b61du0ng lfoa n - classbook
TRANSCRIPT
. vo HO'U BiNH (Chu bien) - NGUYEN BA DANG
B61DU0NG ~·
lfOA_N TAP HAI
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NHA XUAT BAN GIAO DVC Vll;T NAM
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PHAN CONG BIEN SO~N
VU HUu BINH: PHAN DAI SO . NGUYEN BA DANG: PHAN HINH HOC .
-.:1 -_: __ ::: _ ~-~¢i: ~ 9 i: ~~ ~: :~:::::::::::: ::: : _::: :--: :----:-: _::: _:: __ :::: c:_ Nh~m dem d€n cho hQc sinh m(>t cu6n sach Toan chdt
lu(J'ng v~ n(h dung va il?p v~ hinh thuc, cac tac gia da bien SO?n b9 sach Bai dU'O'ng Toan 8 g6m hai t~p .
Day la b(> sach duqc bien SO?n theo djnh hu6ng phat tri€n nang Ive, giup cling c6, nang cao ki€n thuc va b6i duang hQC sinh VUO'n ten kha va gi6i, khai gqi ni€m yeu thich m6n Toan clia hQC sinh va giup cac em bi€t them nhi€u (:mg d\mg clia Toan hQc trong cu(>c s6ng.
N(>i dung b(> sach barn sat Chuang trinh va Sach giao khoa Toan hi~n hanh, m6i chuang g6m nhi€u chli dS, m6i chli dS g6m nhiSu m\}c.
Ml,lc Ki~n thfrc do nhcr h~ thfJng cac ki€n thuc du6i d?ng SO' diJ, VUa giup d~ nh6 ki~n thuc illQt each tflJC quan, vua giup cac em n~m ch~c h~ thfJng cac ki€n thtrc CO' ban.
M\IC Hoi dap nhanh g6m nhfrng cau h6i ng~n gQn yeu cftu hQC sinh tra lai nhanh d€ cling cfJ cac ki~n thtrc CO' ban va tranh nhfrng sai l~m d~ m~c. Trang cac bai tr~c nghi~m c6 nhiSu phuang an 4', ©, (!), @, n~u kh6ng c6 chi thj gi ta quy u6c ch9n phuang an dung.
M\IC HQC giai toan g6m cac vi dl,l di€n hinh tu CO' ban d~n ncing cao nh~m hinh thanh phuang phap giai toan cho h9c sinh. Cac vi d\l ca ban d~t trung khung n€n xanh nhu
'""''· cac vi d\l nang cao d~t trong khung n€n den nhu
'""''· Ben C?nh tuyin chinh la cac lai giai duqc trinh bay mftu mvc nliu m(>t bai lam hoan chinh, con c6 tuyin ph1:1- g6m cac gqi y, nh~n xet, luu y, lai binh, cac ghi nh6, k~t lu~n va ma r(>ng ki~n thuc v6i sv tham gia cua m(>t s6 nhan wjt hog,t hinh nh~m tang tinh h~p dftn cho cu6n sach.
' '---------- ---- -------- ------------------------------------- --- ------ --
0
M\lc Bai t~p g6rn hai lol;li bai t~p ca ban va rn'.ing cao, cac bai t~p duqc ch9n 19c giup h9c sinh nang cao dftn ki niing giai toan va niing hJC tu duy dS vuan len kha, gi6i trong mon Toan, rn?t khac con giup h9c sinh th~y duqc nhi€u i.'.mg d\lng thvc tS cua Toan h9c.
M\lc Em co bi~t? rnang d~n cho h9c sinh nhi€u di€u thu v!, g6rn nhfrng bai toan vui, nhfrng cau d6, nhfrng thong tin v€ cac nha toan h9c c6 lien quan dSn chu d€ dang h9c , nhfrng i.'.:rng d\lng ho?c thS hi~n cua Toan h9c trong dai s6ng.
M\lc Di xa ho·n cung dp them nhfrng kiSn thuc nang cao dS c6 thS hiSu sau han v~n d€ dang h9c.
