// Numbas version: exam_results_page_options {"percentPass": "40", "timing": {"timeout": {"action": "warn", "message": "

il tempo!

"}, "allowPause": false, "timedwarning": {"action": "warn", "message": "

tra cinque minuti consegna

"}}, "showstudentname": true, "name": "test ingresso quarta F", "feedback": {"advicethreshold": 0, "showtotalmark": true, "feedbackmessages": [], "showactualmark": true, "showanswerstate": true, "allowrevealanswer": false, "intro": ""}, "duration": 3600, "question_groups": [{"pickQuestions": 1, "pickingStrategy": "all-shuffled", "name": "Geometria analitica", "questions": [{"name": "riccardo's copy of Julie's copy of Q9 - Coordinate Geometry, Line and Parabola", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "riccardo belle'", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1797/"}], "parts": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "showFeedbackIcon": true, "prompt": "

Qual è il coefficiente angolare di $\\simplify{y={m}x+{yc}}$ ?

\n

Coefficiente =  [[0]]

\n

What is the $y$-coordinate of the point on the line whose $x$-coordinate is $\\var{xval}$?

\n

( $\\var{xval}$ , [[2]] )

\n

\n

", "type": "gapfill", "marks": 0, "showCorrectAnswer": true, "gaps": [{"variableReplacementStrategy": "originalfirst", "type": "numberentry", "marks": "0.5", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "m", "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "mustBeReducedPC": 0, "minValue": "m", "showCorrectAnswer": false, "allowFractions": true, "mustBeReduced": false}, {"variableReplacementStrategy": "originalfirst", "type": "numberentry", "marks": "0.5", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "yc", "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "mustBeReducedPC": 0, "minValue": "yc", "showCorrectAnswer": false, "allowFractions": true, "mustBeReduced": false}, {"variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "type": "jme", "checkvariablenames": false, "marks": 1, "answer": "{xval}*{m}+{yc}", "showpreview": true, "variableReplacements": [], "expectedvariablenames": [], "scripts": {}, "showFeedbackIcon": true, "checkingaccuracy": 0.001, "showCorrectAnswer": false, "vsetrange": [0, 1], "checkingtype": "absdiff"}], "variableReplacements": []}, {"variableReplacementStrategy": "originalfirst", "scripts": {}, "showFeedbackIcon": true, "prompt": "

Fill in the table of values for $y=\\simplify[std]{{a}x^2+{c}}$:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$x$$-3$$-2$$-1$$0$$1$$2$$3$
$y$[[0]][[1]][[2]][[3]][[4]][[5]][[6]]
\n

Slide the points to the correct $y$ values.

\n

\n

$y=\\simplify[std]{{a}x^2+{c}}$

This is the graph you should have obtained.

\n
", "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "variable_groups": [], "ungrouped_variables": ["a", "c", "values", "v1", "m", "yc", "xval"], "tags": [], "variables": {"c": {"definition": "random(-4..4 except 0)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "c"}, "xval": {"definition": "random(-7..7 except [1,0,-1,m])", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "xval"}, "yc": {"definition": "random(-5..5 except [0,m])", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "yc"}, "values": {"definition": "map({a*x^2+c},x,-3..3)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "values"}, "m": {"definition": "random(-2,-1.5,-1,-0.5,0.5,1,1.5,2)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "m"}, "v1": {"definition": "map([x-3,values[x]],x,0..6)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "v1"}, "a": {"definition": "random(-2,-1,-0.5,0.5,1,2)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "a"}}, "metadata": {"description": "

Compute a table of values for a quadratic function. The student input is now disconnected from the graph so that they slide the points on the graph after they input the values and the answer fields are not updated. Now includes a graph in advice.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question"}, {"name": "Unit 4 - Which quadratic function is shown by the parabola?", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Richard Miles", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/882/"}], "rulesets": {}, "parts": [{"showFeedbackIcon": true, "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 0, "type": "1_n_2", "matrix": "[2,0,0,0,0]", "steps": [{"variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "prompt": "

You can narrow down the choices by checking points where the curve intersects the axes, in particular, the $y$-axis intercept. In addition, the $x$-coordinate of the vertex of the parabola is $x=\\frac{-b}{2a}$, when the curve is written in the standard form $y=ax^2+bx+c$. The line $x=\\frac{-b}{2a}$ is also the vertical line of symmetry for the curve.

", "showCorrectAnswer": true, "type": "information"}], "displayType": "radiogroup", "stepsPenalty": "1", "variableReplacements": [], "choices": "[function1,function2,function3,function4,function5]", "scripts": {}, "prompt": "

Decide which of the following expressions for $f(x)$ corresponds to the parabola shown.

