kalviamuthu.blogspot · 12k; tFg;G fzf;F xU kjpg;ngz; tpdhf;fs; ntw;wpf;F top [email protected] - 5 -

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ntw;wpf;F top (Way to Success)

⓬ fzpjk;

muRg; nghJj;Njh;T rpwg;G

ifNaL

jahupg;G

jpU. f. jpNd\; M.Sc., M.Phil., P.G.D.C.A., Ph.D .,

------ghlrk;ke;jkhd tpsf;fk; ngw ------

kpd;dQ;ry; : [email protected]

& [email protected] njhlh;Gf;F : 7418865975 (ghlg;nghUs; rhh;ghf kl;Lk;)> 9787609090 (Gj;jfq;fs; thq;f)

tiyjsk; : www.waytosuccess.org ghl cjtpf; Fwpg;Gfis vkJ ,izajsj;jpypUe;J ,ytrkhf gjptpwf;fpf;nfhs;syhk;


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-2-

muRj;Nju;T tpdhj;jhs; tbtikg;G

gpupT tpdh vz; tpdh tif

Nfl;fg;gLk; tpdhf;fs;

vOj Ntz;bait

kjpg; ngz;fs;

gpupT-m 1 - 40 njupTtpdh (xU kjpg;ngz; tpdhf;fs;) 40 40 40

gpupT-M 41 – 54 6 kjpg;ngz; tpdhf;fs; 14 9 54

55 fl;lha 6 kjpg;ngz; tpdh 2 1 6

gpupT-, 56 – 69 10 kjpg;ngz; tpdhf;fs; 14 9 90

70 fl;lha 10 kjpg;ngz; tpdh 2 1 10

72 60 200

tpdhj;jhs; - gFg;gha;T

Fwpg;G: ,e;j Gj;jfk; jukhf tu Ntz;Lk; vd;w Nehf;fpy; ,ad;wtiu jtWfspd;wp njhFj;J toq;fpAs;Nshk;. mtw;iwAk; kPwp rpy jtWfs; cq;fSf;Fj; njd;glyhk;. mt;thW VNjDk; jtWfs; ,Ue;jhy; vq;fsJ kpd;dQ;ry; Kftupf;F ([email protected]) clNd njuptpf;fTk;. Gj;jfj;jpy; cs;s jtWfSf;fhd jpUj;jq;fs; mt;tg;NghJ vq;fsJ www.waytosuccess.org tiyjsj;jpy; ntspaplg;gl;L mit mt;tg;NghJ update nra;ag;gLk; vd;gijAk;> mLj;jLj;j gjpg;Gfspy; mit rup nra;ag;gl;L tpLk; vd;gijAk; njuptpj;Jf;nfhs;fpNwhk;.

Way to Success Gj;jfq;fs; Ntz;LNthu; 9787609090, 9787201010, 8680810626 Mfpa vz;fisj; njhlu;Gnfhs;Sq;fs;

,ay; vz;

,ay; xU

kjpg;ngz; tpdhf;fs;

MW kjpg;ngz; tpdhf;fs;

gj;J kjpg;ngz; tpdhf;fs;

1 mzpfs; kw;Wk; mzpf;Nfhitfspd; gad;ghLfs; 4 2 1

2 ntf;lh; ,aw;fzpjk; 6 2 2 3 fyg;ngz;fs; 4 2 1 4 gFKiw tbtf;fzpjk; 4 1 3 5 tif Ez;fzpjk; : gad;ghLfs; I 4 2 2 6 tif Ez;fzpjk; : gad;ghLfs; II 2 1 1

7 njhif Ez;fzpjk; : gad;ghLfs; 4 1 2

8 tiff;nfOr;rkd;ghLfs; 4 1 2

9 jdpepiy fzf;fpay; 4 2 1

10 epfo;jfTg; guty; 4 2 1

$Ljy; 40 16 16



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md;ghd khztu;fSf;F

tzf;fk;. 12k; tFg;G fzpjf; ifNaL jw;NghJ cq;fs; ifapy; jto;fpwJ. tof;fkhd topfhl;b E}y; Nghd;W ,J vOjg;gltpy;iy. kw;w Fwpg;NgLfSf;Fk; ,jw;Fk; kp;Fe;j NtWghL cz;L. khztu;fs; fzpj ghlj;ij Gupe;J nfhz;L> vspa Kiwapy; vt;thW tpil mspg;gJ> mNj rkaj;jpy; muRj; Njh;tpy; mjpf kjpg;ngz; ngWk; tifapYk;> nky;yf; fw;Fk; khzth;fspd; gaj;ij Nghf;fp fzpjg; ghlj;jpy; ntw;wp ngWk; tifapYk; ,e;j ifNaL tbtikf;fg;gl;Ls;sJ.

nky;yf; fw;NghUf;fhd MNyhridfs;:

Kaw;rp nra;jhy; fzpjg;; ghlj;jpy; Rygkhf Nju;r;rpngw;W ey;y kjpg;ngz;fSk; ngwKbAk; vd;gij Kjypy; ek;Gq;fs;. gapw;rpfis nra;Jghu;j;jy; kpf mtrpak;.

fzpjg; ghlj;jpy; Mh;tk; kpf Kf;fpak;. gpd;tUk; ghlg;gFjpfSf;F Kf;fpaj;Jtk; nfhLj;J ed;F gapw;rp nra;aTk; xU kjpg;ngz; tpdhf;fs; - njhFjp 1 (121 tpdhf;fs;)>

njhFjp 2 (150 tpdhf;fs;) MW kjpg;ngz; tpdhf;fs; - 1. mzpfs; kw;Wk; mzpf;Nfhitfspd; gad;ghLfs;

3. fyg;ngz;fs; 9. jdpepiyf; fzf;fpay;

gj;J kjpg;ngz; tpdhf;fs; - 2. ntf;lh; ,aw;fzpjk; 3. fyg;ngz;fs; 4. gFKiw tbtf;fzpjk; 6. tif Ez;fzpjk; - gad;ghLfs; II 8. tiff;nfOr;rkd;ghLfs; (gad;ghLfs; tpdhf;fs; 10 kl;Lk;) 9. jdpepiyf; fzf;fpay;

rpwg;ghf nray;gl;lhy; ntw;wp cWjp. muRj;Nju;tpy; 200f;F 200 ngw tho;j;JfpNwhk;.

- ntw;wpf;F top FO

cs;slf;fk; ,ay; jiyg;G gf;fk; vz;

1 kjpg;ngz; Gj;jf tpdhf;fs; (njhFjp I kw;Wk; II) 4 6 kjpg;ngz;fs; mzpfs; kw;Wk; mzpf;Nfhitfspd; gad;ghLfs; 26 6 kjpg;ngz;fs; fyg;ngz;fs; 33 6 kjpg;ngz;fs; jdpepiyf; fzf;fpay; 41

10 kjpg;ngz;fs; ntf;lh; ,aw;fzpjk; 47 10 kjpg;ngz;fs fyg;ngz;fs; 55 10 kjpg;ngz;fs gFKiw tbtpay; 60 10 kjpg;ngz;fs tif Ez;fzpjk; - gad;ghLfs; II 78 10 kjpg;ngz;fs tiff;nfOr;rkd;ghLfs; 82 10 kjpg;ngz;fs jdpepiyf; fzf;fpay; 85

- ,f;Fwpaplg;gl;l tpdhf;fs;> Gj;jfj;jpw;F ntspapy; ,Ue;J Nfl;fg;gl;lit



12k; tFg;G fzf;F xU kjpg;ngz; tpdhf;fs; ntw;wpf;F top

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1. mzpfs; kw;Wk; mzpf;Nfhitfspd; gad;ghLfs;

1. 1 −1 22 −2 44 −4 8

vd;w mzpapd; juk; fhz;f (OCT-11)

(1) 1 (2) 2 (3) 3 (4) 4

2. vd;w %iytpl;l mzpapd; juk; fhz;f (MAR-10, JUN-16)

(1)0 (2)2 (3)3 (4)5

3. 𝐴 = 2 0 1 vdpy; 𝐴𝐴𝑇 ,d; juk; fhz;f (OCT-06,MAR-08,JUN-09,MAR-11, OCT-15)

(1)1 (2)2 (3)3 (4)0

4. 𝐴 = 123 , vdpy; 𝐴𝐴𝑇 ,d; juk; fhz;f (MAR-09, JUN-13,OCT-14, OCT-16)

(1)3 (2)0 (3)1 (4)2

5. 𝜆 −1 00 𝜆 −1

−1 0 𝜆 vd;w mzpapd; juk; 2 vdpy;> 𝜆 d; kjpg;G (JUN-08,OCT-09,JUN-11,OCT-15)

(1) 1 (2)2 (3)3 (4) VNjDk; xU nka;naz;

6. xU jpirapyp mzpapd; thpir 3> jpirapyp 𝑘 ≠ 0, vdpy; 𝐴−1 vd;gJ

(OCT-07,MAR-08,JUN-08,OCT-08,MAR-10,MAR-14,JUN-14,OCT-14)

(1) 1

𝑘2 𝐼 (2) 1

𝑘3 𝐼 (3) 𝟏

𝒌𝑰 (4)kI

7. −1 3 21 𝑘 −31 4 5

vd;w mzpf;F Neh;khW cz;L vdpy; k d; kjpg;Gfs;

(OCT-06,OCT-09,MAR-11,OCT-13,MAR-15)

(1) k VNjDk; xU nka;naz ; (2) 𝑘 = −4 (3) 𝒌 ≠ −𝟒 (4) 𝑘 ≠ 4

8. 𝐴 = 2 13 4

vd;w mzpf;F (adj A)A= (MAR-07,16, JUN-07,15, OCT-08,10,12,15)

(1)

1

50

01

5

(2) 1 00 1

(3) 5 00 −5

(4) 𝟓 𝟎𝟎 𝟓

9. xU rJu mzp A ,d; thpir n vdpy; 𝑎𝑑𝑗 𝐴 vd;gJ (MAR-06,JUN-06,MAR-12,JUN-14)

(1) 𝐴 2 (2) 𝐴 𝑛 (3) 𝑨 𝒏−𝟏 (4) 𝐴

10. 0 0 10 1 01 0 0

vd;w mzpapd; Neh;khW (JUN-12,JUN-13,JUN-16)

(1) 1 0 00 1 00 0 1

(2) 0 0 10 1 0

−1 0 0 (3)

𝟎 𝟎 𝟏𝟎 𝟏 𝟎𝟏 𝟎 𝟎

(4) −1 0 00 −1 00 0 1

11. A vd;w mzpapd; thpir 3 vdpy; det (kA) vd;gJ (OCT-06, 07,JUN-09,JUN-10,JUN-11,MAR-16,17)

(1)𝒌𝟑 𝐝𝐞𝐭(𝑨) (2) 𝑘2 det(𝐴) (3) 𝑘 det(𝐴) (4) det(𝐴)





12. myF mzp I ,d; thpir n, 𝑘 ≠ 0 xU khwpyp vdpy; adj(kI )=

(OCT-10,MAR-11,JUN-12,MAR-13,OCT-13,JUN-15,JUN-16)

(1) 𝑘𝑛 (adj 𝐼) (2) 𝑘(adj 𝐼) (3) 𝑘2 (adj 𝐼) (4) 𝒌𝒏−𝟏 (𝐚𝐝𝐣 𝑰)

13. A , B vd;w VNjDk; ,U mzpfSf;F AB=O vd;W ,Ue;J NkYk; A xU G+r;rpakw;w Nfhit

mzp vdpy;> (MAR-07,OCT-07,MAR-08,MAR-09,MAR-12,MAR-13,OCT-14,MAR-15)

(1)B=0 (2)B xU G+r;rpaf;Nfhit mzp

(3) B xU G+r;rpakw;w Nfhit mzp (4)B=A

14. 𝐴 = 0 00 5

, vdpy;> 𝐴12 vd;gJ (JUN-07,JUN-09,JUN-10, OCT-16, MAR-17)

(1) 0 00 60

(2) 𝟎 𝟎𝟎 𝟓𝟏𝟐 (3)

0 00 0

(4) 1 00 1

15. 3 15 2

vd;gjd; Neh;khW (MAR-06, OCT-07, OCT-08,OCT-09,OCT-11,OCT-12, MAR-14,JUN-14)

(1) 𝟐 −𝟏

−𝟓 𝟑 (2)

−2 51 −3

(3) 3 −1

−5 −3 (4)

−3 51 −2

16. 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 (JUN-06,07,13, MAR-10,12)

(1) xNu xU jPh;T (2) ,uz;L jPh;Tfs;

(3) vz;zpf;ifaw;w jPh;Tfs; (4 ) jPh;T ,y;yhik

17. 𝑎𝑥 + 𝑦 + 𝑧 = 0; 𝑥 + 𝑏𝑦 + 𝑧 = 0; 𝑥 + 𝑦 + 𝑐𝑧 = 0 Mfpa rkd;ghLfspd; njhFg;ghdJ xU

ntspg;gilaw;w jPh;it ngw;wpUg;gpd; 1

1−𝑎+

1

1−𝑏+

1

1−𝑐=

(MAR-07,MAR-09,JUN-10,OCT-12,MAR-14,JUN-15,MAR-16)

(1)1 (2)2 (3) −1 (4)0

18. 𝑎𝑒𝑥 + 𝑏𝑒𝑦 = 𝑐; 𝑝𝑒𝑥 + 𝑞𝑒𝑦 = 𝑑 kw;Wk; ∆1= a bp q

, ∆2= c bd q

, ∆3= a cp d vdpy; (𝑥, 𝑦) ,d;

kjpg;G (JUN-08,OCT-10,MAR-13, OCT-16, MAR-17)

(1) ∆2

∆1,∆3

∆1 (2) 𝐥𝐨𝐠

∆𝟐

∆𝟏, 𝐥𝐨𝐠

∆𝟑

∆𝟏 (3) log

∆1

∆3, log

∆1

∆2 (4) log

∆1

∆2, log

∆1

∆3

19. −2𝑥 + 𝑦 + 𝑧 = 𝑙, 𝑥 − 2𝑦 + 𝑧 = 𝑚, 𝑥 + 𝑦 − 2𝑧 = 𝑛, vd;w rkd;ghLfs; 𝑙 + 𝑚 + 𝑛 = 0 vDkhW

mikAkhapd; mj;njhFg;gpd; jPh;T (MAR-06,JUN-06,JUN-11,OCT-11,JUN-12,OCT-13,MAR-15)

(1) XNu xU G+r;rpakw;w jPh;T (2) ntspg;gilj; jPh;T

(3) vz;zpf;ifaw;w jPh;T (4) jPh;T ,y;yhik ngw;W ,Uf;Fk;

2. ntf;lh; ,aw;fzpjk;

1. 𝑎 xU G+r;rpakw;w ntf;luhfTk; 𝑚 xU G+r;rpakw;w jpirapypahfTk; ,Ug;gpd; 𝑚𝑎 MdJ XuyF

ntf;lh; vdpy; (MAR-07,OCT-12,MAR-14,MAR-16)

(1) 𝑚 = ±1 (2) 𝑎 = 𝑚 (3) 𝒂 =𝟏

𝒎 (4) 𝑎 = 1

2. 𝑎 kw;Wk; 𝑏 ,uz;L XuyF ntf;lh; kw;Wk; 𝜃 vd;gJ mtw;wpw;F ,ilg;gl;l Nfhzk; (𝑎 + 𝑏 )

MdJ XuyF ntf;luhapd; (OCT-06,OCT-07,MAR-08,OCT-09,JUN-10,MAR-11,MAR-12,JUN-15, OCT-15)

(1) 𝜃 =𝜋

3 (2) 𝜃 =

𝜋

4 (3) 𝜃 =

𝜋

2 (4) 𝜽 =

𝟐𝝅

𝟑





3. 𝑎 f;Fk; 𝑏 f;Fk; ,ilg;gl;l Nfhzk; 120°NkYk; mtw;wpd; vz;zsTfs; KiwNa 2, 3 vdpy;

𝑎 .𝑏 MdJ (JUN-07,JUN-09,JUN-14)

(1) 3 (2)− 𝟑 (3)2 (4)− 3

2

4. 𝑢 = 𝑎 × 𝑏 × 𝑐 + 𝑏 × 𝑐 × 𝑎 + 𝑐 × 𝑎 × 𝑏 vdpy; (JUN-06,MAR-07,JUN-08,OCT-08,MAR-12,MAR-15)

(1) u xU XuyF ntf;lh; (2) 𝑢 = 𝑎 +𝑏 + 𝑐 (3) 𝒖 = 𝟎 (4)𝑢 ≠ 0

5. 𝑎 +𝑏 + 𝑐 =0, 𝑎 = 3, 𝑏 = 4, 𝑐 = 5 vdpy;> 𝑎 f;Fk; 𝑏 f;Fk; ,ilg;gl;l Nfhzk;

(JUN-11,MAR-13,MAR-14, OCT-16)

(1) 𝜋

6 (2)

2𝜋

3 (3)

5𝜋

3 (4)

𝝅

𝟐

6. 2𝑖 +3𝑗 +4𝑘 , a𝑖 + 𝑏𝑗 + 𝑐𝑘 Mfpa ntf;lh;fs; nrq;Fj;J ntf;lh;fshapd;> (JUN-07,JUN-13,JUN-14)

(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

7. 3𝑖 + 𝑗 − 𝑘 vd;w ntf;liu xU %iy tpl;lkhfTk; 𝑖 −3𝑗 +4𝑘 I xU gf;fkhfTk; nfhz;l

,izfuj;jpd; gug;G (JUN-08,OCT-09,MAR-10,JUN-11,OCT-12,OCT-14,MAR-16)

(1)10 3 (2)6 30 (3)3

2 30 (4) 3 𝟑𝟎

8. 𝑎 + 𝑏 = 𝑎 − 𝑏 vdpy; (MAR-06,JUN-06,MAR-07,JUN-09,MAR-15)

(1) 𝑎 k; 𝑏 k; ,izahFk; (2) 𝒂 k; 𝒃 k; nrq;Fj;jhFk;

(3) 𝑎 = 𝑏 (4) 𝑎 kw;Wk; 𝑏 XuyF ntf;lh;

9. 𝑝 , 𝑞 kw;Wk; 𝑝 + 𝑞 Mfpait vz;zsT 𝜆 nfhz;l ntf;lh;fshapd; 𝑝 − 𝑞 MdJ

(OCT-08,MAR-09,OCT-13)

(1)2 𝜆 (2) 𝟑𝝀 (3) 2𝜆 (4) 1

10. 𝑎 × 𝑏 × 𝑐 + 𝑏 × 𝑐 × 𝑎 + 𝑐 × 𝑎 × 𝑏 = 𝑥 × 𝑦 vdpy;> (JUN-11,JUN-13,OCT-15)

(1) 𝑥 = 0 (2) 𝑦 = 0 (3) 𝑥 k; 𝑦 k; ,izahFk;

(4) 𝒙 = 𝟎 my;yJ 𝒚 = 𝟎 my;yJ 𝒙 k; 𝒚 k; ,izahFk;

11. 𝑃𝑅 = 2𝑖 + 𝑗 + 𝑘 , 𝑄𝑆 = −𝑖 + 3𝑗 + 2𝑘 vdpy; ehw;fuk; 𝑃𝑄𝑅𝑆 ,d; gug;G

(OCT-06,OCT-07,OCT-10,MAR-13,MAR-15, MAR-17)

(1)5 3 (2)10 3 (3) 𝟓 𝟑

𝟐 (4)

3

2

12. 𝑂𝑄 vd;w myF ntf;lh; kPjhd 𝑂𝑃 ,d; tPoyhdJ OPRQ vd;w ,izfuj;jpd; gug;ig Nghd;W

Kk;klq;fhapd; ∠𝑃𝑂𝑄 MdJ ( JUN-06,MAR-09,JUN-10, MAR-13, JUN-16)

(1)𝐭𝐚𝐧−𝟏 𝟏

𝟑 (2)cos−1

3

10 (3) sin−1

3

10 (4) sin−1

1

3

13. 𝑏 ,d; kPJ 𝑎 ,d; tPoy; kw;Wk; 𝑎 ,d; kPJ 𝑏 ,d; tPoYk; rkkhapd; 𝑎 + 𝑏 kw;Wk; 𝑎 − 𝑏 f;F

,ilg;gl;l Nfhzk; (JUN-07,OCT-09,MAR-11,OCT-11,JUN-14)

(1) 𝝅

𝟐 (2)

𝜋

3 (3)

𝜋

4 (4)

2𝜋

3

14. 𝑎 , 𝑏 , 𝑐 vd;w xU jskw;w ntf;lh;fSf;F 𝑎 × 𝑏 × 𝑐 = 𝑎 × 𝑏 × 𝑐 vdpy;> (MAR-09,OCT-13)

(1) 𝑎 MdJ 𝑏 f;F ,iz (2) 𝑏 MdJ 𝑐 f;F ,iz

(3) 𝒄 MdJ 𝒂 f;F ,iz (4) 𝑎 +𝑏 + 𝑐 =0





15. xU NfhL 𝑥 kw;Wk; 𝑦 mr;RfSld; kpif jpirapy; 45°, 60° Nfhzq;fis Vw;gLj;JfpwJ vdpy; z

mr;Rld; mJ cz;lhf;Fk; Nfhzk; (JUN-12,MAR-14,OCT-14, MAR-17)

(1) 30° (2) 90° (3) 45° (4) 𝟔𝟎°

16. 𝑎 × 𝑏 , 𝑏 × 𝑐 , 𝑐 × 𝑎 = 64 vdpy; 𝑎 , 𝑏 , 𝑐 ,d; kjpg;G (MAR-06,MAR-08,OCT-08,MAR-11,OCT-16)

(1) 32 (2) 8 (3) 128 (4)0

17. 𝑎 + 𝑏 , 𝑏 + 𝑐 , 𝑐 + 𝑎 = 8 vdpy; 𝑎 , 𝑏 , 𝑐 ,d; kjpg;G (JUN-07,MAR-10,MAR-12,OCT-12,MAR-16)

(1) 4 (2) 16 (3) 32 (4)-4

18. 𝑖 + 𝑗 , 𝑗 + 𝑘 , 𝑘 + 𝑖 ,d; kjpg;G (MAR-09,MAR-15, OCT-15, MAR-17)

(1)0 (2)1 (3)2 (4)4

19. (2,10,1) vd;w Gs;spf;Fk; 𝑟 . 3𝑖 − 𝑗 + 4𝑘 = 2 26 vd;w jsj;jpw;Fk; ,ilg;gl;l kpff; Fiwe;j

J}uk; (MAR-07,MAR-08,JUN-09,OCT-09,MAR-11,JUN-15)

(1) 2 26 (2) 26 (3)2 (4)1

26

20. 𝑎 × 𝑏 × (𝑐 × 𝑑 ) vd;gJ (OCT-11,OCT-15)

(1) 𝑎 , 𝑏 , 𝑐 kw;Wk; 𝑑 f;F nrq;Fj;J

(2) 𝑎 × 𝑏 kw;Wk; (𝑐 × 𝑑 ) vd;w ntf;lh;fSf;F ,iz

(3) 𝒂 , 𝒃 I nfhz;l jsKk; 𝒄 , 𝒅 I nfhz;l jsKk; ntl;bf;nfhs;Sk; Nfhl;bw;F ,iz

(4) 𝑎 , 𝑏 I nfhz;l jsKk; 𝑐 ,𝑑 I nfhz;l jsKk; ntl;bf; nfhs;Sk; Nfhl;bw;F nrq;Fj;J.

21. 𝑎 , 𝑏 , 𝑐 vd;gd 𝑎, 𝑏, 𝑐 Mfpatw;iw kl;Lf;fshff; nfhz;L tyf;if mikg;gpy; xd;Wf;nfhd;W

nrq;Fj;jhd ntf;lh;fs; vdpy; 𝑎 , 𝑏 , 𝑐 d; kjpg;G (JUN-08,JUN-12, JUN-16)

(1)𝑎2𝑏2𝑐2 (2) 0 (3) 1

2𝑎𝑏𝑐 (4) 𝒂𝒃𝒄

22. 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 (OCT-11,OCT-14)

(1)2 (2)3 (3)1 (4)0

23. 𝑟 = 𝑠𝑖 + 𝑡𝑗 vd;w rkd;ghL Fwpg;gJ (JUN-08,OCT-10)

(1) 𝑖 kw;Wk; 𝑗 Gs;spfis ,izf;Fk; Neh;f;NfhL (2) 𝒙𝒐𝒚 jsk;

(3) 𝑦𝑜𝑧 jsk; (4) 𝑧𝑜𝑥 jsk;

24. 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 (OCT-10,JUN-12,OCT-13, OCT-16)

(1)1 (2)2 (3)3 (4)4

25. 𝑥−3

1=

𝑦+3

5=

2𝑧−5

3f;F ,izahfTk; (1,3,5) Gs;sp topahfTk; nry;yf;$ba Nfhl;bd; ntf;lh;

rkd;ghL (MAR-17)

(1) 𝑟 = 𝑖 + 5𝑗 + 3𝑘 + 𝑡(𝑖 + 3𝑗 + 5𝑘 ) (2) 𝑟 = 𝑖 + 3𝑗 + 5𝑘 + 𝑡(𝑖 + 5𝑗 + 3𝑘 )

(3) 𝑟 = 𝑖 + 5𝑗 +3

2𝑘 + 𝑡(𝑖 + 3𝑗 + 5𝑘 ) (4) 𝒓 = 𝒊 + 𝟑𝒋 + 𝟓𝒌 + 𝒕 𝒊 + 𝟓𝒋 +

𝟑

𝟐𝒌

26. 𝑟 = 𝑖 − 𝑘 + 𝑡(3𝑖 + 2𝑗 + 7𝑘 ) vd;w NfhLk; 𝑟 . 𝑖 + 𝑗 − 𝑘 = 8 vd;w jsKk; ntl;bf;nfhs;Sk; Gs;sp

(MAR-07,MAR-08, JUN-16)

(1)(8,6,22) (2)( −8, −6, −22) (3)(4,3,11) (4)( −4, −3, −11)





27. (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 (MAR-10,OCT-10,MAR-13,JUN-15)

(1)𝑥 + 4𝑦 − 𝑧 = 0 (2) 𝒙 + 𝟗𝒚 + 𝟏𝟏𝒛 = 𝟎 (3)2 𝑥 + 𝑦 − 𝑧 + 5 = 0 (4) 2𝑥 − 𝑦 + 𝑧 = 0

28. 𝐹 = 𝑖 + 𝑗 + 𝑘 vd;w tpir xU Jfis A(3,3,3) vDk; epiyapypUe;J B(4,4,4) vDk; epiyf;F

efh;j;jpdhy; mt;tpir nra;Ak; NtiyasT. (MAR-06, MAR-13,JUN-13)

(1) 2 myFfs; (2) 3 myFfs; (3)4 myFfs; (4)7 myFfs;

29. 𝑎 = 𝑖 − 2𝑗 + 3𝑘 kw;Wk; 𝑏 = 3𝑖 + 𝑗 + 2𝑘 vdpy; 𝑎 f;Fk; 𝑏 f;Fk; nrq;Fj;jhf cs;s xU XuyF ntf;lh; (OCT-06, JUN-16)

(1)𝑖 +𝑗 +𝑘

3 (2)

𝑖 −𝑗 +𝑘

3 (3)

−𝑖 +𝑗 +2𝑘

3 (4)

𝒊 −𝒋 −𝒌

𝟑

30. 𝑥−6

−6=

𝑦+4

4=

𝑧−4

−8 kw;Wk;

𝑥+1

2=

𝑦+2

4=

𝑧+3

−2vd;w NfhLfs; ntl;bf; nfhs;Sk; Gs;sp

(OCT-07,JUN-09,MAR-11,MAR-12,OCT-12,MAR-14,JUN-14, OCT-16, MAR-17)

(1)(0,0,-4) (2)(1,0,0) (3)(0,2,0) (4)(1,2,0)

31. 𝑟 = −𝑖 + 2𝑗 + 3𝑘 + 𝑡(−2𝑖 + 𝑗 + 𝑘 ) kw;Wk; 𝑟 = 2𝑖 + 3𝑗 + 5𝑘 + 𝑠(𝑖 + 2𝑗 + 3𝑘 )vd;w NfhLfs;

ntl;bf;nfhs;Sk; Gs;sp (OCT-06, JUN-10, JUN-11,JUN-13,MAR-15,MAR-16)

(1) (2,1,1) (2)(1,2,1) (3)(1,1,2) (4)(1,1,1)

32. 𝑥−1

2=

𝑦−2

3=

𝑧−3

4 kw;Wk;

𝑥−2

3=

𝑦−4

4=

𝑧−5

5 vd;w NfhLfSf;fpilNaAs;s kpff; Fiwe;j njhiyT

(MAR-06,OCT-11,OCT-14)

(1)2

3 (2)

𝟏

𝟔 (3)

2

3 (4)

1

2 6

33. 𝑥−3

4=

𝑦−1

2=

𝑧−5

−3 kw;Wk;

𝑥−1

4=

𝑦−2

2=

𝑧−3

3 vd;w ,iz NfhLfSf;fpilNaAs;s kpff; Fiwe;j

njhiyT ( OCT-07, JUN-12, OCT-13,MAR-16)

(1)3 (2)2 (3)1 (4)0

34. 𝑥−1

2=

𝑦−1

−1=

𝑧

1 kw;Wk;

𝑥−2

3=

𝑦−1

−5=

𝑧−1

2 Mfpa ,U NfhLfSk; (JUN-06,MAR-10)

(1) ,iz (2) ntl;bf;nfhs;git

(3) xU jsk; mikahjit (4) nrq;Fj;J

35. 𝑥2 + 𝑦2 + 𝑧2 − 6𝑥 + 8𝑦 − 10𝑧 + 1 = 0 vd;w Nfhsj;jpd; ikak; kw;Wk; Muk;

( OCT-08,JUN-10,OCT-11,JUN-13,MAR-14,JUN-14)

(1) (−3,4, −5),49 (2)( −6, 8, −10),1 (3)(3, −4,5),7 (4)(6, −8,10), 7

3. fyg;ngz;fs;

1. −1+𝑖 3

2

100

+ −1−𝑖 3

2

100

,d; kjpg;G (JUN-10,JUN-11,MAR-16)

(1)2 (2)0 (3) −1 (4)1

2. 𝑒3−𝑖𝜋

4 3

vd;w fyg;ngz;zpd; kl;L tPr;R KiwNa (JUN-07,08, MAR-08,10,15 ,OCT-09,15)

(1)𝑒9,𝜋

2 (2) 𝑒9, −

𝜋

2 (3) 𝑒6 , −

3𝜋

4 (4) 𝒆𝟗,

−𝟑𝝅

𝟒

3. 2𝑚 + 3 + 𝑖(3𝑛 − 2) vd;w fyg;ngz;zpd; ,iznad; 𝑚 − 5 + 𝑖(𝑛 + 4) vdpy; (𝑛, 𝑚) vd;gJ

(MAR-07,OCT-10,JUN-16, MAR-17)

(1) −𝟏

𝟐, −𝟖 (2) −

1

2, 8 (3)

1

2, −8 (4)

1

2, 8





4. 𝑥2 + 𝑦2 = 1vdpy; 1+𝑥+𝑖𝑦

1+𝑥−𝑖𝑦 ,d; kjpg;G (JUN-09,MAR-12,MAR-13,JUN-14, OCT-16)

(1) 𝑥 − 𝑖𝑦 (2) 2𝑥 (3)−2𝑖𝑦 (4) 𝒙 + 𝒊𝒚

5. 2 + 𝑖 3 vd;w fyg;ngz;zpd; kl;L (JUN-12)

(1) 3 (2) 13 (3) 𝟕 (4)7

6. 𝐴 + 𝑖𝐵 = (𝑎1 + 𝑖𝑏1)(𝑎2 + 𝑖𝑏2)(𝑎3 + 𝑖𝑏3) vdpy; 𝐴2 + 𝐵2 ,d; kjpg;G (JUN-15)

(1)𝑎12 + 𝑏1

2 + 𝑎22 + 𝑏2

2 + 𝑎32 + 𝑏3

2 (2)(𝑎1 + 𝑎2 + 𝑎3)2 + (𝑏1 + 𝑏2 + 𝑏3)2

(3)( 𝒂𝟏𝟐 + 𝒃𝟏

𝟐)( 𝒂𝟐𝟐 + 𝒃𝟐

𝟐)( 𝒂𝟑𝟐 + 𝒃𝟑

𝟐) (4)( 𝑎12 + 𝑎2

2 + 𝑎32)( 𝑏1

2 + 𝑏22 + 𝑏3

2)

7. 𝑎 = 3 + 𝑖 kw;Wk; 𝑧 = 2 − 3𝑖 vdpy; cs;s 𝑎𝑧, 3𝑎𝑧 kw;Wk; −𝑎𝑧 vd;gd xU Mh;fd; jsj;jpy; (OCT-06,JUN-08,OCT-14)

(1) nrq;Nfhz Kf;Nfhzj;jpd; Kidg;Gs;spfs;

(2) rkgf;f Kf;Nfhzj;jpd; Kidg;Gs;spfs;

(3) ,U rkgf;f Kf;Nfhzj;jpd; Kidg;Gs;spfs; (4) xNu Nfhliktd

8. fyg;ngz; jsj;jp;y; 𝑧1 , 𝑧2 , 𝑧3 , 𝑧4 vd;w Gs;spfs; KiwNa xU ,izfuj;jpd; Kidg; Gs;spfshf

,Ug;gjw;Fk; mjd; kWjiyAk; cz;ikahf ,Ug;gjw;Fk; cs;s epge;jid (OCT-15,MAR-16)

(1) 𝑧1 + 𝑧4 = 𝑧2 + 𝑧3 (2) 𝒛𝟏 + 𝒛𝟑 = 𝒛𝟐 + 𝒛𝟒

(3) 𝑧1 + 𝑧2 = 𝑧3 + 𝑧4 (4) 𝑧1 − 𝑧2 = 𝑧3 − 𝑧4

9. 𝑧 xU fyg;ngz;izf; Fwpg;gnjdpy; arg 𝑧 + arg(𝑧 ) vd;gJ (OCT-08,MAR-09,MAR-12, OCT-16)

(1) 𝜋

4 (2)

𝜋

2 (3)0 (4)

𝜋

4

10. xU fyg;ngz;zpd; tPr;R 𝜋

2 vdpy; me;j vz; (OCT-10,OCT-11,OCT-13,MAR-14)

(1) Kw;wpYk; fw;gid vz; (2) Kw;wpYk; nka; vz;

(3)0 (4) nka;Aky;y fw;gidAky;y

11. 𝑖𝑧 vd;w fyg;ngz;iz Mjpiag; nghWj;J 𝜋

2 Nfhzj;jpy; fbfhu vjph;jpirapy; Row;Wk;NghJ me;j

vz;zpd; Gjpa epiy (JUN-12,OCT-12,MAR-13,OCT-15)

(1) 𝑖𝑧 (2)−𝑖𝑧 (3)−𝒛 (4) 𝑧

12. fyg;ngz; 𝑖25 3,d; Nghyhh; tbtk; (MAR-06,OCT-06,MAR-07,OCT-07,JUN-09,JUN-15)

(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. P MdJ fyg;G vz; khwp 𝑧 I Fwpf;fpd;wJ 2𝑧 − 1 = 2 𝑧 vdpy; P ,d; epakg;ghij

(JUN-06,MAR-10,MAR-11,JUN-11,OCT-14, MAR-17)

(1) 𝒙 =𝟏

𝟒 vd;w Neh;f;NfhL (2) 𝑦 =

1

4 vd;w Neh;f;NfhL

(3) 𝑧 =1

2 vd;w Neh;f;NfhL (4) 𝑥2 + 𝑦2 − 4𝑥 − 1 = 0 vd;w tl;lk;

14. 1+𝑒−𝑖𝜃

1+𝑒 𝑖𝜃 = (JUN-07,MAR-11,JUN-13)

(1) cos 𝜃 + 𝑖 sin 𝜃 (2) 𝐜𝐨𝐬 𝜽 − 𝒊 𝐬𝐢𝐧 𝜽 (3) sin 𝜃 − 𝑖 cos 𝜃 (4) sin 𝜃 + 𝑖 cos 𝜃

15. 𝑧𝑛 = cos𝑛𝜋

3+ 𝑖 sin

𝑛𝜋

3 vdpy; 𝑧1𝑧2 … . 𝑧6 is vd;gJ (JUN-08,OCT-12,JUN-14,JUN-16)

(1)1 (2) −1 (3) i (4) −𝑖

16. −𝑧 %d;whk; fhy;gFjpapy; mike;jhy; mikAk; fhy;gFjp (OCT-13,MAR-14,MAR-16)

(1) Kjy; fhy;gFjp (2) ,uz;lhk; fhy;gFjp

(3 ) %d;whk; fhy;gFjp (4) ehd;fhk; fhy;gFjp





17. 𝑥 = cos 𝜃 + 𝑖 sin 𝜃 vdpy; 𝑥𝑛 +1

𝑥𝑛 ,d; kjpg;G (MAR-08,OCT-08,MAR-09,MAR-15)

(1) 2 cos 𝒏 𝜽 (2) 2i sin 𝑛𝜃 (3)2 sin 𝑛𝜃 (4) 2i cos 𝑛 𝜃

18. 𝑎 = cos 𝛼 − 𝑖 sin 𝛼, 𝑏 = cos 𝛽 − 𝑖 sin 𝛽, 𝑐 = cos 𝛾 − 𝑖 sin 𝛾 vdpy; (𝑎2𝑐2 − 𝑏2) ∕ 𝑎𝑏𝑐 vd;gJ

(MAR-14,JUN-14)

(1)cos 2(𝛼 − 𝛽 + 𝛾) + 𝑖 sin 2(𝛼 − 𝛽 + 𝛾) (2)−2 cos(𝛼 − 𝛽 + 𝛾)

(3)−𝟐𝒊 𝐬𝐢𝐧 (𝜶 − 𝜷 + 𝜸) (4) 2 cos(𝛼 − 𝛽 + 𝛾)

19. 𝑧1 = 4 + 5𝑖, 𝑧2 = −3 + 2𝑖 vdpy; 𝑧1

𝑧2 vd;gJ (OCT-06,OCT-07,OCT-11,JUN-13,JUN-16)

(1)2

13−

22

13𝑖 (2)−

2

13+

22

13𝑖 (3)−

𝟐

𝟏𝟑−

𝟐𝟐

𝟏𝟑𝒊 (4)

2

13+

22

13𝑖

20. 𝑖 + 𝑖22 + 𝑖23 + 𝑖24 + 𝑖25 ,d; kjpg;G vd;gJ (MAR-06,JUN-06,JUN-07,JUN-09)

(1) 𝒊 (2) −𝑖 (3)1 (4) −1

21. 𝑖13 + 𝑖14 + 𝑖15 + 𝑖16 ,d; ,iz fyg;ngz;

(1) 1 (2) −1 (3)0 (4) −𝑖

22. – 𝑖 + 2 vd;gJ 𝑎𝑥2 − 𝑏𝑥 + 𝑐 = 0vd;w rkd;ghl;bd; xU %ynkdpy; kw;nwhU jPh;T

(MAR-08,OCT-09,MAR-10,MAR-13)

(1) – 𝑖 − 2 (2) 𝑖 − 2 (3) 𝟐 + 𝒊 (4) 2𝑖 + 𝑖

23. ±𝑖 7 vd;w jPh;Tfisf; nfhz;l ,Ugbr; rkd;ghL (OCT-09,JUN-10,MAR-11)

(1) 𝒙𝟐 + 𝟕 = 𝟎 (2) 𝑥2 − 7 = 0 (3) 𝑥2 + 𝑥 + 7 = 0 (4) 𝑥2 − 𝑥 − 7 = 0

24. 4−3i kw;Wk; 4+3i vd;w %yq;fisf; nfhz;l rkd;ghL (MAR-07)

(1)𝑥2 + 8𝑥 + 25 = 0 (2) 𝑥2 + 8𝑥 − 25 = 0 (3) 𝒙𝟐 − 𝟖𝒙 + 𝟐𝟓 = 𝟎 (4) 𝑥2 − 8𝑥 − 25 = 0

25. 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

(OCT-07,JUN-11)

(1) (1,1) (2)(1, −1) (3) (0,1) (4)(1,0)

26. 𝑥2 − 6𝑥 + 𝑘 = 0 vd;w rkd;ghl;bd; xU %yk; −𝑖+3 vdpy; k ,d; kjpg;G (OCT-08,MAR-12)

(1)5 (2) 5 (3) 10 (4)10

27. 𝜔 vd;gJ 1 ,d; Kg;gb %yk; vdpy; (1 − 𝜔 + 𝜔2)4 + (1 + 𝜔 − 𝜔2)4,d; kjpg;G

(MAR-06,JUN-06,OCT-10,JUN-12,OCT-12,OCT-14,JUN-15)

(1)0 (2)32 (3) −16 (4) −32

28. 𝜔 vd;gJ 1 ,d; 𝑛Mk; gb %yk; vdpy; (JUN-10,JUN-13,MAR-15)

(1)1 + 𝜔2 + 𝜔4 + ⋯ = 𝜔 + 𝜔3 + 𝜔5 + ⋯ (2) 𝜔𝑛 = 0

(3) 𝝎𝒏 = 𝟏 (4)𝜔 = 𝜔𝑛−1

29. 𝜔 vd;gJ 1 ,d; Kg;gb %yk; vdpy;> 1 − 𝜔 1 − 𝜔2 1 − 𝜔4 (1 − 𝜔8) ,d; kjpg;G

(MAR-09,OCT-11,OCT-13, OCT-16, MAR-17)

(1) 9 (2) −9 (3)16 (4)32





4. gFKiw tbtf;fzpjk;

1. 𝑦2 − 2𝑦 + 8𝑥 − 23 = 0vd;w gutisaj;jpd; mr;R (OCT-08,MAR-11, OCT-12,JUN-13, OCT-16)

(1) 𝑦 = −1 (2) 𝑥 = −3 (3) 𝑥 = 3 (4) 𝒚 = 𝟏

2. 16𝑥2 − 3𝑦2 − 32𝑥 − 12𝑦 − 44 = 0 vd;gJ (JUN-08,OCT-10)

(1) xU ePs;tl;lk; (2) xU tl;lk; (3) xU gutisak; (4) xU mjpgutisak;

3. 4𝑥 + 2𝑦 = 𝑐 vd;w NfhL 𝑦2 = 16𝑥 vd;w gutisaj;jpd; njhLNfhL vdpy; 𝑐 ,d; kjpg;G (JUN-09,MAR-10,JUN-11,OCT-13,MAR-15)

(1) −1 (2)−2 (3)4 (4) −4

4. 𝑦2 = 8𝑥 vd;w gutisaj;jpy; 𝑡1 = 𝑡 kw;Wk; 𝑡2 = 3𝑡 vd;w Gs;spfspy; tiuag;gl;l njhLNfhLfs;

ntl;bf;nfhs;Sk; Gs;sp (MAR-08,OCT-09,JUN-14)

(1) (𝟔𝒕𝟐, 𝟖𝒕) (2) (8𝑡, 6𝑡2) (3) (𝑡2 , 4𝑡) (4) (4𝑡, 𝑡2)

5. 𝑦2 − 4𝑥 + 4𝑦 + 8 = 0 vd;w gutisaj;jpd; nrt;tfyj;jpd; ePsk;

(MAR-07,MAR-09,MAR-14,JUN-15,JUN-16)

(1)8 (2)6 (3)4 (4)2

6. 𝑦2 = 𝑥 + 4 vd;w gutisaj;jpd; ,af;Ftiuapd; rkd;ghL (JUN-10)

(1)𝑥 =15

4 (2) 𝑥 = −

15

4 (3) 𝒙 = −

𝟏𝟕

𝟒 (4) 𝑥 =

17

4

7. (2, −3) vd;w Kid 𝑥 = 4 vd;w ,af;Ftiuiaf; nfhz;l gutisaj;jpd; nrt;tfy ePsk; (OCT-07,MAR-16)

(1) 2 (2) 4 (3) 6 (4) 8

8. 𝑥2 = 16𝑦 vd;w gutisaj;jpd; Ftpak; (OCT-15)

(1) (4,0) (2) (0,4) (3)(-4,0) (4)(0,-4)

9. 𝑥2 = 8𝑦 − 1 vd;w gutisaj;jpd; Kid (MAR-12)

(1) −1

8, 0 (2)

1

8, 0 (3) 𝟎,

𝟏

𝟖 (4) 0, −

1

8

10. 2𝑥 + 3𝑦 + 9 = 0 vd;wf; NfhL 𝑦2 = 8𝑥 vd;w gutisj;ij njhLk; Gs;sp (MAR-06,MAR-13)

(1)(0,-3) (2)(2,4) (3) −6,9

2 (4)

𝟗

𝟐, −𝟔

11. 𝑦2 = 12𝑥 vd;w gutisaj;jpd; Ftpehzpd; ,Wjpg;Gs;spfspy; tiuag;gLk; njhLNfhLfs; re;jpf;Fk; Gs;sp mikAk; NfhL (JUN-07,OCT-11)

(1) 𝑥 − 3 = 0 (2) 𝒙 + 𝟑 = 𝟎 (3) 𝑦 + 3 = 0 (4) 𝑦 − 3 = 0

12. (-4,4) vd;w Gs;spapypUe;J 𝑦2 = 16𝑥 f;F tiuag;gLk; ,U njhLNfhLfSf;F ,ilNaAs;s Nfhzk; (JUN-08)

(1)45° (2) 30° (3) 60° (4) 𝟗𝟎°

13. 9𝑥2 + 5𝑦2 − 54𝑥 − 40𝑦 + 116 = 0 vd;w $k;G tistpd; ikaj; njhiyj;jfT (𝑒) ,d; kjpg;G

(MAR-07,OCT-11, OCT-16, MAR-17)

(1)1

3 (2)

𝟐

𝟑 (3)

4

9 (4)

2

5

14. 𝑥2

144+

𝑦2

169= 1vd;w ePs;tl;lj;jpd; miu-nel;lr;R kw;Wk; miu-Fw;wr;R ePsq;fs;

(JUN-10,MAR-15)

(1) 26,12 (2)13,24 (3)12,26 (4)13,12





15. 9𝑥2 + 5𝑦2 = 180 vd;w ePs;tl;lj;jpd; Ftpaq;fSf;fpilNa cs;s njhiyT

(JUN-09,JUN-12,MAR-13,JUN-13,OCT-13,MAR-14)

(1) 4 (2) 6 (3) 8 (4) 2

16. xU ePs;tl;lj;jpd; nel;lr;R kw;Wk; mjd; miu Fw;wr;Rfspd; ePsq;fs; KiwNa 8,2 mjd;

rkd;ghLfs; 𝑦 − 6 = 0 kw;Wk; 𝑥 + 4 = 0 vdpy;> ePs;tl;lj;jpd; rkd;ghL

(1) 𝑥+4 2

4+

𝑦−6 2

16= 1 (2)

𝒙+𝟒 𝟐

𝟏𝟔+

𝒚−𝟔 𝟐

𝟒= 𝟏 (3)

𝑥+4 2

16−

𝑦−6 2

4= 1 (4)

𝑥+4 2

4−

𝑦−6 2

16= 1

17. 2𝑥 − 𝑦 + 𝑐 = 0 vd;w Neh;f;NfhL4𝑥2 + 8𝑦2 = 32 vd;w ePs;tl;lj;jpd; njhLNfhL vdpy; 𝑐 ,d; kjpg;G

(OCT-08,OCT-10,MAR-12,OCT-12)

(1)±2 3 (2)±𝟔 (3)36 (4)±4

18. 4𝑥2 + 9𝑦2 = 36 vd;w ePs;tl;lj;jpd; kPJs;s VNjDk; xU Gs;spapypUe;J 5, 0 kw;Wk; (− 5, 0)

vd;w Gs;spfSf;fpilNa cs;s njhiyTfspd; $Ljy; (MAR-06,OCT-09)

(1) 4 (2) 8 (3) 6 (4) 18

19. 9𝑥2 + 16𝑦2 = 144 vd;w $k;G tistpd; ,af;F tl;lj;jpd; Muk; (OCT-06,JUN-08,MAR-11,OCT-14,MAR-16)

(1) 7 (2) 4 (3) 3 (4) 5

20. 16𝑥2 + 25𝑦2 = 400 vd;w tistiuapd; Ftpaj;jpypUe;J xU njhLNfhl;Lf;F tiuag;gLk;

nrq;Fj;Jf; NfhLfspd; mbapd; epakg;ghij (MAR-09,JUN-16)

(1) 𝑥2 + 𝑦2 = 4 (2) 𝒙𝟐 + 𝒚𝟐 = 𝟐𝟓 (3) 𝑥2 + 𝑦2 = 16 (4) 𝑥2 + 𝑦2 = 9

21. 12𝑦2 − 4𝑥2 − 24𝑥 + 48𝑦 − 127 = 0 vd;w mjpgutisaj;jpd; ikaj;njhiyj;jfT

(OCT-09,MAR-10,OCT-15)

(1) 4 (2)3 (3)2 (4)6

22. nrt;tfyj;jpd; ePsk;> Jizar;rpd; ePsj;jpy; ghjp vdf; nfhz;Ls;s mjpguisaj;jpd; ikaj; njhiyj; jfT (JUN-06,JUN-07,MAR-15)

(1) 3

2 (2)

5

3 (3)

3

2 (4)

𝟓

𝟐

23. 𝑥2

𝑎2 −𝑦2

𝑏2 = 1 vd;w mjpgutisaj;jpd; kPJs;s VNjDk; xU Gs;spapypUe;J Ftpaj;jpw;F

,ilNaAs;s njhiyTfspd; tpj;jpahrk; 24 kw;Wk; ikaj;njhiyj;jfT 2 vdpy;

mjpgutisaj;jpd; rkd;ghL (MAR-07,JUN-11)

(1) 𝒙𝟐

𝟏𝟒𝟒−

𝒚𝟐

𝟒𝟑𝟐= 𝟏 (2)

𝑥2

432−

𝑦2

144= 1 (3)

𝑥2

12−

𝑦2

12 3= 1 (4)

𝑥2

12 3−

𝑦2

12= 1

24. 𝑥2 − 4(𝑦 − 3)2 = 16 vd;w mjpgutisaj;jpd; ,af;Ftiu (MAR-06,OCT-09, MAR-16)

(1)𝑦 = ±8

5 (2) 𝒙 = ±

𝟖

𝟓 (3) 𝑦 = ±

5

8 (4) 𝑥 = ±

5

8

25. 4𝑥2 − 𝑦2 = 36 f;F 5𝑥 − 2𝑦 + 4𝑘 = 0 vd;w NfhL xU njhLNfhL vdpy; 𝑘 ,d; kjpg;G (OCT-06,JUN-15)

(1)4

9 (2)

2

3 (3)

𝟗

𝟒 (4)

81

16

26. 𝑥2

16−

𝑦2

9= 1 vd;w mjpgutisaj;jpw;F (2,1)vd;w Gs;spapypUe;J tiuag;gLk; njhLNfhLfspd;

njhLehz; (JUN-13)

(1) 𝟗𝒙 − 𝟖𝒚 − 𝟕𝟐 = 𝟎 (2) 9𝑥 + 8𝑦 + 72 = 0

(3)8𝑥 − 9𝑦 − 72 = 0 (4) 8𝑥 + 9𝑦 + 72 = 0





27. 𝑥2

16−

𝑦2

9= 1 vd;w mjpgutisaj;jpd; njhiyj;njhLNfhLfSf;fpilNaAs;s Nfhzk;

(MAR-08,MAR-12,OCT-13)

(1) 𝜋 − 2 tan−1 3

4 (2) 𝜋 − 2 tan−1

4

3 (3) 𝟐 𝐭𝐚𝐧−𝟏

𝟑

𝟒 (4) 2 tan−1

4

3

28. 36𝑦2 − 25𝑥2 + 900 = 0 vd;w mjpgutisaj;jpd; njhiyj;njhLNfhLfs; (MAR-17)

(1)𝑦 = ±6

5𝑥 (2) 𝒚 = ±

𝟓

𝟔𝒙 (3) 𝑦 = ±

36

25𝑥 (4) 𝑦 = ±

25

36𝑥

29. (8,0)vd;w Gs;spapypUe;J 𝑥2

64−

𝑦2

36= 1 vd;w mjpgutisaj;jpd; njhiyj;njhLfSf;F tiuag;gLk;

nrq;Fj;J J}uq;fspd; ngUf;fy; gyd; (JUN-10,JUN-14,JUN-16)

(1) 25

576 (2)

𝟓𝟕𝟔

𝟐𝟓 (3)

6

25 (4)

25

6

30. 𝑥2

16−

𝑦2

9= 1 vd;w mjpgutisaj;jpd; nrq;Fj;Jj;njhLNfhLfspd; ntl;Lk; Gs;spapd; epakg;ghij

(OCT-08,JUN-12)

(1) 𝑥2 + 𝑦2 = 25 (2) 𝑥2 + 𝑦2 = 4 (3) 𝑥2 + 𝑦2 = 3 (4) 𝒙𝟐 + 𝒚𝟐 = 𝟕

31. 𝑥 + 2𝑦 − 5 = 0,2𝑥 − 𝑦 + 5 = 0 vd;w njhiyj;;;;njhLNfhLfisf; nfhz;l mjpgutisaj;jpd;

ikaj;njhiyj;jfT (OCT-06,OCT-07)

(1) 3 (2) 𝟐 (3) 3 (4) 2

32. 𝑥𝑦 = 8 vd;w nrt;tf gutisaj;jpd; miu FWf;fr;rpd; ePsk; (OCT-10,OCT-12,MAR-14)

(1) 2 (2) 4 (3) 16 (4) 8

33. 𝑥𝑦 = 𝑐2 vd;w nrt;tf mjpgutisaj;jpd; njhiyj;njhLNfhLfs; (MAR-13)

(1)𝑥 = 𝑐, 𝑦 = 𝑐 (2) 𝑥 = 0, 𝑦 = 𝑐 (3) 𝑥 = 𝑐, 𝑦 = 0 (4) 𝒙 = 𝟎, 𝒚 = 𝟎

34. 𝑥𝑦 = 16 vd;w nrt;tf mjpgutisaj;jpd; Kidapd; Maj;njhiyTfs; (MAR-07,OCT-11)

(1) (4,4),(-4,-4) (2) (2,8),(-2,-8) (3)(4,0),(-4,0) (4) (8,0),(-8,0)

35. 𝑥𝑦 = 18 vd;w nrt;tf mjpgutisaj;jpd; xU Ftpak; (MAR-09, MAR-17)

(1)(6,6) (2)(3,3) (3)(4,4) (4)(5,5)

36. 𝑥𝑦 = 32 vd;w nrt;tf mjpgutisaj;jpd; nrt;tfyj;jpd; ePsk; (MAR-08,10, JUN-11, OCT-14,15,16)

(1)8 2 (2)32 (3)8 (4)16

37. 𝑥𝑦 = 72 vd;w jpl;l nrt;tf mjpgutisaj;jpd; kPJs;s VNjDk; xU Gs;spapypUe;J tiuag;gLk; njhLNfhL mjd; njhiyj;njhLNfhLfSld; cz;lhf;Fk; Kf;Nfhzj;jpd; gug;G (MAR-11,JUN-15) (1)36 (2)18 (3)72 (4)144

38. 𝑥𝑦 = 9 vd;w nrt;tf mjpgutisaj;jpd; kPJs;s 6,3

2 vd;w Gs;spapypUe;J tiuag;gLk;

nrq;Fj;J> tistiuia kPz;Lk; re;jpf;Fk; Gs;sp (JUN-12,JUN-14)

(1) 3

8, 24 (2) −24,

−3

8 (3) −

𝟑

𝟖, −𝟐𝟒 (4) 24,

3

8

5. tif Ez;fzpjk; : gad;ghLfs; - I

1. 𝑥 = 2,y; 𝑦 = −2𝑥3 + 3𝑥 + 5 vd;w tistiuapd; rha;T (OCT-16)

(1) −20 (2)27 (3) −16 (4) −21

2. 𝑟 Muk; nfhz;l xU tl;lj;jpd; gug;G A ,y; Vw;gLk; khWk; tPjk; (OCT-08)

(1)2𝜋𝑟 (2) 𝟐𝝅𝒓𝒅𝒓

𝒅𝒕 (3)𝜋𝑟2 𝑑𝑟

𝑑𝑡 (4) 𝜋

𝑑𝑟

𝑑𝑡





3. Mjpap;ypUe;J xU Neh;f;Nfhl;by; 𝑥 njhiytp;y; efUk; Gs;spapd; jpirNtfk; 𝑣 vdTk; 𝑎 + 𝑏𝑣2 = 𝑥2 vdTk; nfhLf;fg;gl;Ls;sJ. ,q;F 𝑎 kw;Wk; 𝑏 khwpypfs;. mjd; KLf;fk; MdJ (MAR-09)

(1)𝑏

𝑥 (2)

𝑎

𝑥 (3)

𝒙

𝒃 (4)

𝑥

𝑎

4. xU cUFk; gdpf;fl;bf; Nfhsj;jpd; fd msT 1 nr.kP 3 / epkplk; vdf; Fiwfpd;wJ. mjd; tpl;lk;

10 nr.kP vd ,Uf;Fk; NghJ tpl;lk; FiwAk; Ntfk; MdJ (MAR-10,MAR-11,JUN-14)

(1)−1

50𝜋 nr.kP / epkplk; (2)

𝟏

𝟓𝟎𝝅 nr.kP / epkplk; (3)

−11

75𝜋 nr.kP / epkplk; (4)

−2

75𝜋 nr.kP / epkplk;

5. 𝑦 = 3𝑥2 + 3 sin 𝑥 vd;w tistiuf;F 𝑥 = 0 tpy; njhLNfhl;bd; rha;T

(MAR-07,JUN-08,JUN-09,MAR-12,JUN-14)

(1)3 (2)2 (3)1 (4)-1

6. 𝑦 = 3𝑥2vd;w tistiuf;F 𝑥,d; Maj;njhiyT 2 vdf; nfhz;Ls;s Gs;spapy; nrq;Nfhl;bd;

rha;thdJ (MAR-06,OCT-06,JUN-07,OCT-09,JUN-13,MAR-15)

(1)1

13 (2)

1

14 (3)

−𝟏

𝟏𝟐 (4)

1

12

7. 𝑦 = 2𝑥2 − 6𝑥 − 4 vDk; tistiuap;y; 𝑥 −mr;Rf;F ,izahfTs;s njhLNfhl;bd;; njhLGs;sp (OCT-10,JUN-15)

(1) 5

2,−17

2 (2)

−5

2,−17

2 (3)

−5

2,

17

2 (4)

𝟑

𝟐,−𝟏𝟕

𝟐

8. 𝑦 =𝑥3

5 vDk; tistiuf;F (−1,

−1

5) vd;w Gs;spapy; njhLNfhl;bd; rkd;ghL (MAR-08)

(1)5𝑦 + 3𝑥 = 2 (2) 𝟓𝒚 − 𝟑𝒙 = 𝟐 (3)3𝑥 − 5𝑦 = 2 (4) 3𝑥 + 3𝑦 = 2

9. 𝜃 =1

𝑡 vDk; tistiuf;F Gs;sp −3,

−1

3 vd;w Gs;spapy; nrq;Nfhl;bd; rkd;ghL (OCT-12,MAR-14)

(1)3𝜃 = 27𝑡 − 80 (2) 5𝜃 = 27𝑡 − 80 (3)𝟑𝜽 = 𝟐𝟕𝒕 + 𝟖𝟎 (4) 𝜃 =1

𝑡

10. 𝑥2

25+

𝑦2

9= 1 kw;Wk;

𝑥2

8−

𝑦2

8= 1 vDk; tistiufSf;F ,ilg;gl;l Nfhzk;

(JUN-07,MAR-09,OCT-11, OCT-16)

(1)𝜋

4 (2)

𝜋

3 (3)

𝜋

6 (4)

𝝅

𝟐

11. 𝑦 = 𝑒𝑚𝑥 kw;Wk; 𝑦 = 𝑒−𝑚𝑥 > 𝑚 > 1 vd;Dk; tistiufSf;F ,ilg;gl;l Nfhzk; (OCT-13,MAR-16)

