fzf;F miuahz;L nghJj;Njh;T - 2016-17 ntw;wpf;F top...

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



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



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



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



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)



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



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



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



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



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



∆= 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



−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



(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)



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



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



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



= 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;



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



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;.



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

fzf;F miuahz;L nghJj;Njh;T - 2016-17 ntw;wpf;F top...

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Transcript of fzf;F miuahz;L nghJj;Njh;T - 2016-17 ntw;wpf;F top...