Tiêu đề Physics (ভেক্টর) physics first paper ACS.pdf ✅

†f±i  Final Revision Batch 1 †f±i Vector wØZxq Aa ̈vq Topicwise CQ Trend Analysis UwcK 2016 2017 2018 2019 2021 2022 2023 †gvU †f±‡ii †hvM-we‡qvM I jwä Ñ 1 1 2 2 2 Ñ 8 AvqZ GKK †f±i 1 1 1 Ñ Ñ 1 2 6 †f±i ivwki wefvRb (Dcvsk) 1 Ñ Ñ 1 Ñ 1 Ñ 3 e„wó I QvZv aiv msμvšÍ Ñ 1 Ñ 1 Ñ 2 2 6 †f±‡ii ̧Y, †ÿÎdj I j¤^ Awf‡ÿc, GKB mgZj, ga ̈eZ©x †KvY 5 4 Ñ 3 7 4 5 28 b`x cvivcvi Ñ 2 Ñ 1 4 4 9 20 †f±i Acv‡iUi Ñ Ñ Ñ 1 Ñ 4 Ñ 5 * we.`a.: 2020 mv‡j GBPGmwm cixÿv AbywôZ nqwb| weMZ mv‡j †ev‡W© Avmv m„Rbkxj cÖkœ 1| 1 km cÖ‡ ̄’i GKwU b`x cvi nIqvi Rb ̈ `yBRb muvZviæ, muvZvi cÖwZ‡hvwMZvq AskMÖnY K‡i| cÖ_g muvZviæ 6 kmh–1 †e‡M † ̄av‡Zi cÖwZK‚‡ji mv‡_ 60 †Kv‡Y Ges wØZxq muvZviæ 6 kmh–1 †e‡M AvovAvwofv‡e muvZvi KvUv ïiæ K‡i| b`x‡Z † ̄av‡Zi †eM 3 kmh–1 | [Xv. †ev. 23] (K) †K.wYK fi‡eM Kx? (L) Kx Kx k‡Z© Kv‡Ri gvb k~b ̈ n‡Z cv‡i? e ̈vL ̈v K‡iv| (M) cÖ_g muvZviæi jwä †eM wbY©q K‡iv| DËi: 5.196 km/h (N) D3 cÖwZ‡hvwMZvq †Kvb muvZviæ Av‡M b`x cvi n‡Z cvi‡e? DËi: t1 = 0.1924 hr ; t2 = 0.167 hr ; t2 < t1 nIqvq wØZxq mvZviæ Av‡M b`x cvi n‡Z cvi‡e| 2| GKwU Mvwoi †cQ‡bi Møvm Qv‡`i mv‡_ 30 †Kv‡Y †njv‡bv| MvwowU v  = 18i  †e‡M GKwU iv ̄Ívq PjwQj| nvVvr e„wó u  = – 12j  †e‡M cov ïiæ n‡jv| [Xv. †ev. 23] (K) wkwkiv1⁄4 Kv‡K e‡j? (L) `iRvi nvZj cÖv‡šÍ †`qv nq †Kb? e ̈vL ̈v K‡iv| (M) Mvwoi mvg‡bi Møv‡m e„wó KZ †e‡M co‡e? DËi: 21.63 GKK (Ans.) (N) DÏxc‡Ki Mvwoi wcQ‡bi Møvm e„wó‡Z wfR‡e wK bvÑ MvwYwZK we‡kølYc~e©K gZvgZ `vI| DËi:  = 33.7;  > 30 nIqvq e„wó Mvwoi wcQ‡bi KvP‡K wfRv‡e| 3| P  = 5i  + 3j  – mk  ; Q  = i  + j  + 4k  ; GLv‡b P  I Q  ci ̄úi j¤^| hw` P  Ges Q  Gi gvb h_vμ‡g †bŠKv Ges GKwU b`xi † ̄av‡Zi `aæwZ wb‡`©k K‡i Z‡e me©wb¤œ c‡_ b`x cvi n‡Z †bŠKvwUi 2 wgwbU mgq jv‡M| [iv. †ev. 23; w`. †ev. 23; g. †ev. 23] (K) mgZjxq †f±‡ii msÁv `vI| (L) †Kv‡bv cÖevnxi AvqZ‡bi cwieZ©b wbY©‡q WvBfvi‡RÝ Gi f~wgKv Av‡Q wK-bv? e ̈vL ̈v Ki| (M) ‘m’ Gi gvb wnmve Ki| DËi: m-Gi gvb 