// Numbas version: finer_feedback_settings
{"name": "Complex numbers Demo", "percentPass": "40", "duration": 0, "question_groups": [{"pickQuestions": 1, "name": "Group", "pickingStrategy": "all-ordered", "questions": [{"name": "Arithmetics of complex numbers II", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Multiplication and addition of complex numbers. Four parts.</p>"}, "parts": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "adaptiveMarkingPenalty": 0, "unitTests": [], "scripts": {}, "prompt": "<p>$\\var{e6*i}(\\simplify[std]{{a}})\\;=\\;$[[0]].</p>", "sortAnswers": false, "useCustomName": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "customName": "", "showFeedbackIcon": true, "gaps": [{"marks": 1, "answerSimplification": "std", "scripts": {}, "checkVariableNames": false, "checkingType": "absdiff", "valuegenerators": [], "failureRate": 1, "notallowed": {"showStrings": false, "partialCredit": 0, "message": "<p>Input all numbers as fractions or integers. 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Also do not include brackets in your answers.</p>", "strings": [".", ")", "("]}, "variableReplacements": [], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "customName": "", "vsetRangePoints": 5, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "unitTests": [], "answer": "({a*z4})", "extendBaseMarkingAlgorithm": true, "type": "jme", "showCorrectAnswer": true, "showPreview": true, "useCustomName": false}]}, {"variableReplacementStrategy": "originalfirst", "marks": 0, "adaptiveMarkingPenalty": 0, "unitTests": [], "scripts": {}, "prompt": "<p>$\\simplify[std,!otherNumbers]{{a}*({a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3)}\\;=\\;$[[0]].</p>", "sortAnswers": false, "useCustomName": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "customName": "", "showFeedbackIcon": true, "gaps": [{"marks": 1, "notallowed": {"showStrings": false, "partialCredit": 0, "message": "<p>Input all numbers as fractions or integers. Also do not include brackets in your answers.</p>", "strings": [".", ")", "("]}, "scripts": {}, "checkVariableNames": false, "checkingType": "absdiff", "valuegenerators": [], "failureRate": 1, "variableReplacements": [], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "customName": "", "vsetRangePoints": 5, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "unitTests": [], "answer": "{(a3 + b3 * i + c3 * i ^ 2 + d3 * i ^ 3)*a}", "extendBaseMarkingAlgorithm": true, "type": "jme", "showCorrectAnswer": true, "showPreview": true, "useCustomName": false}]}, {"variableReplacementStrategy": "originalfirst", "marks": 0, "adaptiveMarkingPenalty": 0, "unitTests": [], "scripts": {}, "prompt": "<p>$(\\simplify[std]{{a}})(\\simplify[std]{ {z1}})(\\simplify[std]{ {z3}})\\;=\\;$[[0]].</p>", "sortAnswers": false, "useCustomName": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "customName": "", "showFeedbackIcon": true, "gaps": [{"marks": 1, "notallowed": {"showStrings": false, "partialCredit": 0, "message": "<p>Input all numbers as fractions or integers. Also do not include brackets in your answers.</p>", "strings": [".", ")", "("]}, "scripts": {}, "checkVariableNames": false, "checkingType": "absdiff", "valuegenerators": [], "failureRate": 1, "variableReplacements": [], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "customName": "", "vsetRangePoints": 5, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "unitTests": [], "answer": "{a*(z1)*(z3)}", "extendBaseMarkingAlgorithm": true, "type": "jme", "showCorrectAnswer": true, "showPreview": true, "useCustomName": false}]}], "ungrouped_variables": ["a", "f6", "s3", "s2", "s1", "d3", "s6", "s5", "s4", "c3", "a3", "b3", "d6", "e6", "z4", "z1", "z2", "z3"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variables": {"a": {"group": "Ungrouped variables", "name": "a", "description": "", "definition": "s1*random(1..9)+s2*random(1..9)*i", "templateType": "anything"}, "s1": {"group": "Ungrouped variables", "name": "s1", "description": "", "definition": "random(1,-1)", "templateType": "anything"}, "s4": {"group": "Ungrouped variables", "name": "s4", "description": "", "definition": "random(1,-1)", "templateType": "anything"}, "z1": {"group": "Ungrouped variables", "name": "z1", "description": "", "definition": "s3*random(1..9)+f6*i", "templateType": "anything"}, "d3": {"group": "Ungrouped variables", "name": "d3", "description": "", "definition": "s4*random(1..9)", "templateType": "anything"}, "a3": {"group": "Ungrouped