// Numbas version: finer_feedback_settings
{"name": "Cubics: Summary (Intermediate steps)", "extensions": [], "custom_part_types": [{"source": {"pk": 2, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/2/edit"}, "name": "List of numbers", "short_name": "list-of-numbers", "description": "<p>The answer is a comma-separated list of numbers.</p>\n<p>The list is marked correct if each number occurs the same number of times as in the expected answer, and no extra numbers are present.</p>\n<p>You can optionally treat the answer as a set, so the number of occurrences doesn't matter, only whether each number is included or not.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "join(\n  if(settings[\"correctAnswerFractions\"],\n    map(let([a,b],rational_approximation(x), string(a/b)),x,settings[\"correctAnswer\"])\n  ,\n    settings[\"correctAnswer\"]\n  ),\n  settings[\"separator\"] + \" \"\n)", "hint": {"static": false, "value": "if(settings[\"show_input_hint\"],\n  \"Enter a list of numbers separated by <code>{settings['separator']}</code>.\",\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "bits:\nlet(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)\n\nexpected_numbers:\nlet(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)\n\nvalid_numbers:\nif(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )\n\nis_sorted:\nassert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )\n\nincluded:\nmap(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )\n\nall_included:\nall(included)\n\nno_extras:\nif(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )\n\ninterpreted_answer:\nif(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)\n\nmark:\nif(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)\n\nnotationStyles:\n[\"en\"]\n\nisSet:\nsettings[\"isSet\"]\n\nextra_numbers:\nfilter(not (x in expected_numbers),x,interpreted_answer)", "marking_notes": [{"name": "bits", "description": "<p>The separate items in the student's answer</p>", "definition": "let(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)"}, {"name": "expected_numbers", "description": "", "definition": "let(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)"}, {"name": "valid_numbers", "description": "<p>Is every number in the student's list valid?</p>", "definition": "if(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )"}, {"name": "is_sorted", "description": "<p>Are the student's answers in ascending order?</p>", "definition": "assert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )"}, {"name": "included", "description": "<p>Is each number in the expected answer present in the student's list the correct number of times?</p>", "definition": "map(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )"}, {"name": "all_included", "description": "<p>Has every number been included the right number of times?</p>", "definition": "all(included)"}, {"name": "no_extras", "description": "<p>True if the student's list doesn't contain any numbers that aren't in the expected answer.</p>", "definition": "if(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "if(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)"}, {"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)"}, {"name": "notationStyles", "description": "", "definition": "[\"en\"]"}, {"name": "isSet", "description": "<p>Should the answer be considered as a set, so the number of times an element occurs doesn't matter?</p>", "definition": "settings[\"isSet\"]"}, {"name": "extra_numbers", "description": "<p>Numbers included in the student's answer that are not in the expected list.</p>", "definition": "filter(not (x in expected_numbers),x,interpreted_answer)"}], "settings": [{"name": "correctAnswer", "label": "Correct answer", "help_url": "", "hint": "The list of numbers that the student should enter. The order does not matter.