(1)tan−1 2𝑚

𝑚2−1 2(2) 𝐭𝐚𝐧−𝟏

𝟐𝒎

𝟏−𝒎𝟐 (3) tan−1 −2𝑚

1+𝑚2 (4) tan−1 2𝑚

𝑚2+1

12. 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

13. 𝑥2

3 + 𝑦2

3 = 𝑎2

3 vd;w tistiuapd; nrq;NfhL 𝑥 −mr;Rld; 𝜃 vd;Dk; Nfhzk; Vw;gLj;Jnkdpy; mr;nrq;Nfhl;bd; rha;T (MAR-11,MAR-13)

(1)− cot 𝜃 (2) 𝐭𝐚𝐧 𝜽 (3) −tan 𝜃 (4) cot 𝜃

14. xU rJuj;jpd; %iytpl;lj;jpd; ePsk; mjpfhpf;Fk; tPjk; 0.1 nr.kP / tpdhb vdpy; gf;f msT 15

2

nr.kP Mf ,Uf;Fk;NghJ mjd; gug;gsT mjpfhpf;Fk; tPjk; (OCT-12,JUN-16)

(1) 𝟏. 𝟓 nr.kP 2/ tpdhb (2) 3 nr.kP 2/ tpdhb (3) 3 2 nr.kP 2/ tpdhb (4) 0.15 nr.kP 2/ tpdhb





15. xU Nfhsj;jpd; fd msT kw;Wk; Muj;jpy; Vw;gLk;;;; khWtPjq;fs; vz;zstpy; rkkhf ,Uf;Fk;NghJ Nfhsj;jpd; tisgug;G (JUN-06,MAR-10)

(1)𝟏 (2) 1

2𝜋 (3) 4𝜋 (4)

4𝜋

3

16. 𝑥3 − 2𝑥2 + 3𝑥+8 mjpfhpf;Fk; tPjkhdJ 𝑥 mjpfhpf;Fk; tPjj;ij Nghy; ,Uklq;F vdpy; 𝑥 ,d;

kjpg;Gfs; (JUN-08,JUN-12)

(1) −1

3, −3 (2)

1

3, 3 (3)

−1

3, 3 (4)

𝟏

𝟑, 𝟏

17. xU cUisapd; Muk; 2 nr.kP ∕tpdhb vd;w tPjj;jp;y; mjpfhpf;fpd;wJ. mjd; cauk; 3 nr.kP ∕ tpdhb vd;w tPjj;jpy; Fiwfpd;wJ. Muk; 3 nr.kP kw;Wk; cauk; 5 nr.kP Mf ,Uf;Fk;NghJ mjd; fd mstpd; khW tPjk; (JUN-07)

(1)23𝜋 (2) 𝟑𝟑𝝅 (3) 43𝜋 (4) 53𝜋

18. 𝑦 = 6𝑥 − 𝑥3 NkYk; 𝑥 MdJ tpdhbf;F 5 myFfs; tPjj;jpy; mjpfhpf;fpd;wJ. 𝑥 = 3 vDk; NghJ

mjd; rha;tpd; khWtPjk; (MAR-13)

(1) - 90 myFfs; ∕tpdhb (2)90 myFfs; ∕tpdhb

(3)180 myFfs; ∕tpdhb (4)-180 myFfs; ∕tpdhb

19. xU fdr;rJuj;jpd; fd msT 4 nr.kP 3∕tpdhb vd;w tPjj;jpy; mjpfhpf;fpd;wJ. mf;fdr;rJuj;jp;d; fd msT 8 f.nr.kP Mf ,Uf;Fk; NghJ mjd; Gwg;gug;gsT mjpfhpf;Fk; tPjk;

(1) 𝟖 nr.kP 2∕tpdhb (2) 16 nr.kP 2∕tpdhb (3) 2 nr.kP 2∕tpdhb (4) 4 nr.kP 2∕tpdhb

20. 𝑦 = 8 + 4𝑥 − 2𝑥2 vd;w tistiu y-mr;ir ntl;;Lk; Gs;spap;y; mikAk; njhLNfhl;bd; rha;T

(1)8 (2) 4 (3)0 (4)-4

21. 𝑦2 = 𝑥 kw;Wk; 𝑥2 = 𝑦 vd;w gutisaq;fSf;fpilNa Mjpapy; mikAk; Nfhzk;

(JUN-06,JUN-10,MAR-14,OCT-14)

(1)2 tan−1 3

4 (2) tan−1

4

3 (3)

𝝅

𝟐 (4)

𝜋

4

22. 𝑥 = 𝑒𝑡 cos 𝑡; 𝑦 = 𝑒𝑡 sin 𝑡 vd;w tistiuapd; njhLNfhL 𝑥-mr;Rf;F ,izahfTs;sJ vdpy; 𝑡 ,d;

kjpg;G (JUN-12)

(1)− 𝝅

𝟒 (2)

𝜋

4 (3)0 (4)

𝜋

2

23. xU tistiuapd; nrq;NfhL 𝑥 - mr;rpd; kpif jpirapy; 𝜃 vd;Dk; Nfhzj;ij Vw;gLj;JfpwJ. mr;nrq;NfhL tiuag;gl;l Gs;spapy; tistiuapd; rha;T (OCT-07)

(1)− 𝐜𝐨𝐭 𝜽 (2) tan 𝜃 (3) −tan 𝜃 (4) cot 𝜃

24. 𝑦 = 3𝑒𝑥kw;Wk; 𝑦 =𝑎

3𝑒−𝑥 vd;Dk; tistiufs; nrq;Fj;jhf ntl;bf;nfhs;fpd;wd vdpy; ‘𝑎’ ,d;

kjpg;G (OCT-10)

(1)−1 (2) 𝟏 (3)1

3 (4) 3

25. 𝑠 = 𝑡3 − 4𝑡2 + 7 vdpy; KLf;fk; G+r;rpakhFk; NghJs;s jpirNtfk; (OCT-06,MAR-07,OCT-09,JUN-15)

(1) 32

3 m/sec (2)

−𝟏𝟔

𝟑 m/sec (3)

16

3 m/sec (4)

−32

3 m/sec

26. xU Neh;f;Nfhl;by; efUk; Gs;spapd; jpirNtfkhdJ> mf;Nfhl;by; xU epiyg;Gs;spapypUe;J efUk; Gs;sp;f;F ,ilapy; cs;s njhiytpd; th;f;fj;jpw;F Neh; tpfpjkhf mike;Js;sJ vdpy; mjd; KLf;fk; gpd;tUk; xd;wpDf;F tpfpjkhf mike;Js;sJ. (OCT-11, JUN-16)

(1)𝑠 (2) 𝑠2 (3) 𝒔𝟑 (4) 𝑠4





27. 𝑦 = 𝑥2 vd;w rhh;gpw;F [−2,2],y; Nuhypd; khwpyp

(1) 2 3

3 (2) 𝟎 (3) 2 (4) −2

28. 𝑎 = 0, 𝑏 = 1 vdf;nfhz;L 𝑓 𝑥 = 𝑥2 + 2𝑥 − 1 vd;w rhh;gpw;F nyf;uhQ;rpapd; ,ilkjpg;Gj;

Njw;wj;jpd;gbAs;s ‘𝑐’ ,d; kjpg;G (MAR-09,OCT-13,OCT-14)

(1)−1 (2) 1 (3)0 (4) 𝟏

𝟐

29. 𝑓 𝑥 = cos𝑥

2 vd;w rhh;gpw;F [𝜋, 3𝜋],y; Nuhy; Njw;wj;jpd;gb mike;j 𝑐 ,d; kjpg;G

(MAR-06, 08, 12, 17)

(1)0 (2) 𝟐𝝅 (3)𝜋

2 (4)

3𝜋

2

30. 𝑎 = 1 kw;Wk; 𝑏 = 4 vdf;nfhz;L> 𝑓 𝑥 = 𝑥 vd;w rhh;gpw;F nyf;uhQ;rpapd; ,ilkjpg;Gj; Njw;wj;jpd;gb mikAk; ‘𝑐’ ,d; kjpg;G (JUN-10,JUN-11, OCT-15, OCT-16)

(1) 𝟗

𝟒 (2)

3

2 (3)

1

2 (4)

1

4

31. lim

𝑥 → ∞𝑥2

𝑒𝑥 d; kjpg;G (OCT-07,OCT-08)

(1)2 (2) 𝟎 (3)∞ (4) 1

32. lim

𝑥 → 0𝑎𝑥−𝑏𝑥

𝑐𝑥−𝑑𝑥 d; kjpg;G (MAR-07,OCT-09)

(1)∞ (2) 0 (3)log𝑎𝑏

𝑐𝑑 (4)

𝐥𝐨𝐠(𝒂/𝒃)

𝐥𝐨𝐠(𝒄/𝒅)

33. 𝑓 𝑎 = 2; 𝑓 ′ 𝑎 = 1; 𝑔 𝑎 = −1; 𝑔′ 𝑎 = 2 vdpy; lim𝑥→𝑎 𝑔 𝑥 𝑓 𝑎 −𝑔 𝑎 𝑓(𝑥)

𝑥−𝑎 ,d; kjpg;G

(JUN-08,MAR-16)

(1) 𝟓 (2)−5 (3)3 (4)−3

34. gpd;tUtdtw;Ws; vJ (0, ∞),y; VWk; rhh;G? (OCT-06,OCT-12,OCT-15)

(1) 𝒆𝒙 (2)1

𝑥 (3)−𝑥2 (4)𝑥−2

35. 𝑓 𝑥 = 𝑥2 − 5𝑥 + 4 vd;w rhh;G VWk; ,ilntsp (MAR-11,JUN-11)

(1)(−∞, 1) (2)(1,4) (3)(𝟒, ∞) (4) vy;yh Gs;spfsplj;Jk;

36. 𝑓 𝑥 = 𝑥2 vd;w rhh;G ,wq;Fk; ,ilntsp (JUN-09,MAR-15)

(1)(−∞, ∞) (2)(−∞, 𝟎) (3)(0, ∞) (4) (−2, ∞)

37. 𝑦 = tan 𝑥 − 𝑥 vd;w rhh;G (JUN-14)

(1) 𝟎,𝝅

𝟐 ,y; VWk; rhh;G (2) 0,

𝜋

2 ,y; ,wq;Fk; rhh;G

(3) 0,𝜋

4 ,y; VWk;

𝜋

4,𝜋

2 ,y; ,wq;Fk;

(4) 0,𝜋

4 ,y; ,wq;Fk;

𝜋

4,𝜋

2 ,y; rhh;G

38. nfhLf;fg;gl;Ls;s miu tl;lj;jpd; tpl;lk; 4 nr.kP. mjDs; tiuag;gLk; nrt;tfj;jpd; ngUk gug;G

(MAR-06,OCT-14)

(1)2 (2)𝟒 (3)8 (4)16

39. 100 kP 2 gug;G nfhz;Ls;s nrt;tfj;jpd; kPr;rpW Rw;wsT (OCT-07,OCT-08,OCT-10,OCT-11,JUN-15)

(1)10 (2)20 (3)40 (4)60

40. 𝑓 𝑥 = 𝑥2 − 4𝑥 + 5vd;w rhh;G [0,3],y; nfhz;Ls;s kPg;ngU ngUk kjpg;G (MAR-12,17, OCT-15)

(1)2 (2)3 (3)4 (4)5





41. 𝑦 = −𝑒−𝑥 vd;w tistiu (MAR-10,MAR-13)

(1) 𝑥 > 0 tpw;F Nky;Nehf;fpf; FopT (2) 𝑥 > 0 tpw;F fPo;Nehf;fpf; FopT

(3) vg;NghJk; Nky;Nehf;fpf; FopT (4) vg;NghJk; fPo;Nehf;fpf; FopT

42. gpd;tUk; tistiufSs; vJ fPo;Nehf;fp FopT ngw;Ws;sJ? (MAR-08,JUN-09,OCT-13)

(1)𝒚 = −𝒙𝟐 (2) 𝑦 = 𝑥2 (3)𝑦 = 𝑒𝑥 (4) 𝑦 = 𝑥2 + 2𝑥 − 3

43. 𝑦 = 𝑥4 vd;w tistiuapd; tisT khw;Wg;Gs;sp (JUN-13,MAR-15)

(1)𝑥 = 0 (2) 𝑥 = 3 (3)𝑥 = 12 (4) vq;Fkpy;iy

44. 𝑦 = 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑 vd;w tistiuf;F 𝑥 = 1,y; xU tisT khw;Wg;Gs;sp cz;nldpy; (1)𝑎 + 𝑏 = 0 (2) 𝑎 + 3𝑏 = 0 (3)𝟑𝒂 + 𝒃 = 𝟎 (4) 3𝑎 + 𝑏 = 1

6. tif Ez;fzpjk; : gad;ghLfs; - II

1. 𝑢 = 𝑥𝑦 vdpy; 𝜕𝑢

𝜕𝑥 f;Fr; rkkhdJ (OCT-08,OCT-10,OCT-15, OCT-16)

(1)𝒚𝒙𝒚−𝟏 (2)𝑢 log 𝑥 (3) 𝑢 log 𝑦 (4) 𝑥𝑦𝑥−1

2. 𝑢 = sin−1 𝑥4+𝑦4

𝑥2+𝑦2 kw;Wk; 𝑓 = sin 𝑢 vdpy;> rkgbj;jhd rhh;G 𝑓 ,d;gb

(1)0 (2)1 (3) 𝟐 (4) 4

3. 𝑢 =1

𝑥2+𝑦2, vdpy; 𝑥

𝜕𝑢

𝜕𝑥+ 𝑦

𝜕𝑢

𝜕𝑦= (MAR-08,JUN-09,JUN-10,JUN-11)

(1)1

2𝑢 (2) 𝑢 (3)

3

2𝑢 (4)−𝒖

4. 𝑦2 𝑥 − 2 = 𝑥2(1 + 𝑥) vd;w tistiuf;F (OCT-09,MAR-13, MAR-17)

(1) 𝑥-mr;Rf;F ,izahd xU njhiyj;njhLNfhL cz;L

(2) 𝒚- mr;Rf;F ,izahd xU njhiyj;njhLNfhL cz;L

(3) ,U mr;RfSf;Fk; ,izahd njhiyj;njhLNfhLfs; cz;L

(4) njhiyj; njhLNfhLfs; ,y;iy

5. 𝑥 = 𝑟 cos 𝜃, 𝑦 = 𝑟 sin 𝜃 vdpy; 𝜕𝑟

𝜕𝑥= (JUN-09,MAR-11,MAR-14)

(1)sec 𝜃 (2)sin 𝜃 (3)𝐜𝐨𝐬 𝜽 (4) cosec 𝜃

6. gpd;tUtdtw;Ws; rhpahd $w;Wfs;: (MAR-12,JUN-16)

(i) xU tistiu Mjpiag; nghWj;J rkr;rPh; ngw;wpUg;gpd; mJ ,U mr;Rfisg; nghWj;Jk; rkr;rPh; ngw;wpUf;Fk;

(ii) xU tistiu ,U mr;Rfisg; nghWj;J rkr;rPh; ngw;wpUg;gpd; mJ Mjpiag; nghWj;Jk; rkr;rPh;; ngw;wpUf;Fk;

(iii) 𝑓 𝑥, 𝑦 = 0 vd;w tistiu 𝑦 = 𝑥 vd;w Nfhl;ilg; nghWj;J rkr;rPh; ngw;Ws;sJ vdpy;

𝑓 𝑥, 𝑦 = 𝑓(𝑦, 𝑥)

(iv) 𝑓 𝑥, 𝑦 = 0 vd;w tistiuf;F 𝑓 𝑥, 𝑦 = 𝑓(−𝑦, −𝑥),cz;ikahapd; mJ Mjpiag; nghWj;J rkr;rPh; ngw;wpUf;Fk;

(1) (ii),(iii) (2)(i),(iv) (3)(i),(iii) (4)(ii),(iv)

7. 𝑢 = log 𝑥2+𝑦2

𝑥𝑦 vdpy; 𝑥

𝜕𝑢

𝜕𝑥+ 𝑦

𝜕𝑢

𝜕𝑦 vd;gJ (JUN-06,MAR-07,OCT-07,MAR-10,OCT-13,JUN-15)

(1) 0 (2) 𝑢 (3) 2𝑢 (4)𝑢−1





8. 28 ,d; 11 Mk; gb%y rjtpfpjg; gpio Njhuhakhf 28 ,d; rjtpfpjg; gpioiag; Nghy; ____

klq;fhFk; (MAR-06,OCT-06,MAR-12,OCT-14)

(1)1

28 (2)

𝟏

𝟏𝟏 (3)11 (4)28

9. 𝑎2𝑦2 = 𝑥2(𝑎2 − 𝑥2) vd;w tistiu (OCT-07,OCT-09,MAR-10,JUN-11,OCT-12,JUN-15)

(1) 𝑥 = 0 kw;Wk; 𝑥 = 𝑎 f;F ,ilapy; xU fz;zp kl;LNk nfhz;Ls;sJ

(2) 𝑥 = 0kw;Wk; 𝑥 = 𝑎 f;F ,ilapy; ,U fz;zpfs; nfhz;L cs;sJ

(3) 𝒙 = −𝒂 kw;Wk; 𝒙 = 𝒂 f;F ,ilapy; ,U fz;zpfs; nfhz;L cs;sJ

(4) fz;zp VJkpy;iy

10. 𝑦2 𝑎 + 2𝑥 = 𝑥2(3𝑎 − 𝑥) vd;w tistiuapd; njhiyj; njhLNfhL (JUN-06,07,08,12, OCT-11,13, 16)

(1) 𝑥 = 3𝑎 (2) 𝒙 = −𝒂/𝟐 (3) 𝑥 = 𝑎/2 (4) 𝑥 = 0

11. 𝑦2 𝑎 + 𝑥 = 𝑥2(3𝑎 − 𝑥) vd;w tistiu gpd;tUtdtw;Ws; ve;jg; gFjpapy; mikahJ?

(MAR-09,JUN-10, 12,14,16)

(1) 𝑥 > 0 (2) 0 < 𝑥 < 3𝑎 (3) 𝒙 ≤ −𝒂 kw;Wk; 𝒙 > 3𝒂 (4)−𝑎 < 𝑥 < 3𝑎

12. 𝑢 = y sin 𝑥, vdpy; 𝜕2𝑢

𝜕𝑥𝜕𝑦= (JUN-07,JUN-08)

(1) 𝐜𝐨𝐬 𝒙 (2)cos 𝑦 (3)sin 𝑥 (4)0

13. 𝑢 = 𝑓 𝑦

𝑥 vdpy;> 𝑥

𝜕𝑢

𝜕𝑥+ 𝑦

𝜕𝑢

𝜕𝑦 ,d; kjpg;G (OCT-08,MAR-09,JUN-13)

(1)0 (2)1 (3)2𝑢 (4) 𝑢

14. 9𝑦2 = 𝑥2(4 − 𝑥2) vd;w tistiu vjw;F rkr;rPh;? (MAR-06,OCT-06,OCT-10,MAR-15,OCT-15)

(1) y – mr;R (2) 𝑥– mr;R (3) 𝑦 = 𝑥 (4) ,U mr;Rfs;

15. 𝑎𝑦2 = 𝑥2(3𝑎 − 𝑥) vd;w tistiu y mr;ir ntl;Lk; Gs;spfs; (MAR-08,MAR-15,MAR-16)

(1) 𝑥 = −3𝑎, 𝑥 = 0 (2)𝑥 = 0, 𝑥 = 3𝑎 (3) 𝑥 = 0, 𝑥 = 𝑎 (4) 𝒙 = 𝟎

7. njhif Ez;fzpjk; : gad;ghLfs;

1. cos 5/3𝑥

cos 5/3𝑥+sin 5/3𝑥

𝜋/2

0𝑑𝑥 ,d; kjpg;G (MAR-12,JUN-13,JUN-15)

(1)𝜋

2 (2)

𝝅

𝟒 (3)0 (4) 𝜋

2. sin 𝑥−cos 𝑥

1+sin 𝑥 cos 𝑥

𝜋/2

0𝑑𝑥 ,d; kjpg;G (JUN-10, MAR-17)

(1)𝜋

2 (2) 𝟎 (3)

𝜋

4 (4) 𝜋

3. 𝑥(1 − 𝑥)4𝑑𝑥1

0 ,d; kjpg;G (MAR-06,MAR-09,JUN-10,MAR-11,OCT-12,JUN-14,MAR-15,OCT-15)

(1)1

12 (2)

𝟏

𝟑𝟎 (3)

1

24 (4)

1

20

4. sin 𝑥

2+cos 𝑥

𝜋/2

−𝜋/2𝑑𝑥 ,d; kjpg;G (JUN-07,OCT-07,OCT-10, OCT-16)

(1)𝟎 (2) 2 (3)log 2 (4)log 4

5. sin4𝑥 𝑑𝑥𝜋

0 ,d; kjpg;G (OCT-06,OCT-09,JUN-11,MAR-14)

(1)3𝜋

16 (2)

3

16 (3)0 (4)

𝟑𝝅

𝟖





6. cos32𝑥 𝑑𝑥𝜋/4

0 ,d; kjpg;G (MAR-07, 08, 10, 14, 17, JUN-09, OCT-08, 11, 14)

(1)2

3 (2)

𝟏

𝟑 (3)0 (4)

2𝜋

3

7. sin2𝑥 cos3𝑥𝑑𝑥𝜋

0 ,d; kjpg;G (JUN-08,MAR-13,OCT-13, JUN-16)

(1)𝜋 (2) 𝜋

2 (3)

𝜋

4 (4) 𝟎

8. 𝑦 = 𝑥 vd;w Nfhl;bw;Fk; 𝑥-mr;R> NfhLfs; 𝑥 = 1 kw;Wk; 𝑥 = 2 Mfpatw;wpw;Fk; ,ilg;gl;l

muq;fj;jpd; gug;G (JUN-07,OCT-08,MAR-09,JUN-12,OCT-15)

(1) 𝟑

𝟐 (2)

5

2 (3)

1

2 (4)

7

2

9. 𝑥 = 0 ,ypUe;J 𝑥 =𝜋

4 tiuapyhd 𝑦 = sin 𝑥 kw;Wk; 𝑦 = cos 𝑥 vd;w tistiufspd; ,ilg;gl;l

gug;G (JUN-06,OCT-07,MAR-10,OCT-13,JUN-14,MAR-16)

(1) 2 + 1 (2) 𝟐 − 𝟏 (3) 2 2 − 2 (4) 2 2 + 2

10. 𝑥2

𝑎2 +𝑦2

𝑏2 = 1 vd;w ePs;tl;lj;jpw;Fk; mjd; Jiz tl;lj;jpw;Fk; ,ilg;gl;l gug;G

(MAR-06,JUN-06,MAR-07,JUN-09)

(1) 𝜋𝑏(𝑎 − 𝑏) (2) 2𝜋𝑎(𝑎 − 𝑏) (3) 𝝅𝒂(𝒂 − 𝒃) (4)2 𝜋𝑏(𝑎 − 𝑏)

11. gutisak; 𝑦2 = 𝑥 f;Fk; mjd; nrt;tfyj;jpw;Fk; ,ilg;gl;l gug;G

(MAR-08,MAR-11,OCT-12,JUN-15)

(1)4

3 (2)

𝟏

𝟔 (3)

2

3 (4)

8

3

12. 𝑥2

9+

𝑦2

16= 1 vd;w tistiuia Fw;wr;ir nghWj;J Row;wg;gLk; jplg;nghUspd; fd msT

(JUN-07,JUN-12,MAR-13,OCT-13,OCT-16)

(1)48𝜋 (2)64𝝅 (3) 32𝜋 (4)128𝜋

13. 𝑦 = 3 + 𝑥2 vd;w tistiu 𝑥 = 0 tpypUe;J 𝑥 = 4 tiu 𝑥- mr;ir mr;rhf itj;Jr; Row;wg;gLk;

jplg;nghUspd; fd msT (MAR-06,OCT-08,OCT-09,MAR-11,JUN-13)

(1)100𝜋 (2)100

9𝜋 (3)

𝟏𝟎𝟎

𝟑𝝅 (4)

100

3

14. NfhLfs; 𝑦 = 𝑥, 𝑦 = 1 kw;Wk; 𝑥 = 0 Mfpait Vw;gLj;Jk; gug;G 𝑦-mr;ir nghWj;Jr; Row;wg;gLk;

jplg;nghUspd; fd msT (JUN-08,OCT-10,JUN-11,JUN-14,OCT-14)

(1)𝜋

4 (2)

𝜋

2 (3)

𝝅

𝟑 (4)

2𝜋

3

15. 𝑥2

𝑎2 +𝑦2

𝑏2 = 1 vd;w ePs;tl;lj;jpd; gug;ig nel;lr;R> Fw;wr;R ,tw;iw nghWj;Jr; Row;wg;gLk;

jplg;nghUspd; fd msTfspd; tpfpjk; (JUN-06,MAR-09,JUN-10,JUN-11,MAR-12,OCT-12,MAR-15, MAR-17)

(1)𝑏2: 𝑎2 (2)𝑎2: 𝑏2 (3) 𝑎: 𝑏 (4) 𝒃: 𝒂

16. (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 (OCT-06, OCT-07,JUN-09,OCT-11,MAR-14,MAR-16)

(1)18𝜋 (2)2𝜋 (3) 36𝜋 (4)𝟗𝝅

17. 𝑥2

3 + 𝑦2

3 = 4 vd;w tistiuapd; tpy;ypd; ePsk; (MAR-08,JUN-08,MAR-12,OCT-14,MAR-15,JUN-16)

(1)48 (2)24 (3)12 (4)96





18. 𝑦 = 2𝑥, 𝑥 = 0 kw;Wk; 𝑥 = 2 ,tw;wpw;F ,ilNa Vw;gLk; gug;G 𝑥-mr;ir nghWj;Jr; Row;wg;gLk;

jplg;nghUspd; tisg;gug;G (OCT-06,MAR-07,OCT-10,OCT-11,MAR-13,JUN-15,OCT-15,MAR-16)

(1)𝟖 𝟓𝝅 (2)2 5𝜋 (3) 5𝜋 (4)4 5𝜋

19. Muk; 5 cs;s Nfhsj;ij jsq;fs; ikaj;jpypUe;J 2 kw;Wk; 4 J}uj;jpy; ntl;Lk; ,U ,izahd

jsq;fSf;F ,ilg;gl;l gFjpapd; tisgug;G (OCT-09,MAR-10,JUN-13,JUN-16, OCT-16)

(1)20𝝅 (2)40𝜋 (3)10𝜋 (4)30𝜋

8. tiff;nfOr; rkd;ghLfs;

1. 𝑑𝑦

𝑑𝑥+ 2

𝑦

𝑥= 𝑒4𝑥vd;w tiff;nfOr; rkd;ghl;bd; njhiff; fhuzp (JUN-09,JUN-11)

(1) log 𝑥 (2) 𝒙𝟐 (3) 𝑒𝑥 (4) 𝑥

2. 𝑑𝑦

𝑑𝑥+ 𝑃𝑦 = 𝑄 vd;w tiff;nfOr; rkd;ghl;bd; njhiff; fhuzp cos 𝑥 vdpy; 𝑃 ,d; kjpg;G

(JUN-07,OCT-08,MAR-09,OCT-15, OCT-16)

(1)− cot 𝑥 (2) cot 𝑥 (3) tan 𝑥 (4) − 𝐭𝐚𝐧 𝒙

3. 𝑑𝑥 + 𝑥𝑑𝑦 = 𝑒−𝑦sec2𝑦𝑑𝑦 ,d; njhiff;fhuzp (OCT-10, MAR-11,OCT-14, JUN-16)

(1) 𝑒𝑥 (2) 𝑒−𝑥 (3) 𝒆𝒚 (4) 𝑒−𝑦

4. 𝑑𝑦

𝑑𝑥+

1

𝑥 log 𝑥 . 𝑦 =

2

𝑥2,d; njhiff;fhuzp (OCT-06,MAR-07,JUN-07,OCT-09,MAR-13,JUN-15)

(1) 𝑒𝑥 (2) 𝐥𝐨𝐠 𝒙 (3) 1

𝑥 (4) 𝑒−𝑥

5. 𝑚 < 0 Mf ,Ug;gpd; 𝑑𝑥

𝑑𝑦+ 𝑚𝑥 = 0,d; jPh;T (MAR-08,10, 12, 14, 17 JUN-09,10, OCT-13)

(1)𝑥 = 𝑐𝑒𝑚𝑦 (2) 𝒙 = 𝒄𝒆−𝒎𝒚 (3) 𝑥 = 𝑚𝑦 + 𝑐 (4) 𝑥 = 𝑐

6. 𝑦 = 𝑐𝑥 − 𝑐2 vd;gjidg; nghJj; jPh;thfg; ngw;w tiff;nfO rkd;ghL (OCT-08,MAR-14)

(1) 𝒚′ 𝟐 − 𝒙𝒚′ + 𝒚 = 𝟎 (2) 𝑦′′ = 0 (3) 𝑦′ = 𝑐 (4) 𝑦′ 2 + 𝑥𝑦′ + 𝑦 = 0

7. 𝑑𝑥

𝑑𝑦

2+ 5𝑦1/3 = 𝑥 vd;w tiff;nfOtpd; (MAR-06, JUN-08,MAR-10, MAR-17)

(1) thpir 2 kw;Wk; gb 1 (2) thpir 1 kw;Wk; gb 2

(3) thpir 1 kw;Wk; gb 6 (4) thpir 1 kw;Wk; gb 3

8. xU jsj;jpy; cs;s 𝑥 -mr;Rf;F nrq;Fj;jy;yhj NfhLfspd; tiff;nfOr; rkd;ghL

(1) 𝑑𝑦

𝑑𝑥= 0 (2)

𝒅𝟐𝒚

𝒅𝒙𝟐 = 𝟎 (3) 𝑑𝑦

𝑑𝑥= 𝑚 (4)

𝑑2𝑦

𝑑𝑥 2 = 𝑚

9. Mjpg;Gs;spia ikakhff; nfhz;l tl;lq;fspd; njhFg;gpd; tiff;nfOr; rkd;ghL

(MAR-09,11,JUN-07,15,16)

(1)𝑥𝑑𝑦 + 𝑦 𝑑𝑥 = 0 (2) 𝑥𝑑𝑦 − 𝑦𝑑𝑥 = 0 (3) 𝒙𝒅𝒙 + 𝒚𝒅𝒚 = 𝟎 (4) 𝑥𝑑𝑥 − 𝑦𝑑𝑦 = 0

10. tiff;nfOr; rkd;ghL 𝑑𝑦

𝑑𝑥+ 𝑝𝑦 = 𝑄 tpd; njhiff; fhuzp (MAR-06)

(1) 𝑝𝑑𝑥 (2) 𝑄 𝑑𝑥 (3) 𝑒 𝑄 𝑑𝑥 (4) 𝒆 𝒑 𝒅𝒙

11. (𝐷2 + 1)𝑦 = 𝑒2𝑥 ,d; epug;Gr; rhh;G (JUN-09,JUN-12,MAR-15)

(1) (𝐴𝑥 + 𝐵)𝑒𝑥 (2) 𝑨 𝐜𝐨𝐬 𝒙 + 𝑩 𝐬𝐢𝐧 𝒙 (3)(𝐴𝑥 + 𝐵)𝑒2𝑥 (4) (𝐴𝑥 + 𝐵)𝑒−𝑥

12. (𝐷2 − 4𝐷 + 4)𝑦 = 𝑒2𝑥 ,d; rpwg;Gj; jPh;T (P.I) (OCT-06,JUN-08,OCT-08,OCT-09,JUN-15)

(1)𝒙𝟐

𝟐𝒆𝟐𝒙 (2) 𝑥𝑒2𝑥 (3) 𝑥𝑒−2𝑥 (4)

𝑥

2𝑒−2𝑥





13. 𝑦 = 𝑚𝑥 vd;w Neh;f;NfhLfspd; njhFg;gpd; tiff;nfOr;rkd;ghL (OCT-07)

(1)𝑑𝑦

𝑑𝑥= 𝑚 (2) 𝒚𝒅𝒙 − 𝒙𝒅𝒚 = 𝟎 (3)

𝑑2𝑦

𝑑𝑥2 = 0 (4) 𝑦𝑑𝑥 + 𝑥𝑑𝑦 = 0

14. 1 + 𝑑𝑦

𝑑𝑥

1/3=

𝑑2𝑦

𝑑𝑥2 vd;w tiff;nfOr; rkd;ghl;bd; gb

(MAR-07,OCT-11,MAR-13,MAR-15,OCT-15,MAR-16)

(1) 1 (2)2 (3)3 (4)6

15. 𝑐 = 1+

𝑑𝑦

𝑑𝑥

3

2/3

𝑑3𝑦

𝑑𝑥3

vd;w tiff;nfOr; rkd;ghl;bd; gb (,q;F 𝑐 xU khwpyp) (JUN-12)

(1) 1 (2)3 (3)-2 (4)2

16. xU fjphpaf;f nghUspd; khWtPj kjpg;G> mk;kjpg;gpd; (p)Neh;tpfpjj;jpy; rpijTWfpwJ. ,jw;F Vw;w

tiff; nfOr; rkd;ghL (k Fiw vz;) (OCT-07,JUN-12,JUN-13)

(1)𝑑𝑝

𝑑𝑡=

𝑘

𝑝 (2)

𝑑𝑝

𝑑𝑡= 𝑘𝑡 (3)

𝒅𝒑

𝒅𝒕= 𝒌𝒑 (4)

𝑑𝑝

𝑑𝑡= −𝑘𝑡

17. 𝑥𝑦 jsj;jpYs;s vy;yh Neh;f;NfhLfspd; njhFg;gpd; tiff;nfOr;rkd;ghL (MAR-16, OCT-16)

(1)𝑑𝑦

𝑑𝑥= xU khwpyp (2)

𝒅𝟐𝒚

𝒅𝒙𝟐 = 𝟎 (3) 𝑦 +𝑑𝑦

𝑑𝑥= 0 (4)

𝑑2𝑦

𝑑𝑥2 + 𝑦 = 0

18. 𝑦 = 𝑘𝑒𝜆𝑥 vdpy; mjd; tiff;nfOr; rkd;ghL (JUN-06,OCT-10,JUN-11,JUN-14, OCT-16, MAR-17)

(1)𝒅𝒚

𝒅𝒙= 𝝀𝒚 (2)

𝑑𝑦

𝑑𝑥= 𝑘𝑦 (3)

𝑑𝑦

𝑑𝑥+ 𝑘𝑦 = 0 (4)

𝑑𝑦

𝑑𝑥= 𝑒𝜆𝑥

19. 𝑦 = 𝑎𝑒3𝑥 + 𝑏𝑒−3𝑥 vd;w rkd;ghl;by; 𝑎 iaAk; 𝑏 iaAk; ePf;fpf; fpilf;Fk; tiff;nfOr; rkd;ghL

(OCT-06,MAR-09,JUN-10,OCT-12,OCT-14)

(1) 𝑑2𝑦

𝑑𝑥2 + 𝑎𝑦 = 0 (2) 𝒅𝟐𝒚

𝒅𝒙𝟐 − 𝟗𝒚 = 𝟎 (3) 𝑑2𝑦

𝑑𝑥2 − 9𝑑𝑦

𝑑𝑥= 0 (4)

𝑑2𝑦

𝑑𝑥2 + 9𝑥 = 0

20. 𝑦 = 𝑒𝑥(𝐴 cos 𝑥 + 𝐵 sin 𝑥) vd;w njhlh;gpy; 𝐴 iaAk; 𝐵 iaAk; ePf;fpg; ngwg;gLk; tiff;nfOr;

rkd;ghL (OCT-09,OCT-11,OCT-13)

(1)𝑦2 + 𝑦1 = 0 (2) 𝑦2 − 𝑦1 = 0

(3) 𝒚𝟐 − 𝟐𝒚𝟏 + 𝟐𝒚 = 𝟎 (4) 𝑦2 − 2𝑦1 − 2𝑦 = 0

21. 𝑑𝑦

𝑑𝑥=

𝑥−𝑦

𝑥+𝑦 vdpy; (MAR-08,MAR-12,OCT-12,JUN-13, JUN-16)

(1)2𝑥𝑦 + 𝑦2 + 𝑥2 = 𝑐 (2) 𝑥2 + 𝑦2 − 𝑥 + 𝑦 = 𝑐

(3) 𝑥2 + 𝑦2 − 2𝑥𝑦 = 𝑐 (4) 𝒙𝟐 − 𝒚𝟐 − 𝟐𝒙𝒚 = 𝒄

22. 𝑓 ′ 𝑥 = 𝑥 kw;Wk; 𝑓 1 = 2 vdpy; 𝑓 𝑥 vd;gJ (OCT-07, MAR-10,JUN-15,OCT-15)

(1)−2

3(𝑥 𝑥 + 2) (2)

3

2(𝑥 𝑥 + 2) (3)

𝟐

𝟑(𝒙 𝒙 + 𝟐) (4)

2

3𝑥( 𝑥 + 2)

23. 𝑥2𝑑𝑦 + 𝑦 𝑥 + 𝑦 𝑑𝑥 = 0 vd;w rkgbj;jhd tiff;nfO rkd;ghl;by; 𝑦 = 𝑣𝑥 vd gpujpaPL nra;Ak;

NghJ fpilg;gJ (JUN-06, MAR-11, JUN-14,OCT-14)

(1)𝒙𝒅𝒗 + 𝟐𝒗 + 𝒗𝟐 𝒅𝒙 = 𝟎 (2)𝑣𝑑𝑥 + 2𝑥 + 𝑥2 𝑑𝑣 = 0

(3)𝑣2𝑑𝑥 − 𝑥 + 𝑥2 𝑑𝑣 = 0 (4)𝑣𝑑𝑣 + 2𝑥 + 𝑥2 𝑑𝑥 = 0

24. 𝑑𝑦

𝑑𝑥− 𝑦 tan 𝑥 = cos 𝑥 vd;w tiff;nfOr; rkd;ghl;bd; njhiff; fhuzp

(MAR-08,JUN-08,JUN-10,OCT-11,MAR-12,OCT-12,OCT-13,MAR-14,JUN-14, MAR-17)

(1)sec 𝑥 (2)𝐜𝐨𝐬 𝒙 (3)𝑒tan 𝑥 (4)cot 𝑥

25. (3𝐷2 + 𝐷 − 14)𝑦 = 13𝑒2𝑥 d; rpwg;G jPh;T (MAR-06,MAR-07,JUN-11,MAR-13)

(1) 26𝑥𝑒2𝑥 (2) 13𝑥𝑒2𝑥 (3) 𝒙𝒆𝟐𝒙 (4)𝑥2/2𝑒2𝑥





26. 𝑓 𝐷 = 𝐷 − 𝑎 𝑔 𝐷 , 𝑔(𝑎) ≠ 0 vdpy; tiff;nfOr; rkd;ghL 𝑓 𝐷 𝑦 = 𝑒𝑎𝑥,d; rpwg;Gj; jPh;T

(JUN-06,OCT-10,JUN-13,MAR-16)

(1) 𝑚𝑒𝑎𝑥 (2) 𝑒𝑎𝑥

𝑔(𝑎) (3) 𝑔(𝑎)𝑒𝑎𝑥 (4)

𝒙𝒆𝒂𝒙

𝒈(𝒂)

9. jdpepiy fzf;fpay;

1. fPo;f;fz;ltw;Ws; vit $w;Wfs;? (MAR-12,JUN-12,MAR-16 )

(i) flTs; cd;id Mrph;tjpf;fl;Lk; (ii) Nuhrh xU G+

(iii) ghypd; epwk; ntz;ik (iv)1 xU gfh vz;

(1)(i),(ii),(iii) (2)(i),(ii),(iv) (3)(i),(iii),(iv) (4)(ii),(iii),(iv)

2. xU $l;Lf; $w;W %d;W jdpf;$w;Wfisf; nfhz;ljhf ,Ug;gpd;> nka;al;ltizapYs;s

epiufspd; vz;zpf;if (OCT-09,OCT-12,MAR-13 )

(1) 8 (2) 6 (3) 4 (4) 2

3. 𝑝 apd; nka;kjpg;G 𝑇 kw;Wk; 𝑞 ,d; nka;kjpg;G 𝐹 vdpy; gpd;tUtdtw;wpy; vit nka;kjpg;G 𝑇 vd

,Uf;Fk;? (OCT-07,MAR-10,OCT-11,MAR-14,MAR-10,JUN-16, MAR-17 )

(i) 𝑝 𝑞 (ii) ∼ 𝑝 𝑞 (iii) 𝑝 ∼ 𝑞 (iv) 𝑝 ∧ ∼ 𝑞

(1)(i),(ii),(iii) (2)(i),(ii),(iv) (3)(i),(iii),(iv) (4)(ii),(iii),(iv)

4. ~[ 𝑝 ⋀(~𝑞)] d; nka; ml;ltizapy; epiufspd; vz;zpf;if (JUN-06, 08,OCT-10, 11, 16 )

(1) 2 (2) 4 (3) 6 (4) 8

5. epge;;jidf; $w;W 𝑝 → 𝑞 f;Fr; rkhdkhdJ

(MAR-06,MAR-09,MAR-11,OCT-13,JUN-14,OCT-14,JUN-15,OCT-15)

(1) 𝑝 ∨ 𝑞 (2) 𝑝 ∨ ~𝑞 (3) ~𝒑 ∨ 𝒒 (4) 𝑝 ∧ 𝑞

6. gpd;tUtdtw;Ws; vJ nka;ikahFk;?

(JUN-07,MAR-08,MAR-09,JUN-09,JUN-10,MAR-12,MAR-13,OCT-13)

(1) 𝑝 ∨ 𝑞 (2) 𝑝 ∧ 𝑞 (3) 𝒑 ∨ ~𝒑 (4) 𝑝 ∧ ~𝑝

7. gpd;tUtdtw;Ws; vJ Kuz;ghlhFk;? (MAR-06,OCT-06,OCT-08,MAR-14,MAR-16)

(1) 𝑝 ∨ 𝑞 (2) 𝑝 ∧ 𝑞 (3) 𝑝 ∨ ~𝑝 (4) 𝒑 ∧ ~𝒑

8. 𝑝 ↔ 𝑞 f;Fr; rkhdkhdJ (MAR-07,JUN-11,MAR-15)

(1)𝑝 → 𝑞 (2) 𝑞 → 𝑝 (3) 𝑝 → 𝑞 ∨ (𝑞 → 𝑝) (4) 𝒑 → 𝒒 ∧ (𝒒 → 𝒑)

9. fPo;fz;ltw;wpy; vJ 𝑅 ,y; <UWg;Gr; nrayp my;y?

(OCT-07,JUN-08,JUN-09,MAR-10,MAR-11,MAR-15)

(1)𝑎 ∗ 𝑏 = 𝑎𝑏 (2)𝑎 ∗ 𝑏 = 𝑎 − 𝑏 (3) 𝒂 ∗ 𝒃 = 𝒂𝒃 (4) 𝑎 ∗ 𝑏 = 𝑎2 + 𝑏2

10. rkdpAila miuf;Fyk;> Fykhtjw;F G+h;j;jp nra;a Ntz;ba tpjpahtJ (MAR-08,JUN-13 )

(1) milg;G tpjp (2) Nrh;g;G tpjp (3) rkdp tpjp (4) vjph;kiw tpjp

11. fPo;f;fz;ltw;Ws; vJ Fyk; my;y? (OCT-08,JUN-10,JUN-16 )

(1)(𝑍𝑛 , +𝑛) (2) (𝑍, +) (3) (𝒁, . ) (4) (𝑅, +)

12. KOf;fspy; * vd;w <UWg;Gr; nrayp 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 − 𝑎𝑏 vd tiuaWf;fg;gLfpwJ vdpy; 3*(4 *5),d; kjpg;G (JUN-06,OCT-12 )

(1)25 (2)15 (3)10 (4)5





13. (𝑍9, +9) ,y; [7],d; thpir (OCT-06,14, MAR-07, 09,11,17, JUN-08,10,14)

(1)9 (2)6 (3)3 (4)1

14. ngUf;fiyg; nghWj;J Fykhfpa xd;wpd; Kg;gb %yq;fspy;> 𝜔2,d; thpir

(JUN-07,OCT-08,JUN-15,OCT-15)

(1)4 (2)3 (3)2 (4)1

15. 3 +11 ([5]+11[6]) ,d; kjpg;G (MAR-06,JUN-09,JUN-13)

(1)[0] (2)[1] (3)[2] (4)[3]

16. nka;naz;fspd; fzk; 𝑅,y; *vd;w <UWg;Gr; nrayp 𝑎 ∗ 𝑏 = 𝑎2 + 𝑏2 . vd tiuaWf;fg;gLfpwJ

vdpy; (3 * 4) * 5,d; kjpg;G (JUN-07,OCT-10,JUN-15, JUN-16)

(1)5 (2)𝟓 𝟐 (3)25 (4)50

17. fPo;f;fz;ltw;Ws; vJ rhp? (OCT-06,JUN-11,OCT-14)

(1) xU Fyj;jpd; xU cWg;gpw;F xd;wpw;F Nkw;gl;l vjph;kiw cz;L

(2) Fyj;jpd; xt;nthU cWg;Gk; mjd; vjph;kiwahf ,Uf;Fnkdpy; mf;Fyk; xU vgPypad; FykhFk;

(3) nka;naz;fis cWg;Gfshff; nfhz;l vy;yh 2 × 2 mzpf;NfhitfSk; ngUf;fy; tpjpapy; FykhFk;

(4) vy;yh 𝑎, 𝑏 ∈ 𝐺 f;Fk; 𝑎 ∗ 𝑏 −1 = 𝑎−1 ∗ 𝑏−1

18. ngUf;fy; tpjpiag; nghWj;J Fykhfpa xd;wpd; ehyhk; %yq;fspy;> – 𝑖 ,d; thpir

(JUN-12, JUN-13,OCT-15, MAR-16, OCT-16)

(1)4 (2)3 (3)2 (4)1

19. ngUf;fiy nghWj;J Fykhfpa xd;wpd; 𝑛Mk; gb %yq;fspy; 𝜔𝑘,d; vjph;kiw ( 𝑘 < 𝑛 )

(JUN-06,MAR-08,OCT-09,OCT-11,MAR-12,OCT-13,JUN-14, MAR-17)

(1) 𝜔1/𝑘 (2) 𝜔−1 (3) 𝝎𝒏−𝒌 (4) 𝜔𝑛/𝑘

20. KOf;fspy; * vd;w <UWg;Gr; nrayp 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 − 1vd tiuaWf;fg;gLfpwJ vdpy; rkdp cWg;G

(MAR-07,MAR-10,OCT-10,JUN-12,MAR-13,MAR-14, OCT-16)

(1)0 (2)1 (3) 𝑎 (4) 𝑏

10. epfo;jfTg; guty;

1. 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 (JUN-08, 09,OCT-13,16)

(1) 1

3 (2)

1

6 (3)

𝟏

𝟗 (4)

1

12

2. 𝑓 𝑥 =𝐴

𝜋

1

16+𝑥2 , −∞ < 𝑥 < ∞ vd;gJ 𝑋 vd;w njhlh; rktha;g;G khwpapd; xU epfo;jfT mlh;j;jpr;

rhh;T ( p.d.f. ) vdpy; 𝐴,d; kjpg;G (MAR-06,MAR-07,OCT-08,OCT-09,MAR-11,OCT-11,MAR-14,JUN-14,JUN-15)

(1)16 (2) 8 (3) 𝟒 (4) 1

3. 𝑋 vd;w rktha;g;G khwpapd; epfo;jfTg; guty; gpd;tUkhW:

X 0 1 2 3 4 5 P(X=𝑥) 1/4 2𝑎 3𝑎 4𝑎 5𝑎 1/4

𝑃(1 ≤ 𝑥 ≤ 4) ,d; kjpg;G (JUN-10,JUN-12, MAR-17)

(1) 10

21 (2)

2

7 (3)

1

14 (4)

𝟏

𝟐





4. 𝑋 vd;w rktha;g;G khwpapd; epfo;jfT epiwr;rhh;G guty; gpd;tUkhW

X −2 3 1

P(X= 𝑥) 𝜆

6

𝜆

4

𝜆

12

𝜆 tpd; kjpg;G (JUN-16 )

(1)1 (2)2 (3)3 (4)4

5. 𝑋 vd;w xU jdpepiy rktha;g;G khwp 0,1,2 vd;w kjpg;Gfisf; nfhs;fpwJ. NkYk;

𝑃 𝑋 = 0 =144

169, 𝑃 𝑋 = 1 =

1

169 vdpy; 𝑃 𝑋 = 2 ,d; kjpg;G (MAR-09,OCT-10,JUN-13 )

(1) 145

169 (2)

𝟐𝟒

𝟏𝟔𝟗 (3)

2

169 (4)

143

169

6. xU rktha;g;G khwp 𝑋,d; epfo;jfT epiwr; rhh;G(p.d.f.) gpd;tUkhW (MAR-16 )

X 0 1 2 3 4 5 6 7 P(X=𝑥) 0 𝑘 2𝑘 2𝑘 3𝑘 𝑘2 2𝑘2 7𝑘2 + 𝑘

𝑘 ,d; kjpg;G

(1)1

8 (2)

𝟏

𝟏𝟎 (3) 0 (4)−1 or

1

10

7. 𝐸 𝑋 + 𝑐 = 8 kw;Wk; 𝐸 𝑋 − 𝑐 = 12 vdpy; 𝑐 ,d; kjpg;G

(OCT-06,OCT-07,JUN-09,MAR-12,OCT-13,MAR-15,OCT-15, MAR-16 )

(1)−𝟐 (2) 4 (3)−4 (4)2

8. 𝑋 vd;w rktha;g;G khwpapd; 3, 4 kw;Wk; 12 Mfpa kjpg;Gfs; KiwNa 1

3,

1

4 kw;Wk;

5

12 Mfpa

epfo;jfTfisf; nfhs;Snkdpy; 𝐸 𝑋 ,d; kjpg;G (OCT-08,JUN-16, OCT-15)

(1) 5 (2) 𝟕 (3) 6 (4) 3

9. 𝑋 vd;w rktha;g;G khwpapd; gutw;gb 4 NkYk; ruhrhp 2 vdpy; 𝐸(𝑋2),d; kjpg;G(JUN-07, 13, MAR-09)

(1) 2 (2) 4 (3) 6 (4) 8

10. xU jdpepiy rktha;g;G khwp 𝑋 f;F 𝜇2 = 20. NkYk; 𝜇2′ = 276 vdpy; 𝑋 ,d; ruhrhpapd; kjpg;G

(OCT-09,MAR-10,JUN-11,MAR-14 )

(1)16 (2)5 (3)2 (4)1

11. 𝑉𝑎𝑟(4𝑋 + 3) ,d; kjpg;G (MAR-06, JUN-06,MAR-08,JUN-08,JUN-15 )

(1)7 (2)𝟏𝟔 𝑽𝒂𝒓(𝑿) (3)19 (4)0

12. xU gfilia 5 Kiw tPRk; NghJ> 1 my;yJ 2 fpilg;gJ ntw;wpnadf; fUjg;gLfpwJ> vdpy;

ntw;wpapd; ruhrhpapd; kjpg;G (OCT-06,JUN-12,MAR-13)

(1) 𝟓

𝟑 (2)

3

5 (3)

5

9 (4)

9

5

13. xU <UWg;Gg; gutypd; ruhrhp 5 NkYk; jpl;ltpyf;fk; 2 vdpy; 𝑛 kw;Wk; 𝑝 ,d; kjpg;Gfs;

(OCT-11,OCT-12,OCT-15, MAR-17)

(1) 4

5, 25 (2) 25,

4

5 (3)

1

5, 25 (4) 𝟐𝟓,

𝟏

𝟓

14. xU <UWg;Gg; gutypd; ruhrhp 12 kw;Wk; jpl;ltpyf;fk; 2 vdpy; gz;gsit 𝑝 ,d; kjpg;G

(OCT-10,MAR-12,MAR-14,OCT-14 )

(1) 1

2 (2)

1

3 (3)

𝟐

𝟑 (4)

1

4

15. xU gfilia 16 Kiwfs; tPRk; NghJ> ,ul;ilg;gil vz; fpilg;gJ ntw;wpahFk; vdpy;

ntw;wpapd; gutw;gb (JUN-07,MAR-10,MAR-11,MAR-16)

(1) 𝟒 (2) 6 (3) 2 (4) 256





16. xU ngl;bapy; 6 rptg;G kw;Wk; 4 nts;isg; ge;Jfs; cs;sd. mtw;wpypUe;J 3 ge;Jfs; rktha;g;G Kiwapy; jpUg;gp itf;fhky; vLf;fg;gl;lhy;> 2 nts;isg; ge;Jfs; fpilf;f epfo;jfT

(JUN-10,OCT-12,OCT-14, JUN-16)

(1) 1

20 (2)

18

125 (3)

4

25 (4)

𝟑

𝟏𝟎

17. ed;F fiyf;fg;gl;l 52 rPl;Lfs; nfhz;l rPl;Lf;fl;bypUe;J 2 rPl;Lfs; jpUg;gp itf;fhkhy; vLf;fg;gLfpd;wd. ,uz;Lk; xNu epwj;jpy; ,Uf;f epfo;jfT (JUN-14,MAR-15 )

(1) 1

2 (2)

26

51 (3)

𝟐𝟓

𝟓𝟏 (4)

25

102

18. xU gha;]hd; gutypy; 𝑃 𝑋 = 0 = 𝑘 vdpy; gutw;gbapd; kjpg;G

(MAR-07,OCT-07,OCT-08,MAR-09,JUN-11,OCT-13,JUN-14)

(1) 𝐥𝐨𝐠𝟏

𝒌 (2)log 𝑘 (3)𝑒𝜆 (4)

1

𝑘

19. xU rktha;g;G khwp 𝑋 gha;]hd; gutiyg; gpd;gw;WfpwJ. NkYk; 𝐸 𝑋2 = 30 vdpy; gutypd; gutw;gb (JUN-08,MAR-11, JUN-16, OCT-16 ) (1)6 (2)5 (3)30 (4)25

20. rktha;g;G khwp 𝑋,d; guty; rhh;G 𝐹(𝑋) xU (MAR-08, JUN-13,OCT-14 )

(1) ,wq;Fk; rhh;G (2) Fiwah (,wq;fh) rhh;G

(3) khwpypr; rhh;G (4) Kjypy; VWk; rhh;G gpd;dh; ,wq;Fk; rhh;G

21. gha;;]hd; gutypd; gz;gsit 𝜆 = 0.25 vdpy; ,uz;lhtJ tpyf;fg; ngUf;Fj; njhif

(OCT-09,OCT-11,OCT-12,MAR-13,MAR-15 )

(1)0.25 (2)0.3125 (3)0.0625 (4)0.025

22. xU gha;]hd; gutypy; 𝑃 𝑋 = 2 = 𝑃 𝑋 = 3 vdpy;> gz;gsit 𝜆,d; kjpg;G

(MAR-06, JUN-06,MAR-08,JUN-09,MAR-12,JUN-15)

(1)6 (2)2 (3)3 (4)0

23. xU ,ay;epiyg; gutypd; epfo;jfT mlh;j;jpr; rhh;G 𝑓(𝑥),d; ruhrhp 𝜇 vdpy; 𝑓(𝑥)𝑑𝑥∞

−∞,d;

kjpg;G (OCT-06,MAR-07,JUN-07,OCT-10 )

(1)1 (2)0.5 (3)0 (4)0.25

24. xU rktha;g;G khwp 𝑋 > ,ay;epiyg; guty; 𝑓 𝑥 = 𝑐𝑒−

12(𝑥−100 )2

25 I gpd;gw;WfpwJ vdpy; 𝑐 ,d;

kjpg;G (JUN-06,MAR-10,JUN-12,MAR-13,MAR-17 )

(1) 2𝜋 (2)1

2𝜋 (3)5 2𝜋 (4)

𝟏

𝟓 𝟐𝝅

25. xU ,ay; epiy khwp 𝑋,d; epfo;jfT mlh;j;jpr; rhh;G 𝑓(𝑥) kw;Wk; 𝑋~𝑁(𝜇, 𝜎2) vdpy;

𝑓(𝑥)𝑑𝑥𝜇

−∞,d; kjpg;G (OCT-16)

(1) tiuaWf;f KbahjJ (2)1 (3) . 5 (4) −.5

26. 400 khzth;fs; vOjpa fzpjj; Njh;tpd; kjpg;ngz;fs; ,ay;epiyg; gutiy xj;jpUf;fpwJ. ,jd; ruhrhp 65. NkYk; 120 khzth;fs; 85 kjpg;ngz;fSf;F Nky; ngw;wpUg;gpd;> kjpg;ngz;fs;

45,ypUe;J 65f;Fs; ngWk; khzth;fspd; vz;zpf;if (OCT-07,JUN-10,JUN-11, MAR-16 )

(1)120 (2)20 (3)80 (4)160



12k; tFg;G fzf;F MW kjpg;ngz; tpdhf;fs; ntw;wpf;F top


6 kjpg;ngz; tpdhf;fs;

1. mzpfSk; mzpf;NfhitfSk;

1. xU G+r;rpakw;w Nfhit mzpahapd;

𝑨𝑻 −𝟏 = 𝑨−𝟏 𝑻vd;gij epWTf (OCT-07)

𝐴𝐴−1 = 𝐼 = 𝐴−1𝐴 .

𝐴𝐴−1 = 𝐼,d; ,UGwKk; epiu epuy; khw;W fhz

𝐴𝐴−1 𝑇 = 𝐼𝑇 epiu epuy; khw;Wf;Fhpa thpir khw;Wg; gz;Gg;gb>

𝐴−1 𝑇𝐴𝑇 = 𝐼………………... 1

,Nj Nghy; 𝐴−1𝐴 = 𝐼 ,d; ,UGwKk; epiu epuy;

khw;W fhz 𝐴𝑇 𝐴−1 𝑇 = 𝐼……………. 2

(1) kw;Wk; (2),ypUe;J

𝐴−1 𝑇𝐴𝑇 = 𝐴𝑇 𝐴−1 𝑇 = 𝐼

vdNt 𝐴−1 𝑇 MdJ 𝐴𝑇 ,d; Neh;khwhFk;

mjhtJ 𝐴𝑇 −1 = 𝐴−1 𝑇

2. Neh;khWfSf;Fhpa thpirkhw;W tpjpia vOjp

epWTf. ( JUN-11,OCT-14)

𝐴 kw;Wk; 𝐵 Mfpait xNu thpir nfhz;l VNjDk; ,U G+r;rpakw;w Nfhit mzpfs; vd;f.

mt;thwhapd; 𝐴𝐵 Ak; xU G+r;rpakw;w Nfhit

mzpahFk;. NkYk; (𝐴𝐵)−1 = 𝐵−1 𝐴−1 mjhtJ ngUf;fypd; Neh;khW mzpahdJ Neh;khW mzpfspd; thpir khw;Wg; ngUf;fYf;Fr; rkkhFk;. ep&gzk;:

𝐴 kw;Wk; 𝐵 G+r;rpakw;w Nfhit mzpfs; vd;f.

𝐴 ≠ 0 kw;Wk; 𝐵 ≠ 0 MFk;

𝐴𝐵 = 𝐴 𝐵

𝐴 ≠ 0, 𝐵 ≠ 0 ⇒ 𝐴 𝐵 ≠ 0 ⇒ 𝐴𝐵 ≠ 0

vdNt 𝐴𝐵 Ak; xU G+r;rpakw;w Nfhit mzpahFk;

∴ 𝐴𝐵 Neh;khW fhzj;jf;fJ.

