["3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n \u0104 C\u00e2u 42 (C\u00e2u 15 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho kh\u1ed1i l\u0103ng tr\u1ee5 c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y B = 2a2 v\u00e0 chi\u1ec1u cao h = a. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng B a3. C 1 a3. D 2a3. A 2 a3. 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 l\u00e0 V = Bh = 2a2 \u00b7 a = 2a3. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 43 (C\u00e2u 26 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho kh\u1ed1i ch\u00f3p c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y B v\u00e0 chi\u1ec1u cao h. Th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i ch\u00f3p \u0111\u00e3 cho \u0111\u01b0\u1ee3c t\u00ednh b\u1edfi c\u00f4ng th\u1ee9c n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A V = 4 B V = Bh. C V = 1 D V = 3Bh. Bh. Bh. 3 3 \u0253 L\u1eddi gi\u1ea3i. 1 Th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i ch\u00f3p c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y B v\u00e0 chi\u1ec1u cao h \u0111\u01b0\u1ee3c t\u00ednh b\u1edfi c\u00f4ng th\u1ee9c V = Bh. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 44 (C\u00e2u 10 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho kh\u1ed1i l\u0103ng tr\u1ee5 c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y B = 4a2 v\u00e0 chi\u1ec1u cao h = a. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng B 4a3. C 4 a3. D 2a3. A 2 a3. 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 l\u00e0 V = Bh = 4a2 \u00b7 a = 4a3. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 45 (C\u00e2u 11 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho kh\u1ed1i ch\u00f3p c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y B v\u00e0 chi\u1ec1u cao h. Th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i ch\u00f3p \u0111\u00e3 cho \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A V = 3Bh. B 4 C V = Bh. D V = 1 V = Bh. Bh. 3 3 \u0253 L\u1eddi gi\u1ea3i. 1 Th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i ch\u00f3p c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y B v\u00e0 chi\u1ec1u cao h l\u00e0 V = Bh. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 46 (C\u00e2u 14 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i ch\u00f3p S.ABC c\u00f3 chi\u1ec1u cao b\u1eb1ng 3, \u0111\u00e1y ABC c\u00f3 di\u1ec7n t\u00edch b\u1eb1ng 10. Th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p S.ABC b\u1eb1ng A 15. B 10. C 2. D 30. VS.ABC = 1 = 1 \u00b7 3.10 = 10. \u0253 L\u1eddi gi\u1ea3i. hB 3 398 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N \u0104 C\u00e2u 47 (C\u00e2u 13 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i ch\u00f3p S.ABC c\u00f3 chi\u1ec1u cao b\u1eb1ng 5, \u0111\u00e1y ABC c\u00f3 di\u1ec7n t\u00edch b\u1eb1ng 6. Th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p S.ABC b\u1eb1ng A 11. B 10. C 15. D 30. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 VS.ABC = 1 \u00b7 S.h = 1 \u00b7 6.5 = 10. 3 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 48 (C\u00e2u 5 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i ch\u00f3p S.ABC c\u00f3 chi\u1ec1u cao b\u1eb1ng 5, \u0111\u00e1y ABC c\u00f3 di\u1ec7n t\u00edch b\u1eb1ng 6. Th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p S.ABC b\u1eb1ng A 30. B 10. C 15. D 11. \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p S.ABC l\u00e0 VS.ABC = 1 \u00b7 5.6 = 10. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 49 (C\u00e2u 8 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i ch\u00f3p c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y B = 7 v\u00e0 chi\u1ec1u cao h = 6. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i ch\u00f3p \u0111\u00e3 cho b\u1eb1ng A 42. B 126. C 14. D 56. \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i ch\u00f3p V = 1 1 \u00b7 6 \u00b7 7 = 14. hB = 33 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 50 (C\u00e2u 21 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i l\u0103ng tr\u1ee5 c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y B v\u00e0 chi\u1ec1u cao h. Th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A V = 1 B 4 C V = 6Bh. D V = Bh. Bh. V = Bh. 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y B v\u00e0 chi\u1ec1u cao h l\u00e0 V = B \u00b7 h. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 51 (C\u00e2u 18 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho kh\u1ed1\u221ai l\u0103ng tr\u1ee5 \u0111\u1ee9ng ABC.A B C c\u00f3 BB = a, \u0111\u00e1y ABC l\u00e0 tam gi\u00e1c vu\u00f4ng c\u00e2n t\u1ea1i B v\u00e0 AC = a 2. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho. A V = a3. B V a3 C V = a3 D V a3 =. . =. 3 6 2 \u0253 L\u1eddi gi\u1ea3i. \u221a Tam gi\u00e1c ABC vu\u00f4ng c\u00e2n t\u1ea1i B v\u00e0 AC = a 2 do \u0111\u00f3 AB = BC = a. Th\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 l\u00e0 V 1 a3 = BB .SABC = a. .a.a = . 2 2 Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 399 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n \u0104 C\u00e2u 52 (C\u00e2u 33 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). Cho t\u1ee9 di\u1ec7n \u0111\u1ec1u ABCD c\u00f3 c\u1ea1nh b\u1eb1ng 4. T\u00ednh di\u1ec7n t\u00edch xung quanh Sxq c\u1ee7a h\u00ecnh tr\u1ee5 c\u00f3 m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y l\u00e0 \u0111\u01b0\u1eddng tr\u00f2n n\u1ed9i ti\u1ebfp tam gi\u00e1c BCD v\u00e0 chi\u1ec1u cao b\u1eb1ng chi\u1ec1u cao c\u1ee7a t\u1ee9 di\u1ec7n AB C D. \u221a \u221a \u221a \u221a B Sxq = 8 2\u03c0. D Sxq = 8 3\u03c0. A Sxq = 16 2\u03c0 C Sxq = 16 3\u03c0 . . 3 3 \u0253 L\u1eddi gi\u1ea3i. \u221a \u221a 2 3 chi\u1ec1u cao t\u1ee9 di\u1ec7n l\u00e0 h = 4\u221a 2 Ta t\u00ednh \u0111\u01b0\u1ee3c b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n n\u1ed9i ti\u1ebfp tam gi\u00e1c BCD l\u00e0 r = , . \u221a 33 16 2\u03c0 T\u1eeb \u0111\u00f3 Sxq = 2\u03c0 \u00b7 r \u00b7 h = . 3 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 53 (C\u00e2u 35 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). \u221a T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i l\u1eadp ph\u01b0\u01a1ng\u221aABCD.A B C D , bi\u1ebft AC = a 3. 3 6a3 \u221a D V = 1 a3. A V = a3. B V= . C V = 3 3a3. 43 \u0253 L\u1eddi gi\u1ea3i. \u221a Kh\u1ed1i l\u1eadp ph\u01b0\u01a1ng ABCD.A B C D c\u00f3 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng ch\u00e9o AC = a 3 n\u00ean c\u00f3 \u0111\u1ed9 d\u00e0i c\u1ea1nh l\u00e0 a. V\u1eady th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i l\u1eadp ph\u01b0\u01a1ng ABCD.A B C D l\u00e0 V = a3. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 54 (C\u00e2u 36 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00ecnh ch\u00f3p t\u1ee9 gi\u00e1c S.AB\u221aCD c\u00f3 \u0111\u00e1y ABCD l\u00e0 h\u00ecnh vu\u00f4ng c\u1ea1nh a, c\u1ea1nh b\u00ean SA vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3\u221ang \u0111\u00e1y v\u00e0 SA = 2a. T\u221a\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i ch\u00f3p S.ABCD. 2a3 2a3 \u221a \u221a2a3 . . C V = 2a3. . AV= B V= D V= 64 3 \u221a \u0253\u221aL\u1eddi gi\u1ea3i. 1 1 a2 a3 2 Ta c\u00f3: V = 3 SABCD \u00d7 SA = a2 \u00d7 = 3 . 3 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 55 (C\u00e2u 16 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ednh th\u1ec3 t\u00edch \u221aV c\u1ee7a kh\u1ed1i l\u1eb7ng tr\u1ee5 tam\u221agi\u00e1c \u0111\u1ec1u c\u00f3 t\u1ea5t c\u1ea3 c\u00e1c \u221ac\u1ea1nh b\u1eb1ng a. \u221a A V = a3 3 a3 3 C V = a3 3 a3 3 . B V = . . D V = 4 . 6 12 2 \u221a\u221a \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3: V = B \u00b7 h = a \u00b7 a2 3 a3 3 =. 44 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 56 (C\u00e2u 36 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00ecnh ch\u00f3p S.ABCD c\u00f3 \u0111\u00e1y l\u00e0 h\u00ecnh vu\u00f4ng c\u1ea1nh a, SA vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t \u0111\u00e1y, SD t\u1ea1o v\u1edbi m\u1eb7t ph\u1eb3ng (\u221aSAB) m\u1ed9t g\u00f3c b\u1eb1ng 30\u25e6. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a k\u221ah\u1ed1i ch\u00f3p S.ABCD. \u221a 6a3 \u221a 6a3 3a3 AV= . B V = 3a3. CV= . D V= . 18 3 3 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 400 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N S A B O DC \u221a G\u00f3c gi\u1eefa SD v\u00e0 mp (SAB) l\u00e0 A\u2019\u221aSD = 30\u25e6 \u21d2 SA = a \u00b7 cot 30\u25e6 = 3a. 1 1 \u221a 3 a3. Khi \u0111\u00f3 V = Bh = a2a 3 = 33 3 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 57 (C\u00e2u 19 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). Cho kh\u1ed1\u221ai ch\u00f3p t\u1ee9 gi\u00e1c \u0111\u1ec1u c\u00f3 t\u1ea5t c\u1ea3 c\u00e1c c\u1ea1nh b\u1eb1ng 2a. T\u221ah\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i ch\u00f3p \u0111\u221a\u00e3 cho b\u1eb1ng 4 2a3 8a3 C8 2a3 2 2a3 A . B . . D . 33 33 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi O l\u00e0 giao \u0111i\u1ec3m hai \u0111\u01b0\u1eddng ch\u00e9o. Do ch\u00f3p t\u1ee9 gi\u00e1c S \u0111\u1ec1u n\u00ean SO \u22a5 (ABCD). \u221a T\u221aa c\u00f3 AC = \u221aBD = 2a \u221a2. Suy ra SO = SD2 \u2212 OD2 = 4a2 \u2212 2a2 = 2a. \u221a Th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p l\u00e0 V = 1 4a2 \u00b7 \u221a = 4 2a3 a2 . 33 AD B O C Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 58 (C\u00e2u 22 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho kh\u1ed1i \u221al\u0103ng tr\u1ee5 \u0111\u1ee9ng ABC.A B C c\u00f3 \u0111\u00e1y l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh a A C v\u00e0 AA = 3a (minh h\u1ecda h\u00ecnh v\u1ebd b\u00ean). Th\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho A C b\u1eb1ng B A 3a3 B 3a3 C a3 D a3 . . . . 4 2 4 2 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 401 B S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n \u221a \u0253 L\u1eddi gi\u1ea3i. a2 3 \u221a Ta c\u00f3 SABC = ; AA = a 3. 4\u221a \u221a 3 3a3 T\u1eeb \u0111\u00f3 suy ra V = a 3 \u00b7 a2 =. 44 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 59 (C\u00e2u 21 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u1ee9ng ABC.A B C c\u00f3 \u0111\u00e1y l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh A C A C a v\u00e0 AA = 2a (minh h\u1ecda nh\u01b0 h\u00ecnh v\u1ebd b\u00ean). Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng B tr\u1ee5 \u0111\u00e3 \u221acho b\u1eb1ng \u221a 3a3 3a3 . A . B 3 \u221a6 \u221a 3a3 C 3a3. D . 2 B \u0253 L\u1eddi gi\u1ea3i. \u221a a2 3 Tam gi\u00e1c ABC \u0111\u1ec1u c\u1ea1nh a n\u00ean S ABC = . 4 Do kh\u1ed1i l\u0103ng tr\u1ee5 ABC.A B C l\u00e0 l\u0103ng tr\u1ee5 \u0111\u1ee9ng n\u00ean \u221a\u0111\u01b0\u1eddng \u221acao c\u1ee7a l\u0103ng tr\u1ee5 l\u00e0 AA = 2a a2 3 3a3 Th\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 l\u00e0 V = AA \u00b7 S = 2a \u00b7 4 = . ABC 2 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 60 (C\u00e2u 25 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u1ee9ng ABC.A B C c\u00f3 \u0111\u00e1y l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh 2a v\u00e0 A C C AA = 3a (minh h\u1ecda nh\u01b0 h\u00ecnh v\u1ebd b\u00ean). Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng \u221a \u221a \u221a \u221a B A 2 3a3. B 3a3. C 6 3a3. D 3 3a3. A B \u0253 L\u1eddi gi\u1ea3i. \u221a Kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho c\u00f3 \u0111\u00e1y l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y l\u00e0 (2a)2 3 v\u00e0 chi\u1ec1u cao l\u00e0 AA = 3a \u221a4 (2a)2 3 \u221a (do l\u00e0 l\u0103ng tr\u1ee5 \u0111\u1ee9ng) n\u00ean c\u00f3 th\u1ec3 t\u00edch l\u00e0 \u00b7 3a = 3 3a3. 4 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 61 (C\u00e2u 26 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). \u221a Cho h\u00ecnh l\u0103ng tr\u1ee5 \u0111\u1ee9ng ABCD.A B C D c\u00f3 \u0111\u00e1y l\u00e0 h\u00ecnh thoi c\u1ea1nh a, BD = a 3, AA = 4a. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng \u221a \u221a \u221a \u221a C2 3a3 D 4 3a3 A 2 3a3. B 4 3a3. . . 3 3 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 402 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N \u0253 L\u1eddi gi\u1ea3i. \u00c1p d\u1ee5ng \u0111\u1ecbnh l\u00ed h\u00e0m s\u1ed1 cosin cho tam gi\u00e1c ta c\u00f3: cos B\u2019AD = AB2 + AD2 \u2212 BD2 = \u22121 \u21d2 B\u2019AD = 120o 2AB.AD \u221a2 a2 \u21d2 SABCD = AB.AD. sin B\u2019AD = 2 3 \u221a \u221a a2 3 2 3a3. \u21d2 VABCD.A B C D = AA .SABCD = 4a. = 2 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 62 (C\u00e2u 23 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u1eadp ph\u01b0\u01a1ng c\u1ea1nh 3a b\u1eb1ng A 27a3. B 3a3. C 9a3. D a3. \u0253 L\u1eddi gi\u1ea3i. Kh\u1ed1i l\u1eadp ph\u01b0\u01a1ng c\u1ea1nh 3a c\u00f3 th\u1ec3 t\u00edch l\u00e0 V = (3a)3 = 27a3. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 63 (C\u00e2u 45 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho kh\u1ed1i h\u1ed9p ch\u1eef nh\u1eadt ABCD.A B C D c\u00f3 \u0111\u00e1y l\u00e0 h\u00ecnh vu\u00f4ng, BD = 2a, g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (A BD)\u221av\u00e0 (ABCD) b\u1eb1ng 60\u25e6. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i h\u1ed9p ch\u221a\u1eef nh\u1eadt \u0111\u00e3 cho b\u1eb1ng A 2 3 a3. \u221a C 2 3 a3. \u221a B 6 3a3. D 2 3a3. 93 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi O l\u00e0 t\u00e2m c\u1ee7a \u0111\u00e1y, ta c\u00f3 A\u2019O\u221aA = 60\u25e6. A D Suy ra A A = AO \u00b7 tan 60\u25e6 = a 3. D Th\u1ec3 t\u00edch kh\u1ed1i h\u1ed9p l\u00e0 V = A A \u00b7 AB2 = \u221a \u00b7 \u00c4 \u221a \u00e42 \u221a B C a3 a2 = 2a3 3. A O B C Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 64 (C\u00e2u 43 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho kh\u1ed1i l\u0103ng tr\u1ee5 tam gi\u00e1c \u0111\u1ec1u ABC.A B C c\u00f3 c\u1ea1nh b\u00ean b\u1eb1ng 2a, g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (A BC)\u221av\u00e0 (ABC) b\u1eb1ng 30\u25e6. Th\u221a\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 \u221acho b\u1eb1ng \u221a A 8 3 a3. B 8 3 a3. C 8 3 a3. D 8 3a3. 9 3 27 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 403 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m BC th\u00ec AM \u22a5 BC (v\u00ec ABC \u0111\u1ec1u). (1) A C (2) B Ta c\u00f3 \u00aeBC \u22a5 AM \u21d2 BC \u22a5 (A M A) \u21d2 BC \u22a5 A M . (3) C BC \u22a5 AA A M B M\u1eb7t kh\u00e1c (A BC) \u2229 (ABC) = BC. T\u1eeb (1), (2) v\u00e0 (3) suy ra ((A BC), (ABC)) = A\u00f7M A = 30\u25e6. X\u00e9t A AM vu\u00f4ng t\u1ea1i A ta c\u00f3 AA \u21d2 AM = AA = \u221a2a \u221a tan A\u00f7M A = = 2a 3. AM tan A\u00f7M A 3 3 V\u00ec ABC \u0111\u1ec1u n\u00ean BC = 2\u221aAM = 4a. 3 1 1 \u221a \u221a Do \u0111\u00f3 S ABC = AM \u00b7 BC = 2 \u00b7 2a 3 \u00b7 4a = 4 3a2. \u221a 2 \u221a V\u1eady VABC.A B C = AA \u00b7 SABC = 2a \u00b7 4 3a2 = 8 3a3. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 65 (C\u00e2u 2 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i l\u0103ng tr\u1ee5 c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y 3a2 v\u00e0 chi\u1ec1u cao 2a. Th\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A a3. B 6a3. C 3a3. D 2a3. \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 l\u00e0 V = B.h = 3a2.2a = 6a3. