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gj;jhk; tFg;G

fzpjk;

nky;y fw;Fk; khzth;fSf;fhd nghJj;Njh;T rpwg;G ifNaL

2012 -2013

gs;spf;fy;tpj;Jiw jpUtz;zhkiy khtl;lk;.

Kd;dpiy : nghd;.mUz;gpurhj;> M.A.,M.Ed.,M.Phil., khtl;l Kjd;ikf; fy;tp mYtyh;> jpUtz;zhkiy khtl;lk;.

njhFg;G : rp.RFkhh;> M.A,B.Sc,B.Ed.,

jiyikahrphpah;> muR cah;epiyg;gs;sp > tpis-Muzp-632 316.

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gj;J kjpg;ngz; tpdhf;fs;

9.nra;Kiw tbtpay; 1. 4.8nr.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.

2) 3nr.kP MuKs;s 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 fzf;fpLf.

fzf;fPL (Calculation) : njhLNfhl;Lj;Jz;bd; ePsk; : PT2 = (op)2 – (OT)2 PT2 = (7)2 – ( 3)2

PT2 = (49) – (9) PT2 = 40 PT = 6.3 cm

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3) 5nr.kP MuKs;s MuKs;s tl;lk; tiuf. tl;lj;jpd; ikaj;jpypUe;J 13 nr.kP njhiytpy; xU Gs;spiaf; Fwpj;J mg;Gs;spapypUe;J tl;lj;jpw;F njhLNfhLfs; tiuf .NkYk;

njhLNfhLfspd; ePsj;ij fzf;fpL f. fzf;fPL (Calculation) : njhLNfhl;Lj;Jz;bd; ePsk; : PA2 = (op)2 – (OA)2 PA2 = (13)2 – (5)2

PA2 = (169) – (25) PA2 = 144 PA = 12 cm 4) 6nr.kP MuKs;s MuKs;s tl;lk; tiuf. tl;lj;jpd; ikaj;jpypUe;J 10 nr.kP njhiytpy; xU Gs;spiaf; Fwpj;J mg;Gs;spapypUe;J tl;lj;jpw;F njhLNfhLfs; tiuf .NkYk; njhLNfhLfspd; ePsj;ij fzf;fpLf.

fzf;fPL (Calculation) : njhLNfhl;Lj;Jz;bd; ePsk; : PA2 = (op)2 – (OA)2 PA2 = (10)2 – (6)2

PA2 = (100) – (36) PA2 = 64 PA = 8 cm

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6. AB =6cm ,AC=7cm,BC=6cm , AD=4.2cm msTfs; nfhz;l tl;lehw;fuk; ABCD tiuf .

7. PQ=4cm , QR = 6cm , PR=7.5cm , QS = 7cm msTfs; nfhz;l tl;lehw;fuk; PQRS tiuf .

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8. EF = 5.2cm , ∠GEF=500 ,FG =6cm , ∠ EGH=400 msTfs; nfhz;l tl;lehw;fuk; EFGH tiuf .

9. AB = 6 cm ,∠ABC=800 ,BC =5.5cm , AD=4.5cm msTfs; nfhz;l tl;lehw;fuk; ABCD tiuf .

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10. AB = 5.8 cm , ∠ABD=350 , AD =4.2cm , AB ||CD vd;w msTfs; nfhz;l tl;lehw;fuk; ABCD tiuf .

11. PQ=4cm , ∠𝑷 = 𝟏𝟎𝟎 , ∠ 𝑷𝑸𝑺 =400 , kw;Wk; ∠𝑺𝑸𝑹=700 vd;w msTfs; nfhz;l tl;lehw;fuk; PQRS tiuf .

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10.tiuglq;fs; 1. tiuglk; %yk; jPh;f;f : x2 -2x-3=0 i) ml;ltiz :

x -3 -2 -1 0 1 2 3 4 X2 9 4 1 0 1 4 9 16 -2x 6 4 2 0 -2 -4 -6 -8 -3 -3 -3 -3 -3 -3 -3 -3 -3 Y 12 5 0 -3 -4 -3 0 5

ii) thpir N[hbfs; : (-3,12) (-2,5)(-1,0) (0,-3) (1,-4) (2,-3) (3,0) (4,5) iii) msT jpl;lk; : x-mr;R 1nrkP = 1 myF y-mr;R 1nrkP = 2 myFfs; iv) jPh;T fzk; : { -1 , 3 }

2. tiuglk; %yk; jPh;f;f : 2x2 +x-6=0 i) ml;ltiz :

x -3 -2 -1 0 1 2 3 X2 9 4 1 0 1 4 9

2x2 18 8 2 0 2 8 18 x -3 -2 -1 0 1 2 3 -6 -6 -6 -6 -6 -6 -6 -6 Y 9 0 -5 -6 -3 4 15

ii) thpir N[hbfs; : (-3,9) (-2, 0)(-1,-5) (0,-6) (1,-3) (2, 4) (3, 15) iii) msT jpl;lk; : x-mr;R 1nrkP = 1 myF y-mr;R 1nrkP = 2 myFfs; iv) jPh;T fzk; : { -2 , 1.5 }

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3) y =2x2 –d; tiuglj;ij tiue;J mjpypUe;J 2x2+x-6=0 vd;w rkd;ghl;ilj; jPh;f;f. ml;ltiiz :

X -3 -2 -1 0 1 2 3 x2 9 4 1 0 1 4 9

Y=2x2 18 8 2 0 2 8 18 ii) thpir N[hbfs; : (-3,18) (-2, 8)(-1,2) (0,0) (1, 2) (2, 8) (3, 18) iii) msT jpl;lk; : x-mr;R 1nrkP = 1 myF y-mr;R 1nrkP = 2 myFfs; y =2x2 0=2x2+x-6 Y= -x +6

x -2 0 2 Y=-x+6 8 6 4

iv) jPh;T fzk; : { -2 , 1.5 } 4) Y = x2+3x+2 d; tiuglj;ij tiue;J mjpypUe;J x2+2x+4 = 0 vd;w rkd;ghl;ilj; jPh;f;f. i) ml;ltiz :

X -3 -2 -1 0 1 2 3 X2 9 4 1 0 1 4 9 3x -9 -6 -3 0 3 6 9 2 2 2 2 2 2 2 2 Y 2 0 0 2 6 12 20

ii) thpir N[hbfs; : (-3,2) (-2,0)(-1,0) (0, 2) (1, 6) (2, 12) (3,20) iii) msT jpl;lk; : x-mr;R 1nrkP = 1 myF y-mr;R 1nrkP = 2 myFfs; Y = x2+3x+2 0=x2+2x+4 Y = x - 2

x -2 0 2 Y=-x-2 -4 -2 0

iv) jPh;T fzk; nka; vz;fs; my;y.

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5) fPo;fhZk; ml;ltizf;F j; jFe;j tiuglk; tiue;J khwpfspd; khWghl;Lj;jd;ikia fhz;f. mk;khWghl;bd; khwpypia fhz;f. NkYk; x=4 vdpy; y-d; kjpg;G fhz;f

X 2 3 5 8 10 y 8 12 20 32 40

----------

X kjpg;G mjpfhpf;Fk; NghJ y mjpfhpf;fpwJ .vdNt Neh;khWghL MFk;. mjhtJ y = kx , k >0, khwpyp=k K=8/2= 12/3=20/5=32/8=40/10 vdNt y=4x, vd;w rkd;ghL Neh;f;Nfhl;il mikf;Fk;. iv) jPh;T :

x=4 vdpy; y=16 vd tiuglj;jpd; %yk; mwpayhk;

6) xU tq;fp %j;j Fbkfdp;d; 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; >

1) &.650 itg;Gj; njhiff;Ff; fpilf;Fk; tl;b kw;Wk; 2) &.45 tl;bahf fpilf;f tq;fpapy; nrYj;jg;gl Ntz;ba itg;Gj;njhif Mfpatw;iw fhz;f. ml;ltiz : itg;Gj; njhif (x)

100 200 300 400 500 600 700

jdptl;b

(y) 10 20 30 40 50 60 70

,e;j ml;ltizapypUe;J Neh;khwy; vd;gij mwpayhk; .mjhtJ y = x/10 vd mwpayhk;;. vdNt tiuglj;jpypUe;J i) &.650 itg;Gj; njhiff;F chpa tl;bj; njhif &.65 vdTk; ii) &.45 tl;bahff; fpilf;f Njitahd itg;Gj;njhif &.450 vdTk; mwpayhk; .

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7) fPo;fhZk; ml;ltizapy; cs;s tptuj;jpw;F tiuglk; tiuglk; tiue;J mjd; %yk; i) X= 4 vdpy; y- d; kjpg;igAk; ii) y = 12 vdpy; x-d; kjpg;igAk; fhz;f.