", "marks": 0, "minMarks": 0, "shuffleChoices": true, "showCorrectAnswer": true}], "functions": {"drawgraph": {"parameters": [], "language": "javascript", "type": "html", "definition": " var miny = Numbas.jme.unwrapValue(scope.variables.miny);\n var maxy = Numbas.jme.unwrapValue(scope.variables.maxy);\n var minx = Numbas.jme.unwrapValue(scope.variables.minx);\n var maxx = Numbas.jme.unwrapValue(scope.variables.maxx);\n var a_inverse = Numbas.jme.unwrapValue(scope.variables.a_inverse);\n var b = Numbas.jme.unwrapValue(scope.variables.b);\n var c = Numbas.jme.unwrapValue(scope.variables.c);\n \n var div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',\n {boundingBox:[minx,maxy,maxx,miny],\n axis:false,\n showNavigation:false,\n grid:true});\n JXG.Options.text.useMathJax = true;\n var brd = div.board; \n var xaxis=brd.createElement('axis', [[minx,0],[maxx,0]]);\n var yaxis=brd.createElement('axis', [[0,miny],[0,maxy]]);\n var xlabel = brd.create('text',[maxx-1,0,function(){return '\\\$x\\\$';}]);\n var ylabel = brd.create('text',[-2,maxy,function(){return '\\\$y\\\$';}]);\n var graph = brd.create('functiongraph',[function(x){ return (1/a_inverse)*x*x+b*x+c;},minx,maxx]);\n\nreturn div;\n "}}, "tags": [], "preamble": {"css": "", "js": ""}, "ungrouped_variables": [], "variables": {"c": {"definition": "random(-7..7 except [b,0,-1,1])", "name": "c", "templateType": "anything", "description": "", "group": "Graph parameters"}, "function4": {"definition": "\"

$\\\\var{a_inverse}\\\\simplify{x^2+{b}}\\\\simplify{x+{c}}$

\"", "name": "function4", "templateType": "long string", "description": "", "group": "Graph parameters"}, "function1": {"definition": "\"

$\\\\frac{1}{\\\\var{a_inverse}}\\\\simplify{x^2+{b}}\\\\simplify{x+{c}}$

\"", "name": "function1", "templateType": "long string", "description": "", "group": "Graph parameters"}, "function3": {"definition": "\"

$\\\\var{a_inverse}\\\\simplify{x^2+{b}}\\\\simplify{x-{c}}$

\"", "name": "function3", "templateType": "long string", "description": "", "group": "Graph parameters"}, "b": {"definition": "random([-4,-3,-2,2,3,4] except a_inverse)", "name": "b", "templateType": "anything", "description": "", "group": "Graph parameters"}, "maxx": {"definition": "20", "name": "maxx", "templateType": "anything", "description": "", "group": "Graph parameters"}, "maxy": {"definition": "15", "name": "maxy", "templateType": "anything", "description": "", "group": "Graph parameters"}, "minx": {"definition": "-20", "name": "minx", "templateType": "anything", "description": "", "group": "Graph parameters"}, "miny": {"definition": "-15", "name": "miny", "templateType": "anything", "description": "", "group": "Graph parameters"}, "function2": {"definition": "\"

$\\\\frac{1}{\\\\var{a_inverse}}\\\\simplify{x^2+{b}*3}\\\\simplify{x+{c}}$

\"", "name": "function2", "templateType": "long string", "description": "", "group": "Graph parameters"}, "function5": {"definition": "\"

$(\\\\simplify{x+{c}})^2+\\\\simplify{x+{b}}$

\"", "name": "function5", "templateType": "long string", "description": "", "group": "Graph parameters"}, "a_inverse": {"definition": "random(2..5)", "name": "a_inverse", "templateType": "anything", "description": "", "group": "Graph parameters"}}, "variable_groups": [{"variables": ["minx", "maxx", "miny", "maxy", "a_inverse", "b", "c", "function1", "function2", "function3", "function4", "function5"], "name": "Graph parameters"}], "advice": "", "statement": "

The curve $y=f(x)$ is shown below.

\n\n\n\n\n\n\n
 {drawgraph()}
", "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"description": "

Tests the ability to match a quadratic function to a given parabola.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question"}]}, {"pickQuestions": 1, "pickingStrategy": "all-shuffled", "name": "Disequazioni", "questions": [{"name": "Arithmetic operations: Inequalities", "extensions": [], "custom_part_types": [], "resources": [["question-resources/drawingresize_grbP9s8.svg", "/srv/numbas/media/question-resources/drawingresize_grbP9s8.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}], "type": "question", "statement": "

Choose the correct symbols to describe the relations between each of these pairs of numbers.

