// Numbas version: exam_results_page_options
{"name": "Alle Beispiele (summative Fassung)", "metadata": {"description": "<p>Alle Beispiele des Tutorials in der&nbsp;summativen Fassung (man kann keine neue Fragen generieren und&nbsp;erst am Ende die L&ouml;sungen anzeigen lassen)</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": true, "showstudentname": true, "question_groups": [{"name": "Lektion 1", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", ""], "questions": [{"name": "Grundrechnungsarten (v1)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "tags": [], "metadata": {"description": "<p>In dieser Aufgabe geht es um das Dddioeren und Subtrahieren von Zufallszahlen.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>L&ouml;se die folgenden Aufgaben zu den Grundrechnungsarten!</p>", "advice": "", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(10..20)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(20..99)", "description": "", "templateType": "anything"}, "sum": {"name": "sum", "group": "Ungrouped variables", "definition": "a+b  ", "description": "", "templateType": "anything"}, "dif": {"name": "dif", "group": "Ungrouped variables", "definition": "b-a", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "sum", "dif"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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": "<p>$\\var{b}+\\var{a}=$</p>", "minValue": "sum", "maxValue": "sum", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}, {"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": "<p>$\\var{b}-\\var{a}=$</p>", "minValue": "dif", "maxValue": "dif", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Grundrechnungsarten (v2)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "tags": [], "metadata": {"description": "<p>In dieser Aufgabe geht es um das Dddioeren und Subtrahieren von Zufallszahlen.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>L&ouml;se die folgenden Aufgaben zu den Grundrechnungsarten!</p>", "advice": "", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(10..20)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(20..99)", "description": "", "templateType": "anything"}, "sum": {"name": "sum", "group": "Ungrouped variables", "definition": "a+b  ", "description": "", "templateType": "anything"}, "dif": {"name": "dif", "group": "Ungrouped variables", "definition": "b-a", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "a*random(2..9)", "description": "", "templateType": "anything"}, "ergebnisse": {"name": "ergebnisse", "group": "Ungrouped variables", "definition": "[c-a,c*a,c/a]", "description": "", "templateType": "anything"}, "operationen": {"name": "operationen", "group": "Ungrouped variables", "definition": "[\"die Differenz\",\"das Produkt\",\"den Quotienten\"]", "description": "", "templateType": "anything"}, "auswahl": {"name": "auswahl", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "", "templateType": "anything"}, "ergebnis": {"name": "ergebnis", "group": "Ungrouped variables", "definition": "ergebnisse[auswahl]", "description": "", "templateType": "anything"}, "operation": {"name": "operation", "group": "Ungrouped variables", "definition": "operationen[auswahl]", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "sum", "dif", "c", "ergebnisse", "operationen", "auswahl", "ergebnis", "operation"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{b}+\\var{a}=$[[0]]&nbsp;</p>", "gaps": [{"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, "minValue": "sum", "maxValue": "sum", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}], "sortAnswers": false}, {"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": "<p>Bestimme {operation} von $\\var{c}$ und $\\var{a}$.</p>", "minValue": "ergebnis", "maxValue": "ergebnis", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Lektion 2", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "questions": [{"name": "Zahlumwandlungen", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "tags": [], "metadata": {"description": "<p>Aufgaben zur Umwandlung von Br&uuml;chen in Dezimalzahlen und umgekehrt.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h3>Zahlumwandlungen</h3>\n<p>Schreibe den gew&ouml;hnlichen Bruch als Dezimalzahl oder umgekehrt.</p>", "advice": "<p><strong>L&ouml;sung zu a):</strong></p>\n<p>Eine M&ouml;glichkeit ist es, durch schriftliche Division das Ergebnis von $\\var{Zaehler}:\\var{nenner}$ zu berechnen.</p>\n<p>Eine andere M&ouml;glichkeit ist es, mit der Umwandlung in einen Dezimalbruch zu arbeiten. Wir m&uuml;ssen jetzt schauen, dass wir eine Zehnertpotenz finden, die ein ganzzahliges Vielfaches von $\\var{Nenner}$ ist. Dazu bestimmen wird die Primfaktorzerlegung von $\\var{Nenner}$ das ist $2^\\var{p2}\\cdot5^\\var{p5}$.</p>\n<p>Wir w&auml;hlen jetzt die Zehnerpotenz $10^\\var{max(p2,p5)}$ aus, die als Exponent das Maximum der Exponenten von $2$ und $5$ hat. Der resultierende Dezimalbruch ist also: $\\frac{\\var{Zaehler}\\cdot\\var{10^max(p2,p5)/Nenner}}{\\var{Nenner}\\cdot\\var{10^max(p2,p5)/Nenner}}=\\frac{\\var{Zaehler*10^max(p2,p5)/Nenner}}{\\var{10^max(p2,p5)}}=\\var{formatnumber(Bruch,\"eu\")}$.</p>\n<p><strong>L&ouml;sung zu b):</strong></p>\n<p>Wir ermitteln eine Zehnerpotenz, die so viele Nullen hat, wie unser Dezimalbruch Nachkommastellen, das ist $1000$. Als Dezimalbruch ergibt sich zun&auml;chst $\\frac{\\var{Zaehler2}}{\\var{Nenner2}}$. Diesen Bruch kann man noch k&uuml;rzen und erh&auml;lt dann als Ergebnis $\\var[fractionNumbers]{Dezimalbruch}$.</p>\n<p><strong>L&ouml;sung zu c):</strong></p>\n<p>Hier musst Du die schriftliche Division bis zur 4. Nachkommastelle durchf&uuml;hren und dann passend auf- oder abrunden.