X 1 3 5 7 8

y 2 6 10 14 16

ml;ltizapypUe;J x-d; kjpg;G mjpfhpf;f y-d; kjpg;Gk; mjpfhpg;gij fhzyhk; .vdNt Neh; khWghl;il Fwpf;fpwJ . vdNt y/x = k , K = 2/1 = 6/3=10/5/=14/716/8 = 2 tiuglj;jpypUe;J i) x=4 vDk; NghJ y=8 ii) y=12 vDk; NghJ x = 6

8) 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) tpfpjrk khwpypiaf; fhz;f . ii) 3 ypl;lh; ghypd; tpiyiaf; fhz;f. ypl;lh;fspd;

vz;zpf;if(x) 1 2 3 4 5

tpiy &.(Y)

15 30 45 60 75

ml;ltizapypUe;J x-d; kjpg;G mjpfhpf;f y-d; kjpg;Gk; mjpfhpg;gij fhzyhk; .vdNt Neh; khWghl;il Fwpf;fpwJ. vdNt y/x = k , K =15/1=10/2=45/3=60/4 =75/5=15, k=15 , y = 15x jPh;T : tiuglj;jpypUe;J i) tpfpjrk khwpyp k=15 ii) ii) 3 ypl;lh; ghypd; tpiy =45,

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9) xU NgUe;J kzpf;F 40 fp.kP. Ntfj;jpy; nry;fpwJ . ,jw;Fwpa Jhu –fhy njhlh;gpw;fhd tiuglk; tiuf. ,ijg; gad;gLj;jp 3 kzp Neuj;jpy; ,g ;NgUe;J gazpj;jj; Jhuj;ijf;

x 1 2 3 4 5 y 40 80 120 160 200

ml;ltizapypUe;J x-d; kjpg;G mjpfhpf;f y-d; kjpg;Gk; mjpfhpg;gij fhzyhk; .vdNt Neh; khWghl;il Fwpf;fpwJ. vdNt y/x = k , K =40/1=80/2= 120/3=160/4 =200/5=40 ; y = 40 x tiuglj;jpypUe;J 3 kzp Neuj;jpy; NgUe;J gadpj;j Jhuk; = 120 fp.kP. 10) thq;fg;gl;l Nehl;Lg; Gj;jfq;fspd; vz;zpf;if kw;Wk; mjw;fhd tpiy tptuk; gpd;tUk; ml;ltidapy; 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; 1) VO Nehl;Lg; Gj;jfq;fspd; tpiyia fhz;f . 2) &.165 –f;F thq;fg;gLk; Nehl;Lg;Gj;jfq;fspd; vz;zpf;ifiaf; fhz;f. jPh;T : ml;ltizapypUe;J x-d; kjpg;G mjpfhpf;f y-d; kjpg;Gk; mjpfhpg;gij fhzyhk; .vdNt Neh; khWghl;il Fwpf;fpwJ. vdNt y/x = k k=30/2=60/4=90/6=120/8=150/10=180/12=15 tiuglj;jpypUe;J ; 1) VO Nehl;Lg; Gj;jfq;fspd; tpiy &. 105 . 2) &.165 –f;F thq;fg;gLk; Nehl;Lg;Gj;jfq;fspd; vz;zpf;if 11 .

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11) xy = 20 , x,y >o vd;gjd; tiuglk; tiuf . mjidg; gad;gLj;jp x= 5 vdpy; y-d; kjpg;igAk; y = 10 vdpy; x - d; kjpg;igAk; fhz;f . jPh;T : xy =20 , x,y >0 , y =20/x

X 1 2 4 5 10 20 y 20 10 5 4 2 1

ml;ltizapypUe;J x-d; kjpg;G mjpfhpf;f y-d; kjpg;G Fiwtij fhzyhk; .vdNt vjph; khWghl;il Fwpf;fpwJ. tiuglj;jpypUe;J x=5 vDk; NghJ y=4 kw;Wk; y=10 vDk; NghJ x=2

12) fPo;fhZk; ml;ltizapy; nfhLf;fg;gl;Ls;s tptuj;jpw;fhd tiuglk; tiuf. mjd; %yk; 12 Ntiyahl;fs; mt;Ntiyia KOJkhf nra;J Kbf;f MFk; ehl;fspd; vz;zpf;ifia fhz;f. Ntiyahl;fspd; vz;zpf;if(x)

3 4 6 8 9 16

ehl;fspd; vz;zpf;if(y)

96 72 48 36 32 18

jPh;T : ml;ltizapypUe;J x-d; kjpg;G mjpfhpf;f y-d; kjpg;G Fiwtij fhzyhk; .vdNt vjph; khWghl;il Fwpf;fpwJ. vdNt K vd;gJ tpfpj rk khwpyp MFk; . K =3x96=4x72 =6x48=8x36 = 9x32 = 16x18 = 288 ; xy =288 ; Y =288/x tiuglj;jpypUe;J 12 Ntiyahl;fs; mt;Ntiyia KOJkhf nra;J Kbf;f MFk; ehl;fs; 24 MFk;

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13) 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;Jhuj;jpid flf;f vLj;Jf; nfhz;l Neuk; Mfpatw;wpidg; gw;wpa tptuq;fs; (Ntf-fhy) gpd;tUk; ml;ltizapy; nfhLf;fg;gl;Ls;sd. Ntfk; (fp.kP/kzp)

2 4 6 10 12

Neuk; (kzpapy;) 60 30 20 12 10

Ntf –fhy tiuglk; tiue;J mjpypUe;J i) mth; kzpf;F 5 fp.kP. Ntfj;jpy; nrd;why; Jhuj;ijf; flf;f MFk; gaz Neuk; . ii) mth; ,f;Fwpg;gpl;l Jhuj;ij 40 kzp Neuj;jpy; flf;f ve;j Ntfj;jpy; gazpf;f Ntz;Lk; Mfpatw;iw fhz;f. jPh;T : ml;ltizapypUe;J x-d; kjpg;G mjpfhpf;f y-d; kjpg;G Fiwtij fhzyhk; .vdNt vjph; khWghl;il Fwpf;fpwJ. vdNt K vd;gJ tpfpj rk khwpyp MFk; . xy = K MFk; . ,q;F xy = 120 MfNt y = 120/x tiuglj;jpypUe;J > i) mth; kzpf;F 5 fp.kP. Ntfj;jpy; nrd;why; Jhuj;ijf; flf;f MFk; gaz Neuk; 24 kzp MFk; ii) mth; ,f;Fwpg;gpl;l Jhuj;ij 40 kzp Neuj;jpy; flf;f mtUila Ntfk; 3 fp.kP /kzp MFk;

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Ie;J kjpg;ngz; tpdhf;fs; fztpay;.

1. A\(BUC) = (A\B) ∩ (A\C) ntz;glk; %yk; rhpghh;f;f .

2. AU(B∩C) = (AUB) ∩ (AUC) vd;gjid ntz;glk; %yk; rhpghh; .

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3. (A U B) ‘ = A’ ∩ B’ vd;gjid ntz;glk; %yk; rhpghh; .

gapw;rpf;fhf :

i) . A\(B∩C) = (A\B) U (A\C) ntz;glk; %yk; rhpghh;f;f

ii) (A∩B) ‘ = A’ U B’ vd;gjid ntz;glk; %yk; rhpghh; .

iii) A∩ (BUC) = (A∩B) U (A∩C) vd;gjid ntz;glk; %yk; rhpghh; .

4) A= {xI -3≤ x< 4 ,x ε R}, B= {x< 5, x ε N } kw;Wk; C= {-5,-3,-1,0,1,3} vd;f

A∩(BUC)=(A∩B)U(A∩C) vdf;fhl;Lf

Answer : LHS : A∩(BUC) i) BUC = {1,2,3,4} U {-5,-3,-1,0,1,3} = {1,2,3,4,-5,-3,-1,0}

ii) A∩ (BUC) = {-3,-2,-1,0,1,2,3}∩{1,2,3,4,-5,-3,-1,0} = {-3,-1,0,1,2,3} ---(1)

RHS : (A∩B)U(A∩C) iii) A∩B = {1,2,3}

iv) A∩C = {-3,-1,0,1,3}

v) (A∩B)U(A∩C) = {-3,-1,0,1,2,3} ----- (2)

(1) = (2)

A∩ (BUC) = (A∩B)U(A∩C) rhpghh;f;fg;gl;lJ

________________________________________________________________

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gapw;rpf;fhf : A = { 1,35,7,9,11} B = { 7,9,13,15,17} C= { 3, 7, 15, 19 21 ,23 } vdpy;

gpd; tUk; tpjpfis rhpghh; .

i) A U (B∩C) = (A U B) ∩(A U C) ii) A\(B∩C) = (A\B) U (A\C) iii) A\(BUC) = (A\B) ∩ (A\C) ___________________________________________________________________________ 4. U = { -2,-1, 0,1,2,3.........10} A= {-2,2,3,4,5 } B= {1,3,5,8,9 } vd;f . bkhh;fdpd; fz epug;gp tpjpfis rhpghh; .

