s DeTT - Jeywin

6 s DeTT
“There is geometry in the humming of the strings, there is music
in the spacing of spheres” - Pythagoras
• |#T+
• b< $T nqTb dDeTTq
bs |d eTTe =qT jT & jTT
$Xw $~ deTs yTq |< >
#|e#T.
6.1 |#T+
sU>D+, >DXdeTT XK. ~ $$< C$TrjT
seTT\ ] *jTCjTTqT. KyTq
=\\T +&H $$< C$TrjT seTT\ +
eTTK+ eV+#T#Tq~. eTT\T m+ q~.
uTeTT\T e+q\qT |{w+> me \eTT eT e#TqT
eT]jTT & T=qTqT ]+#T dVjT|&TqT.
ueqeTT\ sD+ sU>D+ s+&T seTT\T>
|#jTT#Tq~. { ueqeTTqT |{w+> ]+#T
eTs={ n+

`
n

P 1 Q 1 eT]jTT BC \qT \T|eTT.
|&T
AP1 AQ
= P 1 B 41
eT]jTT
1
Q 1 C
eq
AP1 AQ
=
1
P1
B Q 1 C
= 8
2 = 4
1
sK\T P 1 Q 1 eT]jTT BC \T
qe |]o*+#>\eTT
i.e., P1 Q1 < BC
(1)
n
AP2
P2
B
AP3
P3
B
AP4
P4
B
AQ2
Q2C
AQ3
Q3C
2
3
= = eT]jTT P Q ||
3
2
= = eT]jTT P Q ||
AQ4
Q4C
4
1
= = eT]jTT P Q ||
2 2 BC
(2)
3 3 BC
(3)
4 4 BC
(4)
(1), (2), (3) eT]jTT (4) \ qT+& eTqeTT |]o*+q~ @eTq>, @< sK uTeTT
s+&T uTeTT\qT w $u+q n~ eT&e uTeTTq de+seTT> +&TqT.
~XH < d

d) deT~K+&q #jTT sK, DeTT
jTT m (uV+>), $T*q s+&T uTC\ w $u+#TqT.
E
d+ deT~K+&q
#jTT sK AD, BC D e< d+~+#T#Tq~.
s|+#e\dq :
BD
DC
=
AB
AC
sDeTT: CE < DA > +&TqT ^, BA b&+#>
n~ E e< d+~+#TqT.
s|D CE < DA n>T# AC s>K
qT
eT]jTT
+ DAC = + ACE (@+s DeTT\T) (1)
AD nqTq~
+ BAD = + AEC (nqTs| DeTT\T) (2)
+ A qT DdeT~K+&q #jTT# + BAD = + DAC n>TqT. (3)
(1), (2), (3) \ qT+& + ACE = + AEC
eq 3 ACE AE = AC n>TqT (deq DeTT\ m +&T uTeTT\T
deqeTT) 3 BCE CE < DA qT,
(
BD =
BA
DC AE
BD =
AB
DC AC
(< dT# AC s>K
+ ECA = + CAD (@+s DeTT\T) (1)
B
E
A
>
D
|eTT 6.5
>
B C D
A
|eTT 6.6
C
P
182 10th Std. Mathematics

eT]jTT CE
< DA eT]jTT BP s>K n>T#
+ CEA = + DAP (nqTs| DeTT\T) (2)
AD nqTq~ + CAP jTT deT~K+&qeTT n>T#
+ CAD = + DAP
(3)
(1), (2), (3) \ qT+&
+ CEA = + ECA
eq 3 ECA , AC = AE n>TqT (deq DeTT\ m qT+&T uTeTT\T deqeTT)
3 BDA , EC < AD n>TqT
`
(
BD =
BA
DC AE
BD =
BA
DC AC
(< d) deT~K+&q #jTTqT.
E
d+TqT $u+#T#Tq~.
s|+#e\dq : AD nqTq~ + BAC jTT n+s deT~K+&qeTT.
sDeTT : C >T+& CE
i.e., + BAD = + DAC n s|+#eqT.
< AD > +&TqT BA E es b&+#TeTT.
s|D CE < AD n>T# < dTqT (2)
AE
=
AB
n>TqT
AC
3 ACE + ACE = + AEC ( AE = AC ) (3)
B
D
|eTT 6.7
C
Geometry 183

AD eT]jTT CE nqT de+ssK\ AC s>K n>T#
+ DAC = + ACE (@+s DeTT\T deqeTT) (4)
AD eT]jTT CE nqT de+ssK\ BE s>K n>T#
(3), (4), (5) \ qT+& + BAD = + DAC
` AD nqTq~ + BAC jTT deT~K+&qeTT.
eq dT+& CE
< AD > +&TqT ^q n~ BA E e< d+~+#TqT.
s|D CE < AD n>T# < dK n>T#
=
BA
(2)
EA
= ) (3)
+ ACE = + DAC (@+s DeTT\T) (4)
AD eT]jTT CE nqT de+s sK\ BA s>K n>T#
B
E
C
A
P
D
+ PAD = + AEC (nqTs| DeTT\T) (5)
(3), (4), (5) \ qT+&
+ PAD = + DAC
` AD nqTq~ + PAC jTT deT~K+&qeTT. eq + BAC jTT uV deT~K+&qeTT
AD n>TqT.
eq d

5. #TsTeTT ABCD , AB de+seTT> CD q~. AB de+seTT> qT+&TqT
AD P e< eT]jTT BC Q e< d+~+#TqT sK ^jT&q,
AP BQ
= PD
n
QC
s|+#TeTT.
6. |eTT, PC < QK eT]jTT BC < HK . AQ = 6 d+.$, QH = 4 d+.$,
HP = 5 d+.$, KC = 18 d+.$ nsTTq AK eT]jTT PB \qT qT>=qTeTT.
7. |eTT DE < AQ eT]jTT DF < AR
nsTTq EF < QR n s|+#TeTT
8. |eTT DE < AB eT]jTT
DF
< AC nsTTq EF < BC
n s|+#TeTT.
9. 3 ABC , AD nqTq~ + A jTT n+sdeT~K+&qeTT eT]jTT BC D e<
d+~+#TqT.
(i) BD = 2 d+.$, AB = 5 d+.$ eT]jTT DC = 3 d+.$ nsTTq AC qT>=qTeTT.
(ii) AB = 5.6 d+.$, AC = 6 d+.$ eT]jTT DC = 3 d+.$ nsTTq BC qT>=qTeTT..
(iii) AB = x, AC = x–2, BD = x+2 eT]jTT DC = x–1 nsTTq x $\TeqT qT>=qTeTT.
10. 3 ABC , +~ y{ AD nqTq~ + A jTT deT~K+&qeTT n>THjT
|+|eTT.
(i) AB = 4 d+.$, AC = 6 d+.$, BD = 1.6 d+.$ eT]jTT CD = 2.4 d+.$.
(ii) AB = 6 d+.$, AC = 8 d+.$, BD = 1.5 d+.$ eT]jTT CD = 3 d+.$.
11. 3 MNO MP nqTq~ + M jTT uVdeT~K+&qeTT
eT]jTT NO qT b&+#> @s&q sK| P e< d+~+#TqT.
M
Q
MN = 10 d+.$, MO = 6 d+.$, NO = 12 d+.$ nsTTq
6d+.$
OP qT>=qTeTT.
N
P
12. #TsTeTT ABCD + B eT]jTT + D \ ~K+&qeTT\T AC | E e< K+&+#T=q
AB
=
AD
n s|+#TeTT.
BC
13. ABC
DC
T + A jTT n+s ~K+&qeTT BC D e< d+~+#TqT. eT]jTT + A jTT
uV ~K+&qeTT BC b&+#> @s&q sK| E e< d+~+#TqT. nsTTq
BD
=
CD
BE CE
n s|+#TeTT.
14. ABCD #TsTeTT AB =AD. AE eT]jTT AF \T nqTq$ eTeTT> + BAC eT]jTT
+ DACn+s~K+&qeTTq, EF < BD n s|+#TeTT.
188 10th Std. Mathematics
B
A
D
E
P
F
C
Q
E
D
P
A
F
R
10d+.$
12d+.$
B
O
Q
H
P
A
K
C

6.3 ds| uTeTT\T Similar triangles
eTqeTT uTeTT\ dsdeq+ >] 8 e s> neTT C$TrjT |eTT\T seTTqT+& |]eDeTT s+> qT+&qedseTT y{
>] n qT+&TqT
#dTHeTT. qT+& |]eDeTT seTT> +& $e< seTT\T>
qT+&TqT. n{y qT+ rdq |eTT\T seTT
seTT>qT |]eDeTT yssT> qT+&TqT. edTe\
seTT seTT>qT |]eDeTT $e< seTT\T> qT+&TH
ds| edTe\T n n+DeTTqT ^dT |yX|fqT. nqT
| >\ |s$T& mT\qT y{
&\T eT]jTT ds| uTeTT\
deTT\qT|j+ qT>=Hq
#|&T#Tq~. eq ds|
uTeTT\qT|j+ mT\T
eT]jTT DXdy\
qT yT*|q~. nqT .|P 585
ds>VDeTT qT>=HqT.
uTeTT\T eT]jTT \+D uTeTT\ sU>D+ |j+#qT.
dsdeq |eTT\T ds|eTT. B $|seTT deTT> +&qedseTT eTT
ds| uTeTT\ CqeTTqT|j+ deTd\qT k~+#]
u>T> \TdT=qT |j>|&TqT.
eTT
sYusY rdT= n+TqT. u$TqT+ be=\~
_+eTT | +&qqT, uT oseTT\ e< @s&T DeTT\T
m\|&T $ qT+&T >eT+|e#T.
Geometry 189

s#q+
s+&T uTeTT\T ds|eTTq
(i) y{ nqTs| DeTT\T deqeTT ()
(ii) y{ nqTs| uTeTT\ b&e\T w ( nqTbeTT) +&TqT. BH
uTeTTqT $d]+ #|&q< #|e#TqT.
3ABC eT]jTT 3DEF \T ds|yTq
(i) + A = + D, + B = + E, + C + F
(ii)
AB
DE
BC
EF
d#q
3ABC eT]jTT 3DEF \qT d+eTT> 3BCA
+ 3 EFD eT]jTT TCAB + TFDE
n y{ nqTs| oseTT\qT|j+ \|e#TqT.
190 10th Std. Mathematics
CA
FD
= ()
= = > qT+&eqT. B
C E
F
|eTT 6.16 |eTT 6.17
#, oseTT\T A, B eT]jTT C \ eTeTT> D, E, F nqT oseTT\T nqTs|yTq$.
s+&T uTeTT\ ds|qT d+eTT> 3ABC + 3 DEF n yjTe#TqT. eT]jTT B
3ABC, 3DEF ds|eTT> qTq< #TsT ds|eTT> n *jTCjTTqT.
6.3.1 uTeTT\ ds| |b eTs= uTeTT s+&T DeTT\
deqyTq y{ eT&e DeTT deqeTT> qT+&TqT. qT . ks| |b

(i)
(ii)
(iii)
uTeTT\ ks|eTTqT >] s|D\T +& = |*eTT\qT #dTqT.
\+D uTeTT, \+DeTT >\ oseTTqT+& seTTq \+eTT ^q,
\+eTTq sTy|\ >\ uTeTT\T qT+&TqT. eT]jTT
A
s+&T uTeTT\T < uTeTTq ds|eTT>qT+&TqT.
#, (a) T DBA + T ABC
(b) 3 DAC + 3 ABC
(c) 3 DBA + 3 DAC
s+&T u TeTT\T ds|eTT\sTTq, y{ nqTs| u TeTT\
w, y{ nqTs| qT\ w deqeT>TqT.
i.e., TABC TEFG
(iv)
AB
BC
CA
AD
EH
+ nsTTq = = =
EF FG GE
B
s+&T uTeTT\T ds|eTT\sTTq, y{ nqTs| |eTT 6.19 |eTT 6.20
uTeTT\ w y{ nqTs| |]~\ w deqeT>TqT.
T ABC
=qTeTT.
k qTq~
Q
` DPAB
+ D PQR (. |b

\ eTw, |s$T& Zs \&jTTH&T. d+=qTeTT.
k AB
A
eT]jTT DE nqT=qTeTT.
|s$T& eT]jTT eTw &\ b&e\qT
eTeTT> BC eT]jTT EF nqT=qTeTT.
3 ABC eT]jTT DEF
192 10th Std. Mathematics
3 \ qT+&
+ ABC = + DEF =90 o
+ BCA = + EFD
( d+=qTeTT. (eTw, n< eTT eT]jTT >|s b AB eT]jTT DE nqT=qTeTT. n|s
KseTT |&T#TqT C nqT=qTeTT.
E
3 \qT+&
3 ABC eT]jTT EDC
+ ABC = + EDC = 90 o
` ABC + EDC
`
ED =
DC
AB BC
ED =
DC
# AB =
87.
6
# 1.5=
328.5
BC 0.
4
qT, >|seTT mT R 328.5 $.
|eTT 6.22
1.5$
B
1.8$
A
E
D
210$
2.7 $
+ BCA = + DCE
( d+

6 $
G
AB eT]jTT EF nqT=HeTT q&T#T=
y#Tq~. B|eTT u$T qT+& 3.6 $ mT +&q, 4 d+& s yT &
b&eqT qT>=qTeTT?
5
E
z
F
y
6
9
4
x
H
G
C
|eTT 6.25
E
G
y
B 5 D 7 C
x
F
6
Geometry 193

4. u* q +& deTT< rseTTq q~. >&T deTTeT+#qT. |&eTs ~ 50 $ \
q +& y| n]#qT. +& |&e, u* H
10 $ Zs >\&T. +& |&eqT|j+ >&
#s\+f 126 $ \ eTw >& sT~q >\&T. nsTTq
>& b&T ysY |t eTw m+ \TqT
uTeTTAC | D nqT _+\T+& BD
qT>=qTeTT.
7. D ABC DE || BC n>TqT _+ AB eT]jTT AC \|
q$. AB = 3 AD eT]jTT D ABC yX\eTT 72 #.d+.$ nsTTq #TsTeTT DBCE
yX\eTTqT qT>=qTeTT.
D u TeTT\ b&e\T 6 d+.$, 4 d+.$ eT]jTT 9 d+.$. 3PQR
+ 3 ABC n>TqT.
3PQR H= u TeTT b&e 35 d+.$ nsTTq 3PQR b+]w |]~ m+?
8. ABC
9. |eTT DE || BC eT]jTT
AD
=
3
BD
nsTTq, +~ y >D+#TeTT.
5
3 ADE yX\eTT f|jT+ BCED yX\eTT
(i) , (ii)
3 ABC yX\eTT 3 ABC yX\eTT
10. |DeTT sT|j>eTT> qTq uu>eTT
= b]X$T b+eTTqT n_e~ |s#eq
|uTeTT |D #dqT. |eTT #jT yjT&q
u>eTT => H\=\e\dq b]X$T b+eTTqT
*jTCjTTqT. nsTTq => @s&T b]X$T
b+ yX\eTTqT qT>=qTeTT?
11. |eTT #|&qT u\T&T &eT+& seTT >* |eTTqT
#jTT#TH&T. +

12. $

d#q
|sdt dT]+. qT >T]+|DeTTqT \T|#Tq~. |sdt dTqT. B
yTT yTT< *|, s|+qysT jT& n>TqT.
d |e#qeTT +< e&q~.(s|D nudqeTTq e~*yjT&q~.)
d

eeTT\T dss\

eeTT C jTT n+_+

kT#
+ BAQ = + ACB = 62cn>TqT. (dssK ` C dTqT
PB =
PC.
PD
PA
(ii)
(2+x) 2 = 9 4 #
qT x = 8 # 3 = 6. x + 2 = 18 qT x = 16.
4
\ eeTTq uV_+\ dssK eT]jTT AP = 15 d+.$
nsTTq TPCD |]~ qT>=qTeTT?
k

k=qTeTT.
(ii) AP = 12 d+.$, AB = 15 d+.$, CP = PD, nsTTq CD qT>=qTeTT
3. AB eT]jTT CD nqT C\T eeTTq uVeTT> P e< K+&+#T=qT#Tq$.
(i) AB = 4 d+.$ BP = 5 d+.$ eT]jTT PD = 3 d+.$, nsTTq CD qT>=qTeTT.
(ii) BP = 3 d+.$, CP = 6 d+.$ eT]jTT CD = 2 d+.$, nsTTq AB qT>=qTeTT.
4. TABC uTeTT BC eeTTqT P e< d+#TqT. AB eT]jTT AC \qT eTeTT> Q
eT]jTT R es b&+#&q~. nsTTq AQ = AR =
1 (TABC 2
|]~) n s|+#TeTT.
5. de+s #TsTeTT jTT n uTeTT\T eeTTqT d+q, de+s #TsTeTT
s+dt n>Tq s|+#TeTT.
6. =\qT { |]\eTTq 20 d+.$ |q eTs |weTT q~ eT]jTT <
+&| u>eTT beTT> { |]\eTTq |\ >\* Mq|&T |weTT
eTT+\ +&eTT b&e m+ +&TqT?
7. Bs #TsdeTT ABCD n+seTT> qT+&T _+ qT+&q AC
(A)
AD
DB
(B)
200 10th Std. Mathematics
AD
AB
(C)
DE
BC
(D)
AD
EC
AE =
D
A
S
7 cm
R
P
6 cm
T
Q
B
B
C
6.5 cm
P

2. 9ABC DE || BC n>TqT _+ AB eT]jTT AC
| q~. AD R 3 d+.$, DB R 2 d+.$ eT]jTT AE R 2.7 d+.$ nsTTq AC
deqeT>Tq~.
(A) 6.5 d+.$ (B) 4.5 d+.$ (C) 3.5 d+.$ (D) 5.5 d+.$
3. 9PQR RS nqTq~ + R jTT deT~K+&qeTT. PQ R 6 d+.$, QR R 8 d+.$,
P
RP R 4 d+.$ nsTTq PS deqeT>Tq~.
4d+.$
S
(A) 2 d+.$ (B) 4 d+.$ (C) 3 d+.$ (D) 6 d+.$
4. |eTT
AB
AC
=
BD
, + B 40
DC
= eT]jTT 0
+ C = 60 nsTTq 0 + BAD =
(A) 30 0 (B) 50 0 (C) 80 0 (D) 40 0
5. |eTT x $\Te deqyTq~
(A) 4 $ 2
(B) 3 $ 2
(C) 0 $ 8
(D) 0 $ 4
6. uTeTT\T ABC eT]jTT DEF \ + B = + E,
+ C = + F nsTTq
(A)
AB
DE
=
CA
(B)
BC
EF EF
A
x 4
56c
D E
8
56c 10
B
C
=
AB
(C)
AB
FD DE
7. e&q |eTTqT+& |> qT+&T |e#qeTTqT >T]+#TeTT
Q
6d+.$
=
BC
(D)
CA
EF FD
A
B
8d+.$
=
A
40c 60c
D
AB
EF
R
C
(A) T ADB + T ABC (B) T ABD + T ABC
D
(C) T BDC + T ABC (D) T ADB + TBDC
8. 12 $ b&e >\ \Te s 8 $ b&e &qT @ss#TqT. n< deTjTeTTq H\|
>|seTT jTT & 40 $ b&eqT #|q >|seTT mT.
(A) 40 $ (B) 50 $ (C) 75 $ (D) 60 $
9. s+&T ds| uTeTT\ uTeTT\ w 2:3 nsTTq y{ yX\eTT\ w
(A) 9:4 (B) 4:9 (C) 2:3 (D) 3:2
10. uTeTT\T ABC eT]jTT DEF \T ds|eTT\T. y{ yXeTT\T eTeTT> 100
#.d+.$ eT]jTT 49 #.d+.$ eT]jTT BC R 8.2 d+.$ nsTTq EF =
(A) 5.47 d+.$ (B) 5.74 d+.$ (C) 6.47 d+.$ (D) 6.74 d+.$
11. s+&T ds| uTeTT\ |]~\T eTeTT> 24 d+.$ eT]jTT 18 d+.$. yTT

12. eeTT AB eT]jTT CD nqT s+&T C\qT b&+#> n$ P e< d+~+#qT.
AB = 5, AP = 8 eT]jTT CD = 2 nsTTq PD =
(A) 12 (B) 5 (C) 6 (D) 4
13. | |eTT AB eT]jTT CD nqT C\T P e< K+&+#T=qT#Tq$.
AB R16 d+.$, PD R 8 d+.$, PC R 6 d+.$ eT]jTT
AP >PB nsTTq AP =
(A) 8 d+.$ (B) 4 d+.$ (C) 12 d+.$ (D) 6 d+.$
14. eeTT jTT e +\ eeTTq uV _+

