7toyflkf -Ao,afmtpq,lbdlkxumu $ IP[IP’3fpG

Texte intégral

(1)c[[IPo. 7toyflkf -Ao,afmtpq,lbdlkxumu $ IP[IP’3fpG. d;fcdh3fpG. mv’0k; cl’l5]y9ao graflts;ao vko5;q’ s^hk 9aomt;q’ ,t]y9ao vad7t]kf cv39 vkfzkl5d l9D fiD l6omvo rq,,tlvo v5myf my[,tou. ry,g0Qk|hk3fpG mv’0k; cl’l5]y9ao [q;]u cdh;;q’lk ;y]tg]uf ltra’mv’ l5ffkrvo cdh;[q;ltw\ dt-;’lbdlkmydko c]t dy]k lt4k[ao7Qo7;hk;ymtpklkfdkolbdlk @W!#. lD;Dl ,D,Dl ;P’9ao. ,DlD;Dl lD;Dl lD;Dl lD;Dl lD;Dl.

(2) 7eoe xB,c[[IPo7toyflkf ,D$ gsA^,oU wfhIP[IP’0Bogrnjvlvo8k,sâdl6f0v’ 7toyflkf -Ao,afmtpq,lbdlk8vo8Qolt[a[xa[x5’xu @WW_D .o0Ao,afmtpq,lbdlkxumuluj X,D$? cdo0v’ sâdl6f c,joG leo;oxqddt8y c]t vtxqddt8yF 8e]k0Aolv’F w8,5,,y8y.oI6[lk,c9lkdF sâdgdo-yoF sâdgdo3d-yoF g;adg8u c]t dko7eo;oF dkozaoxjPogmy’czjorP’ c]t dkooe.-hF gonhvmuj c]t [=]y,kf0v’I6[mkf]jP, c]t I6[9;pF lt4y8yD xB,c[[IPosq;oUwfhIa[dkoIP[IP’0Bo grnjv.shzh6IPolk,kfIPoIh6 c]t rafmtok 8qogv’F oegvqk7;k,I6hmk’7toyflkfwx.-h.o-u;yfxt9e;ao c]t gxaog7njv’,n.odkoIPoI6h 7toyflkf c]t ;y-kvnjo.o]tfa[l6’0BowxD goNv.omujltcf’vvd.oxB,sq;oUgxaorP’0=h,6o rNo4kogrnjv.shoadIPo c]t 76oe.-hg0Qk.odkoIPo{dkoolvoF faj’oAo 76 c]t oadIPo lk,kf-vdsk0=h,6ogruj,g8u,9kds^kpcs^j’gvdtlko grnjv.shcmfg\ktda[ltrk[dkosao xjPo0v’3]d.oxt95[aoD c8j]t[qf0v’xB,c[[IPosq;oUxtdv[fh;pG dyf9tdeF .97;k, c]t [qfg/ydsafD grnjv-j;p.shoadIPowfhIPoI6hfh;p8qogv’s^kp0BoF .odkoIP[IP’xN,sq;oU9bj’wfh[ao95gvqk 7ecotoe s^n 7e8v[0v’[qfg/ydsafw;hmhkpgsÂ,D 1jk’.fd=8k,dkoIP[IP’xB,sq;oUd7= ’q [+xklt9k0=0h kf8qd[qdrjv’wfhF ltoAo 9bj’0=7;k, Ij;,,noemjkoz6hmujoe.-h 9qj’-j;pg[yj’95f[qdrjv’ s^n 95fzyfrkf c]h;lqj’7e7yf7egsao0v’mjkowxpa’ lt4k[ao7Qo7;hk;ymtpklkfdkolbdlkD r;dgIqk9t4n;jkm5d7e7yfgsao0v’mjkogxao0=h,6omuj,u75o7jk c]t gxaodko-j;pxa[x5’75ootrk[dkolbdlkr;dgIqk.shl6’0BoD lt4k[ao7Qo7;hk;ymtpklkfdkolbdlk.

(3) lk]t[ko |hk. rkdmu @. leo;oxqddt8y c]t vtxqddt8y [qfmu ! leo;oxqddt8y [qfmu @ dko76o c]t dkosko leo;oxqddt8y [qfmu # dkosko rts5rqf [qfmu $ dko[;d c]t dko]q[leo;oxqddt8y [qfmu & dkocdhlq,zqoxqddt8yy [qfmu * vaf8klj;o c]t vaf8klj;orq;rao [qfmu ( gxug-ao c]t fvdg[hp [qfmu ) dkocdh[aoskdjP;da[7;k,w;F w]ptmk’ c]t g;]k [qfmu _ g]dIkd0Ao n [qfmu !W dko7yfw]j\;f7eo;omuj,ug]dIkd [qfmu !! dkocdhlq,zqoIkd0Ao @ [qfmu !@ g]dde]a’ mujde]a’gxao9eo;oxqddt8y 8e]k0Aolv’ [qfmu !# 8e]k0Aolv’.oI6[Ijk’ y  x 2 [qfmu !$ 8e]k0Aolv’.oI6[Ijk’ y  ax 2. 1 1 5 10 13 22 25 32 39 45. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. rkdmu !. [qfmu !& [qfmu !* [qfmu !( [qfmu !) [qfmu !_ [qfmu @W [qfmu @! [qfmu @@ [qfmu @#. 8e]k.oI6[Ijk’. y  a  x  b. 2. 8e]k.oI6[Ijk’ y  ax  p 2  q 8e]k.oI6[Ijk’ y  ax 2  bx  c 7jks^kpl5f c]t 7jk|hvpl5f 0v’8e]k0Aolv’ g7njv’\kp0v’w8rqf0Aolv’ vtlq,zqo0Aolv’ c]t 9eo;o.9zqo0v’lq,zqo0Aolv’ muj8A’leraf]ts;jk’glAo-nj da[glAo0Aolv’ c]t muj8A’leraf]ts;jk’lv’glAo0Aolv’ l6f ;ucvaf g7njv’\kp 0v’.9zqo0Aolv’ c]t ]t[q[vtlq,zqo0Aolv’. 51 58 62 68 68 72 74 77 79 82 86 89 93 97. 100.

(4) rkdmu #. rkdmu $. [qfmu @$ w8,5,,y8y [qfmu @& [qfmu @* [qfmu @( [qfmu @) [qfmu @_ g;adg8u [qfmu #W. ]t[q[lq,zqo0Aolv’muj,ulv’8q;]a[. 105 108. w8,5,,y8y.oI6[lk,c9lkd 3d-yoF -yo c]t 8a’0v’,5,]ts;jk’ sâdgdo-yo sâdgdo3d-yo goNvmuj0v’I6[lk,c9. 108 0. sk 180. 112 122 127 132 136 136 144 150 158. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 7;k,I6hmqj;wxdjP;da[g;dg8uF g;adg8u0t|kodaoF g;adg8ugmqjkdaoF g;adg8udq’dao0hk,dao [qfmu #! dko[;d c]t dko]q[g;adg8u [qfmu #@ dko76og;adg8ufh;pltdkc] [qfmu ## g;adg8u.o]t[q[glAog7Qk8A’lkd [qfmu #$ zqo76oltdkc] Xzqo76o[+,umyf? .oczjorP’ rkdmu & dkozaoxjPogmy’czjorP’ c]t dkooe.-h [qfmu #& dkophko0t|ko c]t dkooe.-h [qfmu #* dkog7yj’7n c]t dkooe.-h [qfmu #( dko\6oIv[ c]t dkooe.-h rkdmu * gonhvmuj c]t [=]y,kf0v’I6[.odk’sk; [qfmu #) goNvmuj0v’I6[mkf]jP, c]t I6[9;p [qfmu #_ [=]y,kf0v’I6[mkf]jP, c]t I6[9;p rkdmu ( lt4y8y [qfmu $W xtgrf0v’0=h,6o [qfmu $! 7jk;afcmdmjkvjP’0v’0=h,6oF 7jkltg]jp [qfmu $@ 4kooypq, [qfmu $# ,afmtpt4ko [qfmu $$ 7jk;afcmddkoc9d1kp0v’0=h,6o 7e8v[[qfg/ydsaf. 165 170 170 178 188 197 197 209 221 221 225 230 234 239 243.

(5) rkdmu ! leo;oxqddt8y c]t vtxqddt8y [qfmu ! leo;oxqddt8y dyf9tde !D 9eo;o c]t leo;oc8d8jk’daoco;.fL .shpqd8q;1jk’D @D lq,,5fgxug-aodew]9kddko0kplyo7hk-toyf|bj’ c,jo. s  20000 , s. g-yj’. s. c,jjo]k7k0kpD 9qj’-vdgxug-aodew]g,njv]k7k0kp c,jo @&WWW du[D. .97;k,. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. !D! oypk,leo;oxqddt8y leo;oxqddt8yc,jozqosko0v’lv’leo;orbf-t7toyf s^n c,joleo;o.oI6[Ijk’ A , B. g-yj’. A. c]t. B. gxaorts5rqfmuj. B  0. B. gvUo;jk “r6f” c]t A “9eo;or6f”D. 7j k 0v’8q ; xj Pomu j g Ia f.sh l eo;oxq ddt8y [+ deoq fc,j o 7jk 0v’8q ; xj P omujgIaf.sh r6fgmqjkda[l6oD 8q;1jk’!G. 1 , x. x2 , 2 x  2x  5. 6 3x 2  8 x  4 , , y 3 5. 4x 2  8x ,  3x  2. x 2  3 xy  y 2 3x 3 y 4. ]h;oc8jgxaoleo;oxqddt8y g,njvr6f0v’c8j]tleo;ogs^qjkoU8jk’l6oD 8q;1jk’@G 9qj’-vdskm5dM7jk0v’8q;xjPo .oc8j]tleo;o8+wxoUmujrk.shgxaoleo;oxqddt8y [+ d eoq f G dD [qfcdhG. 1 x 1. dD .shr6f. 0D x  1 gmqjkda[. [+ d eoq f g,njv. y y 1 2. W c]h;cdhlq,zqo. x  1.. 1. 7D x 1  0. x x 1 2. wfh. x  1.. faj’oAoF. 1 x 1.

(6) 0D .shr6f y y 1 2. y 2  1 gmqjkda[. [+ d eoq f g,njv. W c]h;cdhlq,zqo. x 2  0 le]a[m5dM7jk0v’ x. r6f0v’. x [+gmqjkda[ W x 1 x [+ deoqfD 2 x 1. mujgIaf.sh. wfh. y  1.. faj’oAoF. x 2  1  1D. faj’oAo. y  1.. 7D 9kd. 2. y 2 1  0. mujgxao9eo;o9y’F wfh. le]a[m5dM7jk0v’. x. mujgxao9eo;o9y’ c]t [+,u7jk. !D@ dkoIvo lk,kfIvoleo;oxqddt8y3fp.-hs^da gdo8+wxoUmujgvUo;jk s^da gdorNo4ko0v’leo;o sâdgdorNo4ko 0v’leo;oxqddt8y 4hk;jk. A, B. c]t. K. gxaorts5rqf muj. B0. c]t. K 0. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. xqddt8yD c]h; 9twfh. AK A  . BK B. 8q;1jk’G 9qj’Ivo. [qfcdhG. dD. 2x  6 . 7 x  21. dD. 2 x  6 2( x  3) 2   , 7 x  21 7( x  3) 7. 0D. 5 x  10 5( x  2) 5   , x  2 2 x  x  2 ( x  2)( x  1) x  1. 0D. 5 x  10 . x2  x  2. x  3.. 2. s^n. x  1..

(7) [qfg/ydsaf !D! 9qj’-vd7jk0v’8q;xjPo mujgIaf.shleo;oxqddt8y[+deoqfD 8 x2 3 x $D x (D 2 3 x 5 !WD x 2. !D. 5 x3 &D y 3y  6 )D 22 z 3z  2 !!D 25 y y 9. 2 x *D 2 x x2 _D 1 2 ( x  2) !@D 2 7 y  2y  3. @D. #D. !D@ 9qj’IvoD !D. @D. 9u 5 3u 6. $D. ax  ay bx  by. &D. 20m 4 n6 15m5 n 2. *D. (D. 4 x  8x 6 x  12. )D. 1 1  x y 2 xy. _D. !WD. 2. 1  x 2 1 2  x x. #D. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 14 x3 y 21xy 2. 3x3 6 x5. ax 2  a 2 x bx  ab ab 1 1  a b. !!D. x 2  3x  28 x 2  3x  4. !@D. a 2  b2 a 2  2ab  b 2. !$D. 6a 2  2a  28 a 2  2a. !&D. 15 y 3 ( x  9)3 5 y 4 ( x  9) 2. !)D. 2 x 2  10 x 4 x  20. @!D. x2  9 x2  6x  9. !#D. x 2  2 xy  y 2 x 2  xy  2 y 2. !*D. 2 x3  14 x 2 6 x 2  42 x. !(D. x2  2x 2x  4. !_D. x2  6 x  8 3x 2  12 x. @WD. x2  5x  6 2 x2  6 x. 3.

(8) @@D. a 2  16b 2 4ab  16b 2. @#D. x2  4 x2  4x  4. @$D. 4 x2  9 y 2 4 x 2 y  6 xy 2. @&D. u 2  uv  2u  2v u 2  2uv  v 2. @*D. 6 x 3  28 x 2  10 x 12 x 3  4 x 2. @(D. 12 x 3  78 x 2  42 x 16 x 4  8 x 3. @)D. x3  8 x2  4. @_D. y 3  27 2 y 3  6 y 2  18 y. #WD. x 2  10 x  21 x 2  5 x  14. #!D xudkpF ]qf9ad,u]k7k !W ]hkodu[D xuoU]k7k !@ ]hkodu[D 4k,;jk,ao0Bogmqjk.f gxug-aoL C n  C0  100 F C0. dD gxug-aoxjPocx’ c[[.f I6h;jk]k7kgdqjk0v’]qfgd’ ]k7k.\j C n  85 ]hkodu[L 0D 4k,;jk. C0. 8hv’,u7jkgmqjk.f. Cn  C0  100 C0. C 0  90. ]hkodu[ c]t. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. #@D vu’8k, l6f. 9bj’[+deoqfL. 7D [af ATM 0v’mtok7ko g,njvdjvo[+wfhg-qjkF fP;oU8hv’wfhg-qjk 5 000 du[ 8+ gfnvoD 4k,;jk,ao0Bogmqjk.fgxug-aoL ’D lqj’0=h7;k,zjko3m]tla[,n4ng,njvdjvo 500 du[}7A’F fP;oUlqj’[+glpg’yoD 4k,;jk ,ao]q’gmqjk.fgxug-aoL. 4.

