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If the equation $x^{4}+ax^{3}+bx^{2}+cx+1=0$ (where $a,b,c$ are real number) has no real roots and if at least one of the root is of modulus one, then

  1. $b=c$
  2. $a=c$
  3. $a=b$
  4. none of this
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Option D Should be Correct

  Alternate forms of equation:

x^3 (a + x) + b x^2 + c x + 1

a x^3 + x^2 (b + x^2) + c x + 1

Above equation has no real roots and if at least one of the root is of modulus one (which is both +1 and -1) as stated by question then it should be integer root of the equation.

Putting x= 1 in the main equation we get : c = - a - b - 2............(1)

Putting x= -1 in the main equation we get : c = - a + b + 2.............(2)

Adding (1) and (2) we get

2c = - 2a

c= -a

Correct me if wrong

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