", "variablesTest": {"condition": "", "maxRuns": "100"}, "variables": {"neg": {"description": "

Random negative integers.

", "name": "neg", "group": "Ungrouped variables", "templateType": "anything", "definition": "repeat(random(-300..-1),8)"}, "c": {"description": "", "name": "c", "group": "Ungrouped variables", "templateType": "anything", "definition": "-b"}, "random": {"description": "", "name": "random", "group": "Ungrouped variables", "templateType": "anything", "definition": "repeat(random(0..0.01#0.001),3)"}, "d": {"description": "", "name": "d", "group": "Ungrouped variables", "templateType": "anything", "definition": "-a"}, "pos": {"description": "

Random positive integers.

", "name": "pos", "group": "Ungrouped variables", "templateType": "anything", "definition": "repeat(random(1..300),5)"}, "a": {"description": "", "name": "a", "group": "Ungrouped variables", "templateType": "anything", "definition": "neg[7] + random2"}, "b": {"description": "", "name": "b", "group": "Ungrouped variables", "templateType": "anything", "definition": "neg[7] + 0.9 + random[2]"}, "dec": {"description": "

Random decimals.

", "name": "dec", "group": "Ungrouped variables", "templateType": "anything", "definition": "repeat(random(0..50 #0.01 except 0..50), 7)"}, "random2": {"description": "", "name": "random2", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(0.5..0.8#0.001)"}}, "functions": {}, "tags": ["inequality", "taxonomy"], "variable_groups": [], "parts": [{"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"scripts": {}, "minMarks": 0, "distractors": ["", "", ""], "variableReplacementStrategy": "originalfirst", "displayType": "dropdownlist", "choices": ["

>

", "

<

", "

=

"], "showFeedbackIcon": true, "shuffleChoices": true, "displayColumns": 0, "variableReplacements": [], "marks": 0, "matrix": ["1", 0, 0], "showCorrectAnswer": true, "maxMarks": 0, "type": "1_n_2"}], "showFeedbackIcon": true, "prompt": "

\n

$\\var{dec[6] + 0.001 + random[0]}$  [[0]]  $\\var{dec[6] - random[1]}$

\n

", "marks": 0}, {"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"scripts": {}, "minMarks": 0, "distractors": ["", "", ""], "variableReplacementStrategy": "originalfirst", "displayType": "dropdownlist", "choices": ["

<

", "

>

", "

=

"], "showFeedbackIcon": true, "shuffleChoices": true, "displayColumns": 0, "variableReplacements": [], "marks": 0, "matrix": ["1", 0, 0], "showCorrectAnswer": true, "maxMarks": 0, "type": "1_n_2"}], "showFeedbackIcon": true, "prompt": "

\n

$\\var{neg[7] + random2}$  [[0]]  $\\var{neg[7] + 0.9 + random[2]}$

", "marks": 0}, {"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"scripts": {}, "minMarks": 0, "distractors": ["", "", ""], "variableReplacementStrategy": "originalfirst", "displayType": "dropdownlist", "choices": ["

=

", "

>

", "

<

"], "showFeedbackIcon": true, "shuffleChoices": true, "displayColumns": 0, "variableReplacements": [], "marks": 0, "matrix": [0, "1", 0], "showCorrectAnswer": true, "maxMarks": 0, "type": "1_n_2"}], "showFeedbackIcon": true, "prompt": "

\n

$(\\var{neg[3]}) \\times (\\var{neg[2]})$  [[0]]   $\\var{-neg[3]*neg[2]}$

", "marks": 0}], "ungrouped_variables": ["dec", "neg", "pos", "random2", "random", "a", "b", "c", "d"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Complete the inequality relationships by selecting the correct symbol from a drop down box

"}, "preamble": {"css": "", "js": ""}, "advice": "

\\\begin{align} \\text{Symbol }&\\lt \\text{ denotes \"less than\".} \\\\ \\text{Symbol }&\\gt \\text{denotes \"greater than\".} \\end{align}\

\n

#### a)

\n

$\\var{dec[6] + 0.001 + random[0]}$ is greater than $\\var{dec[6] - random[1]}$ so

\n

\$\\var{dec[6] + 0.001 + random[0]} \\gt \\var{dec[6] - random[1]} \\text{.} \$

\n

#### b)

\n

When both of the numbers that you are comparing are negative, it may be tempting to ignore the negative signs and make an incorrect assumption. For example, when we have -5 and -4 we might ignore the signs and assume -5 is larger than -4 since +5 is larger than +4. This is however wrong, -5 < -4.

