// Numbas version: exam_results_page_options {"name": "Mat 2B - Bab 8 - No 10", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Mat 2B - Bab 8 - No 10", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

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}\$$.

", "advice": "", "rulesets": {}, "extensions": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "0.4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

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\$$.

", "correctAnswer": "matrix([0.8,0.3],[0.2,0.7])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "1.6", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Diketahui

\n
\n
• Nilai eigen dari matriks $A$ adalah $\\lambda_1$ dan $\\lambda_2$ dengan $\\lambda_1\\geq \\lambda_2$.
• \n
• 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 $\\lambda_2$.
• \n
• Populasi awal capung di kedua kolam tersebut adalah \$$\\mathbf{x}_0=\\begin{bmatrix} 600 \\\\ 80 \\end{bmatrix}\$$.
• \n
\n

Tentukan nilai $\\lambda_1$, $\\lambda_2$, $a$, dan $b$ berturut-turut.

", "correctAnswer": "matrix([1,0.5,2,-3])", "correctAnswerFractions": true, "numRows": 1, "numColumns": "4", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

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 $\\dfrac{P_\\infty}{Q_\\infty}$.

", "minValue": "1.5", "maxValue": "1.5", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": true, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Meong Meong Project", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4687/"}]}]}], "contributors": [{"name": "Meong Meong Project", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4687/"}]}