{"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":"f2287d73-820c-5097-bde6-98d5e999333c","task_key":"precalculus--test--f2287d73-820c-5097-bde6-98d5e999333c","task_revision_id":"1","upstream_id":"","short_description":"Let $\\mathbf{v}_0$ be a vector. The vector $\\mathbf{v}_0$ is rotated about the…","config":"precalculus","split":"test","body":"{\"level\":\"Level 4\",\"problem\":\"Let $\\\\mathbf{v}_0$ be a vector.  The vector $\\\\mathbf{v}_0$ is rotated about the origin by an angle of $42^\\\\circ$ counter-clockwise, taking it to vector $\\\\mathbf{v}_1.$  The vector $\\\\mathbf{v}_1$ is then reflected over the line with direction vector $\\\\begin{pmatrix} \\\\cos 108^\\\\circ \\\\\\\\ \\\\sin 108^\\\\circ \\\\end{pmatrix},$ taking it to vector $\\\\mathbf{v}_2.$\\n\\nThe vector $\\\\mathbf{v}_2$ can also be produced by reflecting the vector $\\\\mathbf{v}_0$ over the line with direction vector $\\\\begin{pmatrix} \\\\cos \\\\theta \\\\\\\\ \\\\sin \\\\theta \\\\end{pmatrix},$ where $\\\\theta$ is an acute angle.  Find $\\\\theta.$\",\"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":[]}