Complex Numbers And Quadratic Equations

The correct option is(A): 0.

Given, $%5Calpha$ and $%5Cbeta$ are the roots of equation
$ax%5E%7B2%7D%2Bbx%2Bc%3D0$
$%5Ctherefore%20%5Calpha%2B%5Cbeta%3D-%5Cfrac%7Bb%7D%7Ba%7D$ and $%5Calpha%5Cbeta%20%3D%5Cfrac%7Bc%7D%7Ba%7D$
Now, $S_%7Bn%2B1%7D%3D%5Calpha%5E%7Bn%2B1%7D%2B%5Cbeta%5E%7Bn%2B1%7D$
$%3D%5Calpha%5E%7Bn%2B1%7D%2B%5Cbeta%5E%7Bn%2B1%7D%2B%5Calpha%5E%7Bn%7D%5Cbeta%2B%5Cbeta%5E%7Bn%7D%5Calpha-%5Calpha%5E%7Bn%7D%5Cbeta-%5Cbeta%5E%7Bn%7D%5Calpha$
$%3D%5Calpha%5E%7Bn%7D%5Cleft(%5Calpha%2B%5Cbeta%5Cright)%2B%5Cbeta%5E%7Bn%7D%5Cleft(%5Calpha%2B%5Cbeta%5Cright)-%5Calpha%5Cbeta%20%5Cleft(%5Calpha%5E%7Bn-1%7D%2B%5Cbeta%5E%7Bn-1%7D%5Cright)$
$%3D%5Cleft(%5Calpha%2B%5Cbeta%5Cright)%5Cleft(%5Calpha%5E%7Bn%7D%2B%5Cbeta%5E%7Bn%7D%5Cright)-%5Calpha%5Cbeta%5Cleft(%5Calpha%5E%7Bn-1%7D%2B%5Cbeta%5E%7Bn-1%7D%5Cright)$
$%3D%5Cleft(%5Calpha%2B%5Cbeta%5Cright)%5Cleft(%5Calpha%5E%7Bn%7D%2B%5Cbeta%5E%7Bn%7D%5Cright)-%5Calpha%5Cbeta%20%5Cleft(%5Calpha%5E%7Bn-1%7D%2B%5Cbeta%5E%7Bn-1%7D%5Cright)$
$%3D-%5Cfrac%7Bb%7D%7Ba%7D%20S_%7Bn%7D-%5Cfrac%7Bc%7D%7Ba%7DS_%7Bn-1%7D$
$%5CRightarrow%20S_%7Bn%2B1%7D%3D%5Cfrac%7B-bS_%7Bn%7D-cS_%7Bn-1%7D%7D%7Ba%7D$
$%5Ctherefore%20aS_%7Bn%2B1%7D%2BbS_%7Bn%7D%2BcS_%7Bn-1%7D%3D0$

Given,

%5Cleft(1%2Bx%2Bx%5E%7B2%7D%5Cright)%5E%7Bn%7D%20%3Da_%7B0%7D%2Ba_%7B1%7Dx%20%2Ba_%7B2%7Dx%5E%7B2%7D

%2Ba_%7B3%7Dx%5E%7B3%7D%2B%5Cldots%2Ba_%7Bzn%7D%20x%5E%7B2n%7D

%5Cleft(1%2Bx%2Bx%5E%7B2%7D%5Cright)%5E%7Bn%7D%20%3D%5Cleft(a_%7B0%7D%2Ba_%7B3%7Dx%5E%7B6%7D%2Ba_%7B9%7Dx%5E%7B9%7D%2B%5Cldots%5Cright)

%2Bx%5Cleft(a_%7B1%7D%2Ba_%7B4%7Dx%5E%7B3%7D%2Ba_%7B7%7Dx%5E%7B6%7D%2B%5Cldots%5Cright)%2Bx%5E%7B2%7D%5Cleft(a_%7B2%7D%2Ba_%7B5%7Dx%5E%7B3%7D%2Ba_%7B8%7Dx%5E%7B6%7D%2B%5Cldots%5Cright)

Let

E_%7B1%7D%3Da_%7B0%7D%2Ba_%7B3%7D%2Ba_%7B9%7D%2B%5Cldots

E_%7B2%7D%3Da_%7B1%7D%2Ba_%7B4%7D%2Ba_%7B7%7D%2B%5Cldots

and

E_%7B3%7D%3Da_%7B2%7D%2Ba_%7B5%7D%2Ba_%7B8%7D%2B%5Cldots

put

x%3D1%2C%20%5Comega%2C%20%5Comega%5E%7B2%7D

respectively, we get

%5Cleft(1%2B1%2B1%5Cright)%5E%7Bn%7D%20%3DE_%7B1%7D%2BE_%7B2%7D%2BE_%7B3%7D

%5CRightarrow%203%5E%7Bn%7D%3DE_%7B1%7D%2BE_%7B2%7D%2BE_%7B3%7D%20%5Cldots%5Cleft(i%5Cright)

%5Cleft(1%2B%5Comega%2B%5Comega%5E%7B2%7D%5Cright)%5E%7Bn%7D%20%3DE_%7B1%7D%2B%5Comega%5C%2CE_%7B2%7D%2B%5Comega%5E%7B2%7DE_%7B3%7D

%5CRightarrow%200%3DE_%7B1%7D%2B%5Comega%5C%2C%20E_%7B2%7D%2B%5Comega%5E%7B2%7DE_%7B3%7D%20%5Cldots%5Cleft(ii%5Cright)

and

%5Cleft(1%2B%5Comega%5E%7B2%7D%2B%5Comega%5E%7B4%7D%5Cright)%5E%7Bn%7D%20%3DE_%7B1%7D%2B%5Comega%5E%7B2%7DE_%7B2%7D%2B%5Comega%5E%7B4%7DE_%7B3%7D

%5CRightarrow%200%3DE_%7B1%7D%2B%5Comega%5E%7B2%7DE_%7B2%7D%2B%5Comega%5C%2CE_%7B3%7D%20%5Cldots%5Cleft(iii%5Cright)

On adding Eqs.

%5Cleft(i%5Cright)

%5Cleft(ii%5Cright)

and

%5Cleft(iii%5Cright)

, we get

3%5E%7Bn%7D%3D3E_%7B1%7D%2B%5Cleft(1%2B%5Comega%2B%5Comega%5E%7B2%7D%5Cright)E_%7B2%7D%2B%5Cleft(1%2B%5Comega%5E%7B2%7D%2B%5Comega%5Cright)E_%7B3%7D

%5CRightarrow%203%5E%7Bn%7D%3D3E_%7B1%7D%2B0%2B0

%5CRightarrow%20E_%7B1%7D%3D3%5E%7Bn-1%7D

High-Yield Trend

Chapter Questions
3 MCQs

Official Solution

Official Solution

Official Solution

Complex Numbers And Quadratic Equations

High-Yield Trend

Chapter Questions 3 MCQs

Official Solution

Official Solution

Official Solution

Chapter Questions
3 MCQs