{"id":5339,"date":"2019-05-20T16:53:34","date_gmt":"2019-05-20T14:53:34","guid":{"rendered":"http:\/\/www.mi.uni-koeln.de\/opt\/?page_id=5339"},"modified":"2019-06-25T14:22:50","modified_gmt":"2019-06-25T12:22:50","slug":"or-quiz-teil-b","status":"publish","type":"page","link":"http:\/\/www.mi.uni-koeln.de\/opt\/veranstaltungen\/einfuehrung-in-die-mathematik-des-operations-research-2019\/or-quiz-teil-b\/","title":{"rendered":"OR 2019 Quiz Teil B"},"content":{"rendered":"<p>Frage 3 wurde editiert (Stand 25.6.2019).<\/p>\n\n<script>\n                            if (window.qmn_quiz_data === undefined) {\n                                    window.qmn_quiz_data = new Object();\n                            }\n                    <\/script><script>window.qmn_quiz_data[\"6\"] = {\"quiz_id\":\"6\",\"quiz_name\":\"OR Quiz Teil B\",\"disable_answer\":\"0\",\"ajax_show_correct\":\"0\",\"progress_bar\":0,\"contact_info_location\":\"0\",\"qpages\":[{\"id\":0,\"pagetimer\":0,\"pagetimer_second\":0}],\"skip_validation_time_expire\":0,\"timer_limit_val\":0,\"disable_scroll_next_previous_click\":0,\"disable_scroll_on_result\":0,\"disable_first_page\":0,\"enable_result_after_timer_end\":0,\"enable_quick_result_mc\":1,\"highlight_correct_answer\":1,\"end_quiz_if_wrong\":0,\"form_disable_autofill\":0,\"disable_mathjax\":0,\"enable_quick_correct_answer_info\":0,\"quick_result_correct_answer_text\":\"Correct! 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You have selected wrong answer.\",\"quiz_processing_message\":\"\",\"quiz_limit_choice\":\"Limit of choice is reached.\",\"not_allow_after_expired_time\":0,\"scheduled_time_end\":false,\"prevent_reload\":0,\"limit_email_based_submission\":0,\"total_user_tries\":0,\"is_logged_in\":false,\"error_messages\":{\"email_error_text\":\"Not a valid e-mail address!\",\"number_error_text\":\"This field must be a number!\",\"incorrect_error_text\":\"The entered text is not correct!\",\"empty_error_text\":\"Alle Fragen m\\u00fcssen beantwortet werden\",\"contact_field_required_error_text\":\"Please complete all required fields!\",\"url_error_text\":\"The entered URL is not valid!\",\"minlength_error_text\":\"Required atleast %minlength% characters.\",\"maxlength_error_text\":\"Maximum %maxlength% characters allowed.\",\"recaptcha_error_text\":\"ReCaptcha is missing\",\"phone_error_text\":\"Phone number is invalid\"},\"questions_settings\":[]}\n                    <\/script><style type='text\/css' id='qmn_quiz_template-css'>.mlw_qmn_quiz label {\n\t\t\t\tdisplay: inline;\n\t\t\t}\n\t\t\t.ui-tooltip\n\t\t\t{\n\t\t\t\tmax-width: 500px !important;\n\t\t\t}\n\t\t\t.ui-tooltip-content\n\t\t\t{\n\t\t\t\tmax-width: 500px !important;\n\t\t\t}\n\t\t\t.qmn_error, .qmn_page_error_message\n\t\t\t{\n\t\t\t\tcolor: red;\n\t\t\t}\n\t\t\t.mlw_qmn_hint_link\n\t\t\t{\n\t\t\ttext-decoration:underline;\n\t\t\tcolor:rgb(0,0,255);\n\t\t\t}\n\t\t\t.mlw_qmn_quiz_link\n\t\t\t{\n\t\t\t\tdisplay: inline;\n\t\t\t\tvertical-align:top !important;\n\t\t\t\ttext-decoration: none;\n\t\t\t}\n\t\t\tdiv.mlw_qmn_quiz input[type=radio],\n\t\t\tdiv.mlw_qmn_quiz input[type=submit],\n\t\t\tdiv.mlw_qmn_quiz label {\n\t\t\t\tcursor: pointer;\n\t\t\t}\n\t\t\tdiv.mlw_qmn_quiz input:not([type=submit]):focus,\n\t\t\tdiv.mlw_qmn_quiz textarea:focus {\n\t\t\t\tbackground: #eaeaea;\n\t\t\t}\n\t\t\tdiv.mlw_qmn_quiz {\n\t\t\t\ttext-align: left;\n\t\t\t}\n\t\t\tdiv.quiz_section {\n\n\t\t\t}\n\t\t\t.mlw_horizontal_choice\n\t\t\t{\n\t\t\t\tmargin-right: 20px;\n\t\t\t}\n\t\t\tdiv.mlw_qmn_timer {\n\t\t\t\tposition:fixed;\n\t\t\t\ttop:200px;\n\t\t\t\tright:0px;\n\t\t\t\twidth:130px;\n\t\t\t\tcolor:#00CCFF;\n\t\t\t\tborder-radius: 15px;\n\t\t\t\tbackground:#000000;\n\t\t\t\ttext-align: center;\n\t\t\t\tpadding: 15px 15px 15px 15px\n\t\t\t}\n\t\t\tdiv.mlw_qmn_quiz input[type=submit],\n\t\t\ta.mlw_qmn_quiz_link\n\t\t\t{\n\t\t\t\tborder-radius: 4px;\n\t\t\t\tposition: relative;\n\t\t\t\tbackground-image: linear-gradient(#fff,#dedede);\n\t\t\t\tbackground-color: #eee;\n\t\t\t\tborder: #ccc solid 1px;\n\t\t\t\tcolor: #333;\n\t\t\t\ttext-shadow: 0 1px 0 rgba(255,255,255,.5);\n\t\t\t\tbox-sizing: border-box;\n\t\t\t\tdisplay: inline-block;\n\t\t\t\tpadding: 7px 7px 7px 7px;\n\t\t\t\tmargin: auto;\n\t\t\t\tfont-weight: bold;\n\t\t\t\tcursor: pointer;\n\t\t\t}\n\t\t\t.mlw_qmn_question, .mlw_qmn_question_number, .mlw_qmn_comment_section_text\n\t\t\t{\n\t\t\t\tfont-weight: bold;\n\t\t\t}\n\t\t\t.mlw_next\n\t\t\t{\n\t\t\t\tfloat: right;\n\t\t\t}\n\t\t\t.mlw_previous\n\t\t\t{\n\t\t\t\tfloat: left;\n\t\t\t}\n\t\t\t.mlw_qmn_question_comment, .mlw_answer_open_text, .qmn_comment_section {\n\t\t\t\twidth: 100%;\n\t\t\t\tborder-radius: 7px;\n\t\t\t\tpadding: 2px 10px;\n\t\t\t\t-webkit-box-shadow: inset 0 3px 3px rgba(0,0,0,.075);\n\t\t\t\tbox-shadow: inset 0 3px 3px rgba(0,0,0,.075);\n\t\t\t\tborder: 1px solid #ccc;\n\t\t\t}\n\t\t<\/style>\t\t<div class='qsm-quiz-container qsm-quiz-container-6 qmn_quiz_container mlw_qmn_quiz  quiz_theme_default  '>\n\t\t\t\t\t\t\t\t<form name=\"quizForm6\" id=\"quizForm6\" action=\"\/opt\/wp-json\/wp\/v2\/pages\/5339\" method=\"POST\" class=\"qsm-quiz-form qmn_quiz_form mlw_quiz_form\" novalidate enctype=\"multipart\/form-data\">\n\t\t\t\t<input type=\"hidden\" name=\"qsm_hidden_questions\" id=\"qsm_hidden_questions\" value=\"\">\n\t\t\t\t<input type=\"hidden\" name=\"qsm_nonce\" id=\"qsm_nonce_6\" value=\"a11160b127\">\n\t\t\t\t<input type=\"hidden\" name=\"qsm_unique_key\" id=\"qsm_unique_key_6\" value=\"6ae2ca99d15f6\">\n\t\t\t\t<div id=\"mlw_error_message\" class=\"qsm-error-message qmn_error_message_section\"><\/div>\n\t\t\t\t<span id=\"mlw_top_of_quiz\"><\/span>\n\t\t\t\t\t\t\t<section class=\"qsm-page \">\n\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-4 question-section-id-60 \" data-qid=\"60\">\n\t\t\t\t\t\t\t\t\t\t<span class='mlw_qmn_question_number'>1.&nbsp;<\/span>\n\t\t\t\t\t\t\t<div class='mlw_qmn_question ' >\n\t\t<p>Seien $x_1,\\ldots,x_N\\in\\mathbb{R}^n$ und sei $y=\\sum_{i=1}^N \\alpha_i x_i$. Unter welchen der folgenden Bedingungen ist $y$ eine Konvexkombination der $x_i$?<br \/> <br \/> (a) $\\sum_{i=1}^N\\alpha_i=1$ und $\\alpha_i\\geq 0$ f\u00fcr alle $i=1,\\ldots,N$<br \/> <br \/> (b) $\\sum_{i=1}^N\\alpha_i=1$<br \/> <br \/> (c) $\\alpha_i\\geq 0$ f\u00fcr alle $i=1,\\ldots,N$<br \/> <br \/> (d) $\\alpha_i\\in [0,1]$ f\u00fcr alle $i=1,\\ldots,N$<br \/> <br \/> (e) $\\alpha_i = 1\/N$ f\u00fcr alle $i=1,\\ldots,N$<br \/> <br \/> (f) $\\binom{1}{y}=\\binom{1 \\cdots 1}{x_1 \\cdots x_N}\\alpha$ mit $\\alpha=(\\alpha_1,\\ldots,\\alpha_N)^T\\geq 0$<br \/> <br \/> (g) $\\alpha_i= 0$ f\u00fcr alle $i=1,\\ldots,N$<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class=\"qmn_check_answers