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    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    v;k;15

    izkf;drk

    It is remarkable that a science, which began with the consideration of

    games of chance, should be elevated to the rank of the most important

    subject of human knowledge.

    (mYys[kuh; gSfd og foKku ftldh O;qRifk la;ksx osQ [ksy ls gqbZgS] ogekuo Kku osQ vfr egRoiw.kZfo"k; dh pkb;ksa rd igqp tkrh gSA )

    Pierr e Simon Laplace

    15.1Hkwfedk

    gesavius nSfud thou esa bl izdkj osQ dFku lquusdks feyrsjgrs gSa%(1)laHkor% vkt o"kkZ gksxhA(2)eq>slansggS fd og bl ijh{kk esa mkh.kZgksxkA(3)ok"kd ijh{kk easdfork osQ izFke vkusdhlaHkkouk lcls vf/dgSA(4)Mhty dh dher c

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    izkf;drk 323

    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    ;|fi izkf;drk dh O;qRifk tq, osQ [ksy lsgqbZFkh] fiQj Hkh bldk O;kid iz;ksxHkkSfrd foKku] okf.kT;] tSfod foKku] vk;qoKku] ekSle dk iwokZuqeku vkfn {ks=kksa esagksjgk gSA

    15.2izkf;drk & ,d izk;ksfxd n`f"Vdks.k

    izkf;drk (probability)dh ladYiuk dk fodkl,d vk'p;Ztud

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    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    lkj.kh15.1

    flDdk mNkyus fpr vkus dh iV vkus dhdh la[;k la[;k la[;k

    10

    uhpsnh xbZfHkUuksa osQ eku fyf[k,%

    fpr vku sdh l[a;k

    flDdk mNkyu sdh oQqy

    vkSjiV vku sdh l[a;k

    flDdk mNkyu sdh oQqy

    (ii)flDosQ dks chl ckj mNkfy, vkSj ij dh Hkkfr vki vius izs{k.k fy[k yhft,Aizs{k.kksa osQ bl laxzg osQ fy, ij fn, x, fHkUuksa osQ eku iqu% Kkr dhft,A

    (iii)flDosQ dksvkSj vf/d ckj mNkydj bl iz;ksx dksiqu% dhft, vkSj fpr vkSjiV vkus dh la[;k fy[k yhft,A bldsckn laxr fHkUuksa osQ eku Kkr dhft,A

    vki ns[ksaxsfd vki tSls&tSlsflDdk mNkyusdh la[;k c

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    izkf;drk 325

    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    vkadM+ksaosQ fy, djuk gksxk] vkSj blh izdkj bl izf;k dks vkxsc

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    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    fuEufyf[kr fHkUuksa osQ eku Kkr dhft,%

    ikl sij 1 oQs vku sdh liklk isaQdu sdh oQqy l

    ikl sij 2 oQs vku sdh liklk isaQdu sdh oQqy l

    M

    M

    ikl sij 6 oQs vku sdh liklk isaQdu sdh oQqy l

    (ii) vc iklsdks40 ckj isaQfd,] isz{k.kksadksfy[k yhft, vkSj (i)dh Hkk afr fHkUuksadks ifjdfyr dhft,A

    vki ns[ksaxs fd ikls osQ isaQdus dh la[;k esao`f gksus osQ lkFk&lkFk (i) vkSj(ii)esa ifjdfyr fd, x, izR;sd fHkUu dk eku 1

    6osQ vkSj fudV vkrk tkrk gSA

    blsns[kus osQ fy, vki ,d lkewfgd f;kdyki mlh izdkj dj ldrsgSa ftlizdkj vkius f;kdyki 2 esa fd;k gSA vki viuh d{kk osQ fo|kfFkZ;ksadks NksVs&NksVs oxksesackV nhft,A izR;sd oxZosQ ,d fo|kFkhZdks,d iklk nl ckj isaQdusosQ fy, dfg,Aizs{k.kksa dksfy[k yhft, vkSj lap;h fHkUu ifjdfyr dj yhft,A

    la[;k 1 osQ fy, fHkUuksaosQ eku lkj.kh 15-4 esa fy[ks tk ldrs gSaA bl lkj.kh esagh nwljh la[;kvksalslacaf/r fHkUuksadksHkh fy[kk tk ldrk gS;k vU; la[;kvksaosQ fy,blh izdkj dh vU; lkjf.k;kHkh cukbZtk ldrh gSaA

    lkj.kh15.4

    oxZ ,d oxZ esa,d iklsosQ 1 osQ vkus dh lap;h la[;k

    isaQosQ tkus dh oqQy la[;k ikls osQ isaQosQ tkus dh oqQy la[;k(1) (2) (3)

