Latihan Soal 9 Persiapan PAS Gasal Matematika Wajib Kelas X

 $\begin{array}{l}\\ 76.& \text{Suatu bilangan terdiri atas 3 angka. Jumlah}\\ & \text{ketiga angka tersebut adalah 9. Angka kedua}\\ & \text{dikurangi angka pertama dan angka ketiga }\\ & \text{sama dengan 1. Dua kali angka pertama sama}\\ & \text{dengan jumlah angka kedua dan angka ketiga.}\\ & \text{Angka puluhan pada bilangan tersebut adalah}\\ & ....\\ & \begin{array}{l}\\ \text{a}.& 3\\ \text{b}.& 4\\ \text{c}.& 5\\ \text{d}.& 6\\ \text{e}.& 7\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}& \text{Model matematikanya}\\ & \{\begin{array}{c}A+B+C=9....(1)\\ 2B-A-C=1....(2)\\ 2A=B+C....(3)\end{array}\\ & \text{Saat}(1)+(2),\text{maka}\\ & \begin{array}{l}\\ {\displaystyle A+B+C}& =9\\ {\displaystyle -A+B-C}& =1& +\\ 2B& =10\\ B& =5& ...(4)\end{array}\\ & \text{Jadi},\text{bilangan kedua adalah}=B=5\end{array}\end{array}$

$\begin{array}{l}\\ 77.& (\mathbf{\text{SIMAK UI 2010}})\\ & \text{Jika}x+y+2z=K,x+2y+z=K,\\ & 2x+y+z=K\text{dengan}K\ne 0,\text{maka}\\ & x^2+y^2+z^2\text{bila dinyatakan dalam}K\\ & \text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle \frac{1}{16}{K}^2}\\ \text{b}.& {\displaystyle \frac{3}{16}{K}^2}\\ \text{c}.& {\displaystyle \frac{4}{17}{K}^2}\\ \text{d}.& {\displaystyle \frac{3}{8}{K}^2}\\ \text{e}.& {\displaystyle \frac{2}{3}{K}^2}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{b}}\\ & \begin{array}{rl}& \text{Diketahui sistem persamaan}\\ & \{\begin{array}{c}x+y+2z=K....(1)\\ x+2y+z=K....(2)\\ 2x+y+z=K....(3)\end{array}\\ & \text{maka}\\ & \{\begin{array}{c}z+(x+y+z)=K....(1)\\ y+(x+y+z)=K....(2)\\ x+(x+y+z)=K....(3)\end{array}\\ & \text{Saat}(1)+(2)+(3),\text{maka}\\ & \begin{array}{l}\\ {\displaystyle x+y+2z}& =K\\ {\displaystyle x+2y+z}& =K& \\ {\displaystyle 2x+y+z}& =K& +\\ 4x+4y+4z& =3K\\ x+y+z& ={\displaystyle \frac{3}{4}K}& ...(4)\end{array}\\ & \text{Saat}(4)\text{disubstitusikan ke}(1),(2),\text{dan}(3)\\ & \text{Jelas bahwa akan didapatkan}\\ & x=y=z={\displaystyle \frac{1}{4}K}\\ & \text{Jadi},x^2+y^2+y^2=3{\left({\displaystyle \frac{1}{4}K}\right)}^2={\displaystyle \frac{3}{16}{K}^2}\end{array}\end{array}$