(𝐴𝐵)( 𝐵−1 𝐴−1) = 𝐴(𝐵𝐵−1) 𝐴−1

= 𝐴 𝐼 𝐴−1 = 𝐴 𝐴−1 = 𝐼

,t;thNw ( 𝐵−1 𝐴−1) 𝐴𝐵 = 𝐼 vd epWtyhk;

(𝐴𝐵)( 𝐵−1 𝐴−1) = ( 𝐵−1 𝐴−1) 𝐴𝐵 = 𝐼

𝐴𝐵 ,d; Neh;khW 𝐵−1 𝐴−1MFk;

(𝐴𝐵)−1 = 𝐵−1 𝐴−1

3. 𝑨 = −𝟐 −𝟑𝟓 −𝟔

vdpy; 𝑨−𝟏 𝑻 = 𝑨𝑻 −𝟏 vd;gijr;

rhpghh;f;f ( MAR-10 )

𝐴 = −2 −35 −6

𝐴 = −2 −35 −6

= 12 + 15 = 27 ≠ 0

𝑎𝑑𝑗 𝐴 = −6 3−5 −2

𝐴−1 =1

𝐴 𝑎𝑑𝑗 𝐴 =

1

27 −6 3−5 −2

𝐴−1 𝑇 =1

27 −6 −53 −2

………………. 1

𝐴𝑇 = −2 5−3 −6

,

𝐴𝑇 = −2 −53 −6

= 12 + 15 = 27 ≠ 0

𝐴𝑇 −1 =1

𝐴𝑇 𝑎𝑑𝑗 𝐴𝑇

𝑎𝑑𝑗 𝐴𝑇 = −6 −53 −2

𝐴𝑇 −1 =1

27 −6 −53 −2

……………….. 2

(1) kw;Wk; (2) ypUe;J 𝐴−1 𝑇 = 𝐴𝑇 −1 vd;gJ rhpghh;f;fg;gl;lJ

4. −𝟏 𝟐𝟏 −𝟒

vd;w mzpapd; Neh;khW mzpiaf;

fhz;f. (OCT-07)

𝐴 = −1 21 −4

, vdpy; 𝐴 = −1 21 −4

= 2 ≠ 0

𝐴 xU G+r;rpakw;w Nfhit mzp. vdNt Neh;khW fhzj;jf;fJ. ,izf;fhuzpfspd; mzpahdJ

𝐴𝑖𝑗 = −4 −1−2 −1

adj 𝐴 = −4 −2−1 −1

𝐴−1 =1

𝐴 𝑎𝑑𝑗 𝐴

=1

2 −4 −2−1 −1

= −2 −1

−1

2−

1

2

5. 𝑨 = 𝟑 𝟏 −𝟏𝟐 −𝟐 𝟎𝟏 𝟐 −𝟏

vd;w mzpapd; Neh;khW

mzpiaf; fhz;f. (OCT-09,OCT-11, MAR-17)

𝐴 = 3 1 −12 −2 01 2 −1

𝐴 = 3 1 −12 −2 01 2 −1

= 2 ≠ 0

A G+r;rpakw;w Nfhit mzp. vdNt 𝐴−1 fhz KbAk;

𝐴𝑖𝑗 = 2 2 6

−1 −2 −5−2 −2 −8

adj 𝐴 = 2 −1 −22 −2 −26 −5 −8

𝐴−1 =1






=1

2 2 −1 −22 −2 −26 −5 −8

=

1 −1

2−1

1 −1 −1

3 −5

2−4

6. 𝑨 = 𝟏 𝟐𝟏 𝟏

kw;Wk; 𝑩 = 𝟎 −𝟏𝟏 𝟐

vdpy;>

(𝑨𝑩)−𝟏 = 𝑩−𝟏𝑨−𝟏vd;gij rhpghh; ( JUN-09,JUN-10)

𝐴 = 1 21 1

= 1 − 2 = −1 ≠ 0

𝐵 = 0 −11 2

= 0 + 1 = 1 ≠ 0

A kw;Wk; B G+r;rpakw;w Nfhit mzpfs;. vdNt

𝐴−1 kw;Wk; 𝐵−1fhz KbAk;

𝐴𝐵 = 1 21 1

0 −11 2

= 2 31 1

𝐴𝐵 = 2 31 1

= −1 ≠ 0

AB G+r;rpakw;w Nfhit mzp. vdNt (𝐴𝐵)−1 fhz KbAk;

𝑎𝑑𝑗 𝐴𝐵 = 1 −3

−1 2

(𝐴𝐵)−1 =1

𝐴𝐵 𝑎𝑑𝑗 𝐴𝐵 =

−1 31 −2

………….. 1

𝐴−1 =1


𝑎𝑑𝑗 𝐴 = 1 −2

−1 1

𝐴−1 = −1 21 −1

𝐵−1 =1

𝐵 𝑎𝑑𝑗 𝐵

𝑎𝑑𝑗 𝐵 = 2 1

−1 0

𝐵−1 = 2 1

−1 0

𝐵−1𝐴−1 = 2 1

−1 0

−1 21 −1

= −1 31 −2

……. 2

(1) kw;Wk; (2) ypUe;J (𝐴𝐵)−1 = 𝐵−1𝐴−1 vd;gJ rhpghh;f;fg;gl;lJ

7. 𝑨 = 𝟓 𝟐𝟕 𝟑

kw;Wk; 𝑩 = 𝟐 −𝟏

−𝟏 𝟏 , vdpy;

(𝑨𝑩)−𝟏 = 𝑩−𝟏𝑨−𝟏vd;gij rhpghh;.

( JUN-06,JUN-12,JUN-13,JUN-15)

𝐴 = 5 27 3

; 𝐵 = 2 −1

−1 1

𝐴𝐵 = 5 27 3

2 −1

−1 1 =

8 −311 −4

𝐴−1 –If; fhz

𝐴 = 5 27 3

= 15 − 14 = 1 ≠ 0

𝑎𝑑𝑗 𝐴 = 3 −2

−7 5

𝐴−1 =1


𝐴−1 = 3 −2

−7 5

𝐵−1 –If; fhz

𝐵 = 2 −1

−1 1 = 2 − 1 = 1 ≠ 0

𝑎𝑑𝑗 𝐵 = 1 11 2

𝐵−1 =1

𝐵 𝑎𝑑𝑗 𝐵

𝐵−1 = 1 11 2

𝐵−1𝐴−1 = 1 11 2

3 −2

−7 5

= −4 3−11 8

… … … . (1)

(𝐴𝐵)−1 – If; fhz

𝐴𝐵 = 8 −3

11 4 = −32 + 33 = 1 ≠ 0

𝑎𝑑𝑗 𝐴𝐵 = −4 3−11 8

(𝐴𝐵)−1 =1

𝐴𝐵 𝑎𝑑𝑗 𝐴𝐵

(𝐴𝐵)−1 = −4 3−11 8

… (2)

(1) kw;Wk; (2)ypUe;J (𝐴𝐵)−1 = 𝐵−1𝐴−1 rhpghh;f;fg;gl;lJ

8. 𝑨 = 𝟓 𝟐𝟕 𝟑

kw;Wk; 𝑩 = 𝟐 −𝟏

−𝟏 𝟏 , vdpy;

(𝑨𝑩)𝑻 = 𝑩𝑻𝑨𝑻 vd;gij rhpghh; (JUN-14)

𝐴 = 5 27 3

kw;Wk; 𝐵 = 2 −1

−1 1

𝐴𝐵 = 8 −3

11 −4

(𝐴𝐵)𝑇 = 8 11

−3 −4 …………………… 1

𝐵𝑇 = 2 −1

−1 1 , 𝐴𝑇 =

5 72 3

𝐵𝑇𝐴𝑇 = 2 −1

−1 1

5 72 3

= 10 − 2 14 − 3−5 + 2 −7 + 3

= 8 11

−3 −4 …………………….. 2

(1)kw;Wk; (2) ypUe;J (𝐴𝐵)𝑇 = 𝐵𝑇𝐴𝑇 vd;gJ rhpghh;f;fg;gl;lJ

9. 𝑨 = 𝟏 𝟐𝟑 −𝟓

vd;w mzpapd; Nrh;g;igf; fz;Lgpbj;J

𝑨 𝐚𝐝𝐣 𝑨 = 𝐚𝐝𝐣 𝑨 𝑨 = 𝑨 . 𝑰 vd;gij rhpghh;f;f

(MAR-07,MAR-09,MAR-13)

𝐴 = 1 23 −5

𝐴 = 1 23 −5

= −5 − 6 = −11

adj 𝐴 = −5 −2−3 1

𝐴 adj 𝐴 = 1 23 −5

−5 −2−3 1





= −11 0

0 −11 = −11

1 00 1

= 𝐴 . 𝐼 ………………………………….. 1

adj 𝐴 𝐴 = −5 −2−3 1

1 23 −5

= −11 0

0 −11 = −11

1 00 1

= 𝐴 . 𝐼 ……………………………….... 2

(1) kw;Wk; (2)ypUe;J

𝐴 adj 𝐴 = adj 𝐴 𝐴 = 𝐴 . 𝐼

10. 𝑨 = −𝟒 −𝟑 −𝟑𝟏 𝟎 𝟏𝟒 𝟒 𝟑

d; Nrh;g;G mzp 𝑨 vd epWTf.

(MAR-08,MAR-11,MAR-16, OCT-16)

𝐴 = −4 −3 −31 0 14 4 3

𝐴𝑖𝑗 = −4 1 4−3 0 4−3 1 3

𝑎𝑑𝑗 𝐴 = (𝐴𝑖𝑗 )𝑇 = −4 −3 −31 0 14 4 3

= 𝐴

11. 𝑨 = −𝟏 𝟐 −𝟐𝟒 −𝟑 𝟒𝟒 −𝟒 𝟓

vdpy;> 𝑨 = 𝑨−𝟏 vdf; fhl;Lf.

( MAR-06,MAR-14)

𝐴 = −1 2 −24 −3 44 −4 5

𝐴 = −1 2 −24 −3 44 −4 5

= −1 −15 + 16 − 2 20 − 16 − 2 −16 + 12

= −1 − 8 + 8 = −1

𝐴𝑖𝑗 = 1 −4 −4

−2 3 42 −4 −5

𝑎𝑑𝑗 𝐴 = 1 −2 2

−4 3 −4−4 4 −5

𝐴−1 =1

𝐴 (𝑎𝑑𝑗 𝐴) =

−1 2 −24 −3 44 −4 5

= 𝐴

∴ 𝐴 = 𝐴−1.

12. Neh;khW mzpfhzy; Kiwapy; jPh;f;f

𝒙 + 𝒚 = 𝟑, 𝟐𝒙 + 𝟑𝒚 = 𝟖 (JUN-08,OCT-08,10,12) jug;gl;Ls;s rkd;ghLfis gpd;tUkhW vOj

1 12 3

𝑥𝑦 =

38

𝐴 𝑋 = 𝐵

,q;F, 𝐴 = 1 12 3

= 1 ≠ 0

𝐴 G+r;rpakw;w Nfhit mzp Mjyhy; 𝐴−1 fhzKbAk;

𝐴−1 = 3 −1

−2 1

𝑋 = 𝐴−1𝐵 vd;gJ jPh;thFk;

𝑥𝑦 =

3 −1−2 1

38 =

12

𝑥 = 1, 𝑦 = 2

13. Neh;khW mzp fhzy; Kiwapy; jPh;f;f 𝟐𝒙 − 𝒚 = 𝟕,

𝟑𝒙 − 𝟐𝒚 = 𝟏𝟏 ( JUN-07,MAR-12,15)

2𝑥 − 𝑦 = 7

3𝑥 − 2𝑦 = 11

2 −13 −2

𝑥𝑦 =

711

𝐴 𝑋 = 𝐵, ,q;F

𝐴 = 2 −13 −2

; 𝑋 = 𝑥𝑦 ; 𝐵 =

711

𝑋 = 𝐴−1𝐵 vd;gJ jPh;thFk;

𝐴−1 If;fhz

𝐴 = 2 −13 −2

= −4 + 3 = −1 ≠ 0

𝐴𝑖𝑗 = −2 −31 2

𝑎𝑑𝑗 𝐴 = −2 1−3 2

𝐴−1 =1


𝐴−1 = 2 −13 −2

vdNt 𝑋 = 𝐴−1𝐵

= 2 −13 −2

7

11 =

14 − 1121 − 22

= 3

−1

𝑥 = 3, 𝑦 = −1

14. 𝟒 𝟐 𝟏𝟔 𝟑 𝟒𝟐 𝟏 𝟎

𝟑 𝟕 𝟏

vd;w mzpapd; juk; fhz;f

( MAR-15)

𝐴 = 4 2 16 3 42 1 0

3 7 1

~ 1 2 44 3 60 1 2

3 7 1

𝐶1 ↔ 𝐶3

~ 1 2 40 −5 −100 1 2

3−5 1

𝑅2 → 𝑅2 − 4𝑅1

~ 1 2 40 1 20 1 2

31 1

𝑅2 → −1

5𝑅2

~ 1 2 40 1 20 0 0

31 0

𝑅3 → 𝑅3 + 𝑅2

filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ. ,jpy; ,uz;L G+r;rpakw;w epiufs; cs;sjhy; ∴ 𝜌 𝐴 = 2





15. 𝟑 𝟏 −𝟓𝟏 −𝟐 𝟏𝟏 𝟓 −𝟕

−𝟏−𝟓 𝟐


(OCT-06,MAR-07, OCT-16)

𝐴 = 3 1 −51 −2 11 5 −7

−1−5 2

~ 1 −2 13 1 −51 5 −7

−5−1 2

𝑅1 ↔ 𝑅2

~ 1 −2 10 7 −80 7 −8

−514 7

𝑅2 → 𝑅2 − 3𝑅1

𝑅3 → 𝑅3 − 𝑅1

~ 1 −2 10 7 −80 0 0

−514 −7

𝑅3 → 𝑅3 − 𝑅2

filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ. ,jpy; %d;W G+r;rpakw;w epiufs; cs;sjhy; ∴ 𝜌 𝐴 = 3

16. −𝟐 𝟏 𝟑𝟎 𝟏 𝟏𝟏 𝟑 𝟒

𝟒𝟐𝟕


(OCT-10)

𝐴 = −2 1 30 1 11 3 4

427

~ 1 3 40 1 1

−2 1 3

724

𝑅1 ↔ 𝑅3

~ 1 3 40 1 10 7 11

72

18 𝑅3 → 2𝑅1 + 𝑅3

~ 1 3 40 1 10 0 4

724

𝑅3 → 𝑅3 − 7𝑅2

filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ. ,jpy; %d;W G+r;rpakw;w epiufs; cs;sjhy; , 𝜌 𝐴 = 3

17. 𝟑 𝟏 𝟐𝟏 𝟎 −𝟏𝟐 𝟏 𝟑

𝟎𝟎𝟎 vd;w mzpapd; juk; fhz;f

(JUN -08 )

𝐴 = 3 1 21 0 −12 1 3

000

~ 1 0 −13 1 22 1 3

000 𝑅1 ↔ 𝑅2

~ 1 0 −10 1 50 1 5

000 𝑅2 → 𝑅2 − 3𝑅1

𝑅3 → 𝑅3 − 2𝑅1

~ 1 0 −10 1 50 0 0

000 𝑅3 → 𝑅3 − 𝑅2

filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ.

,jpy; ,uz;L G+r;rpakw;w epiufs; cs;sjhy; ,

𝜌 𝐴 = 2

18. 𝟎 𝟏 𝟐𝟐 −𝟑 𝟎𝟏 𝟏 −𝟏

𝟏−𝟏𝟎


( JUN-11,OCT-12 , MAR-17)

𝐴 = 0 1 22 −3 01 1 −1

1−10

~ 1 1 −12 −3 00 1 2

0−1 1

𝑅1 ↔ 𝑅3

~ 1 1 −10 −5 20 1 2

0−1 1

𝑅2 → 𝑅2 − 2𝑅1

~ 1 1 −10 −5 20 0 12

0−1 4

𝑅3 → 5𝑅3 + 𝑅2

filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ. ,jpy; %d;W G+r;rpakw;w epiufs; cs;sjhy;

∴ 𝜌 𝐴 = 3

19. 𝟏 𝟐 −𝟏𝟐 𝟒 𝟏𝟑 𝟔 𝟑

𝟑−𝟐−𝟕


(OCT-08 , OCT-15)

𝐴 = 1 2 −12 4 13 6 3

3−2−7

~ 1 2 −10 0 30 0 6

3−8−16

𝑅2 → 𝑅2 − 2𝑅1

𝑅3 → 𝑅3 − 3𝑅1

~ 1 2 −10 0 30 0 0

3−80

𝑅3 → 𝑅3 − 2𝑅2


,jpy; ,uz;L G+r;rpakw;w epiufs; cs;sjhy;.

𝜌 𝐴 = 2

20. 𝟏 𝟐 𝟑𝟐 𝟒 𝟔𝟑 𝟔 𝟗

−𝟏−𝟐 −𝟑


(JUN-07)

𝐴 = 1 2 32 4 63 6 9

−1−2 −3

~ 1 2 30 0 00 0 0

−10 0

𝑅2 → 𝑅2 − 2𝑅1

𝑅3 → 𝑅3 − 3𝑅1


,jpy; xNu xU G+r;rpakw;w epiu cs;sjhy;,

𝜌 𝐴 = 1

21. 𝟏 −𝟐 𝟑

−𝟐 𝟒 −𝟏−𝟏 𝟐 𝟕

𝟒

−𝟑𝟔






(MAR-06,JUN-10)

𝐴 = 1 −2 3

−2 4 −1−1 2 7

4

−36

~ 1 −2 30 0 50 0 10

45

10 𝑅2 → 𝑅2 + 2𝑅1

𝑅3 → 𝑅3 + 𝑅1

~ 1 −2 30 0 50 0 0

450 𝑅3 → 𝑅3 − 2𝑅2

filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ. ,jpy; ,uz;L G+r;rpakw;w epiufs; cs;sjhy;

∴ 𝜌 𝐴 = 2

22. 𝟏𝟒 −𝟒 𝟏𝟐 𝟏𝟐𝟎 𝟒 𝟖𝟒 −𝟒 𝟖

𝟒𝟒𝟎

vd;w mzpapd; juk; fhz;f(JUN-12)

𝐴 =1

4 −4 12 120 4 84 −4 8

440

= −1 3 30 1 21 −1 2

110

~ 1 −1 20 1 2

−1 3 3 011

𝑅3 ↔ 𝑅1

~ 1 −1 20 1 20 2 5

011

𝑅3 → 𝑅3 + 𝑅1

~ 1 −1 20 1 20 0 −1

011

𝑅3 → 2𝑅2 − 𝑅3

filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ. ,jpy; %d;W G+r;rpakw;w epiufs; cs;sjhy;

∴ 𝜌 𝐴 = 3

23. mzpf;Nfhit Kiwapy; gpd;tUk; rkd;ghLfspd;

njhFg;Gfis jPh;f;f (OCT-15)

(i) 𝒙 − 𝒚 = 𝟐 (ii) 𝒙 + 𝒚 + 𝟐𝒛 = 𝟎

𝟑𝒚 = 𝟑𝒙 − 𝟕 𝟐𝒙 + 𝒚 − 𝒛 = 𝟎

𝟐𝒙 + 𝟐𝒚 + 𝒛 = 𝟎

(i) 𝑥 − 𝑦 = 2

3𝑦 = 3𝑥 − 7

∆= 1 −13 −3

= 0

∆𝑥= 2 −17 −3

= 1

∆= 0 kw;Wk; ∆𝑥≠ 0 vd;gjhy;> njhFg;G xUq;fikT mw;wJ> ,jw;F jPh;T fpilahJ.

(ii) 𝑥 + 𝑦 + 2𝑧 = 0

2𝑥 + 𝑦 − 𝑧 = 0

2𝑥 + 2𝑦 + 𝑧 = 0

∆= 1 1 22 1 −12 2 1

= 3

∆≠ 0 , Mjyhy; njhFg;G xNu xU jPh;tpidf; nfhz;bUf;Fk;. vdNt Nkw;fz;l rkgbj;jhd njhFg;G ntspg;gilj; jPh;T kl;LNk ngw;wpUf;Fk;

𝑥, 𝑦, 𝑧 = (0,0,0)

24. mzpf;Nfhit Kiwapy; gpd;tUk; rkd;ghL

njhFg;gpid jPh;f;f 𝟐𝒙 + 𝟑𝒚 = 𝟖, 𝟒𝒙 + 𝟔𝒚 = 𝟏𝟔

( JUN-06, MAR-11)

2𝑥 + 3𝑦 = 8………………………………….. 1 4𝑥 + 6𝑦 = 16…………………….………….. 2

∆= 2 34 6

= 12 − 12 = 0

∆𝑥= 8 3

16 6 = 48 − 48 = 0

∆𝑦= 2 84 16

= 32 − 32 = 0

∆= 0 kw;Wk; ∆𝑥= ∆𝑦= 0 vd;gjhYk; ∆ ,d;

Fiwe;jJ xU nfO 𝑎𝑖𝑗 MtJ G+r;rpakw;W

,Ug;gjhy;> njhFg;G xUq;fikT cilajhFk;. vz;zpf;ifaw;w jPh;Tfs; fpilf;Fk;.

vy;yh 2 × 2 rpw;wzpf; Nfhitfs;

G+r;rpaq;fshfTk;> Fiwe;jJ xU (1 × 1) rpw;wzpf;Nfhit G+r;rpakw;wJ Mjyhy; njhFg;G

xNu xU jdpr;rkd;ghl;bw;F FiwAk;. 𝑥 (my;yJ

𝑦 )f;F VNjDk; xU kjpg;gspj;J 𝑦 (my;yJ 𝑥 ),d; kjpg;igf; fhzyhk;.

𝑥 = 𝑡 vdj; ju 𝑦 =1

3(8 − 2𝑡)

vdNt jPh;T fzkhdJ

𝑥, 𝑦 = 𝑡,1

3(8 − 2𝑡) 𝑡 ∈ 𝑅

25. mzpf;Nfhit Kiwapid gad;gLj;jp

𝒙 + 𝒚 + 𝟐𝒛 = 𝟒; 𝟐𝒙 + 𝟐𝒚 + 𝟒𝒛 = 𝟖;

𝟑𝒙 + 𝟑𝒚 + 𝟔𝒛 = 𝟏𝟎 vd;w njhFg;gpid jPh;f;f

( JUN-13, MAR-16)

∆= 1 1 22 2 43 3 6

= 0, ∆𝑥= 4 1 28 2 4

10 3 6 = 0

∆𝑦= 1 4 22 8 43 10 6

= 0, ∆𝑧= 1 1 42 2 83 3 10

= 0

∆= ∆𝑥= ∆𝑦= ∆𝑧= 0 NkYk; ∆ d; vy;yh 2 × 2

rpw;wzpf;Nfhitfspd; kjpg;Gfs; G+r;rpakhtjhYk;

∆𝑥 , ∆𝑦 kw;Wk; ∆𝑧 ,d; rpy rpw;wzpf; Nfhitfs;

G+r;rpakw;wjhAs;sjhy; njhFg;G xUq;fikT mw;wJ. vdNt mjw;F jPh;T fpilahJ.

26. 𝒙 + 𝒚 + 𝟑𝒛 = 𝟒; 𝟐𝒙 + 𝟐𝒚 + 𝟔𝒛 = 𝟕;

𝟐𝒙 + 𝒚 + 𝒛 = 𝟏𝟎 vd;w rkd;ghl;L njhFg;gpid mzpf;Nfhit Kiwapid gad;gLj;jp jPh;T

fhz;f. (MAR-13)

∆= 1 1 32 2 62 1 1

= 0





∆𝑥= 4 1 37 2 6

10 1 1

= 4 2 − 6 − 1 7 − 60 + 3(7 − 20)

= −16 + 53 − 39 = −2 ≠ 0

∆= 0, ∆𝑥 ≠ 0 vdNt njhFg;G xUq;fikT mw;wJ. jPh;T fpilahJ

27. gpd;tUk; rkd;ghLfspd; njhFg;Gfis

mzpf;Nfhit Kiwapy; jPh;f;f. 𝒙 + 𝒚 + 𝟐𝒛 = 𝟎;

𝒙 − 𝒚 − 𝒛 = 𝟓; 𝟐𝒙 + 𝒛 = 𝟔 (MAR-14)

∆= 1 1 21 −1 −12 0 1

= 1 −1 + 0 − 1 1 + 2 + 2 0 + 2 = −1 − 3 + 4 = 0

∆𝑥= 0 1 25 −1 −16 0 1

=0 −1 + 0 − 1 5 + 6 + 2 0 + 6

=0 − 11 + 12 = 1 ≠ 0

∆= 0 kw;Wk; ∆𝑥≠ 0

vdNt njhFg;G xUq;fikT mw;wJ. jPh;T fpilahJ

28. mzpf;Nfhit Kiwapy; jPh;f;f 𝒙 − 𝟐𝒚 = 𝟐;

𝟐𝒙 − 𝟒𝒚 = 𝟒 ( JUN-14)

∆= 1 −22 −4

= −4 + 4 = 0

∆𝑥= 2 −24 −4

= −8 + 8 = 0

∆𝑦= 1 22 4

= 4 − 4 = 0

∆= 0, ∆𝑥= 0, ∆𝑦= 0

jug;gl;l rkd;ghL njhFg;G xUq;fikT cilaJ

NkYk; vz;zpf;ifaw;w jPh;Tfs; fpilf;Fk;. 𝑦 = 𝑘 vdj;ju

𝑥 − 2𝑘 = 2

𝑥 = 2 + 2𝑘

vdNt jPh;T fzkhdJ (2 + 2𝑘, 𝑘),𝑘 ∈ 𝑅

29. gpd;tUk; rkd;ghLfspd; njhFg;gpid mzpf;Nfhit Kiwapy; jPh;f;f

𝒙 + 𝟐𝒚 = 𝟒, 𝟒𝒙 + 𝟖𝒚 = 𝟏𝟔 ( JUN-15)

𝑥 + 2𝑦 = 4………………………………………. 1

4𝑥 + 8𝑦 = 16…………………………………… 2

∆= 1 24 8

= 8 − 8 = 0

∆𝑥= 4 2

16 8 = 32 − 32 = 0

∆𝑦= 1 44 16

= 16 − 16 = 0

∆= 0 kw;Wk; ∆𝑥= ∆𝑦= 0 vd;gjhYk; ∆ ,d;

Fiwe;jJ xU nfO 𝑎𝑖𝑗 MtJ G+r;rpakw;W ,Ug;gjhy;> njhFg;G xUq;fikT cilajhFk;. vz;zpf;ifaw;w jPh;Tfs; fpilf;Fk;. vy;yh 2 × 2 rpw;wzpf; Nfhitfs; G+r;rpaq;fshfTk;> Fiwe;jJ xU (1 × 1) rpw;wzpf;Nfhit G+r;rpakw;wJ Mjyhy; njhFg;G xNu xU jdpr;rkd;ghl;bw;F FiwAk;. 𝑥 (my;yJ 𝑦 )f;F VNjDk; xU kjpg;gspj;J 𝑦 (my;yJ 𝑥 ),d; kjpg;igf; fhzyhk;.

𝑥 = 𝑡 vdj; ju 2𝑦 = 4 − 𝑡 ⇒ 𝑦 =1

2(4 − 𝑡)

vdNt jPh;T fzkhdJ

𝑥, 𝑦 = 𝑡,1

2(4 − 𝑡) 𝑡 ∈ 𝑅

30. mzpf;Nfhit Kiwapy; 𝟐𝒙 + 𝟐𝒚 + 𝒛 = 𝟓, 𝒙 − 𝒚 + 𝒛 = 𝟏, 𝟑𝒙 + 𝒚 + 𝟐𝒛 = 𝟒 vd;w rkd;ghl;Lj;

njhFg;gpid jPh;f;fTk; (MAR-08,MAR-09,MAR-12,OCT-13)

∆= 2 2 11 −1 13 1 2

= 0

∆𝑥= 5 2 11 −1 14 1 2

= −6 ≠ 0

∆= 0 kw;Wk; ∆𝑥≠ 0 vdNt njhFg;G xUq;fikT mw;wJ. jPh;T fpilahJ

31. 𝟒𝒙 + 𝟓𝒚 = 𝟗, 𝟖𝒙 + 𝟏𝟎𝒚 = 𝟏𝟖 vd;w njhFg;gpid

mzpf;Nfhit Kiwapy; jPh;f;f (OCT-06,OCT-09)

∆= 4 58 10

= 40 − 40 = 0

∆𝑥= 9 5

18 10 = 90 − 90 = 0

∆𝑦= 4 98 18

= 72 − 72 = 0

∆= 0 kw;Wk; ∆𝑥= ∆𝑦= 0 vd;gjhYk; ∆ ,d;


,Ug;gjhy;> njhFg;G xUq;fikT cilajhFk;. vz;zpf;ifaw;w jPh;Tfs; fpilf;Fk;.

𝑦 = 𝑘 vd ju 4𝑥 = 9 − 5𝑘 ⇒ 𝑥 =9−5𝑘

4

∴ jPh;T 9−5𝑘

4, 𝑘 , 𝑘 ∈ 𝑅

32. mzpf;Nfhit Kiwapy; gpd;tUk; rkd;ghLfspd; njhFg;igj; jPh;f;f 𝟐𝒙 − 𝟑𝒚 = 𝟕 , 𝟒𝒙 − 𝟔𝒚 = 𝟏𝟒 (JUN-09)

∆= 2 −34 −6

= −12 + 12 = 0

∆𝑥= 7 −3

14 −6 = −42 + 42 = 0

∆𝑦= 2 74 14

= 28 − 28 = 0

∆= 0 kw;Wk; ∆𝑥= ∆𝑦= 0 vd;gjhYk; ∆ ,d;






,Ug;gjhy;> njhFg;G xUq;fikT cilajhFk;. vz;zpf;ifaw;w jPh;Tfs; fpilf;Fk;. vy;yh 2 × 2 rpw;wzpf; Nfhitfs; G+r;rpaq;fshfTk;> Fiwe;jJ xU (1 × 1) rpw;wzpf;Nfhit G+r;rpakw;wJ Mjyhy; njhFg;G xNu xU jdpr;rkd;ghl;bw;F FiwAk;. 𝑥 (my;yJ 𝑦 )f;F VNjDk; xU kjpg;gspj;J 𝑦 (my;yJ 𝑥 ),d; kjpg;igf; fhzyhk;. 2𝑥 − 3𝑦 = 7,y; 𝑦 = 𝑘 vdj; ju

2𝑥 − 3𝑘 = 7 2𝑥 = 7 + 3𝑘

𝑥 =7+3𝑘

2

∴ vdNt jPh;T fzkhdJ

𝑥, 𝑦 = 7+3𝑘

2, 𝑘 , 𝑘 ∈ 𝑅

33. 𝜶 d; vk;kjpg;GfSf;F

𝜶𝒙 + 𝒚 + 𝟑𝒛 = 𝟎, 𝟒𝒙 + 𝟑𝒚 + 𝟖𝒛 = 𝟎

𝟒𝒙 + 𝟐𝒚 + 𝟒𝒛 = 𝟎 vd;w njhFg;G

(i) ntspg;gilahd jPh;T kl;Lk; ngw;wpUf;Fk;

(ii) ntspg;gilahd kw;Wk; ntspg;gilaw;w jPh;T ngw;wpUf;Fk; (mzpf;Nfhit Kiwapid

gad;gLj;Jf) (JUN-16)

𝛼𝑥 + 𝑦 + 3𝑧 = 0

4𝑥 + 3𝑦 + 8𝑧 = 0

4𝑥 + 2𝑦 + 4𝑧 = 0

𝛼 1 34 3 84 2 4

= 𝛼 12 − 16 − 1 16 − 32 + 3(8 − 12)

= 𝛼 −4 − 1 −16 + 3(−4)

= −4𝛼 + 16 − 12

= −4𝛼 + 4

(i) ntspg;gilahd jPh;T

𝛼 ≠ 1

∆≠ 0 vdNt njhFg;G xNu xU jPh;T kl;Lk; nfhz;bUf;Fk;. njhFg;G ntspg;gilj; jPh;T kl;LNk ngw;wpUf;Fk;

𝑥, 𝑦, 𝑧 = (0,0,0)

(ii)ntspg;gilahd kw;Wk; ntspg;gilaw;w jPh;T

𝛼 = 1

∆= 0

∆= 0vd;gjhy; vz;zpf;ifaw;w jPh;Tfs; ,Uf;Fk;.

NkYk; ∆tpd; Fiwe;jJ xU 2 × 2 rpw;wzpf;Nfhit G+r;rpakw;wjha; ,Ug;gjhy; ,j;njhFg;ghdJ ,uz;L rkd;ghLfshff; FiwAk;. vdNt VNjDk; xU khwp VNjDk; xU kjpg;Gk; kw;w ,U khwpfspd; kjpg;gpid ,jd; %yk; fhzyhk;

𝑧 = 𝑘 vd ju 𝑥 + 𝑦 = −3𝑘

4𝑥 + 3𝑦 = −8𝑘

∆= 1 14 3

= 3 − 4 = −1 ≠ 0

∆𝑥= −3𝑘 1−8𝑘 3

= −9𝑘 + 8𝑘 = −𝑘

∆𝑦= 1 −3𝑘4 −8𝑘

= −8𝑘 + 12𝑘 = 4𝑘

fpNukhpd; tpjpg;gb

𝑥 = 𝑘, 𝑦 = −4𝑘

∴ jPh;thdJ 𝑥, 𝑦, 𝑧 = (𝑘, −4𝑘, 𝑘)

34. 𝒙 + 𝒚 + 𝒛 = 𝟕 , 𝒙 + 𝟐𝒚 + 𝟑𝒛 = 𝟏𝟖 , 𝒚 + 𝟐𝒛 = 𝟔 vd;w rkd;ghLfspd; njhFg;G xUq;fikT cilajh vd;gij ju Kiwapy; Muha;f

(OCT-07,MAR-10, JUN-16)

𝑥 + 𝑦 + 𝑧 = 7 𝑥 + 2𝑦 + 3𝑧 = 18 𝑦 + 2𝑧 = 6 jug;gl;l rkd;ghLj;njhFg;gpid gpd;tUkhW mzpr; rkd;ghlhf khw;wp vOjyhk;

1 1 11 2 30 1 2

𝑥 𝑦𝑧 =

7186

𝐴 𝑋 = 𝐵 tphpTg;gLj;jg;gl;l mzpahdJ

[𝐴, 𝐵] = 1 1 11 2 30 1 2

7

186

~ 1 1 10 1 20 1 2

7

116

𝑅2 → 𝑅2 − 𝑅1

~ 1 1 10 1 20 0 0

7

11−5

𝑅3 → 𝑅3 − 𝑅2


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 ,y;yhjJ. vd;gjhy; jPh;T fhz KbahJ

35. ju Kiwapy; gpd;tUk; rkd;ghLfspd;

njhFg;Gfisj; jPh;f;f 𝒙 − 𝒚 + 𝒛 = 𝟑;

𝟐𝒙 + 𝟐𝒚 − 𝒛 = 𝟕; 𝟑𝒙 + 𝒚 = 𝟏𝟏 (OCT-14)

[𝐴, 𝐵] = 1 −1 12 2 −13 1 0

37

11

~ 1 −1 10 −4 30 −4 3

3

−1−2

𝑅2 → 2𝑅1 − 𝑅2

𝑅3 → 3𝑅1 − 𝑅3

~ 1 −1 10 −4 30 0 0

3

−11

𝑅3 → 𝑅2 − 𝑅3

filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ. mJ %d;W G+r;rpakw;w epiufisg; ngw;Ws;sjhy;

𝜌 𝐴, 𝐵 = 3 > kw;Wk; 𝜌 𝐴 = 2,

∴ 𝜌 𝐴, 𝐵 ≠ 𝜌 𝐴


xUq;fikT mw;wJ. vdNt jPh;T fhz KbahJ





36. juKiwia gad;gLj;jp 𝒙 − 𝟒𝒚 + 𝟕𝒛 = 𝟏𝟒, 𝟑𝒙 +

𝟖𝒚 − 𝟐𝒛 = 𝟏𝟑, 𝟕𝒙 − 𝟖𝒚 + 𝟐𝟔𝒛 = 𝟓 vd;w njhFg;gpw;F xUq;fikTj;jd;ikia Muha;f. xUq;fikT cilajhapd;> jPh;f;f

(OCT-11,OCT-13) 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


𝜌 𝐴, 𝐵 = 3 and 𝜌 𝐴 = 2

∴ 𝜌 𝐴, 𝐵 ≠ 𝜌 𝐴


xUq;fikT mw;wJ. vdNt jPh;T fhz KbahJ

3. fyg;ngz;fs;

1. nka;> fw;gid gFjpfis fhz; 𝟏

𝟏+𝒊 (OCT-08)

1

1+𝑖=

1

1+𝑖×

1−𝑖

1−𝑖=

1−𝑖

2=

1

2+ −

1

2 𝑖

nka; gFjp = 1

2. fw;gidg; gFjp = −

1

2

2. 𝟏+𝒊

𝟏−𝒊

𝒏= 𝟏 vdpy; 𝒏 ,d; kPr;rpW kpif KO

vz; kjpg;igf; fhz;f (MAR-16)

1+𝑖

1−𝑖=

1+𝑖

1−𝑖×

1+𝑖

1+𝑖=

1+𝑖 2

2=

2𝑖

2= 𝑖

1+𝑖

1−𝑖

𝑛= 1

𝑖 𝑛 = 1

𝑛 ,d; kPr;rpW kpif KO vz; 4.

3. gpd;tUk; rkd;ghl;bid epiwT nra;Ak; 𝒙

kw;Wk; 𝒚 apd; nka; kjpg;Gfisf; fhz;f

𝟏 − 𝒊 𝒙 + 𝟏 + 𝒊 𝒚 = 𝟏 − 𝟑𝒊 (JUN-13)

𝑥 + 𝑦 + 𝑖(−𝑥 + 𝑦) = 1 − 3𝑖 nka;> fw;gidg; gFjpfis xg;gpl

𝑥 + 𝑦 = 1, −𝑥 + 𝑦 = −3

,r;rkd;ghLfisj; jPh;f;f 𝑥 = 2, 𝑦 = −1

4. gpd;tUk; rkd;ghl;bid epiwT nra;Ak; 𝒙

kw;Wk; 𝒚 apd; nka; kjpg;Gfisf; fhz;f

𝒙𝟐 + 𝟑𝒙 + 𝟖 + 𝒙 + 𝟒 𝒊 = 𝒚(𝟐 + 𝒊)

(OCT-10,JUN-14)

𝑥2 + 3𝑥 + 8 + 𝑥 + 4 𝑖 = 𝑦(2 + 𝑖)

𝑥2 + 3𝑥 + 8 = 2𝑦 + 𝑖(𝑦 − 𝑥 − 4) nka;> fw;gidg; gFjpfis xg;gpl

𝑥2 + 3𝑥 + 8 = 2𝑦

𝑥2 + 3𝑥 + 8 = 4𝑦2………………….. 1

𝑦 − 𝑥 − 4 = 0

𝑥 + 4 = 𝑦……………………………… 2

(2) I (1),y; gpujpapl

𝑥2 + 3𝑥 + 8 = 4(𝑥2 + 16 + 8𝑥)

3𝑥2 + 29𝑥 + 56 = 0

𝑥 + 7 3𝑥 + 8 = 0

𝑥 = −7 my;yJ 𝑥 = −8

3

𝑥 = −7 vdpy; 𝑦 = −7 + 4 = −3

𝑥 = −8

3 vdpy; 𝑦 = −

8

3+ 4 =

4

3

𝑥 = −7, 𝑦 = −3 NkYk; 𝑥 = −8

3, 𝑦 =

4

3

5. fyg;ngz;fspy; Kf;Nfhz rkdpypia vOjp ep&gpf;f

(OCT-09,JUN-10,MAR-12,MAR-14) ,U fyg;ngz;fspd; $Ljypd; kl;L mt;tpU vz;fspd; kl;Lfspd; $LjYf;Ff;

FiwthfNth my;yJ rkkhfTNk ,Uf;Fk;.

𝑧1 + 𝑧2 ≤ 𝑧1 + 𝑧2 ep&gzk;:

𝑧1 kw;Wk; 𝑧2 ,U fyg;ngz;fs; vd;f.

𝑧1 + 𝑧2 2 = (𝑧1 + 𝑧2)(𝑧1 + 𝑧2) ∵ 𝑧 2 = 𝑧𝑧

= 𝑧1 + 𝑧2 (𝑧1 + 𝑧2 )

= 𝑧1𝑧1 + 𝑧1𝑧2 + 𝑧2𝑧1 + 𝑧2𝑧2

= 𝑧1𝑧1 + 𝑧2𝑧2 + 𝑧1𝑧2 + 𝑧1𝑧2

= 𝑧1 2 + 𝑧2 2 + 2𝑅𝑒 (𝑧1𝑧2 )

≤ 𝑧1 2 + 𝑧2 2 + 2 𝑧1𝑧2 (𝑅𝑒 𝑧 ≤ 𝑧 )

= 𝑧1 2 + 𝑧2 2 + 2 𝑧1 𝑧2

= 𝑧1| + | 𝑧2 2

𝑧1 + 𝑧2 2 ≤ 𝑧1| + | 𝑧2 2 ,UGwKk; kpif th;f;f%yk; vLf;f

𝑧1 + 𝑧2 ≤ 𝑧1 + 𝑧2





6. 𝒛𝟏 kw;Wk; 𝒛𝟐 vd;w ,U fyg;ngz;fSf;F ,

(i) 𝒛𝟏𝒛𝟐 = 𝒛𝟏 𝒛𝟐

(ii)𝐚𝐫𝐠 𝒛𝟏. 𝒛𝟐 = 𝐚𝐫𝐠 𝒛𝟏 + 𝐚𝐫𝐠 𝒛𝟐

(OCT-07,JUN-08,JUN-13)

𝑧1 = 𝑟1(cos 𝜃1 + 𝑖 sin 𝜃1) kw;Wk;

𝑧2 = 𝑟2(cos 𝜃2 + 𝑖 sin 𝜃2) vd;f

𝑧1 . 𝑧2 =

𝑟1𝑟2(cos 𝜃1 + 𝑖 sin 𝜃1)(cos 𝜃2 + 𝑖 sin 𝜃2)

= 𝑟1𝑟2[ cos 𝜃1 . cos 𝜃2 − sin 𝜃1 sin 𝜃2

+ 𝑖 sin 𝜃1 . cos 𝜃2 + cos 𝜃1 sin 𝜃2 ]

= 𝑟1𝑟2[cos(𝜃1 + 𝜃2) + 𝑖 sin ( 𝜃1 + 𝜃2)

𝑧1𝑧2 = 𝑟1𝑟2 = 𝑧1 . 𝑧2 kw;Wk;

arg 𝑧1 . 𝑧2 = 𝜃1 + 𝜃2 = arg 𝑧1 + arg 𝑧2

7. 𝒛𝟏, 𝒛𝟐 vd;w VNjDk; ,U fyg;ngz;fSf;F

(i) 𝒛𝟏

𝒛𝟐 =

𝒛𝟏

𝒛𝟐 ,(𝒛𝟐 ≠ 𝟎)

(ii) 𝐚𝐫𝐠 𝒛𝟏

𝒛𝟐 = 𝐚𝐫𝐠 𝒛𝟏 − 𝐚𝐫𝐠 𝒛𝟐 (JUN-14)

𝑧1 = 𝑟1(cos 𝜃1 + 𝑖 sin 𝜃1) kw;Wk;

𝑧2 = 𝑟2(cos 𝜃2 + 𝑖 sin 𝜃2) vd;f

𝑧1 = 𝑟1 , arg 𝑧1 = 𝜃1 and 𝑧2 = 𝑟2 , arg 𝑧2 = 𝜃2 𝑧1

𝑧2=

𝑟1(cos 𝜃1 + 𝑖 sin 𝜃1)

𝑟2(cos 𝜃2 + 𝑖 sin 𝜃2)

=𝑟1 cos 𝜃1+𝑖 sin 𝜃1 cos 𝜃2−𝑖 sin 𝜃2

𝑟2 cos 𝜃2+𝑖 sin 𝜃2 cos 𝜃2−𝑖 sin 𝜃2

=𝑟1

𝑟2

cos 𝜃1 . cos 𝜃2 + sin 𝜃1 sin 𝜃2 + 𝑖 sin 𝜃1 . cos 𝜃2 − cos 𝜃1 sin 𝜃2

cos2 𝜃2 + 𝑖 sin2 𝜃2

=𝑟1

𝑟2 [cos(𝜃1 − 𝜃2) + 𝑖 sin ( 𝜃1 − 𝜃2)]

𝑧1

𝑧2 =

𝑟1

𝑟2 =

𝑧1

𝑧2 and

arg 𝑧1

𝑧2 = 𝜃1 − 𝜃2 = arg 𝑧1 − arg 𝑧2

8. 𝒂𝟏 + 𝒊𝒃𝟏 𝒂𝟐 + 𝒊𝒃𝟐 … (𝒂𝒏 + 𝒊𝒃𝒏) = 𝑨 + 𝒊𝑩 vdpy; ep&gp:

(i) 𝒂𝟏𝟐 + 𝒃𝟏

𝟐 𝒂𝟐𝟐 + 𝒃𝟐

𝟐 … 𝒂𝒏𝟐 + 𝒃𝒏

𝟐 = (𝑨𝟐 + 𝑩𝟐)

(ii) 𝐭𝐚𝐧−𝟏 𝒃𝟏

𝒂𝟏 + 𝐭𝐚𝐧−𝟏

𝒃𝟐

𝒂𝟐 + ⋯ + 𝐭𝐚𝐧−𝟏

𝒃𝒏

𝒂𝒏

= 𝒌𝝅 + 𝐭𝐚𝐧−𝟏 𝑩

𝑨 , 𝒌 ∈ 𝒁 (OCT-13)

nfhs;if:

𝑎1 + 𝑖𝑏1 𝑎2 + 𝑖𝑏2 … (𝑎𝑛 + 𝑖𝑏𝑛) = 𝐴 + 𝑖𝐵

𝑎1 + 𝑖𝑏1 𝑎2 + 𝑖𝑏2 … (𝑎𝑛 + 𝑖𝑏𝑛 | = |𝐴 + 𝑖𝐵|

𝑎1 + 𝑖𝑏1 | | 𝑎2 + 𝑖𝑏2 | … |(𝑎𝑛 + 𝑖𝑏𝑛 | = |𝐴 + 𝑖𝐵|

𝑎12 + 𝑏1

2 𝑎22 + 𝑏2

2 … 𝑎𝑛2 + 𝑏𝑛

2=

𝐴2 + 𝐵2

th;f;fg;gLj;j

𝑎12 + 𝑏1

2 𝑎22 + 𝑏2

2 … 𝑎𝑛2 + 𝑏𝑛

2 = 𝐴2 + 𝐵2

NkYk;,

arg[ 𝑎1 + 𝑖𝑏1 𝑎2 + 𝑖𝑏2 … (𝑎𝑛 + 𝑖𝑏𝑛)]

= arg (𝐴 + 𝑖𝐵)

arg 𝑎1 + 𝑖𝑏1 + arg 𝑎2 + 𝑖𝑏2 + ⋯ + arg(𝑎𝑛 + 𝑖𝑏𝑛)

= arg (𝐴 + 𝑖𝐵)

tan−1 𝑏1

𝑎1 + tan−1

𝑏2

𝑎2 + ⋯ + tan−1

𝑏𝑛

𝑎𝑛

= tan−1 𝐵

𝐴

nghJthf>

tan−1 𝑏1

𝑎1 + tan−1

𝑏2

𝑎2 + ⋯ + tan−1

𝑏𝑛

𝑎𝑛

= 𝑘𝜋 + tan−1 𝐵

𝐴 , 𝑘 ∈ 𝑍

9. 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. ( JUN-07,MAR-10,OCT-14, MAR-17)

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.;

10. fyg;ngz; jsj;jpy; fyg;ngz;fs; 𝟏𝟎 + 𝟖𝒊

−𝟐 + 𝟒𝒊 kw;Wk; (−𝟏𝟏 + 𝟑𝟏𝒊) mikf;Fk; Kf;Nfhzk; xU nrq;Nfhz Kf;Nfhzk; vd

epWTf (OCT-10)





𝐴, 𝐵, 𝐶 vDk; Gs;spfs; KiwNa 10,8 ,

−2,4 , (−11,31) vDk; fyg;ngz;fis Mh;fd; jsj;jpy; Fwpf;fl;Lk;

𝐴𝐵 = 10 + 8𝑖 − (−2 + 4𝑖)

= 12 + 4𝑖

= 144 + 16

= 160

𝐵𝐶 = −2 + 4𝑖 − (−11 + 31𝑖)

= 9 − 27𝑖

= 9 2 + −27 2

= 81 + 729

= 810

𝐶𝐴 = −11 + 31𝑖 − (10 + 8𝑖)

= −21 + 23𝑖

= −21 2 + 23 2

= 970

⇒ 𝐴𝐵2 + 𝐵𝐶2 = 𝐶𝐴2 = 970 nfhLf;fg;gl;l Gs;spfs;> fyg;ngz; jsj;jpy; xU nrq;Nfhz Kf;Nfhzj;ij mikf;Fk;.

11. 𝟕 + 𝟓𝒊 , 𝟓 + 𝟐𝒊 , (𝟒 + 𝟕𝒊) kw;Wk; 𝟐 + 𝟒𝒊 vDk; fyg;ngz;fs; xU ,izfuj;ij

mikf;Fk; vd epWTf. (OCT-16)

𝐴, 𝐵, 𝐶 kw;Wk; 𝐷 vDk; Gs;spfs; KiwNa

2, 4 , 5, 2 , (7, 5) kw;Wk; (4, 7)

𝐴𝐶 d; eLg;Gs;sp = 2+7

2 ,

4+5

2 =

9

2,

9

2

𝐵𝐷 d; eLg;Gs;sp = 5+4

2 ,

2+7

2 =

9

2,

9

2

𝐴𝐶 d; eLg;Gs;spAk;> 𝐵𝐷 apd; eLg;Gs;spAk;

xd;Nw.

∴ vdNt> nfhLf;fg;gl;l Gs;spfs; xU

,izfuj;ij mikf;fpd;wd.

12. (– 𝟖 − 𝟔𝒊) ,d; th;f;f%yk; fhz;f

(MAR-06, OCT-06,JUN-15)

𝑥 + 𝑖𝑦 = – 8 − 6𝑖 vd;f

,UGwKk; th;f;fg;gLj;j>

𝑥2 − 𝑦2 + 2𝑥𝑦𝑖 = −8 − 6𝑖

nka; > fw;gidg; gFjpfis xg;gpl,

𝑥2 − 𝑦2 = −8………………………………. 1 2𝑥𝑦 = −6…………………………………….. 2

𝑥2 + 𝑦2 = 𝑥2 − 𝑦2 2 + (2𝑥𝑦)2

= −8 2 + (−6)2 = 100

𝑥2 + 𝑦2 = 10…………………..………….. 3

(1)+(3)⇒ 2𝑥2 = 2, , 𝑥 = ±1 𝑥 = 1 vdpy; 𝑦 = −3 𝑥 = −1 vdpy; 𝑦 = 3

– 8 − 6𝑖 = 1 − 3𝑖 my;yJ −1 + 3𝑖

13. (– 𝟕 + 𝟐𝟒𝒊) ,d; th;f;f%yk; fhz;f (MAR-07,JUN-09,MAR-15)

𝑥 + 𝑖𝑦 = – 7 + 24𝑖 vd;f

,UGwKk; th;f;fg;gLj;j>

𝑥2 − 𝑦2 + 2𝑥𝑦𝑖 = −7 + 24𝑖

nka;> fw;gidg; gFjpfis xg;gpl,

𝑥2 − 𝑦2 = −7………………………………. 1

2𝑥𝑦 = 24…………………………………….. 2

𝑥2 + 𝑦2 = 𝑥2 − 𝑦2 2 + (2𝑥𝑦)2

= −7 2 + (24)2 = 625

𝑥2 + 𝑦2 = 25…………………..………….. 3

⇒ 𝑥2 = 9, 𝑦2 = 16

𝑥 = ±3, 𝑦 = ±4

𝑥𝑦 kpif vz; vd;gjhy; 𝑥 -k; 𝑦 -k; xNu Fwpahf nfhs;s Ntz;Lk;

𝑥 = 3, 𝑦 = 4 my;yJ 𝑥 = −3, 𝑦 = −4

– 7 + 24𝑖 = 3 + 4𝑖 my;yJ (−3 − 4𝑖)

14. 𝑷 vd;gJ xU fyg;ngz; khwp 𝒛 vdpy;

𝑹𝒆 𝒛 +𝟏

𝒛 −𝒊 = 𝟎 vd;w epge;jidf;F 𝑷 d;

epakg;ghij fhz;f. (OCT-12) 𝑧 = 𝑥 + 𝑖𝑦, 𝑧 = 𝑥 − 𝑖𝑦 vd;f 𝑧 + 1 = 𝑥 − 𝑖𝑦 + 1 = 𝑥 + 1 − 𝑖𝑦

𝑧 − 𝑖 = 𝑥 − 𝑖𝑦 − 𝑖 = 𝑥 − 𝑖(𝑦 + 1)

𝑧 +1

𝑧 −𝑖=

𝑥+1 −𝑖𝑦

𝑥−𝑖(𝑦+1)×

𝑥+𝑖(𝑦+1)

𝑥+𝑖(𝑦+1)

= 𝑥+1 𝑥+𝑖 𝑥+1 𝑦+1 −𝑖𝑦𝑥 +𝑦(𝑦+1)

𝑥2+ 𝑦+1 2

𝑅𝑒 𝑧 +1

𝑧 −𝑖 = 0

𝑥+1 𝑥+𝑦(𝑦+1)

𝑥2+ 𝑦+1 2 = 0

𝑥2 + 𝑥 + 𝑦2 + 𝑦 = 0

𝑥2 + 𝑦2 + 𝑥 + 𝑦 = 0





15. 𝑷 vd;gJ xU fyg;ngz; khwp 𝒛 vdpy;

𝑹𝒆 𝒛+𝟏

𝒛−𝒊 = 𝟎 vd;w epge;jidf;F 𝑷 d;

epakg;ghij fhz;. (MAR-08)

𝑧 = 𝑥 + 𝑖𝑦 vd;f

𝑧 + 1 = 𝑥 + 𝑖𝑦 + 1 = 𝑥 + 1 + 𝑖𝑦

𝑧 − 𝑖 = 𝑥 + 𝑖𝑦 − 𝑖 = 𝑥 + 𝑖(𝑦 − 1)

𝑧+1

𝑧−𝑖=

𝑥+1 +𝑖𝑦

𝑥+𝑖(𝑦−1)×

𝑥−𝑖(𝑦−1)

𝑥−𝑖(𝑦−1)

= 𝑥+1 𝑥−𝑖 𝑥+1 𝑦−1 +𝑖𝑦𝑥 +𝑦(𝑦−1)

𝑥2+ 𝑦−1 2

𝑅𝑒 𝑧+1

𝑧−𝑖 = 0

𝑥+1 𝑥+𝑦(𝑦−1)

𝑥2+ 𝑦−1 2 = 0

𝑥 + 1 𝑥 + 𝑦 𝑦 − 1 = 0

𝑥2 + 𝑥 + 𝑦2 − 𝑦 = 0

𝑥2 + 𝑦2 + 𝑥 − 𝑦 = 0

16. 𝑷 vDk; Gs;sp fyg;ngz; khwp 𝒛 If;

Fwpj;jhy; 𝑷 -,d; epakg;ghijia

𝒁 − 𝟓𝒊 = 𝒁 + 𝟓𝒊 vDk; fl;Lg;ghLfSf;F cl;gl;L fhz;f. (MAR-

15)


𝑥 + 𝑖𝑦 − 5𝑖 = 𝑥 + 𝑖𝑦 + 5𝑖

𝑥 + 𝑖(𝑦 − 5) = 𝑥 + 𝑖(𝑦 + 5)

𝑥2 + (𝑦 − 5)2 = 𝑥2 + (𝑦 + 5)2

,UGwKk; th;f;fg;gLj;j,

𝑥2 + (𝑦 − 5)2 = 𝑥2 + (𝑦 + 5)2

(𝑦 − 5)2 = (𝑦 + 5)2

𝑦2 − 10𝑦 + 25 = 𝑦2 + 10𝑦 + 25

−20𝑦 = 0

⇒ 𝑦 = 0

17. 𝑷 vDk; Gs;sp fyg;G vz; khwp 𝒛 If;

Fwpj;jhy;> 𝑷 ,d; epakg; ghijia

𝟑𝒛 − 𝟓 = 𝟑 𝒛 + 𝟏 vd;Dk; fl;Lg;ghl;Lf;F cl;gl;L fhz;f. (JUN-

11)

𝑧 = 𝑥 + 𝑖𝑦

3(𝑥 + 𝑖𝑦) − 5 = 3 𝑥 + 𝑖𝑦 + 1

3𝑥 + 3𝑖𝑦 − 5 = 3 𝑥 + 1 + 𝑖𝑦

3𝑥 − 5 + 3𝑖𝑦 = 3 𝑥 + 1 + 𝑖𝑦

3𝑥 − 5 2 + (3𝑦)2 = 3 𝑥 + 1 2 + 𝑦2


9𝑥2 + 25 − 30𝑥 + 9𝑦2

= 9[ 𝑥2 + 1 + 2𝑥 + 𝑦2]

9𝑥2 + 25 − 30𝑥 + 9𝑦2 = 9𝑥2 + 9 + 18𝑥 + 9𝑦2

0 = 18𝑥 + 30𝑥 + 9 − 25

0 = 48𝑥 − 16

48𝑥 = 16

𝑥 =1

3

18. 𝑷 vDk; Gs;sp> fyg;G vz; khwp 𝒁 If;

Fwpj;jhy;> 𝑷 ,d; epakg; ghijia

𝒁 − 𝟑𝒊 = 𝒁 + 𝟑𝒊 vd;Dk; epge;jidf;F cl;gl;L fhz;f. (MAR-09)


𝑥 + 𝑖𝑦 − 3𝑖 = 𝑥 + 𝑖𝑦 + 3𝑖

𝑥 + 𝑖(𝑦 − 3) = 𝑥 + 𝑖(𝑦 + 3)

𝑥2 + (𝑦 − 3)2 = 𝑥2 + (𝑦 + 3)2


𝑥2 + (𝑦 − 3)2 = 𝑥2 + (𝑦 + 3)2

(𝑦 − 3)2 = (𝑦 + 3)2

𝑦2 − 6𝑦 + 9 = 𝑦2 + 6𝑦 + 9

−12𝑦 = 0

⇒ 𝑦 = 0

𝑃 d; epakg;ghij 𝑦 = 0

19. 𝑷 vDk; Gs;sp fyg;ngz; khwp 𝒁 If;

Fwpj;jhy; 𝑷 d; epakg;ghijia

𝒁 − 𝟒𝒊 = 𝒁 + 𝟒𝒊 vd;w epge;jidf;Fl;gl;L fhz;f. (OCT-15)


𝑥 + 𝑖𝑦 − 4𝑖 = 𝑥 + 𝑖𝑦 + 4𝑖

𝑥 + 𝑖(𝑦 − 4) = 𝑥 + 𝑖(𝑦 + 4)

𝑥2 + (𝑦 − 4)2 = 𝑥2 + (𝑦 + 4)2


𝑥2 + (𝑦 − 4)2 = 𝑥2 + (𝑦 + 4)2

(𝑦 − 4)2 = (𝑦 + 4)2

𝑦2 − 8𝑦 + 16 = 𝑦2 + 8𝑦 + 16

−16𝑦 = 0

⇒ 𝑦 = 0

20. 𝑷 vd;Dk; Gs;sp fyg;G vz; khwp 𝒛 If;

Fwpj;jhy; 𝑷 ,d; epakg; ghijia

𝟐𝒛 − 𝟏 = 𝒛 − 𝟐 vd;w epge;jidf;F cl;gl;L

fhz;f (MAR-06)

𝑧 = 𝑥 + 𝑖𝑦 vd;f 2(𝑥 + 𝑖𝑦) − 1 = 𝑥 + 𝑖𝑦 − 2

2𝑥 − 1 + 2𝑖𝑦 = 𝑥 − 2 + 𝑖𝑦

2𝑥 − 1 2 + (2𝑦)2 = 𝑥 − 2 2 + (𝑦)2






4𝑥2 + 1 − 4𝑥 + 4𝑦2 = 𝑥2 + 4 − 4𝑥 + 𝑦2

3𝑥2 + 3𝑦2 = 3

÷ 3 𝑥2 + 𝑦2 = 1

𝑃 d; epakg;ghij xU tl;lkhFk;.

