THI TOÁN L6 2011(SPHN)

rm rruu'DAr Hgc LAN vr Nann zorr

Bfr

TRU'ONG EHSP HA NQI

TRUONG THPT CTTUVBN . DHSF

Mdn

thi : TOAN

Thdi gian tdm bdi : I 80 phfit, khong

ki thdi

gian phdt dd

'
diem)
Cho hdm so Y : x3-3mx2 + (nr- l)x+ 2

C6u 1.

(2,0

1. Tim rr d6 hdm sO d3t cgc ti6u tai x = 2. KhAo s6t sq' biOn thi€n

vi

a

vE dO thi (C) cira hirm s6 ftng

vdi gi6 tri cria m tirn dugc.
2. BiQn ludn theo ftsO nghiQm c[ra phuong trinh

:

)-_k
x'-2x-2=
^ L^ ' :-

lx-11

Ciu

'
'.

2. (2,0 di6m)

l. Ciei phLLong trinh : 6sinx - 2cos3x

2

ciai

hQ

phuong

tri'h

:

= 5sin2x.cosx

F:r;.r;j;

{_r_

'..

Irrl,=

'

nr.

I

.'

,

.:

CAu 3. (1,0 di|m )

l.€$x-

Tinh tich phan r =
-a
CAu 4. (1,0 di€m )
Cho tam gi6c cdn MBC c6

a..
.

EMe

:

l20ovd duo'ng cao MH

:

a^12. Trdn duong

thing vudng g6c

vdi m{t phing (MBC) tai M l6y hai di€m A vi D vC hai phia cria di6m M, sao cho ABC le tarn giilc
dAu vd DBC le tam gi6c vu6ng cdn t4i D. Tfnh th€ tfch kh6i cau ngoqi ti€p

til

diQn ABCD.

Cffu 5. (1,0 diem )
Cho c6c sd duong a, b, c thay ddi ludn th6a m6n a + b

* c = 1. Chri'ng

abbcca3
*'b+ca
+

.+"b

r*b.

2

minh ring

:

-

Cdu 6. (2,0 diim )

l. Trong m4t phdng Oxy, cho hinh chir nhat ABCD
phuongtrinh dud'ngthing AB

ldx- 2y+2

,1
c6 giao cli6m c0a hai duong chdo ld M(;;

O),

= 0 vdAB = 2AD. Tim toa d0 c6cdinh A, B, C,

D

biiit ring dinh A c6 hoinh d6 duo.ng.
2. Trong khdng gian Oxyz, cho mit phing (a)

:

3x + 2y

- z* 4:0

vA di€m

d0 didm N sao cho MN vu6ng g6c vdi (a) clOng thd'i N c6ch dAu gOc toa

M(2;2;0).
dQ

X6c dinh toa

O vd rrrit phing (a).

Cdu 7 . (1,0 diem )

-{3 * i, 22: eosloo- i,sin i
Hiy bidu diSn sd ptiLlc z : (?\t'ou.di cl4ng ctai s6.

Cho cdc

si5

phirc z1:

.

\L2/

tsqr ki6n

ki tlri tfe# Bai

leoc ldm fid-lk 7 sE dwvte fA rh{pn ttAn n*}n, t p , a/K./rni

t

;


EAP AN _ TTIANG DIENI
2011
THI TI{TIDII LAN TH{T SAU - NAM

-f*-.151^* -0,*;i,^irc 1iA,r
l.
(+r v = ? khi d6,t'e,\: 0 c+ m =
rruu tni
. Didu:?:t

ki€n cdn : Gi?t su nam so cl4' t'gu
y' = 0<+x=0hodcx=2'
. EiAukiQnd0,fVeu*: IthiY':3x2=6x:3x(x-2) =

Ta c6-y,

r

ru--

oa_-"

'Fi'
tra "x-: 2 lit dl€m cgc ti€u cfra him
(r'rvtr suv
eeJ 'e
ulltr thiAn
UiallB l.,iAn
l Ll !.,inc
o hdrn c6p hai'
Chri f : C6 ih€ ki6m tra cuc tri bing da'

I

y5i nl 1

11'ti

y = x'

+2' ( Hqc sinh tu ve dO tlii)

-3*

=.; ,
{z drcln) 2.{1,0 elia@"
.^

Dat f(x) = x'

-

3xt

X6t phuong trinh

ra c6 | x-r

qx'?
I

-p

2:1x-1X* -zx-21.

x'-zx-Z=

l(

-k,voix*l

ffi o lx t lix'? -2x-2)

-zx-r):{y.;1.r;1'"l-(x' - Lx- l)ix

Suy ra dii thi c0a hdm s6 v

r(x)

n6u x >
1) =
*f(x)
n6u x (
- 1) =

('*).

1"

1

: I x-l I (*t - 2x-2)tr0n midn R\{l } le

l,a0

1) cira dud'ng thdngY
giao cli€m ( v6'i hodnh dQ giao di€m kh6c
SO nghiQm cira pt(*) bang se'
voi dl thi hiin to
I x-t l1x2 -zx-21'

v:

Tr) d6 thi tr6rr ta slry fa

:

- N€u k<-zthi Pt(*) v6 nghiQm
- Nliu k=-|ho[c k> 0 thi p1(*) c6 2 nghiOra phAn biQt
- N€u -2 < 1'-< 0 thl pt(*) co 'l irghi€m phi"n biQi'

:k


l. ( 1,0 iIiA@ . GiAi phuong trinh ...

l0'

Do gi6 tri x md cosx = 0 khdng ld nghiQm cria phuong trinh, n6n xdt cosx

Khi d6

trinh duoc viiit thdnh

phLro-ng

:

sinx
sinx _l- 5.3.-=
cosx

cosrx

€)

il
'2 ili4m)

<+ (tanx

Z4,O tidm).

