Problem 88 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

For which of the following ordered pairs \((\mu, \delta)\), the system of linear equations \(x+2 y+3 z=1\) \(3 x+4 y+5 z=\mu\) \(4 x+4 y+4 z=\delta\) is inconsistent? \(\quad\) [Jan. 8, 2020 (I)] (a) \((4,3)\) (b) \((4,6)\) (c) \((1,0)\) (d) \((3,4)\)

Step-by-Step Solution

Answer

(a) \((4, 3)\) and (c) \((1, 0)\) are inconsistent.

Step 1: Write the system of equations

We are given the following system of linear equations:\[x + 2y + 3z = 1 \3x + 4y + 5z = \mu \4x + 4y + 4z = \delta\]

Step 2: Find condition for inconsistency

To determine when the system is inconsistent, assume the equations don't have a common solution. This typically occurs when augmented matrices lead to a row with zeros on the left and a non-zero value on the right after attempting to solve the system (i.e., no possible solution).

Step 3: Form the augmented matrix

Create the augmented matrix for these equations:\[\begin{bmatrix}1 & 2 & 3 & | & 1 \3 & 4 & 5 & | & \mu \4 & 4 & 4 & | & \delta\end{bmatrix}\]

Step 4: Perform row operations

To simplify the matrix, perform row operations to get it in row-echelon form.First, subtract 3 times the first row from the second row:\[\begin{bmatrix}1 & 2 & 3 & | & 1 \0 & -2 & -4 & | & \mu - 3 \4 & 4 & 4 & | & \delta\end{bmatrix}\]Next, subtract 4 times the first row from the third row:\[\begin{bmatrix}1 & 2 & 3 & | & 1 \0 & -2 & -4 & | & \mu - 3 \0 & -4 & -8 & | & \delta - 4\end{bmatrix}\]

Step 5: Simplify second and third rows

Divide the second row by -2:\[\begin{bmatrix}1 & 2 & 3 & | & 1 \0 & 1 & 2 & | & \frac{3 - \mu}{2} \0 & -4 & -8 & | & \delta - 4\end{bmatrix}\]Now, add 4 times the second row to the third row:\[\begin{bmatrix}1 & 2 & 3 & | & 1 \0 & 1 & 2 & | & \frac{3 - \mu}{2} \0 & 0 & 0 & | & \delta - 4 + 2(3 - \mu)\end{bmatrix}\]The system is inconsistent if:\[\delta - 4 + 2(3 - \mu) eq 0\]

Step 6: Solve for inconsistency condition

Set the inconsistency condition and find expressions:\[\delta - 4 + 6 - 2\mu eq 0\]\[\delta + 2 - 2\mu eq 0\]\[\delta eq 2\mu - 2\]

Step 7: Check each option

Check which pair (\(\mu, \delta\)) satisfies \(\delta = 2\mu - 2\):(a) \((4, 3)\): \(3 eq 2\times4 - 2 = 6\) - Inconsistent(b) \((4, 6)\): \(6 eq 2\times4 - 2 = 6\) - Consistent(c) \((1, 0)\): \(0 eq 2\times1 - 2 = 0\) - Inconsistent(d) \((3, 4)\): \(4 eq 2\times3 - 2 = 4\) - Consistent

Key Concepts

Augmented Matrix

An augmented matrix is a powerful tool in linear algebra that helps us solve systems of equations. In essence, it organizes the coefficients of the variables in an equation system, along with the constants, into a structured matrix format.
Here is how it works:

• Each row of the matrix corresponds to an equation from the system.
• The columns represent the coefficients of each variable and the constants on the right side of the equations.
• The vertical bar in the matrix separates the coefficients from the constants.

For example, in the exercise, the augmented matrix from the system of equations was:\[\begin{bmatrix}1 & 2 & 3 & | & 1 \3 & 4 & 5 & | & \mu \4 & 4 & 4 & | & \delta\end{bmatrix}\]This matrix includes all the necessary information to use row operations to simplify and potentially solve the system.

Row Operations

Row operations are techniques used to manipulate the rows of an augmented matrix to simplify solving systems of equations. These operations help bring the matrix to simpler forms like row-echelon form or reduced row-echelon form.
Three types of row operations can be applied:

• Swapping two rows: This rearranges the equations, which can be helpful to get zeros into desired positions.
• Multiplying a row by a nonzero scalar: It scales the entire row, which maintains the equation's equivalence.
• Adding or subtracting a multiple of one row to another: A key operation for eliminating variables from certain rows.

In the step-by-step solution, specific row operations were performed: - Subtracting 3 times the first row from the second row; - Subtracting 4 times the first row from the third row; - Dividing the second row by -2; and - Adding 4 times the updated second row to the third row. These steps aimed to isolate and identify conditions leading to inconsistency in the system.

Systems of Equations

A system of equations is a set of two or more equations with the same variables. The goal is usually to find a common solution that satisfies all the equations simultaneously.
Consider the system given in the exercise:

• \( x + 2y + 3z = 1 \)
• \( 3x + 4y + 5z = \mu \)
• \( 4x + 4y + 4z = \delta \)

The potential solutions depicted by ordered pairs involve finding values for \(x\), \(y\), and \(z\) that work for all equations.
This particular exercise asks for conditions under which the system is inconsistent, meaning no common solution exists that satisfies all equations.

Consistency Condition

The consistency condition is crucial in determining if a system of equations can be solved. A system is consistent if at least one set of values for the variables exists that satisfies all the equations, otherwise it is inconsistent when no such solutions exist.
For the system in this exercise:

• Inconsistency arises if the augmented matrix, upon simplification using row operations, results in a row that has all zero coefficients but a non-zero constant. This indicates conflicting information that makes the system unsolvable.

In our exercise, the focus was on determining when this condition is met. The derived equation from the matrix simplification was:\[\delta - 4 + 2(3 - \mu) eq 0\]This simplifies to:\[\delta eq 2\mu - 2\]This condition tells us the system is inconsistent when the value of \( \delta \) does not equal \( 2\mu - 2 \). Plugging in each ordered pair option checks for when this equation holds true.