Toàn cảnh đề thi tốt nghiệp THPT môn Toán (2018 – 2022)

["5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit D\u1ef1a v\u00e0o BBT, ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 nghi\u1ec7m duy nh\u1ea5t khi v\u00e0 ch\u1ec9 khi \u00f1m = 4 . m<0 V\u00ec m \u2208 [\u22122017; 2017] v\u00e0 m \u2208 Z n\u00ean ch\u1ec9 c\u00f3 2018 gi\u00e1 tr\u1ecb m nguy\u00ean th\u1ecfa y\u00eau c\u1ea7u l\u00e0 m \u2208 {\u22122017; \u22122016; \u00b7 \u00b7 \u00b7 ; \u22121; 4}. Ch\u00fa \u00fd : Trong, ta \u0111\u00e3 b\u1ecf qua \u0111i\u1ec1u ki\u1ec7n mx > 0 v\u00ec v\u1edbi ph\u01b0\u01a1ng tr\u00ecnh loga f (x) = loga g(x) v\u1edbi 0 < a = 1 ta ch\u1ec9 c\u1ea7n \u0111i\u1ec1u ki\u1ec7n f (x) > 0 (ho\u1eb7c g(x) > 0 ) Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 89 (C\u00e2u 37 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho a > 0, b > 0 th\u1ecfa m\u00e3n log10a+3b+1(25a2 + b2 + 1) + log10ab+1(10a + 3b + 1) = 2. Gi\u00e1 tr\u1ecb c\u1ee7a a + 2b b\u1eb1ng A 5 B 6. C 22. D 11 . . 2 2 \u0253 L\u1eddi gi\u1ea3i. T\u1eeb gi\u1ea3 thuy\u1ebft b\u00e0i to\u00e1n, ta suy ra 25a2 + b2 + 1 > 1, 10\u221aa + 3b + 1 > 1 v\u00e0 10ab + 1 > 1. \u00c1p d\u1ee5ng b\u1ea5t \u0111\u1eb3ng th\u1ee9c C\u00f4-si, ta c\u00f3 25a2 + b2 + 1 \u2265 2 25a2b2 + 1 = 10ab + 1. Khi \u0111\u00f3, log10a+3b+1(25a2 + b2 + 1) + log10ab+1(10a + 3b + 1) \u2265 log10a+3b+1(10ab + 1) + log10ab+1(10a + 3b + 1) \u00bb \u2265 2 log10a+3b+1(10ab + 1) \u00b7 log10ab+1(10a + 3b + 1) = 2. D\u1ea5u \u201c=\u201d x\u1ea3y ra khi ch\u1ec9 khi \u00ae5a = b \u21d4 \u00aeb = 5a log10a+3b+1(10ab + 1) = log10ab+1(10a + 3b + 1) = 1 10ab + 1 = 10a + 3b + 1 \u00aeb = 5a \u21d4 50a2 \u2212 25a = 0 \uf8f1b = 5a \uf8f4 \uf8f4 \uf8eea = 0 (lo\u1ea1i) \uf8f2 \u21d4 \uf8f4\uf8f0 1 \uf8f4 a= (nh\u1eadn) \uf8f3 2 \uf8f15 \uf8f2\uf8f4b = 2 \u21d4 1 \uf8f4\uf8f3a = . 2 \u21d2 a + 2b = 11 . 2 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 90 (C\u00e2u 45 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho ph\u01b0\u01a1ng tr\u00ecnh 3x + m = log3(x \u2212 m) v\u1edbi m l\u00e0 tham s\u1ed1. C\u00f3 bao nhi\u00eau gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a m \u2208 (\u221215; 15) \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 nghi\u1ec7m? A 16. B 9. C 14. D 15. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 248","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT Ta c\u00f3 3x + m = log3(x \u2212 m) \u21d4 3x + x = log3(x \u2212 m) + x \u2212 m (\u2217). X\u00e9t h\u00e0m s\u1ed1 f (t) = 3t + t v\u1edbi t \u2208 R, ta c\u00f3 f (t) = 3t ln 3 + 1 > 0, \u2200t \u2208 R n\u00ean h\u00e0m s\u1ed1 f (t) \u0111\u1ed3ng bi\u1ebfn tr\u00ean t\u1eadp x\u00e1c \u0111\u1ecbnh. M\u1eb7t kh\u00e1c, ph\u01b0\u01a1ng tr\u00ecnh (\u2217) c\u00f3 d\u1ea1ng f (x) = f (log3(x \u2212 m)). Do \u0111\u00f3, f (x) = f (log3(x \u2212 m)) \u21d4 x = log3(x \u2212 m) \u21d4 3x = x \u2212 m \u21d4 3x \u2212 x = \u2212m (\u2217\u2217) X\u00e9t h\u00e0m s\u1ed1 g(x) = 3x \u2212 x v\u1edbi x \u2208 R, ta c\u00f3 g (x) = 3x ln 3 \u2212 1, g (x) = 0 \u21d4 x = log3 \u00c51\u00e3 = a. ln 3 B\u1ea3ng bi\u1ebfn thi\u00ean x \u2212\u221e a +\u221e g (x) \u22120+ +\u221e +\u221e g(x) g(a) T\u1eeb b\u1ea3ng bi\u1ebfn thi\u00ean, ta suy ra ph\u01b0\u01a1ng tr\u00ecnh (\u2217\u2217) c\u00f3 nghi\u1ec7m khi ch\u1ec9 khi m \u2208 (\u2212\u221e; \u2212g (a)). M\u1eb7t kh\u00e1c, m \u2208 Z \u2229 (\u221215; 15) n\u00ean m \u2208 {\u221214; \u221213; \u221212; . . . ; \u22121} (v\u00ec \u2212g(a) \u2248 \u22120,9958452485). Do \u0111\u00f3, c\u00f3 14 gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a m th\u1ecfa m\u00e3n y\u00eau c\u1ea7u b\u00e0i to\u00e1n. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 91 (C\u00e2u 42 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho ph\u01b0\u01a1ng tr\u00ecnh 7x + m = log7(x \u2212 m) v\u1edbi m l\u00e0 tham s\u1ed1. C\u00f3 bao nhi\u00eau gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a m \u2208 (\u221225; 25) \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 nghi\u1ec7m? A 9. B 25. C 24. D 26. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n: x > m. \u0110\u1eb7t t = log7(x \u2212 m) ta c\u00f3 \u00ae7x + m = t \u21d2 7x + x = 7t + t. (1) 7t + m = x Do h\u00e0m s\u1ed1 f (u) = 7u + u \u0111\u1ed3ng bi\u1ebfn tr\u00ean R n\u00ean ta c\u00f3 (1) \u21d4 t = x. T\u1ee9c l\u00e0 7x + m = x \u21d4 m = x \u2212 7x. X\u00e9t h\u00e0m s\u1ed1 g(x) = x \u2212 7x \u21d2 g (x) = 1 \u2212 7x ln 7 = 0 \u21d4 x = \u2212 log7 (ln 7) = x0. B\u1ea3ng bi\u1ebfn thi\u00ean: x \u2212\u221e \u2212 log7(ln 7) +\u221e g (x) +0\u2212 \u2212\u221e g(x) g(x0) \u2212\u221e T\u1eeb \u0111\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 nghi\u1ec7m khi v\u00e0 ch\u1ec9 khi m g (\u2212 log7(ln 7)) \u2248 \u22120,856. (c\u00e1c nghi\u1ec7m n\u00e0y \u0111\u1ec1u th\u1ecfa m\u00e3n \u0111i\u1ec1u ki\u1ec7n v\u00ec x \u2212 m = 7x > 0) Do m nguy\u00ean thu\u1ed9c kho\u1ea3ng (\u221225; 25) n\u00ean m \u2208 {\u221224; \u221216; . . . ; \u22121}. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 92 (C\u00e2u 48 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho ph\u01b0\u01a1ng tr\u00ecnh 2x + m = log2 (x \u2212 m) v\u1edbi m l\u00e0 tham s\u1ed1. C\u00f3 bao nhi\u00eau gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a m \u2208 (\u221218; 18) \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 nghi\u1ec7m? Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 249 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit A 9. B 19. C 17. D 18. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x > m. \u0110\u1eb7t t = log2(x \u2212 m) ta c\u00f3 \u00ae2x + m = t \u21d2 2x + x = 2t + t (1). 2t + m = x Do h\u00e0m s\u1ed1 f (u) = 2u + u \u0111\u1ed3ng bi\u1ebfn tr\u00ean R n\u00ean ta c\u00f3 (1) \u21d4 x = t. Khi \u0111\u00f3 2x + m = x \u21d4 m = x \u2212 2x. X\u00e9t h\u00e0m g(x) = x \u2212 2x \u21d2 g (x) = 1 \u2212 2x ln 2 = 0 \u21d4 x = \u2212 log2(ln 2). B\u1ea3ng bi\u1ebfn thi\u00ean x \u2212\u221e \u2212 log2(ln 2) +\u221e g (x) +0\u2212 g(\u2212 log2(ln 2)) g(x) \u2212\u221e \u2212\u221e T\u1eeb \u0111\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 nghi\u1ec7m khi v\u00e0 ch\u1ec9 khi m \u2264 g(\u2212 log2(ln 2)) \u2248 \u22120,914. Do m nguy\u00ean thu\u1ed9c kho\u1ea3ng (\u221218; 18) n\u00ean m \u2208 {\u221217; \u221216; . . . ; \u22121}. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 93 (C\u00e2u 50 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho a > 0, b > 0 th\u1ecfa m\u00e3n log2a+2b+1 (4a2 + b2 + 1) + log4ab+1 (2a + 2b + 1) = 2. Gi\u00e1 tr\u1ecb c\u1ee7a a + 2b b\u1eb1ng A 15 B 5. C 4. D 3 . . 4 2 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 4a2 + b2 \u2265 4ab, v\u1edbi m\u1ecdi a, b > 0. D\u1ea5u \u201c=\u201d x\u1ea3y ra khi v\u00e0 ch\u1ec9 khi b = 2a (1). Khi \u0111\u00f3 2 = log2a+2b+1 (4a2 + b2 + 1) + log4ab+1 (2a + 2b + 1) \u2265 log2a+2b+1 (4ab + 1) + log4ab+1 (2a + 2b + 1) . M\u1eb7t kh\u00e1c theo b\u1ea5t \u0111\u1eb3ng th\u1ee9c Cauchy ta c\u00f3 log2a+2b+1 (4ab + 1) + log4ab+1 (2a + 2b + 1) \u2265 2. D\u1ea5u \u201c=\u201d x\u1ea3y ra khi v\u00e0 ch\u1ec9 khi log2a+2b+1(4ab + 1) = 1 \u21d4 4ab + 1 = 2a + 2b + 1 (2). 3 3 15 T\u1eeb (1) v\u00e0 (2) ta c\u00f3 8a2 \u2212 6a = 0 \u21d2 a = \u21d2 b = \u21d2 a+ 2b = . 42 4 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 94 (C\u00e2u 48 - M\u0110 104 - BGD&\u0110\u221aT - N\u0103m 2018 - 2019). s\u1ed1 th\u1ef1c). C\u00f3 t\u1ea5t c\u1ea3 bao nhi\u00eau Cho ph\u01b0\u01a1ng tr\u00ecnh 2 log23 x \u2212 log3 x \u2212 1 4x \u2212 m = 0 (m l\u00e0 tham gi\u00e1 tr\u1ecb nguy\u00ean d\u01b0\u01a1ng c\u1ee7a m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 \u0111\u00fang hai nghi\u1ec7m ph\u00e2n bi\u1ec7t? A V\u00f4 s\u1ed1. B 62. C 63. D 64. \u00aex > 0 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 \u0111i\u1ec1u ki\u1ec7n x \u2265 log4 m (*) (v\u1edbi m nguy\u00ean d\u01b0\u01a1ng). Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi (1) \u221a 2 log32 x \u2212 log3 x \u2212 1 4x \u2212 m = 0 \u21d4 \u00f12 log23 x \u2212 log3 x \u2212 1 = 0 (2) 4x = m (3). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 250 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT Ph\u01b0\u01a1ng tr\u00ecnh (2) \uf8ee log3 x =1 \u21d4 \uf8eex = 3 \u21d4\uf8f0 = \u22121 = \u221a \uf8f0 log3 x 2 x 3 . 3 Ph\u01b0\u01a1ng tr\u00ecnh (3) \u21d4 x = log4 m. Do m nguy\u00ean d\u01b0\u01a1ng n\u00ean ta c\u00f3 c\u00e1c tr\u01b0\u1eddng h\u1ee3p sau: TH 1: m = 1 th\u00ec log4 m = 0. Khi \u0111\u00f3 \u0111i\u1ec1u ki\u1ec7n (*) tr\u1edf th\u00e0nh x > 0. Khi \u0111\u00f3 nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh (3) b\u1ecb lo\u1ea1i v\u00e0 nh\u1eadn nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh (2). Do \u0111\u00f3 nh\u1eadn gi\u00e1 tr\u1ecb m = 1. TH 2: m \u2265 2, khi \u0111\u00f3 \u0111i\u1ec1u ki\u1ec7n (*) tr\u1edf th\u00e0nh x \u2265 log4 m (v\u00ec log4 m \u2265 1 ). 2 \u0110\u1ec3 \u221aph\u01b0\u01a1ng tr\u00ecnh (1) c\u00f3 \u0111\u00fang hai nghi\u1ec7m ph\u00e2n bi\u1ec7t 3 \u21d4 3\u221a \u2264 log4 m < 3 3 \u21d4 4 3 \u2264 m < 43 Suy ra m \u2208 {3; 4; 5; . . . ; 63}. V\u1eady t\u1eeb c\u1ea3 2 tr\u01b0\u1eddng h\u1ee3p ta c\u00f3: 63 \u2212 3 + 1 + 1 = 62 gi\u00e1 tr\u1ecb nguy\u00ean d\u01b0\u01a1ng m. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 95 (C\u00e2u 47 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). C\u00f3 bao nhi\u00eau c\u1eb7p s\u1ed1 nguy\u00ean (x; y) th\u1ecfa m\u00e3n 0 \u2264 x \u2264 2020 v\u00e0 log3 (3x + 3) + x = 2y + 9y? A 2019. B 6. C 2020. D 4. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log3 (3x + 3) + x = 2y + 9y \u21d4 [1 + log3 (x + 1)] + x = 2y + 32y \u21d4 log3 (x + 1) + x + 1 = 2y + 32y \u21d4 log3 (x + 1) + 3log3(x+1) = 2y + 32y (*). X\u00e9t h\u00e0m f (t) = t + 3t. Ta c\u00f3 f (t) = 1 + 3t ln 3 > 0, \u2200t \u2208 R. Do \u0111\u00f3, f (t) \u0111\u1ed3ng bi\u1ebfn tr\u00ean R. Khi \u0111\u00f3, (*) \u21d4 f [log3 (x + 1)] = f (2y) \u21d4 log3 (x + 1) = 2y \u21d4 x + 1 = 9y (**). V\u00ec 0 \u2264 x \u2264 2020 n\u00ean 1 \u2264 9y \u2264 2021 \u21d2 0 \u2264 y \u2264 log92021 \u2248 3, 464. V\u00ec y \u2208 Z n\u00ean y \u2208 {0, 1, 2, 3}. V\u1edbi y = 0 \u21d2 x = 0; y = 1 \u21d2 x = 8; y = 2 \u21d2 x = 80; y = 3 \u21d2 x = 728. V\u1eady c\u00f3 4 c\u1eb7p s\u1ed1 nguy\u00ean (x, y) th\u1ecfa m\u00e3n. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 96 (C\u00e2u 48 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Cho h\u00e0m s\u1ed1 f (x) c\u00f3 f (0) = 0. Bi\u1ebft y = f (x) l\u00e0 h\u00e0m s\u1ed1 b\u1eadc b\u1ed1n v\u00e0 y y = f (x) c\u00f3 \u0111\u1ed3 th\u1ecb l\u00e0 \u0111\u01b0\u1eddng cong trong h\u00ecnh b\u00ean. S\u1ed1 \u0111i\u1ec3m c\u1ef1c tr\u1ecb c\u1ee7a h\u00e0m s\u1ed1 g(x) = |f (x3) \u2212 x| l\u00e0 Ox A 5. B 4. C 6. D 3. \u0253 L\u1eddi gi\u1ea3i. X\u00e9t h(x) = f (x3) \u2212 x, ta c\u00f3 h (x) = 3x2f (x3) \u2212 1. X\u00e9t h (x) = 0 \u21d4 3x2f (x3) \u2212 1 = 0 \u21d4 f (x3) = 1 (\u2217), v\u1edbi x = 0. 3x2 \u0110\u1eb7t t = x3, ta c\u00f3 (\u2217) \u21d4 f (t) = \u221a1 (\u2217\u2217). 3 3 t2 X\u00e9t h\u00e0m s\u1ed1 k(t) = \u221a1 , k (t) = \u2212\u221a2 . Ta c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean c\u1ee7a k(t) nh\u01b0 b\u00ean d\u01b0\u1edbi 3 3 t2 9 3 t5 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 251 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit t \u2212\u221e 0 +\u221e k (t) +\u2212 +\u221e +\u221e k(t) 00 D\u1ef1a v\u00e0o \u0111\u1ed3 th\u1ecb c\u1ee7a f (t) ta suy ra ph\u01b0\u01a1ng tr\u00ecnh (\u2217\u2217) c\u00f3 hai nghi\u1ec7m t1 < 0 < t2, v\u1eady (\u2217) c\u00f3 hai nghi\u1ec7m x1 < 0 < x2. Ta c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean c\u1ee7a h(x) nh\u01b0 h\u00ecnh v\u1ebd b\u00ean d\u01b0\u1edbi x \u2212\u221e x1 0 x2 +\u221e h (x) + + 0\u2212 \u22120 h (x1) +\u221e h(x) h(0) \u2212\u221e h (x2) Ta c\u00f3 h(0) = f (0) \u2212 0 = 0, n\u00ean h(x) c\u1eaft Ox t\u1ea1i ba \u0111i\u1ec3m ph\u00e2n bi\u1ec7t, suy ra h\u00e0m s\u1ed1 g(x) = |h(x)| c\u00f3 5 \u0111i\u1ec3m c\u1ef1c tr\u1ecb. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 97 (C\u00e2u 50 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). C\u00f3 bao nhi\u00eau c\u1eb7p s\u1ed1 nguy\u00ean d\u01b0\u01a1ng (m; n) sao cho m +\u221an \u2264 10 v\u00e0 \u1ee9ng v\u1edbi m\u1ed7i c\u1eb7p (m; n) t\u1ed3n t\u1ea1i \u0111\u00fang 3 s\u1ed1 th\u1ef1c a \u2208 (\u22121; 1) th\u1ecfa m\u00e3n 2am = n \u00c4 \u00e4 ln a + a2 + 1 ? A 7. B 8. C 10. D 9. 2am \u00c4 \u0253\u221aL\u1eddi gi\u1ea3i. a \u00e4 + a2 + 1 . Ta th\u1ea5y a = 0 lu\u00f4n th\u1ecfa m\u00e3n = n ln V\u1edbi a \u2208 K = (\u22121; 0) \u222a (0; 1) ta c\u00f3 \u00c4\u221a a2 + 1 \u00e4 2am \u00c4 \u221a \u00e4 \u21d4 n ln a+ n ln a a2 1 = + + am = 2. (1) \u00c4\u221a \u00e4 n ln a + a2 + 1 X\u00e9t h\u00e0m s\u1ed1 f (a) = am v\u1edbi m\u1ecdi a \u2208 K. V\u1edbi m\u1ecdi a \u2208 K, ta c\u00f3 \u221an \u00b7 am \u2212 m \u00b7 am\u22121 \u00b7 n ln \u00c4 + \u221a + \u00e4 a2 + 1 a a2 1 f (a) = (am)2 \u00ef a \u00c4\u221a \u00f2 n \u00b7 am\u22121 \u221a \u2212 m ln a + a2 + 1 \u00e4 = a2 + 1 a2m . X\u00e9t h\u00e0m s\u1ed1 g(a) = \u221a a \u00c4\u221a \u00e4 \u2212 m ln a + a2 + 1 . a2 + 1 Ta c\u00f3 \u221a \u221a a2 a2 + 1 \u2212 a2 +1 \u221am 1\u221a \u221am g (a) = +1 \u2212 a2 + 1 = + 1) a2 \u2212 a2 + 1 a2 (a2 +1 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 252 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT = \u221a1 1 \u00c5 1 1 \u2212 \u00e3 < 0, \u2200a \u2208 (\u22121; 1), m \u2265 1. a2 + a2 + m B\u1ea3ng bi\u1ebfn thi\u00ean c\u1ee7a g(a) tr\u00ean kho\u1ea3ng (\u22121; 1) a \u22121 0 1 g (a) \u2212 g(a) 0 T\u1eeb b\u1ea3ng bi\u1ebfn thi\u00ean tr\u00ean, ta x\u00e9t c\u00e1c tr\u01b0\u1eddng h\u1ee3p sau: N\u1ebfu m \u2208 {2; 4; 6; 8} th\u00ec b\u1ea3ng bi\u1ebfn thi\u00ean c\u1ee7a f (a) nh\u01b0 sau: a \u22121 0 1 f (a) \u2212\u2212 f (1) f (\u22121) +\u221e f (a) \u2212\u221e \u00c4 \u221a\u00e4 V\u00ec f (\u22121) = n ln \u22121 + 2 < 0 v\u1edbi m\u1ecdi n > 0 n\u00ean ph\u01b0\u01a1ng tr\u00ecnh (1) kh\u00f4ng th\u1ec3 c\u00f3 2 nghi\u1ec7m thu\u1ed9c K. N\u1ebfu m \u2208 {3; 5; 7; 9} th\u00ec b\u1ea3ng bi\u1ebfn thi\u00ean c\u1ee7a f (a) nh\u01b0 sau: a \u22121 0 1 f (a) +\u2212 f (1) f (a) +\u221e +\u221e f (\u22121) \u0110\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh (1) c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t thu\u1ed9c K th\u00ec \uf8f1 \u00c4 \u221a\u00e4 \uf8f2 \u2212 n ln \u22121 + 2 <2 \u00aef (\u22121) < 2 \u21d4 \u00c4 \u221a\u00e4 \u21d4n< 2 f (1) < 2 \u00c4 \u221a \u00e4. \uf8f3n ln 1 + 2 < 2 ln 1 + 2 M\u00e0 n nguy\u00ean d\u01b0\u01a1ng n\u00ean n = 1 ho\u1eb7c n = 2. V\u1eady trong tr\u01b0\u1eddng h\u1ee3p n\u00e0y ta c\u00f3 c\u00e1c c\u1eb7p s\u1ed1 nguy\u00ean d\u01b0\u01a1ng (3; 1), (3; 2), (5; 1), (5; 2), (7; 1), (7; 2), (9; 1) th\u1ecfa m\u00e3n y\u00eau c\u1ea7u b\u00e0i to\u00e1n. N\u1ebfu m = 1 th\u00ec b\u1ea3ng bi\u1ebfn thi\u00ean c\u1ee7a f (a) nh\u01b0 sau: a \u22121 0 1 f (a) +\u2212 nn f (a) f (1) f (\u22121) Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 253 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit \u0110\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh (1) c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t thu\u1ed9c K th\u00ec \u00aef (\u22121) < 2 < n (v\u00f4 l\u00ed). f (1) < 2 < n V\u1eady c\u00f3 t\u1ea5t c\u1ea3 7 c\u1eb7p s\u1ed1 nguy\u00ean d\u01b0\u01a1ng (m; n) th\u1ecfa m\u00e3n y\u00eau c\u1ea7u b\u00e0i to\u00e1n. