Exabyte Technology

Bản ghi

1 MvwYwZK m~îvewj RULES OF MATHEMATICS exrmwyz (ALGEBRA) em, Nb, b, Drcv`K, Abywm vš I gvb wby qi m~î (a + b) 2 = a 2 + 2ab + b 2 (a + b) 2 = (a b) 2 + 4ab (a b) 2 = a 2 2ab + b 2 (a b) 2 = (a + b) 2 4ab a 2 + b 2 = (a + b) 2 2ab a 2 + b 2 = (a b) 2 + 2ab a 2 b 2 = (a + b) (a b) 2 (a 2 +b 2 ) = (a + b) 2 + (a b) 2 (a + b + c) 2 = (a 2 + b 2 + c 2 ) + 2 (ab + bc + ca) (a 2 + b 2 + c 2 ) = (a + b + c) 2 2(ab + bc + ca) 2 (ab + bc + ca) = (a + b + c) 2 (a 2 + b 2 + c 2 ) (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 (a + b) 3 = a 3 + b 3 + 3ab (a + b) (a b) 3 = a 3 3a 2 b + 3ab 2 b 3 (a b) 3 = a 3 b 3 3ab (a b) a 3 + b 3 = (a + b) (a 2 ab + b 2 ) a 3 + b 3 = (a + b) 3 3ab (a + b) a 3 b 3 = (a b) (a 2 + ab + b 2 ) a 3 b 3 = (a b) 3 + 3ab (a b) (a + b + c) 3 = a 3 + b 3 + c (a + b) (b + c) (c + a) a 3 + b 3 + c 3 3abc = (a + b + c) (a 2 + b 2 + c 2 ab bc ca) a 3 + b 3 + c 3 3abc = (a + b + c) {(a b) 2 + (b c) 2 + (c a) 2 } 4ab = (a + b) 2 (a b) 2 ab = ( ) ( ) (x + a) (x + b) = x 2 + (a + b) x + ab (x + a) (x b) = x 2 + (a b) x ab (x a) (x + b) = x 2 + (b a) x ab (x a) (x b) = x 2 (a + b) x + ab (x + p) (x + q) (x + r) = x 3 + (p + q + r) x 2 + (pq + qr + rp) x +pqr bc (b c) + ca (c a) + ab (a b) = (b c) (c a) (a b) a 2 (b c) + b 2 (c a) + c 2 (a b) = (b c) (c a) (a b) a (b 2 c 2 ) + b (c 2 a 2 ) + c (a 2 b 2 ) = (b c) (c a) (a b) a 3 (b c) + b 3 (c a) + c 3 (a b) = (b c) (c a) (a b) (a + b + c) b 2 c 2 (b 2 c 2 ) + c 2 a 2 (c 2 a 2 ) + a 2 b 2 (a 2 b 2 ) = (b c)(c a)(a b)(b+c)(c+a)(a+b) (ab + bc + ca) (a + b + c) abc = (a + b) (b + c) (c + a) (b + c) (c + a) (a + b) + abc = (a + b +c) (ab + bc + ca)

2 ev e mgm v mgvav b exrmvwywzk m~î Rb cöwz `q ev cövc q UvKv n j, n R bi `q ev cövc, A = qn UvKv ˆ`wbK m cvw`z Kv Ri cwigvy q n j, d w` b m cvw`z Kv Ri cwigvy, W = qd MwZ em NÈvq q wguvi n j, t NÈvq AwZµvš `~iz, D = qt wguvi q % e w Z ev nªv m a Gi ewa Z ev nªvmk Z gvb, A = a (1 ± ) [ e w i Î + wpý I nªv mi Î wpý cö hvr ] GKK mg q GKK g~ja bi gybvdv r UvKv n j, P UvKv wewb qv M n mgqv š gybvdv I I me w g~jab A n e hlv b, mij gybvdvi Î, I = Pnr UvKv Ges A = P (1 + nr) UvKv Pµe w gybvdvi Î, A = P (1 + r) n UvKv m~pk [ a 0, b 0 Ges m, n mkj c~y msl vi m Ui GKwU Dcv`vb ] a m. a n = a m+n (ab) n = a n b n a n = jmvwi`g [ a > 0 Ges a 1 ] = r ( ) = = a = 1 a > 0 Ges a x = a y n j, x = y = (a am n m ) n = a mn = a 0 = 1 a = a 1 = 0 a = a () = + = b b = = x > 0 Ges a x = b x n j, a = b aviv GLv b, a = cö_g c`, p = kl c`, d = mvavib Aš i, r = mvaviy AbycvZ mgvš i avivi Î, n Zg c` = a + (n 1) d n msl K c `i mgwó = {2a + (n 1) d} c` msl v = () + 1 a I b Gi mgvš i ga K = () n = ()

