"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
www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
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 : <
8
Cty TNHH MTV DWH Khang Viet
Tu ( l ) suy ra w'' =-z^ w 3 - Z Tu {2) suy ra 2 5 , 1
w Z = !=> w = z = 1 Suy ra (2) » w^.|z|^° = z^ <=> = z^ => + = 0 <=> w = 0, w =-1
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
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^
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
www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
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<
www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
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
I a + 2b - 2c I COS(t) =
• 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
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:
10 4 +
( x 2- l ) % 3 = _ 6 x - y
x 2+ 2 3y - X = , 4 x - 3 x ^ y - 9 x y ^
V x + 3y ( x ^ - x ) ^
- 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
www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
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
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
www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
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