    1

    2

    3

    4

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    izkf;drk 327

    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    lHkh oxks esaiz;qDr ikls yxHkx leku :i vkSj leku lkbt osQ gksuk pkfg,A rc ,slkeku fy;k tk,xk fd isaQosQ x, lHkh ikls,d gh ikls}kjk isaQosQ x, gSaA

    bu lkjf.k;ksa esavki D;k ns[krsgSa\vki ns[ksaxs fd tSls&tSls iklk isaQosQ tkus dh la[;k c

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    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    vr%] vfHkiz;ksx ,d f;k gS ftlls ,d ;k vf/d ifj.kke izkIr gksrsgSa Af;kdyki 1 esa laHko ifj.kke fpr vkSj iV Fks] tcfd f;kdyki 3 esalaHko ifj.kke

    1, 2, 3, 4, 5 vkSj6 FksA

    f;kdyki 1 esa] ,d fo'ks"k mNky ij ,d fpr dk vkuk ifj.kke fpr okyh ,d?kVuk (event)gSA blh izdkj] ,d iV dk vkuk ifj.kke iV okyh ,d ?kVuk gSAf;kdyki 2 esa] ,d fo'ks"k la[;k] eku yhft, 1] dk vkuk ifj.kke 1 okyh ,d?kVuk gSA

    ;fn gekjk iz;ksx iklk isaQdusij ,d le la[;k izkIr djuk gks] rks?kVuk esa rhuifj.kke 2] 4 vkSj 6 gksaxsA

    vr%] ,d iz;ksx esa ?kVuk iz;ksx osQ oqQN ifj.kkeksadk laxzg gksrh gS A mPp d{kkvksaesa]vki ?kVuk dh vkSipkfjd ifjHkk"kk dk v;;u djsaaxsA

    vr%] D;k vc vki ;g crk ldrsgSafd f;kdyki 4 esa?kVuk,dkSu&dk Su lh gSa\bl i`"BHkwfe osQ lkFk] vkb, vc ge ns[ksafd izkf;drk D;k gksrh gSA vius vfHkiz;ksxksa

    osQ ifj.kkeksa dkslh/sns[kusij ge izk;ksfxd (experimental);k vkuqHkfod (empirical)izkf;drk izkIr djrsgSaA

    eku yhft, vfHkiz;ksxksa dh oqqQy la[;k ngSA ?kVuk EosQ ?kVus dh vkuqHkfodizkf;drk (empirical probability )fuEu ls ifjHkkf"kr gS %

    P(E) =v f H k i ; z k

    sxkas dh l[a;k ftueas ?kVuk ?kVvfHki;zkxskas dh oqQy l[a;k

    bl v;k; esa] ge vkuqHkfod izkf;drk Kkr djsaxsA rFkk lqfo/k osQ fy, vkuqHkfodizkf;drk osQ LFkku ij osQoy izkf;drk* dk iz;ksx djsaxsA

    vkb, ge oqQN mnkgj.k ysaAvkb, lcls igys ge f;kdyki 2 ij okfil vk tk,vkSj lkj.kh 15-2 ysaA bl

    lkj.kh osQ LraHk (4) esa og fHkUu D;k gSftls vkiusifjdfyr fd;k gS\ tksifjdfyrfd;k gSog vkSj oqQN ugha gS] vfirqfpr izkIr djus dh vkuqHkfod izkf;drk gSA ;kunhft, fd vfHkiz;ksxksa dh la[;k vkSj bu vfHkiz;ksxksa esa fpr vkusdh la[;k osQ vuqlkj

    izkf;drk esaifjorZu gksrk jgrk gSA blh izdkj] iV vkus dh vkuqHkfod izkf;drk lkj.kh15-2 osQ LraHk (5) esa izkIr dh xbZ gSA izkjaHk esa ;g

