// Numbas version: exam_results_page_options {"timing": {"timeout": {"action": "none", "message": ""}, "allowPause": true, "timedwarning": {"action": "none", "message": ""}}, "showQuestionGroupNames": false, "duration": 0, "name": "LTDS demo", "shuffleQuestions": false, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

A collection of questions to show off at a workshop for Newcastle's LTDS, June 2016.

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The derivative of $x^n$ is given by the following:

\\[ \\frac{\\mathrm{d}}{\\mathrm{d}x}(x^n) = n \\times x^{n-1} \\]

Enter the derivatives of each of the three terms in $f(x)$:

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$\\frac{\\mathrm{d}}{\\mathrm{d}x}(\\simplify{{coefficients[2]}*x^{powers[2]}}) =$

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$\\frac{\\mathrm{d}}{\\mathrm{d}x}(\\simplify{{coefficients[1]}*x^{powers[1]}}) =$

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$\\frac{\\mathrm{d}}{\\mathrm{d}x}(\\simplify{{coefficients[0]}*x^{powers[0]}}) =$

Differentiate the following function.

\\[ f(x) = \\simplify[all,!noLeadingMinus]{ {coefficients[2]}*x^{powers[2]} + {coefficients[1]}*x^{powers[1]} + {coefficients[0]}*x^{powers[0]} } \\]

$\\frac{\\mathrm{d}f}{\\mathrm{d}x} = $ [[0]]

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Numbas is really good at creating and marking randomised maths questions. In this question, you're given a random polynomial to differentiate.

Notice how Numbas automatically simplifies the mathematical expressions so they look as if a human wrote them.

See this question in the public editor

", "advice": "

The derivative of $x^n$ is given by the following:

\\[ \\frac{\\mathrm{d}}{\\mathrm{d}x}(x^n) = n \\times x^{n-1} \\]

We can compute the derivative of $f(x)$ by computing the derivatives of each of the three terms, and then adding them together.

\\begin{align}
\\frac{\\mathrm{d}}{\\mathrm{d}x}(\\simplify{{coefficients[2]}*x^{powers[2]}}) &= \\simplify[basic]{{powers[2]}*{coefficients[2]}*x^({powers[2]}-1)} \\\\
&= \\simplify{{coefficients[2]*powers[2]}*x^{powers[2]-1}}
\\end{align}

\\begin{align}
\\frac{\\mathrm{d}}{\\mathrm{d}x}(\\simplify{{coefficients[1]}*x^{powers[1]}}) &= \\simplify[basic]{{powers[1]}*{coefficients[1]}*x^({powers[1]}-1)} \\\\
&= \\simplify{{coefficients[1]*powers[1]}*x^{powers[1]-1}}
\\end{align}

The derivative of a constant is $0$. So,

\\[ \\frac{\\mathrm{d}}{\\mathrm{d}x}(\\var{coefficients[0]}) = 0 \\]

\\begin{align}
\\frac{\\mathrm{d}}{\\mathrm{d}x}(\\simplify{{coefficients[0]}*x^{powers[0]}}) &= \\simplify[basic]{{powers[0]}*{coefficients[0]}*x^({powers[0]}-1)} \\\\
&= \\simplify{{coefficients[0]*powers[0]}*x^{powers[0]-1}}
\\end{align}

Hence,

\\[ \\frac{\\mathrm{d}f}{\\mathrm{d}x} = \\simplify{ {coefficients[2]*powers[2]}*x^{powers[2]-1} + {coefficients[1]*powers[1]}*x^{powers[1]-1} + {coefficients[0]*powers[0]}*x^{powers[0]-1} } \\]

", "functions": {}, "rulesets": {}}, {"name": "Numbas demo: multiple choice", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "parts": [{"extendBaseMarkingAlgorithm": true, "prompt": "

John, Paul, George and Ringo stand in alphabetical order. Who goes first?

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John

", "

Paul

", "

George

", "

Ringo

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Which of the following numbers are congruent to $1$ modulo $3$?

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", "

"], "showCorrectAnswer": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 0, "displayType": "checkbox", "displayColumns": "3", "shuffleChoices": false, "distractors": ["", "2 is not congruent to 1, mod 3", "3 is congruent to 0, mod 3", "", "5 is congruent to 2, mod 3", "6 is congruent to 0, mod 3", "", "8 is congruent to 2, mod 3", "9 is congruent to 0, mod 3"], "maxAnswers": "0", "matrix": ["1", "-1", "-1", "1", "-1", "-1", "1", "-1", "-1"], "type": "m_n_2", "warningType": "none", "minMarks": 0}, {"extendBaseMarkingAlgorithm": true, "adaptiveMarkingPenalty": 0, "maxMarks": 0, "showCellAnswerState": true, "showFeedbackIcon": true, "customName": "Match choices with answers", "unitTests": [], "customMarkingAlgorithm": "", "scripts": {}, "shuffleChoices": true, "useCustomName": true, "choices": ["Dodo", "Blue whale", "Human", "Pterodactyl"], "showCorrectAnswer": true, "variableReplacements": [], "shuffleAnswers": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "layout": {"expression": "", "type": "all"}, "displayType": "checkbox", "minAnswers": 0, "maxAnswers": 0, "matrix": [["1", "-1", "1"], ["1", "-1", "-1"], ["1", "-1", "-1"], ["-1", "1", "1"]], "type": "m_n_x", "warningType": "none", "minMarks": 0, "answers": ["Warm-blooded", "Can fly", "Extinct"]}], "statement": "

Numbas has comprehensive support for multiple choice questions. The order of choices can be randomised, the marking matrix can be calculated from question variables, and you can write specific feedback for each choice.

