Title 2. P1C2. Phy. For FRB-2024_With Solve_Ridoy 05.01.25.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 2024 †gvU †f±‡ii †hvM-we‡qvM I jwä Ñ 1 1 2 2 2 Ñ 3 8 AvqZ GKK †f±i 1 1 1 Ñ Ñ 1 2 1 6 †f±i ivwki wefvRb (Dcvsk) 1 Ñ Ñ 1 Ñ 1 Ñ 2 3 e„wó I QvZv aiv msμvšÍ Ñ 1 Ñ 1 Ñ 2 2 Ñ 6 †f±‡ii ̧Y, †ÿÎdj I j¤^ Awf‡ÿc, GKB mgZj, ga ̈eZ©x †KvY 5 4 Ñ 3 7 4 5 4 29 b`x cvivcvi Ñ 2 Ñ 1 4 4 9 7 20 †f±i Acv‡iUi Ñ Ñ Ñ 1 Ñ 4 Ñ 3 5 *we.`a.: 2020 mv‡j GBPGmwm cixÿv AbywôZ nqwb| weMZ mv‡j †ev‡W© Avmv m„Rbkxj cÖkœ 1| `ywU †f±i v  = 2i ^ + 3j ^ – xk ^ ci ̄úi j¤^| v  Ges u  Gi gvb hw` †b.Kv I GKwU b`x‡Z † ̄av‡Zi †eM wb‡`©k K‡i Z‡e me©wb¤œ c‡_ b`x cvi n‡Z †b.KvwUi 4 wgwbU mgq jv‡M| (K) Ae ̄ vb †f±i Kv‡K e‡j? [g. †ev. 24] (L) wZbwU †f±‡ii jwä KLb k~b ̈ n‡e? e ̈vL ̈v Ki|[g. †ev. 24] (M) DÏxc‡Ki ÔxÕ Gi gvb wbY©q Ki| [g. †ev. 24] mgvavb: Avgiv Rvwb, u  I v  †f±i `ywU ci ̄úi j¤^ n‡j, u  .v  = 0  (3i ^ + 2j ^ + 2k ^ ).(2i ^ + 3j ^ – xk ^ ) = 0  6 + 6 – 2x = 0  x = 6 myZivs, DÏxc‡Ki x Gi gvb 6| (Ans.) (N) b~ ̈bZg mg‡q b`x cvi n‡Z n‡j gvwS‡K b`xi cÖ ̄ A‡cÿv †ewk `~iZ¡ AwZμg Ki‡Z n‡e wK bvÑ MvwYwZKfv‡e hvPvB Ki| [g. †ev. 24] mgvavb: †b.Kvi †eM, |v  | = 2 2 + 32 + (– 6) 2 = 7 ms–1 † ̄av‡Zi †eM, |u  | = 3 2 + 22 + 22 = 17 ms –1 me©wb¤œ c‡_ b`x cvi n‡j, jwä †eM I † ̄av‡Zi †e‡Mi ga ̈eZ©x †KvY n‡e 90|  tan90 = vsin u + vcos  1 0 = vsin u + vcos  u + vcos = 0   = cos –1    –  u v = cos –1     – 17 7 = 126.087 b`xi cÖ ̄ , d = (vsin)t = 7sin(126.087)  (4  60) = 1357.64 m v u w d s (b`xi cvo) b~ ̈bZg mg‡qi †ÿ‡Î cÖ‡qvRbxq mgq, t = d v = 1357.64 7 = 193.9503s  jwä †eM, w = u 2 + v2 = 17 + 72 = 66 ms –1  †b.Kv KZ...©K AwZμvšÍ `~iZ¡, s = wt = 66  193.950 = 1576.65 m > 1357.64 m myZivs, b~ ̈bZg mg‡q b`x cvi n‡Z n‡j gvwS‡K b`xi cÖ ̄ A‡cÿv †ewk `~iZ¡ AwZμg Ki‡Z n‡e| (Ans.) 2| A  = 2i ^ – 3j ^ + k ^ , B  = 3i ^ – j ^ + 5k ^ Ges C  = 3i ^ + 2j ^ – 4k ^ †f±i wZbwU GKwU wÎfz‡Ri wZbwU evû wb‡`©k K‡i| (K) cig