field work centre jtizg; gupl;ir> khu;r; 2016 fwc term...

juk; - 13 (2016) - khu;r ; - 2016 1 ,ize;j fzpjk; - I(A)

mwpTWj;jy;fs;:

gFjp A ,d; vy;yh tpdhf;fSf;Fk; tpil vOJf. xt;nthU tpdhTf;Fk;

tpilfisj; jug;gl;l ,lj;jpy; vOJf. Nkyjpf ,lk; Njitg;gLnkdpd;> ePu;

Nkyjpfj; jhs;fisg; gad;gLj;jyhk;.

gFjp B ,y; cs;s 7 tpdhf;fspy; tpUk;gpa 5 tpdhf;fSf;F khj;jpuk; tpil

vOJf.

xJf;fg;gl;l Neuk; Kbtile;jJk; gFjp A MdJ gFjp B apw;F NkNy ,Uf;ff;

$bajhf ,U gFjpfisAk; ,izj;Jg; guPl;ir kz;lg Nkw;ghHitahsuplk;

ifaspf;f.

tpdhj;jhspd; gFjp B ia khj;jpuk; guPl;ir kz;lgj;jpypUe;J ntspNa vLj;Jr;

nry;tjw;F mDkjpf;fg;gLk;.

,ize;jfzpjk; I

gFjp tpdh vz; fpilj;j Gs;spfs;

A

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B

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tpdhj;jhs; I ,d;; nkhj;jk;

Rl;nlz;

,ize;jfzpjk;I

,ize;jfzpjk;II

,Wjpg; Gs;;spfs;

aho;. tyaf; fy;tpj; jpizf;fsj;jpd; mDruizAld; njhz;ilkhdhW ntspf;fs epiyak; elhj;Jk;

Field Work Centre jtizg; guPl;ir> khu;r;

Term Examination, March

%d;W kzpj;jpahyq;fs; juk; :- 13 (2016) ,ize;j fzpjk; - I

FWC - 2016

- 2016

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gFjp - A

01) fzpjj; njhFj;jwpTf; Nfhl;ghl;ilg; gad;gLj;jp vy;yh 𝑛 𝜖 ℤ+ ,w;Fk; 7𝑛 I 6 ,dhy;

tFf;f tUk; kPjp 1 vdf; fhl;Lf.

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02) 𝑦 = |𝑥 − 1| , 𝑦 = |3𝑥 − 1| vd;Dk; rkdpypfis xNu tupg;glj;jpy; tiuf. ,t;tiuGfis

cgNahfpj;J rkdpyp |3𝑥 − 1| < |𝑥 − 1| I jpUg;jp nra;Ak; 𝑥 ,d; nka;g;ngWkhdj;

njhilapidf; fhz;f.

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03) 𝑧, 𝑤 vd;gd rpf;fnyz;fshFk;.

|𝑧 − 3 − 2𝑖| ≤ 2 , |𝑤 − 7 − 5𝑖| ≤ 1 vDk; rkdpypfis xNu Mfz; tupg;glj;jpy; fhl;b

|𝑧 − 𝑤| ,d; ,opTg; ngWkhdj;ijf;; fhz;f.

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04) 𝑘 ∈ ℝ ,w;F 𝑓(𝑥) = 𝑘𝑥2 − 4𝑘2𝑥 + 4𝑘3 + 𝐾2 − 2 vdf; nfhs;Nthk;. 𝑓(𝑥) I 𝑎(𝑥 − 𝑏)2 + 𝑐

vDk; tbtpy; vLj;Jiuf;f; ,q;F 𝑎, 𝑏, 𝑐 vd;gd 𝑘 ,d; rhu;gpy; Jzpa Ntz;ba khwpypfs;.

,jpypUe;J vy;yh 𝑥 ∈ ℝ ,w;Fk; 𝑓(𝑥) > 2 MFkhW 𝑘 ,d; ngWkhdq;fisf; fhz;f.

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05) lim𝑥−0

√1+sin4𝑥−sin2𝑥−cos𝑥

𝑥= 2 vdf; fhl;Lf.

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06) 𝑦 = 𝑥2 , 𝑦 = 2 − 𝑥2 Mfpa ,U tisapfisAk; xNu tupg;glj;jpy; tiuf. ,U

tisapfshYk; cs;silf;fg;gl;l gpuNjrj;jpd; gug;gsT 8

3 rJu myFfs; vdf; fhl;Lf.

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07) tisap xd;wpd; gukhdr; rkd;ghL 𝑥 = 𝑎 sec3 𝜃 , 𝑦 = 𝑎 tan3 𝜃 ,dhy; jug;gLfpd;wJ; ,q;F

𝑎 > 0, 0 < 𝜃 < 2𝜋 MFk;. tisapapw;F 𝜃 = 𝛼 ,y; tiuag;gl;l njhlypapd; rkd;ghL 𝑥

sec ∝−

𝑦

tan ∝= 𝑎 vdf; fhl;Lf.

