// Numbas version: finer_feedback_settings
{"name": "Complex numbers", "duration": 0, "metadata": {"notes": "", "description": "<p>Questions about&nbsp;complex arithmetic; argument and modulus of complex numbers; complex roots of polynomials; de Moivre's theorem.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "allQuestions": true, "shuffleQuestions": false, "percentPass": 0, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "pickQuestions": 0, "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "showfrontpage": true}, "feedback": {"showtotalmark": true, "advicethreshold": 0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "type": "exam", "questions": [], "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Arithmetics of complex numbers I", "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": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "d3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "name": "d3"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "name": "b3"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..9)+s4*random(1..9)*i", "description": "", "name": "b"}, "s4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s4"}, "z2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)+d6*i", "description": "", "name": "z2"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)+ s4*random(1..9)*i", "description": "", "name": "a1"}, "e6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..4)", "description": "", "name": "e6"}, "s6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s6"}, "z3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s6*random(1..9)+e6*i", "description": "", "name": "z3"}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "a3"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s3"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "name": "c3"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)+s2*random(1..9)*i", "description": "", "name": "a"}, "z1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..9)+f6*i", "description": "", "name": "z1"}, "f6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(1..4)", "description": "", "name": "f6"}, "d6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..4)", "description": "", "name": "d6"}}, "ungrouped_variables": ["a", "b", "f6", "s3", "s2", "s1", "d3", "s6", "s4", "a1", "c3", "a3", "b3", "d6", "e6", "z1", "z2", "z3"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": "<p>$(\\simplify[std]{{a}})(\\simplify[std]{{b}})\\;=\\;$[[0]].</p>\n      <p>&nbsp;</p>\n      <p>&nbsp;</p>", "scripts": {}, "gaps": [{"answer": "{a*b}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "notallowed": {"message": "<p><span class=\"rwRO\" style=\"color: #3f52b8; font-family: Tahoma, 'Sans Serif', Arial; font-size: 11px; background-color: #fdf2c4;\"><span class=\"sq\" style=\"color: #9a9a9a;\"><a style=\"font-family: Verdana, Arial, Helvetica, sans-serif; font-size: medium; background-color: #ffffff;\" href=\"../../32/complex-numbers/\">D</a><span style=\"color: #000000; font-family: Verdana, Arial, Helvetica, sans-serif; font-size: medium; background-color: #ffffff;\">o not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.</span> </span></span></p>", "showStrings": false, "partialCredit": 0, "strings": [".", "(", ")"]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>$(\\simplify[std]{{a1}})^2\\;=\\;$[[0]].</p>", "scripts": {}, "gaps": [{"answer": "({a1^2})", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "notallowed": {"message": "<p><a href=\"../../32/complex-numbers/\">D</a>o not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.</p>", "showStrings": false, "partialCredit": 0, "strings": [".", ")", "("]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>$\\simplify[std,!otherNumbers]{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}\\;=\\;$[[0]].</p>", "scripts": {}, "gaps": [{"answer": "{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "notallowed": {"message": "<p><a href=\"../../32/complex-numbers/\">D</a>o not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.</p>", "showStrings": false, "partialCredit": 0, "strings": [".", ")", "("]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>$(\\simplify[std]{{z1}}) (\\simplify[std]{{z2}}) (\\simplify[std]{{z3}})\\;=\\;$[[0]].</p>", "scripts": {}, "gaps": [{"answer": "{z1*z2*z3}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "notallowed": {"message": "\n            <p><a href=\"../../32/complex-numbers/\">D</a>o not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.</p>\n            <p>&nbsp;</p>\n            ", "showStrings": false, "partialCredit": 0, "strings": [".", ")", "("]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "<p>Express the following in the form $a+bi\\;$ where $a$ and $b$ are real.</p>\n  <p><a href=\"../../32/complex-numbers/\">D</a>o not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.</p>", "tags": ["addition of complex numbers", "checked2015", "complex numbers", "mas1602", "MAS1602", "multiplication of complex numbers", "product of complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>15/07/2015:</strong></p>\n<p>Added tags.</p>\n<p><strong>4/07/2012:</strong></p>\n<p>Added tags.</p>\n<p><strong>16/07/2012:</strong></p>\n<p>Added forbidden strings and warnings about not including decimal points or brackets in the answers as otherwise can just repeat the question and be marked correct.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Elementary examples of multiplication and addition of complex numbers. Four parts.