rct rsr KSCL THr DAr Hoc xAu zltz rAx rH{I1on rnr ivr0x, ioAN, xu6r u

Thdi gian ldm bdi : IB0 philt, kh6ng k€ thdt gian giao diPA tU 96*' 02 trang

rHAN cHUNG cno rAr cA THi srNH (7,0 tliam), 2x-I /-

CAu I. (2,0 ctiiim) Cho hdm s6: y -'4 (H)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; * I4

2. Giei h9 phucrng trinh:ft+x*xy=5yft*"'y'-5y'

tl" *7 - J;+ 3Ciu 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,lv6i 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: 2

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 ChuinCfiu 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 thdngdi6m 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

UiCt n9 s5 cria si5 h4ng thf 3 bing 36.

B. Theo chutrng trinh Nfing caoCdu 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 .

I

thing AB,BC lAn lugt li:.qc, bi6t ring ,tc di qua

xu trong khai tri6n cta nhi thfc [x'Jr

* +)" ,

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Elfp (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 dientich tam gi6c AoB nho nh6t.

C6u VII. b. (1,0 ifiAm) Tim x bi€t rang trong khai tritin "tu(J7*-J--)'r )., , \

s6 cta 3 s6 hpng cu6i bdng 2?,t6ngc6c s6 hang thf 3 vd thri 5 bdng 135.

, t6ng c6c hq

2

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EAp AN-IHANc orfmxV 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 Di6mI

2rA-.tdlem

I

!-.*Q,Q*4i?r0 - -1r T{p x6c dinh : D: m t {t}2. Su bi6n thi6n

Ia) 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)

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b) Cgc tri: Hdm sd kh6ng c6 cgc trfc) 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

:+1" t+l-

ducrngthing x=1

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d) Bing bi6n thi6nIX l-oo

-tr

I

ll

_ll.-

+oo

v

2

0,25

thiaJ.

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?,-.$.'.0*f-Q@.X6t phucmg trinh holrnh dQ giao di€m cria dudng thdng dd cho v6i-, 2x-l fx+l

dd thi @): -=mx-

m+ 2 (1) e {x-l ' l**'-2mx+m-1=0(2) l-0,25

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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 €m>0.

Ysi mr0, (2) c6 hai nghiQm phdn biQt, gi6 sri x,,x, .

' lY'=ffixt-m+2Df;t A(x.yr1, B(xr,!z), ta c6: j ".'

lYr=ffixz-m+2Khi d6 AE =(xr-\)'+nf (xr-x,)' --(*,-*r)'("f +l)

=[(", + \)' - +4x,]1ni + t1

Theo Viet:

fxr+xr-z 1 r_I *-1= AB' =4(m+l)= 8,Ym>0 =+ AB>2J2,Ym>0lxrx, = tnlmMinAB =2J, khi ln = 1.

Yatv m= I ldr niJoi .An-,irn.

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0,25

0,25

CflU II2ra

drem

l. (1,0 didm)

sin3 x(sinx + cosx) + cos'(cosx - sinx) -

<> sino x * cos4 x + sin3 Jccos.r - cos' xsinx

aJ

4.tJ

4 0,25

I - 2sin' xcos' x - sinxcosx(cos' x - sint; : l

I - !rin' 2" - lsin2xcos 2* =1113

-:n- cos4x) - lsin 4x = I4' 4 4

e 0,25

<+ sin4x - cos 4x - 0e sin(4x - 4) = O4'

7t _7r<+x--:-+k-. keZ16 4'

V4y phucmg trinh c6 nghiQm 1A x: * + t:164 ,keZ.0,5

0,25

4

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/iI

Ddt: x+-=Sv

(t- lx+-=-)lS=-5 IVdil- -=1 ,Y ,hQvOnghiQmLP = lo 1".! - toLv

[.- 1 -.[s=3 l"*;-t rx-2 lx=tv6t

t; =;'l*r'=, *t; =i " b--;LY1

VQy h€ phucrng trinh c6 nghiQm 1a.: (x;y) - (2;I) vi (x;y) = (t;t).

4,25

0,5

Cfiu

ilI1'0

tli6m

1=limx+l

<tx+?-.,6+3x'-3x+2

:,1,t+? -2- J71j +z= limx+l 'x' *3x +2

,. (:,1.*t-z G;3-2)-

rlrtl I - ---:-- |x-'r ["' - 3x+2 x' -3x+2 )

0,5

1

60,5

Qnu

ryI

1'0

Vi SAL(ABD)*SCA= fBC L(SAB)+&-300

0,25

Gqi S,4 =x (x > 0). A&4C vu6ng t4i A, na<i.l

I

I

.!.l

I

j5

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c6 SC) - 450 n}n AC = SA =r vd ,SC - xJinsaS ".'*::: 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

t17a ! z (dvtt).-\/J

VOy, Vr.nuro = l* "

'"o'=0,25

CAu V100

(Irenr

l" 1

!=x*{"'+a tren (o;+.o)

,-- IL&7

,.f -1,

X-)/ -L' f-----:

2^lx'* t

Vxtf-'!'=oo{ -zx=2^lr'+!xYx

<+ 1- 2xt = 2x' ^lx' + I eI- 2xtVxh-z*'> o

<+1, .r2 gle=f(t-zx')'=4*'(r'+1)

1r2

X6t hlm s6

Tir b&ng bi6n thi6n cho k6t qu6:

Minv-2 khi "=f(0;+o)" 2

0,5

0,5

CfiuI

VI.al

2rAi

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

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l,\,\l _|,\ilcos-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.

7

?'-&-o-gjEulng trdn (C) c6 t6m

MI -zJi > z= R= M

1(1;3) vd b6n kinh n = Z.

nim ngodi ducrng tron (C).

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Ggi H(x;y). X6t thdy t, M, H thdng hdrrg n€n7fr(a;-2)'1 v-3phuong u6i Ifr(x -1;y- 3) e + = E€) x - Zy - -s

cirng

ta c6

0,5Lpi 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:

f _ _ r

I:-r,= _rol" -t

= I/[,l,li']l2x+!=3 1.._13 --\.5'5)l"-T

i

Y$y H(+,9.]\) 5 /

0,25

Ta c5

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o

tr0

tem:ZCI*

k=0[,'v;'- :)' =fc: (*, J])r[*)' " n 5k

=lClxT.*3k-3nk=0

1lk-6n2*

H0 s6 cria s6 heng thf 3 ld 36, ta ducyc Cl, =3A a n- 9

ek=6.n66Lnx

^ .L .1Ik-54f U YeU CaU Dal toan, ta Co

Z = 6

Vfly s6 hang chria xu trong khai triOn ld

0,5

'CiuVI.b

2'0-.i(Irem

(2x-3v-2=atop 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{^ ' - :+B(a;z)[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

l$i d6 Snou =;OA.OB = 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

lI

I

I

I

I

I

I.I

Jrl I,t./t

suy ra Srou fro nhAt khi vd chi

co:

S yi = tTlx,yol = lx,yol < 3

9v',(z)92a

2"0rdn y6u cdu bdi to6n liJt), *,(*,rr),*^(-+,-

0,5

0,5

CAU

VII.b1r0

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

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0,5

8

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--f6"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.

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