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1.
Math Tutor 1 Aa¨vq
02 ev¯Íe msL¨v (Real Number) 01.01 A¼ (Digit) wnmvewbKvk I MYbvi Kv‡R e¨eüZ wPý ev cÖZxK| MwY‡Z †gvU 10 wU A¼ i‡q‡Q| †hgbt 0, 1, 2, 3, 4, 5, 6, 7, 8 Ges 9| ¯^v_©K A¼t 9 wU | †hgbt 1, 2, 3, 4, 5, 6, 7, 8 Ges 9| AfveÁvcK A¼ t 1 wU| 0 †K mvnvh¨Kvix msL¨v ejv nq| †hgbt 0| 01. ¯^v_©K A¼ KqwU? cÖv_wgK we`¨vjq cÖavb wkÿK-2005 5 wU 9 wU 7 wU 8 wU DËi: L 02. wb‡Pi †KvbwU ¯^v_©K A¼ bq? wbe©vnx Awdmvi, evwYR¨ gš¿Yvj‡qi Avg`vwb-ißvwb Awa`ßi-15 2 5 0 7 DËi: M 03. †Kvb msL¨v‡K mvnvh¨Kvix msL¨v ejv nq? cÖkvmwbK Kg©KZv© I cv‡m©vbvj Awdmvi, wbe©vPb Kwgkb-2004 0 1 2 5 DËi: K 01.02 msL¨v (Number) GK ev GKvwaK A¼ wg‡j msL¨v •Zwi nq Ges msL¨vi †k‡l wU, Uv, Lvbv _v‡K| †hgb- 5 wU Kjg, 13 Lvbv eB | GLv‡b, 5 Ges 13 n‡”Q msL¨v| msL¨vq e¨eüZ A¼¸wji gvb(Value) `yÕfv‡e wPwýZ Kiv hvq| (i) ¯^Kxq gvb (Face Value) (ii) ¯’vbxq gvb (Place Value) 01.03 01 †_‡K 100 ch©šÍ µwgK msL¨v I cÖ‡qvRbxq Z_¨vewj 1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 66 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 ¸iæZ¡c~Y© Z_¨vewjt nv‡Z Kj‡g †kLvi Rb¨ Dc‡iv³ Q‡Ki mv‡_ wb‡Pi Z_¨¸wj wgwj‡q wbb, Zvn‡j mn‡RB g‡b _vK‡e| 1 †_‡K 100 ch©šÍ wjL‡Z ev ¸Y‡Z 0 Av‡Q 11 wU| 1 †_‡K 100 ch©šÍ wjL‡Z ev ¸Y‡Z 1 Av‡Q 21 wU| 1 †_‡K 100 ch©šÍ wjL‡Z ev ¸Y‡Z 2, 3, 4, 5, 6, 7, 8, 9 Av‡Q 20wU K‡i| NMLK NMLK NMLK †ewmK, GgwmwKD I wjwLZ Av‡jvPbv
2.
2Math Tutor 04. 1
†_‡K 100 ch©šÍ wjL‡Z Ô4Õ A¼wU KZevi Av‡m? SouthestBankLtdProbationaryOfficer:12 10 11 19 20 DËi: N 05. 1 †_‡K 100 ch©šÍ ¸Y‡Z, 5 msL¨vwU KZevi Av‡m? 28ZgwewmGm 10 11 28 19 DËi: Tips: cÖ`Ë Ackb¸‡jv‡Z mwVZ DËi †bB| mwVK DËi n‡e 20| 06. 1 †_‡K 100 ch©šÍ ¸Y‡Z, 8 msL¨vwU KZevi Av‡m? BangladeshBank AssistantDirector:13 11 20 80 70 10 DËi: L 07. 1 †_‡K 100 ch©šÍ ¸Y‡Z, 9 msL¨vwU KZevi Av‡m? 28ZgwewmGm(gb¯ÍvwË¡K) 11 14 15 18 20 DËi: O 01.04 ÿz`ªZg I e„nËg msL¨v cÖ`Ë A¼ e„nËg msL¨v ¶z`ªZg msL¨v GK A¼ wewkó msL¨vi 9 1 `yB A¼ wewkó msL¨vi 99 10 wZb A¼ wewkó msL¨vi 999 100 Pvi A¼ wewkó msL¨vi 9999 1000 ¯^vfvweK msL¨vi ÿz`ªZg m`m¨ n‡”Q 1| 08. ¯^vfvweK msL¨vi ÿz`ªZg m`m¨ †KvbwU? cÖavbgš¿xiKvh©vj‡qimnKvixcwiPvjK,M‡elYvKg©KZ©v,†Uwj‡dvbBwÄwbqviImnKvixKw¤úDUvi†cÖvMÖvgvi:13 1 0 Amxg me¸‡jv DËi: K (K) mgvb msL¨K e„nËg msL¨v I ÿz`ªZg msL¨v Ô‡hvMÕ Kivt 9 9 9 9 99 9 999 9 9999 +1 + 10 + 100 + 1000 + 10000 10 10 9 10 99 10 999 10 9999 Dc‡iv³ QK †_‡K eySv hv‡”Q, mgvb msL¨K e„nËg I ÿz`ªZg msL¨vi †hvMdj PvB‡j Rv÷ cÖ_g 9 A¼wU 10 wj‡L evKx †h KqwU 9 _vK‡e †m¸‡jv H Ae¯’vq †i‡L w`‡j †h msL¨vwU Avm‡e †mwUB DËi| †hgb Pvi A‡¼i e„nËg I ÿz`ªZg msL¨vi †hvMdj n‡e- cÖ_g 9 A¼wU 10 G iƒcvšÍi Kiæb Ges evKx wZbwU ÔbqÕ (999) H Ae¯’vq †i‡L w`b, Zvn‡j †hvMdj `uvov‡”Q 10999| A_©vr, †hvMdj n‡e- GKwU Ô`kÕ Ges evKx wZbwU ÔbqÕ| 09. cvuP A‡¼i e„nËg I ¶y`ªZg msL¨vi †hvMdj KZ? ivóªvqË e¨vsK wmwbqi Awdmvi : 00 109999 89999 100009 †KvbwUB bq DËi: K (L) mgvb msL¨K e„nËg msL¨v I ÿz`ªZg msL¨v Ôwe‡qvMÕ Kivt 9 9 9 9 99 9 999 9 9999 - 1 - 10 - 100 - 1000 - 10000 8 8 9 8 99 8 999 8 9999 Dc‡iv³ QK †_‡K eySv hv‡”Q, mgvb msL¨K e„nËg I ÿz`ªZg msL¨vi we‡qvMdj PvB‡j Rv÷ cÖ_g 9 A¼wU 8 wj‡L evKx †h KqwU 9 A¼ _vK‡e †m¸‡jv H Ae¯’vq †i‡L w`‡j †h msL¨vwU Avm‡e †mwUB DËi| †hgb Pvi A‡¼i e„nËg I ÿz`ªZg msL¨vi we‡qvMdj n‡e- cÖ_g 9 A¼wU 8 G iƒcvšÍi Kiæb Ges evKx wZbwU ÔbqÕ (999) H Ae¯’vq †i‡L w`b, Zvn‡j we‡qvMdj `uvov‡”Q 8999| A_©vr, we‡qvMdj n‡e- 1wU Ô8Õ Ges evKx 3wU Ô9Õ| 10. Pvi A‡¼i e„nËg I ¶y`ªZg msL¨vi cv_©K¨ KZ? ivóªvqË e¨vsK Awdmvi t 97 10999 8999 1009 1999 DËi: LNMLK NMLK NMLK ONMLK ONMLK NMLK NMLK
3.
Math Tutor 3 11.
6 A‡¼i e„nËg I ¶z`ªZg msL¨vi cv_©K¨ KZ? †mvbvwj, RbZv I AMÖYx e¨vsK wmwbqi Awdmvi : 08 888889 899999 988888 999888 DËi: L (M) ÿz`ªZg msL¨v †_‡K e„nËg msL¨v we‡qvM Kivi mgq ÿz`ªZg msL¨v A‡cÿv e„nËg msL¨vi GKwU ÔwWwRU/A¼Õ Kg n‡j we‡qvMdj memgq 1 nq| 10 100 1000 10000 100000 - 9 - 99 - 999 - 9999 - 99999 1 1 1 1 1 12. cvuP A‡¼i ¶z`ªZg msL¨v I Pvi A‡¼i e„nËg msL¨vi AšÍi KZ? 29Zg wewmGm 9 10 1 -1 DËi: M (N) weweat 13. `yBkZ bq nvRvi ‡PŠÏ Ges wZivbeŸB nvRvi mvZ kZ bq Gi AšÍi KZ? weweG 92-93 116315 115315 116305 115305 DËi: N 209014 - 93709 = 115305| 14. 0, 1, 2, Ges 3 Øviv MwVZ Pvi A‡¼i e„nËg Ges ÿz`ªZg msL¨vi we‡qvMdj- cjøxwe`y¨Zvqb†ev‡W©imn.mwPe/cwiPvjK2017 3147 2287 2987 2187 0, 1, 2, 3 Øviv Pvi A‡¼i e„nËg msL¨v = 3210 Ges ÿz`ªZg msL¨v= 1023 | myZivs, msL¨v `ywUi we‡qvMdj = 3210 - 1023 = 2187| DËi: N 01.05 e‡M©i AšÍi †R‡b wbb – 01 `ywU µwgK msL¨vi eM©‡K we‡qvM Ki‡j †h msL¨v cvIqv hvq, H µwgK msL¨v `ywU‡K †hvM Ki‡jI GKB msL¨v cvIqv hvq| †hgb-2 I 3 Gi †hvMdj Ges Zv‡`i e‡M©i Aš‘i mgvb| A_©vr, 2 + 3 = 5 2 3 - 2 2 = 9 - 4 = 5| GKBfv‡e, 3 I 4 Gi †hvMdj Ges Zv‡`i e‡M©i AšÍi mgvb| A_©vr, 3 + 4 = 7 2 4 - 2 3 = 16 - 9 = 7 kU©KvU †UKwbK: ÿz`ªZg msL¨v wbY©q = 2 1-AšÍiiM©e‡ e„nËg msL¨v wbY©q = 2 1AšÍiiM©e‡ 15. `ywU µwgK c~Y© msL¨vi e‡M©i AšÍi 47| eo msL¨vwU KZ? 26Zg wewmGm (wkÿv); wdb¨vÝ wgwbw÷ª -2009 24 25 26 30 †h‡nZz `ywU µwgK msL¨vi e‡M©i AšÍi 47 †`qv Av‡Q, †m‡nZz Avgiv ej‡Z cvwi 47 n‡”Q `ywU µwgK msL¨vi †hvMdj| 47 Gi gv‡S `ywU µwgK msL¨v 23 I 24 Av‡Q, hv‡`i gv‡S eo msL¨vwU n‡”Q 24| DËi: K kU© †UKwb‡K mgvavb: eo msL¨vwU = 2 1AšÍiiM©e‡ = 2 147 = 24| 16. `ywU µwgK c~Y©msL¨v wbY©q Kiæb, hv‡`i e‡M©i AšÍi 93|wgwbw÷ª Ae I‡gb - 2007 46, 47 44, 45 43, 44 50, 51 DËi: K Ackb ¸‡jvi gv‡S †h `ywU µwgK msL¨vi †hvMdj 93 †mwUB DËi| GLv‡b Ackb †Z 46 + 47 = 93| 17. `ywU µwgK c~Y© msL¨vi e‡M©i AšÍi 63| †QvU msL¨vwU KZ? 30 31 32 33 63 Gi gv‡S 2wU µwgK 31 I 32 Av‡Q, hv‡`i gv‡S †QvU msL¨vwU n‡”Q 31|mgvavb NMLK Kmgvavb NMLK mgvavb NMLK mgvavb NMLK mgvavb NMLK NMLK NMLK
4.
4Math Tutor kU©
†UKwb‡K mgvavbt 2 1-63 = 2 62 = 31 | myZivs, †QvU msL¨vwU 31| DËit L †R‡b wbb - 02 `ywU msL¨v ej‡Z †h‡Kvb 2wU msL¨v‡K eySvq, ZvB x I y a‡i A¼ Kiv nq| `ywU µwgK msL¨v ej‡Z ci ci `ywU msL¨v‡K eySvq, hv‡`i cv_©K¨ memgq 1 _v‡K, ZvB GKwU msL¨v x I AciwU x + 1 a‡i A¼ Kiv nq| cÖ_gwU‡K ejv nq †QvU msL¨v Ges c‡iiwU‡K ejv nq eo msL¨v| †Kvb mgm¨vi †ÿ‡Î ARvbv ivwk/msL¨v a‡i mgvav‡bi DËg Dcvq n‡”Q ÔkZ©g‡Z/cÖkœg‡ZÕ mgxKiY `vo Kiv‡bv| wjwLZ mgvavb Kivi wbqgt G ai‡Yi A‡¼ `ywU µwgK msL¨vi Ôe‡M©i AšÍiÕ †`qv _vK‡jI ÔµwgK msL¨vÕ `ywU †`qv _v‡K bv| GRb¨ G ai‡Yi mgm¨v mgvav‡bi Rb¨ ïiæ‡ZB µwgK msL¨v `ywU a‡i wb‡q mgvavb Ki‡Z nq| wb‡Pi mgm¨vwU †`Lyb- 18. `yBwU µwgK c~Y© msL¨vi e‡M©i AšÍi 199| eo msL¨vwU KZ? 22Zg wewmGm; cÖvK-cÖv_wgK mnKvix wkÿK-2015 70 80 90 100 DËi: N (cÖ`Ë mgm¨vwU‡Z `ywU µwgK msL¨vi e‡M©i AšÍi 199 †`qv Av‡Q| wKš‘ µwgK msL¨v `ywU †`qv bvB| GRb¨ ïiæ‡ZB msL¨v `ywU a‡i wb‡Z n‡e|) g‡bKwi, †QvU msL¨vwU = x Ges eo msL¨vwU = x +1 (GKevi Ô†R‡b wbb-02Õ c‡o wbb) (GLb cÖ`Ë mgm¨vi Av‡jv‡K GKwU ÔcÖkœg‡Z/kZ©g‡ZÕ mgxKiY `vo Kiv‡Z n‡e| GRb¨ `v‡M hv hv ejv n‡q‡Q, ZvB Kiæb| `v‡M hv hv ejv n‡q‡Q- (1) µwgK msL¨v `ywU eM© n‡e, ZvB eM© K‡i †djyb- x2 Ges (x+1)2 | (2) eM© `ywUi AšÍi n‡e, ZvB Gevi we‡qvM K‡i †djyb- (x+1)2 - x2 . (3) †k‡l ejv n‡q‡Q, GB e‡M©i AšÍi mgvb n‡”Q 199| ZvB AvcwbI †mfv‡e wj‡L †djyb- (x+1)2 - x2 = 199| e¨m&, Gfv‡eB `vwo‡q †Mj ÔcÖkœg‡ZÕ mgxKiYwU!!) cÖkœg‡Z, (x+1)2 - x2 = 199 ev, x2 + 2x +1 - x2 = 199 [(a+b)2 = a2 +2ab +b2 Abymv‡i] ev, 2x = 199 - 1 x = 2 1-199 = 2 198 = 99 AZGe, eo msL¨vwU =x +1 = 99 + 1 = 100| 19. `yBwU µwgK ¯^vfvweK msL¨vi e‡M©i AšÍi 151 n‡j msL¨v `yBwU KZ? WvKI†Uwj‡hvMv‡hvMwefv‡MiAaxbWvKAwa`߇iiwewìs Ifviwkqvi2018 46, 47 75, 76 67, 68 54, 55 DËi: L g‡b Kwi, †QvU msL¨vwU = x Ges eo msL¨vwU = x+1 cÖkœg‡Z, (x + 1)2 - x2 = 151 ev, x2 + 2x + 1 - x2 = 151 ev, 2x = 151 - 1 ev, x = 2 150 = 75 †QvU msL¨vwU = 75 Ges eo msL¨vwU = 75 + 1 = 76 PP©v Kiæb 20. `ywU µwgK ¯^vfvweK msL¨vi e‡M©i AšÍi 45 n‡j, msL¨v `ywU - mvaviYcy‡jiAvIZvqwewfbœ gš¿Yvj‡qimnKvix†cÖvMÖvgviDcmnKvixcÖ‡KŠkjx, cÖkvmwbKKg©KZ©vIe¨w³MZKg©KZ©v:16 21, 22 22, 23 23, 24 20, 21 DËi: L 21. `ywU µwgK msL¨vi e‡M©i AšÍi 37| msL¨v `yBwU wK wK? evsjv‡`k†ijI‡qDcmnKvixcÖ‡KŠkjx(wmwfj):16 12, 13 15, 16 18, 19 20, 21 DËi: M 22. `ywU µwgK msL¨vi e‡M©i AšÍi 25| GKwU msL¨v 12 n‡j, Aci msL¨vwU - ¯^v¯’¨gš¿Yvj‡qiDcmnKvixcÖ‡KŠkjx(wmwfj):16 5 9 11 13 DËi: NNMLK NMLK NMLK mgvavb NMLK mgvavb NMLK
5.
Math Tutor 5 23.
`ywU µwgK msL¨vi e‡M©i AšÍi 11 n‡j, msL¨v `yBwUi e‡M©i mgwó KZ? gwnjvIwkïwelqKgš¿Yvj‡qiAaxbgwnjvwelqKKg©KZ©v:16 16 17 61 71 DËi: M 24. `ywU µwgK c~Y©msL¨v wbY©q Kiæb hv‡`i e‡M©i AšÍi 9 n‡e? Rbkw³,Kg©ms¯’vbIcÖwkÿYey¨‡iviBÝ÷ªv±i:18 4 Ges 5 5 Ges 6 6 Ges 7 7 Ges 8 DËi: K 25. wb‡Pi †Kvb µwgK c~Y© msL¨v؇qi e‡M©i AšÍi 43? L¨v`¨Awa`߇iiLv`¨cwi`k©K/Dc-Lv`¨cwi`k©K:11 21 Ges 22 22 Ges 23 23 Ges 24 24 Ges 25 DËi: K 26. `yBwU µwgK ALÐ msL¨vi e‡M©i AšÍi 49 n‡j, †QvU msL¨vwU n‡e- wewfbœ gš¿Yvjq/wefvM/Awa`߇iie¨w³MZ Kg©KZv© (mvaviY)2018 19 20 24 25 DËi: M 27. ci ci `ywU c~Y© msL¨v wbY©q Ki hv‡`i e‡M©i cv_©K¨ n‡e 53-Rbkw³,Kg©ms¯’vbIcÖwkÿYey¨‡iviDcmnKvixcwiPvjKt01 25 Ges 26 26 Ges 27 27 Ges 28 28 Ges 29 DËi: L 28. `ywU µwgK c~Y©msL¨vi e‡M©i AšÍi 79 n‡j eo msL¨vwU KZ?gv`K`ªe¨wbqš¿YAwa`߇iimnKvixcwiPvjK-2013 40 35 45 100 DËi: K 29. `ywU µwgK c~Y©msL¨vi e‡M©i AšÍi 111 n‡j eo msL¨vwU KZ?ciivóª gš¿Yvj‡qie¨w³MZKg©KZv©-2006 54 55 56 57 DËi: M 30. `ywU µwgK c~Y©msL¨vi e‡M©i AšÍi 197| msL¨vØq KZ?†bŠcwienbgš¿YvjqIcÖwZiÿvgš¿Yvj‡qicÖkvmwbKKg©KZv©-2013 97, 98 96, 97 98, 99 99, 100 DËi: M 01.06 hZ ZZ †R‡b wbb -03 Dc‡ii QKwU jÿ¨ Kiæb, QKwU‡Z †`Lv hv‡”Q,20 msL¨vwU 15 †_‡K 5 †ewk GKBfv‡e 20 msL¨vwU 25 †_‡K 5 Kg| Gevi GKwU cÖkœ `uvo Kiv‡bv hvK-GKwU msL¨v 15 †_‡K hZ eo 25 †_‡K ZZ †QvU| msL¨vwU KZ?QKvbymv‡i, msL¨vwU n‡”Q 20| gRvi e¨vcvi n‡jv, 15 I 25msL¨v `ywU †hvM K‡i 2 Øviv fvM Ki‡jB 20 cvIqv hvq| A_©vr, †Kvb cÖ‡kœ ÒGKwU msL¨v --- †_‡K hZ eo ---- †_‡K ZZ †QvU| msL¨vwU KZ?Ó Giƒc ejv _vK‡j mivmwi cÖ‡kœ cÖ`Ë msL¨v `ywUi Mo Ki‡jB msL¨vwU cvIqv hv‡e| myZivs, msL¨vwU = 2 2515 = 20| 31. GKwU msL¨v 650 n‡Z hZ eo 820 †_‡K ZZ †QvU| msL¨vwU KZ? 22Zg wewmGm 730 735 800 780 DËi: L msL¨v `ywU 650 I 820 †hvM Kiæb Ges 2 Øviv fvM Kiæb, DËi n‡e 735| 32. GKwU msL¨v 553 n‡Z hZ eo 651 †_‡K ZZ †QvU| msL¨vwU KZ? [mve †iwR÷ªvi 1992] 603 601 605 602 DËi: N msL¨v `ywU 553 I 651†hvM Kiæb Ges 2 Øviv fvM Kiæb, DËi n‡e 602| 33. GKwU msL¨v 742 n‡Z hZ eo 830 †_‡K ZZ †QvU| msL¨vwU KZ? [_vbv I †Rjv mgvR‡mev Awdmvi 1999] 780 782 790 786 DËi: N msL¨v `ywU 742 I 830 †hvM Kiæb Ges 2 Øviv fvM Kiæb, DËi n‡e 786| wjwLZ mgvavb Kivi wbqgt 34. GKwU msL¨v 301 †_‡K hZ eo 381 †_‡K ZZ †QvU| msL¨vwU KZ? [30Zg wewmGm] 340 341 342 344 DËi t L (mgm¨vwU‡Z GKwU msL¨vi K_v ejv n‡”Q hv 310 †_‡K hZUzKz eo n‡e, wVK 381 †_‡K ZZUzKzB †QvU n‡e|mgvavb NMLK mgvavb NMLK mgvavb NMLK mgvavb NMLK NMLK NMLK NMLK NMLK NMLK NMLK NMLK NMLK + 5 = 20 2515 + 5 =
6.
