// Numbas version: exam_results_page_options {"name": "Arithmetics of complex numbers II", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Multiplication and addition of complex numbers. Four parts.

"}, "parts": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "adaptiveMarkingPenalty": 0, "unitTests": [], "scripts": {}, "prompt": "

$\\var{e6*i}(\\simplify[std]{{a}})\\;=\\;$[[0]].

", "sortAnswers": false, "useCustomName": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "customName": "", "showFeedbackIcon": true, "gaps": [{"marks": 1, "answerSimplification": "std", "scripts": {}, "checkVariableNames": false, "checkingType": "absdiff", "valuegenerators": [], "failureRate": 1, "notallowed": {"showStrings": false, "partialCredit": 0, "message": "

Input all numbers as fractions or integers. Also do not include brackets in your answers.

", "strings": [".", ")", "("]}, "variableReplacements": [], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "customName": "", "vsetRangePoints": 5, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "unitTests": [], "answer": "{a*e6*i}", "extendBaseMarkingAlgorithm": true, "type": "jme", "showCorrectAnswer": true, "showPreview": true, "useCustomName": false}]}, {"variableReplacementStrategy": "originalfirst", "marks": 0, "adaptiveMarkingPenalty": 0, "unitTests": [], "scripts": {}, "prompt": "

$(\\simplify[std]{{a}})(\\simplify[std]{{z4}})\\;=\\;$[[0]].

", "sortAnswers": false, "useCustomName": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "customName": "", "showFeedbackIcon": true, "gaps": [{"marks": 1, "answerSimplification": "std", "scripts": {}, "checkVariableNames": false, "checkingType": "absdiff", "valuegenerators": [], "failureRate": 1, "notallowed": {"showStrings": false, "partialCredit": 0, "message": "

Input all numbers as fractions or integers. Also do not include brackets in your answers.

", "strings": [".", ")", "("]}, "variableReplacements": [], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "customName": "", "vsetRangePoints": 5, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "unitTests": [], "answer": "({a*z4})", "extendBaseMarkingAlgorithm": true, "type": "jme", "showCorrectAnswer": true, "showPreview": true, "useCustomName": false}]}, {"variableReplacementStrategy": "originalfirst", "marks": 0, "adaptiveMarkingPenalty": 0, "unitTests": [], "scripts": {}, "prompt": "

$\\simplify[std,!otherNumbers]{{a}*({a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3)}\\;=\\;$[[0]].

", "sortAnswers": false, "useCustomName": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "customName": "", "showFeedbackIcon": true, "gaps": [{"marks": 1, "notallowed": {"showStrings": false, "partialCredit": 0, "message": "

Input all numbers as fractions or integers. Also do not include brackets in your answers.

", "strings": [".", ")", "("]}, "scripts": {}, "checkVariableNames": false, "checkingType": "absdiff", "valuegenerators": [], "failureRate": 1, "variableReplacements": [], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "customName": "", "vsetRangePoints": 5, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "unitTests": [], "answer": "{(a3 + b3 * i + c3 * i ^ 2 + d3 * i ^ 3)*a}", "extendBaseMarkingAlgorithm": true, "type": "jme", "showCorrectAnswer": true, "showPreview": true, "useCustomName": false}]}, {"variableReplacementStrategy": "originalfirst", "marks": 0, "adaptiveMarkingPenalty": 0, "unitTests": [], "scripts": {}, "prompt": "

$(\\simplify[std]{{a}})(\\simplify[std]{ {z1}})(\\simplify[std]{ {z3}})\\;=\\;$[[0]].

", "sortAnswers": false, "useCustomName": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "customName": "", "showFeedbackIcon": true, "gaps": [{"marks": 1, "notallowed": {"showStrings": false, "partialCredit": 0, "message": "

Input all numbers as fractions or integers. Also do not include brackets in your answers.

