fzf;f miuahz;l nghjj;njh;t - 2016-17 ntw;wpf;f top...
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12 fzf;F miuahz;L nghJj;Njh;T - 2016-17 ntw;wpf;F top
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fzpjk; gFjp – m
Fwpg;G : (i) midj;J tpdhf;fSf;Fk; fl;lhakhdJ. 𝟒𝟎 × 𝟏 = 𝟒𝟎
(ii) nfhLf;fg;gl;l ehd;F tpilfspy; kpfTk; Vw;Gila tpilapid Njh;e;njLj;J FwpaPl;Lld; tpilapidAk; Nrh;j;J vOJf.
1. xU jpirapyp mzpapd; thpir 3> jpirapyp 𝑘 ≠ 0, vdpy; 𝐴−1 vd;gJ
(1) 1
𝑘2 𝐼 (2) 1
𝑘3 𝐼 (3) 𝟏
𝒌𝑰 (4)kI
2. 3 15 2
vd;gjd; Neh;khW
(1) 𝟐 −𝟏
−𝟓 𝟑 (2)
−2 51 −3
(3) 3 −1
−5 −3 (4)
−3 51 −2
3. kjpg;gpl Ntz;ba %d;W khwpfspy; mike;j %d;W Nehpa mrkgbj;jhd rkd;ghl;Lj; njhFg;gpy;
∆= 0 kw;Wk; ∆𝑥= 0, ∆𝑦≠ 0, ∆𝑧= 0, vdpy;> njhFg;Gf;fhd jPh;T
(1) xNu xU jPh;T (2) ,uz;L jPh;Tfs;
(3) vz;zpf;ifaw;w jPh;Tfs; (4 ) jPh;T ,y;yhik
4. gpd;tUtdtw;wpy;> rkgbj;jhd njhFg;ig nghWj;j tiuapy; vJ rhpahdJ? (1) vg;nghOJNk xUq;fikT mw;wJ
(2) ntspg;gilj; jPh;it kl;LNk ngw;wpUf;Fk; (3) ntspg;gilaw;w jPh;Tfis kl;LNk ngw;wpUf;Fk; (4) nfOf;fs; mzpapd; juk;> khwpfspd; vz;zpf;iff;Fr; rkkhf ,Uf;Fk; NghJ kl;LNk
ntspg;gilj; jPh;tpid kl;Lk; ngw;wpUf;Fk;
5. 2𝑖 +3𝑗 +4𝑘 , a𝑖 + 𝑏𝑗 + 𝑐𝑘 Mfpa ntf;lh;fs; nrq;Fj;J ntf;lh;fshapd;>
(1) a=2, b=3, c=−4 (2)a=4, b=4, c=5 (3) a=4, b=4, c= − 5 (4)a=−2, b=3, c=4
6. 𝑝 , 𝑞 kw;Wk; 𝑝 + 𝑞 Mfpait vz;zsT 𝜆 nfhz;l ntf;lh;fshapd; 𝑝 − 𝑞 MdJ
(1)2 𝜆 (2) 𝟑𝝀 (3) 2𝜆 (4) 1
7. 𝑎 , 𝑏 , 𝑐 vd;gd xU jsk; mikah ntf;lh;fs; NkYk; 𝑎 × 𝑏 , 𝑏 × 𝑐 , 𝑐 × 𝑎 = 𝑎 + 𝑏 , 𝑏 + 𝑐 , 𝑐 + 𝑎 vdpy;
𝑎 , 𝑏 , 𝑐 ,d; kjpg;G
(1)2 (2)3 (3)1 (4)0
8. 𝑖 + 𝑎𝑗 − 𝑘 vDk; tpir 𝑖 + 𝑗 vDk; Gs;sptopNar; nray;gLfpwJ. 𝑗 + 𝑘 vDk; Gs;spiag; nghWj;J
mjd; jpUg;Gj; jpwdpd; msT 8 vdpy; 𝑎 ,d; kjpg;G
(1)1 (2)2 (3)3 (4)4
9. (2,1, −1) vd;w Gs;sp topahfTk;> jsq;fs; 𝑟 . 𝑖 + 3𝑗 − 𝑘 = 0 ; 𝑟 . 𝑗 + 2𝑘 = 0 ntl;bf; nfhs;Sk;
Nfhl;il cs;slf;fpaJkhd jsj;jpd; rkd;ghL
(1)𝑥 + 4𝑦 − 𝑧 = 0 (2) 𝒙 + 𝟗𝒚 + 𝟏𝟏𝒛 = 𝟎 (3)2 𝑥 + 𝑦 − 𝑧 + 5 = 0 (4) 2𝑥 − 𝑦 + 𝑧 = 0
10. 2𝑥 − 𝑦 + 2𝑧 = 5 vd;w jsj;jpd; nrq;Fj;J myF ntf;lh;
(1) 2𝑖 − 𝑗 + 2𝑘 (2)1
3(2𝑖 − 𝑗 + 2𝑘 ) (3)−
1
3(2𝑖 − 𝑗 + 2𝑘 ) (4)±
𝟏
𝟑(𝟐𝒊 − 𝒋 + 𝟐𝒌 )
11. 2 + 𝑖 3 vd;w fyg;ngz;zpd; kl;L
(1) 3 (2) 13 (3) 𝟕 (4)7
12. fyg;ngz; 𝑖25 3,d; Nghyhh; tbtk;
(1) cos 𝜋
2+ 𝑖 sin
𝜋
2 (2)cos 𝜋 + 𝑖 sin 𝜋 (3) cos 𝜋 − 𝑖 sin 𝜋 (4) 𝐜𝐨𝐬
𝝅
𝟐− 𝒊 𝐬𝐢𝐧
𝝅
𝟐
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13. 𝑎𝑥2 + 𝑏𝑥 + 1 = 0 vd;w rkd;ghl;bd; xU jPh;T 1−𝑖
1+𝑖 , 𝑎 Ak; 𝑏 Ak; nka; vdpy; 𝑎, 𝑏 vd;gJ
(1) (1,1) (2)(1, −1) (3) (0,1) (4)(1,0)
14. 𝑧1 , 𝑧2 vd;gd fyg;ngz;fs; vdpy; gpd;tUtdtw;wpy; vJ jtW?
(1) 𝑅𝑒 (𝑧1 + 𝑧2) = 𝑅𝑒 (𝑧1) + 𝑅𝑒 (𝑧2) (2) 𝐼𝑚 (𝑧1 + 𝑧2) = 𝐼𝑚 (𝑧1) + 𝐼𝑚 (𝑧2)
(3) 𝐚𝐫𝐠 (𝒛𝟏 + 𝒛𝟐) = 𝐚𝐫𝐠 𝒛𝟏 + 𝐚𝐫𝐠 𝒛𝟐 (4) 𝑧1𝑧2 = 𝑧1 𝑧2
15. 𝑦2 − 2𝑦 + 8𝑥 − 23 = 0vd;w gutisaj;jpd; mr;R
(1) 𝑦 = −1 (2) 𝑥 = −3 (3) 𝑥 = 3 (4) 𝒚 = 𝟏
16. 9𝑥2 + 5𝑦2 = 180 vd;w ePs;tl;lj;jpd; Ftpaq;fSf;fpilNa cs;s njhiyT
(1) 4 (2) 6 (3) 8 (4) 2
17. 𝑥𝑦 = 16 vd;w nrt;tf mjpgutisaj;jpd; Kidapd; Maj;njhiyTfs;
(1) (4,4),(-4,-4) (2) (2,8),(-2,-8) (3)(4,0),(-4,0) (4) (8,0),(-8,0)
18. 𝑙𝑥 + 𝑚𝑦 + 𝑛 = 0 vd;w NfhL vd;w 𝑥2
𝑎2 −𝑦2
𝑏2 = 1 mjpgutisaj;jpw;F nrq;Nfhlhf mika epge;jid
(1) 𝑎𝑙3 + 2𝑎𝑙𝑚2 + 𝑚2𝑛 = 0 (2) 𝑎2
𝑙2 +𝑏2
𝑚2 = 𝑎2+𝑏2
2
𝑛2
(3) 𝑎2
𝑙2 +𝑏2
𝑚2 = 𝑎2−𝑏2
2
𝑛2 (4) 𝒂𝟐
𝒍𝟐−
𝒃𝟐
𝒎𝟐 = 𝒂𝟐+𝒃𝟐
𝟐
𝒏𝟐
19. 𝑦 = 2𝑥2 − 6𝑥 − 4 vDk; tistiuap;y; 𝑥 −mr;Rf;F ,izahfTs;s njhLNfhl;bd;; njhLGs;sp
(1) 5
2,−17
2 (2)
−5
2,−17
2 (3)
−5
2,
17
2 (4)
𝟑
𝟐,−𝟏𝟕
𝟐
20. 𝑥2
3 + 𝑦2
3 = 𝑎2
3 vDk; tistiuapd; Jiz myFr; rkd;ghLfs;
(1)𝑥 = 𝑎 sin3𝜃 ; 𝑦 = 𝑎 cos3𝜃 (2) 𝒙 = 𝒂 cos𝟑𝜽 ; 𝒚 = 𝒂 sin𝟑𝜽
(3)𝑥 = 𝑎3 sin 𝜃 ; 𝑦 = 𝑎3 cos 𝜃 (4) 𝑥 = 𝑎3 cos 𝜃 ; 𝑦 = 𝑎3 sin 𝜃
21. rhpahd $w;iw Njh;e;njL.