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 98 (C\u00e2u 50 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). C\u00f3 bao nhi\u00eau c\u1eb7p s\u1ed1 nguy\u00ean (m, n) sao cho m +\u221an \u2264 12 v\u00e0 \u1ee9ng v\u1edbi m\u1ed7i c\u1eb7p (m, n) t\u1ed3n t\u1ea1i \u0111\u00fang 3 s\u1ed1 th\u1ef1c \u00e4 \u2208 (\u22121; 2am \u00c4 a2 + 1 ? a 1) th\u1ecfa m\u00e3n = n ln a + A 12. B 10. C 11. D 9. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 a = 0 lu\u00f4n th\u1ecfa m\u00e3n 2am = n ln \u00c4 + \u221a + \u00e4 a a2 1. X\u00e9t a \u2208 D = (\u22121; 0) \u222a (0; 1), ta c\u00f3 \u00c4\u221a \u00e4 a + a2 + 1 2am \u00c4 \u221a \u00e4 \u21d4 n ln am n ln a a2 1 = + + = 2. (1) \u00c4\u221a \u00e4 n ln a + a2 + 1 X\u00e9t h\u00e0m s\u1ed1 g(a) = am . \u00c5 a \u00c4\u221a \u00e3 n \u00b7 am\u22121 \u221a \u2212 m \u00b7 ln a + a2 + 1 \u00e4 Ta c\u00f3 g (a) = a2 + 1 . X\u00e9t h\u00e0m s\u1ed1 h(a) = \u221a a a2m \u221a \u00e4 \u00c4 \u2212 m \u00b7 ln a + a2 + 1 . a2 + 1 Ta c\u00f3 h (a) = 1\u221a \u2212\u221a m =\u221a 1 \u00c51 \u00e3 \u2212 m < 0, \u2200a \u2208 (\u22121; 1), \u2200m \u2265 1. (a2 + 1) a2 + 1 a2 + 1 a2 + 1 a2 + 1 B\u1ea3ng bi\u1ebfn thi\u00ean c\u1ee7a h\u00e0m s\u1ed1 h(a): a \u22121 0 1 h (a) \u2212 h(a) 0 D\u1ef1a v\u00e0o b\u1ea3ng bi\u1ebfn thi\u00ean tr\u00ean ta x\u00e9t c\u00e1c tr\u01b0\u1eddng h\u1ee3p sau \u2014 N\u1ebfu m \u2208 {2; 4; 6; 8; 10} th\u00ec b\u1ea3ng bi\u1ebfn thi\u00ean c\u1ee7a h\u00e0m s\u1ed1 g(a) nh\u01b0 sau a \u22121 0 1 g (a) \u2212\u2212 g(1) g(\u22121) +\u221e g(a) \u2212\u221e Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 254 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT \u221a Do g(\u22121) = n ln(\u22121 + 2) < 0 n\u00ean ph\u01b0\u01a1ng tr\u00ecnh (1) kh\u00f4ng th\u1ec3 c\u00f3 2 nghi\u1ec7m thu\u1ed9c D. \u2014 N\u1ebfu m \u2208 {3; 5; 7; 9; 11} th\u00ec b\u1ea3ng bi\u1ebfn thi\u00ean c\u1ee7a h\u00e0m s\u1ed1 g(a) nh\u01b0 sau a \u22121 0 1 g (a) +\u2212 g(1) g(a) +\u221e +\u221e g(\u22121) \u0110\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh (1) c\u00f3 2 nghi\u1ec7m thu\u1ed9c D th\u00ec \u221a \u00aeg(\u22121) < 2 \u00ae \u2212 n ln(\u22121 + 2) < 2 \u21d4\u221a \u21d4n< 2 \u221a \u21d4 n \u2208 {1; 2}. g(1) < 2 n ln(1 + 2) < 2 ln(1 + 2) \u2014 N\u1ebfu m = 1 th\u00ec b\u1ea3ng bi\u1ebfn thi\u00ean c\u1ee7a h\u00e0m s\u1ed1 g(a) nh\u01b0 sau a \u22121 0 1 g (a) +\u2212 g(1) nn g(a) g(\u22121) \u0110\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh (1) c\u00f3 2 nghi\u1ec7m thu\u1ed9c D th\u00ec \uf8f1 2\u221a \u00aeg(\u22121) < 2 < n \uf8f2n < \u21d4 ln(1 + 2) (v\u00f4 l\u00ed). g(1) < 2 < n \uf8f3n > 2 V\u1eady c\u00f3 10 c\u1eb7p s\u1ed1 nguy\u00ean d\u01b0\u01a1ng (m, n) th\u1ecfa m\u00e3n \u0111\u1ec1 b\u00e0i. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 99 (C\u00e2u 45 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean y sao cho t\u1ed3n t\u1ea1i x \u2208 \u00c51 \u00e3 th\u1ecfa m\u00e3n ;4 3 273x2+xy = (1 + xy)2712x? A 14. B 27. C 12. D 15. \u0253 L\u1eddi gi\u1ea3i. a) Khi y \u2264 0, v\u00ec xy > \u22121 v\u00e0 x > 1 n\u00ean ta c\u00f3 y > \u22123. 3 V\u1edbi y = 0, ph\u01b0\u01a1ng tr\u00ecnh th\u00e0nh 273x2\u221212x \u2212 1 = 0 v\u00f4 nghi\u1ec7m v\u00ec 273x2\u221212x \u2212 1 < 270 \u2212 1 < 0, \u2200x \u2208 \u00c51 \u00e3 ; 4. 3 V\u1edbi y \u2208 {\u22121, \u22122}, ph\u01b0\u01a1ng tr\u00ecnh th\u00e0nh 273x2\u2212(12\u2212y)x \u2212 (1 + xy) = 0, c\u00f3 nghi\u1ec7m v\u00ec: h(x) = 273x2\u2212(12\u2212y)x \u2212 (1 + xy) li\u00ean t\u1ee5c tr\u00ean \u00ef1 \u00f2 v\u00e0 ; 4 3 \u00c51\u00e3 = 31+y\u221212 \u2212 1\u2212 y < 3\u22123 \u2212 1 \u2212 2 < 0 < 274y \u2212 (1 + 4y) = h(4). h 3 23 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 255 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit b) Khi y \u2265 1, x\u00e9t tr\u00ean \u00ef1 \u00f2 ; 4 , ta c\u00f3 3 273x2+xy = (1 + xy)2718x \u21d4 3x2 + xy \u2212 12x = log27(1 + xy) \u21d4 3x \u2212 12 \u2212 log27(1 + xy) + y = 0. x X\u00e9t h\u00e0m g(x) = 3x \u2212 12 \u2212 log27(1 + xy) + y tr\u00ean \u00ef1 \u00f2 ;4 . x3 Ta c\u00f3 g (x) = 3 + ln(1 + xy) \u2212 y 1 3 \u00ef 1 \u00f2 4. x2 ln 27 x(1 + xy) ln 27 >3\u2212 3x2 ln 3 \u22653\u2212 ln 3 > 0, \u2200x \u2208 3 ; \u00ef1 \u00f2 Do \u0111\u00f3, h\u00e0m g(x) \u0111\u1ed3ng bi\u1ebfn tr\u00ean ; 4 . 3 \u00c51 \u00e3 \u00c51\u00e3 V\u00ec th\u1ebf ph\u01b0\u01a1ng tr\u00ecnh g(x) = 0 c\u00f3 nghi\u1ec7m tr\u00ean ; 4 khi v\u00e0 ch\u1ec9 khi g g(4) < 0. 33 \u00c1p d\u1ee5ng b\u1ea5t \u0111\u1eb3ng th\u1ee9c ln(1 + u) < u v\u1edbi m\u1ecdi u > 0, ta c\u00f3 g(4) = \u2212 log27(1 + 4y) + y > \u2212 y + y > 0. 4 ln 27 \u00c51 \u00e3 Do \u0111\u00f3 ph\u01b0\u01a1ng tr\u00ecnh g(x) = 0 c\u00f3 nghi\u1ec7m tr\u00ean ; 4 khi v\u00e0 ch\u1ec9 khi 3 \u00c51\u00e3 < 0 \u21d4 \u2212 log3 y + y \u2212 12 + 1 < 0 \u21d4 1 \u2264 y \u2264 12 (do y l\u00e0 s\u1ed1 nguy\u00ean d\u01b0\u01a1ng, g 1+ 3 3 y u(y) = \u2212 log3 1+ + y \u2212 12 + 1 \u0111\u01a1n \u0111i\u1ec7u t\u0103ng tr\u00ean (0; \u221e) v\u00e0 u(12) \u2264 0 < u(13)). 3 V\u1eady y \u2208 {\u22122; \u22121; 1; 2; . . . ; 12} hay c\u00f3 14 gi\u00e1 tr\u1ecb c\u1ee7a y th\u1ecfa m\u00e3n \u0111\u1ec1 b\u00e0i. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 100 (C\u00e2u 44 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean y sao cho t\u1ed3n t\u1ea1i x \u2208 \u00c51 \u00e3 th\u1ecfa m\u00e3n 273x2+xy = (1 + xy)2715x? ;5 3 A 17. B 16. C 18. D 15. \u0253 L\u1eddi gi\u1ea3i. Tr\u01b0\u1edbc h\u1ebft ta c\u00f3 b\u1ea5t \u0111\u1eb3ng th\u1ee9c sau 27x \u2265 27x ln 3 + 9 \u2212 18 ln 3, \u2200x \u2208 R. (1) Th\u1eadt v\u1eady, x\u00e9t h\u00e0m s\u1ed1 g(x) = 27x \u2212 27x ln 3 \u2212 9 + 18 ln 3, ta c\u00f3 g (x) = 27x \u00b7 3 ln 3 \u2212 27 ln 3 \u21d2 g (x) = 0 \u21d4 x = 2 . 3 x \u2212\u221e 2 +\u221e 3 g (x) \u22120+ g(x) 0 S\u0110T: 0905.193.688 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 256","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT Do \u0111\u00f3 g(x) \u2265 0 \u21d4 27x \u2265 27x ln 3 + 9 \u2212 18 ln 3, \u2200x \u2208 R. Suy ra (1) \u0111\u00fang. \u00c5 1 \u00e3 5, T\u1eeb ph\u01b0\u01a1ng tr\u00ecnh, suy ra xy > \u22121. Do x \u2208 3 ; n\u00ean ta c\u00f3 y > \u22123 hay y \u2265 \u22122. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi f (x) = 273x2\u221215x+xy \u2212 (1 + xy) = 0. Ta c\u00f3 f \u00c51\u00e3 = 3\u221214+y \u2212 1 \u2212 1 f (5) = 275y \u2212 1 \u2212 5y. Ta th\u1ea5y y, 33 f \u00c51 \u00e3 < 0 \u21d4 \u22122 \u2264 y \u2264 \u221215, f (5) > 0 \u2200y = 0. (2) 3 Th\u1eadt v\u1eady, x\u00e9t h\u00e0m s\u1ed1 h(y) = 3\u221214+y \u2212 1 \u2212 1 ta c\u00f3 y, 3 h (y) = 3y\u221214 ln 3 \u2212 1 \u21d2 h (y) = 0 \u21d4 y = y0 = 14 \u2212 log3(3 ln 3) \u2248 12,9. 3 H\u01a1n n\u1eefa do h(\u22122) < 0 < h(\u22123) \u21d2 t\u1ed3n t\u1ea1i y1 \u2208 (\u22123; \u22122) sao cho h (y1) = 0; do h(15) < 0 < h(16) \u21d2 t\u1ed3n t\u1ea1i y2 \u2208 (15; 16) sao cho h (y2) = 0. T\u1eeb \u0111\u00f3 ta c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean y \u22123 y1 y0 y2 +\u221e h (y) \u2212 | \u22120+ | + h(y) 0 0 Suy ra h(y) < 0 \u21d4 y1 < y < y2 \u21d2 \u22122 \u2264 y \u2264 15 (do y \u2208 Z). Ch\u1ee9ng minh t\u01b0\u01a1ng t\u1ef1 ta c\u00f3 (2) \u0111\u00fang. N\u1ebfu y \u2208 {\u22122; \u22121; 1; . . . ; 15}, ta c\u00f3 f \u00c51 \u00e3 \u00b7 f (5) < 0, n\u00ean ph\u01b0\u01a1ng tr\u00ecnh f (x) = 0 lu\u00f4n c\u00f3 nghi\u1ec7m 3 \u00c51 \u00e3 tr\u00ean ; 5 . 3 V\u1edbi y = 0, ta th\u1ea5y f (x) = 0 \u21d4 x = 0, x = 5 n\u00ean lo\u1ea1i. V\u1edbi y \u2265 16, ta c\u00f3 f (x) \u2265 27(3x2 \u2212 15x + xy) ln 3 + 9 \u2212 18 ln 3 \u2212 (1 + xy) = x [27(3x \u2212 15 + y) ln 3 \u2212 y] + 8 \u2212 18 ln 3 \u2265 1 (54 ln 3 \u2212 16) + 8 \u2212 18 ln 3 = 8 > 0. 33 Do \u0111\u00f3, ph\u01b0\u01a1ng tr\u00ecnh f (x) = 0 v\u00f4 nghi\u1ec7m. V\u1eady y \u2208 {\u22122; \u22121; 1; 2; . . . ; 15}. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 257 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit \u0104 C\u00e2u 101 (C\u00e2u 44 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean y sao cho t\u1ed3n t\u1ea1i x \u2208 \u00c51 \u00e3 th\u1ecfa m\u00e3n 273x2+xy = (1 + xy) \u00b7 2718x? ; 6 3 A 19. B 20. C 18. D 21. \u0253 L\u1eddi gi\u1ea3i. \u0110\u1ec3 c\u1eb7p (x; y) th\u1ecfa m\u00e3n ph\u01b0\u01a1ng tr\u00ecnh th\u00ec 1 + xy > 0. Khi \u0111\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi 273x2+xy = (1 + xy) \u00b7 2718x (*) \u21d4 273x2+xy = 27log27(1+xy) \u00b7 2718x \u21d4 3x2 + xy = log27(1 + xy) + 18x \u21d4 3x2 + (y \u2212 18)x \u2212 log27(1 + xy) = 0. \u0110\u1eb7t f (x) = 3x2 + (y \u2212 18)x \u2212 log27(1 + xy), x \u2208 \u00c51 \u00e3 Ta x\u00e9t c\u00e1c tr\u01b0\u1eddng h\u1ee3p sau ; 6. 3 1 TH1. N\u1ebfu y < 0 th\u00ec \u2212y < < 3, suy ra y > \u22123 hay y \u2208 {\u22121; \u22122}. x \u00c51\u00e3 V\u1edbi y = \u22121 th\u00ec ta c\u00f3 f < 0 v\u00e0 lim f (x) = +\u221e n\u00ean theo \u0111\u1ecbnh l\u00ed gi\u00e1 tr\u1ecb trung gian, ph\u01b0\u01a1ng 3 x\u21921\u2212 tr\u00ecnh (\u2217) c\u00f3 nghi\u1ec7m tr\u00ean \u00c51 \u00e3 \u2282 \u00c51 \u00e3 T\u01b0\u01a1ng t\u1ef1, v\u1edbi y = \u22122 ph\u01b0\u01a1ng tr\u00ecnh (\u2217) c\u0169ng c\u00f3 ;1 ;6 . 33 nghi\u1ec7m tr\u00ean \u00c51 1\u00e3 \u2282 \u00c51 \u00e3 ; ; 6. 32 3 TH2. N\u1ebfu y =0 th\u00ec ph\u01b0\u01a1ng tr\u00ecnh (\u2217) tr\u1edf th\u00e0nh 3x2 \u2212 18x = 0 \u00f1x = 0 kh\u00f4ng th\u1ecfa m\u00e3n. \u21d4 , x=6 TH3. N\u1ebfu y \u2265 19, ta c\u00f3 f (x) = 6x + (y \u2212 18) \u2212 y , (1 + xy) ln 27 y2 \u00c5 1 \u00e3 6 f (x) = 6 + (1 + xy)2 ln 27 > 0, \u2200x \u2208 3 ; . \u00c51 \u00e3 Suy ra h\u00e0m s\u1ed1 f (x) \u0111\u1ed3ng bi\u1ebfn tr\u00ean ; 6 . Do \u0111\u00f3 3 f (x) > f \u00c51 \u00e3 = 2 + (y \u2212 18) \u2212 y \u00c5 1 \u00e3 6 3 (3 + y) ln 3 > 0, \u2200y \u2265 19, \u2200x \u2208 3 ; . \u00c51 \u00e3 \u0110i\u1ec1u n\u00e0y d\u1eabn \u0111\u1ebfn f (x) l\u00e0 h\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn tr\u00ean kho\u1ea3ng ; 6 . Suy ra 3 f (x) > f \u00c51\u00e3 = y \u2212 log27 y \u2212 17 3 3 1+ . 3 3 1 X\u00e9t h\u00e0m s\u1ed1 g(t) = t \u2212 log27(1 + t), t > 0. D\u1ec5 th\u1ea5y g (t) = 1 \u2212 (1 + t) ln 27 > 0 (v\u1edbi m\u1ecdi t > 0) n\u00ean h\u00e0m s\u1ed1 g(t) \u0111\u1ed3ng bi\u1ebfn tr\u00ean kho\u1ea3ng (0; +\u221e), k\u00e9o theo \u00c51\u00e3 y \u2212 17 \u2265 g \u00c519\u00e3 \u2212 17 > 0. f =g 3 33 3 3 Nh\u01b0 v\u1eady, ph\u01b0\u01a1ng tr\u00ecnh (\u2217) v\u00f4 nghi\u1ec7m tr\u00ean \u00c51 \u00e3 trong tr\u01b0\u1eddng h\u1ee3p n\u00e0y. ; 6 3 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 258 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT TH4. N\u1ebfu 1 \u2264 y \u2264 18 th\u00ec t\u1eeb t\u00ednh \u0111\u1ed3ng bi\u1ebfn c\u1ee7a h\u00e0m g(t), ta suy ra \u00c51\u00e3 y \u2212 17 < 0, \u2200y \u2208 Z, 1 \u2264 y \u2264 18. f =g 3 33 f (3) = g(3y) \u2212 27 > 0, \u2200y \u2208 Z, 1 \u2264 y \u2264 18. \u00c51 \u00e3 \u0110i\u1ec1u n\u00e0y d\u1eabn \u0111\u1ebfn ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 nghi\u1ec7m tr\u00ean ; 6 . 3 K\u1ebft lu\u1eadn: c\u00f3 t\u1ea5t c\u1ea3 20 gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a y th\u1ecfa m\u00e3n y\u00eau c\u1ea7u b\u00e0i to\u00e1n. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 102 (C\u00e2u 44 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean d\u01b0\u01a1ng y sao cho t\u1ed3n t\u1ea1i s\u1ed1 th\u1ef1c x \u2208 (1; 6) th\u1ecfa m\u00e3n 4(x \u2212 1)ex = y(ex + xy \u2212 2x2 \u2212 3)? A 18. B 15. C 16. D 17. \u0253 L\u1eddi gi\u1ea3i. 4(x \u2212 1)ex = y(ex + xy \u2212 2x2 \u2212 3) \u21d4 4(x \u2212 1)ex \u2212 y(ex + xy \u2212 2x2 \u2212 3) = 0. X\u00e9t h\u00e0m s\u1ed1 y = f (x) = 4(x \u2212 1)ex \u2212 y(ex + xy \u2212 2x2 \u2212 3) li\u00ean t\u1ee5c tr\u00ean [1; 6] c\u00f3 f (x) = 4ex + 4(x \u2212 1)ex \u2212 y(ex + y \u2212 4x) = (ex + y)(4x \u2212 y). Cho f (x) = 0 \u21d4 x = y . 4 Do x \u2208 (1; 6) n\u00ean h\u00e0m s\u1ed1 y = f (x) s\u1ebd t\u1ed3n t\u1ea1i \u0111i\u1ec3m c\u1ef1c tr\u1ecb x = y khi y \u2208 (4; 24). 4 T\u1eeb \u0111\u00f3 ta c\u00f3 c\u01a1 s\u1edf chia c\u00e1c tr\u01b0\u1eddng h\u1ee3p nh\u01b0 sau Tr\u01b0\u1eddng h\u1ee3p 1: y \u2264 4. x1 6 f (x) + f (6) f (x) f (1) \u00aef (1) = \u2212y(e + y \u2212 5) Ta c\u00f3 f (6) = 20e6 \u2212 y(e6 + 6y \u2212 75). \u0110i\u1ec1u ki\u1ec7n c\u1ea7n v\u00e0 \u0111\u1ee7 \u0111\u1ec3 t\u1ed3n t\u1ea1i x l\u00e0 \u00aef (6) > 0 \u21d2 f (1) < 0 \u21d4 \u2212y(e + y \u2212 5) < 0 \u21d4 y > 5 \u2212 e. f (1) \u00b7 f (6) < 0 M\u00e0 y \u2264 4 v\u00e0 y \u2208 N\u2217 n\u00ean y \u2208 {3, 4}. (1) Tr\u01b0\u1eddng h\u1ee3p 2: y \u2265 24. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 259 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit x1 6 f (x) \u2212 f (1) f (x) f (6) \u00aef (1) = \u2212y(e + y \u2212 5) Ta c\u00f3 f (6) = 20e6 \u2212 y(e6 + 6y \u2212 75). \u0110i\u1ec1u ki\u1ec7n c\u1ea7n v\u00e0 \u0111\u1ee7 \u0111\u1ec3 t\u1ed3n t\u1ea1i x l\u00e0 \u00aef (6) < 0 \u21d2 f (1) > 0. f (1) \u00b7 f (6) < 0 M\u1eb7t kh\u00e1c ta th\u1ea5y \u2212y(e + y \u2212 5) < 0, \u2200y \u2265 24 (v\u00f4 l\u00ed) n\u00ean lo\u1ea1i. Tr\u01b0\u1eddng h\u1ee3p 3: 4 < y < 24. x1 y 6 f (x) 4 + f (1) \u22120 f (6) f (x) f y 4 Do f (1) < 0 n\u00ean \u0111\u1ec3 t\u1ed3n t\u1ea1i nghi\u1ec7m x \u2208 (1; 6) th\u00ec f (6) > 0 \u21d4 20e6 \u2212 y(e6 + 6y \u2212 75 > 0 \u00ae \u2212 6y2 \u2212 (e6 \u2212 75)y + 20e6 > 0 \u21d4 y \u2208 N\u2217; y \u2208 (4; 24) \u21d4 y \u2208 {5; 6; . . . ; 18}. (2) T\u1eeb (1) v\u00e0 (2) suy ra y \u2208 {3; 4; 5; 6; . . . ; 18}. V\u1eady c\u00f3 t\u1ea5t c\u1ea3 16 gi\u00e1 tr\u1ecb nguy\u00ean d\u01b0\u01a1ng y th\u1ecfa \u0111\u1ec1 b\u00e0i. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 103 (C\u00e2u 44 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean d\u01b0\u01a1ng y sao cho t\u1ed3n t\u1ea1i s\u1ed1 th\u1ef1c x \u2208 (1; 6) th\u1ecfa m\u00e3n 4(x \u2212 1)ex = y ex + xy + 2x2 \u2212 3 ? A 15. B 18. C 17. D 16. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 4(x \u2212 1)ex = y (ex + xy \u2212 2x2 \u2212 3) \u21d4 4(x \u2212 1)ex \u2212 y (ex + xy \u2212 2x2 \u2212 3) = 0. X\u00e9t f (x) = 4(x \u2212 1)ex \u2212 y (ex + xy \u2212 2x2 \u2212 3) li\u00ean t\u1ee5c tr\u00ean kho\u1ea3ng (1; 6). Ta c\u00f3 f (x) = 4ex + 4(x \u2212 1)ex \u2212 y (ex + y \u2212 4x) = 4xex \u2212 y (ex + y \u2212 4x) = (ex + y) (4x \u2212 y). Tr\u01b0\u1eddng h\u1ee3p 1. V\u1edbi x \u2208 (1; 6) v\u00e0 0 < y \u2264 4 \u21d2 4x \u2212 y > 0, ta c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean sau Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 260 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT x1 6 f (x) + f (6) f (x) f (1) V\u1edbi f (1) = \u2212y(e + y \u2212 5) v\u00e0 f (6) = 20e6 \u2212 y (e6 + 6y \u2212 75) = (20 \u2212 y)e6 + y(75 \u2212 6y) > 0. Suy ra y\u00eau c\u1ea7u b\u00e0i to\u00e1n \u0111\u01b0\u1ee3c th\u1ecfa m\u00e3n khi f (1) < 0 \u21d4 \u2212y(e + y \u2212 5) < 0 \u21d4 e + y \u2212 5 > 0 \u21d4 y > 5 \u2212 e (\u2248 2,28) Do y \u2208 N\u2217, y \u2264 4 n\u00ean y \u2208 {3; 4}. Tr\u01b0\u1eddng h\u1ee3p 2. V\u1edbi x \u2208 (1; 6) v\u00e0 y \u2265 24 \u21d2 4x \u2212 y < 0, ta c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean sau x1 6 f (x) \u2212 f (1) f (x) f (6) Ta th\u1ea5y f (1) = \u2212y(e + y \u2212 5) < 0, y \u2208 N\u2217, y \u2265 24. Suy ra y\u00eau c\u1ea7u b\u00e0i to\u00e1n kh\u00f4ng \u0111\u01b0\u1ee3c th\u1ecfa m\u00e3n. Tr\u01b0\u1eddng h\u1ee3p 3. V\u1edbi x \u2208 (1; 6) v\u00e0 4 < y < 24, ta c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean sau y x1 4 6 f (x) \u22120+ f (1) f (6) f (x) y f 4 Do y > 4 th\u00ec f (1) < 0 n\u00ean \u0111\u1ec3 t\u1ed3n t\u1ea1i nghi\u1ec7m x \u2208 (1; 6) th\u00ec f (6) > 0 \u21d4 20e6 \u2212 y e6 + 6y \u2212 75 > 0 \u00ae \u2212 6y2 \u2212 e6 \u2212 75 y + 20e6 > 0 \u21d4 y \u2208 N\u2217; y \u2208 (4; 24) \u21d4 y \u2208 {5; 6; . . . ; 18}. T\u1eeb 3 tr\u01b0\u1eddng h\u1ee3p tr\u00ean ta c\u00f3 y \u2208 {3; 4; 5; 6; . . . ; 18}. V\u1eady c\u00f3 t\u1ea5t c\u1ea3 16 gi\u00e1 tr\u1ecb y nguy\u00ean d\u01b0\u01a1ng th\u1ecfa m\u00e3n. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 104 (C\u00e2u 44 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean d\u01b0\u01a1ng y sao cho t\u1ed3n t\u1ea1i s\u1ed1 th\u1ef1c x \u2208 (1; 5) th\u1ecfa m\u00e3n 4(x \u2212 1)ex = y(ex + xy \u2212 2x2 \u2212 3)? A 14. B 12. C 10. D 11. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 261","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit X\u00e9t h\u00e0m s\u1ed1 f (x) = 4(x \u2212 1)ex \u2212 yex \u2212 xy2 + 2x2y + 3y. Ta c\u00f3 f (x) = 4ex + 4(x \u2212 1)ex \u2212 yex \u2212 y2 + 4xy = (4x \u2212 y)ex \u2212 y2 + 4xy = (4x \u2212 y)(ex + y). Do y \u2208 N\u2217 n\u00ean f (x) = 0 \u21d4 x = y . 4 TH 1. y \u2208 (1; 5) \u21d4 y \u2208 (4; 20). Ta c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean 4 x1 y 5 f (x) 4 + \u22120 f (1) f (5) f (x) y f 4 Ta c\u00f3 f (1) = \u2212ye \u2212 y2 + 5y = y(5 \u2212 e \u2212 y) < 0, \u2200y \u2208 (4; 20), f (5) = \u22125y2 + (53 \u2212 e5) y + 16e5. Khi \u0111\u00f3 t\u1ed3n t\u1ea1i x \u2208 (1; 5) sao cho f (x) = 0 khi v\u00e0 ch\u1ec9 khi f (5) > 0 \u21d4 \u22125y2 + 53 \u2212 e5 y + 16e5 > 0 \u21d2 \u221233,33 < y < 14,24. Suy ra y \u2208 {5; 6; 7 . . . ; 14}, c\u00f3 10 gi\u00e1 tr\u1ecb th\u1ecfa m\u00e3n. TH 2. 0 < y \u2264 1 \u21d4 0 < y \u2264 4. Ta c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean 4 x1 5 f (x) + f (5) f (x) f (1) Khi \u0111\u00f3 t\u1ed3n t\u1ea1i x \u2208 (1; 5) sao cho f (x) = 0 khi v\u00e0 ch\u1ec9 khi \u00aef (1) < 0 f (5) > 0 \u00ae \u2212 y2 + (5 \u2212 e)y < 0 \u21d4 0 < y < 14,24 \u21d2 y>5\u2212e \u21d2 2,24 < y \u2264 4. Suy ra y \u2208 {3; 4}. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 262 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT TH 3. y \u2265 5 \u21d4 y \u2265 20. Ta c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean 4 x1 5 f (x) \u2212 f (1) f (x) f (5) Khi \u0111\u00f3 t\u1ed3n t\u1ea1i x \u2208 (1; 5) sao cho f (x) = 0 khi v\u00e0 ch\u1ec9 khi \u00aef (1) > 0 f (5) < 0 \u00aey \u2208 (0; 5 \u2212 e) \u21d2 y > 14,24. Tr\u01b0\u1eddng h\u1ee3p n\u00e0y kh\u00f4ng t\u1ed3n t\u1ea1i y. Qua 3 tr\u01b0\u1eddng h\u1ee3p c\u00f3 12 s\u1ed1 y nguy\u00ean d\u01b0\u01a1ng th\u1ecfa m\u00e3n y\u00eau c\u1ea7u \u0111\u1ec1 b\u00e0i. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 105 (C\u00e2u 31 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm gi\u00e1 tr\u1ecb th\u1ef1c c\u1ee7a tham s\u1ed1 m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh 9x \u2212 2.3x+1 + m = 0 c\u00f3 hai nghi\u1ec7m th\u1ef1c x1, x2 th\u1ecfa m\u00e3n x1 + x2 = 1. A m = 6. B m = \u22123. C m = 3. D m = 1. \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t t = 3x > 0. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho tr\u1edf th\u00e0nh: t2 \u2212 6t + m = 0 (*). \uf8f1 \u2265 0 \u21d4 \u00aem \u2264 9 \u21d4 0 < m \u2264 9 (**). \uf8f4\u2206 > 0 \uf8f2 m >0 Ph\u01b0\u01a1ng tr\u00ecnh (*) c\u00f3 hai nghi\u1ec7m d\u01b0\u01a1ng khi S >0 \uf8f3\uf8f4P G\u1ecdi t1, t2 l\u00e0 hai nghi\u1ec7m c\u1ee7a (*). Ta c\u00f3: x1 = log3 t1; x2 = log3 t2. M\u00e0 x1 + x2 = 1 n\u00ean log3 t1 + log3 t2 = 1 \u21d2 t1.t2 = 3 \u21d2 m = 3 (th\u1ecfa (**)). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 106 (C\u00e2u 24 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). M\u1ed9t ng\u01b0\u1eddi g\u1eedi ti\u1ebft ki\u1ec7m v\u00e0o m\u1ed9t ng\u00e2n h\u00e0ng v\u1edbi l\u00e3i su\u1ea5t 7,2%\/n\u0103m. Bi\u1ebft r\u1eb1ng n\u1ebfu kh\u00f4ng r\u00fat ti\u1ec1n ra kh\u1ecfi ng\u00e2n h\u00e0ng th\u00ec c\u1ee9 sau m\u1ed7i n\u0103m s\u1ed1 ti\u1ec1n l\u00e3i s\u1ebd \u0111\u01b0\u1ee3c nh\u1eadp v\u00e0o v\u1ed1n \u0111\u1ec3 t\u00ednh l\u00e3i cho n\u0103m ti\u1ebfp theo. H\u1ecfi sau \u00edt nh\u1ea5t bao nhi\u00eau n\u0103m ng\u01b0\u1eddi \u0111\u00f3 thu \u0111\u01b0\u1ee3c (c\u1ea3 s\u1ed1 ti\u1ec1n g\u1eedi ban \u0111\u1ea7u v\u00e0 l\u00e3i) g\u1ea5p \u0111\u00f4i s\u1ed1 ti\u1ec1n g\u1eedi ban \u0111\u1ea7u, gi\u1ea3 \u0111\u1ecbnh trong kho\u1ea3ng th\u1eddi gian n\u00e0y l\u00e3i su\u1ea5t kh\u00f4ng thay \u0111\u1ed5i v\u00e0 ng\u01b0\u1eddi \u0111\u00f3 kh\u00f4ng r\u00fat ti\u1ec1n ra? A 11 n\u0103m. B 12 n\u0103m. C 9 n\u0103m. D 10 n\u0103m. \u0253 L\u1eddi gi\u1ea3i. Gi\u1ea3 s\u1eed ng\u01b0\u1eddi \u1ea5y g\u1eedi s\u1ed1 ti\u1ec1n M0 v\u00e0o ng\u00e2n h\u00e0ng. Khi \u0111\u00f3, sau n n\u0103m s\u1ed1 ti\u1ec1n c\u1ee7a ng\u01b0\u1eddi \u1ea5y \u0111\u01b0\u1ee3c t\u00ednh b\u1eb1ng c\u00f4ng th\u1ee9c M = M0(1 + 7,2%)n = M0 \u00b7 1, 072n. Theo \u0111\u1ec1 b\u00e0i, ta t\u00ecm n th\u1ecfa m\u00e3n M \u2265 2M0 \u21d4 M0 \u00b7 1,072n \u2265 2M0 \u21d4 n \u2265 log1,072 2 \u2248 9,969602105. V\u1eady sau \u00edt nh\u1ea5t 10 n\u0103m ng\u01b0\u1eddi \u1ea5y m\u1edbi thu \u0111\u01b0\u1ee3c s\u1ed1 ti\u1ec1n nhi\u1ec1u g\u1ea5p \u0111\u00f4i s\u1ed1 ti\u1ec1n v\u1ed1n ban \u0111\u1ea7u. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 263 S\u0110T: 0905.193.688","6. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh m\u0169 v\u00e0 l\u00f4garit B\u00c0I 6. B\u1ea4T PH\u01af\u01a0NG TR\u00ccNH M\u0168 V\u00c0 L\u00d4GARIT \u0104 C\u00e2u 1 (C\u00e2u 16 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh log x \u2265 1 l\u00e0 A (10; +\u221e). B (0; +\u221e). C [10; +\u221e). D (\u2212\u221e; 10). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log x \u2265 1 \u21d4 x \u2265 101 = 10. Do \u0111\u00f3 t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho l\u00e0 S = [10; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 2 (C\u00e2u 1 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh 3x < 2 l\u00e0 A (\u2212\u221e; log3 2). B (log3 2; +\u221e). C (\u2212\u221e; log2 3). D (log2 3; +\u221e). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 3x < 2 \u21d4 x < log3 2. V\u1eady t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho l\u00e0 (\u2212\u221e; log3 2). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 3 (C\u00e2u 26 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh 2x < 5 l\u00e0 A (\u2212\u221e; log2 5). B (log2 5; +\u221e). C (\u2212\u221e; log5 2). D (log5 2; +\u221e). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 2x < 5 \u21d4 x < log2 5. V\u1eady t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho l\u00e0 (\u2212\u221e; log2 5). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 4 (C\u00e2u 20 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh 2x > 3 l\u00e0 A (log3 2; +\u221e). B (\u2212\u221e; log23). C (\u2212\u221e; log32). D (log2 3; +\u221e). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 2x > 3 \u21d4 x > log2 3. T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 S = (log2 3; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 5 (C\u00e2u 6 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh 2x > 5 l\u00e0 A (\u2212\u221e; log2 5). B (log5 2; +\u221e). C (\u2212\u221e; log5 2). D (log2 5; +\u221e). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 2x > 5 \u21d4 x > log2 5. V\u1eady t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 (log2 5; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 6 (C\u00e2u 8 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh log3(2x) > 2 l\u00e0 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 264 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT \u00c5 9 \u00e3 \u00c5 9\u00e3 +\u221e . 0; . A (0; 4). B 2 ; C 2 D (4; +\u221e). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log3(2x) > 2 \u21d4 2x > 32 \u21d4 x > 9 . 2 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 7 (C\u00e2u 3 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh log2 (3x) > 3 l\u00e0 \u00c5 8 \u00e3 \u00c5 8\u00e3 +\u221e . 0; . A (3; +\u221e). B ; C D (0; 3). 33 \u0253 L\u1eddi gi\u1ea3i. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho t\u01b0\u01a1ng \u0111\u01b0\u01a1ng 3x > 23 \u21d4 x > 8 . 3 \u00c5 8 \u00e3 +\u221e . V\u1eady t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 3 ; Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 8 (C\u00e2u 18 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh log5(x + 1) > 2 l\u00e0 A (24; +\u221e). B (9; +\u221e). C (25; +\u221e). D (31; +\u221e). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log5(x + 1) > 2 \u21d4 x + 1 > 52 \u21d4 x > 24. V\u1eady t\u1eadp h\u1ee3p nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 S = (24; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 9 (C\u00e2u 7 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh 2x > 6 l\u00e0 A (log2 6; +\u221e). B (\u2212\u221e; 3). C (3; +\u221e). D (\u2212\u221e; log2 6). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 2x > 6 \u21d4 x > log2 6. V\u1eady t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 S = (log2 6; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 10 (C\u00e2u 16 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00e0m s\u1ed1 f (x) = 2x \u00b7 7x2. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o sau \u0111\u00e2y l\u00e0 kh\u1eb3ng \u0111\u1ecbnh sai? A f (x) < 1 \u21d4 x + x2 log2 7 < 0. B f (x) < 1 \u21d4 x ln 2 + x2 ln 7 < 0. C f (x) < 1 \u21d4 x log7 2 + x2 < 0. D f (x) < 1 \u21d4 1 + x log2 7 < 0. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) = 2x.7x2 < 1 \u21d4 log2 \u00c42x.7x2\u00e4 < 0 \u21d4 x + x2 log2 7 < 0, n\u00ean c\u00e2u A \u0111\u00fang. V\u00e0 f (x) = 2x.7x2 < 1 \u21d4 ln \u00c42x.7x2\u00e4 < 0 \u21d4 x ln 2 + x2 ln 7 < 0, n\u00ean c\u00e2u B \u0111\u00fang. V\u00e0 f (x) = 2x.7x2 < 1 \u21d4 log7 \u00c42x.7x2\u00e4 < 0 \u21d4 x log7 2 + x2 < 0, n\u00ean c\u00e2u C \u0111\u00fang D sai do f (x) = 2x.7x2 < 1 \u21d4 log2 \u00c42x.7x2\u00e4 < 0 \u21d4 x + x2 log2 7 < 0 \u21d4 x (1 + xlog27) < 0. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 265 S\u0110T: 0905.193.688","6. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh m\u0169 v\u00e0 l\u00f4garit \u0104 C\u00e2u 11 (C\u00e2u 15 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh 3x2\u22122x < 27 l\u00e0 A (\u2212\u221e; \u22121). B (3; +\u221e). C (\u22121; 3). D (\u2212\u221e; \u22121) \u222a (3; +\u221e). \u0253 L\u1eddi gi\u1ea3i. 3x2\u22122x < 27 \u21d4 3x2\u22122x < 33 \u21d4 x2 \u2212 2x < 3 \u21d4 x2 \u2212 2x \u2212 3 < 0 \u21d4 \u22121 < x < 3. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 12 (C\u00e2u 21 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh 5x\u22121 \u2265 5x2\u2212x\u22129 l\u00e0 A [\u2212 2 ; 4]. B [\u2212 4 ; 2]. C (\u2212 \u221e ; \u2212 2] \u222a [4 ; + \u221e). D (\u2212 \u221e ; \u2212 4] \u222a [2 ; + \u221e). \u0253 L\u1eddi gi\u1ea3i. 5x \u2212 1 \u2265 5x2 \u2212 x \u2212 9 \u21d4 x \u2212 1 \u2265 x2 \u2212 x \u2212 9 \u21d4 x2 \u2212 2x \u2212 8 \u2264 0 \u21d4 \u2212 2 \u2264 x \u2264 4. V\u1eady t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho l\u00e0 [\u2212 2 ; 4]. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 13 (C\u00e2u 37 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh 3x2\u221223 < 9 l\u00e0 A (\u22125; 5). B (\u2212\u221e; 5). C (5; +\u221e). D (0; 5). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 3x2\u221223 < 9 \u21d4 x2 \u2212 23 < 2 \u21d4 x2 \u2212 25 < 0 \u21d4 \u22125 < x < 5. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 14 (C\u00e2u 30 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh 2x2\u22121 < 8 l\u00e0 A (0; 2). B (\u2212\u221e; 2). C (\u22122; 2). D (2; +\u221e). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 2x2\u22121 < 8 \u21d4 2x2\u22121 < 23 \u21d4 x2 \u2212 1 < 3 \u21d4 x2 \u2212 4 < 0 \u21d4 \u22122 < x < 2. V\u1eady t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 S = (\u22122; 2). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 15 (C\u00e2u 38 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh log3 (18 \u2212 x2) \u2265 2 l\u00e0 A (\u2212\u221e; 3]. B (0; 3]. C [\u22123; 3]. D (\u2212\u221e; \u22123] \u222a [3; +\u221e). \u0110i\u1ec1u ki\u1ec7n x\u00e1c \u0111\u1ecbnh l\u00e0 \u0253 L\u1eddi gi\u1ea3i. \u221a\u221a 18 \u2212 x2 > 0 \u21d4 \u22123 2 < x < 3 2. V\u1edbi \u0111i\u1ec1u ki\u1ec7n x\u00e1c \u0111\u1ecbnh tr\u00ean, b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi 18 \u2212 x2 \u2265 32 \u21d4 9 \u2212 x2 \u2265 0 \u21d4 \u22123 \u2264 x \u2264 3. K\u1ebft h\u1ee3p v\u1edbi \u0111i\u1ec1u ki\u1ec7n x\u00e1c \u0111\u1ecbnh, t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho l\u00e0 S = [\u22123; 3]. Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 266 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT \u0104 C\u00e2u 16 (C\u00e2u 36 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh log3(13 \u2212 x2) \u2265 2 l\u00e0 A (\u2212\u221e; \u22122] \u222a [2; +\u221e). B (\u2212\u221e; 2]. C (0; 2]. D [\u22122; 2]. \u0253 L\u1eddi gi\u1ea3i. V\u00ec c\u01a1 s\u1ed1 a = 3 > 1 n\u00ean log3(13 \u2212 x2) \u2265 2 \u21d4 13 \u2212 x2 \u2265 9 \u21d4 \u2212x2 + 4 \u2265 0 \u21d4 \u22122 \u2264 x \u2264 2. V\u1eady t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 [\u22122; 2]. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 17 (C\u00e2u 36 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh log3(36 \u2212 x2) \u2265 3 l\u00e0 A (\u2212\u221e; \u22123] \u222a [3; +\u221e). B (\u2212\u221e; 3]. C [\u22123; 3]. D (0; 3]. \u0253 L\u1eddi gi\u1ea3i. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi \u00ae36 \u2212 x2 > 0 \u21d4 36 \u2212 x2 \u2265 27 \u21d4 x2 \u2264 9 \u21d4 \u22123 \u2264 x \u2264 3. 36 \u2212 x2 \u2265 27 V\u1eady t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho l\u00e0 [\u22123; 3]. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 18 (C\u00e2u 36 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh log3 (31 \u2212 x2) \u2265 3 l\u00e0 A (\u2212\u221e; 2]. B [\u22122; 2]. C (\u2212\u221e; \u22122] \u222a [2; +\u221e). D (0; 2]. \u0253 L\u1eddi gi\u1ea3i. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho t\u01b0\u01a1ng \u0111\u01b0\u01a1ng 31 \u2212 x2 \u2265 33 \u21d4 x2 \u2264 4 \u21d4 \u22122 \u2264 x \u2264 2. V\u1eady t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 [\u22122; 2]. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 19 (C\u00e2u 24 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh log3(2x) > 4 l\u00e0 \u00c5 81\u00e3 \u00c5 81 \u00e3 0; . +\u221e . A (0; 32). B 2 C (32; +\u221e). D 2 ; \uf8f1x > 0 \u0253 L\u1eddi gi\u1ea3i. \uf8f2 \u00ae2x > 0 81 Ta c\u00f3 log3(2x) > 4 \u21d4 \u21d4 \u21d4x> . 2x > 34 \uf8f3x > 81 2 2 \u00c5 81 \u00e3 +\u221e . V\u1eady t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho l\u00e0 S = 2 ; Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 267 S\u0110T: 0905.193.688","6. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh m\u0169 v\u00e0 l\u00f4garit \u0104 C\u00e2u 20 (C\u00e2u 13 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh 22x < 2x+6 l\u00e0 A (0; 6). B (\u2212\u221e; 6). C (0; 64). D (6; +\u221e). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 22x < 2x+6 \u21d4 2x < x + 6 \u21d4 x < 6. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 21 (C\u00e2u 34 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh 3x2\u221213 < 27 l\u00e0 A (4; +\u221e). B (\u22124; 4). C (\u2212\u221e; 4). D (\u22124; 4). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 3x2\u221213 < 27 \u21d4 3x2\u221213 < 33 \u21d4 x2 \u2212 13 < 3 \u21d4 x2 \u2212 16 < 0 \u21d4 \u22124 < x < 4. V\u1eady b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 t\u1eadp nghi\u1ec7m l\u00e0 (\u22124; 4). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 22 (C\u00e2u 29 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh 2x2\u22127 < 4 l\u00e0 A (\u22123; 3). B (0; 3). C (\u2212\u221e; 3). D (3; +\u221e). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 2x2\u22127 < 4 \u21d4 x2 \u2212 7 < 2 \u21d4 \u22123 < x < 3. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 23 (C\u00e2u 31 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh 9x + 2 \u00b7 3x \u2212 3 > 0 l\u00e0 A [0; +\u221e). B (0; +\u221e). C (1; +\u221e). D [1; +\u221e). \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t t = 3x. \u0110i\u1ec1u ki\u1ec7n t > 0. Khi \u0111\u00f3 b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh tr\u1edf th\u00e0nh t2 + 2t \u2212 3 > 0 \u21d4 \u00f1t < \u22123 t > 1. K\u1ebft h\u1ee3p \u0111i\u1ec1u ki\u1ec7n, ta c\u00f3 t > 1 \u21d2 3x > 1 \u21d4 x > 0. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 24 (C\u00e2u 40 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong n\u0103m 2019, di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi c\u1ee7a t\u1ec9nh A l\u00e0 800 ha. Gi\u1ea3 s\u1eed di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi c\u1ee7a t\u1ec9nh A m\u1ed7i n\u0103m ti\u1ebfp theo \u0111\u1ec1u t\u0103ng 6% so v\u1edbi di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi c\u1ee7a n\u0103m li\u1ec1n tr\u01b0\u1edbc. K\u1ec3 t\u1eeb sau n\u0103m 2019, n\u0103m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 n\u0103m \u0111\u1ea7u ti\u00ean t\u1ec9nh A c\u00f3 di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi trong n\u0103m \u0111\u00f3 \u0111\u1ea1t tr\u00ean 1400 ha. A N\u0103m 2029. B N\u0103m 2028. C N\u0103m 2048. D N\u0103m 2049. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 Sn = 1400 ha; A = 800 ha; r = 6%. \u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c: Sn = A(1 + r)n \u21d2 A(1 + r)n > 1400 \u21d4 n > log1+r \u00c5 1400 \u00e3 \u21d4 n > log1,06 \u00c5 1400 \u00e3 \u21d4 n > 9,609 \u21d2 n = 10. A 800 V\u1eady n\u0103m \u0111\u1ea7u ti\u00ean l\u00e0 n\u0103m 2029. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 268 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT \u0104 C\u00e2u 25 (C\u00e2u 17 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm t\u1eadp nghi\u1ec7m S c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh log22 x \u2212 5 log2 x + 4 \u2265 0. A S = (\u2212\u221e; 2] \u222a [16; +\u221e). B S = [2; 16]. C S = (0; 2] \u222a [16; +\u221e). D S = (\u2212\u221e; 1] \u222a [4; +\u221e). \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n: x > 0. \u0110\u1eb7t t = log2 x, b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho tr\u1edf th\u00e0nh t2 \u2212 5t + 4 \u2265 0 \u21d4 \u00f1t \u2265 4 t \u2264 . 1 \u21d2 \u00f1log2 x \u2265 4 \u21d4 \u00f1x \u2265 16 . log2 x \u2264 1 x\u22642 K\u1ebft h\u1ee3p \u0111i\u1ec1u ki\u1ec7n ta \u0111\u01b0\u1ee3c t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 S = (0; 2] \u222a [16; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 26 (C\u00e2u 27 - M\u0110 104 - 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(a2b) = 4a3. Gi\u00e1 tr\u1ecb c\u1ee7a ab2 b\u1eb1ng A 4. B 2. C 3. D 6. \u0253 L\u1eddi gi\u1ea3i. 9log3(a2b) = 4a3 \u21d4 32 log3(a2b) = 4a3 \u21d4 3log3(a2b)2 = 4a3 \u21d4 (a2b)2 = 4a3 \u21d4 a4b2 = 4a3 \u21d4 ab2 = 4. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 27 (C\u00e2u 40 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean x th\u1ecfa m\u00e3n \u00c43x2 \u2212 9x\u00e4 [log3(x + 25) \u2212 3] \u2264 0? A 24. B V\u00f4 s\u1ed1. C 26. D 25. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x + 25 > 0 \u21d4 x > \u221225. Ta c\u00f3 \u00c43x2 \u2212 9x\u00e4 [log3(x + 25) \u2212 3] = 0 \u21d4 \u00f13x2 \u2212 9x = 0 \u2212 3 = 0 \u21d4 \u00f1x = 0 log3(x + 25) x = 2. B\u1ea3ng x\u00e9t d\u1ea5u v\u1ebf tr\u00e1i x \u221225 0 2 +\u221e 3x2 \u2212 9x +0\u22120+ log3(x + 25) \u2212 3 \u2212 \u22120+ VT \u2212 0 + 0 + \u00f1 \u2212 25 < x \u2264 0 D\u1ef1a v\u00e0o b\u1ea3ng x\u00e9t d\u1ea5u suy ra nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho l\u00e0 x = 2. V\u00ec x \u2208 Z n\u00ean x \u2208 {\u221224, \u221223, . . . , \u22121, 0, 2}, c\u00f3 26 gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a x tho\u1ea3 m\u00e3n. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 28 (C\u00e2u 39 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean d\u01b0\u01a1ng a sao cho \u1ee9ng v\u1edbi m\u1ed7i a c\u00f3 \u0111\u00fang ba s\u1ed1 nguy\u00ean b th\u1ecfa m\u00e3n (3b \u2212 3)(a.2b \u2212 18) < 0? Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 269 S\u0110T: 0905.193.688","6. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh m\u0169 v\u00e0 l\u00f4garit A 72. B 73. C 71. D 74. \u0253 L\u1eddi gi\u1ea3i. Tr\u01b0\u1eddng h\u1ee3p 1. V\u1edbi 3b \u2212 3 > 0 \u21d4 b > 1. Khi \u0111\u00f3 a.2b \u2212 18 < 0 \u21d2 2b < 18 \u00b7 a Suy ra 3 gi\u00e1 tr\u1ecb nguy\u00ean b c\u00f3 th\u1ec3 l\u00e0 b \u2208 {2; 3; 4}. Do \u0111\u00f3 24 < 18 \u2264 25 \u21d2 9 \u2264 a < 9 \u21d2 a = 1. a 16 8 Tr\u01b0\u1eddng h\u1ee3p 2. V\u1edbi 3b \u2212 3 < 0 \u21d4 b < 1. Khi \u0111\u00f3 a.2b \u2212 18 > 0 \u21d2 2b > 18 \u00b7 a Suy ra 3 gi\u00e1 tr\u1ecb nguy\u00ean b c\u00f3 th\u1ec3 l\u00e0 b \u2208 {\u22122; \u22121; 0}. Do \u0111\u00f3 2\u22123 \u2264 18 < 2\u22122 \u21d2 72 < a \u2264 144. a S\u1ed1 gi\u00e1 tr\u1ecb nguy\u00ean d\u01b0\u01a1ng c\u1ee7a a trong tr\u01b0\u1eddng h\u1ee3p n\u00e0y l\u00e0 144 \u2212 73 + 1 = 72. V\u1eady c\u00f3 t\u1ed5ng c\u1ed9ng 1 + 72 = 73 gi\u00e1 tr\u1ecb a th\u1ecfa m\u00e3n. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 29 (C\u00e2u 40 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean d\u01b0\u01a1ng a sao cho \u1ee9ng v\u1edbi m\u1ed7i a c\u00f3 \u0111\u00fang hai s\u1ed1 nguy\u00ean b th\u1ecfa m\u00e3n (5b \u2212 1)(a.2b \u2212 5) < 0? A 20. B 21. C 22. D 19. \u0253 L\u1eddi gi\u1ea3i. \u00f15b \u2212 1 = 0 \uf8eeb = 0 Ta c\u00f3 (5b \u2212 1)(a.2b \u2212 5) = 0 \u21d4 a.2b \u2212 5 = 0 \u21d4 \uf8f0 = log2 5\u00b7 b a 5 TH1: log2 a < b < 0. Khi \u0111\u00f3 \u0111\u1ec3 t\u1ed3n t\u1ea1i \u0111\u00fang hai gi\u00e1 tr\u1ecb c\u1ee7a b th\u00ec b \u2208 {\u22122; \u22121}. Do \u0111\u00f3 \u22123 \u2264 log2 5 < \u22122 \u21d4 1 \u2264 5 < 1 \u21d4 20 < a \u2264 40. a 8 a 4 M\u00e0 a \u2208 N\u2217 n\u00ean a \u2208 {21; 22; . . . ; 40}. TH2: 0 < b < log2 5\u00b7 a Khi \u0111\u00f3 \u0111\u1ec3 t\u1ed3n t\u1ea1i \u0111\u00fang hai gi\u00e1 tr\u1ecb c\u1ee7a b th\u00ec b \u2208 {1; 2}. Do \u0111\u00f3 2 < log2 5 \u2264 3 \u21d4 4 < 5 \u2264 8 \u21d4 5 \u2264 a < 5\u00b7 a a 8 4 M\u00e0 a \u2208 N\u2217 n\u00ean a = 1. V\u1eady c\u00f3 21 s\u1ed1 nguy\u00ean d\u01b0\u01a1ng a th\u1ecfa m\u00e3n y\u00eau c\u1ea7u c\u1ee7a b\u00e0i to\u00e1n. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 30 (C\u00e2u 39 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean x th\u1ecfa m\u00e3n \u00c42x2 \u2212 4x\u00e4 [log2(x + 14) \u2212 4] \u2264 0? A 13. B 14. C V\u00f4 s\u1ed1. D 15. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 270","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT \u0110i\u1ec1u ki\u1ec7n: x > \u221214. Ta c\u00f3 \u00c42x2 \u2212 4x\u00e4 [log2(x + 14) \u2212 4] = 0 \u21d4 \u00f12x2 \u2212 4x = 0 \u2212 4 = 0 \u21d4 \u00f1x = 0 log2(x + 14) x = 2. Do \u0111\u00f3, x = 0, x = 2 l\u00e0 2 nghi\u1ec7m nguy\u00ean c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho. (1) X\u00e9t b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \uf8ee \u00ae2x2 \u2212 4x > 0 \uf8ee \u00aex2 \u2212 2x > 0 \u00c42x2 \u2212 4x\u00e4 [log2(x + 14) \u2212 4] < 0 \u21d4 \uf8ef log2(x + 14) \u2212 4 < 0 \u21d4 \uf8ef x + 14 < 16 \u21d4 x < 0. \uf8ef \uf8ef \uf8ef \uf8ef \u00ae2x2 \u2212 4x < 0 \uf8ef \u00aex2 \u2212 2x < 0 \uf8f0 \uf8f0 log2(x + 14) \u2212 4 > 0 x + 14 > 16 K\u1ebft h\u1ee3p v\u1edbi \u0111i\u1ec1u ki\u1ec7n x > \u221214 v\u00e0 x nguy\u00ean, ta \u0111\u01b0\u1ee3c 13 nghi\u1ec7m nguy\u00ean c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u00c42x2 \u2212 4x\u00e4 [log2(x + 14) \u2212 4] < 0 l\u00e0 x \u2208 {\u221213; \u221212; . . . ; \u22122; \u22121}. (2) T\u1eeb (1) v\u00e0 (2) suy ra b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 15 nghi\u1ec7m. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 31 (C\u00e2u 39 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean x th\u1ecfa m\u00e3n [log3 (x2 + 1) \u2212 log3(x + 31)] (32 \u2212 2x\u22121) \u2265 0? A 27. B 26. C V\u00f4 s\u1ed1. D 28. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n: x > \u221231. \u0110\u1eb7t f (x) = [log3 (x2 + 1) \u2212 log3(x + 31)] (32 \u2212 2x\u22121). Ta c\u00f3 log3 x2 + 1 \u2212 log3(x + 31) = 0 \u21d4 log3 x2 + 1 = log3(x + 31) \u21d4 x2 + 1 = x + 31 \u21d4 x2 \u2212 x \u2212 30 = 0 \u00f1x = 6 (th\u1ecfa m\u00e3n x > \u221231) \u21d4 x = \u22125 (th\u1ecfa m\u00e3n x > \u221231). Ti\u1ebfp \u0111\u1ebfn 32 \u2212 2x\u22121 = 0 \u21d4 2x\u22121 = 32 \u21d4 x \u2212 1 = 5 \u21d4 x = 6 (th\u1ecfa m\u00e3n x > \u221231). B\u1ea3ng x\u00e9t d\u1ea5u c\u1ee7a f (x) nh\u01b0 sau. x \u221231 \u22125 6 +\u221e f (x) + 0 \u2212 0 \u2212 Do \u0111\u00f3, t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho l\u00e0 S = (\u221231; \u22125] \u222a {6}. V\u1eady c\u00f3 t\u1ea5t c\u1ea3 27 s\u1ed1 nguy\u00ean x th\u1ecfa m\u00e3n b\u00e0i to\u00e1n. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 32 (C\u00e2u 39 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean x th\u1ecfa m\u00e3n (4x \u2212 5.2x+2 + 64) 2 \u2212 log(4x) \u2265 0? A 22. B 25. C 23. D 24. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 271 S\u0110T: 0905.193.688","6. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh m\u0169 v\u00e0 l\u00f4garit \u0253 L\u1eddi gi\u1ea3i. \u00ae4x > 0 \u00aex > 0 \u00aex > 0 \u00aex > 0 \u0110i\u1ec1u ki\u1ec7n x\u00e1c \u0111\u1ecbnh: \u21d4 \u21d4 \u21d4 \u21d4 0 < x \u2264 25. 2 \u2212 log(4x) \u2265 0 log10(4x) \u2264 2 4x \u2264 100 x \u2264 25 V\u00ec 2 \u2212 log(4x) \u2265 0 n\u00ean b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u1ec1 b\u00e0i \u0111\u00e3 cho t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi 4x \u2212 5 \u00b7 2x+2 + 64 \u2265 0 \u21d4 4x \u2212 20 \u00b7 2x + 64 \u2265 0 \u21d4 \u00f12x \u2264 4 \u21d4 \u00f1x \u2264 2 2x \u2265 16 x \u2265 4 So l\u1ea1i v\u1edbi \u0111i\u1ec1u ki\u1ec7n x\u00e1c \u0111\u1ecbnh, ta c\u00f3 t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho l\u00e0 S = (0; 2] \u222a [4; 25]. V\u1eady c\u00f3 24 s\u1ed1 nguy\u00ean x th\u1ecfa m\u00e3n y\u00eau c\u1ea7u b\u00e0i to\u00e1n. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 33 (C\u00e2u 44 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). X\u00e9t c\u00e1c s\u1ed1 th\u1ef1c x, y sao cho 499\u2212y2 \u2265 a4x\u2212log7 a2 v\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c d\u01b0\u01a1ng a. Gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a bi\u1ec3u th\u1ee9c P = x2 + y2 + 4x \u2212 3y b\u1eb1ng A 121 B 39 C 24. D 39. . . 