"Luyen gidi de truoc ky thi dai hgc - Tuyen chon vd giai thieu de thi Todn hgc" la mpt trong nhOng cuon thupc bp sach "On luy$n thi Dai hgc", do nhom tac gia chuyen toan THPT bien soan
Trang 1(5^c em H Q C sinh than men!
"Luyen gidi de truoc ky thi dai hgc - Tuyen chon vd giai thieu de thi
Todn hgc" la mpt trong nhOng cuon thupc bp sach "On luy$n thi Dai hgc", do
nhom tac gia chuyen toan THPT bien soan
Voi each viet khoa hpc va sinh dpng giiip ban dpc tiep can voi mon toan
mpt each t y nhien, khong ap luc, ban dpc tro nen t y tin va nang dpng hon;
hieu ro ban chat, biet each phan tich de tim ra trong tarn ciia van de va biet giai
thich, lap luan cho tirng bai toan Sy da dang ciia h^ thong bai tap va tinh
huong giiip ban dpc luon hung thii khi giai toan
Tac gia chii trpng bien soan nhung cau hoi mo, npi dung co ban bam sat
sach giao khoa va cau true de thi Dai hpc, dong thai phan bai tap thanh eac
dang toan co lai giai chi tiet H i ^ n nay de thi Dai hpc khong kho, to hop eua
nhieu van de dan gian, nhung chua nhieu cau hoi mo neu khong nam chae ly
thuye't se lung tiing trong vifc tim 16i giai bai toan V o i mpt bai toan, khong
nen thoa man ngay voi mpt lai giai minh vira tim dupe ma phai co' gang tim
nhieu each giai nhat cho bai toan do, moi mpt each giai se eo them phan kien
thue mai on tap
Mon Toan la mpt mon rat ua phong each tai tu, nhung phai la tai tit mpt
each sang tao va thong minh Khi giai mpt bai toan, thay v i dung thoi gian de
luc Ipi t r i nho, thi ta can phai suy nghT phan tich de tim ra phuong phap giai
quyet bai toan do Do'i voi Toan hpc, khong eo trang sach nao la thua Tung
trang, tung dong deu phai hieu Mon Toan doi hoi phai kien nhan va ben bi
ngay t u nhirng bai tap don gian nhat, nhiing kien thiic co ban nhat V i chinh
nhiing kien thue co ban moi giiip ban dpc hieu dupe nhij'ng kien thuc nang cao
sau nay
Mac du tac gia da danh nhieu tam huyet cho cuon sach, xong sy sai sot la
dieu kho tranh khoi Chung toi rat mong nhan dupe sy phan bi^n va gop y quy
bau eua quy dpc gia de nhirng Ian tai ban sau cuon sach dupe hoan thi^n hon
Thay rnat nhom bien soan
Tac gid: Nguyen Phu Khanh
Cau 4: Tinh tich phan: I = J ^^LLI^ x d x
Cau 5: Cho hinh chop S.ABC eo day ABC la tam giae vuong can tai B, AC = 2a
Tam giae ASC vuong tai S va nkm trong mat phSng vuong goc voi day, SA = a Tinh theo a the tich khoi chop S.ABC va khoang each tix C den mat phiing (SAB)
Cau 6: Cho cac so thye khong am a,b,e thoa a + b + e = l va khong co hai so
nao dong thoi bang 0 T i m gia trj nho nha't ciia bieu thuc:
P = - 1 + ^ r + (e + l ) ( 3 + a + b ) (a + b)(b + e) (e + a)(a + b) ^ '
I I P H A N R I E N G T h i sinh chi dxxtfc chpn lam mpt trong hai phan (phan A hoac B)
A Theo chUorng trinh chuan
C a u 7a: Trong mat phang Oxy cho tam giae ABC npi tiep duong tron (C) ec
phuong trinh: (x + 4)^ + y^ =25, H ( - 6 ; - 1 ) la trye tam tam giac ABC; M ( - 3 ; -2
la trung diem canh BC Xae djnh tpa dp cac dinh A , B , C
Cau 8a: Viet phuong trinh m|it cau (S) co tam nam tren duong than^
d:2iz2 = yzi = £zi va tiep xuc voi hai m^t phSng ( P ) : x + 2 y - 2 z - 2 = 0 v
~3 2 2 ( Q ) : x + 2 y - 2 z + 4 = 0
Trang 2Tuyen chgn & Giai thifu dethi Todu hqc - Nguyen Phii Khdnh , Nguyen Tat Thu
Cau 9a: Chung minh dang thuc sau:
1 „ 2 n - l _ 2 2 " - l 2n ^" 2n + l (n la so nguyen duong, CJ^ la so to hop chap k ciia n phan tu)
B Theo chUorng trinh nang cao
Cau 7b: Trong mat phang Oxy cho elip (E) C6 hai tieu diem I^(W3;0); I^(V3;0)
va di qua diem A sfS;- Lap phuong trinh chinh t5c cua ( E ) va voi moi
V ^/
diem M tren elip, hay tinh bieu thuc: P = F^M^ + FjM^ - 30M^ - F1M.F2M
, X — 1 V z +1
Cau 8b: Trong khong gian Oxyz cho duong thang A: — ^ ~ ^ — ] ~
phang (a):2x + y - 2 z + 3 = 0 Chung minh rang A va (a) cat nhau tai A Lap
phuong trinh mat cau (S) c6 tarn nMm tren A, di qua A va (S) cat mp(a) theo
mgt duong tron c6 ban kinh bang
Cau 9b: Tim cac so phuc z, w thoa + z^ = 0
W ^ Z-5=:1
H\i(}m DAN GIAI
I PHAN C H U N G C H O T A T CA CAC T H I SINH
Cty TNHH MTV DWH Khang Viet
Vay CO 4 diem thoa yeu cau bai toan: A,(2;3), A 2 ( 0 ; l ) , A 3 - ; 0 u , U J , A 4 - ; 4
Cau 2: Dieu ki|n: •
Phuong trinh
sinx ^ 0 sinx + cosx ^ 0
Trang 3TuySii chgn & Giai thifu dethi Toan hqc - Nguyen Phu Khdnh , Nguyen Tat Thu
2 » Suy ra A = Jt-tdt = - t ^ 14 9
Cau 5: Ta c6 A B = B C = ^ - a72, suy ra S^gc = ^ ( ^ ^ j =
Gpi H la chan duang cao ha tu S ciia tarn giac S A C ri> S H 1 ( A B C )
Suy ra SE = VsH^ + HE^ = ^ ^ S , , „ = isE.AB = ^
Vay d(C{SAB)) = 3V S.ABC _ 27213
(x + 4)^ + y2=25
Giai h$ nay ta tim dupe (x;y) = (-4;-5),(-8;3)
VayA{-4;-5) hoac A ( - 8 ; 3 )
Cau 8a: Vi mat cau (S) c6 tarn I e d l { 2 - 3 t ; l + 2t;l + 2t)
Mat cau (S) tiep xuc voi hai mat phang (P) va ( Q ) nen d(l,(P)) = d ( l , ( Q ) ) R
6 - 3 t
- t 2-t|«>t = l = > l ( - l ; 3 ; 3 ) va R = l Vay p h u o n g trinh mat cau (S): (x + i f + (y - 3^ + (z - 3^ = 1
Trang 4Tuyen chqn & Gi&i thifu dethi Todn hgc - Nguyen Phu Khdnh, Nsuuen Tat Thu
Tu (1), (2) va (3) suy ra: ic^„ i c ^ -'-Cl +- + ^ C ^ ^
B Theo chUorng trinh nang cao
Suy ra P = (a + exg )^ + (a - exp )^ - 2(x^ + y2 j _ (a^ - e^x^)
Cau 8b: Xet h^ phuong trinh : <
fz = 0
W = 0:
z = l v6 nghiem • w = - l :
z5 = l (z) =1
Thu lai ta thay cap (w,z) = (-1,1) thoa yeu cau bai toan
OETHITHllfSOZ
I PHAN CHUNG CHO TAT CA CAC THI SINH Cau 1: Cho ham so y = x^ - 3x2 - 3m (m +1) x - 1 a) Khao sat sy bien thien va ve do thi ham so khi m = 0, b) Tim tat ca cac gia tri cua tham so m de ham so ( l ) c6 hai cue tri ciing dau
Cau 5: Cho hinh chop S.ABCD c6 day ABCD la hinh thoi canh a, BAD = 60°
va SA = SB = SD Mat cau ngoai tiep hinh chop S.ABCD c6 ban kinh bang
va SA > a Tinh the tich khoi chop S.ABCD
5
Cau 6: Cho cac sothuc duong a,b,c thoa man a + b + c = 1
^, , , I 2ab 3bc 2ca ^ 5 Chung mmh rang: + ;— + > —
Trang 5Tuyen chiftt b Giai thifu dethi Todn HQC - Nguyen Phu Kh,\nh , Nguyen Tat Thu
Cau 7a: Trong mat phSng Oxy cho tam giac ABC npi tie'p duong tron (C):
(x-1)^ +{y-lf =10 Diem M(0;2) la trung diem canh BC va di^n tich tam
giac ABC bang 12 Tim tpa dp cac dinh cua tam giac ABC
Cau 8a: Trong khong gian Oxyz cho hai duong th^ng:
Viet phuong trinh duong thang A c3t hai duong thang A,, Aj va mat phang
(a) lanluqrttai A , B , M thoa man A M = 2MB dong thoi A l A j
Cau 9a: Gpi zi la nghi^m phuc c6 phan ao am cua phuong trinh z^ - 2z + 5 = 0
Tim tap hp-p cac diem Mcbieu dien so phuc z thoa: 2 z - z ^ + l
