Tài liệu Chuyên đề số phức xác suất

To¸n häc Trung t©m luyÖn thi LuyÖn thi ®¹i häc To¸n häc LuyÖn thi ®¹i häc CHINH PHôC CHUY£N §Ò: S PH C tHÇY HIÕU LIVE HäC VI£N: T¤I QUYÕT T¢M THI §ËU §¹I HäC To¸n häc Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live MỤC LỤC GIÁO ÁN SỐ PHỨC STT BUỔI Kiến thức Trang 1. Kiến thức cơ bản về số phức 2 2. Dạng 1: Các lý thuyết cơ bản về số phức 3 3. Bài toán 1: Công trừ số phức 3 4. Bài toán 2: Nhân hai số phức 3 Bài toán 3: Môđun số phức 3 6. Bài toán 4: Số phức liên hợp 4 7. Bài toán 5: Chia hai số phức 5 5. BUỔI 1 Dạng 2: Tìm số phức z thỏa mãn phương trình 8. 8 Bài toán 1: Tìm số phức z thỏa mãn PT chỉ chứa z hoặc z 8 Bài toán 2: Tìm số phức z thỏa mãn PT chứa z; z; z 11 Bài toán 3: Tìm số phức z thỏa mãn 2 điều kiện PT 19 12. Bài toán 4: Số thuần ảo, số thực 22 13. Bài toán 5: Biến đổi mũ to 27 9. 10. 11. 14. BUỔI 2 BUỔI 3 Dạng 3: Giải phương trình số phức 30 Căn bậc hai số phức 30 16. Bài toán 1: Detal âm và dương 33 17. Bài toán 2: Detal là số phức 37 18. Bài toán 3: Phương trình bậc cao 40 Bài toán 4: Phương trình biết nghiệm 42 15. 19. 20. BUỔI 4 Dạng 4:Tập hợp điểm biểu diễn số phức 43 21. Bài toán 1: Tìm điểm biểu diễn số phức z 43 22. Bài toán 2: Tập hợp điểm biểu diễn số phức z 45 Bài toán 3: Biểu diễn số phức w thông qua z 47 23. 24. BUỔI 5 25. 26. BUỔI 6 Dạng 5:Tìm số phức thỏa mãn môđun nhỏ nhất 48 Đáp án 52 TỔNG ÔN THI THỬ CUỐI CHUYÊN ĐỀ Lì xì giải nhất – nhì – ba – khuyến khích Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 0 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live BÀI TẬP VỀ NHÀ CÁC BUỔI BUỔI BÀI TẬP BUỔI 1 FINISH BUỔI BÀI TẬP H1 – H20 BUỔI 4 H107 – H130 BUỔI 2 H21 – H62 BUỔI 5 H131 – H142 BUỔI 3 H63 – H106 FINISH LÌ XÌ CHĂM CHỈ CÁC BUỔI BUỔI LÌ XÌ CHĂM CHỈ BUỔI 2 10.000 BUỔI 4 10.000 BUỔI 6 10.000 XÁC NHẬN GIÁO VIÊN LÌ XÌ THI CUỐI CHUYÊN ĐỀ QUYẾT TÂM ĐẬU ĐẠI HỌC EM NHÉ! NOTHING IS IMPOSSIBLE  Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 1 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 2 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live DẠNG 1: CÁC LÝ THUYẾT CƠ BẢN VỀ SỐ PHỨC BÀI TOÁN 1: Cộng trừ số phức: Cho z1  a1  b1i; z 2  a 2  b 2 i k.z1  k.a1  k.b1i; z1  z 2   a1  a 2    b1  b 2  i; z1  z 2   a1  a 2    b1  b 2  i Một hành trình ngàn dặm luôn bắt đầu bằng một bước đơn giản đầu tiên. TN 2010 (CB) Cho hai số phức z1  1  2i; z 2  2  3i . a) H1. Xác định phần thực và phần ảo của số phức z1  2z 2 Đáp án: 3  8i b) Xác định phần thực và phần ảo của số phức 3z1  2z 2 Đáp án: 1  12i c) Xác định phần thực và phần ảo của số phức z1  2z 2 Đáp án: 5  4i ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ BÀI TOÁN 2: Nhân hai số phức: Cho z1  a1  b1i; z 2  a 2  b2i z1.z 2  (a1  b1i)(a 2  b2i)   a1.a 2  b1.b 2    a1.b 2  a 2 .b1  i TN 2010 (NC) Cho hai số phức z1  2  5i; z 2  3  4i . H2. a) Xác định phần thực và phần ảo của số phức z1 .z 2 Đáp án: 26  7i b) Xác định phần thực và phần ảo của số phức (z1  i)(z 2  2) Đáp án: 34  22i c) Xác định phần thực và phần ảo của số phức  z1  1 (2i  z 2 ) Đáp án: 39  3i ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ BÀI TOÁN 3: Môđun số phức: Cho z  a  bi  z  a 2  b2 Môđun số phức z là độ dài đoạn OM với M(a;b) Cho hai số phức z1  1  i; z 2  3  2i a) Tính môđun của số phức z1 ; z 2 Đáp án: z1  2; z 2  13 b) Tính mô đun của số phức z1  z 2 ; z1  z 2 ; z1  2z 2 H3. Đáp án: z1  z 2  5; z1  z 2  5; z1  2z 2  74 c) Tính mô đun của số phức z1.z 2 ; z1 (z1  z 2 ); z 2 (z1  2z 2 ) Đáp án: z1.z 2  26; z1 (z1  z 2 )  5 2; z 2 (z1  2z 2 )  442 Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 3 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ BÀI TOÁN 4: Số phức liên hợp cho z  a  bi  z  a  bi Sự học giống như chèo thuyền ngược dòng, nếu không tiến lên sẽ bị kèo lùi lại ngay. H4. TN 2012 Tìm các số phức 2z  z biết z  3  4i Đáp án: 9  4i ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ TN 2010 (NC) Cho hai số phức z1  2  5i; z 2  3  4i . H5. a) Tìm số phức liên hợp của số phức z1.z 2 Đáp án: z  26  7i b) Tìm số phức liên hợp của số phức (z1  i)(z 2  2) Đáp án: z  34  22i c) Tìm số phức liên hợp của số phức  z1  1 (2i  z 2 ) Đáp án: z  39  3i ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 4 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live BÀI TOÁN 5: Chia hai số phức Cho z1  a1  b1i; z 2  a 2  b2i z1 z1.z 2 (a1  b1i)(a 2  b 2i) (a1  b1i)(a 2  b 2i)    z 2 z 2 .z 2 (a 2  b 2i)(a 2  b 2i) a 22  b22 H6. TN 2012 Tìm các số phức 25i ; biết z  3  4i z Đáp án: 4  3i ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ H7. Tìm số phức z biết z  1  9i  5i 1 i Đáp án: 4 ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ Tìm số phức z biết: a)z  3  4i 2i  2  3i  b)z  Đáp án: z  2 3  4i H8. 2 11  i 5 5 c)z   3  4i   5  7i 6  5i Đáp án: z   63 16  i 25 25 Đáp án: z   188 177  i 61 61 d)z  8  5i 2i  1  3  4i 3  2i Đáp án: z  27 411  i 325 325 e)z  1 2  3i Đáp án: z  2 3  i 13 13 ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 5 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live BÀI TẬP TỔNG HỢP DẠNG I: Có trải qua đau khổ, cay đắng con người ta mới hiểu ra được nhiều điều, mới nhìn nhận ra được những sự thật mà nếu như ta sống một cuộc sống êm đềm và hạnh phúc ta không thể nhận ra. Hãy viết các biểu thức sau dưới dạng số phức z = a + bi (a,b  R) A  (3i  4)  (3  2i)  (4  7i) Đáp án: A  55  15i B   7  5i 1  i    3i  2  Đáp án: B  10  i 2 C   3  4i   5  7i  3 2 3 2 D   3  i   1  2i  H9. Đáp án: C  133  169i E   3  i   3  2i  G  1  4i  (1  i)3 H  (1  2i)(3  i)  Đáp án: D  21  30i Đáp án: E  222  346i Đáp án: G  1  2i 2(2  3i)  4  2i 1 i 1  i 3  i 1  2i   1 i 2  i 1 i 1 K  (1  i)(3  2i)  3 i I Đáp án: H  4  2i 1 7  i 10 10 53 9 Đáp án: K   i 10 10 Đáp án: I   ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 6 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live Tìm phần thực ; phần ảo và số phức liên hợp, môđun của mỗi số phức sau: z1  (2i  1)2  3i(i  1)  2i 3 z2  H10. 3  2i  3i i2 Đáp án: z  9i; z  9i; z  9 Đáp án: z  4 22 4 22  i; z   i; z  2 5 5 5 5 5 z 3  (2  3i) 2  (1  i)2 Đáp án: z  5  10i; z  5  10i; z  5 5 z 4  (3  2i)3 Đáp án: z  9  46i; z  9  46i; z  2197 z 5  (2i  1)2  (3  i)2 Đáp án: z  11  10i; z  11  10i; z  221 z 6  (3  4i)(2  i)  5  7i Đáp án: z  3  4i; z  3  4i; z  5 ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 7 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live DẠNG 2: TÌM SỐ PHỨC Z THỎA MÃN PHƯƠNG TRÌNH BÀI TOÁN 1: Tìm số phức z thỏa mãn phương trình chỉ chứa z hoặc z Giải phương trình dạng az  b  0 hoặc az  b  0 trên tập số phức Phương pháp đưa về dạng: (a1  b1i)z  (a 2  b 2 i)  z  a 2  b 2 i  a 2  b 2i  a1  b1i   a1  b1i a12  b12 Những công trình vĩ đại đều khởi nguồn từ một ý nghĩ. Khi một ý nghĩ khi được nung nấu và được lên một kế hoạch hành động cụ thể, với thời gian tự nó sẽ tìm ra những phương cách để biến ý nghĩ đó trở thành hiện thực. Tìm các số phức z sau đó tính môđun và xác định số phức liên hợp biết: H11. a) (3  i)z  4  3i Đáp án: z  9 13  i 10 10 b) (3  i)z  