;ZJF/F TYF AFNAFSLGF lGIDM + – +
+ – –
–
+
;F{5|YD A\G[ ;\bIFDF\YL DM8L ;\bIFGL lGXFGL D}SJLP
;ZBL lGXFGL CMI tIF\ ;ZJF/M SZJMP lJZ]â lGXFGL CMI tIF\ AFNAFSL SZJLP
pNFCZ6 + 2 +3 – 2 –3 + 2 –3 –2 +3
+ sVF5[, A\G[ ;\bIFDF\YL DM8L ;\bIFGL lGXFGL D}SMPf – sVF5[, A\G[ ;\bIFDF\YL DM8L ;\bIFGL lGXFGL D}SMPf – sVF5[, A\G[ ;\bIFDF\YL DM8L ;\bIFGL lGXFGL D}SMPf + sVF5[, A\G[ ;\bIFDF\YL DM8L ;\bIFGL lGXFGL D}SMPf
5 s5|`GDF\ ;ZBL lGXFGL CMJFYL ;ZJF/M + 5 YX[Pf 5 s5|`GDF\ ;ZBL lGXFGL CMJFYL ;ZJF/M – 5 YX[Pf 1 s5|`GDF\ lJZ]â lGXFGL –1 CMJFYL AFNAFSL YX[Pf 1 s5|`GDF\ lJZ]â lGXFGL +1 CMJFYL AFNAFSL YX[Pf
Tushar Soni – The Material Maker [M.Sc., M.Ed.] (Mo. 846006166)
U]6FSFZGF lGIDM + – + –
+ ;F{5|YD VF5[, – A\G[ ;\bIFGM U]6FSFZ – SZJMP +
;ZBL lGXFGL CMI tIF\ ;ZJF/FGL lGXFGL SZJLP lJZ]â lGXFGL CMI tIF\ AFNAFSLGL lGXFGL SZJLP
pNFCZ6 6 sVF5[, A\G[ + s5|`GDF\ ;ZBL lGXFGL . CMJFYL ;ZJF/FGL (+2) (+3) ;\bIFGM U]6FSFZ +6 SZMPf lGXFGL SZMPf 6 sVF5[, A\G[ + s5|`GDF\ ;ZBL lGXFGL . CMJFYL ;ZJF/FGL (–2) (–3) ;\bIFGM U]6FSFZ +6 SZMPf lGXFGL SZMPf 6 sVF5[, A\G[ – s5|`GDF\ lJZ]â lGXFGL . CMJFYL AFNAFSLGL (+2) (–3) ;\bIFGM U]6FSFZ –6 SZMPf lGXFGL SZMPf 6 sVF5[, A\G[ – s5|`GDF\ lJZ]â lGXFGL . CMJFYL AFNAFSLGL (–2) (+3) ;\bIFGM U]6FSFZ –6 SZMPf lGXFGL SZMPf P GM\W o p5ZGF pNFCZ6DF\ VF5[, s f lGXFGL U]6FSFZGL K[P
Tushar Soni – The Material Maker [M.Sc., M.Ed.] (Mo. 846006166)
;FN\]~5GF lGIDM ;FN\]~5DF\ SZJFGL lS|IFVM VF H S|DDF\ H YFIo EFU];AF V[8,[ S[ ;F{5|YD EFUFSFZ tIFZAFN U]6FSFZ4 5KL ;ZJF/M VG[ V\TDF\ AFNAFSLPPP 0FAL AFH] ;ZJF/FGL lGXFGL CMI4 TM HD6L AFH] AFNAFSL YFIP NFPTP A + 2 = 5 A = 5 – 2 A=3 0FAL AFH] AFNAFSLGL lGXFGL CMI4 TM HD6L AFH] ;ZJF/M YFIP NFPTP A – 5 = 10 A = 10 + 5 A = 15 0FAL AFH] U]6FSFZGL lGXFGL CMI4 TM HD6L AFH] EFUFSFZ YFIP NFPTP A X 5 = 10 A = 10 5 A = 2 0FAL AFH] EFUFSFZGL lGXFGL CMI4 TM HD6L AFH] U]6FSFZ YFIP NFPTP A 5 = 10 A = 10 X 5 A = 50 0FAL AFH] JU" VF%IM CMI4 TM HD6L AFH] JU"D}/ YFIP NFPTP √ 0FAL AFH] JU"D}/ VF%I\] CMI4 TM HD6L AFH] JU" YFIP NFPTP √ 0FAL AFH] 3G VF%IM CMI4 TM HD6L AFH] 3GD}/ YFIP NFPTP √ 0FAL AFH] 3GD}/ VF%I\] CMI4 TM HD6L AFH] 3G YFIP NFPTP √
;}+M sVJIJ TYF lJ:TZ6f X – Y = (X + Y)(X – Y) 2 2 2 2 2 2 (X + Y) = X + 2XY + Y (X – Y) = X – 2XY + Y X3 + Y3 = (X + Y)(X2 – XY + Y2) X3 – Y3 = (X – Y)(X2 + XY + Y2) 3 3 3 3 3 3 (X + Y) = X + Y + 3XY(X (X – Y) = X – Y – 3XY(X – Y) DwID 5N = √ √ HIF\4 First Term = 5|YD 5N Last Term = V\lTD 5N 2
2
Tushar Soni – The Material Maker [M.Sc., M.Ed.] (Mo. 846006166)
;}+M M
N
M+N
A XA =A M A AN = AM – N
