Consider the system of linear equations x+y+z=4 μ, x+2 y+2 λ z=10 μ, x+3 y+4 λ² z=μ²+15 where λ, μ ∈ R. Which one of the following statements is NOT correct ?

Solution

Maths sir · 5 steps

The determinant is a perfect square, so only λ = 1/2 needs a closer look at μ.

1. Coefficient determinant. Do R₂ - R₁ and R₃ - R₁, then expand along the first column. The bottom-right block becomes |1, 2λ - 1; 2, 4λ² - 1|.

   Δ = (4λ² - 1) - 2(2λ - 1) = 4λ² - 4λ + 1 = (2λ - 1)²

2. Lambda not one half. Here Δ ≠ 0, so the system has a unique solution for every value of μ. That makes (c) true. (a) is also true: its extra condition μ ≠ 1, 15 is unnecessary, but it does no harm.
3. Lambda equals one half. Now 2λ = 1 and 4λ² = 1, so every z coefficient is 1. Subtracting the first equation from the second gives y = 6μ. Subtracting the second from the third gives y = μ² - 10μ + 15.
4. Match the two. The two values of y must agree: μ² - 16μ + 15 = 0, that is (μ - 1)(μ - 15) = 0. For μ = 1 or μ = 15 the system has infinitely many solutions. For any other μ it is inconsistent.
5. So. (b) is true. (d) says the system is inconsistent whenever μ ≠ 1, but μ = 15 gives infinitely many solutions. So (d) is the statement that is NOT correct.

The idea to keep. When Δ = 0, subtract the equations and find which values of the constants keep them in agreement. An 'inconsistent unless' claim must leave out every such value.

Answer from the NTA final answer key. Solution by PrepWiser.