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NCERT solutions for Class 12 Maths Chapter 4 Determinants are proved here. These NCERT solutions are created by expert team at Careers360 keeping align the latest syllabus of CBSE 2023-24. In this chapter, students will be able to understand the Class 12 Maths Chapter 4 NCERT solutions. If you multiply a matrix with the coordinates of a point, it will give a new point in the space which is explained in NCERT class 12 chapter 4 maths Determinants solutions. In this sense, the matrix is a linear transformation. The determinant of the matrix is the factor by which its volume blows up. You will be familiar with these points after going through ch 4 maths class 12. Interested students can visit chapter wise NCERT solution for math.
The important topics of class 12 maths ch 4 are determinants and their properties, finding the area of the triangle, minor and cofactors, adjoint and the inverse of the matrix, and applications of determinants like solving the system of linear equations, etc are covered in NCERT solutions for Class 12 Maths Chapter 4 Determinants. If you are looking for determinants class 12 solutions then check all NCERT solutions at a single place which will help the students to learn CBSE maths. Here you will get NCERT solutions for class 12 also. Read further to know more about NCERT solutions for Class 12 Maths Chapter 4 PDF download.
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>> Determinant of a Matrix: The determinant is the numerical value of a square matrix.
For a square matrix A of order n, the determinant is denoted by det A or |A|.
Minor and Cofactor of a Matrix:
Minor of an element aij of a determinant is a determinant obtained by deleting the ith row and jth column in which element aij lies.
The cofactor of an element aij of a determinant, denoted by Aij or Cij, is defined as Aij = (-1)(i+j) * Mij, where Mij is the minor of element aij.
Value of a Determinant (2x2 and 3x3 matrices):
For a 2x2 matrix A: |A| = a11 * a22 - a21 * a12
For a 3x3 matrix A: |A| = a11 * |A11| - a12 * |A12| + a13 * |A13|
Singular and Non-Singular Matrix:
If the determinant of a square matrix is zero, the matrix is said to be singular; otherwise, it is non-singular.
Determinant 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|.
Adjoint of a Matrix:
The adjoint of a square matrix A is the transpose of the matrix obtained by cofactors of each element of the determinant corresponding to A. It is denoted by adj(A).
In general, the adjoint of a matrix A = [aij]nĂ—n is a matrix [Aji]nĂ—n, where Aji is a cofactor of element aji.
Properties of Adjoint of a Matrix:
A(adj A) = (adj A)A = |A|In (Identity Matrix)
|adj A| = |A|(n-1)
adj(AT) = (adj A)T (Transpose of the adjoint)
Finding Area of a Triangle Using Determinants:
The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by
Inverse of a Square Matrix:
For a non-singular matrix A (|A| ≠0), the inverse A-1 is defined as A-1 = (1/|A|) * adj(A).
Properties of an Inverse Matrix:
(A-1)-1 = A
(AT)-1 = (A-1)T
(AB)-1 = B-1A-1
(ABC)-1 = C-1B-1A-1
adj(A-1) = (adj A)-1
Solving a System of Linear Equations using Inverse of a Matrix:
Given a system of equations AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
Case I: If |A| ≠0, the system is consistent, and X = A-1B has a unique solution.
Case II: If |A| = 0 and (adj A)B ≠0, the system is inconsistent and has no solution.
Case III: If |A| = 0 and (adj A)B = 0, the system may be either consistent or inconsistent, depending on whether it has infinitely many solutions or no solutions.
Free download Class 12 Determinants NCERT Solutions for CBSE Exam.
NCERT determinants class 12 questions and answers: Excercise- 4.1
Question:1 Evaluate the following determinant-
Answer:
The determinant is evaluated as follows
Question:2(i) Evaluate the following determinant-
Answer:
The given two by two determinant is calculated as follows
Question:3 If , then show that
Answer:
Given determinant then we have to show that ,
So, then,
Hence we have
So, L.H.S. = |2A| = -24
then calculating R.H.S.
We have,
hence R.H.S becomes
Therefore L.H.S. =R.H.S.
Hence proved.
Question:4 If then show that
Answer:
Given Matrix
Calculating
So,
calculating ,
So,
Therefore .
Hence proved.
Question:5(i) Evaluate the determinants.
Answer:
Given the determinant ;
now, calculating its determinant value,
.
Question:5(ii) Evaluate the determinants.
Answer:
Given determinant ;
Now calculating the determinant value;
.
Question:5(iii) Evaluate the determinants.
Answer:
Given determinant ;
Now calculating the determinant value;
Question:5(iv) Evaluate the determinants.
Answer:
Given determinant: ,
We now calculate determinant value:
Question:7(i) Find values of x, if
Answer:
Given that
First, we solve the determinant value of L.H.S. and equate it to the determinant value of R.H.S.,
and
So, we have then,
or or
Question:7(ii) Find values of x, if
Answer:
Given ;
So, we here equate both sides after calculating each side's determinant values.
L.H.S. determinant value;
Similarly R.H.S. determinant value;
So, we have then;
or .