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "allowFractions", "label": "Allow the student to enter fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "correctAnswerFractions", "label": "Display the correct answers as fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "isSet", "label": "Is the answer a set?", "help_url": "", "hint": "If ticked, the number of times an element occurs doesn't matter, only whether it's included at all.", "input_type": "checkbox", "default_value": false}, {"name": "show_input_hint", "label": "Show the input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "separator", "label": "Separator", "help_url": "", "hint": "The substring that should separate items in the student's list", "input_type": "string", "default_value": ",", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Cubics: Summary (Intermediate steps)", "tags": [], "metadata": {"description": "<p>Part (a): Given two cubic functions $g(x)$ and $h(x)$ of the form $ax^3+bx^2+cx+d$, give an expression for the function $f(x)$, where $f(x)=g(x)-2h(x)$.&nbsp;</p>\n<p>Part (b): Solve $f(x)=0$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>\\[g(x)=\\simplify[all]{{m1*A}x^3+{m2*B}x^2+{m3*C}x+{m4*D}},\\quad h(x)=\\simplify[all]{{n1*A}x^3+{n2*B}x^2+{n3*C}x+{n4*D}}\\]</p>", "advice": "<p>a) We are asked to find $f(x)$, where $f(x)=g(x)-2h(x)$.</p>\n<p>Therefore,</p>\n<p>\\[ \\begin{split} f(x) &amp;\\,=&nbsp;\\simplify[all]{{m1*A}x^3+{m2*B}x^2+{m3*C}x+{m4*D}} - 2 \\left( \\simplify[all]{{n1*A}x^3+{n2*B}x^2+{n3*C}x+{n4*D}} \\right) \\\\ &amp;\\,=\\simplify[all]{{m1*A}x^3+{m2*B}x^2+{m3*C}x+{m4*D}} -\\left(\\simplify[all]{{2*n1*A}x^3+{2*n2*B}x^2+{2*n3*C}x+{2*n4*D}}\\right) \\\\&amp;\\,=\\simplify[all,!collectNumbers,!cancelTerms]{{m1*A}x^3+{m2*B}x^2+{m3*C}x+{m4*D} -{2*n1*A}x^3-{2*n2*B}x^2-{2*n3*C}x-{2*n4*D}}.\\end{split} \\]</p>\n<p>Collecting similar terms, we find the expression for $f(x)$:</p>\n<p>\\[ f(x) =&nbsp;\\simplify[all]{{m1*A-2*n1*A}x^3+{m2*B-2*n2*B}x^2+{m3*C-2*n3*C}x+{m4*D-2*n4*D}}.\\]</p>\n<p></p>\n<p>b) We now need to solve $f(x)=0$.</p>\n<p>From part (a), we know that&nbsp;</p>\n<p>\\[ f(x) =&nbsp;\\simplify[all]{{m1*A-2*n1*A}x^3+{m2*B-2*n2*B}x^2+{m3*C-2*n3*C}x+{m4*D-2*n4*D}}.\\]</p>\n<p>Setting this equal to $0$, we now have a cubic equation in terms of $x$:</p>\n<p>\\[ \\simplify[all]{{m1*A-2*n1*A}x^3+{m2*B-2*n2*B}x^2+{m3*C-2*n3*C}x+{m4*D-2*n4*D}} = 0.\\]</p>\n<p>To find the solutions of this equation we want to factorise the left-hand side of the equation into its linear factors.&nbsp;</p>\n<p>To find the first of these factors, we can try appropriate values for $x$ to see if they satisfy $f(x)=0$. A sensible approach to this step is to try the positive and negative factors of the constant term in the equation, which is $\\var{{m4*D-2*n4*D}}$ in this case.&nbsp;</p>\n<p>Starting with $\\var{-b1/a1}$, we find that $f(\\var{-b1/a1})=0$, which tells us that $(\\simplify{x-{-b1/a1}})$ is a factor of $f(x)$. <strong>(Note: The first number you try will often not be a factor, and this is completely fine. It can be helpful to list the factors first and try each one in order until you find one that works.)</strong></p>\n<p></p>\n<p>To find the other factors, we can write $f(x)$ as a product of this factor and a quadratic, and then factorise the quadratic to find the remaining linear factors:</p>\n<p>\\[ \\begin{split} \\simplify[all]{{m1*A-2*n1*A}x^3+{m2*B-2*n2*B}x^2+{m3*C-2*n3*C}x+{m4*D-2*n4*D}} &amp;\\,= (\\simplify{x-{-b1/a1}})(\\simplify{{a2*a3}x^2+{a2*b3+a3*b2}x+{b2*b3}}) \\\\ &amp;\\,= (\\simplify{x-{-b1/a1}})(\\simplify{{a2}x-{-b2}})(\\simplify{{a3}x-{-b3}}) \\end{split}\\]</p>\n<p>Therefore,&nbsp;</p>\n<p>\\[(\\simplify{x-{-b1/a1}})(\\simplify{{a2}x-{-b2}})(\\simplify{{a3}x-{-b3}}) = 0,\\]</p>\n<p>and the solutions for $x$ are \\[ x_1=\\simplify[fractionNumbers]{{sol1}}\\,,x_2=\\simplify[fractionNumbers]{{sol2}}\\,,x_3=\\simplify[fractionNumbers]{{sol3}}\\,.