Let-1-and-13-1-If-a-3-4-4-3-1-and-b-2-5-6-6-5-2-then-the-quadratic-equation-whose-roots-are-a-and-b-is-A-x-2-x-3-0-

Question Number 140981 by EnterUsername last updated on 14/May/21

Let α \neq 1 and α^{13} = 1 . If a = α + α^{3} + α^{4} + α^{- 4} + α^{- 3} +

α^{- 1} and b = α^{2} + α^{5} + α^{6} + α^{- 6} + α^{- 5} + α^{- 2} then the

quadratic equation whose roots are a and b is

(A) x^{2} + x + 3 = 0 (B) x^{2} + x + 4 = 0

(C) x^{2} + x - 3 = 0 (D) x^{2} + x - 4 = 0

Answered by Rasheed.Sindhi last updated on 16/May/21

F O R M U L A E :

F o r n t h r o o t s

α^{k + n} = α^{k}

1 + α + α^{2} + α^{3} + \dots + α^{n} = 0

(S u m o f a l l t h e n t h r o o t s i s z e r o)

α + α^{2} + α^{3} + \dots + α^{n} = - 1

F o r 13 t h r o o t s

α^{k + 13} = α^{k}

1 + α + α^{2} + α^{3} + \dots + α^{12} = 0

(S u m o f a l l t h e 13 t h r o o t s i s z e r o)

α + α^{2} + α^{3} + \dots + α^{12} = - 1

⌣⌢⌣⌢⌣⌢⌣⌢⌣⌢⌣⌢⌣⌢⌣⌢⌣⌣⌢⌣⌢

a = α + α^{3} + α^{4} + α^{- 4} + α^{- 3} + α^{- 1}

= α + α^{3} + α^{4} + α^{9} + α^{10} + α^{12}

b = α^{2} + α^{5} + α^{6} + α^{- 6} + α^{- 5} + α^{- 2}

= α^{2} + α^{5} + α^{6} + α^{7} + α^{8} + α^{11}

a + b = α + α^{2} + α^{3} + \dots + α^{12} = - 1

a b = (α + α^{3} + α^{4} + α^{9} + α^{10} + α^{12})

\times (α^{2} + α^{5} + α^{6} + α^{7} + α^{8} + α^{11})

= α^{3} + α^{6} + α^{7} + α^{8} + α^{9} + α^{12}

+ α^{5} + α^{8} + α^{9} + α^{10} + α^{11} + α^{1}

+ α^{6} + α^{9} + α^{10} + α^{11} + α^{12} + α^{2}

+ α^{11} + α^{1} + α^{2} + α^{3} + α^{4} + α^{7}

+ α^{12} + α^{2} + α^{3} + α^{4} + α^{5} + α^{8}

+ α^{1} + α^{4} + α^{5} + α^{6} + α^{7} + α^{10}

= (α^{1} + α^{2} + α^{3} + \dots + α^{12})

+ (α^{1} + α^{2} + α^{3} + \dots + α^{12})

+ (α^{1} + α^{2} + α^{3} + \dots + α^{12})

= (- 1) + (- 1) + (- 1) = - 3

E q u a t i o n r e q u i r e d :

x^{2} - (a + b) x + a b

x^{2} - (- 1) x + (- 3)

x^{2} + x - 3

S o (C) i s c o r r e c t

Commented by EnterUsername last updated on 19/May/21

Thank you Sir