(9) [qfmu @ dko76o c]t dkoskoleo;oxqddt8y dyf9tde 2 4  . 3 5. 9qj’7yfw]jG !D. @D. 2 3 4   . 3 4 5. #D. 2 4  . 3 5. .97;k, lk,kfoe.-hsâddko dko76o c]t dkosko9eo;oxqddt8y0hk’gmy’oU da[leo;o xqddt8ywfhfaj’oUG @D! sâddko76oleo;oxqddt8y A B. c]t. C D. gxaoleo;oxqddt8yF 3fpmuj. B  0, D  0 F. A C AC   . B D BD. 8q;1jk’G 9qj’76o c]t Ivoleo;o8+wxoUD. c]h;wfh. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 4hk;jk. 5 6x  3 . 3x 7  20 x 2 . 3 5x. @D. 2 x 10 y 2  . y 8x 3. $D. 27 x 3 4y .  2 12 y 15 x 3. &D. 6 x  18 x  2 . 2 x x 9. *D. x 2  x  12 x2 .  x x4. (D. t 2  3t  4 t 2  5t  4  . t 2  3t  4 t 2  5t  4. )D. y 2  3y  2 y 2  6 y  5  . y2  4y  4 y2  2y 1. !D. 6 x2 . @D. 2 x 10 y 2 2 x 5 y  2 y 2y y  3    2  2. 2 5 y 8x 5 y 2x  4x 4x 2x. #D. 4  5 x 2 7 4 28  20 x  2     , 1 x 1 x 5 x3 5x  x. !D #D. 2. [qfcdhG. 7. 5 3x3. . 2. 2  3x 2 5 2 5 10  2    , 1 3x  x 1 x x. 7. 5. x  0.. x  0, y  0. x  0..

(10) $D. 27 x 3 4 y 3  3  3  x3 4y 3 1 3       , x  0, y  0. 2 3 3 12 y 15 x 3 4 y  y 35 x y 5 5y. &D. 6 x  18 x 6( x  3) x 6 1 6  2      , 2 ( x  3)( x  3) x x  3 x( x  3) x x 9 x x. x  0, x  3.. *D. x 2  x  12 x 2 ( x  4)( x  3) x  x x  3 x       x( x  3), 1 1 x x4 x x4 x  0, x  4. t 2  3t  4 t 2  5t  4 (t  4)(t  1) (t  1)(t  4) (t  1) 2     , t  4. t 2  3t  4 t 2  5t  4 (t  4)(t  1) (t  1)(t  4) (t  1) 2. )D. y 2  3 y  2 y 2  6 y  5 ( y  1)( y  2) ( y  1)( y  5) y  5     , y 2  4 y  4 y 2  2 y  1 ( y  2)( y  2) ( y  1)( y  1) y  2. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. (D. y  1, y  2.. @D@ sâddkosko 0v’leo;oxqddt8yD 4hk;jk. A c]t C gxaoleo;oxqddt8yF B D. 3fpmuj. B  0 , C  0, D  0 F. A C A D AD     . B D B C BD. c]h;wfh. 8q;1jk’G 9qj’skoleo;o8+wxoUG !D $D. 35  7. x2 (t  4) 2 t 2  16 .  4t 16t 2. @D. 12 8  . x4 x2. #D. ( y  1) 2 y 2  1 .  3y 15 y. &D. 1 x x2 1 .  y2  y y2  2y 1. [qfcdhG !D. 35 75 1 5 7  2   2 . 2 7 x x x. @D. x2 12 8 3 4 3 1 3  2  2 2  2  2. 4 24 x 2 2x x x x x 6. x0.

(11) #D. ( y  1) 2 y 2  1 ( y  1)( y  1) 5  3y 5 5( y  1) y 1 .       3y 15 y 3y ( y  1)( y  1) 1 y 1 y 1. y  0, y  1. $D. (t  4) 2 t 2  16 (t  4) 2 (4t ) 2 4t (t  4) t  4 4t .       2 4t 4t (t  4)(t  4) 1 t4 t4 16t. t  0, t  4. &D. 1 x ( y  1) 2 x2 1  ( x  1) 1 y 1 y 1 .       2 2 y x 1 y ( x  1) y  y y  2 y  1 y ( y  1) ( x  1)( x  1) x  1, y  1. [qfg/ydsaf 6 . 5x 4 3 11x  . 121x a b  . b a 3x 5y3 .  40 y 3 6 x 2. !D 15 x 2 . @D. #D. $D. &D (D _D. !!D !#D !&D !(D !_D. *D )D. 3 x 2 8x   . 3 x 4 2x  3 3 x   . 2 2x  3 x x 8 x  12 x  1  . x  1 2x  3 1 x 2  3x  2  2 . x2 x  4x  3 t2 s .  2 2t s y y  3x  . 3x  y x. 10  9y5. 3y 2  5y2. 2 10 y 5 c  . c b 4x 6y2 .  24 y 3 7 x 3. !WD. 2 y 3 15 3    . y2 5 y5 y4. !@D. 3y  5 y3 .  3y  5 y 12 x  15 3 x  3  . 4x  5 x 1 2 y 2  3y  4 .  y4 y 1. !$D !*D !)D @WD 7. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. @D! 9qj’76oleo;o]5j,oUD. wv w5 .  w vw a by  2bx  . 2cx  cy ab.

(12) @!D @#D. t 2  3t  2 t 2  5t  6 .  t2  t  2 t2  t  6 4a 2  4a  8 a 2  2a  15 .  a 2  4a  21 4a 2  8a  12. @@D @$D. s 2  3s  10 s 2  2 s  8 .  s 2  3s  10 s 2  2 s  8 3c 2  6c  3 2c 2  12c  18 .  2c 2  10c  12 3c 2  9c  6. @D@ 9qj’skoleo;o8+wxoUD @(D @_D #!D ##D #&D #(D #_D $!D $#D $&D $(D. 45 9  . x2 x b b  . x x2 x  y x2  y2 .  n n 20 5  . n n 3x 4 x  . 4 y 3y. @*D @)D #WD #@D #$D. 12 1  . x3 x2 x x2  . 7 9 3x  3 y x  y  . nm nm n n  . 5 10 5x 6 y  . 6 y 5x. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. @&D. 5( x  1) 2 x 2  1 #*D .  x 5x 2 2 2 #)D ( s  3)  s  9 . 5 5s 2 $WD 32  y  2y  9 . x  3x x  x  6 2 2 $@D d  c  c 2  2cd  d 2 . d  c c  2cd  d x 2  2 x  15 x 2  2 x  15 $$D 2 .  x  8 x  15 x 2  4 x  15 2 2 $*D y 2  2 y  y  4 y  4 . 3y 6 y  3y $)D ab  ac  bd2  cd2 . ab  bc a  c ]ts;jk’goNvmuj0v’I6[9t85]af0t|kf s da[goNvmuj0v’;q’,qoco[.o. ( x  3) 2 x 2  9 .  2x x (t  5) 2 t 2  25 .  3t 3t 2 y y2  4 .  x 2  2x x 2  x  2 a  b a 2  2ab  b 2 .  cd c2  d 2 x 2  4 x  3 x 2  5x  6 .  x 2  4 x  3 x 2  5x  6 y 2  5 y 2 y 2  13 y  15 .  3y 2  y 3y3  y 2 3 x  3a cx  ac  . cy  bc dy  bd. $_D vaf8klj;o c,jo s 2   (s / 2) 2 D 9jq’Ivovaf8klj;ofaj’djk;D &WD vaf8klj;o ]ts;jk’[=]y,kf0v’I6[dhvolkd0t|kf c,jo s 3  (4 / 3) ( s / 2) 3 D 9qj’Ivovaf8klj;ofaj’djk;D 8. s. da[|j;p,qoco[.o.

(13) @D# 9qj’7yfcdh[qfg]d]5j,oUD !D oadIPo7qo|bj’Ivog]dlj;o. 16 1  64 4. phvo0hk * vvdD. g9Qk7yfwfh[+=;jk,ug]dlj;o.f lk,kfIvowfh7nco;oUvudL @D Ivo. 3a 3  4a 4. phvo Ivo. a. vvdD gxaospa’[+cmo. a. fh;p !L 4h k ;j k cmo. a. fh;p W wfh[+L phvospa’L #D Ivo. 11  4 11  1 4 1. 3fp 0hk $ vvdD 4nd[+L phvospa’L. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. $Dvkd-yc,f (Archimedes 278-212djvo7Dl) |bj’.ooad7toyflkf-k;gdiadmuj,u-nj lP’3fj’fa’8ts^vfdkogxaoz6h7Qo7;hkwfhl6f[=]y,kfF l6fgoNvmujma’\qf0v’I6[m+ c]t |j;p,qoD zqo’ko|bj’0v’]k;muj|hklqo.9mujl5fd=c,jovaf8klj;o]ts;jk’[=]y,kf da[[=]y,kf c]t goNvmujda[goNvmju 0v’I6[m+da[|j;p,qoco[.o0v’,aogmqjkdaoD 4   r 3. 3 VC   r 2 h.. .sh [=]y,kf0v’|j;p,qo VS r. [=]y,kf0v’m+ goNvmuj0v’|j;p,qo goNvmuj0v’m+. h. 9qj’oe.-hl6f0hk’gmy’-vdlyj’muj]k;-vdwfh XcotoeG -vd .sh4n r gxao8q;xjPocxj’D &D 9qj’7eo;oD dD. 1 2 3 99    . 2 3 4 100. 0D. 1 2 3 999    . 2 3 4 1000. 7D. 1 2 3 999     0 2 3 4 1000. [+L phvospa’L 9. h. AS  4 r 2 .. AC  2 rh  2 r 2 .. 8k,7jk0v’. r. c]h;.

(14) [qfmu # dkoskorts5rqf dyf9tde !D 9qj’-vdzqosko c]t lj;oglfg,njvsko !* .sh #D @D 9q’j 0Po9eo;o8nj,.lj[jvo9Eg,af 16 = ….  3 + .…. .97;k, #D! dkoskorts5rqfc[[mqj;wx. 8q;1jk’!G sko. B. gxaorts5rqfD g,njv. 2 x 2  5 x  12. .sh. A  BQ  R. gryjo;jk. A. x  4.. 2x 2.  5 x  12. 2x 2.  8x  3x  12. x. 4.  2x  3.   3x  12 0 0. zqosko c,jo glf c,jo. Q( x)  2 x  3. R  0.. 10. sko.sh. B. wfh. Q. c]t. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. .sh A c]t glf R..

(15) 8q;1jk’ @G. sko. .sh. 3  7x  6x 2. 3 x  1.. 6 x2. 7 x 3. 3x. 1. 6 x2. 2 x 9 x 3. 2 x 3. .  9 x 3 0 6. zqosko c,jo glf c,jo. Q( x)  2 x  3. R  6.. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. #D@ dkoskoc[[-yogmmyd (Synthetic Division) 8q;1jk’!G 9qj’skoD dD 0D 7D. 2 x 2  5 x  12 x3  8 3  7x  6x 2. dD ;ymusko. .sh x  4. .sh x  2. .sh 3x  1.. 2 x 2  5 x  12. lextlyf0v’8q;8A’sko0Po c8j-hkpsk0;k c]t 9af]P’ 8k,de]a’s^kpsk|hvpD. zqoskoc,jo glfc,jo. 7jk0v’ x mujgIaf.sh 8q;skogmqjk WD. .sh. x  4.. 2. 5 12 8 12. 2 3. 0. Q( x)  2 x  3. R  0.. 11. 4.

(16) x3  8. .sh. x  2.. lextlyf0v’8q;8A’sko0Po c8j-hkpsk0;k c]t 9af]P’ 8k,de]a’s^kpsk|hvpD. zqoskoc,jo glfc,jo 7D ;ymusko. 1. 0 0 8 2 4 8. 1 2 4. 2. 7jk0v’ x mujgIaf.sh 8q;skogmqjk WD. 0. Q( x)  x 2  2 x  4. R  0.. 3  7 x  6 x 2 .sh 3 x  1  3( x  1 / 3)  1/ 3 6 7 3 6 9 6 2 3. zqoskoc,jo glfc,jo. Q( x)  2 x  3. R  6.. [qfg/ydsaf. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 0D ;ymusko. #D! 9qj’xt8y[afdkoskoma’ @ c[[D @D (2 x 2  x  6) /( x  2). !D (3x 2  5 x  2) /( x  2). #D (2 y 3  5 y 2  y  6) /( y  2). $D ( x 3  5 x 2  x  10) /( x  2). &D (3x 2  11x  1) /( x  4). *D (2 x 2  3x  4) /( x  3). )D (6m 2  5m  6) /(3m  2). (D (8 x 2  14 x  3) /(2 x  3). _D (6 y 2  y  12) /(2 y  3). !WD (6 x 2  11x  12) /(3x  2). !!D ( x 2  4) /( x  2). !@D ( y 2  9) /( y  3). #D@ 9qj’xt8y[afdkoskoc[[mqj;wxD !#D (12 x 2  11x  2) /(3x  2). !$D (9 x 4  2  6 x  x 2 ) /(3x  1). !&D (4 x 4  10 x  9 x 2  10) /(2 x  3). !*D (8 x 2  7  13x  24 x 4 ) /(3x  5  6 x 2 ). !(D (16 x  5 x 3  8  6 x 4  8 x 2 ) /(2 x  4  3x 2 ). !)D (9 x 3  x  2 x 5  9 x 3  2  x) /(2  x 2  3x). !_D (12 x 2  19 x 3  4 x  3  12 x 5 ) /(4 x 2  1). @WD ( x 4  x 3  4 x 2  7 x  2) /( x 2  x  1). 12.