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 66 (C\u00e2u 8 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i ch\u00f3p S.ABC c\u00f3 chi\u1ec1u cao b\u1eb1ng 3, \u0111\u00e1y ABC c\u00f3 di\u1ec7n t\u00edch b\u1eb1ng 10. Th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p S.ABC b\u1eb1ng A 2. B 15. C 10. D 30. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 VS.ABC = 1 = 1 \u00b7 10.3 = 10. B.h 3 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 67 (C\u00e2u 26 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i l\u0103ng tr\u1ee5 c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y 3a2 v\u00e0 chi\u1ec1u cao 2a. Th\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 3a3. B 6a3. C 2a3. D a3. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 th\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 l\u00e0 V = B.h = 3a2.2a = 6a3. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 68 (C\u00e2u 42 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u1ee9ng ABC.A B C c\u00f3 \u0111\u00e1y ABC l\u00e0 tam gi\u00e1c vu\u00f4ng c\u00e2n t\u1ea1i A, AB = a. G\u00f3c gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng BC v\u00e0 m\u1eb7t ph\u1eb3ng (ACC A ) b\u1eb1ng 30\u25e6. Th\u221a\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5\u221a\u0111\u00e3 cho b\u1eb1ng A 1 a3. B 3 a3. C 3 2 a3. D 2 a3. 8 8 2 2 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 404 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N Di\u1ec7n t\u00edch \u0111\u00e1y l\u00e0 SABC = 1 AB2 = a2 A C 2 \u00b7 B C \u00aeAB \u22a5 AC 2 A B Ta c\u00f3 AB \u22a5 AA \u21d2 AB \u22a5 (ACC A ). Suy ra BC , (ACC A ) = (B\u221aC , AC ) = B\u2019C A = 30\u25e6. Khi \u0111\u00f3 AC\u221a = AC. cot 30\u25e6 =a 3\u221a. \u00bb \u221a \u21d2 AA = AC 2 \u2212 A C 2 = (a 3)2 \u2212 a2 = a 2. V\u1eady th\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho l\u00e0 V = SABC .AA = a2 \u221a \u221a 2 \u00b7a 2= 2 \u00b7 a3. 2 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 69 (C\u00e2u 19 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i ch\u00f3p v\u00e0 kh\u1ed1i l\u0103ng tr\u1ee5 c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y, chi\u1ec1u cao t\u01b0\u01a1ng \u1ee9ng b\u1eb1ng nhau v\u00e0 c\u00f3 th\u1ec3 t\u00edch l\u1ea7n l\u01b0\u1ee3t l\u00e0 V1, V2. T\u1ec9 s\u1ed1 V1 b\u1eb1ng V2 A 2 . B 3. C 3 . D 1 . 3 2 3 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi di\u1ec7n t\u00edch \u0111\u00e1y v\u00e0 chi\u1ec1u cao t\u01b0\u01a1ng \u1ee9ng c\u1ee7a kh\u1ed1i ch\u00f3p v\u00e0 kh\u1ed1i l\u0103ng tr\u1ee5 l\u00e0 B v\u00e0 h. \uf8f11 \uf8f2V1 = Bh V1 1\u00b7 Ta c\u00f3 3 \u21d2 = V2 3 \uf8f3V2 = Bh Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 70 (C\u00e2u 6 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i ch\u00f3p v\u00e0 kh\u1ed1i l\u0103ng tr\u1ee5 c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y, chi\u1ec1u cao t\u01b0\u01a1ng \u1ee9ng b\u1eb1ng nhau v\u00e0 c\u00f3 th\u1ec3 t\u00edch l\u1ea7n l\u01b0\u1ee3t l\u00e0 V1, V2. T\u1ec9 s\u1ed1 V1 b\u1eb1ng V2 A 2 . B 3 . C 3. D 1 . 3 2 3 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi \u0111\u01b0\u1eddng cao, di\u1ec7n t\u00edch \u0111\u00e1y l\u1ea7n l\u01b0\u1ee3t l\u00e0 h, B. Khi \u0111\u00f3 \u00e1p d\u1ee5ng c\u00f4ng th\u1ee9c th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p, kh\u1ed1i l\u0103ng tr\u1ee5 ta \u0111\u01b0\u1ee3c V1 = 1 B .h v\u00e0 V2 = B.h. 3 1 Suy ra V1 = 3 B.h = 1 \u00b7 V2 3 B.h Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 71 (C\u00e2u 35 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00ecnh ch\u00f3p S.ABC c\u00f3 \u0111\u00e1y l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh 2a v\u00e0 th\u1ec3 t\u00edch b\u1eb1ng a3. T\u00ednh chi\u1ec1u cao h c\u1ee7a h\u00ecnh ch\u00f3p \u0111\u221a\u00e3 cho. \u221a \u221a \u221a 3a 3a 3a D h = 3a. A h= . B h= . C h= . 6 2 3 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 405 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n (2a)2 \u221a 3 \u221a Do \u0111\u00e1y l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh b\u1eb1ng 2a n\u00ean S ABC = 4 a2 3. = S 1 3V = 3a\u221a3 \u221a H M\u00e0 VS.ABC = S ABC .h \u21d2 h = S ABC = 3a. B 3 a2 3 A C Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 72 (C\u00e2u 32 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh ch\u00f3p S.ABC v\u00e0 c\u00f3 \u0111\u00e1y ABC l\u00e0 tam gi\u00e1c vu\u00f4ng t\u1ea1i S B, AB\u221a= a, BC = 3a; SA vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng \u0111\u00e1y v\u00e0 SA = 30a (tham kh\u1ea3o h\u00ecnh b\u00ean). G\u00f3c gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng SC v\u00e0 m\u1eb7t \u0111\u00e1y b\u1eb1ng AC A 45\u25e6. B 90\u25e6. C 60\u25e6. B D 30\u25e6. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 AC l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a SC l\u00ean mp (ABC) \u21d2 S\u00a4C, (ABC) = S\u2019CA. \u221a\u221a Tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i B c\u00f3 AC = AB2 + BC2 = a 10. \u221a Tam gi\u00e1c SAC vu\u00f4ng t\u1ea1i A c\u00f3 tan S\u2019CA = SA = 3 \u21d2 S\u2019C A = 60\u25e6. AC Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 73 (C\u00e2u 30 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). \u221aCho h\u00ecnh h\u1ed9p ch\u1eef nh\u1eadt ABCD.A B C D c\u00f3 AB = BC = a, AA = A D 6a (tham kh\u1ea3o h\u00ecnh b\u00ean). G\u00f3c gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng A C v\u00e0 m\u1eb7t C ph\u1eb3ng (ABCD) b\u1eb1ng B A 60\u25e6. B 90\u25e6. C 30\u25e6. D 45\u25e6. A D BC \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 406 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N Theo \u0111\u1ec1 b\u00e0i ta th\u1ea5y AC l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a A C l\u00ean m\u1eb7t ph\u1eb3ng A D (ABCD), suy ra C B [A C, (ABCD)] = [AC, A C] = A\u2019CA. X\u00e9t tam gi\u00e1c vu\u00f4ng A AC vu\u00f4ng t\u1ea1i A, ta c\u00f3 tan A\u2019CA = AA = \u221a AA = \u221a = \u221a A D a\u221a6 3, B C AC AB2 + BC2 a 2 Suy ra A\u2019CA = 60\u25e6. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 74 (C\u00e2u 21 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho kh\u1ed1i ch\u00f3p t\u1ee9 gi\u00e1c \u0111\u1ec1u c\u00f3 c\u1ea1nh \u0111\u00e1y b\u1eb1ng a, c\u1ea1nh b\u00ean g\u1ea5p hai l\u1ea7n c\u1ea1nh \u0111\u00e1y. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i ch\u00f3p\u221a\u0111\u00e3 cho. a3 \u221a \u221a \u221a a3 2 = 2 C V = a3 14 a3 14 A V = 2 . B V 6 . 2 . D V = 6 . \u0253 L\u1eddi gi\u1ea3i. C\u1ea1nh \u0111\u00e1y AB = a \u21d2 di\u1ec7n t\u00edch \u0111\u00e1y \u221aSABCD = a2. S \u0110\u01b0\u1eddng ch\u00e9o AC = \u221a HA a2 a 2\u21d2 =. 2\u221a \u221a a 14 C\u1ea1nh b\u00ean SA = 2AB = 2a \u21d2 SH = SA2 \u2212 H A2 = . \u221a\u221a 2 14 a3 B V\u1eady th\u1ec3 t\u00edch V = 1 .a2. a = 14 AH C 32 6 . D Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 75 (C\u00e2u 36 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). \u221a Cho kh\u1ed1i ch\u00f3p S.ABCD c\u00f3 \u0111\u00e1y l\u00e0 h\u00ecnh ch\u1eef nh\u1eadt, AB = a, AD = a 3, SA vu\u00f4ng g\u00f3c v\u1edbi \u0111\u00e1y v\u00e0 m\u1eb7t ph\u1eb3ng (SBC) t\u1ea1o v\u1edbi \u0111\u00e1y m\u1ed9t \u221ag\u00f3c 60\u25e6. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i ch\u00f3p S.ABCD. a3 3a3 A V =. B V= . C V = a3. D V = 3a3. 33 \u0253 L\u1eddi gi\u1ea3i. - T\u1eeb gi\u1ea3 thi\u1ebft ta c\u00f3 S\u2019BA = 60\u25e6 suy ra SH \u221a V\u1eady, V = 1 \u221a \u221a = a3. = AB. tan 60\u25e6 = a 3. .a 3.a2 3 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 76 (C\u00e2u 16 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho kh\u1ed1i ch\u00f3p S.ABC c\u00f3 SA vu\u00f4ng g\u00f3c v\u1edbi \u0111\u00e1y, SA = 4, AB = 6, BC = 10 v\u00e0 CA = 8. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i ch\u00f3p S.ABC. A V = 40. B V = 192. C V = 32. D V = 24. \u0253 L\u1eddi gi\u1ea3i. 1 N\u1eeda chu vi c\u1ee7a tam gi\u00e1c ABC l\u00e0 p = 12 \u21d2 S\u2206ABC = p(p \u2212 6)(p \u2212 10)(p \u2212 8) = 24 \u21d2 V = .24.4 = 3 32 Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 407 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n \u0104 C\u00e2u 77 (C\u00e2u 27 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho kh\u1ed1i ch\u00f3p tam gi\u00e1c \u0111\u1ec1u S.ABC c\u00f3 c\u1ea1nh \u0111\u00e1y b\u1eb1ng a v\u00e0 c\u1ea1nh b\u00ean b\u1eb1ng 2a. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i ch\u00f3\u221ap S.ABC. \u221a 11a3 \u221a 11a3 \u221a 13a3 . . 11a3 AV= . B V= CV= D V= . 12 12 6 4 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi H l\u00e0 tr\u1ecdng t\u00e2m tam gi\u00e1c ABC. S Khi \u0111\u00f3 SH l\u00e0 c\u221ahi\u1ec1u cao c\u1ee7a kh\u1ed1i ch\u00f3p. \u221a A a 3 \u221a 33 H Ta c\u00f3: CH = , SH = SC2 \u2212 CH2 = . 3\u221a \u221a \u221a 3 B 1 a2 3 33 11a3 C Do \u0111\u00f3 V = . . = 12 . 34 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 78 (C\u00e2u 37 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Cho t\u1ee9 di\u1ec7n ABCD c\u00f3 c\u00e1c c\u1ea1nh AB, AC v\u00e0 AD \u0111\u00f4i m\u1ed9t vu\u00f4ng g\u00f3c v\u1edbi nhau; AB = 6a, AC = 7a v\u00e0 AD = 4a. G\u1ecdi M, N, P t\u01b0\u01a1ng \u1ee9ng l\u00e0 trung \u0111i\u1ec3m c\u00e1c c\u1ea1nh BC, CD, DB. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a t\u1ee9 di\u1ec7n A.M N P . B V = 14a3. C V = 28 a3. D V = 7a3. A V = 7 a3. 3 2 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 VABCD = 1 \u00b7 AC \u00b7 AD = 1 \u00b7 6a \u00b7 7a \u00b7 4a = 28a3. D AB 6 N 1 1 6 2 SMNDP 4 SBCD D\u1ec5 th\u1ea5y SMNP = = \u21d2 VAM N P = 1 = 7a3. 4 VABCD PA C M B Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 79 (C\u00e2u 37 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). Cho t\u1ee9 di\u1ec7n ABCD c\u00f3 th\u1ec3 t\u00edch b\u1eb1ng 12 v\u00e0 G l\u00e0 tr\u1ecdng t\u00e2m tam gi\u00e1c BCD. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i ch\u00f3p A.GBC. A V = 3. B V = 4. C V = 6. D V = 5. \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 408 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N 11 Ta c\u00f3 d(G, BC) = d(A, BC) \u21d2 S = S ABC . 3 GBC 3 S VS.GBC = 1 GBC. d(S, (ABC)) .S 3 11 1 = . .S ABC. d(S, (ABC)) = 3 .VS.ABC = 4. 33 A C G B Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 80 (C\u00e2u 38 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). \u221a Cho l\u0103ng tr\u1ee5 tam gi\u00e1c ABC.A B C c\u00f3 \u0111\u00e1y ABC l\u00e0 tam gi\u00e1c vu\u00f4ng c\u00e2n t\u1ea1i A, c\u1ea1nh AC = 2 2. Bi\u1ebft AC t\u1ea1o v\u1edbi m\u1eb7t ph\u1eb3ng (ABC) m\u1ed9t g\u00f3c 60\u25e6 v\u00e0 AC = 4. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n ABCB C . \u221a \u221a A V 8 B 16 C V =8 3 D V 16 3 =. V= . . = . 3 3 3 3 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi H l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a C l\u00ean \u0111\u00e1y \u221a A B (ABC) \u21d2 C\u00f7AH = 60\u25e6 \u21d2 C H = AC sin 60\u25e6 = 2 3. TSA\u1eebB\u0111C\u00f3=ta12c(\u00f32\u221a2)2 = 4 \u221a C \u21d2 VABCB C = 1 16 3 = 2VAC 2C H SABC = . BC 3 3 A H BC Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 81 (C\u00e2u 45 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). Cho hai h\u00ecnh vu\u00f4ng ABCD v\u00e0 ABEF c\u00f3 c\u1ea1nh b\u1eb1ng 1, l\u1ea7n l\u01b0\u1ee3t n\u1eb1m tr\u00ean hai m\u1eb7t ph\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi nhau. G\u1ecdi S l\u00e0 \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9ng v\u1edbi B qua \u0111\u01b0\u1eddng th\u1eb3ng DE. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n ABCDSEF b\u1eb1ng A 7 B 11 C 2 D 5 . . . . 6 12 3 6 \u0253 L\u1eddi gi\u1ea3i. G\u1eafn h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 Oxyz, sao cho A(0; 0; 0) B(1; 0; 0), C(1; 0; 1), D(0; 0; 1), E(1; 1; 0), F (0; 1; 0). \u00c51 4 2\u00e3 Khi \u0111\u00f3 ta c\u00f3 S ; ; . Suy ra th\u1ec3 t\u00edch c\u1ea7n t\u00ednh 33 3 V = 14 12 2 = VS.ABCD + VS.ABEF . .1 + . .1 = . 33 33 3 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 409 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 82 (C\u00e2u 47 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho l\u0103ng tr\u1ee5 ABC.A B C c\u00f3 chi\u1ec1u cao b\u1eb1ng 8 v\u00e0 \u0111\u00e1y l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh b\u1eb1ng 6. G\u1ecdi M , N v\u00e0 P l\u1ea7n l\u01b0\u1ee3t l\u00e0 t\u00e2m c\u1ee7a c\u00e1c m\u1eb7t b\u00ean ABB A , ACC A v\u00e0 BCC B . Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n l\u1ed3i c\u00f3 c\u00e1c\u221a\u0111\u1ec9nh l\u00e0 c\u00e1c \u0111i\u1ec3m A, B,\u221aC, M , N , P b\u1eb1ng \u221a \u221a A 27 3. B 21 3. C 30 3. D 36 3. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi h l\u00e0 chi\u1ec1u cao c\u1ee7a h\u00ecnh l\u0103ng tr\u1ee5 ABC.A B C . A C H V\u00ec ABC \u0111\u1ec1u c\u00f3 \u0111\u1ed9 d\u00e0i c\u1ea1nh b\u1eb1ng 6 n\u00ean E M S ABC = 62 \u00b7 \u221a = \u221a F B 3 9 3. N 4 P Th\u1ec3 t\u00edch l\u1eb7ng tr\u1ee5 ABC.A B C l\u00e0 \u221a\u221a V = h \u00b7 S ABC = 8 \u00b7 9 3 = 72 3. G\u1ecdi E, F , H l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a c\u00e1c c\u1ea1nh AA , A C BB , CC . Th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p A.EM N l\u00e0 B 1 11 1 1 VA.E M N = d(A, (EM N )) \u00b7 S = \u00b7 h \u00b7 S = V. 3 EMN 3 2 4 ABC 24 1 T\u01b0\u01a1ng t\u1ef1, ta c\u00f3 VB.F MP = VC.H N P = V. 24 Th\u1ec3 t\u00edch kh\u1ed1i \u0111a di\u1ec7n ABCM N P l\u00e0 VABC M N P = 1 \u2212 3VA.EMN = 1 \u22123\u00b7 1 = 3 \u221a V V V V = 27 3. 2 2 24 8 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 83 (C\u00e2u 49 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho l\u0103ng tr\u1ee5 ABC.A B C c\u00f3 chi\u1ec1u cao b\u1eb1ng 8 v\u00e0 \u0111\u00e1y l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh b\u1eb1ng 4. G\u1ecdi M , N v\u00e0 P l\u1ea7n l\u01b0\u1ee3t l\u00e0 t\u00e2m c\u00e1c m\u1eb7t b\u00ean ABB A , ACC A v\u00e0 BCC B . Th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n l\u1ed3i c\u00f3 c\u00e1c \u0111\u1ec9nh l\u00e0 c\u00e1c \u0111i\u1ec3m A, B, C, M , N , P b\u1eb1ng \u221a \u221a \u221a \u221a C V = 28 3 3 D V 403 3 A V = 12 3. B V = 16 3. . = . \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 410 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N = 8 \u00b7 42 \u221a 3\u221a = 32 3. Ta c\u00f3 VABC.A B C 4 A C 1 B V\u00e0 ta c\u0169ng c\u00f3 VC .ABC = VA.BC B = 3 VABC.A B C . N Kh\u1ed1i \u0111a di\u1ec7n c\u1ea7n t\u00ecm V = VC.ABP N + VM.ANP B. M Do N , P l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AC v\u00e0 BC n\u00ean P 3 SANP B = 4 SABC T\u1eeb \u0111\u00f3 ta suy ra 31 A C VC.ABP N = 4 VC .ABC = 4 VABC.A B C . B 1 31 VM.ANP B = 2 VB ANP B = 8 VB .ABC = 8 VABC.A B C . V\u1eady th\u1ec3 t\u00edch kh\u1ed1i c\u1ea7n t\u00ecm 11 3 \u221a V = 4 VABC.A B C + 8 VABC.A B C = 8 VABC.A B C = 12 3. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 84 (C\u00e2u 48 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 20\u221a20). 