6Math Tutor Gevi Avcbv‡K
ej‡Z n‡e msL¨vwU KZ? ejyb‡Zv msL¨vwU KZ †mUv Avcwb Rv‡bb? bv Rv‡bb bv| Zvi gv‡b GwU GKwU ARvbv ivwk| ZvB Avcbv‡K ïiæ‡ZB GKwU msL¨v x a‡i wb‡q A¼ Klv ïiæ Ki‡Z n‡e| ) g‡bKwi, msL¨vwU = x (Gevi `vMwUi `y‡qKevi co–b Ges wb‡Pi QKwU †`‡L mgm¨vwU wfZi †_‡K eySvi †Póv Kiæb - cÖ_‡g eySvi †Póv Kiæb- ejv n‡q‡Q GKwU msL¨v (x)301 †_‡K hZUyKz eo n‡e A_©vr, x †_‡K 301 we‡qvM Ki‡j †h gvb ‡ei n‡e , 381 †_‡K H GKwU msL¨v (x) we‡qvM Ki‡j †h gvb †ei n‡e Zvi mgvb| GLb Avgv‡`i x Gi gvb †ei K‡i welqwU cÖgvY Kiv Riæwi| GRb¨ cÖkœvbymv‡i Pjyb GKwU kZ© `uvo Kiv‡bv hvK|) kZ©g‡Z, x - 301 = 381 - xev, x +x = 381 + 301 ev, 2x = 381 + 301 ev, x = 2 301381 ev, x = 2 682 = 341 (DËi) m¤ú~Y© mgm¨vwU wK¬qviwj eySvi Rb¨ Dc‡ii QKwU bZzb K‡i †`Lyb| civgk©: cÖwZwU A¼ evievi we¯ÍvwiZ Kiæb, †`L‡eb hLb wei³ jvM‡Q ZLb g‡bi ARv‡šÍB kU©‡UKwbK •Zwi n‡q †M‡Q! 35. GKwU msL¨v 560 †_‡K hZ Kg, 380 †_‡K Zvi mv‡o wZb¸Y †ewk| msL¨vwU KZ? Dc‡Rjv_vbvwkÿvAwdmvi(AETO):10 450 470 520 500 DËi: M mgm¨vwU‡Z †h msL¨vwU †ei Ki‡Z ejv n‡q‡Q- †mB ÔmsL¨vwUÕ I Ô560ÕGi gv‡S hZUzKz e¨eavb Ges †mB ÔmsL¨vwUÕ I Ô360ÕGi gv‡S hZUzKz e¨eavb , Zv hw` Avgiv Zzjbv Kwi Zvn‡j `ywU e¨eav‡bi cv_©K¨ n‡e mv‡o wZb¸Y †ewk n‡e| QKwU jÿ¨ Kiæb, welqwU wK¬qvi n‡q hv‡e| Gevi Ackb †_‡K ÔmsL¨vwUÕi gvb ewm‡q Df‡qi cv‡k¦©i e¨eavb wbY©q Kiæb Ges Dfq e¨eavb Zzjbv Kiæb †mwU GKwU Av‡iKwUi mv‡o wZb¸Y wKbv? (GLv‡b 110 Gi mv‡o wZb¸Y 70 n‡e bv) (GLv‡b 90 Gi mv‡o wZb¸Y 90 n‡e bv) (GLv‡b 40 Gi mv‡o wZb¸Y 140|) mwVK DËi (GLv‡b 60 Gi mv‡o wZb¸Y 120 n‡e bv) civgk©t cixÿvi LvZvq kU©Kv‡U we‡qvM K‡i wb‡eb| NM LK mgvavb NMLK x 301 381 x cv_©K¨ (x - 301) = cv_©K¨ (381 -x) 341 301 381 341 341 - 301 = 40 eo = 381 - 341 = 40 †QvU Gevi `vMwU co–b †Zv ey‡Sb wKbv? GKwU msL¨v (341) 301 †_‡K hZ (40) eo 381 †_‡K ZZ (40) †QvU| #ey‡S ey‡S mgvavb Kiæb, MwYZ fq `~i Kiæb msL¨vwU 560380 e¨eavb e¨eavb mv‡o wZb¸Y †ewk 470 560380450 560380 e¨eavb 450-380=70 e¨eavb 560 - 450 =110 e¨eavb 470-380=90 e¨eavb 560 - 470 =90 520 560380 500 560380 e¨eavb 500 - 380=120 e¨eavb 560 - 500 = 60 e¨eavb 520 - 380=140 e¨eavb 560 - 520 = 40
7.
Math Tutor 7 36.
765 †_‡K 656 hZ Kg, †Kvb msL¨vi 825 †_‡K ZZUzKz †ewk? ¯^ivóª gš¿YvjqewnivMgbIcvm‡cvU© Awa:mn:cwiPvjK:11;kÖgAwa:kÖg Kg©KZ©vGesRbmsL¨vIcwieviKj¨vYKg©KZ©v:03 932 933 934 935 DËi: M cÖ`Ë cÖ‡kœ ejv n‡q‡Q, 765 I 656 Gi gv‡S hZUzKz e¨eavb, msL¨vwU I 825 Gi gv‡S ZZUzKzB e¨eavb| myZivs, msL¨vwU = 109 + 825 = 934| PP©v Kiæb 37. GKwU msL¨v 31 †_‡K hZ †ewk, 55 †_‡K ZZ Kg, msL¨vwU KZ? cÖwZiÿvgš¿Yvj‡qiwmwfwjqvb÷vdAwdmviGes mnKvixcwiPvjK2016;ciivóª gš¿Yvj‡qimvBdviAwdmvi:12 39 41 43 45 DËi: M 38. GKwU msL¨v 999 †_‡K hZ †QvU 797 †_‡K ZZ eo| msL¨vwU KZ? Lv`¨Awa`߇iiLv`¨/Dc-Lv`¨cwi`k©K-2011 897 898 899 900 DËi: L 39. GKwU msL¨v 742 n‡Z hZ eo 830 n‡Z ZZ †QvU, msL¨vwU KZ? mgvR‡mevAwa`߇iiBDwbqbmgvRKg©xwb‡qvMcixÿv2016 780 782 790 786 DËi: N 40. GKwU msL¨v 470 †_‡K hZ eo 720 †_‡K ZZ †QvU| msL¨vwU KZ? gnv-wnmvewbixÿKIwbqš¿‡KiKvh©vj‡qAwWUi:15 565 595 615 †Kv‡bvwUB bq DËi: L 41. GKwU msL¨v 100 †_‡K hZ eo 320 †_‡K ZZ †QvU| msL¨vwU KZ? cjøxDbœqbImgevqwefv‡MiGKwUevwoGKwULvgvicÖK‡íi Dc‡Rjvmgš^qKvix:17;K…wlm¤úªmviYAwa:mnKvixK…wlKg©KZ©v:16 120 210 220 †Kv‡bvwUB bq DËi: L 01.07 µwgK msL¨vi ¸Ydj †R‡b wbb -04 µwgK msL¨v : x x + 1 x + 2 x + 3 µwgK msL¨v (Gfv‡e a‡i wb‡eb) 1 1 + 1 1 + 2 1 + 3 Dc‡iv³ µwgK msL¨v¸‡jv‡Z x =1 emv‡j 1, 2, 3, 4 1 2 3 4 BZ¨vw` µwgK msL¨v¸‡jv †c‡q hv‡eb| µwgK †Rvo: x x + 2 x + 4 x + 6 µwgK †Rvo msL¨v (Gfv‡e a‡i wb‡eb) 2 2 + 2 2 + 4 2 + 6 Dc‡iv³ µwgK †Rvo msL¨v¸‡jv‡Z x = 2 emv‡j 2, 4, 2 4 6 8 6, 8 BZ¨vw` µwgK †Rvo msL¨v¸‡jv †c‡q hv‡eb| µwgK we‡Rvo: x x + 2 x + 4 x + 6 µwgK we‡Rvo msL¨v (Gfv‡e a‡i wb‡eb) 1 1 + 2 1 + 4 1 + 6 Dc‡iv³ µwgK we‡Rvo msL¨v¸‡jv‡Z x =1 emv‡j 1, 3, 1 3 5 7 5, 7 BZ¨vw` µwgK we‡Rvo msL¨v¸‡jv †c‡q hv‡eb| µwgK †Rvo I µwgK we‡Rvo Dfq‡ÿ‡Î x, x +2, x + 4, x + 6 GKBiKg †`‡L KbwdDRW n‡eb bv, KviY GwU wbf©i K‡i x Gi gv‡bi Dci| x Gi gvb †Rvo wb‡j x, x + 2 … BZ¨vw` †Rvo µwgK msL¨v n‡e Ges x Gi gvb we‡Rvo wb‡j x, x + 2 … BZ¨vw` we‡Rvo µwgK msL¨v n‡e| civgk©: G RvZxq mgm¨vmn MwY‡Zi †h‡Kvb As‡k fv‡jv Kivi Rb¨ 1 †_‡K 25 ch©šÍ bvgZv Aek¨B Rvb‡Z n‡e| (K) `ywU µwgK msL¨vi ¸Ydj 42. `ywU µwgK abvZ¥K we‡Rvo msL¨vi ¸Ydj 255 n‡j msL¨vØq KZ? AgraniBankLtd.SeniorOfficer:13(cancelled) NMLK NMLK NMLK NMLK NMLK mgvavb NMLK 825765656 msL¨vwU e¨eavb 765 - 656 = 109 e¨eavb msL¨vwU - 825 = 109
8.
8Math Tutor 11, 13
13, 15 13, 17 15, 17 DËi: N g‡bKwi, µwgK abvZ¥K we‡Rvo msL¨vØq = x, x + 2 (a‡i †bqv µwgK we‡Rvo msL¨v `ywUi ¸Ydj n‡e 255 Gi mgvb) kZ©g‡Z, x (x + 2) = 255 ev, x2 + 2x – 255 = 0 ev, x2 + 17x – 15x – 255 = 0 ev, x(x + 17) – 15(x + 17) = 0 ev, (x + 17) (x – 15) = 0 x + 17 = 0 A_ev x – 15 = 0 ∴ x = – 17 A_ev x = 15 (x Gi FYvZ¥K gvb MÖnY‡hvM¨ bq) AZGe, µwgK abvZ¥K we‡Rvo msL¨vØq x = 15 I x + 2 = 17 | Ackb †_‡K kU©KvUt Ackb¸‡jvi msL¨vØq ¸Y K‡i hvi ¸Ydj 255 nq †mwUB DËi A_©vr, 15 17 = 255| 43. `ywU µwgK FYvZ¥K †Rvo c~Y©msL¨vi ¸Ydj 24 nq, Z‡e eo msL¨vwU KZ? IslamiBankLtd.ProbationaryOfficer:17 - 4 - 6 4 6 DËi: K 24 = ( 4)( 6)| ( 4) I ( 6) Gi gv‡S eo msL¨vwU n‡”Q 4| 44. `ywU msL¨vi ¸Ydj 162| hw` e„nËg msL¨vwU ÿz`ªZg msL¨vi wظY nq, Z‡e e„nËg msL¨vwU KZ? BangladeshKrishi BankLtd.SeniorOfficer:11 18 15 9 21 DËi: K Ackb Gi 18 †K hw` e„nËg msL¨v wn‡m‡e a‡i †bqv nq, Zvn‡j ÿz`ªZg msL¨vwU n‡e 9| 18 I 9 Gi ¸Ydj n‡e 162 Ges e„nËg msL¨vwU ÿz`ªZg msL¨vi wظY| (L) wZbwU µwgK msL¨vi ¸Ydj 45. wZbwU µwgK msL¨vi ¸Ydj 60 n‡j Zv‡`i †hvMdj KZ n‡e? ¯^ivóª gš¿Yvj‡qigv`K`ªe¨wbqš¿YAwa`߇iiDc-cwi`k©K:13;cwievi cwiKíbvAwa`߇iimnKvixcwiKíbvKg©KZ©v:12;RvZxqivR¯^ †ev‡W©imnKvixivR¯^ Kg©KZ©v:12 20 12 15 14 DËi: L we¯ÍvwiZ wbqgt g‡bKwi, msL¨v wZbwU = x, x + 1, x + 2 kZ©g‡Z, x(x+1) (x+2) = 60 ev, x(x2 + 3x + 2) – 60 = 0 ev, x3 + 3x2 + 2x – 60 = 0 ev, x3 – 3x2 + 6x2 – 18x + 20x – 60 = 0 ev, x2 (x–3) + 6x (x–3) + 20(x–3) = 0 ev, (x – 3) (x2 + 6x + 20) = 0 GLv‡b, x – 3 = 0 ∴ x = 3 µwgK msL¨v wZbwU = 3, 4 I 5 | myZivs, msL¨v wZbwUi †hvMdj = 3 + 4 + 5 = 12 | Drcv`‡K we‡køl‡Yi gva¨‡gt G RvZxq mgm¨v mgvav‡bi Rb¨ GB c×wZwU cvi‡d±| (cÖ_‡g cÖ`Ë msL¨vwU‡K Drcv`‡K we‡kølY K‡i wb‡eb) 2 60 ∴ 60 = 2235 (Gevi GB Drcv`K¸‡jv †_‡K 3 wU µwgK msL¨v •Zwi Ki‡eb) 2 30 = 345 3 15 µwgK msL¨v wZbwUi †hvMdj = 3 + 4 + 5 = 12| 5 46. 3wU µwgK c~Y©msL¨vi ¸Ydj 120| msL¨v 3wUi †hvMdj KZ? 29Zg I 32Zg wewmGm 12 13 14 15 DËi: N 120 = 22235 = 456 myZivs, msL¨v wZbwUi †hvMdj = 4 + 5 + 6 = 15| 47. 3wU µwgK c~Y©msL¨vi ¸Ydj 210| msL¨v 3wUi †hvMdj KZ? evsjv‡`k e¨vsK Gwm÷¨v›U wW‡i±i -04, cÖv_wgK mnKvix wkÿK 2010 (wZ¯Ív)] mgvavb NMLK mgvavb NMLK Kmgvavb NMLK mgvavb NMLK mgvavb NMLK awi, f(x) = x3 + 3x2 + 2x – 60 ∴ f(3) = 33 +(3 32 )+(23)–60 = 27+27 + 6 – 60 = 60 – 60 =0| †h‡nZz x = 3 emv‡j f(x) = 0 nq, †m‡nZz x – 3, f(x) Gi GKwU Drcv`K|
9.
Math Tutor 9 12
14 15 18 DËi: N 210 = 2 35 7 = 56 7| myZivs msL¨v wZbwUi †hvMdj = 5 + 6 + 7 = 18| 48. 3wU µwgK c~Y©msL¨vi ¸Ydj 210| †QvU `ywU msL¨vi †hvMdj KZ? DBBL Assistant officer -09 5 11 20 13 DËi: L 210 = 235 7 = 56 7 | myZivs †QvU `ywU msL¨vi †hvMdj = 5 + 6 = 11| 49. wZbwU wfbœ c~Y©msL¨vi ¸Ydj 6| msL¨v·qi mgwói wظ‡Yi gvb KZ? IBA(MBA):88-89 12 4 18 36 DËi: K 6 = 123| msL¨v·qi mgwó = 1+2+3 = 6| AZGe, msL¨v·qi mgwói wظY = 62 = 12| (6 Ggb GKwU msL¨v hvi Drcv`K·qi †hvMdj I ¸Ydj GKB n‡q _v‡K) (M) cici/ µwgK wZbwU †Rvo ev we‡Rvo msL¨vi ¸Ydj 50. cici wZbwU †Rvo msL¨vi ¸Ydj 192 n‡j, Zv‡`i †hvMdj KZ? wkÿvgš¿Yvj‡qiRywbqiBÝUªv±i (†UK):16 10 18 22 24 DËi: L 192 = 2222223 = 468| ∴ †Rvo msL¨v wZbwUi †hvMdj = 4 + 6 + 8 = 18| 51. wZbwU wfbœ we‡Rvo msL¨vi ¸Ydj 15| ÿz`ªZg msL¨vwU KZ? IBA(MBA):88-89 12 4 18 None DËi: N 15 = 135 | ∴ ÿz`ªZg msL¨vwU = 1| 52. wZbwU µwgK †Rvo c~Y©msL¨vi ÿz`ªZg msL¨vwU e„nËgwUi wZb¸Y A‡cÿv 40 Kg| e„nËg msL¨vwU KZ? PÆMÖvg e›`‡ii wb‡qvM 2017 14 17 18 19 DËi: M g‡bKwi, µwgK msL¨v wZbwU x, x + 2, x + 4 (cÖkœvbymv‡i e„nËg msL¨vwU‡K wZb¸Y Ki‡j cÖvß ¸Ydj I ÿz`ªZg msL¨vi cv_©K¨ 40 n‡e, ZvB kZ©g‡Z e„nËg msL¨vwUi wZb¸Y †_‡K ÿz`ªZg msL¨vwU we‡qvM K‡i mgvb mgvb 40 wjLyb) kZ©g‡Z, 3(x + 4) – x = 40 ev, 3x + 12 – x = 40 ev, 2x = 40 – 12 ev, 2x = 28 ∴ x = 14 myZivs, wb‡Y©q e„nËg msL¨v = x + 4 = 14 + 4 = 18 | 01.08 `ywU msL¨vi †hvMdj, we‡qvMdj I ¸Ydj 53. `ywU msL¨vi mgwó 146 Ges AšÍi 18| msL¨vØq KZ? Agrani BankLtd.Officer(cash):13 74, 62 82, 64 84, 60 80, 62 DËi: L g‡bKwi, eo msL¨vwU = x I †QvU msL¨vwU = y x + y = 146 ---- (1) x y = 18 ---- (2) 1 I 2 bs mgxKiY †hvM K‡i cvB, x + y = 146 x y = 18 2x = 164 ( 146 I 18 Gi †hvMdj) x = 2 164 = 82 (†hvMdj 2 ) (1) †_‡K (2) bs mgxKiY we‡qvM K‡i cvB, x + y = 146 x y = 18 2y = 128 (146 I 18 Gi we‡qvMdj) y = 2 128 = 64 (we‡qvMdj 2) mgvavb NMLK mgvavb NMLK mgvavb NMLK mgvavb NMLK mgvavb NMLK mgvavb NMLK mgvavb NMLK `ywU msL¨vi †hvMdj I we‡qvMdj †`qv _vK‡j eo msL¨v (x) wbY©‡qi wbqg- x = 2 AšÍiqimsL¨v؇mgwóqimsL¨v؇ `ywU msL¨vi †hvMdj I we‡qvMdj †`qv _vK‡j †QvU msL¨v (y) wbY©‡qi wbqg- y = 2 AšÍiqimsL¨v؇-mgwóqimsL¨v؇
10.
10Math Tutor AZGe, msL¨vØq
82 I 64| 54. `ywU msL¨vi †hvMdj 15 Ges we‡qvMdj 13| †QvU msL¨vwU KZ? ivóªvqË¡ e¨vsKAwdmvi:97 1 2 14 18 DËi: K †hvMdj †_‡K we‡qvMdj we‡qvM K‡i 2 Øviv fvM Ki‡j †QvU msL¨vwU cvIqv hv‡e- (15 13) 2 = 1| civgk©t eo msL¨v PvB‡j †hvM Ges †QvU msL¨v PvB‡j we‡qvM K‡i Zvici 2 Øviv fvM Kiæb| 55. `yBwU msL¨vi †hvMdj 60 Ges we‡qvMdj 10 n‡j, eo msL¨vwU KZ? BangladeshkrshiBank(DataEatryOperator):18 35 40 30 45 DËi: K †hvMdj I we‡qvMdj †hvM K‡i 2 Øviv fvM Ki‡j eo msL¨vwU cvIqv hv‡e- (60 + 10) 2 = 35| civgk©t G RvZxq mgm¨v¸‡jv gy‡L gy‡L mgvavb KivB fv‡jv| †hgb- 60 Gi mv‡_ 10 †hvM Ki‡j nq 70 Ges 70 Gi A‡a©K 35| 56. `ywU msL¨vi †hvMdj 33 Ges we‡qvMdj 15| †QvU msL¨vwU KZ? BangladeshBankAsst.Director:14 9 12 15 18 DËi: K 33 †_‡K 15 we‡qvM Ki‡j 18 Ges 18 Gi A‡a©K 9| 57. `ywU msL¨vi †hvMdj 21, we‡qvMdj 7| eo msL¨vi A‡a©K KZ? RbZve¨vsKwmwbqiAwdmvi:11; PubaliBankLtd.JuniorOfficer (cash):12 7 6 9 13 DËi: K 21 Gi mv‡_ 7 †hvM Ki‡j nq 28 Ges 28 Gi A‡a©K 14 n‡”Q eo msL¨v| cÖ‡kœ †P‡q‡Q eo msL¨vi A‡a©K, ZvB 14 Gi A‡a©K n‡e 7| †R‡b wbb -05 `ywU msL¨vi ¸Ydj xy †_‡K x I y msL¨v `ywU †ei Kiv wbqgt cÖ_‡g wPšÍv Ki‡eb ¸YdjwU‡Z x I y KZfv‡e Av‡Q| †hgb- hw` 20 †K a‡i †bqv nq, Zvn‡j 20G x I y Av‡Q- 1 20 = 20, 2 10 = 20, 4 5 = 20 A_©vr, 20 G x I y msL¨vhyMj Av‡Q wZbwU| Gici G‡`i gvS †_‡K cvi‡d± msL¨vhyMjwU Lyu‡R wb‡Z n‡e| cÖkœ n‡”Q cvi‡d± msL¨vhyMj †KvbwU? cvi‡d± msL¨vhyMj n‡”Q †h msL¨vhyMjwU cÖ‡kœi kZ©‡K wm× K‡i| GB AvBwWqvwU GKwU g¨vwRK AvBwWqv, Avcbvi AwfÁZv hZ †ewk n‡e, Avcwb GB AvBwWqvwU e¨envi Ki‡Z ZZ †ewk gRv cv‡eb| 58. `ywU msL¨vi †hvMdj 17 Ges ¸Ydj 72| †QvU msL¨vwU KZ? ivóªvqZe¨vsKwmwbqiAwdmvi:00 8 9 10 11 DËi: K g‡bKwi, eo msL¨vwU x Ges †QvU msL¨vwU y x + y = 17----- (1) xy = 72 ev, x = y 72 ----- (2) (1) G x Gi gvb ewm‡q cvB, y 72 + y = 17 ev, 17 72 2 y y ev, y2 17y + 72 = 0 ev, y2 9y 8y + 72 = 0 ev, y ( y 9) 8 (y 9) = 0 ev, ( y 9) (y 8) = 0 y = 9 A_ev y = 8 hw` y = 9 nq, Zvn‡j x = 9 72 = 8 hw` y = 8 nq, Zvn‡j x = 8 72 = 9 cÖkœvbymv‡i x n‡”Q eo msL¨v Ges y n‡”Q †QvU msL¨v| ZvB x = 9 Ges y = 8-B †hŠw³K| AZGe, †QvU msL¨vwU n‡”Q 8| mgvavb NMLK mgvavb NMLK mgvavb NMLK mgvavb NMLK mgvavb NMLK
11.
