(unit i) chapter : 1 real numbers
TRANSCRIPT
Oswaal CBSE Chapterwise Mind Maps, MATHEMATICS (STANDARD), Class – X [ 1
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(Unit
I) C
HAPT
ER :
1 re
al n
umbe
rs
2 ] Oswaal CBSE Chapterwise Mind Maps, MATHEMATICS (STANDARD), Class – X
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(Unit
II) CH
APTE
R : 2
poly
nomial
s
If(
) an
d(
) are
px
gx
two
po
lyn
om
ials
wit
h
()
0,
then
–g
x≠
()=
()
×(
) +
()
px
gx
qx
rx
wh
ere,
()
= 0
or
rx
deg
ree
of
()
< d
egre
er
x
of
()
gx
Hig
hes
t p
ow
er o
fin
Po
lyn
om
ial,
()
xp
x
Div
isio
n A
lgorithm
Rela
tionship
-Zeroes a
nd
Coeffic
ient
of P
oly
nom
ials
Gra
phic
al R
ep
resenta
tion
Qua
dra
tic P
oly
nom
ial
Degree o
f P
oly
nom
ial
Par
abol
a
yx
x=
–3
–4
2yy
xx
Poly
nom
ials
Case
1-
Gra
ph
cu
ts-a
xis
at
2 p
oin
tsx N
um
ber
of
Zer
oes
2
Case
2-
Gra
ph
cu
ts-a
xis
at
exact
ly o
ne
po
int
x
Nu
mb
er o
fZ
ero
es 1
Case
3-
Gra
ph
do
es n
ot
cut
-ax
isx
Nu
mb
er o
fZ
ero
es 0
Po
lyn
om
ial
Deg
ree
Gen
eral
Fo
rm
Lin
ear
1ax
b+
Qu
ad
rati
c2
ax
bx
c2+
+
0a
≠
Cu
bic
3ax
bx
cxd
32
++
+
0a
≠
��
an
dare
zer
oes
of
Qu
ad
rati
cP
oly
no
mia
l
Th
en,
Su
m o
fzer
oes
,
ax
bx
c2
++
��
+=
–b a
Pro
du
ct o
fzer
oes
��
=c a
Qu
ad
rati
c
��
�,
an
dare
zer
oes
of
Cu
bic
Po
lyn
om
ial
Su
m o
fzer
oes
,
ax
bx
cxd
32
++
+
��
�+
+=
–b a
��
��
��+
+=
c a
��
�=
–d a
Su
m o
fp
rod
uct
s o
fth
ezer
oes
tak
en t
wo
at
a t
ime
Pro
du
ct o
fzer
oes
Cu
bic
Zeroes o
f P
oly
nom
ial
Gra
phic
ally
Typ
es
are
two
po
lyn
om
ials
wit
h
0,
then
–
×(
) +
()
px
gx
qx
) p
xg
xq
x)
×(
px
gx
qx
×(
) p
xg
xq
x)
rx
+(rx
+(
)r
x)
wh
ere,
()
= 0
or
rx
,(rx
,(
) =
0 o
rr
x)
= 0
or
deg
ree
of
()
< d
egre
er
x()
rx()
of
()
gx()
gx()
Div
isio
n A
lgorithm
Degree o
f
Poly
nom
ials C
ase
3-
Gra
ph
do
es n
ot
cut
-ax
isd
oes
no
t cu
t-a
xis
xd
oes
no
t cu
t-a
xis
Po
lyn
om
ial
Lin
ear
Qu
ad
rati
c
Oswaal CBSE Chapterwise Mind Maps, MATHEMATICS (STANDARD), Class – X [ 3
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Chap
ter
: 3 p
air
of
line
ar e
qua
tions
in t
wo v
ariabl
es
Pa
ir o
f Li
nea
r Eq
uations
in T
wo
Va
ria
ble
s
Alg
eb
ra
ic M
eth
od
s
Gra
phic
al R
ep
resenta
tion
So
lve:
2+
3–46 =
0–(i
)x
y
3+
5–74 =
0–(i
i)x
y
So
luti
on
: B
y c
ross
-mu
ltip
lica
tio
n m
eth
od
xy
13 5
–46
–74
2 3
3 5
Then
,=
x3(–
74)–
5(–
46)
y
(–46)(
3)–
(–74
)(2)
=1
2(5
)–3(3
)
=x
–222+
230
=y
–13
8+
148
1
10–
9
=x 8
=1 1
y 10an
d=
x 8
1 1=
y 10
1 1⇔
i.e.
