// Numbas version: exam_results_page_options
{"name": "Solve equations which include a single even power (e.g. x^even=blah)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [{"variables": ["intpower", "intrhs", "intsoln"], "name": "a"}, {"variables": ["bpower", "bsoln", "bxcoeff", "bb", "bc", "brhs"], "name": "b"}, {"variables": ["cpower", "cc", "cb", "cxcoeff", "csolnapprox"], "name": "c"}], "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["dc", "db", "ddenom"], "functions": {}, "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "<p>Questions to test if the student knows the inverse of an even power (and how to solve equations that contain a single power that is even).&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Please complete the following.</p>", "variables": {"bsoln": {"templateType": "anything", "description": "", "definition": "switch(bpower=2, random(2..10), bpower=4, random(2..4), bpower=6, random(2..3 except -1..1), 2)", "group": "b", "name": "bsoln"}, "intpower": {"templateType": "anything", "description": "", "definition": "random(2..8#2)", "group": "a", "name": "intpower"}, "csolnapprox": {"templateType": "anything", "description": "", "definition": "root(((cc-cb)/cxcoeff),(1/cpower))", "group": "c", "name": "csolnapprox"}, "bb": {"templateType": "anything", "description": "", "definition": "random(1..100)", "group": "b", "name": "bb"}, "cxcoeff": {"templateType": "anything", "description": "", "definition": "if(cc-cb>0,random(2..12),random(-12..-2))", "group": "c", "name": "cxcoeff"}, "bxcoeff": {"templateType": "anything", "description": "", "definition": "random(-3..3 except 0..1)", "group": "b", "name": "bxcoeff"}, "db": {"templateType": "anything", "description": "", "definition": "random(-100..100 except -1..1)", "group": "Ungrouped variables", "name": "db"}, "intrhs": {"templateType": "anything", "description": "", "definition": "intsoln^intpower\n", "group": "a", "name": "intrhs"}, "cpower": {"templateType": "anything", "description": "", "definition": "random(2..8#2)", "group": "c", "name": "cpower"}, "bc": {"templateType": "anything", "description": "", "definition": "bsoln^bpower*bxcoeff+bb", "group": "b", "name": "bc"}, "brhs": {"templateType": "anything", "description": "", "definition": "(bc-bb)/bxcoeff", "group": "b", "name": "brhs"}, "intsoln": {"templateType": "anything", "description": "", "definition": "switch(intpower=2, random(2..12), intpower=4, random(2..5), intpower=6, random(2..3), 2)", "group": "a", "name": "intsoln"}, "bpower": {"templateType": "anything", "description": "", "definition": "random(2..8#2)", "group": "b", "name": "bpower"}, "dc": {"templateType": "anything", "description": "", "definition": "random(2..100)", "group": "Ungrouped variables", "name": "dc"}, "cb": {"templateType": "anything", "description": "", "definition": "random(-100..100 except [0,cc])", "group": "c", "name": "cb"}, "ddenom": {"templateType": "anything", "description": "", "definition": "random(2..15)", "group": "Ungrouped variables", "name": "ddenom"}, "cc": {"templateType": "anything", "description": "", "definition": "random(-100..100)", "group": "c", "name": "cc"}}, "tags": [], "parts": [{"customName": "", "customMarkingAlgorithm": "", "sortAnswers": false, "scripts": {}, "prompt": "<p>If &nbsp;$x^\\var{intpower}=\\var{intrhs}$, then $x=$ [[0]], or&nbsp;[[1]].</p>\n<p><em>(Please enter&nbsp;the smaller value in the first gap and the larger value on the second gap)</em></p>", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "unitTests": [], "useCustomName": false, "gaps": [{"customName": "", "customMarkingAlgorithm": "", "scripts": {}, "failureRate": 1, "answer": "{-intsoln}", "valuegenerators": [], "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "vsetRange": [0, 1], "showFeedbackIcon": true, "marks": 1, "vsetRangePoints": 5, "checkVariableNames": false, "unitTests": [], "showPreview": false, "useCustomName": false, "variableReplacements": [], "checkingAccuracy": 0.001}, {"customName": "", "customMarkingAlgorithm": "", "scripts": {}, "failureRate": 1, "answer": "{intsoln}", "valuegenerators": [], "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "vsetRange": [0, 1], "showFeedbackIcon": true, "marks": 1, "vsetRangePoints": 5, "checkVariableNames": false, "unitTests": [], "showPreview": false, "useCustomName": false, "variableReplacements": [], "checkingAccuracy": 0.001}], "variableReplacements": []}, {"customName": "", "customMarkingAlgorithm": "", "sortAnswers": false, "scripts": {}, "prompt": "<p>If &nbsp;$\\simplify{{bxcoeff}y^{bpower}+{bb}}=\\var{bc}$, then $y=$ [[0]], or&nbsp;[[1]].