Ujian Tulis UM UGM 2019 Matematika Dasar Kode 934

1. Jika rata-rata

$a$ ,

$b$ ,

$c$ dan

$a^{2}$ ,

$b^{2}$ ,

$c^{2}$ berturut-turut adalah 2 dan 4, maka rata-rata dari

$ab$ ,

$bc$ ,

$ca$ adalah

$....$

$\frac{10}{3}$

$\frac{11}{3}$

$4$

$\frac{13}{3}$

$\frac{14}{3}$

Pembahasan:

rumus rata-rata atau mean adalah

$\bar{x}=\frac{1}{n}\left (x _{1} \: +\: x_{2}\: +\: ....\: +\: x_{n}\right )$

$\frac{a+b+c}{3}=2\Leftrightarrow a+b+c=6$

$\frac{a^{2}+b^{2}+c^{2}}{3}=4\Leftrightarrow a^{2}+b^{2}+c^{2}=12$

$a+b+c=6$ (kedua ruas dikuadratkan)

$\left ( a+b+c \right )^{2}=6^{2}$

$\left ( a+b+c \right )\left ( a+b+c \right )=36$

$a^{2}+ab+ca+ab+b^{2}+bc+ca+bc+c^{2}=36$

$a^{2}+b^{2}+c^{2}+2ab+2bc+2ca=36$

$a^{2}+b^{2}+c^{2}+2\left ( ab+bc+ca \right )=36$

$\begin{array} \ 12+2\left ( ab+bc+ca \right )=36\end{array}$

$\begin{array} \ 2\left ( ab+bc+ca \right )=24\end{array}$

$ab+bc+ca=12$

rata-rata dari

$ab$ ,

$bc$ ,

$ca$

$\frac{ab+bc+ca}{3}=\frac{12}{3}$

$=4$

___________________________________ C

2. Diketahui

$f\left ( x \right )=x^{2}+1$ dan

$g\left ( x \right )=ax+2$ , dengan

$a\neq 0$ .
Jika

$\left ( f\, o\, g^{-1} \right )\left ( 1 \right )=5$ , maka

$4a^{2}-3=\: ....$
A. -3
B. -2
C. -1
D. 1
E. 2

Pembahasan:

$g\left ( x \right )=ax+b$ maka

$g^{-1}\left ( x \right )=\frac{x-b}{a}$

$f\left ( x \right )=x^{2}+1$

$g\left ( x \right )=ax+2$ maka

$g^{-1}\left ( x \right )\frac{x-2}{a}$

$\begin{aligned}\left ( f\, o\, g^{-1} \right ) \left ( 1 \right )=5\\ f\left ( g^{-1} \left ( 1 \right )\right )=5\\ f\left ( \frac{1-2}{a} \right )=5\\ f\left ( \frac{-1}{a} \right )=5\\ \left ( \frac{-1}{a}^{2} \right )+1=5\\ \frac{1}{a^{2}}=4\\ a^{2}=\frac{1}{4}\end{aligned}$

maka

$4a^{2}-3=4\frac{1}{4}-3$

$=1-3$

$=-2$

___________________________________________ B

Untuk lebih lengkapnya dapat di download di https://drive.google.com/file/d/1UE5PtNu1bUzpI5wdrrCosqEIdv2HI7F7/view?usp=sharing