Cu6i sach la tn\lC Hmrng dfin giai - Dap s6 giup h9c sinh kiSrn tra ll;li bai giai cua minh dS rut ra nhung kinh nghi~rn trong giai toan.
v 6i nhfrng d6i rn&i ca v~ n<)i dung rnn hinh thuc , dap trng yeu du v€ tinh khoa hQC, tinh SU phl;ltn, tinh th\l'C ti€n, tinh h~p d~n, cac tac gia hi VQng r~ng cufin ~ach Se la tnQt ~n ph~rn thiSt thvc va b6 ich khong chi cho cac em h9c sinh, ma con cho cac th~y co giao, cac b~c cha illy h9c sinh, cac nha quan li giao d\lC va t~t ca nhfrng ai quan tarn dSn S\l' nghi~p d6i rn&i giao d\lc.
Cac tac gia chan thanh earn (Jll bl;ln d9c da SU d\lng cu6n sach nay va rnong nh~n duqc nhfrng g6p y cua bl;ln d9c dS cu6n sach ngay cang hoan thi~n han.
M9i y kiSn g6p y xin glri v€: Ban bien ~P sach Toan - Tin, Nha xu~t ban Giao d\lc Vi~t Nam, 187B Giang Vo - D6ng Da -HaN<)i.
cAcrACGIA
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TrU'ac day 6 nam,
tuoi m~ gap 1 O Ian ".,. tuo1 con.
Tuai con hi~n nay?
5
•
Nam nay tuoi m~ gap dung bon Ian
,..., . tuo1 con.
1 O(x - 6) = 4x - 6
PHUONG TRINH B!C NHAT MOT AN
THI -
Con muon mua 40 qua tao.
Gia m(>t qua tao? 3.JOx + 4x = 102
Chu de 1. Phu'O'ng trinh b~c nhat m(>t an
Chu de 2. Phu'ang trinh tlch
Chu de 3. Phu'O'ng trinh chua an iJ mau
Chu de 4. Giai bai toan bang each l~p phu'O'ng trinh
' - .,,,,:' - J PHUONG T:RINH BAC NHAT MOT AN
KIEN THO'C CAN NHO __________________________ _
w Mc)t nghi¢m;
w Nhieu ngh i¢m
(hai, ba, ... );
w V6 s6 nghi¢m.
(ax+ b = 0 J
~
Nhan ho~c chia ca hai v~ v&i cling m9t s6 khac 0.
Chuy~n m9t h1;1ng tu tu v~ nay sang v~ kia va d6i d~u ctia n6 (T6ng quat: C(lng hai v~ v6i cung m(>t da tht'.rc ctia ~n) .
---------------------------------------------
H61 eAP NH ANH ............................................................................. .
0 T~p nghi~m cua phuang trinh 9x2 = 1 la:
~ {±}; © {-±}; (!) {i}; @ M9t dap s6 khac.
8 Cac kh~ng djnh sau dung hay sai?
) 2 l ' ~ h" ~ , h 'nh 9 2 4 a "3 a mQt ng I¥m cua p uang tn x = ;
b) { ~} la t~p nghi~m cua phuang trinh 9x2
= 4;
c) Nghi~m cua phuang trinh x2 = 25 la 5;
d) 0 la t~p nghi~m cua phuang trinh x2 + 1 = 0.
, ~ e Cong thuc doi tir dQ c sang dQ Fla F = 1,8C + 32 (xem ffi\lC A EM co e1tn).
Theo d6, 1'.rng v6i 77°F la:
6) 50°C; @) 5°C; (!) 25°C; @ M9t dap s6 khac .
.,;
.---::.~ HQC GIAI TOAN ···················································································
mdl Cac cijp phuang trinh sau c6 tuang duang khong? Vi sao? 2 2
a) x + 4 = 5 ( 1) va x - 4 = - 3. (2)
(3) b) x 2 + 4 = 5 ( 1) va x 2 + 4 + _l _ = 5 + _l _ . x - 1 x - 1
2 c) x +4=5 (1)
(1)
va 2x2 + 8 = 10. (4)
(5) 2
d) x + 4 = 5 2 va x(x + 4) = 5x.
Giai
a) (1) tuang duang (2) vi c6 cling t~p
nghi~m { 1; -1} ho?c vi cling c<)ng hai vS cua (1) v6i -8 .
b) (1) khong tuang duang (3) vi 1 la nghi~m cua (1) nhung khong la nghi~m cua (3).
Chung minh hai phuong trlnh tuong duong: - Cach 1: Hai phuong trlnh c6 cung m(>t t~pnghi~m.