Math Tutor 11
msL¨vhyMj †ei K‡i `ªæZ mgvavb Kiæb- 72 Gi msL¨vhyMj mg~n- 2 I 36, 3 I 24, 4 I 18, 6 I 12, 8 I 9| GLv‡b cvi‡d± msL¨vhyMj n‡”Q 8 I 9, hv‡`i †hvMdj 17 Ges ¸Ydj 72| myZivs, †QvU msL¨vwU n‡”Q 8| (Avcbvi g‡b n‡Z cv‡i, meKqwU msL¨vhyMj †ei K‡i mgvavb Ki‡Z †Zv mgq †j‡M hv‡e| GiKg fvevi †Kvb my‡hvM †bB| KviY GB bvgZv¸‡jv Avcbvi gyL¯’ Av‡Q, ZvB cÖ‡kœi kZ© †`L‡jB e‡j w`‡Z cvi‡eb †Kvb msL¨vhyMjwU Avcbv‡K P‡qR Ki‡Z n‡e|) 59. `ywU msL¨vi ¸Ydj 189 Ges msL¨v `ywUi †hvMdj 30| msL¨v `ywU KZ? gnvwnmvewbixÿKIwbqš¿‡KiKvh©v.AaxbRywbqiAwWUi:14 9 I 21 7 I 23 8 I 22 22 I 18 DËi: K Ackb Gi msL¨vhyMjwU cÖ‡kœi kZ©‡K wm× K‡i, ZvB GwUB DËi n‡e| 60. †Kvb `ywU msL¨vi †hvMdj 10 Ges ¸Ydj 24? mnKvix_vbvcwievicwiKíbvAwdmvi:98 4, 6 6, 4 12, 2 4, 6 DËi: N Ackb Gi msL¨vhyMjwU cÖ‡kœi kZ©‡K wm× K‡i, ZvB GwUB DËi n‡e| 61. `yBwU msL¨vi AšÍi 7 Ges Zv‡`i MyYdj 60| msL¨v؇qi GKwU- DBBLAssistantofficer:09/BKBofficer:07 4 5 6 7 DËi: L 60 Gi 5 I 12 msL¨vhyMjwU cÖ‡kœi kZ©‡K wm× K‡i| PP©v Kiæb 62. `ywU msL¨vi †hvMdj 23 Ges we‡qvMdj 21| †QvU msL¨vwU KZ? Sonali,JanataandAgraniBankLtd.SeniorOfficer:08 4 3 2 None DËi: N 63. `yBwU msL¨vi ¸Ydj 10 Ges Zv‡`i mgwó 7 n‡j, e„nËg msL¨vwU KZ? EXIMBankLtd.Officer:14 2 2 4 5 DËi: N 64. `ywU msL¨vi ¸Ydj 120 Ges Zv‡`i e‡M©i †hvMdj 289| msL¨v؇qi mgwó KZ? EXIMBankLtd.Officer(IT):13 20 21 22 23 DËi: N 65. `yBwU msL¨vi ¸Ydj 42 Ges we‡qvMdj 1 n‡j msL¨v `y&ÕwU KZ?mve-†iwR÷ªvit03 4, 3 7, 6 8, 6 10, 8 DËi: L 66. 2wU msL¨vi †hvMdj 48 Ges Zv‡`i ¸Ydj 432| Z‡e eo msL¨vwU KZ? cwievicwiKíbvAwa`ßiwb‡qvMcixÿv:14 36 37 38 40 DËi: K 01.09 `ywU msL¨vi †hvMdj, we‡qvMdj I ¸Ydj 67. `ywU msL¨vi †hvMdj Zv‡`i we‡qvMd‡ji wZb¸Y| †QvU msL¨vwU 20 n‡j, eo msL¨vwU KZ? evsjv‡`kK…wle¨vsKAwdmvi:11 5 40 60 80 DËi: L g‡bKwi, eo msL¨vwU = x Ges †QvU msL¨vwU = y cÖkœg‡Z, x + y = 3(x y) (we‡qvMdj‡K 3 ¸Y Ki‡j †hvMd‡ji mgvb n‡e) ev, x + 20 = 3x 320(†QvU msL¨v, y = 20 ewm‡q) ev, x + 20 = 3x 60 ev, 2x = 80 x = 40| Option Test: Ackb mwVK n‡e bv, KviY eo msL¨vwU 20 Gi †P‡q eo n‡e| 40 + 20 = 60 Ges (40 20)3 = 203 = 60 (k‡Z©i mv‡_ wg‡j †M‡Q) I k‡Z©i mv‡_ wgj‡e bv| 68. `ywU msL¨vi AšÍi 2 Ges mgwó 4| Zv‡`i e‡M©i AšÍi KZ? BangladeshBankAsst.Direefor:12 7 8 6 5 DËi: L x = (2 + 4) 2 = 3 Ges y = (4 2) 2 = 1 x2 y2 = 32 12 = 9 1 = 8| A_ev x2 y2 = (x + y) (x y) = 42 =8 (exRMwY‡Z wbq‡g GB mgm¨vwU mgvavb Kiv AwaKZi mnR) 69. `yBwU msL¨vi mgwó 40 Ges Zv‡`i AšÍi 4| msL¨v؇qi AbycvZ KZ? JanataBankLld.ExecutiveOffice(Morring):17 11 : 9 11 : 18 21 : 19 22 : 9 DËi: K x = 40 + 4 = 44 2 = 22, y = 40 4 = 36 2 = 18 (GB MYbv¸‡jv gy‡L gy‡L K‡i †dj‡eb)mgvavb NMLK mgvavb NMLK NM LK mgvavb NMLK NMLK NMLK NMLK NMLK NMLK mgvavb NMLK Nmgvavb NMLK Kmgvavb NMLK
12.
12Math Tutor myZivs, msL¨v؇qi
AbycvZ = 22 : 18 = 11 : 9 | †R‡b wbb-06 (`ye©j‡`i Rb¨) (x + y) 2 1 = 51 ev, x + y = 51 2 = 102 A_ev (x y) 2 1 = 5 ev, x y = 5 2 = 10 Tips: A‡a©K _vK‡j wظY Ki‡jB x+y/ x -y Gi gvb cvIqv hvq| GKBfv‡e GK-Z…Zxqvsk _vK‡j wZb¸Y, GK PZz_©vsk _vK‡j 4¸Y, GK cÂgvsk _vK‡j 5 ¸Y Ki‡j x + y/ x - y Gi gvb cvIqv hvq| 70. `ywU msL¨vi A‡a©‡Ki †hvMdj 51| Zv‡`i cv_©‡K¨i GK PZz_©vsk 13| msL¨vØq KZ? Dc-mnKvixcwiPvjK(kÖg):01 52, 70 26, 27 25, 66 77, 25 DËi: N (we¯ÍvwiZ) ey‡S ey‡S mgvavb: `ywU msL¨vi A‡a©‡Ki †hvMdj 51, Gevi A‡a©K‡K wظY Ki‡j msL¨v `ywUi †hvMdj cvIqv hv‡e| A_©vr, x + y = 51 2 = 102| msL¨v `ywUi cv_©‡K¨i GK PZz_©vsk 13, Gevi GK PZz_©vsk‡K 4 ¸Y Ki‡j msL¨v `ywUi we‡qvMdj cvIqv hv‡e| A_©vr, x - y = 13 4 = 52| x = 102 + 52 = 154 Gi A‡a©K 77 Ges y = 102 - 52 = 50 Gi A‡a©K 25| 71. `ywU msL¨vi cv_©K¨ 11| Zv‡`i †hvMd‡ji GK cÂgvsk 9| msL¨v `ywU wK wK? evsjv‡`k e¨vsK (GwW) 2014 28 Ges 17 29 Ges 18 30 Ges 19 †Kv‡bvwUB bq DËi: K x - y = 11Ges x + y = 9 5 = 45(GK cÂgvsk 9 †K 5 ¸Y Kiv n‡q‡Q) x = 45 + 11 = 56 Gi A‡a©K 28 Ges y =45 - 11 = 34 Gi A‡a©K 17| †R‡b wbb-07 (wb‡Pi mgm¨v¸‡jv exRMvwYwZK m~Î cÖ‡qvM K‡iI mn‡RB mgvavb Ki‡Z cv‡ib) (x+y)2 = x2 + 2xy + y2 (x-y)2 = x2 - 2xy + y2 x2 - y2 = (x + y) (x-y) 72. `ywU msL¨vi e‡M©i mgwó 80 Ges Zv‡`i cv_©‡K¨i eM© 16| msL¨v؇qi ¸Ydj KZ?UCBL wmwbqi Awdmvi 2011 10 16 30 32 DËi: N †`qv Av‡Q, x2 + y2 = 80Ges (x-y)2 = 16 Avgiv Rvwb,(x-y)2 = x2 + y2 - 2xy ev, 16 = 80 - 2xy ev, 2xy = 64 xy = 32| cÖ‡kœ hw` msL¨v `ywU Rvb‡Z PvIqv nZ? Zvn‡j 32 †_‡K x I y msL¨vhyMj‡K †ei K‡i wb‡Z n‡e| 32G x I y Gi Rb¨ wZbwU msL¨vhyMj Av‡Q| †hgb- 1 I 32, 2 I 16 , 4 I 8 (GLv‡b cÖ‡Z¨KwU msL¨vhyM‡ji ¸Ydj 32) GB wZbwU msL¨v hyM‡ji gv‡S ïay 4 I 8 hyMjwU cÖ‡kœi kZ© c~Y© K‡i| A_©vr, 42 + 82 = 80 Ges 8 - 4 = 4 Gi eM© 16| `ywU msL¨vi ¸Ydj †_‡K cvi‡d± msL¨vhyMj †ei Kivi †KŠkjwU fv‡jvfv‡e Avq‡Ë¡ Ki‡Z cvi‡j Avcwb A‡bK RvqMvq `viæY myweav cv‡eb| cvi‡d± msL¨vhyMj n‡”Q H msL¨vhyMj †hwU cÖ‡kœi kZ©‡K c~Y© K‡i| 73. `ywU msL¨vi mgwó 15 Ges Zv‡`i e‡M©i mgwó 113| msL¨v `ywU †ei Kiæb| RbZv e¨vsK wj. (AEO) 2015 6 Ges 9 7 I 8 10 I 5 †Kv‡bvwUB bq DËi: L †`qv Av‡Q, x + y = 15Ges x2 + y2 = 113| Avgiv Rvwb, (x+y)2 = x2 + 2xy + y2 ev, 152 = 113 + 2xy ev, 225 - 113 = 2xy xy = 2 112 = 56(Dc‡ii A‡¼i gZ hw` msL¨v `ywUi ¸Ydj Rvb‡Z PvBZ, Zvn‡j 56 B DËi nZ, wKš‘ msL¨v `ywU †ei Ki‡Z e‡j‡Q ZvB cvi‡d± msL¨vhyMj †ei Ki‡Z n‡e| ) 56 G x I y Gi gvb wn‡m‡e wZbwU msL¨vhyMj 2 I 28, 4 I 14, 7 I 8 Av‡Q| G‡`i gv‡S ïay 7 I 8 msL¨vhyMjwU cÖ‡kœi kZ©‡K wm× K‡i| myZivs msL¨v `ywU n‡”Q 7 I 8| Avcwb GKevi welqwU eyS‡Z cvi‡j †Kvb msL¨v †`Lv gvÎB Zvi gv‡S cvi‡d± msL¨vhyMj †`L‡Z cv‡eb| mewKQz AwfÁZvi Dci wbf©i K‡i| 74. `yBwU msL¨vi AšÍi 5 Ges Zv‡`i e‡M©i cv_©K¨ 65| eo msL¨vwU KZ? evsjv‡`k nvDR wewìs dvBb¨vÝ K‡cv©‡ikb 2017 mgvavb NMLK mgvavb NMLK mgvavb NMLK mgvavb NMLK
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Math Tutor 13 13
11 8 9 DËi: N x - y = 5 Ges x2 - y2 = 65 Avgiv Rvwb, x2 - y2 = (x+y) (x-y) = 65 ev, (x + y)5 = 65 ev, x + y = 5 65 = 13(cÖ‡kœ x - y = 5 †`Iqv Av‡Q Ges Avgiv cvBjvg x + y = 13 | Gevi †hvMdj I we‡qvMdj †hvM K‡i 2 Øviv fvM Ki‡j eo msL¨vwU cvIqv hv‡e) eo msL¨vwU = 2 513 = 9| 01.10 M.mv.¸ †_‡K msL¨v wbY©q †R‡b wbb - 07 †h msL¨vwU `ywU ¸Yd‡ji `ywU‡ZB _v‡K †m msL¨vwUB M.mv.¸| `ywU msL¨vi ¸Ydj †_‡K msL¨vwU †ei Kivi `ÿZv hZ †ewk n‡e G RvZxq mgm¨v mgvavb Kiv ZZ mnR n‡e| 75. cÖ_g I wØZxq msL¨vi ¸Ydj 42 Ges wØZxq I Z…Zxq msL¨vi ¸Ydj 49| wØZxq msL¨vwU KZ? cÖv_wgKmnKvixwkÿK (gyw³‡hv×v)knx`gyw³‡hv×vimšÍvb):10(†ngšÍ) 5 6 7 8 DËi: M cÖ_g wØZxq msL¨v =42 Ges wØZxq I Z…Zxq msL¨vi ¸Ydj = 49| G‡`i M.mv.¸ 7-B n‡e wØZxq msL¨vwU, KviY `ywU ¸Yd‡jB wØZxq msL¨vwU common Av‡Q| wØZxq msL¨vwU 7| 76. cÖ_g I wØZxq msL¨vi ¸Ydj 35 Ges wØZxq I Z…Zxq msL¨vi ¸Ydj 63| wØZxq msL¨vwU KZ? cwiKíbvgš¿YvjqWvUv cÖ‡mwms Acv‡iUi:02 5 6 7 8 DËi: M 35 = 5 7 Ges 63 = 7 9| wØZxq msL¨vwU 7| 77. wZbwU cici †gŠwjK msL¨vi cÖ_g `yBwU msL¨vi ¸Ydj 91, †kl `yBwUi ¸Ydj 143 n‡j, msL¨v wZbwU KZ? moKI Rbc_Awa`߇iiDcmnKvixcÖ‡KŠkjx:10 7, 13, 11 7, 11, 13 11, 7, 13 11, 13, 7 DËi: K 91 = 7 13 Ges 143 = 11 13 G‡`i M.mv.¸ = 13| msL¨v 3wU n‡”Q 7, 13 Ges 11| 01.11 ¯^Kxq gvb I ¯’vbxq gvb (i) ¯^Kxq gvb (Face Value) t †Kvb mv_©K A¼ Avjv`vfv‡e wjL‡j †h msL¨v cÖKvk K‡i, Zv A‡¼i ¯^Kxq gvb| (ii) ¯’vbxq gvb (Place/local Value) t K‡qKwU A¼ cvkvcvwk wjL‡j †Kvb mv_©K A¼ Zvi Ae¯’v‡bi Rb¨ †h msL¨v cÖKvk K‡i, Zv‡K H A‡¼i ¯’vbxq gvb e‡j| Place Value Chart (¯’vbxq gvb wbY©‡qi QK) †KvwU wbhyZ jÿ AhyZ nvRvi kZK `kK GKK ¯’vbxq gvb 9 2 8 3 2 5 4 7 71 = 7 410= 40 5100 = 500 21000 =2000 310000 = 30000 810000 = 800000 21000000 = 2000000 910000000 = 90000000 78. 666 msL¨vwU‡Z me©ev‡gi 6 Gi gvb KZ? cwievicwiKíbvwnmvei¶K/¸`vgi¶K/†Kvlva¨¶:11 60 600 6 DËi: LMLK mgvavb NMLK mgvavb NMLK mgvavb NMLK mgvavb NMLK
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14Math Tutor me©ev‡gi 6
ÔkZKÕ ¯’v‡bi A¼ nIqvq Gi gvb n‡e 600| 79. 3254710 msL¨vwU‡Z 5 Gi ¯’vbxq gvb KZ? QuantitativeAptitudebyS.Chand&Aggarwal 5 10000 50000 54710 DËi: M 80. 458926 msL¨vwU‡Z 8 Gi ¯^Kxq gvb KZ? QuantitativeAptitudebyS.Chand&Aggarwal;Pubali Bank,JuniorOfficer-2019 8 1000 8000 8926 DËi: K 81. 503535 msL¨vwU‡Z 3 Gi ¯’vbxq gvb mg~‡ni mgwó KZ? QuantitativeAptitudebyS.Chand&Aggarwal 6 60 3030 3300 DËi: M mn¯ª ¯’v‡bi 3 Gi gvb 3000 I `k‡Ki ¯’v‡bi 3 Gi gvb 30| mgwó = 3000 + 30 = 3030| 82. 527435 msL¨vwU‡Z 7 I 3 Gi ¯’vbxq gv‡bi g‡a¨ cv_©K¨ KZ? QuantitativeAptitudebyS.Chand&Aggarwal 4 5 45 6970 DËi: N 7 I 3 Gi ¯’vbxq gv‡bi cv_©K¨ = 7000 - 30 = 6970| 83. 32675149 msL¨vwU‡Z 7 Gi ¯’vbxq gvb I ¯^Kxq gv‡bi g‡a¨ cv_©K¨ KZ? QuantitativeAptitudebyS.Chand& Aggarwal; Pubali Bank Ltd. Senior Offficer/Officer :16 5149 64851 69993 75142 DËi: M cÖ`Ë msL¨vwU 32675149 †_‡K 7 Gi ¯^Kxq gvb (face value)I ¯’vbxq gvb (local value) ‡ei Ki‡Z n‡e| Zvici G‡`i †h cv_©K¨ (difference) †ei n‡e ZvB DËi| 32675149msL¨vwU‡Z 7 Gi ¯^Kxq gvb 7 Ges ¯’vbxq gvb 70000. Zv‡`i cv_©K¨ = (70000 - 7) = 69993. 84. cvuP A‡¼i e„nËg I ÿz`ªZg msL¨vi mgwó KZ? QuantitativeAptitudebyS.Chand &Aggarwal 1,110 10,999 109,999 111,110 85. cvuP A¼wewkó ÿz`ªZg msL¨v †_‡K wZb A¼wewkó e„nËg msL¨v we‡qvM Ki‡j KZ Aewkó _v‡K? Quantitative AptitudebyS.Chand&Aggarwal 1 9000 9001 90001 DËi: M 86. 3 w`‡q ïiæ I 5 w`‡q †kl nIqv 5 A¼wewkó ÿz`ªZg msL¨vwU KZ n‡e? QuantitativeAptitudebyS.Chand& Aggarwal 31005 30015 30005 30025 DËi: M 87. 2, 4, 0, 7 A¼¸‡jv Øviv MwVZ 4 A‡¼i ÿz`ªZg msL¨v †KvbwU? QuantitativeAptitudebyS.Chand&Aggarwal 2047 2247 2407 2470 DËi: K 88. GKwU msL¨vi kZK, `kK I GKK ¯’vbxq AsK h_vµ‡g p, q, r n‡j msL¨vwU n‡e-†Rjvwbev©PbAwdmvit04 100r + 10p + q 100p + 10q + r 100q + 10r + p 100pq + r DËi: L p q r r 1 = r q 10 = 10q p 100 = 100p msL¨vwU = 100p + 10q + r 89. 856973 msL¨vwU‡Z 6 Gi ¯’vbxq gvb I ¯^Kxq gv‡bi g‡a¨ cv_©K¨ KZ? Pubali Bank Ltd. Trainee Asst. Teller : 17; Probashi Kallyan Bank Ltd. Senior Officer : 14 973 6973 5994 None of these DËi : M 90. 2, 3 Ges 4 Øviv 3 A‡¼i KZwU we‡Rvo msL¨v MVb Kiv hvq?gv`K`ªe¨ wbqš¿Y Awa`߇ii mnKvix cwiPvjK-2013 2wU 5wU 6wU 7wU 2, 3, 4 Øviv 3 A‡¼i 2wU we‡Rvo msL¨v MVb Kiv hvq| †hgb- 243 Ges 423| DËi: Kmgvavb NMLK NMLK mgvavb NM LK NMLK NMLK NMLK NMLK mgvavb NMLK mgvavb NMLK mgvavb NMLK NMLK NMLK mgvavb
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Math Tutor 15 01.12