3 n = n n = n (n + 1) n 2 = ()() n 3 = () YvËi avivi Î, n Zg c` = ar n 1 n msl K c `i mgwó = ( ) n msl K c `i mgwó = ( ) a + ar + ar 2 + ar n = ; r > 1 ; r < 1 wî KvYwgwZ (Trinmetry) j ^ sin θ = cs θ =. tan θ = j ^ ct θ = sec θ =. csec θ = j ^ j sin θ = cs θ = tan θ = csec θ = θ θ sec θ = θ θ ct θ = θ θ sin 2 θ + cs 2 θ = 1 sin 2 θ = 1 cs 2 θ cs 2 θ = 1 sin 2 θ sec 2 θ tan 2 θ = 1 sec 2 θ = 1 + tan 2 θ tan 2 θ = sec 2 θ 1 csec 2 θ ct 2 θ = 1 csec 2 θ = ct 2 θ + 1 ct 2 θ = csec 2 θ 1 KvY sin cs tan AmsÁvwqZ 3 ct AmsÁvwqZ sec AmsÁvwqZ 3 csec AmsÁvwqZ m KÛ = 1 wgwbu 60 wgwbu = 1 wwwmö 90 wwwmö = 1 mg KvY 0

4 1 = 1 c = ( ) DbœwZ KvY = tan θ AebwZ KvY = sin θ e Ëi e vmva r, K `ª Pv ci iwwqvb KvY θ n j Pv ci ˆ`N, s = rθ GKK 2q PZzf vm 1g PZzf vm 3q PZzf vm 4_ PZzf vm KvY = (n 90 ± θ ) n we Rvo n j, sin θ cs θ, tan θ ct θ, sec θ csec θ 1g PZzf v M cö Z K KvY abvzœk (+) 2q PZzf v M sin θ I csec θ abvzœk (+) Ges evwk jv FYvZœK ( ) 3q PZzf v M tan θ I ct θ abvzœk (+) Ges evwk jv FYvZœK ( ) 4_ PZzf v M cs θ I sec θ abvzœk (+) Ges evwk jv FYvZœK ( ) sin ( θ) = sin θ cs ( θ) = cs θ tan ( θ) = tan θ sec ( θ) = sec θ ct ( θ) = ct θ csec ( θ) = csec θ cwiwgwz (easurement) AvqZ Îi ˆ`N a GKK I cö b GKK n j, Îdj, A = ab em GKK cwimxgv, s = 2(a +b) GKK KY, d = a +b GKK em Îi GK evûi ˆ`N a GKK n j, Îdj, A = a 2 em GKK cwimxgv, s = 4a GKK KY, d = a2 GKK i ^ mi GK evûi ˆ`N a GKK I KY Øq d1, d2 n j, Îdj, A = (d d ) em GKK cwimxgv, s = 4a GKK mvgvš wi Ki a GKK I D PZv h GKK n j, Îdj, A = ah em GKK mvgvš wi Ki `ybwu mwbœwnz evû a, b GKK I Zv `i Aš f z³ KvY θ n j, Îdj, A = ab.sinθ em GKK mvgvš wi Ki GKwU KY d I wecixz kxl we `y n Z K Y i Dci j ^ h n j, Îdj, A = dh em GKK UªvwcwRqv gi mgvš ivj evûøq a, b GKK I D PZv ev j ^ `~iz h GKK n j,