\n

To understand this a bit better, look at the following number line:

\n

\n

Following the number line from left to right, we can see that $\\var{neg[7] + random2}$ is less than $\\var{neg[7] + 0.9 + random[2]}$, so

\n

\$\\var{neg[7] + random2} \\lt \\var{neg[7] + 0.9 + random[2]} \\text{.}\$

\n

\n

#### c)

\n

Multiplying two negative numbers results in a positive number. Therefore we can see without performing any calculation that $(\\var{neg[3]}) \\times (\\var{neg[2]}) \\gt \\var{-neg[3]*neg[2]}$ as positive numbers are always larger than negative numbers.

\n

\$(\\var{neg[3]} \\times \\var{neg[2]}) \\gt \\var{-neg[3]*neg[2]}\$

\n

\$\\var{neg[3] * neg[2]} \\gt \\var{-neg[3]*neg[2]}\$

"}, {"name": "Solve quadratic inequalities", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}], "metadata": {"description": "

This question takes the student through variety of examples of quadratic inequalities by asking them for the range(s) for which $x$ meets the inequality.

", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["a", "b", "c", "d", "g", "f"], "type": "question", "advice": "

### a)

\n

This example is best illustrated by the graph of $\\simplify{f(x)=x^2+({a}-{b})x-{a}{b}}$ below. By finding the roots of the equation we can find the $x$ coordinates where the line crosses the $x$ axis and then we can use a sketch or visualise the graph to work out the set of values for $x$ where $f(x)>0$.\$\\\\[0.1em]\$

\n

{geogebra_applet('Hk5dTptY',[[\"a\",a],[\"b\",b]])}

\n

\\\begin{align} \\simplify{f(x)=x^2+({a}-{b})x-{a}{b}}&>0\\\\ \\simplify{(x+{a})(x-{b})}&=0\\text{.} \\end{align} \

\n

Therefore

\n

\$\\simplify{x<{-a}}\\text{ or }\\simplify{x>{b}}\\text{.}\$

\n

#### b)

\n

We use the same method in this example but this time we use our graph to visualise where $g(x)<0$.\$\\\\[0.1em]\$

\n

{geogebra_applet('PUFStTNa',[[\"c\",c],[\"d\",d]])}

\n

\\\begin{align} \\simplify{f(x)=x^2+({c}-{d})x-{c}{d}}&<0\\\\ \\simplify{(x+{c})(x-{d})}&=0\\text{.} \\end{align} \

\n

Therefore

\n

\$\\simplify{x>{-c}}\\text{ and }\\simplify{x<{d}}\\text{.}\$

\n

#### c)

\n

The notable difference with solving this equation is the requirement to rearrange the inequality before factorisation.\$\\\\[0.1em]\$

\n

{geogebra_applet('CZsSCdH6',[[\"e\",g],[\"f\",f]])}

\n

\\\begin{align} \\simplify{({g}+{f})x-{g}{f}}&>x^2\\\\ \\simplify{x^2-({g}+{f})x+{g}{f}}&<0\\\\ \\simplify{(x-{g})(x-{f})}&=0\\text{.} \\end{align} \

\n

Therefore

\n

\$\\simplify{x>{g}}\\text{ and }\\simplify{x<{f}}\\text{.}\$

\n

Alternatively, we can plot the graph of $x^2$ against $\\simplify{({g}+{f})x-{g}{f}}$ to visualise the same result.

\n

{geogebra_applet('Ez697SNE',[[\"g\",g],[\"f\",f]])}

\n

From this graph we can see that the values of $x$ where $\\simplify{({g}+{f})x-{g}{f}}>x^2$ are the same as the values of $x$ where $\\simplify{x^2-({g}+{f})x+{g}{f}}<0$.

", "variable_groups": [], "rulesets": {}, "statement": "

Solve the following quadratic inequalities by firstly factorising $f(x)$ and then solving for $x$ when $f(x)=0$. It may be helpful to sketch each quadratic.

", "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "(x-{b})(x+{a})", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "{b}", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "-{a}", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"shuffleChoices": false, "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["

AND

", "

OR

"], "scripts": {}, "distractors": ["", ""], "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": [0, "1"], "displayType": "dropdownlist"}], "showFeedbackIcon": true, "prompt": "

Find the range of values for $x$ such that $\\simplify{f(x)=x^2+({a}-{b})x-{a}{b}}>0$.

