// Numbas version: finer_feedback_settings
{"name": "3x3 Determinant Problems", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "3x3 Determinant Problems", "tags": [], "metadata": {"description": "<p>This question tests learner's knowledge of the inverse matrix method for a 3x3 matrix.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<h1>$3 \\times 3$ Determinant Problems</h1>\n<p>Calculate the two values for $x$ that satisfy the equation $|A|=\\var{k}$</p>\n<p>where the matrix $A$ is given by:</p>\n<p>$$<br/>A=\\begin{pmatrix} x&amp;\\var{a}&amp;\\var{b}\\\\ \\var{c}&amp;x&amp;\\var{d}\\\\\\var{e1}&amp;\\var{f}&amp;\\var{g} \\end{pmatrix}<br/>$$</p>", "advice": "<p>We must calculate the determinant as usual, using the expansion of the first row:</p>\n<p>$$<br/>\\begin{aligned}<br/>\\det{A} &amp;= x\\begin{vmatrix} <br/>x &amp;\\var{d} \\\\<br/>\\var{f} &amp; \\var{g} \\\\<br/>\\end{vmatrix} -\\var{a} \\begin{vmatrix}<br/>\\var{c} &amp; \\var{d} \\\\ <br/>\\var{e1} &amp; \\var{g} \\\\<br/>\\end{vmatrix} + \\var{b} \\begin{vmatrix} <br/>\\var{c} &amp; x \\\\ <br/>\\var{e1} &amp; \\var{f} <br/>\\end{vmatrix} \\\\&nbsp;<br/>&amp;= x(x-\\simplify{{d}{f}})-\\var{a}(\\simplify{{c}*{g}}-\\simplify{{e1}*{d}})+\\var{b}(\\simplify{{c}*{f}}-\\var{e1}x) \\\\<br/>&amp;= x^2-\\simplify{{d}*{f}}x-\\simplify{{a}*{c}*{g}}+\\simplify{{a}*{e1}*{d}}+\\simplify{{b}*{c}*{f}}-\\simplify{{b}*{e1}}x \\\\<br/>&amp;= x^2-\\simplify{{d}*{f}+{b}*{e1}}x+\\simplify{{a}*{e1}*{d}+{b}*{c}*{f}-{a}*{c}*{g}}<br/>\\end{aligned}<br/>$$</p>\n<p>Since we know that $\\det A = \\var{k}$:</p>\n<p>$$<br/>\\begin{aligned}<br/>x^2-\\simplify{{d}*{f}+{b}*{e1}}x+\\simplify{{a}*{e1}*{d}+{b}*{c}*{f}-{a}*{c}*{g}} &amp;= \\var{k} \\\\<br/>x^2-\\simplify{{d}*{f}+{b}*{e1}}x+\\simplify{{b}*{e1}*{f}*{d}} &amp;= 0<br/>\\end{aligned}<br/>$$</p>\n<p>This can be solved by formula or by finding factors:</p>\n<p>$$<br/>(x-\\simplify{{f}*{d}})(x-\\simplify{{b}*{e1}})=0<br/>$$</p>\n<p>So we have:</p>\n<p>$$<br/>x=\\simplify{{f}*{d}}\\,\\,\\,\\,\\,or\\,\\,\\,\\,\\,x=\\simplify{{b}*{e1}}<br/>$$</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(6 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(1 .. 5#1)", "description": "", "templateType": "randrange", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1 .. 6#1)", "description": "", "templateType": "randrange", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything", "can_override": false}, "e1": {"name": "e1", "group": "Ungrouped variables", "definition": "random(6 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1 .. 20#1)", "description": "", "templateType": "randrange", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "{a}*{e1}*{d}+{b}*{c}*{f}-{a}*{c}*{g}-{b}*{e1}*{f}*{d}", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1 .. 20#1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "e1", "f", "g", "k"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": 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