// Numbas version: exam_results_page_options
{"name": "Hyperbolic #2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "variable_groups": [], "functions": {}, "ungrouped_variables": ["k", "r", "a", "b"], "extensions": [], "name": "Hyperbolic #2", "variables": {"k": {"definition": "random(30..42#1)", "group": "Ungrouped variables", "name": "k", "templateType": "randrange", "description": ""}, "r": {"definition": "random(2..8#1)", "group": "Ungrouped variables", "name": "r", "templateType": "randrange", "description": ""}, "a": {"definition": "random(10..16#1)", "group": "Ungrouped variables", "name": "a", "templateType": "randrange", "description": ""}, "b": {"definition": "random(2..9#1)", "group": "Ungrouped variables", "name": "b", "templateType": "randrange", "description": ""}}, "statement": "<p>Solve the following equation for \\(x\\):</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{2*{a}}Cosh(\\var{r}x)-\\simplify{2*{b}}Sinh(\\var{r}x)=\\var{k}\\)</p>", "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "variablesTest": {"condition": "k^2>4*(a^2-b^2)", "maxRuns": 100}, "parts": [{"variableReplacementStrategy": "originalfirst", "gaps": [{"precision": "3", "mustBeReducedPC": 0, "strictPrecision": false, "minValue": "ln(({k}+sqrt{{k}^2-4*({a}^2-{b}^2)})/(2*({a}-{b})))/{r}", "precisionPartialCredit": 0, "correctAnswerFraction": false, "allowFractions": false, "type": "numberentry", "mustBeReduced": false, "precisionType": "dp", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": true, "scripts": {}, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "marks": "2", "maxValue": "ln(({k}+sqrt{{k}^2-4*({a}^2-{b}^2)})/(2*({a}-{b})))/{r}"}, {"precision": "3", "mustBeReducedPC": 0, "strictPrecision": false, "minValue": "ln(({k}-sqrt{{k}^2-4*({a}^2-{b}^2)})/(2*({a}-{b})))/{r}", "precisionPartialCredit": 0, "correctAnswerFraction": false, "allowFractions": false, "type": "numberentry", "mustBeReduced": false, "precisionType": "dp", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": true, "scripts": {}, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "marks": 1, "maxValue": "ln(({k}-sqrt{{k}^2-4*({a}^2-{b}^2)})/(2*({a}-{b})))/{r}"}], "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "type": "gapfill", "marks": 0, "showCorrectAnswer": true, "prompt": "<p>Enter&nbsp;the greater of your two answers&nbsp; \\(x=\\)&nbsp;[[0]]</p>\n<p></p>\n<p>Enter&nbsp;the lesser of your two answers&nbsp; &nbsp;\\(x=\\)&nbsp;[[1]]</p>"}], "advice": "<p>\\(\\simplify{2*{a}}Cosh(\\var{r}x)-\\simplify{2*{b}}Sinh(\\var{r}x)=\\var{k}\\)</p>\n<p>\\(\\simplify{2*{a}}.\\frac{1}{2}\\left(e^{\\var{r}x}+e^{-\\var{r}x}\\right)-\\simplify{2*{b}}.\\frac{1}{2}\\left(e^{\\var{r}x}-e^{-\\var{r}x}\\right)=\\var{k}\\)</p>\n<p>\\(\\var{a}e^{\\var{r}x}+\\var{a}e^{-\\var{r}x}-\\var{b}e^{\\var{r}x}+\\var{b}e^{-\\var{r}x}=\\var{k}\\)</p>\n<p>\\(\\simplify{{a}-{b}}e^{\\var{r}x}+\\simplify{{a}+{b}}e^{-\\var{r}x}=\\var{k}\\)</p>\n<p>Multiply across by \\(e^{\\var{r}x}\\)</p>\n<p>\\(\\simplify{{a}-{b}}\\left(e^{\\var{r}x}\\right)^2+\\simplify{{a}+{b}}=\\var{k}e^{\\var{r}x}\\)</p>\n<p>\\(\\simplify{{a}-{b}}\\left(e^{\\var{r}x}\\right)^2-\\var{k}e^{\\var{r}x}+\\simplify{{a}+{b}}=0\\)</p>\n<p>This is in the&nbsp;form of a quadratic equation, hence</p>\n<p style=\"padding-left: 30px;\">\\(e^{\\var{r}x}=\\frac{\\var{k}\\pm \\sqrt{\\var{k}^2-4*\\simplify{{a}-{b}}*\\simplify{{a}+{b}}}}{2*\\simplify{{a}-{b}}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(e^{\\var{r}x}=\\frac{\\var{k}\\pm \\sqrt{\\simplify{{k}^2-4*({a}-{b})*({a}+{b})}}}{\\simplify{2*({a}-{b})}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{r}x=ln\\left(\\frac{\\var{k}\\pm \\sqrt{\\simplify{{k}^2-4*({a}-{b})*({a}+{b})}}}{\\simplify{2*({a}-{b})}}\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(x=\\frac{1}{\\var{r}}ln\\left(\\frac{\\var{k}\\pm \\sqrt{\\simplify{{k}^2-4*({a}-{b})*({a}+{b})}}}{\\simplify{2*({a}-{b})}}\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(x=\\frac{1}{\\var{r}}ln\\left(\\frac{\\var{k}+\\sqrt{\\simplify{{k}^2-4*({a}-{b})*({a}+{b})}}}{\\simplify{2*({a}-{b})}}\\right)\\)&nbsp; &nbsp; or&nbsp; &nbsp; \\(x=\\frac{1}{\\var{r}}ln\\left(\\frac{\\var{k}-\\sqrt{\\simplify{{k}^2-4*({a}-{b})*({a}+{b})}}}{\\simplify{2*({a}-{b})}}\\right)\\)</p>\n<p style=\"padding-left: 30px;\"></p>", "tags": [], "preamble": {"css": "", "js": ""}, "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}