Answer : LHS : (AUB)’ i) (AUB) = {-2,2,3,4,5 } U {1,3,5,8,9 } = {-2,1,2,3,4,5,8,9}

ii) (AUB)’ = { -2,-1, 0,1,2,3......10} \{-2,1,2,3,4,5,8,9} = {-1,0,6,7,10} ----(1)

RHS : ( A’∩B’ )

i) A’ = U \ A ={ -2,-1, 0,1,2,3.........10} \ A= {-2,2,3,4,5 } = {-1,0,1,6,7,8,9,10} ii) B’ = U \ B = ={ -2,-1, 0,1,2,3.........10}\{1,3,5,8,9 } = {-2,-1,0,2,4,67,10 }

iii) A’∩B’ = {-1,0,1,6,7,8,9,10} ∩ {-2,-1,0,2,4,67,10 } = {-1,0,6,7,10} -----(2)

(AUB)’ = A’∩B’ rhpghh;f;fg;gl;Lj

_______________________________________________________________________________________________________

gapw;rpf;fhf : i) U = { -2,-1, 0,1,2,3.........10} A= {-2,2,3,4,5 } B= {1,3,5,8,9 } vdpy; (A∩B)’=

A’UB’ vDk; bkhh;fdpd; fz epug;gp tpjpfis rhpghh;

__________________________________________________________________________________

5) A= { 4,6,8,10}, B= {3,4,5,6,7} kw;Wk; f :A →B f(x) = ½ x+1 vd tiuaWf;fg;gl;lhy; rhh;G

f -ia (i) thpir N[hb fzkhf (ii) ml;ltizahf (iii)mk;Gf;Fwp glkhf (iv) tiuglkhf fhz;f. f(x) = ½ x + 1

f(4) = ½(4) + 1 = 2+1 = 3

f(6) = ½(6) + 1 = 3+1 = 4

f(8) = ½(8) + 1 = 4+1 = 5

f(10) = ½(10) + 1 = 5+1 = 6

i) thpir N[hbfspd; fzkhf : f ={(4,3), (6,4), (8,5), (10,6)}

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ii) ml;ltizahf :

X 4 6 8 10

F(x) 3 4 5 6

iii) mk;Gf;Fwp glkhf : A B

iv) tiuglkhf

6 y .(10>6)

5 .(8>5)

4 .(6>4)

3 .(4>3) 2

1

0 1 2 3 4 5 6 7 8 9 10 x

________________________________________________________________

gapw;rpf;fhf : i) A= { 0,1,2,3 }, B= {1,3, 5,7,9 } kw;Wk; f :A→B f(x) = (x – 3)/3 vd tiuaWf;fg;gl;lhy; rhh;G f -ia (i) thpir N[hb fzkhf (ii) ml;ltizahf (iii)mk;Gf;Fwp glkhf (iv) tiuglkhf fhz;f. ii) A= { 6,9,15,18,21 }, B= {1,2,3,4 , 5 } kw;Wk; f :A→B f(x) = 2x+1 vd tiuaWf;fg;gl;lhy; rhh;G f -ia (i) thpir N[hb fzkhf (ii) ml;ltizahf (iii)mk;Gf;Fwp glkhf (iv) tiuglkhf fhz;f. _________________________________________________________________________________________________________________

4

6

8

10

3

4

5

6

7

7

77

7

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2.nka;naz;fspd; njhlh;thpirfSk; njhlh;fSk; 1) xU $l;Lj;njhlhpy; t10 = 41 t18 =73 vdpy; 27 – tJ cWg;G fhz;f

jPh;T : t18 =a+17d = 73 --------(2) t10 = a+ 9d = 41 --------(1) (2)– (1)= _____________________________________

8d = 32 ; d = 32/8 = 4 , d = 4

d- apd; kjpg;ig rkd;ghL (2) y; gpujpapl a +9(4) = 41 ; a = 41 – 36 = 5 , a=5

t27 = a + 26d = 5 + 26(4) = 5 + 104 = 109 , _________________________________________________________________________________________

2) xU ngUf;Fj; njhlu; thpirapy; KjyhtJ cWg;G 1/3, 6-tJ cWg;G 1/729 vdpy; mj;njhliuf; fhz;f.

jPh;T : t1 = a = 1/3 ----(1) ; t6 = ar5 = 1/729 ----(2) ar5 = 1x3 = r5 = 1/243 , r =1/3 a 1X729

r- apd; kjpg;ig rkd;ghL (2) y; gpujpapl

ar5 = 1/729 = a(1/3)5 = 1/729 = a = 1/3

vdNt mj;njhlh; : 1/3 , 1/27 , 1/81 …………………………….

____________________________________________________________________________________________________________________________

3) 300f;Fk; 500f;Fk; ,ilNa 11My; kPjpapd;wp tFgLk; midj;J ,ay; vz;fspd; $Ljy; fhz;f.

jPh;T : a-it fhz

a = 300/11 = 27 + kPjp 3 (11-3 = 8) ( Kjy; vz;300 cld; 8 I $l;Lf)

a = 300 + 8 = 308

(filrp cWg;G l fhz)

l = 500/11 = 45 + kPjp 5 ( filrp vz; 500 ypUe;J 5 I fopf;f )

l = 500 - 5 = 495

n = l - a +1 = 495-308 +1 = 187 +1 = 17+1 =18 d 11 11 Sn = n/2 (a+l ) = 18/2 (308+495) = 9 (803) = 7227 300f;Fk; 500f;Fk; ,ilNa 11-d; klq;;Ffshd midj;J vz;fspd; $l;Lg;gyd; =7227 4) $Ljy; fhz;f : 122 + 132 +.....+ 352

jPh;T :

(12+22+32+--+352 ) - ( 12+22+32+------+112) (n=35) (n=11) n(n+1)(2n+1) - n(n+1)(2n+1) 2x3 2x3

35x36x71 - 11x12x23

2x3 2x3

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14910 - 506 = 14404

122 + 132 +................+ 352 = 14404

__________________________________________________________________

5) $Ljy; fhz;f : 113+133+143+……………….+283 jPh;T : 113+133+143+……………….+283= (13+23+33+........+283) - (13+23+…………….+103)

= n(n+1) 2 - n(n+1) 2 = (28x29) 2 - (10x11)2 2 2 2 2 = (14x29)2 - (5x11)2 = ( 406)2 - (55)2 = (406+55)(406-55) = ( 461 ) (351) = 161811

= 113+133+143+……………….+283 = 161811 ___________________________________________________________________________________________

gapw;rpf;fhf : i) $Ljy; fhz;f : 162 + 172 + 182 +………………...+ 252 ii)$Ljy; fhz;f : 163 + 173 + 183 +………………...+ 353

_____________________________________________________________________________________________________________________________ ____________________________________________________________________

6) 13+23+33+……………….+k3 = 4356 vdpy; k- d; kjpg;G fhz;f.

jPh;T : 13+23+33+……………….+k3 = 4356

k(k+1) 2 = 4356 = (k(k+1) 2 = 11x11x6x6 = k(k+1) = 66 2 2 2

k2+k -132 =0 = (K +12) (k -11)

k+12 =0 , k-11 = 0

k = -12 my;yJ k = 11 +12 -11

k vd;gJ kpif vz; > vdNt k= 11

______________________________________________________________________________

gapw;rpf;fhf : i) 13+23+33+……………….+k3 = 6084 vdpy; k- d; kjpg;G fhz;f ii) 13+23+33+……………….+k3 = 8281 vdpy; k- d; kjpg;G fhz;f.

7) 6+ 66 +666+ - - - - - vd;w njhlhpy; n cWg;Gf;fspd; $Ljy; fhz;f

jPh;T : Sn = 6+ 66 +666+- - - - - - -- n cWg;Gf;fs; tiu )

Sn = 6( 1 +11 +111+----------- n cWg;Gf;fs; tiu )

6/9 ( 9 + 99 +999+ -------- n cWg;Gf;fs; tiu ) ( 9 My; ngUf;fp 9 My; tFf;f)

2/3 [ (10-1 +100-1 +1000-1) +… n milg;Gf;Fwpfs; tiu] 2/3 [ (10 +102+103+….... n cWg;Gf;fs; tiu) – n ]

Sn = 2/3 (10(10n -1 -n

9 gapw;rpf;fhf :

i) 7+ 77 +777+ - - - - - vd;w njhlhpy; n cWg;Gf;fspd; $Ljy; fhz;f

-132

+1

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3.,aw;fzpjk; 1) fhuzpgLj;Jf : x 3 - 3x2 - 10x + 24

jPh;T : p(x) = x 3 - 3x2 - 10x + 24 vd;f .

,f;Nfhitf;F p(1) ≠ 0 kw;Wk; p(-1) ≠ 0 vdNt (x-1) kw;Wk; (x+1) Mfpad fhuzpfs; my;y . MfNt NtW kjpg;Gfis

gpujpapl;L : p(x) f;F fhuzpfis fhzyhk; .

2 1 -3 -10 +24 0 2 -2 -24

1 -1 -12 0 vdNt p(x) - f;F (x - 2) xU fhuzpahFk;

kw;nwhU fhuzp : x2 –x – 12 +12

mjid fhuzpgLj;j : (x-4) (x+3)

MfNt fhuzpfs; : (x - 2) (x-4) (x+3)

-4 +3

gapw;rpf;fhf : -1

i)fhuzpgLj;Jf : 2x 3 - 3x2 - 3x + 2 ii)fhuzpgLj;Jf : 4x 3 - 5x2 +7x - 6 iii)fhuzpgLj;Jf : x 3 - 5x + 4 2)fhuzp Kiwapy; th;f;f%yk; fhz;f. (6x2-x-2) ( 3x2-5x+2)(2x2-x-1)

jPh;T : (6x2-x-2) = ( 2x+1)(3x-2) ( 3x2-5x+2) = (3x-2) (x-1) (2x2-x-1) = (x-1) ( 2x+1)

√(6x2-x-2) ( 3x2-5x+2)(2x2-x-1) = √( 2x+1)(3x-2) (3x-2) (x-1) (x-1) ( 2x+1) = ( 2x+1)(3x-2) (x-1)

2) th;f;f%yk; fhz;f : tFj;jy; Kiwapy; th;f;f%yk; fhzy; .9x4+12x3+10x2+4x+1 3x2 + 2x +1

3x2 9x4 + 12x3 + 10x2 + 4x +1 i) 9-d; th;f;f%yk; 3 = 3x2

9x4

6x2 +2x + 12x3+ 10x2 ii) 6 cld; 2 I ngUf;f 12 fpilf;Fk ; =2x

+ 12x3+ 4 x2

6x2 +4x +1 +6x2 + 4x +1 iii) 6 cld; 1 I ngUf;f 6 fpilf;Fk; =1

+6x2 + 4x +1 0 √ 9x4+12x3+10x2+4x+1 = | 3x2 + 2x +1|

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3) 9x4+12x3+10x2+ax+b - xU KO th;f;f Nfhit vdpy; a,b -d; kjpg;G fhz; .

