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
Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 8)
Exercise 5.8
1. Verify Rolle’s theorem for
Ans. Consider
(i) Function is continuous in as it is a polynomial function and polynomial function is always continuous.
(ii) exists in , hence derivable.
(iii) and
Conditions of Rolle’s theorem are satisfied, hence there exists, at least one such that
2. Examine if Rolles/ theorem is applicable to any of the following functions. Can you say something about the converse of Rolle’s theorem from these examples:
(i) for
(ii) for
(iii) for
Ans. (i) Being greatest integer function the given function is not differentiable and continuous
hence Rolle’s theorem is not applicable.
(ii) Being greatest integer function the given function is not differentiable and continuous hence Rolle’s theorem is not applicable.
(iii)
Hence, Rolle’s theorem is not applicable.
3. If R is a differentiable function and if does not vanish anywhere, then prove that
Ans. For, Rolle’s theorem, if
(i) is continuous is
(ii) is derivable in
(iii)
Then,
It is given that is continuous and derivable, but
4. Verify Mean Value Theorem if in the interval where and
Ans. (i) Function is continuous in [1, 4] as it is a polynomial function and polynomial function is always continuous.
(ii) exists in [1, 4], hence derivable. Conditions of MVT theorem are satisfied, hence there exists, at least one such that
5. Verify Mean Value Theorem if in the interval where and Find all for which
Ans. (i) Function is continuous in [1, 3] as it is a polynomial function and polynomial function is always continuous.
(ii) exists in [1, 3], hence derivable. Conditions of MVT theorem are satisfied, hence there exists, at least one such that
or
or
or
and other value
Since , therefore the value of does not exist such that .
6. Examine the applicability of Mean Value Theorem for all the three functions being given below:
(i) for
(ii) for
(iii) for
Ans. Mean Value Theorem states that for a function R, if
(i) is continuous on
(ii) is differentiable on
Then there exist some such that
Therefore, the Mean Value Theorem is not applicable to those functions that do not satisfy any of the two conditions of the hypothesis.
(i) for
It is evident that the given function is not continuous at and
Therefore,
is not continuous at
Now let be an integer such that
L.H.L. =
And R.H.L. =
Since, L.H.L. R.H.L.,
Therefore is not differentiable at
Hence Mean Value Theorem is not applicable for for
(ii) for
It is evident that the given function is not continuous at and
Therefore,
is not continuous at
Now let be an integer such that
L.H.L. =
And R.H.L. =
Since, L.H.L. R.H.L.,
Therefore is not differentiable at
Hence Mean Value Theorem is not applicable for for
(iii) for ……….(i)
Here, is a polynomial function of degree 2.
Therefore, is continuous and derivable everywhere i.e., on the real time
Hence is continuous in the closed interval [1, 2] and derivable in open interval (1, 2).
Therefore, both conditions of Mean Value Theorem are satisfied.
Now, From eq. (i),
Again, From eq. (i),
And From eq. (ii),
Therefore, Mean Value Theorem is verified.