7
D$T
“There is perhaps nothing which so occupies the middle position of
mathematics as trigonometry” – J.F. Herbart
• |]#jT+
• dsdMTsDeTT\T
• mT\T eT]jTT jT D$T deTd\qT
k~+#T jT+VDeTT\qT eTT+ \T|
qeTyTq |< , q ds #+<
d

2
nsTTq|{ ( sin i + cos i)
= 1 nqT dMTsDeTT dsdMTsDeTT TqT. , i R 45 0 q|&T ~ deTT ,
204 10th Std. Mathematics
(sin45 0 + cos45 0 ) 2 =
1 1
c + m 2
= 2 ! 1.
2 2
n s+q~> |]>D+|&q~.
eT&T |j>syTq dsdMTsDeTT\qT |]jTH dsdMTsDeTT\T n+,
AB
AB
2
2
2
+
BC
2
=
AB
AC
` 2
AB
j ( a AB ! 0)
2
AB BC
2
` +
AB
j `
AB
j =
AC
` 2
AB
j ( 1 + tan 2 i = sec 2 i. (3)
i = 90c tani eT]jTT seci \qT s+#eTT, dsdMTsD+ (3) qT 0 # i 1 90
\ eT< >\ n i $\Te\T ||s#T#Tq$.
(1) BC 2 , # eTs\ u+#>,
AB
BC
2
2
2
+
BC
2
=
BC
2
AB BC
2
` +
BC
j `
BC
j =
AC
` 2
BC
j ( a BC ! 0)
AC
` 2
BC
j ( cot 2 i + 1= cosec 2 i. (4)
0 0
i = 0 0 coti eT]jTT coseci \qT s+#eTT, dsdMTsD+ (4) 0 0 1 i # 90
0
\ eT< >\ n i $\Te\T ||s#T#Tq$.
A
i
|eTT. 7.1
C
B

(2) qT+& (4) es >\ | dsdMTsDeTT\ = T\ s|eTT\T +~e&q$.
dsdMTsDeTT T\s|eTT\T
(i)
2 2
sin i + cos i = 1
sin 2 i = 1 - cos 2 i () cos 2 i = 1 - sin 2 i
(ii)
2 2
1 + tan i = sec i
sec 2 i - tan 2 i = 1 () tan 2 i = sec 2 i -1
(iii)
2 2
1 + cot i = cosec i
cosec 2 i - cot 2 i = 1 () cot 2 i = cosec 2 i -1
d#q\T
n\DeTT ‘i' , | dsdMTsDeTT\T s|+#&qT. D$T |yTjTeTT\
@y DeTT\T ‘i’ dsdMTsDeTT\T deT>TqT. bs|deTT DeTT i
n\D $\Te\T eyT rdT=H, D$T |yTjTeTT\ L&q D$T dsdMTsDeTT\qT
s|+#T mTe+{ keq|< |&T = yTe\T +~e&q~.
(i) dsdMTsDeTTqT C>> #~$,@$Te&q< eT]jTT k~+#T @~ nedsy
>eT+#e\jTTqT.
keqeTT>, dsdMTsD dT\u u>eTTqT rdT=, $d]+ d]+#T
H ]q u>eTTqT rdT= d]+#&+ eT+.
dsdMTsDeTT sTy|\ ]qeTT> qTqsTT,n< dedeTT |r <
|+> rdT= d++> d]+|eTT.
dedeTT\ d+\qeTTq ;JjT >D yTe\T |j+ y{ _qeTT\qT
\T|eTT.
nedsyT, |r | e]
sTy d]+#T |jT+#TeTT.
2 2 2 2
tan i cot i cosec i sec i | qT+&TqT.
(ii)
(iii)
(iv)
(v)
(vi) , , , ,

nudeTT 7.1
1. +~e&q y{ dsdMTsDe < ssTT+#TeTT.
2 2
2 2
(i) cos i + sec i = 2+sin i (ii) cot i + cos i = sin i
2. +~ dsdMTsDeTT\qT s|+#TeTT.
(i) sec 2 2 2 2
i + cosec i = sec i cosec i (ii)
(iii)
1 - sin i
1 + sin i
2 2
sin i
1 - cos i
= cosec i + cot i
= sec i - tan i (iv)
cos i
= 1 + sin i
sec i - tan i
(v) sec i + cosec i = tan i + cot i (vi)
(vii) sec i ^1
- sin ih^sec i + tan ih = 1 (viii)
3. +~ dsdMTsDeTT\T s|+#TeTT.
(i)
(ii)
(iii)
(iv)
sin^90c
- ih
+
cos i
1 + sin i 1 - cos^90c
- ih = 2sec i
tan i
+
cot i
= 1 + sec i cosec i
1 - cot i 1 - tan i
sin^90
0 - ih
cos^90
0
- ih
+ = cos i + sin i
1 - tan i 1 - cot i
0
tan^90
- ih
+
cosec i + 1 = 2 sec i.
cosec i + 1 cot i
(v)
cot i cosec 1
cosec cot .
cot i
+ i
i i
- cosec i
- = +
+ 1
(vi) ^1 + cot i - cosec ih^1
+ tan i + sec ih
= 2
1 + cos i - sin i
= cot i
sin i ^1
+ cos ih
sin i
= 1 - cos i
cosec i + cot i
2
(vii)
sin i - cos i + 1 =
1
sin i + cos i - 1 sec i - tan i
0
(viii)
tan i sin i sin^90
- ih
2
=
2 0
1 - tan i 2 sin ^90
- ih
- 1
(ix)
1
-
1
=
1
-
1
.
cosec i - cot i sin i sin i cosec i + cot i
2 2
(x)
cot i + sec i
2 2
= ( sin i cos i)
^tan i + cot ih.
tan i + cosec i
2 2 2 2
4. x = a sec i + b tan ieT]jTTy = a tan i + b sec i nsTTq, x y a b
5. tan i = n tan a eT]jTT sin i m sin a
= nsTTq,
cos
2
2
i =
m
2
n
i
- = - s|+#TeTT.
- 1 , n ! 1
- 1
i
! , s|+#TeTT.
6 2
6. sini, cosi eT]jTT tani >TDXDjT+

7.3 mT\T eT]jTT VeTT\ eT<

T]+#TeTT.
(v) q $\Te\qT ||+ eT]jTT *jT y{ k~+#TeTT.
Trigonometry 213

+~ eTT s |=qT |j>|&TqT.
eTT
• eTTqT eT]jTT eTs=
k
es =q |tqT TeTT.
• De eTeTT k eTeTT #sTqT De
=\e>\e! nTe+{ y{
< rdT=qTeTT.
• k =qTeTT. }sDeTTqT ssTT+#T 90c qT+& =\qT
rdyjTTeTT. \ T]+#T=qTeTT.
• edTe jTT mTqT (h) qT>=qT, +~ dMTsDeTT |j+#TeTT.
h = x + y tan i.
TsT#Tq >* |eTT jTT \ =qTeTT. (+\ TqT.
|eTT. 7.7

&| y*jTTq~. >& qT+& 3.5 MT eTTqT ##qT. >|seTT qT+& 30 3
MT |s+ mTqT qT>=qTeTT.
k|seTT mT BD eT]jTT u eTeTT qT+& |]o\ +{ eTeTTq >\ AB qT ^jTTeTT. n$ AB = EC.
AB = EC = 30 3 MT. eT]jTT
AE = BC = 1.5 MT. e&q~
=
3
3
x
Trigonometry 215

\+D 3DEC ,
216 10th Std. Mathematics
tan 30c =
CD
EC
( CD = EC tan 30
` CD = 30 MT.
qT, >|s+ mT, BD = BC + CD
c
=
30 3
3
= 1.
5 + 30 = 31.5 MT.
*# {sT> qTq #T $]q~. #T| u>eTT 30c.DeTT u$T
qT. #T b=qTeTT.
k

k\ _+

& eT]jTT >|s+ \|s+ |u>eTT qT+& >&| u>eTTqT
eT]jTT +~ u>eTTqT eTeTT> 45c eT]jTT60c eTDeTT ##qT. >|s+ mT 90 MT.
nsTTq >& mTqT qT>=qTeTT. ( 3 = 1.
732 |j+)
k& AE eT]jTT >|s+ BD nqT=qTeTT.
AB de+seTT> EC qT ^jTTeTT. n~ AB=EC. qT, AE =
AB = x MTsT, AE = h MTsT. nqT=qTeTT..
BD = 90 MTsT, + DAB = 60c, DEC 45
AE =BC
qT, CD =BD -BC = 90 - h.
\+D 3DAB ,
\+D 3DEC ,
+ = c e&q~.
= h MTsT.
tan 60c =
BD
AB
90
=
90
x
( x =
3
= 30 3
(1)
tan 45c =
DC
=
90 - h
EC x
qT, x = 90 - h
(2)
(1) eT]jT (2) qT+&, 90 - h= 30 3
qT, >& mT, h = 90 - 30 3 = 38.04 MT
VeTT ( lighthouse) | \& u*
sT y|q s+&T |&e\qT ##T#Tq~. |&e\ eTDeTT\T 30c eT]jTT 60c |&e\
eT\ =qTeTT.
kV |u>eTT\T eTeTT> A eT]jTT D nqT=qTeTT.
s+&T |&e\T B eT]jTT C nqT=qTeTT. deTTVeTT |u>eTTq >\

\+D ACD 3
tan 30c =
AD
AC
( AC =
AD
tan
30c ( x + 300
= h
1
c m
3
qT, x + 300 = h 3 (2)
(1) qT (2) |j+#>,
h
300
`
( h
+ = h 3
3
3 -
h
= 300
3
2 h = 300 3 . qT, h = 150 3 .
eq, deTTV mT 150 3 MT.
>* #*+#T#Tq
u\HqT >T]+#qT. ueTeTT qT+& +{ eTeTTq 1.2 MT. >\ZqT. deTjTeTT u\H |jD =qTeTT.
k\ _+\ _+

eTTq C+& d+ueTT \&jTTq~. u$T| _+eTT eT]jTT +~ u>eTTqT #jTT }sDeTT\T eTeTT> 60c eT]jTT 45c
C+&d+ueTT mT 10 MT nsTTq, ueqeTT mT qT>=qTeTT ( 3 = 1.
732 |j+)
k qTq =+&| u>eTTqT eT]jTT 30c eTDeTT =+& |su>eTTqT ##qT. =+&
mTqT qT>=qTeTT.
k qTq =+& mT BD nqT=qTeTT.
z& kqeTTqT A eT]jTT |]o\H _+ EC ^jTTeTT. n~ AB = EC n>TqT.
AE = 14 MT, + ABE = 30c,
+ DEC = 60c e&q~.
\+D 3ABE , tan30c
=
AE
AB
|eTT. 7.18

` AB =
AE
( AB = 14
tan 30c
qT, EC = 14 3 ( a AB = EC )
\+D 3DEC , tan 60c
=
CD
EC
` CD = EC tan 60 c ( CD = (14 3)
3 = 42 MT
qT, =+& mT, BD = BC + CD = 14 + 42 = 56MT.
TsTq|&T < ds mTqT qT>=qTeTT.
kTsT#Tq $eqeTT ds mT\T
D
BE eT]jTT CD nqT=qTeTT
E
+ DAC = 30c , EAB 60
+ = c e&q~..
BE = CD = h MT
AB = x MTsT nqT=q,
15 d+& #]q TsT#Tq $eqeTT jTT ds mT 1500
3 =
1
^x
+ 3000h
3
( 3x = x + 3000 ( x = 1500 MT.
3
A
3 MT.
60
30
x
B
h
|eTT. 7.19
C
h
Trigonometry 221

nudeTT 7.2
1. qT+& ds\T ~+#T |j+#T y\T\eTT }sDeTT 30c. u$T qT+&
y\T \eTT |u>eTT 0.9 MT. nsTTq, y\T \eTT b&eqT qT>=qTeTT.
2. B|d+ueTT eTT+\ u* & b&e 150 3 d+.
MT. B|d+ueTT |u>| }sDyT+?
3. A eT]jTT B nqT s+&T eTT\T n$#jTT XeTT\qT 2 MT. es $q>\e. H\| >\
eTT A, 1 MT &| q dVT&T B qT ##qT. n~ k|sT>T#
qT d +~. A qT+& B }sD+ 30c q|&T A X+ T VseT>TH! < (A |\T| B $q, B |+#T=qTq
nqT=qTeTT.)
4. |]o\&T, s yT|eeTTqT \TdT=qT yT|TeTT| + {sT>
|+|qT. u$T qT+& 1.5MT mT eT]jTT +
|+| |]seTT (Spotlight) q 100 MT =HqT. nsTTq yT|eeTT m+ mT \eT : e mT qTq yT|eeTTqT TqT. $eHXjTeTT q+
|>TsT eT]jTT ~>T q mT yT|TeTT\T
+&eqT. sy + {sT> | |d]+#T T]+#e#TqT.)
5. 40 d+.MT b&e>\ \eTT |P] &\qeTT #jTTq|&T, KseTT q+\ w >\ s+&T #| LsT jTTq$. n$ H\| >\ e&qT eTeTT> 45c
eT]jTT 60c eTDeTT #dTq$. n$ e&qT rdT=qT w esZeTT y|
deTjTeTTq eT]jTT y>eTT m>TsT bs+_+#qT. B @ | >\Tb+=qTeTT. ( 3 = 1.
732 |j+#TeTT)
9. X nqTq&T de+s \eTT q+

ueq || q+ qsTTq, Y qT+& | >\ =qTeTT.
10. s> >~ Ls= jTTq $V e*y&T \ q H (Yacht) eT]jTT |< H ( Barge)qT |]o*+#qT. q H eT]jTT
|< H\ eTDeTT\T eTeTT> 45c eT]jTT 30c. s+&T deTT y{ eT< T\T +&TqT. n$ 300 n&T>T\T +f
yqsTT, e*y&T nbjTd#q < #jTTqT. e*y&T nbjTd#q <
#djTTH&?
13. u$T| \&jTTq u\T&T, de+seTT> ds mT e< >* #*+#Tq
u\HqT >T]+#qT. deTjTeTTq u\T&T qT+& u\H #jTT }sDeTT 60c n<
|]o\H _+ZqT. >* y>eTT
29 3 MT/ nsTTq, ueTeTT qT+& u\H mTqT qT>=qTeTT
14. HsT> qTq sV|s+ beTTqT #sTqT. >|s+ |u>eTTq \&jTTq
eTw 30c eTDeTT yqT (van) >T]+#qT. yqT deT y>+ >|s+qT
#sTqT. 6 $TweTT\ sTy,60c eTDeTT yqTqT qT>=HqT. >|s+qT #sT
yqT + m $TweTT\T rdT=qTqT?
15. s+&T ukeseTT\ qT+& |>VeTT jTT y|qT |]o*+#T }sDeTT
30c eT]jTT 60c s+&T ukeseTT\T eT]jTT |>VeTT n< \+\eTT \
e. ukeseTT\ eT< VeTT eT]jTT u$T >\
=qTeTT. ( 3 1.
732
= |j+#TeTT)
16. 60 MT b&yq >|s+ |u>eTT qT+& ueqeTT | u>eTTqT eT]jTT +~ u>eTTqT
#jTT eTDeTT\T esTd> 30c eT]jTT 60c. ueqeTT mTqT qT>=qTeTT.
17. 40 MT mT >\ >|s+ | u>eTT eT]jTT bV |u>eTTqT eTeTT>
30c eT]jTT60c }sDeTT ##qT.B|>VeTT mTqT qT>=qTeTT. eT]jTT >|s
bV |u>eTTq >\ =qTeTT.
18. dsdT |]\+ qT+& 45 MT mTq _+TsT#Tq
V*|sYqT 30c }sDeTT ##qT. n< _+\ =qTeTT
Trigonometry 223

d]jTq yT\T mqT=qTeTT.
2 2
1. ^1
- sin ih sec i =
nudeTT 7.3
(A) 0 (B) 1 (C) tan 2 i (D) cos 2 i
2 2
2. ^1
+ tan ih sin i =
(A) sin 2 i (B) cos 2 i (C) tan 2 i (D) cot 2 i
2 2
3. ^1 - cos ih^1
+ cot ih =
(A) sin 2 i (B) 0 (C) 1 (D) tan 2 i
4. sin^90c
- ih cos i + cos^90c
- ih sin i =
(A) 1 (B) 0 (C) 2 (D) –1
2
5. 1 -
sin i
=
1 + cos i
(A) cos i (B) tan i (C) cot i (D) cosec i
4 4
6. cos x - sin x =
2
2
(A) 2 sin x - 1 (B) 2 cos x - 1 (C) 1 + 2 sin x (D) 1 - 2 cos x.
7. tan i = xa
nsTTq,
x
$\Te =
2 2
a + x
(A) cos i (B) sin i (C) cosec i (D) sec i
2
8. x = a sec i, y = b tan i nsTTq,
x y
a
2
-
2
$\Te =
b
(A) 1 (B) –1 (C) tan 2 i (D) cosec 2 i
9.
10.
sec i
=
cot i + tan i
(A) cot i (B) tan i (C) sin i (D) – cot i
sin^90c
- ihsin
i cos^90c
- ihcos
i
+
=
tan i
cot i
(A) tan i (B) 1 (C) –1 (D) sin i
11. |qTq |eTT AC =
(A) 25 MT (B) 25 3 MT
(C)
25
MT (D) 25 2 MT
3
12. |qTq |eTT, + ABC =
2
C100 3 MT
2
C
A
60c
25 MT
B
2
(A) 45c
(B) 30c
(C) 60c (D) 50c
A
100 MT
B
224 10th Std. Mathematics

13. >|s+ qT+& &T 28.5 MT |seTTq }sDeTT 45c nsTTq >|s+ mT
(A) 30 MT (B) 27.5 MT (C) 28.5 MT (D) 27 MT
14. |qTq |eTT sin i =
15 . 17
nsTTq BC =
C
(A) 85 MT
(C) 95 MT
(B) 65 MT
(D) 75 MT
A
85 MT
i
B
15. ^1 + tan 2 ih^1 - sin ih^1
+ sin ih =
2 2
2 2
(A) cos i - sin i
(B) sin i - cos i
2 2
(C) sin i + cos i
(D) 0
16. ^1 + cot 2 ih^1 - cos ih^1
+ cos ih =
2 2
2 2
(A) tan i - sec i
(B) sin i - cos i
2 2
(C) sec i - tan i
2 2
(D) cos i - sin i
2 2
17. ^cos
i - 1h^cot
i + 1h + 1 =
18.
19.
(A) 1 (B) –1 (C) 2 (D) 0
2
1 + tan i
2
=
1 + cot i
(A) cos 2 i (B) tan 2 i (C) sin 2 i (D) cot 2 i
2
sin i +
1
2
1 + tan i
=
2 2
(A) cosec i + cot i
2 2
(B) cosec i - cot i
2 2
(C) cot i - cosec i
2 2
(D) sin i - cos i
2 2
20. 9 tan i - 9 sec i =
(A) 1 (B) 0 (C) 9 (D) –9
\Tk?
b msY&dt (26 e], 1913-20 d|+sT, 1996) V+>] >DXdE&T. >DXd
#] |]X #dq |#TsDs\ &q msY&dt, *jHVsY
jT\sY eyT b\>*qysT. n&T q J$\+ dTesT 1,457 >D
s#q\qT #dqT. jT\sY dTesT 800 s#q\T #dqT. nqT >DXdeTT| \yTq
qeTeTT+ eT]jTT de de>H+ k

8
>DqeTT
“Measure what is measurable, and make measurable what is not so”
-Galileo Galilei
• |]#jT+
• |]\ yX\eTT
eT]jTT |Tq|]eDeTT
v d|eTT
v X+KTe
v >eTT
• d+jTT |eTT\T
eT]jTT d s | Tq|]eDeTT\T
]yT&dt
(287 BC - 212 BC)
^dT
bNq X+ ]yT&dt
>=| >D Xd y > >T]+#&qT.
sU>DeTT deT\ |eTT\
y X\eTT\T n e\
(|]\eTT\) y X\eTT\T eT]jTT
| Tq|]eDeTT\qT qT
eTTKeTT> s+#qT.
8.1 |]#jT+
sK\ b&e\T, #TT=\\T eT]jTT deT\ |eTT\
y X\eTT\T, | Tq edTe\ |]\ y X\eTT eT]jTT
| Tq|]eDeTT e+{ =\\qT >] \T| sU>D u>eTTqT
>DqeTT n n+
|j>|&T#Tq$. b< $T sU>DeTT deT\eTT\T, |\T
eTTKeTT\ |]\eTT\T n = | TqeTT\ e\eTT\T
|]>D+|&q$. (eTT\T)
n d XdeTT (Nano Science) nqT >=| ueqqT
|]\ yX\eTT, |Tq|]eDeTT\ w *jTCd+~.
w n dXd d\T eT]jTT k+ |]Cq >TDeTT\
qT \T|, |]eDeTTq d++~+q T]+ HsT=+{sT.
8.2 |]\ yX\eTT (Surface Area)
>eTT | Tq|]eD+ nqTq~ e d|eTT | Tq|]eD+
2/3 e e+T deqeT dd dsLdt
#+~q ^ >D Xd y ]yT&dt s|+#qT.
n&T B eTTKyTq k< q> n_e]+#qT.
n&T |se\jTeTT jTT cdqeTT |<
|j+ B y X\eTTqT qT>=HqT.
|eTT. 8.1
| TqedTe jTT uV y X\eTT
|]\ y X\eTT n>TqT. 3-|]eDeTT
edTe\ uV |]\ y X\eTT < |]\
y X\eT>TqT. = | TqeTT\ |]\ y X\eTT\
|eTT\qT | |eTT #&e#TqT. |eTT 8.2
226

