Question

The number of real values λ, such that the system of linear equations 2x – 3y + 5z = 9, x + 3y – z = –18 and 3x – y + (λ² – |λ|z) = 16 has no solution, is ______.

Options

• 0

• 1

• 2

• 4

Answer 1

The number of real values λ, such that the system of linear equations 2x – 3y + 5z = 9, x + 3y – z = –18 and 3x – y + (λ² – |λ|z) = 16 has no solution, is 2.

Explanation:

Given that system of equations has no solution.

∴ `|(2, -3, 5),(1, 3, -1),(3, -1, λ^2 - |λ|)|` = 0

`\implies` 2(3λ² – 3|λ| – 1) + 3(λ² – |λ| + 3) + 5(–1 –9) = 0

`\implies` 6λ² – 6|λ| – 2 + 3λ² – 3|λ| + 9 – 50 = 0

`\implies` 9|λ|² – 9|λ| – 43 = 0

D = 81 + 4 × 9 × 43 > 0

So λ has two distinct real roots.