// Numbas version: exam_results_page_options
{"name": "HELM Book 1.2.2 exercises", "metadata": {"description": "<p>HELM Book 1.2.2 exercises</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", ""], "variable_overrides": [[], [], []], "questions": [{"name": "1.2.2.1 Write expressions with a single index", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Use the index laws to simplify 3 simple expressions;</p>\n<p>n^a*n^b,&nbsp;n^a/n^b,&nbsp;(n^a)^b, where n is a randomised variable or number, and a and b are randomised nonzero integers.</p>\n<p>Part of HELM Book 1.2</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Rewrite each of the following expressions with a single index. In each case, write the value of the index in the box.</p>", "advice": "<p>(a)</p>\n<p>\\[\\var{abase}^\\var{ex1_idx[0]}\\var{abase}^\\var{ex1_idx[1]}=\\var{abase}^{(\\var{ex1_idx[0]})+(\\var{ex1_idx[1]})}=\\var{abase}^{\\var{ex1_idx[0]+ex1_idx[1]}}\\]</p>\n<p>(b)</p>\n<p>\\[\\frac{\\var{bbase}^\\var{ex1_idx[2]}}{\\var{bbase}^\\var{ex1_idx[3]}}=\\var{bbase}^{(\\var{ex1_idx[2]})-(\\var{ex1_idx[3]})}= \\var{bbase}^{\\var{ex1_idx[2]-ex1_idx[3]} } \\]</p>\n<p>(c)</p>\n<p>\\[\\left(\\var{cbase}^\\var{ex1_idx[4]}\\right)^\\var{ex1_idx[5]}=\\var{cbase}^{(\\var{ex1_idx[4]})\\times (\\var{ex1_idx[5]}) }=\\var{cbase}^{\\var{ex1_idx[4]*ex1_idx[5]} } \\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"alphabet": {"name": "alphabet", "group": "Ungrouped variables", "definition": "['a','b','k','m','n','w','x','y','z']", "description": "", "templateType": "anything", "can_override": false}, "ex1_base": {"name": "ex1_base", "group": "Ungrouped variables", "definition": "[random(string(random(2..29)),random(alphabet)),\n  random(string(random(2..29)),random(alphabet)),\n  random(string(random(2..29)),random(alphabet))]", "description": "", "templateType": "anything", "can_override": false}, "ex1_idx": {"name": "ex1_idx", "group": "Ungrouped variables", "definition": "[random(random(-10..-2),random(2..10)),\n  random(random(-10..-2),random(2..10)),\n  random(random(-10..-2),random(2..10)),\n  random(random(-10..-2),random(2..10)),\n  random(random(-10..-2),random(2..10)),\n  random(random(-10..-2),random(2..10))\n]", "description": "", "templateType": "anything", "can_override": false}, "ex1_base_expr": {"name": "ex1_base_expr", "group": "Ungrouped variables", "definition": "[expression(ex1_base[0]),expression(ex1_base[1]),expression(ex1_base[2])]", "description": "", "templateType": "anything", "can_override": false}, "cbase": {"name": "cbase", "group": "Ungrouped variables", "definition": "expression(ex1_base[2])", "description": "<p>(m^7)^-5 is not accepting 1/m^35 as an answer, but it does accept m^-35</p>", "templateType": "anything", "can_override": false}, "ans1b": {"name": "ans1b", "group": "Ungrouped variables", "definition": "expression(ex1_base[1] + \"^\" + string(ex1_idx[2]-ex1_idx[3]))", "description": "", "templateType": "anything", "can_override": false}, "ans1a": {"name": "ans1a", "group": "Ungrouped variables", "definition": "expression(ex1_base[0] + \"^\" + string(ex1_idx[0]+ex1_idx[1]))", "description": "", "templateType": "anything", "can_override": false}, "abase": {"name": "abase", "group": "Ungrouped variables", "definition": "expression(ex1_base[0])", "description": "", "templateType": "anything", "can_override": false}, "bbase": {"name": "bbase", "group": "Ungrouped variables", "definition": "expression(ex1_base[1])", "description": "", "templateType": "anything", "can_override": false}, "ans1c": {"name": "ans1c", "group": "Ungrouped variables", "definition": "expression(ex1_base[0] + \"^\" + string(ex1_idx[4]*ex1_idx[5]))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["alphabet", "ex1_base", "ex1_idx", "ex1_base_expr", "ans1a", "ans1b", "ans1c", "abase", "bbase", "cbase"], "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>(a) $\\var{abase}^\\var{ex1_idx[0]}$ $\\var{abase}^\\var{ex1_idx[1]} =&nbsp; \\var{abase}^u$, where $u=$[[0]]</p>\n<p>(b) $\\frac{\\var{bbase}^\\var{ex1_idx[2]}}{\\var{bbase}^\\var{ex1_idx[3]}} = \\var{bbase}^u$, where $u=$[[1]]</p>\n<p>(c) $\\left(\\var{cbase}^\\var{ex1_idx[4]}\\right)^\\var{ex1_idx[5]} = \\var{cbase}^u$, where $u=$[[2]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "a index", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ex1_idx[0]+ex1_idx[1]", "maxValue": "ex1_idx[0]+ex1_idx[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "b index", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ex1_idx[2]-ex1_idx[3]", "maxValue": "ex1_idx[2]-ex1_idx[3]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "c index", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ex1_idx[4]*ex1_idx[5]", "maxValue": "ex1_idx[4]*ex1_idx[5]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.2.2 Remove the brackets index question", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Remove the brackets from (na)^k, or from n(a)^kwhere n is a number and a is a variable.