Page 1
OSWAAL ICSE Mind Maps, MATHEMATICS, Class-IX | 1
( 1 )
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CHAP
TER
: 1 R
ATIO
NAL
AND
IRRA
TIONA
L NU
MBE
RS
Ra
tiona
l a
nd
Irra
tiona
l nu
mbers
Irra
tion
aln
um
ber
s
Asu
rdis
am
ixed
surd
,if
ith
asso
me
rati
on
alco
effi
cien
to
ther
than
un
ity.
Ifx
isa
po
siti
ve
rati
on
aln
um
ber
and
kis
ap
osi
tiv
ein
teg
er,
then
xo
r√
xis
1/k
k
call
eda
surd
.
TypeofSurds
Sim
ila
rsu
rds
are
surd
so
fsa
me
ord
eran
dsa
me
rad
ican
ds.
Asu
rdis
ap
ure
surd
,if
itd
oes
no
tco
nta
inan
yo
ther
rati
on
alco
effi
cien
tex
cep
tu
nit
y.
Mixe
d Sim
ilar
3,2
etc.
2 3
, 47
etc.
5, 7
5,10
5 e
tc.
Rat
ion
alan
dir
rati
on
aln
um
ber
s,b
oth
con
stit
ute
sth
ere
aln
um
ber
sR
.Ir
rati
on
aln
um
ber
sca
nb
eex
pre
ssed
asn
on
-ter
min
atin
gn
on
-rec
urr
ing
dec
imal
.�
2=
1.4
14
23
5..
..is
no
n-
term
ina
tin
gn
on
-re
curr
ing
dec
imal
.
Rational numbers
Rat
ion
aln
um
ber
sar
eal
way
sex
pre
ssed
aste
rmin
atin
go
rn
on
-ter
min
atin
gre
curr
ing
dec
imal
s.
How
are
they
exp
ress
ed?
=0.
75is
term
inat
ing
dec
imal
and
=0.
1818
is
no
n-t
erm
inat
ing
recu
rrin
g
dec
imal
.
3 42 11
Exa
mp
le :-
Rat
ion
aliz
e th
e d
eno
min
ato
r:
Step
I :-
4 �6×
�6 �6=
4 �6��
6�3
6
Step
II
:-=
��6
�36
��6 6
=��
6 3(S
imp
lify
if
nee
ded
)N
um
ber
inth
efo
rmo
f,w
he
re
q0
an
d�
p,
qZ
and
p,
qh
ave
no
�co
mm
on
div
iso
r ex
cep
t 1.
p q
Nu
mb
ers
wh
ich
can
no
tb
eex
pre
ssed
inth
efo
rmo
f,w
her
ep
,qZ
,p>
0.�
p q
Howare
they
expre
ssed
?
Page 2
2 | OSWAAL ICSE Mind Maps, MATHEMATICS, Class-IX
( 2 )
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CHAP
TER
: 2 C
OMPO
UND
InTE
REST
Com
pou
nd
inte
rest
Form
ula
for
com
poun
din
tere
st
Co
mp
ou
nd
inte
rest
issi
mp
lein
tere
sto
nth
ep
rin
cip
alp
lus
inte
rest
earn
edo
nin
tere
st.
Th
efo
rmu
lafo
rca
lcu
lati
ng
the
amo
un
tat
com
po
un
din
tere
stis
A=
P(1
+)T
r10
0
Co
mp
ou
nd
inte
rest
=A
mo
un
t–p
rin
cip
al
C.I
.=P
(1+
)Tr
100
–P
=P
(1+
)Tr
100
–1
Application of compound
interest: Growth and depreci
atio
nG
row
th
Depreci
atio
n
VT=
(1+
)Tr
100
V0
VT=
(1–
)Tr
100
V0
With
out using form
ula
Itis
the
dif
fere
nce
bet
wee
nth
efi
nal
amo
un
tan
dth
e(
ori
gin
al)p
rin
cip
al.
�C
om
po
un
din
tere
st=
Fin
alA
mo
un
t—O
rig
inal
pri
nci
pal
CI=
A–
P
Pri
nci
pal
× R
ate
× T
ime
100
SI=
P×
R ×
T
100
SI= Am
ou
nt=
Pri
nci
pal
+ S
imp
le I
nte
rest
A=
P +
SI
For
two
yea
rs
Diff
eren
cebe
twee
nco
mp
oun
d
in
tere
stan
dsi
mp
lein
tere
st
Dif
fere
nce
=P
(R)2
(100
)2
Dif
fere
nce
=
+3×
P(R
)2
(100
)2
P(R
)3
(100
)3
Wh
ere
V=
In
itia
l v
alu
e0
V=
Val
ue
afte
r T
yea
rsT T =
Tim
e p
erio
d
Wh
atis
com
pound
inte
rest
?
