{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"math","formal_name":"MATH","introduction":"MATH organises 12,500 high-school competition problems into seven subjects and five difficulty levels, each with a full worked solution. It underlies much of the later mathematics evaluation work; this catalogue imports the full test split: 5,000 problems across all seven subjects.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://github.com/hendrycks/math","indexing_mode":"index","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[],"history":"","construction":"","limitations":"","saturation":"","scores":[]}},"task_id":"d1489cf2-052a-5534-af33-2326bb934be5","task_key":"precalculus--test--d1489cf2-052a-5534-af33-2326bb934be5","task_revision_id":"1","upstream_id":"","short_description":"The matrix for reflecting through a certain plane $P,$ which passes through the…","config":"precalculus","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"The matrix for reflecting through a certain plane $P,$ which passes through the origin, is given by\\n\\\\[\\\\renewcommand{\\\\arraystretch}{1.5} \\\\begin{pmatrix} \\\\frac{11}{15} & \\\\frac{2}{15} & \\\\frac{2}{3} \\\\\\\\ \\\\frac{2}{15} & \\\\frac{14}{15} & -\\\\frac{1}{3} \\\\\\\\ \\\\frac{2}{3} & -\\\\frac{1}{3} & -\\\\frac{2}{3} \\\\end{pmatrix} \\\\renewcommand{\\\\arraystretch}{1}.\\\\]Find the normal vector of plane $P.$  Enter your answer in the form $\\\\begin{pmatrix} a \\\\\\\\ b \\\\\\\\ c \\\\end{pmatrix},$ where $a,$ $b,$ and $c$ are integers, $a > 0,$ and $\\\\gcd(|a|,|b|,|c|) = 1.$\",\"type\":\"Precalculus\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://github.com/hendrycks/math","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}