Com. Mathematics (VII)

ljm| ofsnfk 4

cª\u]h| L j0fd{ fnfsf sg' sg' cIf/nfO{ laGb' ;dldltsf cfwf/df l7s laraf6
(180° sf] sf0] fdf_ 3d' fpFbf cfsl[ t sG] b| laGb'b]lv a/fa/ b/' Ldf t/ ljk/Lt lbzfdf
xG' 5 <

sg' } klg cfsl[ tnfO{ s'g} lglZrt laGb'df 180° sf] sf]0fdf 3'dfpbF f sG] bb| l] v
a/fa/ b/' Ldf t/ ljk/Lt lbzfdf vlK6g] cj:yf cfpg'nfO{ laGb' ;dldlt ePsf]

elgG5 .

pbfx/0f 2

lbOPsf] lrqnfO{ laGb' ;dldltsf] cfwf/df sG] b| O df 3d' fpFbf s:tf] cfsl[ t aG5 <

;dfwfg A

oxfF lbOPsf] lrqnfO{ laGb' ;dldltsf cwf/df -s]Gb| laGb' O
O df 180°_ 3'dfpFbf klxns] f] lrqsf] pN6f] -ljk/Lt B
lbzfdf_ cfsl[ t aG5 .
O
B

A

cEof; 18.1
1. tnsf lrqx¿df 86 /v] fnfO{ ;dldlt cIf dfg/] k/" f ug'{xf];\ M
-s_ -v_ -u_

ul0ft, sIff & 245

2. tn lbOPsf lrqdf ;dldltsf] cIf / cfwf lrq lbOPsf] 5 . lrq k"/f ugx{' f];\ /
/]vLo ;dldltsf cIfx¿sf] ;ªV\ of klg kQf nufpg'xf];\ M

-s_ -v_ -u_ -3_

3. tnsf k|Tos] laGb' /v] fnfO{ 6«]l;ª ug{'xf];\ . k|To]sdf /v] Lo ;dldltsf] cIf
lvRgx' f];\ . /v] Lo ;dldltsf cIf sltcf]6f 5g\, kQf nufpg'xf];\ M

-s_ -v_ -u_

4. tnsf lrqnfO{ 6]«; u/L O laGb'df 180° sf] sf]0fdf 3'dfpgx' f];\ M

-s_ -v_ -u_ -3_
O
OO O

5. tn lbOPsf lrqdWo] sg' s'g lrqdf laGb' ;dldlt 5g,\ 56' \ofpgx' f];\ M
-s_ -v_ -u_

-3_ -ª_ -r_

246 ul0ft, sIff &

6. lbOPsf] tf;sf kQLx¿df ePsf] cfsl[ t laGb' ;dldlt xf] jf xfO] g\, xf] eg]
lsg <

kl/of]hgf sfo{
ljleGg /ªsf sfuh lnO{ To;nfO{ k6o\ fO{ /v] Lo ;dldlt / laGb' ;dldlt
xg' ] cfs[ltx¿ sF}rLn] sf6]/ sIffdf k:| t't ug'x{ f;] \ .

pQ/
lzIfsnfO{ bv] fpgx' f;] \ .

18.2 6;] ]n];g (Tessellation)

tn lbOPsf lrqx¿ cjnfs] g ug{'xf;] \ . tL lrqx¿df s:tf s:tf cfsl[ tx¿ slt
sltcf6] f /x]sf 5g\ < ;Fu}sf ;fyL;Fu 5nkmn u/L ;r" L tof/ kfg{x' f];\ M

s] tkfOs{F f] 3/df ePsf gfªn\ f], sfk]6{ , 8fs] f,] OF6f jf 9'ª\ufsf] kvfn{ , km6' an cflbdf
o:tf cfsl[ tx¿ bV] ge' Psf] 5 < tL cfsl[ tx¿ s;/L /fv]sf 5g\ < ;uF }sf] ;fyL;uF
5nkmn ug{'xf];\ .

ul0ft, sIff & 247

lj|mofsnfk 1
;a} ;dx" n] lrqdf bv] fP h:t} u/L /ª\uLg sfuhdf Pp6} gfksf lqe'hfsf/
cfsl[ tx¿ agfO{ sFr} Ln] sf6\gx' f];\ . ca lrqdf bv] fP h:t} u/L tL lqeh' fsf/
6j' m| fx¿ rf6k{ k] /df 6fF:gx' f;] \ . o;/L 6j' |mfx¿ 6fF:bf ;txdf vfnL 7fpF g/xg] u/L
/ gvlK6g] u/L 6f:F g'xf];\ / 6;] ]n];gsf af/]df 5nkmn u/L sIffdf k:| t't ug{'xf;] \ .

Ps jf PseGbf a9L HofldtLo cfs[ltx¿ gvK6fOsg / vfnL 7fpF g/fvLsg
;dtn ;tx 9fSg] jf 5f]Kg] k|lj|mofnfO{ 6];n] ];g (Tessellation) elgG5 .

pbfx/0f 1
lbOPsf yf]Knfx¿ hf]8/] lqeh' fsf/ 6;] ]n;] g agfpgx' f];\ / /ª egx{' f];\ .

;dfwfg
dflysf yf]Knfx¿ hf8] /] lgDgfg';f/sf] lqeh' fsf/ 6];]n;] g agfpg ;lsG5 .

248 ul0ft, sIff &

cEof; 18.2
1. lbOPsf lrqx¿ lqeh' fsf/ 6];n] ];gsf x'g\ jf xfO] gg\ / lsg, kQf nufpg'xf];\

.
-s_ -v_ -u_

2. tn bv] fOPsf] h:t} u/L lqe'hfsf/ 6];n] ];g agfO{ k"/f ug{x' f];\ .

3. lbOPsf yfK] nfx¿ hf8] /] lqe'hfsf/ 6];n] ;] g agfpg'xf];\ / /ª eg'{xf];\ .

4. ;dsf]0f lqe'h / ;dafx' lqe'haf6 aGg] Ps Pscf]6f 6];n] ];g agfpgx' f];\ .

kl/ofh] gf sfo{

cfkmg\ f] ljBfno, 3/sf] leQf, 3/df ko| f]u ul/g] afy?dsf 6fon, sfk]{6,
km6' an / elnan cflbdf agfOPsf lrqx¿ cjnf]sg u/L lqe'hfsf/
6];]n;] gsf] lrq cfkmg\ f] sfkLdf agfpg'xf];\ / /ª;d]t e//] sIffdf k|:tt'
ug'x{ f];\ .

pQ/
lzIfsnfO{ b]vfpg'xf;] \ .

ul0ft, sIff & 249

kf7 19 :yfgfGt/0f

(Transformation)

19.0 k'g/jnf]sg (Review)

tn lbOPsf lrqx¿sf] cjnfs] g u/L ;f]lwPsf kZ| gsf af/]df ;fyLx¿lar 5nkmn
u/L lgisif{ kQf nufpg'xf;] \ .

lrq -s_ lrq -v_

-s_ lrq I df s] dflg;sf] cgx' f/ / Pg] fdf bl] vPsf] cg'xf/ P]gfaf6 a/fa/
b'l/df 5g\ <

-v_ lrq I df, s] dflg;sf] cg'xf/ / P]gfdf b]lvPsf] cg'xf/ p:t} / pq} 5g\ <
-u_ lrq II df s] lunf;nfO{ klxn]sf] :yfgaf6 lglZrt lbzfdf ;fbf{ cfsf/df

kl/jtg{ cfPsf] 5 <

sg' } lglZrt lgoddf /xL sg' } j:t'sf] l:ylt (Position) jf gfkdf kl/jtg{
xg' n' fO{ pSt j:ts' f] :yfgfGt/0f (Transformation) elgG5 .

19.1 k/fjtg{ (Reflection)

lj|mofsnfk 1

;a} ljBfyLn{ ] Ps Pscf6] f kk] / lnO{ tn lrqdf bv] fP h:t} u/L laraf6 b'O{
a/fa/ efu x'g] u/L k6\ofpgx' f;] \ .