xy
= 8
an
d=
10
ax
by
c1
11
++
= 0
++
= 0
ax
by
c2
22
,,
,,
,,
–R
eal
nu
mb
ers
ab
ca
bc
11
12
22
Genera
l Fo
rm
Each
so
luti
on
(,
), c
orr
esp
on
ds
xy
to a
po
int
on
th
e li
ne
rep
rese
nti
ng
the
equ
ati
on
an
d v
ice-
ver
sa
Solu
tion G
ra
phic
ally
Gra
ph
ical
Rep
rese
nta
tio
n
Inte
rsec
ting
Lines
Pair
of
Lin
es:
=–2
= 0
xy
3+
4–20 =
0x
y
a1
a2=
1 3,
b1
b2
=–2 4
,c 1 c 2
=0
–20
Co
mp
are
th
e R
ati
os:
=≠
a1
a2
b1
b2
Alg
ebra
ic I
nte
rpre
tati
on
: E
xact
lyo
ne
solu
tio
n–
con
sist
ent
(un
iqu
e)
Pair
of
Lin
es:
= 2
+3
–9 =
0x
y
4+
6–18 =
0x
y
a1
a2=
2 4,
b1
b2
=3 6
,c 1 c 2
=–9
–18
Co
mp
are
th
e R
ati
os
==
a2
b1
b2
Alg
ebra
ic I
nte
rpre
tati
on
= I
nfi
nit
ely
man
y s
olu
tio
ns
–D
epen
den
t=c
1
c2
Gra
phic
al R
epre
senta
tion
Coi
nci
dent
Lines
Pair
of
Lin
es:
=+
2–4 =
0x
y
2+
4–12 =
0x
y
a1
a2=
1 2,
b1
b2
=2 4
,c 1 c 2
=–4
–12
Co
mp
are
th
e R
ati
os
==
a1
a2
b1
b2
Alg
ebra
ic I
nte
rpre
tati
on
: N
o s
olu
tio
n–
Inco
nsi
sten
t
≠
c 1 c 2
Gra
phic
al R
epre
senta
tion
Par
alle
l Li
nes
So
luti
on
: F
rom
eq
uati
on
(ii
),=
3–2
xy
sub
stit
ute
valu
e o
fin
eq
. (i
)x
7(3
–2
)–15
= 2
yy
–29
=–19
=y
y⇔
= 3
–2
x=
19
29
19
29
49
29
So
lve:
7–15
= 2
–(i
)x
y
+2
= 3
–(i
i)x
yBy
Subs
titu
tion
So
lve:
+3
= 6
xy
2+
3=
12
xy
No
w,
Ad
din
g e
qu
ati
on
(i)
an
d (
ii)
3=
18 o
r=
6x
x
Again
, fr
om
(i)
×2–
(ii)
3y =
0 o
r, y
= 0
Hen
ce,
x =
6,
y =
0
By
Elim
inat
ion
No
w,
fro
m=
3–
2x
y
–(i
)
–(i
i)
a1
By C
ross
-Mu
ltip
lica
tio
n
ir o
f Li
nea
r Eq
uaEq
uaEq
tions
Two
Va
ria
ble
s
Alg
eb
ra
ic M
eth
od
s
on
Co
mp
are
th
e R
ati
os:
=
Alg
ebra
ic I
nte
rpre
tati
on
: E
xact
lyo
ne
solu
tio
n–
con
sist
ent
(un
iqu
e)
Pair
of
Lin
es:
= 2
+3
–9 =
0x
y2
+x
y2
+3
–9 =
0x
y3
–9 =
04
+6
–18 =
0x
y4
+x
y4
+6
–18 =
0x
y6
–18 =
0
a1=
2 4,
b1
b2
=3 6
,c 1c 1c c 2
=–9
–18
Co
mp
are
th
e R
ati
os
=a
Alg
ebra
ic I
nte
rpre
tati
on
= I
nfi
nit
ely
a1
4 ] Oswaal CBSE Chapterwise Mind Maps, MATHEMATICS (STANDARD), Class – X
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CHAP
TER
: 4 Q
uadr
atic E
qua
tions
Ro
ots
are
2 3,
–1 2–1 2
2 3x
=o
r=
x
Th
e ro
ots
of
6–
–2=
0x
x2
(3–2)
= 0
or
(2+
1)
= 0
xx
By F
act
ori
zati
on
Fin
d r
oo
ts o
f6
––2=
0x
x2
So
luti
on
: 6
+3
–4
–2 =
0x
xx
2
3(2
+1)–
2(2
+1)
= 0
xx
x
(3–2)(
2+
1)
= 0
xx
Qua
dra
tic
Eq
ua
tions
Eq
uati
on
of
deg
ree
2,
in o
ne
vari
ab
le
Mea
nin
g
ax
bx
c2+
+=
0,
,–
real
nu
mb
ers
ab
c
0a
≠
Genera
l Fo
rm
Qua
dra
tic F
orm
ula R
oo
ts o
f+
+=
0 a
re g
iven
by
ax
bx
c2 –
±b
bac
2–4
2a
Fo
r q
uad
rati
c eq
uati
on
++
= 0
,ax
bx
c2
–4
is D
iscr
imin
an
t (D
)b
ac
2
3.
D <
0N
o r
eal
roo
ts (
imagin
ary
)
So
luti
on
: 2
–5
+3 =
0x
x2
x2–
x+
= 0
5 2
3 22
2
–+
= 0
⇔
⇔
x–
5 4
x–
5 4
5 4
3 2
=±
1 4
x=
or
=x
5 4
1 4+
5 4
1 4–
x=
or
= 1
x3 2
2
–=
0x–
5 4
1 16
2
=1 16
x–
5 4
So
lve:
2–5
+3 =
0x
x2
By
Com
plet
ing
the
Squ
are
Solu
tion o
f a
Qua
dra
tic E
qua
tion
Na
ture o
f R
oots
2.
D =
0T
wo
eq
ual
real
roo
ts
1.
D >
0T
wo
dis
tin
ct r
eal
roo
t
Qua
dra
tic
Eq
ua
Eq
ua
Eq
tions
Qua
dra
tic
Ro
ots
of
= 0
1
1.
D >
0o
dis
tin
ct r
eal
roo
t
Oswaal CBSE Chapterwise Mind Maps, MATHEMATICS (STANDARD), Class – X [ 5
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CHAP
TER
: 5 A
rith
metic
progre
ssions
Arithm
etic
Prog
ressio
ns
Defin
itio
n
Genera
l fo
rm
Com
mon D
ifference
Exa
mp
les
Ho
w m
an
y 2
-dig
it n
um
ber
sare
div
isib
le b
y 3
?
2-d
igit
nu
mb
ers
div
isib
le b
y 3
are
12
, 1
5,
18
, ..
. 9
9=
12
,=
3,
a=
99
ad
n
=+
(–1
)a
an
dn
99
= 1
2 +
(–1
)3n
n=
30
n–1
=i.
e.,
= 2
98
7 3
Su
m o
ffi
rst
n p
osi
tive
inte
ger
sL
et s
= 1
+ 2
+ 3
+ .