Question:8 If , then is equal to
Answer:
Solving the L.H.S. determinant ;
and solving R.H.S determinant;
So equating both sides;
or or
Hence answer is (B).
NCERT determinants class 12 questions and answers: Excercise - 4.2
Question:1 Using the property of determinants and without expanding, prove that
Answer:
We can split it in manner like;
So, we know the identity that If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then the value of the determinant is zero.
Clearly, expanded determinants have identical columns.
Hence the sum is zero.
Question: 2 Using the property of determinants and without expanding, prove that
Answer:
Given determinant
Applying the rows addition then we have;
So, we have two rows and identical hence we can say that the value of determinant = 0
Therefore .
Question:3 Using the property of determinants and without expanding, prove that
Answer:
Given determinant
So, we can split it in two addition determinants:
[ Here two columns are identical ]
and [ Here two columns are identical ]
Therefore we have the value of determinant = 0.
Question:4 Using the property of determinants and without expanding, prove that
Answer:
We have determinant:
Applying we have then;
So, here column 3 and column 1 are proportional.
Therefore, .
Question:5 Using the property of determinants and without expanding, prove that
Answer:
Given determinant :
Splitting the third row; we get,
.
Then we have,
On Applying row transformation and then ;
we get,
Applying Rows exchange transformation and , we have:
also
On applying rows transformation, and then
and then
Then applying rows exchange transformation;
and then . we have then;
So, we now calculate the sum =
Hence proved.
Question:6 Using the property of determinants and without expanding, prove that
Answer:
We have given determinant
Applying transformation, we have then,
We can make the first row identical to the third row so,
Taking another row transformation: we have,
So, determinant has two rows identical.
Hence .
Question:7 Using the property of determinants and without expanding, prove that
Answer:
Given determinant :
As we can easily take out the common factors a,b,c from rows respectively.
So, get then:
Now, taking common factors a,b,c from the columns respectively.
Now, applying rows transformations and then we have;
Expanding to get R.H.S.
Question:8(i) By using properties of determinants, show that:
We have the determinant
Applying the row transformations and then we have:
Now, applying we have:
or
Hence proved.
Question:8(ii) By using properties of determinants, show that:
Answer:
Given determinant :
,
Applying column transformation and then
We get,
Now, applying column transformation , we have:
Hence proved.
Question:9 By using properties of determinants, show that:
Answer:
We have the determinant:
Applying the row transformations and then , we have;
Now, applying ; we have
Now, expanding the remaining determinant;
Hence proved.
Question:10(i) By using properties of determinants, show that:
Answer:
Given determinant:
Applying row transformation: then we have;
Taking a common factor: 5x+4
Now, applying column transformations and
Question:10(ii) By using properties of determinants, show that:
Answer:
Given determinant:
Applying row transformation we get;
[taking common (3y + k) factor]
Now, applying column transformation and
Hence proved.
Question:11(i) By using properties of determinants, show that:
Answer:
Given determinant:
We apply row transformation: we have;
Taking common factor (a+b+c) out.
Now, applying column tranformation and then
We have;
Hence Proved.
Question:11(ii) By using properties of determinants, show that:
Answer:
Given determinant
Applying we get;
Taking 2(x+y+z) factor out, we get;
Now, applying row transformations, and then .
we get;
Hence proved.
Question:12 By using properties of determinants, show that:
Answer:
Give determinant
Applying column transformation we get;
[ after taking the (1+x+x 2 ) factor common out.]
Now, applying row transformations, and then .
we have now,
As we know
Hence proved.
Question:13 By using properties of determinants, show that:
Answer:
We have determinant:
Applying row transformations, and then we have;
taking common factor out of the determinant;
Now expanding the remaining determinant we get;
Hence proved.
Question:14 By using properties of determinants, show that:
Answer:
Given determinant:
Let
Then we can clearly see that each column can be reduced by taking common factors like a,b, and c respectively from C 1, C 2, and C 3.
We then get;
Now, applying column transformations: and
then we have;
Now, expanding the remaining determinant:
.
Hence proved.
Question:15 Choose the correct answer. Let A be a square matrix of order , then is equal to
Answer:
Assume a square matrix A of order of .
Then we have;
( Taking the common factors k from each row. )
Therefore correct option is (C).
Question:16 Choose the correct answer.
Answer:
The answer is (C) Determinant is a number associated to a square matrix.
As we know that To every square matrix of order n, we can associate a number (real or complex) called determinant of the square matrix A, where element of A.
NCERT class 12 maths chapter 4 question answer: Excercise-4.3
Question:1(i) Find area of the triangle with vertices at the point given in each of the following :
Answer:
We can find the area of the triangle with vertices by the following determinant relation:
Expanding using second column
Question:1(ii) Find area of the triangle with vertices at the point given in each of the following :
Answer:
We can find the area of the triangle with given coordinates by the following method:
Question:1(iii) Find area of the triangle with vertices at the point given in each of the following :
Answer:
Area of the triangle by the determinant method:
Hence the area is equal to
Question:2 Show that points are collinear.
