· 10-k; tFg;G fzf;F muRj;Njh;T rpwg;Gf; ifNaL ntw;wpf;F top [email protected] - 5 -

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-1-

ntw;wpf;F top

(Way to Success)

fzf;F

muRj;Njh;T rpwg;G ifNaL

jahupg;G Mrpupau; FO jpU. f. ehfuh[d; M.A.,M.A.,M. Phil., B. Ed., jpUkjp M. yPkh Nuh]; M.Sc., B. Ed.> jpU. f. jpNd\; M.Sc., M. Phil., P.G.D.C.A., (Ph.D.,)

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

kpd;dQ;ry; : [email protected]

& [email protected] njhlh;Gf;F : 7418865975 (ghlg;nghUs; rhu;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;



mailto:[email protected]


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-2-

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

gpupT tpdh vz; tpdh tif Nfl;fg;gLk;

tpdhf;fs; vOj

Ntz;bait kjpg;

ngz;fs; gpupT-m 1 - 15 njupTtpdh (xU kjpg;ngz; tpdhf;fs;) 15 15 15

gpupT-M 16 - 29 2 kjpg;ngz; tpdhf;fs; 14 9 18 30 fl;lha 2 kjpg;ngz; tpdh 2 1 2

gpupT-, 31- 44 5 kjpg;ngz; tpdhf;fs; 14 8 40 45 fl;lha 5 kjpg;ngz; tpdh 2 1 5

gpupT-< 46 nra;Kiw tbtpay; 2 1 10 47 tiuglq;fs; 2 1 10

51 36 100

tpdhj;jhs; - gFg;gha;T

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

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

,ay; vz;

,ay; xU

kjpg;ngz; tpdhf;fs;

,uz;L kjpg;ngz; tpdhf;fs;

Ie;J kjpg;ngz; tpdhf;fs;

1 fzq;fSk; rhh;GfSk; 1 2 2 2 nka;naz;fspd; njhlh;thpirfSk; njhlh;fSk; 2 1 2 3 ,aw;fzpjk; 2 2 3 4 mzpfs; 1 2 1 5 Maj;njhiy tbtpay; 2 2 2 6 tbtpay; 2 1 1

7 Kf;Nfhztpay; 2 2 1

8 mstpay; 1 2 2

11 Gs;spapay; 1 1 1

12 epfo;jfT 1 1 1

$Ljy; 15 16 16

,ay; vz; ,ay; gj;J kjpg;ngz; tpdhf;fs;

9 nra;Kiw tbtpay; 1

10 tiuglq;fs; 1




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-3-

md;ghd khztu;fSf;F>

tzf;fk;. gj;jhk; 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;:

fzpjg;ghlj;jpy; GhpjYf;F Kf;fpaj;Jtk; nfhLj;J kdg;ghlk; nra;Ak; gof;fj;ij iftpLq;fs;.

Nju;T miwf;F EioAk; tiuapy; gbj;Jf; nfhz;bUf;f Ntz;lhk;. Nju;Tf;F Kd;du; ePq;fs; jahupj;J itj;jpUf;Fk; Fwpg;Gfis kl;Lk; mt;tg;NghJ ghu;j;J epidTgLj;jpf; nfhs;tij (Recall) tof;fg;gLj;jpf; nfhs;Sq;fs;.

Nju;tpd;NghJ Kjypy; nfhLf;fg;gLk; 10 epkplq;fspy; ve;;nje;j tpdhf;fis vOJtJ vd;gij KbTnra;J nfhs;Sq;fs;.

nra;Kiw tbtpay; (gFjp IV y;) tpdhTf;F Kjypy; tpilaspAq;fs;. tiuglj;jhs; (GRAPH) filrp gf;fj;jpy; nfhLf;fg;gl;bUf;Fk;. mt;tiuglj;jpw;F ,izahf cs;s gf;fj;jpy; fzpj tpsf;fj;ij nra;J fhl;;Lq;fs;.

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;. Njh;T nra;j ghlg;gFjpapid jpUk;g jpUk;g gapw;rp vLg;gjd; %yk; ntw;wp ngwyhk;. 1 Mark, 2 Marks, Geometry, Graph ,tw;wpy; KOf;ftdk; itj;J nra;Ak; NghJ ntw;wp

epr;rak;. ghlg;Gj;jfj;jpy; nfhLf;fg;gl;Ls;s xU kjpg;ngz; tpdhf;fs; (205 tpdhf;fs;).

midj;ijAk; ed;F gbj;Jf;nfhs;Sq;fs;. ,uz;L kjpg;ngz; kw;Wk; Ie;J kjpg;ngz; tpdhf;fSf;F tpilaspf;f gpd;tUk;

ghlq;fis kl;Lk; ed;F gbj;Jf; nfhs;Sq;fs; 1. fzq;fSk; rhh;GfSk; 3. ,aw;fzpjk; 4. mzpfs; 5. Maj;njhiy tbtpay; 6. tbtpay; (3 Njw;wk; kl;Lk;) 8. mstpay; 12. epfo;jfT 9. nra;Kiw tbtpay; : i. njhLNfhLfs; (8 fzf;Ffs;)

ii. Kf;Nfhzk; tiujy; (8 fzf;Ffs;) 10. tiuglq;fs; : rpy rpwg;G tiuglq;fs; kl;Lk; (9 fzf;Ffs;)

fzpjj;ijg; nghUj;jtiu nfhQ;rk; Mu;tKk; rpwpjsT Kaw;rpAk; gapw;rpAk; ,Ue;jhNy ePq;fs; vspjhf 50 kjpg;ngz;fs; ngw;Wtpl KbAk;. mtutu; jpwikiag; nghUj;J ,d;Dk; $Ljy; kjpg;ngz;fs; ngwTk; cq;fshy; KbAk;.

rpwg;ghf nray;gl;lhy; ntw;wp cWjp. muRj;Nju;tpy; 100f;F 100 ngw tho;j;JfpNwhk;. - Mrpupau; FO



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-4-

cs;slf;fk;

,ay; vz; ,ay; jiyg;G

gf;fk; vz; xU kjpg;ngz; tpdhf;fs;

,uz;L kjpg;ngz; tpdhf;fs;

1 fzq;fSk; rhh;GfSk; 5 19

2 nka;naz;fspd; njhlh;thpirfSk; njhlh;fSk;

6 25

3 ,aw;fzpjk; 7 30 4 mzpfs; 9 36

5 Maj;njhiy tbtpay; 11 43

6 tbtpay; 12 51

7 Kf;Nfhztpay; 14 56 8 mstpay; 15 59 11 Gs;spapay; 17 68 12 epfo;jfT 18 71

9 nra;Kiw tbtpay; 78

10 tiuglq;fs; 88

epidtpw; nfhs;s Ntz;ba Kf;fpa Fwpg;Gfs; 94

FwpaPLfs; rkk; rkkpy;iy

tplf;FiwT FiwT my;yJ rkk;

tpl mjpfk; mjpfk; my;yJ rkk;

rkhdkhd Nrh;g;G

ntl;L cWg;G

cWg;gy;y jF cl;fzk;

cl;fzk; jF cl;fzky;y

cl;fzky;y 𝐴′ 𝑜𝑟 𝐴𝑐 𝐴 d; epug;G fzk;

∅ 𝑜𝑟 ntw;Wf;fzk; 𝑃 𝐴 𝐴 d; mLf;F fzk;

𝑛 𝐴 Mjp vz; |||ly ,Nj Nghd;W

P A 𝐴 vd;w epfo;r;rpapd; epfo;jfT rkr;rPh; tpj;jpahrk;

𝑁 ,ay; vz;fs; 𝑅 nka;naz;fs;

𝑍 KOf;fs; Kf;Nfhzk;

Nfhzk; nrq;Fj;J

,iz czh;j;JfpwJ

vdNt | | jdp kjpg;G

Njhuhakhf rkk; rh;trkk;

𝜋 ig kpif my;yJ Fiw

Njw;wk; KbT | (or) : mjd;gb



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10-k; tFg;G fzf;F muRj;Njh;T rpwg;Gf; ifNaL ntw;wpf;F top

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xU kjpg;ngz; tpdhf;fs; 1. fzq;fSk; rhh;GfSk;

1. A kw;Wk; B vd;gd ,uz;L fzq;fs; vd;f. 𝐴 ∪ 𝐵 = 𝐴 vd;gjw;Fj; Njitahd kw;Wk; NghJkhd fl;Lg;ghL

m) 𝑩 ⊆ 𝑨 M) 𝐴 ⊆ 𝐵 ,) 𝐴 ≠ 𝐵 <) A ∩ 𝐵 = ∅

2. A ⊂ B vdpy;> A ∩ B =

m) 𝐵 M) 𝐴\𝐵 ,) 𝑨 <) 𝐵\𝐴

3. P kw;Wk; Q vd;gd VNjDk; ,uz;L fzq;fs; vdpy;> 𝑃 ∩ 𝑄 =

m) {𝑥: 𝑥 ∈ 𝑃 my;yJ 𝑥 ∈ 𝑄 } M) {𝑥: 𝑥 ∈ 𝑃 kw;Wk; 𝑥 ∉ 𝑄}

,) {𝒙: 𝒙 ∈ 𝑷 kw;Wk; 𝒙 ∈ 𝑸} <) {𝑥: 𝑥 ∉ 𝑃 kw;Wk; 𝑥 ∈ 𝑄}

4. 𝐴 = {𝑝, 𝑞, 𝑟, 𝑠}, 𝐵 = {𝑟, 𝑠, 𝑡, 𝑢} vdpy>; 𝐴\𝐵 =

m) {𝒑, 𝒒} M) {𝑡, 𝑢} ,) {𝑟, 𝑠} <) {𝑝, 𝑞, 𝑟, 𝑠}

5. 𝑛 [𝑝(𝐴)] = 64 vdpy>; 𝑛(𝐴) =

m) 6 M) 8 ,) 4 <) 5

6. A, B kw;Wk; C Mfpa VNjDk; %d;W fzq;fSf;F> 𝐴 ∩ (𝐵 ∪ 𝐶) =

m) (𝐴 ∪ 𝐵) ∪ (𝐵 ∩ 𝐶) M) (𝑨 ∩ 𝑩) ∪ (𝑨 ∩ 𝑪) ,) 𝐴 ∪ (𝐵 ∩ 𝐶) <) (𝐴 ∪ 𝐵) ∩ (𝐵 ∪ 𝐶)

7. A , B Mfpa ,uz;L fzq;fSf;F {(𝐴\𝐵) ∪ (𝐵\𝐴)} ∩ (𝐴 ∩ 𝐵) =

m) ∅ M) 𝐴 ∪ 𝐵 ,) 𝐴 ∩ 𝐵 <) 𝐴’ ∩ 𝐵’

8. fPNo nfhLf;fg;gl;Ls;sitfspy; jtwhd $w;W vJ? m) 𝐴\𝐵 = 𝐴 ∩ 𝐵’ M) 𝑨\𝑩 = 𝑨 ∩ 𝑩 ,) 𝐴\𝐵 = (𝐴 ∪ 𝐵) ∩ 𝐵’ <) A\𝐵 = (𝐴 ∪ 𝐵)\𝐵

9. A, B kw;Wk; C Mfpa VNjDk; %d;W fzq;fSf;F> 𝐵\(𝐴 ∪ 𝐶) =

m) (𝐴\𝐵) ∩ (𝐴\𝐶) M) (𝑩\𝑨) ∩ (𝑩\𝑪) ,) (𝐵\𝐴) ∩ (𝐴\𝐶) <) (𝐴\𝐵) ∩ (𝐵\𝐶)

10. 𝑛(𝐴) = 20, 𝑛(𝐵) = 30 kw;Wk; 𝑛(𝐴 ∪ 𝐵) = 40 vdpy; 𝑛 𝐴 ∩ 𝐵 =

m) 50 M) 10 ,) 40 <) 70

11. {(𝑥, 2), (4, 𝑦)} xU rkdpr; rhh;igf; Fwpf;fpwJ vdpy;> (x, y) = m) (𝟐, 𝟒) M) (4, 2) ,) (2, 2) <) (4, 4)

12. {(7, 11), (5, 𝑎)} xU khwpypr; rhh;igf; Fwpf;fpwJ vdpy;> 𝑎 d; kjpg;G

m) 7 M) 11 ,) 5 <) 9

13. 𝑓(𝑥) = (−1)𝑥 vd;gJ ℕ -ypUe;J ℤ f;F tiuaWf;fg;gl;Ls;sJ. 𝑓 -d; tPr;rfk;

m) {1} M) ℕ ,) {1, −1} <) ℤ

14. 𝑓 = {(6, 3), (8, 9), (5, 3), (−1, 6)}, vdpy; 3-d; Kd; cUf;fs;

m) 5 kw;Wk; -1 M) 6 kw;Wk; 8 ,) 8 kw;Wk; -1 <) 6 kw;Wk; 5

15. 𝐴 = 1, 3, 4, 7, 11 , 𝐵 = {−1, 1, 2, 5, 7, 9} vd;f. 𝑓 = {(1, −1), (3, 2), (4, 1), (7, 5), (11, 9)} vd;wthW mike;j rhh;G 𝑓: 𝐴 → 𝐵 vd;gJ

m) xd;Wf;F xd;whd rhh;G M) Nky; rhh;G ,) ,UGwr;rhh;G <) rhh;G my;y

16. nfhLf;fg;gl;Ls;s glk; Fwpf;Fk; rhh;G. xU m) Nky; rhh;G ,) xd;Wf;F xd;whd rhh;G M) khwpypr; rhh;G <) rhh;G my;y



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17. 𝐴 = {5, 6, 7}, 𝐵 = {1, 2, 3, 4, 5} vd;f. f(𝑥) = 𝑥 − 2 vd;wthW tiuaiw nra;ag;gl;l rhh;G

𝑓: 𝐴 → 𝐵-d; tPr;rfk;

m) {1, 4, 5} M) {1, 2, 3, 4, 5} ,) {2, 3, 4} <) {𝟑, 𝟒, 𝟓}

18. 𝑓(𝑥) = 𝑥2 + 5, vdpy;> 𝑓 (− 4) =

m) 26 M) 21 ,) 20 <) -20

19. xU rhh;gpd; tPr;rfk; XUWg;Gf; fzkhdhy;> mJ xU m) khwpypr;rhh;G M) rkdpr;rhh;G ,) ,UGwr;rhh;G <) xd;Wf;F xd;whd rhh;G

20. 𝑓: 𝐴 → 𝐵 xU ,UGwr; rhh;G kw;Wk; n(𝐴) = 5, vdpy; 𝑛(𝐵) =

m) 10 M) 4 ,) 5 <) 25

2. nka;naz;fspd; njhlh;thpirfSk; njhlh;fSk;

1. gpd;tUtdtw;Ws; vJ nka;ahdf; $w;wy;y? m) ,ay; vz;fspd; fzk; ℕ -y; tiuaiw nra;ag;gl;l nka;naz; kjpg;Gilar; rhh;G xU

njhlh; thpirahFk;. M) xt;nthU rhh;Gk; xU njhlh; thpirapidf; Fwpf;Fk;. ,) xU njhlh; thpir> Kbtpyp vz;zpf;ifapy; cWg;Gfisf; nfhz;bUf;fyhk;. <) xU njhlh;thpir> KbTW vz;zpf;ifapy; cWg;Gfisf; nfhz;bUf;fyhk;.

2. 1, 1, 2, 3, 5, 8, …….. vd;w njhlh;thpirapd; 8 tJ cWg;G

m) 25 M) 24 ,) 23 <) 21

3. 1

2,

1

6,

1

12,

1

20 ……. vd;w njhlh;thpirapy;

1

20 -f;F mLj;j cWg;G

m) 1

24 M)

1

22 ,)

𝟏

𝟑𝟎 <)

1

18

4. 𝑎, 𝑏, 𝑐, 𝑙, 𝑚 vd;gd $l;Lj;njhlh;thpirapy; ,Ug;gpd; 𝑎 − 4𝑏 + 6𝑐 – 4 𝑙 + 𝑚 =

m) 1 M) 2 ,) 3 <) 0

5. a, b, c vd;gd $l;Lj;njhlh;thpirapy; cs;sd vdpy;> 𝑎−𝑏

𝑏−𝑐=

m) 𝑎

𝑏 M)

𝑏

𝑐 ,)

𝑎

𝑐 <) 1

6. 100 𝑛 + 10 vd;gJ xU njhlh;thpirapd; 𝑛-tJ cWg;G vdpy;> mJ

m) xU $l;Lj;njhlh;thpir M) xU ngUf;Fj;njhlh;thpir ,) xU khwpypj; njhlh;thpir <) xU $l;Lj;; njhlh;thpirAk; my;y ngUf;Fj; njhlh;thpirAk; my;y

7. 𝑎1, 𝑎2, 𝑎3, ……. vd;gd xU $l;Lj; njhlh;thpirapYs;sd. NkYk; 𝑎4

𝑎7 =

3

2 vdpy; 13 tJ cWg;G

m) 3

2 M) 0 ,) 12𝑎1 <) 14𝑎1

8. 𝑎1, 𝑎2, 𝑎3,……. vd;gJ xU $l;Lj; njhlh;thpir vdpy; 𝑎5, 𝑎10 , 𝑎15 ….. vd;w njhlh;thpirahdJ m) xU ngUf;Fj;njhlh;thpir M) xU $l;Lj;njhlh;thpir

,) xU $l;Lj;; njhlh;thpirAk; my;y ngUf;Fj; njhlh;thpirAk; my;y

<) xU khwpypj; njhlh;thpir

9. xU $l;Lj; njhlh;thpirapd; mLj;jLj;j %d;W cWg;Gfs; 𝑘 + 2, 4𝑘 − 6, 3𝑘 − 2 vdpy; 𝑘 d; kjpg;G m) 2 M) 3 ,) 4 <) 5



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10. 𝑎, 𝑏, 𝑐, 𝑙, 𝑚, 𝑛 vd;gd $l;Lj; njhlh;thpirapy; mike;Js;sd vdpy;>

3𝑎 + 7, 3𝑏 + 7, 3𝑐 + 7, 3𝑙 + 7, 3𝑚 + 7, 3𝑛 + 7 vd;w njhlh;thpir

m) xU ngUf;Fj;njhlh;thpir M) xU $l;Lj;njhlh;thpir ,) xU khwpypj; njhlh;thpir <) xU $l;Lj;; njhlh;thpirAk; my;y ngUf;Fj; njhlh;thpirAk; my;y

11. xU ngUf;Fj; njhlh;thpirapy; 3 tJ cWg;G 2 vdpy;> mjd; Kjy; 5 cWg;Gfspd; ngUf;fw;gyd;

m) 52 M) 𝟐𝟓 ,) 10 <) 15

12. 𝑎, 𝑏, 𝑐 vd;gd xU ngUf;F njhlh;thpirapy; cs;sd vdpy;> 𝑎−𝑏

𝑏−𝑐 =

m) 𝒂

𝒃 M)

𝑏

𝑎 ,)

𝑎

𝑐 <)

𝑐

𝑏

13. 𝑥, 2𝑥 + 2, 3𝑥 + 3 vd;gd ngUf;Fj; njhlh;thpirapypUg;gpd; 5𝑥, 10𝑥 + 10, 15𝑥 + 15 vd;w njhlh;thpirahdJ

m) xU $l;Lj;njhlh;thpir M) xU ngUf;Fj;njhlh;thpir ,) xU khwpypj; njhlh;thpir <) xU $l;Lj;; njhlh;thpirAk; my;y ngUf;Fj; njhlh;thpirAk; my;y

14. -3, -3, -3, ……… vd;w njhlh;thpirahdJ

m) xU $l;Lj;njhlh;thpir kl;Lk; M) xU ngUf;Fj;njhlh;thpir kl;Lk;

,) xU $l;Lj;; njhlHthpirAk; my;y ngUf;Fj; njhlh;thpirAk; my;y <) xU $l;Lj;njhlh;thpir kw;Wk; ngUf;Fj;njhlh;thpir

15. xU ngUf;Fj; njhlh;thpirapd; Kjy; ehd;F cWg;Gfspd; ngUf;fw;gyd; 256. mjd; nghJtpfpjk; 4 kw;Wk; mjd; Kjy; cWg;G kpif vz; vdpy;> me;jg; ngUf;Fj; njhlh;thpirapd; 3 tJ cWg;G

m) 8 M) 1

16 ,)

1

32 <) 16

16. xU ngUf;Fj; njhlh; thpirapy; 𝑡2 = 3

5 kw;Wk; 𝑡3 =

1

5 vdpy;> mjd; nghJ tpfpjk;

m) 1

5 M)

𝟏

𝟑 ,) 1 <) 5

17. 𝑥 ≠ 0, vdpy; 1 + 𝑠𝑒𝑐 𝑥 + 𝑠𝑒𝑐2 𝑥 + 𝑠𝑒𝑐3 𝑥 + 𝑠𝑒𝑐4 𝑥 + 𝑠𝑒𝑐5𝑥 =

m) (1 + 𝑠𝑒𝑐 𝑥) (𝑠𝑒𝑐2𝑥 + 𝑠𝑒𝑐3𝑥 + 𝑠𝑒𝑐4𝑥) M) (𝟏 + 𝒔𝒆𝒄 𝒙) (𝟏 + 𝒔𝒆𝒄𝟐𝒙 + 𝒔𝒆𝒄𝟒𝒙)

,) (1 − 𝑠𝑒𝑐 𝑥) (𝑠𝑒𝑐 𝑥 + 𝑠𝑒𝑐3𝑥 + 𝑠𝑒𝑐5𝑥) <) (1 + 𝑠𝑒𝑐 𝑥)(1 + 𝑠𝑒𝑐3𝑥 + 𝑠𝑒𝑐4𝑥)

18. 𝑡𝑛 = 3 − 5𝑛 vd;gJ xU $l;Lj; njhlh;thpirapd; 𝑛 tJ cWg;G vdpy;> mf;$l;Lj; njhlh;thpirapd;

Kjy; 𝑛 cWg;Gfspd; $Ljy;

m) 𝒏

𝟐 [𝟏 − 𝟓𝒏] M) 𝑛(1 − 5𝑛) ,)

𝑛

2 (1 + 5𝑛) <)

𝑛

2 (1 + 𝑛)

19. 𝑎𝑚−𝑛 , 𝑎𝑚 , 𝑎𝑚+𝑛 vd;w ngUf;Fj; njhlh;thpirapd; nghJ tpfpjk; =

m) 𝑎𝑚 M) 𝑎−𝑚 ,) 𝒂 𝒏 <) 𝑎−𝑛

20. 1 + 2 + 3 + …. +𝑛 = 𝑘 vdpy;> 13 + 23 +….+𝑛3 vd;gJ

m) 𝒌𝟐 M) 𝑘3 ,) 𝑘(𝑘+1)

2 <) (𝑘 + 1)3

3. ,aw;fzpjk;

1. 6𝑥 − 2 𝑦 = 3, 𝑘𝑥 − 𝑦 = 2 vd;w njhFg;gpw;F xNunahU jPh;T cz;nldpy;>

m) 𝑘 = 3 M) 𝒌 ≠ 𝟑 ,) 𝑘 = 4 <) 𝑘 ≠ 4



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2. ,U khwpfspy; cs;s Nehpay; rkd;ghLfspd; njhFg;G xUq;fikahjJ vdpy;> mtw;wpd; tiuglq;fs; m) xd;wpd; kPJ xd;W nghUe;Jk; M) xU Gs;spapy; ntl;bf;nfhs;Sk;

,) ve;jg;Gs;spapYk; ntl;bf;nfhs;shJ <) x-mr;ir ntl;Lk;

3. 𝑥 − 4𝑦 = 8, 3𝑥 − 12𝑦 = 24 vd;Dk; rkd;ghLfspd; njhFg;gpw;F m) Kbtpyp vz;zpf;ifapy; jPh;Tfs; cs;sd M) jPh;T ,y;iy

,) xNunahU jPh;T kl;Lk; cz;L. <) xU jPh;T ,Uf;fyhk; my;yJ ,y;yhkYk; ,Uf;fyhk;.

4. 𝑝(𝑥) = (𝑘 + 4) 𝑥2 + 13𝑥 + 3𝑘 vd;Dk; gy;YWg;Gf;Nfhitapd; xU G+r;rpak; kw;nwhd;wpd;

jiyfPopahdhy;> 𝑘 d; kjpg;G

m) 2 M) 3 ,) 4 <) 5

5. 𝑓 𝑥 = 2𝑥2 + 𝑝 + 3 𝑥 + 5 vd;Dk; gy;YWg;Gf;Nfhitapd; ,U G+r;rpaq;fspd; $Ljy; G+r;rpak;

vdpy; 𝑝 d; kjpg;G m) 3 M) 4 ,) -3 <) -4

6. 𝑥2 − 2𝑥 + 7 vd;gij 𝑥 + 4 My; tFf;Fk;NghJ fpilf;Fk; kPjp m) 28 M) 29 ,) 30 <) 31

7. 𝑥3 − 5𝑥2 + 7𝑥 − 4 vd;gij 𝑥 − 1 My; tFf;Fk;NghJ fpilf;Fk; <T

m) 𝑥2 + 4𝑥 + 3 M) 𝒙𝟐 − 𝟒𝒙 + 𝟑 ,) 𝑥2 − 4𝑥 − 3 <) 𝑥2 + 4𝑥 − 3

8. (𝑥3 + 1) kw;Wk; 𝑥4 − 1 Mfpadtw;wpd; kP.ngh.t.

m) 𝑥3 − 1 M) 𝑥3 + 1 ,) 𝒙 + 𝟏 <) 𝑥 − 1

9. 𝑥2 − 2𝑥𝑦 + 𝑦2 kw;Wk; 𝑥4 − 𝑦4 Mfpadtw;wpd; kP.ngh.t.

m) 1 M) 𝑥 + 𝑦 ,) 𝒙 − 𝒚 <) 𝑥2 – 𝑦2

10. 𝑥3 – 𝑎3 kw;Wk; (𝑥 − 𝑎)2 Mfpadtw;wpd; kP.ngh.k.

m) (𝑥3 − 𝑎3) (𝑥 + 𝑎) M) (𝑥3 − 𝑎3) (𝑥 − 𝑎)2

,) (𝒙 − 𝒂)𝟐 (𝒙𝟐 + 𝒂𝒙 + 𝒂𝟐) <) (𝑥 + 𝑎)2 (𝑥2 + 𝑎𝑥 + 𝑎2)

11. 𝑘 ∈ ℕ vDk; NghJ a𝑘 , a𝑘+3, a𝑘+5 Mfpatw;wpd; kP.ngh.k.

m) a𝑘+9 M) a𝑘 ,) a𝑘+6 <) 𝒂𝒌+𝟓

12. 𝑥2+5𝑥+6

𝑥2−𝑥−6 vd;Dk; tpfpjKW Nfhitapd; kpfr; RUf;fpa tbtk;

m) 𝑥−3

𝑥+3 M)

𝒙+𝟑

𝒙−𝟑 ,)

𝑥+2

𝑥−3 <)

𝑥−3

𝑥+2

13. 𝑎+𝑏

𝑎−𝑏 kw;Wk;

𝑎3−𝑏3

𝑎3+𝑏3 Mfpad ,U tpfpjKW Nfhitfs; vdpy;> mtw;wpd; ngUf;fw;gyd;

m) 𝒂𝟐+𝒂𝒃+𝒃𝟐

𝒂𝟐−𝒂𝒃+𝒃𝟐 M) 𝑎2−𝑎𝑏+𝑏2

𝑎2+𝑎𝑏+𝑏2 ,) 𝑎2−𝑎𝑏−𝑏2

𝑎2+𝑎𝑏+𝑏2 <) 𝑎2+𝑎𝑏+𝑏2

𝑎2−𝑎𝑏−𝑏2

14. 𝑥2−25

𝑥+3 vd;gij

𝑥+5

𝑥2−9 My; tFf;Fk;NghJ fpilf;Fk; <T

m) (𝒙 − 𝟓) (𝒙 − 𝟑) M) (𝑥 − 5) (𝑥 + 3) ,) (𝑥 + 5) (𝑥 − 3) <) (𝑥 + 5) (𝑥 + 3)

15. 𝑎3

𝑎−𝑏 cld;

𝑏3

𝑏−𝑎 If; $l;l> fpilf;Fk; Gjpa Nfhit

m) 𝒂𝟐 + 𝒂𝒃 + 𝒃𝟐 M) 𝑎2 – 𝑎𝑏 + 𝑏2 ,) 𝑎3 + 𝑏3 <) 𝑎3 – 𝑏3



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16. 49 (𝑥2 − 2𝑥𝑦 + 𝑦2)2 d; th;f;f%yk;

m) 7 𝑥 − 𝑦 M) 7(𝑥 + 𝑦) (𝑥 − 𝑦) ,) 7(𝑥 + 𝑦)2 <) 𝟕(𝒙 − 𝒚)𝟐

17. 𝑥2 + 𝑦2 + 𝑧2 − 2𝑥𝑦 + 2𝑦𝑧 – 2𝑧𝑥 -d; th;f;f%yk;

m) 𝑥 + 𝑦 − 𝑧 M) 𝑥 − 𝑦 + 𝑧 ,) 𝑥 + 𝑦 + 𝑧 <) 𝒙 − 𝒚 − 𝒛

18. 121 𝑥4𝑦8 𝑧6 (𝑙 − 𝑚)2 d; th;f;f %yk;

m) 11𝑥2 𝑦4 𝑧4 𝑙 − 𝑚 M) 11𝑥4 𝑦4 𝑧3(𝑙 − 𝑚)

,) 11𝑥2 𝑦4 𝑧6 𝑙 − 𝑚 <) 𝟏𝟏𝒙𝟐 𝐲𝟒 𝒛𝟑(𝒍 − 𝒎)

19. 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 vd;w rkd;ghl;bd; %yq;fs; rkk; vdpy;> 𝑐 d; kjpg;G

m) 𝑏2

2𝑎 M)

𝒃𝟐

𝟒𝒂 ,) −

𝑏2

2𝑎 <) −

𝑏2

4𝑎

20. 𝑥2 + 5𝑘𝑥 + 16 = 0 vd;w rkd;ghl;bw;F nka;naz; %yq;fs; ,y;iynadpy;>

m) 𝑘 >8

5 M) 𝑘 > −

8

5 ,) −

𝟖

𝟓< 𝒌 <

𝟖

𝟓 <) 0 < 𝑘 <

8

5

21. 3 – I xU %ykhff; nfhz;l ,Ugbr; rkd;ghL

m) 𝑥2 − 6𝑥 − 5 = 0 M) 𝑥2 + 6𝑥 − 5 = 0 ,) 𝑥2 − 5𝑥 − 6 = 0 <) 𝒙𝟐 − 𝟓𝒙 + 𝟔 = 𝟎

22. 𝑥2 –𝑏𝑥 + 𝑐 = 0 kw;Wk; 𝑥2 + 𝑏𝑥 – 𝑎 = 0 Mfpa rkd;ghLfspd; nghJthd %yk;

m) 𝒄+𝒂

𝟐𝒃 M)

𝑐−𝑎

2𝑏 ,)

𝑐+𝑏

2𝑎 <)

𝑎+𝑏

2𝑐

23. 𝑎 ≠ 0, vd mike;j rkd;ghL 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 -d; %yq;fs; 𝛼 kw;Wk; 𝛽 vdpy;> gpd;tUtdtw;Ws; vJ nka;ay;y?

m) 𝛼2 + 𝛽2 =𝑏2−2𝑎𝑐

𝑎2 M) 𝛼𝛽 =𝑐

𝑎 ,) 𝜶 + 𝜷 =

𝒃

𝒂 <)

1

𝛼+

1

𝛽= −

𝑏

𝑐

24. 𝑎𝑥2 𝑏𝑥 + 𝑐 = 0, vd;w ,Ugbr;rkd;ghl;bd; %yq;fs; 𝛼 kw;Wk; 𝛽 vdpy;>1

𝛼 kw;Wk;

1

𝛽 Mfpadtw;iw

%yq;fshff; nfhz;l ,Ugbr; rkd;ghL m) 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 M) 𝑏𝑥2 + 𝑎𝑥 + 𝑐 = 0

,) 𝒄𝒙𝟐 + 𝒃𝒙 + 𝒂 = 𝟎 <) 𝑐𝑥2 + 𝑎𝑥 + 𝑏 = 0

25. 𝑏 = 𝑎 + 𝑐 vd;f. 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 vd;w rkd;ghl;bd; %yq;fs; rkk; vdpy; m) 𝐚 = 𝐜 M) a = −c ,) a = 2c <) a = −2c

4. mzpfs;

1. gpd;tUtdtw;Ws; ve;jf; $w;W nka;ahdjy;y? m) jpirapyp mzpahdJ xU rJu mzpahFk; M) %iytpl;l mzpahdJ xU rJu mzpahFk;. ,) jpirapyp mzpahdJ xU %iytpl;l mzpahFk;. <) %iytpl;l mzpahdJ xU jpirapyp mzpahFk;.

2. 𝐴 = [𝑎𝑖𝑗 ]𝑚 × 𝑛 vd;gJ xU rJu mzp vdpy;>

m) 𝑚 < 𝑛 M) 𝑚 > 𝑛 ,) 𝑚 = 1 <) 𝒎 = 𝒏

3. 3𝑥 + 7 5𝑦 + 1 2 − 3𝑥

= 1 𝑦 − 28 8

vdpy;> 𝑥 kw;Wk; 𝑦 Mfpatw;wpd; kjpg;Gfs; KiwNa

m) −2, 7 M) −1

3, 7 ,) −

1

3, −

2

3 <) 2, -7



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4. 𝐴 = (1 − 2 3) kw;Wk; 𝐵 = −1 2−3

vdpy;> 𝐴 + 𝐵 =

m) (0 0 0) M) 000 ,) (-14) <) tiuaWf;fg;gltpy;iy

5. xU mzpapd; thpir 2 × 3 vdpy; mt;tzpapy; cs;s cWg;Gfspd; vz;zpf;if m) 5 M) 6 ,) 2 <) 3

6. 8 4𝑥 8

= 4 2 11 2

vdpy;> 𝑥 d; kjpg;G

m) 1 M) 2 ,) 1

4 <) 4

7. 𝐴 –d; thpir 3 × 4 kw;Wk; B d; thpir 4 × 3 vdpy;> BA –d; thpir

m) 3 × 3 M) 𝟒 × 𝟒 ,) 4 × 3 <) tiuaWf;fg;gltpy;iy

8. 𝐴 × 1 10 2

= (1 2) vdpy;> 𝐴 d; thpir

m) 2 × 1 M) 2 × 2 ,) 𝟏 × 𝟐 <) 3 × 2

9. 𝐴 kw;Wk; 𝐵 vd;gd rJu mzpfs;. NkYk; 𝐴𝐵 = 𝐼 kw;Wk; 𝐵𝐴 = 𝐼 vdpy;> 𝐵 vd;gJ

m) myF mzp M) G+r;rpa mzp ,) 𝑨-d; ngUf;fy; Neh;khW mzp <) – 𝐴

10. 1 22 1

𝑥𝑦 =

24 vdpy;> 𝑥 kw;Wk; 𝑦 fspd; kjpg;Gfs; KiwNa

m) 2, 0 M) 0, 2 ,) 0, -2 <) 1, 1

11. 𝐴 = 1 −2−3 4

kw;Wk; 𝐴 + 𝐵 = 𝑂 vdpy;> 𝐵 =

m) 1 −2−3 4

M) −𝟏 𝟐 𝟑 −𝟒

,) −1 −2−3 −4

<) 1 00 1

12. 𝐴 = 4 −26 −3

vdpy;> 𝐴2

m) 16 436 9

M) 8 −4

12 −6 ,)

−4 2−6 3

<) 𝟒 −𝟐𝟔 −𝟑

13. A –d; thpir m × n kw;Wk; B –d; thpir p × q vd;f. NkYk;> A kw;Wk; B Mfpadtw;wpd; $Ljy; fhz ,aYnkdpy;>

m) 𝑚 = 𝑝 M) 𝑛 = 𝑞 ,) 𝑛 = 𝑝 <) 𝒎 = 𝒑, 𝒏 = 𝒒

14. 𝑎 31 2

2

−1 =

50 vdpy;> 𝑎 d; kjpg;G

m) 8 M) 4 ,) 2 <) 11

15. 𝐴 = 𝛼 𝛽𝛾 −𝛼

kw;Wk; 𝐴2 = 𝐼, vdpy;>

m) 1 + 𝛼2 + 𝛽𝛾 = 0 M) 1 − 𝛼2 + 𝛽𝛾 = 0 ,) 𝟏 − 𝜶𝟐 − 𝜷𝜸 = 𝟎 <) 1 + 𝛼2 − 𝛽𝛾 = 0

16. 𝐴 = [𝑎𝑖𝑗 ]2× 2 kw;Wk; 𝑎𝑖𝑗 = 𝑖 + 𝑗 vdpy;> 𝐴 =

m) 1 23 4

M) 𝟐 𝟑𝟑 𝟒

,) 2 34 5

<) 4 56 7

17. −1 00 1

𝑎 𝑏𝑐 𝑑

= 1 00 −1

vdpy;> 𝑎, 𝑏, 𝑐 kw;Wk; 𝑑 Mfpatw;wpd; kjpg;Gfs; KiwNa

m) -1, 0, 0, -1 M) 1, 0, 0, 1 ,) -1, 0, 1, 0 <) 1, 0, 0, 0



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18. 𝐴 = 7 21 3

kw;Wk; 𝐴 + 𝐵 = −1 02 −4

vdpy;> mzp 𝐵 =

m) 1 00 1

M) 6 23 −1

,) −𝟖 −𝟐 𝟏 −𝟕

<) 8 2−1 7

19. (5 𝑥 1) 2−1 3

= (20) vdpy;> 𝑥 –d; kjpg;G

m) 7 M) -7 ,) 1

7 <) 0

20. 𝐴 kw;Wk; 𝐵 vd;gd xNu thpirAila rJu mzpfs; vdpy;> fPo;f;fz;litfspy; vJ nka;ahFk;?

m) (𝐴𝐵)𝑇 = 𝐴𝑇 𝐵𝑇 M) (𝐴𝑇𝐵)𝑇 𝐴𝑇 𝐵𝑇 ,) (𝐴𝐵)𝑇 = 𝐵𝐴 <) (𝑨𝑩)𝑻 = 𝑩𝑻 𝑨𝑻

5. Maj;njhiy tbtpay ;

1. (𝑎, −𝑏) (3𝑎, 5𝑏) Mfpa Gs;spfis ,izf;Fk; Neh;f;Nfhl;Lj; Jz;bd; eLg;Gs;sp m) (−𝑎, 2𝑏) M) (2𝑎, 4𝑏) ,) (𝟐𝒂, 𝟐𝒃) <) (−𝑎, −3𝑏)

2. A (1, -3), B (-3, 9) Mfpa Gs;spfis ,izf;Fk; Neh;f;Nfhl;L Jz;il 1:3 vd;w tpfpjj;jpy; gphpf;Fk;

Gs;sp P

m) (2, 1) M) (0, 0) ,) (5

3, 2) <) (1, -2)

3. A (3, 4), B (14, -3) Mfpatw;iw ,izf;Fk; Neh;f;Nfhl;Lj;Jz;L 𝑥 mr;ir 𝑃 ,y; re;jpf;fpd;wJ vdpy;> mf;Nfhl;Lj;Jz;il 𝑃 gphpf;Fk; tpfpjk; m) 4:3 M) 3:4 ,) 2:3 <) 4:1

4. (-2, -5), (-2, 12), (10, -1) Mfpa Gs;spfis Kidfshff; nfhz;l Kf;Nfhzj;jpd; eLf;Nfhl;L ikak; m) (6, 6) M) (4, 4) ,) (3, 3) <) (2, 2)

5. (1, 2), (4, 6), (x, 6), (3, 2) vd;gd ,t;thpirapy; Xh; ,izfuj;jpd; Kidfs; vdpy;> 𝑥-d; kjpg;G

m) 6 M) 2 ,) 1 <) 3

6. (0, 0), (2, 0), (0, 2) Mfpa Gs;spfshy; mikAk; Kf;Nfhzj;jpd; gug;G m) 1 r. myFfs; M) 2 r. myFfs; ,) 4 r. myFfs; <) 8 r. myFfs;

