OP Malhotra Determinants S.Chand ISC Class-12 Maths Solutions Ch-5. Step by step Solutions of OP Malhotra SK Gupta, Anubhuti Gangal S.Chand ISC Class-12 Mathematics with Exe-5(a), Exe-5(b), Exe-5(c), Exe-5(d), Self Revision and Chapter Test Questions. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.
OP Malhotra Determinants S.Chand ISC Class-12 Maths Solutions Ch-5
| Class: | 12th |
| Subject: | Mathematics |
| Chapter : | Ch-5 Determinants of Section -A |
| Board | ISC |
| Writer | OP Malhotra, SK Gupta, Anubhuti Gangal |
| Publications | S.Chand Publications 2020-21 |
-: Included Topics :-
Exe-5(a)
Exe-5(b)
Exe-5(c)
Exe-5(d)
Self Revision
Chapter Test
OP Malhotra Determinants S.Chand ISC Class-12 Maths Solutions Ch-5
Determinant :
Determinant is the numerical value of the square matrix. So, to every square matrix A = [aij] of order n, we can associate a number (real or complex) called determinant of the square matrix A. It is denoted by det A or |A|.
Note :-
- Read |A| as determinant A not absolute value of A.
- Determinant gives numerical value but matrix do not give numerical value.
- A determinant always has an equal number of rows and columns, i.e. only square matrix have determinants.
Exe-5(a)
OP Malhotra Determinants S.Chand ISC Class-12 Maths Solutions Ch-5
Question 1:
Evaluate the following determinants :
(i) …………………….
………………..
Question 2:
Prove that :
……………….
Question 3:
……………………
…………………….
………………………
Question 5:
Determine the value of k for which ………..
Minor :
Minor of an element ay of a determinant, is a determinant obtained by deleting the ith row and jth column in which element ay lies. Minor of an element aij is denoted by Mij.
Note: Minor of an element of a determinant of order n(n ≥ 2) is a determinant of order (n – 1).
Cofactor :
Cofactor of an element aij of a determinant, denoted by Aij or Cij is defined as Aij = (-1)i+j Mij, where Mij is a minor of an element aij.
Exe-5(b)
OP Malhotra Determinants S.Chand ISC Class-12 Maths Solutions Ch-5
Question 1:
Write minors and cofactors of elements of determinant …….
Question 2:
Write down the minors of -2 and 4 in ……………….
Question 3:
………………..
…………………..
………………….
…………………..
Question 12:
If one root of ………………… = 0
Singular and non-singular Matrix :
If the value of determinant corresponding to a square matrix is zero, then the matrix is said to be a singular matrix, otherwise it is non-singular matrix, i.e. for a square matrix A, if |A| ≠ 0, then it is said to be a non-singular matrix and of |A| = 0, then it is said to be a singular matrix.
Theorems.
- If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.
- The determinant of the product of matrices is equal to the product of their respective determinants, i.e. |AB| = |A||B|, where A and B are a square matrix of the same order.
Exe-5(c)
OP Malhotra Determinants S.Chand ISC Class-12 Maths Solutions Ch-5
Question 1:
Find the area of the triangle whose vertices are :
(i)….
(ii) …….
Question 2:
Using determinants, prove that the following points are collinear
(i)….
(ii) …….
Question 3:
…………………..
…………………
Question 5:
If (x, y), (a, 0), (0, b) are collinear, then using determinant prove that ……………… = 1
Adjoint of a Matrix :
The adjoint of a square matrix ‘A’ is the transpose of the matrix which obtained by cofactors of each element of a determinant corresponding to that given matrix. It is denoted by adj(A).
In general, adjoint of a matrix A = [aij]n×n is a matrix [Aji]n×n, where Aji is a cofactor of element aji.
Exe-5(d)
OP Malhotra Determinants S.Chand ISC Class-12 Maths Solutions Ch-5
Question 1:
(a) without evaluate problems (i) to (x) state why each statement is true.
……………………..
(b) Without actually …………………….
Question 2:
Without expanding the determinants show that…………………
Question 3:
…………………..
……………………..
……………………..
Question 28:
Using properties of determinants ……………….. following determinants.
……………….
Properties of Determinants :
To find the value of the determinant, we try to make the maximum possible zero in a row (or a column) by using properties given below and then expand the determinant corresponding that row (or column).
Following are the various properties of determinants:
1- The value of the determinant remains unchanged if the rows and columns of a determinant are interchanged.
2- If any two rows (or columns) of determinants are interchanged, then sign of determinants changes.
3- If any two rows or columns of a determinant are equal or identical, then the value of the determinant is 0.
4- If each element of a row or a column is multiplied by a constant value k, then the value of the determinant originally obtained is multiplied with k.
Self Revision
OP Malhotra Determinants S.Chand ISC Class-12 Maths Solutions Ch-5
Question 1:
Show that the value of the following determinant is negative, if a, b and c are positive and unequal ……..
Question 2:
Use properties of determinant to solve for x :
………………….
…………………
Question 3:
………………….
…………………
…………………..
Question 18:
Using properties of determinants, prove that …………………..
Chapter Test
OP Malhotra Determinants S.Chand ISC Class-12 Maths Solutions Ch-5
Question 1:
(i) If ……………… then find
(ii) Find …………………..
(iii) Write the value ………………….
Question 2:
If A ………….
Question 3:
…………………….
…………………….
…………………….
Question 10:
Using properties …………………..
Question 11:
……………………
-: End of Determinants OP Malhotra S. Chand ISC Class-12 Maths Chapter-5 Solution :-
Return to :- OP Malhotra S. Chand ISC Class-12 Maths Solutions
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