{"id":408,"date":"2026-05-05T04:44:13","date_gmt":"2026-05-05T04:44:13","guid":{"rendered":"https:\/\/79.buffdemo.com\/index.php\/2026\/05\/05\/08-de-on-tap-kiem-tra-giua-hoc-ky-2-mon-toan-11-nam-2024-2025\/"},"modified":"2026-05-05T07:16:57","modified_gmt":"2026-05-05T07:16:57","slug":"08-de-on-tap-kiem-tra-giua-hoc-ky-2-mon-toan-11-nam-2024-2025","status":"publish","type":"post","link":"https:\/\/79.buffdemo.com\/index.php\/2026\/05\/05\/08-de-on-tap-kiem-tra-giua-hoc-ky-2-mon-toan-11-nam-2024-2025\/","title":{"rendered":"08 \u0111\u1ec1 \u00f4n t\u1eadp ki\u1ec3m tra gi\u1eefa h\u1ecdc k\u1ef3 2 m\u00f4n To\u00e1n 11 n\u0103m 2024 \u2013 2025"},"content":{"rendered":"<p><strong>08 \u0111\u1ec1 \u00f4n t\u1eadp ki\u1ec3m tra gi\u1eefa h\u1ecdc k\u1ef3 2 m\u00f4n To\u00e1n 11 n\u0103m 2024 \u2013 2025<\/strong> l\u00e0 t\u00e0i li\u1ec7u quan tr\u1ecdng gi\u00fap h\u1ecdc sinh h\u1ec7 th\u1ed1ng l\u1ea1i ki\u1ebfn th\u1ee9c v\u1ec1 x\u00e1c su\u1ea5t, c\u1ea5p s\u1ed1 c\u1ed9ng \u2013 c\u1ea5p s\u1ed1 nh\u00e2n, logarit v\u00e0 h\u00ecnh h\u1ecdc kh\u00f4ng gian. V\u1edbi h\u1ec7 th\u1ed1ng c\u00e2u h\u1ecfi \u0111a d\u1ea1ng, b\u1ed9 \u0111\u1ec1 n\u00e0y s\u1ebd gi\u00fap c\u00e1c b\u1ea1n l\u1edbp 11 r\u00e8n luy\u1ec7n k\u1ef9 n\u0103ng l\u00e0m b\u00e0i v\u00e0 t\u1ef1 tin \u0111\u1ea1t \u0111i\u1ec3m cao trong k\u1ef3 ki\u1ec3m tra s\u1eafp t\u1edbi! H\u00e3y c\u00f9ng <a href=\"https:\/\/tailieuonthi.edu.vn\/\" rel=\"noopener\" target=\"_blank\"><strong>T\u00e0i Li\u1ec7u Mi\u1ec5n Ph\u00ed<\/strong><\/a> kh\u00e1m ph\u00e1 ngay n\u1ed9i dung chi ti\u1ebft c\u1ee7a b\u1ed9 \u0111\u1ec1 n\u00e0y!<\/p>\n<h2><strong>N\u1ed9i dung <span data-sheets-root=\"1\">08 \u0111\u1ec1 \u00f4n t\u1eadp ki\u1ec3m tra gi\u1eefa h\u1ecdc k\u1ef3 2 m\u00f4n To\u00e1n 11 n\u0103m 2024 \u2013 2025<\/span><\/strong><\/h2>\n<p><strong>08 \u0111\u1ec1 \u00f4n t\u1eadp ki\u1ec3m tra gi\u1eefa h\u1ecdc k\u1ef3 2 m\u00f4n To\u00e1n 11 n\u0103m 2024 \u2013 2025<\/strong> c\u1ee7a tr\u01b0\u1eddng THPT Nho Quan A b\u00e1m s\u00e1t ch\u01b0\u01a1ng tr\u00ecnh h\u1ecdc v\u1edbi h\u1ec7 th\u1ed1ng b\u00e0i t\u1eadp \u0111a d\u1ea1ng. B\u1ed9 \u0111\u1ec1 g\u1ed3m 08 \u0111\u1ec1 thi, m\u1ed7i \u0111\u1ec1 c\u00f3 3 ph\u1ea7n ch\u00ednh, bao g\u1ed3m 12 c\u00e2u tr\u1eafc nghi\u1ec7m nhi\u1ec1u ph\u01b0\u01a1ng \u00e1n l\u1ef1a ch\u1ecdn, 04 c\u00e2u tr\u1eafc nghi\u1ec7m \u0111\u00fang sai v\u00e0 06 c\u00e2u tr\u1eafc nghi\u1ec7m tr\u1ea3 l\u1eddi ng\u1eafn. Th\u1eddi gian l\u00e0m b\u00e0i l\u00e0 90 ph\u00fat, bao g\u1ed3m c\u00e1c b\u00e0i t\u1eadp tr\u1ecdng t\u00e2m nh\u01b0 h\u00ecnh h\u1ecdc kh\u00f4ng gian, logarit, c\u1ea5p s\u1ed1 nh\u00e2n, x\u00e1c su\u1ea5t, l\u00e3i su\u1ea5t k\u00e9p, t\u0103ng tr\u01b0\u1edfng d\u00e2n s\u1ed1 v\u00e0 nhi\u1ec1u ch\u1ee7 \u0111\u1ec1 kh\u00e1c. D\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t s\u1ed1 c\u00e2u h\u1ecfi n\u1ed5i b\u1eadt trong \u0111\u1ec1 thi gi\u1eefa k\u00ec 2 to\u00e1n 11:<\/p>\n<p><strong>PH\u1ea6N I: C\u00c2U TR\u1eaeC NGHI\u1ec6M NHI\u1ec0U PH\u01af\u01a0NG \u00c1N L\u1ef0A CH\u1eccN<\/strong><\/p>\n<p><strong>C\u00e2u 1.