(a) xU njhlh;r;rpahd rhh;ghdJ ,lQ;rhh;e;j ngUkk; ngw;wpUg;gpd; kPg;ngU ngUkKk; ngw;wpUf;Fk; (b) xU njhlh;r;rpahd rhh;ghdJ ,lQ;rhh;e;j rpWkk; ngw;wpUg;gpd; kPr;rpW rpWkKk; ngw;wpUf;Fk;
(c) xU njhlh;r;rpahd rhh;ghdJ kPg;ngU ngUkk; ngw;wpUg;gpd; ,lQ;rhh;e;j ngUkKk; ngw;wpUf;Fk;
(d) xU njhlh;r;rpahd rhh;ghdJ kPr;rpW rpWkk; ngw;wpUg;gpd; ,lQ;rhh;e;j rpWkKk; ngw;wpUf;Fk;
(1) (a) kw;Wk; (b) (2)(a) kw;Wk; (c) (3)(c) kw;Wk; (d) (4) (a),(c) kw;Wk; (d)
22. njhlh;r;rpahd tistiu 𝑦 = 𝑓(𝑥) MdJ (𝑥1 , 𝑦1) vd;w Gs;spapy; 𝑥 → 𝑥1 vDk;NghJ𝑓 ′(𝑥) → ∞ vdpy; 𝑦 = 𝑓(𝑥) f;F
(1) 𝑦 = 𝑥1 vd;w epiyf;Fj;jhd njhLNfhL cz;L (2) 𝑥 = 𝑥1 vd;w fpilkl;l njhLNfhL cz;L
(3) 𝒙 = 𝒙𝟏 vd;w epiyf;Fj;jhd njhLNfhL cz;L (4) 𝒚 = 𝒚𝟏 vd;w fpilkl;l njhLNfhL cz;L
23. 𝑢 = 𝑓 𝑦
𝑥 vdpy;> 𝑥
𝜕𝑢
𝜕𝑥+ 𝑦
𝜕𝑢
𝜕𝑦 ,d; kjpg;G
(1)0 (2)1 (3)2𝑢 (4) 𝑢
24. 𝑦2 = (𝑥 − 1)(𝑥 − 2)2 vd;w tistiu ve;j ,ilntspapy; tiuaWf;fg;gltpy;iy?
(1) 𝑥 ≥ 1 (2)𝑥 ≥ 2 (3)𝑥 < 2 (4)𝒙 < 1
25. sin4𝑥 𝑑𝑥𝜋
0 ,d; kjpg;G
(1) 3𝜋
16 (2)
3
16 (3)0 (4)
𝟑𝝅
𝟖
26. gutisak; 𝑦2 = 𝑥 f;Fk; mjd; nrt;tfyj;jpw;Fk; ,ilg;gl;l gug;G
(1)4
3 (2)
𝟏
𝟔 (3)
2
3 (4)
8
3
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27. Gs;spfs; (0,0),(3,0) kw;Wk; (3,3) Mfpatw;iw Kidg;Gs;spfshff; nfhz;l Kf;Nfhzj;jpd; gug;G
𝑥- mr;ir nghWj;Jr; Row;wg;gLk; jplg;nghUspd; fd msT
(1)18𝜋 (2)2𝜋 (3) 36𝜋 (4)𝟗𝝅
28. 𝐼𝑛 = sinn 𝑥 𝑑𝑥 vdpy; 𝐼𝑛=
(1)−𝟏
𝒏𝐬𝐢𝐧𝐧−𝟏 𝒙 𝐜𝐨𝐬 𝒙 +
𝒏−𝟏
𝒏𝑰𝒏−𝟐 (2)
1
𝑛sinn−1 𝑥 cos 𝑥 +
𝑛−1
𝑛𝐼𝑛−2
(3) −1
𝑛sinn−1 𝑥 cos 𝑥 −
𝑛−1
𝑛𝐼𝑛−2 (4) −
1
𝑛sinn−1 𝑥 cos 𝑥 +
𝑛−1
𝑛𝐼𝑛
29. 𝑑𝑦
𝑑𝑥+ 2
𝑦
𝑥= 𝑒4𝑥vd;w tiff;nfOr; rkd;ghl;bd; njhiff; fhuzp
(1) log 𝑥 (2) 𝒙𝟐 (3) 𝑒𝑥 (4) 𝑥
30. 𝑦 = 𝑒𝑥(𝐴 cos 𝑥 + 𝐵 sin 𝑥) vd;w njhlh;gpy; 𝐴 iaAk; 𝐵 iaAk; ePf;fpg; ngwg;gLk; tiff;nfOr; rkd;ghL
(1)𝑦2 + 𝑦1 = 0 (2) 𝑦2 − 𝑦1 = 0
(3) 𝒚𝟐 − 𝟐𝒚𝟏 + 𝟐𝒚 = 𝟎 (4) 𝑦2 − 2𝑦1 − 2𝑦 = 0
31. (3𝐷2 + 𝐷 − 14)𝑦 = 13𝑒2𝑥 d; rpwg;G jPh;T
(1) 26𝑥𝑒2𝑥 (2) 13𝑥𝑒2𝑥 (3) 𝒙𝒆𝟐𝒙 (4)𝑥2/2𝑒2𝑥
32. 𝑑2𝑦
𝑑𝑥2 = 𝑦 − 𝑑𝑦
𝑑𝑥
3
2
3
vd;w tiff;nfOr; rkd;ghl;bd; thpir kw;Wk; gb
(1)2, 3 (2)3,3 (3)3, 2 (4) 2, 2
33. xU $l;Lf; $w;W %d;W jdpf;$w;Wfisf; nfhz;ljhf ,Ug;gpd;> nka;al;ltizapYs;s epiufspd; vz;zpf;if
(1) 8 (2) 6 (3) 4 (4) 2
34. gpd;tUtdtw;Ws; vJ nka;ikahFk;?
(1) 𝑝 ∨ 𝑞 (2) 𝑝 ∧ 𝑞 (3) 𝒑 ∨ ~𝒑 (4) 𝑝 ∧ ~𝑝
35. KOf;fspy; * vd;w <UWg;Gr; nrayp 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 − 1vd tiuaWf;fg;gLfpwJ vdpy; rkdp cWg;G
(1)0 (2)1 (3) 𝑎 (4) 𝑏
36. 𝑆,∗ tpy; 𝑥 ∗ 𝑦 = 𝑥, 𝑥, 𝑦 ∈ 𝑠 vdpy; ‘∗’vd;gJ
(1) Nrh;g;G tpjpf;F cl;gLk; (2) ghpkhw;W tpjpf;F cl;gLk;
(3) Nrh;g;G kw;Wk; ghpkhw;W tpjpf;F cl;gLk;
(4) Nrh;g;G kw;Wk; ghpkhw;W tpjpf;F cl;glhJ
37. 𝑋 vd;w rktha;g;G khwpapd; gutw;gb 4 NkYk; ruhrhp 2 vdpy; 𝐸(𝑋2),d; kjpg;G
(1) 2 (2) 4 (3) 6 (4) 8
38. xU gha;]hd; gutypy; 𝑃 𝑋 = 0 = 𝑘 vdpy; gutw;gbapd; kjpg;G
(1) 𝐥𝐨𝐠𝟏
𝒌 (2)log 𝑘 (3)𝑒𝜆 (4)
1
𝑘
39. xU ,ay;epiyg; gutypd; epfo;jfT mlh;j;jpr; rhh;G 𝑓(𝑥),d; ruhrhp 𝜇 vdpy; 𝑓(𝑥)𝑑𝑥∞
−∞,d;
kjpg;G
(1)1 (2)0.5 (3)0 (4)0.25
40. 𝑓 𝑥 = 𝑘 𝑥2 , 0 < 𝑥 < 3
0 , kw;nwq;fpYk; vd;gJ epfo;jfT mlh;j;jpr; rhh;G vdpy; 𝑘,d; kjpg;G
(1)1
3 (2)
1
6 (3)
𝟏
𝟗 (4)
1
12
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gFjp – M
Fwpg;G : (i) vitNaDk; gj;J tpdhf;fSf;F tpilaspf;fTk;. 𝟏𝟎 × 𝟔 = 𝟔𝟎
(ii) tpdh vz; 55-f;F fz;bg;ghf tpilaspf;fTk;. gpw tpdhf;fspypUe;J VNjDk; xd;gJ tpdhf;fSf;F tpilaspf;fTk;
41. 𝑩𝑻𝑨𝑻 = 𝟑 𝟏 −𝟏𝟐 −𝟐 𝟎𝟏 𝟐 −𝟏
vdpy; 𝑩−𝟏𝑨−𝟏 fhz;f.