variables", "name": "a3", "description": "", "definition": "s1*random(1..9)", "templateType": "anything"}, "b3": {"group": "Ungrouped variables", "name": "b3", "description": "", "definition": "s2*random(1..9)", "templateType": "anything"}, "s5": {"group": "Ungrouped variables", "name": "s5", "description": "", "definition": "random(1,-1)", "templateType": "anything"}, "z3": {"group": "Ungrouped variables", "name": "z3", "description": "", "definition": "s6*random(1..9)+e6*i", "templateType": "anything"}, "e6": {"group": "Ungrouped variables", "name": "e6", "description": "", "definition": "s5*random(3..9)", "templateType": "anything"}, "z4": {"group": "Ungrouped variables", "name": "z4", "description": "", "definition": "s6*s2*random(1..9)+s3*s5*random(1..9)*i", "templateType": "anything"}, "f6": {"group": "Ungrouped variables", "name": "f6", "description": "", "definition": "s6*random(1..9)", "templateType": "anything"}, "d6": {"group": "Ungrouped variables", "name": "d6", "description": "", "definition": "s4*random(1..9)", "templateType": "anything"}, "c3": {"group": "Ungrouped variables", "name": "c3", "description": "", "definition": "s3*random(1..9)", "templateType": "anything"}, "s3": {"group": "Ungrouped variables", "name": "s3", "description": "", "definition": "random(1,-1)", "templateType": "anything"}, "z2": {"group": "Ungrouped variables", "name": "z2", "description": "", "definition": "s2*random(1..9)+d6*i", "templateType": "anything"}, "s2": {"group": "Ungrouped variables", "name": "s2", "description": "", "definition": "random(1,-1)", "templateType": "anything"}, "s6": {"group": "Ungrouped variables", "name": "s6", "description": "", "definition": "random(1,-1)", "templateType": "anything"}}, "variable_groups": [], "tags": ["addition of complex numbers", "algebra of complex numbers", "checked2015", "complex numbers", "multiplication of complex numbers"], "preamble": {"css": "", "js": ""}, "statement": "<p>Find the following complex numbers in the form $a+bi\\;$ where $a$ and $b$ are real.</p>\n<p>Input all numbers as fractions or integers. Also do not include brackets in your answers.</p>", "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a)</p>\n  <p>The solution is given by:</p>\n  <p><br/>$\\simplify[std]{{e6*i}}(\\simplify[std]{{a}})=\\simplify[std]{{a*e6*i}}$</p>\n  <p><br/>b)</p>\n  <p>$\\simplify[std]{{a}*{z4}={a*z4}}$</p>\n  <p><br/>c) <br/>\\[ \\begin{eqnarray*} \\simplify[std,!otherNumbers]{{a}*({a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3)}&amp;=&amp;\\simplify[std]{{a}*{a3 + b3 * i + c3 * i ^ 2 + d3 * i ^ 3}}\\\\ &amp;=&amp;\\simplify[std]{{a*(a3 + b3 * i + c3 * i ^ 2 + d3 * i ^ 3)}} \\end{eqnarray*} \\]<br/>d)</p>\n  <p>This can be calculated by using the formula twice, firstly to multiply out the first two sets of parentheses,&nbsp;</p>\n  <p>and then to multiply the result of that calculation by the third set of parentheses.</p>\n  <p>So we obtain:<br/>\\[ \\begin{eqnarray*} (\\var{a})(\\var{z1})(\\var{z3})&amp;=&amp;((\\var{a})(\\var{z1}))(\\var{z3})\\\\ &amp;=&amp;(\\var{a*(z1)})(\\var{z3})\\\\ &amp;=&amp;\\var{a*(z1)*(z3)} \\end{eqnarray*} \\]</p>", "type": "question"}, {"name": "Arithmetics of complex numbers III", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "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"], "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", 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"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", "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    "}, {"name": "Arithmetics of complex numbers IV", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "variable_groups": [], "variables": {"rz3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a3=re(z1),a3+random(1,-1),a3)", "description": "", "name": "rz3"}, "s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s1"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(1..9)", "description": "", "name": "c1"}, "z1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(1..9)+s1*random(1..9)*i", "description": "", "name": "z1"}, "a3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(1..9)", "description": "", "name": "a3"}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s3"}, "z2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "re(z1)+s2*random(1,2)+s4*random(1..9)*i", "description": "", "name": "z2"}, "z3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "rz3+s1*random(1..9)*i", "description": "", "name": "z3"}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s2"}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s4"}, "c2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "description": "", "name": "c2"}}, "ungrouped_variables": ["s3", "s2", "s1", "s4", "a3", "rz3", "c2", "c1", "z1", "z2", "z3"], "functions": {}, "parts": [{"prompt": "\n      <p>\\[\\displaystyle z=\\simplify[!collectNumbers]{({z3}*{z2})/{z1}}\\]</p>\n      <p>$z=\\;\\;$[[0]].