z + z f + 2 = 1
B Theo chiToTng trinh nang cao
Cau 7b: Trong m^t phSng voi h^ toa dp Oxy cho hinh vuong ABCD biet
M (2;1); N { 4 ; - 2 ) ; P(2;0); Q ( l ; 2 ) Ian lupt thupc c^inh AB, BC CD, AD Hay
lap phuong trinh cac canh ciia hinh vuong
Cau 8b: Trong khong gian Oxyz cho diem A{3; 2; 3) va hai duong th3ng
, x - 2 y - 3 z - 3 « x - 1 y - 4 z - 3 ^, , • ^ - ^ i
dj : — — = — = — — va d2 : — ^ = ^ = — Chung minh duong thang
di, d2cva diem A ciing n^m trong mpt mat phang Xac dinh toa dp cac dinh B
va C ciia tam giac ABC biet di chua duong cao BH va d2 chua duong trung
tuyen CM ciia tam giac ABC
Cau 9b: Tim m de do thj ham so' y = — tie'p xiic voi Parabol y = x + m
Ham so CO hai eye trj khi va chi khi ( l ) c6 hai nghifm phan bi^t x,,X2
o A'= 1+ m ( m + l ) = m^ + m + 1 > 0 dung voi Vm
10
CtyTNHHMTV DWH Khang Vift
Vi X , langhiemciia ( l ) nen X j - 2 x j = m ( m + l ) Suy ra:
yj = x ^ - 3 x j - 3 m ( m + l ) x i - l = x , ( x ^ - 2 ) » i ) - ( x j - 2 x j j - 2 x j - 3 m ( m + l ) x i - l
= m ( m + l ) x j - m ( m + l ) - 2 x j - 3 m ( m + l ) x j - 1 = |m^ + m + l j ( - 2 x j - l ) Tuongty y2=(m^ + m + l j ( - 2 x 2 - l )
COS X (sin X + COS x)
o 2cos2x - 1 = 2cos3xcosx = cos4x + cos2x o 2cos2 2x -cos2x = 0 cos2x = 0
Trang 6Tuyen chgn & Giai thieu dethi Tomi h^c - Nguyen huu Khdnh , Nguyen TatThu^
Cau 5: Tu gia thiet, suy ra ABD la tarn giac deu nen SABD la hinh chop deu
Goi H, O Ian luot la tarn ciia tarn giac ABD va hinh thoi ABCD
Suy ra S H I ( A B C D )
Mat phSng trung true canh SA cat SH tai I, ta c6 I la tarn mat cau ngoai
tie'p hinh chop S.ABD
Vi ASFI - ASHA, suy ra — = — =^ SA^ = 2SI.SH
SH SA
Ma A H = - A O = ^ ^ S H 2 = S A 2 - ^ 3 3 3
Nen ta c6 phuang trinh
SA^=4Sl2 SA^-^ 2\ 12a' S A ^ - ^ 2^
<::>SA^ = 2 2a' (loai)
SA^ = 2a2 => SA = aV2 SH =
12
Cty TNIIU Af IV DWH Khang Viet
,,2
Mat khac: S^BCD = ^S^^BD =
Vay the tich khoi chop S.ABCD la: V = |SH.SABCD = ^ ^ ' ^ = ~
Cau 6: Bat d3ng thuc can chung minh tuong duong voi
2ab 3bc 2ca ^ 5 (c + a)(c + b)^(a + b)(a + c)^(b + c)(b + c)~3'
o 2ab(l -c) + 3bc(l - a) + 2ca(l - b) > | ( l - a)(l - b)(l - c)
Dang thuc xay ra khi a = i , b = c =
II PHAN RIENG Thi sinh chi dirg-c chpn lam mgt trong hai phan (phan A hoac B)
A Thee chUorng trinh chuan
Cau 7a: Duong tron (C) c6 tam l(l;l)/ suy ra MI = (l;-l)
ViBCdiquaM va vuonggoc voi MI nenBC:x-y + 2 = 0
Toa dp B, C la nghiem ciia he:
"x = 2,y = 4 [x-y + 2 = 0 [x'=4 Lx = -2,y = 0 Suyra B(2;4),C(-2;0) hoac B(-2;0),C(2;4) Gpi A(a;b), suyra ( a - l f + ( b - l f =10 (l)
Trang 7Tuyen chon t-^ Giai thifu aJthi Todn hpc - Nguyen Phu Khanh, Nxiiuen Tat Thu
• a = b - 8 thay vao (l) ta c6: (b - 9)^ + (b -1)^ = 10 v6 nghi^m
V$y A(0;4) hoac A(2;-2)
Cau8a: Vi A e A j , B e A 2 nen A ( l + a;-2 + a;l), B(3 + 2b;4 + b ; l + 3b)
3
z = 2b + l
Vi M € ( a ) nen i ± ^ + l i 2 b + 6^2b + l - l l = 0 o 2 a + 12b-17 = 0 (l)
Mlitkhac A l A j =>AB.Uj =0<=>2a-3b-8 = 0 (2)
Tir (1) va ( 2 ) suyra a = — , b = - =>AB =
GQI M ( X; y) diem bieu dien so' phuc z, suy ra z = x + yi
B Theo chi/orng trinh nang cao
Cau 7b: Gia su duong thing AB qua M va c6 vec to phap tuyen la n(a;b)
^a^ + b^ 7t 0 ) suy ra vec to phap tuyen ciia BC la: fij (-b;a)
Phuong trinh AB c6 dang: ax + by - 2a - b = 0
BC CO dang: - bx + ay + 4b + 2a = 0
14
Cty TNHH MTV DWH Khang Vift
Do ABCD la hinh vuong nen d(P;AB) = d(Q; BC)
A D : - x - y + 3 = 0 ; C D : - x + y + 2 = 0 Cau8b:di qua Mo(2;3;3) covectochi phuang a = ( l ; l ; - 2 )
di qua M j (l;4;3) c6 vecto chi phuong b = ( l ; - 2 ; l )
"^b Taco = (-3; -3; -3) 0, M Q M J = (-1; 1; O) : a,b M o M i = 0 M|it phiing (P) di qua d j , d2 c6 phuang trinh: x + y + z - 8 = 0
De thay t<?a dp cua A thoa phuang trinh (P) A, d i , d2 nam trong mpt
mat phang
t + 5 t + 5 Taco B(2 + t;3 + t;3-2t)=:>M - ; 3 - t
2 ' 2
Do M 6 d 2 = > t = -l=>B(l;2;5), M(2;2;4)
C ( l + c;4-2c;3 + c).Do AC 1 B H : ^ A C i ^ = 0 c = 0=>C(l;4;3) Cau 9b: Hai duong cong da cho tiep xiic nhau <=> h^ phuang trinh sau c6 nghi^m:
Trang 8Tuyeh chgn & Giai thifu dethi Toan hgc - Nguyen Phu Khdtth , Nguyin T^tThu^
O E T H I T H U f s 6 3
I PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1: Cho ham so y = x"* - (3m + 2) x^ + 4m c6 do thi la ( C ^ ) , voi m la tham so
a) Khao sat su bien thien va ve do thi ham so da cho khi m = 0
b) Tim tat ca cac gia trj cua tham so m de do thi (C^) cat Ox tai bo'n diem
phan bift A, B, C, D (x^ < Xg < x^ < x^) thoa BC = 2AB
Cau 2: Giai phuong trinh: cosx + 2\/3cos—sin—= cos3x + —
Cau 3: Giai bat phuong trinh sau: ^Vx'^ + x + 2 < x^ + 3
e X |lnx + ln^ xj dx
Cau 4: Tinh tich phan sau: I = ,
J 1 + Vl + X In X Cau 5: Cho lang try A B C A ' B ' C c6 day A B C la tam giac can A B = A C - a ,
B A C = 120° va A B ' vuong goc voi day ( A ' B ' C ) Gpi M, N Ian lupt la trung
diem cac canh C C va A ' B ' , mat phang ( A A ' C ) tao voi mat phang ( A B C )
mot goc 30" Tinh the tich khoi lang try A B C A ' B ' C va c6 sin ciia goc giua
hai duong thSng A M va C N
Cau 6: Cho cac so thuc a , b , C € [ 0 ; l ] thoa S'^'Us''"^ + S'^"^ = ^ Tim gia trj nho
nha't ciia bieu thuc: P = a^ + b^ + c^ + 3(a.2^ + b.2'' + c^'^
I I PHAN RIENG Thi sinh chi duqc chpn lam mpt trong hai phan (phan A
hoac B)
A Theo chUorng trinh chuan
Cau 7a: Trong mat phSng Oxy cho tam giac ABC c6 M(1;0), N(4;-3) Ian lupt
la trung diem cua AB, AC; D(2;6) la chan duong cao ha Kr A len BC Tim tpa
do cac dinh ciia tam giac ABC
Cau 8a: Trong khong gian Oxyz cho ba duong thMng:
B Theo chUofng trinh nang cao
Cau 7b: Trong mat p h c i n g voi he toa do Oxy cho hai d i e m A ( l ; - l ) va B(4;3) Tim toa dp cac diem C va D sao cho ABCD la hinh vuong
Cau 8b: Trong khong gian voi h^ toa dp Oxyz cho duong thiing A : ^ = ^ = ~
va mat phSng (a): x + 2y - 2z - 1 = 0 Viet phuong trinh mat phang (p) chua A
va tao voi (a) mot goc nho nha't
Cau 9b: Cho cac so phiic p, q (q * O) Chung minh ring neu cac nghi^m cua phuong trinh x^ + px + q^ = 0 c6 modun bang nhau thi ^ la so thuc
Hl/dfNG DANGIAI
I PHAN CHUNG CHO TAT CA CAC THI SINH Cau 1:
a) Ban dpc tu lam b) Phuong trinh hoanh dp giao diem ciia (C^,) va Ox:
x ' * - ( 3 m + 2)x^ + 4m = 0 Dat t = x^, t > 0 T a c 6 p h u o n g t r i n h : t ^ - ( 3 m + 2)t + 4 m = 0 (*)
(C^) ck Ox tai bon diem phan biet khi va chi khi (*) c6 hai nghiem duong
A = 9 m ^ - 4 m + 4 > 0 phanbi^t t j , t2 (t, < t 2 ) » • S = 3m + 2>0 o m > 0 (l)
^3m + 2^'
= m
o 9 m ^ - 1 3 m + 4 = 0<=>m = l , m = - (thoa ( l ) )
4