\n

Factorise $f(x)$:

\n

$\\simplify{f(x)=x^2+({a}-{b})x-{a}{b}}=$ [[0]] $=0$.

\n

Hence,

\n

$x>$ [[1]]

\n

[[3]]

\n

$x<$ [[2]]

"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "(x-{d})(x+{c})", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "{d}", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "-{c}", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"shuffleChoices": false, "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["

AND

", "

OR

"], "scripts": {}, "distractors": ["", ""], "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": ["1", 0], "displayType": "dropdownlist"}], "showFeedbackIcon": true, "prompt": "

Find the range of values for $x$ such that $\\simplify{f(x)=x^2+({c}-{d})x-{c}{d}}<0$.

\n

Factorise $f(x)$:

\n

$\\simplify{f(x)=x^2+({c}-{d})x-{c}{d}}=$ [[0]] $=0$.

\n

Hence,

\n

$x<$ [[1]]

\n

[[3]]

\n

$x>$ [[2]]

"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "(x-{g})(x-{f})", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "{g}", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "{f}", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"shuffleChoices": false, "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["

AND

", "

OR

"], "scripts": {}, "distractors": ["", ""], "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": ["1", 0], "displayType": "dropdownlist"}], "showFeedbackIcon": true, "prompt": "

Find the range of vaues such that: $\\simplify{({g}+{f})x-{g}{f}>x^2}$

\n

Rearrange then factorise the inequality:

\n

[[0]] $<0$.

\n

Use the above result to find the range of values for $x$ such that $\\simplify{({g}+{f})x-{g}{f}>x^2}$.

\n

$x>$ [[1]]

\n

[[3]]

\n

$x<$ [[2]]

"}], "tags": ["factorise a quadratic equation", "factorising", "inequalities", "quadratic inequalities", "range of values for x", "taxonomy"], "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"a": {"description": "", "group": "Ungrouped variables", "definition": "random(1..5)", "name": "a", "templateType": "anything"}, "f": {"description": "", "group": "Ungrouped variables", "definition": "random(5..9)", "name": "f", "templateType": "anything"}, "b": {"description": "", "group": "Ungrouped variables", "definition": "random(1..5)", "name": "b", "templateType": "anything"}, "c": {"description": "", "group": "Ungrouped variables", "definition": "random(3..6)", "name": "c", "templateType": "anything"}, "g": {"description": "", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "g", "templateType": "anything"}, "d": {"description": "", "group": "Ungrouped variables", "definition": "random(2..4)", "name": "d", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}, {"name": "Inequalities that involve a single absolute value", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "ungrouped_variables": ["choice", "sym", "a", "c", "d", "b", "leadCoeff", "prod", "one", "rootList", "r1", "r2", "strict", "backChoice", "backsym", "extremeY", "backprod", "dr", "hole", "nonhole"], "metadata": {"description": "

Inequality involving a single absolute value, question solution uses the piecewise nature of the absolute value function.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

You are given the inequality \$|\\simplify{{a}x+{b}}| \\var{latex(sym)} \\var{c}.\$

\n

one interval.

", "

two intervals.

"], "showFeedbackIcon": true, "distractors": ["", ""], "maxMarks": 0, "variableReplacements": [], "shuffleChoices": false, "marks": 0, "showCorrectAnswer": true, "scripts": {"mark": {"order": "instead", "script": "// apply the normal marking algorithm for this part\nthis.__proto__.mark.apply(this);\n// store whether the student said \"one interval\" in an attribute that's easier to access\nthis.one = this.ticks[0][0];"}}, "displayType": "dropdownlist", "matrix": ["if(one=true,1,0)", "if(one=false,1,0)"], "displayColumns": 0, "type": "1_n_2"}, {"variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "r1", "variableReplacements": [], "marks": 1, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": true, "scripts": {}, "showCorrectAnswer": true, "allowFractions": true, "type": "numberentry", "maxValue": "r1"}, {"minMarks": 0, "variableReplacementStrategy": "originalfirst", "choices": ["

<

", "

"], "showFeedbackIcon": true, "distractors": ["", ""], "maxMarks": 0, "variableReplacements": [], "shuffleChoices": false, "marks": 0, "showCorrectAnswer": true, "scripts": {}, "displayType": "dropdownlist", "matrix": ["if(one=true and strict=true,1,0)", "if(one=true and strict=false,1,0)"], "displayColumns": 0, "type": "1_n_2"}, {"minMarks": 0, "variableReplacementStrategy": "originalfirst", "choices": ["

<

", "