mlwRequiredRadio\">\n\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question60[]\" id=\"question60_1\" value=\"0\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question60_1\">\n\t\t\t\t\t\t\t(a)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question60[]\" id=\"question60_2\" value=\"1\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question60_2\">\n\t\t\t\t\t\t\t(b)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question60[]\" id=\"question60_3\" value=\"2\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question60_3\">\n\t\t\t\t\t\t\t(c)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question60[]\" id=\"question60_4\" value=\"3\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question60_4\">\n\t\t\t\t\t\t\t(d)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question60[]\" id=\"question60_5\" value=\"4\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question60_5\">\n\t\t\t\t\t\t\t(e)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question60[]\" id=\"question60_6\" value=\"5\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question60_6\">\n\t\t\t\t\t\t\t(f)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question60[]\" id=\"question60_7\" value=\"6\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question60_7\">\n\t\t\t\t\t\t\t(g)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_60\" value=\"\" \/>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-4 question-section-id-58 \" data-qid=\"58\">\n\t\t\t\t\t\t\t\t\t\t<span class='mlw_qmn_question_number'>2.&nbsp;<\/span>\n\t\t\t\t\t\t\t<div class='mlw_qmn_question ' >\n\t\t<p>Welche der folgenden Mengen sind konvex?<br \/> <br \/> (a) $\\Big\\{x\\in\\mathbb{R}^2 : \\begin{pmatrix}1&amp;1\\\\0&amp;0\\end{pmatrix}x=\\begin{pmatrix}5\\\\1\\end{pmatrix}\\Big\\}$<br \/> <br \/> (b) $B(0,1) \\setminus \\{e_1\\} \\subseteq \\mathbb{R}^n$<br \/> <br \/> (c) $B(0,1) \\setminus \\{0\\}\\subseteq \\mathbb{R}^n$<br \/> <br \/> (d) $\\operatorname{conv}\\{1,2,3\\}\\subseteq \\mathbb{R}$<br \/> <br \/> (e) $\\{(x,y)\\in \\mathbb{R}^2 : y\\geq x^3\\}$<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class=\"qmn_check_answers mlwRequiredRadio\">\n\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question58[]\" id=\"question58_1\" value=\"0\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question58_1\">\n\t\t\t\t\t\t\t(a)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question58[]\" id=\"question58_2\" value=\"1\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question58_2\">\n\t\t\t\t\t\t\t(b)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question58[]\" id=\"question58_3\" value=\"2\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question58_3\">\n\t\t\t\t\t\t\t(c)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question58[]\" id=\"question58_4\" value=\"3\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question58_4\">\n\t\t\t\t\t\t\t(d)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question58[]\" id=\"question58_5\" value=\"4\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question58_5\">\n\t\t\t\t\t\t\t(e)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_58\" value=\"\" \/>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-4 question-section-id-68 \" data-qid=\"68\">\n\t\t\t\t\t\t\t\t\t\t<span class='mlw_qmn_question_number'>3.&nbsp;<\/span>\n\t\t\t\t\t\t\t<div class='mlw_qmn_question ' >\n\t\t<p>Welche der folgenden Funktionen sind konvex?