kalviamuthu.blogspot · 12k; tFg;G fzf;F xU kjpg;ngz; tpdhf;fs; ntw;wpf;F top [email protected] -...
Transcript of kalviamuthu.blogspot · 12k; tFg;G fzf;F xU kjpg;ngz; tpdhf;fs; ntw;wpf;F top [email protected] -...
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ntw;wpf;F top (Way to Success)
⓬ fzpjk;
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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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muRj;Nju;T tpdhj;jhs; tbtikg;G
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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
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70 fl;lha 10 kjpg;ngz; tpdh 2 1 10
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tpdhj;jhs; - gFg;gha;T
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Way to Success Gj;jfq;fs; Ntz;LNthu; 9787609090, 9787201010, 8680810626 Mfpa vz;fisj; njhlu;Gnfhs;Sq;fs;
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kjpg;ngz; tpdhf;fs;
MW kjpg;ngz; tpdhf;fs;
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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
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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
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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)
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
𝐴 𝑎𝑑𝑗 𝐴
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=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
𝐴 𝑎𝑑𝑗 𝐴
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 𝑎𝑑𝑗 𝐴
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
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= −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 𝑎𝑑𝑗 𝐴
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
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15. 𝟑 𝟏 −𝟓𝟏 −𝟐 𝟏𝟏 𝟓 −𝟕
−𝟏−𝟓 𝟐
vd;w mzpapd; juk; fhz;f
(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
vd;w mzpapd; juk; fhz;f
(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
vd;w mzpapd; juk; fhz;f
( 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
vd;w mzpapd; juk; fhz;f
(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
filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ.
,jpy; ,uz;L G+r;rpakw;w epiufs; cs;sjhy;.
𝜌 𝐴 = 2
20. 𝟏 𝟐 𝟑𝟐 𝟒 𝟔𝟑 𝟔 𝟗
−𝟏−𝟐 −𝟑
vd;w mzpapd; juk; fhz;f
(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
filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ.
,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
vd;w mzpapd; juk; fhz;f
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(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
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∆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;
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;.
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;
Fiwe;jJ xU nfO 𝑎𝑖𝑗 MtJ G+r;rpakw;W
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,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
filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ.
𝜌 𝐴, 𝐵 = 3 kw;Wk; 𝜌 𝐴 = 2
𝜌 𝐴, 𝐵 ≠ 𝜌 𝐴
∴ vdNt jug;gl;l rkd;ghl;Lj; njhFg;G
xUq;fikT ,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
∴ vdNt jug;gl;l rkd;ghl;Lj; njhFg;G
xUq;fikT mw;wJ. vdNt jPh;T fhz KbahJ
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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
filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ.
𝜌 𝐴, 𝐵 = 3 and 𝜌 𝐴 = 2
∴ 𝜌 𝐴, 𝐵 ≠ 𝜌 𝐴
∴ vdNt jug;gl;l rkd;ghl;Lj; njhFg;G
xUq;fikT mw;wJ. vdNt jPh;T fhz KbahJ
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
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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)
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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
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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)
𝑧 = 𝑥 + 𝑖𝑦 vd;f
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
,UGwKk; th;f;fg;gLj;j,
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)
𝑧 = 𝑥 + 𝑖𝑦 vd;f
𝑥 + 𝑖𝑦 − 3𝑖 = 𝑥 + 𝑖𝑦 + 3𝑖
𝑥 + 𝑖(𝑦 − 3) = 𝑥 + 𝑖(𝑦 + 3)
𝑥2 + (𝑦 − 3)2 = 𝑥2 + (𝑦 + 3)2
,UGwKk; th;f;fg;gLj;j,
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)
𝑧 = 𝑥 + 𝑖𝑦 vd;f
𝑥 + 𝑖𝑦 − 4𝑖 = 𝑥 + 𝑖𝑦 + 4𝑖
𝑥 + 𝑖(𝑦 − 4) = 𝑥 + 𝑖(𝑦 + 4)
𝑥2 + (𝑦 − 4)2 = 𝑥2 + (𝑦 + 4)2
,UGwKk; th;f;fg;gLj;j,
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
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,UGwKk; th;f;fg;gLj;j,
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 − (%yq;fspd; $Ljy;)𝑥+%yq;fspd; ngUf;fk;
= 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 − (%yq;fspd; $Ljy;)𝑥+%yq;fspd; ngUf;fk;
= 𝑥2 − 2𝑥 + 5
𝑥4 − 4𝑥3 + 11𝑥2 − 14𝑥 + 10
≡ 𝑥2 − 2𝑥 + 5 (𝑥2 + 𝑝𝑥 + 2)
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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
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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)
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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
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= 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)
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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)
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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;.
