// Numbas version: exam_results_page_options
{"name": "Complex Numbers Ex Sheet 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "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>", "metadata": {"description": "<p>Elementary examples of multiplication and addition of complex numbers. Four parts.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "name": "Complex Numbers Ex Sheet 1", "parts": [{"scripts": {}, "gaps": [{"answer": "{a*b}", "checkingaccuracy": 0.001, "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetrange": [0, 1], "marks": 1, "type": "jme", "expectedvariablenames": [], "notallowed": {"partialCredit": 0, "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, "strings": [".", "(", ")"]}, "checkingtype": "absdiff", "variableReplacements": [], "scripts": {}, "checkvariablenames": false}], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 0, "type": "gapfill", "prompt": "<p>$(\\simplify[std]{{a}})(\\simplify[std]{{b}})\\;=\\;$[[0]].</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>"}, {"scripts": {}, "gaps": [{"answer": "({a1^2})", "checkingaccuracy": 0.001, "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetrange": [0, 1], "marks": 1, "type": "jme", "expectedvariablenames": [], "notallowed": {"partialCredit": 0, "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, "strings": [".", ")", "("]}, "checkingtype": "absdiff", "variableReplacements": [], "scripts": {}, "checkvariablenames": false}], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 0, "type": "gapfill", "prompt": "<p>$(\\simplify[std]{{a1}})^2\\;=\\;$[[0]].</p>"}, {"scripts": {}, "gaps": [{"answer": "{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}", "checkingaccuracy": 0.001, "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetrange": [0, 1], "marks": 1, "type": "jme", "expectedvariablenames": [], "notallowed": {"partialCredit": 0, "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, "strings": [".", ")", "("]}, "checkingtype": "absdiff", "variableReplacements": [], "scripts": {}, "checkvariablenames": false}], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 0, "type": "gapfill", "prompt": "<p>$\\simplify[std,!otherNumbers]{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}\\;=\\;$[[0]].</p>"}, {"scripts": {}, "gaps": [{"answer": "{z1*z2*z3}", "checkingaccuracy": 0.001, "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetrange": [0, 1], "marks": 1, "type": "jme", "expectedvariablenames": [], "notallowed": {"partialCredit": 0, "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, "strings": [".", ")", "("]}, "checkingtype": "absdiff", "variableReplacements": [], "scripts": {}, "checkvariablenames": false}], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 0, "type": "gapfill", "prompt": "<p>$(\\simplify[std]{{z1}}) (\\simplify[std]{{z2}}) (\\simplify[std]{{z3}})\\;=\\;$[[0]].</p>"}], "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"z3": {"definition": "s6*random(1..9)+e6*i", "description": "", "templateType": "anything", "name": "z3", "group": "Ungrouped variables"}, "b": {"definition": "s3*random(1..9)+s4*random(1..9)*i", "description": "", "templateType": "anything", "name": "b", "group": "Ungrouped variables"}, "a1": {"definition": "s1*random(1..9)+ s4*random(1..9)*i", "description": "", "templateType": "anything", "name": "a1", "group": "Ungrouped variables"}, "z1": {"definition": "s3*random(1..9)+f6*i", "description": "", "templateType": "anything", "name": "z1", "group": "Ungrouped variables"}, "b3": {"definition": "s2*random(1..9)", "description": "", "templateType": "anything", "name": "b3", "group": "Ungrouped variables"}, "d3": {"definition": "s4*random(1..9)", "description": "", "templateType": "anything", "name": "d3", "group": "Ungrouped variables"}, "s2": {"definition": "random(1,-1)", "description": "", "templateType": "anything", "name": "s2", "group": "Ungrouped variables"}, "s6": {"definition": "random(1,-1)", "description": "", "templateType": "anything", "name": "s6", "group": "Ungrouped variables"}, "a": {"definition": "s1*random(1..9)+s2*random(1..9)*i", "description": "", "templateType": "anything", "name": "a", "group": "Ungrouped variables"}, "e6": {"definition": "s3*random(1..4)", "description": "", "templateType": "anything", "name": "e6", "group": "Ungrouped variables"}, "s3": {"definition": "random(1,-1)", "description": "", "templateType": "anything", "name": "s3", "group": "Ungrouped variables"}, "d6": {"definition": "s1*random(1..4)", "description": "", "templateType": "anything", "name": "d6", "group": "Ungrouped variables"}, "s4": {"definition": "random(1,-1)", "description": "", "templateType": "anything", "name": "s4", "group": "Ungrouped variables"}, "c3": {"definition": "s3*random(1..9)", "description": "", "templateType": "anything", "name": "c3", "group": "Ungrouped variables"}, "a3": {"definition": "s1*random(1..9)", "description": "", "templateType": "anything", "name": "a3", "group": "Ungrouped variables"}, "f6": {"definition": "s4*random(1..4)", "description": "", "templateType": "anything", "name": "f6", "group": "Ungrouped variables"}, "s1": {"definition": "random(1,-1)", "description": "", "templateType": "anything", "name": "s1", "group": "Ungrouped variables"}, "z2": {"definition": "s2*random(1..9)+d6*i", "description": "", "templateType": "anything", "name": "z2", "group": "Ungrouped variables"}}, "ungrouped_variables": ["a", "b", "f6", "s3", "s2", "s1", "d3", "s6", "s4", "a1", "c3", "a3", "b3", "d6", "e6", "z1", "z2", "z3"], "tags": [], "extensions": [], "preamble": {"js": "", "css": ""}, "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>", "type": "question", "contributors": [{"name": "Joshua Harnwell", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1973/"}]}]}], "contributors": [{"name": "Joshua Harnwell", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1973/"}]}