</p>", "rulesets": {}, "variables": {"p2": {"name": "p2", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "p5": {"name": "p5", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything"}, "Nenner": {"name": "Nenner", "group": "Ungrouped variables", "definition": "2^p2*5^p5", "description": "", "templateType": "anything"}, "Zaehler": {"name": "Zaehler", "group": "Ungrouped variables", "definition": "2*random(1..Nenner/2)-1", "description": "", "templateType": "anything"}, "Bruch": {"name": "Bruch", "group": "Ungrouped variables", "definition": "Zaehler/Nenner", "description": "", "templateType": "anything"}, "Zaehler2": {"name": "Zaehler2", "group": "Ungrouped variables", "definition": "random(41,42,43,44,46,47,49,51,52,53,54,56,57,58,59,61)*6", "description": "", "templateType": "anything"}, "Nenner2": {"name": "Nenner2", "group": "Ungrouped variables", "definition": "1000", "description": "", "templateType": "anything"}, "Dezimalbruch": {"name": "Dezimalbruch", "group": "Ungrouped variables", "definition": "Zaehler2/Nenner2", "description": "", "templateType": "anything"}, "Dezimalzahl": {"name": "Dezimalzahl", "group": "Ungrouped variables", "definition": "formatnumber(Dezimalbruch,\"plain-eu\")", "description": "", "templateType": "anything"}, "Nenner3": {"name": "Nenner3", "group": "Ungrouped variables", "definition": "random(6,7,11,9,13,15,17,19)*random(1..2)", "description": "", "templateType": "anything"}, "Zaehler3": {"name": "Zaehler3", "group": "Ungrouped variables", "definition": "2*random(2..Nenner3/2)-2", "description": "", "templateType": "anything"}, "Bruch3": {"name": "Bruch3", "group": "Ungrouped variables", "definition": "Zaehler3/Nenner3", "description": "", "templateType": "anything"}, "Bruchrund": {"name": "Bruchrund", "group": "Ungrouped variables", "definition": "precround(Bruch3,4)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["p2", "p5", "Nenner", "Zaehler", "Bruch", "Zaehler2", "Nenner2", "Dezimalbruch", "Dezimalzahl", "Nenner3", "Zaehler3", "Bruch3", "Bruchrund"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\frac{\\var{Zaehler}}{\\var{nenner}}=$[[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Dezimalzahl", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "Bruch", "maxValue": "Bruch", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>{Dezimalzahl}=[[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Dezimalbruch", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "Dezimalbruch", "maxValue": "Dezimalbruch", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": true, "mustBeReducedPC": "50", "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\frac{\\var{Zaehler3}}{\\var{nenner3}}=$[[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Gerundet", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "Bruchrund-0.0005", "maxValue": "Bruchrund+0.0005", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Lektion 3", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "questions": [{"name": "Binome & Verwandtes", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "tags": [], "metadata": {"description": "<p>Ein paar Beispiele zum Arbeiten mit mathematischen Ausdr&uuml;cken</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><strong>Binome und Verwandtes</strong></p>\n<p>Verwenden Sie <code>*</code> als Multiplikationszeichen und <code>^</code> f&uuml;r Exponenten.</p>", "advice": "<p><strong>L&ouml;sung zu a):</strong></p>\n<p>$\\left(\\var{a}\\cdot x+\\var{b}\\right)^2=\\left(\\var{a}\\cdot x+\\var{b}\\right)\\cdot\\left(\\var{a}\\cdot x+\\var{b}\\right)$</p>\n<p>$\\quad=\\var{a}\\cdot x\\cdot\\var{a}\\cdot x+\\var{a}\\cdot x\\cdot\\var{b}+\\var{b}\\cdot\\var{a}\\cdot x+\\var{b}\\cdot\\var{b}=\\var{a^2}x^2+\\var{2*a*b}\\cdot x+\\var{b^2}$</p>\n<p><strong>L&ouml;sung zu b):</strong></p>\n<p>$\\left(\\simplify{{a2}*y+{b2}}\\right)^2=\\left((\\var{a2})\\cdot y \\var{if(b2&gt;0,latex('+{b2}'),b2)}\\right)\\cdot\\left((\\var{a2})\\cdot y\\var{if(b2&gt;0,latex('+{b2}'),b2)}\\right)$</p>\n<p>$\\quad=(\\var{a2})\\cdot y\\cdot(\\var{a2})\\cdot y+(\\var{a2})\\cdot y\\cdot\\var{if(b2&gt;0,b2,latex('({b2})'))}+\\var{if(b2&gt;0,b2,latex('({b2})'))}\\cdot(\\var{a2})\\cdot y+\\var{if(b2&gt;0,b2,latex('({b2})'))}\\cdot\\var{if(b2&gt;0,b2,latex('({b2})'))}$</p>\n<p><strong>L&ouml;sung zu c):</strong></p>\n<p>$\\var{d^2}-\\var{c^2}\\cdot z^2=\\var{d}^2-\\left(\\var{c}\\cdot z\\right)^2$ wof&uuml;r wir die dritte binomische Formel $a^2-b^2=(a+b)\\cdot(a-b)$ mit $a=\\var{d},b=\\var{c}\\cdot z$ anwenden k&ouml;nnen und somit erhalten:</p>\n<p>$\\var{d}^2-\\left(\\var{c}\\cdot z\\right)^2=(\\var{d}+\\var{c}\\cdot z)\\cdot(\\var{d}-\\var{c}\\cdot z)$</p>\n<p><strong>L&ouml;sung zu d):</strong></p>\n<p>Wir ben&ouml;tigen die Summenregel $f(x)=g(x)+h(x) \\rightarrow f'(x)=g'(x)+h'(x)$ und die Potenzregel $f(x)=a\\cdot x^n\\rightarrow f'(x)=n\\cdot a\\cdot x^{n-1}$ und erhalten:</p>\n<p>$f'(x)=3\\cdot\\var[fractionNumbers]{b^2/3}x^2+2\\cdot\\var{a*b}x^2+\\var{a^2}=\\var{if(b^2=1,\"\",b^2)}x^2+\\var{2*a*b}x^2+\\var{a^2}=(\\var{if(b^2=1,\"\",b^2)} x+\\var{a})^2$</p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 30px; width: 0px; top: 62px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 15px; width: 50px; top: 47px; height: 20px; font-size: 10px; color: black;\"></span></p>", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2,4,6)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1,3,5)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2,3,4)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1,5,6)", "description": "", "templateType": "anything"}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "-a", "description": "", "templateType": "anything"}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "random(-1,-3,-5,2,4,6)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "a2", "b2", "c", "d"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Multiplizieren Sie aus!