// Numbas version: exam_results_page_options
{"name": "Complex Numbers; Basics 01", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Complex Numbers; Basics 01", "tags": [], "metadata": {"description": "Reinforcing definition of i", "licence": "All rights reserved"}, "statement": "<h4>The symbol $i$ is such that:</h4>\n<h4>\\[&nbsp; &nbsp;i^2=-1&nbsp; \\]</h4>", "advice": "<p>We are asked to evaluate the following, applying the definition that&nbsp;</p>\n<h4>\\[&nbsp; &nbsp;i^2=-1&nbsp; \\]</h4>\n<p></p>\n<p>$\\large i^2$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; That is:&nbsp; $ \\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1} &nbsp;$&nbsp; or&nbsp; $i \\times i$. This one is straightforward from the definition of $i$;</p>\n<p style=\"text-align: center;\">$\\large i^2=-1$&nbsp;</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">$\\large i^3$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; That is:&nbsp; $ \\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1}&nbsp;$&nbsp;</p>\n<p style=\"text-align: left;\">Notice that the first two could be written as&nbsp; $-1$ so this becomes&nbsp; $-1 \\times i$</p>\n<p style=\"text-align: center;\">$\\large i^3=-i$&nbsp;</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: left;\">$\\large i^4$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; That is:&nbsp; $ \\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1}&nbsp;&nbsp;\\times&nbsp; &nbsp;\\sqrt{-1}&nbsp;$&nbsp;&nbsp;</p>\n<p style=\"text-align: left;\">Remember that \"pairs\" become $-1$ and this becomes $-1 \\times -1$</p>\n<p style=\"text-align: center;\">$\\large i^4=1$&nbsp;</p>\n<p style=\"text-align: center;\">&nbsp;&nbsp;</p>\n<p style=\"text-align: left;\">$\\large i^5$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; That is:&nbsp; $ \\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1}&nbsp;&nbsp;\\times&nbsp; &nbsp;\\sqrt{-1}&nbsp;&nbsp;\\times&nbsp; &nbsp;\\sqrt{-1} $&nbsp;&nbsp;</p>\n<p style=\"text-align: left;\">From the previous example, this becomes&nbsp; $1 \\times i$</p>\n<p style=\"text-align: center;\">$\\large i^5=i$&nbsp;</p>\n<p style=\"text-align: center;\">&nbsp;&nbsp;</p>\n<p style=\"text-align: left;\">$\\large i^6$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; That is:&nbsp; $ \\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1}&nbsp;&nbsp;\\times&nbsp; &nbsp;\\sqrt{-1}&nbsp;&nbsp;\\times&nbsp; &nbsp;\\sqrt{-1} &nbsp;&nbsp;\\times&nbsp; &nbsp;\\sqrt{-1}$&nbsp;&nbsp;</p>\n<p style=\"text-align: left;\">Again, thinking in \"pairs\" we have&nbsp; $-1 \\times -1 \\times -1$</p>\n<p style=\"text-align: center;\">$\\large i^6=-1$&nbsp;</p>", "rulesets": {}, "extensions": [], "variables": {"P1": {"name": "P1", "group": "Ungrouped variables", "definition": "random(9 .. 20#1)", "description": "", "templateType": "randrange"}, "i1": {"name": "i1", "group": "Ungrouped variables", "definition": "i^P1", "description": "", "templateType": "anything"}, "P2": {"name": "P2", "group": "Ungrouped variables", "definition": "random(9..20 except P1)", "description": "", "templateType": "anything"}, "i2": {"name": "i2", "group": "Ungrouped variables", "definition": "i^P2", "description": "", "templateType": "anything"}, "P3": {"name": "P3", "group": "Ungrouped variables", "definition": "random(9..20 except P1 except P2)", "description": "", "templateType": "anything"}, "i3": {"name": "i3", "group": "Ungrouped variables", "definition": "i^P3", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["P1", "i1", "P2", "i2", "P3", "i3"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Evaluate the following:</p>\n<p>(<em><strong>Hint:</strong> use previous answers to inform later ones</em>)</p>\n<p></p>\n<p>$\\large i^2$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; That is:&nbsp; $ \\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1} &nbsp;$</p>\n<p style=\"text-align: center;\">$\\large i^2=$ [[0]]</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">$\\large i^3$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; That is:&nbsp; $ \\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1}&nbsp;$&nbsp;</p>\n<p style=\"text-align: center;\">$\\large i^3=$ [[1]]</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">$\\large i^4$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; That is:&nbsp; $ \\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1}&nbsp;&nbsp;\\times&nbsp; &nbsp;\\sqrt{-1}&nbsp;$&nbsp;&nbsp;</p>\n<p style=\"text-align: center;\">$\\large i^4=$ [[2]]</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: left;\">$\\large i^5$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; That is:&nbsp; $ \\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1}&nbsp;&nbsp;\\times&nbsp; &nbsp;\\sqrt{-1}&nbsp;&nbsp;\\times&nbsp; &nbsp;\\sqrt{-1} $&nbsp;&nbsp;</p>\n<p style=\"text-align: center;\">$\\large i^5=$ [[3]]</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: left;\">$\\large i^6$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; That is:&nbsp; $ \\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1} \\times&nbsp; &nbsp;\\sqrt{-1}&nbsp;&nbsp;\\times&nbsp; &nbsp;\\sqrt{-1}&nbsp;&nbsp;\\times&nbsp; &nbsp;\\sqrt{-1} &nbsp;&nbsp;\\times&nbsp; &nbsp;\\sqrt{-1}$&nbsp;&nbsp;</p>\n<p style=\"text-align: center;\">$\\large i^6=$ [[4]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "-1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "-i", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "i", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "-1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Apply this to the following, randomly generated examples, without hints.</p>\n<p>Evaluate:</p>\n<p style=\"text-align: center;\">$ \\large i^{\\var{P1}}=$ [[0]]</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p>Evaluate:</p>\n<p style=\"text-align: center;\">$ \\large i^{\\var{P2}}=$ [[1]]</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p>Evaluate:</p>\n<p style=\"text-align: center;\">$ \\large i^{\\var{P3}}=$ [[2]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{i1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{i2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{i3}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}]}], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}