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b) Vi€t phirong trinh tham s5 cua dirong thang I:i di qua trong tam G cua tam giac ABC va vuong. g6e voi m~t phang ( a ). Tim toa dQ giao diem.[r]
(1)soGD&DT BEN TRE BE KIEM TRA HOC KY II LOP 12 NAM HOC 2011-2012
Mdn Toan Giao due trung hoc ph6 thong
( Thili gian film hili 150 philt, khong ki thiligian giao ttJ) I pHAN BAT BU<)C (7,0 c1iim)
Cau1 (2,5 c1dm) .
11
, 2
f J =2fx2 +Fx+l
dx a)Tinh cac tich phan sau: I= sinx.e'?" dx va
o . x
b) Tinh dien tieh hinh phang gioi han boi d6 thi (C) cua ham s5 y =x3 -x va true Ox
Cau (2,0c1dm)
a) Giai plnrong trinh Z4 - Z2 - 6=°tren t~p s6 phirc
b) Trang matphang toa dQ Oxy cho cac diem A, B, C IfuI hrot la cac di€m bieu di€n cac s5phirc z\ =2 +3i, Z2 =-2 +i va Z3 =-3 +3i Chirng minh tam giac ABC vuong tai B
Call (2,5 c1dm)
Trang khong gian voi h~toa dQ Oxyz cho tam giac ABC voi A( -1;2;1), B(1;-2; 3) va C(1; 2; ~1) a) Vi€tphuong trinh tfmg quat cua mat phang (a) di qua ba diem A, B, C b) Vi€tphirong trinh tham s5 cua dirong thang I:i di qua tam G cua tam giac ABC va vuong
g6e voi m~t phang (a) .
c) Vi€t phirong trinh m~t cftu (S) e6 dirong kinh AB '
II.pHAN TV CHON (3,0 c1iim)
Thi.sinh chon mot hai phdn (phdn A hoac phdn B)
1 •
1 Phan A theo chuong trinh chuan: Ciiu 4A (1,5 c1iim)
2
a) Tinh tieh pharr I , f(2x + 1).lnx.dx
\
b) T'im mocun~ eua so p ire z, : h' = .3+2i+ +1 2·1
1-1
Ciiu 5A (1,5 c1iim) Trang khong gian voi h~toa dQ Oxyz eho m~t phang (a) va dirong thang !::,
Ian trotl I co p uong' h tninh lra: x +2y -2Z+1 ° ' x-I y+2 z-2
=T va - = = -
1 -2
a) Chirng minh rang duong thing !::,.e~t mat phang (a). Tim toa dQ giao diem
b)Tim cac di€m M thuoc dirong thang z; sac eho khoang each ill M d€n m~t phing(a) bang J2. 2.Ph§n B theo chuong trinh nang cao:
Cau 4B (1,5 c1dm)
, x
a) Cho hinhphang Agioi han boi cac duong y=xe2, y=0, x=0, x =1 Tinh th€ tich khoi tron xoay tao quay hinh A quanh true hoanh
b)Vi€t s5 phirc Z =(1- iJ"j)(1 +i) duoi dang IUQTIggiac
Call 5B (1,5 c1iim) Trong khong gian voi M toa de>Oxyz cho mat phang (a) va dirong thing !::,
Ian uotll' co p ironh g tninh la:' x+ 2y- 2z+ 1=°va,x-.1-= =-y+2 z":2
1 -2
a)Chung minh r~ng dirong thing !::,.e~t m~t phang (a) .Tim toa dQ giao di€m
(2)-II€t -HUONG DAN CHAM CUA DE KIEM TRA HQC KY II NAM HQC 2011 -2012
.MON TOA.N - KIIOI 12 - Gia'o due trung hoc phB thong
(Huang din cham gom 05 trang) Call
Call 2.5 d
Dap an
a) Tinh cac tich phan: tr
2
1= fSinx.ecosx.dx D~t t=cosx ~ dt =-sinxdx a
7r
X=- ~t=O
2
Diem
0,25
0,5
0,5 tr
j a I
1= fSinx.ecosx.dx = -fet dt =et ~ e-1
a
2x2+Fx+1 1
J= f dx= f(x+-+-).dx
1 X Fx X
2
= (~+2Fx +mlxl)
2 0,5
=2J2 -.!.+ln2
0,25
.b) Tinh dien tich hinh phang gioi han boo d6thi (C) cua ham sf>y =x3-X vi true Ox
.Phuong trinh hoanh d¢ giao difm CM (C) va true Ox lilx' -X ~ 0¢'> [ : : ~1 0,25
a
G9i S la dien tich cAntim ta c6: S= f(x3._ x).dx + f(x' - x).dx
-1 a
0,25
0,5
0,5
(
X4 X2Ja (X4 X2JI 1 .