Av`a©Zv Kx? [g. †ev. 24] (L) †`LvI †h, i ^  j ^ = k ^ . [g. †ev. 24] (M) A  Ges B  †K mwbœwnZ evû a‡i mvgšÍwi‡Ki †ÿÎdj wbY©q Ki| [g. †ev. 24] mgvavb: Avgiv Rvwb, A  I B  †Kv‡bv mvgvšÍwi‡Ki `ywU mwbœwnZ evû n‡j mvgšÍwi‡Ki †ÿÎdj = |A   B  | eM© GKK  |A   B  | =      i  ^ 2 3 j ^ – 3 – 1 k ^ 1 5 = i ^ (– 15 + 1) – j ^ (10 – 3) + k ^ (– 2 + 9) = – 14i ^ – 7j ^ + 7k ^  mvgvšÍwi‡Ki †ÿÎdj = |A   B  | = (– 14) 2 + (– 7) 2 + 72 = 7 6 eM© GKK (Ans.)
2  HSC Physics 1st Paper Chapter-2 (N) DÏxc‡K wÎfzRwU mg‡KvYx wÎfzR n‡e wK-bvÑ MvwYwZKfv‡e we‡kølY Ki| [g. †ev. 24] mgvavb: DÏxc‡Ki wÎfzRwU mg‡KvYx n‡j †h‡Kv‡bv `ywU evûi ga ̈eZ©x †KvY 90 n‡Z n‡e| G‡ÿ‡Î `ywU †f±‡ii WU ̧Ydj k~b ̈ n‡Z n‡e| GLb, A  .B  = (2i ^ – 3j ^ + k ^ ).(3i ^ – j ^ + 5k ^ ) = 6 + 3 + 5 = 14  0 B  .C  = (3i ^ – j ^ + 5k ^ ).(3i ^ + 2j ^ – 4k ^ ) = 9 – 2 – 20 = – 13  0 C  .A  = (3i ^ + 2j ^ – 4k ^ ).(2i ^ – 3j ^ + k ^ ) = 6 – 6 – 4 = – 4  0 myZivs, wÎfzRwUi †h‡Kv‡bv `ywU evûi ga ̈eZ©x †KvY 90 bv nIqvq wÎfzRwU mg‡KvYx wÎfzR n‡e bv| (Ans.) 3| A  = 2i ^ + 3j ^ + 5k ^ , B  = – i ^ + 2j ^ + 7k ^ Ges C  = i ^ + 7j ^ –k ^ (K) Ae ̄ vb †f±i Kx? (L) i ^ .i ^  0 †Kb? e ̈vL ̈v Ki| [Xv. †ev. 24] (M) A  eivei B  Gi Awf‡ÿc wbY©q Ki| [Xv. †ev. 24] mgvavb: Avgiv Rvwb, A  eivei B  Gi Awf‡ÿc, Bcos = A  .B  | A  |  Bcos = (2i ^ + 3j ^ + 5k ^ ). (– i ^ + 2j ^ + 7k ^ ) 2 2 + 32 + 52 = – 2 + 6 + 35 38 = 6.33 GKK  A  eivei B  Gi Awf‡ÿc 6.33 GKK| (Ans.) (N) DÏxc‡Ki †f±i wZbwU GKB mgZ‡j _vK‡e wKbvÑ MvwYwZKfv‡e hvPvB Ki| [Xv. †ev. 24] mgvavb: Avgiv Rvwb, A  , B  I C  wZbwU †f±i mgZjxq n‡j A  .(B   C  ) = 0 nq|  A  .(B   C  ) =       Ax Bx Cx Ay By Cy Az Bz Cz =       2 – 1 1 3 2 7 5 7 – 1 = 2(– 2 – 49) – 3(1 – 7) + 5(– 7 – 2) = – 102 + 18 – 45 = – 129  0  A  .(B   C  )  0 myZivs, A  .