o;/L bO' c{ f6] f ;dfgfGt/ lqeh' fsf/ cfsl[ tx¿ / tLgcf6] f D A B
cfotfsf/ cfsl[ tx¿ ldn/] ags] f 7f;] cfsl[ tnfO{ lqeh' fsf/ F C
lkH| d (Triangular Prism) elgG5 . ;u“ s} f] lrqdf ADF / BCE
lqeh' fsf/ ;tx xg' \ eg] ABCD, CDFE / ABEF cfotfsf/ E
;tx xg' \ . To; sf/0f of] lqeh' fsf/ lkH| d xf] .
-v_ lk/fld8 (Pyramid)
h;' kfOkx¿ jf l;Gsfx¿ ko| fu] u//] Pp6f lgoldt k~reh' tof/ kf/ .
To;kl5 a/fa/ gfksf kfr“ cf6] f kfOksf 6j' m| f jf l;Gsfx¿ np] m / Ps
5p] k~reh' sf] zLifl{ aGbd' f / csf{] 5p] x¿nfO{ dfly lrqdf bv] fP em“} Ps}
7fpd“ f kg{] u/L hf8] o;/L s:tf] cfsl[ t tof/ xG' 5 x/] / lbOPsf kZ| gx¿sf]
pTt/ kQf nufpm .
-s_ o;df slt cf6] f lqeh' fsf/ ;tx 5g\ <
-v_ o;sf] cfwf/df cfsl[ t sg' cfsf/sf] 5 <
-u_ cfsl[ tsf] gfd s] xfn] f <
sg' } Pp6f axe' h' cfwf/ ePsf] / cGo ;txx¿ lqeh' fsf/ ePsf] HofldtLo 7f;] cfsl[ tnfO{
lk/fld8 (pyramid) elgG5 . cfwf/sf] axe' hsf] eh' fsf] ;ªV\ ofcg;' f/ o;sf] gfd klg km/s
xG' 5 .
– olb cfwf/ cfotsf/ eP o;nfO{ cfotfsf/ cfwf/ lk/fld8 (rectangular based pyramid) elgG5 .
– obL cfwf/ jufs{ f/ eP o;nfO{ jufw{ f/ lk/fld8 (square based pyramid) elgG5 .
– cfwf/ k~reh' eP k~reh' fwf/ lk/fld8 (pentagonal based pyramid) elgG5 .
– cfwf/ if6e\ h' eP if6e\ h' fwf/ lk/fld8 (hexagonal based pyramid) elgG5 .
pbfx/0f 1 O
lbOPsf] if6e\ h' fwf/ lk/fld8sf cfwf/ / cGo ;txx¿sf] gfd nv] .
;dfwfg
if6e\ h' fwf/ lk/fld8sf] cfwf/ if7e\ h' f ABCDEF xf] / o;sf cGo A B
;txx¿ lqeh' x¿ xG' 5g\ / ltgLx¿ o; ks| f/ 5g\ M C
AOB, ΔBOC, ΔCOD, ΔDOE, ΔFOF / ΔFOA
F
ED
46 xfdf| ] ul0ft, sIff *
cEof; 5.1 -u_
-r_
1. lbOPsf 7f;] cfsl[ tx¿sf] gfd nv] M
-s_ -v_
-3_ -ª_
-5_ -h_ -em_
@= tn lbOPsf lkH| d / lk/fld8sf] cfwf/ / cGo ;txx¿sf] gfd nv] M B
-s_ A -v_ O -u_ A C
D
BC
F
PQ
P
RQ SR E
5.2. 7f;] cfsl[ tsf hfnLx¿ (Nets of solid figures)
Pp6f rssf] a66\ f jf d;Lsf] a66\ f np] m . lrqdf bv] fOP em“} o;nfO{ vfn] .
1122334455667788 ud nufpg]
7fp“
-v_ -u_11223344556677889900112233445566778899001122334455 111111111222222222
11111111112222222222
-s_
lrq -s_ / lrq -u_ df s] km/s 5, nv] .
oxf,“ lrq -s_ 7f;] cfsl[ t -3g_ xf] eg] lrq -u_ lrq -s_ sf] ;dtnLo cfsl[ t xf] .
o;/L sg' } klg 7f;] cfsl[ tnfO{ ;dtnLo cfsl[ tdf ¿kfGt/0f ug{ ;lsG5 . pSt ;dtnLo cfsl[ tnfO{
g} lbOPsf] 7f;] cfsl[ tsf] hfnL (Net) elgG5 .
xfdf| ] ul0ft, sIff * 47
tnsf] tflnsfdf sx] L 7f;] cfsl[ tx¿ / ltgLx¿sf hfnLx¿ lbOPsf 5g\ M
cfsl[ tsf] gfd lrq hfnL (nets)
1122334455
3g (cube) 1234567890123 111111
if6d\ v' f (Cuboid) 111111222222
11112222 111111112222222211223344556677889900112233445566177234567
1111111
66] f« x8] g« (tetrahedron) 111111111122222222223333333333444444444455555555556666666666 1122334455667788990011
111111111122222222223333333333444444444455555555556666666666
;fn] L (Cone) 112233445566778899001122334455667788990011223344
an] gf (Cylinder)
cEof; 5.2
1. tn lbOPsf hfnLx¿ sg' 7f;] cfsl[ tsf xg' ,\ nv] M
-s_ 111111111122222222223333333333444444444455555555556666666666 1122334455667788990011 -v_ 123456 -u_
111111111122222222223333333333444444444455555555556666666666 1111111111111122222222222222 112233445566
1111122222
48 xfdf| ] ul0ft, sIff *
-3_ 1111111111111111111122222222222222222222 -ª_ -r_111111111111111112222222222222221212223344556611111117722222228812345671111111111112222222222223344556677889911223344556677889900
11122233344455566677788811111111119992222222222000
11223344
2. tnsf 7f;] cfsl[ tx¿sf] hfnL 6;]« u/ M
-s_ 3g (Cube) -v_ jn] gf (Cylinder) -u_ if8d\ v' f (Cuboid)
-3_ ;fn] L (Cone) -ª_ lqeh' fsf/ lk/fld8 (Triangular pyramid)
3. afSnf] sfuhsf] ko| fu] u/L 3g, if8d\ v' f, 66] f« x8] g, ;fn] L / an] gfsf hfnLx¿ 6;]« u/ . sr“} Ln]
sf6L 7f;] cfsl[ t agfpm / pSt cfsl[ tsf] lrq sfkLdf ptf/ .
4. ltdLx¿n] cfcfkm\ gf] 3/df jf ;db' fodf ePsf jf bv] s] f jf ko| fu] ub{} cfPsf ljleGg 7f;]
cfsl[ t ePsf j:tx' ¿sf] ;r" L tof/ kf/ .
xfdf| ] ul0ft, sIff * 49
kf7
6 lgbz{] fªs\ x¿ (Co-ordinates)
6.0. kg' /jnfs] g (Review)
;u“ s} f] lrqsf] cWoog u/L lbPsf kZ| gsf] pQ/ bp] m M X' Y B
-s_ XOX' nfO{ s] elgG5 < A X
-v_ YOY' nfO{ s] elgG5 < O
-u_ laGb' O af6 laGb' B df kU' g slt PsfO bfof“ uP/
slt PsfO dfly hfgk' 5{ .
-3_ laGb' O, A, B, C / D sf lgbz{] fªs\ x¿ s] s] xg' \ <
-ª_ laGb' (5,-5) nfO{ nv] flrqdf cªs\ g u/ . D C
o;/L X–cIf, Y–cIf laGbx' ¿sf] lgbz{] fªs\ kQf nufpg] / Y'
laGbx' ¿nfO{ nv] flrqdf cªs\ g ug{] tl/sfaf/] cl3Nnf sIffx¿df cWoog ul/;ss] f 5f“} . ca xfdL
kfOyfuf/] ; ;fWo / bO' { laGbl' arsf] b/' Lsf af/d] f cWoog u5f“{} .
6.1 kfOyfuf/] ; ;fWo / ;fs] f] ko| fu] (Pythagoras Theorem and its Application)
kfOyfuf/] ; ;fWo (Pythagoras Theorem)
;dsf0] f lqeh' ABC lvrf“} h;df Α ;dsf0] f A 10cm
5 . AB= 6cm, BC=8cm / AC= 10cm 5 . 8cm C
6cm
lrqdf bv] fP h:t} tLgcf6] } eh' fx¿df jux{ ¿ B
lvrf“} / kT| os] eh' fdf /xs] f jux{ ¿sf] Ifq] kmn lgsfnf“} .
eh' f AB df ePsf] jus{ f] Ifq] kmn = (AB)2 ju{ PsfO
=(6cm)2 =36cm2
eh' f BC df ePsf] jus{ f] Ifq] kmn = (BC)2 ju{ PsfO
eh' f AC df ePsf] jus{ f] Ifq] kmn = 64cm2 -lsg <_
= (AC)2 ju{ PsfO
=100cm2
ca, (AB)2 + (BC)2 = 36 + 64 = 100cm2 = (AC)2
t;y{ AB2 + BC2 = AC2 xG' 5 .
50 xfdf| ] ul0ft, sIff *
bf;] f| ] tl/sf
km/s km/s eh' fx¿sf] gfk ePsf tLgcf6] f ;dsf0] fL lqeh' x¿ lvr M
-s_ -v_ -u_
ca ?n/ ko| fu] u/L tLgcf6] } lqeh' sf eh' fx¿ gfk / tnsf] tflnsfdf e/ M
lrq g=+ AB AB2 BC BC2 AC AC2 AB2 + BC2 kl/0ffd
-s_
-v_
-u_
dflysf ljm| ofsnfkx¿af6 s] lgisif{ lgsfNg ;lsG5, ;fyLx¿;u“ 5nkmn u/ .
sg' } klg ;dsf0] fL lqeh' df nDa / cfwf/df aGg] jus{ f] Ifq] kmnsf] ofu] kmn A h
b
pSt lqeh' sf] s0fd{ f aGg] jus{ f] Ifq] kmn;u“ a/fa/ xG' 5 . lrqdf p
AB= p, BC=b / AC= h eP p2 + b2 = h2 xG' 5, / p, b / h nfO{ B C
kfOyfufl] /og l6k« N; elgG5 . h:t} M 3, 4, 5 kfOyfufl] /og l6k« N; xg' \ . -s;/L <_
pbfx/0f 1 lbOPsf lrqx¿df x sf] dfg kQf nufpm . 24cm
-s_ 26cm
-v_
3cm
xcm xcm
4cm
;dfwfg
oxf“, p = 3cm b = 4cm ;dfwfg b=x=?
oxf,“ p = 24cm
h=x=?
h = 26 cm
ca, xfdLnfO{ yfxf 5, kfOyfuf/] ; ;fWocg;' f/,
kfOyfuf/] ; ;fWocg;' f/, h2 = p2 + b2 xG' 5 . h2 = p2 + b2
cyjf, x2 = 32 + 42
cyjf, x2 = 9 + 16 cyjf, 262 = 242 + x2
cyjf, x2 = 25
cyjf, x2 = (262 - 242 )
cyjf, x 25 cm = 5 cm
cyjf, x2 = (676 - 576)
∴ x = 5cm
cyjf, x2 = 100
∴ x = 10cm
xfdf| ] ul0ft, sIff * 51
pbfx/0f 2
Pp6f 24 ld6/ cUnf] vDafsf] 6K' kfaf6 pSt vDafnfO{ 6j] f lbgsf nflu 25 ld6/ nfdf] tf/ hldgdf
ufl8Psf] 5 eg] pSt tf/ ufl8Psf] :yfg / vDafsf] kmb] larsf] b/' L slt xfn] f <
;dfwfg
oxf,“ vDafsf] nDafO (p) = 24 ld6/
tf/sf] nDafO (h) = 25 ld6/ 25 ld6/ 24 ld6/
tf/ / vDafsf] larsf] b/' L (b) = ?
xfdLnfO{ yfxf 5, p2+b2 = h2
cyjf, b2 = h2 - p2
cyjf, b2 = (25m)2 - (24m2)
= (625- 576) m2
b2 =49m2
∴ b = 7m
ct M vDafsf] km] b / tf/ ufl8Psf] :yfglarsf] b/' L = 7 ld6/
cEof; 6.1
1. tnsf ;dsf0] fL lqeh' x¿df x sf] dfg kQf nufpm M
-s_ 13cm -v_ x 14cm -u_ 72cm
x 21cm
-3_ 48cm x
12cm x -r_ 8cm
x
-ª_ 15ft x
6 in 8 in 25ft
-5_ -h_ 5cm
15cm 8cm -em_ 24cm
x 11m x 7cm
x
52
16m
xfdf| ] ul0ft, sIff *
2. kfOyfuf/] ; ;fWo ko| fu] u//] tnsf lqeh' x¿ ;dsf0] fL xg' \ jf xfO] gg\ hfr“ /] x/] M
-s_ 4 -v_ 3 -u_ 12
7 10
4
10 11
19
-3_ 3
-ª_ -r_
5 9 15 7
34 12 5
3. olb Pp6f cfotsf] nDafO 8cm / rf8} fO 6cm 5 eg] 6
pSt cfotsf] ljs0fs{ f] nDafO slt xfn] f < 6cm
8cm
4. ;u“ s} f] lrqdf x sf] dfg slt xG' 5, kQf nufpm .
