["3. L\u00f4garit \u0104 C\u00e2u 79 (C\u00e2u 49 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Trong kh\u00f4ng gian Oxyz, cho hai \u0111i\u1ec3m A(\u22122; 1; \u22123) v\u00e0 B(1; \u22123; 2). X\u00e9t hai \u0111i\u1ec3m M v\u00e0 N thay \u0111\u1ed5i thu\u221a\u1ed9c m\u1eb7t ph\u1eb3ng (Oxy) sa\u221ao cho M N = 3. Gi\u00e1 tr\u1ecb \u221al\u1edbn nh\u1ea5t c\u1ee7a |AM \u2212 BN\u221a| b\u1eb1ng A 65. B 29. C 26. D 91. C\u00e1ch 1: \u0253 L\u1eddi gi\u1ea3i. A2 A1 B HK NM MDD-109 A G\u1ecdi A1 l\u00e0 \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9ng c\u1ee7#a A\u00bbqua M#(ONx\u00bby\u21d2) \u21d2AA1A1(2\u2212=2; 1; 3). A2 thu\u1ed9c \u0111\u01b0\u1eddng tr\u00f2n (C ) n\u1eb1m trong m\u1eb7t G\u1ecdi A2 l\u00e0 \u0111i\u1ec3m th\u1ecfa m\u00e3n A1A2 = 3 \u21d2 ph\u1eb3ng song song v\u1edbi (Oxy) c\u00f3 t\u00e2m A1, b\u00e1n k\u00ednh R = 3. Khi \u0111\u00f3 |AM \u2212 BN | = |A1M \u2212 BN | = |A2N \u2212 B# N |\u00bb\u2264 A2B. v\u1edbi H# K\u00bb. D\u1ea5u ra v\u00e0 A2B \u0111\u1ea1t gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t \u21d4 A1A2 ng\u01b0\u1ee3c h\u01b0\u1edbng \u221a \u201c=\u201d x\u1ea3y Suy #\u00bb = \u2212 A1A2 H# K\u00bb = \u00c5 9 ; 12 \u00e3 \u21d2 A2 \u00c5 19 ; 17 \u00e3 \u21d2 A2B = 65. HK \u2212 ; 0 \u221a \u2212 ; 3 ra A1A2 5 5 5 5 V\u1eady gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a |AM \u2212 BN | b\u1eb1ng 65. D\u1ea5u \u201c=\u201d x\u1ea3y ra khi v\u00e0 ch\u1ec9 khi M , N n\u1eb1m tr\u00ean giao tuy\u1ebfn (cO\u1ee7axym))\u1eb7tv\u00e0phM#\u1eb3nN\u00bbg c(\u00f9Qn)gvh\u1edb\u01b0i \u1edbmn\u1eb7gtvp\u1edbhi \u1eb3K#ngH\u00bb(.Oxy) (trong \u0111\u00f3 (Q) l\u00e0 m\u1eb7t ph\u1eb3ng \u0111i qua A, B v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi C\u00e1ch 2: A2 B A1 N M MDD-109 A G\u1ecdi A1 l\u00e0 \u0111i\u1ec3m \u0111\u1ed1i x\u1ee9ng v#\u1edbi A\u00bbqua# m\u00bb\u1eb7t ph\u1eb3ng (Oxy) \u21d2 A1(\u22122; 1; 3). v\u00e0 A2, B n\u1eb1m tr\u00ean c\u00f9ng m\u1ed9t G\u1ecdi A2 l\u00e0 \u0111i\u1ec3m th\u1ecfa m\u00e3n A1A2 = M N = (x; y; 0) \u21d2 A2(x \u2212 2; y + 1; 3) Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 198 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\u00eda b\u1edd l\u00e0 m\u1eb7t ph\u1eb3ng (Oxy). V\u00ec M N = 3 \u21d2 x2 + y2 = 3 \u21d4 x2 + y2 = 9. Ta c\u00f3 |M A \u2212 N B| = |M A1 \u2212 N B| = |N A2 \u2212 N B| \u2264 A2B. M\u00e0 \u00bb A2B = (x \u2212 3)2 + (y + 4)2 + 1 = x2 + y2 + 26 \u2212 6x + 8y \u00bb \u2264 35 + (62 + 82) (x2 + y2) \u221a = 65. \u221a V\u1eady gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a |AM \u2212 BN | l\u00e0 65. \uf8f4\uf8f1A2, B, N th\u1eb3ng h\u00e0ng \uf8f4 x y \uf8f2 D\u1ea5u \u201c=\u201d x\u1ea3y ra khi v\u00e0 ch\u1ec9 khi \u22126 = 8 = t \u2265 0 (1) \uf8f4 \uf8f4\uf8f3x2 + y2 = 9. (2) T\u1eeb (1) \u21d2 x = \u22126t, y = 8t. Thay v\u00e0o (2) ta \u0111\u01b0\u1ee3c 36t2 + 64t2 = 9 \u21d4 t2 = 9 \u21d2t= 3 \u21d2 x = \u22129; y = 8t = 12 . 100 10 5 5 \u00c5 19 17 \u00e3 #\u00bb \u00c5 24 \u2212 32 ; \u00e3 Suy ra A2 \u2212 ; ; 3 \u21d2 A2B = ; 5 \u22121 . 5 5 5 \uf8f1 24 \uf8f4\uf8f4x = 1 + t \uf8f4 5 \uf8f4 \uf8f2 32 t Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng A2B : y = \u22123 \u2212 \uf8f4 5 \uf8f4 \uf8f4 \uf8f4\uf8f3z = 2 \u2212 t. \u0110i\u1ec3m N \u2208 A2B \u21d2 N \u00c5 24 32 \u00e3 1 + t; \u22123 \u2212 t; 2 \u2212 t . 55 \u00c5 53 79 \u00e3 \u00c5 62 91 \u00e3 0 0. \u0110i\u1ec3m N \u2208 (Oxy) \u21d2 2\u2212t = 0 \u21d4 t = 2 \u21d2 N ; \u2212 ; \u21d2 M ; \u2212 ; 55 55 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 80 (C\u00e2u 50 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Cho h\u00e0m s\u1ed1 y = f (x) c\u00f3 \u0111\u1ea1o h\u00e0m f (x) = (x \u2212 9)(x2 \u2212 16), \u2200x \u2208 R. C\u00f3 bao nhi\u00eau gi\u00e1 tr\u1ecb nguy\u00ean d\u01b0\u01a1ng c\u1ee7a tham s\u1ed1 m \u0111\u1ec3 h\u00e0m s\u1ed1 g(x) = f (|x3 + 7x| + m) c\u00f3 \u00edt nh\u1ea5t 3 \u0111i\u1ec3m c\u1ef1c tr\u1ecb? A 16. B 9. C 4. D 8. \u0253 L\u1eddi gi\u1ea3i. \uf8eex = 9 Ta c\u00f3 f (x) = (x \u2212 9)(x \u2212 4)(x + 4) \u21d2 f (x) = 0 \u21d4 \uf8efx = 4 \uf8f0 x = \u22124. Ta nh\u1eadn th\u1ea5y r\u1eb1ng, h\u00e0m s\u1ed1 h(x) = |x3 + 7x| \u0111\u1ea1t c\u1ef1c ti\u1ec3u t\u1ea1i x = 0. M\u00e0 g (x) = h (x) \u00b7 f (|x3 + 7x| + m) n\u00ean h\u00e0m s\u1ed1 g(x) c\u00f3 \u00edt nh\u1ea5t 3 \u0111i\u1ec3m c\u1ef1c tr\u1ecb th\u00ec ph\u01b0\u01a1ng tr\u00ecnh f x3 + 7x + m = 0. (1) ph\u1ea3i c\u00f3 \u00edt nh\u1ea5t 2 nghi\u1ec7m b\u1ed9i l\u1ebb kh\u00e1c 0. \u0110\u1eb7t u = x3 + 7x \u21d2 u = 3x2 + 7 > 0, \u2200x \u2208 R \u21d2 u(x) l\u00e0 h\u00e0m \u0111\u1ed3ng bi\u1ebfn. M\u1eb7t kh\u00e1c lim (x3 + 7x) = \u2212\u221e v\u00e0 lim (x3 + 7x) = +\u221e \u21d2 \u1ee9ng v\u1edbi m\u1ed7i gi\u00e1 tr\u1ecb c\u1ee7a u cho duy nh\u1ea5t x\u2192\u2212\u221e x\u2192+\u221e 1 gi\u00e1 tr\u1ecb c\u1ee7a x. \uf8ee|u| + m = 9 \uf8ee|u| = 9 \u2212 m Ph\u01b0\u01a1ng tr\u00ecnh tr\u1edf th\u00e0nh f (|u| + m) = 0 \u21d4 \uf8ef|u| + m = 4 \u21d4 \uf8ef|u| = 4 \u2212 m \uf8f0\uf8f0 |u| + m = \u22124 |u| = \u22124 \u2212 m. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 199 S\u0110T: 0905.193.688","3. L\u00f4garit V\u1edbi m\u1ed7i gi\u00e1 tr\u1ecb d\u01b0\u01a1ng c\u1ee7a |u| s\u1ebd cho 2 gi\u00e1 tr\u1ecb c\u1ee7a u, t\u01b0\u01a1ng \u1ee9ng v\u1edbi 2 gi\u00e1 tr\u1ecb c\u1ee7a x. Do \u0111\u00f3 \u0111\u1ec3 h\u00e0m s\u1ed1 c\u00f3 \u00edt nh\u1ea5t ba c\u1ef1c tr\u1ecb th\u00ec ph\u01b0\u01a1ng tr\u00ecnh f (|u| + m) = 0 ph\u1ea3i c\u00f3 \u00edt nh\u1ea5t 1 nghi\u1ec7m d\u01b0\u01a1ng, suy ra |u| = 9 \u2212 m > 0 \u21d4 m < 9. V\u00ec m \u2208 Z+ n\u00ean m \u2208 {1; 2; 3; 4; 5; 6; 7; 8}. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 81 (C\u00e2u 47 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). 1 \u2212 xy X\u00e9t c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng x, y th\u1ecfa m\u00e3n log3 x + 2y = 3xy + x + 2y \u2212 4. T\u00ecm gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t Pmin c\u1ee7a P = x + y.\u221a \u221a 9 11 \u2212 19 9 11 + 19 A Pmin = . B Pmin = \u221a 9 . \u221a9 11 \u2212 3 C Pmin = 18 11 \u2212 29 D Pmin = 2 . . 3 21 \u0253 L\u1eddi gi\u1ea3i. 1 \u2212 xy V\u1edbi gi\u1ea3 thi\u1ebft b\u00e0i to\u00e1n ta c\u00f3 log3 x + 2y = 3xy + x + 2y \u2212 4 \u21d4 log3 3(1 \u2212 xy) + 3(1 \u2212 xy) = log3(x + 2y) + x + 2y V\u00ec h\u00e0m s\u1ed1 f (x) = x + log3 x \u0111\u1ed3ng bi\u1ebfn tr\u00ean (0; +\u221e) n\u00ean t\u1eeb tr\u00ean ta suy ra 3(1 \u2212 xy) = x + 2y \u21d4 11 = (3x + 2)(3y + 1). \u221a D\u00f9ng b\u1ea5t \u0111\u1eb3ng th\u1ee9c AM \u2212 GM suy ra 3x + 2 + 3y +1\u22652 11. +1 \uf8f1 = \u221a 2 Suy ra x + y \u2265 \u221a th\u1ee9c x\u1ea3y ra 3x + 2 = 3y \uf8f2\uf8f4\uf8f4x 11 \u2212 . 2 11 \u2212 3 \u00ae . \u0110\u1eb3ng khi hay \u221a3 3 3(1 \u2212 xy) = x + 2y \uf8f4\uf8f4\uf8f3y = 11 \u2212 1 3 V\u1eady ph\u01b0\u01a1ng \u00e1n \u0111\u00fang l\u00e0 D. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 82 (C\u00e2u 49 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho h\u00e0m s\u1ed1 y = f (x). \u0110\u1ed3 th\u1ecb c\u1ee7a h\u00e0m s\u1ed1 y = f (x) nh\u01b0 h\u00ecnh y b\u00ean. \u0110\u1eb7t h(x) = 2f (x)\u2212x2. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? 4 A h(4) = h(\u22122) > h(2). B h(4) = h(\u22122) < h(2). 2 C h(2) > h(4) > h(\u22122). D h(2) > h(\u22122) > h(4). \u22122 O2 4x \u22122 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 200 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 Ta c\u00f3 h(x) = 2f (x) \u2212 x2 n\u00ean h (x) = 2 (f (x) \u2212 x). y D\u1ef1a v\u00e0o h\u00ecnh v\u1ebd b\u00ean v\u00e0 t\u00ednh ch\u1ea5t c\u1ee7a t\u00edch ph\u00e2n ta th\u1ea5y h(2) \u2212 h(\u22122) = 4 22 2 h (x) dx = 2 (f (x) \u2212 x) dx > 0 n\u00ean h(2) > h(\u22122). \u22122 \u22122 T\u01b0\u01a1ng t\u1ef1 ta c\u00f3 h(4) > h(\u22122), h(2) > h(4), t\u1eeb \u0111\u00f3 ch\u1ecdn ph\u01b0\u01a1ng \u00e1n C. \u22122 4x O2 \u22122 Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 201 S\u0110T: 0905.193.688","4. H\u00e0m s\u1ed1 m\u0169. H\u00e0m s\u1ed1 L\u00f4garit B\u00c0I 4. H\u00c0M S\u1ed0 M\u0168. H\u00c0M S\u1ed0 L\u00d4GARIT \u0104 C\u00e2u 1 (C\u00e2u 5 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log2 x l\u00e0 A [0; +\u221e). B (\u2212\u221e; +\u221e). C (0; +\u221e). D [2; +\u221e). \u0253 L\u1eddi gi\u1ea3i. H\u00e0m s\u1ed1 x\u00e1c \u0111\u1ecbnh khi x > 0. V\u1eady t\u1eadp x\u00e1c \u0111\u1ecbnh D = (0; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 2 (C\u00e2u 25 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log5 x l\u00e0 A [0; +\u221e). B (\u2212\u221e; 0). C (0; +\u221e). D (\u2212\u221e; +\u221e). \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x > 0. T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log5 x l\u00e0 D = (0; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 3 (C\u00e2u 25 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log6 x l\u00e0 A [0; +\u221e). B (0; +\u221e). C (\u2212\u221e; 0). D (\u2212\u221e; +\u221e). \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x > 0 . V\u1eady t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 l\u00e0 D = (0; +\u221e) . Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 4 (C\u00e2u 22 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log3 x l\u00e0 A (\u2212\u221e; 0). B (0; +\u221e). C (\u2212\u221e; +\u221e). D [0; +\u221e). \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x > 0. V\u1eady t\u1eadp x\u00e1c \u0111\u1ecbnh D = (0; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 5 (C\u00e2u 1 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 log4 x l\u00e0 A (\u2212\u221e; 0). B [0; +\u221e). C (0; +\u221e). D (\u2212\u221e; +\u221e). \u0253 L\u1eddi gi\u1ea3i. T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 log4 x l\u00e0 (0; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 6 (C\u00e2u 2 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = 4x l\u00e0 A R \\\\ {0}. B [0; +\u221e). C (0; +\u221e). D R. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 h\u00e0m s\u1ed1 m\u0169 y = 4x lu\u00f4n x\u00e1c \u0111\u1ecbnh v\u1edbi m\u1ecdi x \u2208 R. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 202 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 7 (C\u00e2u 2 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = 5x l\u00e0 A R. B (0; +\u221e). C R \\\\ {0}. D [0; +\u221e). \u0253 L\u1eddi gi\u1ea3i. H\u00e0m s\u1ed1 m\u0169 y = 5x lu\u00f4n x\u00e1c \u0111\u1ecbnh v\u1edbi m\u1ecdi x \u2208 R. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 8 (C\u00e2u 10 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = 2x l\u00e0 A R. B (0; +\u221e). C [0; +\u221e). D R \\\\ {0}. \u0253 L\u1eddi gi\u1ea3i. T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = 2x l\u00e0 R. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 9 (C\u00e2u 20 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = 3x l\u00e0 A [0; +\u221e). B (0; +\u221e). C R \\\\ {0}. D R. \u0253 L\u1eddi gi\u1ea3i. T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = 3x l\u00e0 D = R. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 10 (C\u00e2u 18 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = 9x l\u00e0 A R. B [0; +\u221e). C R \\\\ {0}. D (0; +\u221e). \u0253 L\u1eddi gi\u1ea3i. T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = 9x l\u00e0 R. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 11 (C\u00e2u 4 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = 7x l\u00e0 A R \\\\ {0}. B [0; +\u221e). C (0; +\u221e). D R. \u0253 L\u1eddi gi\u1ea3i. T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = 7x l\u00e0 D = R. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 12 (C\u00e2u 11 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = 6x l\u00e0 A [0; +\u221e). B R\\\\{0}. C (0; +\u221e). D R. \u0253 L\u1eddi gi\u1ea3i. T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = 6x l\u00e0 D = R. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 203 S\u0110T: 0905.193.688","4. H\u00e0m s\u1ed1 m\u0169. H\u00e0m s\u1ed1 L\u00f4garit \u0104 C\u00e2u 13 (C\u00e2u 18 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = 8x l\u00e0 A R \\\\ {0}. B R. C [0; +\u221e). D (0; +\u221e). \u0253 L\u1eddi gi\u1ea3i. T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = 8x l\u00e0 R. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 14 (C\u00e2u 7 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log3(x \u2212 3) l\u00e0 A (\u2212\u221e; 3]. B (3; +\u221e). C [3; +\u221e). D (\u2212\u221e; \u22123). \u0253 L\u1eddi gi\u1ea3i. H\u00e0m s\u1ed1 x\u00e1c \u0111\u1ecbnh khi v\u00e0 ch\u1ec9 khi x \u2212 3 > 0 \u21d4 x > 3. V\u1eady t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log3(x \u2212 3) l\u00e0 (3; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 15 (C\u00e2u 4 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = 7x l\u00e0 A R \\\\ {0}. B [0; +\u221e). C (0; +\u221e). D R. \u0253 L\u1eddi gi\u1ea3i. H\u00e0m s\u1ed1 y = 7x c\u00f3 t\u1eadp x\u00e1c \u0111\u1ecbnh l\u00e0 D = R. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 16 (C\u00e2u 19 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log3(x \u2212 1) l\u00e0 A (\u2212\u221e; 1]. B [1; +\u221e). C (\u2212\u221e; 1). D (1; +\u221e). \u0253 L\u1eddi gi\u1ea3i. H\u00e0m s\u1ed1 x\u00e1c \u0111\u1ecbnh khi v\u00e0 ch\u1ec9 khi x \u2212 1 > 0 \u21d4 x > 1. V\u1eady t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 l\u00e0 (1; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 17 (C\u00e2u 16 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log3(x \u2212 2) l\u00e0 A (2; +\u221e). B (\u2212\u221e; 2). C [2; +\u221e). D (\u2212\u221e; 2]. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x\u00e1c \u0111\u1ecbnh l\u00e0 x \u2212 2 > 0 \u21d4 x > 2. V\u1eady t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 l\u00e0 D = (2; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 18 (C\u00e2u 16 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log3(x \u2212 4) l\u00e0 A (5; +\u221e). B (\u2212\u221e; +\u221e). C (4; +\u221e). D (\u2212\u221e; 4). \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x \u2212 4 > 0 \u21d4 x > 4. V\u1eady t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 \u0111\u00e3 cho l\u00e0 D = (4; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 204 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 19 (C\u00e2u 20 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log3(x \u2212 4) l\u00e0 A (\u2212\u221e; 4). B (4; +\u221e). C (5; +\u221e). D (\u2212\u221e; +\u221e). \u0253 L\u1eddi gi\u1ea3i. \u0110KX\u0110 x \u2212 4 > 0 \u21d4 x > 4. V\u1eady t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log3(x \u2212 4) l\u00e0 (4; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 20 (C\u00e2u 25 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log2(x \u2212 1) l\u00e0 A (2; +\u221e). B (\u2212\u221e; +\u221e). C (1; +\u221e). D (\u2212\u221e; 1). \u0253 L\u1eddi gi\u1ea3i. H\u00e0m s\u1ed1 x\u00e1c \u0111\u1ecbnh khi x \u2212 1 > 0 \u21d4 x > 1. T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 l\u00e0 D = (1; +\u221e) Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 21 (C\u00e2u 22 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). T\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log2(x \u2212 1) l\u00e0 A (2; +\u221e). B (\u2212\u221e; +\u221e). C (\u2212\u221e; 1). D (1; +\u221e). \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x \u2212 1 > 0 \u21d4 x > 1. V\u1eady t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 \u0111\u00e3 cho l\u00e0 D = (1; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 22 (C\u00e2u 2 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ednh \u0111\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = log x. A y = 1 B y ln 10 Cy= 1 Dy= 1 . =. . . x x x ln 10 10 ln x \u0253 L\u1eddi gi\u1ea3i. \u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c (loga x) 11 = , ta \u0111\u01b0\u1ee3c y = x ln a x ln 10 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 23 (C\u00e2u 18 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). H\u00e0m s\u1ed1 y = 2x2\u2212x c\u00f3 \u0111\u1ea1o h\u00e0m l\u00e0 A (x2 \u2212 x) \u00b7 2x2\u2212x\u22121. B (2x \u2212 1) \u00b7 2x2\u2212x. C 2x2\u2212x \u00b7 ln 2. D (2x \u2212 1) \u00b7 2x2\u2212x \u00b7 ln 2. