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X - MATHEMATICS BY SHORT CUT AND EASY

METHOD

,q;F jug;gl;Ls;s mizj;J fzf;FfSk; Gj;jfj;jpypUe;J khw;W

Kiwapy; nra;ag;gl;Ls;sJ. ,e;j Kiwfs; mizj;Jk; CBSE, ICSE, Ntjpf; fzpjk;>kw;Wk; Neghs ehl;bd; fzpj Gj;jfj;jpy; gpd;gw;w gl;l vspa Kiwfspy; jug;gl;Ls;sJ. ,k;Kiwfs; mizj;Jk; khzth;fSf;Fk;> Mrphpah;fSf;Fk; kpf gaDs;sjhf ,Uf;fk; vd ek;GfpNwd;.khzth;fs; vspjpy; Ghpe;J nfhs;Sk; tifapy; mjpf tpsf;fq;fs; jug;gl;Ls;sJ. ,it khzth;fspd; GhpjYf;fhf kl;LNk. Njh;Tfspy; mjpf tpsf;fq;fs; vOj Ntz;ba mtrpakpy;iy.

10th

MATHEMATICS

BY EASY AND

SHORTCUT

METHOD

BY

K. RAVI MADESWARAN,

M.Sc., M.Phil., M.A., M.A., M.Ed., P.G.D.V.E&S

BT. ASSISTANT,

GOVERNMENT HIGHER SECONDARY SCHOOL,

VALLIPATTU, VANIYAMBADI - 635 752.

VELLORE DISTRICT.

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vLj;Jf;fhl;L 2.6

xU G+e;Njhl;lj;jpy; Kjy; thpirapy; 23 Nuh[hr; nrbfs;> ,uz;lhk; thpirapy; 21 Nuh[hr; nrbfs;> %d;whk; thpirapy; 19 Nuh[hr; nrbfs; vd;w Kiwapy; Nuh[hr; nrbfs; xU njhlh;thpir mikg;gpy; cs;sd. filrp thpirapy; 5 Nuh[hr; nrbfs; ,Ug;gpd;> mg;G+e;Njhl;lj;jpy; vj;jid thpirfs; cs;sd? jug;gl;Ls;s thpirg;gb 23, 21, 19, ..5 , ,tw;iw khw;wp mikj;jhy; fpilg;gJ 5,7,9,..19,21,23, ,e;j njhlhpy; tpj;jpahrk; 2 MFk;. vdNt xt;nthU cUg;gpYk; 3I fopj;jhy; fpilf;Fk; njhlh; 2,4,6,8,20. ,jpypUe;J ehk; vspjhf Ghpe;Jnfhs;syhk;> ,j;njhlhpy; cs;s thpirfspd; vz;zpf;if 10. vLj;Jf;fhl;L 2.7

2010-y; xUth; Mz;L Cjpak; &.30>000 vdg; gzpapy; NrUfpwhh;. NkYk; xt;nthU tUlKk; &.600-I Mz;L Cjpa cah;thfg; ngWfpwhh;. mtUila Mz;L Cjpak; ve;j tUlj;jpy; &. 39>000 Mf ,Uf;Fk; ?

mtUila tUl tUkhzk; xU $l;Lj;njhlhpy; mike;Js;sJ. 30,000, 30,600, 31,200, ..39,000. Kjy; tUl Kbtpy; &.600k;> 2Mk; tUl Kbtpy; &.1200k;> %d;whk; tUl Kbtpy; &.1800k; vdf; nfhz;lhy; filrp tUlj;jpy; mtuJ cah;T &.9>000. ,q;F nghJtpj;jpahrk; 600. vdNt 9,000 = 15, 15 tUlq;fSf;F gpwF mjhtJ 2025k; tUlj;jpy; 600 mtuJ tUkhdk; &.39.000 Mf ,Uf;Fk;. gapw;rp 2.2 - Nfs;tp vz;: 5

4>9>14>.vd;w $l;Lj;njhlh;thpirapy; 17 MtJ cWg;G ahJ? 4>9>14>.vd;w $l;Lj;njhlh;thpirapy; nghJ tpj;jpahrk; 5. vdNt midj;J cWg;GfSld; 1I $l;bdhy; fpilf;Fk; njhlh;thpir 5>10>15> ,j;njhlhpy; 17tJ cWg;G 17*5=85. ,g;nghOJ 85ypUe;J 1I fopf;f Ntz;Lk; (Vw;fdNt ehk; $l;baJ) 17tJ cWg;G 84.

