❙Ð ●❉ ❱⑨ ✣❚ ◗❯❷◆● ❚❘➚
❚❘×❮◆● ❚❍P❚ ❈❍❯❨➊◆ ▲➊ ◗❯Þ ✣➷◆
❑➐ ❚❍■ ❚❍Û ❚❍P❚ ◗❯➮❈ ●■❆ ▲❺◆ ✶ ◆❿▼ ✷✵✶✾
▼➷◆ ❚❖⑩◆
❚❤í✐ ❣✐❛♥ ❧➔♠ ❜➔✐ ✾✵ ♣❤ót✱ ❦❤æ♥❣ ❦➸ 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
✳
❉✳
a3
3
✳
❈➙✉ ✷✳
❈❤♦ ❤➔♠ sè ❜➟❝ ❜❛
y=f(x)
❝â ✤ç t❤à ♥❤÷ ❤➻♥❤ ✈➩✳ ▼➺♥❤ ✤➲ ♥➔♦ ❞÷î✐ ✤➙② ✤ó♥❣❄
❆✳
●✐→ trà ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ sè ❜➡♥❣
−1
✳
❇✳
✣✐➸♠ ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ sè ❧➔
−1
✳
❈✳
✣✐➸♠ ❝ü❝ ✤↕✐ ❝õ❛ ❤➔♠ sè ❧➔
3
✳
❉✳
●✐→ trà ❝ü❝ ✤↕✐ ❝õ❛ ❤➔♠ sè ❧➔
0
✳
x
y
−1
2
3
O
❈➙✉ ✸✳
❈❤♦ sè ♣❤ù❝
z= (1 −2i)2
✳ ❚➼♥❤ ♠æ ✤✉♥ ❝õ❛ sè ♣❤ù❝
1
z
✳
❆✳
1
5
✳
❇✳
√5
✳
❈✳
1
25
✳
❉✳
1
√5
✳
❈➙✉ ✹✳
❚➻♠ ♥❣❤✐➺♠ ❝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+ 3x2−4
✤ç♥❣ ❜✐➳♥ tr➯♥ t➟♣ ❤ñ♣ ♥➔♦ tr♦♥❣ ❝→❝ t➟♣ ❤ñ♣ ✤÷ñ❝ ❝❤♦
❞÷î✐ ✤➙②❄
❆✳
(2; +∞)
✳
❇✳
(0; 2)
✳
❈✳
(−∞; 0) ∪(2; +∞)
✳
❉✳
(−∞; 0)
✳
❈➙✉ ✼✳
❚➼♥❤ t➼❝❤ ♣❤➙♥
I=
2
Z
1
x−1
xdx
✳
❆✳
I= 1 + ln 2
✳
❇✳
I=7
4
✳
❈✳
I= 2 ln 2
✳
❉✳
I= 1 −ln 2
✳
❈➙✉ ✽✳
❑❤è✐ ♥â♥
(N)
❝â ❜→♥ ❦➼♥❤ ✤→② ❜➡♥❣
3
✈➔ ❞✐➺♥ t➼❝❤ ①✉♥❣ q✉❛♥❤ ❜➡♥❣
15π
✳ ❚➼♥❤
t❤➸ t➼❝❤ ❦❤è✐ ♥â♥
(N)
✳
❆✳
12π
✳
❇✳
16π
✳
❈✳
45π
✳
❉✳
36π
✳
❈➙✉ ✾✳
❈❤♦ ❜✐➸✉ t❤ù❝
P=3
v
u
u
t2
3
3
s2
3r2
3
✳ ▼➺♥❤ ✤➲ ♥➔♦ tr♦♥❣ ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ ✤ó♥❣❄
❆✳
P=2
318
✳
❇✳
P=2
3
1
2
✳
❈✳
P=2
3
1
8
✳
❉✳
P=2
3
1
18
✳
❚r❛♥❣ ✶✴✻ ▼➣ ✤➲ ✶✵✶
❈➙✉ ✶✵✳
❈❤♦ sè ♣❤ù❝
z=(2 −3i)(4 −i)
3 + 2i
✳ ❚➻♠ tå❛ ✤ë ✤✐➸♠ ❜✐➸✉ ❞✐➵♥ ❝õ❛ sè ♣❤ù❝
z
tr➯♥
♠➦t ♣❤➥♥❣
Oxy
✳
❆✳
(1; 4)
✳
❇✳
(1; −4)
