maths

khepyf; fy;tpapay; Muha;r;rp kw;Wk; gapw;rp epWtdk; - fw;wy; fl;lfk; +2 fzpjtpay; 2015-16 Page0

SPEED AND ACCURACY BRING SUCCESS IN MATHEMATICS

gd;dpuz;lhk; tFg;Gf; fzpjtpay;

BLUE PRINT

Chapter

No. Chapters No. of Questions Total

Marks 1 Mark 6 Marks 10 Marks

1 mzpfs; kw;Wk; mzpf;Nfhitfs; 4 2 1 26

2 ntf;lh; ,aw;fzpjk; 6 2 2 38

3 fyg;ngz;fs; 4 2 1 26

4 gFKiw tbtf; fzpjk; 4 1 3 40

5 tifEz;fzpjk; gad;ghLfs;-I 4 2 2 36

6 tifEz;fzpjk; gad;ghLfs;-II 2 1 1 18

7 njhifEz;fzpjk; gad;ghLfs; 4 1 2 30

8 tiff;nfOr; rkd;ghLfs; 4 1 2 30

9 jdpepiy fzf;fpay; 4 2 1 26

10 epfo;jfTg; guty; 4 2 1 26

Total Number of Questions 40 16 16 296


; khzth;fSf;fhd vspikahd ghlg; gFjpfSk; mjd; kjpg;ngz;fSk;

Gj;jfj;jpYs;s midj;J 271 xU kjpg;ngz; tpdhf;fspYk; gapw;rp ngw;why;

nghJj; Njh;tpy; 30 kjpg;ngz;fs; ngw KbAk;. Mf nkhj;jk; ml;ltiz I Kjy;

ml;ltiz IV kw;Wk; xU kjpg;ngz; tpdhf;fspy; khzth;fs; KOikahfg; gapw;rp ngw;W

134 kjpg;ngz;fis vspjhfg; ngw KbAk;.

TABLE - I

Chapter

No. Chapters

No. of

6 & 3 Mark

Questions

No. of

10 Mark

Questions

Total

Marks

2 ntf;lh; ,aw;fzpjk; --- 20(2) 20

4 gFKiw tbtf; fzpjk; --- 28(3) 30 2-Mk; ghlj;jpYs;s 20 - gj;J kjpg;ngz; tpdhf;fspYk; 4-Mk;

ghlj;jpYs;s 28 - gj;J kjpg;ngz; tpdhf;fspYk; KOikahfg; gapw;rp ngw;why;

nghJj; Njh;tpy; 50 kjpg;ngz;fs; ngw KbAk;.

TABLE - II

Chapter

No. Chapters

No. of

6 & 3 Mark

Questions

No. of

10 Mark

Questions

Total

Marks

9 jdpepiy fzf;fpay; 33+12(2) 15(1) 22

6 tifEz;fzpjk; gad;ghLfs;-II --- 11(1) 10

3 fyg;ngz;fs; --- 16(1) 10

9-Mk; ghlj;jpYs;s 15 - gj;J kjpg;ngz;;> 33 – MW kjpg;ngz; 12- %d;W

kjpg;ngz;;> 6-Mk; ghlj;jpYs;s 11 - gj;J kjpg;ngz;;;> 3-Mk; ghlj;jpYs;s 16 -

gj;J kjpg;ngz; tpdhf;fspYk; KOikahfg; gapw;rp ngw;why; nghJj; Njh;tpy; 42

kjpg;ngz;fs; ngw KbAk;.

TABLE - III

Chapter

No. Chapters

No. of

6 & 3 Mark

Questions

No. of

10 Mark

Questions

Total

Marks

1 mzpfs; kw;Wk; mzpf;Nfhitfs; 35+13(2) --- 12

10 epfo;jfTg; guty; --- --- --- 1-Mk; ghlj;jpYs;s 35 - MW kjpg;ngz; tpdhf;fspYk; 13 - %d;W kjpg;ngz; tpdhf;fspYk; KOikahfg; gapw;rp ngw;why; nghJj; Njh;tpy; 12 kjpg;ngz;fs; ngw KbAk;.

TABLE - IV

Chapter

No. Chapters

No. of

6 & 3 Mark

Questions

No. of

10 Mark

Questions

Total

Marks

5 tifEz;fzpjk; gad;ghLfs;-I --- --- ---

7 njhifEz;fzpjk; gad;ghLfs; --- --- ---

8 tiff;nfOr; rkd;ghLfs; --- --- ---


fzpjk; - 10 kjpg;ngz; tpdhf;fSk; kw;Wk; vspikahd tpilfSk;

2. ntf;lh; ,aw;fzpjk; (2 x 10 = 20 kjpg;ngz;fs;)

1) Cos(A–B) = CosA CosB + SinA sinB vd epWTf. jPh;T:

P(CosA, SinA)kw;Wk;Q(CosB, SinB)vd;gd

Oia ikakhff; nfhz;l myF tl;lj;jpy; cs;s VNjDk; ,U Gs;spfs;

kw;Wk; vd;w myF

ntf;lh;fis x, ymr;Rj; jpirfspy; vLj;Jf; nfhs;f.

=CosA + SinA

=CosB + SinB

. = Cos(A – B) = Cos(A – B)……...(1)

. = CosA CosB + SinA SinB ……(2)

(1), (2) ypUe;J Cos (A–B) = CosA CosB + SinA SinB

3)ep&gpf;f: Cos(A+B) = CosA CosB – SinA SinB

jPh;T:

P(CosA, SinA)kw;Wk;Q(CosB, –SinB) vd;gd O ia ikakhff; nfhz;l myF tl;lj;jpy; cs;s VNjDk; ,U Gs;spfs;.

kw;Wk; vd;w myF


= CosA + SinA

= CosB – SinB

. = Cos (A+B)= Cos(A+B)………(1)

. = CosA CosB – SinA SinB…….(2)

(1), (2) ypUe;J Cos (A+B) = CosA CosB – SinA SinB

2) Sin(A–B) = SinA CosB – CosA SinB vd ntf;lh; Kiwapy; ep&gp. jPh;T:

P(CosA, SinA) kw;Wk;Q(CosB, SinB) vd;gd

Oia ikakhff; nfhz;l myF tl;lj;jpy; cs;s VNjDk; ,U Gs;spfs;

kw;Wk; vd;w myF

ntf;lh;fis x, y mr;Rj; jpirfspy; vLj;Jf; nfhs;f.

= CosA + SinA

= CosB + SinB

x = Sin (A–B)

= (1) (1) Sin (A–B)

= Sin (A–B)…………..(1)

x =

= (SinA CosB –CosA SinB)………….(2)

(1), (2) ypUe;J Sin (A–B) = SinA CosB – CosA SinB

4) Sin (A+B) = SinA CosB + CosA SinB vd ntf;lh;Kiwapy; ep&gp.

jPh;T:

P(CosA, SinA) kw;Wk; Q (CosB, –SinB) vd;gd Oia ikakhff; nfhz;l myF tl;lj;jpy; cs;s VNjDk; ,U Gs;spfs;

kw;Wk; vd;w myF


= CosA + SinA

= CosB – SinB

x = Sin (A+B)

= (1) (1) Sin (A+B)

= Sin (A+B)…………..(1)

x =

= (SinA CosB + CosA SinB)……….(2)

(1), (2) ypUe;J Sin (A+B)= SinA CosB +CosA SinB


5)xU Kf;Nfhzj;jpd; Fj;Jf;NfhLfs; xNu Gs;spapy; re;jpf;Fk; vd;gjid ntf;lh; Kiwapy; epWTf. jPh;T:

Kf;Nfhzk;ABC –apy; Fj;Jf;

NfhLfs; ADkw;Wk; BEvd;gd

O–apy; ntl;Lfpd;wd.

CO vd;gJAB–f;F nrq;Fj;J vd epWTf.

= , = , =

⇒ = 0

. =0

= 0…………. (1)

= 0⇒ . )= 0

= 0…………. (2)

(1) +(2) ⇒ =0

( ) . =0⇒ . =0

vdNt Kf;Nfhzj;jpd; Fj;Jf; NfhLfs; xNu Gs;spapy; ntl;Lfpd;wd.

7) If = + , = + , = + +

kw;Wk; = + +2 , vdpy;

vd;gijr; rhpghh;f;f. jPh;T:

× =

= (1 – 0) – (1 – 2)+ (0 – 2)

=

× =

= (2 – 1) – (4 – 1)+ (2 – 1)

=

× ) × ( × )=

= (1 – 6) – (1 + 2)+ (–3 –1)

= ….(1)

[ ] =

=1(0 – 1) –1(2 – 2)+1(2 – 0)=1

] =

=1(0 – 1) –1(4 – 1)+ 1(2 – 0)= –2

] – [ ]

= –2( + + ) – 1( + +2 )

= –5 –3 –4 …..(2)

(1) , (2) ypUe;J

6) If = + - , = + ,

= - vdpy;

× × ) =

vdr; rhpghh;f;f. jPh;T:

=

= (0 – 5) – (6 – 0)+ (–2 –0)

=

× ( )=

= (–6 – 6) – (–4 – 5)+ (–12 + 15)

= …………(1)

. = =6

. = )= –9

=6( )+9(

= …………(2)

(1) , (2) ypUe;J

× × )=


8) = = kw;Wk; = = vd;w

NfhLfs; ntl;Lk; vdf; fhl;b mit ntl;Lk; Gs;spiaf; fhz;f.

jPh;T: = + –

= 4 –

= 3 – ,

= 2 + 3

– = 3 –

[ – , , ] = = 0

/ NfhLfs; ntl;bf; nfhs;fpd;wd.

= = = ⇒ Nfhl;by; cs;s VNjDk; xU Gs;sp

( …….(1)

= = = µ ⇒ Nfhl;by; cs;s VNjDk; xU

Gs;sp(2 ……..(2)

(1) , (2)–ypUe;J–λ + 1 = 0 (or) 3µ – 1 = –1

, µ = 0

∴ ntl;Lk; Gs;sp(4, 0 ,–1)

10)(2, –1, –3) topNar; nry;yf;$baJk;

= = kw;Wk; = = Mfpa

NfhLfSf;F ,izahf cs;sJkhd jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f. jPh;T:

= – – 3

= – 4 ;

= 2 – 3 + 2

ntf;lh; rkd;ghL

= + + t

= 2 – –3 + s( – 4 )+t (2 –3 + 2 )

fhh;Brpad; rkd;ghL

=0

=0

(x – 2) (–8) – (y + 1) (14) + (z + 3) (–13) = 0

–8x + 16 – 14y –14–13z –39 = 0

8x + 14y + 13z + 37 = 0

9) = = kw;Wk; = = vd;w

NfhLfs; ntl;bf; nfhs;Sk; vdf; fhl;Lf. NkYk; mit ntl;Lk; Gs;spiaf; fhz;f. jPh;T:

= – ; =2 + –

= – +3 : = +2 –

– = +2 –

[ – , , ] = =0

∴ NfhLfs; ntl;bf; nfhs;fpd;wd

= = = ⇒ Nfhl;by; cs;s VNjDk; xU

Gs;sp( …….(1)

= = = µ Nfhl;by; cs;s VNjDk; xU

Gs;sp( ……...(2)

(1) ,(2)–y; ,Ue;J

⇒ λ – µ = 1 ……….(3)

–λ –1 = 2µ + 1 ⇒ –λ –2µ = 2 ………….(4)

(3) + (4) ⇒µ = –1, λ = 0

∴ ntl;Lk; Gs;sp (1,–1,0)

11)(1, 3, 2) vd;w Gs;sp topr; nry;tJk;

= = kw;Wk; = = vd;w

NfhLfSf;F ,izahdJkhd jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f. jPh;T:

= + +2

= +3 ;

= +2 +2

ntf;lh; rkd;ghL

= + + t

= + + 2 + s( +3 )+ t ( +2 +2 )

fhh;Brpad; rkd;ghL

= 0

= 0

(x – 1) (–8) – (y – 3) (1) + (z – 2) (5) = 0

8x + y – 5z – 1 = 0


12) = = vd;w Nfhl;il cs;slf;fpaJk;

= = vd;w Nfhl;bw;F

,izahdJkhd jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f.

jPh;T: = +2 +

= +3 , = +2 +

ntf;lh; rkd;ghL = + + t

= 2 + 2 + +s( +3 )+t( +2 + )

fhh;Brpad; rkd;ghL =0

=0

(x – 2) (–3) – (y – 2) (–7) + (z – 1) (–5) = 0

–3x + 6 + 7y – 14 –5z + 5 = 0

× by –1⇒ 3x – 7y +5z + 3 = 0

15)(–1, 1, 1) kw;Wk; (1, –1, 1) Mfpa Gs;spfs;

topNar; nry;yf; $baJk; x + 2y +2z =5 vd;w jsj;jpw;F nrq;Fj;jhf miktJkhdjsj;jpd; ntf;lh; kw;Wk; fhh;Brpad;rkd;ghLfisf; fhz;f.

jPh;T: = + +

= + ; = +2 +2

ntf;lh; rkd;ghL =(1–s) + + t

= (1 – s)( + + + s( + ) + t ( + + )


=0

( x + 1) (–4) – (y – 1) (4) + (z – 1) (6) = 0

– 4x – 4 – 4y + 4 + 6z –6 = 0

÷ by –2⇒2x + 2y –3z + 3 = 0

13) (–1, 3, 2) vd;;w Gs;sp topr; nry;tJk;

x + 2y + 2z = 5kw;Wk;3x + y + 2z = 8Mfpa jsq;fSf;Fr; nrq;Fj;jhdJkhd jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f.

jPh;T: = + +2

= +2 ; =3 + +2

ntf;lh; rkd;ghL = + + t

=– + +2 +s( +2 )+ t ( + + )

fhh;Brpad; rkd;ghL = 0

= 0

(x+1)(2) – (y–3) (–4) + (z–2) (–5) = 0

2x + 4y – 5z = 0

16)(1,2,3) kw;Wk; (2,3,1) vd;w Gs;spfs;

top;Na nry;yf;$baJk; 3x – 2y +4z –5 = 0

vd;w jsj;jpw;Fr; nrq;Fj;jhfTk; mike;j jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f.

jPh;T: = + + ;

= + ; = –2 +4

ntf;lh; rkd;ghL = (1 – s) + + t

=(1–s)( + +3 )+s(2 + )+t(3 – + )


= 0

(x – 1) (0) – (y – 2) (10) + (z – 3) (–5) = 0

–10y –5z + 35 = 0

÷ by –5 ⇒2y + z – 7 = 0 14) A (1,–2,3)kw;Wk; B(–1, 2, –1) vd;w Gs;spfs;

top;Na nry;yf;$baJk; = = vd;w

Nfhl;bw;F ,izahdJkhd jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f.

jPh;T: = –2 +3

= – – ; =2 +3 +4

ntf;lh; rkd;ghL =(1 – s) + + t

=(1–s)( –2 +3 )+s( – )+t( + + )

fhh;Brpad; rkd;ghL

=0

= 0

(x – 1) (28) –(y + 2) (0) + (z – 3) (–14) = 0

28x – 14z + 14 = 0 ÷ by 14⇒2x – z + 1 = 0

17) = = vd;w Nfhl;il

cs;slf;fpaJk; (–1, 1, –1) vd;w Gs;sp

top;Na nry;yf; $baJkhd ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f.

jPh;T: = + – ; = + ; = +3 –2

ntf;lh; rkd;ghL = (1 – s) + + t

= (1–s)(– + – ) +s(2 + )+t(2 + – )


=0

(x + 1) (–8) – (y – 1) (–10) + (z + 1) (7) = 0

-8x + 10y + 7z – 11 = 0 × by –1⇒ 8x – 10y – 7z + 11 = 0


18)(2,2,–1), (3,4,2) kw;Wk; (7,0,6) Mfpa Gs;spfs; topNa nry;yf; $ba jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghl;ilf; fhz;f. jPh;T:

= + –

= +

= + 6

(i) ntf;lh; rkd;ghL

= (1–s–t) + + t

= (1–s–t) (2 + – + s(3 + )

+ t (7 + )

(ii) fhh;Brpad; rkd;ghL

= 0

= 0

(x – 2) (20) – (y – 2) (–8) + (z + 1) (–12) = 0

20x – 40 + 8y –16 –12z –12 = 0

20x + 8y – 12z – 68 = 0

÷ by 4⇒5x + 2y – 3z – 17 = 0

20) ntl;Lj;Jz;L tbtpy; xU jsj;jpd; rkd;ghl;ilf; fhz;f. jPh;T:

a, b, c vd;gd x, y, z mr;Rfspy; ntl;Lj;Jz;Lfs;

∴ =

=

=

(i) ntf;lh; rkd;ghL

=(1–s–t) + + t

=(1–s–t) a + + tc

(ii) fhh;Brpad; rkd;ghL

= 0

= 0

(x – a) (bc – 0) – y (– ac – 0) + z (0 + ab) = 0

xbc – abc + yac + zab = 0

xbc + yac + zab = abc

abc –My; tFf;f

19) +4 +2 ,2 kw;Wk; 7

Mfpatw;iw epiy ntf;lh;fshff; nfhz;l Gs;spfs; topNa nry;Yk; jsj;jpd; ntf;lh; kw;Wk; fhh;Brpad; rkd;ghLfisf; fhz;f. jPh;T:

= +4 +2

= 2 –

= 7 ntf;lh; rkd;ghL

= (1 – s – t) + + t

= (1–s–t) +4 +2 )+s(2 – )+ t(7 + )

fhh;Brpad; rkd;ghL

= 0

= 0

(x – 3) (–6) – (y – 4) (13) + (z – 2) (28) = 0

–6x – 13y + 28z + 14 = 0

× by –1⇒ 6x + 13y – 28z – 14 = 0


4. gFKiw tbtf;fzpjk; (3 x 10 = 30 kjpg;ngz;fs;)

1) vd;w gutisaj;jpw;F

mr;R> Kid> Ftpak;> ,af;Ftiu> nrt;tfyj;jpd; rkd;ghL kw;Wk; nrt;tfyj;jpd; ePsk; fhz;f. NkYk; mt;tistiufis tiuf.

jPh;T:.

(y + 3)2 = 8x , [ Y2 = 4aX]

X = x ⇒ x = X Y = y + 3 ⇒ y = Y – 3; a = 2 tyg;Gwk; jpwg;Gila tbtpy; mikfpwJ X, Y Ig;

nghWj;J x, y Ig; nghWj;J

x = X, y = Y–3

mr;R Y =0 y = –3

Kid (0,0) V (0, –3)

Ftpak; (a,0)=(2,0) F(2, –3)

,af;Ftiu X = –a, X = –2 x = –2

nrt;tfyk; X = a, X =2 x =2

nrt;tfyj;jpd; ePsk;

4a 8

(3) vd;w gutisaj;jpw;F

mr;R> Kid> Ftpak;> ,af;Ftiu> nrt;tfyj;jpd; rkd;ghL kw;Wk; nrt;tfyj;jpd; ePsk; Mfpatw;iwf; fhz;f. NkYk; mtw;wpd; tiuglq;fis tiuf.

jPh;T:

, [

X = x – 1 ⇒ x = X + 1; Y = y – 3 ⇒ y = Y + 3; a = 2 ,lg;Gwk; jpwg;Gila tbtpy; mikfpwJ

X, Y Ig; nghWj;J

x, y Ig; nghWj;J

x = X+1, y = Y+3

mr;R Y =0 y =3

Kid (0,0) V (1,3)

Ftpak; (–a,0)=(–2,0) F(–1,3)

,af;Ftiu X = a, X =2 x =3

nrt;tfyk; X = –a, X= –2 x= –1

nrt;tfyj;jpd; ePsk;

4a 8

2) vd;w gutisaj;jpw;F

mr;R> Kid> Ftpak;> ,af;Ftiu> nrt;tfyj;jpd; rkd;ghL kw;Wk; nrt;tfyj;jpd; ePsk; Mfpatw;iwf; fhz;f. NkYk; mtw;wpd; tiuglq;fis tiuf.

jPh;T:

= 12( y +1); [ ]

X = x – 3 ⇒ x = X + 3; Y = y + 1 ⇒y = Y – 1; a = 3

Nky;Nehf;»¤ jpwg;Gila tbtpy; mikfpwJ

X, Y Ig; nghWj;J

x,y Ig; nghWj;J

x= X+3,y = Y–1

mr;R X =0 x =3

Kid (0,0) V (3, –1)

Ftpak; (0, a)=(0,3) F(3,2)

,af;Ftiu Y = –a, Y = –3 y = –4

nrt;tfyk; Y = a, Y =3 y =2

nrt;tfyj;jpd; ePsk;

4a 12

(4) vd;w gutisaj;jpw;F

mr;R> Kid> Ftpak;> ,af;Ftiu> nrt;tfyj;jpd; rkd;ghL kw;Wk; nrt;tfyj;jpd; ePsk; fhz;f. NkYk; mt;tistiufis tiuf.

jPh;T:

= –8( y +2); [ ]

X = x – 1⇒ x = X + 1;Y = y + 2 ⇒ y = Y –2; a = 2

fPo;Nehf;»¤ jpwg;Gila tbtpy; mikfpwJ

X, Y Ig; nghWj;J

x, y Ig; nghWj;J

x=X+1, y = Y–2

mr;R X =0 x =1

Kid (0,0) V (1, –2)

Ftpak; (0,–a)=(0, –2) F(1, –4)

,af;Ftiu Y = a, Y =2 y =0

nrt;tfyk; Y = –a, Y = –2 y = –4

nrt;tfy ePsk;

4a 8


5) vd;w ePs;

tl;lj;jpd; ikaj; njhiyj; jfT> ikak;> Ftpaq;fs;> Kidfs; Mfpatw;iwf; fhz;f. kw;Wk; tiuglk; tiuf. jPh;T:

( + 4(

X = x – 4 ⇒ x = X + 4; Y = y – 2 ⇒y = Y + 2

= 100 , a = 10; = 25 , b = 5

nel;lr;R X mr;R topr; nry;fpwJ

ikaj;njhiyj;jfT = = =

ae =5

X, Y I nghWj;J x, y I nghWj;J

x = X+1,

y = Y–4

ikak; C (0,0) c (1, –4) Ftpaq;fs;

(0, ae)=(0, )

(0, –ae)=(0, )

(1,–4+ )

(1,–4 – )

Kidfs; A (0, a)= A (0,6)

(0, –a)= (0, –6)

A (1,2),

(1,–10)


x = X + 4,

y = Y + 2

ikak; (0,0) C (4,2)

Ftpaq;fs; (ae,0)=(5 ,0)

(–ae,0)=(–5 ,0),

(4+5 ,2)

(4 – 5 ,2)

Kidfs; A (a,0)= A (10,0)