𝐵𝑇𝐴𝑇 𝑇 = 3 1 −12 −2 01 2 −1
𝑇
(𝐴𝐵)𝑇 𝑇 = 𝐴𝐵 = 3 2 11 −2 2
−1 0 −1
𝐴𝐵 −1 = 𝐵−1𝐴−1
𝐴𝐵 −1 =1
𝐴𝐵 adj (𝐴𝐵)
𝐴𝐵 = 3 2 11 −2 2
−1 0 −1 = 3 2 − 0 − 2 −1 + 2 + 1 0 − 2 = 6 − 2 − 2 = 2
adj 𝐴𝐵 = 2 2 6
−1 −2 −5−2 −2 −8
𝐵−1𝐴−1 = 𝐴𝐵 −1 =1
2
2 2 6−1 −2 −5−2 −2 −8
42. 𝒙 − 𝟒𝒚 + 𝟕𝒛 = 𝟏𝟒, 𝟑𝒙 + 𝟖𝒚 − 𝟐𝒛 = 𝟏𝟑, 𝟕𝒙 − 𝟖𝒚 + 𝟐𝟔𝒛 = 𝟓 vd;w rkd;ghLfspd; njhFg;G xUq;fikT cilajh vd;gij juKiwapy; Muha;f.
jug;gl;l rkd;ghLj;njhFg;gpid gpd;tUkhW mzpr; rkd;ghlhf khw;wp vOjyhk;
1 −4 73 8 −27 −8 26
𝑥 𝑦𝑧
= 14135
𝐴 𝑋 = 𝐵 tphpTg;gLj;jg;gl;l mzpahdJ
[𝐴, 𝐵] = 1 −4 73 8 −27 −8 26
14135
~ 1 −4 70 20 −230 20 −23
14
−29−93
𝑅2 → 𝑅2 − 3𝑅1
𝑅3 → 𝑅3 − 7𝑅1
~ 1 −4 70 20 −230 0 0
14
−29−64
𝑅3 → 𝑅3 − 𝑅2
filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ. 𝜌 𝐴, 𝐵 = 3 kw;Wk; 𝜌 𝐴 = 2
∴ 𝜌 𝐴, 𝐵 ≠ 𝜌 𝐴
∴ vdNt jug;gl;l rkd;ghl;Lj; njhFg;G xUq;fikT mw;wJ. vdNt jPh;T fhz KbahJ
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43. 𝑨𝑪 kw;Wk; 𝑩𝑫 I %iytpl;lq;fshff; nfhz;l ehw;fuk; 𝑨𝑩𝑪𝑫 ,d; gug;G 𝟏
𝟐 𝑨𝑪 × 𝑩𝑫 vdf;
fhl;Lf
ehw;fuk; 𝐴𝐵𝐶𝐷 ,d; ntf;lh; gug;G = ∆𝐴𝐵𝐶 ,d; ntf;lh; gug;G + ∆𝐴𝐶𝐷 ,d; ntf;lh; gug;G
=1
2 𝐴𝐵 × 𝐴𝐶 +
1
2 𝐴𝐶 × 𝐴𝐷
= −1
2 𝐴𝐶 × 𝐴𝐵 +
1
2 𝐴𝐶 × 𝐴𝐷
=1
2𝐴𝐶 × −𝐴𝐵 + 𝐴𝐷
=1
2𝐴𝐶 × 𝐵𝐴 + 𝐴𝐷
=1
2 𝐴𝐶 × 𝐵𝐷
ehw;fuk; 𝐴𝐵𝐶𝐷 ,d; gug;G = 1
2 𝐴𝐶 × 𝐵𝐷
44. (i) ,uz;L myF ntf;lh;fspd; $Ljy; Xh; myF ntf;lh; vdpy; mtw;wpd; tpj;jpahrj;jpd; vz;
msT 𝟑 vdf; fhl;Lf.
𝑎 , 𝑏 , 𝑐 vd;git Xh; myF ntf;lh;fs;
𝑎 + 𝑏 = 𝑐
𝑎 + 𝑏 2
= 𝑐 2
𝑎2 + 𝑏2 + 2𝑎𝑏 cos 𝜃 = 𝑐2
1 + 1 + 2 1 (1) cos 𝜃 = 1
2 + 2 cos 𝜃 = 1
2 cos 𝜃 = −1
cos 𝜃 = −1
2
𝑎 – 𝑏 2
= 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝜃
= 1 + 1 − 2 1 1 −1
2 = 3
∴ 𝑎 – 𝑏 = 3
44(ii) 𝒓 𝟐 − 𝒓 . 𝟒𝒊 + 𝟐𝒋 − 𝟔𝒌 − 𝟏𝟏=0 vd;w Nfhsj;jpd; ikak; kw;Wk; Muk; fhz;f
𝑟 = 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘 vd;f
𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘 2
− (𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘 ). 4𝑖 + 2𝑗 − 6𝑘 − 11=0
𝑥2 + 𝑦2 + 𝑧2 − 4𝑥 + 2𝑦 − 6𝑧 − 11 = 0
𝑥2 + 𝑦2 + 𝑧2 − 4𝑥 − 2𝑦 + 6𝑧 − 11 = 0
𝑥2 + 𝑦2 + 𝑧2 + 2𝑢𝑥 + 2𝑣𝑦 + 2𝑤𝑧 + 𝑑 = 0 cld; xg;gpl
2𝑢 = −4 , 2𝑣 = −2 , 2𝑤 = −4
𝑢 = −2, 𝑣 = −1, 𝑤 = 3, 𝑑 = −11
ikak; −𝑢, −𝑣, −𝑤 = (2,1, −3)
Muk; 𝑢2 + 𝑣2 + 𝑤2 − 𝑑 = 4 + 1 + 9 + 11 = 5
45. 𝒏 vd;gJ kpif KO vdpy;> ( 𝟑 + 𝒊)𝒏 + ( 𝟑 − 𝒊)𝒏 = 𝟐𝒏+𝟏 𝐜𝐨𝐬𝒏𝝅
𝟔 vd ep&gp.
3 + 𝑖 = 𝑟(cos 𝜃 + 𝑖 sin 𝜃)
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nka; kw;Wk; fw;gidg; gFjpfis xg;gpl>
𝑟 cos 𝜃 = 3, 𝑟 sin 𝜃 = 1
𝑟 = 3 2
+ 1 2 = 4 = 2
NkYk;, cos 𝜃 = 3
2, sin 𝜃 =
1
2⇒ 𝜃 =
𝜋
6
3 + 𝑖 = 2 cos𝜋
6+ 𝑖 sin
𝜋
6
3 + 𝑖 𝑛
= 2 cos𝜋
6+ 𝑖 sin
𝜋
6
𝑛
= 2𝑛 cos𝑛𝜋
6+ 𝑖 sin
𝑛𝜋
6 ………….. 1
(1),y; 𝑖 f;Fg; gjpy; – 𝑖 I gpujpapl,
3 − 𝑖 𝑛
= 2𝑛 cos𝑛𝜋
6− 𝑖 sin
𝑛𝜋
6 …………… 2
(1) IAk; (2) IAk; $l;l
3 + 𝑖 𝑛
+ 3 − 𝑖 𝑛
= 2𝑛 2cos𝑛𝜋
6
= 2𝑛+1 cos𝑛𝜋
6
46. fyg;ngz;fs; 𝟕 + 𝟗𝒊, −𝟑 + 𝟕𝒊, 𝟑 + 𝟑𝒊 vDk; fyg;ngz;fs; Mh;fd; jsj;jpy; xU nrq;Nfhz Kf;Nfhzj;ij mikf;Fk; vd epWTf.
𝐴, 𝐵 , 𝐶 vDk; Gs;spfs; KiwNa 7 + 9𝑖, −3 + 7𝑖 , 3 + 3𝑖 vDk; fyg;ngz;fis Mh;fd; jsj;jpy; Fwpf;fl;Lk;
𝐴𝐵 = 7 + 9𝑖 − (−3 + 7𝑖) = 10 + 2𝑖 = 10 2 + 2 2 = 104
𝐵𝐶 = −3 + 7𝑖 − (3 + 3𝑖) = −6 + 4𝑖 = −6 2 + 4 2 = 36 + 16 = 52
𝐶𝐴 = 3 + 3𝑖 − (7 + 9𝑖) = −4 − 6𝑖 = −4 2 + −6 2 = 16 + 36 = 52
⇒ 𝐴𝐵2 = 𝐵𝐶2 + 𝐶𝐴2
⇒ ∠𝐵𝐶𝐴 = 90°
vdNt ∆𝐴𝐵𝐶 xU ,U rkgf;f nrq;Nfhz Kf;NfhzkhFk.;
47. (i) 𝒇 𝒙 = 𝒙 − 𝟏 , 𝟎 ≤ 𝒙 ≤ 𝟐 vd;w rhh;gpw;F Nuhypd; Njw;wj;ij rhpghh;f;f.
,e;j rhh;G %ba ,ilntsp [0,2]y; njhlh;r;rpahf cs;sJ. 𝑥 = 1 ∈ (0, 2)tpy; tifaplj;jf;fjy;y.
𝑓 0 = 1 = 𝑓(2)vd ,Ug;gpDk; Nuhypd; vy;yh epge;jidfSk; G+h;j;jp Mftpy;iy. vdNt ,e;j rhh;gpw;F Nuhypd; Njw;wk; Vw;Gilajy;y.
