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1

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a,b,c,d vd;gd bgUf;Fj; bjhlupy; cs;sd vdpy;

2

vdpy;

I fhz;f.

ep\gp

a, ar, ar2, ar3 bgUf;Fj; bjhlupy; cs;sd. a, b = ar, c = ar2, d = ar3 vd;f. LHS =

= RHS xU bgUf;Fj; bjhlupd; Kjy; n, 2n, 3n Mfpa cWg;g[fspd; TLjy; KiwBa vdpy; vd ep\gp

3

4

UD Y

=

rkd;ghL

}yA;fs; rkk; vdpy;

nd;

vd epWt[f.

( ) }yA;fs; rkk; vdpy;,

(

)

(

)

ST

0

OZ Y

4 –My; nUg[wKk; tFf;f,

(m-J) (m-J)

----------①

epWtg;gl;lJ.

5 a, b, c, d Mfpad xU bgUf;Fj; bjhlu;tupirapy; mike;jhy;,

epWt[f.

DO

a, b, c, d Mfpad xU bgUf;Fj; bjhlu; a, b = ar , c = ar2 , d = ar3 vd;f.

----------②

@

①

②

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epWtg;gl;lJ.

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ad – bc ≠0 vd mike;j rkd;ghL

7

nd;}yA;fs; rkbkdpy;

vd epWt[f.

Tl;Lj;bjhlu; tupirapy; nUg;gpd;

Mfpadt[k; Tl;Lj;bjhlu; tupirapy; nUf;Fk; vdf; fhl;Lf. Tl;Lj;bjhlu; tupirapy; cs;sd.

( ) }yA;fs; rkk; vdpy;,

( bghJ tpj;jpahrA;fs; rkk; )

0

UD Y

My; nUg[wKk; tFf;f,

(-1) My; nUg[wKk; bgUf;f ,

⟹ ad – bc =0 ⟹

ST

⟹ ad = bc epWtg;gl;lJ.

@

vdpy;

≠

≠

kw;Wk;

Mfpad xU Tl;Lj; bjhlu; tupirapy;

nUf;Fk; vdf; fhl;Lf. vd;f.

@

9

vd;f.

DO

⇨

10

xU bgUf;Fj; bjhlu;tupirapy;

a, b, c, d Mfpad xU bgUf;Fj; bjhlu;. bgUf;F tpfpjk; r @

vdBt

a, b, c, d Mfpad

mike;jhy;, vd;gita[k; bgUf;Fj; bjhlu;tupirapy; mika[k; vd epWt[f.

OZ Y

≠

8

Tl;Lj;bjhlu; tupirapy; nUf;Fk;.

a, b = ar , c = ar2 , d = ar3 MFk;.

= r

xU Tl;Lj; bjhlu; tupirapy; nUf;Fk;

Tl;Lj;bjhlu; tupirapy; nUg;gpd; vdf; fhl;Lf.

= r

Tl;Lj;bjhlu; tupir

@

tu;f;fg;gLj;j ,

bgUf;F tpfpjA;fs; rkk;. @ mika[k;.

bgUf;Fj;bjhlu; tupirapy;

epWtg;gl;lJ.

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11

12

vd;gd d; }yA;fs; vdpy; Mfpatw;iw }yA;fshff; bfhz;l nUgor; rkd;ghL xd;wpid mikf;f.

vd;gd d; }yA;fs; vdpy; Mfpatw;iw }yA;fshff; bfhz;l nUgor; rkd;ghL xd;wpid mikf;f. -y;

-y; BkYk; ,

BkYk; ,

nUgor;rkd;ghL

@ 13

UD Y

vd;gd d; }yA;fs; vdpy; Mfpatw;iw }yA;fshff; bfhz;l nUgor; rkd;ghL xd;wpid mikf;f. -y;

BkYk; , nUgor;rkd;ghL ( )

( )

*

nUgor;rkd;ghL

@

14

vd;gd d; }yA;fs; vdpy; jiyfPHpfis }yA;fshff; bfhz;l nUgor; rkd;ghL xd;wpid mikf;f.