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a)<br/>The formula for multiplying complex numbers is <br/>\\[\\begin{eqnarray*}\\simplify[]{Re((a + ib)(c + id))} &amp;=&amp; ac -bd \\\\ \\simplify[]{Im((a + ib)(c + id))} &amp;=&amp; ad +bc \\end{eqnarray*} \\]</p>\n  <p>So we have:<br/>\\[\\begin{eqnarray*}\\simplify[]{Re({a}*{b})} &amp;=&amp; \\simplify[]{{Re(a)}*{Re(b)} - {Im( a)}*{Im(b)} = {Re(a*b)}}\\\\ \\simplify[]{Im({a}*{b})} &amp;=&amp; \\simplify[]{{Re(a)}*{Im(b)} + {Im( a)}*{Re(b)} = {Im(a*b)}} \\end{eqnarray*} \\]<br/>Hence the solution is :</p>\n  <p>\\[(\\simplify[std]{{a}})(\\simplify[std]{{b}})=\\var{a*b}\\]<br/>b)</p>\n  <p>This is calculated in a similar way once the expression is written as:</p>\n  <p>$(\\simplify[std]{{a1}})^2=&nbsp;(\\simplify[std]{{a1}}) (\\simplify[std]{{a1}})$ then we find:</p>\n  <p>\\[\\begin{eqnarray*}(\\simplify[std]{{a1}})^2&amp;=&amp;&nbsp;(\\simplify[std]{{a1}}) (\\simplify[std]{{a1}})\\\\ &amp;=&amp; \\simplify[]{({Re(a1)}*{Re(a1)} - {Im(a1)}*{Im(a1)})+ ({Re(a1)}*{Im(a1)} + {Im(a1)}*{Re(a1)})i}\\\\ &amp;=&amp; \\simplify[std]{{a1^2}} \\end{eqnarray*} \\]<br/>c) <br/>We know that $i^2=-1$ which gives $i^3=i^2i=-i$.</p>\n  <p>Hence:<br/>\\[ \\begin{eqnarray*} \\simplify[std,!otherNumbers]{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}&amp;=&amp;\\simplify[std]{{a3} + {b3} * i -{c3} -({d3} * i)}\\\\ &amp;=&amp;\\simplify[std]{ {a3} -{c3} + ({b3} -{d3}) * i}\\\\ &amp;=&amp;\\simplify[std]{{a3 -c3} + {b3 -d3} * i} \\end{eqnarray*} \\]<br/>d) <br/>This can be calculated by using the formula twice, firstly to multiply out the first two sets of parentheses, <br/>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{z1})(\\var{z2})(\\var{z3})&amp;=&amp;((\\var{z1})(\\var{z2}))(\\var{z3})\\\\ &amp;=&amp;(\\var{z1*z2})(\\var{z3})\\\\ &amp;=&amp;\\var{z1*z2*z3} \\end{eqnarray*} \\]</p>"}, {"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. 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*e6*i}", "extendBaseMarkingAlgorithm": true, "type": "jme", "showCorrectAnswer": true, "showPreview": true, "useCustomName": false}]}, {"variableReplacementStrategy": "originalfirst", "marks": 0, "adaptiveMarkingPenalty": 0, "unitTests": [], "scripts": {}, "prompt": "<p>$(\\simplify[std]{{a}})(\\simplify[std]{{z4}})\\;=\\;$[[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. 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  "}, 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[], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>\\[z=(\\var{z2})^4\\]<br/>$z=\\;\\;$[[0]]</p>", "scripts": {}, "gaps": [{"answer": "{z2^4}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>\\[z=(\\var{z3})^{\\var{-d1}}\\]<br/>$z=\\;\\;$[[0]]</p>", "scripts": {}, "gaps": [{"answer": "{re(conj(z3^d1))}/{abs(z3^(2*d1))}+({im(conj(z3^d1))}/{round(abs(z3)^(2*d1))})*i", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answersimplification": "std", "expectedvariablenames": [], "notallowed": {"message": "<p>Make sure that you input the real and imaginary parts as fractions and not as decimals</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>\\[z=i^{\\var{n}}\\]<br/>$z=\\;\\;$[[0]]</p>", "scripts": {}, "gaps": [{"answer": "{i^n}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": 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"Creative Commons Attribution 4.0 International", "description": "<p>Direct calculation of low positive and negative powers of complex numbers. Calculations involving a complex conjugate. Powers of $i$. Four parts.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a)</p>\n  <p>Note that for a complex number $z=a+bi$ we have:</p>\n  <p>$z\\overline{z}=|z|^2=a^2+b^2$.</p>\n  <p>But since $\\var{conj(z1)}=\\overline{\\var{z1}}$ we have:<br/>\\[\\begin{eqnarray*}z&amp;=&amp;(\\var{z1})^{\\var{c1}}(\\var{conj(z1)})^{\\var{c1}}\\\\ &amp;=&amp;((\\var{z1})(\\var{conj(z1)}))^{\\var{c1}}\\\\ &amp;=&amp;\\simplify[]{({re(z1)}^2+{im(z1)}^2)^{c1}}\\\\ &amp;=&amp;\\var{(re(z1)^2+im(z1)^2)^c1} \\end{eqnarray*}\\]</p>\n  <p>b)</p>\n  <p>Note that $(\\var{z2})^4=((\\var{z2})^2)^2$.</p>\n  <p>Since $(\\var{z2})^2=\\simplify[std]{{z2^2}}$ we have:<br/>\\[(\\var{z2})^4=(\\simplify[std]{{z2^2}})^2=\\simplify[std]{{z2^4}}\\]</p>\n  <p>c) <br/>We have<br/>\\[ \\begin{eqnarray*} z&amp;=&amp;(\\var{z3})^{\\var{-d1}}\\\\ &amp;=&amp;\\frac{1}{(\\var{z3})^{\\var{d1}}}\\\\ &amp;=&amp;\\frac{(\\var{conj(z3)})^{\\var{d1}}}{(\\var{z3})^{\\var{d1}}(\\var{conj(z3)})^{\\var{d1}}}\\\\ &amp;=&amp;\\frac{\\var{conj(z3)^d1}}{\\var{abs(z3)^(2*d1)}}\\\\ &amp;=&amp;\\simplify[std]{{re(conj(z3^d1))}/{(abs(z3)^(2*d1))}+({im(conj(z3^d1))}/{round(abs(z3)^(2*d1))})*i} \\end{eqnarray*}\\]<br/>d)<br/>We have $i^2=-1,\\;\\;i^3=-i,\\;\\;i^4=1$.</p>\n  <p>So if $n=4m+r,\\;\\;0\\le r\\le 3$ we have \\[i^n=i^{4m+r}=(i^4)^m \\times i^r=i^r\\]<br/>Hence since $\\var{n}=4\\times\\var{m}+\\var{rem}$ we have:<br/>\\[i^{\\var{n}}=i^{\\var{rem}}=\\simplify{{i^rem}}\\]</p>"}, {"name": "De Moivre's Theorem: $n$th roots of a complex number", "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": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": 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"m1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,q4,t=2,q2,t=3,q3,q1)", "description": "", "name": "m1"}, "argrn": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround((targ1+2*pi*(n-1))/n,3)", "description": "", "name": "argrn"}, "q3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the third quadrant.'", "description": "", "name": "q3"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..5)", "description": "", "name": "b1"}, "q4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the fourth quadrant.'", "description": "", "name": "q4"}, "q1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the first quadrant.'", "description": "", "name": "q1"}, "adj": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(arg(z1)<0,2*pi,0)", "description": "", "name": "adj"}, "q2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the