47 (ii). kjpg;G fhz;f 𝐥𝐢𝐦𝒙→∞
𝐬𝐢𝐧𝟐
𝒙𝟏
𝒙
1
𝑥= 𝑦 vd;f 𝑥 → ∞ vdpy; 𝑦 → 0
lim𝑥→∞
sin2
𝑥1
𝑥
= lim𝑦→0
sin 2𝑦
𝑦
0
0 mikg;G
Nyhgpjhypd; tpjpg;gb
= lim𝑦→0
cos 2𝑦 2
1
=2
1= 2
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48. 𝒇 𝒙 = 𝒙𝟑 − 𝟑𝒙𝟐 + 𝟏 ,−𝟏
𝟐≤ 𝒙 ≤ 𝟒 vd;w rhh;gpd; kPg;ngU ngUkk; kw;Wk; kPr;rpW rpWk kjpg;Gfisf;
fhz;f
𝑓 MdJ −1
2, 4 ,y; njhlh;r;rpahf cs;sJ.
𝑓 𝑥 = 𝑥3 − 3𝑥2 + 1
𝑓 ′ 𝑥 = 3𝑥2 − 6𝑥 = 3𝑥(𝑥 − 2)
𝑥 ,d; vy;yh kjpg;GfSf;F 𝑓′ 𝑥 I fhz Kbtjhy; 𝑓,d; khWepiy vz;fs; 𝑥 = 0 kw;Wk;
𝑥 = 2 kl;LNk MFk;.
,t;tpU khWepiy vz;fSf;Fk; ,ilntsp −1
2, 4 ,y; ,Ug;gjhy;> ,t;ntz;fspy; 𝑓,d; kjpg;G
𝑓 0 = 1 kw;Wk; 𝑓 2 = −3.
,ilntspapd; KbTg;Gs;spfspy; 𝑓 ,d; kjpg;G
𝑓 −1
2 = −
1
2
3− 3 −
1
2
2+ 1 =
1
8 kw;Wk;
𝑓 4 = 43 − 3 42 + 1 = 17
,e;ehd;F vz;fisAk; xg;gpl;L ghh;f;Fk; NghJ kPg;ngU ngUkj;jpd; kjpg;G 𝑓 4 = 17 kw;Wk; kPr;rpW
rpWkj;jpd; kjpg;G 𝑓 2 = −3
49. xU jdp Crypd; ePsk; 𝒍 kw;Wk; KO miyT Neuk; T vdpy; 𝑻 = 𝒌 𝒍 (𝒌 vd;gJ khwpyp). jdp
Crypd; ePsk; 32.1 nr.kP ,ypUe;J 32.0 nr.kPf;F khWk;NghJ> Neuj;jpy; Vw;gLk; rjtPjg; gpioia fzf;fpLf.
𝑇 = 𝑘 𝑙 = 𝑘𝑙1
2 vdpy;
𝑑𝑇
𝑑𝑙= 𝑘
1
2× 𝑙
−1
2 = 𝑘
2 𝑙 kw;Wk;
𝑑𝑙 = 32.0 − 32.1 = −0.1 nr.kP.
𝑇 ,y; cs;sgpio = 𝑇 ,y; Vw;gLk; Njhuhakhd khw;wk;
∆𝑇 ≈ 𝑑𝑇 = 𝑑𝑇
𝑑𝑙 𝑑𝑙 =
𝑘
2 𝑙 (−0.1)
rjtPjg; gpio = ∆𝑇
𝑇 × 100%
=
𝑘
2 𝑙 −0.1
𝑘 𝑙× 100%
=−0.1
2𝑙× 100% =
−0.1
2(32.1)× 100%
=−0.1
64.2× 100%
= −0.156%
miyT Neuj;jpy; Vw;gLk; rjtPjg; gpio 0.156% FiwT MFk;.
50. 𝐬𝐢𝐧 𝒙
𝟗+𝐜𝐨𝐬𝟐 𝒙𝒅𝒙
𝝅
𝟐𝟎
d; kjpg;gpidf; fhz;f.
𝐼 = sin 𝑥 𝑑𝑥
9+cos 2 𝑥
𝜋/2
0
𝑡 = cos 𝑥 vd;f
𝑑𝑡 = − sin 𝑥 𝑑𝑥
sin 𝑥 𝑑𝑥 = −𝑑𝑡
𝑡 = cos 𝑥
𝑥 0 𝜋/2 𝑡 1 0
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𝐼 = −𝑑𝑡
32+𝑡2
0
1=
𝑑𝑡
32+𝑡2
1
0
=1
3 tan−1
𝑡
3
1
0 =
1
3 tan−1
1
3 − tan−1 0 =
1
3 tan−1
1
3 − 0 =
1
3 tan−1
1
3
51. (𝒑 ∨ 𝒒) ∨ (~(𝒑 ∨ 𝒒)) nka;ikah Kuz;ghlh vdf; fhz;f (JUN-08,10, MAR-11)
(𝑝 ∨ 𝑞) ∨ (~(𝑝 ∨ 𝑞)) f;Fhpa nka; ml;ltiz
𝑝 𝑞 𝑝 ∨ 𝑞 ~(𝑝 ∨ 𝑞) (𝑝 ∨ 𝑞) ∨ (~(𝑝 ∨ 𝑞)) 𝑇 𝑇 𝑇 𝐹 𝑇 𝑇 𝐹 𝑇 𝐹 𝑇 𝐹 𝑇 𝑇 𝐹 𝑇
𝐹 𝐹 𝐹 𝑇 𝑇
∴ (𝑝 ∨ 𝑞) ∨ (~(𝑝 ∨ 𝑞)) xU nka;ikahFk;.
52. (i) 𝐺 vd;w Fyj;jpy; cs;s xt;nthU 𝒂 f;F 𝒂−𝟏 −𝟏
= 𝒂 vd ep&gpf;f.
𝑎−1 ∈ 𝐺 vd;gjhy; 𝑎−1 −1 ∈ 𝐺 vd;f
𝑎 ∗ 𝑎−1 = 𝑎−1 ∗ 𝑎 = 𝑒
𝑎−1 ∗ 𝑎−1 −1 = 𝑎−1 −1 ∗ 𝑎−1 = 𝑒
𝑎 ∗ 𝑎−1 = 𝑎−1 −1 ∗ 𝑎−1
𝑎 = 𝑎−1 −1 (tyJ ePf;fy; tpjpg;gb)
52. (ii) xU Fyj;jpd; rkdp cWg;G xUikj; jd;ik tha;e;jJ - ep&gpf;f
𝐺 xU Fyk; vd;f. 𝐺 ,d; rkdp cWg;Gfis 𝑒1 , 𝑒2 vd ,Ug;gjhf nfhs;Nthk;.
𝑒1 I rkdp cWg;ghff; nfhs;Nthkhapd;
𝑒1 ∗ 𝑒2 = 𝑒2…………………………. 1
𝑒2 I rkdp cWg;ghff; nfhs;Nthkhapd;
𝑒1 ∗ 𝑒2 = 𝑒1…………………………… 2
(1) kw;Wk; (2) ,ypUe;J, 𝑒1 = 𝑒2
∴ vdNt> xU Fyj;jpd; rkdp cWg;G xUikj; jd;ik tha;e;jjhFk;.
53. tpkhdj;jpy; gazk; nra;Ak; xU egh; fh];kpf; fjphpaf;fj;jpdhy; ghjpf;fg;gLtJ xU ,ay;epiy
gutyhFk;. ,jd; ruhrhp 𝟒. 𝟑𝟓 𝐦 𝐫𝐞𝐦 MfTk;> jpl;l tpyf;fk; 𝟎. 𝟑𝟗 𝐦 𝐫𝐞𝐦 MfTk;
mike;Js;sJ. xU egh; 𝟓. 𝟐𝟎 𝐦 𝐫𝐞𝐦 f;F Nky; fh];kpf; fjphpaf;fj;jpdhy; ghjpf;fg;gLthh;
vd;gjw;F epfo;jfT fhz;f. ,q;F [ 𝑷 𝟎 < 𝑧 < 1.44 = 𝟎. 𝟒𝟐𝟓𝟏]
,ay;epiy guty; rktha;g;G khwp 𝑋
𝜇 = 4.35, 𝜍 = 0.59
X = 5.2 vdpy;
⇒ 𝑍 =5.2−4.35
0.59
= 1.44
𝑃 𝑋 > 5.2 = 𝑃 𝑍 > 1.44
= 𝑃(1.44 < 𝑍 < ∞)
= 0.5 − 𝑃(0 < 𝑍 < 1.44)
= 0.5 − 0.4251 = 0.0749
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54. xU njhlh;rktha;g;G khwp 𝑿 ,d; epfo;jfT mlh;j;jp rhh;G 𝒇(𝒙) = 𝟑𝒙𝟐, 𝟎 ≤ 𝒙 ≤ 𝟏𝟎, kw;nwq;fpYk;
vdpy;
𝑷(𝑿 ≤ 𝒂) = 𝑷(𝑿 > 𝑎) kw;Wk; 𝑷 𝑿 > 𝑏 = 𝟎. 𝟎𝟓 vd;gjw;fhd 𝒂, 𝒃 d; kjpg;Gfis fhz;f
epfo;jftpd; $Ljy; 1 MFk;
𝑃(𝑋 ≤ 𝑎) = 𝑃(𝑋 > 𝑎)
𝑃 𝑋 ≤ 𝑎 + 𝑃 𝑋 > 𝑎 = 1
2𝑃 𝑋 ≤ 𝑎 = 1
𝑃 𝑋 ≤ 𝑎 =1
2
𝑓(𝑥)𝑑𝑥𝑎
0=
1
2
3𝑥2𝑑𝑥𝑎
0=
1
2
3𝑥3
3
𝑎
0=
1
2
𝑎3 =1
2
𝑎 = 1
2
1
3
𝑷 𝑿 > 𝑏 = 𝟎. 𝟎𝟓
𝑓(𝑥)𝑑𝑥1
𝑏= 0.05
3𝑥2𝑑𝑥1
𝑏= 0.05
3𝑥3
3
1
𝑏= 0.05
1 − 𝑏3 = 0.05
𝑏3 = 1 − 0.05 = 0.95
𝑏3 =95
100
𝑏 = 95
100
1
3
55. (a) 𝟐𝒙𝒚 − 𝟐𝒙 − 𝟖𝒚 − 𝟏 = 𝟎 vd;w nrt;tf mjpgutisaj;jpw;F −𝟐,𝟏
𝟒 vd;w Gs;spapy; njhLNfhL
kw;Wk; nrq;Nfhl;bd; rkd;ghLfisf; fhz;f
𝑥 I nghWj;J tifapl;lhy;
2 𝑥𝑦′ + 𝑦. 1 − 2 − 8𝑦′ = 0
2𝑥𝑦′ + 2𝑦 − 2 − 8𝑦′ = 0
𝑦′ 2𝑥 − 8 = 2 − 2𝑦
𝑦′ =2−2𝑦
2𝑥−8
−2,1
4 vd;w Gs;spapy; 𝑦′ = −
1
8= 𝑚
njhLNfhl;bd; rkd;ghL 𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1 ⇒ 𝑥 + 8𝑦 = 0
nrq;Nfhl;bd; rkd;ghL 𝑦 − 𝑦1 = −1
𝑚 𝑥 − 𝑥1
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𝑦 −1
4= 8 𝑥 + 2
32𝑥 − 4𝑦 + 65 = 0
55. (b) 𝒙𝟐 + 𝟓𝒙 + 𝟕 𝒅𝒚 + 𝟗 + 𝟖𝒚 − 𝒚𝟐𝒅𝒙 = 𝟎 vd;w rkd;ghl;bd; jPh;T fhz;f.
