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
Determinants : NCERT Solutions – Class 12 Maths (Ex 7)
Miscellaneous
1. Prove that the determinant
is independent of
Ans. Let
Expanding along first row,
=
=
which is independent of
2. Without expanding the determinants, prove that: 
Ans. L.H.S. =
Multiplying R1 by R2 by
and R3 by
,
=
= =
[Interchanging C1 and C3]
= =
[Interchanging C2 and C3]
Proved.
3. Evaluate:
Ans. Let
Expanding along first row,
=
=
=
=
= 1
4. If
and
are real numbers and
Show that either
or 
Ans. Given:
Here, Either
……….(i)
Or
[Expanding along first row]
and
and
and
and
……….(ii)
Therefore, from eq. (i) and (ii), either or
5. Solve the equation:
Ans. Given:
Either
……….(i)
Or
But this is contrary as given that .
Therefore, from eq. (i), is only the solution.
6. Prove that:
Ans. L.H.S. =
=
=
=
= =
= =
= R.H.S. Proved.
7. If
and B =
find
Ans. Given: and B =
Since, [Reversal law] ……….(i)
Now
= =
Therefore, exists.
and
and
adj. B =
=
From eq. (i),
=
8. Let A =
verify that:
(i)
(ii)
Ans. Given: Matrix A =
=
Therefore, exists.
and
and
adj. A =
= B (say)
=
………(i)
= =
Therefore, exists.
and
and
adj. B =
=
=
= ….(ii)
Now to find (say), where
C =
=
C =
C = =
=
Therefore, exists.
and
and
adj. A =
= ……….(iii)
Again
=
= = A (given)
(i)
=
[From eq. (ii) and (iii)]
(ii)
=
9. Evaluate:
Ans. Let
=
=
=
=
=
=
=
=
=
10. Evaluate:
Ans. Let
=
= =
=
Using properties of determinants in Exercises 11 to 15, prove that:
11.
Ans. L.H.S. = =
=
=
Expanding along third column, 
=
=
=
=
=
= = R.H.S.
12.
Ans. L.H.S. =
=
= (say) ……….(i)
Now
=
=
=
From eq. (i), L.H.S. =
……….(ii)
Now
=
Expanding along third column,
=
=
=
=
=
From eq. (i), L.H.S. =
= = R.H.S.
13.
Ans. L.H.S. =
=
=
=
=
=
=
=
= = R.H.S.
14.
Ans. L.H.S. =
=
=
=
= = 1 = R.H.S.
15.
= 0
Ans. L.H.S. =
=
=
=
=
= [
C2 and C3 have become identical]
= 0 = R.H.S.
16. Solve the system of the following equations: (Using matrices):
Ans. Putting and
in the given equations,
the matrix form of given equations is
[AX= B]
Here, A = X =
and B =
=
=
exists and unique solution is
……….(i)
Now and
and
adj. A =
=
And
From eq. (i),
=
=
Choose the correct answer in Exercise 17 to 19.
17. If are in A.P., then the determinant
is:
(A) 0
(B) 1
(C)
(D)
Ans. According to question, ……….(i)
Let
=
=
[From eq. (i)] = 0 [ R2 and R3 have become identical]
Therefore, option (A) is correct.
18. If are non-zero real numbers, then the inverse of matrix A =
is:
(A)
(B)
(C)
(D)
Ans. Given: Matrix A =
exists and unique solution is
……….(i)
Now and
and
adj. A =
=
And
=
=
=
Therefore, option (A) is correct.
19. Let A =
where
Then:
(A) Det (A) = 0
(B) Det (A)
(C) Det (A)
(D) Det (A)
Ans. Given: Matrix A =
……….(i)
Since
[
cannot be negative]
Therefore, option (D) is correct.