Tinh th6 tich hinh ch6p S.ABCD... Do d6 ASAD dAu, cqnh a' Theo giA thirit a I sD =t o cat seo theo giao tuy6n ld cluong cao AH cfn NAD.
Trang 1TRIIGNG TF{PT'EAO DUY T{'
CAU nffi:
DE s'Egg g'EgEI&69 F{QC LAN TE{ti\rg { s4/2s9e}
na0rq, ToAN KE{OI A
Thoi gian: 180 phfi; kkhrcg kA thdi gian phtit di.
A pHAN ctrtil{G DANr{ crro rAr cA qAc lni snm:
Cffu I:
Cho hdnr sd Y = xo -Znt'x'+1'
1) Kh&o s6t vd ve dO thihdm sd khi.m = 1'
2) Tim m Ae aO tfrihdm sii c6 ba di€m cuc tri ld ba dinh ctian'rot tam gidc vudng cin.
CSU trI:
2) Crai phuong trinh: 2(1+cosx)(cot'x+1)=*fi*
Tfnhtictrphanl : |r#
,/(;r + t)' (3x +t)
Ciu IV: '
Clro hinh ch6p S.ABCD c6 tlSy ABCD ld nua lpc gi6c ii€u v6i AD = 2a, AB = BC : CD: q dudng cao SO : ar[i vdi O ld trLrng diiim cria AD Tinh th6 tich hinh ch6p S.ABCD Hdy xAc dinh thiOt
di0n do m4t phing qua A vd vu6ng goc vdi SD cat hinh ch6p'
UAU V:
IF-rl'-3.t-& < o
Tim k ae rre uat phuo:rg rrinh sau co nghiem:
1 I,""- ,, *!loe" (x _i), < l.
lZu!3""/
ts I,I{Aid R.IENG (Tt{f S1NH C1iai fXIqC LAM MQT TR6NG E{AI PE{A-I{ ,q HOAC E)
A TF{EO AT{UOT{G TR,}NF{ CO BAN
"ttu Y-;r,g khdng gian v6i hQ toa riQ D0c6c vuong g6c oxyz cho hai duong thing:
a, ,{t - a3 -o
^
o
ua d,,l*$! ,' ;o
1) Tirl a dA hai d'udng thing dr vd dzclt nhau' 2) V6;
":i, "iei pfru"cnig trinh m4t phing
(P) chria dudng thang dz vdL song song v6i
oluoirg thEng,dr Tinh khoing c6ch gita d1 vi dz khi a : 2'
CAU VII a:
Tinr t4p ho.p c6c digm brgu diSn s6 phuc 2z+ -? - i, oii5i rang lSz + ilt < z.Z +9'
Trang 2B.T'F{Eo cFf[toNG g'RiF{Fx roAwc cao
d-4rr 1./T h
l) Trong m[t phing voi hQ tqa dQ DAcdc vudng g6c Oxy cho dudng thing d: x - y + I : 0
vd duong tron (C): x' + 1,' +2x-4y= 0 Tim tga dQ diOm M thuQc ituong thing d mi qira d6 ta ke duoc hai duong thing tii5p xric vcyi ducrng trdn (C) t4i A vd B sao cho g6c AMB bing 600
-
ilTiong kh6ng gian v6i hQ iqadQ DAc6c vu6ng g6c Oxyz cho tludng thing:
()"-)rr-z+l-S
a,1t*-^"-^" *l=:
vdrmdtcAu(S): x'+y'+?'*4x-6y*m=0 Timmdtiducrngthingdcftmgt lx+2y-22-4 0
"au iS) t4i hai ctidrn M, N sao cho khoing cdch gita hai di6m d6 beng 9'
CAu Vntr b:
Giisir x,y,zldbas0ducrngth6am6nx+y- z:-l.Timgi6tr!lcmnhAtcriabi6uthric:
P ' = - (x+yz)(t+rr)("*ry)' *tYt
-Gi6m thicoi thi khdng giii thich gi
Trang 3th€m TR.EI{FNG TE{PT PAO PTTY TtI
BAp AN - TI{ANG BIEM THI THU sAI x{QC tAN vY' Q'4184t281'1}
rvlOrq : Toin, xulii ^l
Cdu/f
CAu 1
2,00
1,00
1
Khi m : t hdm s6 tro thdnh Y: x* - 2{ +1
- Tpp x6c dinh : R
- Chi6u bi6n thi€n :
+Tac6 IY':4x3 -4x,Y'=0(i, ' 1 J ["=O
[x=tl.