8.2.1 eT es d|eTT (Right Circular Cylinder)
= es |\\T, ||sT n\qT s|+ eT]jTT =\ rdT= \Te
esTd neTs> @s&T |Tqs|eTTqT eT es d|eTT n n+ +&TqT. (|eTT 8.3
#&TeTT)
|eTT. 8.3
s#qeTT
Bs#TsdeTT, < uTeTTqT #dT |P] ueTDeTT #dq @s&T |Tq
seTTqT eT es d|eTT n+

(ii)
|Tq eT es d|+ jTT d+|Ps\ yX\eTT (Total Surface Area of a
solid right circular cylinder)
d+|Ps\ yX\eTT, TSA = e\ yX\eTT G 2 I u yX\eTT
228 10th Std. Mathematics
= 2r rh + 2 # rr
d+|Ps\ yX\eTT, = 2r r( h + r)
#.|eDeTT\T.
(iii) eT es u\T d|eTT (Right circular hollow cylinder)
|eTT. 8.7
qT| ||, ssY T yTTT |TqeTT\ seTT\T u\T d|eTT\T n>TqT. u\T
d|+ mT h nqT=q y{ jT{ eT]jTT |* yks\T esTd> R eT]jTT r n>TqT.
e\ yX\eTT CSA = jT{ |]\ yX\+ G |* |]\ yX\eTT
= 2rRh + 2rrh
` CSA = 2r h( R + r)
#.|eDeTT\T.
d+|Ps\ yX\eTT TSA = e\ yX\+ G 2 I u yX\eTT
d#q :
u\T d|eTT eT+eT
= 2 rh( R + r) + 2 # [ rR 2 - rr
2
]
= 2 rh( R + r) + 2 r( R + r)( R - r)
` TSA = 2 r ( R + r)( R - r + h)
#.|eDeTT\T.
- r.
n

=qTeTT. ( r =
22
rdT=qTeTT.)
7
k

k dt+, eTT, |PH {|, yTV+ H yTTT s|eTT\qT #dTHeTT. |q |s=q
edTe\ eT es X+KTe seTTqT *jTTq$.
14 d+.MT

|Tq edTyq X+KTe, deT\

u\T X+KTe jTT e\ yX\eTT (Curved surface area of a hollow cone)
e K+& ykseTT l. eT]jTT +< DeTT ic > rdT=H
eT es X+KTe @s&TqT.
X+KTe ykseTT r nqT=qTeTT.
eq, L = 2rr
(1) qT+&
2r r =
( r = l
(
e K+&eTT y
2rl
#
ic
r l 2
=
360c
A ic
(2)
X+KTe e\ yX\eTT = e K+&eTT yX\eTT
X+KTe e\ yX\eTT A =
(ii)
rl
2
ic
2
c m
360c
= rl
r
`
l
j .
X+KTe e\ yX\eTT = r rl #.|eDeTT\T
eT es |Tq X+KTe jTT d+|Ps\ yX\eTT (Total surface area of the
solid right circular cone)
|Tq X+KTe d+|Ps\ yX\eTT = X+KTe e\ yX\eTT
)
G u y X\eTT
|Tq X+KTe d+|Ps\ yX\eTT = r^l
35 d+.MT eT]jTT 37
d+.MT nsTTq X+KTe e\ y X\eTT eT]jTT d+|Ps \ y X\eTTqT qT>=qTeTT. ( r =
22
)
i
M
l
L
N
M
|eTT. 8.16
360c
d#q
ic
e K+&eTTqT X+KTe> eT&q|&T +~
c m
360c
esT\T d+u$+#TqT:
r =
ic
c m
e K+&eTT X+KTe
l 360c
ykseTT (l) " y\TfT (l)
X\eTT A nqT=qTeTT. n|&T #| b&e (L) " u$T #TT=\ 2rr
yX\eTT " e\ yX\eTT rrl
l
rrl
rr 2
h
r
|eTT. 8.17
l
L
7

k

e K+&eTTqT eT&|> eT es X+KTe
@s&TqT.
X+KTe jTT u |]~ = #|eTT b&e
234 10th Std. Mathematics
( 2r r =
i
2 R
360 # r
c
( r =
i
R
360 c
#
=
120 c #
360c
qT, X+KTe jTT uykseTT, r 21= 7 d+.MT
eT]jTT, X+KTe y\TfT,
eTs= |< :
l = e K+& yks+
X+KTe e\ yX\eTT,
l = R ( l = 21 d+.MT.
CSA = rrl
=
22
7
# 7 # 21= 462.
X+KTe e\ y X\eTT R e K+& y X\eTT
=
i
2
# rR
360 0
=
120
#
22
# 21 # 21
360 7
eq, X+KTe e\ yX\eTT 462 #.d+.MT.
= 462 #.d+.MT
8.2.3 >eTT (Sphere)
es &dtqT < yd+ ueTDeTT #dq|&T @s&T |THseTT >eTT
n+eTT |]eDeTT >\ edTe. B |]\ yX\eTT eT]jTT |Tq|]eDeTT\T
e+&TqT.
(i) >eTT jTT e\ yX\eTT (Curved surface area of a solid sphere)
eTT
es &dt rdT=, < eTT n+ |]\ y X\eTT n< yks eTT >\ e y X\eTTq, H\T>T sT+&Tq +~
eTT dVjTeTT \TdT=qTeTT.
bd + rdT=qTeTT.
+ | u>eTT >T+&T
r
r
d~ neTsTeTT.
+ #T nq> yTTeTT
r
e\ yX\eTT |qT>
rr 2
eTT\T> ]+#TeTT.
|eTT. 8.21
|eTT #|q $

(ii)
>eTT eT]jTT @s&q eeTT\ ykseTT\qT =\TeeTT.
>eTT ykseTT = H\T>T deq eeTT\ ykseTT.
qT, >eTT e\ yX\eTT = 4 I e yX\eTT
= 4 # rr
` >eTT e\ yX\eTT = 4r r
2
#.|eDeTT\T
|Tq ns>eTT (Solid hemisphere)
>+ +T+& be deT\+ > s+&T deTu>\T>
$udT+~. | deTu> |Tq ns>eTT n+ e\ yX\eTT
>eTT e\ yX\eTT
=
=
4 2
rr
2
2
2
=2r r 2
#.|eDeTT\T
ns>eTT jTT d+|Ps\ yX\eTT = e\ yX\eTT G e yX\eTT
(iii)
= 2rr + rr
2 2
= 3rr 2 2rr
#.|eDeTT\T
2
u\T ns>eTT (Hollow hemisphere)
R eT]jTT r u\T ns>eTT jTT jT{ eT]jTT |* yks\T nqT=qTeTT.
e\ yX\eTT = jT{ |]\ yX\eTT G |* |]\ yX\eTT
= 2rR + 2rr
2 2
= 2r(R 2 +r 2 ) #.|eDeTT\T.
d+|Ps\ yX\eTT = jT{ |]\ yX\eTT G |* |]\
yX\eTT + u yX\eTT
2 2 2 2
= 2rR + 2rr + r^R - r h
= 2r(R 2 + r 2 ) + r(R + r)(R - r) #.|eDeTT\T.
\ u\T >eTT >s&
#jTTqT. yVqeTT q&T| n+=qTeTT. ( r =
22
)
7
keTT |* yd+, 2r = 7 MT
y{sT d q&T|y&T yVqeTT q&T| n+eTT jTT |* |]\ y X\eTT
r = r (2 r)
2
=
22
7
= 4 r
2
# 7 2
nqT yVqeTT q&T| n+

eTT jTT d+|Ps\ yX\eTT 675r #.d+.MT. nsTTq < e\
yX\eTT qT>=qTeTT.
keTT d+|Ps\ yX\eTT,
3 r
2
( r 2 = 225
236 10th Std. Mathematics
r = 675r #.d+.MT.
|Tq ns>eTT e\ yX\eTT,
CSA = 2r r 2
= 2r #225 = 450r #.d+.MT.
s b eT+=qTeTT ( r =
22
)
7
k b |* eT]jTT y\T|* yks\T eT]jTT b
eT+=qTeTT.
2. eT es d|+ d+|Ps\ yX\+ 660 d+.MT 2 eT]jTT < =qTeTT.
3. eT es d|eTT jTT e\ yX\eTT eT]jTT u|]~\T esTd> 4400
#.d+.MT. eT]jTT 110 d+.MT. nsTTq < mT eT]jTT ydeTT\qT qT>=qTeTT.
4. ueqeTT 12 eT dbs d+ueTT\T >\e. | < yks+ 50 d+.MT.
eT]jTT mT 3.5 MT. d+ueTT\ s+>T yjTT #.MT. s.20\T KsT nsTTq
yTTeTT 12 d+ueTT\ s+>T yjTT n>T KsT qT>=qTeTT.
5. |Tq eT es d|eTT d+|Ps\ yX\eTT 231 d+.MT 2 < e\ yX\+,
d+|Ps\ yX\eTTq eT&+{ s+&T e+T\T nsTTq < yks+, mT\qT
qT>=qTeTT.
6. eT es d|+ d+|Ps\ yX\+ 1540 d+.MT 2 < mT u yks+q
4 sT nsTTq d|eTT mTqT qT>=qTeTT.`
r
|eTT. 8.25
5 d+.MT
R
|eTT. 8.26

7. s+&T eT es d|eTT\ ykseTT\ w 3:2 eT]jTT y{ mT\ w 5:3
nsTTq y{ e\ yX\eTT\ w qT>=qTeTT.
8. u\T d|eTT e\ yX\eTT 540r #.d+.MT. < |* yd+ 16 d+.MT., mT
15 d+.MT. nsTTq d+|Ps\ yX\+ qT>=qTeTT.
9. dbs| qT| || y\T|* yd+ 25 d+.MT. eT]jTT < b&e 20 d+.MT.
|| eT+=qTeTT.
10. |Tq eT es X+KTe ykseTT eT]jTT mT\T esTd> 7 d+.MT., eT]jTT 24
d+.MT. nsTTq < e\ yX\eTT eT]jTT d+|Ps\ yX\eTT\qT qT>=qTeTT.
11. eT es X+KTe }sDeTT eT]jTT ykseTT esTd> 60c eT]jTT 15 d+.
MT. nsTTq < mT eT]jTT y\TfT\qT qT>=qTeTT.
12. |Tq X+KTe u |]~ 236 d+.MT. eT]jTT y\TfT 12 d+.MT. nsTTq e\
yX\eTTqT qT>=qTeTT.
13. e&| X+KTe s+ qTq~. < yd+ 4.2 d+.MT. eT]jTT mT 2.8 MT
e&| es+ &e+& HdT# |&q~. < nedseT>T HdT >T&
yX\eTTqT qT>=qTeTT.
14. es &dt jTT eK+&eTT eT180c
eT]jTT 21 d+.MT. eK+&eTT jTT =q\T {> *|q u\T X+KTe @s&TqT.
nsTTq X+KTe ykseTT qT>=qTeTT.
15. |Tq X+KTe jTT ykseTT eT]jTT y\TfT\ w 3:5 < e\ yX\eTT 60r
#.d+.MT. nsTTq d+|Ps\ yX\eTTqT qT>=qTeTT.
16. |Tq >eTT e\ yX\eTT 98.56 d+.MT 2 nsTTq >eTT yks+ qT>=qTeTT.
17. |Tq ns>eTT jTT e\ yX\+ 2772 #.d+.MT. nsTTq < d+|Ps\
yX\eTT qT>=qTeTT.
18. s+&T |Tq ns>eTT\ ykseTT\ w 3:5 nsTTq y{ e\ yX\eTT w
eT]jTT d+|Ps\ yX\eTT\ w qT>=qTeTT.
19. u\T ns>eTT jTT uV eT]jTT|* yks\T esTd> 4.2 d+.MT. eT]jTT
2.1 d+.MT nsTTq e\ yX\eTT eT]jTT d+|Ps\ yX\eTTqT qT>=qTeTT.
20. n+s e\ yX\eTT >\ ns>s ueq KseTTq s+>T yjTeqT. u |]~
17.6 MT. s+>T yjTT 1 MT 2 ` 5 KsT nsTTq yTT+ s+>T yjTT n>T KsTqT
qT>=qTeTT.
8.3 |Tq|]eDeTT (Volume)
+es eTqeTT = |TqeTT\ |]\ yX\eTTq d++~+q deTd\qT #$T.
|&T |TqeTT\ |Tq|]eDeTT\qT >D+#TqT \TdT+TqT.
$&Bdq, <
|Tq|]eDeTT y{ |TqeTT\ d+U deqeTT.
Mensuration 237

|eTT qTq |TqeTT jTT |Tq|]eDeTT
R b&e I y&\T I mT
= 1 d+.MT. #1 d+.MT. #1 d+.MT. = 1 d+.MT. 3 .
1d+.MT
edT| jTT |Tq |]eD+ 100 |Tq d+.MT. n #|e#TqT.
1d+.MT
|eTT. 8.27
edTeqT |P]> +| | 1 d+.MT. 3 |Tq|]eDeTTq 100
|TqeTT\T e\jTTqT.
|]\ yX\eTT e |Tq|]eDeTT +&TqT. = |TqeTT\ |Tq|]eDeTT\T +< e&q~.
8.3.1 eT es d|+ |Tq|]eDeTT (Volume of a right circular cylinder)
(i) |Tq es d|+ |Tq|]eD+ (Volume of a solid right circular cylinder)
u yX\eTT eT]jTT mT\ \+qT eT es |Tq d|eTT
|Tq|]eD+ n>TqT.
d|eTT |Tq|]eD+, V = u yX\+ I mT
238 10th Std. Mathematics
= r r 2
# h
eq, d|eTT |Tq|]eDeTT, V = r 2
h
r |T.|eDeTT\T
(ii) u\T d|eTT |Tq|]eDeTT (Volume of a hollow cylinder)
|eTT. 8.28
R eT]jTT r \T eT es u\T d|eTT jTT y\T|* eT]jTT n+s ykseTT\T
r
nqT=qTeTT. h nqTq~ < mT nqT=qTeTT.
|Tq|]eDeTT, V = y\T|* d|eTT
}
n+s d|eTT
|Tq|]eDeTT ` { |Tq|]eDeTT
= rR 2 h - rr 2
h
qT, u\T d|eTT |Tq|]eDeTT,
2 2
V = rh( R - r ) |T.|eDeTT\T
=qTeTT. ( r =
22
)
7
k eT es d|eTT jTT yks eTT eT]jTT mT nqT=qTeTT.
e&q~ h = 8 d+.MT. eT]jTT CSA = 704 #.d+.MT
e\ yX\eTT = 704
( 2r rh = 704
2 #
22
# r # 8 = 704
7
1d+.MT 2
V
= rr 2
h
rr 2
|eTT. 8.29
704d+.MT 2
r
|eTT. 8.30
1d+.MT
h
R
8d+.MT

` r =
704 7
2 # 22 # 8
r r 2
h
d|eTT |Tq|]eDeTT, V =
=
22
7
# = 14 d+.MT
14 14 8
# # #
= 4928 |T.d+.MT
eq, |Tq|]eDeTT = 4.928 sT. (1000 |T.d+.MT R 1 sT)
eq qT| >=+ b&e 28 d+.MT. < jT{ eT]jTT |*
ydeTT\T esTd> 8 d+.MT. eT]jTT 6 d+.MT nsTTq >=+ jTT |Tq|]eDeTT qT>=qTeTT.
1 |T.d+.MT qTeTT sTe 7 >eTT\T nsTTq qT| >=+ sTeqT qT>=qTeTT.( r =
22
)
7
k u\T dbs >=+ jTT |*, y\T|* ykseTT\T
eT]jTT mT nqT=qTeTT.
6d+.MT
e&q~ 2r = 6 d+.MT, 2R = 8 d+.MT , h = 28 d+.MT
>=+ |Tq|]eD+, V = r # h # ( R + r)( R - r)

=
22
# 28 # ( 4 + 3)( 4 - 3)
7
` |Tq|]eD+, V = 616 |T.d+.MT
1 |T.d+.MT. V| sTe = 7 >eTT\T
616 |T.d+.MT V| sTe = 7 # 616 >
8d+.MT
>=+ sTe = 4.312 . >.
|eTT. 8.31
13.86 #.d+.MT. eT]jTT 69.3 |T.d+.MT. nsTTq < mT eT]jTT e\ yX\eTTqT
qT>=qTeTT. ( r =
22
)
7
k

22 # r
2
= 13.86
7
r 2 = 13.86# 7
= 4.41 22
` r = 4.
41 = 2.1 d+.MT.
e\ yX\eTT, CSA =
2rrh
= 2 #
22
# 2.
1 # 5
7
eq, CSA = 66 #.|eDeTT\T
8.3.2 eT es X+KTe |Tq|]eDeTT (Volume of a right circular cone)
r eT]jTT h \T esTd> eT es X+KTe u ykseTT eT]jTT mT nqT=qTeTT.
X+KTe |Tq|]eDeTTq e&q jTT deTT: V =
1 2
# r r h|T.|eDeTT\T
3
+~ eTT eT+|eTT.
eTT
deq mT eT]jTT deq ykseTT\ u\T d|eTT eT]jTT u\T X+KTe\qT
|eTT +< #|q $ jsT#jTTeTT. +< #|+q $+> qT>=H

qT,
1 2
r r h = 4928
3
(
1 22 2
# # r # 24 = 4928
3 7
( r 2 =
4928 # 3 # 7
= 196.
22 # 24
4928d+.MT 3
eq, X+KTe u ykseTT, r = 196 = 14 d+.MT.
r
|eTT. 8.34
8.3.3 X+KTK+&eTT jTT |Tq|]eDeTT (Volume of a Frustum of a Cone)
eT es |Tq X+KTeqT rdT= < s+&T |TqeTT\T> |< ]+q
q eT es X+KTe @s&TqT. X+KTe jTT eTs= u>eTTqT X+KTK+&eTT (frustum)
n+eT+#TeTT.
eTT
=+ +eT{ rdT= < eT es X+KTeqT jsT#jTTeTT. < ]+|eTT q X+KTeqT rdyjTTeTT. @$T $T*jTTq~? | Tq
X+KTe jTT $T*eq u>eTTqT X+KTK+&eTT n n+ @s&T X+KTK+&eTT jTT \
eTT +&TqT. eTTqT
X+KTK+&eTT n+eTT e+&TqT.
eTqeTT |&T X+KTK+&eTT |Tq|]eDeTTqT qT>=HD+#TeTT. (|eTT 3.5 #&TeTT)
R nqTq~ q X+KTe ykseTT r eT]jTT x nqTq~ q X+KTe ykseTT eT]
jTT mT nqT=qTeTT. (q X+KTe K+&qeTT sTy @s&Tq~ q X+KTe n>TqT)
h nqTq~ X+KTK+&eTT mT nqT=qTeTT.
X+KTK+&eTT
}
q X+KTe
}
(V) =
- {
q X+KTe
|Tq|]eDeTT |Tq|]eDeTT |Tq|]eDeTT
=
1
2
R ( x h)
1
2
# r # # + - # r # r # x
3
3
R
Mensuration 241

eT
qT, V = 1 r6 x^R 2 - r 2 h + R 2
h@. (1)
3
|+ 8.36 qT+& DBFE + DDGE
`
BF =
FE
DG GE
(
R =
x + h
r x
( Rx - rx = rh
( x( R - r)
= rh
qT, x =
rh
R - r
(1) qT+& ( V = 1 r x^R 2 - r 2 h + R 2
h
3
6 @
(2)
1 2
( = r6 x^R - rh^R + rh
+ R h@
3
1 2
( = r rh^R + rh
+ R h
3
6 @ (2) qT |j+#>,
eq, X+KTK+&eTT |Tq|]eD+, V =
1 2 2
r h( R + r + Rr)
|Tq |eDeTT\T.
3
* X+KTK+&eTT jTT e\ yX\eTT = r ^R
+ rh l & l h R r
A
h+x
C
= + ^ -
h
F
E
h
2 2
G
R
|eTT. 8.36
x
r
D
B
2 2
* X+KTK+&eTT jTT d+|Ps\ yX\eTT = r^R + rh + rR + rr
& l h R r
242 10th Std. Mathematics
= + ^ -
h
2 2
* ( |

