❑■➎▼ ❚❘❆ ✣➚◆❍ ❑Ý ❍➴❈ ❑Ý ■■ ▼➷◆ ❚❖⑩◆ ✲ ▲❰P ✶✷ ◆❣➔② ❦✐➸♠ tr❛✿ .✴ .✴✷✵✶✽ ❙Ð ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❆◆ ●■❆◆● ❚❘×❮◆● ❚❍P❚ ❱➹ ❚❍⑨◆❍ ❚❘■◆❍ ✖✖✖✖✖✖✖✖✖✖✕ ✣➲ ❝â ✷ tr❛♥❣ ❚❤í✐ ❣✐❛♥ ❧➔♠ ❜➔✐✿ ✹✺ ♣❤ót ▼➣ ✤➲✿ ✶ ❈❹❯ ❍➘■ ❚❘➁❈ ◆●❍■➏▼ ❈➙✉ ✶✳ ❚➻♠ t➟♣ ♥❣❤✐➺♠ S ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ z + z − = tr➯♥ tr÷í♥❣ sè ♣❤ù❝✳ ❆✳ S = {−1 − i; −1 + i}✳ ❇✳ S = {1; − i; + i}✳ ❈✳ S = {1; −1 − i; −1 + i}✳ ❉✳ S = {1}✳ ❈➙✉ ✷✳ ❈❤♦ sè ♣❤ù❝ z = + 3i√✳ ●✐→ trà ❝õ❛ |2iz − z| ❜➡♥❣ ❈✳ 113✳ ❉✳ √113✳ ❆✳ 15✳ ❇✳ 15✳ ❈➙✉ ✸✳ ❈❤♦ sè ♣❤ù❝ z t❤ä❛ ♠➣♥ |iz + 1| = 3✳ ●✐→ trà ♥❤ä ♥❤➜t ❝õ❛ |z| ❜➡♥❣ ❆✳ 4✳ ❇✳ 3✳ ❈✳ 1✳ ❉✳ 2✳ ❈➙✉ ✹✳ ●å✐ z ✱ z ❧❛ ❤❛✐ ♥❣❤✐➺♠ ♣❤ù❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ z −2z +2 = 0✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ P = z1 + z1 ✳ ❆✳ P = 1✳ ❇✳ P = 4✳ ❈✳ P = 0✳ ❉✳ P = √2✳ √ ❈➙✉ ✺✳ ❈❤♦ w = (5 − 2i)(−3 + 2i) ✳ ●✐→ trà ❝õ❛ 2|w| − 377 ❜➡♥❣ ❆✳ −10√377✳ ❇✳ 10√377✳ ❈✳ 7√377✳ ❉✳ −3√377✳ ❈➙✉ √✻✳ ❈❤♦ sè ♣❤ù❝ z t❤ä❛ ♠➣♥ 2z − (3 + 4i) = − 2i✳ √▼æ✲✤✉♥ ❝õ❛ z ❜➡♥❣ ❜❛♦ ♥❤✐➯✉❄ ❇✳ 5✳ ❈✳ 17✳ ❉✳ √29✳ ❆✳ 15✳ ❈➙✉ ✼✳ ❈❤♦ sè ♣❤ù❝ z t❤ä❛ ♠➣♥ 3z − 2(z − 1) + − 5i = 0✳ ❚➼♥❤ ♠æ✲✤✉♥ ❝õ❛ z ✳ ❆✳ |z| = √11✳ ❇✳ |z| = 121✳ ❈✳ |z| = 11✳ ❉✳ |z| = √101✳ ❈➙✉ ✽✳ ❈❤♦ sè ♣❤ù❝ z t❤ä❛ ♠➣♥ z −√(1 − 3i)(−2 + i) = 2i✳ ❚➼♥❤ |z|✳ ❈✳ |z| = √82✳ ❉✳ |z| = 4√5✳ ❆✳ |z| = 2✳ ❇✳ |z| = 2✳ ❈➙✉ ✾✳ ●å✐ z ✱ z ✱ z ✱ z ❧➔ ❝→❝ ♥❣❤✐➺♠ ♣❤ù❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ z − 4z − 27 = 0✳ ●✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ T = |z | + |z |√+ |z | + |z | ❜➡♥❣ ❜❛♦ ♥❤✐➯✉❄ ❆✳ T = + 3✳ ❇✳ T = + 2√3✳ ❈✳ T = 12✳ ❉✳ T = + √6✳ ❈➙✉ ✶✵✳ ❙è ♣❤ù❝ ❧✐➯♥ ❤ñ♣ ❝õ❛ sè ♣❤ù❝ z = − 21 + 32 i ❧➔ ❆✳ z = 32 − 21 i✳ ❇✳ z = − 12 − 32 i✳ ❈✳ z = 12 − 32 i✳ ❉✳ z = 12 + 32 i✳ 2 1 2 3 4 2 ❈➙✉ ✶✶✳ ❈❤♦ ❤❛✐ sè ♣❤ù❝ z = − i ✈➔ z2 = −3 + 2i✳ ●✐→✐ trà ❝õ❛ √ 154 ✳ 13 z1 ❜➡♥❣ ❜❛♦ ♥❤✐➯✉❄ z2 82 ✳ 13 616 ❆✳ ❇✳ 169 ✳ ❈✳ 82 ✳ ❉✳ 13 ❈➙✉ ✶✷✳ ❈❤♦ ❝→❝ sè ♣❤ù❝ z = − 5i ✈➔ z = −2 − 3i✳ ❍➣② t➼♥❤ |z − z |✳ ❇✳ 20✳ ❈✳ 12✳ ❉✳ 2√3✳ ❆✳ 2√5✳ ❈➙✉ ✶✸✳ ❈❤♦ sè ♣❤ù❝ w = + 4i✳ ●✐→ trà ❝õ❛ S = 2|w| − ❜➡♥❣ ❜❛♦ ♥❤✐➯✉❄ ❆✳ S = 10✳ ❇✳ S = 9✳ ❈✳ S = 11✳ ❉✳ S = 5✳ ❈➙✉ ✶✹✳ ❈❤♦ sè ♣❤ù❝ z t❤ä❛ ♠➣♥ z = (3 − i)(4 − 2i)✳ ❍➣② t➼♥❤ ♠æ✲✤✉♥ ❝õ❛ sè ♣❤ù❝ w = iz − 11 −+ ii ✳ ❆✳ |w| = 19✳ ❇✳ |w| = √181✳ ❈✳ |w| = 181✳ ❉✳ |w| = √19✳ ❈➙✉ ✶✺✳ ❇✐➳t r➡♥❣ sè ♣❤ù❝ ❧✐➯♥ ❤ñ♣ ❝õ❛ z ❧➔ z = (2 + 3i) + (4 − 8i)✳ ❚➻♠ sè ♣❤ù❝ z✳ ❆✳ z = −6 − 5i✳ ❇✳ z = + 5i✳ ❈✳ z = −6 + 5i✳ ❉✳ z = − 5i✳ ❈➙✉ ✶✻✳ ❚r➯♥ ♠➦t ♣❤➥♥❣ tå❛ ✤ë Oxy✱ ✤✐➸♠ ❜✐➸✉ ❞✐➵♥ ❝õ❛ sè ♣❤ù❝ z = −5 − 6i ❧➔ ✤✐➸♠ ♥➔♦ s❛✉ ✤➙②❄ ❆✳ P (5; −6)✳ ❇✳ Q(5; 6)✳ ❈✳ M (−5; 6)✳ ❉✳ N (−5; −6)✳ 2 ❚r❛♥❣ ✶✴✷ ▼➣ ✤➲ ✶ ❈➙✉ ✶✼✳ ❚➻♠ sè ♣❤ù❝ z t❤ä❛ ♠➣♥ |z| = ✈➔ ♣❤➛♥ t❤ü❝ ❝õ❛ z ❣➜♣ ❧➛♥ ♣❤➛♥ ↔♦ ❝õ❛ √ ❤❛✐ √ √ ♥â✳ √ ❆✳ z = 2√√5 + i√√5 ❤♦➦❝ z = −2√√5 − √i√5✳ ❇✳ z = −2 + i ❤♦➦❝ z = √ √ √ − i√ 5✳ ❉✳ z = + 5i ❤♦➦❝ z = − − 5i✳ ❈✳ z = − + 5i ❤♦➦❝ z = − 5i✳ ❈➙✉ ✶✽✳ ❚r♦♥❣ ♠➦t ♣❤➥♥❣ tå❛ ✤ë Oxy✱ ❝❤♦ ✤✐➸♠ M ❧➔ ✤✐➸♠ ❜✐➸✉ ❞✐➵♥ ❝❤♦ sè ♣❤ù❝ z = a + bi✳ ❚➼♥❤ S = a + b✳ y O M x ❆✳ S = 4✳ ❇✳ S = 1✳ ❈✳ S = 2✳ ❉✳ S = 3✳ ❈➙✉ ✶✾✳ ❈❤♦ sè ♣❤ù❝ z = − 3i✳ ▼➺♥❤ ✤➲ ♥➔♦ s❛✉ ✤➙② ✤ó♥❣❄ ❆✳ P❤➛♥ t❤ü❝ ❝õ❛ z ❜➡♥❣ ✈➔ ♣❤➛♥ ↔♦ ❝õ❛ z ❜➡♥❣ 3i✳ ❇✳ P❤➛♥ t❤ü❝ ❝õ❛ z ❜➡♥❣ ✈➔ ♣❤➛♥ ↔♦ ❝õ❛ z ❜➡♥❣ 3✳ ❈✳ P❤➛♥ t❤ü❝ ❝õ❛ z ❜➡♥❣ ✈➔ ♣❤➛♥ ↔♦ ❝õ❛ z ❜➡♥❣ −3i✳ ❉✳ P❤➛♥ t❤ü❝ ❝õ❛ z ❜➡♥❣ ✈➔ ♣❤➛♥ ↔♦ ❝õ❛ z ❜➡♥❣ −3✳ ❈➙✉ ✷✵✳ ❈❤♦ sè ♣❤ù❝ w t❤ä❛ ♠➣♥ (3 − 2i)w = + 2i✳ ❚➻♠ sè ♣❤ù❝ ❧✐➯♥ ❤ñ♣ ❝õ❛ w✳ 14 ❆✳ w = − 2i✳ i✳ ❉✳ − i✳ ❇✳ w = + 2i✳ ❈✳ w = 138 + 14 w= 13 13 13 ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❍➌❚✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❚r❛♥❣ ✷✴✷ ▼➣ ✤➲ ✶ ❇❷◆● ✣⑩P ⑩◆ ❈⑩❈ ▼❶ ✣➋ ▼➣ ✤➲ t❤✐ ✶ ✶✳ ❈ ✷✳ ❉ ✸✳ ❉ ✹✳ ❆ ✺✳ ❉ ✻✳ ❈ ✼✳ ❉ ✽✳ ❈ ✾✳ ❇ ✶✶✳ ✶✵✳ ❉ ✶✷✳ ❆ ✶✸✳ ❇ ✶✹✳ ✶✺✳ ❇ ✶✻✳ ✶✼✳ ❆ ✶✾✳ ❇ ❇ ❉ ✶✽✳ ❆ ✷✵✳ ❉ ✶ ❉ ✣⑩P ❈❍■ ❚■➌❚ ▼❶ ✣➋ ✶ ❈➙✉ ✶✳ z + z − = ⇔ (z − 1)(z + 2z + 2) = ⇔ z=1 z−1=0 ⇔ z = −1 ± i z + 2z + = ❱➟② t➟♣ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ S = {1; −1 − i; −1 + i}✳ ❈❤å♥ ✤→♣ →♥ ❈ ❈➙✉ ✷✳ |2iz − z| = |2i(2 + 3i) − (2 − 3i)| = |−8 + 7i| = (−8)2 + 72 = √ 113✳ ❈❤å♥ ✤→♣ →♥ ❉ ❈➙✉ ✸✳ ✣➦t z = x + yi ✈ỵ✐ x, y ∈ R✳ ●å✐ M (x; y) ❧➔ ✤✐➸♠ ❜✐➸✉ ❞✐➵♥ ❝õ❛ sè ♣❤ù❝ z tr➯♥ ♠➦t ♣❤➥♥❣ Oxy ✳ ❑❤✐ ✤â OM = |z|✳ ❚❛ ❝â y Q |iz + 1| = ⇔ |xi − y + 1| = ⇔ x2 + (y − 1)2 = 32 ❙✉② r❛ M ♥➡♠ tr➯♥ ✤÷í♥❣ trá♥ t➙♠ I(0; 1) ❜→♥ ❦➼♥❤ R = 3✳ ●å✐ P (0; −2) ✈➔ Q(0; 4)✳ ❚❛ ❝â OP ≤ OM ≤ OQ✳ ❙✉② r❛ OM ♥❤ä ♥❤➜t ❦❤✐ M ≡ P ✳ ❉♦ ✤â M (0; −2)✳ I O M x −2 P ❱➟② ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ |z| ❜➡♥❣ OM = ❈❤å♥ ✤→♣ →♥ ❉ ❈➙✉ ✹✳ P❤÷ì♥❣ tr➻♥❤ z (0 − 0)2 + (−2 − 0)2 = 2✳ − 2z + = ❝â ∆ = (−1)2 − · = −1 = i2 ✳ ❱➟② ♣❤÷ì♥❣ tr➻♥❤ ❝â ❤❛✐ ♥❣❤✐➺♠ ♣❤ù❝ z1 = − i ✈➔ z2 = + i✳ 1 1 = + = |1| = 1✳ + ❱➟② P = z1 z2 1−i 1+i ❈❤å♥ ✤→♣ →♥ ❆ ❈➙✉ ✺✳ ❚❛ ❝â w = (5 − 2i)(−3 + 2i) =√−11 + 16i✳ + 162 = ❑❤✐ ✤â |w| =√|w| = (−11) 377√ ✳ √ √ ❱➟② 2|w| − 377 = 377 − 377 = −3 377✳ ❈❤å♥ ✤→♣ →♥ ❉ ❈➙✉ ✻✳ 2z − (3 + 4i) = − 2i ⇔ 2z = − 2i + + 4i ⇔ 2z = + 2i ⇔ z = + i √ ❱➟② |z| = 42 + 12 = 17✳ ❈❤å♥ ✤→♣ →♥ ❈ √ ❈➙✉ ✼✳ ✣➦t z = x + yi ✈ỵ✐ x, y ∈ R✳ ❑❤✐ ✤â z = x − yi✳ ✷ ❚❛ ❝â 3z − 2(z − 1) + − 5i = ⇔3(x − yi) − 2(x + yi − 1) + − 5i = ⇔3x − 3yi − 2x − 2yi + + − 5i = ⇔(x + 10) + (−5y − 5)i = ⇔ x + 10 = − 5y − = ⇔ x = −10 y = −1 ❈❤♦ ♥➯♥ z = −10 − i ✈➔ z = √−10 + i✳ ❱➟② |z| = (−10)2 + 12 = 101✳ ❈❤å♥ ✤→♣ →♥ ❉ ❈➙✉ ✽✳ ❚❛ ❝â z√− (1 − 3i)(−2 √ + i) = 2i ⇔ z = 2i + (1 − 3i)(−2 + i) ⇔ z = + 9i✳ ❱➟② |z| = |z| = 12 + 92 = ❈❤å♥ ✤→♣ →♥ ❈ 82✳ ❈➙✉ ✾✳ z = ±3 z2 = √ z − 6z − 27 = ⇔ ⇔ z = −3 z = ±i √ √ √ ❱➟② T = + + + = + 3✳ ❈❤å♥ ✤→♣ →♥ ❇ ❈➙✉ ✶✵✳ ❙è ♣❤ù❝ ❧✐➯♥ ❤ñ♣ ❝õ❛ sè ♣❤ù❝ z = − 12 + 32 i ❧➔ z = − 12 − 32 i✳ ❈❤å♥ ✤→♣ →♥ ❇ ❈➙✉ ✶✶✳ 29 15 9−i z1 = = − − i = z2 −3 + 2i 13 13 29 − 13 15 + − 13 = 82 ✳ 13 ❈❤å♥ ✤→♣ →♥ ❉ ❈➙✉ ✶✷✳ |z − z2 | = |2 − 5i − (−2 − 3i)| = |4 − 2i| = ❈❤å♥ ✤→♣ →♥ ❆ √ ❚❛ ❝â |w| = 32 + 42 = 5✳ ❱➟② S = · − = 9✳ ❈❤å♥ ✤→♣ →♥ ❇ √ 42 + (−2)2 = 5✳ ❈➙✉ ✶✸✳ ❈➙✉ ✶✹✳ ❚❛ ❝â z = (3 − i)(4 − 2i) = 10 − 10i✳ 1+i ❑❤✐ ✤â w = i(10 − 10i) − = 10 + 10i − i = 10 + 9i✳ √1−i √ ❱➟② |w| = 102 + 92 = 181✳ ❈❤å♥ ✤→♣ →♥ ❇ ❈➙✉ ✶✺✳ ❚❛ ❝â z = (2 + 3i) + (4 − 8i) = (2 + 4) + (3 − 8)i = − 5i✳ ❱➟② z = z = + 5i✳ ❈❤å♥ ✤→♣ →♥ ❇ ❈➙✉ ✶✻✳ ❚r➯♥ ♠➦t ♣❤➥♥❣ tå❛ ✤ë Oxy✱ ✤✐➸♠ ❜✐➸✉ ❞✐➵♥ ❝õ❛ sè ♣❤ù❝ z = −5 − 6i ❧➔ ✤✐➸♠ N (−5; −6)✳ ❈❤å♥ ✤→♣ →♥ ❉ ✸ ❈➙✉ ✶✼✳ ✣➦t z = x + yi ✈ỵ✐ x, y ∈ R✳ ❚❤❡♦ ✤➲ ❜➔✐ t❛ ❝â  x = 2y ⇔ x2 + y = 25   x = 2y  ⇔  y2 =  √ x=2 √ y= √ x = −2 √ y = − √ √ √ √ ❱➟② z = + i ❤♦➦❝ z = −2 − i 5✳ ❈❤å♥ ✤→♣ →♥ ❆ ❈➙✉ ✶✽✳ ❚❛ ❝â M (3; 1) ♥➯♥ z = + i✳ ❈❤♦ ♥➯♥ S = + = 4✳ ❈❤å♥ ✤→♣ →♥ ❆ ❈➙✉ ✶✾✳ P❤➛♥ t❤ü❝ ❝õ❛ z ❜➡♥❣ ✈➔ ♣❤➛♥ ↔♦ ❝õ❛ z ❜➡♥❣ −3✳ ❈❤å♥ ✤→♣ →♥ ❉ ❈➙✉ ✷✵✳ (3 − 2i)w = + 2i ⇔ w = 14 + 2i ⇔w= + i − 2i 13 13 14 − i✳ 13 13 ❈❤å♥ ✤→♣ →♥ ❉ ❱➟② w = ❉✉②➺t t trữ rữỡ ũ ủ ợ ♥❣➔② ✶✻ t❤→♥❣ ✵✸ ♥➠♠ ✷✵✶✽ ◆❣÷í✐ s♦↕♥ ❈❛♦ ❚❤➔♥❤ ❚❤→✐ ... 2|w| − 377 = 377 − 377 = −3 377✳ ❈❤å♥ ✤→♣ →♥ ❉ ❈➙✉ ✻✳ 2z − (3 + 4i) = − 2i ⇔ 2z = − 2i + + 4i ⇔ 2z = + 2i ⇔ z = + i √ ❱➟② |z| = 42 + 12 = 17✳ ❈❤å♥ ✤→♣ →♥ ❈ √ ❈➙✉ ✼✳ ✣➦t z = x + yi ✈ỵ✐ x, y ∈ R✳... z = − 12 + 32 i ❧➔ z = − 12 − 32 i✳ ❈❤å♥ ✤→♣ →♥ ❇ ❈➙✉ ✶✶✳ 29 15 9−i z1 = = − − i = z2 −3 + 2i 13 13 29 − 13 15 + − 13 = 82 ✳ 13 ❈❤å♥ ✤→♣ →♥ ❉ ❈➙✉ ✶✷✳ |z − z2 | = |2 − 5i − (−2 − 3i)| = |4 − 2i|... 3i)| = |4 − 2i| = ❈❤å♥ ✤→♣ →♥ ❆ √ ❚❛ ❝â |w| = 32 + 42 = 5✳ ❱➟② S = · − = 9✳ ❈❤å♥ ✤→♣ →♥ ❇ √ 42 + (−2)2 = 5✳ ❈➙✉ ✶✸✳ ❈➙✉ ✶✹✳ ❚❛ ❝â z = (3 − i) (4 − 2i) = 10 − 10i✳ 1+i ❑❤✐ ✤â w = i(10 − 10i) − =