ข้อสอบ PAT 1 - พฤศจิกายน 2557

$$\begin{gathered} \mathrm{กำหนดให้} \mathrm{z}=\mathrm{x}+\mathrm{yi} \left|\mathrm{z}\right|=\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2} \\ \overline{\mathrm{z}}=\mathrm{x}-\mathrm{yi} \left|\overline{\mathrm{z}}\right|=\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2} \\ \mathrm{จากโจทย์} \left|\mathrm{z}\right| + 2\overline{\mathrm{z}} -3\mathrm{z} = 3 - 45\mathrm{i} \\ -\mathrm{แทน} \mathrm{z} , \overline{\mathrm{z}} ,\left|\mathrm{z}\right| \\ \mathrm{จะได้} \sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}+2\left(\mathrm{x}-\mathrm{yi}\right)-3\left(\mathrm{x}+\mathrm{yi}\right)=3-45\mathrm{i} \\ \sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}+2\mathrm{x}-2\mathrm{yi}-3\mathrm{x}-3\mathrm{yi}=3-45\mathrm{i} \\ \left(\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}-\mathrm{x}\right)-5\mathrm{yi}=3-45\mathrm{i} \end{gathered}$$

$$\begin{gathered} \mathbf{เปรียบเทียบ}\mathbf{ }\mathbf{ส}\mathbf{.}\mathbf{ป}\mathbf{.}\mathbf{ส}\mathbf{.}\mathbf{ }\mathbf{ } \\ \mathrm{ส่วนจริงกับส่วนจริง} \mathrm{ส่วนจินตภาพกับส่วนจินตภาพ} \\ \mathrm{จะได้} \sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}-\mathrm{x}=3 \mathrm{และ} -5\mathrm{y}=-45\hspace{0.17em} \\ \sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}=\mathrm{x}+3 \mathrm{y}=9 \end{gathered}$$​
$$\begin{gathered} {\mathrm{x}}^2+{\mathrm{y}}^2={\left(\mathrm{x}+3\right)}^2 \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{แทนค่า} \mathrm{y}=9 \mathrm{ใน} \boxed{1} \\ \cancel{ {\mathrm{x}}^2}+{\mathrm{y}}^2=\cancel{{\mathrm{x}}^2}+6\mathrm{x}+9 \\ {\mathrm{y}}^2=6\mathrm{x}+9\to \boxed{1} \\ \mathrm{จะได้} {\left(9\right)}^2=6\mathrm{x}+9 \\ 72=6\mathrm{x} \\ \mathrm{x}=12 \end{gathered}$$

$$\begin{gathered} \mathrm{ดังนั้น} {\left|\overline{\mathrm{z}}\right|}^2={\left|12-9\mathrm{i}\right|}^2 ; \mathrm{จาก} \left|\overline{\mathrm{z}}\right|=\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2} \\ =(\sqrt{{\left(12\right)}^2+{\left(9\right)}^2}{)}^2 \\ =144+81 \\ =225 \end{gathered}$$​