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08) cw;gj;jp 𝑂 tpypUe;J Neu;NfhL ℓ ,w;F tiuag;gl;l nrq;Fj;jpd; mb 𝑁 ,d; Ms;$Wfs;

(𝑎, 𝑏) MFk;. ℓ ,d; rkd;ghl;ilf; fhz;f.

ℓ MdJ 𝑥, 𝑦 mr;Rf;fis KiwNa 𝐴, 𝐵 ,y; ,ilntl;Lfpd;wJ. ∆ 𝑂𝐴𝑁 = 3 ∆ 𝑂𝐵𝑁 vdpd;

3𝑎2 = 𝑏2 vdf; fhl;Lf.

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09) ikak; 𝐶1 (𝑎, 𝑏) I cila tl;lk; 𝑆1 MdJ ikak; 𝐶2 (𝑐, 𝑑) I cila tl;lk; 𝑆2 I

ntspg;Gwkhfj; njhLfpwJ. njhLg;Gs;spapy; cs;s njhlypahdJ cw;gj;jpapD}L nry;fpwJ.

cw;gj;jpapy; ,Ue;J tiuag;gLk; kw;iwa ,U njhlypfSk; xd;Wf;nfhd;W nrq;Fj;jhapd;;

|𝑏𝑐 − 𝑎𝑑| = |𝑎𝑐 + 𝑏𝑑| vdf; fhl;Lf.

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10) 𝜋

2< 𝐴 < 𝜋 vdTk; 3 tan 𝐴 + 4 = 0 vdTk; nfhs;Nthk;.

2 cot 𝐴 − 5 cos𝐴 + 𝑠𝑖𝑛 𝐴 ,d; ngWkhdj;ijf; fhz;f.

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juk; - 13 (2016) - khu;r ; - 2016 1 ,ize;j fzpjk; - I(B)

gFjp - B

11) (a) 𝑝(𝑥) = 𝑎𝑥3 + 𝑏𝑥2 − 16𝑥 − 12 vdf; nfhs;Nthk;. 𝑝(𝑥) ,d; xU fhuzp 𝑥 + 2 vdTk;

𝑝(𝑥) I 𝑥 − 2 ,dhy; tFf;f tUk; kPjp −48 vdTk; jug;gl;Ls;sd.

(i) 𝑎, 𝑏 ,d; ngWkhdq;fisf; fhz;f.

(ii) 𝑝(𝑥)I Vfgupkhzf; fhuzpfspd; ngUf;fkhf vOJf.

(b) 𝑥2+34𝑥−71

𝑥2+2𝑥−7 vd;w NfhitahdJ 5 ,w;Fk; 9 ,w;Fk; ,ilNa ,Uf;fhJ vdf; fhl;Lf.

(c) 𝑥2 − 𝑏𝑥 + 𝑐 = 0 vd;w ,Ugbr; rkd;ghl;bd; %yq;fs; 𝛼, 𝛽 MFk;.

(i) 𝛼2

𝛽,

𝛽2

𝛼 vd;gtw;iw %yq;fshff; nfhz;l ,Ugbr; rkd;ghl;il 𝑏, 𝑐 ,d;

cWg;Gfspy; fhz;f.

(ii) 𝛼2 = 𝛽 vdpd; 𝑏3 = 𝑐(3𝑏 + 𝑐 + 1) vdf; fhl;Lf.

12) (a) xU guPl;irf;Fj; Njhw;Wk; guPl;rhu;j;jp xUtu; 𝐴, 𝐵, 𝐶 vd;Dk; %d;W gFjpfspd; fPo;

xt;nthU gFjpapYk; KiwNa 5, 4, 3 tpdhf;fs; tPjk; jug;gl;Ls;s ,g;gd;dpuz;L

tpdhf;fspy; MW tpdhf;fspw;F tpil vOj Ntz;Lk;.

(i) xt;nthU gFjpapYk; Kjyhk; tpdh fl;lhakhdJ>

(ii) mtu; ve;j xU gFjpapYk; %d;W tpdhf;fspw;F Nkw;gl tpil vOjtpayhJ>

vdpd; NkNy jug;gl;l xt;nthU tifapYk; mg;guPl;rhu;j;jp MW tpdhf;fisj;

njupe;njLf;fj;jf;f ntt;NtW topfspd; vz;zpf;ifiaf; fhz;f.

(b) 𝑛 xU ,ul;il vz; MFk; NghJ 13 + 3.22 + 33 + 3.42 + 53 + 3.62 + .................. vDk;

njhlupd; Kjy; 𝑛 cWg;Gfspd; $l;Lj;njhif 𝑛

8 [𝑛3 + 4𝑛2 + 10𝑛 + 8] vdf; fhl;Lf.

,jpypUe;J 𝑛 xw;iwahFk; NghJ 𝑛 cWg;Gfspd; $l;Lj;njhif

1

8(𝑛 + 1)[𝑛3 + 7𝑛2 − 3𝑛 − 1] vd;gij ca;j;jwpf.