2 (N) hw` †b.Kvi gvwS b~ ̈bZg mg‡q b`x cvi n‡Z Pvq Z‡e †m b`xi cÖ ̄ A‡cÿv †ewk `~iZ¡ AwZμg Ki‡e wK-bv? MvwYwZK c×wZi mvnv‡h ̈ e ̈vL ̈v Ki| DËi: s = 1.214 d ; hw` †bŠKvi gvwS b~ ̈bZg mg‡q b`x cvi n‡Z Pvq Z‡e †m b`xi cÖ ̄’ A‡cÿv †ewk `~iZ¡ AwZμg Ki‡e| 4| A  = 3i  + 2j  + k  , B  = i  + 2j  + 3k  Ges C  = i  + 2j  + 2k  †f±iÎq wg‡j GKwU wÎgvwÎK †ÿÎ MVb K‡i| [Kz. †ev. 23] (K) wecÖZxc †f±i Kx? (L) †Vjvi †ÿ‡Î jb‡ivjvi‡K fvix g‡b nq †Kb, e ̈vL ̈v K‡iv| (M) B  eivei A  Gi j¤^ Awf‡ÿc wbY©q K‡iv| DËi: 10 14 GKK (N) A  , B  I C  †f±i wZbwU GKB mgZ‡j Aew ̄ Z n‡e wKbvÑ MvwYwZKfv‡e we‡kølY Ki| DËi: A  .(B )   C   0; A  , B  , C  †f±i wZbwU GKB mgZ‡j Aew ̄’Z bq| 5| A  = i  + j  – k  I B  = 2i  – 2j  – 3k  `ywU †f±i| [h. †ev. 23] (K) Ae ̄ vb †f±i Kx? (L) GKB RvZxq `ywU †f±‡ii †hvMdj I we‡qvMdj mgvb n‡Z cv‡i wK? e ̈vL ̈v Ki| (M) DÏxc‡K ewY©Z †f±iØq Øviv MwVZ PZzfz©‡Ri †ÿÎdj wbY©q Ki| DËi: 42 eM© GKK (N) †f±iØq ci ̄úi j¤^ wKbv Zv MvwYwZKfv‡e hvPvB Ki| DËi: A  .B   0; A  I B  ci ̄úi j¤^ bq|
2  HSC Physics 1st Paper Chapter-2 6| C A B AB = 1.2 km BC = 0.5 km wP‡Î BC b`xi wKbviv †Nu‡l Pjv GKwU iv ̄Ív nj AB| b`x‡Z † ̄av‡Zi †eM 2 ms–1 Ges GKRb gvwS 4 ms–1 †e‡M †bŠKv Pvjv‡”Qb| iv ̄Ívq Mvwoi †eM 15 ms–1 | Mvwo B †÷k‡b 40 s hvÎv weiwZ †`q| [h. †ev. 23] (K) †f±i †hv‡Mi wÎfzR m~ÎwU †jL| (L) Uawj e ̈v‡Mi nvZj j¤^v Kivi †h.w3KZv e ̈vL ̈v K‡iv| (M) gvwS me©wb¤œ KZ mg‡q b`x cvwo w`‡Z cvi‡e? DËi: 125 s (N) Mvwo hLb A Ae ̄ v‡b ZLb C Ae ̄ vb †_‡K B Ae ̄ v‡bi D‡Ï‡k ̈ hvÎv Kiv †b.Kvi hvÎxiv Mvwo‡Z DV‡Z cvi‡e wKbv? we‡kølY K‡iv| DËi: †bŠKvi hvÎxiv Mvwo‡Z DV‡Z cvi‡e bv| 7| Y Q(–2, 2, 1) O(0, 0, 0) P(2, – 1, 3) Z X [wm. †ev. 23] (K) † ̄..jvi A‡cÿ‡Ki MÖ ̈vwW‡q‡›Ui msÁv †jL| (L) KvR I UK© Gi GKK AwfbœÑ e ̈vL ̈v K‡iv| (M) PQ  †f±‡ii mgvšÍivj GKwU GKK †f±i wbY©q K‡iv| DËi: 1 29 (– 4i )  + 3j  – 2k  (N) OPQ mg‡KvYx wÎfzR wK bv hvPvB K‡iv| DËi: OP2 + OQ2  PQ2 ; myZivs OPQ mg‡KvYx wÎfzR bq| 8| Kv‡Z©mxq ̄’vbv1⁄4 e ̈e ̄’vq