["1. Nguy\u00ean h\u00e0m Ta c\u00f3 3x \u2212 2 3(x \u2212 2) + 4 3 4 f (x) = (x \u2212 2)2 = (x \u2212 2)2 = x \u2212 2 + (x \u2212 2)2 . Do \u0111\u00f3 3x \u2212 2 \u00c53 4 \u00e3 4 dx = + dx = 3 ln(x \u2212 2) \u2212 + C. (x \u2212 2)2 x \u2212 2 (x \u2212 2)2 x\u22122 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 69 (C\u00e2u 44 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). Cho h\u00e0m s\u1ed1 f (x) li\u00ean t\u1ee5c tr\u00ean R. Bi\u1ebft cos 2x l\u00e0 m\u1ed9t nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x)ex, h\u1ecd t\u1ea5t c\u1ea3 c\u00e1c nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x)ex l\u00e0 A \u2212 sin 2x + cos 2x + C. B \u22122 sin 2x + cos 2x + C. C \u22122 sin 2x \u2212 cos 2x + C. D 2 sin 2x \u2212 cos 2x + C. \u0253 L\u1eddi gi\u1ea3i. f (x)ex dx. V\u00ec cos2x l\u00e0 m\u1ed9t nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x)ex n\u00ean f (x)ex = (cos 2x) = \u22122 sin 2x. Ta c\u00f3 f (x)ex dx = exd (f (x)) = f (x)ex \u2212 f (x)d (ex) = f (x)ex \u2212 M\u00e0 f (x)ex = \u22122 sin 2xdo \u0111\u00f3 f (x)ex dx = \u22122 sin 2x + 2 sin 2x dx. V\u1eady f (x)ex dx = \u22122 sin 2x \u2212 cos2x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 70 (C\u00e2u 40 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Bi\u1ebft F (x) = ex \u2212 2x2 l\u00e0 m\u1ed9t nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) tr\u00ean R. Khi \u0111\u00f3 f (2x) dx b\u1eb1ng A 2ex \u2212 4x2 + C. B 1 e2x \u2212 4x2 + C. C e2x \u2212 8x2 + C. D 1 e2x \u2212 2x2 + C. 2 2 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) = F (x) = ex \u2212 4x. Suy ra f (2x) = e2x \u2212 8x. V\u1eady f (2x) dx = e2x \u2212 8x dx = 1 e2x \u2212 4x2 + C. 2 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 71 (C\u00e2u 39 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). \u00ae2x + 5 khi x \u2265 1 F l\u00e0 nguy\u00ean h\u00e0m c\u1ee7a f tr\u00ean th\u1ecfa Cho h\u00e0m s\u1ed1 f (x) = 3x2 + 4 . Gi\u1ea3 s\u1eed m\u00e3n khi x<1 R F (0) = 2. Gi\u00e1 tr\u1ecb c\u1ee7a F (\u22121) + 2F (2) b\u1eb1ng A 27. B 29. C 12. D 33. \u0253 L\u1eddi gi\u1ea3i. Theo gi\u1ea3 thi\u1ebft F l\u00e0 m\u1ed9t nguy\u00ean h\u00e0m c\u1ee7a f (x) tr\u00ean R n\u00ean ta c\u00f3 \u00ae2x + 5 khi x \u2265 1 \u21d2 F (x) = \u00aex2 + 5x + C1 khi x \u2265 1 f (x) = 3x2 + 4 khi x < 1 x3 + 4x + C2 khi x < 1. V\u00ec F (0) = 0 \u21d2 C2 = 2. M\u1eb7t kh\u00e1c, F (x) li\u00ean t\u1ee5c tr\u00ean R n\u00ean li\u00ean t\u1ee5c t\u1ea1i x = 1 n\u00ean ta c\u00f3 lim F (x) = lim F (x) \u21d4 6 + C1 = 5 + C2 \u21d2 C1 = C2 \u2212 1 = 1. x\u21921+ x\u21921\u2212 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 298 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG \u00aex2 + 5x + 1 khi x \u2265 1 V\u1eady F (x) = x3 + 4x + 2 \u21d2 F (\u22121) + 2F (2) = \u22123 + 2 \u00b7 15 = 27. Ch\u1ecdn \u0111\u00e1p \u00e1n A khi x < 1 \u0104 C\u00e2u 72 (C\u00e2u 40 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = \u00ae2x \u2212 1 khi x \u2265 1 Gi\u1ea3 s\u1eed F l\u00e0 nguy\u00ean h\u00e0m c\u1ee7a f tr\u00ean th\u1ecfa m\u00e3n . 3x2 \u2212 2 khi x < 1 R F (0) = 2. Gi\u00e1 tr\u1ecb c\u1ee7a F (\u22121) + 2F (2) b\u1eb1ng A 9. B 15. C 11. D 6. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 F (x) = \u00aex2 \u2212 x + C1 khi x \u2265 1 x3 \u2212 2x + C2 khi x < 1. M\u00e0 F (0) = 2 n\u00ean C2 = 2. M\u1eb7t kh\u00e1c F (x) c\u00f3 \u0111\u1ea1o h\u00e0m tr\u00ean R n\u00ean F (x) li\u00ean t\u1ee5c tr\u00ean R, do \u0111\u00f3 F (x) li\u00ean t\u1ee5c t\u1ea1i x = 1. Suy ra lim F (x) = lim F (x) \u21d4 C1 = \u22121 + 2 = 1. x\u21921+ x\u21921\u2212 \u00aex2 \u2212 x + 1 khi x \u2265 1 V\u1eady F (x) = x3 \u2212 2x + 2 khi x < 1. Do \u0111\u00f3 F (\u22121) + 2F (2) = 3 + 2 \u00b7 3 = 9. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 73 (C\u00e2u 40 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = \u00ae2x + 3 khi x \u2265 1 Gi\u1ea3 s\u1eed F nguy\u00ean h\u00e0m c\u1ee7a f tr\u00ean th\u1ecfa m\u00e3n . l\u00e0 3x2 + 2 khi x < 1 R F (0) = 2. Gi\u00e1 tr\u1ecb F (\u22121) + 2F (2) b\u1eb1ng A 23. B 11. C 10. D 21. \u0253 L\u1eddi gi\u1ea3i. Do F l\u00e0 nguy\u00ean h\u00e0m c\u1ee7a f tr\u00ean R n\u00ean \uf8f1 (2x + 3) dx = x2 + 3x + C1 khi x \u2265 1 3x2 + 2 dx = x3 + 2x + C2 khi x < 1. \uf8f2\uf8f4\uf8f4F1(x) = F (x) = \uf8f4\uf8f4\uf8f3F2(x) = Theo b\u00e0i ra, ta c\u00f3 F (x) c\u00f3 \u0111\u1ea1o h\u00e0m tr\u00ean R n\u00ean F (x) li\u00ean t\u1ee5c tr\u00ean \u0111\u00f3 v\u00e0 F (0) = 2. \u0110i\u1ec1u n\u00e0y t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi \u00aeF1(1) = F2(1) \u21d4 \u00ae4 + C1 = 3 + C2 \u21d4 \u00aeC1 = 1 F2(0) = 2 C2 = 2 C2 = 2. Do \u0111\u00f3, F1(x) = x2 + 3x + 1, F2(x) = x3 + 2x + 2, suy ra F (\u22121) = F2(\u22121) = \u22121, F (2) = F1(2) = 11. V\u1eady F (\u22121) + 2F (2) = \u22121 + 2 \u00b7 11 = 21. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 74 (C\u00e2u 41 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 299 S\u0110T: 0905.193.688","1. Nguy\u00ean h\u00e0m Cho h\u00e0m s\u1ed1 y = f (x) li\u00ean t\u1ee5c tr\u00ean \u0111o\u1ea1n [\u22121; 6] v\u00e0 c\u00f3 y \u0111\u1ed3 th\u1ecb l\u00e0 \u0111\u01b0\u1eddng g\u1ea5p kh\u00fac ABC trong h\u00ecnh b\u00ean. Bi\u1ebft A2 B F (x) l\u00e0 nguy\u00ean h\u00e0m c\u1ee7a f (x) th\u1ecfa m\u00e3n F (\u22121) = \u22121. \u22121 O Gi\u00e1 tr\u1ecb c\u1ee7a F (5) + F (6) b\u1eb1ng A 23. B 21. C 25. D 19. 6 45 x \u22122 C \u0253 L\u1eddi gi\u1ea3i. \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua hai \u0111i\u1ec3m B(4; 2) v\u00e0 C(6; \u22122) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh y = \u22122x + 10. Do \u0111\u00f3 \u00ae2 khi \u2212 1 \u2264 x \u2264 4 f (x) = \u2212 2x + 10 khi 4 < x \u2264 6. Khi \u0111\u00f3 \u00ae2x \u2212 F (x) = f (x)dx = + C1 + C2 khi \u2212 1 \u2264 x \u2264 4 x2 + 10x khi 4 < x \u2264 6. V\u00ec F (\u22121) = \u22121 n\u00ean ta c\u00f3 2 \u00b7 (\u22121) + C1 = \u22121 \u21d4 C1 = 1. Do h\u00e0m s\u1ed1 F (x) li\u00ean t\u1ee5c t\u1ea1i \u0111i\u1ec3m x = 4 n\u00ean lim F (x) = lim F (x) = F (4) \u21d4 24 + C2 = 8 + C1 \u21d4 C2 = \u221215. x\u21924+ x\u21924\u2212 \u00ae2x + 1 khi \u2212 1 \u2264 x \u2264 4 Do \u0111\u00f3 F (x) = f (x)dx = \u2212 x2 + 10x \u2212 15 khi 4 < x \u2264 6. Suy ra F (5) + F (6) = 10 + 9 = 19. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 75 (C\u00e2u 41 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 y = f (x) li\u00ean t\u1ee5c tr\u00ean \u0111o\u1ea1n [\u22121; 6] v\u00e0 c\u00f3 \u0111\u1ed3 y A2 th\u1ecb l\u00e0 \u0111\u01b0\u1eddng g\u1ea5p kh\u00fac ABC trong h\u00ecnh b\u00ean. Bi\u1ebft F (x) B \u22121 O l\u00e0 nguy\u00ean h\u00e0m c\u1ee7a f (x) th\u1ecfa m\u00e3n F (\u22121) = \u22122. Gi\u00e1 tr\u1ecb 6x 45 c\u1ee7a F (5) + F (6) b\u1eb1ng A 19. B 22. C 17. D 18. \u22122 C \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua hai \u0111i\u1ec3m A(\u22121; 2) v\u00e0 B(4; 2) l\u00e0 y = 2. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m B(4; 2) v\u00e0 C(6; \u22122) l\u00e0 y = \u22122x + 10. \u00ae2 n\u1ebfu \u2212 1 \u2264 x \u2264 4 Do \u0111\u00f3 h\u00e0m s\u1ed1 f (x) = \u2212 2x + 10 n\u1ebfu 4 < x \u2264 6. Do h\u00e0m s\u1ed1 f (x) li\u00ean t\u1ee5c tr\u00ean \u0111o\u1ea1n [\u22121; 6] n\u00ean c\u00f3 nguy\u00ean h\u00e0m tr\u00ean [\u22121; 6]. M\u00e0 2 dx = 2x + C1, (\u22122x + 10) dx = \u2212x2 + 10x + C2. Suy ra F (x) = \u00ae2x + C1 n\u1ebfu \u2212 1 \u2264 x \u2264 4 \u2212 x2 + 10x + C2 n\u1ebfu 4 < x \u2264 6. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 300 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG Ta c\u00f3 F (\u22121) = \u22122 \u21d4 C1 \u2212 2 = \u22122 \u21d4 C1 = 0 n\u00ean n\u1ebfu \u2212 1 \u2264 x \u2264 4 \u00ae2x n\u1ebfu 4 < x \u2264 6. F (x) = \u2212 x2 + 10x + C2 V\u00ec F (x) l\u00e0 nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) tr\u00ean [\u22121; 6] cho n\u00ean F (x) c\u00f3 \u0111\u1ea1o h\u00e0m tr\u00ean [\u22121; 6], suy ra F (x) li\u00ean t\u1ee5c tr\u00ean [\u22121; 6]. L\u1ea1i c\u00f3 lim F (x) = lim 2x = 8, lim F (x) = lim (\u2212x2 + 10x + C2) = C2 + 24. x\u21924\u2212 x\u21924\u2212 x\u21924+ x\u21924+ Suy ra lim F (x) = lim F (x) = F (4) \u21d4 C2 + 24 = 8 \u21d4 C2 = \u221216. x\u21924\u2212 x\u21924+ V\u1eady \u00ae2x n\u1ebfu \u2212 1 \u2264 x \u2264 4 F (x) = \u2212 x2 + 10x \u2212 16 n\u1ebfu 4 < x \u2264 6 v\u00e0 F (5) + F (6) = (\u221225 + 50 \u2212 16) + (\u221236 + 60 \u2212 16) = 17. C\u00e1ch kh\u00e1c: Do h\u00e0m s\u1ed1 y = f (x) li\u00ean t\u1ee5c tr\u00ean \u0111o\u1ea1n [\u22121; 6] n\u00ean t\u1ed3n t\u1ea1i nguy\u00ean h\u00e0m F (x) tr\u00ean \u0111o\u1ea1n [\u22121; 6]. y B A2 M N P6 x \u22121 O 45 Q \u22122 C T\u1eeb \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 ta c\u00f3 Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = f (x), tr\u1ee5c Ox v\u00e0 c\u00e1c \u0111\u01b0\u1eddng th\u1eb3ng x = \u22121, x = 4 b\u1eb1ng di\u1ec7n t\u00edch h\u00ecnh ch\u1eef nh\u1eadt ABN M n\u00ean 44 |f (x)| dx = 10 \u21d4 f (x) dx = 10 \u21d4 F (4) \u2212 F (\u22121) = 10 \u21d4 F (4) = 8. \u22121 \u22121 Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = f (x), tr\u1ee5c Ox v\u00e0 c\u00e1c \u0111\u01b0\u1eddng th\u1eb3ng x = 4, x = 5 b\u1eb1ng di\u1ec7n t\u00edch tam gi\u00e1c vu\u00f4ng N BP n\u00ean 55 |f (x)| dx = 1 \u21d4 f (x) dx = 1 \u21d4 F (5) \u2212 F (4) = 1 \u21d4 F (5) = 9. 44 Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = f (x), tr\u1ee5c Ox v\u00e0 c\u00e1c \u0111\u01b0\u1eddng th\u1eb3ng x = 5, x = 6 b\u1eb1ng di\u1ec7n t\u00edch tam gi\u00e1c vu\u00f4ng P CQ n\u00ean 66 |f (x)| dx = 1 \u21d4 \u2212 f (x) dx = 1 \u21d4 \u2212 [F (6) \u2212 F (5)] = 1 \u21d4 F (6) = 8. 55 V\u1eady F (5) + F (6) = 17. 301 S\u0110T: 0905.193.688 Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t","1. Nguy\u00ean h\u00e0m \u0104 C\u00e2u 76 (C\u00e2u 41 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00e0m s\u1ed1 y = f (x) c\u00f3 \u0111\u1ea1o h\u00e0m l\u00e0 f (x) = 12x2 + 2, \u2200x \u2208 R v\u00e0 f (1) = 3. Bi\u1ebft F (x) l\u00e0 nguy\u00ean h\u00e0m c\u1ee7a f (x) th\u1ecfa m\u00e3n F (0) = 2, khi \u0111\u00f3 F (1) b\u1eb1ng A \u22123. B 1. C 2. D 7. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) = f (x) dx = 4x3 + 2x + C1. V\u00ec f (1) = 3 n\u00ean C1 = \u22123. Khi \u0111\u00f3 f (x) = 4x3 + 2x \u2212 3. Ta c\u00f3 F (x) = f (x) dx = x4 + x2 \u2212 3x + C2. V\u00ec F (0) = 2 n\u00ean C2 = 2. Khi \u0111\u00f3 F (x) = x4 + x2 \u2212 3x + 2. V\u1eady F (1) = 1. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 77 (C\u00e2u 40 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Bi\u1ebft F (x) = ex \u2212 x2 l\u00e0 m\u1ed9t nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) tr\u00ean R. Khi \u0111\u00f3 f (2x) dx b\u1eb1ng A 1 e2x \u2212 2x2 + C. B e2x \u2212 4x2 + C. C 2ex \u2212 2x2 + C. D 1 e2x \u2212 x2 + C. 2 2 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) dx = ex \u2212 x2 + C. 1 1 e2x \u2212 (2x)2 + C = 1 e2x \u2212 2x2 + C. V\u1eady f (2x) dx = f (2x) d(2x) = 22 2 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 78 (C\u00e2u 39 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00e0m s\u1ed1 f (x) = \u221a x . H\u1ecd c\u00e1c nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 g(x) = (x + 1)f (x) l\u00e0 x2 + 2 A x2\u221a+ 2x \u2212 2 + C. B \u221ax \u2212 2 + C. C 2x\u221a2 + x + 2 + C. D \u221ax + 2 + C. 2 x2 + 2 x2 + 2 x2 + 2 2 x2 + 2 Ta c\u00f3 \u0253 L\u1eddi gi\u1ea3i. g(x) dx = (x + 1)f (x) dx = (x + 1)f (x) \u2212 f (x) dx = x\u221a(x + 1) \u2212 \u221a x dx x2 + 2 x2 + 2 = x\u221a(x + 1) \u2212 1 \u221a 1 d x2 + 2 x2 + 2 2 x2 + 2 = x\u221a(x + 1) \u2212 1 \u00b7 \u221a + 2 + C 2 x2 x2 + 2 2 = \u221ax \u2212 2 + C. x2 + 2 Ch\u1ecdn \u0111\u00e1p \u00e1n B 302 S\u0110T: 0905.193.688 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG \u0104 C\u00e2u 79 (C\u00e2u 41 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00e0m s\u1ed1 f (x) = \u221a x . H\u1ecd t\u1ea5t c\u1ea3 c\u00e1c nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 g(x) = (x + 1) \u00b7 f (x) l\u00e0 x2 + 3 A x2\u221a+ 2x \u2212 3 . C 2x\u221a2 + x + 3 . B \u221ax + 3 . D \u221ax \u2212 3 . 2 x2 + 3 2 x2 + 3 x2 + 3 x2 + 3 \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t I = (x + 1) \u00b7 f (x)dx. \u00aeu = x + 1 \u21d2 du = dx \u0110\u1eb7t dv = f (x)dx \u21d2 v = f (x). Khi \u0111\u00f3 I = (x + 1) \u00b7 f (x) \u2212 f (x)dx = (x + 1) \u00b7 \u221a x \u2212 \u221a x dx x2 + 3 x2 + 3 = \u221ax2 + x \u2212 1 (x2 + 3)\u2212 1 d(x2 + 3) 2 x2 + 3 2 1 = \u221ax2 + x \u2212 1 \u00b7 (x2 + 3) 2 +C x2 + 3 2 1 2 \u221a = \u221ax2 + x \u2212 x2 + 3 + C x2 + 3 = x2 +\u221ax \u2212 x2 \u2212 3 + C x2 + 3 = \u221ax \u2212 3 + C. x2 + 3 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 80 (C\u00e2u 42 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho h\u00e0m s\u1ed1 f (x) = \u221a x . H\u1ecd t\u1ea5t c\u1ea3 c\u00e1c nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 g (x) = (x + 1) f (x) l\u00e0 x2 + 1 A x2\u221a+ 2x \u2212 1 + C. B \u221ax + 1 + C. C 2x\u221a2 + x + 1 + C. D \u221ax \u2212 1 + C. 2 x2 + 1 2 x2 + 1 x2 + 1 x2 + 1 \u0253 L\u1eddi gi\u1ea3i. \u221a x x2 + 1 \u2212 x. \u221a x2 + 1 = 1\u221a . Ta c\u00f3 f (x) = x2 + 1 (x2 + 1) x2 + 1 Do \u0111\u00f3 g (x) = (x + 1) f (x) = x +\u221a1 . (x2 + 1) x2 + 1 Ta l\u1ea1i c\u00f3 g (x) dx = x +\u221a1 1 x2 +1 \u2212 3 d x2 + 1 + 1\u221a dx. dx = 2 (x2 + 1) x2 + 1 (x2 + 1) x2 + 1 2 V\u1eady g (x) dx = \u2212 x2 + 1 \u22121 x + C = \u221ax \u2212 1 + C. 2 +\u221a x2 + 1 x2 + 1 Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 303 S\u0110T: 0905.193.688","1. Nguy\u00ean h\u00e0m \u0104 C\u00e2u 81 (C\u00e2u 13 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). 3 . Cho F (x) l\u00e0 m\u1ed9t nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = ex + 2x th\u1ecfa m\u00e3n F (0) = 2 T\u00ecm F (x). A F (x) = ex + x2 + 3 B F (x) = 2ex + x2 \u2212 1 . . 2 2 5 1 C F (x) = ex + x2 + . D F (x) = ex + x2 + . 2 2 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) dx = (ex + 2x) dx = ex + x2 + C \u21d2 F (0) = 1 + C = 3 \u21d2 C = 1 \u21d2 F (x) = ex + x2 + 1 Ch\u1ecdn \u0111\u00e1p \u00e1n D 22 2 \u0104 C\u00e2u 82 (C\u00e2u 40 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). \u22121 Cho h\u00e0m s\u1ed1 f (x) th\u1ecfa m\u00e3n f (2) = 3 v\u00e0 f (x) = x [f (x)]2 v\u1edbi m\u1ecdi x \u2208 R. Gi\u00e1 tr\u1ecb c\u1ee7a f (1) b\u1eb1ng A \u221211. B \u22122. C \u22122. D \u22127. 