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 y = (x2 \u2212 x) \u00b7 2x2\u2212x \u00b7 ln 2 = (2x \u2212 1) \u00b7 2x2\u2212x \u00b7 ln 2. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 24 (C\u00e2u 25 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). H\u00e0m s\u1ed1 y = 3x2\u2212x c\u00f3 \u0111\u1ea1o h\u00e0m l\u00e0 A 3x2\u2212x \u00b7 ln 3. B (2x \u2212 1) \u00b7 3x2\u2212x. C (x2 \u2212 x) \u00b7 3x2\u2212x\u22121. D (2x \u2212 1) \u00b7 3x2\u2212x \u00b7 ln 3. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 205 S\u0110T: 0905.193.688","4. H\u00e0m s\u1ed1 m\u0169. H\u00e0m s\u1ed1 L\u00f4garit \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3: (au) = u \u00b7 au \u00b7 ln a n\u00ean \u00c43x2\u2212x\u00e4 = (2x \u2212 1) \u00b7 3x2\u2212x \u00b7 ln 3. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 25 (C\u00e2u 5 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). \u0110\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = 3x l\u00e0 3x A y = . B y = 3x. C y = x3x\u22121. D y = 3x ln 3. ln 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 y = 3x ln 3. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 26 (C\u00e2u 17 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). \u0110\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = 4x l\u00e0 4x . A y = x \u00b7 4x\u22121. B y = 4x ln 4. C y = ln 4 D y = 4x. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 (4x) = 4x \u00b7 ln 4. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 27 (C\u00e2u 24 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). \u0110\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = 6x l\u00e0 6x . A y = 6x ln 6 . B y = x6x\u22121 . C y = 6x. D y = ln 6 \u0253 L\u1eddi gi\u1ea3i. \u0110\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = 6x l\u00e0 y = 6x ln 6. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 28 (C\u00e2u 27 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). \u0110\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = 5x l\u00e0 5x . A y = 5x. B y = ln 5 C y = 5x ln 5. D y = x5x\u22121. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 y = (5x) = 5x \u00b7 ln 5. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 29 (C\u00e2u 2 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). \u0110\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = x\u22123 l\u00e0 A y = \u2212x\u22124. B y = \u22123x\u22124. C y = \u2212 1 x\u22124. D y = \u2212 1 x\u22122. 3 2 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 y = \u22123x\u22124. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 30 (C\u00e2u 22 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Tr\u00ean kho\u1ea3ng (0; +\u221e), \u0111\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = log2 x l\u00e0 1 1 ln 2 C y = 1 \u00b7. . Ay= . B y =. x D y = 2x x ln 2 x Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 206 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 \u0253 L\u1eddi gi\u1ea3i. \u0110\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = log2 x tr\u00ean kho\u1ea3ng (0; +\u221e) l\u00e0 y = 1 . x ln 2 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 31 (C\u00e2u 28 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ednh \u0111\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = log2 (2x + 1). 2 1 . B y= 2 . 1 Ay= . Cy= Dy= . (2x + 1) ln 2 (2x + 1) ln 2 2x + 1 2x + 1 \u0104 C\u00e2u 32 (C\u00e2u 15 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm t\u1eadp x\u00e1c \u0111\u1ecbnh D c\u1ee7a h\u00e0m s\u1ed1 y = log2(x2 \u2212 2x \u2212 3). A D = (\u2212\u221e; \u22121] \u222a [3; +\u221e). B D = [\u22121; 3]. C D = (\u2212\u221e; \u22121) \u222a (3; +\u221e). D D = (\u22121; 3). \u0253 L\u1eddi gi\u1ea3i. H\u00e0m s\u1ed1 c\u00f3 ngh\u0129a \u21d4 x2 \u2212 2x \u2212 3 > 0 \u21d4 \u00f1x > 3 . x < \u22121 V\u1eady t\u1eadp x\u00e1c \u0111\u1ecbnh l\u00e0 D = (\u2212\u221e; \u22121) \u222a (3; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 33 (C\u00e2u 14 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). S\u1ed1 l\u01b0\u1ee3ng c\u1ee7a lo\u1ea1i vi khu\u1ea9n A trong m\u1ed9t ph\u00f2ng th\u00ed nghi\u1ec7m \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c s(t) = s(0).2t, trong \u0111\u00f3 s(0) l\u00e0 s\u1ed1 l\u01b0\u1ee3ng vi khu\u1ea9n A l\u00fac ban \u0111\u1ea7u, s(t) l\u00e0 s\u1ed1 l\u01b0\u1ee3ng vi khu\u1ea9n A c\u00f3 sau t ph\u00fat. Bi\u1ebft sau 3 ph\u00fat th\u00ec s\u1ed1 l\u01b0\u1ee3ng vi khu\u1ea9n A l\u00e0 625 ngh\u00ecn con. H\u1ecfi sau bao l\u00e2u, k\u1ec3 t\u1eeb l\u00fac ban \u0111\u1ea7u, s\u1ed1 l\u01b0\u1ee3ng vi khu\u1ea9n A l\u00e0 10 tri\u1ec7u con ? A 48 ph\u00fat. B 19 ph\u00fat. C 7 ph\u00fat. D 12 ph\u00fat. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 s(3) = s(0).23 \u21d2 s(0) = s(3) = 78125 23 s(t) = s(0).2t \u21d2 2t = s(t) = 128 \u21d2 t = 7. s(0) Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 34 (C\u00e2u 19 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). Cho ba s\u1ed1 th\u1ef1c d\u01b0\u01a1ng a, b, c kh\u00e1c 1. \u0110\u1ed3 th\u1ecb c\u00e1c h\u00e0m s\u1ed1 y = ax, y = bx, y y = bx y = cx \u0111\u01b0\u1ee3c cho trong h\u00ecnh v\u1ebd b\u00ean. M\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? A a < b < c. B a < c < b. C b < c < a. D c < a < b. y = ax y = cx O x \u0253 L\u1eddi gi\u1ea3i. T\u1eeb \u0111\u1ed3 th\u1ecb ta th\u1ea5y 0 < a < 1 v\u00e0 b, c > 1 \u2200x\u25e6 : \u00aey1 = bx\u25e6 t\u1eeb \u0111\u1ed3 th\u1ecb ta th\u1ea5y y1 > y2 \u21d4 bx\u25e6 > cx\u25e6 \u21d4 b > c. V\u1eady a < c < b. y2 = cx\u25e6 Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 207 S\u0110T: 0905.193.688","4. H\u00e0m s\u1ed1 m\u0169. H\u00e0m s\u1ed1 L\u00f4garit \u0104 C\u00e2u 35 (C\u00e2u 31 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean thu\u1ed9c t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log[(6 \u2212 x)(x + 2)]? A 7. B 8. C 9. D V\u00f4 s\u1ed1. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n (6 \u2212 x)(x + 2) > 0 \u21d4 \u22122 < x < 6 suy ra D = (\u22122; 6). V\u1eady c\u00f3 7 s\u1ed1 nguy\u00ean x thu\u1ed9c t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 \u0111\u00e3 cho. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 36 (C\u00e2u 37 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean thu\u1ed9c t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log [(6 \u2212 x)(x + 2)]? A 7. B 8. C V\u00f4 s\u1ed1. D 9. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x\u00e1c \u0111\u1ecbnh: (6 \u2212 x)(x + 2) > 0 \u21d4 \u22122 < x < 6. M\u00e0 x \u2208 Z \u21d2 x \u2208 {\u22121; 0; 1; 2; 3; 4; 5}. V\u1eady c\u00f3 7 s\u1ed1 nguy\u00ean thu\u1ed9c t\u1eadp x\u00e1c \u0111\u1ecbnh c\u1ee7a h\u00e0m s\u1ed1 y = log [(6 \u2212 x)(x + 2)]. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 37 (C\u00e2u 13 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). 13x T\u00ednh \u0111\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = 13x. . A y = x \u00b7 13x\u22121. B y = 13x \u00b7 ln 13. C y = 13x. D y = ln 13 \u0253 L\u1eddi gi\u1ea3i. C\u00f4ng th\u1ee9c \u0111\u1ea1o h\u00e0m c\u1ee7a y = ax l\u00e0: y = ax ln a. N\u00ean h\u00e0m s\u1ed1 \u0111\u00e3 cho c\u00f3 \u0111\u1ea1o h\u00e0m l\u00e0 y = 13x ln 13. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 38 (C\u00e2u 18 - \u0110TN - BGD&\u0110T\u221a- N\u0103m 2016 - 2017). T\u00ednh \u0111\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = ln 1 + x + 1 . Ay= \u221a 1\u221a . B y = \u221a1 . 2 x+1 1+ x+1 1+ x+1 C y =\u221a 1\u221a . D y =\u221a 2\u221a . x+1 1+ x+1 x+1 1+ x+1 \u221a \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 y = 1 + \u221ax + 1 \u221a1 1\u221a . 1+ x+1 Ch\u1ecdn \u0111\u00e1p \u00e1n A = 2\u221ax+1 = \u221a 1+ x+1 2 x+1 1+ x+1 \u0104 C\u00e2u 39 (C\u00e2u 20 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). H\u00e0m s\u1ed1 f (x) = log2 (x2 \u2212 2x) c\u00f3 \u0111\u1ea1o h\u00e0m l\u00e0 ln 2 1 A f (x) = . B f (x) = . x2 \u2212 2x (x2 \u2212 2x) ln 2 C f (x) = (2x \u2212 2) ln 2 D f (x) = 2x \u2212 2 x2 \u2212 2x . . (x2 \u2212 2x) ln 2 \u0253 L\u1eddi gi\u1ea3i. c\u00f3 f (x) = (x2 \u2212 2x) = 2x \u2212 2 Ta . (x2 \u2212 2x) ln 2 (x2 \u2212 2x) ln 2 Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 208 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 40 (C\u00e2u 19 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). H\u00e0m s\u1ed1 y = 2x2\u22123x c\u00f3 \u0111\u1ea1o h\u00e0m l\u00e0 A (2x \u2212 3) \u00b7 2x2\u22123x \u00b7 ln 2. B 2x2\u22123x \u00b7 ln 2. C (2x \u2212 3) \u00b7 2x2\u22123x. D (x2 \u2212 3x) \u00b7 2x2\u22123x+1. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 y = \u00c42x2\u22123x\u00e4 = (2x \u2212 3) \u00b7 2x2\u22123x \u00b7 ln 2. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 41 (C\u00e2u 26 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). H\u00e0m s\u1ed1 y = 3x2\u22123x c\u00f3 \u0111\u1ea1o h\u00e0m l\u00e0 A (2x \u2212 3) \u00b7 3x2\u22123x. B 3x2\u22123x \u00b7 ln 3. C (x2 \u2212 3x) \u00b7 3x2\u22123x\u22121. D (2x \u2212 3) \u00b7 3x2\u22123x \u00b7 ln 3. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3: y = \u00c43x2\u22123x\u00e4 = (2x \u2212 3) \u00b7 3x2\u22123x \u00b7 ln 3. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 42 (C\u00e2u 28 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). \u0110\u1ea1o h\u00e0m c\u1ee7a h\u00e0m s\u1ed1 y = x\u22123 l\u00e0 B y = \u22121 x\u22122. A y = \u2212x\u22124. 2 C y = \u2212 1 x\u22124. D y = \u22123x\u22124. 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 y = x\u22123 \u21d2 y = \u22123x\u22124. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 43 (C\u00e2u 16 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). M\u1ed9t ng\u01b0\u1eddi g\u1eedi ti\u1ebft ki\u1ec7m v\u00e0o ng\u00e2n h\u00e0ng v\u1edbi l\u00e3i su\u1ea5t 7,5 %\/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 \u0111\u00e3 g\u1eedi, 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 9 n\u0103m. C 10 n\u0103m. D 12 n\u0103m. \u0253 L\u1eddi gi\u1ea3i. \u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c: Sn = A(1 + r)n \u21d2 n = log(1+r) \u00c5 Sn \u00e3 \u21d2 n = log(1+7,5%)(2) \u2248 9,6. A Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 44 (C\u00e2u 25 - M\u0110 103 - 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 6,6%\/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 10 n\u0103m. C 13 n\u0103m. D 12 n\u0103m. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 209","4. H\u00e0m s\u1ed1 m\u0169. H\u00e0m s\u1ed1 L\u00f4garit V\u1edbi s\u1ed1 ti\u1ec1n g\u1eedi ban \u0111\u1ea7u l\u00e0 A, l\u00e3i su\u1ea5t c\u1ed1 \u0111\u1ecbnh l\u00e0 r\/n\u0103m, sau n n\u0103m g\u1eedi ti\u1ec1n, s\u1ed1 ti\u1ec1n c\u00f3 \u0111\u01b0\u1ee3c l\u00e0: Tn = A(1 + r)n. Theo gi\u1ea3 thi\u1ebft: Tn = 2A n\u00ean (1 + r)n = 2. Thay s\u1ed1 ta \u0111\u01b0\u1ee3c: (1 + 0,066)n = 2 \u21d2 n = log1,066 2 \u21d2 n \u2248 10,85. V\u1eady sau \u00edt nh\u1ea5t 11 n\u0103m g\u1eedi ti\u1ec1n s\u1ed1 ti\u1ec1n c\u1ee7a ng\u01b0\u1eddi g\u1eedi \u0111\u1ea1t g\u1ea5p \u0111\u00f4i s\u1ed1 ti\u1ec1n v\u1ed1n ban \u0111\u1ea7u. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 45 (C\u00e2u 39 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). N\u0103m 2020, m\u1ed9t h\u00e3ng xe \u00f4-t\u00f4 ni\u00eam y\u1ebft gi\u00e1 b\u00e1n lo\u1ea1i xe X l\u00e0 750.000.000 \u0111\u1ed3ng v\u00e0 d\u1ef1 \u0111\u1ecbnh trong 10 n\u0103m ti\u1ebfp theo, m\u1ed7i n\u0103m gi\u1ea3m 2% gi\u00e1 b\u00e1n so v\u1edbi gi\u00e1 b\u00e1n c\u1ee7a n\u0103m li\u1ec1n tr\u01b0\u1edbc. Theo d\u1ef1 \u0111\u1ecbnh \u0111\u00f3, n\u0103m 2025 h\u00e3ng xe \u00f4-t\u00f4 ni\u00eam y\u1ebft gi\u00e1 b\u00e1n lo\u1ea1i xe X l\u00e0 bao nhi\u00eau (k\u1ebft qu\u1ea3 l\u00e0m tr\u00f2n \u0111\u1ebfn h\u00e0ng ngh\u00ecn)? A 677.941.000 \u0111\u1ed3ng. B 675.000.000 \u0111\u1ed3ng. C 664.382.000 \u0111\u1ed3ng. D 691.776.000 \u0111\u1ed3ng. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi N0 l\u00e0 gi\u00e1 b\u00e1n xe \u00f4-t\u00f4 n\u0103m 2020, ta c\u00f3 N0 = 750.000.000. Gi\u00e1 xe \u00f4-t\u00f4 b\u00e1n \u1edf n\u0103m 2021 l\u00e0 N1 = N0 \u2212 N0 \u00b7 2% = N0(1 \u2212 2%). Gi\u00e1 xe \u00f4-t\u00f4 b\u00e1n \u1edf n\u0103m 2022 l\u00e0 N2 = N1 \u2212 N1 \u00b7 2% = N1(1 \u2212 2%) = N0(1 \u2212 2%)2. L\u1eadp lu\u1eadn t\u01b0\u01a1ng t\u1ef1, gi\u00e1 xe b\u00e1n \u00f4-t\u00f4 \u1edf n\u0103m 2025 l\u00e0 N5 = N0(1 \u2212 2%)5 = 750.000.000(1 \u2212 2%)5 \u2248 677.941.000 (\u0111\u1ed3ng). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 46 (C\u00e2u 41 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). N\u0103m 2020, m\u1ed9t h\u00e3ng xe \u00f4-t\u00f4 ni\u00eam y\u1ebft gi\u00e1 b\u00e1n lo\u1ea1i xe X l\u00e0 850.000.000 \u0111\u1ed3ng v\u00e0 d\u1ef1 \u0111\u1ecbnh trong 10 n\u0103m ti\u1ebfp theo, m\u1ed7i n\u0103m gi\u1ea3m 2% gi\u00e1 b\u00e1n so v\u1edbi gi\u00e1 b\u00e1n c\u1ee7a n\u0103m li\u1ec1n tr\u01b0\u1edbc. Theo d\u1ef1 \u0111\u1ecbnh \u0111\u00f3, n\u0103m 2025 h\u00e3ng xe \u00f4-t\u00f4 ni\u00eam y\u1ebft gi\u00e1 b\u00e1n lo\u1ea1i xe X l\u00e0 bao nhi\u00eau (k\u1ebft qu\u1ea3 l\u00e0m trong \u0111\u1ebfn h\u00e0ng ngh\u00ecn)? A 768.333.000 \u0111\u1ed3ng. B 765.000.000 \u0111\u1ed3ng. C 752.966.000 \u0111\u1ed3ng. D 784.013.000 \u0111\u1ed3ng. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi M l\u00e0 gi\u00e1 b\u00e1n c\u1ee7a c\u1ee7a xe trong n\u0103m \u0111\u1ea7u ti\u00ean, r% l\u00e0 t\u1ec9 l\u1ec7 gi\u1ea3m gi\u00e1 b\u00e1n theo t\u1eebng n\u0103m li\u1ec1n tr\u01b0\u1edbc. Sau n\u0103m th\u1ee9 nh\u1ea5t: Gi\u00e1 b\u00e1n c\u1ee7a xe l\u00e0 M1 = M \u2212 M \u00d7 r% = M (1 \u2212 r%). Sau n\u0103m th\u1ee9 hai: Gi\u00e1 b\u00e1n c\u1ee7a xe l\u00e0 M2 = M1 \u2212 M1 \u00d7 r% = M1(1 \u2212 r%) = M (1 \u2212 r%)2. Sau n\u0103m th\u1ee9 ba: Gi\u00e1 b\u00e1n c\u1ee7a xe l\u00e0 M3 = M2 \u2212 M2 \u00d7 r% = M2(1 \u2212 r%) = M (1 \u2212 r%)3.. . . Sau n\u0103m th\u1ee9 n: Gi\u00e1 b\u00e1n c\u1ee7a xe l\u00e0 Mn = M (1 \u2212 r%)n. V\u1eady gi\u00e1 b\u00e1n c\u1ee7a xe X trong n\u0103m 2025 (sau 5 n\u0103m l\u01b0u h\u00e0nh) l\u00e0 T = M \u00d7 (1 \u2212 r%)5 = 850.000.000 \u00d7 (1 \u2212 2%)5 \u2248 768.333.000 (\u0111\u1ed3ng). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 47 (C\u00e2u 16 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). x\u22123 T\u00ecm t\u1eadp x\u00e1c \u0111\u1ecbnh D c\u1ee7a h\u00e0m s\u1ed1 y = log5 . x + 2 A D = R\\\\{\u22122}. B D = (\u2212\u221e; \u22122) \u222a [3; +\u221e). C D = (\u22122; 3). D D = (\u2212\u221e; \u22122) \u222a (3; +\u221e). \u0253 L\u1eddi gi\u1ea3i. \uf8f1x+2=0 \uf8f2 H\u00e0m s\u1ed1 x\u00e1c \u0111\u1ecbnh khi x \u2212 3 \u21d4 x \u2208 (\u2212\u221e; \u22122) \u222a (3; +\u221e). \uf8f3x + 2 > 0 V\u1eady D = (\u2212\u221e; \u22122) \u222a (3; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 210 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 48 (C\u00e2u 22 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). M\u1ed9t ng\u01b0\u1eddi g\u1eedi 100 tri\u1ec7u \u0111\u1ed3ng v\u00e0o m\u1ed9t ng\u00e2n h\u00e0ng v\u1edbi l\u00e3i su\u1ea5t 0, 4%\/th\u00e1ng. 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 th\u00e1ng, s\u1ed1 ti\u1ec1n l\u00e3i s\u1ebd \u0111\u01b0\u1ee3c nh\u1eadp v\u00e0o v\u1ed1n ban \u0111\u1ea7u \u0111\u1ec3 t\u00ednh l\u00e3i cho th\u00e1ng ti\u1ebfp theo. H\u1ecfi sau \u0111\u00fang 6 th\u00e1ng, ng\u01b0\u1eddi \u0111\u00f3 \u0111\u01b0\u1ee3c l\u0129nh s\u1ed1 ti\u1ec1n (c\u1ea3 v\u1ed1n ban \u0111\u1ea7u v\u00e0 l\u00e3i) g\u1ea7n nh\u1ea5t v\u1edbi s\u1ed1 ti\u1ec1n n\u00e0o d\u01b0\u1edbi \u0111\u00e2y, n\u1ebfu trong kho\u1ea3ng th\u1eddi gian n\u00e0y ng\u01b0\u1eddi \u0111\u00f3 kh\u00f4ng r\u00fat ti\u1ec1n ra v\u00e0 l\u00e3i su\u1ea5t kh\u00f4ng thay \u0111\u1ed5i? A 102.424.000 \u0111\u1ed3ng. B 102.423.000 \u0111\u1ed3ng. C 102.016.000 \u0111\u1ed3ng. D 102.017.000 \u0111\u1ed3ng. \u0253 L\u1eddi gi\u1ea3i. \u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c t\u00ednh l\u00e3i k\u00e9p th\u00ec s\u1ed1 ti\u1ec1n \u0111\u01b0\u1ee3c l\u0129nh l\u00e0 T = 100 \u00b7 (1 + 0, 4%)6 \u2248 102.424.128, 4 (\u0111\u1ed3ng) Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 49 (C\u00e2u 40 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). ln x Cho h\u00e0m s\u1ed1 y = , m\u1ec7nh \u0111\u1ec1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y \u0111\u00fang? x A 2y + xy = \u2212 1 . 1 C y + xy = \u2212 1 . 1 B y + xy = . x2 D 2y + xy =. x2 x2 x2 \u0253 L\u1eddi gi\u1ea3i. C\u00e1ch 1. y (ln x) \u00b7x\u2212x \u00b7 ln x = 1 \u00b7 x \u2212 ln x = 1 \u2212 ln x \u00b7 = x x2 x2 x2 \u22121 (1 \u2212 ln x) \u00b7 x2 \u2212 (x2) (1 \u2212 ln x) x \u00b7 x2 \u2212 2x (1 \u2212 ln x) \u2212x \u2212 2x (1 \u2212 ln x) \u22121 + 2 (1 \u2212 ln x) y = = = = x4 x4 x4 x3 = \u22123 \u2212 2 ln x \u00b7 x3 1 \u2212 ln x 3 \u2212 2 ln x 2 \u2212 2 ln x \u2212 3 + 2 ln x 1 Suy ra 2y + xy = 2\u00b7 \u2212 x = = \u2212 \u00b7 x2 x3 x2 x2 C\u00e1ch 2. Ta c\u00f3 xy = ln x, l\u1ea5y \u0111\u1ea1o h\u00e0m hai v\u1ebf theo bi\u1ebfn x, ta \u0111\u01b0\u1ee3c y + xy = 1 \u00b7 x \u22121 Ti\u1ebfp t\u1ee5c l\u1ea5y \u0111\u1ea1o h\u00e0m hai v\u1ebf theo bi\u1ebfn x c\u1ee7a bi\u1ec3u th\u1ee9c tr\u00ean ta \u0111\u01b0\u1ee3c y + y + xy = x2 hay 1 2y + xy = \u2212 x2 . Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 50 (C\u00e2u 21 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). X\u00e9t c\u00e1c s\u1ed1 th\u1ef1c a, b th\u1ecfa m\u00e3n a > b > 1. T\u00ecm gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t Pmin c\u1ee7a bi\u1ec3u th\u1ee9c P = a log2a (a2) + 3 logb . b b A Pmin = 19. B Pmin = 13. C Pmin = 14. D Pmin = 15. \u0253 L\u1eddi gi\u1ea3i. P log2a (a2) a \u00ee \u00f32 a Ta c\u00f3 = b + 3 logb b = 2 log a a + 3 logb b b = 4 log a a 2 a. b .b b + 3 logb b a \u00ee \u00f32 + 3 logb = 4 1 + log a b b b \u0110\u1eb7t t = log a b, \u0111i\u1ec1u ki\u1ec7n t > 0 (v\u00ec a > b > 1). b Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 211 S\u0110T: 0905.193.688","4. H\u00e0m s\u1ed1 m\u0169. H\u00e0m s\u1ed1 L\u00f4garit X\u00e9t P = 4(1 + t)2 + 3 = 4t2 + 8t + 3 + 4 = f (t). tt 3 8t3 + 8t2 \u2212 3 (2t \u2212 1)(4t2 + 6t + 3) Ta c\u00f3f (t) = 8t + 8 \u2212 t2 = t2 = 1 t2 . Khi \u0111\u00f3 f (t) = 0 \u21d4 t = 2 Ta c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean: x \u2212\u221e 0 1 +\u221e y 2 y \u22120+ +\u221e +\u221e 15 \u00c51\u00e3 Ta suy ra Pmin = f 2 = 15. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 51 (C\u00e2u 25 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). \u0110\u1ec3 d\u1ef1 b\u00e1o d\u00e2n s\u1ed1 c\u1ee7a m\u1ed9t qu\u1ed1c gia, ng\u01b0\u1eddi ta s\u1eed d\u1ee5ng c\u00f4ng th\u1ee9c S = Aenr; trong \u0111\u00f3 A l\u00e0 d\u00e2n s\u1ed1 c\u1ee7a n\u0103m l\u1ea5y l\u00e0m m\u1ed1c t\u00ednh, S l\u00e0 d\u00e2n s\u1ed1 sau n n\u0103m, r l\u00e0 t\u1ec9 l\u1ec7 t\u0103ng d\u00e2n s\u1ed1 h\u00e0ng n\u0103m. N\u0103m 2017, d\u00e2n s\u1ed1 Vi\u1ec7t Nam l\u00e0 93.671.600 ng\u01b0\u1eddi (T\u1ed5ng c\u1ee5c Th\u1ed1ng k\u00ea, Ni\u00ean gi\u00e1m Th\u1ed1ng k\u00ea n\u0103m 2017, Nh\u00e0 xu\u1ea5t b\u1ea3n Th\u1ed1ng k\u00ea, Tr.79). Gi\u1ea3 s\u1eed t\u1ec9 l\u1ec7 t\u0103ng d\u00e2n s\u1ed1 h\u00e0ng n\u0103m kh\u00f4ng \u0111\u1ed5i l\u00e0 0, 81% d\u1ef1 b\u00e1o d\u00e2n s\u1ed1 Vi\u1ec7t Nam n\u0103m 2035 l\u00e0 bao nhi\u00eau ng\u01b0\u1eddi (k\u1ebft qu\u1ea3 l\u00e0m tr\u00f2n \u0111\u1ebfn ch\u1eef s\u1ed1 h\u00e0ng tr\u0103m)? A 109.256.100. B 108.374.700. C 107.500.500. D 108.311.100. \u0253 L\u1eddi gi\u1ea3i. (2035\u22122017). 0, 81 Ta c\u00f3: S = Aenr = 93671600.e 100 = 108.374.741, 3 \u2248 108.374.700. V\u1eady c\u1ee9 t\u0103ng d\u00e2n s\u1ed1 v\u1edbi t\u1ec9 l\u1ec7 nh\u01b0 v\u1eady th\u00ec \u0111\u1ebfn n\u0103m 2035 d\u00e2n s\u1ed1 n\u01b0\u1edbc ta kho\u1ea3ng 108.374.700 ng\u01b0\u1eddi. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 52 (C\u00e2u 21 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). \u00d4ng A vay ng\u1eafn h\u1ea1n ng\u00e2n h\u00e0ng 100 tri\u1ec7u \u0111\u1ed3ng, v\u1edbi l\u00e3i su\u1ea5t 12%\/n\u0103m. \u00d4ng mu\u1ed1n ho\u00e0n n\u1ee3 cho ng\u00e2n h\u00e0ng theo c\u00e1ch: Sau \u0111\u00fang m\u1ed9t th\u00e1ng k\u1ec3 t\u1eeb ng\u00e0y vay, \u00f4ng b\u1eaft \u0111\u1ea7u ho\u00e0n n\u1ee3; hai l\u1ea7n ho\u00e0n n\u1ee3 li\u00ean ti\u1ebfp c\u00e1ch nhau \u0111\u00fang m\u1ed9t th\u00e1ng, s\u1ed1 ti\u1ec1n ho\u00e0n n\u1ee3 \u1edf m\u1ed7i l\u1ea7n l\u00e0 nh\u01b0 nhau v\u00e0 tr\u1ea3 h\u1ebft ti\u1ec1n n\u1ee3 sau \u0111\u00fang 3 th\u00e1ng k\u1ec3 t\u1eeb ng\u00e0y vay. H\u1ecfi, theo c\u00e1ch \u0111\u00f3, s\u1ed1 ti\u1ec1n m m\u00e0 \u00f4ng A s\u1ebd ph\u1ea3i tr\u1ea3 cho ng\u00e2n h\u00e0ng trong m\u1ed7i l\u1ea7n ho\u00e0n n\u1ee3 l\u00e0 bao nhi\u00eau? Bi\u1ebft r\u1eb1ng, l\u00e3i su\u1ea5t ng\u00e2n h\u00e0ng kh\u00f4ng thay \u0111\u1ed5i trong th\u1eddi gian \u00f4ng A ho\u00e0n n\u1ee3. B (1, 01)3 A m = 100.(1, 01)3 (tri\u1ec7u \u0111\u1ed3ng). m = (1, 01)3 \u2212 1 (tri\u1ec7u \u0111\u1ed3ng). 3 D 120.(1, 12)3 (tri\u1ec7u \u0111\u1ed3ng). C m = 100 \u00d7 1, 03 (tri\u1ec7u \u0111\u1ed3ng). m = (1, 12)3 \u2212 1 3 \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t r l\u00e0 l\u00e3i su\u1ea5t h\u00e0ng th\u00e1ng v\u00e0 m l\u00e0 s\u1ed1 ti\u1ec1n ho\u00e0n n\u1ee3 m\u1ed7i th\u00e1ng. \u2022 S\u1ed1 ti\u1ec1n \u00f4ng A n\u1ee3 ng\u00e2n h\u00e0ng cu\u1ed1i th\u00e1ng th\u1ee9 nh\u1ea5t l\u00e0 T1 = T (1 + r) \u2212 m. \u2022 S\u1ed1 ti\u1ec1n \u00f4ng A n\u1ee3 ng\u00e2n h\u00e0ng cu\u1ed1i th\u00e1ng th\u1ee9 hai l\u00e0 T2 = T1(1 + r) \u2212 m = T (1 + a)2 \u2212 m[1 + (1 + r)]. \u2022 S\u1ed1 ti\u1ec1n \u00f4ng A n\u1ee3 ng\u00e2n h\u00e0ng cu\u1ed1i th\u00e1ng th\u1ee9 ba l\u00e0 T3 = T2(1+r)\u2212m = T (1+r)3\u2212m [1 + (1 + r) + (1 + r)2] Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 212 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 T3 = T (1 + r)3 \u2212 (1 + r)3 \u2212 1 m r . Theo gi\u1ea3 thi\u1ebft c\u00f3 T3 = 0 \u21d2 m = T.r.(1 + r)3 = (1, 01)3 (tri\u1ec7u \u0111\u1ed3ng). (1 + r)3 \u2212 1 (1, 01)3 \u2212 1 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 53 (C\u00e2u 39 - M\u0110 103 - 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 900 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 c\u1ee7a t\u1ec9nh A c\u00f3 di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi trong n\u0103m \u0111\u00f3 \u0111\u1ea1t tr\u00ean 1700 ha? A N\u0103m 2029. B N\u0103m 2051. C N\u0103m 2030. D N\u0103m 2050. \u0253 L\u1eddi gi\u1ea3i. B\u00e0i to\u00e1n tr\u00ean gi\u1ed1ng b\u00e0i to\u00e1n l\u00e3i k\u00e9p khi g\u1eedi ti\u1ec1n v\u00e0o Ng\u00e2n h\u00e0ng: Ta \u0111\u1eb7t S0 = 900ha l\u00e0 di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi c\u1ee7a t\u1ec9nh A n\u0103m 2019, SN = 1700 l\u00e0 di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi sau N n\u0103m (k\u1ec3 t\u1eeb sau n\u0103m 2019) c\u1ee7a t\u1ec9nh A m\u00e0 m\u1ed7i n\u0103m \u0111\u1ec1u t\u0103ng r% = 6% = 0.06 so v\u1edbi di\u1ec7n t\u00edch r\u1eebng tr\u1ed3ng m\u1edbi c\u1ee7a n\u0103m li\u1ec1n tr\u01b0\u1edbc. Sau m\u1ed9t n\u0103m: S1 = (1 + 0.06) \u00b7 S0. Sau hai n\u0103m: S2 = S1 + S1.0.06 = 1.062 \u00b7 S0. ... Sau N n\u0103m: SN = (1 + r%)N \u00b7 S0. \u21d2 1700 < 1.06N \u00b7 900 \u21d4 1.06N > 17 9 \u21d4 N > log1.06 17 \u2248 10.915. 9 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 1700ha l\u00e0 2030. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 54 (C\u00e2u 41 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). N\u0103m 2020, m\u1ed9t h\u00e3ng xe \u00f4 t\u00f4 ni\u00eam y\u1ebft gi\u00e1 b\u00e1n lo\u1ea1i xe X l\u00e0 900.000.000 \u0111\u1ed3ng v\u00e0 d\u1ef1 \u0111\u1ecbnh trong 10 n\u0103m ti\u1ebfp theo, m\u1ed7i n\u0103m gi\u1ea3m 2% gi\u00e1 b\u00e1n so v\u1edbi gi\u00e1 b\u00e1n c\u1ee7a n\u0103m li\u1ec1n tr\u01b0\u1edbc. Theo d\u1ef1 \u0111\u1ecbnh \u0111\u00f3, n\u0103m 2025 h\u00e3ng xe \u00f4 t\u00f4 ni\u00eam y\u1ebft gi\u00e1 b\u00e1n lo\u1ea1i xe X l\u00e0 bao nhi\u00eau (k\u1ebft qu\u1ea3 l\u00e0m tr\u00f2n \u0111\u1ebfn h\u00e0ng ph\u1ea7n ngh\u00ecn)? A 810.000.000 \u0111\u1ed3ng. B 813.529.000 \u0111\u1ed3ng. C 797.258.000 \u0111\u1ed3ng. D 830.131.000 \u0111\u1ed3ng. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi gi\u00e1 xe X n\u0103m 2020 l\u00e0 A = 900.000.000 \u0111\u1ed3ng v\u00e0 r = 2%. Khi \u0111\u00f3 Gi\u00e1 xe X n\u0103m 2021 l\u00e0 A1 = A \u2212 A \u00b7 r = A(1 \u2212 r). Gi\u00e1 xe X n\u0103m 2022 l\u00e0 A2 = A1 \u2212 A1 \u00b7 r = A(1 \u2212 r)2. Gi\u00e1 xe X n\u0103m 2023 l\u00e0 A3 = A2 \u2212 A2 \u00b7 r = A(1 \u2212 r)3. Gi\u00e1 xe X n\u0103m 2024 l\u00e0 A4 = A3 \u2212 A3 \u00b7 r = A(1 \u2212 r)4. Gi\u00e1 xe X n\u0103m 2025 l\u00e0 A5 = A4 \u2212 A4 \u00b7 r = A(1 \u2212 r)5. V\u1eady gi\u00e1 xe X n\u0103m 2025 l\u00e0 A5 = 900.000.000 \u00b7 (1 \u2212 2%)5 \u2248 813.529.000 \u0111\u1ed3ng. Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 213 S\u0110T: 0905.193.688","4. H\u00e0m s\u1ed1 m\u0169. H\u00e0m s\u1ed1 L\u00f4garit \u0104 C\u00e2u 55 (C\u00e2u 41 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). N\u0103m 2020, m\u1ed9t h\u00e3ng xe \u00f4 t\u00f4 ni\u00eam y\u1ebft gi\u00e1 b\u00e1n lo\u1ea1i xe X l\u00e0 800.000.000 \u0111\u1ed3ng v\u00e0 d\u1ef1 \u0111\u1ecbnh trong 10 n\u0103m ti\u1ebfp theo, m\u1ed7i n\u0103m gi\u1ea3m 2% gi\u00e1 b\u00e1n so v\u1edbi gi\u00e1 b\u00e1n c\u1ee7a n\u0103m li\u1ec1n tr\u01b0\u1edbc. Theo d\u1ef1 t\u00ednh \u0111\u00f3, n\u0103m 2025 h\u00e3ng xe \u00f4 t\u00f4 ni\u00eam y\u1ebfu gi\u00e1 b\u00e1n lo\u1ea1i xe X l\u00e0 bao nhi\u00eau (l\u00e0m tr\u00f2n \u0111\u1ebfn h\u00e0ng ngh\u00ecn)? A 708.674.000 \u0111\u1ed3ng. B 737.895.000 \u0111\u1ed3ng. C 723.137.000 \u0111\u1ed3ng. D 720.000.000 \u0111\u1ed3ng. \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t A = 800.000.000 \u0111\u1ed3ng v\u00e0 r = 2%. Gi\u00e1 xe X n\u0103m 2021 : A1 = A \u2212 Ar = A(1 \u2212 r). Gi\u00e1 xe X n\u0103m 2022 : A2 = A(1 \u2212 r) \u2212 A(1 \u2212 r)r = A(1 \u2212 r)2. Gi\u00e1 xe X n\u0103m 2023 : A3 = A(1 \u2212 r)3. Gi\u00e1 xe X n\u0103m 2024 : A4 = A(1 \u2212 r)4. Gi\u00e1 xe X n\u0103m 2025 : A5 = A(1 \u2212 r)5 \u2248 723.137.000 \u0111\u1ed3ng. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 56 (C\u00e2u 35 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). M\u1ed9t ng\u01b0\u1eddi g\u1eedi 50 tri\u1ec7u \u0111\u1ed3ng v\u00e0o m\u1ed9t ng\u00e2n h\u00e0ng v\u1edbi l\u00e3i su\u1ea5t 6%\/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 g\u1ed1c \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 nh\u1eadn \u0111\u01b0\u1ee3c s\u1ed1 ti\u1ec1n nhi\u1ec1u h\u01a1n 100 tri\u1ec7u \u0111\u1ed3ng bao g\u1ed3m g\u1ed1c v\u00e0 l\u00e3i? Gi\u1ea3 \u0111\u1ecbnh trong su\u1ed1t th\u1eddi gian g\u1eedi, l\u00e3i su\u1ea5t kh\u00f4ng \u0111\u1ed5i v\u00e0 ng\u01b0\u1eddi \u0111\u00f3 kh\u00f4ng r\u00fat ti\u1ec1n ra. A 13 n\u0103m. B 14 n\u0103m. C 12 n\u0103m. D 11 n\u0103m. \u0253 L\u1eddi gi\u1ea3i. T\u1ed5ng s\u1ed1 ti\u1ec1n l\u0129nh ra sau n n\u0103m b\u1eb1ng 50.(1, 06)n. D\u00f9ng m\u00e1y t\u00ednh ki\u1ec3m tra th\u1ea5y n = 12 th\u00ec s\u1ed1 ti\u1ec1n l\u1edbn h\u01a1n 100. V\u1eady ch\u1ecdn ph\u01b0\u01a1ng \u00e1n C. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 57 (C\u00e2u 50 - M\u0110 101 - 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 14 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 14. B 12. C 11. D 13. \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 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 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 \u221am = \u221a1 \u00c5 1 \u00e3 \u2200a \u2208 (\u22121; 1). (a2 + 1) a2 + 1 a2 + 1 a2 + 1 a2 + 1 \u2212 m < 0, B\u1ea3ng bi\u1ebfn thi\u00ean c\u1ee7a h\u00e0m s\u1ed1 h(a) nh\u01b0 sau Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 214 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 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; 12} 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(\u22121) +\u221e g(a) \u2212\u221e g(1) \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; 13} 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 \u00aeg(\u22121) < 2 < n (v\u00f4 l\u00ed). g(1) < 2 < n V\u1eady c\u00f3 11 c\u1eb7p s\u1ed1 nguy\u00ean d\u01b0\u01a1ng (m, n) th\u1ecfa m\u00e3n \u0111\u1ec1 b\u00e0i. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 58 (C\u00e2u 49 - M\u0110 102 - 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 + n \u2264 16 v\u00e0 \u1ee9ng v\u1edbi m\u1ed7i c\u1eb7p (m, n) t\u1ed3n t\u1ea1i Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 215 S\u0110T: 0905.193.688","4. H\u00e0m s\u1ed1 m\u0169. H\u00e0m s\u1ed1 L\u00f4garit \u0111\u00fang 3 s\u1ed1 th\u1ef1c a \u2208 (\u22121; 1) th\u1ecfa m\u00e3n 2am = n ln \u00c4 + \u221a + \u00e4 a a2 1? A 16. B 14. C 15. D 13. \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). (a2 + 1) a2 + 1 a2 + 1 a2 + 1 a2 + 1 B\u1ea3ng bi\u1ebfn thi\u00ean c\u1ee7a h\u00e0m s\u1ed1 h(a) nh\u01b0 sau 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; 12; 14} 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(\u22121) +\u221e g(a) \u2212\u221e g(1) \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; 13; 15} 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) Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 216 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 \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 \u00aeg(\u22121) < 2 < n (v\u00f4 l\u00ed). g(1) < 2 < n V\u1eady c\u00f3 13 c\u1eb7p s\u1ed1 nguy\u00ean d\u01b0\u01a1ng (m, n) th\u1ecfa m\u00e3n \u0111\u1ec1 b\u00e0i. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 59 (C\u00e2u 44 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho a > 0, b > 0 th\u1ecfa m\u00e3n log3a+2b+1(9a2 + b2 + 1) + log6ab+1(3a + 2b + 1) = 2. Gi\u00e1 tr\u1ecb c\u1ee7a a + 2b b\u1eb1ng A 6. B 9. C 7 D 5 . . 2 2 \u0253 L\u1eddi gi\u1ea3i. \u00ae(9a2 + b2) + 1 6ab + 1 (b\u1ea5t \u0111\u1eb3ng th\u1ee9c AM-GM) Do a > 0, b > 0 n\u00ean ta c\u00f3 3a + 2b + 1 > 1. \u21d2 log3a+2b+1(9a2 + b2 + 1) log3a+2b+1(6ab + 1) T\u1eeb \u0111\u00f3 log3a+2b+1(9a2 + b2 + 1) + log6ab+1(3a + 2b + 1) log3a+2b+1(6ab + 1) + log6ab+1(3a + 2b + 1) 2 (b\u1ea5t \u0111\u1eb3ng th\u1ee9c AM-GM). \u00ae3a = b > 0 \u21d4 a = 1 v\u00e0 b = 3 D\u1ea5u \u201c=\u201d x\u1ea3y ra khi v\u00e0 ch\u1ec9 khi . 3a + 2b + 1 = 6ab + 1 2 2 17 V\u1eady a + 2b = + 3 = . 22 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 60 (C\u00e2u 47 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). \u221a X\u00e9t c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng a, b, x, y th\u1ecfa m\u00e3n a > 1, b > 1 v\u00e0 ax = by = ab. Gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a bi\u1ec3u th\u1ee9c P = x + 2y thu\u1ed9c t\u1eadp h\u1ee3p n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A (1; 2). B \u00ef 5\u00e3 C [3; 4). D \u00ef5 \u00e3 2; . ;3 . 2 2 \u0253 L\u1eddi gi\u1ea3i. \uf8f1 11 \u221a ax 1 1 ax\u2212 1 1 \uf8f4\uf8f2x \u2212 = 2 \u00b7 loga b Theo b\u00e0i ra ta c\u00f3: ax = by = ab \u21d4 = \u00b7 2 \uf8f3\uf8f4y \u2212 2 = a2 b2 = b2 1 \u21d4 \u21d4 by 1 \u00b7 1 by\u2212 1 1 1 = 2 = 2 \u00b7 logb a. a2 b2 a2 11 31 2 Do \u0111\u00f3: P = x + 2y = 2 + 2 loga b + 1 + logb a = 2 + 2 loga b + logb a. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 217 S\u0110T: 0905.193.688","4. H\u00e0m s\u1ed1 m\u0169. H\u00e0m s\u1ed1 L\u00f4garit \u0110\u1eb7t t = loga b. V\u00ec a, b > 1 n\u00ean loga b > loga 1 = 0. \u2026 \u221a Khi \u0111\u00f3 P = 3 + 1 + 1 \u2265 3 + 2 1 t \u00b7 1 = 3 + 2. t 22 t 2 2 t 2 D\u1ea5u \u0111\u1eb3ng th\u1ee9c x\u1ea3y ra khi v\u00e0 ch\u1ec9 khi t = 1 \u21d4 t = \u221a hay \u221a 2 3 \u221a2 t b = a 2. \u00ef5 \u00e3 V\u1eady P \u0111\u1ea1t gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t l\u00e0 + 2 thu\u1ed9c n\u1eeda kho\u1ea3ng ; 3 . 22 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 61 (C\u00e2u 48 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). X\u00e9t c\u00e1c s\u1ed1 th\u1ef1c kh\u00f4ng \u00e2m x v\u00e0 y th\u1ecfa m\u00e3n 2x + y \u00b7 4x+y\u22121 \u2265 3. Gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a bi\u1ec3u th\u1ee9c P = x2 + y2 + 6x + 4y b\u1eb1ng A 65 B 33 C 49 D 57 . . . . 8 4 8 8 \u0253 L\u1eddi gi\u1ea3i. (1) 2x + y \u00b7 4x+y\u22121 \u2265 3 \u21d4 y \u00b7 22x+2y\u22122 \u2265 3 \u2212 2x \u21d4 2y \u00b7 22y \u2265 (3 \u2212 2x)23\u22122x. X\u00e9t h\u00e0m s\u1ed1 f (t) = t \u00b7 2t tr\u00ean [0; +\u221e) c\u00f3 f (t) = 2t + t \u00b7 2t ln 2 > 0, \u2200t \u2265 0. Suy ra f (t) \u0111\u1ed3ng bi\u1ebfn tr\u00ean [0; +\u221e). (1) \u21d4 2y \u2265 3 \u2212 2x \u21d4 x+y \u2265 3 \u21d4 (x + 3) + (y + 2) \u2265 13 . 22 Ta c\u00f3: P = (x + 3)2 + (y + 2)2 \u2212 13 \u21d2 (x + 3)2 + (y + 2)2 = P + 13. Ta l\u1ea1i c\u00f3: 13 \u2264 (x + 3) + (y + 2) \u2264 2 [(x + 3)2 + (y + 2)2] = 2(P + 13) 2 \u21d4 169 \u2264 2(P + 13) \u21d4 P \u2265 65 . 48 \uf8f11 \uf8f2\uf8f4x = 4 D\u1ea5u x\u1ea3y ra khi v\u00e0 ch\u1ec9 khi 5 \uf8f4\uf8f3y = . 4 65 V\u1eady Pmin = . 8 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 62 (C\u00e2u 49 - M\u0110 103 - 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 127 s\u1ed1 nguy\u00ean y th\u1ecfa m\u00e3n log3 (x2 + y) \u2265 log2 (x + y)? A 89. B 46. C 45. D 90. \u0253 L\u1eddi gi\u1ea3i. C\u00e1ch 1: \u00aex2 + y > 0 \u0110i\u1ec1u ki\u1ec7n x + y > 0. \u0110\u1eb7t k = x + y \u2208 Z+. X\u00e9t h\u00e0m s\u1ed1 f (y) = log3 (x2 + y) \u2212 log2 (x + y) \u2265 0. 11 Suy ra f (y) = (x2 + y) \u00b7 ln 3 \u2212 (x + y) \u00b7 ln 2 < 0 \u21d2 f (y) ngh\u1ecbch bi\u1ebfn. X\u00e9t h\u00e0m s\u1ed1 g (k) = f (k \u2212 x) = log3 (x2 + k \u2212 x) \u2212 log2 k, k \u2208 Z+. Do h\u00e0m s\u1ed1 f ngh\u1ecbch bi\u1ebfn n\u00ean h\u00e0m s\u1ed1 g c\u0169ng ngh\u1ecbch bi\u1ebfn. Gi\u1ea3 s\u1eed k0 l\u00e0 nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh g (k) = 0. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 218 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 Suy ra \u00ae1 \u2264k\u2264 k0 \u21d2 k0 < 128. k \u2208 Z+ N\u00ean g (128) < 0 \u21d4 log3 x2 + 128 \u2212 x < log2 128 \u21d2 x2 \u2212 x + 128 < 3log2 128 \u21d2 \u221244 \u2264 x \u2264 45. V\u1eady c\u00f3 90 s\u1ed1 nguy\u00ean x. C\u00e1ch 2: \u00aex2 + y > 0 \u0110i\u1ec1u ki\u1ec7n x + y > 0. Ta c\u00f3 log3 x2 + y \u2265 log2 (x + y) (1) \u21d4 x2 + y \u2265 3log2(x+y) \u21d4 x2 + y \u2265 (x + y)log2 3 \u21d4 x2 \u2212 x \u2265 (x + y)log2 3 \u2212 (x + y) . \u0110\u1eb7t t = x + y th\u00ec (1) tr\u1edf th\u00e0nh x2 \u2212 x \u2265 tlog2 3 \u2212 t. (2) V\u1edbi m\u1ed7i x nguy\u00ean cho tr\u01b0\u1edbc c\u00f3 kh\u00f4ng qu\u00e1 127 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 127 nghi\u1ec7m t . Ta c\u00f3 h\u00e0m s\u1ed1 f (t) = tlog2 3 \u2212 t \u0111\u1ed3ng bi\u1ebfn tr\u00ean [1; +\u221e) n\u00ean n\u1ebfu x2 \u2212 x > 128log2 3 \u2212 128 = 2059 th\u00ec s\u1ebd c\u00f3 \u00edt nh\u1ea5t 127 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 2059 \u21d4 \u221244 \u2264 x \u2264 45 (do x nguy\u00ean) . V\u1eady c\u00f3 90 s\u1ed1 nguy\u00ean x Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 63 (C\u00e2u 43 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). X\u00e9t c\u00e1c s\u1ed1 th\u1ef1c x, y th\u1ecfa m\u00e3n 2x2+y2+1 \u2264 (x2 + y2 \u2212 2x + 2)4x. Gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a bi\u1ec3u th\u1ee9c 4y P = g\u1ea7n nh\u1ea5t v\u1edbi s\u1ed1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y ? 2x + y + 1 A \u22122. B \u22123. C \u22125. D \u22124. \u0253 L\u1eddi gi\u1ea3i. V\u1edbi c\u00e1c s\u1ed1 th\u1ef1c x, y ta c\u00f3 2x2+y2+1 \u2264 (x2 + y2 \u2212 2x + 2)4x \u21d4 2x2\u22122x+y2+1 \u2264 (x2 + y2 \u2212 2x + 1) + 1. \u0110\u1eb7t t = x2 + y2 \u2212 2x + 1, suy ra t = (x \u2212 1)2 + y2 n\u00ean t \u2265 0. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho tr\u1edf th\u00e0nh 2t \u2264 t + 1 v\u1edbi t \u2265 0. X\u00e9t h\u00e0m s\u1ed1 f (t) = 2t \u2212 t \u2212 1 v\u1edbi t \u2265 0, c\u00f3 f (t) = 2t \u00b7 ln 2 \u2212 1; f (t) = 0 \u21d4 t = t0 = log2(log2 e) \u2248 0,52. H\u00e0m s\u1ed1 f (t) c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean tr\u00ean n\u1eeda kho\u1ea3ng [0; +\u221e) nh\u01b0 sau t0 t0 1 +\u221e f (t) \u2212 0 + 0 +\u221e f (t) 0 f (t0) Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 219 S\u0110T: 0905.193.688","4. H\u00e0m s\u1ed1 m\u0169. H\u00e0m s\u1ed1 L\u00f4garit L\u1ea1i c\u00f3 f (1) = 0, do \u0111\u00f3 t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh f (t) \u2264 0 l\u00e0 [0; 1]. V\u1eady (x \u2212 1)2 + y2 \u2264 1. T\u1eadp h\u1ee3p t\u1ea5t c\u1ea3 c\u00e1c \u0111i\u1ec3m c\u00f3 t\u1ecda \u0111\u1ed9 (x; y) th\u1ecfa m\u00e3n y\u00eau c\u1ea7u 2x2+y2+1 \u2264 (x2 + y2 \u2212 2x + 2)4x n\u1eb1m trong h\u00ecnh tr\u00f2n t\u00e2m I(1; 0), b\u00e1n k\u00ednh R = 1 (\u2217). V\u1edbi d : 2x + y + 1 = 0, suy ra d[I, d] > R n\u00ean 2x + y + 1 = 0 v\u1edbi c\u00e1c c\u1eb7p s\u1ed1 (x; y) th\u1ecfa m\u00e3n (\u2217). Ta c\u00f3 P = 4y \u21d4 (\u2206) : 2P x + (P \u2212 4)y + P = 0. 2x + y + 1 Y\u00eau c\u1ea7u b\u00e0i to\u00e1n suy ra d[I, (\u2206)] \u2264 1 \u21d4 |2P \u00b7 1 + (P \u2212 4) \u00b7 0 + P | \u2264 1 4P 2 + (P \u2212 4)2 \u221a \u21d4 |3P | \u2264 5P 2 \u2212 8P + 16 \u21d4 4P\u221a2 + 8P \u2212 16 \u2264\u221a0 \u21d4 \u2212 5 \u2212 1 \u2264 P \u2264 5 \u2212 1. \u221a Do \u0111\u00f3, min P = \u2212 5 \u2212 1 \u2248 \u22123,2. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 64 (C\u00e2u 44 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). X\u00e9t c\u00e1c s\u1ed1 th\u1ef1c x, y th\u1ecfa m\u00e3n 2x2+y2+1 \u2264 (x2 + y2 \u2212 2x + 2)4x. Gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a bi\u1ec3u th\u1ee9c 8x + 4 P = 2x \u2212 y + 1 g\u1ea7n nh\u1ea5t v\u1edbi s\u1ed1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y ? A 9. B 6. C 7. D 8. \u0253 L\u1eddi gi\u1ea3i. V\u1edbi c\u00e1c s\u1ed1 th\u1ef1c x, y ta c\u00f3 2x2+y2+1 \u2264 (x2 + y2 \u2212 2x + 2)4x \u21d4 2x2\u22122x+y2+1 \u2264 (x2 + y2 \u2212 2x + 1) + 1. \u0110\u1eb7t t = x2 + y2 \u2212 2x + 1, suy ra t = (x \u2212 1)2 + y2 n\u00ean t \u2265 0. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho tr\u1edf th\u00e0nh 2t \u2264 t + 1 v\u1edbi t \u2265 0. X\u00e9t h\u00e0m s\u1ed1 f (t) = 2t \u2212 t \u2212 1 v\u1edbi t \u2265 0, c\u00f3 f (t) = 2t \u00b7 ln 2 \u2212 1; f (t) = 0 \u21d4 t = t0 = log2(log2 e) \u2248 0,52. H\u00e0m s\u1ed1 f (t) c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean tr\u00ean n\u1eeda kho\u1ea3ng [0; +\u221e) nh\u01b0 sau t0 t0 1 +\u221e f (t) \u2212 0 + 0 +\u221e f (t) 0 f (t0) L\u1ea1i c\u00f3 f (1) = 0, do \u0111\u00f3 t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh f (t) \u2264 0 l\u00e0 [0; 1]. V\u1eady (x \u2212 1)2 + y2 \u2264 1. T\u1eadp h\u1ee3p t\u1ea5t c\u1ea3 c\u00e1c \u0111i\u1ec3m c\u00f3 t\u1ecda \u0111\u1ed9 (x; y) th\u1ecfa m\u00e3n y\u00eau c\u1ea7u 2x2+y2+1 \u2264 (x2 + y2 \u2212 2x + 2)4x n\u1eb1m trong h\u00ecnh tr\u00f2n t\u00e2m I(1; 0), b\u00e1n k\u00ednh R = 1 (\u2217). V\u1edbi d : 2x + y + 1 = 0, suy ra d[I, d] > R n\u00ean 2x + y + 1 = 0 v\u1edbi c\u00e1c c\u1eb7p s\u1ed1 (x; y) th\u1ecfa m\u00e3n (\u2217). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 220 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 Ta c\u00f3 P = 8x + 4 \u21d4 (\u2206) : (2P \u2212 8) x \u2212 P y + P \u2212 4 = 0. 2x \u2212 y + 1 Y\u00eau c\u1ea7u b\u00e0i to\u00e1n suy ra d[I, (\u2206)] \u2264 1 \u21d4 |(2P \u2212 8) \u00b7 1 + P \u00b7 0 + P \u2212 4| \u2264 1 \u00bb \u2212 8)2 + P2 (2P \u221a \u21d4 |3P \u2212 12| \u2264 5P 2 \u2212 32P + 64 \u21d4 4P 2 \u221a\u2212 40P + 80 \u2264 0\u221a \u21d4 5 \u2212 5 \u2264 P \u2264 5 + 5. \u221a Do \u0111\u00f3, min P = 5 + 5 \u2248 7,23. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 65 (C\u00e2u 47 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). X\u00e9t c\u00e1c s\u1ed1 th\u1ef1c x, y th\u1ecfa m\u00e3n 2x2+y2+1 \u2264 (x2 + y2 \u2212 2x + 2)4x. Gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t c\u1ee7a bi\u1ec3u th\u1ee9c 4y P = g\u1ea7n nh\u1ea5t v\u1edbi s\u1ed1 n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? 2x + y + 1 A 1. B 0. C 3. D 2. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 2x2+y2+1 \u2264 (x2 + y2 \u2212 2x + 2)4x \u21d4 2x2+y2\u22122x+1 \u2264 (x2 + y2 \u2212 2x + 1) + 1. \u0110\u1eb7t t = x2 + y2 \u2212 2x + 1 = (x \u2212 1)2 + y2, suy ra t \u2265 0. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh tr\u1edf th\u00e0nh 2t \u2264 t + 1 \u21d4 2t \u2212 t \u2212 1 \u2264 0, v\u1edbi t \u2265 0. X\u00e9t h\u00e0m s\u1ed1 f (t) = 2t \u2212 t \u2212 1, v\u1edbi t \u2265 0. Ta c\u00f3 f (t) = 2t \u00b7 ln 2 \u2212 1; f (t) = 0 \u21d4 2t = 1 \u21d4 t = \u00c51\u00e3 = t0 \u2248 0,52877. ln 2 log2 ln 2 V\u00e0 c\u00f3 f (1) = 0. B\u1ea3ng bi\u1ebfn thi\u00ean t 0 t0 1 +\u221e f (t) \u2212 0 + + +\u221e f (t) 0 0 f (t0) T\u1eeb b\u1ea3n bi\u1ebfn thi\u00ean suy ra f (t) \u2264 0 \u21d4 0 \u2264 t \u2264 1. Khi \u0111\u00f3 (x \u2212 1)2 + y2 \u2264 1. (\u2217) Do \u0111\u00f3 t\u1eadp h\u1ee3p c\u00e1c \u0111i\u1ec3m M (x; y) l\u00e0 c\u00e1c \u0111i\u1ec3m n\u1eb1m trong v\u00e0 tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n (C) c\u00f3 t\u00e2m I(1; 0) v\u00e0 b\u00e1n k\u00ednh R = 1. \u00ae(x \u2212 1)2 \u2264 1 \u00aex \u2265 0 T\u1eeb (\u2217), suy ra y2 \u2264 1 \u21d2 v\u00e0 d\u1ea5u \u201c=\u201d kh\u00f4ng \u0111\u1ed3ng th\u1eddi x\u1ea3y ra. Do \u0111\u00f3 2x + y + 1 = 0. y \u2265 \u22121 Khi \u0111\u00f3 P = 4y \u21d4 (2x + y + 1)P = 4y \u21d4 2P x + (P \u2212 4)y + P = 0 (\u2206). 2x + y + 1 \u0110\u1ec3 t\u1ed3n t\u1ea1i (x; y) th\u00ec \u0111\u01b0\u1eddng th\u1eb3ng (\u2206) v\u00e0 \u0111\u01b0\u1eddng tr\u00f2n (C) ph\u1ea3i c\u00f3 \u0111i\u1ec3m chung, ngh\u0129a l\u00e0 d[I, (\u2206)] \u2264 1 \u21d4 |2P + P | \u221a \u2264 1 \u21d4 |3P | \u2264 5P 2 \u2212 8P + 16 \u21d4 4P 2 + 8P \u2212 16 \u2264 0. (2P )2 + (P \u2212 4)2 \u221a\u221a \u221a Gi\u1ea3i b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh ta c\u00f3 \u22121 \u2212 5 \u2264 P \u2264 \u22121 + 5. Suy ra max P = \u22121 + 5 \u2248 1,2361. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 221 S\u0110T: 0905.193.688","4. H\u00e0m s\u1ed1 m\u0169. H\u00e0m s\u1ed1 L\u00f4garit \u0104 C\u00e2u 66 (C\u00e2u 36 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). \u00d4ng A vay ng\u00e2n h\u00e0ng 100 tri\u1ec7u \u0111\u1ed3ng v\u1edbi l\u00e3i su\u1ea5t 1%\/th\u00e1ng. \u00d4ng ta mu\u1ed1n ho\u00e0n n\u1ee3 cho ng\u00e2n h\u00e0ng theo c\u00e1ch: Sau \u0111\u00fang m\u1ed9t th\u00e1ng k\u1ec3 t\u1eeb ng\u00e0y vay, \u00f4ng b\u1eaft \u0111\u1ea7u ho\u00e0n n\u1ee3; hai l\u1ea7n ho\u00e0n n\u1ee3 li\u00ean ti\u1ebfp c\u00e1ch nhau \u0111\u00fang m\u1ed9t th\u00e1ng, s\u1ed1 ti\u1ec1n ho\u00e0n n\u1ee3 \u1edf m\u1ed7i th\u00e1ng l\u00e0 nh\u01b0 nhau v\u00e0 \u00f4ng A tr\u1ea3 h\u1ebft n\u1ee3 sau \u0111\u00fang 5 n\u0103m k\u1ec3 t\u1eeb ng\u00e0y vay. Bi\u1ebft r\u1eb1ng m\u1ed7i th\u00e1ng ng\u00e2n h\u00e0ng ch\u1ec9 t\u00ednh l\u00e3i tr\u00ean s\u1ed1 d\u01b0 n\u1ee3 th\u1ef1c t\u1ebf c\u1ee7a th\u00e1ng \u0111\u00f3. H\u1ecfi s\u1ed1 ti\u1ec1n m\u1ed7i th\u00e1ng \u00f4ng ta c\u1ea7n tr\u1ea3 cho ng\u00e2n h\u00e0ng g\u1ea7n nh\u1ea5t v\u1edbi s\u1ed1 ti\u1ec1n n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A 2,22 tri\u1ec7u \u0111\u1ed3ng. B 3,03 tri\u1ec7u \u0111\u1ed3ng. C 2,25 tri\u1ec7u \u0111\u1ed3ng. D 2,20 tri\u1ec7u \u0111\u1ed3ng. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi s\u1ed1 ti\u1ec1n vay ban \u0111\u1ea7u l\u00e0 M , s\u1ed1 ti\u1ec1n ho\u00e0n n\u1ee3 m\u1ed7i th\u00e1ng l\u00e0 m, l\u00e3i su\u1ea5t m\u1ed9t th\u00e1ng l\u00e0 r. H\u1ebft th\u00e1ng th\u1ee9 nh\u1ea5t, s\u1ed1 ti\u1ec1n c\u1ea3 v\u1ed1n l\u1eabn l\u00e3i \u00f4ng A n\u1ee3 ng\u00e2n h\u00e0ng l\u00e0 M + M r = M (1 + r). Ngay sau \u0111\u00f3 \u00f4ng A ho\u00e0n n\u1ee3 s\u1ed1 ti\u1ec1n m n\u00ean s\u1ed1 ti\u1ec1n \u0111\u1ec3 t\u00ednh l\u00e3i cho th\u00e1ng th\u1ee9 hai l\u00e0 M (1 + r) \u2212 m. Do \u0111\u00f3 h\u1ebft th\u00e1ng th\u1ee9 hai, s\u1ed1 ti\u1ec1n c\u1ea3 v\u1ed1n l\u1eabn l\u00e3i \u00f4ng A n\u1ee3 ng\u00e2n h\u00e0ng l\u00e0 [M (1 + r) \u2212 m] (1 + r) = M (1 + r)2 \u2212 m (1 + r) . Ngay sau \u0111\u00f3 \u00f4ng A l\u1ea1i ho\u00e0n n\u1ee3 s\u1ed1 ti\u1ec1n m n\u00ean s\u1ed1 ti\u1ec1n \u0111\u1ec3 t\u00ednh l\u00e3i cho th\u00e1ng th\u1ee9 ba l\u00e0 M (1 + r)2 \u2212 m (1 + r) \u2212 m. Do \u0111\u00f3 h\u1ebft th\u00e1ng th\u1ee9 ba, s\u1ed1 ti\u1ec1n c\u1ea3 v\u1ed1n l\u1eabn l\u00e3i \u00f4ng A n\u1ee3 ng\u00e2n h\u00e0ng l\u00e0 \u00ee + r)2 \u2212 m (1 + r) \u2212 \u00f3 (1 + r) = M (1 + r)3 \u2212 m(1 + r)2 \u2212 m (1 + r) \u2212 m. M (1 m C\u1ee9 ti\u1ebfp t\u1ee5c l\u1eadp lu\u1eadn nh\u01b0 v\u1eady ta th\u1ea5y sau th\u00e1ng th\u1ee9 n, n \u2265 2, s\u1ed1 ti\u1ec1n c\u1ea3 v\u1ed1n l\u1eabn l\u00e3i \u00f4ng A n\u1ee3 ng\u00e2n h\u00e0ng l\u00e0 M (1 + r)n \u2212 m(1 + r)n\u22121 \u2212 m(1 + r)n\u22122 \u2212 . . . \u2212 m (1 + r) \u2212 m = M (1 + r)n \u2212 m [(1 + r)n \u2212 1] . r Sau th\u00e1ng th\u1ee9 n tr\u1ea3 h\u1ebft n\u1ee3 th\u00ec ta c\u00f3 r)n \u2212 \u00ee + r)n\u22121 \u2212 \u00f3 \u21d4 M (1 + r)nr m (1 r 1 (1 + r)n \u2212 1 . M (1 + = 0 m = Thay s\u1ed1 v\u1edbi M = 100.000.000, r = 1%, n = 5 \u00d7 12 = 60 ta \u0111\u01b0\u1ee3c m \u2248 2, 22 (tri\u1ec7u \u0111\u1ed3ng). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 67 (C\u00e2u 46 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). 1 \u2212 ab X\u00e9t c\u00e1c s\u1ed1 th\u1ef1c d\u01b0\u01a1ng a, b th\u1ecfa m\u00e3n log2 a+b = 2ab + a + b \u2212 3. T\u00ecm gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t Pmin c\u1ee7a P = a + 2b. \u221a \u221a \u221a \u221a A Pmin = 2 10 \u2212 3 3 10 \u2212 7 C Pmin = 2 10 \u2212 1 2 10 \u2212 5 . B Pmin = . . D Pmin = . 2 2 2 2 \u0253 L\u1eddi gi\u1ea3i. - Gi\u1ea3 thi\u1ebft t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi log2(2 \u2212 2ab) + (2 \u2212 2ab) = log2(a + b) + (a + b) \u21d4 2 \u2212 2ab = a + b do h\u00e0m f (t) = log2 t + t \u0111\u1ed3ng bi\u1ebfn tr\u00ean t\u1eadp x\u00e1\u221ac \u0111\u1ecbnh. 2 10 \u2212 3 - R\u00fat a theo b thay v\u00e0o P, khi \u0111\u00f3 Pmin = . 2 Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 222 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 68 (C\u00e2u 41 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). \u0110\u1ea7u n\u0103m 2016, \u00f4ng A th\u00e0nh l\u1eadp m\u1ed9t c\u00f4ng ty. T\u1ed5ng s\u1ed1 ti\u1ec1n \u00f4ng A d\u00f9ng \u0111\u1ec3 tr\u1ea3 l\u01b0\u01a1ng cho nh\u00e2n vi\u00ean trong n\u0103m 2016 l\u00e0 1 t\u1ef7 \u0111\u1ed3ng. Bi\u1ebft r\u1eb1ng c\u1ee9 sau m\u1ed7i n\u0103m th\u00ec t\u1ed5ng s\u1ed1 ti\u1ec1n d\u00f9ng \u0111\u1ec3 tr\u1ea3 l\u01b0\u01a1ng cho nh\u00e2n vi\u00ean trong n\u0103m \u0111\u00f3 t\u0103ng th\u00eam 15% so v\u1edbi n\u0103m tr\u01b0\u1edbc. H\u1ecfi n\u0103m n\u00e0o d\u01b0\u1edbi \u0111\u00e2y l\u00e0 n\u0103m \u0111\u1ea7u ti\u00ean m\u00e0 t\u1ed5ng s\u1ed1 ti\u1ec1n \u00f4ng A d\u00f9ng \u0111\u1ec3 tr\u1ea3 l\u01b0\u01a1ng cho nh\u00e2n vi\u00ean trong c\u1ea3 n\u0103m l\u1edbn h\u01a1n 2 t\u1ef7 \u0111\u1ed3ng? A N\u0103m 2023. B N\u0103m 2022. C N\u0103m 2021. D N\u0103m 2020. \u0253 L\u1eddi gi\u1ea3i. - \u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c (1 + 0, 15)m > 2 \u21d4 m > 4, 9594. V\u1eady sau 5 n\u0103m t\u1ee9c l\u00e0 n\u0103m 2021. Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 223 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit B\u00c0I 5. PH\u01af\u01a0NG TR\u00ccNH M\u0168. PH\u01af\u01a0NG TR\u00ccNH L\u00d4GARIT \u0104 C\u00e2u 1 (C\u00e2u 13 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 3x\u22121 = 27. A x = 9. B x = 3. C x = 4. D x = 10. D S = (\u2212\u221e; \u22122). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 3x\u22121 = 27 \u21d4 3x\u22121 = 33 \u21d4 x \u2212 1 = 3 \u21d4 x = 4 \u221a\u221a Ch\u1ecdn \u0111\u00e1p \u00e1n C D {\u2212 10; 10}. D {\u22124}. \u0104 C\u00e2u 2 (C\u00e2u 3 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm t\u1eadp nghi\u1ec7m S c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh 5x+1 \u2212 1 > 0. 5 A S = (1; +\u221e). B S = (\u22121; +\u221e). C S = (\u22122; +\u221e). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 5x+1 \u2212 1 > 0 \u21d4 5x+1 > 5\u22121 \u21d4 x + 1 > \u22121 \u21d4 x > \u22122. 5 V\u1eady t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho l\u00e0 S = (\u22122; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 3 (C\u00e2u 3 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). T\u1eadp nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log2(x2 \u2212 1) = 3 l\u00e0 A {\u22123; 3}. B {\u22123}. C {3}. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log2(x2 \u2212 1) = 3 \u21d4 x2 \u2212 1 = 23 \u21d4 \u00f1x = 3 x = . \u22123 V\u1eady t\u1eadp nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho l\u00e0 {\u22123; 3}. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 4 (C\u00e2u 13 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). T\u1eadp ng\u00b6hi\u1ec7\u221am c\u1ee7a\u221aph\u00a9\u01b0\u01a1ng tr\u00ecnh log3(x2 \u2212 7) = 2 l\u00e0 A \u2212 15; 15 . B {\u22124; 4}. C {4}. \u0253 L\u1eddi gi\u1ea3i. V\u1edbi \u0111i\u1ec1u ki\u1ec7n x2 \u2212 7 > 0 ta c\u00f3 log3(x2 \u2212 7) = 2 \u21d4 x2 \u2212 7 = 9 \u21d4 \u00f1x = 4 x = \u22124. So v\u1edbi \u0111i\u1ec1u ki\u1ec7n ta nh\u1eadn c\u1ea3 2 nghi\u1ec7m. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 5 (C\u00e2u 4 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 32x\u22121 = 27 l\u00e0 A x = 5. B x = 1. C x = 2. D x = 4. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 32x\u22121 = 27 \u21d4 32x\u22121 = 33 \u21d4 2x \u2212 1 = 3 \u21d4 x = 2. Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 224 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 6 (C\u00e2u 3 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 22x\u22121 = 32 l\u00e0 A x = 3. B 17 C 5 D x = 2. x= . x= . 2 2 \u0253 L\u1eddi gi\u1ea3i. 22x\u22121 = 32 \u21d4 22x\u22121 = 25 \u21d4 2x \u2212 1 = 5 \u21d4 x = 3. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 7 (C\u00e2u 2 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 3x\u22121 = 9 l\u00e0 A x = \u22122. B x = 3. C x = 2. D x = \u22123. \u0253 L\u1eddi gi\u1ea3i. 3x\u22121 = 9 \u21d4 x \u2212 1 = 2 \u21d4 x = 3. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 8 (C\u00e2u 13 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log3(x \u2212 1) = 2 l\u00e0 A x = 8. B x = 9. C x = 7. D x = 10. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x\u00e1c \u0111\u1ecbnh x > 1. log3(x \u2212 1) = 2 \u21d4 x \u2212 1 = 32 \u21d4 x \u2212 1 = 9 \u21d4 x = 10. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 9 (C\u00e2u 13 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 3x\u22122 = 9 l\u00e0 A x = \u22123. B x = 3. C x = 4. D x = \u22124. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 3x\u22122 = 9 \u21d4 3x\u22122 = 32 \u21d4 x \u2212 2 = 2 \u21d4 x = 4. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 10 (C\u00e2u 7 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log2(x \u2212 2) = 3 l\u00e0 A x = 6. B x = 8. C x = 11. D x = 10. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x \u2212 2 > 0 \u21d4 x > 2. X\u00e9t ph\u01b0\u01a1ng tr\u00ecnh log2(x \u2212 2) = 3 \u21d4 x \u2212 2 = 23 \u21d4 x = 10. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 11 (C\u00e2u 10 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 3x+1 = 9 l\u00e0 A x = 1. B x = 2. C x = \u22122. D x = \u22121. Ta c\u00f3 3x+1 = 9 \u21d4 x + 1 = 2 \u21d4 x = 1. \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 Ch\u1ecdn \u0111\u00e1p \u00e1n A 225 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit \u0104 C\u00e2u 12 (C\u00e2u 10 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log2(x + 8) = 5 l\u00e0 A x = 17. B x = 24. C x = 2. D x = 40. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log2(x + 8) = 5 \u21d4 x + 8 = 25 \u21d4 x = 32 \u2212 8 = 24. V\u1eady ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 duy nh\u1ea5t nghi\u1ec7m x = 24. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 13 (C\u00e2u 18 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 22x\u22123 = 2x l\u00e0 A x = 8. B x = \u22128. C x = 3. D x = \u22123. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 22x\u22123 = 2x \u21d4 2x \u2212 3 = x \u21d4 x = 3. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 14 (C\u00e2u 17 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log2(x + 6) = 5 l\u00e0 A x = 4. B x = 19. C x = 38. D x = 26. \u0253 L\u1eddi gi\u1ea3i. log2(x + 6) = 5 \u21d4 x + 6 = 25 \u21d4 x = 26. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 15 (C\u00e2u 15 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log3(5x) = 2 l\u00e0 8 9 A x= . B x = 9. C x= . D x = 8. 5 5 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log3(5x) = 2 \u21d4 \u00ae5x > 0 \u21d4 5x 9 \u21d4 = 9 = x . 5x = 32 5 V\u1eady log3(5x) = 2 nghi\u1ec7m duy x = 9 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 nh\u1ea5t . 5 Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 16 (C\u00e2u 27 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log5(3x) = 2 l\u00e0 25 32 x= . A x = 25. B x= . C x = 32. D 3 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log5(3x) = 2 \u21d4 3x = 52 \u21d4 x = 25 . 25 3 V\u1eady ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 nghi\u1ec7m duy nh\u1ea5t x = . 3 Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 226 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 17 (C\u00e2u 26 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log3(2x) = 2 l\u00e0 9 A x= . B x = 9. C x = 4. D x = 8. 2 \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x\u00e1c \u0111\u1ecbnh: 2x > 0 \u21d4 x > 0. Ph\u01b0\u01a1ng tr\u00ecnh t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi 2x = 32 \u21d4 x = 9 (th\u1ecfa m\u00e3n). 2 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 18 (C\u00e2u 17 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2020 - 2021). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log2(5x) = 3 l\u00e0 8 9 A . B . C 8. D 9. 55 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log2(5x) = 3 \u21d4 5x = 23 \u21d4 x = 8 . 5 8 V\u1eady nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x = . 5 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 19 (C\u00e2u 22 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 5x = 3 l\u00e0 \u221a A x = 3 5. B 3 C x = log3 5. D x = log5 3. x= . 5 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 5x = 3 \u21d4 x log5 3. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 20 (C\u00e2u 28 - M\u0110 101 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). T\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh log2(3x) > 5 l\u00e0 \u00e3 \u00c5 32\u00e3 \u00c5 32 \u00e3 \u00c5 25\u00e3 \u00c5 25 +\u221e . 0; . +\u221e . 0; . A B ; C D ; 33 3 3 \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n 3x > 0 \u21d4 x > 0. 32 \u00c5 32 \u00e3 3 +\u221e . Ta c\u00f3 log2(3x) > 5 \u21d4 3x > 25 \u21d4 x > \u21d4 x \u2208 3 ; \u00c5 32 \u00e3 +\u221e . K\u1ebft h\u1ee3p v\u1edbi \u0111i\u1ec1u ki\u1ec7n, suy ra 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 21 (C\u00e2u 22 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 5x = 2 l\u00e0 \u221a D x = 5. A x = log2 5. B x = log5 2. C 2 x= . 5 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 5x = 2 \u21d4 x = log5 2. Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 227 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit \u0104 C\u00e2u 22 (C\u00e2u 18 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 7x = 2 l\u00e0 \u221a D x = 7. A x = log2 7. B x = log7 2. C x= 2 . 7 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 7x = 2 \u21d4 x = log7 2. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 23 (C\u00e2u 14 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2020 - 2021). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 7x = 3 l\u00e0 \u221a A x= 3 B x = 3 7. C x = log7 3. D x = log3 7. . 7 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 7x = 3 \u21d4 x = log7 3. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 24 (C\u00e2u 12 - M\u0110 101 - 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 (9; +\u221e). B (25; +\u221e). C (31; +\u221e). D (24; +\u221e). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log5(x + 1) > 2 \u21d4 x + 1 > 52 \u21d4 x + 1 > 25 \u21d4 x > 24. V\u1eady t\u1eadp nghi\u1ec7p c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 (24; +\u221e). Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 25 (C\u00e2u 16 - M\u0110 102 - BGD&\u0110T - N\u0103m 2021 - 2022). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 32x+1 = 32\u2212x l\u00e0 A x= 1 B x = 0. C x = \u22121. D x = 1. . 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 32x+1 = 32\u2212x \u21d4 2x + 1 = 2 \u2212 x \u21d4 3x = 1 \u21d4 x = 1 \u00b7 3 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 26 (C\u00e2u 24 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log 1 (2x \u2212 1) = 0 l\u00e0 2 3 2 1 x= . . x= . A x = 1. B 4 C x= 3 D 2 \u0253 L\u1eddi gi\u1ea3i. \u00ae2x \u2212 1 = 1 \uf8f1x = 1 \uf8f2 Ta c\u00f3 log 1 (2x \u2212 1) = 0 \u21d4 2x \u2212 1 > 0 \u21d4 1 \u21d4 x = 1. \uf8f3x > 2 2 V\u1eady nghi\u1ec7m ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho l\u00e0 x = 1. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 27 (C\u00e2u 10 - \u0110MH - BGD&\u0110T - N\u0103m 2021 - 2022). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log2(x + 4) = 3 l\u00e0 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 228 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 A x = 5. B x = 4. C x = 2. D x = 12. \u0253 L\u1eddi gi\u1ea3i. \u00aex + 4 > 0 \u00aex > \u22124 Ta c\u00f3 log2(x + 4) = 3 \u21d4 x + 4 = 23 \u21d4 x = 4 \u21d4 x = 4. V\u1eady x = 4 l\u00e0 nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 28 (C\u00e2u 13 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 32x+1 = 27 l\u00e0 A 2. B 1. C 5. D 4. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 32x+1 = 27 \u21d4 32x+1 = 33 \u21d4 2x + 1 = 3 \u21d4 x = 1. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 29 (C\u00e2u 16 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log2(x + 1) = 1 + log2(x \u2212 1) l\u00e0 A x = 1. B x = \u22122. C x = 3. D x = 2. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n: \u00aex > \u22121 \u21d4 x > 1. x>1 Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi log2(x + 1) = 1 + log2(x \u2212 1) \u21d4 log2(x + 1) = log2 [2 \u00b7 (x \u2212 1)] \u21d4 x + 1 = 2x \u2212 2 \u21d4 x = 3. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 30 (C\u00e2u 27 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log3(2x + 1) = 1 + log3(x \u2212 1) l\u00e0 A x = 4. B x = \u22122. C x = 1. D x = 2. \u0253 L\u1eddi gi\u1ea3i. \u00ae2x + 1 > 0 \u0110i\u1ec1u ki\u1ec7n x \u2212 1 > 0 \u21d4 x > 1. Ta c\u00f3 log3(2x + 1) = 1 + log3(x \u2212 1) \u21d4 log3(2x + 1) = log3[3(x \u2212 1)] \u21d4 2x + 1 = 3x \u2212 3 \u21d4 x = 4 (nh\u1eadn). Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 31 (C\u00e2u 8 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 3x+2 = 27 l\u00e0 A x = \u22122. B x = \u22121. C x = 2. D x = 1. Ta c\u00f3 \u0253 L\u1eddi gi\u1ea3i. Ch\u1ecdn \u0111\u00e1p \u00e1n D 3x+2 = 27 \u21d4 3x+2 = 33 \u21d4 x + 2 = 3 \u21d4 x = 1. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 229 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit \u0104 C\u00e2u 32 (C\u00e2u 15 - M\u0110 103 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 22x\u22121 = 2x l\u00e0 A x = 2. B x = \u22121. C x = 1. D x = \u22122. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 22x\u22121 = 2x \u21d4 2x \u2212 1 = x \u21d4 x = 1. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 33 (C\u00e2u 1 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho ph\u01b0\u01a1ng tr\u00ecnh 4x + 2x+1 \u2212 3 = 0. Khi \u0111\u1eb7t t = 2x, ta \u0111\u01b0\u1ee3c ph\u01b0\u01a1ng tr\u00ecnh n\u00e0o d\u01b0\u1edbi \u0111\u00e2y? A 2t2 \u2212 3 = 0. B t2 + t \u2212 3 = 0. C 4t \u2212 3 = 0. D t2 + 2t \u2212 3 = 0. \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi 22x + 2.2x \u2212 3 = 0. \u0110\u1eb7t t = 2x v\u1edbi t > 0, ta \u0111\u01b0\u1ee3c: t2 + 2t \u2212 3 = 0. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 34 (C\u00e2u 19 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm t\u1ea5t c\u1ea3 c\u00e1c gi\u00e1 tr\u1ecb th\u1ef1c c\u1ee7a tham s\u1ed1 m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh 3x = m c\u00f3 nghi\u1ec7m th\u1ef1c. A m \u2265 1. B m \u2265 0. C m > 0. D m = 0. \u0253 L\u1eddi gi\u1ea3i. V\u00ec 3x > 0 v\u1edbi m\u1ecdi x \u2208 R n\u00ean ph\u01b0\u01a1ng tr\u00ecnh 3x = m c\u00f3 nghi\u1ec7m th\u1ef1c khi m > 0. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 35 (C\u00e2u 12 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh log4(x \u2212 1) = 3. A x = 63. B x = 65. C x = 80. D x = 82. \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho \u21d4 x \u2212 1 = 43 \u21d4 x \u2212 1 = 64 \u21d4 x = 65. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 36 (C\u00e2u 14 - \u0110MH - BGD&\u0110T - N\u0103m 2016 - 2017). Gi\u1ea3i b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh log2(3x \u2212 1) > 3. 1 10 A x > 3. B < x < 3. C x < 3. D x> . 3 3 \u0253 L\u1eddi gi\u1ea3i. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho \u21d4 3x \u2212 1 > 23 \u21d4 3x \u2212 1 > 8 \u21d4 x > 3. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 37 (C\u00e2u 17 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm t\u1eadp nghi\u1ec7m S c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh log 1 (x + 1) < log 1 (2x \u2212 1). 22 \u00c51 \u00e3 A S = (2; +\u221e). B S = (\u2212\u221e; 2). C S = ;2 . D S = (\u22121; 2). 2 \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x > 1 BPT \u21d4 x+1 > 2x \u2212 1 \u21d4 x < 2. . 2 \u00c51 \u00e3 K\u1ebft h\u1ee3p \u0111i\u1ec1u ki\u1ec7n suy ra t\u1eadp nghi\u1ec7m c\u1ee7a BPT l\u00e0 S = ; 2 . 2 Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 230 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 38 (C\u00e2u 22 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm t\u1eadp nghi\u1ec7m S c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log2(x \u2212 1) + log2(x + 1) = 3. A S = {\u22123; 3}. B S = \u00b6{4}\u221a. \u221a \u00a9 C S = {3}. D S = \u2212 10; 10 . \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n: x > 1. Ta c\u00f3 log2(x \u2212 1) + log2(x + 1) = 3 \u21d4 log2(x \u2212 1)(x + 1) = 3 \u21d4 x2 \u2212 1 = 8 \u21d4 \u00f1x = 3 . x = \u22123 So v\u1edbi \u0111i\u1ec1u ki\u1ec7n, ta \u0111\u01b0\u1ee3c: x = 3. V\u1eady ph\u01b0\u01a1ng tr\u00ecnh tr\u00ean c\u00f3 t\u1eadp nghi\u1ec7m S = {3}. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 39 (C\u00e2u 27 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). 2 3 T\u00ednh t\u1ed5ng c\u00e1c nghi\u1ec7m th\u1ef1c c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log3x \u00b7 log9x \u00b7 log27x \u00b7 log81x = b\u1eb1ng A 82 B 80 C 9. D 0. . . 9 9 \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x > 0. Khi \u0111\u00f3: 2 \u00c5 1 1 1 \u00e3 2 3 1\u00d7 2 3 4 3 log3x.log9x.log27x.log81x = \u21d4 \u00d7 \u00d7 (log3x)4 = \u00ef log3x = 2 x=9 log3x = \u22122 1 \u21d4(log3x)4 = 16 \u21d4 \u21d4 x= . 9 1 82 V\u1eady t\u1ed5ng c\u00e1c nghi\u1ec7m l\u00e0 9 + = . 99 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 40 (C\u00e2u 34 - \u0110TK - BGD&\u0110T - N\u0103m 2017 - 2018). C\u00f3 bao nhi\u00eau gi\u00e1 tr\u1ecb nguy\u00ean d\u01b0\u01a1ng c\u1ee7a tham s\u1ed1 m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh sau c\u00f3 nghi\u1ec7m d\u01b0\u01a1ng 16x \u2212 2 \u00b7 12x + (m \u2212 2) \u00b7 9x = 0? A 1. B 2. C 4. D 3. \u0253 L\u1eddi gi\u1ea3i. \u00c5 4 \u00e3x . Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho t\u01b0\u01a1ng \u0111\u01b0\u01a1ng t2 \u2212 2t \u2212 2 + m = 0 v\u1edbi t = 3 Y\u00eau c\u1ea7u b\u00e0i to\u00e1n tr\u1edf th\u00e0nh, t\u00ecm m nguy\u00ean d\u01b0\u01a1ng \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh t2 \u2212 2t \u2212 2 + m = 0 c\u00f3 nghi\u1ec7m l\u1edbn h\u01a1n 1. B\u1eb1ng c\u00e1ch kh\u1ea3o s\u00e1t s\u1ef1 t\u01b0\u01a1ng giao c\u1ee7a hai \u0111\u1ed3 th\u1ecb c\u00e1c h\u00e0m s\u1ed1 y = f (t) = t2 \u2212 2t \u2212 2 v\u00e0 g(x) = \u2212m ta \u0111\u01b0\u1ee3c 0 < m < 3. V\u1eady c\u00f3 hai gi\u00e1 tr\u1ecb c\u1ee7a m th\u1ecfa m\u00e3n l\u00e0 m = 1 ho\u1eb7c m = 2. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 41 (C\u00e2u 14 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). Ph\u01b0\u01a1ng tr\u00ecnh 22x+1 = 32 c\u00f3 nghi\u1ec7m l\u00e0 A x= 5 B x = 2. C 3 D x = 3. . x= . 2 2 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 22x+1 = 32 \u21d4 2x + 1 = 5 \u21d4 x = 2. Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 231 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit \u0104 C\u00e2u 42 (C\u00e2u 14 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). Ph\u01b0\u01a1ng tr\u00ecnh 52x+1 = 125 c\u00f3 nghi\u1ec7m l\u00e0 A x= 3 B 5 C x = 1. D x = 3. . x= . D {1}. 2 2 \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh 52x+1 = 125 \u21d4 2x + 1 = log5 125 \u21d4 2x + 1 = 3 \u21d4 x = 1. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 43 (C\u00e2u 46 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). T\u1eadp nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log2 (x2 \u2212 x + 2) = 1 l\u00e0 A {0}. B {0; 1}. C {\u22121; 0}. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n: x2 \u2212 x + 2 > 0, \u0111\u00fang v\u1edbi m\u1ecdi x \u2208 R. Ta c\u00f3 log2 (x2 \u2212 x + 2) = 1 \u21d4 x2 \u2212 x + 2 = 2 \u21d4 x(x \u2212 1) = 0 \u21d4 \u00f1x = 0 x = 1. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 44 (C\u00e2u 6 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log3 (2x \u2212 1) = 2 l\u00e0 9 7 . x= . A x = 3. B x = 5. C x= 2 D 2 \u0253 L\u1eddi gi\u1ea3i. \uf8f11 \uf8f11 \uf8f2x > \uf8f2x > log3 (2x \u2212 1) = 2 \u21d4 2 \u21d4 2 . \uf8f32x \u2212 1 = 9 \uf8f3x = 5 (T M ) V\u1eady nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x = 5. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 45 (C\u00e2u 8 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log2 (x \u2212 1) = 3 l\u00e0 A 10. B 8. C 9. D 7. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x > 1. Ta c\u00f3 log2 (x \u2212 1) = 3 \u21d4 log2 (x \u2212 1) = log2 23 = 8 \u21d4 x \u2212 1 = 8 \u21d4 x = 9 (th\u1ecfa m\u00e3n x > 1). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 46 (C\u00e2u 22 - M\u0110 104 - BGD&\u0110T - \u0110\u1ee3t 1 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log3 (x \u2212 2) = 2 l\u00e0 A x = 11. B x = 10. C x = 7. D x = 8. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n: x \u2212 2 > 0 \u21d4 x > 2. Ta c\u00f3 log3 (x \u2212 2) = 2 \u21d4 x \u2212 2 = 32 \u21d4 x = 11 (th\u1ecfa m\u00e3n \u0111i\u1ec1u ki\u1ec7n x > 2). V\u1eady ph\u01b0\u01a1ng tr\u00ecnh log3 (x \u2212 2) = 2 c\u00f3 nghi\u1ec7m l\u00e0 x = 11. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 232 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 47 (C\u00e2u 1 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log2(x + 9) = 5 l\u00e0 A x = 41. B x = 23. C x = 1. D x = 16. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 log2(x + 9) = 5 \u21d4 x + 9 = 25 \u21d4 x = 25 \u2212 9 = 23. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 48 (C\u00e2u 24 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 2 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 22x\u22124 = 2x l\u00e0 A x = 16. B x = \u221216. C x = \u22124. D x = 4. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 22x\u22124 = 2x \u21d4 2x \u2212 4 = x \u21d4 x = 4. V\u1eady ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 duy nh\u1ea5t nghi\u1ec7m x = 4. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 49 (C\u00e2u 24 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log2(x + 7) = 5 l\u00e0 A x = 18. B x = 25. C x = 39. D x = 3. \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh log2(x + 7) = 5 \u21d4 x + 7 = 25 \u21d4 x = 25. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 50 (C\u00e2u 21 - M\u0110 101 - BGD&\u0110T - N\u0103m 2021 - 2022). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 32x+1 = 32\u2212x l\u00e0 A x= 1 B x = 0. C x = \u22121. D x = 1. . D 0. 3 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 32x+1 = 32\u2212x \u21d4 2x + 1 = 2 \u2212 x \u21d4 x = 1 \u00b7 3 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 51 (C\u00e2u 11 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). S\u1ed1 nghi\u1ec7m th\u1ef1c c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 2x2+1 = 4 l\u00e0 A 1. B 2. C 3. \u0253 L\u1eddi gi\u1ea3i. 2x2+1 = 22 \u21d4 x2 + 1 = 2 \u21d4 x2 = 1 \u21d4 \u00f1x = 1 x = \u22121. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 52 (C\u00e2u 24 - M\u0110 103 - BGD&\u0110T - N\u0103m 2021 - 2022). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log 1 (2x \u2212 1) = 0 l\u00e0 A x= 3 2 C 1 D 2 . x= . x= . 4 B x = 1. 2 3 \u0253 L\u1eddi gi\u1ea3i. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 233 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit \u0110i\u1ec1u ki\u1ec7n: 2x \u2212 1 > 0 \u21d4 x > 1\u00b7 2 Ta c\u00f3 log 1 (2x \u2212 1) = 0 \u21d4 2x \u2212 1 = 1 \u21d4 x = 1. 2 V\u1eady nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x = 1. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 53 (C\u00e2u 26 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log3(x + 1) + 1 = log3(4x + 1) l\u00e0 A x = 3. B x = \u22123. C x = 4. D x = 2. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x > \u22121. Ta c\u00f3 4 \uf8f1 \u22121 \uf8f1 \u22121 \uf8f2x > \u21d4 \uf8f2x > 4 \u21d4 x = 2. log3(x + 1) + 1 = log3(4x + 1) \u21d4 4 \uf8f33(x + 1) = 4x + 1 \uf8f3x = 2 V\u1eady nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x = 2. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 54 (C\u00e2u 5 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 22x\u22121 = 8 l\u00e0 A x= 3 B x = 2. C x= 5 D x = 1. . . D x = 1. 2 2 \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 22x\u22121 = 8 \u21d4 22x\u22121 = 23 \u21d4 2x \u2212 1 = 3 \u21d4 x = 2. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 55 (C\u00e2u 24 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log2(x + 1) + 1 = log2(3x \u2212 1) l\u00e0 A x = 3. B x = 2. C x = \u22121. \u0253 L\u1eddi gi\u1ea3i. 1 \u0110i\u1ec1u ki\u1ec7n ph\u01b0\u01a1ng tr\u00ecnh x > . 3 log2(x + 1) + 1 = log2(3x \u2212 1) \u21d4 log2 [(x + 1) \u00b7 2] = log2(3x \u2212 1) \u21d4 2(x + 1) = 3x \u2212 1 \u21d4 x = 3 (Th\u1ecfa m\u00e3n \u0111i\u1ec1u ki\u1ec7n ph\u01b0\u01a1ng tr\u00ecnh). V\u1eady nghi\u1ec7m ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x = 3. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 56 (C\u00e2u 3 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 3x+1 = 27 l\u00e0 A x = 4. B x = 3. C x = 2. D x = 1. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 234","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+1 = 27 \u21d4 3x+1 = 33 \u21d4x+1=3 \u21d4 x = 2. V\u1eady nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh l\u00e0 x = 2. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 57 (C\u00e2u 22 - M\u0110 102 - BGD&\u0110T - \u0110\u1ee3t 4 - N\u0103m 2019 - 2020). Nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 22x\u22122 = 2x l\u00e0 A x = \u22122. B x = 2. C x = \u22124. D x = 4. \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh 22x\u22122 = 2x \u21d4 2x \u2212 2 = x \u21d4 x = 2. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 58 (C\u00e2u 9 - M\u0110 104 - BGD&\u0110T - N\u0103m 2021 - 2022). S\u1ed1 nghi\u1ec7m th\u1ef1c c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh 2x2+1 = 4 l\u00e0 A 1. B 2. C 0. D 3. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 2x2+1 = 4 \u21d4 x2 + 1 = 2 \u21d4 x2 = 1 \u21d4 x = \u00b11. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 59 (C\u00e2u 23 - \u0110MH - BGD&\u0110T - N\u0103m 2018 - 2019). T\u1ed5ng t\u1ea5t c\u1ea3 c\u00e1c nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log3(7 \u2212 3x) = 2 \u2212 x b\u1eb1ng A 2. B 1. C 7. D 3. \u0253 L\u1eddi gi\u1ea3i. log3(7 \u2212 3x) = 2\u2212x \u21d4 7 \u2212 3x = 32\u2212x \u21d4 7 \u2212 3x = 9 \u21d4 (3x)2 \u2212 7 \u00b7 3x +9 = 0. (\u2217) 3x \u00ae3x1 + 3x2 = 7 Ph\u01b0\u01a1ng tr\u00ecnh (\u2217) c\u00f3 2 nghi\u1ec7m ph\u00e2n bi\u1ec7t th\u1ecfa m\u00e3n 3x1 \u00b7 3x2 = 9 \u21d2 3x1+x2 = 32 \u21d4 x1 + x2 = 2. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 60 (C\u00e2u 16 - M\u0110 104 - 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 6,1 %\/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 13 n\u0103m. B 10 n\u0103m. C 11 n\u0103m. D 12 n\u0103m. \u0253 L\u1eddi gi\u1ea3i. G\u1ecdi s\u1ed1 ti\u1ec1n ban \u0111\u1ea7u ng\u01b0\u1eddi \u0111\u00f3 g\u1eedi l\u00e0 A0. S\u1ed1 ti\u1ec1n ng\u01b0\u1eddi \u0111\u00f3 thu \u0111\u01b0\u1ee3c sau n n\u0103m g\u1eedi l\u00e0 An = A0 \u00b7 (1 + 6,1%)n. S\u1ed1 ti\u1ec1n ng\u01b0\u1eddi \u0111\u00f3 thu \u0111\u01b0\u1ee3c g\u1ea5p \u0111\u00f4i s\u1ed1 ti\u1ec1n g\u1eedi ban \u0111\u1ea7u khi v\u00e0 ch\u1ec9 khi An = 2A0 \u21d4 A0 \u00b7 (1 + 6,1%)n = 2A0 \u21d4 n = log(1+6,1%) 2 \u2248 11,7. V\u00ec n l\u00e0 s\u1ed1 t\u1ef1 nhi\u00ean n\u00ean n = 12. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 235 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit \u0104 C\u00e2u 61 (C\u00e2u 6 - M\u0110 101 - BGD&\u0110T - N\u0103m 2016 - 2017). Cho a l\u00e0 s\u1ed1 th\u1ef1c d\u01b0\u01a1ng kh\u00e1c 1. T\u00ednh I = log\u221aa a. A I= 1 B I = 0. C I = \u22122. D I = 2. . D x = 5. 2 \u0253 L\u1eddi gi\u1ea3i. I = log\u221aa a = log 1 a = 2 loga a = 2. a2 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 62 (C\u00e2u 9 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log2(1 \u2212 x) = 2. A x = \u22124. B x = \u22123. C x = 3. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n: x < 1. Ta c\u00f3 log2(1 \u2212 x) = 2 \u21d4 1 \u2212 x = 4 \u21d4 x = \u22123. V\u1eady ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 nghi\u1ec7m x = \u22123. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 63 (C\u00e2u 30 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm t\u1eadp nghi\u1ec7m S c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log\u221a2 (x \u2212 1) + log 1 (x + 1) = 1. \u221a\u00a9 \u00b6\u221a \u00a9 2 \u00b6 \u221a A S = 2+ 5 . B S = 2 \u2212 5; 2 + 5 . \u221a \u00ae\u00b4 3+ 13 C S = {3}. D S= . 