Problem 28 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The equations \(x+y+z=6, x+2 y+3 z=10, x+2 y+\) \(m z=n\) give infinite number of values of the triplet \((x,\), \(y, z)\) if (A) \(m=3, n \in R\) (B) \(m=3, n \neq 10\) (C) \(m=3, n=10\) (D) None of these

Step-by-Step Solution

Answer

Option (C) is correct: \( m=3, n=10 \).

Step 1: Understand the problem

We have a system of three equations. The first two equations are \( x+y+z=6 \) and \( x+2y+3z=10 \). The third equation is given by \( x+2y+mz=n \) where \( m \) and \( n \) are parameters. We need to find the conditions on \( m \) and \( n \) such that the system has infinitely many solutions for the triplet \((x, y, z)\).

Step 2: Analyze equations for dependency

In order for the system of equations to have infinitely many solutions, at least one equation must be a linear combination of the others. Let's explore if the third equation can be a linear combination of the first two for certain values of \( m \) and \( n \).

Step 3: Check conditions for linear dependency

Equation 2 can be formed by multiplying equation 1 by a constant and adding. Thus, the third equation is dependent if \( m=3 \) and \( n=10 \). This makes the third equation identical to equation 2, giving an infinite number of solutions.

Step 4: Eliminate options

Now, let's analyze each given option: - (A) If \( m=3 \) and \( n \in \mathbb{R} \), the condition \( n=10 \) must hold for infinite solutions, so this option is not entirely correct.- (B) If \( m=3 \) and \( n eq 10 \), changes to \( n \) that differ from 10 will disrupt the equation's dependency.- (C) \( m=3 \) and \( n=10 \) make the third equation identical to the second, leaving two independent equations with infinitely many solutions.Thus, option (C) is correct.

Key Concepts

infinite solutions

A system of linear equations with infinite solutions implies that there are countless numbers of solutions that satisfy all the given equations simultaneously. Typically, this occurs when the equations in the system are not entirely independent and instead share a proportional relationship. When this happens, it means that you can manipulate one of the equations by multiplying it with a certain factor or adding it to other equations to end up with effectively the same equation. For example, when you have the third equation as a linear combination of the first two.

• To achieve infinite solutions, there must be more variables than independent equations, or at least one redundant equation which does not add new information.
• For practical purposes in analysis, it means the equations describe lines or planes that perfectly overlap, leading to infinite intersection points.
• In practical scenarios, this could represent that any point meeting the criteria of one equation naturally satisfies the other.

So, if you ever detect redundancy or overlapping geometrical representations in your system of equations, you're well on the path to infinite solutions.

linear dependency

Linear dependency in a system of linear equations means that one equation can be derived from the others through addition, subtraction, or scalar multiplication. This is a key concept to verify if your system might have infinitely many solutions. Calculating for linear dependency, you can test if one equation is a result of a linear combination of the others.

• An equation is linearly dependent on others if it doesn’t introduce any new plane or line in the graph of the system.
• This is closely intertwined with the rank of the matrix that represents the system of equations; redundant equations decrease the rank.
• Discovering linear dependencies lets you know that you can eliminate some equations without losing the solutions' validity. This also helps in simplifying the problem.

Exploring the dependencies between equations can often reveal why only specific parameter values (like in our problem: where \(m=3\) and \(n=10\)) give us those infinite solutions.

linear combination

Understanding linear combinations is crucial in solving systems of equations. The idea is that you can add or multiply equations by scalars to form new equations. If an equation in your system can be expressed as a linear combination of other equations, it reinforces the idea of dependency and potentially infinite solutions.

• A linear combination involves taking existing equations, multiplying them by constants, and summing them up.
• Each equation in the system contributes to a part of the resultant equation in its unique way if they are linearly independent.
• When forming linear combinations, you're effectively searching for relationships among the equations that describe the same solution space.

For example, in our exercise, equation two derived through a linear combination can show that the third equation is redundant under particular \(m\) and \(n\) values (i.e., \(m=3\) and \(n=10\)). This simplicity lies in recognizing that the system's space of solutions remains unchanged, leading to infinite possibilities.