6 3 9 6 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) = x [f (x)]2 \u21d4 f (x) = x. f 2(x) Do \u0111\u00f3, f (x) dx = x dx f 2(x) \u21d4\u2212 \u00c51\u00e3 x dx d= f (x) \u21d4 \u2212 1 = 1 x2 + C f (x) 2 \u21d4 f (x) = \u2212 1 . 1 x2 + C 2 Theo gi\u1ea3 thuy\u1ebft, f (2) = \u22121 \u21d2 C = 1 \u21d2 f (x) = \u2212 1 . 3 1 1 x2 + 2 Suy ra f (1) = \u22122. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 83 (C\u00e2u 40 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho F (x) = (x \u2212 1)ex l\u00e0 m\u1ed9t nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x)e2x. T\u00ecm nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x)e2x. A f (x)e2x dx = (4 \u2212 2x)ex + C. B f (x)e2x dx = 2 \u2212 x ex + C. 2 C f (x)e2x dx = (2 \u2212 x)ex + C. D f (x)e2x dx = (x \u2212 2)ex + C. - Ta c\u00f3 f (x)e2x = F (x) = xex. \u0253 L\u1eddi gi\u1ea3i. - Suy ra f (x)e2x dx = e2x.f (x) \u2212 2 f (x)e2x dx = xex \u2212 2(x \u2212 1)ex = (2 \u2212 x)ex + C Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 304 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG B\u00c0I 2. T\u00cdCH PH\u00c2N \u0104 C\u00e2u 1 (C\u00e2u 9 - \u0110MH - BGD\u221a&\u0110T - N\u0103m 2021 - 2022). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = x 2 l\u00e0 A R. B R\\\\{0}. C (0; +\u221e). D (2; +\u221e). \u0253 L\u1eddi gi\u1ea3i. \u221a H\u00e0m s\u1ed1 y = x 2 x\u00e1c \u0111\u1ecbnh khi v\u00e0 ch\u1ec9 khi x > 0. V\u1eady D = (0; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 2 (C\u00e2u 10 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). 5 Tr\u00ean kho\u1ea3ng (0; +\u221e), \u0111\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = x 2 l\u00e0 A y = 27 B y 23 C y = 53 D y = 5 x\u2212 3 . x2. = x2. x2. 2 7 5 2 2 \u0253 L\u1eddi gi\u1ea3i. Tr\u00ean kho\u1ea3ng (0; +\u221e), ta c\u00f3 y \u00c5 5\u00e3 = 5 x 5 \u22121 = 53 = x2 2 x2. 22 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 3 (C\u00e2u 1 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Tr\u00ean kho\u1ea3ng (0; +\u221e), \u0111\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = x5 l\u00e0 4 A y = 4 x 9 . B y = 4 x 1 . C y = 5 x 1 . D y = 5 x\u2212 1 . 4 4 4 4 9 5 4 4 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 y = 5 x 1 . 4 4 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 4 (C\u00e2u 17 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). 4 Tr\u00ean kho\u1ea3ng (0; +\u221e), \u0111\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = x 3 l\u00e0 A y = 4 x\u2212 1 . B y 41 C y = 37 D y 31 3 = x3. x3. = x3. 3 3 7 4 \u0253 L\u1eddi gi\u1ea3i. Tr\u00ean kho\u1ea3ng (0; +\u221e), ta c\u00f3 y = 41 x3. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 5 (C\u00e2u 8 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). 5 Tr\u00ean kho\u1ea3ng (0; +\u221e), \u0111\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = x 3 l\u00e0 A y = 38 B y 52 C y = 5 x\u2212 2 . D y 32 x3. = x3. 3 = x3. 8 3 3 5 Ta c\u00f3 y = 5 x 5 \u22121 = 52 \u0253 L\u1eddi gi\u1ea3i. 3 x3. 33 305 Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t S\u0110T: 0905.193.688","2. T\u00edch ph\u00e2n \u0104 C\u00e2u 6 (C\u00e2u 1 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Tr\u00ean kho\u1ea3ng (0; +\u221e), \u0111\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = x5 l\u00e0 4 A y = 4 x 9 . B y = 4 x 1 . C y = 5 x 1 . D y = 5 x\u2212 1 . 4 4 4 4 9 5 4 4 5 \u0253 L\u1eddi gi\u1ea3i. Tr\u00ean kho\u1ea3ng (0; +\u221e), ta c\u00f3 y = x 1 . 4 4 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 7 (C\u00e2u 24 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = (x \u2212 1) 1 . 3 A D = (\u2212\u221e; 1). B D = (1; +\u221e). C D = R. D D = R \\\\ {1}. 1 \u0253 L\u1eddi gi\u1ea3i. 3 \u0110i\u1ec1u ki\u1ec7n: x\u22121 > 0 (v\u00ec kh\u00f4ng nguy\u00ean) \u21d2 x > 1 \u21d2 t\u1eadp x\u00e1c \u0111\u1ecbnh D = (1; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 8 (C\u00e2u 22 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho hai h\u00e0m s\u1ed1 y = ax, y = bx v\u1edbi a, b l\u00e0 hai s\u1ed1 th\u1ef1c d\u01b0\u01a1ng kh\u00e1c 1, l\u1ea7n l\u01b0\u1ee3t c\u00f3 \u0111\u1ed3 th\u1ecb l\u00e0 (C1) v\u00e0 (C2) y nh\u01b0 h\u00ecnh b\u00ean.M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? (C2) (C1) A 0 < a < b < 1. B 0 < b < 1 < a. C 0 < a < 1 < b. D 0 < b < a < 1. Ox \u0253 L\u1eddi gi\u1ea3i. Theo h\u00ecnh v\u1ebd ta c\u00f3 h\u00e0m y = ax \u0111\u1ed3ng bi\u1ebfn \u21d2 a > 1 v\u00e0 h\u00e0m s\u1ed1 y = bx ngh\u1ecbch bi\u1ebfn \u21d2 b < 1. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 9 (C\u00e2u 11 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm t\u1eadp x\u00e1c \u0111\u1ecbnh D c\u1ee7a h\u00e0m s\u1ed1 y = (x2 \u2212 x \u2212 2)\u22123. A D = R. B D = (0; +\u221e). C D = (\u2212\u221e; \u22121) \u222a (2; +\u221e). D D = R \\\\ {\u22121; 2}. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x\u00e1c \u0111\u1ecbnh: x2 \u2212 x \u2212 2 = 0 \u21d4 x = \u22121 v\u00e0 x = 2. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 10 (C\u00e2u 32 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm t\u1ea5t c\u1ea3 c\u00e1c gi\u00e1 tr\u1ecb th\u1ef1c c\u1ee7a tham s\u1ed1 m \u0111\u1ec3 h\u00e0m s\u1ed1 y = log (x2 \u2212 2x \u2212 m + 1) c\u00f3 t\u1eadp x\u00e1c \u0111\u1ecbnh l\u00e0 R. B m < 0. C m \u2264 2. D m > 2. A m \u2265 0. \u0253 L\u1eddi gi\u1ea3i. H\u00e0m s\u1ed1 y = log (x2 \u2212 2x \u2212 m + 1) x\u00e1c \u0111\u1ecbnh \u21d4 x2 \u2212 2x \u2212 m + 1 > 0 H\u00e0m s\u1ed1 c\u00f3 t\u1eadp x\u00e1c \u0111\u1ecbnh l\u00e0 R \u21d4 b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh x2 \u2212 2x \u2212 m + 1 > 0 x\u1ea3y ra v\u1edbi m\u1ecdi x \u21d4 \u2206 = 4 + 4 (m \u2212 1) < 0 \u21d4 m < 0. Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 306 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG \u0104 C\u00e2u 11 (C\u00e2u 47 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). X\u00e9t c\u00e1c s\u1ed1 th\u1ef1c kh\u00f4ng \u00e2m x v\u00e0 y th\u1ecfa m\u00e3n 2x + y4x+y\u22121 \u2265 3. Gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a bi\u1ec3u th\u1ee9c P = x2 + y2 + 6x + 4y b\u1eb1ng A 33 B 9 C 21 D 41 . . .. . 8 8 4 8 \u0253 L\u1eddi gi\u1ea3i. 1 N\u1ebfu x + y < 3 th\u00ec 2x + y4x+y\u22121 < 2x + y4 2 = 2x + 2y < 3 (lo\u1ea1i). V\u1eady t\u1eeb gi\u1ea3 thi\u1ebft suy ra 2x + 2y \u2265 3. 2 \u00ae2x + 2y \u2265 3 Tr\u00ean m\u1eb7t ph\u1eb3ng t\u1ecda \u0111\u1ed9 mi\u1ec1n nghi\u1ec7m c\u1ee7a h\u1ec7 l\u00e0 ph\u1ea7n kh\u00f4ng b\u1ecb g\u1ea1ch nh\u01b0 h\u00ecnh v\u1ebd x \u2265 0; y \u2265 0 y 2x + 2y = 3 H x O I Ta c\u00f3 P = x2 + y2 + 4x + 2y \u21d4 (x + 2)2 + (y + 1)2 = 5 + P (\u2217) \u221a T\u1eadp h\u1ee3p c\u00e1c \u0111i\u1ec3m (x; y) th\u1ecfa m\u00e3n (\u2217) l\u00e0 \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m I (\u22122; \u22121) b\u00e1n k\u00ednh R = 5 + P , (P > \u22125). \u0110\u1ec3 t\u1ed3n t\u1ea1i c\u1eb7p (x; y) th\u00ec \u0111\u01b0\u1eddng tr\u00f2n ph\u1ea3i c\u00f3 \u0111i\u1ec3m chung v\u1edbi ph\u1ea7n m\u1eb7t ph\u1eb3ng kh\u00f4ng b\u1ecb g\u1ea1ch \u1edf h\u00ecnh tr\u00ean. \u0110i\u1ec1u \u0111\u00f3 x\u1ea3y ra khi b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n kh\u00f4ng b\u00e9 h\u01a1n kho\u1ea3ng c\u00e1ch t\u1eeb t\u00e2m I \u0111\u1ebfn \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh d : 2x + 2y \u2212 3 = 0. \u221a n\u00ean ta ph\u1ea3i c\u00f3 5+P \u2265 \u00c7 \u221a \u00e52 \u21d4 P \u2265 41 B\u1edfi v\u00ec d (I; d) = |\u22122.2 \u2212\u221a1.2 \u2212 3| = 9 2 92 . 22 4 48 D\u1ea5u b\u1eb1ng x\u1ea3y ra khi c\u1eb7p (x; y) l\u00e0 t\u1ecda \u0111\u1ed9 c\u1ee7a \u0111i\u1ec3m H tr\u00ean h\u00ecnh v\u1ebd. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 307 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc B\u00c0I 3. \u1ee8NG D\u1ee4NG C\u1ee6A T\u00cdCH PH\u00c2N TRONG H\u00ccNH H\u1eccC \u0104 C\u00e2u 1 (C\u00e2u 22 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Vi\u1ebft c\u00f4ng th\u1ee9c t\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i tr\u00f2n xoay \u0111\u01b0\u1ee3c t\u1ea1o ra khi quay h\u00ecnh thang cong, gi\u1edbi h\u1ea1n b\u1edfi \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = f (x), tr\u1ee5c Ox v\u00e0 hai \u0111\u01b0\u1eddng th\u1eb3ng x = a, x = b (a < b), xung quanh tr\u1ee5c Ox. b b A V = \u03c0 f 2(x) dx. B V = f 2(x) dx. a a b b C V = \u03c0 f (x) dx. D V = \u03c0 |f (x)| dx. aa \u0253 L\u1eddi gi\u1ea3i. Th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i tr\u00f2n xoay \u0111\u01b0\u1ee3c t\u1ea1o ra khi quay h\u00ecnh thang cong, gi\u1edbi h\u1ea1n b\u1edfi \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = f (x), tr\u1ee5c Ox v\u00e0 hai \u0111\u01b0\u1eddng th\u1eb3ng x = a, x = b; (a < b), xung quanh tr\u1ee5c Ox \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng b th\u1ee9c V = \u03c0 f 2(x) dx. a Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 2 (C\u00e2u 5 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). G\u1ecdi S l\u00e0 di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = ex, y = 0, x = 0, x = 2. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? 2 22 2 A S = \u03c0 e2x dx. B S = ex dx. C S = \u03c0 ex dx. D S = e2x dx. 00 00 \u0253 L\u1eddi gi\u1ea3i. Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = ex, y = 0, x = 0, x = 2 \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c 22 S = |ex| dx = ex dx. 00 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 3 (C\u00e2u 2 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). G\u1ecdi S l\u00e0 di\u1ec7n t\u00edch c\u1ee7a h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = 2x, y = 0, x = 0, x = 2. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? 2 2 2 2 B S = \u03c0 22xdx. C S = 22xdx. D S = \u03c0 2xdx. A S = 2xdx. 0 00 0 \u0253 L\u1eddi gi\u1ea3i. 2 Di\u1ec7n t\u00edch c\u1ee7a h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = 2x, y = 0, x = 0, x = 2 l\u00e0 S = 2x dx. 0 Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 308 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG \u0104 C\u00e2u 4 (C\u00e2u 29 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho h\u00e0m s\u1ed1 f (x) li\u00ean t\u1ee5c tr\u00ean R. G\u1ecdi S l\u00e0 di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng y gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = f (x), y = 0, x = \u22121, x = 2 (nh\u01b0 h\u00ecnh y = f (x) v\u1ebd b\u00ean). M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? \u22121 O 1 2x 12 12 A S = \u2212 f (x) dx \u2212 f (x) dx. B S = \u2212 f (x) dx + f (x) dx. \u22121 1 \u22121 1 12 12 C S = f (x) dx \u2212 f (x) dx. D S = f (x) dx + f (x) dx. \u22121 1 \u22121 1 \u0253 L\u1eddi gi\u1ea3i. 2 12 S = |f (x)| dx = |f (x)| dx + |f (x)| dx \u22121 \u22121 1 Nh\u00ecn h\u00ecnh ta th\u1ea5y h\u00e0m s\u1ed1 f (x) li\u00ean t\u1ee5c v\u00e0 nh\u1eadn gi\u00e1 tr\u1ecb kh\u00f4ng \u00e2m tr\u00ean \u0111o\u1ea1n [\u22121; 1] 11 n\u00ean |f (x)| dx = f (x) dx. \u22121 \u22121 H\u00e0m s\u1ed1 f (x) li\u00ean t\u1ee5c v\u00e0 nh\u1eadn gi\u00e1 tr\u1ecb \u00e2m tr\u00ean \u0111o\u1ea1n [1; 2] 22 n\u00ean |f (x)| dx = \u2212 f (x) dx. 1 1 1 2 V\u1eady S = f (x) dx \u2212 f (x) dx. \u22121 1 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 5 (C\u00e2u 24 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 309 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc Cho h\u00e0m s\u1ed1 f (x) li\u00ean t\u1ee5c tr\u00ean R. G\u1ecdi S l\u00e0 di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi y h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = f (x), y = 0, x = \u22122 v\u00e0 x = 3 (nh\u01b0 h\u00ecnh v\u1ebd y = f (x) b\u00ean). M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? 13 A S = f (x) dx \u2212 f (x) dx. \u22122 1 \u22122 1 1 3 O 3x B S = \u2212 f (x) dx + f (x) dx. \u22122 1 13 C S = f (x) dx + f (x) dx. \u22122 1 1 3 D S = \u2212 f (x) dx \u2212 f (x) dx. \u22122 1 \u0253 L\u1eddi gi\u1ea3i. 3 13 Ta c\u00f3 S = |f (x)| dx = |f (x)| dx + |f (x)| dx. \u22122 \u22122 1 13 Do f (x) \u2265 0 v\u1edbi \u2200x \u2208 [\u22122; 1] v\u00e0 f (x) \u2264 0 v\u1edbi \u2200x \u2208 [1; 3] n\u00ean S = f (x) dx \u2212 f (x) dx. \u22122 1 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 6 (C\u00e2u 29 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi hai \u0111\u01b0\u1eddng y = x2 \u2212 4 v\u00e0 y = 2x \u2212 4 b\u1eb1ng A 36. B 4 C 4\u03c0 D 36\u03c0. . . 3 3 \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m c\u1ee7a hai \u0111\u01b0\u1eddng y = x2 \u2212 4 v\u00e0 y = 2x \u2212 4 l\u00e0 x2 \u2212 4 = 2x \u2212 4 \u21d4 x2 \u2212 2x = 0 \u21d4 \u00f1x = 0 x = 2. Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi hai \u0111\u01b0\u1eddng y = x2 \u2212 4 v\u00e0 y = 2x \u2212 4 l\u00e0 2 S= x2 \u2212 4 \u2212 (2x \u2212 4) 4 dx = . 3 0 4 V\u1eady S = . 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 7 (C\u00e2u 12 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). V\u1edbi c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng a, b b\u1ea5t k\u00ec. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang ? A ln(ab) = ln a + ln b. B ln(ab) = ln a. ln b. C a = ln a D ln a = ln b \u2212 ln a. ln . b b ln b Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 310 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG \u0253 L\u1eddi gi\u1ea3i. V\u1edbi m\u1ecdi s\u1ed1 d\u01b0\u01a1ng a, b ta c\u00f3: ln(ab) = ln a + ln b. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 8 (C\u00e2u 8 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). D 1 log(3a) = log a. V\u1edbi a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng b\u1ea5t k\u00ec, m\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? 3 A log(3a) = 3 log a. B log(a3) = 1 log a. C log(a3) = 3 log a. 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log(a3) = 3 log a. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 9 (C\u00e2u 6 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). V\u1edbi a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd, ln(5a) \u2212 ln(3a) b\u1eb1ng A ln(5a) B ln(2a). C 5 D ln 5 . ln . . ln(3a) 3 ln 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 ln(5a) \u2212 ln(3a) = ln 5a = ln 5 . 3a 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 10 (C\u00e2u 11 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). V\u1edbi a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng tu\u1ef3 \u00fd, log3(3a) b\u1eb1ng A 3 log3 a. B 3 + log3 a. C 1 + log3 a. D 1 \u2212 log3 a. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log3(3a) = log3 3 + log3 a = 1 + log3 a. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 11 (C\u00e2u 43 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). V\u1edbi a v\u00e0 b l\u00e0 hai s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd, log (ab2) b\u1eb1ng A 2 log a + log b. B log a + 2 log b. C 2 (log a + log b). D 1 log a + log b. 2 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log (ab2) = log a + log b2 = log a + 2 log b. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 12 (C\u00e2u 5 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). V\u1edbi a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd, log5 a3 b\u1eb1ng 1 1 A 3 log5 a. B 3 + log5 a. C 3 + log5 a. D 3 log5 a. \u0253 L\u1eddi gi\u1ea3i. log5 a3 = 3 log5 a. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 13 (C\u00e2u 12 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). V\u1edbi a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd, log2 a2 b\u1eb1ng Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 311 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc A 2 log2 a. B 1 C 1 D 2 + log2 a. 2 + log2 a. 2 log2 a. \u0253 L\u1eddi gi\u1ea3i. V\u00ec a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd n\u00ean log2 a2 = 2 log2 a. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 14 (C\u00e2u 11 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). V\u1edbi a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd, log2 (a3) b\u1eb1ng 3 1 A 2 log2 a. B 3 log2 a. C 3 + log2 a. D 3 log2 a. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log2 (a3) = 3 log2 a. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 15 (C\u00e2u 9 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). V\u1edbi a, b l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd v\u00e0 a = 1, loga5 b b\u1eb1ng 1 1 A 5 loga b. B 5 + loga b. C 5 + loga b. D 5 loga b. \u0253 L\u1eddi gi\u1ea3i. 1 loga5 b = 5 loga b. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 16 (C\u00e2u 24 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). V\u1edbi a, b l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd v\u00e0 a = 1, loga3 b b\u1eb1ng 1 1 A 3 + loga b. B 3 loga b. C 3 + loga b. D 3 logab. \u0253 L\u1eddi gi\u1ea3i. 1 Ta c\u00f3 loga3 b = 3 logab. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 17 (C\u00e2u 11 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). V\u1edbi a, b l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd v\u00e0 a = 1 th\u00ec loga4 b b\u1eb1ng 1 1 4 + loga b. A 4 + loga b. B 4 loga b. C 4logab. D \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 loga4 b = 1 loga b. 4 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 18 (C\u00e2u 23 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). V\u1edbi a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd, log4(4a) b\u1eb1ng A 1 + log4 a. B 4 \u2212 log4 a. C 4 + log4 a. D 1 \u2212 log4 a. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log4(4a) = log4 4 + log4 a = 1 + log4 a. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 312 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG \u0104 C\u00e2u 19 (C\u00e2u 33 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). V\u1edbi a, b l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd th\u1ecfa m\u00e3n log2 a\u22122 log4 b = 3, m\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A a = 8b2. B a = 8b. C a = 6b. D a = 8b4. \u0253 L\u1eddi gi\u1ea3i. a a b Ta c\u00f3 log2 a \u2212 2 log4 b = 3 \u21d4 log2 a \u2212 log2 b = 3 \u21d4 log2 b = 3 \u21d4 = 23 \u21d4 a = 8b. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 20 (C\u00e2u 1 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). V\u1edbi a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd, log2(2a) b\u1eb1ng A 1 + log2 a. B 1 \u2212 log2 a. C 2 \u2212 log2 a. D 2 + log2 a. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log2(2a) = log2 2 + log2 a = 1 + log2 a. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 21 (C\u00e2u 17 - M\u0110 102 -\u221aBGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho a > 0 v\u00e0 a = 1. Khi \u0111\u00f3 loga 3 a b\u1eb1ng 1 C \u22121. A \u22123. B . D 3. 33 \u0253 L\u1eddi gi\u1ea3i. \u221a 11 Ta c\u00f3 loga 3 a = loga a 3 = . 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 22 (C\u00e2u 36 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). V\u1edbi m\u1ecdi a, b th\u1ecfa m\u00e3n log2 a3 + log2 b = 5, kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A a3b = 32. B a3b = 25. C a3 + b = 25. D a3 + b = 32. \u0253 L\u1eddi gi\u1ea3i. T\u1eeb gi\u1ea3 thi\u1ebft suy ra a > 0, b > 0. Do \u0111\u00f3 log2 a3 + log2 b = 5 \u21d4 log2(a3b) = 5 \u21d4 a3b = 32. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 23 (C\u00e2u 3 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). V\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c a d\u01b0\u01a1ng, log4(4a) b\u1eb1ng A 1 + log4 a. B 1 \u2212 log4 a. C log4 a. D 4 log4 a. \u0253 L\u1eddi gi\u1ea3i. V\u1edbi a > 0 ta c\u00f3 log4(4a) = log4 4 + log4 a = 1 + log4 a. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 24 (C\u00e2u 21 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). V\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c d\u01b0\u01a1ng a, log3(3a) b\u1eb1ng A 3 log3 a. B 1 \u2212 log3 a. C log3 a. D 1 + log3 a. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log3(3a) = log3 3 + log3 a = 1 + log3 a. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 313 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc \u0104 C\u00e2u 25 (C\u00e2u 4 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). V\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c a d\u01b0\u01a1ng, log2 (2a) b\u1eb1ng A 1 \u2212 log2 a. B 1 + log2 a. C 2 log2 a. D log2 a. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log2 (2a) = log2 2 + log2 a = 1 + log2 a. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 26 (C\u00e2u 12 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). V\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c a d\u01b0\u01a1ng, log5 (5a) b\u1eb1ng A 5 log5 a. B 1 \u2212 log5 a. C 1 + log5 a. D log5 a. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log5 (5a) = log5 5 + log5 a = 1 + log5 a. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 27 (C\u00e2u 17 - M\u0110 101 - B\u221aGD&\u0110T - N\u0103m 2021 - 2022). V\u1edbi a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd, 4 log a b\u1eb1ng A \u22122 log a. B 2 log a. C \u22124 log a. D 8 log a. \u221a \u0253 L\u1eddi gi\u1ea3i. 4 log a Ta c\u00f3 = 4 log a 1 = 4. 1 log a = 2 log a. 2 2 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 28 (C\u00e2u 7 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). V\u1edbi a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng tu\u1ef3 \u00fd, log(100a) b\u1eb1ng A 2 \u2212 log a. B 2 + log a. C 1 \u2212 log a. D 1 + log a. \u0253 L\u1eddi gi\u1ea3i. V\u1edbi a > 0, ta c\u00f3 log(100a) = log 100 + log a = log 102 + log a = 2 + log a. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 29 (C\u00e2u 31 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). 1 V\u1edbi a, b l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd v\u00e0 a = 1, log 1 b\u1eb1ng b3 a 1 3 loga b. A loga b. B \u22123 loga b. C D 3 loga b. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log 1 1 = loga\u22121 b\u22123 = 3 loga b. b3 a Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 30 (C\u00e2u 6 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). Cho h\u00e0m s\u1ed1 y = f (x) li\u00ean t\u1ee5c tr\u00ean \u0111o\u1ea1n [a; b]. G\u1ecdi D l\u00e0 h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi \u0111\u1ed3 th\u1ecb c\u1ee7a h\u00e0m s\u1ed1 y = f (x), tr\u1ee5c ho\u00e0nh v\u00e0 hai \u0111\u01b0\u1eddng th\u1eb3ng x = a, x = b (a < b). Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u00f2n xoay t\u1ea1o th\u00e0nh khi quay D quanh tr\u1ee5c ho\u00e0nh \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 314 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG b b A V = \u03c0 f 2(x) dx. B V = 2\u03c0 f 2(x) dx. a a b b C V = \u03c02 f 2(x) dx. D V = \u03c02 f (x) dx. a a \u0253 L\u1eddi gi\u1ea3i. D\u1ef1a v\u00e0o c\u00f4ng th\u1ee9c t\u00ednh th\u1ec3 t\u00edch kh\u1ed1i tr\u00f2n xoay t\u1ea1o th\u00e0nh khi quay h\u00ecnh ph\u1eb3ng quanh tr\u1ee5c ho\u00e0nh ta b c\u00f3 V = \u03c0 f 2(x) dx. a Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 31 (C\u00e2u 4 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho h\u00ecnh ph\u1eb3ng (H) gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = x2 + 3, y = 0, x = 0, x = 2. G\u1ecdi V l\u00e0 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u00f2n xoay \u0111\u01b0\u1ee3c t\u1ea1o th\u00e0nh khi quay (H) xung quanh tr\u1ee5c Ox. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? 2 2 A V = \u03c0 (x2 + 3)2 dx. B V = \u03c0 (x2 + 3) dx. 0 0 2 2 C V = (x2 + 3)2 dx. D V = (x2 + 3) dx. 00 \u0253 L\u1eddi gi\u1ea3i. 2 Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u00f2n xoay c\u1ea7n t\u00ecm l\u00e0 V = \u03c0 (x2 + 3)2 dx. 0 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 32 (C\u00e2u 13 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho h\u00ecnh ph\u1eb3ng (H) gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng th\u1eb3ng y = x2 + 2, y = 0, x = 1, x = 2. G\u1ecdi V l\u00e0 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u00f2n xoay \u0111\u01b0\u1ee3c t\u1ea1o th\u00e0nh khi quay (H) xung quanh tr\u1ee5c Ox. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? 2 2 A V = \u03c0 (x2 + 2)2 dx. B V = (x2 + 2)2 dx. 1 1 2 2 C V = \u03c0 (x2 + 2) dx. D V = (x2 + 2) dx. 11 \u0253 L\u1eddi gi\u1ea3i. 2 Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u00f2n xoay \u0111\u01b0\u1ee3c t\u1ea1o th\u00e0nh khi quay (H) quanh Ox l\u00e0 V = \u03c0 (x2 + 2)2 dx. 1 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 33 (C\u00e2u 6 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng kh\u00e1c 1. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang v\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c d\u01b0\u01a1ng x, y? Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 315 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc A x = loga x \u2212 loga y. B x loga y loga y = loga x + loga y. C x = loga(x \u2212 y). D loga x = loga x loga y y loga . y \u0253 L\u1eddi gi\u1ea3i. \u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c s\u00e1ch gi\u00e1o khoa loga x = loga x \u2212 loga y . y Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 34 (C\u00e2u 26 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm t\u1eadp x\u00e1c \u0111\u1ecbn\u221ah D c\u1ee7a h\u00e0m s\u1ed1\u221ay = log3(x2 \u2212 4x + 3). A D = (2 \u2212 2; 1) \u222a (3; 2 + 2). B D = (1; 3). \u221a \u221a C D = (\u2212\u221e; 1) \u222a (3; +\u221e). D D = (\u2212\u221e; 2 \u2212 2) \u222a (2 + 2; +\u221e). \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x\u00e1c \u0111\u1ecbnh x2 \u2212 4x + 3 > 0 \u21d4 x \u2208 (\u2212\u221e; 1) \u222a (3; +\u221e) Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 35 (C\u00e2u 24 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho a v\u00e0 b l\u00e0 hai s\u1ed1 th\u1ef1c d\u01b0\u01a1ng th\u1ecfa m\u00e3n a4b = 16. Gi\u00e1 tr\u1ecb c\u1ee7a 4 log2 a + log2 b b\u1eb1ng A 4. B 2. C 16. D 8. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 4 log2 a + log2 b = log2 a4 + log2 b = log2(a4b) = log2 16 = log2 24 = 4. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 36 (C\u00e2u 25 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho a v\u00e0 b l\u00e0 hai s\u1ed1 th\u1ef1c d\u01b0\u01a1ng th\u1ecfa m\u00e3n a3b2 = 32. Gi\u00e1 tr\u1ecb c\u1ee7a 3 log2 a + 2 log2 b b\u1eb1ng A 5. B 2. C 32. D 4. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3: log2 a3b2 = log2 32 \u21d4 3 log2 a + 2 log2 b = 5. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 37 (C\u00e2u 21 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho a v\u00e0 b l\u00e0 hai s\u1ed1 th\u1ef1c d\u01b0\u01a1ng th\u1ecfa m\u00e3n a2b3 = 16. Gi\u00e1 tr\u1ecb c\u1ee7a 2 log2 a + 3 log2 b b\u1eb1ng A 8. B 16. C 4. D 2. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 2 log2 a + 3 log2 b = log2(a2b3) = log2 16 = 4. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 38 (C\u00e2u 3 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). V\u1edbi a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng tu\u1ef3 \u00fd, log5(5a) b\u1eb1ng A 5 + log5 a. B 5 \u2212 log5 a. C 1 + log5 a. D 1 \u2212 log5 a. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log5(5a) = log5 5 + log5 a = 1 + log5 a. Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 316 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG \u0104 C\u00e2u 39 (C\u00e2u 27 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). V\u1edbi a, b l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd th\u1ecfa m\u00e3n log3 a\u22122 log9 b = 2, m\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A a = 9b3. B a = 9b. C a = 6b. D a = 9b2. \u0253 L\u1eddi gi\u1ea3i. = 2 \u21d4 a = 9b. a Ta c\u00f3 log3 a \u2212 2 log9 b = 2 \u21d4 log3 a \u2212 log3 b = 2 \u21d4 log3 b Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 40 (C\u00e2u 18 - M\u0110 103 -\u221aBGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho a > 0 v\u00e0 a = 1, khi \u0111\u00f3 loga a b\u1eb1ng C \u22121. 1 A 2. B \u22122. D . 22 \u0253 L\u1eddi gi\u1ea3i. V\u1edbi a > 0 v\u00e0 a = 1, ta c\u00f3 loga \u221a 11 1 a = loga a 2 = loga a = . 2 2 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 41 (C\u00e2u 12 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). V\u1edbi a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd, log(100a) b\u1eb1ng A 1 \u2212 log a. B 2 + log a. C 2 \u2212 log a. D 1 + log a. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log(100a) = log(100) + log a = 2 + log a. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 42 (C\u00e2u 17 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Cho c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng a, b, v\u1edbi a = 1. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o sau \u0111\u00e2y l\u00e0 kh\u1eb3ng \u0111\u1ecbnh \u0111\u00fang? A loga2(ab) = 1 B loga2(ab) = 2 + 2 loga b. 2 loga b. 1 11 C loga2(ab) = 4 loga b. D loga2(ab) = 2 + 2 loga b. \u0253 L\u1eddi gi\u1ea3i. 11 11 Ta c\u00f3 loga2(ab) = 2 loga(ab) = 2 (1 + loga b) = 2 + 2 loga b, n\u00ean c\u00e2u D \u0111\u00fang. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 43 (C\u00e2u 19 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). \u0110\u1eb7t a = log2 3, b = log5 3. H\u00e3y bi\u1ec3u di\u1ec5n log6 45 theo a v\u00e0 b. 2a2 \u2212 2ab A log6 45 = a + 2ab B log6 45 = . . ab 2a2 \u2212 2ab ab . a + 2ab C log6 45 = . D log6 45 = ab + b ab + b \u0253 L\u1eddi gi\u1ea3i. 1a Ta c\u00f3 b = log3 5 \u21d2 b = log2 3. log3 5 = log2 5. V\u1eady ta \u0111\u01b0a v\u1ec1 c\u01a1 s\u1ed1 2. log2 (32.5) 2a + a 2ab + a log6 45 = log23 + 1 = b = . a+1 ab + b Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 317 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc \u0104 C\u00e2u 44 (C\u00e2u 16 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). V\u1edbi c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng a, b b\u1ea5t k\u00ec. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang ? \u00c5 2a3 \u00e3 \u00c5 2a3 \u00e3 1 b b 3 log2a A log2 = 1 + 3log2a \u2212 log2b. B log2 = 1 + \u2212 log2b. C \u00c5 2a3 \u00e3 D \u00c5 2a3 \u00e3 1 log2 b = 1 + 3log2a + log2b. log2 b = 1 + 3 log2a + log2b. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log2 \u00c5 2a3 \u00e3 = log2 (2a3) \u2212 log2 (b) = log2 (2) + log2 (a3) \u2212 log2 (b) = 1 + 3log2a \u2212 log2b b Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 45 (C\u00e2u 13 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). Cho a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng, a = 1 v\u00e0 P = log \u221a3 a a3. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A P = 1. B P = 1. C P = 9. D P 1 =. 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 P = P = loga1\/3 a3 = 9 loga a = 9. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 46 (C\u00e2u 33 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). \u221a\u221a log \u2026b Cho a, b l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng th\u1ecfa m\u00e3n a = 1, a = b v\u00e0 loga b = 3. T\u00ednh P = \u221a . b a\u221a \u221a\u221a \u221aa A P = \u22125 + 3 3. B P = \u22121 + 3. C P = \u22121 \u2212 3. D P = \u22125 \u2212 3 3. \u0253 L\u1eddi gi\u1ea3i. C\u00e1ch 1: Ph\u01b0\u01a1ng ph\u00e1p t\u1ef1 lu\u1eadn. \u2026b 1 (loga b \u2212 1) 1 \u00c4\u221a \u2212 \u00e4 \u221a \u221a loga \u221aa 2 \u221a 3 1 \u221a3 \u2212 1 = \u22121 \u2212 3. 2 P = b = = = 3\u22122 loga a loga b \u2212 1 1 2 loga b \u2212 1 C\u00e1ch 2: Ph\u01b0\u01a1ng\u221aph\u00e1p tr\u1eafc nghi\u1ec7m. \u221a Ch\u1ecdn a = 2, b = 2 3. B\u1ea5m m\u00e1y t\u00ednh ta \u0111\u01b0\u1ee3c P = \u22121 \u2212 3. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 47 (C\u00e2u 5 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). \u00c53\u00e3 V\u1edbi a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd, log3 a b\u1eb1ng A 1 \u2212 log3 a. B 3 \u2212 log3 a. C 1 D 1 + log3 a. . log3 a \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log3 \u00c53\u00e3 = log3 3 \u2212 log3 a = 1 \u2212 log3 a. a Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 48 (C\u00e2u 12 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). \u0110\u1eb7t log3 2 = a, khi \u0111\u00f3 log16 27 b\u1eb1ng 3a 3 4 4a A . B . C . D . 4 4a 3a 3 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 318 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log16 27 = log24 33 = 3 log2 3 = 3 \u00b7 1 2 = 3 4 4 log3 . 4a Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 49 (C\u00e2u 10 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). V\u1edbi a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd, log2 (a2) b\u1eb1ng 1 1 A 2 + log2 a. B 2 + log2 a. C 2 log2 a. D 2 log2 a. \u0253 L\u1eddi gi\u1ea3i. V\u1edbi a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd, ta c\u00f3 log2 (a2) = 2 log2 a. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 50 (C\u00e2u 20 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). X\u00e9t t\u1ea5t c\u1ea3 c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng a v\u00e0 b th\u1ecfa m\u00e3n log2 a = log8(ab). M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A a = b2. B a3 = b. C a = b. D a2 = b. \u0253 L\u1eddi gi\u1ea3i. log2 a = log8 (ab) \u21d4 log2 a = log23 (ab) 1 \u21d4 log2 a = 3 log2 (ab) \u21d4 3 log2 a = log2 (ab) \u21d4 log2 a3 = log2 (ab) \u21d4 a3 = ab \u21d4 a2 = b. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 51 (C\u00e2u 29 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). X\u00e9t c\u00e1c s\u1ed1 th\u1ef1c a v\u00e0 b th\u1ecfa m\u00e3n log3(3a \u00b7 9b) = log9 3. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A a + 2b = 2. B 4a + 2b = 1. C 4ab = 1. D 2a + 4b = 1. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log3(3a \u00b7 9b) = log9 3 \u21d4 log3(3a \u00b7 32b) = 1 \u21d4 log3 3a+2b = 1 \u21d4 a + 2b = 1 \u21d4 2a + 4b = 1. 2 log9 9 2 log9 9 2 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 52 (C\u00e2u 30 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho a v\u00e0 b l\u00e0 hai s\u1ed1 th\u1ef1c d\u01b0\u01a1ng tho\u1ea3 m\u00e3n 4log2(ab) = 3a. Gi\u00e1 tr\u1ecb c\u1ee7a ab2 b\u1eb1ng A 3. B 6. C 2. D 12. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 4log2(ab) = 2log2(ab) 2 = (ab)2 n\u00ean 4log2(ab) = 3a \u21d4 (ab)2 = 3a \u21d4 ab2 = 3. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 53 (C\u00e2u 30 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Cho a v\u00e0 b l\u00e0 hai s\u1ed1 th\u1ef1c d\u01b0\u01a1ng th\u1ecfa m\u00e3n 9log3(ab) = 4a. Gi\u00e1 tr\u1ecb c\u1ee7a ab2 b\u1eb1ng A 3. B 6. C 2. D 4. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 9log3(ab) = 4a \u21d4 (ab)2 = 4a \u21d4 ab2 = 4. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 319 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc \u0104 C\u00e2u 54 (C\u00e2u 35 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). V\u1edbi a, b l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd th\u1ecfa m\u00e3n log3 a\u22122 log9 b = 3, m\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A a = 27b. B a = 9b. C a = 27b4. D a = 27b2. \u0253 L\u1eddi gi\u1ea3i. = 3 \u21d4 a = 27 \u21d4 a = 27b. a b Ta c\u00f3 log3 a \u2212 2 log9 b = 3 \u21d4 log3 a \u2212 log3 b = 3 \u21d4 log3 b V\u1eady a = 27b. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 55 (C\u00e2u 29 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). V\u1edbi a, b l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd th\u1ecfa m\u00e3n log2 a\u22122 log4 b = 4, m\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A a = 16b2. B a = 8b. C a = 16b. D a = 16b4. \u0253 L\u1eddi gi\u1ea3i. V\u1edbi a, b l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng ta c\u00f3 log2 a \u2212 2 log4 b = 4 \u21d4 log2 a = log2 b + log2 16 \u21d4 log2 a = log2(16b) \u21d4 a = 16b. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 56 (C\u00e2u 37 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). V\u1edbi m\u1ecdi a, b th\u1ecfa m\u00e3n log2 a3 + log2 b = 6, kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A a3b = 64. B a3b = 36. C a3 + b = 64. D a3 + b = 36. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log2 a3 + log2 b = 6 \u21d4 log2 (a3 \u00b7 b) = log2 26 \u21d4 a3b = 26 = 64. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 57 (C\u00e2u 38 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). V\u1edbi m\u1ecdi a, b th\u1ecfa m\u00e3n log2 a3 + log2 b = 8, kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A a3 + b = 64. B a3b = 256. C a3b = 64. D a3 + b = 256. \u0253 L\u1eddi gi\u1ea3i. V\u1edbi a, b > 0, ta c\u00f3 log2 a3 + log2 b = 8 \u21d4 log2 (a3b) = 8 \u21d4 a3b = 28 = 256. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 58 (C\u00e2u 34 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). V\u1edbi m\u1ecdi a, b th\u1ecfa m\u00e3n log2 a3 + log2 b = 7, kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A a3 + b = 49. B a3b = 128. C a3 + b = 128. D a3b = 49. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log2 a3 + log2 b = 7 \u21d4 log2 a3b = 7 \u21d4 a3b = 27 = 128. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 59 (C\u00e2u 19 - M\u0110 104 -\u221aBGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho a > 0 v\u00e0 a = 1, khi \u0111\u00f3 loga 5 a b\u1eb1ng 11 A. B\u2212. C 5. D \u22125. 55 \u221a \u0253 L\u1eddi gi\u1ea3i. 5a Ta c\u00f3 loga = loga a 1 = 1 loga a = 1 5 5 . 5 Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 320 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG \u0104 C\u00e2u 60 (C\u00e2u 37 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). V\u1edbi a > 0, \u0111\u1eb7t log2(2a) = b, khi \u0111\u00f3 log2 (8a4) b\u1eb1ng A 4b + 7. B 4b + 3. C 4b. D 4b \u2212 1. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log2 8a4 = log2 16a4 = log2 (2a)4 = log2(2a)4 \u2212 log2 2 = 4 log2(2a) \u2212 1 = 4b \u2212 1. 2 2 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 61 (C\u00e2u 31 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). V\u1edbi a > 0, \u0111\u1eb7t log2(2a) = b, khi \u0111\u00f3 log2 (4a3) b\u1eb1ng A 3b + 5. B 3b. C 3b + 2. D 3b \u2212 1. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log2(2a) = b \u21d4 1 + log2 a = b suy ra log2 a = b \u2212 1. Khi \u0111\u00f3 log2 (4a3) = log2 4 + log2 a3 = 2 + 3 log2 a = 2 + 3(b \u2212 1) = 3b \u2212 1. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 62 (C\u00e2u 33 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). V\u1edbi a > 0, \u0111\u1eb7t log3 (3a) = b, khi \u0111\u00f3 log3 (9a3) b\u1eb1ng A 3b. B 3b \u2212 1. C 3b + 5. D 3b + 2. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log3 (3a) = b \u21d4 log3 a = b \u2212 1. Suy ra log3 (9a3) = log3 9 + log3 a3 = 2 + 3 log3 a = 2 + 3(b \u2212 1) = 3b \u2212 1. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 63 (C\u00e2u 34 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). V\u1edbi a > 0, \u0111\u1eb7t log3(3a) = b, khi \u0111\u00f3 log3(27a4) b\u1eb1ng A 4b + 3. B 4b. C 4b \u2212 1. D 4b + 7. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log3(3a) = b \u21d4 log3 3 + log3 a = b \u21d4 log3 a = b \u2212 1. M\u1eb7t kh\u00e1c log3(27a4) = log3 27 + log3 a4 = 3 + 4 log3 a = 3 + 4(b \u2212 1) = 4b \u2212 1. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 64 (C\u00e2u 34 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). 1 V\u1edbi a, b l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd v\u00e0 a = 1, log 1 b\u1eb1ng b3 a 1 3 loga b. A 3 loga b. B loga b. C \u22123 loga b. D Ta c\u00f3 log 1 1 = \u2212 loga b\u22123 = 3 loga b. \u0253 L\u1eddi gi\u1ea3i. b3 a 321 Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc \u0104 C\u00e2u 65 (C\u00e2u 31 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). V\u1edbi m\u1ecdi a, b th\u1ecfa m\u00e3n log2 a \u2212 3 log2 b = 2, kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? 4 . A a = 4b3. B a = 3b + 4. C a = 3b + 2. D a= b3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log2 a \u2212 3 log2 b = 2 \u21d4 log2 a = 2 \u21d4 a = 22 \u21d4 a = 4b3. b3 b3 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 66 (C\u00e2u 20 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Cho hai s\u1ed1 th\u1ef1c a v\u00e0 b, v\u1edbi 1 < a < b. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 kh\u1eb3ng \u0111\u1ecbnh \u0111\u00fang? A loga b < 1 < logb a. B 1 < loga b < logb a. C logb a < loga b < 1. D logb a < 1 < loga b. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 1 < a < b \u21d2 \u00ae loga 1 < loga a < loga b \u21d2 \u00ae0 < 1 < loga b \u21d2 logb a < 1 < loga b. logb 1 < logb a < logb b 0 < logb a < 1 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 67 (C\u00e2u 15 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). V\u1edbi a, b l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd v\u00e0 a kh\u00e1c 1, \u0111\u1eb7t P = loga b3 + loga2 b6. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A P = 9 loga b. B P = 27 loga b. C P = 15 loga b. D P = 6 loga b. \u0253 L\u1eddi gi\u1ea3i. P = loga b3 + loga2 b6 = 3 loga b + 1 = 6 loga b. 2 .6 loga b Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 68 (C\u00e2u 29 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho loga b = 2 v\u00e0 loga c = 3. T\u00ednh P = loga (b2c3). A P = 31. B P = 13. C P = 30. D P = 108. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 P = loga (b2c3) = 2 loga b + 3 loga c = 2.2 + 3.3 = 13. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 69 (C\u00e2u 10 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). \u00c5 a2 \u00e3 Cho a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng kh\u00e1c 2. T\u00ednh I = log a . 24 A I= 1 B I = 2. C I = \u22121. D I = \u22122. . 2 2 \u0253 L\u1eddi gi\u1ea3i. a2 a I = log a = 2 log a = 2 (v\u00ec a = 2) 22 22 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 70 (C\u00e2u 8 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng t\u00f9y \u00fd kh\u00e1c 1. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 322 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG A log2 a = loga 2. B log2 a = 1 . C log2 a = 1 D log2 a = \u2212 loga 2. a . log2 loga 2 \u0253 L\u1eddi gi\u1ea3i. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 71 (C\u00e2u 29 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). V\u1edbi m\u1ecdi a, b, x l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng th\u1ecfa m\u00e3n log2 x = 5 log2 a + 3 log2 b, m\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A x = 3a + 5b. B x = 5a + 3b. C x = a5 + b3. D x = a5b3. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log2 x = 5 log2 a + 3 log2 b = log2 a5 + log2 b3 = log2(a5b3) \u21d2 x = a5b3. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 72 (C\u00e2u 31 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). C\u221aho (H) l\u00e0 h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi \u221aparabol y = y 3x2, cung tr\u00f2n c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh y = 4 \u2212 x2 (v\u1edbi \u221a 0 \u2264 x \u2264 2) v\u00e0 tr\u1ee5c ho\u00e0nh (ph\u1ea7n t\u00f4 \u0111\u1eadm trong h\u00ecnh (P1) : y = 3x2 v\u1ebd). Di\u1ec7n t\u00edc\u221ah h\u00ecnh (H) b\u1eb1ng B 4\u03c0 \u2212 \u221a \u221a A 4\u03c0 + 3 3 (P2) : y = 4 \u2212 x2 . . Ox 12 \u221a \u221a6 C 4\u03c0 + 2 3\u22123 D5 3 \u2212 2\u03c0 . . 63 \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m: \u221a\u221a 3x2 = 4 \u2212 x2 \u21d4 3x4 + x2 \u2212 4 = 0 \u21d4 x = 1 (do 0 \u2264 x \u2264 2). Khi \u0111\u00f3 1\u221a 2\u221a S = 3x2dx + 4 \u2212 x2dx = I + J. 01 1\u221a \u221a 1 \u221a 3x2dx = 3x3 3 T\u00ednh I = = . 