</p>\n<p>$\\left(\\var{a}\\cdot x+\\var{b}\\right)^2=$[[0]]</p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 17px; width: 0px; top: 68px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 2px; width: 50px; top: 53px; height: 20px; font-size: 10px; color: black;\"></span></p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "binom1", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a}^2x^2+2*{a}*{b}*x+{b}^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "mustmatchpattern": {"pattern": "$n*$v^2+$n*$v+$n", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Multiplizieren Sie aus!</p>\n<p>$\\left(\\simplify{{a2}* y+{b2}}\\right)^2=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "binom2", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a2}^2*y^2+2*{a2}*{b2}*y+{b2}^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "mustmatchpattern": {"pattern": "$n* $v^2+`+-$n*$v+$n", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "y", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Faktorisieren&nbsp;Sie!</p>\n<p>$\\var{d^2}-\\var{c^2}\\cdot z^2=$[[0]]</p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 53px; width: 0px; top: 65px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 38px; width: 50px; top: 50px; height: 20px; font-size: 10px; color: black;\"></span></p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "binom3", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({d}-{c}*z)*({d}+{c}*z)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "mustmatchpattern": {"pattern": "($n-$n*$v)*($n+$n*$v)", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "z", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Bestimmen Sie die erste Abletung von $f(x)=\\var[fractionNumbers]{b^2/3}x^3+\\var{a*b}x^2+\\var{a^2}\\cdot x$!</p>\n<p>$f'(x)=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "binom4", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({b}x+{a})^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Lektion 4", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", ""], "questions": [{"name": "Gewinnaufteilung", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "tags": [], "metadata": {"description": "<p>Variante der Aufgabe-Nr.&nbsp;<strong><a href=\"https://aufgabenpool.at/ahs/Typ-1/543/Gewinnaufteilung.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">1_759</a></strong>,&nbsp;ehemalige Klausuraufgabe, &ouml;sterreichischer AHS-Maturatermin: 28. Mai 2020</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><strong>Gewinnaufteilung</strong></p>\n<p>Eine Spielgemeinschaft bestehend aus 3 Spieler*innen gewinnt &euro;&nbsp;{Gewinn}. Dieser Gewinn wird wie folgt aufgeteilt: Spieler*in <em>B</em> erh&auml;lt um&nbsp;{bplus}&nbsp;% mehr als Spieler*in <em>A</em>, Spieler*in <em>C</em> erh&auml;lt&nbsp;um {cminus}&nbsp;% weniger als Spieler*in <em>B</em>.</p>\n<p>Mit <em>x</em> wird der Betrag bezeichnet, den Spieler*in <em>A</em> erh&auml;lt (<em>x</em> in &euro;).</p>", "advice": "<p>(1) \"Spieler*in <em>B</em>&nbsp;erh&auml;lt {bplus}&nbsp;% mehr als&nbsp;Spieler*in&nbsp;<em>A</em>\" bedeutet, dass&nbsp;Spieler*in <em>B</em>&nbsp;$100\\,\\%+\\var{bplus}\\,\\%=\\var{100+bplus}\\,\\%$ des Gewinnanteils von Spieler*in <em>A</em> oder das $\\var{1+bplus/100}$-fache von Spieler*in <em>A</em> erh&auml;lt. Sie erh&auml;lt also&nbsp;$\\var{1+bplus/100}\\cdot x$.</p>\n<p>(2) \"Spieler*in&nbsp;<em>C</em>&nbsp;erh&auml;lt {cminus}% weniger als&nbsp;Spieler*in&nbsp;<em>B</em>\" bedeutet, dass Spieler*in <em>C</em>&nbsp;$100\\,\\%-\\var{cminus}\\,\\%=\\var{100-cminus}\\,\\%$ des Gewinnanteils von Spieler*in&nbsp;<em>B</em> oder das $\\var{1-bplus/100}$-fache von Spieler*in&nbsp;<em>B</em> erh&auml;lt. Sie erh&auml;lt also&nbsp;$(\\var{1+bplus/100}\\cdot x)\\cdot\\var{1-cminus/100}$.</p>\n<p>Jetzt addieren wir die drei Gewinnanteile, das muss dann in Summe {Gewinn} ergeben:</p>\n<p style=\"text-align: center;\">$x+\\var{1+bplus/100}\\cdot x+(\\var{1+bplus/100}\\cdot x)\\cdot\\var{1-cminus/100}=\\var{Gewinn}$.</p>\n<p>Wer mag, kann da noch das $x$ herausheben, dann ergibt sich:</p>\n<p style=\"text-align: center;\">$\\var{1+1+bplus/100+(1+bplus/100)*(1-cminus/100)}\\cdot x=\\var{Gewinn}$</p>\n<p>bzw. aufgel&ouml;st nach $x$ dann:&nbsp;</p>\n<p style=\"text-align: center;\">$x=\\var[fractionNumbers]{Gewinn/(1+1+bplus/100+(1+bplus/100)*(1-cminus/100))}$.</p>\n<p style=\"text-align: left;\">Alle drei L&ouml;sungen werden als korrekt akzeptiert.</p>", "rulesets": {}, "variables": {"Gewinn": {"name": "Gewinn", "group": "Ungrouped variables", "definition": "random(1..5)*5000", "description": "", "templateType": "anything"}, "bplus": {"name": "bplus", "group": "Ungrouped variables", "definition": "random(5..10)*5", "description": "", "templateType": "anything"}, "cminus": {"name": "cminus", "group": "Ungrouped variables", "definition": "random(2..5)*10", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["Gewinn", "bplus", "cminus"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><strong>Aufgabenstellung:</strong></p>\n<p>Geben Sie eine Gleichung an, mit der <em>x</em> berechnet werden kann.</p>\n<p>[[0]]</p>\n<p>Verwenden Sie <code>*</code> als Multiplikationszeichen und <code>.</code> als Dezimaltrennzeichen (z.B. $\\frac{1}{2}=0.5$).</p>\n<p><sub><em>Variante der Aufgabe-Nr.&nbsp;<strong><a href=\"https://aufgabenpool.at/ahs/Typ-1/543/Gewinnaufteilung.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">1_759</a></strong>,&nbsp;ehemalige Klausuraufgabe, &ouml;sterreichischer AHS-Maturatermin: 28. Mai 2020</em></sub></p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gleichung", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "x+(1+{bplus}/100)*x+(1+{bplus}/100)*x*(1-{cminus}/100)={Gewinn}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.001", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": ["1", "1000"], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Matrizenrechnung", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "tags": [], "metadata": {"description": "<p>Zwei einfache Aufgaben zur Matrizenrechnung.