Answer:
If the area formed by the points is equal to zero then we can say that the points are collinear.
So, we have an area of a triangle given by,
calculating the area:
Hence the area of the triangle formed by the points is equal to zero.
Therefore given points are collinear.
Question:3(i) Find values of k if area of triangle is 4 sq. units and vertices are
Answer:
We can easily calculate the area by the formula :
or
or or
Hence two values are possible for k.
Question:3(ii) Find values of k if area of triangle is 4 sq. units and vertices are
Answer:
The area of the triangle is given by the formula:
Now, calculating the area:
or
Therefore we have two possible values of 'k' i.e., or .
Question:4(i) Find equation of line joining and using determinants.
Answer:
As we know the line joining , and let say a point on line will be collinear.
Therefore area formed by them will be equal to zero.
So, we have:
or
Hence, we have the equation of line .
Question:4(ii) Find equation of line joining and using determinants.
Answer:
We can find the equation of the line by considering any arbitrary point on line.
So, we have three points which are collinear and therefore area surrounded by them will be equal to zero .
Calculating the determinant:
Hence we have the line equation:
or .
Question:5 If the area of triangle is 35 sq units with vertices and . Then k is
Answer:
Area of triangle is given by:
or
or
Hence the possible values of k are 12 and -2.
Therefore option (D) is correct.
NCERT class 12 maths chapter 4 question answer: Excercise: 4.4
Question:1(i) Write Minors and Cofactors of the elements of following determinants:
Answer:
GIven determinant:
Minor of element is .
Therefore we have
= minor of element = 3
= minor of element = 0
= minor of element = -4
= minor of element = 2
and finding cofactors of is = .
Therefore we have:
Question:1(ii) Write Minors and Cofactors of the elements of following determinants:
Answer:
GIven determinant:
Minor of element is .
Therefore we have
= minor of element = d
= minor of element = b
= minor of element = c
= minor of element = a
and finding cofactors of is = .
Therefore we have:
Question:2(i) Write Minors and Cofactors of the elements of following determinants:
Answer:
Given determinant :
Finding Minors: by the definition,
minor of minor of
minor of minor of
minor of minor of
minor of minor of
minor of
Finding the cofactors:
cofactor of
cofactor of
cofactor of
cofactor of
cofactor of
cofactor of
cofactor of
cofactor of
cofactor of .
Question:2(ii) Write Minors and Cofactors of the elements of following determinants:
Answer:
Given determinant :
Finding Minors: by the definition,
minor of minor of
minor of minor of
minor of minor of
minor of
minor of
minor of
Finding the cofactors:
cofactor of
cofactor of
cofactor of
cofactor of
cofactor of
cofactor of
cofactor of
cofactor of
cofactor of .
Question:3 Using Cofactors of elements of second row, evaluate .
Answer:
Given determinant :
First finding Minors of the second rows by the definition,
minor of
minor of
minor of
Finding the Cofactors of the second row:
Cofactor of
Cofactor of
Cofactor of
Therefore we can calculate by sum of the product of the elements of the second row with their corresponding cofactors.
Therefore we have,
Question:4 Using Cofactors of elements of third column, evaluate
Answer:
Given determinant :
First finding Minors of the third column by the definition,
minor of
minor of
minor of
Finding the Cofactors of the second row:
Cofactor of
Cofactor of
Cofactor of
Therefore we can calculate by sum of the product of the elements of the third column with their corresponding cofactors.
Therefore we have,
Thus, we have value of .
Question:5 If and is Cofactors of , then the value of is given by
Answer:
Answer is (D) by the definition itself, is equal to the product of the elements of the row/column with their corresponding cofactors.
Question:1 Find adjoint of each of the matrices.
Answer:
Given matrix:
Then we have,
Hence we get:
Question:2 Find adjoint of each of the matrices
Answer:
Given the matrix:
Then we have,
Hence we get:
Question:3 Verify .
Answer:
Given the matrix:
Let
Calculating the cofactors;
Hence,
Now,
aslo,
Now, calculating |A|;
So,
Hence we get
Question:4 Verify .
Answer:
Given matrix:
Let
Calculating the cofactors;
Hence,
Now,
also,
Now, calculating |A|;
So,
Hence we get,
.
Question:5 Find the inverse of each of the matrices (if it exists).
Answer:
Given matrix :
To find the inverse we have to first find adjA then as we know the relation:
So, calculating |A| :
|A| = (6+8) = 14
Now, calculating the cofactors terms and then adjA.
So, we have
Therefore inverse of A will be:
Question:6 Find the inverse of each of the matrices (if it exists).
Answer:
Given the matrix :
To find the inverse we have to first find adjA then as we know the relation:
So, calculating |A| :
|A| = (-2+15) = 13
Now, calculating the cofactors terms and then adjA.
So, we have
Therefore inverse of A will be:
Question:7 Find the inverse of each of the matrices (if it exists).
Answer:
Given the matrix :
To find the inverse we have to first find adjA then as we know the relation:
So, calculating |A| :
Now, calculating the cofactors terms and then adjA.
