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
Matrices : NCERT Solutions – Class 12 Maths (Ex 3)
Exercise 3.3
1. Find the transpose of each of the following matrices:
(i)
(ii)
(iii)
Ans. (i) Let A =
Transpose of A = A’ or A^{T} =
(ii)
Transpose of A = A’ or A^{T} =
(iii)
Transpose of A = A’ or A^{T} =
2. If A = and B = then verify that:
(i)
(ii)
Ans. (i) A + B = = =
L.H.S. = (A + B)’ = =
R.H.S. = A’ + B’ = =
= =
L.H.S. = R.H.S. Proved.
(ii) A – B = = =
L.H.S. = (A – B)’ = =
R.H.S. = A’ – B’ = =
= =
L.H.S. = R.H.S. Proved.
3. If A’ = and B = then verify that:
(i)
(ii)
Ans. Given: A’ = and B = then (A’)’ = A =
(i) A + B = =
L.H.S. = (A + B)’ =
R.H.S. = A’ + B’ = =
= =
L.H.S. = R.H.S. Proved.
(ii) A – B = =
L.H.S. = (A – B)’ =
R.H.S. = A’ – B’ = =
= =
L.H.S. = R.H.S. Proved.
4. If A’ = and B = then find (A + 2B)’.
Ans. Given: A’ = and B = then (A’)’ = A =
A +2B = = = =
(A + 2B)’ =
5. For the matrices A and B, verify that (AB)’ = B’A’, where:
(i) A = B =
(ii) A = B =
Ans. (i) AB = =
L.H.S. = (AB)’ = =
R.H.S. = B’A’ = = =
L.H.S. = R.H.S. Proved.
(ii) AB = =
L.H.S. = (AB)’ = =
R.H.S. = B’A’ = = =
L.H.S. = R.H.S. Proved.
6. (i) If A = then verify that A’A = I.
(ii) If A = then verify that A’A = I.
Ans. (i) L.H.S. = A’A =
=
= = = I = R.H.S.
(ii) L.H.S. = A’A = =
= = = I = R.H.S.
7. (i) Show that the matrix A = is a symmetric matrix.
(ii) Show that the matrix A = is a skew symmetric matrix.
Ans. (i) Given: A = ……….(i)
Changing rows of matrix A as the columns of new matrix A’ = = A
A’ = A
Therefore, by definitions of symmetric matrix, A is a symmetric matrix.
(ii) Given: A = ……….(i)
A’ = =
Taking common, A’ = = – A [From eq. (i)]
Therefore, by definition matrix A is a skewsymmetric matrix
8. For a matrix A = verify that:
(i) (A + A’) is a symmetric matrix.
(ii) (A – A’) is a skew symmetric matrix.
Ans. (i) Given: A =
Let B = A + A’ = = =
B’ = = B
B = A + A’ is a symmetric matrix.
(ii) Given:
Let B = A – A’ = = =
B’ =
Taking common, = – B
B = A – A’ is a skewsymmetric matrix.
9. Find (A + A’) and (A – A’) when A =
Ans. Given: A = A’ =
Now, A + A’ = = =
(A + A’) =
Now, A – A’ = = =
(A – A’) = =
10. Express the following matrices as the sum of a symmetric and skew symmetric matrix:
(i)
(ii)
(iii)
(iv)
Ans. (i) Given: A = A’ =
Symmetric matrix = (A + A’) =
= =
And Skew symmetric matrix = (A – A’) =
= =
Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .
(ii) Given: A = A’ =
Symmetric matrix = (A + A’) =
= =
And Skew symmetric matrix = (A – A’) =
= =
Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .
(iii) Given: A = A’ =
Symmetric matrix = (A + A’) =
= =
And Skew symmetric matrix = (A – A’) =
= =
Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .
(iv) Given: A = A’ =
Symmetric matrix = (A + A’) = = =
And Skew symmetric matrix = (A – A’) = =
Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .
Choose the correct answer in Exercises 11 and 12.
11. If A and B are symmetric matrices of same order, AB – BA is a:
(A) Skewsymmetric matrix
(B) Symmetric matrix
(C) Zero matrix
(S) Identity matrix
Ans. Given: A and B are symmetric matrices A = A’ and B = B’
Now, (AB – BA)’ = (AB)’ – (BA)’ (AB – BA)’ = B’A’ – A’B’ [Reversal law]
(AB – BA)’ = BA – AB [From eq. (i)] (AB – BA)’ = – (AB – BA)
(AB – BA) is a skew matrix.
Therefore, option (A) is correct.
12. If A = , then A + A’ = I, if the value of is:
(A)
(B)
(C)
(D)
Ans. Given: A = Also A + A’ = I
Equating corresponding entries, we have
Therefore, option (B) is correct.