- Cach 2: Phuong trlnh nay duqc bien doi W phuong trlnh kia bC1i quy tac chuyen ve ho~c quy tac nhan v6'i cung m(>t so khaco. ---- - - ---------- --- -- -- - -----'
c) ( 1) tmmg duong ( 4) vi cling nhan hai v~ cua (1) v&i 2.
d) (1) kh6ng tuong duong (5) vi 0 la nghi~m cua (5) nhung kh6ng la nghi~m cua (1).
r - - ------- - ---- -- ---- - -- - ---- -,
: Chung minh hai phLlong trlnh kh6ng : : tlio'ng duong: : : - Coch 1: Hai phllong trlnh kh6ng c6 : i cung m(>t t~p nghi~m. : : - Coch 2: Chi ra m(>t nghi~m cua phllong : : trlnh nay kh6ng la nghi~m cua phllong : : trlnh kia. : ~ - ---------------------------~
IU!PI Tim lien h~ gifra cac s6 a, b, c, d d~ phuong trinh
ax3 +bi + ex + d = 0 (1) nh~n :
a) 1 la m(>t nghi~m; b) - 1 la m(>t nghi~m.
Giai a) x = 1 thoa man (1) nen
3 2 a.1 + b.1 + c.l + d = 0.
V~y a + b + c + d = 0.
b) x = -1 thoa man (1) nen 3 2
a.(-1) +b.(-1) + c.(-l) + d=O
hay b + d = a + c.
,---- - ---- ---- -- - - - - - Cl. _ I - Neu ve trai cua phllong trlnh (1) c6
tong cac h~ so bang o thl 1 la m¢t nghi~m cua phllong trlnh. - Neu ve trai cua phllong trlnh (1) c6 tong cac h~ so cua hc;ing tU b~c chiln bang tong cac h~ so cua hc;ing tu b~c le thl -1 la m¢t nghi~m cua phuong trlnh.
MN Giai phucmgtrinh x-20 + x-46 + x-5 + x+6 =0 . (1) 7 33 8 19
<::} x -13 = O<::}x=13 . Nghi~m cua phucmg trinh la x= 13.
Mdl Cho phucmg trinh v&i tham s6 a: a(ax + 1) = 3(ax + 1).
Tim di~u ki~n d~ phucmg trinh:
a) C6 m(>t nghi~m duy nh:it;
b) V 6 nghi~m;
c) V 6 s6 nghi~m.
:'~ -;a~ -p-h~~ -t~~~ -~iu~ -t6~~ -,~~ _t_u_ ~; ': : mau deu bang x - 13. : : a hai phan thuc cuoi, hi~u cua tu va : ~ mau deu bang x - 13. : ' ---------------------------'
(1)
Giai 2 2
(1) <=> a x + a = 3ax + 3 <=> a x - 3ax = 3 - a
<=> a(a - 3)x = 3 - a. (2)
) N J. 0 ' 3 h' 1 a eu a * va a * t 1 x = - - .
a Phuong trinh c6 m<)t nghi~m duy nh&t.
b) NSu a = 0 thi (2) la Ox = 3, vo nghi~m .
Trong m(>t s6 phuong t rlnh c6 nhung : chu bieu thi nhung s6 da biet, nhung : chu d6 duqc g9i la tham s6. T~p : nghi¢m cua phl.lo'ng trlnh thuong ph1,1 : t hu(>c vao cac gia tri cua tham so.
c) NSu a = 3 thi (2) la Ox = 0, vo s6 nghi~m, x tuy y .
.
.................. . ............................... . l!-1=fii~M mJ Giai cac phmmg trinh:
2 2 a) 3(x - 2) + 2(x + 3)(x - 3) = 5(x + 1) ; b) x _ x - 3 + 2x + 1 = 5x .
2 3 6 ,
c) 2x_15 - 2x=7_. 15 10 6 ,
d) ~ - x+lO =2. 20 25
[E Hay viSt m9t phuong trinh c6:
a) Hai nghi~m; b) Vo s6 nghi~m; c) Vo nghi~m.
