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{"id":22904,"date":"2018-03-20T18:01:13","date_gmt":"2018-03-20T11:01:13","guid":{"rendered":"https:\/\/lop12.edu.vn\/?p=22904"},"modified":"2018-03-21T16:09:39","modified_gmt":"2018-03-21T09:09:39","slug":"dai-so-chuong-4-bai-1-so-phuc","status":"publish","type":"post","link":"https:\/\/lop12.edu.vn\/dai-so-chuong-4-bai-1-so-phuc\/","title":{"rendered":"\u0110\u1ea1i s\u1ed1 – Ch\u01b0\u01a1ng 4 – B\u00e0i 1: S\u1ed1 ph\u1ee9c"},"content":{"rendered":"

B\u00e0i 1 (trang 189 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b><\/span><\/p>\n

Cho s\u1ed1 ph\u1ee9c 2+3i;1+2i;2-i<\/p>\n

a) Bi\u1ec3u di\u1ec5n c\u00e1c s\u1ed1 trong m\u1eb7t ph\u1eb3ng s\u1ed1 ph\u1ee9c.<\/p>\n

b) Vi\u1ebft s\u1ed1 ph\u1ee9c li\u00ean h\u1ee3p c\u1ee7a m\u1ed7i s\u1ed1 \u0111\u00f3 v\u00e0 bi\u1ec3u di\u1ec5n ch\u00fang trong m\u1eb7t ph\u1eb3ng ph\u1ee9c.<\/p>\n

c) Vi\u1ebft \u0111\u1ed1i s\u1ed1 c\u1ee7a m\u1ed7i s\u1ed1 ph\u1ee9c v\u00e0 bi\u1ec3u di\u1ebfn ch\u00fang trong m\u1eb7t ph\u1eb3ng ph\u1ee9c.<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

a) C\u00e1c \u0111i\u1ec3m A, B, C trong m\u1eb7t ph\u1eb3ng t\u1ecda \u0111\u1ed9 (h\u00ecnh v\u1ebd) l\u00e0 bi\u1ec3u di\u1ebfn c\u00e1c s\u1ed1 ph\u1ee9c: 1 + 2i; 2 + 3i; 2 \u2013 i.<\/p>\n

\"\"<\/p>\n

b) S\u1ed1 ph\u1ee9c li\u00ean h\u1ee3p c\u1ee7a s\u1ed1 z=2+3i l\u00e0\u00a0z\u2212<\/i>\u00a0=2-3i<\/p>\n

z’=1+2i l\u00e0\u00a0z’\u2212<\/i>=1-2i<\/p>\n

z”=2-i l\u00e0\u00a0z\u2212<\/i>”=2+i<\/p>\n

C\u00e1c \u0111i\u1ec3m M, N P bi\u1ec3u di\u1ec5n cho c\u00e1c s\u1ed1\u00a0z\u2212<\/i>,\u00a0z\u2212<\/i>‘,z\u2212<\/i>” nh\u01b0 sau: (h\u00ecnh v\u1ebd)<\/p>\n

\"Gi\u1ea3i<\/p>\n

c) S\u1ed1 \u0111\u1ed1i c\u1ee7a s\u1ed1 ph\u1ee9c: z=2+3i l\u00e0-z=-2-3i<\/p>\n

z’=1+2i l\u00e0-z=-1-2i<\/p>\n

z”=2-i l\u00e0-z=-2+i<\/p>\n

C\u00e1c \u0111i\u1ec3m P, Q, R l\u1ea7n l\u01b0\u1ee3t bi\u1ec3n di\u1ec5n cho c\u00e1c s\u1ed1 -z, -z\u2019, -z\u2019\u2019.<\/p>\n

\"\"<\/p>\n

B\u00e0i 2 (trang 189 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao)<\/b><\/span><\/p>\n

X\u00e1c \u0111\u1ecbnh ph\u1ea7n th\u1ef1c v\u00e0 ph\u1ea7n \u1ea3o c\u00e1c s\u1ed1 sau:<\/p>\n

a) i+(2-4i)-(3-2i) \u00a0\u00a0\u00a0 b) (\u221a2+3i)2<\/sup><\/p>\n

c) (2+3i)(2-3i) \u00a0\u00a0\u00a0d) i(2-i)(3+i)<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

a) Ta c\u00f3 i+(2-4i)-(3-2i)=-i-1 c\u00f3 ph\u1ea7n th\u1ee9c l\u00e0 -1 v\u00e0 ph\u1ea7n \u1ea3o l\u00e0 -1<\/p>\n

b) Ta c\u00f3 (\u221a2+3i)2<\/sup>=2+6 \u221a2 i+(3i)2<\/sup>=2+6 \u221a2 i-9=-7+6 \u221a2 i c\u00f3 ph\u1ea7n th\u1ee9c l\u00e0 -7 v\u00e0 ph\u1ea7n \u1ea3o l\u00e0 6 \u221a2<\/p>\n

c) Ta c\u00f3 (2+3i)(2-3i)=4-6i+6i-9i2<\/sup>=4+9=13 c\u00f3 ph\u1ea7n th\u1ee9c l\u00e0 13 v\u00e0 ph\u1ea9n \u1ea3o l\u00e0 0.<\/p>\n

d) i(2-i)(3+i)=i(6-i+1)=1+7i c\u00f3 ph\u1ea7n th\u1ee9c l\u00e0 1 v\u00e0 ph\u1ea7n \u1ea3o l\u00e0 7.<\/p>\n