<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class=\"qmn_check_answers \">\n\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question68[]\" id=\"question68_1\" value=\"0\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question68_1\">\n\t\t\t\t\t\t\t$f:\\mathbb{R}^n\\rightarrow \\mathbb{R},\\ f(x)=e^{\\|x\\|}$\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question68[]\" id=\"question68_2\" value=\"1\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question68_2\">\n\t\t\t\t\t\t\t$f:\\mathbb{R}^2\\rightarrow \\mathbb{R},\\ f(x)=x^T \\begin{pmatrix}1 &amp; 0\\\\ 0 &amp; 1\\end{pmatrix}x$\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question68[]\" id=\"question68_3\" value=\"2\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question68_3\">\n\t\t\t\t\t\t\t$f:(0,\\infty)\\rightarrow \\mathbb{R},\\ f(x)=\\sqrt{x}$\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question68[]\" id=\"question68_4\" value=\"3\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question68_4\">\n\t\t\t\t\t\t\t$f:(0,\\infty)\\rightarrow \\mathbb{R},\\ f(x)=\\frac{1}{x}$\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question68[]\" id=\"question68_5\" value=\"4\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question68_5\">\n\t\t\t\t\t\t\t$f:(-\\infty,0)\\rightarrow \\mathbb{R},\\ f(x)=\\frac{1}{x}$\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_68\" value=\"\" \/>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-0 question-section-id-70 \" data-qid=\"70\">\n\t\t\t\t\t\t\t\t\t\t<span class='mlw_qmn_question_number'>4.&nbsp;<\/span>\n\t\t\t\t\t\t\t<div class='mlw_qmn_question ' >\n\t\t<p>Gegeben sei die Menge $A=\\operatorname{conv}\\left\\{\\begin{pmatrix}0\\\\1\\\\-1\\end{pmatrix},\\begin{pmatrix}-1\\\\0\\\\1\\end{pmatrix},\\begin{pmatrix}0\\\\-1\\\\1\\end{pmatrix},\\begin{pmatrix}1\\\\0\\\\-1\\end{pmatrix}\\right\\}\\subseteq \\mathbb{R}^3$.<br \/><br \/>Was ist $\\operatorname{dim}(A)$?<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class='qmn_radio_answers '>\n\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question70-1 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question70\" id=\"question70_1\" value=\"0\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question70_1\">\n\t\t\t\t\t2\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question70-2 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question70\" id=\"question70_2\" value=\"1\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question70_2\">\n\t\t\t\t\t3\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question70-3 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question70\" id=\"question70_3\" value=\"2\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question70_3\">\n\t\t\t\t\t4\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t \t\t\t<label style=\"display: none !important;\" for=\"question70_none\">None<\/label>\n\t\t\t<input type=\"radio\" style=\"display: none;\" name=\"question70\" id=\"question70_none\" checked=\"checked\" value=\"\" \/>\n\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_70\" value=\"\" \/>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-4 question-section-id-59 \" data-qid=\"59\">\n\t\t\t\t\t\t\t\t\t\t<span class='mlw_qmn_question_number'>5.&nbsp;<\/span>\n\t\t\t\t\t\t\t<div class='mlw_qmn_question ' >\n\t\t<p>Welche der folgenden Aussagen sind wahr?<br \/><br \/>(a) Sei $C\\subseteq \\mathbb{R}^n$ konvex, abgeschlossen und nicht leer. Falls $z\\not\\in C$, dann gibt es eine strikte Trennhyperebene von $\\{z\\}$ und $C$.<br \/><br \/>(b) Sei $C\\subseteq \\mathbb{R}^n$ abgeschlossen und nicht leer. Falls $z\\not\\in C$, dann gibt es eine Trennhyperebene von $\\{z\\}$ und $C$.<br \/><br \/>(c) Sei $C\\subseteq \\mathbb{R}^n$ konvex und nicht leer. Falls $z\\not\\in C$, dann gibt es eine Trennhyperebene von $\\{z\\}$ und $C$.