wZbwU we›`y O (0, 0, 0), P (2, 4, 2) Ges Q (2, –4, –4)| [Xv. †ev. 22] (K) mgvb †f±i Kx? (L) †f±i Acv‡iUi † ̄..jvi ivwk‡K †f±i ivwk‡Z iƒcvšÍi K‡iÑ e ̈vL ̈v Ki| (M) PQ  Gi gvb wbY©q Ki| DËi: 10 GKK (N) P I Q Gi Ae ̄ vb †f±iØq ci ̄úi j¤^ n‡e wK-bv hvPvB Ki| DËi: OP  .OQ   0 e‡j †f±i `ywU ci ̄úi j¤^ bq| 9| kvšÍ evZv‡m 6 kmh–1 †e‡M e„wó co‡Q| G mg‡q mvB‡K‡j P‡o Avwe` 8 kmh–1 †e‡M evwo wdi‡Q| nVvr Avwe‡`i Pjvi wecixZ w`‡K 2 kmh–1 †e‡M evZvm cÖevwnZ n‡Z jvMj| Dfq †ÿ‡Î e„wó †_‡K evuP‡Z Avwe` QvZv e ̈envi Kij| [Xv. †ev. 22] (K) gyw3 †eM Kx? [6ô Aa ̈vq] (L) ej I miY k~b ̈ bv n‡jI KvR k~b ̈ n‡Z cv‡i wK? e ̈vL ̈v Ki| [5g Aa ̈vq] (M) w ̄ i evZv‡m e„wói jwä †eM wbY©q Ki| vrc   vr  vc  –vc  DËi: 10 kmh–1 ;  = 53.13 (Dj‡¤^i mv‡_) (N) evZvm cÖevwnZ nIqvi Av‡M I c‡i GKBfv‡e QvZv ai‡j Avwe` e„wó †_‡K iÿv cv‡e wK-bv? MvwYwZKfv‡e hvPvB Ki| DËi: evZvm cÖevwnZ nIqvi c~‡e© 53.19 †Kv‡Y Ges evZvm cÖevwnZ nIqvi c‡i Dj‡¤^i mv‡_ 59.03 †Kv‡Y QvZv ai‡Z n‡e| 10| 500 m cÖ‡ ̄’i GKwU b`x‡Z 6 kmh–1 †e‡M † ̄avZ cÖevwnZ n‡”Q| GB b`xwU gvnxi I wbwa cÖwZ‡hvwMZvi D‡Ï‡k ̈ mvuZvi †K‡U cvi nIqvi wm×všÍ wb‡jv| gvnxi 10 kmh–1 †e‡M † ̄av‡Zi mv‡_  †Kv‡Y Ges wbwa 9 kmh–1 †e‡M † ̄av‡Zi mv‡_ j¤^fv‡e muvZvi KvU‡Z ïiæ Kij| [P. †ev. 22] (K) miY †f±i Kx? (L) †b.Kvi ̧Y Uvbvi mgq `wo hZ j¤^v nq †b.Kv ZZ `aæZ P‡j †Kb? e ̈vL ̈v Ki| (M) -†Kv‡Yi gvb KZ n‡j gvnxi †mvRvmywR b`xi Aci cv‡o †cu.Qv‡e? DËi: 126.87 (N) DÏxcK Abymv‡i †K wRZ‡e? MvwYwZK we‡kølYmn gZvgZ `vI| DËi: t1 = 0.0625 hour ; t2 = 0.0556 hour t2 < t1 nIqvq wbwa cÖwZ‡hvwMZvq wRZ‡e| 11| GKw`b GKwU A‡ji ZvcgvÎv I evZv‡mi †eM cvIqv †M‡jv h_vμ‡g, Q = 2xy2 z 3 – 4xy I v  = (y2 cosx + z3 )i ^ + (2ysinx – 4) j ^ + (3xz2 + 2) k ^ . [P. †ev. 22] (K) WU ̧Yb Kx? (L) N~Y©biZ c„w_ex m~h© n‡Z `~‡i m‡i †M‡j Gi †eM K‡g hvq †Kb? e ̈vL ̈v Ki| [6ô Aa ̈vq] (M) (1, –1, 2) we›`y‡Z H A‡ji ZvcgvÎvi †MÖwW‡q›U wbY©q Ki| DËi: 20i ^ – 36j ^ + 24k ^ (N) Hw`b H A‡ji evZv‡m †Kv‡bv N~Y©b wQ‡jv wKbv Zv