21. nka;naz; Fzfq;fisf; nfhz;l 𝑷 𝒙 = 𝟎 vd;w gy;YWg;Gf; Nfhitr; rkd;ghl;bd; %yq;fs; ,iznaz; ,ul;ilahfj;jhd;

,lk;ngWk; vd ep&gpf;f. (OCT-11)

𝑃 𝑥 = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + ⋯ + 𝑎1𝑥1 + 𝑎0 = 0 vd;gJ nka; Fzfq;fSila xU gy;YWg;Gf; Nfhit.

𝑃 𝑥 = 0 f;F 𝑧 xU %yk; 𝑃 𝑥 = 0f;F 𝑧 k; xU %yk; vd fhl;l Ntz;Lk;.

𝑃 𝑥 = 0f;F 𝑧 xU %yk; Mjyhy;

𝑃 𝑧 = 𝑎𝑛𝑧𝑛 + 𝑎𝑛−1𝑧𝑛−1 + ⋯ + 𝑎1𝑧1 + 𝑎0 = 0 ,UGwKk; fyg;ngz; ,iznaz; fhz>

𝑃 𝑧 = 𝑎𝑛𝑧𝑛 + 𝑎𝑛−1𝑧𝑛−1 + ⋯ + 𝑎1𝑧1 + 𝑎0 = 0

,U fyg;ngz;fspd; $Ljypd; ,iznaz;> mtw;wpd; jdpj;jdp ,izfspd; $LjYf;Fr; rkkhtjhy;

𝑎𝑛𝑧𝑛 + 𝑎𝑛−1𝑧𝑛−1 + ⋯ + 𝑎1𝑧1 + 𝑎0 = 0

𝑎𝑛 𝑧𝑛 + 𝑎𝑛−1 𝑧𝑛−1 + ⋯ + 𝑎1 𝑧1 + 𝑎0 = 0

,q;F 𝑧𝑛 = 𝑧 𝑛 kw;Wk;

𝑎0 , 𝑎1 , 𝑎2 … 𝑎𝑛 nka;naz;fs; Mjyhy; mit xt;nthd;Wk; jdf;Fj;jhNd fyg;ngz; ,izahfpd;wd.

𝑎𝑛𝑧𝑛 + 𝑎𝑛−1𝑧𝑛−1 + ⋯ + 𝑎1𝑧1 + 𝑎0 = 0

⇒ 𝑃 𝑧 = 0

𝑃 𝑥 = 0 f;F 𝑧 k; xU %yk; vd;gNj ,jd; nghUshFk;

22. 𝟐 + 𝟑𝒊 I xU jPh;thf nfhz;l

𝒙𝟒 − 𝟒𝒙𝟐 + 𝟖𝒙 + 𝟑𝟓 = 𝟎 vDk; rkd;ghl;ilj;

jPh; (MAR-16, JUN-16)

2 + 3𝑖 xU %yk;> vdNt 2 − 3𝑖 kw;nwhU %yk;.

%yq;fspd; $Ljy; = 4

%yq;fspd; ngUf;fk; =(2 + 3𝑖)(2 − 3𝑖)

= 4 + 3 = 7

𝑥2 − (%yq;fspd; $Ljy;)𝑥+%yq;fspd; ngUf;fk;

= 𝑥2 − 4𝑥 + 7

𝑥4 − 4𝑥2 + 8𝑥 + 35

≡ 𝑥2 − 4𝑥 + 7 (𝑥2 + 𝑝𝑥 + 5)

𝑥 ,d; nfOit xg;gpl> 8 = 7𝑝 − 20

𝑝 = 4

𝑥2 + 4𝑥 + 5 vd;gJ kw;nwhU fhuzp

𝑥2 + 4𝑥 + 5 = 0 ⇒ 𝑥 = −2 ± 𝑖

vdNt %yq;fs; 2 ± 3𝑖 kw;Wk; −2 ± 𝑖 MFk;

23. (𝟏 − 𝒊)I xU jPh;thf nfhz;l

𝒙𝟑 − 𝟒𝒙𝟐 + 𝟔𝒙 − 𝟒 = 𝟎 vDk; rkd;ghl;il jPh;.

( JUN-07 )

1 − 𝑖 xU %yk;>

1 + 𝑖 kw;nwhU %yk;.

%yq;fspd; $Ljy; = 1 + 𝑖 + 1 − 𝑖 = 2

%yq;fspd; ngUf;fk; = 1 + 𝑖 1 − 𝑖

= 12 + 12 = 2 𝑥2 − (%yq;fspd; $Ljy;)𝑥+%yq;fspd; ngUf;fk;= 0

𝑥2 − 2𝑥 + 2 = 0

𝑥3 − 4𝑥2 + 6𝑥 − 4 = 𝑥2 − 2𝑥 + 2 (𝑥 − 2)

𝑥 − 2 = 0

{1 + 𝑖, 1 − 𝑖, 2}

24. 𝟑 + 𝒊 I xU jPh;thf nfhz;l 𝒙𝟒 − 𝟖𝒙𝟑 + 𝟐𝟒𝒙𝟐 − 𝟑𝟐𝒙 + 𝟐𝟎 = 𝟎 vDk;

rkd;ghl;il jPh;. (MAR-09,MAR-12)

3 + 𝑖 xU %yk;> 3 − 𝑖 kw;nwhU %yk;.

%yq;fspd; $Ljy; =6

%yq;fspd; ngUf;fk; = 10


= 𝑥2 − 6𝑥 + 10

𝑥4 − 8𝑥3 + 24𝑥2 − 32𝑥 + 20

≡ 𝑥2 − 6𝑥 + 10 (𝑥2 + 𝑝𝑥 + 2)

𝑥 ,d; nfOit xg;gpl 10𝑝 − 12 = −32

𝑝 = −2

𝑥2 − 2𝑥 + 2 vd;gJ kw;nwhU fhuzp

𝑥2 − 2𝑥 + 2 = 0 ⇒ 𝑥 = 1 ± 𝑖

vdNt %yq;fs; 3 ± 𝑖, 1 ± 𝑖

25. 𝟏 + 𝟐𝒊 I xU jPh;thff; nfhz;l 𝒙𝟒 − 𝟒𝒙𝟑 + 𝟏𝟏𝒙𝟐 − 𝟏𝟒𝒙 + 𝟏𝟎 = 𝟎 vDk;

rkd;ghl;bd; jPh;Tfisf; fhz;f

(JUN-09,MAR-11)

1 + 2𝑖 xU %yk;, 1 − 2𝑖 kw;nwhU %yk;.

%yq;fspd; $Ljy; = 2

%yq;fspd; ngUf;fk; =5


= 𝑥2 − 2𝑥 + 5

𝑥4 − 4𝑥3 + 11𝑥2 − 14𝑥 + 10

≡ 𝑥2 − 2𝑥 + 5 (𝑥2 + 𝑝𝑥 + 2)





𝑥 ,d; nfOit xg;gpl 5𝑝 − 4 = −14

𝑝 = −2

𝑥2 − 2𝑥 + 2 vd;gJ kw;nwhU fhuzp

𝑥2 − 2𝑥 + 2 ⇒ 𝑥 = 1 ± 𝑖

vdNt %yq;fs; 1 ± 2𝑖, 1 ± 𝑖

26. 𝟏 + 𝒊I xU jPh;thff; nfhz;l 𝒙𝟒 + 𝟒 = 𝟎 vDk; rkd;ghl;bd; jPh;Tfisf; fhz;f.

(JUN-06)

1 + 𝑖 xU %yk;>

1 − 𝑖 kw;nwhU %yk;.

%yq;fspd; $Ljy; =1 + 𝑖 + 1 − 𝑖 = 2

%yq;fspd; ngUf;fk; = 1 + 𝑖 1 − 𝑖

= 12 + 12 = 2 𝑥2 − (%yq;fspd; $Ljy;)𝑥+%yq;fspd; ngUf;fk;= 0

𝑥2 − 2𝑥 + 2 = 0

𝑥4 + 0 𝑥3 + 0 𝑥2 + 0𝑥 + 4

= 𝑥2 − 2𝑥 + 2 𝑥2 + 𝑝𝑥 + 2 = 0

𝑥 ,d; nfOit xg;gpl

0 = 2𝑝 − 4

2𝑝 = 4

𝑝 = 2

𝑥2 + 2𝑥 + 2 = 0 vd;gJ kw;nwhU fhuzp

∴ 𝑥 =−2± 4−8

2=

−2± −4

2=

−2±2𝑖

2= −1 ± 𝑖

∴ {1 + 𝑖, 1 − 𝑖, −1 + 𝑖, −1 − 𝑖}

27. RUf;Ff : 𝐜𝐨𝐬 𝜽+𝒊 𝐬𝐢𝐧𝜽 𝟒

𝐬𝐢𝐧 𝜽+𝒊 𝐜𝐨𝐬 𝜽 𝟓 (OCT-06, 11,16)

cos 𝜃+𝑖 sin 𝜃 4

sin 𝜃+𝑖 cos 𝜃 5 = cos 𝜃+𝑖 sin 𝜃 4

cos 𝜋

2−𝜃 +𝑖 sin

𝜋

2−𝜃

5

= cos 4θ − 5 𝜋

2− 𝜃 +𝑖 sin 4θ − 5

𝜋

2− 𝜃

= cos 9θ −5𝜋

2 +𝑖 sin 9θ −

5𝜋

2

= cos 5𝜋

2− 9θ −𝑖 sin

5𝜋

2− 9θ

= cos 𝜋

2− 9θ −𝑖 sin

𝜋

2− 9θ

= sin 9θ − 𝑖 cos 9θ

28. 𝒏 vd;gJ kpif KO vz; vdpy;

𝟏+𝐬𝐢𝐧 𝜽+𝒊 𝐜𝐨𝐬𝜽

𝟏+𝐬𝐢𝐧 𝜽−𝒊 𝐜𝐨𝐬𝜽

𝒏

= 𝐜𝐨𝐬 𝒏 𝝅

𝟐− 𝜽 + 𝒊 𝐬𝐢𝐧 𝒏

𝝅

𝟐− 𝜽

vd ep&gpf;f (MAR-11)

𝑧 = sin 𝜃 + 𝑖 cos 𝜃 vd;f

1

z= sin 𝜃 − 𝑖 cos 𝜃

1 + sin 𝜃 + 𝑖 cos 𝜃

1 + sin 𝜃 − 𝑖 cos 𝜃

𝑛

= 1 + 𝑧

1 +1z

𝑛

= 𝑧𝑛

= (sin 𝜃 + 𝑖 cos 𝜃)𝑛

= cos 𝜋

2− 𝜃 + 𝑖 sin

𝜋

2− 𝜃

𝑛

= cos 𝑛 𝜋

2− 𝜃 + 𝑖 sin 𝑛

𝜋

2− 𝜃

29. RUf;Ff : 𝐜𝐨𝐬 𝟐𝜽−𝒊 𝐬𝐢𝐧𝟐𝜽 𝟕 𝐜𝐨𝐬𝟑 𝜽+𝒊 𝐬𝐢𝐧 𝟑𝜽 −𝟓

𝐜𝐨𝐬 𝟒𝜽+𝒊 𝐬𝐢𝐧 𝟒𝜽 𝟏𝟐 𝐜𝐨𝐬 𝟓𝜽−𝒊 𝐬𝐢𝐧𝟓𝜽 −𝟔

( JUN-12)

cos 2𝜃−𝑖 sin 2 𝜃 7 cos 3 𝜃+𝑖 sin 3𝜃 −5

cos 4𝜃+𝑖 sin 4𝜃 12 cos 5𝜃−𝑖 sin 5𝜃 −6

=(cos 𝜃+𝑖 sin 𝜃)−14 (cos 𝜃+𝑖 sin 𝜃)−15

(cos 𝜃+𝑖 sin 𝜃)48 (cos 𝜃+𝑖 sin 𝜃)30

= (cos 𝜃 + 𝑖 sin 𝜃)−14−15−48−30

= (cos 𝜃 + 𝑖 sin 𝜃)−107

= cos 107𝜃 − 𝑖 sin 107 𝜃

30. 𝒏 xU kpif KO vz; vdpy;

(𝟏 + 𝐜𝐨𝐬 𝜽 + 𝒊 𝐬𝐢𝐧 𝜽)𝒏 + (𝟏 + 𝐜𝐨𝐬 𝜽 − 𝒊 𝐬𝐢𝐧 𝜽)𝒏

= 𝟐𝒏+𝟏 𝐜𝐨𝐬𝐧 𝜽

𝟐 𝐜𝐨𝐬

𝒏𝜽

𝟐 vd epWTf( JUN-12,MAR-14)

(1 + cos 𝜃 + 𝑖 sin 𝜃)𝑛

= 2 cos2 𝜃

2+ 𝑖 2sin

𝜃

2cos

𝜃

2

𝑛

= 2cos𝜃

2 cos

𝜃

2+ 𝑖 sin

𝜃

2

𝑛

= 2𝑛cosn 𝜃

2 cos 𝑛

𝜃

2+ 𝑖 sin 𝑛

𝜃

2 …….. 1

(1),y; 𝑖 f;Fg; gjpy; – 𝑖 I gpujpapl>

(1 + cos 𝜃 − 𝑖 sin 𝜃)𝑛 = 2𝑛 cosn 𝜃

2 cos 𝑛

𝜃

2− 𝑖 sin 𝑛

𝜃

2

……………. 2

(1)+(2) ⇒

1 + cos 𝜃 + 𝑖 sin 𝜃 𝑛 + 1 + cos 𝜃 − 𝑖 sin 𝜃 𝑛

= 2𝑛 cosn 𝜃

2 2 cos

𝑛𝜃

2

= 2𝑛+1 cosn 𝜃

2 cos

𝑛𝜃

2

31. 𝒏 vd;gJ kpif KO vdpy;>

(𝟏 + 𝒊)𝒏 + (𝟏 − 𝒊)𝒏 = 𝟐𝒏+𝟐

𝟐 𝐜𝐨𝐬𝒏𝝅

𝟒 vd ep&gp.

(OCT-07,MAR-08,MAR-10,OCT-15)

1 + 𝑖 = 𝑟(cos 𝜃 + 𝑖 sin 𝜃) nka; kw;Wk; fw;gidg; gFjpfis xg;gpl>

𝑟 cos 𝜃 = 1, 𝑟 sin 𝜃 = 1

𝑟 = 1 2 + 1 2 = 2

NkYk;, cos 𝜃 =1

2, sin 𝜃 =

1

2⇒ 𝜃 =

𝜋

4

1 + 𝑖 = 2 cos𝜋

4+ 𝑖 sin

𝜋

4





1 + 𝑖 𝑛 = 2 𝑛

cos𝜋

4+ 𝑖 sin

𝜋

4

𝑛

= 2𝑛

2 cos𝑛𝜋

4+ 𝑖 sin

𝑛𝜋

4 ………….. 1

(1),y; 𝑖 f;Fg; gjpy; – 𝑖 I gpujpapl,

1 − 𝑖 𝑛 = 2𝑛

2 cos𝑛𝜋

4− 𝑖 sin

𝑛𝜋

4 …………… 2

(1) IAk; (2) IAk; $l;l

1 + 𝑖 𝑛 + 1 − 𝑖 𝑛 = 2𝑛

2 2cos𝑛𝜋

4

= 2𝑛+2

2 cos𝑛𝜋

4

32. 𝒏 ∈ 𝑵 vdpy;

(𝟏 + 𝒊 𝟑)𝒏 + (𝟏 − 𝒊 𝟑)𝒏 = 𝟐𝒏+𝟏 𝐜𝐨𝐬𝒏𝝅

𝟑

epWTf (JUN-08)

1 + 𝑖 3 = 𝑟(cos 𝜃 + 𝑖 sin 𝜃) nka; kw;Wk; fw;gidg; gFjpfis xg;gpl>

𝑟 cos 𝜃 = 1,𝑟 sin 𝜃 = 3

∴ 𝑟 = 12 + ( 3)2 = 2

NkYk;, cos 𝜃 =1

2, sin 𝜃 =

3

2⇒ 𝜃 =

𝜋

3

1 + 𝑖 3 = 2 cos𝜋

3+ 𝑖 sin

𝜋

3

1 + 𝑖 3 𝑛

= 2𝑛 cos𝑛𝜋

3+ 𝑖 sin

𝑛𝜋

3 ………. 1

(1),y; 𝑖 f;F gjpy; – 𝑖 I gpujpapl,

1 − 𝑖 3 𝑛

= 2𝑛 cos𝑛𝜋

3− 𝑖 sin

𝑛𝜋

3 ……….. 2

(1) IAk; (2) IAk; $l;l

1 + 𝑖 3 𝑛

+ 1 − 𝑖 3 𝑛

= 2𝑛 2 cos𝑛𝜋

3

= 2𝑛+1 cos𝑛𝜋

3

33. 𝒙 +𝟏

𝒙= 𝟐 𝐜𝐨𝐬 𝜽 vdpy;

(i) 𝒙𝒏 +𝟏

𝒙𝒏 = 𝟐 𝐜𝐨𝐬 𝒏𝜽,

(ii) 𝒙𝒏 −𝟏

𝒙𝒏 = 𝟐𝒊 𝐬𝐢𝐧 𝒏𝜽

vd ep&gp (JUN-10)

𝑥 +1

𝑥= 2 cos 𝜃

𝑥2 − 2 cos 𝜃 𝑥 + 1 = 0

𝑥 =2 cos 𝜃± 4 cos 2 𝜃−4

2 =

2 cos 𝜃±2 cos 2 𝜃−1

2

𝑥 = cos 𝜃 ± −sin2 𝜃 = cos 𝜃 ± 𝑖 sin 𝜃

𝑥 = cos 𝜃 + 𝑖 sin 𝜃 vdf; nfhs;f

𝑥𝑛 = cos 𝜃 + 𝑖 sin 𝜃 𝑛

𝑥𝑛 = cos 𝑛𝜃 + 𝑖 sin 𝑛𝜃…….. 1

1

𝑥𝑛 = cos 𝑛𝜃 − 𝑖 sin 𝑛𝜃……….. 2

(1)+ (2) ⇒ 𝑥𝑛 +1

𝑥𝑛 = 2 cos 𝑛𝜃

(1) − (2) ⇒ 𝑥𝑛 −1

𝑥𝑛 = 2𝑖 sin 𝑛𝜃

34. 𝒙 = 𝐜𝐨𝐬 𝜶 + 𝒊 𝐬𝐢𝐧 𝜶 , 𝒚 = 𝐜𝐨𝐬 𝜷 + 𝒊 𝐬𝐢𝐧 𝜷 vdpy;

𝒙𝒎𝒚𝒏 +𝟏

𝒙𝒎𝒚𝒏 = 𝟐 𝐜𝐨𝐬(𝒎𝜶 + 𝒏𝜷) vd ep&gp

(MAR-07,OCT-12) 𝑥𝑚𝑦𝑛 = (cos 𝛼 + 𝑖 sin 𝛼)𝑚 (cos 𝛽 + 𝑖 sin 𝛽)𝑛

= cos 𝑚𝛼 + 𝑖 sin 𝑚𝛼 (cos 𝑛𝛽 + 𝑖 sin 𝑛𝛽)

= cos(𝑚𝛼 + 𝑛𝛽) + 𝑖 sin(𝑚𝛼 + 𝑛𝛽)

1

𝑥𝑚 𝑦𝑛 = cos(𝑚𝛼 + 𝑛𝛽) − 𝑖 sin(𝑚𝛼 + 𝑛𝛽)

𝑥𝑚𝑦𝑛 +1

𝑥𝑚 𝑦𝑛= cos 𝑚𝛼 + 𝑛𝛽 + 𝑖 sin 𝑚𝛼 + 𝑛𝛽 +

cos 𝑚𝛼 + 𝑛𝛽 − 𝑖 sin(𝑚𝛼 + 𝑛𝛽)

𝑥𝑚𝑦𝑛 +1

𝑥𝑚 𝑦𝑛 = 2 cos(𝑚𝛼 + 𝑛𝛽)

35. 𝐜𝐨𝐬 𝜶 + 𝐜𝐨𝐬 𝜷 + 𝐜𝐨𝐬 𝜸 = 𝟎

= 𝐬𝐢𝐧 𝜶 + 𝐬𝐢𝐧 𝜷 + 𝐬𝐢𝐧 𝜸 vdpy;

𝐜𝐨𝐬𝟑 𝜶 + 𝐜𝐨𝐬 𝟑𝜷 + 𝐜𝐨𝐬𝟑 𝜸 = 𝟑𝐜𝐨𝐬( 𝜶 + 𝜷 + 𝜸) kw;Wk;

𝐬𝐢𝐧𝟑 𝜶 + 𝐬𝐢𝐧 𝟑 𝜷 + 𝐬𝐢𝐧 𝟑 𝜸 = 𝟑𝐬𝐢𝐧 ( 𝜶 + 𝜷 + 𝜸)

vd epWTf (MAR-13,OCT-13)

𝑎 = cos 𝛼 + 𝑖 sin 𝛼

𝑏 = cos 𝛽 + 𝑖 sin 𝛽

𝑐 = cos 𝛾 + 𝑖 sin 𝛾

𝑎 + 𝑏 + 𝑐 = cos 𝛼 + cos 𝛽 + cos 𝛾

+𝑖 sin 𝛼 + sin 𝛽 + sin 𝛾

= 0 + 𝑖0 = 0

𝑎 + 𝑏 + 𝑐 = 0 vdpy; 𝑎3 + 𝑏3 + 𝑐3 = 3𝑎𝑏𝑐 cos 𝛼 + 𝑖 sin 𝛼 3 + cos 𝛽 + 𝑖 sin 𝛽 3 + cos 𝛾 + 𝑖 sin 𝛾 3

= 3(cos 𝛼 + 𝑖 sin 𝛼)(cos 𝛽 + 𝑖 sin 𝛽)(cos 𝛾 + 𝑖 sin 𝛾)

⇒ cos 3𝛼 + 𝑖 sin 3𝛼 + cos 3𝛽 + 𝑖 sin 3𝛽

+ (cos3 𝛾 + 𝑖 sin 3𝛾)

= 3[cos( 𝛼 + 𝛽 + 𝛾) + i sin ( 𝛼 + 𝛽 + 𝛾) ]

⇒ cos 3𝛼 + cos 3𝛽 + cos3𝛾 + 𝑖(sin 3𝛼 +sin 3𝛽 + 𝑖 sin 3𝛾)

= 3[cos( 𝛼 + 𝛽 + 𝛾) + i sin ( 𝛼 + 𝛽 + 𝛾) ]

nka; kw;Wk; fw;gid gFjpfis xg;gpl,

cos3 𝛼 + cos 3𝛽 + cos3 𝛾 = 3cos( 𝛼 + 𝛽 + 𝛾) sin3 𝛼 + sin 3 𝛽 + sin 3 𝛾 = 3sin ( 𝛼 + 𝛽 + 𝛾)

36. 𝐜𝐨𝐬 𝜶 + 𝐜𝐨𝐬 𝜷 + 𝐜𝐨𝐬 𝜸 = 𝟎

= 𝐬𝐢𝐧 𝜶 + 𝐬𝐢𝐧 𝜷 + 𝐬𝐢𝐧 𝜸 vdpy;

𝐜𝐨𝐬𝟐 𝜶 + 𝐜𝐨𝐬 𝟐𝜷 + 𝐜𝐨𝐬𝟐 𝜸 = 𝟎 kw;Wk;

𝐬𝐢𝐧𝟐 𝜶 + 𝐬𝐢𝐧 𝟐 𝜷 + 𝐬𝐢𝐧 𝟐 𝜸 = 𝟎

vd epWTf (JUN-06,JUN-11)





𝑎 = cos 𝛼 + 𝑖 sin 𝛼 ⇒ 1

𝑎= cos 𝛼 − 𝑖 sin 𝛼

𝑏 = cos 𝛽 + 𝑖 sin 𝛽 ⇒ 1

𝑏= cos 𝛽 − 𝑖 sin 𝛽

𝑐 = cos 𝛾 + 𝑖 sin 𝛾 ⇒ 1

𝑐= cos 𝛾 − 𝑖 sin 𝛾

,q;F 1

𝑎+

1

𝑏+

1

𝑐= (cos 𝛼 + cos 𝛽 + cos 𝛾) −

(sin 𝛼 + sin 𝛽 + sin 𝛾)

= 0 − 𝑖 0 = 0

𝑎2 + 𝑏2 + 𝑐2 + 2𝑎𝑏 + 2𝑏𝑐 + 2𝑐𝑎 =

(𝑎 + 𝑏 + 𝑐)2

𝑎2 + 𝑏2 + 𝑐2 + 2𝑎𝑏𝑐 1

𝑐+

1

𝑎+

1

𝑏 = 0

𝑎2 + 𝑏2 + 𝑐2 + 2𝑎𝑏𝑐 0 = 0

𝑎2 + 𝑏2 + 𝑐2 = 0

cos 𝛼 + 𝑖 sin 𝛼 2 + cos 𝛽 + 𝑖 sin 𝛽 2 +

cos 𝛾 + 𝑖 sin 𝛾 2 = 0

cos2 𝛼 + 𝑖 sin2 𝛼 + cos 2𝛽 + 𝑖 sin 2 𝛽 +

cos2 𝛾 + 𝑖 sin 2 𝛾 = 0

cos2 𝛼 + cos 2𝛽 + cos2 𝛾 +

𝑖 (sin2 𝛼 + sin 2 𝛽 + sin 2 𝛾) = 0

nka; kw;Wk; fw;gid gFjpfis xg;gpl,

cos2 𝛼 + cos 2𝛽 + cos2 𝛾 = 0 ,

sin2 𝛼 + sin 2 𝛽 + sin 2 𝛾 = 0

37. 𝒊 𝟏

𝟑 vy;yh kjpg;GfisAk; fhz;f

(MAR-13,16)

𝑖 = cos𝜋

2+ 𝑖 sin

𝜋

2

𝑖 1

3 = cos𝜋

2+ 𝑖 sin

𝜋

2

1

3

= cos 2𝑘𝜋 +𝜋

2 + 𝑖 sin 2𝑘𝜋 +

𝜋

2

1

3

= cos 4𝑘 + 1 𝜋

2+ 𝑖 sin 4𝑘 + 1

𝜋

2

13

= cos 4𝑘 + 1 𝜋

6+ 𝑖 sin 4𝑘 + 1

𝜋

6, 𝑘 = 0,1,2

vdNt cis 𝜋

6, cis

5𝜋

6, cis

9𝜋

6 Mfpa kjpg;Gfisg;

ngWk;

38. 𝝎𝟑 = 𝟏, vdpy; −𝟏+𝒊 𝟑

𝟐

𝟓

+ −𝟏−𝒊 𝟑

𝟐

𝟓

= −𝟏 vd

epWTf (OCT-08,JUN-12,MAR-13)

𝜔 xd;wpd; Kg;gb %yk; vdpy;> 𝜔 =−1+𝑖 3

2

𝜔2 =−1−𝑖 3

2

−1+𝑖 3

2

5

+ −1−𝑖 3

2

5

= 𝜔 5 + 𝜔2 5

= 𝜔5 + 𝜔10 = 𝜔2 + 𝜔 = −1

39. 𝝎 vd;gJ xd;wpd; Kg;gb %yk; kw;Wk;

𝒙 = 𝒂 + 𝒃, 𝒚 = 𝒂𝝎 + 𝒃𝝎𝟐, 𝒛 = 𝒂𝝎𝟐 + 𝒃𝝎

vdpy; (i) 𝒙𝒚𝒛 = 𝒂𝟑 + 𝒃𝟑

(ii) 𝒙𝟑 + 𝒚𝟑 + 𝒛𝟑 = 𝟑(𝒂𝟑 + 𝒃𝟑) vd ep&gpf;f

(JUN-15)

(i) 𝑥𝑦𝑧 = 𝑎 + 𝑏 𝑎𝜔 + 𝑏𝜔2 𝑎𝜔2 + 𝑏𝜔

= 𝑎 + 𝑏 𝑎2 + 𝑎𝑏𝜔2 + 𝑎𝑏𝜔 + 𝑏2

= 𝑎 + 𝑏 𝑎2 + 𝑎𝑏(𝜔2 + 𝜔) + 𝑏2

= 𝑎 + 𝑏 𝑎2 − 𝑎𝑏 + 𝑏2 = 𝑎3 + 𝑏3

(ii) 𝑥 + 𝑦 + 𝑧 = 𝑎 + 𝑏 + 𝑎𝜔 + 𝑏𝜔2 + 𝑎𝜔2 + 𝑏𝜔 = 𝑎 1 + 𝜔 + 𝜔2 + 𝑏 1 + 𝜔 + 𝜔2 + 𝑐 1 + 𝜔 + 𝜔2

= 𝑎 0 + 𝑏 0 + 𝑐 0

= 0

𝑥 + 𝑦 + 𝑧 = 0 vdpy; 𝑥3 + 𝑦3 + 𝑧3 = 3𝑥𝑦𝑧

𝑥3 + 𝑦3 + 𝑧3 = 3(𝑎3 + 𝑏3) [ (i) d; %yk;]

40. jPh;f;f 𝒙𝟒 + 𝟒 = 𝟎 (OCT-08,OCT-09,OCT-14)

𝑥4 + 4 = 0

𝑥4 = −4 = 4(−1)

𝑥 = 41

4(−1)1

4

= (22)1

4(cos 𝜋 + 𝑖 sin 𝜋)1

4

= 21

2(cos(2𝑘𝜋 + 𝜋) + 𝑖 sin(2𝑘𝜋 + 𝜋))1

4

= 2 cos 2𝑘 + 1 𝜋

4+ 𝑖 sin 2𝑘 + 1

𝜋

4 ,

𝑘 = 0,1,2,3

2 cis𝜋

4, 2 cis

3𝜋

4, 2 cis

5𝜋

4, 2 cis

7𝜋

4 Mfpa

kjpg;Gfis ngWk;.

41. 𝜶 kw;Wk; 𝜷 vd;git xd;Wf;nfhd;W

,izahdJ. NkYk; 𝜶 = − 𝟐 + 𝒊 vdpy;

𝜶𝟐 + 𝜷𝟐 − 𝜶𝜷 d; kjpg;gpidf; fhz;f. (Mar-17)

𝛼 kw;Wk; 𝛽 Mfpad ,iz vz;fs; MFk;.

𝛼 = − 2 + 𝑖, 𝛽 = − 2 − 𝑖

𝛼2 = − 2 + 𝑖 2

= − 2 2

+ 𝑖2 + 2 − 2 𝑖

= 2 − 1 − 2 2𝑖

= 1 − 2 2𝑖

𝛽2 = − 2 − 𝑖 2

= − 2 2

+ 𝑖2 − 2 − 2 𝑖

= 2 − 1 + 2 2𝑖





= 1 + 2 2𝑖

𝛼𝛽 = − 2 + 𝑖 − 2 − 𝑖

= − 2 𝟐

− 𝑖 2

= 2 + 1 = 3

𝛼2 + 𝛽2 − 𝛼𝛽 = 1 − 2 2𝑖 + 1 + 2 2𝑖 − 3

= 2 − 3

= −1

9. jdpepiyf; fzf;fpay;

1. (a) (𝒑 ∨ 𝒒) ∧ (~𝒒) vd;w $w;Wf;F nka; ml;ltizia mikf;f

(b) 𝒑 ∧ (~𝒑) xU Kuz;ghL vd ep&gp (JUN-06)

(a) (𝑝 ∨ 𝑞) ∧ (~𝑞) f;Fhpa nka; ml;ltiz

𝑝 𝑞 ~𝑞 𝑝 ∨ 𝑞 (𝑝 ∨ 𝑞) ∧ (~𝑞) 𝑇 𝑇 𝐹 𝑇 𝐹 𝑇 𝐹 𝑇 𝑇 𝑇 𝐹 𝑇 𝐹 𝑇 𝐹 𝐹 𝐹 𝑇 𝐹 𝐹

(b) 𝑝 ∧ (~𝑝) f;Fhpa nka; ml;ltiz

𝑝 ~𝑝 𝑝 ∧ ~𝑝 𝑇 𝐹 𝐹 𝐹 𝑇 𝐹

filrp epuypy; 𝐹 kl;LNk cs;sjhy;> xU

𝑝 ∧ (~𝑝) Kuz;ghlhFk;

2. (𝒑 ∧ 𝒒) ∨ [~(𝒑 ∧ 𝒒)] vd;w $w;Wf;F nka;

ml;ltizia mikf;f (JUN-14, MAR-15) (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 𝐹 𝑇 𝑇

𝐹 𝑇 𝐹 𝑇 𝑇

𝐹 𝐹 𝐹 𝑇 𝑇

3. ~( ~𝒑 ∧ ~𝒒 ) vd;w $w;Wf;F nka; ml;ltiz mikf;f (OCT-06)

~( ~𝑝 ∧ ~𝑞 ) f;Fhpa nka; ml;ltiz

𝑝 𝑞 ~𝑝 ~𝑞 (~𝑝) ∧ (~𝑞) ~( ~𝑝 ∧ ~𝑞 ) 𝑇 𝑇 𝐹 𝐹 𝐹 𝑇 𝑇 𝐹 𝐹 𝑇 𝐹 𝑇 𝐹 𝑇 𝑇 𝐹 𝐹 𝑇 𝐹 𝐹 𝑇 𝑇 𝑇 𝐹

4. (𝒑 𝒒) ∨ (~𝒓) f;Fhpa nka; ml;ltizia mikf;f (JUN-07, OCT-11, MAR-13,17)

𝑝 𝑞 𝑟 𝑝 ∧ 𝑞 ~ 𝑟 (𝑝 ∧ 𝑞) ∨ (~𝑟) 𝑇 𝑇 𝑇 𝑇 𝐹 𝑇 𝑇 𝑇 𝐹 𝑇 𝑇 𝑇 𝑇 𝐹 𝑇 𝐹 𝐹 𝐹 𝑇 𝐹 𝐹 𝐹 𝑇 𝑇 𝐹 𝑇 𝑇 𝐹 𝐹 𝐹 𝐹 𝑇 𝐹 𝐹 𝑇 𝑇 𝐹 𝐹 𝑇 𝐹 𝐹 𝐹 𝐹 𝐹 𝐹 𝐹 𝑇 𝑇

5. (𝒑 ∨ 𝑞) ∧ 𝒓 ,d; nka; ml;ltizia mikf;f (MAR-08)

𝑝 𝑞 𝑟 𝑝 ∨ 𝑞 (𝑝 ∨ 𝑞) ∧ 𝑟 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 𝐹 𝑇 𝐹 𝑇 𝐹 𝑇 𝑇 𝑇 𝑇 𝐹 𝐹 𝑇 𝐹 𝐹 𝑇 𝑇 𝑇 𝑇 𝐹 𝑇 𝐹 𝑇 𝐹 𝐹 𝐹 𝑇 𝐹 𝐹 𝐹 𝐹 𝐹 𝐹 𝐹

6. (𝒑 ∧ 𝒒) ∨ 𝒓 ,d; nka; ml;ltizia

mikf;f (OCT-08, 15)

𝑝 𝑞 𝑟 𝑝 ∧ 𝑞 (𝑝 ∧ 𝑞) ∨ 𝑟 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 𝐹 𝑇 𝑇 𝑇 𝐹 𝑇 𝐹 𝑇 𝑇 𝐹 𝐹 𝐹 𝐹 𝐹 𝑇 𝑇 𝐹 𝑇 𝐹 𝑇 𝐹 𝐹 𝐹 𝐹 𝐹 𝑇 𝐹 𝑇 𝐹 𝐹 𝐹 𝐹 𝐹

7. ((~𝒑) ∨ (~𝒒)) ∨ 𝒑 xU nka;ik vdf; fhl;Lf. (OCT-09)

((~𝑝) ∨ (~𝑞)) ∨ 𝑝 f;Fhpa nka; ml;ltiz

𝑝 𝑞 ~𝑝 ~𝑞 (~𝑝) ∨ (~𝑞) ((~𝑝) ∨ (~𝑞)) ∨ 𝑝

𝑇 𝑇 𝐹 𝐹 𝐹 𝑇

𝑇 𝐹 𝐹 𝑇 𝑇 𝑇

𝐹 𝑇 𝑇 𝐹 𝑇 𝑇

𝐹 𝐹 𝑇 𝑇 𝑇 𝑇

filrp epuy; KOtJk; 𝑇 Mjyhy; ((~𝑝) ∨ (~𝑞)) ∨ 𝑝 xU nka;ikahFk;

8. nka; ml;ltiziaf; nfhz;L (𝒑 ∧ ~𝒒 ) ∨ ((~𝒑) ∨ 𝒒) vd;w $w;W nka;ikah

my;yJ Kuz;ghlh vdf; fhz;f. (OCT-09, JUN-13)





𝑝 𝑞 ~𝑝 ~𝑞 𝑝 ∧ (~𝑞) (~𝑝) ∨ 𝑞 (𝑝 ∧ (~𝑞)) ∨ ((~𝑝) ∨ 𝑞)

𝑇 𝑇 𝐹 𝐹 𝐹 𝑇 𝑇

𝑇 𝐹 𝐹 𝑇 𝑇 𝐹 𝑇

𝐹 𝑇 𝑇 𝐹 𝐹 𝑇 𝑇

𝐹 𝐹 𝑇 𝑇 𝐹 𝑇 𝑇

∴ (𝑝 ∧ ~𝑞 ) ∨ ((~𝑝) ∨ 𝑞) xU nka;ikahFk;

9. 𝒒 ∨ 𝒑 ∨ ~𝒒 vd;w $w;W nka;ikah my;yJ

Kuz;ghlh vd;gijf; fhz;f. (MAR-17)

𝑝 𝑞 ~𝑞 𝑝 ∨ (~𝑞) 𝑞 ∨ 𝑝 ∨ ~𝑞

𝑇 𝑇 𝐹 𝑇 𝑇 𝑇 𝐹 𝑇 𝑇 𝑇 𝐹 𝑇 𝐹 𝐹 𝑇

𝐹 𝐹 𝑇 𝑇 𝑇

∴ 𝑞 ∨ 𝑝 ∨ ~𝑞 xU nka;ikahFk;

10. ((~𝒒) ∧ 𝒑) ∧ 𝒒 xU Kuz;ghL vdf; fhl;Lf.

(MAR-12, 16) ((~𝑞) ∧ 𝑝) ∧ 𝑞 f;Fhpa nka; ml;ltiz

𝑝 𝑞 ~𝑞 (~𝑞) ∧ 𝑝 ((~𝑞) ∧ 𝑝) ∧ 𝑞 𝑇 𝑇 𝐹 𝐹 𝐹

𝑇 𝐹 𝑇 𝑇 𝐹

𝐹 𝑇 𝐹 𝐹 𝐹

𝐹 𝐹 𝑇 𝐹 𝐹

filrp epuy; KOtJk; 𝐹 Mjyhy;

((~𝑞) ∧ 𝑝) ∧ 𝑞 xU Kuz;ghlhFk; .

11. 𝒑 ∧ 𝒒 → (𝒑 ∨ 𝒒) vdf; fhl;Lf.

(JUN-06,JUN-12,OCT-13,OCT-14)

𝑝 𝑞 𝑝 ∨ 𝑞 𝑝 ∧ 𝑞 𝑝 ∧ 𝑞 → (𝑝 ∨ 𝑞)

𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 𝐹 𝑇 𝐹 𝑇 𝐹 𝑇 𝑇 𝐹 𝑇 𝐹 𝐹 𝐹 𝐹 𝑇

∴ 𝑝 ∧ 𝑞 → (𝑝 ∨ 𝑞) vd;gJ xU nka;ik

MFk;.

12. ( ~𝒑 ∨ 𝒒) ∨ (𝒑 ∧ (∼ 𝒒)) xU nka;ikah vd;gjid nka; ml;ltiziaf; nfhz;L

jPh;khdpf;f. (OCT-07)

𝑝 𝑞 ~𝑝 ~𝑞 ~𝑝 ∨ 𝑞 𝑝 ∧ (~𝑞) ( ~𝑝 ∨ 𝑞) ∨ (𝑝 ∧ (∼ 𝑞))

𝑇 𝑇 𝐹 𝐹 𝑇 𝐹 𝑇

𝑇 𝐹 𝐹 𝑇 𝐹 𝑇 𝑇

𝐹 𝑇 𝑇 𝐹 𝑇 𝐹 𝑇

𝐹 𝐹 𝑇 𝑇 𝑇 𝐹 𝑇

filrp epuy; KOtJk; 𝑇 Mjyhy; jug;gl;l $w;W xU nka;ikahFk; .

13. (𝒑 ∧ (~𝒑)) ∧ ((~𝒒) ∧ 𝒑) nka;ikah Kuz;ghlh

vdf; fhz;f (JUN-08,10, MAR-11)

(𝑝 ∧ (~𝑝)) ∧ ((~𝑞) ∧ 𝑝) f;Fhpa nka; ml;ltiz

𝑝 𝑞 ~𝑝 ~𝑞 𝑝 (~𝑞) (~𝑞) ∧ 𝑝 (𝑝 ∧ (~𝑝)) ∧ ((~𝑞) ∧ 𝑝)

𝑇 𝑇 𝐹 𝐹 𝐹 𝐹 𝐹

𝑇 𝐹 𝐹 𝑇 𝐹 𝑇 𝐹

𝐹 𝑇 𝑇 𝐹 𝐹 𝐹 𝐹

𝐹 𝐹 𝑇 𝑇 𝐹 𝐹 𝐹

∴ (𝑝 ∧ (~𝑝)) ∧ ((~𝑞) ∧ 𝑝) xU Kuz;ghlhFk;.

14. 𝒑 ↔ 𝒒 ≡ (𝒑 → 𝒒) ∧ (𝒒 → 𝒑) vdf; fhl;Lf

(OCT-06, JUN-09, JUN-15)

𝑝 ↔ 𝑞 f;Fhpa nka; ml;ltiz

𝑝 𝑞 𝑝 ↔ 𝑞 𝑇 𝑇 𝑇 𝑇 𝐹 𝐹 𝐹 𝑇 𝐹 𝐹 𝐹 𝑇

(𝑝 → 𝑞) ∧ (𝑞 → 𝑝) f;Fhpa nka; ml;ltiz

𝑝 𝑞 𝑝 → 𝑞 𝑞 → 𝑝 (𝑝 → 𝑞) ∧ (𝑞 → 𝑝) 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 𝐹 𝐹 𝑇 𝐹 𝐹 𝑇 𝑇 𝐹 𝐹 𝐹 𝐹 𝑇 𝑇 𝑇

,U ml;ltizfSNk xNu khjphpahd filrp epuy;fisg; ngw;Ws;sjhy;

∴ 𝑝 ↔ 𝑞 ≡ (𝑝 → 𝑞) ∧ (𝑞 → 𝑝)

15. ~ 𝒑 ∨ 𝒒 ≡ (~𝒑) ∧ (~𝒒) vdf; fhl;Lf (MAR-06)

~ 𝑝 ∨ 𝑞 f;Fhpa nka; ml;ltiz

𝑝 𝑞 𝑝 ∨ 𝑞 ~ 𝑝 ∨ 𝑞 𝑇 𝑇 𝑇 𝐹 𝑇 𝐹 𝑇 𝐹 𝐹 𝑇 𝑇 𝐹 𝐹 𝐹 𝐹 𝑇

(~𝑝) ∧ (~𝑞) f;Fhpa nka; ml;ltiz

𝑝 𝑞 ~𝑝 ~𝑞 (~𝑝) ∧ (~𝑞) 𝑇 𝑇 𝐹 𝐹 𝐹 𝑇 𝐹 𝐹 𝑇 𝐹 𝐹 𝑇 𝑇 𝐹 𝐹 𝐹 𝐹 𝑇 𝑇 𝑇

filrp epuy;fs; xNu khjphpahdit

~ 𝑝 𝑞 ≡ (~𝑝) ∧ (~𝑞)





16. 𝒑 → 𝒒 ≡ (~𝒑) ∨ 𝒒 vdf; fhl;Lf (MAR-13) 𝑝 → 𝑞 f;Fhpa nka; ml;ltiz

𝑝 𝑞 𝑝 → 𝑞 𝑇 𝑇 𝑇 𝑇 𝐹 𝐹 𝐹 𝑇 𝑇 𝐹 𝐹 𝑇

(~𝑝) ∨ 𝑞 f;Fhpa nka; ml;ltiz

𝑝 𝑞 ~𝑝 (~𝑝) ∨ 𝑞 𝑇 𝑇 𝐹 𝑇

𝑇 𝐹 𝐹 𝐹

𝐹 𝑇 𝑇 𝑇

𝐹 𝐹 𝑇 𝑇

𝑝 → 𝑞 kw;Wk; (~𝑝) ∨ 𝑞 f;Fhpa ml;ltizfspd; filrp epuy;fs; xNu khjphpahapUg;gjhy;

∴ 𝑝 → 𝑞 ≡ (~𝑝) ∨ 𝑞

17. ~ 𝒑 ∧ 𝒒 ≡ ~𝒑 ∨ (~𝒒) vdf;fhl;Lf (OCT-10)

~ 𝑝 ∧ 𝑞 f;Fhpa nka; ml;ltiz

𝑝 𝑞 𝑝 ∧ 𝑞 ~ 𝑝 ∧ 𝑞 𝑇 𝑇 𝑇 𝐹 𝑇 𝐹 𝐹 𝑇 𝐹 𝑇 𝐹 𝑇 𝐹 𝐹 𝐹 𝑇

~𝑝 ∨ (~𝑞) f;Fhpa nka; ml;ltiz

𝑝 𝑞 ~𝑝 ~𝑞 (~𝑝) ∨ (~𝑞) 𝑇 𝑇 𝐹 𝐹 𝐹

𝑇 𝐹 𝐹 𝑇 𝑇

𝐹 𝑇 𝑇 𝐹 𝑇

𝐹 𝐹 𝑇 𝑇 𝑇

,U ml;ltizfspYk; filrp epuy;fs; xNu

khjphpahdit. ∴ ~ 𝑝 ∧ 𝑞 ≡ ~𝑝 ∨ (~𝑞)

18. 𝒑 → 𝒒 kw;Wk; 𝒒 → 𝒑 rkhdkw;wit vdf; fhl;Lf. (MAR-09,JUN-16)

𝑝 → 𝑞 f;Fhpa nka; ml;ltiz 𝑝 𝑞 𝑝 → 𝑞 𝑇 𝑇 𝑇 𝑇 𝐹 𝐹 𝐹 𝑇 𝑇 𝐹 𝐹 𝑇

𝑞 → 𝑝 f;Fhpa nka; ml;ltiz 𝑝 𝑞 𝑞 → 𝑝 𝑇 𝑇 𝑇 𝑇 𝐹 𝑇 𝐹 𝑇 𝐹 𝐹 𝐹 𝑇

𝑝 → 𝑞 kw;Wk; 𝑞 → 𝑝 f;Fhpa epuy;fs; xNu

khjphpahdit my;y. vdNt 𝑝 → 𝑞 kw;Wk; 𝑞 →

𝑝 Mfpait jh;f;f rkhdkhdit my;y.

𝟏𝟗. 𝒑 ↔ 𝒒 ≡ ~𝒑 𝒒 ∧ (∼ 𝒒) ∨ 𝒑 vdf; fhl;Lf.

(MAR-07,10,14, JUN-07,11, OCT-08, 11,12)

𝑝 ↔ 𝑞 f;Fhpa nka; ml;ltiz

𝑝 𝑞 𝑝 ↔ 𝑞 𝑇 𝑇 𝑇 𝑇 𝐹 𝐹 𝐹 𝑇 𝐹 𝐹 𝐹 𝑇

~𝑝 ∨ 𝑞 ∧ (∼ 𝑞) ∨ 𝑝 f;Fhpa nka;

ml;ltiz

𝑝 𝑞 ~𝑝 ~𝑞 ~𝑝 ∨ 𝑞 (∼ 𝑞) ∨ 𝑝 ~𝑝 ∨ 𝑞 ∧ (∼ 𝑞) ∨ 𝑝

𝑇 𝑇 𝐹 𝐹 𝑇 𝑇 𝑇

𝑇 𝐹 𝐹 𝑇 𝐹 𝑇 𝐹

𝐹 𝑇 𝑇 𝐹 𝑇 𝐹 𝐹

𝐹 𝐹 𝑇 𝑇 𝑇 𝑇 𝑇 ,U ml;ltizfspYk; filrp epuy;fs; xNu khjphpahAs;sd.

∴ 𝑝 ↔ 𝑞 ≡ ~𝑝 𝑞 ∧ (∼ 𝑞) ∨ 𝑝

20. 1 ,d; 3Mk; gb %yq;fs; xU Kbthd vgPypad; Fyj;ij ngUf;fypd; fPo; mikf;Fk;

vdf; fhl;Lf. (OCT-14)

𝐺 = {1, 𝜔, 𝜔2}. Nfa;yp ml;ltizahdJ

. 1 𝜔 𝜔2 1 1 𝜔 𝜔2 𝜔 𝜔 𝜔2 1

𝜔2 𝜔2 1 𝜔 ,e;j ml;ltizapypUe;J>

(i) ml;ltizapy; cs;s vy;yh

cWg;GfSk;> 𝐺 ,d; cWg;GfshFk;. vdNt milg;G tpjp cz;ikahfpwJ.

(ii) ngUf;fy; vg;nghOJk; Nrh;g;G

tpjpf;Fl;gLk;.

(iii) rkdpAWg;G 1. mJ rkdp tpjpiag; G+h;j;jp

nra;Ak;.

(iv) 1 ,d; vjph;kiw 1

𝜔 ,d; vjph;kiw 𝜔2

𝜔2 ,d; vjph;kiw 𝜔 kw;Wk; ,J vjph;kiw tpjpiag; G+h;j;jp nra;Ak;

∴ (𝐺, . ) xU FykhFk;.

(v) ghpkhw;W tpjpAk; cz;ikahFk;.





∴ (𝐺, . ) xU vgPypad; FykhFk;.

(vi) 𝐺 xU Kbthd fzk;. Mjyhy; (𝐺, . ) xU Kbthd vgPypad; FykhFk;

21. 1 ,d; 4Mk; gb %yq;fs; ngUf;fypd; fPo; vgpyPad; Fyj;ij mikf;Fk; vd epWTf.

(JUN-11)

1 ,d; 4 Mk; gb %yq;fs; 1, 𝑖, −1, −𝑖

𝐺 = {1, 𝑖, −1, −𝑖 }. Nfa;yp ml;ltizahJ

. 1 −1 𝑖 −𝑖 1 1 −1 𝑖 −𝑖

−1 −1 1 −𝑖 𝑖 𝑖 𝑖 −𝑖 −1 1

−𝑖 −𝑖 𝑖 1 −1 ,e;j ml;ltizapypUe;J,

(i) milg;G tpjp cz;ikahFk;.

(ii) 𝐶 ,y; ngUf;fyhdJ Nrh;g;G

tpjpf;Fl;gLkhjyhy; 𝐺 apYk; mJ cz;ikahFk;.

(iii) rkdp cWg;G 1 ∈ 𝐺 kw;Wk; mJ rkdp tpjpiag; G+h;j;jp nra;fpwJ

(iv) 1 ,d; vjph;kiw 1; 𝑖 ,d; vjph;kiw −𝑖

−1 ,d; vjph;kiw −1; −𝑖 ,d; vjph;kiw 𝑖. vjph; kiw tpjpiaAk; G+h;;j;jp MfpwJ


(v) ml;ltizapypUe;J> ghpkhw;W tpjpAk;

cz;ik.

∴ (𝐺, . ) xU vgPypad; FykhFk;.

22. (𝒁, +) xU Kbtw;w vgPypad; Fyk;; vd epWTf. (OCT-08) (i) milg;G tpjp: ,uz;L KO vz;fspd;

$LjYk; xU KO vz;.

𝑎, 𝑏 ∈ 𝑍 ⇒ 𝑎 + 𝑏 ∈ 𝑍

(ii) Nrh;g;G tpjp: 𝑍 y; $l;ly; vg;nghOJk; Nrh;g;G tpjpf;Fl;gLk;

∀𝑎, 𝑏, 𝑐 ∈ 𝑍, 𝑎 + 𝑏 + 𝑐 = 𝑎 + (𝑏 + 𝑐)

(iii)rkdp tpjp: rkdp cWg;G 𝑂 ∈ 𝑍 kw;Wk;

0 + 𝑎 = 𝑎 + 0 = 𝑎, ∀ 𝑎 ∈ 𝑍 I G+h;j;jp nra;fpwJ. vdNt rkdp tpjp cz;ikahFk;

(iv) vjph;kiw tpjp: xt;nthU 𝑎 ∈ 𝑍 f;Fk;

−𝑎 ∈ 𝑍 I −𝑎 + 𝑎 = 𝑎 + −𝑎 = 0vDkhW fhzyhk;. vdNt vjph;kiw tpjp cz;ikahFk;.

∴ (𝑍, +) xU FykhFk;

(v) ∀𝑎, 𝑏 ∈ 𝑍, 𝑎 + 𝑏 = 𝑏 + 𝑎

∴ $l;ly; ghpkhw;W tpjpf;Fl;gLk;

∴ (𝑍, +) xU vgPypad; FykhFk;

(vi) 𝑍 Kbtw;w fzk; Mjyhy; (𝑍, +) xU KbTw;w vgPypad; FykhFk.;

23. G+r;rpakw;w fyg;ngz;fspd; fzk; fyg;ngz;fspd; tof;fkhd ngUf;fypd; fPo;

xU vgPypad; Fyk; vdf; fhl;Lf. (JUN-16)

(i) milg;G tpjp: 𝐺 = 𝐶 − {0} vd;f. G+r;rpakw;w ,U fyg;ngz;fspd; ngUf;fy; vg;NghJk; G+r;rpakw;w fyg;ngz;zhf ,Uf;Fk;

∴ milg;G tpjp cz;ikahFk;

(ii) Nrh;g;G tpjp: fyg;ngz;fspy; ngUf;fy; Nrh;g;G tpjp vg;NghJk; cz;ikahFk;.

(iii) rkdp tpjp:

1 = 1 + 𝑖0 ∈ 𝐺, 1 rkdp cWg;ghFk;.

NkYk; 1. 𝑧 = 𝑧. 1 = 𝑧 ∀ 𝑧 ∈ 𝐺

∴ rkdp tpjp cz;ik.

(iv) vjph;kiw tpjp:

𝑧 = 𝑥 + 𝑖𝑦 ∈ 𝐺. ,q;F 𝑧 ≠ 0

𝑥 kw;Wk; 𝑦 ,uz;LNk G+r;rpakw;wit my;yJ VNjDk; xd;whtJ G+r;rpakw;wJ.

𝑥2 + 𝑦2 ≠ 0 1

𝑧=

1

𝑥 + 𝑖𝑦=

𝑥 − 𝑖𝑦

𝑥 + 𝑖𝑦 𝑥 − 𝑖𝑦 =

𝑥 − 𝑖𝑦

𝑥2 + 𝑦2

=𝑥

𝑥2+𝑦2 + 𝑖 −𝑦

𝑥2+𝑦2 ∈ 𝐺

NkYk; 𝑧.1

𝑧=

1

𝑧. 𝑧 = 1

∴ 𝑧 MdJ 1

𝑧 vd;w vjph;kiwia 𝐺 ,y;

ngw;Ws;sJ. vjph;kiw tpjp cz;ikahfpwJ.

∴ 𝐺, . xU FykhFk;

(v) ghpkhw;Wg; gz;G:

𝑧1𝑧2 = 𝑎 + 𝑖𝑏 𝑐 + 𝑖𝑑

= 𝑎𝑐 − 𝑏𝑑 + 𝑖(𝑎𝑑 + 𝑏𝑐)

= 𝑐𝑎 − 𝑑𝑏 + 𝑖 𝑑𝑎 + 𝑐𝑏 = 𝑧2𝑧1

∴ ghpkhw;Wg; gz;igAk; mJ G+h;j;jp nra;fpwJ.

∴ 𝐺 MdJ fyg;ngz;fspd; tof;fkhd ngUf;fypd; fPo; xU vgPypad; FykhFk;.

24. 𝟐 × 𝟐 thpir nfhz;l G+r;rpakw;w Nfhit mzpfs; ahTk; Kbtw;w vgPypad; my;yhj Fyj;ij mzp ngUf;fypd; fPo; mikf;Fk; vdf; fhl;Lf. (,q;F mzpapd; cWg;Gfs;

ahTk; 𝑹 Ir; Nrh;e;jit) (OCT-07)

𝐺 vd;gJ 2 × 2 thpir nfhz;l G+r;rpakw;w Nfhit mzpfs; ahTk; mlq;fpa fzk;.

cWg;Gfs; ahTk; 𝑅 Ir; Nrh;e;jit





(i) milg;G tpjp:

,uz;L 2 × 2 thpir nfhz;l G+r;rpakw;w

Nfhit mzpfspd; ngUf;fw;gyd; xU 2 × 2 thpir G+r;rpakw;w Nfhit mzpahFk;. vdNt milg;G tpjp cz;ikahFk;.

𝐴, 𝐵 ∈ 𝐺 ⇒ 𝐴𝐵 ∈ 𝐺

(ii) Nrh;g;G tpjp: mzpg; ngUf;fy; vg;nghOJk; Nrh;g;G tpjpf;Fl;gLk;. vdNt Nrh;g;G tpjp cz;ikahFk;.

𝐴 𝐵𝐶 = 𝐴𝐵 𝐶, ∀𝐴, 𝐵, 𝐶 ∈ 𝐺

(iii) rkdp tpjp:

rkdp cWg;G 𝐼2 = 1 00 1

∈ 𝐺. ,J rkdpg;

gz;ig G+h;j;jp nra;fpwJ.

(iv) vjph;kiw tpjp:

𝐴 ∈ 𝐺 ,d; vjph;kiw 𝐴−1I 𝐺 ,y; fhz

KbAk;. NkYk; mJ 2 × 2 thpir nfhz;lJ.

kw;Wk; 𝐴𝐴−1 = 𝐴−1𝐴 = 𝐼 vdNt> vjph;kiw

tpjp cz;ikahFk;. vdNt 𝐺 xU FykhFk;. nghJthf mzp ngUf;fy; ghpkhw;W

tpjpf;Fl;glhJ. Mjyhy; 𝐺 xU vgPypad;

my;yhj FykhFk;. 𝐺 ,y; vz;zpf;ifaw;w cWg;Gfs; cs;sjhy;> mJ KbTw;w vgPypad; my;yhj FykhFk;.

25. 𝟏 𝟎𝟎 𝟏

, −𝟏 𝟎𝟎 𝟏

, 𝟏 𝟎𝟎 −𝟏

, −𝟏 𝟎𝟎 −𝟏

Mfpa

ehd;F mzpfSk; mlq;fpa fzk; mzpg;ngUf;fypd; fPo; xU vgPypad; Fyj;ij mikf;Fk; vdf; fhl;Lf.

(MAR-11,OCT-13,MAR-15)

𝐼 = 1 00 1

, 𝐴 = −1 00 1

,

𝐵 = 1 00 −1

, 𝐶 = −1 00 −1

𝐺 = {𝐼, 𝐴, 𝐵, 𝐶} vd;f ,t;tzpfis ,uz;L ,uz;lhfg; ngUf;fp ngUf;fy; ml;ltizia mikf;fyhk;

. 𝐼 𝐴 𝐵 𝐶 𝐼 𝐼 𝐴 𝐵 𝐶 𝐴 𝐴 𝐼 𝐶 𝐵 𝐵 𝐵 𝐶 𝐼 𝐴 𝐶 𝐶 𝐵 𝐴 𝐼

(i) milg;G tpjp: ngUf;fy; ml;ltizapd; vy;yh cWg;GfSk; 𝐺 ,d; cWg;Gfs;. 𝐺

MdJ . ,d; fPo; milT ngw;Ws;sJ. vdNt milg;G tpjp cz;ik.

(ii) Nrh;g;G tpjp: mzpg;ngUf;fy; nghJthf Nrh;g;G tpjpf;Fl;gLk;.

(iii) rkdp tpjp: 𝐼 I Kd; itj;J vOjg;gl;Ls;s epiuapd; cWg;Gfs;

vy;yhtw;wpw;Fk; NkNyAs;s epiuAlDk; 𝐼 I NkNy itj;J vOjg;gl;Ls;s epuypy; cs;s cWg;Gfs; ,lg;Gw ,Wjpapy; mike;j

epuYld; xd;wp tpLjyhy;> 𝐼 MdJ rkdp cWg;ghFk;.

(iv) vjph;kiw tpjp:

𝐼. 𝐼 = 𝐼 ⇒ 𝐼 ,d; vjph;kiw 𝐼

𝐴. 𝐴 = 𝐼 ⇒ 𝐴 ,d; vjph;kiw 𝐴

𝐵. 𝐵 = 𝐼 ⇒ 𝐵 ,d; vjph;kiw 𝐵

𝐶. 𝐶 = 𝐼 ⇒ 𝐶 ,d; vjph;kiw 𝐶

ml;ltizapypUe;J . ghpkhw;W tpjpf;Fl;gLk;. vdNt 𝐺 MdJ mzpg;ngUf;fypd; fPo; xU vgPypad; FykhFk;

26. tiuaWf;fg;gl;l FwpaPl;bd; gb (𝒁𝟓 −

𝟎 , .𝟓 ) xU Fyk; vd ep&gp.

(JUN-10)

𝐺 = 𝑍5 − 0 = { 1 , 2 , 3 , [4]} Nfa;yp ml;ltizahJ

.5 [1] [2] [3] [4] [1] [1] [2] [3] [4] [2] [2] [4] [1] [3] [3] [3] [1] [4] [2] [4] [4] [3] [2] [1]

ml;ltizapypUe;J

(i) ngUf;fy; ml;ltizapd; vy;yh

cWg;GfSk; 𝐺-,d; cWg;GfshFk;.


(ii) 5- ,d; kl;Lf;fhd ngUf;fy;> Nrh;g;G

tpjpf;Fl;gLk;.

(iii) rkdpAWg;G[1] ∈ 𝐺 kw;Wk; ,U rkdp tpjpiag; G+h;j;jp nra;Ak;

(iv) [1] ,d; vjph;kiw [1], [2] ,d; vjph;kiw

[3],

[3] ,d; vjph;kiw [2] , [4] ,d; vjph;kiw

[4] vdNt vjph;kiw tpjp g+h;j;jpahfpwJ.

∴ (𝑍5 − 0 , .5 ) xU FykhFk;.

27. (𝒁𝟕 − 𝟎 , .𝟕 ) vd;w Fyj;jpy; cs;s xt;nthU cWg;Gf;Fk; thpiriaf; fhz;f

(MAR-12)





nfhLf;fg;gl;l FykhdJ

( 1 , 2 , 3 , 4 , 5 , 6 , .7 )

𝑂 1 = 1 ; 𝑂 2 = 3

𝑂 3 = 6 ; 𝑂 4 = 3

𝑂 5 = 6 ; 𝑂 6 = 2

28. (𝒛𝟔, +𝟔) vd;w Fyj;jpd; vy;yh cWg;Gfspd;

thpiriaf; fhz;f (JUN-08)

𝑧6 = { 0 , 1 , 2 , 3 , 4 , [5]}

𝑂 0 = 1 ; 𝑂 1 = 6

𝑂 2 = 3 ; 𝑂 3 = 2

𝑂 4 = 3 ; 𝑂 5 = 6

29. Fyj;jpd; ePf;fy; tpjpfis vOjp ep&gpf;f.

(MAR-06,MAR-08,OCT-10, MAR-14)

𝐺 xU Fyk; vd;f. 𝑎, 𝑏, 𝑐 ∈ 𝐺 vd;f

(i) 𝑎 ∗ 𝑏 = 𝑎 ∗ 𝑐 ⇒ 𝑏 = 𝑐 (,lJ ePf;fy; tpjp)

(ii) 𝑏 ∗ 𝑎 = 𝑐 ∗ 𝑎 ⇒ 𝑏 = 𝑐 (tyJ ePf;fy; tpjp)

ep&gzk;:

(i) 𝑎 ∗ 𝑏 = 𝑎 ∗ 𝑐 ⇒ 𝑎−1 ∗ 𝑎 ∗ 𝑏 = 𝑎−1 ∗ 𝑎 ∗ 𝑐

⇒ 𝑎−1 ∗ 𝑎 ∗ 𝑏 = (𝑎−1 ∗ 𝑎) ∗ 𝑐

⇒ 𝑒 ∗ 𝑏 = 𝑒 ∗ 𝑐

⇒ 𝑏 = 𝑐

(ii)𝑏 ∗ 𝑎 = 𝑐 ∗ 𝑎 ⇒ 𝑏 ∗ 𝑎 ∗ 𝑎−1 = 𝑐 ∗ 𝑎 ∗ 𝑎−1

⇒ 𝑏 ∗ 𝑎 ∗ 𝑎−1 = 𝑐 ∗ (𝑎 ∗ 𝑎−1)

⇒ 𝑏 ∗ 𝑒 = 𝑐 ∗ 𝑒

⇒ 𝑏 = 𝑐

30. Fyj;jpd; vjph;kiw tpjpapid vOjp ep&gp. (my;yJ)

𝑮 xU Fyk; vd;f. 𝒂, 𝒃 ∈ 𝑮 vd;f. mt;thwhapd; 𝒂 ∗ 𝒃 −𝟏 = 𝒃−𝟏 ∗ 𝒂−𝟏

(MAR-07,10, JUN-09,12,14,15, OCT-12,

15)

𝑏−1 ∗ 𝑎−1 MdJ (𝑎 ∗ 𝑏) ,d; vjph;kiw vdf; fhl;bdhy; NghJkhdJ

(i) 𝑎 ∗ 𝑏 ∗ 𝑏−1 ∗ 𝑎−1 = 𝑒 kw;Wk;

(ii) 𝑏−1 ∗ 𝑎−1 ∗ 𝑎 ∗ 𝑏 = 𝑒 vd ep&gpf;f Ntz;Lk;

(i) 𝑎 ∗ 𝑏 ∗ 𝑏−1 ∗ 𝑎−1 = 𝑎 ∗ 𝑏 ∗ 𝑏−1 ∗ 𝑎−1

= 𝑎 ∗ 𝑒 ∗ 𝑎−1 = 𝑎 ∗ 𝑎−1 = 𝑒

(ii) 𝑏−1 ∗ 𝑎−1 ∗ 𝑎 ∗ 𝑏 = 𝑏−1 ∗ 𝑎−1 ∗ 𝑎 ∗ 𝑏

= 𝑏−1 ∗ 𝑒 ∗ 𝑏

= 𝑏−1∗ 𝑏 = 𝑒

𝑎 ∗ 𝑏 d; vjph;kiw 𝑏−1 ∗ 𝑎−1

𝑎 ∗ 𝑏 −1 = 𝑏−1 ∗ 𝑎−1

31. “xU Fyj;jpd; rkdp cWg;G xUikj; jd;ik

tha;e;jJ”- ep&gpf;f (JUN-13, OCT-16)

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;.

32. “xU Fyj;jpd; xt;nthU cWg;Gk; xNu xU

vjph;kiwiag; ngw;wpUf;Fk;”- ep&gpf;f

(JUN-13, OCT-16)

𝐺 xU Fyk; vd;f. 𝑎 ∈ 𝐺 vd;f

𝑎 ,d; vjph;kiw cWg;Gfs; 𝑎1 , 𝑎2vd;gjhff; nfhs;Nthk;

𝑎1 I 𝑎 ,d; vjph;kiwahff; nfhs;Nthkhapd;

𝑎 ∗ 𝑎1 = 𝑎1 ∗ 𝑎 = 𝑒

𝑎2 I 𝑎 ,d; vjph;kiwahff; nfhs;Nthkhapd;

𝑎 ∗ 𝑎2 = 𝑎2 ∗ 𝑎 = 𝑒

𝑎1 = 𝑎1 ∗ 𝑒 = 𝑎1 ∗ 𝑎 ∗ 𝑎2

= (𝑎1 ∗ 𝑎) ∗ 𝑎2 = 𝑒 ∗ 𝑎2 = 𝑎2 vdNt xU cWg;gpd; vjph;kiw

xUikj;jd;ik tha;e;jjhFk;.

33. xU Fyj;jpd; xt;nthU cWg;Gk; mjd; vjph;kiwahf ,Uf;Fnkdpy; mf;Fyk; xU vgPypad; FykhFk; vd ep&gpf;fTk;.

(MAR-16)

𝐺 xU Fyk; vd;f

𝑎, 𝑏 ∈ 𝐺 vd;f

nfhLf;fg;gl;lit 𝑎 = 𝑎−1 kw;Wk; 𝑏 = 𝑏−1

𝑥 ∈ 𝐺, 𝑥−1 ∈ 𝐺 vd;f

𝑥 = 𝑎𝑏 vd;f

nfhLf;fg;gl;lJ 𝑥 = 𝑥−1

𝑎𝑏 = 𝑎𝑏 −1 = 𝑏−1𝑎−1 = 𝑏𝑎

ghpkhw;W tpjpia epiwT nra;fpwJ.

𝐺 xU vgPypad; FykhFk;.



12 k; tFg;G fzf;F gj;J kjpg;ngz; tpdhf;fs; ntw;wpf;F top


10 kjpg;ngz; tpdhf;fs;

2. ntf;lh; ,aw;fzpjk;

1. xU Kf;Nfhzj;jpd; Fj;Jf;NfhLfs; xNu Gs;spapy; re;jpf;Fk; vd;gjid ntf;lh; Kiwapy; epWTf.

( OCT-06,JUN-08,OCT-13,MAR-15)

∆𝐴𝐵𝐶 ,y; Fj;Jf;NfhLfs; 𝐴𝐷, 𝐵𝐸 Ak; 𝑂,y; re;jpf;fpd;wd. Fj;Jf;NfhLfs; xNu Gs;sp topNar; nry;Yk; vd;gij epWt 𝐶𝑂 MdJ 𝐴𝐵 f;F nrq;Fj;jhf ,Uf;Fk; vd fhl;bdhy; NghJk;. 𝑂 I Mjpahff; nfhs;f. 𝐴, 𝐵, 𝐶,d;

epiy ntf;lh;fs; KiwNa 𝑎 , 𝑏 , 𝑐

𝑂𝐴 = 𝑎 , 𝑂𝐵 = 𝑏 , 𝑂𝐶 = 𝑐

𝐴𝐷 ⊥ 𝐵𝐶

𝑂𝐴 ⊥ 𝐵𝐶

⇒ 𝑂𝐴 . 𝐵𝐶 = 0

⇒ 𝑎 . 𝑐 − 𝑏 = 0

⇒ 𝑎 . 𝑐 − 𝑎 . 𝑏 = 0……………(1)

𝐵𝐸 ⊥ 𝐶𝐴

𝑂𝐵 ⊥ 𝐶𝐴

⇒ 𝑂𝐵 . 𝐶𝐴 = 0

⇒ 𝑏 . ( 𝑎 − 𝑐 ) = 0

⇒ 𝑏 . 𝑎 − 𝑏 . 𝑐 = 0…………….(2)

(1) kw;Wk; (2) If; $l;l

𝑎 . 𝑐 − 𝑎 . 𝑏 + 𝑏 . 𝑎 − 𝑏 . 𝑐 = 0

𝑎 . 𝑐 − 𝑏 . 𝑐 = 0

(𝑎 − 𝑏 ). 𝑐 = 0

⇒ 𝐵𝐴 . 𝑂𝐶 = 0

⇒ 𝑂𝐶 ⊥ 𝐴𝐵

vdNt %d;W Fj;Jf; NfhLfSk; xNu Gs;spapy; re;jpf;Fk; NfhLfshFk;.

2. 𝒂 = 𝟐𝒊 + 𝟑𝒋 − 𝒌 , 𝒃 = −𝟐𝒊 + 𝟓𝒌 , 𝒄 = 𝒋 − 𝟑𝒌

vdpy; 𝒂 × 𝒃 × 𝒄 = (𝒂 . 𝒄 )𝒃 − 𝒂 . 𝒃 𝒄 vd rhpghh;f;f. ( MAR-07,OCT-08,OCT-09,JUN-16)

𝑏 × 𝑐 = 𝑖 𝑗 𝑘

−2 0 50 1 −3

= −5𝑖 − 6𝑗 − 2𝑘

𝑎 × 𝑏 × 𝑐 = 𝑖 𝑗 𝑘

2 3 −1−5 −6 −2

= −12𝑖 + 9𝑗 + 3𝑘

(𝑎 . 𝑐 ) = 2𝑖 + 3𝑗 − 𝑘 . 𝑗 − 3𝑘

= 2(0) + 3(1) − 1(−3)

= 3 + 3 = 6

(𝑎 . 𝑐 )𝑏 = 6 −2𝑖 + 5𝑘 = −12𝑖 + 30𝑘

𝑎 . 𝑏 = 2𝑖 + 3𝑗 − 𝑘 . −2𝑖 + 5𝑘

= 2(−2) + 3(0) − 1(5)

= −4 − 5 = −9

𝑎 . 𝑏 𝑐 = −9 𝑗 − 3𝑘 = −9𝑗 + 27𝑘

(𝑎 . 𝑐 )𝑏 − 𝑎 . 𝑏 𝑐 = −12𝑖 + 30𝑘 + 9𝑗 − 27𝑘

(𝑎 . 𝑐 )𝑏 − 𝑎 . 𝑏 𝑐 = −12𝑖 + 9𝑗 + 3𝑘

vdNt, 𝑎 × 𝑏 × 𝑐 = (𝑎 . 𝑐 )𝑏 − 𝑎 . 𝑏 𝑐

3. 𝒂 = 𝒊 + 𝒋 + 𝒌 , 𝒃 = 𝟐𝒊 + 𝒌 , 𝒄 = 𝟐𝒊 + 𝒋 + 𝒌 ,

𝒅 = 𝒊 + 𝒋 + 𝟐𝒌 vdpy;

𝒂 × 𝒃 × 𝒄 × 𝒅 = 𝒂 𝒃 𝒅 𝒄 − 𝒂 𝒃 𝒄 𝒅

vd;gijr; rhpghh;f;f. ( MAR-09,OCT-11,OCT-12,MAR-16)

𝑎 × 𝑏 = 𝑖 𝑗 𝑘

1 1 12 0 1

= 𝑖 + 𝑗 − 2𝑘

𝑐 × 𝑑 = 𝑖 𝑗 𝑘

2 1 11 1 2

= 𝑖 − 3𝑗 + 𝑘

𝑎 × 𝑏 × 𝑐 × 𝑑 = 𝑖 𝑗 𝑘

1 1 −21 −3 1

= −5𝑖 − 3𝑗 − 4𝑘 …………….(1)

𝑎 𝑏 𝑐 = 1 1 12 0 12 1 1

= 1

𝑎 𝑏 𝑑 = 1 1 12 0 11 1 2

= −2

𝑎 𝑏 𝑑 𝑐 − 𝑎 𝑏 𝑐 𝑑

= −2 2𝑖 + 𝑗 + 𝑘 − 1(𝑖 + 𝑗 + 2𝑘 )

= −4𝑖 − 2𝑗 − 2𝑘 − 𝑖 − 𝑗 − 2𝑘

= −5𝑖 − 3𝑗 − 4𝑘 ……………………..(2)

(1) kw;Wk; (2) ypUe;J

𝑎 × 𝑏 × 𝑐 × 𝑑 = 𝑎 𝑏 𝑑 𝑐 − 𝑎 𝑏 𝑐 𝑑



12 k; tFg;G fzf;F ntw;wpf;F top


4. 𝐜𝐨𝐬 𝑨 − 𝑩 = 𝐜𝐨𝐬𝑨 𝐜𝐨𝐬𝑩 + 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩 vd

epWTf. (JUN-12,JUN-13)

5. 𝐜𝐨𝐬 𝑨 + 𝑩 = 𝐜𝐨𝐬𝑨 𝐜𝐨𝐬𝑩 − 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩 vd epWTf( MAR-06,08,14,17, JUN-11, OCT-14)

6. 𝐬𝐢𝐧 𝑨 + 𝑩 = 𝐬𝐢𝐧 𝑨 𝐜𝐨𝐬 𝑩 + 𝐜𝐨𝐬𝑨 𝐬𝐢𝐧 𝑩 vd epWTf (OCT-08,MAR-11,13,JUN-14)

7. 𝐬𝐢𝐧 𝑨 − 𝑩

= 𝐬𝐢𝐧 𝑨 𝐜𝐨𝐬𝑩 − 𝐜𝐨𝐬𝑨 𝐬𝐢𝐧 𝑩

vd ntf;lh; Kiwapy; ep&gp

( JUN-07, MAR-12, OCT-07,10, 15,16)

1

2 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

3 ∴ ∠𝑃𝑂𝑄 = ∠𝑃𝑂𝑥 − ∠𝑄𝑂𝑥 = 𝐴 − 𝐵 ∴ ∠𝑃𝑂𝑄 = ∠𝑃𝑂𝑥 + ∠𝑄𝑂𝑥 = 𝐴 + 𝐵 ∴ ∠𝑃𝑂𝑄 = ∠𝑃𝑂𝑥 + ∠𝑄𝑂𝑥 = 𝐴 + 𝐵 ∴ ∠𝑃𝑂𝑄 = ∠𝑃𝑂𝑥 − ∠𝑄𝑂𝑥 = 𝐴 − 𝐵

4 𝑃 , 𝑄 d; Maj;njhiyfs; KiwNa (cos 𝐴, sin 𝐴) kw;Wk; (cos 𝐵, sin 𝐵)

𝑃 , 𝑄 d; Maj;njhiyfs; KiwNa (cos 𝐴, sin 𝐴)kw;Wk; (cos 𝐵, −sin 𝐵)

𝑃 , 𝑄 d; mr;RJ}uq;fs; KiwNa (cos 𝐴, sin 𝐴)kw;Wk;(cos 𝐵, −sin 𝐵)

𝑃 , 𝑄 d; Maj;njhiyfs; KiwNa (cos 𝐴, sin 𝐴) kw;Wk; (cos 𝐵, sin 𝐵).

5 𝑖 , 𝑗 vd;w myF ntf;lh;fis 𝑥, 𝑦 mr;Rj; jpirfspy; vLj;Jf; nfhs;f.

6 𝑂𝑃 = 𝑂𝑀 + 𝑀𝑃 = cos 𝐴𝑖 +sin 𝐴 𝑗

𝑂𝑄 = 𝑂𝐿 + 𝐿𝑄 = cos 𝐵𝑖 +sin 𝐵 𝑗

𝑂𝑃 = 𝑂𝑀 + 𝑀𝑃 = cos 𝐴𝑖 +sin 𝐴 𝑗

𝑂𝑄 = 𝑂𝑁 + 𝑁𝑄 = cos 𝐵𝑖 −sin 𝐵 𝑗


𝑂𝑄 = 𝑂𝑁 + 𝑁𝑄 = cos 𝐵𝑖 −sin 𝐵 𝑗


𝑂𝑄 = 𝑂𝐿 + 𝐿𝑄 = cos 𝐵𝑖 +sin 𝐵 𝑗

7 𝑂𝑃 . 𝑂𝑄 = cos 𝐴𝑖 +sin 𝐴 𝑗 . cos 𝐵𝑖 +sin 𝐵 𝑗 = cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵……. 1

𝑂𝑃 . 𝑂𝑄

= cos 𝐴𝑖 +sin 𝐴 𝑗 . cos 𝐵𝑖 −sin 𝐵 𝑗

= cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵……. 1

𝑂𝑄 × 𝑂𝑃 = 𝑂𝑄 . 𝑂𝑃 sin 𝐴 + 𝐵 𝑘

= sin(𝐴 + 𝐵) 𝑘 ……… 1

𝑂𝑄 × 𝑂𝑃 = 𝑂𝑄 . 𝑂𝑃 sin 𝐴 − 𝐵 𝑘

= sin(𝐴 − 𝐵) 𝑘 ……… 1

8 tiuaiwapd;gb,

𝑂𝑃 . 𝑂𝑄 = 𝑂𝑃 . 𝑂𝑄 cos(𝐴 − 𝐵)

= cos (𝐴 − 𝐵)……….. 2

tiuaiwapd;gb,

𝑂𝑃 . 𝑂𝑄 = 𝑂𝑃 . 𝑂𝑄 cos(𝐴 + 𝐵)

= cos (𝐴 + 𝐵)………….. 2

tiuaiwapd;gb,

𝑂𝑄 × 𝑂𝑃 = 𝑖 𝑗 𝑘

cos 𝐵 − sin 𝐵 0cos 𝐴 sin 𝐴 0

= 𝑘 (sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵)… 2

tiuaiwapd;gb,

𝑂𝑄 × 𝑂𝑃 = 𝑖 𝑗 𝑘

cos 𝐵 sin 𝐵 0cos 𝐴 sin 𝐴 0

= 𝑘 (sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵)....(2)

9 (1) kw;Wk; (2) ypUe;J cos 𝐴 − 𝐵

= cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵


cos 𝐴 + 𝐵 = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵


sin 𝐴 + 𝐵 = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵


sin 𝐴 − 𝐵 = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵





8. 𝒙−𝟏

𝟑=

𝒚−𝟏

−𝟏=

𝒛+𝟏

𝟎 kw;Wk;

𝒙−𝟒

𝟐=

𝒚

𝟎=

𝒛+𝟏

𝟑 vd;w

NfhLfs; ntl;Lk; vdf; fhl;b mit ntl;Lk; Gs;spiaf; fhz;f. ( JUN-07,JUN-09,JUN-15 )

NfhLfs; ntl;bf;nfhs;tjw;fhd epge;jid

𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1

𝑙1 𝑚1 𝑛1

𝑙2 𝑚2 𝑛2

= 0

𝑥−𝑥1

𝑙1=

𝑦−𝑦1

𝑚1=

𝑧−𝑧1

𝑛1 kw;Wk;

𝑥−𝑥2

𝑙2=

𝑦−𝑦2

𝑚2=

𝑧−𝑧2

𝑛2

cld; xg;gpl fpilg;gJ

𝑥1 = 1 𝑥2 = 4 𝑙1 = 3 𝑙2 = 2

𝑦1 = 1 𝑦2 = 0 𝑚1 = −1 𝑚2 = 0

𝑧1 = −1 𝑧2 = −1 𝑛1 = 0 𝑛2 = 3

4 − 1 0 − 1 −1 + 1

3 −1 02 0 3

= 3 −1 03 −1 02 0 3

= 3(−3 − 0) + 1(9 − 0) + 0(0 + 2)

= −9 + 9 = 0

∴ Nkw;Fwpg;gpl;l NfhLfs; xd;iwnahd;W ntl;bf; nfhs;fpd;wd.

ntl;Lk; Gs;sp:

𝑥−1

3=

𝑦−1

−1=

𝑧+1

0= 𝜆 vd;f

𝑥−1

3= 𝜆

𝑦−1

−1= 𝜆

𝑧+1

0= 𝜆

𝑥 − 1 = 3𝜆 𝑦 − 1 = −𝜆 𝑧 + 1 = 0

𝑥 = 3𝜆 + 1 𝑦 = −𝜆 + 1 𝑧 = −1

,e;j Nfhl;by; mike;Js;s VNjDk; xU

Gs;spapd; mikg;G (3𝜆 + 1, −𝜆 + 1, −1)

𝑥−4

2=

𝑦

0=

𝑧+1

3= 𝜇 vd;f

𝑥−4

2= 𝜇

𝑦

0= 𝜇

𝑧+1

3= 𝜇

𝑥 − 4 = 2𝜇 𝑦 = 0 𝑧 + 1 = 3𝜇

𝑥 = 2𝜇 + 4 𝑦 = 0 𝑧 = 3𝜇 − 1


Gs;spapd; mikg;G (2𝜇 + 4,0,3𝜇 − 1)

,it ntl;bf;nfhs;tjhy; VNjDk; 𝜆, 𝜇 f;F

(3𝜆 + 1, −𝜆 + 1, −1) = (2𝜇 + 4,0,3𝜇 − 1)

3𝜆 + 1 = 2𝜇 + 4

−𝜆 + 1 = 0 ⇒ 𝜆 = 1

−1 = 3𝜇 − 1 ⇒ 𝜇 = 0

∴ ntl;Lk; Gs;sp (4,0,−1)

9. 𝒙−𝟏

𝟏=

𝒚+𝟏

−𝟏=

𝒛

𝟑 kw;Wk;

𝒙−𝟐

𝟏=

𝒚−𝟏

𝟐=

−𝒛−𝟏

𝟏 vd;w

NfhLfs; ntl;bf; nfhs;Sk; vdf; fhl;Lf.

NkYk; mit ntl;Lk; Gs;spiaf; fhz;f.

( JUN-06,JUN-10,JUN-11)

NfhLfs; ntl;bf;nfhs;tjw;fhd epge;jid

𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1

𝑙1 𝑚1 𝑛1

𝑙2 𝑚2 𝑛2

= 0

𝑥−𝑥1

𝑙1=

𝑦−𝑦1

𝑚1=

𝑧−𝑧1

𝑛1 kw;Wk;

𝑥−𝑥2

𝑙2=

𝑦−𝑦2

𝑚2=

𝑧−𝑧2

𝑛2

cld; xg;gpl fpilg;gJ,

𝑥1 = 1 𝑥2 = 2 𝑙1 = 1 𝑙2 = 1

𝑦1 = −1 𝑦2 = 1 𝑚1 = −1 𝑚2 = 2

𝑧1 = 0 𝑧2 = −1 𝑛1 = 3 𝑛2 = −1

2 − 1 1 + 1 −1 + 0

1 −1 31 2 −1

= 1 2 −11 −1 31 2 −1

= 1(1 − 6) − 2(−1 − 3) − 1(2 + 1)

= −5 + 8 − 3 = 0

∴ Nkw;Fwpg;gpl;l NfhLfs; xd;iwnahd;W ntl;bf; nfhs;fpd;wd.

ntl;Lk; Gs;sp:

𝑥−1

1=

𝑦+1

−1=

𝑧

3= 𝜆 vd;f

𝑥−1

1= 𝜆

𝑦+1

−1= 𝜆

𝑧

3= 𝜆

𝑥 − 1 = 𝜆 𝑦 + 1 = −𝜆 𝑧 = 3𝜆

𝑥 = 𝜆 + 1 𝑦 = −𝜆 − 1 𝑧 = 3𝜆


Gs;spapd; mikg;G (𝜆 + 1, −𝜆 − 1, 3𝜆)

𝑥−2

1=

𝑦−1

2=

𝑧+1

−1= 𝜇 vd;f

𝑥−2

1= 𝜇

𝑦−1

2= 𝜇

𝑧+1

−1= 𝜇

𝑥 − 2 = 𝜇 𝑦 − 1 = 2𝜇 𝑧 + 1 = −𝜇

𝑥 = 𝜇 + 2 𝑦 = 2𝜇 + 1 𝑧 = −𝜇 − 1


Gs;spapd; mikg;G (𝜇 + 2, 2𝜇 + 1, −𝜇 − 1)

,it ntl;bf;nfhs;tjhy; VNjDk; 𝜆, 𝜇 f;F

(𝜆 + 1, −𝜆 − 1, 3𝜆) = (𝜇 + 2, 2𝜇 + 1, −𝜇 − 1)

𝜆 + 1 = 𝜇 + 2…………….(1)





−𝜆 − 1 = 2𝜇 + 1………….(2)

3𝜆 = −𝜇 − 1………………..(3)

(1) kw;Wk; (2) I jPh;f;f

3𝜇 + 3 = 0

3𝜇 = −3

𝜇 = −1

𝜇 = −1 I (1)y; gpujpapl

𝜆 + 1 = −1 + 2

𝜆 + 1 = 1

𝜆 = 0

∴ ntl;Lk; Gs;sp (1 − 1,0)

10. (𝟐, −𝟏, −𝟑) topNa nry;yf;$baJk;

𝒙−𝟐

𝟑=

𝒚−𝟏

𝟐=

𝒛−𝟑

−𝟒 kw;Wk;

𝒙−𝟏

𝟐=

𝒚+𝟏

−𝟑=

𝒛−𝟐

𝟐Mfpa

NfhLfSf;F ,izahf cs;sJkhd jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf;

fhz;f (MAR-10,OCT-11,16, JUN-16)

ntf;lh; rkd;ghL:

Njitahd jskhdJ 𝐴(2, −1, −3) topNa

nry;;Yk;. NkYk; 𝑢 = 3𝑖 + 2𝑗 − 4𝑘 kw;Wk;

𝑣 = 2𝑖 − 3𝑗 + 2𝑘 f;F ,izahf ,Uf;Fk;

𝑎 = 2𝑖 − 𝑗 − 3𝑘

Njitahd rkd;ghL 𝑟 = 𝑎 + 𝑠𝑢 + 𝑡𝑣

𝑟 = 2𝑖 − 𝑗 − 3𝑘 + 𝑠 3𝑖 + 2𝑗 − 4𝑘

+𝑡(2𝑖 − 3𝑗 + 2𝑘 )

fhh;Brpad; mikg;G:

jsj;jpd; rkd;ghL

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1

𝑙1 𝑚1 𝑛1

𝑙2 𝑚2 𝑛2

= 0

𝑥1 = 2 𝑙1 = 3 𝑙2 = 2

𝑦1 = −1 𝑚1 = 2 𝑚2 = −3

𝑧1 = −3 𝑛1 = −4 𝑛2 = 2

𝑥 − 2 𝑦 + 1 𝑧 + 3

3 2 −42 −3 2

= 0

(𝑥 − 2)(4 − 12) − (𝑦 + 1)(6 + 8)

+(𝑧 + 3)(−9 − 4) = 0

(𝑥 − 2)(−8) − (𝑦 + 1)(14) +(𝑧 + 3)(−13) = 0

−8𝑥 + 16 − 14𝑦 − 14 − 13𝑧 − 39 = 0

−8𝑥 − 14𝑦 − 13𝑧 + 16 − 53 = 0

8𝑥 + 14𝑦 + 13𝑧 + 37 = 0

,JNt Njitahd rkd;ghl;bd; fhh;Brpad;

mikg;G MFk;.

11. (−𝟏, 𝟏, 𝟏) kw;Wk; (𝟏, −𝟏, 𝟏) Mfpa Gs;spfs;

topNar; nry;yf; $baJk; 𝒙 + 𝟐𝒚 + 𝟐𝒛 = 𝟓 vd;w jsj;jpw;F nrq;Fj;jhf miktJkhd jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghl;ilf; fhz;f (MAR-07, 09,JUN-10,OCT-14)

ntf;lh; rkd;ghL:

nfhLf;fg;gl;l ,uz;L Gs;spfs; topNar; nry;yf;$baJk; xU ntf;lUf;F ,izahf cs;sJkhd jsj;jpd; ntf;lh; rkd;ghL

𝑟 = (1 − 𝑠)𝑎 + 𝑠𝑏 + 𝑡𝑣 , ,q;F

𝑎 = −𝑖 + 𝑗 + 𝑘 , 𝑏 = 𝑖 − 𝑗 + 𝑘 , 𝑣 = 𝑖 + 2𝑗 + 2𝑘

𝑟 = (1 − 𝑠) −𝑖 + 𝑗 + 𝑘 + 𝑠 𝑖 − 𝑗 + 𝑘

+𝑡(𝑖 + 2𝑗 + 2𝑘 )

fhh;Brpad; mikg;G:

jsj;jpd; rkd;ghL

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1

𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1

𝑙 𝑚 𝑛 = 0

𝑥1 = −1 𝑥2 = 1 𝑙1 = 1

𝑦1 = 1 𝑦2 = −1 𝑚1 = 2

𝑧1 = 1 𝑧2 = 1 𝑛1 = 2

𝑥 + 1 𝑦 − 1 𝑧 − 1

2 −2 01 2 2

= 0

(𝑥 + 1)(−4 − 0) − (𝑦 − 1)(4 − 0)

+ (𝑧 − 1)(4 + 2) = 0

(𝑥 + 1)(−4) − (𝑦 − 1)(4) + (𝑧 − 1)(6) = 0

−4𝑥 − 4 − 4𝑦 + 4 + 6𝑧 − 6 = 0

−4𝑥 − 4𝑦 + 6𝑧 − 6 = 0

4𝑥 + 4𝑦 − 6𝑧 + 6 = 0

÷ 2 2𝑥 + 2𝑦 − 3𝑧 + 3 = 0

12. (𝟐, 𝟐, −𝟏) , (𝟑, 𝟒, 𝟐) kw;Wk; (𝟕, 𝟎, 𝟔) Mfpa Gs;spfs; topNar; nry;yf; $ba jsj;jpd; ntf;lh; kw;Wk; kw;Wk; fhh;Brpad; rkd;ghl;ilf;

fhz;f (OCT-09)

ntf;lh; rkd;ghL:

xNu Nfhl;likahj nfhLf;fg;gl;l %d;W Gs;spfs; topNar; nry;Yk; jsj;jpd; ntf;lh;

rkd;ghL 𝑟 = (1 − 𝑠 − 𝑡)𝑎 + 𝑠𝑏 + 𝑡𝑐





,q;F 𝑎 = 2𝑖 + 2𝑗 − 𝑘 , 𝑏 = 3𝑖 + 4𝑗 + 2𝑘 ,

𝑐 = 7𝑖 + 6𝑘

𝑟 = (1 − 𝑠 − 𝑡) 2𝑖 + 2𝑗 − 𝑘

+𝑠(3𝑖 + 4𝑗 + 2𝑘 ) + 𝑡(7𝑖 + 6𝑘 )

fhh;Brpad; mikg;G:

jsj;jpd; rkd;ghL

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1

𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1

𝑥3 − 𝑥1 𝑦3 − 𝑦1 𝑧3 − 𝑧1

= 0

𝑥1 = 2 𝑥2 = 3 𝑥3 = 7

𝑦1 = 2 𝑦2 = 4 𝑦3 = 0

𝑧1 = −1 𝑧2 = 2 𝑧3 = 6

𝑥 − 2 𝑦 − 2 𝑧 + 1

1 2 35 −2 7

= 0

(𝑥 − 2)(14 + 6) − (𝑦 − 2)(7 − 15)

+(𝑧 + 1)(−2 − 10) = 0

(𝑥 − 2)(20) − (𝑦 − 2)(−8) +(𝑧 + 1)(−12) = 0

20𝑥 − 40 + 8𝑦 − 16 − 12𝑧 − 12 = 0

20𝑥 + 8𝑦 − 12𝑧 − 68 = 0

÷ 4 5𝑥 + 2𝑦 − 3𝑧 − 17 = 0

13. 𝒙−𝟐

𝟐=

𝒚−𝟐

𝟑=

𝒛−𝟏

𝟑 vd;w Nfhl;il

cs;slf;fpaJk; 𝒙+𝟏

𝟑=

𝒚−𝟏

𝟐=

𝒛+𝟏

𝟏 vd;w

Nfhl;bw;F ,izahdJkhd jsj;jpd; ntf;lh;

kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f(MAR-14)

ntf;lh; rkd;ghL:

𝑢 = 2𝑖 + 3𝑗 + 3𝑘 kw;Wk; 𝑣 = 3𝑖 + 2𝑗 + 𝑘

𝑎 = 2𝑖 + 2𝑗 + 𝑘


𝑟 = 2𝑖 + 2𝑗 + 𝑘 + 𝑠 2𝑖 + 3𝑗 + 3𝑘 + 𝑡(3𝑖 + 2𝑗 + 𝑘 )

fhh;Brpad; mikg;G:

jsj;jpd; rkd;ghL

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1

𝑙1 𝑚1 𝑛1

𝑙2 𝑚2 𝑛2

= 0

𝑥1 = 2 𝑙1 = 2 𝑙2 = 3

𝑦1 = 2 𝑚1 = 3 𝑚2 = 2

𝑧1 = 1 𝑛1 = 3 𝑛2 = 1

𝑥 − 2 𝑦 − 2 𝑧 − 1

2 3 33 2 1

= 0

(𝑥 − 2)(3 − 6) − (𝑦 − 2)(2 − 9)

+(𝑧 − 1)(4 − 9) = 0

(𝑥 − 2)(−3) − (𝑦 − 2)(−7) +(𝑧 − 1)(−5) = 0

−3𝑥 + 6 + 7𝑦 − 14 − 5𝑧 + 5 = 0

−3𝑥 + 7𝑦 − 5𝑧 − 3 = 0

3𝑥 − 7𝑦 + 5𝑧 + 3 = 0

,JNt Njitahd fhh;Brpad; rkd;ghlhFk;.

14. (1, 3, 2) vd;w Gs;sp topr; nry;tJk;

𝒙+𝟏

𝟐=

𝒚+𝟐

−𝟏=

𝒛+𝟑

𝟑 kw;Wk;

𝒙−𝟐

𝟏=

𝒚+𝟏

𝟐=

𝒛+𝟐

𝟐 vd;w

NfhLfSf;F ,izahdJkhd jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf;

fhz;f (JUN-12)

ntf;lh; rkd;ghL:

𝑢 = 2𝑖 − 𝑗 + 3𝑘 kw;Wk; 𝑣 = 𝑖 + 2𝑗 + 2𝑘

𝑎 = 𝑖 + 3𝑗 + 2𝑘


𝑟 = 𝑖 + 3𝑗 + 2𝑘 + 𝑠 2𝑖 − 𝑗 + 3𝑘 + 𝑡(𝑖 + 2𝑗 + 2𝑘 )

fhh;Brpad; mikg;G:

jsj;jpd; rkd;ghL

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1

𝑙1 𝑚1 𝑛1

𝑙2 𝑚2 𝑛2

= 0

𝑥1 = 1 𝑙1 = 2 𝑙2 = 1

𝑦1 = 3 𝑚1 = −1 𝑚2 = 2

𝑧1 = 2 𝑛1 = 3 𝑛2 = 2

𝑥 − 1 𝑦 − 3 𝑧 − 2

2 −1 31 2 2

= 0

(𝑥 − 1)(−2 − 6) − (𝑦 − 3)(4 − 3) +

(𝑧 − 2)(4 + 1) = 0

(𝑥 − 1)(−8) − (𝑦 − 3)(1) +(𝑧 − 2)(5) = 0

−8𝑥 + 8 − 𝑦 + 3 + 5𝑧 − 10 = 0

−8𝑥 − 𝑦 + 5𝑧 + 1 = 0

8𝑥 + 𝑦 − 5𝑧 − 1 = 0





15. (−𝟏, 𝟑, 𝟐) vd;w Gs;sp topr; nry;tJk;

𝒙 + 𝟐𝒚 + 𝟐𝒛 = 𝟓 kw;Wk; 𝟑𝒙 + 𝒚 + 𝟐𝒛 = 𝟖 Mfpa jsq;fSf;Fr; nrq;Fj;jhdJkhd jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad;

rkd;ghLfis fhz;f. (JUN-13)

ntf;lh; rkd;ghL:


𝑢 = 𝑖 + 2𝑗 + 2𝑘 kw;Wk; 𝑣 = 3𝑖 + 𝑗 + 2𝑘

𝑎 = −𝑖 + 3𝑗 + 2𝑘

𝑟 = −𝑖 + 3𝑗 + 2𝑘 + 𝑠 𝑖 + 2𝑗 + 2𝑘 + 𝑡(3𝑖 + 𝑗 + 2𝑘 )

fhh;Brpad; mikg;G: jsj;jpd; rkd;ghL

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1

𝑙1 𝑚1 𝑛1

𝑙2 𝑚2 𝑛2

= 0

𝑥1 = −1 𝑙1 = 1 𝑙2 = 3

𝑦1 = 3 𝑚1 = 2 𝑚2 = 1

𝑧1 = 2 𝑛1 = 2 𝑛2 = 2

𝑥 + 1 𝑦 − 3 𝑧 − 2

1 2 23 1 2

= 0

(𝑥 + 1)(4 − 2) − (𝑦 − 3)(2 − 6) + (𝑧 − 2)(1 − 6) = 0

(𝑥 + 1)(2) − (𝑦 − 3)(−4)

+(𝑧 − 2)(−5) = 0

2𝑥 + 2 + 4𝑦 − 12 − 5𝑧 + 10 = 0

2𝑥 + 4𝑦 − 5𝑧 = 0

16. 𝑨(𝟏, −𝟐, 𝟑) kw;Wk; 𝑩(−𝟏, 𝟐, −𝟏) vd;w Gs;spfs;

topNar; nry;yf;$baJk; 𝒙−𝟐

𝟐=

𝒚+𝟏

𝟑=

𝒛−𝟏

𝟒

vd;w Nfhl;bw;F ,izahdJkhd jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf;

fhz;f ( JUN-14 )

ntf;lh; rkd;ghL:

Njitahd rkd;ghL 𝑟 = (1 − 𝑠)𝑎 + 𝑠𝑏 + 𝑡𝑣

,q;F 𝑎 = 𝑖 − 2𝑗 + 3𝑘 , 𝑏 = −𝑖 + 2𝑗 − 𝑘 ,

𝑣 = 2𝑖 + 3𝑗 + 4𝑘

𝑟 = (1 − 𝑠) 𝑖 − 2𝑗 + 3𝑘 + 𝑠 −𝑖 + 2𝑗 − 𝑘

+𝑡(2𝑖 + 3𝑗 + 4𝑘 )

fhh;Brpad; mikg;G:

jsj;jpd; rkd;ghL

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1

𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1

𝑙 𝑚 𝑛 = 0

𝑥1 = 1 𝑥2 = −1 𝑙1 = 2

𝑦1 = −2 𝑦2 = 2 𝑚1 = 3

𝑧1 = 3 𝑧2 = −1 𝑛1 = 4

𝑥 − 1 𝑦 + 2 𝑧 − 3−2 4 −42 3 4

= 0

(𝑥 − 1)(16 + 12) − (𝑦 + 2)(−8 + 8) +

(𝑧 − 3)(−6 − 8) = 0

(𝑥 − 1)(28) − (𝑦 + 2)(0) + (𝑧 − 3)(−14) = 0 28𝑥 − 28 − 14𝑧 + 42 = 0

28𝑥 − 14𝑧 + 14 = 0

÷ 14 2𝑥 − 𝑧 + 1 = 0

17. (𝟏, 𝟐, 𝟑) kw;Wk; (𝟐, 𝟑, 𝟏) vd;w Gs;spfs;

topNar; nry;yf; $baJk; 𝟑𝒙 − 𝟐𝒚 + 𝟒𝒛 − 𝟓 = 𝟎 vd;w jsj;jpw;Fr; nrq;Fj;jhfTk; mike;j jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f.

(MAR-06,12, OCT-06,07,15, JUN-08, 15)

ntf;lh; rkd;ghL:


,q;F 𝑎 = 𝑖 + 2𝑗 + 3𝑘 , 𝑏 = 2𝑖 + 3𝑗 + 𝑘 ,

𝑣 = 3𝑖 − 2𝑗 + 4𝑘

𝑟 = (1 − 𝑠) 𝑖 + 2𝑗 + 3𝑘 + 𝑠 2𝑖 + 3𝑗 + 𝑘

+𝑡(3𝑖 − 2𝑗 + 4𝑘 )

fhh;Brpad; mikg;G:

jsj;jpd; rkd;ghL

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1

𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1

𝑙 𝑚 𝑛 = 0

𝑥1 = 1 𝑥2 = 2 𝑙1 = 3

𝑦1 = 2 𝑦2 = 3 𝑚1 = −2

𝑧1 = 3 𝑧2 = 1 𝑛1 = 4

𝑥 − 1 𝑦 − 2 𝑧 − 3

1 1 −23 −2 4

= 0

(𝑥 − 1)(4 − 4) − (𝑦 − 2)(4 + 6) +

(𝑧 − 3)(−2 − 3) = 0

(𝑥 − 1)(0) − (𝑦 − 2)(10) + (𝑧 − 3)(−5) = 0





−10𝑦 + 20 − 5𝑧 + 15 = 0

10𝑦 + 5𝑧 − 35 = 0

÷ 5 2𝑦 + 𝑧 − 7 = 0

18. 𝒙−𝟐

𝟐=

𝒚−𝟐

𝟑=

𝒛−𝟏

−𝟐 vd;w Nfhl;il

cs;slf;fpaJk; (−𝟏, 𝟏, −𝟏) vd;w Gs;sp topNar; nry;yf; $baJkhd jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f. (MAR-11,16 )

ntf;lh; rkd;ghL:


,q;F 𝑎 = −𝑖 + 𝑗 − 𝑘 , 𝑏 = 2𝑖 + 2𝑗 + 𝑘 ,

𝑣 = 2𝑖 + 3𝑗 − 2𝑘

𝑟 = (1 − 𝑠) −𝑖 + 𝑗 − 𝑘 + 𝑠 2𝑖 + 2𝑗 + 𝑘

+𝑡(2𝑖 + 3𝑗 − 2𝑘 )

fhh;Brpad; mikg;G:

jsj;jpd; rkd;ghL

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1

𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1

𝑙 𝑚 𝑛 = 0

𝑥1 = −1 𝑥2 = 2 𝑙1 = 2

𝑦1 = 1 𝑦2 = 2 𝑚1 = 3

𝑧1 = −1 𝑧2 = 1 𝑛1 = −2

𝑥 + 1 𝑦 − 1 𝑧 + 1

3 1 22 3 −2

= 0

(𝑥 + 1)(−2 − 6) − (𝑦 − 1)(−6 − 4) +

(𝑧 + 1)(9 − 2) = 0

(𝑥 + 1)(−8) − (𝑦 − 1)(−10) + (𝑧 + 1)(7) = 0

−8𝑥 − 8 + 10𝑦 − 10 + 7𝑧 + 7 = 0

−8𝑥 + 10𝑦 + 7𝑧 − 11 = 0

8𝑥 − 10𝑦 − 7𝑧 + 11 = 0

19. 𝟑𝒊 + 𝟒𝒋 + 𝟐𝒌 , 𝟐𝒊 − 𝟐𝒋 − 𝒌 kw;Wk; 𝟕𝒊 + 𝒌 Mfpatw;iw epiy ntf;lh;fshff; nfhz;l Gs;spfs; topNar; nry;Yk; jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f

( JUN-09, MAR-13,OCT-13, MAR-17)

ntf;lh; rkd;ghL: Njitahd rkd;ghL

𝑟 = (1 − 𝑠 − 𝑡)𝑎 + 𝑠𝑏 + 𝑡𝑐

,q;F 𝑎 = 3𝑖 + 4𝑗 + 2𝑘 , 𝑏 = 2𝑖 − 2𝑗 − 𝑘 ,

𝑐 = 7𝑖 + 𝑘

𝑟 = (1 − 𝑠 − 𝑡) 3𝑖 + 4𝑗 + 2𝑘

+𝑠(2𝑖 − 2𝑗 − 𝑘 ) + 𝑡(7𝑖 + 𝑘 )

fhh;Brpad; mikg;G:

jsj;jpd; rkd;ghL

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1

𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1

𝑥3 − 𝑥1 𝑦3 − 𝑦1 𝑧3 − 𝑧1

= 0

𝑥1 = 3 𝑥2 = 2 𝑥3 = 7

𝑦1 = 4 𝑦2 = −2 𝑦3 = 0

𝑧1 = 2 𝑧2 = −1 𝑧3 = 1

𝑥 − 3 𝑦 − 4 𝑧 − 2−1 −6 −34 −4 −1

= 0

(𝑥 − 3)(6 − 12) − (𝑦 − 4)(1 + 12) +

(𝑧 − 2)(4 + 24) = 0

(𝑥 − 3)(−6) − (𝑦 − 4)(13) +(𝑧 − 2)(28) = 0

−6𝑥 + 18 − 13𝑦 + 52 + 28𝑧 − 56 = 0

−6𝑥 − 13𝑦 + 28𝑧 + 14 = 0

6𝑥 + 13𝑦 − 28𝑧 − 14 = 0

20. ntl;Lj;Jz;L tbtpy; xU jsj;jpd;

rkd;ghl;ilj; jUtpf;f (MAR-10, MAR-15)

Fwpg;G: jsj;jpd; rkd;ghl;il fhh;Brpad; mikg;gpy; my;yJ ntf;lh; mikg;gpy; jUtpf;fyhk;. ,it ,uz;LNk ntf;lh; KiwahFk;.

fhh;Brpad; mikg;G:

𝑎, 𝑏 kw;Wk; 𝑐 vd;gd KiwNa 𝑥, 𝑦 kw;Wk; 𝑧 d; ntl;Lj;Jz;Lfs;

∴jskhdJ (𝑎, 0,0), (0, 𝑏, 0), (0,0, 𝑐) vd;w Gs;spfs; topr; nry;fpwJ.

jsj;jpd; rkd;ghL

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1

𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1

𝑥3 − 𝑥1 𝑦3 − 𝑦1 𝑧3 − 𝑧1

= 0

𝑥1 = 𝑎 𝑥2 = 0 𝑥3 = 0

𝑦1 = 0 𝑦2 = 𝑏 𝑦3 = 0

𝑧1 = 0 𝑧2 = 0 𝑧3 = 𝑐





𝑥 − 𝑎 𝑦 − 0 𝑧 − 0−𝑎 𝑏 − 0 0−𝑎 0 𝑐 − 0

= 0

(𝑥 − 𝑎)(𝑏𝑐) − (𝑦 − 0)(−𝑎𝑐)

+ (𝑧 − 0)(0 + 𝑎𝑏) = 0

(𝑥 − 𝑎)(𝑏𝑐) + 𝑦𝑎𝑐 + 𝑧𝑎𝑏 = 0

𝑥𝑏𝑐 − 𝑎𝑏𝑐 + 𝑦𝑎𝑐 + 𝑧𝑎𝑏 = 0

𝑥𝑏𝑐 + 𝑦𝑎𝑐 + 𝑧𝑎𝑏 = 𝑎𝑏𝑐

÷ 𝑎𝑏𝑐 𝑥𝑏𝑐

𝑎𝑏𝑐+

𝑦𝑎𝑐

𝑎𝑏𝑐+

𝑧𝑎𝑏

𝑎𝑏𝑐=

𝑎𝑏𝑐

𝑎𝑏𝑐

𝑥

𝑎+

𝑦

𝑏+

𝑧

𝑐= 1

fhh;Brpad; mikg;G:

%d;W Gs;spfs; topr; nry;Yk; jsj;jpd;

rkd;ghL 𝑟 = (1 − 𝑠 − 𝑡)𝑎 + 𝑠𝑏 + 𝑡𝑐

𝑟 = (1 − 𝑠 − 𝑡)𝑎𝑖 + 𝑠𝑏𝑗 + 𝑡𝑐𝑘

𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘 = (1 − 𝑠 − 𝑡)𝑎𝑖 + 𝑠𝑏𝑗 + 𝑡𝑐𝑘

𝑥 = (1 − 𝑠 − 𝑡)𝑎; 𝑦 = 𝑠𝑏; 𝑧 = 𝑡𝑐

𝑥

𝑎= 1 − 𝑠 − 𝑡,

𝑦

𝑏= 𝑠,

𝑧

𝑐= 𝑡

𝑥

𝑎+

𝑦

𝑏+

𝑧

𝑐= 1 − 𝑠 − 𝑡 + 𝑠 + 𝑡

𝑥

𝑎+

𝑦

𝑏+

𝑧

𝑐= 1

21. (−𝟏, −𝟐, 𝟏) vd;w Gs;sp topr; nry;tJk;

𝒙 + 𝟐𝒚 + 𝟒𝒛 + 𝟕 = 𝟎 kw;Wk; 𝟐𝒙 − 𝒚 + 𝟑𝒛 + 𝟑 = 𝟎 Mfpa jsq;fSf;F nrq;Fj;jhfTk; cs;s jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad;

rkd;ghLfisf; fhz;f (JUN-06,MAR-08)

ntf;lh; rkd;ghL:

𝑢 = 𝑖 + 2𝑗 + 4𝑘 kw;Wk; 𝑣 = 2𝑖 − 𝑗 + 3𝑘

𝑎 = −𝑖 − 2𝑗 + 𝑘


𝑟 = −𝑖 − 2𝑗 + 𝑘 + 𝑠 𝑖 + 2𝑗 + 4𝑘

+𝑡(2𝑖 − 𝑗 + 3𝑘 )

fhh;Brpad; mikg;G: jsj;jpd; rkd;ghL

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1

𝑙1 𝑚1 𝑛1

𝑙2 𝑚2 𝑛2

= 0

𝑥1 = −1 𝑙1 = 1 𝑙2 = 2

𝑦1 = −2 𝑚1 = 2 𝑚2 = −1

𝑧1 = 1 𝑛1 = 4 𝑛2 = 3

𝑥 + 1 𝑦 + 2 𝑧 − 1

1 2 42 −1 3

= 0

(𝑥 + 1)(6 + 4) − (𝑦 + 2)(3 − 8)

+ (𝑧 − 1)(−1 − 4) = 0

(𝑥 + 1)(10) − (𝑦 + 2)(−5)

+(𝑧 − 1)(−5) = 0

10𝑥 + 10 + 5𝑦 + 10 − 5𝑧 + 5 = 0

10𝑥 + 5𝑦 − 5𝑧 + 25 = 0

÷ 5 2𝑥 + 𝑦 − 𝑧 + 5 = 0

22. (𝟏, 𝟐, −𝟐) topNa nry;yf;$baJk;

𝒙+𝟐

𝟑=

𝒚+𝟏

−𝟐=

𝒛−𝟒

−𝟒vd;w Nfhl;bw;F ,izahfTk;

𝟐𝒙 + 𝟑𝒚 + 𝟑𝒛 = 𝟖 vd;w jsj;jpw;F nrq;Fj;jhfTk; cs;s jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f

(OCT-10)

ntf;lh; rkd;ghL:

𝑢 = 3𝑖 − 2𝑗 − 4𝑘 kw;Wk;𝑣 = 2𝑖 + 3𝑗 + 3𝑘

𝑎 = 𝑖 + 2𝑗 − 2𝑘


𝑟 = 𝑖 + 2𝑗 − 2𝑘 + 𝑠 3𝑖 − 2𝑗 − 4𝑘

+𝑡(2𝑖 + 3𝑗 + 3𝑘 )

fhh;Brpad; mikg;G:

jsj;jpd; rkd;ghL

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1

𝑙1 𝑚1 𝑛1

𝑙2 𝑚2 𝑛2

= 0

𝑥1 = 1 𝑙1 = 3 𝑙2 = 2

𝑦1 = 2 𝑚1 = −2 𝑚2 = 3

𝑧1 = −2 𝑛1 = −4 𝑛2 = 3

𝑥 − 1 𝑦 − 2 𝑧 + 2

3 −2 −42 3 3

= 0

(𝑥 − 1)(−6 + 12) − (𝑦 − 2)(9 + 8)

+ (𝑧 + 2)(9 + 4) = 0

(𝑥 − 1)(6) − (𝑦 − 2)(17)

+(𝑧 + 2)(13) = 0

6𝑥 − 6 − 17𝑦 + 34 + 13𝑧 + 26 = 0

6𝑥 − 17𝑦 + 13𝑧 + 54 = 0





23. 𝒙−𝟏

𝟐=

−𝒚

𝟑=

𝒛+𝟏

𝟏 vd;w Nfhl;il cs;slf;fpaJk;

𝒙 − 𝟐𝒚 + 𝟑𝒛 − 𝟐 = 𝟎 vd;w jsj;jpw;Fk; nrq;Fj;jhfTk; mike;j jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f

(OCT-12)

ntf;lh; rkd;ghL:


𝑎 = 𝑖 + 0𝑗 − 𝑘 , 𝑢 = 2𝑖 − 3𝑗 + 𝑘 , 𝑣 = 𝑖 − 2𝑗 + 3𝑘

𝑟 = 𝑖 + 0𝑗 − 𝑘 + 𝑠 2𝑖 − 3𝑗 + 𝑘 + 𝑡(𝑖 − 2𝑗 + 3𝑘 )

fhh;Brpad; mikg;G:

jsj;jpd; rkd;ghL

𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1

𝑙1 𝑚1 𝑛1

𝑙2 𝑚2 𝑛2

= 0

𝑥1 = 1 𝑙1 = 2 𝑙2 = 1

𝑦1 = 0 𝑚1 = −3 𝑚2 = −2

𝑧1 = −1 𝑛1 = 1 𝑛2 = 3

𝑥 − 1 𝑦 𝑧 + 1

2 −3 11 −2 3

= 0

(𝑥 − 1)(−9 + 2) − (𝑦)(6 − 1)

+ (𝑧 + 1)(−4 + 3) = 0

(𝑥 − 1)(−7) − (𝑦)(5) + (𝑧 + 1)(−1) = 0

−7𝑥 + 7 − 5𝑦 − 𝑧 − 1 = 0

7𝑥 + 5𝑦 + 𝑧 − 6 = 0

3. fyg;ngz;fs;


Fwpj;jhy; 𝑷 ,d; epakg;ghijia

𝑰𝒎.𝟐𝒛+𝟏

𝒊𝒛+𝟏/ = −𝟐 vd;w epge;jidf;F cl;gl;L

fhz;f (MAR-10,JUN-13)


2𝑧+1

𝑖𝑧+1=

2(𝑥+𝑖𝑦 )+1

𝑖(𝑥+𝑖𝑦 )+1=

(2𝑥+1)+𝑖2𝑦

(1−𝑦)+𝑖𝑥

=(2𝑥+1)+𝑖2𝑦

(1−𝑦)+𝑖𝑥×

(1−𝑦)−𝑖𝑥

(1−𝑦)−𝑖𝑥

=(2𝑥+1)(1−𝑦)+2𝑥𝑦 +𝑖[2𝑦(1−𝑦)−𝑥(2𝑥+1)]

(1−𝑦)2+𝑥2

Mdhy;, 𝐼𝑚 .2𝑧+1

𝑖𝑧+1/ = −2

2𝑦(1−𝑦)−𝑥(2𝑥+1)

(1−𝑦)2+𝑥2 = −2

2𝑦 − 2𝑦2 − 2𝑥2 − 𝑥 = −2[1 + 𝑦2 − 2𝑦 + 𝑥2]

2𝑦 − 2𝑦2 − 2𝑥2 − 𝑥 = 2 − 2𝑦2 + 4𝑦 − 2𝑥2

−2𝑦 − 𝑥 + 2 = 0

𝑃 d; epakg;ghij 𝑥 + 2𝑦 − 2 = 0


Fwpj;jhy; 𝑷 ,d; epakg;ghijia 𝑹𝒆.𝒛−𝟏

𝒛+𝒊/ = 𝟏

vd;w epge;jidf;F cl;gl;L fhz;f (JUN-12)


𝑧−1

𝑧+𝑖=

𝑥+𝑖𝑦−1

𝑥+𝑖𝑦+𝑖=

(𝑥−1)+𝑖𝑦

𝑥+𝑖(𝑦+1)

=(𝑥−1)+𝑖𝑦

𝑥+𝑖(𝑦+1)×

𝑥−𝑖(𝑦+1)

𝑥−𝑖(𝑦+1)

=𝑥(𝑥−1)+𝑦(𝑦+1)

𝑥2+(𝑦+1)2 + 𝑖 [fw;gidg; gFjp]

𝑅𝑒 .𝑧−1

𝑧+𝑖/ = 1

𝑥(𝑥−1)+𝑦(𝑦+1)

𝑥2+(𝑦+1)2 = 1

𝑥2 + 𝑦2 − 𝑥 + 𝑦 = 𝑥2 + 𝑦2 + 2𝑦 + 1

−𝑥 − 𝑦 = 1

𝑃 d; epakg;ghij 𝑥 + 𝑦 + 1 = 0



𝑰𝒎.𝟐𝒛+𝒊

𝒊𝒛−𝟏/ = −𝟏 vd;w epge;jidf;F cl;gl;L

fhz;f (MAR-14)


2𝑧+𝑖

𝑖𝑧−1=

2(𝑥+𝑖𝑦 )+𝑖

𝑖(𝑥+𝑖𝑦)−1=

2𝑥+𝑖2𝑦+𝑖

𝑖𝑥−𝑦−1

=2𝑥+𝑖(2𝑦+1)

−(𝑦+1)+𝑖𝑥×

−(𝑦+1)−𝑖𝑥

−(𝑦+1)−𝑖𝑥

=−2𝑥(𝑦+1)−2𝑖𝑥2−𝑖(2𝑦+1)(𝑦+1)+𝑥(2𝑦+1)

(𝑦+1)2+𝑥2

Mdhy;, 𝐼𝑚 .2𝑧+𝑖

𝑖𝑧−1/ = −1

−2𝑥2−(2𝑦+1)(𝑦+1)

(𝑦+1)2+𝑥2 = −1

−2𝑥2 − 2𝑦2 − 3𝑦 − 1 = −𝑦2 − 1 − 2𝑦 − 𝑥2

𝑥2 + 𝑦2 + 𝑦 = 0

𝑃 d; epakg;ghij 𝑥2 + 𝑦2 + 𝑦 = 0



𝐚𝐫𝐠 .𝒛−𝟏

𝒛+𝟏/ =

𝝅

𝟑 vd;w epge;jidf;F cl;gl;L

fhz;f (MAR-13)





arg(𝑧 − 1) − arg(𝑧 + 1) =𝜋

3

arg(𝑥 + 𝑖𝑦 − 1) − arg(𝑥 + 𝑖𝑦 + 1) =𝜋

3

arg((𝑥 − 1) + 𝑖𝑦) − arg (𝑥 + 1) + 𝑖𝑦 =𝜋

3

tan−1 𝑦

𝑥−1− tan−1 𝑦

𝑥+1=

𝜋

3

tan−1 𝑦

𝑥−1−

𝑦

𝑥+1

1+.𝑦

𝑥−1/.