<br \/><br \/>(d) Sei $C\\subseteq \\mathbb{R}^n$ konvex und nicht leer. Falls $z\\not\\in C$, dann gibt es eine strikte Trennhyperebene von $\\{z\\}$ und $C$.<br \/><\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class=\"qmn_check_answers mlwRequiredRadio\">\n\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question59[]\" id=\"question59_1\" value=\"0\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question59_1\">\n\t\t\t\t\t\t\t(a)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question59[]\" id=\"question59_2\" value=\"1\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question59_2\">\n\t\t\t\t\t\t\t(b)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question59[]\" id=\"question59_3\" value=\"2\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question59_3\">\n\t\t\t\t\t\t\t(c)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question59[]\" id=\"question59_4\" value=\"3\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question59_4\">\n\t\t\t\t\t\t\t(d)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_59\" value=\"\" \/>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-4 question-section-id-61 \" data-qid=\"61\">\n\t\t\t\t\t\t\t\t\t\t<span class='mlw_qmn_question_number'>6.&nbsp;<\/span>\n\t\t\t\t\t\t\t<div class='mlw_qmn_question ' >\n\t\t<p>F\u00fcr welche der folgenden Mengen $P$ gilt $P=\\operatorname{conv}(\\operatorname{ext}(P))$?<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class=\"qmn_check_answers mlwRequiredRadio\">\n\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question61[]\" id=\"question61_1\" value=\"0\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question61_1\">\n\t\t\t\t\t\t\t$P$ ist ein Polyeder.\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question61[]\" id=\"question61_2\" value=\"1\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question61_2\">\n\t\t\t\t\t\t\t$P$ ist ein Polytop.\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question61[]\" id=\"question61_3\" value=\"2\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question61_3\">\n\t\t\t\t\t\t\t$P=B(0,1)$.\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question61[]\" id=\"question61_4\" value=\"3\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question61_4\">\n\t\t\t\t\t\t\t$P=(0,1)\\subseteq \\mathbb{R}$.\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question61[]\" id=\"question61_5\" value=\"4\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question61_5\">\n\t\t\t\t\t\t\t$P=\\{x\\in\\mathbb{R}^n : x^Tx=1\\}$.\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_61\" value=\"\" \/>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-0 question-section-id-62 \" data-qid=\"62\">\n\t\t\t\t\t\t\t\t\t\t<span class='mlw_qmn_question_number'>7.&nbsp;<\/span>\n\t\t\t\t\t\t\t<div class='mlw_qmn_question ' >\n\t\t<p>Wieviele Ecken hat das Polyeder $P=\\{x\\in\\mathbb{R}^3 : Ax\\leq b\\}$, wobei<br \/> <br \/> $$ A=\\begin{pmatrix} -1&amp;0&amp;0\\\\0&amp;-1&amp;0\\\\0&amp;0&amp;-1\\\\1&amp;0&amp;0\\end{pmatrix},b=\\begin{pmatrix}0\\\\0\\\\0\\\\5\\end{pmatrix}$$<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class='qmn_radio_answers mlwRequiredRadio'>\n\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question62-1 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question62\" id=\"question62_1\" value=\"0\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question62_1\">\n\t\t\t\t\t4\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question62-2 