8.3.4 >eTT |Tq|]eDeTT (Volume of a sphere)
(i) |Tq>eTT |Tq|]eDeTT (Volume of a Solid Sphere)
eTT
+~ q |j>eTT >eTT |Tq|]eD+ deTTqT HjT|sT#TqT.
V =
4 r r
3
|Tq |eDeTT\T
3
ykseTT R eT]jTT mT h > >\ dbseTT >\ bqT rdT= < {
+|eTT. r ykseTT >\ |Tq >eTTqT n+ +&eqT
eT]jTT jT b]q { |Tq |]eDeTTqT =\TeeTT. y\Te&q sT |Tq >eTT |Tq|]
eDeTTq deqeTT.
qT, >eTT |Tq|]eDeTT,
V =
4 r r
3
|Tq |eDeTT\T
3
r
h
r
y\Te&q sT
(ii)
(iii)
R
|eTT. 8.38
u\T >eTT |Tq|]eDeTT (Volume of a hollow sphere)
u\T >eTT |* eT]jTT y\T|* ykseTT\T esTd> r eT]jTT R nsTTq
} -{
u\T >eTT
}
y\T|* >eTT |* >eTT
=
|Tq|]eDeTT |Tq|]eDeTT |Tq|]eDeTT
=
4
rR -
4
rr
3 3
3 3
` u\T >eTT |Tq|]eDeTT = 4 r( R
3 - r
3
) |Tq |eDeTT\T
3
|Tq ns>eTT |Tq|]eDeTT (Volume of a solid hemisphere)
|Tq ns>eTT |Tq|]eDeTT =
1 # 2
>eTT |Tq|]eDeTT
=
1
# r r
2 3
2 3
= r
3
4 3
R
r |Tq |eDeTT\T
4 3
rr
3
r
2 3
rr
3
R
|eTT. 8.39
|eTT. 8.40
Mensuration 243

(iv)
u\T ns>eTT |Tq|]eDeTT (Volume of a hollow hemisphere)
u\T ns >eTT
}
y\T|* ns >eTT
}
=
- {
|* ns >eTT
|Tq|]eDeTT |Tq|]eDeTT |Tq|]eDeTT
=
2
# r # R -
2
# r # r
3
3
3 3
r
R
244 10th Std. Mathematics
2 3 3
= r^R - r
3
h|Tq |eDeTT\T.
eTT s+ >\ V| c{-|{ ydeTT 8.4 d+.MT. nsTTq < |Tq|]eDeTT
qT>=qTeTT. ( r =
22
)
k=qTeTT.
keTT eT]jTT d|eTT\ eT& ykseTT r n nqT=qTeTT.
X+KTe eT]jTT d|eTT\ eT& mT h nqT=qTeTT.
r
r = h n e&q~.
h
h
X+KTe, ns >eTT, d|eTT\ | Tq|]
eDeTT\T esTd> v 1
, v 2
eT]jTT v 3
nqT=qTeTT.
r
r
= 1
r 2 h : 2
r 3 : r 2
h
1 2 3
r r r
3 3
( =
1 3
r :
2 3 3
r : r
3 3
( V1 : V2 : V = 1
: 2
:
3
1
3 3
|&T, V : V : V
r r r (& r = h)
eq, e\dq |Tq|]eDeTT\ w R 1:2:3
42
10
|eTT. 8.43
9.8 d+.MT
|eTT. 8.42

eTT | Tq|]eDeTT 7421 71
| Tq| d+.MT. nsTTq < yks eTT qT>=qTeTT(
k< q: r eT]jTTV \T esTd> >eTT jTT yks eTT eT]jTT |
V
= 7241 71
|Tq| d+.MT
(
4 3
r r =
50688
3 7
(
4 22 3
# # r =
50688
3 7 7
r 3 =
50688
7
#
3 # 7
4 # 22
= 1728 = 4 3 # 3 3
eq, >eTT ykseTT, r = 12 d+.MT.
eTT |Tq|]eDeTT
>eTT |* ykseTT qT>=qTeTT. (
11352 3
cm
7
r = )
22
7
r =
22
)
7
Tq|]eDeTT\T nqT=qTeTT.
. < y\T|* yks+ 8 d+.MT. nsTTq
keTT y\T|* eT]jTT |* ykseTT nqT=qTeTT.
u\T >eTT |Tq|]eDeTT V nqT=qTeTT
` V =
4 3 3
( r( R - r ) =
3
(
4
# 22 ( 8 3 - r
3
) =
3 7
eq, |* yks+ ,
11352 3
cm
7
11352
7
11352
7
3
512 - r = 387 ( r 3 = 125 = 5 3
r = 5 d+.MT.
nudeTT 8.2
7241 7
1 d+.MT 3
1. d|eTT ykseTT 14 d+.MT. eT]jTT mT 30 d+.MT. nsTTq d|eTT |Tq|]eDeTT
qT>=qTeTT.
2. yT\ 7 d+.MT yd+ >\ dbsb d| |sE
e&TqT. b d| 4 d+.MT. mT es e+&q, 250 eT+~ s>T\ sE
m+ |]eD+ d| yTqT?
3. eT es |Tq d|eTT u ykseTT eT]jTT mT\ yTTeTT 37 d+.MT d|eTT
d+|Ps\ yX\eTT 1628 #.d+.MT. nsTTq d|eTT |Tq|]eDeTT qT>=qTeTT.
Mensuration 245
r
|eTT. 8.44
r
R
8d+.MT
|eTT. 8.45

4. |Tq d|eTT |Tq|]eDeTT 62.37 |Tq| d+.MT. < mT 4.5 d+.MT. nsTTq <
ykseTT qT>=qTeTT.
5. s+&T eT es d|eTT\ ykseTT\ w 2:3 eT]jTT mT\ w 5:3 nsTTq
y{ |Tq|]eDeTT\ w qT>=qTeTT.
6. d|eTT jTT ykseTT eT]jTT mT\ w 5:7 eT]jTT < |Tq|]eD+ 4400
|Tq| d+.MT. nsTTq d|eTT ykseTTqT qT>=qTeTT.
7. Bs #Tsks V| s =\\T 66 d+.MT. I 12 d+.MT. < #T> 12 d+.
MT. mT >\ d|eTT @s&~q~. d|+ |Tq|]eD+ qT>=qTeTT.
8. &| eT es+ qTq~. |\T 28 d+.MT. b&eqT eT]jTT 3 $T.MT
ykseTTqT *eq~. & ykseTT 1 $T.MT.nsTTq | e\dq =jT
|Tq|eDeTTqT qT>=qTeTT.
9. X+KTe jTT ykseTT eT]jTT y\TfT\T esTd> 20 d+.MT. eT]jTT 29 d+.MT.
nsTTq < |Tq|]eDeTT qT>=qTeTT
10. =jT #jT&q |Tq X+KTe u|]~ 44 MT eT]jTT mT 12 MT nsTTq <
|Tq|eDeTTqT qT>=qTeTT.
11. b X+KTK+&eTT s|eTT qTq~. < e] ykseTT eT]jTT mT\T
esTd> 8 d+.MT. eT]jTT 14 d+.MT. < |Tq|]eDeTT
5676
d+.MT 3 nsTTq,
3
eT]jTT es ykseTTqT qT>=qTeTT.
12. X+KTK+&eTT es\ jTT #T=\ =\\T 44 d+.MT. eT]jTT 8.4r . d+.MT. T
14 d+.MT. nsTTq < |Tq|]eD+ qT>=qTeTT.
13. 3ABC \+DuT+ uTeTT 5 d+.MT.,12 d+.MT. eT]jTT 13 d+.MT. nsTTq
uTeTT 12 d+.MT. uTeTTqT |q |TqeTT @s&TqT. @s&q |TqeTT
|Tq|]eD+ qT>=qTeTT.
14. eT es X+KTe ykseTT eT]jTT mT\ w 2:3 < |Tq|]eD+ 100.48
|T.d+.MT. nsTTq y\TfTqT qT>=qTeTT. ( r = 3.14>rdT=qTeTT)
15. es \ X+KTe |Tq|]eD+ 216r |T.d+.MT. u ykseTT 9 d+.MT
nsTTq X+KTe mTqT qT>=qTeTT.
16. >s d \T u u]+> < yks eTT 0.7 d+.MT. d \T k+< 7.95 >eTT/d+.MT.
nsTTq 200 d \T u u]+>\ =qTeTT. ( 12 d+.MT. eT]jTT
10 d+.MT. nsTTq |Tq|]eDeTT qT>=qTeTT.
18. ns>eTT |Tq|]eDeTT1152r |Tq| d+.MT. < e\ yX\eTTqT qT>=qTeTT.
19. 14 d+.MT. uTeTT >\ |TqeTT qT+& ]+q n |< eT es X+KTe
|Tq|]eDeTT qT>=qTeTT.
20. >s u\H >* }~q|&T < ykseTT 7 d+.MT. qT+& 14 d+.
MT.\ |]q~. s+&T d+=qTeTT.
246 10th Std. Mathematics

8.4 d+jTT |TqeTT\T (Combination of Solids)
J$+ eTqeTT ueT\T, yVqeTT\T, b\T, |eTTT yTTT y{ |]
o*+#T#THeTT. n$ s+&T n++f me d+jTT |TqeTT\T> qT+&TqT.
d+jTT |TqeTT\ |]\ yX\eTT eT]jTT |Tq|]eDeTT\qT @$ qT>=qeqT?
|eTT. 8.46
d+jTT |TqeTT\ d+|Ps\ yX\eTT, nqTq~ eT&> qTq n |TqeTT\ d+|Ps\
yX\eTT\ yTTeTTq deqeTT> +&qeds+ sT. | |+ qT+& d+jTT |TqeTT\
d+|Ps\ yX\eTT nqTq~ ns>eTT e\ yX\eTT eT]jTT X+KTe e\ yX\eTT
yTTeTTq deqeTT. d+jTT |TqeTT jTT |Tq|]eDeTT, < >\ n |TqeTT\
|Tq|]eDeTT\ yTTeTTq deqeTT. | |+qT+&,
| TqeTT jTT d+|Ps \ y X\eTT = ns >eTT e\ y X\eTjT G X+KTe e\ y X\eTT
| TqeTT jTT yTT eTT | Tq|]eDeTT = X+KTe | Tq|]eDeTT G ns >eTT | Tq|]eDeTT
eTT| X+KTeqT u]+#TqT> q~. ns>eTT ykseTT
eT]jTT X+KTe u ykseTT 3.5 d+.MT. ueT yTTeTT mT 17.5 d+.MT. nsTTq ueT
jsT#jTT nedseT>T =jT |Tq|]eDeTTqT qT>=qTeTT. ( r =
22 )
7
k u>eTT X+KTe u>eTT ykseTT
ykseTT, r = 3.5 d+.MT.
ykseTT, r = 3.5 d+.MT.
- = 14 d+.MT.
mT , h = 17. 5 3.
5
=jT | Tq|]eDeTT R ns >eTT | Tq|]eDeTT G X+KTe | Tq|]eDeTT
=
2 3
r
1 2
r + rr h
3 3
2
=
rr
^
3 2r + h h
=
22
7
#
3. 5 # 3.
5
# ^2 # 3.
5 + 14h= 269.5
3
eq, ueT #jTT nedseT>T =jT |Tq|]eDeTT = 269.5 |T.d+.MT.
17.5d+.MT
|eTT. 8.47
3.5d+.MT
Mensuration 247

eTT| d|eTTqT +qT> eq~. d|+ u>| mT
8 d+.MT. eT]jTT | jTT yTTeTT mT 11.5 d+.MT. nsTTq | jTT yTTeTT |]\
yX\eTTqT qT>=qTeTT. (r =
22 )
7
keTT u>| d|eTT u>|
ykseTT, r = yTTeTT mT = 8 mT, h = 8 d+.MT
( r = 11.5- 8= 3.5 d+.MT yks eTT, r =3.5 d+.MT = 27
d+.MT
| jTT yTTeTT |]\ yX\eTT = ns>eTT |]\ yX\eTT CSA
|eTT. 8.48
+ d|eTT u>| |]\ yX\eTT CSA
248 10th Std. Mathematics
2
= 2rr
+ 2rrh
= 2 r r( r + h)
= 2 #
22
#
7 7
7 2
` + 8
2
j
` | jTT yTTeTT |]\ yX\eTT = 253 #.d+.MT.
T&seTT, d|eTT| X+KTe eqT> ]+#&q~. >T&seTT jTT yTTeTT
mT 49 MT. u yd+ 42 MT. eT]jTT d|eTT mT 21 MT. HdT T&seTT ]+#T e\dq HdT =qTeTT. ( r =
k|
X+KTe u>|
yd+, 2r = 42 MT. ykseTT, r = 21 MT.
ykseTT, r = 21 MT. mT, h 1
= 49 21
mT , h = 21 MT. y\TfT , l = h + r
- = 28 MT.
2 2
1
2 2
= 28 + 21
22
rdT=qTeTT)
2 2
|eTT. 8.49
= 7 4 + 3 = 35 MT.
e\dq HdT yTTeTT yX\eTT = d|eTT e\ yX\eTT +X+KTe e\ yX\eTT
= 2rrh + rrl
= r r( 2h + l)
=
22
7
# 21^2 # 21 + 35h= 5082
` HdT yX\eTT = 5082 MT 2
HdT

eTT jTT uV eT]jTT n+s ydeTT\T esTd> 8 d+.MT. eT]jTT 4
d+.MT. >eTTqT ]+ eT es X+KTe> eT*q < u ydeTT 8 d+.MT. nsTTq
X+KTe mTqT qT>=qTeTT.
keTT uV eT]jTT n+s ykseTT\T esTd> R eT]jTT r nqT=qTeTT.
X+KTe mT h nqT=qTeTT.
u\T >eTT
2 d+ MT
uV>eTT n+s>eTT X+KTe
4d+.MT
2R = 8 d+.MT. 2r = 4 d+.MT. 2r 1
= 8
( R = 4 d+.MT. ( r = 2 d+.MT. ( r 1
= 4
u\T >eTTqT ]+ X+KTe> jsT#jT&q~.
X+KTe |Tq|]eDeTT = u\T >eTT |Tq|]eDeTT
(
1 2
r r1
h =
4 3 3
r6 R - r @
3 3
(
1 2
# r # 4 # h =
4
3 3
# r # ^4 - 2 h
3
3
( h =
64 - 8 = 14
4
eq, X+KTe mT h = 14 d+.MT.
s s|+ >\ >\ < ydeTT 1.4 d+.MT. nsTTq y{ =+ sT
>\ dbs| ;sT yjTTeTT. dbs| ;sT ydeTT 7 d+.MT. nsTTq { eT+
5.6 d+.MT. |sT>T m >\qT ;sT yjTeqT.
k\ d+K n nqT=qTeTT. > eT]jTT dbs ;sT ykseTT\T esTd>
r 1
eT]jTT r 2
nqT=qTeTT.
>\T
ydeTT 2r 1
= 1.4 d+.MT.
ykseTT r 1
= 0.7 cm
|]q { eT+ mT h nqT=qTeTT.
h = 5.6 d+.MT.
>\qT ;sT ydq sTy
dbs| ;sT
ydeTT,2r 2
= 7 d+.MT.
ykseTT, r 2
= 27
d+.MT.
1.4d+.MT
|]q { |Tq|]eDeTT = n >\ |Tq|]eDeTT
2
( r r2
h = n #
4
rr1
3
3
7d+.MT
5.6d+.MT
|eTT. 8.51
h
8d+.MT
|eTT. 8.50
Mensuration 249

eq, n =
2
3r2
h
3
4r1
3 #
7
#
7
# 5.
6
n = 2 2 = 150.
4 #
7
#
7
#
7
10 10 10
` e\dq >\ d+K = 150.
\ dbs| >=+ + sT Bs#Tsks
={ |eV+#qT. Bs#Tsks ={ b&e 50 MT eT]jTT y&\T 44 MT. nsTTq
={ {eT+ 21 d+.MT. |sT>T m >+\T |TqT. ( r =
22 )
keTT R 15 .MT/>+.
= 15000 MT / >+.
>=+ ydeTT, 2r = 14 d+.MT.
21d+.MT
eq, r = 7
100
MT.
50 MT
|eTT. 8.52
|]q { eTeTT h nqT=qTeTT
qT, h = 21 d+.MT.= 21 MT. 100
1 >+ $&T=+ n&T yX\+ I \eTT I y>eTT
(u yX\+)
= rr 2
# 1 # 15000
=
22 7 7
15000
7 100 100
={ e\dq |]eDeTT { |Tq|]eDeTT,
lbh = 50 # 44 #
# # # |T.MT.
21
100
y>eTT 15 .MT/>+
14 d+.MT
e\dq |]eDeTT { b++\T nedseT>TqT
` T >+\ $&T+\T
eq, e\dq { eT+ |sT>T 2 >+\ deTjT+ |TqT.
7
44 MT
250 10th Std. Mathematics

\ |Tq qT| |\qT ]+
>=+qT jsT#dqT. >=+ y\T|* ydeTT eT]jTT eT+ 8 d+.MT. eT]jTT
1 d+.MT nsTTq >=+ b&eqT qT>=qTeTT. ( r =
22 )
7
k=+ b&e h 1
nqT=qTeTT.
R eT]jTT r \T >=+ jTT y\T|* eT]jTT |* yks eTT\T nqT=qTeTT.
8d+.MT
qT| |\ =\\T lbh = 55#40#15.
qT| >=+ =\\T:
1d+.MT
y\T|* ydeTT, 2R = 8 d+.MT.
` y\T|* ykseTT, R = 4 d+.MT.
15d+.MT
eT+=+ |Tq|]eDeTT = qT| |\ |Tq|]eD+
( r h ( R + r)( R - r)
= lbh
22
7
1
40d+.MT
# h (4 + 3)(4 - 3) = 55 # 40 # 15
1
eq, >=+ b&e, h 1
= 1500 d+.MT. = 15 MT.
nudeTT 8.3
1. u+>seTT ns >eTT| X+KTeqT u]+qT> q~. ns >eTT ydeTT 3.6 d+.MT.
u+>seTT yTT eTT mT 4.2 d+.MT. nsTTq < yTT eTT |]\ y X\eTT qT>=qTeTT.
2. |TqeTT, ns>eTT| d|eTTqT u]+qTq~. |TqeTT jTT ydeTT eT]jTT
mT\T esTd> 21 d+.MT., 25.5 d+.MT. nsTTq < | Tq|]eDeTT qT>=qTeTT.
3. e dbseTT> e+&, < s+&T es\T ns > s|eTT >\=qTeTT.
4. >T&seTT eT es d|eTT| X+KTe u]+qTq~. < yTTeTT mT
eT]jTT u ydeTT\T esTd> 13.5 MT nsTTq 28 MT. dbs| u>| mT 3 MT.
nsTTq >T&seTT jTT d+|Ps\ yX\eTTqT qT>=qTeTT.
5. $\
eT es X+KTeqT jsT#dqT. eT]jTT $eTT> esqT.
nsTTq >eTT ykseTT qT>=qTeTT.
6. |Tq >eTT ykseTT 24 d+.MT. >eTTqT ]+ b&yq r>> eT*q r>
ykseTT 1.2 $T.MT. nsTTq r> b&eqT qT>=qTeTT.
Mensuration 251

