Question

If the following equations

x + y – 3 = 0 

(1 + λ)x + (2 + λ)y – 8 = 0

x – (1 + λ)y + (2 + λ) = 0

are consistent then the value of λ can be ______.

विकल्प

• 1

• –1

• 0

• 2

Answer 1

If the following equations

x + y – 3 = 0 

(1 + λ)x + (2 + λ)y – 8 = 0

x – (1 + λ)y + (2 + λ) = 0

are consistent then the value of λ can be 1.

Explanation:

x + y – 3 = 0 

(1 + λ)x + (2 + λ)y – 8 = 0

x – (1 + λ)y + (2 + λ) = 0

Since the equation in this case has two variables, x and y, involved. Since the third equation should be satisfied by the value of x and y obtained from the first two equations if they are consistent, D = 0, or in other words, the value of D.

⇒ `|(1, 1, -3),(1 + λ, 2 + λ, -8),(1, -1 - λ, 2 + λ)|` = 0

⇒ `|(1, 0, 0),(1 + λ, 1, -5 + 3λ),(1, -2 - λ, 5 + λ)|`

Applying C₂→C₂ – C₁, C₃→C₃ + 3C₁

⇒ (5 + λ) + (2 + λ)(–5 + 3λ) = 0

⇒ 3λ² + 2λ – 5 = 0

⇒ (λ – 1)(3λ + 5) = 0

⇒ λ = 1, `-5/3`

∴ λ = 1