    1

    1gS] blds ckn

    2

    3gS vkSj fiQj

    2

    4gS] vkfn&vkfnA

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    izkf;drk 329

    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    vr%] vkuqHkfod izkf;drk fd, x, vfHkiz;ksxksadh la[;k vkSj bu vfHkiz;ksxkssaesaizkIrgq, ifj.kkeksa dh la[;k ij fuHkZj djrh gSA

    f;kdyki 5 :vkxs v;;u djusls igys] mu lkjf.k;ksadks ns[ksa ftUgsavkiusf;kdyki 3 djrs le; cuk;k FkkA ,d iklsdksvusd ckj isaQdusij 3 osQ vkusdhizkf;drk,Kkr dhft,A lkFk gh] ;g Hkh fn[kkb, fd vfHkiz;ksxksa dh la[;k c

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    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    gy:vkb, ge nksfprksa osQ vkus dh ?kVuk dks E1ls] ,d fpr osQ vkusdh ?kVuk dksE

    2ls vkSj dksbZHkh fpr u vkusdh ?kVuk dks E

    3ls izdV djsaA

    vr%] P(E1) =

    1 0

    5 0= 0.21

    P(E2) =

    2 7

    5 0= 0.55

    P(E3) =

    1 2

    5 0= 0.24

    ;gkvki ;g ns[k ldrs gSafd P(E1) + P(E

    2) + P(E

    3) = 1gSA lkFk gh] E

    1, E

    2

    vkSjE3esa,d vfHkiz;ksx osQ lHkh ifj.kke vk tkrs gSaAmnkgj.k3 :,d ikls dks 1000 ckj isaQdus ij izkIr ifj.kkeksa 1] 2] 3] 4] 5 vkSj 6 dhckjackjrk,lkj.kh 15-6 esanh xbZgSa%

    lkj.kh15.6

    ifj.kke 1 2 3 4 5 6

    ckjackjrk 179 150 157 149 175 190

    izR;sd ifj.kke osQ izkIr gksusdh izkf;drk Kkr dhft,Agy:eku yhft, E

    iifj.kke iosQ izkIr gksus dh ?kVuk dks izdV djrk gS] tgk i= 1,

    2, 3, 4, 5, 6gSA rc]

    ifj.kke1dh izkf;drk= P(E1) =

    1 dh ckjcakjrk

    iklk isaQdu sdh oqQy

    =1 7

    1 0 0= 0.179

    blh izdkj] P(E2) =

    1 5

    1 0 0= 0.15, P(E

    3) =

    1 5

    1 0 0= 0.157, P(E

    4) =

    1 4

    1 0 0= 0.149,

    P(E5) =

    1 7

    1 0 0= 0.175 vkSjP(E

    6) =

    1 9

    1 0 0= 0.19gSA

    ;ku nhft, fdP(E1) + P(E

    2) + P(E

    3) + P(E

    4) + P(E

    5) + P(E

    6) = 1gSA

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    izkf;drk 331

    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    lkFk gh] ;g Hkh nsf[k, fd%(i) izR;sd ?kVuk dh izkf;drk 0 vkSj 1 osQ chp gksrh gSA

    (ii) lHkh izkf;drkvksa dk ;ksxiQy 1 gksrk gSA

    (iii) E1, E

    2, . . ., E

    6esa ,d vfHkiz;ksx osQ lHkh laHko ifj.kke vk tkrs gSaA

    mnkgj.k4 :,d VsyhiQksu funsZf'kdk osQ ,d i`"B ij 200 VsyhiQksu uacj gSaA muosQ bdkbZLFkku okysvad dk ckjackjrk caVu (mnkgj.k osQ fy, la[;k 25828573esa bdkbZosQ LFkkuij vad 3 gS) lkj.kh 15-7 esafn;k x;k gS %

    lkj.kh15.7

    vad 0 1 2 3 4 5 6 7 8 9

    ckjackjrk 22 26 22 22 20 10 14 28 16 20

    i`"Bdks ns[ks fcuk] bu la[;kvksaesalsfdlh ,d la[;k ij viuh isafly j[k nhft,] vFkkZr~la[;k dks;kn`PN;k pquk x;k gSA bdkbZosQ LFkku ij vad 6 osQ gksus dh izkf;drk D;k gksxh\

    gy:bdkbZosQ LFkku ij vad 6 osQ gksus dh izkf;drk

    =

    6 dh ckjcakjrk

    puqh x, VyshiQkus uacjkas dh oQqy

    =1

    2 0= 0.07

    blh izdkj] vki mu la[;kvksaosQ vkus dh izkf;drk izkIr dj ldrs gSa ftuesa bdkbZosQ LFkku ij dksbZ vU; vad gksA

    mnkgj.k5 :,d ekSle osQanzosQ fjdkMZ dks ns[kus ls irk pyrk gS fd fiNys 250ekxr fnuksa esa fd, x, ekSle iwokZuqekuksaesals 175 ckj mldsiwokZuqeku lgh jgsgSaA

