NCERT Solutions Class 12

NCERT SolutionsMathematics
 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 SolutionsChemistry
 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 dand fBlock Elements : NCERT Solutions – Class 12 Chemistry
 The pBlock Elements : NCERT Solutions – Class 12 Chemistry
 The Solid State : NCERT Solutions – Class 12 Chemistry
 Solutions : NCERT Solutions – Class 12 Chemistry

NCERT SolutionsBiology

NCERT SolutionsPhysics
 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 preimage 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 oneone and onto function.
Ans. is oneone: For any R – {+1}, we have
Therefore, is oneone function.
If is oneone, 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 oneone 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 nonempty 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 nonempty 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 oneone function, since element and have the same image 1.
Therefore, F is not oneone 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 nonempty 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 (nonempty) set is a function * : A x A A. We denote by for every ordered pair A x A.
A binary operation on a noempty 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.