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Let $C\subset \mathbb{Z} \times\mathbb{Z}$ be the set of integer pairs $(a, b)$ for which the three complex roots $r_{1}, r_{2}$ and $r_{3}$ of the polynomial $p(x)=x^{3}-2x^{2}+ax-b$ satisfy $r^{3}_{1}+r^{3}_{2}+r^{3}_{3}=0$. Then the cardinality of $C$ is 

  1. $|C| = \infty$
  2. $|C| = 0$ 
  3. $|C| = 1$ 
  4. $1 < |C| < \infty$
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$p(x) = x^{3} - 2x^{2} + ax - b    $

Lets roots = a, b, c.                

 sum of roots = (a+ b+ c) = 2.       
       
$ ab + bc + ca = a       \\

              abc = b   \\

   a^{3} + b^{3} + c^{3} = (a + b + c)^{3} + 3abc - (a + b + c)(ab +bc +ca) \\

        =>                0  = 2^{3} + 3b - 2a  \\

        =>               2a = 8 + 3b.  \\

       = >                 a = 4 + 3b/2       $

 C = (a,b) = ( 1, -2) , ( 4, 0) , ( 7, 2)   .... infinite possibile set (a,b).

So, Ans should be - A.
Answer:

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