    (i) ,d fn, gq, fnu ij iwokZuqeku osQ lgh gksus dh izkf;drk D;k gksxh\

    (ii)fn, gq, fnu ij iwokZuqeku osQ lgh u gksusdh izkf;drk D;k gksxh\

    gy:fnuksadh oqQy la[;k ftuosQ fjdkMZmiyC/ gSa= 250

    http://schools.aglasem.com/
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    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    (i) P(fn, gq, fnu ij iwokZuqeku lgh Fkk)

    =mu fnukas dh l[a;k ftl fnu dk iowkZueqkulghmu fnukas dhoQqy l[a;k ftuoQs fjdkM ZmiyC/

    =1 7

    2 5= 0.7

    (ii) mu fnuksadh la[;k ftl fnu dk iwokZuqeku lgh ughaFkk= 250 175 = 75

    vr%] P(fn, gq, fnu ij iwokZuqeku lgh ugha Fkk)=7

    2 5= 0.3

    ;ku nhft, fd%P(fn, gq, fnu dk iwokZuqeku lgh Fkk)+ P(fn, gq, fnu dk iwokZuqeku lghughaFkk)

    = 0.7 + 0.3 = 1

    mnkgj.k6 :Vk;j cukus okyh ,d oaQiuh r; dh xbZmu nwfj;ksa dk ,d fjdkMZj[krhFkh] ftlosQ igysVk;j dks cny nsusdh vko';drk iM+hA lkj.kh esa 1000 fLFkfr;ksaosQifj.kke fn[kk, x, gSaA

    lkj.kh15.8

    nwjh ( kmesa) 4000lsde 4000ls9000 9001ls14000 14000ls vf/drd rd

    ckjackjrk 20 210 325 445

    ;fn vki bl oaQiuh ls,d Vk;j [kjhnrsgSa] rksbl ckr dh izkf;drk D;k gksxh fd(i) 4000 kmdh nwjh r; djus lsigysgh bls cnyuk vko';d gksxk\(ii);g 9000 kmls Hkh vf/d nwjh rd pysxk\(iii) 4000 kmvkSj 14000 kmosQ chp dh dksbZ nwjh r; djus osQ ckn bls cnyuk

    vko';d gksxk\

    gy:vfHkiz;ksxksadh oqQy la[;k= 1000

    (i) ml Vk;j dh ckjackjrk] ftls4000 kmdh nwjh r; djus ls igyscnyukvko';d gks] 20 gSA

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    izkf;drk 333

    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    vr%]P(4000 kmdh nwjh r; djusls igys Vk;j cnyuk vko';d gks)

    =2 0

    1 0 0= 0.02

    (ii) ml Vk;j dh ckjackjrk tks 9000 kmlsHkh vf/d nwjh r; djsxk= 325 + 445 = 770

    vr%] P(Vk;j 9000 kmlsHkh vf/d nwjh rd pysxk)=7 7

    1 0 0= 0.77

    (iii) ml Vk;j dh ckjackjrk ftls4000 kmvkSj 14000 kmosQ chp dh dksbZ nwjh r;

    dj ysusosQ ckn cnyuk vko';d gksxk = 210 + 325 = 535vr%]P(4000 kmvkSj 14000 kmosQ chp dh dksbZnwjh r; djusosQ ckn Vk;j

    dkscnyuk vko';d gks)=5 3

    1 0 0= 0.535

    mnkgj.k7 :,d fo|kFkhZ}kjk ekfld ;wfuV ijh{kk esa izkIr fd, x, vadksadk izfr'kruhpsfn;k x;k gS%

