NCERT Solutions Class 12 Maths Exercise 1.2

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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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