NCERT Solutions class 12 Maths Exercise 1.3

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)}

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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.

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3. Find  and , if:

(i)  and  

(ii)  and  

Ans. To find:  and

(i)  and

 and  =  =

(ii)  and

 

and  =

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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

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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,

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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, .

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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 .

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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.

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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,

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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

 

 

 

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13. If  : R  R given by  then  is:

(A)

(B)

(C)

(D)  

Ans.  : R  R and

  =

=  =  =

Therefore, option (C) is correct.

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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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