𝑥2 + 5𝑥 + 7 𝑑𝑦 + 9 + 8𝑦 − 𝑦2𝑑𝑥 = 0
𝑥2 + 5𝑥 + 7 𝑑𝑦 = − 9 + 8𝑦 − 𝑦2𝑑𝑥
𝑑𝑦
9+8𝑦−𝑦2= −
𝑑𝑥
𝑥2+5𝑥+7
njhifapl 𝑑𝑦
9+8𝑦−𝑦2= −
𝑑𝑥
𝑥2+5𝑥+7
𝑑𝑦
−[ 𝑦−4 2−16−9]= −
𝑑𝑥
𝑥+5
2
2−
25
4+7
𝑑𝑦
52− 𝑦−4 2= −
𝑑𝑥
𝑥+5
2
2+
3
2
2
sin−1 𝑦−4
5 +
2
3tan−1
𝑥+5
2
3
2
= 𝑐
sin−1 𝑦−4
5 +
2
3tan−1
2𝑥+5
3 = 𝑐
gFjp – ,
Fwpg;G : (i) vitNaDk; gj;J tpdhf;fSf;F tpilaspf;fTk;. 𝟏𝟎 × 𝟏𝟎 = 𝟏𝟎𝟎
(ii) tpdh vz; 70-f;F fz;bg;ghf tpilaspf;fTk;. gpw tpdhf;fspypUe;J VNjDk; xd;gJ
tpdhf;fSf;F tpilaspf;fTk;
56. 𝒙 + 𝟐𝒚 + 𝒛 = 𝟔, 𝟑𝒙 + 𝟑𝒚 − 𝒛 = 𝟑, 𝟐𝒙 + 𝒚 − 𝟐𝒛 = −𝟑 vd;w Nehpa njhFg;gpid mzpf;Nfhit Kiwapy; jPh;f;f.
𝑥 + 2𝑦 + 𝑧 = 6
3𝑥 + 3𝑦 − 𝑧 = 3
2𝑥 + 𝑦 − 2𝑧 = −3
∆= 1 2 13 3 −12 1 −2
= 0
∆𝑥= 1 2 13 3 −1
−3 1 −2 = 0
∆𝑦 = 1 6 13 3 −12 −3 −2
= 0
∆𝑧= 1 2 63 3 32 1 −3
= 0
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∆= 0 kw;Wk; ∆= ∆𝑥= ∆𝑦 = ∆𝑧= 0 NkYk; ∆ tpd; 2 × 2 rpw;wzpf;Nfhitfspd; VNjDk; xd;W
G+r;rpakw;wJ. 1 22 3
≠ 0 Mjyhy;> njhFg;G xUq;fikT cilaJ NkYk; vz;zpf;ifaw;w
jPh;Tfs; ngw;wpUf;Fk;. 𝑧 = 𝑘, 𝑘 ∈ 𝑅 vd;f
(1) kw;Wk; (2)ypUe;J
𝑥 + 2𝑦 = 6 − 𝑘
3𝑥 + 3𝑦 = 3 + 𝑘
∆ = 1 23 3
= 3 − 6 = −3
∆𝑥= 6 − 𝑘 23 + 𝑘 3
= 12 − 5𝑘
∆𝑦= 1 6 − 𝑘3 3 + 𝑘
= 4𝑘 − 15
fpuhkhpd; tpjpg;gb
∴ 𝑥 =∆𝑥
∆=
12−5𝑘
−3=
5𝑘−12
3
𝑦 =∆𝑦
∆=
4𝑘−15
−3=
15−4𝑘
3
∴ jPh;T fzk; 𝑥, 𝑦, 𝑧 = 5𝑘−12
3,
15−4𝑘
3, 𝑘 , 𝑘 ∈ 𝑅
57. 𝐜𝐨𝐬 𝑨 + 𝑩 = 𝐜𝐨𝐬 𝑨 𝐜𝐨𝐬 𝑩 − 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩 vd epWTf
𝑂 I ikakhff; nfhz;l myF tl;lj;jpd; ghpjpapy; 𝑃, 𝑄 vd;w ,U
Gs;spfis vLj;J nfhs;f
𝑂𝑃 kw;Wk; 𝑂𝑄 Mdit 𝑥-mr;Rld; Vw;gLj;Jk; Nfhzk; KiwNa 𝐴 , 𝐵
∴ ∠𝑃𝑂𝑄 = ∠𝑃𝑂𝑥 + ∠𝑄𝑂𝑥 = 𝐴 + 𝐵
𝑃 , 𝑄 d; Maj;njhiyfs; KiwNa (cos 𝐴, sin 𝐴) kw;Wk; (cos 𝐵, −sin 𝐵).
𝑖 , 𝑗 vd;w myF ntf;lh;fis 𝑥, 𝑦 mr;Rj; jpirfspy; vLj;Jf; nfhs;f.
𝑂𝑃 = 𝑂𝑀 + 𝑀𝑃 = cos 𝐴𝑖 +sin 𝐴 𝑗
𝑂𝑄 = 𝑂𝑁 + 𝑁𝑄 = cos 𝐵𝑖 −sin 𝐵 𝑗
𝑂𝑃 . 𝑂𝑄 = cos 𝐴𝑖 +sin 𝐴 𝑗 . cos 𝐵𝑖 −sin 𝐵 𝑗
= cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵…………….(1)
tiuaiwapd;gb, 𝑂𝑃 . 𝑂𝑄 = 𝑂𝑃 . 𝑂𝑄 cos(𝐴 + 𝐵)
= cos (𝐴 + 𝐵)……….. 2
(1) kw;Wk; (2) ypUe;J
cos 𝐴 + 𝐵 = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵
58. 𝒙 + 𝟐𝒚 + 𝟑𝒛 = 𝟔 kw;Wk; 𝒙 + 𝒚 + 𝒛 = 𝟑 Mfpa jsq;fspd; ntl;Lf;NfhL topahfTk;> (𝟎, −𝟏, 𝟏) vd;w Gs;sp topahfTk; nry;Yk; jsj;jpd; fhh;Brpad; kw;Wk; ntf;lh; rkd;ghLfisf; fhz;f.