Vay m = 1, m = - la nhCrng gia trj can tim
Cau 2: Phuong trinh o 2(cos x - cos3\^+ 4Vs c o s : ^ i n - - 3 = 0
' ~ 7 / «
17
Trang 9iux/ni thou Ir Cioi ihicii dc thi Toiin lioc - Ngtll/eii Kluitih , '!'nt Ihu
<=> 4sin 2x.sin x + 2^3 (sin 2x - sin x) - 3 = 0
o2sin2x(2sinx + N/3)-V3(2sinx + >/3J = 0<=>(2sinx + V3)(2sin2x-V3)
Cau 3: Dieu ki?n: x^ + x + 2 > 0 <=> (x + l)|x^ - x + 2j > 0 <=> x > - 1
Bat phuong trinh o 5^(x + l)|x^ - x + 2J < 2(x +1) + 2|x^ - x + 2J
Suy ra
1= | L _ J _ = 2 j ( t ^ - t ) d t =
1 1
Cau 5: Ta c6: BC^ - A B ^ + AC^ - 2AB.ACcos A = Sa^ => BC = aVs
Gpi K la hinh chieu ciia B' len A ' C , suy ra A ' C ' 1 { A B ' K )
Suy ra AB'= B'K.tan30° = | ,
The tich khoi lang try: V = AB'.S^ABC = Gpi E la trung diem cua AB', suy ra M E / / C ' N Nen ( C ' N , A M ) = ( E M , A M )
Cau 6: Xet ham so f (x) = 2" - x - 1 , c6
f (x) = 2 ' ' l n 2 - l = * f ' ( x ) = 0 o x = l o g 2 ^ = xo Lap bang bien thien va ket hop voi f (O) = f (l) = 0 ta suy ra dupe f(x)<0, V x e [ 0 ; l ] hay 2 ' ' - x - l < a V x e [ 0 ; l "
Mat khac \/x,y,zeM, ta c6:
x^ + + z'^ - 3xyz = ^(x + y + z) (x - y)^ + (y - zf + (z - x)^
Do do neu x + y + z<0=>x"' + y'' + z^<3xyz
Tudodan den: 8" - x^ - 1 < 3x.2''o 8" - 1 < x^ + 3x.2^ V x 6 [ 0 ; l "
Suy ra P>8^ +8'' + 8^-3 = 7 Ding thuc xay ra khi va chi khi a = 0,b = 0,c = 1 va cac hoan vj V?y minP = 7
1<
Trang 10Tuye'tt chqn & Giai thifu ttethi Todn hoc - Nguyen Phii Khdnh , Nxm/en Tat Thu
I I P H A N R I E N G Thi sinh chi dirge chpn lam mpt trong hai phan (phan A
hoac B)
A Theo chUcrng trinh chuan
Cau 7a: Ta c6 M N = (3; -3) va MN // BC nen phuong trinh BC:x + y - 8 = 0
Suy ra B ( b ; 8 - b ) Do M la trung diem AB nen A ( 2 - b ; b - 8 )
• Voi x = - = > A B =
3
Suy ra n = -3AB,u = (-14; 11; 5) la VTPT cua (a)
Phuong trinh (a): 14x - l l y - 5z + 25 = 0
Cau 9a: Ta thay z = 0 thoa phuong trinh
Ta xet: z^Q
Tu = 4z => z ^ = 4 = 4 = 2
Do do: = 4z => z^ = 4z.z = 4|z|^ = 16 o | z ^ -4j^z^ + 4J = 0 <=> z = ±2,z = ±2i
Thu 1^1 ta thay bon nghi^m nay thoa phuong trinh
V^y phuong trinh c6 5 nghif m: z = 0, z = ±2, z = +2i
B Theo chi/tfng trinh nang cao
Vay taco C(0;6) va D(-3;2) hoac C(8;0) va D ( 5 ; - 4 )
Cau 8b:Goi PT (fi):ax + by + cz+ d = 0=> nj, = (a;b;c) va n,^^ = ( l ; 2 ; - 2 )
(p) chua A nen -a + 2b + c = 0
a - c + d = 0 Coi (j) la goc giua hai mat phang (p) va (a), suy ra
• Voi T = - phuong trinh c6 nghiem * = •
• Voi T 5-^ ^ de phuong trinh c6 nghiem t khi va chi khi
(2T + 4 f - ( 2 T - l ) ( 5 T - 1 6 ) > 0 o 0 < T < 53
Do do d) nho nhat o t = - — o 13b = -10c
^ 10
Ket luan PT mat ph^ng (p) can tim la : 7x + 10y-13z-20 = 0
Cau 9b: Goi z, = a + bi, Zj = c + di la hai nghiem ciia phuong trinh da cho
Trang 11Tuyi'n chqn Ct Giai tItiC'u dethi Totitt hqc - Nguyen Phu Khanh , Nnuyen Tat Thu
DETHITHUfs64
I PHAN C H U N G C H O T A T C A C A C T H I S I N H
Cau 1: Cho ham so y = x"* - 3x + 1 ( l )
a) Khao sat sy bien thien va ve do thj (C) cua ham so ( l )
b) Xac djnh m de phuong trinh sau c6 4 nghiem thyc phan bi^t:
x|'' -3|x| = m-' - 3 m
Cau 2: Giai p h u o n g trinh: 4cos^ 3xcos2x + cos8x = \/3sin4x + 2cos2x
Cau 3: Giai h ^ p h u o n g trinh:
- d x
Cau 4: Tinh tich phan I =
3 x ' ' - 3 x + 2 Cau 5: Cho hinh chop S.ABCD c6 day ABCD la hinh thoi canh a, BD = a Tren
canh A B lay M sac cho B M = 2 A M Gpi I la giao diem cua A C va D M , SI vuong
goc voi mat phang day va mat ben ( S A B ) tao voi day mpt goc 60"
Tinh the tich cua khoi chop S.IMBC
A Theo chUtfng trinh chuan
Cau 7a: (2 diem) Trong mat ph^ng h ^ tpa dp Oxy, cho hinh thoi A B C D c6 tarn
l ( 2 ; l ) v a A C = 2 B D D i e m M ' 1 ' thupc d u a n g thSng AB; diem N(0; 7) thupc duong thSng CD T i m tpa dp dinh B biet B c6 hoanh dp d u o n g
Cau 8a: Trong khong gian voi h f tpa dp Oxyz, lap p h u o n g trinh d u a n g thSng
d d i qua diem A ( - 1 ; 0 ; - 1 ) va cat duang thang d ' : ^ ^ - ^ ~ ^ i ^ ~
goc giiia d u o n g thang d va d u o n g thang d " : ^^—^ = = - y - " ^ 6 nhat
Cau 9a: T i m phan thuc va phan ao cua so phuc z biet rang z ^ - 1 2 = 2 i ( 3 - z )
B Theo chuomg trinh nang cao
Cau 7b: Viet phuong trinh canh AB phuong trinh duang thang A B c6 h§ so goc
duong), A D cua hinh vuong ABCD biet A ( 2 ; - 1 ) va duong cheo BD c6 phuong trinh: x + 2 y - 5 = 0
Cau 8b: Cho ba diem A ( 5 ; 3 ; - 1 ) , B ( 2 ; 3 ; - 4 ) , C ( 1 ; 2 ; 0 ) C h u n g m i n h r^ng tam
giac A B C la tam giac deu va t i m tpa dp diem D sao cho t i i di#n A B C D la t u di^n deu
Cau 9b: T i m so phuc z sao cho z^ va la hai so phuc lien hpp ciia nhau
Xet ham so: y = x - 3|x| + 1 , ta c6:
23
Trang 12Tuyen chgn & " " f " ''^'t'" T"''" 'wc - Nguyen Phi, Khauh , Nguyen Tat Thu
+) H a m so la mot ham chan nen
( C ) nhan tri,tc Oy lam true doi xung,
dong thoi Vx > 0 thi
y = |xf-3|x| + l = x ^ - 3 x + l
+) Do thi ( C ) la:
+) D y a vao d o thj ( C ) ta suy ra ^
dieu kien cua m de p h u o n g trinh
da cho CO 4 nghiem phan bi^t la:
Cau 2 : Phuong trinh da cho tuong duong v o i
2cos2x(l + cos6x) + cos8x = \/3sin4x + 2cos2x
<=> 2cos2xcos6x + cos8x = \/3sin4x o cos8x + cos4x + cos8x = 7 3 s i n 4 x
o 2cos8x = \/3sin4x - c o s 4 x = 2sin
Cty TNHH MTV DWH Khutig Vipt
Vi X = 0 khong la nghiem cua he, nen ta c6:
1 x^ X
X =
VVs-i
-VN/5-1 Doi chieu dieu kien, ta c6 nghiem cua he da cho la:
Trang 13Cau 5: G(?i H la hinh chieu ciia
1 len AB, suy ra AB 1 (SIH)
=> S H I la goc giua mat ben
( S A B ) va mat day
Do do S H I = 60"
Do tam giac ABD deu nen
Cty TNHH MTV P W H Khang Vift
Khido p = y + -^,khao sat f ( y ) = y + - voi y ^
maxP = f { 7-3S 21 + 375 , dat duQ-c khi
va N ' thuoc canh AB
V
Suy ra M N ' =
nen phuong trinh AB: 4x + 3y - 1 = 0
26
Trang 14Tuyen ch<?n & Gi&i thieu tie thi Toan hoc - Nguyen Phu Klidnh , Nguyen Tii't Thu
7
4 5 2
Phuang trinh A :
7 ' 7 ' 7 ^ Cau 9a: Goi z = a + bi v 6 i a ; b e R
Suy ra phan thuc va phan ao la 3 ; - 1 hoac 3;3
B Theo chi/orng trinh nang cao
Cau 7b: Do A B C D la hinh v u o n g nen p h u o n g t r i n h A C : 2x - y - 5 = 0
Goi I la tarn cua h i n h vuong, suy ra I = B D n A C = > l ( 3 ; l )
Cau 8b: Ta c6 A B = BC = C A = 3N/2 nen tarn giac A B C deu
Gpi G la trong tarn cua tarn giac ABC
' 8 8 5 SuV ra G
3 ' 3 ' 3) va u = A B , A C = (-3; 15; 3) nen p h u o n g t r i n h true cua
duong tron tarn giac A B C la
3 ' 3 ' 3,
-5 5
thoa yeu cau bai toan
C a u 9 b : D a t z = r(cos(p +isincp), (pe [0;27:), t h i = r'''(cos5(p + isin5(p)
1 _ 1 _ cos2(p-isin2(p z^ r^(cos2(p +isin2(p) r^
29