250 ul0ft, sIff &

ca k6o\ fOPsf] efusf lardf sDkf;sf] r'Rrf] jf sndn] Pp6f Kjfn kfg{'xf;] \ .

To;kl5 lrqdf b]vfP h:t} u/L k6o\ fOPsf] efunfO{ vfN] g'xf];\ M

sfuh k6\ofpbF f ag]sf] /v] fb]lv bj' } Kjfn;Ddsf] b'/L gfkL ;fyLx¿;Fu 5nkmn
ug'{xf;] \ .

dflysf] 5nkmnaf6 lgDgfg';f/sf] lgisif{ kQf nufpg ;lsG5 .

● bO' c{ f]6f Kjfnx¿dWo] sg' } Pp6f KjfnnfO{ cfsl[ tdfGbf csf{] Kjfn To;sf]
k|ltlaDa x'G5 .

● sfuh k6\ofpbF f ags] f] /]vf k/fjt{gsf] cIf xf] .

● k/fjtg{ cIfaf6 cfs[lt / k|ltlaDa a/fa/ b/' Ldf /x]sf] x'G5 .

ljm| ofsnfk 2

lbOPsf lrq cjnfs] g ug{'xf];\ / ;fyLx¿;Fu 5nkmn u/L ;f]lwPsf kZ| gsf] pQ/

vfH] gx' f;] \ m

A A'

B B'

1 2 3 4 5 6 7 8 9 10 11 12

-s_ lrqdf /]vf AB nfO{ k/fjt{g u/fpFbf ag]sf] k|ltlaDa (image) sg' xf] <
-v_ s] /]vf AB / /v] f A'B' p:t} / pq} 5g\ <

ul0ft, sIff & 251

-u_ lrqdf k/fjt{gsf] cIf sg' xf] <

-3_ s] k/fjt{gsf] cIfb]lv /]vf AB / /v] f A'B' ;Ddsf] b/' L a/fa/ 5 <
-ª_ s] jf:tljs cfsl[ t -/v] f AB) / kl| tlaDa -/v] f A'B') cg'¿k 5g\ <

● h'g /v] fsf cfwf/df k/fjtg{ ul/G5, To; /]vfnfO{ k/fjtg{ sf] cIf
(axis of reflection) elgG5 .

● jf:tljs j:t' k/fjtg{ eO{ aGg] cfsl[ tnfO{ k|ltlaDa (Image) elgG5 .
● sg' } j:t' jf cfsl[ tnfO{ k/fjt{g ubf{ cfsl[ t / kl| tljDa k/fjtg{ sf]

cIfaf6 a/fa/ b'/Ldf kb{5g\ .

● s'g} klg HoldtLo lrq jf cfs[ltnfO{ k/fjt{g ubf{ jf:tljs cfs[lt /
kl| tlaDa cg¿' k x'G5g\ .

-s_ X – cIfdf k/fjt{g (Reflection on X- axis) Y A (4, 5)
O X
;Fu}sf] u|fkmdf bv] fP h:t} u/L cfcfkmg\ f] u|fkm Y'
sfkLdf X - cIf (XOX' ) / Y- cIf (YOY') A' (4, –5)
lvRg'xf];\ . ca u|fkmdf sg' } laGb' A lnP/
XOX' af6 k/fjtg{ u/fpgx' f];\ / pSt laGbn' fO{
A' gfd lbgx' f;] \ . X- cIfaf6 A ;Ddsf] b/' L /
X- cIfaf6 A' ;Ddsf] b'/L a/fa/ x'gk' b{5 .

To;kl5 u|fkmdf laGb' A' sf] lgbz]{ fª\s ug]/ X'
n]Vgx' f];\ . To;}u/L A sf] lgb]z{ fªs\ slt
xG' 5 < ;fyLx¿;uF 5nkmn ug'{xf;] \ .

oxfF u|fkmdf laGb' A sf] lgb{]zfªs\ (4, 5) / A'
sf] lgb]{zfª\s (4, –5) 5 .

s'g\ } klg laGb' (x, y) nfO{ x–cIfdf k/fjtg{ ubf{ k|ltlaDa (x, –y) xG' 5 . cyft{ \
x lgbz]{ fªs\ pxL /xG5 / y lgb{]zfªs\ sf lrx\g dfq ablnG5 .

252 ul0ft, sIff &

pbfx/0f 1

laGb' P(–3, 2) nfO{ uf| kmdf cªs\ g u/L X- cIfaf6 Y
k/fjt{g u/fO{ u|fkmdf bv] fpg'xf;] \ . OX
Y'
;dfwfg P
B' (3, 4)
oxfF p(x, y) = P (–3, 2) X

ca uf| kmdf P(–3, 2) nfO{ X' X
X- cIfaf6 k/fjt{g ubf,{ M'(4, –5)

P(–3, 2) sf] k|ltlaDa P'(–3, –2) eof] P'

t;y{ P(x', y') = P'(–3, –2)

-v_ Y- cIfdf k/fjtg{ (Reflection on Y- axis) Y

;Fus} f] uf| kmdf bv] fP h:t} u/L cfcfkmg\ f] u|fkm

sfkLdf sg' } laGb' B nfO{ YOY' af6 k/fjtg{ B (–3, 4)
u/fpg'xf];\ o;sf] kl| tlaDa YOY' /]vfaf6 laGb'
B sf] a/fa/ b/' Ldf kb5{ . o;nfO{ B' gfd
lbg'xf;] \ .

ca B / B' sf] lgbz{] fª\s ug/] slt slt X' O
x'G5 < ;fyLx¿;Fu 5nkmn u/L nV] gx' f];\ . Y'
oxfF u|fkmdf laGb' B sf] lgb{]zfªs\ (–3, 4) /
B' sf] lgb]{zfªs\ (3, 4) 5 . Y

pbfx/0f 2
laGb' M(–4, –5) nfO{ uf| kmdf cªsg u/L
Y- cIfaf6 k/fjtg{ u/fO{ uf| kmdf bv] fpg'xf];\ .
;dfwfg

oxfF M(x, y) = M (–4, –5) X' O
M(–4, –5)
ca u|fkmk]k/df M(–4, –5) nfO{ Y- cIfaf6 Y'
k/fjt{g ubf,{ M(–4, –5) sf] k|ltlaDa M'(4, –5)
eof] t;y,{ M'(x', y') = M' (4, –5) xG' 5 .

sg' } klg laGb' (x, y) nfO{ Y–cIfdf k/fjt{g
ubf{ k|ltlaDa (–x, y) xG' 5 . cyf{t\ x
lgb{z] fªs\ sf] lrx\g ablnG5 / y lgb{]zfªs\
pxL /xG5 .

ul0ft, sIff & 253

cEof; 19.1

1. lbOPsf HofldtLo cfs[ltx¿nfO{ m cIfdf k/fjt{g u/L k|ltlaDa lrq lvRg'xf;] \ .
-s_ -v_

m

m

-u_ -3_

mm

2. tnsf HofldtLo cfsl[ tx¿nfO{ k/fjtg{ sf] cIf XX' (X-cIf) df k/fjtg{ ubf{ aGg]
kl| tlaDa nV] g'xf;] \ .

-s_ laGb' A -v_ laGb' M -u_ laGb' R
-3_ /]vf BC -ª_ /v] f PQ -r_ /]vf KL
-5_ RST -h_ UVW

A R TR M
S CK
X' X

P W
U
V
B L
Q

254 ul0ft, sIff &

3. lbOPsf HofldtLo lrqx¿nfO{ Y-cIfdf k/fjt{g ug{x' f];\ / aGg] k|ltlaDa kQf

nufpg'xf;] \ . YJ
F
-s_ laGb' F -v_ laGb' T
-u_ /v] f PQ -3_ /v] f MN Q L K
-ª_ ABC -r_ JKL T

P M
A

BC N

Y'

4. n]vflrqsf] k|of]u u/L lbOPsf lgb{z] fª\sx¿nfO{ X- cIfdf k/fjt{g u/L
kl| tlaDasf] lgbz]{ fªs kQf nufpgx' f];\ M

-s_ (1, 2) -v_ (3, 2) -u_ (–4, 4)

-3_ (–2, 6) -ª_ (8, –7) -r_ (9, –10)

-5_ (–6, –9) -h_ (–5, –8) -em_ (–9, 7)

5. kZ| g (4) sf laGb'x¿nfO{ Y-cIfdf k/fjt{g u/L u|fkmkk] /df egx'{ f];\ .