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∴ (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
∴ (𝐺, . ) xU FykhFk;.
(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
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(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;.
∴ milg;G tpjp cz;ikahFk;
(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)
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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;.
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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
http://kalviamuthu.blogspot.com
12 k; tFg;G fzf;F ntw;wpf;F top
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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
𝑂𝑃 = 𝑂𝑀 + 𝑀𝑃 = cos 𝐴𝑖 +sin 𝐴 𝑗
𝑂𝑄 = 𝑂𝑁 + 𝑁𝑄 = cos 𝐵𝑖 −sin 𝐵 𝑗
𝑂𝑃 = 𝑂𝑀 + 𝑀𝑃 = cos 𝐴𝑖 +sin 𝐴 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
(1) kw;Wk; (2) ypUe;J
cos 𝐴 + 𝐵 = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵
(1) kw;Wk; (2) ypUe;J
sin 𝐴 + 𝐵 = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵
(1) kw;Wk; (2) ypUe;J
sin 𝐴 − 𝐵 = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵
http://kalviamuthu.blogspot.com
12 k; tFg;G fzf;F gj;J kjpg;ngz; tpdhf;fs; ntw;wpf;F top
[email protected] - 49 - www.waytosuccess.org
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
,e;j Nfhl;by; mike;Js;s VNjDk; xU
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
,e;j Nfhl;by; mike;Js;s VNjDk; xU
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
,e;j Nfhl;by; mike;Js;s VNjDk; xU
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)
http://kalviamuthu.blogspot.com
12 k; tFg;G fzf;F gj;J kjpg;ngz; tpdhf;fs; ntw;wpf;F top
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−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
http://kalviamuthu.blogspot.com
12 k; tFg;G fzf;F gj;J kjpg;ngz; tpdhf;fs; ntw;wpf;F top
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,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
Njitahd rkd;ghL 𝑟 = 𝑎 + 𝑠𝑢 + 𝑡𝑣
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
Njitahd rkd;ghL 𝑟 = 𝑎 + 𝑠𝑢 + 𝑡𝑣
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
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12 k; tFg;G fzf;F gj;J kjpg;ngz; tpdhf;fs; ntw;wpf;F top
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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:
Njitahd rkd;ghL 𝑟 = 𝑎 + 𝑠𝑢 + 𝑡𝑣
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:
Njitahd rkd;ghL 𝑟 = (1 − 𝑠)𝑎 + 𝑠𝑏 + 𝑡𝑣
,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
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−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:
Njitahd rkd;ghL 𝑟 = (1 − 𝑠)𝑎 + 𝑠𝑏 + 𝑡𝑣
,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
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12 k; tFg;G fzf;F gj;J kjpg;ngz; tpdhf;fs; ntw;wpf;F top
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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
Njitahd rkd;ghL 𝑟 = 𝑎 + 𝑠𝑢 + 𝑡𝑣
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
Njitahd rkd;ghL 𝑟 = 𝑎 + 𝑠𝑢 + 𝑡𝑣
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
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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:
Njitahd rkd;ghL 𝑟 = 𝑎 + 𝑠𝑢 + 𝑡𝑣
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;
1. 𝑷 vDk; Gs;sp fyg;G vz; khwp 𝒛 If;
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)
𝑧 = 𝑥 + 𝑖𝑦 vd;f
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
2. 𝑷 vDk; Gs;sp fyg;G vz; khwp 𝒛 If;
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 = 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 = 0
3. 𝑷 vDk; Gs;sp fyg;G vz; khwp 𝒛 If;
Fwpj;jhy; 𝑷 ,d; epakg;ghijia
𝑰𝒎.𝟐𝒛+𝒊
𝒊𝒛−𝟏/ = −𝟏 vd;w epge;jidf;F cl;gl;L
fhz;f (MAR-14)
𝑧 = 𝑥 + 𝑖𝑦 vd;f
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
4. 𝑷 vDk; Gs;sp fyg;G vz; khwp 𝒛 If;
Fwpj;jhy; 𝑷 ,d; epakg;ghijia
𝐚𝐫𝐠 .𝒛−𝟏
𝒛+𝟏/ =
𝝅
𝟑 vd;w epge;jidf;F cl;gl;L
fhz;f (MAR-13)
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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/
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𝛽𝑛 = 1 − 𝑖 3 𝑛
= 2𝑛 .cos𝑛𝜋
3− 𝑖 sin
𝑛𝜋
3/
𝛼𝑛 − 𝛽𝑛
= 2𝑛 .cos𝑛𝜋
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
= 2𝑛 .cos5𝑛𝜋
6+ 𝑖 sin
5𝑛𝜋
6/ + 2𝑛 .cos
5𝑛𝜋
6− 𝑖 sin
5𝑛𝜋
6/
= 2𝑛 .cos5𝑛𝜋
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 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