7. (1, 1), (0, 1), (0, 0), (1, 0) Mfpa Gs;spfshy; mikAk; ehw;fuj;jpd; gug;G m) 3 r. myFfs; M) 2 r. myFfs; ,) 4 r. myFfs; <) 1 r. myF

8. 𝑥-mr;Rf;F ,izahd Neh;f;Nfhl;bd; rha;Tf;Nfhzk;

m) 𝟎𝟎 M) 60 0 ,) 450 <) 900

9. (3, -2),(-1, 𝑎) Mfpa Gs;spfis ,izf;Fk; Neh;f;Nfhl;bd; rha;T −3

2, vdpy;> 𝑎-d;kjpg;G

m) 1 M) 2 ,) 3 <) 4

10. (-2, 6), (4, 8) Mfpa Gs;spfis ,izf;Fk; Neh;f;Nfhl;bw;Fr; nrq;Fj;jhd Neh;f;Nfhl;bd; rha;T

m) 1

3 M) 3 ,) −3 <) −

1

3

11. 9𝑥 − 𝑦 − 2 = 0, 2𝑥 + 𝑦 − 9 = 0 Mfpa Neh;f;NfhLfs; re;jpf;Fk; Gs;sp

m) (-1, 7) M) (7, 1) ,) (1, 7) <) (-1, -7)



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12. 4𝑥 + 3𝑦 − 12 = 0 vd;w Neh;f;NfhL 𝑦-mr;ir ntl;Lk; Gs;sp m) (3, 0) M) (0, 4) ,) (3, 4) <) (0, -4)

13. 7𝑦 − 2𝑥 = 11 vd;w Neh;f;Nfhl;bd; rha;T

m) − 7

2 M)

7

2 ,)

𝟐

𝟕 <) −

2

7

14. (2, −7) vd;w Gs;sp topr; nry;tJk; 𝑥-mr;rpw;F ,izahdJkhd Neh;f;Nfhl;bd; rkd;ghL

m) 𝑥 = 2 M) 𝑥 = −7 ,) 𝒚 = −𝟕 <) 𝑦 = 2

15. 2𝑥 − 3𝑦 + 6 = 0, vd;w Neh;f;Nfhl;bd; 𝑥, 𝑦 –ntl;Lj;Jz;Lfs; KiwNa m) 2, 3 M) 3, 2 ,) −3, 2 <) 3, −2

16. xU tl;lj;jpd; ikak; (−6, 4). xU tpl;lj;jpd; xU Kid (−12, 8) vdpy;> mjd; kW Kid m) (−18, 12) M) (−9, 6) ,) (−3, 2) <) (0, 0)

17. Mjpg;Gs;sp topr; nry;tJk; 2𝑥 + 3𝑦 − 7 = 0 vd;w Nfhl;bw;Fr; nrq;Fj;Jkhd Neh;f;Nfhl;bd; rkd;ghL m) 2𝑥 + 3𝑦 = 0 M) 𝟑𝒙 − 𝟐𝒚 = 𝟎 ,) 𝑦 + 5 = 0 <) 𝑦 − 5 = 0

18. 𝑦-mr;rpw;F ,izahdJk; (−2, 5) vd;w Gs;sp topr; nry;tJkhd Neh;f;Nfhl;bd; rkd;ghL

m) 𝑥 − 2 = 0 M) 𝒙 + 𝟐 = 𝟎 ,) 𝑦 + 5 = 0 <) 𝑦 − 5 = 0

19. (2, 5), (4, 6), (a, a) Mfpa Gs;spfs; xNu Neh;f;Nfhl;by; mikfpd;wd vdpy;> 𝑎-d;kjpg;G

m) −8 M) 4 ,) −4 <) 8

20. 𝑦 = 2𝑥 + 𝑘 vd;w Neh;f;NfhL (1, 2), vd;w Gs;sp topr; nry;fpd;wJ vdpy;> 𝑘-d; kjpg;G

m) 0 M) 4 ,) 5 <) -3

21. rha;T 3 MfTk;> 𝑦-ntl;Lj;Jz;L − 4 MfTk; cs;s Neh;f;Nfhl;bd; rkd;ghL m) 𝟑𝒙 − 𝒚 − 𝟒 = 𝟎 M) 3𝑥 + 𝑦 − 4 = 0 ,) 3𝑥 − 𝑦 + 4 = 0 <) 3𝑥 + 𝑦 + 4 = 0

22. 𝑦 = 0 kw;Wk; 𝑥 = −4 Mfpa Neh;f;NfhLfs; ntl;Lk; Gs;sp m) (0, −4) M) (−4, 0) ,) (0, 4) <) (4, 0)

23. 3𝑥 + 6𝑦 + 7 = 0 kw;Wk; 2𝑥 + 𝑘𝑦 = 5 Mfpa Neh;f;NfhLfs; nrq;Fj;jhdit vdpy; 𝑘-d; kjpg;G

m) 1 M) −1 ,) 2 <) 1

2

6. tbtpay; 1. ∆𝐴𝐵𝐶 -d; gf;fq;fs; 𝐴𝐵 kw;Wk; 𝐴𝐶 Mfpatw;iw xU Neh;f;NfhL KiwNa 𝐷 kw;Wk; 𝐸 -fspy;

ntl;LfpwJ.NkYk;> mf;NfhL 𝐵𝐶-f;F ,iz vdpy;> 𝐴𝐸

𝐴𝐶=

m) 𝐴𝐷

𝐷𝐵 M)

𝑨𝑫

𝑨𝑩 ,)

𝐷𝐸

𝐵𝐶 <)

𝐴𝐷

𝐸𝐶

2. ∆𝐴𝐵𝐶- y; 𝐴𝐵 kw;Wk;; 𝐴𝐶- fspYs;s Gs;spfs; 𝐷 kw;Wk; 𝐸 vd;gd 𝐷𝐸 ∥ 𝐵𝐶 vd;wthW cs;sd. NkYk;>

𝐴𝐷 = 3 nr.kP. 𝐷𝐵 = 2 nr.kP. kw;Wk; 𝐴𝐸 = 2.7 nr.kP vdpy;> 𝐴𝐶 =

m) 6.5 nr.kP M) 4.5 nr.kP ,) 3.5 nr.kP <) 5.5 nr.kP

3. ∆𝑃𝑄𝑅-y; 𝑅𝑆 vd;gJ ∠𝑅-d; Nfhz cl;Gw ,Urkntl;b> 𝑃𝑄 = 6nr.kP. 𝑄𝑅 = 8 nr.kP. 𝑅𝑃 = 4 nr.kP.𝑃𝑆 =

m) 2 nr.kP M) 4 nr.kP ,) 3 nr.kP <) 6 nr.kP



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4. glj;jpy; 𝐴𝐵

𝐴𝐶=

𝐵𝐷

𝐷𝐶, ∠𝐵 = 400 , kw;Wk; ∠𝐶 = 600 , vdpy;> ∠𝐵𝐴𝐷 =

m) 300 M) 500 ,) 800 <) 𝟒𝟎𝟎

5. glj;jpy; 𝑥-d; kjpg;ghdJ m) 4.2 myFfs; M) 3.2 myFfs;

,) 0.8 myFfs; <) 0.4 myFfs;

6. ∆𝐴𝐵𝐶 kw;Wk; ∆𝐷𝐸𝐹-fspy;> ∠𝐵 = ∠𝐸, kw;Wk; ∠𝐶 = ∠𝐹 vdpy;>

m) 𝐴𝐵

𝐷𝐸=

𝐶𝐴

𝐸𝐹 M)

𝐵𝐶

𝐸𝐹 =

𝐴𝐵

𝐹𝐷 ,)

𝑨𝑩

𝑫𝑬 =

𝑩𝑪

𝑬𝑭 <)

𝐶𝐴

𝐹𝐷 =

𝐴𝐵

𝐸𝐹

7. nfhLf;fg;gl;l glj;jpw;F nghUe;jhj $w;wpidf; fz;lwpf.

m) ∆𝐴𝐷𝐵~ ∆ 𝐴𝐵𝐶 M) ∆𝑨𝑩𝑫~ ∆𝑨𝑩𝑪

,) ∆𝐵𝐷𝐶~∆ 𝐴𝐵𝐶 <) ∆𝐴𝐷𝐵~ ∆𝐵𝐷𝐶

8. 12 kP ePsKs;s xU Neh;f;Fj;jhd Fr;rp> 8 kP ePsKs;s epoiyj; jiuapy; Vw;gLj;JfpwJ. mNj Neuj;jpy; xU NfhGuk; 40 kP ePsKs;s epoiyj; jiuapy; Vw;gLj;JfpwJ vdpy;> NfhGuj;jpd; cauk; m) 40 kP M) 50 kP ,) 75 kP <) 60 kP

9. ,U tbnthj;j Kf;Nfhzq;fspd; gf;fq;fspd; tpfpjk; 2:3 vdpy;> mtw;wpd; gug;gsTfspd; tpfpjk;

m) 9:4 M) 4:9 ,) 2:3 <) 3:2

10. Kf;Nfhzq;fs; 𝐴𝐵𝐶 kw;Wk; 𝐷𝐸𝐹 tbnthj;jit. mtw;wpd; gug;gsTfs; KiwNa 100 nr.kP2> 49 nr.kP2 kw;Wk; 𝐵𝐶 = 8.2 nr.kP vdpy; 𝐸𝐹 =

m) 5.47 nr.kP M) 5.74 nr.kP ,) 6.47 nr.kP <) 6.74 nr.kP

11. ,U tbnthj;j Kf;Nfhzq;fspd; Rw;wsTfs; KiwNa 24 nr.kP> 18 nr.kP. vd;f. Kjy; Kf;Nfhzj;jpd; xU gf;fk; 8 nr.kP. vdpy;> kw;nwhU Kf;Nfhzj;jpd; mjw;F xj;j gf;fk; m) 4 nr.kP M) 3 nr.kP ,) 9 nr.kP <) 6 nr.kP

12. 𝐴𝐵, 𝐶𝐷 vd;gd xU tl;lj;jpd; ,U ehz;fs;. mit ePl;lg;gLk; NghJ 𝑃 -y; re;jpf;fpd;wd kw;Wk; 𝐴𝐵 = 5 nr.kP> 𝐴𝑃 = 8 nr.kP. 𝐶𝐷 = 2 nr.kP. vdpy;> 𝑃𝐷 =


13. glj;jpy; ehz;fs; 𝐴𝐵 kw;Wk; 𝐶𝐷 vd;gd 𝑃 -y; ntl;Lfpd;wd. 𝐴𝐵 = 16 nr.kP. 𝑃𝐷 = 8 nr.kP. 𝑃𝐶 = 6 kw;Wk; 𝐴𝑃 > 𝑃𝐵 vdpy;> 𝐴𝑃 =


14. 𝑃 vd;Dk; Gs;sp> tl;likak; 𝑂-tpypUe;J 26 nr.kP. njhiytpy; cs;sJ. 𝑃 -apypUe;J tl;lj;jpw;F tiuag;gl;l 𝑃𝑇 vd;w njhLf;Nfhl;bd; ePsk; 10 nr.kP. vdpy;> 𝑂𝑇 =


15. glj;jpy; ∠𝑃𝐴𝐵 = 1200 vdpy;> ∠𝐵𝑃𝑇 =

m) 1200 M) 300 ,) 400 <) 𝟔𝟎 𝟎

16. 𝑂 -it ikakhf cila tl;lj;jpw;F 𝑃𝐴, 𝑃𝐵 vd;gd ntspg;Gs;sp 𝑃-

apypUe;J tiuag;gl;lj; njhLNfhLfs;. ,j;njhLNfhLfSf;F ,ilapy; cs;s Nfhzk; 40 0vdpy;> ∠𝑃𝑂𝐴 =

m) 𝟕𝟎𝟎 M) 80 0 ,) 500 <) 600



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17. glj;jpy;> 𝑃𝐴, 𝑃𝐵 vd;gd tl;lj;jpw;F ntspNaAs;s Gs;sp 𝑃 -apypUe;J tiuag;gl;lj; njhLNfhLfs;. NkYk; 𝐶𝐷 vd;gJ 𝑄 vd;w Gs;spapy;

tl;lj;jpw;F njhLNfhL. 𝑃𝐴 = 8 nr.kP> 𝐶𝑄 = 3 nr.kP vdpy;> 𝑃𝐶 =


18. nrq;Nfhz ∆𝐴𝐵𝐶 –y; ∠𝐵 = 900 kw;Wk; 𝐵𝐷 ⊥ 𝐴𝐶. 𝐵𝐷 = 8 nr.kP> 𝐴𝐷 =4 nr.kP vdpy; 𝐶𝐷 =


19. ,uz;L tbnthj;j Kf;Nfhzq;fspd; gug;gsTfs; KiwNa 16 nr.kP2> 36 nr.kP2. Kjy; Kf;Nfhzj;jpd; Fj;Jauk; 3 nr.kP vdpy;> kw;nwhU Kf;Nfhzj;jpy; mjid xj;j Fj;Jauk; m) 6.5 nr.kP M) 6 nr.kP ,) 4 nr.kP <) 4.5 nr.kP

20. ,U tbnthj;j Kf;Nfhzq;fs; ∆𝐴𝐵𝐶 kw;Wk; ∆𝐷𝐸𝐹 Mfpatw;wpd; Rw;wsTfs; KiwNa 36 nr.kP.

24 nr.kP NkYk;> 𝐷𝐸 = 10 nr.kP. vdpy;> 𝐴𝐵 =


7. Kf;Nfhztpay; 1. (1 − 𝑠𝑖𝑛2𝜃) 𝑠𝑒𝑐2 𝜃 =

m) 0 M) 1 ,) 𝑡𝑎𝑛2 𝜃 <) 𝑐𝑜𝑠 2𝜃

2. (1 + 𝑡𝑎𝑛2 𝜃) 𝑠𝑖𝑛2 𝜃 =

m) 𝑠𝑖𝑛 2𝜃 M) 𝑐𝑜𝑠 2𝜃 ,) 𝒕𝒂𝒏𝟐 𝜽 <) 𝑐𝑜𝑡2 𝜃

3. (1 − 𝑐𝑜𝑠2 𝜃) (1 + 𝑐𝑜𝑡2 𝜃) =

m) 𝑠𝑖𝑛2 𝜃 M) 0 ,) 1 <) 𝑡𝑎𝑛2 𝜃

4. 𝑠𝑖𝑛 (900 − 𝜃) 𝑐𝑜𝑠 𝜃 + 𝑐𝑜𝑠(900 − 𝜃) 𝑠𝑖𝑛 𝜃 =

m) 1 M) 0 ,) 2 <) -1

5. 1 −𝑠𝑖𝑛 2𝜃

1+𝑐𝑜𝑠𝜃 =

m) 𝒄𝒐𝒔 𝜽 M) 𝑡𝑎𝑛 𝜃 ,) 𝑐𝑜𝑡 𝜃 <) 𝑐𝑜𝑠𝑒𝑐 𝜃

6. 𝑐𝑜𝑠4 𝑥 − 𝑠𝑖𝑛4𝑥 =

m) 2𝑠𝑖𝑛2𝑥 − 1 M) 𝟐𝒄𝒐𝒔𝟐𝒙 − 𝟏 ,) 1 + 2𝑠𝑖𝑛2𝑥 <) 1 − 2𝑐𝑜𝑠2𝑥.

7. 𝑡𝑎𝑛 𝜃 =𝑎

𝑥 , vdpy;

𝑥

𝑎2+𝑥2 -d; kjpg;G

m) 𝒄𝒐𝒔 𝜽 M) 𝑠𝑖𝑛 𝜃 ,) 𝑐𝑜𝑠𝑒𝑐 𝜃 <) 𝑠𝑒𝑐 𝜃

8. 𝑥 = 𝑎 𝑠𝑒𝑐𝜃, 𝑦 = 𝑏 𝑡𝑎𝑛𝜃 vdpy;> 𝑥2

𝑎2 −𝑦2

𝑏2 d; kjpg;G =

m) 1 M) -1 ,) 𝑡𝑎𝑛2 𝜃 <) 𝑐𝑜𝑠𝑒𝑐2 𝜃

9. 𝑠𝑒𝑐𝜃

𝑐𝑜𝑡𝜃 +𝑡𝑎𝑛𝜃 =

m) cot 𝜃 M) tan 𝜃 ,) sin 𝜽 <) –cot 𝜃

10. sin 900−𝜃 𝑠𝑖𝑛𝜃

𝑡𝑎𝑛𝜃+

cos 900−𝜃 𝑐𝑜𝑠𝜃

𝑐𝑜𝑡𝜃=

m) 𝑡𝑎𝑛 𝜃 M) 1 ,) -1 <) 𝑠𝑖𝑛 𝜃



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11. glj;jpy; AC =

m) 25 kP M) 𝟐𝟓 𝟑 kP

,) 25

3 kP <) 25 2 kP

12. glj;jpy; ∠𝐴𝐵𝐶 =

m) 450 M) 300

,) 𝟔𝟎𝟎 <) 500

13. xU NfhGuj;jpypUe;J 28.5 kP J}uj;jpy; epd;W nfhz;bUf;Fk; xUth; NfhGuj;jpd; cr;rpia 450 Vw;wf; Nfhzj;jpy; fhz;fpwhh;. mtUila fpilepiyg; ghh;itf; NfhL jiuapypUe;J 1.5 kP cauj;jpy; cs;sJ vdpy;> NfhGuj;jpd; cauk;

m) 30 kP M) 27.5 kP ,) 28.5 kP <) 27 kP

14. glj;jpy; 𝑠𝑖𝑛 𝜃 =15

7 vdpy; BC =

m) 85 kP M) 65 kP ,) 95 kP <) 75 kP

15. (1+tan2 𝜃) (1−sin 𝜃) (1+sin 𝜃) =

m) 𝑐𝑜𝑠2𝜃 − 𝑠𝑖𝑛2𝜃 M) 𝑠𝑖𝑛2 𝜃 − 𝑐𝑜𝑠2𝜃 ,) 𝒔𝒊𝒏𝟐𝜽 + 𝒄𝒐𝒔𝟐𝜽 <) 0

16. (1+cot2 𝜃) (1−cos 𝜃) (1+cos 𝜃) =

m) 𝑡𝑎𝑛2𝜃 − 𝑠𝑒𝑐2 𝜃 M) 𝑠𝑖𝑛2𝜃 − 𝑐𝑜𝑠2𝜃 ,) 𝒔𝒆𝒄𝟐𝜽 – 𝒕𝒂𝒏𝟐𝜽 <) 𝑐𝑜𝑠2𝜃 − 𝑠𝑖𝑛2𝜃

17. (cos2 𝜃 −1) (cot2 𝜃 +1) +1 =

m) 1 M) -1 ,) 2 <) 0

18. 1+𝑡𝑎𝑛2𝜃

1+𝑐𝑜𝑡2𝜃 =

m) 𝑐𝑜𝑠2 𝜃 M) 𝒕𝒂𝒏𝟐 𝜽 ,) 𝑠𝑖𝑛2 𝜃 <) 𝑐𝑜𝑡2 𝜃

19. 𝑠𝑖𝑛2 𝜃 + 1

1+𝑡𝑎𝑛2𝜃 =

m) 𝑐𝑜𝑠𝑒𝑐2𝜃 + 𝑐𝑜𝑡2𝜃 M) 𝒄𝒐𝒔𝒆𝒄𝟐 𝜽 − 𝒄𝒐𝒕𝟐 𝜽 ,) 𝑐𝑜𝑡2 𝜃 − 𝑐𝑜𝑠𝑒𝑐2 𝜃 <) 𝑠𝑖𝑛2𝜃 − 𝑐𝑜𝑠2𝜃

20. 9𝑡𝑎𝑛2 𝜃 – 9𝑠𝑒𝑐2 𝜃 =

m) 1 M) 0 ,) 9 <) -9

8. mstpay;

1. 1 nr.kP Muk; kw;Wk; 1 nr.kP. cauk; nfhz;l xU Neh;tl;l cUisapd; tisgug;gsT m) 𝜋 nr.kP2 M) 𝟐𝝅 nr.kP2 ,) 3𝜋 nr.kP3 <) 2 nr.kP2

2. xU Neh;tl;l cUisapd; MukhdJ mjd; cauj;jpy; ghjp vdpy; mjd; nkhj;jg; Gwg;gug;G

m) 3

2 𝜋 ℎ r.m M)

2

3𝜋 ℎ2 r.m ,)

𝟑

𝟐𝝅𝒉𝟐 r.m <)

2

3𝜋ℎ r.m

3. xU Neh;tl;l cUisapd; mbg;gf;fgug;G 80 r.nr.kP mjd; cauk; 5 nr.kP. vdpy; fd msT

m) 400 nr.kP3 M) 16 nr.kP3 ,) 200 nr.kP3 <) 400

3 nr.kP3

4. xU Neh;tl;l cUisapd; nkhj;jg; gug;G 200 𝜋 r.nr.kP. kw;Wk; mjd; Muk; 5 nr.kP. vdpy; mjd; cauk; kw;Wk; Muj;jpd; $Ljy;




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5. 𝑎 myFfs; MuKk; 𝑏 myFfs; cauKk; nfhz;l xU Neh;tl;l cUisapd; tisgug;G m) 𝜋𝑎2𝑏 r.nr.kP M) 𝟐𝝅𝒂𝒃 r. nr.kP ,) 2𝜋 r.nr.kP <) 2 r.nr.kP

6. xU Neh;tl;lf; $k;G kw;Wk; Neh;tl;l cUisapd; MuKk; cauKk; KiwNa rkk;. cUisapd; fd msT 120 nr.kP3 vdpy $k;gpd; fd msT

m) 1200 nr.kP3 M) 360 nr.kP3 ,) 40 nr.kP3 <) 90 nr.kP3

7. Neh;tl;lf; $k;gpd; tpl;lk; kw;Wk; cauk; KiwNa 12 nr.kP. kw;Wk; 8 nr.kP vdpy; mjd; rhAauk;


8. xU Neh;tl;lf; $k;gpd; mbr;Rw;wsT kw;Wk; rhAauk; KiwNa 120𝜋 nr.kP> 10 nr.kP vdpy; mjd; tisgug;G

m) 1200𝜋 nr.kP 2 M) 𝟔𝟎𝟎𝝅 nr.kP 2 ,) 300𝜋 nr.kP 2 <) 600 nr.kP 2

9. xU Neh;tl;lf; $k;gpd; fd msT kw;Wk; mbg;gf;fg; gug;G KiwNa 48 𝜋 nr.kP3 kw;Wk; 12𝜋 nr.kP2 vdpy; mjd; cauk;


10. 5 nr.kP cauKk;> 48 r.nr.kP. mbg;gf;fg;gug;Gk; nfhz;l xU Neh;tl;l $k;gpd; fd msT m) 240 nr.kP3 M) 120 nr.kP3 ,) 80 nr.kP3 <) 480 nr.kP3

11. ,uz;L cUisfspd; cauq;fs; KiwNa 1:2 kw;Wk; mtw;wpd; Muq;fs; KiwNa 2:1 Mfpa tpfpjq;fspypUg;gpd; fd msTfspd; tpfpjk;

m) 4:1 M) 1:4 ,) 2:1 <) 1:2

12. 2 nr.kP Muk; cs;s xU Nfhsj;jpd; tisgug;gsT

m) 8𝜋 nr.kP2 M) 16 nr.kP2 ,) 12𝜋 nr.kP2 <) 𝟏𝟔𝝅 nr.kP2

13. xU jpz;k miuf; Nfhsj;jpd; tpl;lk; 2 nr.kP vdpy; nkhj;jg; Gwg;gug;G

m) 12 nr.kP2 M) 12 𝜋 nr.kP2 ,) 4𝜋 nr.kP2 <) 𝟑𝝅 nr.kP2

14. 9

16𝜋 f.nr.kP fdmsT nfhz;l Nfhsj;jpd; Muk;

m) 4

3 nr.kP M)

𝟑

𝟒 nr.kP ,)

3

2 nr.kP <)

2

3 nr.kP

15. ,uz;L Nfhsq;fspd; tisgug;Gfspd; tpfpjk; 9:25 vdpy; fd msTfspd; tpfpjk;

m) 81:625 M) 729:15625 ,) 27:75 <) 27:125

16. 𝑎 myFfs; Muk; nfhz;l jpz;k miuf;Nfhsj;jpd; nkhj;jg; Gwg;gug;G

m) 2𝜋 𝑎2 r.m M) 𝟑𝝅 𝒂𝟐 r.m ,) 3𝜋 𝑎 r.m <) 3𝑎2 r.m

17. 100𝜋 r. nr.kP2 tisgug;G nfhz;l Nfhsj;jpd; Muk; m) 25 nr.kP M) 100 nr.kP ,) 5 nr.kP <) 10 nr.kP

18. xU Nfhsj;jpd; tisgug;G 36𝜋 r.nr.kP2 vdpy;> mjd; fd msT m) 12𝜋 nr.kP3 M) 𝟑𝟔𝝅 nr.kP 3 ,) 72𝜋 nr.kP 3 <) 108𝜋 nr.kP 3

19. 12 𝜋 nr.kP2 nkhj;jg;gug;G nfhz;l jpz;k miuf;Nfhsj;jpd; tisgug;G

m) 6𝜋 nr.kP 2 M) 24𝜋 nr.kP 2 ,) 36𝜋 nr.kP 2 <) 𝟖𝝅 nr.kP 2

20. xU Nfhsj;jpd; MukhdJ kw;nwhU Nfhsj;jpd; Muj;jpy; ghjp vdpy; mtw;wpd; fd msTfspd; tpfpjk;

m) 1:8 M) 2:1 ,) 1:2 <) 8:1



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21. xU jpz;k Nfhsj;jpd; tisgug;G 24 nr.kP2. me;j Nfhsj;ij ,uz;L miuf; Nfhsq;fshfg; gphpj;jhy; fpilf;Fk; miuf;Nfhsq;fspy; xd;wpd; nkhj;jg; Gwg;gug;G

m) 12 nr.kP 2 M) 8 nr.kP 2 ,) 16 nr.kP 2 <) 18 nr.kP 2

22. ,uz;L $k;Gfs; rk Muq;fis nfhz;Ls;sd. NkYk; mtw;wpd; rhAauq;fspd; tpfpjk; 4:3 vdpy; tisgug;Gfspd; tpfpjk;

m) 16:9 M) 2:3 ,) 4:3 <) 3:4

11. Gs;spapay;

1. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 vd;w Kjy; 10 gfh vz;fspd; tPr;R

m) 28 M) 26 ,) 29 <) 27

2. njhFg;gpYs;s tptuq;fspy; kpfr;rpwpa kjpg;G 14.1 kw;Wk; mt;tptuj;jpd; tPr;R 28.4 vdpy;> njhFg;gpd; kpfg;nghpa kjpg;G

m) 42.5 M) 43.5 ,) 42.4 <) 42.1

3. njhFg;gpYs;s tptuq;fspy; kpfg;;nghpa kjpg;G 72 kw;Wk;; kpfr;rpwpa kjpg;G 28 vdpy;> mj;njhFg;gpd; tPr;Rf;nfO

m) 44 M) 0.72 ,) 0.44 <) 0.28

4. 11 kjpg;Gfspd; ∑𝑥 =132 vdpy;> mtw;wpd; $l;Lr;ruhrhp

m) 11 M) 12 ,) 14 <) 13

5. 𝑛 cWg;Gfs; nfhz;l ve;j xU vz;fspd; njhFg;gpw;Fk; 𝛴 𝑥 − 𝑥 =

m) 𝛴 𝑥 M) 𝑥 ,) 𝑛𝑥 <) 0

6. 𝑛 cWg;Gfs; nfhz;l ve;j xU vz;fspd; njhFg;gpw;Fk; (Σx) -𝑥 =

m) 𝑛𝑥 M) (𝑛 − 2) 𝑥 ,) (𝒏 − 𝟏) 𝒙 <) 0

7. 𝑥, 𝑦, 𝑧 -d; jpl;ltpyf;fk; 𝑡 vdpy;> 𝑥 + 5, 𝑦 + 5, 𝑧 + 5 –d; jpl;l tpyf;fk;

m) 𝑡

3 M) 𝑡 + 5 ,) 𝒕 <) 𝑥 𝑦 𝑧

8. xU Gs;sp tptuj;jpd; jpl;ltpyf;fk; 1.6 vdpy;> mjd; tpyf;fth;f;fr; ruhrhp (gutw;gb) m) 0.4 M) 2.56 ,) 1.96 <) 0.04

9. xU Gs;sp tptuj;jpd; tpyf;f th;f;fr; ruhrhp 12.25 vdpy;> mjd; jpl;ltpyf;fk; m) 3.5 M) 3 ,) 2.5 <) 3.25

10. Kjy; 11 ,ay; vz;fspd; tpyf;f th;f;fr; ruhrhp

m) 5 M) 10 ,) 5 2 <) 10

11. 10, 10, 10, 10, 10 –d; tpyf;f th;f;fr; ruhrhp

m) 10 M) 10 ,) 5 <) 0

12. 14, 18, 22, 26, 30 –d; tpyf;f th;f;fr; ruhrhp 32 vdpy;> 28, 36, 44, 52, 60 –d; tpyf;f th;f;fr; ruhrhp

m) 64 M) 128 ,) 32 2 <) 32

13. tptuq;fspd; njhFg;G xd;wpd; jpl;ltpyf;fk; 2 2 . mjpYs;s xt;nthU kjpg;Gk; 3 My; ngUf;f fpilf;Fk; Gjpa tptuj; njhFg;gpd; jpl;ltpyf;fk;

m) 12 M) 4 2 ,) 𝟔 𝟐 <) 9 2



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14. ∑(𝑥 − 𝑥 )2 = 48, 𝑥 = 20 kw;Wk; 𝑛 = 12 vdpy;> khWghl;Lf; nfO

m) 25 M) 20 ,) 30 <) 10

15. rpy tptuq;fspd; $l;Lr; ruhrhp kw;Wk; jpl;ltpyf;fk; KiwNa 48, 12 vdpy;> khWghl;Lf;nfO

m) 42 M) 25 ,) 28 <) 48

12. epfo;jfT

1. ∅ vd;gJ xU ,ayh epfo;r;rp vdpy;> 𝑃(∅) =

m) 1 M) 1

4 ,) 0 <)

1

2

2. 𝑆 vd;gJ xU rktha;g;G Nrhjidapd; $Wntsp vdpy;>𝑃(𝑆) =

m) 0 M) 1

8 ,)

1

2 <) 1

3. A vd;w epfo;r;rpapd; epfo;jfT p vdpy;> gpd;tUtdtw;wpy; p vij epiwT nra;Ak; m) 0 < 𝑝 < 1 M) 𝟎 ≤ 𝒑 ≤ 𝟏 ,) 0 ≤ 𝑝 < 1 <) 0 < 𝑝 ≤ 1

4. A kw;Wk; B vd;gd VNjDk; ,U epfo;r;rpfs;. NkYk; S vd;gJ rktha;g;Gr; Nrhjidapd; $Wntsp

vdpy;> P (𝐴 ∩B) =

m) 𝑷(𝑩) – 𝑷(𝑨 ∩ 𝑩) M) 𝑃(𝐴 ∩ 𝐵) – 𝑃(𝐵) ,) 𝑃(𝑆) <) 𝑃[(𝐴 ∪ 𝐵)’]

5. xU khztd; fzpjj;jpy; 100 kjpg;ngz; ngWtjw;fhd epfo;jfT 4

5 . mth; 100 kjpg;ngz; ngwhky;

,Ug;gjw;fhd epfo;jfT

m) 𝟏

𝟓 M)

2

5 ,)

3

5 <)

4

5

6. A kw;Wk; B vd;w ,U epfo;r;rpfspy; P(A) = 0.25, P(B) = 0.05 kw;Wk; P(A∩B) = 0.14, vdpy;> P(A∪B) =

m) 0.61 M) 0.16 ,) 0.14 <) 0.6

7. 20 nghUl;fspy; 6 nghUl;fs; FiwghLilait. rktha;g;G Kiwapy; xU nghUs; Njh;e;njLf;Fk;NghJ mJ Fiwaw;wjhff; fpilg;gjw;fhd epfo;jfT

m) 𝟕

𝟏𝟎 M) 0 ,)

3

10 <)

2

3

8. A kw;Wk; B vd;gd xd;iwnahd;W tpyf;Fk; epfo;r;rpfs; vd;f. me;epfo;r;rpapd; $Wntsp S,

𝑃(𝐴) = 1

3 𝑃(𝐵) kw;Wk; 𝑆 = 𝐴 ∪ 𝐵 vdpy; 𝑃(𝐴) =

m) 𝟏

𝟒 M)

1

2 ,)

3

4 <)

3

8

9. A, B kw;Wk; C vd;gd xd;iwnahd;W tpyf;Fk; %d;W epfo;r;rpfs; vd;f. mtw;wpd; epfo;jfTfs;

KiwNa 1

3,

1

4 kw;Wk;

5

12. vdpy; P(A∪B ∪C) =

m) 19

12 M)

11

12 ,)

7

12 <) 1

10. P(A) = 0.25, P(B) = 0.50 kw;Wk; P(A∩B) = 0.14 vdpy; P (AAk; my;y kw;Wk; BAk; my;y)=

m) 0.39 M) 0.25 ,) 0.11 <) 0.24

11. xU igapy; 5 fUg;G> 4 nts;is kw;Wk; 3 rptg;G epwg;ge;Jfs; cs;sd. rktha;g;G Kiwapy; Njh;e;njLf;fg;gLk; xU ge;J rptg;G epwkhf ,y;yhkypUg;gjw;fhd epfo;jfT

m) 5

12 M)

4

12 ,)

3

12 <)

𝟑

𝟒



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12. xNu Neuj;jpy; ,U gfilfs; cUl;lg;gLfpd;wd. gfilapd; ,uz;L Kfq;fspYk; xNu vz;zhf ,Uf;f epfo;jfT

m) 1

36 M)

1

3 ,)

𝟏

𝟔 <)

2

3

13. xU rPuhd gfil xU Kiw cUl;lg;gLk;NghJ fpilf;Fk; vz; gfh vz; my;yJ gF vz;zhf ,Ug;gjw;fhd epfo;jfT

m) 1 M) 0 ,) 𝟓

𝟔 <)

1

6

14. xU ehzaj;ij %d;W Kiw Rz;Lk; Nrhjidapy; 3 jiyfs; my;yJ 3 G+f;fs; fpilf;f epfo;jfT

m) 1

8 M)

𝟏

𝟒 ,)

3

8 <)

1

2

15. 52 rPl;Lfs; nfhz;l xU rPl;Lf;fl;bypUe;J xU rPl;L vLf;Fk;NghJ mJ xU V]; Mf ,y;yhkYk; kw;Wk; xU ,uhrhthf ,y;yhkypUg;gjw;fhd epfo;jfT

m) 2

13 M)

𝟏𝟏

𝟏𝟑 ,)

4

13 <)

8

13

16. xU nel;lhz;by; 53 nts;spfpoikfs; my;yJ 53 rdpf;fpoikfs; tUtjw;fhd epfo;jfT

m) 2

7 M)

1

7 ,)

4

7 <)

𝟑

𝟕

17. xU rhjhuz tUlkhdJ 53 Qhapw;Wf;fpoikfs; kw;Wk; 53 jpq;fl;fpoikfs; nfhz;bUg;gjw;fhd epfo;jfT

m) 1

7 M)

2

7 ,)

3

7 <) 0

18. 52 rPl;Lfs; xU rPl;Lf;fl;bypUe;J xU rPl;L vLf;Fk;NghJ> mJ `hh;l; murpahf ,Ug;gjw;fhd epfo;jfT

m) 𝟏

𝟓𝟐 M)

16

52 ,)

1

13 <)

1

26

19. xU cWjp epfo;r;rpapd; epfo;jfT m) 1 M) 0 ,) 100 <) 0.1

20. xU rktha;g;Gr; Nrhjidapd; KbthdJ ntw;wpahfNth my;yJ Njhy;tpahfNth ,Uf;Fk;. mr;Nrhjidapy; ntw;wp ngWtjw;fhd epfo;jfT Njhy;tpf;fhd epfo;jftpidg;Nghy; ,Uklq;F vdpy;> ntw;wp ngWtjw;fhd epfo;jfT

m) 1

3 M)

𝟐

𝟑 ,) 1 <) 0

,U kjpg;ngz; tpdhf;fs;

1. fzq;fSk; rhh;GfSk; 1. 𝑨 ⊂ 𝑩 vdpy,; ntd;glj;ijg; gad;gLj;jp 𝑨 ∪ 𝑩 = 𝑩 vdf; fhl;Lf.

𝑨 ⊂ 𝑩

A ∪B 𝑨 ∪ 𝑩 = 𝑩



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2. 𝑨 ⊂ 𝑩 vdpy; , 𝑨 ∩ 𝑩 , 𝑨\𝑩 Mfpatw;iw ntd;glj;jpy; tiuf. (Oct-2014)

𝑨 ⊂ 𝑩

𝑨 ∩ 𝑩 = 𝑨

𝑨\𝑩 = ∅

3. 𝑷 = 𝒂, 𝒃, 𝒄 , 𝑸 = 𝒈, 𝒉, 𝒙, 𝒚 , 𝑹 = {𝒂, 𝒆, 𝒇, 𝒔} vdpy,; (i) 𝑷 \𝑹 (ii) 𝑸 ∩ 𝑹 (iii) 𝑹 \ (𝑷 ∩ 𝑸) (Jun-15)

(i) 𝑃 \𝑅 = {a, b, c}\{a, e, f, s}={b, c}

(ii) 𝑄 ∩ 𝑅 = {g, h, x, y}∩{a, e, f, s} = { }

(iii) 𝑅 \ (𝑃 ∩ 𝑄) ⇒

(𝑃 ∩ 𝑄) = {a, b, c} ∩ g, h, x, y = { }

𝑅 \ (𝑃 ∩ 𝑄) = 𝑎, 𝑒, 𝑓 , 𝑠 \{ }= {𝑎, 𝑒, 𝑓 , 𝑠}

4. 𝑨 = {𝟒, 𝟔, 𝟕, 𝟖, 𝟗}, 𝑩 = {𝟐, 𝟒, 𝟔}, 𝑪 = {𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔}, vdpy; , (i) 𝑨 ∪ (𝑩 ∩ 𝑪) (Apr-14) (ii) 𝑨 ∩ (𝑩 ∪ 𝑪) (Jun-12, Oct-12) (iii) 𝑨\(𝑪\𝑩) fhz;f.

(i) 𝑨 ∪ (𝑩 ∩ 𝑪) ⇒

(𝐵 ∩ 𝐶) = 𝟐, 𝟒, 𝟔 ∩ 1, 𝟐, 3, 𝟒, 5, 𝟔 = {2, 4, 6}

𝐴 ∪ (𝐵 ∩ 𝐶) = 4, 6, 7, 8, 9 ∪ 2, 4, 6 = {2, 4, 6, 7, 8, 9}

(ii) 𝑨 ∩ (𝑩 ∪ 𝑪) ⇒

(𝐵 ∪ 𝐶) = 2, 4, 6 ∪ 1, 2, 3, 4, 5, 6 = {1, 2, 3, 4, 5, 6}

𝐴 ∩ (𝐵 ∪ 𝐶) = 𝟒, 𝟔, 7, 8, 9 ∩ 1, 2, 3, 𝟒, 5, 𝟔 = {4, 6}

(iii) 𝑨\ (𝑪\𝑩) ⇒

(𝐶\𝐵) = 1, 2, 3, 4, 5, 6 \ 2, 4, 6 = {1, 3, 5}

A\ (C\B) = 4, 6, 7, 8, 9 \{1, 3, 5}= {4, 6, 7, 8, 9}

5. A kw;Wk; B vd;gd C-d; cl;fzq;fs;. NkYk; mit ntl;lhf; fzq;fs; vdpy; ntd;glk; tiuf.

6. 𝑨 ∩ (𝑩\𝑪) ntd;glk; tiuf.

(B\C)

A ∩(B\C)



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7. (𝑩 ∪ 𝑪) \𝑨 ntd;glk; tiuf.