</p>\n<p><em>(Please enter&nbsp;the smaller value in the first gap and the larger value on the second gap)</em></p>", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "unitTests": [], "useCustomName": false, "gaps": [{"customName": "", "customMarkingAlgorithm": "", "scripts": {}, "failureRate": 1, "answer": "{-bsoln}", "valuegenerators": [], "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "vsetRange": [0, 1], "showFeedbackIcon": true, "marks": 1, "vsetRangePoints": 5, "checkVariableNames": false, "unitTests": [], "showPreview": false, "useCustomName": false, "variableReplacements": [], "checkingAccuracy": 0.001}, {"customName": "", "customMarkingAlgorithm": "", "scripts": {}, "failureRate": 1, "answer": "{bsoln}", "valuegenerators": [], "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "vsetRange": [0, 1], "showFeedbackIcon": true, "marks": 1, "vsetRangePoints": 5, "checkVariableNames": false, "unitTests": [], "showPreview": false, "useCustomName": false, "variableReplacements": [], "checkingAccuracy": 0.001}], "variableReplacements": []}, {"customName": "", "customMarkingAlgorithm": "", "sortAnswers": false, "scripts": {}, "prompt": "<p>For this question input the&nbsp;<em>exact</em> value by using a <em>fractional power</em> to indicate a root. For example, if the answer was $\\sqrt[3]{\\frac{35}{11}}$, then enter &nbsp;(35/11)^(1/3).</p>\n<p></p>\n<p>If &nbsp;$\\simplify{{cxcoeff}z^{cpower}+{cb}}=\\var{cc}$, then $z=$ [[0]], or&nbsp;[[1]].</p>\n<p><em>(Please enter&nbsp;the smaller value in the first gap and the larger value on the second gap)</em></p>", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "unitTests": [], "useCustomName": false, "gaps": [{"customName": "", "customMarkingAlgorithm": "", "failureRate": 1, "checkingType": "absdiff", "showCorrectAnswer": true, "checkVariableNames": false, "vsetRange": [0, 1], "showFeedbackIcon": true, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "scripts": {}, "answer": "-({(cc-cb)/cxcoeff})^(1/{cpower})", "valuegenerators": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "answerSimplification": "fractionNumbers", "marks": 1, "unitTests": [], "showPreview": true, "useCustomName": false, "variableReplacements": []}, {"customName": "", "customMarkingAlgorithm": "", "failureRate": 1, "checkingType": "absdiff", "showCorrectAnswer": true, "checkVariableNames": false, "vsetRange": [0, 1], "showFeedbackIcon": true, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "scripts": {}, "answer": "({(cc-cb)/cxcoeff})^(1/{cpower})", "valuegenerators": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "answerSimplification": "fractionNumbers", "marks": 1, "unitTests": [], "showPreview": true, "useCustomName": false, "variableReplacements": []}], "variableReplacements": []}, {"customName": "", "customMarkingAlgorithm": "", "sortAnswers": false, "scripts": {}, "prompt": "<p>For this question input the&nbsp;<em>exact</em> value by using a <em>fractional power</em> to indicate a root. For example, if the answer was $\\sqrt[3]{\\frac{35}{11}}$, then enter &nbsp;(35/11)^(1/3).</p>\n<p>If &nbsp;$\\displaystyle{\\simplify{((z+{db})^{bpower})/({ddenom})}}=\\var{dc}$, then $z=$ [[0]], or&nbsp;[[1]].