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
Matrices : NCERT Solutions – Class 12 Maths (Ex 1)
Exercise 3.1
1. In the matrix A = , write:
(i) The order of the matrix.
(ii) The number of elements.
(iii) Write the elements
Ans. (i) There are 3 horizontal lines (rows) and 4 vertical lines (columns) in the given matrix A.
Therefore, Order of the matrix is 3 x 4.
(ii) The number of elements in the matrix A is 3 x 4 = 12.
(iii) Element in first row and third column = 19
Element in second row and first column = 35
Element in third row and third column =
Element in second row and fourth column = 12
Element in second row and third column =
2. If a matrix has 24 elements, what are possible orders it can order? What, if it has 13 elements?
Ans. Since, a matrix having element is of order
(i) Therefore, there are 8 possible matrices having 24 elements of orders 1 x 24, 2 x 12, 3 x 8, 4 x 6, 24 x 1, 12 x 2, 8 x 3, 6 x 4.
(ii) Prime number 13 = 1 x 13 and 13 x 1
Therefore, there are 2 possible matrices of order 1 x 13 (Row matrix) and 13 x 1 (Column matrix).
3. If a matrix has 18 elements, what are the possible orders it can have? What if has 5 elements?
Ans. Since, a matrix having element is of order
(i) Therefore, there are 6 possible matrices having 18 elements of orders 1 x 18, 2 x 9, 3 x 6, 18 x 1, 9 x 2, 6 x 3.
(ii) Prime number 5 = 1 x 5 and 5 x 1
Therefore, there are 2 possible matrices of order 1 x 5 (Row matrix) and 5 x 1 (Column matrix).
4. Construct a 2 x 2 matrix A =
whose elements are given by:
(i)
(ii)
(iii)
Ans. (i) Given: ……….(i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
A2 x 2 =
(ii) Given: ……….(i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
A2 x 2 =
(iii) Given: ……….(i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
A2 x 2 =
5. Construct a 3 x 4 matrix, whose elements are given by:
(i)
(ii)
Ans. (i) Given: ……….(i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
A3 x 4 =
(ii) Given: ……….(i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
Putting in eq. (i)
A3 x 4 =
6. Find the values of
and
from the following equations:
(i)
(ii)
(iii)
Ans. (i)Given:
By definition of Equal matrices,
(ii)
Equating corresponding entries, ……….(i)
……….(ii)
And
[From eq. (i),
]
or
Putting these values of in eq. (i), we have
and
(iii) Given:
Equating corresponding entries, ……….(i)
………. (ii)
And ……….(iii)
Eq. (i) – Eq. (ii) = 9 – 5 = 4
Eq. (i) – Eq. (iii) = 9 – 7 = 2
Putting values of and
in eq. (i),
7. Find the values of
and
from the equation
.
Ans. Equating corresponding entries,
……….(i)
……….(ii)
……….(iii)
……….(iv)
Eq. (i) – Eq. (ii) =
Putting in eq. (i),
Putting in eq. (iii),
Putting in eq. (iv),
8. A =
is a square matrix if:
(A) (B)
(C)
(D) None of these
Ans. By definition of square matrix , option (C) is correct.
9. Which of the given values of
and
make the following pairs of matrices equal:
(A)
(B) Not possible to find
(C)
(D)
Ans. Equating corresponding sides,
And
Also
And
Since, values of are not equal, therefore, no values of
and
exist to make the two matrices equal.
Therefore, option (B) is correct.
10. The number of all possible matrices of order 3 x 3 with each entry 0 or 1 is:
(A) 27
(B) 18
(C) 81
(D) 512
Ans. Since, general matrix of order 3 x 3 is
This matrix has 9 elements.
The number of choices for is 2 (as 0 or 1 can be used)
Similarly, the number of choices for each other element is 2.
Therefore, total possible arrangements (matrices) = times =
Therefore, option (D) is correct.