{"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":"fa336db2-53d3-5fdd-b9b8-f9fd48081442","task_key":"precalculus--test--fa336db2-53d3-5fdd-b9b8-f9fd48081442","task_revision_id":"1","upstream_id":"","short_description":"Let $\\mathbf{a} = \\begin{pmatrix} 1 \\\\ 3 \\\\ 0 \\end{pmatrix},$ $\\mathbf{b} =…","config":"precalculus","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"Let $\\\\mathbf{a} = \\\\begin{pmatrix} 1 \\\\\\\\ 3 \\\\\\\\ 0 \\\\end{pmatrix},$ $\\\\mathbf{b} = \\\\begin{pmatrix} -11 \\\\\\\\ 5 \\\\\\\\ 2 \\\\end{pmatrix},$ and $\\\\mathbf{c} = \\\\begin{pmatrix} 1 + \\\\sqrt{5} \\\\\\\\ 4 \\\\\\\\ -5 \\\\end{pmatrix}.$  Find $k$ if the vectors $\\\\mathbf{a} + \\\\mathbf{b} + \\\\mathbf{c}$ and\\n\\\\[3 (\\\\mathbf{b} \\\\times \\\\mathbf{c}) - 8 (\\\\mathbf{c} \\\\times \\\\mathbf{a}) + k (\\\\mathbf{a} \\\\times \\\\mathbf{b})\\\\]are orthogonal.\",\"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":[]}