See this question in the public editor

", "rulesets": {}, "ungrouped_variables": [], "variables": {}, "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "tags": [], "functions": {}, "advice": "", "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": []}, {"name": "Numbas demo: part types", "extensions": [], "custom_part_types": [{"source": {"pk": 1, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/1/edit"}, "name": "Yes/no", "short_name": "yes-no", "description": "

The student is shown two radio choices: \"Yes\" and \"No\". One of them is correct.

", "help_url": "", "input_widget": "radios", "input_options": {"correctAnswer": "if(eval(settings[\"correct_answer_expr\"]), 0, 1)", "hint": {"static": true, "value": ""}, "choices": {"static": true, "value": ["Yes", "No"]}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\nif(studentanswer=correct_answer,\n correct(),\n incorrect()\n)\n\ninterpreted_answer:\nstudentAnswer=0\n\ncorrect_answer:\nif(eval(settings[\"correct_answer_expr\"]),0,1)", "marking_notes": [{"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=correct_answer,\n correct(),\n incorrect()\n)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "studentAnswer=0"}, {"name": "correct_answer", "description": "", "definition": "if(eval(settings[\"correct_answer_expr\"]),0,1)"}], "settings": [{"name": "correct_answer_expr", "label": "Is the answer \"Yes\"?", "help_url": "", "hint": "An expression which evaluates to true or false.", "input_type": "mathematical_expression", "default_value": "true", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "tags": [], "metadata": {"description": "

Showing off the part types.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Each part of a Numbas question asks the student to enter an answer, and is marked automatically. There are several part types, each with their own input methods and settings.

See this question on the public editor.

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The number entry part asks the student to write a single number. It's marked correct if it's in the accepted range.

I eat 5 apples per day. How many apples do I eat in a week?

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The mathematical expression part type asks the student to write a mathematical expression as their answer. It's marked correct if it's equivalent to the expected answer.

Differentiate the following function:

\\[ f(x) = \\simplify[all,!noLeadingMinus]{{c}x^2+{d}x+{f}} \\]

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The matrix entry part type asks the student to enter the elements of a matrix. It's marked correct if the student's matrix is equal to the expected matrix.

Enter a $3 \\times 3$ identity matrix.

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The match text pattern part type asks the student to enter a short string of text. It's marked correct if it matches the pattern specified by the author.

Write \"Numbas\".

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The choose one from a list part asks the student to choose one item from a list of options. Each option can award a different number of marks.

Which fruit is biggest?

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The choose several from a list part type asks the student to select one or more items from a list. Each option can award or subtract a different number of marks.

Tick every prime number in the list below.

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The match choices with answers part asks the student to match each of a list of 'choices' with a corresponding 'answer'. Each possible pair can award a different number of marks.

Match countries with their capital cities.

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It's possible to create custom part types, or use one somebody else has published. A custom part type consists of settings for question authors, an input widget, and a marking algorithm. This part uses the \"Yes/No\" custom part type, which provides a simple means of asking if the student agrees with a statement.

Was the abacus invented before the mobile phone?

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The gap-fill part type allows you to include multiple input areas in one block of text. They're marked independently of each other, but submitted simultaneously, and all the feedback is shown together.

My dog is 3 years older than my cat, who is half the dog's age. What are their ages?

My dog's age: [[0]]

My cat's age: [[1]]

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Customised for the Numbas demo exam

Motion under gravity. Object is projected vertically with initial velocity $V\\;m/s$. Find time to maximum height and the maximum height. Now includes an interactive plot.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

A Numbas question can include interactive graphics, such as this plot of the trajectory given by the student's answer.

See this question in the public editor

A ball is thrown upwards, and moves according to the equation $\\displaystyle{\\frac{d^2z}{dt^2}=-g}$
(where $z(t)$ is distance in metres measured upwards from the ground and the constant acceleration of gravity, $g$ , is given as $9.81\\;m/s^2$).

The ball is projected upwards with a speed $\\var{v}\\;m/s$.

", "advice": "

a)

Integrating $\\displaystyle{\\frac{\\mathrm{d}^2z}{\\mathrm{d}t^2}=-g}$ once gives the velocity $\\displaystyle{\\frac{\\mathrm{d}z}{\\mathrm{d}t}=-gt+A}$.

But $A=\\var{v}$ as the velocity is $\\var{V}\\;m/s$ at $t=0$.

So the velocity is

\\begin{align} \\frac{\\mathrm{d}z}{\\mathrm{d}t} &= \\var{v}-gt & (1) \\end{align}

Integrating again gives

\\[ z = \\var{v}t-\\frac{g}{2}t^2+B \\]

and $B=0$ as $z=0$ at $t=0$.