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h3>Matrizenrechnung</h3>\n<p>Berechnen Sie die unten angegebenen Matrizen.</p>\n<p><em>Achten Sie bitte darauf, ggf. zun&auml;chst die korrekte Gr&ouml;&szlig;e der Ergebnis-Matrix festzulegen!</em></p>", "advice": "<p><strong>L&ouml;sung zu a)</strong></p>\n<p>Bei der transponierten Matrix werden einfach nur Zeilen und Spalten vertauscht, das Ergebnis ist demnach</p>\n<p>$\\var{B}^T=\\var{BT}$.</p>\n<p><strong>L&ouml;sung zu b):</strong></p>\n<p>Allgemein gilt: Ist $A$ eine $(n,p)$-Matrix und $B$ eine $(p,m)$-Matrix dann ist die Produktmatrix $C = A \\cdot&nbsp; B$ ist eine&nbsp;$(m,n)$-Matrix, und die Elemente $c_{ik}$ ergeben sich als <em>Skalarprodukt</em> der&nbsp;$i$-ten Zeile von $A$ mit der $k$-ten Spalte von $B$, also:<br/> $c_{ik} \\ = \\ \\sum\\limits_{r=1}^p a_{ir} b_{rk} \\qquad(i=1,\\ldots,m; \\ k=1,\\ldots,n).$</p>\n<p>Im konkreten Fall ergibt dies:&nbsp;</p>\n<p>$A$ ist eine $(3,3)$-Matrix und $B$ ist eine $(3,2)$-Matrix, also ist $C=A\\cdot B$ eine $(3,2)$-Matrix, es ergibt sich dann:</p>\n<p>$\\var{A}\\cdot\\var{B}=\\begin{pmatrix}<br/>\\var{A[0][0]}\\cdot \\var{B[0][0]}+&nbsp;\\var{A[0][1]}\\cdot \\var{B[1][0]} +&nbsp; \\var{A[0][2]}\\cdot \\var{B[2][0]} &amp;&nbsp;<br/>\\var{A[0][0]}\\cdot \\var{B[0][1]}+&nbsp;\\var{A[0][1]}\\cdot \\var{B[1][1]} +&nbsp; \\var{A[0][2]}\\cdot \\var{B[2][1]}&nbsp; \\\\<br/>\\var{A[1][0]}\\cdot \\var{B[0][0]}+&nbsp;\\var{A[1][1]}\\cdot \\var{B[1][0]} +&nbsp; \\var{A[1][2]}\\cdot \\var{B[2][0]} &amp;&nbsp;<br/>\\var{A[1][0]}\\cdot \\var{B[0][1]}+&nbsp;\\var{A[1][1]}\\cdot \\var{B[1][1]} +&nbsp; \\var{A[1][2]}\\cdot \\var{B[2][1]}&nbsp; \\\\<br/>\\var{A[2][0]}\\cdot \\var{B[0][0]}+&nbsp;\\var{A[2][1]}\\cdot \\var{B[1][0]} +&nbsp; \\var{A[2][2]}\\cdot \\var{B[2][0]} &amp;&nbsp;<br/>\\var{A[2][0]}\\cdot \\var{B[0][1]}+&nbsp;\\var{A[2][1]}\\cdot \\var{B[1][1]} +&nbsp; \\var{A[2][2]}\\cdot \\var{B[2][1]}&nbsp; \\\\<br/>\\end{pmatrix}=\\var{AxB}$</p>", "rulesets": {}, "variables": {"A": {"name": "A", "group": "Ungrouped variables", "definition": "matrix(\n  repeat(repeat(random(0..9),3), 3)\n)", "description": "", "templateType": "anything"}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "matrix(\n  repeat(repeat(random(0..9),2), 3)\n)", "description": "", "templateType": "anything"}, "AxB": {"name": "AxB", "group": "Ungrouped variables", "definition": "A*B", "description": "", "templateType": "anything"}, "BT": {"name": "BT", "group": "Ungrouped variables", "definition": "transpose(B)", "description": "", "templateType": "anything"}, "square": {"name": "square", "group": "Ungrouped variables", "definition": "[[2,2],[1,1]]", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["A", "B", "AxB", "BT", "square"], "variable_groups": [], "functions": {}, "preamble": {"js": "\n\n", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{B}^T=$ [[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "BT", "correctAnswerFractions": false, "numRows": "2", "numColumns": "3", "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": "matrix", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{A}\\cdot\\var{B}=$</p>", "correctAnswer": "AxB", "correctAnswerFractions": false, "numRows": "1", "numColumns": 1, "allowResize": true, "tolerance": 0, "markPerCell": true, "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}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Quader mit quadratischer Grundfl\u00e4che", "extensions": [], "custom_part_types": [], "resources": [["question-resources/quadrat-quader.png", "/srv/numbas/media/question-resources/quadrat-quader.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "tags": [], "metadata": {"description": "<p>Variante der Aufgabe&nbsp;<a href=\"https://aufgabenpool.at/ahs/Typ-1/271/Quader%20mit%20quadratischer%20Grundfl%C3%A4che.pdf\" target=\"_blank\" rel=\"noopener noreferrer\"><strong>1_562</strong></a> (ehemalige Klausuraufgabe, AHS-Maturatermin: 10. Mai 2017)</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h3>Quader mit quadratischer Grundfl&auml;che</h3>\n<p>Die nachstehende Abbildung zeigt einen Quader, dessen quadratische Grundfl&auml;che in der $xy$-Ebene liegt. Die L&auml;nge einer Grundkante betr&auml;gt {a}&nbsp;L&auml;ngeneinheiten, die K&ouml;rperh&ouml;he&nbsp;betr&auml;gt {a*2}&nbsp;L&auml;ngeneinheiten. Der Eckpunkt $D$ liegt im Koordinatenursprung, der Eckpunkt $C$ liegt auf der positiven <nobr>$y$-Achse.</nobr></p>\n<p>Der Eckpunkt $E$ hat somit die Koordinaten $E = \\left(\\var{a}\\mid 0\\mid\\var{2*a}\\right)$.</p>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/quadrat-quader.png\"/></p>", "advice": "<p>Man ermittelt als Koordinaten der Eckpunkte:</p>\n<p>$A=\\left(\\var{a}\\mid0\\mid0\\right),B=\\left(\\var{a}\\mid\\var{a}\\mid0\\right),H=\\left(0\\mid0\\mid \\var{2a}\\right),G=\\left(0\\mid\\var{a}\\mid\\var{2a}\\right)$</p>\n<p>Somit ergibt sich f&uuml;r den gesuchten Vektor&nbsp;{latex(vektor[auswahl])}$=\\var{loesungen[auswahl]}$</p>", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(3..10)", "description": "", "templateType": "anything"}, "vektor": {"name": "vektor", "group": "Ungrouped variables", "definition": "['\\\\overrightarrow\\{HB\\}','\\\\overrightarrow\\{BH\\}','\\\\overrightarrow\\{AG\\}','\\\\overrightarrow\\{GA\\}']", "description": "", "templateType": "anything"}, "loesungen": {"name": "loesungen", "group": "Ungrouped variables", "definition": "[vector(a,-a,2a),vector(-a,a,-2a),vector(-a,a,2a),vector(a,-a,-2a)]", "description": "", "templateType": "anything"}, "auswahl": {"name": "auswahl", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "vektor", "loesungen", "auswahl"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><strong>Aufgabenstellung:</strong></p>\n<p>Geben Sie die Koordinaten (Komponenten) des Vektors {latex(vektor[auswahl])}&nbsp;an!