$\begin{array}{l}\\ 78.& \text{Diketahui}0,15252525252...={\displaystyle \frac{p}{2q+r}}\\ & \text{Jika jumlah}p\text{dan}q=\text{3 kali}r,\text{maka}\\ & \text{masing-masing harga}p,q,\text{dan}r=....\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle 152,2819,2584}\\ \text{b}.& {\displaystyle 252,{\displaystyle \frac{5638}{7},\frac{8102}{21}}}\\ \text{c}.& {\displaystyle 151,\frac{2819}{7},\frac{1292}{7}}\\ \text{d}.& {\displaystyle 151,{\displaystyle \frac{2819}{7},\frac{2584}{7}}}\\ \text{e}.& {\displaystyle 152,\frac{2819}{14},\frac{1292}{7}}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}& \text{Dari soal diketahui}\\ & \{\begin{array}{ll}0,1\overline{5252}& ={\displaystyle \frac{p}{2q+r}.....(1)}\\ p+q& =3r............(2)\end{array}\\ & \text{dan}\\ & \begin{array}{l}\\ {\displaystyle x}& =0,15252525252...\\ {\displaystyle 1000x}& =152,5252525252...\\ {\displaystyle 10x}& =1,5252525252...& -\\ 990x& =151\\ x& ={\displaystyle \frac{151}{990},\text{maka}}\\ {\displaystyle \frac{p}{2q+r}}& ={\displaystyle \frac{151}{990}}\\ & \{\begin{array}{ll}p& =151.......(3)\\ 2p+r& =990.......(4)\end{array}\end{array}\\ & \text{Dari}(3)\text{diperoleh}:q=3r-p=3r-151....(5)\\ & \text{Dari}(5)\text{disubstitusikan ke}(4)\\ & \begin{array}{rl}2q+r& =990\\ 2(3r-151)+r& =990\\ 6r-302+r& =990\\ 7r& =990+302=1292\\ r& ={\displaystyle \frac{1292}{7}.....(6)}\end{array}\\ & \text{Dari}(3)\mathrm{\&}(6)\text{disubstitusikan ke}(2)\\ & \begin{array}{rl}p+q& =3r\\ 151+q& =3\left({\displaystyle \frac{1292}{7}}\right)\\ q& ={\displaystyle \frac{3876}{7}-151}\\ & ={\displaystyle \frac{3876-1057}{7}}\\ & ={\displaystyle \frac{2819}{7}.....(7)}\end{array}\\ & \text{Jadi},p,q,r\text{adalah}:{\displaystyle 151,\frac{2819}{7},\frac{1292}{7}}\end{array}\end{array}$

$\begin{array}{l}\\ 79.& \text{Perhatikanlah sistem persamaan berikut}\\ & \{\begin{array}{ll}3x+2y-5z& =3\\ 2x-6y+kz& =9\\ 5x-4y-z& =5\end{array}\\ & \text{agar sistem persamaan ini tidak}\\ & \text{memiliki penyelesaian, maka nilai}k=....\\ & \begin{array}{l}\\ \text{a}.& -4\\ \text{b}.& 2\\ \text{c}.& 3\\ \text{d}.& 4\\ \text{e}.& 6\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{d}}\\ & \begin{array}{rl}& \text{Agar sistem persamaan}\\ & \{\begin{array}{ll}3x+2y-5z& =3\\ 2x-6y+kz& =9\\ 5x-4y-z& =5\end{array}\\ & \text{tidak berpenyelesaian, maka}\\ & \text{ingat penyelesaian metode matrik}\\ & \text{buatlah penyebutnya}=0,\text{yaitu}:\\ & \left|\begin{array}{ccc}3& 2& -5\\ 2& -6& k\\ 5& -4& -1\end{array}\right|=0\\ & \text{Selanjutnya}\\ & 3\left|\begin{array}{cc}-6& k\\ -4& -1\end{array}\right|-2\left|\begin{array}{cc}2& k\\ 5& -1\end{array}\right|-5\left|\begin{array}{cc}2& -6\\ 5& -4\end{array}\right|=0\\ & 3(6+4k)-2(-2-5k)-5(-8+30)=0\\ & 18+12k+4+10k+40-150=0\\ & 22x=88\\ & x=4\end{array}\end{array}$