7. eT es X+KTe b |* yks+ 5 d+.MT. b 24 d+.MT. mT es
sT \ < |* yks+ 10 d+.MT. nsTTq
dbs b { eT+ mTqT qT>=qTeTT.
8. 6 d+.MT. ydeTT >\ |Tq >eTTqT 12 d+.MT. ydeTT >\ eT es
dbs b yjT&q~. dbs b =+ u>eTT es sT \eTT
{ |P]> eTTqT>T dbs b { eT+ m+ |s>eqT?
9. 7 d+.MT. |* ykseTT >\ dbs >=+ + >=+ =qTeTT.
10. 4 MT. yd+ eT]jTT 10 MT. mT >\ dbs ={ qT+& 10 d+.MT. yd+ eT]jTT
2.5 .MT/>+ #=|q sT d|s >=+ + es
U n>T m+ deTjT+ |TqT. bs+u + ={ +& sT \< }V+#T=qTeTT.
11. 18 d+.MT. yks+ >\ |Tq >s edTeqT ]+ eT&T |Tq >s
>eTT\qT ysT ysT |]eDeTT\ jsT#d]. n+eTT\ ykseTT\T
2 d+.MT. eT]jTT 12 d+.MT. nsTTq eT&e >eTT ykseTTqT qT>=qTeTT.
12. u\T dbs >=+ b&e 40 d+.MT. < |* eT]jTT uV ykseTT\T
esTd> 4 d+.MT. eT]jTT 12 d+.MT. < ]+ |Tq d|eTT> eT*q < b&e
20 d+.MT. nsTTq @s&q |Tq d|eTT ykseTTqT qT>=qTeTT.
13. eT es qT| X+KTe yd+ 8 d+.MT. eT]jTT mT 12 d+.MT. nsTTq <
]+ 4 $T.MT. yks+ >\ >s dd| >T+&T> jsT#dq m dd| >T+&qT
jsT #jTe#TqT?
14. 12 d+.MT. yd+ eT]jTT 15 d+.MT mT>\ eT es d|eTT +&T> dt+
\\ X+KTe |
u>eTT ns>s+ e#TqT +|e\jTTqT. dt+ n+{ +| m X+KTe\T
e\jTTqT.
15. b&e 4.4 MT. eT]jTT y&\T 2 MT. *q Bs #Tsks| b es| sT
d]+#&q~. b {eT+ mT 4 d+.MT. eT]jTT { 40 d+.MT. yks +
>\ dbs| b ~ #jT&q~. nsTTq dbs b >\ {eT{+ mTqT
qT>=qTeTT.
16. 32 d+.MT. mT eT]jTT 18 d+.MT. ykseTT >\ dbs| T +&T> dT
\ n~ X+Us| | e @s&qT. X+Us | mT
24 d+.MT. nsTTq | jTT y\TfT eT]jTT ykseTTqT qT>=qTeTT.
17. 20 MT. T eT]jTT 14 MT .ydeTT >\ dbs| u$ e&qT. e> $d]
yjT&q eT{ |THs| HqT @s]q < =qTeTT.
252 10th Std. Mathematics

nudeTT 8.4
dsq yTqT mqT=qTeTT.
1. ykseTT 1 d+.MT. eT]jTT mT 1 d+.MT. >\ eT es d|eTT e\
yX\eTT.
(A) r d+.MT 2 (B) 2r d+.MT 2 (C) 3r d+.MT 3 (D) 2 d+.MT 2
2. eT es |Tq d|eTT yks+ < mT d>eTT+&TqT. nsTTq < d+|Ps\
yX\eTT.
(A)
3 rh 2
#.jTT (B)
r h
#.jTT (C)
3 2
r h #.jTT (D)
2 rh
3
2
3
#.jTT
3. eT es d|eTT u yX\eTT 80 d+.MT < mT 5 d+.MT. nsTTq <
2
|Tq|]eDeTT.
(A) 400 d+.MT 3 (B) 16 d+.MT 3 (C) 200 d+.MT 3 (D)
400
d+.MT
3
4. |Tq es d|eTT d+|Ps \ yX\eTT 200rd+.MT 2 eT]jTT ykseTT 5
d+.MT. nsTTq < mT eT]jTT ykseTT\ yTTeTT.
(A) 20 d+.MT. (B) 25 d+.MT. (C) 30 d+.MT. (D) 15 d+.MT.
5. eT es d|eTT ykseTT a jTT eT]jTT mT b jTT nsTTq <
e\ yX\eTT.
(A) r a b #.d+.MT. (B)2rab#.d+.MT. (C) 2r#.d+.MT. (D) 2 #.d+.MT.
6. eT es X+KTe eT]jTT d|eTT\ ykseTT eT]jTT mT\T deqeTT. d|+
|Tq|]eDeTT 120 d+.MT 3 . nsTTq X+KTe |Tq|]eD+
(A) 1200 d+.MT 3 B) 360 d+.MT 3 (C) 40 d+.MT 3 (D) 90 d+.MT 3
7. eT es X+KTe ydeTT eT]jTT mT\T esTd> 12 d+.MT. eT]jTT 8 d+.
MT. nsTTq < y\TfT
(A) 10 d+.MT. (B) 20 d+.MT. (C) 30 d+.MT. (D) 96 d+.MT.
8. eT es X+KTe jTT u |]~ eT]jTT y\TfT\T esTd> 120r d+.MT.
eT]jTT 10 d+.MT. nsTTq X+KTe e\ yX\eTT
(A) 1200r d+.MT 2 (B) 600r d+.MT 2 (C) 300r d+.MT 2 (D) 600 d+.MT 2
9. eT es X+KTe |Tq|]eDeTT eT]jTT u yX\eTT\T esTd> 48r d+.MT 3
eT]jTT 12r d+.MT 3 nsTTq X+KTe mT
(A) 6 d+.MT. (B) 8 d+.MT. (C) 10 d+.MT. (D) 12 d+.MT.
10. eT es X+KTe mT eT]jTT u yX\eTT\T esTd> 5 d+.MT. eT]jTT 48
#.d+.MT. nsTTq X+KTe |Tq|]eDeTT
(A) 240 d+.MT 3 (B) 120 d+.MT 3 (C) 80 d+.MT 3 (D) 480 d+.MT 3
Mensuration 253

11. s+&T d|eTT\ mT\T eT]jTT ykseTT\ w eTeTT\T 1:2 eT]jTT 2:1 nsTTq
y{ |Tq|]eDeTT\ w
(A) 4 : 1 (B) 1 : 4 (C) 2 : 1 (D) 1 : 2
12. >eTT ykseTT 2 d+.MT. nsTTq < e\ yX\eTT
(A) 8r d+.MT 2 (B) 16 d+.MT 2 (C) 12r d+.MT 2 (D) 16r d+.MT 2 .
13. |Tq ns>eTT ydeTT 2 d+.MT. nsTTq < d+|Ps\ yX\eTT
(A) 12 d+.MT 2 (B) 12r d+.MT 2 (C) 4r d+.MT 2 (D) 3r d+.MT 2 .
14. >eTT |Tq|]eDeTT 9 r |T.d+.MT. nsTTq < ykseTT
16
(A) 34
d+.MT. (B) 43
d+.MT. (C) 23
d+.MT. (D) 32
d+.MT.
15. s+&T >eTT\ |]\ yX\eTT\ w 9:25 nsTTq y{ |Tq|]eDeTT\ w
(A) 81 : 625 (B) 729 : 15625 (C) 27 : 75 (D) 27 : 125.
16. |Tq ns>eTT ykseTT a jTT nsTTq < d+|Ps\ yX\eTT
(A) 2r a 2 #.jTT (B) 3ra 2 #.jTT (C) 3ra #.jTT
(D) 3a 2 #.jTT.
17. >eTT |]\ yX\eTT 100r d+.MT 2 , nsTTq < ykseTT
(A) 25 d+.MT (B) 100 d+.MT (C) 5 d+.MT (D) 10 d+.MT.
18. >eTT |]\ yX\eTT36r d+.MT 2 , nsTTq >eTT |Tq|]eDeTT
(A) 12r d+.MT 3 (B) 36r d+.MT 3 (C) 72r d+.MT 3 (D) 108r d+.MT 3
19. |Tq ns>eTT d+|Ps\ yX\eTT 12r d+.MT nsTTq < e\ yX\eTT
2
(A) 6r d+.MT 2 (B) 24r d+.MT 2 (C) 36r d+.MT 2 (D) 8r d+.MT 2
20. >eTT ykseTT += >eTT ykseTT d>eTT nsTTq y{ |Tq|]eDeTT\
w
(A) 1 : 8 (B) 2: 1 (C) 1 : 2 (D) 8 : 1
21. |Tq >eTT e\ yX\eTT 24 d+.MT 2 >eTTqT s+&T ns>eTT\T> $u+q,
< ns>eTT jTT d+|Ps\ yX\eTT.
(A) 12 d+.MT 2 (B) 8 d+.MT 2 (C) 16 d+.MT 2 (D) 18 d+.MT 2
22. s+&T eT es X+KTe\ yks\T deqeTT. y{ y\TfT \ w 4:3 nsTTq
y{ e\ yX\eTT\ w
(A) 16 : 9 (B) 8 : 6 (C) 4 : 3 (D) 3 : 4
254 10th Std. Mathematics

MT \Tk?
+> sYZ jTT @&T e+q\T (The Seven Bridges of Konigsberg) nqTq~
>DXdeTT #] deTd> q~. |c (Prussia) qTe+{ +> sYZ
q>s+ nqTq~ |> q~ (Pregal River) sTy|\ neT]jTTq~. eT]jTT |< Be\T s+&
eTT @&T e+q\ \T|&qT> +&TqT.
k] eyT e+qqT

eTTU+XeTT\T
e.
d+
|sT
|eTT
bs\ ()
e\ yX\eTT
(#.jTT)
d+|Ps\ yX\eTT
(#.jTT)
| Tq|]eDeTT
(| T. jTT)
1
eT es
d|eTT
h
r
2rrh
2r r^h + rh rr 2
h
2
eT es
u\T d|eTT
r
R
h
2r h^R
+ rh 2r ^R + rh^R r + hh
rR h
rh^R
rr h
2 2
r h
2 2
r h^R + rh^R rh
3
eT es
X+KTe
h
r
l
1 2
rrl
r r^l + rh rr h
3
r
1 2 2
h
4 X+KT K+&eTT - - - - - - - - - - - - - - - - - - - - - r h^R + r + Rr
R
3
h
4 2
5 >eTT r
r
4 3
r - - - rr
3
4 3 3
R
6 u\T >eTT - - - - - - r^R r
r
3
h
7 ns >eTT 2rr
2
r
3rr
2
2 3
rr
3
8
u\T ns
>eTT
r
R
2 2
2r ^R
+ r h
2 2
2 2
2r ^R
+ r h+ r^R r h
= r^3R
2 + r
2
h
2 3 3
r^R
3
r h
9 X+KTe l h r
X+KTe e\ yX\eTT R #| yX\eTT
r rl =
i 2
# rr
360
#|eTT b&e R X+KTe u|]~
+
h l r
12 esT 1 MT 3 = 1000 sT , 1 &d.MT 3 = 1 sT , 1000 cm 3 = sT , 1000 sT = 1 .
256 10th Std. Mathematics
2 2
2 2
h l r
2 2
r l h
R
l
L
h
r
10. >=+ eTT # \eTT}
11. ] b+

9
|j> sU>DeTT
• |]#jT+
• dssK\T
• uTeTT\T
• #jT #TsTeTT\T
V>T|
(598-668 AD)
bNq us=| Xdy
usT| VdT
dD+ me |d~q
|*+qT @s]q~.
#jT #TsT+ uTeTT\
b&e p, q, r, s \qT q, nqT
ses$T eT]jTT KyX\+
=s d+qT #qT.
p + r q + s
ses$T yX\+ c
2
mc
2
m.
K yX\+
( t - p)( t - q)( t - r)( t - s)
& 2t = p+q+r+s .
“Give me a place to stand, and I shall move the earth”
-Archimedes
9.1 |]#jT+
sU>DeTT | .|P. 3000 d+eseTT\
eTT+ u$T =\T#T
|j+#T#T+&]. bs+ueTTq sU>DeTT b&e, DeTT,
yX\eTT eT]jTT |Tq|]eDeTT d++~+q deTT\qT
qT>=, y{ ds sDeTT, U>eTT eT]jTT yssT eT\
qT n_e~ #jTT nedseTsTTq~.
= qMq |jTeTT\ desD #dq
bs |D[q ;>DeTT, $XwDeTT yTTT y{ H
sU>DeTTqT e $\Tyq~> |]>D+#&qT. nH
eT+~ >D Xdy\T desD \eTT> $u~+HsT.
>DXdeTT s XK\ >D ueeTT\qT dT\ueTT>
>V+#T sU>DeTT dVjT|&TqT. n

ZsZs> e+&TqT >eT+|eTT. eq K+&qsK\, K+&qsK\T AB,
CD \T eeTTqT _+ L, M \ e<
>\ dssK\T n+eT+|eTT.
eTT 1
eeTTqT ^, eeTT| @< _+T+& be sK\T eeTT| s+&T _+ A, B, C
2 3 4 5
D
eT]jTT D \ e< *jTTqT. l , l , l , l sK\T eeTTq K+&q
2 3 4 5
P
l
sK\T. l 1
sK eeTTqT K+> _+

ks+XeTT: e yks+ R 3.2 d+.MT.
yde |eTT
T |eTT
M
N
Tl
L
Tl
sDeTT
(i) O + 3.2 d+.MT. yks+ eeTTqT ^jTTeTT.
(ii) eeTT | @< _+T]+, OP \T|eTT.
(iii) P qT + rdT= OP | #|eTT L e< K+&+#TqT ^jTTeTT.
!! e+&TqT #|eTT| M, N \qT >T]+|eTT.
(iv) LM = MN
(v) +MPN PT nqT D deT~K+&q sKqT ^jTTeTT.
(vi) TP qT Tl es b&+q,
TlPT nqTq~ P e< dssK
d#q
dssK OP eeTT| _+

nudeTT 9.1
1. 4.2 d+.MT. yks+ eeTTqT ^jTTeTT. e+| @< _+

uuTeTT, osDeTT\ uTeTTqT ]+#T.
uuTeTT eT]jTT osDeTT\qT q uTeTTqT ]+#T >\ $$< kbqeTT\qT
\TdT=H

9.3.1 uuTeTT, osDeTT eT]jTT oseTT qT+& uuTeTTq ^dq q (mT)
e&q uTeTTqT ]+#T.
\ q(mT) b&e 4.2 d+.MT. +&TqT
D ABC ]+|eTT.
ks+XeTT: D ABC , AB = 6 d+.MT, + C = 40c
C qT+ AB >\ q b&e = 4.2 d+.MT.
T |eTT
K
C
yde |eTT
Y
C
H
40c
O
4.2d+.MT
4.2d+.MT
Cl
A
6 d+.MTM
B
A
40c
M
6 d+.MT
B
sDeTT
X
(i) AB = 6 d+.MT >\ sUK+&eTTqT ^jTTeTT.
(ii) + BAX = 40c e+&TqT AX qT ^jTTeTT.
(iii) AY = AX +&TqT ^jTTeTT.
(iv) AB M eT]+#TeTT.
(viii) AB de+s+> eeTTqT C eT]jTT Cl\ e< d+~+#TqT CHCl qT ^jTTeTT.
(ix) AC, BC \qT \|> eTq e\dq T ABC @s&TqT.
d#q
3 ABCl L& e\dq eTs= uTeTT n>TqT.
264 10th Std. Mathematics

9.3.2 uuTeTT, osDeTT eT]jTT oseTT qT+& uuTeTTq >\ eT+ q
uTeTTqT ]+#T.
n~ eeTTqT A eT]jTT
(viii)
Al\ e< d+~+#TqT.
3 ABC TAl BC \T eTq e\dq uTeTT\T.
Practical Geometry 265

\ eTeTT 4.7 d+.MT. =\\
DABC ]+|eTT. eT]jTT A qT+& BC >\ q b&eqT qT>=qTeTT.
ks+XeTT: D ABC , BC = 4.5 d+.MT., + A = 40c, A qT+& BC >\ eT< >+ AM = 4.7 d+.MT.
yde |eTT
K
T |eTT
A
40c
Al
4.7d+.MT
40c
B
M
4.5 d+.MT
C
40c
M
4.5d+.MT
sDeTT
(i) BC = 4.5 d+.MT. sUK+&eTTqT ^jTTeTT.
(ii) + CBX = 40c e+&TqT BX ^jTTeTT.
(iii) BY = BX ^jTTeTT.
(iv) BC \+deT~K+&q sKqT ^jTTeTT. n~ BY eT]jTT BC \qT eTeTT> O eT]jTT
M e< d+~+#TqT.
(v) O + OB yks+ BKC eeTTqT ^jTTeTT.
(vi) Bs e K+&eTT BKC, 40c osDeTTqT *jTT+&TqT.
(vii) M +, 4.7 d+.MT. yks+ e K+&eTTqT ^jT> n~ eeTTqT A eT]jTT
Al\ e< d+~+#TqT.
(viii) 3 ABC TAl BC \T e\dq uTeTT\T.
(ix) CB qT CZ es b&+|eTT.
(x) AE = CZ ^jTTeTT.
(xi) q b&e AE = 3.2 d+.MT.
266 10th Std. Mathematics

nudeTT 9.2
1. AB = 5.2 d+.MT. sUK+&eTT|, 48c DeTT eK+&eTTqT ]+|eTT..
2. uuTeTT PQ = 6 d+.MT, + R = 60c eT]jTT R qT+& PQ >\ q b&e 4 d+.MT
=\\DPQR qT ]+|eTT
3. PQ = 4 d+.MT, + R = 60c, R qT+& PQ >\ q b&e 4.5 d+.MT =\\DPQR qT
]+|eTT.
4. BC = 5d+.MT, + A = 45c eT]jTT A qT+& BC >\ eT+ b&e 4 d+.MT =\\
D ABC ]+|eTT.
5. BC = 5 d+.MT, BAC 40
+ = c eT]jTT A qT+& BC >\ eT+ b&e 6 d+.MT nsTTq
D ABC ]+|eTT. A qT+& q b&eqT qT>=qTeTT.
9.4 #jT #TsT+ (Cyclic quadrilateral)
#TsT+ jTT oseTT\jTT eeTT| neT]q# <
#jT #TsT+ n n+ n_eTTK DeTT\ yTT+ 180c qT #jT #TsT+
A
B
]+#T H\T>T =\\T #\TqT. (= AD eT]jTT CD \qT \T|eTT.
A
B
(v) ABCD e\dq #jT #TsTeTT.
+~ $$< s\ =\\ #jT
#TsTeTTqT ]+#TqT \TdT=H

|< I (eT&T uTeTT\T eT]jTT s+ e&q #jT #TsT+qT ]+#T)
T]+#TeTT. AB = 6 d+.MT. sUK+&eTTqT ^jTTeTT
(ii) A eT]jTT B + esTd> 7 d+.MT. eT]jTT 6.5 d+.MT. ykseTT\ =\\
#|eTT\qT ^jT> n$ C e< K+&+#TqT. AB, AC \qT \T|eTT.
(iii) AB eT]jTT BC \ \+deT~K+&q sK\qT ^jT> n$ O e< *jTTqT.
(iv) O + OA (= OB = OC) yks+ D ABC jTT |]eeTTqT ^jTTeTT.
(v) A + 4.2d+.MT. yks+ |]eeTTqT D e< K+&+#TqT #|eTTqT ^jTTeTT.
(vi)
AD eT]jTT CD \qT \T|eTT.
ABCD nqTq~ eTq e\dq #jT #TsTeTT
268 10th Std. Mathematics

|< II (s+&T uTeTT\T eT]jTT s+&T seTT\T e&q #jT #TsTeTTqT ]+#T)
T]+#TeTT. PQ = 4 d+.MT. sUK+&eTTqT ^jTTeTT.
(ii) P + 7.5 d+.MT. yks+ #|eTTqT ^jTTeTT.
(iii) Q + 6 d+.MT. yks+ eT]jTT #|eTT, eTT+

|< III (eT&T uTeTT\T eT]jTT DeTT q #jT #TsTeTTqT ]+#T)
T]+#TeTT.
AB = 6 d+.MT. sUK+&eTTqT ^jTTeTT.
(ii) B \ #|eTT eeTTqT D e< K+&+#TqT
^jTTeTT.
(vii) AD eT]jTTCD \qT \T|eTT.
(viii) ABCD e\dq #jT #TsTeTT
270 10th Std. Mathematics

|< IV ( s+&T uTeTT\T eT]jTT s+&T DeTT\T q #jT #TsTeTTqT ]+#T)
T]+#TeTT. EF = 5.2 d+.MT. sUK+&eTTqT ^jTTeTT.
+ = c +&TqT EXqT ^jTTeTT.
(ii) E

|< V ( uT+,eT&T DeTT\T e&q, #jT #TsTeTTqT ]+#T)
T]+#TeTT.
PQ = 4 d+.MT. sUK+&eTTqT ^jTTeTT.
(ii) P

|< VI ( s+&T uTeTT\T, DeTT eT]jTT de+s sK q #jT #TsTeTTqT
]+#T)
T]+#TeTT.
AB = 5.8 d+.MT.sUK+&eTTqT ^jTTeTT.
(ii) B n~ BX qT D e< *jTTqT.
AB eT]jTT AD \ \+deT~K+&q sK\T O e< *jTTqT ^jTTeTT.
O + OA (= OB = OD) yks+ D ABD |]eeTTqT ^jTTeTT.
(vi) DY < AB +&TqT DY ^q n~ eeTTqT C e< K+&+#TqT. BC qT \T|eTT.
(vii)
ABCD e\dq #jT #TsTeTT.
Practical Geometry 273