Hence the distance travelled upwards is given by

\\begin{align} z &= \\var{v}t-\\frac{g}{2}t^2 & (2) \\end{align}

{advicegraph()}

b)

The time $t_{\\text{max}}$ taken to reach maximum height is the time satisfying $\\displaystyle{\\frac{dz}{dt}=0}$

$t_{\\text{max}}$ is given from equation $(1)$ by

\\begin{align}
\\var{v} - gt_{\\text{max}} &= 0 \\\\
gt_{\\text{max}} &= \\var{v} \\\\
t_{\\text{max}} &= \\frac{\\var{v}}{g} \\\\[0.5em]
&= \\frac{\\var{v}}{9.81} \\\\[0.5em]
&= \\var{t1}
\\end{align}

(to $2$ decimal places)

This is at the point $A$ in the graph above.

c)

The maximum height $z_{\\text{max}}$ is given from equation $(2)$ by substituting in the value $t_{\\text{max}}= \\var{v}/g$, giving

\\begin{align}
z_{\\text{max}} &= \\var{v} \\times \\frac{\\var{v}}{g} - \\frac{g}{2}\\left(\\frac{\\var{v}}{g}\\right)^2 \\\\
&= \\frac{\\var{v}^2}{g}-\\frac{g\\var{v}^2}{2g^2} \\\\
&= \\frac{\\var{v}^2}{2g} \\\\
&= \\var{mh}\\;m
\\end{align}

(to $2$ decimal places)

This is at the point $B$ in the graph above.

", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"mh": {"name": "mh", "group": "Ungrouped variables", "definition": "precround(V^2/(2*g),2)", "description": "", "templateType": "anything", "can_override": false}, "t1": {"name": "t1", "group": "Ungrouped variables", "definition": "precround(t,2)", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "V/g", "description": "", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "random(50..100)", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "9.81", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["t", "v", "mh", "g", "t1"], "variable_groups": [], "functions": {"advicegraph": {"parameters": [], "type": "html", "language": "javascript", "definition": "var v = Numbas.jme.unwrapValue(scope.variables.v);\nvar mh = Numbas.jme.unwrapValue(scope.variables.mh);\nvar t = Numbas.jme.unwrapValue(scope.variables.t1);\nvar g = Numbas.jme.unwrapValue(scope.variables.g);\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',\n {boundingBox:[-3,mh+50,2*t+2,-50],\n axis:false,\n showNavigation:false,\n grid:false});\nvar brd = div.board; \n var xas = brd.create('line',[[0,0],[1,0]], { strokeColor: 'black',fixed:true});\n var xticks = brd.create('ticks',[xas,1],{\ndrawLabels: true,\nlabel: {offset: [-4, -10]},\nminorTicks: 0\n });\n var yas = brd.create('line',[[0,0],[0,1]], { strokeColor: 'black',fixed:true});\n var yticks = brd.create('ticks',[yas,50],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 0\n }); \nfunction traj(x){return v*x-g*Math.pow(x,2)/2;}\nvar p=brd.create('point',[0,0],{fixed:true,name:'',size:1});\n//var q=brd.create('line',[[0,0],[1,v]],{fixed:true,name:'',dash:2});\nvar r=brd.create('point',[t,0],{fixed:true,name:'A',size:1});\nvar s=brd.create('point',[t,mh],{fixed:true,name:'B',size:1});\nvar maxh=brd.create(\"segment\",[[0,mh],[t,mh]],{dash:1,strokeColor:'green'});\nvar gr=brd.create(\"functiongraph\",[traj,0,2*t],{strokeColor:'blue',strokeWidth:2});\nvar halft=brd.create(\"segment\",[[t,0],[t,traj(t)]],{dash:1,strokeColor:'green'});\nreturn div;"}, "graphsolution": {"parameters": [], "type": "html", "language": "javascript", "definition": "var v = Numbas.jme.unwrapValue(scope.variables.v);\nvar mh = Numbas.jme.unwrapValue(scope.variables.mh);\nvar t = Numbas.jme.unwrapValue(scope.variables.t1);\nvar g = Numbas.jme.unwrapValue(scope.variables.g);\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',\n {boundingBox:[-3,mh+10,2*t+2,-50],\n axis:false,\n showNavigation:false,\n grid:false});\nvar brd = div.board; \nvar xas = brd.create('line',[[0,0],[1,0]], { strokeColor: 'black',fixed:true});\nvar xticks = brd.create('ticks',[xas,1],{\n drawLabels: true,\n label: {offset: [-4, -10]},\n minorTicks: 0\n});\nvar yas = brd.create('line',[[0,0],[0,1]], { strokeColor: 'black',fixed:true});\nvar yticks = brd.create('ticks',[yas,50],{\n drawLabels: true,\n label: {offset: [-20, 0]},\n minorTicks: 0\n}); \nvar p=brd.create('point',[0,0],{fixed:true,name:'',size:1});\nvar q=brd.create('line',[[0,0],[1,v]],{fixed:true,name:'',dash:2});\nvar tree;\n//x is the variable in the equation to be input\nvar nscope = new Numbas.jme.Scope([scope,{variables:{t:new Numbas.jme.types.TNum(0)}}]);\n//create a functiongraph from the student input\nfunction userf(t){\n if(tree) {\n try {\n nscope.variables.t.value = t;\n //the user input is evaluated at x=t\n var val = Numbas.jme.evaluate(tree,nscope).value;\n return val;\n }\n catch(e) {\n return 0;\n }\n }\n else\n return 0;\n}\nvar curve=brd.create('functiongraph',[userf,0,2*t],{strokeColor:'red',strokeWidth:2});\n\n//pick up the student answer and is parsed\nquestion.signals.on('HTMLAttached',function() {\n ko.computed(function(){\n var expr = question.parts[0].gaps[0].display.studentAnswer();\n try {\n tree = Numbas.jme.compile(expr,scope);\n tree = scope.expandJuxtapositions(tree,{singleLetterVariables:true});\n }\n catch(e) {\n tree = null;\n }\n curve.updateCurve();\n\n brd.update();\n });\n}); \n\nreturn div;"}}, "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": "