$\begin{array}{l}\\ 80.& \text{Diketahui}\\ & \left(\begin{array}{ccc}{\displaystyle \frac{1}{5}}& {\displaystyle \frac{1}{5}}& {\displaystyle \frac{1}{5}}\\ {\displaystyle \frac{1}{5}}& {\displaystyle \frac{1}{5}}& -{\displaystyle \frac{4}{5}}\\ -{\displaystyle \frac{2}{5}}& {\displaystyle \frac{1}{10}}& {\displaystyle \frac{1}{10}}\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}1\\ 2\\ 0\end{array}\right)\\ & \text{Nilai}x,y,\text{dan}z\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle \frac{1}{5},\frac{4}{5},-\frac{1}{10}}\\ \text{b}.& -1,5,1\\ \text{c}.& 1,5,-1\\ \text{d}.& -1,1,5\\ \text{e}.& 5,1,-1\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}& \text{Diketahui sistem persamaan}\\ & \{\begin{array}{c}{\displaystyle \frac{1}{5}x+\frac{1}{5}y+\frac{1}{5}z=1....(1)}\\ {\displaystyle \frac{1}{5}x+\frac{1}{5}y-\frac{4}{5}z=2....(2)}\\ -{\displaystyle \frac{2}{5}x+\frac{1}{10}y+\frac{1}{10}z=0....(3)}\end{array}\\ & \text{Saat}(1)+(2),\text{maka}\\ & \begin{array}{l}\\ {\displaystyle \frac{1}{5}x+\frac{1}{5}y+\frac{1}{5}z}& =1& \\ {\displaystyle \frac{1}{5}x+\frac{1}{5}y-\frac{4}{5}z}& =2& -\\ {\displaystyle \frac{5}{5}z}& =-1& \\ {\displaystyle z}& =-1& ...(4)\end{array}\\ & \text{Saat}(1)+(3),\text{maka}\\ & \begin{array}{l}\\ {\displaystyle \frac{1}{5}x+\frac{1}{5}y+\frac{1}{5}z}& =1& |\times 1|& {\displaystyle \frac{1}{5}x+\frac{1}{5}y+\frac{1}{5}z=1}\\ -{\displaystyle \frac{2}{5}x+\frac{1}{10}y+\frac{1}{10}z}& =0& |\times 2|& -{\displaystyle \frac{4}{5}+\frac{1}{5}y+\frac{1}{5}z=0}& -\\ & & & {\displaystyle \frac{5}{5}x=1}& \\ & & & x=-1........(5)\end{array}\\ & \text{Dari persamaan}(4)\mathrm{\&}(5)\text{akan didapatkan}\\ & y=5\\ & \text{Jadi},(x,y,z)=(1,5,-1)\end{array}\end{array}$.

$\begin{array}{l}\\ 81.& \text{Diketahui suatu fungsi kuadrat}\\ & f(x)=a{x}^2+bx+c.\text{Jika fungsi}\\ & (-1,0),(1,4),\text{dan}(2,9),\text{maka}\\ & \text{fungsi yang dimaksud adalah}....\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle f(x)=x^2-2x+3}\\ \text{b}.& f(x)=x^2+2x+3\\ \text{c}.& f(x)=x^2+2x-3\\ \text{d}.& f(x)=x^2-2x-3\\ \text{e}.& f(x)=x^2+2x+1\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{e}}\\ & \begin{array}{rl}& \text{Diketahui sistem persamaan}\\ & \{\begin{array}{c}(-1,0)\Rightarrow f(-1)=a-b+c=0....(1)\\ (1,4)\Rightarrow f(1)=a+b+c=4....(2)\\ (2,9)\Rightarrow f(2)=4a+2b+c=9....(3)\end{array}\\ & \text{Saat}(1)\mathrm{\&}(2),\text{didapatkan}\\ & b=2...............(4)\\ & \text{Saat}(1)\mathrm{\&}(3),\text{didapatkan}\\ & \begin{array}{l}\\ 4a+2b+c& =9& \\ a-b+c& =0& -\\ 3a+3b& =9& \\ a+b& =3& ...(5)\end{array}\\ & \text{Dari persamaan}(4)\mathrm{\&}(5)\text{maka},\\ & \{\begin{array}{ll}a& =1\\ c& =1\end{array}\\ & \text{Jadi},f(x)=a{x}^2+bx+c=x^2+2x+1\end{array}\end{array}$