    lkj.kh15.9

    ;wfuV ijh{kk I II III IV VizkIr vadksa dk 69 71 73 68 74izfr'kr

    bu vkadM+ksa osQ vk/kj ij bl ckr dh izkf;drk Kkr dhft, fd ,d ;wfuV ijh{kk esa ogfo|kFkhZ70%lsvf/d vad izkIr djrk gSA

    gy:yh xbZ ;wfuV ijh{kkvksadh oqQy la[;k 5 gSA

    mu ;wfuV ijh{kkvksadh la[;k] ftuesafo|kFkhZ 70%ls vf/d vad izkIr djrk gS] 3 gSA

    vr%]P(70%ls vf/d vad izkIr djuk)=3

    5= 0.6

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    334 xf.kr

    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    mnkgj.k8 :,d chek oaQiuh usvk;q vkSj nq?kZVukvksaosQ chp osQ laca/ dks Kkr djusosQ fy, ,d fo'ks"k uxj osQ 2000 Mkbojksadk ;n`PN;k p;u fd;k (fdlh Mkboj dksdksbZfo'ks"k ojh;rk fn, fcuk)A izkIr fd, x, vkadM+suhpslkj.kh esa fn, x, gSa%

    lkj.kh15.10

    Mkbojksadh ,d o"kZesa ?kVh nq?kZVuk,

    vk;q(o"kksesa)0 1 2 3 3lsvf/d

    18 - 29 440 160 110 61 35

    30 - 50 505 125 60 22 18

    50ls vf/d 360 45 35 15 9

    uxj ls ;n`PN;k pqusx, ,d Mkboj osQ fy, fuEufyf[kr ?kVukvksa dh izkf;drk,Kkrdhft,%

    (i) 18&29 o"kZ dh vk;qdk ftlosQ lkFk ,d o"kZesaBhd&Bhd 3 nq?kZVuk, ?kVh gSaA

    (ii) 30&50 o"kZdh vk;qdk ftlosQ lkFk ,d o"kZesa ,d ;k vf/d nq?kZVuk,?kVh gSaA

    (iii)ftlosQ lkFk ,d o"kZ esadksbZnq?kZVuk ugha?kVhA

    gy:Mkbojksadh oqQy la[;k= 2000

    (i) mu Mkbojksa dh la[;k] ftudh vk;q 18&29 o"kZgSvkSj ftuosQ lkFk ,d o"kZ esaBhd&Bhd rhu nq?kZVuk,?kVh gSa] 61 gSA

    vr%]P (Mkboj 18&29 o"kZ dk gksftlosQ lkFk Bhd&Bhd rhu nq?kZVuk, ?kVh)

    =6 1

    2 0 0

    = 0.0305 0.031

    (ii)mu Mkbojksa dh la[;k] ftudh vk;q 30&50 o"kZgSvkSj ftuosQ lkFk ,d o"kZ esa,d ;k vf/d nq?kZVuk, ?kVh gSa] 125 + 60 + 22 + 18] vFkkZr~225gSA

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    izkf;drk 335

    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    vr%] P (Mkboj 35&50 o"kZ dk gks vkSj ftlosQ lkFk ,d ;k vf/d nq?kZVuk, ?kVh gSa)

    =2 2

    2 0 0= 0.1125 0.113

    (iii)mu Mkbojksa dh la[;k ftuosQ lkFk ,d o"kZ esa dksbZ nq?kZVuk ugha ?kVh= 440 + 505 + 360 = 130 5

    vr%]P(Mkboj ftuosQ lkFk dksbZ nq?kZVuk ugha ?kVh)=1 3 0

    2 0 0= 0.653

    mnkgj.k9 :ckjackjrk caVu lkj.kh (v;k; 14 osQ mnkgj.k 4 dh lkj.kh 14-3) yhft,ftlesa ,d d{kk osQ 38 fo|kFk;ksa osQ Hkkj fn, x, gSaA

    (i) bl ckr dh izkf;drk Kkr dhft, ftlesa d{kk osQ ,d fo|kFkhZdk Hkkj (kgesa)varjky 46&50 fLFkr gksA