274 10th Std. Mathematics
nudeTT 9.3
1. PQ = 6.5 d+.MT, QR = 5.5 d+.MT, PR = 7 d+.MT eT]jTT PS = 4.5 d+.MT. =\\
#jT #TsT+ PQRS ]+|eTT.
2. AB = 6 d+.MT, AD = 4.8 d+.MT, BD = 8 d+.MT eT]jTT CD = 5.5 d+.MT. =\\
#jT #TsT+ ABCD ]+|eTT.
3. PQ = 5.5 d+.MT, QR = 4.5 d+.MT, + QPR = 45c, PS = 3 d+.MT. =\\ #jT
#TsT+ PQRS ]+|eTT.
4. AB = 7 d+.MT, + A = 80c, AD = 4.5 d+.MT eT]jTT BC = 5 d+.MT. =\\ #jT
#TsT+ ABCD ]+|eTT.
5. KL = 5.5 d+.MT, KM = 5 d+.MT, LM = 4.2 d+.MT eT]jTT LN = 5.3 d+.MT. =\\
#jT #TsT+ KLMN ]+|eTT.
6. EF = 7 d+.MT, EH = 4.8 d+.MT, FH = 6.5 d+.MT eT]jTT EG =6.6 d+.MT. =\\
#jT #TsT+ EFGH ]+|eTT.
7. AB = 6 d+.MT, ABC 70
+ = c, BC = 5 d+.MT eT]jTT + ACD = 30c =\\ #jT
#TsT+ ABCD ]+|eTT.
8. PQ = 5 d+.MT, QR = 4 d+.MT, + QPR = 35c eT]jTT + PRS = 70c =\\ #jT
#TsT+ PQRS ]+|eTT.
9. AB = 5.5 d+.MT + ABC = 50c, BAC 60
+ = c eT]jTT + ACD = 30c =\\ #jT
#TsT+ ABCD ]+|eTT.
10. AB = 6.5 d+.MT, + ABC = 110c, BC = 5.5 d+.MT eT]jTT AB || CD nsTTq ABCD
#jT #TsTeTTqT ]+|eTT.
MT \Tk?
1901 qT+& | d+eseTT uXd eTT, skjTqXd eTT, eqe TsT >D
yTD dyTqeTT# H\T>T d+eseTT\=k] sT>T n+srjT
+>dt du | |< eTT nqT VQeqeTT e&q~. ~jT >DXd eTT =s
e&T Hu VQeqeTT> |]>D+#&T#Tq~.
‚

10
sUeTT\T
“ I think, therefore I am” - Rene Descartes
• |]#jT+
• ~|sU eTT\T
• | sU eTT\T
sD &dtsY
(1596-1650)
bH
&dtsY, y~ eT\jT+TqT >eT+,
wjTH \eTTqT qT>=HqT.
qT yXw sU
>DeTTqT dw+#qT. <
sU
eTT\qT ^jTT esZeTT
#|qT.
10.1 |]#jT+
sUeTT\T de#seTTqT d+#T |eTT\T
n$ mT, sTe >\ d++ s+&T yssT
|]eDeTT\T DeTT }V+#TjT ]qeTT> qT+&e#T.
d+ \T eT]jTT y{ ;>D |D | d]##T |j>eTT L& \>
sUeTT jsT #jTeqT. ^jT&q sUeTT\ qT eT]jTT KeTTq
nqTbeTT>qT +&Tq eTsbseTT\T (x, y) f jTT sUeTT n+\ VQ|< |yTjTeTTqT sU eTT \ ssK
n>TqT.
s+&e n+dT VQ|< ded sUeTT
2
y = f( x) = ax + bx + c, a ! 0 nqT n$q sFjTeTT
esKqT |se\jTeTT n+

y
y
y
O
x
O
x
O
x
y = ( x + 1)( x - 2) ,
VQ|< ded n+dT 2
276 10th Std. Mathematics
y = ( x + 4)( x + 1)( x - 2),
y =
1
( x + 4)( x + 1)( x - 3)( x - 0.5)
14
VQ|< ded n+dT 3 VQ|< ded n+dT 4
IX e s>, y = ax + b,
a ! 0 s|eTTq >\ @| dedeTT\ sUeTT\qT
2
mT ^jTT

2
y = ax + bx + c nqT ~| sUeTTqT ^jTT |

TqT.
>eT
(i) y nqTq~ m\|&
TTDeTT nsTTq+

nudeTT 10.1
1. +~ |yTjTeTT\ sUeTT ^jTTeTT.
(i) y = 3x 2
(ii) y =- 4x 2
(iii) y = ^x + 2h^x
+ 4h
(iv) y = 2x - x + 3
2
2. +~ d$sDeTT\qT sU eTT

kTqT >eT+|e#TqT. eq,
#\eTTqT nqTeT #\eT>TqT.
y = kx nqT=q
(
y = k
x
++
2 4 6 10 12
x
\eTT >+
60 30 20 12 10
y
y>eTT ` \eTT sUeTTqT ^ eT]jTT < |j+
(i) n |jD y>eTT 5 .$/>+ nsTTq|&T rdT=qT |jD \eTTqT,
(ii) nqT 40 >+\ eTTqT dT>=qTeTT.
kTZqT >eT+|eTT. syTq #\eTTqT $eT #\eTT
n+

qT, y =
120 .
x
(2 , 60), (4 , 30), (6 , 20), (10 , 12)
eT]jTT (12 , 10) _+T]+|eTT.
_++ y>eTT |jDeTT
#dq n |jD\eTT
24 >+ n>TqT.
40 >+\ >eTkqeTT #sT
e\dq y>eTT 3 .$/>+ n>TqT.

nudeTT 10.2
1. dT >+ 40 .$ y>eTT |jD+#TqT.

11
• |]#jTeTT
• $dsD =\
‣ y|
‣ eT#\qeTT
‣ $#\qeTT
• $#\q >TDeTT
sY |jTsY dH
(1857-1936)
+>+&
_{wt k+K Xdy sY
|jTsYdH k+K XdeTT jTT
DeTTqT n_e~
#jTT nqT ssTT+#qT.
u XdeTT qT+& \ueTeTT
nqT n+XeTTqT qT |]#jT+
#dqT.
''The Grammer of Science"
nqT n |deTT sTy
HdH d d]+q

n+eT *jTTq$. yTT=qTeTT.
kTDeTT =
L - S =
50. 5 - 42.
4
=
L + S 50. 5 + 42.
4
= 0.087.
8.
1
92.
9

=qTeTT.
k]w $\Te ` w $\Te
>eT
( 7.44 - w $\Te = 2.26
` w $\Te = 7.44-2.26 = 5.18.
11.2.2 eT $#\qeTT (Standard deviation)

(i) | |< (Direct method)
n+XeTT\ jTT esZeTT\T dT\ueTT> \_+qsTT |< |j+#

(iii) CsTT+| n+eTD+#

D XdeTT | 10 eT+~ $=qTeTT.
k rdT=q, d = x A
10 - n>TqT. eq |{qT @s]#

e#T =qTeTT.
k @s&T
=qTeTT.
k

| | $\TeqT 3 # >TD+#> eT$#\qeTTqT 3
# >TD+#&q~.

(ii)
CsTT+| n+eT

k d+K +~ |{
e&q~. =qTeTT.
s> 0–10 10–20 20–30 30–40 40–50 50–60 60–70
n+seTT
bq:|qeTT 8 12 17 14 9 7 4
k eT

eT$#\qeTT v =
298 10th Std. Mathematics
/
/
fd
f
/
/
2 2
fd
- e o
f
=
210
- 10
71 `
-30 2 #
71
j
=
210 -
900 # 10
71 5041
=
14910 - 900 # 10
5041
=
14010
5041
# c
# 10 = 2.
7792 # 10
eT$#\qeTT, v - 16.67.
de]+#&q 40 eTT\ r> b&e\T +~e&q$. $d >D+#TeTT.
b&e (d+.$) 1–10 11–20 21–30 31–40 41–50 51–60 61–70
eTT\ d+K 2 3 8 12 9 5 1
k

d#q\T
(i) s+&T n+H me dsD\ TD+ |j>|&TqT.
(ii) $#\q >TD+ n~eTT> qsTT, TD+ e> qsTT, =qTeTT.
>&T A 38 47 34 18 33
>&T B 37 35 41 27 35
Statistics 299

k&T A
x d = x - xr d 2
18
33
34
38
47
-16
-1
0
4
13
300 10th Std. Mathematics
256
1
0
16
169
170 0 442
x =
v =
170 = 34
5
/d 2
n
=
442 = 88.
4
5
- 9.4..
v
r
$#\q >TDeTT , C.V = 100
x #
=
9.
4
34
=
940
34
= 27.65.
100 #
` >&T A #dq |sT>T\
$#\q >TDeTT R 27.65 (1)
(1) eT]jTT (1) \ qT+, B $#\q >TD+ A $#\q >TD+ H e> q~.
` |sT>T\T #jTT B >&T dseTTqT *jTTH&T.
&T B #dq |sT>T\
$#\q >TDeTT = 13.14 (2)

(
(
2
/ x -18 2 = 9
30
2
/ x -324 = 9
30
( / x 2 - 9720 = 270
/ x 2
= 9990
`
/ x = 540 eT]jTT / x 2
= 9990.
qT>=q&q~. y{ ] F#jTTq|&T n+XeTT 43 qT 53 > ydqT
qT>=q&q~. nsTTq dsq n+eT=qTeTT.
k

s+&e XD n+eT]weTT 3.84 .>. y| 0.46 .> nsTTq w =\qT qT>=qTeTT.
4. 20 |]o\q\ eT$#\qeTT 5 . | |]o\qqT 2 # >TD+#>, |* |]o\q\ eT
$#\qeTT eT]jTT $d qT>=qTeTT.
5. yTT=qTeTT.
6. +~ D+#TeTT.
(i) 10, 20, 15, 8, 3, 4 (ii) 38, 40, 34 ,31, 28, 26, 34
7. +~ |{ eT$#\qeTT >D+#TeTT.
x 3 8 13 18 23
f 7 10 15 10 8
8. |d |

10. |C deTVeTT, uk] esZeTTqT VeTT\ jTeqT\T eT M~ |#=qTeTT.
s=eTT
(`)
0–20 20–40 40–60 60–80 80–100
>VeTT\
jTeqT\ d+K
2 7 12 19 5
12. +~ $uqeTTq $d qT>=qTeTT.
s> n+seTT 20–24 25–29 30–34 35–39 40–44 45–49
bq:|qeTT 15 25 28 12 12 8
13. 100 n+XeTT\ n+eT=qTeTT.
14. 20 n+XeTT\ n+eT 10 eT]jTT 2 jT
qT>=q&q~. #s&jTTq~.
nsTTq dsq n+eTD+#TeTT.
15. n = 10, x = 12 eT]jTT / x 2
= 1530, nsTTq $#\q >TDeTTqT >D+#TeTT.
16. 20, 18, 32, 24, 26 TDeTTqT >D+#TeTT.
17. TDeTT 57 eT]jTT < eT$#\qeTT 6.84 nsTTq
n+eT=qTeTT.
18. deTVeTT 100 eTs~ mT\ dsd] 163.8 d+.$ eT]jTT $#\q >TD+ 3.2
nsTTq y] mT\ eT$#\qeTTqT qT>=qTeTT.
19. / x = 99 , n = 9 eT]jTT / ^x - 10h
2 = 79 n e&q~. nsTTq / x 2
eT]jTT
/ ^x - xh
2
qT qT>=qTeTT.
20. s> sTesT $=qTeTT.
A 58 51 60 65 66
B 56 87 88 46 43
304 10th Std. Mathematics

dsq de

13. @s&T TD+
2
14. ( x - x) = 48,
x = 20
(A) 25 (B) 20 (C) 30 (D) 10
15. TD+
(A) 42 (B) 25 (C) 28 (D) 48
eTTU+XeTT\T
q (i) y| = L-S, nq> ]w, w $\Te\ >\ uTD+ = L + S
eZ]+#&q

12
• |]#jT+
• k+|\ wjT nqTq~.
d+ue
“It is remarkable that a science which began with the consideration
of games of chance should have become the most important object of
human knowledge”
-P.D. Laplace.
12.1 |]#jT+
$+ nH+> eTqeTT #dq~ #jTTq~
neXeTT nqT +,
Te+{ |Tq\T d+u$+#TqT eTT+H V+qfq
y{ y]+ eT]jTT qc Z+ eqy
]q Te+{ d+|Tq\qT eTT+H V+#T eTq
d+ue dT#Tq~.
1654 #y*yT &$TjTsY eBdq y B
|D rdT= qMq d+ue d] *jTCjTTqT. eTq$T|&T
j< |j>eTT, |jTeTT\T, |s| esD eT]jTT
d+ue n

D Xdy\T |j>eTT eT]jTT y\Te&T |*eTT\T e+{ |
|j+TsT. @eTT n n+ nqTq~, HDeTTqT m>TsyjTT, $$< s+>T\ +T\Tq d+ qT+ + rjTT
eT]jTT sE | d\eTT sT>T |eTq$ |j>eTT\
eTTqT sV+#T eTT+< y\Te&T |*eTTqT KeTT> #|eTT.
Te+{ |j>eTTH j< |j>eTT n n+eTT \ n |*eTT\qT mesq #|e#TqT.
j< |j>eTT y\Te&T neXeTT>\ n |*eTT\ d$T |s|
esD n+eTTqT |\TesT sV+#&
|jTeTT n n+ jTTq jT&\, n|&T |Tq A d+u$+q< #|TsyjTT
nqTq$ = |Tq\T
cyTq bqT
{1, 3, 5}, {2, 4, 6}, {3} eT]
S = " 1, 2, 3, 4, 5,
6,
k] TsyjTT ueT d+u$+#T eT]jTT usTdT d+u$+#T nqTq$
deTd+ue |Tq\T.
|sds e] |Tq\T (Mutually exclusive events)
s+&T n+H me |Tq\T |sds e] |Tq\q,
|Tq d+u$+#T nqTq~ $T*q |Tq\T d+u$+#+&
n&T=qTqT. |sds e] |Tq\T k] d+u$+#e. eq A eT]
jTT B nqTq$ s+&T |sds e] |Tqq A + B = z. |eTT 12.1
308 10th Std. Mathematics

HDeTTqT m>Tsydq|&T, ueT d+u$+#T nqTq~ usTdT d+u$+#TqT
n&T=qTqT. n< $ cyTq bqT \ sT
|*eTT\T |sds e]eTT\T. m+ k] s+&T n+H me eTTK$\Te\T
|&T M\TeTT jTT |TqqT E eT]jTT |s|
1 E
E
esDqT S nqT=qTeTT. |s| esD E #+~q |*eTT\T |
2, 4, 6
3
$T*q n |*eTT\ d$T E jTT |Ps |Tq n n+TqT >eT+|eTT.
bqT $d]q|&T E = { 2, 4, 6}
nqTq~ 2 jTT >TDeTT\T d+u$+#T |Tq
nqT=qTeTT. |Tq E jTT |Ps|Tq E = {1, 3, 5} n>TqT. (|eTT 12.2 #&TeTT)
|Ps|Tq\T (Exhaustive events)
|Tq\T E 1 , E 2 , g , E n nqTq$ |Ps|Tq\q y{ dyTqeTT, |s| esD S n>TqT.
K eTT |s| esD @< |Tq\qT K |jTeTT KeTT> |Tq #+~q eT\eTT \_+#eqT.

12.2 d+ue jTT k+|eT
(i)
(ii)
310 10th Std. Mathematics
A nqTL\yTq |*eTT\ d+K
yTT+ |*eTT\ d+K
n( A)
` P( A)
= =
m .
n( S)
n
Mq |*eTT\ d+K nq+yTq eT]jTT |*eTT\T deTd+ueeTT # |
k+|

(iii) 6 jTT |

\ k+| 7 edTe\T |yTq$. j edTeqT rdq
n~ |eTT edTe> qT+&T >\ d+ueqT qT>=qTeTT.
k\ edTe\ d+K = 7.
|eTT edTe\ |TqqT A nqT=qTeTT.
|eTT edTe\ d+K, n(A) = 35 - 7 = 28.
` |eTT edTe mqT=qT >\ d+ue
312 10th Std. Mathematics
n( A)
P( A)
= =
n( S)
28
35
=
4 .
5
b+\
d+ueqT qT>=qTeTT.
k qT+&T |TqqT A nqT=qTeTT.
( 1, 5), ( 1, 6),
( 2, 5), ( 2, 6),
( 3, 5), ( 3, 6),
( 4, 5), ( 4, 6),
( 5, 5)
, ( 5, 6),
(6,5), (6,6) }
` A = {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}.
n(A) = 5.
n( A )
P(A) = = 5 .
n( S)
36
s| +qT b+ qT+&T |TqqT C nqT=qTeTT.
C = {(3,6),(4,5),(4,6),(5,4),(5,5),(5,6),(6,3),(6,4),(6,5),(6,6)}.
n(C) = 10.
n( C)
` P(C) =
n( S)
=
10
=
5 .
36 18
|eTT 12.4

T> \T|&q 52 | eTT\ qT+&, eTTqT j rjT&q~.
nsTTq +~ y{ b+\ d+ueqT qT>=qTeTT.
(i) sE (ii) q\T| sE
(iii) d& (iv) &eT+& 10.
52 | eTT\ eZsD
k\ d+ue P(G) =
& P( G ) = 3 . 7
A
2
3
4
5
6
7
8
9
10
J
Q
K
n( B ) = 20
n( S)
35
n( G ) = 15
n( S)
35
A
2
3
4
5
6
7
8
9
10
J
Q
K
A
2
3
4
5
6
7
8
9
10
J
Q
K
A
2
3
4
5
6
7
8
9
10
J
Q
K
13 13 13 13
Probability 313

\ d+ue 0.76 nsTTq sE eseTT |&+&T >\
d+ueqT qT>=qTeTT.
k\
|Tq.
P( A ) = 0.76
` P( A ) = 1 - 0.
76 a P( A) + P( A) = 1
= 0.24.
` eseTT |&+&T >\ d+ue R 0.24.
T eT]jTT = * s+>T +T\T >\e. * s+>T +
rjTT >\ d+ue, msT| s+>T + rjTT >\ d+ue eT&T ssTTq d+
* s+>T +T\ d+KqT qT>=qTeTT.
kT +T\ d+KqT x nqT=qTeTT.
` yTT+ +T\ d+K, n( S ) =
314 10th Std. Mathematics
5 + x.
* s+>T + rjTT |TqqT B eT]jTT msT| s+>T + rjTT |TqqT R nqT=qTeTT.
P( B ) = 3 P( R)
n e&q~.
n( B )
(
=
n( R)
3
n( S)
n( S)
(
x
5 + x
= 3
5
c 5
m
+ x
( x = 15
` * s+>T +T\ d+K = 15.
=qTeTT.
(i) j mqT=q&q | d+eseTT 53 XyseTT\T qT+&T;
(ii) j mqT=q&q | d+eseTT 52 XyseTT\T eyT qT+&T;
(iii) j mqT=q&q k), eT+>, TTsT, X), (X, X), (X, ~).