{graphsolution()}

Input the vertical distance $z$ as a function of $t$.

Note that at $t=0$ we have $z=0$ and that $\\displaystyle \\frac{dz}{dt}=\\var{v}m/s$.

Input gravitational acceleration as $g$.

$z=$ [[0]]

Your formula is plotted in the graph above. The vertical axis represents $z$ and the horizontal axis represents $t$.

Note that the blue line indicates that:

Your solution should go through $(0,0)$;
Your solution should have this line as the tangent to the curve at $(0,0)$, because $\\displaystyle \\frac{\\mathrm{d}z}{\\mathrm{d}t}=\\var{v}\\; m/s$.

", "gaps": [{"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": "{v}t-1/2g*t^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "g", "value": ""}, {"name": "t", "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": "

Time taken to reach maximum height = [[0]] $s$ (accurate to $2$ decimal places)

Maximum height = [[1]] $m$ (accurate to $2$ decimal places)

", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "t1", "maxValue": "t1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "mh", "maxValue": "mh", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Numbas demo: video", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "variable_groups": [], "parts": [{"stepsPenalty": 1, "gaps": [{"vsetrange": [11, 12], "showpreview": true, "marks": 3, "notallowed": {"partialCredit": 0, "strings": ["."], "message": "

Input all numbers as fractions or integers and not decimals.

", "showStrings": false}, "answer": "({d-a*c}/{b-a})*ln(x+{a})+({d-b*c}/{a-b})*ln(x+{b})+C", "vsetrangepoints": 5, "variableReplacements": [], "showCorrectAnswer": true, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "scripts": {}, "checkingtype": "absdiff", "checkvariablenames": false, "answersimplification": "std", "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "marks": 0, "scripts": {}, "steps": [{"type": "information", "marks": 0, "scripts": {}, "variableReplacements": [], "showCorrectAnswer": true, "prompt": "

First of all, factorise the denominator.

You have to find $a$ and $b$ such that $\\simplify[std]{x^2+{a+b}*x+{a*b}=(x+a)*(x+b)}$

Then use partial fractions to write:
\\[\\simplify[std]{({c}*x+{d})/((x +a)*(x+b)) = A/(x+a)+B/(x+b)}\\]

for suitable integers or fractions $A$ and $B$.

This video solves a similar, simpler example.

", "variableReplacementStrategy": "originalfirst"}], "variableReplacements": [], "showCorrectAnswer": true, "prompt": "

$I=$ [[0]]

Enter the constant of integration as $C$.

Click on Show steps for help if you need it: you'll be given a hint, and see a video which solves a similar example.

", "variableReplacementStrategy": "originalfirst"}], "statement": "

It's easy to include videos in Numbas questions. In this question, if the student gets stuck they can click on \"Show steps\" to be given a hint, and shown a video of someone working through a similar problem.

See this question in the public editor

Find the following integral.

\\[I = \\simplify[std]{Int(({c}*x+{d})/(x^2+{a+b}*x+{a*b}),x )}\\]

", "showQuestionGroupNames": false, "tags": ["2 distinct linear factors", "Calculus", "calculus", "completing the square", "constant of integration", "factorising a quadratic", "indefinite integration", "integrals", "integration", "logarithms", "partial fractions", "Steps", "steps", "two distinct linear factors", "video"], "ungrouped_variables": ["a", "c", "b", "d", "s3", "s2", "s1", "b1", "d1"], "functions": {}, "preamble": {"js": "", "css": ""}, "metadata": {"description": "

Customised for the Numbas demo exam

Factorise $x^2+cx+d$ into 2 distinct linear factors and then find $\\displaystyle \\int \\frac{ax+b}{x^2+cx+d}\\;dx,\\;a \\neq 0$ using partial fractions or otherwise.

Video in Show steps.

", "notes": "\n \t\t \t\t

5/08/2012:

\n \t\t \t\t

Added tags.

\n \t\t \t\t

Added description.

\n \t\t \t\t

Added decimal point as forbidden string.

\n \t\t \t\t

Note the checking range is chosen so that the arguments of the log terms are always positive - could have used abs - might be better?

\n \t\t \t\t

Improved display of Advice.

\n \t\t \t\t

Added information about Show steps, also introduced penalty of 1 mark.

\n \t\t \t\t

Added !noLeadingMinus to ruleset std for display purposes.

\n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "questions": [], "pickQuestions": 0}], "type": "question", "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "

First we factorise $\\simplify[std]{x^2+{a+b}*x+{a*b}=(x+{a})*(x+{b})}$. You can do this by spotting the factors or by completing the square.

