Problem 107 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\mathrm{S}\) is the set of distinct values of ' \(\mathrm{b}\) ' for which the following system of linear equations [2017] \(x+y+z=1\) \(x+a y+z=1\) \(a x+b y+z=0\) has no solution, then \(S\) is : (a) a singleton (b) an empty set (c) an infinite set (d) a finite set containing two or more elements

Step-by-Step Solution

Answer

(b) an empty set

Step 1: Understanding the System of Equations

The given system of equations is:1. \( x + y + z = 1 \)2. \( x + a y + z = 1 \)3. \( a x + b y + z = 0 \)We need to find the values of \( b \) for which this system has no solution.

Step 2: Setting Up the System in Matrix Form

Express the system of equations in matrix form, \( A\vec{x} = \vec{b} \), where:\[A = \begin{bmatrix} 1 & 1 & 1 \ 1 & a & 1 \ a & b & 1 \end{bmatrix}, \, \vec{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}, \, \vec{b} = \begin{bmatrix} 1 \ 1 \ 0 \end{bmatrix}\]

Step 3: Determine Condition for No Solution

For a system of equations to have no solution, the matrix \( A \) must be singular, meaning \( \det(A) = 0 \). Calculate the determinant of matrix \( A \):\[ \det(A) = 1\cdot(a\cdot1 - b\cdot1) - 1\cdot(1\cdot1 - a\cdot1) + 1\cdot(1\cdot b - a\cdot1) \]

Step 4: Simplifying the Determinant

Simplify the expression for the determinant:\[ \det(A) = (a - b) - (1 - a) + (b - a) = 2a - 2 \]

Step 5: Solve for Conditions on \(a\)

Set the simplified determinant equal to zero to find \(a\):\[ 2a - 2 = 0 \]Solving gives:\[ a = 1 \]

Step 6: Examining the System for \(a = 1\)

Substitute \( a = 1 \) into the original set of equations to see the impact on the system:1. \( x + y + z = 1 \)2. \( x + y + z = 1 \)3. \( x + b y + z = 0 \)The first two equations become identical, implying the system depends heavily on the third equation.

Step 7: Analyze for \(b\) when \(a = 1\)

For \(a = 1\), examine when the third equation provides inconsistency:If the system is reduced to: 1. \( x + y + z = 1 \)2. \( x + b y + z = 0 \)Check the condition by solving line for value difference leading to no solutions, for instance if \(1 = 0\).There is no \(b\) making the system consistently inconsistent for every real \(y\).

Step 8: Conclusion

Since there is no specific value of \( b \) that leads to inconsistency when conditions are intended broad, we reconsider integrity. It infers set \(S\) does not have valid subtractive possibilities indicating standard solution scope.

Key Concepts

Matrix Form

In the context of solving systems of linear equations, expressing equations in matrix form is an extremely valuable tool. Matrix form provides a compact and organized way to handle the coefficients of the variables, which can simplify both the theoretical and numerical solutions of the system.

For a given system of equations, each equation's coefficients form a row in a matrix. For instance, given the system:

• x + y + z = 1
• x + ay + z = 1
• ax + by + z = 0

The matrix form is denoted as:\[A\vec{x} = \vec{b}\]Where: \[A = \begin{bmatrix} 1 & 1 & 1 \ 1 & a & 1 \ a & b & 1 \end{bmatrix}, \quad \vec{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad \vec{b} = \begin{bmatrix} 1 \ 1 \ 0 \end{bmatrix}\]
This matrix form makes it easier to apply linear algebra techniques to solve or analyze the consistency of the system, such as checking if the matrix is singular.

Singular Matrix

A matrix's status as singular is central to understanding why a system of linear equations might not have a solution. A matrix is singular if it cannot be inverted. This occurs when the determinant of the matrix is zero. A singular matrix indicates a situation where the equations do not span the space effectively, producing one or more dependent equations that do not contribute new information.

In our scenario, the matrix \( A \) is considered:

• A non-singular (or invertible) matrix has a unique solution.
• A singular matrix means there's either no solution or infinitely many solutions possible.

When \( A \) is singular, the determinant \( \det(A) = 0 \), it's key to conclude if the system under given operations exactly shadows linearly dependent rows. In this exercise, setting \( \det(A) = 2a - 2 = 0 \) leads to the evaluation of consistent variable instabilities across known domains.

Determinant

The determinant is a special number that can be calculated from a square matrix, offering insight into the matrix's properties. Specifically, it can tell us if a matrix is invertible or singular. For a 3x3 matrix like the one explored here, the determinant is calculated using a specific formula involving recursive expansion by minors.

The determinant of matrix \( A \):
\[\det(A) = 1\cdot(a\cdot1 - b\cdot1) - 1\cdot(1\cdot1 - a\cdot1) + 1\cdot(1\cdot b - a\cdot1)\]
Simplifies to:
\[\det(A) = 2a - 2\]
Setting \( \det(A) = 0 \) allows us to derive conditions on \( a \) and subsequently examine the behavior of \( b \). This process reveals singular status - affecting the system by indicating the potential of equations being linear multiples (or nonefficient), notably when there's no unitary solution presence in presented endpoint equations.