NCERT Solutions Class 12
-
NCERT Solutions-Mathematics
- Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 1)
- Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 2)
- Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 3)
- Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 4)
- Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 5)
- Inverse Trigonometric Function : NCERT Solutions – Class 12 Maths (Ex 1)
- Inverse Trigonometric Function : NCERT Solutions – Class 12 Maths (Ex 2)
- Inverse Trigonometric Function : NCERT Solutions – Class 12 Maths (Ex 3)
- Matrices : NCERT Solutions – Class 12 Maths (Ex 1)
- Matrices : NCERT Solutions – Class 12 Maths (Ex 2)
- Matrices : NCERT Solutions – Class 12 Maths (Ex 3)
- Matrices : NCERT Solutions – Class 12 Maths (Ex 4)
- Matrices : NCERT Solutions – Class 12 Maths (Ex 5)
- Determinants : NCERT Solutions – Class 12 Maths (Ex 1)
- Determinants : NCERT Solutions – Class 12 Maths (Ex 2)
- Determinants : NCERT Solutions – Class 12 Maths (Ex 3)
- Determinants : NCERT Solutions – Class 12 Maths (Ex 4)
- Determinants : NCERT Solutions – Class 12 Maths (Ex 5)
- Determinants : NCERT Solutions – Class 12 Maths (Ex 6)
- Determinants : NCERT Solutions – Class 12 Maths (Ex 7)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 1)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 2)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 3)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 4)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 5)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 6)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 7)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 8)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 9)
-
NCERT Solutions-Chemistry
- Aldehydes, Ketones and Carboxylic Acids : NCERT Solutions – Class 12 Chemistry
- Alcohols, Phenols and Ethers : NCERT Solutions – Class 12 Chemistry
- Amines : NCERT Solutions – Class 12 Chemistry
- Biomolecules : NCERT Solutions – Class 12 Chemistry
- Chemical Kinetics : NCERT Solutions – Class 12 Chemistry
- Chemistry in Everyday Life : NCERT Solutions – Class 12 Chemistry
- Coordination Compounds : NCERT Solutions – Class 12 Chemistry
- Electrochemistry : NCERT Solutions – Class 12 Chemistry
- General Principles and Processes of Isolation of Elements : NCERT Solutions – Class 12 Chemistry
- Haloalkanes and Haloarenes : NCERT Solutions – Class 12 Chemistry
- Polymers : NCERT Solutions – Class 12 Chemistry
- Surface Chemistry : NCERT Solutions – Class 12 Chemistry
- The d-and f-Block Elements : NCERT Solutions – Class 12 Chemistry
- The p-Block Elements : NCERT Solutions – Class 12 Chemistry
- The Solid State : NCERT Solutions – Class 12 Chemistry
- Solutions : NCERT Solutions – Class 12 Chemistry
-
NCERT Solutions-Biology
-
NCERT Solutions-Physics
- Electrostatic Potential And Capacitance : NCERT Solutions – Class 12 Physics
- Electric Charges And Fields : NCERT Solutions – Class 12 Physics
- Semiconductor Electronics: Materials, Devices And Simple Circuits : NCERT Solutions – Class 12 Physics
- Ray Optics And Optical Instruments : NCERT Solutions – Class 12 Physics
- Nuclei : NCERT Solutions – Class 12 Physics
- Moving Charges And Magnetism : NCERT Solutions – Class 12 Physics
- Magnetism And Matter : NCERT Solutions – Class 12 Physics
- Electromagnetic Induction : NCERT Solutions – Class 12 Physics
- Dual Nature Of Radiation And Matter : NCERT Solutions – Class 12 Physics
- Current Electricity : NCERT Solutions – Class 12 Physics
- Communication Systems : NCERT Solutions – Class 12 Physics
- Atoms : NCERT Solutions – Class 12 Physics
- Alternating Current : NCERT Solutions – Class 12 Physics
Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 5)
Miscellaneous
1. Let be defined as
Find the function
such that
Ans. Given:
Now and
2. Let be defined as
if
is odd and
if
is even. Show that
is invertible. Find the inverse of
Here, W is the set of all whole numbers.
Ans. Given: defined as
Injectivity: Let be any two odd real numbers, then
Again, let be any two even whole numbers, then
Is is even and
is odd, then
Also, if odd and
is even, then
Hence,
is an injective mapping.
Surjectivity: Let be an arbitrary whole number.
If is an odd number, then there exists an even whole number
such that
If is an even number, then there exists an odd whole number
such that
Therefore, every W has its pre-image in W.
So, is a surjective. Thus
is invertible and
exists.
For :
and
Hence,
3. If
is defined by
find
Ans. Given:
=
4. Show that the function
defined by
R is one-one and onto function.
Ans. is one-one: For any
R – {+1}, we have
Therefore, is one-one function.
If is one-one, let
R – {1}, then
It is cleat that R for all
R – {1}, also
Because
which is not possible.
Thus for each R – {1} there exists R – {1} such that
Therefore is onto function.
5. Show that the function
given by
is injective.
Ans. Let R be such that
Therefore, is one-one function, hence
is injective.
