THI TOÁN L6 2011(ĐDT HN)

TRIIGNG TF{PT'EAO DUY

A. pHAN
Cffu

DE

T{'

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.

ctrtil{G DANr{ crro rAr cA qAc

lni

snm:

I:
Cho hdnr sd Y = xo

-Znt'x'+1'

thihdm sd khi.m = 1'
gidc vudng cin.
2) Tim m Ae aO tfrihdm sii c6 ba di€m cuc tri ld ba dinh ctian'rot tam
1) Kh&o s6t vd ve dO

CSU trI:

2) Crai phuong trinh:

2(1+cosx)(cot'x+1)=*fi*

CAU nffi:

Tfnhtictrphanl

: |r#

,/(;r + t)' (3x

Ciu

IV:

+t)

'

:

CD: q
Clro hinh ch6p S.ABCD c6 tlSy ABCD ld nua lpc gi6c ii€u v6i AD = 2a, AB = BC
Hdy xAc dinh thiOt
dudng cao SO ar[i vdi O ld trLrng diiim cria AD. Tinh th6 tich hinh ch6p S.ABCD.
di0n do m4t phing qua A vd vu6ng goc vdi SD cat hinh ch6p'

:

UAU V:

IF-rl'-3.t-&
Tim k ae rre uat phuo:rg rrinh sau co nghiem:
_i), < l.
1 I,""- ,, *!loe" (x
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:

o

a3
,{t - -o ^ ua d,,l*$!
,' ;o
"''ly-z+1=0
'[.x-32-6:0

a,

thing dr vd dzclt nhau'
pfru"cnig trinh m4t phing (P) chria dudng thang
2) V6;
"iei
:
oluoirg thEng,dr. Tinh khoing c6ch gita d1 vi dz khi a 2'
1)

Tirl

a dA hai d'udng

":i,

CAU

VII

dz

vdL

a:

Tinr t4p ho.p c6c digm brgu diSn s6 phuc 2z+

-?

-

i, oii5i rang

lSz

+ ilt < z.Z +9'

song song v6i


B.T'F{Eo cFf[toNG g'RiF{Fx roAwc
d-4rr

1./T

cao

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*l=:
vdrmdtcAu(S): x'+y'+?'*4x-6y*m=0. Timmdtiducrngthingdcftmgt
a,1t*-^"-^"

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:
*tYt

' =- (x+yz)(t+rr)("*ry)'

P

------------Gi6m thicoi thi khdng giii thich gi th€m----


TR.EI{FNG TE{PT PAO PTTY

BAp AN

TtI

- TI{ANG BIEM THI THU sAI x{QC tAN vY' Q'4184t281'1}
rvlOrq :

PiOm

NQi dung cho tli€m

Cdu/f
CAu

Toin, xulii ^l

2,00

1

Khi m : t hdm s6 tro thdnh
- Tpp x6c dinh : R.
- Chi6u bi6n thi€n

Y:

x*

- 2{

+1.

:

+Tac6 IY':4x3 -4x,Y'=0(i,
'--1 J
["=O
[x=tl.
+Hdms6AOngbi6ntr€nc6ckhoang(-1

;0) u ( i;

+ Hdm s6 nghich bitin trOn c6c khoang (-m;
- Cpc

tri

+oo)'

-t) u (0;1) '

:

+ Hdm

sO dAt c6c

cgc hi t4i

+ Hdm s6 d4 cpc ti6u ,u,

- C6c gi6i h4n t4i vd cuc

- Bang bi€n thiOn

x:0, yco: y(0):

1.

o.
= -1' tcr =
l!-tl=
[x = 1, yrr. = y(l) = 0

{t

',lg =

+co vir

,lill = *-.

