De thi thu thpt quoc gia toan quang tri 1

❙Ð ●❉ ❱⑨ ✣❚ ◗❯❷◆● ❚❘➚ ❑➐ ❚❍■ ❚❍Û ❚❍P❚ ◗❯➮❈ ●■❆ ▲❺◆ ✶ ◆❿▼ ✷✵✶✾ ▼➷◆ ❚❖⑩◆ ❚❘×❮◆● ❚❍P❚ ❈❍❯❨➊◆ ▲➊ ◗❯Þ ✣➷◆ ❚❤í✐ ❣✐❛♥ ❧➔♠ ❜➔✐ ✾✵ ♣❤ót✱ ❦❤ỉ♥❣ ❦➸ t❤í✐ ❣✐❛♥ ❣✐❛♦ ✤➲ ✭ ✣➲ t❤✐ ❝â ✻ tr❛♥❣ ✮ ▼➣ ✤➲ t❤✐ ✶✵✶ ❈➙✉ ✶✳ ❑❤è✐ ❝❤â♣ S.ABCD ❝â ✤→② ABCD ❧➔ ❤➻♥❤ ✈✉æ♥❣ ❝↕♥❤ 3a✱ SA = a, SA ⊥ (ABCD)✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ ❝❤â♣ S.ABCD✳ ❆✳ 6a3 ✳ ❇✳ 9a3 ✳ ❈✳ 3a3 ✳ ❉✳ a ✳ ❈➙✉ ✷✳ ❈❤♦ ❤➔♠ số y = f (x) õ ỗ t ữ ữợ ú y ❆✳ ●✐→ trà ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ sè ❜➡♥❣ −1✳ ❇✳ ✣✐➸♠ ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ sè ❧➔ −1✳ ❈✳ ✣✐➸♠ ❝ü❝ ✤↕✐ ❝õ❛ ❤➔♠ sè ❧➔ 3✳ O x ❉✳ ●✐→ trà ❝ü❝ ✤↕✐ ❝õ❛ ❤➔♠ sè ❧➔ 0✳ −1 ❈➙✉ ✸✳ ❈❤♦ sè ♣❤ù❝ z = (1 − 2i)2 ✳ ❚➼♥❤ ♠æ ✤✉♥ ❝õ❛ sè ♣❤ù❝ ✳ ❆✳ ✳ ❇✳ √ ✳ ❈✳ 25 z ✳ ❉✳ √ ✳ ❈➙✉ ✹✳ ❚➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ log3 (x − 2) = 2✳ ❆✳ x = 11✳ ❇✳ x = 8✳ ❈✳ x = 9✳ ❉✳ x = 10✳ ❈➙✉ ✺✳ ❚➼♥❤ ❞✐➺♥ t➼❝❤ ❝õ❛ ♠➦t ❝➛✉ ❝â ❜→♥ ❦➼♥❤ ❜➡♥❣ 3✳ ❆✳ 9π✳ ❇✳ 18π✳ ❈✳ 12π✳ ❉✳ 36π✳ ❈➙✉ ✻✳ ❍➔♠ sè y = −x3 + 3x2 − ỗ tr t ủ tr t ủ ữủ ữợ (2; +) (0; 2)✳ ❈➙✉ ✼✳ ❚➼♥❤ t➼❝❤ ♣❤➙♥ I = ❆✳ I = + ln 2✳ Z2 ❈✳ (−∞; 0) ∪ (2; +∞)✳ ❉✳ (−∞; 0)✳ x−1 dx x ✳ ❇✳ I = ✳ ❈✳ I = ln 2✳ ❉✳ I = − ln 2✳ ❈➙✉ ✽✳ ❑❤è✐ ♥â♥ (N ) ❝â ❜→♥ ❦➼♥❤ ✤→② ❜➡♥❣ ✈➔ ❞✐➺♥ t➼❝❤ ①✉♥❣ q✉❛♥❤ ❜➡♥❣ 15π✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ ♥â♥ (N )✳ ❆✳ 12π✳ ❇✳ 16π✳ v ❈➙✉ ✾✳ ❆✳ P ❈✳ 45π✳ ❉✳ 36π✳ u s r u 3 ❈❤♦ ❜✐➸✉ t❤ù❝ P = t ✳ ▼➺♥❤ ✤➲ ♥➔♦ tr♦♥❣ ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ ✤ó♥❣❄ =  18 ✳ ❇✳ P 3 =   12 3 ✳ ❈✳ P =   18 ✳ ❉✳ P =   181 ✳ ❚r❛♥❣ ✶✴✻ ▼➣ ✤➲ ✶✵✶ (2 − 3i)(4 − i) ❈❤♦ sè ♣❤ù❝ z = + 2i ♠➦t ♣❤➥♥❣ Oxy ✳ ❆✳ (1; 4)✳ ❇✳ (1; −4)✳ ❈➙✉ ✶✵✳ ✳ ❚➻♠ tå❛ ✤ë ✤✐➸♠ ❜✐➸✉ ❞✐➵♥ ❝õ❛ sè ♣❤ù❝ z tr➯♥ ❈✳ (−1; −4)✳ ❉✳ (−1; 4)✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈ỵ✐ ❤➺ tå❛ ✤ë Oxyz ✱ ❝❤♦ ♠➦t ♣❤➥♥❣ (P )✿ 2x − 2y + z + 2017 = tỡ tr tỡ ữủ ữợ ❧➔ ♠ët ✈➨❝✲tì ♣❤→♣ t✉②➳♥ ❝õ❛ (P )❄ ★✔ ★✔ ★✔ ★✔ ❆✳ n = (4; −4; 2)✳ ❇✳ n = (1; −2; 2)✳ ❈✳ n = (1; −1; 4)✳ ❉✳ n = (−2; 2; 1)✳ ❈➙✉ ✶✶✳ ❈❤♦ ❦❤è✐ ❧➟♣ ♣❤÷ì♥❣ ABCD.A′ B ′ C ′ D′ ❝â ✤ë ❞➔✐ ❝↕♥❤ ❧➔ 3❝♠✳ ❚➼♥❤ t❤➸ t➼❝❤ ❝õ❛ ❦❤è✐ tù ❞✐➺♥ ACB ′ D′ ✳ √ 3 3 ❆✳ 18 2❝♠ ✳ ❇✳ 3❝♠ ✳ ❈✳ 9❝♠ ✳ ❉✳ 18❝♠ ✳ ❈➙✉ ✶✷✳ ❚r♦♥❣ ♠➦t ♣❤➥♥❣ tå❛ ✤ë Oxy ✱ t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ❜✐➸✉ ❞✐➵♥ ❝→❝ sè ♣❤ù❝ z t❤ä❛ ♠➣♥ |z − + 2i| = |z + + 2i| ❧➔ ✤÷í♥❣ t❤➥♥❣ ❝â ♣❤÷ì♥❣ tr➻♥❤ ❆✳ x + 2y = 0✳ ❇✳ x − 2y = 0✳ ❈✳ x − 2y + = 0✳ ❉✳ x + 2y + = 0✳ ❈➙✉ ✶✸✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ trö ❝â ❜→♥ ❦➼♥❤ R = 3, ❝❤✐➲✉ ❝❛♦ h = ❆✳ V = 90π ✳ ❇✳ V = 45π ✳ ❈✳ V = 15π ✳ ❉✳ V = 45✳ ❈➙✉ ✶✹✳ ❈➙✉ ✶✺✳ ❆✳ 2✳ ❈➙✉ ✶✻✳ ❆✳ x2 + x − số ữớ t ỗ t sè y = x − 3x + ❇✳ 3✳ ❈✳ 1✳ ❉✳ 0✳ ❚➻♠ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ sè f (x)= ln(1 − 2x) + C ✳ ❇✳ ln |2x − 1| + C ✳
tr➯♥ −∞; 21 ✳ − 2x 1 ❈✳ ln |2x − 1| + C ✳ ❉✳ − ln |2x − 1| + C ✳ 2 ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ❤➺ tå❛ ✤ë Oxyz ✱ ❝❤♦ ♠➦t ♣❤➥♥❣ (P ) : 2x − 2y + z + = 0✳ ❚➼♥❤ ❦❤♦↔♥❣ ❝→❝❤ d tø ✤✐➸♠ M (1; 2; 1) ✤➳♥ ♠➦t ♣❤➥♥❣ (P )✳ ❈✳ d = 3✳ ❉✳ d = 4✳ ❆✳ d = 1✳ ❇✳ d = ✳ ❈➙✉ ✶✼✳ ❈❤♦ ❤➻♥❤ ❝❤â♣ S.ABC ❝â t❤➸ t➼❝❤ ❜➡♥❣ 1✳ ❚r➯♥ ❝↕♥❤ BC ❧➜② ✤✐➸♠ E s❛♦ ❝❤♦ BE = 2EC ✳ ❚➼♥❤ t❤➸ t➼❝❤ V ❝õ❛ ❦❤è✐ tù ❞✐➺♥ SAEB ✳ 1 ❇✳ V = ✳ ❈✳ V = ✳ ❉✳ V = ✳ ❆✳ V = ✳ ❈➙✉ ✶✽✳ ❈➙✉ ✶✾✳ ❆✳ y′ = ❚➼♥❤ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ sè y = log9 2x ln ✳ x2 + ❇✳ y′ = ln ✳ x2 + x2 +1 ✳ ❈✳
y′ = (x2 x ✳ + 1) ln ❉✳ y′ = (x2 ✳ + 1) ln ●å✐ z1 ✱ z2 ❧➔ ❤❛✐ ♥❣❤✐➺♠ ♣❤ù❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ z − 4z + = 0✳ ❚➼♥❤ w = 1 + + i(z1 z2 + z2 z1 )✳ ❈➙✉ ✷✵✳ z1 z2 w = − + 20i✳ ❉✳ w = + 20i✳ ❈➙✉ ✷✶✳ ❚➼♥❤ tê♥❣ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ log2 (x + 3) = log2 x + + x2 − √ x − + x + 3✳ √ ❆✳ S = 2✳ ❇✳ S = 1✳ ❈✳ S = −1✳ ❉✳ S = − 2✳ ❆✳ w = 20 + i✳ ❇✳ w= + 20i✳ ❈✳ ❚r❛♥❣ ✷✴✻ ▼➣ ✤➲ ✶✵✶ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Oxyz✱ ❝❤♦ ♠➦t ❝➛✉ (S) : x2 + y2 + z2 − 8x + 10y − 6z + 49 = 0✳ ❚➼♥❤ ❜→♥ ❦➼♥❤ R ❝õ❛ ♠➦t ❝➛✉ (S)✳ √ √ ❆✳ R = 151✳ ❇✳ R = 99✳ ❈✳ R = 1✳ ❉✳ R = 7✳ ❈➙✉ ✷✸✳ ❇✐➳t r➡♥❣ ❤➔♠ sè F (x) = mx + (3m + n)x − 4x + ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ sè f (x) = 3x2 + 10x − 4✳ ❚➼♥❤ mn✳ ❆✳ mn = 1✳ ❇✳ mn = 3✳ ❈✳ mn = 2✳ ❉✳ mn = 0✳ ❈➙✉ ✷✷✳ ❈➙✉ ✷✹✳ ❚➼❝❤ ♣❤➙♥ I = Z1 (x − 1)2 dx = a − ln b x2 + ✱ tr♦♥❣ ✤â a; b ❧➔ ❝→❝ sè ♥❣✉②➯♥✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ a + b✳ ❆✳ 0✳ ❇✳ −1✳ ❈✳ 3✳ ❉✳ 1✳ ❈➙✉ ✷✺✳ ❑❤è✐ ❝❤â♣ t❛♠ ❣✐→❝ ✤➲✉ ❝â ♥❤✐➲✉ ♥❤➜t ❜❛♦ ♥❤✐➯✉ ♠➦t ♣❤➥♥❣ ✤è✐ ①ù♥❣❄ ❆✳ 6✳ ❇✳ 9✳ ❈✳ 3✳ ❉✳ 4✳ x+2−m ♥❣❤à❝❤ ❜✐➳♥ ❈➙✉ ✷✻✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❣✐→ trà t❤ü❝ ❝õ❛ t❤❛♠ sè m ✤➸ ❤➔♠ sè y = x+1 tr➯♥ ♠é✐ ❦❤♦↔♥❣ ①→❝ ✤à♥❤ ❝õ❛ ♥â✳ ❆✳ m ≤ −3✳ ❇✳ m < −3✳ ❈✳ m < 1✳ ❉✳ m ≤ 1✳ x ✱ y = 0✱ x = 1✱ x = 4✳ ❚➼♥❤ t❤➸ ❈➙✉ ✷✼✳ ●å✐ (D) ợ ữớ y = t➼❝❤ ✈➟t t❤➸ trá♥ ①♦❛② t↕♦ t❤➔♥❤ ❦❤✐ q✉❛② ❤➻♥❤ (D) q✉❛♥❤ trö❝ Ox✳ 21 21π 15π 15 ❆✳ ✳ ❇✳ ✳ ❈✳ ✳ ❉✳ ✳ 16 16 16 ❈➙✉ ✷✽✳ ❈❤♦ sè ♣❤ù❝ z t❤ä❛ |z − + 2i| = 3✳ ❇✐➳t r➡♥❣ t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ❜✐➵✉ ❞✐➵♥ ❝õ❛ sè ♣❤ù❝ w = 2z + i tr➯♥ ♠➦t ♣❤➥♥❣ (Oxy) ❧➔ ♠ët ✤÷í♥❣ trá♥✳ ❚➻♠ t➙♠ ❝õ❛ ✤÷í♥❣ trá♥ ✤â✳ ❆✳ I(0; 1)✳ ❇✳ I(1; 0)✳ ❈✳ I(1; 1)✳ ❉✳ I(2; −3)✳ ✈➔ ❜✐➸✉ t❤ù❝ P = + ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t✳ ❈➙✉ ✷✾✳ ❈❤♦ x, y > t❤ä❛ ♠➣♥ x + y = x 4y ❚➼♥❤ x2 + y2✳ 