// Numbas version: exam_results_page_options
{"name": "Maria'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": [{"variable_groups": [], "tags": ["addition of complex numbers", "algebra of complex numbers", "checked2015", "complex numbers", "conjugate of a complex number", "division of complex numbers", "mas1602", "MAS1602", "multiplication of complex numbers", "powers of complex numbers"], "showQuestionGroupNames": false, "functions": {}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "question_groups": [{"name": "", "pickQuestions": 0, "questions": [], "pickingStrategy": "all-ordered"}], "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>", "name": "Maria's copy of Arithmetics of complex numbers V", "variables": {"s2": {"name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "templateType": "anything", "description": ""}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(3..5)", "templateType": "anything", "description": ""}, "z1": {"name": "z1", "group": "Ungrouped variables", "definition": "s1*random(1..5)+s3*random(1..5)*i", "templateType": "anything", "description": ""}, "z3": {"name": "z3", "group": "Ungrouped variables", "definition": "s2*random(1..5)+s3*random(1..5)*i", "templateType": "anything", "description": ""}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "templateType": "anything", "description": ""}, "rem": {"name": "rem", "group": "Ungrouped variables", "definition": "random(0..3)", "templateType": "anything", "description": ""}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2000..5000)", "templateType": "anything", "description": ""}, "z2": {"name": "z2", "group": "Ungrouped variables", "definition": "s2*random(1..5)+s4*random(1..5)*i", "templateType": "anything", "description": ""}, "s4": {"name": "s4", "group": "Ungrouped variables", "definition": "random(1,-1)", "templateType": "anything", "description": ""}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "4*m+rem", "templateType": "anything", "description": ""}, "s3": {"name": "s3", "group": "Ungrouped variables", "definition": "random(1,-1)", "templateType": "anything", "description": ""}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(2,3)", "templateType": "anything", "description": ""}}, "ungrouped_variables": ["s3", "s2", "s1", "m", "s4", "d1", "rem", "c1", "n", "z1", "z2", "z3"], "parts": [{"gaps": [{"answersimplification": "std", "vsetrangepoints": 5, "vsetrange": [0, 1], "expectedvariablenames": [], "answer": "{round((z1*conj(z1))^c1)}", "marks": 1, "checkingaccuracy": 0.001, "variableReplacements": [], "showpreview": true, "checkvariablenames": false, "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "notallowed": {"strings": ["."], "message": "<p>Make sure that you input the real and imaginary parts as fractions and not as decimals</p>", "partialCredit": 0, "showStrings": false}, "type": "jme"}], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0, "prompt": "<p>\\[z=(\\var{z1})^{\\var{c1}}(\\var{conj(z1)})^{\\var{c1}}\\]<br/>$z=\\;\\;$[[0]]</p>", "variableReplacements": [], "type": "gapfill"}, {"gaps": [{"answersimplification": "std", "vsetrangepoints": 5, "vsetrange": [0, 1], "expectedvariablenames": [], "answer": "{z2^4}", "marks": 1, "checkingaccuracy": 0.001, "variableReplacements": [], "showpreview": true, "checkvariablenames": false, "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0, "prompt": "<p>\\[z=(\\var{z2})^4\\]<br/>$z=\\;\\;$[[0]]</p>", "variableReplacements": [], "type": "gapfill"}, {"gaps": [{"answersimplification": "std", "vsetrangepoints": 5, "vsetrange": [0, 1], "expectedvariablenames": [], "answer": "{re(conj(z3^d1))}/{abs(z3^(2*d1))}+({im(conj(z3^d1))}/{round(abs(z3)^(2*d1))})*i", "marks": 1, "checkingaccuracy": 0.001, "variableReplacements": [], "showpreview": true, "checkvariablenames": false, "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "notallowed": {"strings": ["."], "message": "<p>Make sure that you input the real and imaginary parts as fractions and not as decimals</p>", "partialCredit": 0, "showStrings": false}, "type": "jme"}], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0, "prompt": "<p>\\[z=(\\var{z3})^{\\var{-d1}}\\]<br/>$z=\\;\\;$[[0]]</p>", "variableReplacements": [], "type": "gapfill"}, {"gaps": [{"answersimplification": "std", "vsetrangepoints": 5, "vsetrange": [0, 1], "expectedvariablenames": [], "answer": "{i^n}", "marks": 1, "checkingaccuracy": 0.001, "variableReplacements": [], "showpreview": true, "checkvariablenames": false, "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0, "prompt": "<p>\\[z=i^{\\var{n}}\\]<br/>$z=\\;\\;$[[0]]</p>", "variableReplacements": [], "type": "gapfill"}], "type": "question", "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>", "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "<p>Direct calculation of low positive and negative powers of complex numbers. Calculations involving a complex conjugate. Powers of $i$. Four parts.</p>", "licence": "Creative Commons Attribution 4.0 International", "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>Question appears to be working correctly.</p>"}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}