5  7i  0 Đáp án: z  11 8  i 5 5 c) ( 3  i)z  4  (5  3i)( 4  5i) Đáp án: d) (5  3i)z  (1  i)2 Đáp án: z   e) (2  3i)2 z  (1  3i) 2  0 f) z  2  3i  4  2i zi 52 39  i 5 5 3 5  i 17 17 32 126 Đáp án: z    i 169 169 Đáp án: z  2 29  i 13 13 ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 8 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live D-2013 Cho số phức z thỏa mãn điều kiện (1  i)(z  i)  2z  2i . Tính môdun của số phức H12. w z  2z  1 z2 Đáp án: w  10 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H13. D-2012. Cho số phức z thỏa mãn  2  i  z  2 1  2i  1 i  7  8i .Tìm mô đun của số phức w  z  i  1 Đáp án: w  5 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H14. A-2010. Tìm phần ảo của số phức z biết z   2   2  i . 1  2.i  Đáp án: b   2 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 9 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live 2 H15. CĐ 2009. Cho số phức z thỏa 1  i  (2  i)z  8  i  1  2i  z .Tìm phần thực và phần ảo của z. Đáp án: a = 2; b = -3 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ A-2010 Cho số phức z thỏa mãn z  H16. (1  3i)3 . Tìm môđun của số phức w = z  iz 1 i Đáp án: w  8 2 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ (TN-2013): Cho số phức z thỏa mãn (1  i)z  2  4i  0 . Tìm số phức liên hợp của z H17. Đáp án: z  3  i ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ (CĐ-2013): Cho số phức z, thỏa mãn điều kiện (3  2i)z  (2  i)2  4  i . Tìm phần thực và phần ảo của H18. số phức w  (1  z)z . Đáp án: a  3; b  1 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 10 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live H19. Tìm môđun của số phức z  (1  i)(2  i) . 1  2i Đáp án: z  2. ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H20. Tìm môđun của số phức: z  1  2i  (1  i)3 1 i Đáp án: z  5 2 2 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ BÀI TOÁN 2: Tìm số phức z thỏa mãn PT có chứa 2 trong 3 thành phần z; z; z  Phương pháp: Gọi z = a + bi (a,b  R)  Nếu đề có z => z = a – bi  Nếu có z => z  a 2  b2 Áp dụng hai số phức bằng nhau: a = a’ ; b = b’ Hai trường hợp xảy ra: Lưu ý: CÔNG THỨC KIẾM HIỆP (THỰC = THỰC ; ẢO = ẢO) - a  a ' Khi biến đổi được a + bi = a’ + b’i  b  b ' a  0 Nếu một bên là a + bi = 0 =>  b  0 Gieo suy nghĩ, gặt hành động. Gieo hành động, gặt thói quen. Gieo thói quen, gặt tính cách. Gieo tính cách, gặt số phận H21. D - 2011.Tìm z thỏa z   2  3i  .z  1  9i Đáp án: z  2  i ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 11 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live 2 CĐ 2011.Cho số phức z thỏa 1  2i  .z  z  4i  20 . Tính z . H22. Đáp án: z  5 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H23. CĐ 2010 Cho số phức z thỏa mãn điều kiện (2  3i)z  (4  i)z  (1  3i)2 . Tìm phần thực và phần ảo của z. Đáp án: a = -2 ; b = 5 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ A – 2011 NC Tính môđun của số phức z, biết: (2z  1)(1  i)  (z  1)(1  i)  2  2i H24. Đáp án: z  2 3 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ Cho số phức z thỏa mãn: z  2z  (1  2i) 2 (1  i) . Tìm modun của số phức z. H25. Đáp án: z  442 3 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 12 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live H26. 3 Tìm phần ảo của z biết: z  3z   2  i   2  i  Đáp án: b = -10 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ Xác định phần ảo z và tính môđun số phức z , biết: z  (1  9i)  (2  3i)z H27. Đáp án: b  1; z  5 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H28.  