(B∪C)

(B∪C) \A

8. 𝑨 ∪ (𝑩 ∩ 𝑪) ntd;glk; tiuf. (Apr-2015)

(B∩C)

A∪(B∩C)

9. 𝑪 ∩ (𝑩\𝑨) ntd;glk; tiuf.

(B\A)

C ∩(B\A)

10. 𝑪 ∩ (𝑩 ∪ 𝑨) ntd;glk; tiuf.

(B∪A)

C ∩(B∪A)

11. 𝑨 = 𝟏, 𝟐, 𝟑, 𝟒 , 𝑩 = −𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟗, 𝟏𝟎, 𝟏𝟏, 𝟏𝟐 vd;f. 𝑹 = 𝟏, 𝟑 , 𝟐, 𝟔 , 𝟑, 𝟏𝟎 , 𝟒, 𝟗

⊆ 𝑨 × 𝑩 xU cwT vdpy; xU rhh;G vdf; fhl;Lf. mjd; kjpg;gfk;> Jizkjpg;gfk;> tPr;rfk; fhz;f. (Jun-2012, Oct-2013)

A apy; cs;s xt;nthU cWg;gpw;Fk; xU epoy; cU B apy; cs;sJ. / ,J xU rhh;G MFk;. kjpg;gfk; = {1, 2, 3, 4}

Jizkjpg;gfk; = {−1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12}

tPr;rfk; = {3, 6, 9, 10}



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12. 𝒇 = {(𝟏𝟐, 𝟐), (𝟏𝟑, 𝟑), (𝟏𝟓, 𝟑), (𝟏𝟒, 𝟐), (𝟏𝟕, 𝟏𝟕)} vd;w rhh;gpy; 2 kw;Wk; 3-d; Kd; cUf;fisf; fhz;f. 2 d; Kd; cUf;fs; = 12, 14

3 d; Kd; cUf;fs; = 13, 15

13. 𝑹 = {𝒂, −𝟐), (−𝟓, 𝒃), (𝟖, 𝒄), (𝒅, −𝟏)} vd;gJ rkdpr; rhh;igf; Fwpf;fpwJ vdpy; , a, b, c , d kjpg;Gfis

fhz;f. (Apr-2013)

𝑎 = −2, 𝑏 = −5, 𝑐 = 8, 𝑑 = −1

14. 𝑨 = {−𝟐, −𝟏, 𝟏, 𝟐} kw;Wk; 𝒇 = { 𝒙,𝟏

𝒙 : 𝒙 ∈ 𝑨} vdpy,; f-d; tPr;rfk; fhz;f. NkYk; f vd;gJ

𝑨 - apypUe;J 𝑨 -f;F xU rhh;ghFkh?

tPr;rfk; = {−1

2, −1, 1,

1

2 }, 𝑓 = −2,

1

−2 , −1,

1

−1 , 1,

1

1 , 2,

1

2

tPr;rfj;jpy; cs;s −1

2 ,

1

2 Mfpa cWg;Gfs; Ay; ,y;yhjjhy; A apypUe;J Af;F rhh;G MfhJ.

15. 𝑨 = {𝟏, 𝟐, 𝟑, 𝟒, 𝟓}, 𝑩 = ℕ kw;Wk; f : A → B MdJ f(x) = x2 vd tiuaWf;fg;gl;Ls;sJ. f d; tPr;rfk;

fhz;f. NkYk; rhh;gpd; tifia fhz;f. (Apr-2015, Jun-2015)

f (x) = x2 f (1) = 12 = 1 f (2) = 22 = 2 f (3) = 32 = 9 f (4) = 42 = 16 f ( 5) = 52 = 25 (i) f - d; tPr;rfk; = {1, 4, 9, 16, 25} (ii) ,J xd;Wf;F xd;whd rhh;G

16. rhh;G 𝒇 = { −𝟏, 𝟐 , −𝟑, 𝟏 , −𝟓, 𝟔 , −𝟒, 𝟑 } I (i) ml;ltiz (ii) mk;Gf;Fwpg;glk; Mfpatw;wpd; %yk; Fwpf;fTk;.

(i) ml;ltiz x −1 −3 −5 −4

f(x) 2 1 6 3

(ii) mk;Gf;Fwpg;glk;

17. 𝒙 = 𝒙, 𝒙 ≥ 𝟎 vDk; NghJ> (Jun-2013)

−𝒙, 𝒙 < 𝟎 vDk; NghJ {(𝒙, 𝒚)| 𝒚 = 𝒙 , 𝒙 ∈ ℝ } vd;w cwT> rhh;ig tiuaWf;fpwjh? mjd; tPr;rfk; fhz;f.

(i) x- d; xt;nthU kjpg;gpw;Fk; 𝑦 = 𝑥 vd;w xU jdpj;j (xNunahU) kjpg;G cs;sJ.

vdNt ,J rhh;ghFk;. (ii) ,jd; tPr;rfk; G+r;rpak; kw;Wk; kpif nka;naz;fs; MFk;.

18. 𝑭 = {(𝟏, 𝟑), (𝟐, 𝟓) (𝟒, 𝟕) (𝟓, 𝟗), (𝟑, 𝟏)} vDk; rhh;gpw;F kjpg;gfk;> tPr;rfk; fhz;f.

(i) kjpg;gfk; = {1, 2, 3, 4, 5}

(ii) tPr;rfk; = {3, 5, 7, 9, 1}



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19. 𝑨 = {𝟏𝟎, 𝟏𝟓, 𝟐𝟎, 𝟐𝟓, 𝟑𝟎, 𝟑𝟓, 𝟒𝟎, 𝟒𝟓, 𝟓𝟎}, 𝑩 = {𝟏, 𝟓, 𝟏𝟎, 𝟏𝟓, 𝟐𝟎, 𝟑𝟎} kw;Wk; 𝑪 = {𝟕, 𝟖, 𝟏𝟓, 𝟐𝟎, 𝟑𝟓, 𝟒𝟓, 𝟒𝟖} vdpy; , A\ (B∩C) fhz;f. (Apr- 2012)

(𝐵 ∩ 𝐶) = {1, 5, 10, 𝟏𝟓, 𝟐𝟎, 30} ∩ {7, 8, 𝟏𝟓, 𝟐𝟎, 35, 45, 48} = {15, 20}

𝐴\ 𝐵 ∩ 𝐶 = 10, 15, 20, 25, 30, 35, 40, 45, 50 \ 15, 20

= {10, 25, 30, 35, 40, 45, 50}

20. 𝑨 = 𝟓, 𝟏𝟎, 𝟏𝟓, 𝟐𝟎 , 𝑩 = {𝟔, 𝟏𝟎, 𝟏𝟐, 𝟏𝟖, 𝟐𝟒} kw;Wk; 𝑪 = {𝟕, 𝟏𝟎, 𝟏𝟐, 𝟏𝟒, 𝟐𝟏, 𝟐𝟖} Mfpa fzq;fSf;F

A\(B\C) = (A\B) \C vd;gJ nka;ahFkh vd Muha;f. (Jun-2013)

LHS 𝐴\(𝐵\𝐶) ⇒

(𝐵\𝐶) = 6, 10, 12, 18, 24 \ 7, 10, 12, 14, 21,28 = {6, 18, 24}

𝐴\(𝐵\𝐶) = 5, 10, 15, 20 \ 6, 18, 24 = {5, 10, 15, 20 …….. (1)

RHS (𝐴\𝐵) \𝐶 ⇒

(𝐴\𝐵) = 5, 10, 15, 20 \ 6, 10, 12, 18, 24 = {5, 15, 20}

(𝐴\𝐵) \𝐶 = 5, 15, 20 \ 7, 10, 12, 14, 21,28 = {5, 15, 20} ……………. (2)

LHS ≠ RHS (1) ≠ (2)

vdNt A\(B\C) ≠ (A\B)\C vd;gJ nka;ay;y.

21. 𝑿 = {𝟏, 𝟐, 𝟑, 𝟒, 𝟓}, 𝒀 = {𝟏, 𝟑, 𝟓, 𝟕, 𝟗} vd;f. X - ypUe;J Y- f;fhd cwT {(1, 1), (1, 3) (3, 5) (3, 7), (5, 7)}

vd tiuaWg;gpd; mJ rhh;G MFkh vd Muha;f. ,y;iy vdpy;> fhuzk; $W. (Apr-2012) (i) rhh;G my;y (ii) Vnddpy; X-y; cs;s 1,3 Mfpa cWg;Gfs; Y-y; cs;s ,uz;L cWg;GfNshL

njhlh;GgLj;jg;gl;Ls;sd. (1, 1), (1, 3) kw;Wk; (3, 5), (3, 7)

22. 𝑨 = {𝒍, 𝒎, 𝒏, 𝒐, 𝟐, 𝟑, 𝟒, 𝟕} kw;Wk; 𝑩 = {𝟐, 𝟓, 𝟑, −𝟐, 𝒎, 𝒏, 𝒐, 𝒑} Mfpatw;wpw;F fzq;fspd; ntl;L>

ghpkhw;Wg; gz;G cilaJ vd;gij rhpghh;f;fTk;. (Apr-2013)

𝐴 ∩ 𝐵 = {𝑙, 𝑚, 𝑛, 𝑜, 2, 3, 4, 7} ∩ {2, 5, 3, −2, 𝑚, 𝑛, 𝑜, 𝑝} = {𝑚, 𝑛, 𝑜, 2, 3} ……… 1 𝐵 ∩ 𝐴 = {2, 5, 3, −2, 𝑚, 𝑛, 𝑜, 𝑝} ∩ {𝑙, 𝑚, 𝑛, 𝑜, 2, 3, 4, 7} = {𝑚, 𝑛, 𝑜, 2, 3} ……… 2

(1) & (2), Mfpatw;wpypUe;J A∩B = B∩A vd;w fzq;fspd; ntl;L> ghpkhw;Wg; gz;G cilaJ vd rhpghh;;f;fg;gl;lJ.

23. 𝑼 = {𝟒, 𝟖, 𝟏𝟐, 𝟏𝟔, 𝟐𝟎, 𝟐𝟒, 𝟐𝟖} , 𝑨 = {𝟖, 𝟏𝟔, 𝟐𝟒} kw;Wk; 𝑩 = {𝟒, 𝟏𝟔, 𝟐𝟎, 𝟐𝟖} vdpy; (𝑨 ∪ 𝑩)’ kw;Wk; (𝑨 ∩ 𝑩)’ Mfpatw;iwf; fhz;f. (Jun-2014) 𝐴 ∪ 𝐵 = 8, 16, 24 ∪ 4, 16, 20, 28 = {4, 8, 16, 20, 24, 28} 𝐴 ∪ 𝐵 ’ = 𝑈\ 𝐴 ∪ 𝐵 = {4, 8, 12, 16, 20, 24, 28}\{4, 8, 16, 20, 24, 28} = {12} 𝐴 ∩ 𝐵 = 8, 𝟏𝟔, 24 ∩ 4, 𝟏𝟔, 20, 28 = {16} 𝐴 ∩ 𝐵 ’ = 𝑈\ 𝐴 ∩ 𝐵 = 4, 8, 12, 16, 20, 24, 28 \ 16 = {4, 8, 12, 20, 24, 28}

24. 𝑨 = {𝟏, 𝟒, 𝟗, 𝟏𝟔} ypUe;J 𝑩 = {−𝟏, 𝟐, −𝟑, −𝟒, 𝟓, 𝟔}f;F

𝒇 = { 𝟏, −𝟏 , 𝟒, 𝟐 , 𝟗, −𝟑 , (𝟏𝟔, −𝟒)} vd;gJ xU rhh;ghFkh? 𝒇 vd;gJ

rhh;G vdpy; mjd; tPr;rfj;ijf; fhz;f.(Oct-12) (i) A apy; cs;s vy;yh cWg;gpw;Fk; xNu xU epoy; cU Bapy; cs;sJ.

/ ,J rhh;G MFk;. (ii) tPr;rfk; = {−1, 2, −3, −4}

25. 𝑨 = {𝟓, 𝟔, 𝟕, 𝟖}, 𝑩 = {−𝟏𝟏, 𝟒, 𝟕, −𝟏𝟎, −𝟕, −𝟗, −𝟏𝟏, −𝟏𝟑} vd;f. 𝒇 = {(𝒙, 𝒚): 𝒚 = 𝟑 − 𝟐𝒙, 𝒙 ∈ 𝑨, 𝒚 ∈ 𝑩 } vd tiuaWf;fg;gl;Ls;sJ. 𝒇-d; tPr;rfk; fhz;f. (Oct-2014) (ii) f d; cWg;Gfis vOJf (Apr-2016)

𝑦 = 3 − 2𝑥



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𝑥 = 5 vdpy;> 𝑦 = 3 − 2 5 = 3 − 10 = −7 𝑥 = 6 vdpy;> 𝑦 = 3 − 2(6) = 3 − 12 = −9 𝑥 = 7 vdpy;> 𝑦 = 3 − 2 (7) = 3 − 14 = −11 𝑥 = 8 vdpy;> 𝑦 = 3 − 2 (8) = 3 − 16 = −13 (i) f -d; tPr;rfk; = {−7, − 9, −11, −13} (ii) 𝑓 = { 5, −7 , 6, −9 , 7, −11 , (8, −13)}

26. 𝑿 = {𝟏, 𝟐, 𝟑, 𝟒}vd;f. 𝒇 = {(𝟐, 𝟑), (𝟏, 𝟒), (𝟐, 𝟏), (𝟑, 𝟐), (𝟒, 𝟒)} vd;w cwT X- ypUe;J X- f;F xU rhh;ghFkh vd Muha;f. cd; tpilf;Nfw;w tpsf;fk; jUf. (Apr-2014) rhh;G my;y. Vnddpy; 𝑋-y; cs;s 2 vd;w cWg;G 𝑌-y; cs;s ,uz;L cWg;GfNshL njhlh;G gLj;jg;gl;Ls;sJ. (2, 3), (2, 1)

27. A, B vd;gd ,U fzq;fs; kw;Wk; U vd;gJ midj;Jf;fzk; vd;f. NkYk; 𝒏 𝑼 = 𝟕𝟎𝟎,

𝒏 (𝑨) = 𝟐𝟎𝟎, 𝒏 (𝑩) = 𝟑𝟎𝟎 kw;Wk; 𝒏 (𝑨 ∩ 𝑩) = 𝟏𝟎𝟎 vdpy;> 𝒏 (𝑨’ ∩ 𝑩 ’) If; fhz;f.

𝑛 (𝐴 ∪ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵) – 𝑛(𝐴 ∩ 𝐵) = 200 + 300 – 100 = 400

𝑛(𝐴’ ∩ 𝐵’) = 𝑛(𝐴 ∪ 𝐵)’ ⇒ 𝑛(𝑈) – 𝑛(𝐴 ∪ 𝐵)= 700 − 400

⇒ 𝑛(𝐴’ ∩ 𝐵’) = 300

28. 𝒏 (𝑨) = 𝟐𝟖𝟓, 𝒏 (𝑩) = 𝟏𝟗𝟓, 𝒏 (𝑼) = 𝟓𝟎𝟎 kw;Wk; 𝒏 (𝑨 ∪ 𝑩) = 𝟒𝟏𝟎 vdpy;> 𝒏 (𝑨’ ∪ 𝑩’) If; fhz;f.

𝑛(𝐴 ∩ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵) – 𝑛(𝐴 ∪ 𝐵)= 285 + 195 − 410 = 480 − 410 = 70

𝑛(𝐴’𝑈𝐵’) = 𝑛 𝐴 ∩ 𝐵 ’ ⇒ 𝑛 𝑈 – 𝑛 𝐴 ∩ 𝐵 = 500 – 70

⇒ 𝑛(𝐴’ ∪ 𝐵’) = 430

29. A, B kw;Wk; C VNjDk; %d;W fzq;fs; vd;f. NkYk;> 𝒏 𝑨 = 𝟏𝟕, 𝒏 𝑩 = 𝟏𝟕,

𝒏 𝑪 = 𝟏𝟕, 𝒏 𝑨 ∩ 𝑩 = 𝟕, 𝒏(𝑩 ∩ 𝑪) = 𝟔, 𝒏(𝑨 ∩ 𝑪) = 𝟓 kw;Wk; 𝒏 𝑨 ∩ 𝑩 ∩ 𝑪 = 𝟐 vdpy;> 𝒏(𝑨 ∪ 𝑩 ∪ 𝑪) – If; fhz;f.

𝑛 (𝐴 ∪ 𝐵 ∪ 𝐶) = 𝑛(𝐴) + 𝑛(𝐵) + 𝑛(𝐶) – 𝑛(𝐴 ∩ 𝐵) − 𝑛(𝐵 ∩ 𝐶) – 𝑛(𝐴 ∩ 𝐶) + 𝑛(𝐴 ∩ 𝐵 ∩ 𝐶)

= 17 + 17 + 17 – 7 – 6 – 5 + 2 = 53 − 18

𝑛 (𝐴 ∪ 𝐵 ∪ 𝐶) = 35

30. fPNo nfhLf;fg;gl;Ls;s ml;ltiz MdJ> 𝑨 = {𝟓, 𝟔, 𝟖, 𝟏𝟎} apypUe;J 𝑩 = {𝟏𝟗, 𝟏𝟓, 𝟗, 𝟏𝟏} f;F

𝒇(𝒙) = 𝟐𝒙 − 𝟏 vd;wthW mike;j xU rhh;G vdpy;> 𝒂 kw;Wk; 𝒃 Mfpadtw;wpd; kjpg;Gfisf; fhz;f.

x 5 6 8 10

f(x) a 11 b 19

𝑓(5) = 2(5) − 1 = 10 − 1 = 9

𝑓(8) = 2(8) − 1 = 16 − 1 = 15

/ 𝑎 = 9 , 𝑏 = 15

31. (𝑨 ∪ 𝑩)’ f;F ntd;glk; tiuf. (Oct-2013)

𝑓(𝑥) = 2 𝑥 − 1



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32. rhh;G 𝒇: (−𝟑, 𝟕) → 𝑹 fPo;f;fz;lthW tiuaWf;fg;gl;Ls;sJ. (Jun -2014)

𝑓 𝒙 = 𝟒𝒙𝟐 − 𝟏 ; −𝟑 ≤ 𝒙 < 2 𝟑𝒙 − 𝟐 ; 𝟐 ≤ 𝒙 ≤ 𝟒

𝟐𝒙 − 𝟑 ; 𝟒 < 𝑥 < 7

𝑥 = {−2, −1,0,1}

vdpy; 𝒇 𝟓 + 𝒇 𝟔 d; kjpg;igf; fhz;f.

𝑓 5 = 2 5 − 3 = 10 − 3 = 7 [4 < 𝑥 < 7 vdpy,; 𝑓 𝑥 = 2𝑥 − 3]

𝑓 6 = 2 6 − 3 = 12 − 3 = 9

𝑓 5 + 𝑓 6 = 7 + 9 = 16

33. 𝑨 = 𝒂, 𝒙, 𝒚, 𝒓, 𝒔 , 𝑩 = {𝟏, 𝟑, 𝟓, 𝟕, −𝟏𝟎} vd nfhLf;fg;gl;Ls;s fzq;fSf;F> fzq;fspd; Nrh;g;G

nrayhdJ> ghpkhw;W gz;G cilaJ vd;gij rhpghh;f;fTk;. (Apr-2016)

𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴 vd;gij rhpghh;g;Nghk;; 𝐴 ∪ 𝐵 = 𝑎, 𝑥, 𝑦, 𝑟, 𝑠 ∪ 1, 3, 5, 7, −10 = {𝑎, 𝑥, 𝑦, 𝑟, 𝑠, 1,3,5,7, −10}………… 1 𝐵 ∪ 𝐴 = 1, 3, 5, 7, −10 ∪ 𝑎, 𝑥, 𝑦, 𝑟, 𝑠 = {𝑎, 𝑥, 𝑦, 𝑟, 𝑠, 1,3,5,7, −10}…………. 2 (1) kw;Wk; (2) ypUe;J 𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴 vdNt> fzq;fspd; Nrh;g;G nrayhdJ> ghpkhw;W gz;G cilaJ

2. nka;naz;fspd; njhlh;thpirfSk; njhlh;fSk; 1. 𝟐, 𝟑 𝟐, 𝟓 𝟐, …… vd;w $l;Lj; njhlh; thpirapd; 12 MtJ cWg;igf; fhz;f. (Apr-2016)

𝑎 = 2, 𝑑 = 3 2 − 2 = 2 2, 𝑛 = 12

𝑡12 = 2 + (12 − 1) × 2 2 = 2 + 22 2 = 23 2

𝑡12 = 23 2

2. 4, 9, 14, ..…… vd;w $l;Lj; njhlh; thpirapd; 17 MtJ cWg;igf; fhz;f. (Apr-2014)

𝑎 = 4, 𝑑 = 9 – 4 = 5, 𝑛 = 17

𝑡17 = 4 + (17 − 1) × 5 = 4 + 16 × 5 = 4 + 80 = 84

𝑡17 = 84

3. 13 My; tFgLk; <hpyf;f kpif KO vz;fspd; vz;zpf;ifiaf; fhz;f. (Apr-2012)

$l;Lj; njhlh; 13, 26, 39,…..91

𝑎 = 13, 𝑑 = 13, 𝑙 = 91

𝑛 = 𝑙−𝑎

𝑑 + 1

𝑛 = 91−13

13 + 1 =

78

13 + 1 = 6 + 1 = 7

𝑛 = 7

4. 5+11+17+…….. +95 vd;w $l;Lj; njhlhpd; $Ljy; fhz;f. (Apr-2013, Jun-2014)

𝑎 = 5, 𝑑 = 11 − 5 = 6, 𝑙 = 95

𝑛 = 𝑙−𝑎

𝑑 + 1

𝑡𝑛 = 𝑎 + (𝑛 − 1)𝑑

𝑡𝑛 = 𝑎 + (𝑛 − 1)𝑑



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𝑛 = 95−5

6 + 1 =

90

6 + 1 = 15 + 1 = 16

𝑆𝑛 = 𝑛

2 𝑎 + 𝑙 =

16

2x [5 + 95]

= 8 × 100

𝑆16 = 800

5. 𝟏𝟑 + 𝟐𝟑 + 𝟑𝟑 + …….. +𝟐𝟎𝟑 $Ljy; fhz;f. (Oct-2014)

13 + 23 + 33 + …….. +𝑛3 = 𝑛(𝑛+1)

2

2

13 + 23 + 33 + …….. +203 = 20(20+1)

2

2=

20x21

2

2= [210]2 = 44100.

6. 𝟏 + 𝟐 + 𝟑 +……. +𝒏 = 𝟏𝟐𝟎 vdpy,; 𝟏𝟑 + 𝟐𝟑 + 𝟑𝟑 +……. 𝒏𝟑 kjpg;G fhz;f. ∑𝑛 = 120 khw;WKiw

∑𝑛3 = ∑𝑛 2 = 1202 = 14400 1 + 2 + 3 + ……… +𝑛 = 𝑛(𝑛+1)

2 = 120

13 + 23 + 33 +……….. + 𝑛3 = 𝑛 𝑛+1

2

2

= 1202 = 14400

7. 𝟏𝟑 + 𝟐𝟑 + 𝟑𝟑 +…… +𝒏𝟑 = 𝟑𝟔𝟏𝟎𝟎 vdpy; , 𝟏 + 𝟐 + 𝟑 +…….+𝒏 kjpg;G fhz;f. (Jun-2013)

∑𝑛3 = (∑𝑛)2 = 36100 = 1902 khw;WKiw

∑𝑛 = 190 13 + 23 + 33 + ……. +𝑛3 = 𝑛(𝑛+1)

2

2= 36100 = 1902

1 + 2 + 3 + ……. +𝑛 = 190 1 + 2 + 3 + ….. +𝑛 = 𝑛(𝑛+1)

2 = 190

8. 𝟏 + 𝟐 + 𝟑 + ……. + 𝑷 = 𝟏𝟕𝟏, vdpy,; 𝟏𝟑 + 𝟐𝟑 + 𝟑𝟑 +…….+𝑷𝟑 kjpg;G fhz;f. ∑𝑃 = 171 khw;WKiw

∑𝑃3 = (∑𝑃)2 1 + 2 + 3 + ….. + 𝑃 = 𝑃(𝑃+1)

2 = 171

= 1712 = 29241 13 + 23 + 33 + …… +𝑃3 = 𝑃(𝑃+1)

2

2= 1712 = 29241

9. 𝟏𝟑 + 𝟐𝟑 + 𝟑𝟑 + … + 𝒌𝟑 = 𝟖𝟐𝟖𝟏, vdpy; , 1+𝟐 + 𝟑 + … . . +𝒌 kjpg;G fhz;f. ∑𝑘3 = (∑𝑘)2 = 8281 = 912 khw;WKiw

∑𝑘 = 91 13 + 23 + 33 + ……. + 𝑘3 = 𝑘(𝑘+1)

2

2 = 8281 = 912

1 + 2 + 3 + ……… + 𝑘 = 91 1 + 2 + 3 +….. + 𝑘 = 𝑘(𝑘+1)

2 = 91

10. xU $l;Lj; njhlh; thpirapy;> Ie;jhk; cWg;gpd; 5 klq;Fk;> Vohk; cWg;gpd; 7 klq;Fk; rkkhf ,Ug;gpd; 12 tJ cWg;G 0 vd epUgpf;f. (Oct-2012)

𝑡𝑛 = 𝑎 + (𝑛 − 1) 𝑑

7𝑡7 = 5𝑡5

7(𝑎 + 6𝑑) = 5(𝑎 + 4𝑑)

7𝑎 + 42𝑑 = 5𝑎 + 20𝑑



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7𝑎 − 5𝑎 + 42𝑑 – 20𝑑 = 0

2𝑎 + 22𝑑 = 0

𝑎 + 11𝑑 = 0

𝑡12 = 0

11. xU $l;Lj; njhlhpy; Sn = 𝟏𝟐𝟕𝟓 kw;Wk; Kjy; cWg;G 𝒂 = 𝟑, nghJ tpj;jpahrk; 𝒅 = 𝟒 vdpy; , 𝒏-d; kjpg;gpidf; fhz;f. (Jun-2012)

𝑆𝑛 = 𝑛

2 [2𝑎 + 𝑛 − 1 𝑑]

𝑆𝑛 = 𝑛

2 2 3 + 𝑛 − 1 4 = 1275

𝑛 6 + 4𝑛 − 4 = 1275 × 2

6𝑛 + 4𝑛2 – 4𝑛 = 2550

4𝑛2 + 2𝑛 – 2550 = 0 (2My; tFf;f)

2𝑛2 + 𝑛 − 1275 = 0

(2𝑛 + 51) (𝑛 − 25) = 0

𝑛 = 25

12. %d;W vz;fspd; tpfpjk; 2:5:7 vd;f. Kjyhk; vz;> ,uz;lhk; vz;zpypUe;J 7-If; fopj;Jg; ngwg;gLk; vz; kw;Wk; %d;whk; vz; Mfpad xU $l;Lj; njhlh;thpiria Vw;gLj;jpdhy;> mt;ntz;fisf; fhz;f. nfhLf;fg;gl;l %d;W vz;fspd; tpfpjk; 2:5:7 me;j vz;fs; KiwNa 2𝑥, 5𝑥 kw;Wk; 7𝑥 vd;f. (𝑥 ≠ 0)

fzf;fpd;gb 2𝑥, 5𝑥 − 7, 7𝑥 vd;gd xU $l;Lj;njhlh;thpir MFk;.

vdNt nghJ tpj;jpahrk; 𝑑 = (5𝑥 − 7) – 2𝑥 = 7𝑥 – (5𝑥 − 7) Fwpg;G:[𝑑 = 𝑡2 − 𝑡1]

3𝑥 − 7 = 2𝑥 + 7

𝑥 = 7 + 7 = 14

vdNt me;j vz;fs; 2𝑥, 5𝑥, 7𝑥 = ( 2 × 14), 5 × 14 , 7 × 14 = 28, 70, 98

13. xU $l;Lj; njhlh;thpirapd; Kjy; cWg;G 6 kw;Wk; nghJtpj;jpahrk; 5 vdpy;> mj;njhlh;thpiriaAk; mjd; nghJ cWg;igAk; fhz;f.

$l;Lj; njhlHthpirapd; nghJ tbtk; ,q;F 𝑎 = 6, 𝑑 = 5

vdNt $l;Lj;njhlh;thpir = 6, (6 + 5), 6 + 2(5), 6 + 3(5), …….

= 6, 11, 16, 21, ……….

$l;Lj;njhlh;thpirapd; nghJtbtk;

= 6 + (𝑛 − 1)(5) = 6 + 5𝑛 − 5 = 5𝑛 + 1

𝑡𝑛 = 5𝑛 + 1

14. gpd;tUk; $l;Lj; njhlh;thpirapd; Kjy; cWg;G kw;Wk; nghJ tpj;jpahrj;ijf; fhz;f.

𝟏

𝟐,𝟓

𝟔,𝟕

𝟔,𝟑

𝟐, … …… ,

𝟏𝟕

𝟔. (Apr-2015)

Kjy; cWg;G 𝑎 =1

2

nghJ tpj;jpahrk; 𝑑 =5

6−

1

2=

5−3

6=

2

6=

1

3

𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, ……….

𝑡𝑛 = 𝑎 + (𝑛 − 1)𝑑

2𝑛2 + 𝑛 − 1275 = 0



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15. xU fbfhuk; xU kzpf;F xU Kiw> 2 kzpf;F ,U Kiw> 3 kzpf;F %d;W Kiw vd;wthW> njhlh;e;J rhpahf xt;nthU kzpf;Fk; xyp vOg;Gk; vdpy;> xU ehspy; mf;fbfhuk; vj;jid Kiw xyp vOg;Gk;? xU fbfhuk; xt;nthU kzpf;Fk; vOg;Gk; xyp xU $l;Lj; njhlh; mf;$l;Lj;njhlh; 1 + 2 + 3 + ……. +12

vdNt 𝑆12 =12

2 [1 + 12] = 6 × 13 = 78

xU ehspy; fbfhuk; vOg;Gk; xyp mstpd; vz;zpf;if = 2 × 78 = 156

16. xUth;> Kjy; khjk; &.640. 2 Mk; khjk; &.720. 3 Mk; khjk; &.800-I Nrkpf;fpwhh;. mth; jd;Dila Nrkpg;ig ,Nj njhlh;thpirapy; njhlh;e;jhy;> 25 MtJ khjk; mth; Nrkpf;Fk; njhifiaf; fhz;f.

Nrkpf;fg;gLk; njhif xU $l;Lj; njhlh; thpir MFk;. 640, 720, 800, ……….

,f;$l;Lj;njhlh; thpirapd; Kjy; cWg;G 𝑎 = 640,

nghJ tpj;jpahrk; 𝑑 = 720 − 640 = 80

25 MtJ khjk; Nrkpf;fg;gLk; njhif 𝑡25 = 640 + (25 − 1) × 80

= 640 + ( 24 × 80 ) = 640 + 1920

= 2560

25 MtJ khjk; Nrkpf;fg;gLk; njhif & 2560

17. 1, 2, 4, 8, ……… vd;w ngUf;Fj;njhlh;thpirapy; 1024 vd;gJ vj;jidahtJ cWg;ghFk;?

1, 2, 4, 8, ……… vd;w ngUf;Fj;njhlhpy;

𝑎 = 1 𝑟 = 𝑡2

𝑡1=

2

1= 2

𝑡𝑛 = 1024

𝑎𝑟𝑛−1 = 1024 1 × 2𝑛−1 = 210 2𝑛−1 = 210

𝑛 − 1 = 10

𝑛 = 10 + 1 = 11

/ 1, 2, 4, 8, ..………. vd;w ngUf;Fj; njhlh; thpirapy; 1024 vd;gJ 11 cWg;ghFk;.

18. 125, 120, 115, 110, …….. vd;w $l;Lj;njhlh; thpirapd; nghJ tpj;jpahrj;ijAk; 15 MtJ cWg;igAk; fhz;f. 125, 120, 115, 110, …… vd;w $l;Lj;njhlh; thpirapd; Kjy; cWg;G 𝑎 = 125

nghJ tpj;jpahrk; 𝑑 = 120 – 125 = − 5

15 MtJ cWg;G 𝑡15 = 125 + (15 − 1) (−5)

= 125 + 14 (−5) = 125 – 70 = 55

19. 𝟏𝟐 − 𝟐𝟐 + 𝟑𝟐 − 𝟒𝟐 +… vd;w njhlhpd; Kjy; 10 cWg;Gfspd; $l;lw;gyidf; fhz;f. (Jun-15)

12 − 22 + 32 − 42 +… Kjy; 10 cWg;Gfs; = (1 − 4) + (9 − 16) + … 5 cWg;Gfs;

𝑆𝑛 =𝑛

2[𝑎 + 𝑙]

𝑡𝑛 = 𝑎 + (𝑛 − 1)𝑑

𝑡𝑛 = 𝑎 + (𝑛 − 1) 𝑑



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= (−3) + (−7) + … 5 cWg;Gfs; = (−3) + (−7) + … 5 cWg;Gfs;

𝑎 = −3,

𝑑 = (−7) – (−3) = −7 + 3 = −4

𝑆𝑛 = 𝑛

2[2𝑎 + 𝑛 − 1 𝑑]

𝑆5 = 5

2[2 −3 + 5 − 1 −4 ]

= 5

2 −6 + −16 =

5

2 −22 = 5 × −11 = −55

20. 𝟏𝟔 − 𝟒𝟖 + 𝟏𝟒𝟒 − 𝟒𝟑𝟐 … vd;w ngUf;Fj;njhlhpy; cs;s Kjy; 25 cWg;Gfspd; $Ljiyf; fhz;f.

,q;F 𝑎 = 16, 𝑟 =−48

16 = −3 ≠ 1

vdNt> 𝑆𝑛 = 𝑎(1−𝑟𝑛 )

1− 𝑟, 𝑟 ≠ 1

𝑆25 = 16[1−(−3)25 ]

1−(−3) =

16[1+325 ]

4

= 4 (1 + 325)

𝒏 (𝒏 + 𝟑), 𝒏 ∈ ℕ kw;Wk; 𝒏 ,ul;ilg;gil vz; vDk; NghJ

21. an =

𝟐𝒏

𝒏𝟐+ 𝟏, 𝒏 ∈ ℕ kw;Wk; 𝒏 xw;iwg;gil vz; vDk; NghJ

vd tiuaWf;fg;gl;l njhlh;thpirapd; 18-tJ kw;Wk; 25tJ cWg;Gfisf; fhz;f. 𝑛 ,ul;ilg;gil vz; vDk; NghJ 𝑎𝑛 = 𝑛 (𝑛 + 3)

vdNt 18-tJ cWg;G 𝑎18 = 18 (18 + 3) = 18 × 21 = 378

𝑛 xw;iwg;gil vz; vDk;NghJ 𝑎𝑛 = 2𝑛

𝑛2+1

vdNt 25-tJ cWg;G 𝑎25 = 2 × 25

252 + 1=

50

625 + 1=

50

626=

25

313

22. njhlh; thpirapYs;s rhh;Gf;F xU vLj;Jf;fhl;L jUf. (Oct-2013)

rhh;G 𝑓: 𝑅 → 𝑅 MdJ 𝑓 𝑥 = 2𝑥 + 1, ∀𝑥 ∈ 𝑅 vd mikAkhdhy;> mJ xU njhlh;thpirahfhJ. Vnddpy;> ,r;rhh;gpd; tPr;rfj;jpd; cWg;Gfisj; njhlh; thpirapy; mikf;f ,ayhJ. NkYk;> ,r;rhh;gpd; kjpg;gfk; MdJ ,ay; vz;fspd; fzk; N MfNth my;yJ ,ay; vz;fspd; cl;fzk; {1,2,….n} MfNth mikag; ngwtpy;iy.

23. 𝒏 tJ cWg;G 𝒂𝒏 = 𝒏(𝒏−𝟐)

𝟑 vdf; nfhLf;fg;gl;l njhlh;thpirapd; Kjy; %d;W cWg;Gfisf;

fhz;f. (Oct-2015)

𝑎𝑛 = 𝑛(𝑛−2)

3

𝑎1 = 1(1−2)

3=

−1

3 , 𝑎2 =

2(2−2)

3=

0

3= 0, 𝑎3 =

3(3−2)

3=

3

3= 1

𝑎𝑛 = 𝑛 𝑛−2

3 nfhLf;fg;gl;l njhlh;thpirapd; Kjy; %d;W cWg;Gfs;

−1

3, 0, 1.

24. 50 kw;Wk; 200 ,tw;wp;w;fpilNaahd 10 My; tFgLk; midj;J KO vz;fspd; $Ljiyf; fhz;f.

50 kw;Wk; 200 ,tw;wp;fpilNaahd 10 My; tFgLk; midj;J KO vz;fspd; $Ljy; 60 + 70 + ……. + 190



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𝑎 = 60, 𝑑 = 10, 𝑙 = 190

𝑛 = 𝑙−𝑎

𝑑+ 1

= 190−60

10+ 1

=130

10+ 1

= 13 + 1

𝑛 = 14

𝑆𝑛 = 𝑛

2[𝑎 + 𝑙]

= 14

2 [60 + 190]

= 7 × 250

𝑆14 = 1750

50 kw;Wk; 200 ,tw;wp;w;fpilNaahd 10 My; tFgLk; midj;J KO vz;fspd; $Ljy;

60 + 70 + ……. +190 = 1750

3. ,aw;fzpjk;

1. jPh;f;f: 𝟑𝒙 + 𝟓𝒚 = 𝟐𝟓, 𝟕𝒙 + 𝟔𝒚 = 𝟑𝟎 (Jun-2012)

𝑥 𝑦 1

5 -25 3 5

6 -30 7 6

𝑥

−150+150=

𝑦

−175+90=

1

18−35

𝑥

0=

𝑦

−85=

1

−17

𝑥 = 0

17 = 0 kw;Wk; 𝑦 =

−85

−17 = 5

2. ePf;fy; Kiwapy; jPh;f;f. 𝟑𝒙 + 𝒚 = 𝟖, 𝟓𝒙 + 𝒚 = 𝟏𝟎

3𝑥 + 𝑦 = 8 ……….. 1

5𝑥 + 𝑦 = 10 …….... 2 − − (−)

(2) – (1) −2𝑥 = − 2

𝑥 = 1 vd (1)-y; gpujpapl> 3(1) + 𝑦 = 8 3 + 𝑦 = 8 𝑦 = 8 − 3 = 5

𝑦 = 5

3. 𝒙 = 𝟏

𝟒 kw;Wk; 𝒙 = −𝟏 vd;w G+r;rpaq;fisf; nfhz;l ,Ugb gy;YWg;Gf;Nfhitiaf; fhz;f.

(Jun-2013) ,Ugb gy;YWg;Gf;Nfhit = 𝑥2 – (𝛼 + 𝛽) 𝑥 + 𝛼𝛽

jPh;T {0,5}

𝑥 = 1

jPh;T {1, 5}



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,q;F 𝛼 = 1

4 , 𝛽 = −1

vdNt> ,Ugb gy;YWg;Gf;Nfhit = 𝑥2 − 1

4 − 1 𝑥 + (

1

4 × −1)

= 𝑥2 – 1−4

4 𝑥 −

1

4

,Ugb gy;YWg;Gf;Nfhit = 𝑥2 +3

4 𝑥 −

1

4

4. 𝟑, 𝟐 Mfpatw;iw KiwNa G+r;rpaq;fspd; $LjyhfTk; kw;Wk; mtw;wpd; ngUf;fw;gydhfTk; nfhz;l gy;YWg;Gf; Nfhitiaf; fhz;f.

,Ugb gy;YWg;Gf; Nfhitapd; G+r;rpaq;fspd; $Ljy; = 𝛼 + 𝛽 = 3

mtw;wpd; ngUf;fw;gyd; = 𝛼𝛽 = 2

vdNt> gy;YWg;Gf;Nfhit = 𝑝 𝑥 = 𝑥2 – 𝛼 + 𝛽 𝑥 + 𝛼𝛽

gy;YWg;Gf;Nfhit= 𝑥2 − 3 𝑥 + 2

5. xU ,Ugb gy;YWg;Gf;Nfhitapd; G+r;rpaq;fspd; $Ljy; -4 kw;Wk; mjd; ngUf;fw;gyd; 3 vdpy;> mf;Nfhitiaf; fhz;f.