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question62\" id=\"question62_2\" value=\"1\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question62_2\">\n\t\t\t\t\t2\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question62-3 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question62\" id=\"question62_3\" value=\"2\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question62_3\">\n\t\t\t\t\t1\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t \t\t\t<label style=\"display: none !important;\" for=\"question62_none\">None<\/label>\n\t\t\t<input type=\"radio\" style=\"display: none;\" name=\"question62\" id=\"question62_none\" checked=\"checked\" value=\"\" \/>\n\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_62\" value=\"\" \/>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-4 question-section-id-67 \" data-qid=\"67\">\n\t\t\t\t\t\t\t\t\t\t<span class='mlw_qmn_question_number'>8.&nbsp;<\/span>\n\t\t\t\t\t\t\t<div class='mlw_qmn_question ' >\n\t\t<p>Seien $A\\in\\mathbb{R}^{m\\times n}$ und $b\\in\\mathbb{R}^m$ gegeben.<br \/><br \/>Welche der folgenden Aussagen sind wahr?<br \/><br \/>(a) Es gibt einen Vektor $x\\in\\mathbb{R}^n$ mit $x\\geq 0$ und $Ax=b$ genau dann, wenn es keinen Vektor $y\\in\\mathbb{R}^m$ gibt, der $A^Ty\\geq 0$ und $b^Ty&lt;0$ erf\u00fcllt.<br \/><br \/>(b) Es gibt einen Vektor $x\\in\\mathbb{R}^n$ mit $Ax\\leq b$ genau dann, wenn es keinen Vektor $y\\in\\mathbb{R}^m$ gibt, der $y\\geq 0$,\u00a0 $A^Ty= 0$ und $b^Ty= 2019$ erf\u00fcllt.<br \/><br \/>(c) Es gibt einen Vektor $x\\in\\mathbb{R}^n$ mit $Ax\\leq b$ genau dann, wenn es keinen Vektor $y\\in\\mathbb{R}^m$ gibt, der $y\\geq 0$,\u00a0 $A^Ty= 0$ und $b^Ty= -2019$ erf\u00fcllt.<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class=\"qmn_check_answers mlwRequiredRadio\">\n\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question67[]\" id=\"question67_1\" value=\"0\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question67_1\">\n\t\t\t\t\t\t\t(a)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question67[]\" id=\"question67_2\" value=\"1\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question67_2\">\n\t\t\t\t\t\t\t(b)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question67[]\" id=\"question67_3\" value=\"2\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question67_3\">\n\t\t\t\t\t\t\t(c)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_67\" value=\"\" \/>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-4 question-section-id-63 \" data-qid=\"63\">\n\t\t\t\t\t\t\t\t\t\t<span class='mlw_qmn_question_number'>9.&nbsp;<\/span>\n\t\t\t\t\t\t\t<div class='mlw_qmn_question ' >\n\t\t<p>Seien $c\\in\\mathbb{R}^n$, $A\\in\\mathbb{R}^{m\\times n}$ und $b\\in\\mathbb{R}^m$ gegeben. Sei<br \/>$$p^*=\\sup\\{c^Tx : x\\in\\mathbb{R}^n, Ax\\leq b\\}$$<br \/>und<br \/>$$d^*=\\inf\\{b^Ty : y\\in\\mathbb{R}^m, y\\geq 0, A^Ty=c\\}.$$<br \/><br \/>Welche der folgenden Aussagen ist wahr?<br \/><br \/>(a) $p^*\\leq d^*$.<br \/><br \/>(b) $p^*\\geq d^*$.<br \/><br \/>(c) Falls es f\u00fcr beide Programme zul\u00e4ssige Vektoren gibt, dann ist $p^*=d^*$.<br \/><br \/>(d) Falls es f\u00fcr beide Programme zul\u00e4ssige Vektoren gibt, dann ist $p^* \\geq d^*$.<br \/><br \/>(e) Es ist $d^*=-\\infty$, falls es keine zul\u00e4ssigen Vektoren f\u00fcr das obige Maximierungsprogramm gibt.