◆❣✉②➵♥ ✣➢❝ ●✐→♣ ✲ ▼ët sè ✤➲ t❤✐ t❤û ✣↕✐ ❤å❝ ✷✵✶✵
✣➋ ❙➮ ✶ ✲ ❚❍■ ❚❍Û ✣❸■ ❍➴❈ ◆❿▼ ✷✵✶✵
❚❘×❮◆● ❚❍P❚ ❈❍❯❨➊◆ ✕ ✣❍❙P
❈➙✉ ✶✳
✭ ✷✱✵ ✤✐➸♠ ✮ ❈❤♦ ❤➔♠ sè
y= 2x3+ 9mx2+ 12m2x+ 1
✱ tr♦♥❣ ✤â
m
❧➔ t❤❛♠ sè✳
✭❛✮ ❑❤↔♦ s→t sü ❜✐➳♥ t❤✐➯♥ ✈➔ ✈➩ ✤ç t❤à ❝õ❛ ❤➔♠ sè ✤➣ ❝❤♦ ❦❤✐
m=−1
✳
✭❜✮ ❚➻♠ t➜t ❝↔ ❝→❝ ❣✐→ trà ❝õ❛
m
✤➸ ❤➔♠ sè ❝â ❝ü❝ ✤↕✐ t↕✐
xCD
✱ ❝ü❝ t✐➸✉ t↕✐
xCT
t❤ä❛
♠➣♥✿
x2CD =xCT .
❈➙✉ ✷✳
✭ ✷✱✵ ✤✐➸♠ ✮
✭❛✮ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✿
√x+ 1 + 1 = 4x2+√3x.
✭❜✮ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✿
5 cos(2x+π
3) = 4 sin(5π
6−x)−9
❈➙✉ ✸✳
✭ ✷✱✵ ✤✐➸♠ ✮
✭❛✮ ❚➻♠ ❤å ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ sè✿
f(x)= xln(x2+1)+x3
x2+1
✳
✭❜✮ ❈❤♦ ❤➻♥❤ ❝❤â♣
❙✳❆❇❈❉
❝â
SA =x
✈➔ t➜t ❝↔ ❝→❝ ❝↕♥❤ ❝á♥ ❧↕✐ ❝â ✤ë ❞➔✐ ❜➡♥❣
❛
✳
❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✤÷í♥❣ t❤➥♥❣
❇❉
✈✉æ♥❣ ❣â❝ ✈î✐ ♠➦t ♣❤➥♥❣
(SAC)
✳ ❚➻♠
①
t❤❡♦
❛
✤➸ t❤➸ t➼❝❤ ❝õ❛ ❦❤è✐ ❝❤â♣
S.ABCD
❜➡♥❣
a3√2
6
✳
❈➙✉ ✹✳
✭ ✷✱✵ ✤✐➸♠ ✮
✭❛✮ ●✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✿
(4x−2.2x−3).log2x−3>4x+1
2−4x
✭❜✮ ❈❤♦ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠
a, b
✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿
(a2 + b+3
4) (b2 + a+3
4)≥(2a+1
2) (2b+1
2
✮✳
❈➙✉ ✺✳
✭ ✷✱✵ ✤✐➸♠ ✮ ❚r♦♥❣ ♠➦t ♣❤➥♥❣ ✈î✐ ❤➺ tå❛ ✤ë
Oxy
✱ ❝❤♦ ❜❛ ✤÷í♥❣ t❤➥♥❣ ✿
d1: 2x+y−3 = 0, d2: 3x+ 4y+ 5 = 0
✈➔
d3: 4x+ 3y+ 2 = 0
✳
✭❛✮ ❱✐➳t ♣❤÷ì♥❣ tr➻♥❤ ✤÷í♥❣ trá♥ ❝â t➙♠ t❤✉ë❝
d1
✈➔ t✐➳♣ ①ó❝ ✈î✐