$\begin{array}{l}\\ 82.& \text{Diketahui persamaan}\{\begin{array}{ll}x-y& =2\\ kx+y& =3\end{array}\\ & \text{memiliki solusi}(x,y)\text{di kuadran I}\\ & \text{Jika dan hanya jika nilai}k\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle k=-1}\\ \text{b}.& k>-1\\ \text{c}.& k<{\displaystyle \frac{3}{2}}\\ \text{d}.& 0<k<{\displaystyle \frac{3}{2}}\\ \text{e}.& -1<k<{\displaystyle \frac{3}{2}}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{e}}\\ & \begin{array}{rl}& \text{Diketahui sistem persamaan}\\ & \{\begin{array}{c}x-y=2....(1)\\ kx+y=3....(2)\end{array}\\ & \text{Dengan metode matriks didapatkan}\\ & x={\displaystyle \frac{\left|\begin{array}{cc}2& -1\\ 3& 1\end{array}\right|}{\left|\begin{array}{cc}1& -1\\ k& 1\end{array}\right|}={\displaystyle \frac{2-(-3)}{1+k}=\frac{5}{k+1}}}\\ & \text{Dengan cara yang sama pula}\\ & y={\displaystyle \frac{\left|\begin{array}{cc}1& 2\\ k& 3\end{array}\right|}{\left|\begin{array}{cc}1& -1\\ k& 1\end{array}\right|}={\displaystyle \frac{3-2k}{k+1}}}\\ & \text{Supaya memiliki solusi di kwadran I},\\ & \text{maka baik}x\text{maupun}y\\ & \text{haruslah positif, akibatnya}:\\ & k+1>0\Rightarrow k>-1\\ & \text{Sebagai akibat yang lain adalah}:\\ & 3-2k>0\Rightarrow k<{\displaystyle \frac{3}{2}}\\ & \text{Jadi},-1<k<{\displaystyle \frac{3}{2}}\end{array}\end{array}$

$\begin{array}{l}\\ 83.& \text{Diketahui sistem persamaan}\\ & y+{\displaystyle \frac{2}{x+z}=4}\\ & 5y+{\displaystyle \frac{18}{2x+y+z}=18}\\ & {\displaystyle \frac{8}{x+z}-\frac{6}{2x+y+z}=3}\\ & \text{Nilai}y+\sqrt{x^2-2xz+y^2}\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle 3}\\ \text{b}.& 5\\ \text{c}.& 7\\ \text{d}.& 9\\ \text{e}.& 11\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}& \text{Diketahui sistem persamaan}\\ & \{\begin{array}{c}y+{\displaystyle \frac{2}{x+z}=4}\\ 5y+{\displaystyle \frac{18}{2x+y+z}=18}\\ {\displaystyle \frac{8}{x+z}-\frac{6}{2x+y+z}=3}\end{array}\\ & \text{Jika disederhanakan beberapa bagian}\\ & \{\begin{array}{ll}y+2A& =4....(1)\\ 5y+18B& =18....(2)\\ 8A-6B& =3....(3)\end{array}\\ & \text{Saat}(1)+(2)\mathrm{\&}(3),\text{maka}\\ & \begin{array}{l}\\ y+2A& =4& |\times 5|& 5y+10A=20\\ 5y+3(8A-3)& =18& |\times 1|& 5y+24A=27& -\\ & & & -14A=-7\\ & & & A={\displaystyle \frac{1}{2}...(4)}\\ \text{maka}B={\displaystyle \frac{1}{6}\mathrm{\&}}& y=3& & \\ \text{akibatnya}\\ \{\begin{array}{ll}x& =1\\ z& =1\end{array}\end{array}\\ & \text{Jadi},y+\sqrt{x^2-2xz+z^2}=3+0=3\end{array}\end{array}$

