{"id":4125,"date":"2015-03-10T21:52:46","date_gmt":"2015-03-10T19:52:46","guid":{"rendered":"https:\/\/kamerpower.com\/?p=4125"},"modified":"2025-06-29T03:37:26","modified_gmt":"2025-06-29T02:37:26","slug":"epreuve-de-2eme-composition-de-mathematiques-issea-yaounde-2014-ensea-abidjan-2014-concours","status":"publish","type":"post","link":"https:\/\/kamerpower.com\/fr\/epreuve-de-2eme-composition-de-mathematiques-issea-yaounde-2014-ensea-abidjan-2014-concours\/","title":{"rendered":"\u00c9preuve de 2\u00e8me composition de Math\u00e9matiques ISSEA Yaounde 2014 \/ ENSEA Abidjan 2014 &#8211; Concours Ing\u00e9nieurs statisticiens \u00e9conomistes &#8211; ISE 2014"},"content":{"rendered":"<p style=\"text-align: center;\"><span style=\"color: #000000;\"><span style=\"color: #3366ff;\">\u00c9COLE NATIONALE SUP\u00c9RIEURE<\/span> &#8212; <span style=\"color: #ff0000;\">INSTITUT SOUS-R\u00c9GIONAL DE STATISTIQUE<\/span><\/span><br \/>\n<span style=\"color: #000000;\"><span style=\"color: #3366ff;\">DE STATISTIQUE ET D\u2019ECONOMIE<\/span> &#8212; <span style=\"color: #ff0000;\">ET D\u2019\u00c9CONOMIE APPLIQU\u00c9E<\/span><\/span><br \/>\n<span style=\"color: #000000;\"><strong><span style=\"color: #3366ff;\">ENSEA &#8211; ABIDJAN<\/span> &#8212; <\/strong><span style=\"color: #ff0000;\"><strong>ISSEA &#8211; YAOUN<\/strong>D\u00c9<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">\u00c9COLE NATIONALE DE LA STATISTIQUE<\/span><br \/>\n<span style=\"color: #000000;\">ET DE L&#8217;ANALYSE \u00c9CONOMIQUE<\/span><br \/>\n<span style=\"color: #000000;\">ENSAE &#8211; S\u00c9N\u00c9GAL<\/span><br \/>\n<span style=\"color: #000000;\">AVRIL 2014<\/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-69d8639058fcc\" 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-69d8639058fcc\"  aria-label=\"Toggle\" \/><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/kamerpower.com\/fr\/epreuve-de-2eme-composition-de-mathematiques-issea-yaounde-2014-ensea-abidjan-2014-concours\/#concours-ingenieurs-statisticiens-economistes\" >CONCOURS ING\u00c9NIEURS STATISTICIENS \u00c9CONOMISTES<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/kamerpower.com\/fr\/epreuve-de-2eme-composition-de-mathematiques-issea-yaounde-2014-ensea-abidjan-2014-concours\/#probleme-1\" >Probl\u00e8me\u00a01<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/kamerpower.com\/fr\/epreuve-de-2eme-composition-de-mathematiques-issea-yaounde-2014-ensea-abidjan-2014-concours\/#probleme-2\" >Probl\u00e8me\u00a02<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/kamerpower.com\/fr\/epreuve-de-2eme-composition-de-mathematiques-issea-yaounde-2014-ensea-abidjan-2014-concours\/#probleme-3\" >Probl\u00e8me\u00a03<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/kamerpower.com\/fr\/epreuve-de-2eme-composition-de-mathematiques-issea-yaounde-2014-ensea-abidjan-2014-concours\/#probleme-4\" >Probl\u00e8me\u00a04<\/a><\/li><\/ul><\/nav><\/div>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"concours-ingenieurs-statisticiens-economistes\"><\/span><span style=\"color: #ff0000;\"><a style=\"color: #ff0000;\" href=\"https:\/\/kamerpower.com\/fr\/issea-cemac-cameroun-concours-epreuves-et-resultats\/\" rel=\"noopener\">CONCOURS ING\u00c9NIEURS STATISTICIENS \u00c9CONOMISTES<\/a><\/span><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">ISE Option \u00c9conomie<\/span><br \/>\n<span style=\"color: #000000;\">2\u00e8me\u00a0COMPOSITION DE MATH\u00c9MATIQUES<\/span><br \/>\n<span style=\"color: #000000;\">(Dur\u00e9e de l\u2019\u00e9preuve : 3\u00a0heures)<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><\/span><\/p>\n<p style=\"text-align: center;\">L\u2019\u00e9preuve est compos\u00e9e de quatre probl\u00e8mes ind\u00e9pendants, qui peuvent \u00eatre trait\u00e9s dans unordre quelconque.<\/p>\n<h2 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"probleme-1\"><\/span><span style=\"color: #3366ff;\">Probl\u00e8me\u00a01<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">Soit a un param\u00e8tre r\u00e9el qui v\u00e9rifie 0 &lt; a &lt; 1. N d\u00e9signe un entier fix\u00e9, N &gt; 1.