TRUO]\G THPT I,N XOAYM6n rhi: TOAN, Khtii A, AII Thdi gian ldm bdi i80 phut, khdng kd thoi gian t giao dd Cffu l.. 2 diern Cho hinh chop S.ABCD c6 driy ABCD ld hinh vuOng cqrnh 2a.. Tinh
Trang 1TRUO]\G THPT I,N XOAY
M6n rhi: TOAN, Khtii A, AII
Thdi gian ldm bdi i80 phut, khdng kd thoi gian t giao dd
Cffu l (2 di€rn) Cho lidrn s6 y " 2(x+i)' =.Flf tr
1 Kh6o s6t vlr v€ dO thi ham sO 1c;
2 Tinr nhii'ng di6rn M trOn (C) sao cho titlp tuydn vcii (Cl) t4i M tao r,6'i hai truc toa c10 mdt tam gi6c c6 trong t0m nim tr6n duong thing 4x* y =' 0.
Cf,u II (3 di6m)
1 Giai phuong trinh
2 ciei h€ phuong trinh {y;,=\li::l (,'/ € R)
x-3iV5-r3,
3 Tinh gi6i han: .I = lim
x-1 Cf,u III (2 di6m)
Trong mpt phdng tga dQ Oxy cho tarn gi6c AI|C co dinh e(z;3), trong lAm
C(0;2)vd irai dinh B,C lAn luot trdn hai dudng thing cl,:x +J/ + 5 = 0 vA
d, :x+2y -7 = 0 Tim toa dQ t4rc tAm cua tam grdc ABC .
CAu IV (2 diern)
Cho hinh chop S.ABCD c6 driy ABCD ld hinh vuOng cqrnh 2a Cdc di6m
AM =2M8, AP =2PD, 1VB = NC,4Sg= SC Cho s,< vudng g6c vcii mflt phi$g c16y
vii goc gifra hai mflt phing (.sBC) vd (ABCD)bing 600
t 1 Tinh di6n tich tam gi6c MNpvit khoAng c6ch tri di6m Qd6n mpt phdng
(uw)
2 Tinh dg ddi dopn vuong goc chung ctsa MP vd sc.
Cfiu V (l di0m) *
Cho c6c sO thqc duong x,!.2 thoa mdn diiiu ki€nt x+21t*; = 5 'I'irn gi6 tri nhr)
nh6t cua bieu thuc, -4x+2y+z 2xy+xz *r-y -'4
FIp vd t6n thi sinh: .: , , , , ,.S0
C6n bO coi thi kh6ng giai thich gi thdm.
Trang 2rRU'oNG rHpr LE xoAy pAp AN - THANG DIfM Of t'ru KHAO SAl IAX t r 'l
','
- ':rr l- " ':''l'' t "' , i,:t :1tt*15
Luu f ' : -Ddp -Hpi sinh-tdm dn g6m 04 il)ng thio trang cdch khdc'vdn cho didm tiit da ' : ' ', ' ':'',1 ;'t;:,.|,r: : 1"';
T
1
2
gh6 hirrr s6 r,= ' 2(x+l)' ,,II|(C) / rcnao sdt vd ve dd th! hdrn sO(C) 1.0
TXD: D = R\{*1}
I
chi€ubii5nthion: / =dD, >o,voi Vxe D
=hdrn stl dong bi6n tr€n rnSi khodng :(-m;-l) vd (-l;+"o)
Cqe tri: hdm s6 khdng c6 cpc tri
Gi6'i h4n, ti0rn cdn :
linty=7, .(+M / x+-a lim y=- ; / x+(-l)' lim y=-e , r+(-l)-Iim !=+q
I .^ ^
= '2y = - ld ti€nr cdn ngang; x = -l ld tiQm cQn ctfng BAng bi6n thi€n:
lrl
., I A*ll / l
"l ,/-ll / L/ ll /
7
l?
