HOTS Zone : Persamaan Nilai Mutlak
Table of Contents
Tipe:
No.
Diketahui\[2016\sqrt{(x+2016)(x-2016)}+2017\sqrt{(y+2017)(y-2017)}=\dfrac12\left(x^2+y^2\right)\]
maka nilai xy = ....
- 4066272
- 4068289
- 5750577,011
- 5756281,95
- 8132544
ALTERNATIF PENYELESAIAN
\(\begin{aligned}
2016\sqrt{(x+2016)(x-2016)}+2017\sqrt{(y+2017)(y-2017)}&=\dfrac12\left(x^2+y^2\right)\\
2\cdot2016\sqrt{x^2-2106^2}+2\cdot2017\sqrt{y^2-2017^2}&=x^2-2016^2+2016^2+y^2-2017^2+2017^2
\end{aligned}\)
Misal \({\sqrt{x^2-2016^2}=p}\) dan \({\sqrt{y^2-2017^2}=q}\)
\(\begin{aligned} 2\cdot2016p+2\cdot2017q&=p^2+2016^2+q^2+2017^2\\ p^2-2\cdot2016p+2016^2+q^2-2\cdot2017q+2017^2&=0\\ (p-2016)^2+(q-2017)^2&=0 \end{aligned}\)
\(\begin{aligned} p-2016&=0\\ p&=2016\\ \sqrt{x^2-2016^2}&=2016\\ x^2-2016^2&=2016^2\\ x^2&=2\cdot2016^2\\ x&=2016\sqrt2 \end{aligned}\)
\(\begin{aligned} q-2017&=0\\ q&=2017\\ \sqrt{y^2-2017^2}&=2017\\ y&=2017\sqrt2 \end{aligned}\)
\(\begin{aligned} xy&=2016\sqrt2\cdot2017\sqrt2\\ &=\boxed{\boxed{8132544}} \end{aligned}\)
Misal \({\sqrt{x^2-2016^2}=p}\) dan \({\sqrt{y^2-2017^2}=q}\)
\(\begin{aligned} 2\cdot2016p+2\cdot2017q&=p^2+2016^2+q^2+2017^2\\ p^2-2\cdot2016p+2016^2+q^2-2\cdot2017q+2017^2&=0\\ (p-2016)^2+(q-2017)^2&=0 \end{aligned}\)
\(\begin{aligned} p-2016&=0\\ p&=2016\\ \sqrt{x^2-2016^2}&=2016\\ x^2-2016^2&=2016^2\\ x^2&=2\cdot2016^2\\ x&=2016\sqrt2 \end{aligned}\)
\(\begin{aligned} q-2017&=0\\ q&=2017\\ \sqrt{y^2-2017^2}&=2017\\ y&=2017\sqrt2 \end{aligned}\)
\(\begin{aligned} xy&=2016\sqrt2\cdot2017\sqrt2\\ &=\boxed{\boxed{8132544}} \end{aligned}\)
Jadi, nilai xy = 8132544.
JAWAB: E
JAWAB: E
No.
Hasil penjumlahan semua solusi persamaan \[\left|x-|2x+6|\right|=99\] adalah....ALTERNATIF PENYELESAIAN
| \begin{aligned}
x-|2x+6|&=-99\\
|2x+6|&=x+99
\end{aligned}
|
\begin{aligned}
x-|2x+6|&=99\\
|2x+6|&=x-99
\end{aligned}
Karena |2x + 6| > x untuk |
|
| \begin{aligned} 2x+6&=-x-99\\ 3x&=-105\\ x&=-35 \end{aligned} | \begin{aligned} 2x+6&=x+99\\ x&=93\\ \end{aligned} | |
Jadi, hasil penjumlahan semua solusi persamaan \[\left|x-|2x+6|\right|=99\] adalah 58.
No.
Hasil penjumlahan semua solusi persamaan \[\left|x-|2x+3|\right|=99\] adalah....ALTERNATIF PENYELESAIAN
| \begin{aligned}
x-|2x+3|&=-99\\
|2x+3|&=x+99
\end{aligned}
|
\begin{aligned}
x-|2x+3|&=99\\
|2x+3|&=x-99
\end{aligned}
Karena |2x + 3| > x untuk |
|
| \begin{aligned} 2x+3&=-x-99\\ 3x&=-102\\ x&=-34 \end{aligned} | \begin{aligned} 2x+3&=x+99\\ x&=96\\ \end{aligned} | |
Jadi, hasil penjumlahan semua solusi persamaan \[\left|x-|2x+3|\right|=99\] adalah 62.
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