    (ii)bl lanHkZ esa,slh nks ?kVuk, crkb, ftuesa ,d dh izkf;drk 0 gks vkSj nwljh dhizkf;drk 1 gksA

    gy: (i)fo|kFk;ksa dh oqQy la[;k 38 gS vkSj 40&50 kgosQ Hkkj okys fo|kFk;ksadhla[;k 3 gSA

    vr%]P (fo|kFkhZdk Hkkj 46&50 kggS)=3

    3= 0.079

    (ii)mnkgj.k osQ fy, og ?kVuk yhft, ftlesafo|kFkhZdk Hkkj 30 kggSA D;ksafdfdlh Hkh fo|kFkhZ dk Hkkj 30 kgughagS] blfy, bl ?kVuk osQ ?kVus dhizkf;drk 0 gksxhA blh izdkj] ,d fo|kFkhZ dk 30 kgls vf/d Hkkj gksusdh

    izkf;drk3

    3= 1gSA

    mnkgj.k10 :chtksaosQ 5 FkSyksa esa ls izR;sd FkSys lsipkl cht ;n`PN;k pqudj mUgsa ,slhekudho`Qr voLFkkvksaesaj[kk x;k tksvaoqQj.k osQ vuqowQy gSaA 20 fnu ckn izR;sd laxzgesavaoqQfjr gq, chtksa dh la[;k fxu dj uhpsn'kkZ, vuqlkj ,d lkj.kh esa fy[kh xbZA

    lkj.kh15.11

    FkSyk 1 2 3 4 5

    vaoqQfjr chtksa 40 48 42 39 41dh la[;k

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    336 xf.kr

    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    fuEufyf[kr chtksa osQ vaoqQj.k dh izkf;drk D;k gS\(i) ,d FkSys esa 40 lsvf/d cht\

    (ii) ,d FkSysesa 49 cht

    (iii) ,d FkSysesa 35 lsvf/d cht

    gy:FkSyksa dh oqQy la[;k 5 gSA

    (i) mu FkSyksadh la[;k] ftuesa 50 chtksaesa ls 40 lsvf/d cht vaoqQfjr gq, gS] 3gSaA

    vr%] P(,d FkSys esa 40 lsvf/d chtksa dk vaoqQj.k)= 53

    = 0.6

    (ii) mu FkSyksa dh la[;k ftuesa 49 cht vaoqQfjr gq, gSa] 0 gSA

    vr%] P(,d FkSysosQ 49 chtksa dk vaoqQj.k) =0

    5= 0

    (iii) mu FkSyksa dh la[;k] ftuesa 35 ls vf/d cht vaoqQfjr gq, gSa] 5gSA

    vr%] visf{kr izkf;drk=5

    5= 1

    fVIi.kh: ij fn, x, lHkh mnkgj.kksaeas bl ckr dh vksj vkiusvo'; ;ku fn;k

    gksxk fd fdlh ?kVuk dh izkf;drk 0 ls 1 rd dh dksbZHkh fHkUu gksldrh gSA

    iz'ukoyh15.1

    1. ,d fosQV eSp esa] ,d efgyk cYysckt [ksyh xbZ30 xsnksa esa 6 ckj pkSdk ekjrh gSApkSdk u ekjstkusdh izkf;drk Kkr dhft,A

    2. 2 cPpksaokys1500 ifjokjksadk ;n`PN;k p;u fd;k x;k gSvkSj fuEufyf[kr vkadM+sfy[k fy, x, gSa%

    ifjokj esa yM+fd;ksa dh la[;k 2 1 0

    ifjokjksadh la[;k 475 814 211

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    izkf;drk 337

    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    ;n`PN;k pqus x, ml ifjokj dh izkf;drk Kkr dhft,] ftlesa

    (i) nksyM+fd;k gksa (ii),d yM+dh gks (iii)dksbZ yM+dh u gks

    lkFk gh] ;g Hkh tkp dhft, fd bu izkf;drkvksadk ;ksxiQy 1 gS;k ughaA

    3. v;k; 14 osQ vuqPNsn 14-4 dk mnkgj.k 5 yhft,A d{kk osQ fdlh ,d fo|kFkhZdktUe vxLr esa gksusdh izkf;drk Kkr dhft,A

    4. rhu flDdksadks ,d lkFk 200 ckj mNkyk x;k gSrFkk buesa fofHkUu ifj.kkeksadhckjackjrk,;sgSa%

    ifj.kke 3fpr 2fpr 1fpr dksbZHkh fpr ugha

    ckjackjrk 23 72 77 28

    ;fn rhuksa flDdksadks iqu% ,d lkFk mNkyk tk,] rksnks fpr osQ vkus dh izkf;drk Kkrdhft,A