,Ugb gy;YWg;Gf; Nfhitapd; G+r;rpaq;fspd; $Ljy; = 𝛼 + 𝛽 = −4

mtw;wpd; ngUf;fw;gyd; = 𝛼𝛽 = 3

vdNt> gy;YWg;Gf;Nfhit = 𝑝 (𝑥) = 𝑥2 – (𝛼 + 𝛽) 𝑥 + 𝛼𝛽 = 𝑥2 − (−4) 𝑥 + 3

6. kP.ngh.t. fhz;f. 𝟐𝟓𝒃𝒄𝟒 𝒅𝟑 , 𝟑𝟓𝒃𝟐 𝒄𝟑 , 𝟒𝟓𝒄𝟑𝒅.

25𝑏𝑐4𝑑3 = 5 × 5 × 𝑏𝑐4𝑑3

35𝑏2𝑐3 = 7 × 5 × 𝑏2𝑐3

45𝑐3𝑑 = 9 × 5 × 𝑐3𝑑

kP.ngh.t = 5𝑐3

7. 𝒎𝟐 − 𝟑𝒎 − 𝟏𝟖, 𝒎𝟐 + 𝟓𝒎 + 𝟔 Mfpa Nfhitfspd; kP.ngh.t. fhz;f. (Apr-2014)

𝑚2 − 3𝑚 − 18 = (𝑚 − 6) (𝑚 + 3)

𝑚2 + 5𝑚 + 6 = (𝑚 + 2) (𝑚 + 3)

kP.ngh.t = (𝑚 + 3)

8. kP.ngh.k. fhz;f. 𝟑 𝒂 − 𝟏 , 𝟐 𝒂 − 𝟏 𝟐, (𝒂𝟐 – 𝟏). (Jun-2015)

3 𝑎 − 1 = 3(𝑎 − 1)

2(𝑎 − 1)2 = 2(𝑎 − 1) (𝑎 − 1)

𝑎2 − 1 = (𝑎 − 1) (𝑎 + 1)

kP.ngh.k. = 3 × 2 (𝑎 − 1) (𝑎 − 1) (𝑎 + 1) = 6(𝑎 − 1)2 (𝑎 + 1)

kP.ngh.k. = 6(𝑎 − 1)2 (𝑎 + 1)

9. kP.ngh.k. fhz;f. 𝒙𝟐𝒚 + 𝒙𝒚𝟐, 𝒙𝟐 + 𝒙𝒚 (Oct-2013)

𝑥2𝑦 + 𝑥𝑦2 = 𝑥𝑦 (𝑥 + 𝑦)

𝑥2 + 𝑥𝑦 = 𝑥 (𝑥 + 𝑦)

kP.ngh.k. = (𝑥 + 𝑦)

gy;YWg;Gf;Nfhit = 𝑥2 + 4 𝑥 + 3



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10. RUf;Ff. 𝒙𝟐−𝟒

𝒂𝟐−𝟏 𝐱

𝒂𝟑−𝒂

𝒙𝟑+𝟐𝒙𝟐

𝑥2−4

𝑎2−1 ×

𝑎3−𝑎

𝑥3+2𝑥2 = 𝑥2−22

𝑎2−1 ×

𝑎(𝑎2−1)

𝑥2(𝑥+2)

= 𝑥+2 (𝑥−2)

𝑎2−1 ×

𝑎(𝑎2−1)

𝑥2(𝑥+2)=

𝑎(𝑥−2)

𝑥2

𝑥2−4

𝑎2−1 ×

𝑎3−𝑎

𝑥3+2𝑥2 = 𝑎(𝑥−2)

𝑥2

11. RUf;Ff. 𝟓𝒙+𝟐𝟎

𝟕𝒙+𝟐𝟖 (Apr-2015)

5𝑥+20

7𝑥+28 =

5(𝑥+4)

7 𝑥+4 ⇒

12. RUf;Ff. 𝒙𝟑−𝟐𝟕

𝒙𝟐−𝟗

𝑥3−27

𝑥2−9=

𝑥3−33

𝑥2−32 = 𝑥−3 (𝑥2+3𝑥+9)

𝑥+3 𝑥−3

⇒ 𝑥3−27

𝑥2−9=

(𝑥2+3𝑥+9)

(𝑥+3)

13. RUf;Ff. 𝒙𝟐−𝟖𝟏

𝒙𝟐−𝟒 𝐱

𝒙𝟐+𝟔𝒙+𝟖

𝒙𝟐−𝟓𝒙−𝟑𝟔

𝑥2−81

𝑥2−4 ×

𝑥2+6𝑥+8

𝑥2−5𝑥−36=

𝑥2−92

𝑥2−22 × 𝑥+2 (𝑥+4)

𝑥−4 (𝑥+9)

= 𝑥−9 𝑥+9

𝑥−2 𝑥+2 ×

𝑥+2 (𝑥+4)

𝑥+4 (𝑥−9)

𝑥2−81

𝑥2−4 ×

𝑥2+6𝑥+8

𝑥2−5𝑥−36 =

𝑥+9

𝑥−2

14. RUf;Ff. 𝑥

𝑥+1÷

𝑥2

𝑥2−1

𝑥

𝑥+1÷

𝑥2

𝑥2−1=

𝑥

𝑥+1÷

𝑥2

𝑥+1 (𝑥−1)=

𝑥

𝑥+1 ×

𝑥+1 (𝑥−1)

𝑥2 = 𝑥−1

𝑥

15. 𝒙𝟑−𝟏

𝒙𝟐+𝟐 cld; ve;j tpfpjKWf; Nfhitiaf; $l;bdhy;

𝟐𝒙𝟑−𝒙𝟐+𝟑

𝒙𝟐+𝟐 fpilf;Fk;?

Njitahd tpfpjKWf; Nfhit 𝑝(𝑥) vd;f.

𝑝 𝑥 = 2𝑥3−𝑥2+3

𝑥2+2−

𝑥3−1

𝑥2+2

= 2𝑥2−𝑥2+3−𝑥3+1

𝑥2+2

= 𝑥3−𝑥2+4

𝑥2+2

5𝑥+20

7𝑥+28=

5

7

𝑥

𝑥+1÷

𝑥2

𝑥2−1=

𝑥−1

𝑥



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16. njhFKiw tFj;jiyg; gad;gLj;jp> <T kw;Wk; kPjp fhz;f.

(𝟑𝒙𝟑 + 𝟒𝒙𝟐 – 𝟏𝟎𝒙 + 𝟔) ÷ (𝟑𝒙 − 𝟐) (Oct-2014)

2/3 3 4 -10 6

0 2 4 -4

3 6 -6 2

<T = 1

3 [3𝑥2 + 6𝑥 − 6] = 𝑥2 + 2𝑥 − 2 , kPjp = 2

17. th;f;f%yk; fhz;f. (𝒙 + 𝟏𝟏)𝟐 – 𝟒𝟒𝒙

(𝑥 + 11)2 − 44𝑥 = 𝑥2 + 22𝑥 + 121 − 44𝑥 = 𝑥2 − 22𝑥 + 121 = (𝑥 − 11)2 = (𝑥 − 11

18. th;f;f%yk; fhz;f. 𝒙𝟒 +𝟏

𝒙𝟒 + 𝟐.

𝑥4 +1

𝑥4 + 2 = 𝑥2 +1

𝑥2 2

= 𝑥2 +1

𝑥2

19. th;f;f%yk; fhz;f. 𝟏𝟐𝟏𝒙𝟖𝒚𝟔 ÷ 𝟖𝟏 𝒙𝟒 𝒚𝟖

121𝑥8𝑦6 ÷ 81 𝑥4 𝑦8 = 121𝑥8𝑦6

81𝑥4𝑦8 = 112𝑥4

92𝑦2 = 11𝑥2

9𝑦 =

11

9

𝑥2

𝑦

20. jPh;f;f: 𝟑𝒙 − 𝟖

𝒙= 𝟐 (Oct-2013)

jPh;T:

3𝑥 − 8

𝑥= 2 ⟹

3𝑥2−8

𝑥= 2

⟹ 3𝑥2 − 8 = 2𝑥

⟹ 3𝑥2 − 2𝑥 − 8 = 0

⟹ (𝑥 − 2) (3𝑥 + 4) = 0

⟹ 𝑥 = 2 my;yJ 𝑥 = − 4

3

21. ,U njhlh;e;j kpif ,ul;ilg; gil vz;fspd; ngUf;fypd; kjpg;G 24 vdpy; mt;ntz;fisf; fhz;f. (Oct-2012)

,U njhlh;e;j kpif ,ul;ilg; gil vz;fs; 𝑥 kw;Wk; 𝑥 + 2 vd;f. fzf;fpd;gb 𝑥 (𝑥 + 2) = 24

𝑥2 + 2𝑥 − 24 = 0

(𝑥 + 6) (𝑥 − 4) = 0

𝑥 = − 6 my;yJ 4

𝑥 kw;Wk; 𝑥 + 2 vd;gJ kpif vz;fs;. vdNt mt;tpU vz;fs; 4 kw;Wk; 6 MFk;.

22. jPh;f;f: 𝒙 + 𝟏

𝒙 =

𝟐𝟔

𝟓 (Apr-2013, Jun-2013)

𝑥 + 1

𝑥 =

26

5 ⟹

𝑥2+1

𝑥=

26

5

⟹ 5𝑥2 + 5 = 26𝑥

⟹ 5𝑥2 – 26𝑥 + 5 = 0

jPh;T = −4

3, 2



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⟹ (𝑥 − 5) (5𝑥 − 1) = 0

⟹ 𝑥 = 5 my;yJ 𝑥 = 1

5

23. 𝒙𝟐 − 𝟏𝟏𝒙 − 𝟏𝟎 = 𝟎 vd;w rkd;ghl;bd; %yq;fspd; jd;ikia Muha;f.

𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 tbtk;

𝑏2 − 4𝑎𝑐 = (−11)2 – 4 × 1 (−10) = 121 + 40 = 161 > 0

/ %yq;fs; nka;;> rkky;y.

24. 𝟐𝒙𝟐 + 𝟓𝒙 + 𝟓 = 𝟎 vd;w rkd;ghl;bd; %yq;fspd; jd;ikia Muha;f.

𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 tbtk; 𝑏2 − 4𝑎𝑐 = (−5)2 – 4 × 2 × 5

= 25 − 40 = −15 < 0

/ %yq;fs; fw;gid

25. 𝒙𝟐 − 𝟖𝒙 + 𝟏𝟐 = 𝟎 vd;w rkd;ghl;bd; %yq;fspd; jd;ikia Muha;f.


𝑏2 − 4𝑎𝑐 = (−8)2 – 4 × 1 × 12

= 64 − 48 = 16 > 0

/ %yq;fs; nka;> rkky;y.

26. 𝟐𝒙𝟐 − 𝟑𝒙 + 𝟒 = 𝟎 vd;w rkd;ghl;bd; %yq;fspd; jd;ikia Muha;f.


𝑏2 − 4𝑎𝑐 = (−3)2 – 4 × 2 × 4

= 9 − 32 = −23 < 0

/ %yq;fs; fw;gid

27. 𝟒𝒙𝟐 − 𝟐𝟖𝒙 + 𝟒𝟗 = 𝟎 vd;w rkd;ghl;bd; %yq;fspd; jd;ikia Muha;f.


𝑏2 − 4𝑎𝑐 = (−28)2 – 4 × 4 × 49

= 784 − 784 = 0

/ %yq;fs; nka;;> rkk;

28. (𝒙 − 𝟐𝒂) (𝒙 − 𝟐𝒃) = 𝟒𝒂𝒃 vd;w ,Ugbr; rkd;ghl;bd; %yq;fspd; jd;ikiaf; fhz;f.

(Jun-14)

(𝑥 − 2𝑎) (𝑥 − 2𝑏) = 4𝑎𝑏 ⟹ 𝑥2 –𝑥 (2𝑎 + 2𝑏) + 4𝑎𝑏 = 4𝑎𝑏 ⟹ 𝑥2 − 2𝑥 (𝑎 + 𝑏) = 0


𝑏2 − 4𝑎𝑐 = [−2(𝑎 + 𝑏)]2 – 4 × 1 × 0

= 4(𝑎 + 𝑏)2 > 0

/ %yq;fs; nka;;> rkky;;y.

jPh;T = 5,1

5

𝑎 = 1, 𝑏 = −11, 𝑐 = −10

𝑎 = 2, 𝑏 = −5, 𝑐 = 5

𝑎 = 1, 𝑏 = −8, 𝑐 = 12

𝑎 = 2, 𝑏 = −3 , 𝑐 = 4

𝑎 = 4, 𝑏 = −28, 𝑐 = 49

𝑎 = 1, 𝑏 = −2 𝑎 + 𝑏 , 𝑐 = 0



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29. nfhLf;fg;gl;Ls;s rkd;ghl;bd; %yq;fs; nka;naz;fs; kw;Wk; rkkhdit vdpy; 𝒌 d; kjpg;igf;

fhz;f. 𝟏𝟐𝒙𝟐 + 𝟒𝒌𝒙 + 𝟑 = 𝟎. (Jun-2015)

12𝑥2 + 4𝑘𝑥 + 3 = 0 vd;w rkd;ghl;bd; %yq;fs; nka; kw;Wk; rkk;.

vdNt ∆= 𝑏2 − 4𝑎𝑐 = 0 ⟹ (4𝑘)2 − 4 × 12 × 3 = 0

16𝑘2 − 144 = 0 16𝑘2 = 144

𝑘2 = 9 k = ± 3

30. 𝟕 + 𝟑, 𝟕 − 𝟑 vd;gij %yq;fshff; nfhz;l ,Ugbr; rkd;ghl;ilf; fhz;f. (Apr-14, Oct-14)

%yq;fspd; $Ljy; = 7 + 3 + 7 − 3 = 14

%yq;fspd; ngUf;fw;gyd; = (7 + 3 ) × (7 − 3 ) = 72 – ( 3)2 = 49 − 3 = 46

rkd;ghL: 𝑥2 - (%yq;fspd; $Ljy;) 𝑥 + %yq;fspd; ngUf;fw;gyd; = 0

31. 𝟑 + 𝟕 , 𝟑 − 𝟕 vd;gij %yq;fshff; nfhz;l ,Ugbr; rkd;ghl;ilf; fhz;f.

%yq;fspd; $Ljy; = 3 + 7 + 3 – 7 = 6

%yq;fspd; ngUf;fw;gyd; = (3+ 7 )× (3 – 7 ) = 32 – ( 7)2 = 9 – 7 = 2

rkd;ghL: 𝑥2- (%yq;fspd; $Ljy;) 𝑥 + %yq;fspd; ngUf;fw;gyd; = 0

𝑥2 − 6𝑥 + 2 = 0

32. 3, 4 vd;gij %yq;fshff; nfhz;l ,Ugbr;rkd;ghl;ilf; fhz;f. (Jun-2012, Oct-2012)

Njitahd rkd;ghL 𝑥2 - (%yq;fspd; $Ljy;) 𝑥 + %yq;fspd; ngUf;fw;gyd; = 0

𝑥2 − (3 + 4) 𝑥 + (3 × 4) = 0

𝑥2 − 7𝑥 + 12 = 0

33. 𝒂𝒙𝟐 − 𝟓𝒙 + 𝒄 = 𝟎 vd;w ,Ugbr; rkd;ghl;bd; %yq;fspd; $Ljy; 10kw;Wk; ngUf;fw;gyd; 10

vdpy; , 𝒂 kw;Wk; 𝒄 Mfpatw;wpd; kjpg;Gfisf; fhz;f. (Apr-2013)

𝑎𝑥2 − 5𝑥 + 𝑐 = 0 vd;w ,Ugbr; rkd;ghl;bd; %yq;fspd; $Ljy; = (−𝑏)

𝑎 =

5

𝑎 = 10

𝑎 =5

10=

1

2

𝑎𝑥2 − 5𝑥 + 𝑐 = 0 vd;w ,Ugbr; rkd;ghl;bd; %yq;fspd; ngUf;fw;gyd; = 𝐶

𝑎 = 10

𝑐 = 10𝑎 = 10 × 1

2 = 5

34. 𝜶, 𝜷 vd;gd 𝟑𝒙𝟐 − 𝟔𝒙 + 𝟒 = 𝟎, vd;Dk; rkd;ghl;bd; %yq;fs; vdpy; 𝜶𝟐 + 𝜷𝟐d; kjpg;G fhz;f.

(Apr-2012, Apr-2016)

3𝑥2 − 6𝑥 + 4 = 0 vd;Dk; rkd;ghl;bd; %yq;fs; 𝛼, 𝛽

𝛼 + 𝛽 = −𝑏

𝑎 =

6

3 = 2 𝛼2 + 𝛽2 = |(𝛼 + 𝛽)2 − 2𝛼𝛽|

𝛼𝛽 = 𝑐

𝑎=

4

3 = (2)2 − 2 ×

4

3

= 4 −8

3 =

12−8

3 =

4

3

𝑥2 – 14𝑥 + 46 = 0

𝑎 = 1

2

𝑐 = 5

𝛼2 + 𝛽2 = 4

3



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35. 𝟑𝒙𝟐 − 𝟓𝒙 + 𝟐 = 𝟎 vd;w rkd;ghl;bd; %yq;fs; 𝜶 , 𝜷 vdpy; 𝜶

𝜷+

𝜷

𝜶 -d; kjpg;G fhz;f. (Apr-15)

3𝑥2 − 5𝑥 + 2 = 0, vd;w rkd;ghl;by; 𝑎 = 3, 𝑏 = −5, 𝑐 = 2

vdNt 𝛼 + 𝛽 = (−𝑏)

𝑎=

−(−5)

3 =

5

3

𝛼𝛽 =𝑐

𝑎 =

2

3

𝛼

𝛽+

𝛽

𝛼=

𝛼2+𝛽2

𝛼𝛽=

(𝛼+𝛽)2−2𝛼𝛽

𝛼𝛽

=

5

3

2−2

2

3

2

3

=

25

9 −

4

3

2

3

= 25−12

9

3

2 =

13

6

36. jPh;. 𝟑𝒙 − 𝟓𝒚 = −𝟏𝟔, 𝟐𝒙 + 𝟓𝒚 = 𝟑𝟏 (Oct-2015)

3𝑥 − 5𝑦 = −16 …………………….. 1

2𝑥 + 5𝑦 = 31 ……………………… 2

(1) + (2) 5𝑥 = 15

𝑥 = 3

𝑥 = 3 vd (1) y; gpujpapl ⟹ 3𝑥 − 5𝑦 = −16

3 3 − 5𝑦 = −16

−5𝑦 = −16 − 9

−5𝑦 = −25

𝑦 = 5

/ jPh;T = {3, 5}

37 . RUf;Ff. 𝒙𝟑

𝒙−𝟐+

𝟖

𝟐−𝒙 (Apr-2016)

𝑥3

𝑥−2+

8

2−𝑥=

𝑥3

𝑥−2−

23

𝑥−2=

𝑥3−23

𝑥−2=

(𝑥−2)(𝑥2+2𝑥+4)

𝑥−2= 𝑥2 + 2𝑥 + 4

4. mzpfs;

1. 𝐀 = 𝟏 𝟐 𝟑𝟐 𝟒 −𝟓𝟑 −𝟓 𝟔

vdpy; (𝑨𝑻)𝑻 = 𝑨 vd;gij rhpghh;.

𝐴 = 1 2 32 4 −53 −5 6

𝐴𝑇 = 1 2 32 4 −53 −5 6

(𝐴𝑇)𝑇 = 1 2 32 4 −53 −5 6

= 𝐴 vd;gJ rhpghh;f;fg;gl;lJ.

2. 𝑨 = 𝟖 𝟓 𝟐𝟏 −𝟑 𝟒

vdpy; 𝑨𝑻 , (𝑨𝑻)𝑻 Mfpatw;iwf; fhz;f. (Jun-12, Apr-13, Jun-14, Apr-15)

𝐴 = 8 5 21 −3 4



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𝐴𝑇 = 8 15 −32 4

(𝐴𝑇)𝑇 = 8 5 21 −3 4

3. 𝑨 = 𝟐 𝟑𝟒 𝟏𝟓 𝟎

vdpy;> A apd; epiu epuy; khw;W mzpiaf; fhz;f.

A apd; epiu epuy; khw;W mzp = 𝐴𝑇 = 2 4 53 1 0

4. 𝑨 = 𝟏 −𝟏 𝟑𝟓 −𝟒 𝟕𝟔 𝟎 𝟗

𝟐 𝟒 𝟖

vdpy; (i) mzpapd; thpiriaf; fhz;f. (ii) 𝒂𝟐𝟒 kw;Wk; 𝐚𝟑𝟐 Mfpa

cWg;Gfisf; fhz;f. (iii) cWg;G 7 mike;Js;s epiu> epuiyf; fhz;f. i) mzpapd; thpir 3 × 4

ii) 𝑎24 = 4, 𝑎32 = 0

iii) cWg;G 7 mike;Js;s epiy 𝑎23

5. 𝑨 = [𝒂𝒊𝒋] =

𝟏𝟔𝟑𝟗

𝟒 𝟐 𝟕 −𝟐

𝟖 𝟓 𝟎−𝟏

vdpy; (i) mzpapd; thpiriaf; fhz;f. (ii) 𝒂𝟏𝟑 kw;Wk; 𝐚𝟒𝟐 Mfpa

cWg;Gfisf; fhz;f. (iii) 2 vd;w cWg;G mike;Js;s epiy Mfpatw;iwf; fhz;f. i) mzpapd; thpir 4 × 3

ii) 𝑎13 = 8, 𝑎42 = −2.

iii) 2 vd;w cWg;gpd; epiy 𝑎22

6. A = 𝟒 −𝟐𝟓 −𝟗

, 𝐁 = 𝟖 𝟐

−𝟏 −𝟑 vdpy;> 𝟔𝑨 − 𝟑𝑩 vd;w mzpiaf; fhz;f. (Apr-2012)

6𝐴 – 3𝐵 = 6 4 −25 −9

− 3 8 2

−1 −3 =

24 −1230 −54

+ −24 −6

3 9

= 0 −18

33 −45

7. 𝑨 = 𝟑 𝟐𝟓 𝟏

, 𝑩 = 𝟖 −𝟏𝟒 𝟑

vdpy;> 𝑪 = 𝟐𝑨 + 𝑩 vd;w mzpiaf; fhz;f. (Jun-2014)

𝐶 = 2𝐴 + 𝐵

= 2 3 25 1

+ 8 −14 3

= 6 4

10 2 +

8 −14 3

= 14 314 5

8. 𝑨 = 𝟐 𝟑

−𝟗 𝟓 −

𝟏 𝟓𝟕 −𝟏

vdpy;> A d; $l;ly; Neh;khW mzpiaf; fhz;f.

(Oct-2012, 2013, Apr- 2015, Jun-2015)

𝐴 = 2 3

−9 5 −

1 57 −1

= 2 3

−9 5 +

−1 −5−7 1

= 1 −2

−16 6

A d; $l;ly; Neh;khW mzp = −1 216 −6



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9. 𝑨 = 𝟖 𝟑 𝟐𝟓 𝟗 𝟏

, 𝐁 = 𝟏 −𝟏𝟑 𝟎

vdpy;> 𝑨 + 𝑩 ,Uf;Fkhdhy; mjidf; fhz;f.

A-d; thpir = 2 × 3

B-d; thpir = 2 × 2

A-d; thpir ≠ B-d; thpir. vdNt A+B fhz ,ayhJ.

10. 𝑨 = 𝟓 𝟔 −𝟐𝟏 𝟎 𝟒

𝟑 𝟐

𝑩 = 𝟑 −𝟏 𝟒𝟐 𝟖 𝟐

𝟕 𝟑

vdpy;> 𝑨 + 𝑩 –If; fhz;f.

𝐴 + 𝐵 = 5 6 −21 0 4

3 2

+ 3 −1 42 8 2

7 3

= 8 5 23 8 6

10 5

11. 𝒙 𝟓 𝟒𝟓 𝟗 𝟏

= 𝟑 𝟓 𝒛𝟓 𝒚 𝟏

vdpy;> 𝒙, 𝒚 kw;Wk; 𝒛 Mfpatw;wpd; kjpg;Gfisf; fhz;f.

𝑥 = 3, 𝑦 = 9, 𝑧 = 4

12. 30 cWg;Gfs; nfhz;l mzpf;F vt;tif thpirfs; ,Uf;f ,aYk;? (Jun-2013)

30 cWg;Gfs; nfhz;l mzpapd; thpir 1 × 30, 2 × 15, 3 × 10, 5 × 6, 6 × 5,

10 × 3, 15 × 2, kw;Wk; 30 × 1 vd ,Uf;f ,aYk;.

13. 8 cWg;Gfs; nfhz;l mzpf;F vt;tif thpirfs; ,Uf;f ,aYk;? 8 cWg;Gfs; nfhz;l mzpapd; thpir 1 × 8, 2 × 4, 4 × 2 kw;Wk; 8 × 1 vd ,Uf;f ,aYk;

14. 𝑨 = 𝒂𝒊𝒋 = 𝒊𝒋 vd;w 𝟐 × 𝟐 thpirf; nfhz;l mzpia mikf;f.

𝐴 = 𝑎11 𝑎12

𝑎21 𝑎22 = 𝑖𝑗

𝑎11 = 1 × 1 = 1, 𝑎12 = 1 × 2 = 2 ,

𝑎21 = 2 × 1 = 2, 𝑎22 = 2 × 2 = 4

∴ 𝐴 = 1 22 4

15. 𝑨 = [𝒂𝒊𝒋] = 𝟐𝒊 – 𝒋 vd;w 𝟐 × 𝟐 thpir nfhz;l mzpia mikf;f.

𝐴 = 𝑎11 𝑎12

𝑎21 𝑎22 = 2𝑖 – 𝑗

𝑎11 = 2 × 1 – 1 = 2 − 1 = 1, 𝑎12 = 2 × 1 – 2 = 2 − 2 = 0

𝑎21 = 2 × 2 – 1 = 4 − 1 = 3, 𝑎22 = 2 × 2 – 2 = 4 − 2 = 2

∴ 𝐴 = 1 03 2

16. 𝑨 = [𝒂𝒊𝒋] = 𝒊−𝒋

𝒊+𝒋 vd;w 𝟐 × 𝟐 thpir nfhz;l mzpia mikf;f.

𝐴 = 𝑎11 𝑎12

𝑎21 𝑎22 =

𝑖−𝑗

𝑖+𝑗

𝑎11 = 1−1

1+1 =

0

2 = 0, 𝑎12 =

1−2

1+2 =

−1

3

𝑎21 = 2−1

2+1 =

1

3 , 𝑎22 =

2−2

2+2 =

0

4 = 0

𝐴 = 0

−1

31

30



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17. (𝟐 − 𝟏) 𝟓𝟒 ngUf;Ff.

(2 − 1) 54 = 10 − 4 = 6

18. 𝑨 = [𝒂𝒊𝒋] = 𝟐𝒊 − 𝟑𝒋 vd;w cWg;Gfs; nfhz;l 𝟐 × 𝟑 thpir cs;s mzp 𝑨 = [𝒂𝒊𝒋] apidf;

fhz;f. (Apr-2014)

𝐴 = 𝑎11 𝑎12 𝑎13

𝑎21 𝑎22 𝑎23 = 2𝑖 − 3𝑗

𝑎11 = 2 1 − 3(1) = 2 − 3 = −1 = 1 𝑎12 = 2 1 − 3(2) = 2 − 6 = −4 = 4 𝑎13 = 2(1) − 3(3) = 2 − 9 = −7 = 7

𝑎21 = 2 2 − 3(1) = 4 − 3 = 1 = 1 𝑎22 = 2 2 − 3(2) = 4 − 6 = −2 = 2 𝑎23 = 2 2 − 3(3) = 4 − 9 = −5 = 5

∴ 𝐴 = 1 4 71 2 5

19. 𝒂 𝟐𝟑 + 𝒃

−𝟏𝟏

= 𝟏𝟎𝟓

vdpy;> 𝒂, 𝒃 kjpg;Gfisf; fhz;f.

𝑎 23 + 𝑏

−11

= 105

2𝑎3𝑎

+ −𝑏𝑏

= 105

2𝑎 − 𝑏3𝑎 + 𝑏

= 105

2𝑎 – 𝑏 = 10………………………………… 1

3𝑎 + 𝑏 = 5……………..…………………… 2

5𝑎 = 15

𝑎 = 15/5

𝑎 = 3

𝑎 = 3 vd (2) y; gpujpapl

(3 × 3) + 𝑏 = 5

9 + 𝑏 = 5

𝑏 = 5 – 9 = −4

𝑏 = −4

20. 𝑨 = 𝟖 −𝟕

−𝟐 𝟒𝟎 𝟑

𝑩 = 𝟗 −𝟑 𝟐𝟔 −𝟏 −𝟓

vdpy; 𝑨𝑩 kw;Wk; 𝑩𝑨 Mfpa mzpfisf; fhz;f.

𝐴𝐵 = 8 −7

−2 40 3

9 −3 26 −1 −5

= 72 − 42 −24 + 7 16 + 35

−18 + 24 6 − 4 −4 − 200 + 18 0 − 3 0 − 15

= 30 −17 516 2 −24

18 −3 −15



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𝐵𝐴 = 9 −3 26 −1 −5

8 −7

−2 40 3

= 72 + 6 + 0 −63 − 12 + 648 + 2 + 0 −42 − 4 − 15

= 78 −6950 −61

21. 𝑨 = 𝟑 𝟐𝟒 𝟎

, 𝑩 = 𝟑 𝟎𝟑 𝟐

vdpy; 𝑨𝑩, 𝑩𝑨 fhz;f. mit rkkhf ,Uf;Fkh?

𝐴𝐵 = 3 24 0

3 03 2

= 9 + 6 0 + 4

12 + 0 0 + 0 =

15 412 0

𝐵𝐴 = 3 03 2

3 24 0

= 9 + 0 6 + 09 + 8 6 + 0

= 9 6

17 6

/ 𝐴𝐵 ≠ 𝐵𝐴

22. 𝑨 = 𝟓 𝟐𝟕 𝟑

, 𝑩 = 𝟑 −𝟐

−𝟕 𝟓 vd;w mzpfs; xd;Wf;nfhd;W ngUf;fy; Neh;khW mzp vd epWTf.

𝐴, 𝐵 vd;w mzpfs; xd;Wf;nfhd;W ngUf;fy; Neh;khW mzpfs; vdpy; gpd;tUk; tpjpia epiwT nra;a Ntz;Lk; 𝐴𝐵 = 𝐵𝐴 = 𝐼

𝐴𝐵 = 5 27 3

3 −2

−7 5 =

1 00 1

= 𝐼

𝐵𝐴 = 3 −2

−7 5

5 27 3

= 1 00 1

= 𝐼

nfhLf;fg;gl;l mzpfs; xd;Wf;nfhd;W ngUf;fy; Neh;khW mzp MFk;.

23. 𝟑 𝟓𝟏 𝟐

, 𝟐 −𝟓

−𝟏 𝟑 Mfpa mzpfs; ngUf;fiyg; nghUj;J xd;Wf;nfhd;W ngUf;fy; Neh;khW

mzp vd epWTf. 𝐴, 𝐵 vd;w mzpfs; xd;Wf;nfhd;W ngUf;fy; Neh;khW mzpfs; vdpy; gpd;tUk; tpjpia epiwT

nra;a Ntz;Lk; 𝐴𝐵 = 𝐵𝐴 = 𝐼>

𝐴 = 3 51 2

, 𝐵 = 2 −5−1 3

𝐴𝐵 = 3 51 2

2 −5

−1 3 =

1 00 1

= I

𝐵𝐴 = 2 −5

−1 3

3 51 2

= 1 00 1

= I

nfhLf;fg;gl;l mzpfs; xd;Wf;nfhd;W ngUf;fy; Neh;khW mzp MFk;.

24. 𝑨 = 𝟏 𝟑𝟗 −𝟔

vdpy; 𝑨𝑰 = 𝑰𝑨 = 𝑨 , vd;gij rhpghh;f;f. ,q;F 𝑰 vd;gJ thpir 2 nfhz;l myF

mzp. (Apr-2014)

𝐴𝐼 = 1 39 −6

1 00 1

= 1 39 −6

= 𝐴

𝐼𝐴 = 1 00 1

1 39 −6

= 1 39 −6

= 𝐴

𝐴𝐼 = 𝐼𝐴 = 𝐴 vd;gJ rhpghh;f;fg;gl;lJ.



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25. 𝟑 −𝟐𝟓 𝟏

𝟒 𝟏𝟐 𝟕

vd;w mzpfspd; ngUf;fiyf; fhz;f. (Apr-2012, Jun-2012)

3 −25 1

4 12 7

= 3 × 4 + (−2 × 2) 3 × 1 + (−2 × 7)

5 × 4 + 1 × 2 5 × 1 + (1 × 7)

= 12 + (−4) 3 + (−14)

20 + 2 5 + (7) =

8 −1122 12

26. 𝟐 𝟗 −𝟑𝟒 −𝟏 𝟎

𝟒 𝟐

−𝟔 𝟕−𝟐 𝟏

ngUf;Ff. (Oct-2014,Apr-2016)

2 9 −34 −1 0

4 2

−6 7−2 1

= 8 − 54 + 6 4 + 63 − 316 + 6 + 0 8 − 7 + 0

= −40 6422 1

27. 𝟔

−𝟑 (𝟐 − 𝟕) ngUf;Ff. (Oct-2013)

6

−3 (2 − 7) =

12 −42−6 21

28. 𝟐𝒙 + 𝒚𝒙 − 𝟑𝒚

= 𝟓𝟏𝟑

vdpy; 𝒙 kw;Wk; 𝒚 Mfpatw;wpd; jPh;T fhz;f. (Oct-2014)

2𝑥 + 𝑦𝑥 − 3𝑦

= 5

13

2𝑥 + 𝑦 = 5…………………… 1 𝑥 − 3𝑦 = 13…………………. 2 (2) + (1) ⟹

(2) ⟹ 𝑥 − 3𝑦 = 13 (1) × 3 ⟹ 6𝑥 + 3𝑦 = 15

7𝑥 = 28 𝑥 = 4

𝑥 = 4 vd rkd;ghL 1y; gpujpapl

⟹ 2 × 4 + 𝑦 = 5

𝑦 = 5 − 8 = −3

29. 𝑨𝟐 × 𝟓 kw;Wk; 𝑩𝟓×𝟒 vd;w mzpfspd; ngUf;fy; tiuaWf;fg;gl;Ls;sjh? Mk; vdpy;> ngUf;fp tUk;

mzpapd; thpiria vOJf. (Jun-2015)

𝐴 -d; epuy;fspd; vz;zpf;ifAk; 𝐵 -d; epiufspd; vz;zpf;ifAk; rkkhf ,Ug;gjhy; ngUf;fw;gyd; mzp 𝐴𝐵 tiuaWf;fg;gLfpwJ. 𝐴𝐵-d; thpir 2 × 4 MFk;.

30. 𝑨𝟏 × 𝟑 kw;Wk; 𝑩𝟒 × 𝟑 vd;w mzpfspd; ngUf;fy; tiuaWf;fg;gl;Ls;sjh? Mk; vdpy;> ngUf;fp tUk; mzpapd; thpiria vOJf.

𝐴 -d; epuy;fspd; vz;zpf;ifAk; 𝐵 -d; epiufspd; vz;zpf;ifAk; rkkhf ,y;iy. vdNt> ,t;tpU mzpfspd; ngUf;fw;gyd; tiuaWf;fg;gltpy;iy.

31. A kw;Wk; B vd;w mzpfSf;F AB fpilf;fg; ngWfpwJ. Mdhy; BA fpilf;fg; ngwtpy;iy vdpy;> A kw;Wk; B-d; thpirfisg; gw;wp $Wf. (Apr-2013)

A kw;Wk; B vd;w mzpfSf;F AB fpilf;fg; ngWfpwJ.

Mdhy; BA fpilf;fg; ngwtpy;iy. vdNt> B-apd; epuy;fspd; vz;zpf;ifAk; A-apd; epiufspd; vz;zpf;ifAk; rkky;y. mjhtJ> A-d; thpir 𝑚 × 𝑛 MfTk;> Bd; thpir 𝑛 × 𝑝 MfTk; ,Uf;f Ntz;Lk;.

jPh;T = {4, −3}



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32. 𝑨 = (𝟏 𝟐 − 𝟏) , B= 𝟐𝟏𝟒 vdpy;> (𝐀𝐁)𝐓 - iaf; fhz;f. (Jun-2013)

𝐴 = (1 2 − 1) , 𝐵 = 214

𝐴𝐵 = (1 2 − 1) × 214 = (2 + 2 − 4) = (0)

(𝐴𝐵)𝑇 = (0)

33. jPh;T fhz;f. 𝟑 𝟐𝟒 𝟓

𝒙𝒚 =

𝟖𝟏𝟑

(Oct-2015)

3 24 5

𝑥𝑦 =

813

3𝑥 + 2𝑦4𝑥 + 5𝑦

= 8

13

3𝑥 + 2𝑦 = 8 ⟹ 3𝑥 + 2𝑦 − 8 = 0

4𝑥 + 5𝑦 = 13 ⟹ 4𝑥 + 5𝑦 − 13 = 0

FWf;Fg; ngUf;fy; Kiw 𝑥 𝑦 1

2 -8 3 2

5 -13 4 5

⟹ 𝑥

−26+40=

𝑦

−32+39=

1

15−8

⟹ 𝑥

14=

𝑦

7=

1

7

⟹ 𝑥

14=

1

7 &

𝑦

7=

1

7

⟹ 𝑥 = 14

7 & 𝑦 =

7

7

⟹ 𝑥 = 2 & 𝑦 = 1

34. 𝒙 + 𝒚𝒚 + 𝒛𝒛 − 𝟓

= 𝟕𝟗𝟎 vdpy; 𝒙, 𝒚 kw;Wk; 𝒛 d; kjpg;Gfis fhz;f. (Apr-2016)

𝑥 + 𝑦𝑦 + 𝑧𝑧 − 5

= 790 𝑧 = 5 vd (2)y; gpujpapl

𝑥 + 𝑦 = 7…………… 1 𝑦 + 5 = 9 ⇒ 𝑦 = 9 − 5 = 4

𝑦 + 𝑧 = 9…………… 2 𝑦 = 4 vd (1) y; gpujpapl

𝑧 − 5 = 0…………….. 3 𝑥 + 4 = 7 ⇒ 𝑥 = 7 − 4 = 3

𝑥 = 3, 𝑦 = 4, 𝑧 = 5



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5. Maj;njhiy tbtpay ;

1. (3, 0), (-1, 4) Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj;Jz;bd; eLg;Gs;spiaf; fhz;f. 𝑥1 , 𝑦1 𝑥2 , 𝑦2

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

eLg;Gs;sp = 𝑥1+ 𝑥2

2,𝑦1+ 𝑦2

2

= 3−1

2,

0+4

2 =

2

2,

4

2 = (1,2)

2. (1, -1), (-5, 3) Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj;Jz;bd; eLg;Gs;spiaf; fhz;f. 𝑥1 , 𝑦1 𝑥2 , 𝑦2

(1, -1), (-5, 3)


2,𝑦1+ 𝑦2

2

= 1−5

2,−1+3

2

= −4

2,

2

2 = (−2, 1)

3. (0, 0), (0, 4) Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj;Jz;bd; eLg;Gs;spiaf; fhz;f. 𝑥1 , 𝑦1 𝑥2 , 𝑦2

(0, 0), (0, 4)


2,𝑦1+ 𝑦2

2

= 0+0

2,

0+4

2

= 0

2,

4

2 = (0,2)

4. xU tl;lj;jpd; ikak; ( − 6, 4). mt;tl;lj;jpd; xU tpl;lj;jpd; xU Kid Mjpg;Gs;sp vdpy;> kWKidiaf; fhz;f. (Jun-2012, 2014, Apr-2015)

𝑥2 , 𝑦2

𝑥1 , 𝑦1 (0, 0)

𝑥1+ 𝑥2

2,𝑦1+ 𝑦2

2 = (−6, 4)

𝑥1+ 0

2,𝑦1+ 0

2 = (−6, 4)

𝑥1

2 = −6 ⟹ 𝑥1 = −12

𝑦1

2 = 4 ⟹ 𝑦1 = 8

kWKid = (−12, 8)

5. (1, 3), (2, 7), (12, −16) Mfpa Gs;spfis Kidfshff; nfhz;l Kf;Nfhzj;jpd; eLf;Nfhl;L ikak; fhz;f.