(–a,0), (–10,0)

A (14,2),

(–6,2)

6) vd;w ePs;

tl;lj;jpd; ikaj; njhiyj; jfT> ikak;> Ftpaq;fs;> Kidfs; Mfpatw;iwf; fhz;f. kw;Wk; tiuglk; tiuf.

jPh;T:

36(

36( + 4 = 144

;

X = x – 1 ⇒ x = X + 1; Y = y + 4 ⇒ y = Y – 4

= 36 a = 6; = 4 b = 2

nel;lr;R Y mr;R topr; nry;fpwJ.

ikaj;njhiyj;jfT= = =

ae =4

7) vd;w ePs;

tl;lj;jpd; ikaj; njhiyj; jfT> ikak;> Ftpaq;fs;> Kidfs; Mfpatw;iwf; fhz;f. kw;Wk; tiuglk; tiuf.

jPh;T

16(

16

X = x + 1 ⇒ x = X–1; Y = y – 2 ⇒ y = Y + 2

; = 9 ⇒ b = 3

nel;lr;R Y mr;R topr; nry;fpwJ

ikaj;njhiyj;jfT

ae = 4× =√7

X, Y I nghWj;J

x, y InghWj;J

x = X–1,

y = Y +2

ikak; C (0,0) C (–1,2)

Ftpaq;fs; (0, ae)=(0, )

(0, –ae)=(0, )

(–1,2+ )

(1,2– )

Kidfs; A (0, a)= A (0,4)

(0, –a)= (0, –4)

A (–1,6)

(–1, –2)


8) =0 vd;w

mjpgutisaj;jpw;F ikaj;njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f. kw;Wk; tiuglk; tiuf. jPh;T:

⇒9( – 16( =199

9 – 16 =144

X = x – 1⇒ x = X +1 ; Y = y +2⇒ y = Y – 2

=16⇒ a =4 ; =9⇒ b =3

FWf;fr;R X–mr;rpy; cs;sJ.


ae =5


x = X + 1,

y = Y – 2

ikak; C (0,0) C (1,–2)

Ftpaq;fs; (ae,0)=(5,0)

(–ae,0)=(–5,0),

(6, –2)

(–4, –2)

cr;rpfs; A (a,0)= A (4,0)

(–a,0) = (–4,0)

A (5, –2),

(–3, –2)

9) vd;w

mjpgutisaj;jpw;F ikaj;njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfis fhz;f. NkYk; tistiuia tiuf. jPh;T:

– 4 =4

;

X = x +3⇒ x = X – 3; Y = y – 2⇒ y = Y +2

=4⇒ a =2, =1⇒ b =1

FWf;fr;R X–mr;rpy; cs;sJ


ae =

10) vd;w

mjpgutisaj;jpw;F ikaj;njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfis fhz;f. NkYk; tistiuia fhz;f. jPh;T:

9( – 16( = –164

9 – 16 = –144

– = 1 ,

X = x + 2 ⇒ x = X – 2, Y = y – 1 ⇒ y = Y + 1

= 9 ⇒ a = 3, = 16⇒ b =4

FWf;fr;R Y–mr;rpy; cs;sJ


ae =3 x =5


x = X–2,

y = Y +1

ikak; C (0,0) c (–2,1)

Ftpaq;fs; (0, ae)= (0,5)

(0,–ae)= (0, )

(–2,6)

(–2, –4)

Kidfs; A (0, a)= A (0,3)

(0, –a)= (0, –3)

A (–2,4),

(–2, –2)


x = X–3,

y = Y +2

ikak; C (0,0) c (–3,2)

Ftpaq;fs; F1(ae,0)=( 0)

F2(–ae,0)=( 0)

( 2)

,2)

cr;rpfs; A (a,0)=(2,0)

(–a,0)=(–2,0)

A (–1,2),

(–5,2)


11) vd;w

mjpgutisaj;jpw;F ikaj;njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfis fhz;f. NkYk; tistiuia fhz;f. jPh;T:

( – 3( = –18

– 3 = –12

– =1

X = x +3⇒ x = X – 3 Y = y – 1⇒ y = Y +1

=4⇒ a =2, =12⇒ b =2

FWf;fr;R Y–mr;rpy; cs;sJ

ikaj;njhiyj;jfT= = =2

ae =4

X, Y I nghWj;J

x, y I nghWj;J

x = X – 3,

y = Y +1

ikak; C (0,0) c (–3,1)

Ftpaq;fs; (0, ae)

= (0,4)

(0, –ae)

= (0 )

(–3,5)

(–3, –3)

Kidfs; A (0, a)

= A (0,2)

(0, –a)

= (0, –2)

A (–3,3),

(–3, –1)

12) xU thy; tpz;kPd; (comet) MdJ #hpaidr;

(sun)Rw;wpg; gutisag; ghijapy; nry;fpwJ. kw;Wk; #hpad; gutisaj;jpd; Ftpaj;jpy;

mikfpwJ. thy; tpz;kPd; #hpadpypUe;J 80 kpy;ypad; fp.kP. njhiytpy; mike;J ,Uf;Fk; NghJ thy; tp;z;kPidAk; #hpaidAk; ,izf;Fk;

NfhL ghijapd; mr;Rld; Nfhzj;jpid

Vw;gLj;Jkhdhy; (i)thy; tpz;kPdpd; ghijapd;

rkd;ghl;ilf; fhz;f. (ii)thy; tpz;kPd; #hpaDf;F vt;TsT mUfpy; tuKbAk; vd;gijAk; fhz;f.

(ghij tyJGwk; jpwg;Giljhf nfhs;f.) jPh;T:

rkd;ghL …….(1)

∆FQP , cos ( =

= ⇒ FQ = 40

= 1 ⇒ FP = PM

80 = 2a + 40

2a = 40⇒ a = 20

(i)thy; tpz;kPdpd; ghijapd; rkd;ghL

(ii) thy; tpz;kPd; #hpaDf;F mUfpy; mikAk;

njhiyT = 20 kpy;ypad; fpkP

13) xU uapy;Nt ghyj;jpd; Nky; tisT gutisaj;jpd; mikg;igf; nfhz;Ls;sJ. me;j

tistpd; mfyk; 100 mbahfTk; mt;tisapd; cr;rpg; Gs;spapd; cauk;

ghyj;jpypUe;J 10 mbahfTk; cs;sJ vdpy;> ghyj;jpd; kj;jpapypUe;J ,lg;Gwk; my;yJ

tyg;Gwk; 10 mb J}uj;jpy; ghyj;jpd; Nky; tisT vt;tsT cauj;jpy; ,Uf;Fk; vdf; fhz;f. jPh;T: ,q;F gutisak; fPo;Nehf;fpj; jpwg;Gilajhf vLj;Jf; bfhŸnth«;. mjhtJ

………….(1)

,J(50, –10)topahfr; nry;fpwJ.

= –4a (–10),4a =250

(1) ⇒

……..(2) BC = y1 vd;f

gutisaj;jpd; Nky; cs;sg; Gs;spB (10, )vd;f.

100

mb ∴

ghyj;jpd; Nky; tisT cauk;=10 –

=9 mb


14) xU uhf;nfl; ntbahdJ nfhSj;Jk;NghJ mJ xU gutisag; ghijapy; nry;fpwJ. mjd; cr;r cauk; 4kPI vl;Lk;NghJ mJ nfhSj;jg;gl;l ,lj;jpypUe;J fpilkl;lj;J}uk; 6 kP njhiytpYs;sJ. ,Wjpahff; fpilkl;lkhf12 kP njhiytpy; jiuia te;jilfpwJ vdpy;, Gwg;gl;l ,lj;jpy; jiuAld; Vw;gLj;jg;gLk; vwpNfhzk; fhz;f. jPh;T:

rkd;ghL ………….(1)

(–6, –4)vd;w Gs;sp tistiuapy; mikfpwJ

= –4a (–4)

a = 9

(1) = –9y

2x = –9

= –

= =

=

=

vwpNfhzk;

15)jiukl;lj;jpypUe;J 7.5kP cauj;jpy; jiuf;F

,izahf¥ nghUj;jg;gl;l xU FohapypUe;J

ntspNaWk; ePh; jiuiaj; njhLk; ghij xU gutisaj;ij Vw;gLj;JfpwJ. NkYk; ,e;j¥

gutisag; ghijapd; Kid Fohapd; thapy;

mikfpwJ. Foha; kl;lj;jpw;F 2.5kP fPNo ePhpd; gha;thdJ Fohapd; Kid topahfr; nry;Yk;

epiy Fj;Jf; Nfhl;bw;F 3 kPl;;lh; J}uj;jpy;

cs;sJ vdpy; Fj;Jf; Nfhl;bypUe;J vt;tsÎ

J}uj;jpw;F mg;ghy; ePuhdJ jiuapy; tpOk; vd;gij¡ fhz;f.

jPh;T:

rkd;ghL = –4ay …..(1)

(3, –2.5)vd;w Gs;sp gutisaj;jpy; mikfpwJ

= –4a (–2.5)

a =

= – y …..(2)

( , –7.5)vd;w Gs;sp gutisaj;jpy; mikfpwJ

= (–7.5)

=27

=3 kP

∴ ePuhdJ jiuapy; tpOk; J}uk; =3 kP

16) xU njhq;F ghyj;jpd; fk;gp tlk; gutisa tbtpYs;sJ. mjd; ghuk; fpilkl;lkhf¢ rPuhf

gutpAs;sJ. mijj; jhq;Fk; ,U J}z;fSf;F

,ilNaAs;s J}uk; 1500 mb. fk;gp tlj;ijj;

jhq;Fk; Gs;spfs; J}zpy; jiuapypUe;J 200 mb cauj;jpy; mike;Js;sd. NkYk; jiuapypUe;J fk;gp tlj;jpd; jho;thd

Gs;spapd; cauk; 70 mb> fk;gptlk; 122 mb cauj;jpy; jhq;Fk; fk;gj;jpw;F ,ilNa cs;s

nrq;Fj;J ePsk; fhz;f. (jiuf;F ,izahf).

jPh;T:

rkd;ghL =4ay …..(1)

(750,130)vd;w Gs;sp gutisaj;jpy; mikfpwJ

= 4a (130)

4a =

(1) ⇒ y…….(2)

P( ,52)vd;w Gs;sp gutisaj;jpy; mikfpwJ

(2) ⇒ = × 52

= 150 mb

jhq;Fk; fk;gj;jpw;F ,ilNa cs;s J}uk;=

=300 mb

17) xU njhq;F ghyj;jpd; fk;gp tlk; gutisa

tbtpypYs;sJ. mjd; ePsk; 40 kPl;lh; MFk;. topg;ghijahdJ. fk;gp tlj;jpd; fPo;kl;lg;

Gs;spapypUe;J 5 kPl;lh; fPNo cs;sJ. fk;gp

tlj;ijj; jhq;Fk; J}z;fspd; cauq;fs; 55

kPl;lh; vdpy;>30 kPl;lh; cauj;jpy; fk;gp tlj;jpw;F

xU Jiz¤ jhq;fp $Ljyhff; nfhLf;fg;gl;lhy;

mj;Jizj; jhq;fpapd; ePsj;ijf; fhz;f. jPh;T:

rkd;ghL = 4ay ………..(1)

(20,50) vd;w Gs;sp gutisaj;jpy; mikfpwJ

=4a(50)

a=2

(1) = 8y ……(2)

( ,25) vd;w Gs;sp gutisaj;jpy; mikfpwJ

(2) ⇒ = 8 × 25

= 200

=

= 10 mb

Jizjhq;fpapd; ePsk; = =20 mb


18)xU Nfh–Nfh tpisahl;L tPuh; tpisahl;Lg;

gapw;rpapd;NghJ mtUf;Fk; Nfh–Nfh Fr;rpfSf;Fk; ,ilNaAs;s J}uk; vg;nghOJk;

8kP Mf ,Uf;FkhW czh;fpwhh;. mt;tpU

Fr;rpfSf;F ,ilg;gl;l J}uk; 6kP våš, mth;

XLk; ghijapd; rkd;ghl;ilf; fhz;f. jPh;T:

glj;jpy; F1 , F2 Nfh–Nfh

Fr;rpfs;. P tpisahl;L tPuh;. rkd;ghL

………..(1)

2a = 8 ⇒ a = 4

× 4e = 6

e =

= 7

(1) ⇒

,JNt XLk; ghijapd; rkd;ghL MFk;

19)#hpad; Ftpaj;jpypUf;FkhW nkh;f;Fhp fpufkhdJ #hpaid xU ePs;tl;lg; ghijapy; Rw;wp tUfpwJ. mjd; miu nel;lr;rpd; ePsk;

36 kpy;ypad; iky;fs; MfTk; ikaj;

njhiyj; jfT 0.206 MfTk; ,Uf;fkhapd;

(i)nkh;f;Fhp fpufkhdJ #hpaDf;F kpf

mUfhikapy; tUk;NghJ cs;s J}uk;

(ii)nkh;f;Fhp fpufkhdJ #hpaDf;F kpfj; njhiytpy; ,Uf;Fk;NghJ cs;s J}uk; Mfpatw;iwf; fhz;f. jPh;T:

rkd;ghL …..(1)

miu nel;lr;rpd; ePsk;=a

=36kpy;ypad; iky;fs;

e=0.206⇒ ae =7.416

A = a –ae =36 – 7.416

=28.584 kpy;ypad; iky;fs

= a +ae =36+7.416

=43.416 kpy;ypad; iky;fs

(i)nkh;f;Fhp fpufj;jpw;Fk; #hpaDf;Fk; ,ilNa cs;s Fiwe;j J}uk;

=28.584 kpy;ypad; iky;fs;

(ii)nkh;f;Fhp fpufj;jpw;Fk; #hpaDf;Fk; ,ilNa cs;s mjpfkhd J}uk;

=43.416kpy;ypad; iky;fs;

20)xU ePs;tl;lg; ghijapd; Ftpaj;jpy; G+kp ,Uf;FkhW xU Jizf;Nfhs; Rw;wp tUfpwJ.

,jd; ikaj; njhiyj;jfT MfTk; G+kpf;Fk;

Jizf; NfhSf;Fk; ,ilg;gl;l kPr;rpW J}uk;

400 fpNyh kPl;lh;fs; MfTk; ,Uf;Fkhdhy; G+kpf;Fk; Jizf;NfhSf;Fk; ,ilg;gl;l mjpfgl;r¤ J}uk; vd;d?

jPh;T:

glj;jpy; F 1 G+kp.

P Jizf;Nfhs;. kPr;rpW J}uk;

CA – C =400⇒ a – ae =400

ae =800× =400

(i) G+kpf;Fk; Jizf;NfhSf;Fk; ,ilg;gl;l mjpfgl;r

J}uk; = A’C = CA’+CF1 = a + ae

=800+400=1200fp.kP

21)xU tisT miu–ePs;tl;l tbtj;jpy; cs;sJ.

mjd; mfyk; 48 mb> cauk; 20 mb.

jiuapypUe;J 10 mb cauj;jpy; tistpd; mfyk; vd;d? jPh;T:

rkd;ghL ………..(1)

2a =48⇒ a =24, b =20

………..(2)

( vd;w Gs;sp ePs;tl;lj;jpy; cs;sJ

= × 576

= × 24

= 12

∴mfyk; = 24 mb


22)xU ghyj;jpd; tisthdJ miu ePs;tl;lj;jpd; tbtpy; cs;sJ.

fpilkl;lj;jpy; mjd; mfyk; 40 mbahfTk;

ikaj;jpypUe;jJ mjd; cauk; 16mbahfTk;

cs;sJ våš, ikaj;jpypUe;jJ tyJ

my;yJ ,lg;Gwj;jpy; 9 mb J}uj;jpy; cs;s jiug;Gs;spapypUe;J ghyj;jpd; cauk; vd;d? jPh;T: miu ePs;tl;lj;jpd; rkd;ghL

……(1)

2a = 40 ⇒ a = 20

b = 16

⇒ ……..(2)

⇒(9, ) vd;w Gs;sp ePs;tl;lj;jpy; cs;sJ

= 1– =

= ft

cauk;= mb

23)xU EioT thapypd; Nkw;$iuahdJ

miu–ePs;tl;l tbtj;jpy; cs;sJ. ,jd;

mfyk; 20 mb ikaj;jpypUe;J mjd;

cauk; 18 mb kw;Wk; gf;fr; Rth;fspd;

cauk; 12 mb våš, VNjDk; xU gf;fr;

RthpypUe;J 4 mb J}uj;jpy; Nkw;$iuapd; cauk; vd;dthf ,Uf;Fk;?

jPh;T: rkd;ghL

………..(1)

2a = 20 ⇒a = 10,

b = 6

(6, vd;w Gs;sp ePs;tl;lj;jpy; cs;sJ

(1) ⇒

= 1–

=

=

= 4.8

Nkw;$iuapd; cauk; =12+4.8 = 16.8mb

24)xU rkjsj;jpd; Nky; nrq;Fj;jhf mike;Js;s

Rthpd; kPJ 15kP ePwKs;s xU VzpahdJ jsj;jpidAk; Rtw;wpidAk; njhLkhW efh;e;J nfhz;L ,Uf;fpwJ vdpy;> Vzpapd; fPo;kl;l

KidapypUe;J 6kP J}uj;jpy; Vzpapy;

mike;Js;s P vd;w Gs;spapd; epakg;ghijiaf; fhz;f. jPh;T:

___PAO = ___BPQ =

∆ PQB-y;

Cos =

∆ ARP-y;

Sin =

Cos 2 + Sin2 = 1

+ = 1

P( –,d; epakghij

,JxU ePs; tl;lk; MFk;.

25)mjpgutisaj;jpd; ikak; (2, 4) NkYk;

(2, 0) topNa nry;;fpwJ. mjd; njhiyj;

njhL NfhLfs; x + 2y – 12 = 0kw;Wk;

x – 2y + 8 = 0Mfpatw;wpw;F ,izahf

,Uf;fpd;wd.

jPh;T: njhiyj;njhL NfhLfs; fPof;;fz;l

NfhLfSf;F ,izahf mikfpwJ.

x + 2y – 12 = 0, x – 2y + 8 = 0

njhiyj;njhL NfhLfspd; tbtk;

x + 2y + = 0 …....(1)

x – 2y + m = 0 ……(2)

,it ikak;(2,4)topahfr; nry;fpwJ.

(1) ⇒ ⇒ = –10

(2) ⇒ ⇒ m = 6

njhiyj;njhL NfhLfspd; rkd;ghL

(1) ⇒ x + 2y – 10 = 0

(2) ⇒ x – 2y + 6 = 0

njhiyj;njhL NfhLfspd; Nrh;g;G¢ rkd;ghL

(x + 2y – 10) (x –2y + 6) = 0

mjpgutisa rkd;ghl;bd; tbtk;


(x + 2y – 10) (x – 2y + 6) + k = 0 ………(3)

(2,0)vd;w Gs;sp topr; nry;fpwJ.

(–8) (8) + k = 0

–64 + k = 0

k = 64

(3) ⇒ (x + 2y – 10)(x – 2y +6) + 64 = 0

,JNt mjpgutisaj;jpd; rkd;ghL

26) x + 2y – 5 = 0–I xU njhiyj;

njhLNfhlhfTk;>(6>0) kw;Wk; (–3>0) vd;w Gs;spfs; topNa nry;yf;$baJkhd nrt;tf mjpgutisaj;jpd; rkd;ghL fhz;f. jPh;T: njhiyj;njhL Nfhl;bd; rkd;ghL

x + 2y –5 = 0

mLj;j bjhiy¤bjhL Nfhl;bd; tbtk;

2x – y + l = 0

njhiyj;njhL NfhLfspd; Nrh;g;G¢ rkd;ghL

(x + 2y – 5) (2x – y+ l ) = 0 nrt;tf mjpgutisaj;jpd; rkd;ghl;bd; tbtk;

(x + 2y – 5) (2x – y + l ) + k = 0 ……….. (1)

(6,0)vd;w Gs;sp topr; nry;fpwJ.

(1) ⇒ (6+0–5) (12–0+l ) + k = 0

l +k = –12 …………(2)

(–3,0) vd;w Gs;sp topahfTk; nry;fpwJ.

(1) ⇒ (–3 + 0 –5) (–6 – 0 + l ) + K = 0

(–8) (–6 + l ) + k = 0

–8 + k = –48 ………….(3)

(2)&(3)–I jPh;f;f =4, k = –16

(1) ⇒ (x + 2y – 5 ) (2x – y + 4) – 16 = 0

27)5x +12y =9 vd;w Neh;;NfhL mjpgutisak;

–Ij; njhLfpwJ vd ep&gpf;f.

NkYk; njhL Gs;spiaAk; fhz;f.

jPh;T:5x + 12y = 9

, ⇒ m = , c =

⇒

⇒

= 9 × – 1 = =

vdNt, NfhlhdJ mjpgutisaj;ij¤ njhLfpwJ.