13) (i) 𝑧1 = 5 + 𝑖 vdTk; 𝑧2 = −2 + 3𝑖 vdTk; jug;gl;Ls;sd.

(a) |𝑧1|2 = 2|𝑧2|2 vdf; fhl;Lf.

(b) arg(𝑧1, 𝑧2) ,idf; fhz;f.

(ii) 16 − 30𝑖 ,d; tu;f;f %yq;fis 𝑎 + 𝑖𝑏 tbtpy; jUf. ,q;F 𝑎, 𝑏 vd;gd nka;naz;fs;.

(iii) 𝑃, 𝑄, 𝑅 Mfpad Mfz; tupg;glj;jpy; KiwNa 𝑧𝑜, 𝑧1, 𝑧2 vd;Dk; rpf;fnyz;fis

tiff;Fwpf;Fk; %d;W NtW Ntwhd Gs;spfshFk;. 𝑃𝑄 = 𝑃𝑅 MfTk; 𝜃 MdJ 𝑃𝑄

,ypUe;J 𝑃𝑅 ,w;F ,lQ;Ropg; Nghf;fpy; msf;fg;gl;l NfhzkhfTk; ,Ug;gpd;

(𝑧2 − 𝑧𝑜) = (𝑧1 − 𝑧𝑜) (cos𝜃 + 𝑖 sin 𝜃) vdf; fhl;Lf.

Mfz; tupg;glj;jpy; |𝑧 − 1| = √2 vDk; tl;lj;ijr; Rw;wp tiuag;gLk; rJuj;jpd; xH

cr;rp 2 + √3 𝑖 vdpd; rJuj;jpd; kw;iwa cr;rpfisf; fhz;f.




FWC

- 2016 - 2016

juk; :- 13 (2016) ,ize;j fzpjk; - I

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14) (a) 𝑦 = 𝑒 tan−1𝑥 + 𝑒−tan−1 vdpd; (1 + 𝑥2)2

𝑑2𝑦

𝑑𝑥2+ 2𝑥(1 + 𝑥2)

𝑑𝑦

𝑑𝑥= 𝑦 vdf; fhl;Lf.

(b) 𝑥 ≠ 1, 4 ,w;F 𝑓(𝑥) = 𝑥

(𝑥−1)(𝑥−4) vdf; nfhs;Nthk;. 𝑓′(𝑥) =

4−𝑥2

(𝑥−1)2 (𝑥−4)2 vdf; fhl;Lf.

jpUk;gw; Gs;spfisAk; mZFNfhLfisAk; fhl;b 𝑦 = 𝑓(𝑥) ,d; tiuig gUk;gbahf

tiuf.

−1 < 𝑘 < −1

9 ,w;F 𝑘(𝑥 − 1)(𝑥 − 4) − 𝑥 = 0 vDk; rkd;ghL nka;j;jPu;Tfisf;

nfhz;buhJ vd;gij ca;j;jwpf.

(c) rJu mbiaf; nfhz;lJk; %b ,y;yhjJk; nkhj;j ntspNkw;gug;gsT 𝑎2 MfTk;

mikAkhW ngl;b xd;iw mikf;f Ntz;bAs;sJ. ngl;bapd; cau;nfhs;ssT 𝑎3

6√3

vdf; fhl;Lf.

15) (a) ∫ 𝑓(𝑥)𝑑𝑥𝑎

𝑜 = ∫ 𝑓(𝑎 − 𝑥)𝑑𝑥𝑎

𝑜 vdf; fhl;Lf. ,jidg; gad;gLj;jp ∫ sin4 𝑥 cos2 𝑥 𝑑𝑥𝜋

2𝑜 =

𝜋

32

vdf; fhl;Lf.

(b) Xu; cfe;j gpujpaPl;ilAk; gFjpfshfj; njhifapLk; KiwiaAk; gad;gLj;jp

∫ 𝑥5 cos𝑥2 𝑑𝑥 If; fhz;f.

(c) gFjpg; gpd;dq;fisg; gad;gLj;jp ∫(𝑥2+2)𝑑𝑥

(𝑥2+1) (𝑥2+4) If; fhz;f.

16) (a) ℓ1 = 𝑎1 𝑥 + 𝑏1𝑦 + 𝑐1 = 0 , ℓ2 = 𝑎2 𝑥 + 𝑏2𝑦 + 𝑐2 = 0 vd;Dk; ,U rkhe;juky;yhj

Neu;NfhLfSf;fpilNa cs;s Nfhzq;fspd; ,U$whf;fpfspd; rkd;ghLfisf; fhz;f.

Kf;Nfhzp 𝐴𝐵𝐶 ,d; gf;fq;fs; 𝐴𝐵, 𝐵𝐶, 𝐶𝐴 ,d; rkd;ghLfs; KiwNa

4𝑥 − 3𝑦 + 2𝑘 = 0, 3𝑥 + 4𝑦 + 𝑘 = 0, 7𝑥 + 𝑦 + 𝑘 = 0 MFk;; ,q;F 𝑘 ∈ ℝ.