    5. ,d oaQiuh us;n`PN;k 2400 ifjokj pqudj ,d ?kj dh vk; Lrj vkSj okguksa dh la[;kosQ chp laca/ LFkkfir djusosQ fy, mudk losZ{k.k fd;kA ,df=kr fd, x, vkadM+suhpslkj.kh esa fn, x, gSa%

    ekfld vk; izfr ifjokj okguksa dh la[;k

    (# esa)0 1 2 2lsvf/d

    7000ls de 10 160 25 0

    7000 10000 0 305 27 2

    10000 13000 1 535 29 1

    13000 16000 2 469 59 25

    16000;k blls vf/d 1 579 82 88

    eku yhft, ,d ifjokj pquk x;k gSA izkf;drk Kkr dhft, fd pqus x, ifjokj

    (i) dh vk; 10000&13000 # izfr ekg gS vkSj mlosQ ikl Bhd&Bhd nks okgu gSaA(ii) dh vk; izfr ekg 16000 # ;k bllsvf/d gSvkSj mlosQ ikl Bhd 1 okgu gSA

    (iii) dh vk; 7000 # izfr ekg lsde gSvkSj mlosQ ikl dksbZokgu ughagSA

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    338 xf.kr

    File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-15\Chap-15.PM65

    (iv) dh vk; 13000&16000 # izfr ekg gSvkSj mlosQ ikl 2 ls vf/d okgu gSA

    (v) ftlosQ ikl 1 ls vf/d okgu ugha gSA

    6. v;k; 14 dh lkj.kh 14-7 yhft,A

    (i) xf.kr dh ijh{kk esa ,d fo|kFkhZ }kjk 20%de vad izkIr djus dh izkf;drk Kkrdhft,A

    (ii) ,d fo|kFkhZ}kjk 60 ;k bllsvf/d vad izkIr djus dh izkf;drk Kkr dhft,A

    7. lkaf[;dh osQ ckjs esa fo|kFk;ksadk er tkuusosQ fy, 200 fo|kfFkZ;ksadk losZ{k.k fd;kx;kA izkIr vkadM+ksadksuhpsnh xbZlkj.kh esa fy[k fy;k x;k gS%

    er fo|kfFkZ;ksadh la[;k

    ilan djrsgSa 135

    ilan ughadjrs gSa 65

    izkf;drk Kkr dhft, fd ;n`PN;k pquk x;k fo|kFkhZs

    (i) lkaf[;dh ilan djrk gS (ii)lkaf[;dh ilan ugha djrk gSA

    8. iz'ukoyh 14-2 dk iz'u 2 nsf[k,A bldh vkuqHkfod izkf;drk D;k gksxh fd bathfu;j

    (i) viusdk;ZLFky ls 7 kmls de nwjh ij jgrh gS\

    (ii) viusdk;ZLFky ls 7 km;k blls vf/d nwjh ij jgrh gS\

    (iii) viusdk;ZLFky ls1

    2 km;k blls de nwjh ij jgrh gS\

    9. f;kdyki:viusfo|ky; osQ xsV osQ lkeusls ,d le;&varjky esaxqtjus okysnksifg;k] rhu ifg;k vkSj pkj ifg;k okguksa dh ckjackjrk fy[k yhft,A vki }kjk ns[ks x,okguksa esalsfdlh ,d okgu dk nksifg;k okgu gksusdh izkf;drk Kkr dhft,A

    10.f;kdyki:vki viuh d{kk osQ fo|kFk;ksa ls,d 3 vad okyh la[;k fy[kusdksdfg,A vki d{kk ls,d fo|kFkhZ dks ;nPN;k pqu yhft,A bl ckr dh izkf;drk D;kgksxh fd mlosQ }kjk fy[kh xbZla[;k 3 lsHkkT; gS\ ;kn jf[k, fd dksbZla[;k 3 ls HkkT;

    gksrh gS] ;fn mlds vadksa dk ;ksx 4 lsHkkT; gksA11. vkVs dh mu X;kjg FkSfy;ksa esa] ftu ij 5 kgvafdr gS] okLro esavkVs ds fuEufyf[kr

    Hkkj (kgesa) gSa%

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