{"id":6777,"date":"2015-06-14T22:22:31","date_gmt":"2015-06-14T20:22:31","guid":{"rendered":"https:\/\/kamerpower.com\/?p=6777"},"modified":"2025-08-11T00:24:58","modified_gmt":"2025-08-10T23:24:58","slug":"epreuves-concours-iut-de-douala-mathematiques-2009-pftin-gi-filiere-1ere-annee","status":"publish","type":"post","link":"https:\/\/kamerpower.com\/fr\/epreuves-concours-iut-de-douala-mathematiques-2009-pftin-gi-filiere-1ere-annee\/","title":{"rendered":"Epreuves concours IUT de Douala Math\u00e9matiques 2009 PFTIN-GI Fili\u00e8re 1\u00e8re ANN\u00c9E IUT de l\u2019Universit\u00e9 de Douala Session juillet 2009"},"content":{"rendered":"<p style=\"text-align: center;\"><span style=\"color: #000000;\">R\u00c9PUBLIQUE DU CAMEROUN<br \/>\nCONCOURS D\u2019ENTREE EN 1\u00e8re ANN\u00c9E<br \/>\n<span style=\"color: #ff0000;\">Session de juillet 2009<\/span>\u00a0<strong>IUT<\/strong>\u00a0de l\u2019Universit\u00e9<br \/>\nde\u00a0Douala\u00a0Fili\u00e8re: <strong>PFTIN-<\/strong><strong>GI<\/strong><br \/>\n<a href=\"https:\/\/kamerpower.com\/fr\/epreuves-concours-iut-douala-mathematiques-2011-pftin-gi-filiere-1ere-annee\/\" target=\"_blank\" rel=\"noopener\"><strong>\u00c9preuve de Math\u00e9matiques<\/strong><\/a><br \/>\nDUREE : 3\u00a0HEURES\u00a0<span style=\"color: #008000;\">KA<\/span><span style=\"color: #ff0000;\">M<\/span><span style=\"color: #ffcc00;\">ER<\/span><span style=\"color: #800080;\">POWER<\/span>.COM<br \/>\n<\/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-69e4dcd652589\" 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-69e4dcd652589\"  aria-label=\"Toggle\" \/><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/kamerpower.com\/fr\/epreuves-concours-iut-de-douala-mathematiques-2009-pftin-gi-filiere-1ere-annee\/#epreuves-concours-iut-de-douala\" >Epreuves concours IUT de Douala \n*******************************<\/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\/epreuves-concours-iut-de-douala-mathematiques-2009-pftin-gi-filiere-1ere-annee\/#epreuves-concours-iut-de-douala-mathematiques-2009-pftin-gi\" >Epreuves concours IUT de Douala Math\u00e9matiques 2009 PFTIN-GI<\/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\/epreuves-concours-iut-de-douala-mathematiques-2009-pftin-gi-filiere-1ere-annee\/#exercice-1-5-points\" >Exercice 1 : 5\u00a0points<\/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\/epreuves-concours-iut-de-douala-mathematiques-2009-pftin-gi-filiere-1ere-annee\/#exercice-2-4-points\" >Exercice 2\u00a0: 4\u00a0points<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/kamerpower.com\/fr\/epreuves-concours-iut-de-douala-mathematiques-2009-pftin-gi-filiere-1ere-annee\/#exercice-3-6-points\" >Exercice 3: 6\u00a0points<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/kamerpower.com\/fr\/epreuves-concours-iut-de-douala-mathematiques-2009-pftin-gi-filiere-1ere-annee\/#exercice-4-5-points\" >Exercice 4: 5\u00a0points<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/kamerpower.com\/fr\/epreuves-concours-iut-de-douala-mathematiques-2009-pftin-gi-filiere-1ere-annee\/#epreuves-concours-iut-de-douala-mathematiques-2009-pftin-gi-filiere-1ere-annee-iut-de-luniversite-de-douala-session-juillet-2009\" >Epreuves concours IUT de Douala Math\u00e9matiques 2009 PFTIN-GI Fili\u00e8re 1\u00e8re ANN\u00c9E IUT de l\u2019Universit\u00e9 de Douala Session juillet 