MATH
Q1) fxpress
S o J ~ L.e-1:.
X
Qzı
-t.he s.e
,,
.- t.ı
•-
X :5
3
X-1
'
b
x ~
~ "d
Le-1; us Make a
-
nuındr~-tor
--..
-
x-1
-
C)
There 1ore,. -t.he
+-
o
roo-ts
o1
x=1•
+
+ --
+
+ -
-
o
-the
1
-1
x+1
dnd
b-9 t.tsiı\j -0\i.s ll\formc\i,on.
-tdbRe
t-
x.=-5
and
x.==-1
-
-
<O
x+s-
?)
lx+1)Cx-1)
5
x+5
()(+!)(x. n 1/j 1//ıı
to, 5] .
>C+1
Cx.- ı)
X-1
Cx-H)
-5
()
İl\-ler'\J;ıl
ı_ _.l:_ -<0
21
x+\
rooi
-lİ1e
_,:,
ı<~
Xi1
X.-1
irıe qua2;-t~
< 2:..
of
de"ornin~i(;) r are
X-t
5
clef ine
5
( x+ 1)(x. -1)
-oo
•
•
-
nwınbe r
; •••
h•• V
VJ\aiW
,r,
, , ..lUC.X• V• ,
=, .3(x+1) -2.Cx-f) < O
ı"-e
o" -the rea f
l\um be-rs
5 ~~----o•.,.,ı,pl...,"w"r:~o.M-.-~~,ıı,,.,-ııı~~A5.,___
SoRve -the
Sol :
dl\
Ô
5
x :,/ O
Sc /
dS
(
dl\J
X qÔ
X. '!5 5
nuınbers x.
real
aU
Set of
us s ho \A.)
>" Ü
X ,S
-!.he
anJ
x ~. o
l
PS-1
101 . 1
-
n
Ct:>
+
t-
sol"iiof\ sei
-
is (-oQ,-5)V(-f,1~"
l,f\e.
1 )(+t \
Sol : *~emember
lxt11
Xt1 =O
lx-t 1 \ =
l..e-l
lx\::: S"x' ><>,o
ı
~nd
x;,-1
l-x.-1,
us
lx+1 l-l x-3»l >0
~nd x.-?> =O ~
=)
x.:-1
S X-+t
x<-1
ma~e
L-x-+~,.
)(.<3
<) -Xf 3
x-3
- 4-
Hence, -i..he
Wri-te
/x+1l-l,<.-3>\
so1-t•f iof\
;ı n
~
2x-2-
=
.sei
equ,Hion
(-2, f)
Söl : ıhe
)( >.., 3
'
~
There-fore 1
pOif\-lS
=$ x.-~
l x-3\
x+1
X+1
x-3
lx-+1 \ -l x.-; \
Q 4-)
x=.3
3
-x-1 o
-1 )(-3l
lf(x)l =iflx ),;C~ )~ o
l-;(x) ' f(x)< O
bj l.fSİAj -lk,s il\\Otma-tion.
tdbie
el
-1
lX4-1\
a"J
l-x. , x< O
> l>l-~\
==>
> \ x-; l.
dıtd
-4.
x~-t
2.x-ı .,-1<x~3 ➔ *•
1
I
4--
'J..>3
/
a ncl >< >1 .
(1,00 ).
is
-ror
the
li,ı e -throu.9h
~he
(2., -2) .
of -the Rif\e passing -{:.hrou.3h
the po ,nts.
(x,, ~") an d ( X:ı, ~2.) ıs founJ b~
the formuJ d
m = ~:ı. - 31
.siope
X:ı. -X1
rn ::: (- 2)- 1
.( 1)-(-ı)
-ıhe equ.ıiiOfl fH
=
-t (
e quO-liof'\
1t-
.
-the tine wi-th s4ope m
(a,b)
is
!i-b :::mCx-a)
pdssiııs -throu3h
lf ıJJe choose -the poınt
~ine is
.ı1 - 1 -=
x +.2. ')
Qine
~
dS
(-1,f l, -the
•
Hence,
3x+ ~j ::- -2.