ST

@

+

15 xU kfpGe;J g[wg;gl Btz;oa Beuj;jpypUe;J 30 epkplk; jhkjkhfg; g[wg;gl;lJ. 150 fp.kP J]uj;jpy; cs;s BrUkplj;ij rupahd Beuj;jpy; brd;wila mjDila tHf;fkhd Btfj;ij kzpf;F 25 fp.kP mjpfg;gLj;j Btz;oapUe;jJ vdpy; kfpGe;jpd; tHf;fkhd Btfj;ijf; fhz;f. kfpGe;jpd; tHf;fkhd Btfk; = x fp.kP/kzp mjpfupf;fg; gl;l gpd; Btfk; = x+25 fp.kP/kzp J]uk; = 150 fp.kP

-y;

OZ Y

BkYk; ,

Beuk;

*

nUgor;rkd;ghL

+

vd;gJ tHf;fkhd Beuk; vd;f. vd;gJ Btfj;ij mjpfg;gLj;jpajhy; vLj;Jf;bfhz;l Beuk; vd;f. @

@

kzp

16 mirtw;w ePupy; xU nae;jpug;glfpd; Btfk; kzpf;F 15 fp.kP

vd;f. mg;glF ePBuhl;lj;jpd; jpirapy; 30 fp.kP J]uk;

Beuk;

DO

brd;W gpwF vjpu; jpirapy;jpUk;gp 4 kzp 30 epkplA;fspy; kPz;Lk; g[wg;gl;l nlj;jpw;F jpUk;gp te;jhy; ePupd; Btfj;ijf; fhz;f. ePupd; Btfk; = x fp.kP/kzp nae;jpug; glfpd; Btfk; = 15 fp.kP/kzp

⟹

⟹

1800 =

9(225-x2

7500 = x2 + 25x x2 + 25x – 7500 = 0 ⟹ (x-75)(x+100)=0 x = 75 my;yJ

x = -100 ( bghUe;jhJ )

@ kfpGe;jpd; tHf;fkhd Btfk; = 75 fp.kP/kzp

)

200 = 225-x2 ⟹ x2 = 225-200 ⟹ x2 = 25 ⟹x = x = 5 my;yJ x = -5 ( bghUe;jhJ ) @ ePupd; Btfk; = 5 fp.kP/kzp

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18 xU tUlj;jpw;F Kd;g[ xUtupd; taJ mtUila

J]uj;ijf; fle;jJ. mjDila Btfk; kzpf;F 15 fp.kP mjpfg;gl;oUe;jhy; gazk; bra;a[k; Beuk; 30 epkplA;fs; Fiwe;jpUf;Fk; vdpy; bjhlu; tz;oapd; rPuhd Btfk; fhz;f. bjhlu; tz;oapd; tHf;fkhd Btfk; = x fp.kP/kzp mjpfupf;fg; gl;l gpd; Btfk; = x+15 fp.kP/kzp J]uk; = 90 fp.kP

⟹

Beuk;

kfdpd; taijg; Bghy; 8 klA;F. jw;BghJ mtUila taJ kfdpd; tajpd; tu;f;fj;jpw;Fr; rkk; vdpy; mtu;fSila jw;Bghija taijf; fhz;f. kfd;

x Mz;Lfs; y Mz;Lfs; xU tUlk; Kd;g[ x-1 Mz;Lfs; y-1 Mz;Lfs; xU tUlk; Kd;g[ , y – 1 = 8(x – 1) y – 1 = 8x – 8 ⟹ y = 8x – 7 ------① jw;Bghija taJ

y = x2 ------②

jw;BghJ ,

② I ①y;

x2 + 15x – 2700 = 0 ⟹ (x-45)(x+60)=0 x = 45 my;yJ

x = -60 ( bghUe;jhJ )

@ bjhlu; tz;oapd; tHf;fkhd Btfk; = 45 fp.kP/kzp

19 xU nuapy; epiyaj;jpypUe;J nuz;L bjhlu; tz;ofs;

20 xU Btiyia bra;a A -f;F B-ia tpl 6 ehl;fs;

Fiwthfj; Bjitg;gLfpwJ. nUtUk; Bru;e;J mt;Btiyiar; bra;jhy; mij 4 ehl;fspy; Kof;f naYk; vdpy;, B jdpBa mt;Btiyia vj;jid ehl;fspy; Kof;f naYk;?