2009<\/a><\/li><\/ul><\/nav><\/div>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"epreuves-concours-iut-de-douala\"><\/span><span style=\"color: #000000;\">Epreuves concours IUT de Douala<br \/>\n<span style=\"color: #ff0000;\"><span style=\"color: #00ff00;\">**********<\/span>***********<span style=\"color: #ffff00;\">**********<\/span><\/span><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"epreuves-concours-iut-de-douala-mathematiques-2009-pftin-gi\"><\/span><span style=\"color: #3366ff;\"><a style=\"color: #3366ff;\" href=\"https:\/\/kamerpower.com\/fr\/concours-iut-douala-2021-2022-cycle-bts-premiere-annee\/\" target=\"_blank\" rel=\"noopener\">Epreuves concours IUT de Douala Math\u00e9matiques 2009 PFTIN-GI<\/a><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"exercice-1-5-points\"><\/span><strong><span style=\"color: #ff0000;\">Exercice 1 : 5\u00a0points<\/span><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"color: #000000;\">Soit f(x ) = e<sup>\u2212|x \u22121|\u00a0<\/sup>une fonction r\u00e9elle de la variable r\u00e9elle x.<\/span><\/p>\n<ol>\n<li><span style=\"color: #000000;\">Calculer les limites de cette fonction . (0.5 pt )<\/span><\/li>\n<li><span style=\"color: #000000;\">D\u00e9montrer que f est maximun au point <em>I<sub>0<\/sub>\u00a0= 1<\/em>\u00a0. (0.5 pt )<\/span><\/li>\n<li><span style=\"color: #000000;\">Calculer la limite \u00e0 gauche et la limite \u00e0 droite au point d\u2019abscisse 1 de <em>h(x)<\/em> = (f(x) &#8211; f(1)) \/ x-1<\/span><br \/>\n<span style=\"color: #000000;\">\u00a0\u00a0( 1 pt )\u00a0kamerpower.com<\/span><\/li>\n<li><span style=\"color: #000000;\"> Que peut-on en d\u00e9duire pour f au point d\u2019abscisse 1 . (0.5 pt )<\/span><\/li>\n<li><span style=\"color: #000000;\">Dresser le tableau de variation et tracer la courbe de f . (1 pt )<\/span><\/li>\n<li><span style=\"color: #000000;\">Calculer <em>I(a) \u00a0 = \u00a0<\/em>\u00a0\u00a0-a -&gt; a \u00a0\u222b \u00a0f(x)dx<\/span><span style=\"color: #000000;\">\u00a0a &gt; 1\u00a0et limite de I(a) quand a \u2192 +\u221e. (1.5 pt)<\/span><\/li>\n<\/ol>\n<h3><span class=\"ez-toc-section\" id=\"exercice-2-4-points\"><\/span><span style=\"color: #ff0000;\"><strong>Exercice 2\u00a0: 4\u00a0points<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">Un d\u00e9 cubique D1\u00a0comprend une face marqu\u00e9e 1, deux faces marqu\u00e9es 3,et une\u00a0face marqu\u00e9e 4.\u00a0kamerpower.com<\/span><br \/>\n<span style=\"color: #000000;\">Un d\u00e9 cubique D2\u00a0comprend deux faces marqu\u00e9es 1, une face marqu\u00e9e 2,et trois\u00a0faces marqu\u00e9es 3.<\/span><br \/>\n<span style=\"color: #000000;\">On suppose que,pour chaque d\u00e9 toutes les faces ont la m\u00eame probabilit\u00e9 d\u2019etre\u00a0obtenue. On lance simultan\u00e9ment les deux soient x le chiffre indiqu\u00e9 par les d\u00e9\u00a0D1 et y le chiffre indiqu\u00e9 par le d\u00e9 D2.<\/span><\/p>\n<ol>\n<li><span style=\"color: #000000;\">Calculer la probabilit\u00e9 pour que l\u2019on ait x = y . (1 pt )<\/span><\/li>\n<li><span style=\"color: #000000;\">Calculer la probabilit\u00e9 que l\u2019on ait x &gt; y . (1.5 pt )<\/span><\/li>\n<li><span style=\"color: #000000;\">Calculer la probabilit\u00e9 pour que l\u2019on ait |x \u2212 y | \u2265 1. (1.5 pt )<\/span><\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-38314 