B\u00e0i 3 (trang 189 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b><\/span><\/p>\n

X\u00e1c \u0111\u1ecbnh c\u00e1c s\u1ed1 ph\u1ee9c bi\u1ec3u di\u1ec5n b\u1edfi c\u00e1c \u0111\u1ec9nh c\u1ee7a m\u1ed9t l\u1ee5c gi\u00e1c \u0111\u1ec1u c\u00f3 t\u00e2m l\u00e0 g\u1ed1c O trong m\u1eb7t ph\u1eb3ng ph\u1ee9c bi\u1ebft r\u1eb1ng m\u1ed9t \u0111\u1ec9nh bi\u1ec3u di\u1ec5n s\u1ed1 i.<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

G\u1ecdi l\u1ee5c gi\u00e1c \u0111\u1ec1u l\u00e0 ABCDEF, trong \u0111\u00f3 A l\u00e0 bi\u1ec3u di\u1ec5n cho s\u1ed1 i.<\/p>\n

Suy ra A(0; 1) v\u00e0 g\u00f3c AOB=60o<\/sup>\u00a0(h\u00ecnh v\u1ebd)<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

V\u1eady s\u00e1u s\u1ed1 ph\u1ee9c c\u1ea7n t\u00ecm l\u00e0:<\/p>\n

\"\"<\/p>\n

B\u00e0i 4 (trang 189 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b><\/span><\/p>\n

Th\u1ef1c hi\u1ec7n ph\u00e9p t\u00ednh:<\/p>\n

\"\"<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

Nh\u00e2n c\u1ea3 t\u1eed v\u00e0 m\u1eabu c\u1ee7a s\u1ed1 \u0111\u00e3 cho v\u1edbi l\u01b0\u1ee3ng li\u00ean h\u1ee3p \u1edf m\u1eabu ta \u0111\u01b0\u1ee3c.<\/p>\n

B\u00e0i 5 (trang 190 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b><\/span><\/p>\n

\"\"<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

B\u00e0i 6 (trang 190 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b><\/span><\/p>\n

Ch\u1ee9ng minh<\/p>\n

a) Ph\u1ea7n th\u1ef1c c\u1ee7a s\u1ed1 ph\u1ee9c z b\u1eb1ng 1\/2(z+z\u2212<\/i>\u00a0), ph\u1ea7n \u1ea3o c\u1ee7a s\u1ed1 ph\u1ee9c z b\u1eb1ng 1\/2i(z-z\u2212<\/i>)<\/p>\n

b) S\u1ed1 ph\u1ee9c z l\u00e0 ph\u1ea7n \u1ea3o khi v\u00e0 ch\u1ec9 khi z=-z\u2212<\/i><\/p>\n

c) V\u1edbi m\u1ecdi s\u1ed1 ph\u1ee9c z, z\u2019 ta c\u00f3<\/p>\n

\"\"<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

a) Gi\u1ea3 s\u1eed z=a+bi ta c\u00f3\u00a0z\u2212<\/i>=a-bi, n\u00ean<\/p>\n

\"\"<\/p>\n

b) Gi\u1ea3 s\u1eed z=a+bi<\/p>\n

Theo b\u00e0i ra z=-z\u2212<\/i>\u00a0<=> a+bi=-(a-bi) <=> a=-a <=> a = 0<\/p>\n

V\u1eady z = bi l\u00e0 m\u1ed9t s\u1ed1 \u1ea3o.<\/p>\n

c) Gi\u1ea3 s\u1eed z=a+bi,z’=a’+b’i. Ta c\u00f3:<\/p>\n

z+z’=(a+a’ )+(b+b’ )i =>\u00a0(z + z’)\u2212<\/i>=(a+a’ )-(b-b’ )i=(a-bi)+(a’-b’ i)=z\u2212<\/i>+z\u2212<\/i>‘\u00a0\u00a0\u00a0(\u0111pcm)<\/p>\n

z.z\u2019=(a+bi)(a’+b’ i)=(aa’-bb’ )+(ab’+a’ b)i<\/p>\n

=>\u00a0zz’\u2212<\/i>=(aa’-bb’ )+(ab’+a’ b)i (1)<\/p>\n

V\u00e0\u00a0z\u2212<\/i>.z\u2212<\/i>‘=(a-bi)(a’-b’ i)=(aa’-bb’ )-(ab’+a’ b)i (2)<\/p>\n