(B   C  )  0 nIqvq †f±i wZbwU GKB mgZ‡j _vK‡e bv| (Ans.) 4| Q M S 100 P † ̄av‡Zi †eM = 3 kmh–1 Dc‡ii wP‡Îi Zgvj ÔPÕ we›`y †_‡K †b.Kv Pvwj‡q b`xi Aci cv‡o hv‡”Q| †m PS eivei 5 kmh–1 †e‡M †b.Kv Pvwj‡q b`xi Aci cv‡oi M we›`y‡Z †cu.Qvq| [b`xi cÖ ̄ PQ = 2 km] (K) †f±i Acv‡iUi Kx? [Xv. †ev. 24] (L) †K›`agyLx ej †Kvb †Kvb wel‡qi Dci wbf©i K‡i?[Xv. †ev. 24] (M) QM Gi `~iZ¡ wbY©q Ki| [Xv. †ev. 24] mgvavb: awi, †b.Kvi †eM v I † ̄av‡Zi †eM u jwä †eM w † ̄av‡Zi mv‡_  †KvY m„wó K‡i|  tan = v sin100 u + v cos100  tan = 5  sin100 3 + 5  cos100   = 66.59 Q v M  u 100 P w GLb, PQM G, tan = PQ QM  QM = PQ tan = 2 tan66.50 = 0.865 km (Ans.) (N) PS eivei †b.Kv Pvwj‡q ÔMÕ we›`y‡Z †c.Qvi mgq Ges P †_‡K † ̄avZnxb Ae ̄ vq mivmwi †b.Kv Pvwj‡q Q-†Z †c.Qvi mg‡qi cv_©K ̈ MvwYwZK we‡kølYc~e©K wbY©q Ki| [Xv. †ev. 24] mgvavb: cÖ_g †ÿ‡Î, cÖ‡qvRbxq mgq, t1 = d v sin = 2 5 sin100 = 0.406 hr wØZxq †ÿ‡Î, cÖ‡qvRbxq mgq, t2 = d v = 2 5 = 0.4 hr  mg‡qi cv_©K ̈ = t1 – t2 = 0.406 – 0.4 = 0.006 hr = 21 s myZivs, PS eivei †b.Kv Pvwj‡q M we›`y‡Z †cu.Qvi mgq Ges P †_‡K † ̄avZnxb Ae ̄ vq mivmwi †b.Kv Pvwj‡q Q †Z †cu.Qvi mg‡qi cv_©K ̈ 21s| (Ans.) 5| wP‡Î 4 kmh–1 †e‡M cÖevwnZ † ̄av‡Zi b`x‡Z †mv‡nj 8 kmh–1 †e‡M AB eivei †b.Kv Pvjv‡bv ïiæ K‡i| 10 wgwb‡U b`xi cÖ ̄ AD eivei D we›`y‡Z †cu.‡Q| wKš‧ †ivnvb AD eivei 10 kmh–1 †e‡M †b.Kv Pvjv‡bv ïiæ K‡i AC eivei C we›`y‡Z †cu.‡Q| Y D C A † ̄av‡Zi w`K
†f±i  Final Revision Batch 3 (K) Ae ̄ vb †f±‡ii msÁv `vI| [iv. †ev. 24] (L) †f±i †hv‡Mi mvgšÍwiK m~Î wewbgq m~Î †g‡b P‡j| e ̈vL ̈v Ki| [iv. †ev. 24] (M) †mv‡n‡ji †ÿ‡Î †b.Kvi †eM I † ̄av‡Zi †e‡Mi jwä †e‡Mi gvb wbY©q Ki| [iv. †ev. 24] mgvavb: †mv‡n‡ji †ÿ‡Î, †b.Kvi jwä †eM, † ̄av‡Zi †e‡Mi mv‡_ 90 †KvY Drcbœ K‡i| awi, †b.Kvi †eM, v = 8 kmh–1 † ̄av‡Zi †eM, u = 4 kmh–1  tan90 = vsin u + vcos  u + vcos = 0   = cos –1    –  u v = cos –1    –  4 8 = 120  jwä †eM, w = u 2 + v2 + 2uvcos = 4 2 + 82 + 2  4  8  cos(120) = 4 3 kmh–1 (Ans.) (N) b`xi cÖ ̄ AD I †mv‡n‡ji •`N© ̈ eivei AwZμvšÍ `~iZ¡ DC mgvb n‡e wK-bv? MvwYwZK we‡kølYc~e©K gZvgZ `vI| [iv. †ev. 24] mgvavb: †mv‡n‡ji †ÿ‡Î, b`xi cÖ ̄ , AD = (v1sin)  t1 = 8  sin (120)  10 60 = 1.154 km †mv‡n‡ji †ÿ‡Î, ADC mg‡KvYx wÎfz‡R, tan = v2 u = 10 4  = tan–1 (2.5) = 68.2 D v2 C  u  A u GLb, ADC mg‡KvYx wÎfz‡R tan = AD DC  DC = 1.154 tan(68.2) = 0.461 km < AD myZivs, b`xi cÖ ̄ AD I †ivnv‡bi b`xi •`N© ̈ eivei AwZμvšÍ `~iZ¡ DC mgvb n‡e bv| (Ans.) 6| wPÎ Abyhvqx O we›`y ̄ w ̄ i e ̄‧KYvi Dci wZbwU ej cÖ‡qvM Kiv nj| 30 O X X Y Y |F2  | |F1  | 60 |F1  | = 11N |F2  | = 22N |F3  | = 33N (K) w ̄ wZ ̄ vcK msNl© Kv‡K e‡j? [iv. †ev. 24] (L) †f±i ̧Yb wewbgq m~Î gv‡b bvÑ e ̈vL ̈v Ki| [iv. †ev. 24] (M) e ̄‧KYvi Dci OX eivei wμqvkxj jwä e‡ji gvb wbY©q Ki| [iv. †ev. 24] mgvavb: 60 |F |3  120 60 |F |2  |F |1  O X Y  R awi, jwä ej R, OX †iLvi mv‡_  †KvY Drcbœ K‡i| myZivs OX eivei wμqvkxj jwä e‡ji gvb, Rcos = |F |1  + |F |2  cos(120) + |F |3  cos(240) = 11 + 22     –  1 2 + 33     –  1 2  – 16.5 N (Ans.) (N) ej cÖ‡qv‡Mi d‡j e ̄‧KYvwU †Kvb w`‡K MwZkxj n‡e? MvwYwZK we‡kølYc~e©K gZvgZ `vI| [iv. †ev. 24] mgvavb: 60 |F |3  120 60 |F |2  |F |1  O X Y  R awi, jwä ej R, X A‡ÿi abvZ¥K w`‡Ki mv‡_  †KvY Drcbœ K‡i| ÔMÕ n‡Z cvB, Rcos = – 16.5 ......(i)  Rsin = |F |1  sin(0) + |F |2  sin(120) + |F |3  sin(240) = 0 + 22  3 2 + 33  – 3 2  Rsin = – 11 3 2 w ......(ii) (ii)  (i) K‡i cvB, tan = – 11 3 2  1 – 16.5   = tan–1     1 3 †h‡nZzsin I cos Df‡qi gvb FYvZ¥K jwä ej Z...Zxq PZzf©v‡M Ae ̄ vb Ki‡e,   = 180 + tan–1     1 3 = 210 A_©vr, OX †iLvi mv‡_ 210 †Kv‡Y wμqv Ki‡e| (Ans.) R 210 X X Y Y O
4  HSC Physics 1st Paper Chapter-2 7| wb‡Pi DÏxcKwU co I mswkøó cÖkœ ̧‡jvi DËi `vI: 30 30 O X X Y Y |F  |1 = 15N |F  |2 = 10N (K) AvqvZ GKK †f±i Kx? [h. †ev. 24] (L) GKwU †f±i ivwk‡K Kxiƒ‡c † ̄..jvi ivwk‡Z iƒcvšÍi Ki‡e? e ̈vL ̈v Ki| [h. †ev. 24] (M) F1 Gi †f±i iƒc wbY©q Ki| [h. †ev. 24] mgvavb: 30 150 |F |1  X X Y Y |F |1  †f±iwU X-A‡ÿi abvZ¥K w`‡Ki mv‡_ (180 – 30) = 150 †KvY Drcbœ K‡i| F1  = |F |1  cos(150)i ^ + |F |1  