Form
ula
for
sim
ple
inte
rest
and
amo
un
t
Page 3
OSWAAL ICSE Mind Maps, MATHEMATICS, Class-IX | 3
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CHAP
TER
: 3 E
XPAN
SIONS
Ex
pa
nsi
ons
Sumofcu
bes
ab
a–
ba+
b²
–²=
()
()
ab
a–
ba
ab+
b³
–³=
()
(²+
²)a
ba
ba
ab
b³
+³=
(+
) (
²–+
²)
If=
0,a+
b +
c
then
³+³+
³ =
3a
bc
abc
(+
)²=
² +
2+
²a
ba
ab
b
(–
)²=
²–
2+
²a
ba
ab
b
()³
=³
+ 3
²+
3²
+³
a +
ba
aab
bb
()³
=³
–3
²+
3²
–³
a–
ba
ab
ab
b
(a+
)² =a²
++
21 a
1 ²a
()²
=²
+²
+²
+2(
)a +
b +
ca
bc
ab +
bc+
ca
(a–
)² =a²
+–
21 a
1 ²a
Page 4
4 | OSWAAL ICSE Mind Maps, MATHEMATICS, Class-IX
( 4 )
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CHAP
TER
: 4 F
ACTO
RIZA
TION
Fa
ctor
iza
tion
Iden
titi
es
Th
ep
roce
sso
fw
riti
ng
an
exp
ress
ion
inth
efo
rmo
fte
rms
or
bra
cke
tsm
ult
ipli
edto
get
her
.
Infa
cto
riza
tio
no
f,w
esp
lit
ax
bx
c2
��
the
coef
fici
ent
of
mid
dle
term
into
b
two
par
tssu
chth
atth
esu
mo
rd
iffe
ren
ceo
ftw
op
arts
iseq
ual
tob
and
the
pro
du
cto
fth
etw
op
arts
iseq
ual
toth
ep
rod
uct
of
and
ac.
xx
² +
7+
10
= 1
,=
7,
= 1
0a
bc
ac
= 1
0 =
5 ×
2
b=
7 =
5+
2
² +
(5+
2)+
10�
xx
=²
+ 5
+2
+10
xx
x
+5)
+ 2
(+
5)=
x(x
x
+2)
(+
5)=
(xx
(–
)³=
³–
³–3
(–
)a
ba
bab
ab
ab
a–
ba+
b²
–²=
()
()
ab
ab
aab
b³
–³=
(–
) (
²++
²)
ab
a+
ba
–ab+
b³
+³=
()
(²
²)
(+
)³=
³ +
³ +
3(
)a
ba
bab
a +
b
ab
cabc=
ab+
ca
bc
ab–
bc–
ca
³ +
³ +
³–3
(+
) (
²+²+
²–
)
Page 5
OSWAAL ICSE Mind Maps, MATHEMATICS, Class-IX | 5
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CHAP
TER
: 5 S
IMUL
TANE
OUS
LIN
EAR
EQUA
TIONS
IN
TWO V
ARIA
BLES
Sim
ult
aneou
s L
inea
r
Eq
ua
tions in t
wo v
aria
ble
s
Wh
entw
oeq
uat
ion
sco
nta
insa
me
two
var
iab
les
(an
din
each
xy
case
),To
get
her
such
equ
atio
ns
are
ca
lle
ds
imu
lta
ne
ou
sli
ne
ar
equ
atio
ns,
e.g
.3
+4
=6
and
8+
5=
3x
yx
y
•R
ead
the
stat
emen
tca
refu
lly
and
iden
tify
the
un
kn
ow
nq
uan
titi
es
•R
ep
rese
nt
the
un
kn
ow
nq
ua
nti
tie
sb
yx,
y,z,
a,b
,c,e
tc.
•F
orm
ula
teth
eeq
uat
ion
sin
term
so
fvar
iab
les
tob
ed
eter
min
edan
dso
lve
the
equ
atio
ns
tog
etth
ev
alu
eso
fth
ere
qu
ired
var
iab
les.
•F
inal
ly,
ver
ify
wit
hth
eco
nd
itio
ns
of
the
ori
gin
alp
rob
lem
.
Cro
ss-
mu
ltip
lica
tio
nm
eth
od
Elimination method
Sub
stit
uti
on
met
ho
d
ax +
by +
c=
0w
her
ean
d c
are
rea
l n
um
ber
s,a, b
ab
���
��
W
ordPro
blems Alg
ebra
icm
eth
ods
toso
lve
simulta
neou
sli
nea
req
uat
ion
s.
e.g
. so
lve
2+
3y
=11
x
and
2=
–1x
–y
Fro
m 2
=–1
,x
–y
=2
+1
�y
x
No
w, S
ub
stit
ute
th
e v
alu
e o
fy
in 2
+ 3
=11
xy
2+
3(2
+ 1
)=11
�x
x
2+
6+
3=
11x
x
8=
8x
=
= 1
No
w, p
ut
the
x�
val
ue
of
in=
2+
1,
=2×
1 +
1=
3x
yx
y�
8 8e.
g. s
olv
e 3
+ 2
= 1
1....
.....(
i)x
y
2+
3=
4...
......
