Pembahasan Matematika IPA SIMAK UI 2015 #9

Soal

Jika diketahui [latex]\sin{\theta} – \cos{\theta}= \frac{\sqrt{5} – \sqrt{3}}{2}[/latex]
dan [latex]\cos^3{\theta} – \sin^3{\theta}= \frac{1}{a}(b\sqrt{5}-c\sqrt{3})[/latex], dengan [latex]a,b,c[/latex] bilangan asli, maka [latex]\ldots[/latex]

(1) [latex] b – c > 0[/latex]

(2) [latex] a – b = 7[/latex]

(3) [latex] a – 2b + c = 0[/latex]

(4) [latex] a + b + c = 12[/latex]

Jawab

Karena [latex]\sin{\theta} – \cos{\theta}= \frac{\sqrt{5} – \sqrt{3}}{2}[/latex], maka [latex]\cos{\theta} – \sin{\theta}= \frac{\sqrt{3} – \sqrt{5}}{2}[/latex],
sehingga berlaku $$ \begin{align} \left( \cos{\theta} – \sin{\theta} \right)^2 &= \left( \dfrac{\sqrt{3} – \sqrt{5}}{2} \right)^2\\ \cos^2{\theta} + \sin^2{\theta} – 2 \sin{\theta}\cos{\theta} &= \dfrac{ 3 + 5 – 2\sqrt{15} }{4}\\ 1 – 2 \sin{\theta}\cos{\theta} &= \dfrac{8 – 2\sqrt{15}}{4}\\ 1 – \dfrac{8 – 2\sqrt{15}}{4} &= 2 \sin{\theta}\cos{\theta}\\ \dfrac{4 – (8 – 2\sqrt{15})}{4} &= 2 \sin{\theta}\cos{\theta}\\ \dfrac{2\sqrt{15} – 4}{4} &= 2 \sin{\theta}\cos{\theta}\\ \dfrac{\sqrt{15} – 2}{4} &= \sin{\theta}\cos{\theta}\\ \end{align} $$

Kemudian ingat bahwa [latex]a^3 – b^3 = (a – b)(a^2 + b^2 + ab)[/latex], sehingga kita peroleh, $$ \begin{align} \cos^3{\theta} – \sin^3{\theta} &= \left(\cos{\theta} – \sin{\theta} \right) \left( \cos^2{\theta} + \sin^2{\theta} + \sin{\theta}\cos{\theta} \right)\\ &= -\left( \dfrac{\sqrt{5} – \sqrt{3}}{2} \right) \left( 1 + \dfrac{\sqrt{15} – 2}{4} \right)\\ &= \left( \dfrac{\sqrt{3} – \sqrt{5}}{2} \right)\left( \dfrac{4 + \sqrt{15} – 2}{4} \right)\\ &= \left( \dfrac{\sqrt{3} – \sqrt{5}}{2} \right)\left( \dfrac{2 + \sqrt{15}}{4} \right)\\ &= \dfrac{ 2\sqrt{3} – 2\sqrt{5} + 3\sqrt{5} – 5\sqrt{3}}{8}\\ \dfrac{1}{a}(b\sqrt{5}-c\sqrt{3}) &= \dfrac{\sqrt{5} – 3\sqrt{3}}{8}\\ \end{align} $$

Dengan demikian [latex]a = 8, b = 1, c = 3[/latex].
Oleh karena itu [latex]b – c = 1 – 3 =-2[/latex], nomor 1 salah. Jadi nomor 3 juga salah. Di samping itu [latex]a – b = 8 – 1 = 7[/latex], nomor 2 benar dan [latex]a+b+c = 8 + 1 + 3 = 12[/latex], nomor 4 benar. Jadi jawabannya adalah C

Leave a Reply

Your email address will not be published. Required fields are marked *