{"id":4312,"date":"2015-03-22T20:07:00","date_gmt":"2015-03-22T18:07:00","guid":{"rendered":"https:\/\/kamerpower.com\/?p=4312"},"modified":"2025-01-29T21:42:07","modified_gmt":"2025-01-29T20:42:07","slug":"epreuve-de-mathematiques-ensp-1996-concours-dentree-a-lecole-nationale-superieure-polytechnique-niveau-baccalaureat-cameroun","status":"publish","type":"post","link":"https:\/\/kamerpower.com\/fr\/epreuve-de-mathematiques-ensp-1996-concours-dentree-a-lecole-nationale-superieure-polytechnique-niveau-baccalaureat-cameroun\/","title":{"rendered":"Epreuve de mathematiques ENSP 1996 concours d&#8217;entr\u00e9e \u00e0 l\u2019Ecole Nationale Sup\u00e9rieure Polytechnique Niveau Baccalaureat Cameroun"},"content":{"rendered":"<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-69d17c5630a98\" 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-69d17c5630a98\"  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-de-mathematiques-ensp-1996-concours-dentree-a-lecole-nationale-superieure-polytechnique-niveau-baccalaureat-cameroun\/#concours-dentree-a-lecole-nationale-superieure-polytechnique-baccalaureat-cameroun\" >Concours d\u2019entr\u00e9e \u00e0 l\u2019Ecole Nationale Sup\u00e9rieure Polytechnique (Baccalaureat Cameroun)<\/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-de-mathematiques-ensp-1996-concours-dentree-a-lecole-nationale-superieure-polytechnique-niveau-baccalaureat-cameroun\/#exercice-1\" >Exercice\u00a01:<\/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-de-mathematiques-ensp-1996-concours-dentree-a-lecole-nationale-superieure-polytechnique-niveau-baccalaureat-cameroun\/#exercice-2\" >Exercice 2:<\/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-de-mathematiques-ensp-1996-concours-dentree-a-lecole-nationale-superieure-polytechnique-niveau-baccalaureat-cameroun\/#exercice-3\" >Exercice 3:<\/a><\/li><\/ul><\/nav><\/div>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"concours-dentree-a-lecole-nationale-superieure-polytechnique-baccalaureat-cameroun\"><\/span><span style=\"color: #ff0000;\">Concours d\u2019entr\u00e9e \u00e0 l\u2019Ecole Nationale Sup\u00e9rieure Polytechnique (Baccalaureat Cameroun)<\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">\u00c9preuve\u00a0de mathematiques ENSP 1996<\/span><br \/>\n(Dur\u00e9e de l\u2019\u00e9preuve : 4\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><\/p>\n<h3 class=\"exercice\"><span class=\"ez-toc-section\" id=\"exercice-1\"><\/span><span lang=\"FR\" style=\"color: #3366ff;\">Exercice\u00a01:<\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><span style=\"color: #000000;\">On place dans un sac 4 boules marqu\u00e9es respectivement des nombres\u00a0:\u00a0<span style=\"color: #ff0000;\">1, -1, \u00bd, et \u20133\/7<\/span>.\u00a0On tire successivement du sac, au hasard 3 boules, marqu\u00e9es respectivement a, b, et c dans l\u2019ordre de leur tirage, et ce, sans remise apr\u00e8s chaque tirage.<\/span><\/p>\n<ol>\n<li style=\"text-align: justify;\"><span style=\"color: #000000;\">Quelle est la probabilit\u00e9 pour que l\u2019\u00e9quation du second degr\u00e9\u00a0: <span style=\"color: #ff0000;\">az\u00b2 + dz + c =0<\/span> \u00a0ait deux solutions complexes, non-r\u00e9elles et dont les images M1 et M2 sont sym\u00e9triques par rapport \u00e0 l\u2019origine O du plan complexe\u00a0?