// Numbas version: exam_results_page_options
{"name": "Resuelve las divisiones de n\u00fameros complejos y exprese el resultado en la forma: $a+bi$.", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Resuelve las divisiones de n\u00fameros complejos y exprese el resultado en la forma: $a+bi$.", "tags": [], "metadata": {"description": "<p>Inverso y divisi&oacute;n de n&uacute;meros complejos.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Resuelve las divisiones de n&uacute;meros complejos y exprese el resultado en la forma: $a+bi$.</p>\n<p>Ingrese $ a $ y $ b $ como fracciones o enteros y no como decimales.</p>", "advice": "<p>La divisi&oacute;n de dos n&uacute;meros complejos se puede realizar multiplicando el numerador y el denominador por el conjugado del denominador.<br/>Suponer que \\[ z = \\frac{a+bi}{c+di},\\;\\; c+di \\neq 0\\] entonces tenemos:<br/>\\[\\begin{eqnarray*} z&amp;=&amp;\\frac{a+bi}{c+di}\\\\ &amp;=&amp;\\frac{(a+bi)(c-di)}{(c+di)(c-di)}\\\\ &amp;=&amp;\\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}\\\\ &amp;=&amp;\\frac{ac+bd}{c^2+d^2}+\\frac{bc-ad}{c^2+d^2}i \\end{eqnarray*} \\]</p>\n<p><br/>Aunque esta es una f&oacute;rmula para el inverso, la mejor manera de encontrar estos n&uacute;meros complejos es recordar multiplicar la parte superior e inferior por el conjugado del denominador.</p>\n<p>(a)</p>\n<p>\\[\\begin{eqnarray*}\\simplify[std]{{c1}/{z1}} &amp;=&amp;\\simplify[std]{({c1}*{conj(z1)})/({z1}*{conj(z1)})}\\\\ &amp;=&amp;\\simplify[std]{{c1*conj(z1)}/{abs(z1)^2}}\\\\ &amp;=&amp; \\simplify[std]{{c1*re(z1)}/{abs(z1)^2}-{c1*im(z1)}/{abs(z1)^2}*i} \\end{eqnarray*} \\]<br/>(b)<br/>\\[\\begin{eqnarray*}\\simplify[std]{{c2}/{z2}} &amp;=&amp;\\simplify[std]{({c2}*{conj(z2)})/({z2}*{conj(z2)})}\\\\ &amp;=&amp;\\simplify[std]{{c2*conj(z2)}/{abs(z2)^2}}\\\\ &amp;=&amp; \\simplify[std]{{c2*re(z2)}/{abs(z2)^2}-{c2*im(z2)}/{abs(z2)^2}*i} \\end{eqnarray*} \\]<br/>(c)<br/>\\[\\begin{eqnarray*}\\simplify[std]{{z1}/{z3}} &amp;=&amp;\\simplify[std]{({z1}*{conj(z3)})/({z3}*{conj(z3)})}\\\\ &amp;=&amp;\\simplify[std]{{z1*conj(z3)}/{abs(z3)^2}}\\\\ &amp;=&amp; \\simplify[std]{{re(z1*conj(z3))}/{abs(z3)^2}+{im(z1*conj(z3))}/{abs(z3)^2}*i} \\end{eqnarray*} \\]<br/>(d)<br/>\\[\\begin{eqnarray*}\\simplify[std]{{z3}/{z2}} &amp;=&amp;\\simplify[std]{({z3}*{conj(z2)})/({z2}*{conj(z2)})}\\\\ &amp;=&amp;\\simplify[std]{{z3*conj(z2)}/{abs(z2)^2}}\\\\ &amp;=&amp; \\simplify[std]{{re(z3*conj(z2))}/{abs(z2)^2}+{im(z3*conj(z2))}/{abs(z2)^2}*i} \\end{eqnarray*} \\]</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "extensions": [], "variables": {"s2": {"name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything"}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything"}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything"}, "z1": {"name": "z1", "group": "Ungrouped variables", "definition": "s2*random(1..9)+s1*random(1..9)*i", "description": "", "templateType": "anything"}, "s3": {"name": "s3", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "z2": {"name": "z2", "group": "Ungrouped variables", "definition": "re(z1)+s2*random(1,2)+s4*random(1..9)*i", "description": "", "templateType": "anything"}, "z3": {"name": "z3", "group": "Ungrouped variables", "definition": "rz3+s1*random(1..9)*i", "description": "", "templateType": "anything"}, "s4": {"name": "s4", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "rz3": {"name": "rz3", "group": "Ungrouped variables", "definition": "if(a3=re(z1),a3+random(1,-1),a3)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["s3", "s2", "s1", "s4", "a3", "rz3", "c2", "c1", "z1", "z2", "z3"], "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>$\\displaystyle \\simplify[std]{{c1}/{z1}}\\;=\\;$[[0]].</p>\n<p>No incluyas corchetes en tu respuesta.</p>\n<p>&nbsp;</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{c1*re(z1)}/{abs(z1)^2}-{c1*im(z1)}/{abs(z1)^2}*i", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": [".", "(", ")"], "showStrings": false, "partialCredit": 0, "message": "<p>Make sure that you input the real and imaginary parts as fractions and not as decimals. Do not include brackets in your answer.</p>"}, "valuegenerators": []}], "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>$\\displaystyle \\simplify[std]{{c2}/{z2}}\\;=\\;$[[0]].</p>\n<p>No incluyas corchetes en tu respuesta.</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{c2*re(z2)}/{abs(z2)^2}-{c2*im(z2)}/{abs(z2)^2}*i", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": [".", "(", ")"], "showStrings": false, "partialCredit": 0, "message": "<p>Make sure that you input the real and imaginary parts as fractions and not as decimals. 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