5 Îdj, A = h(a+b) em GKK wîfz Ri a GKK I D PZv h GKK n j, Îdj, A = ah em GKK wîfz Ri wzb evû a, b, c GKK I a, b Gi Aš f zw³ KvY θ n j, cwimxgv = a + b + c GKK Aa cwimxgv, s = GKK Îdj, A = s (sa)(sb)(sc) em GKK Îdj, A = ab sinθ em GKK mgevû wîfz Ri GKwU evû a GKK n j, cwimxgv = 3a GKK Îdj, A = a em GKK mgwøevû wîfz Ri mgvb evûøq a GKK I Aci evû b GKK n j, cwimxgv = 2a + b GKK Îdj, A = 4a b em GKK e Ëi e vmva r GKK, K `ª Pv ci KvY θ n j, cwiwa, C = 2r GKK Îdj, A = r 2 em GKK e ËKjvi Îdj = θ r em GKK Pv ci ˆ`N, s = θ 2r GKK [ θ = Kv Yi wwwmö cwigvc ] Pv ci ˆ`N, s = rθ GKK [ θ = Kv Yi iwwqvb cwigvc ] AvqZvKvi Nbe i ˆ`N a GKK, cö b GKK I D PZv c GKK n j, KY, d = a +b +c GKK mgmöz ji Îdj = 2(ab + bc + ca) em GKK AvqZb, V = abc NbGKK Nb Ki GK avi a GKK n j, KY, d = a3 GKK c ôz ji K Y i ˆ`N = a2 GKK mgmöz ji Îdj = 6a 2 em GKK AvqZb, V = a 3 NbGKK mge ËK KvY Ki i e vmva r GKK, D PZv h GKK I njvb DbœwZ l n j, njvb DbœwZ, l = h +r GKK eµz ji Îdj = r l em GKK mgmöz ji Îdj = r(l+r) em GKK

6 AvqZb, V = r h NbGKK mge ËK ej bi i e vmva r GKK I D PZv h GKK n j, eµz ji Îdj = 2rh em GKK mgmöz ji Îdj = 2r(h+r) em GKK AvqZb, V = r 2 h NbGKK Mvj Ki e vmva r GKK n j, Z ji Îdj = 4r 2 em GKK AvqZb, V = r NbGKK f±i (Vectr) Avw`we `y A I Aš we `y B n j, H w`kwb ` kk ilvsk AB Øviv m~wpz Kiv nq, Gi ˆ`N AB Ges AB = BA A > B A f±i hv Mi wîfzr wewa t =AB AC +BC f±i we qv Mi wîfzr wewa t AB AC =BC wîfz Ri evûî qi GKB µg Øviv m~wpz f±iî qi hvmdj k~b B C GLv b, AB +BC =AC =(CA ) D C A_ vr, AB +BC +CA =CA CA =0 f±i hv Mi mvgvš wik wewa t =AB AC +AD A B f±i hv Mi wewbgq wewa t h Kv bv u, v f± ii Rb u+v = v+u f±i hv Mi ms hvm wewa t h Kv bv u, v, w Gi Rb (u+v)+w = u+(v+w) f±i hv Mi er b wewa t h Kv bv u, v, w Gi Rb u+v = v+w n j, v=w f± i mvsl wyzk msµvš eèb myî t m, n `ybwu jvi I u, v `ybwu f±i n j (m+n)u=mu + nu Ges m(u+v) = mu + mv Aš we fw³kiy myî t A, B we `yi Ae vb f±i h_vµ g a, b n j Ges AB ilvsk C we `y Z m:n Abycv Z Aš we f³ n j, C we `yi Ae vb f±i, c = O b a B c n C m A Saifuzzaman Antor exabyteantorbd@gmail.com