\\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "A": {"name": "A", "group": "Ungrouped variables", "definition": "a1*a2*a3", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,b1])", "description": "", "templateType": "anything", "can_override": false}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "random(-5..5 except[0,b2,b1])", "description": "", "templateType": "anything", "can_override": false}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "a1*a2*b3+a1*b2*a3+b1*a2*a3", "description": "", "templateType": "anything", "can_override": false}, "C": {"name": "C", "group": "Ungrouped variables", "definition": "a1*b2*b3+b1*a2*b3+b1*b2*a3", "description": "", "templateType": "anything", "can_override": false}, "D": {"name": "D", "group": "Ungrouped variables", "definition": "b1*b2*b3", "description": "", "templateType": "anything", "can_override": false}, "m1": {"name": "m1", "group": "Ungrouped variables", "definition": "random(3..7 #2)", "description": "", "templateType": "anything", "can_override": false}, "m2": {"name": "m2", "group": "Ungrouped variables", "definition": "random(-3..3 #2)", "description": "", "templateType": "anything", "can_override": false}, "m3": {"name": "m3", "group": "Ungrouped variables", "definition": "random(-3..3 #2)", "description": "", "templateType": "anything", "can_override": false}, "m4": {"name": "m4", "group": "Ungrouped variables", "definition": "random(-3..3 #2)", "description": "", "templateType": "anything", "can_override": false}, "n1": {"name": "n1", "group": "Ungrouped variables", "definition": "(m1-1)/2", "description": "", "templateType": "anything", "can_override": false}, "n2": {"name": "n2", "group": "Ungrouped variables", "definition": "(m2-1)/2", "description": "", "templateType": "anything", "can_override": false}, "n3": {"name": "n3", "group": "Ungrouped variables", "definition": "(m3-1)/2", "description": "", "templateType": "anything", "can_override": false}, "n4": {"name": "n4", "group": "Ungrouped variables", "definition": "(m4-1)/2", "description": "", "templateType": "anything", "can_override": false}, "sol1": {"name": "sol1", "group": "Ungrouped variables", "definition": "-b1/a1", "description": "", "templateType": "anything", "can_override": false}, "sol2": {"name": "sol2", "group": "Ungrouped variables", "definition": "-b2/a2", "description": "", "templateType": "anything", "can_override": false}, "sol3": {"name": "sol3", "group": "Ungrouped variables", "definition": "-b3/a3", "description": "", "templateType": "anything", "can_override": false}, "sol1dp": {"name": "sol1dp", "group": "Ungrouped variables", "definition": "precround(sol1,2)", "description": "", "templateType": "anything", "can_override": false}, "sol2dp": {"name": "sol2dp", "group": "Ungrouped variables", "definition": "precround(sol2,2)", "description": "", "templateType": "anything", "can_override": false}, "sol3dp": {"name": "sol3dp", "group": "Ungrouped variables", "definition": "precround(sol3,2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "-b1/a1=round(-b1/a1)", "maxRuns": 100}, "ungrouped_variables": ["a1", "a2", "a3", "b1", "b2", "b3", "A", "B", "C", "D", "m1", "m2", "m3", "m4", "n1", "n2", "n3", "n4", "sol1", "sol2", "sol3", "sol1dp", "sol2dp", "sol3dp"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>If $f(x)=g(x)-2h(x)$, calculate $f(x)$.</p>\n<p>$f(x)\\,=$ [[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{A}x^3+{B}x^2+{C}x+{D}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Solve $f(x)=0$.</p>\n<p></p>\n<p>$x=$[[0]]</p>\n<p><em>(You can give your answers as fractions or to 2 decimal places where necessary)</em></p>\n<p></p>", "gaps": [{"type": "list-of-numbers", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "list-of-numbers", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "settings": {"correctAnswer": "[{sol1dp},{sol2dp},{sol3dp}]", "allowFractions": true, "correctAnswerFractions": true, "isSet": true, "show_input_hint": true, "separator": ","}}], "settings": {"correctAnswer": "[{sol1},{sol2},{sol3}]", "allowFractions": true, "correctAnswerFractions": true, "isSet": true, "show_input_hint": true, "separator": ","}}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}