Tinhthd tich khdi chop S.ABCD tr\eo a.. Theo chucrng trinh Chuin Cfiu VI.a.2,0 itihm 1.. Vi6t phucrng trinh ducrng thdng di6m D1;-3.. Tirn to4 dQ diOm H ldhinh chi6u vudng g6c cua di6m M
Trang 1rct 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)
/-CAu I (2,0 ctiiim) Cho hdm s6: y -'4 x-l (H)\
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" *7 - J;+ 3
Ciu III (1,0 eli€m) Tinh gi6i h4n ' ; 7 = pnlT
r=:-: x+l X'-3X+2
C6u IV (1,0 ititm) Cho hinh ch6p S.ABCD co d6y ABCD le hinh chfr nhflt, s,l
v6i m[t phing d5y, SC tpo v6i m{t phdn g d6y g6c 450 vir tpo v6i m{t phlng
:00 Bitit dO dai c?nh AB = a Tinhthd tich khdi chop S.ABCD tr\eo a.
Cffu V (1,0 diAd Tim gi6 tri nh6 nhdt ctra hdm sd y - x+./r' + ! (x > 0)
Yx
PHAN RIENG (3,0 iti€m)
vuong goc
(srB) g6c
I
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
x+2y-5=0 vd 3x -y+7 =0 Vi6t phucrng trinh ducrng thdng
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 .
thing AB,BC lAn lugt li:
.qc, bi6t ring ,tc di qua
Trang 2Elfp (E) ,
+ * + =1 vd diiSm M thu6c (E) cie str (d) 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
"tu(J7*-J )'
s6 cta 3 s6 hpng cu6i bdng 2?,t6ngc6c s6 hang thf 3 vd thri 5 bdng 135.
, t6ng c6c hq
2
www.VNMATH.com
Trang 3EAp 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)
I
2rA
-.t
dlem
I
!-.*Q,Q*4i?r0 -
-1r T{p x6c dinh : D: m t {t}
2 Su bi6n thi6n
I
a) Chi6u bi6n thi6n Ta c6 : y' * - - < 0 Vx e D
x -l)'
Hdm sd dE cho nghich bi6n trdn c5c khoAng (-*;r) vd (f +o)
0,25
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.
limy - +oo; Limy = -@ , dd thi ctra hdm sd c6 tiQm can dimg h
t+l-ducrngthing x=1
4,25
d) Bing bi6n thi6n
I
X l-oo
-tr
I
ll
_ll.-+oo
v
2
0,25
thi
a
J.
0,25
Trang 4Eulng 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 €
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)
=[(", + \)' - +4x,]1ni + t1
Theo Viet:
I *-1= AB' =4(m+l)= 8,Ym>0 =+ AB>2J2,Ym>0
lm
MinAB =2J, khi ln = 1.
Yatv m= I ldr niJoi .An-,irn
0,25
0,25
0,25
CflU II
2ra
drem
l (1,0 didm)
sin3 x(sinx + cosx) + cos'(cosx - sinx)
-<> sino x * cos4 x + sin3 Jccos.r - cos' xsinx
a
J
4
.t J
I - 2sin' xcos' x - sinxcosx(cos' x - sint; : l
I - !rin' 2" - lsin2xcos 2* =1
113 -:n- cos4x) - lsin 4x = I
<+ sin4x - cos 4x - 0e sin(4x - 4) = O
4' 7t _7r
<+x :-+k- keZ
16 4' V4y phucmg trinh c6 nghiQm 1A x: * + t:
0,5
0,25
4
www.VNMATH.com
Trang 5I
Ddt: x+-=S
v
(t
lS=-5 I
Vdil- -=1 ,Y ,hQvOnghiQm
LP = lo 1".!
- to
Lv
[.- 1
-.
v6t
b ;
LY
1
VQy h€ phucrng trinh c6 nghiQm 1a.: (x;y) - (2;I) vi (x;y) = (t;t).
4,25
0,5
Cfiu
ilI
1'0
tli6m
1=lim
x+l
<tx+?-.,6+3
x'-3x+2
:,1,t+? -2- J71j +z
= lim
x+l 'x' *3x +2
, (:,1.*t-z G;3-2)
- rlrtl I - -: |
x-'r
["' - 3x+2 x' -3x+2 )
0,5
1
6
0,5
Qnu
ry
I
1'0
Vi SAL(ABD)*SCA= f
BC L(SAB)+&-300
www.VNMATH.com
Trang 6c6 SC) - 450 n}n AC = SA =r vd ,SC - xJi
nsaS
".'*::: i : 11 e ::::
::: -" :-1r " : *
LABC vu6ng tu B , c6
2
AB' + BC' - AC' e o' *\- y2 e *= oJi,
2
SA-oJi BC=a
0,25
t17
a ! z (dvtt).