| d+eseTT 53 XyseTT\T b+\ d+ue, | @&T neXeTT\
XyseTT b+\ d+ue fjT>TqT.&,
S = {(~, keT), (keT, eT+>), eT+>, TTsT, X), (X, X), (X, ~)}.
` n(S) = 7.
| @&T neXeTT\ XyseTT b+TsT, X), (X, X)} ` n(A) = 2.
(ii)
(iii)
p(A) =
n( A)
n( S)
=
2 .
7
| d+eseTT 52 XyseTT\T eyT b+, TTqT >eT+|eTT.
k 52 yseTT\T 1 sE.
k, TTsT, X eT]jTT X}.
` n( S ) = 7.
| @&T neXeTT\ sE XyseTT b+=qTeTT.
k 0 .
=
7
12
`
P( A) P( A) 1
+ = n \TdT.
7k
+ 12k= 1 ( 19k = 1.
k = 1
19
P(A) = 7k
=
7 .
19
P( A)
12 P(A) = 7× P( A)
= 7 [1–P(A)]
19 P(A) = 7
qT, P(A) = 19
7
Probability 315

nudeTT 12.1
1. { qT+& qsT es d+K\T yjT&q 100 N{\T >\ d+ qT+& N{ rdq
n~ 10 # u+#&T d+K >\ N{> qT+&T >\ d+ueqT qT>=qTeTT.
2. bqT s+&TksT $ds&q~. yTT+ 9 > qT+&T >\ d+ueqT qT>=qTeTT.
3. s+&T b\T esT $ds&q$. s+&T b\| |&q s+&T d+K\qT #] s+&T
n+eTT\T >\ d+K> @ssq n~ 3 # u+#&T d+K> qT+&T >\ d+ueqT
qT>=qTeTT.
4. 12 eT+ >T& 2 b&q >T&T \dbsTTq$. j >T&T rjT&q, n~
b&q >T&T> qT+&T >\ d+ueqT qT>=qTeTT.
5. s+&T HDeTT\qT m>Tsydq|&T >]w+> ueT b+\ d+ue qT>=qTeTT.
6. u>T> \T|&q 52 | eTT\ qT+& eTTqT rjT> n~ (i) &eT+& (ii)
&eT+& ~ (iii) @dt ~> qT+&T >\ d+ueqT qT>=qTeTT.
7. eT&T HDeTT\qT k] m>TsyjT&q (i) deTT ueT (ii) K+> s+&T
usTdT\T (iii) deTT s+&T ueT\T b+\ d+ue qT>=qTeTT.
8. d+ 1 qT+& 6 es d+K\T *Zq \+T\T eT]jTT 7 qT+& 10 es
d+K\T *Zq msT| +T\T >\e. j + rdq n~ (i) d] d+K >\
+ (ii) \ + b+\ d+ueqT qT>=qTeTT.
9. 1 qT+& 100 es >\ |Ps+eTT\ j d+KqT mqT=q n~ (i)
d+|Ps esZeTT (ii) d+|Ps |TqeTT ~> qT+&T >\ d+ueqT qT>=qTeTT.
10. $, sc eT]jTT n]
j T +T\T >\e. j
+ rdq n~ (i) msT| s+>T (ii) |#s+>T +> qT+&T >\ d+ue qT>=qTeTT.
12. 20 sT\| 1 qT+& 20 es n+\T yjT&jTTq$. j sTqT rdq
n~ (i) 4 >TDeTT (ii) 6 >TDeTT +&T >\ d+ue qT>=qTeTT.
13. n+\T 3, 5, 7 \ s+&+\ d+KqT sb+~+q n~ 57 H me> qT+&T >\
d+ueqT qT>=qTeTT. (|j+q n+\qT eTs=k] |j+#s=qTeTT.
15. s+&T b\qT =q&q~. qT>=q&q
\eTT | qT+&T >\ d+ue @$T?
316 10th Std. Mathematics

16. C& \eTT, |# eT]jTT \T| s+>T\ 54 >\Tq$. \eTT > b+\ d+ue
1
3
eT]jTT |#> b+\ d+ue
4
9
nsTTq C& m
\T| s+>T >\T+&TqT.
17. d+ 100 #=\ 88 u>Tq$. 8 q |eTT eT]jTT 4 me |eTT
*jTTq$. A nqT d+d u>T> q #=\qT eeTT n+^]+#qT. B d+d
me> |eTTq #=\qT n+^]+#\ d+ueqT qT>=qTeTT.
18. 12 +T\T >\ d+ x +T\T \$ (i) j + rdq n~
\ +> qT+&T >\ d+ue @$T? (ii) eTs 6 \ +T\qT d+ ydq,
\ + b+\ d+ue (i) e#T< H s+&+\T> +&TqT. nsTTq
x qT qT>=qTeTT.
19. |^Z VQ+& 100 @u|d\ HDeTT\T, 50 sbsTT HDeTT\T, 20 s+&T sbjT\
HDeTT\T eT]jTT 10 =qTeTT.
12.3 d+ue d+\q deT
n( A , B)
= n( A) + n( B) - n( A + B)
.
sTy|\ n( S ) # u+#>,
n( A , B)
=
n( S)
n( A)
n( B)
n( A + B)
+ - . (1)
n( S)
n( S)
n( S)
j< |j>eTT |d$TT\T A eT]jTT B nqT>TDeTT> A eT]jTT B nqT
|Tq\T eT]jTT d$T S nqT>TDeTT> |s| esD S nsTTq (1) qT+
P( A , B)
= P( A) + P( B) - P( A + B)
n>TqT.
|*eTTH d+ue d+\q d

|*eTT\T (s|D +&)
(i) |s| esD S d++~+q A, B eT]jTT C nqTq$ @y 3 |Tqq
(ii)
P( A , B , C)
= P( A) + P( B) + P( C) - P( A + B) - P( B + C) - P( A + C) + P( A + B + C)
.
A 1
A 2
eT]jTT A 3
nqT eT&T |sds e] |Tqq
P( A , A , A ) = P ( A ) + P ( A ) + P ( A ) .
1 2 3
318 10th Std. Mathematics
1 2 3
(iii) A 1
, A 2
, A 3
, g , A n
nqTq$ |sds e] |Tqq
P( A 1
, A 2
, A 3
, g , A n
) = P( A 1
) + P( A 2
) + P( A 3
) + g + P( A n
).
S
(iv) P( A + B) = P( A) - P( A + B)
,
A
P( A + B) = P( B) - P( A + B)
& A + B nq> A eeTT B < nseTT. n B eeTT A < nseTT.
|eTT 12.6
TsyjT&q$. d+ue d+\q d s+&T usTdT\T deTT ueT |&T >\ d+ueqT qT>=qTeTT.
k

\ d+u qT>=qTeTT. (d+ue d+\q d

(ii)
P( @< X\ eeTT |yXeTT b+=qTeTT. (|j+q nseTT\qT
] |j+#e#T)
k

( 4p + 6p + 3p
= 4
{, p = 13
4 .
qT, P( A ) = 4 . 13
qT+&T >\ d+ueqT qT>=qTeTT.
k sE, VsY eT]jTT msT| eTTqT b+\ d+ueqT qT>=qTeTT.
k \T|, q\T| eT]jTT |# + mqT=qT
|Tq\qT=qTeTT.
n( W)
\T| + b+

322 10th Std. Mathematics
nudeTT 12.2
=
3
5
, P B
=
1
5
1. A eT]jTT B nqTq$ |sds e] |Tq\T P^Ah
^ h nsTTq P( A B)
qT>=qTeTT.
2. A eT]jTT B nqT s+&T |Tq\ P( A) =
1
, P( B)
=
2
eT]jTT P( A , B)
=
1
nsTTq
4 5
2
P( A + B)
qT>=qTeTT.
3. P( A) =
1
, P( B) =
7
, P( A , B) = 1. nsTTq (i) P( A + B)
(ii) P( Al , Bl ) \qT
2 10
qT>=qTeTT.
4. bqT s+&T ksT \ |Ps+eTT\ d+KqT j mqT=q n~ 4
6 # u+|&Tq~> qT+&T >\ d+ueqT qT>=qTeTT.
6. d+ 50 u\T\T eT]jTT 150 qT >\e. y{ d>+ u\T\T eT]jTT d>+ qT
T| |{q$. y{ < j mqT=q, n~ T||{q qT
u\T> qT+&T >\ d+ueqT qT>=qTeTT.
7. s+&T b\qT k] $d]q y{ eTTK $\Te\T yTT+ 3 # eT]jTT 4 # u+|&q~>
qT+&T >\ d+ueqT qT>=qTeTT.
8. T 20 |\T eT]jTT 10 s+C\T >\e. y{ 5 |\T eT]jTT 3 s+\T
b&q$. j |+&TqT rdq n~ | |+&T> eT+ |+&T>
qT+&T >\ d+ueqT qT>=qTeTT.
9. s> 40% $DXdeTT, 30% $ qT+& j
$T C
seTeTT bZqT >\ d+ueqT qT>=qTeTT.
10. u>T> \T|&q 52 eTT\T >\ qT+& eTTqT j rdq n~ d&T>
sE> qT+&T >\ d+ueqT qT>=qTeTT.
11. |f 10 \T|, 6 msT| eT]jTT 10 q\T| +T\Tq$. + j
rdq n~ \T| msT| +> qT+&T >\ d+ueqT qT>=qTeTT.
12. 2, 5, 9 n+\qT|j+ s+&T n+\ d+KqT @s]q (|j+q n+qT ]
|j+#e#T), n~ 2 5 # u+#&T d+K> qT+&T >\ d+ueqT
qT>=qTeTT.
13. ACCOMMODATION nqT | qT+&T >\ d+ueqT
qT>=qTeTT.
,

14. = sT < VQeT b+\ d+ue 0.25, +=qTeTT.
(i) VQeTT\ deTT VQeT b+ 5
4 , 3
2 eT]jTT
3 . A eT]jTT B k~+#T >\ d+ue
7
158
, B eT]jTT C k~+#T >\ d+ue
2
, A eT]jTT C k~+#T >\ d+ue
12
7
35
. eTT>TZs k~+#T >\ d+ue
8
35
nsTTq deTdqT y] deTT sT k~+#T >\ d+ueqT qT>=qTeTT.
dsq de\ d+ue 5
4
nsTTq
q{ qsT esT\T s+&T >\ d+ue
(A) 5
1 (B) 5
2 (C) 5
3 (D) 5
4
6. A eT]jTT B nqTq$ s+&T |Tq\, P(A)=0.25, P(B) = 0.05, P^A + B
nsTTq P^A
, Bh =
(A) 0.61 (B) 0.16 (C) 0.14 (D) 0.6
h=0.14
7. 20 edTe\T >\ e~] 6 |yTq$. j edTe rdq n~ |eTT
~> qT+&T >\ d+ue.
(A) 10
7 (B) 0 (C) 10
3 (D) 3
2
Probability 323

8. A eT]jTT B nqTq$ |sds e] |Tq\T, S nqTq~ |s| esD eT]jTT P( A) P( B)
, S = A , B nsTTq P( A ) =
(A) 1 (B) 1 (C) 3 (D) 3
4 2 4 8
324 10th Std. Mathematics
=
1
3
9. A, B eT]jTT C nqT eT&T |sds e] |Tq\ d+ue\T eTeTT> 31
, 1
4
eT]jTT
5 nsTTq P ^ A , B , C h nqTq~
12
(A) 19 (B) 11 (C) 7 (D) 1
12
12
12
10. P(A) = 0.25, P(B) = 0.50, P^A + Bh = 0.14 nsTTq P(A qT ~ eT]jTT B qT ~) =
(A) 0.39 (B) 0.25 (C) 0.11 (D) 0.24
11. d+ 5 q\T|, 4 \T| eT]jTT 3 msT| +T\Tq$. j +
mqT=q n~ msT| + +&T >\ d+ue.
(A) 5 (B) 4 (C) 3 (D) 3
12 12 12 4
12. s+&T b\T k] $ds&q, s+&T eTTK $\Te\T d+K> qT+&T >\ d+ue
(A) 1 (B) 1 (C) 1 (D) 2
36
3 6 3
13. eTyTq HDeTTqT $d]q|&T | qT+&T >\ d+ue
(A) 1 (B) 0 (C) 5 (D) 1
6 6
14. HDeTTqT 3 ksT m>Tsydq|&T 3 ueT\T 3 usTdT\T b+\ d+ue
(A) 8
1 (B) 4
1 (C) 8
3 (D) 2
1
15. 52 eTT\T >\ qT+& eTTqT rdq n~ @dt eT]jTT sE eTT> +&T >\
d+ue
(A) 2 (B) 11 (C) 4 (D) 8
13 13
13 13
16. |t d+eseTT 53 XyseTT\T 53 XyseTT\T +&T >\ d+ue
(A) 7
2 (B) 7
1 (C) 7
4 (D) 7
3
17. k\ d+ue
(A) 1 (B) 2 (C) 3 (D) 0
7 7 7
18. 52 eTT\Tq qT+& eTTqT rdq n~ VsYsD> qT+&T >\ d+ue
(A) 1 (B) 16 (C) 1 (D) 1
52
52
13 26
19. K |Tq jTT d+ue
(A) 1 (B) 0 (C) 100 (D) 0.1
20. j< |j>eTT y\Te&T |*eTT >\T| z$T> qT+&TqT. >\Tb+\ d+ue, z&be >\ d+ue s+&T sT nsTTq, >\Tb+\ d+ue
(A) 3
1 (B) 3
2 (C) 1 (D) 0

yT\T
1. d$TT\T eT]jTT |yTjTeTT\T
nudeTT 1.1
2. (i) A (ii) z 3. (i) {b, c} (ii) z (iii) {a, e, f, s}
4. (i) {2, 4, 6, 7, 8, 9} (ii) {4, 6} (iii) {4, 6, 7, 8, 9}
10. {–5, –3, –2}, {–5, –3}, dV#seTT e]+#

14. " ^4, 3h, ^6, 4h, ^8, 5h, ^10,
6h,
x 4 6 8 10
f (x) 3 4 5 6
15. (i) 16 (ii) –32 (iii) 5 (iv)
2
3
326 10th Std. Mathematics
16. (i) 23 (ii) 34 (iii) 2
nudeTT 1.5
1 2 3 4 5 6 7 8 9 10
A C C A A B A B B B
11 12 13 14 15 16 17 18 19 20
A B C D A D D B A C
2. yde d+K\ eqT> XDT\T
nudeTT 2.1
1. (i) -
1
, 0,1 (ii) -27, 81, - 243 (iii) -
3
, 2, -
15
3
4 4
2. (i)
9
,
11
(ii) –1536, 18432 (iii) 36, 78 (iv) –21,57
17 21
3. 378,
25
313
4. 195, 256 5. 2, 5, 15, 35, 75 6. 1, 1, 1, 2, 3, 5
nudeTT 2.2
1. A.P : 6, 11, 16, g; keq |TDX&, r =
1
(vi) 2
>TDX&

1. s
20
20
=
15
; 1 -
1
4
`
3
j E 2. s
27
nudeTT 2.5
27
=
1
; 1 -
1
6
`
3
j E 3. (i) 765 (ii) 5 ( 3 12
- 1 )
2
1 -^0.
1h
10
4. (i)
(ii)
10 20
^ 10 - 1 h -
20
5. (i) n = 6 (ii) n = 6 6.
75
; 1 -
4 23
0.
9 81
9
4
`
5
j E
7. 3 + 6 + 12 + g 8. (i)
70 n
6 10 - 1 @ -
7n
(ii) 1 -
2
8 1 -
1 n
81
9 3
`
10
j B
15
9. s
5^4 1
=
- h
15 3
10. s+&e neXeTT, e$T& |+& d+K 1023. 11. r = 2
nudeTT 2.6
1. (i) 1035 (ii) 4285 (iii) 2550 (iv) 17395 (v) 10630 (vi) 382500
2. (i) k = 12 (ii) k = 9 3. 29241 4. 91 5. 3818 cm 2 6. 201825 cm 3
nudeTT 2.7
1 2 3 4 5 6 7 8 9 10
A D C D D A B B B B
11 12 13 14 15 16 17 18 19 20
B A B D A B B A C A
3. ;>DeTT
nudeTT 3.1
1. (4 ,
3 ) 2. (1 , 5) 3. (3 , 2) 4. (
1 ,
2 3
7. (2 , 4) 8. (2 , 1) 9. (5 ,
1 ) 7 10. (6 , – 4)
nudeTT 3.2
1. (i) (4 , 3) (ii) (0.4, 0.3) (iii) (2 , 3) (iv) (
1
,
2
1 ) 5. (1 , 5) 6. (
11 ,
2
23
1 )
3
22 )
31
2. (i) 23 , 7 (ii) ` 18,000 , ` 14,000 (iii) 42 (iv) ` 800 (v) 253 #.d+.MT (vi) 720 .MT
nudeTT 3.3
1. (i) 4, – 2 (ii)
1
,
1
(iii)
3
, -
1
(iv) 0, – 2 (v) 15,
- 15 (vi)
2
, 1
2 2 2 3
3
(vii)
1
,
1
(viii) - 13,
11
2 2
2
2
2
2
2. (i) x - 3x
+ 1 (ii) x - 2x
+ 4 (iii) x + 4 (iv) x - 2 x +
1
5
(v) x
2
-
x
+ 1 (vi) x
3
2
-
x
- 4 (vii) x
2
2
nudeTT 3.4
x 1
2
- - (viii) x - 3 x + 2
3 3
2
2
2
1. (i) x + 2x
- 1, 4 (ii) 3x
- 11x
+ 40, - 125 (iii) x + 2x
- 2, 2
2
2
(iv) x
5
x
5
,
50
3
- + - (v) 2 x -
x
-
3
x +
51
, -
211
3 9 9
2 8 32 32
3 2
(vi) x - 3x - 8x
+
55
, -
41
2 2
Answers 327

=- = XweTT 5 3. p 2, q 0, XweTT -10
nudeTT 3.5
2. a 6, b 11,
=- = -
1. (i) ^x - 1h^x + 2h^x
- 3h (ii) ^x - 1h^2x + 3h^2x
- 1h (iii) ^x - 1h^x - 12h^x
- 10h
2
(iv) ^x - 1h^4x - x + 6h (v) ^x - 1h^x - 2h^x
+ 3h (vi) ^x + 1h^x + 2h^x
+ 10h
(vii) ^x - 2h^x - 3h^2x
+ 1h (viii) ^x - 1h^x + x - 4h (ix) ^x - 1h^x + 1h^x
- 10h
(x) ^x - 1h^x + 6h^2x
+ 1h (xi) ^x - 2h^x + 3x
+ 7h (xii) ^x + 2h^x - 3h^x
- 4h
2
2
1. (i) 7x 2 yz
3 2
(ii) x y
(iii)
nudeTT 3.6
5c
3 (iv) 7xyz
2
2. (i) c - d (ii) x - 3a
(iii) m + 3 (iv) x + 11 (v) x + 2y
(vi)
2x
+ 1 (vii) x - 2 (viii) ^x - 1h^x
+ 1h (ix) 4x ^2x
+ 1h (x) ( a - 1) ( a + 3)
3. (i) x 2 – 4x + 3 (ii) x + 1 (iii) 2^x
1h (iv) x 4
3 2
1. x y z
328 10th Std. Mathematics
2
2
+
nudeTT 3.7
2. 12x 3 y 3 2 2 2
z
3. a b c
2
2
2
+
3 2
4. 264a 4 b 4 c
4 5. a m + 3
2 2
6. xy^x + yh 7. 6( a - 1)
^a
+ 1h 8. 10xy^x + 3yh^x - 3yh( x - 3xy + 9y
)
2 3
3 2
9. ( x + 4) ( x - 3)
^x
- 1h 10. 420x ^3x + yh ^x - 2yh^3x
+ 1h
nudeTT 3.8
2 4 2
1. (i) (x – 3) (x – 2) ( x + 6) (ii) ^x + 2x + 3h^x + 2x + x + 2h
2 3 2
3 3 2
(iii) ^2x + x - 5h^x + 8x + 4x
- 21h (iv) ^x - 5x - 8h^2x - 3x - 9x
+ 5h
2. (i) ^x
+ 1h ( x + 2)
2
3
2 2 4 2 2 4
(ii) ( 3x
- 7)
^4x
+ 5h (iii) ^x - y h^x + x y + y h
(iv) x(x + 2) (5x + 1) (v) (x – 2) ( x – 1) (vi) 2(x + 1) (x + 2)
1. (i)
2x
+ 3
(ii)
x - 4
2
(v) x - x + 1 (vi)
x x
(ix)
^x
- 1h
^x
+ 1h
1. (i) 3x (ii)
2. (i)
x - 1
(ii)
x
x x
(x) 1
x + 9
(iii)
x - 2
- 6
(iii)
- 7
nudeTT 3.9
1
(iii) ^x 2
- 1h (iv)
x + 3x
+ 9
x - 1
x + 3
x + 2
(vii)
x - 1
(viii) (x + 3)
2
+ 2 + 4
x + 1
1
x + 4
nudeTT 3.10
(iv)
(xi)
1
x - 1
^x
+ 1h
^2x
- 1h
(v)
x + 1
(iv)
x - 5
(v) 1 (vi)
x - 5 x - 11
2
(xii) (x – 2)
2x
+ 1
(vi) 1
x + 2
3x
+ 1
4^3x
+ 4h
(vii)
x - 1
x + 1