</p>\n      ", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "customName": "", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{re(conj(z1)*z3*z2)}/{abs(z1)^2}+{im(conj(z1)*z3*z2)}/{abs(z1)^2}*i", "mustmatchpattern": {"message": "Your answer is not in the form $a+bi$.", "pattern": "`+-((integer:$n/integer:$n`?))`? + ((`+-integer:$n`?/integer:$n`?)*i `| `+-i)`?", "partialCredit": 0, "nameToCompare": ""}, "vsetRangePoints": 5, "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": {}, "answerSimplification": "std", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"prompt": "\n      <p>\\[\\displaystyle z=\\simplify[!collectNumbers]{({z2}*{z1})}(\\var{z3})^{-1}\\]</p>\n      <p>$z=\\;\\;$[[0]].</p>\n      ", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "customName": "", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{re(conj(z3)*z1*z2)}/{abs(z3)^2}+{im(conj(z3)*z1*z2)}/{abs(z3)^2}*i", "mustmatchpattern": {"message": "Your answer is not in the form $a+bi$.", "pattern": "`+-((integer:$n/integer:$n`?))`? + ((`+-integer:$n`?/integer:$n`?)*i `| `+-i)`?", "partialCredit": 0, "nameToCompare": ""}, "vsetRangePoints": 5, "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": {}, "answerSimplification": "std", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Express the following complex numbers $z$ in the form $a+bi$.</p>\n<p>Input $a$ and $b$ as fractions and not as decimals.</p>", "tags": ["algebra of complex numbers", "checked2015", "complex arithmetic", "complex numbers", "division of complex numbers", "inverse of complex numbers", "multiplication of complex numbers", "product of complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus", "!collectlikefractions"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Composite multiplication and division of complex numbers. Two parts.</p>"}, "advice": "\n  <p>a) <br/>\\[\\begin{eqnarray*}z=\\simplify[!collectNumbers]{({z3}*{z2})/{z1}} &amp;=&amp;\\simplify[!collectNumbers]{({z3}*{z2}*{conj(z1)})/({z1}*{conj(z1)})}\\\\ &amp;=&amp;\\simplify[!collectNumbers]{({z3*z2}*{conj(z1)})/({abs(z1)^2})}\\\\ &amp;=&amp;\\simplify[!collectNumbers]{{z3*z2*conj(z1)}/{abs(z1)^2}}\\\\ &amp;=&amp; \\simplify[std]{{re(z3*z2*conj(z1))}/{abs(z1)^2}+{im(z3*z2*conj(z1))}/{abs(z1)^2}*i} \\end{eqnarray*} \\]</p>\n  <p>b)<br/>\\[\\begin{eqnarray*}z= \\simplify[!collectNumbers]{({z2}*{z1})}(\\var{z3})^{-1}&nbsp;&amp;=&amp; \\simplify[!collectNumbers]{({z2}*{z1})/{z3}}\\\\ &amp;=&amp;\\simplify[!collectNumbers]{({z2}*{z1}*{conj(z3)})/({z3}*{conj(z3)})}\\\\ &amp;=&amp;\\simplify[!collectNumbers]{({z2*z1}*{conj(z3)})/({abs(z3)^2})}\\\\ &amp;=&amp;\\simplify[!collectNumbers]{{z2*z1*conj(z3)}/{abs(z3)^2}}\\\\ &amp;=&amp; \\simplify[std]{{re(z2*z1*conj(z3))}/{abs(z3)^2}+{im(z2*z1*conj(z3))}/{abs(z3)^2}*i} \\end{eqnarray*} \\]</p>\n  "}, {"name": "Modulus and argument complex numbers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s8*random(1..9)", "description": "", "name": "b3"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=3,-1,1)", "description": "", "name": "s5"}, "arg4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arg(z4),3)", "description": "", "name": "arg4"}, "b4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "b4"}, "s7": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=2,-1,t=3,1,-1)", "description": "", "name": "s7"}, "arg2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arg(z2),3)", "description": "", "name": "arg2"}, "s8": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,1,t=4,-1,t=3,1,-1)", "description": "", "name": "s8"}, "arg3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arg(z3),3)", "description": "", "name": "arg3"}, "m4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,q1,t=2,q3,t=3,q2,q4)", "description": "", "name": "m4"}, "z3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c2+d2*i", "description": "", "name": "z3"}, "z1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a1+b1*i", "description": "", "name": "z1"}, "z4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a3+b3*i", "description": "", "name": "z4"}, "ans3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z3),3)", "description": "", "name": "ans3"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "name": "a2"}, "ans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z1),3)", "description": "", "name": "ans1"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s5*random(1..9)", "description": "", "name": "b2"}, "ans4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z4),3)", "description": "", "name": "ans4"}, "z6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c4+d4*i", "description": "", "name": "z6"}, "m2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,q2,t=2,q1,t=3,q4,q2)", 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"s2"}, "q2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the second quadrant.'", "description": "", "name": "q2"}, "m3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,q3,t=2,q4,t=3,q1,q3)", "description": "", "name": "m3"}, "d2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s7*random(1..9)", "description": "", "name": "d2"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "", "name": "n"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s6*random(1..9)", "description": "", "name": "c2"}, "z5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a4+b4*i", "description": "", "name": "z5"}, "d4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s5*random(1..9)", "description": "", "name": "d4"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "f"}, "z2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a2+b2*i", "description": "", "name": "z2"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(3..9)", "description": "", "name": "b1"}, "m1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,q4,t=2,q2,t=3,q3,q1)", "description": "", "name": "m1"}, "q3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the third quadrant.'", "description": "", "name": "q3"}, "s4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,-1,t=4,-1,1)", "description": "", "name": "s4"}, "ans2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z2),3)", "description": "", "name": "ans2"}, "q4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the fourth quadrant.'", "description": "", "name": "q4"}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "name": "a3"}, "a4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s8*random(1..9)", "description": "", "name": "a4"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,1,t=2,-1,t=3,-1,1)", "description": "", "name": "s3"}, "s6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,-1,t=4,-1,1)", "description": "", "name": "s6"}, "arg1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arg(z1),3)", "description": "", "name": "arg1"}}, "ungrouped_variables": ["ans1", "ans2", "ans3", "ans4", "b4", "b1", "b2", "b3", "d4", "d2", "q1", "q3", "q2", "q4", "s3", "s2", "s1", "s7", "s6", "s5", "s4", "m4", "m1", "z2", "m3", "arg1", "z3", "arg2", "arg3", "tol", "arg4", "m2", "a1", "a3", "s8", "a4", "z4", "z5", "z6", "z1", "c2", "c4", "f", "n", "a2", "t"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": "<p>$|\\var{z1}|=\\;\\;$[[0]], $\\arg(\\var{z1})=\\;\\;$[[1]] radians</p>\n      <p>Input both answers to 3 decimal places.</p>", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans1+tol", "minValue": "ans1-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "arg1+tol", "minValue": "arg1-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>$|\\var{z2}|=\\;\\;$[[0]], $\\arg(\\var{z2})=\\;\\;$[[1]] radians</p>\n      <p>Input both answers to 3 decimal places.</p>", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans2+tol", "minValue": "ans2-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "arg2+tol", "minValue": "arg2-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>$|\\var{z3}|=\\;\\;$[[0]], $\\arg(\\var{z3})=\\;\\;$[[1]] radians</p>\n      <p>Input both answers to 3 decimal places.</p>", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans3+tol", "minValue": "ans3-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "arg3+tol", "minValue": "arg3-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>$|\\var{z4}|=\\;\\;$[[0]], $\\arg(\\var{z4})=\\;\\;$[[1]] radians</p>\n      <p>Input both answers to 3 decimal places.