{"id":9743,"date":"2015-09-11T12:40:12","date_gmt":"2015-09-11T10:40:12","guid":{"rendered":"https:\/\/kamerpower.com\/?p=9743"},"modified":"2025-12-07T23:46:00","modified_gmt":"2025-12-07T22:46:00","slug":"epreuve-concours-ens-yaounde-niveau-bac-geometrie-2012","status":"publish","type":"post","link":"https:\/\/kamerpower.com\/fr\/epreuve-concours-ens-yaounde-niveau-bac-geometrie-2012\/","title":{"rendered":"Epreuve concours ENS Yaounde Niveau BAC G\u00e9om\u00e9trie 2012 Math\u00e9matiques"},"content":{"rendered":"<p style=\"text-align: center;\"><span style=\"color: #000000;\"><strong>R\u00c9PUBLIQUE DU CAMEROUN<\/strong><\/span><br \/>\n<span style=\"color: #000000;\"><strong>Paix \u2013 Travail _ Patrie<\/strong><\/span><br \/>\n<span style=\"color: #000000;\"><strong>UNIVERSITE DE YAOUNDE I<\/strong><\/span><br \/>\n<span style=\"color: #000000;\"><strong>Ecole Normale Sup\u00e9rieure de Yaound\u00e9 ENS<\/strong><\/span><br \/>\n<span style=\"color: #000000;\"><strong>Concours d\u2019entr\u00e9e en premi\u00e8re ann\u00e9e du premier cycle<\/strong><\/span><br \/>\n<strong><span style=\"color: #000000;\">S\u00e9rie: <span style=\"color: #ff0000;\">Math\u00e9matiques<\/span> Epreuve : <span style=\"color: #ff0000;\">G\u00e9om\u00e9trie<\/span> Dur\u00e9e : <span style=\"color: #ff0000;\">3h Session : 2012<\/span><\/span><\/strong><br \/>\n<span style=\"color: #008000; text-shadow: 2px 0 2px #000;\">KA<\/span><span style=\"color: #ff0000; text-shadow: 2px 0 2px #000;\">M<\/span><span style=\"color: #ffcc00; text-shadow: 2px 0 2px #000;\">ER<\/span><span style=\"color: #800080; text-shadow: 2px 0 2px #000;\">POWER<\/span><span style=\"color: #3366ff; text-shadow: 2px 0 2px #000;\">.COM<\/span><\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Contents<\/p>\n<label for=\"ez-toc-cssicon-toggle-item-69da8c1760061\" class=\"ez-toc-cssicon-toggle-label\"><span class=\"ez-toc-cssicon\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #000000;color:#000000\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #000000;color:#000000\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/label><input type=\"checkbox\"  id=\"ez-toc-cssicon-toggle-item-69da8c1760061\"  aria-label=\"Toggle\" \/><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/kamerpower.com\/fr\/epreuve-concours-ens-yaounde-niveau-bac-geometrie-2012\/#epreuve-concours-ens-yaounde-niveau-bac-geometrie-2012-mathematiques\" >Epreuve concours ENS Yaounde Niveau BAC G\u00e9om\u00e9trie 2012\u00a0Math\u00e9matiques\u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/kamerpower.com\/fr\/epreuve-concours-ens-yaounde-niveau-bac-geometrie-2012\/#exercice-1-7-points\" >Exercice 1\/ 7 points<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/kamerpower.com\/fr\/epreuve-concours-ens-yaounde-niveau-bac-geometrie-2012\/#exercice-2-7-points\" >Exercice 2\/ 7 points<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/kamerpower.com\/fr\/epreuve-concours-ens-yaounde-niveau-bac-geometrie-2012\/#exercice-3-6-points\" >Exercice 3\/ 6 points<\/a><\/li><\/ul><\/nav><\/div>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"epreuve-concours-ens-yaounde-niveau-bac-geometrie-2012-mathematiques\"><\/span><span