Question

If `|(1 + x, x, x^2),(x, 1 + x, x^2),(x^2, x, 1 + x)|` = ax⁵ + bx⁴ + cx³ + dx² + λx + µ be an identity in x, where a, b, c, d, λ, µ are independent of x. Then the value of λ is ______.

Options

• 3

• 2

• 4

• –3

Answer 1

If `|(1 + x, x, x^2),(x, 1 + x, x^2),(x^2, x, 1 + x)|` = ax⁵ + bx⁴ + cx³ + dx² + λx + µ be an identity in x, where a, b, c, d, λ, µ are independent of x. Then the value of λ is 3.

Explanation:

Applying C₁→C₁ + C₂ + C₃

Δ = `|((1 + x)^2, x, x^2),((1 + x)^2, 1 + x, x^2),((1 + x)^2, x, 1 + x)|`

Δ = `(1 + x)^2|(1, x, x^2),(1, 1 + x, x^2),(1, x, 1 + x)|`

Differentiating both sides of the given equality w.r.t.x, we get

`2(1 + x)|(1, x, x^2),(1, 1 + x, x^2),(1, x, 1 + x)| + (1 + x)^2 {|(1, 1, x^2),(1, 1, x^2),(1, 1, 1 + x)| + |(1, x, 2x),(1, 1 + x, 2x),(1, x, 1)|}`

= 5ax⁴ + 4bx³ + 3cx² + 2dx + λ

Now putting x = 0

`2|(1, 0, 0),(1, 1, 0),(1, 0, 1)| + |(1, 0, 0),(1, 1, 0),(1, 0, 1)|` = λ

∴ 2(1) + 1(1) = λ

⇒ λ = 3