Next we use partial fractions to find $A$ and $B$ such that

\\[ \\simplify[std]{({c}*x+{d})/((x +{a})*(x+{b})) = A/(x+{a})+B/(x+{b})} \\]

Multiplying both sides of the equation by $\\displaystyle \\simplify[std]{1/((x +{a})*(x+{b}))}$, we obtain

\\begin{align}
&& \\simplify[std]{A*(x+{b})+B*(x+{a})} &= \\simplify[std]{{c}*x+{d}}\\\\
\\Rightarrow && \\simplify[std]{(A+B)*x+{b}*A+{a}*B} &= \\simplify[std]{{c}*x+{d}}
\\end{align}

Coefficients of similar powers of $x$ on each side of the equation must be equal, so we can write down two new equations identifying the coefficients on each side:

Constant term: $\\simplify[std]{{b}*A+{a}*B = {d}}$

Coefficent of $x$: $ \\simplify[std]{A+B={c}}$ which gives $A =\\var{c} -B$

On solving these equations, we obtain $\\displaystyle \\simplify[std]{A = {d-a*c}/{b-a}}$ and $\\displaystyle \\simplify[std]{B={d-b*c}/{a-b}}$, which gives

\\[ \\simplify[std]{({c}*x+{d})/((x +{a})*(x+{b})) = ({d-a*c}/{b-a})*(1/(x+{a}) )+({d-b*c}/{a-b})*(1/(x+{b}))} \\]

\\begin{align}
I &= \\simplify[std]{int(({c}*x+{d})/(x^2+{a+b}*x+{a*b}),x )} \\\\[0.5em]
&= \\simplify[std]{int(({c}*x+{d})/((x +{a})*(x+{b})),x )} \\\\[0.5em]
&= \\simplify[std]{({d-a*c}/{b-a})*(int(1/(x+{a}),x)) +({d-b*c}/{a-b})int(1/(x+{b}),x)} \\\\[0.5em]
&= \\simplify[std]{({d-a*c}/{b-a})*ln(x+{a})+({d-b*c}/{a-b})*ln(x+{b})+C}
\\end{align}

", "variables": {"c": {"name": "c", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)"}, "s2": {"name": "s2", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)"}, "b1": {"name": "b1", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)"}, "a": {"name": "a", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)"}, "d": {"name": "d", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "if(d1=a*c,if(d1+1=b*c,d1+2,d1+1),if(d1=b*c,if(d1+1=a*c,d1+2,d1+1),d1))"}, "b": {"name": "b", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "if(b1=a,b1+s3,b1)"}, "s3": {"name": "s3", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)"}, "d1": {"name": "d1", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..9)"}, "s1": {"name": "s1", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)"}}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}}, {"name": "Adaptive marking: Independent two sample t-test", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "tags": ["average", "data analysis", "differences", "elementary statistics", "hypothesis testing", "mean", "standard deviation", "statistics", "stats", "t-test", "two sample t-test", "variance"], "metadata": {"description": "

Two sample t-test to see if there is a difference between scores on questions between two groups when the questions are asked in a different order.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

An educational psychologist claimed that the order in which questions were asked affected the student’s ability to answer them correctly and hence their total score. In order to test this, $20$ students were randomly divided into two groups of $10$. The first group were given questions in increasing order of difficulty and the second group in decreasing order of difficulty. The ordered test scores obtained were:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

Group 1	{r1[0]}	{r1[1]}	{r1[2]}	{r1[3]}	{r1[4]}	{r1[5]}	{r1[6]}	{r1[7]}	{r1[8]}	{r1[9]}
Group 2	{r2[0]}	{r2[1]}	{r2[2]}	{r2[3]}	{r2[4]}	{r2[5]}	{r2[6]}	{r2[7]}	{r2[8]}	{r2[9]}

Carry out a two-sample t-test to decide if there is evidence of a difference in the average test scores for the two sets of students.

", "advice": "

We test the following hypothesis,

$H_0:\\; \\mu_1=\\mu_2$ versus $H_1:\\; \\mu_1 \\neq \\mu_2$

We find that the mean score of Group 1 is $\\overline{x}_1=\\var{mean1}$ with standard deviation $s_1=\\var{sd1}$ and the mean score of Group 2 is $\\overline{x}_2=\\var{mean2}$ with standard deviation $s_2=\\var{sd2}$.

(All calculated to 3 decimal places.)

Using the formula for the two-sample $t$-statistic as shown above with $n_1=n_2=10$:

The estimate of the pooled variance is calculated to be:

\\[s^2=\\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}= \\frac{\\var{n1-1}\\times \\var{sd1}^2+\\var{n2-1}\\times \\var{sd2}^2}{\\var{n1+n2-2}}=\\var{s^2}.\\]

Hence $s = \\sqrt{\\var{s^2}}=\\var{s}$ to 3 decimal places.

We find that the t-statistic has value:

\\begin{align}
T &= \\frac{(\\overline{x}_1-\\overline{x}_2)-(\\mu_1-\\mu_2)}{s\\sqrt{\\frac{1}{n_1}+\\frac{1}{n_2}}} \\\\
&= \\frac{(\\var{mean1}-\\var{mean2})-(0)}{\\var{s}\\sqrt{\\frac{1}{\\var{n1}}+\\frac{1}{\\var{n2}}}} \\\\
&= \\var{t_statistic}
\\end{align}

Our test statistic is $|T|=\\var{abs(t_statistic)}$.