size-full\" src=\"https:\/\/kamerpower.com\/wp-content\/uploads\/2021\/06\/Concours-IUT-douala-2021-2022-filiere-universite-de-douala.png\" alt=\"Concours IUT douala  fili\u00e8re universit\u00e9 de douala\" width=\"1890\" height=\"945\" srcset=\"https:\/\/kamerpower.com\/wp-content\/uploads\/2021\/06\/Concours-IUT-douala-2021-2022-filiere-universite-de-douala.png 1890w, https:\/\/kamerpower.com\/wp-content\/uploads\/2021\/06\/Concours-IUT-douala-2021-2022-filiere-universite-de-douala-600x300.png 600w, https:\/\/kamerpower.com\/wp-content\/uploads\/2021\/06\/Concours-IUT-douala-2021-2022-filiere-universite-de-douala-300x150.png 300w, https:\/\/kamerpower.com\/wp-content\/uploads\/2021\/06\/Concours-IUT-douala-2021-2022-filiere-universite-de-douala-768x384.png 768w, https:\/\/kamerpower.com\/wp-content\/uploads\/2021\/06\/Concours-IUT-douala-2021-2022-filiere-universite-de-douala-1536x768.png 1536w, https:\/\/kamerpower.com\/wp-content\/uploads\/2021\/06\/Concours-IUT-douala-2021-2022-filiere-universite-de-douala-720x360.png 720w, https:\/\/kamerpower.com\/wp-content\/uploads\/2021\/06\/Concours-IUT-douala-2021-2022-filiere-universite-de-douala-520x260.png 520w, https:\/\/kamerpower.com\/wp-content\/uploads\/2021\/06\/Concours-IUT-douala-2021-2022-filiere-universite-de-douala-320x160.png 320w, https:\/\/kamerpower.com\/wp-content\/uploads\/2021\/06\/Concours-IUT-douala-2021-2022-filiere-universite-de-douala-150x75.png 150w, https:\/\/kamerpower.com\/wp-content\/uploads\/2021\/06\/Concours-IUT-douala-2021-2022-filiere-universite-de-douala-1200x600.png 1200w\" sizes=\"auto, (max-width: 1890px) 100vw, 1890px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"exercice-3-6-points\"><\/span><span style=\"color: #ff0000;\"><strong>Exercice 3: 6\u00a0points<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"color: #000000;\">Soient m et n deux entiers naturels. On consid\u00e8re l\u2019int\u00e9grale suivante: I<sub>m,n\u00a0<\/sub>\u00a0= \u00a0 \u00a00 -&gt;1 \u222b\u00a0 x<sup>n<\/sup>(1-x)<sup>m<\/sup>dx<\/span><\/p>\n<ol>\n<li><span style=\"color: #000000;\">\u00a0 I<sub>m,0<\/sub>\u00a0 \u00a0et \u00a0 I<sub>0,n<\/sub><\/span><\/li>\n<li><span style=\"color: #000000;\"><strong>a.<\/strong> \u00c9tablir la relation de r\u00e9currence entre I<sub>m,n<\/sub>\u00a0 et \u00a0I<sub>m-1,n+1<\/sub><\/span><br \/>\n<span style=\"color: #000000;\">. (0.5 pt)<\/span><br \/>\n<span style=\"color: #000000;\"><strong>b.<\/strong> En d\u00e9duire une formule permettant de calculer I<sub>m,n<\/sub><\/span><br \/>\n<span style=\"color: #000000;\">. (1 pt )\u00a0kamerpower.com<\/span><br \/>\n<span style=\"color: #000000;\"><strong>c.<\/strong> Utiliser cette formule pour calculer I<sub>5,7<\/sub> = I<sub>7,5<\/sub><\/span><br \/>\n<span style=\"color: #000000;\">\u00a0 (0.5 pt )<\/span><\/li>\n<li><span style=\"color: #000000;\"><strong>a.<\/strong> \u00c9tablir la relation de r\u00e9currence entre I<sub>m,n<\/sub> et I<sub>m+1,n-1<\/sub><\/span><br \/>\n<span style=\"color: #000000;\">. (0.5 pt )<\/span><br \/>\n<span style=\"color: #000000;\"><strong>b.<\/strong> En d\u00e9duire une formule permettant de calculer I<sub>m,n\u00a0<\/sub><\/span><span style=\"color: #000000;\">. ( 1 pt )\u00a0kamerpower.com<\/span><br \/>\n<span style=\"color: #000000;\"><strong>c.