.(ii
)x
y
Mu
ltip
ly (
i) b
y 3
an
d (
ii)
by
2 ,
and
su
bst
ance
9+
6=
33
xy
4+
6=
8x
y 5=
25
= 5
xx
�P
ut
= 5
in
(i)
=–2
xy
So
lve
xy
xy
+=
7 a
nd
5+
12
= 7
by
cro
ss-m
ult
ipli
cati
on
, we
hav
e
x
1 12
–7 –7
=y
–7 –7
1 5
=1
1 12
1 5
x
–7–(
–84)
==
112
–5
y
–35–
(–7)
� �x
–7+
84=
=1 7
y
–35+
7
�x 77
==
1 7y
–28
�x
y=
11,
=–
4
Page 6
6 | OSWAAL ICSE Mind Maps, MATHEMATICS, Class-IX
( 6 )
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CHAP
TER
: 6 I
NDIC
ES
Ifn
isa
na
tura
ln
um
be
r,th
en
x=
x.x
.............x
xn
(nti
mes
),w
her
eis
any
real
nu
mb
er.
isca
lled
bas
ean
dn
isth
ex
exp
on
ent
or
ind
ex.
Th
ep
lura
lfo
rmo
fin
dex
isin
dic
es.
Ind
ices
x1/
no
rn �
xis
call
edro
ot
of
.n
xth
Wha
tare
ind
ices
?
Sam
ebas
es
Dividing bases
x.x
=x
;m
nm
+n
wh
ere
m,n
are
po
siti
ve
inte
ger
sx
x=
x;
m>
nm
nm
–n
�
xn
=x
–n
1
=;
nZ
�ba (
)n
ab ( )–
n
(xx
;m
, n
Zm
nm
n�
��
xx
)=
p/q
p1/q
��
qp
√x
(xy
x.y
��
nn
n
=;
0y�
( )
yxn
yxn n
If0
x=
y;
nn
n�
�x=
y;
x>
, y>
00Ifx
=x
;n
=m
nm
�an
dx��
Multi
plyingbase
s
Page 7
OSWAAL ICSE Mind Maps, MATHEMATICS, Class-IX | 7
( 7 )
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CHAP
TER
: 7 L
OGA
RITH
Ms
For
ap
osi
tiv
ere
aln
um
ber
,aa
ab
m≠
1,=
,we
say
that
ism
the
log
arit
hm
of
toth
eb
ase
.b
ao
r,m
=lo
gb
Log
ari
thm
s
Pro
du
ctru
le
Lawsoflo
gari
thm
Som
epr
oper
ties of
logar
ithm
•If
no
bas
eis
ind
icat
edth
enth
eb
ase
isu
nd
erst
oo
dto
be
10.
•B
ase
10lo
gar
ith
mis
kn
ow
nas
com
mo
nlo
gar
ith
m.
•lo
g(–
10)
has
no
mea
nin
gb
ecau
se10
10=
–10
has
no
solu
tio
n.
x
Quot
ient
rule
log
()=
mn
alo
gm
+a
log
n a
log
=a
log
–m a
log
n a
m n( )
log
()
=m
na
log
nm a
log
=m
n
log
m a
log
n a
•lo
g 1
0= 1
[
10=
10]
110
�
•lo
g 1
00=
2
[
10
= 1
00]
210
•lo
g 0
.01=
–2
[ 1
0=
0.0
1]–2
10
•lo
g 0
.001
=–3
[
10=
0.0
01]
–310
•lo
g 1
= 0
[
10
= 1
]0
10
� � � �
log
b=m
a
log
8=
3
[
2³=
8]2
�
�
log
16=
4 [
2
=16
]4
2
�
�
•lo
g=a b
log
b a1
•lo
g=
1a
a
•lo
g
1=0
10
a
Page 8
8 | OSWAAL ICSE Mind Maps, MATHEMATICS, Class-IX
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CHAP
TER
: 8 T
RIAN
GLES
Tria
ngle
s
•G
reat
ersi
de
has
gre
ater
ang
leo
pp
osi
teto
it.
•S
mal
ler
sid
eh
assm
alle
ran
gle
op
po
site
toit
.•
Eq
ual
sid
esh
ave
equ
alan
gle
so
pp
osi
teto
it.
Ifal
lth
esi
des
ofa
tria
ng
lear
eo
fd
iffe
ren
tle
ng
ths
.
Ifan
ytw
osi
des
ofa
tria
ng
lear
eo
feq
ual
len
gth
.
Ifal
lth
eth
ree
sid
eso
fa
tria
ng
lear
eo
feq
ual
len
gth
.
Ifea
chan
gle
of
atr
ian
gle
sis
anac
ute
an
gle
(<9
0°)
an
dth
eir
sum
in18
0°
Ifo
ne
ang
leo
fa
tria
ng
leis
ari
gh
t-an
gle
(=90
°)an
dth
eir
sum
in18
0°
Ifo
ne
ang
leo
fatr
ian
gle
iso
btu
se(>
90°)
and
thei
rsu
min
180°
Right-angled
Obt
use
-an
gle
d
A
B
C
A
BC
A
BC
X
YZ
A
BC
60°
50°
70°
A BC
90°
A
BC
120°
Types
oftr
ian
gles
Conditions fo
r congruency
of triangles.