𝑥1 , 𝑦1 𝑥2 , 𝑦2 𝑥3 , 𝑦3

(1, 3), (2, 7), (12, -16)

eLf;Nfhl;L ikak; G = 𝑥1+ 𝑥2+𝑥3

3,𝑦1+ 𝑦2+𝑦3

3

= 1+2+12

3,

3+7−16

3 =

15

3,−6

3 = (5, −2)



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6. (3, −5), (−7, 4), (10, −-2) Mfpa Gs;spfis Kidfshff; nfhz;l Kf;Nfhzj;jpd; eLf;Nfhl;L ikak; fhz;f.

𝑥1 , 𝑦1 𝑥2 , 𝑦2 𝑥3 , 𝑦3

(3, −5), (−7, 4), (10, −2)

eLf;Nfhl;L ikak; = 𝑥1+ 𝑥2+𝑥3

3,𝑦1+ 𝑦2+𝑦3

3

= 3−7+10

3,−5+4−2

3 =

6

3,−3

3 = (2, −1)

7. A (4, −6), B (3, −2), (5, 2) Mfpa Gs;spfis Kidfshff; nfhz;l Kf;Nfhzj;jpd; eLf;Nfhl;L ikak; fhz;f. (Oct-2012)

𝑥1 , 𝑦1 𝑥2 , 𝑦2 𝑥3 , 𝑦3

(4, −6), (3, −2), (5, 2)


3,𝑦1+ 𝑦2+𝑦3

3

= 4+3+5

3,−6−2+2

3

= 12

3,−6

3

= (4, −2)

8. Gs;sp (1, 3)-I eLf;Nfhl;L ikakhff; nfhz;l Kf;Nfhzj;jpd; ,U Kidfs; (−7, 6) kw;Wk; (8, 5) vdpy;> Kf;Nfhzj;jpd; %d;whtJ Kidiaf; fhz;f. (Apr-2012, Apr-2016)

𝑥1, 𝑦1 𝑥2 , 𝑦2 𝑥3 , 𝑦3

(−7, 6), (8, 5) , (𝑥3 , 𝑦3),


3,𝑦1+ 𝑦2+𝑦3

3 = (1, 3)

−7+8+𝑥3

3,

6+5+𝑦3

3 = (1, 3)

𝑥3+1

3,𝑦3+11

3 = (1,3)

𝑥3+1

3= 1

𝑦3+11

3 = 3

𝑥3 + 1 = 3 𝑦3 + 11 = 9 𝑥3 = 3 − 1 𝑦3 = 9 − 11

𝑥3 = 2 %d;whtJ Kid = (2, -2)

9. (0, 0), (3, 0), (0, 2) Mfpa Gs;spfis Kidfshf nfhz;l Kf;Nfhzj;jpd; gug;gsT fhz;f.

𝐴 = 1

2 { 𝑥1𝑦2 + 𝑥2𝑦3 + 𝑥3𝑦1 − 𝑥2𝑦1 + 𝑥3𝑦2 + 𝑥1𝑦3 } 0 3 0 0

𝐴 = 1

2 0 + 6 + 0 − 0 + 0 + 0 =

1

2 {6 − 0} 0 0 2 0

𝐴 = 1

2 × 6 = 3 r.m.

10. (5, 2), (3, −5), (−5, −1) Mfpa Gs;spfis Kidfshf nfhz;l Kf;Nfhzj;jpd; gug;gsT fhz;f.

𝐴 = 1

2 { 𝑥1𝑦2 + 𝑥2𝑦3 + 𝑥3𝑦1 − 𝑥2𝑦1 + 𝑥3𝑦2 + 𝑥1𝑦3 } 5 3 −5 5

𝐴 = 1

2 −25 − 3 − 10 − 6 + 25 − 5 2 −5 −1 2

𝐴 =1

2 {−38 − 26}

𝑦3 = −2



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𝐴 = 1

2 × −64 = −32 (gug;G vg;nghOJk; kpif vz;zhf ,Uf;f Ntz;Lk;)

𝐴 = 32 r.m.

11. (−4, −5), (4, 5),( −1, −6) Mfpa Gs;spfis Kidfshf nfhz;l Kf;Nfhzj;jpd; gug;gsT fhz;f.

𝐴 = 1

2 { 𝑥1𝑦2 + 𝑥2𝑦3 + 𝑥3𝑦1 − 𝑥2𝑦1 + 𝑥3𝑦2 + 𝑥1𝑦3 } −4 4 −1 −4

𝐴 = 1

2 −20 − 24 + 5 − [−20 − 5 + 24 } −5 5 −6 −5

𝐴 =1

2 {−39 + 1}

𝐴 = 1

2 × −38 = −19

𝐴 = 19 r.m.

12. (1, 2) (−3, 4), (−5, −6) Mfpa Gs;spfis Kidfshf nfhz;l Kf;Nfhzj;jpd; gug;gsT fhz;f.

𝐴 = 1

2 { 𝑥1𝑦2 + 𝑥2𝑦3 + 𝑥3𝑦1 − 𝑥2𝑦1 + 𝑥3𝑦2 + 𝑥1𝑦3 } 1 −3 −5 1

𝐴 = 1

2 4 + 18 − 10 − [−6 − 20 − 6 } 2 4 −6 2

𝐴 =1

2 {12 + 32}

𝐴 = 1

2 × 44 = 22 r. m.

13. A(2, 3), B(4, 0) C(6, −3) Mfpa Gs;spfs; xNu Neh;f;Nfhl;by; mike;Js;sdth vd Muha;f.

𝐴 = 1

2 { 𝑥1𝑦2 + 𝑥2𝑦3 + 𝑥3𝑦1 − 𝑥2𝑦1 + 𝑥3𝑦2 + 𝑥1𝑦3 } 2 4 6 2

𝐴 = 1

2 0 − 12 + 18 − [12 + 0 − 6 } 3 0 −3 3

𝐴 =1

2 {6 − 6}

𝐴 = 1

2 × 0 = 0

/ nfhLf;fg;gl;l Gs;spfs; xNu Neh;f;Nfhl;by; mike;Js;sd.

14. A(4, 3), B(1, 2), C(−2, 1) Mfpa Gs;spfs; xNu Neh;f;Nfhl;by; mike;Js;sdth vd Muha;f.

𝐴 = 1

2 { 𝑥1𝑦2 + 𝑥2𝑦3 + 𝑥3𝑦1 − 𝑥2𝑦1 + 𝑥3𝑦2 + 𝑥1𝑦3 } 4 1 −2 4

𝐴 = 1

2 8 + 1 − 6 − [3 − 4 + 4 } 3 2 1 3

𝐴 =1

2 {3 − 3}

𝐴 = 1

2 × 0 = 0


15. A (−2, −2), B(−6, −2), C(−2,2) Mfpa Gs;spfs; xNu Neh;f;Nfhl;by; mike;Js;sdth vd Muha;f.

𝐴 = 1

2 { 𝑥1𝑦2 + 𝑥2𝑦3 + 𝑥3𝑦1 − 𝑥2𝑦1 + 𝑥3𝑦2 + 𝑥1𝑦3 } −2 −6 −2 −2

𝐴 = 1

2 4 − 12 + 4 − [12 + 4 − 4 } −2 −2 2 −2

𝐴 =1

2 {−4 − 12}

𝐴 = 1

2 × {−16}

𝐴 = 8 ⟹ 𝐴 ≠ 0

/ nfhLf;fg;gl;l Gs;spfs; xNu Neh;f;Nfhl;by; mikahJ.



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16. A −𝟑

𝟐, 𝟑 , B (6, -2), C(-3, 4) Mfpa Gs;spfs; xNu Neh;f;Nfhl;by; mike;Js;sdth vd Muha;f.

𝐴 = 1

2 { 𝑥1𝑦2 + 𝑥2𝑦3 + 𝑥3𝑦1 − 𝑥2𝑦1 + 𝑥3𝑦2 + 𝑥1𝑦3 }

−3

2 6 -3

−3

2

𝐴 = 1

2 3 + 24 − 9 − [18 + 6 − 6 } 3 -2 4 3

𝐴 =1

2 {18 − 18}

𝐴 = 1

2 × 0 = 0


17. 𝒂, 𝟏 , 𝟏, 𝟐 kw;Wk; (𝟎, 𝒃 + 𝟏) Mfpa Gs;spfs; xNu Neh;f;Nfhl;by; mike;jhy; 𝟏

𝒂+

𝟏

𝒃= 𝟏 vd epWTf.

(Apr-2013)

a 1

1 2

0 b+1 = 0

a 1

2𝑎 + 𝑏 + 1 + 0 − 1 + 0 + 𝑎𝑏 + 𝑎 = 0

2𝑎 + 𝑏 + 1 − 1 − 𝑎𝑏 − 𝑎 = 0

𝑎 + 𝑏 – 𝑎𝑏 = 0

𝑎 + 𝑏 = 𝑎𝑏

,UGwKk; "𝑎𝑏" My; tFf;f 𝑎+𝑏

𝑎𝑏 =

𝑎𝑏

𝑎𝑏

𝑎

𝑎𝑏+

𝑏

𝑎𝑏 = 1

1

𝑎+

1

𝑏= 1 vd epWtg;gl;lJ.

18. A (6,7), B (−4,1) C (a, −9) Mfpatw;iw Kidfshff; nfhz;l ∆𝐀𝐁𝐂 d; gug;G 68 r. myFfs; vdpy; 𝒂-d; kjpg;igf; fhz;f. (Jun-2013, 2015)

A = 68

𝐴 = 1

2 { 𝑥1𝑦2 + 𝑥2𝑦3 + 𝑥3𝑦1 − 𝑥2𝑦1 + 𝑥3𝑦2 + 𝑥1𝑦3 = 68 6 −4 a 6

6 + 36 + 7𝑎 − [−28 + 𝑎 − 54 } = 68 × 2 7 1 −9 7

42 + 7𝑎 − [−82 + 𝑎 } = 136

42 + 7𝑎 + 82 – 𝑎 = 1

124 + 6𝑎 = 136

6𝑎 = 12

𝑎 = 12/6

𝑎 = 2

19. ( − 5, − 2) vd;w Gs;sp topr; nry;tJk;> Ma mr;RfSf;F ,izahdJkhd Neh;f;NfhLfspd; rkd;ghLfisf; fhz;f. (Jun-2013)

𝑥 mr;Rf;F ,izahd Nfhl;bd; rkd;ghL 𝑦 mr;Rf;F ,izahd Nfhl;bd; rkd;ghL 𝑦 = 𝑘 𝑥 = 𝑘



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𝑦 = −2 𝑥 = −5 𝑦 + 2 = 0 𝑥 + 5 = 0

20. 𝒙 + 𝟐𝒚 + 𝟏 = 𝟎 , 𝟑𝒙 + 𝟔𝒚 + 𝟐 = 𝟎 Mfpa Neh;f;NfhLfs; ,iz vd epWTf.

𝑥 + 2𝑦 + 1 = 0 d; rha;T 𝑚1 = = −1

2

3𝑥 + 6𝑦 + 2 = 0 d; rha;T 𝑚2 = = −3

6 =

−1

2

𝑚1 = 𝑚2 rha;Tfs; rkk;. vdNt Neh;f;NfhLfs; ,iz.

21. 𝟑𝒙 + 𝟐𝒚 − 𝟏𝟐 = 𝟎 , 𝟔𝒙 + 𝟒𝒚 + 𝟖 = 𝟎 Mfpa Neh;f;NfhLfs; ,iz vd epWTf.

3𝑥 + 2𝑦 − 12 = 0 d; rha;T 𝑚1 = = −3

2

6𝑥 + 4𝑦 + 8 = 0 d; rha;T 𝑚2 = =−6

4 =

−3

2

𝑚1 = 𝑚2 rha;Tfs; rkk;. vdNt Neh;f;NfhLfs; ,iz.

22. 𝒙 + 𝟐𝒚 + 𝟏 = 𝟎 , 𝟐𝒙 − 𝒚 + 𝟓 = 𝟎 Mfpa Neh;f;NfhLfs; nrq;Fj;J vd epWTf.

𝑥 + 2𝑦 + 1 = 0 d; rha;T 𝑚1 = = −1

2

2𝑥 − 𝑦 + 5 = 0 d; rha;T 𝑚2 = = −2

−1 = 2

𝑚1 × 𝑚2 =−1

2 × 2 = −1. vdNt Neh;f;NfhLfs; nrq;Fj;jhdit.

23. 𝟑𝒙 − 𝟓𝒚 + 𝟕 = 𝟎 , 𝟏𝟓𝒙 + 𝟗𝒚 + 𝟒 = 𝟎 Mfpa Neh;f;NfhLfs; nrq;Fj;J vd epWTf.(Apr-2013)

3𝑥 − 5𝑦 + 7 = 0 d; rha;T 𝑚1 = = −3

−5 =

3

5

15𝑥 + 9𝑦 + 4 = 0 d; rha;T 𝑚2 = = −15

9 =

−5

3

𝑚1 × 𝑚2 = 3

5 ×

−5

3 = −1. vdNt Neh;f;NfhLfs; nrq;Fj;jhdit.

24. 𝟓𝒙 − 𝟐𝒚 − 𝟗 = 𝟎 , 𝒂𝒚 + 𝟐𝒙 − 𝟏𝟏 = 𝟎 Mfpa Neh;f;NfhLfs; xd;Wf;nfhd;W nrq;Fj;J vdpy; 𝒂-d; kjpg;igf; fhz;f. (Apr-2016)

5𝑥 − 2𝑦 − 9 = 0 vd;w Neh;f;Nfhl;bd; rha;T 𝑚1 = = −5

−2 =

5

2

2𝑥 + 𝑎𝑦 − 11 = 0 vd;w Neh;f;Nfhl;bd; rha;T 𝑚2 = = −2

𝑎

Neh;f;NfhLfs; xd;Wf;nfhd;W nrq;Fj;J> vdNt 𝑚1 × 𝑚2 = −1

5

2 ×

−2

𝑎 = −1

−𝑎 = −5

𝑎 = 5



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25. rha;Tf;Nfhzk; 𝟒𝟓𝟎 kw;Wk; 𝒚-ntl;Lj;Jz;L 𝟐

𝟓 Mfpatw;iwf; nfhz;l Neh;f;Nfhl;bd; rkd;ghl;ilf;

$Wf. (Oct-2013)

rha;T 𝑚 = 𝑡𝑎𝑛 𝜃 = 𝑡𝑎𝑛450 = 1

𝑦 –ntl;Lj;Jz;L 𝑐 = 2

5

Njitahd Neh;f;Nfhl;bd; rkd;ghL 𝑦 = 𝑚𝑥 + 𝑐

𝑦 = 1𝑥 +2

5

5𝑦 = 5𝑥 + 2 5𝑥 − 5𝑦 + 2 = 0

26. 𝟑𝒙 − 𝒚 + 𝟕 = 𝟎 vd;w Neh;f;Nfhl;bw;F ,izahdJk; (𝟏, −𝟐) vd;w Gs;sp topr; nry;tJkhd

Neh;f;Nfhl;bd; rkd;ghl;ilf; fhz;f. (Jun-2015)

3𝑥 − 𝑦 + 7 = 0 vd;w Neh;f;Nfhl;bw;F ,izahd Neh;f;Nfhl;bd; rkd;ghL 3𝑥 – 𝑦 + 𝑘 = 0

3 𝑥 − 𝑦 + 𝑘 = 0 vd;gJ (1, -2) vd;w Gs;sp topr; nry;fpwJ. vdNt> Neh;f;Nfhl;bd; rkd;ghl;by; 𝑥 = 1, 𝑦 = 2 vdg; gpujpapl

3(1) − (−2) + 𝑘 = 0 ⟹ 3 + 2 + 𝑘 = 0

5 + 𝑘 = 0 ⟹ 𝑘 = −5

Njitahd Neh;f;Nfhl;bd; rkd;ghL 3 𝑥 – 𝑦 – 5 = 0 MFk;.

27. 𝟒 𝒙 − 𝟐𝒚 + 𝟏 = 𝟎 vd;w Neh;f;Nfhl;bd; rha;T kw;Wk; 𝒚-ntl;Lj;Jz;L Mfpatw;iwf; fhz;f. (Apr-15)

nfhLf;fg;gl;l Neh;f;Nfhl;bd; rkd;ghL 4 𝑥 − 2𝑦 + 1 = 0. khw;WKiw

rha;T = =−4

−2= 2 𝑦 = 𝑚𝑥 + 𝑐 ⇒ −2𝑦 = −4𝑥 − 1

𝑦 ntl;Lj;Jz;L = = −1

−2 =

1

2 𝑦 =

−4

−2𝑥 +

−1

−2⇒ 𝑚 =

−4

−2= 2, 𝑐 =

−1

−2=

1

2

rha;T = 2, 𝑦 ntl;Lj;Jz;L = 1

2

28. 𝟐𝒙 – 𝒚 + 𝟏𝟔 = 𝟎 vd;w Neh;f;Nfhl;bd; rkd;ghl;bypUe;J 𝒙, 𝒚 -ntl;Lj;Jz;Lfisf; fhz;f.

nfhLf;fg;gl;l Neh;f;Nfhl;bd; rkd;ghL 2𝑥 – 𝑦 + 16 = 0

𝑥 -ntl;L = = −16

2 = −8

𝑦 -ntl;L = = −16

−1 = 16

𝑥–ntl;Lj; Jz;L = -8, 𝑦 -ntl;Lj;Jz;L = 16

29. (−𝟐, 𝟓) kw;Wk; (𝟑, 𝟔) Mfpa Gs;spfs; topr; nrY;Yk; Neh;f;Nfhl;bd; rkd;ghl;ilf; fhz;f.

,U Gs;spfs; topr; nry;Yk; Nfhl;bd; rkd;ghL

𝑦−𝑦1

𝑦2−𝑦1=

𝑥−𝑥1

𝑥2−𝑥1 (𝑥1 = −2, 𝑦1 = 5, 𝑥2 = 3, 𝑦2 = 6)



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nfhLf;fg;gl;l ,U Gs;spfs; (-2, 5) kw;Wk; (3, 6)

𝑦−5

6−5=

𝑥+2

3+2

𝑦−5

1=

𝑥+2

5

5(𝑦 − 5) = 𝑥 + 2

5𝑦 – 25 = 𝑥 + 2

𝑥 − 5𝑦 + 27 = 0

Njitahd Neh;f;Nfhl;bd; rkd;ghL 𝑥 − 5𝑦 + 27 = 0

30. rha;T −3 kw;Wk; 𝒚 - ntl;Lj;Jz;L 4 nfhz;lJkhd Neh;f;Nfhl;bd; rkd;ghl;ilf; fhz;f. nfhLf;fg;gl;lit rha;T 𝑚 = −3 kw;Wk; 𝑦 - ntl;Lj;Jz;L 𝑐 = 4 Njitahd Neh;f;Nfhl;bd; rkd;ghL 𝑦 = 𝑚𝑥 + 𝑐 𝑦 = − 3𝑥 + 4

3𝑥 – 𝑦 + 4 = 0 Njitahd Neh;f;Nfhl;bd; rkd;ghL 3𝑥 − 𝑦 + 4 = 0

31. (7, 3), (6, 1), (8, 2) kw;Wk; (P, 4) vd;gd Xh; ,izfuj;jpd; thpirg;gb mike;j cr;rpfs; vdpy; P-d; kjpg;igf; fhz;f. (Jun-2012)

,izfuk; ABCD d; cr;rpfs; A(7,3), B(6,1), C(8,2) kw;Wk; D(P,4) vd;f. Xh; ,izfuj;jpd; %iytpl;lq;fs; xd;iwnahd;W ,Urkf;$wpLk; mjhtJ %iytpl;lk; AC-d; eLg;Gs;sp = %iytpl;lk; BD-d; eLg;Gs;sp

/ 7+8

2,

3+2

2 =

6+𝑃

2,

1+4

2

6+P

2 =

7+8

2

6+ P = 15

P = 15 - 6 = 9

P = 9

32. rha;Tf;Nfhzk; 𝟑𝟎𝟎 nfhz;l kw;Wk; (4,2), (3,1) Mfpa Gs;spfis ,izf;Fk; Neh;f;Nfhl;Lj;

Jz;bd; eLg;Gs;sp topr; nry;Yk; Neh;f;Nfhl;bd; rkd;ghl;ilf; fhz;f. (Oct-2012)

rha;Tf;Nfhzk; 𝜃 = 300

rha;T 𝑚 = 𝑡𝑎𝑛 𝜃 = 𝑡𝑎𝑛300 = 1

3

(4, 2), (3, 1) Mfpa Gs;spfis ,izf;Fk; Neh;f;Nfhl;Lj; Jz;bd; eLg;Gs;sp

𝑥1+𝑥2

2,𝑦1+𝑦2

2 =

4+3

2,

2+1

2 =

7

2,

3

2

Njitahd Neh;f;Nfhl;bd; rkd;ghl;bd; rkd;ghL 𝑦 − 𝑦1 = 𝑚( 𝑥 − 𝑥1)

𝑦 − 3

2 =

1

3 𝑥 −

7

2

2𝑦−3

2=

1

3

2𝑥−7

2

3(2𝑦 − 3) = 2 𝑥 − 7

2 3𝑦 − 3 3 − 2𝑥 + 7 = 0

2𝑥 − 2 3 𝑦 + (3 3 − 7) = 0

eLg;Gs;sp fhZk; tha;g;ghL

⇒ 𝑥1+𝑥2

2,𝑦1+𝑦2

2

Fwpg;G: 𝑚 = 1

3

(𝑥1 , 𝑦1) = 7

2,

3

2



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33. 𝟐𝒂 + 𝟐, 𝟑 , 𝟒, 𝟐𝒃 + 𝟏 Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj;Jz;bd; eLg;Gs;spapd; Maj;njhiyTfs; (𝟐𝒂, 𝟐𝒃) vdpy;> 𝒂, 𝒃 Mfpatw;wpd; kjpg;Gfisf; fhz;f. (Apr-2014)

2𝑎 + 2, 3 , 4, 2𝑏 + 1 Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj;Jz;bd; eLg;Gs;sp = (2𝑎, 2𝑏)

𝑥1+𝑥2

2,𝑦1+𝑦2

2 = (2𝑎, 2𝑏)

2𝑎+2+4

2,

3+2𝑏+1

2 = (2𝑎, 2𝑏)

2𝑎+6

2= 2𝑎

2𝑏+4

2 = 2𝑏

2𝑎 + 6 = 4𝑎 2𝑏 + 4 = 4𝑏

2𝑎 = 6 2𝑏 = 4

𝑎 = 3 𝑏 = 2

34. (3, −4) vd;w Gs;sp topr; nry;Yk; kw;Wk; Ma mr;RfSf;F ,izahf mike;j Neh;f;NfhLfspd; rkd;ghLfisf; fhz;f. (Apr-2014)

(3, −4) vd;w Gs;sp topr; nry;Yk; kw;Wk; 𝑥 mr;Rf;F ,izahd Nfhl;bd; rkd;ghL 𝑦 = 𝑐 ,

vdNt> 𝑦 = −4

(3, −4) vd;w Gs;sp topr; nry;Yk; kw;Wk; 𝑦 mr;Rf;F ,izahd Nfhl;bd; rkd;ghL 𝑥 = 𝑐,

vdNt> 𝑥 = 3

35. (−2, 3) vd;w Gs;sp topr; nry;tJk;> rha;T 𝟏

𝟑 cilaJkhd Neh;f;Nfhl;bd; rkd;ghl;ilf; fhz;f.

(Jun- 2014) (𝑥1 , 𝑦1 ) = (−2,3)

rha;T 𝑚 = 1

3

Njitahd Nfhl;bd; rkd;ghL 𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1

𝑦 − 3 =1

3 𝑥 + 2

3𝑦 − 9 = 𝑥 + 2

𝑥 − 3𝑦 + 11 = 0

36. (−3, 5) kw;Wk; (4, −9) Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj;Jz;bid cl;Gwkhf 1:6 vd;w

tpfpjj;jpy; gphpf;Fk; Gs;spapd; mr;Rj; njhiyTfisf; fhz;f. (Oct-2014)

cl;Gwkhf gphpf;Fk; Gs;sp 𝑃 = 𝑙𝑥2+𝑚𝑥 1

𝑙+𝑚,𝑙𝑦2+𝑚𝑦1

𝑙+𝑚

= 1 4 + 6(−3)

1+6,

1 −9 +6(5)

1+6

= 4−18

7,−9+30

7

= −14

7,

21

7

= (−2,3)

cl;Gwkhf gphpf;Fk; Gs;sp 𝑃 = (−2,3)

𝑎 = 3, 𝑏 = 2

𝑥1 = −3, 𝑥2 = 4,

𝑦1 = 5, 𝑦2 = −9

𝑙 = 1, 𝑚 = 6



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37. (2, 1) kw;Wk; (5, −8) vd;w Gs;spfis ,izf;Fk; Nfhl;Lj; Jz;il P,Q vd;w Gs;spfs; %d;W

rkghfq;fshfg; gphpf;fpd;wd. Gs;sp P MdJ 𝟐𝒙 − 𝒚 + 𝒌 = 𝟎 vd;w Nfhl;bd; Nky; cs;sJ vdpy;>

𝒌-d; kjpg;G fhz;f. (Oct-2014)

P,Q vd;w Gs;spfs; AB I Kr;rkf; $wpLk; Gs;spfs; MFk;. mjhtJ P vd;gJ AB I 1:2 vd;w tpfpjj;jpy; gphpf;Fk; Gs;sp MFk;.

vdNt> 𝑃 = 𝑙𝑥2+𝑚𝑥 1

𝑙+𝑚,𝑙𝑦2+𝑚𝑦1

𝑙+𝑚

= 1 5 +2(2)

1+2,

1 −8 + 2(1)

1+2

= 5+4

3,−8+2

3

= 9

3,−6

3

𝑃 = (3, −2)

𝑃 MdJ 2𝑥 − 𝑦 + 𝑘 = 0 vd;w Nfhl;bd; Nky; cs;sJ. vdNt 2 3 + 2 + 𝑘 = 0

8 + 𝑘 = 0

𝑘 = −8

38. rJuk; ABCD-apd; gf;fk; AB MdJ 𝒙 -mr;Rf;F ,izahf cs;sJ. vdpy;> %iytpl;lk; 𝑨𝑪 -apd; rha;T fhz;f. (Oct-2015)

rJuk; ABCD –apd; gf;fk; AB MdJ 𝑥-mr;Rf;F ,izahf cs;sJ. %iytpl;lk; AC MdJ ∠𝐷𝐴𝐵 –I ,Urkf; $wpLk;.

vdNt> ∠𝐵𝐴𝐶 = 450

mjhtJ 𝜃 = 450

MfNt %iytpl;lk; AC-d; rha;T m = tanθ

= tan 450

= 1

6. tbtpay; 1. ∆𝑨𝑩𝑪 –y; ∠𝑨 – d; ntspg;Gw ,Urkntl;b MdJ BC-d; ePl;rpapid E-y; re;jpf;fpwJ. AB = 10 nr.kP> AC = 6 nr.kP kw;Wk; BC = 12 nr.kP vdpy;> CE-If; fhz;f.

∆ABC y; ∠𝐴 – d; ntspg;Gw ,Urkntl;b AE vd;f.

NkYk; AE MdJ BC-d; ePl;rpia E-y; re;jpf;fpwJ. CE = 𝑥 vd;f.

Nfhz ,Urkntl;bj; Njw;wg;gb>

𝐵𝐸

𝐶𝐸=

𝐴𝐵

𝐴𝐶

12+𝑥

𝑥=

10

6 ⟹

12+𝑥

𝑥=

5

3

3(12 + 𝑥) = 5𝑥 ⟹ 36 + 3𝑥 = 5𝑥

36 = 5 𝑥 − 3𝑥

2 𝑥 = 36 ⟹ 𝑥 = 18

/ EC = 18 nr.kP

2. glj;jpy; CD-d; msT fhz;f. (Apr-2015)

EA × EB = EC × ED

8 × 5 = (x + 4) × 4

𝑥1 = 2, 𝑥2 = 5,

𝑦1 = 1, 𝑦2 = −8

𝑙 = 1, 𝑚 = 2



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8×5

4 = 𝑥 + 4

x + 4 = 10

x = 10 - 4 = 6

CD = 6 nr.kP

3. ∆ABC y; DE ∥ BC and 𝑨𝑫

𝑫𝑩=

𝟐

𝟑 , AE = 3.7 nr.kP. vdpy;> EC-If; fhz;f. (Jun- 2012, Jun-2014)

𝐴𝐷

𝐷𝐵 =

𝐴𝐸

𝐸𝐶

2

3 =

3.7

𝐸𝐶

𝐸𝐶 = 3.7×3

2=

11.2

2

= 5.55 nr.kP.

4. D,E vd;w Gs;spfs; ∆ABCy; AB, AC fspy; DE ∥ BC vd;wthW cs;sd. AD = 6 nr.kP> DB = 9 nr.kP kw;Wk; AE = 8 nr.kP> vdpy;> AC- If; fhz;f.

Njy;]; Njw;wg;gb 𝐴𝐷

𝐷𝐵=

𝐴𝐸

𝐸𝐶

6

9 =

8

𝐸𝐶

EC = 8 × 9

6 = 12 nr.kP

AC = AE+EC=12 + 8 = 20 nr.kP

5. D,E vd;w Gs;spfs; ∆ABCy; AB, AC fspy; DE ∥ BC vd;wthW cs;sd. AD = 8 nr.kP> AB = 12 nr.kP> kw;Wk; AE = 12 nr.kP> vdpy;> CE- If; fhz;f.

Njy;]; Njw;wg;gb 𝐴𝐷

𝐷𝐵 =

𝐴𝐸

𝐸𝐶

8

4=

12

𝐸𝐶

EC = 12 × 4

8 =

48

8 = 6

CE = 6 nr.kP

6. ∆ABC y; ∠A vd;w Nfhzj;jpd; cl;Gw ,Urkntl;b AD MdJ> gf;fk; BC I D-y; re;jpf;fpwJ. BD = 2.5 nr.kP> AB = 5 nr.kP. kw;Wk; AC = 4.2 nr.kP. vdpy; DC-If; fhz;f.

(Apr-2012, Oct-2012, 2013, June-2015)

Nfhz ,Urkntl;bj; Njw;wg;gb> 𝐵𝐷

𝐷𝐶 =

𝐴𝐵

𝐴𝐶

2.5

𝐷𝐶=

5

4.2

DC = 2.5 × 4.2

5 =

10.5

5 = 2.1

DC = 2.1 nr.kP



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7. ∆𝑷𝑸𝑹 –d; gf;fq;fs; PQ kw;Wk; PR-fspd; kPJ mike;j Gs;spfs; S kw;Wk; T vd;f. NkYk; ST ∥ QR,

PR = 5.6 nr.kP. kw;Wk; 𝑷𝑺

𝑺𝑸=

𝟑

𝟓 vdpy;> PT-If; fhz;f.

∆PQR -y; PT = x vd;f.

/ TR = PR − PT = 5.6 − x

NkYk; ST ∥ QR Njy;]; Njw;wj;jpd;gb

𝑃𝑆

𝑆𝑄=

𝑃𝑇

𝑇𝑅⟹ 𝑃𝑇 = 𝑇𝑅

𝑃𝑆

𝑆𝑄

x = (5.6 − x) 3

5

5x = 16.8 – 3x

5x + 3x = 16.8 ⇒ 8𝑥 = 16.8

x = 16.8

8 = 2.1

/ PT = 2.1 nr.kP

8. 𝑨𝑫 vd;gJ ∆𝑨𝑩𝑪 -y; ∠A –d; cl;Gw Nfhz ,Urkntl;b. mJ BC-I D-y; re;jpf;fpwJ. BD = 2 nr.kP> AB = 5 nr.kP kw;Wk; DC = 3 nr.kP vdpy; AC-If; fhz;f.


𝐷𝐶=

𝐴𝐵

𝐴𝐶

2

3 =

5

𝐴𝐶

AC = 5×3

2 =

15

2= 7.5

AC = 7.5 nr.kP

9. AD vd;gJ ∆ABC –y; ∠A – d; cl;Gw Nfhz ,Urkntl;b. mJ BC I D-y; re;jpf;fpwJ.

AB = 5.6 nr.kP> AC = 6 nr.kP kw;Wk; DC = 3 nr.kP. vdpy; BC-If; fhz;f.


𝐷𝐶=

𝐴𝐵

𝐴𝐶

𝐵𝐷

3 =

5.6

5

BD = 5.6 × 3

6 = 2.8

BD = 2.8 nr.kP

BC = 2.8 + 3 = 5.8 nr.kP

10. glj;jpy; 𝒙-d; kjpg;igf; fhz;f.

PA × PB = PC × PD

4 × x = 8 × 3

x = 8 × 3

4=

24

4

x = 6

11. xU tl;lj;jpy; AB, CD vd;Dk; ,U ehz;fs; xd;iwnahd;W cl;Gwkhf P-apy; ntl;bf; nfhs;fpd;wd. CP = 4 nr.kP.> AP = 8 nr.kP.> PB = 2 nr.kP.> vdpy;> PD-If; fhz;f. (Apr-2014)

PA × PB = PC × PD

8 × 2 = 4 × x



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8 × 2

4 = x

16

4 = x

x = 4 PD= 4 nr.kP 12. xU tl;lj;jpy; AB, CD vd;Dk; ,U ehz;fs; xd;iwnahd;W cl;Gwkhf P-apy; ntl;bf;

nfhs;fpd;wd. AP= 12 nr.kP> AB = 15 nr.kP> CP = PD vdpy;> CD-If; fhz;f.

PA × PB = PC × PD

12 × 3 = x × x

36 = x2

x = 6

CD = 6+6 = 12 nr.kP

13. glj;jpy; x –d; kjpg;igf; fhz;f.

PA × PB = PC × PD

9 × 4 = (𝑥+2) × 2

9 × 4

2 = 𝑥+2

36

2 = x+2

𝑥+2 = 18

𝑥 = 18 −2= 16

𝑥 = 16

14. AB kw;Wk; CD vd;w ,U ehz;fs; tl;lj;jpw;F ntspNa P vDk; Gs;spapy; ntl;bf; nfhs;fpd;wd. AB = 4 nr.kP> BP= 5 nr.kP kw;Wk; PD = 3 nr.kP vdpy;> CD-If; fhz;f.

PA × PB = PC × PD

9 × 5 = (𝑥+3) × 3

9 × 5

3 = x+3

45

3 = x+3

𝑥+3 = 15

𝑥 = 15 – 3 = 12

CD = 12 nr.kP

15. AB kw;Wk; CD vd;w ,U ehz;fs; tl;lj;jpw;F ntspNa P vDk; Gs;spapy; ntl;bf; nfhs;fpd;wd. BP = 3 nr.kP> CP = 6 nr.kP kw;Wk; CD = 2 nr.kP> vdpy;> AB-If; fhz;f.

PA × PB = PC × PD

(𝑥+ 3) × 3 = 6 × 4

𝑥+3 = 6 × 4

3=

24

3

𝑥+3 = 8

𝑥 = 8 − 3= 5

AB = 5 nr.kP



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16. glj;jpy; TP xU njhLNfhL A,B vd;gd tl;lj;jpd; kPJs;s Gs;spfspy; ∠BTP = 𝟕𝟐𝟎 kw;Wk;

∠ATB = 𝟒𝟑 𝟎vdpy; ∠ABT If; fhz;f. (Apr-2013,Apr-2016)

∠BAT = ∠BTP = 720 [njhLNfhL – ehz; Njw;wk; ]

∆ABT y;

∠ATB + ∠BAT + ∠ABT = 1800

430 + 720 + ∠ABT = 1800

∠ABT = 1800 – 1150 = 650

17. xU tl;lj;jpd; Gs;sp A-y; tiuag;gLk; njhLNfhL PQ vd;f. AB vd;gJ tl;lj;jpd; ehz; vd;f.

NkYk;> ∠BAC = 𝟓𝟒𝟎 kw;Wk; ∠BAQ = 𝟔𝟐𝟎 vd;W mikAkhW tl;lj;jpd; Nky; cs;s Gs;sp C vdpy;> ∠ABC - If; fhz;f.

A-y; PQ xU njhLNfhL kw;Wk; AB xU ehz;.

vdNt ∠BAQ = ∠ACB = 620 (njhLNfhL – ehz;Njw;wk;)

NkYk;> ∠BAC + ∠ACB + ∠ABC = 1800

(xU Kf;Nfhzj;jpd; %d;W Nfhzq;fspd; $Ljy; 1800)

∠ABC = 1800 – (∠BAC + ∠ACB) = 1800 – (540 +620) = 640

∠ABC = 640

18. gpjhfu]; Njw;wj;ij vOJf. (my;yJ) ngsjad; Njw;wj;ij vOJf. xU nrq;Nfhz Kf;Nfhzj;jpy; fh;zj;jpd; th;f;fk; kw;w ,U gf;fq;fspd; th;f;fq;fspd; $LjYf;Fr; rkk;.

𝑩𝑪𝟐 = 𝑨𝑩 𝟐 + 𝑨𝑪𝟐

19. gpjhfu]; Njw;wj;jpd; kWjiyia vOJf. xU Kf;Nfhzj;jpy;> xU gf;fj;jpd; th;f;fk;> kw;w ,U gf;fq;fspd; th;f;fq;fspd; $LjYf;Fr; rkk; vdpy;> Kjy; gf;fj;jpw;F vjpNu cs;s Nfhzk; nrq;Nfhzk; MFk;.

𝑩𝑪𝟐 = 𝑨𝑩 𝟐 + 𝑨𝑪𝟐

20. njhLNfhL ehz;Njw;wj;ij vOJf. tl;lj;jpy; njhLNfhl;bd; njhL Gs;sp topNa xU ehz; tiuag;gl;lhy;> me;j ehz; njhLNfhl;Lld; Vw;gLj;Jk; Nfhzq;fs; KiwNa xt;nthd;Wk; jdpj;jdpahf khw;W tl;lj;Jz;Lfspy; mike;j Nfhzq;fSf;Fr; rkk;.

∠𝑩𝑨𝑻 = ∠𝑩𝑷𝑨 , ∠𝑩𝑨𝑺 = ∠𝑨𝑸𝑩

21. njhLNfhL ehz; Njw;wj;jpd; kWjiyia vOJf. xU tl;lj;jpy; xU ehzpd; xU Kidg;Gs;sp topNa tiuag;gl;l Neh;f;;NfhL me;ehZld; cz;lhf;Fk; NfhzkhdJ kW tl;lj; Jz;bYs;s Nfhzj;jpw;Fr; rkkhdhy;> me;Neh;f;NfhL tl;lj;jpw;F xU njhLNfhlhFk;.

22. MP vd;gJ ∆MNO-y; ∠M-d; ntspg;Gw ,U rkntl;b> NkYk; NO-d; ePl;rpapid P -apy; re;jpf;fpwJ. MN=10 nr.kP> MO = 6 nr.kP NO = 12 nr.kP vdpy; OP If; fhz;f. (Jun-2013)

MP vd;gJ ∆MNO-y; ∠M –d; ntspg;Gw ,U rkntl;b. ∆MNO –y; Nfhz ,Urkntl;b Njw;wj;jpd;gb



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𝑁𝑃

𝑂𝑃 =

𝑀𝑁

𝑀𝑂

12+𝑂𝑃

𝑂𝑃 =

𝑀𝑁

𝑀𝑂

12+𝑂𝑃

𝑂𝑃 =

10

6

72 + 6 𝑂𝑃 = 10 𝑂𝑃

10 𝑂𝑃 − 6 𝑂𝑃 = 72

4 𝑂𝑃 = 72

𝑂𝑃 =72

4= 18 nr.kP

7. Kf;Nfhztpay; 1. Rthpy; rha;j;J itf;fg;gl;l xU VzpahdJ jiuAld; 𝟔𝟎 𝟎Nfhzj;ij Vw;gLj;JfpwJ. Vzpapd; mb

Rtw;wpypUe;J 3.5 kP J}uj;jpy; cs;sJ. vdpy;> Vzpapd; ePsj;ijf; fhz;f.