𝑦

𝑥+1/ =

𝜋

3

2𝑦

𝑥2−1+𝑦2 = tan𝜋

3

2𝑦

𝑥2−1+𝑦2 = 3

2𝑦 = 3(𝑥2 − 1 + 𝑦2)

3𝑥2 + 3𝑦2 − 2𝑦 − 3 = 0 vd;gJ

Njitahd epakg;ghijahFk;.

5. 𝑷 vDk; Gs;sp fyg;G khwp 𝒛 If; Fwpj;jhy;

𝑹𝒆.𝒛+𝟏

𝒛+𝒊/ = 𝟏 vd;w epge;jidf;F cl;gl;L

𝑷 ,d; epakg;ghijia fhz;f (MAR-16) 𝑧 = 𝑥 + 𝑖𝑦 vd;f

𝑧 + 1

𝑧 + 𝑖=

𝑥 + 𝑖𝑦 + 1

𝑥 + 𝑖𝑦 + 𝑖=

(𝑥 + 1) + 𝑖𝑦

𝑥 + 𝑖(𝑦 + 1)

=(𝑥+1)+𝑖𝑦

𝑥+𝑖(𝑦+1)×

𝑥−𝑖(𝑦+1)

𝑥−𝑖(𝑦+1)

=𝑥(𝑥+1)+𝑦(𝑦+1)

𝑥2+(𝑦+1)2 + 𝑖 [fw;gidg; gFjp]

𝑅𝑒 .𝑧−1

𝑧+𝑖/ = 1

𝑥(𝑥 + 1) + 𝑦(𝑦 + 1)

𝑥2 + (𝑦 + 1)2= 1

𝑥2 + 𝑦2 + 𝑥 + 𝑦 = 𝑥2 + 𝑦2 + 2𝑦 + 1

𝑥 − 𝑦 = 1

𝑃 d; epakg;ghij 𝑥 − 𝑦 = 1

6. 𝜶 , 𝜷 vd;git 𝒙𝟐 − 𝟐𝒙 + 𝟐 = 𝟎,d; %yq;fs;

kw;Wk; 𝐜𝐨𝐭 𝜽 = 𝒚 + 𝟏 vdpy;

(𝒚+𝜶)𝒏−(𝒚+𝜷)𝒏

𝜶−𝜷=

𝐬𝐢𝐧𝒏𝜽

𝐬𝐢𝐧𝒏 𝜽 vdf; fhl;Lf.

(MAR-06,OCT-12,OCT-14)

𝑥2 − 2𝑥 + 2 = 0 ,d; %yq;fs; 1 ± 𝑖

𝛼 = 1 + 𝑖, 𝛽 = 1 − 𝑖 vd;f

(𝑦 + 𝛼)𝑛 = ,(cot 𝜃 − 1) + (1 + 𝑖)-𝑛

= ,cot 𝜃 + 𝑖-𝑛

=1

sin 𝑛 𝜃,cos 𝜃 + 𝑖 sin 𝜃-𝑛

(𝑦 + 𝛼)𝑛 =1

sin 𝑛 𝜃[cos 𝑛𝜃 + 𝑖 sin 𝑛𝜃]

,NjNghy,

(𝑦 + 𝛽)𝑛 =1

sin 𝑛 𝜃[cos 𝑛𝜃 − 𝑖 sin 𝑛𝜃]

(𝑦 + 𝛼)𝑛 − (𝑦 + 𝛽)𝑛 =2𝑖 sin 𝑛𝜃

sin 𝑛 𝜃

𝛼 − 𝛽 = (1 + 𝑖) − (1 − 𝑖) = 2𝑖

NkYk;, (𝑦 + 𝛼)𝑛 − (𝑦 + 𝛽)𝑛

𝛼 − 𝛽=

2𝑖 sin 𝑛𝜃

2𝑖 sin𝑛 𝜃=

sin 𝑛𝜃

sin𝑛 𝜃

7. 𝒙𝟐 − 𝟐𝒑𝒙 + 𝒑𝟐 + 𝒒𝟐 = 𝟎 vd;w rkd;ghl;bd;

%yq;fs; 𝜶 , 𝜷 kw;Wk; 𝐭𝐚𝐧 𝜽 =𝒒

𝒚+𝒑 vdpy;

(𝒚+𝜶)𝒏−(𝒚+𝜷)𝒏

𝜶−𝜷= 𝒒𝒏−𝟏 𝐬𝐢𝐧𝒏𝜽

𝐬𝐢𝐧𝒏 𝜽vd epWTf

(MAR-07,OCT-09,16)

𝑥2 − 2𝑝𝑥 + (𝑝2 + 𝑞2) = 0

𝑥 =2𝑝± 4𝑝2−4(𝑝2+𝑞2)

2= 𝑝 ± 𝑖𝑞

𝛼 = 𝑝 + 𝑖𝑞, 𝛽 = 𝑝 − 𝑖𝑞 vd;f

𝛼 − 𝛽 = 2𝑞𝑖

tan 𝜃 =𝑞

𝑦+𝑝 vdf; nfhLf;fg;gl;Ls;sJ

𝑦 + 𝑝 =𝑞

tan 𝜃

𝑦 + 𝑝 = 𝑞 cot 𝜃

𝑦 = 𝑞 cot 𝜃 − 𝑝

𝑦 + 𝛼 = 𝑞 cot 𝜃 − 𝑝 + (𝑝 + 𝑖𝑞)

= 𝑞,cot 𝜃 + 𝑖- = 𝑞cos 𝜃 + 𝑖 sin 𝜃

sin 𝜃

(𝑦 + 𝛼)𝑛 = 𝑞𝑛 (cos 𝜃+𝑖 sin 𝜃)𝑛

sin n 𝜃

(𝑦 + 𝛼)𝑛 =𝑞𝑛

sin n 𝜃[cos 𝑛𝜃 + 𝑖 sin 𝑛𝜃]………..(1)

,Nj Nghy

(𝑦 + 𝛽)𝑛 =𝑞𝑛

sin n 𝜃[cos 𝑛𝜃 − 𝑖 sin 𝑛𝜃]………(2)

(1)−(2) vDk; NghJ

(𝑦 + 𝛼)𝑛 − (𝑦 + 𝛽)𝑛 =𝑞𝑛

sin n 𝜃[2𝑖 sin 𝑛𝜃]

(𝑦+𝛼)𝑛−(𝑦+𝛽)𝑛

𝛼−𝛽=

𝑞𝑛

(2𝑖𝑞) sin n 𝜃,2𝑖 sin 𝑛𝜃-

= 𝑞𝑛−1 sin 𝑛𝜃

sin 𝑛 𝜃

8. 𝒙𝟐 − 𝟐𝒙 + 𝟒 = 𝟎 ,d; %yq;fs; 𝜶 kw;Wk; 𝜷

vdpy; 𝜶𝒏 − 𝜷𝒏 = 𝒊𝟐𝒏+𝟏 𝐬𝐢𝐧𝒏𝝅

𝟑 vd epWTf.

mjpypUe;J 𝜶𝟗 − 𝜷𝟗 d; kjpg;ig ngWf

(OCT-06,OCT-08,MAR-09,MAR-12,JUN-15)

𝑥2 − 2𝑥 + 4 = 0

𝑥 = 1 ± 𝑖 3

𝛼 = 1 + 𝑖 3, 𝛽 = 1 − 𝑖 3

𝛼𝑛 = 1 + 𝑖 3 𝑛

= 2𝑛 .cos𝑛𝜋

3+ 𝑖 sin

𝑛𝜋

3/





𝛽𝑛 = 1 − 𝑖 3 𝑛


3− 𝑖 sin

𝑛𝜋

3/

𝛼𝑛 − 𝛽𝑛


3+ 𝑖 sin

𝑛𝜋

3/ − 2𝑛 .cos

𝑛𝜋

3− 𝑖 sin

𝑛𝜋

3/

𝛼𝑛 − 𝛽𝑛 = 2𝑛 .2𝑖 sin𝑛𝜋

3/

𝛼𝑛 − 𝛽𝑛 = 𝑖2𝑛+1 sin𝑛𝜋

3

𝑛 = 9 vdg; gpujpapl>

𝛼9 − 𝛽9 = 𝑖210 sin9𝜋

3

= 𝑖210(sin3 𝜋) = 0

9. 𝒂 kw;Wk; 𝒃 vd;git 𝒙𝟐 + 𝟐 𝟑𝒙 + 𝟒 = 𝟎 vd;w

rkd;ghl;bd; %yq;fshf ,Ug;gpd; 𝒂𝒏 + 𝒃𝒏 d;

kjpg;gpidf; fhz;f. ,jpypUe;J 𝒂𝟏𝟐 + 𝒃𝟏𝟐d;

kjpg;gpid jUtpf;f (𝒏 vd;gJ xU KO vz;)

(JUN-16)

𝑥2 + 2 3𝑥 + 4 = 0

𝑥 =− 2 3 ± 2 3

2−4(1)(4)

2(1)

=− 2 3 ± 12−16

2

=−2 3± −4

2

=−2 3± 2𝑖

2

=2( − 3± 𝑖)

2

𝑥 = − 3 ± 𝑖

𝑎 = − 3 + 𝑖, 𝑏 = − 3 − 𝑖

𝑎𝑛 = − 3 + 𝑖 𝑛

= 2𝑛 cos5𝜋

6+ 𝑖 sin

5𝜋

6 𝑛

= 2𝑛 .cos5𝑛𝜋

6+ 𝑖 sin

5𝑛𝜋

6/

𝑏𝑛 = − 3 − 𝑖 𝑛

= 2𝑛 cos5𝑛𝜋

6− 𝑖 sin

5𝑛𝜋

6

𝑎𝑛 + 𝑏𝑛


6+ 𝑖 sin

5𝑛𝜋

6/ + 2𝑛 .cos

5𝑛𝜋

6− 𝑖 sin

5𝑛𝜋

6/


6+ 𝑖 sin

5𝑛𝜋

6+ cos

5𝑛𝜋

6− 𝑖 sin

5𝑛𝜋

6/

= 2𝑛 .2cos5𝑛𝜋

6/

𝑎𝑛 + 𝑏𝑛 = 2𝑛+1 cos5𝑛𝜋

6

𝑛 = 12 vd gpujpapl

𝑎12 + 𝑏12 = 213 cos5(12)𝜋

6

= 213(cos 10 𝜋) = 8192(1)

= 8192

10. 𝒙𝟗 + 𝒙𝟓 − 𝒙𝟒 − 𝟏 = 𝟎 vd;w rkd;ghl;ilj; jPh;f;f (JUN-06,11)

𝑥9 + 𝑥5 − 𝑥4 − 1 = 0

𝑥5(𝑥4 + 1) − 1(𝑥4 + 1) = 0

(𝑥5 − 1)(𝑥4 + 1) = 0

𝑥5 − 1 = 0; 𝑥4 + 1 = 0

(i) 𝑥 = (1)1

5 = (cos 0 + 𝑖 sin 0)1

5

= (cos 2𝑘 𝜋 + 𝑖 sin 2𝑘𝜋)1

5

= cos 2𝑘𝜋

5+ 𝑖 sin

2𝑘𝜋

5

𝑘 = 0,1,2,3,4

(ii) 𝑥 = (−1)1

4 = (cos 𝜋 + 𝑖 sin 𝜋)1

4

= (cos(2𝑘 + 1) 𝜋 + 𝑖 sin(2𝑘 + 1)𝜋)1

4

= cos (2𝑘 + 1)𝜋

4+ 𝑖 sin

(2𝑘 + 1)𝜋

4

𝑘 = 0,1,2,3

,t;thW 9 %yq;fs; ngwg;gLfpd;wd.

11. 𝒙𝟕 + 𝒙𝟒 + 𝒙𝟑 + 𝟏 = 𝟎 vd;w rkd;ghl;ilj; jPh;f;f (JUN-09)

𝑥7 + 𝑥4 + 𝑥3 + 1 = 0

𝑥4(𝑥3 + 1) + 1(𝑥3 + 1) = 0

(𝑥4 + 1)(𝑥3 + 1) = 0

𝑥4 = −1, 𝑥3 = −1

(i) 𝑥 = (−1)1


4

= (cos(2𝑘 + 1) 𝜋 + 𝑖 sin(2𝑘 + 1)𝜋)1

4

= cos (2𝑘+1)𝜋

4+ 𝑖 sin

(2𝑘+1)𝜋

4

𝑘 = 0,1,2,3

(ii) 𝑥 = (−1)1


3

= (cos(2𝑘 + 1) 𝜋 + 𝑖 sin(2𝑘 + 1)𝜋)13

= cos (2𝑘+1)𝜋

3+ 𝑖 sin

(2𝑘+1)𝜋

3

𝑘 = 0,1,2

12. 𝒙𝟒 − 𝒙𝟑 + 𝒙𝟐 − 𝒙 + 𝟏 = 𝟎 vd;w rkd;ghl;ilj;

jPh;f;f ( JUN-08,JUN-10,OCT-11, MAR-17)

,e;j gy;YWg;Gf; Nfhitapd; kjpg;Gfs; ngUf;fy; njhlhpy; cs;sd.

𝑟 = −𝑥, 𝑎 = 1, 𝑛 = 5

1 − 𝑥 + 𝑥2 − 𝑥3 + 𝑥4 =𝑎(𝑟𝑛 − 1)

𝑟 − 1=

𝑥5 + 1

𝑥 + 1





,q;F 𝑥 ≠ −1

𝑥5 + 1 = 0 Ij; jPh;j;J 𝑥 = −1 vd;w %yj;ij ePf;fptpl Ntz;Lk;

𝑥5 + 1 = 0 ⇒ 𝑥 = (−1)1

5

= (cos 𝜋 + 𝑖 sin 𝜋)1

5

= (cos(2𝑘 + 1) 𝜋 + 𝑖 sin(2𝑘 + 1)𝜋)15

= cos (2𝑘+1)𝜋

5+ 𝑖 sin

(2𝑘+1)𝜋

5

,q;F 𝑘 = 0,1,2,3,4

cis 𝜋

5, cis

3𝜋

5, cis 𝜋 , cis

7𝜋

5, cis

9𝜋

5 Mfpa

kjpg;Gfisg; ngWk;

,q;F cis 𝜋 = −1 vd;w %yj;ij ePf;fptpl

cis 𝜋

5, cis

3𝜋

5, cis

7𝜋

5, cis

9𝜋

5 Mfpait

rkd;ghl;bd; jPh;TfshFk;.

13. 𝒙 +𝟏

𝒙= 𝟐𝐜𝐨𝐬𝜽 , 𝒚 +

𝟏

𝒚= 𝟐𝐜𝐨𝐬𝝓 vdpy;

(i)𝒙𝒎

𝒚𝒏 +𝒚𝒏

𝒙𝒎 = 𝟐𝐜𝐨𝐬( 𝒎𝜽 − 𝒏𝝓)

(ii)𝒙𝒎

𝒚𝒏 −𝒚𝒏

𝒙𝒎 = 𝟐𝒊 𝐬𝐢𝐧( 𝒎𝜽 − 𝒏𝝓) vdf; fhl;Lf

(JUN-14)

𝑥 +1

𝑥= 2cos 𝜃

𝑥2 − 2cos 𝜃 𝑥 + 1 = 0

𝑥 = cos 𝜃 ± 𝑖 sin 𝜃

𝑥 = cos 𝜃 + 𝑖 sin 𝜃 vdf; nfhs;f

,JNghyNt, 𝑦 = cos 𝜙 + 𝑖 sin 𝜙

𝑥𝑚 = cos 𝑚𝜃 + 𝑖 sin 𝑚𝜃

𝑦𝑛 = cos 𝑛𝜙 + 𝑖 sin 𝑛𝜙

𝑥𝑚

𝑦𝑛=

cos 𝑚𝜃 + 𝑖 sin 𝑚𝜃

cos 𝑛𝜙 + 𝑖 sin 𝑛𝜙

= (cos 𝑚𝜃 + 𝑖 sin 𝑚𝜃)(cos(−𝑛𝜙) + 𝑖 sin(−𝑛𝜙))

𝑥𝑚

𝑦𝑛 = cos( 𝑚𝜃 − 𝑛𝜙) + 𝑖 sin( 𝑚𝜃 − 𝑛𝜙)……(1)

𝑦𝑛

𝑥𝑚 = cos( 𝑚𝜃 − 𝑛𝜙) − 𝑖 sin( 𝑚𝜃 − 𝑛𝜙)……(2)

(1) + (2) ⇒𝑥𝑚

𝑦𝑛 +𝑦𝑛

𝑥𝑚 = 2cos( 𝑚𝜃 − 𝑛𝜙)

(1) − (2) ⇒𝑥𝑚

𝑦𝑛 −𝑦𝑛

𝑥𝑚 = 2𝑖 sin( 𝑚𝜃 − 𝑛𝜙)

14. 𝒂 = 𝐜𝐨𝐬 𝟐𝜶 + 𝒊 𝐬𝐢𝐧 𝟐𝜶 ,

𝒃 = 𝐜𝐨𝐬 𝟐𝜷 + 𝒊 𝐬𝐢𝐧 𝟐𝜷,

𝒄 = 𝐜𝐨𝐬 𝟐𝜸 + 𝒊 𝐬𝐢𝐧 𝟐𝜸 vdpy;

(i) 𝒂𝒃𝒄 +𝟏

𝒂𝒃𝒄= 𝟐 𝐜𝐨𝐬( 𝜶 + 𝜷 + 𝜸)

(ii) 𝒂𝟐𝒃𝟐+𝒄𝟐

𝒂𝒃𝒄= 𝟐 𝐜𝐨𝐬𝟐( 𝜶 + 𝜷 − 𝜸) vd ep&gp

(OCT-10)

𝑎𝑏𝑐 = (cos 𝛼 + 𝑖 sin 𝛼)2(cos 𝛽 + 𝑖 sin 𝛽)2(cos 𝛾 + 𝑖 sin 𝛾)2

𝑎𝑏𝑐 = (cos 𝛼 + 𝑖 sin 𝛼)(cos 𝛽 + 𝑖 sin 𝛽)(cos 𝛾 + 𝑖 sin 𝛾)

𝑎𝑏𝑐 = cos( 𝛼 + 𝛽 + 𝛾) + 𝑖 sin( 𝛼 + 𝛽 + 𝛾)

1

𝑎𝑏𝑐= cos( 𝛼 + 𝛽 + 𝛾) − 𝑖 sin( 𝛼 + 𝛽 + 𝛾)

𝑎𝑏𝑐 +1

𝑎𝑏𝑐= 2cos( 𝛼 + 𝛽 + 𝛾)

(ii) 𝑎2𝑏2+𝑐2

𝑎𝑏𝑐=

𝑎2𝑏2

𝑎𝑏𝑐+

𝑐2

𝑎𝑏𝑐=

𝑎𝑏

𝑐+

𝑐

𝑎𝑏

𝑎𝑏

𝑐=

(cos 2𝛼+𝑖 sin 2𝛼)(cos 2𝛽+𝑖 sin 2𝛽)

(cos 2𝛾+𝑖 sin 2𝛾)

𝑎𝑏

𝑐= cos( 2𝛼 + 2𝛽 − 2𝛾) + 𝑖 sin( 2𝛼 + 2𝛽 − 2𝛾)

𝑐

𝑎𝑏= cos( 2𝛼 + 2𝛽 − 2𝛾) − 𝑖 sin( 2𝛼 + 2𝛽 − 2𝛾)

𝑎𝑏

𝑐+

𝑐

𝑎𝑏= cos( 2𝛼 + 2𝛽 − 2𝛾) +

𝑖 sin( 2𝛼 + 2𝛽 − 2𝛾) + cos( 2𝛼 + 2𝛽 − 2𝛾)

−𝑖 sin( 2𝛼 + 2𝛽 − 2𝛾)

= 2 cos( 2𝛼 + 2𝛽 − 2𝛾)

𝑎2𝑏2+𝑐2

𝑎𝑏𝑐= 2 cos2( 𝛼 + 𝛽 − 𝛾)

15. 𝟑 + 𝒊 𝟐

𝟑 ,d; vy;yh kjpg;GfisAk; fhz;f

(JUN-07)

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

6

3 + 𝑖 2

3 = 22

3 .cos𝜋

6+ 𝑖 sin

𝜋

6/

2

3

= 22

3 .cos𝜋

3+ 𝑖 sin

𝜋

3/

1

3

= 22

3 .cos .2𝑘𝜋 +𝜋

3/ + 𝑖 sin .2𝑘𝜋 +

𝜋

3//

1

3

= 22

3 0cos(6𝑘 + 1)𝜋

9+ 𝑖 sin(6𝑘 + 1)

𝜋

91

,q;F 𝑘 = 0,1,2 kjpg;Gfs;

223 cis .

𝜋

9/ , 2

23 cis

7𝜋

9 , 2

23 cis

13𝜋

9





16. − 𝟑 − 𝒊 𝟐

𝟑 ,d; vy;yh kjpg;GfisAk; fhz;f

(OCT-13)

− 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

223 cis

−5𝜋

9 , 2

23 cis

7𝜋

9 , 2

23 cis

19𝜋

9 (my;yJ)2

23 cis .

𝜋

9/

Mfpa kjpg;Gfisg; ngWk;

17. .𝟏

𝟐− 𝒊

𝟑

𝟐/

𝟑

𝟒 d; vy;yh kjpg;GfisAk; fhz;f

kw;Wk; mjd; kjpg;Gfspd; ngUf;fw;gyd; 1

vdTk; fhl;Lf.

(OCT-07,MAR-08,MAR-11, OCT-15) 1

2− 𝑖

3

2= 𝑟(cos 𝜃 + 𝑖 sin 𝜃) vd;f

𝑟 cos 𝜃 =1

2, 𝑟 sin 𝜃 = −

3

2

𝑟 = 1

4+

3

4= 1

cos 𝜃 =1

2, sin 𝜃 = −

3

2⇒ 𝜃 , 4tJ

fhy;gFjpapYs;sJ

𝜃 =−𝜋

3

1

2− 𝑖

3

2= cos .

−𝜋

3/ + 𝑖 sin .

−𝜋

3/

.1

2− 𝑖

3

2/

3

4= .cos .

−𝜋

3/ + 𝑖 sin .

−𝜋

3//

3

4

= (cos(−𝜋) + 𝑖 sin(−𝜋))1

4

= (cos(2𝑘𝜋 − 𝜋) + 𝑖 sin(2𝑘𝜋 − 𝜋))1

4

= cos .2𝑘−1

4/ 𝜋 + 𝑖 sin .

2𝑘−1

4/ 𝜋

𝑘 = 0,1,2,3

cis .−𝜋

4/ , cis

𝜋

4, cis

3𝜋

4, cis

5𝜋

4 Mfpa kjpg;Gfisg;

ngWk; ,tw;wpd; ngUf;fw;gyd;

cis 0−𝜋

4+

𝜋

4+

3𝜋

4+

5𝜋

41 = cis 0

8𝜋

41 = cis2𝜋

= cos 2𝜋 + 𝑖 sin 2𝜋

= 1

18. .𝟏

𝟐+ 𝒊

𝟑

𝟐/

𝟑

𝟒 d; vy;yh kjpg;GfisAk; fhz;f

kw;Wk; mjd; kjpg;Gfspd; ngUf;fw;gyd; 1

vdTk; fhl;Lf. (MAR-15) 1

2+ 𝑖

3

2= 𝑟(cos 𝜃 + 𝑖 sin 𝜃)

𝑟 cos 𝜃 =1

2, 𝑟 sin 𝜃 =

3

2

𝑟 = 1

4+

3

4= 1

cos 𝜃 =1

2, sin 𝜃 =

3

2⇒ 𝜃 , Kjy;

fhy;gFjpapYs;sJ

𝜃 =𝜋

3

1

2+ 𝑖

3

2= cos

𝜋

3+ 𝑖 sin

𝜋

3

.1

2+ 𝑖

3

2/

3

4= .cos

𝜋

3+ 𝑖 sin

𝜋

3/

3

4

= (cos 𝜋 + 𝑖 sin 𝜋)1

4

= (cos(2𝑘𝜋 + 𝜋) + 𝑖 sin(2𝑘𝜋 + 𝜋))1

4

= cos .2𝑘+1

4/ 𝜋 + 𝑖 sin .

2𝑘+1

4/ 𝜋

𝑘 = 0,1,2,3

∴ cis .𝜋

4/ , cis

3𝜋

4, cis

5𝜋

4, cis

7𝜋

4 Mfpa kjpg;Gfisg;

ngWk;. ,tw;wpd; ngUf;fw;gyd;

cis 0𝜋

4+

3𝜋

4+

5𝜋

4+

7𝜋

41 = cis 0

16𝜋

41

= cis4𝜋

= cos 4𝜋 + 𝑖 sin 4𝜋

= 1





4. gFKiw tbtpay;

1. xU uapy;Nt ghyj;jpd; Nky; tisT gutisaj;jpd; mikg;igf; nfhz;Ls;sJ. me;j tistpd; mfyk; 100 mbahfTk; mt;tistpd; cr;rpg;Gs;spapd; cauk; ghyj;jpypUe;J 10 mbahfTk; cs;sJ vdpy;> ghyj;jpd; kj;jpapypUe;J ,lg;Gwk; my;yJ tyg;Gwk; 10 mb J}uj;jpy; ghyj;jpd; Nky; tisT vt;tsT cauj;jpy; ,Uf;Fk; vdf;

fhz;f. (MAR-09,JUN-16)

,q;F gutisak; fPo;Nehf;fp jpwg;Gilajhf vLj;Jf; nfhs;Nthk;

∴ 𝑥2 = −4𝑎𝑦

,J (50, −10) topahfr; nry;fpwJ

50 × 50 = −4𝑎(−10)

𝑎 =250

4

𝑥2 = −4 .250

4/ 𝑦

𝑥2 = −250𝑦

gutisaj;jpd; Nky; cs;s Gs;sp 𝐵(10, 𝑦1)

100 = −250𝑦1

𝑦1 = −100

250= −

2

5

𝐴𝐵 vd;gJ ghyj;jpd; ikaj;jpypUe;J tyg;Gwj;jpy; 10 mb njhiytpy;> ghyj;jpd; caukhFk;.

𝐴𝐶 = 10 kw;Wk; 𝐵𝐶 =2

5

𝐴𝐵 = 10 −2

5= 9

3

5 mb

mjhtJ> Njitg;gl;l ,lj;jpypUe;J ghyj;jpd;

kpf cauk; 93

5 mb MFk;

2. xU uhf;nfl; ntbahdJ nfhSj;Jk;NghJ mJ xU gutisag; ghijapy; nry;fpwJ. mjd; cr;r cauk; 4kP-I vl;Lk;NghJ mJ nfhSj;jg;gl;l ,lj;jpypUe;J fpilkl;l J}uk; 6 kP njhiytpYs;sJ. ,Wjpahf fpilkl;lkhf 12kP njhiytpy; jiuia te;jilfpwJ vdpy; Gwg;gl;l ,lj;jpy; jiuAld; Vw;gLj;jg;gLk; vwpNfhzk; fhz;f

(MAR-06,JUN-09,JUN-10,JUN-12,OCT-12,MAR-14, MAR-17)

gutisaj;jpd; rkd;ghL

𝑥2 = −4𝑎𝑦 (Kidia Mjpahff; nfhs;f).

,J (6, −4) topr; nry;fpwJ

36 = 16𝑎 ⇒ 𝑎 =9

4

rkd;ghL 𝑥2 = −9𝑦…………………..(1)

(−6, −4) y; rha;itf; fzf;fpl>

(1) I 𝑥-I nghWj;J tiff;nfO fhz

2𝑥 = −9𝑑𝑦

𝑑𝑥

𝑑𝑦

𝑑𝑥= −

2

9𝑥

(−6, −4),y;

𝑑𝑦

𝑑𝑥= −

2

9× −6 =

4

3

tan 𝜃 =4

3

𝜃 = tan−1 .4

3/

Njitahd vwpNfhzk; tan−1 .4

3/

3. 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.(OCT-09,MAR-12,OCT-13)

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.

4. xU thy; tpz;kPd; (comet) MdJ #hpaidr;

(sun) Rw;wp gutisag; ghijapy; nry;fpwJ. kw;Wk; #hpad; gutisaj;jpd; Ftpaj;jpy; mikfpwJ. thy; tpz;kPd; #hpadpypUe;J 80 kpy;ypad; fp.kP njhiytpy; mike;J ,Uf;Fk; NghJ thy; tpz;kPidAk; #hpaidAk;

,izf;Fk; NfhL ghijapd; mr;Rld; 𝝅

𝟑

Nfhzj;jpid Vw;gLj;Jkhdhy; (i) thy; tpz;kPdpd; ghijapd; rkd;ghl;ilf; fhz;f.

(ii)thy; tpz;kPd; #hpaDf;F vt;tsT mUfpy; tuKbAk; vd;gijAk; fhz;f. (ghij tyJGwk; jpwg;Gilajhf nfhs;f) (MAR-08,MAR-13,JUN-13,MAR-16,OCT-15,16)

gutisaj;jpd; ghij tyJgf;fk; jpwg;GilaJ. NkYk; Kidg;Gs;sp MjpapYs;sJ.

thy;tpz;kPdpd; epiy 𝑃

𝐹𝑃 = 80 kpy;ypad; fp.kP.

𝑃 apypUe;J gutisaj;jpd; mr;Rf;F 𝑃𝑄 vd;w nrq;Fj;J tiua

𝐹𝑄 = 𝑥1 vd;f

Kf;Nfhzk; 𝐹𝑄𝑃 apypUe;J

𝑃𝑄 = 𝐹𝑃. sin𝜋

3

= 80 × 3

2

= 40 3

𝐹𝑄 = 𝑥1 = 𝐹𝑃. cos𝜋

3

= 80 ×1

2= 40

𝑉𝑄 = 𝑎 + 40 if 𝑉𝐹 = 𝑎

𝑃 vd;gJ (𝑉𝑄, 𝑃𝑄) = (𝑎 + 40,40 3)

𝑃 vd;gJ gutisak; 𝑦2 = 4𝑎𝑥 Nky; ,Ug;gjhy;

(40 3)2 = 4𝑎(𝑎 + 40)

𝑎 = −60 my;yJ 𝑎 = 20

𝑎 = −60 Vw;Gilajy;y.

∴ ghijapd; rkd;ghL

𝑦2 = 4 × 20 × 𝑥

𝑦2 = 80𝑥 #hpaDf;Fk; thy; tpz;kPDf;Fk; ,ilNaAs;s

kpff; Fiwe;j J}uk; 𝑉𝐹

∴ kpff; Fiwe;j J}uk; 20 kpy;ypad; fp.kP.

5. xU njhq;F ghyj;jpd; fk;gp tlk; gutisa tbtpYs;sJ. mjd; ghuk; fpilkl;lkhf rPuhf gutpAs;sJ. mij jhq;Fk; ,U J}z;fSf;F ,ilNaAs;s J}uk; 1500 mb. fk;gp tlj;ijj; jhq;Fk; Gs;spfs; J}zpy; jiuapypUe;J 200 mb cauj;jpy; mike;Js;sd. NkYk; jiuapypUe;J fk;gp tlj;jpd; jho;thd Gs;spapd; cauk; 70 mb> fk;gptlk; 122 mb cauj;jpy; jhq;Fk; fk;gj;jpw;F ,ilNa cs;s nrq;Fj;J ePsk; (jiuf;F ,izahf) fhz;f.

(OCT-07, OCT-11, JUN-14)

fk;gp tlj;jpd; kPJ kpfj;jho;thd Gs;sp KidahFk;. ,jid Mjpahf nfhs;f. 𝐴𝐵 kw;Wk; 𝐶𝐷 jhq;Fk; J}z;fs;. ,U J}z;fSf;F ,ilNaAs;s njhiyT 1500mb vd;gjhy; 𝑉𝐴′ = 750 mb> 𝐴𝐵 = 200 mb

𝐴′𝐵 = 200 − 70 = 130 mb

𝐵 vd;gJ (750,130)

gutisaj;jpd; rkd;ghL 𝑥2 = 4𝑎𝑦

𝐵 vd;w Gs;sp 𝑥2 = 4𝑎𝑦 ,y; cs;sjhy;

(750)2 = 4𝑎(130)





4𝑎 =75 × 750

13

rkd;ghL 𝑥2 =75×750

13𝑦

fk;gk; 𝑅𝑄 tpypUe;J fk;gp tlj;jpw;F

nrq;Fj;jhd ePsk; 𝑃𝑄 .

𝑅𝑄 = 122, 𝑅𝑅′ = 70 ⇒ 𝑅′𝑄 = 52

𝑉𝑅′ = 𝑥1 ∴ 𝑄 vd;gJ (𝑥1 , 52)

𝑄 gutisaj;jpd; kPJs;s xU Gs;sp.

𝑥12 =

75 × 750

13× 52

𝑥1 = 150 10

𝑃𝑄 = 2𝑥1 = 300 10 mb

6. xU njhq;F ghyj;jpd; fk;gp tlk; gutisa tbtpypYs;sJ. mjd; ePsk; 40 kPl;lh; MFk;. topg;ghijahdJ fk;gp tlj;jpd; fPo;kl;lg; Gs;spapypUe;J 5 kPl;lh; fPNo cs;sJ. fk;gp tlj;ij jhq;Fk; Jhz;fspd; cauq;fs; 55 kPl;lh; vdpy;> 30 kPl;lh; cauj;jpy; fk;gp tlj;jpw;F xU Jiz jhq;fp $Ljyhff; nfhLf;fg;gl;lhy; mj;Jizj;jhq;fpapd;

ePsj;ijf; fhz;f (JUN-06,JUN-15) njhq;F ghyj;jpd; fk;gptlk; Nkw;Gwk; jpwg;Gila gutisa mikg;ig ngw;Ws;sJ.

𝑥2 = 4𝑎𝑦

nfhLf;fg;gl;l tptuq;fspypUe;J tlj;jpd; Kid topg;ghijapy; ,Ue;J 5 kP Nky; mike;Js;sJ. njhq;F ghyj;jpd; tpl;lk; 40

kPl;lh;. Gs;sp 𝐴(20,50) gutisaj;jpd; kPJ mike;J cs;sJ.

400 = 4𝑎(50)

𝑎 = 2

Njitahd rkd;ghL 𝑥2 = 8𝑦

𝑄 (𝑥1 , 25) vd;w Gs;sp gutisaj;jpd; kPJ mike;Js;sJ

𝑥12 = 8(25) = 200 = 10 2

𝑃𝑄 = 2𝑥1 = 20 2 kP. jhq;fpapd; ePsk; MFk;.

7. xU tisT miu-ePs;tl;l tbtpy; cs;sJ. mjd; mfyk; 48 mb> cauk; 20 mb

jiuapypUe;J 10 mb cauj;jpy; tistpd;

mfyk; vd;d? (OCT-06,OCT-13)

jiuapd; eLg;Gs;spia ikak; 𝐶(0,0) Mff; nfhs;s

jiuapd; mfyk; 48 mb. Kidfs; 𝐴(24,0),

𝐴′(−24,0)

2𝑎 = 48 kw;Wk; 𝑏 = 20

∴ ePs;tl;lj;jpd; rkd;ghL

𝑥2

242 +𝑦2

202 = 1………………………………………….(1)

10kP cauKs;s J}zpw;Fk; ikaj;jpw;Fk;

,ilNa cs;s J}uk; 𝑥1

vdNt (𝑥1 , 10) vd;w Gs;sp rkd;ghL (1) I epiwT nra;Ak;.

𝑥1

2

242 +102

202 = 1

𝑥1 = 12 3

∴ jiuapypUe;J 10 mb cauj;jpy; tistpd;

mfyk; vd;gJ 2𝑥1 = 24 3

∴ Njitahd tistpd; mfyk; 24 3 mb.

8. xU EioT thapypd; Nkw;$iuahdJ miu ePs;tl;l tbtj;jpy; cs;sJ. ,jd; mfyk; 20mb. ikaj;jpypUe;J mjd; cauk; 18 mb kw;Wk; gf;fr; Rth;fspd; cauk; 12 mb vdpy; VNjDk; xU gf;fr; RthpypUe;J 4 mb J}uj;jpy; Nkw;$iuapd; cauk; vd;dthf ,Uf;Fk;? (MAR-07,10,17)

gf;fr; RthpypUe;J 4 mb J}uj;jpy; cs;s

Nkw;$iuapd; cauk; 𝑃𝑄𝑅 . glj;jpd; %yk;

𝑃𝑄 = 12 mb. 𝑄𝑅 ,d; ePsj;ij fhz Ntz;Lk;. mfyk; 20 mbahf ,Ug;gjhy;

Kidfs; 𝐴, 𝐴′ ,d; Maj;njhiyfs; KiwNa

(10,0) , (−10,0).

glj;jpd; %yk; 𝐴𝐴′ = 2𝑎 = 20





𝑎 = 10 kw;Wk; 𝑏 = 18 − 12 = 6

𝑥2

100+

𝑦2

36= 1

𝑄𝑅 vd;gJ 𝑦1 vdpy; 𝑅,d; Maj;njhiyfs;

(6, 𝑦1)

ePs;tl;lj;jpd; kPJ 𝑅 miktjhy; S 36

100+

𝑦12

36= 1 ⇒ 𝑦1 = 4.8

𝑃𝑄 + 𝑄𝑅 = 12 + 4.8

∴ Njitahd Nkw;$iuapd; cauk; 16.8

mbahFk;.

9. #hpad; Ftpaj;jpypUf;FkhW G+kpahdJ #hpaid xU ePs;tl;lg; ghijapy; Rw;wp tUfpwJ. mjd; miu-nel;lr;rpd; ePsk; 92.9 kpy;ypad; iky;fs; MfTk;> ikaj;njhiyj; jfT 0.017 MfTk; cs;sJ vdpy; G+kpahdJ #hpaDf;F kpf mUfhikapy; tUk;NghJ cs;s J}uKk; kpfj; njhiytpy; tUk;NghJ cs;s J}uKk; fhz;f.

miu nel;lr;rpd; ePsk; 𝐶𝐴

𝑎 = 92.9 kpy;ypad; iky;fs;

𝑒 = 0.017 vd nfhLf;fg;gl;Ls;sJ #hpaDf;F kpf mUfhikapy; tUk;NghJ

cs;s J}uk; = 𝐹𝐴, kw;Wk; kpfj; njhiytpy; ,Uf;Fk; NghJ cs;s J}uk;= 𝐹𝐴′

𝐶𝐹 = 𝑎𝑒 = 92.9(0.017)

𝐹𝐴 = 𝐶𝐴 − 𝐶𝐹 = 92.9 − 92.9(0.017)

= 92.9,1 − 0.017- = 92.9 × 0.983

= 91.3207 kpy;ypad; iky;fs;

𝐹𝐴′ = 𝐶𝐴′ + 𝐶𝐹 = 92.9 + 92.9(0.017)

= 92.9(1 + 0.017)

= 92.9(1.017) = 94.4793 kpy;ypad; iky;fs;

10. xU rkjsj;jpd; Nky; nrq;Fj;jhf mike;Js;s Rthpd; kPJ 15kP ePsKs;s xU VzpahdJ jsj;jpidAk; Rtw;wpidAk; njhLkhW efh;e;J nfhz;L ,Uf;fpwJ. vdpy;> Vzpapd; fPo;kl;l KidapypUe;J 6kP J}uj;jpy;

Vzpapy; mike;Js;s 𝑷 vd;w Gs;spapd; epakg;ghijiaf; fhz;f.

(OCT-07,OCT-08,MAR-12,JUN-15)

𝐴𝐵 vd;gJ Vzp. Vzpapd; kPJ 𝑃(𝑥1 , 𝑦1) vd;w

Gs;sp 𝐴𝑃 = 6kP ,Uf;FkhW vLj;Jf; nfhs;s

𝑥-mr;Rf;F nrq;Fj;jhf 𝑃𝐷 Ak; 𝑦-mr;Rf;F

nrq;Fj;jhf 𝑃𝐶 Ak; tiua

∆𝐴𝐷𝑃 kw;Wk; ∆𝑃𝐶𝐵 tbnthj;jit

𝑃𝐶

𝐷𝐴=

𝑃𝐵

𝐴𝑃=

𝐵𝐶

𝑃𝐷

𝑥1

𝐷𝐴=

9

6=

𝐵𝐶

𝑦1

𝐷𝐴 =6𝑥1

9=

2𝑥1

3

𝐵𝐶 =9𝑦1

6=

3𝑦1

2

𝑂𝐴 = 𝑂𝐷 + 𝐷𝐴

= 𝑥1 +2𝑥1

3=

5𝑥1

3

𝑂𝐵 = 𝑂𝐶 + 𝐵𝐶

= 𝑦1 +3𝑦1

2=

5𝑦1

2

𝑂𝐴2 + 𝑂𝐵2 = 𝐴𝐵2

25𝑥1

2

9+

25𝑦12

4= 225 ⇒

𝑥12

9+

𝑦12

4= 9

(𝑥1 , 𝑦1) d; epakg;ghij 𝑥2

81+

𝑦2

36= 1. ,J Xh;

ePs;tl;lkhFk;

khw;WKiw:

∠𝑃𝐴𝑂 = ∠𝐵𝑃𝐶 = 𝜃

∆𝑃𝐶𝐵 ,y;

cos 𝜃 =𝑥1

9

∆𝐴𝐷𝑃 ,y;

sin 𝜃 =𝑦1

6

cos2 𝜃 + sin2 𝜃 = 1

𝑥1

2

81+

𝑦12

36= 1

(𝑥1 , 𝑦1) d; epakg;ghij 𝑥2

81+

𝑦2

36= 1. ,J Xh;

ePs;tl;lkhFk;.

11. xU Nfh-Nfh tpiahl;L tPuh; tpisahl;Lg; gapw;rpapd; NghJ mtUf;Fk; Nfh-Nfh Fr;rpfSf;Fk; ,ilNaAs;s J}uk; vg;nghOJk; 8kP Mf ,Uf;FkhW czh;fpwhh;. mt;tpU Fr;rpfSf;F ,ilg;gl;l J}uk; 6kP vdpy; mth; XLk; ghijapd; rkd;ghl;ilf; fhz;f.(MAR-11,15)

Nfh-Nfh Fr;rpfs;

,uz;Lk; 𝐹1 kw;Wk;

𝐹2 ,y; mike;Js;sd.

𝑃(𝑥, 𝑦) vd;w Gs;spahdJ tpisahl;L tPuhpd; epiy





𝐹1𝑃 + 𝐹2𝑃 = 2𝑎 = 8 ⇒ 𝑎 = 4

𝐹1𝐹2 = 2𝑎𝑒 = 6 ⇒ 𝑎𝑒 = 3 ⇒ 4𝑒 = 3 ⇒ 𝑒 =3

4

NkYk; 𝑏2 = 𝑎2(1 − 𝑒2) = 16 .1 −9

16/ ⇒ 𝑏2 = 7

ghijapd; rkd;ghL 𝑥2

16+

𝑦2

7= 1

12. xU ePs;tl;lg; ghijapd; Ftpaj;jpy; G+kp ,Uf;FkhW xU Jizf;Nfhs; Rw;wp tUfpwJ.

,jd; ikaj; njhiyj;jT 𝟏

𝟐MfTk; G+kpf;Fk;

Jizf; NfhSf;Fk; ,ilg;gl;l kPr;rpW J}uk; 400 fpNyh kPl;lh;fs; MfTk; ,Uf;Fkhdhy; G+kpf;Fk; Jizf;NfhSf;Fk; ,ilg;gl;l mjpfgl;r J}uk; vd;d?

( JUN-07,JUN-08,JUN-12,JUN-14,OCT-14)

glj;jpypUe;J G+kpapd; epiy 𝐹1. Jizf;NfhSf;Fk;. G+kpf;Fk; ,ilg;gl;l

kPr;rpW J}uk; 𝐹1𝐴 = 400 fp.kP. G+kpf;Fk; > Jizf;NfhSf;Fk;

,ilg;gl;l mjpfgl;r J}uk; 𝐹1𝐴′ fzf;fpl

Ntz;Lk;

𝐶𝐴 = 𝑎, 𝐶𝐹1 = 𝑎𝑒, 𝐹1𝐴 = 400 fp.kP.

𝐹1𝐴 = 𝐶𝐴 − 𝐶𝐹1 = 𝑎 − 𝑎𝑒

400 = 𝑎(1 − 𝑒)

400 = 𝑎 .1 −1

2/

𝑎 = 800 fp.kP.

𝐶𝐴′ = 800 kw;Wk;

𝐶𝐹1 = 𝑎𝑒 = 800 ×1

2= 400fp.kP

𝐹1𝐴′ = 𝐹1𝐶 + 𝐶𝐴′ = 400 + 800 = 1200fp.kP.

13. #hpad; Ftpaj;jpypUf;FkhW nkh;Fhp fpufkhdJ #hpaid xU ePs;tl;lg;ghijapy; Rw;wp tUfpwJ. mjd; miu nel;lr;rpd; ePsk; 36 kpy;ypad; iky;fs; MfTk; ikaj;njhiyj;jfT 0.206 MfTk;

,Uf;Fkhapd; (i) nkh;f;Fhp fpufkhdJ #hpaDf;F kpf mUfhikapy; tUk;NghJ

cs;s J}uk; (ii) nkh;f;Fhp fpufkhdJ #hpaDf;F kpfj; njhiytpy; ,Uf;Fk;NghJ cs;s J}uk; Mfpatw;iwf; fhz;f.

(OCT-09,JUN-10,OCT-11,MAR-16)

glj;jpypUe;J #hpadpd; epiy 𝐹1

𝐶𝐴 = 36 kpy;ypad; iky;fs;,

𝑒 = 0.206 nkh;f;Fhp fpufkhdJ #hpaDf;F kpf mUfhikapy;> kpfj; njhiytpy; ,Uf;Fk;

epiyfs; 𝐴 kw;Wk; 𝐴′

(i) kpf mUfhikapy; J}uk; 𝐹1𝐴

𝐹1𝐴 = 𝐶𝐴 − 𝐶𝐹1 = 𝑎 − 𝑎𝑒 = 𝑎(1 − 𝑒)

= 36(1 − 0.206) = 36 × 0.794

mUfhik J}uk; = 28.584 kpy;ypad; iky;fs;

(ii) kpfj; njhiytpy; J}uk; 𝐹1𝐴′

𝐹1𝐴′ = 𝐹1𝐶 + 𝐶𝐴′ = 𝑎𝑒 + 𝑎 = 𝑎(𝑒 + 1)

= 36(1 + 0.206) = 1.206 × 36

= 43.416 kpy;ypad; iky;fs;.

14. 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? (OCT-10, JUN-11, MAR-14)

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

ehd;F tifahd gutisaq;fspd; KbTfspd; njhFg;G

15. 𝒚𝟐 − 𝟖𝒙 + 𝟔𝒚 + 𝟗 = 𝟎 vd;w gutisaj;jpd; mr;R> Kid> Ftpak;> ,af;Ftiu> nrt;tfyj;jpd; rkd;ghL kw;Wk; nrt;tfyj;jpd; ePsk; fhz;f. NkYk; mt;tistiuia tiuf

(JUN-08,OCT-10,14)

𝑦2 − 8𝑥 + 6𝑦 + 9 = 0

𝑦2 + 6𝑦 = 8𝑥 − 9

𝑦2 + 6𝑦 + 32 − 32 = 8𝑥 − 9

(𝑦 + 3)2 − 32 = 8𝑥 − 9

(𝑦 + 3)2 = 8𝑥 − 9 + 9

(𝑦 + 3)2 = 8𝑥

𝑌2 = 8𝑋 ,q;F 𝑋 = 𝑥, 𝑌 = 𝑦 + 3

𝑌2 = 4(2)𝑋

𝑎 = 2 gutisak; tyJGwk; jpwg;GilaJ

𝑋, 𝑌 I nghWj;J

𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥,

𝑌 = 𝑦 + 3

mr;R 𝑌 = 0 𝑌 = 0

⇒ 𝑦 + 3 = 0

Kid (0, 0) 𝑋 = 0, 𝑌 = 0

𝑥 = 0, 𝑦 + 3 = 0 𝑉(0, −3)

Ftpak;

(𝑎, 0) (2, 0)

𝑋 = 2, 𝑌 = 0 𝑥 = 2, 𝑦 + 3 = 0

𝐹(2, −3)

,af;Ftiu 𝑋 = −𝑎 𝑋 = −2

𝑋 = −2 𝑥 = −2

nrt;tfyk; 𝑋 = 𝑎 𝑋 = 2

𝑋 = 2 𝑥 = 2

nrt;tfyj;jpd; ePsk; 4𝑎 = 8 8

16. 𝒙𝟐 − 𝟐𝒙 + 𝟖𝒚 + 𝟏𝟕 = 𝟎 vd;w gutisaj;jpd; mr;R> Kid> Ftpak;> ,af;Ftiu> nrt;tfyj;jpd; rkd;ghL kw;Wk; nrt;tfyj;jpd; ePsk; fhz;f. NkYk; mt;tistiuia tiuf

( MAR-15)

𝑥2 − 2𝑥 + 8𝑦 + 17 = 0

𝑥2 − 2𝑥 = −8𝑦 − 17

𝑥2 − 2𝑥 + 12 − 12 = −8𝑦 − 17

(𝑥 − 1)2 − 12 = −8𝑦 − 17

(𝑥 − 1)2 = −8𝑦 − 17 + 1

(𝑥 − 1)2 = −8𝑦 − 16





(𝑥 − 1)2 = −8(𝑦 + 2)

𝑋2 = −8𝑌

,q;F 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 + 2

𝑋2 = −4(2)𝑌

𝑎 = 2 gutisak; fPo;Gwk; jpwg;GilaJ


𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 + 2

mr;R 𝑋 = 0 𝑋 = 0 ⇒ 𝑥 − 1 = 0

⇒ 𝑥 = 1

Kid (0, 0) 𝑋 = 0, 𝑌 = 0

𝑥 − 1 = 0, 𝑦 + 2 = 0 𝑉(1, −2)

Ftpak;

(0, −𝑎) (0, −2)

𝑋 = 0, 𝑌 = −2 𝑥 − 1 = 0, 𝑦 + 2 = −2

𝐹(1, −4)

,af;Ftiu 𝑌 = 𝑎 𝑌 = 2

𝑌 = 2 ⇒ 𝑦 + 2 = 2 𝑦 = 0

nrt;tfyk; 𝑌 = −𝑎 𝑌 = −2

𝑌 = −2 ⇒ 𝑦 + 2 = −2

𝑦 = −4 nrt;tfyj;jpd;

ePsk; 4𝑎 = 8 8

17. 𝒙𝟐 − 𝟒𝒙 + 𝟒𝒚 = 𝟎 vd;w gutisaj;jpd; mr;R> Kid> Ftpak;> ,af;Ftiu> nrt;tfyj;jpd; rkd;ghL kw;Wk; nrt;tfyj;jpd; ePsk; fhz;f.

NkYk; mt;tistiuia tiuf. (JUN-11)

𝑥2 − 4𝑥 + 4𝑦 = 0

𝑥2 − 4𝑥 = −4𝑦

𝑥2 − 4𝑥 + 22 − 22 = −4𝑦

(𝑥 − 2)2 − 4 = −4𝑦

(𝑥 − 2)2 = −4𝑦 + 4

(𝑥 − 2)2 = −4(𝑦 − 1)

(𝑥 − 2)2 = −4(𝑦 − 1)

𝑋2 = −4𝑌 ,q;F 𝑋 = 𝑥 − 2, 𝑌 = 𝑦 − 1

𝑋2 = −4(1)𝑌



𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 − 2, 𝑌 = 𝑦

− 1

mr;R 𝑋 = 0 𝑋 = 0 ⇒ 𝑥 − 2 = 0

⇒ 𝑥 = 2

Kid (0, 0)

𝑋 = 0, 𝑌 = 0 𝑥 − 2 = 0, 𝑦 − 1

= 0 𝑉(2,1)

Ftpak;

(0, −𝑎) (0, −1)

𝑋 = 0, 𝑌 = −1 𝑥 − 2 = 0, 𝑦 − 1= −1

𝐹(2, 0)

,af;Ftiu 𝑌 = 𝑎 𝑌 = 1

𝑌 = 1 ⇒ 𝑦 − 1 = 1 𝑦 = 2

nrt;tfyk; 𝑌 = −𝑎 𝑌 = −1

𝑌 = −1 ⇒ 𝑦 − 1= −2

𝑦 = −1 nrt;tfyj;jpd;

ePsk ; 4𝑎 = 4 4

18. 𝒚𝟐 + 𝟖𝒙 − 𝟔𝒚 + 𝟏 = 𝟎 vd;w gutisaj;jpd; mr;R> Kid> Ftpak;> ,af;Ftiu> nrt;tfyj;jpd; rkd;ghL kw;Wk; nrt;tfyj;jpd; ePsk; fhz;f. NkYk; mt;tistiuia tiuf

(OCT-06,MAR-07,16)

𝑦2 + 8𝑥 − 6𝑦 + 1 = 0

𝑦2 − 6𝑦 = −8𝑥 − 1

𝑦2 − 6𝑦 + 32 − 32 = −8𝑥 − 1

(𝑦 − 3)2 − 9 = −8𝑥 − 1

(𝑦 − 3)2 = −8𝑥 − 1 + 9

= −8𝑥 + 8 = −8(𝑥 − 1)

𝑌2 = −8𝑋 ,q;F 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 − 3

𝑌2 = −4(2)𝑋







𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 − 3

mr;R 𝑌 = 0 𝑌 = 0 ⇒ 𝑦 − 3 = 0

Kid (0, 0) 𝑋 = 0, 𝑌 = 0

𝑥 − 1 = 0, 𝑦 − 3 = 0 𝑉(1,3)

Ftpak;

(−𝑎, 0) (−2, 0)

𝑋 = −2, 𝑌 = 0 𝑥 − 1 = −2, 𝑦 − 3 = 0

𝐹(−1,3)

,af;Ftiu 𝑋 = 𝑎 𝑋 = 2

𝑋 = 2 𝑥 − 1 = 2 𝑥 − 3 = 0

nrt;tfyk; 𝑋 = −𝑎 𝑋 = −2

𝑋 = −2 𝑥 − 1 = −2 𝑥 + 1 = 0


19. 𝒙𝟐 − 𝟔𝒙 − 𝟏𝟐𝒚 − 𝟑 = 𝟎 vd;w gutisaj;jpd; mr;R> Kid> Ftpak;> ,af;Ftiu> nrt;tfyj;jpd; rkd;ghL kw;Wk; nrt;tfyj;jpd; ePsk; fhz;f. NkYk; mt;tistiuia tiuf

(MAR-10)

𝑥2 − 6𝑥 − 12𝑦 − 3 = 0

𝑥2 − 6𝑥 = 12𝑦 + 3

𝑥2 − 6𝑥 + 32 − 32 = 12𝑦 + 3

(𝑥 − 3)2 − 9 = 12𝑦 + 3

(𝑥 − 3)2 = 12𝑦 + 12

(𝑥 − 3)2 = 12(𝑦 + 1)

𝑋2 = 12𝑌 ,q;F 𝑋 = 𝑥 − 3, 𝑌 = 𝑦 + 1

𝑋2 = 4(3)𝑌



𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 − 3, 𝑌 = 𝑦

+ 1

mr;R 𝑋 = 0 𝑋 = 0 ⇒ 𝑥 − 3 = 0

⇒ 𝑥 = 3

Kid (0, 0) 𝑋 = 0, 𝑌 = 0

𝑥 − 3 = 0, 𝑦 + 1 = 0 𝑉(3, −1)

Ftpak;

(0, 𝑎) (0, 3)

𝑋 = 0, 𝑌 = 3 𝑥 − 3 = 0, 𝑦 + 1 = 3

𝐹(3,2)

,af;Ftiu 𝑌 = −𝑎 𝑌 = −3

𝑌 = −3 ⇒ 𝑦 + 1= −3

𝑦 + 4 = 0

nrt;tfyk; 𝑌 = 𝑎 𝑌 = 3

𝑌 = 3 ⇒ 𝑦 + 1 = 3 𝑦 − 2 = 0

nrt;tfyj;jpd; ePsk; 4𝑎 = 12 4𝑎 = 12

20. 𝒚𝟐 − 𝟖𝒙 − 𝟐𝒚 + 𝟏𝟕 = 𝟎 vd;w gutisaj;jpd; mr;R> Kid> Ftpak;> ,af;Ftiu> nrt;tfyj;jpd; rkd;ghL kw;Wk; nrt;tfyj;jpd; ePsk; fhz;f. NkYk; mt;tistiuia tiuf

( JUN-07)

𝑦2 − 8𝑥 − 2𝑦 + 17 = 0

𝑦2 − 2𝑦 = 8𝑥 − 17

𝑦2 − 2𝑦 + 12 − 12 = 8𝑥 − 17

(𝑦 − 1)2 − 1 = 8𝑥 − 17

(𝑦 − 1)2 = 8𝑥 − 17 + 1

(𝑦 − 1)2 = 8𝑥 − 16

(𝑦 − 1)2 = 8(𝑥 − 2)

𝑌2 = 8𝑋 ,q;F 𝑋 = 𝑥 − 2, 𝑌 = 𝑦 − 1

𝑌2 = 4(2)𝑋



𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 − 2, 𝑌 = 𝑦 − 1

mr;R 𝑌 = 0 𝑌 = 0 ⇒ 𝑦 − 1 = 0

Kid (0, 0) 𝑋 = 0, 𝑌 = 0

𝑥 − 2 = 0, 𝑦 − 1 = 0 𝑉(2,1)

Ftpak;

(𝑎, 0) (2, 0)

𝑋 = 2, 𝑌 = 0 𝑥 − 2 = 2, 𝑦 − 1 = 0

𝐹(4,1)





,af;Ftiu 𝑋 = −𝑎 𝑋 = −2

𝑋 = −2 𝑥 − 2 = −2 ⇒ 𝑥 = 0

nrt;tfyk; 𝑋 = 𝑎 𝑋 = 2

𝑋 = 2 𝑥 − 2 = 2 ⇒ 𝑥 = 4


21. 𝒚𝟐 + 𝟒𝒚 + 𝟒𝒙 + 𝟖 = 𝟎 vd;w gutisaj;jpd; mr;R> Kid> Ftpak;> ,af;Ftiu> nrt;tfyj;jpd; rkd;ghL kw;Wk; nrt;tfyj;jpd; ePsk; fhz;f.

