{"id":548,"date":"2026-05-05T04:51:34","date_gmt":"2026-05-05T04:51:34","guid":{"rendered":"https:\/\/79.buffdemo.com\/index.php\/2026\/05\/05\/de-thi-thu-tot-nghiep-thpt-mon-toan-2024-2025-truong-thpt-truc-ninh-nam-dinh\/"},"modified":"2026-05-05T07:25:09","modified_gmt":"2026-05-05T07:25:09","slug":"de-thi-thu-tot-nghiep-thpt-mon-toan-2024-2025-truong-thpt-truc-ninh-nam-dinh","status":"publish","type":"post","link":"https:\/\/79.buffdemo.com\/index.php\/2026\/05\/05\/de-thi-thu-tot-nghiep-thpt-mon-toan-2024-2025-truong-thpt-truc-ninh-nam-dinh\/","title":{"rendered":"\u0110\u1ec1 thi th\u1eed t\u1ed1t nghi\u1ec7p THPT m\u00f4n To\u00e1n 2024 \u2013 2025 tr\u01b0\u1eddng THPT Tr\u1ef1c Ninh Nam \u0110\u1ecbnh"},"content":{"rendered":"<p><strong>\u0110\u1ec1 thi th\u1eed t\u1ed1t nghi\u1ec7p THPT m\u00f4n To\u00e1n 2024 \u2013 2025 tr\u01b0\u1eddng THPT Tr\u1ef1c Ninh Nam \u0110\u1ecbnh<\/strong> \u0111\u01b0\u1ee3c bi\u00ean so\u1ea1n d\u00e0nh cho k\u00ec thi th\u1eed t\u1ed1t nghi\u1ec7p THPT l\u1ea7n 1. T\u00e0i li\u1ec7u n\u00e0y gi\u00fap c\u00e1c b\u1ea1n h\u1ecdc sinh 12 l\u00e0m quen v\u1edbi c\u1ea5u tr\u00fac ra \u0111\u1ec1 v\u00e0 \u0111\u00e1nh gi\u00e1 n\u0103ng l\u1ef1c \u0111\u1ec3 c\u00f3 ti\u1ebfn tr\u00ecnh \u00f4n t\u1eadp hi\u1ec7u qu\u1ea3. H\u00e3y t\u00ecm hi\u1ec3u th\u00eam n\u1ed9i dung c\u1ee7a \u0111\u1ec1 thi th\u1eed n\u00e0y qua b\u00e0i vi\u1ebft sau v\u1edbi <a href=\"https:\/\/tailieuonthi.edu.vn\/\" rel=\"noopener\" target=\"_blank\">tailieuonthi<\/a> n\u00e0o.<\/p>\n<h2><strong>N\u1ed9i dung \u0110\u1ec1 thi th\u1eed t\u1ed1t nghi\u1ec7p THPT m\u00f4n To\u00e1n 2024 \u2013 2025 tr\u01b0\u1eddng THPT Tr\u1ef1c Ninh Nam \u0110\u1ecbnh<\/strong><\/h2>\n<p>N\u1ed9i dung <strong>\u0110\u1ec1 thi th\u1eed t\u1ed1t nghi\u1ec7p THPT m\u00f4n To\u00e1n 2024 \u2013 2025 tr\u01b0\u1eddng THPT Tr\u1ef1c Ninh Nam \u0110\u1ecbnh<\/strong> \u0111\u01b0\u1ee3c \u0111\u00e1nh gi\u00e1 l\u00e0 v\u00f4 c\u00f9ng ph\u00f9 h\u1ee3p v\u1edbi n\u0103ng l\u1ef1c c\u1ee7a c\u00e1c b\u1ea1n h\u1ecdc sinh. \u0110\u1ec1 thi th\u1eed thpt qu\u1ed1c gia m\u00f4n to\u00e1n c\u00f3 t\u1ea5t c\u1ea3 22 c\u00e2u h\u1ecfi v\u1edbi h\u00ecnh th\u1ee9c tr\u1eafc nghi\u1ec7m k\u00e8m t\u1ef1 lu\u1eadn ng\u1eafn. C\u00e1c c\u00e2u h\u1ecfi trong ma tr\u1eadn c\u00f3 \u0111\u1ed9 ph\u00e2n h\u00f3a h\u1ee3p l\u00fd t\u1eeb nh\u1eadn bi\u1ebft \u0111\u1ebfn v\u1eadn d\u1ee5ng cao. Chi ti\u1ebft nh\u01b0 sau:<\/p>\n<p><strong>PH\u1ea6N 1 \u2013 C\u00c2U TR\u1eaeC NGHI\u1ec6M NHI\u1ec0U PH\u01af\u01a0NG \u00c1N L\u1ef0A CH\u1eccN<\/strong><\/p>\n<p>G\u1ed3m 12 