(17) [qfmu $ dko[;d c]t dko]q[leo;oxqddt8y dyf9tde 9qj’7yfw]j 1 2   5 5. 9 4   7 7. 1 2 3    4 5 10. 9 4   7 3. .97;k, $D! dko[;d c]t dko]q[0v’leo;oxqddt8ymuj,ur6ffP;dao. 8q;1jk’G dD 0D. c]t. A C AC   B B B. D. a ba aba b    . a 1 a 1 a 1 a 1. 1 2x  1 1 2x  1  2   . ( x  1)( x  1) x  1 ( x  1)( x  1) ( x  1)( x  1) . 1  (2 x  1) . ( x  1)( x  1). . 1  2x  1 . ( x  1)( x  1). .  2x . ( x  1)( x  1). $D@ mt;u76oIj;,|hvpl5f XmI|l? LCM 8q;1jk’G. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. A C AC   B B B. -vd mI|l 0v’ dD 18 x 3 ; 15 x; 10 x 2 . 0D 2a 2  4a , 3a 2  3a  18 ,. 13. a 2  6a  9..

(18) ;ymu-vdG dD. 18 x 3  2  3 2  x 3 . 15 x  3  5  x.. mI|l c,joG 0D. 10 x 2  2  5  x 2 . 2  3 2  5  x 3  90 x 3 . 2a 2  4a , 3a 2  3a  18 , a 2  6a  9. 2a 2  4a  2a (a  2). 3a 2  3a  18  3(a 2  a  6)  3(a  2)(a  3). a 2  6a  9  (a  3) 2 .. mI|l c,joG. 6a (a  2)(a  3) 2 .. A C AD  BC   . B D BD A C AD  BC   . B D BD. l6f[;d l6f]q[ 8q;1jk’G [qfcdhG. 9qj’7yfw]j. 2 3 2  2  2 . a  2a a  3a a  5a  6 2. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. $D# dko[;d c]t ]q[leo;oxqddt8ymuj,ur6f8jk’dao. 2 3 2 2 3 2    2  2  a  2a a  3a a  5a  6 a(a  2) a (a  3) (a  2)(a  3) 2. 2(a  3) 3(a  2) 2a    a(a  2)(a  3) a(a  2)(a  3) a(a  2)(a  3) 2(a  3)  3(a  2)  2a a(a  2)(a  3). 14. . 2a  6  3a  6  2a . a (a  2)(a  3). . a . a (a  2)(a  3). . 1 . (a  2)(a  3).

(19) $D$ leo;oxqddt8yla[lqo .odhv’4jkpI6[F g-yj’. d. f. w]ptc8jcdh;8kglnvXg]o? sk 95fl5, c,jo. c,jow]ptc8j;af45 sk g]o c]t. 1. f . 1 1  d a. F. c,jo w]ptc8jg’qk sk g]oD. a. a. f. d. 95fl5,. 1. f . 1 1  d a. g]o. g’qk. c,jo8q;1jk’ “leo;oxqddt8yla[lqo” ! c,jo 9eo;or6f c]t. 8q;1jk’ !D dD. 1 1  d a. c,jo r6fD. 9qj’Ivo 2. 3 10 .. 4 1 5. M. 0D. 4. 1 3.. 5 3 6. 7D. f . ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. ;af45. 1 . 1 1  d a. gvUo;jk “7jkltg]jp Ik3,oyd” (harmonic mean) 0v’. [qfcdhG dD ;ymu !G. ’D. M . a. c]t. 3 20  3 10  10  17  5  17  1  17 . 4 4  5 10 9 2 9 18 1 5 5. 2. 15. 2 . 1 1  a b b..

(20) ;ymu @G r6fIj;,. 3 4 ( , )  10. 10 5. 3 10  10  4 10 1 5. 2. 0D ;ymu !G ;ymu @G. 20  3 10  17 . 8  10 18 10. 1 12  1 3  3  11  6  11  2  22 . 5 5  18 3 23 1 23 23 3 6 6. 4. 1 5 ( , )  6. 3 6. r6fIj;,. s^n. LCM (3,6)  6.. 1 12  1 6 3   3  22 . 5 6 5  18 23 3 6 6. 7D ;ymu !G ;ymu @G. ;ymu @G. 1 5 ( , )  ad d a. r6fIj;, f . ’D ;ymu !G. 1 1 ad .   1 1 ad ad  d a ad. f . 1 1 1  d a. M . 2 1 1  a b. . s^n. LCM (d , a)  ad .. 1 ad ad .   ad a  d a  d ad. . 2 2ab .  ba ab ab. r6fIj;,. 1 1 ( , )  ab. s^n L C M (a, b)  ab. a b. M . 2 1 1  a b. . 2 2ab ab .   ab b  a a  b ab. 16. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 4.

(21) 8q;1jk’ @D 9qj’Ivo dD. 2 ab . 1 1  a b. 0D. 1 1  y x . y yx. 7D. 1 2  3 y . 5 1  6 y2. ’D. 1 3  2 x . 7 5  4 x2. 9D. y 1 x . y2 1 x2. dD. 2 2 a  b  a  b  2  ab  2ab . 1 1 a  b a  b a  b ( a  b) 2  a b ab. 0D. 1 1 x y  x  y x  y ( x  y) 2 xy y x  .    y y xy y xy 2 x y yx. 7D. 1 2 y6  2 6y 3 y ( y  6) 3 y 3y   2 2 . 5 1 6y 5y  6 5y2  6  6 y2 6y2. ’D. x6 1 3  2 2 x( x  6) 2 x  4x  2x .  2 2 2 7 5 x x  7 20 x  7 20 4  4 x2 4x 2. 9D. y yx 1 2 x( y  x) x x x .   2 2x 2  2 ( y  x)( y  x) x  y y y x x 1 x2 x2. 17. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. [qfcdhG.

(22) [qfg/ydsaf $D! 9qj’7yfw]jD x 2x  . x2 x2 1 2  . n n a2 b2  . ab a b. !D $D (D. @D &D )D. a b  . a b a b 5a 4a  b  . ab ab x x2 1 .   3x 2 3x 3x 3. #D *D _D. xa ax  . xa xa 5y 3  . xy x x( x  1) 1  . 2 x  x x 1. $D@ 9jq’-vd mI|l a , a 2  a.. @D. t 2  t , t 2.. #D. 6u 2 , 3u (u  3).. $D. a 2  a , 2a  2.. &D. y 2  yx , x 2  xy.. *D. 2t  6 , t 2  9 , t 3  3t 2 .. (D. x  1, x 2  2 x  1, 2 x.. )D. 3  2a , 3  2 a , 9  4a 2 .. _D. 2( x  y ) , 3( x  y ) 2 , 4( x  y ) 2 .. !WD. p 2  q 2 , 2( p  q) , 3( p  q).. $D# 9jq’7yfw]j !D 1  #D. 1 . a 1. @D 1 . 1 1 1 .   2 x  y x  y x  y2. $D. x . x2. a a 2 . a b ab. x3 2x  2 . 2 x  9 x  2x 1  2 . )D 2 1 x  x  12 x  2 x  3 2(5 x  6)  . !WD 1  2 1 x 2 x  x  3 5 x(2 x  3) !@D 1 3  32 2  2 2 . 12 x 8x y 3 xy. a2 a a   a 1 1 a 1 a (D 2 a  1  1 . a a2 a3 3  2 . _D 2 2 x  2 x  1 x  3x  4 !!D 23 2  1  1 3 . 6uv 12v 9u v. &D. !#D. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. !D. *D. 4t  3 3 2t  1   . 18t 3 4t 6t 2. !$D 18. 3y  8 2 y 1 5  .  8y 4y2 y3.

(23) !&D. t 1  1. t 1. !(D. 5. !_D. x 1 . x3. !*D. 2. !)D. 1 2 3 . y2 y2. 2 1 .  2 4( x  5)( x  5) 3( x  5). @WD. 1 3  . 6( x  7)( x  7) 8( x  7) 2. @!D. 3 1  2 . x  4 x  4x  4. @@D. 2 1  2 . x  6x  9 x  9. @#D. 2 1 2x   2 . x3 x3 x 9. @$D. 2x 1 1 .   2 x y x y x y. @&D. x 1 1  2  2 . x  x  2 x  5 x  14 x  8 x  7. @*D. m2 1 1   . 2 m  2m  1 3m  3 6. @)D. 1 1 1 .  2  3m(m  1) m  2m  1 5m 2. #WD. x xy  3 . 2 x  xy  y x  y3. a a  . a 1 a 1. 2. 2. 2. 2. 1 1 1  2  2 . 3x  3x 4 x 3x  6 x  3. @_D. xy 2 y .  2 3 3 x y x  x y2. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. @(D. 2. 2. $D$ 9qj’Ivo !D. *D. !!D. 4 5 . 3 2 10 1 1  6 5. 1 1  4 3 3. b a. a b b a. @D. (D. !@D. 3 8. 1 2 2 1 a b. 1 a b 1. b 2. a b 3 a. #D. )D. !#D. 3 2. 1 2 3 1 x y . 2 x y. 5. 3 1 2  a b. 19. $D. _D. .. !$D. 1 4. 1 6 5 1 a b . 1 a 2b. 3. 5 3 6  a b. .. 1 1  4 5. &D 1 1  2 3 1 5 a . !WD 2 10  a. !&D. 7 3 5  a 2a. ..

(24) x 3 1  x 2x. .. @!D. 2 3  x y . 6 xy. @&D. x 1  y x y . 2 x y. @)D. 1 m2 . 1 m 2. 1. #!D. a 2 b . a2 4 b2. #$D. ab a ab . ab b ab. !(D. @@D. 5 1 3  a a2. .. !)D. 8 4 2  n n2. 3 2  2x 3 y . 1 xy. @#D. @*D. 2 1  ab a . ab ab. @_D. 2 1 n . 1 1  n2 n. #@D. x2 4y2 . x 1 2y. #&D. c 2 d 1 . c 2 d 1. .. !_D. x y  y x . x y x y. @WD. 1 x y . 1 2  x y. @$D. ab a b . 1 1  a b. @(D. 2 1  3 n . 3 1  4 n2. #WD. 1 1  mn mn. 1 2 m  n2. ##D. a 1 ab . b 1 ab. #*D. 1 x 2x  4 . 1 2x  x2. 1. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. !*D. 3 x y . 1 2  x y. #(D 9jq’-vd;jk7jkltg]jpIk3,oyd0v’ * c]t !@ c,jo )D 7eg;Qk “Ik3,oyd” “harmonic” ,aodjP;raoda[lP’fqo8iug-ajoG lkp7uu8k (Quitar) pk; !@ sq;|j;pF g;]kfufgIqk9a[.shlAo]q’Ivf ) sq;|j;p gIqk9twfhlP’mu & “the fifth” g-yj’gxao Ik3,oydda[lP’8Qo8= !@ sq;|j;pD 4hkgIqk9a[.shlAo]q’Ivf * sq;|j;p gIqk9twfh lP’mu ) “Octave” g-yj’gxaoIk3,oyd g-ajofP;daoda[I6[dhvolkdg-yj’,u * |hkF ) 9v, c]t !@ ]jP, 4ndgvUo;jk “dhvolkd Ik3,oyd”. 20.

(25) #)D 9qj’7yfw]j7jkltg]jp Ik3,oyd #_D 7jk.dh7P’0v’  c,jo da[7jk0v’. 3 . 1 1 1   a b c. M  1. 3. 1 7 16   3,141592 654 D. D 9qj’7yfw]jfh;pg7njv’7yfg]d c]h;xP[mP[. $WD gryjo.shG 2. 1 2. 1. ,. c]t. 1. 2. 2. 1 2. 1. D. 1. 2. 1. 2. 2. dD 9jq’7eo;o.shwfh;jk # leo;ofaj’djk;,u7jk. 1 2. 2 5 12 , , 5 12 29. 8k,]efa[D. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 1. 0D gIqklk,kf0Poleo;ofaj’djk; 1jk’[+lUol5fD ,ao,u-nj;jk “g]dlj;o8+gonjv’” g-yj’ltg|u3fpoad7toyflkf-k;va’dyf “9vo ;vo]yl” John Wallis .oxu !*_&D .shI6h;jk7jk0v’g]dlj;o8+gonjv’,uc[[0v’,aoF I6h;jk7jk8+wxc,jo 7jk8+wxvudc,jospa’L 7D 4hkgIqk[;d ! g0Qkda[7jkfaj’djk; gIqk9twfh7jk.dh7P’0v’ faj’djk; fh;pg7njv’7yfg]d c]h;xP[mP[da[ 7jk0v’ 2 D $!D 9qj’Ivo 1 1 2 4. 1. , 2. 1. c]t. 1 4 8. 1. D. 1. 2 4. 1 8. 21. 1 16. 2D. 29 D 70. 4k,;jk. 9qj’-vd7jk.

(26) [qfmu & dkocdhlq,zqoxqddt8yy dyf9tde gInv9ad]e|bj’4hk8k,o=hkmuj[+ws^9twxwfh 5 km 8+-qj;3,’D gInv9ad]eoU4nd.-hh8k,]eo=hk csj’|bj’mujws^3fp.-hw]ptg;]kvaofP;dao0k]jv’wxwfh 9 km lj;o0k0BowxwfhrP’c8j 3 km D .sh c c,jo7;k,w;0v’o=hkmujws^F 9qj’0Polq,zqogrnjv-vdsk c.. .97;k, leo;oxqddt8y. A C  ( B  0, D  0) B D. g,njv. AD  BC.. 8q;1jk’G 9qj’cdhlq,zqo8+wxoU.od5j,9eo;o9y’G dD [qfcdhG dD. 1 2  . x 1 x 1 2  x 1 x. 0D g,njv. x 1 2  . x 1 x. 7D. 2 3 3   . x  1 3x  1 x.  x  1  0, x  0   x  2( x  1). 9kdoUwfhG  x  1  0, x  0   x  2x  2  x  1  0, x  0  x2  x  2.. 0D rkp.8hg’njvow0. x 1 2  x 1 x x( x  1)  2( x  1). x  1  0, x  0;. g,njv. x2  x  2  0   (1)2  4(1)(2)  7  0. faj’oAoF lq,zqo [+,u.9zqomujgxao9eo;o9y’D 22. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. oUgxaosâddkorNo4kogrnjvcdhlq,zqoxqddt8yD.

