NCERT Solutions Class 12
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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)
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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
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NCERT Solutions-Biology
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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
Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 9)
Miscellaneous
Differentiate with respect to the functions in Exercises 1 to 5.
1.
Ans. Let
=
2.
Ans. Let =
=
3.
Ans. Let ……….(i)
4.
Ans. Let =
=
=
=
5.
Ans. Let
[By Quotient Rule]
=
=
Differentiate with respect to the functions in Exercises 6 to 11.
6.
Ans. Let ……….(i)
Now,
= =
And
= =
From eq. (i),
=
=
7.
Ans. Let ……….(i)
=
=
=
8. for some constants
and
Ans. Let for some constants
and
9.
Ans. Let ……….(i)
=
10. for some fixed
and
Ans. Let
= …….(i)
Now taking , let
……….(ii)
=
=
From eq. (ii),
From eq. (i),
11. for
Ans. Let for
Putting and
……….(i)
Now
=
=
……….(ii)
Again
=
=
……….(iii)
Putting the values from eq. (ii) and (iii) in eq. (i),
12. Find if
and
Ans. Given: and
and
=
=
=
13. Find if
Ans. Given:
=
= = 0
14. If for
prove that
Ans. Given:
=
= Proved.
15. If
for some
prove that
Ans. Given: ……….(i)
……….(ii)
Again
[From eq. (ii)
=
= ……….(iii)
Putting values of and
in the given expression,
=
= =
=
which is a constant and is independent of and
16. If
with
prove that
Ans. Given:
=
=
[Taking reciprocal]
17. If
and
find
Ans. Given: and
Differentiating both sides with respect to
and
and
and
and
Now
Again =
= =
=
18. If
show that
exists for all real
and find it.
Ans. Given: =
Now, L.H.D. at
=
=
L.H.D. at
= R.H.D. at
Therefore, is differentiable at
.
Now, L.H.D. at
=
=
And R.H.D. at
=
=
Again L.H.D. at
= R.H.D. at
19. Using mathematical induction, prove that
for all positive integers
Ans. Let be the given statement in the problem.
…..(i)
=
, which is true as
Now we suppose is true.
…….(ii)
To establish the truth of we prove,
=
=
Therefore, is true if
is true but
is true.
By Principal of Induction
is true for all
N.
20. Using the fact that
and the differentiation, obtain the sum formula for cosines.
Ans. Given:
Consider A and B as function of and differentiating both sides w.r.t.
21. Does there exist a function which is continuous everywhere but not differentiable at exactly two points?
Ans. Let us consider the function
is continuous everywhere but it is not differentiable at
and
22. If prove that
Ans. Given:
=
.23. If
show that
Ans. Given:
=
=
Differentiating both sides with respect to
Proved.