$\begin{array}{l}\\ 84.& \text{Diberikan}a,b,\text{dan}c\text{adalah angka-angka}\\ & \text{dari bilangan 3 digit yang memenuhi}\\ & 49a+7b+c=286.\text{Nilai dari}a+b+c\\ & \text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& 16\\ \text{b}.& 17\\ \text{c}.& 18\\ \text{d}.& 19\\ \text{e}.& 20\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}& \text{Diketahui sistem persamaan}\\ & 49a+7b+c=286\\ & \text{Nilai maksimum}a\text{adalah}5\\ & 49\times 5=245,\text{akibatnya}:\\ & 245+7b+c=286\Rightarrow 7b+c=286-245=41\\ & \text{Nilai maksimum}b\text{adalah}5\\ & 7\times 5=35,\text{akibatnya}:\\ & 35+c=41\Rightarrow c=41-35=6\\ & \text{Sehingga}a,b,\text{dan}c\text{adalah}5,5,\text{dan}6\\ & \text{Jadi},\text{nilai}a+b+c=5+5+6=16\end{array}\end{array}$

$\begin{array}{l}\\ 85.& \text{Diketahui sistem persamaan}\\ & (2x+3y{)}^{{.}^{\log (x-y+2z)}}=1\\ & 3^{2x+y+z}\times {27}^{3z+2y+x}=81\\ & 5x+3y+8z=2\\ & \text{Himpunan penyelesaian yang}\\ & \text{memenuhi adalah}....\\ & \begin{array}{l}\\ \text{a}.& \left\{{\displaystyle \frac{17}{12},-\frac{1}{12},\frac{7}{6}}\right\}\\ \text{b}.& \{-{\displaystyle \frac{17}{12},\frac{1}{2},\frac{7}{6}}\}\\ \text{c}.& \{-{\displaystyle \frac{17}{12},-\frac{1}{2},-\frac{7}{6}}\}\\ \text{d}.& \left\{{\displaystyle \frac{17}{12},\frac{1}{12},\frac{7}{6}}\right\}\\ \text{e}.& \{-{\displaystyle \frac{17}{12},-\frac{1}{12},\frac{7}{6}}\}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{e}}\\ & \begin{array}{rl}& \text{Untuk persamaan}(1)\\ & (2x+3y{)}^{{.}^{\log (x-y+2z)}}=(2x+3y{)}^0\\ & \iff (x-y+2z)={10}^0=1\\ & \text{Untuk persamaan}(2)\\ & 3^{2x+y+z}\times {27}^{3z+2y+x}=81\\ & \iff 3^{2x+y+z+3(3z+2y+x)}=3^4\\ & \iff 5x+7y+10z=4\\ & \text{Sehingga sistem persamaan akan terlihat}\\ & \{\begin{array}{c}x-y+2z=1....(1)\\ 5x+7y+10z=4....(2)\\ 5x+3y+8z=2....(3)\end{array}\\ & \text{Saat}(2)\mathrm{\&}(3),\text{maka}\\ & \begin{array}{l}\\ 5x+7y+10z& =4& \\ 5x+3y+8z& =2& -\\ 4y+2z& =2\\ 2y+z& =1...(4)\end{array}\\ & \text{Saat}(1)+(3),\text{maka}\\ & \begin{array}{l}\\ 5x-5y+10z& =5& \\ 5x+7y+10z& =4& -\\ -12y& =1\\ y& =-{\displaystyle \frac{1}{12}...(5)}\end{array}\\ & \text{Dari persamaan}(5)\text{disubstistusikan ke}(4)\\ & \begin{array}{rl}2y+z& =1\\ 2(-{\displaystyle \frac{1}{12}})+z& =1\\ z& =1+{\displaystyle \frac{1}{6}}\\ z& ={\displaystyle \frac{7}{6}}\end{array}\\ & \text{Cukup jelas juga}x=....\\ & \text{Jadi},\text{pilihannya adalah}e\end{array}\end{array}$

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