+Hdms6AOngbi6ntr€nc6ckhoang(-1 ;0) u ( i; +oo)'
+ Hdm s6 nghich bitin trOn c6c khoang (-m; -t) u (0;1) '
- Cpc tri :
+ Hdm sO dAt c6c cgc hi t4i x:0, yco: y(0): 1.
+ Hdm s6 d4 cpc ti6u ,u, {t = -1' tcr = l!-tl= o.
[x = 1, yrr = y(l) = 0
- C6c gi6i h4n t4i vd cuc
',lg = +co vir
,lill = *-.
- Bang bi€n thiOn :
ll-I I , l*E I \0,/ ,/ '\ \ ,/**
I
I
I DO thi:
-Ei6mu5n: y" =17x2*+=4(3x'-1) =$4=11 t',
' o'
L'=-E=uY= g
:> Hai di€m u6n ld: ( t +\ ( -t 4)
u'=[E't
)tY'=[r'oJ
Trang 4Ta c6: l' = 4x3 -4m2 x OC nam s6 c6 ba cgc tri € y'=0 c6 banghiQm ph6n biQt
<=> 4x' - 4m2 x= 0 <=> 4x(x' - *')=Q q=2
l:: : -,= 0 c6 ba nghiQm ph6n biQt +) x2
-m2 = 0 c6 hai nghiQm phAn biQt + 0 € m* 0.
Khi d6 c6 ba cgc tri le A (0;1) ; B ( m; -ma+l ); C ( -m; -mo+l ) Ta c6:
ta(m;-mo), *(-*t-*o).rath6y l*l=lZfl oe A ABC vuong cdn tai A <+
AB,AC =A
(l
(1
-l * 1ly' -14 = x -2 Q)
-2y) (x- y) : 0 Do diAu kiQn: x2 -2y -l > Q q=2 x' -2y>1.
y Thay vno (2) 1u
"6' 2'[; -21s -1+ V"t - 14 = x -2 :>
I > o lr' -2r-l> o
<=>{ ^ =)x'-2x-1=0.
<x-2 lx" -2x-l<0
:> He phuong trinh c6 hai nghiQm
2",1x'-2
)(+(x'
)(+x:
2^
x
-zx-1f ,'-14
2 (1 + cos x) (cot' x * l) = jlT:!
' cos.tr+slnx Diau kicn, ' Itin* * 0
[cosx+sinx*0
(*)
( **
)
Trang 5_ sinx_l .=r _2 sinx _1
<=> 2cos x +2 sin x = sinx +cosx -sin xcos x -i
<=> sinx +cosx+sin xcos x+l =0 D?t t : sinx + cosx
, *r' -t *l = 0 <=> tz +Zt +l= 0 <=> / =-1 TathAy nghiQm t: sinx+ cosx : -1 *0 n0n 2
th6a mdn diAu kien (**) :> 'l-zrrn(**LoJ= -t
( a) -J1-=ri"f-1'l o l" = 2+ukv (Thdamdn(8)visinx:-l)
<+sin[x*
o)=-T=sm[-?J "
l" =tr+Zkn (Lo4ivi sinx:0 n6nvipham("))
=.,E sin [, *Ll,-.,D < t < Ji.=> sinx.cos * =']' Khi d6 ta c6 phucrng finh:
Y!r'r[^ ' +J'
t:J'(".,)ffi
Datt:.ffi => dt -;(#) *=+('-*)
" =
I
Trang 6v-.^^,- ,\ADl.l) =!gp=
3
Di0n tich nira lqc gi6c ddu ABCD ld S = { o' AD :2a.
A
T
{ tDcrn vithc tich.)
2
Ta c6: gp = '!j6 a 6Y =
OA=OB=OC:OD=a)
Vi vu - u ! => Sl =58 = SC: SD =2a Do d6 ASAD dAu, cqnh a' Theo
giA thirit (a) I sD =t (o) cat (seo) theo giao tuy6n ld cluong cao AH cfn NAD.
Vi SO I (ABCD) n6n (a) cit lenCO; theo giao tuy6n h ductng thturg vuong g6c (SAD)
tai A vay thi6t dien can opng x6c dlnh nhu sau: Trong m[t phang (ABCD) dlmg dubng
thing vu6ng g6c vdi AD t4i A, ciit BC tpi F, cit CD t4i E, EH giao SC tpi J, FJ giao SB tpi
N.:> AHJN le thi6t diQn cAn dpg.