✳
❈✳
(−1; −4)
✳
❉✳
(−1; 4)
✳
❈➙✉ ✶✶✳
❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈î✐ ❤➺ tå❛ ✤ë
Oxyz
✱ ❝❤♦ ♠➦t ♣❤➥♥❣
(P)
✿
2x−2y+z+ 2017 = 0
✱
✈➨❝✲tì ♥➔♦ tr♦♥❣ ❝→❝ ✈➨❝✲tì ✤÷ñ❝ ❝❤♦ ❞÷î✐ ✤➙② ❧➔ ♠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′
✳
❆✳
18√2
❝♠
3
✳
❇✳
3
❝♠
3
✳
❈✳
9
❝♠
3
✳
❉✳
18
❝♠
3
✳
❈➙✉ ✶✸✳
❚r♦♥❣ ♠➦t ♣❤➥♥❣ tå❛ ✤ë
Oxy
✱ t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ❜✐➸✉ ❞✐➵♥ ❝→❝ sè ♣❤ù❝
z
t❤ä❛
♠➣♥
|z−1 + 2i|=|z+ 1 + 2i|
❧➔ ✤÷í♥❣ t❤➥♥❣ ❝â ♣❤÷ì♥❣ tr➻♥❤
❆✳
x+ 2y= 0
✳
❇✳
x−2y= 0
✳
❈✳
x−2y+ 1 = 0
✳
❉✳
x+ 2y+ 1 = 0
✳
❈➙✉ ✶✹✳
❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ trö ❝â ❜→♥ ❦➼♥❤
R= 3,
❝❤✐➲✉ ❝❛♦
h= 5.
❆✳
V= 90π
✳
❇✳
V= 45π
✳
❈✳
V= 15π
✳
❉✳
V= 45
✳
❈➙✉ ✶✺✳
❚➼♥❤ sè ✤÷í♥❣ t✐➺♠ ❝➟♥ ❝õ❛ ✤ç t❤à ❤➔♠ sè
y=x2+x−2
x2−3x+ 2
✳
1
1−2x
tr➯♥
−∞;1
2
✳
❆✳
1
2ln(1 −2x) + C
✳
❇✳
ln |2x−1|+C
✳
❈✳
1
2ln |2x−1|+C
✳
❉✳
−1
2ln |2x−1|+C
✳
❈➙✉ ✶✼✳
❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈î✐ ❤➺ tå❛ ✤ë
Oxyz
✱ ❝❤♦ ♠➦t ♣❤➥♥❣
(P) : 2x−2y+z+ 4 = 0
✳
❚➼♥❤ ❦❤♦↔♥❣ ❝→❝❤
d
tø ✤✐➸♠
M(1; 2; 1)
✤➳♥ ♠➦t ♣❤➥♥❣
(P)
✳
❆✳
d= 1
✳
❇✳
d=1
3
✳
❈✳
d= 3
✳
❉✳
d= 4
✳
❈➙✉ ✶✽✳
❈❤♦ ❤➻♥❤ ❝❤â♣
S.ABC
❝â t❤➸ t➼❝❤ ❜➡♥❣
1
✳ ❚r➯♥ ❝↕♥❤
BC
❧➜② ✤✐➸♠
E
s❛♦ ❝❤♦
BE = 2EC
✳ ❚➼♥❤ t❤➸ t➼❝❤
V
❝õ❛ ❦❤è✐ tù ❞✐➺♥
SAEB
✳
❆✳
V=1
3
✳
❇✳
V=2
3
✳
❈✳
V=4
3
✳
❉✳
V=1
6
✳
❈➙✉ ✶✾✳
❚➼♥❤ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ sè
y= log9x2+ 1
✳
❆✳
y′=2xln 9
x2+ 1
✳
❇✳
y′=2 ln 3
x2+ 1
✳
❈✳