- - + - =- (dvdt)
4 ·2
·1 a
Cau2
·,,2.0d I,,· 'c
L - " ~ ~ _ _ .a) Giai phirong trinh tren t~psophirc:
Z4 -"'Z2 -6 =O
. . [t =-2 D~t ~·t=Z2 phuong trinh c6 dang t2 - t - 6=0 T-r'
t=3
2 [z =rJ'2
t=-2 ~ z =-2¢:>
z =-i.J2 t== 3=> Z2 =3¢:>Z = ±J3
b) ZI =2+3i, Z2 =-2+ivi Z3 =-3+3i.
=> A(2;3), B(-2;1), C(-3;3) 0,5
(3), -, - - - - - - - -- - - - --- - ---- - - -- - --------- ------ - -- - - ----- - -- - -- - -- -
AB=,j42 +22 =2J5; BC=,j12+22 =J5; AC=.J52 +02 =5
AB2+ BC2=AC2.V~y tam giac ABC vuorig tai B 0,5
Cau3
2.5<1
Cau4A
-15<1,,
A(-l;2;l),B(1;-2;3) va C(1;2;-l)
a)Vi€t phirong trinh t6ng quat cua m~t phang {a).
,AB(~;-4;2); BC(0;4;-4) ~ [ AB,BC ] =(8;8;8)
, [J 'O~
n(l;l;l) Ta co n, AB,BC cling phirong
-. - -
-M~t phang (a) nhan n lam VTPT va di qua cii€m A
V~y (a): x+y+z-2=0. 0,25
b) Viet phuong trinh tham socua dirong thang ~ di qua HimG cua tam giac
ABC va vuong goc voi mat phang (a).
, 0,25
G(l;%;l) la tam cua tam giac ABC
~-di-~~~-G-~i-~6-~~-~6-~-~&i-~~~-~h~~~-(~)-,--;; -; -~~ -~-ii~-VTCP- - -, -
-1
x=-+t
3
,V~y PTIS cua duong thang ~ y = -:-+t
,
z =1+ t
0,5
c) Viet phirong trinh mat cau (S) co duong kinh AB
1(0;0;2) la trung di€m cua AB 0,25
~X~t c~ii--(sj -c6---duln{g -k,iiih---AB- llh~n i- i~m---t~~--v~ -c6 --b~n iCirili- ~-- --
-R = AB = 216 = 16 . 0,5
'22,
- -_._ -
V~y' (S): X2+y2 +(z-2/ =6 0,25
2 , '
,a) T~ tich pharr 1= f(2x + l).lnx.dx
, -
0,25
,{"
, u =lnx ~ du = - dx '
Bat x '
dv=(2x + 1)dx ~ v=X2+X
-2 -' -2 -. -_.-._ - -
1= f(2X + l).lnx.dx =[(x2 + x)lnxJI:-J(x + l}.dx
1
(4), 0,25
b) T' 'd : ht 3+2i 2·
, im mo un cua so p ire z= -'- + +
1-1
3 +2i t 2· (3 +2i)(1 +i)
Z= + + 1= + +1 0,25
l-i 2
- -
-1+5i
= + + 1=-+-1
2 2 0,25
VAay :Mod0 uun cuan cf Zl'a JP!81- + - '= 3M
4
Cau'5A 1,5d
,,'
Cau4B 1,5d
Trong khong gian voi h~ toa d<)Oxyzcho mat phang (a) va duong thllng b l~n 1trot co pionhtroiuang triri'001'a: x+2y -2Z+1=0 va''~x-I = y+2 = .z-2
'1 -2
a) C~(mg m~nh{~:~ :~(mg thang b.c~t m~t phang (a) .