Trang 15Tuye'n chqn & Giai thi^i dethi Todu hqc - Nguyen Phu Khanh, Nguyen Tai Thu
DETHITHUfSOS
I PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1: Cho ham so y = x'' - 3(m + l)x^ + 3m(m + 2)x - 12m + 8 (C^)
a) Khao sat su bien thien va ve do thi ham so khi m = 0
b) Tim m de do thj (C^) c6 hai diem cue trj A , B sao cho A M + B M nho
nhat voi M(3;3)
Cau 2: Giai phuong trinh: sin^ x + cos^ x = sin 2x cos 2x + tan 2x - 2
Cau 3: Giai h^ phuong trinh : y ^ + ( 4 x - l f = ^4x(8x + l)
40x^ + x = y V l 4 x - l Cau 4: Tinh di^n tich hinh phang gioi han boi cac duong
y = x ; y = x|2 + tan^xj va x = ^
Cau 5: Cho hinh chop S.ABCD c6 day ABCD la hinh thang vuong tai A va D,
tam giac SAD deu c6 canh bang 2a, BC = 3a Cac mat ben tao voi day cac goc
bang nhau Tinh the tich ciia khoi chop S.ABCD
Cau 6: Tim gia trj nho nhat ciia bieu thuc
P = 72x2+2y2-2x + 2y + l+72x2+2y2+2x-2y + l+,y2x2+2y2 + 4x + 4y + 4
II PHAN RIENG Thi sinh chi duQc chpn lam mpt trong hai phan (phan A
hoac B)
A Theo chUcrng frinh chuan
Cau 7a: Trong mat phang Oxy cho tam giac A B C c6 A(1;3),B(-2;0),C 5 3
8'8
Tim tpa dp tam duong tron npi tiep va tam duong tron bang tiep goc A cua
tam giac ABC
Cau 8a: Trong khong gian Oxyz cho ba duong th3ng
fx = -2t x - l y + 1 z - 1 , x + 1 y - 1 z , ,
tpa dp cac dinh ciia hinh vuong M N P Q , biet M triing voi tam cua duong tron
^C); hai dinh N , Q thupc duong tron (C); duong thang P Q di qua E(-3;6) va
>0
Cau 8b: Trong khong gian Oxyz cho hinh chop S.OABC c6 day OABC la hinh
thang vuong tai O va A(3;0;0), AB = OA = ioC, S(0;3;4) va y c > 0 Mpt mlt ph^ng (a) di qua O va vuong goc voi SA cSt SB, SC tgi M va N Tinh the tich khoi chop SOMN
Cau 9b: Tim tap hpp cac diem M trong mat phang phuc bieu dien so'phuc z
sao cho - ^ - i ^ la so'thuc duong
Taco: y' = 3 x ^ - 2 ( m + l)x + m(m + 2) i:>y' = O o Xj = m
X j = m + 2 Suy ra (C^) luon c6 hai diem cue trj A,B voi mpi m
Voi Xj = m => yj = m'^ + 3m^ - 12m + 8 => A^m; m^ + 3m^ - 12m + sj
Voi Xj = m + 2=>y2 = m'^+ 3m^-12m+ 4 B^m+ 2;m^+ 3m^-12m+ 4J
Ta c6: AB = (2; -4) => AB = 2V5 Dodo: A M + B M > A B = 2>/5
D3ng thuc xay ra khi va chi khi AC = kAB, k > 0 (l)
Ma AC = |3-m;-m-'-3m^+12m-6J nen (l) tuong duong voi
3 - m m''+3m^-12m + 6
>0<:> m <3
m^ + 3m2-10m = 0 o m = 0,m = 2,m = -5
31
Trang 16Tuyen chpit b Gi&i thicu de thi Todit hoc - Nguyen Phii Khdnh , Nguyeu Tat Thu
Vay m e {-5,0,2} la nhung gia tri can tim
Cau 2: Dieu kien: cos2x ^0
Phuang trinh da cho tuong duong voi
5 3 1
— +—cos4x = — sin 4x + tan 2x - 2 <=> 3cos4x = 4 sin 4x + 8 tan 2x - 21
8 8 2
Dat t = tan2x, ta c6: sin4x ^
Ta duQC phuang trinh :
Cau 5: Goi I la hinh chieu vuong goc cua S tren(ABCD), tuong t u nhu v i du
tren ta cijng c6 I la tam duong tron npi tiep hinh thang A B C D
Vi t u giac ABCD ngoai tiep nen AB + DC = AD + BC = 5a Di^n tich hinh thang ABCD la S = - ( A B + D C ) A D = -.5a.2a = Sa^
Goi p la nua chu vi va r la ban kinh duong tron npi tiep ciia hinh thang ABCD thi
2
S K ^ ^ = aV3::^SI = VsK2-IK2 ^Vsa^-a^ =a^/2
V a y v 4 s i s , 3 C D 4 a ^ 5 ^ ' ~
-33
Trang 17Tuye'n chgn ۥ* Gicri tltij-u dethi Toan hqc - Nguyen Phti Kliault, Nguyen Tii't Thu
Cau 6: Ta vie't PT duoi dang:
P = >/2 X
Ta CO BDT sau: Voi moi so'thuc a|,a2,bi,b2
+ b? + + b^ > ^(a, + a2)' + (b, + b2)' D
Thatvay (*) tuang dirong voi (ajbj-a2b,)>0 (diing)
Dang thuc xay ra khi va chi khi a^bj = ajbj
Dat s = X + y Ap dung (*) ta duoc:
s + 2
^ ( x l f ( y l f J l ( s 2 f =
Tir (l) va (2) suy ra: P > 2^8^+1 + |s + 2
Dang thuc xay ra khi va chi khi x = y —
Ma s + - 4i >0,dod6: +\>-{s-S'f
Lai c6: S-s + s + 2 >1 + S
(2) (3)
(4) (5) Ket hop (3), (4), (s) suy ra: P > 2 +
Diing thuc xay ra khi va chi khi dSng thuc d (3), (4) va (5) dong thoi xay ra
A Theo chUofng trinh chuan
Cau 7a: Gpi K(x;y) la tam duong tron npi tiep tam giac ABC
Ta c6: KAB = K A C KBC = KBA
AK.AB AK.AC
( A K , A B ) = ( A K , A C J ( B K , B A ) = ( B K , B C ) AK.AB AK.AC
COS(AK,AB) = COS(AK,AC) COS(BK,BA) = COS(BK,BC)
AK.AB AK.AC
BK.BA BK.BC BK.AB BK.BC
AB AC BK.BA BK.BC
Gpi j(a;b) la tam duong tron bang tiep goc A ciia tam giac ABC Ta c6:
( A J , A B ) = ( A J , A C (BJ,BC) = (BJ,AB)
V9y J
AJ.AB AJ.AC
AB AC BJ.BC BJ.AB
I BC AB
2a-b = -l ' |2a + b = -4'
5
r 5 _ 3 ^ 4'"2j
Cau 8a: Ta CO A e dj A(1 + a; -1 + 2a; 1 - a), B 6 dg B(-2b; -1 - 4b; -1 + 2b)
Suy ra AB = (-a-2b-l;-2{a + 2b);a + 2b-2), dat x = a + 2b
Tu AB = >/l3=>(x + l)^+4x2+(x-2f =13<»x = -l,x = |
35
Trang 18• Voi x = l = > A B = {0;2;-3), taco u = ( 2 ; 3 ; - l ) la V T C P cua d2 va A(-1;1;0)
e d2 => A e (a)
= ( 7 ; - 6 ; - 4 ) la V T P T c u a ( a ) Suy ra n = A B , u
B Theo chUorng trinh nang cao
Cau 7b: Ta c6 M ( 2 ; 1 ) va E Q la tiep tuyen cua ( C )
Phuong trinh E Q c6 dang: a(x + 3) + b(y - 6) = 0 <=> ax + by + 3 a - 6 b = 0
5 a - 5 b
= 1
Vi d ( M , E Q ) = N/To nen taco: = N/10
<:>(5a-5bf =10(a^ + b ^ ) » 3 a 2 - 1 0 a b + 3b^ =0<=>a = 3b,b = 3 a
• a = 3b, ta C O phuong trinh E Q : 3x + y + 3 = 0 Khi do tpa dp Q la nghi^m
cua he ( x - 2 ) % ( y - i r = 1 0 ^ x = - l
3x + y + 3 = 0 l y =
Truong hop nay ta loai vi X Q > 0
Khi do tga do Q la nghiem ciia h^
• X = 3, ta C O P ( 6 ; 3 ) , suy ra tarn cua hinh vuong l(4;2) nen N ( 5 ; 0 )
• x = 5 , t a c 6 P (O; 5), suy ra tarn ciia hinh vuong I ( l ; 3) nen N ( - 1 ; 2 ) Vay C O hai bo diem thoa yeu cau bai toan:
M ( 2 ; 1 ) , N ( 5 ; 0 ) , P ( 6 ; 3 ) , Q ( 3 ; 4 ) va M { 2 ; 1 ) , N ( - 1 ; 2 ) , P ( 0 ; 5 ) , Q ( 3 ; 4 ) Cau 8b: Do A B C D la thang vuong tai A va O, dong thoi A e Ox,yQ > 0, O C = 6
Nen ta suy ra dugc C ( 0 ; 6 ; 0 ) Tuong tu ta c6: B ( 3 ; 3 ; 0 )
Ta c6: SB = ( 3 ; - 3 ; - 4 ) , suy ra phuong trinh mat phang ( a ) : 3 x - 3 y - 4 z = 0
Vi SB = (3;0;-4), S C = (0;3;-4) nen ta c6 phuong trinh:
M(3;3;0) va N
7 7 Suy ra OS = (0;3;4), O M = (3;3;0),
0; 96 72
O S A O M = (-12; 12; -9 ),(oS A O M J O N =
3nn
Vay V s o M f , = : ^ ( d v t t )
C a u 9 b : G Q i A ( 0 ; - 1 ) , B ( 0 ; 1 ) la cac diem bieu dien so phuc z^ = - i ; Z g = i
Ta C O A M , B M bieu dien cac so'phuc z + i ; z - i , nen la so'thuc duong
z - i khi va chi khi — = k <=> A M = k B M <r> M A = kMB (k > O ) Do do diem M nam
z - i ^ ' tren duong thang A B va nam ngoai doan thang AB
37
Trang 19Tuyen CHQU b Gicri thij-u tie thi Todn hqc - Nguyen P/iii Klidnh , Nguyen Tat Thu
DETHITHUrSOe
I PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1: Cho ham so y = - 3 m x ^ + ( m - l ) x - m ^ + m (1)
a) Khao sat s^ bien thien va ve do thj ham so (1) khi m = 1