(27) 7D. 2 3 3   . x  1 3x  1 x. g0fdeoqf. 1 D    {1; ; 0}. 3. 2(3 x  1)  3( x  1) 3  ( x  1)(3x  1) x 9x 1 3  ( x  1)(3 x  1) x 3( x  1)(3 x  1)  x(9 x  1) 9 x2  6 x  3  9 x2  x 5x  3 3 x  D 5. x. 3 . 5. [qfg/ydsaf 9qj’cdhlq,zqo.odj5, &D!D. 1  2. x 1. D. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. faj’oAo. &D@D. 4  2. 3 x. &D#D. 5  1 14  0. 3x  5. &D$D. x2  2 x .  2 x 4 x2. &D&D. 5 33  x  2 . x  3 x  6x  9. &D*D. 3x 2  x 5  x 1  .   24 10 40 15. &D(D. 2x 3  x 2  x 1    . 10 14 5 2. &D)D. 5t  22 11 5  2   0. t  6t  9 t  3t t. &D_D. 2x  3 x  6  . 6 x  1 3x  1. &D!WD. 5. 1 1 1   2 . 3x  2 2 x  3 6 x  13x  6. &D!!D. 3x 6   3. 2x x2. &D!@D. &D!#D. 1 1 4   2 . 2x  1 2x  1 4x  1. &D!$D. &D!&D. z 1 z 1 z2    . 5 20 z  2 4. &D!*D 23. 2. 2x 6  . 3 x x 3. 1 x 1  1 . 3x  4 3x  4 2 1 2 1    . x  2 x x 3( x  2).

(28) &D!_D &D@WD &D@@D &D@$D &D@*D &D@)D &D#WD &D#@D &D#$D &D#*D &D#)D &D$WD &D$@D &D$$D &D$*F &D$)D &D&WD. 2 1 1 2    0. &D!)D x  2  x  4  32x  1 . x 1 x  2 x  3 2x  3 x  2 2x  x  6 1 3 1  2  2 . 2 c  c  2 c  2c  3 c  5c  6 5t  22 11 5  .  2 &D@!D 1  3  0. 2 x  5 x  15 t  6t  9 t  3t t 4 8  . &D@#D 2  3 . x 1  x x 1 x 1 1 2  . &D@&D x  1  x  4 . x  1 2x  2 x 3 x3 x2 x2  . &D@(D x  2  x  3 . x  4 x 1 x3 x2 x5 x 4  . &D@_D 4 x  1  6 x  2 . x4 x5 2 x  1 3x  2 3x  1 6x  . &D#!D 2 2  21 . x  1 2x  2 x  x x 1 4 2  2 . &D##D 2 3  2 21 . 2 x  x x 1 t  3t t  3t 2 10  2 . &D#&D 2 1  2 . 2 t  2t t  2t r  r r 1 1 3 . &D#(D x  6  1.  2 x r  r r 1 3 x   4. &D#_D 2 y  1  5 . x 3y 3 2 7 2y   . &D$!D 1  2  212 . 3y 3 x3 x 9 2 4 1  2 . &D$#D 1  1  2 4 . x2 x 4 x2 3 x 4 3 1 5   2 . &D$&D 3  2 11  2. x 1 8 x 1 x  5 x  25 2 4  2  2. &D$(D 2  24  2. x 1 x 1 x 1 x 1 3 12 3 x5 x7  2  1. &D$_D 2  2 .  2 x2 x 4 x  3x  2 x  5 x  6 x  4 x  3 4 x5 x 1  2  2 . 2 x  1 x  x  2 x  3x  2. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. &D!(D. 24.

(29) [qfmu * vaf8klj;o c]t vaf8klj;orq;raoD dyf9tde .oshv’IPooU ,uG Xd? vaf8klj;o9eo;ooadIPopy’ 8+ 9eo;ooadIPo-kpgmqjk.fL X0? vaf8klj;o9eo;ooadIPopy’ 8+ 9eo;ooadIPoma’\qfgmqjk.fL X7? vaf8klj;o9eo;ooadIPo-kp 8+ 9eo;ooadIPoma’\qfgmqjk.fL. .97;k, *D! oypk,vaf8klj;o. 8q;1jk’ !D ,kf8klj;o0v’czomuj. 1 50 000. s^n. 1: 50 000. c]t 8q;9y’D \kp7;k,;jk.oczomuj];’pk;. c,jovaf8klj;o]ts;jk’ czomuj. 2 cm. gmqjk 8q;9y’ 1 km D. 8q;1jk’ @D vkp5r+ $& c]t ]6d !!D vaf8klj;ovkp5r+8+]6dc,jo 8+r+c,jo. 11 45. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. vaf8klj;oc,jog]dlj;o0v’dkoxP[mP[lyj’0v’ @ 1jk’.ofhkoxt]y,ko s^n 0t|kfmuj,usq;|j;pfP;daoD. 45 11. s^n 11 : 45.. s^n. 45 : 11 c]t. vkp5]6d. 8q;1jk’ #D w]ptsjk’c8j3]dsk8kg;ao,u 149,6 ]hkody3]c,af c]t ! xucl’gmqjkda[ 9 500 000 ]hkody3]c,afD vaf8klj;ow]ptsjk’c8j3]dsk8kg;ao8+ ! xucl’c,jo 149, 6  1575 108 9500 000. xucl’D. 8q;1jk’ $D 4hkgvqk3]dgxaosq;|j;p0v’[=]y,kfgIqkwfhfk;rtl5d 0,91 F fk;va’7ko fk;rtsaf 1317 F fk;glqk 762 D wfhvaf8klj;o]ts;jk’[=]y,kf0v’G fk;rtl5d8+fk;rtsaf c,joG fk;va’7ko8+fk;rtsaf c,joG 25. 0,91 91  . 1317 131700 0,15 1  . 1317 8780. 0,15 F.

(30) 762 254  . 1317 439 1 . 1317. fk;glqk8+fk;rtsafc,joG 3]d8+fk;rtsafc,joG. 8q;1jk’ &D ]aflt\u0v’f;’9ao c,joG 3 476 km. ]aflt\u0v’3]d c,joG 6 378 388 m. ]aflt\u0v’8kg;aoc,joG 1391000 km. vaf8klj;o]ts;jk’[=]y,kf0v’G f;’9ao8+8kg;aoc,joG 3]d8+8kg;aoc,joG. 3467 3 )  155 1010. 1391000 6378,388 3 ( )  946 1010. 1391000 (. 8q;1jk’ *D ok.Ijj|bj’ mD 7e lk,kffeleg]afrkp.o. 6 hF. oD ]u leg]af. 3h. c]t mD lu. ;ymucdh !G lq,,5f;jk;Pdc,jo ! sq;|j;pD dD .o 1 h mD 7e fewfh. 1 6. .o 1 h ma’lv’ fewfh. 0v’;Pd c]t mD lu wfh 1 1 46   6 4 6 4. 1 . 4. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. leg]af 4 h D dD 4hk mD 7e c]t mD lu Ij;,daofe9t.-hg;]kgmqjk.f9bj’leg]afL 0D 4hk ma’lk,7qo Ij;,daofe9t.-hg;]kgmqjk.f9bj’leg]afL. 0v’;PdD. 1 6 4   2 52 h. 46 46 6 4 0D .o 1 h mD 7e fewfh 1 0v’;PdF oD ]u 1 c]t mD lu wfh 1 . 4 6 3 1 1 1 3 4  6  4  6  3 0v’;PdD .o 1 h ma’lk,fewfh    6 3 4 6  3 4. ma’lv’Ij;,daofe.-hg;]k. t. ma’lk,Ij;,daofe.-hg;]k t. 1 6  3 4 72    1 13 h. 3  4  6  4  6  3 3  4  6  4  6  3 54 6  3 4. 26.

(31) ;ymucdh @G lq,,5f;jk;Pdc,jo ! sq;|j;pD ;k’ t gxaog;]kmuj-vdD 1 0v’;Pd c]t mD lu f=kwfh 1 0v’;PdD 6 4 1 1 1 h ma’lv’fewfh  0v’;PdD 6 4 1 1 wfhlq,zqo (  ) t  1. 4 6 24 10 t  24  t   2 52 h. cdhlq,zqo 10 1 1 h mD 7e fewfh 0v’;PdF oD ]u fewfh 1 c]t mD lu fewfh 1 . 6 3 4 1 1 1 1 h ma’lk,fewfh   0v’;PdD 6 3 4 1 1 1 wfhlq,zqo (   ) t  1. 4 3 6 54 72 cdhlq,zqo t 1 t   1 13 h. 54 72. .o. 0D .o .o. \kpgsfG { 4hk. a, b , c. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. dD .o 1 h mD 7e fewfh. gxaog;]kleg]af;Pd’ko0v’z6hmu 1, 2 , 3 8k,]efa[D. { 4hk];,daolv’7qo9tleg]af.og;]k { 4hk];,daolk,7qo9tleg]af.og;]k. t. 1. . ab . ab. 1 1  a b 1 abc t  . 1 1 1 bc  ac  ab   a b c. 8q;1jk’ (D Xoe.-h8q;1jk’ *? dHvdoEmu ! c]t @ lk,kf.shoEg8a,4a’rkp.o 30 mn c]t 20 mn 8k,]efa[D dHvdmu # lk,kfxjvpoEvvd\qf4a’ 40 mn D 4hkw0rhv,daoma’ # dHvd9t.-hg;]kgmqjk.f oE9bj’g8a,4a’L [qfcdhG vu’8k,l6f. t. 1 abc  . 1 1 1 bc  ac  ab   a b c. 27.

(32) 1 abc  . 1 1 1 bc  ac  ab   a b c 1 24000 240 120 t     17 17 mn. 1 1 1 800  1200  600 14 7   30 20 40. wfh. t. *D@ oypk,vaf8krq;rao gxaovaf8krq;rao 8+ \kp7;k,;jkG A, B, C. 3, 4, 5. d=8+g,njv. A: B : C  3: 4 : 5.. 3 3 1 A 3 A 3 B 4 A  ,  ,  ,    . B 4 C 5 C 5 A  B  C 3  4  5 12 4. [qfcdhG. lj;omu !G lj;omu @G lj;omu #G. 3  9 000. 357 5 45 000   15 000. 357 7 45 000   21 000. 357 45 000 . 8q;1jk’ @D c[j’ @$W_ gxao # lj;o.ovaf8krq;rao [qfcdhG ,u wfh. 3: 5 : 7.. 1 :1: 1 . 2 3. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 8q;1jk’ !D c[j’g’yo $& WWW du[ gxao # lj;o.ovaf8krq;rao. 1 1 3 6 2 :1:  : :  3: 6 : 2. 2 3 6 6 6. lj;omu !G lj;omu @G lj;omu #G. 3  657. 36 2 6 2409   1314. 3 6 2 2 2409   438. 36 2 2409 . 8q;1jk’ #D 9jq’-vdvaf8klj;orq;rao ]ts;jk’f;’9ao8+3]d8+8kg;aoD [qfcdhG vaf8klj;oc,joG 155 1010 : 946 1010 :1  155 : 946 :1010. 28.

(33) 8q;1jk’ $D vaf8klj;orq;rao ]ts;jk’goNvmuj0v’]k;GwmG;Pfok,Gdex6g9p c,joG 236 800 : 514 000 : 329556 : 181035  1 : 2,17 : 1,39 : 0,76.. [qfg/ydsaf *D! 9qj’-vdvaf8klj;o]5j,oUG @D 12 km / h : 40 km / h. #D 12 km / h : 10 m / sec . !D 8 cm :3 m. $D 2 pence :$12. &D $ 5,50 : $125. *D #W vaf 8+ ! du[D (D 10 m :1 km. )D 4 cm 2 :10 mm 2 . _D 1 g :1 kg. !WD 1 acre :1 hectare. (1 acre  4047 m 2 , 1 hectare 10 000 m 2 ).. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. *D@ g]d39fdjP;da[vaf8klj;o !D 9qj’-vd7jkdt8;’0v’dko9qd.[wrh ! .[9kdwrh ! -5, X&@ .[? .shwfh cvafD @D .ow]pt|bj’xuF xt-kdvo0v’xtgmf|bj’ ,uugfadgduf.\j # _!# WWW 7qo mP[.lj xt-k-qo @$* ]hko7qoD 4k,;jkmP[.ljxt-k-qo !WWW 7qo,ugfadgduf.\jgmqjk.fL vaf8klj;ofaj’djk; gvUo;jkvaf8kdkogduf8+ !WWW 7qoxt9exuoAoD #D .ow]pt|bj’xuF xt-kdvo0v’xtgmf|bj’ ,u7qoglp-u;yf @ !(! WWW 7qomP[.lj xt-k-qo @$* ]hko7qoD 4k,;jkmP[.ljxt-k-qo !WW WWW 7qo,u7qoglp-u;yfgmqjk.fL vaf8klj;ofaj’djk; gvUo;jkvaf8kdkoglp-u;yf8+ !WW WWW 7qoxt9exuoAoD $D oaddy]klto5dgdU7qo|bj’8hv’dko8u]6dfe]q’s^5, g-yj’]6d0k; c]t ]6dfesjk’9kd 0v[38tg[Nv’fP;dao 60 cm c]t 90 cm 8k,]efa[ c]t 95flkplkd.lj0v[38t faj’djk;0v’ma’ @ ]6dsjk’dao 120 cm D 4k,;jk]k;8hv’8u]6d0k; 8e0v[38t sjk’9kd95flkplkd0v’]6 d0k;gmq j k .fgrnj v .sh ] 6 d 0k;ltmh v o9kd0v[38t 8e]6 d fec[[dq ’ |h k L &D oE]yocsj’|bj’]yooEwfh 12  / mn D 4k,;jk1kdwfhoE 390  9t8hv’]=4hk9adokmuL *D s5’g0Qk ! 9vd.ljoE 1 23 9vdD 4k,;jks5’g0Qk # 9vd.ljoEgmqjk.f9vdL (D vaf8klj;o ]ts;jk’dko;afcmdG 1 cm  0,3937 inch D 4k,;jk 1 inch ,u gmqjk.f cm ? 1 foot  0,3048 m D 4k,;jk 1 m ,u gmqjk.f feet ? 1 m 1,0936 yards D 4k,;jk 1 yard ,u gmqjk.f m ? 29.