..n
n
= 1
, la
st t
erm
=a
ln
s=
nn
l(a
+)
2=
nn
(1+
)
2
Nth
term
Wh
en f
irst
& l
ast
ter
ms
are
giv
en :
or
S=
(+
)n
na
an 2
S=
(
+)
na
ln 2
afi
rst
term
�
tota
l te
rms
n�
term
an
n�
th
last
ter
ml
�
Wh
en f
irst
ter
m a
nd
co
mm
on
dif
fern
ce a
re g
iven
:
afi
rst
term
�
com
mo
n d
iffe
ren
ced
�
tota
l te
rms
n�
S=
(2
+(
–1
))
na
nd
n 2
Sum
(S)
If,
,,
are
in
AP,
ab
c
bis
ari
thm
etic
mea
n
b=
ac
+ 2
Arithm
etic m
ea
n
aa
da
da
da
nd
,+
,+
2,
+3
, ..
.+
(–1
)
Lis
t o
fn
um
ber
s in
wh
ich
each
ter
m i
so
bta
ined
by a
dd
ing
a f
ixed
nu
mb
er t
o t
he
pre
ced
ing
term
ex
cep
t th
e fi
rst
term
. F
ixed
nu
mb
er i
sca
lled
co
mm
on
dif
fere
nce
.
•Fix
ed n
um
ber
in
ari
thm
etic
pro
gre
ssio
n w
hic
h p
rov
ides
th
eto
an
d f
ro t
erm
s by a
dd
ing
/su
btr
act
ing
fro
m t
he
pre
sen
t n
um
ber
.•C
an
be
po
siti
ve
or
neg
ati
ve.
Fro
m b
egin
nin
g=
+(
–1
)a
an
dn
firs
t te
rma
�
com
mo
n d
iffe
ren
ced
�
term
an
n�
th
Fro
m t
he
end
=–
(–1
)a
ln
dn
last
ter
ml
�
com
mo
n d
iffe
ren
ced
�
term
an
n�
th
Her
eH
ere
Arithm
etic
og
ressio
ns
Nth
term
Fro
m b
egin
nin
g=
+(
–1
)a
a=
+a
a=
+n
d(
–1
)n
d(
–1
)n
aa
na
a firs
t te
ra
�
com
mo
n d
iffe
ren
ced
�
an
na
nn
an
Her
e
6 ] Oswaal CBSE Chapterwise Mind Maps, MATHEMATICS (STANDARD), Class – X
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(Unit
III)
CHA
PTER
: 6
line
s (in t
wo-d
imen
sions
)
Lines (
In T
wo-
Dim
ensio
ns)
Mea
nin
g
Centr
oid
Coord
ina
te a
xis
Ho
rizo
nta
l =
–ax
is (
Ab
scis
sa)
x
Ver
tica
l =
–ax
is (
Ord
inate
)y
ord
inate
Y
ab
scis
saXX
’
Y’
Dis
tance
form
ula
Are
th
e fo
llo
win
g p
oin
ts v
erti
ces
of
asq
uare
: (
1,
7),
(4
, 2
), (
–1
,–1
), (
–4
, 4
)?A
(1
, 7
); B
= (
4,
2);
C =
(–1
,–1
); D
= (
–4
, 4
)
Sin
ce,
AB
= B
C =
CD
= D
A a
nd
AC
= B
D.
All
fo
ur
sid
es a
nd
dia
go
nals
are
eq
ual
Hen
ce,
AB
CD
is
a s
qu
are
AB
=
BC
=
CD
=
DA
=
AC
=
BD
=68
√
(4 +
4)
+ (
2–
4)
22
√
68
√
(1 +
1)
+ (
7 +
1)
22
√
34
√
(1 +
4)
+ (
7–
4)
22
√
34
√
(–1
+ 4
)+
(–1
–4
)2
2
√
34
√
(4 +
1)
+ (
2 +
1)
22
√
34
√
(1–
4)
+ (
7–
2)
22
√
= = = = = =
Area
of T
ria
ngle
Mid
-poin
t Li
ne S
egm
ent
Section f
orm
ula
Fin
d p
oin
t o
fT
rise
ctio
n o
fli
ne
seg
men
t A
B,
A(2
,–2
) an
d B
(–7
, 4
)
Co
ord
inate
of
P
i.e.
,(–
1,
0)
Co
ord
inate
of
Q
i.e.
,(–
4,
2)
=1
(–7
) +
2(2
) ,1
+ 2
1(4
) +
2(–
2)
1 +
2
2(–
7)
+ 1
(2) ,
2 +
1
2(4
) +
1(–
2)
2 +
1
A (
,)
xy
11
(,
)x
y2
2(
,)
xy
33
BC
G
++
xx
x1
23
3
yy
y1
23
++
,3
G=
xx
12
+R
,
2
yy
12
+ 2
mx
mx
12
21
±
mm
12
±
my
my
12
21
±
mm
12
±
Inte
rnall
y (
+)
Ex
tern
all
y (
–)
PQ
=√
(–
)+
(–
)x
xy
y2
12
21
2
Are
a =
|[
(–
) +
(–
)x
yy
xy
y1
23
23
1+
(–
)]|
xy
y3
12
1 2
Q=
Stu
dy
of
alg
eb
raic
eq
ua
tio
ns
on
gra
ph
s
IQ
uad
ran
t
IVQ
uad
ran
tII
IQ
uad
ran
t
IIQ
uad
ran
tX’
Y’Y
X
(–,+
)(+
,+)
(–,–
)(+
,–)
Ex
am
ple
Ex
am
ple
Lines (
In T
wo-
Dim
ensio
ns) D
ista
nce
form
ula
A (
1,
7);
B =
(4
, 2
); C
= (
–1
,–1
); D
= (
–4
, 4
)
AB
=
BC
=
CD
=
DA
√
CD
=√
CD
=
√
of T
ria
ngle
Tria
ngle
T
(–
)+
(–
)y
y(
–y
y(
–2
21
yy
21
yy
(–
yy
(–
21
(–
yy
(–
2
Oswaal CBSE Chapterwise Mind Maps, MATHEMATICS (STANDARD), Class – X [ 7
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(Unit
Iv) CH
APTE
R : 7
trian
gle
s
Tria
ngle
s
1.