ıJJe
.ıııd
equdt ion
c;;,"
ot
-the
f il\d -the
Q5) For whai vaDue Of k
per p,e fl d ic ı.ı i~r
-lhe lil\e
uıhere.
*
·-
ihe
dr ~
So! : tf
·*
!
k
o-f
€/Pı ~
lirı.e
-t he
-i.o
is -the
~X +~ = 1
1,/\e 2x +k~ :: 3
,~
v;lı.ıe
w h~t
For
l ,nes f>ard lJeJ?
equa-t:.iol'
hcıs
1orm .9=mxtn
-f.he
-ıhe
.slope of -the }ine ,s m.
<ô-Nef\ -two 1,1\es
d1 = ~::'.nı 1 x+n 1 df\d d2 : ~=m2 x+n1
d1 ~"J dı dfe
perpencl,cuJar <=~ m1 .m.2. =--1
d1 ~11d d1 are p;)raJl"J ~..., m1 =m~
We
m, f\
c;ıt\
~ We
wri-te
c~n ~ri-ıe
i i/\ e.s
ff -th e
; re
~f'd
C ( ;.,-4.)
.S o, -the
~=
1 + ! .: : ) m1"' -½
x
~x+ ~ = 1
as
~ =--4-x t1
=> m.2. ::- 4p e r p. , -ih e"
m1 . m1 = -.E:.. . (- ı..) = -1 -) k = - <& ;
k
fnt=m.ı.
fo r -lhe
r~dius
r= 5
)
circ Oe
-½ =-4- ~
UJiih
½f
ceııier
el
shows -lhe ,grcıp h 8 = ıc." .:ıııJ
s hifie J versıof\S
dre as
ft>JloUJS .
e q u ati oAs for i.he .shi f -le d vers iof\5.
lDri-t2
Q 1- )
f ou r
k=
circJe u.Jİ-(h c€rrter C ( a, b) ~f\J
(x - a ):ı... + ( ~ - b) 2. = r .2.. •
equ,Hioıı o f -üıe ci rcle ıs (x-3 l + (~ +4f.::: !;2.,
e qu~iion for
r
( r >o) is
r~ di us
ı;ı s
rJrı equoı-c,orı
Q6) Wri-te
T he
1.ıc-t kj =3
~re p~ra!~e~, -llıe11
(f the fil\e~
So ~ :
-lhen
:;ı
Figvre 1
g
o
~v
,j
~= x.t
Fi.gure i
,'}
X
( ~,-l}
Version
~ 3ı
(;ı)
Ut\¾~
2
c.~r+3 ) =
X
~=Xz.-~f/
Ve,sioll (bJ
➔ 4-
U(\t"t:5
Ver~iorı (c)
J 3 dfld 4J
~ex -aJ"
~:: (X- 4 )2
(~ - .s)
~ = x~ gx+16I'
~::: x'1-6 x +1.ll
V~r sio "
-lı :ı.
(
d)
dfid ➔ 4.
Qj +2.) =(x.-~f ~ = x:ı.- gx+14~
Qı') Ske-ich -fhe curve represeMed
ı:::;c~
Reınem be r
;
at
cE>n-tered
:
Jt.
a
:: 1
1.:
b"
..(he origin.
{he Rtl\g..fh
Here,
t-
1
c).2..
equı¾lOn
elt, ps.e
~
c"2rı
We
vJri-t.e
b:ı.
eqodi,on
minor
are
.1~
col\sider -lhe
uJe
-then
<:1xises
-lh; s
s ~m e
-f-he
is
cerı-ter cu -d"e poi n-t (x., ~o) ·
e qu;ı-tiofl (x- ~ f + (8 tl.)2..:: 1 dS
"t he
4'
9
H.s
bu-t
(8 +ı )ı. : 1 . The
(x:-3 )" t
3"
"
-
cef\ter
of -f.hi5 eJJi pse
( ?,, -2").