Tính môđun số phức z thỏa mãn: (z  1)(4  2i)  (3  i) z  2i  Đáp án: z  2 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ Nghịch lý thường nằm ở chỗ, chúng ta cứ thường phán xét cuộc sống của người khác dựa trên quan điểm cá nhân, rồi lại loay hoay tìm cách sống sao cho hợp với đánh giá theo quan điểm của người khác. H29. Cho số phức z thỏa mãn: z  z 6  7i . Tìm phần thực, phần ảo của số phức z.  1  3i 5 Đáp án: a  1; b  1 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 13 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live H30. Cho số phức z thỏa mãn z  z  2 . Tìm phần thực của số phức w = z2 – z 1  2i Đáp án: a  1 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H31. Tìm mô đun z, biết z thỏa mãn z  (1  i)(3  2i)  5iz 2i Đáp án: z  17 2 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H32. Tìm số phức z thỏa mãn 2  iz z  2i   2z 2  i 1  2i Đáp án: z  1  i ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H33. Tìm mô đun của số phức z, biết rằng z  (4  3i)z  26  6i 2i Đáp án: z  5 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 14 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live H34.   2 Tìm phần ảo của số phức z , biết z  3z  1  2i . Đáp án: b  2 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H35.   2 Tìm phần ảo của số phức z , biết z  1  i  z  1  2i . Đáp án: b  3 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________  5 zi H36. A-2012. Cho số phức z thỏa z 1   2  i .Tính mô đun của số phức w  1  z  z 2 . Đáp án: w  13 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H37. B-2011 CB. Tìm số phức z biết z  5i 3 1  0 . z Đáp án: z  1  i 3; z  2  i 3 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 15 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live H38. 2 A-2011. Tìm tất cả các số phức z thỏa z 2  z  z 1 1 1 1 Đáp án: z    i; z    i; z  0 2 2 2 2 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ z Đáp án: z = 1 z 2 z ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H39. Tìm tất cả các số phức z biết ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H40. Tìm số phức z thỏa mãn z 2  (1  i)z  11i Đáp án: z = 3 + 2i ; z = -2 – 3i ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H41. Tìm số phức z thỏa mãn điều kiện iz  (1  3i)z 2  z 1 i Đáp án: z   45 9  i 26 26 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 16 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live H42. Cho số phức z thỏa mãn z  4  i . Tính A  1  (1  i)z z 1 Đáp án: A  3 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H43. Tìm số phức z thỏa mãn (z  1)(1  i)  z 1 2  z 1 i Đáp án: z  i; z   3 1  i 10 10 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H44.  2 Tìm môđun của số phức z biết z 2 (1  i)  2 z (1  i)  21  i Đáp án: z  5 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 17 Khóa học Luyện Thi Đại Học – Thầy Hiếu Live – 0988 593 390 – Facebook: Hiếu Live H45. Tìm số phức z thỏa mãn điều kiện z.z  2.z  19  4i Đáp án: z  3  2i; z  5  2i ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ H46. Tìm môđun của số phức z – 2i biết (z  2i)(z  2i)  4iz  0 Đáp án: z  2i  2 2 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ 2 Tìm số phức z thỏa mãn 1  z  z  i  (iz  1) 2 và z có phần thực dương. Tính môđun của H47. A  z 4 . z 1 Đáp án: A  5 ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ ________________________________________________________________ ___________________________________________________ Trung Tâm Olympia – Uy Tín – Chất Lượng – Tận Tâm Page 18