// Numbas version: exam_results_page_options
{"name": "Hyperbolic #1", "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", "s"], "variables": {"k": {"group": "Ungrouped variables", "definition": "random(2..9#1)", "name": "k", "templateType": "randrange", "description": ""}, "r": {"group": "Ungrouped variables", "definition": "random(2..18#1)", "name": "r", "templateType": "randrange", "description": ""}, "s": {"group": "Ungrouped variables", "definition": "random(2..21#1)", "name": "s", "templateType": "randrange", "description": ""}}, "name": "Hyperbolic #1", "statement": "<p>Solve the following equation for \\(a\\) and \\(b\\):</p>\n<p style=\"padding-left: 30px;\">\\(aCosh(\\var{k}t)+bSinh(\\var{k}t)=\\var{r}e^{\\var{k}t}+\\var{s}e^{-\\var{k}t}\\)</p>", "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"variableReplacementStrategy": "originalfirst", "gaps": [{"mustBeReducedPC": 0, "variableReplacements": [], "type": "numberentry", "correctAnswerStyle": "plain", "allowFractions": false, "correctAnswerFraction": false, "mustBeReduced": false, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "minValue": "{r}+{s}", "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "marks": "2", "maxValue": "{r}+{s}"}, {"mustBeReducedPC": 0, "variableReplacements": [], "type": "numberentry", "correctAnswerStyle": "plain", "allowFractions": false, "correctAnswerFraction": false, "mustBeReduced": false, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "minValue": "{r}-{s}", "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "maxValue": "{r}-{s}"}], "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "type": "gapfill", "marks": 0, "showCorrectAnswer": true, "prompt": "<p>\\(a=\\)&nbsp;[[0]]</p>\n<p>\\(b=\\)&nbsp;[[1]]</p>"}], "advice": "<p>\\(aCosh(\\var{k}t)+bSinh(\\var{k}t)=\\var{r}e^{\\var{k}t}+\\var{s}e^{-\\var{k}t}\\)</p>\n<p>\\(a.\\frac{1}{2}\\left(e^{\\var{k}t}+e^{-\\var{k}t}\\right)+b.\\frac{1}{2}\\left(e^{\\var{k}t}-e^{-\\var{k}t}\\right)=\\var{r}e^{\\var{k}t}+\\var{s}e^{-\\var{k}t}\\)</p>\n<p>\\(\\frac{a}{2}e^{\\var{k}t}+\\frac{a}{2}e^{-\\var{k}t}+\\frac{b}{2}e^{\\var{k}t}-\\frac{b}{2}e^{-\\var{k}t}=\\var{r}e^{\\var{k}t}+\\var{s}e^{-\\var{k}t}\\)</p>\n<p>\\(\\left(\\frac{a}{2}+\\frac{b}{2}\\right)e^{\\var{k}t}+\\left(\\frac{a}{2}-\\frac{b}{2}\\right)e^{-\\var{k}t}=\\var{r}e^{\\var{k}t}+\\var{s}e^{-\\var{k}t}\\)</p>\n<p>Comparing like with like leads to two linear equations:</p>\n<p>\\(\\left(\\frac{a}{2}+\\frac{b}{2}\\right)e^{\\var{k}t}=\\var{r}e^{\\var{k}t}\\)&nbsp; \\(\\implies&nbsp;\\frac{a}{2}+\\frac{b}{2}=\\var{r}\\) equation (i)</p>\n<p>and</p>\n<p>\\(\\left(\\frac{a}{2}-\\frac{b}{2}\\right)e^{-\\var{k}t}=\\var{s}e^{-\\var{k}t}\\)&nbsp; \\(\\implies&nbsp;\\frac{a}{2}-\\frac{b}{2}=\\var{s}\\)&nbsp;&nbsp;equation (ii)</p>\n<p>Adding equation (i) and&nbsp;equation (ii) gives</p>\n<p style=\"padding-left: 30px;\">\\(a=\\simplify{{r}+{s}}\\)</p>\n<p>Subtracting would give</p>\n<p style=\"padding-left: 30px;\">\\(b=\\simplify{{r}-{s}}\\)</p>", "tags": [], "preamble": {"css": "", "js": ""}, "extensions": [], "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/"}]}