MvwYwZK we‡køl‡Yi gva ̈‡g gZ `vI| DËi: Curl(V)  = 0; A_©vr Hw`b H A‡ji evZv‡m †Kv‡bv N~Y©b wQj bv| 12| B 3kmh–1 AiæY gvwS 8 kmh–1 eiæY gvwS 2 km † ̄avZ AiæY gvwS 8 kmh–1 †e‡M †bŠKv Pvwj‡q b`xi cÖ ̄’ eivei cvi nq| eiæY gvwS GKB †e‡M b`xi cÖ ̄’ eivei †bŠKv Pvjvq| b`xi cÖ ̄’ 2 km| [iv. †ev. 22]
†f±i  Final Revision Batch 3 (K) †f±i †hv‡Mi wÎfzR m~ÎwU †jL| (L) Uawj †e‡Mi nvZj j¤^v ivLvi myweav Kx? (M) DÏxc‡K AiæY gvwS‡K †Kvb w`‡K †b.Kv Pvjv‡Z n‡qwQj? DËi:  = 112.02 (N) DÏxc‡Ki †Kvb gvwS Kg mg‡q b`x cvi n‡e? MvwYwZKfv‡e e ̈vL ̈v Ki| DËi: t1 = 0.27 hr ; t2 = 0.25 hr ; t2 > t1 nIqvq eiæY gvwS Kg mg‡q b`x cvi n‡e| 13| wZbwU †f±i ivwk h_vμ‡g A  = 2i ^ + 2j ^ – k ^ , B  = 6i ^ – 3j ^ +2k ^ Ges C  = (6xy + z3 )i ^ – (3x2 – z)j ^ + (3xz2 – y) k ^ . [iv. †ev. 22] (K) iv ̄Ívi e ̈vswKs Kx? [4_© Aa ̈vq] (L) wZbwU †f±‡ii jwä KLb k~b ̈ nq? (M) A  I B  †f±i؇qi j¤^w`‡K GKK †f±i wbY©q Ki| DËi:  1 5 17 (i ) ^ – 10j ^ – 18k ^ (N) DÏxc‡K C  †f±iwU AN~Y©bkxj wK bv hvPvB Ki| DËi: Curl (C)   0 ; C  †f±iwU AN~Y©bkxj bq| 14| [Kz. †ev. 22] B A † ̄av‡Zi †eM, u = 3 kmh–1 wPÎ-1 muvZviæi †eM, v = 5 kmh–1 120 B A u = 3 kmh–1 wPÎ-2 v = 5 kmh–1 AB = b`xi cÖ ̄’ = 2.2 km 2.2 km cÖ‡ ̄ i GKwU b`x‡Z † ̄av‡Zi †eM 3 kmh–1 | mvuZvi cÖwZ‡hvwMZvq b`x cvwo †`Iqvi j‡ÿ ̈ cÖ_g mvuZviæ 5 kmh–1 †e‡M wPÎ-1 Abymv‡i Ges wØZxq mvuZviæ GKB †e‡M wPÎ-2 Abymv‡i mvuZvi Avi¤¢ Kij| (K) GKK †f±i Kv‡K e‡j? (L) †b.Kvi ̧Y Uvbvi †ÿ‡Î j¤^v iwk w`‡q ̧Y Uvbv nq †Kb? e ̈vL ̈v Ki| (M) cÖ_g mvuZviæi jwä †eM wbY©q Ki| DËi: 4.36 kmh–1 , † ̄av‡Zi mv‡_ 83.41 (N) D3 mvuZvi cÖwZ‡hvwMZvi djvdj MvwYwZKfv‡e we‡kølY Ki| DËi: t1 = 0.51 h ; t2 = 0.44 h ; 2q muvZviæ wRZ‡e| 15| |F | 2  = 8N O |F | 1  = 12N X Y =60  [Kz. †ev. 22] (K) †f±i †ÿ‡Îi WvBfvi‡RÝ Kv‡K e‡j? (L) Ae ̄ vb †f±i GKwU mxgve× †f±iÑ e ̈vL ̈v Ki| (M) ej `ywUi jwä X-A‡ÿi mv‡_ †h †Kv‡Y wμqvkxj Zv wbY©q Ki| DËi: 66.59 (N) ej `ywUi jwäi Abyf~wgK Dcvsk I Dj¤^ Dcvs‡ki g‡a ̈ †KvbwU †ewk? †Zvgvi gZvgZ MvwYwZK hyw3mn `vI| DËi: Fx = 6.92 N; Fy = 16 N Fy > Fx A_©vr Dj¤^ Dcvs‡ki gvb †ewk| 16| †Kv‡bv GKw`b 10 ms–1 †e‡M Lvovfv‡e e„wó cowQj| G mgq GKRb e ̈w3 20 ms–1 †e‡M Mvwo Pvwj‡q hvw”Q‡jb| [wm. †ev. 22] (K) UK© Kv‡K e‡j? (L) AwfKl© ej GKwU msiÿYkxj ej| e ̈vL ̈v Ki| (M) Mvwo PvjK e„wói †eM KZ cwijwÿZ Ki‡eb? DËi: 10 5 m/s (N) hw` Mvwoi MwZi wecixZ w`‡K 25 ms–1 †e‡M evqycÖevn P‡j Z‡e H e ̈w3 `yB †ÿ‡Î e„wó †eu‡K covi cwigvc GKB cwijwÿZ Ki‡eb wK? MvwYwZK we‡køl‡Yi gva ̈‡g gZvgZ Dc ̄ vcb Ki| DËi: evqycÖevn bv n‡j Djø‡¤^i mv‡_ 63.43 †KvY; evqycÖevn n‡j Djø‡¤^i mv‡_ 77.47 †KvY| 17| wb‡Pi DÏxcKwU jÿ ̈ Ki: p(x, y, z) = 2xy4 – x 2 z GKwU † ̄‹jvi ivwk Ges A  = (2x + y)i ^ + (3y + z2 )j ^ + (–5z + x)k ^ GKwU †f±i ivwk Ges B  = (6xy + z3 )i ^ + (3x2 – z)j ^ + (3xz2 – y)k ^ Aci GKwU †f±i ivwk| [wm. †ev. 22] (K) fvi‡K›`a Kx? (L) gnvKl©xq wefe 12 J/kg ej‡Z Kx eyS? (M) (2, –1, –2) we›`y‡Z p Gi †MÖwW‡q›U wbY©q Ki| DËi: 10i ^ – 16j ^ – 4k ^ (N) DÏxc‡K ewY©Z A  I B  †f±i؇qi g‡a ̈ †KvbwU mwjbqWvj Ges †KvbwU AN~Y©bkxj Zv MvwYwZK we‡køl‡Yi gva ̈‡g hvPvB Ki| DËi: A †f±iwU mwjbqWvj, N~Y©bkxj; B †f±iwU mwjbqWvj bq, AN~Y©bkxj 18| F1  = 4N N W E F3  = 4N 1 = 60 F2  = 4N 3 = 45 2 = 30 O S wP‡Î wZbwU mgZjxq †f±i O we›`y‡Z wμqvkxj i‡q‡Q| [e. †ev. 22]
4  HSC Physics 1st Paper Chapter-2 (K) ̄^vaxb †f±i Kx? (L) bvj †f±‡ii w`K e ̈vL ̈v Ki| (M) F1  I F2  †f±i `ywU GKwU mvgšÍwi‡Ki `yÕwU evû wb‡`©k Ki‡j mvgvšÍwiKwUi †ÿÎdj wbY©q Ki| DËi: 8 3 eM© GKK (N) F1  , F2  I F3  †f±i wZbwUi wgwjZ dj †Kvb w`‡K wμqv Ki‡e? MvwYwZK we‡kølYc~e©K gšÍe ̈ Ki| DËi: 124.6 (cwð‡gi mv‡_ DËi eivei) 19| †`Iqv Av‡Q GKwU †f±i †ÿÎ A  = (6xy + z3 )i ^ + (3x2 – z)j ^ + (3xz2 – y)k ^ [e. †ev. 22] (K) Kvj© Kx? (L) •e`y ̈wZK cvLvi evZvm Kxfv‡e wb‡P bv‡g? e ̈vL ̈v Ki| (M) (2, 1, –1) we›`y‡Z A  Gi †MÖwW‡q›U wbY©q Ki| DËi: cÖ‡kœ fzj Av‡Q| (N) MvwYwZK we‡køl‡Yi mvnv‡h ̈ †`LvI †h A  †f±iwU mwjb‡qWvj bvwK msiÿYkxj n‡e? DËi:   . A   0;    A  = 0; ZvB