OZ Y

xBu Beuj;jpy; g[wg;gLfpd;wd. Kjy; tz;o Bkw;F jpiria Behf;fpa[k; , nuz;lhk; tz;o tlf;F jpiria Behf;fpa[k; gazk; bra;fpd;wd. Kjy; tz;oahdJ nuz;lhtJ tz;oia tpl kzpf;F 5 fp.kP

⟹ x2 = 8x – 7 ⟹ x2 – 8 x + 7 = 0 ⟹ (x – 7)(x – 1) = 0 ⟹ x = 7 x = 1 ( bghUe;jhJ ) @ x = 7 vdpy; ② ⟹ y = 72 @ y = 49 @ kfdpd; taJ = 7 Mz;Lfs; @ je;ijapd; taJ = 49 Mz;Lfs; gpujpapl

ST

2700 = x2 + 15x

je;ij

UD Y

17 rPuhd Btfj;jpy; xU bjhlu; tz;oahdJ 90 fp.kP

mjpf Btfj;jpy; bry;fpwJ. nuz;L kzp

Beuj;jpw;Fg; gpwF mtw;wpw;F nilBaa[s;s bjhiyt[ 50 fp.kP vdpy; xt;bthU tz;oapd; ruhrup Btfj;jpidf; fhz;f.

nuz;lhk; tz;oapd; Btfk; = x fp.kP/kzp Kjy; tz;oapd; Btfk; = x + 5 fp.kP/kzp gpjhfu^; Bjw;wg;go ,

B xU Btiyia bra;a MFk; ehl;fs; = x vd;f

@ A xU Btiyia bra;a MFk; ehl;fs; = x – 6 A xU ehspy; bra;a[k; Btiy B xU ehspy; bra;a[k; Btiy

nUtUk; xU ehspy; bra;a[k; Btiy

DO

⟹

8 My; nUg[wKk; tFf;f,

x2 + 5x – 300 = 0 ⟹ (x-15)(x+20)=0 x = 15 my;yJ

x = -20 ( bghUe;jhJ )

@ nuz;lhk; bjhlu; tz;oapd; Btfk; = 15 fp.kP/kzp @ Kjy; bjhlu; tz;oapd; Btfk; = 20 fp.kP/kzp

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4(2x – 6)=x(x – 6) 8x – 24 = x2 – 6x x = 12 my;yJ

⟹ (x-12)(x – 2)=0 x = 2 ( bghUe;jhJ )

@ B xU Btiyia bra;a MFk; ehl;fs; = 12

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ep\gpf;f : mikg;g[ : BE, CD fisr; Bru;.

⊥

⊥

ep\gzk; : 1)

2)

gpjhfu^; (bgsjad;) Bjw;wj;ij Twp epWt[f xU brA;Bfhz Kf;Bfhzj;jpy; fu;zj;jpd; tu;f;fkhdJ mjd; kw;w nU gf;f mst[fspd; tu;f;fA;fspd; TLjYf;Fr; rkk;.

22

jut[ : y; A = 900 2 ep\gpf;f : AB +AC2=BC2 mikg;g[ : AD  BC tiuf. ep\gzk; : 1) –tobthj;jit

AB= BC . BD ------- ①

tiuf. 2)

fSf;F EF Fj;Jauk; MFk;.