NkYk; mt;tistiuia tiuf ( MAR-11 )

𝑦2 + 4𝑦 = −4𝑥 − 8

𝑦2 + 4𝑦 + 22 − 22 = −4𝑥 − 8

(𝑦 + 2)2 − 22 = −4𝑥 − 8

(𝑦 + 2)2 = −4𝑥 − 8 + 4

(𝑦 + 2)2 = −4𝑥 − 4

(𝑦 + 2)2 = −4(𝑥 + 1)

𝑌2 = −4𝑋 ,q;F 𝑋 = 𝑥 + 1, 𝑌 = 𝑦 + 2



𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 + 1, 𝑌 = 𝑦 + 2

mr;R 𝑌 = 0 𝑌 = 0 ⇒ 𝑦 + 2 = 0

Kid (0, 0) 𝑋 = 0, 𝑌 = 0

𝑥 + 1 = 0, 𝑦 + 2 = 0 𝑉(−1, −2)

Ftpak;

(−𝑎, 0) (−1, 0)

𝑋 = −1, 𝑌 = 0 𝑥 + 1 = −1, 𝑦 + 2 = 0

𝐹(−2, −2)

,af;Ftiu 𝑋 = 𝑎 𝑋 = 1

𝑋 = 1 𝑥 + 1 = 1

𝑥 = 0

nrt;tfyk; 𝑋 = −𝑎 𝑋 = −1

𝑋 = −1 𝑥 + 1 = −1 𝑥 + 2 = 0


ePs;tl;lj;jpd; ,U tbtq;fs;

𝑥2

𝑎2+

𝑦2

𝑏2= 1

𝑥2

𝑏2+

𝑦2

𝑎2= 1

ikak; 𝐶(0,0) 𝐶(0,0)

Kidfs; 𝐴(𝑎, 0), 𝐴′(−𝑎, 0)

𝐴(0, 𝑎) 𝐴′ (0, −𝑎)

Ftpaq;fs; 𝐹1(𝑎𝑒, 0), 𝐹2(−𝑎𝑒, 0)

𝐹1(0, 𝑎𝑒) 𝐹2(0, −𝑎𝑒)

22. 𝟑𝟔𝒙𝟐 + 𝟒𝒚𝟐 − 𝟕𝟐𝒙 + 𝟑𝟐𝒚 − 𝟒𝟒 = 𝟎 vd;w ePs;tl;lj;jpd; ikaj;njhiyj;jfT> ikak;> Kidfs;> Ftpaq;fs; Mfpatw;iwf; fhz;f. kw;Wk; mjd; tiuglk; tiuf

( MAR-06,JUN-06,OCT-15 )

36𝑥2 + 4𝑦2 − 72𝑥 + 32𝑦 − 44 = 0

36𝑥2 − 72𝑥 + 4𝑦2 + 32𝑦 − 44 = 0

36(𝑥2 − 2𝑥) + 4(𝑦2 + 8𝑦) − 44 = 0

36(𝑥2 − 2𝑥 + 12 − 12)

+4(𝑦2 + 8𝑦 + 42 − 42) − 44 = 0

36*(𝑥 − 1)2 − 1+ + 4*(𝑦 + 4)2 − 16+ = 44

36(𝑥 − 1)2 − 36 + 4(𝑦 + 4)2 − 64 = 44

36(𝑥 − 1)2 + 4(𝑦 + 4)2 = 44 + 36 + 64

36(𝑥 − 1)2 + 4(𝑦 + 4)2 = 144

36(𝑥−1)2

144+

4(𝑦+4)2

144=

144

144

(𝑥−1)2

4+

(𝑦+4)2

36= 1





𝑋2

4+

𝑌2

36= 1 ,q;F 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 + 4

nel;lr;R 𝑌-mr;Rtopr; nry;fpwJ

𝑎2 = 36, 𝑏2 = 4,

𝑎 = 6, 𝑏 = 2

𝑒 = 1 −𝑏2

𝑎2 = 1 −4

36 =

36−4

36=

32

36

= 16×2

6×6=

4 2

6 =

2 2

3

𝑎𝑒 = 6 ×2 2

3= 4 2


𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 + 4

ikak; (0, 0) 𝑋 = 0, 𝑌 = 0

𝑥 − 1 = 0, 𝑦 + 4 = 0 𝐶(1, −4)

Kidfs; (0, ±𝑎) (0, ±6)

(0, 𝑎) ⇒ (0,6) 𝑋 = 0, 𝑌 = 6

𝑥 − 1 = 0, 𝑦 + 4 = 6 𝐴(1,2)

(0, −𝑎) ⇒ (0, −6) 𝑋 = 0, 𝑌 = −6

𝑥 − 1 = 0, 𝑦 + 4 = −6 𝐴′(1, −10)

Ftpaq;fs; (0, ±𝑎𝑒)

(0, ±4 2)

(0, 4 2)

𝑋 = 0, 𝑌 = 4 2 𝑥 − 1 = 0, 𝑦 + 4 = 4 2

𝑥 = 1, 𝑦 = −4 + 4 2

𝐹1(1, −4 + 4 2)

(0, −4 2)

𝑋 = 0, 𝑌 = −4 2 𝑥 − 1 = 0, 𝑦 + 4 = −4 2

𝑥 = 1, 𝑦 = −4 − 4 2

𝐹2(1, −4 − 4 2)

23. 𝒙𝟐 + 𝟒𝒚𝟐 − 𝟖𝒙 − 𝟏𝟔𝒚 − 𝟔𝟖 = 𝟎 vd;w ePs;tl;lj;jpd; ikaj;njhiyj;jfT> ikak;> Kidfs;> Ftpaq;fs; Mfpatw;iwf; fhz;f. kw;Wk; mjd; tiuglk; tiuf

(𝑥2 − 8𝑥) + (4𝑦2 − 16𝑦) − 68 = 0

(𝑥2 − 8𝑥 + 42 − 42) + 4(𝑦2 − 4𝑦 + 22 − 22) = 68

*(𝑥 − 4)2 − 16+ + 4*(𝑦 − 2)2 − 4+ = 68

(𝑥 − 4)2 + 4(𝑦 − 2)2 = 68 + 16 + 16

(𝑥 − 4)2 + 4(𝑦 − 2)2 = 100

(𝑥−4)2

100+

4(𝑦−2)2

100=

100

100

(𝑥 − 4)2

100+

(𝑦 − 2)2

25= 1

nel;lr;R 𝑥- mr;Rf;F ,izahf cs;sJ

𝑋 = 𝑥 − 4, 𝑌 = 𝑦 − 2

𝑋2

100+

𝑌2

25= 1

𝑎2 = 100, 𝑏2 = 25, 𝑎 = 10, 𝑏 = 5

𝑒 = 1 −25

100=

100−25

100 =

75

100=

25×3

10×10

=5 3

10=

3

2 ⇒ 𝑎𝑒 = 10 ×

3

2= 5 3


𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 − 4, 𝑌 = 𝑦 − 2

ikak; (0,0) 𝑋 = 0, 𝑌 = 0

𝑥 − 4 = 0, 𝑦 − 2 = 0 𝐶(4,2)

Ftpaq;fs; (±𝑎𝑒, 0)

(±5 3, 0)

5 3, 0

𝑋 = 5 3, 𝑌 = 0 𝑥 − 4 = 5 3, 𝑦 − 2 = 0

𝑥 = 4 + 5 3, 𝑦 = 2

𝐹1(4 + 5 3, 2)

−5 3, 0

𝑋 = −5 3, 𝑌 = 0 𝑥 − 4 = −5 3, 𝑦 − 2 = 0

𝑥 = 4 − 5 3, 𝑦 = 2

𝐹2(4 − 5 3, 2)

Kidfs; (±𝑎, 0) (±10,0)

(10, 0) 𝑋 = 10, 𝑌 = 0

𝑥 − 4 = 10, 𝑦 − 2 = 0

𝑥 = 14, 𝑦 = 2

𝐴(14,2) (−10,0)

𝑋 = −10, 𝑌 = 0 𝑥 − 4 = −10, 𝑦 − 2 = 0

𝑥 = −6, 𝑦 = 2

𝐴′(−6,2)





24. 𝟏𝟔𝒙𝟐 + 𝟗𝒚𝟐 + 𝟑𝟐𝒙 − 𝟑𝟔𝒚 = 𝟗𝟐 vd;w ePs;tl;lj;jpd; ikaj;njhiyj;jfT> ikak;> Kidfs;> Ftpaq;fs; Mfpatw;iwf; fhz;f. kw;Wk; mjd; tiuglk; tiuf

(OCT-12, MAR-13, JUN-16)

16𝑥2 + 9𝑦2 + 32𝑥 − 36𝑦 = 92

16𝑥2 + 32𝑥 + 9𝑦2 − 36𝑦 = 92

16(𝑥2 + 2𝑥) + 9(𝑦2 − 4𝑦) = 92

16(𝑥2 + 2𝑥 + 12 − 12) + 9(𝑦2 − 4𝑦 + 22 − 22) = 92

16*(𝑥 + 1)2 − 1+ + 9*(𝑦 − 2)2 − 4+ = 92

16(𝑥 + 1)2 − 16 + 9(𝑦 − 2)2 − 36 = 92

16(𝑥 + 1)2 + 9(𝑦 − 2)2 = 92 + 16 + 36

16(𝑥 + 1)2

144+

9(𝑦 − 2)2

144=

144

144

(𝑥 + 1)2

9+

(𝑦 − 2)2

16= 1

𝑋2

9+

𝑌2

16= 1 ,q;F 𝑋 = 𝑥 + 1, 𝑌 = 𝑦 − 2

nel;lr;R 𝑌 -mr;Rf;F ,iz

𝑎2 = 16, 𝑏2 = 9, 𝑎 = 4, 𝑏 = 3

𝑒 = 1 −𝑏2

𝑎2 = 1 −9

16 =

16−9

16=

7

16=

7

4

𝑎𝑒 = 4 × 7

4= 7


𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 + 1, 𝑌 = 𝑦 − 2

ikak; (0,0) 𝑋 = 0, 𝑌 = 0

𝑥 + 1 = 0, 𝑦 − 2 = 0 𝐶(−1,2)

Kidfs; (0, ±𝑎) (0, ±4)

(0,4) 𝑋 = 0, 𝑌 = 4

𝑥 + 1 = 0, 𝑦 − 2 = 4 𝐴(−1,6) (0, −4)

𝑋 = 0, 𝑌 = −4 𝑥 + 1 = 0, 𝑦 − 2 = −4

𝐴′(−1, −2)


(0, ± 7)

(0, 7)

𝑋 = 0, 𝑌 = 7

𝑥 + 1 = 0, 𝑦 − 2 = 7

𝑥 = −1, 𝑦 = 2 + 7

𝐹1(−1, 2 + 7)

(0, 7)

𝑋 = 0, 𝑌 = − 7

𝑥 + 1 = 0, 𝑦 − 2 = − 7

𝑥 = −1, 𝑦 = 2 − 7

𝐹2(−1, 2 − 7)

25. 𝟏𝟔𝒙𝟐 + 𝟗𝒚𝟐 − 𝟑𝟐𝒙 + 𝟑𝟔𝒚 − 𝟗𝟐 = 𝟎 vd;w ePs;tl;lj;jpd; ikaj;njhiyj;jfT> ikak;> Kidfs;> Ftpaq;fs; Mfpatw;iwf; fhz;f. kw;Wk; mjd; tiuglk; tiuf ( OCT -16)

16𝑥2 + 9𝑦2 − 32𝑥 + 36𝑦 = 92

16𝑥2 − 32𝑥 + 9𝑦2 + 36𝑦 = 92

16(𝑥2 − 2𝑥) + 9(𝑦2 + 4𝑦) = 92

16(𝑥2 − 2𝑥 + 12 − 12) + 9(𝑦2 + 4𝑦 + 22 − 22) = 92

16*(𝑥 − 1)2 − 1+ + 9*(𝑦 + 2)2 − 4+ = 92

16(𝑥 − 1)2 − 16 + 9(𝑦 + 2)2 − 36 = 92

16(𝑥 − 1)2 + 9(𝑦 + 2)2 = 92 + 16 + 36

16(𝑥 − 1)2

144+

9(𝑦 + 2)2

144=

144

144

(𝑥 − 1)2

9+

(𝑦 + 2)2

16= 1

𝑋2

9+

𝑌2

16= 1 ,q;F 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 + 2

nel;lr;R 𝑌 -mr;Rf;F ,iz

𝑎2 = 16, 𝑏2 = 9, 𝑎 = 4, 𝑏 = 3

𝑒 = 1 −𝑏2

𝑎2 = 1 −9

16 =

16−9

16=

7

16=

7

4

𝑎𝑒 = 4 × 7

4= 7






𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 + 2

ikak; (0,0) 𝑋 = 0, 𝑌 = 0

𝑥 − 1 = 0, 𝑦 + 2 = 0 𝐶(1, −2)

Kidfs; (0, ±𝑎) (0, ±4)

(0,4) 𝑋 = 0, 𝑌 = 4

𝑥 − 1 = 0, 𝑦 + 2 = 4 𝐴(1,2) (0, −4)

𝑋 = 0, 𝑌 = −4 𝑥 − 1 = 0, 𝑦 + 2

= −4 𝐴′(1, −6)


(0, ± 7)

(0, 7)

𝑋 = 0, 𝑌 = 7

𝑥 − 1 = 0, 𝑦 + 2 = 7

𝑥 = 1, 𝑦 = −2 + 7

𝐹1(−1, − 2 + 7)

(0, − 7)

𝑋 = 0, 𝑌 = − 7

𝑥 − 1 = 0, 𝑦 + 2 = − 7

𝑥 = 1, 𝑦 = −2 − 7

𝐹2(1, −2 − 7)

26. 𝟗𝒙𝟐 + 𝟐𝟓𝒚𝟐 − 𝟏𝟖𝒙 − 𝟏𝟎𝟎𝒚 − 𝟏𝟏𝟔 = 𝟎 vd;w ePs;tl;lj;jpd; ikaj;njhiyj;jfT> ikak;> Kidfs;> Ftpaq;fs;> Mfpatw;iwf; fhz;f. kw;Wk; mjd; tiuglk; tiuf (MAR-09)

9𝑥2 + 25𝑦2 − 18𝑥 − 100𝑦 − 116 = 0

9𝑥2 − 18𝑥 + 25𝑦2 − 100𝑦 = 116

9(𝑥2 − 2𝑥) + 25(𝑦2 − 4𝑦) = 116

9(𝑥2 − 2𝑥 + 12 − 12)

+25(𝑦2 − 4𝑦 + 22 − 22) = 116

9*(𝑥 − 1)2 − 1+ + 25*(𝑦 − 2)2 − 4+ = 116

9(𝑥 − 1)2 − 9 + 25(𝑦 − 2)2 − 100 = 116

9(𝑥 − 1)2 + 25(𝑦 − 2)2 = 225

9(𝑥 − 1)2 + 25(𝑦 − 2)2 = 225

(𝑥 − 1)2

25+

(𝑦 − 2)2

9= 1

𝑋2

25+

𝑌2

9= 1 ,q;F 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 − 2

nel;lr;R 𝑋 -mr;R topNa nry;fpwJ

𝑎2 = 25, 𝑏2 = 9, 𝑎 = 5, 𝑏 = 3

𝑒 = 1 −9

25 =

25−9

25

= 16

25=

4×4

5×5

=4

5

𝑎𝑒 = 5 ×4

5= 4


𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 − 2

ikak; (0,0) 𝑋 = 0, 𝑌 = 0

𝑥 − 1 = 0, 𝑦 − 2 = 0 𝐶(1,2)

Ftpaq;fs; (±𝑎𝑒, 0) (±4, 0)

(4, 0) 𝑋 = 4, 𝑌 = 0

𝑥 − 1 = 4, 𝑦 − 2 = 0

𝑥 = 5, 𝑦 = 2 , 𝐹1(5,2) (−4, 0)

𝑋 = −4, 𝑌 = 0 𝑥 − 1 = −4, 𝑦 − 2 = 0

𝑥 = −3, 𝑦 = 2 𝐹2(−3,2)

Kidfs; (±𝑎, 0) (±5,0)

(5, 0) 𝑋 = 5, 𝑌 = 0

𝑥 − 1 = 5, 𝑦 − 2 = 0 𝑥 = 6, 𝑦 = 2

𝐴(6, 2) (−5,0)

𝑋 = −5, 𝑌 = 0 𝑥 − 1 = −5, 𝑦 − 2 = 0

𝑥 = −4, 𝑦 = 2

𝐴′(−4,2)





mjpgutisaj;jpd; ,U tbtq;fs;

𝑥2

𝑎2−

𝑦2

𝑏2= 1

𝑥2

𝑏2−

𝑦2

𝑎2= 1

ikak; 𝐶(0,0) 𝐶(0,0)

Kidfs; 𝐴(𝑎, 0), 𝐴′(−𝑎, 0)

𝐴(0, 𝑎) 𝐴′ (0, −𝑎)

Ftpaq;fs; 𝐹1(𝑎𝑒, 0), 𝐹2(−𝑎𝑒, 0)

𝐹1(0, 𝑎𝑒) 𝐹2(0, −𝑎𝑒)

tiuglk;

27. 𝟗𝒙𝟐 − 𝟏𝟔𝒚𝟐 − 𝟏𝟖𝒙 − 𝟔𝟒𝒚 − 𝟏𝟗𝟗 = 𝟎 vd;w mjpgutisaj;jpd; ikaj; njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f NkYk; mjd; tistiuia tiuf.

(OCT-11)

9𝑥2 − 18𝑥 − 16𝑦2 − 64𝑦 = 199

9(𝑥2 − 2𝑥) − 16(𝑦2 + 4𝑦) = 199 9(𝑥2 − 2𝑥 + 12 − 12) − 16(𝑦2 + 4𝑦 + 22 − 22) = 199

9{(𝑥 − 1)2 − 1} − 16*(𝑦 + 2)2 − 4+ = 199

9(𝑥 − 1)2 − 16(𝑦 + 2)2 = 199 + 9 − 64

9(𝑥 − 1)2 − 16(𝑦 + 2)2 = 144

9(𝑥 − 1)2

144−

16(𝑦 + 2)2

144=

144

144

(𝑥 − 1)2

16−

(𝑦 + 2)2

9= 1

𝑋2

16−

𝑌2

9= 1 ,q;F 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 + 2

FWf;fr;R 𝑋-mr;rpw;F ,izahf cs;sJ.

𝑎2 = 16, 𝑏2 = 9, 𝑎 = 4, 𝑏 = 3

𝑒 = 1 +𝑏2

𝑎2 = 1 +9

16 =

16+9

16=

25

16=

5

4

𝑎𝑒 = 4 ×5

4= 5


𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 + 2

ikak; (0,0) 𝑋 = 0, 𝑌 = 0

𝑥 − 1 = 0, 𝑦 + 2 = 0 𝐶(1, −2)

Ftpaq;fs; (±𝑎𝑒, 0) (5, 0)

(±5, 0) 𝑋 = 5, 𝑌 = 0 𝑥 − 1 = 5, 𝑦 + 2 = 0

𝑥 = 6, 𝑦 = −2 𝐹1(6, −2) (−5, 0)

𝑋 = −5, 𝑌 = 0 𝑥 − 1 = −5, 𝑦 + 2 = 0

𝑥 = −4, 𝑦 = −2 𝐹2(−4, −2)

Kidfs; (±𝑎, 0) (±4,0)

(4, 0) 𝑋 = 4, 𝑌 = 0

𝑥 − 1 = 4, 𝑦 + 2 = 0 𝑥 = 5, 𝑦 = −2

𝐴(5, −2) (−4,0)

𝑋 = −4, 𝑌 = 0 𝑥 − 1 = −4, 𝑦 + 2 = 0

𝑥 = −3, 𝑦 = −2

𝐴′(−3, −2)

28. 𝟗𝒙𝟐 − 𝟏𝟔𝒚𝟐 + 𝟑𝟔𝒙 + 𝟑𝟐𝒚 + 𝟏𝟔𝟒 = 𝟎 vd;w mjpgutisaj;jpd; ikaj; njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f NkYk; mjd; tistiuia tiuf. ( JUN-15)

9𝑥2 − 16𝑦2 + 36𝑥 + 32𝑦 + 164 = 0

9(𝑥2 + 4𝑥) − 16(𝑦2 − 2𝑦) = −164 9(𝑥2 + 4𝑥 + 22 − 22) − 16(𝑦2 − 2𝑦 + 12 − 12) = −164

9{(𝑥 + 2)2 − 4} − 16*(𝑦 − 1)2 − 1+ = −164

9(𝑥 + 2)2 − 16(𝑦 − 1)2 = −144

16(𝑦 − 1)2 − 9(𝑥 + 2)2 = 144

16(𝑦 − 1)2

144−

9(𝑥 + 2)2

144=

144

144

(𝑦 − 1)2

9−

(𝑥 + 2)2

16= 1

𝑌2

9−

𝑋2

16= 1 ,q;F 𝑋 = 𝑥 + 2, 𝑌 = 𝑦 − 1

FWf;fr;R 𝑌-mr;rpw;F ,izahf cs;sJ.

𝑎2 = 9, 𝑏2 = 16, 𝑎 = 3, 𝑏 = 4

𝑒 = 1 +𝑏2

𝑎2 = 1 +16

9 =

16+9

9=

25

9=

5

3





𝑎𝑒 = 3 ×5

3= 5


𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 + 2, 𝑌 = 𝑦 − 1

ikak; (0,0) 𝑋 = 0, 𝑌 = 0

𝑥 + 2 = 0, 𝑦 − 1 = 0 𝐶(−2,1)

Kidfs; (0, ±𝑎) (0, ±3)

(0,3) 𝑋 = 0, 𝑌 = 3

𝑥 + 2 = 0, 𝑦 − 1 = 3 𝐴(−2,4) (0, −3)

𝑋 = 0, 𝑌 = −3 𝑥 + 2 = 0, 𝑦 − 1 = −3

𝐴(−2, −2)

Ftpaq;fs; (0, ±𝑎𝑒) (0, ±5)

(0, 5) 𝑋 = 0, 𝑌 = 5

𝑥 + 2 = 0, 𝑦 − 1 = 5

𝑥 = −2, 𝑦 = 1 + 5

𝐹1(−2,6) (0, −5)

𝑋 = 0, 𝑌 = −5 𝑥 + 2 = 0, 𝑦 − 1 = −5

𝑥 = −2, 𝑦 = 1 − 5

𝐹2(−2, −4)

29. 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.

(MAR-10,JUN-13)

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


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 = 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)

30. 𝒙𝟐 − 𝟑𝒚𝟐 + 𝟔𝒙 + 𝟔𝒚 + 𝟏𝟖 = 𝟎 vd;w mjpgutisaj;jpd; ikaj; njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f NkYk; mjd; tistiuia tiuf. (MAR-08,OCT-08,OCT-09,JUN-10,MAR-12,MAR-14)

𝑥2 − 3𝑦2 + 6𝑥 + 6𝑦 + 18 = 0





𝑥2 + 6𝑥 − 3𝑦2 + 6𝑦 + 18 = 0

(𝑥2 + 6𝑥) − 3(𝑦2 − 2𝑦) = −18 (𝑥2 + 6𝑥 + 32 − 32) − 3(𝑦2 − 2𝑦 + 12 − 12) = −18

{(𝑥 + 3)2 − 9} − 3*(𝑦 − 1)2 − 1+ = −18

(𝑥 + 3)2 − 3(𝑦 − 1)2 = −18 + 9 − 3

(𝑥 + 3)2 − 3(𝑦 − 1)2 = −12

3(𝑦 − 1)2 − (𝑥 + 3)2 = 12

3(𝑦−1)2

12−

(𝑥+3)2

12=

12

12

(𝑦−1)2

4−

(𝑥+3)2

12= 1

𝑌2

4−

𝑋2

12= 1 ,q;F 𝑋 = 𝑥 + 3, 𝑌 = 𝑦 − 1


𝑎2 = 4, 𝑏2 = 12,

𝑎 = 2, 𝑏 = 2 3

𝑒 = 1 +𝑏2

𝑎2 = 1 +12

4=

4+12

4=

16

4= 2

𝑎𝑒 = 2(2) = 4


𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 + 3, 𝑌 = 𝑦 − 1

ikak; (0,0) 𝑋 = 0, 𝑌 = 0

𝑥 + 3 = 0, 𝑦 − 1 = 0 𝐶(−3,1)

Kidfs; (0, ±𝑎) (0, ±2)

(0,2) 𝑋 = 0, 𝑌 = 2

𝑥 + 3 = 0, 𝑦 − 1 = 2 𝐴(−3,3) (0, −2)

𝑋 = 0, 𝑌 = −2 𝑥 + 3 = 0, 𝑦 − 1 = −2

𝐴(−3, −1)

Ftpaq;fs; (0, ±𝑎𝑒) (0, ±4)

(0, 4) 𝑋 = 0, 𝑌 = 4

𝑥 + 3 = 0, 𝑦 − 1 = 4

𝑥 = −3, 𝑦 = 1 + 4

𝐹1(−3,5) (0, −4)

𝑋 = 0, 𝑌 = −4 𝑥 + 3 = 0, 𝑦 − 1 = −4

𝑥 = −3, 𝑦 = 1 − 4

𝐹2(−3, −3)

31. 𝟗𝒙𝟐 − 𝟕𝒚𝟐 + 𝟑𝟔𝒙 + 𝟏𝟒𝒚 + 𝟗𝟐 = 𝟎 vd;w mjpgutisaj;jpd; ikaj; njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f NkYk; mjd; tistiuia tiuf.

( JUN-12 )

9𝑥2 − 7𝑦2 + 36𝑥 + 14𝑦 + 92 = 0

9𝑥2 + 36𝑥 − 7𝑦2 + 14𝑦 + 92 = 0

9(𝑥2 + 4𝑥) − 7(𝑦2 − 2𝑦) = −92 9(𝑥2 + 4𝑥 + 22 − 22) − 7(𝑦2 − 2𝑦 + 12 − 12) = −92

9{(𝑥 + 2)2 − 4} − 7*(𝑦 − 1)2 − 1+ = −92

9(𝑥 + 2)2 − 7(𝑦 − 1)2 = −92 + 36 − 7

9(𝑥 + 2)2 − 7(𝑦 − 1)2 = −63

7(𝑦 − 1)2 − 9(𝑥 + 2)2 = 63

7(𝑦−1)2

63−

9(𝑥+2)2

63=

63

63

(𝑦−1)2

9−

(𝑥+2)2

7= 1

𝑌2

9−

𝑋2

7= 1 , 𝑋 = 𝑥 + 2, 𝑌 = 𝑦 − 1


𝑎2 = 9, 𝑏2 = 7, 𝑎 = 3, 𝑏 = 7

𝑒 = 1 +𝑏2

𝑎2 = 1 +7

9 =

7+9

9=

16

9=

4

3

𝑎𝑒 = 3 ×4

3= 4


𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 + 2, 𝑌 = 𝑦 − 1

ikak; (0,0) 𝑋 = 0, 𝑌 = 0

𝑥 + 2 = 0, 𝑦 − 1 = 0 𝐶(−2,1)

Kidfs; (0, ±𝑎) (0, ±3)

(0,3) 𝑋 = 0, 𝑌 = 3

𝑥 + 2 = 0, 𝑦 − 1 = 3 𝐴(−2,4) (0, −3)

𝑋 = 0, 𝑌 = −3 𝑥 + 2 = 0, 𝑦 − 1 = −3

𝐴(−2, −2)

Ftpaq;fs; (0, ±𝑎𝑒) (0, ±4)

(0, 4) 𝑋 = 0, 𝑌 = 4





𝑥 + 2 = 0, 𝑦 − 1 = 4

𝑥 = −2, 𝑦 = 1 + 4

𝐹1(−2,5) (0, −4)

𝑋 = 0, 𝑌 = −4 𝑥 + 2 = 0, 𝑦 − 1 = −4

𝑥 = −2, 𝑦 = 1 − 4

𝐹2(−2, −3)

32. 𝟏𝟔𝒙𝟐 − 𝟗𝒚𝟐 − 𝟑𝟐𝒙 − 𝟑𝟔𝒚 − 𝟏𝟔𝟒 = 𝟎 vd;w

mjpgutisaj;jpd; ikaj; njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f NkYk; mjd; tiuglk; tiuf.

( JUN-14)

16𝑥2 − 9𝑦2 − 32𝑥 − 36𝑦 − 164 = 0

16𝑥2 − 32𝑥 − 9𝑦2 − 36𝑦 − 164 = 0

16(𝑥2 − 2𝑥) − 9(𝑦2 + 4𝑦) = 164 16(𝑥2 − 2𝑥 + 12 − 12) − 9(𝑦2 + 4𝑦 + 22 − 22) = 164

16{(𝑥 − 1)2 − 1} − 9*(𝑦 + 2)2 − 4+ = 164

16(𝑥 − 1)2 − 9(𝑦 + 2)2 = 164 + 16 − 36

16(𝑥 − 1)2 − 9(𝑦 + 2)2 = 144

16(𝑥 − 1)2

144−

9(𝑦 + 2)2

144=

144

144

(𝑥 − 1)2

9−

(𝑦 + 2)2

16= 1

𝑋2

9−

𝑌2

16= 1 ,q;F 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 + 2


𝑎2 = 9, 𝑏2 = 16, 𝑎 = 3, 𝑏 = 4

𝑒 = 1 +𝑏2

𝑎2 = 1 +16

9 =

16+9

9=

25

9=

5

3

𝑎𝑒 = 3 ×5

3= 5


𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 + 2

ikak; (0,0) 𝑋 = 0, 𝑌 = 0

𝑥 − 1 = 0, 𝑦 + 2 = 0 𝐶(1, −2)

Ftpaq;fs; (±𝑎𝑒, 0) (±5, 0)

(5, 0) 𝑋 = 5, 𝑌 = 0

𝑥 − 1 = 5, 𝑦 + 2 = 0

𝑥 = 6, 𝑦 = −2 𝐹1(6, −2) (−5, 0)

𝑋 = −5, 𝑌 = 0 𝑥 − 1 = −5, 𝑦 + 2 = 0

𝑥 = −4, 𝑦 = −2 𝐹2(−4, −2)

Kidfs; (±𝑎, 0) (±3,0)

(3, 0) 𝑋 = 3, 𝑌 = 0

𝑥 − 1 = 3, 𝑦 + 2 = 0 𝑥 = 4, 𝑦 = −2

𝐴(4, −2) (−3,0)

𝑋 = −3, 𝑌 = 0 𝑥 − 1 = −3, 𝑦 + 2 = 0

𝑥 = −2, 𝑦 = −2

𝐴′(−2, −2)

33. 𝟏𝟐𝒙𝟐 − 𝟒𝒚𝟐 − 𝟐𝟒𝒙 + 𝟑𝟐𝒚 − 𝟏𝟐𝟕 = 𝟎 vd;w mjpgutisaj;jpd; ikaj; njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f NkYk; mjd; tistiuia tiuf.

(OCT-07)

12𝑥2 − 4𝑦2 − 24𝑥 + 32𝑦 − 127 = 0

12𝑥2 − 24𝑥 − 4𝑦2 + 32𝑦 = 127

12(𝑥2 − 2𝑥) − 4(𝑦2 − 8𝑦) = 127 12(𝑥2 − 2𝑥 + 12 − 12) − 4(𝑦2 − 8𝑦 + 42 − 42) = 127

12{(𝑥 − 1)2 − 1} − 4*(𝑦 − 4)2 − 16+ = 127

12(𝑥 − 1)2 − 4(𝑦 − 4)2 = 127 − 64 + 12

12(𝑥 − 1)2 − 4(𝑦 − 4)2 = 75

(𝑥−1)2

.75

12/

−(𝑦−4)2

.75

4/

= 1

𝑋2

.75

12/−

𝑌2

.75

4/

= 1 ,q;F 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 − 4


𝑎2 =75

12, 𝑏2 =

75

4,

𝑒 = 1 +𝑏2

𝑎2 = 1 +75

475

12

= 1 +12

4= 1 + 3 = 2





𝑎𝑒 = 75

12× 2

= 3×25

3×4× 2 =

5

2× 2 = 5


𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 − 1, 𝑌 = 𝑦 − 4

ikak; (0,0) 𝑋 = 0, 𝑌 = 0

𝑥 − 1 = 0, 𝑦 − 4 = 0 𝐶(1,4)

Ftpaq;fs; (±𝑎𝑒, 0) (±5, 0)

(5, 0) 𝑋 = 5, 𝑌 = 0

𝑥 − 1 = 5, 𝑦 − 4 = 0

𝑥 = 6, 𝑦 = 4 𝐹1(6,4) (−5, 0)

𝑋 = −5, 𝑌 = 0 𝑥 − 1 = −5, 𝑦 − 4 = 0

𝑥 = −4, 𝑦 = 4 𝐹2(−4,4)

Kidfs;

(±𝑎, 0)

±5

2, 0

.5

2, 0/

𝑋 =5

2, 𝑌 = 0

𝑥 − 1 =5

2, 𝑦 − 4 = 0

𝑥 =7

2, 𝑦 = 4

𝐴 7

2, 4

.−5

2, 0/

𝑋 = −5

2, 𝑌 = 0

𝑥 − 1 = −5

2, 𝑦 − 4 = 0

𝑥 = −3

2, 𝑦 = 4

𝐴′ .−3

2, 4/

34. 𝟓𝒙 + 𝟏𝟐𝒚 = 𝟗 vd;w Neh;f;NfhL mjpgutisak;

𝒙𝟐 − 𝟗𝒚𝟐 = 𝟗 Ij; njhLfpwJ vd ep&gpf;f. NkYk; njhLk; Gs;spiaAk; fhz;f.

( JUN-09,16, MAR-13,OCT-13, 14,16)

𝑦 = 𝑚𝑥 + 𝑐 vd;w NfhL mjpgutisak;

𝑥2

𝑎2 −𝑦2

𝑏2 = 1 f;F njhLNfhlhf ,Uf;f

epge;jid 𝑐2 = 𝑎2𝑚2 − 𝑏2

5𝑥 + 12𝑦 = 9

12𝑦 = −5𝑥 + 9

𝑦 =−5𝑥

12+

9

12

𝑦 =−5

12𝑥 +

3

4

𝑚 =−5

12, 𝑐 =

3

4

𝑥2 − 9𝑦2 = 9

𝑥2

9−

𝑦2

1= 1

𝑎2 = 9, 𝑏2 = 1

𝑐2 =9

16 ;

𝑎2𝑚2 − 𝑏2 = 9 .25

144/ − 1 =

81

144=

9

16

⇒ 𝑐2 = 𝑎2𝑚2 − 𝑏2

mjpgutisaj;jpd; njhLNfhl;bd; rkd;ghL

5𝑥 + 12𝑦 = 9

,J mjpgutisaj;ij njhLk; Gs;sp

.−𝑎2𝑚

𝑐,−𝑏2

𝑐/

−𝑎2𝑚

𝑐= −9 ×

−5

12 ×

4

3= 5

−𝑏2

𝑐= −1 ×

4

3=

−4

3

njhLg;Gs;sp .5,−4

3/

35. 𝒙 − 𝒚 + 𝟒 = 𝟎 vd;w Neh;f;NfhL ePs;tl;lk;

𝒙𝟐 + 𝟑𝒚𝟐 = 𝟏𝟐 f;F njhLNfhlhf cs;sJ vd ep&gpf;f. NkYk; njhLk; Gs;spiaAk; fhz;f.

( JUN-13,MAR-16)





𝑦 = 𝑚𝑥 + 𝑐 vd;w NfhlhdJ 𝑥2

𝑎2 +𝑦2

𝑏2 = 1 f;F

njhLNfhlhf ,Uf;f epge;jid

𝑐2 = 𝑎2𝑚2 + 𝑏2

𝑥 − 𝑦 + 4 = 0

𝑦 = 𝑥 + 4

𝑚 = 1, 𝑐 = 4

𝑥2 + 3𝑦2 = 12

𝑥2

12+

𝑦2

4= 1

𝑎2 = 12, 𝑏2 = 4

𝑐2 = 16 ; 𝑎2𝑚2 + 𝑏2 = 12(1) + 4 = 16

⇒ 𝑐2 = 𝑎2𝑚2 + 𝑏2

𝑥 − 𝑦 + 4 = 0 ePs;tl;lj;jpd; njhLNfhL MFk;

njhLg;Gs;sp .−𝑎2𝑚

𝑐,𝑏2

𝑐/

−𝑎2𝑚

𝑐= −12 × (1) ×

1

4=

−12

4= −3

𝑏2

𝑐= 4 ×

1

4= 1

njhLg;Gs;sp (−3,1)

36. xU mjpgutisaj;jpd; ikak; (𝟐, 𝟒) NkYk;

mJ (𝟐, 𝟎) topNar; nry;fpwJ. ,jd;

njhiyj;njhLNfhLfs; 𝒙 + 𝟐𝒚 − 𝟏𝟐 = 𝟎

kw;Wk; 𝒙 − 𝟐𝒚 + 𝟖 = 𝟎 Mfpatw;wpw;F ,izahf ,Uf;fpd;wd vdpy; mjpgutisaj;jpd; rkd;ghl;ilf; fhz;f.

(MAR-06, 09,15, JUN-06,08,11, OCT-15)

njhiyj;njhLNfhLfspd; ,izf;NfhLfs;

𝑥 + 2𝑦 − 12 = 0 kw;Wk; 𝑥 − 2𝑦 + 8 = 0

∴ njhiyj;njhLNfhLfspd; rkd;ghLfspd;

tbtk;

𝑥 + 2𝑦 + 𝑙 = 0 kw;Wk; 𝑥 − 2𝑦 + 𝑚 = 0

,J mjpgutisaj;jpd; ikak; (2,4) topahf

nry;fpwJ. vdNt,

2 + 8 + 𝑙 = 0 ⇒ 𝑙 = −10

2 − 8 + 𝑚 = 0 ⇒ 𝑚 = 6

∴ njhiyj;njhLNfhLfspd; rkd;ghLfs;

𝑥 + 2𝑦 − 10 = 0 kw;Wk; 𝑥 − 2𝑦 + 6 = 0

njhiyj;njhLNfhLfspd; Nrh;g;Gr; rkd;ghL

(𝑥 + 2𝑦 − 10)(𝑥 − 2𝑦 + 6) = 0

∴ mjpgutisaj;jpd; rkd;ghl;bd; tbtk;

(𝑥 + 2𝑦 − 10)(𝑥 − 2𝑦 + 6) + 𝑘 = 0

,J (2,0) topahf nry;fpwJ

(2 − 10)(2 + 6) + 𝑘 = 0

𝑘 = 64

mjpgutisaj;jpd; rkd;ghL

(𝑥 + 2𝑦 − 10)(𝑥 − 2𝑦 + 6) + 64 = 0

37. 𝒙 + 𝟐𝒚 − 𝟓 = 𝟎 I xU njhiyj;njhL

NfhlhfTk; (𝟔, 𝟎) kw;Wk; (−𝟑, 𝟎)vd;w Gs;spfs; topNa nry;yf;$baJkhd nrt;tf mjpgutisaj;jpd; rkd;ghL fhz;f. (OCT-06, 08,10,12, MAR-07,08,11, JUN-07)

xU njhiyj; njhLNfhL 𝑥 + 2𝑦 − 5 = 0

∴ kw;nwhU njhiyj; njhLNfhl;bd; tbtk;

2𝑥 − 𝑦 + 𝑘 = 0

nrt;tf mjpgutisaj;jpd; rkd;ghL

(𝑥 + 2𝑦 − 5)(2𝑥 − 𝑦 + 𝑘) + 𝑐 = 0

,J (6, 0) topahf nry;fpwJ

(6 − 5)(12 + 𝑘) + 𝑐 = 0

𝑘 + 𝑐 = −12………………………..………(1)

NkYk; (−3,0) topahfTk; nry;fpwJ.

(−3 − 5)(−6 + 𝑘) + 𝑐 = 0

(−8)(−6 + 𝑘) + 𝑐 = 0

48 − 8𝑘 + 𝑐 = 0

−8𝑘 + 𝑐 = −48……………………………(2)

(1) kw;Wk; (2) I jPh;f;f

𝑘 = 4 kw;Wk; 𝑐 = −16

Njitahd nrt;tf mjpgutisaj;jpd;

rkd;ghL (𝑥 + 2𝑦 − 5)(2𝑥 − 𝑦 + 4) − 16 = 0





6. tif Ez;fzpjk; - gad;ghLfs; - I

1. tifaPLfisg; gad;gLj;jp

𝒚 = 𝟏. 𝟎𝟐𝟑

+ 𝟏. 𝟎𝟐𝟒

f;F Njhuha

kjpg;Gfisf; fhz;f. (MAR-2015)

𝑦 = 𝑓(𝑥) = 𝑥1

3

𝑥 = 1, 𝑑𝑥 = ∆𝑥 = 0.02 vd;f

𝑑𝑦 =1

3𝑥−

2

3𝑑𝑥

=1

3(1)−

2

3(0.02)

=1

3(0.02) = 0.0066

𝑓(𝑥 + ∆𝑥) ≈ 𝑦 + 𝑑𝑦 = 𝑓(1) + 0.0066

= 1 + 0.0066

(1.02)1

3 ≅ 1.0066

NkYk; 𝑦 = 𝑓(𝑥) = 𝑥1

4

,q;F 𝑥 = 1, 𝑑𝑥 = ∆𝑥 = 0.02

𝑑𝑦 =1

4𝑥−

3

4𝑑𝑥

=1

4(1)−

3

4(0.02)

=1

4(0.02) = 0.005

𝑓(𝑥 + ∆𝑥) ≈ 𝑦 + 𝑑𝑦 = 𝑓(1) + 0.005

= 1 + 0.005

(1.02)1

4 ≅ 1.005

(1.02)13 + (1.02)

14 ≈ 1.0066 + 1.005 ≈ 2.0116

2. 𝒖 = 𝐬𝐢𝐧 −𝟏 𝒙−𝒚

𝒙− 𝒚 vdpy; A+yhpd; Njw;wj;ijg;

gad;gLj;jp 𝒙𝝏𝒖

𝝏𝒙+ 𝒚

𝝏𝒖

𝝏𝒚=

𝟏

𝟐𝐭𝐚𝐧 𝒖 vdf; fhl;Lf

(MAR-07, MAR-08, JUN-14)

R.H.S. rkgbj;jhd rhh;G my;y. vdNt

𝑓 = sin 𝑢 =𝑥−𝑦

𝑥− 𝑦

𝑓(𝑡𝑥, 𝑡𝑦) =𝑡𝑥−𝑡𝑦

𝑡𝑥− 𝑡𝑦=

𝑡(𝑥−𝑦)

𝑡1/2 ( 𝑥− 𝑦)=

𝑡1−

1 2(𝑥−𝑦)

( 𝑥− 𝑦)

=𝑡

1 2(𝑥−𝑦)

( 𝑥− 𝑦)= 𝑡

1

2𝑓

𝑓 vd;gJ> gb 1

2 cila rkgbj;jhd rhh;G.

∴ A+yhpd; Njw;wj;jpd; gb 𝑥𝜕𝑓

𝜕𝑥+ 𝑦

𝜕𝑓

𝜕𝑦=

1

2𝑓

𝑥𝜕(sin 𝑢)

𝜕𝑥+ 𝑦

𝜕(sin 𝑢)

𝜕𝑦=

1

2(sin 𝑢)

𝑥𝜕𝑢

𝜕𝑥cos 𝑢 + 𝑦

𝜕𝑢

𝜕𝑦cos 𝑢 =

1

2(sin 𝑢)

cos 𝑢 .𝑥𝜕𝑢

𝜕𝑥+ 𝑦

𝜕𝑢

𝜕𝑦/ =

1

2(sin 𝑢)

𝑥𝜕𝑢

𝜕𝑥+ 𝑦

𝜕𝑢

𝜕𝑦=

1

2

sin 𝑢

cos 𝑢

𝑥𝜕𝑢

𝜕𝑥+ 𝑦

𝜕𝑢

𝜕𝑦=

1

2tan 𝑢

3. 𝒖 = 𝐭𝐚𝐧 −𝟏 .𝑥3+ 𝑦3

𝒙−𝒚/ vdpy; A+yhpd; Njw;wj;ijg;

gad;gLj;jp 𝒙𝝏𝒖

𝝏𝒙+ 𝒚

𝝏𝒖

𝝏𝒚= 𝐬𝐢𝐧 𝟐𝒖 vdf; fhl;Lf.

(OCT-09,11, 15)

𝑢 vd;gJ rkgbj;jhd rhh;gy;y.Mdhy; tan 𝑢

vd;gJ xU rkgbj;jhd rhh;ghFk;.

𝑓 = tan 𝑢 =𝑥3+ 𝑦3

𝑥−𝑦 vd tiuaWf;fTk;

𝑓(𝑡𝑥, 𝑡𝑦) =(𝑡𝑥 )3+ (𝑡𝑦 )3

𝑡𝑥−𝑡𝑦=

𝑡3(𝑥3+ 𝑦3)

𝑡(𝑥−𝑦)=

𝑡3−1(𝑥3+ 𝑦3)

(𝑥−𝑦)

=𝑡2(𝑥3+ 𝑦3)

(𝑥−𝑦)= 𝑡2𝑓

𝑓 vd;gJ gb 2 cila rkgbj;jhd rhh;ghFk;

A+yhpd; Njw;wj;jpd; gb

𝑥𝜕𝑓

𝜕𝑥+ 𝑦

𝜕𝑓

𝜕𝑦= 2𝑓

𝑥𝜕(tan 𝑢)

𝜕𝑥+ 𝑦

𝜕(tan 𝑢)

𝜕𝑦= 2(tan 𝑢)

𝑥. sec2 𝑢𝜕𝑢

𝜕𝑥+ 𝑦. sec2 𝑢

𝜕𝑢

𝜕𝑦= 2 tan 𝑢

sec2 𝑢 .𝑥𝜕𝑢

𝜕𝑥+ 𝑦

𝜕𝑢

𝜕𝑦/ = 2 tan 𝑢

𝑥𝜕𝑢

𝜕𝑥+ 𝑦

𝜕𝑢

𝜕𝑦=

2 tan 𝑢

sec 2𝑢=

2 sin 𝑢

cos 𝑢1

cos 2𝑢

= 2 sin 𝑢

cos 𝑢× cos2 𝑢 = 2 sin 𝑢 cos 𝑢 = sin 2𝑢

4. 𝒖 = 𝐬𝐢𝐧 𝒙+𝒚

𝒙+ 𝒚 vdpy;

𝒙𝝏𝒖

𝝏𝒙+ 𝒚

𝝏𝒖

𝝏𝒚=

𝟏

𝟐

𝒙+𝒚

𝒙+ 𝒚 𝐜𝐨𝐬

𝒙+𝒚

𝒙+ 𝒚 vd

ep&gpf;f (JUN-13)

𝑢 = sin .𝑥+𝑦

𝑥+ 𝑦/

sin−1 𝑢 =𝑥+𝑦

𝑥+ 𝑦= 𝑓(𝑥, 𝑦)





𝑥 = 𝑡𝑥, 𝑦 = 𝑡𝑦

𝑓(𝑡𝑥, 𝑡𝑦) =𝑡𝑥+𝑡𝑦

𝑡𝑥+ 𝑡𝑦

=𝑡(𝑥+𝑦)

𝑡( 𝑥+ 𝑦 )

= 𝑡1−1

2(𝑥+𝑦)

𝑥+ 𝑦

= 𝑡1

2(𝑥+𝑦)

( 𝑥+ 𝑦 )= 𝑡

1

2 𝑓

𝑓 d; gb 1

2 , A+yhpd; Njw;wj;jpd; gb

𝑥𝜕𝑓

𝜕𝑥+ 𝑦

𝜕𝑓

𝜕𝑦= 𝑛𝑓

𝑥𝜕( sin−1 𝑢)

𝜕𝑥+ 𝑦

𝜕( sin−1 𝑢)

𝜕𝑦= 𝑛( sin−1 𝑢)

𝑥1

1−𝑢2

𝜕𝑢

𝜕𝑥+ 𝑦

1

1−𝑢2

𝜕𝑢

𝜕𝑦=

1

2 sin−1 sin .

𝑥+𝑦

𝑥+ 𝑦/

1

1−𝑢2.𝑥

𝜕𝑢

𝜕𝑥+ 𝑦

𝜕𝑢

𝜕𝑦/ =

1

2.

𝑥+𝑦

𝑥+ 𝑦/

𝑥𝜕𝑢

𝜕𝑥+ 𝑦

𝜕𝑢

𝜕𝑦=

1

2.

𝑥+𝑦

𝑥+ 𝑦/ 1 − 𝑢2

=1

2.

𝑥+𝑦

𝑥+ 𝑦/ 1 − sin2 .

𝑥+𝑦

𝑥+ 𝑦/

=1

2.

𝑥+𝑦

𝑥+ 𝑦/ cos2 .

𝑥+𝑦

𝑥+ 𝑦/

=1

2.

𝑥+𝑦

𝑥+ 𝑦/ cos .

𝑥+𝑦

𝑥+ 𝑦/

5. 𝒇(𝒙, 𝒚) =𝟏

𝒙𝟐+𝒚𝟐 f;F A+yhpd; Njw;wj;ij

rhpghh;f;f (JUN-06,OCT-10,MAR-14)

𝑓(𝑡𝑥, 𝑡𝑦) =1

𝑡2𝑥2+𝑡2𝑦2=

1

𝑡2(𝑥2+𝑦2)

=1

𝑡 (𝑥2+𝑦2)=

1

𝑡 𝑓(𝑥, 𝑦)

= 𝑡−1𝑓(𝑥, 𝑦)

∴ 𝑓 vd;gJ −1 cila rkgbj;jhd rhh;G. vdNt

A+yhpd; Njw;wj;jpd;gb

𝑥𝜕𝑓

𝜕𝑥+ 𝑦

𝜕𝑓

𝜕𝑦= −𝑓

rhpghh;j;jy;:

𝜕𝑓

𝜕𝑥= −

1

2

2𝑥

(𝑥2+𝑦2)32

= −𝑥

(𝑥2+𝑦2)32

𝑥𝜕𝑓

𝜕𝑥= −

𝑥2

(𝑥2+𝑦2)32

,ijg; Nghy;

𝜕𝑓

𝜕𝑦= −

1

2

2𝑦

(𝑥2+𝑦2)32

= −𝑦

(𝑥2+𝑦2)32

𝑦𝜕𝑓

𝜕𝑦= −

𝑦2

(𝑥2+𝑦2)32

𝑥𝜕𝑓

𝜕𝑥+ 𝑦

𝜕𝑓

𝜕𝑦= −

𝑥2

(𝑥2+𝑦2)32

−𝑦2

(𝑥2+𝑦2)32

= − 𝑥2+𝑦2

(𝑥2+𝑦2)32

= − 1

𝑥2+𝑦2 = −𝑓

A+yhpd; Njw;wk; rhpghh;f;fg;gl;lJ.

6. 𝒖 = 𝐭𝐚𝐧−𝟏 .𝒙

𝒚/ vdpy;

𝝏𝟐𝒖

𝝏𝒙𝝏𝒚=

𝝏𝟐𝒖

𝝏𝒚𝝏𝒙 vd;gij

rhpghh;f;f. (MAR-10)

𝑢 = tan−1 .𝑥

𝑦/

𝜕𝑢

𝜕𝑥=

1

1+.𝑥

𝑦/

2

1

𝑦=

1

1+𝑥2

𝑦2

1

𝑦 =

1

𝑦2+𝑥2

𝑦2

1

𝑦 =

𝑦2

𝑦2+𝑥2

1

𝑦

=𝑦

𝑦2+𝑥2 =𝑦

𝑥2+𝑦2

𝜕𝑢

𝜕𝑦=

1

1+.𝑥

𝑦/

2 𝑥 .−1

𝑦2/ = −

1

1+𝑥2

𝑦2

.𝑥

𝑦2/

= −1

𝑦2+𝑥2

𝑦2

.𝑥

𝑦2/ =𝑦2

𝑦2+𝑥2 .𝑥

𝑦2/

=− 𝑥

𝑥2+𝑦2

𝜕2𝑢

𝜕𝑥𝜕𝑦=

𝜕

𝜕𝑥.𝜕𝑢

𝜕𝑦/ =

𝜕

𝜕𝑥.

− 𝑥

𝑥2+𝑦2/

= − 0(𝑥2+𝑦2).1−(𝑥).2𝑥

(𝑥2+𝑦2)2 1 = − 0𝑥2+𝑦2−2𝑥2

(𝑥2+𝑦2)2 1

= −0−𝑥2+𝑦2

(𝑥2+𝑦2)21 = 0𝑥2−𝑦2

(𝑥2+𝑦2)21…………..(1)

𝜕2𝑢

𝜕𝑦𝜕𝑥=

𝜕

𝜕𝑦.𝜕𝑢

𝜕𝑥/ =

𝜕

𝜕𝑦.

𝑦

𝑥2+𝑦2/

= 0(𝑥2+𝑦2).1−(𝑦).2𝑦

(𝑥2+𝑦2)2 1

= 0𝑥2+𝑦2−2𝑦2

(𝑥2+𝑦2)2 1 = 0𝑥2−𝑦2

(𝑥2+𝑦2)21….(2)

(1) kw;Wk; (2) ypUe;J 𝜕2𝑢

𝜕𝑥𝜕𝑦=

𝜕2𝑢

𝜕𝑦𝜕𝑥





7. 𝒖 =𝒙

𝒚𝟐 −𝒚

𝒙𝟐 vd;w rhh;Gf;F 𝝏𝟐𝒖

𝝏𝒙𝝏𝒚=

𝝏𝟐𝒖

𝝏𝒚𝝏𝒙 vd;gij

rhpghh;f;f (JUN-12, MAR-17)

𝑢 =𝑥

𝑦2 −𝑦

𝑥2

𝜕𝑢

𝜕𝑥=

1

𝑦2 − 𝑦(−2)𝑥−3 =1

𝑦2 +2𝑦

𝑥3

𝜕𝑢

𝜕𝑦= 𝑥. (−2)𝑦−3 −

1

𝑥2 = −2𝑥

𝑦3 −1

𝑥2

𝜕2𝑢

𝜕𝑥𝜕𝑦=

𝜕

𝜕𝑥.𝜕𝑢

𝜕𝑦/ =

𝜕

𝜕𝑥.−

2𝑥

𝑦3 −1

𝑥2/ = −2

𝑦3 −(−2)

𝑥3

=2

𝑥3 −2

𝑦3………………..…(1)

𝜕2𝑢

𝜕𝑦𝜕𝑥=

𝜕

𝜕𝑦.𝜕𝑢

𝜕𝑥/ =

𝜕

𝜕𝑦.

1

𝑦2 +2𝑦

𝑥3/

= −2

𝑦3 +2

𝑥3

=2

𝑥3 −2

𝑦3………………….(2)


𝜕𝑥𝜕𝑦=

𝜕2𝑢

𝜕𝑦𝜕𝑥

8. 𝒖 =𝒙𝟐

𝒚−

𝟐𝒚𝟐

𝒙 vd;w rhh;Gf;F

𝝏𝟐𝒖

𝝏𝒙𝝏𝒚=

𝝏𝟐𝒖

𝝏𝒚𝝏𝒙 vd;gjid rhpghh;f;f. (OCT - 14)

𝑢 =𝑥2

𝑦−

2𝑦2

𝑥

𝜕𝑢

𝜕𝑥=

2𝑥

𝑦−

(−1)2𝑦2

𝑥2 =2𝑥

𝑦+

2𝑦2

𝑥2

𝜕𝑢

𝜕𝑦=

(−1)𝑥2

𝑦2 −2(2𝑦)

𝑥= −

𝑥2

𝑦2 −4𝑦

𝑥

𝜕2𝑢

𝜕𝑥𝜕𝑦=

𝜕

𝜕𝑥.𝜕𝑢

𝜕𝑦/ =

𝜕

𝜕𝑥.−

𝑥2

𝑦2 −4𝑦

𝑥/

= −2𝑥

𝑦2 −(−1)4𝑦

𝑥2 = −2𝑥

𝑦2 +4𝑦

𝑥2

=4𝑦

𝑥2 −2𝑥

𝑦2……………….(1)

𝜕2𝑢

𝜕𝑦𝜕𝑥=

𝜕

𝜕𝑦.𝜕𝑢

𝜕𝑥/ =

𝜕

𝜕𝑦.

2𝑥

𝑦+

2𝑦2

𝑥2 /

=2(−1)𝑥

𝑦2 +2(2𝑦)

𝑥2 =−2𝑥

𝑦2 +4𝑦

𝑥2

=4𝑦

𝑥2 −2𝑥

𝑦2……………………(2)


𝜕𝑥𝜕𝑦=

𝜕2𝑢

𝜕𝑦𝜕𝑥

9. 𝒖 = 𝐬𝐢𝐧 .𝒙

𝒚/ vd;Dk; rhh;Gf;F

𝝏𝟐𝒖

𝝏𝒙𝝏𝒚=

𝝏𝟐𝒖

𝝏𝒚𝝏𝒙

vd;gij rhpghh;f;f (MAR-13)

𝑢 = sin .𝑥

𝑦/

𝜕𝑢

𝜕𝑥=

1

𝑦cos .

𝑥

𝑦/

𝜕𝑢

𝜕𝑦= −

𝑥

𝑦2 cos .𝑥

𝑦/

𝜕2𝑢

𝜕𝑥𝜕𝑦=

𝜕

𝜕𝑥.𝜕𝑢

𝜕𝑦/ =

𝜕

𝜕𝑥.−

𝑥

𝑦2 cos .𝑥

𝑦//

= −1

𝑦2 𝑥. −sin

𝑥

𝑦

1

𝑦 + cos

𝑥

𝑦 (1)

= −1

𝑦2 .−𝑥

𝑦. sin .

𝑥

𝑦/ + cos .

𝑥

𝑦//……..…(1)

𝜕2𝑢

𝜕𝑦𝜕𝑥=

𝜕

𝜕𝑦.𝜕𝑢

𝜕𝑥/ =

𝜕

𝜕𝑦

1

𝑦cos .

𝑥

𝑦/

= 1

𝑦.−sin .

𝑥

𝑦/ .

−𝑥

𝑦2// + cos .

𝑥

𝑦/ .−

1

𝑦2/

= −1

𝑦2 −𝑥

𝑦sin .

𝑥

𝑦/ + cos .

𝑥

𝑦/ ………(2 )


𝜕𝑥𝜕𝑦=

𝜕2𝑢

𝜕𝑦𝜕𝑥

10. 𝒖 = 𝐬𝐢𝐧 𝟑𝒙 𝐜𝐨𝐬 𝟒𝒚 vd;w rhh;Gf;F

𝝏𝟐𝒖

𝝏𝒙𝝏𝒚=

𝝏𝟐𝒖

𝝏𝒚𝝏𝒙vd;gij rhpghh;f;f (MAR-2016)

𝑢 = sin 3𝑥 cos 4𝑦

𝜕𝑢

𝜕𝑥= cos 4𝑦. cos 3𝑥 .3 = 3cos 3𝑥 cos 4𝑦

𝜕𝑢

𝜕𝑦= sin 3𝑥(−sin 4𝑦). 4 = −4sin 3𝑥 sin 4𝑦

𝜕2𝑢

𝜕𝑥𝜕𝑦=

𝜕

𝜕𝑥.𝜕𝑢

𝜕𝑦/ =

𝜕

𝜕𝑥(−4sin 3𝑥 sin 4𝑦)

= −4 sin 4𝑦 cos 3𝑥 . 3

= −12 cos 3𝑥 sin 4𝑦 …..…(1)

𝜕2𝑢

𝜕𝑦𝜕𝑥=

𝜕

𝜕𝑦.𝜕𝑢

𝜕𝑥/ =

𝜕

𝜕𝑦(3cos 3𝑥 cos 4𝑦)

= 3 cos 3𝑥 – sin 4𝑦 . 4

= −12 cos 3𝑥 sin 4𝑦…………(2)


𝜕𝑥𝜕𝑦=

𝜕2𝑢

𝜕𝑦𝜕𝑥



12 Mk; tFg;G fzf;F ntw;wpf;F top


11. 𝒚 = 𝒙𝟑 + 𝟏 vd;fpw tistiuia tiuf

( JUN-09, OCT-13, JUN-16,

OCT-16)

12. 𝒚 = 𝒙𝟑 vd;fpw tistiuia tiuf

(OCT-06, JUN-07,OCT-07, JUN-08, OCT-08, JUN-10, MAR-11, MAR-12)

13. 𝒚𝟐 = 𝟐𝒙𝟑 vd;fpw tistiuia tiuf

(MAR-06, MAR-09, JUN-11, OCT-12,JUN-15)

rhh;gfk;

𝑥-,d; vy;yh nka; kjpg;GfSf;Fk; 𝑓(𝑥) MdJ tiuaWf;fg;gLfpwJ. vdNt 𝑓(𝑥) ,d; rhh;gfk; (−∞,∞) vd;fpw KO ,ilntsp.

𝑥-,d; vy;yh nka; kjpg;GfSf;Fk; 𝑓(𝑥)

MdJ tiuaWf;fg;gLfpwJ. vdNt 𝑓(𝑥) ,d; rhh;gfk; (−∞,∞)

vd;fpw KO ,ilntsp.

