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Install NowNCERT 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.