3x2 + 2x +1

3x2 9x4 + 12x3 + 10x2 + ax +b i) 9-d; th;f;f%yk; 3 = 3x2

9x4

6x2 +2x + 12x3+ 10x2 ii) 6 cld; 2 I ngUf;f 12 fpilf;Fk; =2x

+ 12x3+ 4 x2

6x2+4x +1 +6x2 + ax +b iii) 6 cld; 1 I ngUf;f 6 fpilf;Fk; =1

+6x2 + 4x +1 0 a= 4 , b = 1 ________________________________________________________________________________________________________________

gapw;rpf;fhf : th;f;f%yk; fhz;f kw;Wk; a,b -d; kjpg;G fhz;f.

i) X4-10x3=37x2-60x+36 Ii) 4+25x2-12x-24x3+16x4

ii) 9x4+12x3+28x2-nx+m iii) 4x4 -12x3 +37x2+ax+b

____________________________________________________________________________________________________________________________

4. 2x2-3x-1=0 vd;w rkd;ghl;bd; %yq;fs; α kw;Wk; β vdpy; gpd;tUtdtw;wpd; kjpg;Gf;fis fhz;f .

i) α 2 + β

2 ii) α/ β + β/ α iii) α

2/ β + β

2/ α

---------------------

%yq;fspd; $Ljy; : α +β = -b/a = 3/2 ;

%yq;fspd; ngUf;fy; : α β = c/a = -1/2

i) α 2

+ β2 = (α

+ β)

2 - 2α β =(3/2)

2 – 2(-1/2) = 9/4 +1 = 13/4

ii) α/ β + β/ α = α 2 + β

2 / αβ = (α

+ β)

2-2 α β / αβ = [(3/2)2 -2(-1/2)] / -1/2 = -13/2

iii) α2/ β + β

2/ α = [α

3 + β

3] /αβ = [(α

+β)

3-3 αβ(α

+β)] / αβ = [(27/8)+(9/4)] /-1/2 =-45/4

________________________________________________________________

gapw;rpf;fhf :

6. 2x2-3x-5=0 vd;w rkd;ghl;bd; %yq;fs; α kw;Wk; β vdpy; α2 kw;Wk;

β

2 Mfpatw;iw %yq;fshf nfhz;l

,Ugb rkd;ghl;il fhz;f.

7. 3x2-5x+2=0 vd;w rkd;ghl;bd; %yq;fs; α kw;Wk; β vdpy; gpd;tUtdtw;wpd; kjpg;Gf;fis fhz;f .

i) α - β ii) α/ β + β/ α iii) α2/ β + β

2/ α

____________________________________________________________________________________________________________________________

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4.mzpfs ;; -2

1. A= 4 kw;Wk; B=(1 3 -6)vd;w mzpfSf;F (AB)T =BTAT vd;gij rhpghh;f;f 5 jPh;T : AB = -2 ( 1 3 -6) = -2 -6 12 4 4 12 -24 5 5 15 -30 (AB)T = -2 4 5 -6 12 15 ------------------------(1) 12 -24 -30

BTAT = 1 ( -2 4 5) = -2 4 5 3 -6 12 15 -----------------------(2) -6 12 -24 -30

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

2. A = 1 -1 vdpy; A2 – 4A +5I2 = 0 vd epWTf . 2 3

AxA 1 -1 1 -1 = -1 -4

2 3 2 3 8 7

-4A = -4 1 -1 = -4 4

2 3 -8 -12

5I2 = 5 1 0 5 0 0 1 0 5

A2 – 4A +5I2 = -1 -4 + -4 4 + 5 0 = 0 0

8 7 -8 -12 0 5 0 0 epUgpf;fg;gl;lJ

3. |2i–j| vd;w cWg;Gfisf; nfhz;l> tupir 2 x 3 cs;s mzp A = [aij] apid mikf;f. jPh;T : 2 x 3 tupirAs;s mzp a11 a12 a13 a ij = |2i–j| a21 a22 a23

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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 a22 = | 2 (2) – 3 (2) | = | 4 – 6 | = | - 2 | = 2

a23 = | 2 (2) – 3 (3) | = | 4 – 9 | = | - 5 | = 5 Njitahd mzp A = 1 4 7 1 2 5 _________________________________________________________________________________________________________________

gapw;rpf;fhf :

i) A = 5 2 B = 2 -1 7 3 1 1 vdpy; (AB)T =BTAT rhpghh;f;f .

ii) A = 5 2 B = 3 -2 7 3 -7 5 vd;w mzpfs; xd;Wf;nfhd;W ngUf;fy;

Neh;khW mzp vd epWTf. ( mjhtJ AB =BA = I vd epWTjy; )

iii) aij = ij vdpy; 2 x 2 tupirAila mzp A=[ai j] If; fhz;f.

_________________________________________________________________________________________________

5.Maj;njhiy tbtpay; 1. (1,2) (-3,4) (-5,-6) Mfpatw;iw Kidfshf nfhz;l Kf;Nfhzj;jpd; gug;gsT fhz; .

jPh;T : gug;gsT =1/2 x1 x2 x3 x1

y1 y2 y3 y1

=1/2 1 -3 -5 1

2 4 -6 2

= 1/2 ( 4+18-10) – ( -6-20-6) = 1/2 {(12) -(-32) } = 1

2(12+32) = 22 r.m.

2. (-4,-2) (-3-5) (3,-2 ) (2 , 3) Mfpatw;iw Kidfshf nfhz;l ehw;fuj;jpd; gug;gsT fhz;

jPh;T :gug;gsT= 1/2 x1 x2 x3 x4 x1

y1 y2 y3 y4 y1

-4 -3 3 2 -4

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= 1/2 -2 -5 -2 3 -2

= 1/2 (20 +6+9-4) – ( 6-15-4-12) = 1/2 ( 31) – ( 25) = 1/2 ( 56) = 28 r.m.

3. A(2,3) B(4,0) C(6,3) Mfpad xNu Neh;Nfhl;by; mikAk; Gs;spfs; vd fhl;Lf.

Kf;Nfhzk; ABC- apd; gug;G : .

1/2 x1 x2 x3 x1

Y1 y2 y3 y1

1/2 2 4 6 2

3 0 3 3

1/2 (0 +12+18) –(12+0+6) = 1/2 (30 -30) = 0

________________________________________________________________________________________________________________

gapw;rpf;fhf : i) (-4,-5) (-1,-6) (4,5 ) Mfpa Gs;spfis Kidfshf nfhz;l

Kf;Nfhzj;jpd; gug;G fhz;f ii) (0,0) (4,a) (6,4 ) Mfpa Gs;spfis Kidfshf nfhz;l Kf;Nfhzj;jpd;

gug;G 17 r.m. vdpy; a -apd; kjpg;G fhz; . iii) (-5,-6) (4,-1) (1,2 )(-3,4 ) Mfpa Gs;spfis Kidfshf nfhz;l

ehw;fuj;jpd;; gug;G fhz;f iv) A(6,7) B(-4,1) C(a,9) Mfpa Gs;spfis Kidfshf nfhz;l

Kf;Nfhzj;jpd; gug;G 68 r.m. vdpy; a -apd; kjpg;G fhz; . v) (2> -5)> (3> -4) kw;Wk; (9> k) Mfpa Gs;spfs; xU Nfhliktd vdpy; K-,d;

kjpg;igf; fhz;f. vi) (7>3)>(6>1)>(8>2) kw;Wk; (p>4) vd;gd Xu; ,izfuj;jpy; tupirg;gb

mike;j cr;rpfs; vdpy; p-d; kjpg;igf; fhz;f

6.tbtpay; 1. Njw;wk; 6.1 - mbg;gil tpfpjrkj; Njw;wk; my;yJ Njy;]; Njw;wj;ij epUgp. 2. Njw;wk; 6.3 - Nfhz ,Urkntl;bj; Njw;wj;ij epUgp . 3. gpjhfu]; Njw;wj;ij epUgp.

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8.mstpay; 1. xU ,ilf;fz;l tbtpyhd thspapd; Nkw;Gw Muk; 15nr.kP. cl;Gw Muk; 8nr.kP NkYk; mjd; Mok; 63 nr.kP. vdpy; mjd; nfhs;ssT fhz;f. (ypl;lhpy;). jPh;T :

,ilf;fz;lj;jpd; fd msT = 1/3xπx h(R2+r2+Rr)

= 1/3 x 22/7 X 63(152+82+15x8) = 26994 f.nr.kP.