RH
SA
xio
mA
SA
Axi
om
SS
SA
xio
m
SAS
Axio
m
Pe
rpe
nd
icu
lar
dra
wn
fro
ma
ver
tex
ofa
tria
ng
leto
the
op
po
site
sid
e.
Ali
ne
seg
men
tjo
inin
ga
ver
tex
of
atr
ian
gle
toth
em
id-
po
int
of
the
op
po
site
sid
e.
1.S
um
of
any
two
sid
eso
fa
tria
ng
leg
reat
erth
anth
eth
ird
sid
e2.
Of
all
the
lin
esth
atca
nb
ed
raw
nto
ag
iven
lin
efr
om
ag
iven
po
int
ou
tsi
de
it,t
he
per
pen
dic
ula
ris
the
sho
rtes
t.
A
BC
DE
FA
BC
D
Alt
itu
de
An
yc
lose
dp
lan
efi
gu
reb
ou
nd
ed
by
thre
eli
ne
seg
men
ts.
Iftw
oan
gle
san
da
(no
n-i
ncl
ud
ed)
sid
eo
fo
ne
tria
ng
lear
eeq
ual
totw
oan
gle
san
dco
rres
po
nd
ing
sid
eo
fth
eo
ther
tria
ng
le.P
QR
A
BC
Iftw
oan
gle
san
dth
ein
clu
ded
sid
eo
fo
ne
tria
ng
lear
eeq
ual
totw
oan
gle
san
dth
ein
clu
ded
sid
eo
fth
eo
ther
tria
ng
le.
P
QR
A
BC
Iftw
osi
des
and
the
incl
ud
edan
gle
of
on
etr
ian
gle
are
equ
alto
two
sid
es
an
dth
ein
clu
ded
ang
leo
fth
eo
ther
tria
ng
le.
P
QR
A
BC
P
QR
A
BC
Ifth
ree
sid
eso
fon
etr
ian
gle
are
equ
alto
thre
esi
des
of
oth
ertr
ian
gle
.
Ifth
eh
yp
ote
nu
sean
do
ne
sid
eo
fo
ne
rig
ht
tria
ng
lear
eeq
ual
toth
eh
yp
ote
nu
se
an
dco
rres
po
nd
ing
sid
eo
fth
eo
ther
rig
htt
rian
gle
.
A BC
P
QR
AA
SA
xio
m
Som
eim
por
tan
tte
rms
Inequalitiesoftri
agle
Page 9
OSWAAL ICSE Mind Maps, MATHEMATICS, Class-IX | 9
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CHAP
TER
: 9 M
ID-P
OIN
T TH
EORE
M
Mid
-poi
nt
Theo
rem
Th
eli
ne
seg
men
tjo
inin
gth
em
id-
po
ints
ofa
ny
two
sid
eso
fatr
ian
gle
isp
aral
lelt
oth
eth
ird
sid
ean
dis
equ
alto
hal
fofi
t.In
AB
C,
Dan
dE
are
the
mid
po
ints
of
�A
Ban
dA
Cre
spec
tiv
ely,
then
DE
BC
and
OE
=B
C
Th
eli
ne
thro
ug
hth
em
id–
po
into
fon
esi
de
ofa
tria
ng
lean
dp
aral
lelt
oan
oth
ersi
de
bis
ects
the
thir
dsi
de.
InA
BC
,Dis
the
mid
po
int
�o
fAB
and
DE
BC
,th
enE
isth
em
idp
oin
tofA
Can
dD
E=
BC
Conver
seof m
id–
poi
nt
theo
rem
Inte
rcep
tth
eore
m
Ifa
tran
sver
seli
ne
mak
eseq
ual
inte
rcep
to
nth
ree
or
mo
rep
aral
lel
lin
es,
the
any
oth
erli
ne
cutt
ing
nth
emal
som
akes
equ
alin
terc
epts
.H
ere,
and
AB
,C
Dar
eth
el/
/m//
ntr
ansv
erse
lin
es,t
hen
PQ
:QR
LM
:MN
�
AC
DB
P Q R
L M N
l m n
A
BC
ED
A
BC
ED
1 2
1 2
||
||
Page 10
10 | OSWAAL ICSE Mind Maps, MATHEMATICS, Class-IX
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CHAP
TER
: 10 P
YTHA
GORA
S TH
EORE
M
Py
tha
gor
as
Theo
rem
Ina
rig
ht
–an
gle
dtr
ian
gle
,th
esq
uar
eth
eh
yp
ote
nu
seis
equ
alo
fto
the
sum
of
the
squ
ares
of
the
rem
ain
ing
two
sid
es.