•~
2.2.
are
Also / ~ -lhe ~en9-lh of m;Jor Jf\cl minor ;)(i:ses
~Ad 4,.{2d=.2- ~=6 ~Ad 1. b::. ı.ı:: 4-).
b
J
2.
(')(-5)
T he gr~ph o,:
z.
+ (~ +21 = t
4-
9
-2.
-
ı,..
lx 1+l ~ l ::: 1.
09) Ske-tch -lhe 9rllph of
So~ :1) { f
x.~O
~nd
~~o
/
X-tj:=1
lf
x~O
;rıd
~-< o ,
X-~=1
ı)
3]
'r)
x<O
lf
X
lf
<O
l
(d>b).
-a
rtspecii'le (J.
dnd 1 b
'l..
2.
/.,
e qw a-t ro" ~ - xo) + ( ~ - ~o) _ 1
-
Lr
9
ell,fse
Mo\Jor a/\d
ADso, i f
o~
(x-3)2. + C~+.1)2. _"
b~
dl\d
c:U\cl
~
~o /
~<.o /
-'
+x
x
~
·t
-x+j=1
-X-j = 1
x
~
-1
j
ts
1}
-ih e 3rd p h
of
lx I t Ij ( = 1 .
3)
Q iO) uJha{
r
po.ssess
So~: ·
f
: A fu nc-t,on f Cx)
oJd 'ı f - f (x ) -= f {- xı
Relr'leıtl ber
i .s
eve(\
.
4
f
-f (x')
Q11) Le-1:.
coın pos i-ie
6J f
cı
co Cx)
J
s
tJnd
.2. =
= f(x)
a"
ıs
even
-ytAl'ıl-{ıO il .
2
= X
2 1X
-J
d) 303( x)
D ( f of) =- ~ x ~ 1
I'
= ~ = ı -x) - 2.cX1-x)
foiOOlOincı
-ihe dOMdİ" of each ·
pec if ~
c) .9of (x)
h) fo.9(x)
f of (x:)
So2 : d)
1-X
X
.
Corıs-t.r\.lc,t -lt\e
g(x): :- ~ .
~ ~"d
f CAAliions
fof (~)
;J)
==
~
8-ax is.
uır-e.
s3mme-ltic
is
f
So/
fl-x )-::: -l-X ) t-1= X2.- t-1= f(x) .
f-ıef\leı
f Cx) =f (-x ) ,
i-f
~ven
is
ocid ~
or
or i 3; f'ı.
w i-Lh r es pec.f. -lo
9
-f
f Yff\ funciio~s. are sjmmei<ic
Af;o ., oJJ f ul\C-liol\s Q re s1nı m eiric
~ - axis .
respec~ -io
wH.h
f lx'l =i-+1
does -lhe grdp h ot
(if anj) s~n, metf j
p:Jr-licuQ;r / İS
l11
/
X
*ÔJ
D(fos ~= { xaA l x *o,1f
X/{1
3(x)
=
.9 of(><:)==- f(x) = ı/x
2
1- /x
1-fCx )
2
X-2
X
= 1-x.
d; 303(x) = 1 -3Cx)
3lx) 1- L
/
x
=
I
1- 2 x.
1-X
-t.he. fun,-tion wi-t.h domcHr1 [0,2 ] ~,ıJ
[ô,1] ., whose 3rtıph i~ .shown beOOıJJ. Ske-tc.h -the
_gr~ph of 1+ f C-x/ı.) ancl speci f~ its domQirı ;Jrıc/ r~rı3e .
Q.11 )
rtlr'\Je
f
rete rs
.j
-/;.o
(1,1)
SoJ : 1 +
f
'
:5 .2 ._ 0 ~ - ~
~
.:ı
o
tt*
*♦
><
f ( -;_ )
ıneenS
,
we
4- ~ X ~ Ô
shi ft
IY\l,Jjt
-
.