= 26994/1000 = 26.994 ypl;lh;.

2. 8 nr.kP tpl;lKk; 12 nr.kp. cauKk; nfhz;l xU Neh; tl;l jpz;k ,Uk;G

$k;ghdJ cUf;fg;gl;L 4 kp.kP. MuKs;s jpz;k Nfhs tbt Fz;Lfshf thh;f;fg;gl;lhy; fpilf;Fk; Nfhs tbt Fz;Lfspd; vz;zpf;ifia fhz;f. jPh;T :

fpilf;Fk; Nfhs tbt Fz;Lfspd; vz;zpf;if : 1/3 xπxr2h = 1/3xπ x40x40x120 = 750

4/3πr3 4/3xπx4x4x4 fpilf;Fk; Nfhs tbt Fz;Lfspd; vz;zpf;if = 750

_________________________________________________________________________

3. fspkz;izg; gad;gLj;jp xU khztd; 48nr.kP. cauKk; 12nr.kP MuKs;Neh;tl;l jpz;kf; $k;ig nra;jhh; . mf;$k;ig kw;nwhU khztd; jpz;k Nfhskhf khw;wpdhh; . khw;wg;gl;l Nfhsj;jpd; Muk; vd;d.?

jPh;T :

Nfhsj;jpd; fd msT = $k;gpd; fd msT

4/3xπr3 = 1/3x πr2h 4/3xπr3 = 1/3 xπx12x12x48

r3 = (1/3)x12x12x48x(3/4) Nfhsj;jpd; Muk; = 12 cm ______________________________________________________________________________________________ 4. xU Neu;tl;lf;$k;gpd; ,ilf;fz;lj;jpd; mike;j tl;ltpspk;Gfspd; Rw;wsTfspd; KiwNA 44 nr.kP kw;Wk; 8.4 π nr.kP vd;f. mjd; cauk; 14 nr.kP vdpy; mt;tpilf; fz;lj;jpd; fd msT fhz;f. jPu;T.

,ilf;fz;lj;jpd; fd msT V = 1/3π (R2+r2+Rr) f.m 2R = 44 2r = 8.4

2x22/7 R = 44 r = 8.4/2

R= 44x7/2x22 = 4.2 fd msT V = 1/3(R

2+r

2+Rr)

= 1/3 x 22/7 14x (72x4.22 v(4.2))

=44/3(49x17.64x29.4)

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=44/3(96.04)

=42225.76/3

=1 408.58

=1408.6 nr.kP3

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

gapw;rpf;fhf : 4. 18 nr.kP. MuKs;s jpz;k cNyhf NfhskhdJ cUf;fg;gl;L %d;W rpwpa

ntt;NtW Nfhsq;fshf thh;f;fg;gLfpwJ . thh;f;fg;gl;l ,uz;L jpz;k Nfhsq;fspd; Muq;fs; KiwNa 2 nr.kP. kw;Wk; 12nr.kP. vdpy; %d;whtJ Nfhsj;jpd; Muk; fhz;f.

5. xU jpz;k nghk;ikahdJ miuf;Nfhsj;jpd; kPJ $k;G ,ize;j tbtpy; cs;sJ $k;G kw;Wk; miuf;Nfhsk; Mfpatw;wpd; Muk; 3.5nr.kP. nghk;ikapd; nkhj;j cauk; 17.5 nr.kP. vdpy; mg;nghk;ik jahhpf;fg;gad;gLj;jg;gl;l kuj;jpd; fd msitf; fhz;f .

6. xU Nfhg;igahdJ miuf;Nfhsj;jpd; kPJ cUis ,ize;j tbtpy; cs;sJ .

cUisapd; cauk; 8 nr.kP. kw;Wk; Nfhg;igapd; nkhj;j cauk; 11.5 nr.kP. vdpy; Nfhg;igapd; nkhj;jg;Gwg;gug;ig fhz;f. 7. xU ,ilf;fz;l tbtpyhd thspapd; Nkw;Gw kw;Wk; mbg;Gw Muq;fs; KiwNa 15 nr.kP kw;Wk; 8 nr.kP . NkYk; Mok; 63 nr.kP vdpy; mjd; nfhs;ssit ypl;lupy; fhz;f.

11. Gs;spapay; 1. gpd; tUk; kjpg;Gf;fSf;fhd jpl;l tpsf;fk; fhz;f. 26,28,31,34,34,38,40

jPh;T : $l;Lr;ruhrhp : 26+28+31+34+34+38+40= 231/7 = 33 jpl;l tpyf;fk; : 7

jp.tp. = √∑d2/n

= √154/7

= √154/7

=√22 = 4.69

x d=x-x d2

26 28 31 34 34 38 40

-7 -5 -2 1 1 5 7

49 25 04 01 01 25 49

∑d2=154

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gapw;rpf;fhf : i) gpd; tUk; kjpg;Gf;fSf;fhd jpl;l tpsf;fk; fhz;f. 10, 20, 15, 8, 3 , 4 ii) gpd; tUk; kjpg;Gf;fSf;fhd jpl;l tpsf;fk; fhz;f. 14,16,20,21,25,30 ________________________________________________________________________________________________________________

2. gpd; tUk; kjpg;Gf;fSf;fhd khWghl;Lf;nfO fhz;f. 20,18,32,24,26 jPh;T : $l;Lr;ruhrhp : 20+18+32+24+26 = 120/5 = 24 5 jpl;l tpyf;fk; :

jp.tp. = √120/5 = 24 =4.9

khWghl;Lf;nfO : (σ/ x )x100%

(4.9/24)x100% = 20.42 %

gapw;rpf;fhf :

i) gpd; tUk; kjpg;Gf;fSf;fhd khWghl;Lf;nfO fhz;f. 12,15,18,20,25

12. epfo;jfT 1. ,U gfilfs; xU Nru cUl;lg;gLk; NghJ ,U gfilfspYk; xNu vz; fpilf;f my;yJ Kf vz;fspd; $Ljy; gfh vz;zhf ,Ug;gjw;fhd epfo;jfT fhz;f.

jPh;T :

$Wntsp 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) n(S) = 36 ( 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 )

i) ,U gfilfspYk; xNu vz; fpilf;f epfo;r;rp fzk; A vd;f A = { (1,1) (2,2)(3,3)(4,4)(5,5)(6,6) }

n(A) = 6 , P(A) = n(A)/n(S) = 6/36

ii) Kf vz;fspd; $Ljy; gfh vz;zhf ,Ug;gjw;fhd epfo;r;rp fzk;B vd;f

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

n(B) = 15 P(B) = n(B)/n(S) = 15/36

iii) (AnB) = { (1,1) } , n(AnB) = 1 , P(AnB) = n(AnB)/n(S) = 1/36 P(AUB) = p(A) +P(B) - P(AnB) P(AUB) = (6/36) +(15/36) –(1/36) =20/36= 5/9

_________________________________________________________________________________________________

x d=x-x d2 20

18 32 24 26

-4 -6 8 0 2

16 36 64 0

04

∑d2=120

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28

1. %d;W ehzaq;fs; xU Nru Rz;lg;gLk;NghJ fPo;f;fhZk; epfo;r;rpfSf;fhd epfo;jfTfis fhz;f.

i) Fiwe;jJ ,U jiyfs; fpilf;f ii) rhpahf ,U g+f;fs; fpilf;f iii) %d;W jiyfs; fpilf;f . -------------

$Wntsp S = {HHH, HHT, HTH,THH, TTT,TTH,THT,HTT } n(S) = 8 i) Fiwe;jJ ,U jiyfs; fpilf;f epfo;r;rp fzk; A vd;f.

A= { HHH, HHT, HTH,THH, } , n(A) = 4 , P(A) =n(A)/n(S) =4/8 = 1/2

ii) rhpahf ,U g+f;fs; fpilf;f epfo;r;rp fzk; B vd;f.

B= { TTH,THT,HTT } , n(B) = 3 , P(B) =n(B)/n(S) = 3/8 iii) %d;W jiyfs; fpilf;f epfo;r;rp fzk; C vd;f.

C = { HHH } , n(C) = n(C)/n(S) = 1/8

________________________________________________________________________________

2. 52 rPl;Lf;fisf; nfhz;l rPl;L fl;bypUe;J rk tha;g;G Kiwapy; xU rPl;L Njh;e;njLf;f mJ rPl;L ];NglhfNth my;yJ ,uh[hthfNth ,Ug;gjw;fhd epfo;jftpid fhz;f.

