Problem 81 - A Complete Resource Book in Mathematics for JEE Main

Question

The set of equations : \(\lambda x-y+(\cos \theta) z=0 ; 3 x+y+2 z\) \(=0 ;(\cos \theta) x+y+2 z=0,0 \leq \theta<2 \pi\), has non-trivial solutions. (A) for no values of \(\lambda\) and \(\theta\) (B) for all values of \(\lambda\) and \(\theta\) (C) for all values of \(\lambda\) and only two values of \(\theta\) (D) for only one value of \(\lambda\) and all values of \(\theta\)

Step-by-Step Solution

Answer

C: for all values of \( \lambda \) and only two values of \( \theta \).

Step 1: Identify the System of Equations

The given system of linear equations is: 1) \( \lambda x - y + (\cos \theta) z = 0 \) 2) \( 3x + y + 2z = 0 \) 3) \( (\cos \theta) x + y + 2z = 0 \). We need to find the conditions under which these equations have non-trivial solutions.

Step 2: Determinant and Linear Dependence

For non-trivial solutions, the determinant of the coefficient matrix must be zero. The coefficient matrix is:\[\begin{bmatrix}\lambda & -1 & \cos \theta \3 & 1 & 2 \\cos \theta & 1 & 2\end{bmatrix}\]We calculate the determinant of this matrix and set it equal to zero to find conditions on \( \lambda \) and \( \theta \).

Step 3: Calculate the Determinant

The determinant is:\[\lambda(1 \cdot 2 - 2 \cdot 1) + 1(2 \cdot \cos \theta - 2 \cdot \cos \theta) + \cos \theta(3 \cdot 1 - 1 \cdot \cos \theta)\]This simplifies to:\[2\lambda + 3\cos \theta - \cos^2 \theta\]

Step 4: Condition for Non-Triviality

Setting the determinant equal to zero gives us:\[2\lambda + 3\cos \theta - \cos^2 \theta = 0\]This simplifies to: \[\cos \theta (3 - \cos \theta ) = -2\lambda\]Analyzing this, it's clear that the value of \( \theta \) is limited by the quadratic expression, which equals zero for specific angles.

Step 5: Solve the Quadratic Condition

The quadratic equation \( \cos \theta (3 - \cos \theta ) = c \) can have two valid \( \cos \theta \) roots for real solutions when c is adjusted:\( \cos \theta \) can be \( 0 \) and \( 3 \), but since \( \cos \theta \) must be \( \leq 1 \), valid roots are: \(\cos \theta = 0, \cos \theta = 3 \). \( \cos \theta = 3\) is not possible, hence we find non-trivial solutions for two angles corresponding to \( \cos \theta = 0\).

Step 6: Evaluate the Options

Since the analysis shows that the system has non-trivial solutions for all values of \( \lambda \) and two specific angle values of \( \theta \), the correct answer is C: "for all values of \( \lambda \) and only two values of \( \theta\)."

Key Concepts

Determinant

The concept of a determinant is essential when analyzing linear equations for non-trivial solutions. A determinant is a scalar value that can be computed from the elements of a square matrix, and it provides critical information about the matrix, such as whether the system is invertible or the volume spanned by its vectors.
In our scenario, we are dealing with a coefficient matrix derived from a system of linear equations. For non-trivial solutions to exist, the determinant of this matrix must be zero. This is because a zero determinant indicates that the matrix is singular, meaning it does not have full rank, leading to infinite or more solutions beyond the trivial zero solution.
To calculate the determinant, we use the formula for a 3x3 matrix, which involves a blend of the matrix's elements. Once the determinant is zero, it confirms that the system of linear equations has infinite solutions, hence non-trivial solutions.

Non-Trivial Solutions

The concept of non-trivial solutions is about finding solutions to a set of linear equations that are not the trivial, all-zero solution. These solutions are significant in many mathematical applications as they suggest dependence between variables or parameters in the equations.
In the problem provided, a non-trivial solution occurs when the equations have dependent relationships. This is mathematically determined by analyzing the determinant of the coefficient matrix, which should be zero. When the determinant is zero, it implies there are dependent rows leading to infinite solutions. Such solutions are non-zero and indicate that the system has free variables guiding these solutions.
This concept is crucial in algebra, engineering, and sciences, as it helps identify conditions of equilibrium, behaviors in dynamic systems, and more.

Coefficient Matrix

The coefficient matrix is the foundation of solving systems of linear equations, as it consists of the coefficients of the variables in these equations. Each row corresponds to an equation, and each column corresponds to a variable.
Given the problem, the coefficient matrix was built from the linear equations in the system:

• The first row corresponds to coefficients \( \lambda, -1, \cos \theta \).
• The second row has coefficients representing \( 3, 1, 2 \).
• The third incorporates the terms \( \cos \theta, 1, 2 \).

This matrix is pivotal as it determines the dependency conditions of the system. Manipulating it, such as finding its determinant, enables us to derive insights into the types of solutions inherent in the system, primarily identifying whether solutions are trivial or non-trivial.
Understanding the coefficient matrix equips us with the tools to set systems of equations, explore their solutions, and interpret the underlying relationships of the variables involved.