TGMathM2.pdf

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[qfmu &! dkoltg|u0=h,6ofh;pI6[c8h, X8+? !_*  ດິ ະສ ລຂິ ວນ ຫງ ສະ ສ ວ. ສ. ’  rkdmu ! 9eo;oxqddt8y c]t 9eo;o9y’ [qfmu ! g]dlj;o g;]k @ -qj;3,’ !D 95fxtlq’ { grnjv.shoadIPo,u7;k,leobddjP;da[g]dlj;oS { grnjv.shoadIPoIh67;k,\kpF 7;k,le7ao0v’g]dlj;omujrq;raoda[-u;yfxt9e;aoD { grnjv.shoadIPolk,kfcdh39f[aoskdjP;da[g]dlj;owfhF @D goNv.o8Qo8= { dkoxjPo9eo;omqfltoypq,gxaog]dlj;o c,jogIqkg]uj,9kddko0Po9eo;omqf ltoypq,gxaog]dlj;oly[D a { g]dlj;o ,u7;k,\kp;jk c[j’gxao b lj;oc]h;gvqk a lj;o b a o 9eo; or6f0v' g]dlj; o b o r6f0v' g]dlj; o a - 76o9eo;o|bj’da[g]dlj;o c,jogryjo b ¾ 76oda[ a c]h;sko.sh b s^n sko.sh b c]h;76oda[ a #D lnjdkoIPo{dkolvo { xB,c[[IPo7toyflkf ,@ { ;q’;PoF [aomafpk;F [aomaflkdF [aomafcmd,6, c]t l=lu $D dyf9tdedkoIPo{dkolvo.shxt8y[afgxao & 0Ao8vofaj’oUG 0Ao8vomu !G { 76m;o7no[qfIPogdqjkmujIa[.-h[qfIPo.\j s^n oerkoadIPocdh[qfg/ydsafmuj.sh.o[qfIPodjvoD 1  { 76c[j’oadIPogxao95F 95]t @ sk * 7qo { 76.shoadIPoc8j]t7qovjko39fc]h;fegouodkocdh39f.odyf9tdeD { 76cotoeoadIPoc8j]t95[aombdgvqk7e8v[muj958qogv’ gsao;jk4nd8hv’D 0Ao8vomu @G { 764k,7e8v[0v’c8j]t95 s^n.shc8j]t950Boltg\u7e8v[1j6dt fkorhv,ma’vtmy[kp;ymu c]t gsfzqo.odkocdh39fD 0Ao8vomu #G { 76.shoadIPola’gdf7e8v[ma’\qfmujwfhF c]dxjPo7e7yf7egsao oedao c]h;lts^6[gvqk7e8v[muj4nd8hv’D 0Ao8vomu $G { 76vtmy[kp c]t -Uc9’lyj’mujzyfrkf c]t lyj’muj4nd8hv’c]h;lt ດິ ະສ s^6[95fxtlq’0v’dyf9tdeD 0Ao8vomu &G { 76cotoeoadIPoc8j]t7qogIaf[qfg/ydsaf1j6.oshv’1jk’]t0=h ລຂິ { 76cotoeoadIPo0Bocdh[qfg/ydsaf1j6dtfkoD ວນ { 76cotoeoadIPogIaf[qfg/ydsafmujpa’gs^nv s^n [k’0=h1j6[hkoD &D dkoxtg,uozqo ຫງ xt8y[afgxaolk,I6[c[[7nG ສະ !? la’gdfmafltotmjkmu c]t m5dlyj’mujoadIPoltcf’vvd.oshv’IPog;]koadIPo xt8y[afdyf9tdeF oadIPom5d7qowfhxtdv[lj;og0Qk.odkocdh[aosks^n[+L ສ cdhwfh4nd8hv’-afg9os^n[+L X76vkf.sh7tcooc8j]t95 4hk8hv’dko? ວ. @? la’gdfmafltotmjkmu c]t m5dlyj’mujoadIPoltcf’vvd.oshv’IPog;]koadIPo cdh[qfg/ydsafF ສ. #? la’gdfdkocdh[qfg/ydsafmujoadIPogIaf1j6[hko3fpc8j]tgmnjvdjvo9tg]uj,g0Qk[qf IPo.