2313 25 153 ✳ ❇✳ ✳ ❈✳ ✳ ❉✳ ✳ ❆✳ 100 1156 16 √ a2 ❜➡♥❣ ❈➙✉ ✸✵✳ ❈❤♦ sè t❤ü❝ a > 0, a 6= 1✳ ●✐→ trà log√ a ❆✳ 1✳ ❇✳ ✳ ❈✳ ✳ ❉✳ ✳ 3 M (a; b) tr ỗ t ❝õ❛ ❤➔♠ sè y = x −x s❛♦ ❝❤♦ ❦❤♦↔♥❣ ❝→❝❤ tø M ✤➳♥ ✤÷í♥❣ t❤➥♥❣ d : y = 2x + ♥❤ä ♥❤➜t✳ ❚➼♥❤ (4a + 5)2 + (2b − 7)2✳ ❆✳ 2✳ ❇✳ 0✳ ❈✳ 18✳ ❉✳ 162✳ ❈➙✉ ✸✷✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Oxyz ✱ ❝❤♦ ♠➦t ♣❤➥♥❣ (P ) : x − y + = ✈➔ ❤❛✐ ✤✐➸♠ A(1; 2; 3), B(1; 0; 1)✳ ✣✐➸♠ C(a; b; −2) ∈ (P ) s❛♦ ❝❤♦ t❛♠ ❣✐→❝ ABC ❝â ❞✐➺♥ t➼❝❤ ♥❤ä ♥❤➜t✳ ❚➼♥❤ a + b✳ ❆✳ 2✳ ❇✳ 0✳ ❈✳ 1✳ ❉✳ −3✳ ❈➙✉ ✸✶✳ ❚r❛♥❣ ✸✴✻ ▼➣ ✤➲ ✶✵✶ ❈❤♦ ❤➻♥❤ ♣❤➥♥❣ (D) ữủ ợ ữớ y = 2(x2 − 1); y = − x2✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ trá♥ ①♦❛② t↕♦ t❤➔♥❤ ❞♦ (D) q✉❛② q✉❛♥❤ trö❝ Ox✳ 32 64π 64 32π ❆✳ ✳ ❇✳ ✳ ❈✳ ✳ ❉✳ ✳ 15 15 15 15 ′ ❈➙✉ ✸✹✳ ❈❤♦ ❤➔♠ sè f (x) ❝â ✤↕♦ ❤➔♠ f (x) = (x − 1)(x − 3)(x − 1) ✈ỵ✐ ♠å✐ x t❤✉ë❝ R✳ ❙♦ s→♥❤ f (−2), f (0), f (2)✱ t❛ ✤÷đ❝ ❆✳ f (−2) < f (2) < f (0)✳ ❇✳ f (−2) < f (0) < f (2)✳ ❈✳ f (2) < f (0) < f (−2)✳ ❉✳ f (0) < f (−2) < f (2)✳ √ √ √ √ 2, |w − 2i| = 2✳ ❇✐➳t r➡♥❣ |z − w| ❈➙✉ ✸✺✳ ❈❤♦ ❤❛✐ sè ♣❤ù❝ z, w t❤ä❛ ♠➣♥ |z − 2| = ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t ❦❤✐ z = zo, w = wo✳ ❚➼♥❤ |3zo − wo|✳ √ √ √ ❆✳ 2✳ ❇✳ 2✳ ❈✳ 2✳ ❉✳ 1✳ ❈➙✉ ✸✻✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Oxyz ✱ ❝❤♦ ♠➦t ♣❤➥♥❣ (P ) : x + y + z − = ✈➔ ❜❛ ✤✐➸♠ ✤✐➸♠ ★✥✥✥✥✥✔ ★✥✥✥✥✥✥✔ ★✥✥✥✥✥✔ A(3; 1; 1), B(7; 3; 9) ✈➔ C(2; 2; 2)✳ ✣✐➸♠ M (a; b; c) tr➯♥ (P ) s❛♦ ❝❤♦ |M A + 2M B + 3M C| ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t✳ ❚➼♥❤ 2a − 15b + c✳ ❆✳ 8✳ ❇✳ 1✳ ❈✳ 3✳ ❉✳ 6✳ ❈➙✉ ✸✼✳ ❈❤♦ ❤➻♥❤ ❝❤â♣ S.ABCD ❝â ✤→② ABCD ❧➔ ❤➻♥❤ ✈✉æ♥❣✱ t➙♠ O ✱ ❝↕♥❤ a ✈➔ SO ⊥ √ (ABCD), SA = 2a 2✳ ●å✐ M, N ❧➛♥ ❧÷đt ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ SA, BC ✳ ❚➼♥❤ ❣â❝ ❣✐ú❛ ✤÷í♥❣ t❤➥♥❣ M N ✈➔ ♠➦t ♣❤➥♥❣ (ABCD)✳ π π π ✳ ❇✳ ✳ ❈✳ arctan 2✳ ❉✳ ✳ ❆✳ ❈➙✉ ✸✽✳ ❚➼♥❤ sè ❣✐→ trà ♥❣✉②➯♥ ❝õ❛ t❤❛♠ sè m tr➯♥ ❦❤♦↔♥❣ (−2019; 2019) ✤➸ ❤➔♠ sè y = x4 − 2mx2 − 3m + ỗ tr (1; 2) 2020 ❈✳ 1✳ ❉✳ 2019✳ ❈➙✉ ✸✾✳ ❚➼♥❤ tê♥❣ t➜t ❝↔ tr t số m tỗ t t ởt số ự z tọ ỗ tớ |z| = m ✈➔ |z − 4m + 3mi| = m2✳ ❆✳ 10✳ ❇✳ 9✳ ❈✳ 4✳ ❉✳ 6✳ ❈➙✉ ởt ỏ t ỗ t ố ♥❤❛✉✳ ❍ä✐ ❝â ❜❛♦ ♥❤✐➯✉ ❝→❝❤ ❝➢t ❝❤✐➳❝ ✈á♥❣ ✤â t❤➔♥❤ ✷ ♣❤➛♥ ♠➔ sè ❤↕t ð ♠é✐ ♣❤➛♥ ✤➲✉ ❧➔ sè ❧➫ ❄ ❆✳ 5✳ ❇✳ 180✳ ❈✳ 10✳ ❉✳ 90✳ ❈➙✉ ✸✸✳ ❈➙✉ ✹✶✳ ❈❤♦ ❤➔♠ sè f (x) õ f (x) ỗ t ❤➔♠ sè y = f ′(x) ♥❤÷ ❤➻♥❤ ✈➩ ❜➯♥✳ ❚➼♥❤ sè ✤✐➸♠ ❝ü❝ trà ❝õ❛ √ √ ❤➔♠ sè y = f (x2) tr➯♥ ❦❤♦↔♥❣ (− 5; 5)✳ ❆✳ 2✳ ❇✳ 5✳ ❈✳ 4✳ ❉✳ 3✳ ② y = f ′ (x) ① ❚r❛♥❣ ✹✴✻ ▼➣ ✤➲ ✶✵✶ ❈❤♦ ❤➻♥❤ ❝❤â♣ S.ABCD ❝â SA ⊥ (ABCD)✱ ✤→② ABCD ❧➔ ❤➻♥❤ ❝❤ú ♥❤➟t ✈ỵ✐ √ √ AC = a√ ✈➔ BC = a 2✳ ❚➼♥❤ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ SD ✈➔ BC ✳ √ 2a a 3a ✳ ❇✳ a 3✳ ✳ ❉✳ ✳ ❆✳ ❈✳ ❈➙✉ ✹✷✳ ❈➙✉ ✹✸✳ ◆❣÷í✐ t❛ ❧➔♠ t t ỡ t ữ ợ ❧➔ ❤❛✐ ❦❤è✐ trư ❜➡♥❣ ♥❤❛✉ ✈➔ t❛② ❝➛♠ ❝ơ♥❣ ❧➔ ❦❤è✐ trö✳ ❇✐➳t ❤❛✐ ✤➛✉ ❧➔ ❤❛✐ ❦❤è✐ trö ✤÷í♥❣ ❦➼♥❤ ✤→② ❜➡♥❣ ✶✷✱ ❝❤✐➲✉ ❝❛♦ ❜➡♥❣ ✻✱ ❝❤✐➲✉ ❞➔✐ t↕ ❜➡♥❣ ✸✵ ✈➔ ❜→♥ ❦➼♥❤ t❛② ❝➛♠ ❜➡♥❣ ✷✳ ❍➣② t➼♥❤ t❤➸ t➼❝❤ ✈➟t ❧✐➺✉ ❧➔♠ ♥➯♥ t↕ t❛② ✤â✳ ❆✳ 108π ✳ ❇✳ 504π ✳ ❈✳ 6480π ✳ ❉✳ 502π ✳ ❈➙✉ ✹✹✳ ❙➠♠ ❧è♣ ①❡ æ tỉ ❦❤✐ ❜ì♠ ❝➠♥❣ ✤➦t ♥➡♠ tr➯♥ ♠➦t ♣❤➥♥❣ ♥➡♠ õ ữ ợ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥❤ä R1 = 20cm✱ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ❧ỵ♥ R2 = 30cm ✈➔ ♠➦t ❝➢t ❦❤✐ ❝➢t ❜ð✐ ♠➦t ♣❤➥♥❣ ✤✐ q✉❛ trư❝✱ ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐ ♠➠t ♣❤➥♥❣ ♥➡♠ ♥❣❛♥❣ ❧➔ ❤❛✐ ✤÷í♥❣ trá♥✳ ❇ä q✉❛ ✤ë ❞➔② ❝õ❛ ✈ä s➠♠✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤ỉ♥❣ ❦❤➼ ✤÷đ❝ ❝❤ù❛ ❜➯♥ tr♦♥❣ s➠♠✳ 3 3 ❆✳ 1400πcm ✳ ❇✳ 1250πcm ✳ ❈✳ 2500πcm ✳ ❉✳ 600πcm ✳ ❈❤♦ ❤➔♠ sè f (x) ①→❝ ✤à♥❤ tr➯♥ R✱ ❝â ✤↕♦ ❤➔♠ f ′ (x) = (x + 1)3 (x − 2)5 (x + 3)3 ✳ ❙è ✤✐➸♠ ❝ü❝ trà ❝õ❛ ❤➔♠ sè f (|x|) ❧➔ ❆✳ 3✳ ❇✳ 1✳ ❈✳ 2✳ ❉✳ 5✳ ❈➙✉ ✹✺✳ ✳ ❇✐➳t F ❈➙✉ ✹✻✳ ❈❤♦ F (x) ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ sè f (x) = cos2 x ♠å✐ k ∈ Z✳ ❚➼♥❤ F (0) + F (π) + F (2π) + + F (10π)✳ ❆✳ 45✳ ❇✳ 0✳ ❈✳ 55✳ ❉✳ π  + kπ = k ✈ỵ✐ 44 ởt ữớ ỷ số t 100 tr ỗ ♥❣➙♥ ❤➔♥❣ ✈ỵ✐ ❧➣✐ s✉➜t 0, 5%/t❤→♥❣ ✈➔ ỉ♥❣ t❛ rút ộ t ởt tr ỗ tứ s❛✉ ♥❣➔② ❣û✐ ♠ët t❤→♥❣ ❝❤♦ ✤➳♥ ❦❤✐ ❤➳t t✐➲♥ ✭t❤→♥❣ ❝✉è✐ ❝ị♥❣ ❝â t❤➸ ❦❤ỉ♥❣ ❝á♥ ✤õ ♠ët tr✐➺✉ ỗ ọ ổ t rút t t s t❤→♥❣❄ ❆✳ 100✳ ❇✳ 140✳ ❈✳ 138✳ ❉✳ 139✳ ❈➙✉ ✹✼✳ ❈❤♦ ❤➻♥❤ ❝❤â♣ S.ABCD ❝â ✤→② ❧➔ ❤➻♥❤ ❜➻♥❤ ❤➔♥❤ ✈➔ ❝â t❤➸ t➼❝❤ ❜➡♥❣ 48✳ ❚r➯♥ ❝↕♥❤ SB, SD ❧➜② ✤✐➸♠ ❝→❝ M, N s❛♦ ❝❤♦ SM = M B, SD = 3SN ✳ ▼➦t ♣❤➥♥❣ (AM N ) ❝➢t SC t↕✐ P ✳ ❚➼♥❤ t❤➸ t➼❝❤ V ❝õ❛ ❦❤è✐ tù ❞✐➺♥ SM N P ✳ ❈➙✉ ✹✽✳ ❆✳ V = ✳ ❇✳ V = ✳ ❈✳ V = 2✳ ❚➼♥❤ sè ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ cotx = ❆✳ 2019✳ ❇✳ 2018✳ ❈✳ 1✳ ❈➙✉ ✹✾✳ 2x tr♦♥❣ ❦❤♦↔♥❣ ❉✳ V = 1✳  11π ; 2019π 12 ❉✳ 2020✳  ❚r❛♥❣ ✺✴✻ ▼➣ ✤➲ ✶✵✶ ❈➙✉ ✺✵✳ ❈❤♦ ❤➔♠ sè f (x) ❝â ✤↕♦ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ R ✈➔ t❤ä❛ ♠➣♥ Z1 f (x) dx = 1✱ f (1) = cot 1✳ ❚➼♥❤ t➼❝❤ ♣❤➙♥ I = ❆✳ − ln(cos 1)✳ Z1 ❇✳ 0✳ f (x) tan2 x + f ′ (x) tan x dx✳ ❈✳
−1✳ ❉✳ − cot 1✳ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❍➌❚✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❚r❛♥❣ ✻✴✻ ▼➣ ✤➲ ✶✵✶ ❙Ð ●❉ ❱⑨ ✣❚ ◗❯❷◆● ❚❘➚ ❑➐ ❚❍■ ❚❍Û ❚❍P❚ ◗❯➮❈ ●■❆ ▲❺◆ ✶ ◆❿▼ ✷✵✶✾ ▼➷◆ ❚❖⑩◆ ❚❘×❮◆● ❚❍P❚ ❈❍❯❨➊◆ ▲➊ ◗❯Þ ✣➷◆ ❚❤í✐ ❣✐❛♥ ❧➔♠ ❜➔✐ ✾✵ ♣❤ót✱ ❦❤ỉ♥❣ ❦➸ t❤í✐ ❣✐❛♥ ❣✐❛♦ ✤➲ ✭ ✣➲ t❤✐ ❝â ✻ tr❛♥❣ ✮ ▼➣ ✤➲ t❤✐ ✶✵✷ (2 − 3i)(4 − i) ✳ ❚➻♠ tå❛ ✤ë ✤✐➸♠ ❜✐➸✉ ❞✐➵♥ ❝õ❛ sè ♣❤ù❝ z tr➯♥ + 2i ❈➙✉ ✶✳ ❈❤♦ sè ♣❤ù❝ z = ♠➦t ♣❤➥♥❣ Oxy ✳ ❆✳ (1; 4)✳ ❇✳ (−1; −4)✳ ❈✳ (−1; 4)✳ ❉✳ (1; −4)✳ số ữớ t ỗ t ❤➔♠ sè y = x2 + x − ✳ x − 3x + ❆✳ 3✳ ❇✳ 1✳ ❈✳ 0✳ ❉✳ 2✳ ❈➙✉ ✸✳ ❆✳ P v s u r u 3 t ❈❤♦ ❜✐➸✉ t❤ù❝ P = ✳ ▼➺♥❤ ✤➲ ♥➔♦ tr♦♥❣ ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ ✤ó♥❣❄ = 2 ✳ ❇✳ P = 2 3 ✳ ❈✳ P =   181 ❉✳ P ✳ =  18 ✳ ❈➙✉ ✹✳ số y = x3 + 3x2 ỗ ❜✐➳♥ tr➯♥ t➟♣ ❤đ♣ ♥➔♦ tr♦♥❣ ❝→❝ t➟♣ ❤đ♣ ✤÷đ❝ ữợ (2; +) (; 0) (−∞; 0) ∪ (2; +∞)✳ ❉✳ (0; 2)✳