Ifa l
ine
is d
raw
n p
ara
llel
to
on
e si
de
of
a t
rian
gle
to
in
ters
ect
the
oth
er t
wo
sid
es i
n d
isti
nct
po
ints
,th
e o
ther
tw
o s
ides
are
div
ided
in t
he
sam
e ra
tio.
If,
DE
||
BC
AD
DB
=A
E
EC
then
2.
Ifa l
ine
div
ides
an
ytw
o s
ides
of
a t
rian
gle
in
th
esa
me
rati
o,
then
th
e li
ne
isp
ara
llel
to
th
e th
ird
sid
e.th
en,
DE
||
BC
AD
DB
=A
E
EC
If
3.
Ifin
tw
o t
rian
gle
s,co
rres
po
nd
ing
an
gle
s are
eq
ual,
then
th
eir
corr
esp
on
din
g s
ides
are
in
the
sam
e ra
tio
(o
r p
rop
ort
ion
) an
dh
ence
th
e tw
o t
rian
gle
s are
sim
ilar.
(AA
A c
rite
rio
n)
∆∆
AB
CD
EF
�
IfA
=D
,B
=E
,∠
∠∠
∠
C =
F∠
∠
then
,A
B
DE
=B
C
EF
=A
C
DF
4.
Ifin
tw
o t
rian
gle
s, s
ides
of
on
e tr
ian
gle
are
pro
po
rtio
nal
to (
in t
he
sam
e ra
tio
of
) th
ei.
e.,
sid
es o
fth
e o
ther
tri
an
gle
, th
enth
eir
corr
esp
on
din
g a
ng
les
are
equ
al
an
d h
ence
th
e tw
o t
rian
gle
sare
sim
ilia
r.(S
SS
cri
teri
on
)
then
,A
=D
,B
=E
,∠
∠∠
∠
C =
F∠
∠
AB
DE
=B
C
EF
=C
A
FD
If
AB
C∆
∆D
EF
�
Area
of S
imila
r T
ria
ngle
s
InA
BC
, le
t D
EB
C.
Th
en,
∆
(i)
AD
DB
=A
E
EC
(ii)
(iii
)
AB
DB
=A
C
EC
AD
AB
=A
E
AC
Theorem
s
Sum
mar
y
Sim
ilarity
Rig
ht
ang
led
tria
ngle
theorem
Py
tha
gora
s
(i)
Co
rres
po
nd
ing
an
gle
s are
eq
ual
(ii)
Co
rres
po
nd
ing
sid
es a
re i
n t
he
sam
e ra
tio
A
BC
P
QR
∆∆
AB
CP
QR
~
1.
Ifa p
erp
end
icu
lar
is d
raw
n f
rom
th
e ver
tex
of
the
rig
ht
an
gle
of
a r
igh
t tr
ian
gle
to
th
e h
yp
ote
nu
seth
en t
rian
gle
s o
n b
oth
sid
es o
fth
e p
erp
end
icu
lar
are
sim
ilar
to t
he
wh
ole
tri
an
gle
an
d t
o e
ach
oth
er.
In r
igh
tA
BC
, B
DA
C,
∆⊥
then
,A
DB
AB
C∆
∆~
~∆
∆B
DC
AB
C
∆∆
AD
BB
DC
~
A
B
CD
3.
In a
tri
an
gle
, if
squ
are
of
on
esi
de
is e
qu
al
to t
he
sum
of
the
squ
are
so
fo
ther
tw
o s
ides
, th
en t
he
an
gle
op
po
site
the
firs
t si
de
is a
rig
ht
an
gle
.
IfA
C=
AB
+B
C2
22
then
,B
= 9
0∠
°
A BC
2.
In a
rig
ht
tria
ng
le,
the
squ
are
of
the
hyp
ote
nu
se i
s eq
ual
to t
he
sum
of
the
squ
are
s o
fth
e o
ther
two
sid
es.
In r
igh
tA
BC
,∆
BC
= A
B+
AC
22
2
C
BA
5.
Ifo
ne
an
gle
of
atr
ian
gle
is
equ
al
to o
ne
an
gle
of
the
oth
er t
rian
gle
an
d t
he
sid
es i
ncl
ud
ing
th
ese
an
gle
s are
pro
po
rtio
nal,
then
th
e tw
o t
rian
gle
s are
sim
ilar.
(SA
S c
rite
rio
n)
then
,A
BC
DE
F∆
∆~
AB
DE=
AC
DF
&A
=D
∠∠
If
Th
e ra
tio
of
the
are
as
of
two
sim
ilar
tria
ng
les
is e
qu
al
to t
he
squ
are
of
the
rati
oo
fth
eir
corr
esp
on
din
g s
ides
Her
eA
BC
PQ
R∆
∆~
B
A
CM
Q
P
RN
ar(
AB
C)=
AB
2
ar(
PQ
R)
PQ
=
BC
QR
2 =
CA
RP
2
Tria
ngle
Tria
ngle
Ts
ria
ngle
s
Theorem
s
Sum
mar
y
Sim
ilarity
(i)
Co
rres
po
nd
ing
an
gle
s are
eq
ual
(ii)
Co
rres
po
nd
ing
sid
es a
re i
n t
he
sam
e ra
tio
A
Th
e ra
tio
of
the
are
as
of
two
sim
ilar
tria
ng
les
is e
qu
al
to t
he
squ
are
of
the
rati
oo
fth
eir
corr
esp
on
din
g s
ides
8 ] Oswaal CBSE Chapterwise Mind Maps, MATHEMATICS (STANDARD), Class – X
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CHAP
TER
: 8 c
ircl
es
Cir
cle
s
Defin
itio
n
O
rra
diu
s cen
tre
p
Th
e lo
cus
of
a p
oin
teq
uid
ista
nt
fro
m a
fix
edp
oin
t. F
ixed
Po
int
isca
lled
a c
entr
e &
sep
ara
tio
n o
fp
oin
ts i
s th
e ra
diu
so
fci
rcle
.
Non-i
nte
rsecting lin
e
O
rP
P Q
No
co
mm
on
po
int
bet
wee
nli
ne
PQ
an
dci
rcle
.