So, we have
Therefore inverse of A will be:
Question:8 Find the inverse of each of the matrices (if it exists).
Answer:
Given the matrix :
To find the inverse we have to first find adjA then as we know the relation:
So, calculating |A| :
Now, calculating the cofactors terms and then adjA.
So, we have
Therefore inverse of A will be:
Question:9 Find the inverse of each of the matrices (if it exists).
Answer:
Given the matrix :
To find the inverse we have to first find adjA then as we know the relation:
So, calculating |A| :
Now, calculating the cofactors terms and then adjA.
So, we have
Therefore inverse of A will be:
Question:10 Find the inverse of each of the matrices (if it exists).
Answer:
Given the matrix :
To find the inverse we have to first find adjA then as we know the relation:
So, calculating |A| :
Now, calculating the cofactors terms and then adjA.
So, we have
Therefore inverse of A will be:
Question:11 Find the inverse of each of the matrices (if it exists).
Answer:
Given the matrix :
To find the inverse we have to first find adjA then as we know the relation:
So, calculating |A| :
Now, calculating the cofactors terms and then adjA.
So, we have
Therefore inverse of A will be:
Question:12 Let and . Verify that .
Answer:
We have and .
then calculating;
Finding the inverse of AB.
Calculating the cofactors fo AB:
Then we have adj(AB):
and |AB| = 61(67) - (-87)(-47) = 4087-4089 = -2
Therefore we have inverse:
.....................................(1)
Now, calculating inverses of A and B.
|A| = 15-14 = 1 and |B| = 54- 56 = -2
and
therefore we have
and
Now calculating .
........................(2)
From (1) and (2) we get
Hence proved.
Question:13 If ? , show that . Hence find
Answer:
Given then we have to show the relation
So, calculating each term;
therefore ;
Hence .
[ Post multiplying by , also ]
Question:14 For the matrix , find the numbers and such that .
Answer:
Given then we have the relation
So, calculating each term;
therefore ;
So, we have equations;
and
We get .
Question:15 For the matrix Show that Hence, find .
Answer:
Given matrix: ;
To show:
Finding each term:
So now we have,
Now finding the inverse of A;
Post-multiplying by as,
...................(1)
Now,
From equation (1) we get;
Question:16 If , verify that . Hence find .
Answer:
Given matrix: ;
To show:
Finding each term:
So now we have,
Now finding the inverse of A;
Post-multiplying by as,
...................(1)
Now,
From equation (1) we get;
Hence inverse of A is :
Question:17 Let A be a nonsingular square matrix of order . Then is equal to
Answer:
We know the identity
Hence we can determine the value of .
Taking both sides determinant value we get,
or
or taking R.H.S.,
or, we have then
Therefore
Hence the correct answer is B.
Question:18 If A is an invertible matrix of order 2, then det is equal to
Answer:
Given that the matrix is invertible hence exists and
Let us assume a matrix of the order of 2;
.
Then .
and
Now,
Taking determinant both sides;
Therefore we get;
Hence the correct answer is B.
NCERT determinants class 12 ncert solutions: Excercise- 4.6
Question:1 Examine the consistency of the system of equations.
Answer:
We have given the system of equations:18967
The given system of equations can be written in the form of the matrix;
where , and .
So, we want to check for the consistency of the equations;
Here A is non -singular therefore there exists .
Hence, the given system of equations is consistent.
Question:2 Examine the consistency of the system of equations
Answer:
We have given the system of equations:
The given system of equations can be written in the form of matrix;
where , and .
So, we want to check for the consistency of the equations;
Here A is non -singular therefore there exists .
Hence, the given system of equations is consistent.
Question:3 Examine the consistency of the system of equations.
Answer:
We have given the system of equations:
The given system of equations can be written in the form of the matrix;
where , and .
So, we want to check for the consistency of the equations;
Here A is singular matrix therefore now we will check whether the is zero or non-zero.
So,
As, , the solution of the given system of equations does not exist.
Hence, the given system of equations is inconsistent.
Question:4 Examine the consistency of the system of equations.
Answer:
We have given the system of equations:
The given system of equations can be written in the form of the matrix;
where , and .
So, we want to check for the consistency of the equations;
[ If zero then it won't satisfy the third equation ]
Here A is non- singular matrix therefore there exist .
Hence, the given system of equations is consistent.
Question:5 Examine the consistency of the system of equations.
Answer:
We have given the system of equations:
The given system of equations can be written in the form of matrix;
where , and .
So, we want to check for the consistency of the equations;
Therefore matrix A is a singular matrix.
So, we will then check
As, is non-zero thus the solution of the given system of the equation does not exist. Hence, the given system of equations is inconsistent.
Question:6 Examine the consistency of the system of equations.
Answer:
We have given the system of equations:
The given system of equations can be written in the form of the matrix;
where , and .
So, we want to check for the consistency of the equations;
Here A is non- singular matrix therefore there exist .
Hence, the given system of equations is consistent.
Question:7 Solve system of linear equations, using matrix method.