<\/strong> Utiliser cette formule pour calculer I<sub>7,5<\/sub> = I<sub>5,7<\/sub><\/span><br \/>\n<span style=\"color: #000000;\">. (0.5 pt )<\/span><\/li>\n<li><span style=\"color: #000000;\">Montrer que I<sub>m,n<\/sub> = I<sub>n,m<\/sub><\/span><\/li>\n<\/ol>\n<h3><span class=\"ez-toc-section\" id=\"exercice-4-5-points\"><\/span><span style=\"color: #ff0000;\"><strong>Exercice 4: 5\u00a0points<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ol>\n<li><span style=\"color: #000000;\">Soit z un nombre complexe calculer p = (1 + z + z<sup>2<\/sup> +\u00a0z<sup>3<\/sup>\u00a0+\u00a0z<sup>4<\/sup>\u00a0+\u00a0z<sup>5<\/sup>\u00a0+ z<sup>6<\/sup>)(1-z) \u00a0 \u00a0(0.5 pt)<\/span><\/li>\n<li><span style=\"color: #000000;\">Soit z = cos\u00a02\u03c0\/7 + isin\u00a02\u03c0\/7<\/span><br \/>\n<span style=\"color: #000000;\"><strong>a<\/strong>. D\u00e9terminer la valeur de S = 1 + z + z<sup>2<\/sup> +\u00a0z<sup>3<\/sup>\u00a0+\u00a0z<sup>4<\/sup>\u00a0+\u00a0z<sup>5<\/sup>\u00a0+ z<sup>6<\/sup>. \u00a0(0.5 pt )<\/span><br \/>\n<span style=\"color: #000000;\"><strong>b<\/strong>. En d\u00e9duire la valeur de t = cos\u00a02\u03c0\/7 +\u00a0cos 4\u03c0\/7 +\u00a0cos 6\u03c0\/7. (1 pt )<\/span><\/li>\n<li><span style=\"color: #000000;\">Soit \u03b8 \u2208]0, \u03c0[ u r\u00e9el donn\u00e9. kamerpower.com<\/span><br \/>\n<span style=\"color: #000000;\"><strong>a<\/strong>. R\u00e9soudre l\u2019\u00e9quation u<sup>2<\/sup>\u00a0+ 2(cos \u03b8)u + 1 = 0 \u00a0(1 pt )\u00a0<\/span><br \/>\n<span style=\"color: #000000;\"><strong>b<\/strong>. En d\u00e9duire la r\u00e9solution de l\u2019\u00e9quation<br \/>\nx<sup>4<\/sup>\u00a0+ 2(cos\u03b8)x<sup>2<\/sup>\u00a0+ 1 = 0<sup>\u00a0<\/sup>\u00a0. (2 pts ) <\/span><\/li>\n<\/ol>\n<p><span style=\"color: #000000;\"><strong>NB:<\/strong> <span style=\"color: #ff0000;\">La question 3 est ind\u00e9pendante des autres<\/span><\/span><\/p>\n<h2 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"epreuves-concours-iut-de-douala-mathematiques-2009-pftin-gi-filiere-1ere-annee-iut-de-luniversite-de-douala-session-juillet-2009\"><\/span><span style=\"color: #3366ff;\">Epreuves concours IUT de Douala Math\u00e9matiques 2009 PFTIN-GI Fili\u00e8re 1\u00e8re ANN\u00c9E IUT de l\u2019Universit\u00e9 de Douala Session juillet 2009<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n","protected":false},"excerpt":{"rendered":"<p>&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":38314,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_themeisle_gutenberg_block_has_review":false,"footnotes":""},"categories":[1701],"tags":[1712,1856,1706],"class_list":["post-6777","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-epreuves","tag-epreuve-de-concours-dentree-aux-grandes-ecoles-cameroun","tag-epreuves-de-concours-de-liut-de-douala","tag-mathematiques"],"_links":{"self":[{"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/posts\/6777","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=6777"}],"version-history":[{"count":3,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/posts\/6777\/revisions"}],"predecessor-version":[{"id":70250,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/posts\/6777\/revisions\/70250"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/media\/38314"}],"wp:attachment":[{"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/media?parent=6777"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/categories?post=6777"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/tags?post=6777"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}