// Numbas version: exam_results_page_options
{"name": "Null factor law", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "c", "tworoots", "blist", "d", "f", "g", "h", "j", "k", "l", "plist", "m", "n", "p", "q", "fiveroots"], "name": "Null factor law", "tags": ["factorisation", "Factorisation", "factorised", "null factor law", "product", "solving", "zero divisor"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "1", "prompt": "<p>Given that $\\displaystyle{(\\simplify{x+{a}})(\\simplify{{b}x+{c}})=0}$. Determine the set of possible values of $x$.</p>\n<p>$x=$&nbsp;[[0]]</p>\n<p><strong>Note:</strong>&nbsp;if your answer is $1$ and $2$ input&nbsp;<strong>set(1,2)</strong></p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "fractionNumbers", "scripts": {}, "answer": "{tworoots}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "<p>Notice, this is a quadratic that has already been factorised. You don't need to expand it since we have the null factor law:</p>\n<p>\\[\\text{If } ab=0, \\text{ then } a=0 \\text{ or } b=0.\\]</p>\n<p></p>\n<p>Since</p>\n<p>\\[(\\simplify{x+{a}})(\\simplify{{b}x+{c}})=0,\\]</p>\n<p>this means $\\simplify{x+{a}}=0$ or $\\simplify{{b}x+{c}}=0$. Solving each of these equations gives $x=\\var{-a}$ or $x=\\simplify{-{c}/{b}}$.</p>\n<p>For this question,&nbsp;we input our answer as set$\\left(\\var{-a},\\simplify{-{c}/{b}}\\right)$.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "<p>Solve $\\displaystyle{\\simplify{{l}a}(\\simplify{a+{d}})(\\simplify{{f}a+{g}})\\left(\\simplify{{h}a+{j}/{k}}\\right)\\left(\\simplify{{m}a/{n}+{p}/{q}}\\right)=0}$ for $a$.</p>\n<p>$a=$&nbsp;[[0]]</p>\n<p><strong>Note:</strong>&nbsp;if your answer is $1$, $2$ and $3$ input&nbsp;<strong>set(1,2,3)</strong></p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "fractionNumbers", "scripts": {}, "answer": "{fiveroots}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "<p>Notice, this expression that has already been factorised. You don't need to expand it since we have the null factor law:&nbsp;</p>\n<p>\\[\\text{If } ab=0, \\text{ then } a=0 \\text{ or } b=0.\\]</p>\n<p>Since</p>\n<p>\\[\\simplify{{l}a}(\\simplify{a+{d}})(\\simplify{{f}a+{g}})\\left(\\simplify{{h}a+{j}/{k}}\\right)\\left(\\simplify{{m}a/{n}+{p}/{q}}\\right)=0,\\]</p>\n<p>this means $\\simplify{{l}a}=0$, $\\simplify{a+{d}}=0$, $\\simplify{{f}a+{g}}=0$, $\\simplify{{h}a+{j}/{k}}=0$, or $\\simplify{{m}a/{n}+{p}/{q}}=0$. Solving each of these equations gives $x=0$, $x=\\var{-d}$, $x=\\var[fractionnumbers]{-g/f}$, $x=\\var[fractionnumbers]{-j/(k*h)}$, or $x=\\var[fractionnumbers]{(-p*n)/(q*m)}$.</p>\n<p>For this question,&nbsp;we input our answer as set$\\left(0,\\var{-d},\\var[fractionnumbers]{-g/f},\\var[fractionnumbers]{-j/(k*h)},\\var[fractionnumbers]{(-p*n)/(q*m)}\\right)$.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-12..12 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(-12..12 except [0,a*b])", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..12)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "blist[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "g": {"definition": "blist[2]", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}, "f": {"definition": "blist[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "h": {"definition": "blist[3]", "templateType": "anything", "group": "Ungrouped variables", "name": "h", "description": ""}, "k": {"definition": "abs(j)+1", "templateType": "anything", "group": "Ungrouped variables", "name": "k", "description": ""}, "j": {"definition": "blist[4]", "templateType": "anything", "group": "Ungrouped variables", "name": "j", "description": ""}, "m": {"definition": "(-1)^(random(0,1))*plist[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "fiveroots": {"definition": "set(0,-d,-g/f,-j/(k*h),(-p*n)/(q*m))", "templateType": "anything", "group": "Ungrouped variables", "name": "fiveroots", "description": ""}, "l": {"definition": "blist[5]", "templateType": "anything", "group": "Ungrouped variables", "name": "l", "description": ""}, "q": {"definition": "plist[3]", "templateType": "anything", "group": "Ungrouped variables", "name": "q", "description": ""}, "p": {"definition": "(-1)^(random(0,1))*plist[2]", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "blist": {"definition": "shuffle(-12..12 except 0)[0..6]", "templateType": "anything", "group": "Ungrouped variables", "name": "blist", "description": ""}, "tworoots": {"definition": "set(-a,-c/b)", "templateType": "anything", "group": "Ungrouped variables", "name": "tworoots", "description": ""}, "n": {"definition": "plist[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "plist": {"definition": "shuffle([2,3,5,7,11,13,19,23,29])[0..4]", "templateType": "anything", "group": "Ungrouped variables", "name": "plist", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}