T\u1eeb (1) v\u00e0 (2) suy ra:\u00a0zz’\u2212<\/i>=z\u2212<\/i>.z\u2212<\/i>‘ (\u0111pcm)<\/p>\n

B\u00e0i 7 (trang 190 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b><\/span><\/p>\n

Ch\u1ee9ng minh r\u1eb1ng v\u1edbi m\u1ecdi s\u1ed1 nguy\u00ean m > 0, ta c\u00f3:<\/p>\n

i4m<\/sup>=1;i4m+1<\/sup>;i4m+2<\/sup>=-1;i4m+3<\/sup>=-i<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

Ta c\u00f3: i4m<\/sup>=1=(i2<\/sup>\u00a0)2m<\/sup>=(-1)2m<\/sup>=1,v\u1edbi \u2200m \u2208N*<\/p>\n

i4m+1<\/sup>=i4m<\/sup>.i=1.i=i<\/p>\n

i4m+2<\/sup>=i4m<\/sup>.i2<\/sup>=1.(-1)=-1<\/p>\n

i4m+3<\/sup>=i4m<\/sup>.i3<\/sup>=1.i3<\/sup>=i3<\/sup>=i2<\/sup>.i=-1.i=-i (\u0111pcm)<\/p>\n

\n
\n

B\u00e0i 8 (trang 190 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b><\/span><\/p>\n

Ch\u1ee9ng minh r\u1eb1ng:<\/p>\n

a) N\u1ebfu vect\u01a1\u00a0u\u2192<\/i>\u00a0c\u1ee7a m\u1ed9t m\u1eb7t ph\u1eb3ng ph\u1ee9c bi\u1ec3u di\u1ec5n s\u1ed1 ph\u1ee9c z th\u00ec \u0111\u1ed9 d\u00e0i c\u1ee7a vect\u01a1\u00a0u\u2192<\/i>\u00a0l\u00e0 |u\u2192<\/i>\u00a0|=|z|, t\u1eeb \u0111\u00f3 n\u1ebfu c\u00e1c \u0111i\u1ec3m A1<\/sub>;A2<\/sub>\u00a0theo th\u1ee9c t\u1ef1 bi\u1ec3u di\u1ec5n c\u00e1c s\u1ed1 ph\u1ee9c z1<\/sub>;z2<\/sub>\u00a0th\u00ec |A1<\/sub>A2<\/sub>\u2192<\/i>\u00a0|=|z2<\/sub>-z1<\/sub>\u00a0|<\/p>\n

b) V\u1edbi m\u1ecdi s\u1ed1 ph\u1ee9c z, z\u2019 ta c\u00f3 |zz’ |=|z||z’ | v\u00e0 khi z \u2260 0 th\u00ec<\/p>\n

\"\"<\/p>\n

c) V\u1edbi m\u1ecdi s\u1ed1 ph\u1ee9c z, z\u2019 ta c\u00f3 |z+z’ |\u2264|z|+|z’ |<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

a) N\u1ebfu\u00a0u\u2192<\/i>\u00a0l\u00e0 vect\u01a1 bi\u1ec3u di\u1ec5n s\u1ed1 ph\u1ee9c z = a + bi th\u00ec\u00a0u\u2192<\/i>=(a;b)<\/p>\n

\"\"<\/p>\n

G\u1ecdi A1<\/sub>\u00a0l\u00e0 \u0111i\u1ec3m bi\u1ec3u di\u1ec5n s\u1ed1 ph\u1ee9c Z1<\/sub>=a1<\/sub>+b1<\/sub>\u00a0i=>A1<\/sub>\u00a0(a1<\/sub>;b1<\/sub>)<\/p>\n

A2<\/sub>\u00a0l\u00e0 \u0111i\u1ec3m bi\u1ec3u di\u1ec5n s\u1ed1 ph\u1ee9c Z2<\/sub>=a2<\/sub>+b2<\/sub>\u00a0i=>A2<\/sub>\u00a0(a2<\/sub>;b2<\/sub>)<\/p>\n

\"\"<\/p>\n

b) Ta c\u00f3: z.z\u2019=(a+bi)(a’+b’ i)=(aa’-bb’ )+(ab’+a’ b)i<\/p>\n

\"\"<\/p>\n

V\u1eady |zz’ |=|z||z’ |<\/p>\n

Khi z \u2260 0 ta c\u00f3:<\/p>\n

\"\"<\/p>\n<\/div>\n

\n

\"Gi\u1ea3ic) V\u1edbi m\u1ecdi s\u1ed1 ph\u1ee9c z, z\u2019, ta c\u00f3: z + z\u2019 = (a +a\u2019) + (b +b\u2019)i<\/p>\n