"], "showFeedbackIcon": true, "distractors": ["", ""], "maxMarks": 0, "variableReplacements": [], "shuffleChoices": false, "marks": 0, "showCorrectAnswer": true, "scripts": {}, "displayType": "dropdownlist", "matrix": ["if(one=true and strict=true,1,0)", "if(one=true and strict=false,1,0)"], "displayColumns": 0, "type": "1_n_2"}, {"variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "r2", "variableReplacements": [], "marks": 1, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": true, "scripts": {}, "showCorrectAnswer": true, "allowFractions": true, "type": "numberentry", "maxValue": "r2"}, {"minMarks": 0, "variableReplacementStrategy": "originalfirst", "choices": ["

<

", "

"], "showFeedbackIcon": true, "distractors": ["", ""], "maxMarks": 0, "variableReplacements": [], "shuffleChoices": false, "marks": 0, "showCorrectAnswer": true, "scripts": {}, "displayType": "dropdownlist", "matrix": ["if(one=false and strict=true,1,0)", "if(one=false and strict=false,1,0)"], "displayColumns": 0, "type": "1_n_2"}, {"variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "r1", "variableReplacements": [], "marks": 1, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": true, "scripts": {}, "showCorrectAnswer": true, "allowFractions": true, "type": "numberentry", "maxValue": "r1"}, {"minMarks": 0, "variableReplacementStrategy": "originalfirst", "choices": ["

>

", "

"], "showFeedbackIcon": true, "distractors": ["", ""], "maxMarks": 0, "variableReplacements": [], "shuffleChoices": false, "marks": 0, "showCorrectAnswer": true, "scripts": {}, "displayType": "dropdownlist", "matrix": ["if(one=false and strict=true,1,0)", "if(one=false and strict=false,1,0)"], "displayColumns": 0, "type": "1_n_2"}, {"variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "r2", "variableReplacements": [], "marks": 1, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": true, "scripts": {}, "showCorrectAnswer": true, "allowFractions": true, "type": "numberentry", "maxValue": "r2"}], "prompt": "

The solution to the above inequality

\n

\n
\n

Given by the condition [[1]] [[2]] $x$ [[3]] [[4]].

\n
\n
\n

Given by the condition, $x$ [[5]] [[6]] or $x$ [[7]] [[8]].

\n
", "showFeedbackIcon": true, "variableReplacements": [], "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "rulesets": {}, "advice": "

### Short-cut Method

\n

\n

For questions of this type (where the absolute value is 'less than' or 'less than and equal to' something positive) we can set out our working in a shorter way:

\n

\n

\\begin{alignat}{2} &&|\\simplify{{a}x+{b}}| &\\var{latex(sym)} \\var{c}\\\\ \\var{-c} &\\var{latex(sym)}&\\simplify{{a}x+{b}}&\\var{latex(sym)}\\var{c}\\\\ \\var{-c-b} &\\var{latex(sym)} &\\simplify{{a}x}\\quad&\\var{latex(sym)}\\var{c-b}.\\end{alignat}

\n

\n

\n

### Cases Method

\n

\n

Recall that the absolute value is defined as the piecewise function \$|x|=\\begin{cases}x, &\\text{ for } x\\ge 0\\\\ -x, &\\text{ for } x<0. \\end{cases}\$

\n

This means that $|\\simplify{{a}x+{b}}|$ is actually $\\simplify{{a}x+{b}}$ when $\\simplify{{a}x+{b}}\\ge 0$ but it is $-(\\simplify{{a}x+{b}})$ when $\\simplify{{a}x+{b}}<0$. We have two cases:

\n

\n

Case 1: $\\simplify{{a}x+{b}}\\ge 0$

\n

Rearranging $\\simplify{{a}x+{b}}\\ge 0$ for $x$ gives $x \\ge \\simplify[fractionNumbers]{{-b/a}}$ $x \\le \\simplify[fractionNumbers]{{-b/a}}$ and so case 1 is only relevant for $x \\ge \\simplify[fractionNumbers]{{-b/a}}$ $x \\le \\simplify[fractionNumbers]{{-b/a}}$.

\n

In case 1, our inequality is simply $\\simplify{{a}x+{b}} \\var{latex(sym)} \\var{c}$, and so rearranging it for $x$ gives $x \\var{latex(sym)} \\simplify[fractionNumbers,simplifyFractions,!unitDenominator]{{(c-b)/a}}$ $x \\var{latex(backsym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(c-b)/a}}$

\n

So in conclusion for case 1, we require that $x \\ge \\simplify[fractionNumbers]{{-b/a}}$ $x \\le \\simplify[fractionNumbers]{{-b/a}}$   and  $x \\var{latex(sym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(c-b)/a}}$ $x \\var{latex(backsym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(c-b)/a}}$ which is the same as $\\simplify[fractionNumbers]{-{b/a}}\\le x\\var{latex(sym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(c-b)/a}}$ $\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(c-b)/a}}\\var{latex(sym)} x\\le \\simplify[fractionNumbers]{{-b/a}}$ $x\\var{latex(sym)}\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(c-b)/a}}$ $x\\var{latex(backsym)}\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(c-b)/a}}$.

