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# NCERT Solutions class 12 Maths Exercise 1.3

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## 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  =

### 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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### 7 thoughts on “NCERT Solutions class 12 Maths Exercise 1.3”

1. in q 11 it should be (a,1)

2. thankyou very much these solutions have helped me very much

3. useful

4. thanks for help

5. Thanks

6. Thanks a lot because these solutions help me very much