A¼Ø‡qi ¯’vb cwieZ©b msµvšÍ mgm¨v †R‡b wbb – 08 (AwZ `ye©j‡`i Rb¨) `yB A¼ wewkó msL¨v (Original number) I ¯’vb wewbgqK…Z msL¨v (Reversed number) : xy Gi ¯’vbxq gvb wbY©q Kiv hvK| GLv‡b, x n‡”Q GKK ¯’vbxq A¼ Ges y n‡”Q `kK ¯’vbxq A¼| ∴yx x1 = x y 10 = 10y ∴ yx Gi ¯’vbxq gvb = 1oy + x GLb hw` A¼ `ywU ¯’vb wewbgq K‡i Zvn‡j †h bZzb msL¨vwU MwVZ n‡e Zvi ¯’vbxq gvb cwieZ©b n‡e| ZLb bZzb msL¨vwU‡Z y P‡j hv‡e GKK ¯’v‡b Ges x P‡j hv‡e `kK ¯’v‡b| A_©vr, ∴xy y1 = y x 10 = 10x ∴ yx Gi A¼Øq ¯’vb wewbgq Kivi ci xy Gi ¯’vbxq gvb = 10x + y hw` cÖkœc‡Î Ô`yB A¼ wewkó msL¨vi A¼Ø‡qi mgwóÕ †`qv _v‡K Zvn‡j `yB A¼ wewkó msL¨v (Original number) I ¯’vb wewbgqK…Z msL¨v (Reversed number) wb‡Pi wbq‡g a‡i wb‡eb, Zvn‡j mgm¨vwU mn‡R mgvavb Kiv hv‡e| Ô `yB A¼ wewkó msL¨vi A¼Ø‡qi mgwó 7Õ GB D`vniYwU mvg‡b †i‡L welqwU eySv‡bv hvK| g‡bKwi, GKK ¯’vbxq A¼ x n‡j `kK ¯’vbxq A¼ (7 x) msL¨vwU = 10 (7 x) + x = 70 - 9x ¯’vb wewbgq Ki‡j msL¨vwU `vovq = 10x + (7 x) = 9x + 7| 91. `yB A¼ wewkó msL¨vi A¼Ø‡qi mgwó 10| msL¨vwU †_‡K 18 we‡qvM Ki‡j A¼Øq ¯’vb cwieZ©b K‡i| msL¨vwU KZ? ÷¨vÛvU© e¨vsK wj. (cÖ‡ekbvwi Awdmvi) 2008 64 46 55 73 DËi: K g‡bKwi, GKK ¯’vbxq A¼ x n‡j `kK ¯’vbxq A¼ (10 x) msL¨vwU = 10 (10 x) + x = 100 - 9x ¯’vb wewbgq Ki‡j msL¨vwU `vovq = 10x + (10 x) = 9x + 10 kZ©g‡Z, (100 - 9x) – 18 = 9x + 10 ev, 9x + 9x = 100 – 28 ev, 18x = 72 x = 4 myZivs, wb‡Y©q msL¨vwU = 100 – 9x = 100 – 94 = 100 – 36 = 64 | Option Test : 64 – 18 = 46 92. `yB A¼ wewkó msL¨vi A¼Ø‡qi mgwó 10| msL¨vwU †_‡K 72 we‡qvM Ki‡j A¼Øq ¯’vb cwieZ©b K‡i| msL¨vwU KZ? AvBwmwe A¨vwm‡÷›U †cÖvMÖvgvi 2008 82 91 55 37 DËi: L Option Test : 91 – 72 = 19 93. `yB A¼ wewkó GKwU msL¨vi A¼Ø‡qi ¸Ydj 8| msL¨vwUi mv‡_ 18 †hvM Ki‡j A¼Øq ¯’vb cwieZ©b K‡i| msL¨vwU KZ? hgybv e¨vsK wj. (GgwUI) 2012 mgvavb NMLK mgvavb NMLK
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16Math Tutor 18 24
42 81 DËi: L g‡bKwi, GKK ¯’vbxq A¼ x Ges `kK ¯’vbxq A¼ y msL¨vwU = 10y + x A¼Øq ¯’vb wewbgq Ki‡j msL¨vwU `uvovq = 10x + y 1g kZ©g‡Z, xy = 8 ---- (i) 2q kZ©g‡Z, (10y + x) + 18 = 10x + y ev, 9x – 9y = 18 ev, 9(x – y) = 18 ev, x – y = 2 x = 2 + y ---- (ii) (i) bs mgxKi‡Y x Gi gvb ewm‡q cvB, (2 + y) y = 8 ev, 2y + y2 = 8 ev, y2 + 2y – 8 = 0 ev, y2 + 4y – 2y – 8 = 0 ev, y (y + 4) – 2 ( y + 4) = 0 ev, (y + 4) (y – 2) = 0 y + 4 = 0 A_ev y – 2 = 0 y = – 4 (y Gi FYvZ¥K gvb MÖnY‡hvM¨ bq) y = 2 (ii) bs mgxKi‡Y y = 2 ewm‡q cvB, x = 2 + 2 = 4 myZivs msL¨vwU = 10y + x = 10 2 + 4 = 24 Option Test: me KqwU Ack‡bi A¼Ø‡qi ¸Ydj 8, ZvB 1g kZ©wU cÖgvY Kivi `iKvi †bB| Ackb 24 + 18 = 42 †R‡b wbb – 09 g¨vwRK Z_¨- (Aek¨B Av‡jvPbvwU fv‡jvfv‡e eyS‡eb, cÖ‡qvR‡b GKvwaKevi coyb|) (1) Original number 37 Original number I Reversed number Gi cv_©K¨ memgq 9 Øviv wefvR¨ n‡e| Original number I Reversed number Gi cv_©K¨‡K 9 Øviv fvM K‡i cÖvß fvMdj = msL¨vwUi A¼Ø‡qi AšÍi/cv_©K¨| so, cÖkœc‡Î Original I Reverse number Gi cv_©K¨ †`qv _vK‡j Zv‡K 9 Øviv fvM Ki‡j A¼Ø‡qi AšÍi cvIqv hv‡e| (2) Reversed number 73 (3) Difference 73 37 = 36 (4) Divide by 9 9 36 = 4 (5) Difference of 2 digits 7 3 = 4 Original number Gi GKK ¯’vbxq A¼ eo n‡j reverse Kivi ci gvb e„w× cv‡e| †hgb- 34 (original number) 43(GLv‡b original number Gi GKK ¯’vbxq A¼ 4 eo nIqvq gvb e„w× n‡q‡Q| c~‡e©i 34 †_‡K 9 e„w× †c‡q 43 n‡q‡Q| GRb¨ e„w× cvIqvi K_v ej‡j original number Gi GKK ¯’vbxq A¼ eo _vK‡e|) Ges Original number Gi GKK ¯’vbxq A¼ †QvU n‡j, reverse Kivi ci gvb n«vm cvq| †hgb- 43(original number) 34 (GLv‡b original number Gi GKK ¯’vbxq A¼ 3 †QvU nIqvq gvb n«vm †c‡q‡Q| c~‡e©i 43 †_‡K 9n«vm †c‡q 34 n‡q‡Q| GRb¨ e„w× cvIqvi K_v ej‡j original number Gi GKK ¯’vbxq A¼ †QvU _vK‡e|) (A_ev) reverse Kivi ci gvb e„w× †c‡j reversed number wU eo n‡e wKš‘ original number wU †QvU n‡e Ges GKBfv‡e reverse Kivi ci gvb n«vm †c‡j reversed number wU †QvU n‡e wKš‘ original number wU eo n‡e| (†k‡li wbqgwUB †ek mnR) 94. `yB A¼ wewkó GKwU msL¨vi A¼Ø‡qi mgwó 7| A¼Øq ¯’vb wewbgq Ki‡j †h msL¨v cvIqv hvq, Zv cÖ`Ë msL¨v †_‡K 9 †ewk| msL¨vwU KZ? e¨vsKviÕm wm‡jKkb KwgwU (wmwbqi Awdmvi) 2018 61 25 34 43 DËi: M g‡bKwi,GKK ¯’vbxq A¼ x, Zvn‡j `kK ¯’vbxq A¼ n‡e 7 x | msL¨vwU = 10 (7 x) + x = 70 9x A¼ `ywU ¯’vb wewbgq Ki‡j msL¨vwU = 10x + (7 x) = 9x + 7 cÖkœg‡Z, 9x + 7 9 = 70 - 9x (Reversed msL¨vwU original msL¨v †_‡K 9 †ewk nIqvq 9 we‡qvM K‡i mgvb Kiv n‡q‡Q) ev, 9x + 9x = 70 + 2 ev 18x = 72x = 4 mgvavb NMLK L mgvavb NMLK
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Math Tutor 17 myZivs
msL¨vwU = 70 9x = 70 9 4 = 70 36 = 34| Original number I Reversed number Gi cv_©K¨ 9 ‡K 9 Øviv fvM K‡i 1 cvIqv hv‡”Q, GB 1 n‡”Q original msL¨vwUi A¼Ø‡qi cv_©K¨| Zvn‡j cÖ`Ë Ackb¸‡jvi gv‡S †hwUi A¼Ø‡qi cv_©K¨ 1 Av‡Q †mwUB n‡e wb‡Y©q msL¨vwU| Avgiv Ackb I †Z `ywU‡Z A¼Ø‡qi cv_©K¨ 1 †`L‡Z cvw”Q| Avgiv Rvwb Original number I Reversed number Gi cv_©K¨ †ewk/e„w× †c‡j Original number Gi GKK ¯’vbxq A¼ eo nq| †h‡nZz GB cÖ‡kœ †ewk/ e„w×i K_v ejv n‡q‡Q, ZvB Original number wUi GKK ¯’vbxq A¼ eo n‡e| G Abymv‡i Ackb I Gi gv‡S mwVK DËi n‡e | (GB wbqgwU eyS‡Z mgq jvM‡jI Gi gva¨‡g me‡P‡q Kg mg‡q mgvavb Kiv hvq) (A_ev) 6116 (cv_©K¨ 45, hv mwVK bq) 25 52 (cv_©K¨ 27, hv mwVK bq) 34 43 (cv_©K¨ 9, cÖ‡kœ †h‡nZ z reverse Kivi ci reversed number wU eo n‡”Q †m‡nZz original number wU †QvU n‡e| Ackb -†Z orginal number wU †QvU weavq GwUB mwVK DËi) 43 34 (cv_©K¨ 9 _vK‡jI original number wU eo nIqvq GwU mwVK DËi bq) | 95. `yB AsK wewkó GKwU msL¨v, AsK؇qi ¯’vb wewbg‡qi d‡j 54 e„w× cvq| AsK `ywUi †hvMdj 12 n‡j msL¨vwU KZ? 37Zg wewmGm 57 75 39 93 DËi: M 54 ÷ 9 = 6, wb‡Y©q msL¨vwUi A¼Ø‡qi cv_©K¨ n‡e 6, hv Ackb I †Z Av‡Q| wKš‘ cÖ‡kœ Ôe„w×Õ K_vwU ejv _vKvq msL¨vwUi GKK ¯’vbxq A¼wU ÔeoÕ n‡e †m Abymv‡i mwVK DËi | (A_ev) 39 93 (cv_©K¨ 54, cÖ‡kœ Ôe„w×Õ ejvq reversed number wU eo n‡e Ges original number wU †QvU n‡e, ZvB GwUB mwVK DËi)| 96. `yB A¼ wewkó GKwU msL¨vi A¼Ø‡qi ¯’vb cwieZ©b K‡i cÖvß msL¨v g~j msL¨v A‡cÿv 54 Kg| msL¨vwUi A¼Ø‡qi mgwó 12 n‡j, g~j msL¨vwU KZ? 28 39 82 †KvbwUB bq DËi: N g‡bKwi, GKK ¯’vbxq A¼ x, Zvn‡j `kK ¯’vbxq A¼ n‡e 12 x | msL¨vwU = 10 (12 x) + x = 120 9x A¼ `ywU ¯’vb wewbgq Ki‡j msL¨vwU = 10x + (12 x) = 9x + 12 cÖkœg‡Z, 9x +12 54 = 120 9x (Reversed msL¨vwU originalmsL¨v †_‡K 54Kg nIqvq 54 †hvM K‡i mgvb Kiv n‡q‡Q) ev, 9x + 9x = 120 66 ev, 18x = 54x = 3 myZivs msL¨vwU = 120 93 = 120 27 = 93 cÖkœvbymv‡i A¼Ø‡qi mgwó n‡Z n‡e 12 hv ïay Ackb †Z Av‡Q Ges 54÷9 = 6 Abymv‡i A¼Ø‡qi cv_©K¨ n‡Z n‡e 6| wKš‘ cÖ‡kœ Reversed number g~j msL¨v (Original number) A‡cÿv ÔKgÕ nIqvq msL¨vwUi GKK ¯’vbxq A¼ Ô‡QvUÕ n‡Z n‡e, hv Ack‡b †bB ZvB GB AckbwUI mwVK bq| Z‡e GKK ¯’vbxq A¼ Ô‡QvUÕ (A_©vr, 93 n‡j)n‡j DËiwU mwVK nZ| Ackb , I Gi Original number I Reversed number Gi cv_©K¨ 54, G‡`i gv‡S Ackb I Gi GKK ¯’vbxq A¼ eo nIqvq Giv ev` hv‡e| Ackb Gi GKK ¯’vbxq A¼ Ô‡QvUÕ n‡jI cÖkœvbymv‡i Gi A¼Ø‡qi mgwó 12 bq, ZvB GwUI evwZj| DËi n‡e Ô‡KvbwUB bqÕ| 97. `yB A¼ wewkó †Kvb msL¨vi A¼Ø‡qi mgwó 9| A¼ `ywU ¯’vb wewbgq Ki‡j †h msL¨v cvIqv hvq Zv cÖ`Ë msL¨v n‡Z 45 Kg| msL¨vwU wbY©q Kiæb| gva¨wgK mnKvix cÖavb wkÿK I †Rjv mnKvix wkÿv Awdmvi 2003 54 63 72 81 DËi: M 45 ÷ 9 = 5, wb‡Y©q msL¨vwUi A¼Ø‡qi cv_©K¨ n‡e 5, hv ïay Ackb †Z Av‡Q| (A_ev) 72 27 (cv_©K¨ 45, Ab¨‡Kvb Ack‡bi cv_©K¨ 45 bv _vKvq mivmwi GwUB DËi n‡e) hw` cÖ‡kœ AviI GKwU Ackb 27 _vKZ, Zvn‡jI DËi 72-B nZ| KviY cÖ‡kœ reversed number wU original number †_‡K ÔKg/‡QvUÕ nIqvq original number wU eo n‡e|) 98. `yB A¼ wewkó GKwU msL¨vi A¼Ø‡qi mgwó 8| A¼Øq ¯’vb wewbgq Ki‡j †h msL¨v cvIqv hvq Zv cÖ`Ë msL¨v n‡Z M Mmgvavb NMLK M LKMLK L mgvavb NMLK M M NMmgvavb NMLK N M MLK M NM NM
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18Math Tutor 54 Kg|
msL¨vwU KZ? RbZv e¨vsK (Awdmvi) 2009 71 80 62 53 DËi: K 54÷ 9 = 6, wb‡Y©q msL¨vwUi A¼Ø‡qi cv_©K¨ n‡e 6, hv ïay Ackb †Z Av‡Q| (A_ev) 71 17 (cv_©K¨ 54, hv Ab¨‡Kvb Ack‡b †bB, ZvB mivmwi GwUB DËi) 99. `yB A¼ wewkó GKwU msL¨vi A¼Ø‡qi mgwó 9| msL¨vwU n‡Z 9 we‡qvM Ki‡j Gi A¼Øq ¯’vb wewbgq K‡i| msL¨vwU KZ? cwievi cwiKíbv Awa`ßi cwi`wk©Kv cÖwkÿYv_x© 2013 34 67 54 23 DËi: M 9 ÷ 9 = 1, wb‡Y©q msL¨vwUi A¼Ø‡qi cv_©K¨ n‡e 1, hv me KqwU Ack‡bB Av‡Q| wKš‘ original number †_‡K 9 we‡qvM K‡i reversed number Avm‡e ZvB reversed number wU †QvU n‡e Ges original number wU eo n‡e| G Abymv‡i ïay Ackb †K reverse Ki‡j reversed number wU †QvU n‡e| 100. `yB A¼ wewkó †Kv‡bv msL¨v Ges A¼Øq ¯’vb wewbgq Ki‡j MwVZ msL¨vi cv_©K¨ 36| msL¨vwUi A¼Ø‡qi AšÍi KZ? evsjv‡`k e¨vsK (A¨vwm÷¨v›U wW‡i±i) 2012 4 2 10 16 DËi: K GKK ¯’vbxq A¼ x Ges `kK ¯’vbxq A¼ y n‡j msL¨vwU = 1oy + x, A¼Øq ¯’vb wewbgq Ki‡j msL¨vwU `uvovq 10x + y . (Original number I reversed number Gi cv_©K¨ n‡”Q 36) kZ©g‡Z, (10x + y) – (10y + x) = 36 ev, 9x – 9y = 36 ev, 9 (x – y) = 36 x – y = 36 9 = 4 myZivs msL¨vwUi A¼Ø‡qi AšÍi 4 | kU©Kv‡U mgvavbt msL¨vwUi A¼Ø‡qi AšÍi = 36 9 = 4 101. `yB A¼ wewkó GKwU abvZ¥K c~Y©msL¨v Ges A¼Øq ¯’vb wewbgq Ki‡j MwVZ msL¨vi cv_©K¨ 27| msL¨vwUi A¼Ø‡qi AšÍi KZ? evsjv‡`k K…wl e¨vsK wj. (wmwbqi Awdmvi) 2017 3 4 5 6 DËi: K msL¨vwUi A¼Ø‡qi AšÍi = 27 9 = 3 102. `yB A¼wewkó GKwU msL¨v msL¨vwUi A¼Ø‡qi †hvMd‡ji 4 ¸Y| msL¨vwUi mv‡_ 27 †hvM Ki‡j A¼Øq ¯’vb wewbgq K‡i| msL¨vwU KZ? evsjv‡`k wkwcs K‡cv©‡ikb 2018 12 42 24 36 DËi: N 36 = 3 + 6 = 9 4= 36, 36 + 27 = 63 103. `yB A¼ wewkó GKwU msL¨vi `kK ¯’vbxq A¼wU GKK ¯’vbxq A¼ †_‡K 5 eo| msL¨vwU †_‡K A¼Ø‡qi mgwói 5 ¸Y we‡qvM Ki‡j A¼Øq ¯’vb wewbgq K‡i| msL¨vwU KZ? _vbv mnKvix wkÿv Awdmvi : 2005 61 94 72 83 DËi: M cÖ`Ë Ackbmg~‡ni cÖ‡Z¨KwU‡Z `kK ¯’vbxq A¼wU GKK ¯’vbxq A¼ †_‡K 5 eo, ZvB GB kZ©wU cÖgvY Kivi `iKvi †bB| Avgiv 2q kZ©wU cÖgvY Kie- 61 A¼Ø‡qi mgwó = 6 + 1 = 7, mgwói 5 ¸Y = 75 = 35| 61 - 35 = 26 (GwU mwVK bq) 94 A¼Ø‡qi mgwó = 9 + 4 = 13, mgwói 5 ¸Y = 135 = 65| 94 - 65 = 29 (GwU mwVK bq) 72 A¼Ø‡qi mgwó = 7 + 2 = 9, mgwói 5 ¸Y = 95 = 45 | 72 - 45 = 27 (GwU mwVK) 104. `yB A¼ wewkó GKwU msL¨vi GK‡Ki A¼ `k‡Ki A¼ A‡cÿv 3 †ewk| msL¨vwU Gi A¼Ø‡qi mgwói wZb¸Y A‡cÿv 4 †ewk| msL¨vwU KZ?14Zg wewmGm 47 36 25 14 DËi: M 25 A¼Ø‡qi mgwó = 2 + 5 = 7, mgwói 3 ¸Y = 73 = 21 25 - 21 = 4 †ewk|Mmgvavb NMLK M L K mgvavb NMLK Nmgvavb NMLK mgvavb NMLK mgvavb NMLK M mgvavb NMLK K Kmgvavb NMLK
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Math Tutor 19 01.13
