Ans. **, then it is clear that is 1 – 1 and onto and therefore exists.

Also, and

Hence,

**12. Let ****be an invertible function. Show that the inverse of **** is ****, i.e., **

**Ans. **Let be an invertible function.

Then is one-one and onto

X where is also one-one and onto such that

and

Now, and

**13. If **** : R **** R given by **** then **** is:**

**(A) **

**(B) **

**(C) **

**(D) **** **

**Ans. ** : R R and

=

= = =

Therefore, option (C) is correct.

**14. Let **** : R – **** R be a function defined as **** The inverse of **** is the map **** : Range of **** given by:**

**(A) **

**(B) **

**(C) **

**(D) **

**Ans. **Given: : R – R and

Now, Range of

Let

Therefore, option (B) is correct.

## NCERT Solutions class 12 Maths Exercise 1.3

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes textbook 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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in q 11 it should be (a,1)

thankyou very much these solutions have helped me very much

useful

thanks for help

Thanks

Thanks a lot because these solutions help me very much

Really helpful solutions.

I wanted solutions that i could write in exam with proper language and this has helped me so much.