Ill Tim gia tri cua m dS 3 la m<)t nghi~m cua phuong trinh x3
- mx2
- mx - 3 = 0.
tffiifi&4·' llJ Giai cac phuong trinh:
a) x + 1 + x + 3 = x + 5 + x + 7 . 49 47 45 43 ,
b) ~+ x +l + x+2 + x +3 + x+4 =5 · 15 16 17 18 19 ,
c) x + 1 _ 2x -5 = 3x - 47 _ 4x - 59.
11 15 17 19
Im Cho phuong trinh (m + l)(m - 2)x- m = 1. Tim gia tri cua m dS phuong trinh:
a) C6 m<)t nghi~m duy nh&t; b) Vo s6 nghi~m; c) Vo nghi~m.
[E Giai phuong trinh mx + m- m2
= 2x - 2 (m la tham s6).
fE Giai phuong trinh v&i a, b la tham s6 (a -:t- ±b): x - a + x + b 2ab = 0 a-b a+b a2 -b2 ·
[I] Giai phuong trinh v&i a, b, c la tham s6 (abc t:. 0): x -b- c x-a - c x-a-b ---- + + =3 .
a b c
A ,., EM Cu BIET? ......................................................................................... .
£>(> C VA£>(> F
0 mtoc ta va nhieu nuoc khac, nhi~t d9 duqc tfnh theo d9 C, con a Anh, Mi nhi~t d9 duqc tfnh theo d9 F; C va F Ian luqt la cac chO' cai dau cua Celsius (Xen-xi-Ot-xC1) va Fahrenheit (Phe-ron-hai-tC1).
Nhi~t d9 0°C ung voi 32°F. Khi thang d9 Celsius tang them 1°C thi thang d9
Fahrenheit tang them 1,8°F.
a) L~p h~ thuc giO'a d9 C va d9 F.
b) Nhi~t d9 30°C ung voi bao nhieu d9 F?
c) Nhi~t d9 95°F ung voi bao nhieu d9 C?
d) 0 nhi~t d9 nao thi d9 C va d9 F bieu th! boi cung m9t so?
a) H~ thuc giO'a d9 C va d9 F la F = 1,8C + 32. (1)
b) Thay C = 30 vao (1) thl F = 86. Do d6 30°C ung voi 86°F.
c) Thay F = 95 vao (1) duqc 95 = 1,8C + 32 nen C = 35. Do d6 95°F ung voi 35°C.
d) f)~t C = F = x. Giai phuong trlnh x = 1,8x + 32 duqc x = -40.
Nhi~t d9-40°C ung voi - 40°F. F
120-:::
1C.O :
BO -60
-40
-20
-0
--20 :
-40 -=
Anders Celsius (1701 -1744)
@
c 50
§- 40
; 30
= 20
~ 10
~ 0
§- ·10
= -20
= -30
::... -40
01 XA HON ............................................................................................ .
PHllONG TRINH H~ QUA VA NGHl~M NGOJ;\I LAI
Cho phuong trlnh x - 1 = 3. (1 )
Nhan hai ve cua phuong trlnh (1) voi x - 2 duqc (x - 1 )(x - 2) = 3(x - 2). (2)
Phuong trlnh (1) c6 nghi¢m x = 4, d6 cOng la nghi¢m cua phuong trlnh (2). Ta g9i
phuong trlnh (2) la phU'ctng trinh h~ qua cua phuong trlnh (1 ). T~p nghi¢m cua phuong trlnh h¢ qua chua t~p nghi¢m cua phuong trlnh dacha.
Nhan hai ve cua m<?t phuong trlnh voi cung m<?t da thuc cua an cha m<?t phuong trlnh h¢ qua cua phuong trlnh ban dau:
x - 1 = 3 => (x - 1 )(x - 2) = 3(x - 2).
Blnh phuong hai ve cua m<?t phuong trlnh cOng cha m<?t phuong trlnh h¢ qua cua phuong trlnh ban dau:
x-1=3::::>(x - 1)2 =32. (3)
Nghi¢m cua phuong trlnh ban dau la nghi¢m cua phuong trlnh h¢ qua. Nhung nghi¢m cua phuong trlnh h¢ qua c6 the kh6ng la nghi¢m cua phuong trlnh ban dau.
x = 2 la nghi¢m cua phuong trlnh (2) nhU'ng kh6ng la nghi¢m cua phuong trlnh (1 ).
x = - 2 la nghi¢m cua phuong trlnh (3) nhung khong la nghi¢m cua phuong trlnh (1 ).
