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 2)
Exercise 1.2
1. Show that the function defined by is oneone and onto, where is the set of all nonzero real numbers. Is the result true, if the domain is replaced by N with codomain being same as ?
Ans.
Part I: and
If then
is oneone.
is onto.
Part II: If
where N
is oneone.
But every real number belonging to codomain may not have a preimage in N.
e.g. N is not onto.
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 codomain which have no image in domain N.
e.g. 3 codomain N, but there is no preimage 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 codomain which have no image in domain Z.
e.g. 3 codomain, but domain of is not surjective.
(iii) given by
As
and 1 have same image. is not injective.
e.g. codomain, but domain R of is not surjective.
(iv) given by
If then
i.e., for every N, has a unique image in its codomain. is injective.
There are many such members of codomain of which do not have preimage 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 codomain. is injective.
There are many such members of codomain of which do not have preimage in its domain.
Therefore is not onto.
3. Prove that the Greatest integer Function , given by is neither oneone nor onto, where denotes the greatest integer less than or equal to
Ans. Function , given by
and
is not oneone.
All the images of R belong to its domain have integers as the images in codomain. But no fraction proper or improper belonging to codomain of has any preimage in its domain.
4. Show that the Modulus Function , given by is neither oneone 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 oneone.
Also second set R contains some negative numbers which are not images of any real number.
is not onto.
5. Show that the Signum Function , given by is neither oneone nor onto.
Ans. Signum Function , given by
for
is not oneone.
Except no other members of codomain of has any preimage its domain.
is not onto.
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 oneone.
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 oneone and is not bijective.
7. In each of the following cases, state whether the function is oneone, 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 oneone.
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 oneone.
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.
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.
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 codomain.
So, is not oneone, therefore, is not injective.
(b) Every number of codomain has preimage in its domain e.g., 1 has two preimages 1 and 2.
So, is onto, therefore, is not bijective.
10. Let A = R – {3} and B = R – {1}. Consider the function : A B defined by Is oneone and onto? Justify your answer.
Ans. A = R – {3} and B = R – {1} and
Let A, then and
Now, for
is oneone function.
Now
=
Therefore, is an onto function.
11. Let be defined as Choose the correct answer:
(A) is oneone onto
(B) is manyone onto
(C) is oneone but not onto
(D) is neither oneone nor onto
Ans. and
Let , then and
Therefore, is not oneone function.
Now,
and
Therefore, is not onto function.
Therefore, option (D) is correct.
12. Let be defined as Choose the correct answer:
(A) is oneone onto
(B) is manyone onto
(C) is oneone but not onto
(D) is neither oneone nor onto
Ans. Let R such that
Therefore, is oneone function.
Now, consider R (codomain of ) certainly R (domain of )
Thus for all R (codomain of ) there exists R (domain of ) such that
Therefore, is onto function.
Therefore, option (A) is correct.