</p>", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans4+tol", "minValue": "ans4-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "arg4+tol", "minValue": "arg4-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "<p>Find the modulus and argument (in radians) of the following complex numbers, where the argument lies between $-\\pi$ and $\\pi$.</p>\n  <p>When calculating the argument pay particular attention to the quadrant in which the complex number lies.</p>\n  <p>Input all answers to 3 decimal places.</p>", "tags": ["arctan", "arg", "argument", "argument of complex numbers", "checked2015", "complex number", "complex numbers", "mas1602", "MAS1602", "mod", "modulus", "modulus argument form", "modulus of complex numbers", "quadrants and complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>15/7/2015:</strong></p>\n<p>Added tags.</p>\n<p><strong>5/07/2012:</strong></p>\n<p>Added tags.</p>\n<p>Changed some of the grammar in the advice section.</p>\n<p>Question appears to be working correctly.</p>\n<p>The presentation in IE on using Test Run is not good.</p>\n<p><strong>9/07/2012:</strong></p>\n<p>Display in Advice set out properly.</p>\n<p><strong>13/07/2009:</strong></p>\n<p>Set new tolerance variable tol=0.001 for all numeric input.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p><span style=\"color: #222222; font-family: sans-serif;\"><span style=\"line-height: 20px;\">Finding&nbsp;</span></span>the modulus and argument (in radians) of four complex numbers; the arguments between $-\\pi$ and $\\pi$ and careful with quadrants!</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>Note that the arguments $\\theta$ of the complex numbers are in radians and have to be in the range $-\\pi &lt; \\theta \\le \\pi$.</p>\n    <p>You have to be careful with using a standard calculator when you are finding the argument of a complex number.</p>\n    <p>If $z=a+bi=r(\\cos(\\theta)+i\\sin(\\theta))$ then we have:$r\\cos(\\theta)=a,\\;\\;r\\sin(\\theta)=b$ and so $\\tan(\\theta) = b/a$.</p>\n    <p>Using a calculator to find the argument via $\\arctan(b/a)$ works in the range $-\\pi &lt; \\theta \\le \\pi$ when the complex number is in the first or fourth quadrants &ndash; you get the correct value.</p>\n    <p>However, The calculator gives the wrong value for complex numbers in the other quadrants.</p>\n    <p><strong>Complex number in the Second Quadrant.</strong></p>\n    <p>Since $\\arctan(b/a)$ does not distinguish between the second and fourth quadrants and the calculator gives the argument for the fourth quadrant you have to add $\\pi$ onto the calculator value.</p>\n    <p><strong>Complex number in the Third Quadrant.</strong></p>\n    <p>Since $\\arctan(b/a)$ does not distinguish between the first and third quadrants and the calculator gives the argument for the first quadrant you have to take away $\\pi$ from the calculator value.</p>\n    <p><strong>a)Modulus.</strong></p>\n    <p>\\[ \\begin{eqnarray*} |\\var{z1}|&amp;=&amp;\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &amp;=&amp; \\var{abs(z1)}\\\\ &amp;=&amp;\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.</p>\n    <p><strong>Argument.</strong></p>\n    <p><strong>{m1}</strong></p>\n    <p>Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z1}) &amp;=&amp; \\var{arg(z1)}\\\\ &amp;=&amp; \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n    <p><strong>b)Modulus.</strong></p>\n    <p>\\[ \\begin{eqnarray*} |\\var{z2}|&amp;=&amp;\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &amp;=&amp; \\var{abs(z2)}\\\\ &amp;=&amp;\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.</p>\n    <p><strong>Argument.</strong></p>\n    <p><strong>{m2}</strong></p>\n    <p>Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z2}) &amp;=&amp; \\var{arg(z2)}\\\\ &amp;=&amp; \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n    <p><strong>c)Modulus.</strong></p>\n    <p>\\[ \\begin{eqnarray*} |\\var{z3}|&amp;=&amp;\\sqrt{(\\var{c2})^2+(\\var{d2})^2}\\\\ &amp;=&amp; \\var{abs(z3)}\\\\ &amp;=&amp;\\var{ans3} \\end{eqnarray*} \\] to 3 decimal places.</p>\n    <p><strong>Argument.</strong></p>\n    <p><strong>{m3}</strong></p>\n    <p>Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z3}) &amp;=&amp; \\var{arg(z3)}\\\\ &amp;=&amp; \\var{arg3}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n    <p><strong>d)Modulus.