Q43 / 160 · AP EAMCET 2022 (07 Jul Shift 1) · Mathematics

If the system of simultaneous linear equations $x+y-z=6,4 x+y+z=2$ and $x+k y+z=-8$ has a unique solution $x=2$, $y=\beta, z=\gamma$, then the value of $k$ satisfies the following quadratic equation

  • A
    $x^2-5 x+6=0$
  • B
    $x^2+x-6=0$
  • C
    $x^2-x-6=0$
  • Correct answer
    D
    $x^2+x-2=0$

Solution

$x+y-z=6$ $\begin{aligned} & 4 x+y+z=2 \\ & x+k y+z=-8\end{aligned}$ $x=2, y=\beta, z=\gamma$ $y-3=4 \quad$ ...(i) $y+z=-10$ ...(ii) $2 y=-6$ $\Rightarrow y=-3$ Put in eq no (i) $3=-7$ Now, put values of $x, y$ and $z$, we get $2+k(-3)+(-7)=-8$ $\Rightarrow-3 k=-3$ $\Rightarrow k=1 \longrightarrow$ check all the options which satisfy this value of $k$. we get option (d), $\begin{aligned} & x^2+x-2=0 \\ & \Rightarrow(1)^2+(1)-2=0 \Rightarrow 0=0 \text { verified. }\end{aligned}$

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

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