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{"name": "Vic's copy of Operations on two complex numbers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": ["MAS2103", "checked2015"], "parts": [{"prompt": "<p>$\\lvert z_1 \\rvert=$ [[0]] (Enter your answer to 3 d.p.)</p>", "type": "gapfill", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "gaps": [{"scripts": {}, "showPrecisionHint": false, "type": "numberentry", "showCorrectAnswer": true, "marks": 1, "minValue": "absz1-tol", "maxValue": "absz1+tol", "allowFractions": false, "correctAnswerFraction": false}]}, {"prompt": "<p>$\\lvert z_2 \\rvert=$ [[0]] (Enter your answer to 3 d.p.)</p>", "type": "gapfill", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "gaps": [{"scripts": {}, "showPrecisionHint": false, "type": "numberentry", "showCorrectAnswer": true, "marks": 1, "minValue": "absz2-tol", "maxValue": "absz2+tol", "allowFractions": false, "correctAnswerFraction": false}]}, {"prompt": "<p>$\\lvert z_1z_2 \\rvert=$ [[0]] (Enter your answer to 3 d.p.)</p>", "type": "gapfill", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "gaps": [{"scripts": {}, "showPrecisionHint": false, "type": "numberentry", "showCorrectAnswer": true, "marks": 1, "minValue": "absz1z2-tol", "maxValue": "absz1z2+tol", "allowFractions": false, "correctAnswerFraction": false}]}, {"prompt": "<p>$z_1^\\ast=$ [[0]]</p>", "type": "gapfill", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "gaps": [{"showpreview": true, "scripts": {}, "checkingaccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5, "checkingtype": "absdiff", "vsetrange": [0, 1], "answer": "{conjz1}", "checkvariablenames": false, "expectedvariablenames": []}]}, {"prompt": "<p>$z_1z_1^\\ast=$ [[0]]</p>", "type": "gapfill", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "gaps": [{"showpreview": true, "scripts": {}, "checkingaccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5, "checkingtype": "absdiff", "vsetrange": [0, 1], "answer": "{z1conjz1}", "checkvariablenames": false, "expectedvariablenames": []}]}], "preamble": {"js": "", "css": ""}, "functions": {}, "advice": "<h4>a) and b)</h4>\n<p>The modulus of a complex number $z=a+bi$ is given by</p>\n<p>\\[\\lvert z \\rvert=\\sqrt{a^2+b^2}\\]</p>\n<p>In these parts, $z_1 = \\var{z1}$ and $z_2 = \\var{z2}$, so</p>\n<p>\\begin{align}<br/>\\lvert z_1 \\rvert &amp;=\\sqrt{(\\var{re(z1)})^2+(\\var{im(z1)})^2} \\\\<br/>&amp;= \\var{absz1}<br/>\\end{align}</p>\n<p>and</p>\n<p>\\begin{align}<br/>\\lvert z_2 \\rvert &amp;= \\sqrt{(\\var{re(z2)})^2+(\\var{im(z2)})^2} \\\\<br/>&amp;= \\var{absz2}<br/>\\end{align}</p>\n<p>(both to 3d.p.).</p>\n<h4>c)</h4>\n<p>In general</p>\n<p>\\[\\lvert z_1z_2 \\rvert=\\lvert z_1 \\rvert\\lvert z_2 \\rvert\\]</p>\n<p>so, in this part,</p>\n<p>\\[ \\lvert z_1z_2 \\rvert=\\sqrt{(\\var{re(z1)})^2+(\\var{im(z1)})^2}\\sqrt{(\\var{re(z2)})^2+(\\var{im(z2)})^2}=\\var{absz1z2} \\]</p>\n<p>to 3 d.p.</p>\n<h4>d)</h4>\n<p>If $z=a+bi$, then $z^\\ast=a-bi$, so&nbsp;</p>\n<p>\\[z_1^\\ast=\\var{conjz1}\\]</p>\n<h4>e)</h4>\n<p>In general, $zz^\\ast=(a+bi)(a-bi)=a^2+b^2$, so</p>\n<p>\\[z_1z_1^\\ast=(\\var{z1})(\\var{conjz1})=(\\var{re(z1)})^2+(\\var{im(z1)})^2=\\var{z1conjz1}.\\]</p>", "metadata": {"description": "<p>Calculation of modulus, argument, multiplication by complex conjugate, given two complex numbers.</p>", "notes": "<p><strong>15/7/2012:</strong></p>\n<p>Added tags.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "<p>Given the complex numbers $z_1=\\var{z1}$ and $z_2=\\var{z2}$, calculate the following quantities. &nbsp;In what follows, an asterisk denotes the complex conjugate.</p>", "rulesets": {}, "name": "Vic's copy of Operations on two complex numbers", "question_groups": [{"name": "", "pickQuestions": 0, "pickingStrategy": "all-ordered", "questions": []}], "showQuestionGroupNames": false, "ungrouped_variables": ["absz2", "absz1", "z1", "absz1z2", "tol", "conjz1", "z1conjz1", "z2"], "variables": {"tol": {"name": "tol", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "0.001"}, "absz2": {"name": "absz2", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z2),3)"}, "absz1": {"name": "absz1", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z1),3)"}, "z2": {"name": "z2", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))+random(1..9)*sign(random(-1,1))*i"}, "absz1z2": {"name": "absz1z2", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z1*z2),3)"}, "z1conjz1": {"name": "z1conjz1", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "z1*conj(z1)"}, "z1": {"name": "z1", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))+random(1..9)*sign(random(-1,1))*i"}, "conjz1": {"name": "conjz1", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "conj(z1)"}}, "variable_groups": [], "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Vic Brennan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3173/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Vic Brennan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3173/"}]}