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gapw;rp 2.2 - Nfs;tp vz;: 6

gpd; tUk; $l;Lj ;njhlh;thpirapy; nkhj;jk; vj;jid cWg;Gfs; cs;sd. . 7,13,19,205.

,e;j njhlhpy; nghJ tpj;jpahrk; 6 Mf cs;sJ> vdNt ,j;njhlh; MdJ

6>12>18>..204. vdNt ,j; njhlhpy; cs;s cWg;Gfspd; vz;zpf;if 204/6=34.

gapw;rp 2.2 - Nfs;tp vz;: 11

xU njhiyf;fhl;rpg; ngl;bj;; jahhpg;ghsh; VohtJ Mz;by; 1000 njhiyf;fhl;rpg; ngl;bfisAk;> gj;jhtJ Mz;by; 1450 njhiyf;fhl;rpg; ngl;bfisAk; jahhpj;jhh;. Xt;nthU Mz;Lk; jahhpf;Fk; njhiyf;fhl;rpg; ngl;bfspd; vz;zp;fif rPuhfTk; xU khwpyp vz; msTk; mjpfhpj;jhy;> Kjyhk; Mz;bYk;> 15 MtJ Mz;bYk; jahhpf;fg;gl;l njhiyf;fhl;rpg; ngl;bfspd; vz;zpf;ifiag; fhz;f. 7tJ tUlk; =1000 10tJ tUlk; =1450 vdNt tpj;jpahrk;> 3 tUlk; =450. 1 tUlk; = 150 Kjy; tUlj;jpd; jahhpg;G = 1000 900 (6 tUl jahhpg;G, 6x150=900) = 100

15tJ tUl jahhpg;G = 100 + 14x150 = 100 + 2100

= 2200. gapw;rp 2.2 - Nfs;tp vz;: 12

xUth;> Kjy; khjk; &.640> 2Mk; khjk; &.720> 3Mk; khjk; &.800>-I Nrkpf;fpwhh;. Mth; jd;Dila Nrkpg;ig ,Nj njhlh;thpirapy; njhlh;e;jhy;> 25tJ khjk; mth; Nrkpf;Fk; njhifiaf; fhz;f. nfhLf;fg;gl;l jftypd; gb 640,720,800, ,jd; nghJ tpj;jpahrk; 80. vdNt ,j;njhliu 80ypUe;J Muk;gpf;f xt;nthU cWg;gpYk; 560I fopf;fNtz;Lk;. ,jdhy; fpilf;Fk; njhlh; 80>160>240>25*80 = 2000 ,g;nghOJ ehk; fopj;j 560I 2000I $l;l filrp cWg;G =2560.

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gapw;rp 2.2 - Nfs;tp vz;: 16 xUth; tUlj;jpw;F jdptl;b 14% jUk; KjyPl;by; &.25>000-I KjyPL nra;jhh;. Xt;nthU tUl KbtpYk; fpilf;Fk; mry; kw;Wk; jdptl;b Nrh;e;j nkhj;jj; njhif xU $l;Lj; njhlhthpiria mikf;Fkh? mt;thnwdpy;. 20 Mz;LfSf;Fg; gpwF KjyPl;by; cs;s njhifiaf; fhz;f. S.I. = PNR/100

= 25,000 * 1*14 /100

= 3500.

Kjy; tUl tl;b Rs.3,500, ,uz;lhk; tUl tl;b &.7000> 3k; tUl tl;b &.10>500.. vdpy; 20k; tUl tl;b 20*3,500 = 70,000. vdNt nkhj;j njhif 70,000 + 25,000 = Rs.95,000. vLj;Jf;fhl;L 2.16

5+11+17+.+95 vd;w $l;Lj;njhlhpd; $Ljy; fhz;f.