</p>\n<p>{latex(vektor[auswahl])}$=$[[0]]</p>\n<p><sub><em>Variante der Aufgabe-Nr. <strong><a href=\"https://aufgabenpool.at/ahs/Typ-1/271/Quader%20mit%20quadratischer%20Grundfl%C3%A4che.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">1_562</a></strong>,&nbsp;ehemalige Klausuraufgabe, &ouml;sterreichischer AHS-Maturatermin: 10. Mai 2017</em></sub></p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 54px; width: 0px; top: 132px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 39px; width: 50px; top: 117px; height: 20px; font-size: 10px; color: black;\"></span></p>", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "vektor", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "loesungen[auswahl]", "correctAnswerFractions": false, "numRows": "3", "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}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Lektion 5", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", ""], "questions": [{"name": "Median und Modus", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "tags": [], "metadata": {"description": "<p>Variante der Aufgabe-Nr.&nbsp;<strong><a href=\"https://aufgabenpool.at/ahs/Typ-1/158/Median%20und%20Modus.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">1_450</a></strong>,&nbsp;ehemalige Klausuraufgabe, &ouml;sterreichischer AHS-Maturatermin 15. J&auml;nner 2016</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h3>Median und Modus</h3>\n<p>Gegeben ist eine ungeordnete Liste von 19 nat&uuml;rlichen Zahlen:</p>\n<p>{liste}</p>", "advice": "<p>Es empfiehlt sich, die Liste zu sortieren:</p>\n<p>{sort(liste)}</p>\n<p>Man erkennt dann, dass der mittlere (zehnte) Wert die <strong>{median}</strong> ist, das ist der <strong>Median.<br/></strong>Der am h&auml;ufigsten auftretende Wert ist die <strong>{modus1}</strong>, das ist der <strong>Modus</strong>.</p>", "rulesets": {}, "variables": {"liste": {"name": "liste", "group": "Ungrouped variables", "definition": "shuffle(repeat(random(2..12)*2,6)+repeat(modus1,5)+repeat(notmodus1,4)+repeat(notmodus2,4))", "description": "", "templateType": "anything"}, "modus1": {"name": "modus1", "group": "Ungrouped variables", "definition": "2*random(2..6)-1", "description": "", "templateType": "anything"}, "modus": {"name": "modus", "group": "Ungrouped variables", "definition": "mode(liste)", "description": "", "templateType": "anything"}, "median": {"name": "median", "group": "Ungrouped variables", "definition": "median(liste)", "description": "", "templateType": "anything"}, "notmodus1": {"name": "notmodus1", "group": "Ungrouped variables", "definition": "modus1+4", "description": "", "templateType": "anything"}, "notmodus2": {"name": "notmodus2", "group": "Ungrouped variables", "definition": "modus1+8", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["liste", "modus1", "modus", "median", "notmodus1", "notmodus2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><strong>Aufgabenstellung</strong></p>\n<p>Geben Sie den Median und den Modus dieser Liste an!</p>\n<p>Median:&nbsp;[[0]]</p>\n<p>Modus:&nbsp;[[1]]</p>\n<p><sub>Variante der Aufgabe-Nr.&nbsp;<strong><a href=\"https://aufgabenpool.at/ahs/Typ-1/158/Median%20und%20Modus.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">1_450</a></strong>,&nbsp;ehemalige Klausuraufgabe, &ouml;sterreichischer AHS-Maturatermin 15. J&auml;nner 2016</sub>&nbsp;</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Median", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "Median", "maxValue": "Median", "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": "Modus", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "modus1", "maxValue": "modus1", "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}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Punkt im Viereck", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "tags": [], "metadata": {"description": "<p>Aufgabenstellung zu Koordinaten, die dem Niveau der &ouml;sterreichischen M8-Standards entspricht.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>In einem Koordinatensystem wurde ein Viereck konstruiert.</p>\n<p><em>Bitte etwas Geduld beim Laden der Konstruktion.</em></p>\n<p>{app}</p>", "advice": "<p>Durch &Uuml;berpr&uuml;fen der Lage der einzelnen Punkte:</p>\n<p>{apploesung}</p>", "rulesets": {}, "variables": {"Urliste": {"name": "Urliste", "group": "Ungrouped variables", "definition": "shuffle([1,3,5,7])+shuffle([2,4,6,8])", "description": "", "templateType": "anything"}, "xwerte": {"name": "xwerte", "group": "Ungrouped variables", "definition": "urliste[0..4]", "description": "", "templateType": "anything"}, "ywerte": {"name": "ywerte", "group": "Ungrouped variables", "definition": "urliste[4..8]", "description": "", "templateType": "anything"}, "Eckpunkte": {"name": "Eckpunkte", "group": "Ungrouped variables", "definition": "[vector(-xwerte[0],-ywerte[0]),vector(xwerte[1],-ywerte[1]),vector(xwerte[2],ywerte[2]),vector(-xwerte[3],ywerte[3])]", "description": "", "templateType": "anything"}, "defs": {"name": "defs", "group": "Ungrouped variables", "definition": "[['A',Eckpunkte[0]],['B',Eckpunkte[1]],['C',Eckpunkte[2]],['D',Eckpunkte[3]]]", "description": "", "templateType": "anything"}, "app": {"name": "app", "group": "Ungrouped variables", "definition": "geogebra_applet('https://www.geogebra.org/m/dpk5uxb8',defs)", "description": "", "templateType": "anything"}, "Distraktoren": {"name": "Distraktoren", "group": "Ungrouped variables", "definition": "[vector(floor((-xwerte[0]-xwerte[1])/2),min(-ywerte[0],-ywerte[1])),\n  vector(max(xwerte[1],xwerte[2]),floor((-ywerte[1]+ywerte[2])/2)),\n   vector(ceil((xwerte[2]-xwerte[3])/2),max(ywerte[2],ywerte[3]))]", "description": "", "templateType": "anything"}, "Loesung": {"name": "Loesung", "group": "Ungrouped