Problem 91 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the system of linear equations, \(x+y+z=6\) \(x+2 y+3 z=10\) \(3 x+2 y+\lambda z=\mu\) has more than two solutions, then \(\mu-\lambda^{2}\) is equal to

Step-by-Step Solution

Answer

-10

Step 1: Understand the Condition for Infinitely Many Solutions

For the system to have infinitely many solutions, the equations must be dependent. This requires the determinant of the coefficient matrix to be zero or that all equations describing the same geometric entity (like a plane).

Step 2: Set up the Coefficient Matrix

Extract the coefficients from each equation: Matrix \(A\) is: \[A = \begin{bmatrix}1 & 1 & 1 \1 & 2 & 3 \3 & 2 & \lambda\end{bmatrix}\]

Step 3: Calculate the Determinant of the Coefficient Matrix

Calculate the determinant \(|A|\) for matrix \(A\): \[|A| = 1 \times (2 \times \lambda - 3 \times 2) - 1 \times (1 \times \lambda - 3 \times 3) + 1 \times (1 \times 2 - 2 \times 3)\]Simplify this expression to find de determinant in terms of \(\lambda\).

Step 4: Set Determinant to Zero and Solve for \(\lambda\)

Set \(|A| = 0\) and solve:\[|A| = 2\lambda - 6 - (\lambda - 9) + (2 - 6) = 0\]Simplify to get:\[\lambda = 5\]

Step 5: Use the Condition of Dependency to Find \(\mu\)

For dependent equations, the third equation should be a linear combination of the first two. Given \(\lambda = 5\), substitute and solve:\[3x + 2y + 5z = \mu\ \text{is a linear combination of}\begin{cases}x + y + z = 6 \x + 2y + 3z = 10\end{cases}\]Solving gives \(\mu = 15\).

Step 6: Calculate \(\mu - \lambda^2\)

With \(\lambda = 5\) and \(\mu = 15\), substituting into \(\mu - \lambda^2\) yields:\[\mu - \lambda^2 = 15 - 5^2 = 15 - 25 = -10\]

Key Concepts

Infinitely Many Solutions

In a system of linear equations, having infinitely many solutions means that there are countless sets of values for the variables that satisfy all the equations simultaneously. This usually happens when the system of equations represents the same geometrical space, like overlapping lines or planes. To check if a system has infinitely many solutions, you look for conditions where:

• All equations describe the same line or plane, indicating they are essentially identical.
• The equations are dependent (more on this in the relevant section).
• The determinant of the coefficient matrix is zero, which directly relates to the concept of dependency.

When a system meets these criteria, instead of having a single unique solution, there exists an entire region, line, or plane filling infinite points that can satisfy the system.

Determinant of Matrix

The determinant is a scalar value that can be computed from the elements of a square matrix, offering vital insights into the matrix's properties. For a 3x3 matrix like in our exercise, the determinant can be calculated with a formula involving cross-multiplying and subtracting, like this: If matrix \( A \) is:\[A = \begin{bmatrix}1 & 1 & 1 \1 & 2 & 3 \3 & 2 & \lambda\end{bmatrix}\]The determinant \(|A|\) is calculated as follows:\[|A| = 1\cdot(2\lambda - 6) - 1\cdot(\lambda - 9) + 1\cdot(2 - 6)\]This simplification helps to determine special conditions like dependency or rank of the matrix. A zero determinant implies that the matrix is not full rank, pointing to dependent equations.

Dependent Equations

Equations within a system are dependent when one equation can be expressed as a linear combination of other equations in the system. This means that there is a redundancy in the information that these equations provide about the variables. When dependent, the system is unable to provide a unique solution, leading to either infinitely many solutions or no solutions at all. Geometrically, when equations are dependent, they might represent the same plane or line in higher dimensions. Thus, they do not cover new spatial information.In our exercise:

• The third equation \( 3x + 2y + 5z = \mu \) is dependent on the first two only if \( \lambda = 5 \) and the adjusted equation matches them.
• When this adjustment condition is fulfilled, it indicates the presence of infinite solutions.

Understanding dependency is crucial to analyze the behavior of systems of equations, particularly to determine solvability and the nature of solutions available.