\u00a0<\/span><span style=\"color: #000000;\">Les chances de cet \u00e9v\u00e9nement auraient-elles \u00e9t\u00e9 plus grandes si le tirage des boules avait plut\u00f4t \u00e9t\u00e9 effectu\u00e9 avec remise apr\u00e8s chaque tirage\u00a0?<\/span><\/li>\n<\/ol>\n<h3 class=\"exercice\"><span class=\"ez-toc-section\" id=\"exercice-2\"><\/span><span lang=\"FR\" style=\"color: #3366ff;\">Exercice 2:<\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>On consid\u00e8re la fonction g d\u00e9finie sur l\u2019intervalle [0 , 3] par\u00a0:<\/p>\n<p>g(x) = 3x \/ <span style=\"color: #000000;\">\u221ax+1<\/span><\/p>\n<p><span style=\"color: #000000;\">on note C sa courbe repr\u00e9sentative dans un rep\u00e8re orthonorm\u00e9 (O, i, j) d\u2019unit\u00e9 graphique 2cm. Soit Q la plaque plane d\u00e9finie par la courbe C, l\u2019axe abscisses et la droite d\u2019\u00e9quation x=3.<\/span><\/p>\n<ol>\n<li><span style=\"color: #000000;\">Etudier les variations de g sur <span style=\"color: #ff0000;\">[0\u00a0;3]<\/span>. Tracer la courbe repr\u00e9sentative de g<\/span><\/li>\n<li><span style=\"color: #000000;\">Calculer l\u2019aire de la plaque Q en cm\u00b2 en utilisant une int\u00e9gration par parties.<\/span><\/li>\n<li><span style=\"color: #000000;\">Par rotation de le plaque Q autour de l\u2019axe (O, i), on obtient un solide de r\u00e9volution R. D\u00e9termine le volume en cm\u00b3. <\/span><\/li>\n<li><span style=\"color: #000000;\">Donner une valeur arrondie en mm\u00b3 de R.<\/span><\/li>\n<\/ol>\n<h3 class=\"exercice\"><span class=\"ez-toc-section\" id=\"exercice-3\"><\/span><span lang=\"FR\" style=\"color: #3366ff;\">Exercice 3:<\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"color: #000000;\">L\u2019objectif est de d\u00e9terminer les droites tangentes \u00e0 la fois \u00e0 la courbe repr\u00e9sentant la fonction logarithme n\u00e9p\u00e9rien et \u00e0 celle \u00e0 la fonction exponentielle, puis d\u2019\u00e9tudier la configuration obtenue.\u00a0Le plan est rapport\u00e9 \u00e0 un rep\u00e8re orthonormal (O, i, j), unit\u00e9 graphique 1 cm.<\/span><\/p>\n<p><span style=\"color: #000000;\">On note\u00a0:<\/span><\/p>\n<p><span style=\"color: #000000;\">C1 et C2 les courbes d\u2019\u00e9quations respectives <span style=\"color: #ff0000;\">y =e<sup>x<\/sup>\u00a0et y = lnx\u00a0;<\/span><\/span><br \/>\n<span style=\"color: #000000;\"> T<sub>p<\/sub> la tangente \u00e0 la courbe C1 au point P d\u2019abscisse p, p \u00e9tant un nombre r\u00e9el.<\/span><br \/>\n<span style=\"color: #000000;\"> D<span style=\"font-size: 13.3333330154419px; line-height: 20px;\">1<\/span>\u00a0la tangente \u00e0 la courbe au point T d\u2019abscisse l, l \u00e9tant un nombre r\u00e9el strictement positif.<\/span><\/p>\n<ol>\n<li><span style=\"color: #000000;\">\u00a0Dans cette partie, on cherche le lien entre des droites, tangentes aux deux courbes C1 et C2 et qui sont parall\u00e8les ; puis \u00e0 quelle condition une droite tangente \u00e0 la courbe C1 est \u00e9galement \u00e0 la tangente \u00e0 la courbe C2.<\/span><\/li>\n<\/ol>\n<p><span style=\"color: #000000;\">a)\u00a0D\u00e9terminer une \u00e9quation cart\u00e9sienne de la droite T<sub>p<\/sub>.\u00a0D\u00e9terminer de m\u00eame une \u00e9quation cart\u00e9sienne de la droite D<span style=\"font-size: 13.3333330154419px; line-height: 20px;\">1<\/span>.<\/span><\/p>\n<p><span style=\"color: #000000;\">b)\u00a0D\u00e9terminer l&#8217;en fonction de p pour les droites T<sub>p<\/sub>\u00a0et D<span style=\"font-size: 13.3333330154419px; line-height: 20px;\">1<\/span>\u00a0soient parall\u00e8les. On notera q la valeur obtenu de l&#8217;ainsi obtenue, Q le point de la courbe C2 d\u2019abscisse q et Dq la tangent corresponde.