d2
✈➔
d3
✳
✭❜✮ ❚➻♠ tå❛ ✤ë ✤✐➸♠
▼
t❤✉ë❝
d1
✈➔ ✤✐➸♠
◆
t❤✉ë❝
d2
s❛♦ ❝❤♦
−−→
OM + 4−−→
ON
❂
−→
0
✳
✣➋ ❙➮ ✷ ✲ ❚❘×❮◆● ❚❍P❚ ❈❍❯❨➊◆ ◆●❯❨➍◆ ❍❯➏
❈➙✉ ✶✳
✭✷ ✤✐➸♠✮
❈❤♦ ❤➔♠ sè✿
y=mx2+(m2+1)x+4m3+m
x+m(Cm)
✭❛✮ ❑❤↔♦ s→t sü ❜✐➳♥ t❤✐➯♥ ✈➔ ✈➩ ✤ç t❤à ❝õ❛ ❤➔♠ sè ❦❤✐
m=−1
✭❜✮ ❚➻♠ ❝→❝ ❣✐→ trà ❝õ❛ t❤❛♠ sè ♠ ✤➸ ✤ç t❤à
(Cm)
❝â ❤❛✐ ✤✐➸♠ ❝ü❝ trà
❆✱ ❇
s❛♦ ❝❤♦ ✤♦↕♥
t❤➥♥❣
❆❇
❝➢t ❝↔ trö❝ ❤♦➔♥❤
Ox
✈➔ trö❝ t✉♥❣
Oy
✳
❈➙✉ ✷✳
✭✷ ✤✐➸♠✮
✭❛✮ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✿
tan x+ tan 2x=−sin 3x. cos 2x
✶
◆❣✉②➵♥ ✣➢❝ ●✐→♣ ✲ ▼ët sè ✤➲ t❤✐ t❤û ✣↕✐ ❤å❝ ✷✵✶✵
✭❜✮ ●✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✿
1
√2x2+3x−5>1
2x−1
❈➙✉ ✸✳
✭✶ ✤✐➸♠✮
❚r♦♥❣ ♠➦t ♣❤➥♥❣ ✈î✐ ❤➺ tå❛ ✤ë
Oxy
❝❤♦ ♣❛r❛❜♦❧
(P)
❝â ♣❤÷ì♥❣ tr➻♥❤
y2= 4x
✳ ●å✐
❞
❧➔
✤÷í♥❣ t❤➥♥❣ ✤✐ q✉❛ t✐➯✉ ✤✐➸♠
❋
❝õ❛
(P)
✈➔ ❝➢t
(P)
ð ❤❛✐ ✤✐➸♠
❆✱ ❇
s❛♦ ❝❤♦
F A = 2F B
✳
❚➼♥❤
❆❇
✳
❈➙✉ ✹✳
✭✷ ✤✐➸♠✮
❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈î✐ ❤➺ tå❛ ✤ë
Oxyz
❝❤♦ ❤❛✐ ✤✐➸♠
B(−1,√3,0), C(1,√3,0)
✈➔
M(0,0, a)
✈î✐
a > 0
✳ ❚r➯♥ trö❝
Oz
❧➜② ✤✐➸♠
◆
s❛♦ ❝❤♦ ♠➦t ♣❤➥♥❣
(NBC)
✈✉æ♥❣ ❣â❝ ✈î✐ ♠➦t ♣❤➥♥❣
(MBC)
✳
✭❛✮ ❈❤♦
a=√3
✳ ❚➻♠ ❣â❝ ❣✐ú❛ ♠➦t ♣❤➥♥❣
(NBC)
✈➔ ♠➦t ♣❤➥♥❣
(OBC)
✭❜✮ ❚➻♠
❛
✤➸ t❤➸ t➼❝❤ ❝õ❛ ❦❤è✐ ❝❤â♣
❇❈▼◆
♥❤ä ♥❤➜t✳
❈➙✉ ✺✳
✭✷ ✤✐➸♠✮
✭❛✮ ❚➼♥❤ t➼❝❤ ♣❤➙♥✿
e
∫
1
ln3x
x(ln2x+1) dx
✭❜✮ ❚ø s→✉ ❝❤ú sè ✶✱ ✷✱ ✸✱ ✹✱ ✺✱✻ ❧➟♣ ✤÷ñ❝ ❜❛♦ ♥❤✐➯✉ sè ❝â ♥➠♠ ❝❤ú sè s❛♦ ❝❤♦ tr♦♥❣ sè
❝â ♥➠♠ ❝❤ú sè ✤â ❝â ❤❛✐ ❝❤ú sè ✶ ❝á♥ ❝→❝ ❝❤ú sè ❦❤→❝ ①✉➜t ❤✐➺♥ ❦❤æ♥❣ q✉→ ♠ët ❧➛♥✳