msiÿYkxj| 20| O X P(1, 3, 2) Y Z Q(2, 1, 3) wP‡Îi P I Q we›`yi Ae ̄ vb †f±i h_vμ‡g P  I Q  . [w`. †ev. 22] (K) m`„k †f±i Kv‡K e‡j? (L) `ywU Amgvb †f±‡ii jwä k~b ̈ n‡Z cv‡i bvÑ e ̈vL ̈v Ki| (M) OPQ Gi †ÿÎdj wbY©q Ki| DËi: OPQ-Gi †ÿÎdj 4.33 eM© GKK| (N) P  + Q  I P  – Q  †f±iØq +Y A‡ÿi mv‡_ mgvb †KvY Drcbœ K‡i wK-bv? MvwYwZK we‡kølYc~e©K gZvgZ `vI| DËi: 1 = 55.55; 2 = 35.26 1  2 nIqvq P  + Q  Ges P  – Q  †f±iØq y-A‡ÿi mv‡_ mgvb †KvY Drcbœ Ki‡e bv| 21| B A † ̄av‡Zi †eM, 2 kmh–1 C 110 D wP‡Î † ̄av‡Zi b`x‡Z GKRb †jvK GK cvo n‡Z Aci cv‡o hvIqvi Rb ̈ 4kmh–1 †e‡M AC eivei †bŠKv Pvjv‡bv ïiæ K‡i| †m Aci cv‡o D we›`y‡Z †cuŠ‡Q| BD = 0.5 km, b`xi cÖ ̄’ = AB. [w`. †ev. 22] (K) w ̄ wZ ̄ vcK msNl© Kv‡K e‡j? (L) `yB‡qi AwaK †f±i ivwki †hv‡Mi wbqg e ̈vL ̈v Ki| (M) b`xi cÖ ̄ AB wbY©q Ki| DËi: 2.97 km (N) AC eivei †b.Kv Pvjv‡bv ïiæ Ki‡j Aci cv‡o †cu.Qv‡Z †h mgq jv‡M GKB †e‡M AB eivei †b.Kv Pvjv‡bv kyiæ Ki‡j Aci cv‡o †cu.Qv‡Z Zvi †P‡q Kg bvwK †ewk mgq jvM‡e? MvwYwZK we‡kølYc~e©K Dˇii mc‡ÿ gZvgZ `vI| DËi: tAC = 0.79 hr; tAB = 0.7425 hr 22| wPÎwU jÿ ̈ Ki Ges wb‡Pi cÖkœ ̧‡jvi DËi `vI: X O Y Y A(1,2) B(2,–1) X [h. †ev. 22] (K) GKK †f±i Kv‡K e‡j? (L) `ywU Amgvb †f±‡ii jwä k~b ̈ n‡Z cv‡i bv †Kb? e ̈vL ̈v Ki| (M) DÏxc‡Ki OA  †f±iwU Y A‡ÿi mv‡_ KZ †KvY Drcbœ Ki‡e? DËi: 26.56 (N) DÏxc‡Ki OA  Ges OB  †f±iØq ci ̄úi j¤^ wKbv? MvwYwZKfv‡e hvPvB Ki| DËi: OA  .OB  = 0; OA  Ges OB  †f±i ci ̄úi j¤^| 23| wÎgvwÎK ̄’vbvsK e ̈e ̄’vq `ywU we›`yi ̄’vbv1⁄4 h_vμ‡g P (1, 2, 1) I Q (2, 1, 1)| we›`y `ywUi Rb ̈ m„ó Ae ̄’vb †f±i h_vμ‡g OP  I OQ  | Ae ̄’vb †f±iØq‡K mwbœwnZ evû a‡i mvgvšÍwiK AsKb Ki‡j R we›`yi ̄’vbv1⁄4 R (1, 1, 2) nq| [g. †ev. 22] Q(2, 1, 1) O R(1, 1, 2)  X P(1, 2, 1) Y (K) †f±i Acv‡iUi Kx? (L) •e`y ̈wZK cvLv Nyi‡j Mv‡q evZvm jv‡M †Kb? †f±‡ii mvnv‡h ̈ e ̈vL ̈v Ki| (M)  †Kv‡Yi gvb wbY©q Ki| DËi: 33.56 (N) DÏxc‡Ki PQR mg‡KvYx wÎfzR MVb K‡i wK-bvÑ MvwYwZK e ̈vL ̈v Ki| DËi: PQ2 + QR2  PR2 ; mg‡KvYx wÎfzR MVb K‡i bv|