</p>\n<p>Part of HELM Book 1.2</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Remove the brackets from $\\var{expr}$.</p>", "advice": "<p><span data-jme-visible=\"nv=0\">\\[\\var{expr}=\\var{c}\\var{ve[0]}^{\\var{idx}}\\var{ve[1]}^{\\var{idx}}\\]</span></p>\n<p><span data-jme-visible=\"nv=1\">\\[\\var{expr}=\\var{c}^{\\var{idx}}\\var{ve[0]}^{\\var{idx}}\\]</span></p>\n<p><span data-jme-visible=\"nv=2\">\\[\\var{expr}=\\var{c}^{\\var{idx}}\\var{ve[0]}^{\\var{idx}}\\var{ve[1]}^{\\var{idx}}\\]</span></p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"v": {"name": "v", "group": "Ungrouped variables", "definition": "shuffle(['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','u','v','w','x','y','z'])", "description": "", "templateType": "anything", "can_override": false}, "nv": {"name": "nv", "group": "Ungrouped variables", "definition": "random(0,1,2)", "description": "<p>number of variables to include.</p>\n<p>If nv=0 then 2 variables are used and the constant is not powered.</p>", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "idx": {"name": "idx", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "expr": {"name": "expr", "group": "Ungrouped variables", "definition": "if(nv=0,\n  expression(c+\"*(\"+v[0]+\"*\"+v[1]+\")^\"+idx)\n  ,\n  expression(\n    \"(\"+c+\"*\"+v[0]+ [\"\",\"*\"+v[1]][nv-1] +\")^(\"+idx+\")\"\n    )\n)  ", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "if(nv=0,\n  simplify(expression(c+\"*\"+v[0]+\"^\"+idx+\"*\"+v[1]+\"^\"+idx),\"all\"),\n  simplify(expression(\n    c+\"^\"+idx+\"*\"+v[0]+\"^\"+idx + [\"\",\"*\"+v[1]+\"^\"+idx][nv-1]\n  ),\"all\")\n)", "description": "", "templateType": "anything", "can_override": false}, "ve": {"name": "ve", "group": "Ungrouped variables", "definition": "[expression(v[0]),expression(v[1])]", "description": "<p>for display in the advice</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["v", "nv", "c", "idx", "expr", "ans", "ve"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["(", ")", "You must remove the brackets."], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.2.3 Simplify expression with indices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Simplify n1.v^k1.(n2.v^k2), where n1, n2 are positive integers, v is a random letter variable, and k1 and k2 are nonzero integers.</p>\n<p>The answer should be expressed as n.v^k</p>\n<p>Part of HELM Book 1.2</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Simplify $\\var{c[0]}\\var{v}^\\var{idx[0]}(\\var{c[1]}\\var{v}^\\var{idx[1]})$</p>", "advice": "<p>\\[\\begin{align*}\\var{c[0]}\\var{v}^\\var{idx[0]}(\\var{c[1]}\\var{v}^\\var{idx[1]})&amp;= \\var{c[0]}\\times\\var{c[1]}\\times\\var{ve}^{\\var{idx[0]}}\\times\\var{ve}^{\\var{idx[1]}}\\\\ &amp;=\\var{c[0]*c[1]}\\var{ve}^{(\\var{idx[0]})+(\\var{idx[1]})}\\\\ &amp;=\\var{expr} \\end{align*}\\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"v": {"name": "v", "group": "Ungrouped variables", "definition": "random(['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','u','v','w','x','y','z'])", "description": "<p>the variable to use</p>", "templateType": "anything", "can_override": false}, "idx": {"name": "idx", "group": "Ungrouped variables", "definition": "repeat(random(-12..12 except [0,1]),2)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "repeat(random(2..9),2)", "description": "", "templateType": "anything", "can_override": false}, "expr": {"name": "expr", "group": "Ungrouped variables", "definition": "simplify(expression(c[0]+\"*\"+v+\"^\"+idx[0]+\"(\"+c[1]+\"*\"+v+\"^\"+idx[1]+\")\"),\"all\")", "description": "", "templateType": "anything", "can_override": false}, "ve": {"name": "ve", "group": "Ungrouped variables", "definition": "expression(v)", "description": "<p>for display purposes in the advice.</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["v", "idx", "c", "expr", "ve"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{expr}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.001", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": ["0.5", "0.6"], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "$n*$v`?^?", "partialCredit": 0, "message": "You have not fully factorised the expression.<br />You need to write the expression as a single number times the variable to a single power.", "nameToCompare": ""}, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "menu", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": "", "allowAttemptDownload": false, "downloadEncryptionKey": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "<p>Work through these practice questions. You can try each question as many times as you like. The numbers will change each time. Some of the questions are randomly chosen, so if you re-open this set of exercises you may see some different questions.</p>", "end_message": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "extensions": [], "custom_part_types": [], "resources": []}