(ii) 𝑥 = (−1)1
3 = (cos 𝜋 + 𝑖 sin 𝜋)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
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,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
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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
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12 k; tFg;G fzf;F gj;J kjpg;ngz; tpdhf;fs; ntw;wpf;F top
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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)
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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)
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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
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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
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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
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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
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12 k; tFg;G fzf;F gj;J kjpg;ngz; tpdhf;fs; ntw;wpf;F top
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(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, 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
𝑎 = 1 gutisak; fPo;Gwk; jpwg;GilaJ
𝑋, 𝑌 I nghWj;J
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
𝑎 = 2 gutisak; fPo;Gwk; jpwg;GilaJ
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12 k; tFg;G fzf;F gj;J kjpg;ngz; tpdhf;fs; ntw;wpf;F top
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𝑋, 𝑌 I nghWj;J
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
nrt;tfyj;jpd; ePsk; 4𝑎 = 8 8
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
𝑎 = 3 gutisak; fPo;Gwk; jpwg;GilaJ
𝑋, 𝑌 I nghWj;J
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
𝑎 = 2 gutisak; fPo;Gwk; jpwg;GilaJ
𝑋, 𝑌 I nghWj;J
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)
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12 k; tFg;G fzf;F gj;J kjpg;ngz; tpdhf;fs; ntw;wpf;F top
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,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
nrt;tfyj;jpd; ePsk; 4𝑎 = 8 8
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
𝑎 = 1 gutisak; fPo;Gwk; jpwg;GilaJ
𝑋, 𝑌 I nghWj;J
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
nrt;tfyj;jpd; ePsk; 4𝑎 = 4 4
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
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12 k; tFg;G fzf;F gj;J kjpg;ngz; tpdhf;fs; ntw;wpf;F top
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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, 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, 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)
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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, 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)
Ftpaq;fs; (0, ±𝑎𝑒)
(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
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𝑋, 𝑌 I nghWj;J
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)
Ftpaq;fs; (0, ±𝑎𝑒)
(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, 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)
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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, 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
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𝑎𝑒 = 3 ×5
3= 5
𝑋, 𝑌 I nghWj;J
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
FWf;fr;R 𝑋-mr;rpw;F ,izahf cs;sJ.
𝑎2 = 4, 𝑏2 = 1, 𝑎 = 2, 𝑏 = 1
𝑒 = 1 +𝑏2
𝑎2 = 1 +1
4 =
4+1
4=
5
2
𝑎𝑒 = 2 × 5
2= 5
𝑋, 𝑌 I nghWj;J
𝑥, 𝑦 I nghWj;J 𝑋 = 𝑥 + 3, 𝑌 = 𝑦
− 2
ikak; (0,0) 𝑋 = 0, 𝑌 = 0
𝑥 + 3 = 0, 𝑦 − 2 = 0 𝐶(−3,2)
Ftpaq;fs; (±𝑎𝑒, 0)
(± 5, 0)
5, 0
𝑋 = 5, 𝑌 = 0 𝑥 + 3 = 5, 𝑦 − 2 = 0
𝑥 = −3 + 5, 𝑦 = 2
𝐹1(−3 + 5, 2)
− 5, 0
𝑋 = − 5, 𝑌 = 0 𝑥 + 3 = − 5, 𝑦 − 2 = 0
𝑥 = −3 − 5, 𝑦 = 2
𝐹2(−3 − 5, −2)
Kidfs; (±𝑎, 0) (±2,0)
(2, 0) 𝑋 = 2, 𝑌 = 0
𝑥 + 3 = 2, 𝑦 − 2 = 0 𝑥 = −1, 𝑦 = 2
𝐴(−1,2) (−2,0)
𝑋 = −2, 𝑌 = 0 𝑥 + 3 = −2, 𝑦 − 2 = 0
𝑥 = −5, 𝑦 = 2
𝐴′(−5,2)
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
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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
FWf;fr;R 𝑌-mr;rpw;F ,izahf cs;sJ.
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, 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
FWf;fr;R 𝑌-mr;rpw;F ,izahf cs;sJ.