❈➙✉ ✻✳
✭✶ ✤✐➸♠✮
❈❤♦ ❜è♥ sè ♥❣✉②➯♥
a, b, c, d
t❤❛② ✤ê✐ t❤ä❛ ♠➣♥
1≤a<b<c<d ≤50
✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t
❝õ❛ ❜✐➸✉ t❤ù❝ ✿
S=a
b+c
d
✣➋ ❙➮ ✸
❈➙✉ ✶✳
✭✷ ✤✐➸♠✮✳ ❈❤♦ ❤➔♠ sè
y=2x−1
x+1
✭✶✮✳
✭❛✮ ❑❤↔♦ s→t ✈➔ ✈➩ ✤ç t❤à
(C)
❝õ❛ ❤➔♠ sè ✭✶✮✳
✭❜✮ ❚➻♠ ✤✐➸♠ ▼ t❤✉ë❝ ✤ç t❤à
(C)
✤➸ t✐➳♣ t✉②➳♥ ❝õ❛
(C)
t↕✐ ▼ ✈î✐ ✤÷í♥❣ t❤➥♥❣ ✤✐ q✉❛
▼ ✈➔ ❣✐❛♦ ✤✐➸♠ ❤❛✐ ✤÷í♥❣ t✐➺♠ ❝➟♥ ❝â t➼❝❤ ❤➺ sè ❣â❝ ❜➡♥❣ ✲ ✾✳
❈➙✉ ✷✳
✭✷ ✤✐➸♠✮
✭❛✮ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ s❛✉✿
1
x+1
√2−x2= 2
✳
✭❜✮ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❧÷ñ♥❣ ❣✐→❝✿
sin42x+cos42x
tan( π
4−x).tan( π
4+x)= cos44x
✳
❈➙✉ ✸✳
✭✶ ✤✐➸♠✮ ❚➼♥❤ ❣✐î✐ ❤↕♥ s❛✉✿
L= lim
x→0
ln(2e−e. cos 2x)−3
√1+x2
x2
❈➙✉ ✹✳
✭✷ ✤✐➸♠✮
❈❤♦ ❤➻♥❤ ♥â♥ ✤➾♥❤
❙
❝â ✤ë ❞➔✐ ✤÷í♥❣ s✐♥❤ ❧➔
❧
✱ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ✤→② ❧➔
r
✳ ●å✐
■
❧➔ t➙♠ ♠➦t ❝➛✉ ♥ë✐ t✐➳♣ ❤➻♥❤ ♥â♥ ✭♠➦t ❝➛✉ ❜➯♥ tr♦♥❣ ❤➻♥❤ ♥â♥✱ t✐➳♣ ①ó❝ ✈î✐ t➜t ❝↔ ❝→❝
✤÷í♥❣ s✐♥❤ ✈➔ ✤÷í♥❣ trá♥ ✤→② ❝õ❛ ♥â♥ ❣å✐ ❧➔ ♠➦t ❝➛✉ ♥ë✐ t✐➳♣ ❤➻♥❤ ♥â♥✮✳
✷
◆❣✉②➵♥ ✣➢❝ ●✐→♣ ✲ ▼ët sè ✤➲ t❤✐ t❤û ✣↕✐ ❤å❝ ✷✵✶✵
✭❛✮ ❚➼♥❤ t❤❡♦
r✱ ❧
❞✐➺♥ t➼❝❤ ♠➦t ❝➛✉ t➙♠
■
❀
✭❜✮ ●✐↔ sû ✤ë ❞➔✐ ✤÷í♥❣ s✐♥❤ ❝õ❛ ♥â♥ ❦❤æ♥❣ ✤ê✐✳ ❱î✐ ✤✐➲✉ ❦✐➺♥ ♥➔♦ ❝õ❛ ❜→♥ ❦➼♥❤ ✤→②
t❤➻ ❞✐➺♥ t➼❝❤ ♠➦t ❝➛✉ t➙♠
■
✤↕t ❣✐→ trà ❧î♥ ♥❤➜t❄
❈➙✉ ✺✳
✭✶ ✤✐➸♠✮ ❈❤♦ ❝→❝ sè t❤ü❝
①✱ ②✱ ③
t❤ä❛ ♠➣♥✿
x2+y2+z2= 2
✳
❚➻♠ ❣✐→ trà ❧î♥ ♥❤➜t✱ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝✿
P=x3+y3+z3−3xyz