A 12cm
D
6cm x
B 8cm C 6cm
5. Pp6f ;dafx' lqeh' sf] kT| os] eh' fsf] nDafO 6cm 5 eg] pSt
lqeh' sf] zLifs{ f0] fsf cfwf/df lvlrPsf] nDasf] nDafO slt xfn] f < 6cm
A
10cm 10cm 6cm
B 8cm 8cm D
6. lbOPsf] lrqdf ABCD Pp6f rªu\ f xf] .
h;df AC sf] dfg slt xfn] f < 17cm 17cm
C
7. Pp6f lahn' Lsf] vDafaf6 tf/ em//] Ps 5p] n] hldgdf vDafsf] 25 ld6/
km] bbl] v 7 ld6/ 6f9f laGb' A df 5fo] f] . olb vDafsf] 6K' kfbl] v A
hldg;Ddsf] tf/sf] nDafO 25 ld6/ eP vDafsf] prfO 7 ld6/
slt xfn] f, ;fy} csf{] 5p] n] vDafblv 18m k/ hldgsf] B 18 ld6/ 53
laGb' B df 5fo] f] eg] tf/sf] hDdf nDafO slt xfn] f .
xfdf| ] ul0ft, sIff *
8. tnsf tLg ;ªV\ ofx¿ sg' kfOyfufl] /og l6k« N; xg' \ / sg' xfO] gg\ kQf nufpm M
-s_ 3, 4, 5 -v_ 6, 8, 10 -u_ 12, 13, 14
-3_ 7, 24, 25 -ª_ 10, 12, 14 -r_ 10, 12, 15
6.2 bO' { laGbx' ¿larsf] b/' L (Distance Between two Points)
uf| kmkk] /df bO' c{ f6] f laGbx' ¿ A(x1, y1) / B(x2, y2) np] m .
laGb' A af6 OX df nDa lvr / M gfd bp] m .
To;u} /L laGb' B af6 OX df nDa lvrL N gfd bp] m .
kml] /, A af6 BN df nDa lvr / P gfd bp] m . Y
lrqcg;' f/, OM = x1, AM = PN = y1
B(x2,y2)
ON = x2, NB = y2 A(x ,y )
MN = ON – OM = x2 – x1 (= AP) P
BP = BN – PN = y2 – y1 1 1
X
oxf,“ ΔABP ;dsf0] fL lqeh' xf] h;df cfwf/ N
AP (= X2 – X1), nDa BP (= y2 – y1) / s0f{ (AB) 5 .
xfdLnfO{ yfxf 5, kfOyfuf/] ;sf] ;fWocg;' f/, O M
AB2 = (AP)2 + (BP)2
cyjf, AB2 = (x2-x1)2 +(y2- y1)2
cyjf, AB x2 x12 y2 y12 PsfO
To;sf/0f, sg' } bO' { laGbl' arsf] b/' L kQf nufpg] ;q"
d x2 x12 y2 y12
lrqdf sf7] f ug/] xb] f,{ A(2,4) / B(5,6) 5 .
;q" cg;' f/ AB 5 22 6 42 PsfO
32 22 = 13 PsfO
pbfx/0f 1
olb Pp6f jQ[ sf] sG] b| A(4,6) 5 / pSt jQ[ sf] kl/lwsf] laGb' P(10,8) 5 eg] jQ[ sf] cwJ{ of; slt
xfn] f <
54 xfdf| ] ul0ft, sIff *
;dfwfg P(10,8)
oxf,“ A sG] b| ePsf] Pp6f jQ[ 5 h;df sG] b| A(4,6)
A(4,6) 5 / kl/lwsf] laGb' P(10,8) 5 . ctM
x1 = 4, x2 =10, y1 = 6 / y2= 8 5 . AP= ?
ca, AP x2 x12 y2 y12 PsfO
10 42 8 62 62 22 40 2 10 PsfO
pbfx/0f 2
X– cIfsf] 5 PsfOdf laGb' P / Q laGb' Y– cIfsf] 6 PsfOdf eP P bl] v Q ;Ddsf] b/' L slt xfn] f <
;dfwfg
oxf“ P laGb' X- cIfdf 5 PsfO 5 . t;y{ P(5, 0) xf] . km] l/ Q(0,6)
Q laGb' Y- cIfdf 6 PsfO 5 . t;y{ Q (0, 6) xf] .
-lsgls X-cIfdf y = 0 / Y-cIfdf x = 0 xG' 5 ._
ca (x1, y1) = (5,0) / (x2, y2) =(0, 6)
xfdLnfO{ yfxf 5, PQ x2 x12 y2 y12 PsfO O P(5,0)
0 52 6 02 25 36 61
P bl] v Q ;Ddsf] b/' L 61 PsfO 5 .
pbfx/0f 3
lbOPsf lgbz{] fªs\ x¿ A(3,4) ,B(7,8) / c(11,4) ;dlåafx' lqeh' df zLifl{ aGbx' ¿ xg' \ egL kd| fl0ft
u/ .
;dfwfg
oxf“ , A(3,4)= (x1 ,y1) ; B(7,8)= (x2 ,y2) / C(11,4)= (x3 ,y3) dfGbf,
ca, d(AB) x2 x12 y2 y12 7 32 8 42
16 16 32 4 2 PsfO
d(AC) x3 x12 y3 y12
11 42 4 42 72 0 7 PsfO 55
xfdf| ] ul0ft, sIff *
km] l/, 11 72 4 82 42 42 16 16
32 4 2 PsfO
oxf,“ d(AB) =d(BC) 4 2 PsfO
To; sf/0f, ΔABC ;dlåafx' lqeh' xf] .
cEof; 6.2
1. tn lbOPsf laGbx' ¿larsf] b/' L kQf nufpm M
-s_ (4, -7) / (-1, 5) -v_ (-3, 4) / (4, 3) -u_ (1, -2) / (5, -6)
-r_ (-8, 7) / (-3, 4)
-3_ (1, 7) / (1, 1) -ª_ (2, 7) / (4, 9)
-em_ 4 5,3 3 , 3 5,3 3
-5_ (12, -6) / (6, -8) -h_ (-7, -5) / (-9, 2)
2. olb laGb' A n] X-cIfdf -8 df / laGb' B n] Y-cIfdf 6 df sf6s] f] 5 eg] AB sf] b/' L kQf nufpm .
3. lbOPsf] uf| km kk] /df la/fnf] / d;' fsf] l:ylt lbOPsf] 5 . d';f]
la/fnf] / d;' f] ePsf] laGbs' f] lgbz{] fªs\ kQf nufpm / la/fnf]
ltgLx¿larsf] b/' L lgsfn .
4. laGbx' ¿ A(-4,0), B(-4,-4), C(2,-4) / D(2,0) cfoftsf zLif{ laGbx' ¿ xg' \ egL kd| fl0ft u/ .
5. gS;fdf k:| tt' ubf{ jflnª / sf7df8fs“} f lgbz{] fªs\ laGbx' ¿ jm| dzM (4,7) / (7,3) eP Tof]
laGbl' ardf gS;fdf b/' L slt xfn] f, olb 1 PsfO a/fa/ 55km eP jflnªbl] v sf7df8f;“} Ddsf]
jf:tljs b/' L kQf nufpm .
6. laGbx' ¿ P(1,6), Q(4,1) / R(-4,3) ljifdeh' lqeh' sf zLifl{ aGbx' ¿ xg' \ egL kd| fl0ft u/ .
7. olb A(2,-1) ;B,(3,4); C(-2,3) / D(-3,-2) ;dafx' rte' h{' ABCD sf zLif{ laGbx' ¿ xg' \ eg] o;df
ljs0fx{ ¿ AC / BD sf] b/' L kQf nufpm .
8. olb laGb' P(9,12), laGb' Q(1,6) sG] b| ePsf] jQ[ sf] kl/lwdf k5{ eg] pSt jQ[ sf] cwJ{ of; slt
xfn] f < s] laGb' (-7,0) pSt jQ[ sf] kl/lwdf k5{ <
9. pbu\ d laGb' O af6 laGb' A / laGb' B sf] b/' L kQf nufpm, hxf“ A=(-7,7) 5 / B=(7,-7) 5 .
10. olb P(0,6) / Q (a,0) larsf] b/' L 6 PsfO eP a sf] dfg slt xfn] f <
11. tn lbOPsf laGbx' ¿ /v] Lo laGbx' ¿ xg' \ egL kd| fl0ft u/ M
-s_ (4, 3), (3, 2) / (2, 1) -v_ (5, 1), (3, 2) / (1, 3)
-u_ (24, 3), (0, 2) / (-2, -1) -3_ (3, -1), (1, 1) / (-2, 4)
56 xfdf| ] ul0ft, sIff *
kf7
7 Ifq] kmn / cfotg (Area and Volume)
7.0 kg' /jnfs] g (Review)
ju{ / cfotsf] Ifq] kmn (Area of Square and Rectangles)
Pp6f ABCD cfot np] m . A D
h;df nDafO 7cm / rf8} fO 5cm 5 .
ca o; cfotnfO{ 1cm nDafO / 1cm
rf8} fO ePsf ;fgf jux{ ¿df ljefhg u/ .
slt cf6] f ;fgf ju{ aG5g,\ ug / nv] . b
lbOPsf] cfotdf 7f8fl] t/ 5 cf6] f / B C
t;] fl{] t/ 7 cf6] f ;fgf jux{ ¿ aG5g\ .
/ 35 ;fgf ju{ aG5g\ . o;/L pSt l
cfoftsf] Ifq] kmn 35 ju{ ;l] d
eof] / oxf“ nDafO 7 cm / rf8} fO 5 cm 5 . To; sf/0f 7 cm × 5 cm = 35 cm2 xG' 5 .
cfotsf] Ifq] kmn (A) = nDafO × rf8} fO ju{ PsfO xG' 5 .