ARvbv msL¨v wbY©q †R‡b wbb – 10 Gai‡bi mgm¨v mgvav‡bi †ÿ‡Î ïiæ‡ZB GKwU msL¨v x a‡i wbb, Zvici `v‡M hv hv †hfv‡e ejv Av‡Q †m Abymv‡i GwM‡q hvb| A_©vr, †hvM ej‡j †hvM Kiæb, we‡qvM ej‡j we‡qvM Kiæb...| †k‡li w`‡K GKUv P~ovšÍ dj (†hvMdj/ we‡qvMdj/fvMdj/¸Ydj †h‡KvbwU n‡Z cv‡i) †`qv _vK‡e| Gevi Avcbvi a‡i †bqv cÖvß dj mgvb mgvb P~ovšÍ d‡j wj‡L Zzjbv Kiæb| e¨m&, msL¨vwU P‡j Avm‡e| †hgb- †Kvb msL¨v n‡Z 175 we‡qvM K‡i 130 †hvM Ki‡j †hvMdj 297 n‡e? ivóªvqË e¨vsK Awdmvi - 97 | awi, msL¨vwU = x | Pjyb, `v‡M hv hv ejv Av‡Q †mwU AbymiY Kiv hvK- x – 175 + 130| dvBbvwj, Gevi Zzjbv Kiæb- x – 175 + 130 = 297 x = 342| D‡ëv †g_W AbymiY Kiæb: D‡ëv †g_W n‡”Q ÔP~ovšÍ djÕ †_‡K wcwQ‡q wcwQ‡q ïiæi RvqMvq wd‡i Avmv| A‡bKUv mvg‡bi w`‡K GwM‡q wM‡q bv Ny‡i cybivq Av‡Mi RvqMvq wd‡i Avmv| mnR K_vq, hvevi mgq mvg‡b cv †d‡j‡Qb, Avmvi mgq wcQ‡b cv †dj‡Z n‡e| + 5 – 2 3 = 39 {(GKwU msL¨v + 5) – 2} 3 = 39 D‡ëv †g_‡W Avgiv 39 †_‡K wcwQ‡q wcwQ‡q ÔGKwU msL¨vÕi RvqMvq †cŠQe| GRb¨ 39 Gi Av‡M 3 ¸Y K‡iwQjvg, GLb 39 †K 3 Øviv fvM Kie = 39 3 = 13| 3 Gi Av‡M 2 we‡qvM K‡iwQjvg GLb 13 Gi mv‡_ 2 †hvM Kie = 13 + 2 = 15| 2 Gi Av‡M Avgiv 5 †hvM K‡iwQjvg Gevi 15 †_‡K 5 we‡qvM Kie = 15 – 5 = 10| e¨m&, Avgiv ÔGKwU msL¨vÕi RvqMvq wd‡i Avmjvg| civgk©: D‡ëv †g_WwU P~ovšÍ dj †_‡K ïiæ Ki‡Z n‡e, Zvici ch©vµ‡g †h †h wPý _vK‡e Zvi wecixZ wP‡ýi KvR Ki‡Z n‡e| †hvM _vK‡j we‡qvM, we‡qvM _vK‡j †hvM, ¸Y _vK‡j fvM, fvM _vK‡j ¸Y Ki‡Z n‡e| (K) mgxKiY I D‡ëv †g_W e¨envi K‡i mgvavb hLb GKwU msL¨vi mv‡_ †hvM, we‡qvM, ¸Y, fvM avivevwnKfv‡e GK wbtk¦v‡m e¨envi K‡i P~ovšÍ d‡j †cuŠQv nq ïay ZLbB D‡ëv †g_W e¨envi Kiv hvq| G av‡ci mgxKiY¸‡jvi w`‡K jÿ¨ K‡i †`Lyb, cÖwZwU mgxKi‡Y x GKeviB e¨envi Kiv n‡q‡Q| A_P (L) av‡ci mgxKiY¸‡jv‡Z x GKvwaKevi e¨envi Kiv n‡q‡Q, ZvB (L) av‡c D‡ëv †g_W e¨envi Kiv hv‡e bv|) 105. †Kvb msL¨vi m‡½ 7 †hvM K‡i, †hvMdj‡K 5 w`‡q ¸Y K‡i, ¸Ydj‡K 9 w`‡q fvM K‡i, fvMdj †_‡K 3 we‡qvM Kiv‡Z we‡qvMdj 12 nq| msL¨vwU KZ? eb I cwi‡ek gš¿Yvj‡qi mnKvix cwiPvjK t 95 mgvavb hvÎvi‡¤¢i ¯’vb MšÍe¨¯’j Mr X hvÎvi‡¤¢i ¯’vb †_‡K hvÎv ïiæ K‡i MšÍe¨¯’‡j †cŠQj| MšÍe¨¯’j †_‡K bv Ny‡i D‡ëvfv‡e Avevi hvÎvi‡¤¢i ¯’v‡b wd‡i Avmj| GwUB n‡”Q D‡ëv †g_W! GKwU msL¨v
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20Math Tutor 20 18 22
21 DËi: K awi, msL¨vwU x kZ©g‡Z, 123 9 57 )(x ev, 15 9 57 )(x ev, 57 )(x = 135 ev, 7x = 5 135 ev, 7x = 27 x = 27 - 7 = 20| D‡ëv †g_‡W mgvavb: 12 Gi Av‡M 3 we‡qvM Kiv n‡q‡Q, GLb 12 Gi mv‡_ 3 †hvM Ki‡Z n‡e = 12 + 3 = 15| 3 Gi Av‡M 9 Øviv fvM Kiv n‡q‡Q, GLb 15 Gi mv‡_ 9 ¸Y Ki‡Z n‡e = 15 9 = 135| 9 Gi Av‡M 5 ¸Y Kiv n‡q‡Q, GLb 135 †K 5 Øviv fvM Ki‡Z n‡e = 135 5 = 27| 5 Gi Av‡M 7 †hvM Kiv n‡q‡Q, GLb 7 we‡qvM Ki‡Z n‡e = 27 - 7 = 20| kU©Kv‡U: 12 + 3 = 15 15 9 = 135 135 5 = 27 27 - 7 = 20| (GB AvBwWqv e¨envi K‡i Lye `ªæZ mgvavb Kiv hvq) 106. †Kvb msL¨vi A‡a©‡Ki mv‡_ 4 †hvM Ki‡j †hvMdj nq 14| msL¨vwU KZ? evsjv‡`k e¨vsKAwdmvi -01 10 15 20 25 DËi: M awi, msL¨vwU = x kZ©g‡Z, 2 x + 4 = 14 ev, 2 x = 10 x = 20| D‡ëv †g_‡W mgvavb: 14 Gi Av‡M 4 †hvM Kiv n‡q‡Q, ZvB 14 †_‡K 4 we‡qvM Ki‡Z n‡e= 14 - 4 = 10| 4 Gi Av‡M A‡a©K _vKvq 10†K wظY Ki‡Z n‡e = 102 = 20| gy‡L gy‡L: 14-4 = 10, 102 = 20 civgk©: wb‡Pi mgm¨v¸‡jvi cÖwZwUi mgxKiY •Zwi K‡i †`qv nj| Avcbvi KvR n‡”Q- mgxKiY †_‡K x Gi gvb †ei K‡i msL¨vwU wbY©q Kiv| 107. †Kvb msL¨vi ১ ৩ mv‡_ 6 †hvM Ki‡j †hvMdj 28 nq| msL¨vwU KZ? evsjv‡`k e¨vsK Gwm÷¨v›U wW‡i±i -08 44 66 42 84 DËi: L 28 - 6 = 22 22 3 = 66| mgxKiY: 3 x + 6 = 28 108. †Kvb msL¨vi GK PZz_v©sk †_‡K 4 we‡qvM Ki‡j we‡qvMdj nq 20| msL¨vwU KZ? we‡Kwe Awdmvi - 07 12 24 36 96 DËi: N 20 + 4 = 24 24 4 = 96| mgxKiY: 4 x - 4 = 20 109. †Kvb msL¨vi GK cÂgvsk †_‡K 5 we‡qvM Ki‡j we‡qvMdj nq 10| msL¨vwU KZ?evsjv‡`kK…wle¨vsKwj. (wmwbqiAwdmvi)2011 15 25 50 75 DËi: N 10 + 5 = 15 15 5 = 75| mgxKiY: 5 x - 5 = 10 110. †Kvb msL¨vi wظ‡Yi mv‡_ 2 †hvM Ki‡j †hvMdj 88 n‡e? ¯^v¯’¨gš¿Yvj‡qiAaxb†mevcwi`߇iiwmwbqi÷vdbvm©:16 41 42 44 43 DËi: N 88 - 2 = 86 86 2 = 43 (wظY Gi wecixZ A‡a©K) mgxKiY: 2x + 2 = 88 111. GKwU msL¨vi wظ‡Yi mv‡_ 9 †hvM Kiv nj| hw` cÖvß djvdjwU‡K wZb¸Y Kiv nq Zvn‡j 75 nq| msL¨vwU KZ? iƒcvjx e¨vsK(wmwbqiAwdmvi)2013 3.5 6 8 †Kv‡bvwUB bq DËi: M 75 3 = 25 25-9 = 16 16 2 = 8| mgxKiY: (2x + 9) 3 = 75 112. GKwU msL¨vi e‡M©i mv‡_ 4 †hvM Ki‡j †hvMdj 40 nq| msL¨vwU KZ? GKwUevwoGKwULvgvi(Dc‡Rjvmgš^qKvix)17 4 5 8 6 DËi: N 40 - 4 = 36 36 = 6| (eM© Gi wecixZ eM©g~j) mgxKiY: x2 + 4 = 40 (L) mgxKiY e¨envi K‡i mgvavb mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK
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Math Tutor 21
†R‡b wbb – 11 (AwZ `ye©j‡`i Rb¨) GKwU msL¨v wظY = 2x, wZb¸Y = 3x, Pvi¸Y = 4x cvuP¸Y = 5x BZ¨vw`| GKwU msL¨vi A‡a©K = 2 x , GK-Z…Zxqvsk = 3 x , GK-PZy_©vsk = 4 x , `yB-Z…Zxqvsk = 3 2x , wZb-cÂgvsk = 5 3x GKwU msL¨v I Zvi wecixZ fMœvsk = x I x 1 , 2 I 2 1 , 7 I 7 1 BZ¨vw`| 113. GKwU msL¨vi wZb¸‡Yi mv‡_ wظY †hvM Ki‡j 90 n‡e| msL¨vwU KZ? cÖwZiÿvgš¿Yvjqwmwfwjqvb÷vdAwdmviGes mn:cwi:wb‡qvM:16/cwiKíbvgš¿Yvjqwb‡qvMcixÿv:16 16 18 20 24 DËi: L awi, msL¨vwU x| kZ©g‡Z, 3x + 2x = 90 ev, 5x = 90 x = 18| 114. †Kvb GKwU msL¨vi 13 ¸Y †_‡K 4 ¸Y ev` w`‡j 171 nq, msL¨vwU KZ? cÖv_wgK I MYwkÿv wefv‡M mnKvix cwiPvjK -01 15 17 19 29 DËi: M awi, msL¨vwU x| kZ©g‡Z, 13x - 4x = 171 ev, 9x = 171 x = 19| 115. †Kvb msL¨vi 9 ¸Y †_‡K 15 ¸Y 54 †ewk? AvenvIqv Awa`߇ii mnKvix AvenvIqvwe` -95 9 15 54 6 DËi: K awi, msL¨vwU x| kZ©g‡Z, 15x - 9x = 54 ev, 6x = 54 x = 9| 116. †Kvb msL¨vi 6 ¸Y n‡Z 15¸Y 63 †ewk? Z_¨ gš¿Yvj‡qi Aax‡b mnKvix cwiPvjK, †MÖW-2t03 6 7 3 9 DËi: L awi, msL¨vwU x| kZ©g‡Z, 15x - 6x = 63 ev, 9x = 63 x = 7| 117. GKwU msL¨vi A‡a©K Zvi GK Z…Zxqvs‡ki PvB‡Z 17 †ewk| msL¨vwU KZ? cÖwZiÿv gš¿Yv. Aax‡b mvBdvi Awdmvi- 99 52 84 102 204 DËi: M awi, msL¨vwU x| kZ©g‡Z, 2 x - 3 x = 17 ev, 6 23 xx = 17 ev, x = 102 | 118. GKwU msL¨v I Zvi wecixZ fMœvs‡ki †hvMdj msL¨vwUi wظ‡Yi mgvb| msL¨vwU KZ? weweG : 94-95 1 -1 1 A_ev -1 2 DËi: M awi, msL¨vwU x | kZ©g‡Z, x + x 1 = 2x ev, x 1 = x ev, x2 = 1 x = 1 119. GKwU msL¨vi 5 ¸‡Yi mv‡_ Zvi eM© we‡qvM Ki‡j Ges 6 we‡qvM Ki‡j we†qvMdj k~b¨ nq| msL¨vwU - 13Zg†emiKvixwkÿKwbeÜbIcÖZ¨vqcixÿv(¯‹zj/mgchv©q):16 1 A_ev 2 3 A_ev 4 2 A_ev 3 3 A_ev 4 DËi: M awi, msL¨vwU x | kZ©g‡Z, 5x - x2 - 6 = 0 ev, x2 - 5x + 6 = 0 ev, x2 - 3x - 2x + 6 ev, x (x - 3) - 2(x -3) = 0 ev, (x - 3) (x -2) = 0 x = 3 ev x = 2 | 120. †Kvb msL¨vi wظ‡Yi mv‡_ 3 †hvM Ki‡j †hvMdj msL¨vwUi A‡cÿv 7 †ewk nq| msL¨vwU KZ? evsjv‡`k †c‡Uªvwjqvg G·‡cøv‡ikbGÛ†cÖvWvKkb†Kv¤úvwbwj.(ev‡c·)-17 3 4 5 6 DËi: L awi, msL¨vwU x | kZ©g‡Z, 2x + 3 = x + 7 ev, x = 4 | 121.GKwU msL¨vi wظ‡Yi mv‡_ 20 †hvM K‡i cÖvß djvdj msL¨vwUi 8 ¸Y †_‡K 4 we‡qvM K‡i cÖvß djvd‡ji mgvb| msL¨vwU KZ? c~evjxe¨vsKwj.(†UªBwb A¨vwm‡÷›U)2017 2 3 4 6 DËi: M awi, msL¨vwU x | kZ©g‡Z, 2x + 20 = 8x - 4 ev, 6x = 24 x = 4 | 122. GKwU msL¨vi 4 ¸‡Yi mv‡_ 10 †hvM Ki‡j DËi nq msL¨vwUi 5 ¸Y A‡cÿv 5 Kg| msL¨vwU KZ? Bmjvgx e¨vsKwj.(K¨vk)2017 30 20 25 15 DËi: N awi, msL¨vwU x | kZ©g‡Z, 4x + 10 = 5x - 5mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK
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22Math Tutor x
= 15 | 123. 13 Ges GKwU msL¨vi mgwói GK Z…Zxqvsk, msL¨vwUi wظ‡Yi †P‡q 1 †ewk| msL¨vwU †ei Kiæb| evsjv‡`ke¨vsK (A¨vwm‡÷›UwW‡i±i)2012 6 2 5 3 DËi: L awi, msL¨vwU x [13 Ges GKwU msL¨vi mgwói GK Z…Zxqvsk = (13 + x) 3 1 = 3 13 x ] kZ©g‡Z, 3 13 x = 2x + 1 ev, 6x + 3 = 13 + x ev, 5x = 10 x = 2 01.14 wefvR¨Zv I fvRK msL¨v wbY©q (K) wefvR¨Zvi bxwZ 2 Øviv wefvR¨: †Kv‡bv msL¨vi GKK ¯’vbxq A¼wU k~b¨ (0) A_ev †Rvo n‡j cÖ`Ë msL¨vwU 2 Øviv wefvR¨ n‡e| A_ev msL¨vwUi †k‡l 1wU k~b¨ (0) _vK‡jI 2 Øviv wefvR¨ n‡e| g‡b ivLyb: †h msL¨vi †k‡l 1 wU 0, Zv 2 QvovI 5 I 10 Øviv wefvR¨| †hgb- 24| GLv‡b 4, 2 Øviv wefvR¨| 30 msL¨vwU 2, 5 I 10 Øviv wefvR¨| 4 Øviv wefvR¨: †Kvb msL¨vi GKK I `kK ¯’vbxq A¼ `ywU Øviv MwVZ msL¨v 4 Øviv wefvR¨ n‡j cÖ`Ë msL¨vwU 4 Øviv wefvR¨ n‡e| A_ev msL¨vwUi †k‡l 2 wU k~b¨ (00) _vK‡jI 4 Øviv wefvR¨ n‡e| g‡b ivLyb: †h msL¨vi †k‡l 2 wU k~b¨ (00), Zv 4 QvovI 25 I 100 Øviv wefvR¨| †hgb- 728| GLv‡b 28, 4 Øviv wefvR¨| 500 msL¨vwU 4, 25 I 100 Øviv wefvR¨| 8 Øviv wefvR¨: †Kv‡bv msL¨vi GKK, `kK I kZK ¯’vbxq A¼ wZbwU Øviv MwVZ msL¨v 8 Øviv wefvR¨ n‡j cÖ`Ë msL¨vwU 8 Øviv wefvR¨ n‡e| A_ev msL¨vwUi †k‡l 3wU k~b¨ (000) _vK‡jI 8 Øviv wefvR¨ n‡e| g‡b ivLyb: †h msL¨vi †k‡l 3wU k~b¨ (000), Zv 8 QvovI 125 I 1000 Øviv wefvR¨| †hgb- 7136| GLv‡b †kl wZbwU AsK Øviv MwVZ msL¨v 136, 8 Øviv wefvR¨| 7000 msL¨vwU 8, 125 I 1000 Øviv wefvR¨| †KŠkj: 2 (21 ) Gi †ÿ‡Î †kl 1 wU A¼, 4 (22 ) Gi †ÿ‡Î †kl 2wU A¼ Ges 8 (23 ) Gi †ÿ‡Î †kl 3 wU A¼ fvM Kiv †M‡j cÖ`Ë A¼wU h_vµ‡g 2, 4 I 8 Øviv wefvR¨ n‡e| (cvIqvi †`‡L g‡b ivLyb) 124. wb‡Pi †KvbwU 4 Øviv wefvR¨? evwYR¨gš¿Yvj‡qiAax‡bevsjv‡`kU¨vwidKwgkbwimvm© Awdmvi:10 214133 510056 322569 9522117 DËi: L 125. 91876 * 2 msL¨vwU 8 Øviv wbt‡k‡l wefvR¨ n‡j * Gi RvqMvq †Kvb ÿz`ªZg c~Y©msL¨v e¨envi Kiv hv‡e? evsjv‡`k e¨vsK A¨vwm‡÷›U wW‡i±i 14 1 2 3 4 DËi: M †k‡li wZbwU wWwRU (6 * 2) hw` 8 Øviv wefvR¨ nq Zvn‡j cÖ`Ë msL¨vwU 8 Øviv wefvR¨ n‡e| * wP‡ýi RvqMvq 1, 2, 3, 4 Gi gvS †_‡K 3 emv‡j msL¨vwU (632) 8 Øviv wefvR¨ n‡e| 3 Øviv wefvR¨: †Kv‡bv msL¨vi A¼¸‡jvi †hvMdj 3 Øviv wefvR¨ n‡j, cÖ`Ë msL¨vwU 3 Øviv wefvR¨ n‡e| †hgb- 126 1 + 2 + 6 = 9 9 3 = 3| 126 msL¨vwU 3 Øviv wefvR¨| 9 Øviv wefvR¨: †Kv‡bv msL¨vi A¼¸‡jvi †hvMdj 9 Øviv wefvR¨ n‡j, cÖ`Ë msL¨vwU 9 Øviv wefvR¨ n‡e| †hgb- 1593 1 + 5 + 9 + 3 = 18 18 9 = 2| 1593 msL¨vwU 9 Øviv wefvR¨| 126. wb‡Pi †Kvb msL¨vwU 3 Øviv wb:‡k‡l wefvR¨ bq? cvwbDbœqb†ev‡W©i AwdmmnvqK:15 126 141 324 139 DËi: N 127. 456138 msL¨vwU wb‡¤œi †Kvb msL¨v Øviv wefvR¨? 5 21 9 19 DËi: M 128. 9 w`‡q wefvR¨ 3 A¼ wewkó GKwU msL¨vi cÖ_g A¼ 3, Z…Zxq A¼ 8 n‡j ga¨g A¼wU KZ? kÖgIKg©ms¯’vbgš¿Yvj‡qiAaxb NMLK NMLK mgvavb NMLK NMLK mgvavb NM LK
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Math Tutor 23 KjKviLvbvIcÖwZôvbcwi`k©bcwi`߇iimnKvixcwi`k©K:05 6
7 8 9 DËi: L 1g A¼ I 3q A‡¼i †hvMdj = 3 + 8 = 11, hv 9 Øviv wefvR¨ bq| 11 Gi cieZx© 18 msL¨vwU 9 Øviv wefvR¨, GRb¨ Avgv‡`i 7 †hvM Ki‡Z n‡e| 3 I 8 Gi gv‡S 7 emv‡j msL¨vwU `vuovq 378, hvi A¼¸‡jvi †hvMdj nq 3 + 7 + 8 = 18| Gevi 18 msL¨vwU 9 Øviv wefvR¨, Zvn‡j 378 msL¨vwUI 9 Øviv wefvR¨| 129. 481 * 673 msL¨vwU 9 Øviv wbt‡k‡l wefvR¨ n‡j, * Gi ¯’v‡b †Kvb ÿz`ªZg c~Y©msL¨v n‡e? AMÖYx e¨vsK wj. (wmwbqi Awdmvi) 2017 2 7 5 6 DËi: L 6 Øviv wefvR¨: †Kv‡bv msL¨v‡K 2 Ges 3 Øviv wefvR¨ n‡j msL¨vwU 6 Øviv wefvR¨ n‡e| †KŠkj: 6 Øviv wefvR¨ msL¨vwU Aek¨B †Rvo n‡e, ZvB †mwU Aek¨B 2 Øviv wefvR¨ n‡e| Avcbvi KvR n‡”Q ïay 3 Gi wefvR¨Zv bxwZ cÖ‡qvM K‡i cixÿv K‡i †bqv| 4536 130. 5 * 2 msL¨vwU hw` 6 Øviv wbt‡k‡l wefvR¨ nq, Zvn‡j * ¯’v‡b †Kvb AsKwU em‡e? evsjv‡`k e¨vsK (Awdmvi K¨vk) 16 2 3 6 7 DËi: K msL¨vwU‡K hw` 2 I 3 Øviv fvM Kiv hvq Zvn‡j GwU 6 Øviv wbt‡k‡l wefvR¨ n‡e| msL¨vwUi †kl AsK †Rvo _vKvq GwU 2 Øviv wbt‡k‡l wefvR¨| Gevi 3 Gi wefvR¨Zvi bxwZ Abyhvqx 2, 3, 6 I 7 Gi gvS †_‡K Ggb GKwU AsK 5 * 2 Gi * RvqMvq emv‡Z n‡e †hb AsK¸‡jv †hvM Ki‡j †hvMdj 3 Øviv wefvR¨ nq| G‡ÿ‡Î 2 emv‡j 522 nq, †hLv‡b AsK¸‡jvi mgwó 5 + 2 + 2 = 9, hv 3 Øviv wefvR¨| 7 Øviv wefvR¨: (1) †Kv‡bv msL¨vi GKK ¯’vbxq AsK‡K 5 ¸Y K‡i Aewkó As‡ki mv‡_ †hvM Ki‡j, †hvMdj hw` 7 Øviv wefvR¨ nq Z‡e g~j msL¨vwUI 7 Øviv wefvR¨ n‡e| †hgb- 798 79 (85) 79 + 40 = 119 119 7 = 17| myZivs, 798 msL¨vwU 7 Øviv wefvR¨| (2) †Kv‡bv msL¨vi GKK ¯’vbxq AsK‡K 2 Øviv ¸Y K‡i Aewkó msL¨v †_‡K we‡qvM Kivi ci we‡qvMdj 7 w`‡q wefvR¨ n‡j g~j msL¨vwUI 7 w`‡q wefvR¨ n‡e| †hgb- 861 86 (12) 86 - 2 = 84 84 7 = 12| msL¨vwU 7 Øviv wefvR¨| 13 Øviv wefvR¨: (1) †Kv‡bv msL¨vi GKK ¯’vbxq AsK‡K 4 Øviv ¸Y K‡i Aewkó As‡ki mv‡_ †hvM Ki‡j, †hvMdj hw` 13 Øviv wefvR¨ nq Z‡e g~j msL¨vwUI 13 Øviv wefvR¨ n‡e| †hgb- 14131 1413(14) 1413 + 4 = 1417 1417 13 = 109| msL¨vwU 13 Øviv wefvR¨| 17 Øviv wefvR¨: (1) †Kv‡bv msL¨vi GKK ¯’vbxq AsK‡K 12 Øviv ¸Y K‡i Aewkó As‡ki mv‡_ †hvM Ki‡j, †hvMdj hw` 17 Øviv wefvR¨ nq Z‡e g~j msL¨vwUI 17 Øviv wefvR¨ n‡e| †hgb- 8738 873 (812) 873 + 96 = 969 959 17 = 57 msL¨vwU 17 Øviv wefvR¨| civgk©: 7 Gi wefvR¨Zvi bxwZwU LyeB ¸iæZ¡c~Y©, ZvB gyL¯’ ivLyb| 131. wb‡Pi †KvbwU 2 Ges 7 Øviv wefvR¨? Bangladesh BankOfficer:01 365 362 361 350 DËi: N 132. wb‡Pi †Kvb msL¨vwU 3 Ges 7 Df‡qi Øviv wbt‡k‡l wefvR¨? K‡›Uªvjvi†Rbv‡ijwW‡dÝdvBb¨vÝ-GiKvh©vj‡qiAaxbRywbqiAwWUi2019 303 341 399 406 DËi: M 11 Øviv wefvR¨: (1) †Kv‡bv msL¨vi GKK ¯’vbxq AsK Aewkó AsK¸‡jv †_‡K we‡qvMdj 11 Øviv wefvR¨ n‡j msL¨vwU 11 Øviv wefvR¨ n‡e| †hgb- 1243 124 - 3 = 121 121 11 = 11| 1045 104 - 5 = 99 99 11 = 9| msL¨v `ywU 11 Øviv wefvR¨| (2) msL¨vwUi AsK¸‡jv‡K †kl w`K †_‡K †Rvov †Rvov K‡i †hvM Ki‡j †hvMdj 11 Øviv wefvR¨ n‡e| 1243 12 + 43 = 55 55 11 = 5 | msL¨vwU 11 Øviv wefvR¨| 715 7 + 15 = 22 22 11 = 2| msL¨vwU 11 Øviv wefvR¨| (3) †Kv‡bv msL¨vi we‡Rvo ¯’vbxq As‡Ki mgwó Ges †Rvo ¯’vbxq As‡Ki mgwói cv_©K¨ k~Y¨ n‡j msL¨vwU 11 Øviv wefvR¨ n‡e| †hgb- 1122 (1 + 2) - (1 + 2) = 3 - 3 = 0| msL¨vwU 11 Øviv wefvR¨| NMLK NMLK mgvavb NMLK NMLK mgvavb NMLK