variables", "definition": "vector(random(-1,1)*min(xwerte),random(-1,1)*min(ywerte))", "description": "", "templateType": "anything"}, "apploesung": {"name": "apploesung", "group": "Ungrouped variables", "definition": "geogebra_applet('https://www.geogebra.org/m/dpk5uxb8',defs2)", "description": "", "templateType": "anything"}, "defs2": {"name": "defs2", "group": "Ungrouped variables", "definition": "[['A',Eckpunkte[0]],['B',Eckpunkte[1]],['C',Eckpunkte[2]],['D',Eckpunkte[3]],\n['E',Loesung],['F',Distraktoren[0]],['G',Distraktoren[1]],['H',Distraktoren[2]]]", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["Urliste", "xwerte", "ywerte", "Eckpunkte", "defs", "app", "Distraktoren", "Loesung", "apploesung", "defs2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><strong>Welcher Punkt liegt im Inneren des Vierecks?</strong></p>\n<p>Klicke an.</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": "0", "showCellAnswerState": true, "choices": ["$(\\var{Loesung[0]}\\mid\\var{Loesung[1]})$", "$(\\var{Distraktoren[0][0]}\\mid\\var{Distraktoren[0][1]})$", "$(\\var{Distraktoren[1][0]}\\mid\\var{Distraktoren[1][1]})$", "$(\\var{Distraktoren[2][0]}\\mid\\var{Distraktoren[2][1]})$"], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Quader mit quadratischer Grundfl\u00e4che (GGB)", "extensions": ["geogebra"], "custom_part_types": [], "resources": [["question-resources/quadrat-quader.png", "/srv/numbas/media/question-resources/quadrat-quader.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "tags": [], "metadata": {"description": "<p>Variante der Aufgabe&nbsp;<a href=\"https://aufgabenpool.at/ahs/Typ-1/271/Quader%20mit%20quadratischer%20Grundfl%C3%A4che.pdf\" target=\"_blank\" rel=\"noopener noreferrer\"><strong>1_562</strong></a> (ehemalige Klausuraufgabe, AHS-Maturatermin: 10. Mai 2017)</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h3>Quader mit quadratischer Grundfl&auml;che</h3>\n<p>Die nachstehende Abbildung zeigt einen Quader, dessen quadratische Grundfl&auml;che in der $xy$-Ebene liegt. Die L&auml;nge einer Grundkante betr&auml;gt {a*multi}&nbsp;L&auml;ngeneinheiten, die K&ouml;rperh&ouml;he&nbsp;betr&auml;gt {b*multi}&nbsp;L&auml;ngeneinheiten. Der Eckpunkt $D$ liegt im Koordinatenursprung, der Eckpunkt $C$ liegt auf der positiven <nobr>$y$-Achse.</nobr></p>\n<p>Der Eckpunkt $E$ hat somit die Koordinaten $E = \\left(\\var{a*multi}\\mid 0\\mid\\var{b*multi}\\right)$.</p>\n<p style=\"text-align: center;\">{app}</p>", "advice": "<p>Man ermittelt als Koordinaten der Eckpunkte:</p>\n<p>$A=\\left(\\var{a*multi}\\mid0\\mid0\\right),B=\\left(\\var{a*multi}\\mid\\var{a*multi}\\mid0\\right),H=\\left(0\\mid0\\mid \\var{b*multi}\\right),G=\\left(0\\mid\\var{a*multi}\\mid\\var{b*multi}\\right)$</p>\n<p>Somit ergibt sich f&uuml;r den gesuchten Vektor (im Bild unten rot) {latex(vektor[auswahl])}$=\\var{loesungen[auswahl]}$</p>\n<p>{app2}</p>", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(4..8)", "description": "", "templateType": "anything"}, "vektor": {"name": "vektor", "group": "Ungrouped variables", "definition": "['\\\\overrightarrow\\{HB\\}','\\\\overrightarrow\\{BH\\}','\\\\overrightarrow\\{AG\\}','\\\\overrightarrow\\{GA\\}']", "description": "", "templateType": "anything"}, "loesungen": {"name": "loesungen", "group": "Ungrouped variables", "definition": "[vector(a*multi,-a*multi,b*multi),vector(-a*multi,a*multi,-b*multi),vector(-a*multi,a*multi,b*multi),vector(a*multi,-a*multi,-b*multi)]", "description": "", "templateType": "anything"}, "auswahl": {"name": "auswahl", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(8..12)", "description": "", "templateType": "anything"}, "defs": {"name": "defs", "group": "Ungrouped variables", "definition": "[['a',a],['b',b]]", "description": "", "templateType": "anything"}, "app": {"name": "app", "group": "Ungrouped variables", "definition": "geogebra_applet('https://www.geogebra.org/m/tmwmt8sa',defs)", "description": "", "templateType": "anything"}, "multi": {"name": "multi", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything"}, "defs2": {"name": "defs2", "group": "Ungrouped variables", "definition": "[['a',a],['b',b],['n',auswahl]]", "description": "", "templateType": "anything"}, "app2": {"name": "app2", "group": "Ungrouped variables", "definition": "geogebra_applet('https://www.geogebra.org/m/d7bbz5pz',defs2)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "vektor", "loesungen", "auswahl", "b", "defs", "app", "multi", "defs2", "app2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><strong>Aufgabenstellung:</strong></p>\n<p>Geben Sie die Koordinaten (Komponenten) des Vektors {latex(vektor[auswahl])}&nbsp;an!</p>\n<p>{latex(vektor[auswahl])}$=$[[0]]</p>\n<p><sub><em>Variante der Aufgabe-Nr. <strong><a href=\"https://aufgabenpool.at/ahs/Typ-1/271/Quader%20mit%20quadratischer%20Grundfl%C3%A4che.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">1_562</a></strong>,&nbsp;ehemalige Klausuraufgabe, &ouml;sterreichischer AHS-Maturatermin: 10. Mai 2017</em></sub></p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 54px; width: 0px; top: 132px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 39px; width: 50px; top: 117px; height: 20px; font-size: 10px; color: black;\"></span></p>", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "vektor", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "loesungen[auswahl]", "correctAnswerFractions": false, "numRows": "3", "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}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Potenzfunktionen", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "tags": [], "metadata": {"description": "<p>Variante der Aufgabe-Nr.&nbsp;<strong><a href=\"https://aufgabenpool.at/ahs/Typ-1/193/Potenzfunktionen.