6. laGb' T' (3, –4) nfO{ X– cIfdf k/fjt{g ugx{' f];\ / T sf] lgb{z] fªs\ kQf
nufpgx' f;] \ . /v] f TT' sf] nDafO kQf nufpgx' f;] \ .

7. laGb' P(4, 5) nfO{ Y– cIfdf k/fjtg{ ug'x{ f;] \ . P' sf] lgb{z] fª\s / /]vf PP'
sf] nDafO kQf nufpg'xf];\ .

kl/ofh] gf sfo{

cfˆgf] 3/, ljBfno, af6fdf x'g] u/s] f] k/fjtg{ sf kfFrcf6] f cj:yfx¿sf]
vf]hL u/L l6kf]6 ug{x' f;] \ / sIffdf k|:t't ugx{' f;] \ .

ul0ft, sIff & 255

pQ/

4. -s_ (1, –2) -v_ (3, –2) -u_ (–4, –4)

-3_ (–2, -6) -ª_ (8, 7) -r_ (9, 10)

-5_ (–6, 9) -h_ (–5, 8) -em_ (–9, –7)

5. -s_ (–1, 2) -v_ (–3, 2) -u_ (4, 4)

-3_ (2, 6) -ª_ (–8, –7) -r_ (–9, –10)

-5_ (6, –9) -h_ (5, –8) -em_ (9, 7)

6. T'(3, 4), 8 PsfO 7. P' (–4, 5), 8 PsfO

19.2 lj:yfkg (Translation)

lj|mofsnfk 1

;a} ljBfyL{n] cfcfkm\gf sfkLdf Pp6f l;wf /v] f sfg] x'{ f];\ . lrqdf bv] fP h:t} u/L
Tof] /v] fsf] Ps 5p] df O/h] / /fVgx' f;] \ / O/]h/sf] rf/cf]6} sg' fdf yfK] nf lbP/ gfd
ABCD /fVg'xf;] \ .

B B'

A C C'
D A'

D'

To;kl5 O/h] /nfO{ l3;f//] ToxL /]vfdf cufl8 ;fg'{xf;] \ . O/h] /sf] rf/cf6] } sg' fdf
yfK] nf lbO{ A', B', C' / D' gfdfªs\ g ug'{xf];\ .

ca ;fyLx¿;uF 5nkmn u/L tnsf k|Zgx¿sf] pQ/ vfH] gx' f;] \

-s_ AA', BB', CC' / DD' sf] ;DaGw s] xf]nf <
-v_ s] O/h] /sf] :yfg lglZrt lbzfdf kl/jtg{ eof] <
-u_ s] cfs[lt / k|ltlaDa cg'¿k 5g\ <

ca dflysf lj|mofsnfksf cfwf/df lj:yfkgsf] kl/efiff nv] L cfkmn" ] nv] s] f]
kl/efiffnfO{ ;fyLn] nv] s] f] kl/efiff;uF tn' gf u/L x]g'x{ f];\ .

256 ul0ft, sIff &

● sg' } klg j:t'nfO{ lglZrt lbzf / b'/Ldf ;fg{' jf :yfgfGt/0f ugn'{ fO{
lj:yfkg elgG5 .

● lj:yfkgsf nflu lj:yfkgsf] kl/df0f jf gfk / lbzf pNnv] ugk'{ b5{ .
● lj:yfkgdf cfs[lt / kl| tlaDa cg¿' k xG' 5g\ .
● sg' } klg laGbn' fO{ lj:yfkg ubf{ lbOPsf] kl/df0f / lbzfdf ;dfgfGt/ /v] f

lvRgk' 5{ .

pbfx/0f 1

;Fus} f] /]vfv08 PQ nfO{ ls/0f /v] f ON sf] kl/df0f / P O Q
lbzfdf lj:yflkt ugx{' f];\ . P
P' N Q
;dfwfg Q'
O
-s_ laGb' P af6 ON sf] lbzf / kl/df0f;Fu a/fa/
/ ;dfgfGt/ x'g] u/L PP' lvRgx' f;] \ . N

-v_ laGb' Q af6 ON sf] lbzf / kl/df0f;uF a/fa/
/ ;dfgfGt/ x'g] u/L QQ' lvRgx' f;] \ .

-u_ P' / Q' nfO{ hf]8g\ x' f;] \ . o;/L P'Q' g} /v] fv08 PQ
sf] cfjZos k|ltlaDa xf] .

cEof; 19.2

1. tn lbOPsf laGb' tyf /v] fv08nfO{ lbOPsf lbzf / kl/df0fdf lj:yfkg ug'x{ f;] \ .

-s_ A -v_ B -u_ -3_ X
2 cm 6 cm
P 5 cm
3 cm

P-ª_ -r_ -5_ X

8 cm 3.5 cm 6.5 cm Y
Q MN

ul0ft, sIff & 257

2. /v] f XY nfO{ OK sf] gfk / lbzfdf lj:yfkg ug'x{ f;] \ .

K

XO Y

3. /]vf PQ nfO{ OM sf] gfk / lbzfdf lj:yfkg ug{x' f];\ .

P

OM

Q

4. tnsf k|Tos] HofldtLo cfsl[ tx¿nfO{ lbOPsf] lbzf / kl/df0fdf lj:yfkg ubf{
aGg] kl| tlaDa lvRg'xf];\ .

-s_ A -v_ L -u_ Q Z

BC M XY
N

5. Pp6f aRrf lrKn6] L vN] bf lrKn/] 4m tn cfof] eg] s] of] lj:yfkg xf,] sf/0f
lbgx' f;] \ .

6. eO' Fdf /flvPsf] sfkLsf] kfgfsf] Pp6f dfq 6K' kf] ;dft/] 1 ld6/ cfkml" t/
tfGbf afFsL ;a} 6K' kfx¿ pxL lbzf / kl/df0fdf :yfgfGt/0f xfn] fg\ <

7. Pp6f k:' tsnfO{ Goh' lk|G6 jf 8«Oªkk] / dfly /fv]/ 10 cm cufl8 ;fbf{ s]
k:' tssf afFsL ;a} s'gfx¿ pxL lbzf / kl/df0fdf :yfgfGt/0f xf]nfg < s]

of] lj:yfkg xf] <

kl/ofh] gf sfo{

Pp6f sfuhdf Pp6f l;wf /]vf sf]gx{' f];\ . pSt /v] fsf] dfly Pp6f s'g} laGb'
A lngx' f;] \ . ca To; laGb'nfO{ t;] f]{ /]vfdf bfofFlt/ 15cm sf] kl/df0fdf
lj:yfkg u/L k|ltlaDa kQf nufpgx' f;] \ . pSt cfs[ltnfO{ sIffdf k:| t't
ug{'xf;] \ .

pQ/
lzIfsnfO{ b]vfpgx' f];\ .

258 ul0ft, sIff &

lbzf l:ylt / :sn] 8«Oª

kf7 20 (Bearing and Scale Drawing )

20.0 k'g/jnf]sg (Review)

;Fus} f] sDkf;sf] lrqsf] cjnfs] g u/L tnsf k|Zgx¿df 5nkmn ug{'xf];\ M

-s_ pSt pks/0f s] sfdsf nflu k|of]u ul/G5 <
-v_ pks/0fdf ePsf] N, S, E, W n] s] s] hgfpF5 <
-u_ pks/0fdf l;of] s'g lbzflt/ kms]{sf] 5 <
-3_ pSt pks/0fdf s'g lbzfnfO{ cfwf/ lbzf dflgPsf] xG' 5 <
-ª_ pks/0fdf ePsf] NE, SE, SW, NW n] s] s] hgfp5F <

N ⇒ pQ/ (North) NE ⇒ pQ/ k"j{ (North East)
S ⇒ blIf0f (South) SE ⇒ blIf0f k"j{ (South East)
E ⇒ kj" { (East) SW ⇒ blIf0f klZrd (South West)
W ⇒ klZrd (West) NW ⇒ pQ/ klZrd (North West)

20.1 lbzf l:ylt (Bearing)

ljm| ofsnfk 1

cfˆgf] a]Grdf a;s] f ;fyLx¿sf] Pp6f ;dx" agfpgx' f];\ .
kT| os] ;dx" n] Pp6f Pp6f sfuhsf kfgf lng'xf;] \ / lrqdf b]vfP h:t} u/L
kfgfnfO{ l7s laraf6 bO' { k6s k6\ofpg\xf;] \ .

ul0ft, sIff & 259

km]l/ sg' faf6 pSt kfgfnfO{ b'O{ k6s k6o\ fpg'xf];\ .

