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 4)
Exercise 1.4
1. Determine whether or not each of the definition of * given below gives a binary operation. In the event that * is not a binary operation, given justification for this.
(i) On define * by
(ii) On define * by
(iii) On R, define * by
(iv) On define * by
(v) On define * by
Ans. (i) On = {1, 2, 3, …..},
Let
Therefore, operation * is not a binary operation on .
(ii) On = {1, 2, 3, …..},
Let
Therefore, operation * is a binary operation on .
(iii) on R (set of real numbers)
Let R
Therefore, operation * is a binary operation on R.
(iv) On = {1, 2, 3, …..},
Let
Therefore, operation * is a binary operation on .
(v) On = {1, 2, 3, …..},
Let
Therefore, operation * is a binary operation on .
2. For each binary operation * defined below, determine whether * is commutative or associative:
(i) On define
(ii) On Q, define
(iii) On Q, define
(iv) On define
(v) On define
(vi) On R – {– 1}, define
Ans. (i) For commutativity: and =
For associativity: =
Also, =
Therefore, the operation * is neither commutative nor associative.
(ii) For commutativity: and
For associativity: =
Also, =
Therefore, the operation * is commutative but not associative.
(iii) For commutativity: and =
For associativity: =
Also, =
Therefore, the operation * is commutative and associative.
(iv) For commutativity: and
For associativity: =
Also, =
Therefore, the operation * is commutative but not associative.
(v) For commutativity: and
For associativity: =
Also, =
Therefore, the operation * is neither commutative nor associative.
(vi) For commutativity: and
For associativity: =
Also, =
Therefore, the operation * is neither commutative nor associative.
3. Consider the binary operation on the set {1, 2, 3, 4, 5} defined by Write the operation table of the operation
Ans. Let A = {1, 2, 3, 4, 5} defined by i.e., minimum of and
1  2  3  4  5  
1  1  1  1  1  1 
2  1  2  2  2  2 
3  1  2  3  3  3 
4  1  2  3  4  4 
5  1  2  3  4  5 
4. Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table (table 1.2).
(i) Compute (2 * 3) * 4 and 2 * (3 * 4)
(ii) Is * commutative?
(iii) Compute (2 * 3) * (4 * 5)
(Hint: Use the following table)
Table 1.2
*  1  2  3  4  5 
1  1  1  1  1  1 
2  1  2  1  2  1 
3  1  1  3  1  1 
4  1  2  1  4  1 
5  1  1  1  1  5 
Ans. (i) 2 * 3 = 1 and 3 * 4 = 1
Now (2 * 3) * 4 = 1 * 4 = 1 and 2 * (3 * 4) = 2 * 1 = 1
(ii) 2 * 3 = 1 and 3 * 4 = 1
2 * 3 = 3 * 2 and other element of the given set.
Hence the operation is commutative.
(iii) (2 * 3) * (4 * 5) = 1 * 1 = 1
5. Let *’ be the binary operation on the set {1, 2, 3, 4, 5} defined by H.C.F. of and Is the operation *’ same as the operation * defined in Exercise 4 above? Justify your answer.
Ans. Let A = {1, 2, 3, 4, 5} and H.C.F. of and
*’  1  2  3  4  5 
1  1  1  1  1  1 
2  1  2  1  2  1 
3  1  1  3  1  1 
4  1  2  1  4  1 
5  1  1  1  1  5 
6. Let * be the binary operation on N given by L.C.M. of and Find:
(i) 5 * 7, 10 * 16
(ii) Is * commutative?
(iii) Is * associative?
(iv) Find the identity of * in N.
(v) Which elements of N are invertible for the operation *?
Ans. L.C.M. of and
(i) 5 * 7 = L.C.M. of 5 and 7 = 35
20 * 16 = L.C.M. of 20 and 16 = 80
(ii) L.C.M. of and = L.C.M. of and =
Therefore, operation * is commutative.
(iii) =
=
Similarly,
Thus,
Therefore, the operation is associative.
(iv) Identity of * in N = 1 because = L.C.M. of and 1 =
(v) Only the element 1 in N is invertible for the operation * because
7. Is * defined on the set {1, 2, 3, 4, 5} by L.C.M. of and a binary operation? Justify your answer.
Ans. Let A = {1, 2, 3, 4, 5} and L.C.M. of and
*  1  2  3  4  5 
1  1  2  3  4  5 
2  2  2  6  4  10 
3  3  x  3  12  15 
4  4  4  12  4  20 
5  5  x  15  20  5 
Here, 2 * 3 = 6 A
Therefore, the operation * is not a binary operation.
8. Let * be the binary operation on N defined by H.C.F. of and Is * commutative? Is * associative? Does there exist identity for this binary operation on N?
Ans. H.C.F. of and
(i) H.C.F. of and = H.C.F. of and =
Therefore, operation * is commutative.
(ii) =
=
Therefore, the operation is associative.
Therefore, there does not exist any identity element.
9. Let * be a binary operation on the set Q of rational numbers as follows:
(i) (ii)
(iii) (iv)
(v) (vi)
Find which of the binary operations are commutative and which are associative.
Ans. (i) operation * is not commutative.
And
Here, operation * is not associative.
(ii) operation * is commutative.
And
Here, operation * is not associative.
(iii) and
Therefore, operation * is not commutative.
And
Here, operation * is not associative.
(iv) operation * is commutative.
And
Here, operation * is not associative.
(v) operation * is commutative.
And
Here, operation * is associative.
(vi) and operation * is not commutative.
And
Here, operation * is not associative.
10. Show that none of the operations given above the identity.
Ans. Let the identity be I.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Therefore, none of the operations given above has identity.
11. Let A = N x N and * be the binary operation on A defined by
Show that * is commutative and associative. Find the identity element for * on A, if any.
Ans. A = N x N and * is a binary operation defined on A.
The operation is commutative
Again,
And
Here, The operation is associative.
Let identity function be , then
For identity function
And for
As 0 N, therefore, identityelement does not exist.
12. State whether the following statements are true or false. Justify:
(i) For an arbitrary binary operation * on a set N,
(ii) If * is a commutative binary operation on N, then
Ans. (i) * being a binary operation on N, is defined as
Hence operation * is not defined, therefore, the given statement is false.
(ii) * being a binary operation on N.
Thus, , therefore the given statement is true.
13. Consider a binary operation * on N defined as . Choose the correct answer:
(A) Is * both associative and commutative?
(B) Is * commutative but not associative?
(C) Is * associative but not commutative?
(D) Is * neither commutative nor associative?
Ans. The operation * is commutative.
Again,
And
The operation * is not associative.
Therefore, option (B) is correct.