style=\"color: #3366ff;\"><strong><a style=\"color: #3366ff;\" href=\"https:\/\/kamerpower.com\/fr\/concours-ens-yaounde-2015-2016-1ere-annee-du-2nd-cycle-ens-ecole-normale-superieure-de-luniversite-de-yaounde-l\/\">Epreuve concours ENS Yaounde Niveau BAC G\u00e9om\u00e9trie 2012<\/a>\u00a0Math\u00e9matiques\u00a0<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"exercice-1-7-points\"><\/span><span style=\"color: #ff0000;\"><strong>Exercice 1\/ 7 points<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">Soit O, A, B et C quatre points du plan complexe P tels que ABC est de sens direct et isoc\u00e8le en C ,\u00a0OAC est \u00e9quilat\u00e9ral de sens direct et C est le milieu de [O, B].<\/span><\/p>\n<ol>\n<li><span style=\"color: #000000;\">Faire une figure et placer les points O, A, B et C . [1pt]<\/span><\/li>\n<li style=\"text-align: justify;\"><span style=\"color: #000000;\">Montrer qu\u2019il existe exactement deux isom\u00e9tries qui transforment O en B et A en C . [1pt]<\/span><\/li>\n<li><span style=\"color: #000000;\">Soit r le d\u00e9placement transformant O en B et A en C . [1pt]<br \/>\n<\/span><br \/>\n<span style=\"color: #000000;\"><strong>(a)<\/strong> Montrer que r est une rotation dont on donnera l\u2019angle. [0,5pt+0,5pt]<\/span><br \/>\n<span style=\"color: #000000;\"><span style=\"color: #000000;\"><strong>(b)<\/strong> Construire sur la figure ci-dessus le centre \u03a9 de r. Justifier. [1pt]<\/span><\/span><\/li>\n<li style=\"text-align: justify;\"><span style=\"color: #000000;\">\u00a0Soit S(AC)\u00a0la r\u00e9flexion d\u2019axe (AC ) et S l\u2019antid\u00e9placement appliquant O en B et A en C .<br \/>\n<\/span><br \/>\n<span style=\"color: #000000;\"><strong>(a)<\/strong> Montrer que S est une sym\u00e9trie d\u2019axe (AC ). [1pt]<\/span><br \/>\n<span style=\"color: #000000;\"><span style=\"color: #000000;\"><strong>(b)<\/strong> En d\u00e9duire que S = tAC oSAC\u00a0 \u00a0est la translation de vecteur\u00a0AC. [1pt]<\/span><\/span><\/li>\n<li style=\"text-align: justify;\"><span style=\"color: #000000;\">Soit (O, u, v) un rep\u00e8re orthonorm\u00e9 direct de P tel que A ait pour affixe z = 2. D\u00e9terminer l\u2019affixe\u00a0du point \u03a9.<\/span><\/li>\n<\/ol>\n<h3><span class=\"ez-toc-section\" id=\"exercice-2-7-points\"><\/span><span style=\"color: #ff0000;\"><strong>Exercice 2\/ 7 points<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-9746 size-full\" src=\"https:\/\/kamerpower.com\/wp-content\/uploads\/2015\/09\/Epreuve-concours-ENS-Yaounde-Ecole-normale-sup\u00e9rieure1.png\" alt=\"Epreuve concours ENS Yaounde Ecole normale sup\u00e9rieure\" width=\"1280\" height=\"720\" srcset=\"https:\/\/kamerpower.com\/wp-content\/uploads\/2015\/09\/Epreuve-concours-ENS-Yaounde-Ecole-normale-sup\u00e9rieure1.png 1280w, https:\/\/kamerpower.com\/wp-content\/uploads\/2015\/09\/Epreuve-concours-ENS-Yaounde-Ecole-normale-sup\u00e9rieure1-600x338.png 600w, https:\/\/kamerpower.com\/wp-content\/uploads\/2015\/09\/Epreuve-concours-ENS-Yaounde-Ecole-normale-sup\u00e9rieure1-300x169.png 300w, https:\/\/kamerpower.com\/wp-content\/uploads\/2015\/09\/Epreuve-concours-ENS-Yaounde-Ecole-normale-sup\u00e9rieure1-768x432.png 768w, https:\/\/kamerpower.com\/wp-content\/uploads\/2015\/09\/Epreuve-concours-ENS-Yaounde-Ecole-normale-sup\u00e9rieure1-150x84.png 150w\" sizes=\"auto, (max-width: 1280px) 100vw, 1280px\" \/><\/p>\n<p><span style=\"color: #000000;\"><strong>Donn\u00e9es.