Problem 108 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The number of real values of \(\lambda\) for which the system of linear equations \(2 x+4 y-\lambda z=0\) \(4 x+\lambda y+2 z=0\) \(\lambda x+2 y+2 z=0\) has infinitely many solutions, is: [Online April 8, 2017] (a) 0 (b) 1 (c) 2 (d) 3

Step-by-Step Solution

Answer

(a) 0

Step 1: Determine the conditions for a system to have infinitely many solutions

For a system of linear equations to have infinitely many solutions, the coefficient matrix must have a rank equal to the augmented matrix's rank, which should also be less than the number of variables. Since there are 3 variables, the rank must be less than 3.

Step 2: Write the coefficient matrix

Write the coefficient matrix from the given system of equations:\[A = \begin{pmatrix}2 & 4 & -\lambda \4 & \lambda & 2 \\lambda & 2 & 2\end{pmatrix}\]

Step 3: Compute the determinant of the coefficient matrix

Calculate the determinant of the matrix \( A \). Set this determinant equal to zero to find conditions for \( \lambda \) where the system has infinitely many solutions:\[\det(A) = 2(\lambda \cdot 2 - 2 \cdot 2) - 4(4 \cdot 2 - 2 \cdot \lambda) - \lambda(4 \cdot 2 - \lambda \cdot 4) = 0\]

Step 4: Simplify determinant equation and solve for \( \lambda \)

Simplify the determinant expression to:\[\det(A) = 4 \lambda - 8 - 32 + 8\lambda + 8\lambda - 4\lambda^2 = 0\]Simplifying, we get:\[-4\lambda^2 + 20\lambda - 40 = 0\]Divide the entire equation by \(-4\):\[\lambda^2 - 5\lambda + 10 = 0\]

Step 5: Determine the discriminant to find real solutions

The discriminant \( D \) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( b^2 - 4ac \). For our equation, substitute: \( a = 1, b = -5, c = 10 \).\[D = (-5)^2 - 4 \cdot 1 \cdot 10 = 25 - 40 = -15\]

Step 6: Analyze the discriminant

The discriminant \( D = -15 \) is negative, which means there are no real solutions for \( \lambda \). Hence, there are 0 real values of \( \lambda \) for which the system has infinitely many solutions.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more equations with multiple variables. Each equation represents a line in a geometric space. The solution to a system of equations is the point or points at which these lines intersect.
In cases where the equations represent planes in three-dimensional space, as many as three variables can create:

• No solution: The planes do not meet at any point.
• Exactly one solution: The planes intersect exactly at one point.
• Infinitely many solutions: The planes coincide across a line or the entire plane.

To determine the number of solutions, we often transform the system into a coefficient matrix and analyze its properties.

Determinant of a Matrix

The determinant is a special number that can be calculated from a square matrix. It provides insights into the properties of the matrix, such as whether it's invertible or the volume scaling factor.
For a 3x3 matrix, the determinant helps in understanding the matrix's influence on a given space. It's calculated using a specific formula that multiplies certain elements and subtracts others to yield a single number.
For the system of equations, if the determinant of the coefficient matrix is zero, this suggests the matrix doesn't have full rank. Thus, the system might have infinitely many solutions if the ranks of the coefficient and augmented matrices match but are less than the number of variables.

Discriminant Analysis

The discriminant is part of the quadratic formula used to solve quadratic equations \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants. The formula for the discriminant is given by\[ D = b^2 - 4ac \].
The discriminant determines the nature of the roots of a quadratic equation. Specifically:

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), the equation has two identical real roots.
• If \( D < 0 \), the equation has two complex roots and no real solutions.

In the context of the given system, analyzing the discriminant revealed there were no real values of \( \lambda \) such that the system of equations has infinitely many solutions. This is because a negative discriminant signifies the quadratic doesn't intersect the real line.