\u00a0Montrer que les droites Tp, Dq sont confondues si et seulement si\u00a0:\u00a0<span style=\"color: #ff0000;\">\u00a0q=\u212e<sup>-p<\/sup>\u00a0\u00a0 et\u00a0\u00a0\u00a0 (p+1)\u212e<sup>-p<\/sup> = p-1<\/span><\/span><\/p>\n<p><span style=\"color: #000000;\">\u00a0 <span style=\"color: #ff0000;\">\u00a0 2.<\/span>\u00a0Dans cette partie, on se propose d\u2019\u00e9tudier les solutions de l\u2019\u00e9quation\u00a0:<\/span><\/p>\n<p><span style=\"color: #000000;\">e<sup>-x<\/sup> = (x-1)\/(x+1) \u00a0 &#8212;&#8212;-(1)<\/span><\/p>\n<p><span style=\"color: #000000;\">Pour cela on consid\u00e8re la fonction <em>f<\/em> d\u00e9finie pour tout nombre r\u00e9el x, x \u2260 -1 par\u00a0:<\/span><\/p>\n<p>f(x) = [<span style=\"color: #000000;\">(x-1)\/(x+1)<\/span>].<span style=\"color: #000000;\">e<sup>x<\/sup><\/span><\/p>\n<ul>\n<li><span style=\"color: #000000;\">Montrer que si<span style=\"color: #ff0000;\"> <em>f<\/em> (x)=1<\/span> si et seulement si e<sup>-x\u00a0<\/sup>= (x-1)\/(x+1)\u00a0\u00a0\u00a0<\/span><\/li>\n<\/ul>\n<ul>\n<li><span style=\"color: #000000;\">\u00a0Etudier les variations de <em>f<\/em> sur <span style=\"color: #ff0000;\">I=[0\u00a0;+\u221e[<\/span> et la limite de <em>f<\/em> quand x tend vers +\u221e.<\/span><\/li>\n<li><span style=\"color: #000000;\">D\u00e9montrer que l\u2019\u00e9quation <em>f<\/em> (x)=1 admet , dans <em>I<\/em>, une solution unique m et que m appartient \u00e0 l\u2019intervalle [1,5\u00a0; 1,6].<\/span><\/li>\n<\/ul>\n<p><span style=\"color: #000000;\"><span style=\"color: #ff0000;\">a)<\/span>\u00a0Pour tout nombre r\u00e9el x, diff\u00e9rent de 1 et de \u20131, calculer le produit: \u00a0<span style=\"color: #ff0000;\"><em>f<\/em> (x) \u00d7<em> f<\/em> (-x).<\/span><\/span><\/p>\n<p><span style=\"color: #000000;\"><span style=\"color: #ff0000;\">b)<\/span>\u00a0D\u00e9duire des \u00e9quation pr\u00e9c\u00e9dentes que l\u2019\u00e9quation (1) admet deux solutions oppos\u00e9es.<\/span><\/p>\n<p><span style=\"color: #000000;\"><span style=\"color: #ff0000;\">c)<\/span>\u00a0D\u00e9terminer les tangents communes aux courbes C1 et C2.<\/span><\/p>\n<p><span style=\"color: #000000;\">Tracer dans un rep\u00e8re orthonormal (O, i, j) les courbes C1 et C2. on rappelle que les deux courbes sont sym\u00e9triques par rapport \u00e0 la droite y=x.\u00a0Tracer \u00e9galement les tangentes communes <span style=\"color: #ff0000;\">T<sub>m<\/sub> et T<sub>-m<\/sub><\/span>. On prendra pour m la valeur approch\u00e9e <span style=\"color: #3366ff;\">1,55.<\/span><\/span><\/p>\n<p><span style=\"color: #000000;\"><span style=\"color: #ff0000;\">3.<\/span>\u00a0Etude g\u00e9om\u00e9trique du probl\u00e8me.<\/span><\/p>\n<p><span style=\"color: #000000;\">On consid\u00e8re l\u2019\u00e9quation U du plan dans lui-m\u00eame qui , a tout point M de coordonn\u00e9es (x;y),\u00a0 y non nul, associe le point M\u2019 de coordonn\u00e9es <span style=\"color: #ff0000;\">(x&#8217; ; y&#8217; )<\/span> avec <span style=\"color: #ff0000;\">x&#8217; = -x<\/span> et\u00a0 y&#8217; =\u00a0\\frac{1}{y}.<\/span><\/p>\n<p><span style=\"color: #000000;\">D\u00e9terminer U(M&#8217;). Montrer que, si le point M appartient \u00e0 la courbe C1 alors le point M&#8217; appartient aussi \u00e0 la courbe C1.