,e;j $l;Lj;njhlhpy; nghJ tpj;jpahrk; d=6> vdNt midj;J cWg;Gld; 1I $l;l fpilg;gJ 6+12+18+.+96 ,j;njhlh; 6d; klq;fhFk;. ,j;njhlhpy; cs;s cWg;Gfspd; vz;zpf;if 96/6=16 ,g;NghJ $Ljy; fhz 6I nghJthf vLf;f

6(1+2+3+.+16) = 6(n(n+1)) - 16 2

= 6*16*17 - 16

2

= 816 16 = 800 (,q;F 16I fopf;f Ntz;Lk;. Vnddpy; 16 cWg;GfSld; 1I $l;baJ

vLj;Jf;fhl;L 2.20

8 My; tFgLk; midj;J %d;wpyf;f ,ay; vz;fspd; $Ljy; fhz;f.

8 My; tFgLk; %d;wpyf;f vz;fs; 104,112,120,.992. Xt;nthU cWg;GlDk; 4I $l;b 100I fopf;f fpilf;Fk; njhlh;thpir 8,16,24,.896.

,j;njhlhpy; cs;s cWg;Gfspd; vz;zpf;if 896/8 = 112. ,j;njhlhpy; 8I nghJthf vLf;f

/ 8(1+2+3++112) = 8(n(n+1)) +112*100 -112*4 2

= 8(112*113 ) +112*100- 112*4 2

= 61376 (,q;F ehk; $l;lNtz;baJ 112*100 Vnddpy; 112 cWg;GfspypUe;J 100 fopj;jJ. mj;Jld; fopf;f Ntz;baJ

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112*4> xt;nthU cWg;GlDk; 4I $l;baJ)

gapw;rp 2.4 - Nfs;tp vz;: 3 (i)

$l;Lj;njhlhpd; $l;ly; gyidf; fhz;f. 38+35+32+..+2

,j;njhliu jpUg;gp vOj 2+5+8++32+35+38 ,J xU $l;Lj;njhlh;thpirahFk;. ,jd; n;ghJ tpj;jpahrk; d=3 Xt;nthU cWg;GlDk; 1I $l;l ekf;f fpilg;gJ 3d; klq;F. mjhtJ 3+6+9++39.

3+6+9++39 = 3(1+2+3++13)

= 3(n(n+1)) - 13

2

= 3*13*14 -13

2

= 260.

(,q;F 13 I fopf;f Ntz;Lk; Vnddpy; 13 cWg;GfSld; 1I $l;bNdhk;)

gapw;rp 2.4 - Nfs;tp vz;: 14.

8% tPjk; jdptl;b jUk; epWtdj;jpy; xtpnthU Mz;Lk; &.1000 itg;Gj; njhifahf nrYj;jg;gLfpwJ. Xt;nthU Mz;bd; ,Wjpapy; ngWk; tl;biaf; fzf;fpLf. ngWk; tl;bj;njhiffs; xU $l;Lj; njhlh;thpiria mikf;Fkh? mt;thW mikf;f KbAkhdhy;> 30 Mz;Lfspd; Kbtpy; fpilf;Fk; nkhj;j tl;biaf; fhz;f. 30 tUl tl;bj;njhif xU $l;Lj;njhliu mikf;fpwJ. 80+160+240+.+2400

,jd; $Ljy; 80+160+240+.+2400 = 80(1+2+3++30) = 80(n(n+1))

2

= 80*30*31

2

= 37,200.

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gapw;rp 2.6 - Nfs;tp vz;: 1(iv).

$Ljy; fhz;f. 7+14+21+490

$Ljy; =7(1+2+3++70) = 7(n(n+1))