3a Cho h\u00ecnh ch\u00f3p \u0111\u1ec1u S.ABCD c\u00f3 c\u1ea1nh \u0111\u00e1y b\u1eb1ng a, c\u1ea1nh b\u00ean b\u1eb1ng v\u00e0 O l\u00e0 t\u00e2m c\u1ee7a \u0111\u00e1y. 2 G\u1ecdi M , N , P v\u00e0 Q l\u1ea7n l\u01b0\u1ee3t l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a O tr\u00ean c\u00e1c m\u1eb7t ph\u1eb3ng (SAB), (SBC), (SCD) v\u00e0 (SDA). Th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p O.M N P Q b\u1eb1ng A a3 B 2a3 C a3 D a3 . . . . 48 81 81 96 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi E, F , G v\u00e0 H l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u00e1c c\u1ea1nh AB, BC, S CD v\u00e0 DA. Theo t\u00ednh ch\u1ea5t h\u00ecnh ch\u00f3p \u0111\u1ec1u ta c\u00f3 SO \u22a5 (ABCD), OE, OF , OG, OH l\u1ea7n l\u01b0\u1ee3t vu\u00f4ng g\u00f3c v\u1edbi AB, BC, CD v\u00e0 DA. Q X\u00e9t \u0111\u01b0\u1eddng th\u1eb3ng AB v\u00e0 m\u1eb7t ph\u1eb3ng (SOE), ta c\u00f3 MP \u00aeAB \u22a5 OE N H D AB \u22a5 SO \u21d2 AB \u22a5 (SOE) \u21d2 (SAB) \u22a5 (SOE). A EO G B FC Do \u0111\u00f3 h\u00ecnh chi\u1ebfu M c\u1ee7a O l\u00ean m\u1eb7t ph\u1eb3ng (SAB) ch\u00ednh l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a O l\u00ean giao tuy\u1ebfn SE c\u1ee7a hai m\u1eb7t ph\u1eb3ng (SAB) v\u00e0 (SOE). \u00c7 \u221a \u00e52 \u221a a3 \u2212 a 2 \u221aa . Do tam gi\u00e1c SAB c\u00e2n t\u1ea1i S n\u00ean ta c\u00f3 SE = SA2 \u2212 AE2 = = 2 22 a \u221aa a X\u00e9t tam gi\u00e1c vu\u00f4ng SOE, ta c\u00f3 OE = , SE = suy ra SO = n\u00ean SEO l\u00e0 tam gi\u00e1c vu\u00f4ng c\u00e2n 22 2 t\u1ea1i O v\u00e0 M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a SE. T\u01b0\u01a1ng t\u1ef1, ta c\u0169ng c\u00f3 N , P , Q c\u0169ng l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a SF , SG, SH. \u221a a2 Theo t\u00ednh ch\u1ea5t \u0111\u01b0\u1eddng trung b\u00ecnh trong tam gi\u00e1c, ta c\u00f3 M N P Q l\u00e0 h\u00ecnh vu\u00f4ng c\u1ea1nh . 4 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 411 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n \u00c7 \u221a \u00e52 a2 a2 Di\u1ec7n t\u00edch h\u00ecnh vu\u00f4ng b\u1eb1ng SMNP Q = 4 =. 8 Do OE \u2225 (M N P Q) \u21d2 d (O, (M N P Q)) = d (E, (M N P Q)). M\u1eb7t kh\u00e1c M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a SE n\u00ean d (E, (M N P Q)) = d (S, (M N P Q)) . 1 1a Do \u0111\u00f3 d (O, (M N P Q)) = d (S, (M N P Q)) = d (S, (ABCD)) = SO = . 2 24 V\u1eady th\u1ec3 t\u00edch c\u1ee7a h\u00ecnh ch\u00f3p O.M N P Q b\u1eb1ng VO.MNP Q = 1d (O, (M N P Q)) \u00b7 SMNP Q = 1 \u00b7 a \u00b7 a2 = a3 3 3 4 8 . 96 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 85 (C\u00e2u 44 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho kh\u1ed1i h\u1ed9p ch\u1eef nh\u1eadt ABCD.A B C D c\u00f3 \u0111\u00e1y l\u00e0 h\u00ecnh vu\u00f4ng, BD = 4a, g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (A BD) v\u221a\u00e0 (ABCD) b\u1eb1ng 30\u25e6. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i h\u1ed9p ch\u1eef\u221anh\u1eadt \u0111\u00e3 cho b\u1eb1ng A 16 3 a3. \u221a C 16 3 a3. D \u221a B 48 3a3. 16 3a3. 93 \u0253 L\u1eddi gi\u1ea3i. V\u00ec \u0111\u00e1y ABCD l\u00e0 h\u00ecnh vu\u00f4ng n\u00ean BD = AC = 4a v\u00e0 BD \u22a5 AC. A B D C Do \u0111\u00f3 SABCD = 1 \u00b7 AC \u00b7 BD = 1 \u00b7 4a \u00b7 4a = 8a2. \u221a 2 2 Tam gi\u00e1c A AO vu\u00f4ng t\u1ea1i A, ta c\u00f3 AA = AO \u00b7 tan 30\u25e6 = 2a \u00b7 33 . V\u1eady th\u1ec3 t\u00edch kh\u1ed1i h\u1ed9p ch\u1eef nh\u1eadt l\u00e0 \u221a\u221a A 30\u25e6 3 16 3 a3. V = SABCD \u00b7 AA = 8a2 \u00b7 2a \u00b7 = B 39 O DC Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 86 (C\u00e2u 46 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho kh\u1ed1i h\u1ed9p ch\u1eef nh\u1eadt ABCD.A B C D c\u00f3 \u0111\u00e1y l\u00e0 h\u00ecnh vu\u00f4ng, BD = 4a, g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (A BD) v\u00e0 (ABCD) b\u1eb1ng 60\u25e6. T\u221ah\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i h\u1ed9p ch\u1eef\u221anh\u1eadt \u0111\u00e3 cho b\u1eb1ng \u221a B 16 3 a3. C 16 3 a3. \u221a A 48 3a3. 93 D 16 3a3. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi O l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AC v\u00e0 BD. B A D Ta c\u00f3 BD \u22a5 AC v\u00e0 BD \u22a5 AA n\u00ean BD \u22a5 (A AC) \u21d2 BD \u22a5 B C D A O. X\u00e9t hai m\u1eb7t ph\u1eb3ng (A BD) v\u00e0 (ABCD), c\u00f3 MADD-109 O \uf8f1(A BD) \u2229 (ABCD) = BD C \uf8f4 \uf8f2 A O \u22a5 BD \uf8f4\uf8f3AC \u22a5 BD \u21d2 ((A B\u00a4D), (ABCD)) = (\u2044A O, AC) = A\u2019OA = 60\u25e6. Ta t\u00ednh \u0111\u01b0\u1ee3c Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 412 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N AB = B\u221aD = \u221a4a \u221a 22 = 2a 2. AC BD 4a AO = = = = 2a. 2 22 \u221a AA = AO tan A\u2019OA = 2a tan 60\u25e6 = 2a 3. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i h\u1ed9p ABCD.A B C D b\u1eb1ng V = AB2 \u00b7 AA = \u00c4 \u221a \u00e42 \u00b7 \u221a = \u221a 2a 2 2a 3 16a3 3. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 87 (C\u00e2u 47 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho kh\u1ed1i l\u0103ng tr\u1ee5 tam gi\u00e1c \u0111\u1ec1u ABC.A B C c\u00f3 c\u1ea1nh b\u00ean b\u1eb1ng 4a, g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (A BC) v\u00e0 (ABC) b\u1eb1ng 30\u25e6. Th\u1ec3\u221at\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u221a\u00e3 cho b\u1eb1ng \u221a \u221a B 64 3 a3. C 64 3 a3. D 64 3 a3. A 64 3a3. 3 27 9 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ea1nh BC, khi \u0111\u00f3 AM \u22a5 BC v\u00e0 A M \u22a5 BC. A A Do \u0111\u00f3 g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (A BC) v\u00e0 (ABC) b\u1eb1ng A\u00f7M\u221aA = 30\u25e6. MDD-109 B Tam gi\u00e1c A M A vu\u00f4ng t\u1ea1i A c\u00f3 AM = AA \u00b7 cot 30\u25e6 = 4a 3. B Do tam gi\u00e1c ABC \u0111\u1ec1u n\u00ean C AB \u221a 3 \u21d4 AB = 2\u221aAM = 8a. AM = 23 \u221a\u221a M C AB2 3 64a2 3 = \u221a Khi \u0111\u00f3 di\u1ec7n t\u00edch tam gi\u00e1c ABC b\u1eb1ng = 16a2 3. 44 V\u1eady th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng \u221a\u221a V = AA \u00b7 SABC = 4a \u00b7 16a2 3 = 64a3 3. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 88 (C\u00e2u 48 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho kh\u1ed1i l\u0103ng tr\u1ee5 tam gi\u00e1c \u0111\u1ec1u ABC.A B C c\u00f3 c\u1ea1nh b\u00ean b\u1eb1ng 2a, g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (A BC)\u221av\u00e0 (ABC) b\u1eb1ng 60\u25e6. Th\u221a\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5\u221a\u0111\u00e3 cho b\u1eb1ng \u221a A 8 3 a3. B 8 3 a3. C 8 3 a3. D 8 3a3. 3 9 27 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 413 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n A C G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m BC, g\u00f3c gi\u1eefa m\u1eb7t ph\u1eb3ng (A BC) v\u00e0 B (ABC) l\u00e0 g\u00f3c A\u00f7M A. X\u00e9t tam gi\u00e1c vu\u00f4ng A AM c\u00f3 \u221a AA 2a 2a 3 AM = tan A\u00f7M A = tan 60\u25e6 = . 3 \u221a A 60\u25e6 C 2a 3 M ABC \u0111\u1ec1u c\u00f3 trung tuy\u1ebfn AM = n\u00ean \u221a\u221a 3 B AB = AM : 3 2a 3 \u00b7 \u221a2 4a = =. 2 3 33 S ABC = AB2 \u00b7 \u221a = \u00c5 4a \u00e32 \u00b7 \u221a = \u221a 3 3 3 4 3a2 4 4 . 9 \u221a\u221a ABC = 2a \u00b7 4 3a2 8 3 a3. VABC.A B C = AA \u00b7 S = 99 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 89 (C\u00e2u 45 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho kh\u1ed1i l\u0103ng tr\u1ee5 tam gi\u00e1c \u0111\u1ec1u ABC.A B C c\u00f3 c\u1ea1nh b\u00ean b\u1eb1ng 4a, g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (A BC) v\u221a\u00e0 (ABC) b\u1eb1ng 60\u25e6. Th\u1ec3\u221at\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u221a\u00e3 cho b\u1eb1ng \u221a A 64 3 a3. B 64 3 a3. C 64 3 a3. D 64 3a3. 9 27 3 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi D l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BC. A C B \u00aeAD \u22a5 BC \u21d2 ((A BC), (ABC)) = A\u2019DA = 60\u25e6. Ta c\u00f3 A D \u22a5 BC \u221a AA 4 3a X\u00e9t tam gi\u00e1c ADA vu\u00f4ng t\u1ea1i A, ta c\u00f3 AD = tan 60\u25e6 = . AD 8a 3 X\u00e9t tam gi\u00e1c ABD vu\u00f4ng t\u1ea1i D, ta c\u00f3 AB = sin 6\u221a0\u25e6 = . 3a2 3 . Di\u1ec7n t\u00edch tam gi\u00e1c ABC l\u00e0 S = 1 \u00b7 AD \u00b7 BC = 16 29 A C B D Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho l\u00e0 \u221a\u221a 16 3a2 64 3a3 V = AA \u00b7 S = 4a \u00b7 =. 99 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 90 (C\u00e2u 46 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u1ee9ng ABC.A B C c\u00f3 \u0111\u00e1y ABC l\u00e0 tam gi\u00e1c vu\u00f4ng c\u00e2n t\u1ea1i A, AB = 2a. G\u00f3c gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng BC v\u00e0 m\u1eb7t ph\u1eb3ng (ACC A ) b\u1eb1ng 30\u25e6. Th\u1ec3\u221at\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u221a\u0111\u00e3 cho b\u1eb1ng A 3a3. B a3. C 12 2a3. D 4 2a3. \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 414 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N Ta c\u00f3 BA \u22a5 BC v\u00e0 BA \u22a5 AA \u21d2 BA \u22a5 (ACC A ). A B Suy ra g\u00f3c (BC , (AC A )) = (BC , C A) = B\u2019C A = 30\u25e6. \u221a C Tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A, c\u00f3 AC = A\u221aB. cot A\u2019C B = 2a \u221a3. 30\u25e6 Tam gi\u00e1c CAC vu\u00f4ng t\u1ea1i C, c\u00f3 CC = AC 2 \u2212 AC2 = \u221a2a 2. Th\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 l\u00e0 V = B.h = 1 AB.AC.C C = 4a3 2\u00b7 2 A C 2a 2a B Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 91 (C\u00e2u 43 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u1ee9ng ABC.A B C c\u00f3 \u0111\u00e1y ABC l\u00e0 tam gi\u00e1c vu\u00f4ng c\u00e2n t\u1ea1i A, c\u1ea1nh b\u00ean AA = 2a, g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (A BC) v\u00e0 (ABC) b\u1eb1ng 60\u25e6. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 8 a3. B 8a3. C 8 a3. D 24a3. 9 3 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BC. A C Ta c\u00f3 ABC l\u00e0 tam gi\u00e1c vu\u00f4ng c\u00e2n t\u1ea1i A n\u00ean AI \u22a5 BC v\u00e0 ABC.A B C l\u00e0 A kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u1ee9ng n\u00ean AA \u22a5 BC B C suy ra BC \u22a5 (AA I) \u21d2 BC \u22a5 A I. Do \u0111\u00f3 g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (A BC) v\u00e0 (ABC) b\u1eb1ng g\u00f3c gi\u1eefa A I v\u00e0 AI, I B m\u00e0 tam gi\u00e1c AA I vu\u00f4ng t\u1ea1i A n\u00ean ta c\u00f3 A\u2019IA l\u00e0 g\u00f3c nh\u1ecdn. Suy ra g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (A BC) v\u00e0 (ABC) b\u1eb1ng A\u2019IA = 60\u25e6. Trong tam gi\u00e1c vu\u00f4ng AA I, ta c\u00f3 AI = AA = \u221a2a \u00b7 tan 60\u25e6 3 Ta c\u00f3 ABC l\u00e0 tam gi\u00e1c vu\u00f4ng c\u00e2n t\u1ea1i A n\u221a\u00ean BC = 2AI = \u221a4a , AB = AC = B\u221aC = 2a 6 \u00b7 3 23 V\u1eady th\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho l\u00e0 \u221a = 2a 6 1 1 \u00c7 \u00e52 8a3 \u00b7 2 3 V = AA .S\u2206ABC = AA . AB.AC \u00b7 2a. 3 = 2 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 92 (C\u00e2u 39 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). Cho kh\u1ed1i l\u0103ng tr\u1ee5 ABC.A B C c\u00f3 th\u1ec3 t\u00edch b\u1eb1ng 1. G\u1ecdi M , N l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a c\u00e1c \u0111o\u1ea1n th\u1eb3ng AA v\u00e0 BB . \u0110\u01b0\u1eddng th\u1eb3ng CM c\u1eaft \u0111\u01b0\u1eddng th\u1eb3ng C A t\u1ea1i P , \u0111\u01b0\u1eddng th\u1eb3ng CN c\u1eaft \u0111\u01b0\u1eddng th\u1eb3ng C B t\u1ea1i Q. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n l\u1ed3i A M P B N Q b\u1eb1ng A 1. B 1 C 1 D 2 . . . 3 2 3 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 415 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n C A B M N A C P B Q 1 12 1 2 Ta c\u00f3 VC.ABNM = 2 VC.A B BA = 2 \u00b7 3 VABC.A B C = 3 \u21d2 VCMNA B C =. 3 Do M, N l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a c\u00e1c c\u1ea1nh AA , BB n\u00ean A , B l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a c\u00e1c \u0111o\u1ea1n C P , C Q. Do v\u1eady, tam gi\u00e1c C QP \u0111\u1ed3ng d\u1ea1ng v\u1edbi tam gi\u00e1c C B A v\u1edbi t\u1ec9 s\u1ed1 2 n\u00ean S C QP = 4\u00b7S A B C . Suy ra VC.CQP = 1 \u00b7 d (C, (A B C)) \u00b7 S C QP = 4 \u00b7 1 B C )) \u00b7 S A B C = 4 \u00b7 VC.A B C = 4 3 d (C, (A . 3 3 Khi \u0111\u00f3 VA MP B NQ = VC.C PQ \u2212 VCMNA B C = 4 \u2212 2 = 2 Ch\u1ecdn \u0111\u00e1p \u00e1n D 3 3 . 3 \u0104 C\u00e2u 93 (C\u00e2u 44 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 20\u221a20). 3 3a Cho h\u00ecnh ch\u00f3p \u0111\u1ec1u S.ABCD c\u00f3 c\u1ea1nh \u0111\u00e1y b\u1eb1ng 3a, c\u1ea1nh b\u00ean b\u1eb1ng v\u00e0 O l\u00e0 t\u00e2m c\u1ee7a \u0111\u00e1y. 2 G\u1ecdi M , N , P v\u00e0 Q l\u1ea7n l\u01b0\u1ee3t l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a O tr\u00ean c\u00e1c m\u1eb7t ph\u1eb3ng (SAB), (SBC), (SCD) v\u00e0 (SDA). Th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p O.M N P Q b\u1eb1ng A 9a3 B 2a3 C 9a3 D a3 . . . . 16 3 32 3 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi E, F , G, H l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB, BC, S CD v\u00e0 DA. Do S.ABCD l\u00e0 h\u00ecnh ch\u00f3p \u0111\u1ec1u n\u00ean ABCD l\u00e0 h\u00ecnh vu\u00f4ng c\u1ea1nh 3a, SO \u22a5 (ABCD). Q Do AB \u22a5 OE v\u00e0 AB \u22a5 SO suy ra AB \u22a5 (SOE), (SOE) \u22a5 (SAB) theo giao tuy\u1ebfn SE. M\u1eb7t kh\u00e1c OM \u22a5 IP (SAB) n\u00ean OM \u2282 (SOE) v\u00e0 OM l\u00e0 \u0111\u01b0\u1eddng cao tam gi\u00e1c SOE. M N Tam gi\u00e1c SOA vu\u00f4ng t\u1ea1i O, suy ra A HD \u221a \u00a0 E O G SO = SA2 \u2212 OA2 = 27a2 18a2 3a B F \u2212 =. 4 42 C 3a Suy ra SO = OE = hay tam gi\u00e1c SOE vu\u00f4ng c\u00e2n t\u1ea1i O. T\u1eeb \u0111\u00f3 ta c\u00f3 M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a SE. 2 T\u01b0\u01a1ng t\u1ef1 ta c\u0169ng c\u00f3 N , P , Q l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a SF , SG, SH. Ta c\u0169ng suy ra (M N P Q) \u2225 (ABCD) v\u00e0 I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a SO. Ta c\u00f3 1 11 1 SMNP Q = 4 SEF GH = 4 \u00b7 2 \u00b7 SABCD = 8 SABCD. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 416 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N V\u00ec v\u1eady VO.MNP Q = 1 \u00b7 OI = 1 \u00b7 SABCD \u00b7 SO = 1 \u00b7 9a2 \u00b7 3a = 9a3 3 SMNP Q 3 8 2 3 8 4 . 32 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 94 (C\u00e2u 45 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 202\u221a0). Cho h\u00ecnh ch\u00f3p \u0111\u1ec1u S.ABCD c\u00f3 c\u1ea1nh \u0111\u00e1y b\u1eb1ng 4a, c\u1ea1nh b\u00ean b\u1eb1ng 2 3a v\u00e0 O l\u00e0 t\u00e2m c\u1ee7a \u0111\u00e1y. G\u1ecdi M , N , P v\u00e0 Q l\u1ea7n l\u01b0\u1ee3t l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a O tr\u00ean c\u00e1c m\u1eb7t ph\u1eb3ng (SAB), (SBC), (SCD) v\u00e0 (SDA). Th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p O.M N P Q b\u1eb1ng A 4a3 B 64a3 C 128a3 D 2a3 . . . . 