4 4 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 499\u2212y2 \u2265 a4x\u2212log7 a2 \u21d4 log7 499\u2212y2 \u2265 log7 a4x\u2212log7 a2 (1) \u21d4 (9 \u2212 y2). log7(49) \u2265 (4x \u2212 log7 a2). log7 a. (2) Do \u0111\u00f3 ta \u0111\u01b0\u1ee3c 2.(9 \u2212 y2) \u2265 2.(2x \u2212 log7 a). log7 a. \u0110\u1eb7t t = log7 a th\u00ec (1) tr\u1edf th\u00e0nh t2 \u2212 2x.t + 9 \u2212 y2 \u2265 0. Khi \u0111\u00f3 (1) \u0111\u00fang v\u1edbi m\u1ecdi a > 0 \u21d4 (2) \u0111\u00fang v\u1edbi m\u1ecdi t \u2208 R \u21d4 \u2206 = x2 \u2212 9 + y2 \u2264 0 \u21d4 x2 + y2 \u2264 9. M\u1eb7t kh\u00e1c (4x \u2212 3y)2 \u2264 (16 + 9).(x2 + y2) = 225 \u21d2 4x \u2212 3y \u2264 15. Suy ra P = x2 + y2 + 4x \u2212 3y \u2264 9 + 15 = 24. \uf8f1x y = > 0 12 \u22129\u00b7 \u0110\u1eb3ng th\u1ee9c x\u1ea3y ra khi v\u00e0 ch\u1ec9 khi \uf8f2 \u22123 \u21d2 x = ; y = 4 55 \uf8f3x2 + y2 = 9 V\u1eady gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a P b\u1eb1ng 24. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 34 (C\u00e2u 31 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). Cho h\u00e0m s\u1ed1 y = f (x). H\u00e0m s\u1ed1 y = f (x) c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean nh\u01b0 sau x \u2212\u221e \u22123 1 +\u221e +\u221e \u22123 0 f (x) \u2212\u221e B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh f (x) < ex + m \u0111\u00fang v\u1edbi m\u1ecdi x \u2208 (\u22121; 1) khi v\u00e0 ch\u1ec9 khi A m \u2265 f (1) \u2212 e. B m > f (\u22121) \u2212 1 C m \u2265 f (\u22121) \u2212 1 D m > f (1) \u2212 e. . . ee \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 272 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT f (x) < ex + m \u21d4 f (x) \u2212 ex < m. X\u00e9t h (x) = f (x) \u2212 ex, \u2200x \u2208 (\u22121; 1). h (x) = f (x) \u2212 ex < 0, \u2200x \u2208 (\u22121; 1) (V\u00ec f (x) < 0, \u2200x \u2208 (\u22121; 1) v\u00e0 ex > 0, \u2200x \u2208 (\u22121; 1)). \u21d2 h (x) ngh\u1ecbch bi\u1ebfn tr\u00ean (\u22121; 1) \u21d2 h (\u22121) > h (x) > h (1) , \u2200x \u2208 (\u22121; 1). B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh f (x) < ex + m \u0111\u00fang v\u1edbi m\u1ecdi x \u2208 (\u22121; 1) \u21d4 m \u2265 h (\u22121) \u21d4 m \u2265 f (\u22121) \u2212 1 . e Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 35 (C\u00e2u 39 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean x th\u1ecfa m\u00e3n [log2 (x2 + 1) \u2212 log2(x + 31)] (32 \u2212 2x\u22121) \u2265 0? A 26. B 27. C 28. D V\u00f4 s\u1ed1. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x + 31 > 0 \u21d4 x > \u221231. X\u00e9t c\u00e1c h\u00e0m s\u1ed1 f (x) = log2 (x2 + 1) \u2212 log2(x + 31) v\u00e0 g(x) = 32 \u2212 2x\u22121 tr\u00ean (\u221231; +\u221e). Ta c\u00f3 \u2022 f (x) = 0 \u21d4 log2 x2 + 1 = log2(x + 31) \u21d4 x2 + 1 = x + 31 \u21d4 x2 \u2212 x \u2212 30 = 0 \u21d4 \u00f1x = \u22125 x = 6. \u2022 g(x) = 0 \u21d4 2x\u22121 = 32 \u21d4 x \u2212 1 = 5 \u21d4 x = 6. B\u1ea3ng x\u00e9t d\u1ea5u x \u221231 \u22125 6 +\u221e f (x) + 0 \u2212 0 + g(x) | + | + 0 \u2212 f (x) \u00b7 g(x) +0\u22120\u2212 T\u1eeb b\u1ea3ng x\u00e9t d\u1ea5u, ta c\u00f3 log2 x2 + 1 \u2212 log2(x + 31) 32 \u2212 2x\u22121 \u2265 0 \u21d4 \u221231 < x \u2264 \u22125. K\u1ebft h\u1ee3p \u0111i\u1ec1u ki\u1ec7n x \u2208 Z, ta c\u00f3 x \u2208 {\u221230; \u221229; \u221228; . . . ; \u22125}. V\u1eady c\u00f3 t\u1ea5t c\u1ea3 26 s\u1ed1 nguy\u00ean x th\u1ecfa m\u00e3n y\u00eau c\u1ea7u b\u00e0i to\u00e1n. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 36 (C\u00e2u 39 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean x th\u1ecfa m\u00e3n [log2 (x2 + 1) \u2212 log2(x + 21)] (16 \u2212 2x\u22121) \u2265 0? A V\u00f4 s\u1ed1 . B 17. C 16. D 18. \u0110i\u1ec1u ki\u1ec7n x > \u221221. \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 273","6. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh m\u0169 v\u00e0 l\u00f4garit \uf8ee log2 x2 + 1 \u2212 log2(x + 21) = 0 \uf8ef16 \u2212 2x\u22121 = 0 \uf8ef log2 x2 + 1 \u2212 log2(x + 21) 16 \u2212 2x\u22121 \u2265 0 \u21d4 \uf8ef \u00ae log2 x2 + 1 \u2212 log2(x + 21) > 0 \uf8ef 16 \u2212 2x\u22121 > 0 \uf8ef \uf8ef \uf8ef \uf8ef \u00ae x2 + 1 \uf8ef log2 \u2212 log2(x + 21) < 0 \uf8f0 16 \u2212 2x\u22121 < 0 \uf8eex2 + 1 = x + 21 \uf8eex2 \u2212 x \u2212 20 = 0 \uf8ef16 = 2x\u22121 \uf8ef4 = x \u2212 1 \uf8ef \uf8ef \uf8ef \u00aex2 + 1 > x + 21 \uf8ef \u00aex2 \u2212 x \u2212 20 > 0 \u00f1x = \u22124 ho\u1eb7c x = 5 \uf8ef \uf8ef \u21d4 \uf8ef \u21d4\uf8ef \u21d4 \uf8ef 4>x\u22121 \uf8ef x<5 x < \u22124. \uf8ef \uf8ef \uf8ef\uf8ef \uf8ef \u00aex2 + 1 < x + 21 \uf8ef \u00aex2 \u2212 x \u2212 20 < 0 \uf8f0\uf8f0 4<x\u22121 x>5 So v\u1edbi \u0111i\u1ec1u ki\u1ec7n ta \u0111\u01b0\u1ee3c t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 S = (\u221221; \u22124] \u222a {5}. V\u00ec x l\u00e0 s\u1ed1 nguy\u00ean thu\u1ed9c S n\u00ean x \u2208 {\u221220; \u221219; . . . ; \u22124; 5}. V\u1eady c\u00f3 18 s\u1ed1 nguy\u00ean th\u1ecfa m\u00e3n y\u00eau c\u1ea7u \u0111\u1ec1 b\u00e0i. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 37 (C\u00e2u 39 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean d\u01b0\u01a1ng a sao cho \u1ee9ng v\u1edbi m\u1ed7i a c\u00f3 \u0111\u00fang hai s\u1ed1 nguy\u00ean b th\u1ecfa m\u00e3n 3b \u2212 3 a.2b \u2212 16 < 0? A 34. B 32. C 31. D 33. \u0253 L\u1eddi gi\u1ea3i. Tr\u01b0\u1eddng h\u1ee3p 1 : a = 1 \u21d2 3b \u2212 3 2b \u2212 16 < 0. N\u1ebfu b \u2264 1 ho\u1eb7c b \u2265 4 kh\u00f4ng th\u1ecfa m\u00e3n b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh v\u00e0 b \u2208 {2; 3} th\u1ecfa m\u00e3n. V\u1eady a = 1 th\u1ecfa m\u00e3n. Tr\u01b0\u1eddng h\u1ee3p 2 : a = 2 \u21d2 3b \u2212 3 2.2b \u2212 16 < 0 \u21d4 3b \u2212 3 2b+1 \u2212 16 < 0. N\u1ebfu b \u2264 1 ho\u1eb7c b \u2265 3 kh\u00f4ng th\u1ecfa m\u00e3n b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh v\u00e0 b = 2 th\u1ecfa m\u00e3n. V\u1eady a = 2 kh\u00f4ng th\u1ecfa m\u00e3n. Tr\u01b0\u1eddng h\u1ee3p 3 : a = 3 \u21d2 3b \u2212 3 3.2b \u2212 16 < 0. N\u1ebfu b \u2264 1 ho\u1eb7c b \u2265 3 kh\u00f4ng th\u1ecfa m\u00e3n b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh v\u00e0 b = 2 th\u1ecfa m\u00e3n. V\u1eady a = 3 kh\u00f4ng th\u1ecfa m\u00e3n. Tr\u01b0\u1eddng h\u1ee3p 4 : a > 3. Ta c\u1ea7n t\u00ecm a \u0111\u1ec3 b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh 3b \u2212 3 a.2b \u2212 16 < 0 c\u00f3 2 nghi\u1ec7m b. N\u1ebfu b \u2265 3 \u21d2 3b \u2212 3 a.2b \u2212 16 \u2265 24.(3.8 \u2212 16) > 0 kh\u00f4ng th\u1ecfa m\u00e3n b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh. N\u1ebfu b = 2 \u21d2 3b \u2212 3 a.2b \u2212 16 \u2265 6(4.4 \u2212 16) \u2265 0 kh\u00f4ng th\u1ecfa m\u00e3n b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh. N\u1ebfu b = 1 kh\u00f4ng th\u1ecfa m\u00e3n. N\u1ebfu b < 1 \u21d2 3b \u2212 3 < 0. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh t\u01b0\u01a1ng \u0111\u01b0\u01a1ng a.2b \u2212 16 > 0. Hay a > 16 c\u00f3 hai nghi\u1ec7m b suy ra 33 \u2264 a \u2264 64. 2b K\u1ebft h\u1ee3p l\u1ea1i suy ra c\u00f3 t\u1ea5t c\u1ea3 33 s\u1ed1 nguy\u00ean d\u01b0\u01a1ng a th\u1ecfa m\u00e3n. C\u00e1ch 2: \uf8eeb = 1 X\u00e9t 3b \u2212 3 a.2b \u2212 16 = 0. Do a \u2208 N\u2217 n\u00ean \uf8f0 b = log2 16 . 16 a Tr\u01b0\u1eddng h\u1ee3p 1: log2 a > 1 \u21d4 a < 8. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 274 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT 16 B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 \u0111\u00fang 2 nghi\u1ec7m nguy\u00ean b \u21d4 3 < log2 a \u2264 4 \u21d4 1 \u2264 a < 2 \u21d2 a = 1 (th\u1ecfa m\u00e3n). 16 Tr\u01b0\u1eddng h\u1ee3p 2: log2 a < 1 \u21d4 a > 8. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 \u0111\u00fang 2 nghi\u1ec7m nguy\u00ean b khi v\u00e0 ch\u1ec9 khi \u22122 \u2264 log2 16 < \u221214 \u21d4 32 < a \u2264 64 a Suy ra c\u00f3 32 gi\u00e1 tr\u1ecb a. V\u1eady c\u00f3 33 gi\u00e1 tr\u1ecb c\u1ee7a a th\u1ecfa m\u00e3n. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 38 (C\u00e2u 42 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). \u0110\u1ec3 qu\u1ea3ng b\u00e1 cho s\u1ea3n ph\u1ea9m A, m\u1ed9t c\u00f4ng ty d\u1ef1 \u0111\u1ecbnh t\u1ed5 ch\u1ee9c qu\u1ea3ng c\u00e1o theo h\u00ecnh th\u1ee9c qu\u1ea3ng c\u00e1o tr\u00ean truy\u1ec1n h\u00ecnh. Nghi\u00ean c\u1ee9u c\u1ee7a c\u00f4ng ty cho th\u1ea5y: n\u1ebfu sau n l\u1ea7n qu\u1ea3ng c\u00e1o \u0111\u01b0\u1ee3c ph\u00e1t th\u00ec t\u1ec9 l\u1ec7 1 ng\u01b0\u1eddi xem qu\u1ea3ng c\u00e1o \u0111\u00f3 mua s\u1ea3n ph\u1ea9m A tu\u00e2n theo c\u00f4ng th\u1ee9c P (n) = 1 + 49e\u22120,015n . H\u1ecfi c\u1ea7n ph\u00e1t \u00edt nh\u1ea5t bao nhi\u00eau l\u1ea7n qu\u1ea3ng c\u00e1o \u0111\u1ec3 t\u1ec9 l\u1ec7 ng\u01b0\u1eddi xem mua s\u1ea3n ph\u1ea9m \u0111\u1ea1t tr\u00ean 30%? A 202. B 203. C 206. D 207. Theo b\u00e0i ra ta c\u00f3 \u0253 L\u1eddi gi\u1ea3i. 1 1 + 49e\u22120,015n > 0,3 \u21d4 1 + 49e\u22120,015n < 10 3 \u21d4 e\u22120,015n < 7 147 \u21d4 \u22120,015n < ln 7 147 \u21d4 n > \u2212 1 ln 7 \u2248 202,97 0,015 147 V\u1eady \u00edt nh\u1ea5t 203 l\u1ea7n qu\u1ea3ng c\u00e1o. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 39 (C\u00e2u 41 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong n\u0103m 2019, di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi c\u1ee7a t\u1ec9nh A l\u00e0 600 ha. Gi\u1ea3 s\u1eed di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi c\u1ee7a t\u1ec9nh A m\u1ed7i n\u0103m ti\u1ebfp theo \u0111\u1ec1u t\u0103ng 6% so v\u1edbi di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi c\u1ee7a n\u0103m li\u1ec1n tr\u01b0\u1edbc. K\u1ec3 t\u1eeb sau n\u0103m 2019, n\u0103m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 n\u0103m \u0111\u1ea7u ti\u00ean t\u1ec9nh A c\u00f3 di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi trong n\u0103m \u0111\u00f3 \u0111\u1ea1t tr\u00ean 1000 ha? A N\u0103m 2028. B N\u0103m 2047. C N\u0103m 2027. D N\u0103m 2046. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi P0 l\u00e0 di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi n\u0103m 2019. G\u1ecdi Pn l\u00e0 di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi sau n n\u0103m. G\u1ecdi r% l\u00e0 ph\u1ea7n tr\u0103m di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi t\u0103ng m\u1ed7i n\u0103m. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 275 S\u0110T: 0905.193.688","6. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh m\u0169 v\u00e0 l\u00f4garit Sau 1 n\u0103m, di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi l\u00e0 P1 = P0 + P0r = P0 (1 + r). Sau 2 n\u0103m, di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi l\u00e0 P2 = P1 + P1r = P0 (1 + r)2. ... Sau n n\u0103m, di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi l\u00e0 Pn = P0 (1 + r)n. Theo gi\u1ea3 thi\u1ebft: P0 = 600, r = 0, 06, ta c\u00f3 600 (1 + 0, 06)n > 1000 \u21d4 (1, 06)n > 10 \u21d4 n > log1,06 10 \u2248 8, 8 6 6 Do \u0111\u00f3 n = 9. V\u1eady sau 9 n\u0103m (t\u1ee9c n\u0103m 2028) th\u00ec t\u1ec9nh A c\u00f3 di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi trong n\u0103m \u0111\u00f3 \u0111\u1ea1t tr\u00ean 1000 ha. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 40 (C\u00e2u 42 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Trong n\u0103m 2019, di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi c\u1ee7a t\u1ec9nh A l\u00e0 1000 ha. Gi\u1ea3 s\u1eed di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi c\u1ee7a t\u1ec9nh A m\u1ed7i n\u0103m ti\u1ebfp theo \u0111\u1ec1u t\u0103ng 6% so v\u1edbi di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi c\u1ee7a n\u0103m li\u1ec1n tr\u01b0\u1edbc. K\u1ec3 t\u1eeb sau n\u0103m 2019, n\u0103m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 n\u0103m \u0111\u1ea7u ti\u00ean t\u1ec9nh A c\u00f3 di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi trong n\u0103m \u0111\u00f3 \u0111\u1ea1t tr\u00ean 1400 ha? A N\u0103m 2043. B N\u0103m 2025. C N\u0103m 2024. D N\u0103m 2042. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi Sn l\u00e0 di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi c\u1ee7a t\u1ec9nh A sau n n\u0103m. r l\u00e0 ph\u1ea7n tr\u0103m di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi t\u0103ng th\u00eam sau m\u1ed7i n\u0103m. S l\u00e0 di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi n\u0103m 2019. Khi \u0111\u00f3 Sn = S(1 + r)n. V\u1edbi S = 1000 ha, r = 6% = 0,06 suy ra Sn = 1000 (1 + 0,06)n = 1000 (1,06)n. \u0110\u1ec3 Sn \u2265 1400 \u21d4 1000 (1,06)n \u2265 1400 \u21d4 n \u2265 log1,06 \u00c57\u00e3 \u2248 5,77. 5 V\u1eady n\u0103m \u0111\u1ea7u ti\u00ean t\u1ec9nh A c\u00f3 di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi trong n\u0103m \u0111\u00f3 \u0111\u1ea1t tr\u00ean 1400 ha l\u00e0 n\u0103m 2025. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 41 (C\u00e2u 49 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean x sao cho \u1ee9ng v\u1edbi m\u1ed7i x c\u00f3 kh\u00f4ng qu\u00e1 242 s\u1ed1 nguy\u00ean y th\u1ecfa m\u00e3n log4(x2 + y) \u2265 log3(x + y)? A 55. B 28. C 29. D 56. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x + y > 0 v\u00e0 x2 + y > 0. Khi \u0111\u00f3 log4(x2 + y) \u2265 log3(x + y) \u21d4 x2 + y \u2265 4log3(x+y) \u21d4 x2 + y \u2265 (x + y)log3 4 \u21d4 x2 \u2212 x > (x + y)log3 4 \u2212 (x + y). (1) \u0110\u1eb7t t = x + y th\u00ec (1) \u0111\u01b0\u1ee3c vi\u1ebft l\u1ea1i l\u00e0 x2 \u2212 x > tlog3 4 \u2212 t. (2) V\u1edbi m\u1ed7i x nguy\u00ean cho tr\u01b0\u1edbc c\u00f3 kh\u00f4ng qu\u00e1 242 s\u1ed1 nguy\u00ean y th\u1ecfa m\u00e3n b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh (1). T\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh (2) c\u00f3 kh\u00f4ng qu\u00e1 242 nghi\u1ec7m t. Nh\u1eadn th\u1ea5y f (t) = tlog3 4 \u2212 t \u0111\u1ed3ng bi\u1ebfn tr\u00ean [1; +\u221e) n\u00ean n\u1ebfu x2 \u2212 x > 243log3 4 \u2212 243 = 781 th\u00ec s\u1ebd c\u00f3 \u00edt nh\u1ea5t 243 nghi\u1ec7m nguy\u00ean t \u2265 1. Do \u0111\u00f3 y\u00eau c\u1ea7u b\u00e0i to\u00e1n t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi x2 \u2212 x \u2264 781 \u21d4 \u221227 \u2264 x \u2264 28 (do x nguy\u00ean). V\u1eady c\u00f3 t\u1ea5t c\u1ea3 28 + 28 = 56 s\u1ed1 nguy\u00ean x th\u1ecfa y\u00eau c\u1ea7u b\u00e0i to\u00e1n. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 276 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT \u0104 C\u00e2u 42 (C\u00e2u 44 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). X\u00e9t t\u1ea5t c\u1ea3 c\u00e1c s\u1ed1 th\u1ef1c x, y sao cho a4x\u2212log5 a2 \u2264 2540\u2212y2 v\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c d\u01b0\u01a1ng a. Gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a bi\u1ec3u th\u1ee9c P = x2 + y2 + x \u2212 3y b\u1eb1ng A 125 B 80. C 60. D 20. . 