(I njhL Gs;sp= =

njhL Gs;sp=(5, )

28) x – y + 4 = 0 vd;w Neh;;NfhL ePs;tl;lk;

–f;F njhLNfhlhf cs;sJ vd

ep&gpf;f. NkYk; njhLk; Gs;spiaAk; fhz;f. jPh;T:

x – y +4 = 0 ⇒ y = x + 4⇒m = 1, c = 4

⇒

⇒ ⇒

,

= 12 × 1 + 4, = 16 =

∴vdNt, NfhlhdJ mjpgutisaj;ij¤njhLfpwJ.

njhL Gs;sp=

njhL Gs;sp=(–3,1)


6. tifEz; fzpjk; gad;ghLfs; - II (1 x 10 = 10 kjpg;ngz;fs;)

1) y = -Ïd; tistiu tiuf. 2) y = -Ïd; tistiutiuf. 3) -Ïd; tistiu

tiuf.

rhh;gfk;: (-

ePl;bg;G

fpilkl;l ePl;bg;G : (-

epiyf; Fj;J ePl;;bg;G: (-

ntl;Lj;Jz;L

x ntl;Lj;Jz;L = -1

y ntl;Lj;Jz;L= 1

Mjp Mjp topahfr; nry;yhJ

rhh;gfk;: (-

ePl;bg;G

fpilkl;l ePl;bg;G : (-

epiyf; Fj;J ePl;;bg;g: (-

ntl;Lj;Jz;L

x ntl;Lj;Jz;L = 0

y ntl;Lj;Jz;L= 0

Mjp Mjp topahfr; nry;Yk;

rhh;gfk;: [0

ePl;bg;G Kjy; kw;Wk; ehd;fhk; fhy;; gFjpapy; fhzg;ngWk; ntl;Lj;Jz;L

x ntl;Lj;Jz;L = 0

y ntl;Lj;Jz;L= 0

Mjp Mjp topahfr; nry;Yk;.

rkr;rPh; ve;j mr;Rf;Fk; rkr;rPh; fpilahJ

rkr;rPh;

Mjpiag; nghWj;J¢ rkr;rPh;

mikAk;

rkr;rPh;

xmr;Rf;F¢ rkr;rPh; MFk;.

njhiyj;njhL NfhLfs; njhiyj;njhL NfhLfs; mikahJ



Xhpay;Gj; jd;ik

( VWKfkhfr; nry;Yk;

Xhpay;Gj; jd;ik

( VWKfkhfr; nry;Yk;

Xhpay;Gj; jd;ik

y = VWKfkhfr; nry;Yk;

y = - ,wq;FKfkhfr; nry;Yk;

rpwg;Gg; Gs;spfs;

(- -Ïy; fPo;Gwkhf FopthFk;

(0, -Ïy; Nkw;Gwkhf FopthFk;

(0, 1) -Ïy; tisT khw;Wg; Gs;sp

rpwg;Gg; Gs;spfs;

(- -Ïy; fPo;Gwkhf FopthFk;

(0, -Ïy; Nkw;Gwkhf FopthFk;

(0, 0) -Ïy; tisT khw;Wg; Gs;sp

rpwg;Gg; Gs;spfs;

(0, 0) tisT khw;Wg; Gs;sp my;y


4) u = vdpy; A+yhpd; Njw;wj;ijg;

gad;gLj;jp tan u vdf; fhl;Lf.

jPh;T:

u = ⇒ = =f (x, y)

f vd;gJ gb cila rkgbj;jhd rhh;G

∴A+yhpd; Njw;wj;jpd; gb

f

x sin u

xcosu + ycos u = sin u

x tan u

7) u = –f;F vd;gij¢

rhpghh;f;f.

jPh;T:

= . = . =

= =

=

= = ……(1)

=

= = …….(2)

1)=(2)

5) A+yhpd; Njw;wj;ijg; gad;gLj;jp ep&gpf;f.

, if

jPh;T:

u = ⇒ tan u = =f (x, y)

⇒ f vd;gJ gb cila rkgbj;jhd rhh;G

∴A+yhpd; Njw;wj;jpd; gb

= 2f

= 2 tan u

+ = 2 tan u

8) u = sin 3x cos 4y –f;F vd;gij¢

rhpghh;f;f. jPh;T:

u = sin 3x cos 4y

= 3 cos 3x cos 4y

= –4sin3x sin 4y

= = –4 cos 3x 3 sin 4y

= –12 cos 3x sin 4y ……….(1)

= = 3 cos 3x (–sin 4y) 4

= –12 cos 3x sin 4y ……….(2)

(1),(2)ypUe;J

6) f(x,y)= – f;F A+yhpd; Njw;wj;ij

ep&gpf;f. jPh;T:

f(tx, ty) = = = f (x, y)

⇒ f vd;gJ gb –1 cila rkgbj;jhd rhh;G

∴ A+yhpd; Njw;wj;ij rhpghh;f;f

= –f,ij ep&gpf;f

= =

= –

= – = = f

∴A+yhpd; Njw;wk; ep&gpf;fg;gl;lJ.

9)u= –f;F vd;gij¢ rhpghh;f;f.

jPh;T:

=

=

= = ……(1)

= = ……(2)

(1),(2)ypUe;J


10) w = vd;w rhh;gpy; u = kw;Wk; v = y log x,

vDkhW ,Ug;gpd; kw;Wk; fhz;f.

jPh;T:

w =

u =

v = y log x ⇒

= Since

=

=

=

11)tifaPLfisg; gad;gLj;jp

y = + -d; Njhuha kjpg;Gid fzf;fpLf. jPh;T:

= 1,

X = 1 vdpy; y = 1+1 = 2

X = 1 vdpy;

= 0.0116

+ =

2 + 0.0116

= 2.0116

+ 2.0116


9. jdpepiy¡ fzf;fpay; (1 x 10 = 10 kjpg;ngz;fs;)

1)G+r;rpakw;w fyg;ngz;fspd; fzkhd C–

tiuaWf;fg;gl;l

kw;Wk; vd;w rhh;Gfs;

ahTk; mlq;fpa fzk; MdJ

rhh;Gfspd; Nrh;g;gpd; fPo; xU vgPypad; Fyk; mikf;Fk; vd epWTf.

jPh;T: G = vd;f.

o – rhh;Gfspd; Nrh;g;G

o

1)milg;G tpjp ml;ltizapy; ,Ue;J milg;G tpjp cz;ik MFk;.

2)Nrh;g;G tpjp rhh;Gfspd; Nrh;g;G nghJthf Nrh;g;G tpjpf;F cl;gLk;.

3)rkdp tpjp: G rkdp cWg;G MFk;

4)vjph;kiw tpjp

vd;gd –d; vjph;kiw MFk;.

5) ghpkhw;W tpjp ml;ltizapy; ,Ue;J ghpkhw;W tpjp cz;ikMFk;.

∴(G, o)xU vgPypad; Fyk; MFk;

3)( xU Fyj;ij mikf;Fk;

vdf;fhl;Lf.

jPh;T: G =

∙7d; ngUf;fy; kl;L 7

∙7 [1] [2] [3] [4] [5] [6]

[1] [1] [2] [3] [4] [5] [6]

[2] [2] [4] [6] [1] [3] [5]

[3] [3] [6] [2] [5] [1] [4]

[4] [4] [1] [5] [2] [6] [3]

[5] [5] [3] [1] [6] [4] [2]

[6] [6] [5] [4] [3] [2] [1]

1)milg;G tpjp ml;ltizapy; ,Ue;J milg;G tpjp cz;ik MFk;.

2)Nrh;g;G tpjp

ngUf;fy; kl;L 7nghJthf¢ Nrh;g;G tpjpf;F

cl;gLk;.

3)rkdp tpjp:[1] G rkdp cWg;G MFk;

4)vjph;kiw tpjp

vd;gd

vjph;kiw MFk;.

∴(G, ∙7)xU Fyk; MFk;

2)

vd;fpw fzk; mzpg;ngUf;fypd; fPo; xU

Fyj;ij mikf;Fk; vdf;fhl;Lf.

jPh;T:

I = , A = , B = ,

C = , D = E =

G =

I A B C D E

I I A B C D E

A A B I E C D

B B I A D E C

C C D E I A B

D D E C B I A

E E C D A B I

1)milg;G tpjp : ml;ltizapy; ,Ue;J milg;G tpjp cz;ik MFk;.

2)Nrh;g;G tpjp:rhh;Gfspd; Nrhh;g;G nghJthf Nrh;g;G tpjpf;F cl;gLk;.

3)rkdp tpjp:I= G rkdp cWg;G MFk;

4)vjph;kiw tpjp: I, A, B, C, D, E vd;gd I, B, A, C,

D, E –d; vjph;kiw MFk;.∴G mzpg;ngUf;fiyg; nghWj;j xU Fyk; MFk;

4)11–Ïd; kl;Lf;F fhzg;ngw;w ngUf;fypd; fPo;

vd;w fzk; xU vgPypad; Fyj;ij mikf;Fk; vdf; fhl;Lf

jPh;T: G =

∙11 – ngUf;fy; kl;L 11

∙11 [1] [3] [4] [5] [9]

[1] [1] [3] [4] [5] [9]

[3] [3] [9] [1] [4] [5]

[4] [4] [1] [5] [9] [3]

[5] [5] [4] [9] [3] [1]

[9 [9] [5] [3] [1] [4]

1) milg;G tpjp:ml;ltizapy; ,Ue;J milg;G tpjp cz;ik MFk;.

2)Nrh;g;G tpjp

ngUf;fy; kl;L 11nghJthf¢ Nrh;g;G tpjpf;F

cl;gLk;.

3)rkdp tpjp:[1] G rkdp cWg;G MFk;

4)vjph;kiw tpjp

vd;gd

vjph;kiw MFk;.

5)ghpkhw;W tpjp: ml;ltizapy; ,Ue;J ghpkhw;W tpjp cz;ik MFk;

∴(G,∙11 xU vgPypad; Fyk; MFk;


5) ; x , vd;w mikg;gpy; cs;s

mzpfs; ahTk; mlq;fpa fzk; G MdJ mzpg;ngUf;fypd; fPo; xU Fyk; vdf; fhl;Lf.

jPh;T:G = vd;f

1) milg;G tpjp: X =

Y = G, x ≠ 0, y ≠ 0

XY = G, [∵2xy ≠ 0]

∴milg;G tpjp cz;ik MFk;. 2) Nrh;g;G tpjp : mzpg; ngUf;fy; vg;nghOJk; Nrh;g;G tpjpf;F cl;gl;lJ

3) rkdp tpjp

E = rkdp cWg;G vd;f.

⇒XE=X ⇒2xe = x, e =

E= G rkdp cWg;G MFk;.

4) vjph;kiw tpjp: = vd;gJ

X–Ïd; vjph;kiw vd;f.

⇒ = , =

–Ïd; Neh;khW G

∴ G mzpg;ngUf;fiyg; nghWj;J xU Fyk; MFk;.

6) , a R – mikg;gpy; cs;s vy;yh

mzpfSk; mlq;fpa fzk; mzpg;ngUf;fypd; fPo; xU vgPypad; Fyj;ij mikf;Fk; vdf;fhl;Lf.

jPh;T: G =

1)milg;G tpjp: A= , B= G,

a, b R – .

AB = G[∵ ab ≠ 0]

∴ milg;G tpjp cz;ik MFk;.

2)Nrh;g;G tpjp : mzpg; ngUf;fy; vg;nghOJk; Nrh;g;G tpjpf;F cl;gl;lJ

3)rkdp tpjp:

E = rkdp cWg;G vd;f

AE = ⇒ ae = a ⇒ e =1

G. rkdp cWg;G MFk;.

4)vjph;kiw tpjp: = vd;gJ

A–Ïd; vjph;kiw vd;f

= E ⇒ =1⇒

–Ïd; Neh;khW G

5)ghpkhw;W tpjp

AB = = = BA

∴ ghpkhw;W tpjp cz;ik MFk;.

∴G mzpg;ngUf;fiyg; nghWj;J xU vgPypad; Fyk; MFk;.

7)(Z, xU Kbtw;w vgPypad; Fyk; vd fhl;Lf.

,q;F vd;gJ a b = a+b+2vDkhW tiuaWf;fg;gl;Ls;sJ.

jPh;T: Z = KOf;fspd; fzk;

a b = a+b+2

1) milg;G tpjp

∀ a, b Z ⇒ a b = a+b+2 Z

/milg;G tpjp cz;ik MFk;.

2) Nrh;g;G tpjp:⩝ a, b, c Z

a (b c)= a (b+ c+2)

= a +(b+ c+2)+2

= a + b + c +4

(a b c =(a + b +2) c

= a + b + c +4

⇒ a (b c)=(a b c

∴ Nrh;g;G tpjp cz;ik MFk;.. 3) rkdp tpjp:

e–I rkdp cWg;G vd;f.

e = ⇒ a + e +2= a ⇒ e = –2

–2 Z .rkdp tpjp cz;ik MFk;.

4) vjph;kiw tpjp

a–d; vjph;kiw

= –2⇒

⇒ Z vjph;kiw tpjp cz;ik MFk;.

5) ghpkhw;W tpjp:⩝ a, b Z

a b = a+b+2= b + a +2=

ghpkhw;W tpjp cz;ik MFk;

Z xU Kbtw;w fzk;. Mjyhy;

∴(Z, Kbtw;w vgPypad; Fyk; MFk;.


8) G vd;gJ kpif tpfpjKW vz; fzk; vd;f.

a = vDkhkW tiuaWf;fg;gl;l nrayp

,d; fPo; G xU Fyj;ij mikf;Fk; vdf;fhl;Lf.

jPh;T:G vd;gJ kpif tpfpjKW vz; fzk; vd;f.

a =

1)milg;G tpjp: ∀ a, b G , a = G

∴milg;G tpjp cz;ik MFk;.

2)Nrh;g;G tpjp: ⩝ a, b, c G

= = =

=

⇒ a (b c)=(a b c ∴Nrh;g;G tpjp cz;ik MFk;.

3)rkdp tpjp: e–I rkdp cWg;G vd;f

⇒ ⇒ = a, e =3

G rkdp tpjp cz;ik MFk;.

4)vjph;kiw tpjp:a–d; vjph;kiw

∴ =3⇒ =3⇒ = , G

vjph;kiw tpjp cz;ik MFk;. ∴(G, xU FykhFk;

9)1 Ij; jtpu kw;w vy;yh tpfpjKW vz;fSk;

mlq;fpa fzk; G vd;f. G –y; -I

a, b G. vDkhW

tiuaWg;Nghk;. (G, xU Kbtw;w vgPypad;

Fyk; vdf;fhl;Lf. jPh;T:

G =1–I jtpu kw;w vy;yh tpfpjKW vz;fs; mlq;fpa fzk;

1)milg;G tpjp:a, b, G, a ≠ 1, b ≠ 1

khwhf

a =1(or) b =1⇒⇐ to a, b, G,

G

2)Nrh;g;G tpjp:⩝ a, b, c G

=

=

=

=(

=

=

∴ Nrh;g;G tpjp cz;ik MFk;

3) rkdp tpjp:e–I rkdp cWg;G vd;f.

⇒

e(1–a)=0⇒ e =0, since a ≠ 1

0 ∴rkdp tpjp G+h;j;jpahfpwJ.

4) vjph;kiw tpjp

a–d; vjph;kiw ⇒

a ≠ 1, d; vjph;kiw

5) ghpkhw;W tpjp

ghpkhw;W tpjp cz;ik MFk;. G xU

Kbtw;w fzk; ∴(G, ) xU Kbtw;w vgPypad; Fyk; MFk;

10) –1–I jtpu kw;w vy;yh tpjpjKW vz;fSk;

cs;slf;fpa fzk; G MdJ

= vDkhW tiuaWf;fg;gl;l

nrayp - d; fPo; xU vgPypad; Fyj;ij mikf;Fk; vdf; fhl;Lf.

jPh;T G =–1-I jtpu kw;w vy;yh tpfpjKW

vz;fs; mlq;fpa fzk; =

1) milg;G tpjp

a, b, G, a ≠ –1 and b ≠ –1

khwhf

a = –1(or) b = –1⇒⇐ to a, b, G,

∴ G

milg;G tpjp cz;ik MFk;.

2) Nrh;g;G tpjp: ∀ a, b, c G

=

=

=

=(

=(

=

=

⇒ =

∴Nrh;g;G tpjp cz;ik MFk; 3) rkdp tpjp:

e –I rkdp cWg;G vd;f.

=

e(1+a)⇒ e =0, [ / a ≠ –1]

0 G rkdp tpjp G+h;j;jpahfpwJ 4) vjph;kiw tpjp

a–d; vjph;kiw ⇒

d; vjph;kiw G

vjph;kiw tpjp cz;ik MFk; 5) ghpkhw;W tpjp

a, b, G,

G xU Kbtw;w fzk;

∴(G, ) Kbtw;w vgPypad; Fyk; MFk;


11) G = vd;gJ $l;liyg;

nghWj;J xU Kbtw;w vgPypad; Fyk; vdf; fhl;Lf. jPh;T: 1)milg;G tpjp

∀ a + , c + d G, a, b, c, d Q

a + c, b + d G

∴milg;G tpjp cz;ik MFk;. 2) Nrh;g;G tpjp $l;ly; Nrh;g;G tpjp vg;NgHJk; cz;ik MFk;. 3) rkdp tpjp:

∀ a + G,vd;f

0=0+0 G

0 G ∴ rkdp tpjp G+h;j;jpahfpwJ

4) vjph;kiw tpjp

∀ a + G,vd;f.

G

= +

–d; vjph;kiw

vjph;kiw tpjp cz;ik MFk;. 5) ghpkhw;W tpjp

∀ a + , c+ G

(a + )+(c+ =(a + c)+(b + d)

=(c + a)+(d + b)

=(c+ )+(a + )

ghpkhw;W tpjp cz;ik MFk;.

G xU Kbtw;w fzk;.

∴(G,+) Kbtw;w vgPypad; Fyk; MFk;

12) G = vd;w fzkhdJ ngUf;fypd; fPo; xU vgPypad; Fyj;ij mikf;Fk; vdf; fhl;Lf. jPh;T:

G =

1) milg;G tpjp

∀ G, a, b z

= G, a+b z

∴milg;G tpjp cz;ik MFk;. 2) Nrh;g;G tpjp

G

)= . 2 b+c

=

= . =

∴ Nrh;g;G tpjp cz;ik MFk;. 3) rkdp tpjp:

∀ G,vd;f

20 =1 G⇒ 2a .1=1.2a

= 2a

∴1 G rkdp tpjp G+h;j;jpahfpwJ.

4) vjph;kiw tpjp

∀ G,vd;f G

= = =1

– d; vjph;kiw vjph;kiw tpjp cz;ik MFk

5) ghpkhw;W tpjp

∀ G

= = = . ghpkhw;W tpjp cz;ik MFk;.

∴(G, .) xU vgPypad; Fyk; MFk;

13) =1 vDkhW cs;s fyg;ngz;fs; ahTk;

mlq;fpa fzk; M MdJ fyg;ngz;fspd; ngUf;fypd; fPo; xU Fyj;ij mikf;Fk;

vdf; fhl;Lf.. jPh;T:

M =kl;L kjpg;G 1 cs;s midj;J¡

fyg;ngz;fspd; fzk; 1) milg;G tpjp

∀ M ⇒ M

= =1

∴ milg;G tpjp cz;ik MFk;. 2) Nrh;g;G tpjp

fyg;ngz; ngUf;fy; vg;nghOJk; Nrh;g;G tpjpf;F cl;gl;lJ

3) rkdp tpjp:

∀ M vd;f1 M ⇒ z.1=1 . z=z

1 G rkdp cWg;G MFk;. 4) vjph;kiw tpjp

∀ z M,vd;f M ⇒ z. = . z=1

[∵

∴z–d; vjph;kiw

(M, .) xU Fyk; MFk;


14)tof;fkhd ngUf;fypd; fPo; 1– ,d; n–Mk; gb %yq;fs; Kbthd Fyj;ij mikf;Fk; vdf;fhl;Lf.

jPh;T: G = ,

1) milg;G tpjp

, G,0 , m

, = G vd ep&gpf;f Ntz;Lk;

epiy(i) if + m n

If + m n then G

epiy(ii) if +m n tFj;jy; Nfhl;ghl;bd;gb

+ m =(q.n)+ r where 0 r < n

= = . = . G

∴ milg;G tpjp cz;ik MFk;. 2) Nrh;g;G tpjp tof;fkhd ngUf;fy; vg;nghOJk; Nrh;g;G tpjpf;F cl;gl;lJ 3) rkdp tpjp:

∀ G,vd;f 1 G ⇒ .1 = 1. =

∴1 G rkdp cWg;G MFk;. 4) vjph;kiw tpjp

∀ G,vd;f G

⇒ . = =l

–d; vjph;kiw vjph;kiw tpjp cz;ik MFk;. 5) ghpkhw;W tpjp

∀ G

. = = =

∴ ghpkhw;W tpjp cz;ik MFk;.

G–y; n cWg;Gfs; kl;LNk cs;sjhy;

∴(G, .). Kbthd vgPypad; Fyk; MFk;

15) xU Fyk; vdf; fhl;Lf.

Zn = { [0],[1],[2]…….[n-1] }

1) milg;G tpjp

[l], [m] Zn

[l] + n [m] = [l+m] if l+m < n

[r] if l+m ≥ n

l + m = qn + r, 0 ≤ r < n

[l] + n [m] Zn

2) Nrh;g;G tpjp

+ n vg;NghJk; Nrh;g;G tpjpia epiwT nra;Ak;

3) rkdp tpjp:

rkdp cWg;G [0] Zn

4) vjph;kiw tpjp

[l] Zn d; vjph;kiw [n-l] Zn

( xU Fyk; MFk;


3. fyg;ngz;fs; (1 x 10 = 10 kjpg;ngz;fs;) 1) P vDk; Gs;sp fyg;ngz; khwp z If; Fwpj;jhy;

P–d; epakg;ghijia fhz;f. Im = –2

jPh;T:

z = x + iyvd;f

= =

= ×

Im

= –2

–x(2x + 1) +2y(1 – y) = – 2[(1 – y)2 + x2 ]

–2x2 – x + 2y – 2y2 = –2(1 + y2 – 2y + x2)

–x + 2y = –2 + 4y

x + 2y – 2 = 0

,JNt P d; epakg;ghij MFk;.

4) P vd;Dk; Gs;sp fyg;G vz; khwp z If;

Fwpj;jhy; P d; epakg; ghijia gpd;tUk;

fl;Lghl;LfSf;F cl;gl;L¡ fhz;f.arg =

jPh;T:z=x+ iyvd;f

arg = arg (z – 1) – arg(z + 1) =

⇒ arg (x + iy – 1) – arg(x + iy + 1) =

⇒ arg [(x – 1) + iy] – arg[(x + 1)+ iy] =

– =

=

= tan

=

2y = ( )

= 0

,JNt P Ïd; epakg;ghij MFk;

2) P vDk; Gs;sp fyg;ngz; khwp zIf; Fwpj;jhy;

Pd; epakg;ghijia¡ fhz;f.

Re =1

jPh;T z = x + iyvd;f.

= =

= ×

Re = 1

(x – 1)x + y(y + 1) = x2 + (y + 1)2

x2 – x + y2 + y = x2 + y2 + 1 + 2y

x + y + 1 = 0

,JNtP d; epakg;ghij MFk; .