Given that we have $n_1+n_2-2=18$ degrees of freedom, we look up this value on the T-distribution table for $t_{18}$

\\[\\begin{array}{r|rrrrr}&0.10&0.05&0.01&0.001\\\\\\hline18&1.734&2.101&2.878&3.922\\end{array}\\]

We see that the t-statistic {t_statistic_range} and the table tells us that the $p$ value {p_value_range}.

Hence we conclude that we {reject} the null hypothesis. There is {evidence_strength} evidence of a difference between the average scores of the two groups.

", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"p_value_range": {"name": "p_value_range", "group": "Advice messages", "definition": "['is less than $0.001$','lies between $0.001$ and $0.01$','lies between $0.01$ and $0.05$','lies between $0.05$ and $0.10$','is greater than $0.10$'][scenario]", "description": "

Describe where the p-value lies in relation to the critical values

", "templateType": "anything", "can_override": false}, "sigma2": {"name": "sigma2", "group": "Setup", "definition": "random(8..10#0.2)", "description": "

Population standard deviation of sample 2

", "templateType": "anything", "can_override": false}, "t99": {"name": "t99", "group": "Critical t-values", "definition": "2.878", "description": "", "templateType": "anything", "can_override": false}, "n2": {"name": "n2", "group": "Setup", "definition": "10", "description": "

Size of sample 2

", "templateType": "anything", "can_override": false}, "t999": {"name": "t999", "group": "Critical t-values", "definition": "3.922", "description": "", "templateType": "anything", "can_override": false}, "t_statistic_range": {"name": "t_statistic_range", "group": "Advice messages", "definition": "['is greater than $\\\\var{t999}$','lies between $\\\\var{t99}$ and $\\\\var{t999}$','lies between $\\\\var{t95}$ and $\\\\var{t99}$','lies between $\\\\var{t90}$ and $\\\\var{t95}$','is less than $\\\\var{t90}$'][scenario]", "description": "

Describe where the t-statistic lies in relation to the critical values

", "templateType": "anything", "can_override": false}, "mu2": {"name": "mu2", "group": "Setup", "definition": "random(65..75#0.5)", "description": "

Population mean of sample 2

", "templateType": "anything", "can_override": false}, "evidence_strength": {"name": "evidence_strength", "group": "Advice messages", "definition": "['very strong','strong','slight','no','no'][scenario]", "description": "

How much evidence is there against the null hypothesis?

", "templateType": "anything", "can_override": false}, "n1": {"name": "n1", "group": "Setup", "definition": "10", "description": "

Size of sample 1

", "templateType": "anything", "can_override": false}, "mu1": {"name": "mu1", "group": "Setup", "definition": "random(55..75#0.5)", "description": "

Population mean of sample 1 (we'll generate samples from different distributions to produce different outcomes)

", "templateType": "anything", "can_override": false}, "reject": {"name": "reject", "group": "Advice messages", "definition": "if(scenario<2,'do reject','do not reject')", "description": "

Do we reject the null hypothesis?

", "templateType": "anything", "can_override": false}, "p_value": {"name": "p_value", "group": "Stats", "definition": "ttest(abs(t_statistic),19,2)", "description": "

p-value corresponding to the t-statistic

", "templateType": "anything", "can_override": false}, "t_statistic": {"name": "t_statistic", "group": "Stats", "definition": "(mean1-mean2)*sqrt(n1*n2)/(s*sqrt(n1+n2))", "description": "", "templateType": "anything", "can_override": false}, "t90": {"name": "t90", "group": "Critical t-values", "definition": "1.734", "description": "", "templateType": "anything", "can_override": false}, "decision_marking_matrix": {"name": "decision_marking_matrix", "group": "Advice messages", "definition": "[\n [1,0,0,0,0],\n [0,1,0,0,0],\n [0,0,1,0,0],\n [0,0,0,1,0],\n [0,0,0,0,1]\n][scenario]", "description": "

Marking matrix for the multiple choice questions

", "templateType": "anything", "can_override": false}, "sd2": {"name": "sd2", "group": "Stats", "definition": "precround(pstdev(r2),3)", "description": "

Sample standard deviation of sample 2

", "templateType": "anything", "can_override": false}, "t95": {"name": "t95", "group": "Critical t-values", "definition": "2.101", "description": "", "templateType": "anything", "can_override": false}, "r1": {"name": "r1", "group": "Samples", "definition": "repeat(round(normalsample(mu1,sigma1)),n1)", "description": "

Sample 1

", "templateType": "anything", "can_override": false}, "mean1": {"name": "mean1", "group": "Stats", "definition": "mean(r1)", "description": "

Sample mean of sample 1

", "templateType": "anything", "can_override": false}, "mean2": {"name": "mean2", "group": "Stats", "definition": "mean(r2)", "description": "

Sample mean of sample 1

", "templateType": "anything", "can_override": false}, "r2": {"name": "r2", "group": "Samples", "definition": "repeat(round(normalsample(mu2,sigma2)),n2)", "description": "

Sample 2

", "templateType": "anything", "can_override": false}, "scenario": {"name": "scenario", "group": "Advice messages", "definition": "sum(map(award(1,abs(t_statistic)Which scenario are we in - how many critical values of the t distribution does t_statistic exceed?

