Q53 / 160 · TS EAMCET 2020 (10 Sep Shift 2) · Mathematics

$p$ is non-zero real number. If the equation whose roots are the squares of the roots of the equation $x^3-p x^2+p x-1=0$ is identical with the given equation, then $p=$

  • A
    $\frac{1}{2}$
  • B
    2
  • Correct answer
    C
    3
  • D
    -1

Solution

Let $\alpha, \beta, \gamma$ are roots of equation $x^3-p x^2+p x-1=0$ $\because \quad \alpha+\beta+\gamma=p$ $\alpha \beta+\beta \gamma+\alpha \gamma=p$ $\alpha \beta \gamma=1$ Also given, $\quad \alpha^2+\beta^2+\gamma^2=p$ $\alpha^2 \beta^2+\beta^2 \gamma^2+\alpha^2 \gamma^2=p$ $\alpha^2 \beta^2 \gamma^2=1$ $\therefore(\alpha+\beta+\gamma)^2=\alpha^2+\beta^2+\gamma^2+2(\alpha \beta+\beta \gamma+\gamma \alpha)$ $p^2=p+2 p$ $p^2=3 p$ $p=3$

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

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