// Numbas version: exam_results_page_options
{"name": "Arithmetics of complex numbers III", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s1", "description": ""}, "rz3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a3=re(z1),a3+random(1,-1),a3)", "name": "rz3", "description": ""}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(1..9)", "name": "c1", "description": ""}, "z1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(1..9)+s1*random(1..9)*i", "name": "z1", "description": ""}, "a3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(1..9)", "name": "a3", "description": ""}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s3", "description": ""}, "z2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "re(z1)+s2*random(1,2)+s4*random(1..9)*i", "name": "z2", "description": ""}, "z3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "rz3+s1*random(1..9)*i", "name": "z3", "description": ""}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s2", "description": ""}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s4", "description": ""}, "c2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "name": "c2", "description": ""}}, "ungrouped_variables": ["s3", "s2", "s1", "s4", "a3", "rz3", "c2", "c1", "z1", "z2", "z3"], "name": "Arithmetics of complex numbers III", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Inverse and division of complex numbers. &nbsp;Four parts.</p>"}, "functions": {}, "parts": [{"prompt": "<p>$\\displaystyle \\simplify[std]{{c1}/{z1}} = $ [[0]]</p>", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "customName": "", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{c1*re(z1)}/{abs(z1)^2}-{c1*im(z1)}/{abs(z1)^2}*i", "mustmatchpattern": {"nameToCompare": "", "pattern": "`+-((integer:$n/integer:$n`?))`? + ((`+-integer:$n`?/integer:$n`?)*i `| `+-i)`?", "partialCredit": 0, "message": "Your answer is not in the form $a+bi$."}, "answerSimplification": "std", "useCustomName": false, "checkingType": "absdiff", "valuegenerators": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showPreview": true, "customName": "", "checkVariableNames": false, "unitTests": [], "scripts": {}, "vsetRangePoints": 5, "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"prompt": "<p>$\\displaystyle \\simplify[std]{{c2}/{z2}}\\;=\\;$[[0]]</p>", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "customName": "", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{c2*re(z2)}/{abs(z2)^2}-{c2*im(z2)}/{abs(z2)^2}*i", "mustmatchpattern": {"nameToCompare": "", "pattern": "`+-((integer:$n/integer:$n`?))`? + ((`+-integer:$n`?/integer:$n`?)*i `| `+-i)`?", "partialCredit": 0, "message": "Your answer is not in the form $a+bi$."}, "answerSimplification": "std", "useCustomName": false, "checkingType": "absdiff", "valuegenerators": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showPreview": true, "customName": "", "checkVariableNames": false, "unitTests": [], "scripts": {}, "vsetRangePoints": 5, "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"prompt": "<p>$\\displaystyle \\simplify[std]{{z1}/{z3}}\\;=\\;$[[0]].</p>\n      <p>Do not include brackets in your answer.</p>", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "customName": "", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{re(z1*conj(z3))}/{abs(z3)^2}+{im(z1*conj(z3))}/{abs(z3)^2}*i", "mustmatchpattern": {"nameToCompare": "", "pattern": "`+-((integer:$n/integer:$n`?))`? + ((`+-integer:$n`?/integer:$n`?)*i `| `+-i)`?", "partialCredit": 0, "message": "Your answer is not in the form $a+bi$."