33 00 x=1\u21d2t= \u03c0 2\u221a \uf8f1 \u03c06 \uf8f2 . T\u00ednh J = 4 \u2212 x2dx : \u0110\u1eb7t x = 2 sin t \u21d2 dx = 2 cos tdt v\u00e0 x = 2 \u21d2 t = \uf8f3 2 1 Khi \u0111\u00f3 \u03c0 \u03c0\u03c0 \u00c51 \u00e3 \u221a 2 22 (1 + cos 2t) dt = 2 t + sin 2t J= 4 \u2212 4 sin2 t.2 cos tdt = 4 cos2 tdt = 2 2 \u03c0 = 2\u03c0 \u2212 3 2 3 . \u03c0 \u03c0 6 6 2 \u03c0\u03c0 66 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 323 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc \u221a \u221a\u221a 3 + 2\u03c0 \u2212 3 4\u03c0 \u2212 3 V\u1eady S = 332 = (\u0111vdt). 6 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 73 (C\u00e2u 36 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho hai h\u00e0m s\u1ed1 f (x) = ax3 + bx2 + cx \u2212 2 v\u00e0 g(x) = dx2 + ex + 2 y (a, b, c, d, e \u2208 R). Bi\u1ebft r\u1eb1ng \u0111\u1ed3 th\u1ecb c\u1ee7a h\u00e0m s\u1ed1 y = f (x) v\u00e0 y = g(x) c\u1eaft nhau t\u1ea1i ba \u0111i\u1ec3m c\u00f3 ho\u00e0nh \u0111\u1ed9 l\u1ea7n l\u01b0\u1ee3t l\u00e0 \u22122; \u22121; 1 (tham kh\u1ea3o h\u00ecnh v\u1ebd). H\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi hai \u0111\u1ed3 th\u1ecb \u0111\u00e3 cho c\u00f3 di\u1ec7n t\u00edch b\u1eb1ng x 1 A 37 B 13 C 9 D 37 \u22122 \u22121 O . . . . 6 2 2 12 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) \u2212 g(x) = ax3 + (b \u2212 d)x2 + (c \u2212 e)x \u2212 4 (1). M\u1eb7t kh\u00e1c ph\u01b0\u01a1ng tr\u00ecnh f (x) \u2212 g(x) = 0 c\u00f3 3 nghi\u1ec7m ph\u00e2n bi\u1ec7t x = \u22122, x = \u22121, x = 1 n\u00ean f (x) \u2212 g(x) = 0 \u21d4 (x + 2)(x + 1)(x \u2212 1) = 0 \u21d4 x3 + 2x2 \u2212 x \u2212 2 = 0 (2). T\u1eeb (1) v\u00e0 (2), suy ra f (x) \u2212 g(x) = 2x3 + 4x2 \u2212 2x \u2212 4. Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 \u22121 1 S= (2x3 + 4x2 \u2212 2x \u2212 4) dx \u2212 (2x3 + 4x2 \u2212 2x \u2212 4) dx = 37 . 6 \u22122 \u22121 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 74 (C\u00e2u 43 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho hai h\u00e0m s\u1ed1 f (x) = ax3 + bx2 + cx \u2212 1 v\u00e0 g(x) = dx2 + ex + y 1 (a, b, c, d, e \u2208 R). Bi\u1ebft r\u1eb1ng \u0111\u1ed3 th\u1ecb c\u1ee7a h\u00e0m s\u1ed1 y = f (x) v\u00e0 2 y = g(x) c\u1eaft nhau t\u1ea1i ba \u0111i\u1ec3m c\u00f3 ho\u00e0nh \u0111\u1ed9 l\u1ea7n l\u01b0\u1ee3t \u22123; \u22121; 2 (tham kh\u1ea3o h\u00ecnh v\u1ebd). H\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi hai \u0111\u1ed3 th\u1ecb \u0111\u00e3 cho c\u00f3 di\u1ec7n t\u00edch b\u1eb1ng A 253 B 125 C 253 D 125 . . . . 12 12 48 48 2x \u22123 \u22121 O \u0253 L\u1eddi gi\u1ea3i. Do (C) : y = f (x) v\u00e0 (C ) : y = g(x) c\u1eaft nhau t\u1ea1i 3 \u0111i\u1ec3m ph\u00e2n bi\u1ec7t c\u00f3 ho\u00e0nh \u0111\u1ed9 \u22123; \u22121; 2 n\u00ean f (x) \u2212 g(x) = A(x + 3)(x + 1)(x \u2212 2). Do f (0) \u2212 g(0) = \u22123 n\u00ean \u22126A = \u22123 \u21d2 A = \u22121. 2 24 1 1 (x3 T\u1eeb \u0111\u00f3 f (x) \u2212 g(x) = (x + 3)(x + 1)(x \u2212 2) = + 2x2 \u2212 5x \u2212 6). 44 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 324 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG V\u1eady di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 \u22121 2 S= 1 (x3 + 2x2 \u2212 5x \u2212 6) dx + 1 (x3 + 2x2 \u2212 5x \u2212 6) dx 253 =. 4 4 48 \u22123 \u22121 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 75 (C\u00e2u 29 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho h\u00e0m s\u1ed1 f (x) li\u00ean t\u1ee5c tr\u00ean R. G\u1ecdi S l\u00e0 di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = f (x), y = 0, x = \u22121 v\u00e0 x = 4 (nh\u01b0 h\u00ecnh v\u1ebd b\u00ean d\u01b0\u1edbi). M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? y y = f (x) 1 4x \u22121 O 14 14 A S = \u2212 f (x) dx + f (x) dx. B S = f (x) dx \u2212 f (x) dx. \u22121 1 \u22121 1 14 1 4 C S = f (x) dx + f (x) dx. D S = \u2212 f (x) dx \u2212 f (x) dx. \u22121 1 \u22121 1 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 h\u00e0m s\u1ed1 f (x) \u2265 0\u2200x \u2208 [\u22121; 1]; f (x) \u2264 0\u2200x \u2208 [1; 4], n\u00ean 4 1 4 14 S = |f (x)| dx = |f (x)| dx + |f (x)| dx = f (x) dx \u2212 f (x) dx. \u22121 \u22121 1 \u22121 1 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 76 (C\u00e2u 45 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho \u0111\u01b0\u1eddng th\u1eb3ng y = x v\u00e0 parabol y = 1 x2 + a (a l\u00e0 tham s\u1ed1 th\u1ef1c y x2 2 y= +a S1 d\u01b0\u01a1ng). G\u1ecdi S1 v\u00e0 S2 l\u1ea7n l\u01b0\u1ee3t l\u00e0 di\u1ec7n t\u00edch c\u1ee7a hai h\u00ecnh ph\u1eb3ng \u0111\u01b0\u1ee3c g\u1ea1ch O 2 y=x ch\u00e9o trong h\u00ecnh v\u1ebd d\u01b0\u1edbi \u0111\u00e2y. Khi S1 = S2 th\u00ec a thu\u1ed9c kho\u1ea3ng n\u00e0o d\u01b0\u1edbi S2 \u0111\u00e2y? x \u00c5 3 1 \u00e3 \u00c5 1 \u00e3 \u00c5 1 2 \u00e3 \u00c5 2 3 \u00e3 . 0; . . . A ; B C ; D ; 72 3 35 57 \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m: 1 x2 + a = x \u21d4 x2 \u2212 2x + 2a = 0 (1) \uf8f1\u2206 > 0 \uf8f11 \u2212 2a > 0 2 \uf8f4\uf8f4 \u21d4 0 < a < 1 \uf8f2\uf8f2 . 2 Ph\u01b0\u01a1ng tr\u00ecnh tr\u00ean c\u00f3 2 nghi\u1ec7m d\u01b0\u01a1ng ph\u00e2n bi\u1ec7t \u21d4 S > 0 \u21d4 2 > 0 \uf8f3\uf8f4P > 0 \uf8f3\uf8f42a > 0 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 325 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc 1 Khi 0 < a < 2 ph\u01b0\u01a1ng tr\u00ecnh (1) c\u00f3 hai nghi\u1ec7m d\u01b0\u01a1ng ph\u00e2n bi\u1ec7t x1 < x2, x1 x2 \u00c5 \u00c5 1 x2 \u00e3 \u2212 \u00e3 S1 = S2 \u21d4 + a \u2212 x dx = 1 x2 \u2212 a + x dx 2 2 0 x1 \u21d4 1 x13 + ax1 \u2212 1 x21 = \u2212 1 x32 \u2212 ax2 + 1 x22 + 1 x31 + ax1 \u2212 1 x21 6 2 6 2 6 2 \u21d4 \u2212 1 x32 \u2212 ax2 + 1 x22 = 0 \u21d4 x22 + 6a \u2212 3x2 = 0. 6 2 T\u1eeb (1) suy ra 2a = \u2212x22 + 2x2 \uf8eex2 = 0 (lo\u1ea1i) 3 \u00c51 2\u00e3 ; . Th\u1ebf v\u00e0o (2) ta \u0111\u01b0\u1ee3c: 2x22 \u2212 3x2 = 0 \u21d4 \uf8f0 3 \u21d2 a = = 0, 375 \u2208 x2 = 2 8 35 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 77 (C\u00e2u 43 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho \u0111\u01b0\u1eddng th\u1eb3ng y = 3 v\u00e0 parabol y = 1 x2 + y y = 1 x2 + a x 2 42 3 a, (a l\u00e0 tham s\u1ed1 th\u1ef1c d\u01b0\u01a1ng). G\u1ecdi S1, S2 l\u1ea7n y= x S1 4 l\u01b0\u1ee3t l\u00e0 di\u1ec7n t\u00edch c\u1ee7a hai h\u00ecnh ph\u1eb3ng \u0111\u01b0\u1ee3c g\u1ea1ch S2 x ch\u00e9o trong h\u00ecnh v\u1ebd b\u00ean. Khi S1 = S2 th\u00ec a thu\u1ed9c O kho\u1ea3ng n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A \u00c51 9 \u00e3 B \u00c53 7\u00e3 ; . ; . 4 32 16 32 \u00c5 3 \u00e3 \u00c5 7 1\u00e3 0; . ;. C 16 D 32 4 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m l\u00e0 1 x2 \u2212 3 + a = 0 \u21d4 2x2 \u2212 3x + 4a = 0. x 24 \uf8f13 (\u2217) Theo \u0111\u1ec1 b\u00e0i ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m 0 < x1 < x2 th\u1ecfa m\u00e3n \uf8f2x1 + x2 = 2 (\u2217\u2217). \uf8f3x1x2 = 2a T\u1eeb \u0111\u1ed3 th\u1ecb \u0111\u1ec1 b\u00e0i, ta c\u00f3 x1 x2 \u00c5 1 x2 \u00e3 \u00c5 1 x2 \u00e3 S1 \u2212 S2 = 0 \u21d4 \u2212 3 + a dx + \u2212 3 + a dx = 0 x x 24 24 0 x1 x2 3\u00e3 dx = 0 \u21d4 \u00c5 1 x3 \u2212 3 \u00e3 x2 x+a ax \u21d4 \u00c5 1 x2 \u2212 68 x2 + =0 24 0 0 \u21d4 1 x32 \u2212 3 x22 + ax2 = 0 \u21d4 a = \u2212 x22 + 3x2 . (\u2217 \u2217 \u2217) 6 8 6 8 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 326 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG 3 T\u1eeb (\u2217) ta suy ra x1 = 2 \u2212 x2, thay v\u00e0o (\u2217\u2217) ta \u0111\u01b0\u1ee3c \u00c53 \u00e3 = \u2212 x22 + 3x2 \u21d4 2x22 \u2212 3x2 = 0 \u21d2 x2 = 9 \u21d2 a = 27 2 \u2212 x2 x2 3 4 3 4 8 . 128 \u00c5 3 7 \u00e3 . V\u1eady a \u2208 ; 16 32 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 78 (C\u00e2u 46 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho hai h\u00e0m s\u1ed1 f (x) = ax4 + bx3 + cx2 + 2x v\u00e0 g(x) = mx3 + nx2 \u2212 x v\u1edbi a, b, c, m, n \u2208 R. Bi\u1ebft h\u00e0m s\u1ed1 y = f (x) \u2212 g(x) c\u00f3 ba \u0111i\u1ec3m c\u1ef1c tr\u1ecb l\u00e0 \u22121, 2 v\u00e0 3. Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi hai \u0111\u01b0\u1eddng y = f (x) v\u00e0 y = g (x) b\u1eb1ng A 71 B 32 C 16 D 71 . . . . 6 3 3 12 \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t h(x) = f (x) \u2212 g(x) = ax4 + (b \u2212 m)x3 + (c \u2212 n)x2 + 3x. Suy ra h (x) = 4ax3 + 3(b \u2212 m)x2 + 2(c \u2212 n)x + 3.(1) V\u00ec h\u00e0m s\u1ed1 h(x) c\u00f3 ba \u0111i\u1ec3m c\u1ef1c tr\u1ecb \u22121, 2, 3 n\u00ean ph\u01b0\u01a1ng tr\u00ecnh h (x) = 0 c\u00f3 ba nghi\u1ec7m ph\u00e2n bi\u1ec7t \u22121, 2 v\u00e0 3. Suy ra, h (x) c\u00f3 d\u1ea1ng h (x) = r(x + 1)(x \u2212 2)(x \u2212 3) (2). T\u1eeb (1), l\u1ea5y x = 0, suy ra h (0) = 3. Do \u0111\u00f3, t\u1eeb (2) suy ra 3 = r \u00b76 \u21d4 r = 1 . 2 1 V\u1eady h (x) = (x + 1)(x \u2212 2)(x \u2212 3). 2 Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 33 S= |h (x)| dx = 1 |(x + 1)(x \u2212 2)(x \u2212 3)| dx = 71 . 2 12 \u22121 \u22121 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 79 (C\u00e2u 44 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = ax4 + bx3 + cx2 + 3x v\u00e0 g(x) = mx3 + mx2 \u2212 x v\u1edbi a, b, c, m, n \u2208 R. Bi\u1ebft h\u00e0m s\u1ed1 y = f (x) \u2212 g(x) c\u00f3 ba \u0111i\u1ec3m c\u1ef1c tr\u1ecb l\u00e0 \u22121; 2; 3. Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi \u0111\u01b0\u1eddng y = f (x) v\u00e0 y = g (x) b\u1eb1ng A 32 B 71 C 71 D 64 . . . . 3 9 6 9 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) = 4ax3 + 3bx2 + 2cx + 3, g (x) = 3mx2 + 2nx \u2212 1. Do h\u00e0m s\u1ed1 y = f (x) \u2212 g(x) c\u00f3 ba \u0111i\u1ec3m c\u1ef1c tr\u1ecb l\u00e0 \u22121; 2; 3 n\u00ean ta suy ra a = 0 v\u00e0 f (x) \u2212 g (x) = 4ax3 + (3b \u2212 3m)x2 + (2c \u2212 2n)x + 4 = 4a(x + 1)(x \u2212 2)(x \u2212 3). L\u1ea1i c\u00f3 f (0) \u2212 g (0) = 24a = 4 \u21d2 a = 1 Suy ra f (x) \u2212 g (x) = 2 + 1)(x \u2212 2)(x \u2212 3). . (x 63 V\u1eady di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 3 S= 2 (x + 1)(x \u2212 2)(x \u2212 3) 71 dx = . 39 \u22121 Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 327 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc \u0104 C\u00e2u 80 (C\u00e2u 47 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho hai h\u00e0m s\u1ed1 f (x) = ax4 + bx3 + cx2 + x v\u00e0 g(x) = mx3 + nx2 \u2212 2x; v\u1edbi a, b, c, m, n \u2208 R. Bi\u1ebft h\u00e0m s\u1ed1 y = f (x) \u2212 g(x) c\u00f3 ba \u0111i\u1ec3m c\u1ef1c tr\u1ecb l\u00e0 \u22121, 2 v\u00e0 3. Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi hai \u0111\u01b0\u1eddng y = f (x) v\u00e0 y = g (x) b\u1eb1ng A 32 B 16 C 71 D 71 . . . . 3 3 12 6 \u0253 L\u1eddi gi\u1ea3i. X\u00e9t h\u00e0m s\u1ed1 y = f (x) \u2212 g(x), ta c\u00f3 y = f (x) \u2212 g (x) = 4ax3 + 3bx2 + 2cx + 1 \u2212 (3mx2 + 2nx \u2212 2). = 4ax3 + 3(b \u2212 m)x2 + 2(c \u2212 n)x + 3. V\u00ec h\u00e0m s\u1ed1 c\u00f3 ba \u0111i\u1ec3m c\u1ef1c tr\u1ecb l\u00e0 \u22121, 2 v\u00e0 3 n\u00ean 4ax3 + 3(b \u2212 m)x2 + 2(c \u2212 n)x + 3 = 4a(x + 1)(x \u2212 2)(x \u2212 3) \u21d4 4ax3 + 3(b \u2212 m)x2 + 2(c \u2212 n)x + 3 = 4ax3 \u2212 16ax2 + 4ax + 24a. \uf8f13(b \u2212 m) = \u221216a \uf8f13(b \u2212 m) = \u22122 \uf8f4 \uf8f4 Suy ra \uf8f2 \u21d4 \uf8f4 \u2212 n) = 1 \uf8f4 2 2(c \u2212 n) = 4a \uf8f4 \uf8f22(c \uf8f4\uf8f33 = 24a \uf8f4 = 1 \uf8f4 . \uf8f3\uf8f4\uf8f4a 8 Do \u0111\u00f3 f (x) \u2212 g (x) = 1 x3 \u2212 2x2 + 1 + 3. x 22 Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng c\u1ea7n t\u00ecm 33 S = |f (x) \u2212 g (x)| dx = 1 x3 \u2212 2x2 + 1 + 3 71 x dx = . 22 12 \u22121 \u22121 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 81 (C\u00e2u 42 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho hai h\u00e0m s\u1ed1 f (x) = ax4 + bx3 + cx2 + 2x v\u00e0 g(x) = mx3 + nx2 \u2212 2x v\u1edbi a, b, c, m, n \u2208 R. Bi\u1ebft r\u1eb1ng h\u00e0m s\u1ed1 y = f (x) \u2212 g(x) c\u00f3 ba \u0111i\u1ec3m c\u1ef1c tr\u1ecb l\u00e0 \u22121, 2 v\u00e0 3. Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi hai \u0111\u01b0\u1eddng y = f (x) v\u00e0 y = g (x) b\u1eb1ng A 32 B 71 C 71 D 64 . . . . 3 9 6 9 \u0253 L\u1eddi gi\u1ea3i. H\u00e0m s\u1ed1 y = f (x) \u2212 g(x) l\u00e0 h\u00e0m \u0111a th\u1ee9c n\u00ean x\u00e1c \u0111\u1ecbnh v\u00e0 li\u00ean t\u1ee5c tr\u00ean R, c\u00f3 y = f (x) \u2212 g (x) = 4ax3 + 3bx2 + 2cx + 2 \u2212 3mx2 + 2nx \u2212 2 = 4ax3 + 3(b \u2212 m)x2 + 2(c \u2212 n)x + 4. V\u00ec h\u00e0m s\u1ed1 y = f (x) \u2212 g(x) c\u00f3 ba \u0111i\u1ec3m c\u1ef1c tr\u1ecb l\u00e0 \u22121, 2 v\u00e0 3 n\u00ean ch\u00fang l\u00e0 nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh f (x) \u2212 g (x) = 0, do \u0111\u00f3 ta c\u00f3 \uf8f12 \uf8f4\uf8f44a = \uf8f1(\u22121) \u00b7 4a + 3(b \u2212 m) + (\u22121) \u00b7 2(c \u2212 n) + 4 = 0 \uf8f4 3 \uf8f4 \uf8f4 \u21d4 \uf8f4 \u2212 m) = \u2212 8 \uf8f2 \uf8f2 8 \u00b7 4a + 4 \u00b7 3(b \u2212 m) + 2 \u00b7 2(c \u2212 n) + 4 = 0 3(b \uf8f43 \uf8f3\uf8f427 \u00b7 4a + 9 \u00b7 3(b \u2212 m) + 3 \u00b7 2(c \u2212 n) + 4 = 0 \uf8f4 \uf8f4 2 \uf8f3\uf8f4\uf8f42(c \u2212 n) = . 3 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 328 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG Suy ra f (x) \u2212 g (x) = 2 x3 \u2212 8 x2 + 2 + 4. x 333 V\u1eady di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u1ed3 th\u1ecb y = f (x) v\u00e0 y = g (x) l\u00e0 3 23 S = |f (x) \u2212 g (x)| dx = |f (x) \u2212 g (x)| dx + |f (x) \u2212 g (x)| dx \u22121 \u22121 2 23 = \u00c5 2 x3 \u2212 8 x2 + 2 + \u00e3 dx + \u00c5 2 x3 \u2212 8 x2 + 2 + \u00e3 dx x 4 x 4 333 333 \u22121 2 \u00c5 1 x4 23 8 x3 1 x2 \u00e3 \u00c5 1 x4 8 x3 1 x2 \u00e3 = \u2212 + + 4x + \u2212 + + 4x 693 693 \u22121 2 = 44 47 + 9 \u2212 44 71 + =. 9 18 2 9 9 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 82 (C\u00e2u 41 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Bi\u1ebft F (x) v\u00e0 G(x) l\u00e0 hai nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) tr\u00ean R v\u00e0 3 f (x) dx = F (3) \u2212 G(0) + a (a > 0). G\u1ecdi S l\u00e0 di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng 0 y = F (x), y = G(x), x = 0 v\u00e0 x = 3. Khi S = 15 th\u00ec a b\u1eb1ng? A 15. B 12. C 18. D 5. \u0253 L\u1eddi gi\u1ea3i. Gi\u1ea3 thi\u1ebft F (x), G(x) \u0111\u1ec1u l\u00e0 nguy\u00ean h\u00e0m c\u1ee7a f (x) n\u00ean ta c\u00f3 F (x) = G(x) + C \u21d2 F (0) = G(0) + C. 33 Ta c\u00f3 f (x) dx = F (x) = F (3) \u2212 F (0) = F (3) \u2212 (G(0) + C) = F (3) \u2212 G(0) \u2212 C. 00 3 Theo gi\u1ea3 thi\u1ebft f (x) dx = F (3) \u2212 G(0) + a n\u00ean C = \u2212a. 0 Suy ra F (x) = G(x) \u2212 a \u21d4 F (x) \u2212 G(x) = \u2212a. 33 3 0 Ta c\u00f3 S = |F (x) \u2212 G(x)| dx = | \u2212 a| dx = ax = 3a. 00 M\u00e0 S = 15 n\u00ean ta c\u00f3 a = 5. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 83 (C\u00e2u 41 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). 5 Bi\u1ebft F (x) v\u00e0 G(x) l\u00e0 hai nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) tr\u00ean R v\u00e0 f (x) dx = F (5) \u2212 G(0) + a 0 (a > 0). G\u1ecdi S l\u00e0 di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = F (x), y = G(x), x = 0 v\u00e0 x = 5. Khi S = 20 th\u00ec a b\u1eb1ng A 4. B 15. C 25. D 20. \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t G(x) = F (x) + C (v\u1edbi C l\u00e0 h\u1eb1ng s\u1ed1). 5 Ta c\u00f3 f (x) dx = F (5) \u2212 F (0) = F (5) \u2212 G(0) \u2212 C = F (5) \u2212 G(0) + C. 0 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 329 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc Suy ra C = a. Do \u0111\u00f3 5 55 S = |F (x) \u2212 G(x)| dx = |a| dx = a dx = 5a. 0 00 Theo gi\u1ea3 thi\u1ebft S = 20 \u21d4 5a = 20 \u21d4 a = 4. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 84 (C\u00e2u 39 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Bi\u1ebft F (x); G(x) l\u00e0 hai nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) tr\u00ean R v\u00e0 4 f (x)dx = F (4) \u2212 G(0) + a(a > 0). 0 G\u1ecdi S l\u00e0 di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = F (x); y = G(x); x = 0; x = 4. Khi S = 8 th\u00ec a b\u1eb1ng A 8. B 4. C 12. D 2. \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t F (x) = G(x) + c. 4 T\u1eeb gi\u1ea3 thi\u1ebft suy ra S = |F (x) \u2212 G(x)| dx = 8 \u21d2 |F (x) \u2212 G(x)| = 2 hay |c| = 2. 0 4 Ta c\u00f3 f (x)dx = F (4) \u2212 G(0) + a \u21d4 F (4) \u2212 F (0) = F (4) \u2212 G(0) + a \u21d4 \u2212G(0) \u2212 c = \u2212G(0) + a \u21d4 a = \u2212c 0\u21d2 a = \u00b12. M\u00e0 a > 0 \u21d2 a = 2 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 85 (C\u00e2u 28 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). \u00d4ng An c\u00f3 m\u1ed9t m\u1ea3nh v\u01b0\u1eddn h\u00ecnh Elip c\u00f3 \u0111\u1ed9 d\u00e0i tr\u1ee5c l\u1edbn b\u1eb1ng 16m v\u00e0 \u0111\u1ed9 d\u00e0i tr\u1ee5c b\u00e9 b\u1eb1ng10m. \u00d4ng mu\u1ed1n tr\u1ed3ng hoa tr\u00ean 8m m\u1ed9t d\u1ea3i \u0111\u1ea5t r\u1ed9ng 8m v\u00e0 nh\u1eadn tr\u1ee5c b\u00e9 c\u1ee7a elip l\u00e0m tr\u1ee5c \u0111\u1ed1i x\u1ee9ng (nh\u01b0 h\u00ecnh v\u1ebd). Bi\u1ebft kinh ph\u00ed \u0111\u1ec3 tr\u1ed3ng hoa l\u00e0 100.000 \u0111\u1ed3ng\/1m2. H\u1ecfi \u00f4ng An c\u1ea7n bao nhi\u00eau ti\u1ec1n \u0111\u1ec3 tr\u1ed3ng hoa tr\u00ean d\u1ea3i \u0111\u1ea5t \u0111\u00f3? (S\u1ed1 ti\u1ec1n \u0111\u01b0\u1ee3c l\u00e0m tr\u00f2n \u0111\u1ebfn h\u00e0ng ngh\u00ecn). A 7.862.000 \u0111\u1ed3ng. B 7.653.000 \u0111\u1ed3ng. C 7.128.000 \u0111\u1ed3ng. D 7.826.000 \u0111\u1ed3ng. \u0253 L\u1eddi gi\u1ea3i. X\u00e9t h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 Oxy \u0111\u1eb7t g\u1ed1c t\u1ecda \u0111\u1ed9 v\u00e0o t\u00e2m c\u1ee7a khu v\u01b0\u1eddn, khi \u0111\u00f3 khu v\u01b0\u1eddn c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x2 y2 + = 1. 64 25 \u2026 x2 Ph\u1ea7n \u0111\u1ed3 th\u1ecb ph\u1ea7n ph\u00eda tr\u00ean tr\u1ee5c Ox c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 y = f (x) = 5 1 \u2212 . 64 4\u00a0 Do v\u1eady di\u1ec7n t\u00edch c\u1ee7a d\u1ea3i \u0111\u1ea5t l\u00e0 S = 2 5 1 \u2212 x2 dx. 64 \u22124 \u0110\u1eb7t x = 8 sin t \u2212\u03c0 t \u03c0 \u21d2 dx = 8 cos t dt v\u00e0 cos t 2 \u03c02 0. \u0110\u1ed5i c\u1eadn: x = \u22124 \u21d2 t = \u2212 ; x = 4 \u21d2 t = \u03c0 66 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 330 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG \u03c0\u03c0 66 \u03c0 \u221a 20 3 \u21d2 S = 80 cos2 t dt = 40 \u00c5 sin 2t \u00e3 6 = 40\u03c0 + (m2). (1 + cos 2t) dt = 40 t + 2 \u03c0 3 \u03c0 \u03c0 \u2212 6 6 6 \u2212 \u2212 Do \u0111\u00f3, s\u1ed1 ti\u1ec1n c\u1ea7n d\u00f9ng l\u00e0 100.000S \u2248 7.653.000 \u0111\u1ed3ng. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 86 (C\u00e2u 28 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). K\u00ed hi\u1ec7u (H) l\u00e0 h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = 2(x \u2212 1)ex, tr\u1ee5c tung v\u00e0 tr\u1ee5c ho\u00e0nh. T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i tr\u00f2n xoay thu \u0111\u01b0\u1ee3c khi quay h\u00ecnh (H) xung quanh tr\u1ee5c Ox. A V = 4 \u2212 2e. B V = (4 \u2212 2e)\u03c0. C V = e2 \u2212 5. D V = (e2 \u2212 5)\u03c0. \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m c\u1ee7a \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = 2(x \u2212 1)ex v\u00e0 tr\u1ee5c ho\u00e0nh l\u00e0 2(x \u2212 1)ex = 0 \u21d4 x = 1 Th\u1ec3 t\u00edch V c\u1ee7a kh\u1ed1i tr\u00f2n xoay thu \u0111\u01b0\u1ee3c khi quay h\u00ecnh (H) xung quanh tr\u1ee5c Ox l\u00e0 11 V = [2(x \u2212 1)ex]2 dx = 4 (x \u2212 1)2e2x dx 00 1 X\u00e9t t\u00edch ph\u00e2n I = (x \u2212 1)2e2x dx 0 \u00aeu = (x \u2212 1)2 \uf8f1 du = 2(x \u2212 1) dx \uf8f2 \u0110\u1eb7t dv = e2x dx \u21d2 \uf8f3v = 1 e2x , 2 1 1 Ta c\u00f3: I = 1 (x \u2212 1)2e2x 1 (x \u2212 1)e2x dx = \u2212 1 \u2212 (x \u2212 1)e2x dx \u2212 20 2 00 \u00aeu1 = (x \u2212 1) \uf8f1 du1 = dx dv1 = e2x dx \uf8f2 \u0110\u1eb7t \u21d2 \uf8f3v1 = 1 e2x , 2 \u00d1 1 \u00e9 dx Do \u0111\u00f3 I = \u22121 \u2212 1 1 1 \u22121 \u2212 \u00c71 1 e2x 1\u00e5 \u22121 \u00c51 e2 1\u00e3 (x \u2212 1)e2x \u2212 e2x = \u2212 = \u2212 \u2212 + = 22 02 2 24 0 2 244 0 e2 \u2212 5 4 V\u1eady V = 4I = 4 \u00b7 e2 \u2212 5 = e2 \u2212 5. 4 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 87 (C\u00e2u 34 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ednh th\u1ec3 t\u00edch V c\u1ee7a ph\u1ea7n v\u1eadt th\u1ec3 gi\u1edbi h\u1ea1n b\u1edfi hai m\u1eb7t ph\u1eb3ng x = 1 v\u00e0 x = 3, bi\u1ebft r\u1eb1ng khi c\u1eaft v\u1eadt th\u1ec3 b\u1edfi m\u1eb7t ph\u1eb3ng t\u00f9y \u00fd vu\u00f4ng g\u00f3c v\u1edbi tr\u1ee5c Ox t\u1ea1\u221ai \u0111i\u1ec3m c\u00f3 ho\u00e0nh \u0111\u1ed9 x (1 x 3) th\u00ec \u0111\u01b0\u1ee3c thi\u1ebft di\u1ec7n l\u00e0 m\u1ed9t h\u00ecnh ch\u1eef nh\u1eadt c\u00f3 hai c\u1ea1nh l\u00e0 3x v\u00e0 3x2 \u2212 2. \u221a 124\u03c0 A V = 32 + 2 15. B V =. 3 124 \u00c4 \u221a\u00e4 C . D V = 32 + 2 15 \u03c0. V = 3 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 331 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc \u221a Di\u1ec7n t\u00edch thi\u1ebft di\u1ec7n l\u00e0 S(x) = 3x 3x2 \u2212 2. 3 3\u221a Suy ra th\u1ec3 t\u00edch v\u1eadt th\u1ec3 t\u1ea1o th\u00e0nh l\u00e0: V = S(x) dx = 3x 3x2 \u2212 2 dx. 1 1 124 S\u1eed d\u1ee5ng MTCT ta \u0111\u01b0\u1ee3c : V = . 3 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 88 (C\u00e2u 27 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). M\u1ed9t ch\u1ea5t \u0111i\u1ec3m A xu\u1ea5t ph\u00e1t t\u1eeb O, chuy\u1ec3n \u0111\u1ed9ng th\u1eb3ng v\u1edbi v\u1eadn t\u1ed1c bi\u1ebfn thi\u00ean theo th\u1eddi gian b\u1edfi quy lu\u1eadt v(t) = 1 t2 + 13 (m\/s), trong \u0111\u00f3 t (gi\u00e2y) l\u00e0 kho\u1ea3ng th\u1eddi gian t\u00ednh t\u1eeb l\u00fac A b\u1eaft \u0111\u1ea7u t 100 30 chuy\u1ec3n \u0111\u1ed9ng. T\u1eeb tr\u1ea1ng th\u00e1i ngh\u1ec9, m\u1ed9t ch\u1ea5t \u0111i\u1ec3m B c\u0169ng xu\u1ea5t ph\u00e1t t\u1eeb O, chuy\u1ec3n \u0111\u1ed9ng th\u1eb3ng c\u00f9ng h\u01b0\u1edbng v\u1edbi A nh\u01b0ng ch\u1eadm h\u01a1n 10 gi\u00e2y so v\u1edbi A v\u00e0 c\u00f3 gia t\u1ed1c b\u1eb1ng a (m\/s2) (a l\u00e0 h\u1eb1ng s\u1ed1). Sau khi B xu\u1ea5t ph\u00e1t \u0111\u01b0\u1ee3c 15 gi\u00e2y th\u00ec \u0111u\u1ed5i k\u1ecbp A. V\u1eadn t\u1ed1c c\u1ee7a B t\u1ea1i th\u1eddi \u0111i\u1ec3m \u0111u\u1ed5i k\u1ecbp A b\u1eb1ng A 15 (m\/s). B 9 (m\/s). C 42 (m\/s). D 25 (m\/s). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 vB(t) = a dt = at + C. Do vB(0) = 0 n\u00ean C = 0 \u21d2 vB(t) = at. Qu\u00e3ng \u0111\u01b0\u1eddng ch\u1ea5t \u0111i\u1ec3m A \u0111i \u0111\u01b0\u1ee3c trong 25 gi\u00e2y l\u00e0 25 25 375 =. \u00c5 1 13 \u00e3 \u00c5 1 13 \u00e3 t t2 02 SA = t2 + dt = t3 + 100 30 300 60 0 Qu\u00e3ng \u0111\u01b0\u1eddng ch\u1ea5t \u0111i\u1ec3m B \u0111i \u0111\u01b0\u1ee3c trong 15 gi\u00e2y l\u00e0 15 at2 15 225a SB = at dt = =. 20 2 0 Ta c\u00f3 375 = 225a \u21d4a = 5 . 22 3 5 V\u1eadn t\u1ed1c c\u1ee7a B t\u1ea1i th\u1eddi \u0111i\u1ec3m \u0111u\u1ed5i k\u1ecbp A l\u00e0 vB (15) = 3 \u00b7 15 = 25 (m\/s). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 89 (C\u00e2u 27 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). M\u1ed9t ch\u1ea5t \u0111i\u1ec3m A xu\u1ea5t ph\u00e1t t\u1eeb O, chuy\u1ec3n \u0111\u1ed9ng th\u1eb3ng v\u1edbi v\u1eadn t\u1ed1c bi\u1ebfn thi\u00ean theo th\u1eddi gian b\u1edfi quy lu\u1eadt v (t) = 1 t2 + 58 (m\/s), trong \u0111\u00f3 t (gi\u00e2y) l\u00e0 kho\u1ea3ng th\u1eddi gian t\u00ednh t\u1eeb l\u00fac A b\u1eaft \u0111\u1ea7u t 120 45 chuy\u1ec3n \u0111\u1ed9ng. T\u1eeb tr\u1ea1ng th\u00e1i ngh\u1ec9, m\u1ed9t ch\u1ea5t \u0111i\u1ec3m B c\u0169ng xu\u1ea5t ph\u00e1t t\u1eeb O, chuy\u1ec3n \u0111\u1ed9ng th\u1eb3ng c\u00f9ng h\u01b0\u1edbng v\u1edbi A nh\u01b0ng ch\u1eadm h\u01a1n 3 gi\u00e2y so v\u1edbi A v\u00e0 c\u00f3 gi\u00e1 t\u1ed1c b\u1eb1ng a (m\/s2) ( a l\u00e0 h\u1eb1ng s\u1ed1). Sau khi B xu\u1ea5t ph\u00e1t \u0111\u01b0\u1ee3c 15 gi\u00e2y th\u00ec \u0111u\u1ed5i k\u1ecbp A. V\u1eadn t\u1ed1c c\u1ee7a B t\u1ea1i th\u1eddi \u0111i\u1ec3m \u0111u\u1ed5i k\u1ecbp A b\u1eb1ng A 25 (m\/s). B 36 (m\/s). C 30 (m\/s). D 21 (m\/s). \u0253 L\u1eddi gi\u1ea3i. Th\u1eddi \u0111i\u1ec3m ch\u1ea5t \u0111i\u1ec3m B \u0111u\u1ed5i k\u1ecbp ch\u1ea5t \u0111i\u1ec3m A th\u00ec ch\u1ea5t \u0111i\u1ec3m B \u0111i \u0111\u01b0\u1ee3c 15 gi\u00e2y, ch\u1ea5t \u0111i\u1ec3m A \u0111i \u0111\u01b0\u1ee3c 18 gi\u00e2y. Bi\u1ec3u th\u1ee9c v\u1eadn t\u1ed1c c\u1ee7a ch\u1ea5t \u0111i\u1ec3m B c\u00f3 d\u1ea1ng vB(t) = a dt = at + C m\u00e0 vB(0) = 0 \u21d2 vB(t) = at. Do t\u1eeb l\u00fac ch\u1ea5t \u0111i\u1ec3m A b\u1eaft \u0111\u1ea7u chuy\u1ec3n \u0111\u1ed9ng cho \u0111\u1ebfn khi ch\u1ea5t \u0111i\u1ec3m B \u0111u\u1ed5i k\u1ecbp th\u00ec qu\u00e3ng \u0111\u01b0\u1eddng hai ch\u1ea5t \u0111i\u1ec3m b\u1eb1ng nhau do \u0111\u00f3 18 15 \u00c5 1 t2 + 58 \u00e3 dt = at dt \u21d4 225 = a \u00b7 225 \u21d4 a = 2. 120 45 2 00 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 332 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG V\u1eady v\u1eadn t\u1ed1c c\u1ee7a ch\u1ea5t \u0111i\u1ec3m B t\u1ea1i th\u1eddi \u0111i\u1ec3m \u0111u\u1ed5i k\u1ecbp A b\u1eb1ng vB(t) = 2 \u00b7 15 = 30 (m\/s). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 90 (C\u00e2u 21 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00ecnh ph\u1eb3ng D gi\u1edbi h\u1ea1n b\u1edfi \u0111\u01b0\u1eddng cong y = ex, tr\u1ee5c ho\u00e0nh v\u00e0 c\u00e1c \u0111\u01b0\u1eddng th\u1eb3ng x = 0, x = 1. Kh\u1ed1i tr\u00f2n xoay t\u1ea1o th\u00e0nh khi quay D quanh tr\u1ee5c ho\u00e0nh c\u00f3 th\u1ec3 t\u00edch V b\u1eb1ng bao nhi\u00eau? A V \u03c0e2 B V \u03c0 (e2 + 1) C V = e2 \u2212 1 D V \u03c0 (e2 \u2212 1) =. =. . =. 2 2 2 2 \u0253 L\u1eddi gi\u1ea3i. V = \u03c0 1 (ex)2 dx = \u03c0 e2x 1 = \u03c0 (e2 \u2212 1) 20 2 0 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 91 (C\u00e2u 35 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). vI M\u1ed9t ng\u01b0\u1eddi ch\u1ea1y trong th\u1eddi gian 1 gi\u1edd, 8 v\u1eadn t\u1ed1c v (km\/h) ph\u1ee5 thu\u1ed9c th\u1eddi gian t(h) c\u00f3 \u0111\u1ed3 th\u1ecb l\u00e0 m\u1ed9t ph\u1ea7n c\u1ee7a \u0111\u01b0\u1eddng \u00c51 \u00e3 parabol v\u1edbi \u0111\u1ec9nh I ; 8 v\u00e0 tr\u1ee5c \u0111\u1ed1i x\u1ee9ng song song v\u1edbi tr\u1ee5c tung nh\u01b0 h\u00ecnh b\u00ean. 2 T\u00ednh qu\u00e3ng s \u0111\u01b0\u1eddng ng\u01b0\u1eddi \u0111\u00f3 ch\u1ea1y \u0111\u01b0\u1ee3c trong kho\u1ea3ng th\u1eddi gian 45 ph\u00fat, k\u1ec3 t\u1eeb khi b\u1eaft \u0111\u1ea7u ch\u1ea1y. A s = 4, 0 km. B s = 2, 3 km. C s = 4, 5 km. D s = 5, 3 km. O 11t 2 \u0253 L\u1eddi gi\u1ea3i. 3 T\u1eeb gi\u1ea3 thi\u1ebft ta c\u00f3 h\u00e0m v\u1eadn t\u1ed1c l\u00e0 v(t) = \u221232t2 + 32t. V\u1eady s = 4 \u221232t2 + 32t dt = 4, 5 km. 0 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 92 (C\u00e2u 41 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho h\u00e0m s\u1ed1 f (x) = ax3 + bx2 + cx \u2212 1 v\u00e0 g(x) = dx2 + ex + y 2 1 (a, b, c, d, e \u2208 R). Bi\u1ebft r\u1eb1ng \u0111\u1ed3 th\u1ecb c\u1ee7a h\u00e0m s\u1ed1 y = f (x) v\u00e0 y = g(x) c\u1eaft nhau t\u1ea1i ba \u0111i\u1ec3m c\u00f3 ho\u00e0nh \u0111\u1ed9 l\u1ea7n l\u01b0\u1ee3t l\u00e0 \u22123; \u22121; 1 (tham kh\u1ea3o h\u00ecnh v\u1ebd). H\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi hai \u0111\u1ed3 th\u1ecb \u0111\u00e3 cho c\u00f3 di\u1ec7n t\u00edch b\u1eb1ng A 9 B 8. C 4. D 5. . 2 \u22123 \u22121 O 1 x \u0253 L\u1eddi gi\u1ea3i. Do (C) : y = f (x) v\u00e0 (C ) : y = g(x) c\u1eaft nhau t\u1ea1i 3 \u0111i\u1ec3m ph\u00e2n bi\u1ec7t c\u00f3 ho\u00e0nh \u0111\u1ed9 \u22123; \u22121 v\u00e0 1 n\u00ean f (x) \u2212 g(x) = A(x + 3)(x + 1)(x \u2212 1). T\u1eeb gi\u1ea3 thi\u1ebft ta c\u00f3 f (0) \u2212 g(0) = \u22123 n\u00ean \u22123A = \u22123 \u21d4 A= 1 . 2 22 \u21d2 f (x) \u2212 g(x) = 1 + 3)(x + 1)(x \u2212 1) = 1 x3 + 3 x2 \u2212 1 \u2212 3 (x x . 2 2 2 22 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 333 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 \u22121 1 S = [f (x) \u2212 g(x)] dx + [g(x) \u2212 f (x)] dx \u22123 \u22121 . \u22121 1 = \u00ef 1 x3 + 3 x2 \u2212 1 \u2212 3\u00f2 dx \u2212 \u00ef 1 x3 + 3 x2 \u2212 1 \u2212 3\u00f2 dx = 2 \u2212 (\u22122) = 4. x x 2 2 22 2 2 22 \u22123 \u22121 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 93 (C\u00e2u 40 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho hai h\u00e0m s\u1ed1 f (x) = ax3 + bx2 + cx + 3 v\u00e0 g (x) = dx2 + ex \u2212 3 y 44 1 (a, b, c, d, e \u2208 R). Bi\u1ebft r\u1eb1ng \u0111\u1ed3 th\u1ecb c\u1ee7a h\u00e0m s\u1ed1 y = f (x) v\u00e0 y = g (x) \u22122 O c\u1eaft nhau t\u1ea1i ba \u0111i\u1ec3m c\u00f3 ho\u00e0nh \u0111\u1ed9 l\u1ea7n l\u01b0\u1ee3t l\u00e0 \u22122; 1; 3 (tham kh\u1ea3o h\u00ecnh v\u1ebd). H\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi hai \u0111\u1ed3 th\u1ecb \u0111\u00e3 cho c\u00f3 di\u1ec7n t\u00edch b\u1eb1ng 3 A 253 B 125 C 125 D 253 x . . . . 48 24 48 24 \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m ax3 +bx2 +cx+ 3 = dx2 +ex\u2212 3 \u21d4 ax3 +(b\u2212d)x2 +(c\u2212e)x+ 3 = 0. 44 2 3 \u0110\u1eb7t h(x) = ax3 + (b \u2212 d)x2 + (c \u2212 e)x + . 