Seca
nt
Tangent
and
ta
ngent
poin
t
O
r
P
A
Q
On
ly o
ne
com
mo
np
oin
t b
etw
een
cir
cle
an
d P
Q l
ine.
Theorem
s
Facts
2.
Th
ere
is o
ne
an
d o
nly
on
e ta
ngen
t to
a c
ircl
ep
ass
ing
th
rou
gh
a p
oin
tly
ing
on
th
e ci
rcle
.
P
1.
Th
ere
is n
o t
an
gen
tto
a c
ircl
e p
ass
ing
thro
ug
h a
po
int
lyin
gin
sid
e th
e ci
rcle
.
P
3.
Th
ere
are
ex
act
ly t
wo
tan
gen
ts t
o a
cir
cle
thro
ug
ha p
oin
t ly
ing
ou
tsid
eth
e ci
rcle
.
T1
P
T2
Q
X
P
Y
O
OP
Q =
90
°
Th
e ta
ngen
t at
an
yp
oin
t o
fa c
ircl
e is
per
pen
dic
ula
r to
the
rad
ius
thro
ug
hth
e p
oin
t o
fco
nta
ct
PQ
= P
R
O
Q R
P
Th
e le
ng
th o
fta
ngen
ts d
raw
nfr
om
an
ex
tern
al
po
int
to a
cir
cle
are
eq
ual
O
r
Q
B
AP
Tw
o c
om
mo
np
oin
ts b
etw
een
lin
e P
Q a
nd
circ
le.
Cir
cle
s
Seca
nt
Facts
Oswaal CBSE Chapterwise Mind Maps, MATHEMATICS (STANDARD), Class – X [ 9
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CHAP
TER
: 9 c
ons
truc
tions
Const
ructions
Tangent
to c
ircle
Ste
p 3
. Jo
in P
Qan
d P
R,
req
uir
edta
ngen
ts t
oth
e ci
rcle
Ste
p 1
. Jo
in P
Oan
d b
isec
t it
at
mid
-po
int
M
Defin
itio
n
To
dra
wgeo
met
rica
l sh
ap
es u
sin
gco
mp
ass
es,
rule
r et
c
Line S
egm
ent
Div
isio
nof lin
e in r
atios
Giv
en:
Lin
e se
gm
ent,
rati
o (
3 :
2)
Ste
p 3
. T
hro
ug
hA
(=
3),
dra
w3
m
lin
e p
ara
llel
to
BA
5
cutt
ing
AB
at
CA
C :
CB
= 3
:2
Ste
p 2
. L
oca
te 5
po
ints
A,
A,
A,
A,
Aat
equ
al
12
34
5
dis
tan
ces
(A=
A=
A=
A=
A).
12
34
5
Join
BA
5
Ste
p 1
. D
raw
an
y r
ay A
X,
Mak
ing
acu
te a
ng
lew
ith
lin
e se
gm
ent
AB
Giv
en:
Lin
e se
gm
ent,
rati
o (
3 :
2)
1.
Dra
w a
ny r
ay A
X m
ak
ing
an
acu
te a
ng
le w
ith
lin
ese
gm
ent
AB
2.
Dra
w r
ay B
Y |
| A
X
3.
Lo
cate
A,
A,
A1
23
(=
3)
on
AX
an
dm
B,
B(
= 2
) o
n B
Y1
2n
Join
AB
, in
ters
ecti
ng
32
AB
at
CA
C :
CB
= 3
: 2
Tria
ngle
sim
ilar t
o g
iven
tria
ngle
3.
Dra
w l
ine
para
llel
to
BC
4
fro
m B
inte
rsec
tin
g B
C a
t C
'.3
Dra
w l
ine
||
to
AC
fro
m C
'in
ters
ecti
ng
AB
at
A'
A'B
C'
is t
he
req
uir
ed t
rian
gle
Ste
p 2
. M
as
cen
tre
an
dra
diu
s =
MO
dra
w a
circ
le,
inte
rsec
tin
gg
iven
cir
cle
at
Q a
nd
R
Giv
en:
Cir
cle
wit
h c
entr
e O
an
d p
oin
t P
ou
tsid
e it
.
1.
Dra
w a
ny
ray B
X m
ak
ing
an
acu
te a
ng
lew
ith
BC
Co
nst
ruct
a t
rian
gle
sim
ilar
to a
giv
en∆
AB
C w
ith
sid
es
o
fth
e co
rres
po
nd
ing
sid
es o
fA
BC
∆3 4
2.
Lo
cate
4 p
oin
ts (
gre
ate
r o
f3
dis
tan
ce f
rom
each
oth
er
(BB
= B
B=
BB
= B
B)
11
22
33
4
Join
BC
4
an
d 4
) o
n B
X a
t eq
ual
Met
hod
1
Met
hod
2
Const
rnst
rns
uctions
Tangent
to c
ircle
Giv
en:
Lin
e se
gm
ent,
rati
o (
3 :
2)
1.
Dra
w a
ny r
ay A
X m
ak
ing
an
acu
te a
ng
le w
ith
lin
ese
gm
ent
AB
2.
