![](\"resources/question-resources/quadrat-quader.png\"/)
Ein Probe-Test für dieses Tutorial.
", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "duration": 0, "percentPass": "50", "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", ""], "questions": [{"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": "In dieser Aufgabe geht es um das Dddioeren und Subtrahieren von Zufallszahlen.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Löse die folgenden Aufgaben zu den Grundrechnungsarten!
", "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": "$\\var{b}+\\var{a}=$[[0]]
", "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": "Bestimme {operation} von $\\var{c}$ und $\\var{a}$.
", "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": "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": "Zwei einfache Aufgaben zur Matrizenrechnung.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Berechnen Sie die unten angegebenen Matrizen.
\nAchten Sie bitte darauf, ggf. zunächst die korrekte Größe der Ergebnis-Matrix festzulegen!
", "advice": "Lösung zu a)
\nBei der transponierten Matrix werden einfach nur Zeilen und Spalten vertauscht, das Ergebnis ist demnach
\n$\\var{B}^T=\\var{BT}$.
\nLösung zu b):
\nAllgemein gilt: Ist $A$ eine $(n,p)$-Matrix und $B$ eine $(p,m)$-Matrix dann ist die Produktmatrix $C = A \\cdot B$ ist eine $(m,n)$-Matrix, und die Elemente $c_{ik}$ ergeben sich als Skalarprodukt der $i$-ten Zeile von $A$ mit der $k$-ten Spalte von $B$, also:
$c_{ik} \\ = \\ \\sum\\limits_{r=1}^p a_{ir} b_{rk} \\qquad(i=1,\\ldots,m; \\ k=1,\\ldots,n).$
Im konkreten Fall ergibt dies:
\n$A$ 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:
\n$\\var{A}\\cdot\\var{B}=\\begin{pmatrix}
\\var{A[0][0]}\\cdot \\var{B[0][0]}+ \\var{A[0][1]}\\cdot \\var{B[1][0]} + \\var{A[0][2]}\\cdot \\var{B[2][0]} &
\\var{A[0][0]}\\cdot \\var{B[0][1]}+ \\var{A[0][1]}\\cdot \\var{B[1][1]} + \\var{A[0][2]}\\cdot \\var{B[2][1]} \\\\
\\var{A[1][0]}\\cdot \\var{B[0][0]}+ \\var{A[1][1]}\\cdot \\var{B[1][0]} + \\var{A[1][2]}\\cdot \\var{B[2][0]} &
\\var{A[1][0]}\\cdot \\var{B[0][1]}+ \\var{A[1][1]}\\cdot \\var{B[1][1]} + \\var{A[1][2]}\\cdot \\var{B[2][1]} \\\\
\\var{A[2][0]}\\cdot \\var{B[0][0]}+ \\var{A[2][1]}\\cdot \\var{B[1][0]} + \\var{A[2][2]}\\cdot \\var{B[2][0]} &
\\var{A[2][0]}\\cdot \\var{B[0][1]}+ \\var{A[2][1]}\\cdot \\var{B[1][1]} + \\var{A[2][2]}\\cdot \\var{B[2][1]} \\\\
\\end{pmatrix}=\\var{AxB}$
$\\var{B}^T=$ [[0]]
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Die nachstehende Abbildung zeigt einen Quader, dessen quadratische Grundfläche in der $xy$-Ebene liegt. Die Länge einer Grundkante beträgt {a} Längeneinheiten, die Körperhöhe beträgt {a*2} Längeneinheiten. Der Eckpunkt $D$ liegt im Koordinatenursprung, der Eckpunkt $C$ liegt auf der positiven
Der Eckpunkt $E$ hat somit die Koordinaten $E = \\left(\\var{a}\\mid 0\\mid\\var{2*a}\\right)$.