<\/strong> M\u1ed9t h\u1ed9p \u0111\u1ef1ng 70 t\u1ea5m th\u1ebb, \u0111\u00e1nh s\u1ed1 t\u1eeb 1 \u0111\u1ebfn 70. R\u00fat ng\u1eabu nhi\u00ean m\u1ed9t t\u1ea5m th\u1ebb. K\u00ed hi\u1ec7u  a  l\u00e0 s\u1ed1 ghi tr\u00ean th\u1ebb. G\u1ecdi  A  l\u00e0 bi\u1ebfn c\u1ed1: \u201c a  l\u00e0 \u01b0\u1edbc c\u1ee7a 28\u201d,  B  l\u00e0 bi\u1ebfn c\u1ed1: \u201c a  l\u00e0 \u01b0\u1edbc c\u1ee7a 70\u201d. T\u00ednh s\u1ed1 ph\u1ea7n t\u1eed c\u1ee7a bi\u1ebfn c\u1ed1  A cap B .<\/p>\n<p>A. 4\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0B. 6\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0C. 8\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0D. 7<\/p>\n<p><strong>C\u00e2u 2.<\/strong> Cho h\u00ecnh ch\u00f3p  S.ABC  c\u00f3  SA perp (ABC) . X\u00e1c \u0111\u1ecbnh g\u00f3c gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng  SC  v\u00e0 m\u1eb7t ph\u1eb3ng  (ABC) .<\/p>\n<p>A.  ShatBC  \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 B.  ShatAC  \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 C.  ShatCA  \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 D.  ShatCB <\/p>\n<p><strong>PH\u1ea6N II: C\u00c2U TR\u1eaeC NGHI\u1ec6M \u0110\u00daNG SAI<\/strong><\/p>\n<p><strong>C\u00e2u 1.<\/strong> M\u1ed9t h\u1ed9p \u0111\u1ef1ng 9 t\u1ea5m th\u1ebb c\u00f9ng lo\u1ea1i \u0111\u01b0\u1ee3c \u0111\u00e1nh s\u1ed1 t\u1eeb 1 \u0111\u1ebfn 9. R\u00fat ng\u1eabu nhi\u00ean \u0111\u1ed3ng th\u1eddi hai t\u1ea5m th\u1ebb t\u1eeb h\u1ed9p. X\u00e9t c\u00e1c bi\u1ebfn c\u1ed1 sau:<\/p>\n<p>$A$: \u201cC\u1ea3 hai t\u1ea5m th\u1ebb \u0111\u1ec1u ghi s\u1ed1 ch\u1eb5n\u201d.<br \/>\n$B$: \u201cCh\u1ec9 c\u00f3 m\u1ed9t t\u1ea5m th\u1ebb ghi s\u1ed1 ch\u1eb5n\u201d.<br \/>\n$C$: \u201cT\u00edch hai s\u1ed1 ghi tr\u00ean hai t\u1ea5m th\u1ebb l\u00e0 m\u1ed9t s\u1ed1 ch\u1eb5n\u201d.<br \/>\n$D$: \u201cT\u1ed5ng hai s\u1ed1 ghi tr\u00ean hai t\u1ea5m th\u1ebb l\u00e0 m\u1ed9t s\u1ed1 ch\u1eb5n\u201d.<\/p>\n<p>X\u00e1c \u0111\u1ecbnh t\u00ednh \u0111\u00fang sai c\u1ee7a c\u00e1c m\u1ec7nh \u0111\u1ec1 sau:<\/p>\n<p>A. $B \\cap D = \\emptyset$.\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0B. $C = A \\cup B$.<br \/>\nC. $P(A \\cup B) = P(A) + P(B)$.\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0D. Bi\u1ebfn c\u1ed1 $A$ v\u00e0 $D$ \u0111\u1ed9c l\u1eadp.<\/p>\n<p><strong>C\u00e2u 2.