Question

Let α and β be two roots of the equation x² + 2x + 2 = 0, then  α¹⁵ +  β¹⁵ is equal to

(1)   256 

(2)    512

(3)   –256

(4)   –512

Answer 1

The correct option is (3) -256

x² + 2x + 2 = 0 and its roots are α, β.

x² + 2x + 1 + 1 = 0

⇒ (x+1)² + 1 = 0 [Since, a² + 2ab + b² = (a + b)²]

⇒ (x+1)² = −1

⇒ x + 1 = √−1

⇒ x¹ = −1+ i ; x² = −1 −i  [Since, √−1= i]

Assuming α = −1 + i and β = −1 − i

Squaring on both sides, we get

⇒ α² = (−1+i)² and β² = (−1−i)²

⇒ α² = 1 + i² − 2i and β² = 1 + i² + 2i

⇒ α² = 1 − 1 − 2i and β² = 1 − 1 + 2i [Since, i² = −1]

∴ α² = −2i and β² = 2i

Now consider, α¹⁵ + β¹⁵ = (α¹⁴ × α)+(β¹⁴ × β)

= [(α²)⁷ × α] + [(β²)⁷ × β)]

= (−2i)⁷ × (−1 + i) + (2i)⁷ × (−1 −i)

= −2⁷i⁷(−1 + i) − 2⁷i⁷(1 + i)

=−2⁷i⁷(−1 + i + 1 + i)

= −128i⁷(2i)

= (−128 × 2) × i8

= −256(i²)⁴

= −256(−1)⁴

∴ α¹⁵ + β¹⁵ = −256

Answer 2

Correct option  (3)  -256

Explanation: