Q46 / 160 · TS EAMCET 2023 (12 May Shift 1) · Mathematics

If the expression $x^3+3 x^2-9 x+\lambda$ is of the form $(x-\alpha)^2$ $(x-\beta)$, then the values of $\lambda$ are

  • A
    $27,-5$
  • B
    $-27,-5$
  • C
    $27,5$
  • Correct answer
    D
    $-27,5$

Solution

$x^3+3 x^2-9 x+\lambda=(x-\alpha)^2(x-\beta)$ $\begin{aligned} & 2 \alpha+\beta=-3 ... (i) \\ & 2 \alpha \beta+\alpha^2=-9 ... (ii) \\ & \text { From (i), } \beta=-3-2 \alpha \\ & \qquad \quad 2 \alpha(-3-2 \alpha)+\alpha^2=-9 \\ & \Rightarrow \quad-6 \alpha-4 \alpha^2+\alpha^2=-9 \\ & \Rightarrow \quad 3 \alpha^2+6 \alpha-9=0 \\ & \Rightarrow \quad \alpha^2+2 \alpha-3=0 \\ & \Rightarrow \quad(\alpha+3)(\alpha-1)=0 \\ & \therefore \quad \alpha=+1,-3 \\ & \quad \beta=-5,+3 \\ & \therefore \quad \alpha \cdot \alpha \cdot \beta=-\lambda \\ & \quad \lambda=-\alpha^2 \beta=-(+1)^2-5 \text { or }-(3)^2 \cdot(+3) \\ & \quad \lambda=+5 \text { or }-27 .\end{aligned}$

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

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