3 81 81 3 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi E, F , G, H l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB, BC, S CD v\u00e0 DA. Do S.ABCD l\u00e0 h\u00ecnh ch\u00f3p \u0111\u1ec1u n\u00ean ABCD l\u00e0 h\u00ecnh vu\u00f4ng c\u1ea1nh 4a v\u00e0 SO \u22a5 (ABC\u221aD). \u221a Q AC 4a 2 Do \u0111\u00f3 OA = = = 2a 2 v\u00e0 OE = 2a. I P 22 MN Do AB \u22a5 OE v\u00e0 AB \u22a5 SO suy ra AB \u22a5 (SOE) hay m\u1eb7t ph\u1eb3ng (SOE) vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (SAB) theo A H D giao tuy\u1ebfn SE. M\u1eb7t kh\u00e1c OM \u22a5 (SAB) n\u00ean OM \u22a5 SE. E OG Tam gi\u00e1c SOA vu\u00f4ng t\u1ea1i O, suy ra \u221a\u221a BF C SO = SA2 \u2212 OA2 = 12a2 \u2212 8a2 = 2a. Suy ra SO = OE = 2a hay tam gi\u00e1c SOE vu\u00f4ng c\u00e2n t\u1ea1i O. T\u1eeb \u0111\u00f3 ta c\u00f3 M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a SE. T\u01b0\u01a1ng t\u1ef1 ta c\u0169ng c\u00f3 N , P , Q l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a SF , SG, SH. Ta c\u0169ng suy ra (M N P Q) \u2225 (ABCD), k\u00e9o theo SO \u22a5 (M N P Q). Khi \u0111\u00f3 g\u1ecdi I l\u00e0 giao \u0111i\u1ec3m c\u1ee7a SO v\u00e0 m\u1eb7t ph\u1eb1ng (M N P Q) th\u00ec OI \u22a5 (M N P Q). Do M I \u2225 OE n\u00ean I l\u00e0 trung \u0111i\u1ec3m SO. Ta c\u00f3 SMNP Q = 1 = 1 \u00b7 1 \u00b7 SABCD = 1 4 SEF GH 4 2 8 SABCD. V\u00ec v\u1eady VO.MNP Q = 1 \u00b7 OI = 1 \u00b7 SABCD \u00b7 SO = 1 \u00b7 16a2 \u00b7 2a = 2a3 3 SMNP Q 3 8 2 3 8 2 . 3 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 95 (C\u00e2u 48 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho kh\u1ed1i h\u1ed9p ch\u1eef nh\u1eadt ABCD.A B C D c\u00f3 \u0111\u00e1y l\u00e0 h\u00ecnh vu\u00f4ng, BD = 2a, g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (A BD) v\u00e0 (ABCD) b\u1eb1ng 30\u25e6. T\u221ah\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i h\u1ed9p ch\u1eef nh\u1eadt \u0111\u00e3 cho b\u1eb1ng \u221a \u221a B 2 3 a3. \u221a D 2 3 a3. A 6 3a3. C 2 3a3. 93 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 417 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n G\u1ecdi O l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AC. A D Do ABCD l\u00e0 h\u00ecnh vu\u00f4ng n\u00ean AO \u22a5 BD (1). D D\u1ec5 th\u1ea5y A BD c\u00e2n t\u1ea1i A do A B v\u00e0 A D l\u00e0 hai \u0111\u01b0\u1eddng ch\u00e9o c\u1ee7a hai B C A h\u00ecnh ch\u1eef nh\u1eadt m\u1eb7t b\u00ean b\u1eb1ng nhau n\u00ean A O \u22a5 BD (2). \uf8f1(A BD) \u2229 (ABCD) = BD 30\u25e6 \uf8f4 \uf8f2 Ta c\u00f3 AO \u22a5 BD \uf8f4\uf8f3A O \u22a5 BD. Do \u0111\u00f3 g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (A BD) v\u00e0 (ABCD) ch\u00ednh l\u00e0 g\u00f3c gi\u1eefa O hai \u0111\u01b0\u1eddng th\u1eb3ng AO v\u00e0 A O hay A\u2019OA = 30\u25e6. \u221aB C X\u00e9t A AO c\u00f3 A\u2019AO = 90\u25e6 n\u00ean AA = AO \u00b7 tan A\u2019OA = a \u00b7 tan 30\u25e6 = 3a . 3\u221a L\u1ea1i c\u00f3 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng ch\u00e9o c\u1ee7a h\u00ecnh vu\u00f4ng \u0111\u00e1y b\u1eb1ng 2a n\u00ean AB = A\u221aD = a\u221a2. \u00c4 \u221a \u00e42 a 3 2 3 a3. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i h\u1ed9p ch\u1eef nh\u1eadt l\u00e0 V = SABCD \u00b7 AA =a2 \u00b7 = 3 3 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 96 (C\u00e2u 42 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i ch\u00f3p \u0111\u1ec1u S.ABCD c\u00f3 AC = 4a, hai m\u1eb7t ph\u1eb3ng (SAB) v\u00e0 (SCD) vu\u00f4ng g\u00f3c v\u1edbi nhau. Th\u1ec3 t\u00edch \u221ac\u1ee7a kh\u1ed1i ch\u00f3p \u0111\u00e3 cho b\u221a\u1eb1ng D 16 a3. A 16 2 a3. B 8 2 a3. 3 33 C 16a3. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 S l\u00e0 \u0111i\u1ec3m chung c\u1ee7a hai m\u1eb7t ph\u1eb3ng (SAB) v\u00e0 (SCD). S M\u1eb7t kh\u00e1c AB \u2225 CD n\u00ean giao tuy\u1ebfn c\u1ee7a (SAB) v\u00e0 (SCD) l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng d qua \u0111i\u1ec3m S v\u00e0 song song v\u1edbi AB, CD. G\u1ecdi O l\u00e0 t\u00e2m h\u00ecnh vu\u00f4ng suy ra SO \u22a5 (ABCD). G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m AB, J l\u00e0 trung \u0111i\u1ec3m CD. Khi \u0111\u00f3 SI \u22a5 AB \u21d2 SI \u22a5 d. SJ \u22a5 CD \u21d2 SJ \u22a5 d. I A D Suy ra g\u00f3c gi\u1eefa (SAB) v\u00e0 (SCD) l\u00e0 I\u2018SJ = 90\u25e6 O J \u221a B C 2 2a. Ta c\u00f3 AD = A\u221aC = 2 11 \u221a V\u00ec ISJ vu\u00f4ng t\u1ea1i S n\u00ean SO = IJ = AD = 2a. 22 \u221a \u221a Th\u1ec3 t\u00edch S.AB C D l\u00e0 V = 1 = 1 \u00b7 2a \u00b7 8a2 = 8 2 a3. 3 \u00b7 SO \u00b7 SABCD 3 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 97 (C\u00e2u 34 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). M\u1ed9t chi\u1ebfc b\u00fat ch\u00ec c\u00f3 d\u1ea1ng kh\u1ed1i l\u0103ng tr\u1ee5 l\u1ee5c gi\u00e1c \u0111\u1ec1u c\u00f3 c\u1ea1nh \u0111\u00e1y b\u1eb1ng 3 mm v\u00e0 chi\u1ec1u cao b\u1eb1ng 200 mm. Th\u00e2n b\u00fat ch\u00ec \u0111\u01b0\u1ee3c l\u00e0m b\u1eb1ng g\u1ed7 v\u00e0 ph\u1ea7n l\u00f5i c\u00f3 d\u1ea1ng kh\u1ed1i tr\u1ee5 c\u00f3 chi\u1ec1u cao b\u1eb1ng chi\u1ec1u d\u00e0i c\u1ee7a b\u00fat v\u00e0 \u0111\u00e1y l\u00e0 h\u00ecnh tr\u00f2n c\u00f3 b\u00e1n k\u00ednh b\u1eb1ng 1 mm. Gi\u1ea3 \u0111\u1ecbnh 1 m3 g\u1ed7 c\u00f3 gi\u00e1 a (tri\u1ec7u \u0111\u1ed3ng); 1 m3 than ch\u00ec c\u00f3 gi\u00e1 9a (tri\u1ec7u \u0111\u1ed3ng). Khi \u0111\u00f3 gi\u00e1 nguy\u00ean v\u1eadt li\u1ec7u l\u00e0m m\u1ed9t chi\u1ebfc b\u00fat ch\u00ec nh\u01b0 tr\u00ean g\u1ea7n nh\u1ea5t v\u1edbi k\u1ebft qu\u1ea3 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A 97,03a \u0111\u1ed3ng. B 10,33a \u0111\u1ed3ng. C 9,7a \u0111\u1ed3ng. D 103,3a \u0111\u1ed3ng. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 418","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N T\u00ednh theo \u0111\u01a1n v\u1ecb \u0111o chi\u1ec1u d\u00e0i l\u00e0 mm, ta c\u00f3: \u221a \u221a Di\u1ec7n t\u00edch l\u1ee5c gi\u00e1c \u0111\u1ec1u c\u00f3 c\u1ea1nh b\u1eb1ng 3 l\u00e0 6 \u00b7 32 3 27 3 = . 42 L\u0103ng tr\u1ee5 l\u1ee5c gi\u00e1c \u0111\u1ec1u c\u00f3 c\u221a\u1ea1nh \u0111\u00e1y b\u1eb1ng 3 v\u00e0 chi\u1ec1u cao b\u1eb1ng 200 c\u00f3 th\u1ec3 t\u00edch 27 3 \u221a V1 = 2 \u00b7 200 = 2700 3. Ph\u1ea7n l\u00f5i c\u00f3 d\u1ea1ng kh\u1ed1i tr\u1ee5 c\u00f3 chi\u1ec1u cao b\u1eb1ng 200 v\u00e0 b\u00e1n k\u00ednh r = 1 c\u00f3 th\u1ec3 t\u00edch V2 = \u03c0 \u00b7 12 \u00b7 200 = 200\u03c0. Th\u1ec3 t\u00edch ph\u1ea7n th\u00e2n b\u00fat ch\u00ec l\u00e0m b\u1eb1ng g\u1ed7\u221al\u00e0 \u00c4 V3 = V1 \u2212 V2 = 2700 3 \u2212 \u00e4 mm3. 200\u03c0 Khi \u0111\u00f3 gi\u00e1 nguy\u00ean v\u1eadt li\u1ec7u l\u00e0m 1 chi\u1ebfc b\u00fat ch\u00ec l\u00e0 V3 \u00b7 a \u00b7 10\u22129 + V2 \u00b7 9a \u00b7 10\u22129 \u2248 9,703a (\u0111\u1ed3ng). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 98 (C\u00e2u 32 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). \u00d4ng A d\u1ef1 \u0111\u1ecbnh s\u1eed d\u1ee5ng h\u1ebft 5,5m2 k\u00ednh \u0111\u1ec3 l\u00e0m m\u1ed9t b\u1ec3 c\u00e1 b\u1eb1ng k\u00ednh c\u00f3 d\u1ea1ng h\u00ecnh h\u1ed9p ch\u1eef nh\u1eadt kh\u00f4ng n\u1eafp, chi\u1ec1u d\u00e0i g\u1ea5p \u0111\u00f4i chi\u1ec1u r\u1ed9ng (c\u00e1c m\u1ed1i gh\u00e9p c\u00f3 k\u00edch th\u01b0\u1edbc kh\u00f4ng \u0111\u00e1ng k\u1ec3). B\u1ec3 c\u00e1 c\u00f3 dung t\u00edch l\u1edbn nh\u1ea5t b\u1eb1ng bao nhi\u00eau (k\u1ebft qu\u1ea3 l\u00e0m tr\u00f2n \u0111\u1ebfn h\u00e0ng ph\u1ea7n tr\u0103m)? A 1,17 m3. B 1,01 m3. C 1,51 m3. D 1,40 m3. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi x; 2x; h l\u1ea7n l\u01b0\u1ee3t l\u00e0 chi\u1ec1u r\u1ed9ng, d\u00e0i v\u00e0 cao c\u1ee7a b\u1ec3 c\u00e1. Ta c\u00f3 di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n c\u1ee7a b\u1ec3 c\u00e1 l\u00e0 2x2 + 2(xh + 2xh) = 5,5 \u21d4 h = 5,5 \u2212 2x2 . 6x \u2026 5,5 \u0110i\u1ec1u ki\u1ec7n 0 < x < . 2 2x2 Th\u1ec3 t\u00edch b\u1ec3 c\u00e1 V (x) = 2x2 \u00b7 5,5 \u2212 = 1 \u2212 2x3). (5,5x 6x 3 \u2026 5,5 Khi \u0111\u00f3 V (x) = 1 (5,5 \u2212 6x2) \u21d2 V (x) = 0 \u21d4 x = . 3\u221a 6 11 33 Do \u0111\u00f3 Vmax = 54 \u2248 1,17 m3. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 99 (C\u00e2u 12 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho t\u1ee9 di\u1ec7n ABCD c\u00f3 tam gi\u00e1c BCD vu\u00f4ng t\u1ea1i C, AB vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (BCD), AB = 5a, BC\u221a= 3a v\u00e0 CD = 4a. T\u00ednh\u221ab\u00e1n k\u00ednh R c\u1ee7a m\u1eb7t c\u1ea7\u221au ngo\u1ea1i ti\u1ebfp t\u1ee9 di\u1ec7n AB\u221aCD. A R = 5a 2 5a 3 5a 2 5a 3 . B R= . C R= . D R= . 3322 \u0253 L\u1eddi gi\u1ea3i. \u221a \u00a0 DB \u00e32 \u00c5 AB2 \u00e3 5a 2 \u00c5 C\u00f3 DB = 5a \u21d2 R = += 2 22 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 100 (C\u00e2u 34 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho kh\u1ed1i ch\u00f3p S.ABCD c\u00f3 \u0111\u00e1y l\u221a\u00e0 h\u00ecnh vu\u00f4ng c\u1ea1nh a, SA vu\u00f4ng g\u00f3c v\u1edbi \u0111\u00e1y v\u00e0 kho\u1ea3ng c\u00e1ch t\u1eeb a2 A \u0111\u1ebfn m\u1eb7t ph\u1eb3ng (SBC) b\u1eb1ng . T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i ch\u00f3p \u0111\u00e3 cho. 2\u221a a3 3a3 a3 A V =. B V = a3. CV= . D V =. 2 93 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 419 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n G\u1ecdi H l\u00e0 ch\u00e2n \u0111\u01b0\u1eddng vu\u00f4ng g\u00f3c c\u1ee7a A l\u00ean c\u1ea1nh SB Ta c\u00f3 SA \u22a5 BC v\u00e0 BC \u22a5 AB \u21d2 BC \u22a5 (SAB) \u21d2 BC \u22a5 AH\u221a S M\u00e0 AH \u22a5 SB \u21d2 AH \u22a5 (SBC) \u21d2 d (A, (SBC)) = AH = a 2 2 H 1 1 1 A X\u00e9t \u2206SAB vu\u00f4ng t\u1ea1i A c\u00f3 AH l\u00e0 \u0111\u01b0\u1eddng cao \u21d2 = + AH2 SA2 AB2 1 11 \u21d2 a2\/2 = a2 + SA2 \u21d2 SA = a D BC Di\u1ec7n t\u00edch \u0111\u00e1y ABCD l\u00e0 SABCD = a2 \u21d2 th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p S.ABCD V = 1 = 1 .a.a2 = a3 3 .SA.SABCD 3 . 3 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 101 (C\u00e2u 50 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). Cho h\u00e0m s\u1ed1 f (x) c\u00f3 \u0111\u1ea1o h\u00e0m li\u00ean t\u1ee5c tr\u00ean \u0111o\u1ea1n [0; 1] th\u1ecfa m\u00e3n 11 f (1) = 0, [f (x)]2 dx = 7, x2f (x) dx = 1 . 3 00 1 T\u00edch ph\u00e2n f (x) dx b\u1eb1ng 0 A 7 B 1. C 7 D 4. . . 5 4 \uf8f1 \u0253 L\u1eddi gi\u1ea3i. \u00aeu = f (x) \uf8f2 du = f (x) dx C\u00e1ch 1. \u0110\u1eb7t dv = x2 dx \u21d2 x3 \uf8f3v = . 3 1 11 1 \u00ef x3 \u00f2 x3 Khi \u0111\u00f3: = (x) f (x) dx. x2f (x) dx = f \u2212 3 33 0 00 1 Suy ra x3f (x) dx = \u22121. (1) (2) 0 1 M\u1eb7t kh\u00e1c, do [f (x)]2 dx = 7 n\u00ean 0 1 [f (x)]2 + 14x3f (x) + 49x6 dx = 0 = 7 \u2212 14 + 7 = 0. 0 1 Hay f (x) + 7x3 2 dx = 0. 0 Suy ra f (x) + 7x3 = 0 \u21d2 f (x) = \u22127x3 \u21d2 f (x) = \u2212 7 x4 + C. 4 1 M\u00e0 f (1) = 0 n\u00ean C = 7 suy ra f (x) = 7 \u2212 x4). Khi \u0111\u00f3 7 , (1 f (x) dx = . 44 5 0 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 420 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N L\u01b0u \u00fd. C\u00f3 th\u1ec3 gi\u1ea3i th\u00edch v\u00ec sao t\u1eeb (2) suy ra f (x) + 7x3 = 0, \u2200x \u2208 [0; 1] nh\u01b0 sau: Theo gi\u1ea3 thi\u1ebft ta c\u00f3 h\u00e0m s\u1ed1 y = (f (x) + 7x3)2 li\u00ean t\u1ee5c v\u00e0 kh\u00f4ng \u00e2m tr\u00ean \u0111o\u1ea1n [0; 1] (do \u0111\u00f3, \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 n\u00e0y l\u00e0 m\u1ed9t \u0111\u01b0\u1eddng n\u00e9t li\u1ec1n tr\u00ean \u0111o\u1ea1n [0; 1] v\u00e0 kh\u00f4ng c\u00f3 \u0111i\u1ec3m n\u00e0o n\u1eb1m b\u00ean d\u01b0\u1edbi tr\u1ee5c Ox). T\u00edch ph\u00e2n \u1edf (2) c\u00f3 gi\u00e1 tr\u1ecb b\u1eb1ng di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = (f (x) + 7x3)2, tr\u1ee5c ho\u00e0nh, \u0111\u01b0\u1eddng th\u1eb3ng x = 0, \u0111\u01b0\u1eddng th\u1eb3ng x = 1. M\u00e0 theo (2) th\u00ec h\u00ecnh ph\u1eb3ng n\u00e0y c\u00f3 di\u1ec7n t\u00edch b\u1eb1ng 0 n\u00ean f (x) + 7x3 = 0, \u2200x \u2208 [0; 1]. C\u00e1ch 2 (ti\u1ebfp n\u1ed1i t\u1eeb (1)). Ta c\u00f3 b\u1ea5t \u0111\u1eb3ng th\u1ee9c Bunhiacovski \u0111\u1ed1i v\u1edbi t\u00edch ph\u00e2n: N\u1ebfu hai h\u00e0m s\u1ed1 f (x), g(x) li\u00ean t\u1ee5c tr\u00ean \u0111o\u1ea1n [a; b] th\u00ec ta lu\u00f4n c\u00f3 \uf8eeb \uf8f92 \u00d1 b \u00e9b \uf8f0 f (x).g(x)dx\uf8fb \u2264 f 2(x)dx . g2(x)dx. a aa D\u1ea5u \\\"=\\\" x\u1ea3y ra khi v\u00e0 ch\u1ec9 khi g(x) = kf (x), \u2200x \u2208 [a; b]. Tr\u1edf l\u1ea1i b\u00e0i to\u00e1n. T\u1eeb (1), ta c\u00f3: \u00d11 \u00e92 1 1 1 = x3f (x)dx \u2264 x6dx \u00b7 [f (x)]2dx = 1 \u00b7 7 = 1. 7 0 00 Nh\u01b0 v\u1eady d\u1ea5u \\\"=\\\" x\u1ea3y ra, t\u1ee9c l\u00e0 f (x) = kx3. Thay tr\u1edf l\u1ea1i v\u00e0o (1), ta \u0111\u01b0\u1ee3c: 1 k x6dx = \u22121 \u21d2 k = \u22121 \u21d2 k = \u22127. 7 0 V\u1eady f (x) = \u22127x3 \u21d2 f (x) = \u2212 7 x4 + C do f\u21d2(1)=0 f (x) = \u2212 7 x4 + 7 . 4 44 1 7 Do \u0111\u00f3 f (x) dx = . 5 0 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 102 (C\u00e2u 44 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho t\u1ee9 di\u1ec7n \u0111\u1ec1u ABCD c\u00f3 c\u00e1c c\u1ea1nh b\u1eb1ng a. G\u1ecdi M , N l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a c\u00e1c c\u1ea1nh AB, BC v\u00e0 E l\u00e0 \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9ng v\u1edbi B qua D. M\u1eb7t ph\u1eb3ng (M N E) chia kh\u1ed1i t\u1ee9 di\u1ec7n ABCD th\u00e0nh hai kh\u1ed1i \u0111a di\u1ec7n,\u221atrong \u0111\u00f3 kh\u1ed1i \u0111a di\u1ec7n ch\u221a\u1ee9a \u0111\u1ec9nh A c\u00f3 th\u1ec3 t\u00edch V\u221a. T\u00ednh V . \u221a 7 2a3 11 2a3 C V = 13 2a3 2a3 A V = . B V= . . D V= . 216 216 216 18 \u0253 L\u1eddi gi\u1ea3i. \u221a a3 2 Ta c\u00f3 th\u1ec3 t\u00edch kh\u1ed1i t\u1ee9 di\u1ec7n ABCD b\u1eb1ng = X. 12 A G\u1ecdi P, Q l\u1ea7n l\u01b0\u1ee3t l\u00e0 giao \u0111i\u1ec3m c\u1ee7a N E v\u1edbi CD v\u00e0 M E 2 v\u1edbi AD. D\u1ec5 th\u1ea5y AQ = CP = a. 3 M 1 Ta d\u1ec5 d\u00e0ng t\u00ednh \u0111\u01b0\u1ee3c VE .BM N = X. \u00c1p d\u1ee5ng t\u1ec9 s\u1ed1 th\u1ec3 B Q 2 N D VE.P QD 2 2 C t\u00edch ta c\u00f3 VE .BM N = . Suy ra VE.P QD = 9 .VE.BMN \u21d2 P E 9 77 VBM N E QP = 9 .VE.BMN = .X 18 T\u1ee9c l\u00e0 ph\u1ea7n kh\u1ed1i \u0111a di\u1ec7n kh\u00f4ng ch\u1ee9a \u0111i\u1ec3m A c\u00f3 th\u1ec3 7 t\u00edch b\u1eb1ng X, n\u00ean ph\u1ea7n ch\u1ee9a \u0111i\u1ec3m A c\u00f3 th\u1ec3 t\u00edch l\u00e0 \u221a18 11 11 2a3 X= . 18 216 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 421 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 103 (C\u00e2u 42 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho kh\u1ed1i l\u0103ng tr\u1ee5 ABC.A B C , kho\u1ea3ng c\u00e1ch t\u1eeb C \u0111\u1ebfn \u0111\u221a\u01b0\u1eddng th\u1eb3ng BB b\u1eb1ng 2, kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn c\u00e1c \u0111\u01b0\u1eddng th\u1eb3ng BB v\u00e0 CC l\u1ea7n l\u01b0\u1ee3t b\u1eb1ng 1 v\u00e0 \u221a3, h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a A l\u00ean m\u1eb7t 23 ph\u1eb3ng (A B C ) l\u00e0 trung \u0111i\u1ec3m M c\u1ee7a B C v\u00e0 A M = . Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho 3 b\u1eb1ng \u221a 2\u221a 3 A 2. C 3. . B 1. D 3 \u0253 L\u1eddi gi\u1ea3i. D\u1ef1ng AEF nh\u01b0 h\u00ecnh v\u1ebd sao cho AA \u22a5 (AEF ). AC N Khi \u0111\u00f3 VABC.A B C b\u1eb1ng v\u1edbi th\u1ec3 t\u00edch c\u1ee7a l\u0103ng tr\u1ee5 (T ) c\u00f3 m\u1eb7t \u0111\u00e1y AEF v\u00e0 c\u1ea1nh b\u00ean AA B T\u1ee9c l\u00e0 VABC.A B C = S AEF \u00b7 AA . \u221a T\u1eeb c\u00e1ch d\u1ef1ng ta suy ra AE = 1, AF = 3 v\u00e0 EF = 2. Suy ra AEF vu\u00f4ng t\u1ea1i A.