Einige Beispiele zum Umgang mit mehrdeutigen Lösungen in NUMBAS.
", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "Die unten stehenden Aufgaben haben jeweils verschiedene Lösungen. Um diese zu prüfeen, ist es in NUMBAS erforderlich, mit einem \"Custom Marking Algorithm\" zu arbeiten.
", "advice": "Lösung zu a):
\nMan erkennt leicht, dass der Flächeninhalt $\\var{n}$ ein Vielfaches von Fünf ist (letzte Stelle ist ein 5er), somt wählt man $a=5,b=\\var{n/5}$.
\nEs gibt andere Lösungen, z.B. $a=\\var{a},b=\\var{b}$.
\nFür die triviale Lösung $a=1,b=\\var{n}$ gibt es nur die halbe Punktzahl.
\nLösung zu b):
\nNach der \"Kipp-Regel\" findet man jedenfalls die Lösungen $\\vec{n_1}=\\var{vector(-p[1],p[0])}$ und $\\vec{n_2}=\\var{vector(p[1],-p[0])}$.
\nAuch alle Vielfachen dieser Vektoren sind korrekte Lösungen.
\nFür die triviale Lösung $\\vec{n_0}=\\vec{0}=\\begin{pmatrix}0\\\\0\\end{pmatrix}$ gibt es nur die halbe Punktzahl.
\nLösung zu c):
\nGemäß dem Lemma von Bezout existieren jedenfalls Zahlen $x,y\\in\\mathbb{Z}$, so dass $x\\cdot\\var{other}+y\\cdot\\var{prim}=\\operatorname{ggT}(\\var{other},\\var{prim})$ gilt.
\nSolche Zahlen findet man sicher mit dem erweiterten Euklidischen Algorithmus.
\nFängt man mit diesem an, so sieht man allerdings bereits im ersten Schritt, dass $\\var{other}=\\var{-sschlau}\\cdot\\var{prim}+\\var{rest}\\quad\\Leftrightarrow\\quad 1\\cdot\\var{other}\\var{sschlau}\\cdot\\var{prim}=\\var{rest}$ gilt, somit ist also $r=1, s=\\var{sschlau}$.
\nMan kann auch stumpf bis zum Ende den Algorithmus weiterrechnen, das führt zur Gleichung $\\var{bezout[1]}\\cdot\\var{other}+(\\var{bezout[2]})\\cdot\\var{prim}=1$, die man mit $\\var{rest}$ multipliziert und somit als weitere mögliche Lösung $r=\\var{rdumm}, s=\\var{sdumm}$ erhält.
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\nGeben Sie mögliche ganzzahlige Seitenlängen $a,b$ an, die dieses Rechteck haben kann.
\n$a=$[[0]] $\\qquad b=$[[1]]
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "a", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a", "maxValue": "a", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": true, "customName": "b", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b", "maxValue": "b", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "mark:\n correctif(dot(interpreted_answer[0],p) = 0); \n assert(interpreted_answer[0]<>matrix([0],[0]),sub_credit(0.5,\"Nullvektor is fad!\"))", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Gegeben ist der Vektor $\\vec{p}=\\var{p}$.
\nGeben Sie einen Normalvektor $\\vec{n}$ zu $p$ an!
\n$\\vec{n}=$[[0]]
", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "Normalvektor", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "q", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "mark:\n assert(interpreted_answer[0] = floor(interpreted_answer[0]) and\n interpreted_answer[1] = floor(interpreted_answer[1]),\n fail(\"Nur ganze Zahlen erlaubt.\"));\n correctif(interpreted_answer[0] * other + interpreted_answer[1] * prim = rest)", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Bestimmen Sie Zahlen $r,s\\in\\mathbb{Z}$, so dass $r\\cdot\\var{other}+ s\\cdot\\var{prim}=\\var{rest}$ gilt.
\n$r=$[[0]] ,$\\qquad s=$[[1]]
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