✳
❈➙✉ ✻✳
✭✶ ✤✐➸♠✮ ❚r♦♥❣ ♠➦t ♣❤➥♥❣ tå❛ ✤ë
Oxy
❝❤♦ ❤➻♥❤ ❝❤ú ♥❤➟t
ABCD
❝â t➙♠
I(1
2; 0)
✣÷í♥❣ t❤➥♥❣
❆❇
❝â ♣❤÷ì♥❣ tr➻♥❤✿
x−2y+ 2 = 0, AB = 2AD
✈➔ ❤♦➔♥❤ ✤ë ✤✐➸♠
❆
➙♠✳
❚➻♠ tå❛ ✤ë ❝→❝ ✤➾♥❤ ❝õ❛ ❤➻♥❤ ❝❤ú ♥❤➟t ✤â✳
❈➙✉ ✼✳
✭✶ ✤✐➸♠✮ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤
2009y2−x2=x2+2010
y2+2010
3log3(x+ 2y+ 6) = 2log2(x+y+ 2) + 1
✣➋ ❙➮ ✹ ✲ ❚❍P❚ ❈❍❯❨➊◆ ▲×❒◆● ❱❿◆ ❈❍⑩◆❍
P❍❺◆ ❈❍❯◆● ❈❍❖ ❚❻❚ ❈❷ ❚❍➑ ❙■◆❍ ✭✼✱✵ ✤✐➸♠✮
❈➙✉ ✶✳
✭✷✱✵ ✤✐➸♠✮
❈❤♦ ❤➔♠ sè
y=x4−2x2+2
✭❛✮ ❑❤↔♦ s→t sü ❜✐➳♥ t❤✐➯♥ ✈➔ ✈➩ ✤ç t❤à
(C)
❝õ❛ ❤➔♠ sè✳
✭❜✮ ❱✐➳t ♣❤÷ì♥❣ tr➻♥❤ ❝→❝ t✐➳♣ t✉②➳♥ ❦➫ ✤➳♥ ✤ç t❤à
(C)
✱ ❜✐➳t r➡♥❣ ❝→❝ t✐➳♣ t✉②➳♥ ♥➔② ✤✐
q✉❛ ✤✐➸♠
A(0; 2)
❈➙✉ ✷✳
✭✷✱✵ ✤✐➸♠✮
✭❛✮ ●✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✿
(2x+ 3.2−x)2log2x−log2(x+6) >1
✭❜✮ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✿
(sin x+cos x)2−2sin2x
1+cot2x=√2
2sin π
4−x−sin π
4−3x
❈➙✉ ✸✳
✭✶✱✵ ✤✐➸♠✮ ❚➼♥❤ t➼❝❤ ♣❤➙♥✿
I=R2
1
x√x−1
x−5dx
❈➙✉ ✹✳
✭✶✱✵ ✤✐➸♠✮ ❈❤♦ ❤➻♥❤ ❝❤â♣
❙✳❆❇❈
❝â ✤→② ❧➔ t❛♠ ❣✐→❝ ✤➲✉ ❝↕♥❤
❛
✱ t❛♠ ❣✐→❝
❙❆❈
❝➙♥ t↕✐
❙
✱ ❣â❝
❙❇❈
❜➡♥❣
600
✱ ♠➦t ♣❤➥♥❣
(SAC)
✈✉æ♥❣ ❣â❝ ✈î✐ ♠➦t ♣❤➥♥❣
(ABC)
✳ ❚➼♥❤ t❤❡♦
❛
t❤➸ t➼❝❤ ❝õ❛ ❦❤è✐ ❝❤â♣
S.ABC
✳
❈➙✉ ✺✳
✭✶✱✵ ✤✐➸♠✮ ❚➻♠
♠
✤➸ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ❝â ♥❣❤✐➺♠ t❤ü❝✿
x3+x2+x−m(x2+1)2= 0
P❍❺◆ ❘■➊◆● ✭✸✱✵ ✤✐➸♠✮
❚❤➼ s✐♥❤ ❝❤➾ ✤÷ñ❝ ❧➔♠ ♠ët tr♦♥❣ ❤❛✐ ♣❤➛♥ ✭♣❤➛♥ ✶ ❤♦➦❝ ♣❤➛♥ ✷✮
✶✳ ❚❤❡♦ ❝❤÷ì♥❣ tr➻♥❤ ❈❤✉➞♥✿
❈➙✉ ✻✳
✭✷✱✵ ✤✐➸♠✮ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈î✐ ❤➺ t♦↕ ✤ë