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">1_484</a></strong>,&nbsp;ehemalige Klausuraufgabe, &ouml;sterreichischer AHS-Maturatermin: 10. Mai 2016</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h3>Potenzfunktionen</h3>\n<p>Gegeben sind die Graphen von vier verschiedenen Potenzfunktionen $f$ mit $f(x)=a \\cdot x^z$ sowie sechs Bedingungen f&uuml;r den Parameter $a$ und den Exponenten $z$. Dabei ist $a$ eine reelle, $z$ eine nat&uuml;rliche Zahl.</p>\n<p><em>Bitte etwas Geduld beim Laden der Graphen.</em></p>", "advice": "<p>Die erste Entscheidung sollte das $z$ betreffen: Bei einer Gerade ist es $z=1$, U-f&ouml;rmiger Verlauf ist $z=2$ und S-f&ouml;rmiger Verlauf ist $z=3$</p>\n<p>F&uuml;r $z=1$ und $z=3$ kann man jetzt pr&uuml;fen: Ist der Graph streng monoton wachsend, dann ist $a&gt;0$ korrekt, ist er streng monoton fallend, ist $a&lt;0$ korrekt.</p>\n<p>F&uuml;r $z=2$ pr&uuml;ft man: Ist der Graph oben offen ($\\cup$), dann ist $a&gt;0$, ist der Graph unten offen ($\\cap$), dann ist $a&lt;0$.</p>", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "[-random(3..9)/20,+random(5..10)/25]", "description": "", "templateType": "anything"}, "z": {"name": "z", "group": "Ungrouped variables", "definition": "[1,2,3]", "description": "", "templateType": "anything"}, "ggburl": {"name": "ggburl", "group": "Ungrouped variables", "definition": "'https://www.geogebra.org/m/wvb8ker2'", "description": "", "templateType": "anything"}, "graphs": {"name": "graphs", "group": "Ungrouped variables", "definition": "[geogebra_applet(ggburl,[[\"a\",a[0]],[\"n\",z[0]]]),geogebra_applet(ggburl,[[\"a\",a[1]],[\"n\",z[0]]]),\n  geogebra_applet(ggburl,[[\"a\",a[0]],[\"n\",z[1]]]),geogebra_applet(ggburl,[[\"a\",a[1]],[\"n\",z[1]]]),\n  geogebra_applet(ggburl,[[\"a\",a[0]],[\"n\",z[2]]]),geogebra_applet(ggburl,[[\"a\",a[1]],[\"n\",z[2]]])]", "description": "", "templateType": "anything"}, "answers": {"name": "answers", "group": "Ungrouped variables", "definition": "['a<0,z=1','a>0,z=1',\n  'a<0,z=2','a>0,z=2',\n  'a<0,z=3','a>0,z=3']", "description": "", "templateType": "anything"}, "choices": {"name": "choices", "group": "Ungrouped variables", "definition": "deal(6)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "z", "ggburl", "graphs", "answers", "choices"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "useCustomName": true, "customName": "Aufgabe", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<table>\n<tbody>\n<tr>\n<td style=\"padding-left: 30px;\">\n<h3>A&nbsp;&nbsp;</h3>\n</td>\n<td>{graphs[choices[0]]}</td>\n<td></td>\n<td style=\"text-align: right;\">\n<h3>B&nbsp;&nbsp;</h3>\n</td>\n<td>{graphs[choices[1]]}</td>\n</tr>\n<tr>\n<td style=\"padding-left: 30px;\">&nbsp;</td>\n<td></td>\n<td></td>\n<td style=\"text-align: right;\"></td>\n<td></td>\n</tr>\n<tr>\n<td style=\"padding-left: 30px;\">\n<h3>C&nbsp;&nbsp;</h3>\n</td>\n<td>{graphs[choices[2]]}</td>\n<td></td>\n<td style=\"padding-left: 30px;\">\n<h3>D&nbsp;&nbsp;</h3>\n</td>\n<td>{graphs[choices[3]]}</td>\n</tr>\n</tbody>\n</table>\n<p><strong>Aufgabenstellung:</strong></p>\n<p>Kreuzen&nbsp;Sie f&uuml;r jeden der vier Graphen (A bis D) jeweils die entsprechende Bedingung f&uuml;r den Parameter $a$ und den&nbsp;Exponenten $z$ an!</p>", "minMarks": 0, "maxMarks": 0, "minAnswers": "4", "maxAnswers": "4", "shuffleChoices": false, "shuffleAnswers": true, "displayType": "radiogroup", "warningType": "prevent", "showCellAnswerState": true, "choices": ["A", "B", "C", "D"], "matrix": [["1", 0, 0, 0, 0, 0], [0, "1", 0, 0, 0, 0], [0, 0, "1", 0, 0, 0], [0, 0, 0, "1", 0, 0]], "layout": {"type": "all", "expression": ""}, "answers": ["{latex(answers[choices[0]])}", "{latex(answers[choices[1]])}", "{latex(answers[choices[2]])}", "{latex(answers[choices[3]])}", "{latex(answers[choices[4]])}", "{latex(answers[choices[5]])}"]}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><sub><em>Variante der Aufgabe-Nr.&nbsp;<strong><a href=\"https://aufgabenpool.at/ahs/Typ-1/193/Potenzfunktionen.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">1_484</a></strong>,&nbsp;ehemalige Klausuraufgabe, &ouml;sterreichischer AHS-Maturatermin: 10. Mai 2016</em></sub></p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 784px; width: 0px; top: 31px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 769px; width: 50px; top: 16px; height: 20px; font-size: 10px; color: black;\"></span></p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 27px; width: 0px; top: 28px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 12px; width: 50px; top: 13px; height: 20px; font-size: 10px; color: black;\"></span></p>"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Lektion 6", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "questions": [{"name": "Mehrdeutige L\u00f6sungen", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "tags": [], "metadata": {"description": "<p>Einige Beispiele zum Umgang mit mehrdeutigen L&ouml;sungen in NUMBAS.</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<h3>Mehrdeutige L&ouml;sungen</h3>\n<p>Die unten stehenden Aufgaben haben jeweils verschiedene L&ouml;sungen. Um diese zu pr&uuml;feen, ist es in NUMBAS erforderlich, mit einem \"Custom Marking Algorithm\" zu arbeiten.</p>", "advice": "<p><strong>L&ouml;sung zu a):</strong></p>\n<p>Man erkennt leicht,&nbsp; dass der Fl&auml;cheninhalt $\\var{n}$ ein Vielfaches von F&uuml;nf ist (letzte Stelle ist ein 5er), somt w&auml;hlt man $a=5,b=\\var{n/5}$.</p>\n<p>Es gibt andere L&ouml;sungen, z.B. $a=\\var{a},b=\\var{b}$.</p>\n<p>F&uuml;r die triviale L&ouml;sung $a=1,b=\\var{n}$ gibt es nur die halbe Punktzahl.</p>\n<p><strong>L&ouml;sung zu b):</strong></p>\n<p>Nach der \"Kipp-Regel\" findet man jedenfalls die L&ouml;sungen $\\vec{n_1}=\\var{vector(-p[1],p[0])}$ und&nbsp;$\\vec{n_2}=\\var{vector(p[1],-p[0])}$.&nbsp;</p>\n<p>Auch alle Vielfachen dieser Vektoren sind korrekte L&ouml;sungen.