</strong></p>\n    <p>\\[ \\begin{eqnarray*} |\\var{z4}|&amp;=&amp;\\sqrt{(\\var{a3})^2+(\\var{b3})^2}\\\\ &amp;=&amp; \\var{abs(z4)}\\\\ &amp;=&amp;\\var{ans4} \\end{eqnarray*} \\] to 3 decimal places.</p>\n    <p><strong>Argument.</strong></p>\n    <p><strong>{m4}</strong></p>\n    <p>Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z4}) &amp;=&amp; \\var{arg(z4)}\\\\ &amp;=&amp; \\var{arg4}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>"}, {"name": "De Moivre's Theorem: Positive Powers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, 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"showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans1+tol", "minValue": "ans1-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "arg1+tol", "minValue": "arg1-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"answer": "{a3}+{b3}*i", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>Find the modulus and argument of $\\var{z1}$ to 3 decimal places.</p>\n      <p>(i) $|\\var{z1}|\\;=\\;$ [[0]], to 3 decimal places.</p>\n      <p>(ii) 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"marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>Find the modulus and argument of $\\var{z2}$ to 3 decimal places.</p>\n      <p>(i) $|\\var{z2}|\\;=\\;$ [[0]], to 3 decimal places.</p>\n      <p>(ii) $\\arg(\\var{z2})\\;=\\;$[[1]] radians, to 3 decimal places.</p>\n      <p>Hence find:</p>\n      <p>(iii) $(\\var{z2})^{\\var{n4}}\\;=\\;$[[2]]</p>\n      <p>Input as a complex number, with real and imaginary parts integral values.</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Use de Moivre's theorem to write the following complex numbers in the form $a+bi$.</p>\n  <p>Note that for these questions, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.</p>\n  <p>Important: When calculating the final answer in part (iii) of each question, you must use non-truncated values for the modulus and argument calculated in parts (i) and (ii) and not the approximated values, otherwise the final answer may not be correct.</p>", "tags": ["arctan", "argument of a complex number", "argument of complex number", "argument of complex numbers", "checked2015", "complex numbers", "de Moivre's theorem", "de Moivre's Theorem", "de moivre's theorem", "MAS1602", "modulus of complex numbers", "quadrants", "quadrants in the complex plane"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>15/7/2015:</strong></p>\n<p>Added tags.</p>\n<p><strong>5/07/2012:</strong></p>\n<p>Added tags.</p>\n<p>The question doesn't really make sense. In the instruction we are asked to find modulus and argument of (a+i*b)^n but the question that is displayed is to find the modulus and argument of (a+i*b). Does the question need to be rewrittten to avoid this conflicting instruction?</p>\n<p><strong>9/07/2012:</strong></p>\n<p>Changed prompt instructions to make this question clearer.</p>\n<p>Corrected request from 2dps to 3 dps for last question.</p>\n<p>Also set new tolerance variable, tol=0.001 for all numeric answers.</p>\n<p><strong>13/07/2012:</strong></p>\n<p><strong><br /></strong>Not a good question as can be done without using de Moivre. Needs to be recast.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find modulus and argument of two complex numbers. Then use De Moivre's Theorem to find positive powers of the complex numbers.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>Given a complex number $z=r(\\cos(\\theta)+i\\sin(\\theta))$ de Moivre's theorem states that $z^n=r^n(\\cos(n\\theta)+i\\sin(n\\theta))$ for an integer power $n$.<br/>So if we know the modulus $r$ and the argument $\\theta$ for $z$ then the theorem provides a way of calculating $z^n$.</p>\n  <p>As usual, you must be careful that the argument is calculated correctly by paying attention to the quadrant of the complex plane in which the complex number lies.</p>\n  <p>Also remember that for this question, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.</p>\n  <p>With the above in mind we can now answer the questions:</p>\n  <p>a)</p>\n  <h4>Modulus</h4>\n  <p>\\[ \\begin{eqnarray*} |\\var{z1}|&amp;=&amp;\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &amp;=&amp; \\var{abs(z1)}\\\\ &amp;=&amp;\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.</p>\n  <p>Note that $r^{\\var{n2}}=|(\\var{z1})^{\\var{n2}}| =\\var{abs(z1)}^{\\var{n2}}=\\var{abs(z1)^n2}$ which we will use in the calculation for $(\\var{z1})^{\\var{n2}}$</p>\n  <h4>Argument</h4>\n  <p>{m1}.