1. www.VNMATH.com
rct rsr KSCL THr DAr Hoc xAu zltz rAx rH{I1
on rnr ivr0x, ioAN, xu6r u
Thdi gian ldm bdi : IB0 philt, kh6ng k€ thdt gian giao di
PA tU 96*' 02 trang
rHAN cHUNG cno rAr cA THi srNH (7,0 tliam)
, 2x-I (H)/-
CAu I. (2,0 ctiiim) Cho hdm s6: y -'4
x-l
1. Kh6o s6t ss bi6n thi6n vd vE dO thi (H) crtahdm s6.
2. Tim cfrc giStri cria * dQ duong thing ! =mx-m+z cht OO ttri @) tai hai di6m phAn
bil.t a,B sao cho dopn AB c6 dO dei nho nhAt.
C6u II. (2,0 cfiAm)
1. Gi6i phucmg trinh: sin'x(sinx + cosx) + cos'x(cosx - sinx; * I
4
2. Giei h9 phucrng trinh:
ft+x*xy=5y
ft*"'y'-5y'
tl" *7r=:-:--
Ciu III. (1,0 eli€m) Tinh gi6i h4n ; 7 = pnlT - J;+
3
' x+l X'-3X+2
C6u IV. (1,0 ititm) Cho hinh ch6p S.ABCD co d6y ABCD le hinh chfr nhflt, s,l vuong goc
v6i m[t phing d5y, SC tpo v6i m{t phdn g d6y g6c 450 vir tpo v6i m{t phlng (srB) g6c
:00. Bitit dO dai c?nh AB = a . Tinhthd tich khdi chop S.ABCD treo a. I
Cffu V. (1,0 diAd Tim gi6 tri nh6 nhdt ctra hdm sd y - x+./r' + !
Yx
(x > 0)
: 2
PHAN RIENG (3,0 iti€m)
Thi sinh chi ctwqc ldm mQt trong hai phdn (phfrn A hogc B)
A. Theo chucrng trinh Chuin
Cfiu VI.a.(2,0 itihm)
1. Cho tam gi6c ABC cdn tai d,bi6t phuong trinh ducmg thing AB,BC lAn lugt li:
x+2y-5=0 vd 3x -y+7 =0. Vi6t phucrng trinh ducrng thdng .qc, bi6t ring ,tc di qua
di6m D(1;-3).
2. Trong mdt phdng v6i hQ trpc to4 dQ Oxy, cho dulng tron (C) co phucrng trinh:
*'+ y'*2x-6y+6=0 vd di6m M(-3;r).Gqi A vir B IdctrctiOp di6m ke tir M ddn e).
Tirn to4 dQ diOm H ldhinh chi6u vudng g6c cua di6m M tr}n AB .
Cf,u VII.a. (1,0 ili€m) Tim sO hang chira xu trong khai tri6n cta nhi thfc * ,
[x'Jr +)"
UiCt n9 s5 cria si5 h4ng thf 3 bing 36.
B. Theo chutrng trinh Nfing cao
Cdu VI.b. Q,0 meryl
1. Trong m{t phSng v6i hQ truc to4 d6 Oxy, eho tam giSc ABC, dinh B ndm trOn ducrng
thdng (A): zx -3y+14=0, c?nh .4C song songvdi (A), dudrng cao AH cophuongtrinh:
x-2y-1= 0. Gqi M(-3;0 lir trung di6m cria c4nh BC .X6c dinh toa dQ c6c dinh A,B,C .
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Elfp (E) , * vd diiSm M thu6c (E). cie str (d)
+ + =1
td ttu<rng thsng ti6p
xirc v6i (E) tai M vit (Q chttruc ox, oy ldnluqt t4i A, B. Tim top d0 di6m M dC dien
tich tam gi6c AoB nho nh6t.
C6u VII. b. (1,0 ifiAm) Tim x bi€t rang trong khai tritin , t6ng c6c hq
r )., ,
"tu(J7*-J--)'
s6 cta 3 s6 hpng cu6i bdng 2?,t6ngc6c s6 hang thf 3 vd thri 5 bdng 135.
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EAp AN-IHANc orfm
xV nu KscL THr DAr Hec NAnn zal: - lAn thrn, I
MOn: Tofn; ftr6it n
(D,ip dn - thang dtd*t gdm 07 trang)
Ciu D6p 6n Di6m
I !-.*Q,Q*4i?r0 - -
2rA 1r T{p x6c dinh : D: m t {t}
-.t
dlem 2. Su bi6n thi6n
I
a) Chi6u bi6n thi6n. Ta c6 : y' * - ----l)' < 0. Vx e D
0,25
I
x
Hdm sd dE cho nghich bi6n trdn c5c khoAng (-*;r) vd (f +o)
b) Cgc tri: Hdm sd kh6ng c6 cgc trf
c) Gi6i h4n vd ti6m cfln:
tliy - Z; IY_! = 2, d6 thi cria hdm s6 c6 tiQm cfn ngang ld ducmg
thdng ! =2. 4,25
limy - +oo; Limy = -@ , dd thi ctra hdm sd c6 tiQm can dimg h
:+1" t+l-
ducrngthing x=1
d) Bing bi6n thi6n
X l-ooI
I +oo
-tr
ll
0,25
v
a
_ll.- 2
J. thi
0,25
?,-.$.'.0*f-Q@.
