// Numbas version: exam_results_page_options
{"name": "Polynomials: equating coefficients 1 [L4 Randomised]", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"n": {"definition": "random(3..5)", "name": "n", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "j": {"definition": "shuffle(-12..12 except 0)[0..7]", "name": "j", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "k": {"definition": "repeat(random(0,random(-12..12 except 0)),n)+[random(-12..12 except 0)]+[0,0,0,0]", "name": "k", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "otherpoly": {"definition": "'\\$\\\\simplify{C_0+{j[0]}+({j[1]}C_1+{j[2]})x+C_2*x^2+({j[3]}C_3)x^3+(C_4/{j[4]})x^4+(C_5/{j[5]}+{j[6]})x^5}\\$'", "name": "otherpoly", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "poly": {"definition": "if(n=2,'\\$\\\\simplify{{k[0]}+{k[1]}x+{k[2]}x^2}\\$',\nif(n=3, '\\$\\\\simplify{{k[0]}+{k[1]}x+{k[2]}x^2+{k[3]}x^3}\\$',\nif(n=4, '\\$\\\\simplify{{k[0]}+{k[1]}x+{k[2]}x^2+{k[3]}x^3+{k[4]}x^4}\\$',\nif(n=5, '\\$\\\\simplify{{k[0]}+{k[1]}x+{k[2]}x^2+{k[3]}x^3+{k[4]}x^4+{k[5]}x^5}\\$',''\n))))", "name": "poly", "description": "", "templateType": "anything", "group": "Ungrouped variables"}}, "extensions": [], "rulesets": {}, "variable_groups": [], "parts": [{"showCorrectAnswer": true, "unitTests": [], "marks": 0, "gaps": [{"maxValue": "k[0]-j[0]", "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "mustBeReducedPC": 0, "unitTests": [], "marks": 1, "scripts": {}, "variableReplacements": [], "customMarkingAlgorithm": "", "allowFractions": true, "extendBaseMarkingAlgorithm": true, "minValue": "k[0]-j[0]", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry"}, {"maxValue": "(k[1]-j[2])/j[1]", "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "mustBeReducedPC": 0, "unitTests": [], "marks": 1, "scripts": {}, "variableReplacements": [], "customMarkingAlgorithm": "", "allowFractions": true, "extendBaseMarkingAlgorithm": true, "minValue": "(k[1]-j[2])/j[1]", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry"}, {"maxValue": "k[2]", "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "mustBeReducedPC": 0, "unitTests": [], "marks": 1, "scripts": {}, "variableReplacements": [], "customMarkingAlgorithm": "", "allowFractions": true, "extendBaseMarkingAlgorithm": true, "minValue": "k[2]", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry"}, {"maxValue": "k[3]/j[3]", "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "mustBeReducedPC": 0, "unitTests": [], "marks": 1, "scripts": {}, "variableReplacements": [], "customMarkingAlgorithm": "", "allowFractions": true, "extendBaseMarkingAlgorithm": true, "minValue": "k[3]/j[3]", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry"}, {"maxValue": "k[4]*j[4]", "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "mustBeReducedPC": 0, "unitTests": [], "marks": 1, "scripts": {}, "variableReplacements": [], "customMarkingAlgorithm": "", "allowFractions": true, "extendBaseMarkingAlgorithm": true, "minValue": "k[4]*j[4]", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry"}, {"maxValue": "(k[5]-j[6])*j[5]", "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "mustBeReducedPC": 0, "unitTests": [], "marks": 1, "scripts": {}, "variableReplacements": [], "customMarkingAlgorithm": "", "allowFractions": true, "extendBaseMarkingAlgorithm": true, "minValue": "(k[5]-j[6])*j[5]", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry"}], "scripts": {}, "variableReplacements": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "<p>What can be said about the value of $C_i$?</p>\n<p>$C_0=$&nbsp;[[0]], $C_1=$&nbsp;[[1]], $C_2=$&nbsp;[[2]], $C_3=$&nbsp;[[3]], $C_4=$&nbsp;[[4]], $C_5=$&nbsp;[[5]]&nbsp;</p>", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "type": "gapfill"}], "advice": "<p>Two polynomials are equal if and only if their corresponding coefficients&nbsp;are also&nbsp;equal.</p>\n<p>So we 'equate the coefficients' of the polynomials given in this question, this means that the constant terms must be equal, ie</p>\n<p>\\[\\simplify{{k[0]}=C_0+{j[0]}}\\]</p>\n<p>the coefficients of the $x$ terms must be equal, ie</p>\n<p>\\[\\simplify{{k[1]}={j[1]}C_1+{j[2]}}\\]</p>\n<p>the&nbsp;coefficients of the $x^2$ terms must be equal, ie</p>\n<p>\\[\\simplify{{k[2]}=C_2}\\]</p>\n<p>the&nbsp;coefficients of the $x^3$ terms must be equal, ie</p>\n<p>\\[\\simplify{{k[3]}={j[3]}C_3}\\]</p>\n<p>the&nbsp;coefficients of the $x^4$ terms must be equal, ie</p>\n<p>\\[\\simplify{{k[4]}=C_4/{j[4]}}\\]</p>\n<p>and the&nbsp;coefficients of the $x^5$ terms must be equal, ie</p>\n<p>\\[\\simplify{{k[5]}=C_5/{j[5]}+{j[6]}}.\\]</p>\n<p>Solving equal of these equations for $C_i$ gives our required values.</p>", "tags": ["l4", "L4"], "preamble": {"css": "", "js": ""}, "metadata": {"description": "<p>Equating coefficients of a polynomial. Basic ones that don't require simultaneous equations.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "ungrouped_variables": ["k", "n", "poly", "otherpoly", "j"], "statement": "<p>Suppose that during some working, you find that&nbsp;</p>\n<p style=\"text-align: center;\">{poly} $=$ {otherpoly}</p>\n<p style=\"text-align: left;\">for all $x$.</p>", "name": "Polynomials: equating coefficients 1 [L4 Randomised]", "functions": {}, "type": "question", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Paul Hancock", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1738/"}, {"name": "Abbi Mullins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2466/"}, {"name": "Matthew James Sykes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2582/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Paul Hancock", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1738/"}, {"name": "Abbi Mullins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2466/"}, {"name": "Matthew James Sykes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2582/"}]}