Dra
w r
ay B
Y |
| A
X
A1
23
Met
hod
2
10 ] Oswaal CBSE Chapterwise Mind Maps, MATHEMATICS (STANDARD), Class – X
To know about more useful books for class-10 click here
(Unit
v) C
HAPT
ER :
10 in
trodu
ction
to t
rigono
metr
y an
d tr
igono
metr
ic id
entities
Intr
oduc
tion
toTr
igonom
etr
ya
nd
Trig
onom
etr
icId
entitie
s
Stu
dy o
fre
lati
on
ship
s b
etw
een
the
sid
es &
an
gle
s o
fa r
igh
t tr
ian
gle
Trig
on
om
etr
y
Trig
on
om
etr
y R
atio
Sin
e o
fA
∠B
C
AC
Co
sin
e o
fA
∠A
B
AC
Tan
gen
t o
fA
∠B
C
AB
Co
seca
nt
of
A∠
AC
BC
Sec
an
t o
fA
∠A
C
AB
Co
tan
gen
t o
fA
∠A
B
BC
======
Sid
eo
pp
osi
te
toA
∠
Sid
e o
pp
osi
te t
oC
∠A
C B
Hyp
oten
use
∠A
30
°0
°4
5°
60
°9
0°
1 21
0si
n A
1 2√
3 2√
01
cos
A3 2√
1 2√
1 2
10
No
t (
∞)
def
ined
tan
A1 3
√3
√
No
t (∞
)d
efin
ed
2 3√
21
cose
c A
2√
21
No
t (
∞)
def
ined
sec
A2 3
√2
√10
No
t (∞
)d
efin
edco
t A
3√
1 3√
Va
lues
Com
ple
menta
ry
Angle
s
sin
(9
0°
–A
) =
co
s A
cos
(90
°–
A)
= s
in A
tan
(9
0°
–A
) =
co
t A
cot
(90
°–
A)
= t
an
A
sec
(90
°–
A)
= c
ose
c A
cose
c (9
0°
–A
) =
sec
A
No
te:
Ho
w t
o l
earn
th
e re
lati
on
“S
om
e p
eop
le h
ave”
“C
url
y B
row
n H
air
”
“th
rou
gh
pro
per
Bru
shin
g”
sin
=�
P H
cos
=�
B H
tan
=�
P B
Trig
on
om
etr
ic Id
entities
cos
A +
sin
A =
12
2
1 +
tan
A =
sec
A;0
A9
02
2≤
≤°
≤≤
°co
tA
+ 1
= c
ose
cA
;0A
90
22
Exa
mp
le
Ex
pre
ss t
an
A,
cos
A i
n t
erm
s o
fsi
n A
So
luti
on
: S
ince
, co
sA
+ s
inA
= 1
22
tan
A =
=
sin
A
cos
A
sin
A
√1
–si
nA
2
cos
A =
1–
sin
Aco
s A
=2
2i.
e.√
1–
sin
A2
Intr
Intr
Inod
trod
truc
tiuc
tiuc
on
toTr
igonom
etr
yd
Trig
od
Trig
od
Tnom
etr
metr
me
icId
entitie
s
∠A
0°
1 20
sin
A
1co
s A
3 2√
0ta
n A
1
√
sec
A
No
t (∞
)d
efin
edco
t A
Exa
mp
le
1–
sin
A2
sin
A2
sin
A
Oswaal CBSE Chapterwise Mind Maps, MATHEMATICS (STANDARD), Class – X [ 11
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CHAP
TER
: 11 he
ight
s a
nd d
ista
nces
Heig
hts
and
Dis
tance
sExa
mp
les
Det
erm
ine
hei
gh
t o
fo
bje
ct A
B
In�
AB
C,
B =
90
°,C
= 6
0°
��
Her
e, t
an
60
° =
AB
BC
i.e.
,A
B =
15√
3m
=A
B
15
√
3
Mea
surin
g A
ngle
s
An
gle
of
Ele
vati
on
is
equ
al
to A
ng
le o
fD
epre
ssio
n
A
n
g
l
e
o
f
D
e
p
r
e
s
s
i
o
n
Ap
plic
ation–T
rig
on
om
etr
icR
atios (
To d
ete
rm
ine)
A
30
°
60
°
B
20
0m
D
C
Fin
dan
dx
h
x
h
Dis
tan
ce b
etw
een
tw
o o
bje
cts
90
°
A
BCD
(i)
BD
is
a t
ree
AC
= D
Cif
CD
is
bro
ken
(ii)
��
A
B
D
F
l
a
g
C
Fin
d fla
g le
ng
th
x
h
hei
ght
/ le
ngt
h o
f an
obj
ect
Det
erm
ine
wid
th A
B
Fro
m f
igu
re,
AB
= A
D +
DB
In r
igh
tA
PD
,A
30
°D
90
°∆
��
==
,
√3
mta
n 3
0°
=i.
e.,
AD
= 3
PD
AD
In r
igh
tB
PD
,B
45
°D
90
°∆
��
==
,
tan
45
° =
i.e.
,B
D =
3P
D
BD
∴A
B =
(3
+ 3
)m =
3(
+ 1
)m√
3√
3
h 200
—ta
n 6
0º
=
...
(i)
hx+
200
—ta
n 6
0º
=
...
(ii)
h x
—ta
n=
.
..(i
)�
h+
DC
x
—ta
n=
.