(34) 4k,;jk 1 mile ,u gmqjk.f km ? 1 gram  0, 03527 ounce D 4k,;jk 1 ounce ,u gmqjk.f grams ? 1 kg  2,2046 pounds D 4k,;jk 1 pound ,u gmqjk.f kg ? 1 gallon  3,78541liters D 4k,;jk 1 liter ,u gmqjk.f gallon ? 1 hectare  2, 470966 acres D 4k,;jk 1 acre ,u gmqjk.f hectare ? 1 hectare 10 000 m 2 D 4k,;jk 1 acre ,u gmqjk.f m 2 ? )D .oczomuj muj,u,kf8klj;o 1 cm 8+ 150 km D dD 4hk;jklv’g,nv’sjk’dao 3, 2 cm .oczomujF 8q;9y’sjk’dao9ad km ? 0D 4hk]qf7ao|bj’vvd9kdg,nv’|bj’ 1kdIvfg,nv’mu@ .og;]k17.00 h F 8hv’vvd gfuomk’g;]k.fL I6h;jk7;k,w;ltg]jp0v’]qfc,jo 50 km / h. 7D 4hk]qf7aofaj’djk;dyooE,ao 12 km /  D 4k,;jk9t\qfg’yogmqjk.fdu[L I6h;jk]k7k oE,ao 1  / 8000 k. _D g7njv’vafgvdtlkogdqjkvafwfh @W |hk 8+okmuF g7njv’.\jvafwfh )W |hk 8+okmuD dD vafgvdtlko )WW |hkD g7njv’gdqjk.-hg;]kgmqjk.fL g7njv’.\j.-hg;]kgmqjk.fL 0D 4hk.-h ma’ @ g7njv’rhv,dao9t.-hg;]kgmqjk.fL !WD gInv9ad]e|bj’.-hg;]kgmqjkdao]jv’8k,dtcloEws^ 20 km c]t 0Bom;odtcloE. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 1 km  0, 621 mile D. 10 km.. dD 4hkdtcloEws^ gmqjk 2 km / h D 4k,sk7;k,w; v 0v’gInvgmy’oE[+ws^D 0D c]jogInvm;odtcloE 16 km .-hg;]kgmqjk.fL 7D 4k,;jk.-hg;]kgmqjk.fgrnjvc]jowx 8 km da[ 8 km 3fp0qog7njv’g0Qk{vvdgInv 15 mn ?. !!D lv’dHvdlk,kfw0oEg0Qkg8a,4a’ wfh 20 mn c]t 15 mn 8k,]efa[D dHvdmu # lk,kfxjvpoEvvd4a’\qfwfh 5 mn D 4hk4a’,uoEg7yj’4a’F 4k,;jkw0rhv,daoma’ # dHvd.-hg;]kgmqjk.foE\qf4a’L !@D oaddy]k7qo|bj’.-hg;]k.f|bj’4u[]qfl;odtcl]q,muj,u7;k,w; 5 km / h wfh 5 km D 4hk]k;4u[8k,dtcl]q,.og;]kfP;daowxwfh 15 km D 4k,;jk.ovaf8k7;k,w; faj’djk;]k;9t.-hg;]kgmqjk.f4u[]qf 15 km l;odtcl]q,muj,u7;k,w; 4 km / h L. 30.

(35) *D# vaf8klj;orq;raoD !D 9qj’c[j’ @$* gxao # lj;o.ovaf8krq;rao 1 12 : 2 :3 13 . @D 9qj’c[j’g’yo !& WWW du[ gxao # lj;o.ovaf8klj;orq;rao. 1 1 1 : : . 2 3 6. #D ];’Iv[0v’I6[lk,c9|bj’c,jo 3 600 m D ];’pk;0v’0hk’gxaovaf8klj;o rq;rao da[ 3: 4 :5. 9qj’-vd];’pk;0v’c8j]t0hk’D $D g’yo !# du[ xtdv[,us^Po &W vafF !W vaf c]t & vaf g-yj’,u9eo;ogmqjkdaoD 4k,;jk9eo;os^Poma’\qf,ugmqjk.fL &D 9qj’c[j’g’yo @! WWW du[ gxao @ lj;og-yj’lj;omu ! gmqjk. 3 4. 0v’lj;omu @D. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. *D 9qj’c[j’\kdw,h 12 kg 8k,vaf8klj;o0v’vkp5 $&F $) c]t $@D (D vkp50v’c,jg4Qk gmqjk @ gmnjv]6dlk; c]t gmqjk $ gmnjv0v’s^kolk;F ma’ # ];,dao,u _! xuD 4k,skvkp50v’c8j]t7qo,ugmqjk.fL )D 9qj’c[j’g’yo !$ WWW du[ .shgfadpy’ # 7qo c]t gfad-kp ! 7qo 3fp.shgfadpy’c8j ]t7qowfh @ gmnjv0v’gfad-kpD _D 4hk oD xk /kdg’yo $WW WWW du[ .o _ gfnvo c]t oD x6 /kdg’yo #WW WWW du[ .o ) gfnvo g-yj’wfhfvdg[hp];,daogxao @& WWW du[D 4k,;jkc8j]t7qowfhg’yogmqjk.fL !WD 4hk x : y  3 12 : 2 13 c]t y : z 1 14 : 2 17 . 9qj’-vd x : y : z. !!D 4hk & 7qogIaf;Pd|bj’rkp.o 2 h leg]afD 4k,;jk !@ 7qo.-hg;]kgmqjk.f9bj’ leg]afL. 31.

(36) [qfmu ( gxug-ao c]t fvdg[hp dyf9tde !D. @D #D. .oshv’IPooU,uG Xd? vaf8klj;o9eo;ooadIPopy’ 8+ -kp gmqjk.fL gxao gxug-aoL X0? vaf8klj;o9eo;ooadIPopy’ 8+ ma’\qf gmqjk.fL gxao gxug-aoL X7? vaf8klj;o9eo;ooadIPo-kp 8+ ma’\qf gmqjk.fL gxao gxug-aoL /kdg’yo,udeoqf c]t /kdg’yoxt1af8jk’daoco;.fL fk;g7njv’ c]t 1n,g’yo8jk’daoco;.fL. .97;k, gxug-ao c,jovaf8klj;o lj;o !WW g-ajoG 40%. gmqjkda[vaf8klj;o $WG!WW s^n @G&D 40 100. 2 . 5 40% gmqjkda[9eo;omqfltoypq, 0,04. 40%. gmqjkda[g]dlj;o. s^n. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. (D! gxug-ao. g]dlj;ogxaogxug-aomuj7;o9njG 3  75%  0,75. 4 1  50%  0,50. 2 1  25%  0,25. 4 1  10%  0,01. 10. 2  66 23 %  0, (66). 3 1  33 13 %  0, (33). 3 1  12 12 %.  12,5%  0,125. 8 1  5%  0,05. 20 56 : 4280 D. 9qj’7yfgxaogxug-aoD. 8q;1jk’ !D. lq,,5fvaf8kdkogduf c,jo. [qfcdhG. 56 560 140  100    1,3084...  1,31%. 4280 428 107 32.

(37) 8q;1jk’ @D. goNvmujo= hkmtg],uxt,ko. 2 3. 0v’goNvmuj3]dD 4k,;jkgoNvmuj0v’fyo,u9ad. gxug-ao0v’goNvmuj3]dL [qfcdhG. 1. 8q;1jk’ #F. dew]. 5%. 2 1   33 13 %. 3 3. 0v’]k7k-N gxao9adgxug-ao0v’]k7k0kpL. 0.05 5   4,76%. 1.05 105. [qfcdhG. (D@ fvdg[hp/kd,udeoqf (Simple interest) mtok7ko7yfw]jfvdg[hpg’yo/kd,udeoqf gmnjvfP;3fp[+gvqkwx[;d.lj8QomboD g,njv.-hg7njv’\kp I: A: R: T:. l6f. I. 8QomboD fvdg[hpD 9eo;og’yoma’|qfD vaf8kfvdg[hpD g;]kD. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. P:. PTR  A  P. 100. 8q;1jk’ !D 9jq’7yfw]jfvdg[hp/kd,udeoqf #@F& ]hkodu[ .ovaf8kfvdg[hp !&: 8+xu rkp.o $ xuD [qfcdhG. I. PRT 32, 5  15  4  19, 5  100 100. ]hkodu[D. 8q;1jk’ @D 9qj’7yfw]jvaf8kfvdg[hp8+xu0v’g’yo/kd,udeoqf &@F& ]hkodu[F g-yj’4vowfh7no gxaog’yo &)F) ]hkodu[rkp.o # xuD. 33.

(38) [qfcdhG. ,u. I  A  P  58, 8  52 ,5  6 ,3. 100 I PTR I R . 100 PT wfh R  100 I  100  6,3  100  21  4  21  4%. 52,5  3 525 21 PT. (D# fvdg[hp/kdxt1af (Compound interest). P: I: A: r: n:. l6f. 8QomboD fvdg[hpD 9eo;og’yo.o[ao-uD vaf8kfvdg[hpD g;]kXxu?. A  P(1  r ) n. ryl6fD 9eo;og’yo.o[ao-umhkpxu mu ! @ #. A  P(1  r ).. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. mtok7ko7yfw]jfvdg[hpg’yo/kdxt1afgxao’;f c]h;[;d.lj 8QomboD g,njv.-hg7njv’\kpG. A  P(1  r )(1  r )  P(1  r ) 2 . A  P(1  r ) 3 .. ..................... n. A  P (1  r ) n .. 8q;1jk’ !D 9jq’-vd9eo;og’yo.o[ao-u9kddko/kdxt1af &W ]hkodu[ .ovaf8kfvdg[hp $: .o # xuD [qfcdhG. A  P (1  r ) n. ]hkodu[D A  50(1,04) 3  56, 2432 ]hkodu[D. A  50(1  0, 04)3. 34.

(39) 8q;1jk’ @D 9qj’-vdvaf8kfvdg[hp0v’g’yo/kdxt1af $F& ]hkodu[ .o @ xu ,ug’yo.o[ao-u $F)*(@ ]hkodu[D [qfcdhG ,u A  P (1  r ) n A  (1  r ) 2 . P 4,8672  (1  r ) 2 . 4,5000 48672  (1  r ) 2 . 45000 2. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ.  26  2    (1  r ) .  25  26  1  r. 25 26  1  r. 25 1  r  r  4%. 25. (D$ Ij;,mbo s^n fk;g7njv’. 8q;1jk’ !D -kpz6j|bj’Ij;,mboda[[=]ylafcsj’|bj’ !W ]hkodu[ m5d;aomu ! ,a’dvo0v’m5dxu .ovaf8kfvdg[hp $: 8+xuD 4k,;jk.o;aomu #! mao;kxumu # ]k;,ug’yoIj;,mbo gmqjk.fL [qfcdhG. 8Qombo xumu !G. !W ]hkoD. 8Qombo xumu @G. 10  1,04  10  20,4. 8Qombo xumu #G. 20,4  1,04  10  31,216. g’yombo #! mao;k xumu #G. 31,216  1,04  32,46. ]hkoD ]hkoD. ]hkoD. 8q;1jk’ @D py’7qo|bj’fk;]qfgd’ _W ]hkodu[ 3fp9jkplqf $W:D lj;opa’gs^nv9jkp’;f.ovaf 8kfvdg[hp #: 8+xuF 4hk]k;9jkpgfnvo]t & ]hkodu[D 4k,;jk deoqf9adgfnvo 9bj’9t\qf c]t gfnvol5fmhkp9jkpgmqjk.fL 35.

(40) [qfcdhG |Umuj8hv’9jkp. 90  0, 6  54 ]hkodu[D. vaf8kfvdg[hp gfnvo]t. 3%  0, 25%. 12. [qfg/ydsaf. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 54 1, 0025  5  49,135 |U.o8Qogfnvomu !G 49,135 1, 0025  5  44, 2578 @G #G 44, 2578 1, 0025  5  39,3685 $G 39,3685 1, 0025  5  34, 4669 &G 34, 4669 1, 0025  5  29,5531 *G 29,55311, 0025  5  24, 6270 (G 24, 6270 1, 0025  5  19, 6886 )G 19, 6886 1, 0025  5  14, 7378 _G 14, 7378 1, 0025  5  9, 7746 !WG 9, 7746 1, 0025  5  4, 7990 gfnvol5fmhkpgfnvomu !! ]k;8hv’9jkp 4,7990  1,0025  4,811 ]hkodu[D. (D!D c8j0=h ! { * 9qj’7yfw]jgxaogxug-aoD !D _ .o )@ 7qo8kpdjvovkp5 $& phvofnj,gs^Qks^kpD @D & .o !$$ 7qo8kpdjvovkp5 &W phvof6f1ks^kpD #D rklu]kpwfhc,jo 0,47 12 du[.o ! du[D $D rklugs^Qkoeg0Qkc,jo !&W vaf.o @ du[D &D oadIPopy’ &WW 7qo.ooadIPo _*W 7qoD *D dew] & du[ 8+ mbo !@& du[D (D .o[hkocsj’|bj’,uc,jpy’ 32% F r+-kp 30% D 4k,;jkgfad,u9adgxug-aoL )D rtoad’ko7qo|bj’9jkp7jkg-qjkgInvo 12% 0v’g’yogfnvoF 7jkvksko 52% 0v’lj;og’yomujgs^nvD 4k,;jkg’yogfnvo]k;gs^nv9adgxug-aoL _D dew] 2% 0v’]k7k0kpgxao9adgxug-ao0v’]k7k-ND !WD 4k,;jk 15% 0v’ @)F&W gxaogmqjk.fL 36.