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24Math Tutor g‡b
ivLyb: (3) bs wbqg †Rvo msL¨K A‡¼i †ÿ‡Î mwVK DËi w`‡jI we‡Rvo msL¨K A‡¼i †ÿ‡Î A‡bK mgq mwVK DËi †`q bv| †hgb- 209, 726, 759 BZ¨vw` 11 Øviv wefvR¨ n‡jI (3) bs wbqgvbyhvqx cÖgvY Ki‡Z mÿg n‡eb bv| 133. wb‡Pi †Kvb msL¨vwU 11 Øviv wbt‡k‡l wefvR¨? c~evjx e¨vsK wj. †UªBwb A¨vwm‡÷›U †Ujvi) 2017 235641 245642 315624 415624 DËi: N (L) fvRK msL¨v †R‡b wbb – 12 fvRK : †h †h ivwk Øviv †Kvb msL¨v‡K fvM Kiv hvq, †m †m ivwk H msL¨vi fvRK| †hgb- 20 †K 1, 2, 4, 5, 10, 20 Øviv fvM Kiv hvq, ZvB 1, 2, 4, 5, 10, 20 n‡”Q 20 Gi fvRK| fvR¨ : fvRK Øviv †h msL¨v‡K fvM Kiv hvq, H msL¨v‡K fvR¨ e‡j| †hgb- Dc‡ii D`vni‡Y 20 n‡”Q fvR¨| g‡b ivLyb- 1 †h‡Kvb msL¨vi fvRK, KviY 1 Øviv mKj msL¨v wefvR¨| fvRK/Drcv`K/¸YbxqK GKB wRwbm| fvR¨/¸wYZK GKB wRwbm| fvRK msL¨v wbY©‡qi mvaviY wbqgt 32 Gi fvRK msL¨v wbY©q Kiv hvK| 32 Gi fvRK mg~n n‡”Q 32 †K †h †h msL¨v Øviv fvM Kiv hvq| 24 †K 1, 2, 3, 4, 6, 8, 12 I 24 Øviv fvM Kiv hvq| A_©vr, 24 Gi fvRKmg~n = 1, 2, 3, 4, 6, 8, 12 I 24| 24 Gi fvRKmsL¨v n‡”Q 8 wU| GB c×wZ‡Z eo msL¨vi fvRK msL¨v wbY©q Kiv KwVb I mgqmv‡cÿ, ZvB fvRK msL¨v wbY©‡q Avgiv kU©KvU wbqg AbymiY Kie| fvRK msL¨v wbY©‡qi kU©KvU wbqgt cÖ_‡g †h msL¨vi fvRK msL¨v wbY©q Kie, †m msL¨vwU‡K †gŠwjK Drcv`‡K we‡kølY Kie| 2 24 24 Gi †gŠwjK Drcv`Kmg~n = 2 2 2 3| 2 12 GLv‡b Drcv`Kmg~‡ni gv‡S 2 Av‡Q 3wU Ges 3 Av‡Q 1wU| GLb m~P‡Ki wbqgvbyhvqx 2 Gi cvIqvi 3 Ges 2 6 3 Gi cvIqvi 1 wjLyb Gfv‡e- 23 31 | Zvici wfwË 2 I 3 †K †Ku‡U w`b- 23 31 | Gevi wfwË ev` 3 w`‡q cÖwZwU cvIqvi Gi mv‡_ 1 K‡i †hvM Kivi ci ¸Y Kiæb- (3 + 1) (1 + 1) = 4 2 = 8| e¨m&, GLv‡b cÖvß 8 n‡”Q 24 Gi †gvU fvRK msL¨v| 134. 36 msL¨vwUi †gvU KZ¸‡jv fvRK i‡q‡Q? cjøxDbœqb †ev‡W©iwnmvemnKvix:14 6wU 8wU 9wU 10wU DËi: M 36 Gi †gŠwjK Drcv`Kmgg~n = 2233 = 22 32 fvRK msL¨v = 22 32 = (2 + 1) (2 + 1) = 3 3 = 9 wU | 135. 72 Gi fvRK msL¨v KZ? 26ZgwewmGm 7 8 12 13 DËi: M 72Gi †gŠwjK Drcv`Kmg~n = 22233 = 23 32 fvRK msL¨v = 23 32 = (3+1) (2+1) = 43 = 12 wU| 136. 540 msL¨vwUi KZ¸‡jv fvRK Av‡Q? AvenvIqvAwa`߇ii mnKvixAvenvIqvwe`:04] 18 20 22 24 DËi: N 540 Gi †gŠwjK Drcv`Kmg~n = 2233 35 = 22 33 51 fvRK msL¨v = 22 33 51 = (2+1)(3+1)(1+1) = 342 = 24 wU| 137. 1008 msL¨vwUi KqwU fvRK Av‡Q? Dc‡RjvI_vbvwkÿv Awdmvit05/_vbvwbev©PbAwdmvit04 20 24 28 30 DËi: N 1008 Gi †gŠwjK Drcv`Kmg~n = 2222 337 = 24 32 71 fvRK msL¨v = 24 32 71 = (4+1)(2+1) (1+1) = 532= 30wU| 138. wb‡Pi †Kvb c~Y© msL¨vwU mev©waK msL¨K fvRK Av‡Q? 29ZgwewmGm 88 91LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK NMLK
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Math Tutor 25 95
99 DËi: K KvQvKvwQ msL¨vi gv‡S †Rvo msL¨vi fvRK msL¨v memgq †ewk _v‡K| GLv‡b 88 Gi fvRK msL¨v †ewk| 139. 32 Ges 64 Gi fvRK msL¨vi cv_©K¨ KZ? IBA:88-89 3 2 1 †Kv‡bvwUB bq DËi: M 32 Gi fvRK msL¨v 6wU Ges 64 Gi fvRK msL¨v 7wU| fvRK msL¨vi cv_©K¨ = 7 - 6 = 1| 140. wb¤œwjwLZ msL¨v¸‡jvi g‡a¨ †KvbwUi fvRK msL¨v †e‡Rvo? 16ZgwewmGm 2048 1024 512 48 DËi: L c~Y©eM© msL¨vi fvRK msL¨v memgq †e‡Rvo nq| cÖ`Ë Ackb¸‡jvi gv‡S 1024 n‡”Q c~Y©eM© msL¨v, 1024 Gi fvRK msL¨v †e‡Rvo| (M) x I y Gi gv‡S ---- Øviv wefvR¨ fvRK msL¨v wbY©q †R‡b wbb – 13 1 †_‡K 25 Gi gv‡S 5 Øviv wefvR¨ msL¨v KqwU? GLv‡b 5 Øviv wefvR¨ ej‡Z eySv‡”Q 1 †_‡K 25 Gi gv‡S GiKg KqwU msL¨v Av‡Q hv‡`i‡K 5 Øviv fvM Kiv hvq| GiKg msL¨vmg~n n‡”Q 5, 10, 15, 20 I 25 | 1 †_‡K 25 Gi gv‡S GB 5wU msL¨v Av‡Q hv‡`i‡K 5 Øviv fvM Kiv hvq| Zvn‡j cÖ`Ë cÖ‡kœi DËi n‡”Q 5wU| gRvi welq n‡”Q- 25 †K 5 Øviv fvM Ki‡j Avgiv mivmwi GB GKB DËi 5 †c‡q hvB| A_©vr, G ai‡Yi mgm¨vi mgvavb fvM K‡i KivB me‡P‡q mnR - 25 5 = 5| g‡b ivLyb- 5 Øviv wefvR¨ msL¨vmg~ni w`‡K jÿ¨ Kiæb- cÖwZwU msL¨vB 5 Gi ¸wYZK| Zvi gv‡b 1 †_‡K 25 Gi gv‡S 5 Øviv wefvR¨ msL¨v KqwU? GB K_vwUi Av‡iv GKwU A_© Av‡Q, †mwU n‡”Q- 1 †_‡K 25 Gi gv‡S 5 Gi ¸wYZK KqwU? 141. 1 †_‡K 80 ch©šÍ 4 Øviv wefvR¨ msL¨v KqwU? 19 20 21 22 DËi: L 1 †_‡K 80 ch©šÍ 4 Gi ¸wYZK/ 4 Øviv wefvR¨ msL¨v = 80 4 = 20 wU| 142. 12 I 96 Gi g‡a¨ (GB `ywU msL¨vmn) KqwU msL¨v 4 Øviv wefvR¨? evsjv‡`kcjøxwe`y¨Zvqb†ev‡W©imnKvixmwPe/mnKvixcwiPvjK (cÖkvmb):16;cÖvK-cÖv_wgKmnKvixwkÿK:14;18ZgwewmGm 21 23 24 22 DËi: N cÖ‡kœ 12 †_‡K 96 ch©šÍ 4 Gi KqwU ¸wYZK †mwU †ei Ki‡Z ejv n‡q‡Q| Avgiv hw` welqwU †f‡½ †f‡½ †`wL- 1 †_‡K 96 ch©šÍ 4 Gi ¸wYZK/ 4 Øviv wefvR¨ msL¨vmg~n = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96 | †gvU 24wU| wKš‘ cÖ‡kœ 4 Gi ¸wYZK 12 †_‡K ïiæ n‡Z n‡e ejv n‡q‡Q, ZvB 4 Gi cÖ_g 2wU ¸wYZK 4 I 8 ev` w`‡Z n‡e| Zvn‡j 12 †_‡K 96 ch©šÍ 4 Øviv wefvR¨ msL¨v/ 4 Gi ¸wYZK n‡e 22 wU| GB mgm¨vwU fvM c×wZ mgvavb Kiv hvK| cÖ_‡g 96 †K 4 Øviv fvM Kiv hvK- 96 4 = 24wU| eyS‡Z cvi‡Qb †Zv? GB 24 wKš‘ G‡m‡Q 1 †_‡K 96 ch©šÍ Gwiqvi Rb¨ | wKš‘ cÖ‡kœ ejv n‡q‡Q 4 Gi ¸wYZK ïiæ n‡e 12 †_‡K| GRb¨ g‡b g‡b wn‡me K‡i 12 Gi Av‡Mi `ywU ¸wYZK 4 I 8 †K ev` w`‡Z n‡e| A_©vr, fvRK msL¨v = 24 - 2 = 22 wU| 143. 5 I 95 Gi g‡a¨ 5 I 3 Øviv wefvR¨ msL¨v KZwU? cjøx mÂq e¨vsK (K¨vk) 2018; ivóªvqË¡ e¨vsK (wmwbqi Awdmvi) 1998 6wU 9wU 7wU 15wU DËi: K 5 I 95 Gi g‡a¨ 5 I 3 Øviv wefvR¨ msL¨vmg~n- 15, 30, 45, 60, 75, 90| †`Lv hv‡”Q 5 I 3 Øviv wefvR¨ cÖ_g msL¨v 15, hv 5 I 3 Gi j.mv.¸| evKx msL¨v¸‡jv 5 I 3 Gi j.mv.¸Õi ¸wYZK| GRb¨ GKvwaK msL¨v Øviv wefvR¨ msL¨v PvIqv n‡j H GKvwaK msL¨vi j.mv.¸ †ei K‡i †mwU Øviv fvM Ki‡Z n‡e| †hgb- 5 I 3 Gi j.mv.¸ 15 95 15 = 6.33 (DËi `kwgK Qvov wb‡Z n‡e) fvRK msL¨v = 6 wU| (N) KZ †hvM ev we‡qvM Ki‡j wbt‡k‡l wefvR¨ n‡e 144. 1056 Gi mv‡_ me©wb¤œ KZ †hvM Ki‡j †hvMdj 23 Øviv wb:‡k‡l wefvR¨ n‡e? evsjv‡`kK…wlDbœqbK‡c©v‡ik‡bi mnKvixcÖkvmwbKKg©KZ©v:17 2 3LK mgvavb NMLK mgvavb NMLK mgvavb NMLK mgvavb NM LK mgvavb NM LK mgvavb NM
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26Math Tutor 18 21
DËi: K 1056 †K 23 Øviv fvM K‡i cvB, 23) 1056 ( 45 92 136 115 21 cÖ`Ë msL¨vi mv‡_ (23 - 21) = 2 †hvM Ki‡j cÖvß msL¨vwU 23 Øviv wefvR¨ n‡e| g‡b ivLyb: cÖ‡kœ †hvM ej‡j ÔfvRK I fvM‡klÕ Gi cv_©K¨‡K †hvM Ki‡Z nq| 145. 4456 Gi mv‡_ †Kvb ÿz`ªZg msL¨v †hvM Ki‡j †hvMdj 6 Øviv wb:‡k‡l wefvR¨ n‡e? EXIMBankLtd. Officer :13 2 3 4 5 DËi: K 146. 105 Gi m‡½ KZ †hvM Ki‡j †hvMdjwU 23 Øviv wbt‡k‡l wefvR¨ n‡e? cyevjx e¨vsK wj. (Rywbqi Awdmvi) 2013 3 18 21 10 DËi: N 147. mvZ A‡¼i e„nËg msL¨v wbY©q Kiæb hv 6 Øviv wefvR¨| 9999999 †K 6 Øviv fvM K‡i cvB, 6) 9999999 ( 1666666 9999996 3 fvM‡kl 3 ev` w`‡j cÖvß 9999996 msL¨vwU wbt‡k‡l wefvR¨| cÖkœwU hw` Gfv‡e ejv nZ- mvZ A‡¼i e„nËg msL¨v †_‡K †Kvb ÿz`ªZg msL¨v we‡qvM w`‡j we‡qvMdj 6 Øviv wbt‡k‡l wefvR¨? - G‡ÿ‡Î DËi nZ 3| we‡qvM ejv _vK‡j fvM‡klB DËi nq| 148. cuvP A‡¼i ÿz`ªZg †Kvb msL¨v 41 Øviv wb:‡k‡l wefvR¨? JanataBankLtd.Asst.ExecutiveOff.:(Teller):15 10004 10025 10041 10045 DËi: K 10000 †K 41 Øviv fvM K‡i cvB, 41) 10000 ( 243 82 180 164 160 123 37 cÖ`Ë msL¨vi mv‡_ (41 - 37) = 4 †hvM Ki‡Z n‡e- 10000 + 4 = 10004 | AZGe, cvuP A‡¼ ÿz`ªZg msL¨v 10004, 41 Øviv wbt‡k‡l wefvR¨| †R‡b wbb – 14 †hvM ev we‡qvM ejv bv _vK‡j ÿz`ªZg ev e„nËg msL¨v‡K wbt‡k‡l wefvR¨ Ki‡Z KLb ÔfvM‡klÕ we‡qvM Ki‡eb A_ev KLb ÔfvRK I fvM‡klÕ Gi cv_©K¨‡K †hvM Ki‡eb? Reve: GwU m¤ú~Y© wbf©i K‡i cÖ`Ë msL¨vi Dci| †hgb - 24 bs mgm¨vi †ÿ‡Î fvM‡kl 3 we‡qvM K‡iwQ KviY ÔfvRK I fvM‡klÕGi cv_©K¨ †hvM Ki‡j msL¨vwU `uvovZ- 9999999 + ( 6- 3) = 10000002, Zvn‡j ZLb msL¨vwU Avi mvZ A‡¼i e„nËg msL¨v _vKZ bv| wKš‘ hLb ÔfvM‡klÕ we‡qvM K‡iwQ ZLb cÖvß 9999996 msL¨vwU mvZ A‡¼I e„nËg msL¨v wn‡m‡e wU‡K †M‡Q| Avevi, 25 bs mgm¨v †ÿ‡Î ÔfvRK I fvM‡klÕ Gi cv_©K¨‡K †hvM K‡iwQ, KviY ÔfvM‡klÕ we‡qvM Ki‡j msL¨vwU `uvovZ- 9963, Zvn‡j ZLb msL¨vwU Avi cvuP A‡¼i ÿz`ªZg msL¨v _vKZ bv| wKš‘ hLb ÔfvRK I fvM‡klÕ Gi cv_©K¨ †hvM K‡iwQ ZLb cÖvß 10004 msL¨vwU cvuP A‡¼i ÿz`ªZg msL¨v wn‡m‡e wU‡K †M‡Q| g‡b ivLyb: wefvR¨Zvi cÖ‡kœ †hvM ev we‡qvM ejv bv _vK‡j GKwU kU©KvU g‡b ivLyb- e„nËg msL¨vi †ÿ‡Î ÔfvM‡kl we‡qvM Ki‡Z nq Ges ÿz`ªZg msL¨vi †ÿ‡Î ÔfvRK I fvM‡klÕGi cv_©K¨‡K †hvM Ki‡Z nq| 149. GKwU msL¨v‡K 45 w`‡q fvM Ki‡j fvM‡kl 23 _v‡K| hw` H msL¨vwU‡K 9 w`‡q fvM Kiv nq Z‡e fvM‡kl KZ n‡e? cÖavbgš¿xi Kvh©vjq : IqvPvi Kb‡÷ej: 2019 3 4 5 100 DËi: M 45 Øviv †h msL¨v‡K fvM Kiv hvq 9 ØvivI H msL¨v‡K fvM Kiv hvq| Avgiv cÖ‡kœ †`L‡Z cvw”Q 45 Øviv GKwU msL¨v‡K fvM Kivq fvM‡kl 23 Av‡Q| Avgiv hw` fvM hvIqv AskUzKz x awi, Zvn‡j fvM‡klmn msL¨vwU n‡e- x + 23| Avgiv Gevi msL¨vwU‡K 9 Øviv fvM Kie- 9 23x = 9 23 9 x 45 †h‡nZz 9 Gi ¸wYZK, †m‡nZz x AskUzKz 45 Øviv †hgb fvM hv‡e GKBfv‡e 9 ØvivI fvM hv‡e| evKx _vKj 23 †K 9 Øviv fvM Kiv| Pjyb fvM Kiv hvK- mgvavb NM LK mgvavb NM LK mgvavb NM LK NM LK mgvavb NM
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Math Tutor 27 9