2

= 7*70*71

2

= 17,395

vLj;Jf;fhl;L 3.17

2x3 3x

2 3x +2 vDk; gy;YWg;Gf;Nfhitia xUgbf; fhuzpfshf

fhuzpg;gLj;Jf. 2-3-3+2 0 vdNt (x-1) xU fhuzp my;y. 2 -3 = -3+2 vdNt (x+1) xU fhuzp. 2x3 3x2 3x +2 = (x+1) (2x2 +2) ,q;F 2x3 x = 2x2 , kj;jpapy; cs;s xd; nfOit ngw x3 d; nfOit -1My; ngUf;fp 2*-1=-2, x2 nfOTld; $l;l -5 fpilf;Fk; ,J xd; nfOthFk;. gy;YWg;Gf; Nfhitapd; filrp cWg;ig 1My; tFf;f fpilg;gJ 2. 2x3 3x2 3x +2 = (x+1) (2x2 5x +2) ,g;NghOJ (2x2 5x +2) I vspa Kiwapy; fhuzpg;gLj;Jk; Kiwia gw;wp ghh;g;Nghk;. (2x2 5x +2) y; kj;jpapy;cs;s cWg;ig ,uz;lhf gphpf;f Ntz;Lk;. mt;thW gphpf;ifapy; Kjy; ,uz;L cWg;Gf;fs; kw;Wk; filrp ,U cWg;gfSk; xNu tpfpjj;jpy; ,Uf;f Ntz;Lk;. mjhtJ . 2:4 = 1:2 ,jid vOJk;NghJ 2x2 4x x +2 ,g;;N:;;;;; ;ghJ filrp cWg;gpypUe;J () I nghJthf vLf;f Ntz;Lk;. i.e. (x-2)

2x2 5x -+2 = (x-2)(2x 1) ,g;NghJ 2x2 x = 2x kw;Wk; 2/-2= -1 vdNt 2x3 3x2 3x +2 = (x+1)(x-2)(2x -1)

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gapw;rp 3.5 - Nfs;tp vz;: 1(i). fhuzpg;gLj;Jf. x3 -2x2 5x + 6

nfOf;fspd; $Ljy; 1-2-5+6=0 vdNt (x-1) xU fhuzp. Nkw;$wpa Kiwapy; x3 -2x2 -5x +6 = (x -1)(x2 x 6)

mjhtJ x3 x = x2 NkYk; x3d; nfOit 1 My; ngUf;f fpilg;gJ x2d; nfO. NkYk; x3 -2x2 -5x +6 d; filrp cWg;ig (x-1)d; filrp cWg;ghy; tFf;f Ntz;Lk;. mjhtJ -1My; tFf;f Ntz;Lk;.

NkYk; (x2-x-6) I fhuzpg;gLj;j x

3 x

2 5x +6 = (x-1)(x -3)(x+2).

gapw;rp 3.5 - Nfs;tp vz;: 1(ii). fhuzpg;gLj;Jf. x3 23x2 + 142x 120

1-23+142-120=0 vdNt (x-1) xU fhuzp. X

3 23x

2 +142x 120 = (x 1)(x

2 22x +120)

= (x-1) (x-10)(x-12) x3 x = x2 NkYk; x3d; nfOit 1My; ngUf;fp -23 cld; $l;l fpilg;gJ -22 kw;Wk; -120 -1 =120

gapw;rp 3.5 - Nfs;tp vz;: 1(iv). fhuzpg;gLj;Jf 4x3 5x2 + 7x 6 4-5+7-6=0 vdNt (x-1) xU fhuzp. 4x

3-5x

2+7x-6 = (x-1)( 4x

2 x+6) ,q;F 4x3 x = 4x2 kw;Wk; x3d; nfOit1

My; ngUf;f fpilg;gJ 4 ,ij -5cld; $l;l fpilg;gJ -1 mjhtJ x filrp cWg;G -6 I -1My; tFf;f Ntz;Lk;.

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vLj;Jf;fhl;L 3.20

x4+3x

3-x-3 kw;Wk; x3+x2-5x+3 Mfpa gy;YWg;Gf; Nfhitfspd; kP.ngh.t. fhz;f.

,jid $l;ly; fopj;jy; Kiwapy; nra;Nthk;. x4+3x3+0x2-x-3 ..(1) x

3+x

2-5x+3......(2) vd vLj;Jf; nfhs;f.