To;kl5 k6o\ fPsf] efunfO{ vf]Ngx' f];\ pQ/ klZrd pQ/ (N) k"j{ (NE)
/ lrqdf b]vfP h:t} u/L k6o\ fOPsf (NW)
7fpdF f ags] f /v] fsf 5p] df gfdfª\sg pQ/
u/L ;fyLx¿;Fu 5nkmn u/L tnsf
kZ| gx¿sf] pQ/ vfH] g'xf];\ M 045°

klZrd kj" { (E)
(W)

-s_ cfkmn" ] agfPsf] gd'gf -lrq_ df blIf0f klZrd blIf0f kj" { (SE)
sltcf]6f lbzfx¿ 5g\ < ltgLx¿
s] s] xg' \ < (SW) blIf0f (S)

-v_ pQ/ k"j{ lbzf bv] fpg] /]vfn] slt l8u|Lsf] sf]0f agfPsf] 5 <

-u_ pQ/ / pQ/ k"j{ lbzf bv] fpg] /]vflar slt l8u|Lsf] sf0] f 5, gfKg'xf];\ .

-3_ s] pQ/ / klZrd, klZrd / blIf0f tyf blIf0f / k"j{ b]vfpg] /v] fx¿lar
klg 90° sf sf]0f ags] f 5g\ <

-ª_ s] pQ/ / pQ/ klZrd, klZrd / blIf0f klZrd tyf blIf0f kj" { b]vfpg]
;a} /]vfx¿n] cfk;df 45°/45° sf sf0] f agfPsf 5g\ <

pQ/ lbzf hgfpg] /]vfnfO{ cfwf/ dfg/] 38Lsf] ;'O{sf] lbzfdf sg' }
b'O{ :yfglarsf] cjl:yltnfO{ tLg cªs\ sf] sf0] fsf ¿kdf k:| t't ug]{
tl/sfnfO{ lbzf l:ylt (Bearing) elgG5 .

260 ul0ft, sIff &

20.1.1 gS;f k9fO (Map Reading) a;kfs{ dlGb/
ljBfno
ljm| ofsnfk 2
c:ktfn gu/kflnsf ejg
;Fus} f] lrqdf sg' } gu/kflnsfsf sx] L
:yfgx¿ b]vfOPsf] 5 . pSt lrqnfO{ rft} f/f
sfkLdf 6«]; ug'x{ f];\ . gu/kflnsf
ejgnfO{ cfwf/ dfgL ;fyLx¿;uF k|x/L sfof{no
5nkmn u//] lgDglnlvt :yfgx¿sf]
lbzfl:ylt kQf nufpgx' f;] \ / sIffdf kfgL 6o\ fªs\ L
k:| t't ug'{xf];\ .

-s_ ljBfno -v_ dlGb/

-u_ a;kfs{ -3_ k|x/L sfof{no

-ª_ c:ktfn -r_ kfgL 6o\ fªs\ L

20.2 :sn] 8O« ª (Scale Drawing)

lj|mofsnfk 3

;a} ljBfyL{ pko'St ;dx" df a:g'xf;] \ . lbOPsf] cj:yfsf] cWoog u/L tnsf
k|Zgx¿af/] ;dx" df 5nkmn ugx{' f];\ / sIffdf k:| t't ug'x{ f;] \ .

elnan sf6] {

9 cm

4.5 cm

|kxf/ ]/vf
dWo ]/vf
|kxf/ ]/vf
aflx/L lsgf/f

3 cm 1.5 cm 1.5 cm 3 cm

:s]n 1:200 (1 cm = 200 cm)

ul0ft, sIff & 261

j|m=;= /v] fsf] gfd gS;fsf] jf:tljs jf:tljs sf6] { lgisif{
/]vfsf] gfk sf6] {sf] gfk / gS;fsf]
1 sf]6{sf] nDafO
2 . sf6] {sf] rf8} fO gfksf] cgk' ft
3 . dWo /v] fbl] v
9 cm 18 m 1:200
kx| f/ /v] f–
;Ddsf] b'/L

● jf:tljs sf6] s{ f] rf8} fO / gS;fsf] rf8} fOsf] cg'kft slt 5 <
● jf:tljs sf]6s{ f] nDafO / gS;fsf] nDafOsf] cg'kft slt 5 <
● gS;fsf] dWo /]vfb]lv kx| f/ /v] f;Ddsf] b'/L 6 cm 5 eg] jf:tljs b/' L

slt xfn] f <

● dflysf] elnan sf]6n{ fO{ :sn] 1 cm = 1 OGr lnP/ 8O« ª k]k/df gS;f
tof/ ug{'xf;] \ .

● dflysf] 5nkmnsf cfwf/df s] lgisif{ lgsfNg ;lsG5 <

1. lglZrt :s]nsf] k|ofu] u/L Hofb} 7'nf / Hofb} ;fgf j:tn' fO{ /]vfªs\ g ug{
;lsG5 .

2. :s]ndf jf:tljs j:t' / lrq -gS;f_ lvr/] cfjZostfcg';f/ 7'nf] jf
;fgf] gfk lnP/ lglZrt cg'kft agfOG5 .

3. lglZrt :s]nsf] ko| f]u u/L gS;faf6 j:t'sf] jf:tljs gfk lgsfNg
;lsG5 .

pbfx/0f 1

1 cm = 500 m sf] :s]n ko| f]u u/L Pp6f gS;f tof/ ubf{ bO' { :yfglarsf] gS;fsf]
b/' L 9cm eP pSt b'O{ :yfglarsf] jf:tljs b'/L slt xfn] f <

;dfwfg
oxfF :s]n 1 cm = 500 m jf:tljs b/' L
:sn] 9cm = (9 × 500) m = 4500 m
t;y,{ bO' { 7fpF larsf] jf:tljs b'/L= 4500m

262 ul0ft, sIff &

cEof; 20

1. sIffsf ljBfyL{ lrqdf b]vfP h:t} u/L kd| fb]
rp/df plePsf 5g\ eg] tnsf k|Zgx¿sf] k]Daf bfh] ]{

pQ/ lbgx' f;] \ .

-s_ bf]h{] plePsf] :yfgsf] lbzfl:ylt sg' hfls/ k|ltef
xf] <

-v_ hfls/ plePsf] lbzfl:ylt s'g xf] < rdn] L lht'

;'s'dfof

2. tn lbOPsf] lrqsf cfwf/df ;f]lwPsf k|Zgsf]
hjfkm n]Vgx' f];\ M

-s_ :yfg O af6 laGb' A sf] lbzfl:ylt kQf nufpgx' f;] \ .

-v_ :yfg O af6 laGb' C sf] lbzf l:ylt kQf C A
nufpgx' f];\ . E

-u_ :yfg O af6 laGb' F sf] lbzf l:ylt kQf nuf- F
pg'xf];\ .
O
-3_ :yfg O af6 laGb' B sf] lbzf l:ylt kQf
nufpg'xf];\
B

3. gk] fnsf] jf:tljs gS;f cjnf]sg u/L tn ;f]lwPsf D
kZ| gsf] pQ/ nV] g'xf];\ M

-s_ kf]v/faf6 gk] fnuGhsf] lbzfl:ylt nV] g'xf;] \ .