+1 wıe~ns wc
u p -lo 1 uoi-t..
MUS, t
Jt
:t:*.!J( , uJe d r~w -l~is.
-i:he fvf\ (tion
~=1+ -x
coıı~iJer
-dıe s3mr neirj ot -lhe fun ciion wr i .
3 -Gxi s ➔ ¼ we cO/\s ;Jer * , -ıteı<
~f1cl
For doındill:
(-xı.2.) ,
• •
_,,,.,
•
-~4-- ---;_
Do~irı : C-~,oJ
-
t
Pı;k,ge :
-ol- -.. . -;.
1
i
1
><
Q12~ f xpres.s -ihe
or
iero
de8ree -t.h~n {he JeAo-mi~ f.
SmJtQer
has
~s. -dıe
sum. Of
1uflc-lion uJho.se nvmeTa--t.or
~fl(.(JOI)
ancl ~"o--{"ıer f'd"tiof\dj
a poi~Aom;~i
,~ e i-(hrr
r~iion;.Q
x.3
x2-+1x+,;
use
Sc Q: UJe mus-(
divislo"'
~i9or iirttl'l .
Xı.+lx 3
X3
l t-ıl·+;x. X - 2._
- -1x.
;x.
2.. -
_
- 2X2. - 4,..x -
6
Xt6
Q 13J Express
5 0 .Q;
*~ We
1(-t* lf
lf
➔ Fo r
2
~
alj
u;e
co/\si Je r
x as
acu-i.e f:lrıgle .
eli\
sın ( 3;'
- ><- / ,
~
n
<3} - x
<
¾[ .
So' it
cha,ıges- C S11\ ~ Cos).
i-'>
Mg 0-tıve .
Hence,
-X) = -coS)(.
Q 14-; Express
s 0 Q:
cosx .
de-fermi"e -ine s;grı o1 i.he fL<l\c-i.ion ( + or - )
k = 1t, 2.rc , -the -tuf\c-iion does l\oi chaAge.
k.= 4, 3:f' , ,the func.-liOn chdAgff.(si f\~Co.s / -ta~ co-c )
ARso, 1J,e f•tııc-tlo/l
Sif\ (
.sinx ~Ad
cos(kıı+xı, -l::>"(k1t:rx'\,. co-f.(krı~x~
sin(kn+x) ,
~ Fi r s-i
ın -t.erlfls of
sil\ ( 3 "- - x /
co.s3x ıf\ -ierrns of >İf\X ~lld cosx.
~ si n ( a+ b).= sina . cosb+ cos~ . sirıb
* .sin.2.x =2 . sinx.ros;ıc.
2
1t cos Ca +b1:: cosa.cos b :t sil\a. sin b
co~2x.::-2.cos x-1
*
cos3x = ros(2.x-txı
= cos2x.rosx -s1nlx . .sir1X
==
(2..cosrı.x - t ') co sx
- (.ı . s;ııx.cosx') .s;nx
:::: 2cos3 x - cosx - 2. s;,{2-·x.cos>t;
-==-
2 c-os3x. - cosx - ..2. (1- cos:ıx)cosx
= .2cos~x- cosx +2ros3 x -2cosx
=- 4cos 3 X.
-&osx.
~ef\ce,
.
ı~
sın
v
= 1. 7
0
4--
CT) S1
Thı.ıs,,
r
5cos4 ~
cose
rıo-t
A!so/
1-cos2. (1rcsz. ~
=- 4-
€Jn d
İS
ne1;).tjve
be
pos iti ve.
-l~n
e=
= 1
Ol\
ıt-
~
~
cos v
'+-- 4-ccs2. &
= cos2 &
As' //
=-2..
-the İ/\"terva1
[n:, 3_;]-
1-t
c-~
-1
~
_ 1
siıı G- _ j_ =;> :Sif\Ô __
=;> Sirn7 = ır: ıfl
cos
f
li b
1.
-.2 / ıf5
::ı._
e