Answer:
The given system of equations
can be written in the matrix form of AX =B, where
, and
we have,
.
So, A is non-singular, Therefore, its inverse exists.
as we know
So, the solutions can be found by
Hence the solutions of the given system of equations;
x = 2 and y =-3 .
Question:8 Solve system of linear equations, using matrix method.
Answer:
The given system of equations
can be written in the matrix form of AX =B, where
, and
we have,
.
So, A is non-singular, Therefore, its inverse exists.
as we know
So, the solutions can be found by
Hence the solutions of the given system of equations;
Question:9 Solve system of linear equations, using matrix method.
Answer:
The given system of equations
can be written in the matrix form of AX =B, where
, and
we have,
.
So, A is non-singular, Therefore, its inverse exists.
as we know
So, the solutions can be found by
Hence the solutions of the given system of equations;
Question:10 Solve system of linear equations, using matrix method.
Answer:
The given system of equations
can be written in the matrix form of AX =B, where
, and
we have,
.
So, A is non-singular, Therefore, its inverse exists.
as we know
So, the solutions can be found by
Hence the solutions of the given system of equations;
Question:11 Solve system of linear equations, using matrix method.
Answer:
The given system of equations
can be written in the matrix form of AX =B, where
, and
we have,
.
So, A is non-singular, Therefore, its inverse exists.
as we know
Now, we will find the cofactors;
So, the solutions can be found by
Hence the solutions of the given system of equations;
Question:12 Solve system of linear equations, using matrix method.
Answer:
The given system of equations
can be written in the matrix form of AX =B, where
,
we have,
.
So, A is non-singular, Therefore, its inverse exists.
as we know
Now, we will find the cofactors;
So, the solutions can be found by
Hence the solutions of the given system of equations;
Question:13 Solve system of linear equations, using matrix method.
Answer:
The given system of equations
can be written in the matrix form of AX =B, where
,
we have,
.
So, A is non-singular, Therefore, its inverse exists.
as we know
Now, we will find the cofactors;
So, the solutions can be found by
Hence the solutions of the given system of equations;
Question:14 Solve system of linear equations, using matrix method.
Answer:
The given system of equations
can be written in the matrix form of AX =B, where
,
we have,
.
So, A is non-singular, Therefore, its inverse exists.
as we know
Now, we will find the cofactors;
So, the solutions can be found by
Hence the solutions of the given system of equations;
Question:15 If , find . Using solve the system of equations
Answer:
The given system of equations
can be written in the matrix form of AX =B, where
,
we have,
.
So, A is non-singular, Therefore, its inverse exists.
as we know
Now, we will find the cofactors;
So, the solutions can be found by
Hence the solutions of the given system of equations;
Answer:
So, let us assume the cost of onion, wheat, and rice be x , y and z respectively.
Then we have the equations for the given situation :
We can find the cost of each item per Kg by the matrix method as follows;
Taking the coefficients of x, y, and z as a matrix .
We have;
Now, we will find the cofactors of A;
Now we have adjA;
s
So, the solutions can be found by
Hence the solutions of the given system of equations;
Therefore, we have the cost of onions is Rs. 5 per Kg, the cost of wheat is Rs. 8 per Kg, and the cost of rice is Rs. 8 per kg.
NCERT solutions for class 12 maths chapter 4 Determinants: Miscellaneous exercise
Question:1 Prove that the determinant is independent of .
Answer:
Calculating the determinant value of ;
Clearly, the determinant is independent of .
Question:2 Without expanding the determinant, prove that
Answer:
We have the
Multiplying rows with a, b, and c respectively.
we get;
= R.H.S.
Hence proved. L.H.S. =R.H.S.
Question:4 If and are real numbers, and
Answer:
We have given
Applying the row transformations; we have;
Taking out common factor 2(a+b+c) from the first row;
Now, applying the column transformations;
we have;
and given that the determinant is equal to zero. i.e., ;
So, either or .
we can write as;
are non-negative.
Hence .
we get then
Therefore, if given = 0 then either or .
Question:5 Solve the equation
Answer:
Given determinant
Applying the row transformation; we have;
Taking common factor (3x+a) out from first row.
Now applying the column transformations; and .
we get;
as ,
or or
Question:6 Prove that .
Answer:
Given matrix
Taking common factors a,b and c from the column respectively.
we have;
Applying , we have;
Then applying , we get;
Applying , we have;
Now, applying column transformation; , we have
So we can now expand the remaining determinant along we have;
Hence proved.
Question:7 If and , find .
Answer:
We know from the identity that;
.
Then we can find easily,
Given and
Then we have to basically find the matrix.
So, Given matrix
Hence its inverse exists;
Now, as we know that
So, calculating cofactors of B,
Now, We have both as well as ;
Putting in the relation we know;
Question:8(i) Let . Verify that,
Answer:
Given that ;
So, let us assume that matrix and then;
Hence its inverse exists;
or ;
so, we now calculate the value of
Cofactors of A;
Finding the inverse of C;
Hence its inverse exists;
Now, finding the ;
or
Now, finding the R.H.S.