<\/strong> L\u1edbp 11A c\u00f3 40 h\u1ecdc sinh, trong \u0111\u00f3 c\u00f3 22 h\u1ecdc sinh nam v\u00e0 18 h\u1ecdc sinh n\u1eef. Gi\u00e1o vi\u00ean l\u1ea7n l\u01b0\u1ee3t ch\u1ecdn 2 h\u1ecdc sinh l\u00ean b\u1ea3ng \u0111\u1ec3 gi\u1ea3i b\u00e0i t\u1eadp. G\u1ecdi $A$ l\u00e0 bi\u1ebfn c\u1ed1 \u201cL\u1ea7n th\u1ee9 nh\u1ea5t ch\u1ecdn \u0111\u01b0\u1ee3c h\u1ecdc sinh nam\u201d, $B$ l\u00e0 bi\u1ebfn c\u1ed1 \u201cL\u1ea7n th\u1ee9 hai ch\u1ecdn \u0111\u01b0\u1ee3c h\u1ecdc sinh nam\u201d.<\/p>\n<p>a) Bi\u1ebfn c\u1ed1 \u201c\u00cdt nh\u1ea5t 1 l\u1ea7n ch\u1ecdn \u0111\u01b0\u1ee3c h\u1ecdc sinh n\u1eef\u201d \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n l\u00e0 $\\bar{A} \\cup \\bar{B}$.<br \/>\nb) Bi\u1ebfn c\u1ed1 \u201cC\u00f3 kh\u00f4ng qu\u00e1 1 l\u1ea7n ch\u1ecdn \u0111\u01b0\u1ee3c h\u1ecdc sinh n\u1eef\u201d \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n l\u00e0: $\\bar{A}B \\cup A\\bar{B}$.<br \/>\nc) $n(A B \\cup \\bar{A} B) = 768$.<br \/>\nd) $n(A \\cup B) = 462$.<\/p>\n<p><strong>PH\u1ea6N III: C\u00c2U TR\u1eaeC NGHI\u1ec6M TR\u1ea2 L\u1edcI NG\u1eaeN<\/strong><\/p>\n<p><strong>C\u00e2u 1.<\/strong> Cho h\u00ecnh ch\u00f3p  S.ABC  c\u00f3  angle ASB = 120^\u00b0 ,  angle BSC = 60^\u00b0 ,  angle ACB = 90^\u00b0  v\u00e0  SA = SB = SC . G\u1ecdi  I  l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a  S  l\u00ean m\u1eb7t ph\u1eb3ng  (ABC) . T\u1ec9 s\u1ed1  AI\/BI  l\u00e0<\/p>\n<p><strong>C\u00e2u 2.<\/strong> Ch\u1ecb Hoa m\u1edbi \u0111\u01b0\u1ee3c nh\u1eadn v\u00e0o m\u1ed9t T\u1eadp \u0111o\u00e0n v\u00e0 nh\u1eadn l\u01b0\u01a1ng kh\u1edfi \u0111i\u1ec3m l\u00e0 9 tri\u1ec7u \u0111\u1ed3ng\/th\u00e1ng. Bi\u1ebft r\u1eb1ng c\u1ee9 sau ba n\u0103m l\u01b0\u01a1ng m\u1ed7i th\u00e1ng c\u1ee7a ch\u1ecb t\u0103ng th\u00eam 15% so v\u1edbi m\u1ee9c l\u01b0\u01a1ng hi\u1ec7n t\u1ea1i. H\u1ecfi sau 8 n\u0103m l\u00e0m vi\u1ec7c t\u1ed5ng ti\u1ec1n l\u01b0\u01a1ng ch\u1ecb Hoa nh\u1eadn \u0111\u01b0\u1ee3c l\u00e0 bao nhi\u00eau tri\u1ec7u \u0111\u1ed3ng (<em>l\u00e0m tr\u00f2n k\u1ebft qu\u1ea3 \u0111\u1ebfn h\u00e0ng \u0111\u01a1n v\u1ecb<\/em>)?<\/p>\n<h2><strong>T\u1ea3i <span data-sheets-root=\"1\">08 \u0111\u1ec1 \u00f4n t\u1eadp ki\u1ec3m tra gi\u1eefa h\u1ecdc k\u1ef3 2 m\u00f4n To\u00e1n 11 n\u0103m 2024 \u2013 2025<\/span><\/strong><\/h2>\n<p>B\u1ea1n \u0111ang t\u00ecm ki\u1ebfm t\u00e0i li\u1ec7u \u00f4n t\u1eadp ch\u1ea5t l\u01b0\u1ee3ng \u0111\u1ec3 chu\u1ea9n b\u1ecb cho k\u1ef3 ki\u1ec3m tra gi\u1eefa k\u1ef3 2 m\u00f4n To\u00e1n 11? B\u1ed9 <strong>08 \u0111\u1ec1 \u00f4n t\u1eadp ki\u1ec3m tra gi\u1eefa h\u1ecdc k\u1ef3 2 m\u00f4n To\u00e1n 11 n\u0103m 2024 \u2013 2025<\/strong> s\u1ebd gi\u00fap b\u1ea1n n\u1eafm v\u1eefng ki\u1ebfn th\u1ee9c, r\u00e8n luy\u1ec7n t\u01b0 duy logic v\u00e0 l\u00e0m