(0,5 diem) diem)
Ta CO sin(2a + p) = 3sinp » sin ((a + p) + a) = 3sin ((a + P) - a)( 1) VT( 1) = sin(a + P)cosa + cos(a + P)sina VT( 1) = sin(a + P)cosa + cos(a + P)sina
VP( i ) = 3(sin(a + P)cosa - cos(a + p)sina)
0,25
( i ) o 4cos((x + P)sina = 2sin(a + P)cosa (2)
Do tan(a + P) va lana c6 nghTạncn la chia hai vc ciia (2) cho 2cos(a + p)cosa 2cos(a + p)cosa
Ta sc difdc lan(a + P) = 2tana (dpcm).
0,25
D E S O 3 9 D E THCT S O C H O C KI II M O N T O A N L6P 1 0
Thdi gian lam bai: 90 phut
Cau I . (3,0 diem) Giiii cac bat phiftlng trinh sau:
X - 5 x + 6
2. 7x" - 2 x + l <x*' - 4 x + 5.
3. 3 V x + -—p= < 2x + 7.
2Vx 2x
C f i u I I . (1,5 diem)
1. Tim m dc bat phifdng Irinh X" - 8 X + 20 • < 0, V X €
mx" + 2(m + 1 )x + 9m + 4
2. Tim m di! bal phiTdng Irinh : (2 - m)x + m" - 3 m + 2 > 0, Vx € [0; 1].
MA
fiu I I I . (2,0 diem) Trong mat phIng Oxy cho A ( - 2 ; 0), B(0; 1), C( l ; 2). 1. Vicl phirong irinh ciia diTctng Iron (C) di qua 3 diem A, B, C . 1. Vicl phirong irinh ciia diTctng Iron (C) di qua 3 diem A, B, C .
2. Viel phúdng lilnh licp luyen d ciia di/cfng iron (C) lai M ( - 2 ; 0). 3. Tim diem N ihuoc difiJng Iron (C) sao cho lam giac NAB can lai N . 3. Tim diem N ihuoc difiJng Iron (C) sao cho lam giac NAB can lai N . Ifiu IV. (1,0 diem) Chifng minh rang nc'u x > y > 1 ihi x + 1 + > 6
( x - y ) ( y + l)
fiu V. (2,0 diem) Cho parabol (P):y^ =8x, diem 1(2; 4) thuoc (P). Xet goc
-6ng thay đi quay quanh I , hai canh goc vuong cat (P) lai M , N (khac diem I). Chufng minh duTtng lhang M N hion di qua mot diem co djnh. Chufng minh duTtng lhang M N hion di qua mot diem co djnh.
| j C a u V I . (0,5 diem) Gia siir a, p la hai giii Irj khac nhau cua x lh6a man jicosx + bsinx = c vdi er +b~ * 0. Chiang minh: cos" —cos^ — = ^'^^ jicosx + bsinx = c vdi er +b~ * 0. Chiang minh: cos" —cos^ — = ^'^^
2 2 4(a2 + b2)
D A P A N T H A M K H A O
edit Dap an Diem
I (3,0 (3,0 diem)
1. (1,0 diem) Giai bat phUdng trinh...
, + 4 x + 4 (x + 2 r Ta CO — < 0 <=> — < 0 <=> x~ - 5 x + 6 X ' - 5 x + 6 x = -2 fx It -2 x^ - 5 x + 6 < 0 X = -2 2 < X < 3
Vay lap nghiem ciia BPTda cho la S=(2; 3 ) u ( - 2
0,5
0,5
1. (1,0 diem) Giai bat phrftfng trinh..
Ta CO - 2 x + l < x ^ - 4 x + 5 o \](x-l)^ < x^ - 4 x + 5 x - 1 <x^ - 4 x + 5 (*) x - 1 <x^ - 4 x + 5 (*) VI X ' - 4x + 5 = (X - 2)- + 1 > 0, Vx 6 R nen X - 1 < X - 4x + 5 <=> - ( X " - 4 x + 5 ) < x - l x^ - 5 x + 6 > 0 3 X — 2 .1>0 4 x<2 X >3
Vay tap nghiem ciia BPT da cho la S = (-oo; 2) u (3; + oo).
0,25
0,5
0,25
3. (1,0 diem) Giai bat phrfcftis trinh...