\"\"<\/p>\n

Theo y\u00eau c\u1ea7u b\u00e0i to\u00e1n ta c\u1ea7n ch\u1ee9ng minh:<\/p>\n

\"\"<\/p>\n

Theo Bu-nhi-c\u1ed1p-xki ta c\u00f3 b\u1ea5t \u0111\u1eb3ng th\u1ee9c (*) \u0111\u00fang v\u1edbi \u2200a,b,a’,b’\u2208R n\u00ean |z+z’ | \u2264 ||z|+|z’ | (\u0111pcm)<\/p>\n

B\u00e0i 9 (trang 190 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b><\/span><\/p>\n

X\u00e1c \u0111\u1ecbnh t\u1eadp h\u1ee3p c\u00e1c \u0111i\u1ec3m trong m\u1eb7t ph\u1eb3ng ph\u1ee9c bi\u1ec3u di\u1ec5n c\u00e1c s\u1ed1 ph\u1ee9c z th\u1ecfa m\u00e3n t\u1eebng \u0111i\u1ec1u ki\u1ec7n sau:<\/p>\n

\"\"<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

a) Gi\u1ea3 s\u1eed z=a+bi => z \u2013 I = a+(b-1)i<\/p>\n

\"\"<\/p>\n

V\u1eady qu\u1ef9 t\u00edch \u0111i\u1ec3m M(a; b) bi\u1ec3u di\u1ec5n s\u1ed1 ph\u1ee9c z=a+bi th\u00f5a m\u00e3n |z \u2013 i| =1 l\u00e0 \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m I(0; 1), b\u00e1n k\u00ednh R = 1.<\/p>\n

b) Gi\u1ea3 s\u1eed z=a+bi<\/p>\n

\"\"<\/p>\n

Theo b\u00e0i 8b ta c\u00f3:<\/p>\n

\"\"<\/p>\n

Theo b\u00e0i ra ta c\u00f3:<\/p>\n

\"\"<\/p>\n

<=> a2<\/sup>+(b-1)2<\/sup>=a2<\/sup>+(b+1)2<\/sup><\/p>\n

<=> b = 0. V\u1eady z = a, hay t\u1eadp h\u1ee3p c\u00e1c \u0111i\u1ec3m c\u1ea7n t\u00ecm l\u00e0 tr\u1ee5c th\u1ef1c<\/p>\n

Gi\u1ea3 s\u1eed z=a+bi =>\u00a0z\u2212<\/i>=a-bi\u00a0z\u2212<\/i>-3+4i=a-bi-3+4i=(a-3)+(4-b)i<\/p>\n

Theo b\u00e0i ra, ta c\u00f3 |z|=|z\u2212<\/i>-3+4i|<=> |z|=|(a-3)+(4-b)i|<\/p>\n

\"\"<\/p>\n

V\u1eadt qu\u1ef9 t\u00edch \u0111i\u1ec3m c\u00e1c \u0111i\u1ec3m c\u1ea7n t\u00ecm tr\u00eam \u0111\u01b0\u1eddng th\u1eb3ng c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh 6a+8b=25 trong m\u1eb7t ph\u1eb3ng ph\u1ee9c (Oxy)<\/p>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

B\u00e0i 1 (trang 189 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao): Cho s\u1ed1 ph\u1ee9c 2+3i;1+2i;2-i a) Bi\u1ec3u di\u1ec5n c\u00e1c s\u1ed1 trong m\u1eb7t ph\u1eb3ng s\u1ed1 ph\u1ee9c. b) Vi\u1ebft s\u1ed1 ph\u1ee9c li\u00ean h\u1ee3p c\u1ee7a m\u1ed7i s\u1ed1 \u0111\u00f3 v\u00e0 bi\u1ec3u di\u1ec5n ch\u00fang trong m\u1eb7t ph\u1eb3ng ph\u1ee9c. c) Vi\u1ebft \u0111\u1ed1i s\u1ed1 c\u1ee7a m\u1ed7i s\u1ed1 ph\u1ee9c v\u00e0 bi\u1ec3u di\u1ebfn ch\u00fang trong […]<\/p>\n","protected":false},"author":3,"featured_media":22905,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mi_skip_tracking":false,"tdm_status":"","tdm_grid_status":""},"categories":[1302],"tags":[1392,1393],"yoast_head":"\n\u0110\u1ea1i s\u1ed1 - Ch\u01b0\u01a1ng 4 - B\u00e0i 1: S\u1ed1 ph\u1ee9c<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/lop12.edu.vn\/dai-so-chuong-4-bai-1-so-phuc\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u0110\u1ea1i s\u1ed1 - 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