second quadrant.'", "description": "", "name": "q2"}, "z1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a1+b1*i", "description": "", "name": "z1"}, "md1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround((abs(z1))^(1/n),3)", "description": "", "name": "md1"}, "arg1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(targ1,3)", "description": "", "name": "arg1"}, "argr1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(targ1/n,3)", "description": "", "name": "argr1"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,-1,t=3,-1,1)", "description": "", "name": "s2"}}, "ungrouped_variables": ["q1", "argrn", "q3", "q2", "q4", "md1", "arg1", "s2", "s1", "targ1", "n", "a1", "m1", "t", "tol", "ans1", "gap", "adj", "z1", "argr1", "b1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": "<p>Find the modulus and argument of $\\var{z1}$ to 3 decimal places.</p>\n<p><strong>(i)</strong> $|\\var{z1}|\\;=\\;$ [[0]], to 3 decimal places.</p>\n<p><strong>(ii)</strong> $\\arg(\\var{z1})\\;=\\;$[[1]] radians, to 3 decimal places.&nbsp;</p>\n<p>Hence find the following $\\var{n}$th roots of $\\var{z1}$ i.e. solve for $z$, $z^\\var{n}=\\var{z1}$.</p>\n<p>How many roots are there? [[2]]</p>\n<p>All the roots have the same modulus.</p>\n<p>Input the modulus here: [[3]] (to 3 decimal places).</p>\n<p>What is the argument of the root with the least argument? [[4]] radians (to 3 decimal places)</p>\n<p>What is the argument of the root with the greatest argument? [[5]] radians (to 3 decimal places).</p>\n<p>If the roots are ordered in terms of their increasing arguments, what is the angle between successive roots? [[6]] radians (to 3 decimal places).</p>", "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "ans1-tol", "maxValue": "ans1+tol", "precision": "3", "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "arg1-tol", "maxValue": "arg1+tol", "precision": "3", "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}, {"allowFractions": false, "integerAnswer": true, "showCorrectAnswer": true, "minValue": "n", "integerPartialCredit": 0, "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "marks": 0.5, "maxValue": "n"}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "md1-tol", "maxValue": "md1+tol", "precision": "3", "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "argr1-tol", "maxValue": "argr1+tol", "precision": "3", "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "argrn-tol", "maxValue": "argrn+tol", "precision": "3", "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "gap-tol", "maxValue": "gap+tol", "precision": "3", "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "<p>Find the $\\var{n}$th roots of $\\var{z1}$.&nbsp;</p>\n  <p>Important: When calculating the roots, 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>\n  <p><strong>For the purposes of this question all arguments of complex numbers are between $0$ and $2\\pi$ radians.</strong></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", "MAS1602", "modulus of complex numbers", "quadrants", "quadrants in the complex plane", "roots of a complex number"], "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>27/08/2012:</strong></p>\n<p>Added tags.</p>\n<p>Added description.</p>\n<p>Based &nbsp;on question using DM's theorem for positive powers.</p>\n<p><strong>&nbsp;</strong></p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find modulus and argument of the complex number $z_1$ and find the $n$th roots of $z_1$ where $n=5,\\;6$ or $7$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>To be completed.</p>"}, {"name": "De Moivre's Theorem: Negative Powers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, 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{"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(ta4,3)", "description": "", "name": "a4"}, "q2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the second quadrant.'", "description": "", "name": "q2"}, "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": ["co", "ans1", "arg2", "ans3", "ans4", "tb4", "b4", "b1", "b2", "b3", "d4", "d2", "z6", "q1", "q3", "q2", "q4", "s3", "s2", "s1", "s7", "s6", "s5", "s4", "m4", "m1", "m3", "arg1", "z3", "tb3", "ta4", "arg3", "tol", "ta3", "arg4", "m2", "a1", "a3", "s8", "a4", "z4", "z5", "ans2", "z1", "z2", "c4", "f", "a2", "t", "n2", "n4", "c2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "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": "all,!collectNumbers,!noLeadingMinus", "marks": 2, "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>"}, {"name": "De Moivre's Theorem: Positive Powers", "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": {"c4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a4=f,f+1,f)", "description": "", "name": "c4"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tb3,3)", "description": "", "name": "b3"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=3,-1,1)", "description": "", "name": "s5"}, "arg4": {"templateType": "anything", "group": "Ungrouped variables", 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"q1", "q3", "q2", "q4", "s3", "s2", "s1", "s7", "s6", "s5", "s4", "m4", "m1", "m3", "arg1", "z3", "tb3", "ta4", "arg3", "tol", "ta3", "arg4", "m2", "a1", "a3", "s8", "a4", "z4", "z5", "ans2", "z1", "z2", "c4", "f", "a2", "t", "n2", "n4", "c2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "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) $\\arg(\\var{z1})\\;=\\;$[[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 integral values.