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?
(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: Analyze the determinant of the coefficient matrix and use Cramer's rule or Gaussian elimination to determine the consistency of the system based on the values of \lambda and \mu.
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, which means (2\lambda - 1)^2 \neq 0, so \lambda \neq 1/2. If \lambda = 1/2, the system becomes 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 = \mu^2 - 10\mu + 15. Thus, 6\mu = \mu^2 - 10\mu + 15, or \mu^2 - 16\mu + 15 = 0, which factors to (\mu - 1)(\mu - 15) = 0. So, if \lambda = 1/2, the system is consistent only if \mu = 1 or \mu = 15. If \lambda = 1/2 and \mu \neq 1 and \mu \neq 15, the system is inconsistent. Option (2) states it is inconsistent if \lambda = 1/2 and \mu \neq 1, which is incorrect because it is consistent if \mu = 15.
Correct Answer: 2