["1. Kh\u00e1i ni\u1ec7m v\u1ec1 m\u1eb7t tr\u00f2n xoay 16 \u221a 3\u03c0 C8 \u221a 3\u03c0 . . A 8\u03c0. B 3 3 D 16\u03c0. \u0253 L\u1eddi gi\u1ea3i. S 60\u25e6 Ta c\u00f3 SAB l\u00e0 tam gi\u00e1c \u0111\u1ec1u n\u00ean \u0111\u01b0\u1eddng sinh c\u1ee7a h\u00ecnh n\u00f3n l\u00e0 = SA = AB = 2r = 2 \u00b7 2 = 4. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n l\u00e0 Sxq = \u03c0r = \u03c0 \u00b7 2 \u00b7 4 = 8\u03c0. r=2 B AO Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 55 (C\u00e2u 36 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y b\u1eb1ng 5 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 100 \u221a 3\u03c0 C 50 \u221a 3\u03c0 A 50\u03c0. . . B 3 3 D 100\u03c0. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 sin 30\u25e6 = r \u21d2 l = r = 5 = 10. S l sin 30\u25e6 1 h 30\u25e6 l 2 Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n l\u00e0 Sxq = \u03c0rl = \u03c0.5.10 = 50\u03c0. AB O r=5 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 56 (C\u00e2u 27 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh b\u1eb1ng 3 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 B 36\u03c0. \u221a \u221a A 18\u03c0. C 6 3\u03c0. D 12 3\u03c0. \u0253 L\u1eddi gi\u1ea3i. Tam gi\u00e1c SAB c\u00f3 SA = SB = , A\u2019SB = 60\u25e6 \u21d2 SAB \u0111\u1ec1u c\u00f3 r = OA = 3. \u21d2 SA = AB = 2OA = 6. Khi \u0111\u00f3 di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n l\u00e0 Sxq = \u03c0r = \u03c0 \u00b7 3 \u00b7 6 = 18\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 57 (C\u00e2u 32 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y b\u1eb1ng 4 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 448 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U 64 \u221a 3\u03c0 32 \u221a 3\u03c0 . . A 3 B 32\u03c0. C 64\u03c0. D 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 A\u2019SB = 60\u25e6 \u21d2 H\u2019SB = 30\u25e6; HB = 4. S \u00c1p d\u1ee5ng t\u1ec9 s\u1ed1 l\u01b0\u1ee3ng gi\u00e1c cho SHB ta c\u00f3 AB sin 30\u25e6 = HB \u21d2 SB = HB = 4 = 8. H SB sin 30\u25e6 1 2 V\u1eady Sxq = \u03c0rl = \u03c0 \u00b7 HB \u00b7 SB = \u03c0 \u00b7 8 \u00b7 4 = 32\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 58 (C\u00e2u 34 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). C\u1eaft h\u00ecnh tr\u1ee5 (T ) b\u1edfi m\u1ed9t m\u1eb7t ph\u1eb3ng qua tr\u1ee5c c\u1ee7a n\u00f3, ta \u0111\u01b0\u1ee3c thi\u1ebft di\u1ec7n l\u00e0 m\u1ed9t h\u00ecnh vu\u00f4ng c\u1ea1nh b\u1eb1ng 7. Di\u1ec7n t\u00edch xung quanh c\u1ee7a (T ) b\u1eb1ng A 49\u03c0 B 49\u03c0 C 49\u03c0. D 98\u03c0. . . 4 2 \u0253 L\u1eddi gi\u1ea3i. Gi\u1ea3 s\u1eed h\u00ecnh tr\u1ee5 c\u00f3 t\u00e2m hai \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y l\u00e0 O v\u00e0 O c\u00f9ng v\u1edbi thi\u1ebft di\u1ec7n D O qua tr\u1ee5c l\u00e0 h\u00ecnh vu\u00f4ng ABCD nh\u01b0 h\u00ecnh v\u1ebd. A 17 Ta c\u00f3 h = AD = 7, r = AO = AB = . 22 7 C 2 Di\u1ec7n t\u00edch xung quanh Sxq = 2\u03c0rh = 2 \u00b7 \u03c0 \u00b7 \u00b7 7 = 49\u03c0. O B Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 59 (C\u00e2u 29 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). C\u1eaft h\u00ecnh tr\u1ee5 (T ) b\u1edfi m\u1ed9t m\u1eb7t ph\u1eb3ng qua tr\u1ee5c c\u1ee7a n\u00f3, ta \u0111\u01b0\u1ee3c thi\u1ebft di\u1ec7n l\u00e0 m\u1ed9t h\u00ecnh vu\u00f4ng c\u1ea1nh b\u1eb1ng 1. Di\u1ec7n t\u00edch xung quanh c\u1ee7a (T ) b\u1eb1ng \u03c0 \u03c0 A \u03c0. B . C 2\u03c0. D . 2 4 \u0253 L\u1eddi gi\u1ea3i. Thi\u1ebft di\u1ec7n qua tr\u1ee5c c\u1ee7a (T ) l\u00e0 h\u00ecnh vu\u00f4ng ABCD c\u00f3 c\u1ea1nh 1. D 1 O C Do \u0111\u00f3 \u0111\u01b0\u1eddng cao c\u1ee7a h\u00ecnh tr\u1ee5 l\u00e0 h = 1 v\u00e0 b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y r = . 2 Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 (T ) l\u00e0 1 A Sxq = 2\u03c0rh = 2\u03c0 \u00b7 1 \u00b7 2 = \u03c0. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 449 O B S\u0110T: 0905.193.688","1. Kh\u00e1i ni\u1ec7m v\u1ec1 m\u1eb7t tr\u00f2n xoay Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 60 (C\u00e2u 28 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). C\u1eaft h\u00ecnh tr\u1ee5 (T ) b\u1edfi m\u1ed9t m\u1eb7t ph\u1eb3ng qua tr\u1ee5c c\u1ee7a n\u00f3, ta \u0111\u01b0\u1ee3c thi\u1ebft di\u1ec7n l\u00e0 h\u00ecnh vu\u00f4ng c\u1ea1nh b\u1eb1ng 3. Di\u1ec7n t\u00edch khung quanh c\u1ee7a (T ) b\u1eb1ng A 9\u03c0 B 18\u03c0. C 9\u03c0. D 9\u03c0 . . 4 2 \u0253 L\u1eddi gi\u1ea3i. Gi\u1ea3 s\u1eed ta c\u00f3 h\u00ecnh tr\u1ee5 v\u00e0 thi\u1ebft di\u1ec7n nh\u01b0 h\u00ecnh b\u00ean, thi\u1ebft di\u1ec7n l\u00e0 ph\u1ea7n g\u1ea1ch O ch\u00e9o. A Khi \u0111\u00f3, di\u1ec7n t\u00edch xung quanh c\u1ee7a (T ) \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c Sxq = 2\u03c0ON \u00b7 AA 9 = 9\u03c0. = 2\u03c0 2 M A O N Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 61 (C\u00e2u 26 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). C\u1eaft h\u00ecnh tr\u1ee5 (T ) b\u1edfi m\u1ed9t m\u1eb7t ph\u1eb3ng qua tr\u1ee5c c\u1ee7a n\u00f3, ta \u0111\u01b0\u1ee3c thi\u1ebft di\u1ec7n l\u00e0 m\u1ed9t h\u00ecnh vu\u00f4ng c\u1ea1nh b\u1eb1ng 5. Di\u1ec7n t\u00edch xung quanh c\u1ee7a (T ) b\u1eb1ng A 25\u03c0 B 25\u03c0. C 50\u03c0. D 25\u03c0 . . 2 4 \u0253 L\u1eddi gi\u1ea3i. Gi\u1ea3 s\u1eed thi\u1ebft di\u1ec7n c\u1eaft h\u00ecnh tr\u1ee5 (T ) b\u1edfi m\u1eb7t ph\u1eb3ng qua tr\u1ee5c OO l\u00e0 h\u00ecnh vu\u00f4ng B ABB A (nh\u01b0 h\u00ecnh v\u1ebd b\u00ean). Khi \u0111\u00f3 ta c\u00f3 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh l\u00e0 = AA = 5, O 5 A b\u00e1n k\u00ednh \u0111\u00e1y l\u00e0 r = OA = . 2 V\u1eady di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 (T ) l\u00e0 Sxq = 2\u03c0r = 2\u03c0 \u00b7 5 \u00b7 5 = 25\u03c0. 2 B O A Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 62 (C\u00e2u 10 - M\u0110 101 - 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 4\u03c0. B 2\u03c0. C 3\u03c0. D 6\u03c0. \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 \u0111\u00e3 cho l\u00e0 Sxq = 2\u03c0.r.h = 2\u03c0.2.1 = 4\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 450 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U \u0104 C\u00e2u 63 (C\u00e2u 24 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho tam gi\u00e1c OIM vu\u00f4ng t\u1ea1i I c\u00f3 OI = 3 v\u00e0 IM = 4. Khi quay tam gi\u00e1c OIM quanh c\u1ea1nh g\u00f3c vu\u00f4ng OI th\u00ec \u0111\u01b0\u1eddng g\u1ea5p kh\u00fac OIM t\u1ea1o th\u00e0nh h\u00ecnh n\u00f3n c\u00f3 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh b\u1eb1ng A 7. B 3. C 5. D 4. \u0253 L\u1eddi gi\u1ea3i. X\u00e9t tam gi\u00e1c OIM vu\u00f4ng t\u1ea1i I, ta c\u00f3 O OM 2 = OI2 + IM 2 \u21d4 OM 2 = 32 + 42 = 25 \u21d4 OM = 5. Khi quay tam gi\u00e1c OIM quanh c\u1ea1nh g\u00f3c vu\u00f4ng OI th\u00ec \u0111\u01b0\u1eddng g\u1ea5p kh\u00fac I M OIM t\u1ea1o th\u00e0nh h\u00ecnh n\u00f3n c\u00f3 \u0111\u01b0\u1eddng sinh l\u00e0 c\u1ea1nh huy\u1ec1n OM . V\u1eady \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh c\u1ee7a h\u00ecnh n\u00f3n l\u00e0 5. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 64 (C\u00e2u 25 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho tam gi\u00e1c OIM vu\u00f4ng t\u1ea1i I c\u00f3 OI = 3 v\u00e0 IM = 4. Khi quay tam gi\u00e1c OIM quanh c\u1ea1nh g\u00f3c vu\u00f4ng OI th\u00ec \u0111\u01b0\u1eddng g\u1ea5p kh\u00fac OM I t\u1ea1o th\u00e0nh h\u00ecnh n\u00f3n c\u00f3 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh b\u1eb1ng A 4. B 3. C 5. D 7. \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh n\u00f3n c\u00f3 chi\u1ec1u cao h = OI = 3\u221a, b\u00e1n k\u00ednh \u0111\u00e1y r\u221a= IM = 4 n\u00ean O \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh l\u00e0 = OM = IM 2 + OI2 = 32 + 42 = 5. M I Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 65 (C\u00e2u 44 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00ecnh n\u00f3n c\u00f3 g\u00f3c \u1edf \u0111\u1ec9nh b\u1eb1ng 120\u25e6 v\u00e0 chi\u1ec1u cao b\u1eb1ng 2. G\u1ecdi (S) l\u00e0 m\u1eb7t c\u1ea7u \u0111i qua \u0111\u1ec9nh v\u00e0 ch\u1ee9a \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y c\u1ee7a h\u00ecnh n\u00f3n \u0111\u00e3 cho. Di\u1ec7n t\u00edch c\u1ee7a (S) b\u1eb1ng A 16\u03c0 B 64\u03c0 C 64\u03c0. D 48\u03c0. . . 3 3 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi h\u00ecnh n\u00f3n \u0111\u1ec9nh A, \u0111\u01b0\u1eddng k\u00ednh \u0111\u00e1y h\u00ecnh n\u00f3n l\u00e0 BC. B A G\u1ecdi I l\u00e0 t\u00e2m m\u1eb7t c\u1ea7u (S). 120 \u25e6 Ta c\u00f3 ABC c\u00e2n t\u1ea1i A c\u00f3 B\u2019AC = 120\u25e6 v\u00e0 AI \u22a5 BC t\u1ea1i O n\u00ean C B\u2018AI = 60\u25e6 suy ra IAB \u0111\u1ec1u. O Tam gi\u00e1c IAB \u0111\u1ec1u v\u00e0 OB \u22a5 IA t\u1ea1i O suy ra OB l\u00e0 \u0111\u01b0\u1eddng trung tuy\u1ebfn I c\u1ee7a \u2206IAB. M\u00e0 OA = 2 suy ra AI = 2OA = 4. V\u1eady di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u (S) l\u00e0 S = 4\u03c0AI2 = 64\u03c0. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 451 S\u0110T: 0905.193.688","1. Kh\u00e1i ni\u1ec7m v\u1ec1 m\u1eb7t tr\u00f2n xoay Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 66 (C\u00e2u 40 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00ecnh l\u0103ng tr\u1ee5 tam gi\u00e1c \u0111\u1ec1u ABC.A B C c\u00f3 \u0111\u1ed9 d\u00e0i c\u1ea1nh \u0111\u00e1y b\u1eb1ng a v\u00e0 chi\u1ec1u cao b\u1eb1ng h. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i tr\u1ee5 ngo\u1ea1i ti\u1ebfp l\u0103ng tr\u1ee5 \u0111\u00e3 cho. A V = \u03c0a2h B V \u03c0a2h C V = 3\u03c0a2h. D V \u03c0a2h . =. =. 9 3 9 \u0253 L\u1eddi gi\u1ea3i. Kh\u1ed1i tr\u1ee5 ngo\u1ea1i ti\u1ebfp h\u00ecnh l\u0103ng tr\u1ee5 \u0111\u00e3 cho c\u0169ng c\u00f3 chi\u1ec1u cao l\u00e0 A h = OO , trong \u0111\u00f3 O, O l\u1ea7n l\u01b0\u1ee3t l\u00e0 t\u00e2m c\u1ee7a tam gi\u00e1c ABC v\u00e0 O tam gi\u00e1c A B C . B\u00e1n k\u00ednh \u0111\u00e1y c\u1ee7a kh\u221a\u1ed1i tr\u1ee5 ch\u00ednh l\u00e0 b\u00e1n k\u00ednh B C 3 \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp c\u1ee7a m\u1eb7t \u0111\u00e1y l\u00e0 a . 3 \u03c0a2h V\u1eady th\u1ec3 t\u00edch l\u0103ng tr\u1ee5 l\u00e0 V = . 3 A O C B Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 67 (C\u00e2u 28 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i tr\u1ee5 ngo\u1ea1i ti\u1ebfp h\u00ecnh l\u1eadp ph\u01b0\u01a1ng c\u00f3 c\u1ea1nh b\u1eb1ng a. A V = \u03c0a3 B V = \u03c0a3. C V = \u03c0a3 D V \u03c0a3 . . =. 4 6 2 \u0253 L\u1eddi gi\u1ea3i. \u221a V\u00ec kh\u1ed1i tr\u1ee5 ngo\u1ea1i ti\u1ebfp h\u00ecnh l\u1eadp ph\u01b0\u01a1ng c\u1ea1nh b\u1eb1ng a n\u00ean \uf8f1 a2 Do \u0111\u00f3 V = \u03c0R2h = \u03c0a3 \uf8f2R = 2 . . \uf8f3h = a 2 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 68 (C\u00e2u 27 - M\u0110 101 - 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 1,2 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 c\u00f3 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 1,8 m. B 1,4 m. C 2,2 m. D 1,6 m. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi R1, R2, R l\u1ea7n l\u01b0\u1ee3t l\u00e0 b\u00e1n k\u00ednh c\u1ee7a tr\u1ee5 th\u1ee9 nh\u1ea5t, th\u1ee9 hai v\u00e0 d\u1ef1 ki\u1ebfn s\u1ebd l\u00e0m, ta c\u00f3 V = V1 + V2 = \u03c0R2h \u21d4 \u03c0R12h + \u03c0R22h \u21d4 R2 = R12 + R22 \u00bb\u00bb \u21d2 R = R12 + R22 = 12 + (1,2)2 \u2248 1,56 (m). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 452 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U V\u1eady gi\u00e1 tr\u1ecb c\u1ea7n t\u00ecm l\u00e0 1,6 m. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 69 (C\u00e2u 18 - M\u0110 102 - 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 1m v\u00e0 1,4m. 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 c\u00f3 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 1,7m. B 1,5m. C 1,9m. D 2,4m. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi chi\u1ec1u cao c\u1ee7a c\u00e1c h\u00ecnh tr\u1ee5 l\u00e0 h. G\u1ecdi V1, V2 l\u1ea7n l\u01b0\u1ee3t l\u00e0 th\u1ec3 t\u00edch c\u1ee7a h\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y R1 = 1m, R2 = 1,4m. G\u1ecdi V l\u00e0 th\u1ec3 t\u00edch c\u1ee7a h\u00ecnh tr\u1ee5 d\u1ef1 \u0111\u1ecbnh l\u00e0m v\u00e0 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y l\u00e0 R. \u221a 96 \u2248 1,72 m. Ta c\u00f3 V = V1 + V2 \u21d4 \u03c0R2h = \u03c0R12h + \u03c0R22h \u21d4 R2 = R12 + R22 \u21d4 R = 2, Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 70 (C\u00e2u 11 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). \u221a T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i tr\u1ee5 c\u00f3 b\u00e1n k\u221a\u00ednh \u0111\u00e1y r = 4 v\u00e0 chi\u1ec1u cao h = 4 2. A V = 128\u03c0. \u221a B V = 64 2\u03c0. C V = 32\u03c0. D V = 32 2\u03c0. \u0253 L\u1eddi gi\u1ea3i. \u221a\u221a Th\u1ec3 t\u00edch kh\u1ed1i tr\u1ee5 V = \u03c0r2.h = \u03c0.42.4 2 = 64 2\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 71 (C\u00e2u 43 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho t\u1ee9 di\u1ec7n \u0111\u1ec1u ABCD c\u00f3 c\u1ea1nh b\u1eb1ng 3a. H\u00ecnh n\u00f3n (N ) c\u00f3 \u0111\u1ec9nh A v\u00e0 \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y l\u00e0 \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c BCD. T\u00ednh d\u221ai\u1ec7n t\u00edch xung quanh Sxq c\u1ee7a (N ). \u221a A Sxq = 6\u03c0a2. B Sxq = 3 3\u03c0a2. C Sxq = 12\u03c0a2. D Sxq = 6 3\u03c0a2. \u0253 L\u1eddi gi\u1ea3i. \u221a 2 3a 3 \u221a - B\u00e1n k\u00ednh \u0111\u00e1y R = . = a 3. 32 \u221a\u221a - Suy ra di\u1ec7n t\u00edch xung quanh Sxq = \u03c0Rl = \u03c0a 3.3a = \u03c0a23 3. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 72 (C\u00e2u 40 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). T\u1eeb m\u1ed9t t\u1ea5m t\u00f4n h\u00ecnh ch\u1eef nh\u1eadt k\u00edch th\u01b0\u1edbc 50 cm \u00d7 240 cm, ng\u01b0\u1eddi ta l\u00e0m c\u00e1c th\u00f9ng \u0111\u1ef1ng n\u01b0\u1edbc h\u00ecnh tr\u1ee5 c\u00f3 chi\u1ec1u cao b\u1eb1ng 50 cm, theo hai c\u00e1ch sau (xem h\u00ecnh minh h\u1ecda d\u01b0\u1edbi \u0111\u00e2y): C\u00e1ch 1: G\u00f2 t\u1ea5m t\u00f4n ban \u0111\u1ea7u th\u00e0nh m\u1eb7t xung quanh c\u1ee7a th\u00f9ng. C\u00e1ch 2: C\u1eaft t\u1ea5m t\u00f4n ban \u0111\u1ea7u th\u00e0nh hai t\u1ea5m b\u1eb1ng nhau, r\u1ed3i g\u00f2 m\u1ed7i t\u1ea5m \u0111\u00f3 th\u00e0nh m\u1eb7t xung quanh c\u1ee7a m\u1ed9t th\u00f9ng. K\u00ed hi\u1ec7u V1 l\u00e0 th\u1ec3 t\u00edch c\u1ee7a th\u00f9ng g\u00f2 \u0111\u01b0\u1ee3c theo c\u00e1ch 1 v\u00e0 V2 l\u00e0 t\u1ed5ng th\u1ec3 t\u00edch c\u1ee7a hai th\u00f9ng g\u00f2 \u0111\u01b0\u1ee3c theo c\u00e1ch 2. T\u00ednh t\u1ec9 s\u1ed1 V1 . V2 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 453 S\u0110T: 0905.193.688","1. Kh\u00e1i ni\u1ec7m v\u1ec1 m\u1eb7t tr\u00f2n xoay A V1 = 1 B V1 = 1. C V1 = 2. D V1 = 4. . V2 V2 V2 V2 2 \u0253 L\u1eddi gi\u1ea3i. R Ban \u0111\u1ea7u b\u00e1n k\u00ednh \u0111\u00e1y l\u00e0 R, sau khi c\u1eaft v\u00e0 g\u00f2 ta \u0111\u01b0\u1ee3c 2 kh\u1ed1i tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y l\u00e0 . 2 \u0110\u01b0\u1eddng cao c\u1ee7a c\u00e1c kh\u1ed1i tr\u1ee5 kh\u00f4ng thay \u0111\u1ed5i. Ta c\u00f3: V1 = Sd \u00b7h = \u03c0R2 \u00b7 h; V2 = 2 (Sd1 \u00b7 h) = \u00c5 R \u00e32 \u00b7h = \u03c0R2h . 2\u03c0 2 2 Khi \u0111\u00f3: V1 = 2 . V2 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 73 (C\u00e2u 42 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). Cho hai h\u00ecnh vu\u00f4ng c\u00f3 c\u00f9ng c\u1ea1nh b\u1eb1ng 5 \u0111\u01b0\u1ee3c x\u1ebfp ch\u1ed3ng l\u00ean nhau Q P sao cho \u0111\u1ec9nh X c\u1ee7a m\u1ed9t h\u00ecnh vu\u00f4ng l\u00e0 t\u00e2m c\u1ee7a h\u00ecnh vu\u00f4ng c\u00f2n l\u1ea1i (nh\u01b0 h\u00ecnh v\u1ebd). T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a v\u1eadt th\u1ec3 tr\u00f2n xoay khi quay m\u00f4 h\u00ecnh tr\u00ean xung \u00c4quan\u221ah tr\u1ee5c XY . \u00c4 \u221a\u00e4 X \u00e4 AV= 125 1 + 2 \u03c0 B V= 125 5 + 2 2 \u03c0 \u00c4 6 \u221a\u00e4 . \u00c4 12\u221a \u00e4 . CV= 125 5 + 4 2 \u03c0 DV= 125 2 + 2 \u03c0 24 . 4 . M N A B Y \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 454 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U Ta th\u1ea5y r\u1eb1ng khi xoay h\u00ecnh xung quanh tr\u1ee5c XY th\u00ec h\u00ecnh vu\u00f4ng 5 \u1edf tr\u00ean s\u1ebd t\u1ea1o th\u00e0nh h\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y l\u00e0 v\u00e0 chi\u1ec1u cao l\u00e0 2 \u00c5 5 \u00e32 125\u03c0 5, khi \u0111\u00f3 th\u1ec3 t\u00edch c\u1ee7a n\u00f3 l\u00e0 V1 = 5\u03c0 2 =. 4 x y H\u00ecnh vu\u00f4ng \u1edf d\u01b0\u1edbi s\u1ebd t\u1ea1o th\u00e0nh hai h\u00ecnh n\u00f3n c\u00f3 chung m\u1eb7t \u0111\u00e1y v\u00e0 c\u00f3 \u0111\u01b0\u1eddng k\u00ednh \u0111\u00e1y l\u00e0 AB\u221a nh\u01b0 h\u00ecnh b\u00ean. Chi\u1ec1u cao v\u00e0 b\u00e1n k\u00ednh 52 \u0111\u00e1y c\u1ee7a h\u00ecnh n\u00f3n n\u00e0y l\u00e0 n\u00ean th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i hai n\u00f3n gh\u00e9p 2 \u221a \u00e52 \u221a 1 \u00c7 52 125\u03c0 2 l\u1ea1i l\u00e0 V2 = 2 \u00b7 \u03c0 2 =. 