6. Give examples of two functions
and
such that
is injective but
is not injective.
(Hint: Consider and
)
Ans. Given: two functions and
Let and
Therefore, is injective but
is not injective.
7. Give examples of two functions
and
such that
is onto but
is not onto.
(Hint: Consider and
)
Ans. Let
These are two examples in which is onto but
is not onto.
8. Given a non empty set X, consider P (X) which is the set of all subsets of X.
Define the relation AR in P (X) as follows:
For subsets A, B in P (X), ARB if and only if AB. Is R an equivalence relation on P (X)? Justify your answer.
Ans. (i) A A
R is reflexive.
(ii) A B
B
A
R is not commutative.
(iii) If A B, B
C, then A
C
R is transitive.
Therefore, R is not equivalent relation.
9. Given a non-empty set X, consider the binary operation * : P (X) x P (X) P (X) given by A * B = A
B
A, B in P (X), where P (X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P (X) with respect to the operation *.
Ans. Let S be a non-empty set and P(S) be its power set. Let any two subsets A and B of S.
A
B
S
A
B
P(S)
Therefore, is an binary operation on P(S).
Similarly, if A, B P(S) and A – B
P(S), then the intersection of sets
and difference of sets are also binary operation on P(S) and A
S = A = S
A for every subset A of sets
A
S = A = S
A for all A
P(S)
S is the identity element for intersection
on P(S).
10. Find the number of all onto functions from the set {1, 2, 3, ……., } to itself.
Ans. The number of onto functions that can be defined from a finite set A containing elements onto a finite set B containing
elements =
11. Let S =
and T = {1, 2, 3}. Find
of the following functions F from S to T, if it exists.
(i) F =
(ii) F =
Ans. S = and T = {1, 2, 3}
(i) F =
(ii)
F is not one-one function, since element and
have the same image 1.
Therefore, F is not one-one function.
12. Consider the binary operation * : R x R
R and o = R x R
R defined as
and
R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that
R,
[If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.
Ans. Part I: also
operation * is commutative.
Now,
And
Here,
operation * is not associative.
Part II: R
And,
operation
is not commutative.
Now and
Here
operation
is associative.
Part III: L.H.S. =
R.H.S. =
= L.H.S. Proved.
Now, another distribution law:
L.H.S.
R.H.S.
As L.H.S. R.H.S.
Therefore, the operation does not distribute over.
13. Given a non-empty set X, let * : P (X) x P (X)
P (X) be defined as A * B = (A – B)
(B – A),
A, B
P (X). Show that the empty set
is the identity for the operation * and all the elements A of P (X) are invertible with A-1 = A. (Hint:
and
)
Ans. For every A P(X), we have
=
And =
is the identity element for the operation * on P(X).
Also A * A = (A – A) (A – A) =
Every element A of P(X) is invertible with
= A.
14. Define binary operation * on the set {0, 1, 2, 3, 4, 5} as
Show that zero is the identity for this operation and each element of the set is invertible with
being the inverse of
Ans. A binary operation (or composition) * on a (non-empty) set is a function * : A x A A. We denote
by
for every ordered pair
A x A.
A binary operation on a no-empty set A is a rule that associates with every ordered pair of elements
(distinct or equal) of A some unique element
of A.
* | 0 | 1 | 2 | 3 | 4 | 5 |
0 | 0 | 1 | 2 | 3 | 4 | 5 |
1 | 1 | 2 | 3 | 4 | 5 | 0 |
2 | 2 | 3 | 4 | 5 | 0 | 1 |
3 | 3 | 4 | 5 | 0 | 1 | 2 |
4 | 4 | 5 | 0 | 1 | 2 | 3 |
5 | 5 | 0 | 1 | 2 | 3 | 4 |
For all A, we have
(mod 6) = 0
And and
0 is the identity element for the operation.
Also on 0 = 0 – 0 = 0 *
2 * 1 = 3 = 1 * 2
15. Let A = {–1, 0, 1, 2}, B = {–4, –2, 0, 2} and
be the functions defined by
A and
A. Are
and
equal? Justify your answer.
(Hint: One may note that two functions and
such that
A, are called equal functions).
Ans. When then
and
At
and
At
and
At
and
Thus for each A,
Therefore, and
are equal function.
16. Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is:
(A) 1
(B) 2
(C) 3
(D) 4
Ans. It is clear that 1 is reflexive and symmetric but not transitive.
Therefore, option (A) is correct.
17. Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is:
(A) 1
(B) 2
(C) 3
(D) 4
Ans. 2
Therefore, option (B) is correct.
18. Let
be the Signum Function defined as
and
be the Greatest Function given by
where
is greatest integer less than or equal to
Then, does
and
coincide in (0, 1)?
Ans. It is clear that and
Consider which lie on (0, # 1)
Now,
And
in (0, 1]
Therefore, option (B) is correct.
19. Number of binary operation on the set
are:
(A) 10
(b) 16
(C) 20
(D) 8
Ans. A =
A x A =
= 4
Number of subsets = = 16
Hence number of binary operation is 16.
Therefore, option (B) is correct.