y′=x
(x2+ 1) ln 3
✳
❉✳
y′=1
(x2+ 1) ln 9
✳
❈➙✉ ✷✵✳
●å✐
z1
✱
z2
❧➔ ❤❛✐ ♥❣❤✐➺♠ ♣❤ù❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
z2−4z+ 5 = 0
✳ ❚➼♥❤
w=
1
z1
+1
z2
+i(z12z2+z22z1)
✳
❆✳
w= 20 + 4
5i
✳
❇✳
w=4
5+ 20i
✳
❈✳
w=−4
5+ 20i
✳
❉✳
w= 4 + 20i
✳
❈➙✉ ✷✶✳
❚➼♥❤ tê♥❣ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
1
2log2(x+ 3) = log2x+ 1 + x2−
x−4 + 2√x+ 3
✳
❆✳
S= 2
✳
❇✳
S= 1
✳
❈✳
S=−1
✳
❉✳
S= 1 −√2
✳
❚r❛♥❣ ✷✴✻ ▼➣ ✤➲ ✶✵✶
❆✳
2
✳
❇✳
3
✳
❈✳
1
✳
❉✳
0
✳
❈➙✉ ✶✻✳
❚➻♠ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ sè
f(x)=
❈➙✉ ✷✷✳
❚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) = mx3+ (3m+n)x2−4x+ 3
❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠
sè
f(x) = 3x2+ 10x−4
✳ ❚➼♥❤
mn
✳
❆✳
mn = 1
✳
❇✳
mn = 3
✳
❈✳
mn = 2
✳
❉✳
mn = 0
✳
❈➙✉ ✷✹✳
❚➼❝❤ ♣❤➙♥
I=
1
Z
0
(x−1)2
x2+ 1 dx=a−ln b
✱ 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
✳
❈➙✉ ✷✻✳
❚➻♠ t➜t ❝↔ ❝→❝ ❣✐→ trà t❤ü❝ ❝õ❛ t❤❛♠ sè
m
✤➸ ❤➔♠ sè
y=x+ 2 −m
x+ 1
♥❣❤à❝❤ ❜✐➳♥
tr➯♥ ♠é✐ ❦❤♦↔♥❣ ①→❝ ✤à♥❤ ❝õ❛ ♥â✳
❆✳
m≤ −3
✳
❇✳
m < −3
✳
❈✳
m < 1
✳
❉✳
m≤1
✳
❈➙✉ ✷✼✳
●å✐
(D)
❤➻♥❤ ♣❤➥♥❣ ❣✐î✐ ❤↕♥ ❜ð✐ ❝→❝ ✤÷í♥❣
y=x
4
✱
y= 0
✱
x= 1
✱
x= 4
✳ ❚➼♥❤ t❤➸
t➼❝❤ ✈➟t t❤➸ trá♥ ①♦❛② t↕♦ t❤➔♥❤ ❦❤✐ q✉❛② ❤➻♥❤
(D)
q✉❛♥❤ trö❝
Ox
✳
❆✳
21
16
✳
❇✳
21π
16
✳
❈✳
15π
8
✳
❉✳
15
16
✳
❈➙✉ ✷✽✳
❈❤♦ sè ♣❤ù❝
z
t❤ä❛
|z−1 + 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)
✳
❈➙✉ ✷✾✳
❈❤♦
x, y > 0
t❤ä❛ ♠➣♥
x+y=3
2
✈➔ ❜✐➸✉ t❤ù❝
P=4
x+1
4y
✤↕t ❣✐→ trà ♥❤ä ♥❤➜t✳