2
1 . (i) x + 2x
+ 4 (ii)
2
x + 1
2.
(v)
x + 1
(vi)
x - 2
3 2
2x
+ 2x
+ 5
3.
2
x + 2
4
x + 4
2
nudeTT 3.11
(iii)
(vii)
2( x + 4)
x + 3
2
x + 1
5x
- 7x
+ 6
4. 1
2x
- 1
nudeTT 3.12
2 3
1. (i) 14 a 3 b 4 c
5 (ii) 17 ( a - b) ( b - c)
(iii) x - 11
(iv) x + y (v)
11
9
2
x
y
(vi)
8
5
(iv)
2
x - 5
(viii) 0
2. (i) 4x
- 3 (ii) ( x + 5)( x - 5)( x + 3)
(iii) 2x - 3y - 5z
(iv) x
2
2 4 3
( a + b) ( x - y) ( b - c)
( x + y) ( a - b) ( b + c)
2 3 5
+
1
(v) ^2x 3 3x 2 2x
1
2
+ h^ - h^
+ h (vi) ^2x - 1h^x - 2h^3x
+ 1h
x
nudeTT 3.13
2 2 2 2
1. (i) x - 2x
+ 3 (ii) 2x + 2x
+ 1 (iii) 3x - x + 1 (iv) 4x - 3x
+ 2
2. (i) a =- 42,
b = 49 (ii) a = 12,
b = 9 (iii) a = 49,
b =- 70 (iv) a = 9, b =- 12
nudeTT 3.14
1. "-6,
3, 2. -
4 $ , 3
3 . 3. 5,
3
'-
1
5
4. -
3
, 5
2
6. 5,
1
$ 5
. 7.
5
,
3
$ - . 8.
1
,
1
2 2 ' 2 2 1
b a
9. -
5
, 3
2
nudeTT 3.15
$ . 5.
-
4 $ ,2
3 .
$ . 10. 7,
8
$ 3
.
1. (i) "-7,
1, (ii) '- 3 + 5
, -3 - 5
1 (iii) -3,
1
$
2 2
2
.
(iv) $ a - b 2 , - `
a + b
2
j. (v) " 3,
1, (vi) "-1,
3,
2. (i) " 4,
3, (ii)
2
,
1
$ . (iii)
1
$ , 2 . (iv)
2b
,
b
5 3
2
$ -
3a
a
.
(v)
1
$ ,
a a . (vi)
a + b
,
a - b ^9 $ . (vii)
+ 769h
^9 769
,
- h
6 6
8 8
1. 8 8
1
nudeTT 3.16
2
(viii) ,
b
)-
3 a
2. 9 eT]jTT 6 3. 20 MT, 5MT 10MT, 10MT 4.
1
2
3 MT
2
5. 45 .MT/>+ 6. 5 .MT/>+ 7. 49 d+\T, 7 d+\T 8. 24 d+.MT 9. 12 sE\T
10. yTTeTT 20 .MT/>+ eT]jTT s+&e s\T y>eTT 15 .MT/>+
Answers 329

nudeTT 3.17
1. (i) yde+ (ii) nyde+ (iii) ydeeTT\T eT]jTT deqeTT\T (iv) ydeeTT\T eT]jTT
deqeTT\T (v) nydeeTT\T (vi) yde+
2. (i)
25
2
(ii) ! 3 (iii) – 5 or 1 (iv) 0 or 3
nudeTT 3.18
1. (i) 6,5 (ii) -
r
, p (iii)
5 , 0 (iv) 0,
-
25
k
3
8
2. (i) x
2 - 7x
+ 12 = 0 (ii) x
2 - 6 x + 2 = 0 (iii) 4 x
2 - 1 6 x + 9 = 0
3. (i)
13
6
(ii)
!
1
(iii)
35
3
18
4. 3
4
2
5. 4x - 29x
+ 25 = 0
6. x 2 2 2
+ 3x + 2 7. x - 11x
+ 1 = 0 8. (i) x - 6x
+ 3 = 0
(ii) 27x 2 - 18x
+ 1 = 0 (iii) 3x
2 - 18x
+ 25 = 0 9. x 2
+ 3 x - 4 = 0
10. k =- 18 11. a = !24 12. p = ! 3 5
1.
400
200
300
330 10th Std. Mathematics
nudeTT 3.19
1 2 3 4 5 6 7 8 9 10
B C A A C D B C C C
11 12 13 14 15 16 17 18 19 20
D B A A A D D D B C
21 22 23 24 25
D A C C A
500
250
400
f p ,
4. e\T
nudeTT 4.1
c 400 200 300
500 250 400
m , 3 # 2 , 2 # 3 2.
3. (i) 2 # 3 (ii) 3 # 1 (iii) 3 # 3 (iv) 1 # 3 (v) 4 # 2
4. 1 # 8, 8 # 1, 2 # 4, 4 # 2
6
8
13
f p , ^ 6 8 13
5. 1 # 30,30 # 1, 2 # 15,
15 # 2, 3 # 10,10 # 3, 5 # 6,
6 # 5, 10 # 1, 1 # 10, 15 # 1,
1 # 15
1
6. (i) c
2
2
4
m (ii) c
1
3
0
2
J
K 0
m (iii) K
K
1
L
3
1 N
J
-
3
O
K 1
O
0 O 7. (i) K
K 2
P
K 3
L
1 N
2
O
1 O
3
O
O
2
P
J1
K
2
(ii) K 0
K
K
1
L
2
8. (i) 3 4 # (ii) 4, 0 (iii) 2e n&T esTd eT]jTT 3e \Te esTd 9.
9 N
2
O
2 O
1
O
O
2
P
J1
K 2
K
(iii)
1
K 2
K 3
L2
h
N
2O
O
1O
0O
P

nudeTT 4.2
1. x = 2,
y =- 4, z =- 1 2. x = 4, y =- 3
3.
-1
2
14
e o 4.
16 - 6
3
14 5
0 - 18
e o
33 - 45
6. a = 3, b = –4
7.
J 2
X 5
12 N J 2
= K
5
O
K
K - 11 O
3 O , Y K
= K
5
K
14
L
5
P L
5
TV DVD Video CD
55 27 20 16
11. 72 30 25 27
47 33 18 22 s
13 N
5
O
O
-2
O
P

3. (i) @sFjTeTT\T (ii) @sFjTeTT\T

6. sU>D+
nudeTT 6.1
1. (i) 20d+.MT (ii) 6d+.MT (iii) 1 2. 7.5d+.MT 3. (i)

334 10th Std. Mathematics
8. >DqeTT
nudeTT 8.1
1. 704#.d+.MT, 1936 #.d+.MT 2. h = 8 d+.MT, 352 #.d+.MT 3. h = 40 d+.MT, d = 35 d+.MT
4. ` 2640 5. r = 3.5 d+.MT, h = 7 d+.MT 6. h = 28 d+.MT
7. C 1
: C 2
= 5 : 2 8. 612r#.d+.MT 9. 3168 #.d+.MT
10. 550 #.d+.MT, 704 #.d+.MT 11. h = 15 3 d+.MT, l = 30 d+.MT 12. 1416 #.d+.MT
13. 23.1 m 2 14. 10.5 d+.MT 15. 301 7 5 #.d+.MT 16. 2.8 d+.MT
17. 4158#.d+.MT 18. C 1
: C 2
= 9 : 25, T 1
: T 2
= 9 : 25
19. 44.1r #.d+.MT, 57.33r #.d+.MT 20. ` 246 .40
nudeTT 8.2
1. 18480 |T.d+.MT 2. 38.5 sT 3. 4620 |T.d+.MT 4. r = 2.1 d+.MT
5. V : V = 20:
27 6. 10 d+.MT 7. 4158 |T.d+.MT 8. 7.04 |T.d+.MT
1 2
9. 8800 |T.d+.MT 10. 616 |T.d+.MT 11. 5d+.MT 12. 1408.6 |T.d+.MT
13. 314 2 7
|T.d+.MT 14. 2 13 d+.MT 15. 8 d+.MT 16. 2.29 .>
17. 3050 2 3
|T.d+.MT 18. 288r#.d+.MT 19.
nudeTT 8.3
718
2
|T.d+.MT 20. 1: 8
3
1. 11.88r #.d+.MT 2. 7623|T.d+.MT 3. 220 #.d+.MT 4. 1034 #.MT
5. 12 d+.MT 6. 12.8 .MT 7. 2 d+.MT 8. 1 d+.MT
9. 1386 sT 10. 3 >+\T. 12 $TweTT\T. 11. 16 d+.MT 12. 16 d+.MT
13. 750 dd| >T+&T 14. 10 X+KTe\T 15. 70 d+.MT
16. r = 36 d+.MT, l = 12 13 d+.MT 17. 11MT
nudeTT 8.4
1 2 3 4 5 6 7 8 9 10 11
B C A A B C A B D C C
12 13 14 15 16 17 18 19 20 21 22
D D B D B C B D A D C
10. sUeTT\T
nudeTT 10.1
2. (i) "-2,
2, (ii) "-2,
5, (iii) " 5,
1, (iv) -
1 $ , 3.
3. {–1, 5} 4. {–2, 3} 5. {–2.5, 2} 6. {–3, 5} 7. k

11. k+U XdeTT
nudeTT 11.1
1. (i) 36, 0.44 (ii) 44, 0.64 2. 71 3. 3.38 .> 4. 2 5 , 20 5. 3.74
6. (i) 5.97 (ii) 4.69 7. 6.32 8. 1.107 9. 15.08
10. 36.76, 6.06 11. 416, 20. 39 12. 54.19 13. 4800, 240400 14. 10.2, 1.99
15. 25 16. 20.42 17. 12 18. 5.24 19. 1159, 70
20. A me dsyTq~
nudeTT 11.2
1 2 3 4 5 6 7 8 9 10
D A C B D C C B A D
11 12 13 14 15
D B C D B
12. d+ue
nudeTT 12.1
1. 1
10
2. 1
9
3. 1
3
4. 1
5
5. 3
4
6. (i) 1
4
(ii) 3
4
(iii) 12
13
7. (i) 7
8
(ii) 3
8
(iii) 1
2
8. (i) 1
2
(ii) 3
5
9. (i) 1
10
(ii) 24
25
10. 1
2
11. (i) 1 (ii) 2
4 3
12. (i) 1
4
(ii) 17
20
13. 1
3
14. 1
36
15. 1
6
16. 12
17. (i) 22
25
(ii) 24
25
18. (i) , (ii) 3 19. (i) 5
9
(ii) 17
18
1.
6.
nudeTT 12. 2
4 2. 3 3. (i) 1 (ii) 4 4.
5 20
5 5
5 7.
8
11. 13
8 12.
4
9
8.
2 13. 5
,
3
9 9. 3 10. 4
10
5 13
13
5 5.
9
4 14. (i) 0.45 (ii) 0.3 15.
13
nudeTT 12. 3
1 2 3 4 5 6 7 8 9 10
C D B A A B A A D A
11 12 13 14 15 16 17 18 19 20
D C C B B C D A A B
8
25
101
105
Answers 335

1. f^xh =
x - 1 , x 1
x + 1
336 10th Std. Mathematics
! nsTTq, f( x)
2 =
3f( x)
1
f( x)
+ 3
+ n s|+#TeTT.
2. yde $\Te\ x ( x - 1)( x - 2)( x - 3)( x - 4)
= 15 dMTsDeTTqT k~+#TeTT..
=
5 ! 2 )
2
(yT:x
3.
x
x jTT @ $\Te log102,
log10^2 - 1h eT]jTT log 2
x
10 ^ + 3 h nqT eT&T d+K\qT
eTeTT> rdT=q A.P. @ssT#TqT? (yT: x = log 5 2)
4. keq w r qT *q G.P. yTTT | 1
n n - 1 n +
nsTTqsTTq, b
1
" n , nqT esTd G.P. n>Tq s|+|eTT.
6. 17, 21 ....... eT]jTT 16, 21....... nqT s+&T n+X&\ = d+K\T q&TqT.
s+&T XD\ q&T yTT=qTeTT.
7.
a + a
n - 1 n + 1
a =
, n > 1 nsTTqsTTq, a
n 2
" n , nqT esTd A.P. n>Tq s|+|eTT.
8.
6 6 2 2
sin a + cos a + 3 sin a cos a = 1 n s|+|eTT.
9.
sin x + cos x 3 2
= tan x + tan x + tan x + 1 n s|+|eTT.
cos
2
x
10. s+&+\ d+KqT y{ n+\ yTTeTT u+q, u>|\eTT 4 eT]jTT XweTT 3
e#TqT. n< s+&+\ d+KqT y{ n+\ \eTT u+q, u>|\eTT 3 eT]jTT
XweTT 5 e#TqT. nsTTq s+&+\ d+KqT qT>=qTeTT. (yT : 23)
11. 4 # u+#> XweTT 1 > e#T s+&+\ d+K\ yTTeTTqT qT>=qTeTT.
1 1
2 2 2
12.
a b c
# (1
b c a
)( a b c)
1
-
1
2bc
a b + c
+ + 2
+ + - + + - dedeTTqT d]+|eTT.
(yT : 1210)
( yT :
2
13. ax + bx + c = 0 nqT ~| dMTsDeTTq yde eT\eTT\T e eT]jTT
14. f x
1 )
2bc
a + b + c < 0 > e+&TqT. nsTTq c nqT d+K >TsTqT qT>=qTeTT. (d#q: f( x ) = 0
q yde eT\eTT\T qsTTq, f( x ) jT+TsTqT *jTT+&TqT)
(yT: c < 0)
^ h
x - 1 n>TqT, x jTT n yde d+K\qT qT>=qTeTT.
= 2
x - x + 6
> 0
$TXeT deTd\T
(|

2
15. k~+#TeTT: 1 a a g a
x
+ + + + =( 1 a)( 1 a 2 )( 1 a 4 )( 1 a
8
)
+ + + +
(yT: x = 15)
16. >D+#TeTT: 6 2
x x 4 3
x 6
2 3
x x 4x
1 2 - 1 + 1 2 - 2
jT+=qTeTT. (yT: +f\ d+K 36)
19. deTy>eTT be s\T 30 .MT eTT
4
5
>ZqT. |*eTT> s\T >eT+qT #sT 45 $TweTT\T
\deTjTqT. n< + 18 MTsT |jD+q s ]jTT+&q 9 $TweTT\
eTT+H #]jTT+&TqT. nsTTq s\T y>eTTqT eT]jTT |jD =qTeTT.
(yT: s\T y>eTT 30 .MT/>+ eT]jTT |jD

27. d]+|eTT: (1 +
2
)(1 +
2
)(1 +
2
) g(1 +
2
). (yT : ( n 1 )( n 2 )
338 10th Std. Mathematics
2
3
4
n
+ + )
6
28. eT&T es &dt\T >\e. y{ s+&+{ ykseTT\T r n+>TeTT\T eT]jTT eT&e
< ykseTT 2 r n+>TeTT\T. eT&T &dT\qT \eTT| \ 6 es &dt\qT H\| eseTT neT]q, eT<
u>eTT sT &dT\+{ qT @&e &dTqT neTsTeTT. | &dT eTs= s+&T
&dT\qT qT+&eqT. sT &dT\ +eTT jTT yX\eTTqT
qT>=qTeTT. (yT : 192 3 #.n+>TeTT\T)
30. 4 d+.MT ykseTT eT]jTT 5 d+.MT mT >\ dbs =jT eTTqT+& n<
ykseTT eT]jTT 3 d+.MT mT >\ e X+KTeqT ]+ rdyjT&q~. $T*q
=jTeTT |]\ yX\eTT 76 d+.MT2 n s|+|eTT.
31.
1 2 3
g
n
1
1
2! 3! 4! ( n 1)! ( n 1)!
= 1 # 2 # 3 # g # n.
+ + + + +
= - +
n #|eTT.
+

seTT: >D+
|X| |D[
\eTT: 2.30 >+\T
s>: X yTT+ esT\T: 100
nudq eTT-B
q yT
u>eTT-C
|< yT
u>eTT-D
# |< yT
yTTeTT
|X\ d+K 15 10 9 2 36
esT\T 15 20 45 20 100
\eTT
($TweTT\)
20 35 65 30 2.30 >+
|X\ ksTT
ksTT
esT\ XeTT
]qeTT 12
eT

u>eTT\T
u>eTT\T eT]jTT mqT=qT
|X\ d+K\T
|X\ d+K
qT+& es
yT\T
ee\dq |X\T
bs+X
d+K
A 1 15 15 15
B 16 30
C 31 45
D
46
47
16
30e |X KyTq~ eT]jTT
s+&+{ { yjTeqT
16
45e |X KyTq~ eT]jTT
s+&+{ { yjTeqT
2
s+&+{ {
2
s+&+{ {
$wjT d q esT\ $uq
bs+XeTT
|X\ d+K
1 esT 2 esT\T 5 esT\T 10 esT\T
1
d$TT\T eT]jTT
|yTjTeTT\T
1 2 2 15
2
yde d+K\ eqT>
XDT\T
2 1 2 14
3 ;>DeTT 2 2 3 21
10
9
1
1
yTT+
esT\T
4 e\T 1 2 1 10
5 s|C$T 2 2 2 16
6 sU>D+ 2 1 1 9
7 D$T 2 2 1 11
8 >DqeTT 1 2 2 15
9 |j> sU>D+ 2 20
10 sUeTT\T 2 20
11 k+U XdeTT 1 1 1 8
12 d+ue 1 1 1 8
yTT+ 15 16 16 4 167
340 10th Std. Mathematics

s|deTTqTq
eTT B
(2 esT\T)
u>eTT C
(5 esT\T)
u>eTT D
(10 esT\T)
yTT+
esT\T
XeTT
--- 6 (2) 6 (5) 1 (10) 52 31
nud \ qT+& 10 (1) 8 (2) 8 (5) 3 (10) 96 58
s+|&q n\ d+K = |X >\ esT\qT \T|qT.
u>eTT - A
1. 1 qT+& 15 es d+K >\ 15 |X\+{ K+> y{ q yTqT mqT=qT |X\>TqT. = |X
@>qT esT 4 yT\qT *jTT+&TqT. = |X esT.
2. 15 |X\ 10 |X\T bs|deTT e&q d]jTq yTqT mqT=qT |X\>TqT. $T*q |X\T 2, 3,
5, 6 eT]jTT 7 yTTT bsu>eTT\ qT+& jsT#jT&q |X\>TqT. $ bs|d+ d\ |X\ 10 |X\ yT*eeqT. = |X 2 esT\T.
2. yTTTqT. ~ s+&+{ {
nqT seTTqqT+&TqT.
3. 14 |X\T bs|deTT >\ bsu> esTdqT+&TqT.
4. 14 |X\ 6 |X\T \ s+&T |X\T jsT#jT&q |X\T n>TqT. n$ 2, 3, 5 eT]jTT 8 bsu>eTT\q+\
eTT\qT+& n&T>&TqT.
u>eTT - C
1. 31 qT+& 45 es d+K >\ |X\ qT+& 9 |X\ yT*eeqT. = |X 5 esT\T.
2. yTTTqT. n~ s+&+{ {
nqT seTT qT+&TqT.
3. yTT\ bsu> esTd qT+&TqT..
4. yTTTqT. n$ 2, 3, 5 eT]jTT 8 e bsu>eTT\+\
eTT\ qT+& n&T>&TqT.
6. 30(m), 30(_), 45(m) eT]jTT 45(_) d+K\ |X\T 2, 3, 5 eT]jTT 8 bsu>eTT >\ eTT qT+& n&T>&TqqT +eTT - D
1. u>eTTq+eTT eT]jTT 10 e bsu>eTT\ qT+& = bsu>eTT qT+& s+&T |X\T
n&T>&TqT. n$ s+&+{ { seTTqqT+&TqT. = |X 10 esT\T.
2. s+&T |X\ yT*eeqT.
3. 46(m), 47(m), 46(_) eT]jTT 47(_) d+K *q |X\ |X, bs|deTT >\ &TqT. $T*q eT&T |X\T nudeTT qT+& n&T>&TqT.
10th Std. Mathematics
341

BLUE PRINT - X Std.
XDT\T
CqeTT ns+#dT=qT nqsTT+#T keTseTT yTT+
VSA SA LA VLA VSA SA LA VLA VSA SA LA VLA VSA SA LA VLA esT\T
1(1) 2(1) 5(1) 2(1) 5(1) 15
2(1) 5(1) 1(1) 1(1) 5(1) 14
;>DeTT 2(1) 5(1) 1(1) 1(1) 2(1) 5(1) 5(1) 21
e\T 4(2) 5(1) 1(1) 10
s|C$T 2(1) 1(1) 2(1) 5(1) 1(1) 5(1) 16
sU>D+ 1(1) 2(1) 5(1) 1(1) 9
D$T 1(1) 2(1) 5(1) 1(1) 2(1) 11
>DqeTT 1(1) 2(1) 5(1) 2(1) 5(1) 15
|j>
10(2) 20
sU>D+
sUeTT\T 10(2) 20
k+U XdeTT 5(1) 2(1) 1(1) 8
d+ue 2(1) 5(1) 1(1) 8
yTT+ 2(2) 10(5) 20(4) 5(5) 16(8) 30(6) 8(8) 6(3) 25(5) 5(1) 40(4) 167
• +&sDeTT>\ d+K |X\ d+KqT *jTCjTTqT.
• eTs= d+K esT\qT *jTCjTTqT.
VSA ` # q yT SA ` q yT
LA ` |< yT VLA ` # |< yT
342 10th Std. Mathematics