5) P vDk; Gs;sp fyg;ngz; khwp z If; Fwpj;jhy;

P d; epakg;ghijia¡ fhz;f. arg =

jPh;T:

z = x + iy vd;f.

arg =

⇒ arg (z – 1) – arg(z + 3) =

⇒ arg (x + iy – 1) – arg(x + iy + 3) =

arg [(x – 1 + iy] – arg[(x + 3) + iy] =

– =

=

= tan

= ∞

0 =

= 0

,JNt P Ïd; epakg;ghij MF

3) P vd;Dk; Gs;sp fyg;G vz; khwp z If; Fwpj;jhy;

P d; epakg;ghijia gpd;tUk; fl;Lghl;LfSf;F

cl;gl;L fhz;f. Re =1

jPh;T:

z = x + iy

= =

= ×

Re = 1

(x + 1)x + y(y + 1) = x2 + (y + 1)2

x2 + x + y2 + y = x2 + y2 + 1 + 2y

x – y – 1 = 0 ,JNt P d; epakg;ghij MFk;

tan = ∞


6) α , β vd;git x2–2x+2=0 –d; %yq;fs;

kw;Wk; cot θ = y+1 vdpy; =

vdf; fhl;Lf.

jPh;T:x2 –2x +2=0

x=1 i, α =1+I, = l–i vd;f

⇒ α–β =2i

cot θ = y+1 ⇒ y = cot θ –1= – 1

(y+α)n= =

(y+α)n= n =

= =

9) –,d; vy;yh kjpg;GfisAk; fhz;f.

jPh;T:

2 ( cos + i sin )

=

=

=

= [ cos( ) + i sin( )], k = 0, 1, 2

7)x2–2px+(p2+q2)=0 vd;w rkd;ghl;bd; %yq;fs;

α , β kw;Wk; tan θ = vdpy;

= qn–1 vd epWTf.

jPh;T: x2–2px +(p2+q2)=0⇒ x = p ± qi

α = p + qi, = p – qi ⇒ α–β =2iq

(y + α)n = = qn

(y + )n = (

(y + β)n = (

=

= qn–1

10) –,d; vy;yh kjpg;GfisAk; fhz;f.

jPh;T:

2 ( cos + i sin )

=

=

=

=

= ;k = 0, 1, 2

8) x2 – 2x + 4=0–,d; Kyq;fs; α ,β vdpy;

αn – βn = i2n+1sin mjpypUe;J α9– β9–d; kjpg;ig¥

ngWf.

Ôh;T: x2 – 2x + 4 = 0

⇒ x = 1 ± i

α = 1+ i = 2 ( cos + i sin )

= 1– i = 2 ( cos – i sin )

αn = 2n (cos + isin )

n = 2n (cos – i sin )

αn – n = 2n 2i sin

= i 2n+1sin

n = 9 α9 – β9 = i 29+1 sin = 0

11) – ,d; vy;yh kjpg;GfisAk; fhz;f.

kw;Wk; mjd; kjpg;Gfspd; ngUf;fw;gyd; 1 vdTk; fhl;Lf.

jPh;T –i = cos ( ) + i sin ( )

=

=

=

= , k = 0, 1, 2, 3

k = 0 vdpy;,

k = 1 vdpy;,

k = 2 vdpy;,

k = 3 vdpy;,

kjpg;Gfspd; ngUf;fw;gyd;

=

= = 1


12) x9+x5–x4–1=0 vd;w rd;ghl;ilj; jPh;f;f

jPh;T:

x9 + x5 – x4 – 1 = 0

x5(x4 + 1) – 1(x4 + 1) = 0

(x5 – 1)(x4 + 1) = 0

x5 – 1 = 0 ⇒ x = =

=

, k = 0, 1, 2, 3, 4

x4 + 1 = 0 ⇒ x =

=

=

, k = 0, 1, 2, 3

15) =2 cos , =2 cos vdpy;

(i) – nф)

(ii) – nф) vdf;fhl;Lf.

jPh;T:

x= cos ;

y= cosф + i sinф

=

= cos(m –nф)+ isin (m – nф)….(1)

∴ = cos(m –nф) – isin (m – nф)…..(2)

(1)+(2)⇒ = – nф)

(1) – (2)⇒ – nф)

13)x7+ x4 + x3 + 1 = 0 vd;w rkd;ghl;ilj; jPh;f;f.

jPh;T:

x7 + x4 + x3 + 1 = 0

x4(x3 + 1)+1(x3 + 1) = 0

(x4 + 1)(x3 + 1) = 0

x4 + 1= 0 ⇒x =

=

x =

, k = 0, 1, 2, 3

x3 + 1= 0 ⇒x =

=

x = [

, k = 0, 1, 2

16) a = cos 2 + isin 2 , b= cos 2 + isin 2 kw;Wk;

C = cos 2 + isin 2 vdpy;

(i) + =2 cos (

(ii) =2 cos ( vd epUgp.

jPh;T:

i)abc=(cos 2 + isin 2 ( cos 2 + isin 2

(cos 2 + isin 2

= cos 2( +i sin 2(

= cos ( +i sin ( ….(1)

∴ = cos ( –i sin ( …(2)

∴(1)+(2)⇒

+ =2 cos (

(ii) = +

=

= cos 2( + i sin 2( ….(3)

= cos 2( i sin2( …(4)

∴(3)+(4)⇒

+ =2 cos 2(α+β-γ)

=2 cos2(α+β-γ)

14)x4– x3 + x2 – x + 1 = 0 vd;w rkd;ghl;ilj; jPh;f;f.

jPh;T:

x4 – x3 + x2 – x +1 = 0

= 0

= –1, x ≠–1

x =

x =

=

x = cos + isin , k = 0, 1, 3, 4; x ≠ –1

k = 0 vdpy;, x = + i

k = 1 vdpy;, x = + i

k = 3 vdpy;, x = + i

k = 4 vdpy;, x = + i

(k=2My; Vw;gLk; %yj;ij ePf;f> fhuzk;x ≠ –1)


1. mzpfSk; mzpf;NfhitfSk; 6 kw;Wk; 3 kjpg;ngz; tpdhf;fs;

1) A = ,vdpy; A(adjA)=(adjA) A=

vd;gij¢ rhpghh;f;fTk;.

jPh;T:

|A|= =2

|A| I2 = …………(1)

adj A =

A (adjA)= = ……(2)

(adj A) A = = ……(3)

(1),(2),(3) ypUe;J A(adjA)(adjA) A=

2) A = vdpy;A (adj A)=(adj A) A

= vd;gij¢ rhpghh;f;fTk;.

jPh;T:

= = -11

= …………(1)

adj A =

A (adjA)= = ……(2)

(adj A) A = = ...…(3)

(1),(2),(3) ypUe;J A(adjA)(adjA) A=

3) A = kw;Wk; B = vdpy;

= vd;gij¢ rhpghh;f;fTk;.

jPh;T:

AB = =

= ………..(1)

= = …(2)

(1) , (2) ypUe;J =

4) A = kw;Wk; B

= vdpy; = vd;gij¢ rhpghh;f;fTk;

AB = =

= = 1 0

Adj(AB)=

= Adj(AB)= = ....(1)

= =1 0 Adj A =

= Adj(A) =

B = = = 1 0

Adj B =

= Adj(B) = =

= = .......(2)

(1),(2) ypUe;J =

5) A = kw;Wk; B = vdpy;

( = vd;gij¢ rhpghh;f;fTk;.

jPh;T:

AB = =

= =2–3= –1 ≠ 0

= = _____(1)

= = –1 ≠ 0

= =

= =1 ≠ 0

= adj (AB)


=

= ______(2)

(1)=(2)⇒( =

6) Nrh;g;G mzp fhz;f

A = adj A =

7) Nrh;g;G mzp fhz;f

A =

adj A =

8) Nrh;g;G mzp fhz;f A =

[Aij] =

adj A =

=

9) Nrh;g;G mzp fhz;f.

A =

[Aij] =

adj A =

=

10) Nrh;g;G mzp fhz;f.

A = [Aij] =

= =

adj A = =

11) A= vdpy; A apd; Nrh;g;G mzp A

vd epWTf.

A=

[Aij] =

adj A = = = A

12) A = –f;F A= vdf; fhl;Lf.

A =

A * A = X

= = I ∴ A =

13) vd;w mzpapd; Neh;khWfisf;

fhz;f.

jPh;T: A = =

=1(1–1) – 0+3(–2–1)= –9

adj A =

adj A =

= adj A


14 .gpd;tUk; mzpapd; Neh;khW fhz;f.

A =

= = = 1

adj A = =

= = adj A

=

15 .gpd;tUk; mzpapd; Neh;khW fhz;f.

A =

= = -1

adj A = =

= = adj A

=

=

6. gpd;tUk; mzpapd; Neh;khW fhz;f.

i)

A =

= = 35

adj A = =

= = adj A

17 . gpd;tUk; mzpapd; Neh;khW fhz;f.

A =

= = 5

adj A = =

= = adj A


A =

= = 2

adj A = =

= = adj A



A=

= =

vdNt> Neh;khW fhz ,ayhJ.


A=

= =

adj A =

= adj A

=

=


A=

= = 2

adj A =

= adj A

=

=

22) Neh;khW mzpfhzy; Kiwapy; jPh;f;f.

7x+3 y = -1, 2x + y = 0

Neh;khW cz;L

⇒ x = -1 kw;Wk; y = 2


2x - y = 7, 3x – 2y = 11

Neh;khW cz;L

⇒ x = 3 kw;Wk; y = -1

X =

X =



x+y = 3, 2x +3 y = 8

Neh;khW cz;L

⇒ x = 1 kw;Wk; y = 2

25) vd;w mzpapd; juk; fhz;f.

jPh;T:A =

~ ,

,e;j mzp VWgb tbtj;jpy; cs;sJ.

,q;F G+[;[pakw;w epiufspd; vz;zpf;if = 1

(A)=1


jPh;T:A =

~



(A)=2


jPh;T:

A =

~

~

~



(A)=3

28) vd;w mzpapd; juk; fhz;f

jPh;T:

A =

~

~


,q;F G+[;[pakw;w epiufspd; vz;zpf;if= 2

(A)=2

X =



jPh;T:

A =

~

~

~



(A)=2


jPh;T:

A =

~

~



(A)=2


jPh;T:

A =

~

~

~



(A)=3


jPh;T:

A =

~ ,

~



(A)=1



jPh;T:

A =

~

~



(A)=2


jPh;T:

A =

~

~

~

~



(A)=2


jPh;T:

A =

~

~



(A)=2


jPh;T:

A = ~

~

~

~



(A)=2

37) mzpf;Nfhit Kiwapy; jPh;f;f.

2x +3y =5,4x +6y =12

jPh;T:

i)

∴njhFg;G xUq;fikT ,y;yhjJ. vdNt, jPh;T fhz

,ayhJ.


38)3x +2y =5 , x +3y =4 mzpf;Nfhit Kiwapy; jPh;f;f.

jPh;T:

vd;gjhy; njhFjpf;F XNu xU jPh;T cz;L

fpuhkhpd; tpjpg;gb x =

= = 1

y =

= = 11

∴ jPh;T x = 1, y=1

39) x +y =3 , 2 x +3y =7 mzpf;Nfhit Kiwapy; jPh;f;f.

jPh;T:

vd;gjhy; njhFjpf;F XNu xU jPh;T cz;L

fpuhkhpd; tpjpg;gb x =

= = 2

y =

= = 1

∴ jPh;T x = 2, y=1

40) x -y = 2 , 3y =3x-7 mzpf;Nfhit Kiwapy; jPh;f;f.

jPh;T: jug;gl;l njhFjpapid fPo;f;fz;lthW vOjyhk;.

x -y = 2 , 3x -3y= 7

, vd;gjhy; njhFjpf;F jPh;T ,y;iy


x +y +2z =0, 2x+ y- z =0,2x + 2y +z =0

jPh;T: ≠ 0

vd;gjhy; njhFjpf;F XNu xU jPh;T cz;L. NkYk; njhFjpahdJ rkgbj;jhd rkd;ghLfspd; njhFjp vd;gjhy; njhFjpf;F ntspg;gilj; jPh;T kl;Lk; cz;L.

jPh;T: (x,y,z) == (0,0,0)


4x +5y =9,8x +10y =18

jPh;T:

(i)

(ii)

(iii) Fiwe;jjJ xU of

∴ njhFg;ghdJ xUq;fikTg; ngw;W vz;zpf;ifaw;w jPh;Tfis ngWfpd;wJ.

jPh;T fhz y = t vd;f. t R

⇒4x = 9-5t x =

∴(x, y)= t R


43)2x +3y =8,4x +6y =16 mzpf;Nfhit Kiwapy; jPh;f;f.

jPh;T:

(i)

(ii)

(iii) Fiwe;jjJ xU of

∴ njhFg;ghdJ xUq;fikTg; ngw;W vz;zpf;ifaw;w jPh;Tfis ngWfpd;wJ.

jPh;T fhz y = t vd;f. t R

⇒2x +3t =8 x =

∴(x, y)= t R


2x +2y +z =5, x– y + z =1,3x + y +2z =4

jPh;T:

≠ 0

(i)

(ii)

∴ njhFg;G xUq;fikT ,y;yhjJ vdNt jPh;T fhz ,ayhJ.

45) jPh;f;f x + y +2z =4,2x +2y +4z =8,

3x +3y +6z =10

jPh;T:

(i)

(ii)

(iii) –y; cs;s midj;J2x2rpw;wzpf;

Nfhitapd; kjpg;G

(iv) –y; VNjDk; xU 2x2rpw;wzpf;

Nfhitapd; kjpg;G 0

∴njhFg;G xUq;fikT ,y;yhjJ. vdNt, jPh;T fhz ,ayhJ.

46) jug;gl;Ls;s njhFjp xUq;fikT cilajh vdf;fhz;f. xUq;fikT cilaJ vdpy; jPh;T fhz;f.

x + y +z =7, x +2y +3z =18, y +2z =6

jug;gl;Ls;s njhFjpapd; mzpr;rkd;ghL

=

AX = B

=


(A)=2

NkYk; (A,B)=3 (A) ≠ (A,B)

∴njhFg;G xUq;fikT ,y;yhjJ. vdNt, jPh;T fhz ,ayhJ.


47) jug;gl;Ls;s njhFjp xUq;fikT cilajh vdf;fhz;f. xUq;fikT cilaJ vdpy; jPh;T fhz;f.

x -4 y +7z =14, 3x +8y -2z =13, 7x-8 y +26z =5

jug;gl;Ls;s njhFjpapd; mzpr;rkd;ghL

=

AX = B

=


(A)=2

NkYk; (A,B)=3

(A) ≠ (A,B)

∴njhFg;G xUq;fikT ,y;yhjJ vdNt jPh;T fhz ,ayhJ.

48) Neu;khWfSf;Fupa tupirkhw;W tpjp.

tpjp: Akw;Wk; B xNu tupir nfhz;l g+r;rpakw;w

Nfhit mzpfs; vdpy;

epUgzk;:A kw;Wk; B xNu tupir nfhz;l g+r;rpakw;w

Nfhit mzpfs;.

∴ AB Ak; g+r;rpakw;w Nfhit mzp

(AB) = = = I ….(1)

,NjNghy (AB) = I ………….(2)

(1), (2) ypUe;J (AB) = (AB) = I

∴ epUgpf;fg;gl;lJ.


9. jdpepiyf; fzf;fpay; - 6 kjpg;ngz; tpdhf;fSk; tpilfSk; f;F xU F ,Ue;jhYk; F tuntz;Lk;;. f;F xU T ,Ue;jhYk; T tuntz;Lk;.

f;F T F f;F F ……….. kw;wjw;F T f;F T T f;F T nkYk; F F f;F T … kw;wjw;F F

1) (iii) (pq)(~q)

p q (pq) ~q (pq)(~q)

T T T F F

T F T T T

F T T F F

F F F T F

2) (iv) ~ [(~ p)(~ )]

p q ~p ~q T T F F F T T F F T F T F T T F F T F F T T T F

3) 9.5: (pq)(~ r )

p q r (pq) (pq)(~ r )

T T T T F T

T T F T T T

T F T F F F

T F F F T T

F T T F F F

F T F F T T

F F T F F F

F F F F T T

4) 9.6: (pq) r

p q r (pq) (pq) r

T T T T T

T T F T F

T F T T T

T F F T F

F T T T T

F T F T F

F F T F F

F F F F F

5) 9.2 – 7 (p q ) [~ (pq )]

p q (p q ~ (pq )] (p q ) [~ (pq )]

T T T F T T F F T T F T F T T F F F T T

6) 9.2 - 9 (p q ) r

p q r (p q ) (p q ) r

T T T T T

T T F T T

T F T T T

T F F T T

F T T T T

F T F T T

F F T F T

F F F F F


7) 9.2 - 10 (pq ) r

p q r (pq ) (pq ) r

T T T T T

T T F T T

T F T F T

T F F F F

F T T F T

F T F F F

F F T F T

F F F F F

8) Example 9.7: ~ (p q ) (~ p) ( ~q )

p q p q ~ (p q )

T T T F

T F T F

F T T F

F F F T

p q T T F F F T F F T F F T T F F F F T T T

9) Example 9.10: (i) [ (~ p ) (~ q ) ] p

p q

T T F F F T T F F T T T F T T F T T F F T T T T

filrp epuy; KGtJk; T Mjyhy; bka;ik MFk;.

10) Example 9.10: (ii) [(~ q)p) ]q

p q T T F F F T F T T F F T F F F F F T F F

filrp epuy; KGtJk; F Mjyhy; MFk;.

11) Example 9.11: [(~ p)q] [ p (~q )]

p q

T T F F T F T

T F F T F T T

F T T F T F T

F F T T T F T



12) EXERCISE 9.3 (i) [(~ p) q ) ] p

p q (~ p) q ) (~ p) q ) ] p T T F F F T F F F F F T T T F F F T F F


13) EXERCISE 9.3 (ii) (p q ) [~ (p

p q T T T F T T F T F T F T T F T F F F T T


14) EXERCISE 9.3 (iii) [ p (~q ) ] [ ( ~ p ) q ]

p q T T F F F T T

T F F T T F T

F T T F F T T

F F T T F T T


15) EXERCISE 9.3 ( iv) q [p (~q )

p q T T F T T

T F T T T

F T F F T

F F T T T


16) EXERCISE 9.3 (v) [ p ( ~p ) ] [ (~ q) p ]

p q (~ q) p [ p ( ~p ) ] [ (~ q) p ]

T T F F F F F

T F F T F T F

F T T F F F F

F F T T F T F


17) EXERCISE 9.3 - 2. p q (~ p ) q vdf; fhl;Lf

p q T T T

T F F

F T T

F F T


p q T T F T

T F F F

F T T T

F F T T

p q (~ p ) q MFk;.

18) EXERCISE 9.3 - 3. p q ( p q ) ( q p )

p q p q

T T T

T F F

F T F

F F T

p q p q q p ( p q ) ( q p )

T T T T T

T F F T F

F T T F F

F F T T T

p q ( p q ) ( q p ) MFk;.

19) EXERCISE 9.3 - 4. p q[ (~ p ) q ) [ (~q ) p )

p q p q

T T T

T F F

F T F

F F T

p q T T F F T T T

T F F T F T F

F T T F T F F

F F T T T T T

,U ml;ltizfspypUe;J p q [ (~ p ) q ) [ (~q ) p ) MFk;.

20) EXERCISE 9.3 - 5. ~ (pq ) (~p)(~q ) vdf; fhl;Lf

p q (pq ~ (pq ) T T T F T F F T F T F T F F F T

p q

T T F F F T F F T T F T T F T F F T T T

~ (pq ) (~p)(~q ) MFk;.


21) EXERCISE 9.3 - 6.

p q T T T T

T F F T

F T T F

F F T T

vdnt p q q p

22) EXERCISE 9.3 - 7. (pq) ( p q)

p q pq p q (pq) ( p q)

T T T T T

T F F T T

F T F T T

F F F F T

filrp epuy; KGtJk; T (pq) ( p q) MFk;.

23) Fyk;:

24) Fyj;jpd; ePf;fy; tpjpapid vGjp epU:gp/

G a,b,c

i. (,lJ ePf;fy; tpjp)

ii. (tyJ ePf;fy; tpjp)

epU:gzk; (,lJ ePf;fy; tpjp)

(tyJ ePf;fy; tpjp)

25) Fyj;jpd; vjph;kiw tpjpapid vGjp epU:gp

xU Fyk; vd;f. a,b,

epU:gzk;

I.


II)

vdnt

26) Example 9.12: (Z, +)

(i) tpjp: ,uz;L KG vz;fspd; TLjy; xU KG vz;

(ii) tpjp: z-y; $l;ly; vg;nghOJk; Nrh;g;G tpjpf;F cl;gl;lJ

(iii) tpjp: Tl;ly; rkdp 0 kw;Wk; mJ 0+a = a+0 = a

I G+h;j;jp nra;fpwJ. vdNt, rkdp tpjp cz;ikahFk; (iv) tpjp: Xt;bthU f;Fk; I

vDkhW fhzyhk;. vdNt, vjph;kiw tpjp cz;ikahFk;.

(v) tpjp:

Tl;ly; tpjp ghpkhw;W tpjpf;F cl;gl;lJ.

Z xU Kbtw;w fzk;. ∴ (Z, +)

27) Example 9.13: (R – {0}, . ) ‘ ’

(i) tpjp: ,uz;L G+[;[pakw;w nka;naz;fspd; ngUf;fy; xU G+[;[pakw;w nka;naz; MFk;.

0}

(ii) tpjp:

R - {0} y;ngUf;fy; vg;nghOJk; Nrh;g;G tpjpf;F cl;gLk; ∴Nrh;g;G tpjp cz;ikahfpwJ (iii) tpjp: rkdp cWg;G 1 0} kw;Wk; 1.a = a.1 = a, a 0}

vdNt, rkdp tpjp cz;ikahfpwJ.

(iv) v tpjp:

Xt;bthU f;Fk; I

vDkhW fhzyhk;. vdNt, vjph;kiw tpjp cz;ikahfpwJ.


(v) tpjp: ghpkhw;W tpjpf;F cz;ikahfpwJ.

∴ ( , )

R – {0} xU Kbtw;w fzk;. ∴ (R – {0), .)

28) Example 9.14: (cube roots of unity

g;

1 1 1

1

1

(i) tpjp: ml;ltidapy; cs;s vy;yh cWg;g[fSk; -d; cWg;g[fs; MFk;.

vdnt milt[ tpjp cz;ikahfpwJ

(ii) tpjp: bgUf;fy; vg;nghJk; nrh;g;g[ tpjpf;F cl;gl;lJ (iii) tpjp: rkdp cWg;G 1. mJ rkdp tpjpiag; G+h;j;jp nra;Ak;

(iv) tpjp: 1

(v) tpjp: ghpkhw;W tpjpa[k; cz;ikahFk;.

G xU Kbthd fzk;. ∴ (G, .)

29) Example 9.15: 1 4 (fourth roots of unity )

1 -1 i -i

1 1 -1 i -i

-1 -1 1 -i i

i i -i -1 1

-i -i i 1 -1

bgUf;fy; nkw;fz;l ml;ltidapy; ,Ue;J

(i) tpjp: ml;ltidapy; cs;s vy;yh cWg;g[fSk; -d; cWg;g[fs; MFk;.

vdnt, milt[ tpjp cz;ikahfpwJ

(ii) tpjp:

C y; bgUf;fy; vg;nghJk; nrh;g;g[ tpjpf;F cl;gLk;;. Mjyhy; GYk; cz;ikahFk;. (iii) tpjp: rkdp cWg;G 1 G. mJ rkdp tpjpiag;G+h;j;jp nra;Ak.;

(iv) tpjp: 1

(v) tpjp: nkw; fz;l ml;ltidapy; ,Ue;J ghpkhw;W tpjpa[k; cz;ikahFk;.

vdnt (G, .) xU vgPypad; Fyk; MFk;.