(i) 𝐴𝐵 MdJ 𝐵𝐶 ,w;Fr; nrq;Fj;J vdf; fhl;Lf.

(ii) 𝐴B𝐶 ,d; ,U$whf;fpapd; rkd;ghl;il 𝑘 ,d; rhu;gpy; fhz;f.

(iii) 𝐴B𝐶 ,d; ,U$whf;fpahdJ 𝐴𝐶 I 𝐸(−4, 3) ,y; re;jpg;gpd; 𝑘 = 25 vdf; fhl;Lf.

(iv) 𝑘 = 25 vdpd; 𝐴𝐵𝐶𝐷 xU rJuk; MFkhW 𝐷 ,d; Ms;$Wfisf; fhz;f.

(b) (𝑥 − 1)2 + (𝑦 − 3) 2 = 4, (𝑥 + 3)2 + (𝑦 − 1) 2 = 9 Mfpa tl;lq;fs; ,U Gs;spfspy;

,ilntl;Lk; vdf; fhl;b ,Utl;lq;fSf;Fkhd nghJj; njhlypfspd; rkd;ghLfisf;

fhz;f.

17) (a) tan 3𝐴 ,id tan 𝐴 rhu;gpy; jUf. tan 3𝐴

tan 𝐴= 𝑘 vdpd;

sin3𝐴

sin𝐴 ,id 𝑘 rhu;gpy; fhz;f.

(b) sin−1 𝑥 + tan−1 𝑥 = 𝜋

2 vdpd; 2𝑥2 + 1 = √5 vdf; fhl;Lf.

(c) Nfhird; newpiaf; $Wf.

tof;fkhd Fwpg;gPl;by; ahJnkhU Kf;Nfhzpapd; gug;gsT 1

2 𝑏𝑐 sin 𝐴 vd epWTf.

xU tl;l ehw;gf;fypd; mLj;Js;s ,U gf;fq;fspd; ePsq;fs; KiwNa 5𝑐𝑚, 2𝑐𝑚

MFk;. mtw;wpw;F ,ilNa cs;s Nfhzk; 60𝑜 MFk;. mj;Jld; tl;l ehw;gf;fypd;

gug;gsT 4√3 𝑐𝑚2 vdpd; vQ;rpa ,U gf;fq;fspd; ePsq;fisf; fhz;f.

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mwpTWj;jy;fs;:

gFjp A ,d; vy;yh tpdhf;fSf;Fk; tpil vOJf. xt;nthU tpdhTf;Fk;

tpilfisj; jug;gl;l ,lj;jpy; vOJf. Nkyjpf ,lk; Njitg;gLnkdpd;> ePu;

Nkyjpfj; jhs;fisg; gad;gLj;jyhk;.

gFjp B ,y; cs;s 7 tpdhf;fspy; tpUk;gpa 5 tpdhf;fSf;F khj;jpuk; tpil

vOJf.

xJf;fg;gl;l Neuk; Kbtile;jJk; gFjp A MdJ gFjp B apw;F NkNy ,Uf;ff;

$bajhf ,U gFjpfisAk; ,izj;Jg; guPl;ir kz;lg Nkw;ghHitahsuplk;

ifaspf;f.

tpdhj;jhspd; gFjp B ia khj;jpuk; guPl;ir kz;lgj;jpypUe;J ntspNa vLj;Jr;

nry;tjw;F mDkjpf;fg;gLk;.

,ize;jfzpjk; II

gFjp tpdh vz; fpilj;j Gs;spfs;

A

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5

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B

11

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17

tpdhj;jhs; II ,d;; nkhj;jk;

Rl;nlz;

,ize;jfzpjk;I

,ize;jfzpjk;II

,Wjpg; Gs;;spfs;




%d;W kzpj;jpahyq;fs; juk; :- 13 (2016) ,ize;j fzpjk; - II

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gFjp - A

01) ,uz;L Jzpf;iffs; xNu Gs;spapy; ,Ue;J xNu Ntfj;Jld; NtW Ntwhd jpirfspy;

vwpag;gLfpd;wd. xt;nthU tifapYk; tPr;R 𝑅 vdTk; gwg;G Neuq;fs; 𝑡1 , 𝑡2 vdTk;

,Ug;gpd;> 𝑅 =1

2𝑔𝑡1𝑡2 vdf; fhl;Lf.

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02) 100 𝑘𝑚ℎ−1 ,y; fpof;fhf nry;Yk; xU Gifapuj;jpy; cs;s xU gazp> xU Neuhd

tPjpapy; xU fhu; nry;tij ghu;f;fpd;whu;. fhu; 125 𝑘𝑚ℎ−1 ,y; nj 30𝑜 Nk jpirapy;

nry;tjhfj; Njhd;WfpwJ. fhupd; cz;ikahd Ntfk; ahJ?