X6t phucmg trinh holrnh dQ giao di€m cria dudng thdng dd cho v6i
-,thi @): 2x-l
dd
x-l
-=mx-
m+ 2 (1) e
'
fx+l
{
l**'-2mx+m-1=0(2) l-
0,25

4. www.VNMATH.com
Eulng thlng y=mx-m+2 cdt (H) tqi hai ili6m ph6n biQt e (1) c6
hai nghiQm ph6n biQt e (2) c6 hai nghiCm phAn biQt kh6c 1 € 0,25
m>0.
Ysi mr0, (2) c6 hai nghiQm phdn biQt, gi6 sri x,,x, .
' lY'=ffixt-m+2
Df;t A(x.yr1, B(xr,!z), ta c6: j ".'
lYr=ffixz-m+2
Khi d6 AE =(xr-)'+nf (xr-x,)' --(*,-*r)'("f +l) 0,25
=[(", + )' - +4x,]1ni + t1
Theo Viet:
fxr+xr-z 1 r_
I *-1= AB' =4(m+l)= 8,Ym>0 =+ AB>2J2,Ym>0
0,25
lxrx, = tn
lm
MinAB =2J, khi ln = 1.
Yatv m= I ldr niJoi .An-,irn.
CflU II l. (1,0 didm)
a
2ra J
sin3 x(sinx + cosx) + cos'(cosx - sinx) - 4
drem .t
J
<> sino x * cos4 x + sin3 Jccos.r - cos' xsinx 0,25
4
I - 2sin' xcos' x - sinxcosx(cos' x - sint; : l
e I - !rin' 2" - lsin2xcos 2* =1 0,25
113
-:n- cos4x) - lsin 4x = I
4' 4 4
<+ sin4x - cos 4x - 0e sin(4x - 4) = O
4'
7t _7r
<+x--:-+k-. keZ
16 4' 0,5
V4y phucmg trinh c6 nghiQm 1A x: * + t: ,keZ.
164
0,25
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5. /i www.VNMATH.com
I
Ddt: x+-=S
v
4,25
(t
lx+-=-)
-
lS=-5
Vdil- -=1 ,Y
I
= lo
,hQvOnghiQm
LP 1".! - to
Lv
[.- 1 -. 0,5
v6t
[s=3 rx-2
*t; =i " lx=t l"*;-t
t; =;'l*r'=, b--;
LY
1
VQy h€ phucrng trinh c6 nghiQm 1a.: (x;y) - (2;I) vi (x;y) = (t;t).
Cfiu <tx+?-.,6+3
1=lim
ilI x+l x'-3x+2
1'0 :,1,t+? -2- J71j +z
= lim
x+l 'x' *3x +2 0,5
tli6m
,. (:,1.*t-z G;3-2)
rlrtl I ---:--
x-'r
["' - 3x+2 x' -3x+2 )
|
- -
1
6
0,5
Qnu Vi SAL(ABD)*SCA= f
ry BC L(SAB)+&-300
I
1'0
0,25
Gqi S,4 =x (x > 0). A&4C vu6ng t4i A, na<
.l i
I
j
5
I
I
.!
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6. www.VNMATH.com
c6 SC) - 450 n}n AC = SA =r vd ,SC - xJi
e :::: :::
" :*
nsaS
".'*::: i : 11 -" :-1r
LABC vu6ng tu B , c6
2
AB' + BC' - AC' e o' *-
2
y2 e *= oJi, 0,25
SA-oJi. BC=a
a t17 (dvtt).
!z
VOy, Vr.nuro =
l* " '"o'= J -/ 0,25
CAu V
X6t hlm s6 !=x*{"'+a
l" 1
tren (o;+.o)
100
(Irenr I
,--
L&7
,.f X-
)/ -1,
-L' f-----:
t
2^lx'*
Vx
tf-' 0,5
!'=oo{ -zx=2^lr'+!
xYx
<+ 1- 2xt = 2x'
^lx' +
I eI- 2xt
Vx
h-z*'> o
<+1, .r2 gle= 1r
f(t-zx')'=4*'(r'+1) 2
0,5
Tir b&ng bi6n thi6n cho k6t qu6:
Minv-2 khi
(0;+o)" "=f2
Cfiu L,-.Qr-q.-4i-Q4-
I
VI.a Gqi,vdc tcr phfp tuyiin cna AB tiii;ij; ililr'uptuy6ffi';a-it I
n ,t (a;b),(o, *b, *A)
l
2rA
i
fit:;-fl vd v6c to ph6p tuy6n cua AC
| 0,5
GIEIn g-T-L?ij,.193e.le
-9g 449-9 s9e-.g-,-g *s*-yl lgle$s*
'"y_p ]
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7. www.VNMATH.com
l,,l
cos-B-cosc<+ffi _|,il
-t=-t=
lryllryl lryllryl
t
<+f -J
lta-ul
e22d +2b' LSab=0(*)
./S
'ld +b'
Giei (*), ta dugc 2a =b ho{c tta =2b .
