Question

Consider the system of linear equations \( x + y + z = 4\mu \), \( x + 2y + 2\lambda z = 10\mu \), \( x + 3y + 4\lambda^2 z = \mu^2 + 15 \), where \( \lambda, \mu \in \mathbb{R} \). Which one of the following statements is NOT correct?

• A. The system has a unique solution if \( \lambda \neq \frac{1}{2} \) and \( \mu \neq 1, 15 \).
• B. The system has an infinite number of solutions if \( \lambda = \frac{1}{2} \) and \( \mu = 15 \).
• C. The system is inconsistent if \( \lambda = \frac{1}{2} \) and \( \mu \neq 1 \).
• D. The system is consistent if \( \lambda \neq \frac{1}{2} \).

Answer 1

The Correct Option is C

To identify the incorrect statement, we analyze the provided system of linear equations:

• \( x + y + z = 4\mu \)
• \( x + 2y + 2\lambda z = 10\mu \)
• \( x + 3y + 4\lambda^2 z = \mu^2 + 15 \)

We will examine the conditions for a unique solution, infinite solutions, or inconsistency, aligning with the given options.

1. Statement 1: The system has a unique solution if \( \lambda eq \frac{1}{2} \) and \( \mu eq 1, 15 \):

A unique solution exists if the determinant of the coefficient matrix is non-zero. The coefficient matrix is:

\( \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 2\lambda \\ 1 & 3 & 4\lambda^2 \end{vmatrix} \)

The determinant calculation is:

\( \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 2\lambda \\ 1 & 3 & 4\lambda^2 \end{vmatrix} = 1(2 \cdot 4\lambda^2 - 2\lambda \cdot 3) - 1(1 \cdot 4\lambda^2 - 2\lambda \cdot 1) + 1(1 \cdot 3 - 2 \cdot 1) \)

This simplifies to:

\( = 8\lambda^2 - 6\lambda - 4\lambda^2 + 2\lambda + 3 - 2 \)

\( = 4\lambda^2 - 4\lambda + 1 \)

Which factors to

\( (2\lambda - 1)^2 \).

The determinant is zero when \( \lambda = \frac{1}{2} \). Therefore, if \( \lambda eq \frac{1}{2} \), the determinant is non-zero, implying a unique solution, independent of \( \mu \)'s specific values in this context. This statement appears plausible.

1. Statement 2: The system has an infinite number of solutions if \( \lambda = \frac{1}{2} \) and \( \mu = 15 \):

When \( \lambda = \frac{1}{2} \), the determinant is zero (\( (2 \times \frac{1}{2} - 1)^2 = 0 \)), indicating the possibility of infinite solutions. Substituting \(\lambda = \frac{1}{2}\) and \(\mu = 15\) into the original equations:

• \( x + y + z = 4(15) = 60 \)
• \( x + 2y + 2(\frac{1}{2}) z = 10(15) \Rightarrow x + 2y + z = 150 \)
• \( x + 3y + 4(\frac{1}{2})^2 z = (15)^2 + 15 \Rightarrow x + 3y + z = 225 + 15 = 240 \)

Checking consistency with these values: The modified equations are \( x+y+z=60 \), \( x+2y+z=150 \), and \( x+3y+z=240 \). Subtracting the first from the second yields \( y = 90 \). Subtracting the second from the third yields \( y = 90 \). Substituting \( y=90 \) into the first two equations gives \( x+90+z=60 \Rightarrow x+z=-30 \) and \( x+180+z=150 \Rightarrow x+z=-30 \). Since \( x+z \) is consistent, the system has infinite solutions when \( \lambda = \frac{1}{2} \) and \( \mu = 15 \). This statement appears correct.

1. Statement 3: The system is inconsistent if \( \lambda = \frac{1}{2} \) and \( \mu eq 1 \):

For \( \lambda = \frac{1}{2} \), the determinant is zero. Let's re-examine the system with \( \lambda = \frac{1}{2} \):

• \( x + y + z = 4\mu \)
• \( x + 2y + z = 10\mu \)
• \( x + 3y + z = \mu^2 + 15 \)

Subtracting the first equation from the second gives: \( y = 6\mu \).

Subtracting the second equation from the third gives: \( y = \mu^2 + 15 - 10\mu \).

For the system to be consistent, these two expressions for \( y \) must be equal:

\( 6\mu = \mu^2 + 15 - 10\mu \)

\( \mu^2 - 16\mu + 15 = 0 \)

\( (\mu - 1)(\mu - 15) = 0 \)

This implies consistency only when \( \mu = 1 \) or \( \mu = 15 \). If \( \lambda = \frac{1}{2} \) and \( \mu \) is neither 1 nor 15, the system is inconsistent. The statement claims inconsistency if \( \lambda = \frac{1}{2} \) and \( \mu eq 1 \). This is not entirely accurate, as inconsistency requires \( \mu \) to be different from *both* 1 and 15. For instance, if \( \lambda = \frac{1}{2} \) and \( \mu = 15 \), the system is consistent (as shown in statement 2). Therefore, this statement is incorrect.

1. Statement 4: The system is consistent if \( \lambda eq \frac{1}{2} \):

If \(\lambda eq \frac{1}{2}\), the determinant of the coefficient matrix is non-zero. A non-zero determinant guarantees a unique solution for any values of \( \mu \). Therefore, the system is always consistent when \(\lambda eq \frac{1}{2}\). This statement is correct.

The statement that is NOT correct is option 3.