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Edited By Ramraj Saini | Updated on Sep 13, 2023 09:18 PM IST | #CBSE Class 12th

**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.

This Story also Contains

- NCERT Determinants Class 12 Questions And Answers
- NCERT Determinants Class 12 Questions And Answers PDF Free Download
- NCERT Class 12 Maths Chapter 4 Question Answer - Important Formulae
- NCERT Class 12 Maths Chapter 4 Question Answer (Intext Questions and Exercise)
- NCERT determinants class 12 solutions: Excercise: 4.5
- NCERT solutions for class 12 Maths - Chapter wise
- NCERT solutions for class 12 - subject wise
- NCERT Solutions - Class Wise
- NCERT Books and NCERT Syllabus

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.

**Also read:**

- Class 12 Maths Chapter 4 Determinants Notes
- NCERT Exemplar Solutions For Class 12 Maths Chapter 4 Determinants

>> 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 A_{ij} or C_{ij}, is defined as A_{ij} = (-1)(i+j) * M_{ij}, where M_{ij} is the minor of element a_{ij}.

Value of a Determinant (2x2 and 3x3 matrices):

For a 2x2 matrix A: |A| = a_{11} * a_{22} - a_{21 }* a_{12}

For a 3x3 matrix A: |A| = a_{11} * |A_{11}| - a_{12} * |A_{12}| + a_{13} * |A_{13}|

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 = [a_{ij}]_{n×n} is a matrix [A_{ji}]_{n×n}, where A_{ji} is a cofactor of element a_{ji}.

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 (x_{1}, y_{1}), (x_{2}, y_{2}), and (x_{3}, y_{3}) 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^{-1}A^{-1}

(ABC)^{-1} = C^{-1}B^{-1}A^{-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^{-1}B 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

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.**

- Determinants Class 12 NCERT Solutions Exercise 4.1
- Determinants Class 12 NCERT Solutions Exercise 4.2
- Determinants Class 12 NCERT Solutions Exercise 4.3
- Determinants Class 12 NCERT Solutions Exercise 4.4
- Determinants Class 12 NCERT Solutions Exercise 4.5
- Determinants Class 12 NCERT Solutions Exercise 4.6
- Determinants Class 12 NCERT Solutions Miscellaneous Exercise

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 × 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.

**Minors and Cofactors**

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.

**Applications of determinants and matrix**

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 Exemplar Class 12 Chemistry Solutions____NCERT Exemplar Class 12 Mathematics Solutions____NCERT Exemplar Class 12 Biology Solutions____NCERT Exemplar Class 12 Physics Solutions__

- NCERT solutions for class 12 mathematics
- NCERT solutions class 12 chemistry
- NCERT solutions for class 12 physics
- NCERT solutions for class 12 biology

- NCERT solutions for class 12
- NCERT solutions for class 11
- NCERT solutions for class 10
- NCERT solutions for class 9

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.

1. What are the key themes discussed in chapter 4 maths class 12 ncert solutions?

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

2. What is the weightage of the chapter determinants for CBSE board exam?

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.

3. How are the NCERT solutions helpful in the board exam?

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.

4. Which is the best book for CBSE class 12 Maths ?

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.

5. What applications of determinants are discussed in ncert solutions class 12 maths chapter 4?

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.

6. Where can I find the complete solutions of NCERT class 12 Maths ?

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.

Application Date:20 November,2023 - 19 December,2023

Application Date:20 November,2023 - 19 December,2023

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9 minHave a question related to CBSE Class 12th ?

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.

If you'll do hard work then by hard work of 6 months you can achieve your goal but you have to start studying for it dont waste your time its a very important year so please dont waste it otherwise you'll regret.

Yes, you can take admission in class 12th privately there are many colleges in which you can give 12th privately.

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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.

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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.

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The career as a Talent Agent is filled with responsibilities. A Talent Agent is someone who is involved in the pre-production process of the film. It is a very busy job for a Talent Agent but as and when an individual gains experience and progresses in the career he or she can have people assisting him or her in work. Depending on one’s responsibilities, number of clients and experience he or she may also have to lead a team and work with juniors under him or her in a talent agency. In order to know more about the job of a talent agent continue reading the article.