2 \u0253 L\u1eddi gi\u1ea3i. T\u1eadp x\u00e1c \u0111\u1ecbnh D = (1; +\u221e). V\u1edbi x \u2208 D, ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi log\u221a2 (x \u2212 1) + log 1 (x + 1) = 1 2 \u21d42 log2 (x \u2212 1) \u2212 log2 (x + 1) = 1 \u21d4 log2 (x \u2212 1)2 = 1 (x + 1) \u21d4x2 \u2212 2x + 1 = 2x + 2 \u21d4x2 \u2212 4x \u2212 1 = 0 \u221a \u00f1x = 2 + 5 (ch\u1ecdn) \u21d4\u221a x = 2 \u2212 5 (lo\u1ea1i) Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 64 (C\u00e2u 4 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). 1 T\u00ecm nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log25(x + 1) = . 2 23 A x = \u22126. B x = 6. C x = 4. D x= . 2 \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x > \u22121. Ph\u01b0\u01a1ng tr\u00ecnh t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi x+1 = 25 1 = 5 \u21d2 x = 4 2 Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 236 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 65 (C\u00e2u 11 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u1eadp nghi\u1ec7m S c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh log3(2x + 1) \u2212 log3(x \u2212 1) = 1. A S = {4}. B S = {3}. C S = {\u22122}. D S = {1}. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x > 1. 2x + 1 2x + 1 x\u22121 x\u22121 Ph\u01b0\u01a1ng tr\u00ecnh t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi log3 = 1 \u21d4 = 3 \u21d4 2x + 1 = 3x \u2212 3 \u21d2 x = 4 Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 66 (C\u00e2u 39 - M\u0110 101 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho ph\u01b0\u01a1ng tr\u00ecnh log9 x2 \u2212 log3(3x \u2212 1) = \u2212 log3 m (m l\u00e0 tham s\u1ed1 th\u1ef1c). C\u00f3 t\u1ea5t c\u1ea3 bao nhi\u00eau gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a tham s\u1ed1 m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 nghi\u1ec7m? A 2. B 4. C 3. D V\u00f4 s\u1ed1. \u0253 L\u1eddi gi\u1ea3i. 1 \u0110i\u1ec1u ki\u1ec7n x > v\u00e0 m > 0. 3 1 x1 Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho t\u01b0\u01a1ng \u0111\u01b0\u01a1ng: log3 x \u2212 log3(3x \u2212 1) = log3 m \u21d4 3x \u2212 1 = m x1 X\u00e9t h\u00e0m s\u1ed1 f (x) = v\u1edbi x > . 3x \u2212 1 3 C\u00f3 f (x) = \u2212 1 < 0, \u2200x > 1 (3x \u2212 1)2 3 x 1 +\u221e f (x) 3 1 \u2212 3 +\u221e f (x) D\u1ef1a v\u00e0o b\u1ea3ng bi\u1ebfn thi\u00ean, ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 nghi\u1ec7m khi 1 > 1 \u21d4 0 < m < 3. m3 Do m \u2208 Z \u21d2 m \u2208 {1, 2}. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 67 (C\u00e2u 37 - M\u0110 102 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho ph\u01b0\u01a1ng tr\u00ecnh log9 x2 \u2212 log3(6x \u2212 1) = \u2212 log3 m (m l\u00e0 tham s\u1ed1 th\u1ef1c). C\u00f3 t\u1ea5t c\u1ea3 bao nhi\u00eau gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 nghi\u1ec7m? A 6. B 5. C V\u00f4 s\u1ed1. D 7. \u0253 L\u1eddi gi\u1ea3i. X\u00e9t ph\u01b0\u01a1ng tr\u00ecnh log9 x2 \u2212 log3(6x \u2212 1) = \u2212 log3 m. \uf8f11 \uf8f2x > \u0110i\u1ec1u ki\u1ec7n: 6 \uf8f3m > 0. log9 x2 \u2212 log3(6x \u2212 1) = \u2212 log3 m \u21d4 log3 x + log3 m = log3(6x \u2212 1) \u21d4 mx = 6x \u2212 1 \u21d4 x(6 \u2212 m) = 1 (1) Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 237 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit \u2022 V\u1edbi m = 6, ph\u01b0\u01a1ng tr\u00ecnh (1) tr\u1edf th\u00e0nh 0 = 1 (v\u00f4 l\u00fd). \u2022 V\u1edbi m = 6, ph\u01b0\u01a1ng tr\u00ecnh (1) c\u00f3 nghi\u1ec7m x = 1 n\u00ean 6\u2212m 1 > 1 \u21d4 1 \u2212 1 > 0 \u21d4 m > 0 \u21d4 0 < m < 6 (th\u1ecfa m\u00e3n). 6\u2212m 6 6\u2212m 6 6\u2212m M\u00e0 m \u2208 Z \u21d2 m \u2208 {1; 2; 3; 4; 5}. V\u1eady c\u00f3 5 gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a m th\u1ecfa m\u00e3n. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 68 (C\u00e2u 39 - M\u0110 102 - 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 [log2(x + 30) \u2212 5] \u2264 0? A 30. B V\u00f4 s\u1ed1. C 31. D 29. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x + 30 > 0 \u21d4 x > \u221230. \u00c43x2 \u2212 9x\u00e4 = 0 (1) (2). \u0110\u1eb7t f (x) = \u00c43x2 \u2212 9x\u00e4 [log2(x + 30) \u2212 5]. X\u00e9t f (x) = 0 \u21d4 [log2(x + 30) \u2212 5] = 0 Ph\u01b0\u01a1ng tr\u00ecnh (1) \u21d4 3x2 = 32x \u21d4 x2 = 2x \u21d4 \u00f1x = 0 x = 2. Ph\u01b0\u01a1ng tr\u00ecnh (2) \u21d4 log2(x + 30) = 5 \u21d4 x + 30 = 32 \u21d4 x = 2. T\u1eeb \u0111\u00f3 ta c\u00f3 b\u1ea3ng x\u00e9t d\u1ea5u sau: x \u221230 0 2 +\u221e 3x2 \u2212 9x +0\u22120+ log2(x+30)\u22125 \u2212 \u22120+ f (x) \u22120+0+ T\u1eeb \u0111\u00f3 ta suy ra \u0111\u01b0\u1ee3c t\u1eadp nghi\u1ec7m c\u1ee7a b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh ban \u0111\u1ea7u l\u00e0 S = (\u221230; 0] \u222a {2}. V\u1eady c\u00f3 31 s\u1ed1 nguy\u00ean th\u1ecfa m\u00e3n b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh ban \u0111\u1ea7u. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 69 (C\u00e2u 42 - \u0110TK - BGD\u221a&\u0110T - N\u0103m 2017 - 2018). Cho d\u00e3y s\u1ed1 (un) th\u1ecfa m\u00e3n log u1 + 2 + log u1 \u2212 2 log u10 = 2 log u10 v\u00e0 un+1 = 2un v\u1edbi m\u1ecdi n \u2265 1. Gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t c\u1ee7a n \u0111\u1ec3 un > 5100 b\u1eb1ng A 247. B 248. C 229. D 290. \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 un = u1.2n\u22121 n\u00ean u10 = u1.29. \u0110\u1eb7t t = log u1, ta c\u00f3 t+ 2 + t \u2212 2t \u2212 18 log 2 = 18 log 2+2t \u21d4 18+t = 2 \u2212 t \u2212 18 log 2 \u21d4 t = \u221218 log 2+1 \u21d4 u1 = 2\u221217.5. Suy ra un > 5100 \u21d4 2n\u221218.5 > 5100 \u21d4 2n\u221218 > 599 \u21d4 n > 18 + 99. log2 5. T\u1eeb \u0111\u00f3 ta c\u00f3 nmin = 248. Ch\u1ecdn \u0111\u00e1p \u00e1n B 238 S\u0110T: 0905.193.688 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t","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 70 (C\u00e2u 34 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). G\u1ecdi S l\u00e0 t\u1eadp h\u1ee3p t\u1ea5t c\u1ea3 c\u00e1c gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a tham s\u1ed1 m sao cho ph\u01b0\u01a1ng tr\u00ecnh 16x \u2212 m \u00b7 4x+1 + 5m2 \u2212 45 = 0 c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t. H\u1ecfi S c\u00f3 bao nhi\u00eau ph\u1ea7n t\u1eed? A 13. B 3. C 6. D 4. \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t t = 4x, t > 0. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho tr\u1edf th\u00e0nh t2 \u2212 4mt + 5m2 \u2212 45 = 0. (\u2217) V\u1edbi m\u1ed7i nghi\u1ec7m t > 0 c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh (\u2217) s\u1ebd t\u01b0\u01a1ng \u1ee9ng v\u1edbi duy nh\u1ea5t m\u1ed9t nghi\u1ec7m x c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh ban \u0111\u1ea7u. Do \u0111\u00f3, y\u00eau c\u1ea7u b\u00e0i to\u00e1n t\u01b0\u01a1ng \u0111\u01b0\u01a1ng ph\u01b0\u01a1ng tr\u00ecnh (\u2217) c\u00f3 hai nghi\u1ec7m d\u01b0\u01a1ng ph\u00e2n bi\u1ec7t. Khi \u0111\u00f3 \uf8f1\u221a \u221a \uf8f4\u22123 5<m<3 5 \uf8f1\u2206 > 0 \uf8f1 \u2212 m2 + 45 > 0 \uf8f4 \u221a \uf8f4 \uf8f4\uf8f4 \uf8f2\uf8f4m > 0 \uf8f2\uf8f2 S > 0 \u21d4 4m > 0 \u21d4 \uf8f4 \u00f1m < \u22123 \u21d4 3 < m < 3 5. \uf8f4 \uf8f3\uf8f4P > 0 \uf8f4\uf8f35m2 \u2212 45 > 0 \uf8f4 \uf8f4 m>3 \uf8f3 Do m \u2208 Z n\u00ean m \u2208 {4; 5; 6}. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 71 (C\u00e2u 35 - M\u0110 102 - BGD&\u0110T - N\u0103m 2017 - 2018). G\u1ecdi S l\u00e0 t\u1eadp h\u1ee3p t\u1ea5t c\u1ea3 c\u00e1c gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a tham s\u1ed1 m sao cho ph\u01b0\u01a1ng tr\u00ecnh 25x \u2212 m \u00b7 5x+1 + 7m2 \u2212 7 = 0 c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t. H\u1ecfi S c\u00f3 bao nhi\u00eau ph\u1ea7n t\u1eed? A 7. B 1. C 2. D 3. \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t t = 5x, \u0111i\u1ec1u ki\u1ec7n t > 0. Ph\u01b0\u01a1ng tr\u00ecnh tr\u1edf th\u00e0nh t2 \u2212 5mt + 7m2 \u2212 7 = 0 (\u2217). Y\u00eau c\u1ea7u b\u00e0i to\u00e1n tr\u1edf th\u00e0nh: t\u00ecm m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh (\u2217) c\u00f3 hai nghi\u1ec7m d\u01b0\u01a1ng ph\u00e2n bi\u1ec7t \uf8f1 \u2026 28 \u2026 28 3 <m< \uf8f1\u2206 = \u22123m2 + 28 > 0 \uf8f4 \u2212 \uf8f4 \uf8f4 3 \uf8f2 \uf8f4 \u21d4 1 < m < \u202628. \uf8f4 \u21d4 5m > 0 \uf8f4 \uf8f2 \u21d4 m>0 3 \uf8f3\uf8f47m2 \u2212 7 > 0 \uf8f4 \u00f1m > 1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 m < \u22121 Suy ra S = {2; 3}. V\u1eady c\u00f3 2 gi\u00e1 tr\u1ecb tham s\u1ed1 m th\u1ecfa m\u00e3n y\u00eau c\u1ea7u b\u00e0i to\u00e1n. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 72 (C\u00e2u 33 - M\u0110 103 - BGD&\u0110T - N\u0103m 2017 - 2018). G\u1ecdi S l\u00e0 t\u1ea5t c\u1ea3 c\u00e1c gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a tham s\u1ed1 m sao cho ph\u01b0\u01a1ng tr\u00ecnh 4x \u2212 m \u00b7 2x+1 + 2m2 \u2212 5 = 0 c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t. H\u1ecfi S c\u00f3 bao nhi\u00eau ph\u1ea7n t\u1eed? A 3. B 5. C 2. D 1. \u0253 L\u1eddi gi\u1ea3i. (1) Ta c\u00f3 4x \u2212 m \u00b7 2x+1 + 2m2 \u2212 5 = 0 \u21d4 4x \u2212 2m \u00b7 2x + 2m2 \u2212 5 = 0. \u0110\u1eb7t t = 2x, t > 0. Ph\u01b0\u01a1ng tr\u00ecnh (1) th\u00e0nh: t2 \u2212 2m \u00b7 t + 2m2 \u2212 5 = 0. (2) Y\u00eau c\u1ea7u b\u00e0i to\u00e1n \u21d4 (2) c\u00f3 2 nghi\u1ec7m d\u01b0\u01a1ng ph\u00e2n\u221abi\u1ec7t \u221a \uf8f1 \u2212 5<m< 5 \uf8f1\u2206 > 0 \uf8f1m2 \u2212 2m2 + 5 > 0 \uf8f4 \u221a 10 \uf8f4 \u221a 5. \uf8f4\uf8f4 \uf8f4 \u21d4 \uf8f2m > 0 \uf8f2\uf8f2 \u2026 \u20265 \u21d4 2 < m < \u21d4 S > 0 \u21d4 2m > 0 \u2212 2 \uf8f3\uf8f4P > 0 \uf8f3\uf8f42m2 \u2212 5 > 0 \uf8f4 < 5 \u2228 m > \uf8f4\uf8f3\uf8f4m 2 Do m l\u00e0 s\u1ed1 nguy\u00ean n\u00ean m = 2. V\u1eady S ch\u1ec9 c\u00f3 m\u1ed9t ph\u1ea7n t\u1eed duy nh\u1ea5t. Ch\u1ecdn \u0111\u00e1p \u00e1n D Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 239 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit \u0104 C\u00e2u 73 (C\u00e2u 28 - M\u0110 104 - BGD&\u0110T - N\u0103m 2017 - 2018). G\u1ecdi S l\u00e0 t\u1eadp h\u1ee3p c\u00e1c gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a tham s\u1ed1 m sao cho ph\u01b0\u01a1ng tr\u00ecnh 9x \u2212m3x+1 +3m2 \u221275 = 0 c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t. H\u1ecfi S c\u00f3 bao nhi\u00eau ph\u1ea7n t\u1eed? A 8. B 4. C 19. D 5. \u0253 L\u1eddi gi\u1ea3i. Ph\u01b0\u01a1ng tr\u00ecnh 9x \u2212 m3x+1 + 3m2 \u2212 75 = 0 \u21d4 (3x)2 \u2212 3m \u00b7 3x + 3m2 \u2212 75 = 0. \u0110\u1eb7t t = 3x, t > 0. Ph\u01b0\u01a1ng tr\u00ecnh tr\u1edf th\u00e0nh t2 \u2212 3mt + 3m2 \u2212 75 = 0 (1). Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t khi v\u00e0 ch\u1ec9 khi ph\u01b0\u01a1ng tr\u00ecnh (1) c\u00f3 hai nghi\u1ec7m d\u01b0\u01a1ng ph\u00e2n bi\u1ec7t \uf8f1\u2206 = 300 \u2212 3m2 > 0 \uf8f1 \u2212 10 < m < 10 \uf8f4 \uf8f4 \uf8f4 \uf8f2\uf8f4m > 0 \uf8f2 \u21d4 3m > 0 \u21d4 \uf8f4 \u00f1m < \u22125 \u21d4 5 < m < 10. \uf8f4\uf8f33m2 \u2212 75 > 0 \uf8f4 \uf8f4 \uf8f3 m>5 Do m nguy\u00ean n\u00ean S = {6; 7; 8; 9}. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 74 (C\u00e2u 47 - M\u0110 102 - BGD&\u0110T\u221a- N\u0103m 2018 - 2019). s\u1ed1 th\u1ef1c). C\u00f3 t\u1ea5t c\u1ea3 bao nhi\u00eau Cho ph\u01b0\u01a1ng tr\u00ecnh 2 log22 x \u2212 3 log2 x \u2212 2 3x \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 \u0111\u00e3 cho c\u00f3 \u0111\u00fang hai nghi\u1ec7m ph\u00e2n bi\u1ec7t? A 79. B 80. C v\u00f4 s\u1ed1. D 81. \u0253 L\u1eddi gi\u1ea3i. \u00aex > 0 \u00aex > 0 \u00aex > 0 \u0110i\u1ec1u ki\u1ec7n 3x \u2212 m \u2265 0 \u21d4 3x \u2265 m \u21d4 . x \u2265 log3 m TH1: V\u1edbi m = 1, ph\u01b0\u01a1ng tr\u00ecnh tr\u1edf th\u00e0nh 2 log22 x \u2212 3 log2 x \u2212 2 \u221a \u2212 1 = 0 \u21d4 \u00f12 log22 x \u2212 3 log2 x \u2212 2 = 0 3x 3x \u2212 1 = 0 \uf8ee log2 x = 2 \uf8eex = 4 \uf8eex = 4 \u21d4 \uf8ef log2 x = \u22121 \u21d4 \uf8ef = \u221a1 \u21d4 \uf8f0 \u221a1 \uf8ef 2 \uf8efx x \uf8f0 \uf8ef 2 = . \uf8f0 2 3x = 1 x=0 V\u1eady nh\u1eadn gi\u00e1 tr\u1ecb m = 1. TH2: V\u1edbi m > 1, ph\u01b0\u01a1ng tr\u00ecnh tr\u1edf th\u00e0nh 2 log22 x \u2212 3 log2 x \u2212 2 \u221a \u2212 m = 0 \u21d4 \u00f12 log22 x \u2212 3 log2 x \u2212 2 = 0 3x 3x \u2212 m = 0 \uf8ee log2 x = 2 \uf8eex = 4 \u21d4 \uf8ef log2 x = \u2212 1 \u21d4 \uf8ef = \u221a1 \uf8ef 2 \uf8efx 2 \uf8f0 \uf8ef \uf8f0 3x = m x = log3 m. Ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t khi \u221a1 \u2264 log3 m < 4 \u21d4 \u221a1 \u2264 m < 34. 2 32 M\u00e0 m > 1 n\u00ean ta c\u00f3 m \u2208 {3, 4, . . . , 80}, c\u00f3 78 gi\u00e1 tr\u1ecb c\u1ee7a m. V\u1eady c\u00f3 79 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. Ch\u1ecdn \u0111\u00e1p \u00e1n A Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 240 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 75 (C\u00e2u 20 - \u0110TN - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm t\u1eadp h\u1ee3p c\u00e1c gi\u00e1 tr\u1ecb c\u1ee7a tham s\u1ed1 th\u1ef1c m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh 6x + (3 \u2212 m)2x \u2212 m = 0 c\u00f3 nghi\u1ec7m thu\u1ed9c kho\u1ea3ng (0; 1). A [3; 4]. B [2; 4]. C (2; 4). D (3; 4). \u0253 L\u1eddi gi\u1ea3i. Ta c\u00f3 6x + (3 \u2212 m)2x \u2212 m = 0 \u21d4 m = 6x + 3.2x 2x + 1 6x + 3.2x X\u00e9t h\u00e0m s\u1ed1 f (x) = 2x + 1 + TX\u0110: D1=2xR. ln 3 + f (x) = + 6x. ln 6 + 3.2x. ln 2 > 0, \u2200x \u2208 n\u00ean h\u00e0m s\u1ed1 f (x) \u0111\u1ed3ng bi\u1ebfn tr\u00ean R. (2x + 1)2 R Suy ra 0 < x < 1 \u21d4 f (0) < f (x) < f (1) \u21d4 2 < f (x) < 4 v\u00ec f (0) = 2, f (1) = 4. V\u1eady ph\u01b0\u01a1ng tr\u00ecnh (1)c\u00f3 nghi\u1ec7m thu\u1ed9c kho\u1ea3ng (0; 1) khi m \u2208 (2; 4). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 76 (C\u00e2u 35 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). H\u1ecfi ph\u01b0\u01a1ng tr\u00ecnh 3x2 \u2212 6x + ln(x + 1)3 + 1 = 0 c\u00f3 bao nhi\u00eau nghi\u1ec7m ph\u00e2n bi\u1ec7t? A 2. B 1. C 3. D 4. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n: x > \u22121. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi 3x2 \u2212 6x + 3 ln(x + 1) + 1 = 0. X\u00e9t h\u00e0m s\u1ed1 y = 3x2 \u2212 6x + 3 ln(x + 1) + 1 li\u00ean t\u1ee5c tr\u00ean kho\u1ea3ng (\u22121; +\u221e). y = 6(x \u2212 1) + 3 6x2 \u2212 3 =. x + 1 x + 1\u221a 2 y = 0 \u21d4 2x2 \u2212 1 = 0 \u21d4 x = \u00b1 2 (th\u1ecfa \u0111i\u1ec1u ki\u1ec7n). x \u22121 \u221a \u221a +\u221e y \u22122 2 +\u221e 2 y 2 0 \u2212\u221e +0\u2212 + \u00c7 \u221a \u00e5 \u2212 2 f \u00c7 \u221a \u00e5 2 2 f 2 \u00c7 \u221a \u00e5 \u00c7 \u221a \u00e5 2 2 V\u00ec f \u2212 > 0, f < 0 v\u00e0 lim y = +\u221e n\u00ean \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 c\u1eaft tr\u1ee5c ho\u00e0nh t\u1ea1i 3 \u0111i\u1ec3m ph\u00e2n 2 2 x\u2192+\u221e bi\u1ec7t. Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 77 (C\u00e2u 46 - M\u0110 103 - 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 log32 x \u2212 log3 x \u2212 1 5x \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 \u0111\u00e3 cho c\u00f3 \u0111\u00fang hai nghi\u1ec7m ph\u00e2n bi\u1ec7t? A 123. B 125. C V\u00f4 s\u1ed1. D 124. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t \u0253 L\u1eddi gi\u1ea3i. S\u0110T: 0905.193.688 241","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit \u00aex > 0 \u00aex > 0 \u0110i\u1ec1u ki\u1ec7n: \u21d4 2 log23 x \u2212 5x \u2212 m \u2265\u221a0 (m > 0) x \u2265 log5 m. log3 x \u2212 1 5x \u2212 m = 0 (1) \u21d4 \u00f12 log32 x \u2212 log3 x \u2212 1 = 0 \u21d4 \uf8ee = 3, x = \u221a1 5x \u2212 m = 0 x = 3 \uf8f0 log5 m. x TH 1. N\u1ebfu m = 1 th\u00ec x = log5 m = 0 (lo\u1ea1i) n\u00ean ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 2 nghi\u1ec7m ph\u00e2n bi\u1ec7t. TH 2. N\u1ebfu m > 1 th\u00ec ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 \u0111\u00fang 2 nghi\u1ec7m ph\u00e2n bi\u1ec7t khi v\u00e0 ch\u1ec9 khi \u221a1 \u2264 log5 m < 3 \u21d4 \u221a1 \u2264 m < 125. Do m \u2208 Z \u21d2 m \u2208 {3; 4; 5; . . . ; 124}. N\u00ean c\u00f3 123 3 53 gi\u00e1 tr\u1ecb m tho\u1ea3 m\u00e3n. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 78 (C\u00e2u 43 - \u0110TK - BGD&\u0110T - l\u1ea7n 1 - N\u0103m 2019 - 2020). Cho ph\u01b0\u01a1ng tr\u00ecnh log22 (2x) \u2212 (m + 2) log2 x + m \u2212 2 = 0 (m l\u00e0 tham s\u1ed1 th\u1ef1c). T\u1eadp h\u1ee3p t\u1ea5t c\u1ea3 c\u00e1c gi\u00e1 tr\u1ecb c\u1ee7a m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t thu\u1ed9c \u0111o\u1ea1n [1; 2] l\u00e0 A (1; 2). B [1; 2]. C [1; 2). D [2; +\u221e). \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x\u00e1c \u0111\u1ecbnh: x > 0. Ta c\u00f3 log22 (2x) \u2212 (m + 2) log2 x + m \u2212 2 = 0 \u21d4 (log2 x + 1)2 \u2212 m log2 x \u2212 2 log2 x + m \u2212 2 = 0 \u21d4 log22 x \u2212 1 \u2212 m log2 x + m = 0 \u21d4 (log2 x \u2212 1) (log2 x + 1) \u2212 m (log2 x \u2212 1) = 0 \u21d4 (log2 x \u2212 1) (log2 x + 1 \u2212 m) = 0 \u00f1 log2 x \u00f1x \u21d4 log2 x = 1 \u2212 1 \u21d4 x = 2 \u2208 [1; 2] = m = 2m\u22121 . \u00ae1 \u2264 2m\u22121 \u2264 2 \u21d41\u2264 Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 hai nghi\u1ec7m ph\u00e2n bi\u1ec7t thu\u1ed9c \u0111o\u1ea1n [1; 2] khi v\u00e0 ch\u1ec9 khi 2m\u22121 = 2 2m\u22121 < 2 \u21d4 0 \u2264 m \u2212 1 < 1 \u21d4 1 \u2264 m < 2 Hay m \u2208 [1; 2). Ch\u1ecdn \u0111\u00e1p \u00e1n C \u0104 C\u00e2u 79 (C\u00e2u 45 - M\u0110 102 - 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. \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. \u0110\u1eb7t f (x) = (4x \u2212 4 \u2212 y) ex \u2212 (xy2 \u2212 2x2y \u2212 3y). Ta c\u00f3 f (x) = (4x \u2212 y) ex + y (4x \u2212 y) = (4x \u2212 y) (ex + y). y . V\u00ec y l\u00e0 s\u1ed1 nguy\u00ean d\u01b0\u01a1ng n\u00ean ex + y > 0. Do \u0111\u00f3 f (x) = 0 \u21d4 4x \u2212 y =0\u21d4x= 4 f (x) < 0 \u21d4 x < y v\u00e0 f (x) > 0 \u21d4 x > y . 44 V\u1edbi f (1) = \u2212ye \u2212 y2 + 5y, f (5) = (16 \u2212 y) e5 \u2212 5y2 + 53y. Tr\u01b0\u1eddng h\u1ee3p 1. N\u1ebfu 0 < y \u2264 4 th\u00ec 4x \u2212 y > 0 \u21d2 f (x) > 0 v\u1edbi m\u1ecdi x \u2208 (1; 5), ta c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 242 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 5 f (x) + f (5) f (x) f (1) T\u1eeb y\u00eau c\u1ea7u b\u00e0i to\u00e1n, ta c\u00f3 \u00aef (1) < 0 \u00ae \u2212 ye \u2212 y2 + 5y < 0 \u00ae \u2212 5y2 \u2212 e5 \u2212 53 y + 16e5 > 0 \u21d4 (16 \u2212 y) e5 \u2212 5y2 + 53y > 0 \u21d4 \u2212 y (e + y \u2212 5) < 0 f (5) > 0 \u00ae \u2212 33, 33120491 < y < 14, 24857309 \u21d4 \u21d4 5 \u2212 e < y < 14, 24857309. y > 5\u2212e Suy ra 5 \u2212 e < y \u2264 4. M\u00e0 y nguy\u00ean d\u01b0\u01a1ng n\u00ean y \u2208 {3; 4}. C\u00f3 2 gi\u00e1 tr\u1ecb nguy\u00ean d\u01b0\u01a1ng c\u1ee7a y th\u1ecfa m\u00e3n. y Tr\u01b0\u1eddng h\u1ee3p 2. N\u1ebfu 4 < y < 20 th\u00ec 1 < < 5, ta c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean 4 x1 y 5 4 f (x) \u22120 + f (1) f (5) y f (x) f 4 Do \u2212y (e + y \u2212 5) < 0, \u2200y \u2208 (4; 20) n\u00ean \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 nghi\u1ec7m x \u2208 (1; 5) th\u00ec 16e5 \u2212 y e5 + 5y \u2212 53 > 0 \u21d4 \u22125y2 \u2212 e5 \u2212 53 y + 16e5 > 0 \u21d4 \u221233, 33120491 < y < 14, 24857309. Suy ra 4 < y \u2264 14, 24857309. M\u00e0 y nguy\u00ean d\u01b0\u01a1ng n\u00ean y \u2208 {5; 6; 7; . . . ; 14}. C\u00f3 10 gi\u00e1 tr\u1ecb nguy\u00ean d\u01b0\u01a1ng c\u1ee7a y th\u1ecfa m\u00e3n. Tr\u01b0\u1eddng h\u1ee3p 3. N\u1ebfu y \u2265 20 th\u00ec y \u2265 5, ta c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean 4 x1 y 4 f (x) \u2212 f (1) y f (x) f 4 Do f (1) = \u2212y (e + y \u2212 5) < 0, \u2200y \u2265 20 n\u00ean ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho kh\u00f4ng c\u00f3 nghi\u1ec7m x \u2208 (1; 5). V\u1eady k\u1ebft h\u1ee3p 3 tr\u01b0\u1eddng h\u1ee3p tr\u00ean ta c\u00f3 12 gi\u00e1 tr\u1ecb nguy\u00ean d\u01b0\u01a1ng c\u1ee7a y th\u1ecfa m\u00e3n y\u00eau c\u1ea7u b\u00e0i to\u00e1n. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 80 (C\u00e2u 31 - M\u0110 102 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm t\u1ea5t c\u1ea3 c\u00e1c gi\u00e1 tr\u1ecb th\u1ef1c c\u1ee7a tham s\u1ed1 m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh 4x \u2212 2x+1 + m = 0 c\u00f3 hai nghi\u1ec7m th\u1ef1c ph\u00e2n bi\u1ec7t. A m \u2208 (\u2212\u221e; 1). B m \u2208 (0; +\u221e). C m \u2208 (0; 1]. D m \u2208 (0; 1). Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 243 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit \u0253 L\u1eddi gi\u1ea3i. X\u00e9t ph\u01b0\u01a1ng tr\u00ecnh 4x \u2212 2x+1 + m = 0. \u0110\u1eb7t 2x = t > 0, ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho tr\u1edf th\u00e0nh t2 \u2212 2t + m = 0. Ta c\u00f3 \u2206 = 1 \u2212 m. Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 2 nghi\u1ec7m th\u1ef1c ph\u00e2n bi\u1ec7t khi ph\u01b0\u01a1ng tr\u00ecnh t2 \u2212 2t + m = 0 c\u00f3 2 nghi\u1ec7m d\u01b0\u01a1ng ph\u00e2n bi\u1ec7t, khi \u0111\u00f3 \uf8f1\u2206 > 0 \uf8f1m < 1 \uf8f4\uf8f4 \uf8f2\uf8f2 P > 0 \u21d4 m > 0 \u21d4 0 < m < 1. \uf8f4 S > 0 \uf8f4 2 > 0 \uf8f3 \uf8f3 Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 81 (C\u00e2u 42 - M\u0110 103 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm t\u1ea5t c\u1ea3 c\u00e1c gi\u00e1 tr\u1ecb th\u1ef1c c\u1ee7a tham s\u1ed1 m \u0111\u1ec3 b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh log22 x \u2212 2 log2 x + 3m \u2212 2 < 0 c\u00f3 nghi\u1ec7m th\u1ef1c. A m < 1. B 2 C m < 0. D m \u2264 1. m< . 3 \u0253 L\u1eddi gi\u1ea3i. \u0110\u1eb7t t = log2 x. V\u1edbi m\u1ed7i gi\u00e1 tr\u1ecb c\u1ee7a t, lu\u00f4n c\u00f3 m\u1ed9t gi\u00e1 tr\u1ecb x t\u01b0\u01a1ng \u1ee9ng. B\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho tr\u1edf th\u00e0nh t2 \u2212 2t + 3m \u2212 2 < 0; \u2206 = 3 \u2212 3m. V\u00ec h\u1ec7 s\u1ed1 a = 1 > 0, b\u1ea5t ph\u01b0\u01a1ng tr\u00ecnh t2 \u2212 2t + 3m \u2212 2 < 0 c\u00f3 nghi\u1ec7m \u21d4 \u2206 > 0 \u21d4 m < 1. Ch\u1ecdn \u0111\u00e1p \u00e1n A \u0104 C\u00e2u 82 (C\u00e2u 40 - M\u0110 104 - BGD&\u0110T - N\u0103m 2016 - 2017). T\u00ecm t\u1ea5t c\u1ea3 c\u00e1c gi\u00e1 tr\u1ecb th\u1ef1c c\u1ee7a tham s\u1ed1 m \u0111\u1ec3 h\u00e0m s\u1ed1 y = ln(x2 \u2212 2x + m + 1) c\u00f3 t\u1eadp x\u00e1c \u0111\u1ecbnh l\u00e0 R. B 0 < m < 3. A m = 0. C m < \u22121 ho\u1eb7c m > 0. D m > 0. \u0253 L\u1eddi gi\u1ea3i. x2 \u2212 2x + m + 1 > 0 v\u1edbi m\u1ecdi x \u2208 R \u21d0\u21d2 \u2206 = 1 \u2212 m \u2212 1 < 0 \u21d0\u21d2 m > 0. Ch\u1ecdn \u0111\u00e1p \u00e1n D \u0104 C\u00e2u 83 (C\u00e2u 50 - M\u0110 101 - 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 4 log22 x + log2 x \u2212 5 7x \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 \u0111\u00e3 cho c\u00f3 \u0111\u00fang hai nghi\u1ec7m ph\u00e2n bi\u1ec7t? A 49. B 47. C V\u00f4 s\u1ed1. D 48. \u0253 L\u1eddi gi\u1ea3i. \u00aex > 0 \u00aex > 0 \u0110i\u1ec1u ki\u1ec7n 7x \u2212 m \u2265 0 \u21d4 7x \u2265 m. V\u1edbi m nguy\u00ean d\u01b0\u01a1ng ta c\u00f3 \u221a \u00f14\u221alog22 x + log2 \u2212 \uf8eex = 2 7x 7x \u2212 m = 0 4 log22 x + log2 x \u2212 5 \u2212 m = 0 \u21d4 x 5 = 0 \u21d4 \uf8ef 2\u2212 5 \uf8efx 4 \uf8f0 = x = log7 m. \u0110\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 \u0111\u00fang 2 nghi\u1ec7m ph\u00e2n bi\u1ec7t c\u00f3 hai tr\u01b0\u1eddng h\u1ee3p 2 > log7 m \u2265 2\u2212 5 \u21d4 5 \u2264 m < 72. 4 72\u2212 4 Tr\u01b0\u1eddng h\u1ee3p n\u00e0y m \u2208 {3; 4; 5; . . . ; 48}, c\u00f3 46 gi\u00e1 tr\u1ecb nguy\u00ean d\u01b0\u01a1ng c\u1ee7a m. Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 244 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 log7 m = 0 \u21d4 m = 1. Tr\u01b0\u1eddng h\u1ee3p n\u00e0y c\u00f3 1 gi\u00e1 tr\u1ecb c\u1ee7a m th\u1ecfa m\u00e3n. V\u1eady c\u00f3 t\u1ea5t c\u1ea3 47 gi\u00e1 tr\u1ecb c\u1ee7a m th\u1ecfa m\u00e3n y\u00eau c\u1ea7u. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 84 (C\u00e2u 36 - M\u0110 103 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho ph\u01b0\u01a1ng tr\u00ecnh log9 x2 \u2212 log3(5x \u2212 1) = \u2212 log3 m (m l\u00e0 tham s\u1ed1 th\u1ef1c). C\u00f3 t\u1ea5t c\u1ea3 bao nhi\u00eau gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 nghi\u1ec7m? A V\u00f4 s\u1ed1. B 5. C 4. D 6. \u0253 L\u1eddi gi\u1ea3i. \uf8f11 \uf8f2x > \u0110i\u1ec1u ki\u1ec7n: 5 \uf8f3m > 0. X\u00e9t ph\u01b0\u01a1ng tr\u00ecnh: log9 x2 \u2212 log3(5x \u2212 1) = \u2212 log3 m (1). C\u00e1ch 1. 5x \u2212 1 5x \u2212 1 1 x x x (1) \u21d4 log3 x \u2212 log3(5x \u2212 1) = \u2212 log3 m \u21d4 log3 = log3 m \u21d4 = m \u21d4 5 \u2212 = m (2). 1 \u00c51 \u00e3 X\u00e9t f (x) = 5 \u2212 tr\u00ean kho\u1ea3ng ; +\u221e . x5 1 \u00c5 1 \u00e3 \u00c5 1\u00e3 +\u221e 5 C\u00f3 f (x) = x2 > 0, \u2200x \u2208 5 ; v\u00e0 lim f (x) = lim \u2212 x = 5. x\u2192+\u221e x\u2192+\u221e Ta c\u00f3 b\u1ea3ng bi\u1ebfn thi\u00ean c\u1ee7a h\u00e0m s\u1ed1 f (x) x \u22121 +\u221e 5 + y 5 y 0 1 Ph\u01b0\u01a1ng tr\u00ecnh (1) c\u00f3 nghi\u1ec7m khi v\u00e0 ch\u1ec9 ph\u01b0\u01a1ng tr\u00ecnh (2) c\u00f3 nghi\u1ec7m x > . 5 T\u1eeb b\u1ea3ng bi\u1ebfn thi\u00ean suy ra ph\u01b0\u01a1ng tr\u00ecnh (1) c\u00f3 nghi\u1ec7m khi v\u00e0 ch\u1ec9 khi 0 < m < 5. M\u00e0 m \u2208 Z v\u00e0 m > 0 n\u00ean m \u2208 {1; 2; 3; 4}. V\u1eady c\u00f3 4 gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 nghi\u1ec7m. C\u00e1ch 2. \uf8f11 \uf8f2x > V\u1edbi 5 , ta c\u00f3 \uf8f3m > 0 (1) \u21d4 log3 x \u2212 log3(5x \u2212 1) = \u2212 log3 m \u21d4 log3 5x \u2212 1 = log3 m x \u21d4 5x \u2212 1 = m x \u21d4 (5 \u2212 m)x = 1 (2). V\u1edbi m = 5, ph\u01b0\u01a1ng tr\u00ecnh (2) th\u00e0nh 0 \u00b7 x = 1 (v\u00f4 nghi\u1ec7m). V\u1edbi m = 5, (2) \u21d4 x = 5 1 \u2212 m. 1 1 1 m X\u00e9t x > 5 \u21d4 5 \u2212m > 5 \u21d4 5 \u00b7 (5 \u2212 m) > 0 \u21d4 0 < m < 5. M\u00e0 m \u2208 Z v\u00e0 m > 0 n\u00ean m \u2208 {1; 2; 3; 4}. V\u1eady c\u00f3 4 gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 nghi\u1ec7m. Ch\u1ecdn \u0111\u00e1p \u00e1n C Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 245 S\u0110T: 0905.193.688","5. Ph\u01b0\u01a1ng tr\u00ecnh m\u0169. Ph\u01b0\u01a1ng tr\u00ecnh L\u00f4garit \u0104 C\u00e2u 85 (C\u00e2u 36 - M\u0110 104 - BGD&\u0110T - N\u0103m 2018 - 2019). Cho ph\u01b0\u01a1ng tr\u00ecnh log9 x2 \u2212 log3(4x \u2212 1) = \u2212 log3 m (m l\u00e0 tham s\u1ed1 th\u1ef1c). C\u00f3 t\u1ea5t c\u1ea3 bao nhi\u00eau gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a m \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 nghi\u1ec7m? A 5. B 3. C V\u00f4 s\u1ed1. D 4. \u0253 L\u1eddi gi\u1ea3i. \uf8f11 \uf8f2x > \u0110i\u1ec1u ki\u1ec7n: 4 \uf8f3m > 0. \u21d4 log3 x \u2212 log3(4x \u2212 = \u2212 log3 m \u21d4 x1 Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho 1) = . 4x \u2212 1 m x \u22121 1 X\u00e9t h\u00e0m s\u1ed1 f (x) = , ta c\u00f3 f (x) = < 0, \u2200x > . 4x \u2212 1 (4x \u2212 1)2 4 Suy ra b\u1ea3ng bi\u1ebfn thi\u00ean: x 1 +\u221e 4 y\u2212 +\u221e y1 4 Do \u0111\u00f3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 nghi\u1ec7m khi 1 > 1 \u21d4 m < 4. V\u1eady m \u2208 {1, 2, 3}. m4 Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 86 (C\u00e2u 46 - M\u0110 101 - BGD&\u0110T - N\u0103m 2017 - 2018). Cho ph\u01b0\u01a1ng tr\u00ecnh 5x + m = log5(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 (\u221220; 20) \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 cho c\u00f3 nghi\u1ec7m? A 20. B 19. C 9. D 21. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n: x > m. Ta c\u00f3 5x + m = log5(x \u2212 m) \u21d4 5x + x = x \u2212 m + log5(x \u2212 m). (1) X\u00e9t h\u00e0m s\u1ed1 f (t) = 5t + t, f (t) = 5t ln 5 + 1 > 0, \u2200t \u2208 R. Do \u0111\u00f3 t\u1eeb (1) suy ra x = log5(x \u2212 m) \u21d4 m = x \u2212 5x. 1 ln 5 X\u00e9t h\u00e0m s\u1ed1 g(x) = x \u2212 5x, g (x) = 1 \u2212 5x \u00b7 ln 5, g (x) = 0 \u21d4 x = log5 = \u2212 log5 ln 5 = x0. B\u1ea3ng bi\u1ebfn thi\u00ean x \u2212\u221e \u2212 log5 ln 5 +\u221e g (x) +0\u2212 \u2212\u221e g(x) g(x0) \u2212\u221e Do \u0111\u00f3 \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 nghi\u1ec7m th\u00ec m g(x0) \u2248 \u22120, 92. C\u00e1c gi\u00e1 tr\u1ecb nguy\u00ean c\u1ee7a m \u2208 (\u221220; 20) l\u00e0 {\u221219; \u221218; . . . ; \u22121}, c\u00f3 19 gi\u00e1 tr\u1ecb m th\u1ecfa m\u00e3n. Ch\u1ecdn \u0111\u00e1p \u00e1n B Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 246 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 87 (C\u00e2u 50 - \u0110TK - BGD&\u0110T - l\u1ea7n 2 - N\u0103m 2019 - 2020). C\u00f3 bao nhi\u00eau s\u1ed1 nguy\u00ean x sao cho t\u1ed3n t\u1ea1i s\u1ed1 th\u1ef1c y th\u1ecfa m\u00e3n log3(x + y) = log4(x2 + y2)? A 3. B 2. C 1. D V\u00f4 s\u1ed1. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n x + y > 0. \u0110\u1eb7t t = log3(x + y) = log4(x2 + y2) \u21d2 \u00aex + y = 3t x2 + y2 = 4t \u00c1p d\u1ee5ng b\u1ea5t \u0111\u1eb3ng th\u1ee9c Cauchy, ta c\u00f3 9t = (x + y)2 \u2264 2(x2 + y2) = 2 \u00b7 4t \u21d4 \u00c5 9 \u00e3t \u2264 2 \u21d4 t \u2264 log 9 2. 44 Ta l\u1ea1i c\u00f3 x2 + y2 = 4t \u21d2 x2 \u2264 4t \u2264 4log 9 2 \u2248 3,27. Do x nguy\u00ean n\u00ean x \u2208 {\u22121; 0; 1}. 4 \u00aey = 3t \u00aet = 0 x=0\u21d2 y2 = 4t \u21d2 . y=1 \u00aey = 3t \u2212 1 \u00aet = 0 x=1\u21d2 y2 = 4t \u2212 1 \u21d2 . y=0 x = \u22121 \u21d2 \u00aey = 3t +1 \u2265 1 \u21d2 \u00aet \u2265 0 +1 \u2265 2 \u21d2 x2 + y2 \u2265 5. (lo\u1ea1i) y2 + 1 = 4t y = 3t (v\u00ec m\u00e2u thu\u1eabn v\u1edbi x2 + y2 \u2264 4log 9 2 \u2248 3,27) 4 V\u1eady x \u2208 {\u22121; 0}. Ch\u1ecdn \u0111\u00e1p \u00e1n B \u0104 C\u00e2u 88 (C\u00e2u 45 - \u0110TK - BGD&\u0110T - N\u0103m 2016 - 2017). H\u1ecfi c\u00f3 bao nhi\u00eau gi\u00e1 tr\u1ecb m nguy\u00ean trong [\u22122017; 2017] \u0111\u1ec3 ph\u01b0\u01a1ng tr\u00ecnh log(mx) = 2 log(x + 1) c\u00f3 nghi\u1ec7m duy nh\u1ea5t? A 2017. B 4014. C 2018. D 4015. \u0253 L\u1eddi gi\u1ea3i. \u0110i\u1ec1u ki\u1ec7n: x > \u22121 v\u00e0 x = 0. log(mx) = 2 log(x + 1) \u21d4 mx = (x + 1)2 \u21d4 m = (x + 1)2 \u00f1x = 1 x (x + 1)2 x2 \u2212 1 X\u00e9t h\u00e0m s\u1ed1 f (x) = x (x > \u22121, x = 0); f (x) = x2 = 0 \u21d4 x = \u22121( lo\u1ea1i) L\u1eadp b\u1ea3ng bi\u1ebfn thi\u00ean: x \u22121 0 1 +\u221e y\u2212 \u22120+ 0 +\u221e +\u221e y \u2212\u221e 4 Th.S Nguy\u1ec5n Ho\u00e0ng Vi\u1ec7t 247 S\u0110T: 0905.193.688"]
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