</p>\n<p><em>(Please enter&nbsp;the smaller value in the first gap and the larger value on the second gap)</em></p>", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "unitTests": [], "useCustomName": false, "gaps": [{"customName": "", "customMarkingAlgorithm": "", "failureRate": 1, "checkingType": "absdiff", "showCorrectAnswer": true, "checkVariableNames": false, "vsetRange": [0, 1], "showFeedbackIcon": true, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "scripts": {}, "answer": "-({dc*ddenom})^(1/{bpower})-{db}", "valuegenerators": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "answerSimplification": "basic", "marks": 1, "unitTests": [], "showPreview": true, "useCustomName": false, "variableReplacements": []}, {"customName": "", "customMarkingAlgorithm": "", "failureRate": 1, "checkingType": "absdiff", "showCorrectAnswer": true, "checkVariableNames": false, "vsetRange": [0, 1], "showFeedbackIcon": true, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "scripts": {}, "answer": "({dc*ddenom})^(1/{bpower})-{db}", "valuegenerators": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "answerSimplification": "basic", "marks": 1, "unitTests": [], "showPreview": true, "useCustomName": false, "variableReplacements": []}], "variableReplacements": []}], "rulesets": {}, "advice": "<p>Suppose you wanted to solve $x^2=4$, that is, you wanted the number that squares to give four. The number $2$ comes to mind, however, $-2$ is also a solution since $(-2)^2=(-2)\\times(-2)=4$. &nbsp;Recall the product of two negatives is a positive, so the product of any even number of negative numbers is positive. This means when we find a positive solution to an equation like $x^4=10\\,000$ there will also be a negative solution, in particular, the solution to&nbsp;$x^4=10\\,000$ would be&nbsp;$x=\\pm\\sqrt[4]{10\\,000}=\\pm10$.</p>\n<p></p>\n<p>a) Since the power in $x^\\var{intpower}=\\var{intrhs}$ is even we will take the plus or minus $\\var{intpower}$<span data-jme-visible=\"intpower=2\">nd</span> <span data-jme-visible=\"intpower=3\">rd</span> <span data-jme-visible=\"intpower&gt;3\">th</span> &nbsp;root to get two solutions.</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$x^\\var{intpower}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\var{intrhs}$&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">&nbsp;</td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\sqrt[\\var{intpower}]{x^\\var{intpower}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\pm\\sqrt[\\var{intpower}]{\\var{intrhs}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">&nbsp;</td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$x$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\pm\\var{intsoln}$</td>\n</tr>\n</tbody>\n</table>\n<p>That is, $x$ equals $-\\var{intsoln}$ or $\\var{intsoln}$.</p>\n<p></p>\n<p>b) Given&nbsp;$\\simplify{{bxcoeff}y^{bpower}+{bb}}=\\var{bc}$, we can rearrange the equation&nbsp;to get $y^\\var{bpower}$ by itself and then we can take the plus or minus&nbsp;$\\var{bpower}$<span data-jme-visible=\"bpower=2\">nd</span> <span data-jme-visible=\"bpower=3\">rd</span> <span data-jme-visible=\"bpower&gt;3\">th</span> &nbsp;root to get $y$ by itself.</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$\\simplify{{bxcoeff}y^{bpower}+{bb}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\var{bc}$&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">&nbsp;</td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\simplify{{bxcoeff}y^{bpower}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\simplify[basic]{{bc}-{bb}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">&nbsp;</td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\simplify{{bxcoeff}y^{bpower}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{bc-bb}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$y^\\var{bpower}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\simplify[!basic]{{bc-bb}/{bxcoeff}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$y^\\var{bpower}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{bc-bb}/{bxcoeff}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\sqrt[\\var{bpower}]{y^\\var{bpower}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\pm\\sqrt[\\var{bpower}]{\\var{brhs}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$y$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\pm\\var{bsoln}$</td>\n</tr>\n</tbody>\n</table>\n<p>That is, $y$ equals $-\\var{bsoln}$, or $\\var{bsoln}$.