(M > N) M N A A = (M < N) 3FTF\SGF lGIDM M N A A =1 (M = N) RMZ;G\] 1F[+O/ = l2 ,\ARMZ;G\] 1F[+O/ =lXb ;D3GG\] 3GO/ = l3 ,\A3GG\] 3GO/ =lXbXh B]<,F G/FSFZGL JS|;5F8LG\] 1F[+O/ = 2 rh OR dh A\W G/FSFZGL S],;5F8LG\] 1F[+O/ = 2 r (h + r) 1F[+O/ TYF JT]"/GM 5lZ3 =2 r 3GO/ JT]"/G\] 1F[+O/ = r2 2 VW"UM/FGL JS|;5F8LG\] 1F[+O/ =2 r VW"UM/FGL S],;5F8LG\] 1F[+O/ = 3 r2 UM/FGL JS|;5F8LG\] 1F[+O/ = 4 r2 2 G/FSFZG\] 3GO/ = rh l+HIF = jIF; Z V[8,[ S[ r = d 2 jIF; = 2 X l+HIF V[8,[ S[ d = 2r X\S]GL +F\;L µ\RF. (l) =√ X\S]GL l+HIF (r) =√ X\S]GL µ\RF. (h) =√ X\S]GL JS|;5F8LG\] 1F[+O/ = rl X\S]GL S],;5F8LG\] 1F[+O/ = r (l + r) X\S]G\] 3GO/ = ⁄ r2h VW"UM/FG\] 3GO/ UM/FG\] 3GO/
= ⁄
r3
= ⁄
3
r
Tushar Soni – The Material Maker [M.Sc., M.Ed.] (Mo. 846006166)
~5F\TZ ! DL8Z = !__ ;[PDLP ! RMPDLP = !_4___ RMP;[PDLP ! VZ = !__ RMPDLP ! C[S8ŸZ = !__ VZ ! lSPU|FP = !___ U|FD ! lSJg8, = !__ lSPU|FP ! l,8Z = !___ lDPl,P ! 3G;[PDLP = ! lDl,l,8Z ! l,8Z = !___ 3G;[PDLP GOM v BM8 !P 50TZ lS\DT D}/lS\DT BZFÔT ZP GOM J[RF6lS\DT D}/lS\DT #P BM8 D}/lS\DT J[RF6lS\DT $P GOM s8SFDF\f @ GOM D}PlS\ 2 !__ 5P BM8 s8SFDF\f @ BM8 D}PlS\ 2 !__ AC]]5NL !P H[ AC]5NLDF\ V[S H 5N CMI4 T[ AC]5NLG[ V[S5NL SC[ K[P 2 2 3 3 NFPTP 3x, 5a b, p qr, 4a b , 15 ZP H[ AC]5NLDF\ A[ 5N CMI4 T[ AC]5NLG[ lä5NL SC[ K[P 2 2 2 3 NFPTP x + 3, a 1, p q + 5, a 15 #P H[ AC]5NLDF\ +6 5N CMI4 T[ AC]5NLG[ l+5NL SC[ K[P 2 3 3 NFPTP 3x + y + 2, a + a + 1, a b + 2 Tushar Soni – The Material Maker [M.Sc., M.Ed.] (Mo. 846006166)
JU" VG[ 3G ;\bIF 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
JU" 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225
;\bIF
JU"
JU"D}/s√f
10 20 30 40 50 60 70 80 90
100 400 900 1600 2500 3600 4900 6400 8100
10 20 30 40 50 60 70 80 90
100 10000
;\bIF 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
JU"D}/s√f 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
100
JU" 256 289 324 361 400 441 484 529 576 625 676 729 784 841 900
JU"D}/s√f 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
;\bIF 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
3G 3GD}/s f 1 1 8 2 27 3 64 4 125 5 216 6 343 7 512 8 729 9 1000 10 1331 11 1728 12 2197 13 2744 14 3375 15
;\bIFGM 3G VG]DFGGL ZLTYL 3GD}/ XMWJ\] ;\bIFGF SZJFYL V[SDGM D/TL pNFCZ6 o $)!# V\S ;\bIFGF V[SDGM V\S ;DH}TL o 1 1 1. VF5[, ;\bIFGF A[ EFU 5F0MP ;M4 NXS VG[ 2 8 V[SDYL AGTL ;\bIFGM V[S EFU VG[ AFSLGF V\SYL AGTL ;\bIFGM ALÔ[ EFUP 3 7 VCL\ $)!# DF\ )!# V[S EFU VG[ $ ALÔ[ 4 4 EFU 5 5 6 6 2. VCL\ 5|YD EFU )!# K[P H[GF V[SDGM V\S 7 3 # K[P VCL\ ! YL ) JrR[ S. ;\bIFGF 3G 8 2 SZJFYL V[SDGF V\SDF\ # VFJ[m VF ;\bIF * 9 9 K[P sAFH]GF SMQ8SDF\ H]VMf
0
0
3. ALÔ EFUGL ;\bIF S[ H[ A[ ;\bIFGF 3GGL JrR[ CMI T[ ;\bIFVM 5{SL GFGL ;\bIF K[P VCL\ ALÔ[ EFU $ K[P T[ !# ! VG[ Z# (GL JrR[GL ;\bIF K[P ! VG[ Z 5{SL ! GFGL ;\bIF K[P ! V[ 5}6"3G ;\bIFGF 3GD}/GF NXSGF[ V\S AGX[P $)!# G\] 3GD}/ !* YX[P
Tushar Soni – The Material Maker [M.Sc., M.Ed.] (Mo. 846006166)
Tushar Soni – The Material Maker [M.Sc., M.Ed.] (Mo. 846006166)