(Oct-2012, Apr-2013, Jun-2014, Jun-2015)

nrq;Nfhzk; ∆ABC –y;

cos 600 = 𝐵𝐶

𝐴𝐶

1

2 =

3.5

𝐴𝐶

𝐴𝐶 = 3.5 × 2 = 7

Vzpapd; ePsk; = 7kP

2. 30 kP ePsKs;s xU fk;gj;jpd; epoypd; ePsk; 𝟏𝟎 𝟑 kP vdpy; #hpadpd; Vw;wf;Nfhzj;jpd; (jiu

kl;lj;jpypUe;J Vw;wf;Nfhzk;) mstpidf; fhz;f. (Apr-2012, Apr-2014)

nrq;Nfhz ∆ABC –y;

𝑡𝑎𝑛𝜃 = 𝐴𝐵

𝐵𝐶

= 30

10 3 =

3

3

= 3× 3

3

= 3

tan 600 = 3 / 𝜃 = 600

#hpadpd; Vw;wf;Nfhzk; = 600

3. xU Rik Ch;jpapypUe;J Rikia ,wf;f VJthf 𝟑𝟎𝟎 Vw;wf; Nfhzj;jpy; xU rha;T jsk; cs;sJ. rha;T jsj;;jpd; cr;rp jiuapypUe;J 0.9 kP cauj;jpy; cs;sJ vdpy; rha;T jsj;jpd; ePsk; ahJ? (Oct-2014, Apr-2015)


sin 300 = 𝐴𝐵

𝐴𝐶

1

2 =

0.9

𝐴𝐶

𝐴𝐶 = 0.9 × 2 = 1.8 rha;T js ePsk; = 1.8 kP



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4. cauk; 150 nr.kP. cs;s xU rpWkp xU tpsf;Ff; fk;gj;jpd; Kd; epd;wthW 𝟏𝟓𝟎 𝟑 nr.kP. ePsKs;s epoiy Vw;gLj;Jfpwhs; vdpy; tpsf;Ff; fk;gj;jpd; cr;rpapd; Vw;wf;Nfhzk; fhz;f.

(Jun-12)


𝑡𝑎𝑛 𝜃 = 𝐴𝐵

𝐵𝐶

= 150

150 3

= 1

3

tan 300 =1

3

/ 𝜃 = 300

Vw;wf;Nfhzk; = 𝟑𝟎𝟎

5. 𝟏−𝒔𝒊𝒏𝜽

𝟏+𝒔𝒊𝒏𝜽 = 𝒔𝒆𝒄𝜽 − 𝒕𝒂𝒏𝜽 vd epWTf. (Oct-2012, Jun-2014)

1−𝑠𝑖𝑛𝜃

1+𝑠𝑖𝑛𝜃=


1+𝑠𝑖𝑛𝜃x


1−𝑠𝑖𝑛𝜃 =

(1−𝑠𝑖𝑛𝜃 )2

1−𝑠𝑖𝑛2𝜃

= (1−𝑠𝑖𝑛𝜃 )2

𝑐𝑜 𝑠2𝜃 =

1−𝑠𝑖𝑛 𝜃

𝑐𝑜𝑠𝜃

2

1

2

=1−𝑠𝑖𝑛 𝜃

𝑐𝑜𝑠𝜃

= 1

𝑐𝑜𝑠𝜃 −

𝑠𝑖𝑛𝜃

𝑐𝑜𝑠𝜃= 𝑠𝑒𝑐𝜃 − 𝑡𝑎𝑛𝜃 vd epWtg;gl;lJ.

6. 𝒔𝒆𝒄𝟐𝜽 + 𝒄𝒐𝒔𝒆𝒄𝟐𝜽 = 𝐭𝐚𝐧 𝜽 + 𝐜𝐨𝐭 𝜽 vd epWTf. (Apr-2014, 2015)

𝑠𝑒𝑐2𝜃 + 𝑐𝑜𝑠𝑒𝑐2𝜃 = 1 + 𝑡𝑎𝑛2𝜃 + 1 + 𝑐𝑜𝑡2𝜃

= 𝑡𝑎𝑛2𝜃 + 𝑐𝑜𝑡2𝜃 + 2

= 𝑡𝑎𝑛2𝜃 + 𝑐𝑜𝑡2𝜃 + 2𝑡𝑎𝑛𝜃. 𝑐𝑜𝑡𝜃 = (𝑡𝑎𝑛𝜃 + 𝑐𝑜𝑡𝜃)2

= 𝑡𝑎𝑛 𝜃 + 𝑐𝑜𝑡 𝜃 vd epWtg;gl;lJ.

7. 𝒔𝒊𝒏𝜽

𝒄𝒐𝒔𝒆𝒄𝜽 +

𝒄𝒐𝒔𝜽

𝒔𝒆𝒄𝜽 = 𝟏 vd;gij epWTf. (Jun- 2012, Oct-2014)

𝑠𝑖𝑛𝜃

𝑐𝑜𝑠𝑒𝑐𝜃 +

𝑐𝑜𝑠𝜃

𝑠𝑒𝑐𝜃=

𝑠𝑖𝑛𝜃1

𝑠𝑖𝑛𝜃

+𝑐𝑜𝑠𝜃

1

𝑐𝑜𝑠𝜃

= 𝑠𝑖𝑛𝜃 × 𝑠𝑖𝑛𝜃 + ( 𝑐𝑜𝑠𝜃 × 𝑐𝑜𝑠𝜃)

= 𝑠𝑖𝑛2𝜃 + 𝑐𝑜𝑠2𝜃

= 1 vd epWtg;gl;lJ.

8. 𝒄𝒐𝒔𝜽

𝒔𝒆𝒄𝜽−𝒕𝒂𝒏𝜽 = 𝟏 + 𝐬𝐢𝐧𝜽 vd;w Kw;nwhUikia epWTf. (Jun-2013)

𝑐𝑜𝑠𝜃

𝑠𝑒𝑐𝜃 −𝑡𝑎𝑛𝜃 =

𝑐𝑜𝑠𝜃

𝑠𝑒𝑐𝜃 −𝑡𝑎𝑛𝜃 ×

𝑠𝑒𝑐𝜃 +𝑡𝑎𝑛𝜃

𝑠𝑒𝑐𝜃 +𝑡𝑎𝑛𝜃

= 𝑐𝑜𝑠𝜃 (𝑠𝑒𝑐𝜃 +𝑡𝑎𝑛𝜃 )

𝑠𝑒𝑐 2𝜃−𝑡𝑎𝑛 2𝜃 = 𝑐𝑜𝑠𝜃 ×

1

𝑐𝑜𝑠𝜃 + 𝑐𝑜𝑠𝜃 ×

𝑠𝑖𝑛𝜃

𝑐𝑜𝑠𝜃

= 1+ sin𝜃 vd epWtg;gl;lJ.

𝑠𝑒𝑐2𝜃 – 𝑡𝑎𝑛2𝜃 = 1



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9. 𝟏

𝒔𝒊𝒏𝟐𝜽−

𝟏−𝒔𝒊𝒏𝟐𝜽

𝟏−𝒄𝒐𝒔𝟐𝜽 = 𝟏 vd epWTf. (Apr-2013)

1

𝑠𝑖𝑛 2𝜃−

1−𝑠𝑖𝑛 2𝜃

1−𝑐𝑜𝑠 2𝜃= 𝑐𝑜𝑠𝑒𝑐2𝜃 −

𝑐𝑜𝑠 2𝜃

𝑠𝑖𝑛 2𝜃

= cosec2 𝜃 − cot2 𝜃

= 1 vd epWtg;gl;lJ.

10. (𝒔𝒊𝒏𝟔 𝜽 + 𝒄𝒐𝒔𝟔 𝜽) = 𝟏 − 𝟑𝒔𝒊𝒏𝟐 𝜽 𝒄𝒐𝒔𝟐 𝜽 vd epWTf. (Apr-2012)

𝑠𝑖𝑛6 𝜃 + 𝑐𝑜𝑠6 𝜃 = (𝑠𝑖𝑛2 𝜃)3 + (𝑐𝑜𝑠2 𝜃)3 (𝑎 = 𝑠𝑖𝑛2 𝜃, 𝑏 = 𝑐𝑜𝑠2 𝜃)

= (𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠2 𝜃)3 – 3 𝑠𝑖𝑛2 𝜃 𝑐𝑜𝑠2 𝜃 (𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠2 𝜃)

= 1 − 3𝑠𝑖𝑛2 𝜃 𝑐𝑜𝑠2 𝜃 vd epWtg;gl;lJ.

11. 𝟏+𝒔𝒆𝒄𝜽

𝒔𝒆𝒄𝜽 =

𝒔𝒊𝒏𝟐𝜽

𝟏−𝒄𝒐𝒔𝜽 vd epWTf. (Oct-2013, Apr-2016)

1+𝑠𝑒𝑐𝜃

𝑠𝑒𝑐𝜃 =

1+1

𝑐𝑜𝑠𝜃1

𝑐𝑜𝑠𝜃

= 𝑐𝑜𝑠𝜃 +1

𝑐𝑜𝑠𝜃×

𝑐𝑜𝑠𝜃

1

= 1 + 𝑐𝑜𝑠 𝜃

= (1 + 𝑐𝑜𝑠 𝜃) × (1−𝑐𝑜𝑠𝜃 )

(1−𝑐𝑜𝑠𝜃 )

= (1 + 𝑐𝑜𝑠 𝜃) × (1−𝑐𝑜𝑠𝜃 )

(1−𝑐𝑜𝑠𝜃 )

= 1−𝑐𝑜𝑠 2𝜃

1−𝑐𝑜𝑠𝜃

= 𝑠𝑖𝑛 2𝜃


12. 200 kP ePsKs;s E}ypdhy; xU fhw;whb fl;lg;gl;L gwe;Jf; nfhz;bUf;fpwJ. me;j E}y;

jiukl;lj;Jld; 𝟑𝟎𝟎 Nfhzj;ij Vw;gLj;jpdhy;> fhw;whb jiukl;lj;jpypUe;J vt;tsT cauj;jpy; gwf;fpwJ. vdf; fhz;f. (,q;F E}y; xU Neh;f;Nfhl;by; cs;sjhff; fUJf). glj;jpy; C vd;gJ fhw;whb vd;f. jiukl;lj;jpypUe;J fhw;whbf;F cs;s J}uk; BC = h vd;f.

E}ypd; ePsk; AC = 200 kP

nrq;Nfhz ∆ABC y; sin 300 = 𝐵𝐶

𝐴𝐶

1

2 =

ℎ

200

ℎ = 200

2 = 100

jiukl;lj;jpypUe;J fhw;whbf;F cs;s J}uk;

BC = h = 100 kP

13. 𝟏−𝒄𝒐𝒔𝜽

𝟏+𝒄𝒐𝒔𝜽 = 𝒄𝒐𝒔𝒆𝒄𝜽 − 𝒄𝒐𝒕𝜽 vd epWTf.


1+𝑐𝑜𝑠𝜃 =


1+𝑐𝑜𝑠𝜃×


1−𝑐𝑜𝑠𝜃 =

(1−𝑐𝑜𝑠𝜃 )2

1−𝑐𝑜𝑠2𝜃

= (1−𝑐𝑜𝑠𝜃 )2

𝑠𝑖𝑛2𝜃

= 1−𝑐𝑜𝑠 𝜃

𝑠𝑖𝑛𝜃

2

1

2

𝑐𝑜𝑠𝑒𝑐2 𝜃 – 𝑐𝑜𝑡2 𝜃 = 1.

1 − 𝑐𝑜𝑠2𝜃 = 𝑠𝑖𝑛2𝜃

𝑎3 + 𝑏3

= (𝑎 + 𝑏)3 – 3𝑎𝑏(𝑎 + 𝑏)

𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠2 𝜃 = 1



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=1−𝑐𝑜𝑠 𝜃

𝑠𝑖𝑛𝜃

= 1

𝑠𝑖𝑛𝜃 –

𝑐𝑜𝑠𝜃

𝑠𝑖𝑛𝜃

= 𝑐𝑜𝑠𝑒𝑐𝜃 − 𝑐𝑜𝑡𝜃

14. xU ∆𝑨𝑩𝑪 –apy; nrq;Nfhzk; ‘C’-apy; mikag;ngw;why; 𝐜𝐨𝐬(𝑨 + 𝑩)kw;Wk; 𝐬𝐢𝐧 (𝑨 + 𝑩) d; kjpg;Gfisf; fhz;f (Apr-2016)

xU Kf;Nfhzj;jpd; midj;J Nfhzq;fspd; $Ljy; 1800

C = 900 , vdNt> 𝐴 + 𝐵 = 900

cos(𝐴 + 𝐵) = cos 900 = 0

sin (𝐴 + 𝐵) = sin 900 = 1

15. 5 mb cauKs;s xU rpWtd; xU J}zpypUe;J 100 mb J}uj;jpy; epd;W mjd; cr;rpia 𝟒𝟓𝟎 Vw;wf; Nfhzj;jpy; ghh;j;jhy; J}zpd; cauk; vd;d? (Jun-2013) glj;jpy; AB =rpWtd; = 5 mb

BC = rpWtDf;Fk; J}zpw;Fk; ,ilNaahd J}uk; = 100 mb

J}zpd; cauk; = CE = CD + DE = 5 + DE (CD = AB = 5 mb)

nrq;Nfhz ∆ADE-y;

𝑡𝑎𝑛450 = 𝐷𝐸

𝐴𝐷=

𝐷𝐸

100= 1

/ 𝐷𝐸 = 100

/ J}zpd; cauk; 𝐶𝐸 = 5 + 100 = 105 mb

16. 𝜽 xU FWq;Nfhzk; kw;Wk; 𝒔𝒊𝒏𝜽 = 𝒄𝒐𝒔𝜽 vdpy; 𝒕𝒂𝒏𝟐𝜽 − 𝟐𝒄𝒐𝒔𝟐𝜽 = 𝟎 vd ep&gpf;f. (Oct-2013) 𝜃 xU FWq;Nfhzk; kw;Wk; 𝑠𝑖𝑛𝜃 = cos 𝜃 vdpy; 𝜃 = 45° LHS

𝑡𝑎𝑛2𝜃 − 2𝑐𝑜𝑠2𝜃 = 𝑡𝑎𝑛245° − 2𝑐𝑜𝑠245°

= 12 − 2 1

2

2

= 1 − 2 ×1

2

= 1 − 1= 0 RHS = 0 vd ep&gpf;fg;gl;lJ.

8. mstpay; 1. xU jpz;k Neh;tl;l cUisapd; Muk; 7 nr.kP. kw;Wk; cauk; 20 nr.kP vdpy; mjd; tisgug;G>

nkhj;j Gwg;gug;G Mfpatw;iwf; fhz;f. jpz;k Neh;tl;l cUis: r = 7 nr.kP. h = 20 nr.kP. tisgug;G = 2𝜋𝑟ℎ r.m nkhj;j Gwg;gug;G = 2𝜋𝑟(ℎ + 𝑟) r.m

= 2 × 22

7 × 7 × 20 = 2 ×

22

7 × 7 (20 + 7)

= 880 r.nr.kP = 1188 r.nr.kP

2. xU jpz;k Neh;tl;l cUisapd; Muk; 14 nr.kP. kw;Wk; cauk; 8 nr.kP. vdpy; mjd; tisgug;G> nkhj;j Gwg;gug;G Mfpatw;iwf; fhz;f. (Apr-2014) jpz;k Neh;tl;l cUis: r =14 nr.kP h = 8 nr.kP

𝑡𝑎𝑛450 = 1



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tisgug;G = 2𝜋𝑟ℎ r.m nkhj;j Gwg;gug;G = 2𝜋𝑟(ℎ + 𝑟) r.m

= 2 × 22

7 × 14 × 8 = 2 ×

22

7 × 14 (8 + 14)

= 704 r.nr.kP = 1936 r.nr.kP

3. ,U Neh;tl;l cUisfspd; Muq;fspd; tpfpjk; 3 : 2 vd;f. NkYk; cauq;fspd; tpfpjk; 5 : 3 vdpy;

tisgug;Gfspd; tpfpjk; fhz;f. (June-2015)

Neh;tl;l cUis: 𝑟1 ∶ 𝑟2 = 3 ∶ 2, ℎ1 ∶ ℎ2 = 5 ∶ 3

tisgug;Gfspd; tpfpjk; = 2𝜋𝑟1ℎ1: 2𝜋𝑟2ℎ2

= 3 × 5 : (2 × 3) = 5 : 2

4. xU jpz;k cUisapd; Muk; 14 nr.kP. mjd; cauk; 30 nr.kP. vdpy; cUisapd; fd msitf; fhz;f. (Oct-2012)

jpz;k cUis: r =14 nr.kP h=30 nr.kP

fd msT = 𝜋𝑟2ℎ f.m

= 22

7 × 14 × 14 × 30

=18480 f.nr.kP.

5. ,U Neh;tl;l cUisfspd; Muq;fspd; tpfpjk; 2 : 3 vd;f. NkYk; cauq;fspd; tpfpjk; 5 : 3 vdpy; fdmsTfspd;; tpfpjk; fhz;f.

Neh;tl;l cUis: 𝑟1 ∶ 𝑟2 = 2 ∶ 3, ℎ1 ∶ ℎ2 = 5 ∶ 3

fdmsTfspd;; tpfpjk; = 𝜋𝑟12ℎ1: 𝜋𝑟2

2ℎ2

= 2 × 2 × 5 : (3 × 3 × 3)

= 20 : 27

6. xU jpz;k Neh;tl;lf; $k;gpd; Muk; kw;Wk; rhAauk; KiwNa 35 nr.kP. kw;Wk; 37 nr.kP. vdpy; mjd; tisgug;G kw;Wk; nkhj;jg; gug;Gf; fhz;f. jpz;k Neh;tl;lf; $k;G: r = 35 nr.kP 𝑙 = 37 nr.kP

tisgug;G = 𝜋𝑟 𝑙 r.m

= 22

7 × 35 × 37

= 4070 r.nr.kP nkhj;jGwg;gug;G = 𝜋𝑟 (𝑙 + 𝑟) r.m

= 22

7 × 35 (37 + 35)

= 22

7 × 35 × 72

= 7920 r.nr.kP 7. xU jpz;k Neh;tl;lf; $k;gpd; mbr;Rw;wsT 236 nr.kP. mjd; rhAauk; 12 nr.kP. vdpy; mjd;

tisgug;Gf; fhz;f. (Apr-2013, Jun-2013)

jpz;k Neh;tl;lf; $k;G: l =12 nr.kP

2𝜋𝑟 = 236

𝜋r = 236

2 = 118



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tisgug;G = 𝜋 𝑟𝑙 r.m

= 118 × 12 = 1416 r.nr.kP

8. xU jpz;k Neh;tl;lf; $k;gpd; Muk; kw;Wk; cauk; KiwNa 7 nr.kP. kw;Wk; 24 nr.kP. vdpy; mjd; tisgug;G kw;Wk; nkhj;jg; gug;Gf; fhz;f. (Jun-12)

jpz;k Neh;tl;lf; $k;G: r =7 nr.kP h= 24 nr.kP

𝑙 = 𝑟2 + ℎ2

= 72 + 242

= 49 + 576 = 625

= 25 nr.kP

tisgug;G = 𝜋𝑟𝑙 r.m

= 22

7 × 7 × 25

=550 r.nr.kP nkhj;j Gwg;gug;G = 𝜋𝑟(𝑙 + 𝑟) r.m

= 22

7 × 7 × (25 + 7) =704 r.nr.kP

9. kuj;jpyhd xU jpz;kf; $k;gpd; mbr;Rw;wsT 44 nr.kP. mjd; cauk; 12 nr.kP. vdpy; mjd; fd msT fhz;f. (Oct-2013)

jpz;kf; $k;G: h =12 nr.kP 2𝜋r = 44 nr.kP

2 × 22

7 × 𝑟 = 44

𝑟 = 44 × 1

2 ×

7

22

r = 7 nr.kP

fd msT = 1

3 𝜋𝑟2ℎ f.m

= 1

3 ×

22

7 × 7 × 7 × 12

= 616 f.nr.kP

10. xU jpz;kf; $k;gpd; Muk; kw;Wk; rhAauk; KiwNa 20 nr.kP. kw;Wk; 29 nr.kP. vdpy; mjd; fd

msitf; fhz;f. (Apr-2012).

jpz;kf; $k;G: r = 20 nr.kP l = 29 nr.kP

h = 𝑙2 − 𝑟2 = 292 − 202

= 841 − 400 = 441 = 21

h = 21 nr.kP

fd msT = 1

3 𝜋𝑟2ℎ f.m

= 1

3 ×

22

7 × 20 × 20 × 21

= 8800 f.nr.kP

11. Xh; miuf;Nfhsj;jpd; fd msT 1152 𝝅 f.nr.kP. vdpy; mjd; tisg;gug;igf; fhz;f. miuf;Nfhsj;jpd; fd msT = 1152 𝜋 f.nr.kP.

2

3 𝜋r3 = 1152 𝜋



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𝑟3 = 1152 × 3

2

𝑟3 = 1728

r = 12 nr.kP

tisgug;G = 2𝜋𝑟2 r.m

= 2 × 𝜋 × 12 × 12 = 288 𝜋 r.nr.kP

12. xU jpz;k Neh; tl;l cUisapd; nkhj;jg; Gwg;gug;G 660 r.nr.kP. mjd; mbg;gf;fj;jpd; tpl;lk; 14 nr.kP. vdpy;> mt;TUisapd; cauk;> tisg;gug;igf; fhz;f. (Oct-2012)

jpz;k Neh; tl;l cUis: tpl;lk; = 14 nr.kP

Muk; = 7 nr.kP nkhj;jg;gug;G = 660 r.nr.kP

2𝜋𝑟 (ℎ + 𝑟) = 660

2 × 22

7 × 12 × (ℎ + 7) = 660

(ℎ + 7) = 660 × 1

2 ×

7

22 ×

1

7

ℎ + 7 = 15

ℎ = 15 − 7 = 8 nr.kP h = 8 nr.kP

tisg;gug;G = 2𝜋𝑟ℎ r.m

= 2 × 22

7 × 7 × 8

= 352 r.nr.kP

13. 7 nr.kP. Muk; nfhz;l Nfhs tbt gY}dpy; fhw;W nrYj;jg;gLk; NghJ Muk; 14 nr.kP. Mf mjpfhpj;jhy;> mt;tpU epiyfspy; gY}dpd; fd msTfspd; tpfpjj;ijf; fhz;f.

Nfhsk; : 𝑟1 = 7 nr.kP 𝑟2 = 14 nr.kP

fd msTfspd; tpfpjk; 4

3 𝜋r1

3 : 4

3 𝜋r2

3

= 7 × 7 × 7 : 14 × 14 × 14

= 1: 8

14. 14nr.kP. gf;f msT nfhz;l xU fd rJuj;jpy; ,Ue;J ntl;br; vLf;fg;gLk; kpfg;nghpa $k;gpd; fd msT fhz;f. $k;gpd; tpl;lk; = 14 nr.kP Muk; (r) = 7 nr.kP cauk; (h) = 14 nr.kP

fd msT = 1

3 𝜋𝑟2ℎ f.m

= 1

3 ×

22

7 × 7 × 7 × 14

= 718.67 f.nr.kP

15. 8.4 nr.kP tpl;lk; nfhz;l xU Nfhs tbt jpz;k cNyhf vwpf;Fz;bd; fd msitf; fhz;f. (Apr-2013, Jun-2015)

Nfhsj;jpd; tpl;lk; = 8.4 nr.kP

Muk; (r) = 4.2 nr.kP = 42

10



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fd msT = 4

3 𝜋𝑟3 f.m

= 4

3 ×

22

7 ×

42

10 ×

42

10 ×

42

10

= 38808

125

fd msT = 310.464 f.nr.kP 16. xU jpz;k miuf;Nfhsj;jpd; tisgug;G 2772 r.nr.kP. vdpy; mjd; nkhj;jg; gug;igf; fhz;f.

miuf;Nfhsj;jpd; tisgug;G = 2𝜋𝑟2 = 2772 r.nr.kP.

𝜋𝑟2 = 2772

2

𝜋𝑟2 = 1386

miuf;Nfhsj;jpd; nkhj;jg;gug;G = 3𝜋𝑟2 = 3 × 1386 = 4158 r.nr.kP. 17. ,U jpz;k miuf;Nfhsq;fspd; Muq;fs; 3:5 vd;w tpfpjj;jpy; cs;sd. mtw;wpd; tisgug;Gfspd;

tpfpjk; kw;Wk; nkhj;jg; gug;Gfspd; tpfpjk; fhz;f. jpz;k miuf;Nfhsk;: 𝑟1: 𝑟2 = 3 ∶ 5

tisgug;Gfspd; tpfpjk; = 2𝜋𝑟12 ∶ 2𝜋𝑟2

2 = 3 × 3 : ( 5 × 5) = 9 ∶ 25

nkhj;jg; gug;Gfspd; tpfpjk; = 3𝜋𝑟12: 3𝜋𝑟2

2 = 3 × 3 : (5 × 5 ) = 9 ∶ 25

18. 98.56 r.nr.kP. Gwg;gug;G nfhz;l xU jpz;kf; Nfhsj;jpd; Muj;ijf; fhz;f. (Oct-2013)

jpz;kf; Nfhsk;: Gwg;gug;G = 98.56 r.nr.kP.

4𝜋𝑟2 = 98.56

4 ×22

7× 𝑟2 = 98.56

𝑟2 = 98.56 × 1

4 ×

7

22

𝑟2 = 7.84

Muk; r = 2.8 nr.kP

19. xU jpz;k miuf;Nfhsj;jpd; nkhj;jg;gug;G 𝟔𝟕𝟓𝝅 r.nr.kP. vdpy; mjd; tisg;gug;igf; fhz;f.

(Jun-2014)

miuf;Nfhsj;jpd; nkhj;jg;gug;G = 3𝜋𝑟2 = 675𝜋

𝜋r2 =675𝜋

3 = 225𝜋

tisg;gug;G = 2𝜋𝑟2 = 225𝜋 × 2 = 450𝜋 r.nr.kP

20. xU cs;sPlw;w cUisapd; cs; kw;Wk; ntsp Muq;fs; KiwNa 12 nr.kP. kw;Wk; 18 nr.kP.

NkYk; mjd; cauk; 14 nr.kP. vdpy; mjd; tisgug;G kw;Wk; nkhj;jg;gug;G fhz;f. (Apr-2012)

cs;sPlw;w cUis:

r = 12 nr.kP R= 18 nr.kP h=14 nr.kP tisgug;G = 2𝜋ℎ (𝑅 + 𝑟) r.m nkhj;jg;gug;G = 2𝜋(𝑅 + 𝑟) (𝑅 − 𝑟 + ℎ) r.m

= 2 × 22

7× 14 × (18 + 12) = 2 ×

22

7 × (18 + 2)(18 − 12 + 14)

= 2 × 22 × 2 × 30 = 2 × 22

7× 30 × 20

= 2640 r.nr.kP = 3771.43 r.nr.kP



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21. 14 kP. MoKs;s xU fpzw;wpd; Muk; 5 kP. xU rJu kPl;lUf;F &.2 tPjk; mf;fpzw;wpd; cl;Gwr; Rtw;wpw;F rpnkz;l; G+r MFk; nrytpidf; fhz;f. (Oct-2014)

cUis : r = 5 kP h=14 kP

fpzw;wpd; cl;Gwg; gug;G (cUisapd; tisgug;G) = 2𝜋𝑟ℎ = 2 × 22

7 × 5 × 14

= 2 × 22 × 5 × 2

= 440 r.kP xU rJu kPl;lUf;F rpnkz;l; G+r MFk; nryT = &.2

440 rJu kPl;lUf;F rpnkz;l; G+r MFk; nryT = &. 440 × 2 = &.880

22. KiwNa 3 nr.kP. kw;Wk; 4 nr.kP. Muq;fshff; nfhz;l ,U Nfhsq;fspd; fd msTfspd; tpfpjj;jpidf; fhz;f. (Jun-2012)

Nfhsk; : R = 4 nr.kP r = 3 nr.kP

,U Nfhsq;fspd; fd msTfspd; tpfpjk; = 4

3 𝜋𝑅3 ∶

4

3 𝜋𝑟3

= 𝑅3: 𝑟3

= 43: 33 = 64: 27

23. 62.37 f.nr.kP. fd msT nfhz;l xU jpz;k Neh;tl;l cUisapd; cauk; 4.5 nr.kP. vdpy;

mt;TUisapd; Muj;ijf; fhz;f. (Jun-2013)

cUisapd; cauk; h = 4.5 nr.kP fd msT V = 62.37 f.nr.kP. Muk; r =?

fd msT 𝑉 = 𝜋𝑟2ℎ = 62.37 f.nr.kP

22

7 × r2 × 4.5 = 62.37

𝑟2 = 62.37 × 7

22 × 4.5 = 4.41

𝑟 = 4.41 = 2.1 nr.kP Muk; r = 2.1 nr.kP

24. miuf;Nfhs tbt fpz;zj;jpd; jbkd; 0.25 nr.kP kw;Wk; mjd; cl;Gw Muk; 5 nr.kP vdpy; mf;fpz;zj;jpd; ntspg;Gw tisgug;igf; fhz;f. (Apr-2015)

miuf;Nfhsk;: cl;Gw Muk; r = 5 nr.kP jbkd; w = 0.25 nr.kP

∴ ntspg;Gw Muk; R = 5+ 0.25 = 5.25 nr.kP

ntspg;Gw tisgug;G = 2𝜋𝑅2 = 2 × 22

7 × 5.25 × 5.25

= 173.25 r.nr.kP

25. xU cs;sPlw;w Nfhsj;jpd; ntsp kw;Wk; cs; Muq;fs; KiwNa 12 nr.kP. kw;Wk; 10 nr.kP. vdpy; mf;Nfhsj;jpd; fd msitf; fhz;f. cs;sPlw;w Nfhsk;: ntspg;Gw Muk; R = 12 nr.kP

cl;Gw Muk; r = 10 nr.kP

cs;sPlw;w Nfhsj;jpd; fd msT 𝑉 = 4

3 𝜋 (𝑅3 − 𝑟3) =

4

3 ×

22

7 (123 − 103)



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= 88

21 1728 − 1000

= 88

21 × 728

= 3050 2

3 f.nr.kP

26. xU jpz;k Neh;tl;lf; $k;gpd; fd msT 4928 f.nr.kP. kw;Wk; mjd; cauk; 24 nr.kP vdpy; mf;$k;gpd; Muj;ijf; fhz;f. $k;gpd; fd msT = 4928 nr.kP kw;Wk; cauk; h = 24 nr.kP

1

3𝜋r2h = 4928

1

3 ×

22

7 × r2 × 24 = 4928

𝑟2 = 4928 × 3× 7

22× 24 = 196

𝑟 = 196 =14 nr.kP

27. xU Neh; tl;lf; $k;gpd; fd msT 𝟐𝟏𝟔 𝝅 f.nr.kP kw;Wk; mf;$k;gpd; Muk; 9 nr.kP vdpy;> mjd; cauj;ijf; fhz;f. $k;gpd; Muk; r = 9 nr.kP $k;gpd; fd msT = 216 𝜋 f.nr.kP

$k;gpd; fd msT =1

3 𝜋𝑟2ℎ = 216 𝜋

1

3 × 𝜋 × 9 2 × ℎ = 216 𝜋

ℎ = 216 × 3

9×9 = 8 nr.kP

28. 7 kP cs;tpl;lKs;s xU Nfhsj;jpDs; xU rh;f;f]; tPuh; Nkhl;lhh; irf;fpspy; rhfrk; nra;fpwhh;. me;j rhfr tPuh; rhfrk; nra;af; fpilj;jpLk; cs;sPlw;wf; Nfhsj;jpd; cl;Gwg;gug;igf; fhz;f. cs;sPlw;w Nfhsj;jpd; cs; tpl;lk; 2r = 7kP Nkhl;lhh; irf;fps; tPuh;> rhfrk; nra;af; fpilj;jpLk; gug;G = Nfhsj;jpd; cl;Gw tisg;gug;G

= 4 𝜋𝑟2 = 𝜋(2𝑟)2

=22

7 × 72 =

22

7 × 7 × 7 = 154 nr.kP2

29. xU $k;G> xU miuf;Nfhsk; kw;Wk; xU cUis Mfpad rk mbg;gug;gpidf; nfhz;Ls;sd. $k;gpd; cauk;> cUisapd; cauj;jpw;F rkkhfTk;> NkYk; ,t;Tauk;> mtw;wpd; Muj;jpw;F rkkhfTk; ,Ue;jhy; ,k;%d;wpd; fd msTfSf;fpilNa cs;s tpfpjj;ijf; fhz;f.

$k;G> miuf;Nfhsk;> cUis Mfpatw;wpd; nghJthd Muk; r vd;f. $k;G> cUis Mfpatw;wpd; nghJthd cauk; h vd;f. fzf;fpd;gb r = h

$k;G> miuf;Nfhsk;> cUis Mfpatw;wpd; fd msTfspd; tpfpjk;

1

3 𝜋r2h:

2

3 𝜋r3: 𝜋r2h =

1

3∶

2

3∶ 1 [r = h]

= 1 : 2 : 3

30. 4.2 kP tpl;lKs;s miuf;Nfhs tbtj; njhl;bapy; epug;gg;gLk; ePhpd; fd msitf; fhz;f.

(Apr-2014)

miuf;Nfhs tbtj; njhl;bapd; tpl;lk; d= 4.2 kP



www.Padasalai.Net



Muk; r = 2.1 m

miuf;Nfhs tbtj; njhl;bapd; fd msT = 2

3𝜋𝑟3

=2

3 x

22

7 x 2.1 x 2.1 x 2.1

miuf;Nfhs tbtj; njhl;bapd; fd msT = 19.404 kP 3 [1kP 3 = 1000 ypl;lh;]

= 19404 ypl;lh;

31. 14 nr.kP. gf;f msT nfhz;l xU fdr; rJuj;jpypUe;J> kpfg;nghpa (kPg;ngU fd msTs;s) Nfhsk; ntl;b vLf;fg;gLfpwJ. vdpy;> mf;Nfhsj;jpd; fd msT fhz;f. (Jun- 2014)

fdr;rJuj;jpd; gf;f msT = Nfhsj;jpd; tpl;lk; d= 14 nr.kP Nfhsj;jpd; Muk; r=7 nr.kP

Nfhsj;jpd; fd msT = 4

3𝜋𝑟3

=4

3×

22

7× 7 × 7 × 7

=4312

3= 1437

1

3 nr.kP

32. 21 nr.kP MuKs;s xU tl;lj;jpypUe;J 𝟏𝟐𝟎𝟎 ikaf;Nfhzk; nfhz;l xU tl;lf; Nfhzg;gFjpia ntl;bnaLj;J> mjd; Muq;fis xd;wpizj;J xU $k;ghf;fpdhy;> fpilf;Fk; $k;gpd; tisgug;igf; fhz;f. (Oct-2014)

tl;lf; Nfhzg;gFjpapd; Muk; R = 21 nr.kP

ikaf;Nfhzk; 𝜃 = 1200

tpy;ypd; ePsk; 𝑙 =𝜃

3600 × 2𝜋𝑅

xU tl;lf; Nfhzg;gFjpapd; Muq;fis xd;wpizj;J xU $k;ghf;fpdhy;> $k;gpd; mbr;Rw;wsT = tl;lf; Nfhzg;gFjpapd; tpy;ypd; ePsk;

2𝜋𝑟 = 1200

3600 × 2𝜋 × 21 7

3

𝑟 = 7 nr.kP $k;gpd; rhAauk; 𝑙 = tl;lf; Nfhzg;gFjpapd; Muk; = 21 nr.kP

$k;gpd; tisgug;G = 𝜋𝑟𝑙

=22

7 × 7 × 21 = 462 nr.kP2

khw;W Kiw: $k;gpd; tisgug;G = tl;lf; Nfhzg;gFjpapd; gug;G

=𝜃

3600 × 𝜋𝑅2

7

=1200

3600 × 22

7× 21 × 21 = 462 nr.kP2

3

33. 24 nr.kP cauKila xU $k;gpd; mbg;gug;G 550 nr.kP2 vdpy; mjd; fd msT ahJ? (Oct-15)

$k;gpd; cauk; h = 24 nr.kP

mjd; mbg;gug;G 𝜋𝑟2 = 550 nr.kP2



www.Padasalai.Net



$k;gpd; fd msT = 1

3 𝜋𝑟2ℎ

= 1

3× 𝜋𝑟2 × ℎ

= 1

3× 550 × 24

= 4400 nr.kP3

34. xU cUis tbt ,Uk;Gf; Fohapd; ntspg;Gw tpl;lk; 25 nr.kP mjd; ePsk; 20 nr.kP. kw;Wk;

jbkd; 1 nr.kP. vdpy; mf;Fohapd; nkhj;jg;Gwg;gug;igf; fhz;f. (Oct-2015)

cUis tbt ,Uk;Gf; Fohapd; ntspg;Gw tpl;lk; D = 25 nr.kP

/ ntspg;Gw Muk; R = 12.5 nr.kP

mjd; jbkd; w = 1 nr.kP

/ cl;Gw Muk; r = R − w

= 12.5 – 1

r = 11.5 nr.kP

cUis tbtf; Fohapd; ePsk; (cauk;) h = 20 nr.kP

Fohapd; nkhj;jg;Gwg;gug;G = 2𝜋 𝑅 + 𝑟 (𝑅 − 𝑟 + ℎ) r.m

= 2𝜋 12.5 + 11.5 (12.5 − 11.5 + 20)

= 2 x 22

7x 24 x 21

= 3168 nr.kP2

35. xU jpz;k Neh;tl;l cUisapd; tisgug;G kw;Wk; mbr;Rw;wsT KiwNa 4400 r.nr.kP. kw;Wk; 110 nr.kP vdpy; mt;TUisapd; cauj;ijAk;> tpl;lj;ijAk; fhz;f. (Apr-2016)

cUisapd; tisgug;G = 4400r.nr.kP.

xU jpz;k Neh;tl;l cUisapd; mbr;Rw;wsT 2𝜋𝑟 = 110 nr.kP

2 ×22

7× 𝑟 = 110

tpl;lk; 2𝑟 =110×7

22=35cm

tisgug;G 2𝜋𝑟ℎ = 110 × ℎ = 4400

vdNt> cUisapd; cauk; ℎ =4400

110= 40 nr.kP

36. 𝟏𝟖𝟎°ikaf; NfhzKk; 21 nr.kP. MuKk; nfhz;l tl;lNfhz tbtpyhd ,Uk;Gj; jfl;bd; Muq;fis ,izj;J xU $k;G cUthf;fg;gLfpwJ vdp;y; mf;$k;gpd; Muj;ijf; fhz;f.

(Apr-2016) 180°ikaf; NfhzKk; 21 nr.kP. MuKk; nfhz;l tl;lNfhz tbtpyhd ,Uk;Gj; jfl;bd; Muq;fis ,izj;J xU $k;G cUthf;fg;gLfpwJ. R ;$k;gpd; Muk; vd;f.



www.Padasalai.Net



$k;gpd; mbr;Rw;wsT = tl;ltpy;ypd; ePsk;

2𝜋𝑅 =𝜃

360× 2𝜋𝑟

𝑅 =180

360× 21 = 10.5 nr.kP.

11. Gs;spapay;

1. 43, 24, 38, 56, 22, 39, 45 Mfpa Gs;sp tptuq;fspd; tPr;R kw;Wk; tPr;Rf; nfO fhz;f. (Apr-2012, 2015)

L = 56, S = 22

tPr;R = L−S = 56 −22 = 34

tPr;Rf; nfO = 𝐿−𝑆

𝐿+𝑆 =

56−22

56+22

= 34

78 =

17

39= 0.435

2. xU tFg;gpYs;s 13 khzth;fspd; vil (fp.fp) gpd;tUkhW: 42.5, 47.5, 48.6, 50.5, 49, 46.2, 49.8, 45.8,

43.2, 48, 44.7, 46.9, 42.4. ,tw;wpd; tPr;R> tPr;Rf;nfO fhz;f.