-\/
J
VOy, Vr.nuro = l*
" '"o'=
0,25 CAu V
100
(Irenr
l" 1
!=x*{"'+a tren (o;+.o)
, I
L&7 ,.f
-1,
2^lx'* t
Vx
tf-'
<+ 1- 2xt = 2x'
^lx' + I eI- 2xt
Vx
h-z*'> o
f(t-zx')'=4*'(r'+1)
1r 2
X6t hlm s6
Tir b&ng bi6n thi6n cho k6t qu6:
Minv-2 khi
"=f
0,5 0,5
Cfiu
I
VI.a
l
2rA
i
GIEIn
L,-.Qr-q.-4i-Q4-Gqi,vdc tcr phfp tuyiin cna AB tiii;ij; ililr'uptuy6ffi';a-it I
fit:;-fl vd v6c to ph6p tuy6n cua AC n ,t (a;b),(o, *b, *A)
|
-9g 449-9 g-T-L?ij,.193e.le s9e-.g-,-g *s*-yl lgle$s* '"y_p ]
0,5
6
www.VNMATH.com
Trang 7l,\,\l _|,\il
cos-B-cosc<+ffi -t=-t=
lryllryl lryllryl
t lta-ul
<+f -J 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
MI -zJi > z= R= M
1(1;3) vd b6n kinh n = Z
nim ngodi ducrng tron (C)
0,25
Ggi H(x;y) X6t thdy t, M, H thdng hdrrg n€n7fr(a;-2)
'1 v-3
phuong u6i Ifr(x -1;y- 3) e + = E€) x - Zy - -s
cirng
ta c6
0,5
Lpi c6 NAM - NHA= IA' = IM.IH md IM.IH - IMJH ,
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:
_ r
-t
Trang 8tr0
k=0
[,'v;'- :)' =fc: (*, J])r[*)' " n 5k
=lClxT.*3k-3n
k=0
1lk-6n
2*
H0 s6 cria s6 heng thf 3 ld 36, ta ducyc Cl, =3A a n- 9
ek=6.
n66
Lnx
f U YeU CaU Dal toan, ta Co
Z = 6 Vfly s6 hang chria xu trong khai triOn ld
0,5
'Ciu
VI.b
2'0
-.i
(Irem
(2x-3v-2=a top dO di6m A thobmdn h0 phucrng !) trinh: 4'^ "
l*-2y-l=0 "-\-)-/
!,1-1.'8 4i-'.@
Vi BC L AH n6n BC c6 phuong trinh: 2x + y* c = 0
Do M(-3;0) e BC n€n c = 6
Vfly phuong trinh BC Id 2x + y + 6 = 0
Ma B (A), to4 dQ B tho6 mdn hd phucrng trinh:
(2, -3v +I4 = 0
[2x+ y*6=0
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
YQy A(r;0), B(-4;2), C(-2;-2).
0,5
0,5
?.' Q'.Q-.Fi-c*)
Gei M(xo;%) € (n)* +i',
phuong trinh (d): Ij!- + l''
9t
1
O
|",y,I l6n nh6t.
TG.;bfidds thii; clt
,
36 = 44 +9y1' >
6ia, rc-rr xhy ra g 4xt
Tt (1) vd (2), ta dugc
V4y c6 b6n di6m M thoi
*,(+,ll),*,(+
l l khi ]
I I
l
I I I I
I
I I
.I
,t
./t
suy ra Srou fro nhAt khi vd chi
co:
S yi = tTlx,yol
= lx,yol < 3
9v',(z) 92a
2"0
rdn y6u cdu bdi to6n li Jt),
*,(*,rr),*^(-+,-0,5
0,5
CAU
VII.b
1r0
Tdng c6c hQ s6 cria ba s6 hang cu6i bdng 22, n€n
9-i.l :l 9u ' ! 9= :??:9 )9i rly*g jf'h lg duec n = 6
( , \6 : - :.:_,/- ; V'- khi d6, ta c6 kirai tri6n
[Jr # ) =Lc:(Jr ) t#J
0,25
0,5
8
www.VNMATH.com
Trang 9f6"e;a;;6-hds iii i 3 ;tttiti i uB",e' it5;e'
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.
0,25