Ina
tria
ng
le,i
fth
esq
uar
eo
fo
ne
sid
eis
equ
alto
the
sum
oft
he
squ
ares
ofo
ther
two
sid
es,
then
the
tria
ng
leis
rig
ht
ang
led
tria
ng
le.
Mat
hemat
ical
ly
BC
A
Co
nv
erse
of
Py
thag
ora
sth
eore
m
Statement
BC
A
In
AB
C, i
f A
C²
= A
B²
+B
C²
then
B=
90°
∠A
BC
is
rig
ht-
ang
led
tri
ang
le∴
In
AB
C,
B =
90°
∠A
C²=
AB
² +
C²
B
Page 11
OSWAAL ICSE Mind Maps, MATHEMATICS, Class-IX | 11
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CHAP
TER
: 11 RE
CTIL
INEA
R FI
GURE
S
Recti
linea
r
Fig
ures
Pla
ne
fig
ure
bo
un
ded
by
lin
e se
gm
ents
.
Inre
gu
lar
po
lyg
on
,we
hav
e,(i
)A
llsi
des
are
ofe
qu
alle
ng
th(i
i)A
llin
teri
or
ang
les
are
of
equ
alm
easu
res.
(iii
)A
llex
teri
or
ang
les
are
of
equ
alm
easu
res.
Ac
lose
dp
lan
efi
gu
reen
clo
sed
by
fou
rst
raig
ht
lin
ese
gm
ents
.
AB
DC
Aq
uad
rila
tera
lh
avin
go
ne
pai
ro
fop
po
site
sid
esp
aral
lel
and
oth
erp
air
of
op
po
site
sid
esar
en
on
–p
aral
lel.
AB
DC
Atr
apez
ium
hav
ing
no
n-
par
alle
lsid
eseq
ual
.A
B
DC
Aq
uad
rila
tera
lin
wh
ich
bo
thp
air
of
op
po
site
ssi
des
are
par
alle
l.an
dal
lver
tex
ang
les
are
no
teq
ual
to90
°A
B
DC
O
Ifal
lo
fth
ean
gle
so
fa
par
alle
log
ram
isa
rig
hta
ng
le.
AB
DC
Ifal
lth
esi
de
of
ap
ar
all
elo
gr
am
(o
th
er
th
an
re
ct
an
gle
)a
re
equ
al.
AB
DC
Iftw
oa
dja
ce
nt
sid
eso
fa
rect
ang
lear
eeq
ual
.
AB
DCA
qu
adri
late
rali
nw
hic
htw
op
air
so
fa
dja
cen
tsi
des
are
equ
al.
A
CB
D
(i)
Bo
thp
airs
ofo
pp
osi
tesi
des
are
equ
al.
(ii)
Bo
thp
airs
ofo
pp
osi
tean
gle
sar
eeq
ual
.(i
ii)
On
ep
air
ofo
pp
osi
tesi
des
iseq
ual
and
par
alle
l.(i
v)
Dia
go
na
lsb
isec
tea
cho
ther
an
db
isec
tth
ep
aral
lelo
gra
m.
(v)
Rh
om
bu
sas
asp
ecia
lp
aral
lelo
gra
mw
ho
sed
iag
on
als
mee
tatr
igh
tan
gle
s.(v
i)In
are
ctan
gle
,dia
go
nal
sar
eeq
ual
,in
asq
uar
e,th
eyar
eeq
ual
and
mee
tatr
igh
tan
gle
s.
Isosceles trapezium
Kit
e
Sq
uar
e
Rh
om
bu
s
Typ
eso
fq
uad
rila
tera
l
Rec
tan
gle
Som
eth
eore
ms on
parallelogram
Co
nst
ruct
ion
s o
f p
oly
go
n
Tria
ng
leQ
uad
rila
tera
lP
aral
lelo
gra
mTr
apez
ium
Rec
tan
gle
Rh
om
bu
sS
qu
are
Page 12
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CHAP
TER
: 12
THEO
REMS
ON
AREA
Theo
rem
s
onre
aA
Th
ere
gio
nb
ou
nd
edb
yth
ep
erim
eter
oft
he
pla
ne
fig
ure
.
Are
a
IfA
B=
un
its
and
BC
=l
bu
nit
sb
eth
ele
ng
than
dth
eb
read
tho
fre
ctan
gle
AB
CD
,
�A
rea
=sq
-un
its
lb
ABC
D
lu
nit
s
bu
nit
s
L
MN
P
QR
If
LM
NP
QR
,�
then
ar
(
LM
N)
ar (
P
QR
)�
IfR
,a
nd
Ra
retw
op
oly
go
n1
2
reg
ion
s,w
hic
hh
ave
no
reg
ion
inco
mm
on
or
wh
ose
inte
rsec
tio
nis
afi
nit
en
um
ber
of
po
ints
and
ali
ne
seg
me
nt
an
dre
gio
nR
=R
R1
2�
(st
and
sfo
ru
nio
n)t
hen
ar� (R
)=ar
(R)
+ar
(R)
12
R
PQ
SR1
R2
Area addition axiom
Congruence Area AxiomTheore
mson
Are
as
•A
dia
go
nal
of
par
alle
log
ram
sd
ivid
esit
into
two
tria
ng
les
ofe
qu
alar
ea.