..(i
i)�
Dis
tance
Obj
ect
Hei
ght
Heig
hts
an
Heig
hts
an
Heig
hd
Dis
tance
s
An
gle
of
Ele
vati
on
is
equ
al
to A
ng
le o
fD
epre
ssio
n
A
n
g
l
e
o
f
D
e
p
r
e
s
s
i
o
n
12 ] Oswaal CBSE Chapterwise Mind Maps, MATHEMATICS (STANDARD), Class – X
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(Unit
vI) CH
APTE
R : 12
Area
s r
elat
ed t
o c
ircl
es
Area
s R
ela
ted
to
Cir
cle
s
Cir
cle
Cir
cum
fere
nce
=×
dia
met
er =
2�
�r
Are
a =
�r2
Secto
r
Mea
nin
g
Fo
rmu
la
Po
rtio
n o
fth
eci
rcu
lar
reg
ion
en
clo
sed
by t
wo
rad
ius
an
d t
he
corr
esp
on
din
g a
rc
Are
a
×are
a o
fci
rcle
�
36
0°
A =
A =
�
r2�
36
0°
Len
gth
of
arc
× c
ircu
mfe
ren
ce�
36
0°
L =
L =
× 2
�r
�
36
0°
Segm
ent
Are
a =
Are
a o
fth
e co
rres
po
nd
ing
sec
tor–
Are
a o
fth
e co
rres
po
nd
ing
tri
an
gle
×–
are
a o
fO
AB
�r2
∆
�
36
0°
= =–
��
r2
36
0°
1 2r2
sin�
Q
P
Are
a o
fS
ecto
r =
1 2×
L×
r
Fo
rmu
la
Area
- C
om
bin
ation o
f f
igures
Are
a o
fT
= A
rea o
fP
+ A
rea o
fQ
PQ
T
Mea
nin
g
Fin
d A
rea o
fsh
ad
ed r
egio
n
14
cm
Are
a o
fsq
uare
AB
CD
= 1
4×
14
cm
2
= 1
96
cm
2
Dia
met
er o
fea
ch c
ircl
e,D
=
= 7
cm
14 2
Fo
r ea
ch c
ircl
e, r
ad
ius
()
=r
cm7 2
Are
a o
f1
cir
cle
=�
r2
Are
a o
f4
cir
cles
= 4
×
2 cm2
22 7
7 2(
)
×
=15
4 4×
4cm
2
= 1
54 c
m2
Are
a o
fsh
ad
ed r
egio
n =
Are
a o
fA
BC
D–
Are
a o
f4
cir
cles
= (
19
6–
15
4)
= 4
2 c
m2
Ex
am
ple
Po
rtio
n o
fth
e ci
rcu
lar
reg
ion
encl
ose
d b
etw
een
a c
ho
rd a
nd
th
eco
rres
po
nd
ing
arc
Mea
nin
g Se
gm
en
t
Min
or
Seg
men
t
ea
s R
ela
ted
to
Cir
cle
sF
o
Segm
ent
ina
tion o
f f
igures
Po
rtio
n o
fth
e ci
rcu
lar
reg
ion
encl
ose
d b
etw
een
a c
ho
rd a
nd
th
eco
rres
po
nd
ing
arc
Mea
nin
g
Oswaal CBSE Chapterwise Mind Maps, MATHEMATICS (STANDARD), Class – X [ 13
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CHAP
TER
: 13
sur
face
are
as a
nd v
olu
mes
Surfa
ce A
rea
s a
nd
Volu
mes
Su
m o
fsu
rface
are
as
of
the
face
s o
fso
lid
.
Qu
an
tity
of
3-D
sp
ace
encl
ose
d b
y a
ho
llo
w/
clo
sed
so
lid
.
Volu
me
Surfa
ce A
rea
Com
bin
ation o
f S
olid
s
Cu
rved
Su
rface
Are
a
r 2
r 1
lh
Frust
um
of
Cone
5 c
m
5 c
m
Giv
en–
Inn
er d
iam
eter
of
the
Cyli
nd
rica
l g
lass
= 5
cm
Hei
gh
t =
5 c
m
Fin
d–
Act
ual
cap
aci
ty o
fC
yli
nd
rica
l g
lass
So
luti
on
:–
Ap
pare
nt
cap
aci
ty
of
the
gla
ss =
�r
h2
=.
×.
×.
×3
14
25
25
5cm
3
=.
98
125
cm3
Act
ual
cap
aci
ty =
Ap
pare
nt
cap
aci
ty–
Vo
lum
e o
fh
emis
ph
ere
= 9
8.1
25
–32.7
1=
65.4
2 c
m3
Vo
lum
e o
fh
emis
ph
ere
=π
,if
=2.5
cmr
r3
= ×
3.1
4 ×
(2.5
)cm
= 3
2.7
1 c
m3
33
2 – 32 – 3
Conversio
n o
f S
olid
s
1 2
A c
op
per
ro
d–
Dia
met
er 1
cm,
len
gth
8cm
con
ver
ted
in
to a
wir
e o
fle
ngth
18m
Fin
d t
he
thic
kn
ess
of
the
wir
e.
So
luti
on
: V
olu
me
of
the
rod
=×
8cm
�3
=2
3�
cm
Let
be
the
rad
ius
of
cro
ss-s
ecti
on
of
the
r
wir
e, v
olu
me
=1800
cm�
××
r23
∴×
×=
��
r21800
2
=r2
=r
cm
=-
Th
ick
nes
sD
iam
eter
of
the
cro
ssse
ctio
n
==
.cm
007
cm
(
(2
1 — 900
1 — 30
1 — 15
Volu
me
CS
A =
π(
+)
lr
r1
2
wh
ere
=l
hr
r+
(–
)2
12
2
TS
A =
(+
)+�
�l
rr
rr
l2
2 l2 2
(+
)
To
tal
Su
rface
Are
a
l
r 2
r 1
r 2
r 1
V =
(+
+)
�h
rr
rr
2 12 2
12
1 — 3
rfa
ce A
rea
s a
nd
Volu
mes
rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr 222222222222222222222222222222222222
r 1
lhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
Frust
uust
uus
m o
fCone
So
luti
on
:–
Ap
pare
nt
cap
aci
ty
1 2
A c
op
per
ro
d–
Dia
met
er 1
cm,
len
gth
8cm
con
ve
con
ve
con
rted
in
to a
wir
e o
fle
ngth
18m
Fin
d t
he
thic
kn
ess
of
the
wir
e.