<\/strong><\/span><br \/>\n<span style=\"color: #000000;\">OABC est un carr\u00e9 d\u2019ar\u00eate 1.<\/span><br \/>\n<span style=\"color: #000000;\">(C 1) est tangent \u00e0 (OC) et \u00e0 (OA).<\/span><br \/>\n<span style=\"color: #000000;\">(C 2) est tangent \u00e0 (AB) et \u00e0 (BC).<\/span><br \/>\n<span style=\"color: #000000;\">(C 1) et (C 2) sont tangents en F . kamerpower.com<\/span><br \/>\n<span style=\"color: #000000;\">(C 1) a pour centre D et pour rayon x.<\/span><br \/>\n<span style=\"color: #000000;\">(C 2) a pour centre E et pour rayon y.<\/span><br \/>\n<span style=\"color: #000000;\">x &lt; y et la somme des aires de (C 1) et (C 2) est maximale.<\/span><\/p>\n<ol>\n<li><span style=\"color: #000000;\">On voudrait d\u00e9terminer OD et OE.<br \/>\n<\/span><br \/>\n<span style=\"color: #000000;\"><strong>(a)<\/strong> Montrer que OD = x\u00a0\u221a2, BE = y\u00a0\u221a2 et x + y = 2 \u2212\u00a0\u221a2. \u00a0[0,5pt+0,5pt+1pt]<\/span><br \/>\n<span style=\"color: #000000;\"><span style=\"color: #000000;\"><strong>(b)<\/strong> D\u00e9terminer la somme des aires de (C 1) et (C 2) ; puis OD et OE. [0,5pt+1pt]<\/span><\/span><\/li>\n<li style=\"text-align: justify;\"><span style=\"color: #000000;\">Soit P la parabole de foyer F et de directrice (BC).<br \/>\n<\/span><br \/>\n<span style=\"color: #000000;\"><strong>(a)<\/strong> Montrer que E \u2208 P et d\u00e9terminer la param\u00e8tre de P . [0,5pt+1pt]<\/span><br \/>\n<span style=\"color: #000000;\"><strong>(b)<\/strong> Reproduire la figure pr\u00e9c\u00e9dente pour OA = 2cm. [1pt]<\/span><br \/>\n<span style=\"color: #000000;\"><strong>(c)<\/strong> Construire sur la figure pr\u00e9c\u00e9dente la parabole P , son sommet et son axe. [1pt]<\/span><\/li>\n<\/ol>\n<p><span style=\"color: #000000;\">\u00a0<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"exercice-3-6-points\"><\/span><span style=\"color: #ff0000;\"><strong>Exercice 3\/ 6 points<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ol>\n<li style=\"text-align: justify;\"><span style=\"color: #000000;\">Le plan (P ) \u00e9tant rapport\u00e9 \u00e0 un rep\u00e8re orthonorm\u00e9 (O, <em>i\u00a0,\u00a0j<\/em>), on consid\u00e8re l\u2019application F qui \u00e0\u00a0tout point M (x, y) de (P ) associe le point M\u2032(x\u2032, y\u2032) tel que x\u2032= x \u2212 y + 2 et y\u2032= 2y \u2212 1.<br \/>\n<\/span><br \/>\n<span style=\"color: #000000;\"> <strong>(a)<\/strong> L\u2019application F est-elle une isom\u00e9trie ? Justifier. [1pt]<\/span><br \/>\n<span style=\"color: #000000;\"> <strong>(b)<\/strong> L\u2019application F admet-elle de point invariant ? Justifier. [1pt]<\/span><br \/>\n<span style=\"color: #000000;\"><span style=\"color: #000000;\"> <strong>(c)<\/strong> D\u00e9terminer les droites globalement invariantes par F . [1pt]<\/span><\/span><\/li>\n<li style=\"text-align: justify;\"><span style=\"color: #000000;\">Etant donn\u00e9 un rep\u00e8re (O, <em>i, j, k<\/em>)\u00a0de l\u2019espace (E ), on consid\u00e8re le plan (Q) : x = y + z + 1 et la\u00a0droite (d) de rep\u00e8re (O,\u00a0i\u00a0\u2212 j \u00a0\u2212 k).