Ta g9i nhO'ng nghi¢m d6 la nghi~m ngof!i /ai.
Khi giai m<?t phuong trlnh, c6 khi ta can bien doi n6 thanh phuong trlnh h¢ qua.
Khi d6, can thit lf!i cac nghi¢m cua phuong t rlnh h¢ qua vao phuong trlnh ban dau de loc;ii b6 nhO'ng nghi¢m ngoc;ii lai.
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~ ~~CHU f>E t41 A PHUONG TUINH TiCH
KIEN THO'C CAN NHC1 - - - - - - - - - - - - - - - - - - - - - - - - - - - I
F(x) = 0
[
A(x) = 0
A(x)B(x)C(x) = 0 ¢=> B(x) = 0
C(x) =0
H61 E>AP N HANH ............................................................................ .
0 T~p nghi~m cua phuong trinh (x - 1)(2x + 1) = (x - l)(x + 3) la:
~ 2; @ {2}; (i lva2; @ {1;2} .
8 Phuong trinh (x + 3)(x2
- 4) = 0 c6 cac nghi~m la:
~ 3 va4; © 3 va2; (j -3 va 2; @ M('>t dap s6 khac.
C) DiSn vao ch6 chAm ( .. . ):
a) N~u a+ c = ... thi -1 la m()t nghi~m cua pliuong trinh ax2 +bx+ c = 0.
b) N~u a+ c = ... thi 1 la m()t nghi~m cua phuong trinh ax2
+bx+ c = 0.
@
. . --r --~ HQC GIAI TOAN ................................................................................. .
mdl Giai phuong trinh x2 - (x + 3)(3x + 1) = 9.
Giai 2
x - (x + 3)(3x + 1) = 9 2
¢:::> (x - 9) - (x + 3)(3x + 1) = 0
¢:::> (x + 3)(x - 3) - (x + 3)(3x + 1) = 0
¢:::> (x + 3)(x - 3 - 3x - 1) = 0
¢:::> (x + 3)(-2x - 4) = 0 ¢:::> [x + 3 = O ¢:::> [ x = - 3
- 2x - 4 = 0 x = - 2.
V~y S = {-3; -2}.
ldttfJ Giai phuong trinh x3
+ 4x + 5 = 0.
Giai 3 3
x + 4x+5 = 0¢=>(x +1)+(4x+4) = 0 2
¢:::> (x + l)(x - x + 1) + 4(x + 1) = 0 2
¢=>(x+l)(x -x+5) = 0. (1)
Ta thiiy x 2
- x + 5 ~ ( x - ~ )' + 4~ > 0 nen ( 1)
¢:::> x + 1 = 0 ¢:::> x = -1. V~y S = { -1 }.
------------- -- -------,
: Dung khai trien ngay. : : Hay chu y den : x2
- 9 = (x + 3)(x - 3). I
I ' I : [x + 3= 0 , Ki hi¢u I -2X-4 = 0 I
I I
: c6 nghia la x+ 3 = o : ho~c - 2x - 4 = 0.
--------- --~-,... '' 1+4=5. I
Tong cac h¢ s6 cua hc;ing tu : b~c le bang tong cac h¢ s6 : cua hc;ing tu b~c chan, : phi.fang trlnh c6 nghi¢m - 1. :
'--------- --------- -- - '
Id@ Giai phmmg trinh (x + 14)3
- (x + 12)3
= 1352. (1)
Giai
D~t x + 13 = y. Ta c6 3 3
(1) ¢:::> (y + 1) - (y - 1) = 1352
, '
: f)~t an ph1,1 x + 13 = y, ta dl.fqc : : phi.fang trlnh vai an y th~t : : don gian! :
3 2 3 2 ¢:::> (y + 3 y + 3 y + 1) - (y - 3 y + 3 y - 1) = 13 5 2
2 2 2 [y = 15 ¢:::> 6y + 2 = 1352 ¢:::> 6y = 1350 ¢:::> y = 225 ¢:::>
J:=-15.
V&i y = 15 thi x + 13 = 15 ¢:::> x = 2.
V&i y = -15 thi x + 13 = -15 ¢:::> x = - 28.
V~y S = {2; -28}.