\u221a F 1 3 T\u1eeb \u0111\u00f3 S AEF = AE \u00b7 AF = . EH 2 2 G\u1ecdi N l\u00e0 trung \u0111i\u1ec3m BC v\u00e0 H = EF \u2229 M N th\u00ec M N \u2225 AA ; H l\u00e0 trung \u0111i\u1ec3m EF v\u00e0 AH \u22a5 M N A C 1 M T\u1eeb \u0111\u00f3 AH = EF = 1. 2 B AM N vu\u00f4ng t\u1ea1i A c\u00f3 1 = 1 \u2212 1 = 1 \u21d2 AM = 2. A\u221aM 2 AH2 AN 2 4 = AM 2 + AN 2 = \u221a4 . Cu\u1ed1i c\u00f9ng AA = MN \u221a3 V\u1eady VABC.A B C = S AEF \u00b7 AA = 3 \u00b7 \u221a4 = 2. 23 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 104 (C\u00e2u 45 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho kh\u1ed1i l\u0103ng tr\u1ee5 ABC.A B C , kho\u1ea3ng c\u00e1ch t\u1eeb C \u0111\u1ebfn \u0111\u01b0\u1edd\u221ang th\u1eb3ng BB b\u1eb1ng 2, kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn c\u00e1c \u0111\u01b0\u1eddng th\u1eb3ng BB v\u00e0 CC l\u1ea7n l\u01b0\u1ee3t b\u1eb1ng 1 v\u00e0 3, h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a A l\u00ean m\u1eb7t ph\u1eb3ng (A B C ) l\u00e0 trung \u0111i\u1ec3m M c\u1ee7a B C v\u00e0 A M = 2. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng C2 \u221a 3 \u221a . B 2. 3 D 1. A 3. \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 422 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N D\u1ef1ng AEF nh\u01b0 h\u00ecnh v\u1ebd sao cho AA \u22a5 (AEF ). AC N Khi \u0111\u00f3 VABC.A B C b\u1eb1ng v\u1edbi th\u1ec3 t\u00edch c\u1ee7a l\u0103ng tr\u1ee5 (T ) c\u00f3 m\u1eb7t \u0111\u00e1y AEF v\u00e0 c\u1ea1nh b\u00ean AA . B T\u1ee9c l\u00e0 VABC.A B C = S AEF \u00b7 AA . \u221a T\u1eeb c\u00e1ch d\u1ef1ng ta suy ra AE = 1, AF = 3 v\u00e0 EF = 2. Suy ra AEF vu\u00f4ng t\u1ea1i A.\u221a F 1 3 T\u1eeb \u0111\u00f3 S AEF = AE \u00b7 AF = . EH 2 2 G\u1ecdi N l\u00e0 trung \u0111i\u1ec3m BC v\u00e0 H = EF \u2229 M N th\u00ec M N \u2225 AA ; H l\u00e0 trung \u0111i\u1ec3m EF v\u00e0 AH \u22a5 M N . A C 1 M T\u1eeb \u0111\u00f3 AH = EF = 1. 2 B AM N vu\u00f4ng t\u1ea1i A c\u00f3 1 = 1 \u2212 1 = 3 \u21d2 AM = \u221a2 . AM 2 AH2 AN 2 4 3 \u221a Cu\u1ed1i c\u00f9ng AA = MN = AM 2 + AN 2 = \u221a4 . \u221a3 3 \u00b7 \u221a4 = 2. V\u1eady VABC.A B C = S AEF \u00b7 AA = 23 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 105 (C\u00e2u 39 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). \u221a Cho kh\u1ed1i l\u0103ng tr\u1ee5 ABC.A B C , kho\u1ea3ng c\u00e1ch t\u1eeb C \u0111\u1ebfn \u0111\u01b0\u1eddng th\u1eb3ng BB b\u1eb1ng 5, kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn c\u00e1c \u0111\u01b0\u1eddng th\u1eb3ng BB v\u00e0 CC l\u1ea7n l\u01b0\u1ee3t b\u1eb1ng 1 v\u221a\u00e0 2, h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a A l\u00ean m\u1eb7t ph\u1eb3ng (A B C ) l\u00e0 trung \u0111i\u1ec3m M c\u1ee7a B C v\u00e0 A M = 5. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng \u221a 5 2\u221a 15 \u221a \u221a 15 A2 . . C 5. . 3 B 3 D 3 \u0253 L\u1eddi gi\u1ea3i. H AC N B F K J A C M B G\u1ecdi J, K l\u1ea7n l\u01b0\u1ee3t l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a A n\u00ean BB v\u00e0 CC , H l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a C l\u00ean BB . Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 423 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n Ta c\u00f3 AJ \u22a5 BB v\u00e0 AK \u22a5 CC \u21d2 AK \u22a5 BB \u21d2\u221aBB \u22a5 (AJK). Suy ra BB \u22a5 JK \u21d2 JK \u2225 CH \u21d2 JK = CH = 5. X\u00e9t AJK c\u00f3 JK2 = AJ2 + AK2 = 5 \u21d2 AJK vu\u221a\u00f4ng t\u1ea1i A. 5 G\u1ecdi F l\u00e0 trung \u0111i\u1ec3m JK khi \u0111\u00f3 AF = JF = F K = . \u221a2 G\u1ecdi N l\u00e0 trung \u0111i\u1ec3m BC, khi \u0111\u00f3 A M = AN = 5. X\u00e9t tam gi\u00e1c vu\u00f4ng AN F cos N\u2019AF = AF = 1 \u21d2 N\u2019AF = 60\u25e6. AN 2 V\u1eady ta c\u00f3 S 1 \u00b7 AK = 1 \u21d2 S AJK = S ABC \u00b7 cos 60\u25e6 \u21d2 S = S AJK = AJ cos 6\u221a0\u25e6 = 2. AJ K 2 ABC X\u00e9t AM A vu\u00f4ng t\u1ea1i M c\u00f3 M\u00f7AA = A\u00f7M F = 30\u25e6 hay AM = A M \u00b7 tan 30\u25e6 = 5 \u221a3 . V\u1eady th\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 l\u00e0 V = AM \u00b7 S ABC = 2 15 . 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 106 (C\u00e2u 49 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho l\u0103ng tr\u1ee5 ABC.A B C c\u00f3 chi\u1ec1u cao b\u1eb1ng 6 v\u00e0 \u0111\u00e1y l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh b\u1eb1ng 4. G\u1ecdi M, N, P l\u1ea7n l\u01b0\u1ee3t l\u00e0 t\u00e2m c\u00e1c m\u1eb7t b\u00ean ABB A , ACC A , BCC B . Th\u1ec3 t\u00edch kh\u1ed1i \u0111a di\u1ec7n l\u1ed3i c\u00f3 c\u00e1c \u0111\u1ec9nh l\u00e0 c\u00e1c \u0111i\u1ec3m\u221aA, B, C, M, N, P b\u1eb1ng \u221a \u221a \u221a A 9 3. B 10 3. C 7 3. D 12 3. \u0253 L\u1eddi gi\u1ea3i. AC B F N P D M E AC B G\u1ecdi DEF l\u00e0 thi\u1ebft di\u1ec7n c\u1ee7a l\u0103ng tr\u1ee5 c\u1eaft b\u1edfi m\u1eb7t ph\u1eb3ng (M N P ). D\u1ec5 ch\u1ee9ng minh \u0111\u01b0\u1ee3c (DEF ) \u2225 (ABC) v\u00e0 D, E, F l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a c\u00e1c \u0111o\u1ea1n 1\u221a th\u1eb3ng AA , BB , CC suy ra VABC.DEF = 2 VABC.A B C = 12 3. Ta c\u00f3 VABCP NM = VABC.DEF \u2212 VADMN \u2212 VBMP E \u2212 VCP MF . 3 \u221a 1 4 VABC.DEF 9 3. M\u1eb7t kh\u00e1c VADM N = VBMP E = VCP MF = 12 VABC.DEF \u21d2 VABCP NM = = Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 107 (C\u00e2u 46 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho l\u0103ng tr\u1ee5 ABC.A B C c\u00f3 chi\u1ec1u cao b\u1eb1ng 4 v\u00e0 \u0111\u00e1y l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh b\u1eb1ng 4. G\u1ecdi M, N v\u00e0 P l\u1ea7n l\u01b0\u1ee3t l\u00e0 t\u00e2m c\u1ee7a c\u00e1c m\u1eb7t b\u00ean ABB A , ACC A v\u00e0 BCC B . Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n l\u1ed3i Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 424 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N c\u00f3 c\u00e1c \u0111\u1ec9n\u221ah l\u00e0 c\u00e1c \u0111i\u1ec3m A, B, C, M, N, P b\u1eb1ng \u221a B \u221a A 14 3 8 3. \u221a D 20 3 . C 6 3. . 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 ABC.A B C l\u00e0 A C B C V =4\u00b7 \u221a \u221a 42 \u00b7 3 = 16 3. M A 4 B G\u1ecdi th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n l\u1ed3i c\u00f3 c\u00e1c \u0111\u1ec9nh l\u00e0 c\u00e1c \u0111i\u1ec3m A, B, N P C, M , N , P l\u00e0 V1. Ta c\u00f3: V1 = VAMNCB + VBMNP + VBNP C . 13 D\u1ec5 th\u1ea5y VA = V v\u00e0 VAMNCB = 4 VA ABC . ABC 3 VAM N C B = 1 Suy ra V . 4 11 VBA B C =V v\u00e0 VBMNP = 8 VBA B C . 3 1 Suy ra VBM N P = V. 24 11 VA BCB = VA B CC =V v\u00e0 VBNP C = 4 VBA B C . 3 1 Suy ra VBNP C = V . 12 3\u221a V\u1eady V1 = VAMNCB + VBMNP + VBNP C = V =6 3. 8 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 108 (C\u00e2u 49 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). Cho kh\u1ed1i ch\u00f3p S.ABCc\u00f3 \u0111\u00e1y ABC l\u00e0 tam gi\u00e1c vu\u00f4ng c\u00e2n t\u1ea1i A, AB = a, S\u2019BA = S\u2019CA = 90\u25e6, g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (SAB) v\u00e0 (SAC) b\u1eb1ng 60\u25e6. Th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p \u0111\u00e3 cho b\u1eb1ng A a3. B a3 C a3 D a3 . . . 3 2 6 \u0253 L\u1eddi gi\u1ea3i. S I A B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t H S\u0110T: 0905.193.688 C 425","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n - G\u1ecdi I v\u00e0 H l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a SA v\u00e0 BC. Theo gi\u1ea3 thi\u1ebft S\u2019BA = S\u2019CA = 90\u25e6 n\u00ean \u0111i\u1ec3m I c\u00e1ch \u0111\u1ec1u 4 \u0111i\u1ec3m S, A, B, C, suy ra I l\u00e0 t\u00e2m m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p S.ABC. - M\u1eb7t kh\u00e1c, H l\u00e0 t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c ABC, suy ra IH l\u00e0 tr\u1ee5c \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c ABC, khi \u0111\u00f3 IH\u22a5 (ABC). -\u21d2NBgoC\u00e0i=raa, \u221ata2mv\u00e0giA\u00e1cHA=B Ca\u221av2u.\u00f4n\u0110g\u1eb7tc\u00e2InHt\u1ea1=i A n\u00ean AH\u22a5BC; AB = a x, x > 0. 2 - D\u1ef1ng h\u1ec7 tr\u1ee5c\u00c7ta\u1ecd\u221aa 2\u0111\u1ed9 Oxyz nh\u01b0 h\u00ecnh\u221av\u1ebd sao cho \u221a \u00e5 \u00c7 a2 \u00e5 \u00c7 a2 \u00e5 H (0; 0; 0), A ; 0; 0 , B 0; ; 0 , C 0; \u2212 ; 0 , I (0; 0; x). 2 2\u221a 2 \u00c7\u00e5 \u2212a 2 Do I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a SA n\u00ean ta c\u00f3 S ; 0; 2x . 2 - Ta c\u00f3 A# B\u00bb = \u00c7 \u221a ; \u221a \u00e5 A# S\u00bb = \u00c4\u221a 0; \u00e4 \u2212 a2 a2 0, 2x , \u2212a 2; ; 22 \u00eeA# B\u00bb, A# S\u00bb\u00f3 \u00c4\u221a \u221a suy ra vC#eAc\u00bbt\u01a1=p\u00c7h\u00e1ap\u221at2u;y\u1ebfan\u221ac2\u1ee7;a0\u00e5m,\u1eb7tC# pS\u00bbh=\u1eb3n\u00c7g \u2212(SaA\u221aB2); l\u00e0\u221an#\u00bb1 = \u00e5 = ax 2; ax 2; a2\u00e4. -Ta c\u00f3 a2 ; 2x , 22 22 sn#u\u00bb2y=ra\u00eeC#veA\u00bbc,t\u01a1C#pS\u00bbh\u00f3\u00e1p=t\u00c4uayx\u1ebf\u221an 2c\u1ee7; a\u2212max\u1eb7\u221at 2p;ha\u1eb32n\u00e4g. (SAC) l\u00e0 Theo gi\u1ea3 thi\u1ebft g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (SAB) v\u00e0 (SAC) b\u1eb1ng 60\u25e6 n\u00ean ta c\u00f3 |n#\u00bb1.n#\u00bb2| cos60\u25e6 = |cos (n#\u00bb1, n#\u00bb2)| = |n#\u00bb1| . |n#\u00bb2| =\u221a |2a2x2 \u2212 2a\u221a2x2 + a4| 2a2x2 + 2a2x2 + a4. 2a2x2 + 2a2x2 + a4 \u21d2 1 = a4 \u21d4 4a2x2 + a4 = 2a4 \u21d4 x2 = a2 \u21d4 x = a 2 2a2x2 + 2a2x2 + a4 42 (do x, a > 0). a2 Khi \u0111\u00f3 d (S, (ABC)) = 2d (I, (ABC)) = 2.IH = 2x = a; S\u2206ABC = 2 V\u1eady th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p S.ABC l\u00e0 1 1 a2 a3 VS.ABC = .d (S, (ABC)) .S\u2206ABC = .a. = (\u0111vtt). 3 3 2 6 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 109 (C\u00e2u 49 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). Cho h\u00ecnh h\u1ed9p ABCD.A B C D c\u00f3 chi\u1ec1u cao b\u1eb1ng 8 v\u00e0 di\u1ec7n t\u00edch \u0111\u00e1y b\u1eb1ng 9. G\u1ecdi M , N , P , Q l\u1ea7n l\u01b0\u1ee3t l\u00e0 t\u00e2m c\u00e1c m\u1eb7t b\u00ean ABB A , BCC B , CDD C , DAA D . T\u00ednh th\u1ec3 t\u00edch kh\u1ed1i \u0111a di\u1ec7n l\u1ed3i c\u00f3 c\u00e1c \u0111\u1ec9nh l\u00e0 A, B, C, D, M , N , P , Q. A 27. B 30. C 18. D 36. \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 426 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N G\u1ecdi E, F , G, H l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AA , BB , CC , A D DD . Khi \u0111\u00f3 B E 11 C VABCD.EF GH = 2 VABCD.A B C D = \u00b7 9 \u00b7 8 = 36. M 2 N Q H P F G\u1ecdi V l\u00e0 th\u1ec3 t\u00edch kh\u1ed1i t\u1ee9 di\u1ec7n l\u1ed3i c\u1ea7n t\u00ednh. A G khi \u0111\u00f3 D V = VABCD.EF GH \u2212 VE.AMQ \u2212 VF.BMN \u2212 VG.CNP \u2212 VH.DP Q. Trong \u0111\u00f3 BC VE .AM Q = VF.BM N = VG.C N P = VH.DP Q = EQ \u00b7 EM \u00b7VE.AH F = 1 \u00b7 1 \u00b7VABC D.EF GH = 36 = 3 EH EF 4 6 24 . 2 \u21d2 V = 36 \u2212 4 \u00b7 3 = 30. 2 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 110 (C\u00e2u 47 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh ch\u00f3p \u0111\u1ec1u S.ABCD c\u00f3 c\u1ea1nh \u0111\u00e1y b\u1eb1ng a, c\u1ea1nh b\u00ean b\u1eb1ng 2a v\u00e0 O l\u00e0 t\u00e2m c\u1ee7a \u0111\u00e1y. G\u1ecdi M , N , P , Q l\u1ea7n l\u01b0\u1ee3t l\u00e0 c\u00e1c \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9ng v\u1edbi O qua tr\u1ecdng t\u00e2m c\u1ee7a c\u00e1c tam gi\u00e1c SAB, SBC, SCD, SD\u221aA v\u00e0 S l\u00e0 \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9\u221ang v\u1edbi S qua O. Th\u1ec3 t\u00edc\u221ah c\u1ee7a kh\u1ed1i ch\u00f3p S .M N\u221aP Q b\u1eb1ng 20 14a3 40 14a3 C 10 14a3 2 14a3 A . B . . D . 81 81 81 9 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 427 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n S Q MP N K D G I H K A I GO C BH S G\u1ecdi G , H , I v\u00e0 K l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u00e1c c\u1ea1nh AB, BC, CD v\u00e0 DA. Ta c\u00f3 SG H I K = 1 1 a2. 2 SABCD = 2 G\u1ecdi G, H, I v\u00e0 K l\u1ea7n l\u01b0\u1ee3t l\u00e0 tr\u1ecdng t\u00e2m c\u00e1c tam gi\u00e1c SAB, SBC, SCD v\u00e0 SDA. Hai h\u00ecnh vu\u00f4ng GHIK v\u00e0 G H I K 24 = 2 a2. \u0111\u1ed3ng d\u1ea1ng t\u1ec9 s\u1ed1 b\u1eb1ng 3 n\u00ean SGHIK = 9 \u00b7 SG H I K 9 8 a2. Hai h\u00ecnh vu\u00f4ng MNPQ v\u00e0 GH I K \u0111\u1ed3ng d\u1ea1ng t\u1ec9 s\u1ed1 b\u1eb1ng 2 n\u00ean SMNP Q = 4 \u00b7 SGH I K = 9 \u221a \u221a \u2026 2a2 14 Tam gi\u00e1c SAO vu\u00f4ng t\u1ea1i O n\u00ean SO = SA2 \u2212 AO2 = 4a2 \u2212 = a. 42 \u221a Ta c\u00f3 d(O, (M N P Q)) = 2 \u00b7 d(M, (GHIK)) = 2 \u21d2 d(S , (M N P Q)) = 5 = 5 14 SO SO a. 3 36 V\u1eady th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p S .M N P Q l\u00e0 \u221a\u221a 1 1 8 a2 5 14 20 14a3 VS = 3 \u00b7 SMNP Q \u00b7 d(S , (M N P Q)) = 3 \u00b7 9 \u00b7 a = . .M N P Q 6 81 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 111 (C\u00e2u 43 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019\u221a- 2020). Cho h\u00ecnh ch\u00f3p \u0111\u1ec1u S.ABCD c\u00f3 c\u1ea1nh \u0111\u00e1y b\u1eb1ng a, c\u1ea1nh b\u00ean b\u1eb1ng 3a v\u00e0 O l\u00e0 t\u00e2m c\u1ee7a \u0111\u00e1y. G\u1ecdi M , N , P , Q l\u1ea7n l\u01b0\u1ee3t l\u00e0 c\u00e1c \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9ng v\u1edbi O qua tr\u1ecdng t\u00e2m c\u1ee7a c\u00e1c tam gi\u00e1c SAB, SBC, SCD, SD\u221aA v\u00e0 S l\u00e0 \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9\u221ang v\u1edbi S qua O. Th\u1ec3 t\u00edc\u221ah c\u1ee7a kh\u1ed1i ch\u00f3p S .M N\u221aP Q b\u1eb1ng A 40 10a3 10 10a3 C 20 10a3 2 10a3 . B . . D . 81 81 81 9 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 428 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N S Q MP N K D G I H K A GO I C BH S G\u1ecdi G , H , I v\u00e0 K l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u00e1c c\u1ea1nh AB, BC, CD v\u00e0 DA. Ta c\u00f3 SG H I K = 1 = 1 a2. 2 SABCD 2 G\u1ecdi G, H, I v\u00e0 K l\u1ea7n l\u01b0\u1ee3t l\u00e0 tr\u1ecdng t\u00e2m c\u00e1c tam gi\u00e1c SAB, SBC, SCD v\u00e0 SDA. Hai h\u00ecnh vu\u00f4ng GHIK v\u00e0 G H I K \u0111\u1ed3ng d\u1ea1ng t\u1ec9 s\u1ed1 b\u1eb1ng 2 n\u00ean SGH I K = 4 \u00b7 SG H I K = 2 a2. 3 9 9 8 a2. Hai h\u00ecnh vu\u00f4ng MNPQ v\u00e0 GH I K \u0111\u1ed3ng d\u1ea1ng t\u1ec9 s\u1ed1 b\u1eb1ng 2 n\u00ean SMNP Q = 4 \u00b7 SGHIK = 9 \u221a \u221a \u2026 2a2 10 Tam gi\u00e1c SAO vu\u00f4ng t\u1ea1i O n\u00ean SO = SA2 \u2212 AO2 = 3a2 \u2212 = a. 42 \u221a Ta c\u00f3 d(O, (M N P Q)) = 2 \u00b7 d(O, (GHIK)) = 2 \u21d2 d(S , (M N P Q)) = 5 = 5 10 SO SO a. 3 36 V\u1eady th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p S .M N P Q l\u00e0 \u221a\u221a 1 1 8 a2 5 10 20 10a3 VS.MNP Q = 3 \u00b7 SMNP Q \u00b7 d(S , (M N P Q)) = 3 \u00b7 9 \u00b7 a = . 6 81 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 112 (C\u00e2u 47 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 -\u221a2020). Cho h\u00ecnh ch\u00f3p \u0111\u1ec1u S.ABCD c\u00f3 c\u1ea1nh \u0111\u00e1y b\u1eb1ng a, c\u1ea1nh b\u00ean b\u1eb1ng a 2 v\u00e0 O l\u00e0 t\u00e2m c\u00f9a \u0111\u00e1y. G\u1ecdi M , N , P , Q l\u1ea7n l\u01b0\u1ee3t l\u00e0 c\u00e1c \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9ng v\u1edbi O qua tr\u1ecdng t\u00e2m c\u1ee7a c\u00e1c tam gi\u00e1c SAB, SBC, SCD, S\u221aDA v\u00e0 S l\u00e0 \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9\u221ang c\u1ee7a S qua O. Th\u1ec3 t\u00edc\u221ah c\u1ee7a kh\u1ed1i ch\u00f3p S .M N\u221aP Q b\u1eb1ng A 2 6 a3. B 40 6 a3. C 10 6 a3. D 20 6 a3. 