(41) ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. (D@ fvdg[hp/kd,udeoqfD !D 9qj’-vdsk9eo;og’yomuj4vowfh9kdg’yo/kd &FW& ]hkodu[.ovaf8kfvdg[hp ): deoqf $ xuD @D 9qj’-vdsk9eo;og’yomuj8hv’.-h7no9kddko1n, @F@ ]hkodu[.ovaf8kfvdg[hp ): deoqf 2 12 xuD #D 4hk;jkfvdg[hpg’yo/kd,udeoqf !_@ WWW du[ .o 2 12 xuc,jo !@ WWW du[D 9qj’-vd vaf8kfvdg[hp 8+ xuD $D 4hk;jkfvdg[hp g’yo/kd,udeoqf @W& WWW du[ .o $ xu c,jo &( $WW du[D 9qj’-vd vaf8kfvdg[hp 8+ xuD &D 4hk/kd,udeoqf @!W WWW du[ 4vowfh @#& @WW du[.o *: 8+ xuD 9qj’-vdg;]kD *D 4hk/kd,udeoqf !*) WWW du[ 4vowfh !)_ WWW du[ .o &: 8+ xuD 9qj’-vd g;]kD (D /kdgmqjk.f .o 2 12 % 8+xu deoqf * xu wfh fvdg[hp !@ (@W du[L )D 1n,gmqjk.f .o 8% 8+xu rkp.o # xu 9jkp7no )F*) ]hkodu[L _D /kdgmqjk.f .o 7 12 % 8+xu deoqf 2 12 xuwfh fvdg[hp !FW#& ]hkodu[L !WD 1n,gmqjk.f .o *: 8+xu rkp.o 3 14 xu wfh.-hfvdg[hp !&& @@W du[L (D# fvdg[hp/kdxt1afD !D 9qj’-vdsk9eo;og’yo.o[ao-u9kddko/kdxt1af &FW& ]hkodu[.ovaf8kfvdg[hp $: .o $ xuD @D 9qj’-vdsk9eo;og’yo.o[ao-u9kdko/kdxt1af @F@ ]hkodu[.ovaf8kfvdg[hp $: .o 2 12 xuD #D 4hk;jkfvdg[hpg’yo/kdxt1af !_@ WWW du[ .o 2 12 xu c,jo !@ WWW du[D 9qj’-vd vaf8kfvdg[hp 8+ xuD $D 4hk;jkfvdg[hp g’yo/kdxt1af @W& WWW du[ .o $ xu c,jo @) (WW du[D 9qj’-vd vaf8kfvdg[hp 8+ xuD &D 4hk/kdxt1af !W ]hkodu[ 4vowfh !WF$W$ ]hkodu[ .o @: 8+ xuD 9qj’-vd g;]kD *D 4hk/kdxt1af ! ]hkodu[ 4vowfh !F!&(*@& ]hkodu[ .o &: 8+ xuD 9qj’-vd g;]kD 37.

(42) ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. (D /kdxt1afgmqjk.f .o 2 12 % 8+xu rkp.o * xu wfhfvdg[hp !@ (@W du[L )D /kdxt1afgmqjk.f .o 4% 8+xu rkp.o # xu ,ug’yo.o[ao-u )$ WWW du[L _D /kdxt1afgmqjk.f .o 2 12 % 8+xu rkp.o 2 12 xu wfh fvdg[hp !W #(W du[L !WD /kdxt1afgmqjk.f.o *: 8+xurkp.o 3 14 xuwfhfvdg[hp &_ (&W du[L (D$D Ij;,mbo s^n fk;g7njv’G !D -kp7qo|bj’Ij;,mbo @W ]hkodu[da[mtok7ko.o8Qogfnvo,a’dvom5dxu.ovaf8kfvd g[hp &: 8+xuD 4k,;jksâ’9kd9jkpgmnjvmu $ ]k;,umbo.o[ao-ugmqjk.fL @D py’7qo|bj’fk;]qf9ad & ]hkodu[.ovaf8kfvdg[hp *: 8+xu c]t 9jkp’;f & clodu[ 8+gfnvoD 4k,;jk9adgfnvo9jkp\qf c]t gfnvol5fmhkp9jkpgmqjk.fL #D oadlbdlk7qo|bj’ 0v’7totcrflkfxumu|bj’wfhIa[mbodkolbdlk &W ]hkodu[ 9kd mtok7ko gonjv’9kd]k;-totgd,4k,8v[,tskdedy]kgv-Pogd,7A’mu @& ;P’9ao _{!) mao;k @WW_D mtok7kog]uj,7yfw]jfvdg[hp.sh]k;c8j;aomu ! ,a’dvo @W!W .ovaf8kfvdg[hp &: 8+xuD m5dmhkpgfnvomao;k0v’c8j]txu ]k;8hv’4vo g’yovvd & ]hkodu[ grnjv9jkp7jkgmu,xu8+wxD 4k,;jksâ’9kd9q[xumu ( 9kd7tot crflkfg’yombodkolbdlk.o[ao-u]k;gs^nvgmqjk.fL XcotoeG ]k;g]uj,9jkp7jkgmu, c8jxumu @ gxao8Qowx?D $D oadIPo7qo|bj’fk;7v,ru;g8u,n4n (Laptop) # ]hkodu[ 3fp9jkplqf #W:D lj;opa’ gs^nv.sh9jkp’;f * gfnvoD 4k,;jk9jkp’;fgfnvo]tgmqjk.fL gfnvol5fmhkp9jkp gmqjk.fL &D py’7qo|bj’fk;gInvo !)W ]hkodu[3fp9jkplqf #W: c]t 9jkp’;fgfnvo]t * ]hkodu[D 9jkp,kwfh & gfnvo]k;0kp7no.sh[]= ly af 3fpwfhdew] &: 0v’9eo;og’yomujwfh9jkp c]h;D 4k,;jk[=]ylaf8hv’9jkpg’yo.sh]k;gmqjk.fL. 38.

(43) [qfmu ) dkocdh[aoskdjP;da[7;k,w;F w]ptmk’ c]t g;]k dyf9tde !D @D. dkog7jnvomujlt\jeltg\uc,jodkog7njvomujco;.fL 9jq’0Pol6fdkorq;rao]ts;jk’7;k,w;F w]ptmk’ c]t g;]kD. .97;k, dkogfuomk’fh;ppkorkskot.f|bj’,aodjP;0hv’da[7;k,w;F w]ptmk’ c]t g;]kD )D! 7;k,w;. v vG. 7;k,w;ltg]jpF. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 7;k,w; wfh,k9kdg7njv’;afcmd7;k,w;g;]k;af46g7njvomujD g-ajoG xu !_)# 7;k,w;l6’l5f0v’]qf Jet  powered car c,jo 1019 km / h. xu !__W 7;k,w;l6’l5f0v’]qfw2 TGV c,jo 515 km / h. .o-6,xu !_WW 7;k,w;l6’l5f0v’pqo Concord c,jo @ gmnjv0v’7;k,w; 0v’lP’D 7;k,w;ltg]jpD 7;k,w;ltg]j p c,j o zq o sko0v’w]ptmk’da [ g;]kG S , t. S:. w]ptmk’F. g-ajo G 7;k,w;ltg]jp0v’ cl’c,jo 7;k,w;ltg]jp0v’lP’c,jo 7;k,w;ltg]jp0v’3]dxyjovhv,8kg;ao 7;k,w;ltg]jp 0v’ 3]d xyjovhv,8q;gv’ 8q;1jk’ !D .sh [qfcdhG. S  92 34 km, t  2h10mn. F. t:. g;]kD. 300 000 km / s. 340m / s.. ;ao}Iv[D 23 14 15 h }Iv[D. 365 14. 9qj’-vd7;k,w;ltg]jp. 371 S 92 34 92 34 371 6 21 km / h. v   10  1  4    42 26 13 t 2 60 26 4 13 6. 39. v..

(44) 8q;1jk’ @D -kp7qo|bj’4u[]qf4u[w]ptmk’ 10 km 0kwx0Bo7hvpfh;p7;k,w; 10 km / h. 0kda[]q’7hvpfh;p7;k,w; 15 km / h. 9qj’-vd7;k,w;ltg]jpD [qfcdhG. ,u. S  2  10  20 km. 10 10 2 5 t   1   h. 3 3 10 15 wfh v  S  20  20  3  12 km / h. 5 5 t 3. 8q;1jk’ #D 9qj’-vd7;k,w;ltg]jp0v’3]dxyjovhv,8kg;aoI6h;jkw]ptmk’0v’3]dxyjovhv, 8kg;aoc,jo 149 597 910 km D v. S 149597910 47   17113 km / h  17113, 235 km / h. 1 14 t 200 365  23 4. 15. 8q;1jk’ $D 9qj’-vd7;k,w;ltg]jp0v’3]dxyjovhv,8qogv’F. I6h;jk];’vhv,0v’3]dc,jo. 40 009152 m.. [qfcdhG. v. 40 009152 23. 14. 69.  1671. km / h  1671, 69 km / h.. 100. 15. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. [qfcdhG. 8q;1jk’ &D xu !_W# 16jltst]afvkg,]ydklv’vhkpohv’8td5owi X Wright ? gxaoz6hmevyf mujzt]yfpqowfh.oltst]afvkg,]ydk3fp[yowfhwd 120 feet  37 m .og;]k 12 ;yokmuD 9qj’-vd7;k,w;ltg]jpD [qfcdhD. v. S 37 1   3 m / s. 12 t 12. )D@ dik2 0v’ w]ptmk’{g;]k 8q;1jk’ !G ]qfg,7ao|bj’vvd9kdot7vos^;’;P’9aog;]k 11.00 3,’ Ivfg0njvooE’bj, 13.00 3,’ w]ptmk’ 95 km D ]qfg,vud7aovvd9kdg0njvooE’bj, 12.00 3,’ Ivfot7vos^;’;P’9ao 14.00 3,’D 9qj’oe.-hdik2-vd95f c]t g;]kl;o mk’daoD . (0=h,6o9kd Wright brother-Wikipedia, the free encyclopedia.mht). 40.

(45) dik2G w]ptmk’ 95km. (km). g0njvooE’bj,. 71km. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 50km. g;]k. 7o 11.00. 12.00. 95fl;omk’dao c,jo. 12.30. km 71. 13.00. ( h). 14.00. c]t g;]k 12.30 h. 8q;1jk’ @G ]qfg,7ao|bj’vvd9kdot7vos^;’;P’9aog;]k 08.00 3,’F Ivfdklug;]k 12.00 3,’ g-yj’sjk’9kdot7vos^;’;P’9ao 200 km F raddyog0QkmjP’ 40 mn c]t Ivfs^;’rt[k’g;]k 17.00 3,’ g-yj’sjk’9kdot7vos^;’;P’9ao 400 km D ]qfgd’7ao|bj’vvd9kds^;’rt[k’g;]k 10.00 3,’ Ivflk]kr676o g;]k 12.00 3,’ g-yj’sjk’9kds^;’rt[k’ 140 km F raddyog0QkmjP’ 60mn c]t Ivfot7vos^;’;P’9aog;]k 16.00 3,’D 9qj’oe.-hdik2-vd95f c]t g;]kl;omk’daoD 41.

(46) dik2G w]ptmk’. (km). 400km. 300km 260km 230km 200km. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 100km. g;]k. 8.00. 10.00. 95fl;omk’dao c,jo. 12.40. km 230. ( h). 16.00 17.00. 13.00 13.20. c]t g;]k13.20 h D. [qfg/ydsaf )D! 7;k,w; c]t 7;k,w;ltg]jpD !D -kp7qo|bj’9tgfuomk’ 10 km 3fppjk’vvdxkdmk’ c]h;0uj]qfg,. 1 hD 2. 2 km. fh;p7;k,w; 4 km / h. 9qj’-vd7;k,w;ltg]jp 0v’ dkogfuomk’D. @D py’7qo|bj’0a[]qf 12 km wxgfyjo[yofh;p7;k,w; 60 km / h F 0ujgInv[yo 1500 km fh;p7;k,w; 1000 km / h c]h;0uj]qfggmd-u 40 km fh ; p7;k,w; 80 km / h. 9qj’-vd7;k,w;ltg]jp0v’dkogfuomk’L #D -kp7qo|bj’dt;jk9t0a[]qf 300 km fh;pg;]k 3h F c8j8q;9y’w]pt 100 km mevyf]k;c]jowfhrP’ 50 km / h D 4k,;jklj;opa’gs^nv]k;8hv’c]jofh;p7;k,w; gmqjk.f grnjv.shIvf8k,g;]kmujdtw;hL 42.

(47) $D 767qo|bj’gfuomk’wxxt9edko.\jmujg,nv’leraoF zQ’lk]uD ]k;0uj]qfg, 400 km fh;pg;]k 7 h F ovocI,7nomujs^;’rt[k’ c]t 4hkgInv3fplko 15 h F 0ujgInv c]t radcI,7now]ptmk’ 150 km fh;pg;]k 30 h c]t 0uj]qf8Jvd8Jvd 5 km fh;pg;]k 30 mn D 9qj’-vd7;k,w;ltg]jpdkogfuomk’D &D oaddy]k,to5fgsâd7qo|bj’g]uj,9kd]vpoE 1 km fh;pg;]k 15 mn F 4u[]qf 50 km fh;p7;k,w; 40 km / h c]t c]jo 10 km fh;pg;]k 30 mn D 9qj’-vd7;k,w; ltg]jp0v’oaddy]k7qooAoD. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. )D@ dik2 0v’w]ptmk’{g;]kD !D ]qf[aom5d7ao|bj’vvdgfuomk’9kdot7vos^;’;P’9ao Xo7D ;P’9ao? g;]k 6.00h Ivfxkd-ao g-yj’sjk’9kd o7D ;P’9ao 150 km g;]k 8.00h F raddyog0Qk 30 mn F Ivfmjkc0d g-yj’sjk’9kd o7D ;P’9ao 300 km g;]k 10.00h F radfnj, dkg2 15 mn c]t Ivflts;aootg0f g-yj’sjk’9kd o7D ;P’9ao 500 km g;]k 13h. ]qfg,7ao|bj’vvd9kd o7D ;P’9ao g;]k 6.30h Ivfxkddtfy’ g-yj’sjk’ 9kd o7D ;P’9ao 200 km g;]k 9.00h raddyog0Qk 30 mn F Ivflts;aootg0f g;]k 12.30h F 9qj’oe.-hdik2-vd95f c]t g;]k8k,maodaoD @D o7D ;P’9ao c]t ;a’;P’sjk’dao 150 km D 0t[;oc0j ’ 0a o ]q f 4u [ vvd9kd o7D ;P’9aog;]k 7.00h Ivfclo-5,g;]k 9.00h g-yj’sjk’9kd o7D ;P’9ao 80 km 9kdoAod=4[ u 0Bor6rtfh;p7;k,w; 30 km / h ,5j’|hkl6j;a’;P’D ]qf86h7ao|bj’ vvd9kd;a’;P’g;]k 8.00h Ivfsuogsu[ g;] 9.00h g-yj’sjk’9kd;a’;P’ 50 km F raddyog0Qkg-Qk 30 mn c]t Ivf o7D ;P’9ao g;]k 10.30h D 9qj’oe.-h dik2-vd95f c]t g;]kl;omk’daoD )D# ]qfg, # -toyf7n ]qfg,ohvpF ]qfg,dk’ c]t ]qfg,.spj vvd9kdlt4kou 8ts^kf g-Qk X8s^-? wx{da[,tsk;ymtpkw]csj’-kf X,-? g-yj’,uw]ptmk’ 12 km 3fpzjkolt4kou]qfg,lkp.8h Xl8? muj,u8k8t]k’g;]kgfuo]qffaj’oUG. 43.