\n

\n

Case 2: $\\simplify{{a}x+{b}}< 0$

\n

Rearranging $\\simplify{{a}x+{b}}< 0$ for $x$ gives $x < \\simplify[fractionNumbers]{{-b/a}}$ $x > \\simplify[fractionNumbers]{{-b/a}}$ and so case 2 is only relevant for $x < \\simplify[fractionNumbers]{{-b/a}}$ $x > \\simplify[fractionNumbers]{{-b/a}}$.

\n

In case 2, our inequality is actually $-(\\simplify{{a}x+{b}}) \\var{latex(sym)} \\var{c}$, and so rearranging it for $x$ gives $x \\var{latex(backsym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(-c-b)/a}}$ $x \\var{latex(sym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(-c-b)/a}}$

\n

So in conclusion for case 2, we require that $x < \\simplify[fractionNumbers]{-{b/a}}$ $x > \\simplify[fractionNumbers]{-{b/a}}$  and  $x \\var{latex(backsym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(-c-b)/a}}$ $x \\var{latex(sym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(-c-b)/a}}$ which is the same as $\\simplify[fractionNumbers]{{-(b+c)/a}}\\var{latex(sym)} x< \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{-b}/{a}}$ $\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{-b/a}}< x \\var{latex(sym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{-{(b+c)/a}}$ $x\\var{latex(backsym)}\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(-c-b)/a}}$ $x\\var{latex(sym)}\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(-c-b)/a}}$.

\n

\n

Therefore the solution to our original inequality $|\\simplify{{a}x+{b}}| \\var{latex(sym)} \\var{c}$ is

\n

\$\\var[fractionNumbers]{r1}<x<\\var[fractionNumbers]{r2}.\$

\n

\$\\var[fractionNumbers]{r1}\\le x \\le\\var[fractionNumbers]{r2}.\$

\n

\$x< \\var[fractionNumbers]{r1}, \\, x> \\var[fractionNumbers]{r2}. \$

\n

\$x\\le \\var[fractionNumbers]{r1}, \\, x\\ge\\var[fractionNumbers]{r2}. \$

\n

\n

\n

", "tags": [], "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {"graph": {"parameters": [], "definition": "a = Numbas.jme.unwrapValue(scope.variables.leadcoeff);\nr1 = Numbas.jme.unwrapValue(scope.variables.r1);\nr2 = Numbas.jme.unwrapValue(scope.variables.r2);\nprod = Numbas.jme.unwrapValue(scope.variables.backprod);\ndr=Numbas.jme.unwrapValue(scope.variables.dr);\nexy=Numbas.jme.unwrapValue(scope.variables.extremey);\nhole = Numbas.jme.unwrapValue(scope.variables.hole);\nnonhole = Numbas.jme.unwrapValue(scope.variables.nonhole);\n\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{boundingBox:[r1-dr,1.2*exy,r2+dr,-1.2*exy],grid:true,axis:false});\nvar board = div.board;\n\n// create the x-axis.\nvar xaxis = board.create('line',[[0,0],[1,0]], { strokeColor: 'black', fixed: true});\nvar xticks = board.create('ticks',[xaxis,2],{\n drawLabels: true,\n label: {offset: [-4, -10]},\n minorTicks: 0\n});\n\n// create the y-axis\nvar yaxis = board.create('line',[[0,0],[0,1]], { strokeColor: 'black', fixed: true });\nvar yticks = board.create('ticks',[yaxis,10],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 0,\n});\n\n\n\nif(prod==2 || prod==4 || prod==-2 || prod==-4){ptColour = 'green';} else {ptColour = 'red';};\n\nboard.create('point',[hole,0],{fixed:true,withLabel:false,strokeColour:'red', fillColor:'red'});\nboard.create('point',[nonhole,0],{fixed:true,withLabel:false,strokeColor:ptColour,fillColor:ptColour});\n\nif(prod==2 || prod==-4 || prod== 1 || prod==-3){ outColour = 'green';} else {outColour = 'red';};\nif(prod==-2 || prod==4 || prod== -1 || prod==3){inColour = 'green';} else {inColour = 'red';};\n\nboard.create('functiongraph',[function(x){return a*(x**2-(r1+r2)*x+r1*r2);},r1,r2],{strokeColor:inColour,strokeWidth:2});\nboard.create('functiongraph',[function(x){return a*(x**2-(r1+r2)*x+r1*r2);},r1-dr,r1],{strokeColor:outColour,strokeWidth:2});\nboard.create('functiongraph',[function(x){return a*(x**2-(r1+r2)*x+r1*r2);},r2,r2+dr],{strokeColor:outColour,strokeWidth:2});\n\nreturn div;", "type": "html", "language": "javascript"}}, "variables": {"a": {"description": "", "templateType": "anything", "definition": "random(-12..12 except 0)", "group": "Ungrouped variables", "name": "a"}, "choice": {"description": "", "templateType": "anything", "definition": "random(1..4)", "group": "Ungrouped variables", "name": "choice"}, "hole": {"description": "", "templateType": "anything", "definition": "-b/d", "group": "Ungrouped variables", "name": "hole"}, "c": {"description": "", "templateType": "anything", "definition": "random(1..12)", "group": "Ungrouped variables", "name": "c"}, "d": {"description": "

