Question

If c < 1 and the system of equations x + y – 1 = 0, 2x – y – c = 0 and – bx+ 3by – c = 0 is consistent, then the possible real values of b are

विकल्प

• `b ∈ (-3, 3/4)`

• `b ∈ (-3/2, 4)`

• `b ∈ (-3/4, 3)`

• None of these

Answer 1

`b ∈ (-3/4, 3)`

Explanation:

Given system is consistent `|(1, 1, -1),(2, -1, -c),(-b, 3b, -c)|` = 0

⇒ c + bc – 6b + b + 2c + 3bc = 0

⇒ 3c + 4bc – 5b = 0

⇒ c = `(5b)/(3 + 4b)`

Now c < 1 ⇒ `(5b)/(3 + 4b) < 1` ⇒ `(5b)/(3 + 4b) - 1 < 0`

⇒ `(b - 3)/(3 + 4b) < 0`

⇒ `b ∈ (-3/4, 3)`