0,50
CAU
v
1,00
ll"-tl'-3x-ft<o (1)
f;'"*" ,' *ilog, (x-l)' < l Q)
Khix> 1(2) (}logrx+logr(x-1)(1<=)x(x-1) 12 <=>xz -x-2(0<=> -1<x<2'
Vi x> 1:> 1 <x<2 B6tphuongtrinh(1) e (t-t)'-:;<ft'D?t(x)=(x-i)3-3x'
f (x): 3(x-1)2 - 3 :3x (x-Z).
V6i 1< x12:)f (x)= 3x(x- 2) <0=>Hdm sO f1x; nghichbii5nh6n (1;21
:tT,ll f @)= f(2)=-s EehQc6nghiemtaphiic6k> Tli/t"l =-5 viyk>-5 thi hc
co nghi6m
1,00
I
dr vi dz c6t nhau € hO sau c6 nghiQm duy nhit e
(x-az-a=0
)r',+r =o
lax+3y-3=0
Khri x, y, ztaduo.c a2 -3a+12: 0 A = 9-48 < 0
tOn t?i gi6 tri ctra a d0 dr cit dz.
Phucrng hinh v0 nghiQm VflY kh6ng
2
Vdi a:2 ta c6
(x-22-2=0 .lzx+3y-3=0
o"
t, -z+t=o to'\*-32-6=o Tac6:
q =lz;t;tl, M r e d, =) M ,(2; -! 0) ,l* =lttt;l1;Mz € d, =2 *rr(O;t;-z).
I,00
Trang 73x + y -l -7 (z+2):0 $ 3x + Y
d( d' ; d' ) : d( Mr/e) ) (vi (P)//
Mat phing (P) chira d2 vd // d1 :> V,,q)= (:;t;-z):> phuong trinh (p) :
o;1;-2) l5:0.
lt.z -t- 7.0 - I sl l o 1P.,t rr ri1.1
\- /
Jl, +12 +72 J59
y,=
lr,(
-'72
-d2 vd
t di€m bi6u di6n ld N(x; Y)' Do
z' = 22+ 3 - I :) ct - lt -t)tu= |(l*r)'(1)' TiI gin thi€t ltz + il' < zz +9'
+r (:a)' +(:a+ 1)' o' +bz +9.(2).Thay c6c gi6tri 4 b tu (1) va (2) ta c6: Tpp hqp c6c
di€m bi6u di6n s0 phirc z' ld hinh trdn tdm ,[',*), '\-' + )' ban kinh R =+ 4 '
Vitit l4i (C) dudi dang ( x+i)2 + (y-2)' :5 :> (C) c6 tim I Gi;2) b6n kinh R=.F' Theo gi6
thiiSt goc MAB:600:> g6c AMI = 300:> MI = 2iA:2R:216 VaV M thuQc dudng trdn
tAm I b5n kinh 2 6co phucrng trinh: (x+l)' * (y-2)' : QJt)z = 20' Do M e (d)n6n tga dQ
crlra M th6a mdn nghiQm tem cua cta hQ: nu't(" L -\2 <=> {
+ l)' + (y -2)' =2a ''l(' + r)' + (x -t)' = zg
:> *2 :9 :) x: t 3 VOi x:3:) y = 4 V6i x: -3 :) y:'2.
Viy c6 hai di6m M cAntim ld Mr( 3;4); Mz (3; -2)'
(d):x-y+l:0
Trang 8viet tai (S): (x +z)' +(t-3)2 + z' =r3
| (-2;3;0); R: ll3-* = IN MN : 9
-m.
n€n
@iAu kiQn m< 13.) M4t ciu (S).An ti* c6 tim
FrN = 29 .:,
IH
d.
(d):
v-[2
Ix
I
-t
0
0'
,(o
D[t x: t:>
;1;-r) ed.d(t t
65
9=)m=-? (
2x
x+
2
IH=
CAU
VIIb
1,00
,'y'
a (x+1)3 (r+i)''
Md x +1 =L*Ll = riff => (r + r)' ,-! *' .Tucrng to (v * 1)' ,!u,=>pcf3)' - 4/ \27)
Ddu ":t'xdv ' ra € {lz=5 Maxnr:
[;)'
I,00