30) Example 9.16: (C, +)

(i) tpjp: vdnt milt[ tpjp cz;ikahfpwJ

(ii) tpjp:

C y; Tl;ly; vg;nghJk; nrh;g;g[ tpjpf;F cl;gLk;;. Mjyhy; CYk; cz;ikahFk;. (iii) tpjp: rkdp cWg;G 0 = 0 + i C. mJ rkdp tpjpiag; G+h;j;jp nra;Ak;.

(iv) tpjp: xt;nthU Z f;Fk; Z I Z + (-Z) = -Z + Z = 0 vDkhW fhzyhk;.


(v) tpjp:

C xU Kotw;w fzk;. (C, +) xU Kotw;w vgPypad; Fyk; MFk;.

31) Example 9.17:

(i) tpjp: G+[;[pakw;w ,uz;L fyg;ngz;fspd; ngUf;fy; vg;nghOJk; G+[;[pakw;w fyg;ngz;zhf ,Uf;Fk;

(ii) Nrh;g;G tpjp: fyg;ngz;fspy; ngUf;fy; Nrh;g;G vg;nghOJk; cz;ikahFk;

(iii) rkdp tpjp: 1 = 1 + 0i G NkYk; 1.Z = Z.1 = Z Z G

(iv) tpjp: Z = x + iy G ∴xt;nthU Z G f;Fk; G NkYk; (v) tpjp: fyg;ngz; ngUf;fy; ghpkhw;W tpjpia epiwT nra;Ak;

(C-{0} , .) xU vgPypad; Fyk; MFk;.

32) Example 9.19: 2x2

( R )

jPh;t[:

G = 2x2

(i) tpjp: ,uz;L 2X2 thpir bfhz;l g{r;rpakw;w nfhit mzpfspd; bgUf;fy; gyd; xU 2X2 thpir bfhz;l g{r;rpakw;w nfhit mzpahFk;. vdnt, milt[ tjpp cz;ikahFk ;.

(ii) Nrh;g;G tpjp: mzpg;; ngUf;fy; Nrh;g;G tpjpia epiwT nra;Ak;.

(iii) rkdp tpjp:

vd;gJ rkdp cWg;ghFk;.

(iv) tpjp: xt;nthU G+[;[pakw;w A vd;w mzpf;Fk; mjd; Neh;khW A-1

cz;L.

A . A-1

= A-1

A . = I

(v) tpjp: mzpg;ngUf;fy; ghpkhw;W tpjpia epiwT nra;ahJ.

G xU Kbtw;w fzk;. (G,.) xU Kbtw;w vgPypad; Fyk;.

33) Example 9.20:

10

01,

10

01,

10

01,

10

01

jPh;t[

vd;f. mzpg;bgUf;fy;

I A B C

I I A B C

A A I C B

B B C I A

C C B A I

apd; vy;yhcWg;g[fSk;G-d;; cWg;g[fs; MFk; vdnt bgUf;fy; milt[

tpjpia epiwt[ bra;fpwJ.

mzp bgUf;fy; vg;nghJk; nrh;g;g[ tpjpf;F cl;gl;lJ.

I ; ;

ml;ltidapypUe;J . ghpkhw;W tpjpf;F cl;gl;lJ .

vdnt G MdJ mzpg;bgUf;fypd; fPH; xU vgPypad; Fyk;.


3 kjpg;ngz; tpdhf;fSk; tpilfSk

34. xU Fyk; G–y; , .

ep&gzk; : vd;gjhy; .

a–d; vjpu;; kiw vd;f.

–d; vjpu;; kiw vd;f. = = e

(1), (2) ypUe;J = (tyJ ePf;fy; tpjp)

35. xU Fyj;jpd; rkdp cWg;G xUik jd;ik tha;e;jJ.

ep&gzk; : G xU Fyk; vd;f. G –d; rkdp cWg;Gfs; kw;Wk; vd;f.

–I rkdp cWg;ghf¡ nfhs;Nthkhapd; =

I rkdp cWg;ghf nfhs;Nthkhapd; =

(1) kw;Wk; (2)–y; ,Ue;J

∴ xU Fyj;jpd; rkdp cWg;G xUik jd;ik tha;e;jJ.

36. xU Fyj;jpd; xt;nthU cWg;Gk; xNu xU vjph;kiwiag; ngw;wpUf;Fk;.

ep&gzk; : G xU Fyk; vd;f. vd;f. –d; vjph;kiw cWg;Gfs; , vd;f.

–I a –d; vjph;kiw vd;f. = = ………….. (1)

–I a –d; vjph;kiw vd;f. = = ………….. (2)

,g;NghJ =

xU Fyj;jpd; xt;nthU cWg;Gk; xNu xU vjph;kiwiag; ngw;wpUf;Fk;.

37 to 45 nra;Jghh;f;f:

Example 9.4: gpd;tUk; $w;WfSf;F nka; ml;ltizfs; mikf;f:

(i) qp ~~ (ii) qp ~~

EXERCISE 9.2 gpd;tUk; $w;WfSf;F nka; ml;ltizfs; mikf;f:

(1) qp ~ (2) qp ~~ (3) qp~

(4) pqp ~ (5) qqp ~ (6) qp ~~

(8) qqp ~


Gj;jf Fwpf;Nfhs; tpdhக்கள்

njhFjp – I CHAPTER – 1 (அணிகள் மற்றும் அணிக்ககோவைகளின் பயன்போடுகள்)

01.

1 1 2

2 2 4

4 4 8

vd;w mzpapd; juk; fhz;f

a) 1 b) 2 c) 3 d) 4

02.

1

2

0

4

0

vd;w %iy tpl;l mzpapd; juk; fhz;f

a) 0 b) 2 c) 3 d) 5

03. A = [ 2 0 1 ]vdpy; TAA d; juk; fhz;f

a) 1 b) 2 c) 3 d) 0

04.

1

2

3

A

vdpy; TAA d; juk; fhz;f

a) 3 b) 0 c) 1 d) 2

05.

1 0

0 1

1 0

vd;w mzpapd; juk; 2 vdpy; tpd; kjpg;G

a) 1 b) 2 c) 3 d) VNjDk; xU nka;naz;.

06. xU jpirapyp mzpapd; thpir 3, 0k vdpy; 1A vd;gJ

a) 2k

1I b)

3k

1I c)

k

1 I d) k I

07.

1 3 2

1 3

1 4 5

k

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

a) k VNjDk; xU nka;naz; b) 4k c) 4k d) 4k

08. 2 1

3 4A

vd;w mzpf;F ( adjA ) A =

a)

15

15

0

0

b) 1 0

0 1

c) 5 0

0 5

d)

5 0

0 5


09. xU rJu mzp A d; thpir n vdpy; Aadj vd;gJ

a) 2A b) nA c)

1nA

d) A

10.

0 0 1

0 1 0

1 0 0

vd;w mzpapd; Neh;khW

a)

100

010

001

b)

0 0 1

0 1 0

1 0 0

c)

0 0 1

0 1 0

1 0 0

d)

1 0 0

0 1 0

0 0 1

11. A vd;w mzpapd; thpir 3 vdpy; det (kA) vd;gJ

a) )Adet(k3 b) )Adet(k2 c) )Adet(k d) det (A)

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

a) )(Iadjk n b) )(Iadjk c) )(2 Iadjk d) )(1 Iadjk n

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

a) B = O b) B xU G+r;rpaf; Nfhit mzp c) B xU G+r;rpakw;w Nfhit mzp d) B = A.

14. A = 0 0

0 5

vdpy;> 12A vd;gJ

a) 0 0

0 60

b) 12

0 0

0 5

c) 0 0

0 0

d) 1 0

0 1

15. 3 1

5 2

vd;gjd; Neh;khW

a) 2 1

5 3

b) 2 5

1 3

c)

3 1

5 3

d) 3 5

1 2

16. kjpg;gpl Ntz;ba %d;W khwpfspy; mike;j %d;W Nehpa mrkgbj;jhd rkd;ghl;Lj; njhFg;gpy; 0 kw;Wk;

0x , 0y kw;Wk; 0z vdpy; njhFg;Gf;fhd jPh;T

a) xNu xU jPh;T b) ,uz;L jPh;Tfs; c) vz;zpf;ifaw;w jPh;Tfs; d) jPh;T ,y;yhik

17. 000 czyx;zbyx;zyax Mfpa rkd;ghLfspd; njhFg;ghdJ xU ntspg;gilaw;w jPh;it

ngw;wpUg;gpd;

c1

1

b1

1

a1

1

a) 1 b) 2 c) –1 d) 0

18. dqepe;cbeae yxyx kw;Wk; qp

ba1 ;

qd

bc2 ;

dp

ca3 vdpy; (x,y) d; kjpg;G

a)

1

3

1

2 , b)

1

3

1

2 log,log c)

2

1

3

1 log,log d)

3

1

2

1 log,log

19. nzyx;mzyx;lzyx 222 vd;w rkd;ghLfs; 0 nml vDkhW mikAkhapd;

mj;njhFg;gpd; jPh;T a) xNu xU G+r;rpkw;w jPh;T b) ntspg;gil¤ jPh;T

c) vz;zpf;ifaw;w jPh;T d) jPh;T ,y;yhik ngw;W ,Uf;Fk;


CHAPTER – 2 (வைக்டர் இயற்கணிதம்)

20. a xU G+r;rpakw;w ntf;luhfTk; m G+r;rpakw;w xU jpirapypahfTk; ,Ug;gpd; am MdJ XuyF ntf;lh; vdpy;

a) 1m b) ma c) m

1a d) a = 1

21. akw;Wk; b

,uz;L XuyF ntf;lH kw;Wk; vd;gJ mtw;wpw;F ,ilg;gl;l Nfhzk; ba MdJ XuyF

ntf;luhapd;

a) 3

b)

4

c)

2

d)

3

2

22. a f;Fk; b f;Fk; ,ilg;gl;l Nfhzk; 120° NkYk; mtw;wpd; vz;zsTfs; KiwNa 32, a . b MdJ

a) 3 b) 3 c) 2 d) 2

3

23. )ba(c)ac(b)cb(au

vdpy;

a) u xU XuyF ntf;lH b) cbau

c) 0u

d) 0u

24. 0cba

, 3a

, 4b

, 5c

vdpy; a f;Fk; b f;Fk; ,ilg;gl;l Nfhzk;

a) 6

b)

3

2 c)

3

5 d)

2

25. k4j3i2

, kcjbia

Mfpa ntf;lh;fs; nrq;Fj;J ntf;lHfshapd;>

a) a = 2, b = 3, c = -4 b) a = 4, b = 4, c = 5 c) a = 4, b = 4, c = -5 d)a = -2, b =3, c =4

26. kji 3 vd;w ntf;liu xU %iy tpl;lkhfTk; kji 43 I xU gf;fkhfTk; nfhz;l ,izfuj;jpd; gug;G

a) 310 b) 306 c) 302

3 d) 303

27. baba vdpy;

a) a k; b k; ,izahFk; b) a k; b k; nrq;Fj;jhFk; c) ba

d) a kw;Wk; b XuyF ntf;lH

28. q,p kw;Wk; qp Mfpait vz;zsT nfhz;l ntf;lh;fshapd; qp

MdJ.

a) 2 b) 3 c) 2 d) 1

29. yxx)bxa(xc)axc(xb)cxb(xa vdpy;

a) 0x b) 0y

c) x k; y k; ,izahFk; d) 0x my;yJ 0y my;yJ x k; y k; ,izahFk;

30. kjiPR 2 , kjiSQ 23 vdpy; ehw;fuk; PQRS d; gug;G

a) 35 b) 310 c) 2

35 d)

2

3

31.

OQ vd;w myF ntf;lH kPjhd

OP d; tPoyhdJ OPRQ vd;w ,izfuj;jpd; gug;ig¥ Nghd;W Kk;klq;fhapd; POQ MdJ.

a) 3

1tan 1 b)

10

3cos 1 c)

10

3sin 1 d)

3

1sin 1


32. b d; kPJ a d; tPoy; kw;Wk; a d; kPJ b d; tPoYk; rkkhapd; a + b kw;Wk; a - b f;F ,ilg;gl;l Nfhzk;

a) 2

b)

3

c)

4

d)

3

2

33. c,b,a vd;w jskw;w ntf;lHfSf;F cx)bxa()cxb(xa vdpy;

a) a MdJ b f;F ,iz b) b MdJ c f;F ,iz c) c MdJ a f;F ,iz d) 0 cba

34. xU NfhL x kw;Wk; y mr;Rf;fSld; kpif jpirapy; 45°, 60° Nfhzq;fis Vw;gLj;JfpwJ vdpy; z mr;Rld; mJ cz;lhf;Fk; Nfhzk;

a) 30 b) 90 c) 45 d) 60

35. ac,cb,ba

=64vdpy; c,b,ad; kjpg;G

a) 32 b) 8 c) 128 d) 0

36. ac,cb,ba

=8vdpy; c,b,ad; kjpg;G

a) 4 b) 16 c) 32 d) -4

37. ik,kj,ji d; kjpg;G

a) 0 b) 1 c) 2 d) 4

38. (2,10,1) vd;w Gs;spf;Fk; 262)43( kjir

vd;w jsj;jpw;Fk; ,ilg;gl;l kpff; Fiwe;j J}uk;

a) 262 b) 26 c) 2 d) 26

1

39. )dxc(x)bxa( vd;gJ

a) c,b,a kw;Wk; d f;F nrq;Fj;J b) )bxa( kw;Wk; )dxc( vd;w ntf;lHfSf;F ,iz.

c) ba, I nfhz;l jsKk; c , d nfhz;l jsKk; ntl;bf; nfhs;Sk; Nfhl;bw;F ,iz.

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

40. c,b,a vd;gd a,b,c Mfpatw;iw kl;Lfshff; nfhz;L tyf;if mikg;gpy; xd;Wf;nfhd;W nrq;Fj;jhd

ntf;lHfs; vdpy; c,b,a,d; kjpg;G

a) 222 cba b) 0 c) abc2

1 d) abc

41. c,b,a vd;gd xU jsk; mikah ntf;lHfs; NkYk; [ , , ] [ , , ]a x b b x c c x a a b b c c a vdpy;

c,b,a d; kjpg;G

a) 2 b) 3 c) 1 d) 0

42. jtisr vd;w rkd;ghL Fwpg;gJ

a) i kw;Wk; j Gs;spfis ,izf;Fk; Neh;;NfhL b) xoy jsk; c) yoz jsk; d) zox jsk;

43. kjai vDk; tpir ji vDk; Gs;sp topNa nray;gLfpwJ. kj vDk; Gs;spia¥ nghWj;J mjd; jpUg;Gj;

jpwdpd; msT 8 vdpy; a d; kjpg;G.

a) 1 b) 2 c) 3 d) 4


44. 3

5z2

5

3y

1

3x

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

a) k5j3itk3j5ir

b) k3j5itk5j3ir

c) k5j3itk2

3j5ir

d)

k

2

3j5itk5j3ir

45. kjitkir 723 vd;w NfhLk; 8. kjir vd;w jsKk; ntl;bf;nfhs;Sk; Gs;sp

a) (8, 6, 22) b) (-8, - 6, -22) c) (4, 3, 11) d) (-4, -3, -11)

46. (2,1,-1) vd;w Gs;sp topahfTk; jsq;fs; 0)kj3i(r

; 0)k2j(r

ntl;bf;nfhs;Sk; Nfhl;il cs;slf;fpaJkhd jsj;jpd; rkd;ghL

a) x+4y-z=0 b) x+9y+11z=0 c) 2 x+y-z+5=0 d) 2x-y+z=0

47. kjiF vd;w tpir xU Jfis A (3,3,3) vDk; epiyapypUe;J B (4,4,4) vDk; epiyf;F efh;j;jpdhy;

mt;tpir nra;Ak;NtiyasT

a) 2 myFfs; b) 3 myFfs; c) 4 myFfs; d) 7 myFfs;

48. kjia 32 kw;Wk; kjib 23 vdpy; a f;Fk; b f;Fk; nrq;Fj;jhf cs;s xU XuyF ntf;lH

a) 3

kji

b)

3

kji

c)

3

k2ji

d)

3

kji

49. 8

4

4

4

6

6

zyx kw;Wk; 2

3z

4

2y

2

1x

vd;w NfhLfs; ntl;bf;nfhs;Sk; Gs;sp

a) (0, 0, -4) b) (1, 0, 0) c) (0, 2, 0) d) (1, 2, 0)

50. kji2tk3j2ir

kw;Wk; k3j2isk5j3i2r

vd;w NfhLfs; ntl;bf; nfhs;Sk; Gs;sp

a) (2, 1, 1) b) (1, 2, 1) c) (1, 1, 2) d) (1, 1, 1)

51. 4

3z

3

2y

2

1x

kw;Wk;

5

5z

4

4y

3

2x

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

a) 3

2 b)

6

1 c)

3

2 d)

62

1

52. 3

5z

2

1y

4

3x

kw;Wk;

3

3z

2

2y

4

1x

vd;w ,izNfhLfSf;fpilNaAs;s kpff; Fiwe;j njhiyT

a) 3 b) 2 c) 1 d) 0

53. 1

z

1

1y

2

1x

kw;Wk;

2

1z

5

1y

3

2x

Mfpa ,U NfhLfSk;

a) ,iz b) ntl;bf;nfhs;git c) xU jsk; mikahjit d) nrq;Fj;J.

54. 01z10y8x6zyx 222 vd;w Nfhsj;jpd; ikak; kw;Wk; Muk;

a) (-3, 4, -5) , 49 b) (-6, 8, -10) , 1 c) (3, -4, 5) , 7 d) (6, -8, 10) , 7


CHAPTER – 3 (கலப்வபண்கள்)

55.

100100

2

31

2

31

iid; kjpg;G a) 2 b) 0 c) - 1 d) 1

56. 343 /ie vd;w fyg;ngz;zpd; kl;L tPr;R KiwNa

a) 2

,9 e b)

2,9

e c) 4

3,6

e d) 4

3,9

e

57. (2m + 3) + i( 3n-2) vd;w fyg;ngz;zpd; ,iznad; (m-5) + i ( n+4) vdpy; (n,m) vd;gJ.

a)

8,

2

1 b)

8,

2

1 c)

8,

2

1 d)

8,

2

1

58. 122 yx vdpy; iyx

iyx

1

1d; kjpg;G

a) iyx b) x2 c) iy2 d) iyx

59. 32 i vd;w fyg;ngz;zpd; kl;L

a) 3 b) 13 c) 7 d) 7

60. )iba()iba()iba(iBA 332211 vdpy; 22 BA d; kjpg;G

a) 23

23

22

22

21

21 bababa b) 2321

2321 bbbaaa

c) 23

23

22

22

21

21 bababa d) 2

32

22

12

32

22

1 bbbaaa

61. a = 3 + i kw;Wk; z = 2 – 3i vdpy; cs;s az, 3az kw;Wk; -az vd;gd xU Mh;fd; jsj;jpy;

a) nrq;Nfhz Kf;Nfhzj;jpd; Kidg;Gs;spfs; b) rkgf;f Kf;Nfhzj;jpd; Kidg;Gs;spfs;

c) ,U rkgf;f Kf;Nfhzj;jpd; Kidg;Gs;spfs; d) xNu Nfhliktd.