</p>\n<p>F&uuml;r die triviale L&ouml;sung $\\vec{n_0}=\\vec{0}=\\begin{pmatrix}0\\\\0\\end{pmatrix}$ gibt es nur die halbe Punktzahl.</p>\n<p><strong>L&ouml;sung zu c):</strong></p>\n<p>Gem&auml;&szlig; 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.</p>\n<p>Solche Zahlen findet man sicher mit dem erweiterten Euklidischen Algorithmus.</p>\n<p>F&auml;ngt 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 $r=1, s=\\var{sschlau}$.</p>\n<p>Man kann auch stumpf bis zum Ende den Algorithmus weiterrechnen, das f&uuml;hrt zur Gleichung&nbsp;$\\var{bezout[1]}\\cdot\\var{other}+(\\var{bezout[2]})\\cdot\\var{prim}=1$, die man mit $\\var{rest}$ multipliziert und somit als weitere m&ouml;gliche L&ouml;sung&nbsp;$r=\\var{rdumm}, s=\\var{sdumm}$ erh&auml;lt.</p>\n<p></p>", "rulesets": {}, "variables": {"n": {"name": "n", "group": "Aufgabenteil a)", "definition": "a*b", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Aufgabenteil a)", "definition": "random(3,7,11)^2*5", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Aufgabenteil a)", "definition": "random(7,11,13)", "description": "", "templateType": "anything"}, "p": {"name": "p", "group": "Aufgabenteil b)", "definition": "vector(shuffle([-5,-4,-3,-2,-1,1,2,3,4,5])[0..2])", "description": "", "templateType": "anything"}, "q": {"name": "q", "group": "Aufgabenteil b)", "definition": "vector(p[1],-p[0])", "description": "", "templateType": "anything"}, "prim": {"name": "prim", "group": "Aufgabenteil c)", "definition": "random(47,53,59,61,67,71,73,79,83,89)", "description": "", "templateType": "anything"}, "other": {"name": "other", "group": "Aufgabenteil c)", "definition": "-sschlau*prim+rest", "description": "", "templateType": "anything"}, "rest": {"name": "rest", "group": "Aufgabenteil c)", "definition": "random(3,5,7)", "description": "", "templateType": "anything"}, "bezout": {"name": "bezout", "group": "Aufgabenteil c)", "definition": "euclid_gcd(other,prim)", "description": "", "templateType": "anything"}, "rdumm": {"name": "rdumm", "group": "Aufgabenteil c)", "definition": "rest*bezout[1]", "description": "", "templateType": "anything"}, "sdumm": {"name": "sdumm", "group": "Aufgabenteil c)", "definition": "rest*bezout[2]", "description": "", "templateType": "anything"}, "sschlau": {"name": "sschlau", "group": "Aufgabenteil c)", "definition": "-random(2..4)", "description": "", "templateType": "anything"}, "rschlau": {"name": "rschlau", "group": "Aufgabenteil c)", "definition": "1", "description": "", "templateType": "anything"}, "ggT": {"name": "ggT", "group": "Aufgabenteil c)", "definition": "bezout[0]", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Aufgabenteil a)", "variables": ["n", "a", "b"]}, {"name": "Aufgabenteil b)", "variables": ["p", "q"]}, {"name": "Aufgabenteil c)", "variables": ["prim", "other", "rest", "bezout", "rdumm", "sdumm", "sschlau", "rschlau", "ggT"]}], "functions": {"euclid_gcd": {"parameters": [["a", "number"], ["b", "number"]], "type": "anything", "language": "javascript", "definition": "  a = +a;\n  b = +b;\n  if (a !== a || b !== b) {\n    return [NaN, NaN, NaN];\n  }\n  \n  if (a === Infinity || a === -Infinity || b === Infinity || b === -Infinity) {\n    return [Infinity, Infinity, Infinity];\n  }\n  // Checks if a or b are decimals\n  if ((a % 1 !== 0) || (b % 1 !== 0)) {\n    return false;\n  }\n  var signX = (a < 0) ? -1 : 1,\n    signY = (b < 0) ? -1 : 1,\n    x = 0,\n    y = 1,\n    u = 1,\n    v = 0,\n    q, r, m, n;\n  a = Math.abs(a);\n  b = Math.abs(b);\n\n  while (a !== 0) {\n    q = Math.floor(b / a);\n    r = b % a;\n    m = x - u * q;\n    n = y - v * q;\n    b = a;\n    a = r;\n    x = u;\n    y = v;\n    u = m;\n    v = n;\n  }\n  return [b, signX * x, signY * y];"}}, "preamble": {"js": "", "css": ""}, "parts": [{"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(\"Bitte nur ganzzahlige L\u00f6sungen!\"));\n  assert(interpreted_answer[0] > 0 and interpreted_answer[1] > 0,\n         fail(\"Seitenl\u00e4ngen d\u00fcrfen nicht negativ sein!\"));\n  correctif(interpreted_answer[0] * interpreted_answer[1] = n);\n  assert(interpreted_answer[0] > 1 and\n         interpreted_answer[1] > 1,\n         sub_credit(0.5, \"Eine Seitenl\u00e4nge 1 ist fad!\"))\n  ", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Ein Rechteck hat den Fl&auml;cheninhalt $A=\\var{n}\\,$Fl&auml;cheneinheiten.</p>\n<p>Geben Sie m&ouml;gliche ganzzahlige Seitenl&auml;ngen $a,b$ an, die dieses Rechteck haben kann.</p>\n<p>$a=$[[0]] $\\qquad b=$[[1]]</p>", "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": "<p>Gegeben ist der Vektor $\\vec{p}=\\var{p}$.</p>\n<p>Geben Sie einen Normalvektor $\\vec{n}$ zu $p$ an!</p>\n<p>$\\vec{n}=$[[0]]</p>", "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": "<p>Bestimmen Sie Zahlen $r,s\\in\\mathbb{Z}$, so dass $r\\cdot\\var{other}+&nbsp;s\\cdot\\var{prim}=\\var{rest}$ gilt.</p>\n<p>$r=$[[0]] ,$\\qquad s=$[[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "r", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "rschlau", "maxValue": "rschlau", "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": "s", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "sschlau", "maxValue": "sschlau", "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}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": false, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": false, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": false, "advicethreshold": 0, "intro": "<p>In dieser Version k&ouml;nnen Antworten erst ganz am Schluss angezeigt werden.</p>\n<p>Viel Erfolg!</p>", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "extensions": ["geogebra", "stats"], "custom_part_types": [], "resources": [["question-resources/quadrat-quader.png", "/srv/numbas/media/question-resources/quadrat-quader.png"]]}