// Numbas version: exam_results_page_options
{"name": "Harry's copy of John's copy of Complex Numbers Ex Sheet 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "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>", "showQuestionGroupNames": false, "name": "Harry's copy of John's copy of Complex Numbers Ex Sheet 3", "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    ", "question_groups": [{"questions": [], "pickingStrategy": "all-ordered", "name": "", "pickQuestions": 0}], "functions": {}, "tags": ["checked2015", "complex numbers", "conjugate of a complex number", "division of complex numbers", "inverse of complex numbers", "mas1602", "MAS1602", "multiplication of complex numbers"], "parts": [{"gaps": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "scripts": {}, "vsetrangepoints": 5, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "vsetrange": [0, 1], "notallowed": {"message": "<p>Make sure that you input the real and imaginary parts as fractions and not as decimals. Do not include brackets in your answer.</p>", "strings": [".", "(", ")"], "showStrings": false, "partialCredit": 0}, "checkingtype": "absdiff", "type": "jme", "expectedvariablenames": [], "answer": "{c1*re(z1)}/{abs(z1)^2}-{c1*im(z1)}/{abs(z1)^2}*i", "showpreview": true, "answersimplification": "std", "checkvariablenames": false}], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0, "prompt": "<p>$\\displaystyle \\simplify[std]{{c1}/{z1}}\\;=\\;$[[0]].</p>\n      <p>&nbsp;Do not include brackets in your answer.</p>\n      <p>&nbsp;</p>", "type": "gapfill", "variableReplacements": [], "scripts": {}}, {"gaps": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "scripts": {}, "vsetrangepoints": 5, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "vsetrange": [0, 1], "notallowed": {"message": "<p>Make sure that you input the real and imaginary parts as fractions and not as decimals. Do not include brackets in your answer.</p>", "strings": [".", "(", ")"], "showStrings": false, "partialCredit": 0}, "checkingtype": "absdiff", "type": "jme", "expectedvariablenames": [], "answer": "{c2*re(z2)}/{abs(z2)^2}-{c2*im(z2)}/{abs(z2)^2}*i", "showpreview": true, "answersimplification": "std", "checkvariablenames": false}], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0, "prompt": "<p>$\\displaystyle \\simplify[std]{{c2}/{z2}}\\;=\\;$[[0]].</p>\n      <p>&nbsp;Do not include brackets in your answer.</p>", "type": "gapfill", "variableReplacements": [], "scripts": {}}, {"gaps": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "scripts": {}, "vsetrangepoints": 5, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "vsetrange": [0, 1], "notallowed": {"message": "<p>Make sure that you input the real and imaginary parts as fractions and not as decimals. Also do not include brackets in your answer.</p>", "strings": [".", "(", ")"], "showStrings": false, "partialCredit": 0}, "checkingtype": "absdiff", "type": "jme", "expectedvariablenames": [], "answer": "{re(z1*conj(z3))}/{abs(z3)^2}+{im(z1*conj(z3))}/{abs(z3)^2}*i", "showpreview": true, "answersimplification": "std", "checkvariablenames": false}], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0, "prompt": "<p>$\\displaystyle \\simplify[std]{{z1}/{z3}}\\;=\\;$[[0]].</p>\n      <p>Do not include brackets in your answer.</p>", "type": "gapfill", "variableReplacements": [], "scripts": {}}, {"gaps": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "scripts": {}, "vsetrangepoints": 5, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "vsetrange": [0, 1], "notallowed": {"message": "<p>Make sure that you input the real and imaginary parts as fractions and not as decimals. &nbsp;Do not include brackets in your answer.</p>", "strings": [".", "(", ")"], "showStrings": false, "partialCredit": 0}, "checkingtype": "absdiff", "type": "jme", "expectedvariablenames": [], "answer": "{re(z3*conj(z2))}/{abs(z2)^2}+{im(z3*conj(z2))}/{abs(z2)^2}*i", "showpreview": true, "answersimplification": "std", "checkvariablenames": false}], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0, "prompt": "<p>$\\displaystyle \\simplify[std]{{z3}/{z2}}\\;=\\;$[[0]].</p>\n      <p>&nbsp;Do not include brackets in your answer.</p>", "type": "gapfill", "variableReplacements": [], "scripts": {}}], "ungrouped_variables": ["s3", "s2", "s1", "s4", "a3", "rz3", "c2", "c1", "z1", "z2", "z3"], "type": "question", "preamble": {"js": "", "css": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variables": {"c2": {"name": "c2", "definition": "random(1..5)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "s4": {"name": "s4", "definition": "random(1,-1)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "s3": {"name": "s3", "definition": "random(1,-1)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "rz3": {"name": "rz3", "definition": "if(a3=re(z1),a3+random(1,-1),a3)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "z1": {"name": "z1", "definition": "s2*random(1..9)+s1*random(1..9)*i", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "c1": {"name": "c1", "definition": "s3*random(1..9)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "z3": {"name": "z3", "definition": "rz3+s1*random(1..9)*i", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "z2": {"name": "z2", "definition": "re(z1)+s2*random(1,2)+s4*random(1..9)*i", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "a3": {"name": "a3", "definition": "s3*random(1..9)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "s2": {"name": "s2", "definition": "random(1,-1)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "s1": {"name": "s1", "definition": "random(1,-1)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}}, "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>Question a</strong> - sometimes the complex number is generated as a/(b+i*c) but sometimes the complex number is displayed as a decimal, i.e. <span id=\"prompt\"><span><span id=\"MathJax-Element-42-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-1241\" class=\"math\">0.0975609756+0.1219512195i if this happens then the question is invalid.</span></span></span></span></p>\n<p><span id=\"prompt\"><span><span id=\"MathJax-Element-42-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-1241\" class=\"math\"><strong>16/07/2012:</strong></span></span></span></span></p>\n<p><span><span><span class=\"MathJax\"><span class=\"math\">The above issue has been resolved,</span></span></span></span></p>\n<p>&nbsp;Also forbid brackets in the answers as otherwise can repeat the question and be marked as correct.</p>\n<p><span id=\"prompt\"><span><span id=\"MathJax-Element-42-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-1241\" class=\"math\"><span style=\"display: inline-block; position: relative; width: 14.327em; height: 0px; font-size: 130%;\"><span style=\"position: absolute; clip: rect(1.862em, 1000em, 2.909em, -0.394em); top: -2.683em; left: 0em;\"><span id=\"MathJax-Span-1242\" class=\"mrow\"><span id=\"MathJax-Span-1243\" class=\"mstyle\"><span id=\"MathJax-Span-1244\" class=\"mrow\"><span id=\"MathJax-Span-1245\" class=\"mn\" style=\"font-family: MathJax_Main;\">0.0975609756</span><span id=\"MathJax-Span-1246\" class=\"mo\" style=\"font-family: MathJax_Main; padding-left: 0.222em;\">+</span><span id=\"MathJax-Span-1247\" class=\"mn\" style=\"font-family: MathJax_Main; padding-left: 0.222em;\">0.1219512195</span><span id=\"MathJax-Span-1248\" class=\"mi\" style=\"font-family: MathJax_Math; font-style: italic;\">i</span></span></span></span></span></span></span></span></span></span></p>\n<p>&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Inverse and division of complex numbers. &nbsp;Four parts.</p>"}, "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}]}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}