fhh;Brpad; rkd;ghL:
nfhLf;fg;gl;l ,uz;L jsq;fspd; ntl;Lf;NfhL topNa nry;Yk; jsj;jpd; rkd;ghL
𝑥 + 2𝑦 + 3𝑧 − 6 + 𝜆(𝑥 + 𝑦 + 𝑧 − 3) = 0
(0, −1, 1)top nry;tjhy; 0 + 2(−1) + 3(1) − 6 + 𝜆(0 − 1 + 1 − 3) = 0
−2 + 3 − 6 + 𝜆(−3) = 0
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−3𝜆 = 5
𝜆 = −5
3
𝑥 + 2𝑦 + 3𝑧 − 6 −5
3(𝑥 + 𝑦 + 𝑧 − 3) = 0
3𝑥 + 6𝑦 + 9𝑧 − 18 − 5𝑥 − 5𝑦 − 5𝑧 + 15 = 0
−2𝑥 + 𝑦 + 4𝑧 − 3 = 0
2𝑥 − 𝑦 − 4𝑧 + 3 = 0
ntf;lh; rkd;ghL:
𝑟 . 𝑛1 − 𝑞1 + 𝜆 𝑟 . 𝑛2 − 𝑞2 = 0
𝑟 . 𝑛1 − 𝑞1 = 𝑟 . 𝑖 + 2𝑗 + 3𝑘 − 6 ,
𝑟 . 𝑛2 − 𝑞2 = 𝑟 . 𝑖 + 𝑗 + 𝑘 − 3
𝑟 . 𝑖 + 2𝑗 + 3𝑘 − 6 + 𝜆 𝑟 . 𝑖 + 𝑗 + 𝑘 − 3 = 0
𝑟 . 𝑖 + 2𝑗 + 3𝑘 − 6 −5
3 𝑟 . 𝑖 + 𝑗 + 𝑘 − 3 = 0
59. − 𝟑 − 𝒊 𝟐
𝟑 ,d; vy;yh kjpg;GfisAk; fhz;f
− 3 − 𝑖 = 𝑟(cos 𝜃 + 𝑖 sin 𝜃)
𝑟 cos 𝜃 = − 3, 𝑟 sin 𝜃 = −1
𝑟 = ( 3)2 + 12 = 2
cos 𝜃 = − 3
2, sin 𝜃 = −
1
2⇒ 𝜃 = −𝜋 +
𝜋
6=
−5𝜋
6
− 3 − 𝑖 = 2 cos −5𝜋
6 + 𝑖 sin
−5𝜋
6
− 3 − 𝑖 2
3 = 22
3 cos −5𝜋
6 + 𝑖 sin
−5𝜋
6
2
3
= 223 cos 2𝑘𝜋 −
5𝜋
6 + 𝑖 sin 2𝑘𝜋 −
5𝜋
6
23
= 223 cos 12𝑘 − 5
𝜋
9+ 𝑖 sin 12𝑘 − 5
𝜋
9
𝑘 = 0,1,2 vdNt
22
3 cis −5𝜋
9 , 2
2
3 cis 7𝜋
9 , 2
2
3 cis 19𝜋
9 (my;yJ)2
2
3 cis 𝜋
9 Mfpa kjpg;Gfisg; ngWk;
60. 𝒙𝟐 − 𝟒𝒚𝟐 + 𝟔𝒙 + 𝟏𝟔𝒚 − 𝟏𝟏 = 𝟎 vd;w mjpgutisaj;jpd; ikaj; njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f NkYk; mjd; tistiuia tiuf.
𝑥2 − 4𝑦2 + 6𝑥 + 16𝑦 − 11 = 0
𝑥2 + 6𝑥 − 4𝑦2 + 16𝑦 = 11
(𝑥2 + 6𝑥 + 32 − 32) − 4(𝑦2 − 4𝑦 + 22 − 22) = 11
{(𝑥 + 3)2 − 9} − 4 𝑦 − 2 2 − 4 = 11
(𝑥 + 3)2 − 4 𝑦 − 2 2 = 4
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(𝑥+3)2
4−
𝑦−2 2
1= 4
𝑋2
4−
𝑌2
1= 1 ,q;F 𝑋 = 𝑥 + 3, 𝑌 = 𝑦 − 2
FWf;fr;R 𝑋-mr;rpw;F ,izahf cs;sJ.
𝑎2 = 4, 𝑏2 = 1, 𝑎 = 2, 𝑏 = 1
𝑒 = 1 +𝑏2
𝑎2 = 1 +1
4 =
4+1
4=
5
2
𝑎𝑒 = 2 × 5
2= 5
𝑋, 𝑌 I nghWj;J
𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 + 3, 𝑌 = 𝑦 − 2
ikak; (0,0) 𝑋 = 0, 𝑌 = 0
𝑥 + 3 = 0, 𝑦 − 2 = 0 𝐶(−3,2)
Ftpaq;fs; (±𝑎𝑒, 0)
(± 5, 0)
5, 0
𝑋 = 5, 𝑌 = 0 𝑥 + 3 = 5, 𝑦 − 2 = 0
𝑥 = −3 + 5, 𝑦 = 2
𝐹1(−3 + 5, 2)
− 5, 0
𝑋 = − 5, 𝑌 = 0 𝑥 + 3 = − 5, 𝑦 − 2 = 0
𝑥 = −3 − 5, 𝑦 = 2
𝐹2(−3 − 5, −2)
Kidfs; (±𝑎, 0) (±2,0)
(2, 0) 𝑋 = 2, 𝑌 = 0
𝑥 + 3 = 2, 𝑦 − 2 = 0 𝑥 = −1, 𝑦 = 2
𝐴(−1,2) (−2,0)
𝑋 = −2, 𝑌 = 0 𝑥 + 3 = −2, 𝑦 − 2 = 0
𝑥 = −5, 𝑦 = 2
𝐴′(−5,2)
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61. jiukl;lj;jpypUe;J 7.5 kP cauj;jpy; jiuf;F ,izahf nghUj;jg;gl;;l xU FohapypUe;J ntspNaWk; ePh; jiuiaj; njhLk; ghij xU gutisaj;ij Vw;gLj;JfpwJ. NkYk; ,e;j gutisag; ghijapd; Kid Fohapd; thapy; mikfpwJ. Foha; kl;lj;jpw;F 2.5 kP fPNo ePhpd; gha;thdJ Fohapd; Kid topahfr; nry;Yk; epiy Fj;Jf;Nfhl;bw;F 3 kPl;lh; J}uj;jpy; cs;sJ vdpy; Fj;Jf; Nfhl;bypUe;J vt;tsT J}uj;jpw;F mg;ghy; ePuhdJ jiuapy; tpOk; vd;gijf; fhz;f.
nfhLf;fg;gl;l tptuq;fspd; gb gutisak; fPo;Nehf;fp jpwg;Gilajhf mikfpwJ
𝑥2 = −4𝑎𝑦
𝑃 vd;w Gs;sp gutisag; ghijapy; Foha; kl;lj;jpw;F 2.5 kP fPNoAk;> Fohapd; Kid topNa nry;Yk; epiy Fj;Jf; Nfhl;bw;F 3 kP mg;ghYk; cs;sJ.
𝑃 vd;gJ(3, −2.5)
vdNt, 9 = −4𝑎(−2.5)
𝑎 =9
10
∴ gutisaj;jpd; rkd;ghL
𝑥2 = −4 ×9
10𝑦
Fj;Jf; Nfhl;bd; mbg;Gs;spapypUe;J 𝑥1 J}uj;Jf;F mg;ghy; ePuhdJ jiuapy; tpOtjhf nfhs;f. Mdhy; FohahdJ jiukl;lj;jpypUe;J 7.5kP cauj;jpy; mike;Js;sJ.
(𝑥1 , −7.5) vd;w Gs;sp gutisaj;jpYs;sJ.
𝑥12 = −4 ×
9
10× −7.5 = 27
𝑥1 = 3 3
∴ vdNt> jz;zPh; jiuiaj; njhLk; ,lj;Jf;Fk; Fohapd; KidapypUe;J tiuag;gLk;
Fj;Jf;Nfhl;bw;Fk; ,ilg;gl;l J}uk; 3 3 kP.
62. xU ghyj;jpd; tisthdJ miu ePs;tl;lj;jpd; tbtpy; cs;sJ. fpilkl;lj;jpy; mjd; mfyk; 40 mbahfTk; ikaj;jpypUe;J mjd; cauk; 16 mbahfTk; cs;sJ vdpy; ikaj;jpypUe;J tyJ my;yJ ,lg;Gwj;jpy; 9 mb J}uj;jpy; cs;s jiug;Gs;spapypUe;J ghyj;jpd; cauk; vd;d?
ghyj;jpd; eLg;Gs;spia ikak; 𝐶(0,0)Mf vLj;Jf; nfhs;Nthk;. fpilkl;lk; 40 mb vdNt
Kidfs; 𝐴(20,0) kw;Wk; 𝐴′(−20,0)
2𝑎 = 40 ⇒ 𝑎 = 20, 𝑏 = 16
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rkd;ghL
𝑥2
400+
𝑦2
256= 1
ikaj;jpypUe;J 9 mb tyg;Gwj;jpy; cauj;ij 𝑦1 vd;f. vdNt (9, 𝑦1) vd;w Gs;sp rkd;ghl;by; cs;sJ.
92
400+
𝑦12
256= 1
𝑦1
2
256= 1 −
81
400=
319
400
𝑦12 = 256
319
400
⇒ 𝑦1 =16 319
20=
4 319
5
∴ ikaj;jpypUe;J tyJ my;yJ ,lJGwj;jpy; 9 mb J}uj;jpy; cs;s jiug;Gs;spapypUe;J
ghyj;jpd; cauk; 4 319
5mb
63. 𝒙𝟐 + 𝒚𝟐 = 𝟓𝟐 vd;w tl;lj;jpw;F 𝟐𝒙 + 𝟑𝒚 = 𝟔 vd;w Neh;f;Nfhl;bw;F ,izahf tiuag;gLk;
njhLNfhLfspd; rkd;ghLfisf; fhz;f
𝑥2 + 𝑦2 = 52 nfhLf;fg;gl;l tistiu Gs;sp ( 𝑥1 , 𝑦1)y; tiuag;gLk; njhLNfhL 2𝑥 + 3𝑦 = 6 f;F ,izahf cs;sJ.