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03) xt;nthd;Wk; 𝑚 jpzpTs;s 𝐴, 𝐵 vd;Dk; ,uz;L Jzpf;iffs; KiwNa 3𝑢, 𝑢 fjpfSld;

xU Neu;Nfhl;bd; topNa xNu jpirapy; ,aq;fp NkhJfpd;wd. NkhJifapd; gpd;du; mit

mNj jpirapy; KiwNa 𝑢, 𝑘𝑢 fjpfSld; ,aq;Ffpd;wd. kPsikTf; Fzfk; 𝑒 MFk;.

𝑒 = 1 vdf; fhl;Lf.

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04) epiyahd fg;gpapd; Nkyhfr; nry;Yk; ePsh ,ioapd; Edpfspy; 𝑚, 3𝑚 jpzpTs;s

Jzpf;iffs; ,izf;fg;gl;L fg;gpapy; ,Ue;J KiwNa 𝑎, 𝑏 vd;Dk; J}uq;fspy; ,io

,Wf;fkhf ,Uf;FkhW gpbf;fg;gl;L RahjPdkhf tplg;gLfpd;wJ. rf ;jpf; f hg ;G Nf hl;ghl;il

khj;jpuk; gad;gLj;jp> njhFjpapd; Mu;KLfy; 𝑔

2 vdf; fhl;Lf.

,ioapy; cs;s ,Oitiaf; fhz;f.

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05) ,aw;if ePsk; ℓ ck; kPs; jd;ik kl;L 2 𝑚𝑔 ck; cila Xu; ,io mjd; xU Edp Gs;sp

𝐴 ,y; epiyg;gLj;jg;gl;L kW Edp 𝐵 ,w;F 𝑚 jpzpTs;s xU Jzpf;if

,izf;fg;gl;Ls;sJ. Jzpf;if 𝑤 vd;Dk; Nfhzf; fjpAld; xU fpil tl;lj;jpy;

,aq;Ffpd;wJ. ,Oit 3 𝑚𝑔 MFk; NghJ ,io mjd; kPs; jd;ik vy;iyia milfpwJ.

,e;epiy milag;gLk;NghJs;s Nfhzf; fjpiaf; fhz;f.

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06) 6 𝑚 ePsKk; 20 𝑘𝑔 jpzpTk; cila xU rPuhd Vzp 𝐴𝐵 mjd; mb 𝐴 xU fulhd

fpilj;jiu kPJ Xa;tpy; ,Uf;fpd;wJ. Vzp Gs;sp 𝐶 ,y; xU xg;gkhd epiyf;Fj;J Rtupd;

cr;rpf;F vjpuhf Xa;tpy; ,Uf;fpd;wJ; ,q;F 𝐴𝐶 ,d; ePsk; 5𝑚. Vzpiaf; nfhz;l

epiyf;Fj;Jj; jsk; RtUf;Fr; nrq;Fj;jhdJ. mj;Jld; Vzpf;Fk; jiuf;Fk; ,ilNaahd

Nfhzk; 60𝑜 MFk;. 80 𝑘𝑔 jpzpTila xU kdpjd; Vzpapd; kPJ Gs;sp 𝐷 ,y; epw;fpd;whd;;

,q;F 𝐴𝐷 MdJ 4𝑚 MFk;. Vzp tOf;Fk; jWthapy; ,Uf;fpd;wJ. Vzpf;Fk; jiuf;Fk;

,ilapyhd cuha;Tf; Fzfj;ijf; fhz;f.

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07) 𝑎, 𝑏, 𝑐 vd;gd 𝑎 + 𝑏 + 𝑐 = 0 , |𝑎| = 1 , |𝑏| = 2, |𝑐| = 3 MFkhWs;s fhtpfs; vdpd;

𝑎 . 𝑏 + 𝑏. 𝑐 + 𝑐. 𝑎 = −7 vdf; fhl;Lf.

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08) 𝐴, 𝐵 vd;gd xU khjpupntspapy; cs;s ,U epfo;r;rpfs;, 𝑃(𝐵) > 0 vdpd;

(i) 𝑃(𝐴/𝐵) ≥ 𝑃(𝐴)+ 𝑃(𝐵)− 1

𝑃(𝐵) vdf; fhl;Lf.

(ii) 𝐴, 𝐵 rhuhjit vdpd; 𝑃(𝐴 ∪ 𝐵) = 1 − 𝑃( 𝐴 )𝑃 ( 𝐵 ) vd epWTf.

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09) xU ngl;bapy; cs;s 16 ehzaq;fspy; 4 ,y; ,Ugf;fKk; jiy cz;L. kpFjp

Nfhlhjit. xU ehzak; vLf;fg;gl;L Rz;lg;gbd; jiy Njhd;Wk; epfo;jfT ahJ?