</p>", "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": "<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": "Modulus and argument complex 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"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", 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"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": "Roots of a cubic real polynomial", "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/"}], "tags": ["checked2015", "complex numbers", "complex roots of real  polynomials", "conjugate roots", "roots of polynomial equations", "roots of polynomials", "roots of real polynomials"], "metadata": {"description": "<p>Using a given list of four complex numbers, find by inspection the one that is a root of a given cubic real polynomial and hence find the other roots.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>From the list of complex numbers, choose the one which is a root of the given equation $f(z)=0$ , and hence find all roots.</p>", "advice": "<p><strong>a) Finding a root</strong>.</p>\n    <p>In order to find which one of the four choices is a root you have to evaluate $f(z)$ for each choice.If you find for a choice of $z$ that $f(z)=0$ then that choice of $z$ is a root of the equation.</p>\n    <p>Note that\\[\\begin{eqnarray*} \\simplify{f({z1})} &amp;=&amp;\\simplify[std]{{z1}^3+{-2*a1 -c1}*{z1} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z1} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\ &amp;=&amp;\\simplify[std]{{z1^3}+{-2*a1 -c1}{z1 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z1} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\ &amp;=&amp;\\simplify[std]{{z1^3}+{(( -2) * a1 -c1)*z1^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z1}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\ &amp;=&amp;0 \\end{eqnarray*}. \\]So of the list of choices $z_1=\\var{z1}$ is a root.</p>\n    <p><strong>b) The other roots</strong></p>\n    <p>Now that you have found a complex root it is very simple to find another complex root.</p>\n    <p>Since $f(z)$ is a polynomial with real coefficients then if $z=z_0$ is a root we have that the conjugate $z=\\overline{z_0}$ is also a root.</p>\n    <p>Hence the complex number $z_2=\\overline{\\var{z1}}=\\var{conj(z1)}$ is a root.</p>\n    <p>To find the real root $z_3=c$ we 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"description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "x1a2": {"name": "x1a2", "group": "Ungrouped variables", "definition": "if(tx1a2=a2,a2+1,tx1a2)", "description": "", "templateType": "anything", "can_override": false}, "tx3a2": {"name": "tx3a2", "group": "Ungrouped variables", "definition": "s6*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "z1": {"name": "z1", "group": "Ungrouped variables", "definition": "a1+ b1*i", "description": "", "templateType": "anything", "can_override": false}, "tx1a2": {"name": "tx1a2", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "tx2a1": {"name": "tx2a1", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "if(abs(a1)=f1,f1+1,f1)", "description": "", "templateType": "anything", "can_override": false}, "z4": {"name": "z4", "group": "Ungrouped variables", "definition": "x3a1+x3b1*i", "description": "", "templateType": "anything", "can_override": false}, "x1b2": {"name": "x1b2", "group": "Ungrouped variables", "definition": "xs2*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "z2": {"name": "z2", "group": "Ungrouped variables", "definition": "x1a1+x1b1*i", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "s5*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "x2b1": {"name": "x2b1", "group": "Ungrouped variables", "definition": "xs2*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "x2a2": {"name": "x2a2", "group": "Ungrouped variables", "definition": "if(tx2a2=a2,a2+1,tx2a2)", "description": "", "templateType": "anything", "can_override": false}, "x3a1": {"name": "x3a1", "group": "Ungrouped variables", "definition": "if(tx3a1=a1,a1+1,tx3a1)", "description": "", "templateType": "anything", "can_override": false}, "tx3a1": {"name": "tx3a1", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "x3a2": {"name": "x3a2", "group": "Ungrouped variables", "definition": "if(tx3a2=a2,a2+1,tx3a2)", "description": "", "templateType": "anything", "can_override": false}, "s3": {"name": "s3", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "s6": {"name": "s6", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "xs1": {"name": "xs1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "xs3": {"name": "xs3", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "x3b1": {"name": "x3b1", "group": "Ungrouped variables", "definition": "xs3*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "x1b1": {"name": "x1b1", "group": "Ungrouped variables", "definition": "xs1*random(1..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["tx3a1", "f1", "tx3a2", "b2", "tx2a2", "x1b2", "x1b1", "b1", "tx1a1", "tx2a1", "xs2", "xs3", "xs1", "x3a1", "x3a2", "s3", "s2", "s1", "s6", "s5", "s4", "x2a1", "x2a2", "x1a2", "x1a1", "a1", "a2", "z4", "c2", "c1", "z1", "z2", "z3", "x3b1", "x3b2", "x2b1", "x2b2", "tx1a2"], "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>Given &nbsp;$\\displaystyle f(z) = \\simplify[std]{z ^ 3 + {( -2) * a1 -c1}*z ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * z -{c1 * (a1 ^ 2 + b1 ^ 2)}}$, one of the following complex numbers is a root $z_1$ of the equation $f(z)=0$.</p>\n        <p>Choose the correct value for $z_1$:[[0]]</p>", "gaps": [{"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, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 4, "showCellAnswerState": true, "choices": ["<p>$\\simplify{{a1}+{b1}i}$</p>", "<p>$\\simplify{{x1a1}+{x1b1}i}$</p>", "<p>$\\simplify{{x2a1}+{x2b1}i}$</p>", "<p>$\\simplify{{x3a1}+{x3b1}i}$</p>"], "matrix": [1, 0, 0, 0], "distractors": ["\n            \n            \n            <p>Correct!