•P
ara
lle
log
ram
on
the
sam
eb
ase
or
eq
ua
lb
ase
sa
nd
bet
wee
nth
esa
me
par
alle
lli
nes
are
equ
alin
area
.•
Are
ao
ftr
ian
gle
ish
alf
of
ap
aral
lelo
gra
mo
nth
esa
me
bas
ean
db
etw
een
sam
ep
aral
lell
ines
•A
med
ian
oft
rian
gle
div
ides
itin
totw
otr
ian
gle
so
feq
ual
area
•Tr
ian
gle
so
nth
esa
me
bas
ean
db
etw
een
the
sam
ep
aral
lel
lin
esar
eeq
ual
inar
ea.
Tria
ngl
esw
ithth
esa
me verte
x
and
base
sal
ong
the sa
me
line
Are
a o
f
XY
ZA
rea
of
Y
TZ
=X
YY
T
Are
a o
f
XY
ZA
rea
of
X
TZ
=X
YX
T
Are
a o
f
YT
ZA
rea
of
X
TZ
=Y
TX
T
Z
XN
YT
• • •
Page 13
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CHAP
TER
: 13
CIRC
LE
Cir
cle
A s
et o
f th
ose
po
ints
(lo
cus)
in
ap
lan
e th
at a
re a
t a
giv
en c
on
stan
td
ista
nce
fro
m a
fix
ed p
oin
t.
Th
efi
xed
po
int
Cis
call
edth
ece
ntr
ean
dth
eco
nst
ant
dis
tan
cer
isca
lled
the
rad
ius.
Th
ese
to
fal
lpo
ints
Po
fth
ep
lan
e,su
chth
atC
P<
,fo
rmth
ein
teri
or
ro
fth
eci
rcle
.
Inth
eg
iven
fig
.,C
P’>
,th
eref
ore
,r
P’l
ies
ou
tsid
eth
eci
rcle
.
Ch
ord
of
circ
lep
assi
ng
thro
ug
hit
sce
ntr
ele
ng
tho
fdia
met
er=
2×ra
diu
s.
Ali
ne
join
ing
any
two
po
ints
ofa
circ
le.
rC
P
rC
O
P’
P
CQ
PC
QP
C
QS
Tan
gen
tP
•A
lin
ew
hic
hm
eets
aci
rcle
ino
ne
and
on
lyo
ne
po
int(
SQ
).•
Pis
the
po
into
fco
nta
ct.
CQ
POn
eh
alf
of
the
wh
ole
arc
oft
he
circ
leis
call
edse
mi-
circ
le.
Ali
nes
wh
ich
inte
rsec
tsa
circ
lein
two
po
ints
(PQ
).
CQ
P CQ
P
Seca
nt
Seca
nt
Tangent
•Tw
oci
rcle
sar
eca
lled
equ
al(o
rco
ng
ruen
t)if
and
on
lyif
they
hav
esa
me
rad
ius.
•A
stra
igh
tli
ne
dra
wn
form
the
cen
ter
of
aci
rcle
tob
isec
ta
cho
rdw
hic
his
no
tad
iam
eter
isat
rig
hta
ng
leto
the
cho
rd.
•T
he
per
pen
dic
ula
rto
ach
ord
fro
mth
ece
nte
rb
isec
tsth
ech
ord
.•
Eq
ual
cho
rds
sub
ten
deq
ual
ang
les
atth
ece
nte
r.
•If
two
arcs
are
equ
al,t
hen
thei
rco
rres
po
nd
ing
cho
rds
are
equ
al.
Inte
rio
ran
dE
xter
ior
po
int
Dia
mete
r
Chord
Arc
Som
eIm
portant Theore
ms and results
Inth
eg
iven
fig
,P
Qd
eno
tes
the
arc
PQ
of
the
circ
lew
ith
cen
ter
C.