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10) Kf;Nfhzp 𝐴𝐵𝐶 ,y; 𝑃, 𝑄, 𝑅 vd;w tpirfs; 𝐵𝐶, 𝐶𝐴, 𝐴𝐵 topNa jhf;Ffpd;wd. tpisAs;

Kf;Nfhzp 𝐴𝐵𝐶 ,d; Rw;Wtl;l ikaj;jpD}lhfr; nrd;why; 𝑃 cos𝐴 + 𝑄 cos𝐵 + 𝑅 cos𝐶 = 0

vdf; fhl;Lf.

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juk; - 13 (2016) - khu;r ; - 2016 1 ,ize;j fzpjk; - II(B)

gFjp - B

11) (a) fpilj;jiu kPJ cs;s Gs;sp 𝑂 ,y; ,Ue;J fjp 2𝑢 cld; epiyf;Fj;jhf Nky;

Nehf;fp Jzpf;if 𝑃 vwpag;gLfpd;wJ. Jzpf;if 𝑃 vwpag;gl;L 𝑢

𝑔 Neuj;jpd;gpd;

Jzpf;if 𝑄 MdJ mNj Gs;sp 𝑂 ,y; ,Ue;J epiyf;Fj;jhf Nky; Nehf;fp 𝑣

Ntfj;Jld; vwpag;gLfpd;wJ. (𝑣 > 𝑢). ,U Jzpf;iffSf;Fkhd Ntf-Neu tiuGfis

xNu tupg;glj;jpy; tiuf.

Ntf - Neu tiuig khj;jpuk; gad;gLj;jp

(i) Jzpf;if 𝑃 mjpAau; cauj;ij mila vLf;Fk; Neuk; 2𝑢

𝑔 vdf; fhl;Lf.

(ii) Jzpf;if 𝑃 milAk; mjpAau; cauk; 2𝑢2

𝑔 vdf; fhl;Lf.

(iii) Jzpf;if 𝑄 MdJ 𝑃 I mjd; mjpAau; Gs;spapy; fle;J nry;Ynkdpd;>

(𝛼) 𝑣 =5𝑢

2 vdf; fhl;Lf.

(𝛽) 𝑃 MdJ jiuia mile;J 2𝑢

𝑔 Neuj;jpd; gpd; 𝑄 jiuia milAk; vdf;

fhl;Lf.

(iv) Jzpf;if 𝑄 mjpAau; Gs;spia mile;j gpd;G 𝑃 MdJ vwpaw; Gs;spia milAk;

vd;gij ca;j;jwpf.

(b) ,uz;L glFfs; 𝐴,𝐵 khwhf; fjpfSld; Neu;topNa nry;fpd;wd. ez;gfy; 𝐴,𝐵

vd;gtw;wpd; jhdf;fhtpfs; ntspr;r tPL njhlu;ghf KiwNa 𝑖 + 2𝑗 , − 𝑖 + 𝑗 MFk;.

Kg;gJ epkplq;fspd; gpd;du; 𝐴, 𝐵 ,d; jhdf;fhtpfs; KiwNa (−𝑖 + 3𝑗) , (2𝑖 − 𝑗)

MFk;.

(i) 𝐴, 𝐵 Mfpatw;wpd; Ntfq;fis fhtp tbtpy; fhz;f.

(ii) ez;gfypd; gpd;du; 𝑡 kzpj;jpahy Neuj;jpy; 𝐴,𝐵 Mfpatw;wpd; jhdf; fhtpfisf;

fhz;f.

(iii) 𝑡 Neuj;jpd; gpd; 𝐵 njhlu;ghf 𝐴 ,d; jhdf;fhtp 𝑟 = (2 − 10𝑡)𝑖 + (1+ 6𝑡)𝑗 vdf;

fhl;Lf.

(iv) 𝐴 Ak; 𝐵 Ak; kpf neUf;fkhf cs;s NghJ Neuk; 𝑡 iaf; fhz;f.

12) (a) 𝑀 jpzpit cila Xu; xg;gkhd Mg;G xU xg;gkhd fpilahd Nkir kPJ cs;sJ.

Mg;gpd; FWf;Fntl;L xU rkgf;f Kf;Nfhzp MFk;. cr;rpapD}lhfr; nryYk;

,ionahd;wpd; Edpfspy; 𝑚1, 𝑚2 jpzpTfs; cs;s Jzpf;iffs; ,izf;fg;gl;L

mit Xg;gkhd rha;Kfq;fspd; kPJ itf;fg;gl;Ls;sd. Mg;gpd; nghJ FWf;F ntl;Lj;

jsj;jpy;> ,io ,Wf;fkhf ,Uf;f njhFjp Xa;tpy; ,Ue;J tpLtpf;fg;gLfpd;wJ.

,ize;j fzpjk; - II




juk; :- 13 (2016)

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- 2016 - 2016

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(i) 𝑚1 > 𝑚2 vdpd;> Mg;gpd; Mu;KLfy; √3(𝑚1−𝑚2)𝑔

3 𝑚1+3𝑚2+4𝑀 vdf; fhl;Lf.

