Consider the system of linear equation x + y + z = 4μ, x + 2y + 2λz = 10μ, x + 3y + 4λ²z = μ² + 15 where λ, μ ∈ R. Which one of the following statements is NOT correct ? [JEE (Main) 2024]
(1) The system has unique solution if λ ≠ 1/2 and μ ≠ 1,15
(2) The system is inconsistent if λ = 1/2 and μ ≠ 1
(3) The system has infinite number of solution if λ = 1/2 and μ = 15
(4) The system is consistent if λ ≠ 1/2
Step-by-Step Solution
Key Concept: The system of linear equations has a unique solution if the determinant of the coefficient matrix is non-zero. If the determinant is zero, the system may have infinite solutions or be inconsistent depending on the consistency of the augmented matrix.
The coefficient matrix is A = [[1, 1, 1], [1, 2, 2\lambda], [1, 3, 4\lambda^2]]. The determinant |A| = 1(8\lambda^2 - 6\lambda) - 1(4\lambda^2 - 2\lambda) + 1(3 - 2) = 4\lambda^2 - 4\lambda + 1 = (2\lambda - 1)^2. For a unique solution, |A| \neq 0, so \lambda \neq 1/2. If \lambda = 1/2, the determinant is 0. Checking the augmented matrix for consistency at \lambda = 1/2, the equations become x + y + z = 4\mu, x + 2y + z = 10\mu, x + 3y + z = \mu^2 + 15. Subtracting the first from the second gives y = 6\mu. Subtracting the second from the third gives y = \mu^2 + 15 - 10\mu. Thus 6\mu = \mu^2 - 10\mu + 15, or \mu^2 - 16\mu + 15 = 0, which gives (\mu - 1)(\mu - 15) = 0. So for \lambda = 1/2, the system is consistent if \mu = 1 or \mu = 15. It is inconsistent if \mu \neq 1 and \mu \neq 15. Option (2) states it is inconsistent if \lambda = 1/2 and \mu \neq 1, which is not necessarily true as it could be \mu = 15.
Correct Answer: (2)