(48) w]ptmk’9kd 8s^-. 9. !@ _. g;]kgfuo]qf ]qfg,ohvp ]qfg,dk’. (km). 8s^- vvd l8 Ivf vvd ,- Ivf vvd l8 Ivf vvd 8s^- Ivf. ]qfg,.spj. 7.00. 7.10. 7.20. 7.30. 7.25. 7.40. 7.35. 7.30. 7.45. 7.45. 7.35. 7.50. 7.50. 7.40. 7.55. 7.55. 7.45. 8.00. 8.00. 7.50. 8.05. 8.30. 8.05. 8.25. 9qj’oe.-h dik2-vd95f c]t g;]k8k,mao s^n l;omk’daoD. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. )D$ ot7vos^ ; ’;P’9a o c]t mjk ]kfsj k ’da o 90 km D ]q f g,7a o |b j ’ vvd9kd ot7vos^ ; ’;P’9ao g;]k 6.00 h Ivfmjk]kf 8.00 h D 4k,;jk]qfg,vud7ao|bj’ vvd9kdmjk]kfg;]k 6.00 h 1kdl;omk’da[]qfg,mujvvd9kdot7vos^;’16j km 45 8hv’c]jofh;pw;gmqjk.fL Xoe.-hdik2?. 44.

(49) [qfmu _ g]dIkd0Ao n dyf9tde 1 9 2.. !D @D #D. 2 3. 9qj’7eo;o dD 0D 27 . 7D | 3| leo;o7kf76j 0v’ (a  b) c,joleo;o.fL 9qj’.-h[aomaflkdlhk’[aofkmjvo-njmuj,u];’pk; 2 ,. 3 ,.... .97;k, _D! oypk, Ikd0Ao n .sh. gxao9eo;o4h;o[;dD y n  x c]t 0Po.oI6[. gvUo;jkIkd0Ao. y. n. n. 0v’9eo;o9y’ x g,njv. y x. d=]tou 8q;1jk’G. F. n  2. 16. 3. (2) 4. 3. 2. 4. 1 3 3  [(2) ]. 1 42. 1 2 2  (2 ). 1 4.  (2) 2. #D. 3. 1 2. . 1 3.  2.. 2.. 1 2 2  [(3) ]. 2 sn^  3 | 3 | 3. _D@ 75o]adltot Ikd0Ao n !D n xy  n x  n y . d=]tou. n. x  x2.  2.. 2. 1 2 2 2 (3)  [(3) ]. @D. 1. Ikd0Aolv’0Po. x0. 1 4 4  (2 ). 4. xn.. 2. 3. 1 2.  3.. x  0, y  0. x nx  . y ny. m n x m  n xm  x n ..  . 45. D. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 1. n.

(50) 8q;1jk’G !D. n. &D. n p. *D. kn km. (D.  n x n  | x | , n  2k    n x n  x , n  2k  1. x. x . xp. np. x.. km m kn  x  xn.. x. (k  ). (k  ).. 2  8  2  8  16  4 2  4.. 8 2. 2 2. . ( 2)  2  4.  2.. 2. 4. 4 22.  22  4.. 6. 2  23 .. 16  4 24  2. (3)2  | 3 | 3. 3. 16  3 8  2  3 8  3 2  2 3 2.. )D. 3. 3 8 8 2 3  . 27 27 3. _D.  2 3. 6. 3.  2  6. 6 23.  22  4.. 32 2. !WD !!D. 3. !@D. 3. !#D. 4 (3) 4  | 3 | 3.  5 (3) 5  3. . 2. 2  6 4.. 64 . 3. 32. 2. . 23 6. 6. 2  26  2.. 32 2 3.  22  4.. 46. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 8  2. @D #D $F &D *D (D. np. $D.

(51) _D# dko4vo Ikd0Ao @ 8q;1jk’G. 625. 625.. . 6 2 5  2 5  . 4 5 5  2 2 5. 4 2 2 5 2 2 5 0 0 0. @D 9qj’4voIkd 0v’ 6, 25. 6,25.. . 6, 2 5  2, 5  . #D 9qj’4voIkd 0v’ 62,50 . 9kd0;ksk-hkp0v’rkdlj;o4h;o c]t 9kd-hkpsk0;k0v’rkdlj;omqfltoypq, 9afgxao\;f]tlv’8q;g]d. 4. 4. 2. 2 5. 2 0. 2 5 0 0. 62,50. 5 5  2 2 5. .o7jk-afg9omqfltoypq, 102.. 6 2, 5 0 0 0 0 0  7, 9 0 5 4 9 1 4 9  9  1 3 5 0 1 5 8 0  0 1 3 4 1 1 5 8 0 5  0 0 0 9 0 0  0 0 0 9 0 0 0. 0.  7 9 0 2 1 0 9 7. 5 5. 47. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. !D 9qj’4voIkd0v’. 9kd0;ksk-hkp9afgxao \;f]tlv’8q;g]d.  7, 9 1 1 3 4 1  0 0 0 0 5  7 9 0 2 5.

(52) $D 9qj’4voIkd 0v’ 138384 138384 . 3 8 3 8 4  3 7 2 9 6 7  7  4 6 9. 1 . 7 4 2  2  1 4 8 4. 0 4 8 3  4 6 9 0 1 4 8 4  1 4 8 4 0 0 0 0. &D 9qj’4voIkd 0v’ 1383,84 3 8 3, 8 4  3 7, 2. 1 . 9. 0 4 8 3  4 6 9 0 1 4 8 4. 6. 7  7  4 6 9. 7. 4. 2  2  1 4 8 4.  1 4 8 4 0 0 0 0. *D 9qj’4voIkd0v’ 138,384 .o7jk-afg9omqfltoypq, 102. 138,384 . ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 1383,84 . 1 3 8, 3 8 4 0  1 1, 7 6 2 1  1  2 1 1 0 3 8 2 2 7  7  1 5 8 9  . 2 1 1 0. 1 2 3 4 6  6  1 4 0 7 6 7 3 8 5 8 9 1 4 9 4 0.  1 4 0 7 6 0 0 8 6 4. 48.

(53) (D 9qj’4voIkd 0v’ 5. 5. .o7jk-afg9omqfltoypq, 102.. 5, 0 0 0 0 0 0  2, 2 4 1 0 0  8 4 1 6 0 0  1 3 2 9. 3 4 2 2. 6 . 4 4. 3. 3. 4 4. 6. 6. . 2, 2 4. 8 . 4 1 3 2 9. 6  2 6 7 9 6. 2 7 1 0 0  2 6 7 9 6 0 0 3 0 4. )D 9jq’4voIkd0v’. .o7jk-afg9omqfltoypq, 102.. 5 0, 0 0 0 0 0 0  7, 0 7 1  7, 0 7 4 9 1 4 0  0  0 0 0 0 1 0 0 1 4 0 7  7  9 8 4 9  0 0 0 1 4 1 4 1  1  1 4 1 4 1 1 0 0 0 0  9 8 4 9 0. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 50 . 50. 0 1 5 1 0 0  1 4 1 4 1 0 0 9 5 9. [qfg/ydsaf _D! 9qj’IvoD !D. 36 .. @D. *D. 75.. (D. !!D. 3. 27 . 125. !@D. 3. 64.. #D. 64.. )D. 3. 64 . 343. !#D. 3. 81.. $D. 216.. _D. 4. !$D. 32.. 49. 3. 121.. &D. 32 .. 54.. !WD. 3. 192.. 4. !&D. 4. 16 . 81. 162..

(54) !*D. 256 . 625. !(D. 4x .. !)D. @!D. 18a 4 b . c6. @@. 64a 5 . a8. @#D. 27 x 4 . y9. @(D. 16 x 4 y 4 .. @)D. @*D. 3. 2. 4. 2. 8x y .. !_D. 3. x3 y 6 .. @$D. 4. 81x 8 . y 12. @_D. 4. 16 x 4 . y2. 4x3 . y4. @WD. 3. x7 y9 .. @&D. 3. . 5.  32 x 5 . y5. #WD. 5. 243x 5 y 6 . z 10. 8x 6 . y3. _D@ 9qj’4voIkd 841.. @D. 1225. $D. 10609. &D. 21609. #D. 5476. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. !D. 50.

(55) [qfmu !W dko7yfw]j\;f7eo;omuj,ug]dIkd .97;k, !WD! dko[;d c]t ]q[ 8q;1jk’G !D @D. 5 3  3  2 3  4 3. 27  108  12  32  3  62  3  22  3.  3 3  6 3  2 3.  7 3. 2. 3. xy 2  7. 3 x 2. 3. 3. xy 2  5. x 4 x 7. 3. 3. xy 2 .. x  7 x 9. 4 8  2 18  4 4  2  2 9  2 .  4  2 2  2  3 2.  8 2  6 2.  2 2.. *D. 2 12 . 1 1 3  2 43   . 3 3 3.  2 2 3 . 4 3. (D. 3 . 9. 3. . 9 3 4 3 . 3 1    4   3. 3  11 3  . 3 1 1 3 3 3 81  3  33  3  3  . 9 9 3. 51. 3. x.. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. #D $D &D.

(56) 3 333 3 33. 1 3. 3 33 3. .. 3.. 1  (3  ) 3 3. 3 8 3  3. 3. !WD@ dko76o 8q;1jk’G !D 3. . 3 5 2  3 3 5 2 3  32  5 2  3. . 5 2. . . 3  2  5  3  5  2  2  3  2  2.. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. @D.  3  5 6..  15  10  6  2.. #D. . $D. 3. 3 5.    3 2. 2.  2 3  5 .  5. 2.  3  5  2 15. 5 2.   3 5    2  2. 2. 2.   2  .. 2 3 5.  9(5)  2  6 10 .  45  2  6 10 .  47  6 10 .. &D. 3. *D.  3 m  3 n   3 m2  3 mn  3 n2   3. (D )D. 3. 2  3 4  3 2  4  3 8  23  2.. m3  3 m 2 n  3 mn 2  3 m 2 n  3 mn 2  3 n3  m  n. 3. 6. 6. 6. 6. 6. 2  22  23  24  23  24  26  2  2 2 . 3. 6. 6. 6. 6. 2  6 8  3 2  23  3 2  2  22  23  25 . 52.

(57) _D. 5. 3. n m. 3. 3. m 5 3  n   m  15  n   n           n m  n  m m. 1.   15. 2. n   . m. !WD# dko76o @ 9eo;o7kf76j leo;o7kf76j 0v’ 3. c,jo. a3b. c,jo. a  b. 3. a 2  3 ab  3 b .. !D. (5  2 )(5  2 )  5 2  ( 2 ) 2  25  2  23.. @D. ( 7  3 )( 7  3 )  ( 7 ) 2  ( 3 ) 2  7  3  4.. #D. (2 3  1)(2 3  1)  (2 3 ) 2  1  12  11  1.. $D. ( 3 3  3 2)( 3 32  3 6  3 22 )  ( 3 3)3  ( 3 2)3  3  2  1.. !WD$ dko7afIkd vvd9kdr6f 8. 8q;1jk’G!D @D #D $D &D *D (D. . 2 3. 8. . 2. 3. . 2. . 2. 8 2  4 2. 2. . 52. . 3( 5  2)  3( 5  2). 54. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 8q;1jk’G. a b. 52 3 2 3 2   3  2.   32 3 2 3 2 3 2 1 1 2 3 1 2 3 1 2 3 1 2 3 1      . 2 2 12  1 11 2 3  1 2 3  1 2 3  1 (2 3 )  1 52 1. 1 3. 2 2. 3. 2.  . 1 3. 2 2. 3. 2. 1 3. 2 3 3. 52 1.    . 3. 22. 3. 22. 3. 22. 3. 22.  . 3. 6. 4 . 2 23  6 2 4 2 6 2 6   2 .. 2 2. 1. . 3. 2 2  3 6  3 32. 3. 2 2  3 6  3 32. 3. 2 3 3. 3. 2 2  3 6  3 32 (3 2 ) 3  (3 3 ) 3. . 53. ..

(58) . 3. 2 2  3 6  3 32 . 23.  (3 2 2  3 6  3 3 2 ).. !WD& dkosko 8q;1jk’G. 52 3 1. . 32 52. . 3 1 52. . 52. . 32 52.  . 3  1 ( 3  1)  1 54  . 3  1  ( 3  1) 1  . 3 1 3 1 1 3 1 . .   2 3 1 3 1. [qfg/ydsaf !WD! 9qj’7yfw]j !D 4 3  2 3  5 3. #D 8  32. &D 27  108  12. (D 125  5  80. _D 18  50. !!D 75  32  27 . !#D 18  50  12  !&D !(D !_D @!D @#D @&D. 1  8. 8 128  3 18  162.. 4 45  3 54  6 5. 7 24  4 5  2 6 . 12  2 48. 147  4 3.. 75.. @D 7 2  9 2  2 2. $D 20  125  45. *D 12  27 . )D 63  28  112. !WD 128  32  27 . !@D 75  48. !$D 8  20  12. !*D !)D @WD @@D @$D @*D. 180  28  2 5. 4 48  5 27 . 3 24  5 54 . 3 32  2 50  8 18. 128  4 75  162. 2 363  5 243  192. 54. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. . 52.