<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class=\"qmn_check_answers mlwRequiredRadio\">\n\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question63[]\" id=\"question63_1\" value=\"0\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question63_1\">\n\t\t\t\t\t\t\t(a)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question63[]\" id=\"question63_2\" value=\"1\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question63_2\">\n\t\t\t\t\t\t\t(b)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question63[]\" id=\"question63_3\" value=\"2\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question63_3\">\n\t\t\t\t\t\t\t(c)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question63[]\" id=\"question63_4\" value=\"3\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question63_4\">\n\t\t\t\t\t\t\t(d)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question63[]\" id=\"question63_5\" value=\"4\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question63_5\">\n\t\t\t\t\t\t\t(e)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_63\" value=\"\" \/>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-4 question-section-id-72 \" data-qid=\"72\">\n\t\t\t\t\t\t\t\t\t\t<span class='mlw_qmn_question_number'>10.&nbsp;<\/span>\n\t\t\t\t\t\t\t<div class='mlw_qmn_question ' >\n\t\t<p>Gegeben seien folgende lineare Programme:<br \/>\\begin{equation}p^*=\\sup\\left\\{(5,8)x : \\begin{pmatrix}1 &amp; 2 \\\\ 3 &amp; 4 \\end{pmatrix}x\\leq \\begin{pmatrix}3\\\\3\\end{pmatrix}\\right\\}\\qquad (PLP)\\end{equation}<br \/>und<br \/>\\begin{equation}d^*=\\inf\\left\\{(3,3)y : y\\geq 0, \\begin{pmatrix}1 &amp; 3 \\\\ 2 &amp; 4 \\end{pmatrix}y=\\begin{pmatrix}5\\\\8\\end{pmatrix}\\right\\}.\\qquad (DLP)\\end{equation}<br \/><br \/>Welche der folgenden Aussagen ist wahr?<br \/><br \/>(a) $p^*= d^*$.<br \/><br \/>(b) $p^*\\leq \\inf\\left\\{(3,3)y : y\\geq 0, \\begin{pmatrix}1 &amp; 3 \\\\ 2 &amp; 4 \\\\5 &amp; 6\\end{pmatrix}y=\\begin{pmatrix}5\\\\8\\\\10\\end{pmatrix}\\right\\}$.<br \/><br \/>(c) $p^*=\\inf\\left\\{(3,3)y : y\\geq 0, \\begin{pmatrix}1 &amp; 3 \\\\ 2 &amp; 4 \\\\5 &amp; 6\\end{pmatrix}y=\\begin{pmatrix}5\\\\8\\\\10\\end{pmatrix}\\right\\}$.<br \/><br \/>(d) Seien $x^*$ optimal f\u00fcr (PLP) und $y^*$ optimal f\u00fcr (DLP), dann gilt: <br \/>$$\\begin{pmatrix}3 &amp; 4 \\end{pmatrix}x^*-3\\neq 0 \\Rightarrow y^*_2=0.$$<br \/><br \/>(e) Seien allgemein $x^*$ optimal f\u00fcr ein primales LP $\\sup\\{c^Tx : x\\in\\mathbb{R}^n, Ax\\leq b\\}$ und $y^*$ optimal f\u00fcr das zugeh\u00f6rige duale LP, dann gilt f\u00fcr alle Zeilen $i$: <br \/>$$\\left(Ax^*-b\\right)_i= 0 \\Rightarrow y^*_i\\neq 0.$$<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class=\"qmn_check_answers mlwRequiredRadio\">\n\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question72[]\" id=\"question72_1\" value=\"0\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question72_1\">\n\t\t\t\t\t\t\t(a)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question72[]\" id=\"question72_2\" value=\"1\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question72_2\">\n\t\t\t\t\t\t\t(b)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question72[]\" id=\"question72_3\" value=\"2\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question72_3\">\n\t\t\t\t\t\t\t(c)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question72[]\" id=\"question72_4\" value=\"3\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question72_4\">\n\t\t\t\t\t\t\t(d)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question72[]\" id=\"question72_5\" value=\"4\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question72_5\">\n\t\t\t\t\t\t\t(e)\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_72\" value=\"\" \/>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-4 question-section-id-64 \" data-qid=\"64\">\n\t\t\t\t\t\t\t\t\t\t<span class='mlw_qmn_question_number'>11.