---------

jPh;T : $Wntspapd; vz;zpf;if n(S) = 52 A vd;gJ ];NgL rPl;L fpilf;Fk; epfo;r;rp vd;f. n(A) =13 P(A) =n(A)/n(S) = 13/52 B vd;gJ uh[h rPl;L fpilf;Fk; epfo;r;rp vd;f.n(B) = 4 P(B) = n(A)/n(S) = 4/52 ];NgL uh[h rPl;L fpilf;Fk; epfo;r;rpapd; vz;zpf;if n(AnB)=1 P(AnB) = n(AnB) = 1/52 P(AUB) = P(A) +P(B) – P(AnB) = (13/52)+(4/52)- (1/52) = 16/52 =4/13 _________________________________________________________________________________________________________________

gapw;rpf;fhf : i) Kjy; 100 tiuapyhd KO vz;fspypUe;J rk tha;g;G Kiwapy; Njh;njLf;fg;gLk; xU

vz; m) xU KO th;f;f vz;zhf ,Uf;f M) KO fdkhf ,y;yhky; ,Uf;f epfo;jfTfis fhz;f.

ii) XU igapy; 10 nts;is > 6 rptg;G 10 fUg;G ge;Jfs; cs;sd. Rk tha;g;G Kiwapy; xU ge;jpid vLf;Fk;NghJ mJ nts;is my;yJ rptg;G epwg; ge;jhf ,Ug;gjw;fhd epfo;jftpid fhz;f.

iii) 1 Kjy; 50 tiuapyhd KOf;fspypUe;J rk tha;g;G Kiwapy; Xh; vz; Njh;e;njLf;fg;gLk; NghJ mt;ntz; 4 my;yJ 6 My; tFgLtjw;fhd epfo;jfT fhz;f.

________________________________@@@@@@@@@@@@______________________________________________

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29

,Ukjpg;ngz; tpdhf;fs; :

1. fztpay;

1.ntz;glk; %yk; fhl;Lf. A∩(BUC)

______________________________________________________________________________________________________________________

B

2. ACB vdpy; A∩B =A kw;Wk; A\B = Φ

Mfpatw;iw ntz;glj;jpy; fhl;Lf. A∩B=A

B

_______________________________________________________________________________________________________________________

3. f = { (12,2) (13,3) ( 15,3) ( 14,2) (17,17) vd;w rhh;gpy; 2 kw;Wk; 3 Mfpatw;wpd; Kd; cUf;fis fhz;f.

2 kw;Wk; 3 Mfpatw;wpd; Kd; cUf;fs; : 2-d; Kd; cUf;fs; = 12>14

: 3 –d; Kd; cUf;fs; = 13>15

____________________________________________________________________________________________________________________

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

kjpg;gfk; = { 1,2,3,4,5 } , tPr;rfk; = { 1,3,5,7,9 }

5. A= {1,2,3,4,5}, B=N kw;Wk; f : A B f(x) =x2 என

tiuaWf;fg;gl;lhy; rhh;G f- d; tPr;rfj;ij fhz;f. kw;Wk; rhh;gpd; tifia fhz;f.

A= {1,2,3,4,5} B= {1,2,3,4,5,…………………..}

f(x) = x2

f(1) = 12 =1

f(2) = 22 =4

A

A

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30

f(3) = 32 =9

f(4) = 42 =16

f(5) = 52 =25

tPr;rfk; = {1,4,9,16,25} , rhh;gpd; tif : xd;W-xd;W

________________________________________________________________

6. fPNo cs;s ml;ltiz MdJ A= {5, 6, 8, 10} ypUe;J B = {19,15, 9, 11} f;F

f(x) = 2x-1 vd;wthW mike;j xU rhu;G vdpy; a, b fisf; fhz;f.

X 5 6 8 10

F(x) a 11 B 19

jPh;T :

f(5) = 2(5) – 1 = 9 f(8) = 2(8)-1 = 15 Ans : a = 9, b = 15 ________________________________________________________________________

6.nfhLf;fg;gl;Ls;s f = {(1,3), (2,5), (4,7), (5,9), (3,1)} vd;w rhu;Gf;F kjpg;gfk; kw;Wk; tPr;rfk; fhz;f. jPh;T : kjpg;gfk; = { 1, 2, 4, 5, 3 } tPr;rfk; = { 3, 5, 7, 9, 1} ____________________________________________________________________________________________________________

7. A, B, C vd;gd VNjDk; %d;W fzq;fs; vd;f. NkYk; n (A) = 17,

n(B)= 17, n (17), n(AB) 7,n(BC) 6,n(AC) 5 kw;Wk; n(ABC) 2 vdpy;

n(ABC) fhz;f jPh;T :

= n(ABC) n(A) n(B) n(C) -n(AB) -n(BC) -n(AC) n(ABC) = 17 + 17 + 17 – 7 – 6 – 5 + 2 = 51 – 18 + 2 = 53 – 18 = 35

8. n (U) = 500, n (A) = 285, n (B) = 195, n (AUB) = 410, vdpy; ' n( A’ B’) If; fhz;f.

n( A’) U− n( A) = 500 – 285 = 215

n( B’) U− n( B ) = 500 - 195 = 305

n (AB) n(A) n(B) n(AB) 410 = 285 + 195 - n (AB)

n (AB) 285 + 195 – 410 = 480 – 410 = 70

(AB)' A'B'

n (A'B' ) n () - n (AB)

= 500 – 70 = 430

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

1. 4>9>14> …………….vd;w $l;Lj;njhlh;thpirapd; 17tJ cWg;G fhz;f. tn = a + (n-1)d t17 = 4+(17-1)5 = 4+(16)5 = 84.

_____________________________________________________________________________________________________ 2. 7>13> 19>………….>205 vd;w $l;Lj;njhlhpy; cs;s cWg;Gf;fspd; vz;zpf;ifia

fhz;f. n = l-a +1 = 205-7+ 𝟏 = 198+𝟏 = 33+1 +34 d 6 6

______________________________________________________________________________________________________ 3. $Ljy; fhz;f. 12+22+32+ ………………..+ 252

n(n+1)(2n+1) = 25(26)(51) = 5525 2x3 2x3 ______________________________________________________________________________________________________ 4. 1+3+5+…………………..25 cWg;Gfs; tiu $Ljy; fhz;f .

,q;F n=25 , n2 = 252 = 25x25 = 625

______________________________________________________________________________________________________

5. a = 5, n = 30 l= 121 vdpy; Sn = ?

Sn = n/2 [ a+l]

= 30/2 [5+121] = 15 (126) = 630 __________________________________________________________________________ 6. 1 + 3 + 5 + 7 + 9 + ................. + 21 = ?

n = [( l-a ) /d ] +1 n = (21 – 1 /2 ) +1 = 10 + 1 = 11 , n = 11

n2 = 11x11 = 121 ; 1 + 3 + 5 + 7 + 9 + ................. + 21 = 121

__________________________________________________________________________

gapw;rpf;fhf i) 9 + 99 + 999 + ............. n cWg;Gfs; tiu $Ljy; fhz;f. ii) 11 + 12 + ............... 20 = ? iii) 1 + 3 + 5 + ................. vd;w xw;iw ,yf;f vz; njhlupd; 20 cWg;Gfs; tiu $Ljy; fhz;f.

iv. 13+23+33+ ............. +203 = ?

__________________________________________________________________________________

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3.,aw;fzpjk; 1. njhFKiw tFj;jiy gad;gLj;jp <T kw;Wk; kPjp fhz; : x3+x2-7x-3 ÷(x-3)

3 1 1 -7 -3 0 3 12 15 1 4 5 12 <T : x2+4x+5 , kPjp = 12 ___________________________________________________________________________________________________

2. X3 - 6x2+11x - 6 vd;w gy;YWg;Gf;Nfhitf;F x—1 xU fhuzpah vd Muha;f. P(x) = X3 - 6x2+11x – 6 P(1) = 1-6+11-6 = 0 kPjp = 0 vdNt (x – 1) xU fhuzp MFk; ( NkYk; ,q;F nfOf;fspd; $Ljy; = 0 vd;gijAk; mwpf )

_______________________________________________________________________________________________________

3. R+j;jpuj;ij gad;gLj;jp jPh;T fhz;f . 15x2 -11x +2 = 0 . ,q;F a=15 , b =-11 , c= 2 , vdNt R+j;jpuk; -b± √b2-4ac =11± √(-11)2 -4(15)(2) 2a 30 11√121-120 = 11± 1 = 12/30 , 10/30 = 2/5 or 1/3 30 30 ___________________________________________________________________________________ 4. X2-6x+5 = 0 vd;w rkd;ghl;bd; %yq;fspd; $Ljy; kw;Wk; ngUf;fw;gyd; fhz;f. ,q;F a= 1 , b = -6 , c = 5

%yq;fspd; $Ljy; α+β = -b/a = 6/1 = 6

%yq;fspd; ngUf;fw;gyd αβ ; = c/a = 5

_________________________________________________________ 5. 3+√7, 3-√7 ,tw;iw %yq;fshf nfhz;l ,Ugb rkdlghl;il fhz;f.

%yq;fspd; $Ljy; α+β = 3+√7 + 3-√7 =6 %yq;fspd; ngUf;fw;gyd; αβ ; = (3+√7)(3-√7) = 9-7 = 2 Njitahd ,Ugb rkd;ghL x2 –(%yq;fspd; $Ljy;)x + (%yq;fspd; ngUf;fw;gyd;) =0 x2 –(6)x +(2) = 0 ; x2 –6x+2 =0 ___________________________________________________________________________________________________________

6. 9x2+12x+4 =0 vd;w rkd;ghl;bd %yq;fspd; jd;ikia Muha;f.; ,q;F a= 9 , b = 12 , c = 4

Δ = b2 – 4ac = (12)2 - (4)(9)(4) = 144 - 144 = 0 vdNt %yq;fs; nka; kw;Wk; rkkhdit . _________________________________________________________________________________________________

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7. x3+x2-7 x -3 vd;gij (x-3) My; tFf;Fk;NghJ <T kw;Wk; kPjp fhz;f.