BPT da cho c6 the viel lai
<2 x + 1 ^ Dat u = N/X + Dat u = N/X + 4x 1 X >0 -7 (*) => u - 1 = X + • 2N / X ' [ U > > ^ 1 (Ihco BDT Co-si) 4x (*)tnnhanh 2u--3u-9>0=:> u >3 x > 0<x < 8 + 3>/7 > 3 o x >0 2x-6>Ac+ 1 >0 3 + sfl 0<yf\ 3-V7 8-3\/7
Vay lap nghiem cua BPT da cho la T = 0; 8-3N/7^ '8 + 3>/7 - ; + 0 0 T = 0; 8-3N/7^ '8 + 3>/7 - ; + 0 0
II
(1,5 diem) diem)
1. (1,0 diem) Tim m de bflt phrftfng trinh.
Vi X" - 8x + 20 = (x - 4)" + 4 > 0, Vx e R , BPT da cho tiTdng difdng vdi mx" +2(m + l)x + 9m + 4<0, Vx € R (*) difdng vdi mx" +2(m + l)x + 9m + 4<0, Vx € R (*)
' Néu m = 0: 2x + 4<0<=>x<-2 (khong thoa man moi x). • Neu m ^0: (*)•» |m < 0 • Neu m ^0: (*)•» |m < 0
|Á = -8m^ -2m + l<0
m<0 1 1 <=> m < 1
m < — hoac m > — 2 2 4
2. (0,5 diem) Tim m de bS't phifdng trinh
Dat l(x) = ( 2 - m)x + m2 -3m+ 2
De f(x) > 0, Vx € (0; 1 ] Ihi f(0)>0 ^ j m 2 - 3 m + 2>0 f(l)>0 lm2-4m + 4 > 0
176
u y iNMM M i v u v v H Rnanfl vỉi m < 1 m > 2 <=> m^2 m < 1 m > 2 0,25 III (2,0 diem)
1. (0,5 diem) \i&'t phtf</ng trinh cua dtfc/ng tron,
(C) CO diing x- + y- -2ax-2by+ c= 0 , vdi â - o O Thco giii ihicl (C) qua ba diem A, B, C nen Thco giii ihicl (C) qua ba diem A, B, C nen
4 + 2a + c = 0 1 - 2b + c = 0 » <; 1 - 2b + c = 0 » <; 5 - 2 a - 4 b + c = 0 7 2 2 c = to Vay (C): x" + y^ + 7x -1 ly +10 = 0. 0,5
2. (0,5 diem) Vifít phif(/ng trinh tiep tuyen d cua di/(/ng tron.. (C) C O lam I 7 n 2' 2 va ban kinh R = (C) C O lam I 7 n 2' 2 va ban kinh R =
d licp xiic (C) tai M d CO VTCP MI = 3 n
2' 2
0,25
d qua M va c6 VTCP MI nen phifdng Irinh cija d la
-3(x + 2) + 1 ly = 0 3x - lly + 6 = 0. 0,25
3. (1,0 diem) Tim diem N thuoc drfc/ng tron (C) sao cho tarn giac...
Trung diem E ciia doan AB la E -1; -2 , AB = (2; 1). 0,25
PhiTcJng trinh difcJng trung Irifc d' ciia doan AB di qua E va c6
VTPT ABla 2(x + 1) + y - 0 < = > 2 x + y +1 = 0. 0,25 N e ( O va ANAB can tai N, nen loa do diem N ia nghiem cua N e ( O va ANAB can tai N, nen loa do diem N ia nghiem cua
he 2x + y + | = 0 (1)
x2+ y 2 + 7 x - l l y + 10 = 0 (2)
Tir (Ij => y = -2x - - (3) 0,25
The (3) vao (2) va rut gon lai la di/dc:
V a y CO hai d i e m can l i m N , '-l+sjlb 11 , N . IV (1,0 diem) ChiJnK minh... Ta CO X + 1 + • ( x - y ) ( y + l) = ( x- y ) + ( y + 1) + 8 ( x - y ) ( y + 1)
A p diing B D T Co-si cho ba so diTdng 8 (x - y ) . ( y + 1). Ta CO X + 1 + • ( x - y ) ( y + l) 8 ( x - y K y + 1) > 3 3 ( x - y ) ( y + l ) 8 ( x - y ) ( y + l) = 6. D a n g ihuTc xay ra <=> X > y > 0 X - y = y + 1 = ( x - y ) ( y + l) x = 3 y=r V (2,0 diem)
Cliurng minh drf&ns thang MN luon di qua mpt diem códjnh.