<br/>Hence we see that: <br/>\\[\\begin{eqnarray*} \\arg(\\var{z1}) &amp;=&amp; \\var{arg(z1)}\\\\ &amp;=&amp; \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n  <p>We have $\\arg((\\var{z1})^{\\var{n2}})=\\var{n2}\\times \\var{arg(z1)} = \\var{n2*arg(z1)}$ radians.</p>\n  <p>Hence we have \\[\\begin{eqnarray*}(\\var{z1})^{\\var{n2}} &amp;=&amp; \\var{abs(z1)^n2}(\\cos(\\var{n2*arg(z1)})+\\sin(\\var{n2*arg(z1)})i)\\\\ &amp;=&amp; \\var{abs(z1)^n2}\\cos(\\var{n2*arg(z1)})+\\var{abs(z1)^n2}\\times\\sin(\\var{n2*arg(z1)})i\\\\ &amp;=&amp; \\simplify[std]{{a3}+{b3}i}. \\end{eqnarray*} \\]&nbsp;</p>\n  <p>b)</p>\n  <h4>Modulus</h4>\n  <p>\\[ \\begin{eqnarray*} |\\var{z2}|&amp;=&amp;\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &amp;=&amp; \\var{abs(z2)}\\\\ &amp;=&amp;\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.</p>\n  <p>Note that $r^{\\var{n4}}=|(\\var{z2})^{\\var{n4}}| =\\var{abs(z2)}^{\\var{n4}}=\\var{abs(z2)^n4}$ which we will use in the calculation for $(\\var{z2})^{\\var{n4}}$</p>\n  <h4>Argument</h4>\n  <p>{m2}.<br/>Hence we see that: <br/>\\[\\begin{eqnarray*} \\arg(\\var{z2}) &amp;=&amp; \\var{arg(z2)}\\\\ &amp;=&amp; \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n  <p>We have $\\arg((\\var{z2})^{\\var{n4}})=\\var{n4}\\times \\var{arg(z2)} = \\var{n4*arg(z2)}$ radians.</p>\n  <p>Hence we have \\[\\begin{eqnarray*}(\\var{z2})^{\\var{n4}} &amp;=&amp; \\var{abs(z2)^n4}(\\cos(\\var{n4*arg(z2)})+\\sin(\\var{n4*arg(z2)})i)\\\\ &amp;=&amp; \\var{abs(z2)^n4}\\cos(\\var{n4*arg(z2)})+\\var{abs(z2)^n4}\\times\\sin(\\var{n4*arg(z2)})i\\\\ &amp;=&amp; \\simplify[std]{{a4}+{b4}i}. \\end{eqnarray*} \\]&nbsp;</p>\n  <p>&nbsp;</p>"}, {"name": "De Moivre's Theorem: Negative 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"vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n<p>Find the modulus and argument of $\\var{z1}$ to 3 decimal places.</p>\n  <p>(i) $|\\var{z1}|\\;=\\;$ [[0]], to 3 decimal places.</p>\n  <p>(ii) $\\arg(\\var{z1})\\;=\\;$[[1]] radians, to 3 decimal places.</p>\n  <p>Hence find:</p>\n  <p>(iii) $(\\var{z1})^{\\var{n2}}\\;=\\;$[[2]]</p>\n  <p>Input as a complex number, with real and imaginary parts to 3 decimal places.</p>\n", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans2+tol", "minValue": "ans2-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "arg2+tol", "minValue": "arg2-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"answer": "{a4}+{b4}*i", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n<p>Find the modulus and argument of $\\var{z2}$ to 3 decimal places.</p>\n  <p>(i) $|\\var{z2}|\\;=\\;$ [[0]], to 3 decimal places.</p>\n  <p>(ii) $\\arg(\\var{z2})\\;=\\;$[[1]] radians, to 3 decimal places.</p>\n  <p>Hence find:</p>\n  <p>(iii) $(\\var{z2})^{\\var{n4}}\\;=\\;$[[2]]</p>\n  <p>Input as a complex number, with real and imaginary parts to 3 decimal places.</p>\n", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Use de Moivre's theorem to write the following complex numbers in the form $a+bi$.</p>\n<p>Note that for these questions, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.</p>\n<p>Important: When calculating the final answer in part (iii) of each question, you must use non-truncated values for the modulus and argument calculated in parts (i) and (ii) and not the approximated values, otherwise the final answer will not be correct to three decimal places.</p>", "tags": ["arctan", "argument of a complex number", "argument of complex number", "argument of complex numbers", "checked2015", "complex numbers", "de Moivre's theorem", "de Moivre's Theorem", "de moivre's theorem", "MAS1602", "modulus of complex numbers", "quadrants", "quadrants in the complex plane"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>15/7/2015:</strong></p>\n<p>Added tags.</p>\n<p><strong>26/11/2013</strong></p>\n<p>Turn off fractionNumbers in the answer to part a) 2.</p>\n<p><strong>5/07/2012:</strong></p>\n<p>Added tags.</p>\n<p>The question doesn't really make sense. In the instruction we are asked to find modulus and argument of (a+i*b)^n but the question that is displayed is to find the modulus and argument of (a+i*b). Does the question need to be rewrittten to avoid this conflicting instruction?</p>\n<p><strong>9/07/2012:</strong></p>\n<p>Changed prompt instructions to make this question clearer.</p>\n<p>Corrected request from 2dps to 3 dps for last question.