3 1 +1 -7 -3 0 +3 +12 +15 1 +4 +5 +12 <T = x2 + 4x +5, kPjp = 12 ____________________________________________________________________________________________________________

8. x3-6x2+11x-6 vd;w gy;YWg;Gf; Nfhitf;F (x-1) vd epWTf.

1 1 -6 +11 -6 0 +1 -5 +6 1 -5 +6 0 P (x) f;F (x-1) xU fhuzp.

__________________________________________________________________________ 9. 35 x5 y3 z4, 49 x2 y z3, 14x y2 z2 Mfpatw;wpd; kP.ngh.t fhz;f.

35, 49, 14 ,d; kP.ngh.t = 7

x5 y3 z4, x2 y z3, x y2 z2 ,d; kP.ngh.t = x y z 2

35 x5 y3 z4, 49 x2 y z3, 14x y2 z2 ,d; kP.ngh.t = 7 x y z

2

_______________________________________________________________________________________________

10. 3 x2 y z, 4 x3 y 3 ,d; kP.ngh.k fhz;f. jPh;T : 3, 4 ,d; kP.ngh.k = 3 x 4 = 12 x2 y z, x3 y 3 ,d; kP.ngh.k = x3 y3 z 3x2 y z, 4x3 y 3 ,d; kP.ngh.k = 12 x3 y3 z

__________________________________

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4. mzpfs; .

1. A = 8 5 2 1 -3 4 vdpy; (AT)T = A vd epWTf .

AT = 8 1 (AT)T = 8 5 2 5 -3 1 -3 4

2 4 _______________________________________________________________________________________________________ 2. 3 5 2 -5

1 2 -1 3 Mfpad mzpg;ngUf;fiyg; nghUj;Jg; xd;Wf;nfhd;W Neh;khW mzp vd epWTf .

3 5 2 -5 = 6-5 15-15 = 1 0 = I

1 2 -1 3 2-2 -5+6 0 1 2 -5 3 5 = 6-5 15-15 = 1 0 = I

-1 3 1 2 2-2 -5 +6 0 1

vdNt nfhLf;fg;gl;Ls;s mzpfs; mzpg;ngUf;fiyg; nghUj;Jg; xd;Wf;nfhd;W Neh;khW mzpfshf mikfpd;wd. _______________________________________________________________________________________________________ 3. A = 2 3 - 1 5 vdpy; A ,d; $l;ly; Neh;khW mzpiaf; fhz;f.

-9 5 7 -1

A = 2 3 - 1 5 1 -2 -9 5 7 -1 = -16 6 mzp A – apd; $l;ly; Neh;khW = -1 2

16 -6 ________________________________________________________________________________________________________

4. ngUf;Ff. : 6 (2 -7) = 6x2 6x-7 = 12 -42 -3 -3x2 -3x-7 -6 21 _________________________________________________________________________________________________________________

gapw;r;rpf;fhf i) 1 2 x 0 x 0 vdpy; x, y ,d; kjpg;Gfisf; fhz;f.

3 3 0 y 9 0

ii) 2x+Y 5 vdpy; x, y ,d; jPh;Tfisf; fhz;f. X – 37 13

iii) aij = ij vdpy; 2 x 2 tupirAila mzp A=[ai j] If; fhz;f.

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35

5.Maj;njhiy tbtpay;

1. (3>0) (-1>4) Mfpa Gs;spfis ,izf;Fk; Nfhl;bd; eLg;Gs;sp fhz;f . M (x,y) = x1+x2 , y1+y2 =3-1 , 0+4 = M(1,2)

2 2 2 2

_______________________________________________________________________________________________ 2. A(4,-6) B(3,-2) C(5,2) Mfpatw;iw cr;rpfshff; nfhz;l Kf;Nfhzj;jpd;

eLf;Nfhl;L ikak; fhz;f. G(x,y) = x1+x2+x3 , y1+y2 +y3 =4+3+5 -6-2+2 = G (4,-2 ) 3 3 3 3

_______________________________________________________________________________________________________

3. A(2,3) B(-4,0) C(6,-3) Mfpa Gs;spfs; xNu Neh;Nfhl;by; mike;Js;sd vd fhl;Lf. ΔABC – apd; gug;G = ½ {( 0-12+18) – ( 12+0-6) } = ½(6-6) =0

_________________________________________________________________________________________________________

4. (3,2) kw;Wk; (-1,4) Mfpa Gs;spfis ,izf;Fk; Neh;Nfhl;bd; rha;T fhz;f. m = y2 –y1 = 4 - 2 = 2/-4 = -1/2 x2-x1 -1-3 _________________________________________________________________________________________________________________

5. (-2>3) vd;w Gs;sp topr; nry;tJk; > rha;T 1/3 cilaJkhd Neh;Nfhl;bd; rkd;ghL fhz;f.

,q;F m = 1/3 , (x1,y1) = (-2,3) rha;T – Gs;sp tbtk; rkd;ghL : y - y1 = m (x - x1) y – 3 = 1/3(x+2) 3(Y-3) = (x+2) 3y-9 = x+2 = x – 3y +11 = 0

_______________________________________________________________________________________________________________ 6. 2x –y +16 = 0 vd;w Neh;Nfhl;bd; rkd;ghl;bypUe;J x > y ntl;Lj;Jz;Lfis fhz;f

2x –y +16 = 0 ; 2x –y = - 16 ÷ -16 , (x/-8) - ( y/-16) =1 , x ntl;Lj;Jz;L = -8 y ntl;Lj;Jz;L =16

_______________________________________________________________________________________________________________

7. 3x+2y-12 =0 , 6x+4y+8=0 Mfpa Neh;f;NfhLfs; xd;Wf;nfhd;W ,iz vd epWTf. 3x+2y-12 =0 –d; rha;T m1 = -xd; nfO/y-d; nfO = -3/2 6x+4y+8=0 –d; rha;T m2 = -xd; nfO/y-d; nfO = -3/2 m1 = m2 , MfNt ,U Neh;NfhLfSk; ,izahFk; . ( Fwpg;G : m1 x m2 = -1 Mdhy; xd;Wf;nfhd;W nrq;Fj;J MFk; )

________________________________________________________________________________________________________________

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6.tbtpay;

1. Δ ABC –y; DE|| BC kw;Wk; AD/DB = 2/3 , AE = 3.7 cm vdpy; EC – ia fhz;f . A jPh;T : ΔABC –y; DE|| BC

𝐴𝐷

𝐷𝐵 =

𝐴𝐸

𝐸𝐶 (Njy;]; Njw;wk;) D E

EC = AExDB/AD = 3.7x3/2 = 5.55 cm B C

___________________________________________________________________________________________________________________________

2. Δ ABC –y; DE|| BC , AD = 6 , DB = 9 , AE = 8 vdpy; AC – If; fhz;f. (gapw;rpf;fhf ) 3. Δ ABC –y; DE|| BC , AD=8, AB=12, AE =12 vdpy; CE – If; fhz;f. (gapw;rpf;fhf )

_______________________________________________________________________________________________________________ 4. xU tl;lj;jpy; AB,CD vDk; ,U C C

ehz;fs; xd;iwnahd;W cl;Gwkhf A P-apy; ntl;bf;nfhs;fpd;wd. AP=4, B CP=8, DP=3, vdpy; PB ia fhz;f.

jPh;T :

PAXPB = PCXPD D

PB = PCXPD = 8x3 =6 PA 4

_______________________________________________________________________________________________________________ 5. xU tl;lj;jpy; AB,CD vDk; ,U A B P

ehz;fs; xd;iwnahd;W ntspg;Gwkhf P-apy; ntl;bf;nfhs;fpd;wd. AB=5, BP=4, DP=2, vdpy; CD ia fhz;f D C jPh;T :

PAXPB = PCXPD 9 x 4 = (2 + x ) 2 2+x = 36/2 ; x = 18 – 2 = 16

_________________________________________________________________________________________________

11.Gs;spapay ; 1. 43,24,38,56,22,39,45 Mfpa Gs;sp tptuq;fSf;F tPr;R kw;Wk; tPr;Rf;nfO fhz;f.

jPh;T : VWthpirapy; vOj : 22,24,38,39,43,45,56

tPr;R = L – S = 56 – 22 = 34

tPr;Rf;nfO = (L-S)/L+S) = (56-22)/(56+22) = 34/78 = 0.436

2. xU Gs;sp tptuj;jpd; kPr;rpW kjpg;G 12 > mjd; tPr;R 58 vdpy; mg;Gs;sp tptuj;jpd; kPg;ngU kjpg;G fhz;f.

jPh;T : kPg;ngU kjpg;G = kPr;rpW kjpg;G + tPr;R

L = S +R ; L = 12+58 =70 kPg;ngU kjpg;G = 70

____________________________________________________________________________________________________________

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3. Kjy; 13 ,ay; vz;fspd; jpl;ltpyf;fk; fhz;f . jPh;T : Kjy; n ,ay; vz;fspd; jpl;ltpyf;fk; = √(n2-1)/12 Kjy; 13 ,ay; vz;fspd; jpl;ltpyf;fk; = √(169 -1) /12 = √14 = 3.76 ___________________________________________________________________________________________________________ 4. xU Gs;sp tptuj;jpd; khWghl;Lf;nfO 57 kw;Wk; jpl;l tpyf;fk; 6.84 vdpy; mjd;