Oxyz
✱ ❝❤♦ ❝→❝ ✤✐➸♠ ✿
A(−1; −1; 0) , B (1; −1; 2) , C (2; −2; 1) , D (−1; 1; 1)
✳
✸
◆❣✉②➵♥ ✣➢❝ ●✐→♣ ✲ ▼ët sè ✤➲ t❤✐ t❤û ✣↕✐ ❤å❝ ✷✵✶✵
✭❛✮ ❚➼♥❤ ❣â❝ ✈➔ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ❝→❝ ✤÷í♥❣ t❤➥♥❣
❆❇
✈➔
❈❉
✳
✭❜✮ ●✐↔ sû
(α)
❧➔ ♠➦t ♣❤➥♥❣ ✤✐ q✉❛ ❉ ✈➔ ❝➢t ❜❛ trö❝ t♦↕ ✤ë
Ox, Oy, Oz
t÷ì♥❣ ù♥❣ t↕✐
❝→❝ ✤✐➸♠
▼✱ ◆✱ P
❦❤→❝ ❣è❝
❖
s❛♦ ❝❤♦
❉
❧➔ trü❝ t➙♠ ❝õ❛ t❛♠ ❣✐→❝
▼◆P
✳ ❍➠② ✈✐➳t
♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ♠➦t ♣❤➥♥❣
(α)
❈➙✉ ✼✳
✭✶✱✵ ✤✐➸♠✮
❈❤♦
a, b, c
❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤♦↔ ♠➣♥
ab +bc +ca = 3
✳
❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿
1
1+a2(b+c)+1
1+b2(a+c)+1
1+c2(b+a)≤1
abc
✷✳ ❚❤❡♦ ❝❤÷ì♥❣ tr➻♥❤ ◆➙♥❣ ❝❛♦✿
❈➙✉ ✽✳
✭✷✱✵ ✤✐➸♠✮
❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈î✐ ❤➺ t♦↕ ✤ë
Oxyz
✱ ❝❤♦ ❝→❝ ✤✐➸♠
A(−1; −1; 0) , B (1; −1; 2) , C (2; −2; 1) , D (−1;
✳
✭❛✮ ❚➼♥❤ ❣â❝ ✈➔ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ❝→❝ ✤÷í♥❣ t❤➥♥❣
❆❇
✈➔
❈❉
✳
✭❜✮ ●✐↔ sû
(α)
❧➔ ♠➦t ♣❤➥♥❣ ✤✐ q✉❛
❊
✈➔ ❝➢t t✐❛
Ox
t↕✐
▼
✱ t✐❛
Oy
t↕✐
◆
✱ t✐❛
Oz
t↕✐
P
✳
❱✐➳t ♣❤÷ì♥❣ tr➻♥❤ ♠➦t ♣❤➥♥❣
(α)
❦❤✐ tù ❞✐➺♥ ❖▼◆P ❝â t❤➸ t➼❝❤ ♥❤ä ♥❤➜t✳
❈➙✉ ✾✳
✭✶✱✵ ✤✐➸♠✮ ❚➻♠ ❤➺ sè ❝õ❛
x10
tr♦♥❣ ❦❤❛✐ tr✐➸♥
1+1
x+x310 (x6= 0)
✣➲ sè ✺ ✲❚❍P❚ ❈❍❯❨➊◆ ▲×❒◆● ❱❿◆ ❈❍⑩◆❍
P❍❺◆ ❈❍❯◆● ❈❍❖ ❚❻❚ ❈❷ ❈⑩❈ ❚❍➑ ❙■◆❍ ✭✼ ✤✐➸♠✮
❈➙✉ ✶✳
✭✷ ✤✐➸♠✮ ❈❤♦ ❤➔♠ sè
y=x−1
x+1
✳
✭❛✮ ❑❤↔♦ s→t sü ❜✐➳♥ t❤✐➯♥ ✈➔ ✈➩ ✤ç t❤à