3) ① ② fisf; Tl;l , AB2+AC2=BC . BD + BC . DC = BC . (BD+DC) = BC . BC = BC2 Bjw;wk; epWtg;gl;lJ.

fSf;F DG Fj;Jauk; MFk;.

xU rpWtd; ituj;jpd; FWf;F btl;Lj; Bjhw;w totpy;, glj;jpy; fhl;oathW xU gl;lk; bra;jhd;. nA;F AE =16 br.kP , EC =81 br.kP . mtd; BD vd;w FWf;Ff; Fr;rpapidg; gad;gLj;j tpUk;g[fpwhd;. mf;Fr;rpapd; ePsk; vt;tst[ nUf;fBtz;Lk;? bfhLf;fg;gl;l glj;jpy; xU brA;Bfhz Kf;Bfhzk;. BkYk; DE AC tobthj;jit.

bfhz;Lk;,

(1), (2), (3) fspd; go vd epWtg;gl;lJ. 23 Bfhz nUrkbtl;oj; Bjw;wj;ij Twp epWt[f.

ST

xBu mog;gf;fk; vdt[k; cs;sd.

- tobthj;jit

AC= BC . DC ------- ②

24 3)

UD Y

mog;gil tpfpj rk Bjw;wj;ij epWt[f. ( my;yJ ) Bjy;^; Bjw;wj;ij epWt[f. xU Beu;f;BfhL xU Kf;Bfhzj;jpd; xU gf;fj;jpw;F nizahft[k; kw;w nuz;L gf;fA;fis btl;LkhWk; tiuag;gl;lhy; mf;BfhL mt;tpUg; gf;fA;fisa[k; rk tpfpjj;jpy; gpupf;Fk;. jut[ : y;

21

OZ Y

xU Kf;Bfhzj;jpy; xU Bfhzj;jpd; cl;g[w (btspg;g[w) nUrkbtl;oahdJ mf;Bfhzj;jpd; vjpu;gf;fj;ij cl;g[wkhf (btspg;g[wkhf) mf;Bfhzj;jpd; mlf;fpa gf;f'A;fspd; tpfpjj;jpy; gpupf;Fk;. jut[ : y; AD vd;gJ BAC -d; cs; nUrk btl;o. mJ BC I D-y; re;jpf;fpwJ. ep\gpf;f :

DO

mikg;g[ DA f;F nizahf C tHpahf tiuag;gl;l nizBfhl;il BA -d; ePl;rp E –y; re;jpf;fpwJ. ep\gzk; : 1) , BkYk; AC xU FWf;Fbtl;o BAD = AEC (xj;j BfhzA;fs; rkk;) DAC = ACE (xd;W tpl;lBfhzA;fs; rkk;) BAD = DAC (AD Bfhz nUrk btl;o)

2) 3)

@ AEC = ACE AE = AC (rk BfhzA;fSf;F vjpu;gf;fAfs; rkk;) Bjy;^; Bjw;wj;jpd;go,

(

AE = AC )

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@ ED2 = EA x EC = 16 x 81 @ ED = √ = 4 x 9 = 36 xU nUrkgf;f brA;Bfhz Kf;Bfhzk;. BkYk; AE BD. @ BE = ED BD = 2 ED = 2 x 36 =72 br.kP

25 xU jhkiug; g{thdJ

jz;zPu; kl;lj;jpw;F Bky; 20br.kP cauj;jpy; cs;sJ. kPjpg;gFjp jz;zPu; kl;lj;jpw;F fPBH cs;sJ. fhw;W tPRk;BghJ jz;L js;sg;gl;L jhkiug;g{thdJ jz;od; Muk;g epiyapypUe;J 40br.kP J]uj;jpy; jz;zPiuj; bjhLfpwJ. Muk;g epiyapy; jz;zPu; kl;lj;jpw;F fPBH cs;s jz;od; ePsk; fhz;f.. B = jhkiug;g{ vd;f. AB, OA vd;gJ ePu;kl;lj;jpw;F Bky;, ePu;kl;lj;jpw;F fPH; jz;od; ePsk; vd;f. OA = x vd;f. fhw;W tPRk;BghJ jz;L jz;zPupd; Bkw;gug;ig bjhLk; g[s;sp C vd;f. OC = OA + AB = x + 20 gpjhfu^; Bjw;wg;go , ( x+20 )2 = x2 + 402 x2 + 40x + 400 = x2+1600 40x =1200 @ x =30 Muk;g epiyapy; ePu; kl;lj;jpw;F fPH; jz;od; ePsk; 30 br.kP http://doozystudy.blogspot.in/