\j.shgvqkoadIPo,kcdh[qfg/ydsaf[k’0=h1j6dtfkoD \kpgsfG le]a[[qf8+Mwxd=.shxt8y[af7ndao s^n \6o.-h8k,ltrk[7;k, g\ktlq,0v’c8j]t[qfD 2 *D 7ecotoe s^n 7e8v[dyf9tde dyf9tdemu ! 85 850 17 75 750 3 8,5 0,75 10 100 2 100 1000 4 2 200 1 125 25 5 0,2 1,25 10 1000 5 100 20 4 dyf9tdemu @ { 76cotoeoadIPoc8j]t95c8h,I6[7no.\jc]h;mklurkdlj;omuj8hv’dkoD { 76vkfc[j’7e4k,.shc8j]t95c8d8jk’daod=wfhg-ajoG 7e4k, dD.sh95|bj’F 7e4k, 0D.shvud95|bj’ c]t 7e4k, 7D d=.shvud95vnjogrnjvxt1afg;]kD dyf9tdemu # 3 2 dD u 560 210 g 0D u 450 180 g 8 5 dyf9tdemu $ 3 3 dD u 630 236,25 g 0D u 630 378 g 8 5 (D 7ecotoe s^n 7e8v[[qfg/ydsaf 21 34 105 !D 1) 2,1 2) 3,4 3) 10,5 10 10 10 214 10811 2201 4) 2,14 5) 10,811 6) 2,201 100 1000 1000 5 18 831 @D 1) 0,625 2) 0,72 3) 55,4 8 25 15 2 6 10 4) 0,016 5) 0,08 6) 0,08 125 75 125 14 42 5 11,5 #D 1) u3 2,8 2) 2,3 u 2,875 15 15 4 4 10 1 5 3) 0,1u 0, 001 4) 21 u 15 1000 1000 7 17 3 5) 3,12 u 4,42 6) 4,03 u 0,01209 12 1000 4 1 $D dD 0D 5 7D 7 ’D 12 3 9D 1 16 4 16 16 20 5 4 3 &D A D C B *D dD 45 D 0D 30 D 7D 36 D ’D 15D (D g]dlj;o|hvpdj;k ! g]dlj;ogmqjkda[ ! g]dlj;os^kpdj;k ! 21 15 17 17 19 19 21 ; ; ; ; 23 19 19 17 19 17 19 3 )D 9eo;o\kd[umujglpwx u 210 90 |j;pS pa’gs^nv 210  90 120 |j;p 7 _D 9eo;o\kdoafmuj0kp 3 u 240 144 kg S pa’gs^nv 240  144 96 kg ດິ ະສ 5 !WD dD 9eo;og0Qk|q,dyoc]h; 5 u 96 40 dhvoS pa’gs^nv 96  40 56 dhvo 12 0D 9eo;og0Qk|q,dyoc]h; 3 u 96 36 dhvoS pa’gs^nv 96  36 60 dhvo ລຂິ 8 ວນ !!D dD 4hk[ao95c8j 2 9t,uoE 2 u12 8 A 0D 4hk[ao95c8j 3 9t,uoE 3 u12 9 A 3 3 4 4 ຫງ !@D ]qflu2hk,u 1 u 720 240 7ao 3 ສະ ]qflucf’,u 1 u 720 180 7ao 4 ]qflu0k;,u 5 u 720 300 7ao ສ 12 mbo0v’mhk; d=]tdao,u 5 u11 016 000 2 295 000 du[ ວ. !#D 24 ສ. mbo0v’mhk; la’mv’,u 1 u11 016 000 1 377 000 du[ 8 2 mbo0v’ok’ verk,u u11 016 000 7 344 000 du[ 3 !$D [=]y,kfoE\kddhP’,u 1 u 20 5 A S [=]y,kfoE\kdoaf,u 3 u 20 6 A 4 10 [=]y,kfoE\kd,j;’,u 2 u 20 8 A S [=]y,kfoEvhvp,u 1 u 20 1 A 5 20 4  [qfmu @ mt;u76o c]t v5xt76o0v’g]dlj;o.