DO thi: di qua c6c di€m (0; - Z I : GZ; i)
,l
Nh{n,giao di6n crha hai tiqm c&n ICl; i) ldm tAm cl6i xfng
I
I I'
It
/ llr
; tl | ,
ll 'lv tA
l,r
0.2s 0.25
0.25
0.25
,'firn nhfing di€m M t'r€n (C) sao cho fiOp.tupi€n v6i (C) tgi M tpo vd'i hai trr,rc tpa d0 mOt tam gi6c c6 trgng'tAt: nirn trdn dub'ng thang 4x + y = 6
* _l
Gsi M(x";4 ) e (C) li di€rn cAn tirn
2(xo + l) ' Cqi A titlp tuy6n v6-i (C) t4iM ta c6 phuong trinh
0.25
Trang 3L: y = "f'Go)6-ro) + ' =lu -t, ) | =t-f. -(x x^)+ ro -l
2(xn+l)' ' (xn + l)' t* *o' ' 21xo+11
coi A: Anox:)a1-r" ' 2 )-)- -Zxt-t ;0) B: Anoy= ufo, 'zffil {-)v -7
Ifti d6 At4o v6'i hai trpc toa dQ AOAB c6 trgng tdm ld:
^,( xl -zxo- I *ot - 2xo - 1)
\irr -.-""_r.'1 6 ' 6(xo +1)' )
:4x + Y = 0=) -O'4 -z:'-t* "i, tt',;t = O
l)o Ge dtrdng thing o o(.ro +
r)-.t 3 4 = (.r, + tr l)- (vi A, B *O ndn xo2 -Zxo-1*0)
Voi xo =-)-uel;-|) ;v6i *o=-1-ueltll
trt-t
| *o *l=t I *o =-t
jxo+l=
lr'= L z L'o z
0.25
4.25
0.25
II
I
GiAi phuong trinh s"(+-z')+z =O'i"(;.") (1) 1.0
/ -\
Ddr rr=|**L i, 6) lo"= r,-4,dodo: 6'
a /.- -\
(r) <> ,lsinl \/ \3 lL-2u+il*z 3) = Dsin u o Jt;nz, +z = Jisinu
'- t: i ;'),i i r= 2 si n 2', o
{i'r'i n r'.1r,, + 2 cos2 a = 0 €
L}!:'r'= -l
+ Voi coszr = 0 e sin u =!l= sin u =1 e " =t,* kLn e, =!+ k2r,k e Z
+ V6i tatlu = -7 e u = -L 4" + ktrk6thoo v6'i
sinu > 0= u = -L+Qt +t\, =?o +l2r a *=7o +l,n.l e z
Vdy PT co hai ho nghi€rn: *=!+k\n,keZvit x=?+l2n,l eZ
0.25
0.25
0.25 0.25
7
(x,y e R) Giai lrQ phuong trinn Jx{ = x +7 Y^+1
l"tYt =loYz -1
(1)
riL (r)r \ / f, = U+(do y-l y = tttri (l)v6 nghi€rn), thay vdo
"f7u*t)'
(z) + ),tl '\Y-t) *!+l = t0/' -t e y' (7 y+r)' =(roy, -r)(y-i)'
<* 39;,0 + 34rr3 *By' -2y.+l =o e (.v+l)(:y+t) =(tly'? *6y+t) = o
Iv=*l:?x=3
* I ; = -i:; 3-" =, vay hQ c6 hQ c6 trai nghidm: (";y)= (:;-t), (t'-i)
0.2s 0.25
0,25
0.25
3 Tinh gi6'ihan: 1= limx -3 + i.6.3t
Trang 45
f
_-4
0.2s 0.25'
' :'
0.25
0.25
TII
Trong m{t phing tga dQ Oxy cho tam gi6c
hai dinh B,C lAn lugt tr6n hai tluo'ng thing
tqa d0 trtlc ttn cila lam giitc ABC .