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25 27 700 kP cauj;jpy; gwe;J bfhz;oUf;Fk; xU b#ypfhg;lupypUe;J xUtu; Xu; Mw;wpd; nU fiufspy; Bebujpuhf cs;s nU bghUl;fis nwf;f BfhzA;fspy; fhz;fpwhu; vdpy; Mw;wpd; mfyj;ijf; fhz;f. CD = b#ypfhg;lu; gwf;Fk; cauk; = 700 kP AB = nU bghUl;fSf;F nilBa[s;s J]uk;

UD Y

http://doozystudy.blogspot.in/ Beu; f;Fj;jhd xU kuj;jpd; Bky;ghfk; fhw;wpdhy; 26 Kwpe;J mk;Kwpe;j gFjp fPBH tpGe;Jtplhky;, kuj;jpd; cr;rp jiua[ld; Bfhzj;ij Vw;gLj;JfpwJ. kuj;jpd; cr;rp mjd;moapypUe;J 30 kP bjhiytpy; jiuiaj; bjhLfpwJ vdpy; kuj;jpd; KG cauj;ija[k; fhz;f. AB kuj;jpd; cilahj gFjp BC kuj;jpd; cile;j gFjp brA;Bfhz ABC y; ,

brA;Bfhz CDB y; ,

⟹

√ √ √ √

√

⟹

brA;Bfhz ADC y; ,

√

⟹

√ √

⟹

√

kuj;jpd; KG cauk;

√

Mw;wpd; mfyk;

√

√

√

√

√

(√

kP

)

x 2.732

kP

28 tFg;giwapy; mku;e;J bfhz;oUf;Fk; xU

25 29 xU Bfhg[uj;jpd; moapypUe;J xU Fd;wpd; cr;rpapd; Vw;wf;Bfhzk; vd;f. Fd;wpd; moapypUe;J Bfhg[uj;jpd; cr;rpapd; Vw;wf;Bfhzk; kw;Wk; Bfhg[uj;jpd; cauk; 50 kP vdpy; Fd;wpd; cauj;ijf; fhz;f. AD= Bfhg[uk; =50 kP BC =Fd;W = x kP vd;f brA;Bfhz DAB y; ,

brA;Bfhz ACD y; ,

√

⟹

√

)

kP

DO

vdpy;

(√

√

khztd; efu;e;j J]uk; =

30

⟹

√

√ brA;Bfhz CAB y; ,

⟹

√

OZ Y

ST

khztd; fUk;gyifapy; fpilepiyg; ghu;itf; Bfhl;oypUe;J 1.5kP cauj;jpy; cs;s Xtpaj;ij Vw;w Bfhzj;jpy; fhz;fpwhd;. Xtpak; mtDf;Fj; bjspthfj; bjupahjyhy; Beuhf fUk;gyifia Behf;fp efu;e;J kPz;Lk; me;j Xtpaj;ij Vw;w Bfhzj;jpy; bjspthf fhz;fpwhd; vdpy; mtd; efu;e;j J]uj;ijf; fhz;f. CD = fUk;gyifapy; Xtpaj;jpd; cauk; = 1.5kP AB = khztd; efu;e;j J]uk; = xkP brA;Bfhz BCD y; ,

BkYk;

≠

√ ⟹

√

⟹ √ √ Fd;wpd; cauk;

√ x √ = 150 kP

31

vd ep\gp.

vdf; fhl;Lf.

x

=L.H.S

√

√

√

√

√

√

√ @

=R.H.S

L.H.S==R.H.S epWtg;gl;lJ.