</p>\n            \n            \n            \n            <p>\\[\\begin{eqnarray*}\n            \n            \\simplify{f({z1})}\t&=&\\simplify[std]{{z1}^3+{-2*a1 -c1}*{z1} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z1} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z1^3}+{-2*a1 -c1}{z1 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z1} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z1^3}+{(( -2) * a1 -c1)*z1^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z1}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\t\n            \n            &=&0\t\t\t\t\t\t\t\t\n            \n            \\end{eqnarray*}\n            \n            \\] Hence is a root.</p>\n            \n            ", "\n            \n            \n            <p>Not the correct choice as :\\[\\begin{eqnarray*}\n            \n            \\simplify{f({z2})}\t&=&\\simplify[std]{{z2}^3+{-2*a1 -c1}*{z2} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z2} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z2^3}+{-2*a1 -c1}{z2 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z2} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z2^3}+{(( -2) * a1 -c1)*z2^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z2}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\t\n            \n            &=&\\simplify[std]{{z2^3+(( -2) * a1 -c1)*z2^2+ (2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z2-c1 * (a1 ^ 2 + b1 ^ 2)}}\\neq 0\t\t\t\t\t\t\t\t\n            \n            \\end{eqnarray*}\\]<br/>Hence not a root</p>\n            \n            ", "\n            \n            \n            <p>Not the correct choice as :\\[\\begin{eqnarray*}\n            \n            \\simplify{f({z3})}\t&=&\\simplify[std]{{z3}^3+{-2*a1 -c1}*{z3} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z3} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z3^3}+{-2*a1 -c1}{z3 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z3} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z3^3}+{(( -2) * a1 -c1)*z3^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z3}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\t\n            \n            &=&\\simplify[std]{{z3^3+(( -2) * a1 -c1)*z3^2+ (2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z3-c1 * (a1 ^ 2 + b1 ^ 2)}}\\neq 0\t\t\t\t\t\t\t\t\n            \n            \\end{eqnarray*}\\]<br/>Hence not a root</p>\n            \n            ", "\n            \n            \n            <p>Not the correct choice as :\\[\\begin{eqnarray*}\n            \n            \\simplify{f({z2})}\t&=&\\simplify[std]{{z4}^3+{-2*a1 -c1}*{z4} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z4} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z4^3}+{-2*a1 -c1}{z4 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z4} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n            \n            &=&\\simplify[std]{{z4^3}+{(( -2) * a1 -c1)*z4^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z4}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\t\n            \n            &=&\\simplify[std]{{z4^3+(( -2) * a1 -c1)*z4^2+ (2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z4-c1 * (a1 ^ 2 + b1 ^ 2)}}\\neq 0\t\t\t\t\t\t\t\t\n            \n            \\end{eqnarray*}\\]<br/>Hence not a root</p>\n            \n            "]}], "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>The remaining roots of $f(z)$ are:</p>\n        <p>$z_2=\\;\\;$[[0]] (enter the complex root here)</p>\n        <p>$z_3=\\;\\;$[[1]] (enter the real root here)</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": "{a1}-{b1}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, "caseSensitive": false, "valuegenerators": []}, {"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}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Roots of a quartic real polynomial", "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": "ch(a3,mz1*mz2,random(-5..5 except 0),-5,5)", "name": "b3", "description": ""}, "mz3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(z3)^2", "name": "mz3", "description": ""}, "z2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a2+b2*i", "name": "z2", "description": ""}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-6..6 except [0,-a1,a1])", "name": "b1", "description": ""}, "r12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "re(z1+z2)", "name": "r12", "description": ""}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "a1", "description": ""}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..6 except round(b1*a2/a1))", "name": "b2", "description": ""}, "mz1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(z1)^2", "name": "mz1", "description": ""}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,a1,a2])", "name": "a3", "description": ""}, "z3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a3+b3*i", "name": "z3", "description": ""}, "mz2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(z2)^2", "name": "mz2", "description": ""}, "z1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a1+ b1*i", "name": "z1", "description": ""}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3-a1..3-a1)", "name": "a2", "description": ""}}, "ungrouped_variables": ["r12", "mz1", "mz2", "mz3", "a1", "a3", "a2", "b1", "b2", "b3", "z1", "z2", "z3"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "functions": {"ch": {"type": "number", "language": "jme", "definition": "if(round(a^2+b^2)|round(s),ch(a,s,random(r..t except 0),r,t),b)", "parameters": [["a", "number"], ["s", "number"], ["b", "number"], ["r", "number"], ["t", "number"]]}}, "showQuestionGroupNames": false, "parts": [{"prompt": "<p>Given &nbsp;$\\displaystyle f(z) = \\simplify[std]{z ^ 4+ {( -2) * r12}*z ^ 3+ {mz1+mz2+4*re(z1)*re(z2)} * z^2 -{2*(re(z2)*mz1+re(z1)*mz2)}z+{mz1*mz2}}$, one of the following complex numbers is a root $z_1$ of the equation $f(z)=0$.