quen v\u1edbi c\u1ea5u tr\u00fac \u0111\u1ec1 thi m\u1ed9t c\u00e1ch hi\u1ec7u qu\u1ea3. V\u1edbi h\u1ec7 th\u1ed1ng c\u00e2u h\u1ecfi \u0111a d\u1ea1ng t\u1eeb c\u01a1 b\u1ea3n \u0111\u1ebfn n\u00e2ng cao, \u0111\u00e2y ch\u00ednh l\u00e0 ch\u00eca kh\u00f3a gi\u00fap b\u1ea1n chinh ph\u1ee5c \u0111i\u1ec3m 9+ trong k\u1ef3 thi s\u1eafp t\u1edbi. \u0110\u1ec3 kh\u00f4ng b\u1ecf l\u1ee1 c\u01a1 h\u1ed9i \u00f4n luy\u1ec7n v\u1edbi nh\u1eefng \u0111\u1ec1 thi ch\u1ea5t l\u01b0\u1ee3ng, h\u00e3y t\u1ea3i ngay qua \u0111\u01b0\u1eddng link d\u01b0\u1edbi \u0111\u00e2y.<\/p>\n<p><iframe height=\"480\" loading=\"lazy\" src=\"https:\/\/drive.google.com\/file\/d\/1o3znQpN7KyvK8SdjIHU1GKno5EF02BMk\/preview\" width=\"640\"><\/iframe><br \/><a class=\"migration-iframe-fallback\" href=\"https:\/\/drive.google.com\/file\/d\/1o3znQpN7KyvK8SdjIHU1GKno5EF02BMk\/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><a class=\"custom-button\" href=\"https:\/\/drive.google.com\/file\/d\/1o3znQpN7KyvK8SdjIHU1GKno5EF02BMk\/view\" rel=\"noopener nofollow\" target=\"_blank\">T\u1ea3i 08 \u0111\u1ec1 \u00f4n t\u1eadp ki\u1ec3m tra gi\u1eefa h\u1ecdc k\u1ef3 2 m\u00f4n To\u00e1n 11 n\u0103m 2024 \u2013 2025<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>08 \u0111\u1ec1 \u00f4n t\u1eadp ki\u1ec3m tra gi\u1eefa h\u1ecdc k\u1ef3 2 m\u00f4n To\u00e1n 11 n\u0103m 2024 \u2013 2025 l\u00e0 t\u00e0i li\u1ec7u quan tr\u1ecdng gi\u00fap h\u1ecdc sinh h\u1ec7 th\u1ed1ng l\u1ea1i ki\u1ebfn th\u1ee9c v\u1ec1 x\u00e1c su\u1ea5t, c\u1ea5p s\u1ed1 c\u1ed9ng \u2013 c\u1ea5p s\u1ed1 nh\u00e2n, logarit v\u00e0 h\u00ecnh h\u1ecdc kh\u00f4ng gian. V\u1edbi h\u1ec7 th\u1ed1ng c\u00e2u h\u1ecfi \u0111a d\u1ea1ng, b\u1ed9 \u0111\u1ec1 [&#8230;]\n","protected":false},"author":1,"featured_media":1607,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[60],"tags":[74,61],"class_list":["post-408","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-tong-hop","tag-thay-dang-viet-dong","tag-tong-hop"],"_links":{"self":[{"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/posts\/408","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=408"}],"version-history":[{"count":1,"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/posts\/408\/revisions"}],"predecessor-version":[{"id":1608,"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/posts\/408\/revisions\/1608"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/media\/1607"}],"wp:attachment":[{"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/media?parent=408"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/categories?post=408"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/79.buffdemo.com\/index.php\/wp-json\/wp\/v2\/tags?post=408"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}