If α and β are the roots of the equation x2 + x + 1 = 0, then whic

1.	If α and β are the roots of the equation x2 + x + 1 = 0, then which of the following are the roots of the equation x2 - x + 1 = 0 ?1. α7 and β13 2. α13 and β7 3. α20 and β20 4. None of the above
Answer» Correct Answer - Option 1 : α⁷ and β¹³ Concept: 1, ω and ω² are cube root of unity, where, ω = \(\rm \frac{-1+\sqrt3i}{2}\) and ω² = \(\rm \frac{-1-\sqrt3i}{2}\), ω³ = 1 1 + ω + ω2 = 0 In quadratic equation, \(\rm ax^2+bx+c=0\), \(\rm x = {-b \pm \sqrt{b^2-4ac} \over 2a}\) Consider a quadratic equation: ax² + bx + c = 0. Let, α and β are the roots. Sum of roots = α + β = -b/a Product of the roots = α × β = c/a Calculation: Here, α and β are the roots of the equation x2 + x + 1 = 0 α \(\rm ={-1 +\sqrt{1^2-4} \over 2}\)=\(\rm \frac{-1+\sqrt3i}{2}\) and β = \(\rm \frac{-1-\sqrt3i}{2}\)......(Using \(\rm x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)) α = ω and β = ω2 In equation x2 - x + 1 = 0 Roots = \(\rm {1 \pm\sqrt{1^2-4} \over 2}\)\(=\rm \frac{1\pm\sqrt3i}{2}\) So, roots are -β and -α, i.e., - ω2 and -ω Sum of roots in quadratic equation x2 - x + 1 = 0, is 1 Assume roots are α7 and β13 α⁷ + β13 = ((-ω)⁷ + (- ω2 )¹³) = -((ω³)²+ (ω³)⁸ω²) = - (ω + ω2 ) = -(-1) = 1 Hence, option (1) is correct.

If α and β are the roots of the equation x2 + x + 1 = 0, then which of the following are the roots of the equation x2 - x + 1 = 0 ?1. α7 and β13 2. α13 and β7 3. α20 and β20 4. None of the above

Answer» Correct Answer - Option 1 : α⁷ and β¹³

Concept:

1, ω and ω² are cube root of unity, where, ω = \(\rm \frac{-1+\sqrt3i}{2}\) and ω² = \(\rm \frac{-1-\sqrt3i}{2}\),

ω³ = 1
1 + ω + ω2 = 0

In quadratic equation, \(\rm ax^2+bx+c=0\), \(\rm x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

Consider a quadratic equation: ax² + bx + c = 0.

Let, α and β are the roots.

Sum of roots = α + β = -b/a
Product of the roots = α × β = c/a

Calculation:

Here, α and β are the roots of the equation x2 + x + 1 = 0

α \(\rm ={-1 +\sqrt{1^2-4} \over 2}\)=\(\rm \frac{-1+\sqrt3i}{2}\) and β = \(\rm \frac{-1-\sqrt3i}{2}\)......(Using \(\rm x = {-b \pm \sqrt{b^2-4ac} \over 2a}\))

α = ω and β = ω2

In equation x2 - x + 1 = 0

Roots = \(\rm {1 \pm\sqrt{1^2-4} \over 2}\)\(=\rm \frac{1\pm\sqrt3i}{2}\)

So, roots are -β and -α, i.e., - ω2 and -ω

Sum of roots in quadratic equation x2 - x + 1 = 0, is 1

Assume roots are α7 and β13

α⁷ + β13 = ((-ω)⁷ + (- ω2 )¹³)

= -((ω³)²+ (ω³)⁸ω²)

= - (ω + ω2 )

= -(-1)

= 1

Hence, option (1) is correct.

If α and β are the roots of the equation x2 + x + 1 = 0, then which of the following are the roots of the equation x2 - x + 1 = 0 ?1. α7 and β13 2. α13 and β7 3. α20 and β20 4. None of the above

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