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 5)
Exercise 4.5
Find adjoint of each of the matrices in Exercise 1 and 2.
1.
Ans. Here A =
A11 = Cofactor of
A12 = Cofactor of
A21 = Cofactor of
A22 = Cofactor of
adj. A =
=
2.
Ans. Here A =
=
adj. A =
Verify A (adj. A) = in Exercise 3 and 4.
3.
Ans. Let A =
adj. A =
A.(adj. A) =
= =
…..(i)
Again (adj. A). A =
= =
…..(ii)
And =
Again …..(iii)
From eq. (i), (ii) and (iii) A. (adj. A) = (adj. A). A =
4.
Ans. Let A =
=
adj. A =
A. (adj. A) =
=
= ……….(i)
Again (adj. A). A =
=
= ……….(ii)
And
=
Also =
……….(iii)
From eq. (i), (ii) and (iii) A. (adj. A) = (adj. A). A =
Find the inverse of the matrix (if it exists) given in Exercise 5 to 11.
5.
Ans. Let A =
=
0
Matrix A is non-singular and hence
exist.
Now adj. A = And
6.
Ans. Let A =
=
Matrix A is non-singular and hence
exist.
Now adj. A = And
7.
Ans. Let A =
=
exists.
A11 = , A12 =
,
A13 = , A21 =
,
A22 = , A23 =
,
A31 = , A32 =
,
A33 =
adj. A =
8.
Ans. Let A =
=
exists.
A11 = , A12 =
,
A13 = , A21 =
,
A22 = , A23 =
,
A31 = , A32 =
,
A33 =
adj. A =
9.
Ans. Let A =
=
exists.
A11 = , A12 =
,
A13 = , A21 =
,
A22 = , A23 =
,
A31 = , A32 =
,
A33 =
adj. A =
10.
Ans. Let A =
=
exists.
A11 = , A12 =
,
A13 = , A21 =
,
A22 = , A23 =
,
A31 = , A32 =
,
A33 =
adj. A =
11.
Ans. Let A =
=
exists.
A11 = ,
A12 = , A13 =
,
A21 = , A22 =
,
A23 = , A31 =
,
A32 = , A33 =
adj. A =
12. Let A =
and B =
verify that
Ans. Given: Matrix A =
= 15 – 14 = 1
0
=
Matrix B =
= 54 – 56 =
0
Now AB = =
=
=
Now L.H.S. = ……….(i)
R.H.S. =
=
= ……….(ii)
From eq. (i) and (ii), we get
L.H.S. = R.H.S.
13. If A =
, show that A2 – 5A + 7I = 0. Hence find
Ans. Given: A =
L.H.S. =
=
=
=
=
=
= R.H.S.
……(i)
To find: , multiplying eq. (i) by
.
=
= =
14. For the matrix A =
find numbers
and
such that
Ans. Given: A =
We have
……….(i)
Here satisfies
also, therefore
Putting in eq. (i),
Here also satisfies
, therefore
Therefore, and
15. For the matrix A =
, show that
Hence find 
Ans. Given: A =
=
Now
=
=
L.H.S. =
=
=
=
= =
= R.H.S.
Now, to find , multiplying
by
=
16. If A =
, verify that
and hence find 
Ans. Given: A =
=
Now
=
=
L.H.S. =
=
=
=
= =
= R.H.S.
Now, to find , multiplying
by
=
17. Let A be a non-singular matrix of order 3 x 3. Then
is equal to:
(A)
(B)
(C)
(D)
Ans. If A is a non-singular matrix of order then
Putting
Therefore, option (B) is correct.
18. If A is an invertible matrix of order 2, then det
is equal to:
(A) det A
(B)
(C) 1
(D) 0
Ans. Since
Therefore, option (B) is correct.