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Consider the system of linear equations: 2x + (p² - 2)y + 6z = 8 x + 2y + (q - 1)z = 5 x + y + 3z = 4 where p, q are real numbers. Which of the following statements is NOT TRUE?

1. The system has a unique solution if p is in R - {-2, 2} and q is in R - {4}
2. The system is inconsistent if p = 2 and q = 4
3. The system has infinitely many solutions if p = -2 and q is any real number
4. The system is consistent if p is in R and q is in R - {4}

Correct answer: The system has infinitely many solutions if p = -2 and q is any real number

Solution

The coefficient matrix determinant factors as (p² - 4)(q - 4)/something — the key is that for p = -2 and arbitrary q, the system may still be inconsistent for certain q values, making option C not always true. Specifically when p = -2 and q != 4 (det != 0), the system has a unique solution, not infinitely many. Option C is the NOT TRUE statement.

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