-v_ sf7df8fF}af6 dgfªsf] lbzfl:ylt nV] gx' f;] \ .

-u_ hgsk/' af6 tfKn]h'ªsf] lbzfl:ylt nV] gx' f];\ .

4. tnsf kZ| gdf b'O{ :yfglarsf] jf:tljs b/' L kQf nufpg'xf];\ M

-s_ b'O{ :yfglarsf] gS;fsf] b/' L = 9 cm [:sn] 1 cm = 200m]

-v_ bO' { :yfglarsf] gS;fsf] b/' L = 3.5 cm, [:s]n 1 cm = 500 ft]

5. Pp6f rp/sf] jf:tljs nDafO 125 m / 75 m 5 . 1 cm : 10 cm sf]
cg'kft lnP/ rp/sf] /]vfªs\ g ug{x' f];\ .

6. 1cm n] jf:tljs 2 lkm6 hgfpg] u/L 20 ft nDafO / 18 ft rf8} fO ePsf]
sIffsf]7fsf] lrq agfP/ b]vfpg'xf];\ .

ul0ft, sIff & 263

7. lrqdf lbOPsf] lahn' Lsf] vDaf 1:100 cm sf] :sn] df agfOPsf]
5 . ?n/n] gfk/] vDafsf] jf:tljs prfO kQf nufpg'xf;] \ .

8. ljBfnosf] rp/df uP/ k|Zg g= ! sf] lrqdf bv] fP h:t} u/L kT| o]s
lbzfdf Ps Ps hgf ;fyL pleP/ lbzf l:ylt kQf nufpgx' f];\ .
k|Tos] ;fyLnfO{ lbzfl:yltsf] gfdn] afn] fpgx' f;] \ .

kl/of]hgf sfo{
(1) g]kfnsf] gS;f lnP/ s'g} k|bz] sf] ;b/ds' fdnfO{ sG] b| dfgL cf7cf6] f
lbzf (N, S, E, W, SE, SW, NW, NE) df kg]{ Ps Pscf6] f :yfgsf] gfd
n]Vg'xf];\ .
(2) ;/sf/n] af9L klx/fa] f6 kLl8t ufpFsf afl;Gbfx¿nfO{ ;/' lIft :yfgdf
;fg{'kg]{ eP5 . tkfOxF{ ¿sf % hgfsf] ;dx" nfO{ lj1 ;d"x dfg/] Jojl:yt
tyf ;/' lIft a;f]af;sf nfuL vfnL hUufsf] gS;f lbP5 . pSt hUufdf
s;/L a;fa] f;sf] ofh] gf agfpgx' G' 5 . lrq;lxtsf] ;/sf/nfO{ ;e' mfj
lbg'xf;] \ .

pQ/
lzIfsnfO{ b]vfpg'xf;] \ .

264 ul0ft, sIff &

ldl>t cEof;

1. lbOPsf] lrqsf cfwf/df x / y sf] dfg kQf nufpg'xf];\ M

-s_ A D

85°
yo o

xo

CB

-v_ lrq -s_ df ags] f s'g s'g sf0] fx¿ cg'¿k 5g\ nV] gx' f;] \ .
-u_ lrq -s_ df laGb' ;dldlt 5 ls /]vLo ;dldlt 5 <

2. tnsf k|Tos] laGb'x¿nfO{ nv] flrq agfO{ cªs\ g ug{x' f];\ . kT| o]s laGbn' fO{ jm| dzM
hf]8\b} hfg'xf;] \ . o;/L aGg] cfs[ltsf] gfd klg n]Vg'xf];\ .

A(–4, 2) B(4, 3) C(2, 5)

-s_ pSt cfsl[ t ABC nfO{ X-cIfdf k/fjt{g ubf{ aGg] cfsl[ t A'B'C' sf]
lgbz{] fªs\ n]Vg'xf;] \ .

-v_ cfs[lt A'B'C' cg¿' k cfs[lt x'g,\ lsg < sf/0f lbgx' f];\ .

-u_ cfs[lt ABC cfs[lt A'B'C' df k/fjt{g x'Fbf sg' } lglZrt lgod kfngf
ePsf] 5 < To;af6 sg' } ;q" lgsfNg ;Dej xfn] f < lzIfs;uF 5nkmn
ug'x{ f;] \ .

3. olb Pp6f cfotsf] nDafO 3 cm / rf8} fO 4 cm 5 eg] pSt cfotsf] ljs0fs{ f]
nDafO slt xfn] f < pSt cfotsf] ljs0f{ /v] Lo ;dldltsf] cIf xf,] sf/0f lbg'xf;] \ .

4. 1 cm n] jf:tljs 10 m hgfpg] u/L 90 m nDafO / 45 m rf}8fO ePsf]
km' 6an d}bfgsf] lrq agfpgx' f;] \ . o;/L ag]sf] cfotfsf/ k'm6an d}bfgsf
rf/cf]6f s'gfdf s:tf sf0] fx¿ aG5g\ < To; km'6an d}bfgsf las0f{nfO{ cIf
dfg]/ To;nfO{ :yfgfGt/0f ubf{ s:tf] cfs[lt aGnf, 5nkmn ug'{xf;] \ .

5. /]vLo ;dldlt x'g] cª\u|h] L cIf/x¿dWo] V df Pscf]6f / H df b'O{cf6] f /v] f
;dldltsf cIf x'G5g\ . o:t} c¿ /]vLo ;dldlt xg' ] cªu\ h]| L cIf/x¿ kQf
nufO{ ;"rL tof/ kfgx'{ f];\ . To:tf cIf/x¿dWo] laGb' ;dldlt xg' ] cIf/x¿
s'g s'g 5g\ <

ul0ft, sIff & 265

6. tn lbOPsf laGb'x¿sf] lgbz{] fª\sx¿ nV] gx' f;] \ M

Y

X' O X

Y'

kfrF cf]6f ;fgf juf{sf/ sf7] f a/fa/ Ps PsfO dfGbf

k|To]s laGb'nfO{ x-cIf / Y-cIfdf :yfgfGt/ ubf{ aGg] laGbs' f] lgb]z{ fª\s klg
nV] g'xf;] \ .
7. Pp6f uf8L T :yfgaf6 200 km k"js{ f] :yfg S df ku' ]sf] 5 . uf8Ln] S af6

csf{] klZrddf kg]{ :yfg U df 300 km sf] ofqf u/5] . of] hfgsf/LnfO{
:sn] 8«Oª u/L lgDglnlvt s/' f kQf nufpg'xf;] \ M
-s_ T af6 U ;Ddsf] b'/L -v_ U af6 T ;Ddsf] lbzfl:ylt

266 ul0ft, sIff &

8. ;Fus} f] lrqdf sx] L :yfgx¿ bv] fOPsf] 5 . lrqnfO{ cEof; kl' :tsfdf 6«;]
u/L / sf7df8f}nF fO{ cfwf/ dfgL lgDglnlvt :yfgx¿sf] lbzfl:ylt kQf

nufpgx' f];\ .

bj] 3f6 uf;] fOs'08 tftf]kfgL

sf7df8fF} bf]nvf
kgf}tL
knfGrf]s
eujtL

hgsk'/

-s_ uf;] fOs0' 8
-v_ tftf]kfgL
-u_ bj] 3f6
-3_ bf]nvf
-ª_ knfGrf]s eujtL
-r_ kgf}tL
-5_ hgsk'/
-h_ olb uf];fOs'08, tftf]kfgL / bf]nvfnfO{ AB-cIf dflg k/fjt{g ubf{ pSt

:yfgx¿sf] lbzf l:yltdf s:tf] kl/jt{g cfpF5, 5nkmn ugx{' f];\ .

ul0ft, sIff & 267

9. lrqdf Pp6f ufpsF f d'Vo 7fpFx¿ xf6ahf/ dlGb/ xn' fs
b]vfOPsf] 5 . olb :sn] 1 cm = sfofn{ o
100 m eP ?n/ ko| fu] u/L :j:Yo :jf:Yo
rfs} Laf6 lgDglnlvt 7fpFsf] rf}sL

jf:tljs b'/L kQf nufpg'xf;] \ M

-s_ dlGb/ kfgLsf]

-v_ kfgLsf] dx' fg d'xfg
-u_ ljBfno ljBfno

-3_ xf6ahf/

-ª_ x'nfs sfofn{ o

-r_ :jf:Yo rf}lsaf6 ljBfno, xf6ahf/ / xn' fs sfo{nosf] lbzfl:ylt klg

n]Vgx' f];\ .

