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"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Solve the following simultaneous equations:</p>\n<table style=\"width: 449px;\">\n<tbody>\n<tr>\n<td style=\"text-align: right; width: 38.825px;\">$y$</td>\n<td style=\"text-align: center; width: 41.8625px;\">$=$</td>\n<td style=\"text-align: left; width: 319.312px;\">$\\simplify{{grad}x+{yint}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right; width: 38.825px;\">$y$</td>\n<td style=\"text-align: center; width: 41.8625px;\">$=$</td>\n<td style=\"text-align: left; width: 319.312px;\">$\\simplify{x^2+{quadxcoeff}x+{quadccoeff}}$</td>\n</tr>\n</tbody>\n</table>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"minroot": {"name": "minroot", "group": "Ungrouped variables", "definition": "min(root1,root2)", "description": "", "templateType": "anything", "can_override": false}, "quadxcoeff": {"name": "quadxcoeff", "group": "Ungrouped variables", "definition": "sroots+grad", "description": "", "templateType": "anything", "can_override": false}, "ansyvalue2": {"name": "ansyvalue2", "group": "Ungrouped variables", "definition": "grad*maxroot+yint", "description": "", "templateType": "anything", "can_override": false}, "sroots": {"name": "sroots", "group": "Ungrouped variables", "definition": "-root1-root2", "description": "", "templateType": "anything", "can_override": false}, "root2": {"name": "root2", "group": "Ungrouped variables", "definition": "random(-12..12)", "description": "", "templateType": "anything", "can_override": false}, "root1": {"name": "root1", "group": "Ungrouped variables", "definition": "random(-12..12)", "description": "", "templateType": "anything", "can_override": false}, "quadccoeff": {"name": "quadccoeff", "group": "Ungrouped variables", "definition": "proots+yint", "description": "", "templateType": "anything", "can_override": false}, "maxroot": {"name": "maxroot", "group": "Ungrouped variables", "definition": "max(root1, root2)", "description": "", "templateType": "anything", "can_override": false}, "proots": {"name": "proots", "group": "Ungrouped variables", "definition": "root1*root2", "description": "", "templateType": "anything", "can_override": false}, "yint": {"name": "yint", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "grad": {"name": "grad", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "ansyvalue": {"name": "ansyvalue", "group": "Ungrouped variables", "definition": "grad*minroot+yint", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["root1", "root2", "grad", "sroots", "proots", "yint", "quadxcoeff", "quadccoeff", "ansyvalue", "ansyvalue2", "minroot", "maxroot"], "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>Input $x_1$ as the smallest $x$ solution, and $x_2$ as the biggest.&nbsp;</p>\n<p></p>\n<p>$x_1 =$ [[0]]&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $y_1 =$ [[1]]</p>\n<p>$x_2 =$ [[2]]&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $y_2 =$ [[3]]</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "x1", "marks": "1/4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "minroot", "maxValue": "minroot", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "y1", "marks": "1/4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ansyvalue", "maxValue": "ansyvalue", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "x2", "marks": "1/4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "maxroot", "maxValue": "maxroot", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "y2", "marks": "1/4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ansyvalue2", "maxValue": "ansyvalue2", "correctAnswerFraction": true, "allowFractions": true, "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": "Partial fractions with repeated factor", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Chris Knapp", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14987/"}], "tags": ["algebra", "Algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions", "common denominator"], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n    <p>&nbsp;</p>\n  <p>&nbsp;</p>\n    ", "advice": "", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"nb": {"name": "nb", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "description": "", "templateType": "anything", "can_override": false}, "c3": {"name": "c3", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "new": {"name": "new", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "s1_": {"name": "s1_", "group": "Ungrouped variables", "definition": "if(c_<0,-1,1)", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c_": {"name": "c_", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "a_": {"name": "a_", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "if(c<0,-1,1)", "description": "", "templateType": "anything", "can_override": false}, "d_": {"name": "d_", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything", "can_override": false}, "q1": {"name": "q1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except p1)", "description": "", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "nb_": {"name": "nb_", "group": "Ungrouped variables", "definition": "if(c_<0,'taking away','adding')", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a2_": {"name": "a2_", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything", "can_override": false}, "a1_": {"name": "a1_", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except p)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-9..9 except 0 except a)", "description": "", "templateType": "anything", "can_override": false}, "b_": {"name": "b_", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "b1+b3=0\n", "maxRuns": "100000"}, "ungrouped_variables": ["a", "c", "b", "nb", "s1", "a_", "c_", "b_", "nb_", "a1_", "a2_", "s1_", "new", "a1", "a2", "a3", "d", "d_", "b1", "b2", "b3", "p", "q", "c1", "c2", "c3", "p1", "q1"], "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>Write \\[\\simplify{(({a1+a3})x^2+{a1*a+a1*b+a2+2*a*a3} * x + {a1*a*b + a2*b + a3*a^2})/ ((x + {a})^2 * (x + {b}))}\\] as the sum of partial fractions.</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "0", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a1} / (x + {a}) + {a2}/(x + {a})^2 + {a3} / (x + {b})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 1e-05, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [10, 11], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": [")(", ")*("], "showStrings": false, "partialCredit": 0, "message": "<p>Input as the sum of partial fractions.</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Partial fractions with linear factors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Chris Knapp", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14987/"}], "tags": ["algebra", "Algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions", "common denominator"], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n    <p>&nbsp;</p>\n  <p>&nbsp;</p>\n    ", "advice": "", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a1": {"name": "a1", "group": "Ungrouped variables", "definition": " 1", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,round(b*a2/a1)])", "description": "", "templateType": "anything", "can_override": false}, "d_": {"name": "d_", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,round(b*a2/a1)])", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything", "can_override": false}, "a2_": {"name": "a2_", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "s1_": {"name": "s1_", "group": "Ungrouped variables", "definition": "if(c_<0,-1,1)", "description": "", "templateType": "anything", "can_override": false}, "nb_": {"name": "nb_", "group": "Ungrouped variables", "definition": "if(c_<0,'taking away','adding')", "description": "", "templateType": "anything", "can_override": false}, "c_": {"name": "c_", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b_": {"name": "b_", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "if(c<0,-1,1)", "description": "", "templateType": "anything", "can_override": false}, "a1_": {"name": "a1_", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything", "can_override": false}, "nb": {"name": "nb", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "description": "", "templateType": "anything", "can_override": false}, "a_": {"name": "a_", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "d", "nb", "a1", "a2", "s1", "a_", "c_", "b_", "d_", "nb_", "a1_", "a2_", "s1_"], "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>Write \\[\\simplify{({a*a2 + &nbsp;c*a1} * x + {a * d + &nbsp;c * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))}\\] as the sum of partial fractions.</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "0", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a} / ({a1}*x + {b}) + ({c} / ({a2}*x + {d}))", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 1e-05, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [10, 11], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": [")(", ")*("], "showStrings": false, "partialCredit": 0, "message": "<p>Input as the sum of partial fractions.</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Algebraic fractions with factorised denominators", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Chris Knapp", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14987/"}], "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions", "common denominator"], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p></p>\n<p>&nbsp;</p>", "advice": "", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"], "std1": ["std", "collectNumbers"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,round(b*a2/a1)])", "description": "", "templateType": "anything", "can_override": false}, "nb": {"name": "nb", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(-9..9 except [round(b/a1),round(d/a2)])", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "templateType": "anything", "can_override": false}, "s1": {"name": "s1", "group": 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