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Question

The equation x2+2x+2=0 has roots α and β. Then value of α15+β15 is


A

512

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B

256

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C

-512

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D

-256

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Solution

The correct option is D

-256


Explanation for the correct option:

Step 1: Find the roots of the given equation.

A quadratic equation x2+2x+2=0 is given.

Compute the roots using the quadratic formula.

x=-2±22-4·1·22·1⇒x=-2±4-82⇒x=-2±-42⇒x=-2±2i2⇒x=-1-i,-1+i

The roots of the quadratic equation x2+2x+2=0 are -1-i and -1+i.

Step 2: Find the value of the given expression.

It is given that, the roots of x2+2x+2=0 are α,β.

Therefore, α=-1-i and β=-1+i.

According to Euler's representation of the complex number, a complex number z=r(cosθ+isinθ) can be represented as z=r·eiθ.

Where, eiθ=cosθ+isinθ.

We can rewrite α as follows:

α=2-12-i12⇒α=2cos-3π4+isin-3π4

Use Euler's representation.

α=2ei-3π4

Similarly, β=2ei3π4

Now, evaluate the given expression as follows:

α15+β15=2ei-3π415+2ei3π415⇒α15+β15=215ei-45π4+ei45π4⇒α15+β15=215cos-45π4+isin-45π4+cos45π4+isin45π4⇒α15+β15=215cos-45π4-isin45π4+cos45π4+isin45π4⇒α15+β15=2152cos45π4⇒α15+β15=215-22⇒α15+β15=-28⇒α15+β15=-256

Therefore, the value of α15+β15 is -256.

Hence, the option D is correct.


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