G o i M ( 1 \ / m " , N ; m , N V / V -: n (m * n, m ^ 4, n 4 ) T a c o 1M = m " - 2 ; m - 4 ' m 2 - 1 6 ; m - 4 IN = 8 - 2 ; n - 4 8 ; n - 4
Thco gill I h i e l I M 1 I N HCMI I M . I N = 0 m - 1 6 8 n - 1 6 + ( m - 4 ) ( n - 4 ) = 0 <=> ( m - 4)(n - 4 ) ( ( m + 4)(n + 4) + 64) = 0 <=> mn = - 4 ( m + n ) - 8 0 (1) Phifdng trinh cíia M N l i i : 8x - (m + n)y + mn = 0 (2) The (1) vao (2), la diTdc: 8x - (m + n)y - 4(m + n) - 80 = 0 hay 8(x - 10) - ( m + n)(y + 4) = 0.
Toa do ciia d i e m co dinh ma M N di qua l i i nghiem cua he
I no
Uty I N H H M I V U V V H KhanglTi^r
U-10 = 0 fx = 10 y + 4 = 0 ' ^ | y = - 4
V a y dir("tng lhang M N d i qua d i e m (10; - 4 ) .
Chitng minh...
Tit acosx + bsinx = c => (acosx - c)^ = ( - b s i n x ) ^ ãcos"x - 2accosx + c^ = b^( 1 - c o s ^ x ) <=>(a" + b ' ^ ) c o s - x - 2 a c c o s x + c" - b ^ = 0 (1) Suy ra cosa, cosp la hai nghiem cua (1) T h c o djnh l i V i - e l :
cosa + cosp = 2ac ; cosacosp =
a 2 + b 2
p. ^, 2 a ? P 1 + cosa 1 + cosP
D o d o : cos —cos - = t l 2 2 2 2
_ 1 + (cosa + cosP) + cosa cosP
4 2ac 1 + — ^ + c 2 - b 2 4 (a + c) .^2 4 ( a - + b ^ ) — (dpcm). 0,25 0,25
Phdn II
M O T S6 BE T H I HQC K ) , T H A M K H A O MdN T O A N L(SfP 1 0 B A M I E N B A C - T R U N G - N A M T R O N G NHITNG N A M GAN DAY B A M I E N B A C - T R U N G - N A M T R O N G NHITNG N A M GAN DAY
D± SO 1 DE THI HOC Ki I MON TOAN L6P 10
•
TRUdNG THPT CHUYEN HA Ndl - AMSTERDAM
Cfiu Ị (3,0 diem) Cho ham so y = ax^ + bx + c (a ;t 0) c6 do thi la parabol (P).
1. T i m cac gia tri cua a, b, c sao cho parabol (P) co dinh I ( - l ; 4) va c^t Oy tai diem c6 tung do bang 3.
2. V d i cac gia trj a, b, c vi/a tim difcJc d cau 1) hay:
a) Lap bang bic'n Ihien ham só va ve do thj (P).
b) T i m cac gia tri cua m de phiTdng trinh sau co hai nghiem phan biet: ax^ + b x + c = 5 - m ^ .
Cfiu I I . (2,0 diem) Cho he phiTdng trinh mx + y = 2 ; vdi m la tham sọ
X + my = 1 + m 1. Giai va bien luan he Iren theo cac gia tri cua m.
2. T i m va cac so nguyen m de he co dung mot nghiem (x; y) va nghỉm nay thoa man dieu kien (x + y) la so nguyen.
Cfiu I I I . (4,0 diem) Cho hinh thoi A B C D canh a, BCD = 60". Goi G 1^ trong
tam lam giac BCD, K la trung diem ciia A D . Cho diem I , J thoa man h ^ thtfc:
i K + 2IB = 6, 5 J B - 3 J C = o".
1. Bieu dien vectd BG theo vectrt AB, A D . Tinh dp dai doan th^ng C I . 2. Chilng minh BG vuong goc IJ.
3. Xac djnh vi tri ciia diem P tren dúdng thang BD sao cho bieu thufc PK^ +2PB^ lii nho nhát. Tinh gia trj nho nhat do theo ạ
1
4. T i m tap hdp diem M thoa man - M B ( M C + M D ) - — = M J . M B - - M B '
Cfiu I V . (1,0 diem) Cho a, b, c > 0 thoa man hp thuTc 3a + 3b + c = 12.
ChiJng minh rang — + — + — > 4. a b c
Hl/dNG D A N - D A P SO
:flu Ị
[. Tim cac ^ici tri ciia a, h, c sao cho parabol (P) co dinh.. (?) cii Oy tai C((); 3) nen c = 3.
(P) CO dinh I ( - l ; 4 ) nen < 2a
a - b + c = 4 Giai he ta diTdc a = - 1 , b = - 2 .