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, 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
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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
FWf;fr;R 𝑋-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, 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
FWf;fr;R 𝑋-mr;rpw;F ,izahf cs;sJ.
𝑎2 =75
12, 𝑏2 =
75
4,
𝑒 = 1 +𝑏2
𝑎2 = 1 +75
475
12
= 1 +12
4= 1 + 3 = 2
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𝑎𝑒 = 75
12× 2
= 3×25
3×4× 2 =
5
2× 2 = 5
𝑋, 𝑌 I nghWj;J
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)
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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
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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)
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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
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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)
(1) kw;Wk; (2) ypUe;J 𝜕2𝑢
𝜕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)
(1) kw;Wk; (2) ypUe;J 𝜕2𝑢
𝜕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 )
(1) kw;Wk; (2) ypUe;J 𝜕2𝑢
𝜕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)
(1) kw;Wk; (2) ypUe;J 𝜕2𝑢
𝜕𝑥𝜕𝑦=
𝜕2𝑢
𝜕𝑦𝜕𝑥
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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 < ∞
epiyf;Fj;J ePl;bg;G −∞ < 𝑦 < ∞
fpilkl;l ePl;bg;G 0 ≤ 𝑥 < ∞
epiyf;Fj;J ePl;bg;G −∞ < 𝑦 < ∞
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.
tistiuf;F ve;j xU njhiyj;njhLNfhLfSk;
,y;iy.
tistiuf;F ve;j xU njhiyj;njhLNfhLfSk;
,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;
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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
𝑑𝑡𝛼𝐴 ⇒
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
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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
𝑑𝑡𝛼𝐴 ⇒
𝑑𝐴
𝑑𝑡= 𝑘𝐴 ⇒ 𝐴 = 𝑐𝑒𝑘𝑡
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
𝑑𝑡𝛼𝐴 ⇒
𝑑𝐴
𝑑𝑡= 𝑘𝐴 ⇒ 𝐴 = 𝑐𝑒𝑘𝑡
𝑡 = 0 vdpy; 𝐴 = 𝐴0
/ 𝐴0 = 𝑐𝑒0 = 𝑐
/ 𝐴 = 𝐴0𝑒𝑘𝑡
50 tUlq;fspy; Nubak; Muk;g epiyapypUe;J
5 % rpijTWfpwJ.
𝑡 = 50 vdpy; , 𝐴 = .95 𝐴0
/ 0.95 𝐴0 = 𝐴0𝑒50𝑘 ⇒ 𝑒50𝑘 = 0.95
NkYk; 𝑡 = 100 vdpy;
𝐴 = 𝐴0𝑒100𝑘 = 𝐴0 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
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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
𝑑𝑡𝛼𝐴 ⇒
𝑑𝐴
𝑑𝑡= 𝑘𝐴 ⇒ 𝐴 = 𝑐𝑒𝑘𝑡
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
𝑑𝑡= 𝑘𝐴 ⇒ 𝐴 = 𝑐𝑒𝑘𝑡
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;.
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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;.
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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;
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(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
∴ milg;G tpjp cz;ikahFk;
(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:
(i) ngUf;fy; ml;ltizapd; vy;yh
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;
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𝜔𝑙 . 𝜔𝑛−𝑙 = 𝜔𝑛−𝑙 . 𝜔𝑙 = 𝜔𝑛 = 1 ,t;thwhf vjph;kiw tpjp cz;ikahfpwJ.
∴ (𝐺, . ) xU FykhFk;.
(v) ghpkhw;W tpjp:
𝜔𝑙 . 𝜔𝑚 = 𝜔𝑙+𝑚 = 𝜔𝑚+𝑙 = 𝜔𝑚 . 𝜔𝑙 ∀𝜔𝑙 , 𝜔𝑚 ∈ 𝐺
∴ (𝐺, . ) xU vgPypad; FykhFk;. 𝐺 ,y; 𝑛
cWg;Gfs; kl;LNk cs;sjhy; (𝐺, . )MdJ 𝑛 thpir nfhz;;l Kbthd vgPypad; FykhFk;.
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
(i) ngUf;fy; ml;ltizapd; vy;yh
cWg;GfSk; G,d; cWg;GfshFk;.
/ milg;G tpjp cz;ikahFk;.
(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]
[4] ,d; vjph;kiw [3]
[5] ,d; vjph;kiw [9]
[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;:
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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
/ milg;G tpjp cz;ikahFk;.
(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
/ 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;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.
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(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;.
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(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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