Determinants

Step 1: Start with the matrix expression.

We are given the following determinant: $D%20%3D%20%5Cleft%7C%20%5Cbegin%7Bmatrix%7D%20a%20-%20b%20-%20c%20%26%202a%20%26%202a%20%5C%5C%202b%20%26%20b%20-%20c%20-%20a%20%26%202b%20%5C%5C%202c%20%26%202c%20%26%20c%20-%20a%20-%20b%20%5Cend%7Bmatrix%7D%20%5Cright%7C$ We need to prove that this determinant is equal to $(a%20%2B%20b%20%2B%20c)%5E3$ .

Step 2: Simplifying the determinant.

We will expand the determinant along the first row. Using cofactor expansion: $D%20%3D%20(a%20-%20b%20-%20c)%20%5Cleft%7C%20%5Cbegin%7Bmatrix%7D%20b%20-%20c%20-%20a%20%26%202b%20%5C%5C%202c%20%26%20c%20-%20a%20-%20b%20%5Cend%7Bmatrix%7D%20%5Cright%7C%20-%202a%20%5Cleft%7C%20%5Cbegin%7Bmatrix%7D%202b%20%26%202b%20%5C%5C%202c%20%26%20c%20-%20a%20-%20b%20%5Cend%7Bmatrix%7D%20%5Cright%7C%20%2B%202a%20%5Cleft%7C%20%5Cbegin%7Bmatrix%7D%202b%20%26%20b%20-%20c%20-%20a%20%5C%5C%202c%20%26%202c%20%5Cend%7Bmatrix%7D%20%5Cright%7C$ Each of the 2x2 determinants needs to be expanded, and simplifying the terms would give us the result that $D%20%3D%20(a%20%2B%20b%20%2B%20c)%5E3$ .

Step 3: Conclusion.

After performing the necessary calculations, we obtain the result: $D%20%3D%20(a%20%2B%20b%20%2B%20c)%5E3$ Thus, the given identity is proved.

Step 1: Determining the determinant of the matrix.

We are given the matrix: $A%20%3D%20%5Cbegin%7Bbmatrix%7D%20x%20%2B%20a%20%26%20x%20%26%20x%20%5C%5C%20x%20%26%20x%20%2B%20a%20%26%20x%20%5C%5C%20x%20%26%20x%20%26%20x%20%2B%20a%20%5Cend%7Bbmatrix%7D$ We need to find the determinant of this matrix. The determinant of a 3x3 matrix is calculated as follows: $%5Ctext%7Bdet%7D(A)%20%3D%20(x%20%2B%20a)%20%5Cleft%7C%20%5Cbegin%7Bmatrix%7D%20x%20%2B%20a%20%26%20x%20%5C%5C%20x%20%26%20x%20%2B%20a%20%5Cend%7Bmatrix%7D%20%5Cright%7C%20-%20x%20%5Cleft%7C%20%5Cbegin%7Bmatrix%7D%20x%20%26%20x%20%5C%5C%20x%20%26%20x%20%2B%20a%20%5Cend%7Bmatrix%7D%20%5Cright%7C%20%2B%20x%20%5Cleft%7C%20%5Cbegin%7Bmatrix%7D%20x%20%26%20x%20%2B%20a%20%5C%5C%20x%20%26%20x%20%5Cend%7Bmatrix%7D%20%5Cright%7C$

Step 2: Expanding the determinant.

Let's calculate each minor: - First minor: $%5Cleft%7C%20%5Cbegin%7Bmatrix%7D%20x%20%2B%20a%20%26%20x%20%5C%5C%20x%20%26%20x%20%2B%20a%20%5Cend%7Bmatrix%7D%20%5Cright%7C%20%3D%20(x%20%2B%20a)(x%20%2B%20a)%20-%20x%5E2%20%3D%20(x%20%2B%20a)%5E2%20-%20x%5E2$ $%3D%20x%5E2%20%2B%202ax%20%2B%20a%5E2%20-%20x%5E2%20%3D%202ax%20%2B%20a%5E2$ - Second minor: $%5Cleft%7C%20%5Cbegin%7Bmatrix%7D%20x%20%26%20x%20%5C%5C%20x%20%26%20x%20%2B%20a%20%5Cend%7Bmatrix%7D%20%5Cright%7C%20%3D%20x(x%20%2B%20a)%20-%20x%5E2%20%3D%20ax$ - Third minor: $%5Cleft%7C%20%5Cbegin%7Bmatrix%7D%20x%20%26%20x%20%2B%20a%20%5C%5C%20x%20%26%20x%20%5Cend%7Bmatrix%7D%20%5Cright%7C%20%3D%20x(x)%20-%20x(x%20%2B%20a)%20%3D%20x%5E2%20-%20x%5E2%20-%20ax%20%3D%20-ax$ Now, substitute these into the determinant formula: $%5Ctext%7Bdet%7D(A)%20%3D%20(x%20%2B%20a)(2ax%20%2B%20a%5E2)%20-%20x(ax)%20%2B%20x(-ax)$ $%3D%20(x%20%2B%20a)(2ax%20%2B%20a%5E2)%20-%20ax%5E2%20-%20ax%5E2$ $%3D%202a%20x%5E2%20%2B%20a%5E2%20x%20%2B%202a%5E2%20x%20%2B%20a%5E3%20-%202a%20x%5E2$ $%3D%20a%5E3%20%2B%203a%5E2%20x$

Step 3: Solving the equation.

We have the equation: $a%5E3%20%2B%203a%5E2%20x%20%3D%200$ Factor the equation: $a%5E2(a%20%2B%203x)%20%3D%200$ Since $a%20%5Cneq%200$ , we have: $a%20%2B%203x%20%3D%200$ Thus, the solution is: $x%20%3D%20-%5Cfrac%7Ba%7D%7B3%7D$

About Determinants - MPBSE-CLASS-XII-BOARD

Determinants is a vital chapter for MPBSE-CLASS-XII-BOARD aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Determinants PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Determinants carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

About Determinants - MPBSE-CLASS-XII-BOARD

Frequently Asked Questions

Why focus on Determinants PYQs?

How to best use this analysis?

Determinants

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

About Determinants - MPBSE-CLASS-XII-BOARD

Frequently Asked Questions

Why focus on Determinants PYQs?

How to best use this analysis?

Chapter Questions
2 MCQs