<\/span><\/p>\n<p><span style=\"color: #000000;\">Soit P le point d\u2019abscisse m (m \u00e9tant le point d\u00e9fini en II.1.c) de la courbe C1, T<sub>m<\/sub> est donc la tangente \u00e0 la courbe C1 au point P et \u00e0 la courbe C2 au point Q d\u2019abscisse \u212e<sup>-m<\/sup>.<\/span><\/p>\n<ul>\n<li><span style=\"color: #000000;\">D\u00e9terminer en fonction de m les coordonn\u00e9es de P\u2019 = U(P).<\/span><\/li>\n<li><span style=\"color: #000000;\">V\u00e9rifier que la droite T<sub>-m<\/sub> est tangente \u00e0 la courbe C1 au point P\u2019 et qu\u2019elle est aussi tangente \u00e0 la courbe C2 en un point Q<sub>1<\/sub>, dont on donnera les coordonn\u00e9es en fonction de m.<\/span><\/li>\n<li><span style=\"color: #000000;\">\u00a0Compl\u00e9ter la figure en pla\u00e7ant les points P,Q<sub>1<\/sub>, P&#8217;, et Q.<\/span><\/li>\n<li><span style=\"color: #000000;\">Justifier les coordonn\u00e9es suivantes\u00a0<\/span><\/li>\n<\/ul>\n<p>P(m; <span style=\"color: #000000;\">(m+1)\/(m-1)<\/span>) ; Q(<span style=\"color: #000000;\">(m-1)\/(m+1) ; -m<\/span>) ; P(-m; <span style=\"color: #000000;\">(m-1)\/(m+1)<\/span>) ; Q1(<span style=\"color: #000000;\">(m+1)\/(m-1) ; m;\u00a0<\/span>)<\/p>\n<ul>\n<li><span style=\"color: #000000;\">En d\u00e9duire que les droites T<sub>m<\/sub> et T<sub>-m<\/sub> sont sym\u00e9triques par rapport \u00e0 la droite d\u2019\u00e9quation y=x . D\u00e9terminer la nature du quadrilat\u00e8re <span style=\"color: #ff0000;\">PQ<sub>1<\/sub>QP&#8217;.<\/span><\/span><\/li>\n<li><span style=\"color: #000000;\">Montrer que l\u2019aire du domaine limit\u00e9 par le segment [PP\u2019] et l\u2019arc de la courbe C1 d\u2019extr\u00e9mit\u00e9 P et P&#8217; est \u00e9gales \u00e0 2<em>m<\/em>. On admettra que cet arc est situ\u00e9 en-dessous du segment [PP&#8217;].<\/span><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><span style=\"color: #3366ff;\"><a style=\"color: #3366ff;\" href=\"https:\/\/kamerpower.com\/fr\/epreuves-concours-iut-douala-mathematiques-2011-pftin-gi-filiere-1ere-annee\/\" rel=\"noopener\">\u00a0Epreuve de mathematiques<\/a> ENSP 1996 concours d&#8217;entr\u00e9e \u00e0 l\u2019Ecole Nationale Sup\u00e9rieure Polytechnique Niveau Baccalaureat Cameroun<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":4313,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_themeisle_gutenberg_block_has_review":false,"footnotes":""},"categories":[1701],"tags":[1712,1727,1706],"class_list":["post-4312","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-epreuves","tag-epreuve-de-concours-dentree-aux-grandes-ecoles-cameroun","tag-epreuve-ensp-concours-cameroun","tag-mathematiques"],"_links":{"self":[{"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/posts\/4312","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=4312"}],"version-history":[{"count":1,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/posts\/4312\/revisions"}],"predecessor-version":[{"id":67826,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/posts\/4312\/revisions\/67826"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/media\/4313"}],"wp:attachment":[{"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/media?parent=4312"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/categories?post=4312"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kamerpower.com\/fr\/wp-json\/wp\/v2\/tags?post=4312"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}