Cofactors of B;
Hence L.H.S. = R.H.S. proved.
Question:8(ii) Let , Verify that
Answer:
Given that ;
So, let us assume that
Hence its inverse exists;
or ;
so, we now calculate the value of
Cofactors of A;
Finding the inverse of B ;
Hence its inverse exists;
Now, finding the ;
Hence proved L.H.S. =R.H.S. .
Question:9 Evaluate
Answer:
We have determinant
Applying row transformations; , we have then;
Taking out the common factor 2(x+y) from the row first.
Now, applying the column transformation; and we have ;
Expanding the remaining determinant;
.
Question:10 Evaluate
Answer:
We have determinant
Applying row transformations; and then we have then;
Taking out the common factor -y from the row first.
Expanding the remaining determinant;
Question:11 Using properties of determinants, prove that
Answer:
Given determinant
Applying Row transformations; and , then we have;
Expanding the remaining determinant;
hence the given result is proved.
Question:12 Using properties of determinants, prove that
Answer:
Given the determinant
Applying the row transformations; and then we have;
Applying row transformation we have then;
Now we can expand the remaining determinant to get the result;
hence the given result is proved.
Question:13 Using properties of determinants, prove that
Answer:
Given determinant
Applying the column transformation, we have then;
Taking common factor (a+b+c) out from the column first;
Applying and , we have then;
Now we can expand the remaining determinant along we have;
Hence proved.
Question:14 Using properties of determinants, prove that
Answer:
Given determinant
Applying the row transformation; and we have then;
Now, applying another row transformation we have;
We can expand the remaining determinant along , we have;
Hence the result is proved.
Question:15 Using properties of determinants, prove that
Answer:
Given determinant
Multiplying the first column by and the second column by , and expanding the third column, we get
Applying column transformation, we have then;
Here we can see that two columns are identical.
The determinant value is equal to zero.
Hence proved.
Question:16 Solve the system of equations
Answer:
We have a system of equations;
So, we will convert the given system of equations in a simple form to solve the problem by the matrix method;
Let us take, ,
Then we have the equations;
We can write it in the matrix form as , where
Now, Finding the determinant value of A;
Hence we can say that A is non-singular its invers exists;
Finding cofactors of A;
, ,
, ,
, ,
as we know
Now we will find the solutions by relation .
Therefore we have the solutions
Or in terms of x, y, and z;
Question:17 Choose the correct answer.
If are in A.P, then the determinant
is
Answer:
Given determinant and given that a, b, c are in A.P.
That means , 2b =a+c
Applying the row transformations, and then we have;
Now, applying another row transformation, , we have
Clearly we have the determinant value equal to zero;
Hence the option (A) is correct.
Question:18 Choose the correct answer.
If x, y, z are nonzero real numbers, then the inverse of matrix is
Answer:
Given Matrix ,
As we know,
So, we will find the ,
Determining its cofactor first,
Hence
Therefore the correct answer is (A)
Question:19 Choose the correct answer.
Answer:
Given determinant
Now, given the range of from
Therefore the .
Hence the correct answer is D.
If you are interested in Determinants Class 12 NCERT Solutions exercises then these are listed below.
The six exercises of NCERT Class 12 Maths solutions chapter 4 Determinants covers the properties of determinants, co-factors and applications like finding the area of triangle, solutions of linear equations in two or three variables, minors, consistency and inconsistency of system of linear equations, adjoint and inverse of a square matrix, and solution of linear equations in two or three variables using inverse of a matrix. You can also check Determinants NCERT solutions if you are facing any problems during practice.
What are the Determinants?
To every square matrix of order n, we can associate a number (real or complex) called determinant of the square matrix A. Let's take a determinant (A) of order two-
If A is a then the determinant of A is written as |A|
matrix ,
The six exercises of this chapter determinants covers the properties of determinants, co-factors and applications like finding the area of triangle, solutions of linear equations in two or three variables, minors, consistency and inconsistency of system of linear equations, adjoint and inverse of a square matrix, and solution of linear equations in two or three variables using inverse of a matrix.
Topics and sub-topics of NCERT class 12 maths chapter 4 Determinants
4.1 Introduction
4.2 Determinant
4.2.1 Determinant of a matrix of order one
4.2.2 Determinant of a matrix of order two
4.2.3 Determinant of a matrix of order 3 Ă— 3
4.3 Properties of Determinants
4.4 Area of a Triangle
4.5 Minors and Cofactors
4.6 Adjoint and Inverse of a Matrix
4.7 Applications of Determinants and Matrices
4.7.1 Solution of a system of linear equations using the inverse of a matrix
Also read,
NCERT exemplar solutions class 12 maths chapter 4
Topics of NCERT Class 12 Maths Chapter Determinants
The main topics covered in chapter 4 maths class 12 are:
Determinants
Ch 4 maths class 12 includes concepts of calculation of determinants with respect to their order one, two, three. Also class 12 NCERT topics discuss concepts related to the expansion of the matrix to calculate the determinant. there are good quality questions in Determinants class 12 solutions.