</p>\n<p>Also set new tolerance variable, tol=0.001 for all numeric answers.</p>\n<p><strong>13/07/2012:</strong></p>\n<p><strong><br /></strong>Not a good question as can be done without using de Moivre. Needs to be recast.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find modulus and argument of two complex numbers. Then use De Moivre's Theorem to find negative powers of the complex numbers.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>Given a complex number $z=r(\\cos(\\theta)+i\\sin(\\theta))$ de Moivre's theorem states that $z^n=r^n(\\cos(n\\theta)+i\\sin(n\\theta))$ for an integer power $n$.<br/>So if we know the modulus $r$ and the argument $\\theta$ for $z$ then the theorem provides a way of calculating $z^n$.</p>\n  <p>As usual, you must be careful that the argument is calculated correctly, by paying attention to the quadrant of the complex plane in which lies.</p>\n  <p>Also remember that for this question, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.</p>\n  <p>With the above in mind we can now answer the questions:</p>\n  <p>a)</p>\n  <h4>Modulus</h4>\n  <p>\\[ \\begin{eqnarray*} |\\var{z1}|&amp;=&amp;\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &amp;=&amp; \\var{abs(z1)}\\\\ &amp;=&amp;\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.</p>\n  <p>Note that $r^{\\var{n2}}=|(\\var{z1})^{\\var{n2}}| =\\var{abs(z1)}^{\\var{n2}}=\\var{abs(z1)^n2}$ which we will use in the calculation for $(\\var{z1})^{\\var{n2}}$</p>\n  <h4>Argument</h4>\n  <p>{m1}.<br/>Hence we see that: <br/>\\[\\begin{eqnarray*} \\arg(\\var{z1}) &amp;=&amp; \\var{arg(z1)}\\\\ &amp;=&amp; \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n  <p>We have $\\arg((\\var{z1})^{\\var{n2}})=\\var{n2}\\times \\var{arg(z1)} = \\var{n2*arg(z1)}$ radians.</p>\n  <p>Hence we have \\[\\begin{eqnarray*}(\\var{z1})^{\\var{n2}} &amp;=&amp; \\var{abs(z1)^n2}(\\cos(\\var{n2*arg(z1)})+\\sin(\\var{n2*arg(z1)})i)\\\\ &amp;=&amp; \\var{abs(z1)^n2}\\cos(\\var{n2*arg(z1)})+\\var{abs(z1)^n2}\\times\\sin(\\var{n2*arg(z1)})i\\\\ &amp;=&amp; \\simplify[std]{{a3}+{b3}i} \\end{eqnarray*} \\] to 3 decimal places for real and imaginary parts.</p>\n  <p>b)</p>\n  <h4>Modulus</h4>\n  <p>\\[ \\begin{eqnarray*} |\\var{z2}|&amp;=&amp;\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &amp;=&amp; \\var{abs(z2)}\\\\ &amp;=&amp;\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.</p>\n  <p>Note that $r^{\\var{n4}}=|(\\var{z2})^{\\var{n4}}| =\\var{abs(z2)}^{\\var{n4}}=\\var{precround(abs(z2)^n4,6)}$ which we will use in the calculation for $(\\var{z2})^{\\var{n4}}$</p>\n  <h4>Argument</h4>\n  <p>{m2}.<br/>Hence we see that: <br/>\\[\\begin{eqnarray*} \\arg(\\var{z2}) &amp;=&amp; \\var{arg(z2)}\\\\ &amp;=&amp; \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n  <p>We have $\\arg((\\var{z2})^{\\var{n4}})=\\var{n4}\\times \\var{arg(z2)} = \\var{n4*arg(z2)}$ radians.</p>\n  <p>Hence we have \\[\\begin{eqnarray*}(\\var{z2})^{\\var{n4}} &amp;=&amp; \\var{precround(abs(z2)^n4,6)}(\\cos(\\var{n4*arg(z2)})+\\sin(\\var{n4*arg(z2)})i)\\\\ &amp;=&amp; \\var{precround(abs(z2)^n4,6)}\\cos(\\var{n4*arg(z2)})+\\var{precround(abs(z2)^n4,6)}\\times\\sin(\\var{n4*arg(z2)})i\\\\ &amp;=&amp; \\simplify[std]{{a4}+{b4}i} \\end{eqnarray*} \\] to 3 decimal places for real and imaginary parts.</p>\n  <p>&nbsp;</p>"}]}], "showQuestionGroupNames": false, "navigation": {"browse": true, "showresultspage": "oncompletion", "reverse": true, "preventleave": true, "allowregen": false, "onleave": {"action": "warnifunattempted", "message": "<p>You have not attemoted this question!</p>"}, "showfrontpage": true}, "metadata": {"description": "<p>Content: complex arithmetic; argument and modulus of complex numbers; de Moivre's theorem.</p>\n<p>This homework counts 10% towards your final Engineering Mathematics mark.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "feedback": {"intro": "", "showanswerstate": true, "showtotalmark": true, "showactualmark": true, "advicethreshold": 0, "allowrevealanswer": true, "feedbackmessages": [], "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "showstudentname": true, "timing": {"timedwarning": {"action": "none", "message": ""}, "timeout": {"action": "none", "message": ""}, "allowPause": true}, "type": "exam", "contributors": [{"name": "seb SEE", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1836/"}], "extensions": [], "custom_part_types": [], "resources": []}