:

1

I
ll-

v'!

o + o - o

I' \0,/g'l

I

I

-l

, l*E-r

II

0

x l--

+co

+

,/ '\ \ ,/**
^A'

I

I . DO thi:
I

-Ei6mu5n: y" =17x2*+=4(3x'-1)

[r=+ =rr=!
=$4=11 t',
'

L'=-E=uY=
:>

Hai di€m u6n

( t +\ -- ( -t 4)
ld:u'=[E't
)tY'=[r'oJ

o'
g

1,00


Ta c6:

l'

<=> 4x'

=

-

-4m2 x. OC nam s6 c6 ba cgc tri

4x3

4m2x= 0

<=> 4x(x' - *')=Q q=2

m2 = 0 c6 hai nghiQm phAn biQt + 0

€

m*

€

y'=0 c6 banghiQm ph6n biQt

l:: : -,=

0.

c6

ba nghiQm ph6n biQt

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

=

2",1x'-2

xz

-l

y

+2xy

r--:-

* 1ly' -14 = x -2.

(l )(+(x' -2y) (x(1

(1)

y)

:

,^.

DiAu kiQn:

Q)

0. Do diAu kiQn: x2

-2y

-l

>

Q

q=2

x' -Zy >1.

x' -2y>1.

)(+x: y. Thay vno (2) 1u "6' 2'[; -21s -1+ V"t - 14 = x -2 :>

x2^-zx- I > o
o
lr'
<=>{ ^-2r-l> =)x'-2x-1=0.
1f ,'-14 lx" -2x-l<0

:>

He phuong trinh c6 hai nghiQm

2 (1 + cos

x) (cot' x *

l)

'

=

*0
jlT:!
. Diau kicn,
Itin*
'
cos.tr+slnx

[cosx+sinx*0

(*)
( ** )

<+

+)

x2

-


--rt"r,

_ sinx_l

-rin*+cosJc

.=r _2__

1-cos'x cosx+sinx 1-cosx

-sin xcos x -i
:
<=> sinx +cosx+sin xcos x+l =0. D?t t sinx + cosx
t2-1
/
-\
Khi
sinx.cos * =']'

t
<
<
Ji.=>
sin [, *Ll,-.,D
=.,E
Y!r'r[^
' +J'

sinx _1
Sinx+cosJf

<=> 2cos x +2 sin x = sinx +cosx

,

*r' -t *l
2

= 0 <=> tz +Zt

th6a mdn diAu

kien (**)

+l=

:>

0 <=> /

=-1.

'l-zrrn(**LoJ=

d6 ta c6 phucrng

TathAy nghiQm

finh:

t: sinx+ cosx : -1 *0 n0n

-t
(Thdamdn(8)visinx:-l)

( a) -J1-=ri"f-1'l o l" =--2+ukv
<+sin[x*
o)=-T=sm[-?J "
(Lo4ivi sinx:0 n6nvipham("))
l" =tr+Zkn

pl
t:J'(".,)ffi
dx

Datt:.ffi

=> dt

-;(#) *=+('-*) "

=

I

--


Di0n tich nira lqc gi6c ddu ABCD ld S =

v-.^^,,\ADl.l) =!gp=
3

Ta c6: gp =

tDcrn

{

{
A

o' . AD

:2a.

T

vithc tich.)

'!j6 a 6Y =
vu

- ! Sl
ViOA=OB=OC:OD=a)
=58 = SC: SD =2a. Do d6 ASAD dAu, cqnh a' Theo
=>
so L(ABCD)
u

)

giA

2

thirit (a)

Vi SO

I

I sD =t (o) cat (seo) theo giao tuy6n ld cluong cao AH cfn NAD.

(ABCD) n6n

(a) cit lenCO; theo giao tuy6n h ductng

0,50

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.

CAU

1,00

v

ll"-tl'-3x-ft{,
1
f;'"*"

(1)

-

,' *ilog, (x-l)' < l.

Di€ukiQn:x-l

>O:)x)l'

Q)

Khix> 1(2) (}logrx+logr(x-1)(1<=)x(x-1) 12
Vi x> 1:> 1
-x-2(0<=> -1
e (t-t)'-:;
f (x): 3(x-1)2 - 3 :3x (x-Z).
V6i 1< x12:)f (x)= 3x(x- 2) <0=>Hdm

:tT,ll f @)= f(2)=-s.

