Problem 79 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The system of linear equations \(x+y+z=2\) \(2 x+3 y+2 z=5\) \(2 x+3 y+\left(a^{2}-1\right) z=a+1 \quad\) [Jan 09 2019I] (a) is inconsistent when \(\mathrm{a}=4\) (b) has a unique solution for \(|\mathrm{a}|=\sqrt{3}\) (c) has infinitely many solutions for \(\mathrm{a}=4\) (d) is inconsistent when \(|\mathrm{a}|=\sqrt{3}\)

Step-by-Step Solution

Answer

(c) has infinitely many solutions for \\ a=4.

Step 1: Understanding the Problem

We have a system of three linear equations with parameters that can change the behavior of the solutions. The goal is to determine conditions for inconsistency or types of solutions based on different values of 'a'.

Step 2: Write the System of Equations

Write the given system of linear equations as: 1. \( x + y + z = 2 \) 2. \( 2x + 3y + 2z = 5 \) 3. \( 2x + 3y + (a^2 - 1)z = a + 1 \)

Step 3: Check Consistency for a = 4

Substitute \(a = 4\) into the third equation: \[(a^2 - 1)z = (4^2 - 1)z = 15z\] and \[(a + 1) = (4 + 1) = 5\]So the third equation becomes \(2x + 3y + 15z = 5\). Compare with equations 1 and 2 to check for contradictions.

Step 4: Solve for a = 4

Substitute \(a = 4\) and solve the system:1. \( x + y + z = 2 \)2. \( 2x + 3y + 2z = 5 \)3. \( 2x + 3y + 15z = 5 \)Subtract equation 2 from equation 3: \[ (2x + 3y + 15z) - (2x + 3y + 2z) = 5 - 5 \]\[13z = 0 \Rightarrow z = 0 \]Substitute \(z = 0\) in equations 1 and 2 to find \(x\) and \(y\).

Step 5: Solving for a Solution when a = 4

Substitute \(z = 0\) into equations 1 and 2:Equation 1 becomes \(x + y = 2\).Equation 2 becomes \(2x + 3y = 5\).Solve this system of 2 equations in 2 variables with substitution or elimination to find a consistent solution, indicating infinite solutions exist due to a leading zero row appearing only in a third equation trial test.

Step 6: Check Consistency for |a| = sqrt(3)

Substitute \(|a| = \sqrt{3}\) into the third equation:\((a^2 - 1)z = (3 - 1)z = 2z\)and\(a + 1 = \sqrt{3} + 1\)The third equation becomes \(2x + 3y + 2z = \sqrt{3} + 1\). Compare and analyze for solution consistency with equations 1 and 2.

Step 7: Determine Solutions for |a| = sqrt(3)

The system 1. \( x + y + z = 2 \)2. \( 2x + 3y + 2z = 5 \)3. \( 2x + 3y + 2z = \sqrt{3} + 1 \) has a conflict when substituting equations meaning there is consistency only when \(|a| = \sqrt{3} = 1.732\), leaving a reduced possibility for unique or no solutions.

Key Concepts

System of Equations

A system of equations is like a set of mathematical puzzles all tied together. Each equation represents a rule that relates different variables. In a linear system, these rules are linear, meaning each variable gets multiplied by a constant and then summed up. Linear relationships look like straight lines when plotted on a graph.

Our given system consists of three equations:

• \(x + y + z = 2\)
• \(2x + 3y + 2z = 5\)
• \(2x + 3y + (a^2 - 1)z = a + 1\)

Here, the equations incorporate a parameter \(a\) which affects the overall behavior of the system. Depending on the value of \(a\), the solutions to the system can change significantly. Understanding this setup is crucial as it lays the foundation for analyzing the range of potential solutions.

Consistency in Systems of Equations

Consistency in a system of equations refers to whether there is at least one solution that satisfies all equations simultaneously. When all equations in a system can share at least one common solution, the system is consistent. Otherwise, it's inconsistent.

For example, when \(a = 4\), the system becomes:

• \(x + y + z = 2\)
• \(2x + 3y + 2z = 5\)
• \(2x + 3y + 15z = 5\)

By substituting \(a = 4\), we notice that the third equation simplifies to match the second, except with different coefficients for \(z\). This results in conditions where the system appears consistent, offering solutions for \(x\), \(y\), and \(z\). Meanwhile, substituting other values might introduce contradictions and inconsistencies, such as \(|a| = \sqrt{3}\), where the equations conflict and offer no solution.

Solutions to Linear Systems

Solutions to a linear system depend on how the equations in that system intersect:

• Unique solution: Occurs when the lines or planes intersect at a single point. This happens when the number of independent equations matches the number of unknowns, and they aren't parallel or coincide.
• Infinitely many solutions: This is seen when the lines or planes overlap exactly; in 3D, this means an entire line or plane is common to all and the system cannot pinpoint one particular solution.
• No solution: Describes a system where the lines or planes never meet, such as parallel planes in three dimensions.

When \(a = 4\), solving the system leaves us with consistent equations, suggesting infinitely many solutions rather than one specific solution. Conversely, when \(|a| = \sqrt{3}\), the system becomes inconsistent, meaning there isn't any solution that satisfies all equations. Understanding these different types of solutions is important for interpreting results in practical situations.