2 D\u1ef1a v\u00e0o \u0111\u1ed3 th\u1ecb ta c\u00f3 h(x) = 0 c\u00f3 ba nghi\u1ec7m l\u00e0 x = \u22122; x = 1; x = 3. Khi \u0111\u00f3 ta c\u00f3 h\u1ec7 \uf8f1 \u2212 \u2212 \u2212 \u2212 \u22123 \uf8f11 8a + 4(b d) 2(c e) = 2 \uf8f4\uf8f4a = \uf8f4 \uf8f4 4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 + (b \u2212 d) + (c \u2212 e) = \u22123 \u21d4 \uf8f4 \u2212 d = \u2212 1 \uf8f2 \uf8f2 a b \uf8f4 2 \uf8f42 \uf8f4 \uf8f4 \uf8f4 + 9(b \u2212 d) + 3(c \u2212 e) = \u22123 \uf8f4 \u2212 e = \u2212 5 . \uf8f4\uf8f3\uf8f427a \uf8f4\uf8f3\uf8f4c 24 Khi \u0111\u00f3 di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng c\u1ea7n t\u00ednh l\u00e0 31 3 S= |f (x) \u2212 g(x)| dx = 1 x3 \u2212 1 x2 \u2212 5 + 3 dx + 1 x3 \u2212 1 x2 \u2212 5 + 3 dx x x 4 2 42 4 2 42 \u22122 \u22122 1 63 4 253 = += . 16 3 48 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 94 (C\u00e2u 41 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 334 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG Cho \u0111\u01b0\u1eddng th\u1eb3ng y = 3x v\u00e0 parabol y = 2x2 +a (a l\u00e0 tham s\u1ed1 y y = 3x th\u1ef1c d\u01b0\u01a1ng). G\u1ecdi S1 v\u00e0 S2 l\u1ea7n l\u01b0\u1ee3t l\u00e0 di\u1ec7n t\u00edch c\u1ee7a hai h\u00ecnh ph\u1eb3ng \u0111\u01b0\u1ee3c g\u1ea1ch ch\u00e9o trong h\u00ecnh v\u1ebd b\u00ean. Khi S1 = S2 th\u00ec a thu\u1ed9c kho\u1ea3ng n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A \u00c54 9 \u00e3 B \u00c5 4\u00e3 C \u00c5 9\u00e3 D \u00c59 \u00e3 S2 ; . 0; . 1; . ;1 . 5 10 5 8 10 y = 2x2 + a S1 O x \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m 2x2 + a = 3x \u21d4 2x2 \u2212 3x + a = 0 (1) c\u00f3 hai nghi\u1ec7m d\u01b0\u01a1ng ph\u00e2n bi\u1ec7t \uf8f1\u2206 = 9 \u2212 8a > 0 \uf8f4 \uf8f19 a \u21d4 \uf8f4 = 2 > 0 \u21d4 \uf8f2a < 8 \u21d40 < a< 9 \uf8f4 . \uf8f2P \uf8f4 3 \uf8f3a > 0 8 \uf8f4 \uf8f3\uf8f4S = > 0 2\u221a 3 \u00b1 9 \u2212 8a Ta \u0111\u01b0\u1ee3c nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x = . \u221a \u221a4 3 \u2212 9 \u2212 8a 3 + 9 \u2212 8a G\u1ecdi x1 = 4 ; x2 = . 4 Ta c\u00f3 S1 = S2. x2 x1 \u21d4 (2x2 + a \u2212 3x) dx = \u2212 (2x2 + a \u2212 3x) dx. 0 x1 x1 x2 \u21d4 (2x2 + a \u2212 3x) dx + (2x2 + a \u2212 3x) dx = 0. 0 x1 x2 \u21d4 (2x2 \u2212 3x + a) dx = 0. 0 \u21d4 \u00c5 2 x3 \u2212 3 x2 + \u00e3 x2 = 0. ax 32 0 \u21d4 2 (x2)3 \u2212 2 (x2)2 + a (x2) = 0. 3 3 \u21d4 2 (x2)2 \u2212 3 \u00b7 x2 + a = 0 ( do x2 = 0 ). 3 2 Ta l\u1ea1i c\u00f3 x2 l\u00e0 nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh (1) n\u00ean x2 l\u00e0 nghi\u1ec7m c\u1ee7a h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh sau \uf8f12 (x2)2 \u2212 3 + a = 0 \uf8f2 2 x2 3 \uf8f32 (x2)2 \u2212 3x2 + a = 0. \u21d4 \uf8f12 (x2)2 \u2212 3 \u2212 2 (x2)2 + 3x2 = 0 \uf8f2 2 x2 3 \uf8f3a = \u22122 (x2)2 + 3x2. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 335 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc \u21d4 \uf8f1 \u22124 (x2)2 + 3 = 0 \uf8f2 2 x2 3 \uf8f3a = \u22122 (x2)2 + 3x2. \uf8f19 \uf8f2\uf8f4x2 = 8 \u21d4 27 \uf8f4\uf8f3a = . 32 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 95 (C\u00e2u 41 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho \u0111\u01b0\u1eddng th\u1eb3ng y = 3 v\u00e0 parabol y = x2 +a ( a l\u00e0 tham s\u1ed1 th\u1ef1c x 2 y = x2 + a d\u01b0\u01a1ng). G\u1ecdi S1, S2 l\u1ea7n l\u01b0\u1ee3t l\u00e0 di\u1ec7n t\u00edch hai h\u00ecnh ph\u1eb3ng \u0111\u01b0\u1ee3c g\u1ea1ch ch\u00e9o y y = 3 x 2 trong h\u00ecnh v\u1ebd b\u00ean. Khi S1 = S2 th\u00ec a thu\u1ed9c kho\u1ea3ng n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? \u00c51 \u00e3 \u00c52 9\u00e3 \u00c5 1\u00e3 \u00c5 2\u00e3 ; 9 . ; . 9 ;. 0; . A B C D 2 16 5 20 20 2 5 S1 S2 x \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m: x2 + a = 3 \u21d4 2x2 \u2212 3x + 2a = 0. x 2 \uf8f1a > 0 \u00aea > 0 \uf8f2 \u0110\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 2 nghi\u1ec7m d\u01b0\u01a1ng th\u00ec \u21d4 9 \u2206 > 0 \uf8f3a < . \u221a 16 3 + 9 \u2212 16a G\u1ecdi hai nghi\u1ec7m \u0111\u00f3 l\u00e0 0 < x1 < x2 th\u00ec x2 = . 4 \u0110\u1ec3 S1 = S2 khi v\u00e0 ch\u1ec9 khi x1 x2 x2 + a \u2212 3 dx = x2 + a \u2212 3 dx x x 22 0 x1 x1 x2 \u00c5 3\u00e3 \u00c5 3\u00e3 \u21d4 x2 + a \u2212 x dx = \u2212 x2 + a \u2212 x dx 22 0 x1 x1 x2 \u00c5 3\u00e3 \u00c5 3\u00e3 \u21d4 x2 + a \u2212 x dx + x2 + a \u2212 x dx = 0 22 0 x1 x2 \u00c5 3\u00e3 x2 x \u21d4 + a \u2212 2 dx = 0. 0 Ta c\u00f3 x2 + a \u2212 3\u00e3 = 0 \u21d4 x23 + ax2 \u2212 3 x22 = 0 x dx 3 4 \u00c5 x2 2 0 \u21d4 1 \u00c5 \u2212 9 + \u00e3 = 0 \u21d4 3 \u2212 a \u2212 9 + 3a = 0 \u21d4 \u22123x2 + 8a = 0 3 x22 4 x2 3a 2 x2 4 x2 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 336 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG \u21d4 8a = 3 \u00b7 3 + \u221a \u2212 16a \u21d4 \u221a \u2212 16a = 32a \u2212 9 \u21d4 \uf8f19 < a < 9 9 39 \uf8f2 16 32 4 \uf8f31024a2 \u2212 432a = 0 \u21d4 a= 27 \u2208 \u00c52; 9 \u00e3 . 64 5 20 \u00c52 9 \u00e3 C\u00f3 th\u1ec3 gi\u1ea3i nhanh b\u1eb1ng m\u00e1y t\u00ednh cho k\u1ebft qu\u1ea3 a = 0,421875 thu\u1ed9c kho\u1ea3ng ; . 5 20 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 96 (C\u00e2u 48 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). Cho h\u00e0m s\u1ed1 f (x) li\u00ean t\u1ee5c tr\u00ean R v\u00e0 th\u1ecfa m\u00e3n xf (x3) + f (1 \u2212 x2) = \u2212x10 + x6 \u2212 2x, \u2200x \u2208 R. Khi 0 \u0111\u00f3 f (x) dx b\u1eb1ng \u22121 B \u221213. C 17 D \u22121. 4 . A \u221217. 4 20 \u0253 L\u1eddi gi\u1ea3i. +Ta c\u00f3 xf (x3) + f (1 \u2212 x2) = \u2212x10 + x6 \u2212 2x \u21d4 x2f (x3) + xf (1 \u2212 x2) = \u2212x11 + x7 \u2212 2x2 00 0 \u21d4 x2f x3 dx + xf 1 \u2212 x2 dx = \u2212x11 + x7 \u2212 2x2 dx \u22121 \u22121 \u22121 00 = \u221217 24 \u21d4 1 f x3 d x3 \u2212 1 f 1 \u2212 x2 d 1 \u2212 x2 32 \u22121 \u22121 01 \u21d4 1 f (x) dx \u2212 1 f (x) dx = \u221217 (1) 32 24 \u22121 0 Ta c\u00f3 x2f (x3) + xf (1 \u2212 x2) = \u2212x11 + x7 \u2212 2x2 11 1 \u21d4 x2f x3 dx + xf 1 \u2212 x2 dx = \u2212x11 + x7 \u2212 2x2 dx 0 1 0 1 0 \u21d41 f x3 d x3 \u2212 1 f 1 \u2212 x2 d 1 \u2212 x2 = \u2212 5 3 2 8 00 10 \u21d4 1 f (x) dx \u2212 1 f (x) dx = \u22125 32 8 01 1 \u21d4 f (x) dx = \u22123 4 0 0 \u221213 T\u1eeb (1) \u21d2 f (x) dx = 4 \u22121 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 97 (C\u00e2u 43 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 y = f (x) = x3 + ax2 + bx + c v\u1edbi a, b, c l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c. Bi\u1ebft h\u00e0m s\u1ed1 g(x) = f (x) + f (x) + f (x) c\u00f3 hai gi\u00e1 tr\u1ecb c\u1ef1c tr\u1ecb l\u00e0 \u22124 v\u00e0 2. Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 337 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc f (x) y = v\u00e0 y = 1 b\u1eb1ng g(x) + 6 A 2 ln 2. B ln 6. C 3 ln 2. D ln 2. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) = 3x2 + 2ax + b, f (x) = 6x + 2a. Suy ra g(x) = x3 + (a + 3)x2 + (2a + b + 6)x + (2a + b + c), suy ra g (x) = 3x2 + (2a + 6)x + (2a + b + 6). Theo \u0111\u1ec1 b\u00e0i g(x) c\u00f3 hai gi\u00e1 tr\u1ecb c\u1ef1c tr\u1ecb, suy ra g (x) = 0 c\u00f3 hai nghi\u1ec7m x1 v\u00e0 x2. X\u00e9t ph\u01b0\u01a1ng tr\u00ecnh f (x) = 1 \u21d4 f (x) = g(x) + 6 \u21d4 3x2 + (2a + 6)x + 2a + b + 6 = 0. g(x) + 6 Suy ra di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 x2 f (x) \u2212 g(x) \u2212 6 S = dx g(x) + 6 x1 x2 g (x) = dx g(x) + 6 x1 x2 = ln |g(x) + 6| = ln 8 \u2212 ln 2 = 2 ln 2. x1 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 98 (C\u00e2u 46 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = x3+ax2+bx+c v\u1edbi a, b, c l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c. Bi\u1ebft h\u00e0m s\u1ed1 g(x) = f (x)+f (x)+f (x) c\u00f3 hai gi\u00e1 tr\u1ecb c\u1ef1c tr\u1ecb l\u00e0 \u22125 v\u00e0 3. Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = f (x) v\u00e0 g(x) + 6 y = 1 b\u1eb1ng A 2 ln 3. B ln 2. C ln 15. D 3 ln 2. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 g (x) = f (x) + f (x) + f (x) = f (x) + f (x) + 6. Do h\u00e0m s\u1ed1 c\u00f3 hai c\u1ef1c tr\u1ecb, n\u00ean ph\u01b0\u01a1ng tr\u00ecnh g (x) = 0 \u21d4 f (x) + f (x) + 6 = 0 c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t x1, x2. Ph\u01b0\u01a1ng tr\u00ecnh ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m f (x) = 1 \u21d4 f (x) + f (x) + 6 = 0 \u21d4 x = x1, x = x2. g(x) + 6 Suy ra di\u1ec7n t\u00edch c\u1ea7n t\u00ednh x2 f (x) + f (x) + 6 S = dx g(x) + 6 x1 x2 g (x) = dx g(x) + 6 x1 x2 = ln |g(x) + 6| x1 = ln g(x1) + 6 = ln 9 = 2 ln 3. g(x2) + 6 Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 338 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG \u0104 C\u00e2u 99 (C\u00e2u 47 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = x3+ax2+bx+c v\u1edbi a, b, c l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c. Bi\u1ebft h\u00e0m s\u1ed1 g(x) = f (x)+f (x)+f (x) c\u00f3 hai gi\u00e1 tr\u1ecb c\u1ef1c tr\u1ecb l\u00e0 \u22125 v\u00e0 2. Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = f (x) v\u00e0 g(x) + 6 y = 1 b\u1eb1ng A ln 3. B 3 ln 2. C ln 10. D ln 7. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) = 3x2 + 2ax + b, f (x) = 6x + 2a v\u00e0 f (x) = 6. Do \u0111\u00f3 g (x) = f (x) + f (x) + 6. Ta c\u00f3 g(x) l\u00e0 h\u00e0m s\u1ed1 b\u1eadc ba v\u1edbi h\u1ec7 s\u1ed1 c\u1ee7a x3 b\u1eb1ng 1 n\u00ean g(x) c\u00f3 hai gi\u00e1 tr\u1ecb c\u1ef1c tr\u1ecb l\u00e0 \u22125 v\u00e0 2 th\u00ec g(x) c\u00f3 hai \u0111i\u1ec3m c\u1ef1c tr\u1ecb x1, x2 v\u1edbi x1 < x2 v\u00e0 g (x1) = 2, g (x2) = \u22125. Ph\u01b0\u01a1ng tr\u00ecnh ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m gi\u1eefa hai \u0111\u01b0\u1eddng y = f (x) v\u00e0 y = 1 l\u00e0 g(x) + 6 f (x) = 1 \u21d4 f (x) = g(x) + 6 g(x) + 6 \u21d4 f (x) = f (x) + f (x) + f (x) + 6 \u21d4 f (x) + f (x) + 6 = 0 \u21d4 g (x) = 0 \u21d4 \u00f1x = x1 x = x2. V\u1eady di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng c\u1ea7n t\u00ecm l\u00e0 x2 x2 g(x) + 6 \u2212 f (x) x2 S = 1\u2212 f (x) dx = dx = f (x) + f (x) + 6 dx g(x) + 6 g(x) + 6 g(x) + 6 x1 x1 x1 x2 x2 = g (x) dx = (ln |g(x) + 6|) = | ln 1 \u2212 ln 8| = 3 ln 2. g(x) + 6 x1 x1 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 100 (C\u00e2u 47 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00e0m s\u1ed1 b\u1eadc b\u1ed1n y = f (x). Bi\u1ebft r\u1eb1ng h\u00e0m s\u1ed1 g(x) = ln(f (x)) c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean nh\u01b0 sau: x \u2212\u221e x1 x2 x3 +\u221e +\u221e ln 6 +\u221e g(x) 43 ln 8 ln 2 Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = f (x) v\u00e0 y = g (x) thu\u1ed9c kho\u1ea3ng n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A (5; 6). B (4; 5). C (2; 3). D (3; 4). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 339","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc Ta c\u00f3 g(x) = ln(f (x)) \u21d4 f (x) = eg(x). T\u1eeb b\u1ea3ng bi\u1ebfn thi\u00ean ta \u0111\u01b0\u1ee3c g(x1) = ln 43 \u21d2 f (x1) = 43 \u00b7 8 8 g(x2) = ln 6 \u21d2 f (x2) = 6. g(x3) = ln 2 \u21d2 f (x3) = 2. Ta c\u00f3 f (x) \u2212 g (x) = g (x).eg(x) \u2212 g (x) = g (x) eg(x) \u2212 1 . \u00f1g (x) = 0. f (x) \u2212 g (x) = 0 \u21d4 eg(x) \u2212 1 = 0 \u00f1g (x) = 0. \u21d4 g(x) = 0 (v\u00f4 nghi\u1ec7m) \u21d4 x \u2208 {x1, x2, x3} . Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = f (x) v\u00e0 y = g (x) l\u00e0. x3 x2 x3 S = |f (x) \u2212 g (x)| dx = |f (x) \u2212 g (x)| dx + |f (x) \u2212 g (x)| dx x1 x1 x2 x2 x3 = [f (x) \u2212 g (x)] dx + [f (x) \u2212 g (x)] dx x1 x2 = |[f (x) \u2212 g(x)]|xx21 | + |[f (x) \u2212 g(x)] x3 | x2 = f (x2) \u2212 g(x2) \u2212 f (x1) \u2212 g(x1) + | f (x3) \u2212 g(x3) \u2212 f (x2) \u2212 g(x2) = (6 \u2212 ln 6) \u2212 43 \u2212 ln 43 + |(2 \u2212 ln 2) \u2212 (6 \u2212 ln 6)| \u2248 3, 42 \u2208 (3; 4). 88 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 101 (C\u00e2u 48 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00e0m s\u1ed1 b\u1eadc b\u1ed1n y = f (x). Bi\u1ebft r\u1eb1ng h\u00e0m s\u1ed1 g(x) = ln f (x) c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean nh\u01b0 sau: x \u2212\u221e x1 x2 x3 +\u221e +\u221e +\u221e ln 42 g(x) ln 37 ln 10 Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = f (x) v\u00e0 y = g (x) thu\u1ed9c kho\u1ea3ng n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A (38; 39). B (25; 26). C (28; 29). D (35; 36). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 g(x) = ln f (x) \u21d2 g (x) = f (x)\u00b7 f (x) T\u1eeb b\u1ea3ng bi\u1ebfn thi\u00ean ta th\u1ea5y g(x) > 0, \u2200x \u2208 R suy ra f (x) = eg(x) > 1, \u2200x \u2208 R. \uf8eex = x1 Ph\u01b0\u01a1ng tr\u00ecnh f (x) = g (x) \u21d4 g (x).f (x) = g (x) \u21d4 g (x). [f (x) \u2212 1] = 0 \u21d4 g (x) = 0 \u21d4 \uf8efx = x2 \uf8f0 x = x3. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 340 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = f (x) v\u00e0 y = g (x) l\u00e0: x3 x2 \u00c5 f (x) \u00e3 x3 \u00c5 f (x) \u00e3 S = |f (x) \u2212 g (x)| dx = f (x) \u2212 f (x) dx + f (x) \u2212 f (x) dx x1 x1 x2 t==f (x) 42 1 37 1 1 \u2212 dt + 1 \u2212 dt 10 t 42 t \u2248 35, 438 \u2208 (35; 36). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 102 (C\u00e2u 43 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00e0m s\u1ed1 b\u1eadc b\u1ed1n y = f (x). Bi\u1ebft r\u1eb1ng h\u00e0m s\u1ed1 g(x) = ln f (x) c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean x \u2212\u221e x1 x2 x3 +\u221e +\u221e ln 35 +\u221e y ln 30 ln 3 Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = f (x) v\u00e0 y = g (x) thu\u1ed9c kho\u1ea3ng n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A (33; 35). B (37; 40). C (29; 32). D (24; 26). \u0253 L\u1eddi gi\u1ea3i. T\u1eeb b\u1ea3ng bi\u1ebfn thi\u00ean h\u00e0m s\u1ed1 g(x) = ln f (x) ta c\u00f3 ln f (x) \u2265 ln 3, \u2200x \u2208 R \u21d4 f (x) \u2265 3, \u2200x \u2208 R. Ta c\u00f3 g (x) = f (x)\u00b7 f (x) T\u1eeb b\u1ea3ng bi\u1ebfn thi\u00ean ta c\u00f3 \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = g(x) c\u00f3 3 \u0111i\u1ec3m c\u1ef1c tr\u1ecb l\u00e0 A x1; ln 30 , B x2; ln 35 , C x3; ln 3 n\u00ean f x1 = f x2 = f x3 = 0 v\u00e0 f x1 = 30, f x2 = 35, f x3 = 3. Do y = f (x) l\u00e0 h\u00e0m s\u1ed1 b\u1eadc 3 n\u00ean ph\u01b0\u01a1ng tr\u00ecnh f (x) = 0 ch\u1ec9 c\u00f3 3 nghi\u1ec7m x1, x2, x3. X\u00e9t ph\u01b0\u01a1ng tr\u00ecnh ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m c\u1ee7a f (x) v\u00e0 g (x) ta c\u00f3 f (x) \u00f1f (x) = 0 \uf8eex = x1 f (x) = g (x) \u21d4 f (x) = f (x) \u21d4 f (x) = 1(v\u00f4 nghi\u1ec7m) \u21d4 \uf8efx = x2 \uf8f0 x = x3. Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = f (x) v\u00e0 y = g (x) l\u00e0: x3 x3 f (x) x3 \u00c5 1 \u00e3 S = |g (x) \u2212 f (x)| dx = \u2212 f (x) dx = f (x). \u2212 1 dx x1 f (x) x1 f (x) x1 x2 \u00c5 1 \u00e3 x3 \u00c5 1 \u00e3 = f (x). \u2212 1 dx + f (x). \u2212 1 dx. x1 f (x) x2 f (x) + T\u00ednh I1 = x2 \u00c5 1 \u00e3 x2 \u00c5 1 \u00e3 x1; x2 ) f (x). \u2212 1 dx = f (x). 