d

", "templateType": "anything", "definition": "random(-12..12 except -1..1)", "group": "Ungrouped variables", "name": "d"}, "r1": {"description": "

r1

", "templateType": "anything", "definition": "rootList[0]", "group": "Ungrouped variables", "name": "r1"}, "prod": {"description": "", "templateType": "anything", "definition": "leadCoeff*choice", "group": "Ungrouped variables", "name": "prod"}, "backprod": {"description": "", "templateType": "anything", "definition": "leadCoeff*backchoice", "group": "Ungrouped variables", "name": "backprod"}, "leadCoeff": {"description": "", "templateType": "anything", "definition": "sign(a)", "group": "Ungrouped variables", "name": "leadCoeff"}, "r2": {"description": "

r2

", "templateType": "anything", "definition": "rootList[1]", "group": "Ungrouped variables", "name": "r2"}, "extremeY": {"description": "

approx of the furthest we need to plot in the y direction

", "templateType": "anything", "definition": "(r2-r1)^2/4\n\n//abs((c*b-a)*b-(b*(d*d-1)+d*(c*b-a))^2/(4*d*(c*d-1)))", "group": "Ungrouped variables", "name": "extremeY"}, "nonhole": {"description": "", "templateType": "anything", "definition": "(a-b*c)/(c*d-1)", "group": "Ungrouped variables", "name": "nonhole"}, "one": {"description": "", "templateType": "anything", "definition": "if(choice=3 or choice=4,true,false)", "group": "Ungrouped variables", "name": "one"}, "sym": {"description": "", "templateType": "anything", "definition": "['>','\\\\ge','<','\\\\le'][choice-1]", "group": "Ungrouped variables", "name": "sym"}, "dr": {"description": "", "templateType": "anything", "definition": "r2-r1", "group": "Ungrouped variables", "name": "dr"}, "strict": {"description": "", "templateType": "anything", "definition": "if(choice=1 or choice=3,true,false)", "group": "Ungrouped variables", "name": "strict"}, "rootList": {"description": "", "templateType": "anything", "definition": "sort([(-b-c)/a,(-b+c)/a])", "group": "Ungrouped variables", "name": "rootList"}, "b": {"description": "", "templateType": "anything", "definition": "random(-12..12 except 0)", "group": "Ungrouped variables", "name": "b"}, "backChoice": {"description": "", "templateType": "anything", "definition": "mod(choice+1,4)+1", "group": "Ungrouped variables", "name": "backChoice"}, "backsym": {"description": "", "templateType": "anything", "definition": "['>','\\\\ge','<','\\\\le'][backChoice-1]", "group": "Ungrouped variables", "name": "backsym"}}, "type": "question"}]}], "navigation": {"showfrontpage": true, "reverse": true, "browse": true, "showresultspage": "oncompletion", "allowregen": false, "onleave": {"action": "warnifunattempted", "message": "

Devi rispospondere ancora a una domanda

"}, "preventleave": true}, "metadata": {"description": "

Test di ingresso su geometria analitica (retta e parabola) e disequazioni

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": true, "type": "exam", "contributors": [{"name": "riccardo belle'", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1797/"}], "extensions": ["geogebra", "jsxgraph"], "custom_part_types": [], "resources": [["question-resources/drawingresize_grbP9s8.svg", "/srv/numbas/media/question-resources/drawingresize_grbP9s8.svg"]]}