A = l × b ju{ PsfO
k]ml/, xfdLnfO{ yfxf 5, ;a} e'hfx¿ a/fa/ ePsf] cfot g} ju{ xf] . ju{df l
nDafO = rf8} fO xG' 5 . To; sf/0f, jus{ f] Ifq] kmn = -nDafO × nDafO_ ju{ PsfO
(A) = l × l = l2 ju{ PsfO eof] . l
7.1 rte' h{' / lqeh' sf] Ifq] kmn (Area of Quadrilaterals and Triangles)
(I) ;dfgfGt/ rte' h{' sf] Ifq] kmn (Area of a Parallelogram) AE B A'
D C
afSnf] sfuhsf] kGgfdf Pp6f ;dfgfGt/ rte' h{' ABCD
lvr / laGb' D af6 AB df nDa lvr . To;kl5 pSt ;=r=nfO{
sr“} Ln] sf6 . km] l/ pSt ;dfgfGt/ rte' h' s{ f] DE af6 sf6L
ΔADE / rte' h' { BCDE nfO{ 56' o\ fpm .
xfdf| ] ul0ft, sIff * 57
lrqdf bv] fP em“} ΔADE nfO{ ;=r=sf] csfk{] 6l\ 6 hf8] . E B A'
cfot EA'CD tof/ eof] . h;sf] Ifq] kmn ;dfgfGt/
rte' h{' ABCD ;u“ a/fa/ xG' 5 .
ca, ;dfgfGt/ rte' h{' ABCD sf] Ifq] kmn CD
= cfot CDAE sf] Ifq] kmn
= CD × CE ju{ PsfO
= cfwf/ × prfO ju{ PsfO
olb cfwf/ (base)= b / prfO (height) = h eP, ;=r= sf] Ifq] kmn A= b × h xG' 5 .
(II) lqeh' sf] Ifq] kmn (Area of Triangle)
sf8a{ f8] s{ f] ko| fu] u//] Pp6f ΔABC agfpm . o;df cfwf/ BC (b) / prfO (h) 5 . lrq
-s_ df bv] fP h:t} zLifl{ aGb' A nfO{ cfwf/df kg{] u/L k6o\ fpm / k6o\ fOPsf] To; 7fpa“ f6 sf6 .
ΔAPQ aG5 . ca, ΔAPQ sf] cfwf/ PQ df zLifl{ aGb' A af6 nDa lvr / To; nDaaf6 sf6 . lrq
-u_ df bv] fP h:t} rte' h{' PQCB sf bO' t{ km{ hf8] . cfot MNCB tof/ eof] .
AA A
PQ PQ MP QN
B C B CB C
ca, ΔABC sf] Ifq] kmn = cfot MNCB sf] Ifq] kmn = MB x BC ju{ PsfO
= 1 h x b = 1 b x h ju{ PsfO MB 1 prfO
2 2 2
∴ lqeh' sf] Ifq] kmn (A) = 1 b x h ju{ PsfO
2
(III) ;dafx' lqeh' sf] Ifq] kmn (Area of Equilateral Triangle)
;u“ s} f] lrq ;dafx' lqeh' xf] . o;df eh' fsf] a A
nDafO 'a' 5 . A af6 BC df nDa AD B a
lvrf“+} h;n] cfwf/ BC nfO{ cfwf u5{ .
kfOyfuf/] ; ;fWocg;' f/, AD2 = AC2 -CD2 D C
a
cyjf, AD AC2 CD2
2
58 xfdf| ] ul0ft, sIff *
xfdf| ] ul0ft, sIff * 59
60 xfdf| ] ul0ft, sIff *
xfdLnfO{ yfxf 5, lqeh' sf] Ifq] kmn (A) =
= 1 9 3.6cm2
2
9 1.8cm2 = 16.2 cm2
-v_ oxf“ ;dnDa rte' h{' ABCE df, b1 = AB = 4cm
b2 = CE = 6cm
h = AD = 3cm
;dnDa rte' h{' ABCE sf] Ifq] kmn (A) = ?
xfdLnfO{ yfxf 5, ;dnDa rte' h{' sf] Ifq] kmn (A)= 1 hb1 b2
2
= 1 3 4 6 cm2
2
= 1 3 10 cm2 = 15 cm2
2
-u_ oxf“ ljs0f{ (FH) = 11cm
EK= p1 = 7cm
GL = p2 = 5cm
Ifq] kmn A 1 ljs0f{p1 p2
2
pbfx/0f 2
;u“ s} f] lrqdf 5fof kf/s] f] efusf] Ifq] kmn kQf nufpm .
;dfwfg
oxf,“ ;dfgfGt/ rte' h{' ABCD 5 A B
cfwf/ (CD)= 9cm / P 5.5cm Q
prfO (h) = 7cm 5
ABCD sf] Ifq] kmn (A1) = b × h 3cm 7cm
S 4.5cm R
= 9 × 7 cm2 D 9cm C
= 63cm2
xfdf| ] ul0ft, sIff * 61
km] l/, ;dnDa rte' h{' PQRS 5 h;df
prfO (h) = 3cm
PQ = b1 = 5.5 cm
RS = b2 = 4.5 cm
PQRS sf] Ifq] kmn (A2) =
ca, 5fof kfl/Psf] efusf] Ifq] kmn = A1- A2
= 63 cm2- 15cm2
= 48cm2
pbfx/0f 3
Pp6f 240 ft nDafO ePsf] / 180 ft rf8} fO ePsf] cfoftsf/ 16ft 180ft
vt] sf] lardf Pp6f 16 ft lsgf/f ePsf] Pp6f jufs{ f/ kfv] /L
5 eg] kfv] /Lafxs] sf] vt] sf] Ifq] kmn slt xfn] f <
;dfwfg 240ft
cfoftsf/ vt] sf] nDafO (l) =240 ft
cfoftsf/ vt] sf] rf8} fO (b) =180 ft
cfoftsf/ vt] sf] Ifq] kmn l × b = 240 ×180 ju{ lkm6
= 43200 ju{ lkm6
km] l/, jufs{ f/ kfv] /Lsf] Ifq] kmn = l2 = 162 ju{ lkm6 = 256 ju{ lkm6
ca, kfv] /Lafxs] sf] vt] sf] Ifq] kmn (A) = vt] sf] Ifq] kmn – kfv] /Lsf] Ifq] kmn
= (43200- 256) ju{ lkm6
= 42944 ju{ lkm6
cEof; 7.1
1. tnsf HofldtLo cfsl[ tx¿sf] Ifq] kmn lgsfn M
-s_ -v_ 1.4ft -u_
3.6cm 4cm
5cm 6cm
62 xfdf| ] ul0ft, sIff *
-3_ -ª_ -r_ 6in
-em_ 7.5in
4m 6cm
9m -h_ 9cm
-5_ 4.5cm
6cm 9cm 3.6cm 5.5cm
7.5cm
13cm
-`_ -6_ -7_
4.5cm
12cm
8cm 6.5cm
6.4cm
8.8cm
8cm 2.2cm
2. tn lbOPsf lrqx¿df 5fof kfl/Psf] efusf] Ifq] mkmn lgsfn M
SP111111111111111111111111112222222222222222222222222233333333333333333333333333D444444444444444444444444445555555555555555555555555566666666666666666666666666A777777777777777777777777778888888888888888888888888839999999999999999999999999900000000000000000000000000.1111111111111111111111111152222222222222222222222222261m33333333333333333333333333m44444444444444444444444444055555555555555555555555555m6666666666666666666666666677777777777777777777777777888888888888888888888888889999999999999999999999999900000000000000000000000000C111111111111111111111111112222222222222222222222222233333333333333333333333333B444444444444444444444444445555555555555555555555555566666666666666666666666666Q7777777777777777777777777710m E11111111111111111112222222222222222222D33333333333333333334444444444444444444555555555555555555566666666666666666667777777777777777777A888888888888888888899999999999999999990000000000000000000811111111111111111112222222222222222222.233333333333333333334444444444444444444c5555555555555555555m666666666666666666677777777777777777778888888888888888888999999999999999999940000000000000000000.1111111111111111111822222222222222222223333333333333333333c4444444444444444444m555555555555555555566666666666666666667777777777777777777888888888888888888899999999999999999990000000000000000000111111111111111111122222222222222222221111111111111111111FB222222222222222222233333333333333333333444444444444444444455555555555555555554.66666666666666666665Cccmm 12cm
R
E 111111111111111111122222222222222222223333333333333333333H444444444444444444455555555555555555556666666666666666666777777777777777777788888888888888888889999999999999999999000000000000000000061111111111111111111c11111111111111111112222222222222222222J2222222222222222222m333333333333333333344444444444444444445555555555555555555666666666666666666677777777777777777778888888888888888888G99999999999999999990000000000000000000111111111111111111122222222222222222223333333333333333333 F
7cm
3. tnsf lrqx¿df x sf] dfg kQf nufpm M SM N
AP
7cm x R x
B DC
QT PO
x 9.2cm A = 44cm2
A = 56cm2
A = 92cm2
xfdf| ] ul0ft, sIff * 63
4. Pp6f 120 ld6/ nDafO / 110 ld6/ rf8} fO ePsf] cfoftsf/ aur“} fsf] lardf 18 ld6/ nfdf]
/ 9 ld6/ rf8} fO ePsf] elnan sf6] { agfOPsf] 5 . elnan sf6] a{ fxs] aur“} fsf] Ifq] kmn slt
xfn] f <
5. 9 lkm6 nfdf] / 7 lkm6 rf8} f ePsf] kvfn{ df slt cf6] f 1 ju{ lkm6sf af8] x{ ¿ gvK6fOs{ g agfpg
;lsPnf <
6. ;u“ s} f] lrqdf Pp6f ufps“ f ljleGg dxŒjk0" f{ 7fpx“ ¿ lbOPsf] 5 . lrq x/] L lgDglnlvt
7fp“ x¿sf] Ifq] kmn kQf nufpm M 218m
-s_ ;fjh{ lgs zfr} fnosf] Ifq] kmn kfgL 6o\ fªs\ L ljBfno 25m
-v_ kfv] /Lsf] Ifq] kmn
2m
-u_ ljBfnosf] Ifq] kmn 6m 4m
-3_ rft} f/Lsf] Ifq] kmn
4m 4m zzzffjf}r}r{hffnlngoos rft} f/L
-ª_ kfgL 6o\ fªs\ Lsf] Ifq] kmn 8m
11m
kf]v/L 6m
7.2. 3g / if8d\ v' fsf] cfotg (Volume of Cube and Cuboids)
ljm| ofsnfk 1 cm
1 cm
1. 1 cm nDafO, 1cm rf8} fO / 1 cm prfO ePsf] 3gsf] cfotg 1 cm
slt xG' 5 <
2. Pp6f 5 cm nDafO / 4 cm rf8} fO ePsf] cfoftsf/ afs;sf]
cfwf/df gvK6fOsg sltcf6] f 1 cm3 sf a66\ fx¿ /fVg ;lsPnf <
3. o:t} ltg cf6] f txx¿;u“ } Ps dfly csf{] ub{} /fVbf 5 cm 4 cm
slt cf6] f PsfO 3gx¿ c6fpnfg\ / s:tf] 7f;] 3 cm
cfsl[ t aG5 <
o;df hDdf 60 cf6] f PsfO 3gx¿ xG' 5g\ .
t;y{ 60 = 5 x 4 x 3 xG' 5 hxf“ if8d\ v' fsf] nDafO 5cm,
rf8} fO 4 cm / prfO 3cm 5 .
To; sf/0f, if8d\ v' fsf] cfotg = l x b x h 3g PsfO xG' 5 . 4 cm
64 5 cm
xfdf| ] ul0ft, sIff *
xfdf| ] ul0ft, sIff * 65
xfdLnfO{ yfxf 5,
cfotg (V) = l x b x h wg PsfO
= (9 x 5 x 3) cm3 = 135 cm3
-v_ ;dfwfg
oxf“ lbOPsf] 7f;] sf] cfotg egs] f] if8d\ v' fsf/ efux¿sf] cfotgsf] ofu] kmn xf] .
