// Numbas version: exam_results_page_options
{"name": "Blathnaid's copy of Arithmetics of complex numbers V", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Blathnaid's copy of Arithmetics of complex numbers V", "variables": {"s4": {"name": "s4", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(1,-1)"}, "m": {"name": "m", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(2000..5000)"}, "s2": {"name": "s2", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(1,-1)"}, "z1": {"name": "z1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "s1*random(1..5)+s3*random(1..5)*i"}, "n": {"name": "n", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "4*m+rem"}, "z3": {"name": "z3", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "s2*random(1..5)+s3*random(1..5)*i"}, "d1": {"name": "d1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(2,3)"}, "c1": {"name": "c1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(3..5)"}, "s3": {"name": "s3", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(1,-1)"}, "s1": {"name": "s1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(1,-1)"}, "rem": {"name": "rem", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(0..3)"}, "z2": {"name": "z2", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "s2*random(1..5)+s4*random(1..5)*i"}}, "functions": {}, "ungrouped_variables": ["s3", "s2", "s1", "m", "s4", "d1", "rem", "c1", "n", "z1", "z2", "z3"], "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": [], "variable_groups": [], "preamble": {"css": "", "js": ""}, "extensions": [], "metadata": {"licence": "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>"}, "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>", "parts": [{"gaps": [{"checkVariableNames": false, "unitTests": [], "failureRate": 1, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "showPreview": true, "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "answer": "{z2^4}", "type": "jme", "marks": 1, "answerSimplification": "std", "customMarkingAlgorithm": "", "showCorrectAnswer": true, "scripts": {}, "checkingAccuracy": 0.001, "checkingType": "absdiff", "vsetRangePoints": 5, "vsetRange": [0, 1]}], "unitTests": [], "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>\\[z=(\\var{z2})^4\\]<br/>$z=\\;\\;$[[0]]</p>", "variableReplacementStrategy": "originalfirst", "type": "gapfill", "marks": 0, "sortAnswers": false, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "scripts": {}}, {"gaps": [{"checkVariableNames": false, "unitTests": [], "failureRate": 1, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "showPreview": true, "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "answer": "{i^n}", "type": "jme", "marks": 1, "answerSimplification": "std", "customMarkingAlgorithm": "", "showCorrectAnswer": true, "scripts": {}, "checkingAccuracy": 0.001, "checkingType": "absdiff", "vsetRangePoints": 5, "vsetRange": [0, 1]}], "unitTests": [], "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>\\[z=i^{\\var{n}}\\]<br/>$z=\\;\\;$[[0]]</p>", "variableReplacementStrategy": "originalfirst", "type": "gapfill", "marks": 0, "sortAnswers": false, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "scripts": {}}], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "type": "question", "contributors": [{"name": "Blathnaid Sheridan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/447/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Blathnaid Sheridan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/447/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}