9 81 81 81 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 429 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n Do S.ABCD l\u00e0 h\u00ecnh ch\u00f3p \u0111\u1ec1u n\u00ean suy ra M N P Q l\u00e0 S N h\u00ecnh vu\u00f4ng. G\u1ecdi K l\u00e0 t\u00e2m h\u00ecnh ch\u1eef nh\u1eadt M N P Q, ta c\u00f3 \u221a M K P 2 5a 6 A Q C S K = S O + OK = SO + SO = . 36 SMNP Q = 4\u00b7 1 \u00b7 4 = 8 a2. G1 G4 G2 G3 2 9 SABCD 9 D \u221a B 1 20 6a3 \u00b7 O V\u1eady VS .M N P Q = 3 \u00b7 SMNP Q \u00b7 S K = 81 S Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 113 (C\u00e2u 45 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh ch\u00f3p \u0111\u1ec1u S.ABCD c\u00f3 t\u1ea5t c\u1ea3 c\u00e1c c\u1ea1nh b\u1eb1ng a v\u00e0 O l\u00e0 t\u00e2m \u0111\u00e1y. G\u1ecdi M , N , P , Q l\u1ea7n l\u01b0\u1ee3t l\u00e0 c\u00e1c \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9ng v\u1edbi O qua tr\u1ecdng t\u00e2m c\u1ee7a c\u00e1c tam gi\u00e1c SAB, SBC, SCD, SDA v\u00e0 S l\u00e0 \u0111i\u1ec3m \u221a\u0111\u1ed1i x\u1ee9ng v\u1edbi S qua O. Th\u221a\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i ch\u00f3p S \u221a.M N P Q b\u1eb1ng \u221a A2 2a3 20 2a3 C 40 2a3 10 2a3 . B . . D . 9 81 81 81 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 430 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N S Q M P N D G G1 I G2 A E F A O BB C S Ta c\u00f3 S.ABCD l\u00e0 h\u00ecnh ch\u00f3p \u0111\u1ec1u c\u00f3 t\u1ea5t c\u1ea3 c\u00e1c c\u1ea1nh \u0111\u1ec1u b\u1eb1ng a \u21d2 SO = a \u221a 2 . 2 G\u1ecdi G, I l\u1ea7n l\u01b0\u1ee3t l\u00e0 tr\u1ecdng t\u00e2m c\u00e1c tam gi\u00e1c SDA, SDC. G\u1ecdi E, F l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m DA\u221a, DC. \u221a 2 2 1a 2 \u21d2 GI = a . Ta c\u00f3 GI = EF , EF = AC = 3 22 3 \u221a M\u00e0 G, I l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a OQ, OP \u21d2 QP = 2GI = 2 3 2a \u221a . T\u1eeb gi\u1ea3 thi\u1ebft cho d\u1ec5 d\u00e0ng suy ra \u0111\u01b0\u1ee3c MNPQ l\u00e0 h\u00ecnh vu\u00f4ng c\u1ea1nh PQ = 2 2a \u21d2 SMNP Q = 8a2 3 . 9 G\u1ecdi O l\u00e0 t\u00e2m h\u00ecnh vu\u00f4ng M N P Q k\u1ebb GH \u2225 QO (H \u2208 OO ) \u21d2 H l\u00e0 trung \u0111i\u1ec3m OO (v\u00ec G l\u00e0 trung \u0111i\u1ec3m OQ). \u221a \u221a = 2OH = 2 \u00b7 1 \u00b7 SO = a \u221a 2 2a \u00b7 2 2a 2 Ta c\u00f3 QO = = v\u00e0 OO . 323 \u221a3 3 a2 V\u00ec S v\u00e0 S \u0111\u1ed1i x\u1ee9ng nhau qua O n\u00ean S O = SO = . \u221a\u221a \u221a2 a 2 a 2 5a 2 Ta c\u00f3 S O = S O + OO = + = . 2 3\u221a 6 \u221a 1 1 \u00b7 5a 2 8a2 20 2a3 VS .MNP Q = 3 \u00b7SO \u00b7 SMNP Q = 36 \u00b7 9 = . 81 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 114 (C\u00e2u 46 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2\u221a020). Cho h\u00ecnh ch\u00f3p \u0111\u1ec1u S.ABCD c\u00f3 c\u1ea1nh \u0111\u00e1y b\u1eb1ng 2a, c\u1ea1nh b\u00ean b\u1eb1ng 3a v\u00e0 O l\u00e0 t\u00e2m c\u1ee7a \u0111\u00e1y. G\u1ecdi M , N , P v\u00e0 Q l\u1ea7n l\u01b0\u1ee3t l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a O tr\u00ean c\u00e1c m\u1eb7t ph\u1eb3ng (SAB), (SBC), Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 431 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n (SCD) v\u00e0 (SDA). Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i ch\u00f3p O.M N P Q b\u1eb1ng A 8a3 B a3 C a3 D 16a3 . . . . 81 6 12 81 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi E l\u00e0 trung \u0111i\u1ec3m AB. G\u1ecdi M l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a O S l\u00ean SE, suy ra OM \u22a5 (SAB). Tam gi\u00e1c SOB c\u00f3 \u221a\u221a Q P SO = SB2 \u2212 OB2 = 3a2 \u2212 2a2 = a. M N Tam gi\u00e1c SOE c\u00f3 SE2 = SO2 + OE2 = a2 + a2 = 2a2 v\u00e0 HD G SM \u00b7 SE = SO2 \u21d4 SM = SO2 = a2 = 1 A O SE SE2 2a2 . E C 2 BF Do \u0111\u00f3 M l\u00e0 trung \u0111i\u1ec3m SE. G\u1ecdi H, G, F l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AD, DC, CB. T\u01b0\u01a1ng t\u1ef1 ta ch\u1ee9ng minh \u0111\u01b0\u1ee3c N , P , Q l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m\u221ac\u1ee7a SF\u221a, SG, SH. AC 2a 2 a 2 Suy ra M N P Q l\u00e0 h\u00ecnh vu\u00f4ng c\u00f3 \u0111\u1ed9 d\u00e0i c\u1ea1nh b\u1eb1ng = =. 44 2 Ta c\u00f3 d(O, (M N P Q)) = d(S, (M N P Q)) = SO = a. 22 V\u1eady VO.MNP Q = 1 \u00b7 d(O, (M N P Q)) \u00b7 SMNP Q = 1 \u00b7 a \u00c7 \u221a \u00e52 = a3 3 3 2 \u00b7 a2 . 2 12 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 115 (C\u00e2u 48 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u1ee9ng ABC.A B C c\u00f3 \u0111\u00e1y ABC l\u00e0 tam gi\u00e1c vu\u00f4ng c\u00e2n t\u1ea1i A, c\u1ea1nh b\u00ean AA = 2a, g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng A BC v\u00e0 (ABC) b\u1eb1ng 30\u25e6. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 24a3. B 8 a3. C 8a3. D 8 a3. 3 9 \u0253 L\u1eddi gi\u1ea3i. K\u1ebb AH \u22a5 BC, ta c\u00f3 AA \u22a5 (ABC) n\u00ean AA \u22a5 BC. A C AH \u22a5 BC v\u00e0 AA \u22a5 BC suy ra BC \u22a5 AA H \u21d2 A H \u22a5 BC. B Suy ra g\u00f3c gi\u1eefa A BC v\u00e0 (ABC) l\u00e0 A\u00f7HA \u21d2 A\u00f7HA = 30\u25e6. \u2206A AH vu\u00f4ng t\u1ea1i A c\u00f3 \u221a 2a 3\u00b7 tan A\u00f7HA = AA \u21d2 tan 30\u25e6 = 2a \u21d4 AH = 2a = AH AH t\u221aan 30\u25e6 \u2206ABC vu\u00f4ng c\u00e2n t\u1ea1i A n\u00ean BC = 2AH = 4a 3. 1\u221a \u221a \u21d2 SABC = 1 = 2a 3 \u00b7 4a 3 = 12a2\u00b7 AH.BC 2 AC 2 H V\u1eady th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 \u0111\u1ee9ng ABC.A B C l\u00e0 24a3. B Ch\u1ecdn \u0111\u00e1p \u00e1n A 432 S\u0110T: 0905.193.688 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N \u0104 C\u00e2u 116 (C\u00e2u 50 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). Cho kh\u1ed1i t\u1ee9 di\u1ec7n c\u00f3 th\u1ec3 t\u00edch b\u1eb1ng V . G\u1ecdi V l\u00e0 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n c\u00f3 c\u00e1c \u0111\u1ec9nh l\u00e0 c\u00e1c trung V \u0111i\u1ec3m c\u1ee7a c\u00e1c c\u1ea1nh c\u1ee7a kh\u1ed1i t\u1ee9 di\u1ec7n \u0111\u00e3 cho, t\u00ednh t\u1ec9 s\u1ed1 . V V 1 V 1 CV 2 V 5 A =. B =. =. D =. V2 V4 V3 V8 \u0253 L\u1eddi gi\u1ea3i. A QP E B D F N M C C\u00e1ch 1. \u0110\u1eb7c bi\u1ec7t h\u00f3a t\u1ee9 di\u1ec7n cho l\u00e0 t\u1ee9 di\u1ec7n \u0111\u1ec1u c\u1ea1nh a. H\u00ecnh \u0111a di\u1ec7n c\u1ea7n t\u00ednh c\u00f3 \u0111\u01b0\u1ee3c b\u1eb1ng c\u00e1ch c\u1eaft a 4 g\u00f3c c\u1ee7a t\u1ee9 di\u1ec7n, m\u1ed7i g\u00f3c l\u00e0 c\u0169ng l\u00e0 m\u1ed9t t\u1ee9 di\u1ec7n \u0111\u1ec1u c\u00f3 c\u1ea1nh b\u1eb1ng . 2 VV Do \u0111\u00f3 th\u1ec3 t\u00edch ph\u1ea7n c\u1eaft b\u1ecf l\u00e0 V = 4. = . 82 \u00c5 1 \u00e33 1 (V\u00ec v\u1edbi t\u1ee9 di\u1ec7n c\u1ea1nh gi\u1ea3m n\u1eeda th\u00ec th\u1ec3 t\u00edch gi\u1ea3m =) 28 V\u1eady V V \u21d4V 1 = =. 2 V2 C\u00e1ch 2. Kh\u1ed1i \u0111a di\u1ec7n l\u00e0 hai kh\u1ed1i ch\u00f3p t\u1ee9 gi\u00e1c (gi\u1ed1ng nhau) c\u00f3 c\u00f9ng \u0111\u00e1y l\u00e0 h\u00ecnh b\u00ecnh h\u00e0nh \u00fap l\u1ea1i. = 2VN.MEP F 11 1 Suy ra: V = 4.VN.MEP = 4.VP.MNE = 4. . V = V 24 2 ( Do chi\u1ec1u cao gi\u1ea3m m\u1ed9t n\u1eeda, c\u1ea1nh \u0111\u00e1y gi\u1ea3m m\u1ed9t n\u1eeda n\u00ean di\u1ec7n t\u00edch gi\u1ea3m 4 ) V = V \u2212 VA.QEP \u2212 VB.QMF \u2212 VC.MNE \u2212 VD.NP F C\u00e1ch 3. Ta c\u00f3 VV VA.QEP \u2212 VB.QMF VC.M N E VD.NP F 111 111 111 111 1 =1\u2212 \u2212 \u2212 =1\u2212 .. \u2212 .. \u2212 .. \u2212 .. = . VV VV 222 222 222 222 2 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 117 (C\u00e2u 46 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). \u221a Cho kh\u1ed1i l\u0103ng tr\u1ee5 ABC.A B C , kho\u1ea3ng c\u00e1ch t\u1eeb \u0111i\u1ec3m C \u0111\u1ebfn \u0111\u01b0\u1eddng th\u1eb3ng BB b\u1eb1ng 5, kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn c\u00e1c \u0111\u01b0\u1eddng th\u1eb3ng BB v\u00e0 CC l\u1ea7n l\u01b0\u1ee3t b\u1eb1ng 1 v\u00e0\u221a2, h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a A 15 l\u00ean m\u1eb7t ph\u1eb3ng (A B C ) l\u00e0 trung \u0111i\u1ec3m M c\u1ee7a B C v\u00e0 A M = . Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i l\u0103ng tr\u1ee5 3 \u0111\u00e3 cho\u221ab\u1eb1ng \u221a \u221a \u221a 15 2 5 C 5. 2 15 A . B . D . 3 3 3 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 433 S\u0110T: 0905.193.688","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n K\u1ebb AI \u22a5 BB , AK \u22a5 CC . Kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn BB , AB CC l\u1ea7n l\u01b0\u1ee3t l\u00e0 1; 2 n\u00ean AI = 1, AK = 2. G\u1ecdi F l\u00e0 tr\u221aung \u0111i\u1ec3m c\u1ee7a B\u221aC. F 15 \u21d2 AF = 15 V\u00ec A M = . CI 33 KE \u00aeAI \u22a5 BB AB Ta c\u00f3 \u21d2 BB \u22a5 (AIK) BB \u22a5 AK \u21d2 BB \u22a5 IK. V\u00ec CC \u2225 BB n\u00ean \u221a M d(C, BB ) = d(K, BB ) = IK = 5 C \u21d2 AIK vu\u00f4ng t\u1ea1i A. G\u1ecdi E l\u00e0 trung \u0111i\u1ec3m c\u1ee7a IK \u21d2 EF \u2225 BB \u21d2 EF \u22a5 (AIK) \u21d2 EF \u22a5 AE. M\u00e0 AM \u22a5 (ABC) n\u00ean ((ABC), (AIK)) = E\u00ffF, AM = A\u00f7M E = F\u2019AE (E# F\u00bb, A# M\u00bb l\u00e0 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a c\u1ee7a (AKI), (AB\u221aC)). \u221a Ta c\u00f3 5 3 AE \u221a2 = \u21d2 F\u2019AE = 30\u25e6 (AE l\u00e0 \u0111\u01b0\u1eddng trung tuy\u1ebfn c\u1ee7a tam gi\u00e1c AK I cos F\u2019AE = = AF 15 2 3 vu\u00f4ng t\u1ea1i A). H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a tam gi\u00e1c ABC l\u00ean m\u1eb7t ph\u1eb3ng (AIK) l\u00e0 tam gi\u00e1c AIK n\u00ean S AIK = S ABC \u00b7 cos E\u2019AF \u21d2 S ABC = S AIK = \u221a2 . cos E\u2019AF 3 X\u00e9t tam gi\u00e1c AM F vu\u00f4ng t\u1ea1i A, ta c\u00f3 tan \u00f7AM F = AF \u21d2 AM = AF \u221a = 5. \u221a AM tan \u00f7AM F \u221a \u221a2 2 15 V\u1eady VABC.A B C = 5 \u00b7 3 = . 3 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 118 (C\u00e2u 49 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). \u221a X\u00e9t kh\u1ed1i t\u1ee9 di\u1ec7n ABCD c\u00f3 c\u1ea1nh AB = x v\u00e0 c\u00e1c c\u1ea1nh c\u00f2n l\u1ea1i \u0111\u1ec1u b\u1eb1ng 2 3. T\u00ecm x \u0111\u1ec3 th\u1ec3 t\u00edch kh\u1ed1i t\u1ee9 di\u1ec7\u221an ABCD \u0111\u1ea1t gi\u00e1 tr\u1ecb l\u1edbn\u221anh\u1ea5t. \u221a \u221a A x = 6. B x = 14. C x = 3 2. D x = 2 3. \u0253 L\u1eddi gi\u1ea3i. - G\u1ecdi M, N l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a CD, AB. Khi \u0111\u00f3 ta t\u00ednh \u0111\u01b0\u1ee3c AM = BM = 3, suy ra MN = \u2026 x2 9\u2212 . 4 \u2026 x2 x. 9\u2212 - G\u1ecdi h l\u00e0 chi\u1ec1u cao c\u1ee7a kh\u1ed1i ch\u00f3p h\u1ea1 t\u1eeb \u0111\u1ec9nh A, ta c\u00f3 h = 3 4 \u221a v\u00e0 hmax khi x = 3 2. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 119 (C\u00e2u 44 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). X\u00e9t kh\u1ed1i ch\u00f3p S.ABC c\u00f3 \u0111\u00e1y l\u00e0 tam gi\u00e1c vu\u00f4ng c\u00e2n t\u1ea1i A, SA vu\u00f4ng g\u00f3c v\u1edbi \u0111\u00e1y, kho\u1ea3ng c\u00e1ch t\u1eeb A \u0111\u1ebfn m\u1eb7t ph\u1eb3ng (SBC) b\u1eb1ng 3. G\u1ecdi \u03b1 l\u00e0 g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (SBC) v\u00e0 (ABC). T\u00ednh cos \u03b1 khi th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p S.ABC nh\u221a\u1ecf nh\u1ea5t. \u221a 1 3 2 2 A cos \u03b1 = . B cos \u03b1 = 3 . C cos \u03b1 = . D cos \u03b1 = . 3 2 3 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 434 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 1. KH\u1ed0I \u0110A DI\u1ec6N G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m BC v\u00e0 H l\u00e0 ch\u00e2n \u0111\u01b0\u1eddng cao k\u1ebb t\u1eeb A xu\u1ed1ng SI, theo gi\u1ea3 thi\u1ebft ta c\u00f3 AH = 3 AH 3 v\u00e0 S\u2018IA = \u03b1. X\u00e9t c\u00e1c tam gi\u00e1c vu\u00f4ng AHI, SAI (t\u01b0\u01a1ng \u1ee9ng vu\u00f4ng t\u1ea1i H v\u00e0 A), c\u00f3 AI = =, sin \u03b1 sin \u03b1 AH 3 AI 2 3 1 HI = = . Suy ra SI = = . Di\u1ec7n t\u00edch tam gi\u00e1c SBC l\u00e0 SI \u00b7 BC = tan \u03b1 tan \u03b1 HI cos \u03b1 \u00b7 sin \u03b1 2 1 9 SI \u00b7 2AI = sin2 \u03b1\u00b7 . cos \u03b1 2 \u0110\u1ec3 th\u1ec3 t\u00edch kh\u1ed1i ch\u00f3p S.ABC nh\u1ecf nh\u1ea5t th\u00ec S SBC ph\u1ea3i nh\u1ecf nh\u1ea5t, t\u1ee9c l\u00e0 sin2 \u03b1 \u00b7 cos \u03b1 l\u1edbn nh\u1ea5t. \u0110\u1eb7t t = cos \u03b1, v\u1edbi 0 \u2264 t \u2264 1, khi \u0111\u00f3 sin2 \u03b1 \u00b7 cos \u03b1 = (1 \u2212\u221acos2 \u03b1) cos \u03b1 = t \u2212 t3 = f (t). Kh\u1ea3o s\u00e1t h\u00e0m s\u1ed1 f (t) tr\u00ean [0; 1], ta t\u00ecm \u0111\u01b0\u1ee3c max f (t) = \u221a1 \u21d4 cos \u03b1 = 3 [0;1] 3 3 . Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 120 (C\u00e2u 31 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). \u00d4ng A d\u1ef1 \u0111\u1ecbnh s\u1eed d\u1ee5ng h\u1ebft 6,5 m2 k\u00ednh \u0111\u1ec3 l\u00e0m m\u1ed9t b\u1ec3 c\u00e1 b\u1eb1ng k\u00ednh c\u00f3 d\u1ea1ng h\u00ecnh h\u1ed9p ch\u1eef nh\u1eadt kh\u00f4ng n\u1eafp, chi\u1ec1u d\u00e0i g\u1ea5p \u0111\u00f4i chi\u1ec1u r\u1ed9ng (c\u00e1c m\u1ed1i gh\u00e9p c\u00f3 k\u00edch th\u01b0\u1edbc kh\u00f4ng \u0111\u00e1ng k\u1ec3). B\u1ec3 c\u00e1 c\u00f3 dung t\u00edch l\u1edbn nh\u1ea5t b\u1eb1ng bao nhi\u00eau (k\u1ebft qu\u1ea3 l\u00e0m tr\u00f2n \u0111\u1ebfn h\u00e0ng ph\u1ea7n tr\u0103m)? A 2,26 m3. B 1,61 m3. C 1,33 m3. D 1,50 m3. \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t chi\u1ec1u r\u1ed3ng l\u00e0 x (m); chi\u1ec1u cao l\u00e0 h (m) (v\u1edbi x, h > 0). Ta c\u00f3 2x2 + 2xh + 4xh = 6, 5 \u21d4 h = 6,5 \u2212 2x2 . 6x \u221a 13 h Do h > 0, x > 0 n\u00ean 6,5 \u2212 2x2 > 0 \u21d4 0 < x < . x 2 \u221a 6,5x \u2212 2x3 \u00c7 13 \u00e5 L\u1ea1i c\u00f3 V = 2x2h = = f (x), v\u1edbi x \u2208 0; 2 3\u221a f (x) = 13 \u2212 2x2, f (x) = 0 \u21d4 x = \u00b1 39 . 2x 66 x 0 \u221a \u221a f (x) 39 13 2 6 f (x) +0\u2212 \u221a 13 39 54 \u00c7 \u221a \u00e5 \u221a 39 13 39 V\u1eady V f = \u2248 1,50 m3. 6 54 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 121 (C\u00e2u 26 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). \u00d4ng A d\u1ef1 \u0111\u1ecbnh s\u1eed d\u1ee5ng h\u1ebft 6,7 m2 k\u00ednh \u0111\u1ec3 l\u00e0m m\u1ed9t b\u1ec3 c\u00e1 b\u1eb1ng k\u00ednh c\u00f3 d\u1ea1ng h\u00ecnh h\u1ed9p ch\u1eef nh\u1eadt kh\u00f4ng n\u1eafp, chi\u1ec1u d\u00e0i g\u1ea5p \u0111\u00f4i chi\u1ec1u r\u1ed9ng (c\u00e1c m\u1ed1i gh\u00e9p c\u00f3 k\u00edch th\u01b0\u1edbc kh\u00f4ng \u0111\u00e1ng k\u1ec3). B\u1ec3 c\u00e1 c\u00f3 dung t\u00edch l\u1edbn nh\u1ea5t b\u1eb1ng bao nhi\u00eau (k\u1ebft qu\u1ea3 l\u00e0m tr\u00f2n \u0111\u1ebfn h\u00e0ng ph\u1ea7n tr\u0103m)? A 1,57 m3. B 1,11 m3. C 1,23 m3. D 2,48 m3. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 435","3. Kh\u00e1i ni\u1ec7m v\u1ec1 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i \u0111a di\u1ec7n G\u1ecdi a, b, c l\u1ea7n l\u01b0\u1ee3t l\u00e0 chi\u1ec1u d\u00e0i, chi\u1ec1u r\u1ed9ng v\u00e0 chi\u1ec1u cao c\u1ee7a b\u1ec3 c\u00e1 (a, b, c > 0). \u00aea = 2b \uf8f1a = 2b \uf8f2 Theo \u0111\u1ec1 b\u00e0i, ta c\u00f3 2ac + 2bc + ab = 6,7 \u21d4 6,7 \u2212 2b2 . \uf8f3c = 6b Th\u1ec3 t\u00edch b\u1ec3 c\u00e1 l\u00e0 V = abc = 2b2(6,7 \u2212 2b2) 6,7b \u2212 2b3 = = f (b). 