) 23 ( 2 18 5 A_©vr, H msL¨vwU‡K 9 w`‡q fvM Ki‡j 5 fvM‡kl _vK‡e| GKevi eyS‡Z cvi‡j A¼wU †`Lv gvÎ †m‡K‡ÛB mgvavb Ki‡Z cvi‡eb| 01.15 †gŠwjK msL¨v †R‡b wbb – 15 †gŠwjK msL¨v: †h msL¨vi †Kvb cÖK…Z Drcv`K †bB Zv‡K †gŠwjK msL¨v e‡j| A_ev †h msL¨v‡K 1 I H msL¨v e¨ZxZ Ab¨ †Kvb msL¨v Øviv fvM Kiv hvq bv, Zv‡K †gŠwjK msL¨v e‡j| †hgb- 2, 3, 5, 7 BZ¨vw`| 2, 3, 5 I 7 G 1 I Zviv wb‡Riv e¨ZxZ Ab¨ †Kvb Drcv`K †bB, ZvB Giv †gŠwjK msL¨v| †hŠwMK msL¨v: †h msL¨vq 1 I H msL¨v e¨ZxZ Av‡iv Ab¨ †Kvb Drcv`K _v‡K Zv‡K †hŠwMK msL¨v e‡j| †hgb- 4| 4 Gi Dcrcv`Kmg~n- 1, 2, 4| A_©vr, 4-G 1 I 4 QvovI Av‡iv GKwU Drcv`K 2 Av‡Q, ZvB 4 n‡”Q †hŠwMK msL¨v| †gŠwjK msL¨v m¤úwK©Z wKQz ¸iæZ¡c~Y© Z_¨: (K) 2 e¨ZxZ me †Rvo msL¨v †hŠwMK msL¨v| 2 -B GKgvÎ †Rvo †gŠwjK msL¨v I †QvU †gŠwjK msL¨v| (L) †gŠwjK w؇RvU ev †Rvo †gŠwjK: `ywU †gŠwjK msL¨vi AšÍi 2 n‡j, Zv‡`i †gŠwjK w؇RvU e‡j| †hgb- 5, 7| (M) †gŠwjK w·RvU: wZbwU †gŠwjK msL¨vi µwgK AšÍi 2 n‡j, Zv‡`i †gŠwjK w·RvU e‡j| †hgb- 3, 5, 7| (N) 1 †_‡K 100 ch©šÍ †gŠwjK msL¨v 25wU Ges G‡`i †hvMdj 1060| 101 †_‡K 200 ch©šÍ †gŠwjK msL¨v 21wU| 1 †_‡K 500 ch©šÍ †gŠwjK msL¨v 95wU| 1 †_‡K 1000 ch©šÍ †gŠwjK msL¨v 168wU| 1 †_‡K 5000 ch©šÍ †gŠwjK msL¨v 669 wU| 150. me‡P‡q †QvU †gŠwjK msL¨v †KvbwU? PubaliBankLtd.(SeniorOfficer) 2017 0 1 2 3 DËi: M (K) †gŠwjK msL¨vi ZvwjKv 1 †_‡K 100 ch©šÍ †gŠwjK msL¨vi QK: cÖ`Ë msL¨v †gŠwjK msL¨v me©‡gvU 1-10 ch©šÍ 2, 3, 5, 7 4 wU 1 50 ch©šÍ 15 wU 11-20 ch©šÍ 11, 13, 17, 19 4 wU 21-30 ch©šÍ 23, 29 2 wU 31- 40 ch©šÍ 31, 37 2 wU 41-50 ch©šÍ 41, 43, 47 3 wU 51-60 ch©šÍ 53, 59 2 wU 51 100 ch©šÍ 10 wU 61-70 ch©šÍ 61, 67 2 wU 71-80 ch©šÍ 71, 73, 79 3 wU 81-90 ch©šÍ 83, 89 2 wU 91-100 ch©šÍ 97 1 wU 1 100 ch©šÍ 25 wU NMLK
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Math Tutor 27
g‡b ivLyb : 44 22 3 22 3 21 101 †_‡K 200 ch©šÍ †gŠwjK msL¨vi QK: cÖ`Ë msL¨v †gŠwjK msL¨v me©‡gvU 101-110 ch©šÍ 101, 103, 107, 109 4 wU 101 150 ch©šÍ 10 wU 111-120 ch©šÍ 113 1 wU 121-130 ch©šÍ 127 1 wU 131-140 ch©šÍ 131, 137, 139 3 wU 141-150 ch©šÍ 149 1 wU 151-160 ch©šÍ 151, 157 2 wU 151 200 ch©šÍ 11 wU 161-170 ch©šÍ 163, 167 2 wU 171-180 ch©šÍ 173, 179 2wU 181-190 ch©šÍ 181 1 wU 191-200 ch©šÍ 191, 193, 197, 199 4 wU 101 200 ch©šÍ 21 wU g‡b ivLyb : 41 1 31 22 21 4 151. wb‡Pi †Kvb msL¨vwU †gŠwjK? ivóªvqË¡ e¨vsKwmwbqiAwdmvi:00 49 51 53 55 DËi: M (L) †gŠwjK msL¨v wbY©q †R‡b wbb – 16 (†gŠwjK msL¨v wbY©‡qi †KŠkj) †gŠwjK msLv mn‡R wbY©‡qi Dcvq n‡”Q- 7, 11, 13, 17 N‡ii bvgZv m¤ú‡K© Lye fv‡jv Avq‡Ë¡ ivLv| cvkvcvwk 3, 7, 11 Gi wefvR¨Zvi bxwZI fv‡jvfv‡e AvqË¡ ivLv PvB| 2 I 5 Gi wefvR¨Zvi bxwZi e¨vcv‡i aviYv: 2 e¨ZxZ †Kvb †Rvo msL¨v †gŠwjK msL¨v nq bv, ZvB memgq †e‡Rvo msL¨vi gv‡S †gŠwjK msL¨v LyuR‡Z n‡e, GRb¨ †gŠwjK msL¨v †ei Ki‡Z KL‡bv 2 Gi wefvR¨Zvi bxwZi `iKvi n‡e bv| Gevi †e‡Rvo msL¨vi gv‡S 5 LyeB ¸iæZ¡c~Y©| 5 †gŠwjK msL¨v, wKš‘ evKx †h‡Kvb msL¨vi GKK ¯’v‡b 5 _vK‡j †mwU †hŠwMK msL¨v| †hgb- 15, 55, 75, 105 BZ¨vw`| GRb¨ 5 Gi wefvR¨Zvi bxwZ wb‡qI gv_v Nvgv‡Z n‡e bv, KviY †Kvb msL¨vi GKK ¯’v‡b 5 †`L‡jB eySv hv‡e GwU †hŠwMK msL¨v| 3 Gi wefvR¨Zvi bxwZi e¨vcv‡i aviYv: 3 Gi wefvR¨Zvi bxwZwU fv‡jvfv‡e AvqË¡ ivLv PvB, KviY 3 Gi wefvR¨Zvi bxwZ w`‡q A‡bK †hŠwMK msL¨v Lye mn‡RB †ei Kiv hvq| †Kvb msL¨v †gŠwjK wKbv, †mwU wbY©‡qi avc¸‡jv‡Z cÖ‡e‡ki ïiæ‡ZB 3 Gi wefvR¨Zvi bxwZwU cÖ_‡g cÖ‡qvM K‡i †`L‡eb| †gŠwjK msL¨v wbY©‡qi avcmg~n: avc-01: cÖ_‡g 3 Gi wefvR¨Zvi bxwZ w`‡q hvQvB Ki‡eb, msL¨vwU †hvwMK wKbv? hw` †hŠwMK nq, Zvn‡j Avi G‡Mv‡bvi `iKvi †bB| †hgb- 117 msL¨vwU †gŠwjK wKbv? 3 Gi wefvR¨Zvi bxwZ Abyhvqx hvQvB Kiv hvK- 1 + 1 + 7 = 9, †h‡nZz †hvMdj 9, 3 Øviv wefvR¨, †m‡nZz 117 msL¨vwUI 3 Øviv wefvR¨ A_©vr, 117 msL¨vwU †hŠwMK| ZvB Avi 2q av‡c hvIqvi `iKvi †bB| Gevi Av‡iv GKwU msL¨v †bqv hvK- 143 msL¨vwU †gŠwjK wKbv? cÖ_‡g 3 Gi wefvR¨Zvi bxwZ Abyhvqx †`Lv hvK- 1 + 4 + 3 = 8, hv 3 Øviv wefvR¨ bq| A_©vr, 143 msL¨vwU 3 Øviv wefvR¨ bq | mveavb! Zvi gv‡b 143 †gŠwjK msL¨v bq| fv‡jv K‡i g‡b ivLyb, 3 Øviv fvM bv †M‡j msL¨vwU‡K wØZxq av‡c wb‡q †h‡Z n‡e| avc-02: 2q av‡c cÖ`Ë msL¨vwUi KvQvKvwQ GKwU eM©g~j wb‡Z n‡e Ges D³ eM©g~‡ji c~‡e© †gŠwjK msL¨v †ei Ki‡Z n‡e| 143 2,3, 5, 7, 11 12 2, 3, 5 †jLvi `iKvi †bB, KviY 2q av‡c Avmvi Av‡MB Avcwb 2, 3, 5 hvQvB K‡i wb‡q‡Qb| Gevi 7 I 11 Gi NMLK
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28Math Tutor †KvbwU Øviv
hw` 143 †K fvM Kiv hvq, Zvn‡j msL¨vwU †hŠwMK Avi hw` fvM Kiv bv hvq Zvn‡j msL¨vwU †gŠwjK| 143 †K 11 Øviv fvM Kiv hvq, ZvB 143 †gŠwjK msL¨v bq| PP©v Kiæb: 133, 127, 119, 141 | 152. wb‡Pi †KvbwU †gŠwjK msL¨v? 30ZgwewmGm 91 87 63 59 DËi: N 3 Gi wefvR¨Zvi bxwZ Abyhvqx I ev`| 91 = 7 13 Abyhvqx ev`| 153. wb‡Pi †Kvb msL¨vwU †gŠwjK? 10gwewmGm 91 143 47 87 DËi: M 91 I 143 c~‡e© cÖgvY Kiv n‡q‡Q| 3 Gi wefvR¨Zvi bxwZ Abyhvqx ev`| †QvU †QvU msL¨v _vK‡j mivmwi DËi Kiv hvq| 154. wb‡Pi †KvbwU †gŠwjK? ivóªvqËe¨vsKwmwbqiAwdmvi:00 49 51 53 55 DËi: M 155. †KvbwU †gŠwjK msL¨v bq? cvewjKmvwf©mKwgk‡bmnKvixcwiPvjK:04 221 227 223 229 DËi: K cÖ_g av‡c 3 Gi wefvR¨Zvi bxwZ Abyhvqx GKwU‡KI ev` †`qv hv‡”Q bv| Gevi wØZxq av‡c hvIqv hvK- cÖ`Ë me KqwU msL¨v KvQvKvwQ nIqvq Avgvi me KqwUi Rb¨ GKwU eM©g~j wb‡Z cvwi| KvQvKvwQ eM©g~j 15 †bqv hvK| 221 227 223 229 7, 11, 13 15 7, 11, 13 15 7, 11, 13 15 7, 11, 13 15 (221 = 1317) (227, 223, 229 Gi †KvbwUB‡K 7, 11, 13 Øviv fvM Kiv hvq bv) 221 †gŠwjK msL¨v bq| †gŠwjK msL¨v wbY©‡qi †ÿ‡Î †Kvb †UKwb‡Ki `iKvi n‡e bv, hw` Avcwb bvgZv ev wefvR¨Zvq `ÿ _v‡Kb| (M) x †_‡K y ch©šÍ †gŠwjK msL¨v wbY©q †R‡b wbb – 17 x †_‡K y ch©šÍ fvRK msL¨v wbY©q Ki‡Z wM‡q Avgiv cÖvq fvlvMZ RwUjZvq c‡i hvB| Pjyb fvlvMZ RwUjZv `~i Kiv hvK- 2 †_‡K 31 ch©šÍ †gŠwjK msL¨v = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 = 11 wU| (+) (+) †_‡K ch©šÍ g‡b ivLyb- †Kvb msL¨v Ô†_‡KÕ gv‡b H msL¨vwU starting point, ZvB H msL¨vmn †gŠwjK msL¨v MYbv Ki‡Z n‡e Ges †Kvb msL¨v Ôch©šÍÕ gv‡b H msL¨vwU ending point, ZvB Ôch©šÍÕ _vK‡j H msL¨vmn †gŠwjK msL¨v MYbv Ki‡Z n‡e| A_©vr, Ô‡_‡KÕ I Ôch©šÍÕ _vK‡j starting I ending point mn wn‡me Ki‡Z n‡e| 2 †_‡K 31 Gi g‡a¨ †gŠwjK msL¨v = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 = 10 wU| (+) (-) †_‡K Gi g‡a¨ g‡b ivLyb- Ô‡_‡KÕ _vK‡j H msL¨vmn Ges †Kvb msL¨vi Ôg‡a¨Õ _vK‡j H msL¨v e¨ZxZ wn‡me Ki‡Z nq| Wv‡bi Q‡K †`Lyb, 31 Gi g‡a¨ gv‡b 31 bq Zvi Av‡Mi msL¨v¸‡jv‡K wb‡`©k Ki‡Q| A_©vr, Ô‡_‡KÕ I Ôg‡a¨Õ _vK‡j cÖ_gUv wn‡me Ki‡Z n‡e wKš‘ †k‡liUv MYbvq Avm‡e bv| 2 Ges 31 Gi g‡a¨ †gŠwjK msL¨v = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 = 9 wU| (-) (-) I/Ges Gi g‡a¨ g‡b ivLyb- ÔGesÕ I ÔGi g‡a¨Õ _vK‡j ÔïiæÕ I Ô‡klÕ ev` hv‡e| Q‡K †`Lyb, 2 I 31 ev‡` Zv‡`i g‡a¨ Ae¯’vbiZ msL¨v¸‡jvi gv‡S †gŠwjK msL¨v †ei Ki‡Z ejv n‡q‡Q| 156. 1 †_‡K 10 ch©šÍ msL¨vi g‡a¨ †gŠwjK msL¨v KZwU? BankersSelectionCommittee(SeniorOfficer)2018; 10g mgvavb NMLK NMLK Nmgvavb NMLK KMLmgvavb NMLK
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30Math Tutor wewmGm 4 3 6
5 DËi: K 4wU : 2, 3, 5, 7 | 157. 1 †_‡K 31 ch©šÍ KqwU †gŠwjK msL¨v Av‡Q? wd‡gj †m‡KÛvix GwmmU¨v›UAwdmvi:99 10 wU 11wU 12 wU 13 wU DËi: L 11wU : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 158. 2 Ges 32 -Gi g‡a¨ †gŠwjK msL¨v KqwU? 24Zg wewmGm 11wU 9wU 8wU 10wU DËi: N ÔGesÕ I Ôg‡a¨Õ _vKvq 2 I 32 ev‡` wn‡me Ki‡Z n‡e- 3, 5,7, 11, 13, 17, 19, 23, 29, 31| 159. 10 I 30 Gi g‡a¨ KqwU †gŠwjK msL¨v Av‡Q? gv`K`ªe¨wbqš¿YAwa`߇iimnKvixcwiPvjK:99 4wU 6wU 9wU 5wU DËi: L 6wU : 11,13,17,19,23, Ges 29| 160. 50 -Gi †P‡q †QvU KqwU †gŠwjK msL¨v Av‡Q? Dc‡Rjv I _vbv wkÿv Awdmvi: 05 10wU 12wU 14 wU 15wU DËi: N 15wU: 2, 3 , 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 , 41, 43 Ges 47| 161. 20 Gi †P‡q eo Ges 200 Gi †P‡q †QvU KZ¸wj †gŠwjK msL¨v Av‡Q? AvenvIqv Awa`߇ii mnKvix AvenvIqvwe`: 04 35 37 38 40 DËi: M 1 †_‡K 200 ch©šÍ †gŠwjK msL¨v 46 wU Ges 1 †_‡K 20 Gi g‡a¨ †gŠwjK msL¨v 8 wU| GLb 46 †_‡K 8 ev` w`‡j _v‡K 38| myZivs 20 - 200 ch©šÍ †gŠwjK msL¨v 38 wU| 162. 22 Ges 72 Gi g‡a¨ KqwU †gŠwjK msL¨v i‡q‡Q? cwiKíbvgš¿YvjqGescÖevmxKj¨vYI•e‡`wkKKg©ms¯’vb gš¿Yvj‡qimn: cwiPvjK:06 12wU 9wU 11wU 10wU DËi: K 12 wU: 44 22 3 22 3 21 21 †_‡K 70 ch©šÍ 11wU I 1 wU 71 = 11 + 1 = 12 wU| 163. 25 †_‡K 55-Gi g‡a¨ †gŠwjK msL¨v Av‡Q? Dc‡RjvI _vbvwkÿvAwdmvi:05 4wU 6 wU 7 wU 9 wU DËi: M 7wU: 29, 31, 37, 41, 43, 47 Ges 53| 164. 43 †_‡K 60-Gi g‡a¨ †gŠwjK msL¨v - 26Zg wewmGm 5 3 7 4 DËi: N 4wU: 43, 47, 53, 59| 165. 45 †_‡K 72 -Gi g‡a¨ KqwU †gŠwjK Av‡Q? Sonali, Janata and Agrani Bank senior officer: 08 5 6 7 8 DËi: L 6wU: 47, 53, 59, 61, 67 Ges 71| 166. 56 †_‡K 100 Gi g‡a¨ †gŠwjK msL¨v KqwU? EXIM BankLtd. (TraineeAsst.Officer)2018 8 9 10 11 DËi: L 167. 50 Gi †P‡q †QvU KZwU †gŠwjK msL¨v Av‡Q? Janata BankLtd. (Asst.Officer)2015 14 15 16 18 DËi: L 168. 50 †_‡K 103 ch©šÍ KZwU †gŠwjK msL¨v Av‡Q? cÖwZiÿvgš¿Yvj‡qiAaxbGWwgwb‡÷ªkbAwdmviIcv‡m©vbvjAwdmvi:06 10wU 11wU 12wU 13wU DËi: M 169. 90 †_‡K 100 Gi g‡a¨ KqwU †gŠwjK msL¨v Av‡Q? kÖgIKg©ms¯’vbgš¿Yvj‡qiAaxbKjKviLvbvIcÖwZôvb cwi`k©bcwi`߇ii mnKvix cwi`k©K:05 2wU 1wU 3wU GKwUI bq DËi: L 170. 100 †_‡K 110 ch©šÍ msL¨v¸‡jvi g‡a¨ KqwU †gŠwjK msL¨v i‡q‡Q? evsjv‡`k†ijI‡qnvmcvZvjmg~nmn:mvR©b:05; PviwU GKwU `yBwU wZbwU DËi: K 171. 100 -Gi ‡P‡q eo Ges 150-Gi †P‡q †QvU KqwU †gŠwjK msL¨v Av‡Q? ivóªvqËe¨vsKAwdmvi:97 7wU 8wU 9wU 10wU DËi: N (N) x I y msL¨vi g‡a¨ e„nËg I ÿz`ªZg †gŠwjK msL¨vi AšÍi wbY©q I Ab¨vb¨ 172. 60 †_‡K 80 -Gi ga¨eZx© e„nËg I ÿz`ªZg †gŠwjK msL¨vi AšÍi n‡e- 27 Zg wewmGm NM LK NM LK NM LK NM LK NM LK NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK
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Math Tutor 31 8
12 18 140 DËi: M 60 I 80 Gi gv‡S †gŠwjK msL¨vmg~n : 61, 67, 71, 73, 79| G‡`i gv‡S ÿz`ªZg †gŠwjK msL¨v 61 I e„nËg †gŠwjK msL¨v 79| G‡`i cv_©K¨ = 79 - 61 = 18| 173. 30 †_‡K 80 Gi ga¨eZ©x e„nËg I ÿz`ªZg †gŠwjK msL¨vi e¨eavb KZ? RajshahiKrishiUnnayanBank (cashier) :17;mgevq Awa. wØZxq †kÖYxi †M‡R‡UW Awdmvi: 97 35 42 48 55 DËi: M cv_©K¨ = 79 - 31 = 48| 174. 30 †_‡K 90 Gi ga¨eZx© e„nËg I ÿz`ªZg †gŠwjK msL¨vi AšÍi KZ? _vbv I †Rjv mgvR‡mev Awdmvi:99 58 42 68 62 DËi: K 30 31(†gŠwjK)... (†gŠwjK)89 90| myZivs †gŠwjK msL¨v `ywUi AšÍi = 89-31 = 58| 175. 40 †_‡K 100 ch©šÍ e„nËg I ÿz`ªZg †gŠwjK msL¨vi AšÍi KZ? Lv`¨Awa`߇iiAax‡bLv`¨cwi`k©K:00 59 56 60 70 DËi: L 40 41(†gŠwjK msL¨v) ..... (†gŠwjK msL¨v) 97 100| cv_©K¨ = 97-41 = 56| 176. 10 †_‡K 60 ch©šÍ †h mKj †gŠwjK msL¨vi GKK ¯’vbxq A¼ 9 Zv‡`i mgwó KZ? RajshahiKrishi UnnayanBank (Supervisor):17 146 99 105 107 DËi: N 177. cÖ_g 9wU †gŠwjK msL¨vi Mo- BangladeshBank Asst. Director:14 9 11 11 9 1 11 9 2 DËi: M cÖ_g 9 wU †gŠwjK msL¨vi †hvMdj = 2+3+5+7+11+13+17+19+23 = 100 Mo = 9 100 = 11 9 1 | 178. wb‡Pi †KvbwU cÖ_g 5wU †gŠwjK msL¨vi Mo? BangladeshHouseBuildingFinanceCorporation(so):17 4.5 5.6 7.5 8.6 DËi: L 179. 30 †_‡K 50 Gi g‡a¨ mKj †gŠwjK msL¨vi Mo KZ? AgraniBankLtd.SeniorOfficer:17(Cancelled) 37 37.8 39.8 39 DËi: M 30 †_‡K 50 Gi gv‡Si †gŠwjK msL¨vi †hvMdj = 31+37+41+43+47 = 199 Mo = 5 199 = 39 5 4 = 39.8| 01.16 †Rvo msL¨v I we‡Rvo msL¨v †R‡b wbb – 18 ( †Rvo I we‡Rvo msL¨v msµvšÍ mgvav‡bi †KŠkj) µwgK †Rvo I we‡Rvo m¤ú‡K© †ewmK Av‡jvPbv Ô‡R‡b wbb-00Õ †_‡K c‡o wbb| µwgK †e‡Rvo/AhyM¥/ abvZ¥K we‡Rvo : cÖ‡kœ µwgK we‡Rvo/AhyM¥/abvZ¥K we‡Rvo _vK‡j 1 ewm‡q mgvavb Kiæb| µwgK FYvZ¥K we‡Rvo: cÖ‡kœ µwgK FYvZ¥K we‡Rvo _vK‡j -1 ewm‡q mgvavb Kiæb| µwgK †Rvo/hyM¥: cÖ‡kœ µwgK †Rvo/hyM¥/abvZ¥K †Rvo _vK‡j 2 ewm‡q mgvavb Kiæb| µwgK FYvZ¥K †Rvo: cÖ‡kœ µwgK FYvZ¥K †Rvo _vK‡j -2 ewm‡q mgvavb Kiæb| abvZ¥K c~Y©msL¨v: 1, 2, 3, 4, 5, 6, 7, 8, 9 BZ¨vw`| FYvZ¥K c~Y©msL¨v: -1, -2, -3, -4, -5, -6, -7, -8, -9 BZ¨vw`| GKvwaK we‡Rvo msL¨vi ¸Ydj me mgq we‡Rvo nq| †hgb- 357 = 105| 180. x I y DfqB we‡Rvo msL¨v n‡j †Rvo msL¨v n‡e? 32Zg wewmGm(we‡kl) x+y+1 xy xy + 2 x+y DËi: N x = 1 I y = 1| †Rvo msL¨v = ? 1+1+1 = 3 (mwVK bq) 11 = 1 (mwVK bh) 1 1 + 2 = 3 (mwVK bq) 1 + 1 = 2 (mwVK) 181. hw` x GKwU FYvZ¥K we‡Rvo c~Y©msL¨v nq Ges y GKwU abvZ¥K †Rvo msL¨v nq, Z‡e xy Aek¨B- GgweGg: 06 †Rvo Ges FYvZ¥K we‡Rvo Ges FYvZ¥K †Rvo Ges abvZ¥K we‡Rvo Ges abvZ¥KNM LK N ML Kmgvavb NM LK mgvavb NM LK NM LK mgvavb NM LK NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK mgvavb NM LK