(1) (2)x =( x4+3x

3+0x

2-x-3) (x

4+x

3-5x

2+3x)

= 2x3+5x

2-4x -3 ..(3)

(3) + (2) = ( 2x3+5x

2-4x -3) + (x

3+x

2-5x+3)

= 3x3+6x2-9x = 3x(x2+2x-3) vdNt kP.ngh.t.= (x2+2x-3)

vLj;Jf;fhl;L 3.21

3x4+6x

3-12x

2-24x kw;Wk; 4x4+14x3+8x2-8x. Mfpa gy;YWg;Gf; Nfhitfspd;

kP.ngh.t. fhz;f. Kjy; Nfhitia 4My; ngUf;f fpilg;gJ 12x4+24x3-48x2-96x (1) 2Mk; Nfhitia 3My; ngUf;f fpilg;gJ ; 12x4+42x3+24x2-24x ..(2)

(1) (2) = -18x3-72x2-72x = -18x(x2+4x+4) vdNt kP.ngh.t.= x(x2+4x+4)

gapw;rp 3.6 - Nfs;tp vz;: 3(i). fPo;fz;l gy;YWg;Gf; Nfhitfspd; kP.ngh.t. fhz;f.

X3-9x

2+23x-15, 4x

2-16x+12

X3-9x

2+23x-15 .(1)

4x2-16x+12 .(2) vd;f.

cah;e;j mLf;Ffis rkd; nra;tjd;%yk; ePf;fyhk;. vdNt rkd;ghL (2) I x My; ngUf;fp 4My; tFf;f fpilg;gJ x3-4x2+3x (3)

(1) (3) =-5x2+20x-15

= -5(x2-4x+3) vdNt kP.ngh.t.= (x2-4x+3).

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gapw;rp 3.6 - Nfs;tp vz;: 3(ii).

3x3+18x

2+33x+18, 3x

2+13x+10 kP.ngh.t. fhz;f.

cah;e;j mLf;if ePf;f 2Mk; cWg;ig xMy; ngUf;f Ntz;Lk;. (3x

3+18x

2+33x+18) (3x

3+13x

2+10x)

= 5x2+23x+18(1)

3x2+13x+10.(2) vd;f.

(1) I 3 MYk; kw;Wk; (2)I 5MYk; ngUf;fp fopf;f Ntz;Lk;. (15x

2+69x+54) (15x

2+65x+50) = 4x+4

= 4(x+1)

vdNt kP.ngh.t.= (x+1) gapw;rp 3.6 - Nfs;tp vz;: 3(iii).

2x3+2x

2+2x+2, 6x

3+12x

2+6x+12 kP.ngh.t. fhz;f.

Kjy; Nfhitapy; 2k; ,uz;lhk; Nfhitapy; 6k; nghJthf vLj;J fopf;f (x

3+x

2+x+1) (x

3+2x

2+x+2) = -x

2-1

= -(x2+1) NkYk; 2>6y; kP.ngh.t.=2

vdNt kP.ngh.t.=2(x2+1).

gapw;rp 3.6 - Nfs;tp vz;: 3(iv). X

3-3x

2+4x-12, x

4+x

3+4x

2+4x kP.ngh.t. fhz;f.

,uz;lhk; Nfhitapy; xI ntspNa vLj;J Kjy; NfhitapypUe;J fopf;f (X

3-3x

2+4x-12) (x

3+x

2+4x+4)

= -4x2-16

= -4(x2+4) vdNt kP.ngh.t= (x2+4)

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gapw;rp 3.10 - Nfs;tp vz;: 2(vii). RUf;Ff. 2x2+5x-3 2x2+x-1

2x2+9x+9 2x

2+x-3

,q;F 2x2+5x-3 y; cs;s ika cWg;ig ,uz;lhf gphpf;f Ntz;Lk;> mjpy; Kjy; ,U vz; kw;Wk; mLj;j ,U cWg;gfspd; tpfpjq;fs; rkkhf ,Uf;f Ntz;Lk;. mjhtJ 2x2+6x-x-3 =(x+3)(2x-1) filrp ,U cWg;Gfspy; (-)I vLf;f Ntz;Lk; ,g;nghOJ 2x2 /x=2x NkYk; -3/3=-1 ,jd; %yk; fpilg;gJ (2x-1) mNjnghy 2x2+9x+9 = 2x2+6x+3x+9 3 nghJthf vLf;fNtz;Lk; = (x+3)(2x+3)

2x2+x-1 = 2x

2+2x- x -1 (-) nghJthf vLf;fNtz;Lk;

