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SIMAK UI 2010

Solusi SIMAK UI 2010 Kode 209 Nomor 15


Jika\ diketahui\ x-y=\frac{1}{2-\sqrt{3}},\ y-z=\frac{1}{2+\sqrt{3}},\ maka
x^2+y^2+z^2-xy-yz-xz=....
(A)\quad 5\\(B)\quad 10\\(C)\quad 15\\(D)\quad 20\\(E)\quad 25
Jawaban: C

x-y=\frac{1}{2-\sqrt{3}}\quad .......................\quad (1)
y-z=\frac{1}{2+\sqrt{3}}\quad .......................\quad (2)
Jumlah (1) dan (2) menghasilkan:

x-z=\frac{1}{2-\sqrt{3}}+\frac{1}{2+\sqrt{3}}=\frac{2+\sqrt{3}+2-\sqrt{3}}{4-3}=4
sehingga:

(x-z)^2=16\\x^2+z^2-2xz=16\quad .......................\quad (*)
Perkalian (1) dan (2) menghasilkan:

(x-y)(y-z)=\frac{1}{2-\sqrt{3}}\times \frac{1}{2+\sqrt{3}}=\frac{1}{4-3}=1
xy-xz-y^2+yz=1\\
y^2-xy+xz-yz=-1\quad .......................\quad (**)
Jumlah (*) dan (**) menghasilkan:

x^2+y^2+z^2-xy-yz-xz=16+(-1)=15

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