∴ V = (6 x 4 x 3 + 6 x 3 x 3) 3g PsfO
= (72+54) cm3
= 126 cm3
pbfx/0f 3
Pp6f 10 m nDafO, 9 m rf8} fO / 8 m prfO ePsf] 6o\ fªs\ Ldf slt k6] f« n] c6fpnf, kQf nufpm .
;dfwfg
oxf,“ 6o\ fªs\ Lsf] nDafO (l) = 10 m
6o\ fªs\ Lsf] rf8} fO (b) = 9 m
6o\ fªs\ Lsf] prfO (h) = 8 m
cfotg (V) = l x b x h 3g PsfO
= (10 x 9 x 8) m3
= 720 m3
To;sf/0f, pSt 6o\ fªs\ Ldf 720 m3 k6] f« n] c6fp5“ .
cEof; 7.2
1. tnsf gfk ePsf if8d\ v' fx¿sf] cfotg kQf nufpm M
nDafO rf}8fO prfO
-s_ 10 cm
-v_ 4 cm 6 cm 5 cm
-u_ 50 cm 2 cm
-3_ 6 cm 40 cm 3 cm
-ª_ 20 cm 2 cm
10 cm 30 cm
5 cm
2
3.15 cm
66 xfdf| ] ul0ft, sIff *
xfdf| ] ul0ft, sIff * 67
kf7
8 :yfgfGt/0f (Transformation)
8.0 kg' /jnfs] g (Review)
tn lbOPsf lrqx¿ x/] / kT| os] lrqsf af/d] f ;fyLx¿;u“ 5nkmn u/L lgisifd{ f ku' L, nv] M
m B'
C' B
BQ
AP A'
A
75°
CR C
-s_ -v_ -u_
8.1 k/fjtg{ (Reflection) Y
A' (3,4)
dflysf] klxnf] lrqdf s] kfof} < o;df
Pp6f emG8fsf] k|ltlaDa /]vf m ;“u OX
bv] fOPsf] 5 . o;nfO{ m ;u“ emG8fsf] A' (3,-4)
k/fjtg{ elgG5 . of] xfdLn] sIff 7 df
kl9;ss] f 5f“} . ca xfdL lgbz{] fªs\ af6 Y'
k/fjtg{ sf af/d] f hfgsf/L lnG5f“} M
-s_ X- cIfaf6 k/fjtg{
lrqdf laGb' A np] m . A nfO{ XOX' X'
af6 k/fjtg{ u/ / pSt laGbn' fO{
A' gfd bp] m . o;df X- cIfaf6 A
;Ddsf] b/' L / X-cIfaf6 A' ;Ddsf]
b/' L a/fa/ xG' 5 . ca uf| kmdf laGb'
A sf] lgb]{zfª\s ug]/ n]v .
To;u} /L A' sf] lgbz{] fªs\ slt xG' 5
oxf“ A sf] lgbz{] fªs\ (3,4) 5 / A'
sf] lgbz{] fªs\ (3,-4) 5 .
68 xfdf| ] ul0ft, sIff *
pbfx/0f 1
laGb' (-4,5) nfO{ X- cIfaf6 k/fjtg{ u/L kl| tlaDa laGbs' f] lgbz{] fªs\ nv] .
;dfwfg Y
oxf,“ A(x,y) = A(-4,5) A'
x = –4, y = 5
ca, X- cIfaf6 k/fjtg{ ubf,{ X' O X
(-4,5) sf] kl| tlaDa (-4,-5) eof] .
A
(x', y') = (-4,-5) Y'
-v_ Y- cIfaf6 k/fjtg{ (Refelction by Y-axis)
;u“ s} f] lrqdf laGb' B nfO{ YOY' af6 k/fjtg{ B Y
u//] x/] . o;sf] kl| tljDa YOY' /v] faf6 laGb' B'
B sf] a/fa/ b/' Ldf k5{ . o;nfO{ B' gfd bp] m .
ca B / B' sf] lgbz{] fªs\ ug/] slt slt 5 x/] < X' X
ltgLx¿ jm| dzM (-6,5) / (6,5) xG' 5g\ . O
t;y{ (-6,5) nfO{ Y -cIfaf6 k/fjtg{ ubf{ (6,5) xG' 5 . Y'
pbfx/0f 2
laGb' (-5,-7) nfO{ uf| km kk] /df cªs\ g u/L Y- cIfaf6 k/fjtg{ u/fO{ uf| km kk] /df bv] fpm .
;dfwfg
oxf“ Q(-5,-7) lbPsf] 5 . Y
hxf“ x= -5, / y= -7 5 .
ca, uf| km kk] /df Q(-5,-7) nfO{ Y- cIfaf6
k/fjtg{ ubf,{
(-5,-7) sf] kl| tlaDa (5,-7) eof] . X' O X
, Q'(x', y') = (5,-7) xG' 5 .
xfdf| ] ul0ft, sIff * Q Q'
Y'
69
pbfx/0f 3
A(2,2), B(4,6) / C(6,3) Pp6f lqeh' sf zLifl{ aGbx' ¿ xg' \ . ΔABC nfO{ nv] flrqdf k:| tt' u/L pSt
lqeh' nfO{ Y-cIf;u“ k/fjtg{ u/ / kl| tlaDa lqeh' sf zLifl{ aGbx' ¿sf lgbz{] fªs\ x¿ nv] .
;dfwfg B' Y B
oxf,“ nv] flrqdf xb] f{ ΔABC sf] kl| tlaDa
ΔA'B'C' xf] .
ca, laGb' A(2,2), B(4,6) / C(6,3) nfO{ C' A' C
Y-cIf;u“ k/fjtg{ ubf,{ X' A X
A' (-2,2) O
B' (-4,6) Y'
C' (-6,3)
cfjZos kl| tlaDa ΔA'B'C' xf,] h;df zLifl{ aGbx' ¿sf] lgbz{] fªs\ x¿ A'(-2,2), B'(-4,6) / C'(-6,3) 5g\ .
cEof; 8.1
1. lbOPsf HofldtLo cfsl[ tx¿nfO{ lbOPsf] cIf m ;u“ k/fjtg{ u/L kl| tlaDa lrq lvr M
-s_ -v_ -u_
m
mm
-3_ -ª_ m -r_
m
m
2. n]vflrqsf] k|of]u u/L lbOPsf lgb]{zfª\sx¿nfO{ X- cIf;“u k/fjt{g u/L k|ltlaDasf]
lgbz{] fªs\ nv] M
-s_ (1,2) -v_ (-2,3) -u_ (4,-5) -3_ (-6,6) -ª_ (-5,-4)
-r_ (-2,5) -5_ (9,-8) -h_ (-3,-9) -em_ (-10,12) -`_ (7,8)
3. kZ| g g= 2 sf laGbx' ¿nfO{ Y- cIfaf6 k/fjtg{ u/L uf| km kk] /df nv] .
4. laGb' P(5,-6) nfO{ Y-cIf af6 k/fjtg{ u/ / P' sf] lgbz{] fªs\ kQf nufpm . /v] f PP' sf]
b/' L kQf nufpm .
70 xfdf| ] ul0ft, sIff *
5. P(4,3), Q(7,3) / R(4,-3) Pp6f ;dsf0] f lqeh' sf zLifl{ aGbx' ¿ xg' \ . pSt lqeh' nfO{ nv] flrqdf
cªs\ g u/L X- cIf;u“ k/fjtg{ u/ .
6. A(2,-2), B(2,3) C(5,3) / D(5,-2) Pp6f cfotsf zLifl{ aGbx' ¿ xg' \ . ca pSt cfotnfO{
nv] flrqdf cªs\ g u/L Y-cIfaf6 k/fjtg{ u/L nv] flrqdf k:| tt' u/ .
7. A(-2,3), B(-5,2) / C(-4,5) nfO{ nv] flrqdf cªs\ g u/L klxn] X-cIfaf6 k/fjtg{ u/L ΔA'B'C'
kQf nufpm . km] l/ ΔA'B'C' nfO{ Y- cIfaf6 k/fjtg{ u/L clGtd kl| tlaDanfO{ nv] flrqdf
k:| tt' u/ .
8.2 kl/jm| d0f (Rotation)
sg' } laGb' jf lrqnfO{ sg' } laGba' f6 lbOPsf] lbzfdf / lbOPsf] sf0] fdf :yfgfGt/0f u/fpgn' fO{
kl/jm| d0f (Rotation) elgG5 . kl/jm| d0fsf nflu lgDglnlvt ltg cj:yfx¿ cfjZos 5g\ M
• kl/jm| d0fsf] sG] b| ( Center of Rotation)
• kl/jm| d0fsf] sf0] f (Angle of Rotation)
• kl/jm| d0fsf] lbzf (Direction of Rotation)
38Lsf] ;O' sf] lbzfnfO{ kl/jm| d0fsf] C0ffTds (Negative) lbzf / 38Lsf] ;O' s{ f] ljk/Lt
lbzfnfO{ kl/jm| d0fsf] wgfTds (Positive) lbzf elgG5 .
-s_ pbu\ d laGb' O(0,0) af6 +90° df kl/jm| d0f
O pbu\ d laGb' xf] . XOX' / YOY' bO' { cIfx¿ xg' \ . P'(x',y') Y P(x,y)
P(x,y) Pp6f laGb' xf] . ca P laGbn' fO{ O af6 90° X' O (0,0) X
df 3d' fpm / P'(x',y') df k¥' ofpm, hxf“ ∠POP' =
90° xG' 5 / OP=OP' xG' 5 .
nv] flrqaf6 laGb' P / P' sf] lgbz{] fªs\ x/] f“} .
P(x,y)=(4,6) 5 / P'(x',y')=(-6,4) 5 .
Y'
xfdf| ] ul0ft, sIff * 71
To;u} /L nv] flrqaf6 xb] f{ pSt laGbn' fO{ X' Y P(4,6)
38Lsf] ;O' s{ f] lbzfdf jf C0ffTds lbzfdf
kl/jm| d0f u/fpb“ f P(4,6) af6 P'(6,-4) eof] . (0,0) X
To;nfO{ lgDgfg;' f/ bv] fpg ;lsG5 . O
P(4,6) +90° P1(-6,4) P'(6,-4)
P(4,6) –90° P1(6,-4) Y'
Y
A(x,y)
-v_ pbu\ d laGb' O(0,0) af6 180° df kl/jm| d0f
uf| km kk] /df laGb' (x, y) np] m . laGb' A nfO{ pbu\ d (0,0) X
laGb' O(0,0) af6 wgfTds lbzfdf 180° df X' O
kl/jm| d0f u/fpm / laGb' A'(x',y') gfdn] hgfpm . hxf“
0A=OA' 5 . uf| kmdf laGb' A / A' sf] lgbz{] fªs\ ugL
nv] . hxf“ A(x,y)= (7,4) 5 / A(x',y')= (-7,-4) 5 .
A'(x',y')
Y'
To:t} (7,4) nfO{ 180° af6 C0ffTds lbzfdf kl/jm| d0f u/fpb“ f (-7,-4) g} xG' 5 .
pbfx/0f 1
zLifl{ aGbx' ¿ A(6,6), B(4,5) / C(6,2) ePsf] Pp6f lqeh' nfO{ nv] flrqdf k:| tt' u/ . . pSt lqeh' nfO{
pbu\ d laGb' (0,0) af6 -s_ 90° wgfTds lbzfdf / -v_ 180° df kl/jm| d0f u/ .
