Question

If a and b are the roots of the equation x^2 +x+2=0, then ( a^10 + b^10) / [a^-10 + b^-10)

A 4096
B. 2048
C. 1024
D. 512
E. 256

Answer 1

EXPLANATION.

α,β are the roots of the equation,

⇒ x² + x + 2 = 0.

As we know that,

Sum of zeroes of quadratic equation,

⇒ α + β = -b/a.

⇒ α + β = -1.

Products of zeroes of quadratic equation,

⇒ αβ = c/a.

⇒ αβ = 2.

To find values of = (α¹⁰ + β¹⁰)/(α⁻¹⁰ + β⁻¹⁰).

We can also write as,

⇒ (α⁻¹⁰ + β⁻¹⁰) = 1/(α¹⁰ + β¹⁰).

Put the values in equation, we get.

⇒ (α¹⁰ + β¹⁰)/(1/α¹⁰ + 1/β¹⁰).

⇒ (α¹⁰ + β¹⁰)/(α¹⁰ + β¹⁰)/α¹⁰β¹⁰.

⇒ α¹⁰β¹⁰.

⇒ (αβ)¹⁰.

⇒ (2)¹⁰.

⇒ 1024.

Option [C] is correct answer.

Answer 2

$\huge\underline{\underline{\texttt{{Question:}}}}$

If a and b are the roots of the equation x^2 +x+2=0, then ( a^10 + b^10) / [a^-10 + b^-10)

• A 4096
• B. 2048
• C. 1024
• D. 512
• E. 256

$\huge\underline{\underline{\texttt{{Solution:}}}}$

• Option C. 1024

$\huge\underline{\underline{\texttt{{Explanation:}}}}$

According to the question

Here α and β are the roots of the equation,

⇒ x² + x + 2 = 0.

Therefore,

⇒α + β = -1/1

or,

⇒ α + β = -b/a.

⇒ α + β = -1.

Products of zeroes of quadratic equation,

⇒ αβ = c/a.

⇒ αβ = 2.

Now,

We have to find the value of

(α¹⁰+β¹⁰)/{α^(-10)+β^(-10)}

=(α¹⁰+β¹⁰)/(1/α¹⁰ +1/β¹⁰)

={(α¹⁰+β¹⁰)/(α¹⁰+β¹⁰)}×(αβ)¹⁰

=(αβ)¹⁰

=(2)¹⁰

=1024

Hence, option C is correct

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