-5_ :jf:Yo rf}sLaf6 dlGb/ / ljBfno sg' lbzfl:yltdf 5g\ <

pQ/
lzIfsnfO{ bv] fpg'xf;] \ .

268 ul0ft, sIff &

kf7 21 tYofªs\ sf] k|:t'tLs/0f

(Presentation of Data)

21.0 k'g/jnfs] g (Review)

Pp6f k':ts k;ndf Ps dlxgfdf kT| o]s lbg lajm| L ePsf k':tssf] ;ª\VofTds
ljj/0f o;k|sf/ 5 M

13, 14, 13, 16, 18, 20
13, 25, 10, 18, 12, 10
28, 25, 12, 15, 17, 15
25, 24, 20, 10, 25, 18

20, 22, 18, 15, 13, 20

dflysf] hfgsf/LnfO{ tnsf] h:t} tflnsf agfP/ e/L tflnsf k"/f ugx'{ f;] \ .

k':tssf] ldnfg lrxg\ af/Daf/tf ;l~rt af/Daf/tf
;ª\Vof
3 3
10 III 2 3+2=5

12 II

dflysf] tflnsfnfO{ s:tf] tflnsf elgG5, ;fyLx¿lar 5nkmn u/L lgisif{ lgsfNgx' f;] \ .

ul0ft, sIff & 269

ul0ft kf \7o 'k:tss ]f ; \ªVof21.1 /]vflrq (Line Graph)

ljm| ofsnfk 1
Pp6f k:' ts ljjm| ]tfn] Ps xKtfdf lajm| L u/]sf ul0ft ljifosf] kf7\ok:' tsx¿sf]
;ªV\ ofnfO{ 8fN] dfn] lgDg /v] flrqdf k|:t't ul/g\ . lbOPsf] /v] flrqsf] cWoog u/L
lgDglnlvt k|Zgsf af/d] f 5nkmn ug{'xf;] \ M

Y

45
40
35
30
25
20
15
10
5
O cfOtaf/ ;fd] af/ dªu\ naf/ aw' af/ laxLaf/ zj' m| af/ zlgaf/ X

af/x¿

-s_ ;f]daf/ sltcf6] f ul0ft k:' ts lajm| L eP5g\ <
-v_ z'j|maf/ sltcf]6f ul0ft k:' ts laj|mL eP5g\ <
-u_ s'g s'g af/ a/fa/ ;ªV\ ofdf ul0ft k:' ts laj|mL eP5 <
-3_ sg' af/ ;ae} Gbf a9L ul0ft k':ts laj|mL eP5 <

lbOPsf r/ d"No / To;sf] af/Daf/tfnfO{ u|fkmdf cªl\ st u/L j|mdzM l;wf
/v] fn] hf8] b\ f aGg] lrqnfO{ /]vflrq (Line Graph) elgG5 .
/v] flrq agfpFbf ckgfpgk' g]{ r/0fx¿
1. lbOPsf] tYofªs\ nfO{ tflnsfdf /fVg'k5{ .
2. af/Daf/tfnfO{ Y – cIfdf / r/ d"No -kl/df0f_ nfO{ X – cIfdf /fVg'k5{ .
3. pko'St :sn] sf] 5gf6] ug{k' 5{ .
4. laGb'x¿nfO{ uf| kmdf k|:t't u/L k|Tos] laGbn' fO{ /v] fv08n] hf]8\gk' 5{ .

270 ul0ft, sIff &

pbfx/0f 1 la |jmL ; \ªVof
c; ]fh
Pp6f dfa] fOn k;ndf 5 dlxgfdf laj|mL ePsf] df]afOnsf] ljj/0f tnsf] /v] flrqdfsflQs
bv] fOPsf] 5 . pSt /]vflrqsf] cWoog u/L lgDglnlvt kZ| gx¿sf] pQ/ lbgx' f];\ .d \ªl;/

df]afOnsf] lajm| L ljj/0f'k;
df3
Ykmf 'ug
35
30
25
20
15
10
5

OX

-s_ sg' dlxgfdf ;a}eGbf a9L dfa] fOn lajm| L ePsf] /x]5 <
-v_ s\g' dlxgfdf ;a}eGbf sd dfa] fOn lajm| L ePsf] 5 <
-u_ c;fh] dlxgfdf sltcf]6f dfa] fOn laj|mL ePsf] /x5] <
-3_ k:| t't /]vflrqsf cfwf/df af/Daf/tf tflnsf agfpgx' f];\ .

;dfwfg
lbOPsf] /]vflrqsf cfwf/df dfly lbOPsf k|Zgsf] pQ/ o;ks| f/ 5 M
-s_ kmfu'g dlxgfdf ;ae} Gbf a9L (35 cf]6f_ df]afOn lajm| L ePsf] 5 .
-v_ k;' dlxgfdf ;ae} Gbf sd (10 cf6] f_ df]afOn lajm| L ePsf] 5 .
-u_ c;fh] dlxgfdf 20 cf]6f df]afOn lajm| L ePsf] 5 .

dlxgf c;fh] sflQs dªl\ ;/ k'; df3 kmfug'
dfa] fOn ;ª\Vof 20 30 15 10 20 35

ul0ft, sIff & 271

pbfx/0f 2

jiff{ofddf s'g} 7fpdF f nuftf/ 5 lbg;Dd k/s] f] jiffs{ f] dfkg o;k|sf/ 5 M

lbg cfOtaf/ ;f]daf/ dªu\ naf/ aw' af/ laxLaf/ z'j|maf/
jiff{ 20 mm 15 mm 25 mm 10 mm 28 mm 30 mm
dflysf] tYofª\snfO{ /v] flrqdf k|:tt' ug'x{ f];\ .
;dfwfg
oxfF ;donfO{ X – cIfdf / jiff{nfO{ Y – cIfdf /fvL /]vflrq lvRbf,

^ lbgsf] jiff{ ljj/0f

Yjif {f -mm df_ X
35
30 cfOtaf/
25 ;f]daf/
20 d \ªunaf/
15 a'waf/
10 laxLaf/
5 z'j|maf/

O

lbgx¿

272 ul0ft, sIff &

cEof; 21.1

1. lbOPsf /v] flrqdf sf7df8f}sF f] ^ Ytfk |jmd (°C)
lbgsf] tfkj|md ljj/0f bv] fOPsf]
30 cfOtaf/
5 . o;sf] cWoog u/L 25 ; ]faf/
20
lgDglnlvt k|Zgx¿sf] pQ/ 15 d \ªunaf/
'awaf/
n]Vgx' f];\ .
laxLaf/
-s_ sg' af/ ;aeGbf a9L tfkj|md 10 X
/x]5 < 'z |jmaf/5

-v_ sg' af/ ;aeGbf sd tfkj|md O

/x]5 <

-u_ aw' af/sf] tfkjm| d slt /x5] < af/x¿
-3_ lbOPsf] /]vflrqnfO{ af/Daf/tf

tflnsfdf b]vfpg'xf;] \ .

2. zf/bf cfwf/et" ljBfnosf] sIff 7 df ljut 6 jifs{ f] egf{b/ o;ks| f/ 5 M

jif{ 2074 2075 2076 2077 2078
egfb{ / 20 26 18 14 12

lbOPsf] cfFs8fnfO{ /]vflrqdf k:| tt' ug'{xf];\ .

3. sIff 7 sf ljBfyLn{ fO{ sf]le8–19 af6 aRg sg' s'g ;'/Iffsf ;fwgx¿ k|ofu]
ug{'k5{ egL ;fl] wPsf] kZ| gdf kl| tlj|mof o;k|sf/ kfOof] M

;fwg df:s ;l] g6fOh/ km];l;N8 kGhf ;fag'
ljBfyL{ ;ªV\ of 30 22 18 8 25

dflysf] hfgsf/LnfO{ /]vflrqdf k:| tt' ug{x' f];\ .