Properties of determinants
This ch 4 maths class 12 comprehensively and elaborately discussed the properties of determinants, which are vastly used. To get a good hold of these concepts you can refer to NCERT solutions for class 12 maths chapter 4.
Area of triangle
This ch 4 maths class 12 also includes concepts of the area of a triangle in which vertices are given. You can refer to class 12 NCERT solutions for questions about these concepts.
Maths class 12 chapter 4 discussed the minors and cofactors. To get command of these concepts you can go through the NCERT solution for class 12 maths chapter 4.
Adjoint and Inverse of a matrix
concepts related to adjoints and inverse of the matrix are detailed in maths class 12 chapter 4. And it also concerns conditions for the existence of the inverse of a matrix. Determinants class 12 solutions include quality questions to understand the concepts.
ch 4 maths class 12 deliberately discussed the applications of determinants and matrices. it also includes the terms consistent system inconsistent system. concepts related to the solution of a system of linear equations using the inverse of a matrix. For questions on these concepts, you can browse NCERT solutions for class 12 chapter 4.
Topics mentioned in class 12 NCERT are very important and students are suggested to go through all the concepts discussed in the topics. Questions related to all the above topics are covered in the NCERT solutions for class 12 maths chapter 4
NCERT Class 12 Maths solutions chapter 4 will assist the students in the exam preparation in a strategic way.
Class 12 Maths Chapter 4 NCERT solutions are prepared by the experts, therefore, students can rely upon the same without any second thought .
NCERT solutions for Class 12 Maths Chapter 4 provides the detailed solution for all the questions. This will help the students in analysing and understanding the questions in a better way.
NCERT Solutions for Class 12 Maths Chapter 4 primarily focuses on the topic of determinants. This chapter covers the following key themes:
Definition of determinants
Properties of determinants
Area of a parallelogram and a triangle
The inverse of a matrix
Adjoint and inverse of a matrix
Solutions of linear equations using matrices
Determinant as scaling factor
The topic algebra which contains two topics matrices and determinants which has 13 % weightage in the maths CBSE 12th board final examination. students can prioritise their subjects according to respective weightage and study accordingly.
Only knowing the answer does not guarantee to score good marks in the exam. One should know how to answer in order to get good marks. NCERT solutions are provided by the experts who know how best to write answers in the board exam in order to get good marks in the board exam.
NCERT textbook is the best book for CBSE class 12 maths. Most of the questions in CBSE class 12 board exam are directly asked from NCERT textbook. So you don't need to buy any supplementary books for CBSE class 12 maths.
According to NCERT Solutions for Class 12 Maths Chapter 4, determinants play a crucial role in algebra and have multiple practical applications. The concept of determinants is valuable in solving systems of linear equations. With determinants, students can explore concepts such as changes in area, volume, and variables through integrals. Additionally, determinants can be used to determine the values of square matrices. Interested students can study determinants class 12 ncert pdf both online and offline.
Here you will get the detailed NCERT solutions for class 12 maths by clicking on the link. also you can find these in official web page of careers360.
Hello aspirant,
The purpose of graphic design extends beyond the brand's look. Nevertheless, by conveying what the brand stands for, it significantly aids in the development of a sense of understanding between a company and its audience. The future in the field of graphic designing is very promising.
There are various courses available for graphic designing. To know more information about these courses and much more details, you can visit our website by clicking on the link given below.
https://www.careers360.com/courses/graphic-designing-course
Thank you
Hope this information helps you.
hello,
Yes you can appear for the compartment paper again since CBSE gives three chances to a candidate to clear his/her exams so you still have two more attempts. However, you can appear for your improvement paper for all subjects but you cannot appear for the ones in which you have failed.
I hope this was helpful!
Good Luck
Hello dear,
If you was not able to clear 1st compartment and now you giving second compartment so YES, you can go for your improvement exam next year but if a student receives an improvement, they are given the opportunity to retake the boards as a private candidate the following year, but there are some requirements. First, the student must pass all of their subjects; if they received a compartment in any subject, they must then pass the compartment exam before being eligible for the improvement.
As you can registered yourself as private candidate for giving your improvement exam of 12 standard CBSE(Central Board of Secondary Education).For that you have to wait for a whole year which is bit difficult for you.
Positive side of waiting for whole year is you have a whole year to preparing yourself for your examination. You have no distraction or something which may causes your failure in the exams. In whole year you have to stay focused on your 12 standard examination for doing well in it. By this you get a highest marks as a comparison of others.
Believe in Yourself! You can make anything happen
All the very best.
Hello Student,
I appreciate your Interest in education. See the improvement is not restricted to one subject or multiple subjects  and  we cannot say if improvement in one subject in one year leads to improvement in more subjects in coming year.
You just need to have a revision of all subjects what you have completed in the school. have a revision and practice of subjects and concepts helps you better.
All the best.
Hi,
You just need to give the exams for the concerned two subjects in which you have got RT. There is no need to give exam for all of your subjects, you can just fill the form for the two subjects only.
Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.
The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.
A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.