Page 14
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CHAP
TER
: 14
MEN
SURA
TION
(PLA
NE F
IGUR
ES) — P
art-
1
Mensu
ra
tion
(P
lane f
igu
res)
•A
rea=
×
Bas
e×H
eig
ht
1 2 A
BC
A BC
Hei
gh
t
Bas
e
Hei
gh
t
Bas
e
or
•H
eron
’s f
orm
ula
: Are
a=√
()(
)()
ss-
as-
bs-
cw
her
e a,
b, c
are
th
e si
des
of
tria
ngl
ean
da+
b+c
2
√3
a²4
Area of Is
osce
les
trian
gle
1 4√
4²–
²a
bb
A
BC
aa
b
wh
ere
isth
eb
bas
ean
da
isth
ee
qu
al
sid
eo
ftr
ian
gle
Are
a =
Len
gth
x B
read
th
AB
DC
Len
gth
Bre
adth
Are
aof
equi
late
ral
A
BC
aa
a
AB C
D
sid
e
sid
e
sid
e
sid
e
Are
a =
sid
e ×
sid
eRectangle
Square
AB
DC
Bas
e
Hei
gh
t
Are
a =
Bas
e× H
eig
ht
AB
DC
d1 d2
Are
a =
(
d×
d)
12
1 2
Rhombus
Parallelogram
Trap
eziu
m
Are
a =
(
sum
of
par
alle
lsi
des
) ×
hei
gh
t
1 2
Areaand CircumferenceofcircleCir
cum
fere
nce
Area
orpe
rim
eter
rO
C
C=
2r�
A=
²�r
A
BC
ab
c
Peri
met
er =
a+b+
cr
OB
R
C
A=
(R²–
r²)
�
A
BC
aa
a
A
BC
aa
b
Equilateral
Isos
cele
s
Peri
met
er =
3a
Peri
met
er =
2a+
b
AB
CD
sid
e
sid
e
sid
e
sid
e
Rec
tan
gle
Squar
e
AB
DC
l
b
Peri
met
er =
2(
)l+
b
Peri
met
er =
4×
sid
e
AB
DC
a
a
aa
Peri
met
er =
4×
a
Rhombus
tria
ngle
AB
DC
Hei
gh
t
Qu
arte
rci
rcle
A=
�r²
2
A=
�r²
4
Or
Or
s =
=A
=A
Page 15
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CHAP
TER
: 14
MEN
SURA
TION
(SOLI
DS 3
–D)—
Par
t-2
Mensu
ra
tion
(S
oli
ds) 3
–D
Cubo
id
Cube
Are
ctan
gu
lar
soli
dw
ith
six
face
sal
lre
ctan
gu
lar
insh
ape.
Len
gth
Bre
adth
Hei
gh
t
Dia
gonal
Volume
2()
lb+
bh+
hl
2(
+)
hl
b
�(+
²+)
l²b
h²
Len
gth
()
× b
read
th (
)×h
eig
ht
()
lb
h
a
a
a
What is cube?
Volum
e
Tota
lsu
rfac
eA
rea
Aso
lid
inw
hic
hea
chfa
ceis
asq
uar
e.i.
e.le
ng
th,
bre
adth
and
hei
gh
tar
eeq
ual
say
un
it.
a
Len
gth
× b
read
th ×
hei
gh
t³=
(ed
ge)
³=
a ×
a ×
a=a
6²a
4²a
�(+
²+)
=a²
aa²
�3a
Page 16
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CHAP
TER
: 15
TRIG
NOMEN
TRY
Tri
gnom
entr
y
Fo
r ri
gh
t-an
gle
d t
rian
gle
.
A BC
90°
Hei
gh
tH
yp
ote
nu
se
•si
nθ
=
=
Hei
gh
tH
yp
ote
nu
seA
BA
C
•co
sθ
=
=
Bas
eH
yp
ote
nu
seB
CA
C
Bas
e
•ta
nθ
=
=
Hei
gh
tB
ase
AB
BC
•co
tθ
=
=
Hei
gh
tB
ase
BC
AB
•se
cθ
=
=
Bas
e
Hy
po
ten
use
BC
AC
•co
sec
θ=
=
Hei
gh
t
Hy
po
ten
use
AB
AC
Recipro
cal Relat
ions
•co
sec
or
sin
θ =
θ =
1si
nθ
1co
sec
θ
•se
co
r c
os
θ =
θ =
1co
sθ
1se
cθ
•co
to
r t
anθ
=θ
=1
tan
θ1
cot
θ
•ta
nθ
=si
nθ
cos
θ
•co
t θ=
sin
θco
sθ
•si
n²
cos²
θ +
θ=1
•se
c²–
tan
²θ
θ=1
•co
sec²
–co
t²θ
θ=1
θ0°
30°
45°
60°
90°
sin
θ
cos
θ
tan
θ
cot
θ
sec
θ
cose
cθ
0 1 0 ∞ 1 ∞
1 21 0 0 1∞∞
111 √2 1 √2
1 21 √3
√3
√2
√2
√3
2 √3 2
2 √3
21 √32√3
2√3
Trig
onom
etri
c rat
ios of
som
est
anda
rdan
gles
Trigonometric Ratio of
complementary angle
•si
n–
cos
(90°
θ) =
θ•
cos
–si
n(9
0°θ)
=θ
•co
sec
–se
c(9
0°θ)
=θ
•se
c–
cose
c(9
0°θ)
=θ
•ta
n–
cot
(90°
θ) =
θ•
cot
–ta
n(9
0°θ)
=θ
θ
Page 17
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CHAP
TER
: 16
COORD
INAT
E GE
OMET
RY
Coo
rdin
ate
Geo
met
ry
O(0
,0)
(–v
e)-a
xis
y
x-a
xis
(–v
e)-a
xis
x
(+a,
+b
)(–
a, +
b)
(–a,
–b)
(+a,
–b)
1st
Qu
adra
nt
2nd
Qu
adra
nt
4th
Qu
adra
nt
3rd
Qu
adra
nt
5 4 3 2 1 –1 –2 –3 –4 –5
12
34
5–1
–2–3
–4–5
(–2,
4)
Plo
t a
po
int
(–2,
4)
•W
hen
th
e li
ne
isto
-axi
s, i
ts s
lop
e is
��x
zero
or
= t
an0°
= 0
m
•W
hen
th
e li
ne
isto
-axi
s, i
ts s
lop
e is
���
y
or
tan
90°
= n
ot
def
ined
.m
=
xx’
y y’
B
A
c
O
Th
eg
rap
ho
flin
ear
equ
atio
nis
alw
ays
ast
raig
htl
ine.