c\u00e2u h\u1ecfi v\u1edbi n\u1ed9i dung v\u1ec1 \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1, h\u00ecnh h\u1ecdc kh\u00f4ng gian, ph\u01b0\u01a1ng tr\u00ecnh logarit, x\u00e1c su\u1ea5t th\u1ed1ng k\u00ea. V\u00ed d\u1ee5 c\u00e1c c\u00e2u h\u1ecfi trong <a href=\"https:\/\/tailieuonthi.edu.vn\/de-thi-thu-thpt\/de-thi-thu-thpt-mon-toan\/\" rel=\"noopener\" target=\"_blank\">\u0111\u1ec1 thi th\u1eed thpt qu\u1ed1c gia m\u00f4n to\u00e1n<\/a> nh\u01b0 sau:<\/p>\n<p><strong>C\u00e2u 1<\/strong>: Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh  cos x = -1\/2 l\u00e0<\/p>\n<p>A.  x = \u00b1 \u03c0\/6 + k\u03c0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0B.  x = \u00b1 \u03c0\/6 + 2k\u03c0 <\/p>\n<p>C.  x = \u00b1 \u03c0\/3 + 2k\u03c0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 D.  x = \u00b1 2\u03c0\/3 + 2k\u03c0 <\/p>\n<p><strong>C\u00e2u 2<\/strong>: Trong kh\u00f4ng gian Oxyz, cho tam gi\u00e1c  ABC  v\u1edbi  A(1;2;-1), B(2;-1;3), C(-3;5;1) T\u1ecda \u0111\u1ed9 tr\u1ecdng t\u00e2m c\u1ee7a tam gi\u00e1c  ABC  l\u00e0<\/p>\n<p>A.  (0;2;1) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 B.  (0;2;-1) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 C.  (0;-2;1) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0D.  (0;-2;-1) <\/p>\n<p><strong>PH\u1ea6N 2 \u2013 C\u00c2U TR\u1eaeC NGHI\u1ec6M \u0110\u00daNG SAI<br \/>\n<\/strong><br \/>\nC\u00f3 4 c\u00e2u, th\u00ed sinh c\u1ea7n \u0111\u1ecdc k\u0129 \u0111\u1ec1 v\u00e0 c\u00e1c c\u00e2u tr\u1ea3 l\u1eddi, ch\u1ecdn m\u1ec7nh \u0111\u1ec1 <strong>\u0110\u00daNG\/SAI<\/strong> cho m\u1ed7i c\u00e2u tr\u1ea3 l\u1eddi.<\/p>\n<p><strong>C\u00e2u 1<\/strong>: Cho m\u1eabu s\u1ed1 li\u1ec7u d\u01b0\u1edbi d\u1ea1ng b\u1ea3ng sau<\/p>\n<p>beginarray|c|c|c|c|c|c| hline S\u1ed1 c\u00e2u tr\u1ea3 l\u1eddi \u0111\u00fang &amp; [16;21) &amp; [21;26) &amp; [26;31) &amp; [31;36) &amp; [36;41) hline T\u1ea7n s\u1ed1 &amp; 4 &amp; 6 &amp; 8 &amp; 10 &amp; 12 hline endarray N = 40<\/p>\n<p>a) Gi\u00e1 tr\u1ecb \u0111\u1ea1i di\u1ec7n c\u1ee7a l\u1edbp  [36;41)  l\u00e0  38,5 .<\/p>\n<p>b) C\u00f4ng th\u1ee9c t\u00ednh s\u1ed1 trung b\u00ecnh m\u1eabu l\u00e0  overlinex = (18,5 \u00b7 4 + 23,5 \u00b7 6 + 28,5 \u00b7 8 + 33,5 \u00b7 10 + 38,5 \u00b7 12)\/(40)\n<\/p>\n<p>c) S\u1ed1 trung b\u00ecnh l\u00e0  30 .<\/p>\n<p>d) Ph\u01b0\u01a1ng sai c\u1ee7a m\u1eabu s\u1ed1 li\u1ec7u l\u00e0  S\u00b2 = 32,75 .