2 \u0253 L\u1eddi gi\u1ea3i. Do a d\u01b0\u01a1ng n\u00ean a4x\u2212log5 a2 \u2264 2540\u2212y2 \u21d4 a4x\u22122 log5 a \u2264 52(40\u2212y2). (1) \u0110\u1eb7t log5 a = t th\u00ec a = 5t. Tac\u00f3 (C2) (C1) I O (1) \u21d4 5t(4x\u22122t) \u2264 52(40\u2212y2) (2) \u21d4 2tx \u2212 t2 \u2264 40 \u2212 y2 \u21d4 t2 \u2212 2tx + 40 \u2212 y2 \u2265 0. (1) \u0111\u00fang v\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c d\u01b0\u01a1ng a khi v\u00e0 ch\u1ec9 khi (2) \u0111\u00fang v\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c t khi v\u00e0 ch\u1ec9 khi \u2206 = x2 + y2 \u2212 40 \u2264 0 \u21d4 x2 + y2 \u2264 40. Theo b\u1ea5t \u0111\u1eb3ng th\u1ee9c Bunhia - C\u1ed1pxki, ta c\u00f3 (x \u2212 3y)2 \u2264 10(x2 + y2) \u2264 10.40 = 400. Suy ra x \u2212 3y \u2264 20. Khi \u0111\u00f3 P = x2 + y2 + x \u2212 3y \u2264 40 + 20 = 60. \uf8f1x2 + y2 = 40 \u00aex = 2 \uf8f2 D\u1ea5u b\u1eb1ng x\u1ea3y ra khi y\u21d4 y = \u22126 . \uf8f3x = \u22123 > 0 V\u1eady gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a P b\u1eb1ng 60. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 43 (C\u00e2u 45 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). X\u00e9t t\u1ea5t c\u1ea3 c\u00e1c s\u1ed1 th\u1ef1c x, y sao cho 89\u2212y2 \u2265 a6x\u2212log2 a3 v\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c d\u01b0\u01a1ng a. Gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a bi\u1ec3u th\u1ee9c P = x2 + y2 \u2212 6x \u2212 8y b\u1eb1ng A \u221221. B \u22126. C \u221225. D 39. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 y A 4 89\u2212y2 \u2265 a6x\u2212log2 a3 , \u2200a > 0 \u21d4 3 9 \u2212 y2 \u2265 6x \u2212 3 log2 a log2 a, \u2200a > 0 M \u21d4 log22 a \u2212 2x log2 a + 9 \u2212 y2 \u2265 0, \u2200a > 0 \u21d4 \u2206 = x2 + y2 \u2212 9 \u2264 0. G\u1ecdi M (x; y) thu\u1ed9c h\u00ecnh tr\u00f2n (C) t\u00e2m O, b\u00e1n k\u00ednh R = O 3x 3. G\u1ecdi A(3; 4), ta c\u00f3 OA = 5 > R. Do \u0111\u00f3 A n\u1eb1m ngo\u00e0i h\u00ecnh tr\u00f2n (C). Khi \u0111\u00f3 P = (x \u2212 3)2 + (y \u2212 4)2 \u2212 25 = M A2 \u2212 25 \u2265 (OA \u2212 R)2 \u2212 25 = \u221221. V\u1eady min P = \u221221 khi O, M, A theo th\u1ee9 t\u1ef1 th\u1eb3ng h\u00e0ng. Ch\u1ecdn \u0111\u00e1p \u00e1n A 277 S\u0110T: 0905.193.688 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t","6. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh m\u0169 v\u00e0 l\u00f4garit \u0104 C\u00e2u 44 (C\u00e2u 49 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean x sao cho \u1ee9ng v\u1edbi m\u1ed7i x c\u00f3 kh\u00f4ng qu\u00e1 728 s\u1ed1 nguy\u00ean y th\u1ecfa m\u00e3n log4 (x2 + y) \u2265 log3(x + y)? A 59. B 58. C 116. D 115. \u0253 L\u1eddi gi\u1ea3i. \u00aex2 + y > 0 \u0110i\u1ec1u ki\u1ec7n x + y > 0. \u0110\u1eb7t k = x + y, suy ra k \u2208 Z+. X\u00e9t h\u00e0m s\u1ed1 f (y) = log4 (x2 + y) \u2212 log3(x + y) \u2265 0. (*) Ta c\u00f3 f (y) = 1 \u22121 < 0 (v\u00ec x \u2208 Z+ n\u00ean x2 \u2265 x \u21d2 x2 + y \u2265 x+y hay (x2 + y) ln 4 (x + y) ln 3 11 x2 + y \u2212 x + y > 0 v\u00e0 ln 4 > ln 3 > 0). Suy ra f (y) ngh\u1ecbch bi\u1ebfn tr\u00ean m\u1ed7i kho\u1ea3ng m\u00e0 f (y) x\u00e1c \u0111\u1ecbnh. X\u00e9t g(k) = f (k \u2212 x) = log4 (x2 + k \u2212 x) \u2212 log3 k, k \u2208 Z+. Do f ngh\u1ecbch bi\u1ebfn n\u00ean g c\u0169ng ngh\u1ecbch bi\u1ebfn. Gi\u1ea3 s\u1eed k0 l\u00e0 m\u1ed9t nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh g(k) = 0. Khi \u0111\u00f3 k0 l\u00e0 nghi\u1ec7m duy nh\u1ea5t c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh g(k) = 0. Suy ra (*) tr\u1edf th\u00e0nh g(k) \u2265 g (k0) \u21d4 \u00ae1 \u2264k\u2264 k0 \u21d2 k0 \u2264 728. k \u2208 Z+ Khi \u0111\u00f3 g(728) \u2264 0 \u21d4 log4 x2 \u2212 x + 728 \u2264 log3 728 \u21d4 x2 \u2212 x + 728 < 4089 \u21d4 x2 \u2212 x \u2212 3361 < 0 \u21d4 \u221257,476 \u2264 x \u2264 58,478. V\u00ec x nguy\u00ean n\u00ean x \u2208 {\u221257; \u221256; . . . ; 58}. Khi \u0111\u00f3 c\u00f3 116 gi\u00e1 tr\u1ecb x th\u1ecfa b\u00e0i to\u00e1n. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 45 (C\u00e2u 49 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean x sao cho \u1ee9ng v\u1edbi m\u1ed7i x c\u00f3 kh\u00f4ng qu\u00e1 255 s\u1ed1 nguy\u00ean y th\u1ecfa m\u00e3n log3 (x2 + y) \u2265 log2(x + y)? A 80. B 79. C 157. D 158. \u0253 L\u1eddi gi\u1ea3i. \u00aex + y > 0 \u00aey > \u2212x \u0110i\u1ec1u ki\u1ec7n x2 + y > 0 \u21d4 y > \u2212x2 . V\u00ec x \u2208 Z n\u00ean suy ra x2 \u2265 x \u21d4 \u2212x2 \u2264 \u2212x do \u0111\u00f3 c\u00f3 \u0111i\u1ec1u ki\u1ec7n y > \u2212x \u21d2 y \u2265 1 \u2212 x. X\u00e9t h\u00e0m s\u1ed1 f (y) = log3 (x2 + y) \u2212 log2 (x + y). 1 1 (x + y) ln 2 \u2212 (x2 + y) ln 3 Ta c\u00f3 f (y) = \u2212 = (x2 + y) (x + y) ln 3. ln 2 . (x2 + y) ln 3 (x + y) ln 2 V\u00ec x \u2264 x2 n\u00ean 0 < x + y \u2264 x2 + y. H\u01a1n n\u1eefa, 0 < ln 2 < ln 3. Do \u0111\u00f3 (x + y) ln 2 < (x2 + y) ln 3 \u21d2 f (y) < 0. Nh\u1eadn x\u00e9t:f (1 \u2212 x) = log3 (x2 \u2212 x + 1) \u2212 log2 1 \u2265 0, \u2200x \u2208 Z. \u00aef (y) = 0 Gi\u1ea3 s\u1eed ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 nghi\u1ec7m \u21d2 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 nghi\u1ec7m duy nh\u1ea5t y = m. f (y) < 0 C\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean: Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 278 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT y 1\u2212x m +\u221e f (y) \u2212 \u2212 f (y) 0 N\u00ean b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh f (y) \u2265 0 \u21d4 1 \u2212 x \u2264 y \u2264 m do \u0111\u00f3 \u0111\u1ec3 b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 kh\u00f4ng qu\u00e1 255 gi\u00e1 tr\u1ecb th\u00ec m \u2264 255 \u2212 x n\u00ean f (256 \u2212 x) < 0 \u21d4 log3 x2 \u2212 x + 256 \u2212 log2 256 < 0 \u21d4 x2 \u2212\u221ax + 256 < 38 \u221a 1 \u2212 25221 1 + 25221 \u21d4 <x< . 22 V\u00ec x \u2208 Z n\u00ean \u221278 \u2264 x \u2264 79 \u21d2 c\u00f3 158 gi\u00e1 tr\u1ecb x th\u1ecfa m\u00e3n. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 46 (C\u00e2u 47 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean y sao cho t\u1ed3n t\u1ea1i x \u2208 \u00c51 \u00e3 th\u1ecfa m\u00e3n 273x2+xy = (1 + xy)279x? ; 3 3 A 27. B 9. C 11. D 12. \u0253 L\u1eddi gi\u1ea3i. \u0110\u1ec3 c\u1eb7p (x, y) th\u1ecfa m\u00e3n ph\u01b0\u01a1ng tr\u00ecnh th\u00ec 1 + xy > 0. Khi \u0111\u00f3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho t\u01b0\u01a1ng \u0111\u01b0\u01a1ng 3x2 + (y \u2212 9) x \u2212 log27 (1 + xy) = 0. \u00c51 \u00e3 \u0110\u1eb7t f (x) = 3x2 + (y \u2212 9) x \u2212 log27 (1 + xy), x \u2208 3 ; 3. Ta x\u00e9t c\u00e1c tr\u01b0\u1eddng h\u1ee3p sau TH1. N\u1ebfu y < 0 th\u00ec \u2212y < 1 < 3, suy ra y > \u22123 hay y \u2208 {\u22121; \u22122}. x V\u1edbi y = \u22121 \u21d2 f \u00c51\u00e3 < 0 v\u00e0 lim f (x) = +\u221e n\u00ean theo \u0111\u1ecbnh l\u00ed gi\u00e1 tr\u1ecb trung gian, ph\u01b0\u01a1ng 3 x\u21921\u2212 tr\u00ecnh c\u00f3 nghi\u1ec7m tr\u00ean \u00c51 \u00e3 \u2282 \u00c51 \u00e3 ;1 ;3 . 33 T\u01b0\u01a1ng t\u1ef1, v\u1edbi y = \u22122, ph\u01b0\u01a1ng tr\u00ecnh c\u0169ng c\u00f3 nghi\u1ec7m tr\u00ean \u00c51 ; 1\u00e3 \u2282 \u00c51 ; \u00e3 3. 32 3 TH2. N\u1ebfu y = 0 th\u00ec ph\u01b0\u01a1ng tr\u00ecnh tr\u1edf th\u00e0nh 3x2 \u2212 9x = 0 hay \u00f1x = 0 (kh\u00f4ng th\u1ecfa m\u00e3n). x=3 TH3. N\u1ebfu y \u2265 10, ta c\u00f3 y , f (x) = 6x + (y \u2212 9) \u2212 3 (1 + xy) ln 3 y2 \u00c5 1 \u00e3 (x) = 6 + 3 (1 + xy)2 ln 3 3 f > 0, \u2200x \u2208 3 ; . Suy ra f (x) \u0111\u1ed3ng bi\u1ebfn tr\u00ean \u00c51 \u00e3 Do \u0111\u00f3 f (x) > f \u00c51 \u00e3 = 2 + (y \u2212 9) \u2212 y > 0. ;3 . 3 3 (3 + y) ln 3 \u00c51 \u00e3 \u0110i\u1ec1u n\u00e0y d\u1eabn \u0111\u1ebfn f (x) l\u00e0 h\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn tr\u00ean ; 3 . 3 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 279 S\u0110T: 0905.193.688","6. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh m\u0169 v\u00e0 l\u00f4garit Suy ra f (x) > f \u00c51\u00e3 = y \u2212 log27 y \u2212 8 3 3 1+ . 3 3 X\u00e9t h\u00e0m s\u1ed1 g(t) = t \u2212 log27 (1 + t) , t > 0. D\u1ec5 th\u1ea5y g (t) = 1 \u2212 1 > 0 n\u00ean g(t) l\u00e0 h\u00e0m \u0111\u1ed3ng bi\u1ebfn tr\u00ean (0; +\u221e), k\u00e9o theo 3 (3 + t) ln 3 \u00c51\u00e3 y \u2212 8 \u2265 g \u00c510\u00e3 \u2212 8 > 0. f =g 3 33 33 \u00c51 \u00e3 Nh\u01b0 v\u1eady, ph\u01b0\u01a1ng tr\u00ecnh v\u00f4 nghi\u1ec7m tr\u00ean ; 3 trong tr\u01b0\u1eddng h\u1ee3p n\u00e0y. 3 TH4. N\u1ebfu 1 \u2264 y \u2264 9 th\u00ec t\u1eeb t\u00ednh \u0111\u1ed3ng bi\u1ebfn c\u1ee7a g(t), ta suy ra \u00c51\u00e3 y \u2212 8 \u2264 g \u00c59\u00e3 \u2212 8 = 3 \u2212 log27 4 \u2212 8 < 0, \u2200y \u2208 [1; 9]. f =g 3 3 3 3 3 3 M\u1eb7t kh\u00e1c f (3) = 3y \u2212 log27(1 + 3y) = g(3y) \u2265 g(3) = 3 \u2212 log27 4 > 0 \u21d2 f (3) > 0, \u2200y \u2208 [1; 9]. \u00c51 \u00e3 \u0110i\u1ec1u n\u00e0y d\u1eabn \u0111\u1ebfn ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 nghi\u1ec7m tr\u00ean ; 3 , \u2200y \u2208 [1; 9]. 3 K\u1ebft lu\u1eadn c\u00f3 t\u1ea5t c\u1ea3 11 gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a y th\u1ecfa m\u00e3n y\u00eau c\u1ea7u. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 47 (C\u00e2u 41 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean d\u01b0\u01a1ng a sao cho \u1ee9ng v\u1edbi m\u1ed7i a c\u00f3 \u0111\u00fang hai s\u1ed1 nguy\u00ean b th\u1ecfa m\u00e3n 4b \u2212 1 a.3b \u2212 10 < 0? A 182. B 179. C 180. D 181. \u0253 L\u1eddi gi\u1ea3i. Theo \u0111\u1ec1 b\u00e0i a \u2208 Z; a \u2265 1 v\u00e0 b \u2208 Z. \u00df 4b \u2212 1 < 0 b<0 Tr\u01b0\u1eddng h\u1ee3p 1: a3b \u2212 10 > 0 \u21d4 b > log3 10 \u00b7 a V\u00ec c\u00f3 \u0111\u00fang hai s\u1ed1 nguy\u00ean b th\u1ecfa m\u00e3n n\u00ean b \u2208 {\u22122; \u22121}. Do \u0111\u00f3 \u22122 > log3 10 \u2265 \u22123 \u21d4 270 \u2265 a > 90 n\u00ean a \u2208 {91; 92; ...; 270} \u00b7 Suy ra c\u00f3 180 gi\u00e1 tr\u1ecb c\u1ee7a a tho\u1ea3 a m\u00e3n tr\u01b0\u1eddng h\u1ee3p 1. \u00df 4b \u2212 1 > 0 b>0 Tr\u01b0\u1eddng h\u1ee3p 2: a3b \u2212 10 < 0 \u21d4 b < log3 10 \u00b7 a V\u00ec c\u00f3 \u0111\u00fang hai s\u1ed1 nguy\u00ean b th\u1ecfa m\u00e3n n\u00ean b \u2208 {1; 2}. Do \u0111\u00f3 3 \u2265 log3 10 > 2 \u21d4 10 > a \u2265 10 n\u00ean a = 1. a 9 27 Suy ra c\u00f3 1 gi\u00e1 tr\u1ecb c\u1ee7a a tho\u1ea3 m\u00e3n tr\u01b0\u1eddng h\u1ee3p 2. V\u1eady c\u00f3 180 + 1 = 181 gi\u00e1 tr\u1ecb c\u1ee7a a tho\u1ea3 m\u00e3n y\u00eau c\u1ea7u b\u00e0i to\u00e1n. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 48 (C\u00e2u 44 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). X\u00e9t t\u1ea5t c\u1ea3 s\u1ed1 th\u1ef1c x, y sao cho 275\u2212y2 \u2265 a6x\u2212log3 a3 v\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c d\u01b0\u01a1ng a. Gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a bi\u1ec3u th\u1ee9c P = x2 + y2 \u2212 4x + 8y b\u1eb1ng A \u221215. B 25. C \u22125. D \u221220. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 280","Ch\u01b0\u01a1ng 2. H\u00c0M S\u1ed0 L\u0168Y TH\u1eeaA. H\u00c0M S\u1ed0 M\u0168 V\u00c0 H\u00c0M S\u1ed0 L\u00d4GARIT Gi\u1ea3 s\u1eed x, y th\u1ecfa 275\u2212y2 \u2265 a6x\u2212log3 a3 v\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c d\u01b0\u01a1ng a. Ta c\u00f3 P = x2 + y2 \u2212 4x + 8y \u21d4 x2 + y2 \u2212 4x + 8y \u2212 P = 0. Suy ra \u0111i\u1ec3m M (x; y) thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m I(2; \u22124) v\u00e0 b\u00e1n k\u00ednh \u00bb\u221a R1 = 22 + (\u22124)2 + P = 20 + P M\u1eb7t kh\u00e1c, 275\u2212y2 \u2265 a6x\u2212log3 a3 \u21d4 5 \u2212 y2 .3 \u2265 6x \u2212 log3 a3 log3 a. Suy ra, 275\u2212y2 \u2265 a6x\u2212log3 a3 \u21d4 5 \u2212 y2 .3 \u2265 6x \u2212 3 log3 a log3 a. \u0110\u1eb7t t = log3 a, t \u2208 R. Suy ra 5 \u2212 y2 .3 \u2265 (6x \u2212 3t)t \u21d4 \u22123t2 + 6xt \u2212 15 + 3y2 \u2264 0. Theo \u0111\u1ec1 b\u00e0i ta c\u00f3 275\u2212y2 \u2265 a6x\u2212log3 a3 \u0111\u00fang v\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c d\u01b0\u01a1ng a n\u00ean \u22123t2 + 6xt \u2212 15 + 3y2 \u2264 0 \u0111\u00fang v\u1edbi m\u1ecdi t \u2208 R. \u00df \u22123 < 0 \u22640 \u21d4 9x2 + 9y2 \u2212 45 \u2264 0 \u21d4 x2 + y2 \u2264 5. Do \u0111\u00f3 (3x)2 + 3 \u221215 + 3y2 \u221a Suy ra t\u1eadp h\u1ee3p c\u00e1c \u0111i\u1ec3m M (x; y) l\u00e0 h\u00ecnh tr\u00f2n t\u00e2m O(0; 0) v\u00e0 b\u00e1n\u221ak\u00ednh R2 = 5. V\u1eady \u0111\u1ec3 t\u1ed3n t\u1ea1i c\u1eb7p\u221a(x; y) th\u00ec \u0111\u01b0\u1eddng tr\u00f2n \u221aI; R1 v\u00e0 h\u00ec\u221anh tr\u00f2\u221an O; \u221a5 ph\u1ea3i c\u00f3 \u0111i\u1ec3m chung. Do \u0111\u00f3 IO \u2264 R1 + 5 \u21d4 22 + (\u22124)2 \u2264 20 + P + 5 \u21d4 5 \u2264 20 + P \u21d4 P \u2265 \u221215. V\u1eady gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a P l\u00e0 \u221215. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 49 (C\u00e2u 48 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean a sao cho \u1ee9ng v\u1edbi m\u1ed7i a, t\u1ed3n t\u1ea1i \u00edt nh\u1ea5t b\u1ed1n s\u1ed1 nguy\u00ean b \u2208 (\u221212; 12) th\u1ecfa m\u00e3n 4a2+b \u2264 3b\u2212a + 65? A 4. B 6. C 5. D 7. \u0253 L\u1eddi gi\u1ea3i. 4a2+b \u2264 3b\u2212a + 65 \u21d4 3b + 65 \u2265 4a2 \u00b7 4b \u21d4 \u00c5 3 \u00e3b + 65 \u00b7 3a \u00c5 1 \u00e3b \u2212 4a2 \u00b7 3a \u2265 0. (1) \u00b7 3a 4 4 H\u00e0m s\u1ed1 f (b) = \u00c5 3 \u00e3b + 65 \u00b7 3a \u00b7 \u00c5 1 \u00e3b \u2212 4a2 \u00b7 3a. 44 Ta c\u00f3 f (b) = \u00c5 3 \u00e3b ln 3 + 65 \u00b7 3a \u00b7 \u00c5 1 \u00e3b ln 1 < 0, \u2200b. 44 44 B\u1ea3ng bi\u1ebfn thi\u00ean x \u2212\u221e a +\u221e f (b) \u22120\u2212 +\u221e y=0 f (b) \u22124a2 \u00b7 3a Ta \u0111\u01b0\u1ee3c t\u1eadp nghi\u1ec7m S = (\u2212\u221e; \u03b1]. S ch\u1ee9a \u00edt nh\u1ea5t 4 s\u1ed1 nguy\u00ean t\u1ed1 b \u2208 (\u221212; 12) \u21d4 {\u221211; \u221210; \u22129; \u22128} \u2282 (\u2212\u221e; \u03b1] \u21d4 f (\u22128) \u2265 0 \u21d4 \u00c5 4 \u00e38 + 65 \u00b7 3a \u00b7 48 \u2212 4a2 \u00b7 3a \u2265 0 \u21d4 a \u2208 {\u22123; \u22122; . . . ; 3} (TABLE \u22125 \u2192 5). 3 Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 281 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG 3 B\u00c0I 1. NGUY\u00caN H\u00c0M \u0104 C\u00e2u 1 (C\u00e2u 9 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). H\u1ecd nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = 3x2 + 1 l\u00e0 A x3 + C. B x3 C 6x + C. D x3 + x + C. + x + C. 