(C)
❝õ❛ ❤➔♠ sè✳
✭❜✮ ❚➻♠
❛
✈➔
❜
✤➸ ✤÷í♥❣ t❤➥♥❣ ✭❞✮✿
y=ax +b
❝➢t
(C)
t↕✐ ❤❛✐ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ✤è✐ ①ù♥❣
♥❤❛✉ q✉❛ ✤÷í♥❣ t❤➥♥❣ ✭❞✬✮✿
x−2y+3 = 0
✳
❈➙✉ ✷✳
✭✷ ✤✐➸♠✮
✭❛✮ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤
√x2+6y=y+3
√x+y+√x−y= 4
✳
✭❜✮ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤
sin 3x−4 cos(x−π
6)−3
sin 3x−1= 0
✳
❈➙✉ ✸✳
✭✶ ✤✐➸♠✮ ❚➼♥❤ t➼❝❤ ♣❤➙♥
3ln2
∫
0
e2xdx
1+√3ex+1
✳
❈➙✉ ✹✳
✭✶ ✤✐➸♠✮ ❈❤♦ ❤➻♥❤ ❤ë♣ ✤ù♥❣
ABCDA′B′C′D′
❝â ✤→② ❧➔ ❤➻♥❤ t❤♦✐ ❝↕♥❤
❛
✱ ❣â❝
❆❇❈
❜➡♥❣
600
✱ ❣â❝ ❣✐ú❛ ♠➦t ♣❤➥♥❣
(A′BD)
✈➔ ♠➦t ♣❤➥♥❣ ✤→② ❜➡♥❣
600
✳
✭❛✮ ❚➼♥❤ t❤❡♦
❛
t❤➸ t➼❝❤ ❤➻♥❤ ❤ë♣✳
✭❜✮ ❚➼♥❤ t❤❡♦
❛
❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ✤÷í♥❣ t❤➥♥❣
❈❉✬
✈➔ ♠➦t ♣❤➥♥❣
(A′BD)
✳
✹
◆❣✉②➵♥ ✣➢❝ ●✐→♣ ✲ ▼ët sè ✤➲ t❤✐ t❤û ✣↕✐ ❤å❝ ✷✵✶✵
❈➙✉ ✺✳
✭✶ ✤✐➸♠✮ ❚➻♠ ❣✐→ trà ❧î♥ ♥❤➜t ✈➔ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❤➔♠ sè
y=sin(x−π
4)
sin x+√1+2cos2x, x ∈hπ
2;πi
P❍❺◆ ❘■➊◆● ✭✸ ✤✐➸♠✮
❚❤➼ s✐♥❤ ❝❤➾ ✤÷ñ❝ ❧➔♠ ♠ët tr♦♥❣ ❤❛✐ ♣❤➛♥ ✭♣❤➛♥ ✶ ❤♦➦❝ ♣❤➛♥ ✷✮✳
✶✳ ❚❤❡♦ ❝❤÷ì♥❣ tr➻♥❤ ❝❤✉➞♥
❈➙✉ ✻✳
✭✷ ✤✐➸♠✮
✭❛✮ ❚r♦♥❣ ♠➦t ♣❤➥♥❣
Oxy
✱ ❝❤♦ t❛♠ ❣✐→❝
❆❇❈
❜✐➳t
A(1; 4)
✱ ♣❤÷ì♥❣ tr➻♥❤ ✤÷í♥❣ ❝❛♦
❇❍
❧➔
x−2y+ 9 = 0
✱ ♣❤÷ì♥❣ tr➻♥❤ ✤÷í♥❣ ♣❤➙♥ ❣✐→❝ tr♦♥❣
❈❉
❧➔
x+y−3 = 0
✳ ❚➻♠
❤❛✐ ✤➾♥❤
❇
✈➔
❈
✳
✭❜✮ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥
Oxyz
✱ ❝❤♦ ✤÷í♥❣ t❤➥♥❣ ✭❞✮✿
x−1
1=y−1
2=z+2
−2
✈➔ ♠➦t ❝➛✉ ✭❙✮✿
(x−1)2+(y+1)2+(z+3)2= 9
✳