L= 50.5, S= 42.4

tPr;R = L − S

= 50.5 − 42.4 = 8.1


𝐿+𝑆 =

50.5−42.4

50.5+42.4=

8.1

92.9 =

81

929= 0.0871

3. gpd;tUk; kjpg;GfSf;F tPr;R kw;Wk; tPr;Rf; nfO fhz;f. 59, 46, 30, 23, 27, 40, 52,35, 29.(Oct-2014)

L = 59, S = 23

tPr;R = L − S

= 59 − 23

= 36


𝐿+𝑆 =

59−23

59+23

= 36

82 =

18

41= 0.4390

4. gpd;tUk; kjpg;GfSf;F tPr;R kw;Wk; tPr;Rf; nfO fhz;f. 41.2, 33.7, 29.1, 34.5, 25.7, 24.8, 56.5, 12.5.

L= 56.5, S=12.5

tPr;R =L − S

= 56.5 − 12.5

= 44


𝐿+𝑆 =

56.5−12.5

56.5+12.5=

44

69= 0.6376

5. xU Gs;sp tptuj;jpd; kPr;rpW kjpg;G 12. mjd; tPr;R 59 vdpy; mg;Gs;sp tptuj;jpd; kPg;ngU kjpg;igf; fhz;f. tPr;R = 59, S = 12

tPr;R = L − S



www.Padasalai.Net



59 = L − 12

59+12 = L

71 = L

L = 71

6. 50 msTfspy; kpfg;nghpa kjpg;G 3.84 fp.fp. mjd; tPr;R 0.46 fp.fp. vdpy;> mitfspd; kPr;rpW

kjpg;igf; fhz;f. (Apr-2013)

L = 3.84, tPr;R = 0.46

tPr;R = L − S

0.46 = 3.84 –S

S = 3.84 – 0.46

S = 3.38 fp.fp.

7. xU Gs;sp tptuj; njhFg;gpd; kPg;ngU kjpg;G 7.44 kw;Wk; tPr;R 2.26 vdpy;> mj;njhFg;gpd; kPr;rpW kjpg;igf; fhz;f. (Oct-2012)

L = 7.44, tPr;R = 2.26

tPr;R = L − S

2.26 = 7.44 –S

S = 7.44 – 2.26

S = 5.18

8. fz;lwpe;j Gs;sp tptuj; njhFg;gpYs;s 20 kjpg;Gfspd; jpl;ltpyf;fk; 𝟓 vd;f. Gs;sp tptuj;jpd;

xt;nthU kjpg;igAk; 2 My; ngUf;fpdhy; fpilf;Fk; Gjpa Gs;sp tptuq;fspd; jpl;ltpyf;fk; kw;Wk; tpyf;f th;f;fruhrhp fhz;f.

jpl;ltpyf;fk; = 5

Gjpa jpl;ltpyf;fk; = 2 × 5 = 2 5

tpyf;f th;f;fruhrhp = (2 5)2 = 4 × 5 = 20

9. Kjy; 13 ,ay; vz;fspd; jpl;l tpyf;fj;ijf; fzf;fpLf. (Oct-2013, Jun-2015)

𝑛 = 13

jpl;ltpyf;fk; = 𝑛2−1

12 =

132−1

12 =

169−1

12=

168

12 = 14 = 3.7

10. Kjy; 10 ,ay; vz;fspd; jpl;l tpyf;fj;ijf; fzf;fpLf. (Apr-2014, Jun-2014)

𝑛 = 10

jpl;ltpyf;fk; = 𝑛2−1

12 =

102−1

12 =

100−1

12 =

99

12 = 2.87

11. xU Gs;sp tptuj;jpd; khWghl;Lf; nfO 57 kw;Wk; jpl;ltpyf;fk; 6.84 vdpy; mjd;; $l;;Lruhrhp fhz;f. (Jun-2012, Jun-2013, Apr-2016)

C.V. = 57, 𝜎 = 6.84

𝐶. 𝑉. = 𝜎

𝑥 × 100

57 = 6.84

𝑥 × 100

𝑥 = 6.84

57 × 100



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𝑥 = 6.84

57 = 12

𝑥 = 12

12. xU FOtpy; 100 Ngh; cs;sdh;. mth;fspd; cauq;fspd; $l;Lruhrhp 163.8 nr.kP. kw;Wk; khWghl;Lf;

nfO 3.2 vdpy; jpl;ltpyf;fk; fhz;f. 𝑥 = 163.8 C.V. = 3.2, 𝜎= ?

C. V. = 𝜎

𝑥 × 100

3.2 = 𝜎

163.8 × 100

𝜎 =3.2 x 163.8

100= 15.24

𝜎 = 5.24

13. xU Gs;sp tptuj;jpy; 30 kjpg;Gfspd; $l;Lruhrhp kw;Wk; jpl;ltpyf;fk; KiwNa 18 kw;Wk; 3 MFk;. mtw;wpd; $l;Lj;njhifiaAk;> NkYk; mtw;wpd; th;f;fq;fspd; $l;Lj;njhifiaAk; fhz;f. 𝑛 = 30, 𝑥 = 18, 𝜎 = 3 ∑𝑥 = n × 𝑥 = 30 × 18 = 540

𝜎2 = ∑𝑥2

𝑛 −

∑𝑥

𝑛

2 = 32

∑𝑥2

𝑛 − 182 = 9

∑𝑥2

30 − 324 = 9

∑𝑥2 – 9720 = 270 ∑𝑥2 = 270 + 9720 = 9990

14. 𝒏 = 𝟏𝟎, 𝒙 = 𝟏𝟐 kw;Wk; ∑ 𝒙𝟐 = 𝟏𝟓𝟑𝟎 vdpy;> khWghl;Lf; nfOitf; fzf;fpLf.

𝑛 = 10, 𝑥 = 12, ∑𝑥2 = 1530

jpl;ltpyf;fk; 𝜎 = ∑𝑥2

𝑛−

∑𝑥

𝑛

2 =

1530

10− 122

= 153 − 144 = 9 = 3

khWghl;Lf; nfO C.V. = 𝜎

𝑥 × 100

= 3

12 × 100

= 1

4 × 100 = 25

C.V. = 25

15. xU Gs;sp tptuj; njhFg;gpYs;s 100 kjpg;Gfspd; ruhrhp kw;Wk; jpl;ltpyf;fk; KiwNa 48 kw;Wk; 10 MFk;. midj;J kjpg;Gfspd; $l;Lj;njhif kw;Wk; mitfspd; th;f;fq;fspd; $l;Lj;njhif Mfpatw;iwf; fhz;f.

100 kjpg;Gfspd; ruhrhp 𝑥 = 48

100 kjpg;Gfspd; $l;Lj;njhif ∑ 𝑥 = 48 × 100 = 4800

jpl;ltpyf;fk; 𝜎 = 10

vdNt> tpyf;f th;f;fruhrhp 𝜎2 = ∑𝑥2

𝑛−

∑𝑥

𝑛

2= 102 = 100

∑𝑥 = 540 ∑𝑥2 = 9990



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∑𝑥2

𝑛−

4800

100

2 = 100 ⟹

∑𝑥2

𝑛 − (48)2 = 100

∑𝑥2

𝑛 − 2304 = 100 ⟹

∑𝑥2

𝑛 = 100 + 2304 = 2404

100 kjpg;Gfspd; th;f;fq;fspd; $l;Lj;njhif ∑𝑥2 = 100 × 2404 = 2,40, 400

12. epfo;jfT

1. xU Fwpg;gpl;l ehspy; kio ngWtjw;fhd epfo;jfT 0.76 mf;Fwpg;gpl;l ehspy; kio tuhky; ,Uf;f epfo;jfT ahJ?

P(A) = 0.76

P A’ = 1− P (A)

= 1− 0.76

P A’ = 0.24

2. Kjy; 20 ,ay; vz;fspypUe;J xU KO vz; rktha;g;G Kiwapy; Njh;e;njLf;fg;gLfpwJ. me;j vz; gfh vz;zhf ,Uf;f epfo;jfT fhz;f. (Apr-2012)

S = {1, 2, 3,……..20

n(S) = 20

gfh vz;zhf ,Uf;Fk; epfo;r;rp A = {2, 3, 5, 7, 11, 13, 17, 19}

n(A) = 8

P(A) = 𝑛(𝐴)

𝑛(𝑆) =

8

20 =

2

5

P(A) = 2

5

3. 35 nghUl;fs; mlq;fpa njhFg;G xd;wpy; 7 nghUl;fs; FiwghLilad. mjpypUe;J xU nghUs; rk tha;g;G Kiwapy; Njh;e;njLf;f> mJ Fiwghlw;w nghUshf ,Uf;f epfo;jfT ahJ?

(Jun-12, Apr-13)

n(S) = 35

n(A) = 28 ( 35 −7= 28)

P(A) = 𝑛(𝐴)

𝑛(𝑆) =

28

35 =

4

5

P(A) = 4

5

4. rktha;g;G Kiwapy; Njh;e;njLf;fg;gLk; nel;lhz;by; 53 nts;spf;fpoikfs; ,Uf;f epfo;jfT ahJ?

S = {(QhapW> jpq;fs;)> (jpq;fs;> nrt;tha;)> (nrt;tha;> Gjd;)> (Gjd;> tpahod;)> (tpahod;> nts;sp)>

(nts;sp> rdp)> (rdp>QhapW)}

n(S) = 7

A ={(tpahod;> nts;sp)> (nts;sp> rdp)}

n(A) = 2

P(A) = 𝑛(𝐴)

𝑛(𝑆) =

2

7

P(A) = 2

7



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5. rktha;g;G Kiwapy; Njh;e;njLf;fg;gLk; nel;lhz;by; 52 nts;spf;fpoikfs; ,Uf;f epfo;jfT ahJ? S = {(QhapW> jpq;fs;)> (jpq;fs;> nrt;tha;)> (nrt;tha;> Gjd;)> (Gjd;> tpahod;)> (tpahod;> nts;sp)>

(nts;sp> rdp)> (rdp>QhapW)}

n(S) = 7

A = {(QhapW> jpq;fs;)> (jpq;fs;> nrt;tha;)> (nrt;tha;> Gjd;)> (Gjd;> tpahod;)> (rdp>QhapW)}

n(A) = 5

P(A) = 𝑛(𝐴)

𝑛(𝑆) =

5

7

P(A) = 5

7

6. rktha;g;G Kiwapy; Njh;e;njLf;fg;gLk; rhjhuz Mz;by; 53 nts;spf;fpoikfs; ,Uf;f epfo;jfT ahJ?

S = {QhapW> jpq;fs;> nrt;tha;> Gjd;> tpahod;> nts;sp> rdp}

n(S) = 7

A= { nts;sp }

n(A) = 1

P(A) = 𝑛(𝐴)

𝑛(𝑆) =

1

7

P(A) = 1

7

7. xU rPuhd gfil ,uz;L Kiw cUl;lg;gLfpwJ. Kf vz;fspd; $Ljy; 9 fpilf;f epfo;jfT ahJ? (Oct-2012, 2014)

𝑆 = { 1,1 , 1,2 , 1,3 , 1,4 , 1,5 , 1,6 ,

2,1 , 2,2 , 2,3 , 2,4 , 2,5 , 2,6 ,

3,1 , 3,2 , 3,3 , 3,4 , 3,5 , 3,6 ,

4,1 , 4,2 , 4,3 , 4,4 , 4,5 , 4,6 ,

5,1 , 5,2 , 5,3 , 5,4 , 5,5 , 5,6 ,

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

n(S) = 36

A = {(3, 6), (4, 5), (5, 4), (6, 3)}

n(A) = 4

P(A) = 𝑛(𝐴)

𝑛(𝑆) =

4

36 =

1

9

P(A) = 1

9

8. 12 ey;y Kl;ilfSld; 3 mOfpa Kl;ilfs; fye;Js;sd. rktha;g;G Kiwapy; Njh;e;njLf;fg;gLk; xU Kl;il mOfpajhf ,Uf;f epfo;jfT ahJ?

n(S) = 15

n(A) = 3



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P(A) = 𝑛(𝐴)

𝑛(𝑆)

= 3

15 =

1

5

P(A) = 1

5

9. ,U ehzaq;fis xNu rkaj;jpy; Rz;Lk; NghJ> mjpfgl;rk; xU jiy fpilf;f epfo;jfT fhz;f.

(Jun-2015)

S = {HH, HT, TH, TT}

n(S) = 4

A = {TT, HT, TH}

n(A) = 3

P(A) = 𝑛(𝐴)

𝑛(𝑆) =

3

4

P(A) = 3

4

10. mh;n[z;bdh> gq;fshNj\;> rPdh> mq;Nfhyh> U\;ah> my;[Phpah Mfpa ehLfspd; ngah;fisf; nfhz;l gl;baypypUe;J xU Rw;Wyhg; gazp rktha;g;G Kiwapy; xU ehl;bd; ngaiuj; Njh;e;njLf;fpwhh;. ‘m’ vd;w vOj;jpy; Muk;gkhFk; ehl;bd; ngaiuj; Njh;e;njLf;f epfo;jfT vd;d?

n(S) = 6

n(A) = 3

P(A) = 𝑛(𝐴)

𝑛(𝑆) =

3

6 =

1

2

P(A) = 1

2

11. ed;F fiyj;J itf;fg;gl;l 52 rPl;Lfisf; nfhz;l rPl;Lf; fl;bypUe;J rk tha;g;Gr; Nrhjid Kiwapy; xU rPl;L vLf;fg;gLfpwJ. me;j rPl;L gpd;tUtdthf ,Uf;f epfo;jfTfisf; fhz;f. m) fUg;G ,uhrh M) ];NgL fhh;L. (Oct-2013)

n(S) = 52 fUg;G ,uhrh n(A) = 2

P(A) = 𝑛(𝐴)

𝑛(𝑆) =

2

52 =

1

26

n(S) = 52 ];NgL fhh;L n(B) = 13

P(B) = 𝑛(𝐴)

𝑛(𝑆) =

13

52 =

1

4

12. ed;F fiyj;J itf;fg;gl;l 52 rPl;Lfisf; nfhz;l rPl;Lf; fl;bypUe;J rk tha;g;Gr; Nrhjid Kiwapy; xU rPl;L vLf;fg;gLfpwJ. me;j rPl;L gpd;tUtdthf ,Uf;f epfo;jfTfisf; fhz;f. m) ,uhrh M) lakz;l; 10.

n(S) = 52 ,uhrh n(A) = 4

P(A) = 𝑛(𝐴)

𝑛(𝑆) =

4

52 =

1

13

n(S) = 52 lakz;l; 10 n(B) = 1

P(B) = 𝑛(𝐵)

𝑛 (𝑆) =

1

52



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13. %d;W gfilfs; xNu Neuj;jpy; cUl;lg;gLk;NghJ %d;W gfilfspYk; xNu vz; fpilg;gjw;fhd epfo;jfT ahJ? (Apr-2014)

S = 1, 1, 1 , 1, 1, 2 , …….. 6, 6, 6 n(S) = 216 A = {(1, 1, 1), (2, 2, 2), (3, 3, 3), (4, 4, 4), (5, 5, 5), (6, 6, 6)} n(A) = 6

P(A) = 𝑛(𝐴)

𝑛(𝑆) =

6

216 =

1

36

14. %d;W ehzaq;fs; xNu Neuj;jpy; Rz;lg;gLfpd;wd. gpd;tUk; epfo;r;rpfSf;F epfo;jfT fhz;f.

(i) Fiwe;jJ xU jiy fpilg;gJ (ii) ,U G+f;fs; kl;Lk; fpilg;gJ (iii) Fiwe;jJ ,U jiyfs; fpilg;gJ

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} n(S) = 8

(i) Fiwe;jJ xU jiy fpilg;gJ A = {HHH, HHT, HTH, HTT, THH, THT, TTH} n(A) = 7

P(A) = 𝑛(𝐴)

𝑛(𝑆) =

7

8

(ii) ,U G+f;fs; kl;Lk; fpilg;gJ B = {HTT, TTH, THT} n(B) = 3

P(B) = 𝑛(𝐵)

𝑛 (𝑆) =

3

8

(iii) Fiwe;jJ ,U jiyfs; fpilg;gJ C = {HHH, HTH, HHT, THH} n(C) = 4

P(C) = 𝑛(𝐶)

𝑛 (𝑆) =

4

8 =

1

2

P(A) = 7

8 , P(B) =

3

8 , P(C) =

1

2

15. xU tFg;gpy; cs;s 35 khzth;fspy; 20 Ngh; Mz;fs; kw;Wk; 15 Ngh; ngz;fs;. rktha;g;G Kiwapy; xU khzth; Njh;e;njLf;fg;gLfpwhh; vdpy;> gpd;tUk; epfo;r;rpfspd; epfo;jfTfisf; fhz;f. (i) Njh;e;njLf;fg;gLgth; khztdhf ,Uj;jy;> (ii) Njh;e;njLf;fg;gLgth; khztpahf ,Uj;jy;.

tFg;gpy; cs;s khzth;fspd; vz;zpf;if n(S) = 35

(i) khztd; Njh;e;njLf;fg;gl epfo;jfT n(A) = 7

P(A) = 𝑛(𝐴)

𝑛(𝑆) =

20

35 =

4

7

(ii) khztp Njh;e;njLf;fg;gl epfo;jfT n(B) = 15

P(B) = 𝑛(𝐵)

𝑛 (𝑆) =

15

35 =

3

7

16. xU igapy; 5 rptg;G kw;Wk; rpy ePy epwg;ge;Jfs; cs;sd. mg;igapypUe;J xU ePy epwg; ge;ij vLg;gjw;fhd epfo;jfT> xU rptg;G epwg; ge;ij vLg;gjw;fhd epfo;jftpd; %d;W klq;F vdpy;> mg;igapYs;s ePy epwg; ge;Jfspd; vz;zpf;ifiaf; fhz;f. igapy; 5 rptg;G epwg;ge;Jfs;> 𝑥 ePy epwg; ge;JfSk; cs;sd. vdNt> n(S) = 5 + 𝑥 B vd;gJ xU ePy epwg;ge;ij vLf;Fk; epfo;r;rp kw;Wk; R vd;gJ xU rptg;G epwg;ge;ij vLf;Fk; epfo;r;rp vd;f. fzf;fpd;gb P(B) = 3P(R)



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𝑛(𝐵)

𝑛(𝑆) = 3

𝑛 𝑅

𝑛(𝑆)

𝑥

5+𝑥 = 3

5

5+𝑥

⇒ 𝑥 = 3 × 5 = 15

ePy epwg; ge;Jfspd; vz;zpf;if 𝑥 = 15

17. xU rktha;g;Gr; Nrhjidapy; xU epfo;r;rp A vd;f. me;;epfo;r;rpapd; epug;G epfo;r;rp 𝑨 vd;f.

P(A) : P(𝑨 ) = 7:12, vdpy;> P(A)- If; fhz;f.

P(A) : P(𝐴 ) = 7:12 vdNt 𝑃(𝐴)

P(𝐴 ) =

7

12

12 × P(A) = 7× P(𝐴 )

12P(A) = 7[1−P(A) ] = 7 − 7P(A)

19P(A) = 7

P (A) = 7

19

18. A kw;Wk; B vd;gd xd;iwnahd;W tpyf;Fk; epfo;r;rpfs;. NkYk; P(A) = 𝟑

𝟓 kw;Wk; P(B) =

𝟏

𝟓 , vdpy;

P(A∪B)-If; fhz;f. A kw;Wk; B vd;gd xd;iwnahd;W tpyf;Fk; epfo;r;rpfs; vdNt> P(A∪B) = P(A) + P(B)

P(A∪B) = 3

5 +

1

5 =

𝟒

𝟓 [P(A ∩ B) = 0]

19. A kw;Wk; B vd;w ,uz;L epfo;r;rpfspy; P(A) = 𝟏

𝟒 , P(B) =

𝟐

𝟓 kw;Wk; P(A∪B) =

𝟏

𝟐 vdpy; , P(A∩B)-If;

fhz;f. P(A∪B) = P(A) + P(B) – P(A∩B)

P(A∩B) = P(A) +P(B) – P (A∪B) = 1

4 +

2

5−

1

2 =

5+8 − 10

20 =

𝟑

𝟐𝟎

20. xU igapy; 10 nts;is> 6 rptg;G kw;Wk; 10 fUg;G epwg;ge;Jfs; cs;sd. rktha;g;G Kiwapy; xU ge;jpid vLf;Fk;NghJ mJ nts;is my;yJ rptg;G epwg; ge;jhf ,Ug;gjw;fhd epfo;jftpidf; fhz;f. nts;isg; ge;Jfspd; vz;zpf;if n(W) = 10, rptg;Gg; ge;Jfspd; vz;zpf;if n(R) = 6, fUg;Gg; ge;Jfspd; vz;zpf;if n(B) =10 /$Wntsp n(S) = 10+6+10=26 nts;is my;yJ rptg;G epwg;ge;J vLf;f epfo;jfT

P(W∪R) = P(W) + P(R) = 10

26 +

6

26 =

16

26=

8

13 [P(W∩R)= 0]

21. 1 Kjy; 100 tiuapyhd KO vz;fspypUe;J rk tha;g;G Kiwapy; Njh;e;njLf;fg;gLk; xU vz; xU KO th;f;fkhf ,Uf;f epfo;jfT fhz;f.

$Wntsp S = 1, 2, 3,…..100 n(S) = 100

KO th;f;f vz;fs; A = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100} n(A) = 10

P(A) = 𝑛(𝐴)

𝑛(𝑆) =

10

100 =

1

10

22. 1 Kjy; 100 tiuapyhd KO vz;fspypUe;J rk tha;g;G Kiwapy; Njh;e;njLf;fg;gLk; xU vz; xU KO fdkhf ,y;yhky; ,Uf;f epfo;jfT fhz;f. $Wntsp S = 1, 2, 3, …….100



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n (S) = 100 KO fd vz;fs; A = {1, 8, 27, 64}

n(A) = 4

epfo;jfT P(A) = 𝑛(𝐴)

𝑛(𝑆) =

4

100 =

1

25

xU KO fdkhf ,y;yhky; ,Uf;f epfo;jfT P A’ = 1−P(A)

= 1− 1

25 =

25−1

25 =

24

25

23. 20 rPl;Lfspy; 1 Kjy; 20 tiuAs;s KO vz;fs; Fwpf;fg;gl;Ls;sd. rktha;g;G Kiwapy; xU rPl;L

vLf;fg;gLfpd;wJ. mt;thW vLf;fg;gl;l rPl;bYs;s vz; (1) 4-d; klq;fhf ,Uf;f epfo;jfT fhz;f. (2) 6-d; klq;fhf ,y;yhky; ,Uf;f epfo;jfT fhz;f. (Jun-2014, Apr-2016) $Wntsp S = 1, 2, 3, …….20

n(S)= 20 (1) 4 -d; klq;Ffs; A = {4, 8, 12, 16, 20}

n(A) = 5

4-d; klq;fhf ,Uf;f epfo;jfT P(A) = 𝑛(𝐴)

𝑛(𝑆) =

5

20 =

1

4

(2) 6 –d; klq;Ffs; B={6,12,18}

𝑛 𝐵 = 3, 𝑃 𝐵 = 𝑛(𝐵)

𝑛(𝑆)=

3

20

6-d; klq;fhf ,y;yhky; ,Uf;f epfo;jfT 𝑃 𝐵 = 1 − 𝑃 𝐵 = 1 −3

20=

17

20

24. xU igapy; cs;s 1 Kjy; 100 tiuAs;s vz;fshy; Fwpf;fg;gl;l rPl;LfspypUe;J xU rPl;L vLf;fg;gLfpwJ. mt;thW vLf;fg;gLk; rPl;bd; vz; 10 My; tFgLk; vz;zhf ,Ug;gjw;fhd epfo;jftpidf; fhz;f. $Wntsp S = 1, 2, 3, …….100

n(S) = 100 10 My; tFgLk; vz;fs; A = 10, 20, 30,….100

n(A) = 10


𝑛(𝑆) =

10

100 =

1

10

25. ,U gfilfs; xU Nru cUl;lg;gLfpd;wd. Kf vz;fisf; nfhz;L mikf;fg;gLk; <hpyf;f vz; 3 My; tFgLk; vz;zhf ,Ug;gjw;fhd epfo;jfT fhz;f. $Wntsp S = 1,1 , 1,2 , 1,3 ,….. 6,6

n(S) = 36 Kf vz;fisf; nfhz;L mikf;fg;gLk; <hpyf;f vz; 3 My; tFgLk; vz;zhf ,Uf;Fk; epfo;r;rp A vd;f. A = {12, 15, 21, 24, 33, 36, 42, 45, 51, 54, 63, 66} n(A) = 12


𝑛(𝑆) =

12

36 =

1

3

26. ed;F fiyj;J mLf;fp itf;fg;gl;l 52 rPl;Lfisf; nfhz;l rPl;Lf;fl;bypUe;J rk tha;g;G Kiwapy; xU rPl;L vLf;fg;gLfpwJ. me;j rPl;L ];NglhfNth (spade) my;yJ ,uhrhthfNth (king) ,Ug;gjw;fhd epfo;jftpidf; fhz;f. (Jun-2013)

nkhj;j rPl;Lfspd; vz;zpf;if 𝑛 𝑆 = 52 ];NgL rPl;Lfspd; vz;zpf;if 𝑛 𝐴 = 13

epfo;jfT 𝑃 𝐴 = 𝑛(𝐴)

𝑛(𝑆)=

13

52



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,uhrh rPl;Lfspd; vz;zpf;if 𝑛 𝐵 = 13

epfo;jfT 𝑃 𝐵 =𝑛(𝐵)

𝑛 (𝑆)=

4

52

];NgL rPl;lhfTk; ,uhrh rPl;lhfTk; ,Uf;Fk; rPl;Lfspd; vz;zpf;if 𝑛 𝐴 ∩ 𝐵 = 1

epfo;jfT 𝑃 𝐴 ∩ 𝐵 = 𝑛(𝐴∩𝐵)

𝑛 (𝑆)=

1

52

/ rPl;L ];NglhfNth my;yJ ,uhrhfNth ,Ug;gjw;fhd epfo;jfT 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵)

=13

52+

4

52−

1

52=

17−1

52=

16

52=

4

13

27. 2 nr.kP. MuKs;s tl;lj;jpd; cs;Ns cs;s Gs;spfspy; rktha;g;G Kiwapy; xU Gs;spiaj; Njh;T nra;a mJ tl;lj;ij (ghpjp) tpl ikaj;Jf;F mUfpy; ,Uf;f epfo;jfT vd;d? (Oct-15)

2nr.kP. MuKs;s tl;lj;jpd; cs;Ns cs;s Gs;spfs; vd;gJ mt;tl;lj;jpd; gug;gsthFk;. vdNt> $Wntsp 𝑛 𝑆 = 𝜋𝑟2 = 𝜋 × 22 = 4𝜋

Njh;e;njLf;fg;gLk; Gs;sp tl;lj;ij (ghpjp) tpl ikaj;Jf;F mUfpy; ,Uf;f Ntz;Lnkdpy; 1 nr.kP MuKs;s tl;lj;jpd; gFjpf;Fs; mikAk; Gs;spfshFk;. vdNt epfo;r;rpfspd; vz;zpf;if 𝑛 𝐴 = 𝜋𝑟2 = 𝜋 𝑥 12 = 𝜋

epfo;jfT 𝑃 𝐴 =𝑛(𝐴)

𝑛(𝑆)=

𝜋

4𝜋=

1

4



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9. nra;Kiw tbtpay;;;;;;;; tpdhj;jhs; gFg;gha;T

10 kjpg;ngz;

nkhj;jk;

1 10

gy;NtW tbtpay; cUtq;fis Jy;ypakhd msTfspd; cjtpAld; fw;wwpAk; fzpjj;jpd; Xh; gphpNt nra;Kiw tbtpay; MFk;.

tl;lj;Jld; njhlh;Gs;s xU Neh;f;NfhL mt;tl;lj;jpid xNunahU Gs;spapy; kl;Lk; njhLkhdhy;> mf;NfhL tl;lj;jpw;F xU njhLNfhL vdg;gLk;.

njhLNfhL tiuaiw:

glj;jpy; Neh;f;NfhL PQ-Ak; tl;lKk; xNu xU nghJg;Gs;spiaf; nfhz;Ls;sd. mjhtJ> Neh;f;NfhL tl;lj;jpid xNu xU Gs;spapy; njhLfpwJ. Neh;f;NfhL PQ MdJ tl;lj;jpw;F Gs;sp A-y; njhLNfhL vdg;gLk;. A vd;gJ njhLg;Gs;sp. PQ vd;gJ njhLNfhL.

tl;lq;fs; kw;Wk; njhLNfhLfis mbg;gilahff; nfhz;l $w;Wfs;:

1. tl;lj;jpd; VNjDk; xU Gs;spapy; tiuag;gl;lj; njhLNfhL njhLGs;sp topr;nry;Yk; Muj;jpw;Fr; nrq;Fj;jhFk;.

2. tl;lj;jpd; Nky; cs;s xU Gs;spapy; xNu xU njhLNfhL kl;LNk tiua KbAk;. 3. tl;lj;jpw;F ntspNa cs;s Gs;spapypUe;J> mt;tl;lj;jpw;F ,U njhLNfhLfs; tiua KbAk;. 4. tl;lj;jpw;F ntspapYs;s VNjDk; xU Gs;spapypUe;J mt;tl;lj;jpw;F tiuag;gl;l ,U

njhLNfhLfspd; ePsq;fs; rkk;. 5. tpl;lq;fs;> tl;lg;ghpjpapy; 900 ia Vw;gLj;Jk;.

njhLNfhL tiuAk; Kiwfs;: 1. ikaj;ijg; gad;gLj;jp tl;lj;jpd; Nky; cs;s Gs;spapy; njhLNfhL tiujy;. 2. njhLNfhL ehz; Njw;wj;ijg; gad;gLj;jp tiujy;. 3. tl;lj;jpw;F ntspNa cs;s Gs;spapypUe;J njhLNfhLfs; tiujy;.

I. ikaj;ijg; gad;gLj;jp tl;lj;jpd; Nky; cs;s Gs;spapy; njhLNfhL tiujy;:

v.fh. 9.1> gapw;rp:9.1 (1) 1. v.fh. 9.1: 3.2 nr.kP MuKs;s tl;lk; tiuf. tl;lj;jpd; Nky; P vd;w xU Gs;spiaf; Fwpj;J mg;Gs;sp topNa xU njhLNfhL tiuf.

cjtpg;glk;



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tiuKiw:

1. O it ikakhff; nfhz;L

3.2 nr.kP. MuKs;s tl;lk; tiue;Njd;.

2. tl;lj;jpd; Nky; P vd;w Gs;spia Fwpj;J OP tiue;Njd;.

3. OP I L tiu ePl;bg;Gr; nra;Njd;.

4. P I ikakhf itj;J xU

Fwpg;gpl;l mstpy; OL vd;w fjphpy; ,U tpy;fs; ntl;b M,N vdg; ngahpl;Nld;.

5. Nfhl;Lj;Jz;L MN-

w;F ikaf;Fj;J NfhL tiue;Njd;.

6. mf;Nfhl;bw;F TT′

vdg; ngahpl;Nld;. 7. P topNa nry;Yk;

TPT′ Njitahd njhLNfhlhFk;.

gapw;rp Nkw;nfhs;s…. 2. gapw;rp 9.1 (1) 4.2 nr.kP MuKs;s xU tl;lk; tiue;J mt;tl;lj;jpd; Nky; VNjDk; xU Gs;spiaf; Fwpf;f. tl;lj;jpd; ikaj;ijg; gad;gLj;jp mg;Gs;sp topNa njhLNfhL tiuf. II. njhLNfhL – ehz; Njw;wk;:

xU tl;lj;jpYs;s ehz; kw;Wk; mjd; xU Kidapy; mike;j njhLNfhL Mfpatw;wpw;fpilNaAs;s Nfhzk;> ehzpd; xd;W tpl;l tl;lj;Jz;by; mikAk; Nfhzj;jpw;Fr; rkk;.

v.fh. 9.2> gapw;rp 9.1- (2)

3. v.fh. 9.2: 3.2 nr.kP MuKs;s tl;lk; tiuf. tl;lj;jpd; Nky; P vd;w Gs;spiaf; Fwpj;J

mg;Gs;spapy; njhLNfhL - ehz; Njw;wj;ijg; gad;gLj;jp njhLNfhL tiuf.

cjtpg;glk;



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tiuKiw:

1. O it ikakhff; nfhz;L 3.2 nr.kP MuKs;s tl;lk; tiue;Njd;. 2. tl;lj;jpd; Nky; P vd;w Gs;spiaf; Fwpj;Njd;. 3. tl;lj;jpd; Nky; Q,R vd;w Gs;spfis Fwpj;Njd;.

4. PQ, QR, PR I ,izj;Njd;.

5. P -y; ∠QPT = ∠PRQ vd ,Uf;FkhW tiue;Njd;.

6. LM mstpy; PQ y; tpy; ntl;Lk; ,lj;jpypUe;J kw;nwhU tpy; ntl;bNdd;. 7. TP I T′ tiu ePl;bj; Njitahd njhLNfhL T′ PT I tiue;Njd;. 8.TPT′ vd;gJ Njitahd njhLNfhL MFk;.

gapw;rp Nkw;nfhs;s…. 4. gapw;rp: 9.1 (2) 4.8 nr.kP MuKs;s xU tl;lk; tiuf. tl;lj;jpd; Nky; VNjDk; xU Gs;spiaf; Fwp. njhLNfhL - ehz; Njw;wj;ijg; gad;gLj;jp mg;Gs;sp topNa njhLNfhL tiuf. III. tl;lj;jpw;F ntspNa mike;j Gs;spapypUe;J tl;lj;jpw;F njhLNfhl;L Nrhbfs; tiujy;.

v.fh. 9.3> gapw;rp 9.1 , (3)> (4)> (5) 5. v.fh. 9.3: 3 nr.kP MuKs;s tl;lk; tiuf. tl;lj;jpd; ikaj;jpypUe;J 7 nr.kP njhiytpy; xU Gs;spiaf; Fwpj;J mg;Gs;spapypUe;J tl;lj;jpw;F njhLNfhLfs; tiuf. NkYk; njhLNfhLfspd; ePsj;ij mse;J vOJf. (Apr-2014)

rhpg;ghh;j;jy;

nrq;Nfhz ∆𝑂𝑃𝑇 y; 𝑃𝑇 = 𝑂𝑃2 − 𝑂𝑇2 = 72 − 32 = 49 − 9 = 40 𝑃𝑇 = 6.3 nr.kP [Njhuhakhf]

cjtpg;glk;



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tiuKiw:

1. O it ikakhff; nfhz;L 3 nr.kP MuKs;s tl;lk; tiue;Njd;. 2. tl;likak; O-tpypUe;J 7 nr.kP njhiytpy; P vd;w Gs;spiaf; Fwpj;Njd;.

3. OP f;F ikaf;Fj;Jf;NfhL tiue;Njd;. 4.ikaf;Fj;Jf;NfhL OP I ntl;Lk; Gs;spf;F M vdg; ngahpl;Nld;.

5. M I ikakhfTk;> MO it MukhfTk; nfhz;L kw;nwhU tl;lk; tiue;Njd;. 6. mt;tl;lk; Kd;G tiue;j tl;lj;ij T, T′ Gs;spfspy; ntl;LfpwJ.

7. PT, PT’ I ,izj;Njd;. 8. PT, PT’ Njitahd njhLNfhLfshFk;.

gapw;rp Nkw;nfhs;s….

6. gapw;rp 9.1 3) 10 nr.kP tpl;lKs;s xU tl;lk; tiuf. tl;lj;jpd; ikaj;jpypUe;J 13 nr.kP njhiytpy; P vd;w Gs;spiaf; Fwpj;J mg;Gs;spapypUe;J tl;lj;jpw;F PA kw;Wk; PB vd;w njhLNfhLfs; tiue;J mjd; ePsq;fisf; fzf;fpLf. (June-2013, Apr-2015,June-2015)

7. gapw;rp 9.1 - 4). 6 nr.kP MuKs;s xU tl;lk; tiue;J mjd; ikaj;jpypUe;J 10 nr.kP njhiytpYs;s xU Gs;spiaf; Fwpf;f. mg;Gs;spapypUe;J tl;lj;jpw;F njhLNfhLfs; tiue;J mjd; ePsq;fis fzf;fpLf. (Oct-2014)

8. gapw;rp 9.1 5). 3 nr.kP MuKs;s tl;lj;jpd; ikaj;jpypUe;J 9 nr.kP njhiytpy; xU Gs;spiaf; Fwpf;f. mg;Gs;spapypUe;J tl;lj;jpw;F njhLNfhLfs; tiue;J> mjd; ePsq;fis fzf;fpLf. (Oct-2012, June-2013, Oct-2013, Oct-2015, Apr-2016)

cjtpg;glk;



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rhpghh;j;jy;: 𝑃𝐴 = 92 − 32

= 81 − 9

= 72

= 8.4 nr.kP

tiuKiw:

1. O it ikakhff; nfhz;L 3 nr.kP MuKs;s tl;lk; tiue;Njd;. 2. tl;likak; O-tpypUe;J 9 nr.kP njhiytpy; P vd;w Gs;spiaf; Fwpj;Njd ;.

3. OP f;F ikaf;Fj;Jf;NfhL tiue;Njd;. 4. ikaf;Fj;Jf;NfhL OP I ntl;Lk; Gs;spf;F M vdg; ngahpl;Nld;.

5. M I ikakhfTk;> MO it MukhfTk; nfhz;L kw;nwhU tl;lk; tiue;Njd;. 6. mt;tl;lk; Kd;G tiue;j tl;lj;ij T, T′ Gs;spfspy; ntl;LfpwJ.

7. PT, PT’ I ,izj;Njd;. 8. PT, PT’ Njitahd njhLNfhLfshFk;.

Kf;Nfhzq;fs; tiujy;: gf;f msTfisAk;> Nfhzq;fisAk; nfhz;L Kf;Nfhzk; tiuAk; tpjk; ehk; mwpe;jNj. ,g;gFjpapy; ,U Kiwfspy; tl;l gFjpapy; Kf;Nfhzk; tiuAk; tpjj;jpid ehk; mwpe;J nfhs;syhk;. i) mbg;gf;fk;> cr;rpf;Nfhzk; kw;Wk; cr;rpapypUe;J mbg;gf;fj;jpw;F tiuag;gl;l Fj;Jf;Nfhl;bd; ePsk; Mfpa %d;W msTfSk; nfhLf;fg;gl;lhy; ∆ tiuAk; tpjk;. ii) mbg;gf;fk;> cr;rpf;Nfhzk; kw;Wk; cr;rpapypUe;J mbg;gf;fj;jpw;F tiuag;gl;l eLf;Nfhl;bd; ePsk; nfhLf;fg;gl;lhy; ∆ tiuAk; tpjk; Mfpatw;iw fhz;Nghk;. Kjypy; tl;lf;Nfhzg;gFjpia mikf;Fk; tpjj;ij vLj;Jf;fhl;bd; %yk; fhzyhk;.

gapw;rp Nkw;;nfhs;s…. 9. gapw;rp: 9.2 1) AB=5.2 nr.kP ePsKs;s Nfhl;Lj;Jz;bd; kPJ 480 Nfhzk; Vw;gLj;Jk; tl;lg; gFjpia mikf;f.



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10. v.fh: 9.4 AB=6 nr.kP> ∠C = 𝟒𝟎𝟎 kw;Wk; cr;rp C apypUe;J AB-f;F tiuag;gl;l Fj;Jf;Nfhl;bd; ePsk; 4.2 nr.kP nfhz;l ∆ABC tiuf. (Apr-2013, June-2013, June-2014)

tiuKiw:

1. AB=6 nr.kP ePsKs;s xU Nfhl;Lj;Jz;L tiue;Njd;.

2.∠BAX=500 ∠BAY =400,Uf;FkhW ,U fjph;fs; tiue;Njd;. 3. ABf;F ikaf;Fj;J NfhL tiue;Njd;. 4. mf;NfhL AX I O y; ntl;LfpwJ.