6b 3 6,7b \u2212 2b3 X\u00e9t h\u00e0m s\u1ed1 f (b) = v\u1edbi b > 0. 3 6,7 \u2212 6b2 \u2026 67 Ta c\u00f3 f (b) = , f (b) = 0 \u21d4 b = , f (b) \u2248 1,57. 3 60 B\u1ea3ng bi\u1ebfn thi\u00ean b0 \u2026 67 +\u221e f (b) 60 f (b) +0\u2212 0 1,57 \u2212\u221e D\u1ef1a v\u00e0o b\u1ea3ng bi\u1ebfn thi\u00ean, dung t\u00edch l\u1edbn nh\u1ea5t c\u1ee7a b\u1ec3 c\u00e1 g\u1ea7n b\u1eb1ng 1,57. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 436 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U 2 B\u00c0I 1. KH\u00c1I NI\u1ec6M V\u1ec0 M\u1eb6T TR\u00d2N XOAY \u0104 C\u00e2u 1 (C\u00e2u 3 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u1ee5 tr\u00f2n xoay c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r v\u00e0 chi\u1ec1u cao h b\u1eb1ng A 1 \u03c0r2h. B 2\u03c0rh. C 4 \u03c0r2h. D \u03c0r2h. 3 3 \u0253 L\u1eddi gi\u1ea3i. C\u00f4ng th\u1ee9c t\u00ednh th\u1ec3 t\u00edch kh\u1ed1i tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r v\u00e0 chi\u1ec1u cao h l\u00e0 V = \u03c0r2h. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 2 (C\u00e2u 8 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n c\u00f3 chi\u1ec1u cao h v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y r l\u00e0 A 1 \u03c0r2h. B \u03c0r2h. C 4 \u03c0r2h. D 2\u03c0r2h. 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n c\u00f3 chi\u1ec1u cao h v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y r l\u00e0 V = 1 \u03c0r2h. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 3 (C\u00e2u 12 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Th\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y B v\u00e0 c\u00f3 chi\u1ec1u cao h l\u00e0 A 3Bh. B Bh. C 4 D 1 Bh. Bh. 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i l\u0103ng tr\u1ee5 c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y B v\u00e0 c\u00f3 chi\u1ec1u cao h l\u00e0 V = Bh. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 4 (C\u00e2u 3 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n c\u00f3 chi\u1ec1u cao h v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y r l\u00e0 A \u03c0r2h. B 2\u03c0r2h. C 1 \u03c0r2h. D 4 \u03c0r2h. 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n c\u00f3 chi\u1ec1u cao h v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y r l\u00e0 V = 1 \u03c0r2h. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 437 S\u0110T: 0905.193.688","1. Kh\u00e1i ni\u1ec7m v\u1ec1 m\u1eb7t tr\u00f2n xoay \u0104 C\u00e2u 5 (C\u00e2u 6 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n c\u00f3 chi\u1ec1u cao h v\u00e0 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r l\u00e0 A \u03c0r2h. B 4 \u03c0r2h. C 2\u03c0r2h. D 1 \u03c0r2h. 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n c\u00f3 chi\u1ec1u cao h v\u00e0 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r l\u00e0 V = 1 \u03c0r2h. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 6 (C\u00e2u 13 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). Th\u1ec3 t\u00edch kh\u1ed1i n\u00f3n c\u00f3 chi\u1ec1u cao h v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y r l\u00e0 A 2\u03c0r2h. B \u03c0r2h. C 1 \u03c0r2h. D 4 \u03c0r2h. 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i n\u00f3n c\u00f3 chi\u1ec1u cao h v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y r l\u00e0 V = 1 \u03c0r2h. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 7 (C\u00e2u 7 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). Cho kh\u1ed1i n\u00f3n c\u00f3 chi\u1ec1u cao h = 3 v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y r = 4. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n \u0111\u00e3 cho b\u1eb1ng A 16\u03c0. B 48\u03c0. C 36\u03c0. D 4\u03c0. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 V = 1 \u00b7 h = 1 \u03c0r2h = 1 \u03c0 \u00b7 42 \u00b7 3 = 16\u03c0. B 3 3 3 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 8 (C\u00e2u 12 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho kh\u1ed1i n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 5 v\u00e0 chi\u1ec1u cao h = 2. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n \u0111\u00e3 cho b\u1eb1ng A 10\u03c0 B 10\u03c0. C 50\u03c0 D 50\u03c0. . . 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n \u0111\u00e3 cho b\u1eb1ng V = 1\u03c0r2h = 1 \u03c052 \u00b7 2 = 50\u03c0 33 . 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 9 (C\u00e2u 2 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho kh\u1ed1i n\u00f3n c\u00f3 b\u00e1n k\u00ednh r = 2, chi\u1ec1u cao h = 5. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n \u0111\u00e3 cho b\u1eb1ng A 20\u03c0 B 20\u03c0. C 10\u03c0 D 10\u03c0. . . 33 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch V = 1\u03c0r2h = 1 \u03c0 \u00b7 22 \u00b7 5 = 20\u03c0 33 . 3 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 10 (C\u00e2u 5 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho kh\u1ed1i c\u1ea7u c\u00f3 b\u00e1n k\u00ednh r = 2. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i c\u1ea7u \u0111\u00e3 cho b\u1eb1ng A 16\u03c0. B 32\u03c0 C 32\u03c0. D 8\u03c0 . . 3 3 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 438 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U \u0253 L\u1eddi gi\u1ea3i. 4\u03c0r3 4\u03c0 \u00b7 23 32\u03c0 Th\u1ec3 t\u00edch kh\u1ed1i c\u1ea7u V = = =. 3 33 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 11 (C\u00e2u 9 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho kh\u1ed1i n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 2 v\u00e0 chi\u1ec1u cao h = 4. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n \u0111\u00e3 cho b\u1eb1ng A 8\u03c0. B 8\u03c0 C 16\u03c0 D 16\u03c0. . . 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i n\u00f3n: V = 1 \u00b7 \u03c0 \u00b7 r2 \u00b7 h = 1 \u00b7 \u03c0 \u00b7 22 \u00b7 4 = 16\u03c0 . 33 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 12 (C\u00e2u 17 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Cho kh\u1ed1i tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 4 v\u00e0 chi\u1ec1u cao h = 3. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 48\u03c0. B 4\u03c0. C 16\u03c0. D 24\u03c0. \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u1ee5 l\u00e0 V = \u03c0r2h = \u03c0 \u00b7 42 \u00b7 3 = 48\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 13 (C\u00e2u 8 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Cho kh\u1ed1i tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 5 v\u00e0 chi\u1ec1u cao h = 3. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 5\u03c0. B 30\u03c0. C 25\u03c0. D 75\u03c0. \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 5 v\u00e0 chi\u1ec1u cao h = 3 l\u00e0 V = \u03c0r2h = \u03c0 \u00b7 52 \u00b7 3 = 75\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 14 (C\u00e2u 13 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). Cho kh\u1ed1i tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 3 v\u00e0 chi\u1ec1u cao h = 5. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 45\u03c0. B 5\u03c0. C 15\u03c0. D 30\u03c0. \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i tr\u1ee5 l\u00e0 V = \u03c0r2h = \u03c0 \u00b7 32 \u00b7 5 = 45\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 15 (C\u00e2u 24 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho kh\u1ed1i tr\u1ee5 b\u00e1n k\u00ednh \u0111\u00e1y r = 6 v\u00e0 chi\u1ec1u cao h = 3. Th\u1ec3 t\u00edch kh\u1ed1i tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 108\u03c0. B 36\u03c0. C 18\u03c0. D 54\u03c0. \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i tr\u1ee5 V = \u03c0r2h = \u03c0 \u00b7 62 \u00b7 3 = 108\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 439 S\u0110T: 0905.193.688","1. Kh\u00e1i ni\u1ec7m v\u1ec1 m\u1eb7t tr\u00f2n xoay \u0104 C\u00e2u 16 (C\u00e2u 28 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho kh\u1ed1i tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 4 v\u00e0 chi\u1ec1u cao h = 3. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 16\u03c0. B 48\u03c0. C 36\u03c0. D 12\u03c0. \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i tr\u1ee5 \u0111\u00e3 cho l\u00e0 V = \u03c0r2h = \u03c0 \u00b7 42 \u00b7 3 = 48\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 17 (C\u00e2u 28 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho kh\u1ed1i tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 2 v\u00e0 chi\u1ec1u cao h = 3. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 12\u03c0. B 18\u03c0. C 6\u03c0. D 4\u03c0. \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u1ee5 \u0111\u00e3 cho l\u00e0 V = h \u00b7 \u03c0 \u00b7 r2 = 12\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 18 (C\u00e2u 14 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho kh\u1ed1i l\u0103ng tr\u1ee5 c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y B = 5a2 v\u00e0 chi\u1ec1u cao l\u00e0 h = a. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng B 5a3. C 5 a3. D 5 a3. A 5 a3. 6 2 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 V = Bh = 5a2 \u00b7 a = 5a3. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 19 (C\u00e2u 20 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i n\u00f3n c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y 3a2 v\u00e0 chi\u1ec1u cao 2a. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n \u0111\u00e3 cho l\u00e0 A 3a3. B 6a3. C 2a3. D 2 a3 . 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n \u0111\u00e3 cho l\u00e0 V = 1 B.h = 1 \u00b7 3a2.2a = 2a3. 3 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 20 (C\u00e2u 12 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 tr\u00f2n xoay c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh l b\u1eb1ng A \u03c0rl. B 4\u03c0rl. C 2\u03c0rl. D 4 \u03c0rl. 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 tr\u00f2n xoay Sxq = 2\u03c0rl. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 21 (C\u00e2u 3 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n c\u00f3 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh l v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y r b\u1eb1ng A 4\u03c0rl. B 2\u03c0rl. C \u03c0rl. D 1 \u03c0rl. 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n c\u00f3 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh l v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y r l\u00e0: Sxq = \u03c0rl. Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 440 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U \u0104 C\u00e2u 22 (C\u00e2u 12 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 c\u00f3 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh l v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y r b\u1eb1ng A 4\u03c0rl. B \u03c0rl. C 1 D 2\u03c0rl. \u03c0rl. 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 l\u00e0 Sxq = 2\u03c0rl. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 23 (C\u00e2u 32 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian, cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A, AB = a v\u00e0 AC = 2a. Khi quay tam gi\u00e1c ABC xung quanh c\u1ea1nh g\u00f3c vu\u00f4ng AB th\u00ec \u0111\u01b0\u1eddng g\u1ea5p kh\u00fac ACB t\u1ea1o th\u00e0nh m\u1ed9t h\u00ecnh n\u00f3n. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n\u221a\u0111\u00f3 b\u1eb1ng \u221a A 5\u03c0a2. B 5\u03c0a2. C 2 5\u03c0a2. D 10\u03c0a2. \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng sinh\u221ac\u1ee7a h\u00ecnh n\u00f3n l\u00e0 \u221a B = BC = AB2 + AC2 = a2 + (2a)2 = 5a. a Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n c\u00f3 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh 2a , b\u00e1n k\u00ednh \u0111\u00e1y r l\u00e0 \u221a\u221a A Sxq = \u03c0r = \u03c0 \u00b7 AC \u00b7 BC = \u03c0 \u00b7 2a \u00b7 5a = 2 5\u03c0a2. C Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 24 (C\u00e2u 7 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 8 v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh = 3. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 24\u03c0. B 192\u03c0. C 48\u03c0. D 64\u03c0. \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 Sxq = 2\u03c0r = 2\u03c0 \u00b7 8 \u00b7 3 = 48\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 25 (C\u00e2u 3 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 4 v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh = 3. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 48\u03c0. B 12\u03c0. C 16\u03c0. D 24\u03c0. \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 4 v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh = 3 th\u00ec c\u00f3 di\u1ec7n t\u00edch xung quanh l\u00e0 Sxq = 2\u03c0r = 2\u03c0 \u00b7 4 \u00b7 3 = 24\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 26 (C\u00e2u 1 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 5 v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh l = 3. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 15\u03c0. B 25\u03c0. C 30\u03c0. D 75\u03c0. \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch xung quanh Sxq = 2\u03c0rl = 30\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 441 S\u0110T: 0905.193.688","1. Kh\u00e1i ni\u1ec7m v\u1ec1 m\u1eb7t tr\u00f2n xoay \u0104 C\u00e2u 27 (C\u00e2u 2 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 7 v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh l = 3. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 42\u03c0. B 147\u03c0. C 49\u03c0. D 21\u03c0. \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 l\u00e0 S = 2\u03c0 \u00b7 r \u00b7 l = 2\u03c0 \u00b7 7 \u00b7 3 = 42\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 28 (C\u00e2u 25 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Cho h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 2 v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh l = 5. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n \u0111\u00e3 cho b\u1eb1ng B 20\u03c0 C 10\u03c0. D 10\u03c0 A 20\u03c0. . . 3 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 Sxq = \u03c0rl = \u03c0 \u00b7 2 \u00b7 5 = 10\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 29 (C\u00e2u 13 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Cho h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 2 v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh = 7. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n \u0111\u00e3 cho b\u1eb1ng B 14\u03c0. C 14\u03c0 D 98\u03c0 A 28\u03c0. . . 3 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n b\u00e1n k\u00ednh \u0111\u00e1y r = 2 v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh = 7 l\u00e0 Sxq = \u03c0r = \u03c0 \u00b7 2 \u00b7 7 = 14\u03c0. Ghi ch\u00fa: C\u00e2u h\u1ecfi n\u00e0y \u0111\u00e3 \u0111\u01b0\u1ee3c ch\u1ec9nh s\u1eeda so v\u1edbi \u0111\u1ec1 g\u1ed1c, \u0111\u1ed5i gi\u1ea3 thi\u1ebft r = 7, = 2 th\u00e0nh r = 2, = 7. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 30 (C\u00e2u 13 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Cho m\u1eb7t c\u1ea7u c\u00f3 b\u00e1n k\u00ednh r = 4. Di\u1ec7n t\u00edch c\u1ee7a m\u1eb7t c\u1ea7u \u0111\u00e3 cho b\u1eb1ng A 16\u03c0. B 64\u03c0. C 64\u03c0 D 256\u03c0 . . 