3 6 Tuy nhi\u00ean, hai h\u00ecnh n\u00e0y c\u00f3 chung ph\u1ea7n h\u00ecnh n\u00f3n t\u1ea1o th\u00e0nh khi xoay ph\u1ea7n m\u00e0u cam xung quanh XY . D\u1ec5 th\u1ea5y ph\u1ea7n chung n\u00e0y 5 c\u0169ng l\u00e0 h\u00ecnh n\u00f3n nh\u01b0ng chi\u1ec1u cao v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y l\u00e0 . Do \u0111\u00f3, \u221a 2 1 \u00c5 5 \u00e32 125\u03c0 2 th\u1ec3 t\u00edch ph\u1ea7n chung l\u00e0 V3 = \u03c0 = . 3 2\u221a \u00e4 24 \u00c4 V\u1eady V = V1 + V2 \u2212 V3 = 125 5 + 4 2 \u03c0 . 24 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 74 (C\u00e2u 40 - \u0110TK - BGD\u221a&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh n\u00f3n c\u00f3 chi\u1ec1u cao b\u1eb1ng 2 5. M\u1ed9t m\u1eb7t ph\u1eb3n\u221ag \u0111i qua \u0111\u1ec9nh h\u00ecnh n\u00f3n v\u00e0 c\u1eaft h\u00ecnh n\u00f3n theo m\u1ed9t thi\u1ebft di\u1ec7n l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u00f3 di\u1ec7n t\u00edch b\u1eb1ng 9 3. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n gi\u1edbi h\u1ea1n b\u1edfi h\u00ecnh n\u00f3n \u0111\u00e3 ch\u221ao b\u1eb1ng \u221a 32 5\u03c0 C 32 5\u03c0. A . B 32\u03c0. D 96\u03c0. 3 \u0253 L\u1eddi gi\u1ea3i. \u2206ABC \u0111\u1ec1u, \u0111\u1eb7t AB = BC\u221a= CA = x. \u221a x2 3 \u221a Ta c\u00f3: S\u221a\u2206ABC = 93 \u21d4 4 = 93 \u21d4 x = 6. 6. 3 \u221a AI = = 3 3 (\u2206ABC \u0111\u1ec1u). 2 \u221a\u221a \u2206AOI vu\u00f4ng t\u1ea1i O: OI = AI\u221a2 \u2212 AO2 = 7. \u2206OIB vu\u00f4ng t\u1ea1i I:R = OB = OI2 + BI2 = 4. \u221a 1 .\u03c0R2.h = 1\u221a 32\u03c0 5 V\u1eady th\u1ec3 t\u00edch kh\u1ed1i n\u00f3n: V = \u03c0.16.2 5 = 33 3 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 75 (C\u00e2u 44 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). Cho h\u00ecnh tr\u1ee5 c\u00f3 chi\u1ec1u cao b\u1eb1ng 6a. Bi\u1ebft r\u1eb1ng khi c\u1eaft h\u00ecnh tr\u1ee5 \u0111\u00e3 cho b\u1edfi m\u1ed9t m\u1eb7t ph\u1eb3ng song song v\u1edbi tr\u1ee5c v\u00e0 c\u00e1ch tr\u1ee5c m\u1ed9t kho\u1ea3ng b\u1eb1ng 3a, thi\u1ebft di\u1ec7n thu \u0111\u01b0\u1ee3c l\u00e0 m\u1ed9t h\u00ecnh vu\u00f4ng. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u1ee5 \u0111\u01b0\u1ee3c gi\u1edbi h\u1ea1n b\u1edfi h\u00ecnh tr\u1ee5 \u0111\u00e3 cho b\u1eb1ng A 216\u03c0a3. B 150\u03c0a3. C 54\u03c0a3. D 108\u03c0a3. \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 455 S\u0110T: 0905.193.688","1. Kh\u00e1i ni\u1ec7m v\u1ec1 m\u1eb7t tr\u00f2n xoay G\u1ecdi thi\u1ebft di\u1ec7n thu \u0111\u01b0\u1ee3c l\u00e0 h\u00ecnh vu\u00f4ng ABCD. Khi \u0111\u00f3 h = AD = C AB = 6a v\u00e0 d (OO , (ABCD)) = d (O, (ABCD)) = OM = 3a. D B AB 6a O G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a c\u1ea1nh AB n\u00ean AM = = = 3a. 22 Ta c\u00f3 AM O vu\u00f4ng t\u1ea1i M v\u00e0 OM \u221a= AM = 3a n\u00ean AM O vu\u00f4ng c\u00e2n t\u1ea1i M . Suy ra r = AO = 3a 2. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u1ee5 l\u00e0 V = \u03c0r2 \u00b7h= \u00c4 \u221a \u00e42 \u00b7 108\u03c0a3. \u03c0 3a 2 6a = \/MO \/ A Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 76 (C\u00e2u 36 - M\u0110 102 - \u221aBGD&\u0110T - N\u0103m 2018 - 2019). Cho h\u00ecnh tr\u1ee5 c\u00f3 chi\u1ec1u cao b\u1eb1ng\u221a4 2. C\u1eaft h\u00ecnh tr\u1ee5 \u0111\u00e3 cho b\u1edfi m\u1ed9t m\u1eb7t ph\u1eb3ng song song v\u1edbi tr\u1ee5c v\u00e0 c\u00e1ch tr\u1ee5c m\u1ed9t kho\u1ea3ng b\u1eb1ng 2, thi\u1ebft di\u1ec7n thu \u0111\u01b0\u1ee3c c\u00f3 di\u1ec7n t\u00edch b\u1eb1ng 16. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh\u221atr\u1ee5 \u0111\u00e3 cho b\u1eb1ng \u221a \u221a \u221a A 24 2\u03c0. B 8 2\u03c0. C 12 2\u03c0. D 16 2\u03c0. \u0253 L\u1eddi gi\u1ea3i. Gi\u1ea3 s\u1eed h\u00ecnh tr\u1ee5 c\u00f3 hai \u0111\u00e1y l\u00e0 c\u00e1c h\u00ecnh tr\u00f2n t\u00e2m O v\u00e0 t\u00e2m O . C\u1eaft h\u00ecnh C tr\u1ee5 b\u1edfi m\u1ed9t m\u1eb7t ph\u1eb3ng song song v\u1edbi tr\u1ee5c, ta \u0111\u01b0\u1ee3c thi\u1ebft di\u1ec7n l\u00e0 h\u00ecnh ch\u1eef O nh\u1eadt ABCD (v\u1edbi AB l\u00e0 d\u00e2y cung c\u1ee7a h\u00ec\u221anh tr\u00f2n \u0111\u00e1y t\u00e2m O). Do h\u00ecnh tr\u1ee5\u221ac\u00f3 chi\u1ec1u cao l\u00e0 h = OO = 4 2 \u21d2 n\u00ean c\u00f3 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh D = AD = 4 2. Theo b\u00e0i ra, di\u1ec7n t\u00edch h\u00ecnh ch\u1eef nh\u1eadt ABCD b\u1eb1ng 16 n\u00ean AB \u00b7 CD = 16 \u21d4 AB = 16 = 1\u221a6 \u221a B = 2 2. K AD 4 2 O A G\u1ecdi K l\u00e0 trung \u0111i\u1ec3m \u0111o\u1ea1n AB th\u00ec OK \u22a5 AB, m\u00e0 OK\u221a \u22a5 AD n\u00ean OK \u22a5 (ABCD). Suy ra kho\u1ea3ng c\u00e1ch gi\u1eefa OO v\u00e0 (ABCD) l\u00e0 OK = 2. X\u00e9t tam gi\u00e1c vu\u00f4ng AOK c\u00f3 \u221a \u00a0 \u00c5 AB \u00e32 \u2026\u00c4\u221a \u00e42 \u00c4\u221a \u00e42 OK2 OK2 2 2 R = OA = + AK 2 = + 2 = + = 2. \u221a\u221a Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 l\u00e0 S = 2\u03c0R = 2\u03c0 \u00b7 2 \u00b7 4 2 = 16\u03c0 2. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 77 (C\u00e2u 37 - M\u0110 103 - B\u221aGD&\u0110T - N\u0103m 2018 - 2019). Cho h\u00ecnh tr\u1ee5 c\u00f3 chi\u1ec1u cao b\u1eb1ng 3 2. C\u1eaft h\u00ecnh tr\u1ee5 \u0111\u00e3 cho b\u1edfi m\u1eb7t ph\u221a\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 12 2. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh\u221atr\u1ee5 \u0111\u00e3 cho b\u1eb1ng \u221a \u221a \u221a A 6 10\u03c0. B 6 34\u03c0. C 3 10\u03c0. D 3 34\u03c0. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 456","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U \u221a\u221a B Ta c\u00f3: SABCD = 12 2 = 3 2 \u00b7 CD O \u21d2 CD = 4 A \u21d2 CI = 2\u221a \u221a \u21d2 CO = CI2 + I\u221aO2 = 5 = r. C V\u1eady Sxq = 2\u03c0rl = 6 10\u03c0. OI D Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 78 (C\u00e2u 39 - M\u0110 104 - B\u221aGD&\u0110T - N\u0103m 2018 - 2019). Cho h\u00ecnh tr\u1ee5 c\u00f3 chi\u1ec1u cao b\u1eb1ng 3 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 18. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh\u221atr\u1ee5 \u0111\u00e3 cho b\u1eb1ng \u221a \u221a \u221a A 6 3\u03c0. B 6 39\u03c0. C 3 39\u03c0. D 12 3\u03c0. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi chi\u1ec1u cao c\u1ee7a h\u00ecnh tr\u1ee5 l\u00e0 h. A O Thi\u1ebft di\u1ec7n c\u1ee7a h\u00ecnh tr\u1ee5 c\u1eaft b\u1edfi m\u1eb7t ph\u1eb3ng song song v\u1edbi tr\u1ee5c l\u00e0 h\u00ecnh ch\u1eef B nh\u1eadt ABB A . A G\u1ecdi H l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a O tr\u00ean AB th\u00ec OH l\u00e0 kho\u1ea3ng c\u00e1ch t\u1eeb O \u0111\u1ebfn m\u1eb7t HO ph\u1eb3ng (ABB A ) n\u00ean OH = 1. \u221a B Di\u1ec7n t\u00edch thi\u1ebft di\u1ec7n l\u00e0: Std = AB \u00b7 AA trong \u0111\u00f3 AA =h=3 3 n\u00ean AB = Std = 1\u221a8 = \u221a 2 3. AA 3 3 Do tam gi\u00e1c OAB c\u00e2n n\u00ean OH2 = OB2 \u2212 HB2 = OB2 \u2212 AB2 \u00c4 4\u221a \u00e42 \u21d2 OB2 = OH2 + AB2 = 1 + 2 3 = 4 44 \u21d2 OB = 2 V\u1eady di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 l\u00e0 \u221a\u221a Sxq = 2\u03c0 \u00b7 R \u00b7 h = 2\u03c0 \u00b7 2 \u00b7 3 3 = 12 3\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 79 (C\u00e2u 42 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). C\u1eaft h\u00ecnh n\u00f3n (N ) b\u1edfi m\u1eb7t ph\u1eb3ng \u0111i qua \u0111\u1ec9nh v\u00e0 t\u1ea1o v\u1edbi m\u1eb7t ph\u1eb3ng \u0111\u00e1y m\u1ed9t g\u00f3c 60\u25e6 ta \u0111\u01b0\u1ee3c thi\u1ebft di\u1ec7\u221an l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh\u221a4a. Di\u1ec7n t\u00edch xung quan\u221ah c\u1ee7a (N ) b\u1eb1ng \u221a A 8 7\u03c0a2. B 4 13\u03c0a2. C 8 13\u03c0a2. D 4 7\u03c0a2. \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 457 S\u0110T: 0905.193.688","1. Kh\u00e1i ni\u1ec7m v\u1ec1 m\u1eb7t tr\u00f2n xoay Gi\u1ea3 s\u1eed \u0111\u1ec9nh c\u1ee7a h\u00ecnh n\u00f3n l\u00e0 S, t\u00e2m c\u1ee7a \u0111\u00e1y l\u00e0 O. Thi\u1ebft di\u1ec7n S l\u00e0 tam gi\u00e1c SAB \u0111\u1ec1u c\u1ea1nh b\u1eb1ng 4a. B OM G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB, ta c\u00f3 \u00aeSM \u22a5 AB , suy ra A OM \u22a5 AB ((SAB), (OAB\u221a)) = S\u2019M O = 60\u25e6. \u221a Ta c\u00f3 SM = SA2 \u2212 M A2 = (4a)2 \u2212 (2a)2 = 2a 3. Trong tam gi\u00e1c SOM vu\u00f4ng t\u1ea1i O, ta c\u00f3 SO = SM \u00b7 sin 60\u25e6 = 3a. \u221a \u221a Suy ra OA = SA2 \u2212 SO2 =\u221a (4a)2 \u2212 (3a\u221a)2 = a 7. V\u1eady Sxq = \u03c0OA \u00b7 SA = \u03c0 \u00b7 a 7 \u00b7 4a = 4a2\u03c0 7. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 80 (C\u00e2u 47 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). C\u1eaft h\u00ecnh n\u00f3n (N ) b\u1edfi m\u1eb7t ph\u1eb3ng \u0111i qua \u0111\u1ec9nh v\u00e0 t\u1ea1o v\u1edbi m\u1eb7t ph\u1eb3ng ch\u1ee9a \u0111\u00e1y m\u1ed9t g\u00f3c b\u1eb1ng 60\u25e6, ta \u0111\u01b0\u1ee3\u221ac thi\u1ebft di\u1ec7n l\u00e0 tam gi\u00e1c \u221a\u0111\u1ec1u c\u1ea1nh 2a. Di\u1ec7n t\u00edch xu\u221ang quanh c\u1ee7a (N ) b\u1eb1ng\u221a A 7\u03c0a2. B 13\u03c0a2. C 2 7\u03c0a2. D 2 13\u03c0a2. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi tam gi\u00e1c thi\u1ebft di\u1ec7n l\u00e0 \u2206SAB v\u1edbi S l\u00e0 \u0111\u1ec9nh h\u00ecnh n\u00f3n, S I l\u00e0 t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y, M l\u00e0 trung\u221a\u0111i\u1ec3m AB. V\u00ec \u2206SAB l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh 2a n\u00ean SM = 3a v\u00e0 SM , IM c\u00f9ng vu\u00f4ng g\u00f3c v\u1edbi AB n\u00ean S\u2019M I = 60\u25e6. T\u1eeb \u0111\u00f3 suy ra \u221a 3a , M I = cos 60\u25e6 \u00b7 SM = 60\u25e6 A I 2 M v\u00e0 3a B . SI = sin 60\u25e6 \u00b7 SM = 2 Theo \u0111\u1ecbnh l\u00fd Pitago, ta \u0111\u01b0\u1ee3c \u221a IB2 = M I2 + M B2 = 3a2 + a2 = 7a2 \u21d4 IB = 7a 44 2 v\u00e0 SB2 = IB2 + SI2 = 4a2 \u21d4 SB = 2a. \u221a T\u1eeb \u0111\u00f3 ta \u0111\u01b0\u1ee3c di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n l\u00e0 S = \u03c0Rl = 7\u03c0a2. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 81 (C\u00e2u 47 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). C\u1eaft h\u00ecnh n\u00f3n (\u2135) b\u1edfi m\u1eb7t ph\u1eb3ng \u0111i qua \u0111\u1ec9nh v\u00e0 t\u1ea1o v\u1edbi m\u1eb7t ph\u1eb3ng ch\u1ee9a \u0111\u00e1y m\u1ed9t g\u00f3c b\u1eb1ng 30\u25e6, ta \u0111\u01b0\u1ee3c\u221athi\u1ebft di\u1ec7n l\u00e0 tam gi\u00e1c \u0111\u221a\u1ec1u c\u1ea1nh 4a. Di\u1ec7n t\u00edch xu\u221ang quanh c\u1ee7a (\u2135) b\u1eb1ng\u221a A 4 7\u03c0a2. B 8 7\u03c0a2. C 8 13\u03c0a2. D 4 13\u03c0a2. \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 458 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U S Gi\u1ea3 s\u1eed h\u00ecnh n\u00f3n (\u2135) c\u00f3 \u0111\u1ec9nh S, \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y c\u00f3 t\u00e2m O, thi\u1ebft A di\u1ec7n n\u00f3i tr\u00ean l\u00e0 tam gi\u00e1c \u0111\u1ec1u SAB. G\u1ecdi H l\u00e0 trung \u0111i\u1ec3m \u0111o\u1ea1n O AB. Khi \u0111\u00f3 AB \u22a5 (SHO) n\u00ean H S\u2019HO = ((SAB), (OAB)) = 30\u25e6. B Do \u0111\u00f3 SO = SH sin 30\u25e6 = \u221a \u00b7 1 = \u221a 3SA 3a, \u221a 2 2\u221a 3SA \u00b7 3 OH = SH cos 30\u25e6 = 22 = 3a, \u221a \u2026 AB2 \u221a OA = OH2 + HA2 = OH2 + = 13a. 4 V\u1eady di\u1ec7n t\u00edch xung quay c\u1ee7a h\u00ecnh n\u00f3n (\u2135) l\u00e0 \u221a Sxq = \u03c0OA \u00b7 SA = 4 13\u03c0a2. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 82 (C\u00e2u 42 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). C\u1eaft h\u00ecnh n\u00f3n (N ) b\u1edfi m\u1eb7t ph\u1eb3ng \u0111i qua \u0111\u1ec9nh v\u00e0 t\u1ea1o v\u1edbi m\u1eb7t ph\u1eb3ng ch\u1ee9a \u0111\u00e1y m\u1ed9t g\u00f3c b\u1eb1ng 30\u25e6, ta \u0111\u01b0\u1ee3\u221ac thi\u1ebft di\u1ec7n l\u00e0 tam gi\u00e1c \u221a\u0111\u1ec1u c\u1ea1nh 2a. Di\u1ec7n t\u00edch xu\u221ang quanh c\u1ee7a (N ) b\u1eb1ng\u221a A 7\u03c0a2. B 13\u03c0a2. C 2 13\u03c0a2. D 2 7\u03c0a2. \u0253 L\u1eddi gi\u1ea3i. S MDD-109 A O 30\u25e6 H B X\u00e9t h\u00ecnh n\u00f3n (N ) v\u00e0 m\u1eb7t ph\u1eb3ng (SAB) \u0111i qua \u0111\u1ec9nh, c\u1eaft (O) t\u1ea1i A, B. G\u1ecdi H l\u00e0 trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n\u221ath\u1eb3ng AB.\u221a AB 3 2a \u00b7 3 \u221a Do SAB \u0111\u1ec1u n\u00ean SH = = = a 3. 22 \u00aeSH \u22a5 AB Do (SAB) \u2229 (OAB) = AB v\u00e0 n\u00ean ((SAB), (OAB)) = (SH, OH) = S\u2019HO = 30\u25e6. OH \u22a5 AB \u21d2 SO = SH \u00b7 sin S\u2019HO = \u221a \u00b7 sin 30\u25e6 = \u221a a3 a3 2 \u221a \u00c3 \u00c7 \u221a \u00e52 \u221a SB2 (2a)2 a3 13 \u21d2 OB = \u2212 SO2 = \u2212 = a. 22 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 459 S\u0110T: 0905.193.688","1. Kh\u00e1i ni\u1ec7m v\u1ec1 m\u1eb7t tr\u00f2n xoay \u221a \u221a a 13 13\u03c0a2. V\u1eady Sxq = \u03c0 \u00b7 SB \u00b7 OB = \u03c0 \u00b7 2a \u00b7 2 = Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 83 (C\u00e2u 48 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). C\u1eaft h\u00ecnh tr\u1ee5 (T ) b\u1edfi m\u1ed9t m\u1eb7t ph\u1eb3ng song song v\u1edbi tr\u1ee5c v\u00e0 c\u00e1ch tr\u1ee5c m\u1ed9t kho\u1ea3ng b\u1eb1ng 2a, ta \u0111\u01b0\u1ee3c thi\u1ebft di\u1ec7\u221an l\u00e0 m\u1ed9t h\u00ecnh vu\u00f4ng c\u00f3 d\u221ai\u1ec7n t\u00edch b\u1eb1ng 36a2. Di\u221a\u1ec7n t\u00edch xung quanh c\u1ee7a\u221a(T ) b\u1eb1ng A 4 13\u03c0a2. B 12 13\u03c0a2. C 6 13\u03c0a2. D 8 13\u03c0a2. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi r v\u00e0 l l\u1ea7n l\u01b0\u1ee3t l\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y l\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh c\u1ee7a h\u00ecnh tr\u1ee5 (T ). B O C\u1eaft h\u00ecnh tr\u1ee5 (T ) b\u1edfi m\u1ed9t m\u1eb7t ph\u1eb3ng song song v\u1edbi tr\u1ee5c OO ta \u0111\u01b0\u1ee3c thi\u1ebft A di\u1ec7n l\u00e0 h\u00ecnh vu\u00f4ng ABCD c\u00f3 di\u1ec7n t\u00edch b\u1eb1ng 36a2. Suy ra SABCD = CD2 = 36a2 \u21d2 CD = AD = 6a = l. G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a CD. Ta c\u00f3 \u00aeOI \u22a5 CD \u21d2 OI \u22a5 (ABCD). OI \u22a5 AD 2a C O I D Suy ra OI = d (O, (ABCD)) = d (OO , (ABCD)) = 2a. CD Trong OID vu\u00f4ng t\u1ea1i I, c\u00f3 ID = = 3a; OI = 2a. 2\u221a Suy ra OD2 = OI2 + ID2 = 13a2 \u21d2 OD = a 13 = r. \u221a \u221a Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 (T ) l\u00e0 Sxq = 2\u03c0rl = 2\u03c0 \u00b7 a 13 \u00b7 6a = 12 13\u03c0a2. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 84 (C\u00e2u 42 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). C\u1eaft h\u00ecnh tr\u1ee5 (T ) b\u1edfi m\u1eb7t ph\u1eb3ng song song v\u1edbi tr\u1ee5c v\u00e0 c\u00e1ch tr\u1ee5c m\u1ed9t kho\u1ea3ng b\u1eb1ng 3a, ta \u0111\u01b0\u1ee3c thi\u1ebft di\u1ec7n\u221al\u00e0 m\u1ed9t h\u00ecnh vu\u00f4ng c\u00f3 di\u1ec7n t\u00edch b\u1eb1ng 16a2. Di\u221a\u1ec7n t\u00edch xung quanh c\u1ee7a (T ) b\u1eb1ng A 16 13 \u03c0a2. B \u221a C 8 13 \u03c0a2. \u221a 4 13\u03c0a2. 33 D 8 13\u03c0a2. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi (P ) l\u00e0 m\u1eb7t ph\u1eb3ng song song v\u1edbi tr\u1ee5c OO v\u00e0 c\u1eaft h\u00ecnh tr\u1ee5 (T ). C O Theo gi\u1ea3 thi\u1ebft ta c\u00f3, m\u1eb7t ph\u1eb3ng (P ) c\u1eaft h\u00ecnh tr\u1ee5 (T ) theo thi\u1ebft di\u1ec7n l\u00e0 B h\u00ecnh vu\u00f4ng ABCD. D MDD-109 Khi \u0111\u00f3, di\u1ec7n t\u00edch c\u1ee7a h\u00ecnh vu\u00f4ng SABCD = 16a2 \u21d2 AB = CD = 4a. I O \u00aeOI \u22a5 AB A G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m AB \u21d2 \u21d2 OI \u22a5 (ABCD). OI \u22a5 AD Do \u0111\u00f3 OI = 3a. \u221a \u221a \u221a M\u1eb7t kh\u00e1c r = OA = OI2 + IA2 = 9a2 + 4a2 = a 13. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 (T ) b\u1eb1ng \u221a\u221a Sxq = 2\u03c0 \u00b7 OA \u00b7 AD = 2\u03c0a 13 \u00b7 4a = 8 13\u03c0a2. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 460 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U \u0104 C\u00e2u 85 (C\u00e2u 45 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). C\u1eaft h\u00ecnh tr\u1ee5 (T ) b\u1edfi m\u1eb7t ph\u1eb3ng song song v\u1edbi tr\u1ee5c v\u00e0 c\u00e1ch tr\u1ee5c m\u1ed9t kho\u1ea3ng b\u1eb1ng 2a, ta \u0111\u01b0\u1ee3c thi\u1ebft di\u1ec7n l\u00e0 m\u1ed9t h\u00ecnh vu\u00f4ng c\u00f3 di\u1ec7n t\u00edch b\u1eb1ng 16a2. Di\u1ec7\u221an t\u00edch xung quanh c\u1ee7a (\u221aT ) b\u1eb1ng \u221a \u221a C 16 2 \u03c0a2. D 32 2 \u03c0a2. A 8 2\u03c0a2. B 16 2\u03c0a2. 33 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi thi\u1ebft di\u1ec7n l\u00e0 h\u00ecnh vu\u00f4ng ABCD v\u00e0 O, O l\u1ea7n l\u01b0\u1ee3t l\u00e0 t\u00e2m c\u1ee7a hai \u0111\u00e1y v\u00e0 I l\u00e0 A O trung \u0111i\u1ec3m CD. B Theo b\u00e0i ra, ta c\u00f3 D \u2022 OI = 2a. 1 I CD \u2022 SABCD = CD2 = 16a2 \u21d4 CD = 4a \u21d2 IC = ID = = 2a. O 2 C \u2022 h = OO = AD =\u221aCD = 4a. \u221a \u221a Khi \u0111\u00f3 R = OD = OI2 + I\u221aC2 = 4a2 +\u221a4a2 = 2 2a. V\u1eady Sxq = 2\u03c0 \u00b7 R \u00b7 h = 2\u03c0 \u00b7 2 2a \u00b7 4a = 16 2\u03c0a2. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 86 (C\u00e2u 46 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). C\u1eaft h\u00ecnh tr\u1ee5 (T ) b\u1edfi m\u1eb7t ph\u1eb3ng song song v\u1edbi tr\u1ee5c v\u00e0 c\u00e1ch tr\u1ee5c m\u1ed9t kho\u1ea3ng b\u1eb1ng 3a, ta \u0111\u01b0\u1ee3c thi\u1ebft di\u1ec7n\u221al\u00e0 m\u1ed9t h\u00ecnh vu\u00f4ng c\u00f3 d\u221ai\u1ec7n t\u00edch b\u1eb1ng 36a2. Di\u1ec7\u221an t\u00edch xung quanh c\u1ee7a (\u221aT ) b\u1eb1ng A 12 2\u03c0a2. B 36 2\u03c0a2. C 24 2\u03c0a2. D 18 2\u03c0a2. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi d = M O l\u00e0 kho\u1ea3ng c\u00e1ch t\u1eeb tr\u1ee5c t\u1edbi m\u1eb7t ph\u1eb3ng thi\u1ebft di\u1ec7n. O Di\u1ec7n t\u00edch h\u00ecnh vu\u00f4ng: AB2 = 36a2 \u21d2 AB = 6a \u21d2 AM = AB = 3a. 2 X\u00e9t tam gi\u00e1c AM O vu\u00f4ng t\u1ea1i M , ta c\u00f3: \u221a\u221a \u221a r = AM 2 + d2 = 9a2 + 9a2 = 3 2a. V\u1eady di\u1ec7n t\u00edch xung quanh c\u1ee7a (T ) l\u00e0 B \u221a\u221a M O Sxq = 2\u03c0r \u00b7 h = 2\u03c0 \u00b7 3 2a \u00b7 6a = 36 2\u03c0a2. Ar Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 87 (C\u00e2u 30 - M\u0110 104 - 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 3 mm v\u00e0 chi\u1ec1u cao 200 mm. Th\u00e2n b\u00fat ch\u00ec \u0111\u01b0\u1ee3c l\u00e0m b\u1eb1ng g\u1ed7 v\u00e0 ph\u1ea7n l\u00f5i \u0111\u01b0\u1ee3c l\u00e0m b\u1eb1ng than ch\u00ec. 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 1 mm. Gi\u1ea3 \u0111\u1ecbnh 1 m3 g\u1ed7 c\u00f3 gi\u00e1 \u03b1 (tri\u1ec7u \u0111\u1ed3ng), 1m3 than ch\u00ec c\u00f3 gi\u00e1 7\u03b1 (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 84,5 \u00b7 \u03b1 (\u0111\u1ed3ng). B 9,07 \u00b7 \u03b1 (\u0111\u1ed3ng). C 8,45 \u00b7 \u03b1 (\u0111\u1ed3ng). D 90,07 \u00b7 \u03b1 (\u0111\u1ed3ng). \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch ph\u1ea7n l\u00f5i than ch\u00ec V1 = \u03c0 \u00b7 0,0012 \u00b7 0,2 = 2\u03c0 \u00b7 10\u22127 m3. S\u1ed1 ti\u1ec1n l\u00e0m l\u00f5i than ch\u00ec T1 = V1 \u00b7 7\u03b1 \u00b7 106 = 1,4\u03c0a (\u0111\u1ed3ng). Th\u1ec3 t\u00edch ph\u1ea7n th\u00e2n b\u1eb1ng g\u1ed7 c\u1ee7a b\u00fat ch\u00ec l\u00e0 \u221a V2 = 6 \u00b7 0,0032 \u00b7 3 \u00b7 0,2 \u2212 2\u03c0 \u00b7 10\u22127 = \u221a \u00b7 27 \u00b7 10\u22127 \u2212 2\u03c0 \u00b7 10\u22127)m3. 4 (3 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 461 S\u0110T: 0905.193.688","1. Kh\u00e1i ni\u1ec7m v\u1ec1 m\u1eb7t tr\u00f2n xoay \u221a S\u1ed1 ti\u1ec1n l\u00e0m ph\u1ea7n th\u00e2n b\u1eb1ng g\u1ed7 c\u1ee7a b\u00fat ch\u00ec T2 = V2 \u00b7 \u03b1 \u00b7 106 = (2,7 3 \u2212 \u03c0 \u00b7 0,2)\u03b1 (\u0111\u1ed3ng). Gi\u00e1 v\u1eadt li\u1ec7u l\u00e0m b\u00fat tr\u00ec T = T1 + T2 \u2248 8,45\u03b1 (\u0111\u1ed3ng). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 88 (C\u00e2u 23 - M\u0110 103 - 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 1m v\u00e0 1, 8m. 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 c\u00f3 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 2, 8m. B 2, 6m. C 2, 1m. D 2, 3m. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi hai b\u1ec3 n\u01b0\u1edbc h\u00ecnh tr\u1ee5 ban \u0111\u1ea7u l\u1ea7n l\u01b0\u1ee3t c\u00f3 chi\u1ec1u cao l\u00e0 h, b\u00e1n k\u00ednh r1, r2, th\u1ec3 t\u00edch l\u00e0 V1, V2. Ta c\u00f3 m\u1ed9t b\u1ec3 n\u01b0\u1edbc m\u1edbi c\u00f3 chi\u1ec1u cao h, V = V1 + V2. \u21d2 \u03c0r2h = \u03c0r12h + \u03c0r22h \u21d2 \u03c0r2h = \u03c0 \u00b7 12 \u00b7 h + \u03c0 \u00b7 1, 82 \u00b7 h \u21d4 r = 1 + 1, 82 \u2248 2, 1m. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 89 (C\u00e2u 31 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 201\u221a7). Cho h\u00ecnh ch\u00f3p t\u1ee9 gi\u00e1c \u0111\u1ec1u S.ABCD c\u00f3 c\u00e1c c\u1ea1nh \u0111\u1ec1u b\u1eb1ng a 2. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i n\u00f3n c\u00f3 \u0111\u1ec9nh S v\u00e0 \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y l\u00e0 \u0111\u01b0\u1edd\u221ang tr\u00f2n n\u1ed9i ti\u1ebfp t\u1ee9 gi\u00e1c ABCD. \u221a \u03c0a3 2\u03c0a3 \u03c0a3 2\u03c0a3 A V = . B V= . C V = . D V= . 2 66 2 \u0253 L\u1eddi gi\u1ea3i. \u221a AB a 2 B\u00e1n k\u00ednh \u0111\u00e1y c\u1ee7a h\u00ecnh n\u00f3n r = = . S 2 2\u221a \u0110\u01b0\u1eddng ch\u00e9o\u221ah\u00ecnh vu\u00f4ng A\u221aBCD c\u00f3 AC = AB 2 = 2a \u21d2 HA = a. C\u1ea1nh B b\u00ean SA = a 2 \u21d2 SH = SA2 \u2212\u221aHA2 = a \u21d2 \u0111\u01b0\u1eddng cao h\u00ecnh n\u00f3n h = a. H 1\u03c0r2h = 1 a 2 2 a3\u03c0 \u21d2 th\u1ec3 t\u00edch V = \u03c0. .a = . 3 32 6 C A D Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 90 (C\u00e2u 32 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00ecnh h\u1ed9p ch\u1eef nh\u1eadt ABCD.A B C D c\u00f3 AD = 8, CD = 6, AC = 12. T\u00ednh di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n Stp c\u1ee7a h\u00ecnh tr\u1ee5 c\u00f3 hai \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y l\u00e0 hai \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp hai h\u00ecnh ch\u1eef nh\u1eadt ABCD v\u00e0 A B C D . \u221a A Stp = 576\u03c0. B Stp = 10(\u221a2 11 + 5)\u03c0. D Stp = 5(4 11 + 5)\u03c0. C Stp = 26\u03c0. \u0253 L\u1eddi gi\u1ea3i. B C C \u221a A D Ta c\u00f3 A\u221aC = AB2 + BC\u221a2 = 10, B CC = AC 2 \u2212 AC2 = 2 11. AC Do \u0111\u00f3 h\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y l\u00e0 r = = 5, \u221a2 \u0111\u01b0\u1eddng sinh l = CC = 2 11. \u221a V\u1eady Stp = 2\u03c0rl + 2\u03c0r2 = 10(2 11 + 5)\u03c0. A D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 462 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 91 (C\u00e2u 49 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). Cho m\u1eb7t c\u1ea7u t\u00e2m O, b\u00e1n k\u00ednh R. X\u00e9t m\u1eb7t ph\u1eb3ng (P ) thay \u0111\u1ed5i c\u1eaft m\u1eb7t c\u1ea7u theo giao tuy\u1ebfn l\u00e0 \u0111\u01b0\u1eddng tr\u00f2n (C). H\u00ecnh n\u00f3n (N ) c\u00f3 \u0111\u1ec9nh S n\u1eb1m tr\u00ean m\u1eb7t c\u1ea7u, c\u00f3 \u0111\u00e1y l\u00e0 \u0111\u01b0\u1eddng tr\u00f2n (C) v\u00e0 c\u00f3 chi\u1ec1u cao l\u00e0 h(h > R). T\u00ednh h \u0111\u1ec3 th\u1ec3 t\u00edch kh\u1ed1i n\u00f3n \u0111\u01b0\u1ee3c t\u1ea1o n\u00ean b\u1edfi (N ) c\u00f3 gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t. \u221a \u221a A h = 3R. B h = 2R. C h= 4R D 3R . h= . 32 \u0253 L\u1eddi gi\u1ea3i. S O R d I rM Ta bi\u1ebft r\u1eb1ng khi cho tr\u01b0\u1edbc \u0111\u01b0\u1eddng tr\u00f2n (C) b\u1ea5t k\u1ef3 n\u1eb1m tr\u00ean m\u1eb7t c\u1ea7u, h\u00ecnh n\u00f3n (N ) c\u00f3 \u0111\u00e1y l\u00e0 (C) s\u1ebd \u0111\u1ea1t gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t khi v\u00e0 ch\u1ec9 khi \u0111i\u1ec3m S th\u1ecfa m\u00e3n SO vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng ch\u1ee9a (C). V\u1eady trong b\u00e0i to\u00e1n n\u00e0y ta ch\u1ec9 x\u00e9t c\u00e1c h\u00ecnh n\u00f3n \u0111\u1ec9nh S v\u1edbi \u0111i\u1ec3m S th\u1ecfa SO vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng ch\u1ee9a \u0111\u01b0\u1eddng tr\u00f2n giao tuy\u1ebfn (C). Th\u1ec3 t\u00edch kh\u1ed1i n\u00f3n \u0111\u01b0\u1ee3c t\u1ea1o n\u00ean b\u1edfi (N ) l\u00e0 V = 1 = 1 h.\u03c0.r2 = 1 [R2 \u2212 (h \u2212 R)2] = 1 \u03c0(\u2212h3 + 2h2R). 3 h.S(C) 3 h.\u03c0. 3 3 X\u00e9t h\u00e0m f (h) = \u2212h3 + 2h2R, h \u2208 (R, 2R), c\u00f3 f (h) = \u22123h2 + 4hR. f (h) = 0 \u21d4 \u22123h2 + 4hR = 0 \u21d4 h = 0 ho\u1eb7c h = 4R L\u1eadp b\u1ea3ng bi\u1ebfn thi\u00ean ta t\u00ecm \u0111\u01b0\u1ee3c max f (h) = . 3 32 R3, 4R 1 \u03c0 32 R3 t\u1ea1i h= . V\u1eady th\u1ec3 t\u00edch kh\u1ed1i n\u00f3n \u0111\u01b0\u1ee3c t\u1ea1o n\u00ean b\u1edfi (N ) c\u00f3 gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t l\u00e0 V = 3 27 = 27 3 32 \u03c0R3 4R khi h = . 81 3 C.\u00e1ch kh\u00e1c: G\u1ecdi O l\u00e0 t\u00e2m m\u1eb7t c\u1ea7u, I v\u00e0 r l\u00e0 b\u00e1n k\u00ednh c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n (C). Ta c\u00f3 OI = h \u2212 R v\u00e0 r2 = R2 \u2212 OI2 = 2Rh \u2212 h2. Th\u1ec3 t\u00edch kh\u1ed1i n\u00f3n \u0111\u01b0\u1ee3c t\u1ea1o n\u00ean b\u1edfi (N ) l\u00e0 V = 1 = 1 h.\u03c0.r2 = 1 [R2 \u2212 (h \u2212 R)2] = 1 \u03c0h2(2R \u2212 h). 3 h.S(C) 3 h.\u03c0. 3 3 Ta c\u00f3 h.h.(4R \u2212 2h) \u00c5 h + h + 4R \u2212 2h \u00e33 = \u00c5 4R \u00e33 \u21d2 h2(2R \u2212 h) 1 \u00c5 4R \u00e33 33 23 Do \u0111\u00f3 V l\u1edbn nh\u1ea5t khi h = 4R \u2212 2h \u21d4 h = 4R 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 92 (C\u00e2u 50 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00ecnh n\u00f3n \u0111\u1ec9nh S c\u00f3 chi\u1ec1u cao h = a v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y r = 2a. M\u1eb7t ph\u1eb3ng (P ) \u0111i qua S c\u1eaft Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 463 S\u0110T: 0905.193.688","1. Kh\u00e1i ni\u1ec7m v\u1ec1 m\u1eb7t tr\u00f2n xoay \u221a \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y t\u1ea1i A v\u00e0 B sao cho AB = 2 3a. T\u00ednh kho\u1ea3ng c\u00e1ch d t\u1eeb t\u00e2m c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y \u0111\u1ebfn (P ). \u221a 3a \u221a 5a \u221a 2a . . . A d= 2 B d = a. C d= 5 D d= 2 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi O l\u00e0 t\u00e2m c\u1ee7a \u0111\u00e1y h\u00ecnh n\u00f3n, I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB, H S l\u00e0 ch\u00e2n \u0111\u01b0\u1eddng cao c\u1ee7a tam gi\u00e1c SOI. Khi \u0111\u00f3 ta c\u221a\u00f3 d = OH. H 2a D\u1ec5 d\u00e0ng t\u00ednh \u0111\u01b0\u1ee3c OS = OI = a n\u00ean d = OH = . 2 O B A I Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 93 (C\u00e2u 49 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong t\u1ea5t c\u1ea3 c\u00e1c h\u00ecnh ch\u00f3p t\u1ee9 gi\u00e1c \u0111\u1ec1u n\u1ed9i ti\u1ebfp m\u1eb7t c\u1ea7u c\u00f3 b\u00e1n k\u00ednh b\u1eb1ng 9, t\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i ch\u00f3p c\u00f3 th\u1ec3 t\u00edch l\u1edbn nh\u1ea5t. \u221a \u221a C V = 576 2. D V = 144 6. A V = 144. B V = 576. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi OH = x. S Khi \u0111\u00f3 \u0111\u1ed9 d\u00e0i c\u1ea1nh \u0111\u00e1y c\u1ee7a ch\u00f3p l\u00e0 2(81 \u2212 x2). O R V\u1eady V = f (x) = 2 (9 + x)(81 \u2212 x2). AB 3 2 H f (x) = (x + 9)(9 \u2212 3x). DC 3 V\u1eady th\u1ec3 t\u00edch l\u1edbn nh\u1ea5t khi x = 3, v\u1edbi V = 576 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 94 (C\u00e2u 27 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). M\u1ed9t chi\u1ebfc b\u00fat ch\u00ec kh\u1ed1i l\u0103ng tr\u1ee5 l\u1ee5c gi\u00e1c \u0111\u1ec1u c\u00f3 c\u1ea1nh \u0111\u00e1y 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 \u0111\u01b0\u1ee3c l\u00e0m b\u1eb1ng than ch\u00ec. 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 ch\u00ec v\u00e0 \u0111\u00e1y l\u00e0 h\u00ecnh tr\u00f2n b\u00e1n k\u00ednh 1 mm. Gi\u1ea3 \u0111\u1ecbnh 1 m3 g\u1ed7 c\u00f3 gi\u00e1 tr\u1ecb a (tri\u1ec7u \u0111\u1ed3ng), 1 m3 than ch\u00ec c\u00f3 gi\u00e1 tr\u1ecb 8a (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 sau \u0111\u00e2y? A 9,7a (\u0111\u1ed3ng). B 97,03a (\u0111\u1ed3ng). C 90,7a (\u0111\u1ed3ng). D 9,07a (\u0111\u1ed3ng). \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch ph\u1ea7n l\u00f5i \u0111\u01b0\u1ee3c l\u00e0m b\u1eb1ng than ch\u00ec: Vr = \u03c0R2h = \u03c0 \u00b7 10\u22126 \u00b7 0, 2 = 0, 2 \u00b7 10\u22126\u03c0 m3. Th\u1ec3 t\u00edch chi\u1ebfc\u221ab\u00fat ch\u00ec kh\u1ed1i l\u0103ng tr\u1ee5 l\u1ee5c g\u221ai\u00e1c \u0111\u1ec1u: V = B \u00b7 h = 3 3 \u00b7 (3 \u00b7 10\u22123)2 \u00b7 (0,2) = 27 3 \u00b7 10\u22126 m3. 2 10 \u221a 27 3 Th\u1ec3 t\u00edch ph\u1ea7n th\u00e2n b\u00fat ch\u00ec \u0111\u01b0\u1ee3c l\u00e0m b\u1eb1ng g\u1ed7: Vt = V \u2212 Vr = 10 \u00b7 10\u22126 \u2212 0,2 \u00b7 10\u22126\u03c0 m3. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 464 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U Gi\u00e1 nguy\u00ean v\u1eadt li\u1ec7u l\u00e0\u221am m\u1ed9t chi\u1ebfc b\u00fat ch\u00ec: \u00c7 \u00e5 0,2 \u00b7 10\u22126\u03c0 \u00b7 8a + 27 3 \u00b7 10\u22126 \u2212 0,2 \u00b7 10\u22126\u03c0 a \u2248 9,07 \u00b7 10\u22126a (tri\u1ec7u \u0111\u1ed3ng). 10 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 95 (C\u00e2u 31 - M\u0110 102 - 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 3mm v\u00e0 chi\u1ec1u cao b\u1eb1ng 200mm. Th\u00e2n b\u00fat ch\u00ec \u0111\u01b0\u1ee3c l\u00e0m b\u1eb1ng g\u1ed7 v\u00e0 ph\u1ea7n l\u00f5i \u0111\u01b0\u1ee3c l\u00e0m b\u1eb1ng than ch\u00ec. 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 1mm. Gi\u1ea3 \u0111\u1ecbnh 1m3 g\u1ed7 c\u00f3 gi\u00e1 a (tri\u1ec7u \u0111\u1ed3ng), 1m3 than ch\u00ec c\u00f3 gi\u00e1 6a (tri\u1ec7u \u0111\u1ed3ng). Khi \u0111\u00f3 gi\u00e1 nguy\u00ean 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 84,5a (\u0111\u1ed3ng). B 78,2a (\u0111\u1ed3ng). C 8,45a (\u0111\u1ed3ng). D 7,82a (\u0111\u1ed3ng). \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch ph\u1ea7n ph\u1ea7n l\u00f5i \u0111\u01b0\u1ee3c l\u00e0m b\u1eb1ng than ch\u00ec l\u00e0 Vr = \u03c0 \u00b7 (10\u22123)2 \u00b7 0,2 = 0,2 \u00b7 10\u22126\u03c0 m3. Th\u1ec3 t\u00edch chi\u1ebfc b\u00fat ch\u00ec kh\u1ed1i l\u0103ng tr\u1ee5 l\u1ee5c gi\u00e1c \u0111\u1ec1u l\u00e0 V =B\u00b7h= 3 \u221a\u221a 3 27 3 \u00b7 10\u22126 m3. (3 \u00b7 10\u22123)2 \u00b7 0,2 = 2 10 Th\u1ec3 t\u00edch ph\u1ea7n th\u00e2n b\u00fat ch\u00ec \u0111\u01b0\u1ee3c l\u00e0m b\u1eb1ng g\u1ed7 l\u00e0 \u221a 27 3 Vt = V \u2212 Vr = 10 \u00b7 10\u22126 \u2212 0,2 \u00b7 10\u22126\u03c0 m3. Gi\u00e1 nguy\u00ean v\u1eadt li\u1ec7u l\u00e0m m\u1ed9t chi\u1ebfc b\u00fat ch\u00ec l\u00e0 \u00c7 \u221a \u00e5 27 3 10\u22126\u03c0 0,2 \u00b7 10\u22126\u03c0 \u00b7 6a + \u00b7 10\u22126 \u2212 0,2 \u00b7 a \u2248 7,82 \u00b7 10\u22126a (tri\u1ec7u \u0111\u1ed3ng). 10 Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 465 S\u0110T: 0905.193.688","2. M\u1eb7t c\u1ea7u B\u00c0I 2. M\u1eb6T C\u1ea6U \u0104 C\u00e2u 1 (C\u00e2u 10 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). Di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u b\u00e1n k\u00ednh R b\u1eb1ng A 4\u03c0R2. B 2\u03c0R2. C 4\u03c0R2. D \u03c0R2. 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u b\u00e1n k\u00ednh R b\u1eb1ng 4\u03c0R2. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 2 (C\u00e2u 9 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). Th\u1ec3 t\u00edch kh\u1ed1i c\u1ea7u b\u00e1n k\u00ednh R b\u1eb1ng A 4\u03c0R3. B 4\u03c0R3. C 2\u03c0R3. D 3\u03c0R3. 3 4 \u0253 L\u1eddi gi\u1ea3i. C\u00f4ng th\u1ee9c t\u00ednh th\u1ec3 t\u00edch kh\u1ed1i c\u1ea7u c\u00f3 b\u00e1n k\u00ednh R l\u00e0 V = 4 \u03c0R3. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 3 (C\u00e2u 45 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). Th\u1ec3 t\u00edch kh\u1ed1i c\u1ea7u b\u00e1n k\u00ednh a b\u1eb1ng A 4\u03c0a3 B 4\u03c0a3. C \u03c0a3 D 2\u03c0a3. . . 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i c\u1ea7u b\u00e1n k\u00ednh a l\u00e0 V = 4 \u03c0a3. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 4 (C\u00e2u 9 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). Cho m\u1eb7t c\u1ea7u c\u00f3 b\u00e1n k\u00ednh R = 2. Di\u1ec7n t\u00edch c\u1ee7a m\u1eb7t c\u1ea7u \u0111\u00e3 cho b\u1eb1ng A 32\u03c0 B 8\u03c0. C 16\u03c0. D 4\u03c0. . 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u S = 4\u03c0 \u00b7 r2 = 4\u03c0 \u00b7 22 = 16\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 5 (C\u00e2u 8 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho kh\u1ed1i c\u1ea7u c\u00f3 b\u00e1n k\u00ednh r = 4. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i c\u1ea7u \u0111\u00e3 cho b\u1eb1ng A 256\u03c0 B 64\u03c0. C 64\u03c0 D 256\u03c0. . . 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i c\u1ea7u V = 4 \u03c0r3 = 4 \u03c0 \u00b7 43 = 256 33 \u03c0. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 466 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U \u0104 C\u00e2u 6 (C\u00e2u 21 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho kh\u1ed1i c\u1ea7u c\u00f3 b\u00e1n k\u00ednh r = 4. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i c\u1ea7u \u0111\u00e3 cho b\u1eb1ng A 64\u03c0. B 64\u03c0 C 256\u03c0. D 256\u03c0 . . 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i c\u1ea7u l\u00e0 V = 4\u03c0r3 = 256 \u03c0. 33 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 7 (C\u00e2u 5 - M\u0110 101 - 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 256\u03c0 B 64\u03c0 C 16\u03c0. D 64\u03c0. . . 3 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch c\u1ee7a m\u1eb7t c\u1ea7u c\u00f3 b\u00e1n k\u00ednh r = 4 l\u00e0 S = 4\u03c0r2 = 4\u03c0 \u00b7 42 = 64\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 8 (C\u00e2u 11 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Cho m\u1eb7t c\u1ea7u c\u00f3 b\u00e1n k\u00ednh r = 5. Di\u1ec7n t\u00edch c\u1ee7a m\u1eb7t c\u1ea7u \u0111\u00e3 cho b\u1eb1ng A 25\u03c0. B 500\u03c0 C 100\u03c0. D 100\u03c0 . . 