4. tnsf af/Daf/tf tflnsfnfO{ /]vflrqdf k:| tt' ug'{xf;] \ M

-s_ sIff 1234567

ljBfyL{ ;ª\Vof 26 24 28 40 35 45 50

ul0ft, sIff & 273

-v_ kl/df0f (kg) 2 4 6 9 12
d"No (?=) 500 1000 1500 2250 3000

5. ljBfnodf Ps lbg laxfg 10:00 ahb] l] v 4:00 ah];Dd 2/2 306fsf] km/sdf
ydfl{] d6/n] tfkjm| d gfk/] /]vflrq agfO{ sIffdf k:| t't ug{x' f];\ .

kl/of]hgf sfo{

/]l8of], 6l] nlehg jf kqklqsfnufotsf cGo ljleGg dfWodaf6 Ps
xKtf;Ddsf] tfkjm| dsf] l6kf]6 ugx'{ f;] \ . To;nfO{ /]vflrqdf bv] fO{
sIffsf]7fdf k:| t't ug'{xf;] \ .

pQ/

lzIfsnfO{ b]vfpg'xf];\ .

21.2 ax':tDe lrq (Multiple Bar Diagram)

sIffsf ljBfyL{nfO{ s'g vn] a9L dg k/fpg'xG' 5 egL ;fl] wPsf kZ| gdf kf| Kt
k|ltlj|mofcg;' f/ tof/ ul/Psf] :tDe lrq lbOPsf] 5 . pSt :tDe lrqsf] cWoog
u/L lgDglnlvt kZ| gx¿sf af/d] f 5nkmn ugx{' f];\ M

ljBfyLn{ ] dg k/fPsf] vn]

Y

50
40
30
20
10

OX
ljBfy {L ; \ªVof
'km6an
elnan
l |jm ]s6
Aof8ldG6g
]6 'an ]6lg;

vn] x¿

274 ul0ft, sIff &

-s_ lbOPsf] :tDe lrq s:tf] :tDe lrq xf] <
-v_ ;a}eGbf a9L ljBfyL{n] dg k/fPsf] v]n s'g xf] <
-u_ ;a}eGbf yf/] } ljBfyLn{ ] dg k/fPsf] vn] sg' xf] <
-3_ slt hgf ljBfyLn{ ] lj|ms6] vn] dg k/fpbF f /x5] g\ <

lj|mofsnfk 1

;/:jtL dfWolds ljBfnosf sIff 6 bl] v 10 ;Ddsf ljBfyLs{ f] 5fq / 5fqf ;ªV\ ofnfO{
tnsf] :tDe lrqdf k:| tt' ul/Psf] 5 . ;fyL;uF 5nkmn u/L ;fl] wPsf kZ| gsf] pQ/
vfH] gx' f;] \ M

>L ;/:jtL dfWolds ljBfnosf ljBfyL{sf] ljj/0f

ljBfy {L ; \ªVof Y ;ªs\ t]
70 5fqf
60 6 78 5fq
50
40 sIffx¿ 9 10 X
30
20
10

O

-s_ ;aeGbf a9L / ;aeGbf sd ljBfyL{ s'g s'g sIffdf /x5] g\ <
-v_ s'g sg' sIffdf 5fqeGbf 5fqf a9L /x5] g\ <
-u_ s'g s'g sIffdf 5fqfeGbf 5fq a9L /x]5g\ <
-3_ sg' s'g sIffdf 5fq / 5fqf a/fa/ /x5] g\ <
-ª_ of] s:tf] :tDe lrq xf] <

● PseGbf a9L cfk;df ;DalGwt ;"rgf tyf tYofªs\ nfO{ k:| tt' ul/Psf]
:tDe lrqnfO{ ax:' tDe lrq (Multiple Bar Diagram) elgG5 .

● ax':tDe lrqsf] lgdf{0f ubf{ ;fwf/0f :tDe lrqdf h:t} k|To]s :tDesf]
rf8} fO a/fa/ xg' 'k5{ .

● ax':tDe lrqsf] prfOn] ;ªV\ of hgfpF5 .

ul0ft, sIff & 275

pbfx/0f 1

gu/ ;'wf/ ;ldltsf] cfof]hgfdf ;~rfng ePsf] cfFvf, sfg, 3f6F L / bfFt k/LIf0f
lzlj/df btf{ eO{ :jf:Yo k/LIf0f u/fpg] JolStx¿sf] tYofªs\ o;k|sf/ kfOof] .

:jf:Yo k/LIf0f cfFvf sfg 3fF6L bfFt
;]jf
dlxnf k?' if dlxnf k?' if dlxnf k?' if dlxnf k?' if
k/LIf0f 15 20 17 22 65 73 32 40

;ªV\ of

pSt tYofª\ssf] ax:' tDe lrqnfO{ u|fkmdf b]vfpgx' f;] \ .
;dfwfg
oxfF ;fgf] sf]7f a/fa/ bz hgf JolSt dfg]/ ax:' tDe lrq agfpFbf,

la/fdL ; \ªVof :jf:Yo lzlj/ btf{ tYofªs\ ;ªs\ t]
dlxnf
Y k?' if
80
70 X
60
50
40
30
20
10

O cfvF f sfg 3f6F L bftF

:jf:Yo k/LIf0f ;j] f

pbfx/0f 2

Pp6f ljBfnosf] sIff 6 bl] v 8 ;Ddsf] jflifs{ k/LIffsf] glthf tnsf] ax:' tDe
lrqdf k:| tt' ul/Psf] 5 . pSt :tDe lrqsf cfwf/df tnsf k|Zgsf] pQ/
nV] g'xf];\ .

276 ul0ft, sIff &

ljBfy {L ; \ªVof Y jflif{s k/LIffsf] glthf ;ªs\ t]
A u8]|
40 B u8]|
35 C u8]|
30
25 X
20
15
10
5

O6 7 8

sIffx¿

-s_ sIff 8 df slt hgfn] u|8] A NofPsf /x]5g\ <

-v_ sIff 6 bl] v 8 ;Dd slt hgfn] u8]| B NofPsf /x]5g\ <

-u_ s'g sIffdf ;a}eGbf a9L ljBfyL{n] u]8| A NofPsf /x]5g\ <

-3_ s'g sIffsf ;a}eGbf sd ljBfyL{n] u|8] C NofPsf /x5] g\ <

;dfwfg

oxfF lbOPsf] ax':tDe lrqsf cfwf/df dfly lbOPsf k|Zgsf] pQ/ o;k|sf/ 5 M

-s_ sIff 8 df 10 hgfn] u]|8 A NofPsf /x5] g\ .

-v_ sIff 6 bl] v 8 ;Dd u8]| B Nofpg] ljBfyL{sf] ;ªV\ of = 25 + 30 + 35 = 90
hgf

-u_ sIff 7 df ;a}eGbf a9L ljBfyLn{ ] u|]8 A NofPsf /x]5g\ .

-3_ sIffdf 6 df ;ae} Gbf sd ljBfyLn{ ] u8]| C NofPsf /x5] g\ .

ul0ft, sIff & 277

cEof; 21.2

1. tnsf] tYofªs\ nfO{ ax':tDe lrqdf bv] fpg'xf;] \ M

-s_ lbgx¿ cfOtaf/ ;f]daf/ dªu\ naf/ a'waf/ laxLaf/ zj' |maf/

lajm| L kl/df0f df5f df;' df5f df;' df5f df;' df5f df;' df5f df;' df5f df;'
-lsnfu] f| ddf 10 18 15 12 17 20 19 22 20 25 27 20

-v_ ljBfno A B C D E

ljBfyL{ 5fqf 5fq 5fqf 5fq 5fqf 5fq 5fqf 5fq 5fqf 5fq

250 210 125 175 310 350 425 400 520 550

2. sg' } ljBfnodf ljBfyL{ ljBfno hfbF f ;jf/L ;fwg k|ofu] u/L tyf lxF8]/
cfpg] u/]sf] ljj/0f o;k|sf/ 5 M

ljBfno cfpg] ;fwg lx8F ]/ a; df6] /;fOsn ;fOsn

ljBfyL{ ;ªV\ of 5fqf 5fq 5fqf 5fq 5fqf 5fq 5fqf 5fq

25 30 40 45 20 15 55 60

dflysf] tYofª\snfO{ ax:' tDe lrqdf b]vfpg'xf;] \ .