If you want to know more about talent agent meaning, how to become a Talent Agent, or Talent Agent job description then continue reading this article.

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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.

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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.

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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.

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Individuals who opt for a career as a talent director are professionals who work in the entertainment industry. He or she is responsible for finding out the right talent through auditions for films, theatre productions, or shows. A talented director possesses strong knowledge of computer software used in filmmaking, CGI and animation. A talent acquisition director keeps himself or herself updated on various technical aspects such as lighting, camera angles and shots.

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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.

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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.

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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.

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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 cost got reduced the viewership for these types of content has increased on a large scale. Therefore, the career as vlogger has a lot to offer. If you want to know more about the career as vlogger, how to become a vlogger, so on and so forth then continue reading the article. Students can visit Jamia Millia Islamia, Asian College of Journalism, Indian Institute of Mass Communication to pursue journalism degrees.

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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.

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Content writing is meant to speak directly with a particular audience, such as customers, potential customers, investors, employees, or other stakeholders. The main aim of professional content writers is to speak to their targeted audience and if it is not then it is not doing its job. There are numerous kinds of the content present on the website and each is different based on the service or the product it is used for.

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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.

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Linguistic meaning is related to language or Linguistics which is the study of languages. A career as a linguistic meaning, a profession that is based on the scientific study of language, and it's a very broad field with many specialities. Famous linguists work in academia, researching and teaching different areas of language, such as phonetics (sounds), syntax (word order) and semantics (meaning).

Other researchers focus on specialities like computational linguistics, which seeks to better match human and computer language capacities, or applied linguistics, which is concerned with improving language education. Still, others work as language experts for the government, advertising companies, dictionary publishers and various other private enterprises. Some might work from home as freelance linguists. Philologist, phonologist, and dialectician are some of Linguist synonym. Linguists can study French, German, Italian.

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Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.

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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.

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A career as Production Engineer is crucial in the manufacturing industry. He or she ensures the functionality of production equipment and machinery to improve productivity and minimize production costs in order to drive revenues and increase profitability.

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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.

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Individuals who opt for a career as product designers are responsible for designing the components and overall product concerning its shape, size, and material used in manufacturing. They are responsible for the aesthetic appearance of the product. A product designer uses his or her creative skills to give a product its final outlook and ensures the functionality of the design.

Students can opt for various product design degrees such as B.Des and M.Des to become product designers. Industrial product designer prepares 3D models of designs for approval and discusses them with clients and other colleagues. Individuals who opt for a career as a product designer estimate the total cost involved in designing.

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A career as R&D Personnel requires researching, planning, and implementing new programs and protocols into their organization and overseeing new products’ development. He or she uses his or her creative abilities to improve the existing products as per the requirements of the target market.

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A Commercial Manager negotiates, advises and secures information about pricing for commercial contracts. He or she is responsible for developing financial plans in order to maximise the business's profitability.

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ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.

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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

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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.

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Individuals in the computer systems analyst career path study the hardware and applications that are part of an organization's computer systems, as well as how they are used. They collaborate closely with managers and end-users to identify system specifications and business priorities, as well as to assess the efficiency of computer systems and create techniques to boost IT efficiency. Individuals who opt for a career as a computer system analyst support the implementation, modification, and debugging of new systems after they have been installed.

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A Test Manager is a professional responsible for planning, coordinating and controlling test activities. He or she develops test processes and strategies to analyse and determine test methods and tools for test activities. The test manager jobs involve documenting tests that have been carried out, analysing and evaluating software quality to determine further recommended procedures.

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A career as Azure Developer comes with the responsibility of designing and developing cloud-based applications and maintaining software components. He or she possesses an in-depth knowledge of cloud computing and Azure app service.

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A Deep Learning Engineer is an IT professional who is responsible for developing and managing data pipelines. He or she is knowledgeable about analyzing and storing data collected from various sources. A Career as a Deep Learning Engineer needs to help the data scientists and analysts to create effective data sets.

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