<\/p>\n<p><strong>C\u00e2u 2<\/strong>: Trong kh\u00f4ng gian Oxyz, cho tam gi\u00e1c  triangle ABC , bi\u1ebft  A(-1;0;3), B(4;2;0), C(3;1;-3)\n<\/p>\n<p>a)  M(a;b;c)  th\u1ecfa m\u00e3n  AM\u20d7 = 3CB\u20d7 . Khi \u0111\u00f3  a + b + c = -13 <\/p>\n<p>b)  N(a;b;c) in (Oxy)  sao cho  BN  vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng  AC . Khi \u0111\u00f3  4a\u00b2 + b\u00b2 + c\u00b2 = 162 <\/p>\n<p>c)  D(-2;-1;0)  l\u00e0 m\u1ed9t \u0111\u1ec9nh c\u1ee7a h\u00ecnh b\u00ecnh h\u00e0nh  ABCD <\/p>\n<p>d)  G(2;-1;0)  l\u00e0 tr\u1ecdng t\u00e2m tam gi\u00e1c  ABC <\/p>\n<p><strong>PH\u1ea6N 3 \u2013 C\u00c2U TR\u1eaeC NGHI\u1ec6M TR\u1ea2 L\u1edcI NG\u1eaeN<\/strong><\/p>\n<p>C\u1ea5u tr\u00fac c\u00e2u h\u1ecfi \u1edf ph\u1ea7n 3 t\u01b0\u01a1ng t\u1ef1 nh\u01b0 d\u1ea1ng c\u00e2u h\u1ecfi t\u1ef1 lu\u1eadn c\u1ee7a c\u00e1c \u0111\u1ec1 thi n\u0103m tr\u01b0\u1edbc. \u0110\u1ec3 chinh ph\u1ee5c tr\u1ecdn s\u1ed1 \u0111i\u1ec3m b\u1ea1n c\u1ea7n n\u1eafm v\u1eefng ki\u1ebfn th\u1ee9c, c\u00e1c l\u1eadp lu\u1eadn v\u00e0 c\u00e1ch tr\u00ecnh b\u00e0y h\u1ee3p l\u00ed \u0111\u00fang v\u1edbi barem \u0111i\u1ec3m \u0111\u1ec1 \u0111\u01b0a ra. V\u00ed d\u1ee5:<\/p>\n<p><strong>C\u00e2u 1<\/strong>: Bi\u1ebft \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1  y = 1\/4x\u2074 + 3x\u00b2 &#8211; 9x &#8211; 1  c\u00f3 hai c\u1ef1c tr\u1ecb  A  v\u00e0  B . Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng  AB  l\u00e0  y = ax + b, (a,b in mathbbR) . T\u00ednh t\u1ed5ng  a + b .<\/p>\n<p><strong>C\u00e2u 2<\/strong>: S\u1ed1 d\u00e2n c\u1ee7a m\u1ed9t th\u1ecb tr\u1ea5n sau  t  n\u0103m k\u1ec3 t\u1eeb n\u0103m 1970 \u0111\u01b0\u1ee3c \u01b0\u1edbc t\u00ednh b\u1edfi c\u00f4ng th\u1ee9c:<br \/>\n f(t) = (26t + 10)\/(t + 5) \u00b7 f(t) (\u0111\u01a1n v\u1ecb t\u00ednh b\u1eb1ng ngh\u00ecn ng\u01b0\u1eddi) (Ngu\u1ed3n: Gi\u1ea3i t\u00edch 12 n\u00e2ng cao, NXBGD Vi\u1ec7t Nam, 2020). Xem  y = f(t)  l\u00e0 m\u1ed9t h\u00e0m s\u1ed1 x\u00e1c \u0111\u1ecbnh tr\u00ean n\u1eeda kho\u1ea3ng  [0; +\u221e) . \u0110\u1ed3 th\u1ecb h\u00e0m s\u1ed1  y = f(t)  c\u00f3 \u0111\u01b0\u1eddng ti\u1ec7m c\u1eadn ngang l\u00e0  y = a . Gi\u00e1 tr\u1ecb c\u1ee7a  a  l\u00e0 bao nhi\u00eau?<\/p>\n<h2><strong>T\u1ea3i \u0110\u1ec1 thi th\u1eed t\u1ed1t nghi\u1ec7p THPT m\u00f4n To\u00e1n 2024 \u2013 2025 tr\u01b0\u1eddng THPT Tr\u1ef1c Ninh Nam \u0110\u1ecbnh<\/strong><\/h2>\n<p>\u0110\u1ec3 chu\u1ea9n b\u1ecb th\u1eadt t\u1ed1t cho k\u00ec thi thpt qu\u1ed1c gia m\u00f4n to\u00e1n s\u1eafp t\u1edbi, c\u00e1c b\u1ea1n h\u00e3y t\u1ea3i<strong> \u0110\u1ec1 thi th\u1eed t\u1ed1t nghi\u1ec7p THPT m\u00f4n To\u00e1n 2024 \u2013 2025 tr\u01b0\u1eddng THPT Tr\u1ef1c Ninh Nam \u0110\u1ecbnh<\/strong> qua \u0111\u01b0\u1eddng link ph\u00eda d\u01b0\u1edbi \u0111\u1ec3 \u00f4n luy\u1ec7n nh\u00e9. \u0110\u00e2y s\u1ebd l\u00e0 ch\u00eca kh\u00f3a gi\u00fap b\u1ea1n \u00f4n t\u1eadp hi\u1ec7u qu\u1ea3 h\u01a1n v\u00e0 chinh ph\u1ee5c \u0111i\u1ec3m s\u1ed1 cao nh\u1ea5t trong k\u00ec thi ch\u00ednh th\u1ee9c s\u1eafp t\u1edbi.