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Geometric Progression

If α, β are...

Question

If $α, β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0$

A

α^{3} + β^{3} = \frac{3 a b c - b^{3}}{a^{3}}

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B

α^{3} + β^{3} = \frac{b^{3} - 3 a b c}{a^{3}}

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C

α^{2} + β^{2} = \frac{2 a c - b^{2}}{a^{2}}

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D

\frac{1}{α^{2}} + \frac{1}{β^{2}} = \frac{b^{2} - 2 a c}{c^{2}}

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Solution

The correct options are
A $α^{3} + β^{3} = \frac{3 a b c - b^{3}}{a^{3}}$
D $\frac{1}{α^{2}} + \frac{1}{β^{2}} = \frac{b^{2} - 2 a c}{c^{2}}$

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Similar questions

α, β

are the roots of

a x^{2} + b x + c = 0

then match the elements of list I with elements of list II:

List I	List II
A) $\frac{α}{β} + \frac{β}{α} =$	1) $\frac{c^{2}}{a^{2}}$
B) $\frac{α^{2} + β^{2}}{α^{- 2} + β^{- 2}} =$	2) $\frac{c^{5} [3 a b c - b^{3}]}{a^{8}}$
C) $α^{3} + β^{3}$	3) $\frac{b^{2} - 2 a c}{a c}$
D) $α^{5} β^{8} + α^{8} β^{5} =$	4) $\frac{3 a b c - b^{3}}{a^{3}}$

The correct match from List-I to List-II:

A, B, C, D

Q. In the quadratic equation

a x^{2} + b x + c = 0

Δ = b^{2} - 4 a c

α + β, α^{2} + β^{2}, α^{3} + β^{3}

Q. In the quadratic equation

a x^{2} +

+ c = 0

Δ = b^{2} - 4 a c

α + β, α^{2} + {β^{2}, α}^{3} + β^{3}

are in G.P. where

α, β

are the roots of

a x^{2} +

+ c = 0

α, β

be roots of the equation

a x^{2} + b x + c = 0

Δ = b^{2} - 4 a c

α + β, α^{2} + β^{2}, α^{3} + β^{3}

G P

α, β

are the roots of

a x^{2} + b x + c = 0

α + β, α^{2} + β^{2}, α^{3} + β^{3}

are in G.P., where

△ = b^{2} - 4 a c

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