;dfwfg
-s_ oxf,“ ΔABC sf zLifl{ aGbx' ¿ jm| dzM (6,6), A' C' Y A
(4,5) / (6,2) 5g\ . B'
B
ca laGbx' ¿ A, B / C nfO{ jm| dzM 90° df X'
kl/j|md0f u/fp“bf aGg] k|ltljDa lqe'hnfO{ C
nv] flrqdf 5fof kf/L bv] fOPsf] 5 . OX
hxf“,
72 xfdf| ] ul0ft, sIff *
-v_ ΔABC nfO{ nv] flrqaf6 180° df kl/jm| d0f YA
u/fpb“ f aGg] ΔA'B'C' nfO{ nv] flrqdf 5fof B
kf/L bv] fOPsf] 5 h;df,
O C
X' X
C'
B'
A'
Y'
cEof; 8.2
1. tnsf lrqx¿nfO{ nv] flrqdf ;f//] 56' 6\ f56' 6\ } laGb' O(0,0) af6 3gfTds lbzfdf 90° / 180°
df kl/jm| d0f u//] bv] fpm M
-s_ Y A -v_ YA
X' O X X' O X
Y' Y'
YA
-u_ -3_
YA
X' O X X' O X
Y' Y'
xfdf| ] ul0ft, sIff * 73
2. kZ| g 1 sf lrqx¿nfO{ C0ffTds lbzfdf 90° / 180° df kl/jm| d0f u/L nv] flrqdf k:| tt' u/ .
3. tnsf laGbx' ¿nfO{ nv] flrqdf cªs\ g u/L +90°, -90° / 180° df kl/jm| d0f u/L nv] flrqdf
56' 6\ f 56' 6\ } k:| tt' u/ .
-s_ (-4,7) -v_ (4,-7) -u_ (5,9) -3_ (3,0) -ª_ (-4,-8)
-r_ (2,-5) -5_ (10,-10) -h_ (0,6) -em_ (0,0) -`_ (-9,-9)
4. A(0,0), B(3,0), C(3,3) / D(0,3) zLifl{ aGbx' ¿ ePsf] Pp6f jun{ fO{ nv] flrqdf lvrL To;nfO{
pbu\ d laGb' O(0,0) af6 -s_ +90° / -v_ -90° df kl/jm| d0f u/fO{ nv] flrqdf k:| tt' u/ .
5. tn lbOPsf zLifl{ aGbx' ¿af6 aGg] HofldtLo cfsl[ tnfO{ nv] flrqdf cªl\ st u/L pbu\ d laGb'
O(0,0) af6 -i_ 90° / -ii_ -90° df kl/jm| d0f u/fO{ 56' 6\ f56' 6\ } nv] flrqdf k:| tt' u/ .
-s_ (2,7), (3,3), / (6,7)
-v_ (3,2), (-2,2), (6,5) / (1,5) YA
-u_ (10,6), / (12,6)
6. ;u“ s} f] lrqnfO{ pbu\ d laGb' (0,0) af6 X' X
-90°, +90° / 180° df kl/jm| d0f u/fO{
56' 6\ f56' 6\ } nv] flrqdf k:| tt' u/ .
O
Y'
74 xfdf| ] ul0ft, sIff *
8.3 lj:yfkg (Displacement)
6a] n' df Pp6f lstfa /fv . To;nfO{ l3;f//] sx] L afof“ A B A' B'
;f/ . klxns] f] lstfasf] :yfg ABCD lyof] / clxnsf] D C'
:yfg A'B'C'D' eof] . lstfasf] :yfg lglZrt lbzfdf →
kl/jt{g eof] . o;nfO{ pSt lstfasf] lj:yfkg
elgG5 . ca AA', BB', CC' / DD' hf8] / gfk / C D'
ltgLx¿sf] ;DaGw s:tf] kfof} nv] . oxf,“ AA', BB',
CC' / DD' a/fa/ / k/:k/ ;dfgfGt/ 5g\ .
sg' } klg laGb' jf j:tn' fO{ lbOPsf] lbzfdf lglZrt b/' Ldf ;fg{' jf :yfgfGt/0f ugn{' fO{ lj:yfkg
(translation) elgG5 . lj:yfkgsf nflu lj:yfkgsf] kl/df0f / lbzf pNnv] ug{' cfjZos 5 .
sg' } lgbz{] fªs\ nfO{ bfof“ lj:yfkg ubf{ ±, afof“ lj:yfkg ubf{ —, dfly lj:yfkg ubf{ ± / tn
lj:yfkg ubf{ — nl] vG5 .
pbfx/0f 1 Y A'
C'
;u“ s} f] lrqdf ΔABC lbOPsf] 5 / pSt lqeh' nfO{ B'
5 PsfO bfof“ / 4 PsfO dfly lj:yfkg u/L cfsl[ t A
A'B'C' k¥' ofOPsf] 5 . ca ΔABC / ΔA'B'C'
sf zLif{ laGbx' ¿sf lgbz{] fªs\ x¿ x/] f“} .
BC
X
oxf,“ lj:yfkg cufl8 / lj:yfkg k5fl8sf] x / y lgbz{] fªs\ x/] f“} .
tLgcf6] } zLifl{ aGbx' ¿df x- sf] dfgdf lj:yfkgkl5 5 ylkPsf] 5 . To:t,} y sf] dfgdf klg
lj:yfkgkl5 4 ylkPsf] 5 . lj:yfkgkl5 kl| tlaDa lqeh' nfO{ 5fof kf/L bv] fOPsf] 5 .
xfdf| ] ul0ft, sIff * 75
pbfx/0f 2 A'(-5,6) D'(-1,6) Y
A(-2,2), B(-3,-2), C(3,-2) / D(2,2) Pp6f rte' h{' sf B'(-6,2) A(-2,2) C'(0,2)
zLifl{ aGbx' ¿ xg' \ . pSt rte' h{' nfO{ nv] flrqdf k:| tt' D(2,2)
u/L 3 PsfO afo“ f / 4 PsfO dfly lj:yfkg u/L X'
nv] flrqdf k:| tt' u/ . O X
;dfwfg B(-3,-2) C(3,-2)
oxf,“ A(-2,2), B(-3,-2) , C(3,-2) / D(2,2) 5 . o;nfO{ Y'
;u“ s} f] nv] flrqdf bv] fOPsf] 5 . o;df kl| tlaDa
rt'e'{hsf zLif{laGb'sf lgb]{zfª\sx¿ A(-5,6),
B(-6,2), C(0,2) / D(-1,6) 5g\ .
cEof; 8.3
1. tn lbOPsf cfsl[ tx¿nfO{ lbPsf] lbzf / kl/df0fdf lj:yfkg u/ M
-s_ -v_ -u_
2cm 2cm 2cm
2. laGb' (4,-5) nfO{ nvf] lrqdf cªs\ g u/L 3 PsfO bfof“ / 4 PsfO dfly lj:yfkg u/L k:| tt'
u/ .
3 tnsf lgbz{] fªs\ x¿nfO{ nv] flrqdf cªs\ g u/L 3 PsfO afof“ / 3 PsfO dfly lj:yfkg
u/L nv] flrqdf k:| tt' u/ M
-s_ (4,9) -v_ (-3,6) -u_ (-2,2) -3_ (-5,5)
-ª_ (2,-3) -r_ (4,-7) -5_ (-4,8) -h_ (-5,-6)
4. P(-1,-3) / Q(4,5) nfO{ 2 PsfO bfof“ / 3 PsfO dfly lj:yfkg u/L nv] flrqdf k:| tt' u/ .
5. zLifl{ aGbx' ¿ A(1,0), B(4,5) / C(7,-2) ePsf] ΔABC nfO{ nv] flrqdf cªs\ g u/L 3 PsfO bfof“
/ 5 PsfO tn lj:yfkg u/L nv] flrqdf k:| tt' u/ .
6. laGbx' ¿ (4,6), (7,5), (5,1) / (2,2) nfO{ nv] flrqdf lvrL aGg] cfsl[ tnfO{ 4 PsfO afof“ / 5
PsfO tn lj:yfkg u/L nv] flrqdf k:| tt' u/ .
7. A(4,1) nfO{ nv] flrqdf cªs\ g u/L 5 PsfO bfof“ / 4 PsfO dfly lj:yfkg u/ . km] l/ pSt
kl| tlaDa laGbn' fO{ 2 PsfO bfof“ / 5 PsfO tn lj:yfkg u/L nv] flrqdf k:| tt' u/ .
8. laGb' (-3,5) nfO{ slt PsfOdf lj:yfkg ubf{ (4,5) aG5, nv] flrqdf bv] fpm .
76 xfdf| ] ul0ft, sIff *
9kf7 lbzfl:ylt / :sn] 8O« ª
(Bearing and Scale Drawing)
9.0. kg' /jnfs] g (Review)
tn lbOPsf] gS;fdf kfv] /fnfO{ sG] b| dfgL lgDglnlvt :yfgx¿ hf8] M
hD' nf, ;v' t{] , sf7df8f,“} Onfd, lj/f6gu/, jL/uGh / dxG] bg| u/
p=
9.1. lbzfl:ylt (Bearing)
lof;u“ s} f] lrq x/] /] lbOPsf kZ| gx¿sf pQ/x¿sf af/d] f 5nkmn u/f“} M
N nfO{ cfwf/ /v] f dfg/] dfkg ubf,{ 000º
-s_ NE sf] dfkg slt xG' 5 < N
-v_ E sf] dfkg slt 5 <
315º NW NE 045º
E 090º
-u_ SE sf] dfkg slt 5 . 270º W 0
-3_ S sf] dfkg slt xfn] f < ESGE 135º
-ª_ SW sf] dfkg slt 5 < SW S 77
-r_ ;a} sf0] fx¿ sg' lbzfdf lnOPsf] 5 < 225º 180º
-5_ ;as} f] dfkgnfO{ slt cªs\ sf ¿kdf k:| tt' ul/Psf] 5 <
xfdf| ] ul0ft, sIff *
dflysf] 5nkmnaf6 lgDglnlvt tLg s/' fx¿ yfxf kfpg ;lsG5 M
-s_ N nfO{ cfwf/ /v] f dfGg] -v_ 38Lsf] ;O' s{ f] lbzfdf gfKg]
-u_ dfkg tLg cªs\ df k:| tt' ug{]
t;y,{ pQ/ lbzf hgfpg] /v] fnfO{ cfwf/ /v] f dfg/] 38Lsf] ;O' s{ f] lbzfdf sg' } bO' { :yfglarsf]
b/' LnfO{ dfkg u/L tLg cªs\ sf] sf0] fsf ¿kdf k:| tt' ug{] tl/sfnfO{ lbzfl:ylt (Bearing)
elgG5 . csf{] zAbdf o;nfO{ sDkf; lbzfl:ylt (Compass Bearing) klg elgG5 . pbfx/0fsf
nflu dfly lbOPsf] lrqdf O af6 E sf] lbzfl:ylt 090º 5 .
pbfx/0f 1
:yfg P af6 :yfg Q sf] lbzfl:ylt 075º 5 eg] :yfg Q af6 P sf] lbzfl:ylt slt xfn] f <
;dfwfg N1
oxf,“ Q sf] lbzfl:ylt = ∠NPQ = 075º N
∠NPQ + ∠PQN1 = 180º [∴ PN || QN ]
1
or, 075º + ∠PQN1 = 180º
or, ∠PQN1 = 180º – 075° = 105º 075° Q
P
∴ Q af6 P sf] lbzfl:ylt = 360º – ∠PQN1
= 360º – 105º = 255º
pbfx/0f 2
olb kfv] /fsf] dxG] bu| k' mfaf6 s=] cfO={ l;x+ kn' sf] lbzfl:ylt 155º 5 eg] s=] cfO={ l;x+ kn' af6 dxG] bu| k' mfsf]
lbzfl:ylt slt xfn] f <
;dfwfg N
dfgf,“} dxG] bu| k' mf = A A 155º
s=] cfO={ l;x+ kn' = B
kZ| gfg;' f/,
:yfg B sf] lbzfl:ylt =∠NAB = 155º
∠NAB + ∠ABN1 = 180º [∴ AN || BN1] N2
cyjf, 155º + ∠ABN1 = 180º B
cyjf, ∠ABN1 = 180º - 155º = 25º xfdf| ] ul0ft, sIff *
ca, s=] cfO={ l;x+ kn' (B) af6 dxG] bu| k' mf (A) sf] lbzfl:ylt = ?
ca, B af6 A sf] lbzfl:ylt
= 360º - ∠ABN1
= 360º - 25º = 325º
78
cEof; 9.1
1. tn lbOPsf lrqx¿df :yfg P af6 :yfg B sf] lbzfl:ylt pNnv] u/ M
-s_ -v_ -u_ -3_
B B
P
P B PB P
2. pQ/ hgfpg] /v] fnfO{ cfwf/ dfg/] lbOPsf lbzfl:yltnfO{ sf0] fdf k:| tt' u/ .