5. O it ikakhfTk;> OA I MukhfTk; nfhz;L xU tl;lk; tiue;Njd;. 6. tl;lg;gFjp ACB vd;gJ Nfhzk; 400 I nfhz;bUf;Fk;. 7. ikaf;Fj;Jf;NfhL MO tpy; MH = 4.2 nr.kP ,Uf;FkhW xU tpy; tiue;Njd;.

8.ABf;F ,izahf CHC′ tiue;Njd;. mJ tl;lj;ij C kw;Wk; C’ y; re;jpf;Fk;. 9. AC, BC I ,izj;Njd;. 10. ABC Njitahd Kf;Nfhzk; MFk;.

cjtpg;glk;

90° − 40° = 50°

∠𝐵𝐴𝑋 = 50°

40°



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gapw;rp Nkw;;nfhs;s….

11. gapw;rp 9.2 (2) ∆PQR y; mbg;gf;fk; PQ = 6 nr.kP> ∠𝑅 = 600kw;Wk; cr;rp R-ypUe;J PQ-f;F tiuag;gl;l Fj;Jf;Nfhl;bd; ePsk; 4 nr.kP vd ,Uf;FkhW ∆PQR tiuf.

12. gapw;rp 9.2 3) PQ = 4 nr.kP> ∠𝑅 = 250kw;Wk; cr;rp R-ypUe;J PQ-f;F tiuag;gl;l Fj;Jf;Nfhl;bd;

ePsk; 4.5 nr.kP vd;w msTfs; nfhz;l ∆PQR tiuf.

13. v.fh. 9..5 mbg;gf;fk; BC = 5.5 nr.kP> ∠𝑨 = 𝟔𝟎𝟎kw;Wk; cr;rp A-apypUe;J tiuag;gl;l eLf;NfhL AM d; ePsk; =4.5 nr.kP nfhz;l ∆ABC tiuf. (Apr-2012)

tiuKiw:

1. BC =5.5 nr.kP ePsKs;s xU Nfhl;Lj;Jz;L tiue;Njd;.

2.∠CBX =600 , ∠CBY =600 ,Uf;FkhW fjph;fs; BX, BY; tiue;Njd;. 3. 𝐵𝑌 ⊥ 𝐵𝑋 tiue;Njd;. BCf;F ikaf;Fj;J NfhL tiue;Njd;.

4. mf;NfhL BY I O y; ntl;LfpwJ. BC I M y; ntl;LfpwJ 5. O it ikakhfTk;> OBI MukhfTk; nfhz;L xU tl;lk; tiue;Njd;. 6. tl;lg;gFjp BKC vd;gJ Nfhzk; 600 I nfhz;bUf;Fk;.

7. M I ikakhff; nfhz;L 4.5 nr.kP. MuKs;s ,U tpy;fs; tiue;Njd;. mJ tl;lj;ij A kw;Wk; A′ y; re;jpf;Fk;. 8. AB, AC Mfpatw;iw ,izj;Njd; 9. ABC Njitahd Kf;Nfhzk; MFk;.

cjtpg;glk;

90° − 60° = 30°

∠𝐶𝐵𝑌 = 30°



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14. gapw;rp 9.2 4) ∆ABC y; BC = 5 nr.kP> ∠𝑨 = 𝟒𝟓𝟎 kw;Wk; cr;rp A apypUe;J BC f;F tiuag;gl;l eLf;Nfhl;bd; ePsk; 4 nr.kP vd ,Uf;Fk;gb ∆ABC tiuf.(June-2012, June-2015, Oct-2015)

tiuKiw:

1. BC =5 nr.kP ePsKs;s xU Nfhl;Lj;Jz;L tiue;Njd;.

2.∠CBX =450 ,Uf;FkhW fjph; BX tiue;Njd;. 3. 𝐵𝑌 ⊥ 𝐵𝑋 tiue;Njd;. BCf;F ikaf;Fj;J NfhL tiue;Njd;.

4. mf;NfhL BY I O y; ntl;LfpwJ. BC I M y; ntl;LfpwJ 5. O it ikakhfTk;> OBI MukhfTk; nfhz;L xU tl;lk; tiue;Njd;. 6. tl;lg;gFjp BKC vd;gJ Nfhzk; 450 I nfhz;bUf;Fk;.

7. M I ikakhff; nfhz;L 4nr.kP. MuKs;s ,U tpy;fs; tiue;Njd;. mJ tl;lj;ij A kw;Wk; A′ y; re;jpf;Fk;. 8. AB, AC Mfpatw;iw ,izj;Njd; 9. ABC Njitahd Kf;Nfhzk; MFk;.

cjtpg;glk;

90° − 45° = 45°

∠𝐶𝐵𝑋 = 45°



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15. v.fh: 9.6 BC = 4.5 nr.kP ∠𝑨 = 𝟒𝟎𝟎 kw;Wk; cr;rp A apypUe;J BC f;F tiuag;gl;l eLf;Nfhl;bd; ePsk; AM = 4.7 nr.kP vd ,Uf;Fk;gb ∆ ABC tiuf. NkYk; A-apypUe;J BC f;F tiuag;gl;l Fj;Jf;Nfhl;bd; ePsk; fhz;f.

tiuKiw:

1. BC=4.5 nr.kP msTs;s xU Nfhl;Lj;Jz;L tiue;Njd;. 2. ∠𝐶𝐵𝑋 = 400 , vd ,Uf;Fk; gb 𝐵𝑋 tiue;Njd; 3. 𝐵𝑌 ⊥ 𝐵𝑋 tiue;Njd; 4. 𝐵𝐶 f;F ikaf;Fj;Jf;NfhL tiue;Njd;. mJ 𝐵𝑌 kw;Wk; 𝐵𝐶 fis KiwNa 𝑂 kw;Wk; 𝑀 Gs;spfspy; re;jpf;fl;Lk;

5. 𝑂 it ikakhfTk;> 𝑂𝐵 I MukhfTk; nfhz;L xU tl;lk; tiue;Njd;. mjpy; Gs;sp 𝐾 If; Fwpj;Njd;. nghpa tpy; 𝐵𝐾𝐶 MdJ cr;rpf;Nfhzk; 40° f; nfhz;bUf;Fk;

6. 𝑀I ikakhff; nfhz;L 4.7 nr.kP MuKs;s ,U tpy;fs; tiue;Njd;. mJ tl;lj;ij 𝐴 kw;Wk; 𝐴′ y; re;jpf;Fk; 7. 𝐴𝐵 kw;Wk; 𝐴𝐶 Mfpadtw;iw ,izj;Njd;. ∆𝐴𝐵𝐶 Njitahd Kf;Nfhzk; MFk; 8. 𝐶𝐵 I 𝐶𝑍 tiu ePl;bNdd;

9. 𝐴𝐸 ⊥ 𝐶𝑍 tiue;Njd; 10. Fj;Jf;Nfhl;L 𝐴𝐸 d; ePsk; 3.2 nr.kP

cjtpg;glk;

90° − 40° = 50°

∠𝐶𝐵𝑋 = 50°



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16. gapw;rp 9.2 5) BC = 5 nr.kP> ∠𝑩𝑨𝑪 = 𝟒𝟎𝟎 kw;Wk; cr;rp AapypUe;J BCf;F tiuag;gl;l eLf;Nfhl;bd; ePsk; 6 nr.kP msTfs; nfhz;l ∆ABC tiuf. NkYk; cr;rp AapypUe;J tiuag;gl;l Fj;Jf;Nfhl;bd; ePsk; fhz;f. (Apr -2015)

tiuKiw:

1. BC=5 nr.kP msTs;s xU Nfhl;Lj;Jz;L tiue;Njd;. 2. ∠𝐶𝐵𝑋 = 500 , ∠𝐶𝐵𝑌 = 400 , vd ,Uf;FkhW ,U fjph;fs; tiue;Njd;. 3. 𝐵𝑌 ⊥ 𝐵𝑋 tiue;Njd; 4. 𝐵𝐶 f;F ikaf;Fj;Jf;NfhL tiue;Njd;. mJ 𝐵𝑌 kw;Wk; 𝐵𝐶 fis KiwNa 𝑂 kw;Wk; 𝑀 Gs;spfspy; re;jpf;fl;Lk;

5. O it ikakhfTk;> OB I MukhfTk; nfhz;L xU tl;lk; tiue;Njd;. 6. M ypUe;J 6 nr.kP njhiytpy; tl;lg;ghpjpapy; 𝐴, 𝐴’ Gs;spfspy; ntl;LkhW tiue;Njd;. ;

6. AB, BC I ,izj;Njd;. ABC Njitahd ∆ MFk;. 7. CB I CZ tiu ePl;bg;G nra;Njd;. ;

9. 𝐴𝐸 ⊥ 𝐶𝑍 tiue;Njd; 10. Fj;Jf;Nfhl;L 𝐴𝐸 d; ePsk; 5 nr.kP

cjtpg;glk;

90° − 40° = 50°

∠𝐶𝐵𝑋 = 50



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10. tiuglq;fs; tpdhj;jhs; gFg;gha;T

10 kjpg;ngz;

nkhj;jk;

1 10

tiuglq;fs; vd;gd jfty;fisf; fhl;Lk; glq;fshFk;. ,g;gFjpapy;

i) ,Ugbf;; Nfhitfspd; tiuglq;fs; ii) rpy rpwg;G tif tiuglq;fs;

vd ,uz;L gphpTfs; cs;sd. ,jpy; ,uz;lhk; tif gw;wpa njspthd tpsf;fq;fSk;> tiuglq;fSk; fPNo jug;gl;Ls;sd. ,ij njspthf fw;W gapw;rp ngw;why; tiuglj;jpy; KO kjpg;ngz; ngw ,aYk;.

gy;YWg;Gf; Nfhitapd; khwpfSf;fpilNa cs;s njhlh;G

i) Neh; khWghL 𝑦

𝑥= 𝑘

ii) vjph; khWghL 𝑥𝑦 = 𝑘 vd;wthW mikAk; NghJ> gy;YWg;Gf; Nfhitapd; tiuglj;ij tiuAk; tpjj;ij fw;Wf; nfhs;Nthk;.

Neh;khWghl;bd; tiuglk; xU Neh;f;NfhlhFk;.

vjph;khWghl;bd; tiuglk; Neh;Nfhlw;w tistiuahf mikAk;. ,J nrt;tf mjpgutisak; vdg;gLk;. ,g;gFjpapy; tiuglk; tiua>

ml;ltiz nfhLf;fg;gl;l fzf;Ffs;: v.fh. 10.7> 10.8> gapw;rp: 10.2 (2)> (3)> (6)

ml;ltiz jahhpf;fg;gl Ntz;ba fzf;Ffs;:

v.fh. 10.9> gapw;rp: 10.2 (1)> (4)> (5)

Neh;khWghL tiuglq;fs;: v.fh. 10.7> 10.9> gapw;rp 10.2 (1)> (2)> (3)> (4)

vjph;khWghL tiuglq;fs;: v.fh. 10.8> gapw;rp 10.2 (5)> (6)

1. v.fh. 10.7 fPo;f;fhZk; ml;ltizf;Fj; jFe;j tiuglk; tiue;J> khwpfspd; khWghl;Lj; jd;ikiaf; fhz;. mk;khWghl;bd; khwpypiaAk;. fhz;f

𝒙 2 3 5 8 10 𝒚 8 12 20 32 40

NkYk; 𝒙 = 𝟒 vdpy; 𝒚d; kjpg;igf;fhz;f.

Gs;spfs; (2, 8) (3, 12) (5, 20) (8, 32) (10, 40)

,t;tiuglk; Neh;khWghl;ilf; Fwpf;fpwJ.

/ 𝑦

𝑥= 𝑘



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jPh;T:

𝑘 = 8

2=

12

3=

20

5=

32

8=

40

10= 4

i) Neh;khWghl;bd; khwpyp 4

ii) 𝑥 = 4 vdpy;

𝑦

4= 4 ⇒ 4 4 = 𝑦

/ y=16 , (4, 16)

2. v.fh: 10.8: xU kpjptz;b Xl;Lgth; A vd;w ,lj;jpypUe;J B vd;w ,lj;jpw;F xU rPuhd Ntfj;jpy; xNu topapy; ntt;NtW ehl;fspy; gazk; nra;fpwhh;. mth; gazk; nra;j Ntfk;> mj;J}uj;jpidf; flf;f vLj;Jf;nfhz;l Neuk; Mfpadtw;iwg; gw;wpa tptuq;fs; (Ntf-fhy) gpd;tUk; ml;ltizapy; nfhLf;fg;gl;Ls;sd.

Ntfk; (fp.kP / kzp) 𝒙 2 4 6 10 12 Neuk; (kzpapy;) 𝒚 60 30 20 12 10

Ntf-fhy tiuglk; tiue;J mjpypUe;J i) mth; kzpf;F 5 fp.kP. Ntfj;jpy; nrd;why; J}uj;ijf; flf;f MFk; gaz Neuk;. ii) mth; ,f;Fwpg;gpl;l J}uj;ij 40 kzp Neuj;jpy; flf;f ve;j Ntfj;jpy; gazpf;f Ntz;Lk;

Mfpatw;iwf; fhz;. (Oct-2014) jPh;T: Gs;spfs; (2,60) (4,30) (6,20) (10,12) (12,10) ,t;tiuglk; vjph; khWghl;ilf; Fwpf;fpwJ 𝑥𝑦 = 𝑘

𝑘 = 2 × 60 = 4 × 30 = 6 × 20 = 10 × 12 = (12 × 10)

𝑘 = 120 i) 𝑥 = 5 vdpy; 5 × 𝑦 = 120

𝑦 =120

5= 24 kzp

ii) 𝑦 = 40 vdpy; 𝑥 × 40 = 120

⇒ 𝑥 =120

40= 3 fp.kP / kzp MFk;.



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3. gapw;rp: 10.2 2). thq;fg;gl;l Nehl;Lg; Gj;jfq;fspd; vz;zpf;if kw;Wk; mjw;fhd tpiy tptuk; gpd;tUk; ml;ltizapy; jug;gl;Ls;sJ.

Nehl;Lg;Gj;jfq;fspd; vz;zpf;if 𝒙 2 4 6 8 10 12 tpiy 𝒚 30 60 90 120 150 180

,jw;fhd tiuglk; tiue;J mjd; %yk; i) VO Nehl;Lg;Gj;jfq;fspd; tpiyiaf; fhz;f. ii) 165 f;F thq;fg;gLk; Nehl;Lg;Gj;jfq;fspd; vz;zpf;ifiaf; fhz;f. (Oct-2015) jPh;T: Gs;spfs; (2, 30) (4, 60) (6, 90) (8, 120) (10, 150)(12, 180)

𝑦

𝑥= 𝑘

𝑘 =30

2=

60

4=

90

6=

120

8=

150

10=

180

12 = 15

𝑘 = 15 ,t;tiuglk; Neh;khWghl;ilf; Fwpf;fpwJ. i) 𝑥 = 7 vdpy;

𝑦

7= 15

⇒ 𝑦 = 15 × 7 ⇒ 𝑦 = 105 (7,105)

ii) 𝑦 = 165 vdpy; 165

𝑥= 15

⇒ 15𝑥 = 165,

𝑥 =165

15= 11

(11,165)

4. gapw;rp 10.2 3)

Nkw;fz;l ml;ltizapy; cs;s tptuj;jpw;F tiuglk; tiue;J> mjd;%yk; i) 𝒙 = 𝟒 vdpy; 𝒚-d; kjpg;igf; fhz;f. ii) 𝒚 = 𝟏𝟐 vdpy; 𝒙-d; kjpg;igf; fhz;f. (Apr-2012, June-2012, Apr-2013) jPh;T: Gs;spfs; (1, 2) (3, 6) (5, 10) (7, 14) (8, 16)

𝑦

𝑥= 𝑘

𝑘 =2

1=

6

3=

10

5=

14

7=

16

8= 2

𝑘 = 2 ,t;tiuglk; Neh;khWghl;ilf; Fwpf;fpwJ. i) 𝑥 = 4 vdpy;

𝑦

4= 2 ⇒ 𝑦 = 8

(4,8) ii) 𝑦 = 12 vdpy;

12

𝑥= 2 ⇒ 2𝑥 = 12, 𝑥 =

12

2= 6

(6,12)

𝒙 1 3 5 7 8 𝒚 2 6 10 14 16



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5. gapw;rp 10.2 6)

Ntiyahl;fspd; vz;zpf;if 𝒙 3 4 6 8 9 16 ehl;fspd; vz;zpf;if 𝒚 96 72 48 36 32 18

ml;ltizapy; nfhLf;fg;gl;Ls;s tptuj;jpw;fhd tiuglk; tiuf. mjd;%yk; 12 Ntiyahl;fs; mt;Ntiyia KOtJkhf nra;J Kbf;f MFk; ehl;fspd; vz;zpf;ifiaf; fhz;f. (Apr-2016) jPh;T: Gs;spfs; (3, 96) (4, 72) (6, 48) (8, 36) (9, 32) (16, 18)

𝑥𝑦 = 𝑘 𝑘 = (3 × 96) = (4 × 72) = (6 × 48)

= (8 × 36) = (9 × 32) = (16 × 18) 𝑘 = 288

,t;tiuglk; vjph;khWghl;ilf; Fwpf;fpwJ. 𝑥 = 12 vdpy; 𝑦 =? 12𝑥𝑦 = 288

𝑦 =288

12= 24 (12, 24)

ml;ltiz jahhpf;fg;gl Ntz;ba fzf;Ffs;: 6. v.fh: 10.9: xU tq;fp> %j;j Fbkfdpd; itg;Gj;njhiff;F 10% jdptl;b jUfpwJ.

itg;Gj;njhiff;Fk; mjw;F Xh; Mz;Lf;Ff; fpilf;Fk; tl;bf;Fk; ,ilNaahd njhlh;gpidf; fhl;l xU tiuglk; tiuf. mjd;%yk;> i) 650 itg;Gj; njhiff;Ff; fpilf;Fk; tl;b kw;Wk; ii) 45 tl;bahff; fpilf;f tq;fpapy; nrYj;jg;gl Ntz;ba itg;Gj;njhif Mfpatw;iwf; fhz;f. (Oct-2013, Apr-2015)

jPh;T: ml;ltiz

itg;Gj; njhif 𝑥 100 200 300 400 500 600 700 jdptl;b 𝑦 10 20 30 40 50 60 70

ml;ltizapypUe;J 𝑦 =1

10𝑥

,jd; tiuglk; xU Neh;f;NfhL MFk;. vdNt ,J Neh;khWghl;il Fwpf;Fk;. i) & 650 itg;Gj; njhiff;F chpa tl;bj;njhif & 65

𝑦 =650

10= 65

(650, 65) ii) & 45 tl;bahff; fpilf;f Njitahd itg;Gj;njhif & 450 45 =

𝑥

10⇒ 450 = 𝑥

⇒(450, 45)



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7. gapw;rp: 10.2 1). xU NgUe;J kzpf;F 40 fp.kP Ntfj;jpy; nry;fpwJ. ,jw;Fhpa J}u-fhy

njhlh;gpw;fhd tiuglk; tiuf. ,ijg;gad;gLj;jp 3 kzp Neuj;jpy; ,g;NgUe;J gazpj;j J}uj;ijf; fz;Lgpb. (June-2013,June-2014)

jPh;T: ml;ltiz

Neuk; (kzp) 𝑥 1 2 3 4 5 J}uk; (fp.kP) 𝑦 40 80 120 160 200

Gs;spfs;: (1,40) (2,80) (3,120) (4,160) (5,200)

𝑦

𝑥= 𝑘

⇒40

1=

80

2=

120

3=

160

4=

200

5= 40

k=40 ,t;tiuglk; Neh;khWghl;ilf; Fwpf;fpwJ. 𝑥 = 3 vdpy;

𝑦

3= 40

𝑦=120fp.kP (3,120)

8. gapw;rp: 10.2 4). xU ypl;lh; ghypd; tpiy 15 vd;f. ghypd; msTf;Fk; tpiyf;Fk; cs;sj;

njhlh;gpidf; fhl;Lk; tiuglk; tiuf. mjidg; gad;gLj;jp> i) tpfpj rk khwpypiaf; fhz;f. ii) 3 ypl;lh; ghypd; tpiyiaf; fhz;f. (Apr -2014) jPh;T: ml;ltiz:

ghypd; msT 𝑥 1 2 3 4 5 tpiy 𝑦 15 30 45 60 75

Gs;spfs;: (1,15) (2,30) (3,45) (4,60) (5,75)

𝑦

𝑥= 𝑘

,t;tiuglk; Neh;khWghL nfhz;lJ. 15

1=

30

2=

45

3=

60

4=

75

5= 15

(i) tpfpjrk khwpyp 𝑘 = 15

(ii) 3 ypl;lh; ghypd; tpiy 𝑦

3= 15

⇒ 𝑦 = 45 > = & 45 (3,45)



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9. gapw;rp: 10.2 5) 𝒙𝒚 = 𝟐𝟎, 𝒙, 𝒚 > 0 vd;gjd; tiuglk; tiuf. mjidg;gad;gLj;jp 𝒙 = 𝟓 vdpy; 𝒚d; kjpg;igAk;> 𝒚 = 𝟏𝟎 vdpy; 𝒙d; kjpg;igAk; fhz;f. (Oct-2012, June-2015) jPh;T:

ml;ltiz:

𝑥 1 2 4 5 10 𝑦 20 10 5 4 2

𝑥𝑦 = 20 Gs;spfs; (1,20) (2,10) (4,5) (5,4) (10,2) ,t;tiuglk; vjph;khWghL nfhz;lJ. 𝑘 = 20

𝑥 = 4 vdpy;

4 × 𝑦 = 20 ⇒ 𝑦 = 20

4= 5 (4,5)

𝑦 = 10 vdpy;

𝑥 × 10 = 20 ⇒ 𝑥 =20

10= 2 (2,10)



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epidtpw; nfhs;s Ntz;ba Kf;fpa Fwpg;Gfs; ,ay; 1: fzq;fSk; rhh;GfSk;

ghpkhw;W tpjpfs; : i) 𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴 ii) 𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴 Nrh;g;G tpjpfs; : i) 𝐴 ∪ 𝐵 ∪ 𝐶 = (𝐴 ∪ 𝐵) ∪ 𝐶 ii) 𝐴 ∩ 𝐵 ∩ 𝐶 = (𝐴 ∩ 𝐵) ∩ 𝐶 gq;fPl;L tpjpfs;: i) 𝐴 ∪ 𝐵 ∩ 𝐶 = 𝐴 ∪ 𝐵 ∩ (𝐴 ∪ 𝐶) ii) 𝐴 ∩ 𝐵 ∪ 𝐶 = 𝐴 ∩ 𝐵 ∪ (𝐴 ∩ 𝐶)

fz tpj;jpahrj;jpw;fhd b khh;fdpd; tpjpfs;: i) 𝐴\ 𝐵 ∪ 𝐶 = 𝐴\𝐵 ∩ 𝐴\𝐶

ii) 𝐴\(𝐵 ∩ 𝐶) = (𝐴\𝐵) ∪ (𝐴\𝐶)

fz epug;gpf;fhd b khh;fdpd; tpjpfs;:

i) 𝐴 ∪ 𝐵 ′ = 𝐴′ ∩ 𝐵′ ii) 𝐴 ∩ 𝐵 ′ = 𝐴′ ∪ 𝐵′

fzq;fspd; Nrh;g;gpd; Mjp vz;izf; fz;lwpAk; #j;jpuq;fs;: i) 𝑛 𝐴 ∪ 𝐵 = 𝑛 𝐴 + 𝑛 𝐵 − 𝑛(𝐴 ∩ 𝐵) ii) 𝑛 𝐴 ∪ 𝐵 ∪ 𝐶 = 𝑛 𝐴 + 𝑛 𝐵 + 𝑛 𝐶 − 𝑛 𝐴 ∩ 𝐵 − 𝑛 𝐵 ∩ 𝐶 − 𝑛 𝐴 ∩ 𝐶 + 𝑛(𝐴 ∩ 𝐵 ∩ 𝐶)

,ay; 2: nka;naz;fspd; njhlh;thpirfSk; njhlh;fSk; i) xU $l;Lj; njhlh;thpirapy; 𝑛tJ cWg;G fhZk; tha;ghL 𝑡𝑛 = 𝑎 + (𝑛 − 1)𝑑 ii) xU $l;Lj; njhlh; thpirapy; 𝑛 cWg;Gfs; tiu $Ljy; fhZk; tha;g;ghL

𝑆𝑛 = 𝑛

2 2𝑎 + 𝑛 − 1 𝑑 𝑜𝑟

𝑛

2[𝑎 + 1]

iii) xU ngUf;Fj; njhlh; thpirapy; 𝑛 cWg;Gfs; tiu $Ljy; fhZk; tha;g;ghL

𝑆𝑛 = 𝑎[𝑟𝑛− 1]

𝑟−1=

𝑎[1−𝑟𝑛 ]

1−𝑟 (if 𝑟 ≠ 1),

𝑆𝑛 = 𝑛𝑎 ( if 𝑟 = 1) iv) Kjy; 𝑛 ,ay; vz;fspd; $Ljy;

= 1 + 2 + 3 + … + 𝑛 = 𝑛(𝑛 + 1)

2

v) Kjy; 𝑛 ,ay; vz;fspd; th;f;fq;fspd; $Ljy;

= 12 + 22 + 32 + … + 𝑛2 = 𝑛 𝑛+1 (2𝑛+1)

6

vi) Kjy; 𝑛 ,ay; vz;fspd; fdq;fspd; $Ljy;

= 13 + 23 + 33 + ⋯ + 𝑛3 = 𝑛(𝑛 + 1)

2

2

vii) Kjy; 𝑛 xw;iwg;gil ,ay; vz;fspd; $Ljy; = 1 + 3 + 5 + 7 + … + (2𝑛 − 1) cWg;Gfs; tiu = 𝑛2 viii) Kjy; 𝑛 xw;iwg;gil ,ay; vz;fspd; $Ljy; (filrp cWg;G 𝑙 nfhLf;fg;gl;lhy;)

= 1 + 3 + 5 + 7 + ……….. + 𝑙 = 𝑙+1

2

2

,ay; 3: ,aw;fzpjk; i) 𝑝 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 vd;w gy;YWg;Gf;

Nfhitapy; G+r;rpaq;fspd; $Ljy; =−𝑏

𝑎, G+r;rpaq;fspd;

ngUf;fw;gyd; =𝑐

𝑎

ii) 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 vd;w ,Ugbr; rkd;ghl;bd; %yq;fspd; jd;ikf;fhl;b ∆= 𝑏2 − 4𝑎𝑐 iii) 𝑏2 − 4𝑎𝑐 = 0 vdpy;> %yq;fs; nka;naz;fs;> NkYk; rkkhdit. iv) 𝑏2 − 4𝑎𝑐 > 0 vdpy;> %yq;fs; nka;naz;fs;> Mdhy; rkky;y. v) 𝑏2 − 4𝑎𝑐 < 0 vdpy;> %yq;fs; fw;gid vz;fs;.

𝜶 kw;Wk; 𝜷 Mfpatw;iw nfhz;l rpy KbTfs; gpd;tUkhW jug;gl;Ls;sd.

(i) 𝛼 − 𝛽 = (𝛼 + 𝛽)2 − 4𝛼𝛽 (ii) 𝛼2 + 𝛽2 = [(𝛼 + 𝛽)2 − 2𝛼𝛽 ] (iii) 𝛼2 − 𝛽2 = 𝛼 + 𝛽 𝛼 − 𝛽 =

(𝛼 + 𝛽)[ (𝛼 + 𝛽)2 − 4𝛼𝛽 ] only if 𝛼 ≥ 𝛽 (iv) 𝛼3 + 𝛽3 = (𝛼 + 𝛽)3 − 3𝛼𝛽(𝛼 + 𝛽) (v) 𝛼3 − 𝛽3 = (𝛼 − 𝛽)3 + 3𝛼𝛽(𝛼 − 𝛽) (vi) 𝛼4 + 𝛽4 = 𝛼2 + 𝛽2 2 − 2𝛼2𝛽2 = [(𝛼 + 𝛽)2 − 2𝛼𝛽 ]2 − 2(𝛼𝛽)2 (vii) 𝛼4 − 𝛽4 = 𝛼 + 𝛽 𝛼 − 𝛽 (𝛼2 + 𝛽2)

,ay; 4: mzpfs; i) xU mzpapy; 𝑚 epiufSk; 𝑛 epuy;fSk; cs;sd. vdpy;> mjd; thpir 𝑚 × 𝑛 MFk;. ii) ,uz;L mzpfspd; thpirfs; rkkhf ,Ug;gpd; me;j mzpfisf; $l;lNth my;yJ fopf;fNth ,aYk;. iii) mzp A-apd; thpir 𝑚 × 𝑛 kw;Wk; B-d; thpir 𝑛 × 𝑝 vdpy;> ngUf;fw;gyd; mzp AB d; thpir 𝑚 × 𝑝

iv) nghJthf mzpfspd; ngUf;fy; ghpkhw;Wg; gz;G cilajy;y. mjhtJ AB ≠ BA v) mzpfspd; ngUf;fy; Nrh;g;Gg; gz;G cilaJ. mjhtJ (AB)C = A(BC) vi) (𝐴𝑇)𝑇 = 𝐴, (𝐴 + 𝐵)𝑇 = 𝐴𝑇 + 𝐵𝑇 kw;Wk;

(𝐴𝐵)𝑇 = 𝐵𝑇𝐴𝑇



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,ay; 5: Maj;njhiy tbtpay; i) (𝑥1, 𝑦1), (𝑥2, 𝑦2) Mfpa Gs;spfis ,izf;Fk; Neh;f;Nfhl;Lj; Jz;bd; eLg;Gs;sp

= 𝑥1+𝑥2

2,𝑦1+𝑦2

2

ii) 𝐴(𝑥1, 𝑦1), 𝐵(𝑥2, 𝑦2), kw;Wk; C(𝑥3, 𝑦3). Kf;Nfhzk; 𝐴𝐵𝐶d; eLf;Nfhl;L ikak;

𝐺 = 𝑥1+𝑥2+𝑥3

3,

𝑦1+𝑦2+𝑦3

3

iii) gphpT #j;jpuk; (cl;Gwkhf)= 𝑙𝑥2+𝑚𝑥1

𝑙+𝑚 ,

𝑙𝑦2+𝑚𝑦1

𝑙+𝑚

iv) (𝑥1, 𝑦1), 𝑥2, 𝑦2 , (𝑥3, 𝑦3) Mfpa Gs;spfis cr;rpfshf nfhz;l Kf;Nfhzj;jpd; gug;gsT

= 1

2 {(x1 y2 + x2 y3 + x3 y1) – (x2 y1 +x3 y2 + x1 y3)} r.m

v) (𝑥1, 𝑦1), 𝑥2,𝑦2 , (𝑥3, 𝑦3) kw;Wk; (𝑥4, 𝑦4) Mfpa Gs;spfis cr;rpfshf nfhz;l ehw;fuj;jpd;

gug;gsT = 1

2 {(x1 y2 + x2 y3 + x3y4 + x4 y1)

– (x2 y1 +x3 y2 +x4y3 + x1 y4)} r.m

vi) 𝑥 -mr;rpd; rkd;ghL 𝑦 = 0 vii) 𝑦 -mr;rpd; rkd;ghL 𝑥 = 0 viii) 𝑥 -mr;rpw;F ,izahd Nfhl;bd; rkd;ghL

𝑦 = 𝑘

ix) 𝑦 - mr;rpw;F ,izahd Nfhl;bd; rkd;ghL

𝑥 = 𝑘

x) 𝑎 𝑥 + 𝑏𝑦 + 𝑐 = 0 vd;w Nfhl;bw;F ,izahd Nfhl;bd; rkd;ghL 𝑎𝑥 + 𝑏𝑦 + 𝑘 = 0

xi) 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 vd;w Nfhl;bw;F nrq;Fj;jhd Nfhl;bd; rkd;ghL 𝑏𝑥 − 𝑎𝑦 + 𝑘 = 0 xii) Mjpg;Gs;sp topr; nry;Yk; Nfhl;bd; rkd;ghL 𝑦 = 𝑚𝑥 xiii) rha;T 𝑚 kw;Wk; 𝑦- ntl;Lj; Jz;L 𝑐 nfhz;l Nfhl;bd; rkd;ghL 𝑦 = 𝑚𝑥 + 𝑐 xiv) rha;T 𝑚 nfhz;L 𝑥1, 𝑦1 topahfr; nry;Yk; Nfhl;bd; rkd;ghL 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1) xv) (𝑥1, 𝑦1), (𝑥2, 𝑦2) Mfpa Gs;spfs; topahfr; nry;Yk; Nfhl;bd; rkd;ghL

𝑦−𝑦1

𝑦2−𝑦1=

𝑥−𝑥1

𝑥2−𝑥1

xvi) 𝑥 - ntl;Lj;Jz;L 𝑎 kw;Wk; 𝑦 -ntl;Lj; Jz;L 𝑏

nfhz;l Nfhl;bd; rkd;ghL 𝑥

𝑎+

𝑦

𝑏= 1

xvii) fpilepiyf; Nfhl;bd; rha;T G+r;rpakhFk;. Neh;f;Fj;Jf;Nfhl;bd; rha;T tiuaWf;f ,ayhjJ. xviii) ,U NfhLfspd; rha;Tfs; rkk; vdpy; mf;NfhLfs; xd;Wf;nfhd;W ,izahFk;. xix) Neh;f;Fj;jw;w ,U Neh;f;NfhLfspd; rha;Tfspd; ngUf;fw;gyd; −1 (𝑚1 × 𝑚2 = −1) vdpy;> mf;NfhLfs; xd;Wf;nfhd;W nrq;Fj;jhFk;

,ay; 7: Kf;Nfhztpay; 1) 𝑠𝑖𝑛2𝜃 + 𝑐𝑜𝑠2𝜃 = 1 2) 𝑠𝑖𝑛2𝜃 = 1 − 𝑐𝑜𝑠2𝜃

3) 𝑠𝑖𝑛𝜃 = 1 − 𝑐𝑜𝑠2𝜃

4) 𝑐𝑜𝑠2𝜃 = 1 − 𝑠𝑖𝑛2𝜃

5) cosθ = 1 − 𝑠𝑖𝑛2𝜃

6) 𝑠𝑒𝑐2𝜃 − 𝑡𝑎𝑛2𝜃 = 1 7) 𝑠𝑒𝑐2𝜃 = 1 + 𝑡𝑎𝑛2𝜃

8) 𝑠𝑒𝑐𝜃 = 1 + 𝑡𝑎𝑛2𝜃 9) 𝑡𝑎𝑛2𝜃 = 𝑠𝑒𝑐2𝜃 − 1

10) tanθ = sec2θ − 1

11) 𝑐𝑜𝑠𝑒𝑐2𝜃 = 1 + 𝑐𝑜𝑡2𝜃

12)𝑐𝑜𝑠𝑒𝑐𝜃 = 1 + 𝑐𝑜𝑡2𝜃 13) 𝑐𝑜𝑠𝑒𝑐2𝜃 − 𝑐𝑜𝑡2𝜃 = 1 14) 𝑐𝑜𝑡2𝜃 = 𝑐𝑜𝑠𝑒𝑐2𝜃 − 1

15) cotθ = 𝑐𝑜𝑠𝑒𝑐2𝜃 − 1

,ay; 8: mstpay;

i) xU cUisapd; tisgug;G 𝐶𝑆𝐴 = 2𝜋𝑟ℎ r.m ii) xU cUisapd; nkhj;jg; gug;gsT 𝑇𝑆𝐴 = 2𝜋𝑟(ℎ + 𝑟) r.m iii) xU cUisapd; fd msT 𝑉 = 𝜋𝑟2 ℎ f.m

iv) xU $k;gpd; rhAauk; 𝑙 = 𝑟2 + ℎ2

v) xU $k;gpd; cauk; ℎ = 𝑙2 − 𝑟2

vi) xU $k;gpd; Muk; 𝑟 = 𝑙2 − ℎ2 vii) xU $k;gpd; tisgug;G 𝐶𝑆𝐴 = 𝜋𝑟𝑙 r.m viii) xU $k;gpd; nkhj;jg; gug;gsT 𝑇𝑆𝐴 = 𝜋𝑟(𝑙 + 𝑟) r.m

ix) xU $k;gpd; fd msT 𝑉 =1

3 𝜋𝑟2ℎ f.m



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x) ,ilf;fz;lj;jpd; fd msT 𝑉 =1

3 𝜋ℎ(𝑅2 + 𝑟2 + 𝑅𝑟) f.m

xi) xU Nfhsj;jpd; tisgug;G 𝐶𝑆𝐴 = 4𝜋𝑟2 r.m

xii) xU Nfhsj;jpd; fd msT 𝑉 =4

3 𝜋𝑟3 f.m

xiii) xU miuf;Nfhsj;jpd; tisgug;G 𝐶𝑆𝐴 = 2𝜋𝑟2 r.m xiv) xU miuf;Nfhsj;jpd; nkhj;jg;gug;gsT 𝑇𝑆𝐴 = 3𝜋𝑟2 r.m

xv) xU miuf;Nfhsj;jpd; fd msT 𝑉 = 4

3𝜋𝑟3 f.m

xvi) xU cs;sPlw;w cUisapd; tisgug;G 𝐶𝑆𝐴 = 2𝜋ℎ(𝑅 + 𝑟) r.m xvii) xU cs;sPlw;w cUisapd; nkhj;jg;gug;gsT 𝑇𝑆𝐴 = 2𝜋 (𝑅 + 𝑟) (𝑅 − 𝑟 + ℎ) r.m xviii) xU cs;sPlw;w cUisapd; fd msT 𝑉 = 𝜋ℎ (𝑅 + 𝑟) (𝑅 − 𝑟) f.m

xix) xU cs;sPlw;w Nfhsj;jpd; fd msT

𝑉 = 4

3𝜋(𝑅3 − 𝑟3) f.m

,ay; 11: Gs;spapay; i ) tPr;R = L – S

ii) tPr;Rf;nfO = 𝐿−𝑆

𝐿+𝑆

iii) njhFf;fg;glhj tptuq;fspd; jpl;ltpyf;fk;

1. 𝜎 = ∑𝑥2

𝑛−

∑𝑥

𝑛

2

2. 𝜎 = ∑𝑑2

𝑛 (𝑑 = 𝑥 − 𝑥 )

3. 𝜎 = ∑𝑑2

𝑛−

∑𝑑

𝑛

2

(𝑑 = 𝑥 − 𝐴)

4. 𝜎 = ∑𝑑2

𝑛−

∑𝑑

𝑛

2

× 𝑐 ( 𝑑 =𝑥−𝐴

𝑐 )

iv) njhFf;fg;gl;l tptuq;fspd; jpl;ltpyf;fk;

1. 𝜎 = ∑𝑓𝑑2

∑𝑓 (𝑑 = 𝑥 − 𝑥 ) 2. 𝜎 =

∑ 𝑓𝑑2

∑ 𝑓−

∑𝑓𝑑

∑𝑓

2

(𝑑 = 𝑥 − 𝐴)

3. 𝜎 = ∑𝑓𝑑2

∑𝑓−

∑𝑓𝑑

∑𝑓

2

× 𝑐 ( 𝑑 =𝑥−𝐴

𝑐 )

v) nfhLf;fg;gl;l Gs;sp tptuj;jpy; cs;s xt;nthU vz;ZlDk; (kjpg;G)VNjDk; xU Fwpg;gpl;l vz;izf; $l;bdhNyh my;yJ fopj;jhNyh fpilf;Fk; Gjpa tptuj;jpd;jpl;ltpyf;fk; khwhJ. vi) nfhLf;fg;gl;l tptuj;jpYs;s xt;nthU vz;izAk; (kjpg;G) xU khwpyp k My; ngUf;f my;yJ tFf;f fpilf;Fk; Gjpa kjpg;Gfspd; jpl;l tpyf;fkhdJ> gioa jpl;ltpyf;fj;ij khwpyp k My; ngUf;f my;yJ tFf;f fpilf;Fk; vz;zhf ,Uf;Fk;.

vii) Kjy; n ,ay; vz;fspd; jpl;ltpyf;fk; 𝜎 = 𝑛2−1

12

viii) khWghl;Lf; nfO C. V = 𝜎

𝑥 × 100



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