", "templateType": "anything", "can_override": false}, "sigma1": {"name": "sigma1", "group": "Setup", "definition": "random(8..10#0.2)", "description": "

Population standard deviation of sample 1

", "templateType": "anything", "can_override": false}, "sd1": {"name": "sd1", "group": "Stats", "definition": "precround(pstdev(r1),3)", "description": "

Sample standard deviation of sample 1

", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Stats", "definition": "precround(sqrt(((n1-1)*sd1^2+(n2-1)*sd2^2)/(n1+n2-2)),3)", "description": "

Used in the formula for the t statistic

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Find the mean and standard deviations of the scores of the two groups. Round your answers to 3 decimal places.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

	Mean	Standard deviation
Group 1	[[0]]	[[1]]
Group 2	[[2]]	[[3]]

Now find the two sample t-test statistic $T$ using the values you have just calculated and enter it here: [[4]]

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The two-sample t-statistic for two independent sets of data where one set has $n_1$ data points and the other set $n_2$ data points is calculated as follows:

\\[T = \\frac{(\\overline{x}_1-\\overline{x}_2)-(\\mu_1-\\mu_2)}{s\\times\\sqrt{\\frac{1}{n_1}+\\frac{1}{n_2}}}\\;\\;\\;\\]

where $\\overline{x}_1,\\;\\overline{x}_2$ are the sample means and

\\[s^2=\\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}\\]

where $s_1,\\;s_2$ are the sample standard deviations.

Use the values you calculated to 3 decimal places in order to find $T$.

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You have not given your answer to the correct precision.

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Given the value $|T|$ of the t-statistic you have found, choose the range for the $p$ value by looking up the t tables:

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$p$ is less than $0.1\\%$

", "

$p$ lies between $0.1\\%$ and $1\\%$

", "

$p$ lies between $1 \\%$ and $5\\%$

", "

$p$ lies between $5 \\%$ and $10\\%$

", "

$p$ is greater than $10\\%$

"], "matrix": "decision_marking_matrix"}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "scenario", "part": "p1", "must_go_first": false}], "variableReplacementStrategy": "alwaysreplace", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Given the $p$-value and the range you have found, what is the strength of evidence against the null hypothesis that there is no difference in the average times for the left and right hands?

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Very Strong Evidence

", "

Strong Evidence

", "

Evidence

", "

Weak Evidence

", "

No Evidence

"], "matrix": "decision_marking_matrix"}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "scenario", "part": "p2", "must_go_first": false}], "variableReplacementStrategy": "alwaysreplace", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

What do you decide based on the above analysis?

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": "1", "showCellAnswerState": true, "choices": ["

We reject the null hypothesis at the $0.1\\%$ level

", "

We reject the null hypothesis at the $1\\%$ level.

", "

We reject the null hypothesis at the $5\\%$ level.

", "

We do not reject the null hypothesis but consider further investigation.

", "

We do not reject the null hypothesis.

"], "matrix": "decision_marking_matrix"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "GeoGebra test - motion on a slope", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "functions": {}, "ungrouped_variables": [], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "

Either construct the intersection of two circles centred at $\\mathbf{a}$ and $\\mathbf{b}$, or use the Regular Polygon tool.

(I could embed another GeoGebra applet here)

", "rulesets": {}, "parts": [{"stepsPenalty": 0, "precisionType": "dp", "prompt": "

What is the total force acting on the mass, along the slope? Enter your answer in $\\mathrm{N}$, to 2 decimal places.

", "precisionMessage": "

You have not given your answer to the correct precision.

", "allowFractions": false, "variableReplacements": [], "maxValue": "mass*acceleration", "strictPrecision": false, "minValue": "mass*acceleration", "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

There are two forces acting on the mass: gravity and friction.

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}, {"precisionType": "dp", "prompt": "

What is the force due to gravity, in the direction of the slope? Enter your answer in $\\mathrm{N}$, to 2 decimal places.

", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "mass*a_gravity", "strictPrecision": false, "minValue": "mass*a_gravity", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "

What is the force due to friction, in the direction of the slope? Enter your answer in $\\mathrm{N}$, to 2 decimal places.

", "precisionMessage": "

You have not given your answer to the correct precision.

", "allowFractions": false, "variableReplacements": [], "maxValue": "mass*a_friction", "strictPrecision": false, "minValue": "mass*a_friction", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": "3", "type": "numberentry", "showPrecisionHint": false}, {"displayColumns": 0, "prompt": "

What happens to the mass next?

", "matrix": "if(acceleration=0,[0,1,0],[1,0,0])", "shuffleChoices": false, "variableReplacements": [], "choices": ["

It moves down the slope.

", "

It moves up the slope.

", "

It remains stationary.

"], "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}, {"precisionType": "dp", "prompt": "

At what time does the mass reach the ground? Enter your answer in seconds to 2 decimal places, or $0$ if the mass never reaches the ground.

", "precisionMessage": "

You have not given your answer to the correct precision.