N
-s_pQ/ – klZrd (NW) NW NE
-v_ blIf0f kj" { (SE) WNW 0 ENE
-u_klZrd – pQ/ – klZrd (WNW) W E
-3_ kj" { – pQ/ – kj" { (ENE) SW SE
S
3. lbOPsf lrqx¿df :yfg X af6 :yfg Y sf] lbzfl:ylt lbOPsf] 5 eg] :yfg Y af6 :yfg X
sf] lbzfl:ylt kQf nufpm .
-s_ -v_ -u_
4. lbOPsf lbzfl:yltnfO{ lrq agfP/ bv] fpm M
-s_ laGb' A af6 Pp6f hxfhsf] lbzfl:ylt 120º 5 .
-v_ ufps“ f] kw“ /] fa] f6 dlGb/sf] lbzfl:ylt 280º 5 .
-u_ Pp6f 8f8“ fsf] 6K' kfaf6 uf7] sf] lbzfl:ylt 075º 5 .
5. Pp6f ufps“ f] dlGb/af6 ljBfnosf] lbzfl:ylt 062º eP pSt ljBfnoaf6 dlGb/sf] lbzfl:ylt
slt xfn] f, lrqåf/f bv] fpm .
6. Pp6f vx/] vfn] f 120º sf] lbzfl:yltdf alu/xs] f] lyof] . kmf6“ df ku' k] l5 jifft{ s\ f] en] ;u“ } pSt
vfn] f 200º sf] lbzfl:yltdf aUg yfNof] eg] pSt vfn] fn] slt l8uL| sf] sf0] fdf lbzf kl/jtg{
u¥of] xfn] f <
xfdf| ] ul0ft, sIff * 79
7. ;u“ s} f] lrqdf ljleGg :yfgsf laGbx' ¿ lbOPsf 5g\ . N M
lgDglnlvt laGbx' ¿sf] lbzfl:ylt kQf nufpgx' f;] \ M A B
-s_ laGb' A af6 M
-v_ laGb' A af6 B X C
-u_ laGb' A af6 C
-3_ laGb' A af6 X
-ª_ laGb' A af6 A
9.2. :sn] 8O« ª (Scale Drawing)
ltdf| ] 3/af6 ljBfno;Ddsf] b/' LnfO{ sfkLdf /v] f lvr/] bv] fpg ;S5f} ls ;Sbg} f,} To;nfO{
sfkLdf bv] fpg ;lsb“ g} lsgls 3/af6 ljBfno;Ddsf] b/' L sfkLsf] nDafOeGbf w/] } 5 .
t;y,{ sg' } bO' { 7fpx“ ¿larsf] b/' L h;nfO{ ld6/ (m), lsnfl] d6/ (km) jf dfOn (mile) df gflkG5,
To;nfO{ gS;fdf bv] fpg ;Dej xb“' g} . pSt b/' LnfO{ plrt b/' Ldf ¿kfGt/0f u/L jf:tljs
b/' LnfO{ gS;fdf k:| tt' ug{ ;Dej kfg{] dfkgsf] tl/sfnfO{ g} :sn] 8O« ª (scale drawing)
elgG5 . pbfx/0fsf nflu sf7df8fb“} l] v kfv] /f;Ddsf] jf:tljs b/' L 200km 5 . olb :sn]
1cm = 100 km eP sf7df8fb“} l] v kfv] /fsf] b/' LnfO{ gS;fdf 2cm df bv] fpg ;lsG5 . 200 km
= 2cm df nl] vGG5 .
[:sn] M 1 cm = 100 km jf 1:100]
pbfx/0f 1
1 cm = 600 m sf] :sn] ko| fu] u/L Pp6f gS;f tof/ ubf{ bO' { :yfglarsf] gS;fsf] b/' L 6cm eP
pSt bO' { :yfgx¿larsf] jf:tljs b/' L slt xfn] f <
;dfwfg
oxf“, :sn] 1 cm = 600m jf:tljs b/' L
:sn] 6 cm = (6 × 600) m = 3600 m
To;n} ] bO' { 7fpl“ arsf] jf:tljs b/' L = 3600 m
80 xfdf| ] ul0ft, sIff *
pbfx/0f 2
Pp6f hxfh ;?' sf] :yfgaf6 045º lbzfl:yltdf 600 dfOn (miles) / 135º lbzfl:yltdf 800 dfOn
(miles) p8fg ub5{ eg] pSt hxfhsf] ;?' sf] :yfg / clGtd :yfglarsf] jf:tljs b/' L slt xfn] f, ;fy}
clGtd :yfgaf6 ;?' sf] :yfgsf] lbzfl:ylt kQf nufpm . :sn] 1 cm = 200 mile
;dfwfg N N1 N2
dfgf,“} hxfhsf] ;?' sf] :yfg = P 135° R
hxfhsf] clGtd :yfg = R Q
045°
∠NPQ = 045º 3cm 4cm 4.5cm
∠N1QR = 135º P
5cm
lrqcg;' f/, PR = 5cm
To;n} ,] ;?' sf] :yfgaf6 clGtd :yfglarsf] jf:tljs b/' L = 5 × 200 = 1000 mile
km] l/, lrqdf kf| 6] S] 6/ ko| fu] u/L gfKbf ∠PRN2 = 079º
clGtd :yfgaf6 ;?' sf] :yfgsf] lbzfl:ylt = 360º - ∠PRN1
= 360º - 079º = 281º
pbfx/0f 3
/dz] kT| os] lbg 060º sf] lbzfl:yltdf 120 ld6/ / 150º sf] lbzfl:yltdf 180 ld6/ lx8“ g\ ] ub5{ .
1 cm = 40 m sf] :sn] ko| fu] u//] p;n] ;?' u/s] f] 7fp“ / clGtd 7fp;“ Ddsf] jf:tljs b/' L kQf
nufpm . ;fy} clGtd :yfgaf6 ;?' :yfgsf] lbzfl:ylt slt xfn] f < N N1
N2
;dfwfg
lrqdf ?n/ ko| fu] u//] gfKbf,
;?' sf] :yfg = A, clGtd :yfg = C dfGbf 150°
;?' :yfgbl] v clGtd :yfgsf] gS;fsf] b/' L = 5.2 cm B
1 cm = 40 ld6/ jf:tljs b/' L 060°
3cm
5.2 cm = (40 × 5.2) ld6/ jf:tljs b/' L
= 208 ld6/
A
5.2cm
km] l/, lrqdf kf| 6] S] 6/ ko| fu] u/L gfKbf ∠ACN2 = 65º C
clGtd :yfg (C) af6 ;?' :yfg (A) sf] lbzfl:ylt = 360º - 065º
= 295º
xfdf| ] ul0ft, sIff * 81
cEof; 9.2
1. tnsf kZ| gx¿df bO' { :yfglarsf] jf:tljs b/' L kQf nufpm M
-s_ bO' { :yfglarsf] gS;fsf] b/' L = 7 cm, [:sn] 1 cm = 750 m]
-v_ bO' { :yfglarsf] gS;fsf] b/' L = 6.5 cm [:sn] 1 cm = 1000 miles]
-u_ dxG] b| uk' mf / rd/] f] uk' mflarsf] gS;fsf] b/' L = 3 cm [:sn] 1 cm = 250 m]
2. sf7df8fs“} f] rjm| kysf] nDafO 27km 5 . olb :sn] 1cm = 12km eP pSt rjm| kysf] gS;fsf]
nDafO slt xfn] f <
3. dfwjn] :yfg A af6 030º lbzfl:yltdf 5km lx8“ k] l5 :yfg B df kU' 5 . To;kl5 B af6 140º
sf] lbzfl:yltdf 3 km lx8“ k] l5 :yfg C df kU' 5 / cGTodf l;wf C af6 A df kmsG{ 5 eg,]
-s_ plrt :sn] 5fgL :sn] 8O« ª u/ .
-v_ :yfg C af6 :yfg A ;Ddsf] :sn] b/' L slt xfn] f <
-u_ :yfg C af6 :yfg A ;Dd l;wf kmlsb“{ f p;n] jf:tljs b/' L slt kf/ u5{ <
-3_ :yfg C af6 A :yfgsf] lbzfl:ylt kQf nufpm .
4. Pp6f ;x/sf] a;kfsa{ f6 500 ld6/ blIf0fdf Pp6f dlGb/ kb5{ / kf8} Lkfv] /L dlGb/af6 065º
lbzfl:yltdf k5{ . a;kfsa{ f6 kf8} Lkfv] /L 145º lbzfl:yltdf k5{ eg] kf8} Lkfv] /L / dlGb/larsf]
jf:tljs b/' L slt xfn] f < 1 cm = 100 m sf] :sn] ko| fu] u/L bv] fpm .
5. :yfg B af6 :yfg A 400 ld6/ klZrddf k5{ . :yfg A af6 :yfg C sf] lbzfl:ylt 050º
5 / :yfg B af6 C sf] lbzfl:ylt 290º 5 eg,]
-s_1 cm = 40 m :sn] lnO{ :sn] 8O« ª u/ .
-v_ :yfg B / :yfg C larsf] jf:tljs b/' L slt xfn] f <
-u_ :yfg C sf cfwf/df :yfg B sf] lbzfl:ylt kQf nufpm .
6. lrqdf Pp6f ufps“ f] dV' o 7fpx“ ¿ bv] fOPsf] 5 . xn' fs sfofn{ o ljBfno
olb :sn] 1 cm = 150 ld6/ eP ?n/ ko| fu] dlGb/
u/L xn' fs sfofn{ oaf6 lgDglnlvt 7fpx“ ¿sf]
jf:tljs b/' L kQf nufpm M
-s_ dlGb/ -v_ kfgLsf] dx' fg
-u_ :jf:Yo rfs} L -3_ ljBfno kfgLsf] :jf:Yo rfs} L
-ª_ xf6ahf/ d'xfg xf6ahf/
82 xfdf| ] ul0ft, sIff *
kf7
10 ;dx" (Sets)
10.0. kg' /jnfs] g (Review)
tnsf ;dx" x¿sf] cWoog u/L lbOPsf kZ| gx¿sf pQ/x¿sf] vfh] L u/ M
U = { 1 bl] v 20 ;Dddf ;ªV\ ofx¿ }
A= { 20 eGbf ;fgf 3 sf ckjTox{ ¿ }, / B= { 20 eGbf ;fgf 4 sf ckjTox{ ¿} eP
-s_ ;dx" x¿ U, A, / B sf ;b:ox¿ ;r" Ls/0f u/ .
-v_ ;dx" x¿ U, A, / B nfO{ eg] lrqdf k:| tt' u/ .
-u_ eg] lrqsf] ko| fu] u/L tnsf ;dx" x¿ kQf nufpm M
(i) A∪B (ii) A∩B
-3_ A / B ;dx" U sf s:tf ;dx" x¿ xg' \ <
-ª_ ;dx" A df kg{] t/ ;dx" B df gkg{] ;b:ox¿sf] ;dx" lgdf0{ f u/ .
-r_ ;dx" U df kg{] t/ A ∪ B df gkg{] ;b:ox¿sf] ;dx" kQf nufpm .
dflysf kZ| gx¿sf af/d] f xfdLn] sIff 7 df g} kl9;ss] f 5f“} . ca xfdL ;dx" x¿sf] km/s
/ ;dx" sf k/" ssf af/d] f cWoog ub5{ f“} .
