Problem 71 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(\left|\begin{array}{ccc}x+3 & x+2 & (x+2)^{3} \\ x+2 & x+3 & (x+2)^{3} \\\ (x+2)^{3} & x+2 & x+3\end{array}\right|\) \(=a x^{7}+b x^{6}+c x^{5}+d x^{4}+e x^{3}+f x^{2}+g x+h\) be an iden- tity in \(x\), where \(a, b, c, d, e, f, g, h\) are independent of \(x\), then the value of \(g\) is (A) \(-213\) (B) 213 (C) 0 (D) None of these

Step-by-Step Solution

Answer

The value of \(g\) is 213.

Step 1: Identify the Matrix

We are given a 3x3 determinant \(\left|\begin{array}{ccc}x+3 & x+2 & (x+2)^{3} \ x+2 & x+3 & (x+2)^{3} \ (x+2)^{3} & x+2 & x+3 \end{array}\right|\). This matrix has three distinct rows, each containing expressions of \(x\).

Step 2: Apply Properties of Determinants

For a determinant of a matrix to be zero, if any two rows or columns of a determinant are identical, the determinant is zero. In our matrix, observe that the second and third columns differ only in the constant terms, but they aren't exact repeats or scalar multiples.

Step 3: Use Row Reduction for Simplification

Perform row operations to simplify the calculation: Subtract the second row from the first row, the second row from the third row, and then swap rows if needed to make calculations more straightforward. The aim is to create as many zeros as possible, facilitating easier determinant computation.

Step 4: Compute the Determinant

Using row operations, compute the determinant of the transformed (if applicable) matrix using cofactor expansion. Substitute back if you performed substitution, and multiply combinations of minor matrices.

Step 5: Expand and Simplify the Expression

Use algebraic manipulation to expand the matrix determinant into a polynomial of \(x\), then simplify the expression to identify the coefficient of each term.

Step 6: Identify Coefficients

Focus specifically on expanding terms again that arise during manipulation to get the specific coefficient \(g\) of \(x^2\). Simplify further to confirm the consistency of the expression, making careful note of terms appearing with \(x^2\).

Step 7: Verify and Extract the Coefficient for \(x^2\)

Finally, confirm through computation of each term that contributes to the coefficient for \(x^2\) in the polynomial expression. This will be your answer for \(g\), ensuring consistency across all manipulative steps.

Key Concepts

Matrix Operations

Matrix operations are essential in simplifying complex problems in mathematics, such as computing determinants. They involve various procedures like addition, subtraction, and transformation of rows and columns to simplify the matrix structure. In the given exercise, which starts with a 3x3 matrix, the key is performing row operations.

• Row operations: These include techniques such as subtracting one row from another to create zeros in the matrix. This step reduces complexity and prepares the matrix for further calculations.
• Identifying identical rows: Recognize if any rows or columns are similar, as this can imply a zero determinant, simplifying the process.

Row operations help transform a matrix into an easier form for further computation, like cofactor expansion.

Cofactor Expansion

Cofactor expansion, or Laplace expansion, is a method for computing the determinant of a matrix. This technique involves expanding a determinant down a row or column by breaking it into minors and corresponding cofactors. In our context, after simplifying the matrix with row operations, we apply cofactor expansion to calculate the determinant.

• Cofactors are calculated by removing the row and column of each element, determining the minor's determinant, and then alternately changing signs.
• The determinant is the sum of the product of elements and their respective cofactors.

Cofactor expansion is crucial in efficiently finding determinants, especially for larger matrices.

Polynomial Expansion

Once the determinant of the matrix is computed using cofactor expansion, the next step is polynomial expansion. This involves expressing the determinant as a polynomial in terms of the variable, in our case, x. This step is essential to understand and manipulate expressions algebraically.

• Express each term involving x: Break down the matrix determinant completely in terms of the powers of x.
• Ensure each term is clearly defined to the power of x it corresponds to, making it easier to identify coefficients later.

By expanding the polynomial, we derive the coefficients associated with each power of x, which is key for solving the problem.

Algebraic Manipulation

Algebraic manipulation involves reworking equations or expressions to simplify or solve them. In this exercise, once the determinant is expanded into a polynomial, algebraic manipulation helps to isolate and identify specific coefficients from the polynomial expansion.

• Distribution: Carefully distributing terms within the polynomial to reveal the structure of each coefficient.
• Combining like terms: Group like powers of x together and simplify the expression to identify the coefficient's value.

For the solution, the focus is on determining the coefficient of the \(x^2\) term, which is extracted through detailed algebraic manipulation, to find the value of g.