Vay (P) CO phúdng trinh: y = - x * - 2 x + 3.
2. a) L^p bdn^i bien thien ham sovci ve do thi (P)-
+ Tap xiic djnh: D = M.
+ Bang bien thien:
X - G O - 1 +00
4 y
— 0 0 —00
+ Do thi doc gia tif vẹ
b) Tim cac gid tri ciia m de
phU(m^ trinh sau co hai n^hi^m...
So nghiem cua phiWng trinh - x ^ - 2 x + 3 = 5 - m ' ( * ) l a so diem chung cua do thi ham so
(C):y = th' - X - 2 x + 3 va di/clng lang A C O phúdng trinh y = 5 - m 2 . De phúdng trinh (*) c6 hai
nghiem phan biet Ihi A cat
(C) tai hai diem phan biet
<=> 5 - m ' = 0 5 - m ^ > 4
m = ±V5
TuySn chpn 39 JS W s J c hpc kl m6n loAn Bp 10 Ning cao - Phgm IrprigTfar
C&u 11.
1. Gicii \'d bi('n lucin h(' tren then cdc aid trf cua m.
H$ phiTdng Irinh da cho la he bac nhát hai an, ta c6 cac djnh thtfc:
D = m 1 1 m 1 m 2 1 = m - l ; D = m 2 = m - l ; D = 1 1 + m 1 + m m y 1 1 + m
* Neu D 0 <=> m 5^ ± 1 , he da cho c6 nghiem duy nhát:
* Neu D = 0<=>m = ±l, = m + m - 2 . 1 X =• m + l m + 2" m +1 + m = 1: D = D = 0, he da cho v6 so nghiem: X y + m = - 1 : = -2 5>t 0. he da cho v6 nghiem. Kel luan: x e l y = 2 - x "
• m ;^ ± 1 : HPT da cho c6 nghiem duy nhát:
X = • 1 y = m + l m + 2 m + 1 m = 1: HPT da cho v6 so nghiSm: x e R [y = 2 - x • m = - 1 : HPT da cho v6 nghiem.
2. Tim cdc soiii^uyen m de he cd dun}' mot rif^hi^m (x; y) vd nahi^m...
1
X = —
Vdi m ^ ±1 : HPT dii cho c6 nghiem duy nhát: m + l m + 2 m + l
^ . m + 3 , 2
Ta co: X + y = =m + l m 1 + - + l
De m va x + y nguyC-n Ihi m + l e | - l ; l;-2; 2|; suy ra m 6 | - 2 ; 0 ; - 3 ; 1
VI m^±l n e n m G l - 3 ; - 2 ; ( )
ICSu IIỊ
I . Bieu dien vect(r B G theo vect(f...
Ta c6:
B G = | B E = | ( B C + C E )
3 A D + - C D = - A D - - A B
3 3
(vdi E la irung diem ciia CD).
Tir lK + 2IB = 0 ^ B i = i B k ^ B I = — •
3 6
V i tam giac BCD deu c6 DBC = 60" va lam giac ABD dcu c6 IBD - len lie = 90".
:^30"
i Ta CO I C = >/lB^ + B C ' = faV3
+ a2 = a>/39
Ị Chvtriii minh BG vuonii i^dc IJ.
Taco I J - A J - A I
Ma A I = AB + B l = AB + - f BA + A K ) = - A B + - AD 3V / 3 6 Tir 5JG - 3JC = ()', la suy ra AJ = ^ AC = ^ ( A B + A D )
The (2) va (3) vao (I) ta di/rtc IJ = AB ^ IJ//AB
(1) (2) (2) (3) (4) (5) Ma A B I B G rir (4) va (5) suy ra IJ 1 BG.
^dc djnh \'i tri ciia diem P tren dKifn}" thdn^i BD sao cho bieu thtJtc...
Ta c6:
- -y -2 / • •\ / . -.^2
(0 = P K ' + 2 P B ' = p k +2PB = ( p i + l k ) + 2 ( p i + IB) = 3 P I ' + 2 P i ( l K + 2IB + 1 K ' + 2 I B
VII, B, K CO dinh nen IK, IB khong doị m nho nhal khi va chi khi PI nho
nhal. Do P chay trcn BD c6' dinh nen de PI nho nhál thi P Irilng vdi hinh chieu H
cua I len BD. Khi do: to = HK^ + 2HB^ = —â + 2
16 v4y 16
4. Tim tap hfp diem M thoa man...
, 1-Tacd -MB(MC + M D ) - y = M J . M B — M B '