3 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch c\u1ee7a m\u1eb7t c\u1ea7u c\u00f3 b\u00e1n k\u00ednh r = 5 l\u00e0 S = 4\u03c0r2 = 4\u03c0 \u00b7 52 = 100\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 9 (C\u00e2u 5 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x \u2212 1)2 + (y + 2)2 + (z + 3)2 = 4. T\u00e2m c\u1ee7a (S) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (\u22121; 2; 3). B (2; \u22124; \u22126). C (\u22122; 4; 6). D (1; \u22122; \u22123). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) : (x \u2212 a)2 + (y \u2212 b)2 + (z \u2212 c)2 = R2 th\u00ec c\u00f3 t\u00e2m l\u00e0 (a; b; c). V\u1eady t\u00e2m c\u1ee7a (S) l\u00e0 (1; \u22122; \u22123). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 10 (C\u00e2u 16 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). Cho m\u1eb7t c\u1ea7u c\u00f3 b\u00e1n k\u00ednh r = 5. Di\u1ec7n t\u00edch c\u1ee7a m\u1eb7t c\u1ea7u \u0111\u00e3 cho b\u1eb1ng A 500\u03c0 B 25\u03c0. C 100\u03c0 D 100\u03c0. . . 3 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u c\u1ea7n t\u00ecm l\u00e0 S = 4\u03c0r2 = 4\u03c0 \u00b7 52 = 100\u03c0 (\u0111vdt). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 11 (C\u00e2u 19 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Di\u1ec7n t\u00edch S c\u1ee7a m\u1eb7t c\u1ea7u b\u00e1n k\u00ednh R \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 467 S\u0110T: 0905.193.688","2. M\u1eb7t c\u1ea7u A S = 16\u03c0R2. B S = 4\u03c0R2. C S = \u03c0R2. D S = 4\u03c0R2. 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch c\u1ee7a m\u1eb7t c\u1ea7u b\u00e1n k\u00ednh R l\u00e0 S = 4\u03c0R2. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 12 (C\u00e2u 6 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Di\u1ec7n t\u00edch S c\u1ee7a m\u1eb7t c\u1ea7u b\u00e1n k\u00ednh R \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A S = 4\u03c0R2. B S = 16\u03c0R2. C S = 4\u03c0R2. D S = \u03c0R2. 3 \u0253 L\u1eddi gi\u1ea3i. C\u00f4ng th\u1ee9c di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u b\u00e1n k\u00ednh R l\u00e0 S = 4\u03c0R2. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 13 (C\u00e2u 6 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Di\u1ec7n t\u00edch S c\u1ee7a m\u1eb7t c\u1ea7u b\u00e1n k\u00ednh R \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c n\u00e0o sau \u0111\u00e2y? A S = \u03c0R2. B S = 4\u03c0R2. C S = 4\u03c0R2. D S = 16\u03c0R2. 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch S c\u1ee7a m\u1eb7t c\u1ea7u b\u00e1n k\u00ednh R \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c S = 4\u03c0R2. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 14 (C\u00e2u 25 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Di\u1ec7n t\u00edch S c\u1ee7a m\u1eb7t c\u1ea7u b\u00e1n k\u00ednh R \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A S = \u03c0R2. B S = 16\u03c0R2. C S = 4\u03c0R2. D S = 4\u03c0R2. 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 di\u1ec7n t\u00edch S c\u1ee7a m\u1eb7t c\u1ea7u b\u00e1n k\u00ednh R l\u00e0 S = 4\u03c0R2. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 15 (C\u00e2u 9 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i c\u1ea7u b\u00e1n k\u00ednh 4a b\u1eb1ng A 4 \u03c0a3. B 256 \u03c0a3. C 256\u03c0a3. D 64 \u03c0a3. 3 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i c\u1ea7u b\u00e1n k\u00ednh 4a b\u1eb1ng 4 \u03c0(4a)3 = 256 \u03c0a3. 33 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 16 (C\u00e2u 6 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Di\u1ec7n t\u00edch S c\u1ee7a m\u1eb7t c\u1ea7u b\u00e1n k\u00ednh R \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A S = 4\u03c0R2. B S = 16\u03c0R2. C S = 4\u03c0R2. D S = \u03c0R2. 3 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch S c\u1ee7a m\u1eb7t c\u1ea7u b\u00e1n k\u00ednh R l\u00e0 S = 4\u03c0R2. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 468 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U \u0104 C\u00e2u 17 (C\u00e2u 18 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i c\u1ea7u b\u00e1n k\u00ednh 2a l\u00e0 A 4 \u03c0a3. B 32 \u03c0a3. D 8 \u03c0a3. 33 C 32\u03c0a3. 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i c\u1ea7u b\u00e1n k\u00ednh 2a l\u00e0 V = 4 \u03c0(2a)3 = 32 \u03c0a3. 33 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 18 (C\u00e2u 13 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i c\u1ea7u b\u00e1n k\u00ednh 4a b\u1eb1ng A 4 \u03c0a3. B 256 \u03c0a3. C 64\u03c0a3. D 64 \u03c0a3. 3 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i c\u1ea7u b\u00e1n k\u00ednh 4a l\u00e0 V = 4 \u03c0R3 = 4 \u00b7 (4a)3 = 256 \u03c0a3. \u03c0 33 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 19 (C\u00e2u 15 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i c\u1ea7u b\u00e1n k\u00ednh 2a b\u1eb1ng A 8\u03c0a3. B 4 \u03c0a3. C 8 \u03c0a3. D 32 \u03c0a3. 3 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i c\u1ea7u l\u00e0 V = 4 \u03c0(2a)3 = 32 \u03c0a3. 33 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 20 (C\u00e2u 22 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho \u0111i\u1ec3m M n\u1eb1m ngo\u00e0i m\u1eb7t c\u1ea7u S(O; R). Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A OM \u2264 R. B OM > R. C OM = R. D OM < R. \u0253 L\u1eddi gi\u1ea3i. Kh\u1eb3ng \u0111\u1ecbnh \u0111\u00fang l\u00e0 OM > R. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 21 (C\u00e2u 15 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho \u0111i\u1ec3m M n\u1eb1m ngo\u00e0i m\u1eb7t c\u1ea7u S(O; R). Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A OM < R. B OM = R. C OM > R. D OM \u2264 R. \u0253 L\u1eddi gi\u1ea3i. M n\u1eb1m ngo\u00e0i m\u1eb7t c\u1ea7u S(O; R) \u21d4 OM > R. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 22 (C\u00e2u 4 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i c\u1ea7u b\u00e1n k\u00ednh r \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A V = 1\u03c0r3. B V = 2\u03c0r3. C V = 4\u03c0r3. D V = 4\u03c0r3. 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i c\u1ea7u c\u00f3 b\u00e1n k\u00ednh r l\u00e0 V = 4 \u03c0r3. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 469 S\u0110T: 0905.193.688","2. M\u1eb7t c\u1ea7u \u0104 C\u00e2u 23 (C\u00e2u 19 - M\u0110 104 - 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 kh\u1ed1i c\u1ea7u \u0111\u00e3 cho l\u00e0 32\u03c0 8\u03c0 A . B 16\u03c0. C 32\u03c0. D . 3 3 \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch kh\u1ed1i c\u1ea7u b\u00e1n k\u00ednh r =2 l\u00e0 V = 4 \u03c0r3 = 4 \u00b7 \u03c0 \u00b7 23 = 32\u03c0 . 33 3 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 24 (C\u00e2u 22 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho m\u1eb7t c\u1ea7u b\u00e1n k\u00ednh R ngo\u1ea1i ti\u1ebfp \u221am\u1ed9t h\u00ecnh l\u1eadp ph\u01b0\u01a1ng c\u1ea1nh a. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2\u221ay \u0111\u00fang? \u221a B a= 3R Ca = 2R. . D 2 3R A a = 2 3R. a= . 33 \u221a \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh l\u1eadp \u221aph\u01b0\u01a1ng c\u00f3 \u0111\u1ed9 d\u00e0i \u0111\u221a\u01b0\u1eddng ch\u00e9o l\u00e0 a 3. T\u1eeb \u0111\u00f3 b\u00e1n k\u00ednh c\u1ee7a m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh l\u1eadp ph\u01b0\u01a1ng a 3 2 3R l\u00e0 R = . Do v\u1eady a = . 23 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 25 (C\u00e2u 40 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A, AB = a v\u00e0 A\u2019CB = 30\u25e6. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i n\u00f3n nh\u221a\u1eadn \u0111\u01b0\u1ee3c khi quay tam gi\u00e1c ABC quanh c\u1ea1nh A\u221aC. 3\u03c0a3 \u221a 3\u03c0a3 AV= . B V = 3\u03c0a3. CV= . D V = \u03c0a3. 39 \u0253 L\u1eddi gi\u1ea3i. Kh\u1ed1i n\u00f3n\u221anh\u1eadn \u0111\u01b0\u1ee3c khi quay tam gi\u00e1c ABC quanh c\u1ea1nh AC c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y l\u00e0 AB = a, \u0111\u01b0\u1eddng cao l\u00e0 AC = 3a. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 26 (C\u00e2u 30 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00ecnh ch\u00f3p S.ABCD c\u00f3 \u0111\u00e1y l\u00e0 h\u00ecnh ch\u1eef nh\u1eadt v\u1edbi AB = 3a, BC = 4a, SA = 12a v\u00e0 SA vu\u00f4ng g\u00f3c v\u1edbi \u0111\u00e1y. T\u00ednh b\u00e1n k\u00ednh R c\u1ee7a m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p S.ABCD. A R= 5a B 17a C R= 13a D R = 6a. . R= . . 2 2 2 \u0253 L\u1eddi gi\u1ea3i. Theo gi\u1ea3 thi\u1ebft ta suy ra S\u2019AC = S\u2019BC = S\u2019DC = 90\u25e6. S AD Do \u0111\u00f3 m\u1eb7t c\u1ea7\u221au ngo\u1ea1i ti\u1ebfp S.ABCD c\u00f3 \u0111\u221a\u01b0\u1eddng k\u00ednh l\u00e0 SC. Ta c\u00f3 AC = AB2 + BC2 = 5a, SC = SA2 + AC2 = 13a. SC 13a V\u1eady R = = 22 BC Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 27 (C\u00e2u 42 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00ecnh ch\u00f3p S.ABC c\u00f3 \u0111\u00e1y ABC l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh b\u1eb1ng 1, m\u1eb7t b\u00ean SAB l\u00e0 tam gi\u00e1c \u0111\u1ec1u v\u00e0 n\u1eb1m trong m\u1eb7t ph\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng \u0111\u00e1y. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i c\u1ea7u ngo\u1ea1i ti\u1ebfp Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 470 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U h\u00ecnh ch\u00f3p \u0111\u00e3\u221acho. \u221a C V =4 \u221a 5 15\u03c0 5 15\u03c0 3\u03c0 5\u03c0 A V = . B V= . . D V = . 18 54 27 3 \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t R l\u00e0 b\u00e1n k\u00ednh m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp kh\u1ed1i ch\u00f3p. S G D\u1ef1ng h\u00ecnh nh\u01b0 h\u00ecnh b\u00ean v\u1edbi IG l\u00e0 tr\u1ee5c \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c ABC v\u00e0 I\u221aG l\u00e0 tr\u1ee5c \u0111\u221a\u01b0\u1eddng tr\u00f2n ngo\u221a\u1ea1i ti\u1ebfp tam gi\u00e1c SAB. 3 3 \u21d2 IH = 6 Ta c\u00f3: G H = ; GH = . 6 6\u221a 6 \u221a \u221a 15 \u21d2 V = 4 \u03c0R3 = 5 15\u03c0 Do v\u1eady R = IH2 + HA2 = . I 6 3 54 A C HG B Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 28 (C\u00e2u 41 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00ecnh h\u1ed9p ch\u1eef nh\u1eadt ABCD.A B C D c\u00f3 AB = a, AD = 2a v\u00e0 AA = 2a. T\u00ednh b\u00e1n k\u00ednh R c\u1ee7a m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp t\u1ee9 di\u1ec7n ABB C . A R = 3a. B 3a C R= 3a D R = 2a. R= . . 4 2 \u0253 L\u1eddi gi\u1ea3i. B\u00e1n k\u00ednh m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfpABB C b\u1eb1ng v\u1edbi b\u00e1n k\u00ednh m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh h\u1ed9p ch\u1eef nh\u1eadt \u0111\u00e3 cho \u221a v\u00e0 c\u0169ng b\u1eb1ng n\u1eeda \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng ch\u00e9o d\u00e0i nh\u1ea5t c\u1ee7a h\u00ecnh h\u1ed9p. Suy ra R = 1 AB 2 + AD2 + AA 2 = 3a . 22 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 29 (C\u00e2u 43 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 20\u221a17). Cho h\u00ecnh ch\u00f3p t\u1ee9 gi\u00e1c \u0111\u1ec1u S.ABCD c\u00f3 c\u1ea1nh \u0111\u00e1y b\u1eb1ng 3 2a, c\u1ea1nh b\u00ean b\u1eb1ng 5a. T\u00ednh b\u00e1n k\u00ednh R c\u1ee7a m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p S.ABCD. \u221a\u221a A R = 3a. B R = 2a. C R= 25a D R = 2a. . 8 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi O l\u00e0 t\u00e2m h\u00ecnh vu\u00f4ng ABCD, G l\u00e0 trung \u0111i\u1ec3m SD, S GI \u22a5 SD, I \u2208 SO. \u221a \u221a\u221a Ta c\u00f3 c\u1ea1nh \u0111\u00e1y b\u1eb1ng 3 2a n\u00ean BD =\u221a3 2a. 2 = 6a, OD = 3a. X\u00e9t SOD vu\u00f4ng t\u1ea1i O ta c\u00f3: SO = SD2 \u2212 OD2 = 4a. G Ta c\u00f3 SGI \u0111\u1ed3ng d\u1ea1ng v\u1edbi SOD (g-g), suy ra I D A SO = SD \u21d2 4a.R = 1 (5a)2 \u21d2 R = 25a \u00b7 SG SI 2 8 O BC Ch\u1ecdn \u0111\u00e1p \u00e1n C 471 S\u0110T: 0905.193.688 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t","2. M\u1eb7t c\u1ea7u \u0104 C\u00e2u 30 (C\u00e2u 42 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh ch\u00f3p S.ABC c\u00f3 \u0111\u00e1y l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh 4a, SA vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng \u0111\u00e1y, g\u00f3c gi\u1eefa m\u1eb7t ph\u1eb3ng (SBC) v\u00e0 m\u1eb7t \u0111\u00e1y b\u1eb1ng 60\u25e6. Di\u1ec7n t\u00edch c\u1ee7a m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p S.ABC b\u1eb1ng A 172\u03c0a2 B 76\u03c0a2 C 84\u03c0a2. D 172\u03c0a2 . . . 3 3 9 \u0253 L\u1eddi gi\u1ea3i. \u221a \u221a Tam gi\u00e1c ABC \u0111\u1ec1u c\u1ea1nh 4a, AM = 4a 3 = 2a 3 v\u1edbi M l\u00e0 trung 2 S N \u0111i\u1ec3m BC. Do (SAM ) \u22a5 BC n\u00ean g\u00f3c gi\u1eefa 2 m\u1eb7t ph\u1eb3n\u221ag (S\u221aBC) v\u00e0 A d (ABC) l\u00e0 S\u2019M A = 60\u25e6. Khi \u0111\u00f3 SA = AM. tan 60\u25e6 = 2a 3. 3 = 6a. I Qua t\u00e2m G c\u1ee7a tam gi\u00e1c \u0111\u1ec1u ABC d\u1ef1ng tr\u1ee5c Gx vu\u00f4ng g\u00f3c m\u1eb7t ph\u1eb3ng (ABC) th\u00ec G c\u00e1ch \u0111\u1ec1u A, B, C v\u00e0 t\u00e2m m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp S.ABC n\u1eb1m tr\u00ean Gx. T\u1eeb trung \u0111i\u1ec3m E c\u1ee7a SA d\u1ef1ng \u0111\u01b0\u1eddng th\u1eb3ng d song song v\u1edbi AM c\u1eaft Gx t\u1ea1i I th\u00ec IS = IA n\u00ean I l\u00e0 t\u00e2m m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp ch\u00f3p S.ABC. C G M B Theo \u0111\u1ecbnh l\u00fd Pytago cho tam gi\u00e1c vu\u00f4ng IAG ta c\u00f3 \u221a \u00a0 S A \u00e32 \u00c5 2 \u00e32 \u00c3 \u00c7 \u221a \u00e52 \u2026 43 I G2 \u00c5 AM (3a)2 4a 3 a R = IA = + GA2 = + = + = 23 33 V\u1eady S = 4\u03c0R2 = 4\u03c0 \u00b7 43 a2 = 172 \u03c0a2. 33 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 31 (C\u00e2u 40 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh ch\u00f3p S.ABC c\u00f3 \u0111\u00e1y l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh 4a, SA vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng \u0111\u00e1y, g\u00f3c gi\u1eefa m\u1eb7t ph\u1eb3ng (SBC) v\u00e0 m\u1eb7t ph\u1eb3ng \u0111\u00e1y b\u1eb1ng 30\u25e6. Di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p S.ABC b\u1eb1ng B 172\u03c0a2 C 76\u03c0a2 D 76\u03c0a2 A 52\u03c0a2. . . . 3 9 3 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 472 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a c\u1ee7a BC. S d N Ta c\u00f3 \u00aeBC \u22a5 AM \u21d2 BC \u22a5 (SAM ) \u21d2 BC \u22a5 SM . A BC \u22a5 SA T\u1eeb \u0111\u00f3 suy ra (S\u00a4BC) , (ABC) = S\u00ffM, AM = S\u2019M A = 30\u25e6. I Ta c\u00f3 AM \u221a SA = AM. tan 30\u25e6 = \u221a \u221a1 = 2a. HM C = 2a 3; 2a 3. 3 B G\u1ecdi H l\u00e0 tr\u1ecdng t\u00e2m tam gi\u00e1c ABC, d\u1ef1ng \u0111\u01b0\u1eddng th\u1eb3ng d \u0111i qua H v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (ABC). \u0110\u01b0\u1eddng th\u1eb3ng d l\u00e0 tr\u1ee5c c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c ABC. M\u1eb7t ph\u1eb3ng trung tr\u1ef1c c\u1ee7a \u0111o\u1ea1n SA \u0111i qua trung \u0111i\u1ec3m N c\u1ee7a SA, c\u1eaft \u0111\u01b0\u1eddng th\u1eb3ng d t\u1ea1i \u0111i\u1ec3m I. Khi \u0111\u00f3 I l\u00e0 t\u00e2m m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p S.ABC v\u00e0 b\u00e1n k\u00ednh m\u1eb7t c\u1ea7u n\u00e0y l\u00e0 R = AI. \u221a SA 2 4a 3 L\u1ea1i c\u00f3 IH = AN = = a; AH = AM = ; 2 3\u221a 3 \u221a \u2026 16a2 a 57 AI = AH 2 + IH2 + a2 = =. 33 Di\u1ec7n t\u00edch t\u00edch m\u1eb7t c\u1ea7u c\u1ea7n t\u00ecm l\u00e0 S = 4\u03c0R2 = 19a2 = 76\u03c0a2 4\u03c0. . 33 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 32 (C\u00e2u 40 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh ch\u00f3p S.ABC c\u00f3 \u0111\u00e1y l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh 2a, SA S vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng \u0111\u00e1y, g\u00f3c gi\u1eefa m\u1eb7t ph\u1eb3ng (SBC) v\u00e0 H m\u1eb7t ph\u1eb3ng \u0111\u00e1y b\u1eb1ng 60\u25e6. Di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh I ch\u00f3p S.ABC b\u1eb1ng AC GM A 43\u03c0a2 B 19\u03c0a2 C 43\u03c0a2 B . . . 3 3 9 D 21\u03c0a2. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ea1nh BC, H l\u00e0 trung \u0111i\u1ec3m c\u1ea1nh SA v\u00e0 G l\u00e0 tr\u1ecdng\u221at\u00e2m ta\u221am gi\u00e1c ABC. 2 2 (2a) 32 3a Do tam gi\u00e1c ABC l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh 2a n\u00ean AG = AM = \u00b7 = . 3 32 3 ((S\u00a4BC) , (ABC)) = S\u2019M A = 60\u25e6. \u221a\u221a Trong tam gi\u00e1c vu\u00f4ng SAM , ta c\u00f3 SA = AM. tan S\u2019M A = a 3. 3 = 3a. SA 3a Suy ra AH = = . 22 D\u1ef1ng \u0111\u01b0\u1eddng th\u1eb1ng d vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (ABC) t\u1ea1i \u0111i\u1ec3m G. Khi \u0111\u00f3, d l\u00e0 tr\u1ee5c c\u1ee7a tam gi\u00e1c AB C . Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 473 S\u0110T: 0905.193.688","2. M\u1eb7t c\u1ea7u D\u1ef1ng \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a c\u1ea1nh SA c\u1eaft \u0111\u01b0\u1eddng th\u1eb1ng d t\u1ea1i \u0111i\u1ec3m I. Khi \u0111\u00f3, I l\u00e0 t\u00e2m m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p S.ABC v\u00e0 b\u00e1n k\u00ednh R = IA. Trong tam gi\u00e1c vu\u00f4ng I GA, ta c\u00f3 I A2 = I G2 + AG2 = 43a2 \u21d2 R2 = I A2 = 43a2 . 