&nbsp;<\/span>\n\t\t\t\t\t\t\t<div class='mlw_qmn_question ' >\n\t\t<p>Mit dem Verfahren von Fourier und Motzkin kann man:<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class=\"qmn_check_answers mlwRequiredRadio\">\n\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question64[]\" id=\"question64_1\" value=\"0\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question64_1\">\n\t\t\t\t\t\t\tlineare Programme l\u00f6sen.\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question64[]\" id=\"question64_2\" value=\"1\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question64_2\">\n\t\t\t\t\t\t\tlineare Projektionen entlang von Koordinatenachsen berechnen.\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question64[]\" id=\"question64_3\" value=\"2\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question64_3\">\n\t\t\t\t\t\t\tbestimmen, ob ein System von linearen Ungleichungen erf\u00fcllbar ist.\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question64[]\" id=\"question64_4\" value=\"3\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question64_4\">\n\t\t\t\t\t\t\tbestimmen, ob ein gegebenes Polyeder leer ist.\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_64\" value=\"\" \/>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-4 question-section-id-71 \" data-qid=\"71\">\n\t\t\t\t\t\t\t\t\t\t<span class='mlw_qmn_question_number'>12.&nbsp;<\/span>\n\t\t\t\t\t\t\t<div class='mlw_qmn_question ' >\n\t\t<p>Welche Aussagen zum Simplex-Verfahren sind korrekt?<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class=\"qmn_check_answers \">\n\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question71[]\" id=\"question71_1\" value=\"0\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question71_1\">\n\t\t\t\t\t\t\tMit dem Simplex-Verfahren findet man alle Optimall\u00f6sungen eines Linearen Programms.\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question71[]\" id=\"question71_2\" value=\"1\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question71_2\">\n\t\t\t\t\t\t\tMit dem Simplex-Verfahren kann man bestimmen, ob der Optimalwert eines Programms unbeschr\u00e4nkt ist.\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question71[]\" id=\"question71_3\" value=\"2\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question71_3\">\n\t\t\t\t\t\t\tFalls das LP zul\u00e4ssig und beschr\u00e4nkt ist, findet das Verfahren immer eine ganzzahlige Optimall\u00f6sung\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm_check_answer  mrq_checkbox_class\">\n\t\t\t\t\t\t<input type=\"checkbox\" class=\"qsm-multiple-response-input\"  name=\"question71[]\" id=\"question71_4\" value=\"3\" \/>\n\t\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question71_4\">\n\t\t\t\t\t\t\tIn jeder Iteration gilt, dass $u_k$ eine zul\u00e4ssige L\u00f6sung des dualen Programms ist.\t\t\t\t\t\t<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_71\" value=\"\" \/>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<\/section>\n\t\t\t\t\t<input type=\"hidden\" name=\"qmn_question_list\" value=\"60Q58Q68Q70Q59Q61Q62Q67Q63Q72Q64Q71Q\" \/>\n\t\t\t\t\t\t\t<div id=\"mlw_error_message_bottom\" class=\"qsm-error-message qmn_error_message_section\"><\/div>\n\t\t\t\t\t<input type=\"hidden\" name=\"qmn_all_questions_count\" id=\"qmn_all_questions_count\" value=\"12\" 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