// Numbas version: exam_results_page_options {"name": "Set Soal Matematika IIB - Bab 8 Bagian 2", "metadata": {"description": "", "licence": "None specified"}, "duration": 2700, "percentPass": 0, "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "PG + Isian", "pickingStrategy": "all-shuffled", "pickQuestions": 1, "questionNames": ["", "", "", ""], "questions": [{"name": "Mat 2B - Bab 8 - No 6", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Meong Meong Project", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4687/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Pillihlah semua matriks yang memenuhi syarat Teorema Perron-Frobenius.

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Pilih semua matriks yang mempunyai invers.

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Matriks \$$A=\\begin{bmatrix} \\var{2+3*a} & \\var{-2-2*a}\\\\ \\var{3+3*a} & \\var{-3-2*a} \\end{bmatrix}\$$ mempunyai vektor eigen \$$\\mathbf{v}=\\begin{bmatrix} 2\\\\3 \\end{bmatrix}\$$ yang bersesuaian dengan nilai eigen \$$\\lambda=\\ldots\$$.

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Diketahui \$$\\mathbf{v}=\\begin{bmatrix} \\var{a*0.01} \\\\ a \\\\ b \\end{bmatrix} \$$ merupakan suatu vektor eigen yang berkaitan dengan nilai eigen \$$\\lambda = 5\$$ dari matriks \$$A=\\begin{bmatrix} 1 & -4 & 0 \\\\ -4 & 1 & 0 \\\\ 0 & 0 & 2 \\end{bmatrix}\$$. Maka nilai \$$a+b=\\ldots\$$.

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Penyebaran populasi capung yang berada di dua kolam \$$P\$$ dan \$$Q\$$ dapat dimodelkan sebagai berikut. Setiap hari, \$$20\\%\$$ capung yang berada dalam kolam \$$P\$$ akan pindah ke kolam \$$Q\$$. Sebaliknya, \$$30\\%\$$ capung yang berada dalam kolam \$$Q\$$ akan pindah ke kolam \$$P\$$. Misalkan \$$P_t\$$ dan \$$Q_t\$$ berturut-turut menyatakan banyaknya capung pada kolam \$$P\$$ dan \$$Q\$$ setiap waktu. Maka dapat diperoleh sistem persamaan beda \$$\\begin{cases} P_{t+1}&=0.8P_t+0.3Q_t \\\\ Q_{t+1}&=0.2P_t+0.7Q_t \\end{cases}\$$.

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Misalkan \$$\\mathbf{x}_t=\\begin{bmatrix} P_t \\\\Q_t \\end{bmatrix}\$$. Bentuk iterasi model tersebut dapat ditulis sebagai \$$\\mathbf{x}_{t+1}=A\\mathbf{x}_t\$$. Tentukan matriks \$$A\$$.

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Diketahui

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• Nilai eigen dari matriks $A$ adalah $\\lambda_1$ dan $\\lambda_2$ dengan $\\lambda_1\\geq \\lambda_2$.
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• Vektor \$$\\mathbf{v}_1=\\begin{bmatrix} 3 \\\\ a \\end{bmatrix}\$$ merupakan vektor eigen yang bersesuaian dengan $\\lambda_1$, dan vektor \$$\\mathbf{v}_2=\\begin{bmatrix} 3 \\\\ b \\end{bmatrix}\$$ merupakan vektor eigen yang bersesuaian dengan $\\lambda_2$.
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• Populasi awal capung di kedua kolam tersebut adalah \$$\\mathbf{x}_0=\\begin{bmatrix} 600 \\\\ 80 \\end{bmatrix}\$$.
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Tentukan nilai $\\lambda_1$, $\\lambda_2$, $a$, dan $b$ berturut-turut.

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Dalam waktu yang cukup lama, misalkan banyaknya capung pada kolam $P$ sebanyak $P_\\infty$, sedangkan banyaknya capung pada kolam $Q$ sebanyak $Q_\\infty$. Tentukan nilai dari $\\dfrac{P_\\infty}{Q_\\infty}$.

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Waktunya 5 menit lagi ya.

Selamat datang!

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Silakan gunakan set soal ini dengan bijak. Kalian bisa mencoba set soal ini terus-menerus.

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Set soal ini dapat digunakan oleh siapapun secara gratis.

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Petunjuk pengerjaan soal:

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• Jawaban dapat dituliskan dalam bentuk desimal maupun pecahan.
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• Jika jawaban dituliskan dalam bentuk desimal, pisahkan dengan tanda titik dan bulatkan hingga dua angka di belakang titik. Contoh: ketik 0.67 untuk menjawab \$$\\frac{2}{3}\$$.
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• Jika jawaban dituliskan dalam bentuk pecahan, pisahkan pembilang dan penyebut dengan garis miring dan jawab dengan pecahan paling sederhana. Contoh: ketik 2/3 untuk menjawab \$$\\frac{2}{3}\$$.
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Jika ada kunci jawaban yang salah, kalian dapat memberitahu kami di Instagram @meongmeongproject atau OA Line @eog7710d.

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