(ii) 𝑀 = 3𝑚, 𝑚1 = 2𝑚 , 𝑚2 = 𝑚 vdpd; ,ioapy; css ,Oitiaf; fhz;f.

(b) cs;shiu 𝑎 I cila Xu; xg;gkhd epiyg;gLj;jg;gl;l Nfhstbtg; ghj;jpuk; xd;wpd;

cl;Gwkhf RahjPdkhf ,aq;fj;jf;f xU Jzpf;if ghj;jpuj;jpd; mjpjho; Gs;sp 𝐴 ,y;

,Ue;J fpilahf fjp 2√𝑎𝑔 cld; vwpag;gLfpwJ.

(i) 𝐴 ,w;F NkNy Xu; cauk; 5𝑎

3 ,y; Jzpf;if ghj;jpuj;jpd; Nkw;gug;ig tpl;L

ntspNaWnkdf; fhl;Lf.

(ii) NkYk; Jzpf;if milAk; mjpAau; cauk; 50𝑎

27 vdTk; fhl;Lf.

13) 𝑚 jpzpTs;s xU Jzpf;if 𝑃 MdJ kPs;jd;ik kl;L 2 𝑚𝑔 Mf cs;s Xu; ,ioapd; xU

EdpAld; ,izf;fg;gl;L ,ioapd; kw;iwa Edp xU rPypq;fpd; xU epiyj;j Gs;sp 𝑂 ,w;F

,izf;fg;gl;L ePl;rp ℓ Mf ,Uf;f ehg;gj;jpy; njhq;Ffpd;wJ. ,ioapd; ,aw;if ePsk; 2ℓ

vdf; fhl;Lf.

𝑂𝑃 epiyf;Fj;jhf ,Uf;f ,ioapd; ePsk; 5ℓ ,w;F rkkhf ,Uf;fj;jf;fjhf ,io

<u;f;fg;gl;L Jzpf;if 𝑃 Xa;tpy; ,Ue;J tpLtpf;fg;gLfpd;wJ. ,ioapd; ePl;rp ℓ + 𝑥 Mf

,Uf;Fk; NghJ (−ℓ < 𝑥 < 2ℓ) Jzpf;if 𝑃 ,d; ,af;fr; rkd;ghl;il vOjp> toikahd

Fwpg;gPl;by; �� =−𝑔

ℓ𝑥 vdf; fhl;Lf.

Nkw;Fwpj;j rkd;ghl;bd; jPu;T 𝑥 = 𝐴cos𝑤𝑡 + 𝐵 sin𝑤𝑡 vdf; nfhz;L 𝐴,𝐵, 𝑤 If; fhz;f.

,jpypUe;J Jzpf;if kPz;Lk; ,aw;if ePsj;ij mila vLf;Fk; Neuj;ijf; fhz;f.

14) (a) Kf;Nfhzp 𝐴𝐵𝐶 ,y; 𝐴𝐷 vd;gJ 𝐵A𝐶 ,d; ,U$whf;fp MFkhW 𝐷 vd;gJ 𝐵𝐶 kPJs;s

Gssp. 𝐵𝐷 = 𝜆 𝐷𝐶 vdf; nfhs;f. 𝐴𝐵 = 𝑎, 𝐴𝐶 = 𝑏 MFk;.

(i) 𝐵𝐷 , 𝐷𝐶 vd;gtw;iw 𝑎 , 𝑏 rhu;ghff; fhz;f.

(ii) 𝐴𝐷 I 𝑎 , 𝑏 rhu;ghff; fhz;f.

(iii) vz;zpg; ngUf;fj;ij fUJtjd; %yk; 𝐴𝐵

𝐴𝐶=

𝐵𝐷

𝐷𝐶 vdf; fhl;Lf.

(b) 𝑥𝑦 jsj;jpy; cs;s ehd;F tpirfisf; nfhz;l xU njhFjp fPNo jug;gl;Ls;sJ.

tpir Gs;spfs;

−2𝑖 (3,3)

−5𝑗 (4,0)

𝑝𝑖 + 𝑗 (−8,−2)

−3𝑗 − 4𝑗 (0, 𝑞)

(i) ,t;tpirfspd; tpisAis 𝑝 ,d; rhu;ghf fhz;f.

(ii) tpirj;njhFjp 𝑦 mr;Rf;Fr; rkhe;jukhfj; njhopw;gLk; xU jdp tpir 𝑭 ,w;Fr;

rktYthdJ vdj; jug;gl;bUf;f 𝑝 ,d; ngWkhdj;ijf; fhz;f.

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(iii) 𝑭 ,d; jhf;ff;NfhL 𝑥 = 3 vd NkYk; jug;gl;bUf;f> 𝑞 ,d; ngWkhdj;ijf;

fhz;f.

(iv) tpirj;njhFjp cw;gj;jpf;$lhfj; njhopw;gLk; xU jdp tpirAld; Xu; ,iz 𝑮

My; gpujpaPL nra;ag;gl KbAk; vdpd; 𝑮 ,d; gUkid vOjp mjd; Nghf;iff;

Fwpg;gpLf.