Clai

3tanx(l + tan2x)

-

l)(3tan2x

3tanx

-1 = 0 <+3tan3x-2tanx-l =0
+ l) = 6 o tanx: I <+ x=I+kn
,kez.
4

5tanx

, { , J',1,= ? - lx
[(x + 2Y)z - 2(x- Y) = 4L

d6 ta c6 hQ

:

r:-

rac6

III
Xdt hdni

.1

(i,-

sO

de

f1x; = ln(x +

,IWI)

.p

thi f(x)

Urtn

: afr

Do AABC

IV
(1

t,

Jf #

c(i thd

n€n MB = MC =

d€r.r

^/

46T. tr4p

= 4a

+fry.E

-ry1.

0,50

6Qo.

2^l2avdBC:2rl-6a.

n€n AB = AC BC
=

di€nl
=

0,50

itgt x = tant.

=2{6a

Do ADBC vr-rdng cdn ndn DB = DC
y716

T).

fi,*

Tu gi6 thiiit suy ra ABMC cdn tai M+ EffF[ =

Vi MH

0,50

L\

frr-z+ ro(i)=-vFilf . I,ri;#ill
* l, .1f# =fr-#.ff #

1,
= ln[(2
;711 = ln(x +
lf
2
vay t=^[76 +ln[(2+/3XV7-tlt.
Ltut it

0-50

Tfnh tich phdn ..

lf

suy.a

* {;=?

Gz+1 l./3

:--l

Q didm)

="n,

[;J-. {r, _],

Giai he ndy ta dugc hai nghiQm (x, y) = (S,t);

tli6m).

0,50

+ zvl

JT1 20,v= lx+2ylzo tnihetrdthanh {"r:;;,

D{r u=

(1,0

=0

trffiuong trinh .....

HQ pr da cho dusc vir5t thdnh

rir

*

-

0,50

vd gg

:BC+

=

2

=2,13a.

,ffifiT--frfi

= 2a.
1,00

Suyra AD=MA+MD=6a vd
AB2 + BD2 = ADz

"+ TED:90o .
Tuong tu ta cfing c6 frD = 90o .
4_

Vdy m4t cAu ngoai tiOp trl'dien ABCD c6 dLrbng kinh ld AD, n€n V1p = -'irRt = 36nat.
J

2


L. (1,0 aiiiml. Chirng minh ....
Tuongtqr cirngc6

ab

:

tr6' thdnh

a*bc:
+

G+c)G+c)

(a+ b)(a+ c), b + ac= (b + a)(b + c). Khi d6 bat ddngthrlcddcho

bc

ca'-3

+

G;ft+c) *;*"

ab(a+b) +bc(b+c) +ca(c+a)

t1
o
-t4*

z

1

4

(a+b)(b+c)(a+c)

- ?ab9:1
-4
(a+b)(b+c)(a+c)
(a+b)lD+cj(

(a+b)(b+c)(ctqi

zabc

€t n_
r - ;lnlt6*ti1'ta3\= ,14 =--jE-- (a+b)(b+c)(c+a) -

1 ili€m\

\'-'

_ji\-

_

8'

-,\-

MdtkhectheoBDTC6- si tac6

: a +b>2{{b, b+cZ2rffi' c+ aZ 2'lla''

Tird6suyraBDTduqcchfuigminh.EingthricxAyraklria=b=c'
hodnh dQ duong.n€n'a >
ViAthuQcdu'bngth6ng:x-2y+2:U nenA(/a-z;a),\t\) AUUrruatrrtuvuuvrrE!rwl"s(2;'I)'
trung di6m crha AC n€n C(3 -7a; a). Vi BC I .A'B ndn ffi -6 =

Do

N4 ld

Suy ra phuong rrinli BC

:

2x* y + 5a-

6

:

Vi M cfing ld trung di€m cria BD n€n D(Za

0. Do B ld giao cira AB

vi BC n€n B(2 -2E

.2

1'

-

a)'

- l; a'2)

*

l)'
D"
-4)'+(2-2a)2 =zo azaaz -.40a=0
[: 2 =u- z(.via>
^B:t^D
A(2;2),8(-2;0),
= C(4;a) vdD(3;0)'
Thaya:2vitotgad6cacdi6m,tatimdu'-occ6cdinh

*(*

W
2 ttidmJ GqiN(x; y; z), do MN

Ta

c6

+

ON2 = 72

Do do

ON:

d(1V,

3t)2

(u))

,'"
ffi
' -re" 4td = i*a

N rd

[i=-lit;,
\z: _t

+(2+2t)2 +tz vikhoiLngc6chtilN d€n(ci) ld:d(t\

{2+3t)2 +(2+2t)2 +t2:14(t+

l)'?

. 7 1 3,

vey N(-

;;;;1)

(l,O aiiimj. Bitlu di6n s6 phiic .. ..-

2(-* . ir: 2(cos 11 + ;.sinT) vit z2 cos(- fl + ;ti" t- f)
N€n a=ztcos(f *f l*rsin(f *ilt :z(cosff+i.r,nff)

Ta

w!
I

Arcfitl

c6

siv,*
J

zt =

i3)
\a

\L2 /

,

LrtL

13ft
\
/. -.t
= ?':r'cns .2 * i Sin- i = - ^1"

:'

(u)=lja lt+ t

I