= (x+1)(2x -1)

2x2+x-3 = 2x

2+3x-2 x -3 (-) nghJthf vLf;fNtz;Lk ;

= (2x+3)(x-1)

2x2+5x-3 2x

2+x-1 = (x+3)(2x-1) X (2x+3)(x-1)

2x2+9x+9 2x

2+x-3 (x+3)(2x+3) (x+1)(2x -1)

= (x-1) (x+1). gapw;rp 3.12 - Nfs;tp vz;: 2(v). th;f;f %yk; fhz;f. (6x2+5x-6)(6x2-x-2)(4x2+8x+3) Nkw;$wpa Kiwapy; fhuzpgLj;j 6x

2+5x-6 = 6x

2+9x-4x-6 = (2x+3)(3x-2)

6x2-x-2 = 6x

2-4x+3x-2 = (3x-2)(2x+1)

4x2+8x+3= 4x

2+6x+2x+3= (2x+3)(2x+1)

vdNt th;f;f%yk; = (2x+3)(3x-2)(2x+1).

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vLj;Jf;fhl;L 3.33

th;f;f %yk; fhz;f. x4-10x3+37x2-60x+36

________X2 -5x +6____________

X2 X

4 10x

3 + 37x

2 60x +36

+ X2 x

4________________________

2x2 -5x -10x

3 + 37x

2

-5x -10x3 + 25x

2____________

2x2-10x +6 12x

2 60x + 36

12x2 60x + 36

0 ,q;F

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Nfhz ,Urk ntl;b Njw;wk ;

xU Kf;Nfhzj;jpy; xU Nfhzj;jpd; cl;Gw (ntspg;Gw) ,Urkntl;bahdJ mf;Nfhzj;jpd; vjh; gf;fj;ij clGwkhf (ntspgGwkhf)> mf;Nfhzj;jpid mlf;fpa gf;fq;fspd; tpfpj;jpy; gphpf;Fk;. cl;Gw Nfhz ,Urkntl;b nfhLf;fg;gl;Ls;sit: ABC y; BAC clGw ,Urkntl;b AD, mJ BC-I Dy; re;jpf;fpwJ. epUgpf;f: BD = AB DC AC

mikg;G: DA-f;F ,izahfTk; C-apd; topahfTk; xU ,izNfhL tiuf. mf;NfhL BA-tpd; ePl;rpia Gs;sp Ey; re;jpf;fl;Lk; vdNt> CE II DA

1= BAD, 2= DAC, 3= ACE, 4= AEC, vd;f. ( glj;ij Gj;jfj;jpypUg;gij tiue;J fhs;sTk; )

1= 4 xj;jNfhzq;fs; rkk;. .. (1) 2= 3 xd;W tpl;l Nfhzq;fs; rkk;..(2) 1= 2 AD Nfhz ,Urkntl;b .(3) (1), (2), (3) ypUe;J ekf;F fpilg;gJ 3= 4

ACE ,Urkgf;f Kf;Nfhzk; AE=AC

,g;.NghJ BCE y; CE II DA BD BA

---- = ----

DC AE (Njy;]; Njw;wk;)

BD AB

---- = ----

DC AC (AE=AC) Njw;wk; epUg;gpf;fg;gl;lJ..

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13

gpjhfu]; Njw;wk; xU Kf;Nfhzj;jpy; xU gf;fj;jpd; th;f;fk;> kw;w ,U gf;fq;fspd; th;f;fq;fspd; $LjYf;Fr; rkk;. nfhLf;fg;gl;Ls;sit: ABC-y; A=90 epUgpf;f: BC2 = AB2 + AC2 mikg;G: AD BC tiuf. Gj;jfj;jpy; cs;s glj;ij tiue;J nfhs;f. ABD ABC Kf;Nfhzq;fspy; gug;Gfspd; tpfpjkhdJ mtw;wpd; gf;fq;fspd; tpfpjq;fspd; tpfpjj;jpw;F rkkhFk;.