<\/p>\n<p><iframe height=\"480\" loading=\"lazy\" src=\"https:\/\/drive.google.com\/file\/d\/1yF6c5J64kU2yrkbMMLoiV8WcpD-MjT-C\/preview\" width=\"640\"><\/iframe><br \/><a class=\"migration-iframe-fallback\" href=\"https:\/\/drive.google.com\/file\/d\/1yF6c5J64kU2yrkbMMLoiV8WcpD-MjT-C\/view\" rel=\"nofollow noopener\" target=\"_blank\">M\u1edf t\u00e0i li\u1ec7u n\u1ebfu khung xem tr\u01b0\u1edbc kh\u00f4ng hi\u1ec3n th\u1ecb<\/a><\/p>\n<p><strong><a class=\"custom-button\" href=\"https:\/\/drive.google.com\/file\/d\/1yF6c5J64kU2yrkbMMLoiV8WcpD-MjT-C\/view\" rel=\"nofollow noopener\" target=\"_blank\">T\u1ea3i \u0110\u1ec1 thi th\u1eed t\u1ed1t nghi\u1ec7p THPT m\u00f4n To\u00e1n 2024 \u2013 2025 tr\u01b0\u1eddng THPT Tr\u1ef1c Ninh Nam \u0110\u1ecbnh <\/a><\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0110\u1ec1 thi th\u1eed t\u1ed1t nghi\u1ec7p THPT m\u00f4n To\u00e1n 2024 \u2013 2025 tr\u01b0\u1eddng THPT Tr\u1ef1c Ninh Nam \u0110\u1ecbnh \u0111\u01b0\u1ee3c bi\u00ean so\u1ea1n d\u00e0nh cho k\u00ec thi th\u1eed t\u1ed1t nghi\u1ec7p THPT l\u1ea7n 1. T\u00e0i li\u1ec7u n\u00e0y gi\u00fap c\u00e1c b\u1ea1n h\u1ecdc sinh 12 l\u00e0m quen v\u1edbi c\u1ea5u tr\u00fac ra \u0111\u1ec1 v\u00e0 \u0111\u00e1nh gi\u00e1 n\u0103ng l\u1ef1c \u0111\u1ec3 c\u00f3 ti\u1ebfn tr\u00ecnh [&#8230;]\n","protected":false},"author":1,"featured_media":1747,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[8],"class_list":["post-548","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-de-thi-thu-thpt-mon-toan","tag-de-thi-thu-thpt-mon-toan"],"_links":{"self":[{"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/posts\/548","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/comments?post=548"}],"version-history":[{"count":1,"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/posts\/548\/revisions"}],"predecessor-version":[{"id":1748,"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/posts\/548\/revisions\/1748"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/media\/1747"}],"wp:attachment":[{"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/media?parent=548"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/categories?post=548"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/tags?post=548"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}