b) T i m tat ca cac gia trj ciia tham so m de do thj ham so (1) cit duong thing
y = -2m^ tai ba diem phan biet A, B, C (voi < Xg < x^ ) sao cho doan thang
A C CO dp dai nho nha't
-C a u 3: Giai p h u a n g trinh: Vx^ +3x + 6 + V2x^ - 1 = 3x + 1
^ ^, , , , , , , V x l n x ,
Cau 4: T m h tich phan: 1= ^ d x
^•^(lnx + x + 1)
Cau 5: Cho hinh chop S.ABCD c6 day la hinh thang vuong tai A, D, A B = A D = a,
CD = 2a Canh ben SD 1 ( A B C D ) va SD = a Goi E la t r u n g d i e m cua DC Xac
djnh tam va tinh ban kinh mat cau ngoai tiep hinh chop S.BCE
Cau 6: Cho x , y , z > 0 thoa x^ + y^ + + 2xyz = 1 C h u n g m i n h rang:
8(x + y + z)^ < lOlx-* + y-' + z-*) +11(1 + 4xyz)(x + y + z) - 12xyz
II PHAN RIENG T h i sinh chi dupe chpn lam mpt trong hai phan (phan A
hoac B)
A Theo chuorng trinh chuan
Cau 7a: Trong mat ph5ng ( O x y ) cho tam giac A B C v o i A(2; - 1 ) va p h u a n g
trinh d u o n g phan giac trong cua B va C Ian lupt la: d j : x - 2 y + l = 0 va
d j : x + y + 3 = 0 Viet p h u a n g trinh canh BC
x - 1 y + 1 z - 2
Cau 8a: Trong khong gian Oxyz cho hai d u o n g thang Aj : —y— = ^ = — —
va A2 : ^ = ^^—^ = Y • Viet p h u a n g trinh mat phang (P) chua d u o n g thang
1
va tao vai d u o n g thang A2 mpt goc cp thoa coscp =
7 3
-Cau 9a: Cho so phuc z thoa dieu ki^n: |z - 2 + 3i| = 5
T i m tap h p p d i e m bieu dien so phuc vv = 2z - i + 3
38
Cty TNHH MTV DWH Khang Vift
B Theo chUcrng trinh nang cao
Cau 7b: Trong mat p h l n g Oxy cho cho hinh thoi A B C D c6 A ( l ; 2), phucmg trinh B D la: x - y - 1 = 0 T i m toa dp cac dinh con lai ciia h i n h thoi, biet rSng
BD = 2AC va B c6 tung dp am
Cau Sb: Trong khong gian Oxyz cho diem A(3; 2; 3) va hai d u o n g t h i n g :
d , : • va d , : x - 2 y - 3 z - 3 , , x - 1 y - 4 z - 3
C h u n g m i n h d u o n g thang d i , d2 va diem A ciing nam trong mpt mat phSng
Xac djnh toa dp cac d i n h B va C cua tam giac ABC biet d i chua d u o n g cao B H
va d2 chua d u o n g t r u n g tuyen C M cua tam giac ABC
Cau 9b: Ti'nh gia t r i bieu thuc:
c _ r " - ^ r ^ 4-q2p4 , , / i\kp2k , , ol004 p2008 01005^-2010 i»-*-2010 ' ^ * - 2 0 1 0 * - 2 0 1 0 + - + V~U * - 2 0 1 0 + - + '^ ^2010"-^ *-2010
Hi;(3rNGDiiNGiiii
I PHAN CHUNG CHO TAT CA CAC THI SINH Cau 1:
a) Ban dpc t u lam b) Phuang trinh hoanh dp giao diem cua hai do thj
- 3 m x ^ + ( m - l ) x - m ^ + m = - 2 m ' '
o x ^ - 3 m x ^ + ( m - l ) x + 2 m - ^ - m ^ + m = 0
< » ( x - m ) ( x ^ - 2 m x - 2 m ^ + m - l ) = 0
X = m _ x ^ - 2 m x - 2 m ^ + m - l = 0 (*) Gpi f ( x ) la v e t r a i ciia (*)
Do thj ham so (1) c i t d u o n g t h i n g y = - 2 m ^ tai hai diem phan bi^t k h i va chi k h i (*) CO hai nghi^m phan bipt X i , X 2 khac m
Trang 20Tuye'n chQtt & Gi&i thifu dethi Toan h,n \'guycti Phu Khanh , Nguyen Tat Thu
7 x 2 - 4 x - 8 = 0 ^ nen x + 2 + 2V2x^ - 1 0, Vx € K D o do p h u a n g t r i n h (3) t u a n g d u o n g v o i
(b)<=>
1
X >
-2 4(2x2 - 1 ) = 4 x 2 - 4 x + l
> d t = i ^ ^ d x D o i can: x = l = > t = l , x = e = > t =
-Suy ra I V d t _ 1 1
1
1 r e2 l ] iCt + l f 3 ( t + l ) 2 2 3 l ( e + 2)2 4 j Cau 5: V i A B = D E = A D = a va
D A B = l v nen A B E D la h i n h
v u o n g Tarn giac B D C c6 EB =
ED = EC = a nen v u o n g tai B,
BE 1 C D nen t r u n g d i e m M ciia BC la tam d u o n g t r o n ngoai tie'p t a m giac EBC D u n g
A la true d u o n g t r o n ngoai tie'p t a m giac EBC t h i A song song v o i SD
D y n g m a t p h a n g
t r u n g t r y c canh SC, m a t phang d o cSt A tai I
D i e m I la tam m a t cau ngoai tie'p h i n h chop S.BCE
Ke S N // D M Cc4t M I tai N ta c6 S D M N la h i n h c h u nhat v a i SD = a va
41
Trang 212 4 2 4 2 '
Ta CO SI^ = SN^ + N I ^ = SN^ + ( N M - IM)^ = ^a^ + (a - IM)^
Ma IC^ = I M ^ + MC^ = I M ^ + — va R = IC = IS
2 nen - a ^ + ( a - I M ) ^ = I M ^ + — o I M = - a 2
• Chung minh (1): Khong giam tinh tong quat la gia su x = min {x, y, z}
x^ + y"' + z^ + 6xyz - (x + y + z)(xy + yz + zx)
= x ( x - y ) ( x - z ) + (y + z - x ) ( y - z ) ^ > 0 Suy ra dieu phai chiing minh Dang thiic xay ra khi va chi khi x = y = z > 0
• Chung minh (2): Nhgn thay trong ba so x , y , z luon ton tai hai so sao cho
chung cimg Ion han hoac cung nho hon i Khong giam tinh tong quat ta
gia sir hai so do la x, y
z(2x - l ) ( 2 y - 1 ) > 0 o 4xyz > 2zx + 2zy - z (3)
Ta chi can chiing minh: 1 + 2xz + 2zy - z > 2(xy + yz + zx)
hay z + 2xy < 1 (4)
Theo gia thie't x^ + y^ + z^ + 2xyz - 1 suy ra xy < ^ ^ ^ < 1 => 1 - xy > 0
Do do (4) tuong duong voi: (z + xy)^ ^ ( l - xy)^ <=> 1 - (z^ + 2xyzj - 2xy > 0
o ( x - y ) ^ > 0 (dung)
Do do (2) dupe chiing minh Dang thuc xay ra khi va chi khi x = y = z = ^
Sir dung cac BDT (1) va (2) tn c6:
10(x^ + y ^ + z^) + n ( l + 4xyz)(x + y + z ) - 1 2 x y z
Cty TNHHMTV DWH KHangVm
= 8(x"' 4-^^ +z-') + 2(x-' H-y-' +6xyz) + n ( l + 4xyz)(x + y + z)-24xyz
2 8(x'' + y"* + z^) + 24 (xy + yz + zx)(x + y + z) - 24xyz = 8(x + y + z)^
Suy ra dieu phai chirng minh Dang thiic xay ra khi va chi khi x = y = z = i
II PHAN RIENG Thi sinh chi dupe chpn lam mpt trong hai phan (phan A hoac B)
Theo chUorng trinh chuan Cau 7a: Gpi A ] , A2 Ian lupt la cac diem dol xirng voi A qua d^ va d2 , ta tim dupe A , ( 0 ; 3 ) va A2(-2;-5)
Theo rinh chat duong phan giac ta suy ra A , , A2 G BC Phuong trinh BC: 4x - y + 3 = 0
Cau 8a: Duong thang Aj di qua A ( l ; - 1 ; 2 ) va c6 u7 = ( l ; 2 ; - l ) la VTCP
Duong thiing A2 c6 U j = (2;-2;l) la VTCP Gpi n = (a;b;c) l a m p t V T P T c i i a (a)
o 3 a 2 =2a2 +4ab + 5b^ » a ^ - 4 a b - 5 b ^ =0c:>a = -b,a = 5b
• a = - b , ta chpn a = 1 => b = - l , c = - 1 Phuong trinh ( a ) la: x - y - z = 0
• a = 5b, ta chpn b = 1 =>a =5,c = 7 Phuong trinh (a) la: 5x+y+7z-18=0 Cau 9a: Ta c6 z = ——^—- nen dieu ki^n bai toan dupe viet lai nhu sau:
| w - 7 + 7i| = 10
Gpi M ( x ; y ) la diem bieu dien so phiic w, ta eo: (x - 7 ) ^ + ( y + 7)^ = 100
Do do tap hpp ciia diem M la duong tron tam I ( 7 ; - 7 ) , ban kinh R = 10
B Theo chUorng trinh nang cao Cau 7b:Ta c6 AC 1 BD nen phuong trinh A C : x + y - 3 = 0
Gpi I la giao ciia hai duong cheo AC va BD, suy ra tpa dp ciia I la nghipm ciia h f
[ x - y - l = 0 x = 2
x + y - 3 = 0 [ y = l
.I(2;1)=>C(3;0)
4 ]
Trang 22TuifS'ti chpn & Gi&i thifu dethi Todtt hqc - Nguyen Phti Kltdnh , Nguyen Tat Tttu
Gpi B(b; b - 1 ) , ta c6 BD = 2AC IB = 2IA
» ( b - 2 ) ^ + ( b - 2 ) ^ = 8 c 5 b = 4,b = 0 Do b - l < 0 = > b = 0
Vay B ( 0 ; - 1 ) , D ( 4 ; 3 )
