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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 ______.
Options
1
–1
0
2
Solution
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