</p>\n      <p>Choose the correct value for $z_1$:[[0]]</p>", "stepsPenalty": 0, "scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["<p>$\\simplify[std]{{a1}+{b1}i}$</p>", "<p>$\\simplify[std]{{z3}}$</p>"], "showCorrectAnswer": true, "displayColumns": 4, "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": true, "scripts": {}, "maxMarks": 0, "type": "1_n_2", "minMarks": 0, "variableReplacementStrategy": "originalfirst", "matrix": [1, 0], "marks": 0}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"showCorrectAnswer": true, "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "prompt": "<p>Since you are given that $f(z)$ has a complex root $z_1$ and since $f(z)$ is a polynomial with real coefficients then the complex conjugate $\\overline{z_1}$ must also be a root.</p>\n          <p>Since $(z-z_1)(z-\\overline{z_1})=(z^2-2\\operatorname{Re}(z)+|z_1|^2)$ we have that:\\[f(z)=(z^2-2\\operatorname{Re}(z)+|z_1|^2)(z^2+az+b)=\\simplify{z ^ 4+ {( -2) * r12}*z ^ 3+&nbsp; {mz1+mz2+4*re(z1)*re(z2)} * z^2&nbsp;&nbsp;-{2*(re(z2)*mz1+re(z1)*mz2)}z+{mz1*mz2}}\\]&nbsp;where $a$ and $b$ are real.</p>\n          <p>Looking at the constant term we see that :</p>\n          <p>\\[|z_1|^2b = \\var{mz1*mz2}\\]</p>\n          <p>Hence $|z_1|^2$ divides $ \\var{mz1*mz2}$.</p>\n          <p>An easy test to see if one of the complex numbers given is <span style=\"text-decoration: underline;\">not</span> a root is to see if its modulus squared does <span style=\"text-decoration: underline;\">not</span> divide&nbsp;$ \\var{mz1*mz2}$. If it does not divide then the other must be the root.&nbsp;</p>", "variableReplacements": [], "marks": 0}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>Write down the quadratic factor with real coefficients, $q_1(z)$, of $f(z)$ which has $z_1$ as a root:</p>\n      <p>$q_1(z)=\\;$[[0]]</p>\n      <p>Apart from $z_1$, $q_1(z)$ has another root $z_2$, which is also a root of $f(z)$.</p>\n      <p>$z_2=\\;$[[1]]</p>", "stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "z^2-{2*a1}*z+{mz1}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "{a1}-{b1}i", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"showCorrectAnswer": true, "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "prompt": "<p>If $z_1$ is a root then its conjugate $z_2$= $\\overline{z_1}$ &nbsp;is also a root.</p>", "variableReplacements": [], "marks": 0}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>Since $q_1(z)$ is a factor of $f(z)$ the other roots are given by finding the other quadratic factor $q_2(z)$ &nbsp;of $f(z)=q_1(z)q_2(z)$</p>\n        <p>$q_2(z)\\;=$[[0]]</p>\n        <p>Find the roots of $q_2(z)$ and hence the remaining two roots $z_3,\\;z_4$ of $f(z)$</p>\n        <p>$z_3=\\;$[[1]] (imaginary part negative)</p>\n        <p>$z_4=\\;$[[2]] (imaginary part positive).</p>", "stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "z^2-{2*a2}*z+{mz2}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "{a2}-{b2}*i", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "{a2}+{b2}*i", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"showCorrectAnswer": true, "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "prompt": "<p>$q_1(z)q_2(z)=f(z)$.</p>\n          <p>Once you have found $q_1(z)$ then the easiest way to find $q_2(z)$ is to compare the terms in $z^3$ and the constant terms.</p>", "variableReplacements": [], "marks": 0}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "<p>Given two complex numbers, find by inspection the one that is a root of a given quartic real polynomial $f(z)$ and hence find the other roots.</p>", "tags": ["checked2015", "MAS1602", "mas1602"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>15/07/2015:</strong></p>\n<p>Added tags.</p>\n<p><strong>25/08/2012:</strong></p>\n<p>Copied question finding roots of a cubic in order to create new question finding roots of a quartic with 4 complex roots.</p>\n<p>Function ch finds the imaginary part of the complex number $z_3$ and ensures that $z_3$ is not a solution by insisting that $|z_3|^2$ does not divide the constant term of the polynomial. This is a simple way for the students to test to see which one of $z_1$ and $z_2$ is a solution.</p>\n<p>Added tags.</p>\n<p>Added description.</p>\n<p>Checked calculation.OK.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given two complex numbers, find by inspection the one that is a root of a given quartic real polynomial and hence find the other roots.&nbsp;</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p><strong>a)&nbsp;</strong>We use the method given in Show steps for part a).</p>\n  <p>Note that $|\\var{z1}|^2=\\var{mz1}$ divides the constant term $\\var{mz1*mz2}$,</p>\n  <p>but that $|\\var{z3}|^2=\\var{mz3}$ does not divides the constant term $\\var{mz1*mz2}$.&nbsp;</p>\n  <p>Hence $\\var{z1}$ is the root we are looking for.</p>\n  <p><strong>b) A quadratic factor of $f(z)$.</strong></p>\n  <p>Since $f(z)$ is a polynomial with real coefficients then if $z=z_1$ is a root we have that the conjugate $z=\\overline{z_1}$ is also a root.</p>\n  <p>Hence the complex number $z_2=\\overline{\\var{z1}}=\\var{conj(z1)}$ is a root.</p>\n  <p>Hence $q_1(z) = (z-(\\var{z1}))(z-(\\var{conj(z1)}))=\\simplify[std]{z^2-{2*a1}*z+{abs(z1)^2}}$ is a factor of $f(z)$.</p>\n  <p><strong>c)The other quadratic factor and the other roots.</strong></p>\n  <p>We have that $f(z)=q_1(z)q_2(z)$, where $q_1(z)$ is as above and we have to find the quadratic $q_2(z)=z^2+az+b$ with real coefficients $a$ and $b$.