}, "answerSimplification": "std", "useCustomName": false, "checkingType": "absdiff", "valuegenerators": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showPreview": true, "customName": "", "checkVariableNames": false, "unitTests": [], "scripts": {}, "vsetRangePoints": 5, "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"prompt": "<p>$\\displaystyle \\simplify[std]{{z3}/{z2}}\\;=\\;$[[0]].</p>", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "customName": "", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{re(z3*conj(z2))}/{abs(z2)^2}+{im(z3*conj(z2))}/{abs(z2)^2}*i", "mustmatchpattern": {"nameToCompare": "", "pattern": "`+-((integer:$n/integer:$n`?))`? + ((`+-integer:$n`?/integer:$n`?)*i `| `+-i)`?", "partialCredit": 0, "message": "Your answer is not in the form $a+bi$."}, "answerSimplification": "std", "useCustomName": false, "checkingType": "absdiff", "valuegenerators": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showPreview": true, "customName": "", "checkVariableNames": false, "unitTests": [], "scripts": {}, "vsetRangePoints": 5, "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Express the following in the form $a+bi$.</p>\n    <p>Input $a$ and $b$ as fractions or integers and not as decimals.</p>", "tags": ["checked2015", "complex numbers", "conjugate of a complex number", "division of complex numbers", "inverse of complex numbers", "multiplication of complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus", "!collectLikeFractions"]}, "preamble": {"css": "", "js": ""}, "type": "question", "extensions": [], "advice": "\n    \n    \n    <p>Division of two complex numbers can be performed by mutiplying both the numerator and denominator by the conjugate of the denominator.<br/>Suppose that \\[ z = \\frac{a+bi}{c+di},\\;\\; c+di \\neq 0\\] then we have:<br/>\\[\\begin{eqnarray*}\n    \n    z&#38;=&#38;\\frac{a+bi}{c+di}\\\\\n    \n    &#38;=&#38;\\frac{(a+bi)(c-di)}{(c+di)(c-di)}\\\\\n    \n    &#38;=&#38;\\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}\\\\\n    \n    &#38;=&#38;\\frac{ac+bd}{c^2+d^2}+\\frac{bc-ad}{c^2+d^2}i\n    \n    \\end{eqnarray*}\n    \n    \\]<br/>Although this is a formula for the inverse, the best way to find these complex numbers is to remember to multiply top and bottom by the conjugate of the denominator.<br/>(a)<br/>\\[\\begin{eqnarray*}\\simplify[std]{{c1}/{z1}} &#38;=&#38;\\simplify[std]{({c1}*{conj(z1)})/({z1}*{conj(z1)})}\\\\\n    \n    &#38;=&#38;\\simplify[std]{{c1*conj(z1)}/{abs(z1)^2}}\\\\\n    \n    &#38;=&#38; \\simplify[std]{{c1*re(z1)}/{abs(z1)^2}-{c1*im(z1)}/{abs(z1)^2}*i}\n    \n    \\end{eqnarray*} \\]<br/>(b)<br/>\\[\\begin{eqnarray*}\\simplify[std]{{c2}/{z2}} &#38;=&#38;\\simplify[std]{({c2}*{conj(z2)})/({z2}*{conj(z2)})}\\\\\n    \n    &#38;=&#38;\\simplify[std]{{c2*conj(z2)}/{abs(z2)^2}}\\\\\n    \n    &#38;=&#38; \\simplify[std]{{c2*re(z2)}/{abs(z2)^2}-{c2*im(z2)}/{abs(z2)^2}*i}\n    \n    \\end{eqnarray*} \\]<br/>(c)<br/>\\[\\begin{eqnarray*}\\simplify[std]{{z1}/{z3}} &#38;=&#38;\\simplify[std]{({z1}*{conj(z3)})/({z3}*{conj(z3)})}\\\\\n    \n    &#38;=&#38;\\simplify[std]{{z1*conj(z3)}/{abs(z3)^2}}\\\\\n    \n    &#38;=&#38; \\simplify[std]{{re(z1*conj(z3))}/{abs(z3)^2}+{im(z1*conj(z3))}/{abs(z3)^2}*i}\n    \n    \\end{eqnarray*} \\]<br/>(d)<br/>\\[\\begin{eqnarray*}\\simplify[std]{{z3}/{z2}} &#38;=&#38;\\simplify[std]{({z3}*{conj(z2)})/({z2}*{conj(z2)})}\\\\\n    \n    &#38;=&#38;\\simplify[std]{{z3*conj(z2)}/{abs(z2)^2}}\\\\\n    \n    &#38;=&#38; \\simplify[std]{{re(z3*conj(z2))}/{abs(z2)^2}+{im(z3*conj(z2))}/{abs(z2)^2}*i}\n    \n    \\end{eqnarray*} \\]</p>\n    \n    ", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}