3 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch c\u1ee7a m\u1eb7t c\u1ea7u l\u00e0 S = 4\u03c0r2 = 64\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 31 (C\u00e2u 16 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Cho h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 2 v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh = 5. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n \u0111\u00e3 cho b\u1eb1ng A 10\u03c0 B 50\u03c0 C 20\u03c0. D 10\u03c0. . . 3 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch xung quanh h\u00ecnh n\u00f3n l\u00e0 Sxq = \u03c0r = \u03c0 \u00b7 2 \u00b7 5 = 10\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 442 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U \u0104 C\u00e2u 32 (C\u00e2u 21 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). Cho h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 2 v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh = 7. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n \u0111\u00e3 cho b\u1eb1ng A 28\u03c0 B 14\u03c0. C 28\u03c0. D 14\u03c0 . . 3 3 \u0253 L\u1eddi gi\u1ea3i. \u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c t\u00ednh di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n Sxq = \u03c0r = \u03c0 \u00b7 2 \u00b7 7 = 14\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 33 (C\u00e2u 4 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh l. Di\u1ec7n t\u00edch xung quanh Sxq c\u1ee7a h\u00ecnh n\u00f3n \u0111\u00e3 cho \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A Sxq = \u03c0rl. B Sxq = 2\u03c0rl. C Sxq = 4\u03c0rl. D Sxq = 4 \u03c0rl. 3 \u0253 L\u1eddi gi\u1ea3i. Theo c\u00f4ng th\u1ee9c Sxq = \u03c0rl. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 34 (C\u00e2u 20 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh . Di\u1ec7n t\u00edch xung quanh Sxq c\u1ee7a h\u00ecnh n\u00f3n \u0111\u00e3 cho \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A 4 B Sxq = \u03c0r . C Sxq = 4\u03c0r . D Sxq = 2\u03c0r . Sxq = \u03c0r . 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n l\u00e0 Sxq = \u03c0r . Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 35 (C\u00e2u 11 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh l. Di\u1ec7n t\u00edch xung quanh Sxq c\u1ee7a h\u00ecnh m\u00f3n \u0111\u00e3 cho \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A Sxq = 2\u03c0rl. B Sxq = 4 C Sxq = \u03c0rl. D Sxq = 4\u03c0rl. \u03c0rl. 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c l\u00e0 Sxq = \u03c0rl. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 36 (C\u00e2u 25 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh . Di\u1ec7n t\u00edch xung quanh Sxq c\u1ee7a h\u00ecnh n\u00f3n \u0111\u00e3 cho \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A Sxq = 4\u03c0r . B Sxq = \u03c0r . C Sxq = 4 . D Sxq = 2\u03c0r . \u03c0r 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch xung quanh Sxq c\u1ee7a h\u00ecnh n\u00f3n \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c Sxq = \u03c0r . Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 443 S\u0110T: 0905.193.688","1. Kh\u00e1i ni\u1ec7m v\u1ec1 m\u1eb7t tr\u00f2n xoay \u0104 C\u00e2u 37 (C\u00e2u 13 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00ecnh tr\u1ee5 c\u00f3 chi\u1ec1u cao h = 1 v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y r = 2. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 3\u03c0. B 4\u03c0. C 2\u03c0. D 6\u03c0. \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch xung quanh Sxq = 2\u03c0rl = 4\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 38 (C\u00e2u 24 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh l. Di\u1ec7n t\u00edch xung quanh Sxq c\u1ee7a h\u00ecnh tr\u1ee5 \u0111\u00e3 cho \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A Sxq = 4\u03c0rl. B Sxq = 2\u03c0rl. C Sxq = 3\u03c0rl. D Sxq = \u03c0rl. \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 l\u00e0 Sxq = 2\u03c0rl. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 39 (C\u00e2u 19 - M\u0110 102 - BGD\u221a&\u0110T - N\u0103m 2016 - 2017). Cho kh\u1ed1i n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 3 v\u00e0 chi\u1ec1u cao h = 4. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i n\u00f3n \u0111\u00e3 cho. \u221a \u221a A V = 16\u03c0 3 C V = 16\u03c0 3. 3 . B V = 4\u03c0. D V = 12\u03c0. \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i n\u00f3n \u0111\u00e3 cho l\u00e0 V = 1 \u03c0r2.h = 1 \u03c0.3.4 = 4\u03c0. 33 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 40 (C\u00e2u 18 - M\u0110 104 - BG\u221aD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 3 v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh l = 4. T\u00ednh di\u1ec7n t\u00edch xung quanh Sxq c\u1ee7a h\u00ecnh n\u00f3n \u0111\u00e3 cho. \u221a \u221a \u221a C Sxq = 39\u03c0. D Sxq = 8 3\u03c0. A Sxq = 12\u03c0. B Sxq = 4 3\u03c0. \u221a \u0253 L\u1eddi gi\u1ea3i. Sxq = \u03c0rl = 4 3\u03c0 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 41 (C\u00e2u 39 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). \u221a Trong kh\u00f4ng gian, cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A, AB = a v\u00e0 AC = 3a. T\u00ednh \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh c\u1ee7a h\u00ecnh n\u00f3n, nh\u1eadn \u0111\u01b0\u1ee3c kh\u221ai quay tam gi\u00e1c ABC x\u221aung quanh tr\u1ee5c AB. A = a. B = 2a. C = 3a. D = 2a. \u0253 L\u1eddi gi\u1ea3\u221ai. \u0110\u01b0\u1eddng sinh c\u1ee7a h\u00ecnh n\u00f3n c\u00f3 \u0111\u1ed9 d\u00e0i b\u1eb1ng \u0111o\u1ea1n BC = AB2 + AC2 = 2a. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 42 (C\u00e2u 39 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). Cho kh\u1ed1i (N ) c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y b\u1eb1ng 3 v\u00e0 di\u1ec7n t\u00edch xung quanh b\u1eb1ng 15\u03c0. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i n\u00f3n (N ) Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 444 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U A V = 12\u03c0. B V = 20\u03c0. C V = 36\u03c0. D V = 60\u03c0. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 Sxq =\u221a\u03c0rl n\u00ean 15\u03c0 = 3\u03c0l \u21d2 l = 5. Suy ra h = l2 \u2212 r2 = 4. 1 = 1 h\u03c0r2 = 12\u03c0. Do \u0111\u00f3 V = hS 3 2 \u0110\u00e1y Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 43 (C\u00e2u 17 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). Cho kh\u1ed1i n\u00f3n c\u00f3 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh b\u1eb1ng 2a v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y b\u1eb1ng a. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n \u0111\u00e3 cho b\u1eb1n\u221ag 3\u03c0a3 \u221a 3\u03c0a3 2\u03c0a3 \u03c0a3 . . . . A 3 B 2 C 3 D 3 \u0253 L\u1eddi gi\u1ea3i. \u221a\u221a Ta c\u00f3 l = 2a; r = a, suy ra h = l2 \u2212 r2\u221a= 3a. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n l\u00e0 V = 1 \u03c0r2h = 3\u03c0a3 . 33 lh r Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 44 (C\u00e2u 24 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). M\u1ed9t kh\u1ed1i \u0111\u1ed3 ch\u01a1i g\u1ed3m hai kh\u1ed1i tr\u1ee5 (H1), (H2) x\u1ebfp ch\u1ed3ng l\u00ean nhau, l\u1ea7n l\u01b0\u1ee3t 1 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y v\u00e0 chi\u1ec1u cao t\u01b0\u01a1ng \u1ee9ng l\u00e0 r1, h1, r2, h2 th\u1ecfa m\u00e3n r2 = 2 r1, h2 = 2h1 (tham kh\u1ea3o h\u00ecnh v\u1ebd b\u00ean). Bi\u1ebft r\u1eb1ng th\u1ec3 t\u00edch c\u1ee7a to\u00e0n b\u1ed9 kh\u1ed1i \u0111\u1ed3 ch\u01a1i b\u1eb1ng 30 cm3, th\u1ec3 t\u00edch kh\u1ed1i tr\u1ee5 (H1) b\u1eb1ng A 24 cm3. B 15 cm3. C 20 cm3. D 10 cm3. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi V1, V2 l\u1ea7n l\u01b0\u1ee3t l\u00e0 th\u1ec3 t\u00edch c\u1ee7a hai kh\u1ed1i tr\u1ee5 (H1), (H2). Ta c\u00f3 V2 = h2\u03c0r22 = 2h1\u03c0 1 r12 = 1 h1\u03c0r12 = 1 M\u00e0 V1 + 1 = 30 n\u00ean V1 = 20. 4 2 2 V1. 2 V1 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 45 (C\u00e2u 10 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho kh\u1ed1i n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 4 v\u00e0 chi\u1ec1u cao h = 2. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n \u0111\u00e3 cho b\u1eb1ng Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 445 S\u0110T: 0905.193.688","1. Kh\u00e1i ni\u1ec7m v\u1ec1 m\u1eb7t tr\u00f2n xoay A 8\u03c0 B 8\u03c0. C 32\u03c0 D 32\u03c0. . . 3 3 1r2\u03c0h = 1 \u00b7 42 \u00b7 \u03c0 \u00b7 2 = 32\u03c0 \u0253 L\u1eddi gi\u1ea3i. . Ta c\u00f3 V = 33 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 46 (C\u00e2u 23 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Cho kh\u1ed1i tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 3 v\u00e0 chi\u1ec1u cao h = 4. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 4\u03c0. B 12\u03c0. C 36\u03c0. D 24\u03c0. \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i tr\u1ee5 l\u00e0 V = \u03c0r2h = \u03c0 \u00b7 32 \u00b7 4 = 36\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 47 (C\u00e2u 10 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho kh\u1ed1i n\u00f3n c\u00f3 di\u1ec7n t\u00edch \u0111\u00e1y b\u1eb1ng 3a2 v\u00e0 chi\u1ec1u cao 2a. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n \u0111\u00e3 cho b\u1eb1ng A 3a3. B 6a3. C 2a3. D 2 a3 . 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n \u0111\u00e3 cho b\u1eb1ng V = 1 \u00b7 3a2.2a = 2a3. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 48 (C\u00e2u 41 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian, cho h\u00ecnh ch\u1eef nh\u1eadt ABCD c\u00f3 AB = 1 v\u00e0 AD = 2. G\u1ecdi M, N l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AD v\u00e0 BC. Quay h\u00ecnh ch\u1eef nh\u1eadt \u0111\u00f3 xung quanh tr\u1ee5c M N , ta \u0111\u01b0\u1ee3c m\u1ed9t h\u00ecnh tr\u1ee5. T\u00ednh di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n Stp c\u1ee7a h\u00ecnh tr\u1ee5 \u0111\u00f3. A Stp = 4\u03c0. B Stp = 2\u03c0. C Stp = 6\u03c0. D Stp = 10\u03c0. \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 1, chi\u1ec1u cao h = 1 n\u00ean c\u00f3 Stp = 2\u03c0r2 + 2\u03c0rh = 4\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 49 (C\u00e2u 26 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00ecnh n\u00f3n c\u00f3 di\u1ec7n t\u00edch xung quanh b\u1eb1ng 3\u03c0a2 v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y b\u1eb1ng a. T\u00ednh \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh l c\u1ee7a \u221ah\u00ecnh n\u00f3n \u0111\u00e3 cho. \u221a 3a 5a l = 2 2a. . A l= . B C l = D l = 3a. 2 2 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n: Sxq = \u03c0rl = \u03c0al = 3\u03c0a2 \u21d2 l = 3a. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 50 (C\u00e2u 14 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). Cho h\u00ecnh n\u00f3n c\u00f3 di\u1ec7n t\u00edch xung quanh b\u1eb1ng 3\u03c0a2 v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y b\u1eb1ng a. \u0110\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh c\u1ee7a h\u00ecnh n\u00f3n \u0111\u00e3 cho b\u1eb1ng B 3a. C 2a. D 3a \u221a . 2 A 2 2a. = Sxq = 3\u03c0a2 = 3a. \u0253 L\u1eddi gi\u1ea3i. \u03c0r \u03c0a Ta c\u00f3 Sxq = \u03c0r \u21d2 Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 446 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U \u0104 C\u00e2u 51 (C\u00e2u 38 - M\u0110 101 - B\u221aGD&\u0110T - N\u0103m 2018 - 2019). Cho h\u00ecnh tr\u1ee5 c\u00f3 chi\u1ec1u cao b\u1eb1ng 5 3. C\u1eaft h\u00ecnh tr\u1ee5 \u0111\u00e3 cho b\u1edfi m\u1eb7t ph\u1eb3ng song song v\u1edbi tr\u1ee5c v\u00e0 c\u00e1ch tr\u1ee5c m\u1ed9t kho\u1ea3ng b\u1eb1ng 1, thi\u1ebft di\u1ec7n thu \u0111\u01b0\u1ee3c c\u00f3 di\u1ec7n t\u00edch b\u1eb1ng 30. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh\u221atr\u1ee5 \u0111\u00e3 cho b\u1eb1ng \u221a \u221a \u221a A 10 3\u03c0. B 5 39\u03c0. C 20 3\u03c0. D 10 39\u03c0. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi O, O l\u1ea7n l\u01b0\u1ee3t l\u00e0 t\u00e2m c\u1ee7a hai \u0111\u00e1y v\u00e0 ABCD l\u00e0 thi\u1ebft di\u1ec7n song song v\u1edbi B tr\u1ee5c v\u1edbi A, B \u2208 (O); C, D \u2208 (O ). G\u1ecdi H l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB \u21d2 OH = d(OO , (ABCD)) = 1. H O A V\u00ec SABCD = 30 \u21d4 AB \u00b7 BC = 30. Suy ra AB = 3\u221a0 \u221a HB = \u221a 3. = 2 3 \u21d2 HA = 53 \u221a \u221a B\u00e1n k\u00ednh c\u1ee7a \u0111\u00e1y l\u00e0 r = OH2 + HA2 = 3 + 1 = 2. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 C \u221a\u221a O Sxq = 2\u03c0rh = 2\u03c0 \u00b7 2 \u00b7 5 3 = 20 3\u03c0. D Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 52 (C\u00e2u 22 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). M\u1ed9t c\u01a1 s\u1edf s\u1ea3n xu\u1ea5t c\u00f3 hai b\u1ec3 n\u01b0\u1edbc h\u00ecnh tr\u1ee5 c\u00f3 chi\u1ec1u cao b\u1eb1ng nhau, b\u00e1n k\u00ednh \u0111\u00e1y l\u1ea7n l\u01b0\u1ee3t b\u1eb1ng 1 m v\u00e0 1, 5 m. Ch\u1ee7 c\u01a1 s\u1edf d\u1ef1 \u0111\u1ecbnh l\u00e0m m\u1ed9t b\u1ec3 n\u01b0\u1edbc m\u1edbi, h\u00ecnh tr\u1ee5, c\u00f3 c\u00f9ng chi\u1ec1u cao v\u00e0 th\u1ec3 t\u00edch b\u1eb1ng t\u1ed5ng th\u1ec3 t\u00edch c\u1ee7a hai b\u1ec3 n\u01b0\u1edbc tr\u00ean. B\u00e1n k\u00ednh \u0111\u00e1y c\u1ee7a b\u1ec3 n\u01b0\u1edbc d\u1ef1 \u0111\u1ecbnh l\u00e0m g\u1ea7n nh\u1ea5t v\u1edbi k\u1ebft qu\u1ea3 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A 1, 6 m. B 2, 5 m. C 1, 8 m. D 2, 1 m. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi h l\u00e0 chi\u1ec1u cao c\u1ee7a c\u00e1c b\u1ec3 n\u01b0\u1edbc v\u00e0 r l\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y c\u1ee7a b\u1ec3 n\u01b0\u1edbc d\u1ef1 \u0111\u1ecbnh l\u00e0m. Theo gi\u1ea3 thi\u1ebft, ta c\u00f3 \u03c0r2h = \u03c0 \u00b7 12 \u00b7 h + \u03c0 \u00b7 (1, 5)2 \u00b7h \u21d4 r2 = 1 + 9 = 13 . \u221a 44 Suy ra r = 13 \u2248 1, 8 m. 2 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 53 (C\u00e2u 22 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y b\u1eb1ng 3. Bi\u1ebft r\u1eb1ng khi c\u1eaft h\u00ecnh tr\u1ee5 \u0111\u00e3 cho b\u1edfi m\u1ed9t m\u1eb7t ph\u1eb3ng qua tr\u1ee5c, thi\u1ebft di\u1ec7n thu \u0111\u01b0\u1ee3c l\u00e0 m\u1ed9t h\u00ecnh vu\u00f4ng. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 18\u03c0. B 36\u03c0. C 54\u03c0. D 27\u03c0. \u0253 L\u1eddi gi\u1ea3i. Theo gi\u1ea3 thi\u1ebft m\u1eb7t ph\u1eb3ng \u0111i qua tr\u1ee5c OO c\u1eaft h\u00ecnh tr\u1ee5 theo thi\u1ebft di\u1ec7n m\u1ed9t h\u00ecnh vu\u00f4ng. Do \u0111\u00f3: AD = AB = 2R = 6. H\u00ecnh tr\u1ee5 \u0111\u00e3 cho c\u00f3 R = 3, chi\u1ec1u cao h = 6, do \u0111\u00f3 c\u00f3 di\u1ec7n t\u00edch xung quanh l\u00e0: Sxq = 2\u03c0.R.h = 36\u03c0 (\u0111vdt). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 54 (C\u00e2u 35 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y b\u1eb1ng 2 v\u00e0 g\u00f3c \u1edf \u0111\u1ec9nh b\u1eb1ng 60\u25e6. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n \u0111\u00e3 cho b\u1eb1ng Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 447 S\u0110T: 0905.193.688"]
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