Cau 8b: Ta c6: d j qua M Q (2;3;3) c6 vecto chi phuang a = ( l ; l ; - 2 )
d2 qua M i ( l ; 4 ; 3 ) c6 vecta chi phuang b = ( l ; - 2 ; l )
Ta CO a,b ?i 0 va a, b MoMi' = 0
G Q I ( a ) la mat phcing di qua hai duong thang d],d2
Ta CO p h u a n g trinh cua ( a ) la: x + y + z - 8 = 0 Suy ra A € ( a )
a) Khao sat su bien thien va ve do thj ham so khi m = 3
b) T i m ta't ca cac gia trj cua tham so so m de ( C ^ ) ck d u a n g thSng y = -14
tai ba diem c6 hoanh do khong nho hon - 9
Cau 2: Giai p h u a n g trinh: ( c o s 2 x - 5 c o s x 3 ) ( 2 s i n x - l ) ^ _ ^ _
Cau 5: Cho lang try A B C A ' B ' C c6 day ABC la tam giac vuong tai A , A B = a,
/i^C = a 7 3 ; A ' A = A ' B = A ' C M a t ph5ng ( A ' A B ) tao v o i mat ph^ng (ABC)
inpt goc 60" Tinh the tich khoi lang tru va c6 sin ciia goc giira hai d u o n g th^ng A C va A ' B
Cau 6: Cho cac so thuc d u o n g x, y, z thay doi
Tim gia trj Ian nhat cua bieu thuc P =
X' + yz yJ5y^ + zx >/3z^ + xy
I I PHAN R I E N G T h i sinh chi du<?c chpn lam mpt trong hai phan (phan A
hole B)
A Theo chUtfng trinh c h u a n
Cau 7a: Trong mat phang voi h ^ toa do Oxy, cho tam giac ABC biet A ( 5 ; 2 )
Phuong trinh d u o n g trung true canh BC, d u a n g trung tuyen C C Ian lupt la
x + y - 6 = 0 v a 2 x - y + 3 = 0 T i m toa do cac dinh cua tam giac ABC
Cau 8a: Trong khong gian voi he toa do Oxyz cho d u o n g thang
:lzl = l^ = ^ v a m a t p h S n g ( P ) : 2 x + y - 2 z + 9 = 0
Viet phuang trinh d u a n g th^ng A nam trong (P) cat va vuong goc voi d
Cau 9a: Xac djnh so hang khong phu thupe vao x k h i khai trien bieu thuc
voi n la so'nguyen duong thoa man C„ + 2n = A ^ + i
B Theo chUtfng trinh n a n g cao
Cau 7b: Cho tam giac A B C nhpn, viet phuang trinh duang thing A C , biet toa
dp chan cac duong cao ha tu cac dinh A, B, C Ian lupt la: A i ( - l ; - 2 ) , Bi (2;2),
C, ( - l ; 2 )
Cau 8b: Trong h ^ toa dp O x y z , cho d u o n g thang d : x - 1 y + 2 _ z
- 1 = — va hai
<liem A ( 1 ; 2 ; 4 ) , B ( - 1 ; 2 ; 4 ) Viet phuang trinh d u o n g th5ng A d i qua A va cat
d sao cho khoang each t u B den A Ion nhat
22x-y ^2'< =2'""^
log2 x(log4 y - 1 ) = 4 Cau 9b: Gia le phuang trinh:
45
Trang 23Tuye'n ch(,m b Giai thiftt dethi Toiin liQC - Nguyen Phu Khanh, Nguyen TA't Thu
Yeu cau bai toan o (*) c6 ba nghi^m phan bi^t khong nho Hon - 9
Dieu do xay ra khi va chi khi 12 < m < 62
Cau2:Dieu ki^n: cosx?t — o x ^ t i — + k27t
cit Suy ra I = J - y
Gpi K la trung diem AB, suy ra HK 1 AB Suy ra AB ± ( A ' K H ) nen A I C H
la goc giOa hai mat phang (A'AB) voi (ABC) nen A ' K H = 60"
GQ'I E la trung diem ciia A'C A
suy ra HE // A'B nen goc giiia hai duong thcing A'B va A C la goc giiia hai duong thang HE va AE
^ AE^ + EH^ — AH^
Ap dune dinh l i Co sin ta c6: cos AEH =
^ • ^ 2AE.HE
17
47 itmiliiriu.;
Trang 24Tuye'ti chgn b Giai thi?u dethi Toan hpc - Nguyen Phu Khanh , Nguyen Tat Thu
Cau6:Dat a = = = ^ => abc = 1 P = 1 1 1
• + — = = + V3 + a 73+ b 73+ c
Ap dving BDT Co si ta c6: , = , < +
-737a ^4(3 +a) 3 + a 4
1 1 1 3 27 + 6(a + b + c) + ab + bc + ca 3 Suy ra P < + + + - = - i — + -
3 + a 3 + b 3 + c 4 28 + 9(a + b + c) + 3(ab + be + ca) 4
^ , , , 27 + 6(a + b + c) + ab + bc + ca 3
Ta chune minn —^— < —
^ 28 + 9(a + b + c) + 3(ab + bc + ca) 4
<»3(a + b + c) + 5(ab + bc + ca)>24 (*)
A Theo chi/cTng t r i n h chuan
Cau 7a; Goi C = (c;2c + 3) va I = ( m ; 6 - m) la trung diem cua BC
A c ( P ) va A i d = > U A = n,u Phuong trinh tham so'ciia A : Cau 9a:
= (5;0;5)
x = t
y = - l Z=:4 + t
Ta c6: Cl + 2n = A^+j <=>
n e N , n > 3 <=> n = 8
n 2 - 9 n + 8 = 0 Theo nhj thuc Newton ta c6:
Trong do c6 hai so' hang khong phy thupc vao X la: —Cg.C3 va Cg.C4
Do do so hang khong phu thupc vao x la: -Cg.C^ + Cg.C^ = -98
B Theo chUoTng t r i n h nang cao
(1) (2) (3) (4)
Cau 7b: Ta c6 CB^x = A B ^ j Tii giac ABjHCj npi tiep => A B ^ j = AHCj
AHCi=V5C
Tu giac AiHBiC noi tiep => AjHC = AjBjC
Tu (1), (2), (3) va (4) suy ra: CBpc = A ^ B ^ => AC la phan giac ngoai goc Bj cua tarn giac AjB^Ci
Taco: AjBj : 4 x - 3 y - 2 = 0;BiCi : y - 2 = 0 Phuang trinh duang phan giac ciia goc tgo boi hai dt A j B j , BjCi
Trang 254 x - 3 y - 2 = |y-2|« "x-2y + 2 = 0
2 x + y - 6 = 0 Vay phuong trinh canh A C : 2x + y - 6 = 0
Cau 8b: Gia su A la duong thing di qua A va d tgi M ( l - 1 ; -2 + t;2t) € A
[22''-y + 2" = 2^''y f2^(''"yW2''"y-2 = 0 [2''"y=l
< o<! " log2x(log4y-l) = 4 [log2x(log2y-2) = 8 [log2x(log2y-2) = 8
a) Khao sat sy bien thien va ve do thi ham so (1) khi m = 0
b) Tim tat ca cac gia trj m de do thj ham so' (1) cat Ox t^i bon diem phan bi^t
CO hoanh dp Ion hon - 3
Cau 2: Giai phuong trinh: sin x (N/2 sin 2x + V2 +1) = sin 5x - >/2 sin x cos 2x
Cau 3: Giai phuong trinh : 8x^ - 13x + 7 = (1 + -)^2x^ - 2
Cau 4: Tinh tich phan: 1 = f G ^ d x
n^Vx^+l
50
C Cau 5: Cho hinh lang try ABCD.A'B'C'D' c6 day ABCD la hinh thoi, canh bSng a, ABC = 60" Hinh chie'u cua A' len mat phang (ABCD) la giao diem cua
AC va BD Mat phSng (A'B'BA) tao voi mat day (ABCD) mpt goc 60° Tinh
the tich cua khol chop va khoang each giiia hai duong thing BD va A'C
Cau 6: Cho cac so thyc a,b,c>0 thoa a + b + c = 1 Tim gia trj Ion nha't ciia
A Theo chUorng trinh chuan
Cau 7a: Trong mat ph3ng Oxy cho tam giac ABC, duong cao xuat phat tu A c6
4
phuong trinh x + 2 y - 3 = 0, trung diem BC thupc Ox va G(0;-) la trpng tam
57
tam giac ABC Tim tpa dp cac dinh ciia tam giac biet S^pc - ~Y'
Cau 8a: Trong khong gian tpa dp Oxyz, lap phuong trinh mat phSng (a) di qua hai diem A ( 0 ; - l ; 2 ) , B(l;0;3) va tiep xiic voi mat cau (S) c6 phuong trinh:
( x - l ) 2 + ( y - 2 ) 2 + ( z + l ) 2 = 2
Cau 9a: Mpt hpp dung 40 vien bi trong do c6 20 vien bi do, 10 vien bi xanh, 6 vien bi vang, 4 vien bi trang Lay ngau nhien 2 bi, tinh xac suat de 2 vien bi lay
ra CO cung mau
B Theo chUcrng trinh nang cao
Cau 7b: Trong mat phSng voi h$ tpa dp Oxy cho 2 duong tron (Cj): x^ + y^ = 13
va (C2): (x - 6)^ + y^ = 25 Gpi A la giao diem ciia (Ci) va (C2) vai yA < 0 Viet phuong trinh duong thSng di qua A va cat (Ci), (C2) theo 2 day cung c6 dp dai
bang nhau
Cau 8b: Trong khong gian voi h^ tpa dp Oxyz cho hai diem A(1;1;0), B(2;l;-1)
va duong thing d : ^^-^ = - ^ - ^ ^ ~ ^ ' ^ ™ ^^^^ ^ thupc duong thang
d sao cho A ABC c6 dipn tich nho nha't
Cau 9b: Cho z = Tinh gia trj ciia bieu thuc:
P = r i f (2z + — + 1 ^ 3 f^ + 3 3 1 ' 4 + z +f 4 ^ n 4
V zj 1^ z ; V z > ^ z J
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Trang 26HirOTNGDANGlAt
