Problem 118 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The number of values of \(\mathrm{k}\) for which the homogeneous system of linear equations \(4 x+k y+2 z=0 ; k x+4 y+z=0 ; 2 x+2 y+z=0\) possess a non- zero solution is (A) 2 (B) 1 (C) Zero (D) 3

Step-by-Step Solution

Answer

Zero values for \(k\) allow non-zero solutions as the system is never singular.

Step 1: Understand the System of Equations

We have three linear equations: 1. Equation 1: \(4x + ky + 2z = 0\) 2. Equation 2: \(kx + 4y + z = 0\) 3. Equation 3: \(2x + 2y + z = 0\)We're looking for values of \(k\) that allow non-trivial solutions.

Step 2: Write in Matrix Form

These equations can be rewritten in matrix form as follows:\[\begin{bmatrix}4 & k & 2 \k & 4 & 1 \2 & 2 & 1 \\end{bmatrix}\begin{bmatrix}x \y \z \\end{bmatrix}= \begin{bmatrix}0 \0 \0 \\end{bmatrix}\]

Step 3: Conditions for Non-Trivial Solutions

A homogeneous system has non-trivial solutions when the determinant of the coefficient matrix is zero.

Step 4: Calculate the Determinant

Compute the determinant of the matrix:\[\begin{vmatrix}4 & k & 2 \k & 4 & 1 \2 & 2 & 1 \\end{vmatrix}\]Using cofactor expansion along the first row:\(4(4 \cdot 1 - 1 \cdot 2) - k(k \cdot 1 - 1 \cdot 2) + 2(k \cdot 2 - 4 \cdot 2)\)

Step 5: Simplify the Determinant Expression

Simplify the expressions:\[4(4 - 2) - k(k - 2) + 2(2k - 8)\]This simplifies to \[8 - k^2 + 2k - 16 \] resulting in the equation:\[-k^2 + 2k - 8 = 0\]

Step 6: Solve the Quadratic Equation for k

Solve \[-k^2 + 2k - 8 = 0\] by factoring or using the quadratic formula:\[k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] where \(a = -1\), \(b = 2\), \(c = -8\).Calculate the discriminant \(2^2 - 4(-1)(-8) = 4 - 32 = -28\).

Step 7: Conclusion from Discriminant

The discriminant is negative, \(-28\). Thus, there are no real solutions for \(k\) where the determinant is zero.

Key Concepts

Homogeneous System

In mathematics, a homogeneous system of linear equations is one where all the constant terms are zero. This means that every equation in the system can be expressed in the form:

• \( a_1 x_1 + a_2 x_2 + \ldots + a_n x_n = 0 \)

Here, each side of the equation equals zero. These types of systems are interesting because they always have at least one solution, the trivial solution. In the trivial solution, every variable is set to zero.
However, more intriguing are the non-trivial solutions, where at least one variable is not zero. The system needs to meet specific conditions for these solutions to exist. When examining a homogeneous system, such as the one described in the exercise, determining the existence of non-trivial solutions involves analyzing the coefficient matrix associated with the system.

Non-Trivial Solutions

Non-trivial solutions in a homogeneous system of equations occur when there is at least one value in the solution set that is not zero.

• To find such solutions, the determinant of the system's coefficient matrix should equal zero.

This condition signals that the system has infinitely many solutions, including non-trivial ones.
In the given problem, when checking for non-trivial solutions, the step-by-step solution highlights solving the determinant to identify possible values of \( k \). Non-trivial solutions provide insight into dependencies between variables, which could imply a relationship beyond simply all variables being zero. This characteristic is fundamental when determining the linear dependence of different vectors represented by the equations.
Effectively, non-trivial solutions are significant in fields like engineering, physics, or computer science because they often signify meaningful real-world phenomena being represented mathematically.

Matrix Determinant

The concept of a matrix determinant is fundamental in understanding solutions to systems of linear equations.

• A determinant is a special number that is calculated from a square matrix.
• It provides information about the matrix such as whether it is invertible and how many solutions the associated system of equations might have.

For a homogeneous system, a zero determinant indicates the system has non-trivial solutions, meaning the equations don't just intersect at the origin but blend along a line or plane of solutions instead.
In our exercise, we use the determinant of the coefficient matrix to evaluate the conditions under which \( k \) allows such non-trivial solutions to exist.
The calculation of the determinant, as shown, involves utilizing a method such as cofactor expansion that helps simplify the understanding of the relationship between variables in a matrix. Recognizing when and how to calculate the determinant is critical both for solving equations and understanding the geometric interpretations of these solution sets.