Question

Let S be the set of all λ ∈ R for which the system of linear equations

2x – y + 2z = 2

x – 2y + λz = –4

x + λy + z = 4

has no solution. Then the set S ______.

पर्याय

• is an empty set.

• is a singleton.

• contains more than two elements.

• contains exactly two elements.

Answer 1

Let S be the set of all λ ∈ R for which the system of linear equations

2x – y + 2z = 2

x – 2y + λz = –4

x + λy + z = 4

has no solution. Then the set S contains exactly two elements.

Explanation:

D = `|(2, -1, 2),(1, -2, λ),(1, λ, 1)|`

D = 2(–2 – λ²) + 1(1 – λ) + 2(λ + 2)

= –4 – 2λ² + 1 – λ + 2λ + 4

= –2λ² + λ + 1

= –(2λ² – λ – 1)

= –(2λ + 1)(λ – 1)

For no solution D = 0 and atleast one of D_x, D_y, D_z ≠ 0

D = 0 ⇒ λ = `(-1)/2, 1`

D_x = `|(2, -1, 2),(-4, -2, λ),(4, λ, 1)|`

= 2(–2 – λ²) + 1(–4 – 4λ) + 2(–4λ + 8)

= –4 – 2λ² – 4 – 4λ – 8λ + 16

= –2λ² – 12λ + 8

For both λ = `(-1)/2` and λ = 1

D_x ≠ 0

So no solution for λ = `(-1)/2, 1` S contains two elements