∴ 𝑑𝑦
𝑑𝑥
( 𝑥1 ,𝑦1)= −
2
3
2𝑥 + 2𝑦𝑑𝑦
𝑑𝑥= 0
𝑑𝑦
𝑑𝑥= −
𝑥
𝑦
𝑑𝑦
𝑑𝑥
( 𝑥1 ,𝑦1)= −
𝑥1
𝑦1
−2
3= −
𝑥1
𝑦1
𝑥1 =2
3𝑦1
( 𝑥1 , 𝑦1) tistiu 𝑥2 + 𝑦2 = 52 y; mike;Js;sJ
∴ 𝑥12 + 𝑦1
2 = 52
2
3𝑦1
2+ 𝑦1
2 = 52
13𝑦12 = 9 × 52
𝑦12 = 36
𝑦1 = ±6
𝑦1 = 6 vdpy; 𝑥1 = 4 kw;Wk;
𝑦1 = −6 vdpy; 𝑥1 = − 4
(4,6) kw;Wk; (−4, −6) Njitahd Gs;spfshFk;.
(4, 6) y; njhLNfhl;bd; rkd;ghL
𝑦 − 6 = −2
3(𝑥 − 4)
3𝑦 − 18 = −2𝑥 + 8
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2𝑥 + 3𝑦 − 26 = 0
(−4, −6) y; njhLNfhl;bd; rkd;ghL
𝑦 + 6 = −2
3(𝑥 + 4)
3𝑦 + 18 = −2𝑥 − 8
2𝑥 + 3𝑦 + 26 = 0
64. 𝒘 = 𝒖𝟐𝒆𝒗 vd;w rhh;gpy; 𝒖 =𝒙
𝒚 kw;Wk; 𝒗 = 𝒚 𝐥𝐨𝐠 𝒙 vDkhW ,Ug;gpd;
𝝏𝒘
𝝏𝒙kw;Wk;
𝝏𝒘
𝝏𝒚 I rq;fpyp
tpjpfisg; gad;gLj;jpf fhz;f
𝜕𝑤
𝜕𝑥=
𝜕𝑤
𝜕𝑢 𝜕𝑢
𝜕𝑥+
𝜕𝑤
𝜕𝑣 𝜕𝑣
𝜕𝑥 kw;Wk;
𝜕𝑤
𝜕𝑦=
𝜕𝑤
𝜕𝑢 𝜕𝑢
𝜕𝑦+
𝜕𝑤
𝜕𝑣 𝜕𝑣
𝜕𝑦
𝜕𝑤
𝜕𝑢= 2𝑢𝑒𝑣
𝜕𝑤
𝜕𝑣= 𝑢2𝑒𝑣
𝜕𝑢
𝜕𝑥=
1
𝑦
𝜕𝑢
𝜕𝑦= −
𝑥
𝑦2
𝜕𝑣
𝜕𝑥=
𝑦
𝑥
𝜕𝑣
𝜕𝑦= log 𝑥
𝜕𝑤
𝜕𝑥=
2𝑢𝑒𝑣
𝑦+ 𝑢2𝑒𝑣 𝑦
𝑥= 𝑥𝑦 𝑥
𝑦2 (2 + 𝑦)
𝜕𝑤
𝜕𝑦= 2𝑢𝑒𝑣 −
𝑥
𝑦2 + 𝑢2𝑒𝑣 log 𝑥
=𝑥2
𝑦3 𝑥𝑦 [𝑦 log 𝑥 − 2]
65. 𝒚𝟐 = 𝒙 − 𝟓 𝟐(𝒙 − 𝟔) vd;w tistiu kw;Wk; KiwNa (i) 𝒙 = 𝟓 kw;Wk; 𝒙 = 𝟔 (ii) 𝒙 = 𝟔 kw;Wk;
𝒙 = 𝟕 Mfpa NfhLfSf;F ,ilNaahd gug;Gfisf; fhz;f.
(i) 𝑦2 = 𝑥 − 5 2(𝑥 − 6)
𝑦 = 𝑥 − 5 𝑥 − 6
tistiuahdJ 𝑥 mr;ir 𝑥 = 5 kw;Wk; 𝑥 = 6,y; ntl;LfpwJ. 𝑥 MdJ 5 f;Fk; 6 f;Fk;
,ilapy; ve;j xU kjpg;ig ngw;whYk; 𝑦2 ,d; kjpg;G Fiwahf cs;sJ.
∴ 5 < 𝑥 < 6 vd;w ,ilntspapy; tistiu mikahJ. ∴ 𝑥 = 5 kw;Wk; 𝑥 = 6 vd;w
,ilntspapy; gug;G 0 MFk;.
(ii) Njitahd gug;G 𝑦𝑑𝑥𝑏
𝑎
tistiu 𝑥 mr;ir nghWj;J rkr;rPuhf ,Ug;gjhy;
gug;G = 2 𝑥 − 5 𝑥 − 6 𝑑𝑥7
6
= 2 𝑡 + 1 𝑡 𝑑𝑡1
0
= 2 𝑡3/2 + 𝑡1/2 𝑑𝑡1
0
= 2 𝑡
52
5
2
+𝑡
32
3
2
1
0
𝑡 = 𝑥 − 6
𝑑𝑡 = 𝑑𝑥
𝑥 6 7 𝑡 0 1
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= 2 2
5+
2
3
= 2 6+10
15
=32
15 rJu myFfs;
66. 𝒚𝟐 = 𝟒𝒂𝒙 vd;w gutisaj;jpd; mjd; nrt;tfyk; tiuapyhd gug;gpid 𝒙 -mr;rpd; kPJ Row;Wk;NghJ fpilf;Fk; jplg; nghUspd; tisgug;igf; fhz;f.
𝑦2 = 4𝑎𝑥, 𝑥 = 0, 𝑥 = 𝑎 kw;Wk; 𝑥-mr;R ,tw;why; milag;gLk; gug;ig 𝑥-mr;irg; nghWj;J Row;Wk; NghJ Njitahd jplg; nghUspd; tisg;gug;G fpilf;Fk;
𝑆 = 2𝜋 𝑦 1 + 𝑑𝑦
𝑑𝑥
2𝑑𝑥
𝑎
0
𝑦2 = 4𝑎𝑥
𝑥 I nghWj;J tifapl
2𝑦𝑑𝑦
𝑑𝑥= 4𝑎
𝑑𝑦
𝑑𝑥=
2𝑎
𝑦
1 + 𝑑𝑦
𝑑𝑥
2= 1 +
2𝑎
𝑦
2
= 1 +4𝑎2
𝑦2
= 𝑦2+4𝑎2
𝑦2 = 4𝑎𝑥 +4𝑎2
𝑦2
= 4𝑎(𝑥+𝑎)
𝑦2
𝑦 1 + 𝑑𝑦
𝑑𝑥
2= 𝑦.
4𝑎 . 𝑥+𝑎
𝑦= 2 𝑎. 𝑥 + 𝑎
𝑆 = 2𝜋 𝑦 1 + 𝑑𝑦
𝑑𝑥
2𝑑𝑥
𝑎
0
= 2𝜋 2 𝑎. 𝑥 + 𝑎 𝑑𝑥𝑎
0
= 4𝜋 𝑎 𝑥 + 𝑎 𝑑𝑥𝑎
0
= 4𝜋 𝑎 𝑥 + 𝑎 1/2 𝑑𝑥𝑎
0
= 4𝜋 𝑎 𝑥+𝑎
32
3
2
𝑎
0
=8𝜋 𝑎
3 𝑥 + 𝑎
3
2 𝑎
0
=8𝜋 𝑎
3 ( 𝑎 + 𝑎
3
2 − 0 + 𝑎 3
2)
=8𝜋 𝑎
3 ( 2𝑎
3
2 − 𝑎3
2)
=8𝜋 𝑎
3 ( 2𝑎
3
2 − 𝑎3
2)
=8𝜋 𝑎
3 2 2𝑎
3
2 − 𝑎3
2
=8𝜋𝑎2
3 2 2 − 1 rJu myFfs;
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67. jPh;f;f 𝟐 𝒙𝒚 − 𝒙 𝒅𝒚 + 𝒚𝒅𝒙 = 𝟎
nfhLf;fg;gl;Ls;s rkd;ghL 𝑑𝑦
𝑑𝑥=
−𝑦
2 𝑥𝑦 −𝑥
𝑦 = 𝑣𝑥
𝐿. 𝐻. 𝑆 = 𝑣 + 𝑥𝑑𝑣
𝑑𝑥 ; 𝑅. 𝐻. 𝑆 =
−𝑣
2 𝑣−1=
𝑣
1− 2 𝑣
𝑣 + 𝑥𝑑𝑣
𝑑𝑥=
𝑣
1− 2 𝑣
𝑥𝑑𝑣
𝑑𝑥=
𝑣
1− 2 𝑣− 𝑣
𝑥𝑑𝑣
𝑑𝑥=
𝑣−𝑣+2𝑣 𝑣
1− 2 𝑣
𝑥𝑑𝑣
𝑑𝑥=
2𝑣 𝑣
1− 2 𝑣
1− 2 𝑣
𝑣 𝑣 𝑑𝑣 = 2
𝑑𝑥
𝑥
𝑣− 3
2 − 21
𝑣 𝑑𝑣 = 2
𝑑𝑥
𝑥
njhifapl −2𝑣−1/2 − 2 log 𝑣 = 2 log 𝑥 + 2 log 𝑐
−𝑣−1/2 = log(𝑣𝑥𝑐)
− 𝑥
𝑦= log 𝑐𝑦
𝑐𝑦 = 𝑒−
𝑥
𝑦
𝑦𝑒
𝑥
𝑦 = 𝑐
68. Nubak; rpijAk; khWtPjkhdJ> mjpy; fhzg;gLk; mstpw;F tpfpjkhf mike;Js;sJ. 50 tUlq;fspy; Muk;g mstpypUe;J 5 rjtPjk; rpije;jpUf;fpwJ vdpy; 100 tUl Kbtpy;
kPjpapUf;Fk; msT vd;d? [𝑨𝟎 I Muk;g msT vdf; nfhs;f]
t vd;w Neuj;jpy; Nubaj;jpd; msT A vd;f
𝐴 = 𝐴(𝑡) 𝑑𝐴
𝑑𝑡𝛼𝐴 ⇒
𝑑𝐴
𝑑𝑡= 𝑘𝐴 ⇒ 𝐴 = 𝑐𝑒𝑘𝑡
𝑡 = 0 vdpy; 𝐴 = 𝐴0
/ 𝐴0 = 𝑐𝑒0 = 𝑐
/ 𝐴 = 𝐴0𝑒𝑘𝑡 50 tUlq;fspy; Nubak; Muk;g epiyapypUe;J 5 % rpijTWfpwJ.