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"> 1) On consid\u00e8re la suite\u00a0u_{{n}}\u00a0de nombres r\u00e9els v\u00e9rifiant les relations suivantes<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">\u00a0u_{{0}}=\u00a00<\/span><br \/>\n<span style=\"color: #000000;\">u_{{N}}= 1<\/span><br \/>\n<span style=\"color: #000000;\"> u_{{n}} = a\u00a0. \u00a0u_{{n+1}} + (1 &#8211; a)\u00a0u_{{n-1}}, pour tout n entier &gt;\u00a00<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">Exprimer \u00a0en fonction de n, N et a (on discutera selon les valeurs du param\u00e8tre a).<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">2) On consid\u00e8re la suite () de nombres r\u00e9els v\u00e9rifiant les relations suivantes<\/span><\/p>\n<p style=\"text-align: justify;\">u_{{0}}= 1<br \/>\nu_{{N}}= 0<br \/>\n<span style=\"color: #000000;\">\u00a0= a\u00a0.\u00a0 + (1 &#8211; a), pour tout n entier &gt;\u00a00<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">Exprimer \u00a0en fonction de n, N et a.<\/span><\/p>\n<h2 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"probleme-2\"><\/span><span style=\"color: #3366ff;\">Probl\u00e8me\u00a02<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">Le Calife appelle son Grand Vizir, et lui tient ce discours:<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">\u00abCher ami, tout le monde sait que tu r\u00eaves de prendre ma succession quand je me retirerai,\u00a0et m\u00eame peut-\u00eatre avant.<\/span><br \/>\n<span style=\"color: #000000;\"> Alors, pour que tout soit clair entre nous, je te propose le jeu suivant.<\/span><br \/>\n<span style=\"color: #000000;\"> J\u2019ai dans les mains deux sacs de formes et de couleurs identiques. Celui que je tiens dans ma\u00a0main droite contient deux boules rouges, celui que j\u2019ai dans ma main gauche en contient trois.\u00a0Tu peux v\u00e9rifier.<\/span><br \/>\n<span style=\"color: #000000;\"> Voici six boules bleues : je te laisse les mettre dans ces sacs comme tu le souhaites. Quand tu\u00a0auras proc\u00e9d\u00e9 \u00e0 la totale r\u00e9partition de ces six boules bleues entre les deux sacs, tu fermeras\u00a0les yeux, je tirerai au sort, enti\u00e8rement au hasard, un sac, et tu choisiras une boule au hasard\u00a0dans le sac que je te proposerai.<\/span><br \/>\n<span style=\"color: #000000;\"> Si la boule est bleue, tu prendras ma succession instantan\u00e9ment.<\/span><br \/>\n<span style=\"color: #000000;\"> Mais si elle est rouge, tu seras banni \u00e0 jamais et condamne \u00e0 l\u2019exil\u00bb<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">Comment le Grand Vizir doit-il r\u00e9partir ses six boules bleues entre les deux sacs de fa\u00e7on \u00e0\u00a0<\/span><span style=\"color: #000000;\">maximiser ses chances de devenir Calife ? Le r\u00e9sultat doit \u00eatre justifi\u00e9.<\/span><\/p>\n<h2 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"probleme-3\"><\/span><span style=\"color: #3366ff;\">Probl\u00e8me\u00a03<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">C d\u00e9signe le corps des nombres complexes.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #ff0000;\">Partie A<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">Soit f l\u2019application de C dans C qui, \u00e0 tout complexe z, , associe f(z) d\u00e9fini par<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">M d\u00e9signe le point courant d\u2019affixe z.<\/span><\/p>\n<ol style=\"text-align: justify;\">\n<li><span style=\"color: #000000;\">D\u00e9terminer l\u2019ensemble, not\u00e9 U, des points M tels que le module |f(z)|\u00a0de f(z) soit \u00e9gal \u00e0 1.<\/span><\/li>\n<li><span style=\"color: #000000;\">D\u00e9terminer l\u2019ensemble V des points M tels que f(z) soit un nombre r\u00e9el strictement n\u00e9gatif.<\/span><\/li>\n<li><span style=\"color: #000000;\">D\u00e9terminer l\u2019ensemble W des points M tels que |f(z)|\u00a0soit un nombre imaginaire pur.<\/span><\/li>\n<\/ol>\n<p style=\"text-align: justify;\"><span style=\"color: #ff0000;\">Partie B<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">Soit g l\u2019application de C dans C qui, \u00e0 tout complexe z, \u00a0, associe g(z) d\u00e9fini par:<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"> M d\u00e9signe le point courant d\u2019affixe z.