-hg;]k ! -qj;3,’ !D 95fxtlq’ ¾ grnjv.shoadIPo,u7;k,lk,kfG { -vdmt;u76oIj;,|hvpl5f c]t v5xt76oIj;,s^kpl5f0v’lv’ s^n s^kp9eo;o4h;oD { -vdmt;u76oIj;,|hvpl5f c]t v5xt76oIj;,s^kpl5f0v’lv’ s^n s^kpg]dlj;oD { oe.-hmt;u76oIj;,|hvpl5f c]t v5xt76oIj;,s^kpl5f0v’g]dlj;ogrnjvG D -vdskr6fIj;,0v’g]dlj;oS D 7af9hvog]dlj;oD @D goNv.o8Qo8= { mt;u76oIj;,|hvpl5f0v’lv’ s^n s^kpg]dlj;o gmqjkda[zqosko]ts;jk’mt;u76o Ij;,|hvpl5f0v’9eo;or6f c]t v5xt76oIj;,s^kpl5f0v’r6fD { v5xt76oIj;,s^kpl5f0v’lv’ s^n s^kpg]dlj;o gmqjkda[zqosko]ts;jk’v5xt 76oIj;,s^kpl5f0v’9eo;or6f c]t mt;u76oIj;,|hvpl5f0v’r6fD #D lnjdkoIPo{dkolvo xB,c[[IPo7toyflkf ,@D { [aomafpk;F [aomaflkd c]t l=luD $D dyf9tdedkoIPo{dkolvo.shxt8y[af8k,0Ao8vo7nda[[qfmu !D le]a[[qfoU dko-vdskmt;u76oIj;,|hvpl5f c]t v5xt76oIj;,s^kpl5f0v’lv’ s^n s^kp9eo;o4h;o]kp]tvPf.shg[yj’[qfmu (.oxB,c[[IPo7toyflkf ,D! 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C 90 S D 4 9 u 45 ສ. !WD dD F 8 u150 1200 0D F  32 113 5 48 u 20 220 7D P 160 ’D I 2, 75 6 80 1 152 225 1125 9D V u 3,14 u 62 u 5 188, 4 lD h 3 2 u 9,8 19, 6 98 66  [qfmu !_ dkocpdlj;o76o.oleo;o g;]k $ -qj;3,’ !D 95fxtlq’ ™ grnjv.shoadIPolk,kfG { xjPozqo[;d.oleo;o.shgxaoleo;omujgxaozqo76o3fpdkocpdgvqklj;o76oIj;, 0v’r;d,aovvdgxaolj;olj;o76oD { xjPoleo;omujgxaozqo76o.shgxaoleo;omujgxaozqo[;d @D goNv.o8Qo8+ ™ dkocpdlj;o76o.oleo;o 3fpdkocpdgvqklj;o76oIj;,0v’r;d,aovvd gxaolj;o76oD #D lnjdkoIPo{dkolvo ™ xN,c[[IPoD $D dyf9tdedkoIPo{dkolvo {.shxt8y[af8k,7ecotoe.o0=hmu $ 16j[qfmu !D { xt8y[afc8j]tdyf9tdegxaod5j,F d5j,]t @{$ 7qoD { 76gvqk.9.lj8nj, 0Ao8vodkolvo.97;k, X76vkfpqd8q;1jk’vnjomujrq;rao da[c8j]t[aosk grnjv.shoadIPog0Qk.9c9h’?D { cdh[qfg/ydsaf X[k’0=h? c8j0=hmu !