sin(150)j ^ = 15 cos(150)i ^ + 15 sin(150)j ^ = – 15 3 2 i ^ + 7.5j ^ myZivs, F1 Gi †f±i iƒc, F1   – 15 3 2 i ^ + 7.5j ^ (Ans.) (N) DÏxc‡K F  1 + F  2 Ges F  1 – F  2 ci ̄úi j¤^ wK-bv? e ̈vL ̈v Ki| [h. †ev. 24] mgvavb: ÔMÕ n‡Z cvB, F1  = – 15 3 2 i ^ + 7.5j ^ wPÎ n‡Z, F2  = |F |2  cos(30) i ^ + |F |2  sin(30) j ^ = 5 3i ^ + 5j ^ GLb, F1  + F2  =     – 15 3 2 + 5 3 i ^ + (7.5 + 5)j ^ = – 5 3 2 i ^ + 12.5j ^  F1  – F2  =     – 15 3 2 – 5 3 i ^ + (7.5 – 5)j ^ = – 25 3 2 i ^ + 2.5j ^ (F1 )  + F2  .(F1 )  – F2  =    – 5 3  2 i ^ + 12.5j ^ .     – 25 3 2 i ^ + 2.5j ^ = 93.75 + 31.25 = 125 myZivs, (F1 )  + F2  .(F1 )  – F2   0 nIqvq †f±i `ywU ci ̄úi j¤^ bq| (Ans.) 8| † ̄av‡Zi AbyK‚‡j †b.Kvi †eM 20 kmh–1 Ges † ̄av‡Zi cÖwZK‚‡j †b.Kvi †eM 10 kmh–1 | b`xi we ̄Ívi 2 km| B P 2 km (K) Kvj© Gi msÁv `vI| [w`. †ev. 24] (L) r  = 5i ^ + 3j ^ GKwU mxgve× †f±iÑ e ̈vL ̈v Ki|[w`. †ev. 24] (M) A n‡Z †mvRv Aci cv‡oi B we›`y‡Z †c.Q‡Z n‡j †b.KvwU‡K †Kvb w`‡K Pvjbv Ki‡Z n‡e? [w`. †ev. 24] mgvavb: awi, †b.Kvi †eM, v Ges † ̄av‡Zi †eM, u cÖkœg‡Z, v + u = 20 ......(i) v – u = 10 ......(ii) (i) + (ii) K‡i, 2v = 30  v = 15 kmh–1 GLb, v Gi gvb (i) G ewm‡q, u = 5 kmh–1 awi, A n‡Z †mvRv Aci cv‡oi B we›`y‡Z †cu.Qv‡Z †b.KvwU‡K † ̄av‡Zi w`‡Ki mv‡_  †Kv‡Y Pvjbv Ki‡Z n‡e|  tan90 = vsin u + vcos  u + vcos = 0  = cos –1    –  u v = cos –1    –  5 15 = 109.47 (Ans.) (N) b`x cvi nIqvi Rb ̈ †b.KvwU hw` A we›`y n‡Z †mvRvmywR iIqvbv Ki‡Zv Zvn‡j b`x cvi n‡Z c~‡e©i †P‡q mgq Kg bv †ewk jvM‡Zv? MvwYwZK we‡kølY K‡i gZvgZ `vI| [w`. †ev. 24] mgvavb: cÖ_g †ÿ‡Î cÖ‡qvRbxq mgq, t1 = d vsin = 2 15 sin(109.47) = 0.141 hr ÔMÕ n‡Z cvB, 1g †ÿ‡Î,  = 109.47 †b.Kvi †eM, v = 15 kmh– 1 † ̄av‡Zi †eM, u = 5 kmh–1 b`xi cÖ ̄ , d = 2 km 2q †ÿ‡Î,  = 90 wØZxq †ÿ‡Î cÖ‡qvRbxq mgq, t2 = d vsin = 2 15 sin90 = 0.133 hr  t2 < t1 myZivs, b`x cvi nIqvi Rb ̈ †b.KvwU hw` †mvRvmywR iIbv Ki‡Zv Zvn‡j b`x cvi n‡Z c~‡e©i †P‡q Kg mgq jvM‡Zv| (Ans.)