", "allowFractions": false, "variableReplacements": [], "maxValue": "t_ground", "strictPrecision": false, "minValue": "t_ground", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "

A mass of $\\var{mass}\\,\\mathrm{kg}$ is resting on a plane inclined at $\\var{incline}^{\\circ}$ to the horizontal. The distance along the plane from the ground to the mass is $\\var{distance}\\mathrm{m}$.

A gravitational force of $9.8\\,\\mathrm{N/kg}$ is acting on the mass, and the coefficient of friction between the plane and the mass is $\\mu = \\var{c_friction}$.

{geogebra_applet('xn3p5x73',[[\"height\",height],[\"c_\\{friction\\}\",c_friction],[\"mass\",mass]],[])}

", "variable_groups": [{"variables": ["height", "c_friction", "mass", "incline", "distance"], "name": "Setup"}, {"variables": ["slope", "gravity", "a_gravity", "a_friction", "acceleration_naive", "acceleration"], "name": "Acceleration"}, {"variables": ["t_ground"], "name": "Answer"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"slope": {"definition": "vector(cos(radians(incline)),sin(radians(incline)))", "templateType": "anything", "group": "Acceleration", "name": "slope", "description": ""}, "acceleration": {"definition": "if(acceleration_naive<0,acceleration_naive,0)", "templateType": "anything", "group": "Acceleration", "name": "acceleration", "description": ""}, "incline": {"definition": "precround(degrees(arctan(height/50)),1)", "templateType": "anything", "group": "Setup", "name": "incline", "description": ""}, "t_ground": {"definition": "if(acceleration=0,0,sqrt(-2*distance/acceleration))", "templateType": "anything", "group": "Answer", "name": "t_ground", "description": ""}, "distance": {"definition": "45", "templateType": "anything", "group": "Setup", "name": "distance", "description": ""}, "a_friction": {"definition": "let(f,dot(matrix([[0,1],[-1,0]])*slope,gravity)*c_friction,\n min(f,-a_gravity)\n)", "templateType": "anything", "group": "Acceleration", "name": "a_friction", "description": ""}, "c_friction": {"definition": "random(0.01..0.5#0.01)", "templateType": "anything", "group": "Setup", "name": "c_friction", "description": ""}, "gravity": {"definition": "vector(0,-9.8)", "templateType": "anything", "group": "Acceleration", "name": "gravity", "description": ""}, "a_gravity": {"definition": "dot(gravity,slope)", "templateType": "anything", "group": "Acceleration", "name": "a_gravity", "description": ""}, "mass": {"definition": "random(5..35#5)", "templateType": "anything", "group": "Setup", "name": "mass", "description": ""}, "acceleration_naive": {"definition": "a_gravity+a_friction", "templateType": "anything", "group": "Acceleration", "name": "acceleration_naive", "description": ""}, "height": {"definition": "random(10..40)", "templateType": "anything", "group": "Setup", "name": "height", "description": ""}}, "metadata": {"description": "

Given the gradient of a slope and the coefficient of friction for a mass resting on it, use the equations of motion to calculate how it moves.

Includes a GeoGebra rendering of the model.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Embed geogebra with deployggb.js", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "functions": {}, "ungrouped_variables": ["a", "a_def", "b", "b_def"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "

Either construct the intersection of two circles centred at $\\mathbf{a}$ and $\\mathbf{b}$, or use the Regular Polygon tool.

Here's one way of doing it:

{geogebra_applet('eG6ezhKk',[[\"A\",a_def],[\"B\",b_def]],[])}

", "rulesets": {}, "parts": [{"prompt": "

Construct an equilateral triangle with corners at the two given points, using the Polygon tool (it's not sufficient to create line segments between the corners).

{geogebra_applet('HjFhq32N',[[\"A\",a_def],[\"B\",b_def]],[[\"Tool1\",\"p0g0\"]])}

When your construction is complete, this part will be marked correct.

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "extension"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

In this question, the position of point $\\mathbf{a}$ is picked randomly, and $\\mathbf{B}$ is 3 units to its left.

$\\mathbf{a} = \\var{a}$

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "vector(random(3..5#0.25),random(1..2#0.25))", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b_def": {"definition": "\"(\"+b[0]+\",\"+b[1]+\")\"", "templateType": "anything", "group": "Ungrouped variables", "name": "b_def", "description": ""}, "b": {"definition": "a-vector(3,0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "a_def": {"definition": "\"(\"+a[0]+\",\"+a[1]+\")\"", "templateType": "anything", "group": "Ungrouped variables", "name": "a_def", "description": ""}}, "metadata": {"description": "

A sneak peek at a really clever integration of GeoGebra with Numbas.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}], "pickingStrategy": "all-ordered", "name": "", "pickQuestions": 0}], "allQuestions": true, "navigation": {"showresultspage": "oncompletion", "onleave": {"action": "none", "message": ""}, "browse": true, "preventleave": false, "allowregen": true, "reverse": true, "showfrontpage": true}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": ["geogebra", "jsxgraph", "stats"], "custom_part_types": [{"source": {"pk": 1, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/1/edit"}, "name": "Yes/no", "short_name": "yes-no", "description": "

The student is shown two radio choices: \"Yes\" and \"No\". One of them is correct.