{* 16jshv’D { g]nvdgvqk[qfg/ydsaf @ 0=h9kd0=hmu & c]t * grnjvd;fdkD { d;fdk @W okmuD {.sh;Pd[hko[qfg/ydsafmujpa’gs^nvD 67 &D 7ecotoe s^n 7e8v[0v’dyf9tde dyf9tdemu ! dD < AMNPQ AMJIN  AJQPI 5u 4  2u 4 20  8 28 < AMNPQ (5  2) u 4 7u4 28 < zqowfhIa[9kddko7yfw]jma’lv’d=]tou gmqjkdaoD ດິ ະສ < -vdskgoNvmuj0v’ MNPQ 8k, a, b c]t k < AMNPQ AMJIN  AJQPI ka  kb ລຂິ ວນ < AMNPQ k ( a  b) faj’oAoF k a  b ka  kb ຫງ 0D < AMNPQ AMLEJ  ALEIN  AJQKE  AEKPI ສະ 5u 4  5u 2  3u 4  3u 2 20  10  12  6 48 ສ < AMNPQ (5  3)(4  2) ວ. 8u 6 ສ. 48 < -vdskgoNvmuj0v’ MNPQ 8k, a , b, c c]t d < AMNPQ AMLEJ  ALEIN  AJQKE  AEKPI ad  ac  bd  bc < AMNPQ (a  b)(c  d ) ad  ac  bd  bc faj’oAo ab cd ad  ac  bd  bc 68  dyf9tdemu @ < 70 u11  70 u 22  70 u17 70 11  22  17 70 11  22  17 350 < 2a u 3  2a u 5  2a u 6 2a 3  5  6 4a < 2 a 1  3 a 1  4 a 1 a 1 2  3  4 3 a 1 < m2  m2 m  2  m2 m  3 m 2 ª¬1  m  2  m  3 º¼ m 2 >1  m  2  m  3@ 0 dyf9tdemu # A x  1 3x  1 ; B 2x 2x  2 C x  2 3 x D 2x 1 2x  3 dyf9tdemu $ lj;o76oIj;,0v’[aofkleo;o]5j,oUc,joG < 3u 5  3u 7  3u11  3 u13 c,jo 3 < 4  8  12  16  20  24 c,jo 4 < 2ab  2ac  2ad  2ae  2af c,jo 2a < (1  n)(m  n)  (m  n)(2  n) c,jo m  n < b  b 2  b3  b 4 c,jo b < ab  (ab) 2 c,jo ab *D 7ecotoe s^n 7e8v[0v’[qfg/ydsaf !D dD 5x  5 0D 2a 2  2a 69  2 1  y3 7D 7a  7 ’D 1  y3 1 9D 1 lD 6 -D a 2  b2 pD 1  x3 fD m3  n3 @D dD 5 x  4 y 0D 2bx 7D 13a ’D 7m  2m2 9D 3a  9b lD n3  18n2  9n  15 -D a 2  b2 5a 2 2b3 m  n ດິ pD fD 8D ະສ 36 25 #D dD 5 0D 3 7D 11 ’D 4 9D 2a ລຂິ lD 2mn ວນ -D ab pD 3ac ຫງ fD 4 x 2 y 2 8D x 4D x  y mD n m  n ສະ $D dD 7 x  7 y 7 x  y 0D 3a  3b  3c 3 a  b  c ສ 7D 2ab  2ac  2ad 2a(b  c  d ) ວ. ’D 4n  6mn 2n 2  3m ສ. 9D 22ab  33ac  44ad 11a(2b  3c  4d ) lD 10 xy  5 x 2 y 2 5 xy 2  xy -D 36b 2  9b 9b 4b  1 pD 5a 2  10a 3  15a 4 5a 2 1  2a  3a 2 fD 15m4  25m2 5m 2 (3m 2  5) 70 &D dD 3m5  3m 4  12m3  9m 2 3m 2 m3  m 2  4m  3 0D 4a 2 b 2 c 2  8ab 2 c  12a 2 b3 c3 4ab 2 c ac  2  3abc 2 7D 9 x3 y  6 x 2 y 2  3 xy 3 3 xy 3 x 2  2 xy  y 2 ’D 5n3  10n 2 m  15nm 2 5n n 2  2mn  3m 2 *D dD ax  ay  bx  by ax  ay  bx  by a x y b x y x y ab 0D pm  pn  mq  nq pm  pn  mq  nq pm  pn  mq  nq p mn q mn mn pq 7D x 2  2 x  5 x  10 x 2  2 x  5 x  10 x 2  2 x  5 x  