ABC c6 dinh A(Zfi)treng ta; c(OJ+,d
dr:x+y+5=0 vit dr:x+2y*7=0 Tim 2.0
t
Xet B(h;-1, -5) e d, C (t -Zc;c)e d" Theo cdng tlrrl'c trong r6m ta c6:
{e{e{ 1.3-r-5+c=3.2 lc-b=8 lc=-7
Do cto 3 =(-7;Z), C = (S;t)
Cgi /1(r";"yr)ld trpc tdrn cira tam gi6c ABC .Taco
';7,
= (* o * 2; yo - r,';'rt = (ro + 7 ; I o -.- z),W = ( t z; - r),'ib = 1z; - z1
"l'iL tinh ch6t truc tdm
r LlLnt Ll"I'l t ,
= I!r# trn: : * tlif *;l';[ -']) =i * I',1; - i;,==',',-(' ; v ) =8 4)
vi' rrLrc ' t.t,' H(6J'i&.1 (21'7 )
0.2s 0.25 0.25 0.25
0.2-s
0.25 0.25 0.25
cho hinh clrop S.ABCD co d6y AIICD ld hinh vudng canh 2o c6c di6m
A'{,N,P,Othil ttt' nim tr0n circ c4nh AB,BC,DA,SCsao cho:
.41'{ = 2ll47, 4P = ZPD, NB = NC, 4,\Q =,9c cho ,!l vudng g6c v6'i rn{t phing d6y vd
goc giil'a hai rndt phing (SAC)va (anCn)bdng 600.
l Tfnh di€n tich tam gidc MNPvit, khoirng c6ch tir di€rn Qd0n m4t phing
(urvr)
2 T(nh dO ddi doan vudns c6c MP vd SC .
2.4
IV
C
'fir gii thi€t suy ra:
Sool,ri, = Srr,r, -Sn,,o* -So*, -S*.rn R
i I 4a 4a I 2a t(2a ) - toaz /i\\
=4o ,??": l -+al)o 1: : e li\\ /i\\
Ta c6 (SBl) t nc + SBA= 600Id g6c giira hai m4t phi{ i iO \
(sBC) va (ttnco)=.TBr = 6o0 I i , 'l\ \
= rar 60o , = a AB (O,e,wn)) _,t l{ * sA=Zalj (o.(,a.nur)) ec 3 I I :i I \', \ I i' ,' i I \',, \ I \t, \
Matl<hr{c: \-' '1 - \4\ /'' =-, i , I \,', \
rvr(r.r\rraL';6@il- sr =n=ol ,- i I V" \
=rt(e,(uur'))=]sa=iryJi'W
f , '., |4li"" "\,
/,
l {.:',,,, ,,,,., ' 1, ''t''' \
N
0.2s 0.25
0.25
0.25
Trang 5Goi / = A[p olC Dlrng // r sC (l), (/ e SC).
Ta co MP / I BD = MP L AC tlit MP L SA* MP f (S/C) :> Mp -L IJ (Z)
"l"ir (t),(Z)= l/ Id ctoqn vudng goc chung crta Mp vd SC
Mdt l<hirc:
)-i.zalz.taltJ
ffi
.h/_ IJ SA IJ
stnJL.4-_=_e.:_= rc sc ')
:-.AC
-1
Clro c6c s6 thLlc duo:rg x,1t,z lhba rr:rfin cli
, ,.; - 4x+2v+z x-v z
cua bten tlluc /'= Zxy+xz 2 '
4
Nguy€n Thi Hanh
-q'./fu
4s
Ll clLrq.c ]!qt qud tu'ong trr.
T6 tru6'ng chu5,Qn rndn thim tlinh
4 I ix 5 ( 4 5-x\ (t \ +_+ _ _l _+ l+l _+x l s
'r'rreo BDr c6-sita co r ) ,E+.2x[1 -]=+-i=i
Dirr cling rhflc xay ,u Lni {t = ] - c ( x =1
lx +2y * z = 5 t
\r, +, = 4(*'!'t t o)
(-.
vdy rnin ,=Lo{;;l ,=4(*,y,2>o)
Luu y: co thc clirng ilao hdm: X6r f (*)=+x{s-.x) x(s-x) a 'z a - 5-x4
ki€n: x+ 2y + z = 5 Tim gi6 tri nh6
x e (0;5)ta cfing
l4nh I'udng, ngdy I9 thdng 07 ndm 201I Girto vi€n ra tld vi lgp ttr{p rin
BiliHAi Quang
0.25 0.25 0.25
0.25
0.25
0.25 0.25
0.25