✐✳ ❈❤ù♥❣ ♠✐♥❤ ✭❞✮ ✈➔ ✭❙✮ ❝â ❤❛✐ ✤✐➸♠ ❝❤✉♥❣
❆✱ ❇
♣❤➙♥ ❜✐➺t✳
✐✐✳ ❱✐➳t ♣❤÷ì♥❣ tr➻♥❤ ♠➦t ♣❤➥♥❣
(α)
❜✐➳t
(α)
q✉❛
❆✱ ❇
✈➔ ❝➢t ✭❙✮ t❤❡♦ ♠ët ❣✐❛♦
t✉②➳♥ ❧➔ ♠ët ✤÷í♥❣ trá♥ ❧î♥ ❝õ❛ ✭❙✮✳
❈➙✉ ✼✳
✭✶ ✤✐➸♠✮ ❚ø ❝→❝ ❝❤ú sè ✶✱ ✷✱ ✸✱ ✹✱ ✺ ❝â t❤➸ ❧➟♣ ✤÷ñ❝ ❜❛♦ ♥❤✐➯✉ sè tü ♥❤✐➯♥ ❝â ♥➠♠ ❝❤ú
sè✱ tr♦♥❣ ✤â ❝❤ú sè ✸ ❝â ♠➦t ✤ó♥❣ ❜❛ ❧➛♥✱ ❝→❝ ❝❤ú sè ❝á♥ ❧↕✐ ❝â ♠➦t ❦❤æ♥❣ q✉→ ♠ët ❧➛♥✳
❚r♦♥❣ ❝→❝ sè tü ♥❤✐➯♥ ♥â✐ tr➯♥✱ ❝❤å♥ ♥❣➝✉ ♥❤✐➯♥ ♠ët sè✱ t➻♠ ①→❝ s✉➜t ✤➸ sè ✤÷ñ❝ ❝❤å♥
❝❤✐❛ ❤➳t ❝❤♦ ✸✳
✷✳ ❚❤❡♦ ❝❤÷ì♥❣ tr➻♥❤ ♥➙♥❣ ❝❛♦
❈➙✉ ✽✳
✭✷ ✤✐➸♠✮
✭❛✮ ❚r♦♥❣ ♠➦t ♣❤➥♥❣
Oxy
✱ ❝❤♦ ✤÷í♥❣ trá♥
(C)
✿
(x−1)2+(y+1)2= 4
✳ ▼ët ✤÷í♥❣ trá♥
✭❈✬✮ t✐➳♣ ①ó❝ ✈î✐
Oy
✈➔ t✐➳♣ ①ó❝ ♥❣♦➔✐ ✈î✐
(C)
✳ ❚➻♠ t➙♠ ❝õ❛ ✭❈✬✮ ❜✐➳t t➙♠ t❤✉ë❝
✤÷í♥❣ t❤➥♥❣ ✭❞✮✿
2x−y= 0
✳
✭❜✮ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥
Oxyz
✱ ❝❤♦ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ ✭❛✮ ✈➔ ✭❜✮ ❝â ♣❤÷ì♥❣ tr➻♥❤ ❧➛♥ ❧÷ñt ❧➔
x+2
4=y+1
−1=z
1,x−2
4=y−1
−1=z+2
1
✳
✐✳ ❈❤ù♥❣ ♠✐♥❤ ✭❛✮ s♦♥❣ s♦♥❣ ✈î✐ ✭❜✮✱ t➼♥❤ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ❝❤ó♥❣✳
✐✐✳ ❱✐➳t ♣❤÷ì♥❣ tr➻♥❤ ♠➦t ♣❤➥♥❣
(α)
q✉❛ ✭❛✮ ✈➔ ✈✉æ♥❣ ❣â❝ ✈î✐ ♠♣✭❛✱ ❜✮✳
❈➙✉ ✾✳
✭✶ ✤✐➸♠✮ ❚➻♠
♥
♥❣✉②➯♥ ❞÷ì♥❣ ❜✐➳t✿
C1
n
2−2C2
n
22+3C3
n
23+...+(−1)n−1nCn
n
2n=1
32
✳
✣➋ ❙➮ ✻ ✲ ❚❘×❮◆● ❚❍P❚ ❈❍❯❨➊◆ ▲➊ ◗❯Þ ✣➷◆
▼æ♥ t❤✐✿ ❚♦→♥ ✭ ❑❤è✐ ❆✮
❈➙✉ ✶✳
✭✷ ✤✐➸♠✮
✭❛✮ ❑❤↔♦ s→t sü ❜✐➳♥ t❤✐➯♥ ✈➔ ✈➩ ✤ç t❤à ❤➔♠ sè
y=x3−4x2
✭
C1)
✭❜✮ ❍➠② ✈✐➳t ♣❤÷ì♥❣ tr➻♥❤ t✐➳♣ t✉②➳♥ ❝❤✉♥❣ ❝õ❛ ✭
C1
✮ ✈➔ ♣❛r❛❜♦❧
(P) : y=x2−8x+4
✳
❈➙✉ ✷✳
✭✷ ✤✐➸♠✮
✺