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Soal No. 6 OSK Matematika SMA 2019

Berikut ini adalah Soal No. 6 OSK Matematika SMA 2019 (Soal No. 6 Kemampuan Dasar)

6. Misalkan $a = 2 \sqrt{2} − \sqrt{8-  4 \sqrt{2}}$ dan $b = 2  \sqrt{2} + \sqrt{8 -  4 \sqrt{2}}$. Jika $\frac{a}{b}+ \frac{b}{a}=x+ y \sqrt{2}$ dengan x, y bulat, maka nilai $x + y$ adalah...

Jawaban:

$a+b=2\sqrt{2}-\sqrt{8-4\sqrt{2}}+2\sqrt{2}+\sqrt{8-4\sqrt{2}}$ $a+b=4\sqrt{2}$

$a.b=\left( 2\sqrt{2}-\sqrt{8-4\sqrt{2}} \right)\left( 2\sqrt{2}+\sqrt{8-4\sqrt{2}} \right)$

$a.b=8-(8-4\sqrt{2})$

$a.b=4\sqrt{2}$

$\frac{a}{b}+\frac{b}{a}=x+y\sqrt{2}$

$\frac{{{a}^{2}}+{{b}^{2}}}{ab}=x+y\sqrt{2}$

$\frac{{{(a+b)}^{2}}-2ab}{ab}=x+y\sqrt{2}$

$\frac{{{(4\sqrt{2})}^{2}}-2.4\sqrt{2}}{4\sqrt{2}}=x+y\sqrt{2}$

$4\sqrt{2}-2=x+y\sqrt{2}$

$-2+4\sqrt{2}=x+y\sqrt{2}$

$x=-2$ dan $y=4$

$x+y=-2+4=2$

Sumber Jawaban: https://www.catatanmatematika.com/2019/03/pembahasan-osk-matematika-sma-tahun-2019.html

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