10.1. ;dx" x¿sf] km/s ( Difference of Sets)
tnsf eg] lrqx¿ cWoog u/ / s] s] bl] vG5, ;an} ] cfcfkm\ gf sfkLdf nv] M
U U U
AB Aa B e 4
bd
l;tf cfOt 8f]Ndf cf B 1 A
xl/ Zofd 5 2 6
3
;a} eg] lrqdf ;dx" B afxs] A sf] efu dfq 5fof kfl/Psf] 5 . ;dx" A sf] dfq efudf 5fof
kfl/Psf] 5 . B sf] sg' } klg efudf 5fof kfl/Psf] 5g} / 5fof kf/s] f] efun] ;dx" A df kg{] t/
B df gkg{] ;b:ox¿sf] ;dx" nfO{ hgfp5“ . of] g} ;dx" A af6 ;dx" B sf] km/s xf] .
xfdf| ] ul0ft, sIff * 83
olb ;dx" A / ;dx" B ;jJ{ ofks ;dx" U sf pk;dx" x¿ xg' \ eg] ;dx" A df kg{] t/ ;dx" B df
gkg{] ;b:ox¿sf] ;dx" nfO{ jf ;dx" A df dfq kg{] ;b:ox¿sf] ;dx" nfO{ A km/s B elgG5 /
o;nfO{ A-B n] hgfOG5 . A-B = {x:x ∈ A / x ∉ B}
dflysf] klxnf] eg] lrqdf A = { ;Ltf, xl/, cfOt } 5 / B= {cfOt, 8fN] df, Zofd} 5 .
A df dfq kg{] ;b:ox¿ A–B = { ;Ltf, xl/, cfOt } – {cfOt, 8fN] df, Zofd}
= { ;Ltf, xl/ } eof] .
To:t} bf;] f| df A -B= {a, b, c} / t;] f| df A–B= {4, 5, 6} eof] -s;/L <_
pbfx/0f 1
olb, U = {cfbz{ df= lj=sf ;Dk0" f{ ljBfyLx{ ¿sf] ;dx" }
A = { cfbz{ df=lj=sf sIff 8 sf ljBfyLx{ ¿sf] ;dx" } /
B = { cfbz{ df=lj=sf ;Dk0" f{ 5fqfx¿sf] ;dx" } eP A-B / B-A kQf nufpm / eg] lrqdf
5fof kf//] bv] fpm .
;dfwfg A U
oxf,“ U= {cfbz{ df= lj=sf ;Dk0" f{ ljBfyLx{ ¿sf] ;dx" } A–B B
A = { cfbz{ df=lj=sf sIff 8 sf ljBfyLx{ ¿sf] ;dx" }
B = { cfbz{ df=lj=sf ;Dk0" f{ 5fqfx¿sf] ;dx" }
ca, A–B = { x: cfbz{ df=lj=sf sIff 8 sf ljBfyLx{ ¿ xg' \ t/ 5fqf xfO] gg\ )}
A–B ={ cfbz{ df=lj=sf sIff 8 sf 5fqx¿sf] ;dx" } U B
A
km/] L, B–A = {x: cfbz{ df=lj=sf 5fqf t/ sIff 8 sf xfO] g} B–A
= { cfbz{ df=lj=sf sIff 8 sfafxs] sf 5fqfx¿sf] ;dx" }
gf6] M 1. A-B ≠ B-A
2. A– B ∪ B – A ePdf o;nfO{ ;dldtLo km/s (Symmetrical Difference) elgG5 .
pbfx/0f 2
olb U = {a, b, c, d, e, f, g, h, i, o, u} , A= {a, b, c, d, e}, B = {a, e, i,o, u} /
C = { d, e, f, i, } eP tnsf ;dx" x¿ kQf nufpm / eg] lrqdf k:| tt' u/ M
-s_ A-B -v_ B-C -u_ A∪(B-C)
-3_ U-(A∪B) -ª_ (A∪B) - (A ∩B) -r_ (A-B) ∪ (B-A)
84 xfdf| ] ul0ft, sIff *
;dfwfg Ab a BU
oxf,“ U = { a, b, c, d, e, f, g, h, i, o, u} , A = { a, b, c, d, e } c e
d i
B = { a, e, i,o, u } / C = { d, e, f, i } o
-s_ A– B = {x:x ∈ A / x ∉ B} u
= {a, b, c, d, e} - {a, e, i, o, u} A-B
= {b,c, d}
BU
-v_ B-C = {x: x ∈ B / x ∉ C} a
= { a, e, i,o, u } - { d, e, f, i, } o e d
= {a, o, u} u f
C
-u_ A∪(B-C) = {x∈A cyjf x ∈ B-C}
B-C
= {a, b, c, d, e} ∪ {a, o, u}
= {a, b, c, d, e,o,u } A Bb- C a U
c e oB
d u
C
AU
B
-3_ U-(A∪B) = {x ∈U / x∉A∪B} C
oxf,“ A∪B = { a, b, c, d, e} ∪ { a, b, c, d, e, i,o, u } A U-(A∪B)
= { a, b, c, d, e, i, o, u} xG' 5 . BU
= { a, b, c, d, e, f, g, h, i, o, u} - { a, b, c, d, e, i, o, u }
= {f,g, h}
-ª_ A ∩ B = { a, b, c, d, e} ∩ {a, e, i, o, u} = {a, e} 5fof kfl/Psf] efu
(A∪B) - (A ∩ B)]
t;y,{ (A∪B) - (A ∩ B) = {a, b, c, d, e, i, o, u} - {a, e}
1111111111111111222222222222222233333333333333334444444444444444A55555555555555556666666666666666777777777777777788888888888888881111111111111111999999999999999922222222222222220000000000000000333333333333333311111111111111114444444444444444222222222222222255555555555555553333333333333333666666666666666644444444444444447777777777777777555555555555555588888888888888886666666666666666999999999999999900000000000000001111111111111111B222222222222222233333333333333334444444444444444555555555555555566666666666666667777777777777777 U
= {b, c, d, i, o, u} (A−B) ∪ (B − A)
-r_ oxf,“ B-A = { x ∈ B / x ∉ A} 85
= {a, e, i, o, u } - {a, b, c, d, e }
= {i, o, u }
/ -s_ af6 xfdLnfO{ yfxf 5 A-B = { b, c, d}
ca, (A-B) ∪(B-A) = { b, c, d} ∪ {i, o, u }
= {b, c, d, i, o, u}
-ª_ / -r_ af6 s] kfof} < nv] .
xfdf| ] ul0ft, sIff *
cEof; 10.1
1. olb A = {2, 4, 6, 8,10} / B = { 4, 8, 10, 12, 14} eP tnsf ;dx" x¿ kQf nufpm M
-s_ A∪B -v_ A∩B -u_ A-B
-3_ B-A -ª_ (A-B)∪(B-A)
2. kZ| g g= 1 sf ;dx" x¿nfO{ 56' 6\ f56' 6\ } eg] lrqdf k:| tt' u/ .
3. olb P = {20 eGbf ;fgf hf/] k0" f{ ;ªV\ ofx¿sf] ;dx" } / Q={20 eGbf ;fgf 4 sf ckjTox{ ¿sf]
;dx" } eP P-Q / Q-P nfO{ eg] lrqdf k:| tt' u/ .
4. lbOPsf] lrqdf tnsf ;dx" x¿nfO{ 56' 6\ f56' 6\ } 5fof kf//] bv] fpm .
-s_ A-B -v_ B-C A BU
-u_ C-A -3_ (A∪B)-C
-ª_ A-(B∪C) -r_ (B∩C)-(C∩A) C
5. olb U={1,2,3,4,5,6,7,8,9}, A={2,4,6,8}, B={3,5,7,9} / C={3,6,9} eP
-s_ (A∪B)-C -v_ (B∪C)-A -u_ (C∪A)-B kQf nufO{ eg] lrqdf k:| tt' u/ .
6. tnsf eg] lrqx¿sf] 5fof kf/s] f] efusf] gfd ;ªs\ t] df nv] M
P QU A BU
AB U
n
C
7. olb A = {a, e, i, o, u } , B = {i, o, u,w} / C = {e, i, o} eP (A-B)-C / A-(B-C) nfO{
56' 6\ f56' 6\ } eg] lrqdf k:| tt' u/ .
8. U = { ltdf| ] ljBfnodf sIff 8 sf ;Dk0" f{ ljBfyLx{ ¿sf] ;dx" }
A = { sIff 8 sf skbL{ vN] g dg k/fpgs] f] ;dx" }
B = { sIff 8 sf 8G8Llaof] vN] g dg k/fpgs] f] ;dx" }
C = { sIff 8 sf km' 6an vN] g dg k/fpgs] f] ;dx" }
olb A, B / C ;a} kl| tR5l] bt ;dx" x¿ eP A, B, C sf] ;DaGwnfO{ egl] rqdf k:| tt' u/ .
9. kZ| g 8 sf cfwf/df tnsf ;dx" x¿nfO{ 5fof kf//] bv] fpm M xfdf| ] ul0ft, sIff *
-s_ skbL{ dg k/fpg] ljBfyLx{ ¿sf] ;dx"
-v_ skbL{ / 8G8Llaof] dg k/fpg] ljBfyLs{ f] ;dx"
-u_ skbL,{ 8G8Llaof] jf km' 6an vN] g dg k/fpgs] f] ;dx"
-3_ tLg cf6] } vn] vN] g dg k/fpgs] f] ;dx"
86
U
A
1A 11 U
3 2 4 13
6 8 15
17
5 10 12 19
7 14 16
18
9A
xfdf| ] ul0ft, sIff * 87
88 xfdf| ] ul0ft, sIff *
xfdf| ] ul0ft, sIff * 89
10.3 eg] lrqsf] ko| fu] ( Use of Venn Diagrams)
ul0ft1 John Venn n] la;f“} ztfAbLdf ;dx" sf ljm| ofx¿nfO{ lrqåf/f k:| tt' u/s] f lyP . pg}
ul0ft1sf] gfdaf6 eg] lrq (Venn- Diagram) elgPsf] xf] . ;dx" sf zflAbs ;d:ofx¿nfO{
eg] lrqsf] ko| fu] u/L ;dfwfg ug{ ;lsG5 .
tnsf] pbfx/0f cWoog u/f“} M
u08sL c~rnsf lhNnfx¿sf] ;dx" = { sf:sL, :ofªh\ f, tgx,'“ ndh'ª, uf/] vf, dgfª }
wfn} flul/ c~rnsf lhNnfx¿sf] ;dx" = {jfUnª' , DofUbL, kjt{ , d:' tfª}
u08sL c~rnsf lhNnfx¿sf] ;dx" nfO{ G dfGbf, G = { sf:sL, :ofªh\ f, tgx,“' ndhª' , uf/] vf, dgfª} .
u08sL c~rnsf 6 cf]6f lhNnfx¿ ;d"x G sf ;b:ox¿ 5g\ . o;nfO{ G ;d"xsf]
u0fgfTdstf elgG5 . To:t,} wfn} flul/ c~rnsf lhNnfx¿sf] ;dx" D sf] u0fgfTdstf 4 eof] .
sg' } klg ;dx" df ePsf] hDdf ;b:ox¿sf] ;ªV\ ofnfO{ pSt ;dx" sf] u0fgfTdstf -cardinality_ elgG5 .
oxf“ M dflysf] ;dx" D sf] u0fgfTdstf 4 5 . o;nfO{ n(D) = 4 nl] vG5 .