</p>\n  <p>\\[\\begin{eqnarray*}f(z) &amp;=&amp; \\simplify[std]{z ^ 4+ {( -2) * r12}*z ^ 3+&nbsp; {mz1+mz2+4*re(z1)*re(z2)} * z^2&nbsp;&nbsp;-{2*(re(z2)*mz1+re(z1)*mz2)}z+{mz1*mz2}}\\\\&amp;=&amp;q_1(z)q_2(z)\\\\&amp;=&amp;(\\simplify[std]{z^2-{2*a1}*z+{mz1}})(z^2+az+b)\\\\&amp;=&amp;\\simplify[std]{z^4+(a-{2*a1})z^3+(b-{2*a1}*a+{mz1})*z^2+({mz1}a-{2*a1}b)*z+{mz1}*b}\\end{eqnarray*}\\]</p>\n  <p>Identifying the constant terms and the coefficients of $z^3$ on both sides of this equation gives:</p>\n  <p>$a=\\var{-2*a2},\\;\\;b=\\var{mz2}$</p>\n  <p>Hence $q_2(z)=\\simplify[std]{z^2-{2*a2}*z+{mz2}}$</p>\n  <p>You can then find the roots of this quadratic, giving the other roots of $f(z)$:&nbsp;</p>\n  <p>$z_3=\\simplify[std]{{a2}-{b2}*i}$ &nbsp; (negative imaginary part)</p>\n  <p>$z_4=\\simplify[std]{{a2}+{b2}*i}$ &nbsp; (positive imaginary part)</p>\n  <p>&nbsp;</p>"}, {"name": "The distance between two 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": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(d2=t2,t2+random(1..4),t2)", "description": "", "name": "b3"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s5"}, "t2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s5*random(1..9)", "description": "", "name": "t2"}, "c4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "name": "c4"}, "b4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "b4"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "a1"}, "ans2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z3-z4),3)", "description": "", "name": "ans2"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s6*random(1..9)", "description": "", "name": "c2"}, "d2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s7*random(1..9)", "description": "", "name": "d2"}, "t3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s7*random(1..9)", "description": "", "name": "t3"}, "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"}, "z6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c4+d4*i", "description": "", "name": "z6"}, "z5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a4+b4*i", "description": "", "name": "z5"}, "ans3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z5-z6),3)", "description": "", "name": "ans3"}, "d4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b4=t3,t3+random(1..4),t3)", "description": "", "name": "d4"}, "z2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a2+b2*i", "description": "", "name": "z2"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b2=t1,t1+random(1..4),t1)", "description": "", "name": "b1"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "name": "a2"}, "ans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z1-z2),3)", "description": "", "name": "ans1"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s5*random(1..9)", "description": "", "name": "b2"}, "s4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s4"}, "t1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "name": "t1"}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "a3"}, "a4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s6*random(1..9)", "description": "", "name": "a4"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s3"}, "s6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s6"}, "s7": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s7"}}, "ungrouped_variables": ["ans1", "ans2", "ans3", "b4", "b1", "b2", "b3", "d4", "d2", "s3", "s2", "s1", "s7", "s6", "s5", "s4", "z2", "tol", "z3", "a1", "a3", "a2", "a4", "z4", "z5", "z6", "z1", "c2", "c4", "t2", "t3", "t1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "parts": [{"showCorrectAnswer": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans1+tol", "minValue": "ans1-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "\n      \n      \n      <p>Find the distance between $\\var{z1}$ and $\\var{z2}$.</p>\n      \n      \n      \n      <p>Distance = [[0]]</p>\n      \n      \n      ", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans2+tol", "minValue": "ans2-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "\n      \n      \n      <p>Find the distance between $\\var{z3}$ and $\\var{z4}$.</p>\n      \n      \n      \n      <p>Distance = [[0]]</p>\n      \n      \n      ", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans3+tol", "minValue": "ans3-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "\n      \n      \n      <p>Find the distance between $\\var{z5}$ and $\\var{z6}$.</p>\n      \n      \n      \n      <p>Distance = [[0]]</p>\n      \n      \n      ", "variableReplacements": [], "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Find the distance between the following complex numbers, leaving your answer in decimal form to 3 decimal places:</p>", "tags": ["checked2015", "complex numbers", "distance between complex numbers", "mas1602", "MAS1602", "modulus", "modulus of a complex number", "modulus of complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>15/07/2015:</strong></p>\n<p>Added tags.</p>\n<p><strong>5/07/2012:</strong></p>\n<p>Added tags.</p>\n<p>Perhaps more steps are needed in the solutions? It isn't explained how to find the modulus of the complex number.&nbsp;<strong>Explanation included in Advice.</strong></p>\n<p>Question appears to be working correctly.</p>\n<p><strong>13/07/2012:</strong></p>\n<p>Set new variable tol=0 for all numeric input.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Finding the distance between two complex numbers using the modulus of their difference. Three parts.</p>"}, "advice": "\n  \n  \n  <p>The distance D between two complex numbers $z_1=a+bi$ and $z_2=c+di$ is given by the modulus of the difference i.e.<br/>\\[D=|z_1-z_2| = |(a-c)+(b-d)i|=\\sqrt{(a-c)^2+(b-d)^2}\\]<br/>Applying to the questions we have:\t\t\t\t\t<br/>a) \\[ \\begin{eqnarray*} D&#38;=&#38;|(\\var{z1})-(\\var{z2})|\\\\\n  \n  &#38;=&#38;|\\var{z1-z2}|\\\\\n  \n  &#38;=&#38; \\var{abs(z1-z2)}\\\\\n  \n  &#38;=&#38;\\var{ans1}\n  \n  \\end{eqnarray*} \\] to 3 decimal places.