62. fyg;ngz; 4321 z,z,z,z vd;w Gs;spfs; KiwNa xU ,izfuj;jpd; Kidg;Gs;spfshf ,Ug;gjw;Fk; mjd;

kWjiyAk; cz;ikahf ,Ug;gjw;Fk; cs;s epge;jid

a) 3241 zzzz b) 4231 zzzz c) 4321 zzzz d) 4321 zzzz

63. z fyg;ngz;izf; Fwpg;gnjdpy; )z(arg)z(arg vd;gJ

a) 4

b)

2

c) 0 d)

3

64. xU fyg;ngz;zpd; tPr;R 2

vdpy; me;j vz;

a) Kw;wpYk; fw;gid vz; b) Kw;wpYk; nka; vz; c) 0 d) nka;Aky;y fw;gidAky;y

65. iz vd;w fyg;ngz;iz Mjpia¥ nghWj;J 2

Nfhzj;jpy; fbfhu vjph; jpirapy; Row;Wk; NghJ me;j vz;zpd;

Gjpa epiy

a) iz b) -iz c) -z d) z

66. fyg;ngz; 325 )i( d; Nghyhh; tbtk;

a) 2

sin2

cos

i b) sincos i c) sincos i d) 2

sin2

cos

i


67. P MdJ fyg;G vz; khwp z I Fwpf;fpd;wJ zz 212 vdpy; P d; epakg;ghij

a)4

1x vd;w Nehf;NfhL b)

4

1y vd;w Nehf;NfhL c)

2

1z vd;w Nehf;NfhL d) 01422 xyx vd;w tl;lk;

68.

i

i

e

e

1

1= a) sincos i b) sincos i c) cossin i d) cossin i

69. 3

sin3

cos n

in

zn vdpy; 621 z.........zz vd;gJ

a) 1 b) -1 c) i d) -i

70. z %d;whk; fhy;gFjpapy; mike;jhy; z mikAk; fhy;gFjp

a) Kjy; fhy;gFjp b) ,uz;lhk; fhy;gFjp c) %d;whk; fhy;gFjp d) ehd;fhk; fhy;gFjp

71. sinicosx vdpy; n

n

xx

1 d; kjpg;G

a) ncos2 b) ni sin2 c) nsin2 d) ni cos2

72. sinicoscsinicosb,sinicosa vdpy; abc/)bca( 222 vd;gJ

a) )sin()(2cos i b) )cos(2

c) - )sin(2 i d) )cos(2

73. iz,iz 2354 21 vdpy; 2

1

z

z vd;gJ

a) i13

22

13

2 b) i

13

22

13

2 c) i

13

22

13

2 d) i

13

22

13

2

74. 25242322 iiiii d; kjpg;G vd;gJ

a) i b) - i c) 1 d) – 1

75. 16151413 iiii d; ,iz fyg;ngz;

a) 1 b) - 1 c) 0 d) -i

76. –i +2 vd;gJ 02 cbxax vd;w rkd;ghl;bd; xU %ynkdpy; kw;nwhU jPHT

a) 2i b) 2i c) i2 d) ii 2

77. 7i vd;w jPh;Tfisf; nfhz;l ,Ugb¢ rkd;ghL

a) 072 x b) 072 x c) 072 xx d) 072 xx

78. 4-3i kw;Wk; 4+3i vd;w %yq;fisf; nfhz;l rkd;ghL

a) 02582 xx b) 02582 xx c) 02582 xx d) 02582 xx

79. 012 bxax vd;w rkd;ghl;bd; xU jPh;T i

i

1

1aAk; >bAk; nka; vdpy; (a,b) vd;gJ.

a) (1,1) b) (1,-1) c) (0,1) d) (1,0)

80. 062 kxx vd;w rkd;ghl;bd; xU %yk; -i +3 vdpy; kd; kjpg;G

a) 5 b) 5 c) 10 d) 10

81. vd;gJ 1d; Kg;gb %ynkdpy; 4242 11 )()( d; kjpg;G

a) 0 b) 32 c) -16 d) -32

82. vd;gJ 1d; nk; gb %ynkdpy;

a) ..................1 5342

b) 0n c) 1n d) 1 n

83. vd;gJ 1d; Kg;gb %ynkdpy; )()()()( 842 1111 d; kjpg;G

a) 9 b) -9 c) 16 d) 32


CHAPTER – 4 (பகுமுவை ைடிைக்கணிதம்)

84. 023x8y2y2 vd;w gutisaj;jpd; mr;R

a) 1y b) 3x c) 3x d) 1y

85. 0441232316 22 yxyx vd;gJ

a) xU ePs;tl;lk; b) xU tl;lk;. c) xU gutisak; d) xU mjpgutisak;

86. cy2x4 vd;w NfhL x16y2 vd;w gutisaj;jpd; njhLNfhL vdpy; c d; kjpg;G.

a) 1 b) -2 c) 4 d) -4

87. x8y2 vd;w gutisaj;jpd; tt1 kw;Wk; t3t2 vd;w Gs;spfspy; tiuag;gl;l njhLNfhLfs; ntl;bf;nfhs;Sk; Gs;sp

a) )t8,t6( 2 b) )t6,t8( 2 c) )t4,t( 2 d) )t,t4( 2

88. 08y4x4y2 vd;w gutisaj;jpd; nrt;tfyj;jpd; ePsk;

a) 8 b) 6 c) 4 d) 2

89. 4xy2 vd;w gutisaj;jpd; ,af;Ftiuapd; rkd;ghL

a) 4

15x b)

4

15x c)

4

17x d)

4

17x

90. (2,-3) vd;w Kid x = 4 vd;w ,af;Ftiuiaf; nfhz;l gutisaj;jpd; nrt;tfy ePsk;

a) 2 b) 4 c) 6 d) 8

91. y16x2 vd;w gutisaj;jpd; Ftpak;

a) (4,0) b) (0,4) c) (-4,0) d) (0,-4)

92. 1y8x2 vd;w gutisaj;jpd; Kid

a)

0,

8

1 b)

0,

8

1 c)

8

1,0 d)

8

1,0

93. 2x + 3y + 9 = 0 vd;w NfhL x8y2 vd;w gutisaj;ijj; njhLk; Gs;sp .

a)(0,-3) b) (2,4) c)

2

9,6 d)

6,

2

9

94. x12y2 vd;w gutisaj;jpd; Ftpehzpd; ,Wjpg;Gs;spfspy; tiuag;gLk; njhLNfhLfs; re;jpf;Fk; Gs;sp mikAk; NfhL

a) x - 3 =0 b) x+ 3 =0 c) y + 3 =0 d) y – 3 = 0

95. (-4, 4) vd;w Gs;spapypUe;J x16y2 f;F tiuag;gLk; ,U njhLNfhLfSf;F ,ilNaAs;s Nfhzk;

a) 45 b) 30 c) 60 d) 90

96. 0116y40x54y5x9 22 vd;w $k;G tistpd; ikaj;njhiyj;jfT (e)d; kjpg;G

a) 3

1 b)

3

2 c)

9

4 d)

5

2

97. 1169

y

144

x 22

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

a) 26, 12 b) 13 ,24 c) 12, 26 d) 13, 12

98. 180y5x9 22 vd;w ePs;tl;lj;jpd; Ftpaq;fSf;fpilNa cs;s njhiyT.

a) 4 b) 6 c) 8 d) 2.


99. xU ePs;;tl;lj;jpd; nel;lr;R kw;Wk; mjd; miu Fw;wr;Rfspd; ePsq;fs; 8,2 KiwNa mjd; rkd;ghLfs; y – 6 = 0

kw;Wk; x + 4 =0 vdpy; ePs;tl;lj;jpd; rkd;ghL

a) 116

)6y(

4

)4x( 22

b) 14

)6y(

16

)4x( 22

c) 14

)6y(

16

)4x( 22

d) 116

)6y(

4

)4x( 22

100. 0cyx2 vd;w Neh;;NfhL 32y8x4 22 vd;w ePs;tl;lj;jpd;; njhLNfhL vdpy; cd; kjpg;G.

a) 32 b) 6 c) 36 d) 4

101. 36y9x4 22 vd;w ePs;tl;lj;jpd; kPJs;s VNjDk; xU Gs;spapypUe;J )0,5( kw;Wk; )0,5( vd;w

Gs;spfSf;fpilNa cs;s njhiyTfspd; $Ljy; .

a) 4 b) 8 c) 6 d) 18

102. 144y16x9 22 vd;w $k;G tistpd; ,af;F tl;lj;jpd; Muk;

a) 7 b) 4 c) 3 d) 5.

103. 400y25x16 22 vd;w tistiuapd; Ftpaj;jpypUe;J xU njhLNfhl;Lf;F tiuag;gLk; nrq;Fj;J NfhLfspd; mbapd;epakg;ghij

a) 4yx 22 b) 25yx 22 c) 16yx 22 d) 9yx 22

104. 0127y48x24x4y12 22 vd;w mjpgutisaj;jpd; ikaj;njhiy¤jfT.

a) 4 b) 3 c) 2 d) 6

105. nrt;tfyj;jpd; ePsk; Jizar;rpd; ePsj;jpy; ghjp vdf; nfhz;Ls;s mjpgutisaj;jpd; ikaj;njhiyj;jfT

a) 2

3 b) 3

5 c)

2

3 d)

2

5

106. 1b

y

a

x

2

2

2

2

vd;w mjpgutisaj;jpd; kPJs;s VNjDk; xU Gs;spapypUe;J Ftpaj;jpw;F ,ilNaAs;s

njhiyTfspd; tpj;jpahrk; 24 kw;Wk; ikaj;njhiyj;jfT 2 vdpy; mjpgutisaj;jpd; rkd;ghL

a) 1432

y

144

x 22

b) 1144

y

432

x 22

c) 1312

y

12

x 22

d) 112

y

312

x 22

107. 16)3y(4x 22 vd;w mjpgutisaj;jpd; ,af;Ftiu

a) 5

8y b)

5

8x c)

8

5y d)

8

5x

108. 36yx4 22 f;F 0k4y2x5 vd;w NfhL xU njhLNfhL vdpy; k d; kjpg;G

a) 9

4 b)

3

2 c)

4

9 d)

16

81

109. 19

y

16

x 22

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

a) 072y8x9 b) 072y8x9 c) 072y9x8 d) 072y9x8

110. 19

y

16

x 22

vd;w mjpgutisaj;jpd; njhiynjhLNfhLfSf;fpilNaAs;s Nfhzk;

a)

4

3tan2 1 b)

3

4tan2 1 c)

4

3tan2 1 d)

3

4tan2 1

111. 0900x25y36 22 vd;w mjpgutisaj;jpd; njhiynjhLNfhLfs;

a) x5

6y b) x

6

5y c) x

25

36y d) x

36

25y


112. (8,0) vd;w Gs;spapypUe;J 136

y

64

x 22

vd;w mjpgutisaj;jpypUe;J njhiyj;njhLNfhLfSf;F tiuag;gLk;

nrq;Fj;J J}uq;fspd; ngUf;fy; gyd;

a) 576

25 b)

25

576 c)

25

6 d)

6

25

113. 19

y

16

x 22

vd;w mjpgutisaj;jpd; nrq;Fj;J¤ njhLNfhLfspd; ntl;Lk; Gs;spapd; epakg;ghij.

a) 25yx 22 b) 4yx 22 c) 3yx 22 d) 7yx 22

114. x+2y – 5 =0 , 2 x – y + 5 =0 vd;w njhiyj;njhLNfhLfisf; nfhz;l mjpgutisaj;jpd; ikaj;njhiyj;jfT

a) 3 b) 2 c) 3 d) 2

115. xy = 8 vd;w nrt;tf gutisaj;jpd; miu FWf;fr;rpd; ePsk;

a) 2 b) 4 c) 16 d) 8

116. 2cxy vd;w nrt;tf mjpgutisaj;jpd; njhiynjhLNfhLfs;.

a) cy,cx b) cy,0x c) 0y,cx d) 0y,0x

117. xy = 16 vd;w nrt;tf mjpgutisaj;jpd; Kidapd; Maj;njhiyTfs;.

a) (4,4) (-4, -4) b) (2,8) , (-2,-8) c) (4,0), (-4,0) d) (8,0 ) (-8,0).

118. xy = 18 vd;w nrt;tf mjpgutisaj;jpd; nrt;tfyj;jpd; xU Ftpak;

a) (6,6) b) (3,3) c) (4,4) d) (5,5)

119. xy = 32 vd;w nrt;tf mjpgutisaj;jpd; nrt;tfyj;jpd; ePsk;.

a) 28 b) 32 c) 8 d) 16

120. xy = 72 vd;w jpl;l nrt;tf mjpgutisaj;jpd; kPJs;s VNjDk; xU Gs;spapypUe;J tiuag;gLk; njhLNfhL

mjd; njhiyj; njhL NfhLfSld; cz;lhf;Fk; Kf;Nfhzj;jpd; gug;G.

a) 36 b) 18 c) 72 d) 144

121. xy = 9 vd;w nrt;tf mjpgutisaj;jpd; kPJs;s

2

3,6 vd;w Gs;spapypUe;J tiuag;gLk; nrq;Fj;J>

tistiuia kPz;Lk; re;jpf;Fk; Gs;sp

a)

24,

8

3 b)

8

3,24 c)

24,

8

3 d)

8

3,24


njhFjp – II

CHAPTER – 5 (ைவக நுண்கணிதம் பயன்போடுகள் I)

01. 2x y; 532 3 xxy vd;w tistiuapd; rha;T

a) -20 b) 27 c) -16 d) -21

02. r Muk; nfhz;l xU tl;lj;jpd; gug;G A y; Vw;gLk; khWk; tPjk;

a) r2 b) dt

drr2 c)

dt

drr 2 d)

dt

dr

03. MjpapypUe;J xU Neh;;Nfhl;by; x njhiytpy; efUk; Gs;spapd; jpirNtfk; v vdTk; a+bv2=x

2vdTk;

nfhLf;fg;gl;Ls;sJ. ,q;F a kw;Wk; b khwpypfs; mjd; KLf;fkhdJ

a)x

b b)

x

a c)

b

x d)

a

x

04. XU cUFk; gdpf;fl;b Nfhsj;jpd; fd msT 1nrkP³ / epkplk; vdf; Fiwfpd;wJ. mjd; tpl;lk; 10cm vd ,Uf;Fk; NghJ tpl;lk; FiwAk; NtfkhdJ.

a) min/50

1cm

b) min/

50

1cm

c) min/

75

11cm

d) min/

75

2cm

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

a) 3 b) 2 c) 1 d) -1

06. y = 3x² vd;w tistiuf;F x d; Maj;njhiyT 2 vdf; nfhz;Ls;s Gs;spapy; nrq;Nfhl;bd; rha;thdJ

a) 1 / 13 b) 1 / 14 c) -1 / 12 d) 1 / 12

07. y = 2x² -6x-4 vd;w tistiuapy; x mr;Rf;F ,izahfTs;s njhLNfhl;bd; njhLGs;sp

a)

2

17,

2

5 b)

2

17,

2

5 c)

2

17,

2

5 d)

2

17,

2

3

08. y = x³ / 5 vDk; tistiuf;F (-1,-1/5) vd;w Gs;spapy; njhLNfhl;bd; rkd;ghL

a) 5y + 3x = 2 b) 5y – 3x = 2 c) 3x – 5y = 2 d) 3x+3y = 2

09. = 1 / t vDk; tistiuf;F¥ Gs;sp (-3, -1/3 ) vd;w Gs;spapy; nrq;Nfhl;bd; rkd;ghL

a) 80273 t b) 80275 t c) 80273 t d) t1

10. 19

y

25

x 22

kw;Wk; 18

y

8

x 22

vDk; tistiufSf;F ,ilg;gl;l Nfhzk;

a) 4 b) 3 c) 6 d) 2

11. y = emxkw;Wk; y = e

-mx , m > 1 vd;Dk; tistiufSf;F ,ilgl;l Nfhzk;.

a) tan-1

1m

m2

2 b) tan-1

2m1

m2

c) tan-1

2m1

m2

d) tan-1

1m

m2

2

12. x2/3

+ y2/3

= a2/3vDk; tistiuapd; Jiz myFr; rkd;ghLfs;

a) 33 cos;sin ayax b) 33 sin;cos ayax

c) cos;sin 33 ayax d) sin;cos 33 ayax

13. x2/3

+ y2/3

= a2/3vd;w tistiuapd; nrq;NfhL x mr;Rld; vd;Dk; Nfhzk; Vw;gLj;Jnkdpy;, mr;nrq;Nfhl;bd;

rha;T .

a) cot b) tan c) tan d) cot

14. xU rJuj;jpd; %iy tpl;lj;jpd; ePsk; mjpfhpf;Fk; tPjk; 0.1 nr.kP/tpehb vdpy; , gf;f msT cm2

15 Mf ,Uf;Fk;

NghJ mjd; gug;gsT mjpfhpf;Fk; tPjk;?

a) 1.5 nr.kP ² / tpehb. b) 3 nr.kP ² / tpehb. c) 3 2 nr.kP ² / tpehb. d) 0.15 nr.kP ² / tpehb..


15. xU Nfhsj;jpd; fdmsT kw;Wk; Muj;jpNyw;gLk; khWtPjq;fs; vz;zstpy; rkkhf ,Uf;Fk; NghJ Nfhsj;jpd; tisgug;G

a) 1 b) 21 c) 4 d) 34

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

a)

3,

3

1 b)

3,

3

1 c)

3,

3

1 d)

1,

3

1

17. xU cUisapd; Muk; 2 nrkP / tpdhb vd;w tPjj;jpy; mjpfhpf;fpd;wJ. mjd; cauk; 3 nrkP / tpdhb vd;w tPjj;jpy; Fiwfpd;wJ. Muk; 3cm kw;Wk; cauk; 5 cm Mf ,Uf;Fk; NghJ mjd; fd mstpd; khWtPjk;.

a) 23 b) 33 c) 43 d) 53

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

khWtPjk;

a) – 90 myFfs; / tpdhb. b) 90 myFfs; / tpdhb. c) 180 myFfs; / tpdhb d) – 180 myFfs; / tpdhb.

19. xU fdrJuj;jpd; fdmsT 4 nrkP ³ / tpdhb myFfs; tPjj;jpy; mjpfhpf;fpd;wJ. mf;fdrJuj;jpd; fd msT 8 f.nr.kP Mf ,Uf;Fk; NghJ mjd; Gwg;gug;gsT mjpfhpf;Fk; tPjk;

a) 8 nrkP² / tpdhb. b) 16nrkP² / tpehb. c) 2 nrkP²/ tpehb d) 4 nrkP² / tpehb.

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

a) 8 b) 4 c) 0 d) -4

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

a) 2 tan-1

4

3

b) tan-1

3

4

c) 2 d) 4

22. x = et cos t ; y = e

t sin t vd;w tistiuapd; njhLNfhL x mr;Rf;F ,izahfTs;sJ. vdpy; t d; kjpg;G

a) – 4 b) 4 c) 0 d) 2

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


24. y = 3exkw;Wk; y =

3

ae

-xvd;Dk; tistiufs; nrq;Fj;jhf ntl;bf; nfhs;fpd;wd vdpy; , ‘a’ d; kjpg;G

a) -1 b) 1 c) 1/ 3 d) 3.

25. s = t³ - 4t² + 7 vdpy; KLf;fk; G+r;rpakhFk; NghJs;s jpirNtfk;

a) 3

32m / sec b)

3

16m / sec c)

3

16m / sec d)

3

32 m / sec

26. xU Neh;f;Nfhl;by; efUk; Gs;spapd; jpirNtfkhdJ> mf;Nfhl;by; xU epiyg;Gs;spapypUe;J efUk; Gs;spf;F

,ilapy; cs;s njhiytpd; th;f;fj;jpw;F Neh; tpfpjkhf mike;Js;snjdpy; , mjd; KLf;fk; gpd;tUk; xd;wpDf;F tpfpjkhf mike;Js;sJ.

a) s b) s² c) s³ d) s4.

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

a) 332 b) 0 c) 2 d) -2.

28. a = 0, b = 1 vdf; nfhz;L f(x) = x² + 2x -1 vd;w rhh;gpw;F nyf;uhQ;rpapd; ,ilkjpg;G Njw;wj;jpd; gbAs;s ‘c’ d; kjpg;G .

a) -1 b) 1 c) 0 d) 1/ 2

29. f(x) = cos x / 2 vd;w rhh;gpw;F 3, y; Nuhy; Njw;wj;jpd;gb mike;j c d; kjpg;G

a) 0 b) 2 c) 2 d) 23


30. a = 1 kw;Wk; b = 4 vdf; nfhz;L xxf vd;w rhh;gpw;F nyf;uhQ;rpapd; ,ilkjpg;G Njw;wj;jpd;gb mikAk;

‘c’ d; kjpg;G

a) 9 / 4 b) 3 / 2 c) 1 / 2 d) 1 / 4

31. x

limx

2

e

x d; kjpg;G

a) 2 b) 0 c) d) 1

32. 0x

lim xx

xx

dc

ba

d; kjpg;G

a) b) 0 c) log cd

ab d)

)d/clog(

)b/alog(

33. f(a) = 2 ; f’ (a) = 1 ; g (a) = -1 ; g’ (a) = 2 vdpy; ax

lim ax

)x(f)a(g)a(f)x(g

d; kjpg;G

a) 5 b) -5 c) 3 d) -3

34. gpd;tUtdtw;Ws; vJ ,0 y; VWk; rhh;G ?

a) ex b) 1/ x c) – x² d) x

-2

35. f(x) = x² -5x + 4 vd;w rhh;G ,wq;Fk; ,ilntsp .

a) 1, b) (1,4) c) ,4 d) vy;yh Gs;spfsplj;Jk;

36. f(x) = x² vd;w rhh;G ,wq;Fk; ,ilntsp

a) , b) 0, c) ,0 d) ,2

37. y = tan x – x vd;w rhh;G

a)

2,0 y; VWk; rhh;G b)

2,0 y; ,wq;Fk; rhh;G

c)

4,0 y; VWk;

2,

4y; ,wq;Fk; d)

4,0 y; ,wq;Fk;

2,

4y; VWk;

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

a) 2 b) 4 c) 8 d) 16

39. 100 kP² gug;G nfhz;Ls;s nrt;tfj;jpd; kPr;rpW Rw;wsT

a) 10 b) 20 c) 40 d) 60

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

a) 2. B) 3 c) 4 d) 5

41. y = -e-xvd;w tistiu

a) x > 0 tpw;F Nky;Nehf;fp¡ FopT b) x > 0 tpw;F fPo;Nehf;fpf; FopT

c) vg;NghJk; Nky;Nehf;fpf; FopT. D) vg;NghJk; fPo;Nehf;fpf; FopT

42. gpd;tUk; tistiufSs; vJ fPo;Nehf;fp¡ FopT ngw;Ws;sJ?

a) y = - x² b) y = x² c) y = ex. d) y = x² + 2x – 3.

43. y = x4vd;w tistiuapd; tisT khw;Wg;Gs;sp

a) x = 0 b) x = 3. c) x = 12 d) vq;Fkpy;iy

44. y = ax³ + bx² + cx + d vd;w tistiuf;F x = 1 y; xU tisT khw;Wg;Gs;sp cz;nldpy;

a) a+b = 0 b) a+ 3b = 0 c) 3 a + b = 0 d) 3a + b = 1


CHAPTER – 6 (ைவக நுண்கணிதம் பயன்போடுகள் II)

45. u = xyvdpy;

x

u

f;F rkkhdJ

a) yxy-1

b) u log x c) u log y d) xyx-1

46. u = sin-1

22

44

yx

yxkw;Wk; f = sin u vdpy; rkgbj;jhd rhh;G f d; gb

a) 0 b) 1 c) 2 d) 4

47. u = ,

yx

1

22

vdpy; x x

u

+ y

y

u

a) u2

1 b) u c) u

2

3 d) – u

48. )x(x)x(y 12 22 vd;w tistiuf;F

a) x mr;Rf;F ,izahd xU njhiynjhLNfhL cz;L

b) y mr;Rf;F ,izahd xU njhiynjhLNfhL cz;L

c) ,U mr;RfSf;Fk; ,izahd njhiynjhLNfhLfs; cz;L

d) njhiynjhLNfhLfs; ,y;iy

49. sin,cos ryrx vdpy;, x

r

=

a) sec b) sin c) cos d) eccos

50. gpd;tUtdtw;Ws; rhpahd $w;Wfs; :

i) xU tistiu Mjpia¥ nghWj;J¢ rkr;rPh; ngw;wpUg;gpd;, mJ ,U mr;Rfisg; nghWj;Jk; rkr;rPh;

ngw;wpUf;Fk;.

ii) xU tistiu ,U mr;Rfis¥ nghWj;J rkr;rPh; ngw;wpUg;gpd,; mJ Mjpia¥ nghWj;Jk; rkr;rPh;

ngw;wpUf;Fk;..

iii) f(x,y) =0 vd;w tistiu y = x vd;w Nfhl;il¥ nghWj;J rkr;rPh; ngw;Ws;snjdpy; f(x,y) = f(y, x)

iv) f(x,y) =0, vd;w tistiuf;F f(x,y) = f(-y,-x), cz;ikahapd; mJ Mjpia¥ nghWj;J¢ rkr;rPh; ngw;wpUf;Fk.;

a) (ii), (iii) b) (i), (iv) c) (i), (iii) d) (ii), (iv)

51. u = log

xy

yx 22

vdpy;y

uy

x

ux

vd;gJ

a) 0 b) u c) 2 u d) u-1

.

52. 28 d; 11k;gb %y rjtpfpj¥ gpio Njhuhakhf 28 d; rjtpfpj¥ gpioia¥ Nghy; …………. klq;fhFk;.

a) 1/28 b) 1 / 11 c) 11 d) 28

53. a²y² = x² (a²-x²) vd;w tistiu

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

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

c) x = - a kw;Wk; x = a f;F ,ilapy; ,U fz;zpfs; nfhz;L cs;sJ.

d) fz;zp VJkpy;iy.