<=>xz

sO

1,00

f1x; nghichbii5nh6n (1;21

EehQc6nghiemtaphiic6k>

Tli/t"l =-5. viyk>-5

thi hc

co nghi6m.
2,00

Ciu VIa

(x-az-a=0
dr

vi

dz c6t nhau

€

hO sau c6 nghiQm duy

nhit

e )r',+r

=o

lax+3y-3=0

lr-u-6

I

Khri x, y, ztaduo.c a2 -3a+12: 0. A = 9-48 < 0

tOn t?i gi6

tri ctra a d0 dr cit

q

I,00

hinh v0 nghiQm. VflY kh6ng

dz.

(x-22-2=0 .lzx+3y-3=0
to'\*-32-6=o . Tac6:
t, -z+t=o
=lz;t;tl, M r e d, =) M ,(2; -! 0) ,l* =lttt;l1;Mz € d, =2 *rr(O;t;-z).

Vdi a:2 ta c6
2

Phucrng

=o

o"

.

I,00

Mat phing (P) chira
3x + y

-l

-7

d2 vd

//

(z+2):0 $

d1

:>

y,= V,,q)=

(:;t;-z):>

phuong trinh (p)

:

lr,( o;1;-2)

3x +

Y

-'72

- l5:0.

d( d' ; d' ) : d( Mr/e) ) (vi (P)// d2 vd 1P.,t
\- / rr ri1.1

lt.z

-t-

Jl,

7.0

+12

-

I

sl

+72

lo

J59

t di€m bi6u di6n ld N(x; Y)' Do

z' = 22+

3

-

I

:) lt.
ct

+r (:a)' +(:a+ 1)' . o'
di€m bi6u di6n s0 phirc

-t)tu=

|(l*r)'(1)'

TiI gin thi€t ltz + il' < zz +9'

+bz +9.(2).Thay c6c gi6tri

z'

ld hinh trdn tdm

4

b tu (1) va (2) ta c6: Tpp hqp c6c

,[',*), ban kinh R =+4
'\-'
+ )'

'

(d):x-y+l:0

Vitit l4i (C) dudi dang ( x+i)2 + (y-2)'
thiiSt goc

MAB:600:>

tAm I b5n kinh

2..6co

g6c

:5 :> (C) c6 tim I Gi;2) b6n kinh R=.F' Theo gi6

AMI = 300:> MI = 2iA:2R:216. VaV M

phucrng trinh:

(x+l)' * (y-2)'

:

QJt)z

= 20' Do

thuQc dudng trdn

M e (d)n6n

(x-v+l=0
lY=x+\
<=> {
nu't("
cta hQ:
tem cua
crlra M th6a mdn nghiQm
L + -\2
l)' + (y -2)' =2a--''l(' + r)' + (x -t)' = zg
:>

*2

Viy

:9 :) x: t

c6 hai di6m

3. VOi

x:3:)

y = 4. V6i

x:

-3

M cAntim ld Mr( 3;4); Mz (3; -2)'

:) y:'2.

tga

dQ


viet tai (S): (x +z)' +(t-3)2
| (-2;3;0);

R: ll3-*

+

z' =r3 -m.

= IN. MN

:

9 n€n FrN

=

fn
l---+

,3_m_t;l

IH

@iAu kiQn m< 13.) M4t

29
4

ciu (S).An

ti*

c6

tim

.:,
1,00

@i6u kiQn m < -3) ld khoing c6ch tu I dtin

d.

(d):

2x

0

[2
Ixx+

0'

I

v- -t
2

D[t

x: t:>

,(o ;1;-r) ed.d(t t
65

9=)m=-?

IH=

(

CAU

1,00

VIIb

,'y'
a (x+1)3 (r+i)''
I,00

Md x +1

-,

Ddu

=L*Ll = riff

":t'xdv' ra €

=>

(r + r)'

,-!

fx=y-Z . Maxnr:

{

lz=5

*' .Tucrng to (v *

[;)'

1)'

,!u,=>pcf3)'.
4/

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