</p>\n  \n  \n  \n  <p>b) \\[ \\begin{eqnarray*} D&#38;=&#38;|(\\var{z3})-(\\var{z4})|\\\\\n  \n  &#38;=&#38;|\\var{z3-z4}|\\\\\n  \n  &#38;=&#38; \\var{abs(z3-z4)}\\\\\n  \n  &#38;=&#38;\\var{ans2}\n  \n  \\end{eqnarray*} \\] to 3 decimal places.</p>\n  \n  \n  \n  <p>c) \\[ \\begin{eqnarray*} D&#38;=&#38;|(\\var{z5})-(\\var{z6})|\\\\\n  \n  &#38;=&#38;|\\var{z5-z6}|\\\\\n  \n  &#38;=&#38; \\var{abs(z5-z6)}\\\\\n  \n  &#38;=&#38;\\var{ans3}\n  \n  \\end{eqnarray*} \\] to 3 decimal places.</p>\n  \n  \n  "}, {"name": "The modulus of 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": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "name": "tol", "description": ""}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..4)", "name": "b3", "description": ""}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s5", "description": ""}, "c4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a4=f,f+1,f)", "name": "c4", "description": ""}, "b4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "name": "b4", "description": ""}, "s4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s4", "description": ""}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s2", "description": ""}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "name": "a1", "description": ""}, "z5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a4+b4*i", "name": "z5", "description": ""}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s6*random(1..9)", "name": "c2", "description": ""}, "d2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s7*random(1..9)", "name": "d2", "description": ""}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..5)", "name": "n", "description": ""}, "z3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c2+d2*i", "name": "z3", "description": ""}, "z1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a1+b1*i", "name": "z1", "description": ""}, "z4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a3+b3*i", "name": "z4", "description": ""}, "ans2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z2*z3),3)", "name": "ans2", "description": ""}, "ans3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z4)^n,3)", "name": "ans3", "description": ""}, "d4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s5*random(1..9)", "name": "d4", "description": ""}, "f": {"templateType": "anything", "group": 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"a4", "z4", "z5", "z6", "z1", "c2", "z3", "f", "n"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "functions": {}, "showQuestionGroupNames": false, "parts": [{"showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "ans1-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "maxValue": "ans1+tol"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "<p>$|\\var{z1}|=\\;\\;$[[0]]</p>", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "ans2-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "maxValue": "ans2+tol"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "<p>$|(\\var{z2})(\\var{z3})|=\\;\\;$[[0]]</p>", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "ans3-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "maxValue": "ans3+tol"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "<p>$|(\\var{z4})^{\\var{n}}|=\\;\\;$[[0]]</p>", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "ans4-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "maxValue": "ans4+tol"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "<p>Let \\[z=\\frac{\\var{z5}}{\\var{z6}}\\]<br/>$|z|=\\;\\;$[[0]]</p>", "variableReplacements": [], "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Find the modulus of each of the following complex numbers, leaving your answer in decimal form to 3 decimal places:</p>", "tags": ["checked2015", "complex number", "complex numbers", "division of complex numbers", "mas1602", "MAS1602", "modulus of a complex number", "modulus of complex numbers", "modulus of the division of complex numbers", "modulus of the power of complex numbers", "modulus of the product of complex numbers", "multiplication of complex numbers", "multiply complex numbers", "product of complex numbers", "properties of the modulus of complex numbers", "rationalise the denominator", "rationalising the denominator"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>15/07/2015:</strong></p>\n<p>Added tags.</p>\n<p><strong>5/07/2012:</strong></p>\n<p>Added tags.</p>\n<p>Question appears to be working correctly. &nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Finding the modulus of four complex numbers; includes finding the modulus of a product, a power and a quotient.</p>"}, "advice": "<p>Recall that $|a+bi|=\\sqrt{a^2+b^2}$ and that:</p>\n  <p>1. $ |z^n| = |z|^n$</p>\n  <p>2. $ |z_1z_2|=|z_1|\\;|z_2|$</p>\n  <p>3. $ |z_1/z_2|=|z_1|/|z_2|$</p>\n  <p>a) \\[ \\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>b) \\[ \\begin{eqnarray*} |(\\var{z2})(\\var{z3})|&amp;=&amp;|\\var{z2}|\\;|\\var{z3}|\\\\ &amp;=&amp; \\var{abs(z2)}\\times \\var{abs(z3)}\\\\ &amp;=&amp;\\var{abs(z2*z3)}\\\\ &amp;=&amp;\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.</p>\n  <p>c) \\[ \\begin{eqnarray*} |(\\var{z4})^{\\var{n}}|&amp;=&amp;|\\var{z4}|^{\\var{n}}\\\\ &amp;=&amp; \\var{abs(z4)}^{\\var{n}}\\\\ &amp;=&amp; \\var{abs(z4)^n}\\\\ &amp;=&amp;\\var{ans3} \\end{eqnarray*} \\] to 3 decimal places.</p>\n  <p>d) \\[ \\begin{eqnarray*} \\left|\\frac{\\var{z5}}{\\var{z6}}\\right|&amp;=&amp;\\frac{|\\var{z5}|}{|\\var{z6}|}\\\\ &amp;=&amp; \\frac{\\var{abs(z5)}}{\\var{abs(z6)}}\\\\ &amp;=&amp; \\var{abs(z5/z6)}\\\\ &amp;=&amp;\\var{ans4} \\end{eqnarray*} \\] to 3 decimal places.</p>"}]}], "contributors": [{"name": "Henrik Skov Midtiby", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/829/"}], "extensions": [], "custom_part_types": [], "resources": []}