54. y² (a+2x) = x² (3a-x) vd;w tistiuapd; njhiynjhL¡NfhL

a) x = 3a b) x = - a / 2 c) x = a / 2 d) x = 0.


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

a) x > 0 b) 0 < x < 3a c) ax kw;Wk; ax 3 d) – a < x < 3a

56. u = y sin x , vdpy;yx

u2

=

a) cos x b) cos y c) sin x d) 0

57. u = f

x

yvdpy;

y

uy

x

ux

d; kjpg;G.

a) 0 b) 1 c) 2 u d) u

58. 9y² = x² (4-x²) vd;w tistiu vjw;F¢ rkr;rPh; ?.

a) y mr;R. b) x mr;R. c) y = x d) ,U mr;Rfs;.

59. ay² = x² (3a-x) vd;w tistiu y mr;ir ntl;Lk; Gs;spfs;.

a) x = - 3a , x = 0 b) x =0, x = 3a. c) x = 0, x = a. d) x = 0

CHAPTER – 7 (வதோவக நுண்கணிதம் பயன்போடுகள்)

60.

2/

03/53/5

3/5

xsinxcos

xcosdx d; kjpg;G

a) 2 b) 4 c) 0 d)

61.

2/

0 xcosxsin1

xcosxsin dx d; kjpg;G.

a) 2 b) 0 c) 4 d)

62. 1

0

4)x1(x dx d; kjpg;G.

a) 1/ 12 b) 1/30 c) 1/ 24 d) 1/ 20

63.

2/

2/ xcos2

xsindx d; kjpg;G.

a) 0 b) 2 c) log 2 d) log 4

64.

0

4 xdxsin d; kjpg;G

a) 163 b) 3 / 16 c) 0 d) 83

65. 4/

0

3 xdx2cos d; kjpg;G

a) 2 / 3 b) 1/ 3 c) 0 d) 32

66.

0

32 xdxcosxsin

d; kjpg;G

a) b) 2 c) 4 d) 0.

67. y = x, vd;w Nfhl;bw;Fk; x mr;R, NfhLfs; x = 1 , x = 2 Mfpatw;wpw;Fk; ,ilg;gl;l muq;fj;jpd; gug;G.

a) 3 / 2. b) 5 / 2. c) 1 / 2 d) 7 / 2


68. x = 0 ypUe;J x = / 4 tiuapyhd y = sin x kw;Wk; y = cos x vd;w tistiufspd; ,ilg;gl;l gug;G.

a) 12 b) 12 c) 122 d) 222

69. 1b

y

a

x

2

2

2

2

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

a) bab b) baa 2 c) baa d) bab 2

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

a) 4 / 3 b) 1 / 6 c) 2 / 3 d) 8 / 3

71. 116

y

9

x 22

vd;w tistiuia¡ Fw;wr;ir nghWj;J¢ Row;wg;gLk; jplg;nghUspd; fdmsT.

a) 48 b) 64 c) 32 d) 128

72. y = 2x3 vd;w tistiu x = 0 ypUe;J x = 4 tiu x mr;ir mr;rhf itj;J¢ Row;wg;gLk; jplg;nghUspd;

fdmsT.

a) 100 b) 9

100 c)

3

100 d) 100 / 3

73. NfhLfs; y = x, y = 1 kw;Wk; x = 0 Mfpait Vw;gLj;Jk; gug;G y mr;ir¥ nghWj;J Row;wg;gLk; jplg;nghUspd; fdmsT

a) 4 b) 2 c) 3 d) 32

74. 1b

y

a

x

2

2

2

2

vd;w ePs;tl;lj;jpd; gug;ig nel;lr;R> Fw;wr;R ,tw;iw¥ nghWj;J¢ Row;wg;gLk; jplg;nghUspd;

fdmsTfspd; tpfpjk;

a) b² : a² b) a² : b² c) a : b d) b : a

75. (0,0), (3,0) kw;Wk; (3,3) Mfpatw;iw Kidg; Gs;spfshff; nfhz;l Kf;Nfhzj;jpd; gug;G x mr;ir¥ nghWj;J Row;wg;gLk; jplg;nghUspd; fd msT

a) 18 b) 2 c) 36 d) 9

76. x2/3

+ y2/3

= 4 vd;w tistiuapd; tpy;ypd; ePsk;

a) 48 b) 24 c) 12 d) 96

77. y = 2x, x = 0 kw;Wk; x = 2 ,tw;wpw;F ,ilNa Vw;gLk; gug;G x mr;ir¥ nghWj;J¢ Row;wg;gLk; jplg;nghUspd; tisg;gug;G

a) 58 b) 52 c) 5 d) 54

78. Muk; 5 cs;s Nfhsj;ij¤ jsq;fs; ikaj;jpypUe;J 2 kw;Wk; 4 J}uj;jpy; ntl;Lk; ,U ,izahd jsq;fSf;fpilg;gl;;l gFjpapd; tisg;gug;G

a) 20 b) 40 c) 10 d) 30

CHAPTER – 8 (ைவகக்வகழு சமன்போடுகள்)

79. x4ex

y2

dx

dy vd;w tiff;nfO rkd;ghl;bd; njhiff; fhuzp

a) log x. b) x² c) ex. d) x.

80. QPydx

dy vd;w tiff;nfO rkd;ghl;bd; njhiff; fhuzp cos x vdpy; P d; kjpg;G

a) – cot x. b) cot x c) tan x d) – tan x.


81. dx + xdy = e-y

sec² y dy d; njhifaPl;Lf; fhuzp

a) ex. b) e

-x c) ey

. d) e-y

82. 2x

2y.

xlogx

1

dx

dy d; njhifaPl;Lf; fhuzp

a) ex. b) log x. c) 1/ x d) e

-x

83. m < 0 Mf ,Ug;gpd; 0dx

mxdy

d; jPh;T

a) x = cemy

b) x = ce-my

c) x = my + c d) x = c

84. y = cx – c² vd;gij nghJj; jPh;thf¥ ngw;w tiff;nfO rkd;ghL

a) (y’)² - xy’ +y =0 b) y’’ = 0 c) y’=c. d) (y’)² + xy’ + y = 0

85. xy5dy

dx 3/12

vd;w tiff;nfOtpd;

a) thpir 2 kw;Wk; gb 1 b) thpir 1 kw;Wk; gb 2

c) thpir 1 kw;Wk; gb 6 d) thpir 1 kw;Wk; gb 3.

86. xU jsj;jpy; cs;s x mr;Rf;F¢ nrq;Fj;jy;yhj NfhLfspd; tiff;nfO rkd;ghL

a) 0dx

dy b) 0

dx

yd

2

2

c) mdx

dy d) m

dx

yd

2

2

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

a) x dy + y d x = 0 b) x dy – y d x = 0 c) x d x + y dy = 0 d) x d x – y d y =0

88. tiff;nfO rkd;ghL dy

Py Qdx

tpd; njhiff; fhuzp

a) Pdx b) Qdx c) Qdx

e d) Pdx

e

89. (D² + 1 ) y = e2xd; epug;G¢ rhh;G

a) (Ax+B)ex. b) A cos x + B sin x . c) (Ax+B)e

2x. d) (Ax+B)e

-x.

90. (D² -4D+4) y = e2xd; rpwg;Gj; jPh;T (PI).

a) x22

e2

x b) xe

2x. c) xe

-2x. d) x2e

2

x

91. y = mx vd;w Neh;;NfhLfspd; njhFg;gpd; tiff;nfO rkd;ghL

a) mdx

dy b) y d x – x dy = 0 c) 0

dx

yd

2

2

d) y d x + x dy = 0

92. 2

23/1

dx

yd

dx

dy1

vd;w tiff;nfO rkd;ghl;bd;gb

a) 1 b) 2 c) 3 d) 6

93.

3/2

3

3

3

dx

yd

dx

dy1

c

vd;w tiff;nfO rkd;ghl;bd;gb ( ,q;F c xU khwpyp )

a) 1 b) 3 c) -2 d) 2.


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

tiff;nfO rkd;ghL ( k Fiw vz;)

a) p

k

dt

dp b) kt

dt

dp c) kp

dt

dp d) kt

dt

dp

95. xy jsj;jpYs;s vy;yh Neh;;NfhLfspd; njhFg;gpd; tiff;nfO rkd;ghL

a) dx

dy xU khwpyp b) 0dx

yd

2

2

c) 0dx

dyy d) 0y

dx

yd

2

2

96. xkey vdpy; mjd; tiff;nfO rkd;ghL.

a) ydx

dy b) ky

dx

dy c) 0ky

dx

dy d) xe

dx

dy

97. y = ae3x

+ be-3xvd;w rkd;ghl;by; a iaAk; b iaAk; ePf;fpf; fpilf;Fk; tiff;nfO rkd;ghL.

a) 0aydx

yd

2

2

b) 0y9dx

yd

2

2

c) 0dx

dy9

dx

yd

2

2

d) 0x9dx

yd

2

2

98. y = e x (A cos x + B sin x ) vd;w njhlh;gpy; A iaAk; B iaAk; ePf;fpf; fpilf;Fk; tiff;nfO rkd;ghL.

a) y2 + y1 = 0 b) y2 - y1 = 0 c) y2 -2y1 +2y = 0 d) y2 -2 y1 -2y = 0

99. yx

yx

dx

dy

vdpy;

a) 2xy + y² - x² = c. b) x²+y²-x +y = c. c) x²+y²-2xy = c. d) x²-y²-2xy = c.

100. xxf ' kw;Wk; f(1) = 2 vdpy; f(x) vd;gJ

a) )2xx(3

2 b) )2xx(

2

3 c) )2xx(

3

2 d) )2x(x

3

2

101. x² dy + y (x+y) dx= 0 vd;w rkg;gbj;jhd tiff;nfO rkd;ghl;by; y = vx vd gpujpaPL nra;Ak; NghJ fpilg;gJ.

a) x dv + (2v +v²) dx = 0 b) v dx + (2x +x²) dv = 0 c) v³ dx - (x +x²) dv = 0 d) v dv + (2x +x²) dx = 0

102. xcosxtanydx

dy vd;w tiff;nfO rkd;ghl;bd; njhiff;fhuzp

a) sec x . b) cos x. c) etanx

. d)cot x.

103. (3D²+D-14) y = 13e2x d; rpwg;G¤ jPh;T

a) 26 x e2x

. b) 13 x e2x

. c) x e2x

. d) x22

e2

x

104. f(D) = (D-a) g(D), g(a) 0 vdpy; tiff;nfO rkd;ghL f(D) y = eaxd; rpwg;Gj; jPh;T.

a) meax

b) eax

/ g(a) c) g(a) eax

d) xeax

/ g(a)

CHAPTER – 9 (jdpepiy fzf;fpaல் ;;;)

105. fPo;f;fz;ltw;Ws; vit $w;Wfs; ?

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

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

a) (i), (ii), (iii) b) (i), (ii),(iv) c) (i), (iii), (iv) d) (ii), (iii), (iv)

106. xU $l;Lf; $w;W %d;W jdpf;$w;Wfisf; nfhz;ljhf ,Ug;gpd;> nka;al;ltizapYs;s epiufspd; vz;zpf;if

a) 8 b) 6 c) 4 d) 2


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

(i) p v q (ii) ~ p v q iii) p v ~q iv) p ^ ~ q.


108. ~ [ p ^ (~q) ] d; nka; ml;ltizapy; epiufspd; vz;zpf;if

a) 2 b) 4 c) 6 d) 8.

109. epge;jidf; $w;W qp f;F rkhdkhdJ

a) p v q. b) p v ~ q c) ~p v q. d) p ^ q.

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

a) p v q b) p ^ q c) p v ~p d) p ^ ~p

111. gpd;tUtdtw;Ws; vJ Kuz;ghlhFk; ?

a) p v q b) p ^ q c) p v ~q d) p ^ ~q

112. qp f;F rkkhdJ

a) qp b) pq c) p q q p d) pqqp

113. fPo;f;fz;ltw;wpy; vJ Ry; <UWg;Gr; nrayp my;y ?

a) a * b = ab. b) a * b = a – b c) a * b = ab d) a * b = 22 ba

114. rkdpAila miuf;Fyk; Fykhtjw;F¥ G+Hj;jp nra;a Ntz;ba tpjpahtJ

a) milg;G tpjp b) NrHg;G tpjp c) rkdp tpjp d) vjpHkiw tpjp

115. fPo;f;fz;ltw;Ws; vJ Fyk; my;y ?

a) (Zn, +n). b) (Z, +) c) (Z, .) d) (R, +)

116. KOf;fspy; * vd;w <UWg;G nrayp a * b = a+ b – ab, vd tiuaWf;fg;gLfpwJ vdpy; 3 * ( 4 * 5) d; kjpg;G

a) 25 b) 15 c) 10 d) 5

117. (Z9, +9) y; [7] d; thpir

a) 9 b) 6 c) 3 d) 1.

118. ngUf;fiy¥ nghWj;J¡ Fykhfpa xd;wpd; Kg;gb %yq;fspy; 2 d; thpir

a) 4 b) 3 c) 2 d) 1

119. [ 3 ] + 11( [5] + 11[6]) d; kjpg;G

a) [0] b) [1] c) [2] d) [3]

120. nka;naz;fspd; fzk; R y; * vd;w <UWg;G nrayp a * b = 22 ba vd tiuaWf;fg;gLfpwJ vdpy; (3 *4)* 5 d; kjpg;G

a) 5 b) 25 c) 25 d) 50

121. fPo;f;fz;ltw;Ws; vJ rhp ?

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

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

c) nka;naz;fis cWg;Gfshff; nfhz;l vy;yh 2 x 2 mzpf;NfhitfSk; ngUf;fy; tpjpapy; FykhFk;

d) vy;yh a, b G f;Fk; (a*b)-1

= a-1

* b-1

.

122. ngUf;fy; tpjpia¥ nghWj;J¡ Fykhfpa xd;wpd; ehyhk; %yq;fspy; – i d; thpir

a)4 b) 3 c) 2 d) 1


123. ngUf;fiy¥ nghWj;J¡ Fykhfpa xd;wpd; nk; gb %yq;fspy; k d; vjph;kiw (k<n)

a) k1 b) 1 c) kn d) kn

124. KOf;fspy; * vd;w <UWg;G nrayp a * b = a + b – 1 vd tiuaWf;fg;gLfpwnjdpy; rkdp cWg;G

a) 0 b) 1 c) a d) b

CHAPTER – 10 (நிகழ்தகவுப் பரைல்)

125 . f(x) =

elsewhere,0

3x0,kx 2

vd;gJ epfo;jfT mlh;j;jp¢ rhh;G vdpy; k d; kjpg;G

a) 1/ 3 b) 1/ 6 c) 1/ 9 d) 1/12.

126. f(x) =

xx

A,

16

12

vd;gJ X vd;w njhlh; rktha;g;G khwpapd; xU epfo;jfT mlh;j;jp rhh;G

(p.d.f) vdpy; A d; kjpg;G

a) 16 b) 8 c) 4 d) 1

127. X vd;w rktha;g;G khwpapd; epfo;jfT guty; gpd;tUkhW

x 0 1 2 3 4 5

P(X = x) ¼ 2a 3a 4a 5a ¼

41 xP d; kjpg;G

a) 10 / 21 b) 2 / 7 c) 1/ 14 d) 1/ 2

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

X – 2 3 1

P(X = x) 6 4 12

d; kjpg;G

a) 1 b) 2 c) 3 d) 4

129. X vd;w xU jdpepiy rktha;g;G khwp 0, 1, 2 vd;w kjpg;Gfisf; nfhs;fpwJ. NkYk; P(X = 0) = 144 / 169,

P(X = 1) = 1/169 vdpy; P(X = 2) d; kjpg;G

a) 145 / 169 b) 24 / 169 c) 2 / 169 d) 143 /169

130. xU rktha;g;G khwp X d; epfo;jfT epiwr; rhh;G (p.d.f) gpd;tUkhW

x 0 1 2 3 4 5 6 7

P(X = x) 0 K 2k 2k 3k K² 2k² 7k²+k

k d; kjpg;G

a) 1/ 8 b) 1/ 10 c) 0 d) – 1 or 1/10

131. E(X + c) = 8 kw;Wk; E (X – c) = 12 vdpy; c d; kjpg;G.

a) – 2 b) 4 c) – 4 d) 2.

132. X vd;w rktha;g;G khwpapd; 3,4 kw;Wk; 12 Mfpa kjpg;Gfs; KiwNa 1/3, 1 / 4 kw;Wk; 5 / 12 Mfpa epfo;jfTfis¡

nfhs;Snkdpy; E(X) d; kjpg;G.

a) 5 b) 6 c) 7 d) 3


133. X vd;w rktha;g;G khwpapd; gutw;gb 4 NkYk; ruhrhp 2 vdpy; E(X²) d; kjpg;G

a) 2 b) 4 c) 6 d) 8

134. xU jdpepiy rktha;g;G khwp X f;F 202 NkYk; 276'2 vdpy; X d; ruhrhpapd; kjpg;G

a) 16 b) 5 c) 2 d) 1

135. Var (4X + 3) d; kjpg;G

a) 7 b) 16 Var (X) c) 19 d) 0

136. xU gfilia 5 Kiw tPRk; NghJ 1 my;yJ 2 fpilg;gJ ntw;wpnadf; fUjg;gLfpwnjdpy; ntw;wpapd;

ruhrhpapd; kjpg;G

a) 5/3 b) 3/5 c) 5/9 d) 9/5

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

a)

25,

5

4 b)

5

4,25 c)

25,

5

1 d)

5

1,25

138. xU <UWg;G gutypd; ruhrhp 12 kw;Wk;jpl;l tpyf;fk; 2 vdpy; gz;gsit p d; kjpg;G

a) 1/ 2 b) 1 / 3 c) 2 / 3 d) 1 / 4.

139. xU gfilia 16 Kiw tPRk; NghJ ,ul;ilgil vz; fpilg;gJ ntw;wpahFk; vdpy; ntw;wpapd; gutw;gb

a) 4 b) 6 c) 2 d) 256

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

a) 1/20 b) 18 / 125 c) 4 /25 d) 3 /10

141. ed;F fiyf;fg;gl;l 52 rPl;Lfs; nfhz;l rPl;Lf;fl;bypUe;J 2 rPl;Lfs; jpUg;gp itf;fhky; vLf;fg;gLfpd;wd. ,uz;Lk; xNu epwj;jpy; ,Uf;f epfo;jfT.

a) 1 / 2. b) 26 / 51 c) 25 /51 d) 25 /102

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

a) log 1/ k b) log k c) e d) 1 / k

143. xU rktha;g;G khwp X gha;]hd; gutiy¥ gpd;gw;WfpwJ. NkYk;, E(X²) = 30 vdpy; gutypd; gutw;gb

a) 6 b) 5 c) 30 d) 25

144. rktha;g;G khwp X d; guty; rhh;G F(X) xU

a) ,wq;Fk; rhh;G b) Fiwah¢ (,wq;fh) rhh;G

c) khwpyp¢ rhh;G d) Kjypy; VWk; rhh;G gpd;dH ,wq;Fk; rhh;G

145. gha;]hd; gutypy; gz;gsit = 0.25 vdpy;, Mjpiag; nghWj;J ,uz;lhtJ tpyf;fg; ngUf;Fj; njhif.

a) 0.25 b) 0.3125 c) 0.0625 d) 0.025

146. xU gha;]hd; gutypy; P(X = 2 ) = P(X = 3) vdpy; gz;gsit d; kjpg;G

a) 6 b) 2 c) 3 d) 0

147. xU ,ay;epiy gutypd; epfo;jfT mlh;j;jp¢ rhh;G f(x) d; ruhrhp vdpy;

dx)x(f d; kjpg;G.

a) 1 b) 0.5 c) 0 d) 0.25

148. xU rktha;g;G khwp X ,ay;epiy¥ guty; f(x) = 25

)100x(2/1 2

ce

I gpd;gw;Wfpwnjdpy; c d; kjpg;G

a) 2 b) 2

1 c) 25 d) 25

1

149. xU ,ay;epiy khwp X d; epfo;jfT mlh;j;jp¢ rhh;G f(x) kw;Wk; 2,~ NX vdpy;

µ

dx)x(f d; kjpg;G

a) tiuaWf;fg;glhjJ. b) 1 c) 0.5 d) – 0.5

150. 400 khzth;fs; vOjpa fzpjj; Njh;tpd; kjpg;ngz;fs; ,ay;epiy¥ gutiy xj;jpUf;fpwJ. ,jd; ruhrhp 65.

NkYk; 120 khzth;fs; 85 kjpg;ngz;fSf;F Nky; ngw;wpUg;gpd; kjpg;ngz;fs; 45 ypUe;J 65 f;Fs; ngWk; khzth;fspd; vz;zpf;if.

a) 120 b) 20 c) 80 d) 160


nghJj; Njh;tpy; [_d; 2015 tiu Nfl;fg;glhj 10 kjpg;ngz; tpdhf;fs;

1. mzpfs; kw;Wk; mzpf;Nfhitfspd; gad;ghLfs; - 18 tpdhf;fs;

1) A =

110

432

433

våš, A(adj A) = (adj A)A = |A| I v‹gij¢ rçgh®¡f. (1.1 – 3)

2) A =

110

432

433

- Ï‹ ne®khW fh©f. k‰W« A3 = A

- 1 #¢ rçgh®¡f. (1.1 – 6)

3) A =

122

212

221

- Ï‹ nr®¥ò mâ 3AT.vd ãWÎf. (1.1 – 7)

4) A = 3

1

221

212

122

våš , A- 1 = AT vd ãWÎf. (1.1 – 9)

5) A =

312

321

111

våš , A(adj A) = (adj A)A = |A| I3 v‹gij¢ rçgh®¡f. (1.4)

ne® khW mâ¡ fhzš Kiwæš Ã‹ tU« neça rk‹gh£L¤ bjhF¥òfis¤ Ô®¡fÎ«.

6) x + y + z = 9 ; 2x + 5y + 7z = 52 ; 2x + y – z = 0. (1.2 – 3)

7) 2x – y + z = 7 ; 3x + y – 5z = 13 ; x + y + z = 5 (1.2 – 4)

ÑnH bfhL¡f¥g£LŸs mrkgo¤jhd neça¢ rk‹gh£L¤ bjhF¥Ãid mâ¡nfhit Kiwæš Ô®¡f

8) x + 2y + z = 7 9) x + y + 2z = 6 10) x + y + 2z = 4

2x – y + 2z = 4 3x + y – z = 2 2x + 2y + 4z = 8

x + y – 2z = – 1 4x + 2y + z = 8 3x + 3y + 6z = 12

(1.18 – 1) (1.18 – 2) (1.18 – 4)

11) x + y + z = 4 12) 3x + y – z = 2 13) x + 2y + z = 6

x – y + z = 2 2x – y + 2z = 6 3x + 3y – z = 3

2x + y – z = 1 2x + y – 2z = – 2 2x + y – 2z = – 3

(1.4 – 4) (1.4 – 6) (1.4 – 7)


Ã‹tU« rk‹gh£L¤ bjhF¥ò xU§fikÎ cilajh v‹gij MuhŒf. xU§fikÎ

cilajhæ‹mjid¤ Ô®¡f.