<br \/>\n<\/span><br \/>\n<span style=\"color: #000000;\"> <strong>(a)<\/strong> Montrer que (Q) et (d) sont orthogonaux en un point I \u00e0 d\u00e9terminer. [1pt]<\/span><br \/>\n<span style=\"color: #000000;\"> <strong>(b)<\/strong> D\u00e9terminer l\u2019expression analytique de la reflexion S par rapport \u00e0 (Q). [1pt]<\/span><br \/>\n<span style=\"color: #000000;\"> <strong>(c)<\/strong> D\u00e9terminer sans calcul la nature de S \u25e6 s o\u00f9 s est le demi-tour d\u2019axe (d). [1pt]<\/span><\/li>\n<\/ol>\n<p style=\"text-align: center;\"><span style=\"color: #3366ff;\"><strong>Epreuve concours ENS Yaounde Niveau BAC G\u00e9om\u00e9trie 2012\u00a0Math\u00e9matiques\u00a0<\/strong><\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-9745 size-full\" src=\"https:\/\/kamerpower.com\/wp-content\/uploads\/2015\/09\/Epreuve-concours-ENS-Yaounde-Ecole-normale-sup\u00e9rieure.png\" alt=\"Epreuve concours ENS Yaounde Ecole normale sup\u00e9rieure\" width=\"1280\" height=\"720\" srcset=\"https:\/\/kamerpower.com\/wp-content\/uploads\/2015\/09\/Epreuve-concours-ENS-Yaounde-Ecole-normale-sup\u00e9rieure.png 1280w, https:\/\/kamerpower.com\/wp-content\/uploads\/2015\/09\/Epreuve-concours-ENS-Yaounde-Ecole-normale-sup\u00e9rieure-600x338.png 600w, https:\/\/kamerpower.com\/wp-content\/uploads\/2015\/09\/Epreuve-concours-ENS-Yaounde-Ecole-normale-sup\u00e9rieure-300x169.png 300w, https:\/\/kamerpower.com\/wp-content\/uploads\/2015\/09\/Epreuve-concours-ENS-Yaounde-Ecole-normale-sup\u00e9rieure-768x432.png 768w, https:\/\/kamerpower.com\/wp-content\/uploads\/2015\/09\/Epreuve-concours-ENS-Yaounde-Ecole-normale-sup\u00e9rieure-150x84.png 150w\" sizes=\"auto, (max-width: 1280px) 100vw, 1280px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":9745,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_themeisle_gutenberg_block_has_review":false,"footnotes":""},"categories":[1701],"tags":[1934,1712,1706],"class_list":["post-9743","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-epreuves","tag-epreuve-concours-ens-yaounde-ancienne-epreuve","tag-epreuve-de-concours-dentree-aux-grandes-ecoles-cameroun","tag-mathematiques"],"_links":{"self":[{"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/posts\/9743","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/comments?post=9743"}],"version-history":[{"count":2,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/posts\/9743\/revisions"}],"predecessor-version":[{"id":72119,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/posts\/9743\/revisions\/72119"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/media\/9745"}],"wp:attachment":[{"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/media?parent=9743"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/categories?post=9743"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/tags?post=9743"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}