❚➼♥❤
x2+y2
✳
❆✳
153
100
✳
❇✳
5
4
✳
❈✳
2313
1156
✳
❉✳
25
16
✳
❈➙✉ ✸✵✳
❈❤♦ sè t❤ü❝
a > 0, a 6= 1
✳ ●✐→ trà
log√a3
3
√a2
❜➡♥❣
❆✳
1
✳
❇✳
2
3
✳
❈✳
4
9
✳
❉✳
9
4
✳
❈➙✉ ✸✶✳
●å✐
M(a;b)
❧➔ ✤✐➸♠ tr➯♥ ✤ç t❤à ❝õ❛ ❤➔♠ sè
y=x−2
x
s❛♦ ❝❤♦ ❦❤♦↔♥❣ ❝→❝❤ tø
M
✤➳♥ ✤÷í♥❣ t❤➥♥❣
d:y= 2x+ 6
♥❤ä ♥❤➜t✳ ❚➼♥❤
(4a+ 5)2+ (2b−7)2
✳
❆✳
2
✳
❇✳
0
✳
❈✳
18
✳
❉✳
162
✳
❈➙✉ ✸✷✳
❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥
Oxyz
✱ ❝❤♦ ♠➦t ♣❤➥♥❣
(P) : x−y+ 2 = 0
✈➔ ❤❛✐ ✤✐➸♠
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)
✤÷ñ❝ ❣✐î✐ ❤↕♥ ❜ð✐ ❤❛✐ ✤÷í♥❣
y= 2(x2−1); y= 1 −x2
✳ ❚➼♥❤
t❤➸ t➼❝❤ ❦❤è✐ trá♥ ①♦❛② t↕♦ t❤➔♥❤ ❞♦
(D)
q✉❛② q✉❛♥❤ trö❝
Ox
✳
❆✳
32
15
✳
❇✳
64π
15
✳
❈✳
64
15
✳
❉✳
32π
15
✳
❈➙✉ ✸✹✳
❈❤♦ ❤➔♠ sè
f(x)
❝â ✤↕♦ ❤➔♠
f′(x) = (x−1)(x2−3)(x4−1)
✈î✐ ♠å✐
x
t❤✉ë❝
R
✳
❙♦ s→♥❤
f(−2), f(0), f(2)
✱ t❛ ✤÷ñ❝
❆✳
f(−2) < f(2) < f(0)
✳
❇✳
f(−2) < f(0) < f(2)
✳
❈✳
f(2) < f(0) < f(−2)
✳
❉✳
f(0) < f(−2) < f(2)
✳
❈➙✉ ✸✺✳
❈❤♦ ❤❛✐ sè ♣❤ù❝
z, w
t❤ä❛ ♠➣♥
|z−3√2|=√2,|w−4√2i|= 2√2
✳ ❇✐➳t r➡♥❣
|z−w|
✤↕t ❣✐→ trà ♥❤ä ♥❤➜t ❦❤✐
z=zo, w =wo
✳ ❚➼♥❤
|3zo−wo|
✳
❆✳
6√2
✳
❇✳
2√2
✳
❈✳
4√2
✳
❉✳
1
✳
❈➙✉ ✸✻✳
❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥
Oxyz
✱ ❝❤♦ ♠➦t ♣❤➥♥❣
(P) : x+y+z−3 = 0
✈➔ ❜❛ ✤✐➸♠ ✤✐➸♠
A(3; 1; 1), B(7; 3; 9)
✈➔
C(2; 2; 2)
✳ ✣✐➸♠
M(a;b;c)
tr➯♥
(P)
s❛♦ ❝❤♦
|
★✥✥✥✥✥✔
MA + 2
★✥✥✥✥✥✥✔
MB + 3
★✥✥✥✥✥✔
MC|
✤↕t
❈➙✉ ✸✼✳
❈❤♦ ❤➻♥❤ ❝❤â♣
S.ABCD
❝â ✤→②
ABCD
❧➔ ❤➻♥❤ ✈✉æ♥❣✱ t➙♠
O
✱ ❝↕♥❤
a
✈➔
SO ⊥
(ABCD), SA = 2a√2
✳ ●å✐