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Fwpg;ghttp://doozystudy.blogspot.in/ pl;l nilbtspapy; cs;sd. Bfhg[uj;jpd; cr;rpapypUe;J ghu;f;Fk; BghJ Rtw;wpd; cr;rp kw;Wk; mo Mfpatw;wpd; nwf;ff;BfhzA;fs; KiwBa kw;Wk; MFk;. Bfhg[uj;jpd; cauk; 90 kP vdpy; Rtw;wpd; cauj;ijf; fhz;f. √ Vw;wf;Bfhzk; = nwf;ff;Bfhzk; AE= Rtu; = h kP vd;f. BD = Bfhg[uk; = 90 kP @ CD = 90 – h brA;Bfhz ECD y; ,

33 flw;fiuapy; cs;s brA;Fj;jhd ghiw xd;wpd;

kPJ fl;lg;gl;Ls;s xU fyA; fiu tpsf;fj;jpy; epd;W http://doozystudy.blogspot.in/ bfhz;oUf; Fk; xU rpWkp fpHf;F jpirapy; nU glFfisg; ghu;f;fpwhs;. mg;glFfspd; nwf;ff;BfhzA;fs; KiwBa kw;Wk; MFk;. nU glFfSf;F nilBa[s;s J]uk; 300 kP vdpy; fly; kl;lj;jpypUe;J fyA;fiu tpsf;fj;jpd; cr;rpapd; J]j;ijf; fhz;f. Vw;wf;Bfhzk; = nwf;ff;Bfhzk; AD = fly; kl;lj;jpypUe;J fyA;fiu tpsf;fj;jpd; cauk; = h kP vd;f. BC = glFfSf;F nilBa[s;s J]uk; = 300 kP AB = x vd;f. brA;Bfhz DAB y; ,

⟹ brA;Bfhz DBA y; ,

√

⟹

√ ⟹

x

√

√

√

√

AB = EC vd;gjhy; ,

√

√

ST

⟹ ⟹

√

√

√ √

√ My; nUg[wKk; bgUf;f ,

Rtw;wpd; cauk;

kP

34 40 kP cauKs;s xU Bfhg[uj;jpd; cr;rp kw;Wk;

⟹ ⟹

√

√

√

fly; kl;lj;jpypUe;J fyA;fiu tpsf;fj;jpd; cr;rpapd;

cauk; √ kP (m) 259.80 kP 60 kP cauKs; s xU Bfhg[uj;jpypUe;J xU 35 fl;llj;jpd; cr;rp kw;Wk; mo Mfpatw;wpd; nwf;ff; BfhzA;fs; KiwBa kw;Wk; vdpy; fl;llj;jpd; cauk; fhz;f. Vw;wf;Bfhzk; = nwf;ff;Bfhzk; AE= fl;llk; = h kP vd;f. BD = Bfhg[uk; = 60 kP @ CD = 60 – h brA;Bfhz ECD y; ,

OZ Y

mo Mfpatw;wpypUe;J xU fyA;fiu tpsf;fj;jpd; cr;rpapd; Vw;wf;BfhzA;fs; KiwBa kw;Wk; vdpy; fyA;fiu tpsf;fpd; cr;rpapypUe;J Bfhg[uj;jpd; mof;F cs;s J]uj;ijf; fhz;f. Vw;wf;Bfhzk; = nwf;ff;Bfhzk; AE = Bfhg[uk; = 40 kP vd;f. BD = fyA;fiu tpsf;fk; AE = BC = 40 @ BD = 40 + x brA;Bfhz ABD y; ,

DO

√

⟹

√

⟹

√

fyA;fiu tpsf;fj;jpd; cauk; = 40+20 =60 kP

⟹

√

=

√

⟹

√ Bfhg[uj;jpd; moapypUe;J fyA;fiu tpsf;fpd; cr;rpf;F kP

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√

√

√

40+x=3x 40=2x x = 20

√

√

√

brA;Bfhz DBA y; ,

brA;Bfhz ECD y; ,

cs;s J]uk;

√

brA;Bfhz CAD y; , √

⟹ AB

UD Y

32 xU brA;Fj;jhd RtUk; xU Bfhg[uKk; xU

⟹

√

x

√

√

√

√

AB = EC vd;gjhy; ,

√

√

√

= 40 fl;llj;jpd; cauk;

kP

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