Question

Let a, b, c, d be in arithmetic progression with common difference λ. If `|(x + a - c, x + b, x + a),(x - 1, x + c, x + b),(x - b + d, x + d, x + c)|` = 2, then value of λ² is equal to ______.

Options

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Answer 1

Let a, b, c, d be in arithmetic progression with common difference λ. If `|(x + a - c, x + b, x + a),(x - 1, x + c, x + b),(x - b + d, x + d, x + c)|` = 2, then value of λ² is equal to 1.

Explanation:

`|(x + a - c, x + b, x + a),(x - 1, x + c, x + b),(x - b + d, x + d, x + c)|` = 2

Given, a, b, c and d are in AP

(b – a) = (c – b) = (d – c)

⇒ a + c = 2b, b + d = 2c

C₂→C₂ – C₃

⇒ `|(x - 2λ, λ, x + a),(x - 1, λ, x + b),(x + 2λ, λ, x + c)|` = 2

R₂→R₂ – R₁, R₃→R₃ – R₁

⇒ `|(x - 2λ, λ, x + a),(2λ - 1, 0, λ),(4λ, 0, 2λ)|` = 2

⇒ –λ((2λ – 1)2λ – 4λ²) = 2

⇒ 2λ² = 2

λ² = 1