15) (a) xt;nthd;Wk; epiw 𝑤 it cila 𝐴𝐵,𝐵𝐶, 𝐶𝐷,𝐷𝐴 vd;Dk; ehd;F rPuhd Nfhy;fs; xU

rl;lg;gly; 𝐴𝐵𝐶𝐷 ia Mf;FkhW mtw;wpd; Kidfspy; xg;gkhf %lg;gl;Ls;sd. Nfhy;

𝐴𝐵 apd; eLg;Gs;spAk; %l;L 𝐶 Ak; Xu; ,Nyrhd kPs;jd;ikapd;wpa ,ioapdhy;

njhLf;fg;gl;bUf;Fk;; mNjNtis Nfhy; 𝐴𝐷 apd; eLg;Gs;spAk; %l;L 𝐶 Ak; ,d;Ndhu;

,Nyrhd kPs;jd;ikapd;wpa rk ,ioapdhy; njhLf;fg;gl;Ls;sd. rl;lg;gly; %l;L 𝐴

apypUe;J njhq;ftplg;gl;L> 𝐵A𝐷 = 120𝑜 MfTk; ,Uf;FkhW ehg;gj;jpy;

Ngzg;gLfpd;wJ. xt;nthU ,ioapYk; cs;s ,Oit √3 𝑤 vdf; fhl;Lf.

(b)

tupg;glj;jpy; fhzg;gLk; ,Nyrhd rl;lg;gly; xU epiyf;Fj;Jj; jsj;jpy; ,Uf;Fk;

mNjNtis fpilj;J}uk; 4𝑎 apy; ,Uf;Fk; 𝐴, 𝐶 Mfpatw;wpy; cs;s Mjhuq;fspd; kPJ

ehg;gj;jpy; ,Uf;fpd;wJ. 𝐶B𝐷 = 120𝑜, 𝐴B𝐷 = 90𝑜, 𝐵𝐶 = 2𝑎 , 𝐵𝐷 = 𝑎 MFk;. 𝐵𝐶

fpilahdJ. Gs;sp 𝐵 apypUe;J 12 𝑁 vd;Dk; Rik njhq;ftplg;gbd;> Nghtpd; Fwpg;gPl;g;

gad;gLj;jp xU jifg;G tupg;glj;ij tiue;J> ,jpypUe;J> Nfhy; 𝐴𝐵 ahdJ gUkd;

9 𝑁 cs;s ,Oitia cilanjdf; fhl;Lf.

Vida Nfhy;fs; xt;nthd;wpYk; cs;s jifg;ig mJ ,Oitah> cijg;gh vdf;

fhl;bj; Jzpf.

16) (i) glj;jpYs;s jpz;k $k;gpd;

mbj;Jz;bd; GtpaPu;g;G ikaj;ij

njhifaplyhy; fhz;f.

𝟔𝒓

𝟏𝟎𝒓

𝟐𝒓

𝑫

𝒂

𝟐𝒂

𝟏𝟐 𝑵

𝑩

𝑨

𝑪

1200

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(ii) tpir 𝑃 gpuNahfpf;fg;gLk; NghJ

Jz;lk; tOf;fhJ ftpOk; epiyapy;

,Ug;gpd; Jz;lj;jpd; epiw 𝑤 Mf

𝑃 =3𝑤

2(cos𝜃+sin𝜃) vdf; fhl;Lf.

(iii) 𝑃 ,d; ,opTg; ngWkhdj;ijAk; mg;NghJ 𝜃 ,d; ngWkhdj;ijAk; fhz;f.

17) (a) xU khjpupntsp 𝑆 ,y; cs;s 𝐴, 𝐵 vd;w epfo;r;rpfs; rhuhjit vdpd; 𝐴′ ,𝐵 rhuhjit

vdf; fhl;Lf.

(b) ‘ASSISTANT’ vd;w nrhy;ypy; ,Ue;J Xu; vOj;Jk; ‘STATISTICS’ vd;w nrhy;ypy;

,Ue;J Xu; vOj;Jk; vLf;fg;gLfpd;wJ. mt; vOj;Jf;fs; ,uz;Lk;

(i) xNu vOj;jhf ,Ug;gjw;fhd

(ii) ,U vOj;Jf;fSk; ntt;Ntwhd capnuOj;jhf ,Ug;gjw;fhd

(iii) ,U vOj;Jf;fSk; capnuOj;Jf;fshf ,Ug;gjw;fhd

(iv) ,U vOj;Jf;fSk; xNu nka;naOj;jhf ,Ug;gjw;fhd

epfo;jfTfisf; fhz;f.

𝟓𝒓

𝟑𝒓

𝟐𝒓

𝜽 𝑷

www.aswarp

hysic

s.wee

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www.aswarp

hysic

s.wee

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field work centre jtizg; gupl;ir> khu;r; 2016 fwc term...

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