// Numbas version: exam_results_page_options
{"name": "Expand complex conjugates", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Expand complex conjugates", "tags": ["complex conjugates", "complex numbers"], "metadata": {"description": "<p>Simple multiplication of complex conjugates. Complex numbers are of the form $a+bi$ where $a$ and $b$ are randomised between 1 and 9 inclusive.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Question 1</p>", "advice": "<p>To expand $(\\var{z1})(\\var{z2})$ you expand the brackets as follows</p>\n<p></p>\n<p>\\[(\\var{z1})(\\var{z2})=(\\var{re(z1)})^2+(\\simplify{{imz1}i})(\\simplify{{imz2}i})+(\\simplify{{imz1}i})(\\var{re(z2)})+(\\simplify{{imz2}i})(\\var{re(z1)})\\]</p>\n<p></p>\n<p>Notice that $(\\simplify{{imz1}i})(\\var{re(z2)})+(\\simplify{{imz2}i})(\\var{re(z1)})=0$</p>\n<p></p>\n<p>Also remembering that $i^2=-1$ you can simplify $(\\simplify{{imz1}i})(\\simplify{{imz2}i})=(\\var{im(z1)})(\\var{im(z2)})(i^2)=(\\var{im(z1)})(\\var{im(z2)})(-1)$ and now get</p>\n<p></p>\n<p>\\begin{align}<br/>(\\var{z1})(\\var{z2}) &amp;=(\\var{re(z1)})^2+(\\var{im(z1)})(\\var{im(z2)})(-1) \\\\<br/>&amp;=\\var{z1z2}<br/>\\end{align}</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"z1": {"name": "z1", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))+sign(random(-1,1))*random(1..9)*i", "description": "", "templateType": "anything", "can_override": false}, "z2": {"name": "z2", "group": "Ungrouped variables", "definition": "conj(z1)", "description": "", "templateType": "anything", "can_override": false}, "z1z2": {"name": "z1z2", "group": "Ungrouped variables", "definition": "z1*z2", "description": "", "templateType": "anything", "can_override": false}, "rez1": {"name": "rez1", "group": "Ungrouped variables", "definition": "(z1+z2)/2", "description": "", "templateType": "anything", "can_override": false}, "rez2": {"name": "rez2", "group": "Ungrouped variables", "definition": "(z1+z2)/2", "description": "", "templateType": "anything", "can_override": false}, "imz1": {"name": "imz1", "group": "Ungrouped variables", "definition": "(z1-rez1)/i", "description": "", "templateType": "anything", "can_override": false}, "imz2": {"name": "imz2", "group": "Ungrouped variables", "definition": "(z2-rez2)/i", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["z1", "z2", "z1z2", "rez1", "rez2", "imz1", "imz2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Simplify as far as possible $(\\var{z1})(\\var{z2})$</p>", "minValue": "z1z2", "maxValue": "z1z2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Mike Phipps", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14139/"}]}]}], "contributors": [{"name": "Mike Phipps", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14139/"}]}