ข้อสอบคณิต O-NET - 2558

$$\begin{gathered} \mathrm{จากสูตร} \\ {\left(น\pm ล\right)}^2 = {น}^2\pm 2\mathrm{นล}+{ล}^2 \\ \left(น-ล\right)\left(น+ล\right) = {น}^2-{ล}^2 \end{gathered}$$

$$\begin{gathered} \mathrm{จะได้} a^2 = {\left(\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}\right)}^2 \\ = \frac{{\left(\sqrt{3}+\sqrt{2}\right)}^2}{{\left(\sqrt{3}-\sqrt{2}\right)}^2} \\ = \frac{3+2\sqrt{3}\sqrt{2}+2}{3-2\sqrt{3}\sqrt{2}+2} \\ = \frac{5+2\sqrt{6}}{5-2\sqrt{6}} \end{gathered}$$

$$\begin{gathered} \mathrm{จะได้} a^2+\frac{1}{a^2} = \frac{5+2\sqrt{6}}{5-2\sqrt{6}}+\frac{5-2\sqrt{6}}{5+2\sqrt{6}} \\ = \frac{{\left(5+2\sqrt{6}\right)}^2+{\left(5-2\sqrt{6}\right)}^2}{\left(5-2\sqrt{6}\right)\left(5+2\sqrt{6}\right)} \\ = \frac{25+20\sqrt{6}+24+25-20\sqrt{6}+24}{25-24} \\ = 98 \\ \mathrm{ดังนั้น} \mathrm{ตอบข้อ} 5 \end{gathered}$$