I PHAN C H U N G CHO TAT CA CAC THI SINH
Cau 1:
a) Ban doc tu lam
b) Phuang trinh hoanh dQ giao diem: x"* - 2(2m + l)x^ + 5m - 1 = 0
Hoanh dp bon giao diem: xj = -yji^, \2 = ^3 = sjh' ^4 = \lh
Yeu cau bai toan tuong duong voi -,Jt^ > -3 « t2 < 9
Cau 2: Phuang trinh
o 2 sin x sin 2x + 2 sin X + 2 sin xcos 2x + V2 (sin x - sin 5x) = 0
<=> sin X sin 2x + sin X (1 + cos 2x) - V2 cos 3x sin 2x = 0
o sin X sin 2x + cos x sin 2x - V2 sin 2x cos 3x = 0 ,
o sin 2x (sin X + cos x - V2 cos 3x) = 0
2j + 4 x ^ + 2 ( 2 x - l ) +5 = 0 phuong trinh v6 nghi^m
Vay phuang trinh da cho c6 hai nghi^m: x = 1; x = - —
8 2dt
Cau 4: Dat xVx = tan t => - N/ x d x = (l + tan^ t)dt => Vxdx = — ,
Cau 5: GQ\ la giao diem ciia AC va BD, ta c6 A ' 0 1 (ABCD)
Goi K la hinh chieu cua O len AB, suy ra A' KO = 60'
The tich khoi lang tru la: V = A ' O S ^ B C D - ^ - ^
Ve O H I A ' C H e A ' C t a c o BD 1 (A'AC) => BD 1 O H , hay d(BD,A'C) = O H
Trong tam giac vuong A U C ta co: OH =
V O A ' ^ O C ^ " 2 V T 3 3a
-V | y d ( B D A ' C ) = j ^
Trang 27Tuyen chqtt &• Gidi thifu dethi Todn hgc - Nguyen Phu Khdnh , Nguyen Tat Thu
I I PHAN R I E N G T h i s i n h chi dupe chpn l a m mpt trong hai phan (phan A
h o l e B)
A Theo chUorng trinh chuan
Cau 7a: Gpi M la t r u n g diem BC, suy ra M ( m ; 0 )
Mat cau (S) c6 tam I ( l ; 2 ; - 1 ) , ban kinh R = ^
Do ( a ) tiep xiic v o i (S) nen ta c6:
H a i vien bi lay ra c6 cijng m a u nen ta c6 cac t r u a n g hop sau
H a i vien bi lay ra c6 ciing mau do: So'each lay C20 = 190;
Hai vien bi lay ra c6 cung mau xanh: So each lay Cfo = 45;
H a i vien bi lay ra c6 cimg mau vang: So each lay C^ = 1 5 ;
H a i vien b i lay ra c6 eung mau trang: So each lay C 4 = 6 ;
So each lay hai vien bi cung mau la: 256
Do do, xae suat can tinh la: P =
B Theo chi/orng trinh nSng cao Cau 7b: Xet h?:
• A = A B thoa yeu cau bai toan
• A A B gia su A eat hai d u o n g tron ( C j ) , (C2) Ian l u p t tai M , N
Phep doi x u n g tam A bien M thanh N v a ( C i ) thanh ( C 3 )
V i M € ( C , ) ^ N e ( C 3 ) = ^ N € ( C 2 ) n ( C 3 ) Phuang trinh ( C 3 ) : (x - 4 ) ^ + (y + 6 ^ = 13
( x - 4 ) ^ + { y + 6 f =13 ( x - 6 ) ^ + y 2 = 2 5
Trang 28a) K h a o sat s u bien thien va ve do t h i h a m so (1) k h i m = - 1
b) T i m m de d o thj h a m so (1) c6 ba d i e m cue trj tao t h a n h m p t tarn giac
v u o n g
Cau 2: Giai h ^ p h u o n g trinh: ( x - y) 2 + l 1 = y ( 8 y 2 - 3 x y + 2)
3x^ +4y^ + 2 = 3y(x + 4) Cau 3: Giai p h u o n g trinh: sin4x + cos 3x + — = sm 71
Cau 5: C h o h l n h c h o p S.ABCD c6 day A B C D la h i n h t h o i tarn O canh b a n g a ,
^^BC = 60" H i n h chieu ciia S len mat day la trung diem ciia OB SC tao v o i day
m p t goc 6 0 ° G o i M la t r u n g d i e m canh C D T i n h the tich k h o i c h o p S.ABCD
va k h o a n g each gii>a hai d u o n g t h a n g SB va A E Cau 6: C h o cac so t h u c d u o n g a, b , c T i m gia trj n h o nhat ciia b i e u t h u c
A Theo chi/orng trinh chuan
Cau 7a: T r o n g mat phang he toa dp Oxy, cho hinh thoi A B C D c6 tam 1(2; 1) va
A C = 2BD D i e m M ^ 1 ^ thuoc d u o n g thang A B ; d i e m N ( 0 ; 7) thuoc d u a n g
thang C D T i m tpa d o d i n h B biet B c6 h o a n h d o d u o n g Cau 8a: T r o n g h | toa dp O x y z , cho hai d u o n g thang:
Viet p h u o n g t r i n h mat p h i n g chiVa ( d j ) va song song v o i ( d 2 ) , xac d j n h tpa
dp cac d i e m A , B Ian l u p t thuoc ( d j ) va ( d 2 ) sao cho dp dai doan A B nho nhat
Cau 9a: Gpi Zi,Z2 la hai nghi^m cua p h u o n g trinh 2010z^ - 2 0 0 9 z +2010 = 0
- 1 — ry 'y
T i n h gia t r i M =11 - Zj.Z2 I - l z j - Z 2 l
B Theo chUorng trinh nang cao
Cau 7b: T r o n g mat phSng v o i h^ tpa dp Oxy, cho e l i p ( E ) c6 dp dai tryc Ion
bang 4\[2 , cac d i n h tren true nho va cac tieu d i e m c i i n g thuoc m p t d u a n g tron
H a y lap p h u o n g t r i n h c h i n h tac ciia ( E )
Cau 8b: T i m tham so thuc m sao cho d u o n g thSng d : x = 2 ( y - l ) = z + 1 cat
mat cau (S) :x^ + y^ + z^ + 4x - 6y + m = 0 tai 2 d i e m p h a n b i ^ t M , N sao cho dp
Trang 29Do thj ham so c6 ba diem eye trj khi va chi khi m + 2 > 0 o m > -2
Khi do, ba diem cue tr| la:
A(0;m + 1), B(>/m + 2 ; - m 2 - 3 m - 3 ) , C(-VmT2;-m2 - 3 m - 3 )
Suy ra AB = (Vm + 2;-(m+ 2)^), AB = (-Vm + 2;-(m+ 2)^)
Do tam giac ABC can tai A nen tam giac ABC vuong khi va chi khi
AB.AC = 0
<»-(m + 2) + (m + 2)'* = 0 o m + 2 = l<=>m = -l la gia tn can tim
Cau 2: Giai h§ phirong trinh: _ y{ Y Y ) ^ )
[ 3 x 2 + 4 y ^ + 2 = 3y(x + 4) (2)
(1) x(x2 - 2xy + y2 +1) - 8y^ - 3xy^ + 2y
<=> x^ - Sy-' - 2x^y + 4xy^ + x - 2y = 0
o ( x - 2 y ) ( x ^ +2xy + 4y^)-2xy(x-2y) + x-2y = 0
Vaynghifmcua h^'la: (x;y) = (2;l) -5±733 -5±N/33'
Cau 3: Phuong trinh o sin 4x + cos 3X + ^
6) -cos x
6J = 0
o sin'4x -2sin2x.sin 7t
X + — 6J
^ABCD -2S 'ABC - •
>/7 Vay the tich khol chop: V = -SH.SABCD = o
59
Trang 30Tuyen chgii & Gi&i thieu tie thi Todn hgc - Nguyen Phii Kftanh , Nguyen Tat Thu
Do do: P DSng thuc xay ra khi a = b = c Vay m i n P = ^
I I PHAN R I E N G Thi sinh chi du(?c chgn lam mpt trong hai phan (phan A
ho|c B)
A Theo chiTcrng trinh chuan
Cau 7a: Goi N ' la diem doi xung cua N qua tam I thi ta c6 N ' ( 4 ; - 5 ) va N '
thupc c?nh AB Suy ra M N ' = 4 ; - — nen phuong trinh A B : 4x + 3y - 1 = 0
Cty TNHH MTVDVVniatang Vi?t
Ca\i 8a: D u o n g thSng di qua diem M ( l ; - 1 ; 2 ) va c6 VTCP u j = ( 1 ; - 1 ; 0 )
p u a n g t h S n g d j d i qua diem N ( l ; - 2 ; 0 ) va c6 VTCP U 2 = ( - l ; l ; 2 )
|itph3ng (P) chii-a d , va song song voi 62 c6 vec to phap tuyen
^ = (-2;-2;0) = - 2 ( l ; l ; 0 ) 'huong trinh mat phSng ( P ) : x + y = 0
;ai diem A , B Ian luot thupc d , va d 2 nen
A ( l + a ; - l - a ; 2 ) ; B ( l - b ; - 2 + b;b) =^ A B = ( - a - b ; a + b - l ; b - 2 )
Dp dai doan A B nho nhat o A B la doan vuong goc chung cua hai duong
' A B U ] ' = 0 r2a + 4b = 5
2 a + 2 b = l thSng d i va d2 <=>
B Theo chiTorng trinh nang cao
Cau 7b: Gpi (E): - y + ^ = 1 (a > b > 0) la phuong trinh elip can t i m
a b Theo gia thiet a = 2 ^ 2 , cac dinh tren Oy la B i ( 0 ; - b ) , B 2 ( 0 ; b ) , F i ( - c ; 0 ) , F2(c;0) Tii giac F1B1F2B2 la hinh thoi nen va c6 4 d i n h nSm tren mpt duong tron nen FiBjFjBj la hinh vuong suy ra b = c ma a^ = b^ + c^ b = 2
Vay phuong trinh (E): — + = 1
8 4 Cau 8b: Thay phuong trinh cua duong thang d vao phuong trinh (S) 4t2 + ( l + t ) ^ + ( 2 t - l ) ^ + 4 2 t - 6 ( l + t) + m = 0
O t ^ = ^ — — < = > t i 7 =±-yj4-m v 6 i m < 4