M, N
❧➛♥ ❧÷ñt ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛
SA, BC
✳ ❚➼♥❤ ❣â❝ ❣✐ú❛ ✤÷í♥❣
t❤➥♥❣
MN
✈➔ ♠➦t ♣❤➥♥❣
(ABCD)
✳
❆✳
π
3
✳
❇✳
π
4
✳
❈✳
arctan 2
✳
❉✳
π
6
✳
❈➙✉ ✸✽✳
❚➼♥❤ sè ❣✐→ trà ♥❣✉②➯♥ ❝õ❛ t❤❛♠ sè
m
tr➯♥ ❦❤♦↔♥❣
(−2019; 2019)
✤➸ ❤➔♠ sè
y=x4−2mx2−3m+ 1
✤ç♥❣ ❜✐➳♥ tr➯♥ ❦❤♦↔♥❣
(1; 2)
✳
❆✳
2
✳
❇✳
2020
✳
❈✳
1
✳
❉✳
2019
✳
❈➙✉ ✸✾✳
❚➼♥❤ tê♥❣ t➜t ❝↔ ❝→❝ ❣✐→ trà ❝õ❛ t❤❛♠ sè
m
✤➸ tç♥ t↕✐ ❞✉② ♥❤➜t ♠ët sè ♣❤ù❝
z
t❤ä❛ ♠➣♥ ✤ç♥❣ t❤í✐
|z|=m
✈➔
|z−4m+ 3mi|=m2
✳
❆✳
10
✳
❇✳
9
✳
❈✳
4
✳
❉✳
6
✳
❈➙✉ ✹✵✳
▼ët ❝❤✐➳❝ ✈á♥❣ ✤❡♦ t❛② ❣ç♠ ✷✵ ❤↕t ❣✐è♥❣ ♥❤❛✉✳ ❍ä✐ ❝â ❜❛♦ ♥❤✐➯✉ ❝→❝❤ ❝➢t ❝❤✐➳❝
✈á♥❣ ✤â t❤➔♥❤ ✷ ♣❤➛♥ ♠➔ sè ❤↕t ð ♠é✐ ♣❤➛♥ ✤➲✉ ❧➔ sè ❧➫ ❄
❆✳
5
✳
❇✳
180
✳
❈✳
10
✳
❉✳
90
✳
❈➙✉ ✹✶✳
❈❤♦ ❤➔♠ sè
f(x)
❝â ✤↕♦ ❤➔♠ ❧➔
f′(x)
✳ ✣ç t❤à ❝õ❛ ❤➔♠
sè
y=f′(x)
♥❤÷ ❤➻♥❤ ✈➩ ❜➯♥✳ ❚➼♥❤ sè ✤✐➸♠ ❝ü❝ trà ❝õ❛
❤➔♠ sè
y=f(x2)
tr➯♥ ❦❤♦↔♥❣
(−√5; √5)
✳
❆✳
2
✳
❇✳
5
✳
❈✳
4
✳
❉✳
3
✳①
②
0 5
y=f′(x)
2
❚r❛♥❣ ✹✴✻ ▼➣ ✤➲ ✶✵✶
❣✐→ trà ♥❤ä ♥❤➜t✳ ❚➼♥❤
2a−15b+c
✳
❆✳
8
✳
❇✳
1
✳
❈✳
3
✳
❉✳
6
✳
❈➙✉ ✹✷✳
❈❤♦ ❤➻♥❤ ❝❤â♣
S.ABCD
❝â
SA ⊥(ABCD)
✱ ✤→②
ABCD
❧➔ ❤➻♥❤ ❝❤ú ♥❤➟t ✈î✐
AC =a√5
✈➔
BC =a√2
✳ ❚➼♥❤ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛
SD
✈➔
BC
✳
❆✳
a√3
2
✳
❇✳
a√3
✳
❈✳
3a
4
✳
❉✳
2a
3
✳
❈➙✉ ✹✸✳
◆❣÷í✐ t❛ ❧➔♠ t↕ t➟♣ ❝ì t❛② ♥❤÷ ❤➻♥❤ ✈➩ ✈î✐ ❤❛✐ ✤➛✉ ❧➔ ❤❛✐ ❦❤è✐ trö ❜➡♥❣
♥❤❛✉ ✈➔ t❛② ❝➛♠ ❝ô♥❣ ❧➔ ❦❤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 tæ ❦❤✐ ❜ì♠ ❝➠♥❣ ✤➦t ♥➡♠ tr➯♥ ♠➦t ♣❤➥♥❣ ♥➡♠ ♥❣❛♥❣ ❝â