</p>\n<p></p>\n<p>c) Given&nbsp;$\\simplify{{cxcoeff}z^{cpower}+{cb}}=\\var{cc}$, we can rearrange the equation&nbsp;to get $z^\\var{cpower}$ by itself and then we can take the plus or minus $\\var{cpower}$<span data-jme-visible=\"cpower=2\">nd</span> <span data-jme-visible=\"cpower=3\">rd</span> <span data-jme-visible=\"cpower&gt;3\">th</span> &nbsp;root to get $z$ by itself.</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$\\simplify{{cxcoeff}z^{cpower}+{cb}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\var{cc}$&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">&nbsp;</td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\simplify{{cxcoeff}z^{cpower}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\simplify[basic]{{cc}-{cb}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">&nbsp;</td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\simplify{{cxcoeff}z^{cpower}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{cc-cb}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$z^\\var{cpower}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\simplify[!basic]{{cc-cb}/{cxcoeff}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$z^\\var{cpower}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\simplify[fractionNumbers]{{(cc-cb)/cxcoeff}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\sqrt[\\var{cpower}]{z^\\var{cpower}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\pm\\sqrt[\\var{cpower}]{\\simplify[fractionNumbers]{{(cc-cb)/cxcoeff}}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$z$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\pm\\simplify[fractionNumbers]{{(cc-cb)/cxcoeff}^{1/{cpower}}}$</td>\n</tr>\n</tbody>\n</table>\n<p>That is, $z$ equals $-\\sqrt[\\var{cpower}]{\\simplify[fractionNumbers]{{(cc-cb)/cxcoeff}}}$, or&nbsp;$\\sqrt[\\var{cpower}]{\\simplify[fractionNumbers]{{(cc-cb)/cxcoeff}}}$.</p>\n<p></p>\n<p>d) Given $\\displaystyle{\\simplify{((z+{db})^{bpower})/({ddenom})}}=\\var{dc}$, we can rearrange the equation&nbsp;to get $\\simplify{(z+{db})^{bpower}}$ by itself, then we can take the plus or minus $\\var{bpower}$<span data-jme-visible=\"cpower=2\">nd</span> <span data-jme-visible=\"bpower=3\">rd</span> <span data-jme-visible=\"bpower&gt;3\">th</span> &nbsp;root of both sides to get $\\simplify{z+{db}}$ by itself, and then rearrange to get $z$ by itself.</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$\\displaystyle{\\simplify{((z+{db})^{bpower})/({ddenom})}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\var{dc}$&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">&nbsp;</td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\simplify{(z+{db})^{bpower}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\simplify[basic]{{dc}*{ddenom}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">&nbsp;</td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\simplify{(z+{db})^{bpower}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\var{dc*ddenom}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\sqrt[\\var{bpower}]{\\simplify{(z+{db})^{bpower}}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\pm\\sqrt[\\var{bpower}]{\\var{dc*ddenom}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\simplify{z+{db}}$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\pm\\sqrt[\\var{bpower}]{\\var{dc*ddenom}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$z$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\pm\\simplify{root({dc*ddenom},{bpower})-{db}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td></td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$z$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\pm\\simplify{{dc*ddenom}^(1/{bpower})-{db}}$</td>\n</tr>\n</tbody>\n</table>\n<p>That is, $z$ equals $-\\simplify{root({dc*ddenom},{bpower})-{db}}$, or $\\simplify{root({dc*ddenom},{bpower})-{db}}$.</p>", "name": "Solve equations which include a single even power (e.g. x^even=blah)", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}