~TTS cua b y =-2+t Z= 2-2t
Xet phirong trinh: ( + t + 2(- + t) - 2(2-~t) + = <=>7t -6= <=>t = % 0,25 13
x=
-7
-8
~ Y
=-7 '
Z=~ 0,25
,7
~~;;-~~~~-~~ -~-~~;;:;;-p~h~-~- ~;~;-~l¥-;-+n -- - -- -- - -~:;~
-b) Tim diem M thuoc dirong thang b sac chokhoang each illM den mat
phang(a) bang .fi .
,MEb.~M(1+t;-2+t;2-2t)
0,25
~ d(M;(a)) =.fi <=>17t;61 ~J2 <=>17t- 61=3J2
6±3J2
<=>t= -7
0,25
V~y:
M(~3+3J2 -8+3.fi. 2-6.fiJ
, 7' ' '
, ,
M(13-3.fi. -8-3.fi. 2+6fiJ
7 " ' 0,25
(5). -,
'Call 5B
1,5.d,
0~25
,''''
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x
a) Chohinh phang A gioi han boi cac dirong y=xe2, y= 0, x=0, x =1 Tinh th€ tich kh6i trim xoay tao quay hinh A quanh true hoanh
1
GQi V la th€ tich dn tim ta co V = nfx2
ex dx
o
{
u =X2 =>du = 2xdx
Bat
dv=edx => v=eX
1
V =nfx2ex dx=n(x2eX I~-2fx.e".dx)
o'
,0,25
=n(e -2J)
-,1 {u =X =>du =dx
Tinh J= I='.dx Bat
dv=eXdx => v=eX
o
1 1
J= I='.dx =xet - fex dx =1
,0
0,25
V=7r(e-2) (dvdt) 0,25
V~y:
b) Vi~t s6 phirc z =(1- iF3 )(1+i) duoi dang IUQ1lg giac,
(1-iF3J =2[COS( -;) +iSin( - ;) 1
..
Trang khong gian voi Mtoa d(>Oxyz cho mat phang (a) va duong thang !:: Ifin Itrot co p irong tnxoh inh lsa: x+ 2y- 2z.+1=° '.va -1-x-I =-1-y+2 = =2z-2
a) Chirng minh rangdtrong thang t3.c~t mat phang (a)
{
X =l+t
PTTS cua t3 y =-2+t 0,25
z=2-2t
Xet, Phuang trinh: ( 1+t+2(- +t) - (2 - 2t).+1= °¢:> 7t -6 = °¢:> t= ~7
-13
X=-7
-8 0,25
=>,
y=-7
(6)
, -, -- -- - --- - -- - - . -- ----- --- ---- - - --- -- - - -_.-- - ---- - ------ - ----- ----- - -- - - - -- -- . - -
-Vay:A f)i'rong t anh'" g A; - h'" () - A (13 8 2)
ucat mat pang a tar ' 7;-7; 7 0;25
b) Tim diem M thuoc dirong thang sac cho khoangcach nr M den mat
phang (a) bang fi
ME 6.=>M(l +t;-2+t;2-2t) 0,25
-~-~~(~~~;-;~-Ji-~; -li~+-t-~4+2t~4+4t+-11-~-h -- --- -- ---- - - --
- -
~ 17t -61=3 fi 0,25
6±3 fi
~t= -7
-_. _. -. - -
-vsv. M(13 +3 fi -8 +3 fi.2 - 6 fiJ. M(i3 -3 fi -8 - 3 fi.2+ 6 fiJ
.Y 7' ' '- 7' '7 0,25
N~u hoc sinh lam bai khong thee hu6ng d~n cham nhung dung v~ cho du diem thee tirng cau
- HET -,