12 12 43a2 43\u03c0a2 Di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p S.ABC l\u00e0 S = 4\u03c0R2 = 4\u03c0 \u00b7 = . 12 3 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 33 (C\u00e2u 39 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t \u221a2 - N\u0103m 2019 - 2020). Cho h\u00ecnh n\u00f3n (N ) c\u00f3 \u0111\u1ec9nh S, b\u00e1n k\u00ednh \u0111\u00e1y b\u1eb1ng 2a v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh b\u1eb1ng 4a. G\u1ecdi (T ) l\u00e0 m\u1eb7t c\u1ea7u\u221a\u0111i qua \u0111\u1ec9nh S v\u00e0 \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y c\u1ee7a (N ). B\u00e1n\u221ak\u00ednh c\u1ee7a (T ) b\u1eb1ng \u221a \u221a D 8 14a A4 2a C4 14a . B 14a. . . 3 77 \u0253 L\u1eddi gi\u1ea3i. X\u00e9t m\u1eb7t c\u1ea7u (O; R) v\u00e0 h\u00ecnh n\u00f3n c\u00f3 \u0111\u00e1y l\u00e0 h\u00ecnh tr\u00f2n t\u00e2m A n\u1ed9i ti\u1ebfp m\u1eb7t c\u1ea7u nh\u01b0 h\u00ecnh v\u1ebd. \u0110\u1eb7t h = CA, r = AB. \u221a S Theo gi\u1ea3 thi\u1ebft ta c\u00f3 AB = 2a, CB = 4a. X\u00e9t tam gi\u00e1c vu\u00f4ng CBD c\u00f3 111 R AB2 = BC2 + BD2 O 11 1 =+ 2a2 16a2 BD2 R \u221a 4 7a Khi \u0111\u00f3 BD = . Ar 7 \u221a B = AB 4 7a Ta c\u00f3 BC \u00b7 BD \u00b7 CD \u21d4 4a \u00b7 = \u221a \u221a7 2a \u00b7 CD. Suy ra CD = 8 14a . D 7\u221a 4 14a V\u1eady b\u00e1n k\u00ednh m\u1eb7t c\u1ea7u c\u1ea7n t\u00ecm l\u00e0 . 7 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 34 (C\u00e2u 41 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t \u221a2 - N\u0103m 2019 - 2020). Cho h\u00ecnh n\u00f3n (N ) c\u00f3 \u0111\u1ec9nh S, b\u00e1n k\u00ednh \u0111\u00e1y b\u1eb1ng 3a v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh b\u1eb1ng 4a. G\u1ecdi (T ) l\u00e0 m\u1eb7t c\u1ea7u\u221a\u0111i qua S v\u00e0 \u0111\u01b0\u1eddng tr\u00f2n \u221a\u0111\u00e1y c\u1ee7a (N ). B\u00e1n k\u00ednh\u221ac\u1ee7a (T ) b\u1eb1ng \u221a A2 10a 16 13a C8 13a D 13a. . B . . 3 13 13 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 474 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U G\u1ecdi O, AB l\u1ea7n l\u01b0\u1ee3t l\u00e0 t\u00e2m v\u00e0 \u0111\u01b0\u1eddng k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y c\u1ee7a h\u00ecnh n\u00f3n (N ). S I G\u1ecdi I l\u00e0 t\u00e2m m\u1eb7t c\u1ea7u (T ). B Suy ra IA = IB = IS v\u00e0 I \u2208 SO. Do \u0111\u00f3 I l\u00e0 t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i O ti\u1ebfp tam gi\u00e1c \u221aSAB. \u221a \u221a Ta c\u00f3 SO = SA\u221a2 \u2212 OA2 \u221a= 16a2 \u2212 3a2 = 13a v\u00e0 sin S\u2019AB = SO 13a 13 sin S\u2019AO = = =. SA 4a 4 V\u1eady b\u00e1n k\u00ednh c\u1ee7a m\u1eb7t c\u1ea7u (T ) l\u00e0 A \u221a SB 4\u221aa 8 13a R= = =. 2 sin S\u2019AB 2 \u00b7 13 13 4 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 35 (C\u00e2u 42 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Cho h\u00ecnh n\u00f3n (N ) c\u00f3 \u0111\u1ec9nh S, b\u00e1n k\u00ednh \u0111\u00e1y b\u1eb1ng a v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh b\u1eb1ng 4a. G\u1ecdi (T ) l\u00e0 m\u1eb7t c\u1ea7u \u0111i q\u221aua \u0111\u1ec9nh S v\u00e0 \u0111\u01b0\u1eddng tr\u00f2n\u221a\u0111\u00e1y c\u1ee7a (N ). B\u00e1n k\u00ednh\u221ac\u1ee7a (T ) b\u1eb1ng \u221a A2 6a 16 15a 8 15a D 15a. . B . C . 3 15 15 \u0253 L\u1eddi gi\u1ea3i. N\u1ebfu c\u1eaft m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp kh\u1ed1i n\u00f3n (N ) b\u1edfi m\u1eb7t ph\u1eb3ng qua S tr\u1ee5c (SAB), ta \u0111\u01b0\u1ee3c m\u1ed9t h\u00ecnh tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c SAB. M I Khi \u0111\u00f3 b\u00e1n k\u00ednh m\u1eb7t c\u1ea7u (T ) b\u1eb1ng b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i OB ti\u1ebfp SAB. G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a SB. K\u1ebb \u0111\u01b0\u1eddng vu\u00f4ng g\u00f3c v\u1edbi SB t\u1ea1i M , c\u1eaft SO t\u1ea1i I. Khi \u0111\u00f3 I l\u00e0 t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp SAB v\u00e0 r = SI l\u00e0 b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp SAB. Ta c\u00f3 SIM SBO \u21d2 SI SM \u21d2 SI = SM \u00b7 SB. = SB SO SO \uf8f1SM = 2a \uf8f4 \u221a A \uf8f2 8 15a Trong \u0111\u00f3 = 4a \u21d2 r = SI = . SB = \u221a \u221a \u2212 OB2 = a 15 15 \uf8f4 SB2 \uf8f3SO Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 36 (C\u00e2u 39 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). \u221a Cho h\u00ecnh n\u00f3n (N ) c\u00f3 \u0111\u1ec9nh S, b\u00e1n k\u00ednh \u0111\u00e1y b\u1eb1ng a v\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh b\u1eb1ng 2 2a. G\u1ecdi (T ) l\u00e0 m\u1eb7t c\u1ea7u\u221a\u0111i qua S v\u00e0 \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y c\u1ee7a (N ). B\u00e1n k\u00ednh\u221ac\u1ee7a (T ) b\u1eb1ng \u221a A4 7a 4a 8 7a D 7a. . B . C . 73 7 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 475 S\u0110T: 0905.193.688","2. M\u1eb7t c\u1ea7u Gi\u1ea3 s\u1eed m\u1eb7t ph\u1eb3ng (\u03b1) qua tr\u1ee5c c\u1ee7a h\u00ecnh n\u00f3n, c\u1eaft h\u00ecnh n\u00f3n theo thi\u1ebft di\u1ec7n S l\u00e0 tam gi\u00e1c SAB c\u00e2n t\u1ea1i S. Khi \u0111\u00f3, (\u03b1) c\u1eaft m\u1eb7t c\u1ea7u (T ) theo \u0111\u01b0\u1eddng tr\u00f2n l\u1edbn l\u00e0 \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp \u221a tam gi\u00e1c SAB. 2a 2 Suy ra b\u00e1n k\u00ednh c\u1ee7a (T ) l\u00e0 b\u00e1n k\u00ednh R c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c SAB. B a H A \u221a\u221a G\u1ecdi H l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB, ta c\u00f3 HA = HB = a v\u00e0 SH = SA2 \u2212 IA2 = a 7. \u221a \u221a Khi \u0111\u00f3, ta c\u00f3 S = 1 \u00b7 AB = 1 a7 \u00b7 2a = 7a2. SH SAB 2 2 SA \u00b7 SB \u00b7 AB M\u00e0 S SAB = 4R n\u00ean suy ra SA \u00b7 SB \u00b7 AB 2 \u221a\u221a \u221a R= = 2a \u00b7 2 \u221a2a \u00b7 2a = 4a 7 4S SAB 4 \u00b7 a2 7 7 . Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 37 (C\u00e2u 43 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00ecnh n\u00f3n c\u00f3 g\u00f3c \u1edf \u0111\u1ec9nh b\u1eb1ng 120\u25e6 v\u00e0 chi\u1ec1u cao b\u1eb1ng 4. G\u1ecdi (S) l\u00e0 m\u1eb7t c\u1ea7u \u0111i qua \u0111\u1ec9nh v\u00e0 ch\u1ee9a \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y c\u1ee7a h\u00ecnh n\u00f3n \u0111\u00e3 cho. Di\u1ec7n t\u00edch c\u1ee7a (S) b\u1eb1ng A 64\u03c0. B 256\u03c0. C 192\u03c0. D 96\u03c0. \u0253 L\u1eddi gi\u1ea3i. S S 120\u25e6 AH B A H I B S G\u1ecdi S l\u00e0 \u0111\u1ec9nh c\u1ee7a h\u00ecnh n\u00f3n v\u00e0 g\u1ecdi I l\u00e0 t\u00e2m m\u1eb7t c\u1ea7u. G\u1ecdi \u0111\u01b0\u1eddng k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y c\u1ee7a h\u00ecnh n\u00f3n l\u00e0 AB; H l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB. Ta c\u00f3 A\u2019SH = 1 A\u2019SB = 60\u25e6\u00b7 2 \u00aeAI = AS V\u00ec n\u00ean AIS l\u00e0 tam gi\u00e1c \u0111\u1ec1u. A\u2018SI = 60\u25e6 Suy ra AI = R = 2SH = 8. V\u1eady Smc = 4\u03c0R2 = 256\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 38 (C\u00e2u 43 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00ecnh n\u00f3n c\u00f3 g\u00f3c \u1edf \u0111\u1ec9nh b\u1eb1ng 120\u25e6 v\u00e0 chi\u1ec1u cao b\u1eb1ng 1. G\u1ecdi (S) l\u00e0 m\u1eb7t c\u1ea7u \u0111i qua \u0111\u1ec9nh v\u00e0 ch\u1ee9a \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y c\u1ee7a h\u00ecnh n\u00f3n \u0111\u00e3 cho. Di\u1ec7n t\u00edch c\u1ee7a (S) b\u1eb1ng A 16\u03c0. B 12\u03c0. C 4\u03c0. D 48\u03c0. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 476","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U X\u00e9t SM O vu\u00f4ng c\u00f3 S M tan M\u2019SO = OM \u21d2 OM = OS. tan 60\u25e6 = \u221a 3\u00b7 O OS N K\u1ebb \u0111\u01b0\u1eddng k\u00ednh SS c\u1ee7a m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh n\u00f3n. Tam gi\u00e1c SM S vu\u00f4ng t\u1ea1i M c\u00f3 M O \u22a5 SS \u221a. 3)2 OM 2 ( = 3. Suy ra OM 2 = OS.OS \u21d4 OS = = OS 1 Do \u0111\u00f3 b\u00e1n k\u00ednh m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh n\u00f3n l\u00e0 OS + OS 1 + 3 S R = = = 2. 22 V\u1eady di\u1ec7n t\u00edch c\u1ee7a (S) l\u00e0 S = 4\u03c0R2 = 4\u03c0.22 = 16\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 39 (C\u00e2u 26 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ednh b\u00e1n k\u00edn\u221ah R c\u1ee7a m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp m\u1ed9t h\u00ecnh l\u1eadp ph\u01b0\u01a1ng c\u00f3 c\u1ea1nh b\u1eb1ng 2a. A a3 B R = a. \u221a D R= \u221a R= . C R = 2 3a. a 3. 3 \u0253 L\u1eddi gi\u1ea3i. \u221a C\u221a\u1ea1nh h\u00ecnh l\u1eadp ph\u01b0\u01a1ng b\u1eb1ng 2a \u21d2 \u0111\u01b0\u1eddng ch\u00e9o h\u00ecnh l\u1eadp ph\u01b0\u01a1ng b\u1eb1ng 2a 3 \u21d2 b\u00e1n k\u00ednh m\u1eb7t c\u1ea7u b\u1eb1ng a 3. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 40 (C\u00e2u 50 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho m\u1eb7t c\u1ea7u (S) c\u00f3 b\u00e1n k\u00ednh b\u1eb1ng 4, h\u00ecnh tr\u1ee5 (H) c\u00f3 chi\u1ec1u cao b\u1eb1ng 4 v\u00e0 hai \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y n\u1eb1m tr\u00ean (S). G\u1ecdi V1 l\u00e0 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u1ee5 (H) v\u00e0 V2 l\u00e0 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i c\u1ea7u (S). T\u00ednh t\u1ec9 s\u1ed1 V1 . V2 V1 = 9 V1 1 V1 3 V1 2 A . B = . C = . D = . V2 16 V2 3 V2 16 V2 3 \u0253 L\u1eddi gi\u1ea3i. 256\u03c0 - Ta c\u00f3 V2 = . 3 \u221a \u221a \u221a - B\u00e1n k\u00ednh \u0111\u00e1y c\u1ee7a tr\u1ee5 r = 42 \u2212 22 = 2 3, suy ra V1 = 4.\u03c0(2 3)2 = 48\u03c0. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 41 (C\u00e2u 47 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00ecnh n\u00f3n (N ) c\u00f3 \u0111\u01b0\u1eddng sinh t\u1ea1o v\u1edbi \u0111\u00e1y m\u1ed9t g\u00f3c 60\u25e6. M\u1eb7t ph\u1eb3ng qua tr\u1ee5c c\u1ee7a (N ) c\u1eaft (N ) \u0111\u01b0\u1ee3c thi\u1ebft di\u1ec7n l\u00e0 m\u1ed9t tam gi\u00e1c c\u00f3 b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n n\u1ed9i ti\u1ebfp b\u1eb1ng 1. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i n\u00f3n gi\u1edbi h\u1ea1n\u221ab\u1edfi (N ). B V = 9\u03c0. \u221a D V = 3\u03c0. A V = 9 3\u03c0. C V = 3 3\u03c0. \u0253 L\u1eddi gi\u1ea3i. Thi\u1ebft di\u1ec7n qua tr\u1ee5c c\u1ee7a (N ) l\u00e0 tam gi\u00e1c c\u00e2n \u1edf \u0111\u1ec9nh, \u0111\u01b0\u1eddng sinh t\u1ea1o v\u1edbi \u0111\u00e1y m\u1ed9t g\u00f3c 60\u25e6 suy ra thi\u1ebft di\u1ec7n tam gi\u00e1c \u0111\u1ec1u. Tam gi\u00e1c \u0111\u1ec1u c\u00f3 t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n n\u1ed9i ti\u1ebfp tr\u00f9ng v\u1edbi tr\u1ecdng t\u00e2m, t\u1eeb \u0111\u00f3 t\u00ednh \u0111\u01b0\u1ee3c c\u1ea1nh c\u1ee7a tam gi\u00e1c thi\u1ebft di\u1ec7n l\u00e0 \u221a6 v\u00e0 \u0111\u01b0\u1eddng cao l\u00e0 3. V = 1\u03c0 \u00b7 \u00c51 \u00b7 \u221a6 \u00e32 3 3 23 \u00b7 3 = 3\u03c0. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 477 S\u0110T: 0905.193.688","2. M\u1eb7t c\u1ea7u Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 42 (C\u00e2u 41 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00ecnh ch\u00f3p S.ABC c\u00f3 \u0111\u00e1y l\u00e0 tam gi\u00e1c \u0111\u1ec1u c\u1ea1nh 2a, SA vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng \u0111\u00e1y, g\u00f3c gi\u1eefa m\u1eb7t ph\u1eb3ng (SBC) v\u00e0 m\u1eb7t ph\u1eb3ng \u0111\u00e1y b\u1eb1ng 30\u25e6. Di\u1ec7n t\u00edch c\u1ee7a m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p S.ABC b\u1eb1ng A 43\u03c0a2 B 19\u03c0a2 C 19\u03c0a2 D 13\u03c0a2. . . . 3 3 9 \u0253 L\u1eddi gi\u1ea3i. S d E I AC 30\u25e6 G M B G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BC, ta c\u00f3 g\u00f3c S\u2019M A l\u00e0 g\u00f3c gi\u1eefa (SBC) v\u00e0 (ABC) \u21d2 S\u2019M A = 30\u25e6. G\u1ecdi G l\u00e0 t\u221ar\u1ecdng t\u00e2m c\u1ee7a tam gi\u00e1c ABC \u221akhi \u0111\u00f3 ta c\u00f3: 2a 3\u221a 2 2a 3 \u221a \u221a1 AM = = a 3, AG = AM = , SA = AM \u00b7 tan 30\u25e6 = a3 \u00b7 = a. 2 33 3 Qua G k\u1ebb \u0111\u01b0\u1eddng th\u1eb3ng d vu\u00f4ng g\u00f3c v\u1edbi (ABC) \u21d2 d \u2225 SA. \u00df (P ) \u22a5 SA (P ) \u2229 d = G\u1ecdi E l\u00e0 trung \u0111i\u1ec3m c\u1ee7a SA, qua E k\u1ebb m\u1eb7t ph\u1eb3ng (P ) sao cho: {I } Khi \u0111\u00f3 I l\u00e0 t\u00e2m c\u1ee7a m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp kh\u1ed1i ch\u00f3p S.ABC v\u00e0 kh\u221a\u1ed1i c\u1ea7u \u0111\u00f3 c\u00f3 b\u00e1n k\u00ednh l\u00e0: = a 57 \u221a \u00a0 SA \u00e32 \u2026 a2 4a2 I G2 \u00c5 . R = IA = + AG2 = + AG2 = + 6 2 43 V\u1eady di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp h\u00ecnh ch\u00f3p S.ABC l\u00e0: S = 4\u03c0R2 = 19\u03c0a2 . 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 43 (C\u00e2u 42 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00ecnh n\u00f3n c\u00f3 g\u00f3c \u1edf \u0111\u1ec9nh b\u1eb1ng 120\u25e6 v\u00e0 chi\u1ec1u cao b\u1eb1ng 3. G\u1ecdi (S) l\u00e0 m\u1eb7t c\u1ea7u \u0111i qua \u0111\u1ec9nh v\u00e0 ch\u1ee9a \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y c\u1ee7a h\u00ecnh n\u00f3n \u0111\u00e3 cho. Di\u1ec7n t\u00edch c\u1ee7a (S) b\u1eb1ng A 144\u03c0. B 108\u03c0. C 48\u03c0. D 96\u03c0. \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 478 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. M\u1eb6T N\u00d3N. M\u1eb6T TR\u1ee4. M\u1eb6T C\u1ea6U G\u1ecdi H l\u00e0 t\u00e2m \u0111\u00e1y, AB l\u00e0 \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a \u0111\u00e1y h\u00ecnh n\u00f3n v\u00e0 SC l\u00e0 A S 120 \u25e6 \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a m\u1eb7t c\u1ea7u (S). Khi \u0111\u00f3 SH = 3 v\u00e0 A\u2019SC = 60\u25e6. SH H SA = cos 60\u25e6 = 6 (\u0111v\u0111d). B SA2 = SH.SC \u21d4 62 = 3.SC \u21d4 SC = 12. B\u00e1n k\u00ednh c\u1ee7a m\u1eb7t c\u1ea7u (S) l\u00e0 R = 6 n\u00ean di\u1ec7n t\u00edch c\u1ee7a (S) l\u00e0 S = 4\u03c0.62 = 144\u03c0 (\u0111vdt). C Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 44 (C\u00e2u 49 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x \u2212 4)2 + (y + 3)2 + (z + 6)2 = 50 v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng x y+2 z\u22123 d: = = \u22121 . C\u00f3 bao nhi\u00eau \u0111i\u1ec3m M thu\u1ed9c tr\u1ee5c ho\u00e0nh, v\u1edbi ho\u00e0nh \u0111\u1ed9 l\u00e0 s\u1ed1 nguy\u00ean, m\u00e0 2 4 t\u1eeb M k\u1ebb \u0111\u01b0\u1ee3c \u0111\u1ebfn (S) hai ti\u1ebfp tuy\u1ebfn c\u00f9ng vu\u00f4ng g\u00f3c v\u1edbi d? A 29. B 33. C 55. D 28. \u221a \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(4; \u22123; \u22126), R = 5 2. Ta c\u00f3 M \u2208 Ox \u21d2 M (a; 0; 0). G\u1ecdi (P ) l\u00e0 m\u1eb7t ph\u1eb3ng ch\u1ee9a hai ti\u1ebfp tuy\u1ebfn t\u1eeb M \u0111\u1ebfn (S). Khi \u0111\u00f3 (P ) \u0111i qua M (a; 0; 0), vu\u00f4ng g\u00f3c v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng d, ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (P ) l\u00e0 2(x \u2212 a) + 4y \u2212 z = 0 \u21d4 2x + 4y \u2212 z \u2212 2a = 0. Ta c\u00f3 M l\u00e0 \u0111i\u1ec3m n\u1eb1m ngo\u00e0i m\u1eb7t c\u1ea7u, suy ra IM > R \u21d4 (a \u2212 4)2 + 9 + 36 > 50 \u21d4 (a \u2212 4)2 > 5 (1) (2) d (I , (P )) < R \u21d4 |8 \u2212 12\u221a+ 6 \u2212 2a| < \u221a \u21d4 |2 \u2212 2a| < \u221a 52 5 42 21 T\u1eeb (1) v\u00e0 (2), suy ra \u00ae(a \u2212 4)2 > 5 \uf8f1a2 \u2212 8a + 11 > 0 \uf8f1 \u00f1a \u2265 7 \u00f1 \u2212 15 \u2264 a \u2264 1 \u221a \uf8f2 \uf8f4 \u21d4 \uf8f2 |2 \u2212 2a| < 5 42 \u21d4 \uf8f3a2 \u2212 350 \u21d4 7 \u2264 a \u2264 17. 3 a\u22641 2a + 1 < \uf8f4 \u2212 15 \u2264 a \u2264 17 \uf8f3 V\u00ec a \u2208 Z, suy ra c\u00f3 28 \u0111i\u1ec3m M tho\u1ea3 m\u00e3n. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 479 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN 3 B\u00c0I 1. H\u1ec6 T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 1 (C\u00e2u 12 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A(2; \u22124; 3) v\u00e0 B(2; 2; 7). Trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n AB c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (1; 3; 2). B (2; 6; 4). C (2; \u22121; 5). D (4; \u22122; 10). \u0253 L\u1eddi gi\u1ea3i. \uf8f1 = xA + xB = 2 2 \uf8f4\uf8f4xM \uf8f4 \uf8f4 G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n th\u1eb3ng AB. Khi \u0111\u00f3 \uf8f4 = yA + yB = \u22121 \u21d2 M (2; \u22121; 5). \uf8f2 2 yM \uf8f4 \uf8f4 = zA + zB = 5 \uf8f4 2 \uf8f4\uf8f3\uf8f4zM Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 2 (C\u00e2u 10 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). A# B\u00bb c\u00f3 to\u1ea1 \u0111\u1ed9 l\u00e0 Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A(1; 1; \u22122) v\u00e0 B(2; 2; 1). V\u00e9c-t\u01a1 A (3; 3; \u22121). B (\u22121; \u22121; \u22123). C (3; 1; 1). D (1; 1; 3). #\u00bb \u0253 L\u1eddi gi\u1ea3i. AB 1; 3). Ta c\u00f3 = (2 \u2212 1; 2 \u2212 1; 1 \u2212 (\u22122)) = (1; Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 3 (C\u00e2u 3 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). 