IU'tfCI Giai phucmg trinh x3
+ (x - 3)3
= (2x - 3l Giai
D~t x - 3 = y. Ta c6 x3
+ y3 = (x + y)3
3 3 3 3 ¢=> x + y + 3xy(x + y) = x + y
¢=> 3xy(x + y) = 0 ¢=> l;: ~ ¢=> l: : ~ = 0
x + y = 0 2x - 3 = 0.
Vay S = {O· 3· i }. . ' ' 2
----- - - - -- - - -~- -, : Hay chu <; den cac ca s6: : x + (x- 3) = 2x- 3. '
IU'N Giai phucmg trinh (x - 1)4
+ (x + 3)4
= 32. (1)
Giai 1- - - - - - - - - - - - - - - - - I
: Voi d<,mg (x + a)4 + (x + b)4 = c, : I I
' • d " ~ h 1· a+ b ' ' nen atan p u a y=x+ - . ' : . . 2 : 1 _ - - - - - - - - - - - - - - - - - - - - - - _1
D~t x + 1 = y. Ta c6 4 4
(1) ¢:::> (y - 2) + (y + 2) = 32 4 3 2 4 3 2
¢:::> (y - 8y + 24y - 8y + 16) + (y + 8y + 24y + 8y + 16) = 32 4 2 2 2
¢:::> 2y + 48y + 32 = 32 ¢:::> y (y + 24) = 0.
Do y2 + 24 > 0 nen y = 0 ~ x = -1.
V~y S = { - 1}.
'U@I Giai phucmg trinh x(x + l)(x + 2)(x + 3) = 360. (1)
Giai 2 2
(1) ¢=> (x + 3x)(x + 3x + 2) = 360.
D~tx2 + 3x + 1 =y. Tac6(y- l)(y+ 1)=360
2 2 [y=19 ¢:::> y - 1 = 360 ¢:::> y = 361 ¢:::>
y = -19.
V 6i y = 19 thi x2 + 3x + 1 = 19 ¢=> x2 + 3x - 18 = 0
,----- - ----------------, : Nhan hqp If ve trai lam xuat : : hi~n bieu thuc chua an gi6ng :
: nhau (la x2 + 3x). : \ I , ___________________ __ _ ,
[x = 3
¢=> (x - 3)(x + 6) = 0 ¢=> x = _ 6_
2 2 ( 3 )2
71 V 6i y = - 19 thi x + 3x + 1 = -19 ¢=> x + 3x + 20 = 0 ¢=> x + 2 + 4 = 0 ,
vo nghi~m. V~y S = {3; -6}.
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~ ••••••• ••••••••••• Bil.I Til.P ••• ••••••• •••••••••••••••••••••••
CO BAN
fll Giai cac phuong trinh: 3 2 a) x - 2x + 3x - 6 = O; 2
c) x +9x+20 = 0;
fll Giai cac phuong trinh: 3 2
a) x + 5x + 3x - 9 = O; 3 2
c) x - 7x + 8x + 16 = O;
fl] Giai cac phuong trinh: 3 2
a) x + x -2 = O; 3
c) x - 7x + 6 = O;
NANG CAO
fm Giai cac phuong trinh: 3 3 3
a) (x + l) +(2x -1 ) = 27x; 2 2
c) ( x - 16) = l 6x + 1 ;
fll Giai cac phuong trinh: 2 2
a)(x - x + l)(x - x + 2) = 2;
2 c) (x - 1) (2x - 1)(2x - 3) = 3;
Pia Giai phuong trinh:
3 2 b) x - 4x + x -4 = O;
2 d) x -x-12=0.
3 b) x + (x - 2)(2x+ 1)=8; 3 2
d) x - 3x + x + 5 = 0.
3 b) x + 2x - 3 = O;
3 2 d) x -6x +llx-6 = 0.
4 4 b) (x-3) +(x+l) =256;
4 2 d) x = 5x + 8x + 12.
b) x(x + 2)(x + 3)(x + 5) = 280;
d) (x + 3)(x + 4)(x + 5) = x. , ... - - - - - - - - - - - -------- - - -- ' : E>~t x + 4 = y. : , _______________________ }
5 5 (x - 3) + (5 - x) = 32 . .,
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L<'.11 GIAI "£>QC £>AO"
Khi giai phuong trlnh x(x + 1) = 12, b;:in Toan dua ra loi giai sau:
x(x + 1) = 12 <=> x(x + 1) = 3.4.