Class 9 Mathematics Chapter 11 (Hindi Medium) PDF

i`"Bh; {ks=kiQy vkSj vk;ru 157 vè;k; 11 i`"Bh; {ks=kiQy vkSj vk;ru 11.1 ,d yac o`Ùkh; 'kaoqQ dk i`"Bh; {ks=kiQy vHkh rd ge lok±xle vkÑfr;ksa dks ,d osQ...

i`"Bh; {ks=kiQy vkSj vk;ru 157 vè;k; 11 i`"Bh; {ks=kiQy vkSj vk;ru 11.1 ,d yac o`Ùkh; 'kaoqQ dk i`"Bh; {ks=kiQy vHkh rd ge lok±xle vkÑfr;ksa dks ,d osQ Åij ,d j[k dj Bksl tfur dj jgs FksA la;ksx ls bu vkÑfr;ksa dks fizTe ( prism) dgrs gSaA vc ,d vU; izdkj osQ Bkslksa dks ns[ksa tks fizTe ugha gSaA (bl izdkj osQ Bksl fijkfeM (pyramids) dgykrs gSaA) vkb, ns[ksa fd budks fdl izdkj tfur fd;k (cuk;k) tkrk gSA fØ;kdyki % ,d ledks.k f=kHkqt ABC ftldk dks.k B ledks.k gks] dkV yhft,A nksuksa yac Hkqtkvksa esa ls fdlh ,d] eku yhft, AB, osQ vuqfn'k ,d yach vkSj eksVh Mksjh fpidk nhft, [nsf[k, vkÑfr 11.1(a)]A Mksjh dks nksuksa gkFkksa ls f=kHkqt osQ nksuksa vksj ls idM+s gq,] f=kHkqt dks Mksjh osQ vuqfn'k dbZ ckj ?kqekb,A vki D;k ns[krs gS\a tc f=kHkqt Mksjh osQ vuqfn'k ?kwe jgk gS] rks tks og vkÑfr cuk jgk gS] D;k vki mls igpkurs gSa [nsf[k, vkÑfr 11.1(b)]\ D;k vkidks bl ckr dh ;kn fnykrh gS fd blh vkdkj osQ ,d NksVs crZu (ik=k) esa Hkjh vkius dHkh vkblØhe [kkbZ Fkh [nsf[k, vkÑfr 11.1 (c) vkSj (d)]\ vkÑfr 11.1 Rationalised 2023-24 158 xf.kr ;g vkÑfr ,d yac o`Ùkh; 'kaoqQ (right circular cone) dgykrh gSA vkÑfr 11.1(c) esa fcUnq A bl yEc o`Ùkh; 'kaoqQ dk 'kh"kZ (vertex) dgykrk gS] AB bldh Å¡pkbZ dgykrh gS vkSj BC vk/kj dh f=kT;k dgykrh gSA AC bl 'kaoqQ dh fr;Zd Å¡pkbZ (slant height) dgykrh gSA ;gk¡ B o`Ùkh; vk/kj dk oasQnz (centre) gSA 'kaoqQ dh Å¡pkbZ] f=kT;k vkSj fr;Zd Å¡pkbZ dks izk;% Øe'k% h, r vkSj l ls O;Dr fd;k tkrk gSA ,d ckj iqu% ns[ksa fd fdl izdkj osQ 'kaoqQ dks ge vkÑfr 11.2 yac o`Ùkh; 'kaoqQ ugha dg ldrs gSaA vki vkÑfr 11.2 dks nsf[k,A buesa tks vki 'kaoqQ ns[k jgs gSa os yac o`Ùkh; 'kaoqQ ugha gSaA (a) esa] 'kh"kZ dks vk/kj osQ oasQnz ls feykus okyh js[kk vk/kj ij yac ugha gS vkSj (b) esa] vk/kj o`Ùkh; ugha gSA tSlk fd csyu dh fLFkfr esa Fkk] tc rd vU;Fkk u dgk tk,] ^'kaoqQ* ls gekjk rkRi;Z yac o`Ùkh; ^'kaoqQ* ls gh gksxkA fØ;kdyki % (i) ,d lkiQ cus gq, dkx”k osQ 'kaoqQ dks mlosQ 'kh"kZ ls tkus okyh fdlh Hkqtk ;k fdukjs osQ vuqfn'k dkfV, ftlesa dksbZ vfrO;kfidrk u gks rFkk [kksy dj nsf[k, fd fdl vkdkj osQ dkx”k ls 'kaoqQ dk i`"B cuk FkkA (ftl Hkqtk ;k fdukjs osQ vuqfn'k vki 'kaoqQ dks dkVsaxs og mldh fr;Zd Å¡pkbZ gksxh ftls l ls O;Dr fd;k tkrk gSA) [kksyk gqvk dkx”k vkidks ,d xksy osQd osQ Hkkx dh rjg fn[kkbZ nsxkA (ii) ;fn vki mu Hkqtkvksa] ftuosQ fljksa ij A vkSj B vafdr gS] dks eksM+ dj feyk ysa] rks vki ns[ksaxs fd vkÑfr 11-3 (c) dk ofØr Hkkx 'kaoqQ dk o`Ùkh; vkèkkj cukrk gSA vkÑfr 11.3 Rationalised 2023-24 i`"Bh; {ks=kiQy vkSj vk;ru 159 (iii) ;fn vkÑfr 11.3 (c) esa fn, dkx”k dks O ls tkrh gqbZ js[kkvksa }kjk lSdM+ksa NksVs&NksVs VqdM+ksa esa foHkkftr dj fy;k tk,] rks ;s dVs gq, Hkkx yxHkx f=kHkqt osQ vkdkjksa osQ gSa vkSj buesa ls izR;sd dh Å¡pkbZ 'kaoqQ dh fr;Zd Å¡pkbZ l osQ cjkcj gSA 1 (iv) vc izR;sd f=kHkqt dk {ks=kiQy = × izR;sd f=kHkqt dk vk/kj × l 2 vr%] iwjs dkx”k dk {ks=kiQy = lHkh f=kHkqtksa osQ {ks=kiQyksa dk ;ksx 1 1 1 1 = b1l + b2l + b3l + = l ( b1 + b2 + b3 + ) 2 2 2 2 1 = × l × [vkÑfr 11.3(c) dh iwjh ofØr ifjlhek dh yackbZ] 2 (pw¡fd b1 + b2 + b3 +... feydj bl vkÑfr osQ ofØr Hkkx dks cukrs gSa) ijUrq bl ofØr Hkkx ls 'kaoqQ dk vk/kj curk gSA lkFk gh] bl vk/kj dh ifjf/ = 2πr, tgk¡ r vk/kj dh f=kT;k gSA 1 blfy,] 'kaoqQ dk oØ i`"Bh; {ks=kiQy = πr = πrl × l × 2π 2 tgk¡ r vk/kj dh f=kT;k gS vkSj l fr;Zd Å¡pkbZ gSaA è;ku nhft, fd l2 = r2 + h2 gksrk gS] ftls ge vkÑfr 11.4 ls ns[k ldrs gSa (ikbFkkxksjl izes; ls)A ;gk¡ h 'kaoqQ dh Å¡pkbZ gSA vkÑfr 11.4 vr%] l = r 2 + h2 gksxkA vc ;fn 'kaoqQ osQ vk/kj dks can j[kk tkrk gS] rks