𝑡 = 50 vdpy; , 𝐴 = .95 𝐴0
/ 0.95 𝐴0 = 𝐴0𝑒50𝑘 ⇒ 𝑒50𝑘 = 0.95 NkYk; 𝑡 = 100 vdpy;
𝐴 = 𝐴0𝑒100𝑘 = 𝐴0 𝑒50𝑘 2
= 𝐴0(.95)2 = 0.9025𝐴0
100 tUl Kbtpy; kPjpapUf;Fk; Nubaj;jpd; msT 0.9025𝐴0
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69. xU NgUe;J epiyaj;jpy; xU epkplj;jpw;F cs;Ns tUk; NgUe;Jfspd; vz;zpf;if gha;]hd;
gutiyg; ngw;wpUf;fpwJ vdpy; 𝝀 = 𝟎. 𝟗 vdf; nfhz;L (i) 5 epkpl fhy ,ilntspapy; rhpahf 9
NgUe;Jfs; cs;Ns tu (ii) 8 epkpl fhy ,ilntspapy; 10f;Fk; Fiwthf NgUe;Jfs; cs;Ns tu
(iii) 11 epkpl fhy ,ilntspapy; Fiwe;jgl;rk; 14 NgUe;Jfs; cs;Ns tu> epfo;jfT fhz;f
(i) xU epkplj;jpy; cs;Ns tUk; NgUe;JfSf;fhd 𝜆 = 0.9
∴ 5 epkplq;fspy; cs;Ns tUk; NgUe;JfSf;fhd 𝜆 = 0.9 × 5 = 4.5
𝑃 rhpahf 9 NgUe;Jfs; cs;Ns tu =𝑒−𝜆𝜆9
9!
𝑃(𝑋 = 9) =𝑒−4.5(4.5)9
9!
(ii) 𝑃 8 epkplq;fspy; 10 NgUe;JfSf;F
Fiwthf cs;Ns tu = 𝑃 𝑋 < 10
,q;F 𝜆 = 0.9 × 8 = 7.2
Njitahd epfo;jfT = 𝑒−7.2(7.2)𝑥
𝑥!
9
𝑥=0
(iii) 𝑃 11 epkplq;fspy; Fiwe;jgl;rk; 14 NgUe;Jfs; cs;Ns tu
= 𝑃 𝑋 ≥ 14
= 1 − 𝑃 𝑋 < 14
,q;F 𝜆 = 0.9 × 11 = 9.9
Njitahd epfo;jfT = 1 − 𝑒−9.9(9.9)𝑥
𝑥!
13
𝑥=0
70(a). tof;fkhd ngUf;fypd; fPo; 1,d; 𝒏Mk; gb %yq;fs; Kbthd vgPypad; Fyj;ij mikf;Fk; vdf;fhl;Lf
1 ,d; 𝑛Mk; gb %yq;fshtd 1, 𝜔, 𝜔2 … 𝜔𝑛−1
𝐺 = { 1, 𝜔, 𝜔2 … 𝜔𝑛−1} vd;f.
,q;F 𝜔 = cis 2𝜋
𝑛
(i) milg;G tpjp:
𝜔𝑙 , 𝜔𝑚 ∈ 𝐺, 0 ≤ 𝑙, 𝑚 ≤ (𝑛 − 1)
𝜔𝑙𝜔𝑚 = 𝜔𝑙+𝑚 ∈ 𝐺 vd ep&gpf;f Ntz;Lk;
epiy (i)
𝑙 + 𝑚 < 𝑛 vd;f
𝑙 + 𝑚 < 𝑛 vdpy; 𝜔𝑙+𝑚 ∈ 𝐺
epiy(ii)
𝑙 + 𝑚 ≥ 𝑛 vd;f tFj;jy; Nfhl;ghl;bd;gb>
𝑙 + 𝑚 = 𝑞. 𝑛 + 𝑟, 0 ≤ 𝑟 < 𝑛, kpif KO vz;.
𝜔𝑙+𝑚 = 𝜔𝑞𝑛 +𝑟 = 𝜔𝑛 𝑞 . 𝜔𝑟
= 1 𝑞 . 𝜔𝑟 = 𝜔𝑟 ∈ 𝐺 ∵ 0 ≤ 𝑟 < 𝑛
milg;G tpjp cz;ikahFk;.
(ii) Nrh;g;G tpjp: fyg;ngz;fspd; fzj;jpy; ngUf;fyhdJ vg;nghOJk; Nrh;g;G tpjpia cz;ikahf;Fk;.
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𝜔𝑙 . 𝜔𝑝 . 𝜔𝑚 = 𝜔𝑙 . 𝜔 𝑝+𝑚 = 𝜔𝑙+ 𝑝+𝑚
= 𝜔 𝑙+𝑝 +𝑚 = 𝜔𝑙 . 𝜔𝑝 . 𝜔𝑚 ∀ 𝜔𝑙 , 𝜔𝑝 , 𝜔𝑚 ∈ 𝐺
(iii) rkdp tpjp:
rkdp cWg;G 1 ∈ 𝐺 kw;Wk; mJ
1. 𝜔𝑙 = 𝜔𝑙 . 1 = 𝜔𝑙 ∀𝜔𝑙 ∈ 𝐺 vd;gij G+h;j;jp nra;fpwJ.
(iv) vjph;kiw tpjp:
𝜔𝑙 ∈ 𝐺 f;F 𝜔𝑛−𝑙 ∈ 𝐺 kw;Wk;
𝜔𝑙 . 𝜔𝑛−𝑙 = 𝜔𝑛−𝑙 . 𝜔𝑙 = 𝜔𝑛 = 1 ,t;thwhf vjph;kiw tpjp cz;ikahfpwJ.
∴ 𝐺, . xU FykhFk;.
(v) ghpkhw;W tpjp:
𝜔𝑙 . 𝜔𝑚 = 𝜔𝑙+𝑚 = 𝜔𝑚+𝑙 = 𝜔𝑚 . 𝜔𝑙 ∀𝜔𝑙 , 𝜔𝑚 ∈ 𝐺
∴ 𝐺, . xU vgPypad; FykhFk;. 𝐺 ,y; 𝑛 cWg;Gfs; kl;LNk cs;sjhy; 𝐺, . MdJ 𝑛 thpir nfhz;;l Kbthd vgPypad; FykhFk;.
70 (b). (𝟎, 𝝅),ilntspapy; 𝒇 𝜽 = 𝐬𝐢𝐧 𝟐𝜽 vd;w rhh;G ve;j ,ilntspfspy; FopT milfpd;wd
vd;gijAk; kw;Wk; tisT khw;Wg; Gs;spfisAk; fhz;f.
𝑓 𝜃 = sin 2𝜃 ; (0, 𝜋)apy;
𝑓′ 𝜃 = 2 cos 2𝜃
𝑓 ′′ 𝜃 = −4 sin 2𝜃
𝑓 ′′ 𝜃 = 0 ⇒ 2𝜃 = 0, 𝜋, 2𝜋
𝜃 = 0,𝜋
2, 𝜋
Mdhy; 0, 𝜋 ∉ (0, 𝜋)
∴ 𝜃 =𝜋
2∈ (0, 𝜋)
,q;F 𝜋
2, (0, 𝜋) ia 0,
𝜋
2 kw;Wk;
𝜋
2, 𝜋 vd gphpf;fpd;wJ
𝜃 ∈ 0,𝜋
2 tpy; 𝑓 ′′ 𝜃 < 0 kw;Wk;
𝜋
2, 𝜋 apy; 𝑓 ′′ 𝜃 > 0 MFk;.
vdNt 𝑓 𝜃 vd;gJ 0,𝜋
2 tpy; fPo;Nehf;fpf; FopthfTk;>
𝜋
2, 𝜋 y; Nky; Nehf;fp FopthfTk;
cs;sJ. 𝜋
2, 𝑓
𝜋
2 mjhtJ
𝜋
2, 0 xU tisT khw;Wg; Gs;spahFk;.
jahhpg;G
f. jpNd\; M.Sc., M.Phil., P.G.D.C.A., (Ph.D.,) ntw;wpf;F top FO