<\/span><\/p>\n<ol style=\"text-align: justify;\">\n<li><span style=\"color: #000000;\">\u00a0Ecrire g(i) sous forme can\u00e9sienne et sous forme trigonom\u00e9trique.<\/span><\/li>\n<li><span style=\"color: #000000;\">\u00a0R\u00e9soudre l\u2019\u00e9quation g(z) = 2i<\/span><\/li>\n<li><span style=\"color: #000000;\">D\u00e9terminer l\u2019ensemble A des points M tels que |g(z)|\u00a0= 2.<\/span><\/li>\n<li><span style=\"color: #000000;\">D\u00e9terminer l\u2019ensemble B des points M tels que l\u2019argument arg(g(z)) soit \u00e9gal \u00e0 \u00a0,\u00a0<\/span><span style=\"color: #000000;\">module .<\/span><\/li>\n<li><span style=\"color: #000000;\">\u00a0Etudier l\u2019intersection de A et B.<\/span><\/li>\n<li><span style=\"color: #000000;\">R\u00e9soudre l\u2019\u00e9quation f(z) = g(z).<\/span><\/li>\n<\/ol>\n<h2 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"probleme-4\"><\/span><span style=\"color: #3366ff;\">Probl\u00e8me\u00a04<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">Le symbole Ln d\u00e9signe 1e logarithme n\u00e9p\u00e9rien, de base e = 2,718.<\/span><br \/>\n<span style=\"color: #000000;\"> On consid\u00e8re la famille de fonctions \u00a0, o\u00f9 a et b sont deux param\u00e8tres r\u00e9els, d\u00e9finie sur\u00a0<\/span><span style=\"color: #000000;\">\u00a0\u00a0&#8211; {1} , par:<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><span style=\"color: #ff0000;\"> 1<\/span> &#8211; D\u00e9terminer les r\u00e9els a et b pour que la courbe C repr\u00e9sentant graphiquement \u00a0dans le\u00a0<\/span><span style=\"color: #000000;\">rep\u00e8re orthonorm\u00e9 usuel coupe l\u2019axe des abscisses au point E (e, 0), et pour que la tangente \u00e0\u00a0<\/span><span style=\"color: #000000;\">C au point E soit parall\u00e8le \u00e0 la droite y = 2x.<\/span><br \/>\n<span style=\"color: #000000;\"> Dans la suite du probl\u00e8me, on notera par f la fonction correspondant aux valeurs ainsi<\/span><br \/>\n<span style=\"color: #000000;\"> trouv\u00e9es de a et b.<\/span><br \/>\n<span style=\"color: #000000;\"><span style=\"color: #ff0000;\"> 2<\/span> &#8211; Etudier tr\u00e8s pr\u00e9cis\u00e9ment les variations de f (d\u00e9riv\u00e9es, concavit\u00e9, limites, asymptotes<\/span><br \/>\n<span style=\"color: #000000;\"> \u00e9ventuelles, intersection avec les axes, etc&#8230;).<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><span style=\"color: #ff0000;\">3<\/span> &#8211; Soit la fonction g d\u00e9finie sur l\u2019intervalle \u00a0&#8211; { l } par :\u00a0<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\"><span style=\"color: #ff0000;\"> 3a)<\/span> Etudier les positions respectives des courbes F et G repr\u00e9sentant les fonctions f et g.<\/span><br \/>\n<span style=\"color: #000000;\"><span style=\"color: #ff0000;\"> 3b)<\/span> Calculer une primitive de g(x).<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"color: #3366ff;\">\u00c9preuve de 2\u00e8me composition de Math\u00e9matiques ISSEA Yaounde 2014 \/ ENSEA Abidjan 2014 &#8211; Concours Ing\u00e9nieurs statisticiens \u00e9conomistes &#8211; ISE 2014<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21563,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_themeisle_gutenberg_block_has_review":false,"footnotes":""},"categories":[1701],"tags":[1702,1712,1706],"class_list":["post-4125","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-epreuves","tag-epreuve-concours-issea-yaounde-ensea-abidjan","tag-epreuve-de-concours-dentree-aux-grandes-ecoles-cameroun","tag-mathematiques"],"_links":{"self":[{"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/posts\/4125","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=4125"}],"version-history":[{"count":2,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/posts\/4125\/revisions"}],"predecessor-version":[{"id":67827,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/posts\/4125\/revisions\/67827"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/media\/21563"}],"wp:attachment":[{"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/media?parent=4125"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/categories?post=4125"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/tags?post=4125"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}