olb, U ={ ;fs{ /fi6x« ¿sf] ;dx" } A BU11111111111111222222222222223333333333333344444444444444555555555555556666666666666677777777777777888888888888889999999999999900000000000000111111111111112222222222222233333333333333111111111111112222222222222233333333333333444444444444445555555555555566666666666666777777777777778888888888888899999999999999000000000000001111111111111122222222222222
A ={ gk] fn, ef/t, kfls:tfg }
B = { e6' fg, aªu\ fnfbz] , >Lnªs\ f, dflNbE;, ckmuflg:yfg } eP,
A∪B = { gk] fn, ef/t, kfls:tfg, e6' fg, aªu\ fnfbz] , >Lnªs\ f, ckmuflg:yfg } xG' 5 . n(A∪B) slt xG' 5 <
oxf,“ A∪B df hDdf 8 cf6] f ;b:ox¿ 5g\ . To;sf/0f n(A∪B) = 8 eof] .
kml] /, n(A) = 3 / n(B) = 5 5 . n(A) + n(B) = 3 + 5 = 8 eof] .
n(A∪B)=n(A) + n(B) eof] .
bO' c{ f6] f ;dx" x¿ cnlUuPsf 5g\ eg] n(A∪B)=n(A) + n(B) xG' 5 . A B U
To:t,} lbOPsf] eg] lrqdf x/] . 1 1111111111111122222222222222333333333333335444444444444444555555555555556666666666666677777777777777 6
2 7
A = {1, 2, 3, 4, 5}; B = { 4, 5, 6, 7 } n(A) = 5, n(B) = 4 5 . 3
km] l/, A∪B = {1, 2, 3, 4, 5,6,7} / A∩B = {4, 5} 5 .
n(A∪B) = 7 / n(A∩B) = 2
To;sf/0f, eg] lrqsf] ko| fu] ubf{ n(A) + n(B) = 9 eof] t/ n(A∪B) = 7 5 .
hg' n(A) + n(B) eGbf 2 jf n(A∩B) n] sd 5 .
To;sf/0f, n(A∪B) = n(A) + n(B) - n(A∩B) eof] .
bO' c{ f6] f ;dxÒ A / B kl| tR5l] bt ;dxÒ x¿ eP n(A∪B) = n(A) + n(B) - n(A∩B) xG' 5 . ;fy,}
;dxÒ A df dfq kg{] ;b:ox¿sf] ;ªV\ of 3 5 . t;y,{ n0(A) = 3 5 .
n0(A) = n(A) - n(A∩B)
;dxÒ B df dfq kg{] ;b:ox¿sf] ;ªV\ of 2 5 . t;y,{ n (B) = 2 5 .
0
n0(B) = n(B) - n(A∩B) xfdf| ] ul0ft, sIff *
90
xfdf| ] ul0ft, sIff * 91
92 xfdf| ] ul0ft, sIff *
kf7
11 k0" f{ ;ªV\ ofx¿ (Whole Numbers)
11.0 kg' /jnfs] g (Review)
‘ul0ft1 Kroncker sf cg;' f/ eujfgn\ ] kf| sl[ ts ;ªV\ ofx¿ dfq ;l[ i6 u/s] f xg' \ / afs“ L ;a}
;ªV\ ofx¿ dflg;n] g} kl| tkfbg u/s] f xg' \ .’ ul0ftdf ;ªV\ ofx¿sf] ;?' jft uGtLsf ;ªV\ ofx¿ 1,
2,3, 4, .....af6 ;'?cft ePsf] xf] . b'O{cf]6f uGtLsf ;ª\Vofx¿ hf]8\bf uGtLsf] ;ª\Vof g}
aG5, h:t} M 3+3 = 6 xG' 5 . t/ 3 - 3 slt xG' 5 < o;nfO{ hgfpgsf nflu yk ;ªV\ ofsf] cfjZostf
dx;;' ul/of] / kf| sl[ ts ;ªV\ ofx¿df zG" o (0) yk eof] . o;/L k0" f{ ;ªV\ ofsf] ;dx" sf] ljsf;
eof] . o;nfO{ (W) n] hgfOG5 . W = { 0,1, 2,3, 4, .......} xG' 5 . ca, tnsf] ljm| ofsnfk u/f“} .
27 cf6] f l;Gsfx¿ np] m . 11111111111111111222222222222222221111333333333333333332222114444441144111111111111111222222212122222222223322233333333333333311111111111111111122222221212222222222111111133333333333113333331111112222222221111122223233223333332222313133333111111111111111322222222221212112222233332323233323333333311111111111144144411441422221221221222122222323332322333333333331111111111111111122222222222221212223333333332233333331111111111111111112222222222222222221111111133333333333333331111111111212222212222222222222233333333333133311311131132222222211111111111333333322222232222233333333 111111111222221211222111111111111333131232233222322222222222311111111113333333213333333122222322223313233132311111132111111222241442442141142423222333321232132232111111313331333132222222222233333333
o;nfO{ 10 3ftsf] ;dx" df ljefhg u/ .
2 x 10 + 7 = 2 x 101 + 7 x 100
o;nfO{ bzdnj ;ªV\ ofªs\ g k4lt (decimal numeration system) elgG5 .
kml] /, 27 nfO{ 5 3ftsf] ;dx" df ljefhg u/ . 11111111212122222233333332321111111112222222221111113333311333211222222222333333331111111112222222223333333311111111122222222233333333 111111111111111111222222222222222222113131333333333333311111133221111222222212111111122223222222333333322111111111111133111133333332221212212222122222223333333333333332222331111111111111111111222222222222222222233333333333333333111111111111111111122222222222222222223333333333333333311111111122222222211113333311133133222221222233333333111111112222122221333133331111311222222221133333333221414444444111111222222122233333333 111111112222222112332323333311111111212222222333333332
111111111212222222223333333311111111144444444222212212223333333223311111111112222222222111111333313333113222222222133333333321111111112222222211222333333333
5 x 5 + 2 = 1 x 52+ 2 x 50
= 1 x 52+ 0 X51 + 2 x 50
o;nfO{ k~rcfwf/ ;ªV\ ofªs\ g k4lt (quinary numeration system) elgG5 .
cGTodf, 27 nfO{ 2 3ftsf] ;dx" df ljefhg u/ .
2 sf] 3ftfªs\ sf] ;dx" df 11111111221222222111111113333312333222122222233333333234444444441111111122222212233333332311111111122212222223333333332 1111111121222222211233333333111111112222222222333333333444444444 11111111222122222233333333111111111222222222333333333 111111111111111111222222222222222222111111113133333133333333333221211212211212111122222222223323232333333333333111111111111111112222222122112122222222232323333333333333331111111111111111122222222222222222133333333333333332 1111111112222222223333313311111113222222122233333333 11111111122222222233333333
ljefhg ubf,{ 1111111122222212223333333311111111222221222323333333 111111112221212222333333223311111111122222222233333333 1111111112222222221313111331311333222222212333332333 1111111121222222232333333311111111122222222233333333
16 + 8 +2 + 1
= 1 X 24+ 1 X 23 + 1X 21 + 1 X 20
=1 X 24+ 1 X 23 + 0 X 22 + 1X 21 + 1 X 20
o;nfO{ låcfwf/ ;ªV\ ofªs\ g k4lt (binary numeration system) elgG5 .
xfdf| ] ul0ft, sIff * 93
11.1 bzdnj ;ªV\ ofªs\ g k4lt (Decimal Numeration System)
;u“ s} f] tflnsfdf 1256 nfO{ :yfgdfg tflnsfdf bv] fPsf] 5 .
xhf/ ;o b; Ps
103 102 101 100
1 256
1256 nfO{ lj:tfl/t ¿kdf nV] bf
1256 = 1 x 1000 + 2 x 100 + 5 x 10 + 6 x 1
= 1 x 103 + 2 x 102 + 5 x 101 + 6 x 100
sg' } klg ;ªV\ ofnfO{ 10 sf] 3ftsf ¿kdf nl] vG5 / :yfgdfg tflnsfdf Ps, b;, ;o, xhf/,
===== jf 100 ,101, 102, 103, 104, ……. xG' 5 eg] To:tf] ;ªV\ ofªs\ g k4ltnfO{ bzdnj ;ªV\ ofªs\ g
k4lt (decimal numeration system) elgG5 . o;df 0,1,2,3,4,5,6,7,8 / 9 u/L 10 cf6] f
cªs\ x¿ ko| fu] ul/G5 .
pbfx/0f 1
35731 nfO{ lj:tfl/t ¿kdf nv] M
10
;dfwfg
oxf“, 35731 = 3 x 10000 + 5 x1000 + 7 x 100 + 3 x 10 + 1 x 1
10 = 3 x 104 + 5 x 103 +7 x102 + 3 x 101 + 1 x 100
tnsf] pbfx/0f x/] f“} M
179 nfO{ 5 sf] 3ftsf] ;dx" df ljefhg ubf,{
179 = 125 + 50 + 4
= 125 + 2 × 25 + 4
= 5 × 5 × 5 + 2 × 5 × 5 + 4 ×1
= 53 + 2 × 52 + 4 × 50
= 1 × 53 + 2 × 52 + 0 × 51 + 4 × 50
oxf,“ 179 nfO{ cfwf/ 5 sf] 3ftsf ¿kdf 1204 JoSt ul/Psf] 5 . o;nfO{ k~rcfwf/ k4lt elgG5 .
bzdnj ;ªV\ ofªs\ g k4ltdf 0,1,2,3,4,5,6,7,8 / 9 u/L 10 cf6] f cªs\ x¿ ko| fu] u/] h:t} sg' }
klg ;ªV\ ofnfO{ 0,1, 2, 3 / 4 sf] dfq ko| fu] u/L 5 sf] 3ftsf ¿kdf nl] vG5 . :yfgdfg tflnsfdf
Ps, kfr“ , klRr;, Ps ;o klRr;, 5 ;o klRr;, ==== jf 50, 51, 52, 53, 54, …. xG' 5 eg] To:tf]
;ªV\ ofªs\ g k4ltnfO{ k~rfwf/ ;ªV\ ofªs\ g k4lt (quinery numeration system) elgG5 .
dflysf] pbfx/0fnfO{ 12045 nl] vG5 .
94 xfdf| ] ul0ft, sIff *
pbfx/0f 2
33 nfO{ k~rcfwf/ ;ªV\ ofdf nv] .
33 nfO{ 5 sf] 3ftsf] ;dx" df ljefhg ubf,{
;dfwfg
oxf“ 33 = 25 + 5+ 3
=1 × 52 + 1 × 51 + 3 × 50 nl] vG5 .
= 1135 xG' 5 .
11.3 bzdnj k4ltnfO{ k~rcfwf/df ¿kfGt/
tl/sf,
– lbOPsf] ;ªV\ ofnfO{ 5 n] efu ub{} hfg]
– zi] fnfO{ bfofl“ t/ nV] g]
– bfofl“ t/ nl] vPsf zi] fsf cªs\ x¿nfO{ tnaf6 dfly jm| ddf ldnfpg] / nV] g]
tnsf] pbfx/0f x/] f“} M
pbfx/0f 3
tnsf ;ªV\ ofnfO{ k~rcfwf/ k4ltdf ¿kfGt/ u/ .
-s_ 512
;dfwfg -v_ 7521
;dfwfg
5 512 zi] f ( 5 n] efu ubf_{
5 7521 z]if
5 102 2
5 20 2 5 1504 1
54 0 5 300 4
04 5 60 0
ca zi] fnfO{ tnaf6 dflylt/ ldnfP/ 5 12 0
/fVbf xG' 5 . t;y,{ xG' 5 . 52 2
4022 512 =4022 02
10 5
:yfgdfg tflnsfdf bv] fpb“ f, t;y,{ 752110 = 2200415 xG' 5 .
:yfgdfg tflnsfdf bv] fpb“ f,
53 52 51 50
40 2 2
55 54 53 52 51 50
22 0 0 4 1
xfdf| ] ul0ft, sIff * 95
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