GIS officer work on various GIS software to conduct a study and gather spatial and non-spatial information. GIS experts update the GIS data and maintain it. The databases include aerial or satellite imagery, latitudinal and longitudinal coordinates, and manually digitized images of maps. In a career as GIS expert, one is responsible for creating online and mobile maps.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.
If you are intrigued by the programming world and are interested in developing communications networks then a career as database architect may be a good option for you. Data architect roles and responsibilities include building design models for data communication networks. Wide Area Networks (WANs), local area networks (LANs), and intranets are included in the database networks. It is expected that database architects will have in-depth knowledge of a company's business to develop a network to fulfil the requirements of the organisation. Stay tuned as we look at the larger picture and give you more information on what is db architecture, why you should pursue database architecture, what to expect from such a degree and what your job opportunities will be after graduation. Here, we will be discussing how to become a data architect. Students can visit NIT Trichy, IIT Kharagpur, JMI New Delhi.
Individuals who opt for a career as a remote sensing technician possess unique personalities. Remote sensing analysts seem to be rational human beings, they are strong, independent, persistent, sincere, realistic and resourceful. Some of them are analytical as well, which means they are intelligent, introspective and inquisitive.
Remote sensing scientists use remote sensing technology to support scientists in fields such as community planning, flight planning or the management of natural resources. Analysing data collected from aircraft, satellites or ground-based platforms using statistical analysis software, image analysis software or Geographic Information Systems (GIS) is a significant part of their work. Do you want to learn how to become remote sensing technician? There's no need to be concerned; we've devised a simple remote sensing technician career path for you. Scroll through the pages and read.
Budget analysis, in a nutshell, entails thoroughly analyzing the details of a financial budget. The budget analysis aims to better understand and manage revenue. Budget analysts assist in the achievement of financial targets, the preservation of profitability, and the pursuit of long-term growth for a business. Budget analysts generally have a bachelor's degree in accounting, finance, economics, or a closely related field. Knowledge of Financial Management is of prime importance in this career.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.
Individuals who opt for a career as a stock analyst examine the company's investments makes decisions and keep track of financial securities. The nature of such investments will differ from one business to the next. Individuals in the stock analyst career use data mining to forecast a company's profits and revenues, advise clients on whether to buy or sell, participate in seminars, and discussing financial matters with executives and evaluate annual reports.
A Researcher is a professional who is responsible for collecting data and information by reviewing the literature and conducting experiments and surveys. He or she uses various methodological processes to provide accurate data and information that is utilised by academicians and other industry professionals. Here, we will discuss what is a researcher, the researcher's salary, types of researchers.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.
Individuals who opt for a career as an environmental engineer are construction professionals who utilise the skills and knowledge of biology, soil science, chemistry and the concept of engineering to design and develop projects that serve as solutions to various environmental problems.
A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.
A Conservation Architect is a professional responsible for conserving and restoring buildings or monuments having a historic value. He or she applies techniques to document and stabilise the object’s state without any further damage. A Conservation Architect restores the monuments and heritage buildings to bring them back to their original state.
A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.
Highway Engineer Job Description: A Highway Engineer is a civil engineer who specialises in planning and building thousands of miles of roads that support connectivity and allow transportation across the country. He or she ensures that traffic management schemes are effectively planned concerning economic sustainability and successful implementation.
Are you searching for a Field Surveyor Job Description? A Field Surveyor is a professional responsible for conducting field surveys for various places or geographical conditions. He or she collects the required data and information as per the instructions given by senior officials.
Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.
A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.
Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.
The audiologist career involves audiology professionals who are responsible to treat hearing loss and proactively preventing the relevant damage. Individuals who opt for a career as an audiologist use various testing strategies with the aim to determine if someone has a normal sensitivity to sounds or not. After the identification of hearing loss, a hearing doctor is required to determine which sections of the hearing are affected, to what extent they are affected, and where the wound causing the hearing loss is found. As soon as the hearing loss is identified, the patients are provided with recommendations for interventions and rehabilitation such as hearing aids, cochlear implants, and appropriate medical referrals. While audiology is a branch of science that studies and researches hearing, balance, and related disorders.
An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.
Are you searching for an ‘Anatomist job description’? An Anatomist is a research professional who applies the laws of biological science to determine the ability of bodies of various living organisms including animals and humans to regenerate the damaged or destroyed organs. If you want to know what does an anatomist do, then read the entire article, where we will answer all your questions.
For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.
Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.
Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.
Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.
Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.
A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.
The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.
A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.
Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.
An individual who is pursuing a career as a producer is responsible for managing the business aspects of production. They are involved in each aspect of production from its inception to deception. Famous movie producers review the script, recommend changes and visualise the story.
They are responsible for overseeing the finance involved in the project and distributing the film for broadcasting on various platforms. A career as a producer is quite fulfilling as well as exhaustive in terms of playing different roles in order for a production to be successful. Famous movie producers are responsible for hiring creative and technical personnel on contract basis.
In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.
In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion.
Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article.
For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.
Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.
Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.
Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.
Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.
A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.
The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.