2). V\u00e9c-t\u01a1 A# B\u00bb c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A (1; 1; \u22121) v\u00e0 B (2; 3; A (1; 2; 3). B (\u22121; \u22122; 3). C (3; 5; 1). D (3; 4; 1). #\u00bb \u0253 L\u1eddi gi\u1ea3i. AB Ta c\u00f3 = (2 \u2212 1; 3 \u2212 1; 2 + 1) = (1; 2; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 4 (C\u00e2u 10 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m M (2; 1; \u22121) tr\u00ean tr\u1ee5c Oz c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (2; 1; 0). B (0; 0; \u22121). C (2; 0; 0). D (0; 1; 0). \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m M (x0; y0; z0) tr\u00ean tr\u1ee5c Oz l\u00e0 M (0; 0; z0). Suy ra h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m M (2; 1; \u22121) tr\u00ean tr\u1ee5c Oz l\u00e0 (0; 0; \u22121). Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 480 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 5 (C\u00e2u 10 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m M (2; 1; \u22121) tr\u00ean tr\u1ee5c Oy c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (0; 0; \u22121). B (2; 0; \u22121). C (0; 1; 0). D (2; 0; 0). \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m M (2; 1 \u2212 1) tr\u00ean tr\u1ee5c Oy c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 (0; 1; 0). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 6 (C\u00e2u 6 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m M (3; 1; \u22121) tr\u00ean tr\u1ee5c Oy c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (0; 1; 0). B (3; 0; 0). C (0; 0; \u22121). D (3; 0; \u22121). \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m M (3; 1; \u22121) tr\u00ean tr\u1ee5c Oy c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 (0; 1; 0). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 7 (C\u00e2u 22 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m M (2; 1; \u22121) tr\u00ean m\u1eb7t ph\u1eb3ng (Ozx) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (0; 1; 0). B (2; 1; 0). C (0; 1; \u22121). D (2; 0; \u22121). \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m M (2; 1; \u22121) tr\u00ean m\u1eb7t ph\u1eb3ng (Oxz) l\u00e0 \u0111i\u1ec3m M (2; 0; \u22121). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 8 (C\u00e2u 24 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t ph\u1eb3ng (P ) : 2x + 3y + z + 2 = 0. V\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng (P ) A #n\u00bb3 = (2; 3; 2). B #n\u00bb1 = (2; 3; 0). C #n\u00bb2 = (2; 3; 1). D #n\u00bb4 = (2; 0; 3). \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh t\u1ed5ng qu\u00e1t c\u1ee7a m\u1eb7t ph\u1eb3ng Ax + By + Cz + D = 0 (v\u1edbi A2 + B2 + C2 > 0) c\u00f3 v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn (A; B; C). V\u1eady v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng (P ) l\u00e0 #n\u00bb2 = (2; 3; 1). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 9 (C\u00e2u 17 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A(3; 2; 1) tr\u00ean tr\u1ee5c Ox c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (0; 2; 1). B (3; 0; 0). C (0; 0; 1). D (0; 2; 0). \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A(3; 2; 1) l\u00ean tr\u1ee5c Ox l\u00e0 A (3; 0; 0). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 10 (C\u00e2u 2 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A (1; 2; 5) tr\u00ean tr\u1ee5c Ox c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (0; 2; 0). B (0; 0; 5). C (1; 0; 0). D (0; 2; 5). \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A (1; 2; 5) tr\u00ean tr\u1ee5c Ox c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 (1; 0; 0). Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 481 S\u0110T: 0905.193.688","1. H\u1ec7 t\u1ecda \u0111\u1ed9 trong kh\u00f4ng gian \u0104 C\u00e2u 11 (C\u00e2u 6 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A(3; 5; 2) tr\u00ean tr\u1ee5c Ox c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (0; 5; 2). B (0; 5; 0). C (3; 0; 0). D (0; 0; 2). \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A(3; 5; 2) tr\u00ean tr\u1ee5c Ox c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 (3; 0; 0) Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 12 (C\u00e2u 7 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A (8 ; 1 ; 2) tr\u00ean tr\u1ee5c Ox c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (0 ; 1 ; 0). B (8 ; 0 ; 0). C (0 ; 1 ; 2). D (0 ; 0 ; 2). \u0253 L\u1eddi gi\u1ea3i. T\u1ecda \u0111\u1ed9 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a A (8 ; 1 ; 2) l\u00ean tr\u1ee5c Ox l\u00e0 (8 ; 0 ; 0). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 13 (C\u00e2u 7 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, \u0111i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A(1; 4; 2) tr\u00ean m\u1eb7t ph\u1eb3ng (Oxy)? A N (0; 4; 2). B P (1; 4; 0). C Q(1; 0; 2). D M (0; 0; 2). \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A(1; 4; 2) tr\u00ean m\u1eb7t ph\u1eb3ng (Oxy) l\u00e0 \u0111i\u1ec3m c\u00f3 t\u1ecda \u0111\u1ed9 (1; 4; 0). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 14 (C\u00e2u 25 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, \u0111i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A(1; 2; 3) tr\u00ean m\u1eb7t ph\u1eb3ng (Oxy)? A Q(1; 0; 3). B P (1; 2; 0). C M (0; 0; 3). D N (0; 2; 3). \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A(1; 2; 3) tr\u00ean m\u1eb7t ph\u1eb3ng (Oxy) l\u00e0 P (1; 2; 0). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 15 (C\u00e2u 22 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, \u0111i\u1ec3m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A(3; 5; 2) tr\u00ean m\u1eb7t ph\u1eb3ng (Oxy)? A M (3; 0; 2). B Q(0; 0; 2). C P (0; 5; 2). D N (3; 5; 0). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (Oxy) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh z = 0. Do \u0111\u00f3 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a \u0111i\u1ec3m A(3; 5; 2) tr\u00ean m\u1eb7t ph\u1eb3ng (Oxy) l\u00e0 N (3; 5; 0). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 16 (C\u00e2u 12 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 -# 2\u00bb021). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(\u22122; 3; 5). T\u1ecda \u0111\u1ed9 c\u1ee7a v\u00e9c-t\u01a1 OA l\u00e0 A (\u22122; 3; 5). B (2; \u22123; 5). C (\u22122; \u22123; 5). D (2; \u22123; \u22125). Oxyz, # \u0253\u00bb L\u1eddi gi\u1ea3i. A. VTr\u1eadoynOg# Ak\u00bbh=\u00f4n(g\u2212g2i;a3n; 5). t\u1ecda \u0111\u1ed9 c\u1ee7a v\u00e9c-t\u01a1 OA ch\u00ednh l\u00e0 t\u1ecda \u0111\u1ed9 c\u1ee7a \u0111i\u1ec3m Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 482 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 17 (C\u00e2u 14 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 -# 2\u00bb021). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(4; \u22121; 3). T\u1ecda \u0111\u1ed9 c\u1ee7a v\u00e9c-t\u01a1 OA l\u00e0 A O# A\u00bb = (\u22124; 1; 3). B O# A\u00bb = (4; \u22121; 3). C O# A\u00bb = (\u22124; 1; \u22123). D O# A\u00bb = (4; 1; 3). Ta c\u00f3 O# A\u00bb = (4; \u22121; 3). \u0253 L\u1eddi gi\u1ea3i. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 18 (C\u00e2u 19 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 -O# 2A\u00bb02l\u00e01). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(3; 2; \u22124). T\u1ecda \u0111\u1ed9 c\u1ee7a v\u00e9c-t\u01a1 A (3; \u22122; \u22124). B (\u22123; \u22122; 4). C (3; 2; \u22124). D (3; 2; 4). T\u1ecda \u0111\u1ed9 v\u00e9c-t\u01a1 O# A\u00bb = (3; 2; \u22124). \u0253 L\u1eddi gi\u1ea3i. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 19 (C\u00e2u 9 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2# 0\u00bb21). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(2; \u22121; 4). T\u1ecda \u0111\u1ed9 c\u1ee7a v\u00e9c-t\u01a1 OA l\u00e0 A (\u22122; 1; 4). B (2; \u22121; 4). C (2; 1; 4). D (\u22122; 1; \u22124). #\u00bb \u0253 L\u1eddi gi\u1ea3i. OA Ta c\u00f3 = (2; \u22121; 4). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 20 (C\u00e2u 12 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho hai v\u00e9c-t\u01a1 #u\u00bb = (1; \u22122; 3) v\u00e0 #v\u00bb = (\u22121; 2; 0). T\u1ecda \u0111\u1ed9 v\u00e9c-t\u01a1 #u\u00bb + #v\u00bb l\u00e0 A (0; 0; \u22123). B (0; 0; 3). C (\u22122; 4; \u22123). D (2; \u22124; 3). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 #u\u00bb + #v\u00bb = (0; 0; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 21 (C\u00e2u 11 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz cho hai v\u00e9c-t\u01a1 #u\u00bb(\u22121; 2; 0) v\u00e0 #v\u00bb(1; \u22122; 3). T\u1ecda \u0111\u1ed9 c\u1ee7a v\u00e9c-t\u01a1 #u\u00bb+ #v\u00bb l\u00e0 A (\u22122; 4; \u22123). B (2; \u22124; 3). C (0; 0; 3). D (0; 0; \u22123). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 #u\u00bb + #v\u00bb = (0; 0; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 22 (C\u00e2u 2 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho hai v\u00e9c-t\u01a1 #u\u00bb = (\u22121; 2; \u22125) v\u00e0 #v\u00bb = (0; \u22122; 3). T\u1ecda \u0111\u1ed9 c\u1ee7a v\u00e9c-t\u01a1 #u\u00bb + #v\u00bb l\u00e0 A (1; \u22124; 8). B (\u22121; 0; \u22122). C (\u22121; 4; \u22128). D (1; 0; 2). Ta c\u00f3 #u\u00bb + #v\u00bb = (\u22121; 0; \u22122). \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 Ch\u1ecdn \u0111\u00e1p \u00e1n B 483 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t","1. H\u1ec7 t\u1ecda \u0111\u1ed9 trong kh\u00f4ng gian \u0104 C\u00e2u 23 (C\u00e2u 4 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho hai v\u00e9c-t\u01a1 #u\u00bb = (0; \u22122; 3) v\u00e0 #v\u00bb = (\u22121; 2; \u22125). T\u1ecda \u0111\u1ed9 c\u1ee7a v\u00e9c-t\u01a1 #u\u00bb + #v\u00bb l\u00e0 A (1; \u22124; 8). B (\u22121; 0; \u22122). C (\u22121; 4; \u22128). D (1; 0; 2). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 #u\u00bb + #v\u00bb = (0 \u2212 1; \u22122 + 2; 3 \u2212 5) = (\u22121; 0; \u22122). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 24 (C\u00e2u 7 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(1; 2; \u22123). H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a A l\u00ean m\u1eb7t ph\u1eb3ng (Oxy) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (0; 2; \u22123). B (1; 0; \u22123). C (1; 2; 0). D (1; 0; 0). \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh chi\u1ebfu c\u1ee7a \u0111i\u1ec3m A(a; b; c) l\u00ean m\u1eb7t ph\u1eb3ng (Oxy) l\u00e0 \u0111i\u1ec3m A (a; b; 0) n\u00ean h\u00ecnh chi\u1ebfu c\u1ee7a \u0111i\u1ec3m A(1; 2; \u22123) l\u00ean m\u1eb7t ph\u1eb3ng (Oxy) l\u00e0 \u0111i\u1ec3m A (1; 2; 0). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 25 (C\u00e2u 23 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(1; 2; \u22123). H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a A l\u00ean m\u1eb7t ph\u1eb3ng (Oxy) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (1; 0; \u22123). B (1; 0; 0). C (1; 2; 0). D (0; 2; \u22123). \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a A(1; 2; \u22123) l\u00ean m\u1eb7t ph\u1eb3ng (Oxy) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 (1; 2; 0). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 26 (C\u00e2u 27 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz. Cho hai vect\u01a1 #u\u00bb = (1; \u22124; 0) v\u00e0 #v\u00bb = (\u22121; \u22122; 1). Vect\u01a1 #u\u00bb + 3 #v\u00bb c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (\u22122; \u22126; 3). B (\u22124; \u22128; 4). C (\u22122; \u221210; \u22123). D (\u22122; \u221210; 3). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3: #u\u00bb = (1; \u22124; 0); 3 #v\u00bb = (\u22123; \u22126; 3). Suy ra #u\u00bb + 3 #v\u00bb = (\u22122; \u221210; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 27 (C\u00e2u 14 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho hai vect\u01a1 #u\u00bb = (1; 3; \u22122) v\u00e0 #v\u00bb = (2; 1; \u22121). T\u1ecda \u0111\u1ed9 c\u1ee7a vect\u01a1 #u\u00bb \u2212 #v\u00bb l\u00e0 A (3; 4; \u22123). B (\u22121; 2; \u22123). C (\u22121; 2; \u22121). D (1; \u22122; 1). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 #u\u00bb \u2212 #v\u00bb = (\u22121; 2; \u22121). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 28 (C\u00e2u 8 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, t\u00ecm t\u1ecda \u0111\u1ed9 t\u00e2m I v\u00e0 b\u00e1n k\u00ednh R c\u1ee7a m\u1eb7t c\u1ea7u (x \u2212 1)2 + (y + 2)2 + (z \u2212 4)2 = 20. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 484 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u221a \u221a A I(\u22121; 2; \u22124), R = 5 2. B I(\u22121; 2; \u22124), R =\u221a2 5. C I(1; \u22122; 4), R = 20. D I(1; \u22122; 4), R = 2 5. \u0253 L\u1eddi gi\u1ea3i. Pt m\u1eb7t c\u1ea7u (x \u2212 x\u25e6)2 + (y \u2212 y\u25e6)2 + (z \u2212 z\u25e6)2 = R2 c\u00f3 t\u00e2m l\u00e0 I(x\u25e6; y\u25e6; z\u25e6), b\u00e1n k\u00ednh l\u00e0: R. Do \u0111\u00f3 m\u1eb7t c\u1ea7u (x\u22121)2 +(y \u2212 (\u22122))2 +(z \u22124)2 = \u221a c\u00f3 t\u00e2m I (1; \u22122; 4) v\u00e0 b\u00e1n k\u00ednh R = \u221a (2 5)2 2 5. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 29 (C\u00e2u 2 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho m\u1eb7t c\u1ea7u (S) : x2 + (y + 2)2 + (z \u2212 2)2 = 8. T\u00ecm b\u00e1n k\u00ednh R c\u1ee7a (S). B R = 4. \u221a D R = 64. A R = 8. C R = 2 2. \u221a\u221a \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) c\u00f3 b\u00e1n k\u00ednh R = 8 = 2 2. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 30 (C\u00e2u 9 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x + 3)2 + (y + 1)2 + (z \u2212 1)2 = 2. T\u00e2m c\u1ee7a (S) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (3; 1; \u22121). B (3; \u22121; 1). C (\u22123; \u22121; 1). D (\u22123; 1; \u22121). \u0253 L\u1eddi gi\u1ea3i. T\u00e2m c\u1ee7a (S) : (x + 3)2 + (y + 1)2 + (z \u2212 1)2 = 2 c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 (\u22123; \u22121; 1). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 31 (C\u00e2u 8 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong \u221akh\u00f4ng gian Oxyz, m\u1eb7t c\u1ea7\u221au (S) : (x \u2212 5)2 + (y \u2212 1)2 + (z + 2)2 = 3 c\u00f3 b\u00e1n k\u00ednh b\u1eb1ng A 3. B 2 3. C 3. D 9. \u221a \u0253 L\u1eddi gi\u1ea3i. B\u00e1n k\u00ednh m\u1eb7t c\u1ea7u R = 3. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 32 (C\u00e2u 11 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m I(1; 1; 1) v\u00e0 A(1; 2; 3). Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t c\u1ea7u t\u00e2m I v\u00e0 \u0111i qua A l\u00e0 A (x + 1)2 + (y + 1)2 + (z + 1)2 = 29. B (x \u2212 1)2 + (y \u2212 1)2 + (z \u2212 1)2 = 5. C (x \u2212 1)2 + (y \u2212 1)2 + (z \u2212 1)2 = 25. D (x + 1)2 + (y + 1)2 + (z + 1)2 = 5. \u0253 L\u1eddi gi\u1ea3i. \u221a M\u1eb7t c\u1ea7u t\u00e2m I(1; 1; 1), b\u00e1n k\u00ednh R = IA = 5 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 (x \u2212 1)2 + (y \u2212 1)2 + (z \u2212 1)2 = 5. Ch\u1ecdn \u0111\u00e1p \u00e1n B 485 S\u0110T: 0905.193.688 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t","1. H\u1ec7 t\u1ecda \u0111\u1ed9 trong kh\u00f4ng gian \u0104 C\u00e2u 33 (C\u00e2u 26 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : x2 + y2 + z2 + 2y \u2212 2z \u2212 7 = 0. B\u00e1n k\u00ednh c\u1ee7a m\u1eb7t c\u1ea7u \u0111\u00e3 cho b\u1eb1ng \u221a \u221a D 3. A 9. B 15. C 7. \u0253 L\u1eddi gi\u1ea3i. M\u221a\u1eb7t c\u1ea7u \u0111\u00e3 cho c\u00f3 ph\u221a\u01b0\u01a1ng tr\u00ecnh d\u1ea1ng x2 + y2 + z2 + 2ax + 2by + 2cz + d = 0 c\u00f3 b\u00e1n k\u00ednh l\u00e0 a2 + b2 + c2 \u2212 d = 12 + 12 + 7 = 3. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 34 (C\u00e2u 18 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : x2 + y2 + z2 \u2212 2y + 2z \u2212 7 = 0. B\u00e1n k\u00ednh c\u1ee7a m\u1eb7t c\u1ea7u \u0111\u00e3 cho b\u1eb1ng B 3. C 15. \u221a A 9. D 7. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 R = 12 + (\u22121)2 \u2212 (\u22127) = 3. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 35 (C\u00e2u 23 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S): (x \u2212 2)2 + (y + 4)2 + (z \u2212 1)2 = 9. T\u00e2m S c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0: A (\u22122; 4; \u22121). B (2; \u22124; 1). C (2; 3; 1). D (\u22122; \u22124; \u22121). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh (x \u2212 a)2 + (y \u2212 b)2 + (z \u2212 c)2 = R2 c\u00f3 t\u00e2m I(a; b; c) n\u00ean t\u00e2m c\u1ee7a m\u1eb7t c\u1ea7u (S) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 (2; \u22124; 1). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 36 (C\u00e2u 10 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : x2 + y2 + (z + 2)2 = 9. B\u00e1n k\u00ednh c\u1ee7a (S) b\u1eb1ng A 6. B 18. C 9. D 3. \u0253 L\u1eddi gi\u221a\u1ea3i. M\u1eb7t c\u1ea7u (S) : x2 + y2 + (z + 2)2 = 9 c\u00f3 b\u00e1n k\u00ednh r = 9 = 3. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 37 (C\u00e2u 7 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : x2 + (y \u2212 2)2 + z2 = 9. B\u00e1n k\u00ednh (S) b\u1eb1ng A 6. B 18. C 3. D 9. \u221a \u0253 L\u1eddi gi\u1ea3i. B\u00e1n k\u00ednh R = 9 = 3. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 38 (C\u00e2u 20 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz , cho m\u1eb7t c\u1ea7u (S) : x2 + y2 + (z \u2212 1)2 = 16. B\u00e1n k\u00ednh c\u1ee7a (S) l\u00e0: A 32. B 8. C 4. D 16. \u221a \u0253 L\u1eddi gi\u1ea3i. B\u00e1n k\u00ednh m\u1eb7t c\u1ea7u (S) l\u00e0 R = 16 = 4. Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 486 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 39 (C\u00e2u 12 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : x2 + y2 + (z \u2212 2)2 = 16. B\u00e1n k\u00ednh c\u1ee7a (S) b\u1eb1ng A 4. B 32. C 16. D 8. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) : x2 + y2 + (z \u2212 2)2 = 16 c\u00f3 b\u00e1n k\u00ednh R = 4. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 40 (C\u00e2u 14 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x + 1)2 + (y \u2212 2)2 + (z + 3)2 = 4. T\u00e2m c\u1ee7a (S) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (\u22121; 2; \u22123). B (2; \u22124; 6). C (1; \u22122; 3). D (\u22122; 4; \u22126). \u0253 L\u1eddi gi\u1ea3i. Trong kh\u00f4ng gian Oxyz, m\u1eb7t c\u1ea7u (S) : (x \u2212 a)2 + (y \u2212 b)2 + (z \u2212 c)2 = R2 c\u00f3 t\u00e2m l\u00e0 I(a, b, c). Khi \u0111\u00f3 m\u1eb7t c\u1ea7u (S) : (x + 1)2 + (y \u2212 2)2 + (z + 3)2 = 4 c\u00f3 t\u00e2m I(\u22121, 2, \u22123). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 41 (C\u00e2u 6 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x + 1)2 + (y + 2)2 + (z \u2212 3)2 = 9. T\u00e2m c\u1ee7a (S) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (\u22122; \u22124; 6). B (2; 4; \u22126). C (\u22121; \u22122; 3). D (1; 2; \u22123). \u0253 L\u1eddi gi\u1ea3i. T\u00e2m c\u1ee7a m\u1eb7t c\u1ea7u (S) : (x + 1)2 + (y + 2)2 + (z \u2212 3)2 = 9 c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 (\u22121; \u22122; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 42 (C\u00e2u 14 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x \u2212 1)2 + (y \u2212 2)2 + (z + 3)2 = 9. T\u00e2m c\u1ee7a (S) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (\u22121; \u22122; 3). B (\u22122; \u22124; 6). C (1; 2; \u22123). D (2; 4; \u22126). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(1; 2; \u22123). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 43 (C\u00e2u 3 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(1; \u22124; 0) v\u00e0 b\u00e1n k\u00ednh b\u1eb1ng 3. Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a (S) l\u00e0 A (x + 1)2 + (y \u2212 4)2 + z2 = 9. B (x \u2212 1)2 + (y + 4)2 + z2 = 9. C (x \u2212 1)2 + (y + 4)2 + z2 = 3. D (x + 1)2 + (y \u2212 4)2 + z2 = 3. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(1; \u22124; 0) v\u00e0 b\u00e1n k\u00ednh b\u1eb1ng 3 n\u00ean (S) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 (x \u2212 1)2 + (y + 4)2 + z2 = 9. Ch\u1ecdn \u0111\u00e1p \u00e1n B 487 S\u0110T: 0905.193.688 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t","1. H\u1ec7 t\u1ecda \u0111\u1ed9 trong kh\u00f4ng gian \u0104 C\u00e2u 44 (C\u00e2u 24 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(0; \u22122; 1) v\u00e0 b\u00e1n k\u00ednh b\u1eb1ng 2. Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a (S) l\u00e0 A x2 + (y + 2)2 + (z \u2212 1)2 = 2. B x2 + (y \u2212 2)2 + (z + 1)2 = 2. C x2 + (y \u2212 2)2 + (z + 1)2 = 4. D x2 + (y + 2)2 + (z \u2212 1)2 = 4. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(0; \u22122; 1) v\u00e0 b\u00e1n k\u00ednh R = 2 c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x2 + (y + 2)2 + (z \u2212 1)2 = 22 \u21d4 x2 + (y + 2)2 + (z \u2212 1)2 = 4. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 45 (C\u00e2u 8 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(0; 1; \u22122) v\u00e0 b\u00e1n k\u00ednh b\u1eb1ng 3. Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a (S) l\u00e0 A x2 + (y \u2212 1)2 + (z + 2)2 = 9. B x2 + (y + 1)2 + (z \u2212 2)2 = 9. C x2 + (y \u2212 1)2 + (z + 2)2 = 3. D x2 + (y + 1)2 + (z \u2212 2)2 = 3. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u c\u00f3 t\u00e2m (a; b; c) v\u00e0 b\u00e1n k\u00ednh R c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 (x \u2212 a)2 + (y \u2212 b)2 + (z \u2212 c)2 = R2. Do \u0111\u00f3, ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t c\u1ea7u (S) l\u00e0 x2 + (y \u2212 1)2 + (z + 2)2 = 9. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 46 (C\u00e2u 5 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(\u22121; 3; 0) v\u00e0 b\u00e1n k\u00ednh b\u1eb1ng 2. Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a (S) l\u00e0 A (x \u2212 1)2 + (y + 3)2 + z2 = 2. B (x \u2212 1)2 + (y + 3)2 + z2 = 4. C (x + 1)2 + (y \u2212 3)2 + z2 = 4. D (x + 1)2 + (y \u2212 3)2 + z2 = 2. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(\u22121; 3; 0) v\u00e0 b\u00e1n k\u00ednh R = 2. Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t c\u1ea7u (S) l\u00e0 (x + 1)2 + (y \u2212 3)2 + z2 = 4. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 47 (C\u00e2u 25 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x + 1)2 + (y \u2212 3)2 + z2 = 9. T\u00e2m m\u1eb7t c\u1ea7u (S) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (1; \u22123; 0). B (\u22121; 3; 0). C (1; 3; 0). D (\u22121; \u22123; 0). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) : (x + 1)2 + (y \u2212 3)2 + z2 = 9 c\u00f3 t\u00e2m l\u00e0 (\u22121; 3; 0). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 48 (C\u00e2u 9 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x \u2212 1)2 + (y + 3)2 + z2 = 9. T\u00e2m c\u1ee7a (S) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (1; \u22123; 0). B (1; 3; 0). C (\u22121; 3; 0). D (\u22121; \u22123; 0). \u0253 L\u1eddi gi\u1ea3i. T\u00e2m c\u1ee7a m\u1eb7t c\u1ea7u (S) : (x \u2212 1)2 + (y + 3)2 + z2 = 9 l\u00e0 I(1; \u22123; 0). Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 488 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 49 (C\u00e2u 14 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x \u2212 1)2 + y2 + (z + 2)2 = 4. T\u00e2m c\u1ee7a (S) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (\u22121; 0; 2). B (1; 0; 2). C (1; 0; \u22122). D (\u22121; 0; \u22122). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) : (x \u2212 1)2 + y2 + (z + 2)2 = 4 c\u00f3 t\u1ecda \u0111\u1ed9 t\u00e2m l\u00e0 (1; 0; \u22122). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 50 (C\u00e2u 5 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x + 1)2 + y2 + (z \u2212 2)2 = 4. T\u00e2m c\u1ee7a (S) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (1; 0; \u22122). B (\u22121; 0; 2). C (1; 0; 2). D (\u22121; 0; \u22122). \u0253 L\u1eddi gi\u1ea3i. T\u1ecda \u0111\u1ed9 t\u00e2m m\u1eb7t c\u1ea7u (S) : (x + 1)2 + y2 + (z \u2212 2)2 = 4 l\u00e0 I(\u22121; 0; 2). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 51 (C\u00e2u 24 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : x2 + (y \u2212 2)2 + (z + 1)2 = 6. \u0110\u01b0\u1eddng k\u00ednh c\u1ee7a (S) b\u1eb1ng \u221a \u221a D 12. A 3. B 6. C 2 6. \u221a \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng k\u00ednh c\u1ee7a (S) b\u1eb1ng 2R = 2 6. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 52 (C\u00e2u 18 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x \u2212 2)2 + (y + 1)2 + (z \u2212 3)2 = 4. T\u00e2m c\u1ee7a (S) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (\u22124; 2; \u22126). B (4; \u22122; 6). C (2; \u22121; 3). D (\u22122; 1; \u22123). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) : (x \u2212 2)2 + (y + 1)2 + (z \u2212 3)2 = 4 c\u00f3 t\u00e2m l\u00e0 (2; \u22121; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 53 (C\u00e2u 26 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x \u2212 2)2 + (y + 1)2 + (z \u2212 3)2 = 4. T\u00e2m c\u1ee7a (S) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (\u22122; 1; \u22123). B (\u22124; 2; \u22126). C (4; \u22122; 6). D (2; \u22121; 3). \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u (S) : (x \u2212 2)2 + (y + 1)2 + (z \u2212 3)2 = 4 c\u00f3 t\u00e2m I(2; \u22121; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 54 (C\u00e2u 2 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, m\u1eb7t c\u1ea7u (S) : (x + 1)2 + (y \u2212 2)2 + z2 = 9 c\u00f3 b\u00e1n k\u00ednh b\u1eb1ng A 3. B 81. C 9. D 6. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 R2 = 9 n\u00ean b\u00e1n k\u00ednh m\u1eb7t c\u1ea7u R = 3. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 489 S\u0110T: 0905.193.688","1. H\u1ec7 t\u1ecda \u0111\u1ed9 trong kh\u00f4ng gian \u0104 C\u00e2u 55 (C\u00e2u 10 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, v\u00e9c-t\u01a1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng (Oxy)? B #\u00bb C #j\u00bb = (0; 1; 0). D m#\u00bb = (1; 1; 1). A #i\u00bb = (1; 0; 0). k = (0; 0; 1). \u0253 L#\u00bb\u1eddi gi\u1ea3i. M\u1eb7t ph\u1eb3ng (Oxy) c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 k = (0; 0; 1). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 56 (C\u00e2u 10 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(3; \u22121; 1). H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a A tr\u00ean m\u1eb7t ph\u1eb3ng (Oyz) l\u00e0 \u0111i\u1ec3m A M (3; 0; 0). B N (0; \u22121; 1). C P (0; \u22121; 0). D Q(0; 0; 1). \u0253 L\u1eddi gi\u1ea3i. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a A(3; \u22121; 1) tr\u00ean m\u1eb7t ph\u1eb3ng (Oyz) l\u00e0 \u0111i\u1ec3m N (0; \u22121; 1). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 57 (C\u00e2u 12 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). x\u22122 y\u22121 z Trong kh\u00f4ng gian Oxyz, cho \u0111\u01b0\u1eddng th\u1eb3ng d : \u22121 = = . \u0110\u01b0\u1eddng th\u1eb3ng d c\u00f3 m\u1ed9t vect\u01a1 2 1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 B u#\u00bb2 = (2; 1; 0). C u#\u00bb3 = (2; 1; 1). D u#\u00bb4 = (\u22121; 2; 0). A u#\u00bb1 = (\u22121; 2; 1). \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng d : x\u22122 = y \u22121 = z c\u00f3 m\u1ed9t v\u00e9c-t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 u#\u00bb1 = (\u22121; 2; 1). \u22121 2 1 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 58 (C\u00e2u 43 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho hai \u0111i\u1ec3m A(3; \u22122; 3) v\u00e0 B(\u22121; 2; 5). T\u00ecm t\u1ecda \u0111\u1ed9 trung \u0111i\u1ec3m I c\u1ee7a \u0111o\u1ea1n th\u1eb3ng AB. A I(\u22122; 2; 1). B I(1; 0; 4). C I(2; 0; 8). D I(2; \u22122; \u22121). \u0253 L\u1eddi gi\u1ea3i. Trung \u0111i\u1ec3m AB l\u00e0 (1; 0; 4). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 59 (C\u00e2u 17 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho c\u00e1c \u0111i\u1ec3m A(3; \u22124; 0), B(\u22121; 1; 3), C(3; 1; 0). T\u00ecm t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m D tr\u00ean tr\u1ee5c ho\u00e0nh sao cho AD = BC. A D(\u22122; 0; 0) ho\u1eb7c D(\u22124; 0; 0). B D(0; 0; 0) ho\u1eb7c D(\u22126; 0; 0). C D(6; 0; 0) ho\u1eb7c D(12; 0; 0). D D(0; 0; 0) ho\u1eb7c D(6; 0; 0). \u0253 L\u1eddi gi\u1ea3i. Do D \u2208 Oy n\u00ean D = (d; 0; 0). Khi \u0111\u00f3 AD = (d \u2212 3)2 + (16), BC = 5. Theo gi\u1ea3 thi\u1ebft AD = BC \u21d4 (d \u2212 3)2 + (16) = 5 \u21d4 (d \u2212 3)2 + 16 = 25 \u21d4 (d \u2212 3)2 = 9 \u21d4 \u00f1d \u2212 3 = \u22123 \u00f1d = 0 \u00f1D(0; 0; 0) \u21d4\u21d2 . d\u22123=3 d = 6 D(6; 0; 0) Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 490 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 60 (C\u00e2u 13 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, h\u00ecnh chi\u1ebfu c\u1ee7a \u0111i\u1ec3m M (2; \u22122; 1) tr\u00ean m\u1eb7t ph\u1eb3ng (Oxy) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (2; 0; 1). B (2; \u22122; 0). C (0; \u22122; 1). D (0; 0; 1). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 h\u00ecnh chi\u1ebfu c\u1ee7a \u0111i\u1ec3m M (xo; yo; zo) tr\u00ean m\u1eb7t ph\u1eb3ng (Oxy) l\u00e0 \u0111i\u1ec3m M (xo; yo; 0). Do \u0111\u00f3h\u00ecnh chi\u1ebfu c\u1ee7a \u0111i\u1ec3m M (2; \u22122; 1) tr\u00ean m\u1eb7t ph\u1eb3ng (Oxy) l\u00e0 \u0111i\u1ec3m M (2; \u22122; 0). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 61 (C\u00e2u 17 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho hai v\u00e9c-t\u01a1 #u\u00bb = (1; \u22124; 0) v\u00e0 #v\u00bb = (\u22121; \u22122; 1). V\u00e9c-t\u01a1 #u\u00bb + 3 #v\u00bb c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (\u22122; \u221210; 3). B (\u22122; \u22126; 3). C (\u22124; \u22128; 4). D (\u22122; \u221210; \u22123). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 3 #v\u00bb = (\u22123; \u22126; 3). Do \u0111\u00f3 #u\u00bb + 3 #v\u00bb = (\u22122; \u221210; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 62 (C\u00e2u 32 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m #b\u00bb20=19(\u2212-22;022; 50)).. T\u00edch v\u00f4 h\u01b0\u1edbng #a\u00bb \u00b7 \u00c4 #a\u00bb + #b\u00bb\u00e4 Trong kh\u00f4ng gian Oxyz, cho c\u00e1c vect\u01a1 #a\u00bb = (1; 0; 3) v\u00e0 b\u1eb1ng A 25. B 23. C 27. D 29. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 #a\u00bb + #\u00bb = (#b\u00bb\u2212\u00e41;=2;18.)(.\u22121) b Suy ra #a\u00bb. \u00c4 #a\u00bb + + 0.2 + 3.8 = 23. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 63 (C\u00e2u 44 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x + 1)2 + (y \u2212 2)2 + (z \u2212 1)2 = 9. T\u00ecm t\u1ecda \u0111\u1ed9 t\u00e2m I v\u00e0 t\u00ednh b\u00e1n k\u00ednh R c\u1ee7a (S). A I(\u22121; 2; 1) v\u00e0 R = 3. B I(1; \u22122; \u22121) v\u00e0 R = 3. C I(\u22121; 2; 1) v\u00e0 R = 9. D I(1; \u22122; \u22121) v\u00e0 R = 9. \u0253 L\u1eddi gi\u1ea3i. D\u1ef1a v\u00e0o d\u1ea1ng t\u1ed5ng qu\u00e1t c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t c\u1ea7u (S) : (x \u2212 a)2 + (y \u2212 b)2 + (z \u2212 c)2 = R2. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 64 (C\u00e2u 46 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi d\u00e2y l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t c\u1ea7u c\u00f3 t\u00e2m I(1; 2; \u22121) v\u00e0 ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t ph\u1eb3ng (P ) : x \u2212 2y \u2212 2z \u2212 8 = 0? A (x + 1)2 + (y + 2)2 + (z \u2212 1)2 = 3. B (x \u2212 1)2 + (y \u2212 2)2 + (z + 1)2 = 3. C (x \u2212 1)2 + (y \u2212 2)2 + (z + 1)2 = 9. D (x + 1)2 + (y + 2)2 + (z \u2212 1)2 = 9. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 491","1. H\u1ec7 t\u1ecda \u0111\u1ed9 trong kh\u00f4ng gian G\u1ecdi m\u1eb7t c\u1ea7u c\u1ea7n t\u00ecm l\u00e0 (S). Ta c\u00f3 (S) l\u00e0 m\u1eb7t c\u1ea7u c\u00f3 t\u00e2m I(1; 2; \u22121) v\u00e0 b\u00e1n k\u00ednh R. V\u00ec (S) ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t ph\u1eb3ng (P ) : x \u2212 2y \u2212 2z \u2212 8 = 0 n\u00ean ta c\u00f3 R = d(I; (P )) = |1 \u2212 2.2 \u2212 2.(\u22121) \u2212 8| = 3. 12 + (\u22122)2 + (\u22122)2 V\u1eady ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t c\u1ea7u c\u1ea7n t\u00ecm l\u00e0 (x \u2212 1)2 + (y \u2212 2)2 + (z + 1)2 = 9. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 65 (C\u00e2u 21 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : x2 + y2 + z2 + 2x \u2212 2z \u2212 7 = 0. B\u00e1n k\u00ednh c\u1ee7a m\u1eb7t c\u1ea7u \u0111\u00e3 cho\u221ab\u1eb1ng B 9. C 3. \u221a A 7. D 15. \u0253 L\u1eddi gi\u1ea3i. x2 + y2 + z2 + 2x \u2212 2z \u2212 7 = 0 \u21d4 x2 + y2 + z2 \u2212 2 \u00b7 (\u22121) \u00b7 x + 2 \u00b7 0 \u00b7 y \u2212 2 \u00b7 1 \u00b7 z \u2212 7 = 0. Suy ra a = \u22121, b = 0, c = 1, d = \u22127. \u221a V\u1eady t\u00e2m m\u1eb7t c\u1ea7u I(\u22121; 0; 1) b\u00e1n k\u00ednh R = a2 + b2 + c2 \u2212 d = (\u22121)2 + 02 + 12 + 7 = 3. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 66 (C\u00e2u 22 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : x2 + y2 + z2 \u2212 2x + 2y \u2212 7 = 0. B\u00e1n k\u00ednh c\u1ee7a m\u1eb7t c\u1ea7u \u0111\u00e3 cho b\u1eb1ng B 9. \u221a \u221a A 3. C 15. D 7. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 (S) : x2 + y2 + z2 \u2212 2x + 2y \u2212 7 = 0 \u21d4 (x \u2212 1)2 + (y + 1)2 + z2 = 9 V\u1eady b\u00e1n k\u00ednh c\u1ee7a m\u1eb7t c\u1ea7u b\u1eb1ng 3. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 67 (C\u00e2u 14 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x \u2212 1)2 + (y + 2)2 + (z \u2212 3)2 = 16. T\u00e2m c\u1ee7a (S) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 A (\u22121; \u22122; \u22123). B (1; 2; 3). C (\u22121; 2; \u22123). D (1; \u22122; 3). \u0253 L\u1eddi gi\u1ea3i. T\u1eeb ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t c\u1ea7u (S), suy ra t\u00e2m c\u1ee7a m\u1eb7t c\u1ea7u (S) c\u00f3 t\u1ecda \u0111\u1ed9 l\u00e0 (1; \u22122; 3). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 68 (C\u00e2u 33 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m l\u00e0 \u0111i\u1ec3m I (0 ; 0 ; \u22123) v\u00e0 \u0111i qua \u0111i\u1ec3m M (4 ; 0 ; 0). Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a (S) l\u00e0 A x2 + y2 + (z + 3)2 = 25. B x2 + y2 + (z + 3)2 = 5. C x2 + y2 + (z \u2212 3)2 = 25. D x2 + y2 + (z \u2212 3)2 = 5. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3: IM = 5. M\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I (0 ; 0 ; \u22123) \u0111i qua \u0111i\u1ec3m M (4 ; 0 ; 0) c\u00f3 b\u00e1n k\u00ednh l\u00e0 R = IM = 5 V\u1eady ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t c\u1ea7u (S) l\u00e0 x2 + y2 + (z + 3)2 = 25. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 492 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 69 (C\u00e2u 6 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) : x2 + (y \u2212 2)2 + (z + 1)2 = 6. \u0110\u01b0\u1eddng k\u00ednh c\u1ee7a (S) b\u1eb1ng \u221a B 12. \u221a D 3. A 6. C 2 6. \u0253 L\u1eddi gi\u1ea3i. \u221a\u221a Ta c\u00f3 b\u00e1n k\u00ednh c\u1ee7a (S) l\u00e0 6 n\u00ean \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a (S) b\u1eb1ng 2 6. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 70 (C\u00e2u 35 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho \u0111i\u1ec3m A(1; 2; 3). Ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t c\u1ea7u t\u00e2m A v\u00e0 ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t ph\u1eb3ng x \u2212 2y + 2z + 3 = 0 l\u00e0 A (x \u2212 1)2 + (y \u2212 2)2 + (z \u2212 3)2 = 2. B (x + 1)2 + (y + 2)2 + (z + 3)2 = 2. C (x + 1)2 + (y + 2)2 + (z + 3)2 = 4. D (x \u2212 1)2 + (y \u2212 2)2 + (z \u2212 3)2 = 4. \u0253 L\u1eddi gi\u1ea3i. M\u1eb7t c\u1ea7u t\u00e2m A ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t ph\u1eb3ng \u0111\u00e3 cho c\u00f3 b\u00e1n k\u00ednh R = |1 \u2212\u221a2.2 + 2.3 + 3| = 2. 1+4+4 V\u1eady ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t c\u1ea7u c\u1ea7n t\u00ecm l\u00e0 (x \u2212 1)2 + (y \u2212 2)2 + (z \u2212 3)2 = 4. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 71 (C\u00e2u 29 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho \u0111i\u1ec3m M (1; \u22122; 3). G\u1ecdi I l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a M tr\u00ean tr\u1ee5c Ox. Ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t c\u1ea7u t\u00e2m I b\u00e1n k\u00ednh IM ? A (x \u2212 1)2 + y2 + z2 = \u221a13. B (x + 1)2 + y2 + z2 = 13. C (x \u2212 1)2 + y2 + z2 = 13. D (x + 1)2 + y2 + z2 = 17. \u0253 L\u1eddi gi\u1ea3i. \u221a H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a M tr\u00ean Ox l\u00e0 I(1; 0; 0). M\u1eb7t kh\u00e1c IM = 13 \u21d2 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t c\u1ea7u l\u00e0 (x \u2212 1)2 + y2 + z2 = 13. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 72 (C\u00e2u 7 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho \u0111i\u1ec3m A(2; 2; 1). T\u00ed\u221anh \u0111\u1ed9 d\u00e0i \u0111o\u1ea1n th\u1eb3ng OA. A OA = 3. B OA = 9. C OA = 5. D OA = 5. \u221a\u221a \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 OA = 22 + 22 + 12 = 9 = 3. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 73 (C\u00e2u 16 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, t\u00ecm t\u1ea5t c\u1ea3 c\u00e1c gi\u00e1 tr\u1ecb c\u1ee7a m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh x2 + y2 + z2 \u2212 2x \u2212 2y \u2212 4z + m = 0 l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1ed9t m\u1eb7t c\u1ea7u. A m > 6. B m \u2265 6. C m \u2264 6. D m < 6. \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1ed9t m\u1eb7t c\u1ea7u khi 1 + 1 + 4 \u2212 m > 0 \u21d4 m < 6. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 493 S\u0110T: 0905.193.688","1. H\u1ec7 t\u1ecda \u0111\u1ed9 trong kh\u00f4ng gian \u0104 C\u00e2u 74 (C\u00e2u 6 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho m\u1eb7t c\u1ea7u (S) : (x \u2212 5)2 + (y \u2212 1)2 + (z + 2)2 = 9. T\u00ednh b\u00e1n k\u00ednh R c\u1ee7a (S). A R = 3. B R = 18. C R = 9. D R = 6. \u221a \u0253 L\u1eddi gi\u1ea3i. B\u00e1n k\u00ednh R = 9 = 3 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 75 (C\u00e2u 26 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). v\u00e0 #\u00bb = (\u22121; 0; \u22122). T\u00ednh cTorso\u00c4ng#a\u00bb,k#hb\u00bb\u00f4\u00e4n. g gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho hai v\u00e9c-t\u01a1 #a\u00bb = (2; 1; 0) b A cos \u00c4 #a\u00bb, #b\u00bb\u00e4 = 2 B cos \u00c4 #a\u00bb, #b\u00bb\u00e4 = \u22122. . 5 25 #\u00bb\u00e4 2 #\u00bb\u00e4 2 C cos \u00c4 #a\u00bb, b = \u2212 . D cos \u00c4 #a\u00bb, b = . 25 5 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 cos \u00c4 #a\u00bb, #\u00bb\u00e4 = #a\u00bb. #b\u00bb = 2. (\u22121) +\u221a1.0\u221a+ 0. (\u22122) = \u22122 . b | #a\u00bb| . #b\u00bb 5. 5 5 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 76 (C\u00e2u 12 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho ba \u0111i\u1ec3m M (2; 3; \u22121), N (\u22121; 1; 1) v\u00e0 P (1; m \u2212 1; 2). T\u00ecm m \u0111\u1ec3 tam gi\u00e1c M N P vu\u00f4ng t\u1ea1i N . A m = \u22126. B m = 0. C m = \u22124. D m = 2. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 N# M\u00bb = (3; 2; \u22122) v\u00e0 N# P\u00bb = (2; mch\u1ec9\u2212k2h;i1N#).M\u00bb.N# P\u00bb = 0 \u21d4 m = 0. Tam gi\u00e1c M N P vu\u00f4ng t\u1ea1i N khi v\u00e0 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 77 (C\u00e2u 48 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho m\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(2; 1; 1) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x + y + 2z + 2 = 0. Bi\u1ebft m\u1eb7t ph\u1eb3ng (P ) c\u1eaft m\u1eb7t c\u1ea7u (S) theo giao tuy\u1ebfn l\u00e0 m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n c\u00f3 b\u00e1n k\u00ednh b\u1eb1ng 1. Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a m\u1eb7t c\u1ea7u (S). A (S): (x + 2)2 + (y + 1)2 + (z + 1)2 = 8. B (S): (x + 2)2 + (y + 1)2 + (z + 1)2 = 10. C (S): (x \u2212 2)2 + (y \u2212 1)2 + (z \u2212 1)2 = 8. D (S): (x \u2212 2)2 + (y \u2212 1)2 + (z \u2212 1)2 = 10. \u0253 L\u1eddi gi\u1ea3i. kho\u1ea3ng c\u00e1ch t\u1eeb t\u00e2m I \u0111\u1ebfn m\u1eb7t ph\u1eb3ng (P ) l\u00e0 d = 3. \u221a\u221a b\u00e1n k\u00ednh m\u1eb7t c\u1ea7u l\u00e0 R = 32 + 12 = 10. ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t c\u1ea7u l\u00e0 (x \u2212 2)2 + (y \u2212 1)2 + (z \u2212 1)2 = 10. Ch\u1ecdn \u0111\u00e1p \u00e1n D 494 S\u0110T: 0905.193.688 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN \u0104 C\u00e2u 78 (C\u00e2u 48 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) c\u00f3 t\u00e2m I(1; 2; 3) v\u00e0 \u0111i qua \u0111i\u1ec3m A(5; \u22122; \u22121). X\u00e9t c\u00e1c \u0111i\u1ec3m B, C, D thu\u1ed9c (S) sao cho AB, AC, AD \u0111\u00f4i m\u1ed9t vu\u00f4ng g\u00f3c v\u1edbi nhau. Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i t\u1ee9 di\u1ec7n ABCD c\u00f3 gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t b\u1eb1ng A 256. B 128. C 256 D 128 . . 3 3 \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t AB = a, AC = b, AD = c th\u00ec ABCD l\u00e0 t\u1ee9 di\u1ec7n vu\u00f4ng \u0111\u1ec9nh D P A, n\u1ed9i ti\u1ebfp m\u1eb7t c\u1ea7u (S) b\u00e1n k\u00ednh R. C Khi \u0111\u00f3 ABCD l\u00e0 t\u1ee9 di\u1ec7n \u0111\u1eb7t \u1edf g\u00f3c A c\u1ee7a h\u00ecnh h\u1ed9p ch\u1eef nh\u1eadt t\u01b0\u01a1ng \u1ee9ng c\u00f3 c\u00e1c c\u1ea1nh AB, AC, AD v\u00e0 \u0111\u01b0\u1eddng ch\u00e9o AN l\u00e0 \u0111\u01b0\u1eddng k\u00ednh M c N Ib c\u1ee7a c\u1ea7u. Ta c\u00f3 a2 + b2 + c2 = 4R2. 1 1 a2b2c2. X\u00e9t V = VABCD = abc \u21d4V2 = 36 6 aA B E\u221a M\u00e0 a2 + b2 + c2 \u221a \u21d4 \u00c5 a2 + b2 + c2 \u00e33 a2b2c2 \u21d4 \u00c5 4R2 \u00e33 36V 2 \u21d4 V R3 \u00b7 4 3 3 3 a2b2c2 . 33 27 \u221a 256 Do R = IA = 4 3 n\u00ean ta c\u00f3 V (D\u1ea5u \u201c=\u201d x\u1ea3y ra \u21d4 a = b = c = 8). 3 256 V\u1eady max VABCD = . 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 79 (C\u00e2u 33 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, cho \u0111i\u1ec3m I (1; 2; 3) v\u00e0 m\u1eb7t ph\u1eb3ng (P ) : 2x \u2212 2y \u2212 z \u2212 4 = 0. M\u1eb7t c\u1ea7u t\u00e2m I ti\u1ebfp x\u00fac v\u1edbi (P ) t\u1ea1i \u0111i\u1ec3m H. T\u00ecm t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m H. A H (\u22121; 4; 4). B H (\u22123; 0; \u22122). C H (3; 0; 2). D H (1; \u22121; 0). \u0253 L\u1eddi gi\u1ea3i. H l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a \u0111i\u1ec3m I l\u00ean m\u1eb7t ph\u1eb3ng (P ). X\u00e9t (P ) c\u00f3 m\u1ed9t vtpt l\u00e0 #n\u00bb = (2; \u22122; \u22121) x\u22121 y\u22122 z\u22123 Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng qua I v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi (P ) l\u00e0 d : 2 = \u22122 = \u22121 \uf8f1x = 3 \uf8f12x \u2212 2y \u2212 z \u2212 4 = 0 \uf8f4 \uf8f2 \uf8f2 T\u1ecda \u0111\u1ed9 H l\u00e0 nghi\u1ec7m c\u1ee7a h\u1ec7 x \u2212 1 y \u2212 2 z \u2212 3 \u21d4 y = 0 \u21d2 H (3; 0; 2) \uf8f3 2 = \u22122 = \u22121 \uf8f4\uf8f3z = 2 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 80 (C\u00e2u 50 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). Trong kh\u00f4ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, x\u00e9t c\u00e1c \u0111i\u1ec3m A(0; 0; 1), B(m; 0; 0), C(0; n; 0), D(1; 1; 1) v\u1edbi m > 0; n > 0 v\u00e0 m + n = 1. Bi\u1ebft r\u1eb1ng khi m, n thay \u0111\u1ed5i, t\u1ed3n t\u1ea1i m\u1ed9t m\u1eb7t c\u1ea7u c\u1ed1 \u0111\u1ecbnh ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t ph\u1eb3ng (ABC) v\u00e0 \u0111i qua D.\u221aT\u00ednh b\u00e1n k\u00ednh R c\u1ee7a m\u1eb7t c\u1ea7u \u0111\u00f3? \u221a 2 3 3 A R = 1. B R= . C R= . D R= 2 . 2 2 \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi I(1; 1; 0) l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a D l\u00ean m\u1eb7t ph\u1eb3ng (Oxy). xy Ta c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh theo \u0111o\u1ea1n ch\u1eafn c\u1ee7a m\u1eb7t ph\u1eb3ng (ABC) l\u00e0: + + z = 1. mn Suy ra ph\u01b0\u01a1ng tr\u00ecnh t\u1ed5ng qu\u00e1t c\u1ee7a (ABC) l\u00e0 nx + my + mnz \u2212 mn = 0. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 495 S\u0110T: 0905.193.688","1. H\u1ec7 t\u1ecda \u0111\u1ed9 trong kh\u00f4ng gian M\u1eb7t kh\u00e1c d(I; (ABC)) = \u221a |1 \u2212 mn| = 1 (v\u00ec m + n = 1) v\u00e0 ID = 1. m2 + n2 + m2n2 \u21d2 ID = d((I; (ABC)). N\u00ean t\u1ed3n t\u1ea1i m\u1eb7t c\u1ea7u t\u00e2m I (l\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a D l\u00ean m\u1eb7t ph\u1eb3ng Oxy) ti\u1ebfp x\u00fac v\u1edbi (ABC) v\u00e0 \u0111i qua D. Khi \u0111\u00f3 R = 1. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 81 (C\u00e2u 49 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) t\u00e2m I(4; 1; 2) b\u00e1n k\u00ednh b\u1eb1ng 2. G\u1ecdi M , N l\u00e0 hai \u0111i\u1ec3m l\u1ea7n l\u01b0\u1ee3t thu\u1ed9c hai tr\u1ee5c Ox, Oy sao cho \u0111\u01b0\u1eddng th\u1eb3ng M N ti\u1ebfp x\u00fac v\u1edbi (S), \u0111\u1ed3ng th\u1eddi m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp t\u1ee9 di\u1ec7n OIM N c\u00f3 b\u00e1n k\u00ednh b\u1eb1ng 7 \u00b7 G\u1ecdi A l\u00e0 ti\u1ebfp \u0111i\u1ec3m c\u1ee7a M N v\u00e0 (S), gi\u00e1 tr\u1ecb AM.AN 2 b\u1eb1ng \u221a \u221a A 6 2. B 14. C 8. D 9 2. \u0253 L\u1eddi gi\u1ea3i. C\u00e1ch 1: Ta c\u00f3 d I, (Oxy) = 2 n\u00ean m\u1eb7t c\u1ea7u (S) ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t ph\u1eb3ng (Oxy) t\u1ea1i \u0111i\u1ec3m A(4; 1; 0), \u0111\u1ed3ng th\u1eddi \u0111\u01b0\u1eddng th\u1eb3ng M N (t0i\u1ebf;pn;x0\u00fa)ctva\u1edbci\u00f3(SA#)M\u00bbc\u0169=ng(mt\u1ea1i\u2212\u01114i\u1ec3;m\u22121A; (04); v1\u00e0; 0A#) Nd\u00bbo=M(\u2212N4\u2282; n(\u2212Ox1y; 0).). G\u1ecdi M (m; 0; 0), N Do A \u2208 MN n\u00ean A# M\u00bb = k A# N\u00bb \u00aem \u2212 4 = \u22124k \u21d2 (m \u2212 4)(n \u2212 1) = 4 \u21d4m = 4n n\u22121= 0. \u21d2 n \u2212 1, \u2212 1 = k(n \u2212 1) Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng trung tr\u1ef1c \u0111o\u1ea1n OI : 4x + y + 2z \u2212 21 = 0. 2 Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng trung tr\u1ef1c \u0111o\u1ea1n OM : x = m\u00b7 n2 Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng trung tr\u1ef1c \u0111o\u1ea1n ON : y = \u00b7 2 \u00c5 m n \u2212n2 + 6n \u2212 21 \u00e3 Do \u0111\u00f3 t\u00e2m m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp t\u1ee9 di\u1ec7n OIM N l\u00e0 J ; ; . 22 4n \u2212 4 77 Theo gi\u1ea3 thuy\u1ebft c\u1ea7u ngo\u1ea1i ti\u1ebfp t\u1ee9 di\u1ec7n OIM N c\u00f3 b\u00e1n k\u00ednh b\u1eb1ng n\u00ean OJ = 22 \u21d4 OJ2 = 49 4 4n2 n2 n2 \u2212 6n + 21 2 49 \u21d4 (n \u2212 1)2 + 4 + = 16(n \u2212 1)2 4 \u21d4 n4 \u2212 4n3 \u2212 10n2 + 28n + 49 = 0 \u221a \u21d4 n = 1 \u00b1 2 2. \u221a\u221a V\u00ec n > 0 n\u00ean ch\u1ecdn n\u221a= 1 + 2 2, suy ra m = 4 + 2. Khi \u0111\u00f3 AM.AN = 6 2. C\u00e1ch 2: D\u1ec5 th\u1ea5y m\u1eb7t c\u1ea7u (S) ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t ph\u1eb3ng (Oxy) t\u1ea1i \u0111i\u1ec3m A(4; 1; 0), \u0111\u1ed3ng th\u1eddi \u0111\u01b0\u1eddng th\u1eb3ng M N ti\u1ebfp x\u00fac v\u1edbi (S) Nc\u0169(n0g; nt;\u1ea10i )\u0111ti\u1ec3amc\u00f3AA#(1M;\u00bb4;=0)(mdo\u2212M4N; \u2212\u22821; 0(O) vx\u00e0y)A#. N\u00bb = (\u22124; n \u2212 1; 0). G\u1ecdi M (m; 0; 0); Do A \u2208 MN n\u00ean #\u00bb = #\u00bb \u00aem \u2212 4 = \u22124k \u21d2 1 4 = 1. AM k AN \u21d2 + \u2212 1 = k(n \u2212 1) n m G\u1ecdi J l\u00e0 trung \u0111i\u1ec3m M N \u21d2 J mn v\u00e0 I(4; 1; 2) thu\u1ed9c \u0111\u01b0\u1eddng th\u1eb3ng \u2206 vu\u00f4ng g\u00f3c v\u1edbi (Oxy) t\u1ea1i ; ;0 \uf8f1 2 m2 \uf8f4x = \uf8f4 2 \uf8f4 \u0111i\u1ec3m J. Ph\u01b0\u01a1ng tr\u00ecnh \u2206 l\u00e0 \uf8f2 = n y 2 \uf8f4 \uf8f4 \uf8f4\uf8f3z = t. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 496 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. PH\u01af\u01a0NG PH\u00c1P T\u1eccA \u0110\u1ed8 TRONG KH\u00d4NG GIAN mn Suy ra t\u00e2m c\u1ee7a m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp t\u1ee9 di\u1ec7n OIM N l\u00e0 \u0111i\u1ec3m K ; ; t . 22 \uf8f11 4 \uf8f11 4 + = 1 + = 1 \uf8f4 \uf8f4 \uf8f4 n m \uf8f4 n m \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 7 \uf8f4 m2 n2 49 \uf8f2 \uf8f2 Theo gi\u1ea3 thi\u1ebft ta c\u00f3 h\u1ec7 OK = \u21d4 + + t2 = \uf8f4 2 \uf8f44 4 4 \uf8f4 \uf8f4 \uf8f4 = 7 \uf8f4 m 2 n \u2212 1 2 + (t \u2212 2)2 = 49 \uf8f4\uf8f3\uf8f4I K \uf8f4 \u22124 + \uf8f4 \uf8f3 2 22 4 \uf8f1 4n \uf8f1 4n \uf8f4\uf8f4m = \uf8f4\uf8f4m = \uf8f4 b \u2212 1 \uf8f4 n \u2212 1 \uf8f4 \uf8f4 \uf8f4 n2 \u2212 6n + 21 \uf8f2 \uf8f4 \uf8f2 \u21d4 4m + n + 4t \u2212 21 = 0 \u21d4 t = 4(n \u2212 1) \uf8f4 m2 n2 49 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 + + t2 = \uf8f4 m2 n2 49 \u00b7 \uf8f4 \uf8f3 \uf8f4 + + t2 = 44 4 \uf8f4 \uf8f3 44 4 n2 4n2 n2 \u2212 6n + 21 2 49 Suy ra 4 + (n \u2212 1)2 + = 16(n \u2212 1)2 4 \u21d4 4n2 + 64 1 + 1 2 n\u22125+ 16 2 n\u22121 + n\u22121 = 196 \u21d4 4n2 + 64 + 128 + (n 64 + (n \u2212 5)2 + 32(n \u2212 1 + 256 = 196 n\u22121 \u2212 1)2 5).n \u2212 1 (n \u2212 1)2 \u21d4 5n2 \u2212 10n + 25 + 320 + 32(n \u2212 5 + 4). 1 = 132 \u21d4 (n \u2212 1)2 + 64 = 16 (n \u2212 1)2 n\u22121 (n \u2212 1)2 \u00f1n \u221a = 1\u22122 2 \u21d4 (n \u2212 1)2 \u22128 2 = 0 \u21d4 (n \u2212 1)2 = 8 \u21d4 \u221a n = 1 + 2 2. \u221a \u221a\u221a V\u1edbi n = 1 \u2212 2\u221a2 ta \u0111\u01b0\u1ee3c m = 4 \u2212 \u221a2 \u21d2 AM.AN = 6\u221a2. V\u1edbi n = 1 + 2 2 ta \u0111\u01b0\u1ee3c m = 4 + 2 \u21d2 AM.AN = 6 2. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 82 (C\u00e2u 50 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Trong kh\u00f4ng gian Oxyz cho m\u1eb7t c\u1ea7u (S) t\u00e2m I(9; 3; 1) b\u00e1n k\u00ednh b\u1eb1ng 3. G\u1ecdi M, N l\u00e0 hai \u0111i\u1ec3m l\u1ea7n l\u01b0\u1ee3t thu\u1ed9c 2 tr\u1ee5c Ox, Oz sao cho \u0111\u01b0\u1eddng th\u1eb3ng M N ti\u1ebfp x\u00fac v\u1edbi (S), \u0111\u1ed3ng th\u1eddi m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp t\u1ee9 di\u1ec7n OI M N c\u00f3 b\u00e1n k\u00ednh b\u1eb1ng 13 \u00b7 G\u1ecdi A l\u00e0 ti\u1ebfp \u0111i\u1ec3m c\u1ee7a MN v\u00e0 (S), gi\u00e1 tr\u1ecb AM.AN 2 b\u1eb1ng \u221a B 18. \u221a D 39. A 12 3. C 28 3. \u0253 L\u1eddi gi\u1ea3i. Do I(9; 3; 1) n\u00ean d I, (Oxz) = 3 = R. Suy ra (S) ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t ph\u1eb3ng (Oxz). G\u1ecdi M (a; 0; 0) \u2208 Ox, N (0; 0; b) \u2208 Oz. \u0110\u01b0\u1eddng M N ti\u1ebfp x\u00fac v\u1edbi (S) t\u1ea1i A n\u00ean A l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a I l\u00ean (Oxz). Suy ra A(9; 0; 1). ab G\u1ecdi K l\u00e0 trung \u0111i\u1ec3m M N . Khi \u0111\u00f3, K ; 0; . 22 13 G\u1ecdi H l\u00e0 t\u00e2m m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp t\u1ee9 di\u1ec7n OI M N . Theo gi\u1ea3 thuy\u1ebft, ta suy ra OH = 2 . Suy ra HK \u22a5 M N . G\u1ecdi T l\u00e0 trung \u0111i\u1ec3m, ta c\u00f3 OM \u21d2 OM \u22a5 KT \u00b4 \u21d2 OM \u22a5 (KHT ) \u21d2 OM \u22a5 HK \u21d2 HK \u22a5 (OM N ). OM \u22a5 HT M\u00e0 IA \u22a5 (OM N ) n\u00ean HK \u2225 IA. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 497 S\u0110T: 0905.193.688"]
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