❤➻♥❤ ❝❤✐➳✉ ❜➡♥❣ ♥❤÷ ❤➻♥❤ ✈➩ ✈î✐ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥❤ä
R1= 20cm
✱
❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ❧î♥
R2= 30cm
✈➔ ♠➦t ❝➢t ❦❤✐ ❝➢t ❜ð✐ ♠➦t ♣❤➥♥❣
✤✐ q✉❛ trö❝✱ ✈✉æ♥❣ ❣â❝ ✈î✐ ♠➠t ♣❤➥♥❣ ♥➡♠ ♥❣❛♥❣ ❧➔ ❤❛✐ ✤÷í♥❣ trá♥✳ ❇ä
q✉❛ ✤ë ❞➔② ❝õ❛ ✈ä s➠♠✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤æ♥❣ ❦❤➼ ✤÷ñ❝ ❝❤ù❛ ❜➯♥ tr♦♥❣
s➠♠✳
❆✳
1400πcm3
✳
❇✳
1250πcm3
✳
❈✳
2500πcm3
✳
❉✳
600πcm3
✳
❈➙✉ ✹✺✳
❈❤♦ ❤➔♠ 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
✳
❈➙✉ ✹✻✳
❈❤♦
F(x)
❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ sè
f(x) = 1
cos2x
✳ ❇✐➳t
Fπ
4+kπ=k
✈î✐
♠å✐
k∈Z
✳ ❚➼♥❤
F(0) + F(π) + F(2π) + ... +F(10π)
✳
❆✳
45
✳
❇✳
0
✳
❈✳
55
✳
❉✳
44
✳
❈➙✉ ✹✼✳
▼ët ♥❣÷í✐ ❣û✐ sè t✐➲♥
100
tr✐➺✉ ✤ç♥❣ ✈➔♦ ♥❣➙♥ ❤➔♥❣ ✈î✐ ❧➣✐ s✉➜t
0,5%/
t❤→♥❣ ✈➔
æ♥❣ t❛ rót ✤➲✉ ✤➦♥ ♠é✐ t❤→♥❣ ♠ët tr✐➺✉ ✤ç♥❣ ❦➸ tø s❛✉ ♥❣➔② ❣û✐ ♠ët t❤→♥❣ ❝❤♦ ✤➳♥ ❦❤✐
❤➳t t✐➲♥ ✭t❤→♥❣ ❝✉è✐ ❝ò♥❣ ❝â t❤➸ ❦❤æ♥❣ ❝á♥ ✤õ ♠ët tr✐➺✉ ✤ç♥❣✮✳ ❍ä✐ æ♥❣ t❛ rót ❤➳t t✐➲♥
s❛✉ ❜❛♦ ♥❤✐➯✉ t❤→♥❣❄
❆✳
100
✳
❇✳
140
✳
❈✳
138
✳
❉✳
139
✳
❆✳
V=1
3
✳
❇✳
V=1
2
✳
❈✳
V= 2
✳
❉✳
V= 1
✳
x
tr♦♥❣ ❦❤♦↔♥❣
11π
12 ; 2019π
❆✳
2019
✳
❇✳
2018
✳
❈✳
1
✳
❉✳
2020
✳
❚r❛♥❣ ✺✴✻ ▼➣ ✤➲ ✶✵✶
❈➙✉ ✹✽✳
❈❤♦ ❤➻♥❤ ❝❤â♣
S.ABCD
❝â ✤→② ❧➔ ❤➻♥❤ ❜➻♥❤ ❤➔♥❤ ✈➔ ❝â t❤➸ t➼❝❤ ❜➡♥❣
48
✳ ❚r➯♥
❝↕♥❤
SB, SD
❧➜② ✤✐➸♠ ❝→❝
M, N
s❛♦ ❝❤♦
SM =MB, SD= 3SN
✳ ▼➦t ♣❤➥♥❣
(AMN)
❝➢t
SC
t↕✐
P
✳ ❚➼♥❤ t❤➸ t➼❝❤
V
❝õ❛ ❦❤è✐ tù ❞✐➺♥
SMNP
✳
❈➙✉ ✹✾✳
❚➼♥❤ sè ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
cotx = 2