So
luti
on
: V
olu
me
of
the
rod
=×
2=
×2
=×
=×
8cm
�=
�
=×
3
=2
3�
cm
Let
be
the
rad
ius
of
cro
ss-s
ecti
on
of
the
wir
e, v
olu
me
=1800
cmw
ire,
vo
lum
e =
1800
cm�
wir
e, v
olu
me
=1800
cmw
ire,
vo
lum
e =
1800
cm×
×w
ire,
vo
lum
e =
1800
cmw
ire,
vo
lum
e =
1800
cmr
wir
e, v
olu
me
=1800
cmw
ire,
vo
lum
e =
1800
cm×
×w
ire,
vo
lum
e =
1800
cmr
wir
e, v
olu
me
=1800
cm×
×w
ire,
vo
lum
e =
1800
cm2
wir
e, v
olu
me
=1800
cm2
wir
e, v
olu
me
=1800
cmw
ire,
vo
lum
e =
1800
cm×
×w
ire,
vo
lum
e =
1800
cm2
wir
e, v
olu
me
=1800
cm×
×w
ire,
vo
lum
e =
1800
cmw
ire,
vo
lum
e =
1800
cm×
×w
ire,
vo
lum
e =
1800
cmr
wir
e, v
olu
me
=1800
cm×
×w
ire,
vo
lum
e =
1800
cm2
wir
e, v
olu
me
=1800
cm×
×w
ire,
vo
lum
e =
1800
cmr
wir
e, v
olu
me
=1800
cm×
×w
ire,
vo
lum
e =
1800
cm3
��
1800
2×
=1800
2×
=
��
1800
2�
�×
=
��
×=
1800
2×
=
��
×=
Th
ick
nes
sD
iam
eter
of
the
cro
ssse
ctio
n
(=
×(
=× (
=× (
=×
2
14 ] Oswaal CBSE Chapterwise Mind Maps, MATHEMATICS (STANDARD), Class – X
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(Unit
VII)
CHA
PTER
: 14 s
tatistics
Sta
tist
ics
A c
oll
ecti
on
,an
aly
sis,
in
terp
reta
tio
no
fq
uan
tita
tive
data
Defin
itio
n
Fre
qu
ency
ob
tain
ed b
y a
dd
ing
the
freq
uen
cies
of
all
th
e cl
ass
esp
rece
din
g t
he
giv
ing
cla
ss
Cum
ula
tive F
req
uency
Cla
ss M
ark
Group
ed
Da
ta
Mea
n
3 M
edia
n =
Mo
de
+ 2
Mea
n
Og
ive
Em
peric
al
Rela
tionship
Up
per
cla
ssli
mit
Lo
wer
cla
ssli
mit
+ 2[
[
Med
ian
=l
+
n – 2–
cf
(
(
f
×h
Mod
e=l
+
ff
10
–
(
(×h
2–
–f
ff
10
2
A C
um
ula
tive
Fre
qu
ency
Gra
ph
Rep
rese
nta
tio
n o
fcu
mu
lati
ve
freq
uen
cies
wit
hre
spec
t to
giv
en c
lass
inte
rvals
Mea
nin
g
Ass
um
ed M
ean
(S
ho
rt c
ut)
xa
=+
–
He
re,
d=
x–
ai
i
Dir
ect
Met
ho
d (
Lo
ng
cu
t)x
=–
∑f i
n
i=
1
i=
1
n
∑f
x ii
——
Ste
p D
evia
tio
nx
a=
+–
×h
Her
e,=
ui
xa
i–
—— h
∑f i
n
i=
1
i=
1
n
∑f
u ii
——
∑f i
n
i=
1
i=
1
n
∑f
d ii
——
Sta
tist
iis
tiis
cs
Ass
um
ed M
ean
(S
ho
rt c
ut)
He
re
Dir
ect
Met
ho
d (
Lo
ng
cu
t)
Ste
p D
evia
tio
nx
a=
+x
a=
+x
a–
n
Oswaal CBSE Chapterwise Mind Maps, MATHEMATICS (STANDARD), Class – X [ 15
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CHAP
TER
: 15
proba
bility
Prob
ab
ility
Theoretica
l P
rob
ab
ility
Exp
erim
enta
l P
rob
ab
ility
Va
lue
Defin
itio
ns
Su
re o
r C
erta
in E
ven
t
Wh
en e
ven
t h
avin
gp
rob
ab
ilit
y t
o o
ccu
r as
1
Exa
mp
les
Dic
e
Tw
o d
ice
are
ro
lled
, w
hat
is p
rob
ab
ilit
yo
fget
tin
g 1
2 a
s a s
um
?
So
luti
on
: N
um
ber
of
po
ssib
le
ou
tco
me
= 6
=3
62
Nu
mb
er o
ffa
vo
ura
ble
ou
tco
mes
= 1
Co
inW
hen
a c
oin
is
toss
ed,
wh
at
wo
uld
be
the
pro
bab
ilit
y o
fap
pea
rin
g h
ead
?
So
luti
on
: T
ota
l o
utc
om
es =
2F
avo
ura
ble
ou
tco
mes
= 1
Req
uir
ed P
rob.
P(E
) =
1 — 2
Card
Wh
at
is t
he
pro
bab
ilit
y o
fget
tin
gan
ace
fro
m a
pack
of
52
card
s?
So
luti
on
: N
um
ber
of
favo
ura
ble
ou
tco
mes
= 4
Nu
mb
er o
fp
oss
ible
ou
tco
mes
= 5
2
4P
(E)
=— 52
1=
— 13
Even
t h
avin
go
nly
on
e o
utc
om
e o
fth
eex
per
imen
t
Ele
men
tary
Even
t
Su
m o
fp
rob
ab
ilit
ies
of
all
ele
men
tary
even
ts i
s 1
.F
or
even
ts A
, B
, C
; P
(A)
+ P
(B)
+ P
(C)
= 1
Nu
mb
er o
fo
utc
om
esfa
vo
ura
ble
to
EN
um
ber
of
all
po
ssib
leo
utc
om
es o
fth
eex
per
imen
t
P(E
) =W
hat
act
uall
y h
ap
pen
sin
an
ex
per
imen
t
Nu
mb
er o
ftr
ials
in
P(E
) =
wh
ich
th
e ev
ent
hap
pen
ed
To
tal
nu
mb
er o
ftr
ials
Wh
at
we
exp
ect
toh
ap
pen
in
an
ex
per
imen
t
0≤
P(E
)≤
1
Fo
r ev
ent
E,
com
ple
men
t ev
ent,
P(E–
) =
1–
P(E
)
Co
mp
lem
enta
ry E
ven
t
1 — 36
P(E
) =
ob
ab
ility
Va
lue
o d
ice
are
ro
lled
, w
hat
is p
rob
ab
ilit
y
Even
t h
avin
go
nly
on
e o
utc
om
e o
fth
eex
per
imen
t
Ele
men
tary
Even
t
Fo
r ev
ents
A,