Q133 / 200 · TS EAMCET 2001 · Mathematics

The biquadratic equation, two of whose roots are $1+i, 1-\sqrt{2}$, is

  • Correct answer
    A
    $x^4-4 x^3+5 x^2-2 x-2=0$
  • B
    $x^4+4 x^3-5 x^2+2 x+2=0$
  • C
    $x^4+4 x^3-5 x^2+2 x-2=0$
  • D
    $x^4+4 x^3+5 x^2-2 x+2=0$

Solution

When $1+i, 1-i$ are the roots, then Sum of the roots $=1+i+1-i=2$ and product of the roots $=1+1=2$ The equation is $x^2-2 x+2=0$ When $1-\sqrt{2}, 1+\sqrt{2}$ are the roots, then Sum of the roots $=1-\sqrt{2}+1+\sqrt{2}=2$ Product of the roots $=1-2=-1$ The equation is $x^2-2 x-1=0$ The bi-quadratic equation is $\begin{array}{r}\left(x^2-2 x+2\right)\left(x^2-2 x-1\right)=0 \\ \left(x^4-(2+2) x^3=(-1+4+2) x^2\right. \\ =(2-4) x-2=0 \\ \Rightarrow \quad x^4-4 x^3+5 x^2-2 x-2=0\end{array}$

Hence, the correct answer is option: 'A'.

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