tr➯♥ −∞; 12 ✳ − 2x 1 − ln |2x − 1| + C ✳ ❈✳ ln |2x − 1| + C ✳ 2 ❈➙✉ ✺✳ ❚➻♠ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ sè f (x) = ❆✳ ln |2x − 1| + C ✳ ❇✳ ❉✳ ln(1 − 2x) + C ✳ ❈➙✉ ✻✳ ❑❤è✐ ♥â♥ (N ) ❝â ❜→♥ ❦➼♥❤ ✤→② ❜➡♥❣ ✈➔ ❞✐➺♥ t➼❝❤ ①✉♥❣ q✉❛♥❤ ❜➡♥❣ 15π✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ ♥â♥ (N )✳ ❆✳ 36π✳ ❇✳ 12π✳ ❈✳ 45π✳ ❉✳ 16π✳ ❈➙✉ ✼✳ ❚➼♥❤ ❞✐➺♥ t➼❝❤ ❝õ❛ ♠➦t ❝➛✉ ❝â ❜→♥ ❦➼♥❤ ❜➡♥❣ 3✳ ❆✳ 36π✳ ❇✳ 9π✳ ❈✳ 12π✳ ❉✳ 18π✳ ❈➙✉ ✽✳ ❈❤♦ sè ♣❤ù❝ z = (1 − 2i)2 ✳ ❚➼♥❤ ♠æ ✤✉♥ ❝õ❛ sè ♣❤ù❝ ✳ ❆✳ ✳ 25 ❇✳ √ ✳ ❈✳ √ z 5✳ ❉✳ ✳ ❈➙✉ ✾✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ trö ❝â ❜→♥ ❦➼♥❤ R = 3, ❝❤✐➲✉ ❝❛♦ h = ❆✳ V = 15π✳ ❇✳ V = 45π✳ ❈✳ V = 90π✳ ❉✳ V = 45✳ ❈➙✉ ✶✵✳ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ❤➺ tå❛ ✤ë Oxyz ✱ ❝❤♦ ♠➦t ♣❤➥♥❣ (P )✿ 2x − 2y + z + 2017 = 0✱ ✈➨❝✲tì ♥➔♦ tr♦♥❣ ❝→❝ tỡ ữủ ữợ ởt tỡ t ❝õ❛ (P )❄ ❆✳ ★✔ n = (−2; 2; 1)✳ ❇✳ ★✔ n = (4; −4; 2)✳ ❈✳ ★✔ n = (1; −1; 4)✳ ❉✳ ★✔ n = (1; −2; 2)✳ ❈➙✉ ✶✶✳ ❚➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ log3 (x − 2) = 2✳ ❆✳ x = 10✳ ❇✳ x = 8✳ ❈✳ x = 9✳ ❉✳ x = 11✳ ❚r❛♥❣ ✶✴✻ ▼➣ ✤➲ ✶✵✷ ❈➙✉ ✶✷✳ ❚➼♥❤ t➼❝❤ ♣❤➙♥ I = Z2 x−1 dx x ✳ ✳ ❇✳ I = + ln 2✳ ❈✳ I = ✳ ❉✳ I = ln 2✳ ❈➙✉ ✶✸✳ ❈❤♦ ❤➔♠ sè ❜➟❝ ❜❛ y = f (x) ❝â ỗ t ữ ữợ ✤ó♥❣❄ y ❆✳ ✣✐➸♠ ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ sè ❧➔ −1✳ ❇✳ ●✐→ trà ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ sè ❜➡♥❣ −1✳ ❈✳ ●✐→ trà ❝ü❝ ✤↕✐ ❝õ❛ ❤➔♠ sè ❧➔ 0✳ O x ❉✳ ✣✐➸♠ ❝ü❝ ✤↕✐ ❝õ❛ ❤➔♠ sè ❧➔ 3✳ −1 ❆✳ I = − ln ❑❤è✐ ❝❤â♣ S.ABCD ❝â ✤→② ABCD ❧➔ ❤➻♥❤ ✈✉æ♥❣ ❝↕♥❤ 3a✱ SA = a, SA ⊥ (ABCD)✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ ❝❤â♣ S.ABCD✳ a3 3 ❆✳ 6a ✳ ❇✳ 9a ✳ ❈✳ 3a ✳ ❉✳ ✳ ′ ′ ′ ′ ❈➙✉ ✶✺✳ ❈❤♦ ❦❤è✐ ❧➟♣ ♣❤÷ì♥❣ ABCD.A B C D ❝â ✤ë ❞➔✐ ❝↕♥❤ ❧➔ 3❝♠✳ ❚➼♥❤ t❤➸ t➼❝❤ ❝õ❛ ❦❤è✐ tù ❞✐➺♥ ACB ′D′✳ √ 3 3 ❆✳ 3❝♠ ✳ ❇✳ 9❝♠ ✳ ❈✳ 18❝♠ ✳ ❉✳ 18 2❝♠ ✳ ❈➙✉ ✶✻✳ ❚r♦♥❣ ♠➦t ♣❤➥♥❣ tå❛ ✤ë Oxy ✱ t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ❜✐➸✉ ❞✐➵♥ ❝→❝ sè ♣❤ù❝ z t❤ä❛ ♠➣♥ |z − + 2i| = |z + + 2i| ❧➔ ✤÷í♥❣ t❤➥♥❣ ❝â ♣❤÷ì♥❣ tr➻♥❤ ❆✳ x + 2y = 0✳ ❇✳ x − 2y + = 0✳ ❈✳ x − 2y = 0✳ ❉✳ x + 2y + = 0✳ ❈➙✉ ✶✼✳ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ❤➺ tå❛ ✤ë Oxyz ✱ ❝❤♦ ♠➦t ♣❤➥♥❣ (P ) : 2x − 2y + z + = 0✳ ❚➼♥❤ ❦❤♦↔♥❣ ❝→❝❤ d tø ✤✐➸♠ M (1; 2; 1) ✤➳♥ ♠➦t ♣❤➥♥❣ (P )✳ ❇✳ d = 3✳ ❈✳ d = 4✳ ❉✳ d = 1✳ ❆✳ d = ✳ 3 ❈➙✉ ✶✽✳ ❇✐➳t r➡♥❣ ❤➔♠ sè F (x) = mx + (3m + n)x − 4x + ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ sè f (x) = 3x2 + 10x − 4✳ ❚➼♥❤ mn✳ ❆✳ mn = 3✳ ❇✳ mn = 2✳ ❈✳ mn = 1✳ ❉✳ mn = 0✳ x ❈➙✉ ✶✾✳ ●å✐ (D) ợ ữớ y = ✱ y = 0✱ x = 1✱ x = 4✳ ❚➼♥❤ t❤➸ t➼❝❤ ✈➟t t❤➸ trá♥ ①♦❛② t↕♦ t❤➔♥❤ ❦❤✐ q✉❛② ❤➻♥❤ (D) q✉❛♥❤ trö❝ Ox✳ 15 21π 21 15π ❆✳ ✳ ❇✳ ✳ ❈✳ ✳ ❉✳ ✳ 16 16 16 ❈➙✉ ✶✹✳ ❚➼♥❤ tê♥❣ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ 21 log2(x + 3) = log2 x + + x2 − √ x − + x + 3✳ √ ❆✳ S = 2✳ ❇✳ S = 1✳ ❈✳ S = − 2✳ ❉✳ S = −1✳ x+2−m ❈➙✉ ✷✶✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❣✐→ trà t❤ü❝ ❝õ❛ t❤❛♠ sè m ✤➸ ❤➔♠ sè y = ♥❣❤à❝❤ ❜✐➳♥ x+1 tr➯♥ ♠é✐ ❦❤♦↔♥❣ ①→❝ ✤à♥❤ ❝õ❛ ♥â✳ ❆✳ m ≤ −3✳ ❇✳ m < −3✳ ❈✳ m ≤ 1✳ ❉✳ m < 1✳ ❈➙✉ ✷✵✳ ❚r❛♥❣ ✷✴✻ ▼➣ ✤➲ ✶✵✷ ❈➙✉ ✷✷✳ ❆✳ 3✳ ❈➙✉ ✷✸✳ ❆✳ 1✳ ❈➙✉ ✷✹✳ ❑❤è✐ ❝❤â♣ t❛♠ ❣✐→❝ ✤➲✉ ❝â ♥❤✐➲✉ ♥❤➜t ❜❛♦ ♥❤✐➯✉ ♠➦t ♣❤➥♥❣ ✤è✐ ①ù♥❣❄ ❇✳ 6✳ ❈✳ 9✳ ❉✳ 4✳ √ ❈❤♦ sè t❤ü❝ a > 0, a 6= 1✳ ●✐→ trà log√a3 a2 ❜➡♥❣ ❇✳ ❚➼❝❤ ♣❤➙♥ I = Z1 ✳ ❈✳ ✳ 9 ✳ ❉✳ (x − 1)2 dx = a − ln b✱ tr♦♥❣ ✤â a; b ❧➔ ❝→❝ sè ♥❣✉②➯♥✳ ❚➼♥❤ ❣✐→ x2 + trà ❝õ❛ ❜✐➸✉ t❤ù❝ a + b✳ ❆✳ 0✳ ❈➙✉ ✷✺✳ ❆✳ y′ = ❇✳ 1✳ ❈✳ 3✳ ❉✳ −1✳ ❉✳ y′ = ❚➼♥❤ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ sè y = log9 x2 + ✳ (x2 ✳ + 1) ln ❇✳ y′ = 2x ln ✳ x2 + ❈✳
y′ = (x2 x ✳ + 1) ln ln ✳ x2 + ●å✐ z1 ✱ z2 ❧➔ ❤❛✐ ♥❣❤✐➺♠ ♣❤ù❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ z − 4z + = 0✳ ❚➼♥❤ w = 1 + + i(z1 z2 + z2 z1 )✳ ❈➙✉ ✷✻✳ z1 ❆✳ z2 4 w = − + 20i✳ ❈✳ w = + 20i✳ ❉✳ w = + 20i✳ 5 ❈❤♦ ❤➻♥❤ ❝❤â♣ S.ABC ❝â t❤➸ t➼❝❤ ❜➡♥❣ 1✳ ❚r➯♥ ❝↕♥❤ BC ❧➜② ✤✐➸♠ E s❛♦ ❝❤♦ w = 20 + i✳ ❈➙✉ ✷✼✳ ❇✳ BE = 2EC ✳ ❚➼♥❤ t❤➸ t➼❝❤ V ❝õ❛ ❦❤è✐ tù ❞✐➺♥ SAEB ✳ ❇✳ V = ✳ ❈✳ V = ✳ ❉✳ V = ✳ ❆✳ V = ✳ 3 ❈➙✉ ✷✽✳ ❈❤♦ sè ♣❤ù❝ z t❤ä❛ |z − + 2i| = 3✳ ❇✐➳t r➡♥❣ t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ❜✐➵✉ ❞✐➵♥ ❝õ❛ sè ♣❤ù❝ w = 2z + i tr➯♥ ♠➦t ♣❤➥♥❣ (Oxy) ❧➔ ♠ët ✤÷í♥❣ trá♥✳ ❚➻♠ t➙♠ ❝õ❛ ✤÷í♥❣ trá♥ ✤â✳ ❆✳ I(2; −3)✳ ❇✳ I(0; 1)✳ ❈✳ I(1; 0)✳ ❉✳ I(1; 1)✳ ❈➙✉ ✷✾✳ ❈❤♦ x, y > t❤ä❛ ♠➣♥ x + y = ❚➼♥❤ x2 + y ✳ 25 ✳ ❆✳ 16 ❈➙✉ ✸✵✳ ✈➔ ❜✐➸✉ t❤ù❝ P = + ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t✳ x 4y 2313 153 ✳ ❈✳ ✳ ❉✳ ✳ 1156 100 ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Oxyz ✱ ❝❤♦ ♠➦t ❝➛✉ (S) : x2 + y + z − 8x + 10y − 6z + 49 = 0✳ ❇✳ ❚➼♥❤ ❜→♥ ❦➼♥❤ R ❝õ❛ ♠➦t ❝➛✉ (S)✳ ❆✳ R = 7✳ ❇✳ R = 1✳ ❈✳ R= √ 99✳ ❉✳ R= √ 151✳ ❈❤♦ ❤➻♥❤ ❝❤â♣ S.ABCD ❝â ✤→② ABCD ❧➔ ❤➻♥❤ ✈✉æ♥❣✱ t➙♠ O✱ ❝↕♥❤ a ✈➔ SO ⊥ √ (ABCD), SA = 2a 2✳ ●å✐ M, N ❧➛♥ ❧÷đt ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ SA, BC ✳ ❚➼♥❤ ❣â❝ ❣✐ú❛ ✤÷í♥❣ t❤➥♥❣ M N ✈➔ ♠➦t ♣❤➥♥❣ (ABCD)✳ π π π ✳ ❇✳ arctan 2✳ ❈✳ ✳ ❉✳ ✳ ❆✳ ❈➙✉ ✸✶✳ ❈➙✉ ✸✷✳ ❈❤♦ F (x) ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ sè f (x) = ✳ ❇✐➳t F cos2 x ♠å✐ k ∈ Z✳ ❚➼♥❤ F (0) + F (π) + F (2π) + + F (10π)✳ ❆✳ 55✳ ❇✳ 45✳ ❈✳ 44✳ ❉✳ π  + kπ = k ✈ỵ✐ 0✳ ❚r❛♥❣ ✸✴✻ ▼➣ ✤➲ ✶✵✷ ❚➼♥❤ sè ❣✐→ trà ♥❣✉②➯♥ ❝õ❛ t❤❛♠ sè m tr➯♥ ❦❤♦↔♥❣ (−2019; 2019) ✤➸ ❤➔♠ sè y = x4 2mx2 3m + ỗ tr (1; 2)✳ ❆✳ 1✳ ❇✳ 2020✳ ❈✳ 2019✳ ❉✳ 2✳ ởt ỏ t ỗ t ❣✐è♥❣ ♥❤❛✉✳ ❍ä✐ ❝â ❜❛♦ ♥❤✐➯✉ ❝→❝❤ ❝➢t ❝❤✐➳❝ ✈á♥❣ ✤â t❤➔♥❤ ✷ ♣❤➛♥ ♠➔ sè ❤↕t ð ♠é✐ ♣❤➛♥ ✤➲✉ ❧➔ sè ❧➫ ❄ ❆✳ 5✳ ❇✳ 90✳ ❈✳ 10✳ ❉✳ 180✳   11π x ; 2019π ❈➙✉ ✸✺✳ ❚➼♥❤ sè ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ cotx = tr♦♥❣ ❦❤♦↔♥❣ 12 ❆✳ 2018✳ ❇✳ 1✳ ❈✳ 2020✳ ❉✳ 2019✳ 2 ❈➙✉ ✸✻✳ ❈❤♦ ❤➻♥❤ ♣❤➥♥❣ (D) ✤÷đ❝ ợ ữớ y = 2(x 1); y = − x ✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ trá♥ ①♦❛② t↕♦ t❤➔♥❤ ❞♦ (D) q✉❛② q✉❛♥❤ trö❝ Ox✳ 64π 32 32π 64 ❆✳ ✳ ❇✳ ✳ ❈✳ ✳ ❉✳ ✳ 15 15 15 15 ❈➙✉ ✸✼✳ ❚➼♥❤ tê♥❣ t➜t ❝↔ ❝→❝ ❣✐→ trà ❝õ❛ t❤❛♠ sè m ✤➸ tỗ t t ởt số ự z tọ ỗ tớ |z| = m |z 4m + 3mi| = m2✳ ❆✳ 6✳ ❇✳ 9✳ ❈✳ 4✳ ❉✳ 10✳ ❈➙✉ ✸✸✳ ❈➙✉ ✸✽✳ ◆❣÷í✐ t❛ ❧➔♠ t↕ t➟♣ ỡ t ữ ợ ❦❤è✐ trư ❜➡♥❣ ♥❤❛✉ ✈➔ t❛② ❝➛♠ ❝ơ♥❣ ❧➔ ❦❤è✐ trư✳ ❇✐➳t ❤❛✐ ✤➛✉ ❧➔ ❤❛✐ ❦❤è✐ trư ✤÷í♥❣ ❦➼♥❤ ✤→② ❜➡♥❣ ✶✷✱ ❝❤✐➲✉ ❝❛♦ ❜➡♥❣ ✻✱ ❝❤✐➲✉ ❞➔✐ t↕ ❜➡♥❣ ✸✵ ✈➔ ❜→♥ ❦➼♥❤ t❛② ❝➛♠ ❜➡♥❣ ✷✳ ❍➣② t➼♥❤ t❤➸ t➼❝❤ ✈➟t ❧✐➺✉ ❧➔♠ ♥➯♥ t↕ t❛② ✤â✳ ❆✳ 6480π ✳ ❇✳ 502π ✳ ❈✳ 108π ✳ ❉✳ 504π ✳ ❈➙✉ ✸✾✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Oxyz ✱ ❝❤♦ ♠➦t ♣❤➥♥❣ (P ) : x + y + z − = ✈➔ ❜❛ ✤✐➸♠ ✤✐➸♠ ★✥✥✥✥✥✔ ★✥✥✥✥✥✥✔ ★✥✥✥✥✥✔ A(3; 1; 1), B(7; 3; 9) ✈➔ C(2; 2; 2)✳ ✣✐➸♠ M (a; b; c) tr➯♥ (P ) s❛♦ ❝❤♦ |M A + 2M B + 3M C| ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t✳ ❚➼♥❤ 2a − 15b + c✳ ❆✳ 1✳ ❇✳ 6✳ ❈✳ 3✳ ❉✳ 8✳ √ √ √ √ ❈➙✉ ✹✵✳ ❈❤♦ ❤❛✐ sè ♣❤ù❝ z, w t❤ä❛ ♠➣♥ |z − 2| = 2, |w − 2i| = 2✳ ❇✐➳t r➡♥❣ |z − w| ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t ❦❤✐ z = zo, w = wo✳ ❚➼♥❤ |3zo − wo|✳ √ √ √ ❆✳ 2✳ ❇✳ 1✳ ❈✳ 2✳ ❉✳ 2✳ ởt ữớ ỷ số t 100 tr ỗ ✈➔♦ ♥❣➙♥ ❤➔♥❣ ✈ỵ✐ ❧➣✐ s✉➜t 0, 5%/t❤→♥❣ ✈➔ ỉ♥❣ t rút ộ t ởt tr ỗ tø s❛✉ ♥❣➔② ❣û✐ ♠ët t❤→♥❣ ❝❤♦ ✤➳♥ ❦❤✐ ❤➳t t✐➲♥ ✭t❤→♥❣ ❝✉è✐ ❝ị♥❣ ❝â t❤➸ ❦❤ỉ♥❣ ❝á♥ ✤õ ♠ët tr ỗ ọ ổ t rút t t s ♥❤✐➯✉ t❤→♥❣❄ ❆✳ 139✳ ❇✳ 140✳ ❈✳ 100✳ ❉✳ 138✳ x−2 s❛♦ ❝❤♦ ❦❤♦↔♥❣ ❝→❝❤ tø M ❈➙✉ ✹✷✳ ●å✐ M (a; b) tr ỗ t sè y = x ✤➳♥ ✤÷í♥❣ t❤➥♥❣ d : y = 2x + ♥❤ä ♥❤➜t✳ ❚➼♥❤ (4a + 5)2 + (2b − 7)2✳ ❚r❛♥❣ ✹✴✻ ▼➣ ✤➲ ✶✵✷ ❆✳ 90✳ ❇✳ 10✳ ❈✳ 180✳ ❉✳ 5✳ ❚➼♥❤ sè ❣✐→ trà ♥❣✉②➯♥ ❝õ❛ t❤❛♠ sè m tr➯♥ ❦❤♦↔♥❣ (−2019; 2019) ✤➸ ❤➔♠ sè y = x4 − 2mx2 − 3m + ỗ tr (1; 2) 2019 ❇✳ 1✳ ❈✳ 2✳ ❉✳ 2020✳ ❈➙✉ ✹✽✳ ❚➼♥❤ sè ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ cotx = 2x tr♦♥❣ ❦❤♦↔♥❣ ❆✳ 2019✳ ❇✳ 1✳ ❈✳ 2020✳ ❈➙✉ ✹✾✳  11π ; 2019π 12 ❉✳ 2018✳  ❈❤♦ ❤➔♠ sè f (x) ❝â ✤↕♦ ❤➔♠ f ′ (x) = (x − 1)(x2 − 3)(x4 − 1) ✈ỵ✐ ♠å✐ x t❤✉ë❝ R✳ ❙♦ s→♥❤ f (−2), f (0), f (2)✱ t❛ ✤÷đ❝ ❆✳ f (2) < f (0) < f (−2)✳ ❇✳ f (−2) < f (0) < f (2)✳ ❈✳ f (0) < f (−2) < f (2)✳ ❉✳ f (−2) < f (2) < f (0)✳ ❈➙✉ ✺✵✳ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❍➌❚✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❚r❛♥❣ ✻✴✻ ▼➣ ✤➲ ✶✵✻ ❙Ð ●❉ ❱⑨ ✣❚ ◗❯❷◆● ❚❘➚ ❑➐ ❚❍■ ❚❍Û ❚❍P❚ ◗❯➮❈ ●■❆ ▲❺◆ ✶ ◆❿▼ ✷✵✶✾ ▼➷◆ ❚❖⑩◆ ì P ị ✾✵ ♣❤ót✱ ❦❤ỉ♥❣ ❦➸ t❤í✐ ❣✐❛♥ ❣✐❛♦ ✤➲ ✭ ✣➲ t❤✐ ❝â ✻ tr❛♥❣ ✮ ▼➣ ✤➲ t❤✐ ✶✵✼ ❈➙✉ ✶✳ ❈❤♦ ❦❤è✐ ❧➟♣ ♣❤÷ì♥❣ ABCD.A′ B ′ C ′ D′ ❝â ✤ë ❞➔✐ ❝↕♥❤ ❧➔ 3❝♠✳ ❚➼♥❤ t❤➸ t➼❝❤ ❝õ❛ ❦❤è✐ tù ❞✐➺♥ ACB ′ D′ ✳ √ ❆✳ 18 2❝♠3 ✳ ❇✳ 18❝♠3 ✳ ❈✳ 3❝♠3 ✳ ❉✳ 9❝♠3 ✳ ❈➙✉ ✷✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ trö ❝â ❜→♥ ❦➼♥❤ R = 3, ❝❤✐➲✉ ❝❛♦ h = ❆✳ V = 15π✳ ❇✳ V = 90π✳ ❈✳ V = 45π✳ ❉✳ V = 45✳ ❈➙✉ ✸✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈ỵ✐ ❤➺ tå❛ ✤ë Oxyz ✱ ❝❤♦ ♠➦t ♣❤➥♥❣ (P ) : 2x − 2y + z + = 0✳ ❚➼♥❤ ❦❤♦↔♥❣ ❝→❝❤ d tø ✤✐➸♠ M (1; 2; 1) ✤➳♥ ♠➦t ♣❤➥♥❣ (P )✳ ❆✳ d = 1✳ ❇✳ d = ✳ ❈✳ d = 3✳ ❉✳ d = 4✳ ❈➙✉ ✹✳ ❍➔♠ sè y = x3 + 3x2 ỗ tr t ủ tr t ủ ữủ ữợ ❆✳ (2; +∞)✳ ❇✳ (0; 2)✳ ❈✳ (−∞; 0)✳ ❉✳ (−∞; 0) ∪ (2; +∞)✳ (2 − 3i)(4 − i) ✳ ❚➻♠ tå❛ ✤ë ✤✐➸♠ ❜✐➸✉ ❞✐➵♥ ❝õ❛ sè ♣❤ù❝ z tr➯♥ + 2i ❈➙✉ ✺✳ ❈❤♦ sè ♣❤ù❝ z = ♠➦t ♣❤➥♥❣ Oxy ✳ ❆✳ (1; 4)✳ ❇✳ (1; −4)✳ ❈➙✉ ✻✳ ❚➼♥❤ t➼❝❤ ♣❤➙♥ I = Z2 ❈✳ (−1; −4)✳ ❉✳ (−1; 4)✳ ❈✳ I = + ln 2✳ ❉✳ I = ln 2✳ x−1 dx✳ x ❆✳ I = − ln 2✳ ❇✳ I = ✳ ❈➙✉ ✼✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈ỵ✐ ❤➺ tå❛ ✤ë Oxyz ✱ ❝❤♦ ♠➦t ♣❤➥♥❣ (P )✿ 2x − 2y + z + 2017 = tỡ tr tỡ ữủ ữợ ❧➔ ♠ët ✈➨❝✲tì ♣❤→♣ t✉②➳♥ ❝õ❛ (P )❄ ❆✳ ★✔ n = (1; −2; 2)✳ ❇✳ ★✔ n = (4; −4; 2)✳ ❈✳ ★✔ n = (1; −1; 4)✳ ❉✳ ★✔ n = (−2; 2; 1)✳ ❈➙✉ ✽✳ ❚➼♥❤ số ữớ t ỗ t số y = x2 + x − ✳ x − 3x + ❆✳ 3✳ ❇✳ 1✳ ❈✳ 2✳ ❉✳ 0✳ ❈➙✉ ✾✳ ❈❤♦ sè ♣❤ù❝ z = (1 − 2i)2 ✳ ❚➼♥❤ ♠æ ✤✉♥ ❝õ❛ sè ♣❤ù❝ ✳ ❆✳ ✳ ❇✳ ❈➙✉ ✶✵✳ ❆✳ P = √ ✳ ❈✳ z ✳ 25 ❉✳ √ 5✳ v s u r u 3 t ✳ ▼➺♥❤ ✤➲ ♥➔♦ tr♦♥❣ ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ ✤ó♥❣❄ ❈❤♦ ❜✐➸✉ t❤ù❝ P = 2 18 ✳ ❇✳ P =  18 3 ✳ ❈✳ P =   18 ✳ ❉✳ P =   12 ✳ ❚r❛♥❣ ✶✴✻ ▼➣ ✤➲ ✶✵✼ ❚➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ log3(x − 2) = 2✳ ❆✳ x = 11✳ ❇✳ x = 8✳ ❈✳ x = 10✳ ❉✳ x = 9✳ ❈➙✉ ✶✷✳ ❑❤è✐ ❝❤â♣ S.ABCD ❝â ✤→② ABCD ❧➔ ❤➻♥❤ ✈✉æ♥❣ ❝↕♥❤ 3a✱ SA = a, SA ⊥ (ABCD)✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ ❝❤â♣ S.ABCD✳ a3 3 ❆✳ 3a ✳ ❇✳ 9a ✳ ❈✳ 6a ✳ ❉✳ ✳ ❈➙✉ ✶✸✳ ❑❤è✐ ♥â♥ (N ) ❝â ❜→♥ ❦➼♥❤ ✤→② ❜➡♥❣ ✈➔ ❞✐➺♥ t➼❝❤ ①✉♥❣ q✉❛♥❤ ❜➡♥❣ 15π ✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ ♥â♥ (N )✳ ❆✳ 36π ✳ ❇✳ 12π ✳ ❈✳ 16π ✳ ❉✳ 45π ✳