9 ''^ 3 Khi do: M ( 2 t , ; l + t i ; - l + 2 t i ) , N ( 2 t 2 ; l + t 2 ; - l + 2t2) Suy ra M N ^ = 9 ( t , - t 2 f = 4 ( 4 - m )
Nen M N = 8 <» M N ^ = 64 o m = - 1 2 Vgy m = -12 la gia trj can t i m
61
Trang 31Cau 9b: Dat t = |z|, t > 0 thi ta c6 \zf (a.z + b) = -c.z
I PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1: Cho ham so y = - 3mx^ + 4m^ c6 do thj (Cm)
a) Khao sat sy bien thien va ve do thi ham so' khi m = 1,
b) Xac djnh m de hai diem eye trj cua do thi ham so' doi xiing nhau qua
duong thing y = x
Cau 2: Giai phuong trinh: (sin^ x +1 j +1 = 73 sin 2x + 4 sin
Cau 3: Giai hf phuong trinh
n
x + —
6J -xy = 1
6 ( y - 2 ) 2 + ^ = x3y[(2y + 3)2-6'
dx Cau 4: Tinh tich phan: I = [ -
- i l + x + Vl + x^
Cau 5: Cho hinh chop S.ABCD c6 day ABCD la hinh chu nhat eo tam O va
AB = a, A D = a73 ; SO = SD Mat phSng (SBD) vuong goc voi mat day, mat
phing (SAC) tao voi day mpt goc 60° Tinh the tich khoi chop S.ABCD
Cau6: Chung minh rang ne'u a,b,c>0 t h i :
a + b b + c ic + a ^
(a+b Vb+c ya+c
I I PHAN RIENG Thi sinh chi dugc chpn lam mpt trong hai phan (phan A
ho^c B)
A Theo chUorng trinh chuan
Cau 7a: Trong mp voi h? tpa dp Oxy cho duong tron (C) c6 phuong trinh
x^ + y^ - 2x + 6y -15 = 0 Viet PT duong thing A vuong goc voi duong thing
d : 4x - 3y + 2 = 0 va cat duong tron (C) t^i A, B sao cho AB = 6
62
Cau 8a: Trong khong gian voi h# tpa dp Oxyz cho hai duong thang:
d j : 2 ^ = X = ^ ; d 2 : ^ = ^ = ^ va hai diem A(l;-1;2), B(3;-4;-2) Xet vi tri tuong doi ciia d] va d2 Tim tpa dp diem I tren duong thing d j sao
cho lA + IB dgt gia trj nho nhat
Cau 9a: Cho Z j , Z2 la cae nghi#m phuc ciia phuong trinh 2z^ - 4z + l l = 0
Cau 8b: Trong khong gian Oxyz, tim tren Ox diem A each deu duong , x — 1 V z + 2 '
thang d : — ^ = = va mat phang (a): 2x - y - 2z = 0 Cau 9b: (1 diem) Giai h^ phuong trinh sau:
2logi_x(-xy - 2x + y + 2) + log2+y (x^ - 2x +1) = 6 logi-x (y + 5) - log2+y (x + 4) = 1
Cac diem eye tri ciia (C^,) la M(0; 4m^)va N(2m; 0 ) Trung diem cua doan M N la l ( m ; 2m^ jva M N = (2m;-4m3) Duong thing d : y = x c6 vecto chi phuong la u = ( l ; l )
M , N doi xung nhau qua duong thing (d) <=> M N 1 (d) va I G (d)
M N u = 0 2m'' = m Ke't hpp dieu kif n ta dupe m = ±
Trang 32" ^
Cau 2: Phuong trinh t u o n g d u o n g voi :
sin"* X + 2sin^ x + 2 - 2\/3 sin xcosx - 2 V 3 s i n x - 2 c o s x = 0
12y^ - 48y + 63 = 8x^y^ + 24x^y^ + 6x^y
T r u hai p h u o n g t r i n h cvia h? ta c6 duoc:
(8y^ + 24y^ + 6y +1) = - ( y ^ - Uy^ + 48y - 64)
o x ^ ( 2 y + l f = - ( y - 4 ) ^ <=>x(2y + l ) = - y + 4 o x + y + 2xy = 4
T ^ n u •u'^^ , [x^ + y ^ - x y - l
Ket hgip voi h^ ta co: <
[x + y + 2xy = 4 Dat S = X + y , P = xy ta c6:
Gpi K la hinh chieu cua H len canh A C , suy ra goc H K S chinh la goc giira hai mat phang ( S A C ) va mat day nen H K S - 60°
1 n 7?
7 2 U 7 b j 72l7^ 7 ? j 72l7^ 7 b J 1
Trang 33A Theo chUtfng trinh chuan
Cau 7a: Duong tron (C) c6 tam 1(1;-3), ban kinh R = 5 Gpi H la trung diem
Suyra l A + IB dat gia trj nho nha't bang A j B , dat dup-c khi A i , I , B thang hang
I la giao diem ciia A j B va d
B Theo chUtfng trinh nang cao
Cau 7b: Duong thang AB c6 phuong trinh dang: a(x + 2) + b ( y - 2) = 0
<=> ax + by + 2a - 2b = 0 vai a^ + b^ > 0 Duong thang CD c6 phuong trinh dang:
Suy ra phuong trinh A B : x - y + 4 = 0, C D : x - y - 4 = 0
Phuong trinh BC va D A c6 dang x + y + c = 0
Khoang each t u A deh m|it phang (a) : d ( A ; a ) = - j = = = = =
( a ) qua M o ( l ; 0 ; - 2 ) va c6 vecto chi phuong u = ( l ; 2; 2 ) D3t M Q M I = u
Do do: d ( A ; ( a ) ) la ihionj'; cao ve tu A trong tam giac A M Q M I
j/ A ^ 2.S,^MoMi ['•'^^o;"] V8a2-24a + 36
= > d ( A ; A ) = 1- = -^;— i =
Theo gia thiet: d ( A ; ( a ) ) - C I ( A ; A )
|2a| Vsa^ - 24a + 36 , 2 o 2 -.^ A 2 r -.^ n
o ! — = <::> 4a = Sa"^ - 24a + 36 <=> 4a - 24a + 36 = 0
3 3
Trang 34l o g i - x ( y + 5)-log2+y(x + 4) = l
j l o g i - x ( y + 2) + log2+y (1 - x) - 2 = 0 (1)
l l o g i - x ( y + 5)-Iog2^y(x + 4) = 1 ( 2 ) ' Dat l o g 2 + y ( l - x ) = t thi (1) tro thanh: t + ^ - 2 = 0 < » ( t - l ) ^ = O o t = l
Vol t = 1 ta c6: 1 - X = y + 2 o y = - X - 1 (3).The vao (2) ta c6:
K i e m tra dieu k i ^ n ta thay chi c6 x = - 2 , y = 1 thoa man dieu k i | n tren
Vay h? CO nghipm day nhat x = -2, y = 1
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I PHAN C H U N G C H O T A T CA C A C T H I S I N H
Cau 1: Cho ham so y = x^ - Sx^ + 1 , c6 do thj la ( C )
a) Khao sat su bien thien va ve do thj (C) ciia ham so
b) T i m cac diem A, B thupc do thi (C) sao cho tiep tuyen ciia (C) tai A , B
song song v o i nhau va A B = 4V2
Cau 2: Giai p h u o n g trinh: 5 cos 2x + - - 4 sin rilTt >
x + 9
Cau 3: Giai p h u o n g trinh: 2(x^ + 2) = 5Vx^ + l
dx Cau 4: T i n h tich phan: I = f-
i x ^ + 2 x
63
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Cau 5: Cho t u d i ^ n ABCD c6 ABC la tarn giac deu canh b i n g 2a, A D vuong g6c
day, A D = ay/3 Gpi E, F Ian lupt trung diem ciia cac doan BC, DE Tinh the
jfch hinh chop F.ABC Chung m i n h AF vuong goc vol CD
Cau 6: Cho so thuc d u o n g a, b thoa man: 6|a^ + b^ j + 20ab = 5(a + b)(ab + 3)
Tim gia trj nho nhat ciia bieu thuc
A Theo chi/orng trinh chuan
Cau 7.a: Trong mat phJing tpa dp Oxy, cho hinh chii nhat ABCD, voi toa dp cac
f 4^
dinh A ( l ; l ) G p i G 2 ; - la trpng tarn tam giac ABD T i m tpa dp cac dinh con
l^i ciia hinh chu nhat bie't D nam tren duong thang c6 phuong trinh: x - y - 2 = 0
Cau 8.a: Trong mat phSng tpa dp Oxyz, cho hai mat phang ( P ) : x + ^ - 2 z + 5 = 0
va ( Q ) : X + 2y - 2z - 1 3 = 0 Viet phuong trinh mat cau (S) d i qua goc tpa dp
O, qua diem A (5; 2; 1 ) dong thoi tiep xiic voi ca hai mat phSng ( P ) va ( Q ) Cau 9.a: Tinh m o d u n ciia so phuc z , bie't z^ + 12i = z va z c6 phan thuc duong
B Theo chUorng trinh nang cao
Cau 7.b Trong mat phang tpa dp Oxyz, cho elip (E): — + ^ = 1 va duong
thSng d : X + y + 2013 = 0 Lap phuong trinh d u o n g thang A vuong goc voi d va
c^t ( E ) tai hai diem M , N sao cho M N = —
3 Cau 8.b Trong mat phSng tpa dp O x y z , cho mat phang ( P ) : x - 2y + 2z + 2 = 0
va duong th5ng ( d ) : = = Mat cau (S) c6 tarn I nam tren duong thang (d) va giao voi mat phang ( P ) theo mpt d u o n g tron, d u o n g tron nay Voi tam I tao thanh mpt hinh non c6 the tich Ion nhat Viet p h u o n g trinh mat
cau (S), bie't ban kinh mgt cau bSng 3N/3
Cau 9.b: Giai h$ p h u o n g trinh sau: 2x'^ + 2 x y - 3 x - y + l = 0
2 = 0