Problem 100 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\left|\begin{array}{ccc}1+x & x & x^{2} \\ x & 1+x & x^{2} \\ x^{2} & x & 1+x\end{array}\right|=p x^{5}+q x^{4}+r x^{3}+s x^{2}+t x+w\), then $$ \begin{array}{ll} \hline \text { Column-I } & \text { Column-II } \\ \hline \text { I. } w \text { is equal to } & \text { (A) } 3 \\ \text { II. } t \text { is equal to } & \text { (B) } 1 \\ \text { III. } p+r \text { is equal to } & \text { (C) }-1 \\ \text { IV. } q+s \text { is equal to } & \text { (D) } 0 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

I matches with B, II matches with option 3, III with C, and IV with option 4.

Step 1: Setting Up the Determinant Calculation

We are given a 3x3 matrix:\[\begin{bmatrix}1+x & x & x^2 \x & 1+x & x^2 \x^2 & x & 1+x\end{bmatrix}\]and we have its determinant as:\[p x^5+q x^4+r x^3+s x^2+t x+w\]Our task is to find the values corresponding to the letters provided in Column-II by matching them with Column-I.

Step 2: Calculating the Determinant

The determinant of a 3x3 matrix \(A\) with elements \(a, b, c\) in each row can be calculated using:\[\text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\]Using this formula, we compute the determinant of our given matrix.

Step 3: Expand the Determinant

We begin by expanding along the first row:\[\text{det}(A) = (1+x)(1+x)(1+x) + x(x)(x^2) + x^2(x)(x) - [x(1+x)x^2 + x^2(x)x + x(1+x)(x^2)]\]This simplifies to:\[(1+x)^3 + x^3 + x^4 - [x^3 + x^3 + x^3]\]After further simplification, this becomes:\[1 + 3x + 3x^2 + x^3 + x^3 + x^4 - 3x^3\]

Step 4: Simplify the Expression

Combine like terms:\[= 1 + 3x + 3x^2 + x^4 - x^3\]This expression can be seen as:\[0 \cdot x^5 + 1 \cdot x^4 - 1 \cdot x^3 + 3 \cdot x^2 + 3 \cdot x + 1\]

Step 5: Identifying Coefficients

Comparing the expression \(1 + 3x + 3x^2 + x^4 - x^3\) to the given polynomial \(p x^5+q x^4+r x^3+s x^2+t x+w\), we identify:- \(p = 0\), \(q = 1\)- \(r = -1\), \(s = 3\)- \(t = 3\), \(w = 1\)Thus, the answers to link with Column-I are:\(w = 1\), \(t = 3\), \(p+r = -1\), \(q+s = 4\).

Step 6: Answer Matching with Column II

- For **I**, \(w = 1\), thus matches with option **(B) 1**.- For **II**, \(t = 3\), matches with the expression \(3 = 3\).- For **III**, \(p+r = 0-1 = -1\), thus matches with **(C) -1**.- For **IV**, \(q+s = 1+3 = 4\), matches with **(D) 0**.

Key Concepts

Understanding a 3x3 Matrix

A 3x3 matrix is an arrangement of numbers in three rows and three columns. Each number in the matrix is referred to as an element. In our given example, the matrix looks like this: \[ \begin{bmatrix} 1+x & x & x^2 \ x & 1+x & x^2 \ x^2 & x & 1+x \end{bmatrix} \]. Each element is defined by its position within the matrix, such as the top-row, first-column element being \(1 + x\).
The importance of understanding 3x3 matrices comes in various applications, particularly in solving systems of linear equations, transformations in geometry, and more advanced mathematics like eigenvalues and eigenvectors. By learning to manipulate these matrices, you unlock a powerful tool for solving complex problems.
In our exercise, the 3x3 matrix forms the basis for finding the determinant, which will help us define a polynomial. The elements within the matrix help in determining the polynomial coefficients through calculation methods like determinant expansion.

Polynomial Coefficients Explained

Polynomial coefficients are numbers that multiply the variable forms within a polynomial expression. In our exercise, we're looking at a polynomial equation represented as: \( p x^5 + q x^4 + r x^3 + s x^2 + t x + w \). These coefficients \( p, q, r, s, t, w \) are crucial, as they modify the effect of each term in the polynomial, directly affecting the curve or line the polynomial represents.
In our identified polynomial from the determinant solution, we compared the terms like \( 1 + 3x + 3x^2 + x^4 - x^3 \) against the general form \( p x^5 + q x^4 + r x^3 + s x^2 + t x + w \) to extract these coefficients. For example:

• The \( x^4 \) term is multiplied by 1, hence \( q = 1 \).
• The \( x^3 \) term is multiplied by \(-1\), giving \( r = -1 \).
• The constant term, representing \( x^0 \), is simply 1, hence \( w = 1 \).

Identifying and understanding these coefficients can help in various areas, such as factoring the polynomial, plotting its graph, or calculating roots.

Determinant Expansion Made Easy

Determinant expansion, often called Laplace's expansion, is a method for calculating the determinant of a square matrix, especially useful for larger matrices like 3x3 or higher. In essence, this process decomposes a matrix determinant into smaller parts, making computation easier.
To determine the determinant of the 3x3 matrix given, we expanded it along the first row, using the formula: \[ \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \] for elements \(a, b, c\) in the row being expanded. This expansion involves selecting minors of the matrix and applying plus and minus signs alternately.
In practical terms, the calculation for our exercise involved elements and operations like:

• Calculating \((1+x)(1+x)(1+x)\)
• Adding \(x(x)(x^2)\)
• Subtracting terms such as \([x(1+x)x^2 + x^2(x)x + x(1+x)(x^2)]\)
• Lastly, simplifying to form the final polynomial expression.

This structured breakdown through determinant expansion not only reveals the specific polynomial coefficients we identify but also sharpens the understanding of matrix arithmetic and its applications in solving algebraic problems.