", "strings": [".", ")", "("]}, "scripts": {}, "checkVariableNames": false, "checkingType": "absdiff", "valuegenerators": [], "failureRate": 1, "variableReplacements": [], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "customName": "", "vsetRangePoints": 5, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "unitTests": [], "answer": "{a*(z1)*(z3)}", "extendBaseMarkingAlgorithm": true, "type": "jme", "showCorrectAnswer": true, "showPreview": true, "useCustomName": false}]}], "ungrouped_variables": ["a", "f6", "s3", "s2", "s1", "d3", "s6", "s5", "s4", "c3", "a3", "b3", "d6", "e6", "z4", "z1", "z2", "z3"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variables": {"a": {"group": "Ungrouped variables", "name": "a", "description": "", "definition": "s1*random(1..9)+s2*random(1..9)*i", "templateType": "anything"}, "s1": {"group": "Ungrouped variables", "name": "s1", "description": "", "definition": "random(1,-1)", "templateType": "anything"}, "s4": {"group": "Ungrouped variables", "name": "s4", "description": "", "definition": "random(1,-1)", "templateType": "anything"}, "z1": {"group": "Ungrouped variables", "name": "z1", "description": "", "definition": "s3*random(1..9)+f6*i", "templateType": "anything"}, "d3": {"group": "Ungrouped variables", "name": "d3", "description": "", "definition": "s4*random(1..9)", "templateType": "anything"}, "a3": {"group": "Ungrouped variables", "name": "a3", "description": "", "definition": "s1*random(1..9)", "templateType": "anything"}, "b3": {"group": "Ungrouped variables", "name": "b3", "description": "", "definition": "s2*random(1..9)", "templateType": "anything"}, "s5": {"group": "Ungrouped variables", "name": "s5", "description": "", "definition": "random(1,-1)", "templateType": "anything"}, "z3": {"group": "Ungrouped variables", "name": "z3", "description": "", "definition": "s6*random(1..9)+e6*i", "templateType": "anything"}, "e6": {"group": "Ungrouped variables", "name": "e6", "description": "", "definition": "s5*random(3..9)", "templateType": "anything"}, "z4": {"group": "Ungrouped variables", "name": "z4", "description": "", "definition": "s6*s2*random(1..9)+s3*s5*random(1..9)*i", "templateType": "anything"}, "f6": {"group": "Ungrouped variables", "name": "f6", "description": "", "definition": "s6*random(1..9)", "templateType": "anything"}, "d6": {"group": "Ungrouped variables", "name": "d6", "description": "", "definition": "s4*random(1..9)", "templateType": "anything"}, "c3": {"group": "Ungrouped variables", "name": "c3", "description": "", "definition": "s3*random(1..9)", "templateType": "anything"}, "s3": {"group": "Ungrouped variables", "name": "s3", "description": "", "definition": "random(1,-1)", "templateType": "anything"}, "z2": {"group": "Ungrouped variables", "name": "z2", "description": "", "definition": "s2*random(1..9)+d6*i", "templateType": "anything"}, "s2": {"group": "Ungrouped variables", "name": "s2", "description": "", "definition": "random(1,-1)", "templateType": "anything"}, "s6": {"group": "Ungrouped variables", "name": "s6", "description": "", "definition": "random(1,-1)", "templateType": "anything"}}, "variable_groups": [], "tags": ["addition of complex numbers", "algebra of complex numbers", "checked2015", "complex numbers", "multiplication of complex numbers"], "preamble": {"css": "", "js": ""}, "statement": "

Find the following complex numbers in the form $a+bi\\;$ where $a$ and $b$ are real.

\n

Input all numbers as fractions or integers. Also do not include brackets in your answers.

", "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

a)

\n

The solution is given by:

\n


$\\simplify[std]{{e6*i}}(\\simplify[std]{{a}})=\\simplify[std]{{a*e6*i}}$

\n


b)

\n

$\\simplify[std]{{a}*{z4}={a*z4}}$

\n


c)
\\[ \\begin{eqnarray*} \\simplify[std,!otherNumbers]{{a}*({a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3)}&=&\\simplify[std]{{a}*{a3 + b3 * i + c3 * i ^ 2 + d3 * i ^ 3}}\\\\ &=&\\simplify[std]{{a*(a3 + b3 * i + c3 * i ^ 2 + d3 * i ^ 3)}} \\end{eqnarray*} \\]
d)

\n

This can be calculated by using the formula twice, firstly to multiply out the first two sets of parentheses, 

\n

and then to multiply the result of that calculation by the third set of parentheses.

\n

So we obtain:
\\[ \\begin{eqnarray*} (\\var{a})(\\var{z1})(\\var{z3})&=&((\\var{a})(\\var{z1}))(\\var{z3})\\\\ &=&(\\var{a*(z1)})(\\var{z3})\\\\ &=&\\var{a*(z1)*(z3)} \\end{eqnarray*} \\]

", "name": "Arithmetics of complex numbers II", "extensions": [], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}