tr➯♥ −∞; 21 ✳ ❈➙✉ ✶✹✳ ❚➻♠ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ sè f (x) = − 2x 1 ❆✳ − ln |2x − 1| + C ✳ ❇✳ ln |2x − 1| + C ✳ ❈✳ ln |2x − 1| + C ✳ ❉✳ ln(1 − 2x) + C ✳ 2 ❈➙✉ ✶✺✳ ❈❤♦ ❤➔♠ sè ❜➟❝ ❜❛ y = f (x) õ ỗ t ữ ữợ ú y ỹ t ❤➔♠ sè ❧➔ −1✳ ❇✳ ●✐→ trà ❝ü❝ ✤↕✐ ❝õ❛ ❤➔♠ sè ❧➔ 0✳ ❈✳ ✣✐➸♠ ❝ü❝ ✤↕✐ ❝õ❛ ❤➔♠ sè ❧➔ 3✳ O x ❉✳ ●✐→ trà ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ sè ❜➡♥❣ −1✳ −1 ❈➙✉ ✶✶✳ ❚r♦♥❣ ♠➦t ♣❤➥♥❣ tå❛ ✤ë Oxy✱ t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ❜✐➸✉ ❞✐➵♥ ❝→❝ sè ♣❤ù❝ z t❤ä❛ ♠➣♥ |z − + 2i| = |z + + 2i| ❧➔ ✤÷í♥❣ t❤➥♥❣ ❝â ♣❤÷ì♥❣ tr➻♥❤ ❆✳ x + 2y + = 0✳ ❇✳ x − 2y + = 0✳ ❈✳ x − 2y = 0✳ ❉✳ x + 2y = 0✳ ❈➙✉ ✶✼✳ ❚➼♥❤ ❞✐➺♥ t➼❝❤ ❝õ❛ ♠➦t ❝➛✉ ❝â ❜→♥ ❦➼♥❤ ❜➡♥❣ 3✳ ❆✳ 18π ✳ ❇✳ 9π ✳ ❈✳ 12π ✳ ❉✳ 36π ✳ log2 (x + 3) = log2 x + + x2 − ❈➙✉ ✶✽✳ ❚➼♥❤ tê♥❣ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ √ x − + x + 3✳ √ 2✳ ❈✳ S = −1✳ ❉✳ S = 2✳ ❆✳ S = 1✳ ❇✳ S = − ❈➙✉ ✶✾✳ ❈❤♦ sè ♣❤ù❝ z t❤ä❛ |z − + 2i| = 3✳ ❇✐➳t r➡♥❣ t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ❜✐➵✉ ❞✐➵♥ ❝õ❛ sè ♣❤ù❝ w = 2z + i tr➯♥ ♠➦t ♣❤➥♥❣ (Oxy) ❧➔ ♠ët ✤÷í♥❣ trá♥✳ ❚➻♠ t➙♠ ❝õ❛ ✤÷í♥❣ trá♥ ✤â✳ ❆✳ I(1; 0)✳ ❇✳ I(0; 1)✳ ❈✳ I(1; 1)✳ ❉✳ I(2; −3)✳ ❈➙✉ ✷✵✳ ❑❤è✐ ❝❤â♣ t❛♠ ❣✐→❝ ✤➲✉ ❝â ♥❤✐➲✉ ♥❤➜t ❜❛♦ ♥❤✐➯✉ ♠➦t ♣❤➥♥❣ ✤è✐ ①ù♥❣❄ ❆✳ 4✳ ❇✳ 3✳ ❈✳ 6✳ ❉✳ 9✳ 2 ❈➙✉ ✷✶✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Oxyz ✱ ❝❤♦ ♠➦t ❝➛✉ (S) : x + y + z − 8x + 10y − 6z + 49 = 0✳ ❚➼♥❤ ❜→♥ ❦➼♥❤ R ❝õ❛ ♠➦t ❝➛✉ (S)✳ √ √ ❆✳ R = 1✳ ❇✳ R = 99✳ ❈✳ R = 7✳ ❉✳ R = 151✳ ❈➙✉ ✶✻✳ ❚r❛♥❣ ✷✴✻ ▼➣ ✤➲ ✶✵✼ √ ❈❤♦ sè t❤ü❝ a > 0, a 6= 1✳ ●✐→ trà log√a a2 ❜➡♥❣ ❆✳ ✳ ❇✳ ✳ ❈✳ ✳ ❉✳ 1✳ ❈➙✉ ✷✸✳ ❈❤♦ ❤➻♥❤ ❝❤â♣ S.ABC ❝â t❤➸ t➼❝❤ ❜➡♥❣ 1✳ ❚r➯♥ ❝↕♥❤ BC ❧➜② ✤✐➸♠ E s❛♦ ❝❤♦ BE = 2EC ✳ ❚➼♥❤ t❤➸ t➼❝❤ V ❝õ❛ ❦❤è✐ tù ❞✐➺♥ SAEB ✳ 1 ❆✳ V = ✳ ❇✳ V = ✳ ❈✳ V = ✳ ❉✳ V = ✳ 3 ❈➙✉ ✷✹✳ ❇✐➳t r➡♥❣ ❤➔♠ sè F (x) = mx + (3m + n)x − 4x + ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ sè f (x) = 3x2 + 10x − 4✳ ❚➼♥❤ mn✳ ❆✳ mn = 0✳ ❇✳ mn = 3✳ ❈✳ mn = 2✳ ❉✳ mn = 1✳ ✈➔ ❜✐➸✉ t❤ù❝ P = + ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t✳ ❈➙✉ ✷✺✳ ❈❤♦ x, y > t❤ä❛ ♠➣♥ x + y = x 4y ❚➼♥❤ x2 + y2✳ 2313 153 25 ✳ ❇✳ ✳ ❈✳ ✳ ❉✳ ✳ ❆✳ 16 1156 100 x ❈➙✉ ✷✻✳ ●å✐ (D) ❤➻♥❤ ợ ữớ y = y = 0✱ x = 1✱ x = 4✳ ❚➼♥❤ t❤➸ t➼❝❤ ✈➟t t❤➸ trá♥ ①♦❛② t↕♦ t❤➔♥❤ ❦❤✐ q✉❛② ❤➻♥❤ (D) q✉❛♥❤ trö❝ Ox✳ 15π 15 21 21π ✳ ❇✳ ✳ ❈✳ ✳ ❉✳ ✳ ❆✳ 16 16 16
❈➙✉ ✷✼✳ ❚➼♥❤ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ sè y = log9 x + ✳ ln x 2x ln ′ ′ ′ ′ ❆✳ y = ✳ ❇✳ y = ✳ ❈✳ y = ✳ ❉✳ y = ✳ 2 x +1 (x + 1) ln (x + 1) ln x2 + ❈➙✉ ✷✷✳ ❈➙✉ ✷✽✳ ❚➼❝❤ ♣❤➙♥ I = Z1 (x − 1)2 dx = a − ln b x2 + ✱ tr♦♥❣ ✤â a; b ❧➔ ❝→❝ sè ♥❣✉②➯♥✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ a + b✳ ❆✳ −1✳ ❇✳ 0✳ ❈✳ 3✳ ❉✳ 1✳ ❈➙✉ ✷✾✳ ●å✐ z1 ✱ z2 ❧➔ ❤❛✐ ♥❣❤✐➺♠ ♣❤ù❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ z − 4z + = 0✳ ❚➼♥❤ w 1 + + i(z1 z2 + z2 z1 )✳ z1 z2 4 ❈✳ w = + 20i✳ ❉✳ w = − + 20i✳ ❆✳ w = + 20i✳ ❇✳ w = 20 + i✳ 5 = ❚➻♠ t➜t ❝↔ ❝→❝ ❣✐→ trà t❤ü❝ ❝õ❛ t❤❛♠ sè m ✤➸ ❤➔♠ sè y = x +x 2+−1 m ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ ♠é✐ ❦❤♦↔♥❣ ①→❝ ✤à♥❤ ❝õ❛ ♥â✳ ❆✳ m ≤ −3✳ ❇✳ m < 1✳ ❈✳ m ≤ 1✳ ❉✳ m < −3✳ ❈➙✉ ✸✶✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Oxyz ✱ ❝❤♦ ♠➦t ♣❤➥♥❣ (P ) : x − y + = ✈➔ ❤❛✐ ✤✐➸♠ A(1; 2; 3), B(1; 0; 1)✳ ✣✐➸♠ C(a; b; −2) ∈ (P ) s❛♦ ❝❤♦ t❛♠ ❣✐→❝ ABC ❝â ❞✐➺♥ t➼❝❤ ♥❤ä ♥❤➜t✳ ❚➼♥❤ a + b✳ ❆✳ 0✳ ❇✳ 1✳ ❈✳ −3✳ ❉✳ 2✳ ❈➙✉ ✸✵✳ ❚r❛♥❣ ✸✴✻ ▼➣ ✤➲ ✶✵✼ ❈➙✉ ✸✷✳ ◆❣÷í✐ t❛ ❧➔♠ t↕ t➟♣ ❝ì t❛② ♥❤÷ ❤➻♥❤ ✈➩ ✈ỵ✐ ❤❛✐ ✤➛✉ ❧➔ ❤❛✐ ❦❤è✐ trư ❜➡♥❣ ♥❤❛✉ ✈➔ t❛② ❝➛♠ ❝ơ♥❣ ❧➔ ❦❤è✐ trư✳ ❇✐➳t ❤❛✐ ✤➛✉ ❧➔ ❤❛✐ ❦❤è✐ trư ✤÷í♥❣ ❦➼♥❤ ✤→② ❜➡♥❣ ✶✷✱ ❝❤✐➲✉ ❝❛♦ ❜➡♥❣ ✻✱ ❝❤✐➲✉ ❞➔✐ t↕ ❜➡♥❣ ✸✵ ✈➔ ❜→♥ ❦➼♥❤ t❛② ❝➛♠ ❜➡♥❣ ✷✳ ❍➣② t➼♥❤ t❤➸ t➼❝❤ ✈➟t ❧✐➺✉ ❧➔♠ ♥➯♥ t↕ t❛② ✤â✳ ❆✳ 504π ✳ ❇✳ 6480π ✳ ❈✳ 108π ✳ ❉✳ 502π ✳ 2 (D) ữủ ợ ❜ð✐ ❤❛✐ ✤÷í♥❣ y = 2(x − 1); y = − x ✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ trá♥ ①♦❛② t↕♦ t❤➔♥❤ ❞♦ (D) q✉❛② q✉❛♥❤ trö❝ Ox✳ 64 32 64π 32π ❆✳ ✳ ❇✳ ✳ ❈✳ ✳ ❉✳ ✳ 15 15 15 15 ′ ❈➙✉ ✸✹✳ ❈❤♦ ❤➔♠ sè f (x) ❝â ✤↕♦ ❤➔♠ f (x) = (x − 1)(x − 3)(x − 1) ✈ỵ✐ ♠å✐ x t❤✉ë❝ R✳ ❙♦ s→♥❤ f (−2), f (0), f (2)✱ t❛ ✤÷đ❝ ❆✳ f (−2) < f (0) < f (2)✳ ❇✳ f (2) < f (0) < f (−2)✳ ❈✳ f (−2) < f (2) < f (0)✳ ❉✳ f (0) < f (−2) < f (2)✳ ❈➙✉ ✸✺✳ ❈❤♦ ❤➔♠ sè f (x) ❝â ✤↕♦ ❤➔♠ ❧➔ f (x) ỗ t số y = f (x) ♥❤÷ ❤➻♥❤ ✈➩ ❜➯♥✳ ❚➼♥❤ sè ✤✐➸♠ ❝ü❝ trà ❝õ❛ √ √ ❤➔♠ sè y = f (x2) tr➯♥ ❦❤♦↔♥❣ (− 5; 5)✳ ❆✳ 2✳ ❇✳ 3✳ ❈✳ 4✳ ❉✳ 5✳ ② y = f ′ (x) ởt ữớ ỷ số t 100 tr ỗ ♥❣➙♥ ❤➔♥❣ ✈ỵ✐ ❧➣✐ s✉➜t 0, 5%/t❤→♥❣ ✈➔ ỉ♥❣ t❛ rút ộ t ởt tr ỗ tứ s❛✉ ♥❣➔② ❣û✐ ♠ët t❤→♥❣ ❝❤♦ ✤➳♥ ❦❤✐ ❤➳t t✐➲♥ ✭t❤→♥❣ ❝✉è✐ ❝ị♥❣ ❝â t❤➸ ❦❤ỉ♥❣ ❝á♥ ✤õ ♠ët tr✐➺✉ ỗ ọ ổ t rút t t s t❤→♥❣❄ ❆✳ 140✳ ❇✳ 138✳ ❈✳ 100✳ ❉✳ 139✳ ❈➙✉ ✸✼✳ ❈❤♦ ❤➻♥❤ ❝❤â♣ S.ABCD ❝â ✤→② ❧➔ ❤➻♥❤ ❜➻♥❤ ❤➔♥❤ ✈➔ ❝â t❤➸ t➼❝❤ ❜➡♥❣ 48✳ ❚r➯♥ ❝↕♥❤ SB, SD ❧➜② ✤✐➸♠ ❝→❝ M, N s❛♦ ❝❤♦ SM = M B, SD = 3SN ✳ ▼➦t ♣❤➥♥❣ (AM N ) ❝➢t SC t↕✐ P ✳ ❚➼♥❤ t❤➸ t➼❝❤ V ❝õ❛ ❦❤è✐ tù ❞✐➺♥ SM N P ✳ 1 ❈✳ V = ✳ ❉✳ V = 2✳ ❆✳ V = 1✳ ❇✳ V = ✳ ❈➙✉ ✸✻✳ ❚r❛♥❣ ✹✴✻ ▼➣ ✤➲ ✶✵✼ ❈➙✉ ✸✽✳ ❙➠♠ ❧è♣ ①❡ æ tæ ❦❤✐ ❜ì♠ ❝➠♥❣ ✤➦t ♥➡♠ tr➯♥ ♠➦t ♣❤➥♥❣ õ ữ ợ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥❤ä R1 = 20cm✱ ❜→♥ ❦➼♥❤ ữớ trỏ ợ R2 = 30cm t t ❝➢t ❜ð✐ ♠➦t ♣❤➥♥❣ ✤✐ q✉❛ trư❝✱ ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐ ♠➠t ♣❤➥♥❣ ♥➡♠ ♥❣❛♥❣ ❧➔ ❤❛✐ ✤÷í♥❣ trá♥✳ ❇ä q✉❛ ✤ë ❞➔② ❝õ❛ ✈ä s➠♠✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤æ♥❣ ❦❤➼ ✤÷đ❝ ❝❤ù❛ ❜➯♥ tr♦♥❣ s➠♠✳ 3 3 ❆✳ 2500πcm ✳ ❇✳ 1400πcm ✳ ❈✳ 600πcm ✳ ❉✳ 1250πcm ✳ ❈➙✉ ✸✾✳ ❆✳ ❚➼♥❤ sè ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ cotx = 2x tr♦♥❣ ❦❤♦↔♥❣ 2018✳ ❇✳ 1✳ ❈✳ 2020✳  11π ; 2019π 12 ❉✳ 2019✳  x−2 ●å✐ M (a; b) tr ỗ t sè y = s❛♦ ❝❤♦ ❦❤♦↔♥❣ ❝→❝❤ tø M x ✤➳♥ ✤÷í♥❣ t❤➥♥❣ d : y = 2x + ♥❤ä ♥❤➜t✳ ❚➼♥❤ (4a + 5)2 + (2b − 7)2 ✳ ❈➙✉ ✹✵✳ ❆✳ 18✳ ❇✳ 0✳ ❈✳ 162✳ √ √ ❉✳ √ 2✳ √ ❈❤♦ ❤❛✐ sè ♣❤ù❝ z, w t❤ä❛ ♠➣♥ |z − 2| = 2, |w − 2i| = 2✳ ❇✐➳t r➡♥❣ |z − w| ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t ❦❤✐ z = zo , w = wo ✳ ❚➼♥❤ |3zo − wo |✳ ❈➙✉ ✹✶✳ ❆✳ √ 2✳ ❇✳ √ 2✳ ❈✳ 1✳ ❉✳ √ 2✳ ▼ët ❝❤✐➳❝ ✈á♥❣ ✤❡♦ t ỗ t ố ọ õ ❝→❝❤ ❝➢t ❝❤✐➳❝ ✈á♥❣ ✤â t❤➔♥❤ ✷ ♣❤➛♥ ♠➔ sè ❤↕t ð ♠é✐ ♣❤➛♥ ✤➲✉ ❧➔ sè ❧➫ ❄ ❈➙✉ ✹✷✳ ❆✳ 90✳ ❇✳ 10✳ ❈✳ 180✳ ❉✳ 5✳ ❚➼♥❤ sè ❣✐→ trà ♥❣✉②➯♥ ❝õ❛ t❤❛♠ sè m tr➯♥ ❦❤♦↔♥❣ (−2019; 2019) ✤➸ ❤➔♠ sè y = x4 − 2mx2 3m + ỗ tr (1; 2) ❈➙✉ ✹✸✳ ❆✳ 2020✳ ❇✳ 1✳ ❈✳ 2019✳ ❉✳ 2✳ ❈❤♦ ❤➻♥❤ ❝❤â♣ S.ABCD ❝â ✤→② ABCD ❧➔ ❤➻♥❤ ✈✉æ♥❣✱ t➙♠ O✱ ❝↕♥❤ a ✈➔ SO ⊥ √ (ABCD), SA = 2a 2✳ ●å✐ M, N ❧➛♥ ❧÷đt ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ SA, BC ✳ ❚➼♥❤ ❣â❝ ❣✐ú❛ ✤÷í♥❣ t❤➥♥❣ M N ✈➔ ♠➦t ♣❤➥♥❣ (ABCD)✳ ❈➙✉ ✹✹✳ ❆✳ π ✳ ❇✳ arctan 2✳ ❈✳ π ✳ ❉✳ π ✳ ❈❤♦ ❤➻♥❤ ❝❤â♣ S.ABCD ❝â SA ⊥ (ABCD)✱ ✤→② ABCD ❧➔ ❤➻♥❤ ❝❤ú ♥❤➟t ✈ỵ✐ √ AC = a ✈➔ BC = a 2✳ ❚➼♥❤ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ SD ✈➔ BC ✳ √ ❈➙✉ ✹✺✳ √ ❆✳ 3a ✳ ❇✳ 2a ✳ ❈✳ √ a 3✳ ❉✳ a ✳ ❚➼♥❤ tê♥❣ t➜t tr t số m tỗ t t ởt số ự z tọ ỗ t❤í✐ |z| = m ✈➔ |z − 4m + 3mi| = m2 ✳ ❈➙✉ ✹✻✳ ❆✳ 6✳ ❇✳ 4✳ ❈✳ 9✳ ❉✳ 10✳ ❚r❛♥❣ ✺✴✻ ▼➣ ✤➲ ✶✵✼ ❈➙✉ ✹✼✳ ❈❤♦ ❤➔♠ sè f (x) ❝â ✤↕♦ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ R ✈➔ t❤ä❛ ♠➣♥ Z1 ✱ f (x) dx = f (1) = cot ✳ ❚➼♥❤ t➼❝❤ ♣❤➙♥ I = Z1 f (x) tan2 x + f ′ (x) tan x dx
✳ ✳ ❇✳ 0✳ ❈✳ − ln(cos 1)✳ ❉✳ −1✳ ❈➙✉ ✹✽✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Oxyz ✱ ❝❤♦ ♠➦t ♣❤➥♥❣ (P ) : x + y + z − = ✈➔ ❜❛ ✤✐➸♠ ✤✐➸♠ ★✥✥✥✥✥✔ ★✥✥✥✥✥✥✔ ★✥✥✥✥✥✔ A(3; 1; 1), B(7; 3; 9) ✈➔ C(2; 2; 2)✳ ✣✐➸♠ M (a; b; c) tr➯♥ (P ) s❛♦ ❝❤♦ |M A + 2M B + 3M C| ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t✳ ❚➼♥❤ 2a − 15b + c✳ ❆✳ 8✳ ❇✳ 1✳ ❈✳ 6✳ ❉✳ 3✳ π  ✳ ❇✐➳t F + kπ = k ✈ỵ✐ ❈➙✉ ✹✾✳ ❈❤♦ F (x) ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ sè f (x) = cos2 x ♠å✐ k ∈ Z✳ ❚➼♥❤ F (0) + F (π) + F (2π) + + F (10π)✳ ❆✳ 55✳ ❇✳ 44✳ ❈✳ 45✳ ❉✳ 0✳ ′ ❈➙✉ ✺✵✳ ❈❤♦ ❤➔♠ sè f (x) ①→❝ ✤à♥❤ tr➯♥ R✱ ❝â ✤↕♦ ❤➔♠ f (x) = (x + 1) (x − 2) (x + 3) ✳ ❙è ✤✐➸♠ ❝ü❝ trà ❝õ❛ ❤➔♠ sè f (|x|) ❧➔ ❆✳ 5✳ ❇✳ 1✳ ❈✳ 3✳ ❉✳ 2✳ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❍➌❚✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❆✳ − cot ❚r❛♥❣ ✻✴✻ ▼➣ ✤➲ ✶✵✼ ❙Ð ●❉ ❱⑨ ✣❚ ◗❯❷◆● ❚❘➚ ❑➐ ❚❍■ ❚❍Û ❚❍P❚ ◗❯➮❈ ●■❆ ▲❺◆ ✶ ◆❿▼ ✷✵✶✾ ▼➷◆ ❚❖⑩◆ ❚❘×❮◆● ❚❍P❚ ❈❍❯❨➊◆ ▲➊ ◗❯Þ ✣➷◆ ❚❤í✐ ❣✐❛♥ ❧➔♠ ❜➔✐ ✾✵ ♣❤ót✱ ❦❤ỉ♥❣ ❦➸ t❤í✐ ❣✐❛♥ ❣✐❛♦ ✤➲ ✭ ✣➲ t❤✐ ❝â ✻ tr❛♥❣ ✮ ▼➣ ✤➲ t❤✐ ✶✵✽ ❈➙✉ ✶✳ ❈❤♦ ❤➔♠ số y = f (x) õ ỗ t ữ ữợ ú y ❆✳ ✣✐➸♠ ❝ü❝ ✤↕✐ ❝õ❛ ❤➔♠ sè ❧➔ 3✳ ❇✳ ●✐→ trà ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ sè ❜➡♥❣ −1✳ ❈✳ ●✐→ trà ❝ü❝ ✤↕✐ ❝õ❛ ❤➔♠ sè ❧➔ 0✳ O x ❉✳ ✣✐➸♠ ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ sè ❧➔ −1✳ −1 ❈➙✉ ✷✳ ❚➼♥❤ ❞✐➺♥ t➼❝❤ ❝õ❛ ♠➦t ❝➛✉ ❝â ❜→♥ ❦➼♥❤ ❜➡♥❣ 3✳ ❆✳ 9π✳ ❇✳ 36π✳ ❈✳ 12π✳ ❉✳ 18π✳ ❈➙✉ ✸✳ ❚➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ log3 (x − 2) = 2✳ ❆✳ x = 8✳ ❇✳ x = 9✳ ❈✳ x = 10✳ ❉✳ x = 11✳ ❈➙✉ ✹✳ ❚➼♥❤ sè ✤÷í♥❣ t✐➺♠ ỗ t số y = x2 + x − ✳ x − 3x + ❆✳ 0✳ ❇✳ 1✳ ❈✳ 3✳ ❉✳ 2✳ ❈➙✉ ✺✳ ❈❤♦ ❦❤è✐ ❧➟♣ ♣❤÷ì♥❣ ABCD.A′ B ′ C ′ D′ ❝â ✤ë ❞➔✐ ❝↕♥❤ ❧➔ 3❝♠✳ ❚➼♥❤ t❤➸ t➼❝❤ ❝õ❛ ❦❤è✐ tù ❞✐➺♥ ACB ′D′✳ √ ❆✳ 18❝♠3 ✳ ❇✳ 18 2❝♠3 ✳ ❈✳ 9❝♠3 ✳ ❉✳ 3❝♠3 ✳ ❈➙✉ ✻✳ ❚r♦♥❣ ♠➦t ♣❤➥♥❣ tå❛ ✤ë Oxy✱ t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ❜✐➸✉ ❞✐➵♥ ❝→❝ sè ♣❤ù❝ z t❤ä❛ ♠➣♥ |z − + 2i| = |z + + 2i| ❧➔ ✤÷í♥❣ t❤➥♥❣ ❝â ♣❤÷ì♥❣ tr➻♥❤ ❆✳ x − 2y + = 0✳ ❇✳ x − 2y = 0✳ ❈✳ x + 2y + = 0✳ ❉✳ x + 2y = 0✳ ❈➙✉ ✼✳ ❈❤♦ sè ♣❤ù❝ z = (2 − 3i)(4 − i) ✳ ❚➻♠ tå❛ ✤ë ✤✐➸♠ ❜✐➸✉ ❞✐➵♥ ❝õ❛ sè ♣❤ù❝ z tr➯♥ + 2i ♠➦t ♣❤➥♥❣ Oxy✳ ❆✳ (1; −4)✳ ❇✳ (−1; −4)✳ ❈✳ (1; 4)✳ ❉✳ (−1; 4)✳ ❈➙✉ ✽✳ ❑❤è✐ ❝❤â♣ S.ABCD ❝â ✤→② ABCD ❧➔ ❤➻♥❤ ✈✉æ♥❣ ❝↕♥❤ 3a✱ SA = a, SA ⊥ (ABCD)✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ ❝❤â♣ S.ABCD ✳ a ❆✳ 3a3 ✳ ❈✳ 9a3 ✳ ❇✳ ✳ ❉✳ 6a3 ✳ ❈➙✉ ✾✳ ❚➻♠ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ sè f (x) = ❆✳ ln |2x − 1| + C ✳ ❇✳ ln(1 − 2x) + C ❈➙✉ ✶✵✳ ❚➼♥❤ t➼❝❤ ♣❤➙♥ I = Z2 x−1 dx x ✳ tr➯♥
−∞; 12 − 2x ❈✳ ln |2x − 1| + C ✳ ✳ ❉✳ − 12 ln |2x − 1| + C ✳ ✳ ❚r❛♥❣ ✶✴✻ ▼➣ ✤➲ ✶✵✽ ❆✳ I = ln 2✳ ❇✳ I = − ln 2✳ ❈✳ I = + ln 2✳ ❉✳ I= ✳ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ❤➺ tå❛ ✤ë Oxyz ✱ ❝❤♦ ♠➦t ♣❤➥♥❣ (P ) : 2x − 2y + z + = 0✳ ❚➼♥❤ ❦❤♦↔♥❣ ❝→❝❤ d tø ✤✐➸♠ M (1; 2; 1) ✤➳♥ ♠➦t ♣❤➥♥❣ (P )✳ ❇✳ d = 4✳ ❈✳ d = 1✳ ❉✳ d = 3✳ ❆✳ d = ✳ ❈➙✉ ✶✶✳ ❑❤è✐ ♥â♥ (N ) ❝â ❜→♥ ❦➼♥❤ ✤→② ❜➡♥❣ ✈➔ ❞✐➺♥ t➼❝❤ ①✉♥❣ q✉❛♥❤ ❜➡♥❣ 15π ✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ ♥â♥ (N )✳ ❆✳ 36π ✳ ❇✳ 12π ✳ ❈✳ 16π ✳ ❉✳ 45π ✳ ❈➙✉ ✶✷✳ ❈➙✉ ✶✸✳ ❆✳ √ ✳ z ❈❤♦ sè ♣❤ù❝ z = (1 − 2i)2 ✳ ❚➼♥❤ ♠æ ✤✉♥ ❝õ❛ sè ♣❤ù❝ ✳ ❇✳ ✳ 25 ❈✳ √ 5✳ ❉✳ ✳ ❍➔♠ số y = x3 + 3x2 ỗ tr➯♥ t➟♣ ❤đ♣ ♥➔♦ tr♦♥❣ ❝→❝ t➟♣ ❤đ♣ ✤÷đ❝ ❝❤♦ ữợ (0; 2) (2; +) (; 0)✳ ❉✳ (−∞; 0) ∪ (2; +∞)✳ ❈➙✉ ✶✹✳ ❈➙✉ ✶✺✳ ❆✳ P = v s u r u 3 t ❈❤♦ ❜✐➸✉ t❤ù❝ P = ✳ ▼➺♥❤ ✤➲ ♥➔♦ tr♦♥❣ ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ ✤ó♥❣❄  18 ✳ ❇✳ P =   12 3 ✳ ❈✳ P =   18 ✳ ❉✳ P =   181 ✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ trö ❝â ❜→♥ ❦➼♥❤ R = 3, ❝❤✐➲✉ ❝❛♦ h = ❆✳ V = 90π ✳ ❇✳ V = 45✳ ❈✳ V = 45π ✳ ❉✳ V = 15π ✳ ❈➙✉ ✶✻✳ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ❤➺ tå❛ ✤ë Oxyz ✱ ❝❤♦ ♠➦t ♣❤➥♥❣ (P )✿ 2x − 2y + z + 2017 = 0✱ ✈➨❝✲tì ♥➔♦ tr♦♥❣ ❝→❝ tỡ ữủ ữợ ởt tỡ t ❝õ❛ (P )❄ ★✔ ★✔ ★✔ ★✔ ❆✳ n = (4; −4; 2)✳ ❇✳ n = (−2; 2; 1)✳ ❈✳ n = (1; −1; 4)✳ ❉✳ n = (1; −2; 2)✳ ❈➙✉ ✶✼✳ ❈❤♦ ❤➻♥❤ ❝❤â♣ S.ABC ❝â t❤➸ t➼❝❤ ❜➡♥❣ 1✳ ❚r➯♥ ❝↕♥❤ BC ❧➜② ✤✐➸♠ E s❛♦ ❝❤♦ BE = 2EC ✳ ❚➼♥❤ t❤➸ t➼❝❤ V ❝õ❛ ❦❤è✐ tù ❞✐➺♥ SAEB ✳ ❆✳ V = ✳ ❇✳ V = ✳ ❈✳ V = ✳ ❉✳ V = ✳ ❈➙✉ ✶✽✳ 3 ❚➼♥❤ tê♥❣ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ log2 (x + 3) = log2 x + + x2 − √ x − + x + 3✳ √ 2✳ ❉✳ S = 2✳ ❆✳ S = −1✳ ❇✳ S = 1✳ ❈✳ S = − ❈➙✉ ✶✾✳ x ●å✐ (D) ❤➻♥❤ ♣❤➥♥❣ ❣✐ỵ✐ ❤↕♥ ❜ð✐ ❝→❝ ✤÷í♥❣ y = ✱ y = 0✱ x = 1✱ x = 4✳ ❚➼♥❤ t❤➸ t➼❝❤ ✈➟t t❤➸ trá♥ ①♦❛② t↕♦ t❤➔♥❤ ❦❤✐ q✉❛② ❤➻♥❤ (D) q✉❛♥❤ trö❝ Ox✳ 15 21 15π 21π ❆✳ ✳ ❇✳ ✳ ❈✳ ✳ ❉✳ ✳ ❈➙✉ ✷✵✳ 16 16 16 ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Oxyz ✱ ❝❤♦ ♠➦t ❝➛✉ (S) : + + ❚➼♥❤ ❜→♥ ❦➼♥❤ R ❝õ❛ ♠➦t ❝➛✉ (S)✳ √ √ 151✳ ❈✳ R = 99✳ ❆✳ R = 7✳ ❇✳ R = ❈➙✉ ✷✶✳ x2 y2 z2 − 8x + 10y − 6z + 49 = 0✳ ❉✳ R = 1✳ ❚r❛♥❣ ✷✴✻ ▼➣ ✤➲ ✶✵✽ ❈➙✉ ✷✷✳ ❈❤♦ x, y > t❤ä❛ ♠➣♥ x + y = ❚➼♥❤ x2 + y ✳ 2313 ❆✳ ✳ 1156 ❈➙✉ ✷✸✳ ❆✳ 6✳ ❇✳ ✈➔ ❜✐➸✉ t❤ù❝ P = + ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t✳ x 4y 153 ✳ 100 ❈✳ 25 ✳ 16 ❉✳ ✳ ❑❤è✐ ❝❤â♣ t❛♠ ❣✐→❝ ✤➲✉ ❝â ♥❤✐➲✉ ♥❤➜t ❜❛♦ ♥❤✐➯✉ ♠➦t ♣❤➥♥❣ ✤è✐ ①ù♥❣❄ ❇✳ 4✳ ❈✳ 9✳ ❉✳ 3✳ ❈❤♦ sè ♣❤ù❝ z t❤ä❛ |z − + 2i| = 3✳ ❇✐➳t r➡♥❣ t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ❜✐➵✉ ❞✐➵♥ ❝õ❛ sè ♣❤ù❝ w = 2z + i tr➯♥ ♠➦t ♣❤➥♥❣ (Oxy) ❧➔ ♠ët ✤÷í♥❣ trá♥✳ ❚➻♠ t➙♠ ❝õ❛ ✤÷í♥❣ trá♥ ✤â✳ ❆✳ I(1; 1)✳ ❇✳ I(2; −3)✳ ❈✳ I(1; 0)✳ ❉✳ I(0; 1)✳ ❈➙✉ ✷✹✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❣✐→ trà t❤ü❝ ❝õ❛ t❤❛♠ sè m ✤➸ ❤➔♠ sè y = tr➯♥ ♠é✐ ❦❤♦↔♥❣ ①→❝ ✤à♥❤ ❝õ❛ ♥â✳ ❆✳ m ≤ −3✳ ❇✳ m < 1✳ ❈✳ m ≤ 1✳ ❈➙✉ ✷✺✳ x+2−m ♥❣❤à❝❤ ❜✐➳♥ x+1 ❉✳ m < −3✳ ❇✐➳t r➡♥❣ ❤➔♠ sè F (x) = mx3 + (3m + n)x2 − 4x + ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ sè f (x) = 3x2 + 10x − 4✳ ❚➼♥❤ mn✳ ❆✳ mn = 0✳ ❇✳ mn = 1✳ ❈✳ mn = 2✳ ❉✳ mn = 3✳ ❈➙✉ ✷✻✳ ❈➙✉ ✷✼✳ ❚➼❝❤ ♣❤➙♥ I = Z1 (x − 1)2 dx = a − ln b✱ tr♦♥❣ ✤â a; b ❧➔ ❝→❝ sè ♥❣✉②➯♥✳ ❚➼♥❤ ❣✐→ x2 + trà ❝õ❛ ❜✐➸✉ t❤ù❝ a + b✳ ❆✳ −1✳ ❇✳ 0✳ ❈✳ 3✳ ❉✳ 1✳ ●å✐ z1 ✱ z2 ❧➔ ❤❛✐ ♥❣❤✐➺♠ ♣❤ù❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ z − 4z + = 0✳ ❚➼♥❤ w = 1 + + i(z1 z2 + z2 z1 )✳ ❈➙✉ ✷✽✳ z1 ❆✳ z2 w = − + 20i✳ ❈➙✉ ✷✾✳ ❆✳ 1✳ ❈➙✉ ✸✵✳ ❆✳ y′ = + 20i✳ w = + 20i✳ √ ❈❤♦ sè t❤ü❝ a > 0, a 6= 1✳ ●✐→ trà log√a3 a2 ❜➡♥❣ ✳ ❈✳ ✳ ❇✳ ❇✳ w= ❈✳ ❉✳ w = 20 + i✳ ❉✳ ✳ ❉✳ y′ = ❚➼♥❤ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ sè y = log9 x2 + ✳ ln ✳ x2 + ❇✳ y′ = x ✳ (x2 + 1) ln ❈✳
y′ = ✳ (x2 + 1) ln 2x ln ✳ x2 + ❈❤♦ ❤➻♥❤ ❝❤â♣ S.ABCD ❝â ✤→② ❧➔ ❤➻♥❤ ❜➻♥❤ ❤➔♥❤ ✈➔ ❝â t❤➸ t➼❝❤ ❜➡♥❣ 48✳ ❚r➯♥ ❝↕♥❤ SB, SD ❧➜② ✤✐➸♠ ❝→❝ M, N s❛♦ ❝❤♦ SM = M B, SD = 3SN ✳ ▼➦t ♣❤➥♥❣ (AM N ) ❝➢t SC t↕✐ P ✳ ❚➼♥❤ t❤➸ t➼❝❤ V ❝õ❛ ❦❤è✐ tù ❞✐➺♥ SM N P ✳ ❈➙✉ ✸✶✳ ❆✳ V = 1✳ ❇✳ V = ✳ ❈✳ V = 2✳ ❉✳ V = ✳ ❚➼♥❤ sè ❣✐→ trà ♥❣✉②➯♥ ❝õ❛ t❤❛♠ sè m tr➯♥ ❦❤♦↔♥❣ (−2019; 2019) ✤➸ ❤➔♠ sè y = x4 2mx2 3m + ỗ ❜✐➳♥ tr➯♥ ❦❤♦↔♥❣ (1; 2)✳ ❆✳ 2020✳ ❇✳ 2✳ ❈✳ 1✳ ❉✳ 2019✳ ❈➙✉ ✸✷✳ ❚r❛♥❣ ✸✴✻ ▼➣ ✤➲ ✶✵✽ (D) ữủ ợ ữớ y = 2(x2 − 1); y = − x2✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ trá♥ ①♦❛② t↕♦ t❤➔♥❤ ❞♦ (D) q✉❛② q✉❛♥❤ trö❝ Ox✳ 64π 32π 32 64 ❆✳ ✳ ❇✳ ✳ ❈✳ ✳ ❉✳ ✳ 15 15 15 15 ❈➙✉ ✸✹✳ ❈❤♦ ❤➻♥❤ ❝❤â♣ S.ABCD ❝â ✤→② ABCD ❧➔ ❤➻♥❤ ✈✉æ♥❣✱ t➙♠ O ✱ ❝↕♥❤ a ✈➔ SO ⊥ √ (ABCD), SA = 2a 2✳ ●å✐ M, N ❧➛♥ ❧÷đt ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ SA, BC ✳ ❚➼♥❤ ❣â❝ ❣✐ú❛ ✤÷í♥❣ t❤➥♥❣ M N ✈➔ ♠➦t ♣❤➥♥❣ (ABCD)✳ π π π ❆✳ arctan 2✳ ❇✳ ✳ ❈✳ ✳ ❉✳ ✳ ❈➙✉ ✸✺✳ ▼ët ♥❣÷í✐ ❣û✐ sè t 100 tr ỗ ợ st 0, 5%/t❤→♥❣ ✈➔ ỉ♥❣ t❛ rót ✤➲✉ ✤➦♥ ♠é✐ t❤→♥❣ ởt tr ỗ tứ s ỷ ởt t ❝❤♦ ✤➳♥ ❦❤✐ ❤➳t t✐➲♥ ✭t❤→♥❣ ❝✉è✐ ❝ò♥❣ ❝â t❤➸ ổ ỏ ởt tr ỗ ọ ổ t rút ❤➳t t✐➲♥ s❛✉ ❜❛♦ ♥❤✐➯✉ t❤→♥❣❄ ❆✳ 139✳ ❇✳ 140✳ ❈✳ 138✳ ❉✳ 100✳ ❈➙✉ ✸✸✳ ❈➙✉ ✸✻✳ ❈❤♦ ❤➔♠ số f (x) õ f (x) ỗ t❤à ❝õ❛ ❤➔♠ sè y = f ′(x) ♥❤÷ ❤➻♥❤ ✈➩ ❜➯♥✳ ❚➼♥❤ sè ✤✐➸♠ ❝ü❝ trà ❝õ❛ √ √ ❤➔♠ sè y = f (x2) tr➯♥ ❦❤♦↔♥❣ (− 5; 5)✳ ❆✳ 3✳ ❇✳ 4✳ ❈✳ 5✳ ❉✳ 2✳ ② y = f ′ (x) ① ❚➼♥❤ tê♥❣ t➜t ❝↔ ❝→❝ ❣✐→ trà ❝õ❛ t❤❛♠ sè m tỗ t t ởt số ự z tọ ỗ tớ |z| = m |z 4m + 3mi| = m2✳ ❆✳ 6✳ ❇✳ 9✳ ❈✳ 4✳ ❉✳ 10✳ √ √ √ √ ❈➙✉ ✸✽✳ ❈❤♦ ❤❛✐ sè ♣❤ù❝ z, w t❤ä❛ ♠➣♥ |z − 2| = 2, |w − 2i| = 2✳ ❇✐➳t r➡♥❣ |z − w| ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t ❦❤✐ z = zo, w = wo✳ ❚➼♥❤ |3zo − wo|✳ √ √ √ ❇✳ 2✳ ❈✳ 2✳ ❉✳ 1✳ ❆✳ 2✳ ❈➙✉ ✸✼✳ ❈➙✉ ✸✾✳ ❙➠♠ ❧è♣ ①❡ ỉ tỉ ❦❤✐ ❜ì♠ ❝➠♥❣ ✤➦t ♥➡♠ tr➯♥ ♠➦t ♣❤➥♥❣ ♥➡♠ ♥❣❛♥❣ ❝â ❤➻♥❤ ❝❤✐➳✉ ❜➡♥❣ ♥❤÷ ❤➻♥❤ ✈➩ ợ ữớ trỏ ọ R1 = 20cm ữớ trỏ ợ R2 = 30cm t t ❦❤✐ ❝➢t ❜ð✐ ♠➦t ♣❤➥♥❣ ✤✐ q✉❛ trư❝✱ ✈✉ỉ♥❣ ❣â❝ ợ t ữớ trỏ ọ q✉❛ ✤ë ❞➔② ❝õ❛ ✈ä s➠♠✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤æ♥❣ ❦❤➼ ✤÷đ❝ ❝❤ù❛ ❜➯♥ tr♦♥❣ s➠♠✳ 3 3 ❆✳ 2500πcm ✳ ❇✳ 1250πcm ✳ ❈✳ 1400πcm ✳ ❉✳ 600πcm ✳ ❚r❛♥❣ ✹✴✻ ▼➣ ✤➲ ✶✵✽ ❈➙✉ ✹✵✳ ◆❣÷í✐ t t t ỡ t ữ ợ ❤❛✐ ✤➛✉ ❧➔ ❤❛✐ ❦❤è✐ trö ❜➡♥❣ ♥❤❛✉ ✈➔ t❛② ❝➛♠ ❝ơ♥❣ ❧➔ ❦❤è✐ trư✳ ❇✐➳t ❤❛✐ ✤➛✉ ❧➔ ❤❛✐ ❦❤è✐ trư ✤÷í♥❣ ❦➼♥❤ ✤→② ❜➡♥❣ ✶✷✱ ❝❤✐➲✉ ❝❛♦ ❜➡♥❣ ✻✱ ❝❤✐➲✉ ❞➔✐ t↕ ❜➡♥❣ ✸✵ ✈➔ ❜→♥ ❦➼♥❤ t❛② ❝➛♠ ❜➡♥❣ ✷✳ ❍➣② t➼♥❤ t❤➸ t➼❝❤ ✈➟t ❧✐➺✉ ❧➔♠ ♥➯♥ t↕ t❛② ✤â✳ ❆✳ 6480π ✳ ❇✳ 108π ✳ ❈✳ 504π ✳ ❉✳ 502π ✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Oxyz✱ ❝❤♦ ♠➦t ♣❤➥♥❣ (P ) : x + y + z − = ✈➔ ❜❛ ✤✐➸♠ ✤✐➸♠ ★✥✥✥✥✥✔ ★✥✥✥✥✥✥✔ ★✥✥✥✥✥✔ A(3; 1; 1), B(7; 3; 9) ✈➔ C(2; 2; 2)✳ ✣✐➸♠ M (a; b; c) tr➯♥ (P ) s❛♦ ❝❤♦ |M A + 2M B + 3M C| ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t✳ ❚➼♥❤ 2a − 15b + c✳ ❆✳ 8✳ ❇✳ 6✳ ❈✳ 1✳ ❉✳ 3✳ ❈➙✉ ✹✷✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Oxyz ✱ ❝❤♦ ♠➦t ♣❤➥♥❣ (P ) : x − y + = ✈➔ ❤❛✐ ✤✐➸♠ A(1; 2; 3), B(1; 0; 1)✳ ✣✐➸♠ C(a; b; −2) ∈ (P ) s❛♦ ❝❤♦ t❛♠ ❣✐→❝ ABC ❝â ❞✐➺♥ t➼❝❤ ♥❤ä ♥❤➜t✳ ❚➼♥❤ a + b✳ ❆✳ −3✳ ❇✳ 1✳ ❈✳ 2✳ ❉✳ 0✳ ❈➙✉ ✹✸✳ ❈❤♦ ❤➻♥❤ ❝❤â♣ S.ABCD ❝â SA ⊥ (ABCD)✱ ✤→② ABCD ❧➔ ❤➻♥❤ ❝❤ú ♥❤➟t ✈ỵ✐ √ √ AC = a ✈➔ BC = a 2✳ ❚➼♥❤ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ SD √ ✈➔ BC ✳ √ a 3a 2a ❆✳ a 3✳ ✳ ❈✳ ✳ ❉✳ ✳ ❇✳ ởt ỏ t ỗ ❤↕t ❣✐è♥❣ ♥❤❛✉✳ ❍ä✐ ❝â ❜❛♦ ♥❤✐➯✉ ❝→❝❤ ❝➢t ❝❤✐➳❝ ✈á♥❣ ✤â t❤➔♥❤ ✷ ♣❤➛♥ ♠➔ sè ❤↕t ð ♠é✐ ♣❤➛♥ ✤➲✉ ❧➔ sè ❧➫ ❄ ❆✳ 180✳ ❇✳ 90✳ ❈✳ 10✳ ❉✳ 5✳ x−2 s❛♦ ❝❤♦ ❦❤♦↔♥❣ ❝→❝❤ tø M ❈➙✉ ✹✺✳ ●å✐ M (a; b) ❧➔ ✤✐➸♠ tr➯♥ ỗ t số y = x ữớ t❤➥♥❣ d : y = 2x + ♥❤ä ♥❤➜t✳ ❚➼♥❤ (4a + 5)2 + (2b − 7)2✳ ❆✳ 0✳ ❇✳ 162✳ ❈✳ 2✳ ❉✳ 18✳ ❈➙✉ ✹✶✳ ❈➙✉ ✹✻✳ ❈❤♦ ❤➔♠ sè f (x) ❝â ✤↕♦ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ R ✈➔ t❤ä❛ ♠➣♥ Z1 ✱ f (x) dx = f (1) = cot ✳ ❚➼♥❤ t➼❝❤ ♣❤➙♥ I = Z1 f (x) tan2 x + f ′ (x) tan x dx
✳ ✳ ❇✳ 0✳ ❈✳ − cot 1✳ ❉✳ −1✳ ′ ❈➙✉ ✹✼✳ ❈❤♦ ❤➔♠ sè f (x) ①→❝ ✤à♥❤ tr➯♥ R✱ ❝â ✤↕♦ ❤➔♠ f (x) = (x + 1) (x − 2) (x + 3) ✳ ❙è ✤✐➸♠ ❝ü❝ trà ❝õ❛ ❤➔♠ sè f (|x|) ❧➔ ❆✳ 3✳ ❇✳ 2✳ ❈✳ 5✳ ❉✳ 1✳  11π  x ❈➙✉ ✹✽✳ ❚➼♥❤ sè ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ cotx = tr♦♥❣ ❦❤♦↔♥❣ ; 2019π 12 ❆✳ 2020✳ ❇✳ 2019✳ ❈✳ 1✳ ❉✳ 2018✳ ❆✳ − ln(cos 1) ❚r❛♥❣ ✺✴✻ ▼➣ ✤➲ ✶✵✽ ❈❤♦ ❤➔♠ sè f (x) ❝â ✤↕♦ ❤➔♠ f ′ (x) = (x − 1)(x2 − 3)(x4 − 1) ✈ỵ✐ ♠å✐ x t❤✉ë❝ R✳ ❙♦ s→♥❤ f (−2), f (0), f (2)✱ t❛ ✤÷đ❝ ❆✳ f (2) < f (0) < f (−2)✳ ❇✳ f (−2) < f (0) < f (2)✳ ❈✳ f (0) < f (−2) < f (2)✳ ❉✳ f (−2) < f (2) < f (0)✳ ❈➙✉ ✹✾✳ ❈➙✉ ✺✵✳ ❈❤♦ F (x) ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ sè f (x) = ✳ ❇✐➳t F cos2 x ♠å✐ k ∈ Z✳ ❚➼♥❤ F (0) + F (π) + F (2π) + + F (10π)✳ ❆✳ 44✳ ❇✳ 45✳ ❈✳ 55✳ ❉✳ π  + kπ = k ✈ỵ✐ 0✳ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❍➌❚✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❚r❛♥❣ ✻✴✻ ▼➣ ✤➲ ✶✵✽ ✣⑩P ⑩◆ ❇❷◆● ✣⑩P ⑩◆ ❈⑩❈ ▼❶ ✣➋ ▼➣ ✤➲ t❤✐ ✶✵✶ ✶✳ ❈ ✷✳ ❆ ✸✳ ❆ ✹✳ ❆ ✺✳ D ✻✳ ❇ ✼✳ ❉ ✽✳ ❆ ✾✳ ❇ ✶✵✳ ❈ ✶✶✳ A ✶✷✳ ❈ ✶✸✳ ❇ ✶✹✳ ❇ ✶✺✳ A ✶✻✳ ❉ ✶✼✳ ❆ ✶✽✳ ❇ ✶✾✳ ❈ ✷✵✳ ❇ ✷✶✳ ❇ ✷✷✳ ❈ ✷✸✳ ❈ ✷✹✳ ❈ ✷✺✳ ❆ ✷✻✳ ❈ ✷✼✳ ❇ ✷✽✳ ❉ ✷✾✳ ❆ ✸✵✳ ❈ ✸✶✳ ❈ ✸✷✳ ❇ ✸✸✳ ❇ ✸✹✳ ❈ ✸✺✳ ❆ ✸✻✳ ❆ ✸✼✳ ❆ ✸✽✳ ❇ ✸✾✳ ❆ ✹✵✳ ❆ ✹✶✳ ❉ ✹✷✳ ❇ ✹✸✳ ❇ ✹✹✳ ❇ ✹✺✳ ❆ ✹✻✳ ❉ ✹✼✳ ❉ ✹✽✳ ❉ ✹✾✳ ❇ ✺✵✳ ❇ ▼➣ ✤➲ t❤✐ ✶✵✷ D ✸✳ ❇ ✹✳ ❉ ✺✳ ❇ ✻✳ ❇ B ✶✶✳ ❉ ✶✷✳ ❆ ✶✸✳ ❇ ✶✹✳ ❈ ✶✾✳ ❇ ✷✵✳ ❇ ✷✶✳ ❉ ✷✷✳ ❇ ✷✽✳ ❆ ✷✾✳ ❉ ✸✵✳ ❇ ✸✼✳ ❉ ✸✽✳ ❉ ✹✻✳ ❆ ✹✼✳ ❈ ✶✳ ❇ ✶✵✳ ✷✳ A ✽✳ ❉ ✾✳ ❇ ✶✺✳ ❇ ✶✻✳ ❈ ✶✼✳ ❉ ✶✽✳ ❇ ✷✸✳ ❈ ✷✹✳ ❈ ✷✺✳ ❈ ✷✻✳ ❈ ✷✼✳ ❉ ✸✶✳ ❈ ✸✷✳ ❈ ✸✸✳ ❇ ✸✹✳ ❆ ✸✺✳ ❆ ✸✻✳ ❆ ✸✾✳ ❉ ✹✵✳ ❆ ✹✶✳ ❆ ✹✷✳ ❉ ✹✸✳ ❉ ✹✹✳ ❈ ✹✺✳ ❇ ✹✽✳ ❈ ✹✾✳ ❉ ✺✵✳ ❇ ✼✳ ▼➣ ✤➲ t❤✐ ✶✵✸ D ✸✳ ❆ ✹✳ ❈ ✶✵✳ ❆ ✶✶✳ ❈ ✶✷✳ ❈ ✶✸✳ ❆ ✶✹✳ ✶✾✳ ❉ ✷✵✳ ❈ ✷✶✳ ❇ ✷✽✳ ❆ ✷✾✳ ❉ ✸✼✳ ❉ ✹✻✳ ❇ ✶✳ ❉ ✷✳ A ✽✳ ❈ ✾✳ ❆ ✶✺✳ ❈ ✶✻✳ ❈ ✶✼✳ ❈ ✶✽✳ ❈ ✷✸✳ ❈ ✷✹✳ ❆ ✷✺✳ ❆ ✷✻✳ ❈ ✷✼✳ ❇ ✸✶✳ ❉ ✸✷✳ ❆ ✸✸✳ ❈ ✸✹✳ ❆ ✸✺✳ ❈ ✸✻✳ ❆ ✸✾✳ ❉ ✹✵✳ ❆ ✹✶✳ ❆ ✹✷✳ ❈ ✹✸✳ ❈ ✹✹✳ ❆ ✹✺✳ ❇ ✹✽✳ ❉ ✹✾✳ ❇ ✺✵✳ ❈ ✺✳ ❉ ✻✳ ❇ B ✷✷✳ ❇ ✸✵✳ ❉ ✸✽✳ ❇ ✹✼✳ ❆ ✼✳ ▼➣ ✤➲ t❤✐ ✶✵✹ ✶✳ ❆ ✷✳ ❆ ✸✳ ❆ ✹✳ ❉ ✺✳ ❉ ✻✳ ❆ ✶✵✳ ❇ ✶✶✳ ❈ ✶✷✳ ❉ ✶✸✳ ❈ ✶✹✳ ❇ ✶✺✳ ✶✾✳ ❉ ✷✵✳ ❇ ✷✶✳ ❉ ✷✷✳ ❇ ✷✸✳ ❉ ✷✽✳ ❆ ✷✾✳ ❉ ✸✵✳ ❆ ✸✶✳ ❉ ✸✼✳ ❆ ✸✽✳ ❉ ✸✾✳ ❉ ✹✵✳ ❆ ✹✻✳ ❇ ✹✼✳ ❆ ✹✽✳ ❈ ✹✾✳ ❇ ✽✳ ❆ ✾✳ D D ✶✼✳ ❈ ✶✽✳ ❈ ✷✹✳ ❆ ✷✺✳ ❇ ✷✻✳ ❉ ✷✼✳ ❆ ✸✷✳ ❉ ✸✸✳ ❉ ✸✹✳ ❇ ✸✺✳ ❉ ✸✻✳ ❆ ✹✶✳ ❉ ✹✷✳ ❆ ✹✸✳ ❈ ✹✹✳ ❉ ✹✺✳ ❉ ✺✵✳ ❈ ✶ B ✼✳ ❇ ✶✻✳ ▼➣ ✤➲ t❤✐ ✶✵✺ C ✺✳ ❇ ✶✷✳ ❇ ✶✸✳ ❈ ✶✹✳ ❆ ✶✺✳ ✷✵✳ ❇ ✷✶✳ ❉ ✷✷✳ ❆ ✷✽✳ ❇ ✷✾✳ ❈ ✸✵✳ ❈ ✸✼✳ ❆ ✸✽✳ ❉ ✹✻✳ ❉ ✹✼✳ ❇ ✶✳ ❆ ✷✳ ❆ ✸✳ ❇ ✶✵✳ ❇ ✶✶✳ ❈ ✶✾✳ ❉ ✹✳ ✾✳ D ✻✳ ❉ ✼✳ ❆ ✽✳ ❇ D ✶✻✳ ❆ ✶✼✳ ❈ ✶✽✳ ❇ ✷✸✳ ❆ ✷✹✳ ❈ ✷✺✳ ❇ ✷✻✳ ❆ ✷✼✳ ❇ ✸✶✳ ❈ ✸✷✳ ❆ ✸✸✳ ❇ ✸✹✳ ❆ ✸✺✳ ❉ ✸✻✳ ❈ ✸✾✳ ❇ ✹✵✳ ❈ ✹✶✳ ❆ ✹✷✳ ❆ ✹✸✳ ❆ ✹✹✳ ❈ ✹✺✳ ❇ ✹✽✳ ❈ ✹✾✳ ❉ ✺✵✳ ❉ ▼➣ ✤➲ t❤✐ ✶✵✻ ✶✳ ❉ ✷✳ ❉ ✸✳ ❈ ✹✳ ❈ ✺✳ ❆ ✶✵✳ ❆ ✶✶✳ ❉ ✶✷✳ ✶✾✳ ❆ ✻✳ ❇ ✼✳ ❉ ✽✳ ❆ ✾✳ ❆ B ✶✸✳ ❈ ✶✹✳ ❉ ✶✺✳ D ✶✼✳ ❈ ✶✽✳ ❇ ✷✵✳ ❉ ✷✶✳ ❉ ✷✷✳ ❇ ✷✸✳ ❇ ✷✹✳ ❉ ✷✺✳ ❈ ✷✻✳ ❇ ✷✼✳ ❇ ✷✽✳ ❇ ✷✾✳ ❆ ✸✵✳ ❉ ✸✶✳ ❇ ✸✷✳ ❆ ✸✸✳ ❆ ✸✹✳ ❇ ✸✺✳ ❈ ✸✻✳ ❆ ✸✼✳ ❇ ✸✽✳ ❈ ✸✾✳ ❈ ✹✵✳ ❈ ✹✶✳ ❇ ✹✷✳ ❆ ✹✸✳ ❆ ✹✹✳ ❆ ✹✺✳ ❆ ✹✻✳ ❈ ✹✼✳ ❉ ✹✽✳ ❉ ✹✾✳ ❉ ✺✵✳ ❆ D ✶✻✳ ▼➣ ✤➲ t❤✐ ✶✵✼ B ✽✳ C ✾✳ ❆ ✶✺✳ ❉ ✶✻✳ ❈ ✶✼✳ D ✶✽✳ ❆ ✷✸✳ ❉ ✷✹✳ ❈ ✷✺✳ ❉ ✷✻✳ ❆ ✷✼✳ ❇ ✸✶✳ ❆ ✸✷✳ ❆ ✸✸✳ ❈ ✸✹✳ ❇ ✸✺✳ ❇ ✸✻✳ ❉ ✸✾✳ ❆ ✹✵✳ ❆ ✹✶✳ ❇ ✹✷✳ ❉ ✹✸✳ ❆ ✹✹✳ ❉ ✹✺✳ ❈ ✹✽✳ ❆ ✹✾✳ ❇ ✺✵✳ ❈ ✶✳ ❉ ✷✳ ❈ ✸✳ ❆ ✹✳ ❇ ✺✳ ❈ ✻✳ ❆ ✶✵✳ ❉ ✶✶✳ ❆ ✶✷✳ ❆ ✶✸✳ ❇ ✶✹✳ ❆ ✶✾✳ ❉ ✷✵✳ ❈ ✷✶✳ ❆ ✷✷✳ ❈ ✷✽✳ ❈ ✷✾✳ ❈ ✸✵✳ ❇ ✸✼✳ ❆ ✸✽✳ ❉ ✹✻✳ ❉ ✹✼✳ ❇ ✼✳ ▼➣ ✤➲ t❤✐ ✶✵✽ B ✸✳ ❉ D ✺✳ ❈ ✻✳ ❇ ✼✳ ❇ ✶✵✳ ❇ ✶✶✳ ❈ ✶✷✳ ❇ ✶✸✳ ❉ ✶✹✳ ❆ ✶✺✳ ❇ ✶✻✳ ❈ ✶✼✳ ✶✾✳ ❇ ✷✵✳ ❉ ✷✶✳ ❉ ✷✷✳ ❇ ✷✸✳ ❆ ✷✹✳ ❇ ✷✽✳ ❇ ✷✾✳ ❈ ✸✵✳ ❇ ✸✶✳ ❆ ✸✷✳ ❆ ✸✼✳ ❉ ✸✽✳ ❈ ✸✾✳ ❇ ✹✵✳ ❈ ✹✶✳ ❆ ✹✻✳ ❇ ✹✼✳ ❆ ✹✽✳ ❉ ✹✾✳ ❆ ✺✵✳ ❆ ✶✳ ❇ ✷✳ ✹✳ ✷ ✽✳ ❆ ✾✳ ❉ A ✶✽✳ ❇ ✷✺✳ ❇ ✷✻✳ ❈ ✷✼✳ ❈ ✸✸✳ ❆ ✸✹✳ ❈ ✸✺✳ ❆ ✸✻✳ ❆ ✹✷✳ ❉ ✹✸✳ ❆ ✹✹✳ ❉ ✹✺✳ ❉ ... (D) q✉❛♥❤ trö❝ Ox✳ 15 21? ? 21 15π ❆✳ ✳ ❇✳ ✳ ❈✳ ✳ ❉✳ ✳ ❈➙✉ ✷✺✳ 16 16 16 ●å✐ z1 ✱ z2 ❧➔ ❤❛✐ ♥❣❤✐➺♠ ♣❤ù❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ z − 4z + = 0✳ ❚➼♥❤ w = 1 + + i(z1 z2 + z2 z1 )✳ ❈➙✉ ✷✻✳ z1 ❆✳ z2 w = − + 20i✳... ❆✳ 2 019 ✳ ❇✳ 2 018 ✳ ❈✳ 1? ?? ❈➙✉ ✹✾✳ 2x tr♦♥❣ ❦❤♦↔♥❣ ❉✳ V = 1? ??  11 π ; 2 019 π 12 ❉✳ 2020✳  ❚r❛♥❣ ✺✴✻ ▼➣ ✤➲ ✶✵✶ ❈➙✉ ✺✵✳ ❈❤♦ ❤➔♠ sè f (x) ❝â ✤↕♦ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ R ✈➔ t❤ä❛ ♠➣♥ Z1 f (x) dx = 1? ?? f (1) ... −3)✳ ❇✳ I(0; 1) ✳ ❈✳ I (1; 0)✳ ❉✳ I (1; 1) ✳ ❈➙✉ ✷✾✳ ❈❤♦ x, y > t❤ä❛ ♠➣♥ x + y = ❚➼♥❤ x2 + y ✳ 25 ✳ ❆✳ 16 ❈➙✉ ✸✵✳ ✈➔ ❜✐➸✉ t❤ù❝ P = + ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t✳ x 4y 2 313 15 3 ✳ ❈✳ ✳ ❉✳ ✳ 11 56 10 0 ❚r♦♥❣ ❦❤æ♥❣