\n
Man ermittelt als Koordinaten der Eckpunkte:
\n$A=\\left(\\var{a}\\mid0\\mid0\\right),B=\\left(\\var{a}\\mid\\var{a}\\mid0\\right),H=\\left(0\\mid0\\mid \\var{2a}\\right),G=\\left(0\\mid\\var{a}\\mid\\var{2a}\\right)$
\nSomit ergibt sich für den gesuchten Vektor {latex(vektor[auswahl])}$=\\var{loesungen[auswahl]}$
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\nGeben Sie die Koordinaten (Komponenten) des Vektors {latex(vektor[auswahl])} an!
\n{latex(vektor[auswahl])}$=$[[0]]
\nVariante der Aufgabe-Nr. 1_562, ehemalige Klausuraufgabe, österreichischer AHS-Maturatermin: 10. Mai 2017
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Variante der Aufgabe-Nr. 1_484, ehemalige Klausuraufgabe, österreichischer AHS-Maturatermin: 10. Mai 2016
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Gegeben sind die Graphen von vier verschiedenen Potenzfunktionen $f$ mit $f(x)=a \\cdot x^z$ sowie sechs Bedingungen für den Parameter $a$ und den Exponenten $z$. Dabei ist $a$ eine reelle, $z$ eine natürliche Zahl.
\nBitte etwas Geduld beim Laden der Graphen.
", "advice": "Die erste Entscheidung sollte das $z$ betreffen: Bei einer Gerade ist es $z=1$, U-förmiger Verlauf ist $z=2$ und S-förmiger Verlauf ist $z=3$
\nFür $z=1$ und $z=3$ kann man jetzt prüfen: Ist der Graph streng monoton wachsend, dann ist $a>0$ korrekt, ist er streng monoton fallend, ist $a<0$ korrekt.
\nFür $z=2$ prüft man: Ist der Graph oben offen ($\\cup$), dann ist $a>0$, ist der Graph unten offen ($\\cap$), dann ist $a<0$.
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| \n | \n | \n | \n | \n |
\nC\n | \n{graphs[choices[2]]} | \n\n | \nD\n | \n{graphs[choices[3]]} | \n
Aufgabenstellung:
\nKreuzen Sie für jeden der vier Graphen (A bis D) jeweils die entsprechende Bedingung für den Parameter $a$ und den Exponenten $z$ an!
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\n\n
"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"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": "
Aufgaben zur Umwandlung von Brüchen in Dezimalzahlen und umgekehrt.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Schreibe den gewöhnlichen Bruch als Dezimalzahl oder umgekehrt.
", "advice": "Lösung zu a):
\nEine Möglichkeit ist es, durch schriftliche Division das Ergebnis von $\\var{Zaehler}:\\var{nenner}$ zu berechnen.
\nEine andere Möglichkeit ist es, mit der Umwandlung in einen Dezimalbruch zu arbeiten. Wir müssen 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 $2^\\var{p2}\\cdot5^\\var{p5}$.
\nWir wählen 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: $\\frac{\\var{Zaehler}\\cdot\\var{10^max(p2,p5)/Nenner}}{\\var{Nenner}\\cdot\\var{10^max(p2,p5)/Nenner}}=\\frac{\\var{Zaehler*10^max(p2,p5)/Nenner}}{\\var{10^max(p2,p5)}}=\\var{formatnumber(Bruch,\"eu\")}$.
\nLösung zu b):
\nWir ermitteln eine Zehnerpotenz, die so viele Nullen hat, wie unser Dezimalbruch Nachkommastellen, das ist $1000$. Als Dezimalbruch ergibt sich zunächst $\\frac{\\var{Zaehler2}}{\\var{Nenner2}}$. Diesen Bruch kann man noch kürzen und erhält dann als Ergebnis $\\var[fractionNumbers]{Dezimalbruch}$.
\nLösung zu c):
\nHier musst Du die schriftliche Division bis zur 4. Nachkommastelle durchführen und dann passend auf- oder abrunden.
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