10 x x2 5 x2 x2 x5 ’D 4ax  6ay  4bx  6by 4ax  4bx  6ay  6by 4ax  4bx  6ay  6by 4x a  b  6 y a  b a  b 4x  6 y 2 a  b 2x  3y 9D x 2  3x  2 x  6 x 2  3x  2 x  6 x 2  3x  2 x  6 x x 3 2 x 3 x3 x2 71 lD a 2c 2  acd  acd  d 2 a 2c 2  acd  acd  d 2 a 2c 2  acd  acd  d 2 ac ac  d  d ac  d ac  d ac  d 2 ac  d -D 3n(n  2)  3n(3  2n) 3n ª¬ n  2  3  2n º¼ 3n ª¬ n  2  3  2n º¼ 3n > n  2  3  2n @ ດິ ະສ 3n 3n  1 pD ( x  5)(n  m)  ( x  5)(n  2m) x  5 ª¬ n  m  n  2m º¼ ລຂິ x  5 ª¬ n  m  n  2m º¼ x  5 > n  m  n  2m @ ວນ 3m x  5 ຫງ fD (1  3x)(2  3 x)  (3 x  1)(1  x) (1  3 x)(2  3 x)  1 (1  3 x)(1  x) 1  3 x ª¬ 2  3 x  1  x º¼ ສະ 1  3x > 2  3x  1  x @ 1  3x 4 x  3 ສ ວ. 8D (2 x  4)(3x  5)  2(3x  6)( x  1) 2 x  2 3x  5  2 3 x  2 x 1 2 x  2 3x  5  2 3 x  2 x  1 ສ. x  2 ª¬ 2 3x  5  6 x  1 º¼ x  2 > 6 x  10  6 x  6@ 4 x  2 mn m 2 n mn 2 mn § m n · 4D   ¨1   ¸ 5 10 15 5 © 2 3¹ a3 a 2b a 2c a2 § b c· mD   3¨ a  ¸ 2 x3 4 x4 6 x3 2x © 2x 3 ¹ 72  rkdmu $ lq,zqo c]t vtlq,zqo0Ao|bj’mju,u|bj’8q;]a[ [qfmu @W ltg\uzqo g;]k @ -qj;3,’ !D 95fxtlq’ ™ grnjv.shoadIPolk,kfG { oe.-hoypk,F 75o]adltot0v’ltg\uzqocdh[qf/ndsaf c]t g]d39f { cdh39f[aosk.o-u;yf9y’0v’la’7q,wfh @D goNv.o8Qo8= { ltg\uzqoc,jolv’9eo;omuj8+daofh;pg7njv’\kpgmjqkda[ " " c]t gxao9y’m5dM7jk0v’8q;|a’ln X pqdg;Ao7jk8hv’shk, ? {dkoxjPoltg\uuzqo|bj’gxaoltg\uuzqo.\jmP[gmqjkda[ltg\uuzqomuj.shdjvoD #D lnjdkoIPo { dkolvo { xnh,c[[IPo { -y’ , \kd38ho-y’ X4hk,ug’njvow0rhv,? $D dyf9tdedkoIPo {dkolvo {.shxt8y[af8k,dkocotoe.o0=hmu $ 1j6[qfmu !D {.sh76gvqk.9.ljcotoe]tvPfdyf9tdemu @ c]t.97;k,D &D 7ecotoe s^n 7e8v[dyf9tde dyf9tdemu ! ltg\uuzqo 2a 10 { 4hk a 6 ltg\uuzqogxaoltg\uuzqozyf { 4hk a 5 ltg\uuzqogxaoltg\uuzqo4nd 73  dyf9tdemu @ d? x 3 1 4 gxao ltg\uuzqo4nd x 3 1  2 42 gxaoltg\uuzqo4nd x 3 1  a 4a gxaoltg\uuzqo4nd x a b faj’oAo a7 b7 gxaoltg\uuzqo4nd x a b faj’oAo ac bc gxaoltg\uuzqo4nd 0? ດິ ະສ x a b faj’oAo ac bc x a b faj’oAo auc buc x a b faj’oAo ລຂິ au 1 bu 1 cz0 ວນ c c 7? ຫງ a 1 4  a  7 10