HOTS Zone : Aljabar [2]

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Tipe:

Standar SBMPTN HOTS

No.

Diketahui $x,y\in R$, $x>2016$ dan $x>2017$. Jika $2016\sqrt{(x+2016)(x-2016)}+2017\sqrt{(x+2017)(x-2017)}={\displaystyle \frac{1}{2}}(x^2+y^2)$
maka nilai $xy=$

1. $4066272$
2. $4068289$
3. $\mathrm{5750577,011}$

4. $5756281,$
5. $8132544$

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ALTERNATIF PENYELESAIAN

Misal $a=\sqrt{(x+2016)(x-2016)}$
$$\begin{array}{rl}{a}^2& =x^2-{2016}^2\\ x^2& =a^2+{2016}^2\end{array}$$ Misal $b=\sqrt{(y+2017)(y-2017)}$
$$\begin{array}{rl}{b}^2& =y^2-{2017}^2\\ y^2& =b^2+{2017}^2\end{array}$$ $$\begin{array}{rl}2016a+2017b& ={\displaystyle \frac{1}{2}}(a^2+{2016}^2+b^2+{2017}^2)\\ 2\cdot 2016a+2\cdot 2017b& =a^2+{2016}^2+b^2+{2017}^2\\ a^2-2\cdot 2016a+{2016}^2+b^2-2\cdot 2017b+{2017}^2& =0\\ (a-2016{)}^2+(b-2017{)}^2& =0\end{array}$$ didapat $a=2016$ dan $b=2017$
$$\begin{array}{rl}{x}^2& ={2016}^2+{2016}^2\\ & =2\cdot {2016}^2\\ x& =2016\sqrt{2}\end{array}$$ $$\begin{array}{rl}{y}^2& ={2017}^2+{2017}^2\\ & =2\cdot {2017}^2\\ x& =2017\sqrt{2}\end{array}$$ $$\begin{array}{rl}xy& =2016\sqrt{2}\cdot 2017\sqrt{2}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle 8132544}}}}\end{array}$$

Jadi, $xy=8132544$.
JAWAB: E

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No.

Jika $n-{\displaystyle \frac{1}{n}}=x$, maka berapakah $n^2+{\displaystyle \frac{1}{n^2}}$ jika dinyatakan dalam $x$?

1. $x^2+1$
2. $x+1$
3. $x^3+1$

4. $x^2+2$
5. $\sqrt{x}+\sqrt{1}$

Matematika Zone Id

ALTERNATIF PENYELESAIAN

$\begin{array}{rl}n-{\displaystyle \frac{1}{n}}& =x\\ {(n-{\displaystyle \frac{1}{n}})}^2& =x^2\\ n^2-2(n)\left({\displaystyle \frac{1}{n}}\right)+{\left({\displaystyle \frac{1}{n}}\right)}^2& =x^2\\ n^2-2+{\displaystyle \frac{1}{n^2}}& =x^2\\ n^2+{\displaystyle \frac{1}{n^2}}& =\boxed{{\displaystyle \boxed{{\displaystyle x^2+2}}}}\end{array}$

Jadi, $n^2+{\displaystyle \frac{1}{n^2}}=x^2+2$.
JAWAB: D

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No.

Jika ${\displaystyle \frac{3a+4b}{2a-2b}}=5$ maka tentukan nilai dari ${\displaystyle \frac{a^2+2b^2}{ab}}$

Diktat Pembinaan Olimpiade

ALTERNATIF PENYELESAIAN

$$\begin{array}{rl}{\displaystyle \frac{3a+4b}{2a-2b}}& =5\\ 3a+4b& =10a-10b\\ 7a& =14b\\ {\displaystyle \frac{a}{b}}& =2\end{array}$$

$$\begin{array}{rl}{\displaystyle \frac{a^2+2b^2}{ab}}& ={\displaystyle \frac{a}{b}}+2{\displaystyle \frac{b}{a}}\\ & =2+2\left({\displaystyle \frac{1}{2}}\right)\\ & =\boxed{{\displaystyle \boxed{{\displaystyle 3}}}}\end{array}$$

Jadi, ${\displaystyle \frac{a^2+2b^2}{ab}}=3$.

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No.

Nilai dari ${\displaystyle \frac{(1945+2011{)}^2+(2011-1945{)}^2}{{1945}^2+{2011}^2}}$ adalah ....

Diktat Pembinaan Olimpiade

ALTERNATIF PENYELESAIAN

Misal $a=1945$ dan $b=2011$
$$\begin{array}{rl}{\displaystyle \frac{(1945+2011{)}^2+(2011-1945{)}^2}{{1945}^2+{2011}^2}}& ={\displaystyle \frac{(a+b{)}^2+(b-a{)}^2}{a^2+b^2}}\\ & ={\displaystyle \frac{a^2+2ab+b^2+b^2-2ab+a^2}{a^2+b^2}}\\ & ={\displaystyle \frac{2(a^2+b^2)}{a^2+b^2}}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle 2}}}}\end{array}$$

Jadi, ${\displaystyle \frac{(1945+2011{)}^2+(2011-1945{)}^2}{{1945}^2+{2011}^2}}=2$.

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No.

Bilangan-bilangan real $a$, $b$, dan $c$ memenuhi sistem persamaan $a+b=8$ dan $ab=c^2+16$. Hasil dari $a+b+c=$ ....

1. $4$
2. $5$
3. $6$

4. $7$
5. $8$

Matematika Zone Id

ALTERNATIF PENYELESAIAN

$$\begin{array}{rl}ab& \le {\displaystyle \frac{(a+b{)}^2}{4}}\\ & \le {\displaystyle \frac{8^2}{4}}\\ & \le 16\end{array}$$ $$\begin{array}{rl}{c}^2+16& \ge 16\\ ab& \ge 16\end{array}$$ Didapat $ab=16$ dan $c=0$

$a+b+c=8+0=\boxed{{\displaystyle \boxed{{\displaystyle 8}}}}$

Jadi, $a+b+c=8$.
JAWAB: E

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No.

Jika $x+{\displaystyle \frac{1}{x}}=4$ maka nilai $x^3+x^{-3}=$ ....

ALTERNATIF PEMBAHASAN

$\begin{array}{rl}{(x+{\displaystyle \frac{1}{x}})}^2& =4^2\\ x^2+2+{\displaystyle \frac{1}{x^2}}& =16\\ x^2+{\displaystyle \frac{1}{x^2}}& =14\end{array}$

$\begin{array}{rl}(x^2+{\displaystyle \frac{1}{x^2}})(x+{\displaystyle \frac{1}{x}})& =(14)(4)\\ x^3+x+{\displaystyle \frac{1}{x}}+{\displaystyle \frac{1}{x^3}}& =56\\ x^3+4+{\displaystyle \frac{1}{x^3}}& =56\\ x^3+{\displaystyle \frac{1}{x^3}}& =\boxed{{\displaystyle \boxed{{\displaystyle 52}}}}\end{array}$

Jadi, $x^3+x^{-3}=52$.

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No.

ALTERNATIF PEMBAHASAN

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No.

ALTERNATIF PEMBAHASAN

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