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 3x2 + 1 dx = x3 + x + C, v\u1edbi C l\u00e0 h\u1eb1ng s\u1ed1. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 2 (C\u00e2u 7 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). Nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = x3 + x l\u00e0 A x4 + x2 + C. B 3x2 + 1 + C. C x3 + x + C. D 1 x4 + 1 x2 + C. 42 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 (x3 + x) dx = 1 x4 + 1 x2 + C. 42 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 3 (C\u00e2u 14 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). Nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = x4 + x2 l\u00e0 A 4x3 + 2x + C. B 1 x5 + 1 x3 + C. C x4 + x2 + C. D x5 + x3 + C. 53 Ta c\u00f3 f (x) dx = \u0253 L\u1eddi gi\u1ea3i. Ch\u1ecdn \u0111\u00e1p \u00e1n B (x4 + x2) dx = 1 x5 + 1 x3 + C. 53 \u0104 C\u00e2u 4 (C\u00e2u 48 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). H\u1ecd nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = ex + x l\u00e0 A ex + x2 + C. B ex + 1 x2 + C. C 1 ex + 1 x2 + C. 2 x+1 2 D ex + 1 + C. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 282","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG Ta c\u00f3 f (x) dx = (ex + x) dx = ex dx + x dx = ex + 1 x2 + C, v\u1edbi C l\u00e0 h\u1eb1ng s\u1ed1. 2 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 5 (C\u00e2u 1 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). H\u1ecd t\u1ea5t c\u1ea3 c\u00e1c nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = 2x + 6 l\u00e0 A x2 + 6x + C. B 2x2 + C. C 2x2 + 6x + C. D x2 + C. \u0253 L\u1eddi gi\u1ea3i. (2x + 6) dx = x2 + 6x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 6 (C\u00e2u 12 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). H\u1ecd t\u1ea5t c\u1ea3 c\u00e1c nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = 2x + 3 l\u00e0 A 2x2 + C. B x2 + 3x + C. C 2x2 + 3x + C. D x2 + C. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 (2x + 3) dx = x2 + 3x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 7 (C\u00e2u 8 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). H\u1ecd t\u1ea5t c\u1ea3 nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = 2x + 4 l\u00e0 A 2x2 + 4x + C. B x2 + 4x + C. C x2 + C. D 2x2 + C. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) dx = (2x + 4) dx = x2 + 4x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 8 (C\u00e2u 11 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). D \u2212 sin x + C. H\u1ecd t\u1ea5t c\u1ea3 c\u00e1c nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = cos x + 6x l\u00e0 A sin x + 3x2 + C. B \u2212 sin x + 3x2 + C. C sin x + 6x2 + C. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 (cos x + 6x) dx = sin x + 3x2 + C. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 9 (C\u00e2u 8 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). H\u00e0m s\u1ed1 F (x) l\u00e0 m\u1ed9t nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) tr\u00ean kho\u1ea3ng K n\u1ebfu A F (x) = \u2212f (x), \u2200x \u2208 K. B f (x) = F (x), \u2200x \u2208 K. C F (x) = f (x), \u2200x \u2208 K. D f (x) = \u2212F (x), \u2200x \u2208 K. \u0253 L\u1eddi gi\u1ea3i. H\u00e0m s\u1ed1 F (x) l\u00e0 m\u1ed9t nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) tr\u00ean kho\u1ea3ng K n\u1ebfu F (x) = f (x), \u2200x \u2208 K. Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 283 S\u0110T: 0905.193.688","1. Nguy\u00ean h\u00e0m \u0104 C\u00e2u 10 (C\u00e2u 14 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). x2 dx b\u1eb1ng A 2x + C. B 1 x3 + C. C x3 + C. D 3x3 + C. 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 x2 dx = 1 x3 + C. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 11 (C\u00e2u 14 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). x3 dx b\u1eb1ng A 4x4 + C. B 3x2 + C. C x4 + C. D 1 x4 + C. 4 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 x3 dx = 1 x4 + C. 4 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 12 (C\u00e2u 25 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). x4 dx b\u1eb1ng A 1 x5 + C. B 4x3 + C. C x5 + C. D 5x5 + C. 5 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 x4 dx = 1 x5 + C 5 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 13 (C\u00e2u 6 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). 5x4 dx b\u1eb1ng A 1 x5 + C. B x5 + C. C 5x5 + C. D 20x3 + C. 5 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 5x4 dx = 5 \u00b7 x5 + C = x5 + C. 5 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 14 (C\u00e2u 14 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). 6x5 dx b\u1eb1ng A 6x6 + C. B x6 + C. C 1 x6 + C. D 30x4 + C. 6 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 6x5 dx = 6 \u00b7 x6 + C = x6 + C. 6 Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 284 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 15 (C\u00e2u 24 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). 3x2 dx b\u1eb1ng A 3x3 + C. B 6x + C. C 1 x3 + C. D x3 + C. 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 3x2 dx = x3 + C. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 16 (C\u00e2u 5 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). 4x3 dx b\u1eb1ng A 4x4 + C. B 1 x4 + C. C 12x2 + C. D x4 + C. 4 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 4x3 dx = x4 + C. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 17 (C\u00e2u 11 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = x2 + 4. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) dx = 2x + C. B f (x) dx = x2 + 4x + C. C x3 D f (x) dx = x3 + 4x + C. f (x) dx = + 4x + C. 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) dx = x2 + 4 x3 Ch\u1ecdn \u0111\u00e1p \u00e1n C dx = + 4x + C. 3 \u0104 C\u00e2u 18 (C\u00e2u 27 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = ex + 2. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o sau \u0111\u00e2y \u0111\u00fang? A f (x) dx = ex\u22122 + C. B f (x) dx = ex + 2x + C. C f (x) dx = ex + C. D f (x) dx = ex \u2212 2x + C. Ta c\u00f3 f (x) dx = \u0253 L\u1eddi gi\u1ea3i. (ex + 2) dx = ex + 2x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 19 (C\u00e2u 11 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = x2 + 3. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) dx = x2 + 3x + C. B x3 f (x) dx = + 3x + C. 3 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 285 S\u0110T: 0905.193.688","C f (x) dx = x3 + 3x + C. 1. Nguy\u00ean h\u00e0m D f (x) dx = 2x + C. \u0253 L\u1eddi gi\u1ea3i. x3 Ta c\u00f3 f (x) dx = + 3x + C. 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 20 (C\u00e2u 20 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = ex + 1. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o sau \u0111\u00e2y \u0111\u00fang? A f (x) dx = ex\u22121 + C. B f (x) dx = ex \u2212 x + C. C f (x) dx = ex + x + C. D f (x) dx = ex + C. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) dx = ex + x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 21 (C\u00e2u 9 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = x2 + 1. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) dx = x3 + x + C. B x3 f (x) dx = + x + C. 3 C f (x) dx = x2 + x + C. D f (x) dx = 2x + C. Ta c\u00f3 f (x) dx = \u0253 L\u1eddi gi\u1ea3i. Ch\u1ecdn \u0111\u00e1p \u00e1n B (x2 + 1) dx = x3 + x + C. 3 \u0104 C\u00e2u 22 (C\u00e2u 14 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = ex + 3. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x)dx = ex + 3x + C. B f (x)dx = ex + C. C f (x)dx = ex\u22123 + C. D f (x)dx = ex \u2212 3x + C. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x)dx = (ex + 3)dx = ex + 3x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 23 (C\u00e2u 13 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = x2 + 2. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) dx = 2x + C. B x3 f (x) dx = + 2x + C. 3 C f (x) dx = x2 + 2x + C. D f (x) dx = x3 + 2x + C. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 286 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 f (x) dx = x2 + 2 x3 Ch\u1ecdn \u0111\u00e1p \u00e1n B dx = + 2x + C. 3 \u0104 C\u00e2u 24 (C\u00e2u 23 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = ex + 4. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) dx = ex + 4x + C. B f (x) dx = ex + C. C f (x) dx = ex\u22124 + C. D f (x) dx = ex \u2212 4x + C. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) dx = (ex + 4) dx = ex + 4x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 25 (C\u00e2u 15 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = 4x3 \u2212 3. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) dx = x4 \u2212 3x + C. B f (x) dx = x4 + C. C f (x) dx = 4x3 \u2212 3x + C. D f (x) dx = 12x2 + C. Ta c\u00f3 f (x) dx = \u0253 L\u1eddi gi\u1ea3i. Ch\u1ecdn \u0111\u00e1p \u00e1n A 4x3 \u2212 3 dx = x4 \u2212 3x + C. \u0104 C\u00e2u 26 (C\u00e2u 18 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = 4 + cos x. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) dx = \u2212 sin x + C. B f (x) dx = 4x + sin x + C. C f (x) dx = 4x \u2212 sin x + C. D f (x) dx = 4x + cos x + C. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) dx = 4x + sin x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 27 (C\u00e2u 23 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = 2 + cos x. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) dx = 2x + sin x + C. B f (x) dx = 2x + cos x + C. C f (x) dx = \u2212 sin x + C. D f (x) dx = 2x \u2212 sin x + C. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) dx = (2 + cos x) dx = 2x + sin x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 287 S\u0110T: 0905.193.688","1. Nguy\u00ean h\u00e0m \u0104 C\u00e2u 28 (C\u00e2u 25 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = 4x3 \u2212 2. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o sau \u0111\u00e2y \u0111\u00fang? A f (x) dx = x4 \u2212 2x + C. B f (x) dx = 4x3 \u2212 2x + C. C f (x) dx = 12x2 + C. D f (x) dx = x4 + C. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) dx = (4x3 \u2212 2) dx = x4 \u2212 2x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 29 (C\u00e2u 21 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = 1 + cos x. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) dx = \u2212 sin x + C. B f (x) dx = x \u2212 sin x + C. C f (x) dx = x + cos x + C. D f (x) dx = x + sin x + C. Ta c\u00f3 \u0253 L\u1eddi gi\u1ea3i. f (x) dx = (1 + cos x) dx = x + sin x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 30 (C\u00e2u 9 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = 4x3 \u2212 4. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) dx = 12x2 + C. B f (x) dx = 4x3 \u2212 4x + C. C f (x) dx = x4 \u2212 4x + C. D f (x) dx = x4 + C. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 4x3 \u2212 4 dx = x4 \u2212 4x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 31 (C\u00e2u 1 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00e0m s\u1ed1 f (x) = ex + 2x. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) dx = ex + 2x2 + C. B f (x) dx = ex \u2212 x2 + C. C f (x) dx = ex + C. D f (x) dx = ex + x2 + C. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) dx = ex + x2 + C. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 32 (C\u00e2u 12 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho f (x)dx = \u2212 cos x + C. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 288 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG A f (x) = \u2212 sin x. B f (x) = cos x. C f (x) = sin x. D f (x) = \u2212 cos x. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) = (\u2212 cos x + C) = sin x. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 33 (C\u00e2u 36 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00e0m s\u1ed1 f (x) = 1 \u2212 1 \u00b7 Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? cos2 2x 1 A f (x) dx = x + cos 2x + C. B f (x) dx = x + tan 2x + C. 2 C 1 D f (x) dx = x \u2212 1 tan 2x + C. f (x) dx = x + tan 2x + C. 2 2 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) dx = \u00c5 1 \u00e3 dx = x \u2212 1 tan 2x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n D 1\u2212 cos2 2x 2 \u0104 C\u00e2u 34 (C\u00e2u 4 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A exdx = xex + C. B exdx = ex+1 + C. C exdx = \u2212ex+1 + C. D exdx = ex + C. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 exdx = ex + C. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 35 (C\u00e2u 14 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). \u03c0 H\u00e0m s\u1ed1 F (x) = cot x l\u00e0 m\u1ed9t nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y tr\u00ean kho\u1ea3ng 0; 2 A f2(x) = 1 . B f1(x) = \u22121 . C f4(x) = 1 D f3(x) = \u2212 1 . x cos2 x . sin2 x sin2 cos2 x \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 1 dx = \u2212 cot x + C suy ra F (x) = cot x tr\u00ean kho\u1ea3ng \u03c0 l\u00e0 m\u1ed9t nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m sin2 x 0; 2 s\u1ed1 f3(x) = \u2212 1 \u00b7 sin2 x Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 36 (C\u00e2u 11 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). H\u00e0m s\u1ed1 F (x) = cot x l\u00e0 m\u1ed9t nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y tr\u00ean kho\u1ea3ng 0; \u03c0 ? 2 A f2(x) = 1 x . B f1(x) = \u2212 1 x. C f3(x) = \u2212 1 x. D f4(x) = 1 x . sin2 cos2 sin2 cos2 \u0253 L\u1eddi gi\u1ea3i. 1 Ta c\u00f3 \u2212 sin2 x dx = cot x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 289 S\u0110T: 0905.193.688","1. Nguy\u00ean h\u00e0m \u0104 C\u00e2u 37 (C\u00e2u 16 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o sau \u0111\u00e2y \u0111\u00fang? A exdx = ex + C. B exdx = xex + C. C exdx = \u2212ex+1 + C. D exdx = ex+1 + C. \u0253 L\u1eddi gi\u1ea3i. exdx = ex + C. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 38 (C\u00e2u 32 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00e0m s\u1ed1 f (x) = 1 + e2x. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) dx = x + 1 ex + C. B f (x) dx = x + 2e2x + C. 2 C f (x) dx = x + e2x + C. D f (x) dx = x + 1 e2x + C. 2 Ta c\u00f3 f (x) dx = \u0253 L\u1eddi gi\u1ea3i. Ch\u1ecdn \u0111\u00e1p \u00e1n D 1 + e2x dx = x + 1 e2x + C. 2 \u0104 C\u00e2u 39 (C\u00e2u 5 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Tr\u00ean kho\u1ea3ng (0; +\u221e), h\u1ecd nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = x3 l\u00e0 2 A 31 B 52 f (x)dx = x 2 + C. f (x)dx = x 5 + C. 2 2 C 25 D 21 f (x)dx = x 2 + C. f (x)dx = x 2 + C. 5 3 H\u1ecd nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = x3 l\u00e0 \u0253 L\u1eddi gi\u1ea3i. 2 23 5 f (x) dx = x 2 dx = x 2 + C. 5 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 40 (C\u00e2u 27 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00e0m s\u1ed1 f (x) = 1 + sin x. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x)dx = x \u2212 cos x + C. B f (x)dx = x + sin x + C. C f (x)dx = x + cos x + C. D f (x)dx = cos x + C. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x)dx = x \u2212 cos x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 41 (C\u00e2u 2 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = cos 3x. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 290 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG A cos 3x dx = 3 sin 3x + C. B cos 3x dx = sin 3x + C. C cos 3x dx = \u2212sin 3x + C. 3 3 D cos 3x dx = sin 3x + C. \u0253 L\u1eddi gi\u1ea3i. 1 cos 3x d(3x) = sin 3x +C cos 3x dx = 3 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 42 (C\u00e2u 23 - \u0110MH - BGD&\u0110\u221aT - N\u0103m 2016 - 2017). T\u00ecm nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = 2x \u2212 1. \u221a \u221a A f (x) dx = 2 \u2212 1) 2x \u2212 1 + C. B f (x) dx = 1 \u2212 1) 2x \u2212 1 + C. (2x (2x 33 \u221a \u221a C f (x) dx = \u22121 (2x \u2212 1) 2x \u2212 1 + C. D f (x) dx = 1 \u2212 1) 2x \u2212 1 + C. (2x 32 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) dx = \u221a \u2212 1 dx = 1 (2x \u2212 1) 1 d(2x \u2212 1) 2x 2 2 = 1 \u00b7 2 \u2212 3 + C = 1 \u2212 \u221a \u2212 1 + C (2x (2x 1) 2x 1) 2 23 3 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 43 (C\u00e2u 22 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = cos 2x. A 1 B f (x) dx = \u22121 sin 2x + C. . f (x) dx = sin 2x + C. 2 2 C f (x) dx = 2 sin 2x + C. . D f (x) dx = \u22122 sin 2x + C. 1 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) dx = 1 cos 2x d(2x) = sin 2x + C. 2 2 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 44 (C\u00e2u 24 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). 1 Bi\u1ebft F (x) l\u00e0 m\u1ed9t nguy\u00ean h\u00e0m c\u1ee7a f (x) = x \u2212 1 v\u00e0 F (2) = 1. T\u00ednh F (3). 7 1 F (3) = . A F (3) = ln 2 \u2212 1. B F (3) = ln 2 + 1. C F (3) = . D 4 2 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 F (x) = ln |x \u2212 1| + C. Do F (2) = 1 n\u00ean C = 1 \u21d2 F (x) = ln |x \u2212 1| + 1. Khi \u0111\u00f3 F (3) = ln 2 + 1. Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 291 S\u0110T: 0905.193.688","1. Nguy\u00ean h\u00e0m \u0104 C\u00e2u 45 (C\u00e2u 10 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). 2 T\u00ecm nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = x2 + x2 . A f (x) dx = x3 \u2212 2 + C. B f (x) dx = x3 \u2212 1 + C. 3x 3x C x3 2 D x3 1 f (x) dx = + + C. f (x) dx = + + C. 3x 3x \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 \u00c5 + 2\u00e3 dx = x3 \u2212 2 + C. x2 x2 3 x Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 46 (C\u00e2u 4 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). Nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = x4 + x l\u00e0 A x4 + x2 + C. B 4x3 + 1 + C. C x5 + x2 + C. D 1 x5 + 1 x2 + C. 52 \u0253 L\u1eddi gi\u1ea3i. D x3 + x2 + C. Ta c\u00f3 x4 + x dx = 1 x5 + 1 x2 + C. D x2 + C. 52 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 47 (C\u00e2u 6 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). Nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = x3 + x2 l\u00e0 A x4 + x3 + C. B 1 x4 + 1 x3 + C. C 3x2 + 2x + C. 43 Ta c\u00f3 f (x) dx = \u0253 L\u1eddi gi\u1ea3i. Ch\u1ecdn \u0111\u00e1p \u00e1n B (x3 + x2) dx = 1 x4 + 1 x3 + C. 43 \u0104 C\u00e2u 48 (C\u00e2u 15 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). H\u1ecd t\u1ea5t c\u1ea3 c\u00e1c nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = 2x + 5 l\u00e0 A x2 + 5x + C. B 2x2 + 5x + C. C 2x2 + C. \u0253 L\u1eddi gi\u1ea3i. H\u1ecd t\u1ea5t c\u1ea3 c\u00e1c nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = 2x + 5 l\u00e0 F (x) = x2 + 5x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 49 (C\u00e2u 24 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). x + 2 H\u1ecd t\u1ea5t c\u1ea3 c\u00e1c nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = x \u2212 1 tr\u00ean kho\u1ea3ng (1; +\u221e) l\u00e0 A x + 3 ln (x \u2212 1) + C. B x \u2212 3 ln (x \u2212 1) + C. C x\u2212 3 + C. D x \u2212 (x 3 1)2 + C. (x \u2212 1)2 \u2212 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 292","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG Ta c\u00f3: x+2 x\u22121+3 \u00c5 3\u00e3 F (x) = f (x) dx = x \u2212 1 dx = x \u2212 1 dx = 1+ dx x\u22121 = x + 3 ln |x \u2212 1| + C \u2212x\u2212\u2208\u2212(1\u2212;+\u2212\u221e\u2212\u2192) F (x) = x + 3 ln (x \u2212 1) + C. V\u1eady F (x) = x + 3 ln (x \u2212 1) + C. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 50 (C\u00e2u 25 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = 4x3 \u2212 1. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) dx = x4 \u2212 x + C . B f (x) dx = 12x2 + C . C f (x) dx = 4x3 \u2212 x + C . D f (x) dx = x4 + C . Ta c\u00f3 \u0253 L\u1eddi gi\u1ea3i. f (x) dx = 4x3 \u2212 1 dx = x4 \u2212 x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 51 (C\u00e2u 26 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 f (x) = 3 + cos x. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o du\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) dx = 3x \u2212 sin x + C. B f (x) dx = 3x + sin x + C. C f (x) dx = \u2212 sin x + C. D f (x) dx = 3x + cos x + C. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) dx = 3x + sin x + C. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 52 (C\u00e2u 27 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00e0m s\u1ed1 f (x) = ex + 2x. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) dx = ex + x2 + C. B f (x) dx = ex + C. C f (x) dx = ex \u2212 x2 + C. D f (x) dx = ex + 2x2 + C. Ta c\u00f3 f (x) dx = \u0253 L\u1eddi gi\u1ea3i. Ch\u1ecdn \u0111\u00e1p \u00e1n A ex + 2x dx = ex + x2 + C. \u0104 C\u00e2u 53 (C\u00e2u 35 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00e0m s\u1ed1 f (x) = 1 \u2212 1 2x \u00b7 Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? cos2 1 A f (x) dx = x + tan 2x + C. B f (x) dx = x + cot 2x + C. 2 C f (x) dx = x \u2212 1 tan 2x + C. D 1 2 f (x) dx = x + tan 2x + C. 2 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 293 S\u0110T: 0905.193.688","1. Nguy\u00ean h\u00e0m \u0253 L\u1eddi gi\u1ea3i. f (x) dx = \u00c5 1 \u00e3 dx = x \u2212 1 tan 2x + C. 1\u2212 cos2 2x 2 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 54 (C\u00e2u 30 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Cho h\u00e0m s\u1ed1 f (x) = 1 + e2x. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 \u0111\u00fang? A f (x)dx = x + 1 ex + C. B f (x)dx = x + 1 e2x + C. 2 2 C f (x)dx = x + 1 e2x + C. D f (x)dx = x + 1 e2x + C. 2 2 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 1 + e2x dx = x + 1 e2x + C. 2 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 55 (C\u00e2u 34 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). 3x \u2212 1 H\u1ecd t\u1ea5t c\u1ea3 c\u00e1c nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = (x \u2212 1)2 tr\u00ean kho\u1ea3ng (1; +\u221e) l\u00e0 A 3 ln(x \u2212 1) \u2212 x 2 1 + C. B 3 ln(x \u2212 1) + x 1 1 + C. \u2212 \u2212 1 2 C 3 ln(x \u2212 1) \u2212 x \u2212 1 + C. D 3 ln(x \u2212 1) + x \u2212 1 + C. \u0253 L\u1eddi gi\u1ea3i. 3x \u2212 1 3(x \u2212 1) + 2 3 2 Ta c\u00f3 f (x) = (x \u2212 1)2 = (x \u2212 1)2 = x \u2212 1 + (x \u2212 1)2 V\u1edbi x > 1 ta c\u00f3 \u00c53 2\u00e3 d(x \u2212 1) d(x \u2212 1) = 3 ln(x \u2212 1) \u2212 2 + C. f (x) dx = + dx = 3 +2 x \u2212 1 (x \u2212 1)2 x\u22121 (x \u2212 1)2 x\u22121 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 56 (C\u00e2u 42 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Bi\u1ebft F (x) = ex + 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 2ex + 2x2 + C. B 1 e2x + x2 + C. C 1 e2x + 2x2 + C. D 2e2x + 4x2 + C. 2 2 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (2x) dx = 1 \u00b7 f (2x) d(2x) 2 = 1 \u00b7 F (2x) + C 2 = 1 \u00b7 e2x + (2x)2 + C 2 = 1 e2x + 2x2 + C. 2 Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 294 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 57 (C\u00e2u 40 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). Bi\u1ebft F (x) = ex + 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 e2x + 8x2 + C. B 2ex + 4x2 + C. C 1 e2x + 2x2 + C. D 1 e2x + 4x2 + C. 2 2 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 1 f (2x) d(2x) = 1 + C = 1 e2x + 4x2 + C. Ch\u1ecdn \u0111\u00e1p \u00e1n D f (2x) dx = F (2x) 2 22 \u0104 C\u00e2u 58 (C\u00e2u 25 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). H\u1ecd nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = 4x (1 + ln x) l\u00e0 A 2x2 ln x + 3x2. B 2x2 ln x + x2. C 2x2 ln x + 3x2 + C. D 2x2 ln x + x2 + C. \u00aeu = 1 + ln x \uf8f11 \u0253 L\u1eddi gi\u1ea3i. \uf8f2 du = dx 2x dx = 2x2 (1 + ln x) \u2212 x2 + C = 2x2 ln x + x2 + C. \u0110\u1eb7t \u21d2 x dv = 4x dx \uf8f3v = 2x2. Khi \u0111\u00f3 f (x) dx = 2x2 (1 + ln x) \u2212 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 59 (C\u00e2u 27 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00e0m s\u1ed1 f (x) th\u1ecfa f (x) = 3 \u2212 5 sin x v\u00e0 f (0) = 10. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A f (x) = 3x + 5 cos x + 5. B f (x) = 3x + 5 cos x + 2. C f (x) = 3x \u2212 5 cos x + 2. D f (x) = 3x \u2212 5 cos x + 15. \u0253 L\u1eddi gi\u1ea3i. f (x) = (3 \u2212 5 sin x) dx = 3x + 5 cos x + C. f (0) = 10 \u21d2 5 + C = 10 \u21d2 C = 5. V\u1eady h\u00e0m s\u1ed1 c\u1ea7n t\u00ecm: f (x) = 3x + 5 cos x + 5. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 60 (C\u00e2u 2 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). 1 T\u00ecm nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = . 5x \u2212 2 dx 1 dx \u22121 A 5x \u2212 2 = 5 ln |5x \u2212 2| + C. B 5x \u2212 2 = 2 ln(5x \u2212 2) + C. C dx 2 = 5 ln |5x \u2212 2| + C. D dx = ln |5x \u2212 2| + C. 5x \u2212 5x \u2212 2 \u0253 L\u1eddi gi\u1ea3i. dx 1 d(5x \u2212 2) = 1 ln |5x \u2212 2| + C. Ta c\u00f3 5x \u2212 2 = 5(5x \u2212 2) 5 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 61 (C\u00e2u 8 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = 2 sin x. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 295 S\u0110T: 0905.193.688","1. Nguy\u00ean h\u00e0m A 2 sin x dx = 2 cos x + C. B 2 sin x dx = sin2 x + C. C 2 sin x dx = sin 2x + C. D 2 sin x dx = \u22122 cos x + C. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 2 sin x dx = 2 sin x dx = \u22122 cos x + C Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 62 (C\u00e2u 9 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = 7x. B 7x dx = 7x + C. A 7x dx = 7x ln 7 + C. ln 7 C 7x dx = 7x+1 + C. D 7x dx = 7x+1 + C. x+1 \u0253 L\u1eddi gi\u1ea3i. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 63 (C\u00e2u 28 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). \u03c0 T\u00ecm nguy\u00ean h\u00e0m F (x) c\u1ee7a h\u00e0m s\u1ed1 f (x) = sin x + cos x th\u1ecfa m\u00e3n F = 2. 2 A F (x) = cos x \u2212 sin x + 3. B F (x) = \u2212 cos x + sin x + 3. C F (x) = \u2212 cos x + sin x \u2212 1. D F (x) = \u2212 cos x + sin x + 1. \u0253 L\u1eddi gi\u1ea3i. = 2 n\u00ean C = 1. Ta c\u00f3 F (x) = f (x) dx = \u2212 cos x + sin x + C. M\u00e0 F \u03c0 2 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 64 (C\u00e2u 37 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho F (x) = \u2212 1 l\u00e0 m\u1ed9t nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) . T\u00ecm nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) ln x. 3x3 x A ln x 1 B f (x) ln x dx = ln x \u2212 1 + C. f (x) ln x dx = + + C. x3 5x5 x3 5x5 C ln x 1 D f (x) ln x dx = \u2212ln x + 1 + C. f (x) ln x dx = x3 + 3x3 + C. x3 3x3 \u0253 L\u1eddi gi\u1ea3i. f (x) \u00c5 1 \u00e3 1 1 =\u2212 . Suy ra f (x) = . T\u1eeb gi\u1ea3 thi\u1ebft, = (F (x)) = x3 x 3x3 x4 \u0110\u1ec3 t\u00ednh f (x) ln x dx, d\u00f9ng t\u00edch ph\u00e2n t\u1eebng ph\u1ea7n v\u1edbi u = ln x v\u00e0 dv = f (x) dx. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 65 (C\u00e2u 31 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). H\u1ecd t\u1ea5t c\u1ea3 c\u00e1c nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = 2x \u2212 1 tr\u00ean kho\u1ea3ng(\u22121; +\u221e) l\u00e0 (x + 1)2 A 2 ln(x + 1) + 2 + C. B 2 ln(x + 1) + 3 + C. x+1 x+1 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 296 S\u0110T: 0905.193.688","Ch\u01b0\u01a1ng 3. NGUY\u00caN H\u00c0M. T\u00cdCH PH\u00c2N V\u00c0 \u1ee8NG D\u1ee4NG C 2 ln(x + 1) \u2212 2 + C. D 2 ln(x + 1) \u2212 3 + C. x+1 x+1 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 2x \u2212 1 2(x + 1) \u2212 3 f (x) dx = (x + 1)2 dx = (x + 1)2 dx \u00ef2 3\u00f2 3 = \u2212 dx = 2 ln(x + 1) + + C. x + 1 (x + 1)2 x+1 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 66 (C\u00e2u 34 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). H\u1ecd t\u1ea5t c\u1ea3 c\u00e1c nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = 2x + 1 tr\u00ean kho\u1ea3ng (\u22122; +\u221e) l\u00e0 (x + 2)2 A 2 ln(x + 2) + 1 + C. B 2 ln(x + 2) \u2212 1 + C. x+2 x+2 C 2 ln(x + 2) \u2212 3 + C. D 2 ln(x + 2) + 3 + C. x+2 x+2 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) dx = 2x + 4 \u2212 3 \u00ef2 \u2212 3 \u00f2 3 + C. Ch\u1ecdn \u0111\u00e1p \u00e1n D dx = dx = 2 ln |x + 2| + x + 2 (x + 2)2 x+2 (x + 2)2 \u0104 C\u00e2u 67 (C\u00e2u 35 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). \u03c0 4 Cho h\u00e0m s\u1ed1 f (x). Bi\u1ebft f (0) = 4 v\u00e0 f (x) = 2 sin2 x + 1, \u2200x \u2208 R, khi \u0111\u00f3 f (x) dx b\u1eb1ng 0 A \u03c02 + 15\u03c0 B \u03c02 + 16\u03c0 \u2212 16 C \u03c02 + 16\u03c0 \u2212 4 D \u03c02 \u2212 4 . . . . 16 16 16 16 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 f (x) = 2 sin2 x + 1 dx = (2 \u2212 cos 2x) dx = 2x \u2212 1 sin 2x + C. 2 V\u00ec f (0) = 4 \u21d2 C = 4 Hay f (x) = 2x \u2212 1 sin 2x + 4. 2 \u03c0\u03c0 \u03c0 4 4 \u00c5 1 \u00e3 1 4 \u03c02 1 \u03c02 + 16\u03c0 \u2212 4 2x 4 . Suy ra f (x) dx = \u2212 sin 2x + dx = x2 + cos 2x + 4x = +\u03c0\u2212 4 = 2 4 0 16 16 00 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 68 (C\u00e2u 35 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). 3x \u2212 2 H\u1ecd t\u1ea5t c\u1ea3 c\u00e1c nguy\u00ean h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 f (x) = (x \u2212 2)2 tr\u00ean kho\u1ea3ng (2; +\u221e) l\u00e0 A 3 ln(x \u2212 2) + 4 B 3 ln(x \u2212 2) + x 2 2 + C. x \u2212 2 + C. \u2212 2 4 C 3 ln(x \u2212 2) \u2212 x\u22122 + C. D 3 ln(x \u2212 2) \u2212 x \u2212 2 + C. \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 297 S\u0110T: 0905.193.688"]