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{"name": "Mat 2B - Bab 10 - No 5", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Mat 2B - Bab 10 - No 5", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p><strong>Hemodialisis</strong> adalah proses yang dilakukan oleh suatu mesin untuk menyaring urea dari darah pasien jika ginjal pasien bermasalah. Banyaknya urea saat dialisis seringkali dimodelkan dengan memisalkan terdapatnya dua buah tempat tersimpannya urea: pada darah pasien, yang secara langsung disaring oleh mesin dialisis, dan tempat lain yang tidak bisa disaring secara langsung oleh mesin dialisis. Sistem persamaan diferensial yang mendeskripsikan model tersebut adalah<br/>\\[\\begin{cases} \\dfrac{dc}{dt} &amp;= -\\dfrac{K}{V}c+ap-bc \\\\ \\dfrac{dp}{dt} &amp;= -ap+bc \\end{cases}\\]<br/>dengan \\(c\\) dan \\(p\\) adalah konsentrasi urea di darah pasien dan di tempat lain (dalam mg/mL) dan semua konstanta lainnya bernilai positif. Misal \\(K=3\\), \\(V=2\\), \\(a=b=1\\), konsentrasi urea awal \\(c(0)=5\\) mg/mL, dan \\(p(0)=5\\) mg/mL.</p>", "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": "<p>Jika \\(\\mathbf{x}(t)=\\begin{bmatrix} c(t) \\\\ p(t) \\end{bmatrix}\\), tentukan matriks \\(A\\) agar model dapat dituliskan menjadi masalah nilai awal \\(\\mathbf{x}'=A\\mathbf{x}\\), \\(\\mathbf{x}(0)=\\begin{bmatrix} c_0 \\\\ p_0 \\end{bmatrix}\\).</p>", "correctAnswer": "matrix([-2.5,1],[1,-1])", "correctAnswerFractions": true, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "1.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Diketahui</p>\n<ul>\n<li>Matriks \\(A\\) mempunyai dua buah nilai eigen berbeda, \\(\\lambda_1\\) dan \\(\\lambda_2\\), dengan \\(\\lambda_1&gt;\\lambda_2\\).</li>\n<li>Vektor eigen yang berkorespondensi dengan nilai eigen \\(\\lambda_1\\) adalah \\(\\mathbf{v}_1=\\begin{bmatrix} a \\\\ 2 \\end{bmatrix}\\).</li>\n<li>Vektor eigen yang berkorespondensi dengan nilai eigen \\(\\lambda_2\\) adalah \\(\\mathbf{v}_2=\\begin{bmatrix} b \\\\ 1 \\end{bmatrix}\\).</li>\n</ul>\n<p>Tentukan nilai $\\lambda_1$, $\\lambda_2$, $a$, dan $b$ berturut-turut.</p>", "correctAnswer": "matrix([-0.5,-3,1,-2])", "correctAnswerFractions": true, "numRows": "1", "numColumns": "4", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "0.8", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Solusi umum dari model tersebut dapat dituliskan menjadi $\\mathbf{x}(t)=C_1\\mathbf{v}_1e^{\\lambda_1t} + C_2\\mathbf{v}_2e^{\\lambda_2t}$.</p>\n<p>Tentukan $C_1$ dan $C_2$.</p>", "correctAnswer": "matrix([3,-1])", "correctAnswerFractions": true, "numRows": 1, "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.6", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Tentukan \\(\\lim\\limits_{t\\to\\infty}c(t)\\).</p>", "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "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/"}]}