We
can
fin
dth
ev
alu
eo
fif
val
ue
x
of
yis
giv
enso
isd
epen
den
tx
va
ria
ble
an
dis
ind
ep
en
de
nt
y
var
iab
le.
We
can
fin
dv
alu
eo
fif
val
ue
y
of
isg
iven
soy
isd
epen
den
tx
var
iab
lean
dis
ind
epen
den
tx
var
iab
le.
xy
=+
–b a
–c a
Ifa
lin
eax
+b
y+
c=o
then
,
Ifa
lin
eax
+b
y+
c=o
then
,y=
–a b
–c b
x+
Ste
p1:
Dra
wth
eg
rap
ho
fea
chli
nea
req
uat
ion
on
the
sam
eg
rap
hp
aper
.S
tep
2:
Th
en
we
de
term
ine
the
coo
rdin
ates
oft
he
po
ints
ofi
nte
rsec
tio
no
ftw
oli
nes
dra
wn
.S
tep
3:
Th
eco
ord
inat
eso
fth
ep
oin
to
fin
ters
ecti
on
of
two
lin
esis
the
req
uir
edso
luti
on
.
Gra
phic
also
lutio
n
of simulta
neous linear equatio
n
Dis
tance
between
two
points
dx
xy
y=
√(
–)²
+ (
–)²
12
12
dx
y=
√(
–0)²
+ (
–0)²
11
dx
y=
√²
+²
11
y-a
xis
(+v
e)
(+v
e)
Plo
ttin
ga
Poin
t
y=
mx+
c
y=
x t
an
+c
�
Inclination/Slope of a straight line
Graphoflineareq
uatio
n
O
y'
x'
x
y
�
Page 18
18 | OSWAAL ICSE Mind Maps, MATHEMATICS, Class-IX
( 18 )
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CHAP
TER
: 17
STAT
ISTI
CS
Sta
tist
ics
Dis
con
tin
uo
us
(Dis
cret
e)
Av
aria
ble
wh
ich
can
tak
ean
yv
alu
ew
ith
ina
cert
ain
ran
ge.
Av
ari
ab
lew
hic
hca
nn
ot
assu
me
any
val
ue
bet
wee
ntw
og
iven
val
ues
.
Ase
to
fn
um
eric
al
fact
sco
llec
ted
wit
hso
me
def
init
eo
bje
ct.
Dat
a
Dat
a co
llec
ted
in
its
ori
gin
al f
orm
.
Ori
gin
ald
ata
pre
sen
ted
ina
par
ticu
lar
ord
er.
(i.e
.asc
end
ing
or
des
cen
din
go
rder
)•
Bar
gra
ph
•H
isto
gra
m•
Fre
qu
ency
po
lyg
on
Tab
ula
rar
ran
gem
ent
of
the
giv
enn
um
eric
ald
ata
sho
win
gth
efr
equ
ency
ofd
iffe
ren
tvar
iate
s.
Cla
ss-m
ark
s
Cla
ss-i
nte
rval
s
Cla
ssin
terv
alar
e1-
10,1
1-20
,21
-0,
31-4
0,41
-50
ing
iven
3e
xa
mp
leo
ffr
eq
ue
nc
yd
istr
ibu
tio
n.
Th
em
id-p
oin
tofa
clas
sis
the
clas
sm
ark
so
fth
ecl
ass.
�C
lass
-mar
k=
Up
per
cla
ss l
imit
+ l
ow
er c
lass
lim
it
2
Aq
ua
nti
tyw
hic
his
mea
sure
din
exp
erim
ent.
Variable
Mea
n and
med
ian o
f
ungrouped
data
Array
ed
Mea
n(f
orit
ems)
n
Median(fo
rNite
ms)
IfN
iso
dd
Med
ian
=v
alu
e o
f
(
)N
+1
2
th
item
Mea
n,
=x
�n i=1
xi
n
Med
ian
=(
)N 2
th
item
+1 2
(
)N
+1
2
th
item
[
]
e.g
.fr
equ
ency
dis
trib
uti
on
tab
le
Cla
ss i
nte
rval
Tell
y m
ark
sF
req
uen
cy
1-10
11-2
0
21-3
0
31-4
0
41-5
0
Tota
l
4 7 6 6 7 30