(59) @(D 5 a 6  4 5 a 6  2 5 a 6 . @_D 2 3 x 2  3 6 x 4  9 x 6 . !WD@ 9jq’7yfw]j. 3 3 u 5  2 6 u 10  2u 5 5 y 2. 7  3. 4 5  2 2.. 2 3  3 2  3. !!D 27  1 3. 3 !#D 2 8  2 .. . 7 3 7.  2 ..  3  6 2 . 2 !_D  3 2  . @!D 2 6  3 3  5 2  . @#D  2  3  5  3  . 2 @&D  3  2  . 2 @(D  7  2  . @_D #!D ##D #&D #(D #_D $!D $#D. 2 7. (2 3  2 )(3 3  2 ). (3 5  2 3 ) . 2. (2 5  3)(2 5  3). (3 5  2 2 )(2 5  2 3 ). 3. y  3 y.. 5. 2. 3 3  5 3.. &D !(D. u.. @D  5  . $D 5  6. *D 2  3 6. )D 10 3  5 11. !WD 2 7  63.. 3  3.. . 3. 4 (3 2  3 16 ).. (5 u 2  5 v 3 )(5 u 3  5 v 2 ).. !@D. 8  5  125.. !$D !*D !)D. 3. . . 32 3 .. . . 2 3 4 3 5 3 . 2. . . 8  32 .. . . @WD @@D @$D. 3 3. @*D. ( 5  3) 2 .. @)D #WD #@D #$D #*D #)D $WD. .  . ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. !D #D &D (D _D. @)D #WD. 82 2 .. 34. . 7 2. 5 2. . 32 .. . . . 3 5 .. . 5 2 .. (2 7  1) . 2. ( 7  1)( 7  1). ( 7  2 3 )( 5  3 2 ). ( 3 4  3 9)( 3 2  3 3). (3 x  3 y 2 )(3 x 2  2 3 y ).. (3 x  3 y )(3 x 2  3 xy  3 y 2 ). (3 x  3 y )(3 x 2  3 xy  3 y 2 ). $@D 3 48  75  1 147 . 7 2 (2 3  5)(5  2 3 )  (4  5 )  8 5.. 55.

(60) !WD# 9qj’7afIkdvvd9kdr6f @D. 4. .. #D. 1. .. &D. *D. . 2 5 7 3. )D. . 6 3 2. . 14 2 5. 2 4. $D. .. 2 2 10. (D !WD !#D. . 2 6 3 2 6 1. !*D !_D @@D @&D @)D. 2 1 3. #(D. !$D. .. !(D. .. 5 3 1 . 2 5 1 2. @WD. .. @#D. 3 8  12 3 2 1 . 3 2 1. .. 1 3. 3. x 3 y 1 . a 2b. @*D @_D. x2 . y3. #!D #$D. !!D. #@D #&D. .. #*D. 5 3. _D. . 8 2 1. . 3 3 2 5 2 5 1 2 3 2. . 3 10 3 1. !&D. .. !)D. .. 2 2 1 5 52 6 3 2 . 3 2 3 2 5. .. @!D. .. @$D @(D. 2 5 1 1. !@D. . 3 7. #WD. .. 3. y. .. ##D. 3. 8x . y2. #*D. 1 1 2  3. .. #_D. . 3 4 32 2 3 2 1 3 2 6 . 3 1 3 . 4 35 7 2. .. .. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 3. !D. . 7 2 4 7 2 . 2 7 1 3. 2y . x. 1 x y z 1 3. 4 3 6 3 9. . .. !WD$ 9qj’7yfw]j s^n Ivo !D #D. . . 7   3    3 7 . 7  3 33 1 0,4 2 2 a 3 . 4 a. @D $D. 1300  2 52  12 1 94  5 3 14 . 3. 2a 2  6 32a 4 .. 56.

(61) &D. 5. (D.  a 3 2  3 ab  6a b b. )D. 3  3  2m 4  3m 4  .    . 5  2  10 5  2 . 3. *D. 4x. 3. x2  5 y 2.  a 8 a 2b  a 4b3    b. 6. 3. xy  xy. 3. y2.   2xy. 3. . xy .. ab 2 . 2. 2. _D.  1 1 1  x 2  x 6  .   2  . 3. !@D !#D !&D !(D !_D. 3. !$D. 7.. 4x3 y 2 3. 2 xy 2.  2x. 3. x y 3. x y. .. !*D. 3y 2 . 4x. !)D. .. @WD. 3 3. ( 2a ) 9 .. 8u 3 v 5 3. 4u 2 v 2. 6c. 3. .. 2ab . 9c 2. 1. .. 3. ( x  y) 2. 3. 8 16 x 6 y 4 .. @!D. 4. 16 x 4  3 16 x 24 y 4 .. @@D. @#D. 4. 3m 2 n 2  4 3m 3 n 2 .. @$D. @&D. 3. x 3n ( x  y ) 3n  6. @*D. n. x 2n y n. @(D. 3. 23 3 2 . 3 2 3. @)D. 4. 2 3 2 2.. @_D. 6. a3. #WD. 3. !WD& 9qj’7eo;o. a a. 2 x 5 y 3  3 16 x  7 y  7 . 2. n. .. x x x x.. 2 2 2 2 ... 57. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. !WD. 2 4  2  4  2   2 3 9 9 3 3 a a  . !!D  a  1  a  a  1  a 9  1 .             1  1 1 1 1  1 1  1  a 6  b 6  a 3  a 6 b 6  b 3   a 2  b 2  .        .

(62) [qfmu !! dkocdhlq,zqoIkd0Ao @ dyf9tde Iv[;Po0v’38ho3,’ (pendulum) 0Boda[g;]kmuj8hv’dko0v’38ho3,’wd; wx{da[ |bj’Iv[g-yj’g;]kgxao;yokmu c]t ];’pk;0v’38ho3,’ l gxao[kf8uo  30 cm  g-yj’,ul6f faj’oUG. T  2. 9qj’-vd7jk0v’ l g,njv. l 32. D. T 1 s. X! ;yokmu?D. .97;k, A. g,njv B  0, A  B 2 . B g,njv B  0, A  B.. 8q;1jk’G. 9qj’-vd.9zqo 0v’lq,zqo dD 2 x  1  2  x 0D 2 x  1  2  x 7D x  7  8  2 x  1.. [qfcdhG. dD. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. AB. 2x 1  2  x 2 x  1  (2  x)2 , 2  x  0 2 x 1  4  4 x  x2. x2  6 x  5  0 x  1, x  5.. c8j;jk x  5 [+8v[lt|v’g’njvow0 faj’oAoF .9zqoc,jo x  1.. 58. 2 x  0D.

(63) 0D. 2x 1  2  x 2 x  1  2  x, 2  x  0 3x  3, 2  x  0 x  1, 2  x  0.. faj’oAoF .9zqoc,jo. x  7  8  2x  1 x  7  1 8  2x. . x7.   1  2. 8  2x. . 2. .. x  7  1 8  2x  2 8  2x 3x  2  2 8  2 x ..  3 x  2 2   2. 8  2x. . 2. .. c8j;jk8hv’wfhd;f7no. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. 7D. x  1.. 9 x 2  12 x  4  4(8  2 x).. 9 x 2  12 x  4  32  8 x. 9 x 2  4 x  28  0. ( x  2)(9 x  14)  0. x  2, x  . d;f7no x  2. wfh. 14 . 9 2  7  8  2  2  1. 9  4  1. 3 2 1. x. 14 9. wfh. . 4nd8hv’D. 14 28 7  8  1. 9 9.  14  63 72  28   1. 9 9 49 100   1. 9 9. 59.

(64) 7 10   1. 3 3. [+4nd8hv’D.  1  1.. faj’oAoF d5j,.9zqoc,jo. S  {2}.. [qfg/ydsaf 9qj’-vdd5j,.9zqo 0v’lq,zqoD !!D!D m  8  0. !!D@D. 5 x  1  6  10.. !!D#D !!D&D !!D(D. !!D$D !!D*D !!D)D. x  7  x  5.. x  7  3x  1. x  3  x  3.. x  1  x  2. x  12  x  2.. !!D_D  !!D!!D !!D!#D. 4 x  1  x  2  x  3. x  1  a.. !!D!WD !!D!@D !!D!$D. !!D!&D. 4  2 x  x 2  x  2.. !!D!*D 11  2 x . !!D!(D. 20  x 20  x   6. x x. !!D!)D. !!D!_D !!D@!D !!D@#D !!D@&D !!D@(D !!D@_D !!D#!D !!D##D !!D#&D. x 2  21  x  3.. !!D@WD !!D@@D !!D@$D !!D@*D !!D@)D !!D#WD !!D#@D !!D#$D !!D#*D. 2 x  4  1. 2 x  3  0. 5 x  1  1  4. 2 x  1  1  2. 3 5 x  1  3  2. x  6  x. 15  2 x  x. 8 x  12  2 x. 5 x  6  x. 60. 3 x  4  3 x  5  9.. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. r 2  5  r.. x  7  x  2  6 x  13. x  3  a  x.. 22  x ..  2 x  x  9. 3 x  3  5. x  1  2  2. 6 5  x  2  8.. 4 x  3  7  17. 6 4 x  3  5  3. x  20  x. 3  2 x  x. 7  6 x  4 x. 7 x  12  x..

(65) !!D#(D 2 x  9 x  2. !!D#_D 2  x  1  1 !!D$!D 1  x  3  5 !!D$#D 4  x  2  x. !!D$&D x  4  2  x. !!D$(D 2  x  3  x  1. !!D$_D x  1  5  x  6.. !!D#)D !!D$WD !!D$@D !!D$$D !!D$*D !!D$)D !!D&WD. !!D&!D. | x | 1  | x |  a.. !!D&@D. !!D&#D. xa  xa. 3 x  11x  2 . 4  2x  4  2 6  5  x  7. 5  x  5  x. x  9  3  x. 3  x  2  x.. x  5  4  x  3. x 1  x 1. x 1 3  . x 1 2. xa  3. xa. T  2. m k. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. !!D&$D 4k,;jk ];’pk; l 0v’ 38ho3,’ g,njv T  0, 5 s. !!D&&D oad2u-ydlkf“3ig[uf I5d” Robert Hooke (1635-1703) rq[;jkl6f lk,kf[vdIv[;Po0v’;a f 45 m u j , u , ;olko m. (kg ). muj1nf0Bo1nf]q’8k,lkpg-nvdmuj,af8yf,aog,njv,ao4ndxjvp]q’D 9qj’-vd7jk0v’ m .shI6h;jk T  1 s c]t k  10 D !!D&*D l6f7;k,w;0v’9ts^;f Rocket muj9tlk,kf0BoIvfvk;tdkfwfh3fp[+,cu I’ fb’f6f0v’3]dc,jo v  c]t. Re. G, Me. 2GM e Re. F 3fp. G. c,jocI’4j;’|ad0v’3]dF. c,jo,;olko c]t ]aflt\u0v’3]dD 9qj’-vd7jk0v’ c]t v D. 61. Re. Me. 8k,.

(66) [qfmu !@ g]dde]a’ mujde]a’gxao9eo;oxqddt8y dyf9tde !D. vjkoco;.fL c,jog]dspa’L c8j]tlj;o gvUoco;.fL 3 ,u 7jkgmqjk.fL 30 ,u 7jkgmqjk.fL 2 2 ,u 7jkgmqjk.fL 34 4. 1. 0PovudI6[|bj’wfhc[[. fL ,u7jkgmqjk.fL .sh a, b, m, n gxao9eo;o4h;o8jk’l6oD 9qj’8nj,.lj[jvo,ug7njv’\kp4k, 42. 1. a0  ?.. n. a am. a m  a n  a?.. a m n. ?. n.  a?.. (ab) n  a ?b ? .. (a )  a .. n. n. #D.  a?.. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. @D. ?. a? a a b  .     .   ? b b a b 4 x 2  4 x 1  1. 9qj’Ivo. 2 x  2  x 1. .sh,u8q;de]a’gxao9eo;o4h;o[;dD. .97;k, !@D! g]dde]a’muj,u8q;de]a’ gxao9eo;oxqddt8yD oypk,G. .sh. a, p. c]t. p q. a. \kpgsfG 8q;1jk’ !G. 1 83. a  0, q 3. gxao9eo;o4h;o8jk’l6oD. q. 76j p 7ud. . q. s^n. 1 83. 1 3 3  (2 ).  8 2 2. 1 (8) 3. 3. 3. q.  ap..  8  (2)  2 3. 3. 3. ap. s^n 62. [+deoqfD 3. 2. 1 (8) 3. 1 3.  2.. 1 3 3  [(2) ].  (2). 3. 1 3.  2..

(67) !@D@ 75o]adltot0v’ g]dde]a’muj,u8q;de]a’gxao9eo;oxqddt8yD a , b    {0}, p, m  , q , n   p q. !D. a a. m n. p q. @D. a a p q. a. m n. a. p m  q n. m. (a ) n  a. $D. (ab) q  a q b q .. p. .. p m  q n. #D. p. .. .. p. p q. p q. 8q;1jk’G !D. 2 23. 2. @D. 2 23. 4 6 2. #D. 2 6 3 (2 ) 4. $D. 1 (8  27) 3. 8 ( 27. 2 )3. 8 27. . 4 6. . . 2 4  3 2 6. . 2 6  3 2 4 1  83. 2 3 3 (3 ). (. (D. 1 3 6 (27a ). 8.  20  1.. . 1 27 3 ( ). *D.  20  1.. 1  21  . 2. 1  27 3. 2 83. . 1  ) 3. 2 2  3 2 3. ສະ ສ. ຫງວ ວ. ສ ນລຂິ ະສ ດິ. a a ( )  p. b bq p p a q b q *D ( )  ( ) . b a. &D. &D. p m  q n.  2  3  6.. 2 3 3 (2 ) 2 3 3 (3 ). . . 1 3 3 (3 ) 1 3 3 (2 ). 1 3 6  [(3a) ]. 22. 4  . 9 3 2. 3  . 2. 3 6  (3a). 1  (3a) 2. 63.  3a ..

7toyflkf -Ao,afmtpq,lbdlkxumu $ IP[IP&rsquo;3fpG

7toyflkf -Ao,afmtpq,lbdlkxumu $ IP[IP’3fpG