1 \u2212 dx (do f (x) \u2265 0, \u2200x \u2208 x1 f (x) x1 f (x) \u0110\u1eb7t t = f (x) \u21d2 dt = f (x)dx. \u0110\u1ed5i c\u1eadn: x = x1 \u21d2 t = f x1 = 30. x = x2 \u21d2 t = f x2 = 35. Suy ra I1 = 35 1 \u2212 1 dt = (t \u2212 ln |t|) 35 = 35 \u2212 ln 35 \u2212 30 + ln 30 = 5 + ln 6 \u00b7 30 t 30 7 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 341 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc + T\u00ednh I2 = x3 \u00c5 1 \u00e3 x3 \u00c5 1 \u00e3 f (x). \u2212 1 dx = \u2212 f (x). 1 \u2212 dx (do f (x) \u2264 0). x2 f (x) x2 f (x) \u0110\u1eb7t t = f (x) \u21d2 dt = f (x)dx. \u0110\u1ed5i c\u1eadn: x = x2 \u21d2 t = f x2 = 35. x = x3 \u21d2 t = f x3 = 3. Suy ra I2 = \u2212 3 1\u2212 1 3 = \u2212(3 \u2212 ln 3 \u2212 35 + ln 35) = 32 \u2212 ln 35\u00b7 35 t dt = \u2212(t \u2212 ln |t|) 35 3 6 32 \u2212 ln 35 = 37 + ln 18 \u2248 34, 39 \u2208 (33; 35). V\u1eady S = 5 + ln + 7 3 245 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 103 (C\u00e2u 41 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). 2 Bi\u1ebft F (x) v\u00e0 G(x) l\u00e0 hai nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) tr\u00ean R v\u00e0 f (x)dx = F (2) \u2212 G(0) + a 0 (a > 0). G\u1ecdi S l\u00e0 di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = F (x), y = G(x), x = 0 v\u00e0 x = 2. Khi S = 6 th\u00ec a b\u1eb1ng A 4. B 6. C 3. D 8. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 F (x) v\u00e0 G(x) l\u00e0 hai nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) tr\u00ean R n\u00ean ta c\u00f3 \u2200x \u2208 R : F (x) = G(x) + C (v\u1edbi C l\u00e0 h\u1eb1ng s\u1ed1). Do \u0111\u00f3 F (0) = G(0) + C (1). 2 L\u1ea1i c\u00f3 f (x)dx = F (2) \u2212 F (0) 0 \u21d4 F (2) \u2212 G(0) + a = F (2) \u2212 F (0) \u21d4 F (0) = G(0) \u2212 a (2). T\u1eeb (1) v\u00e0 (2) suy ra C = \u2212a. Khi \u0111\u00f3 F (x) = G(x) \u2212 a, \u2200x \u2208 R \u21d4 |F (x) \u2212 G(x)| = a, \u2200x \u2208 R. Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = F (x), y = G(x), x = 0 v\u00e0 x = 2 l\u00e0 22 S = |F (x) \u2212 G(x)| dx = a. dx = 2a = 6 \u21d2 a = 3. 00 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 104 (C\u00e2u 46 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00e0m s\u1ed1 b\u1eadc b\u1ed1n y = f (x). Bi\u1ebft r\u1eb1ng h\u00e0m s\u1ed1 g(x) = ln f (x) c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean nh\u01b0 sau x \u2212\u221e x1 x2 x3 +\u221e +\u221e +\u221e g(x) ln 196 16 ln 12 ln 12 Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = f (x) v\u00e0 y = g (x) thu\u1ed9c kho\u1ea3ng n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A (7; 8). B (6; 7). C (8; 9). D (10; 11). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 342 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG \u0253 L\u1eddi gi\u1ea3i. T\u1eeb BBT c\u1ee7a g(x) ta c\u00f3 ln f (x) \u2265 ln 4 \u21d4 f (x) \u2265 4; \u2200x \u2208 R. Ta c\u00f3 g (x) = f (x)\u00b7 f (x) \u00ef f (x) = 0 (\u2217) f (x) = 1 (\u2217\u2217). X\u00e9t ph\u01b0\u01a1ng tr\u00ecnh f (x) = g (x) \u21d4 Do f (x) \u2265 4; \u2200x \u2208 R suy ra ph\u01b0\u01a1ng tr\u00ecnh (\u2217\u2217) v\u00f4 nghi\u1ec7m. \uf8ee x = x1 T\u1eeb \u0111\u00f3 suy ra f (x) = 0 \u21d4 g (x) = 0 \u21d4 \uf8f0 x = x2 x = x3. \u00ef 1\u00f2 M\u1eb7t kh\u00e1c f (x) \u2212 g (x) = f (x). 1 \u2212 . f (x) Ta c\u00f3 b\u1ea3ng x\u00e9t d\u1ea5u x \u2212\u221e x1 x2 x3 +\u221e f (x)\u2212 \u22120+0\u22120+ g (x) x3 x2 x3 V\u1eadyS = |f (x) \u2212 g (x)| dx = [f (x) \u2212 g (x)] dx \u2212 [f (x) \u2212 g (x)] dx x1 x1 x2 x2 x3 = [f (x) \u2212 g(x)] \u2212 [f (x) \u2212 g(x)] x1 x2 = 2f x2 \u2212 f x1 \u2212 f x3 \u2212 2 ln f x2 + ln f x1 + ln f x3 = 199 \u2212 12 \u2212 4 \u2212 2 ln 199 + ln 12 + ln 4 \u2248 7,704 \u2208 (7; 8). 2. 16 16 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 105 (C\u00e2u 45 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00e0m s\u1ed1 f (x) = 3x4 + ax3 + bx2 + cx + d(a, b, c, d \u2208 R) c\u00f3 ba \u0111i\u1ec3m c\u1ef1c tr\u1ecb l\u00e0 \u22122, \u22121 v\u00e0 1. G\u1ecdi y = g(x) l\u00e0 h\u00e0m s\u1ed1 b\u1eadc hai c\u00f3 \u0111\u1ed3 th\u1ecb \u0111i qua ba \u0111i\u1ec3m c\u1ef1c tr\u1ecb c\u1ee7a \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = f (x). Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi hai \u0111\u01b0\u1eddng y = f (x) v\u00e0 y = g(x) b\u1eb1ng A 500 B 36 C 2932 D 2948 . . . . 81 5 405 405 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) = 12x3 + 3ax2 + 2bx + c. (1) M\u1eb7t kh\u00e1c, v\u00ec y = f (x) l\u00e0 h\u00e0m s\u1ed1 b\u1eadc b\u1ed1n v\u00e0 c\u00f3 ba \u0111i\u1ec3m c\u1ef1c tr\u1ecb \u22122, \u22121, 1 n\u00ean suy ra f (x) = 12(x + 3)(x + 1)(x \u2212 1) = 12(x3 + 2x2 \u2212 x \u2212 2) = 12x3 + 24x2 \u2212 12x \u2212 24. (2) \uf8f13a = 24 \uf8f1a = 8 \uf8f4\uf8f4 \uf8f2\uf8f2 T\u1eeb (1) v\u00e0 (2), ta c\u00f3 h\u1ec7 ph\u01b0\u01a1ng tr\u00ecnh 2b = \u221212 \u21d4 b = \u22126 \uf8f4\uf8f3c = \u221224 \uf8f4\uf8f3c = \u221224. Suy ra f (x) = 3x4 + 8x3 \u2212 6x2 \u2212 24x + d. C\u00e1ch 1: \u00c51 1\u00e3 x Ta c\u00f3 f (x) = f (x) + \u2212 7x2 \u2212 16x + d + 4. 46 Khi \u0111\u00f3 \u0111\u1ed3 th\u1ecb \u0111i qua ba \u0111i\u1ec3m c\u1ef1c tr\u1ecb c\u1ee7a f (x) l\u00e0 g(x) = \u22127x2 \u2212 16x + d + 4. Do \u0111\u00f3 ta c\u00f3 1 1 S = |f (x) \u2212 g(x)| dx = 3x4 + 8x3 + x2 \u2212 8x \u2212 4 2948 dx = . \u22122 405 \u22122 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 343 S\u0110T: 0905.193.688","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc C\u00e1ch 2: X\u00e9t ph\u01b0\u01a1ng tr\u00ecnh ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m c\u1ee7a f (x), g(x) l\u00e0 f (x) = g(x) \u21d4 f (x) \u2212 g(x) = 0. Nh\u1eadn x\u00e9t r\u1eb1ng f (x) \u2212 g(x) l\u00e0 h\u00e0m s\u1ed1 b\u1eadc b\u1ed1n v\u00e0 theo gi\u1ea3 thi\u1ebft, ph\u01b0\u01a1ng tr\u00ecnh tr\u00ean c\u00f3 3 nghi\u1ec7m \u22122, \u22121, 1. Khi \u0111\u00f3 f (x) \u2212 g(x) = 3(x2 \u2212 1)(x + 2)(mx + n) = 3x3 + 6x2 \u2212 3x \u2212 6 (mx + n) = 3mx4 + 3nx3 + 6mx3 + 6nx2 \u2212 3mx2 \u2212 3nx \u2212 6mx \u2212 6n = 3mx4 + 3(n + 2m)x3 + 3(2n \u2212 m)x2 \u2212 3(n + 2m)x \u2212 6n. V\u00ec f (x) l\u00e0 h\u00e0m s\u1ed1 b\u1eadc b\u1ed1n v\u00e0 g(x) l\u00e0 h\u00e0m s\u1ed1 b\u1eadc hai, n\u00ean ta c\u00f3 th\u1ec3 \u0111\u1ed3ng nh\u1ea5t h\u1ec7 s\u1ed1 b\u1eadc 4 v\u00e0 b\u1eadc 3 c\u1ee7a f (x) v\u00e0 f (x) \u2212 g(x). Suy ra m = 1 v\u00e0 n = 2 . 3 Khi \u0111\u00f3 f (x) \u2212 g(x) = (x + 2)(x2 \u2212 1)(3x + 2). Do \u0111\u00f3 11 S = |f (x) \u2212 g(x)| dx = (x + 2)(x2 \u2212 1)(3x + 2) 2948 dx = . 405 \u22122 \u22122 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 106 (C\u00e2u 38 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). M\u1ed9t bi\u1ec3n qu\u1ea3ng c\u00e1o c\u00f3 d\u1ea1ng h\u00ecnh elip v\u1edbi b\u1ed1n \u0111\u1ec9nh A1, A2, B1, B2 nh\u01b0 h\u00ecnh v\u1ebd b\u00ean. Bi\u1ebft chi ph\u00ed \u0111\u1ec3 s\u01a1n ph\u1ea7n t\u00f4 \u0111\u1eadm l\u00e0 200.000 \u0111\u1ed3ng\/m2 v\u00e0 ph\u1ea7n c\u00f2n B2 l\u1ea1i l\u00e0 100.000 \u0111\u1ed3ng\/m2. H\u1ecfi s\u1ed1 ti\u1ec1n \u0111\u1ec3 s\u01a1n theo c\u00e1ch tr\u00ean g\u1ea7n nh\u1ea5t v\u1edbi s\u1ed1 ti\u1ec1n n\u00e0o d\u01b0\u1edbi \u0111\u00e2y, bi\u1ebft A1A2 = 8m, B1B2 = 6m v\u00e0 t\u1ee9 gi\u00e1c M N A2 M N P Q l\u00e0 h\u00ecnh ch\u1eef nh\u1eadt c\u00f3 M Q = 3 m? A1 QP A 7.322.000 \u0111\u1ed3ng. B 7.213.000 \u0111\u1ed3ng. C 5.526.000 \u0111\u1ed3ng. B1 D 5.782.000 \u0111\u1ed3ng. \u0253 L\u1eddi gi\u1ea3i. Ch\u1ecdn h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 Oxy sao cho tr\u1ee5c ho\u00e0nh tr\u00f9ng v\u1edbi tr\u1ee5c l\u1edbn, y B2 tr\u1ee5c tung tr\u00f9ng v\u1edbi tr\u1ee5c b\u00e9 c\u1ee7a bi\u1ec3n qu\u1ea3ng c\u00e1o. N Khi \u0111\u00f3, \u0111\u01b0\u1eddng vi\u1ec1n c\u1ee7a bi\u1ec3n qu\u1ea3ng c\u00e1o c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh c\u1ee7a A2 x2 y2 M Ox d\u1ea1ng elip sau (E) : a2 + b2 = 1. Theo gi\u1ea3 thi\u1ebft ta c\u00f3 \u00aeA1A2 = 8 \u21d4 \u00ae2a = 8 \u21d4 \u00aea = 4 \u21d2 (E) : A1 P B1B2 = 6 2b = 6 b=3 Q B1 \u221a x2 + y2 = 1 \u21d2 y = \u00b1 3 16 \u2212 x2 . 16 9 4 Ta c\u00f3: MQ = 3 \u21d2 \u00aeM = d\u2229 (E) v\u1edbi d: y = 3 \u21d2 M \u00c5\u221a 3\u00e3 v\u00e0 N \u00c5\u221a 3\u00e3 \u22122 3; 2 3; . N = d \u2229 (E) 2 22 Do Elip nh\u1eadn tr\u1ee5c Ox v\u00e0 Oy l\u00e0m tr\u1ee5c \u0111\u1ed1i x\u1ee9ng n\u00ean di\u1ec7n t\u00edch ph\u1ea7n t\u00f4 m\u00e0u g\u1ea5p 4 di\u1ec7n t\u00edch h\u00ecnh \u221a \u221a ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi y = 3 16 \u2212 x2 v\u00e0 c\u00e1c \u0111\u01b0\u1eddng th\u1eb3ng x = 2 3, tr\u1ee5c tung, tr\u1ee5c ho\u00e0nh, ch\u00ednh l\u00e0 \u221a 4\u221a 2 3 2 3\u00c4\u221a \u00e4 \u221a 16 \u2212 x2 dx. S=4 \u00c5 3 \u00e3 x2 4 16 \u2212 dx = 3 00 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 344 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG \u0110\u1eb7t x = 4 sin t, khi \u0111\u00f3 dx = 4 cos t dt. V\u00e0 v\u1edbi x = 0 \u21d2 t = 0; v\u1edbi \u221a \u03c0 x=2 3\u21d2t= . 3 \u03c0 \u03c0\u03c0 3 33 \u03c0 \u00c4 \u2212 sin2 \u00b7 \u00b7 \u00e4 cos2 t dt = 24 (1 + cos 2t) dt = (24t + 12 sin 2t) 3 = t S=3 16 16 t 4 cos dt = 48 0 0\u221a 0 0 8\u03c0 + 6 3 m2. \u00c4 \u221a\u00e4 \u00c4 \u221a\u00e4 S\u1ed1 ti\u1ec1n \u0111\u1ec3 s\u01a1n theo y\u00eau c\u1ea7u b\u00e0i to\u00e1n l\u00e0 T = 100.000\u00d7 4\u03c0 \u2212 6 3 +200.000\u00d7 8\u03c0 + 6 3 \u2248 7.322.000 \u0111\u1ed3ng. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 107 (C\u00e2u 46 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = x3+ax2+bx+c v\u1edbi a, b, c l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c. Bi\u1ebft h\u00e0m s\u1ed1 g(x) = f (x)+f (x)+f (x) c\u00f3 hai gi\u00e1 tr\u1ecb c\u1ef1c tr\u1ecb l\u00e0 \u22123 v\u00e0 6. Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi c\u00e1c \u0111\u01b0\u1eddng y = f (x) v\u00e0 g(x) + 6 y = 1 b\u1eb1ng A 2 ln 3. B ln 3. C ln 18. D 2 ln 2. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) = 3x2 + 2ax + b. f (x) = 6x + 2a. f (x) = 6. X\u00e9t h\u00e0m s\u1ed1 g(x) = f (x) + f (x) + f (x), ta c\u00f3 g (x) = f (x) + f (x) + f (x) = f (x) + f (x) + 6. \u00aeg(m) = \u22123 Theo gi\u1ea3 thi\u1ebft ta c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh g (x) = 0 c\u00f3 hai nghi\u1ec7m m, n v\u00e0 g(n) = 6. X\u00e9t ph\u01b0\u01a1ng tr\u00ecnh f (x) \u00f1x = m = 1 \u21d4 g(x) + 6 \u2212 f (x) = 0 \u21d4 f (x) + f (x) + 6 = 0 \u21d4 g(x) + 6 x = n. Di\u1ec7n t\u00edch h\u00ecnh ph\u1eb3ng c\u1ea7n t\u00ednh l\u00e0 n nn \u00c5 f (x) \u00e3 g(x) + 6 \u2212 f (x) f (x) + f (x) + 6 S= 1 \u2212 dx = dx = dx = g(x) + 6 g(x) + 6 g(x) + 6 m mm n ln |g(x) + 6| = |ln |g(n) + 6| \u2212 ln |g(m) + 6|| = |ln 12 \u2212 ln 3| = ln 4 = 2 ln 2. m Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 108 (C\u00e2u 44 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). 1 Cho h\u00e0m s\u1ed1 f (x) c\u00f3 \u0111\u1ea1o h\u00e0m li\u00ean t\u1ee5c tr\u00ean R. Bi\u1ebft f (6) = 1 v\u00e0 xf (6x) dx = 1, khi \u0111\u00f3 0 6 x2f (x) dx b\u1eb1ng 0 A 107 B 34. C 24. D \u221236. . 3 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 345","3. \u1ee8ng d\u1ee5ng c\u1ee7a t\u00edch ph\u00e2n trong h\u00ecnh h\u1ecdc 1 Theo b\u00e0i ra xf (6x) dx = 1. 0 \u0110\u1eb7t t = 6x \u21d2 dt = 6 dx. \u0110\u1ed5i c\u1eadn x = 0 \u21d2 t = 0 ; x = 1 \u21d2 t = 6 16 66 Do \u0111\u00f3 xf (6x) dx = 1 \u21d4 1 \u00b7 f (t) dt = 1 \u21d4 1 t \u00b7 f (t) dt = 1 \u21d4 t \u00b7 f (t) dt = 36. t 66 36 0 0 00 6 T\u00ednh I = x2f (x) dx. 0 \u00aeu = x2 \u00ae du = 2x dx \u0110\u1eb7t \u21d2 dv = f (x) dx v = f (x). 66 \u21d2 I = x2f (x) 6 \u2212 2xf (x) dx = 36f (6) \u2212 2 xf (x) dx = 36 \u00b7 1 \u2212 2 \u00b7 36 = \u221236. 0 00 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 109 (C\u00e2u 48 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00e0m s\u1ed1 y = f (x). \u0110\u1ed3 th\u1ecb c\u1ee7a h\u00e0m s\u1ed1 y = f (x) nh\u01b0 h\u00ecnh b\u00ean. y \u0110\u1eb7t g(x) = 2f (x) \u2212 (x + 1)2. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? 4 A g(\u22123) > g(3) > g(1). B g(1) > g(\u22123) > g(3). C g(3) > g(\u22123) > g(1). D g(1) > g(3) > g(\u22123). 2 \u22123 O1 3x \u22122 \u0253 L\u1eddi gi\u1ea3i. - Ta c\u00f3 g (x) = 2 (f (x) \u2212 (x + 1)) . 33 - T\u1eeb g(3) \u2212 g(1) = g (x) dx = 2 (f (x) \u2212 (x + 1)) dx < 0 suy ra g(3) < g(1). 11 33 - T\u01b0\u01a1ng t\u1ef1 g(3) \u2212 g(\u22123) = g (x) dx = 2 (f (x) \u2212 (x + 1)) dx > 0 suy ra g(\u22123) < g(3). \u22123 \u22123 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 110 (C\u00e2u 46 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00e0m s\u1ed1 y = f (x). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 346 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG \u0110\u1ed3 th\u1ecb h\u00e0m s\u1ed1 y = f (x) nh\u01b0 h\u00ecnh b\u00ean. y \u0110\u1eb7t g(x) = 2f (x) + x2. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? 3 O1 3 \u22123 \u22121 x \u22123 A g(3) < g(\u22123) < g(1). B g(1) < g(3) < g(\u22123). C g(1) < g(\u22123) < g(3). D g(\u22123) < g(3) < g(1). \u0253 L\u1eddi gi\u1ea3i. y 3 Ta c\u00f3 g (x) = 2f (x) + 2x = 0 \u21d4 f (x) = \u2212x. T\u1eeb h\u00ecnh b\u00ean suy ra g (x) = 0 t\u1ea1i x = \u22123, x = 1 ho\u1eb7c x = 3. \u22123 \u2212O1 1 3 x H\u01a1n n\u1eefa, trong kho\u1ea3ng (\u22123; 1) \u0111\u1ed3 th\u1ecb y = f (x) n\u1eb1m d\u01b0\u1edbi \u0111\u1ed3 th\u1ecb y = \u2212x n\u00ean g (x) \u00e2m trong kho\u1ea3ng (\u22123; 1). X\u00e9t t\u01b0\u01a1ng t\u1ef1 trong kho\u1ea3ng (1; 3), ta \u0111\u01b0\u1ee3c b\u1ea3ng bi\u1ebfn thi\u00ean c\u1ee7a g(x) nh\u01b0 sau. \u22123 y = \u2212x y = f (x) x \u22123 1 3 g (x) \u22120+ g(\u22123) g(3) g(x) g(1) C\u1ea7n so s\u00e1nh g(\u22123) v\u1edbi g(3). Ta c\u00f3: 33 g(3) \u2212 g(\u22123) = g (x) dx = 2 [f (x) + x] dx = \u22123 \u22123 1 3 = \u22122 [(\u2212x) \u2212 f (x)] dx + 2 [f (x) \u2212 (\u2212x)] dx = 2(\u2212S1 + S2) < 0 \u21d2 g(3) < g(\u22123), \u22123 1 trong \u0111\u00f3 S1, S2 l\u00e0 di\u1ec7n t\u00edch ph\u1ea7n h\u00ecnh ph\u1eb3ng gi\u1edbi h\u1ea1n b\u1edfi hai \u0111\u01b0\u1eddng y = f (x) v\u00e0 y = \u2212x, t\u01b0\u01a1ng \u1ee9ng khi \u22123 < x < 1 v\u00e0 1 < x < 3. Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 347 S\u0110T: 0905.193.688"]
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