𝑥 ≥ 0 vd ,Uf;Fk; NghJ 𝑦 ed;F tiuaWf;fg;gl;Ls;sJ. [0,∞)

ePl;bg;G

fpilkl;l ePl;bg;G −∞ < 𝑥 < ∞

epiyf;Fj;J ePl;bg;G −∞ < 𝑦 < ∞

fpilkl;l ePl;bg;G −∞ < 𝑥 < ∞


fpilkl;l ePl;bg;G 0 ≤ 𝑥 < ∞


ntl;Lj; Jz;Lfs;

𝑥 = 0 vdpy; 𝑦 = 1 𝑦 = 0 vdpy; 𝑥 = −1

𝑥 = 0, vdpy; 𝑦 = 0 𝑦 = 0, vdpy; 𝑥 = 0

𝑥 = 0, vdpy; 𝑦 = 0 𝑦 = 0, vdpy; 𝑥 = 0

Mjp tistiuahdJ Mjp topr; nry;yhJ

tistiuahdJ Mjp topr; nry;Yk;

tistiuahdJ Mjp topr; nry;Yk;

rkr;rPh; Nrhjid

tistiuahdJ rkr;rPh; jd;ikia ngwtpy;iy

tistiuahdJ Mjpia nghWj;J

rkr;rPuhdJ

tistiuahdJ 𝑥 −mr;ir nghWj;J rkr;rPuhdJ.

njhiyj; njhL

NfhLfs;

tistiuf;F ve;j xU njhiyj;njhLNfhLfSk;

,y;iy.


,y;iy.


,y;iy.

Xhpay;G jd;ik

vy;yh 𝑥 f;Fk; 𝑦′ ≥ 0 Mjyhy;> tistiuahdJ

(−∞,∞) KOtJkhf VWKfkhf nry;Yk;

vy;yh 𝑥 f;Fk; 𝑦′ ≥ 0 Mjyhy;>

tistiuahdJ (−∞,∞) KOtJkhf VWKfkhf nry;Yk;

𝑦 = 2𝑥3

2 vd;w fpisapy; tistiu VWKfkhf ,Uf;Fk;.

𝑦 = − 2𝑥3

2 vd;w fpisapy; tistiu ,wq;F Kfkhf ,Uf;Fk;.

rpwg;Gg; Gs;spfs;

(−∞, 0) vd;w ,ilntspapy; fPo;Nehf;fp FopthfTk; kw;Wk; (0,∞) vd;w ,ilntspapy; Nky; Nehf;fp FopthfTk;

,Uf;Fk;. (0, 1) vd;gJ tisT khw;Wg;

Gs;sp

(−∞, 0) vd;w ,ilntspapy; fPo;Nehf;fp

FopthfTk; kw;Wk; (0,∞) vd;w

,ilntspapy; Nky; Nehf;fp FopthfTk;

,Uf;Fk;. (0, 0) vd;gJ tisT

khw;Wg; Gs;sp

(0,0) vd;gJ tisT khw;Wg; Gs;spay;y.

tiuglk;





8. tiff;nfOr;rkd;ghLfs;

1. xU ,urhad tpistpy;> xU nghUs; khw;wk;

milAk; khW tPjkhdJ t Neuj;jpy; khw;wkilahj mg;nghUspd; mstpw;F tpfpjkhf cs;sJ. xU kzp Neu Kbtpy; 60 fpuhKk; kw;Wk; 4 kzp Neu Kbtpy; 21 fpuhKk; kPjkpUe;jhy;> Muk;g epiyapy; > mg;nghUspd; vilapidf; fhz;f.

(MAR-11, OCT-15)

t vd;w Neuj;jpy; nghUspd; ,Ug;G A vd;f

𝑑𝐴

𝑑𝑡𝛼𝐴 ⇒

𝑑𝐴

𝑑𝑡= 𝑘𝐴 ⇒ 𝐴 = 𝑐𝑒𝑘𝑡

𝑡 = 1 vdpy; 𝐴 = 60 ⇒ 𝑐𝑒𝑘 = 60 … …… . . (1)

𝑡 = 4 vdpy; 𝐴 = 21 ⇒ 𝑐𝑒4𝑘 = 21 … …… . (2)

(1) ⇒ 𝑐4𝑒4𝑘 = 604 …… … (3)

(3)

(2)⇒ 𝑐3 =

604

21

⇒ 𝑐 = 85.15 (klf;ifiag; gad;gLj;jp)

Muk;gj;jpy; 𝑡 = 0tpy; 𝐴 = 𝑐 = 85.15 fpuhk; (Njhuhakhf)

∴ Muk;gj;jpy; nghUspd; vil 85.15 fpuhk; (Njhuhakhf).

2. xU tq;fpahdJ njhlh; $l;L Kiwapy; tl;biaf; fzf;fpLfpwJ. mjhtJ tl;b tPjj;ij me;je;j Neuj;jpy; mrypd; khW tPjj;jpy; fzf;fpLfpwJ. xUtuJ tq;fp ,Ug;gpy; njhlh;r;rpahd $l;L tl;b %yk;

Mz;nlhd;Wf;F 8 % tl;b ngUFfpwJ vdpy;> mtuJ tq;fpapUg;gpd; xU tUl fhy mjpfhpg;gpd; rjtPjj;ijf; fzf;fpLf.

[ 𝒆.𝟎𝟖 ≈ 𝟏. 𝟎𝟖𝟑𝟑 vLj;Jf; nfhs;f]. (OCT-07)

t vDk; Neuj;jpy; mry; A vd;f 𝑑𝐴


𝑑𝐴

𝑑𝑡= 𝑘𝐴 ⇒

𝑑𝐴

𝑑𝑡= 0.08 𝐴, ,q;F 𝑘 = 0.08

⇒ 𝐴(𝑡) = 𝑐𝑒0.08𝑡

xU tUl mjpfhpg;G rjtPjk; = 𝐴(1)−𝐴(0)

𝐴(0)× 100

= 𝐴(1)

𝐴(0)− 1 × 100 =

𝑐. 𝑒0.08

𝑐− 1 × 100

= 8.33% vdNt xU Mz;by; mjpfhpf;Fk; rjtPjk;

= 8.33%

3. xU ,we;jth; cliy kUj;Jth; ghpNrhjpf;Fk; NghJ> ,we;j Neuj;ij Njhuhakhf fzf;fpl Ntz;bAs;sJ. ,we;jthpd; clypd;

ntg;gepiy fhiy 10.00 kzpastpy; 𝟗𝟑. 𝟒𝒐 F

vd Fwpj;Jf; nfhs;fpwhh;. NkYk; 2 kzp Neuk;

fopj;J ntg;g epiy msit 𝟗𝟏. 𝟒𝒐 F vdf; fhz;fpwhh;. miwapd; ntg;gepiy msT

(epiyahdJ) 𝟕𝟐𝒐 F vdpy;> ,we;j Neuj;ij fzf;fpLf. (xU kdpj clypd; rhjhuz

c\;z epiy 𝟗𝟖. 𝟔𝒐 F vdf; nfhs;f)

[𝒍𝒐𝒈𝒆𝟏𝟗.𝟒

𝟐𝟏.𝟒= −𝟎. 𝟎𝟒𝟐𝟔 × 𝟐. 𝟑𝟎𝟑 kw;Wk;

𝒍𝒐𝒈𝒆𝟐𝟔.𝟔

𝟐𝟏.𝟒= 𝟎. 𝟎𝟗𝟒𝟓 × 𝟐. 𝟑𝟎𝟑] (JUN-11,JUN-16)

t vd;w Neuj;jpy; clypd; ntg;gepiyapid T vd;f epA+l;ldpd; Fsph;r;rp tpjpg;gb

𝑑𝑇

𝑑𝑡𝛼(𝑇 − 72) [Vnddpy; 𝑆 = 72𝑜𝐹 ]

𝑑𝑇

𝑑𝑡= 𝑘(𝑇 − 72) ⇒ 𝑇 − 72 = 𝑐𝑒𝑘𝑡

my;yJ 𝑇 = 72 + 𝑐𝑒𝑘𝑡

𝑡 = 0Mf ,Uf;Fk; NghJ ,

𝑇 = 93.4

93.4 = 72 + 𝑐𝑒𝑘(0)

93.4 = 72 + 𝑐

𝑐 = 93.4 − 72

𝑐 = 21.4

[Kjypy; Fwpf;fg;gl;l Neuk; fhiy 10 kzp

vd;gJ 𝑡 = 0 vd;f ]

/𝑇 = 72 + 21.4𝑒𝑘𝑡 [Njhuhaj;jpd; Jy;ypaj;jd;ikia mjpfhpf;f kzpahdJ epkplkhf vLj;Jf;

nfhs;sg;gLfpwJ]

𝑡 = 120 vdpy; , 𝑇 = 91.4 ⇒ 𝑒120𝑘 = 19.4

21.4

⇒ 𝑘 =1

120log𝑒

19.4

21.4

= 1

120(−0.0426 × 2.303)

𝑡1 vd;gJ ,we;j Neuj;jpw;Fg; gpd; fhiy 10

kzpf;F cs;shd Neuk; vd;f

𝑡 = 𝑡1vDk; NghJ

𝑇 = 98.6 ⇒ 98.6 = 72 + 21.4𝑒𝑘𝑡1

⇒ 𝑡1 =1

𝑘𝑙𝑜𝑔𝑒

26.6

21.4 =

−120 × 0.0945 × 2.303

0.0426 × 2.303

= −266 epkplk;

Kjy; mstPlhd fhiy 10 kzpf;F Kd;djhf

4 kzp 26 epkplk;.

∴,we;j Neuk; Njhuhakhf





10.00 kzp – 4 kzp 26 epkplk;. ,we;j Neuk; Njhuhakhf 5.34 A.M.

4. Ez;Zaph;fspd; ngUf;fj;jpy;> ghf;Bhpahtpd; ngUf;ftPjkhdJ mjpy; fhzg;gLk; ghf;Bhpahtpd; vz;zpf;if tpfpjkhf mike;Js;sJ. ,g;ngUf;fj;jhy; ghf;Bhpahtpd; vz;zpf;if 1 kzp Neuj;jpy; Kk;klq;fhfpwJ vdpy; Ie;J kzp Neu Kbtpy; ghf;Bhpahtpd; vz;zpf;if Muk;g epiyiaf; fhl;bYk;

𝟑𝟓klq;fhFk; vdf; fhl;Lf.

(JUN-06, MAR-09, OCT-11)

t Neuj;jpy; ghf;Bhpahf;fspd; vz;zpf;if A vd;f.

𝑑𝐴


𝑑𝐴


Muk;gj;jpy; 𝑡 = 0 vDk; NghJ 𝐴 = 𝐴0

/𝐴0 = 𝑐𝑒𝑜 = 𝑐

/𝐴 = 𝐴0𝑒𝑘𝑡

𝑡 = 1 vDk;NghJ 𝐴 = 3𝐴0 ⇒ 3𝐴0 = 𝐴0𝑒𝑘

⇒ 𝑒𝑘 = 3

𝑡 = 5vDk; NghJ

𝐴 = 𝐴0𝑒5𝑘 = 𝐴0(𝑒𝑘)5 = 35 . 𝐴0

/ 5 kzp Neu Kbtpy; ghf;Bhpaq;fspd;

vz;zpf;if35 klq;fhFk;.

5. 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]

(MAR-06, JUN-09,MAR-10,OCT-12, OCT-16)

t vd;w Neuj;jpy; Nubaj;jpd; msT A vd;f

𝐴 = 𝐴(𝑡)

𝑑𝐴


𝑑𝐴


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 𝑒

50𝑘 2

= 𝐴0(.95)2

= 0.9025𝐴0

100 tUl Kbtpy; kPjpapUf;Fk; Nubaj;jpd;

msT 0.9025𝐴0

6. &.1000 vd;w njhiff;F njhlh;r;rp $l;L tl;b fzf;fplg;gLfpwJ tl;b tPjk; Mz;nlhd;Wf;F 4 rjtPjkhf ,Ug;gpd;> mj;njhif vj;jid Mz;Lfspy; Muk;gj; njhifiag; Nghy; ,U

klq;fhFk;? (𝒍𝒐𝒈𝒆𝟐 = 𝟎. 𝟔𝟗𝟑𝟏). (MAR-15,JUN-07,08,12,OCT-06,10)

t vd;w Neuj;jpy; mry; A vd;f 𝑑𝐴

𝑑𝑡 𝛼 𝐴 ⇒

𝑑𝐴

𝑑𝑡= 𝑘𝐴 ⇒

𝑑𝐴

𝑑𝑡= 0.04 𝑡, ,q;F 𝑘 = 0.04

⇒ 𝐴 = 𝑐. 𝑒 .04𝑡

𝑡 = 0 vdpy;

𝐴 = 1000 ⇒ 1000 = 𝑐𝑒0 ⇒ 𝑐 = 1000

/ 𝐴 = 1000𝑒 .04𝑡

𝐴 = 2000 Mf ,Uf;Fk; NghJ t If; fhz;f

2000 = 1000𝑒 .04𝑡

⇒ 𝑡 =log 2

0.04=

0.6931

0.04= 17 tUlq;fs; (Njhuhakhf)

7. 𝟏𝟓𝟎𝑪 ntg;gepiy cs;s xU miwapy;

itf;fg;gl;Ls;s NjePhpd; ntg;gepiy 𝟏𝟎𝟎𝟎𝑪

MFk;. mJ 5 epkplq;fspy; 𝟔𝟎𝟎𝑪 Mf Fiwe;J

tpLfpwJ. NkYk; 5 epkplk; fopj;J NjePhpd; ntg;gepiyapidf; fhz;f.

(OCT-09, OCT-13, JUN-15, MAR-17)

t vd;w Neuj;jpy; NjePhpd; ntg;gepiy T vd;f epA+l;ldpd; Fsph;r;rp tpjpg;gb

𝑑𝑇

𝑑𝑡𝛼(𝑇 − 𝑆) ⇒

𝑑𝑇

𝑑𝑡= 𝑘(𝑇 − 𝑆)

⇒ (𝑇 − 𝑆) = 𝑐𝑒𝑘𝑡 ⇒ 𝑇 = 15 + 𝑐𝑒𝑘𝑡 ,

,q;F 𝑆 = 150𝐶

𝑡 = 0 vDk;NghJ 𝑇 = 100

100 = 15 + 𝑐𝑒𝑘(0)

100 − 15 = 𝑐

𝑐 = 85

∴ 𝑇 = 15 + 85𝑒𝑘𝑡

𝑡 = 5 vDk; NghJ

𝑇 = 60 ⇒ 60 = 15 + 85𝑒5𝑘 ⇒ 𝑒5𝑘 =45

85

𝑡 = 10 vDk; NghJ

𝑇 = 15 + 85𝑒10𝑘 = 15 + 85 45

85

2

= 38.820





8. xU efuj;jpy; cs;s kf;fs; njhifapd; tsh;r;rptPjk; me;Neuj;jpy; cs;s kf;fs;

njhiff;F tpfpjkhf mike;Js;sJ. 1960

Mk; Mz;by; kf;fs; njhif 1,30,000 vdTk;

1990,y; kf;fs; njhif 1,60,000 MfTk;

,Ug;gpd; 2020 Mk; Mz;by; kf;fs; njhif vt;tsthf ,Uf;Fk;?

0𝒍𝒐𝒈𝒆 0𝟏𝟔

𝟏𝟑1 =. 𝟐𝟎𝟕𝟎; 𝒆.𝟒𝟐 = 𝟏. 𝟓𝟐1

(MAR-08, JUN-10, MAR-14, JUN-14)

t vDk; Neuj;jpy; kf;fs; njhif A vd;f

𝑑𝐴


𝑑𝐴


1960k; Mz;L kf;fs; njhifapid njhlf;f

kf;fs; njhifahff; nfhs;f.

𝑡 = 0 tpy;

𝐴 = 130000

130000 = 𝑐𝑒0 = 𝑐

𝐴 = 130000𝑒𝑘𝑡

1990 Mk; Mz;by;

𝑡 = 30 tpy; , 𝐴 = 160,000

/160,000 = 130000 × 𝑒30𝑘 ⇒ 𝑒30𝑘 =16

13

2020 k; Mz;by; A If; fhz 𝑡 = 60 apy;

𝐴 = 130000 × 𝑒60𝑘

= 130,000 × 016

131

2~197000

2020 y; kf;fs;njhif Njhuhakhf 197000.

9. xU fjphpaf;fg; nghUs; rpijAk; khWtPjkhdJ> mjd; vilf;F tpfpjkhf mike;Js;sJ. mjd; vil 10 kp.fpuhk; Mf ,Uf;Fk;NghJ rpijAk; khWtPjk; ehnshd;Wf;F 0.051 kp.fpuhk; vdpy; mjd; vil 10 fpuhkpypUe;J 5 fpuhkhff; Fiwa vLj;Jf; nfhs;Sk; fhy msitf; fhz;f.

(𝒍𝒐𝒈𝒆𝟐 = 𝟎. 𝟔𝟗𝟑𝟏) (MAR-12, JUN-13,OCT-14) t vDk; Neuj;jpy; fjphpaf;fg; nghUspd; vil

A

𝑑𝐴

𝑑𝑡𝛼 𝐴 ⇒

𝑑𝐴


𝑡 = 0 vdpy; 𝐴 = 10 ⇒ 𝑐 = 10

⇒ 𝐴 = 10𝑒𝑘𝑡

kWgbAk; 𝑑𝐴

𝑑𝑡= 𝑘𝐴

𝐴 = 10vd ,Uf;Fk; NghJ 𝑑𝐴

𝑑𝑡= −0.051 vdf;

nfhLf;fg;gl;Ls;sJ. [rpijTWtjhy;]

⇒ −0.051 = 10𝑘 ⇒ 𝑘 = −0.0051

/ 𝐴 = 10𝑒−0.0051

𝐴 = 5 vDk; NghJ 𝑡 If; fhz

5 = 10𝑒−0.0051𝑡 ⇒1

2= 𝑒−0.0051𝑡 ⇒ 2 = 𝑒0.0051𝑡

⇒ 𝑙𝑜𝑔2 = 0.0051𝑡

⇒ 𝑡 =𝑙𝑜𝑔 2

0.0051~136 ehl;fs;.

10. xU Nehahspapd; rpWePhpypUe;J Ntjpg;nghUs; ntspNaWk; mstpid njhlh;r;rpahf Nfj;Njlh; vd;w fUtpapd; %yk; fz;fhzpf;fg;gLfpwJ. 𝒕 = 𝟎 vd;w Neuj;jpy; Nehaspf;F 10kp.fpuhk; Ntjpg;nghUs;

nfhLf;fg;gLfpwJ. ,J – 𝟑𝒕𝟏/𝟐kp.fpuhk; / kzp vd;Dk; tPjj;jpy; ntspNaWfpwJ vdpy;

(i) Neuk; 𝒕 > 0 vDk; NghJ> Nehahspapd; clypYs;s Ntjpg;nghUspd; msitf; fhZk;

nghJr; rkd;ghL vd;d ?

(ii) KOikahf Ntjpg;nghUs; ntspNaw vLj;Jf; nfhs;Sk; Fiwe;jgl;r fhy msT

vd;d?

(i) A vd;w Neuj;jpy; Ntjpg;nghUspd; vil t vd;f

Ntjpg;nghUs; ntspNaWk; tPjk; −3𝑡1

2

𝑑𝐴

𝑑𝑡= −3𝑡

12 ⇒ 𝐴 = −2𝑡

32 + 𝑐

𝑡 = 0 vdpy;, 𝐴 = 10 ⇒ 𝑐 = 10

𝑡 vDk; Neuj;jpy; 𝐴 = 10 − 2𝑡3

2

(ii) 𝐴 = 10 vdpy;> Ntjpg;nghUs; KOikahf

ntspNawptpl;lJ vdg; nghUs;

0 = 10 − 2𝑡3

2 ⇒ 5 = 𝑡3

2

⇒ 𝑡3 = 25 ⇒ 𝑡 = 2.9 kzp.

vdNt Nehahspapd; clypUe;J 2.9 kzp

my;yJ 2 kzp 54 epkplj;jpy; Ntjpg;nghUs;

KOikahf ntspNaWk;.





9. jdpepiyf; fzf;fpay;

1. (𝒁,∗) xU Kbtw;w vgpyPad; Fyk; vdf; fhl;Lf. ,q;F ∗ MdJ 𝒂 ∗ 𝒃 = 𝒂 + 𝒃 + 𝟐 vDkhW tiuaWf;fg;gl;Ls;sJ.

(OCT-08,JUN-10,MAR-16) (i) milg;G tpjp:

𝑎, 𝑏 kw;Wk; 2 KO vz;fs; Mjyhy;

𝑎 + 𝑏 + 2 k; xU KO vz;

∴ 𝑎 ∗ 𝑏 ∈ 𝑧, ∀𝑎, 𝑏 ∈ 𝑧

milg;G tpjp cz;ikahFk;.

(ii) Nrh;g;G tpjp:

𝑎, 𝑏, 𝑐 ∈ 𝑧 vd;f

(𝑎 ∗ 𝑏) ∗ 𝑐 = (𝑎 + 𝑏 + 2) ∗ 𝑐

= (𝑎 + 𝑏 + 2) + 𝑐 + 2 = 𝑎 + 𝑏 + 𝑐 + 4

𝑎 ∗ (𝑏 ∗ 𝑐) = 𝑎 ∗ (𝑏 + 𝑐 + 2)

= 𝑎 + (𝑏 + 𝑐 + 2) + 2 = 𝑎 + 𝑏 + 𝑐 + 4

(𝑎 ∗ 𝑏) ∗ 𝑐 = 𝑎 ∗ (𝑏 ∗ 𝑐) Nrh;g;G tpjp cz;ikahFk;

(iii) rkdp tpjp:

𝑒 rkdp cWg;G vd;f.

𝑒 d; tiuaiwapypUe;J 𝑎 ∗ 𝑒 = 𝑎

∗ ,d; tiuaiwapypUe;J 𝑎 ∗ 𝑒 = 𝑎 + 𝑒 + 2

𝑎 + 𝑒 + 2 = 𝑎

𝑒 = −2

−2 ∈ 𝑍. rkdp tpjp cz;ikahFk;.

(iv) vjph;kiw tpjp:

𝑎 ∈ 𝐺 vd;f. 𝑎 d; vjph;kiw 𝑎−1 vdf; nfhz;lhy;

𝑎−1 ,d; tiuaiwg;gb , 𝑎 ∗ 𝑎−1 = 𝑒 = −2

∗,d; tiuaiwg;gb , 𝑎 ∗ 𝑎−1 = 𝑎 + 𝑎−1 + 2

𝑎 + 𝑎−1 + 2 = −2

𝑎−1 = −𝑎 − 4

−𝑎 − 4 ∈ 𝑍.

∴ vjph;kiw tpjp cz;ikahFk;

∴ (𝑍,∗) xU FykhFk;.

(v) ghpkhw;Wg; gz;G:

𝑎, 𝑏 ∈ 𝐺 vd;f

𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 2 = 𝑏 + 𝑎 + 2 = 𝑏 ∗ 𝑎 ∴ ∗ ghpkhw;W tpjpf;Fl;lgl;lJ.

∴ (𝑍,∗) xU vgPypad; FykhFk;. NkYk; 𝑍 xU Kbtw;w fzkhjyhy; ,f;Fyk; Kbtw;w vgPypad; FykhFk;.

2. .𝒙 𝒙𝒙 𝒙

/ , 𝒙 ∈ 𝑹 − {𝟎} vd;w mikg;gpy; cs;s

mzpfs; ahTk; mlq;fpa fzk; 𝑮 MdJ mzpg;ngUf;fypd; fPo; xU Fyk; vdf; fhl;Lf

(JUN-13, MAR-15)

𝐺 = .𝑥 𝑥𝑥 𝑥

/ / 𝑥 ∈ 𝑅 − *0+ vd;f.

mzpg;ngUf;fypd; fPo; 𝐺 xU Fyk; vd fhl;LNthk;

(i) milg;G tpjp:

𝐴 = .𝑥 𝑥𝑥 𝑥

/ ∈ 𝐺, 𝐵 = .𝑦 𝑦𝑦 𝑦/ ∈ 𝐺

𝐴𝐵 = 2𝑥𝑦 2𝑥𝑦2𝑥𝑦 2𝑥𝑦

∈ 𝐺, (∵ 𝑥 ≠ 0, 𝑦 ≠ 0 ⇒ 2𝑥𝑦 ≠ 0)

𝐺 MdJ mzpg;ngUf;fypd; fPo; milT

ngw;Ws;sJ.

(ii) mzpg;ngUf;fy; vg;nghOJNk Nrh;g;G

tpjpf;Fl;gLk;.

(iii) 𝐸 = .𝑒 𝑒𝑒 𝑒

/ ∈ 𝐺 vd;gJ𝐴𝐸 = 𝐴 , 𝐴 ∈ 𝐺 vd;f

𝐴𝐸 = 𝐴 ⇒ .𝑥 𝑥𝑥 𝑥

/ .𝑒 𝑒𝑒 𝑒

/ = .𝑥 𝑥𝑥 𝑥

/

.2𝑥𝑒 2𝑥𝑒2𝑥𝑒 2𝑥𝑒

/ = .𝑥 𝑥𝑥 𝑥

/

⇒ 2𝑥𝑒 = 𝑥 ⇒ 𝑒 =1

2 (∵ 𝑥 ≠ 0)

vdNt> 𝐸 = 1/2 1/21/2 1/2

∈ 𝐺 vd;gJ

𝐴𝐸 = 𝐴, ∀𝐴 ∈ 𝐺 vDkhW cs;sJ

,Nj Nghy; 𝐸𝐴 = 𝐴 , 𝐴 ∈ 𝐺 vdf; fhl;lyhk;

∴ 𝐺 ,d; rkdp cWg;G 𝐸 MFk;. vdNt rkdp tpjp cz;ikahFk;.

(iv) 𝐴−1 = .𝑦 𝑦𝑦 𝑦/ ∈ 𝐺 vd;gJ 𝐴−1𝐴 = 𝐸

,t;thwhapd; 2𝑥𝑦 2𝑥𝑦2𝑥𝑦 2𝑥𝑦

= 1/2 1/21/2 1/2

2𝑥𝑦 =1

2⇒ 𝑦 =

1

4𝑥

𝐴−1 =

1

4𝑥

1

4𝑥1

4𝑥

1

4𝑥

∈ 𝐺 vd;gJ 𝐴−1𝐴 = 𝐸 vDkhW

cs;sJ.

,Nj Nghy; 𝐴𝐴−1 = 𝐸

∴ 𝐴 d; vjph;kiw 𝐴−1 MFk;

∴ mzpg;ngUf;fypd; fPo; 𝐺 xU FykhFk;.





3. 1 Ij; jtpu kw;w vy;yh tpfpjKW vz;fSk;

mlq;fpa fzk; 𝑮 vd;f. 𝑮 y; ∗ I

𝒂 ∗ 𝒃 = 𝒂 + 𝒃 − 𝒂𝒃 ∀𝒂, 𝒃 ∈ 𝑮 vDkhW

tiuaWg;Nghk;. (𝑮,∗) xU Kbtw;w vgPypad;

Fyk; vdf;fhl;Lf. (JUN-08,15, MAR-12)

𝐺 = 𝑄 − {−1} vd;f

𝑎, 𝑏 ∈ 𝐺. 𝑎 kw;Wk; 𝑏 tpfpjKW vz;fs;

𝑎 ≠ 1, 𝑏 ≠ 1

(i) milg;G tpjp: 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 − 𝑎𝑏 xU

tpfpjKW vz; MFk;.

𝑎 ∗ 𝑏 ∈ 𝐺 vdf; fhl;Ltjw;F 𝑎 ∗ 𝑏 ≠ 1 vd

ep&gpf;f Ntz;Lk;. khwhf 𝑎 ∗ 𝑏 = 1 vdf;

nfhz;lhy; 𝑎 + 𝑏 − 𝑎𝑏 = 1

⇒ 𝑏 − 𝑎𝑏 = 1 − 𝑎

⇒ 𝑏(1 − 𝑎) = 1 − 𝑎

⇒ 𝑏 = 1 (∵ 𝑎 ≠ 1 ⇒ 1 − 𝑎 ≠ 0 )

,J rhj;jpakpy;iy. Vnddpy; 𝑏 ≠ 1.

/ ekJ jw;Nfhs; jtwhdJ.

/ 𝑎 ∗ 𝑏 ≠ 1 vdNt 𝑎 ∗ 𝑏 ∈ 𝐺

/ milg;G tpjp cz;ikahFk;.

(ii) Nrh;g;G tpjp:

𝑎 ∗ (𝑏 ∗ 𝑐) = 𝑎 ∗ (𝑏 + 𝑐 − 𝑏𝑐)

= 𝑎 + (𝑏 + 𝑐 − 𝑏𝑐) − 𝑎(𝑏 + 𝑐 − 𝑏𝑐)

= 𝑎 + 𝑏 + 𝑐 − 𝑏𝑐 − 𝑎𝑏 − 𝑎𝑐 + 𝑎𝑏𝑐

(𝑎 ∗ 𝑏) ∗ 𝑐 = (𝑎 + 𝑏 − 𝑎𝑏) ∗ 𝑐

= 𝑎 + 𝑏 − 𝑎𝑏 + 𝑐 − (𝑎 + 𝑏 + 𝑎𝑏)𝑐

= 𝑎 + 𝑏 + 𝑐 − 𝑎𝑏 − 𝑎𝑐 − 𝑏𝑐 + 𝑎𝑏𝑐

/ 𝑎 ∗ (𝑏 ∗ 𝑐) = (𝑎 ∗ 𝑏) ∗ 𝑐, ∀ 𝑎, 𝑏, 𝑐 ∈ 𝐺

/ Nrh;g;G tpjp cz;ikahFk;.

(iii) rkdp tpjp: 𝑒 vd;gJ rkdp cWg;G vd;

∗ d; tiuaiwg;gb , 𝑎 ∗ 𝑒 = 𝑎 + 𝑒 − 𝑎𝑒

𝑒 d; tiuaiwg;gb , 𝑎 ∗ 𝑒 = 𝑎

⇒ 𝑎 + 𝑒 − 𝑎𝑒 = 𝑎

⇒ 𝑒(1 − 𝑎) = 0

⇒ 𝑒 = 0 Vnddpy; 𝑎 ≠ 1

𝑒 = 0 ∈ 𝐺

/ rkdp tpjp G+h;j;jpahfpwJ.

(iv) vjph;kiw tpjp:

𝑎 ∈ 𝐺 ,d; vjph;kiw 𝑎−1 vd;f.

vjph;kiwapd; tiuaiwg;gb 𝑎 ∗ 𝑎−1 = 𝑒 = 0

∗d; tiuaiwg;gb , 𝑎 ∗ 𝑎−1 = 𝑎 + 𝑎−1 − 𝑎𝑎−1

⇒ 𝑎 + 𝑎−1 − 𝑎𝑎−1 = 0

⇒ 𝑎−1(1 − 𝑎) = −𝑎

⇒ 𝑎−1 =𝑎

𝑎−1∈ 𝐺, Vnddpy; 𝑎 ≠ 1

/ vjph;kiw tpjp G+h;j;jpahFk; . / (G,*) xU FykhFk;.

(v) ghpkhw;W tpjp:

𝑎, 𝑏 ∈ 𝐺 f;F

𝑎 ∗ 𝑏 = 𝑎 + 𝑏 − 𝑎𝑏

= 𝑏 + 𝑎 − 𝑏𝑎

= 𝑏 ∗ 𝑎

/ 𝐺,y; * ghpkhw;W tpjpf;Fl;gLfpwJ. vdNt

(𝐺,∗) xU vgPypad; FykhFk;. 𝐺

KbTw;wjhjyhy; (𝐺,∗) KbTw;w vgPypad; FykhFk;.

4. G+r;rpakw;w fyg;ngz;fspd; fzkhd 𝑪 − *𝟎+

,y; tiuaWf;fg;gl;l 𝒇𝟏(𝒛) = 𝒛, 𝒇𝟐(𝒛) = −𝒛,

𝒇𝟑(𝒛) =𝟏

𝒛, 𝒇𝟒(𝒛) = −

𝟏

𝒛∀ 𝒛 ∈ 𝑪 − {𝟎} vd;w

rhh;Gfs; ahTk; mlq;fpa fzk; {𝒇𝟏, 𝒇𝟐, 𝒇𝟑, 𝒇𝟒} MdJ rhh;Gfspd; Nrh;g;gpd; fPo; xU vgPypad;

Fyk; mikf;Fk; vd epWTf. (OCT-06,09,15)

𝐺 = {𝑓1 , 𝑓2 , 𝑓3 , 𝑓4} vd;f

(𝑓1°𝑓1)(𝑧) = 𝑓1 𝑓1(𝑧) = 𝑓1(𝑧)

𝑓1°𝑓1 = 𝑓1 , 𝑓2°𝑓1 = 𝑓2, 𝑓3°𝑓1 = 𝑓3 , 𝑓4°𝑓1 = 𝑓4

NkYk; (𝑓2°𝑓2)(𝑧) = 𝑓2 𝑓2(𝑧)

= 𝑓2(−𝑧) = −(−𝑧) = 𝑧 = 𝑓1(𝑧)

𝑓2°𝑓2 = 𝑓1

,NjNghy;> 𝑓2°𝑓3 = 𝑓4 , 𝑓2°𝑓4 = 𝑓3

(𝑓3°𝑓2)(𝑧) = 𝑓3 𝑓2(𝑧) = 𝑓3(−𝑧) = −1

𝑧= 𝑓4(𝑧)

𝑓3°𝑓2 = 𝑓4

,NjNghy;> 𝑓3°𝑓3 = 𝑓1 , 𝑓3°𝑓4 = 𝑓2

(𝑓4°𝑓2)(𝑧) = 𝑓4 𝑓2(𝑧) = 𝑓4(−𝑧)

= −1

−𝑧=

1

𝑧= 𝑓3(𝑧)

𝑓4°𝑓2 = 𝑓3

,Nj Nghy; 𝑓4°𝑓3 = 𝑓2 , 𝑓4°𝑓4 = 𝑓1 Nkw;fz;ltw;iw gad;gLj;jp ngUf;fy; ml;ltizia mikf;f

° 𝑓1 𝑓2 𝑓3 𝑓4 𝑓1 𝑓1 𝑓2 𝑓3 𝑓4 𝑓2 𝑓2 𝑓1 𝑓4 𝑓3 𝑓3 𝑓3 𝑓4 𝑓1 𝑓2 𝑓4 𝑓4 𝑓3 𝑓2 𝑓1

ngUf;fy; ml;ltizapypUe;J

(i) ml;ltizapd; vy;yh cWg;GfSk; 𝐺,d; cWg;Gfshjyhy;> milg;G tpjp cz;ikahFk;





(ii) rhh;Gfspd; Nrh;g;G nghJthf Nrh;g;G

tpjpf;Fl;gLk;.

(iii) 𝐺 d; rkdp cWg;G 𝑓1 MFk;. vdNt rkdp tpjp cz;ikahfpwJ.

(iv) ml;ltizapypUe;J

𝑓1 ,d; vjph;kiw 𝑓1 ; 𝑓2 ,d; vjph;kiw 𝑓2

𝑓3 ,d; vjph;kiw 𝑓3 ; 𝑓4 ,d; vjph;kiw 𝑓4

vjph;kiw tpjp cz;ikahfpwJ.

(𝐺, °) xU FykhFk;

(v) ml;ltizapypUe;J ghpkhw;W tpjp cz;ikahfpwJ.

∴ (𝐺, °) xU vgPypad; FykhFk;.

5. (𝒁𝒏, +𝒏) xU Fyk; vdf; fhl;Lf. (JUN-11, MAR-14)

𝑍𝑛 = {,0-, ,1-, ,2-, … [𝑛 − 1]} vd;gJ 𝑛,d; kl;Lf;F fhzg;ngw;w rh;trkj; njhFg;Gfs; vd;f.

,𝑙-, ,𝑚- ∈ 𝑍𝑛0 ≤ 𝑙, 𝑚 < 𝑛 vd;f

(i) milg;G tpjp: tiuaiwg;gb>

,𝑙-+𝑛 ,𝑚- = ,𝑙 + 𝑚- , 𝑙 + 𝑚 < 𝑛 vdpy;,𝑟- , 𝑙 + 𝑚 ≥ 𝑛 vdpy;

,q;F 𝑙 + 𝑚 = 𝑞. 𝑛 + 𝑟 0 ≤ 𝑟 < 𝑛

,U epiyfspYk;, ,𝑙 + 𝑚- ∈ 𝑍𝑛 kw;Wk; [𝑟] ∈ 𝑍𝑛


(ii) 𝑛 ,d; kl;Lf;Fhpa $l;ly; vg;nghOJk; Nrh;g;G tpjpf;Fl;gLk;. Mjyhy; Nrh;g;G tpjp cz;ikahFk;.

(iii) rkdp cWg;G [0] ∈ 𝑍𝑛mJ rkdp tpjpiag; G+h;j;jp nra;fpwJ.

(iv) [𝑙] ∈ 𝑍𝑛 ,d; vjph;kiw [𝑛 − 𝑙] ∈ 𝑍𝑛

,𝑙-+𝑛 ,𝑛 − 𝑙- = [0]

,𝑛 − 𝑙-+𝑛 ,𝑙- = [0]

∴ vjph;kiw tpjp cz;ikahFk;.

(𝑍𝑛 , +𝑛) xU FykhFk;.

6. (𝒁𝟕 − *,𝟎-+, .𝟕 ) xU Fyj;ij mikf;Fk; vdf; fhl;Lf. (MAR-10, OCT-16) 𝐺 = {,1-, ,2-, … [6]} vd;f Nfa;yp ml;ltizahdJ

.7 [1] [2] [3] [4] [5] [6] [1] [1] [2] [3] [4] [5] [6] [2] [2] [4] [6] [1] [3] [5] [3] [3] [6] [2] [5] [1] [4] [4] [4] [1] [5] [2] [6] [3] [5] [5] [3] [1] [6] [4] [2] [6] [6] [5] [4] [3] [2] [1]

ml;ltizapypUe;J:


cWg;GfSk; 𝐺,d; cWg;GfshFk;

∴ milg;G tpjp cz;ikahFk;.

(ii) 7-d; kl;Lf;fhd ngUf;fy;> Nrh;g;G tpjpf;Fl;gLk;.

(iii) rkdpAWg;G [1] ∈ 𝐺 kw;Wk; ,J rkdp tpjpiag; G+h;j;jp nra;Ak;.

(iv) [1] ,d; vjph;kiw [1]; [2] ,d; vjph;kiw [4];

[3] ,d; vjph;kiw [5];[4] ,d; vjph;kiw [2];

[5] ,d; vjph;kiw [3]; [6] ,d; vjph;kiw [6] vdNt vjph;kiw tpjp G+h;j;jpahfpwJ.

∴ (𝑍7 − *,0-+, .7 ) xU Fyj;ij mikf;Fk;

7. tof;fkhd ngUf;fypd; fPo; 1,d; 𝒏Mk; gb %yq;fs; Kbthd Fyj;ij mikf;Fk;

vdf;fhl;Lf (MAR-11)

1 ,d; 𝑛Mk; gb %yq;fshtd vd;f

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.


(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;.

8. 11 ,d; kl;Lf;F fhzg;ngw;w ngUf;fypd; fPo;

*,𝟏-, ,𝟑-, ,𝟒-, ,𝟓-, ,𝟗-+ vd;w fzk; xU vgPypad; Fyj;ij mikf;Fk; vdf; fhl;Lf.

(MAR-07, JUN-09)

𝐺 = *,1-, ,3-, ,4-, ,5-, ,9-+ vd;f nfa;ypapd; ml;ltizahdJ

.11 ,1- ,3- ,4- ,5- ,9- ,1- ,1- ,3- ,4- ,5- ,9- ,3- ,3- ,9- ,1- ,4- ,5- ,4- ,4- ,1- ,5- ,9- ,3- ,5- ,5- ,4- ,9- ,3- ,1- ,9- ,9- ,5- ,3- ,1- ,4-

Nkw;fz;l ml;ltizapypUe;J


cWg;GfSk; G,d; cWg;GfshFk;.


(ii) 11 ,d; kl;Lf;fhd ngUf;fy; vg;nghOJk;

Nrh;g;G tpjpf;Fl;gLk;.

(iii) rkdpAWg;G ,1- ∈ 𝐺

(iv) ,1- ,d; vjph;kiw [1]

[3] ,d; vjph;kiw [4]



[9] ,d; vjph;kiw ,5-

/ vdNt jug;gl;l fzk; 11f;Fl;gl;L fhzg;ngw;w ngUf;fypd; fPo; xU Fyj;ij

mikf;Fk;.

(v) ml;ltizapd; %yk; ghpkhw;W tpjpAk; cz;ikahfpwJ.

/ vdNt xU vgPypad; FykhFk;.

9. 𝑮 vd;gJ kpif tpfpjKW vz;fspd; fzk;

vd;f. 𝒂 ∗ 𝒃 =𝒂𝒃

𝟑 vDkhW tiuaWf;fg;gl;l

nrayp∗ ,d; fPo; 𝑮 xU Fyj;ij mikf;Fk; vdf;fhl;Lf. (MAR-06, JUN-06, OCT-07,10,12,14)

(i) milg;G tpjp:

𝑎, 𝑏 ∈ 𝐺 vd;f. 𝑎, 𝑏 xU kpif tpfpjKW

vz;fs; Mjyhy; 𝑎𝑏 k; xU kpif tpfpjKW

vz;. vdNt 𝑎𝑏

3k; xU kpif tpfpjKW vz;

∴𝑎𝑏

3∈ 𝐺, 𝑎 ∗ 𝑏 ∈ 𝐺

∴ milg;G tpjp cz;ik.

(ii) Nrh;g;G tpjp:

𝑎, 𝑏, 𝑐 ∈ 𝐺 vd;f

(𝑎 ∗ 𝑏) ∗ 𝑐 =𝑎𝑏

3∗ 𝑐 =

.𝑎𝑏

3/𝑐

3=

𝑎𝑏𝑐

9

𝑎 ∗ (𝑏 ∗ 𝑐) = 𝑎 ∗𝑏𝑐

3=

𝑎.𝑏𝑐

3/

3=

𝑎𝑏𝑐

9

∴ (𝑎 ∗ 𝑏) ∗ 𝑐 = 𝑎 ∗ (𝑏 ∗ 𝑐)

∴ vdNt Nrh;g;G tpjp cz;ikahFk;.

(iii) rkdp tpjp:

rkdpAWg;G 𝑒 vd;f.

𝑒 ,d; tiuaiwg;gb , 𝑎 ∗ 𝑒 = 𝑎

∗ ,d; tiuaiwg;gb, 𝑎 ∗ 𝑒 =𝑎𝑒

3

𝑎𝑒

3= 𝑎 ⇒

𝑒

3= 1 ⇒ 𝑒 = 3 ∈ 𝐺

vdNt rkdp tpjp cz;ikahFk;.

(iv) vjph;kiw tpjp:

𝑎 ∈ 𝐺 vd;f. 𝑎 ,d; vjph;kiw 𝑎−1

𝑎−1 ,d; tiuaiwg;gb , 𝑎 ∗ 𝑎−1 = 𝑒 = 3

∗ ,d; tiuaiwg;gb , 𝑎 ∗ 𝑎−1 =𝑎𝑎−1

3

𝑎𝑎−1

3= 3

𝑎 ∗ 𝑎−1 = 9

𝑎−1 =9

𝑎∈ 𝐺

∴ vdNt vjph;kiw tpjp cz;ikahFk;. ∴ (𝐺,∗) xU FykhFk;.

10. 0𝟏 𝟎𝟎 𝟏

1 , 0𝝎 𝟎𝟎 𝝎𝟐1 , 𝝎

𝟐 𝟎𝟎 𝝎

, 0𝟎 𝟏𝟏 𝟎

1 , 𝟎 𝝎𝟐

𝝎 𝟎 , 0

𝟎 𝝎𝝎𝟐 𝟎

1

vd;fpw fzk; mzpg;ngUf;fypd; fPo; xU

Fyj;ij mikf;Fk; vdf;fhl;Lf. (𝝎𝟑 = 𝟏 ),

(JUN-12,MAR-13,17)

𝐼 = 01 00 1

1 , 𝐴 = 0𝜔 00 𝜔21 , 𝐵 = 𝜔

2 00 𝜔

,

𝐶 = 00 11 0

1 , 𝐷 = 0 𝜔2

𝜔 0 , 𝐸 = 0

0 𝜔𝜔2 0

1

𝐺 = {𝐼, 𝐴, 𝐵, 𝐶, 𝐷, 𝐸} vd;f ,t;tzpfis ,uz;L ,uz;lhf vLj;J ngUf;fp> ngUf;fy; ml;ltizia gpd;tUkhW

mikf;fyhk;:





I A B C D E I I A B C D E A A B I E C D B B I A D E C C C D E I A B D D E C B I A E E C D A B I

(i) ngUf;fy; ml;ltizapy; cs;s vy;yh

cWg;GfSk; G ,d; cWg;GfshFk;. vdNt G MdJ mzpg;ngUf;fypd; fPo; milT ngw;Ws;sJ. mjhtJ Nrh;g;G tpjp cz;ikahFk;.

(ii) nghJthf> mzp ngUf;fy; Nrh;g;G

tpjpf;Fl;gLkhjyhy; ,q;F ‘.’ MdJ Nrh;g;G tpjpf;Fl;gLk;.

(iii) ml;ltizapypUe;J njspthf I MdJ rkdp cWg;G MFk;.

(iv) 𝐼. 𝐼 = 𝐼 ⇒ 𝐼 ,d; vjph;kiw 𝐼

𝐴. 𝐵 = 𝐵. 𝐴 = 𝐼

⇒ 𝐴 Ak; B Ak; xd;Wf;nfhd;W vjph;kiwahFk;.

𝐶. 𝐶 = 𝐼 ⇒ 𝐶 ,d; vjph;kiw 𝐶

𝐷. 𝐷 = 𝐼 ⇒ 𝐷 ,d; vjph;kiw 𝐷.

𝐸. 𝐸 = 𝐼 ⇒ 𝐸 ,d; vjph;kiw 𝐸

vdNt mzpg;ngUf;fypd; fPo; 𝐺xU FykhFk;

11. 𝒛 = 𝟏 vDkhW cs;s fyg;ngz;fs; ahTk;

mlq;fpa fzk; M MdJ fyg;ngz;fspd; ngUf;fypd; fPo; xU Fyj;ij mikf;Fk; vdf;

fhl;Lf. (OCT-11,JUN-14)

M = *𝑧 ∈ 𝐶/ 𝑧 = 1+

(i) milg;G tpjp: z1 , z2 ∈ 𝑀 vd;f

𝑧1𝑧2 = 𝑧1 𝑧2 = 1.1 = 1 ⇒ 𝑧1 , 𝑧2 ∈ 𝑀


(ii) Nrh;g;G tpjp: fyg;ngz;fspd; ngUf;fy; vg;nghOJk; Nrh;g;G tpjpf;Fl;gLk;

𝑧1 . (𝑧2 . 𝑧3) = (𝑧1 . 𝑧2). 𝑧3

(iii) rkdp tpjp: xt;nthU 𝑧 ∈ 𝑀 f;Fk;

1 = 1 ∈ ℂ I vDkhW fhzyhk;

𝑧. 1 = 1. 𝑧 = 𝑧

∴ 1 rkdp cWg;G. vdNt rkdp tpjp cz;ikahFk;.

(iv) vjph;kiw tpjp: 𝑧 ∈ 𝑀 vd;f

𝑧 = 1

,q;F 1

𝑧 =

1

𝑧 =

1

1= 1 ⇒

1

𝑧∈ 𝑀

kw;Wk; 𝑧.1

𝑧=

1

𝑧. 𝑧 = 1

/ 𝑧 ,d; vjph;kiw 1

𝑧∈ 𝑀

vdNt vjph;kiw tpjp cz;ikahFk;.

/ vdNt fyg;ngz;fs; ngUf;fypd; fPo; 𝑀 xU FykhFk;.

12. − 1I jtpu kw;w vy;yh tpfpjKW vz;fSk;

cs;slf;fpa fzk; 𝑮 MdJ

𝒂 ∗ 𝒃 = 𝒂 + 𝒃 + 𝒂𝒃 vDkhW tiuaWf;fg;gl;l

nrayp * ,d; fPo; xU vgPypad; Fyj;ij mikf;Fk; vdf; fhl;Lf. (JUN-07, MAR-09,JUN-16)

𝐺 = 𝑄 − {−1} vd;f.

𝑎, 𝑏 ∈ 𝐺 vd;f. vdNt 𝑎 kw;Wk; 𝑏 tpfpjKW

vz;fs; 𝑎 ≠ −1, 𝑏 ≠ −1.

(i) milg;G tpjp: 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 𝑎𝑏 vd;gJ xU tpfpjKW vz;.

𝑎 ∗ 𝑏 ∈ 𝐺 vd epWt 𝑎 ∗ 𝑏 ≠ −1 > vdTk; fhl;l

Ntz;Lk;. khwhf 𝑎 ∗ 𝑏 = −1 vdf; nfhs;Nthk;. ,t;thwhapd;

𝑎 + 𝑏 + 𝑎𝑏 = −1

⇒ 𝑏 + 𝑎𝑏 = −1 − 𝑎

⇒ 𝑏(1 + 𝑎) = −(1 + 𝑎)

⇒ 𝑏 = −1 (∵ 𝑎 ≠ −1 ⇒ 1 + 𝑎 ≠ 0)

Mdhy; 𝑏 ≠ −1 vdNt> ,J rhj;jpakpy;iy.

/ vdNt> ek; jw;Nfhs; jtwhdjhFk;.

/ 𝑎 ∗ 𝑏 ≠ −1 ∴ 𝑎 ∗ 𝑏 ∈ 𝐺


(ii) Nrh;g;G tpjp:

𝑎 ∗ (𝑏 ∗ 𝑐) = 𝑎 ∗ (𝑏 + 𝑐 + 𝑏𝑐)

= 𝑎 + (𝑏 + 𝑐 + 𝑏𝑐) + 𝑎(𝑏 + 𝑐 + 𝑏𝑐)

= 𝑎 + 𝑏 + 𝑐 + 𝑏𝑐 + 𝑎𝑏 + 𝑎𝑐 + 𝑎𝑏𝑐)

(𝑎 ∗ 𝑏) ∗ 𝑐 = (𝑎 + 𝑏 + 𝑎𝑏) ∗ 𝑐

= 𝑎 + 𝑏 + 𝑎𝑏 + 𝑐 + (𝑎 + 𝑏 + 𝑎𝑏)𝑐

= 𝑎 + 𝑏 + 𝑐 + 𝑎𝑏 + 𝑎𝑐 + 𝑏𝑐 + 𝑎𝑏𝑐

/ 𝑎 ∗ (𝑏 ∗ 𝑐) = (𝑎 ∗ 𝑏) ∗ 𝑐, ∀ 𝑎, 𝑏, 𝑐 ∈ 𝐺

/ Nrh;g;G tpjp cz;ikahFk;.

(iii) rkdp tpjp: 𝑒 vd;gJ rkdp cWg;G vd;f.

∗ ,d; tiuaiwg;gb , 𝑎 ∗ 𝑒 = 𝑎 + 𝑒 + 𝑎𝑒

𝑒 ,d; tiuaiwg;gb, 𝑎 ∗ 𝑒 = 𝑎

⇒ 𝑎 + 𝑒 + 𝑎𝑒 = 𝑎

⇒ 𝑒(1 + 𝑎) = 0

⇒ 𝑒 = 0 𝑎 ≠ −1 Mjyhy;

𝑒 = 0 ∈ 𝐺

/ rkdp tpjp G+h;j;jpahfpwJ.





(iv) vjph;kiw tpjp:

𝑎 ∈ 𝐺 ,d; vjph;kiw 𝑎−1 vd;f

vjph;kiwapd; tiuaiwg;gb> 𝑎 ∗ 𝑎−1 = 𝑒 = 0

∗ ,d; tiuaiwg;gb 𝑎 ∗ 𝑎−1 = 𝑎 + 𝑎−1 + 𝑎𝑎−1

⇒ 𝑎 + 𝑎−1 + 𝑎𝑎−1 = 0

⇒ 𝑎−1(1 + 𝑎) = −𝑎

⇒ 𝑎−1 =−𝑎

1+𝑎∈ 𝐺, [Vnddpy; 𝑎 ≠ −1]

/ vjph;kiw tpjp G+h;j;jpahfpwJ . / (G,*) xU FykhFk;.

(v) ghpkhw;W tpjp:

VNjDk; 𝑎, 𝑏 ∈ 𝐺 f;F

𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 𝑎𝑏

= 𝑏 + 𝑎 + 𝑏𝑎

= 𝑏 ∗ 𝑎

/ G,y; * ghpkhw;W tpjpf;Fl;gLtjhy; (𝐺,∗) xU vgPypad; FykhFk;.

13. .𝒂 𝒐𝒐 𝒐

/ , 𝒂 ∈ 𝑹 − *𝟎+ mikg;gpy; cs;s vy;yh

mzpfSk; mlq;fpa fzk; mzpg;ngUf;fypd; fPo; xU vgPypad; Fyj;ij mikf;Fk; vdf;

fhl;Lf. (MAR-08)

G vd;gJ .𝑎 𝑜𝑜 𝑜

/ , 𝑎 ∈ 𝑅 − *0+ vd;w

mikg;gpy; cs;s mzpfs; ahTk; mlq;fpa

fzk; vd;f.

(i) milg;G tpjp:

A = .𝑎 𝑜𝑜 𝑜

/ ∈ 𝐺, 𝐵 = .𝑏 𝑜𝑜 𝑜

/ ∈ 𝐺

AB = .𝑎𝑏 𝑜𝑜 𝑜

/ ∈ 𝐺(∵ 𝑎 ≠ 0, 𝑏 ≠ 0 ⇒ 𝑎𝑏 ≠ 0)

G MdJ mzpg;ngUf;fypd; fPo; milTg;

ngw;Ws;sJ.

(ii) Nrh;g;G tpjp:

mzpg; ngUf;fyhdJ vg;nghOJk; Nrh;g;G

tpjpf;Fl;gLk;.

(iii) rkdp tpjp:

𝐸 = .𝑒 𝑜𝑜 𝑜

/ ∈ 𝐺 MdJ xt;nthU 𝐴 ∈ 𝐺 f;Fk;

𝐴𝐸 = 𝐴 vd;gjhf mikfpwJ vdf; nfhs;f.

𝐴𝐸 = 𝐴 ⇒ .𝑎 𝑜𝑜 𝑜

/ .𝑒 𝑜𝑜 𝑜

/ = .𝑎 𝑜𝑜 𝑜

/

⇒ .𝑎𝑒 𝑜𝑜 𝑜

/ = .𝑎 𝑜𝑜 𝑜

/

⇒ 𝑎𝑒 = 𝑎

⇒ 𝑒 = 1

vdNt> 𝐸 = .1 𝑜𝑜 𝑜

/ ∈ 𝐺 MdJ xt;nthU

𝐴 ∈ 𝐺 f;Fk; 𝐴𝐸 = 𝐴 vd;gjhf mikfpwJ.

,Nj Nghy; xt;nthU 𝐴 ∈ 𝐺 f;Fk; 𝐸𝐴 = 𝐴

vdf; fhl;lyhk;

∴ vdNt 𝐺,y; 𝐸 MdJ rkdp cWg;G MFk;

vdNt rkdp tpjp cz;ikahfpwJ.

(iv) vjph;kiw tpjp:

𝐴−1 = .𝑥 𝑜𝑜 𝑜

/ ∈ 𝐺 MdJ 𝐴−1𝐴 = 𝐸

.𝑥 𝑜𝑜 𝑜

/ .𝑎 𝑜𝑜 𝑜

/ = .1 𝑜𝑜 𝑜

/

.𝑥𝑎 𝑜𝑜 𝑜

/ = .1 𝑜𝑜 𝑜

/

𝑥𝑎 = 1

𝑥 =1

𝑎

𝐴−1 = .1/𝑎 𝑜𝑜 𝑜

/ ∈ 𝐺 MdJ 𝐴−1𝐴 = 𝐸

vDkhW cs;sJ.

∴ ,Nj Nghy; 𝐴𝐴−1 = 𝐸 vdTk; fhl;lyhk;.

vdNt 𝐴−1MdJ 𝐴 ,d; vjph;kiw MFk;.

vdNt vjph;kiw tpjp cz;ikahfpwJ.

mzpg;ngUf;fypd; fPo; 𝐺 xU FykhFk;

(v) ghpkhw;W tpjp:

𝐴, 𝐵 ∈ 𝐺

𝐴𝐵 = .𝑎𝑏 𝑜𝑜 𝑜

/ = .𝑏𝑎 𝑜𝑜 𝑜

/ = 𝐵𝐴

∴ mzpg;ngUf;fypd; fPo; 𝐺 xU vgPypad;

FykhFk;.

14. 𝑮 = {𝟐𝒏/ 𝒏 ∈ 𝒁} vd;w fzkhdJ ngUf;fypd; fPo; xU vgPypad; Fyj;ij mikf;Fk; vdf;

fhl;Lf. (OCT-13)

𝐺 = {2𝑛/ 𝑛 ∈ 𝑍}

(i) milg;G tpjp: 𝑥 = 2𝑟 , 𝑦 = 2𝑠 ∈ 𝐺, 𝑟, 𝑠 ∈ 𝑍 vd;f.

𝑥𝑦 = 2𝑟 . 2𝑠 = 2𝑟+𝑠 ∈ 𝐺 (∵ 𝑟, 𝑠 ∈ 𝑍 ⇒ 𝑟 + 𝑠 ∈ 𝑍)

∴ milg;G tpjp cz;ikahFk;.





(ii) Nrh;g;G tpjp:

𝑥 = 2𝑟 , 𝑦 = 2𝑠 , 𝑧 = 2𝑡 ∈ 𝐺 , 𝑟, 𝑠, 𝑡 ∈ 𝑍 vd;f

(𝑥. 𝑦). 𝑧 = (2𝑟 . 2𝑠)2𝑡 = 2𝑟+𝑠 . 2𝑡 = 2(𝑟+𝑠)+𝑡

= 2𝑟+(𝑠+𝑡) = 2𝑟(2𝑠 . 2𝑡) = 𝑥. (𝑦. 𝑧)

∴ Nrh;g;G tpjp cz;ikahFk;.

(iii) rkdp tpjp:

xt;nthU 𝑥 = 2𝑟 ∈ 𝐺 f;Fk; 1 = 20 ∈ 𝐺 MdJ

𝑥. 1 = 2𝑟 . 1 = 2𝑟 = 𝑥 vDkhWk;

1. 𝑥 = 1.2𝑟 = 2𝑟 = 𝑥 vDkhWk; cs;sjhy; 1

MdJ rkdp cWg;G MFk;. vdNt rkdp tpjp

cz;ikahFk;.

(iv) vjph;kiw tpjp:

xt;nthU 𝑥 = 2𝑟 ∈ 𝐺 f;Fk; 𝑥−1 = 2−𝑟 ∈ 𝐺

MdJ

𝑥. 𝑥−1 = 2𝑟 . 2−𝑟 = 2𝑟+(−𝑟) = 20 = 1 vdTk;

𝑥−1 . 𝑥 = 2−𝑟 . 2𝑟 = 2(−𝑟)+𝑟 = 20 = 1 vdTk;

cs;sJ

∴ 2𝑟 ,d; vjph;kiw 2−𝑟 ∈ 𝐺

,t;thwhf vjph;kiw tpjpAk; cz;ikahfpwJ.

vdNt> 𝐺 xU FykhFk;.

(v) ghpkhw;W tpjp:

𝑥 = 2𝑟 , 𝑦 = 2𝑠 ∈ 𝐺

𝑥. 𝑦 = 2𝑟 . 2𝑠 = 2𝑟+𝑠 = 2𝑠+𝑟 = 2𝑠2𝑟 = 𝑦. 𝑥

(Vnddpy; 𝑍,y; $l;ly; ghpkhw;W tpjpf;Fl;gLk; )

∴(𝐺, . ) xU vgPypad; FykhFk;.





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