ABD = AB2 ABC BC

2 .(1)

mNjNghy AD C ABC

AD C = AC2

ABC BC2

...(2)

(1)+(2) AB2

+ AC2 = ABD + AD C

BC2 BC

2 ABC ABC

= ABC =1

ABC

AB2+AC

2 = BC

2

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14

cauq;fSk; J}uq;fSk;

,U nghUl;fs; (Gs;spfs;) Vw;w Nfhzj;jpy; ,Uf;Fk; NghJ a

h = ___________ I h= NfhGuj;jpd; cauk; Tan - Tan a= ,U nghUl;fSf;F .,ilgl;l J}uk;

,U nghUl;fs; (Gs;spfs;) ,wf;f Nfhzj;jpy; xNu gf;fj;jpy; ,Uf;Fk; NghJ

a

h = __________.II

Cot - Cot

,U nghUl;fs; (Gs;spfs;) ,wf;f Nfhzj;jpy; vjpnujph; gf;fj;jpy; ,Uf;Fk; NghJ

a

h = __________.III

Cot + Cot

vLj;Jf;fhl;L 7.19

Xh; mjpNtfg; Nghh; tpkhdk;> jiu kl;lj;jpypUe;J 3000 kP cauj;jpy;> kw;nwhU mjpNtfg; Nghh; tpkhdj;ij Neh; Nkyhff; flf;fpwJ. Mt;thW flf;Fk; NghJ jiu kl;lj;jpy; xU Fwpg;gpl;l Gs;spapypUe;J mtw;wpd; Vw;wf; Nfhzq;fs; KiwNa 60 kw;Wk; 45 vdpy;> me;j Neuj;jpy; ,uz;lhtJ Nghh; tpkhdk; kw;Wk; Kjy; Nghh; tpkhdk; Mfpatw;wpw;F ,ilg;gl;l J}uj;ijf; fzf;fpLf.

a

h = ___________ I h= NfhGuj;jpd; cauk; Tan - Tan a= ,U nghUl;fSf;F .,ilgl;l J}uk;

3000 - h = h

Tan 60 tan 45

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15

3000 - h = h

3 1

h= 1268 m

vLj;Jf;fhl;L 7.21

xU nrq;Fj;jhd RtUk;> xU NfhGuKk; xU Fwpg;gpl;l ,ilntspapy; cs;sd. NfhGuj;jpd; cr;rpapypUe;J ghh;f;Fk; NghJ> Rtw;wpd; cr;rp kw;Wk; mb Mfpatw;wpd; ,wf;ff; Nfhzq;fs; KiwNa 45 kw;Wk; 60 MFk;. NfhGuj;jpd; cauk; 90 kP vdpy;> Rtw;wpd; cauj;ijf; fhz;f.

h= a

cot - cot

90= a

cot 45 - cot 60

90= a

1 1/3

a = 38.04m

vLj;Jf;fhl;L7.22

flw;fiuapy; cs;s nrq;Fj;jhdg; ghiw xd;wpd; kPJ fl;lg;gl;Ls;s xU fyq;fiu tpsf;fj;jpy; epd;Wf;nfhz;bUf;Fk; xU rpWkp> fpof;fjpirapy; ,U glFfisg; ghh;f;fpwhs;. Mg;glFfspd; ,wf;ff;Nfhzq;fs; KiwNa 30>60 kw;Wk; ,U glFfSf;fpilNaAs;s J}uk; 300kP vdpy;> fly ;kl;lj;jpypUe;J fyq;fiu tpsf;fj;jpd; cr;rpapd; J}uj;ijf; fhz;f. ( glFfSk;> fyq;fiu tpsf;fKk; xNu Neh;Nfhl;by; cs;sd) h= a

cot - cot h= 300

cot 30 - cot 60

h= 300

3 - 1/3

h= 1503 m

Fwpg;G: vLj;Jf;fhl;L fzf;Ffs ; 7.23, 7.24, 7.26, I formula I I gad;gLj;jp nra;ayhk;.

gapw;rp7.2; fzf;Ffs; 10, 11,12,13, 14, 15, Mfpatw;iw formula II gad;gLj;jp nra;ayhk;. gapw;rp7.2; fzf;Ffs; 8I formula III I gad; g:Lj;jp nra;ayhk;

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16

fzf;FfSf;F Njitahd glq;fis Gj;jfj;jpypUe;J tiue;J nfhs;sTk;