- V6i 2a=b, chgn a=l suy ra b=2 thi ,tr(UZ).
Do D e AC n6n phucrng trinh AC ld: l(x-l) +2(y+3) = g
hay x+2y+5 = 0 ( loai do AC ll AB)
- Voi lIa = zb, chgn a = Z svy ra b =l 1 thi ,a1Z;tt1 .
Do D e AC n€n phucrng trinh AC ld: z(x -L)+ il(y + 3) : 0
hay 2x +lly +31 = 0 (nhan).
Vfly, AC: 2x+lly+31:0.
?'-&-o-gj
Eulng trdn (C) c6 t6m 1(1;3) vd b6n kinh n = Z.
MI -zJi > z= R= M nim ngodi ducrng tron (C).
0,25
Ggi H(x;y). X6t thdy t, M, n€n7fr(a;-2) cirng
H thdng hdrrg
phuong u6i Ifr(x -1;y- 3) e
'1 v-3
+ = E€) x - Zy - -s 0,5
Lpi c6 NAM - NHA= IA' = IM.IH md IM.IH - IMJH , ta c6
IM.IH = IA' e -4(x- 1) - 2(y -3) = 4 e Zx * y =3
'---_--___---___--_-T
to4 tlQ di6m H r}roh mdn hO phuong trinh:
I:-r,= _rol"
-t I/[,l,li']
f_ _ r
= --.5'5)
l2x+!=3 1.._13 0,25
i
l"-T
Y$y H(+,9.]
) 5 /
Ta c5
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tr0 n 5k 1lk-6n
=fc: (*, J])r[*)' "
:ZCI* 2*
o tem [,'v;'- :)' =lClxT.*3k-3n
k=0 k=0
H0 s6 cria s6 heng thf 3 ld 36, ta ducyc Cl, =3A a n- 9
^ .L
f U YeU CaU Dal toan, ta
.1Ik-54 6 ek=6.
Co
Z = 0,5
n66
Vfly s6 hang chria xu trong khai triOn ld Lnx
'Ciu !,1-1.'8 4i-'.@
VI.b Vi BC L AH n6n BC c6 phuong trinh: 2x + y* c = 0
Do M(-3;0) e BC n€n c = 6.
2'0
-.i
Vfly phuong trinh BC Id 2x + y + 6 = 0
(Irem
Ma B. (A), to4 dQ B tho6 mdn hd phucrng trinh: 0,5
(2, -3v +I4 = 0
' -
{^ y*6=0 :+B(a;z)
[2x+
Y-iY-f.1,.9)1t-tryle--{.ri-rp-g.t_T__qL?it)........_..
Cenh AC ll (A) va di qua C ndn AC co phucmg trinh:
2(x +2) -3(y +2) = 0 hay 2x -3y + 2 = 0 .
(2x-3v-2=a 0,5
top dO di6m !)
A thobmdn h0 phucrng trinh: 4'^ " "--)-/
l*-2y-l=0
YQy A(r;0), B(-4;2), C(-2;-2).
?.'..Q'.Q-.Fi-c*)
Gei M(xo;%) € (n)* +i',
phuong trinh (d): Ij!- + l''
9t l
l 0,5
1
l$i d6 Snou =;OA.OB = suy ra Srou fro nhAt khi vd chi khi ]
O I
|",y,I l6n nh6t.
I
l
--TG.;bfidds thii; clt , co: I
36 = 44 +9y1' > yi = tTlx,yol lx,yol < 3
I
S
= I
6ia, rc-rr xhy ra g 4xt 9v',(z) I
92a
Tt (1) vd (2), ta dugc I
0,5
2"0 I
V4y c6 b6n di6m M thoi rdn y6u cdu bdi to6n li .I I
*,(+,ll),*,(+ Jt), *,(*,rr),*^(-+,- Jrl
,t I
./t
CAU Tdng c6c hQ s6 cria ba s6 hang cu6i bdng 22, n€n
0,25
VII.b 9-i.l :l 9u ' ! 9= :??:9..)9i rly*g jf'h lg duec n = 6
1r0
( , 6 --:---. :.:_,/- ;--V'- --
khi d6, ta c6 kirai tri6n . =Lc:(Jr. )
[Jr. # ) t#J 0,5
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9. www.VNMATH.com
--f6"e;a;;6-hds iii i 3 ;tttiti i uB",e' it5;e'
0,25
V$y * = -1 vd x= 2 thod mdn y6u cdu cria bdi to6n.
Q!rt-!: Hgc sinh ldm theo cdch khdc drtng phdn ndo thi vfin cho iti6m phdn tuong drng.