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32Math Tutor x =
- 1, y = 2. xy = - 12 = - 2| xy Aek¨B †Rvo I FYvZ¥K n‡e | DËi: K 182. hw` 𝒂 & 𝑏 DfqB abvZ¥K †Rvo c~Y©msL¨v nq, Z‡e wb‡Pi †KvbwU Aek¨B †Rvo msL¨v n‡e? MBA : 06 1) ab 2) (a + 1)b 3) ab+1 1 only 1& 2 1 & 3 1, 2 & 3 DËi: M a = 2, b = 2. †Rvo msL¨vi Ackb Lyu‡R †ei Ki‡Z n‡e| 1) ab = 22 = 4 (†Rvo) 2) (a + 1)b = (2+1)2 = 9 (we‡Rvo) 3) ab+1 = 22+1 = 8 (†Rvo) 1 I 3 bs-G †Rvo msL¨v G‡m‡Q, hv Ackb †Z Av‡Q| 183. wb‡Pi †KvbwU Aek¨B we‡Rvo msL¨v n‡e? BGgweG (Xvwe): GwcÖj -07 1) `yBwU †Rvo msL¨vi ¸Ydj 2) `yBwU we‡Rvo msL¨vi ¸Ydj 3) GKwU †Rvo Ges GKwU we‡Rvo msL¨vi †hvMdj 1, 2 & 3 1 only 2 & 3 only 1 & 3 only DËi: †Rvo = 2, we‡Rvo = 1| cÖkœvbyhvqx we‡Rvo msL¨v †ei Ki‡Z n‡e| 1) 22= 4 (†Rvo) 2) 11 = 1 (we‡Rvo) 3) 2+ 1= 3 (we‡Rvo) 2 I 3 bs-G we‡Rvo msL¨v G‡m‡Q, hv Ackb †Z Av‡Q| 184. hw` n Ges p `ywU AhyM¥ msL¨v nq, Z‡e wb‡¤œi †KvbwU Aek¨ hyM¥ msL¨v n‡e? c~evjx e¨vsKt 06/ _vbv wkÿv Awdmvit 99 n+p np np+2 n+p+1 DËi: K 185. hw` n Ges p `ywU †Rvo msL¨v nq, Z‡e wb‡¤œi †KvbwU Aek¨B we‡Rvo msL¨v n‡e? AMÖYxe¨vsKAwdmvi:08 n+2p np+1 n + p 2n+p DËi: L 186. hw` m GKwU †Rvo c~Y©msL¨v Ges n GKwU we‡Rvo c~Y©msL¨v nq Ges Dfq msL¨vB abvZ¥K nq, Z‡e wb‡Pi †KvbwU Aek¨B abvZ¥K †Rvo msL¨v n‡e? IBA(MBA):87-88 m2 +n2 mn + n2 m3 +n3 mn+𝑚2 DËi: N 187. hw` x GKwU abvZ¥K †Rvo msL¨v nq, Z‡e wb‡Pi †KvbwU e¨ZxZ Ab¨ mKj DËi we‡Rvo n‡e? IBA (MBA):05-06 (x+3) (x+5) x2 + 5 x2 + 6x +9 3x2 + 4 DËi: N 188. hw` m I n `ywU FYvZ¡K c~Y©msL¨v nq, Z‡e wb‡¤œi †KvbwU Aek¨B mwVK? DutchBanglaBankLtd. :17 m + n < 0 m – n < 0 mn < 0 None DËi: K †h‡Kvb `ywU FYvZ¥K c~Y©msL¨v a‡i †bqv hvK: m = -2 I n = -3 m + n < 0 ev, (-2) + (-3) < 0 = -5 < 0 GLv‡b -5, 0 Gi †P‡q †QvU, ZvB GwUB mwVK| 189. hw` 2x – 3 we‡Rvo msL¨v nq Z‡e cieZ©x †Rvo msL¨v †ei Kiæb|FirstSecurityIslamiBnakLtd.Officer:14 2x - 5 2x - 4 2x - 2 4x + 1 DËi: M we‡Rvo Gi mv‡_ 1 †hvM Ki‡j cieZx© †Rvo msL¨v cvIqv hvq| Avevi †Rv‡oi mv‡_ 1 †hvM Ki‡j cieZx© we‡Rvo msL¨v cvIqv hvq| cÖ‡kœ 2x – 3 n‡”Q GKwU we‡Rvo msL¨v, Gi cieZx© †Rvo msL¨v †ei Kivi Rb¨ 1 †hvM Ki‡Z n‡e- 2x – 3 + 1 = 2x - 2 | 190. hw` 3x+1GKwU we‡Rvo msL¨v wb‡`©k K‡i, Z‡e wb‡Pi †KvbwU Zvi cieZx© we‡Rvo msL¨v n‡e? MBA : 05 3(x+1) 3(x+2) 3(x+3) 3x+2 DËi: K GKwU we‡Rvo msL¨v †_‡K cieZx© we‡Rvo msL¨v †ei Ki‡Z n‡j 2 †hvM Ki‡Z nq| cÖ‡kœ cÖ`Ë 3x+1 n‡”Q GKwU we‡Rvo msL¨v| cieZx© we‡Rvo msL¨v †ei Ki‡Z n‡j 3x+1 Gi mv‡_ 2 †hvM Ki‡Z n‡e A_©vr, (3x+1) + 2 = 3x + 1 + 2 = 3x + 3 = 3 ( x+ 1) | 191. hw` n – 5 GKwU †Rvo c~Y©msL¨v nq, Z‡e cieZ©x †Rvo µwgK c~Y©msL¨v †KvbwU? BangladeshHouse BuildingFinanceCorporation (SO):17 n - 7 n - 3 n - 4 n - 2 DËi: L GKwU †Rvo msL¨v †_‡K cieZx© †Rvo msL¨v †ei Ki‡Z n‡j 2 †hvM Ki‡Z nq| cieZx© †Rvo msL¨v †ei Ki‡Z n‡j n – 5 Gi mv‡_ 2 †hvM Ki‡Z n‡e mgvavb NM LK mgvavb NM LK mgvavb NM LK K mgvavb NM LK NM LK NM LK NM LK NM LK M mgvavb NM LK M mgvavb NM LK mgvavb
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Math Tutor 33 A_©vr,
n – 5 + 2 = n – 3 | 192. cvuPwU c~Y© msL¨vi ¸Ydj hw` we‡Rvo msL¨v nq Zvn‡j D³ cvuPwU c~Y©msL¨vi wVK KqwU we‡Rvo n‡e? kÖ: cwi: 05 2 3 4 5 DËi: 5 GKvwaK msL¨vi ¸Ydj †e‡Rvo n‡Z n‡j GKvwaK msL¨vi cÖwZwUB †e‡Rvo n‡Z n‡e, ZvB GLv‡b D³ cvuPwU c~Y© msL¨vi me KqwU we‡Rvo| 01.17 g~j` I Ag~j` msL¨v †R‡b wbb – 19 g~j` msL¨v(Rational Number)t k~Y¨ I mKj ¯^vfvweK msL¨v g~j` msL¨v| †hgb: 0, 1, 2, 3 BZ¨vw` | cÖK…Z I AcÖK…Z mKj fMœvsk g~j` msL¨v | †hgb: 2 1 , 5 11 8 7 , BZ¨vw`| `kwg‡Ki c‡ii Ni¸‡jv mmxg n‡j msL¨vwU g~j` msL¨v| †hgb: 4. 678 | mKj c~Y© eM© ¯^vfvweK msL¨vi eM©g~j g~j` msL¨v| †hgb- 49 = 7, 64 = 8, 121 = 11 BZ¨vw`| mKj c~Y© Nb ¯^vfvweK msL¨vi Nbg~j g~j` msL¨v| †hgb- 3 27 = 3, 3 125 = 5 BZ¨vw`| `kwg‡Ki c‡ii Ni¸‡jv †cŠY‡cŠwYK AvKv‡i Amxg n‡j| †hgb: 3 4 = 1.33333... = 1. 3 , 3 10 = 3.3333.., = 3. 3 BZ¨vw`| Ag~j` msL¨v(Irrational Number)t `kwg‡Ki c‡ii Ni¸‡jv hw` wfbœ wfbœ AvKv‡i Amxg nq, Zvn‡j msL¨vwU Ag~j` msL¨v| †hgb: 3.142857... mKj †gŠwjK msL¨v, c~Y©eM© I c~Y©Nb bq Ggb mKj msL¨vi eM©g~j Ges Nbg~j me mgq Ag~j` msL¨v| †hgb: 33 11532 ,,, , 12 , 3 22 BZ¨vw`| K‡qKwU weL¨vZ Ag~j` msL¨vt cvB t 𝜋 GKwU Ag~j` msL¨v| GLv‡b, 𝜋 = 3.14285... | Aqjvi msL¨v t e GKwU Ag~j` msL¨v| GLv‡b, e = 2.71828....| dvB (†mvbvjx AbycvZ) t 𝜑 n‡”Q GKwU Ag~j` msL¨v| GLv‡b, 𝜑 = 1.618033... 193. hw` p GKwU †gŠwjK msL¨v nq Z‡e P - 26Zg wewmGm GKwU ¯^vfvweK msL¨v GKwU c~Y©msL¨v GKwU g~j` msL¨v GKwU Ag~j` msL¨v DËi:N 194. 2 msL¨vwU wK msL¨v ? 25Zg wewmGm GKwU ¯^vfvweK msL¨v GKwU c~Y© msL¨v GKwU g~j` msL¨v GKwU Ag~j` msL¨v DËi:N 195. 5 wK ai‡bi msL¨v?AvbmviIwfwWwcAwa:mv‡K©jA¨vWRyU¨v:05 GKwU ¯^vfvweK msL¨v GKwU c~Y© msL¨v GKwU g~j` msL¨v GKwU Ag~j` msL¨vDËi:N 196. 7 3 msL¨v †Kvb ai‡bi msL¨v? 12Zg wbeÜb RwUj msL¨v g~j` msL¨v Ag~j` msL¨v ev¯Íe msL¨ DËi: M 197. wb‡Pi †KvbwU g~j` msL¨v? 9gwkÿKwbeÜb:13 2 3 8 3 9 2 8 DËi: L 8 n‡”Q c~Y© Nb msL¨v ZvB 3 8 n‡”Q g~j` msL¨v| 3 8 = 3 3 2 = 2 (GKwU g~j` msL¨v) 198. †h msL¨v‡K `ywU c~Y© msL¨vi fvMdj AvKv‡i cÖKvk Kiv hvq bv Zv‡K wK e‡j? cÖavbgš¿xiKvh©vj‡qimnKvixcwiPvjK, M‡elYvKg©KZ©v,†Uwj‡dvbBwÄwbqviImnKvixKvw¤úDUvi†cÖvMÖvgvi:13 g~j` msL¨v ¯^vfvweK msL¨v Ag~j` msL¨v RwUj msL¨v DËi: M 199. g~j` msL¨vi †mU †evSvq wb‡Pi †KvbwU‡K? cÖevmxKj¨vY I •e‡`wkKKg©ms¯’vbgš¿Yvj‡qimnKvixcwiPvjK:12 Z Q P N DËi: L Z n‡”Q c~Y© msL¨vi †mU, Q n‡”Q g~j` msL¨vi †mU, P n‡”Q †gŠwjK msL¨vi †mU Ges N n‡”Q ¯^vfvweK msL¨vi †mU| 200. wb‡Pi †KvbwU Ag~j` msL¨v? lôcÖfvlKwbeÜbIcÖZ¨qb:10 mgvavb NM LK NM LK mgvavb NM LK NM LK NM LK NM LK NM LK mgvavb NM LK
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34Math Tutor 9 16 2 4 49 26 64 DËi: L
I N 201. wb‡Pi †KvbwU g~j` msL¨v? WvK,†Uwj‡hvMv‡hvMIZ_¨cÖhyw³ gš¿Yvj‡qimnKvix†cÖvMÖvgvi:17 243 3 343 3 392 3 676 3 DËi: L 202. wb‡Pi †KvbwU Ag~j` msL¨v? wewfbœ gš¿YvjqmnKvix†gBb‡Ub¨vÝ BwÄwbqcvi:17 27 3 125 3 5 81 4 4 32 5 8 DËi: K 203. wb‡Pi †KvbwU Ag~j` msL¨v? 18Zg†emiKvixwkÿKwbeÜb(¯‹zj mgch©vq):17 𝜋 2 11 me¸‡jv DËi: N 204. `yB A¼wewkó GKwU msL¨v‡K A¼Ø‡qi ¸Ydj Øviv fvM Ki‡j fvMdj 3 nq| H msL¨vwUi mv‡_ 18 †hvM Ki‡j A¼Øq ¯’vb wewbgq K‡i| msL¨vwU KZ? 34Zg wewmGm wjwLZ g‡bKwi, GKK ¯’vbxq A¼ = x Ges `kK ¯’vbxq A¼ = y msL¨vwU = 10y + x A¼Øq ¯’vb wewbgq Ki‡j = 10x + y 1g kZ©vbymv‡i, xy xy10 = 3 ev, 10y + x = 3xy ………(i) 2q kZ©vbymv‡i, 10y + x + 18 = 10x + y ev, 9x = 9y + 18 ev, 9x - 9y = 18 ev, 9 (x - y) = 18 ev, x - y = 2 ev, x = 2 + y ………………(ii) (i) bs mgxKi‡Y x = 2 + y ewm‡q cvB, 10y + 2 + y = 3(2+y)y ev, 11y + 2 = 6y + 3y2 ev, 3y2 + 6y - 11y -2 = 0 ev, 3y2 - 5y - 2 = 0 ev, 3y2 - 6y + y - 2 = 0 ev, 3y (y-2) + 1(y-2) = 0 ev, (y-2) (3y + 1) = 0 y = 2 A_ev y = - 3 1 FYvZ¥K gvb MÖnY‡hvM¨ bv nIqvq y = 2 n‡e| (ii) bs mgxKi‡Y y = 2 ewm‡q cvB, x = 2 + 2 = 4 myZivs, wb‡Y©q msL¨vwU = 10y + x = 102 + 4 = 24. (DËi) 205. `yB A¼wewkó †Kvb msL¨vi A¼Ø‡qi mgwó 9| A¼Øq ¯’vb wewbgq Ki‡j †h msL¨v cvIqv hvq Zv cÖ`Ë msL¨v n‡Z 45 Kg| msL¨vwU wbY©q Kiæb| 31Zg wewmGm wjwLZ g‡bKwi, GKK ¯’vbxq A¼ x n‡j `kK ¯’vbxq A¼ (9 - x) msL¨vwU = 10(9-x) + x = 90 - 9x ¯’vb wewbgq Ki‡j msL¨vwU = 10x + (9 -x) = 10x + 9 -x = 9x + 9 kZ©g‡Z, 9x + 9 + 45 = 90 - 9x ev, 9x + 9x = 90 - 54 ev, 18x = 36 x = 18 36 = 2 myZivs, msL¨vwU = 90 - 92 = 90 - 18 = 72. 206. `yB A¼wewkó †Kvb msL¨vi `kK ¯’vbxq A¼wU GKK ¯’vbxq A¼ n‡Z 5 eo| msL¨vwU †_‡K A¼Ø‡qi mgwói cuvP¸Y we‡qvM Ki‡j A¼Ø‡qi ¯’vb wewbgq nq| msL¨vwU KZ? 23Zg wewmGm wjwLZ g‡bKwi, GKK ¯’vbxq A¼ x Ges `kK ¯’vbxq A¼ x + 5 msL¨vwU = 10 (x+5) + x kZ©g‡Z, 10 (x+5) + x-5(x+5+x) = 10x + x + 5 ev, 10x + 50 + x-5x -25-5x = 11x +5 ev, x+25 = 11x + 5 ev, 10x = 20 x = 2 myZivs, wb‡Y©q msL¨vwU= 10 (x+5) + x mgvavb mgvavb mgvavb NM LK NM LK NM LK NM LK ïay wjwLZ Av‡jvPbv
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2Math Tutor = 10
(2+5) + 2 = 70 + 2 = 72 (DËi) 207. `yB A¼wewkó GKwU msL¨vi GKK ¯’vbxq A¼ `kK ¯’vbxq A‡¼i wZb¸Y A‡cÿv GK †ewk| A¼Øq ¯’vb wewbgq Ki‡j †h msL¨v cvIqv hvq, Zv A¼Ø‡qi mgwói AvU¸‡Yi mgvb| msL¨vwU KZ? cÖwZiÿvgš¿Yvj‡qi AwdmmnKvixKvg-Kw¤úDUvigy`ªvÿwiK2019; beg`kg†kÖwYiMwYZ: Abykxjbx 12.4Gi12bscÖkœ g‡bKwi, `kK ¯’vbxq A¼ = x Ges GKK ¯’vbxq A¼ = 3x +1 msL¨vwU = x10 + 3x + 1 = 10x + 3x + 1 = 13x + 1 A¼Øq ¯’vb wewbgq Ki‡j = x + (3x +1) 10 = x + 30x + 10 = 31x + 10 cÖkœg‡Z, 31x + 10 = (x+3x+1) 8 ev, 31x + 10 = 8x + 24x + 8 ev, 10 - 8 = 32x - 31x ev, x = 2 msL¨vwU = 132 + 1 = 26 + 1 = 27 (DËi) 208. `ywU msL¨v Ggb †h, cÖ_g msL¨v wØZxq msL¨v †_‡K 30 MÖnY Ki‡j Zv‡`i AbycvZ 2 : 1 nq| wKš‘ hw` wØZxq msL¨v cÖ_g msL¨v †_‡K 50 MÖnY K‡i Z‡e Zv‡`i AbycvZ nq 1 : 3| msL¨v `ywU KZ? evsjv‡`k †ijI‡qieywKsmnKvix2029 g‡b Kwi, cÖ_g msL¨v = x Ges wØZxq msL¨v = y 1g kZ©vbymv‡i, 1 2 30y 30x ev, x + 30 = 2y - 60 ev, x = 2y - 90 ……… (i) 2q kZ©vbymv‡i, 3 1 50y 50x ev, 3x - 150 = y + 50 ev, 3x - y = 200 …….. (ii) (ii) bs mgxKi‡Y x = 2y - 90 ewm‡q cvB, 3 (2y -90) - y = 200 ev, 6y - 270 - y = 200 ev, 5y = 200 + 270 ev, 5y = 470 y = 5 470 = 94 y Gi gvb (ii) bs mgxKi‡Y ewm‡q cvB, x = 294 - 90 = 188 - 90 = 98 AZGe, cÖ_g msL¨vwU 98 Ges wØZxq msL¨vwU 94 (DËi) 209. `yB A¼ wewkó GKwU msL¨vi A¼Ø‡qi AšÍi 4; msL¨vwUi A¼Øq ¯’vb wewbgq Ki‡j †h msL¨v cvIqv hvq, Zvi I g~j msL¨vwUi †hvMdj 110; msL¨vwU wbY©q Ki| ; beg`kg†kÖwYiMwYZ:Abykxjbx 12.4Gi12bscÖkœ g‡b Kwi, GKK ¯’vbxq A¼ x Ges `kK ¯’vbxq A¼ y. msL¨vwU = x + 10y 1g kZ©vbymv‡i, x - y = 4 ……. (i) [ hLb, x>y] Avevi, y- x = 4 …… (ii) [hLb, y>x] A¼Øq ¯’vb wewbgq Ki‡j msL¨vwU n‡e = 10x + y 2q kZv©bymv‡i, 10x + y + x + 10y = 4 + 10 ev, 11x + 11y = 110 ev, 11 (x + y) = 110 x + y = 10 ………... (iii) (i) I (iii) bs mgxKiY †hvM K‡i cvB, x - y = 4 x + y = 10 2x = 14 x = 7 Avevi, (ii) bs †_‡K (iii) bs we‡qvM K‡i cvB, y - x - x - y = 4 -10 ev, -2x = -6 x = 3 (iii) bs mgxKi‡Y x = 7 ewm‡q cvB, 7 + y = 10 y = 10 - 7 = 3. Avevi, (iii) bs mgxKi‡Y x = 3 ewm‡q cvB, 3 + y = 10 y = 10 - 3 = 7. AZGe, x= 7 Ges y = 3 n‡j, msL¨vwU = x + 10y = 7 + 103 = 37 A_ev, x = 3 Ges y = 7 n‡j. msL¨vwU = = x + 10y = 3 + 107 = 73. DËi: wb‡Y©q msL¨vwU 37 A_ev 73 mgvavb mgvavb mgvavb