$l;Lr;ruhrhpia fhz;f. jPh;T : khWghl;Lf;nfO = (σ/x) x100

57 = (6.84 /x)100 = x = 684/57 = 12 $l;Lr;ruhrhp x = 12

_________________________________________________________________________________________

12.epfo;jfT 1. xU rPuhd gfil cUl;lg;gLk;NghJ fpilf;fg; ngWk; vz;fs; m) xw;iw vz;fshf jPh;T : $Wntsp S = { 1,2,3,4,5,6 } n(S) = 6 m) xw;iw vz;fs; fpilf;Fk; epfo;r;rp fzk; A vd;f . A = {1,3,5 } , n(A) = 3 , P(A) = n(A) /n(S) =3/6 =1/2

M) ,ul;il vz;fs; fpilf;Fk; epfo;r;rp fzk; B vd;f B = { 2,4,6 } , n(B) = 3 , P(B) = n(B) /n(S) =3/6 =1/2

,) gfh vz;fs; fpilf;Fk; epfo;r;rp fzk; C vd;f C = {2,3,5 } , n(c ) = 3 P(C) = n(c ) /n(S) = 3/6 =1/2 ________________________________________________________________________________________________________________ 2. ,U ehzaq;fs; xU Nru Rd;lg;gLk; NghJ fPo;f;fhZk; epfo;r;rpfSf;fhd epfo;jfT

fhz;f. i) ,U jiyfs; fpilj;jy; ii) Fiwe;jJ xU jiy fpilj;jy; iii) xU g+ kl;Lk; fpilj;jy; .

jPh;T : $Wntsp S = { HH , HT , TH , TT } , n(S) =4 i) ,U jiyfs; fpilf;f epfo;r;rp fzk; A vd;f .

A = { HH } , n(A) = 1 , P(A) = n(A)/n(S) = 1/4 ii) Fiwe;jJ xU jiy fpilf;Fk; epfo;r;rp fzk; B vd;f

B = { HH , HT , TH } , n(B) = 3 , P(B) = n(B)/n(S) = 3/4 iii) xU g+ kl;Lk; fpilf;Fk; epfo;r;rp fzk; C vd;f C = { HT , TH } , n(c ) = 2 P(C) = n(C ) = 2/4 =1/2 _________________________________________________________________________________________________________________

3. ,U ehzaq;fis xNu rkaj;jpy; Rz;Lk; NghJ mjpfgl;rkhf xU jiy fpilg;gjw;fhd epfo;jfT fhz;f.

jPh;T : $Wntsp S = { HH , HT , TH , TT } , n(S) =4 mjpfgl;rkhf xU jiy fpilg;gjw;fhd epfo;r;rp fzk; A vd;f A = { HH , HT , TH } n(A) = 3 P(A) = n(A)/n(S) =3/4 _______________________________________________________________________________ @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

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fztpay; : 1. fz tpj;jpahrk; njhlh;ghd bkhh;;fd; tpjpfs; :

i) A\(B∩C) = (A\B)U(A\C)

ii) A\(BUC) = (A\B)∩(A\C)

2. fz epug;gpf;fhd njhlh;ghd bkhh;;fd; tpjpfs; i) (AUB)’ = A’∩B’ ii) (A∩B)’ = A’UB’ 3. gq;fPl;L tpjpfs; : i) A U(B∩C) = (AUB)∩ (AUC)

ii) A∩(BUC) = (A∩B)U(A∩C)

nka;naz;fspd; njhlh;thpirfSk; njhlh;fSk; 1. $l;Lj;njhlhpd; n-tJ cWg;G : tn = a+(n-1)d

2. $l;Lj;njhlhpd; cWg;Gf;fspd; vz;zpf;if : n =( l-a)/d +1

3. $l;Lj;njhlhpd; n cWg;Gf;fs; tiu $Ljy; : Sn = n/2[2a+(n-1)]d)

4. $l;Lj;njhlhpd; n cWg;Gf;fs; tiu $Ljy; : Sn = n/2 (a+ l)

5. ngUf;Fj;njhlhpd; n cWg;Gf;fs; tiu $Ljy; : Sn= a (rn-1/r-1) if r>1

6. ngUf;Fj;njhlhpd; n cWg;Gf;fs; tiu $Ljy; : Sn= a (1-rn/1-r) if r<1 7. Kjy; n ,ay; vz;fspd; $Ljy; : n(n+1)/2

8. Kjy; n xw;iw ,ay; vz;fspd; $Ljy; : n2 , n= (l+1)/2 9. Kjy; n ,ay; vz;fspd; th;f;fq;fspd; $Ljy; : n(n+1)(2n+1)/2x3

10. Kjy; n ,ay; vz;fspd; fdq;fspd; $Ljy; : [n(n+1)/2]2 3.,aw;fzpjk ;

1. G+[;[paq;fspd; $Ljy; (%yq;fspd; $Ljy;) : -b/a 2. G+[;[paq;fspd; ngUf;fy; (%yq;fspd; ngUf;fy;) : c/a 3. %yq;fspd; jd;ikia Muha;jy; :

t.vz; jd;ikf;fhl;b %yq;fspd; jd;ik

1 b2 -4ac > 0 nka;naz;fs;; > rkkpy;iy

2 b2 -4ac = 0 nka;naz;fs; > rkk;

3 b2 -4ac < 0 fw;gid

R+j;jpuq;fs;:

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5.Maj;njhiy tbtpay ; 1. ,U Gs;spfSf;fpilNa cs;s njhiyT fhz : √(x1-x2)2-(y1-y2)2 2. Nfhl;Lj;Jz;bd; eLg;;;;Gs;sp fhz : [x1+x2]/2 , [y1+y2]/2 3. Kf;Nfhzj;jpd; eLf;Nfhl;Lr;re;jp : [x1+x2+x3]/3 , [y1+y2+y3]/3 4. ,U NfhLfs; xd;Wf;nfhd;W ,izahf epge;jid : m1 = m2

5. ,U NfhLfs; xd;Wf;nfhd;W nrq;Fj;jhf epge;jid : m1 xm2 = -1

6. Neh;f;Nfhl;bd; rkd;ghLfs;

t.vz; mikg;G rkd;ghL 1 rha;T( m) – yntl;L c Y = mx + c 2 rha;T( m) - Gs;sp (x1,y1) y–y1 = m(x-x1) 3 ,U Gs;spfs; (x1,y1) (x2,y2) (y-y1)/(y2-y1) =( x-x1) /(x2-x1)

4 ntl;Lj;Jz;Lfs; xntl;L= a, yntl;L=b (x/a) + (Y/b) = 1

8.mstpay ; fd cUtq;fs ; : t.vz; cUtk; fd msT

(f.m) tisgug;G (r.m)

nkhj;jg;gug;G (r.m)

1 jpz;k cUis Πr2h 2π rh 2π r(h+r)

2 jpz;k $k;G 1/3 Πr2h Πrl πr(l+r)

3 jpz;k Nfhsk; 4/3 Πr3 4πr

2 4πr

2

4 jpz;k miuf;Nfhsk; 2/3 Πr3 2πr

2 3πr

2

11.Gs;spapay;

1. tPr;R = L – S (kPg;ngU kjpg;G – kPr;rrpW kjpg;G )

2. tPr;Rf;nfO = (L-S)/(L+S)

3. jpl;ltpyf;fk; = √∑d2/n 4. khWghl;Lf;nfO = (σ/x )x100%

12.epfo;jfT $Wntsp (S) :

1. rPuhd ehzaj;ij xU Kiw Rz;Ljy; = S={H,T} [ n(S) =2 ] 2. rPuhd ehzaj;ij ,U Kiw Rz;Ljy; = S={HH,HT,TH,TT} [ n(S) =4

]

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3. rPuhd gfilia xU Kiw cUl;Ljy; = S={1,2,3,4,5,6} [ n(S) =6 ]

4. rPuhd gfilia ,U Kiw cUl;Ljy; = S={(1,1)…(6,6)} [ n(S) =36]

5. xU cWjpahd epfo;r;rpapd; epfo;jfT = 1 6. ,ayhj epfo;r;rpapd; epfo;jfT = 0 7. xU epfo;r;rpapd; epfo;jfT = 0 ≤ P(A) ≤ 1 8. epfo;jftpd; $l;ly; Njw;wk; = P(AUB) = P(A) +P(B) – P(AnB) 9. ,U epfo;r;rpfs; xd;iwnahd;W tpyf;Fk;

epfo;r;rpfs; vdpy; = P(AUB) = P(A) +P(B) 10. ,U epfo;r;rpfs; xd;iwnahd;W tpyf;fh

epfo;r;rpfs; vdpy; = P(AUB) = P(A) +P(B) – P(AnB) epfo;jfT P(A) = n(A)/n(B) P(A) = 1 - P(A’) P(A’) = 1 – P(A) @@@@@@@@@@@@@@&&&&&&&&&&&&&&&&&&@@@@@@@@@@@@@@@@

Wish You All Success And All the best

By :

C . Sugumar , Headmaster , Govt .High School ,

Vilai -632 316. Tv malai District .