3. lbOPsf] ax':tDe lrqdf g]kfnsf] kfrF cf6] f k|b]zdf pTkfbg xg' ] wfg / uxF'sf]
kl/df0f -xhf/ dl] 6«s6gdf_ lbOPsf] 5 . pSt ax':tDe lrqsf] cjnfs] g u/L
tnsf kZ| gsf] pQ/ n]Vg'xf;] \ M

kfrF cf6] f kb| z] df pTkflbt wfg / uxsF' f] kl/df0f

90 Y
80
70
60
50
40
30
20
10

O
pTkflbt kl/df0f -xhf/ ]dl «6s6gdf_ ;ªs\ t]
|k ]bz g=! wfg
|k ]bz g=@ uxF'

afUdtL X
u08sL
'nlDagL

kb| z] x¿

278 ul0ft, sIff &

-s_ sg' kb| ]zdf ;a}eGbf a9L kl/df0fdf wfgsf] pTkfbg ePsf] /x]5 <
-v_ s'g kb| ]zdf ;ae} Gbf a9L uxF'sf] pTkfbg ePsf] /x5] <
-u_ s'g kb| z] df a/fa/ kl/df0fdf wfg / uxs'F f] pTkfbg ePsf] /x5] <
-3_ s'g s'g k|b]zdf ;aeGbf sd wfgsf] pTkfbg ePsf] /x5] <

4. Pp6f ljBfnosf] tLg jifs{ f] P;OO{sf] glthf tnsf] ax':tDe lrqdf k:| t't
ul/Psf] 5 . pSt :tDe lrqsf cfwf/df tnsf k|Zgsf] pQ/

nV] g'xf];\ M

ljBfy {L ; \ªVof 18 Y P;OOs{ f] glthf ;ªs\ t]

16 A+
14 A
12

10 B

8

6

4

2

O 2075 2076 2077 X

;fn -lj=;=+ _ df

-s_ lj= ;= 2075 df slt hgf ljBfyLn{ ] u]8| A+ NofPsf /x]5g\ <

-v_ lj= ;= 2076 df slt hgf ljBfyLn{ ] u8]| A NofPsf /x5] g\ <

-u_ s'g ;fndf ;ae} Gbf a9L ljBfyL{n] u]8| A+ NofPsf /x5] g\ <

-3_ s'g ;fndf ;ae} Gbf sd ljBfyL{n] u|8] B NofPsf /x5] g\ <

kl/ofh] gf sfo{

tkfO{sF f] ljBfnosf] jflifs{ pT;j tyf cleefjs lbj;df ;~rfng ePsf
Dol' hsnlro/, !)) ld= bf}8, @)) ld= bf8} , rDrf bf}8, lx;fa bf}8df
;xefuL ePsf ljBfyLs{ f] ljj/0fsf cfwf/df ax':tDe lrq agfpgx' f;] \ .

pQ/
lzIfsnfO{ bv] fpgx' f];\ .

ul0ft, sIff & 279

jflif{s Aofhb/ - |kltztdf_ldl>t cEof;

1. sg' } ;x/sf] jz} fv dlxgfsf] Ps xKtfsf] clwstd tfkjm| d lgDglnlvt tflnsfdf
lbOPsf] 5 . pSt tYofª\ssf cfwf/df :tDelrq agfpg'xf];\ .

ut] clwstd tfkjm| d (°C df_

16 35.5
17 37
18 36
19 32
20 30
21 33
22 34.5

o; tYofª\snfO{ /]vflrqdf k:| tt' u/L lgDgfg;' f/ kZ| gsf] hjfkm lxgx' f;] \ M

-s_ ;a}eGbf a9L udL{ slt ut] lyof] <
-v_ 20 ut]sf] clwstd tfkj|md slt /x]5 <
-u_ Ps xKtfdf tfkjm| ddf slt kl/jtg{ eP5, 5nkmn ugx{' f];\ .
-3_ clwstd / Gog" td tfkjm| dlarsf] km/s nV] g'xf;] \ .

2. sg' } a}ªs\ n] d'2tL vftfdf /sd hDdf ug]{ vftfjfnfnfO{ lbPsf] Aofhb/nfO{
tnsf] /]vflrqdf bv] fOPsf] 5 . pSt /v] flrqsf cfwf/df lgDglnlvt k|Zgsf]
pQ/ n]Vgx' f];\ .

vftfjfnfnfO{ lbPsf] jflifs{ Aofhb/sf] ljj/0f

Y

16
14
12
10
8
6
4
2

O 2074 2075 2076 2077 2078 X

;fn -lj=;+=df_

280 ul0ft, sIff &

-s_ sg' jif{sf] Aofhb/ ;aeGbf a9L 5 <
-v_ s'g jif{sf] Aofb/ ;aeGbf sd 5 <
-u_ sg' s'g jifs{ f] Aofhb/ a/fa/ 5g\ <
-3_ o; tYofªs\ sf cfwf/df :tDelrq agfpgx' f;] \ .
3. sg' } ufpFsf] dlxnf / k?' ifsf] kfrF jifs{ f] hg;ª\Vof lgDgfg';f/ /x]sf] 5 M

;fn -lj=;_ 2073 2074 2075 2076 2077

hg;ªV\ of dlxnf k?' if dlxnf k'?if dlxnf k'?if dlxnf k?' if dlxnf k'?if

12 11 13 14 14 15 17 13 17 11

o; tYofªs\ sf cfwf/df lgDgfg';f/ kZ| gx¿sf] hjfkm n]Vg'xf;] \ M

-s_ dlxnfsf] eGbf k'?ifsf] ;ªV\ of a9L ePsf] ;fn pNnv] ugx{' f];\ .
-v_ kfFr jif{df dlxnfsf] hg;ªV\ of k?' ifsf] eGbf slt kl| tztn] a9L /xs] f] 5 <
-u_ s'g s'g jifd{ f dlxnf / k'?ifsf] ;ªV\ of a/fa/ /x]sf] 5 <

4. Pp6f t/sf/L k;ndf rf/ xKtfdf laj|mL ePsf] cfn' / sfpnLsf] ljj/0f lgDg
ax:' tDe lrqdf bv] fOPsf] 5 . pSt ax:' tDe lrqsf cfwf/df tnsf k|Zgsf] pQ/
n]Vg'xf;] \ .

cfn' / sfpnLsf] lajm| L ljj/0f ;ªs\ t]
cfn'
Y sfpnL
16
14 X
12
10
8
6
4
2

O
t/sf/Ls ]f kl/df0f -kg df_
klxn ]f xKtf
b ]f |; ]f xKtf

]t |; ]f xKtf
r }fy ]f xKtf

;do -xKtfdf_

ul0ft, sIff & 281

-s_ klxnf] xKtfdf slt lsnfu] |fd cfn' lajm| L ePsf] /x5] <
-v_ sg' xKtfdf cfn' / sfpnL a/fa/ kl/df0fdf lajm| L ePsf] /x]5 <
-u_ ;aeGbf a9L s'g xKtfdf sfpnL lajm| L ePsf] /x]5 <
-3_ ;aeGbf sd cfn' s'g xKtfdf lajm| L ePsf] /x5] <
-ª_ rfy} f] xKtfdf slt slt cfn' / sfpnL laj|mL ePsf] /x]5 <
-r_ bf;] |f] xKtfsf t'ngfdf t;] |f] xKtf sfpnL slt kl| tzt a9L laj|mL ePsf]

/x]5 <
-5_ laj|mLb/ w]/} 36a9 ePsf] t/sf/L s'g xf] <
pQ/
lzIfsnfO{ bv] fpgx' f;] \ .

282 ul0ft, sIff &