14) x – 3y – 8z = – 10 ; 3x + y – 4z = 0 ; 2x + 5y + 6z – 13 = 0. (1.5 – 1 (ii))

Ã‹tU« rkgo¤jhd neça rk‹ghLfis¤ Ô®¡fÎ« :

15) x + y + 2z = 0 ; 3x + 2y + z = 0 ; 2 x + y – z = 0. (1.21)

16) x + 2y – 5z = 0 ; 3x + 4y + 6z = 0 ; x + y + z = 0 (1.27)

17) Ã‹tU« rk‹gh£Lfë‹ xU§fikÎ¤ j‹ikia MuhŒf.

2x – 3y + 7z = 5 ; 3x + y – 3z = 13 ; 2x + 19y – 47z = 32. (1.23)

18) ju¥g£l rk‹gh£L¤ bjhF¥ò xU§fikÎ cilajh vd¢ rçgh®¤J, m›thW xU§fikÎ

cilajhæ‹

mjid¤ Ô®¡fÎ« .

x – y + z = 5 ; – x + y – z = – 5 ; 2x – 2y + 2z = 10. (1.25)

2. ntf;lh; ,aw;fzpjk; - 0 tpdh

3. fyg;ngz;fs; - 2 tpdhf;fs;

P vD« òŸë fy¥bg© kh¿ z I¡ F¿¤jhš, P Ï‹ ãak¥ghijia¥ Ã‹tUtdt‰¿‰F fh©f.

1) Re 11

iz

z. (3.11 – (i)) 2) arg

23

1

z

z. (3.2 – 8(v))

4.gFKiw tbtfzpjk; - 1 tpdh

1) 0681684 22 yxyx vD«ÚŸt£l¤Â‹ ika¤ bjhiy¤jfÎ, ika«, Féa§fŸ, KidfŸ

M»at‰iw¡ fh©f. tiugl«tiuf (4.2 – 6 (ii) )

5. tif Ez;fzpjk; gad;ghLfs;-I - 15 tpdhf;fs;

1) xU éir ÏG¥gh‹ _y« brY¤j¥gL« fU§fš #šèfŸ édho¡F 30 f.mo Åj« nkèUªJ ÑnH

bfh£l¥gL« nghJ, mit T«ò tot¤ij¡ bfhL¡»wJ. vªneu¤ÂY« m¡T«Ã‹ é£lK« cauK«

rkkhfnt ÏU¡Fkhdhš, T«Ã‹ cau« 10 moahf ÏU¡F« nghJ cau« v‹d Åj¤Âš ca®»wJ

v‹gij¡ fh©f. (5.1 – 9)

2) x = a ( + sin ), y = a (1 + cos ) v‹w JizayF rk‹ghLfis¡ bfh©l tistiu¡F 2

Ïš

bjhLnfhL, br§nfhL M»at‰¿‹ rk‹ghLfis¡ fh©f. (5.13)

3) 144916 22 yx v‹w tistiu¡F x1 = 2 k‰w« y1 > 0.vd ÏU¡FkhW (x1, y1) v‹w òŸëæš

tiua¥gL« bjhLnfhL, br§nfhL Ït‰¿‹ rk‹ghLfis¡ fh©f. (5.14)


4) x = acos , y = bsin v‹w JizayF¢ rk‹ghLfis¡ bfh©l ÚŸt£l¤Â‰F = 4

. Ïš

tiua¥gL« bjhLnfhL, br§nfhL Ït‰¿‹ rk‹ghLfis¡ fh©f. (5.15)

5) x = a cos4 , y = a sin4 , 0 2

v‹w Jiz myF¢ rk‹ghLfis¡ bfh©l tistiu¡F

tiua¥g£l

vªjbthU bjhLnfhL« V‰gL¤J« Ma m¢R¤ J©Lfë‹ TLjš a vd¡ fh£Lf. (5.2 – 5)

6) kÂ¥ò¡ fh©f: x

x

x sin

0lim

(5.35)

7) 20,sin2 xxxxf v‹gj‹ Û¥bgU bgUk« k‰W« Û¢ÁW ÁWk« M»at‰iw¡ fh©f. (5.48 (a))

Ã‹tU« rh®òfS¡F ÏlŠrh®ªj bgUk k‰W« ÁWk kÂ¥òfis¡ fh©f.

8) x4 – 6x2 . (5.9 – 3 (iii)) 9) (x2 – 1)3. (5.9 – 3 (iv)

10) sin2 [0, ]. (5.9 – 3 (v)) 11) t + cos t . (5. 9 – 3 (vi))

12) xU étrhæ br›tf totkhd taY¡F ntèæl nt©oÍŸsJ. m›taè‹ xU g¡f¤Âš MW x‹W

ne®nfh£oš xL»wJ. m¥g¡f¤Â‰F ntè njitæšiy. mt® 2400 mo¡F ntèæl¡ fUÂÍŸsh®.

m›tifæš bgUk gu¥gsÎ bfhŸSkhW cŸs Ús, mfy msÎfŸ v‹d? (5.52)

13) y = x4 – 4x3 v‹w tistiu¡F¡ FêÎ k‰W« tisÎ kh‰W¥ òŸëfisÍ« gçnrhÂ¡f. (5.63)

14) fhìa‹ tistiu y =2xevªj Ïilbtëfëš FêÎ, FéÎ mil»wJ v‹gijÍ« tisÎ kh‰W¥

òŸëfisÍ« fh©f. (5.64)

15) ;2sin f ,0 v‹w tistiu¡F FêÎ k‰W« tisÎ kh‰W¥ òŸëfisÍ« gçnrhÂ¡f.

(5.11 – 5)

6. tif Ez;fzpjk; gad;ghLfs;-II - 2 tpdhf;fs;

1) w = u2ev, v‹w rh®Ãš u =y

x k‰W« v = y log x vDkhW ÏU¥Ã‹

x

w

k‰W«

y

w

fh©f. (6.18)

2) yxu 4cos3sin v‹w rh®Ã‰F

xy

u

yx

u

22

v‹gij¢ rçgh®¡f. 6.3-1 (iii)


7. njhif Ez;fzpjk; gad;ghLfs; - 4 tpdhf;fs;

1) y = x + 1 v‹w nfh£o‰F« sy = x2 – 1 v‹w tistiu¡F« Ïilna cŸs mu§f¤Â‹ gu¥ig¡ fh©f.

(7.26)

2) tistiu y2 = x k‰W« y = x – 2 v‹w nfh£odhš milgL« gu¥Ãid¡ fh©f. (7.28)

3) 12

2

2

2

b

y

a

x v‹w ÚŸt£l¤Âdhš cUthF« mu§f¤Â‹ gu¥ig¡ fh©f. (7.31)

4) y2 = (x – 5)2 (x – 6) v‹w tistiu¡F Kiwna (i) x = 5 k‰W« x = 6

(ii) x = 6 k‰W« x = 7 M»a nfhLfS¡F Ïilnaahd gu¥òfis¡ fh©f. (7.32)

8. tiff;nfOr; rkd;ghLfs; - 4 tpdhf;fs;

1) Ô®¡f : 02 dxydyxxy (8.13)

2) x = 0 Mf ÏU¡F« nghJ y = 1 vd ÏU¡Fkhdhš (1 + ex/y) dx + ex/y (1 – x/y) dy = 0 v‹w rk‹gh£o‹

Ô®Î fh©f. (8.3 -15)

3) Ô®¡f: )(sin 2xx

y

dx

dy (8.4 -5)

4) xU nehahëæ‹ ÁWÚçèUªJ ntÂ¥bghUŸ btënaW« mséid¤ bjhl®¢Áahf¡ nf¤njl® v‹w

fUéæ‹ _y« f©fhâ¡f¥gL»wJ t = 0 v‹w neu¤Âš nehahë¡F 10 ä.»uh« ntÂ¥bghUŸ

bfhL¡f¥gL»wJ.

ÏJ –3t1/2 ä.»uh«/kâ v‹D« Åj¤Âš btënaW»wJ våš,

(i) neu« t > 0 vD« nghJ, nehahëæ‹ clèYŸs ntÂ¥bghUë‹ msit¡ fhQ« bghJ¢ rk‹ghL

v‹d?

(ii) KGikahf ntÂ¥ bghUŸ btënaw vL¤J¡ bfhŸS« Fiwªjg£r¡ fhy msÎ v‹d? (8.38)

9. jdpepiyf; fzf;fpay; - 1 tpdh

1) QbabaG ,/2 v‹gJ T£liy¥ bghW¤J xU Kot‰w vÕèa‹ Fy« vd¡ fh£Lf. (9.22)

10. epfo;jfTg; guty; - 0 tpdh

**********************************************************


10 kjpg;ngz; khjphpj; Njh;Tfs;

TEST - 1

1. vd ntf;lH Kiwapy; ep&gp.

2. vd;w Gs;sp topr; nry;tJk; kw;Wk; vd;w NfhLfSf;F

,izahdJkhd jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f. 3. kw;Wk; vd;w Gs;spfs; topNa; nry;yf; $baJk; vd;w jsj;jpw;Fr;

nrq;Fj;jhfTk; mike;j jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f

4.

vdpy; vd rhpghHf;f.

5. ntl;Lj;Jz;L tbtpy; xU jsj;jpd; rkd;ghl;ilj; jUtpf;f.

TEST – 2


7. vd;w Nfhl;il cs;slf;fpaJk; vd;w Nfhl;bw;F ,izahdJkhd jsj;jpd;

ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f.

8. vd;w Nfhl;il cs;slf;fpaJk; vd;w Gs;sp topNa nry;yf; $baJkhd jsj;jpd;

ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f.

9. kw;Wk; vd;w NfhLfs; ntl;bf; nfhs;Sk; vdf; fhl;Lf. NkYk; mit

ntl;Lk; Gs;spiaf; fhz;f.

10. kw;Wk; Mfpatw;iw epiy ntf;lHfshff; nfhz;l Gs;spfs;

topNa nry;Yk; jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f.

TEST – 3


12. topNa nry;yf;$baJk; kw;Wk; Mfpa NfhLfSf;F

,izahf cs;sJkhd jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f.

13. kw;Wk; Mfpa Gs;spfs; têNa nry;yf; $baJk; vd;w jsj;jpw;F¢ nrq;Fj;jhf miktJkhd jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghl;ilf; fhz;f.

14. vdpy;,

vd¢ rhpghHf;f.

15. xU Kf;Nfhzj;jpd; Fj;Jf;NfhLfs; xNu Gs;spapy; re;jpf;Fk; vd;gjid ntf;lH Kiwapy; epWTf.

TEST – 4


17. vd;w Gs;sp topr; nry;tJk; kw;Wk; Mfpa jsq;fSf;Fr;

nrq;Fj;jhdJkhd jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f.

18. kw;Wk; vd;w Gs;spfs; topNa nry;yf;$baJk; vd;w Nfhl;bw;F

,izahdJkhd jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f.

19. kw;Wk; vd;w NfhLfs; ntl;Lk; vdf; fhl;b, mit ntl;Lk; Gs;spiaf;

fhz;f.

20. kw;Wk; Mfpa Gs;spfs; topNa nry;yf;$ba jsj;jpd; ntf;lH kw;Wk; fhHBrpad;

rkd;ghl;ilf; fhz;f.


TEST - 5

21. vd;w gutisaj;jpd; mr;R> Kid> Ftpak;> ,af;Ftiuapd; rkd;ghL> nrt;tfyj;jpd;

rkd;ghL> nrt;tfyj;jpd; ePsk; Mfpatw;iwf; fhz;f. NkYk; mjd; tiuglk; tiuf.

22. vd;w gutisaj;jpd; mr;R> Kid> Ftpak;> ,af;Ftiuapd; rkd;ghL>

nrt;tfyj;jpd; rkd;ghL> nrt;tfyj;jpd; ePsk; Mfpatw;iwf; fhz;f. NkYk; mjd; tiuglk; tiuf.

23. vd;w ePs;tl;lj;jpd; ikaj; njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfs;

Mfpatw;iwf; fhz;f. NkYk; mjd; tistiuia tiuf.

24. vd;w mjpgutisaj;jpd; ikaj; njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfs;

Mfpatw;iwf; fhz;f. NkYk; mjd; tistiuia tiuf.

25. vd;w NeHNfhL ePs;tl;lk; f;F njhLNfhlhf cs;sJ vd ep&gpf;f. NkYk; njhLk;

Gs;spiaAk; fhz;f

TEST – 6

26. vd;w gutisaj;jpd; mr;R> Kid> Ftpak;> ,af;Ftiuapd; rkd;ghL>

nrt;tfyj;jpd; rkd;ghL> nrt;tfyj;jpd; ePsk; Mfpatw;iwf; fhz;f. NkYk; mjd; tiuglk; tiuf 27. vd;w ePs;tl;lj;jpd; ikaj; njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfs;

Mfpatw;iwf; fhz;f. NkYk; mjd; tistiuia tiuf. 28. vd;w mjpgutisaj;jpd; ikaj; njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfs;

Mfpatw;iwf; fhz;f. NkYk; mjd; tistiuia tiuf. 29. vd;w mjpgutisaj;jpd; ikaj; njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfs;

Mfpatw;iwf; fhz;f. NkYk; mjd; tistiuia tiuf. 30. mjpgutisaj;jpd; ikak; NkYk; topNa nry;fpwJ. ,jd; njhiyj; njhLNfhLfs;

kw;Wk; Mfpatw;wpw;F ,izahf ,Uf;fpd;wd. vdpy;, mjpgutisaj;jpd; rkd;ghL; fhz;f.

TEST – 7

31. vd;w gutisaj;jpd; mr;R> Kid> Ftpak;> ,af;Ftiuapd; rkd;ghL> nrt;tfyj;jpd; rkd;ghL> nrt;tfyj;jpd; ePsk; Mfpatw;iwf; fhz;f. NkYk; mjd; tiuglk; tiuf.

32. vd;w ePs;tl;lj;jpd; ikaj; njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f. NkYk; mjd; tistiuia tiuf.

33. vd;w mjpgutisaj;jpd; ikaj; njhiyj;jfT> ikak;> Ftpaq;fs;> cr;rpfs; Mfpatw;iwf; fhz;f. NkYk; mjd; tistiuia tiuf.

34. vd;w NeH;NfhL mjpgutisak; -Ij; njhLfpwJ vd ep&gpf;f NkYk; njhLk; Gs;spiaAk; fhz;f.

35. -I xU njhiyj; njhLNfhlhfTk;> kw;Wk; vd;w Gs;spfs; topNa nry;yf;$baJkhd nrt;tf mjpgutisaj;jpd; rkd;ghL fhz;f.

TEST – 8

36. xU uapy;Nt ghyj;jpd; Nky; tisT gutisaj;jpd; mikg;igf; nfhz;Ls;sJ. me;j tistpd; mfyk;

mbahfTk; mt;tistpd; cr;rpg;Gs;spapd; cauk; ghyj;jpypUe;J mbahfTk; cs;sJ vdpy;> ghyj;jpd;

kj;jpapypUe;J ,lg;Gwk; my;yJ tyg;Gwk; mb J}uj;jpy; ghyj;jpd; Nky; tisT vt;tsT cauj;jpy; ,Uf;Fk; vdf; fhz;f.

37. xU tisT miu-ePs;tl;l tbtj;jpy; cs;sJ. mjd; mfyk; mb cauk; mb jiuapypUe;J mb cauj;jpy; tistpd; mfyk; vd;d?

38. xU ePs;tl;lg; ghijapd; Ftpaj;jpd; G+kp ,Uf;FkhW xU Jizf;Nfhs; Rw;wp tUfpwJ. ,jd; ikaj; njhiyj;

jfT MfTk; G+kpf;Fk; Jizf; NfhSf;Fk; ,ilg;gl;l kPr;rpW J}uk; fpNyh kPl;lHfs; MfTk;

,Uf;Fkhdhy; G+kpf;Fk; Jizf;NfhSf;Fk; ,ilg;gl;l mjpfgl;r J}uk; vd;d? 39. xU Nfh-Nfh tpisahl;L tPuH tpisahl;Lg; gapw;rpapd;NghJ mtUf;Fk; Nfh-Nfh Fr;rpfSf;Fk; ,ilNaAs;s

J}uk; vg;nghOJk; 8kP Mf ,Uf;FkhW czHfpwhH. mt;tpU Fr;rpfSf;F ,ilg;gl;l J}uk; kP vdpy; mtH XLk; ghijapd; rkd;ghl;ilf; fhz;f.

40. xU njhq;F ghyj;jpd; fk;gp tlk; gutisa tbtpypYs;sJ. mjd; ePsk; kPl;lH MFk;. topg;ghijahdJ

fk;gp tlj;jpd; fPo;kl;lg; Gs;spapypUe;J kPl;lH fPNo cs;sJ. fk;gp tlj;ijj; jhq;Fk; J}z;fspd; cauq;fs;

kPl;lH vdpy;> kPl;lH cauj;jpy; fk;gp tlj;jpw;F xU Jiz¤jhq;fp $Ljyhff; nfhLf;fg;gl;lhy; mj;Jizj;jhq;fpapd; ePsj;ijf; fhz;f.


TEST – 9

41. xU uhf;nfl; ntbahdJ nfhSj;Jk;NghJ mJ xU gutisag; ghijapy; nry;fpwJ. mjd; cr;r cauk; kP-I

vl;Lk;NghJ mJ nfhSj;jg;gl;l ,lj;jpypUe;J fpilkl;l J}uk; kP njhiytpYs;sJ. ,Wjpahf fpilkl;lkhf

kP njhiytpy; jiuia te;jilfpwJ. vdpy;, Gwg;gl;l ,lj;jpy; jiuAld; Vw;gLj;jg;gLk;vwpNfhzk; fhz;f. 42. xU thy; tpz;kPd; MdJ #hpaidr; Rw;wp gutisag; ghijapy; nry;fpwJ. kw;Wk; #hpad; gutisaj;jpd;

Ftpaj;jpy; mikfpwJ. thy; tpz;kPd; #hpadpypUe;J 80 kpy;ypad; fp.kP njhiytpy; mike;J ,Uf;Fk; NghJ

thy; tpz;kPidAk; #hpaidAk; ,izf;Fk; NfhL ghijapd; mr;Rld; Nfhzj;jpid Vw;gLj;Jkhdhy; thy;

tpz;kPdpd; ghijapd; rkd;ghl;ilf; fhz;f. thy; tpz;kPd; #hpaDf;F vt;tsT mUfpy; tuKbAk;

vd;gijAk; fhz;f. (ghij tyJGwk; jpwg;Gilajhf nfhs;f). 43. xU ghyj;jpd; tisthdJ miu ePs;tl;lj;jpd; tbtpy; cs;sJ. fpilkl;lj;jpy; mjd; mfyk; mbahfTk;

ikaj;jpypUe;J mjd; cauk; mbahfTk; cs;sJ vdpy;, ikaj;jpypUe;J tyJ my;yJ ,lg;Gwj;jpy; mb J}uj;jpy; cs;s jiug;Gs;spapypUe;J ghyj;jpd; cauk; vd;d?

44. xU tisT miu-ePs;tl;l tbtj;jpy; cs;sJ. mjd; mfyk; mb cauk; mb jiuapypUe;J mb cauj;jpy; tistpd; mfyk; vd;d?

TEST – 10

45. jiukl;lj;jpypUe;J kP cauj;jpy; jiuf;F ,izahf nghUj;jg;gl;l xU FohapypUe;J ntspNaWk; ePH

jiuiaj; njhLk; ghij xU gutisaj;ij Vw;gLj;JfpwJ. NkYk;,,e;j¥ gutisag; ghijapd; Kid

Fohapd; thapy; mikfpwJ. Foha; kl;lj;jpw;F kP fPNo ePhpd; gha;thdJ Fohapd; Kid topahfr; nry;Yk;

epiy¡ Fj;Jf;Nfhl;bw;F kPl;lH J}uj;jpy; cs;sJ vdpy; Fj;Jf; Nfhl;bypUe;J vt;tsT J}uj;jpw;F mg;ghy; ePuhdJ jiuapy; tpOk; vd;gijf; fhz;f.

46. xU njhq;F ghyj;jpd; fk;gp tlk; gutisa tbtpYs;sJ. mjd; ghuk; fpilkl;lkhf rPuhf¥ gutpAs;sJ.

mijj; jhq;Fk; ,U J}z;fSf;F ,ilNaAs;s J}uk; mb. fk;gp tlj;ijj; jhq;Fk; Gs;spfs; J}zpy;

jiuapypUe;J mb cauj;jpy; mike;Js;sd. NkYk; jiuapypUe;J fk;gp tlj;jpd; jho;thd Gs;spapd;

cauk; mb> fk;gptlk; mb caj;jpy; jhq;Fk; fk;gj;jpw;F ,ilNa cs;s nrq;Fj;J ePsk; fhz;f. (jiuf;F ,izahf)

47. #hpad; Ftpaj;jpypUf;FkhW nkHf;Fhp fpufkhdJ #hpaid xU ePs;tl;lg; ghijapy; Rw;wp tUfpwJ. mjd; miu nel;lr;rpd; ePsk; kpy;ypad; iky;fs; MfTk; ikaj; njhiyj; jfT MfTk; ,Uf;Fkhapd;

nkHf;Fhp fpufkhdJ #hpaDf;F kpf mUfhikapy; tUk;NghJ cs;s J}uk;

nkHf;Fhp fpufkhdJ #hpaDf;F kpfj; njhiytpy; ,Uf;Fk;NghJ cs;s J}uk; Mfpatw;iwf; fhz;f.

48. xU rkjsj;jpd; Nky; nrq;Fj;jhf mike;Js;s Rthpd; kPJ kP ePsKs;s xU VzpahdJ jsj;jpidAk;>

Rtw;wpidAk; njhLkhW efHe;J nfhz;L ,Uf;fpwJ vdpy;> Vzpapd; fPo;kl;l KidapypUe;J kP J}uj;jpy;

Vzpapy; mike;Js;s vd;w Gs;spapd; epakg;ghijiaf; fhz;f.


6 kjpg;ngz; khjphpj; Njh;Tfs;

TEST - 11

1.

11

21A

kw;Wk;

21

10B vdpy;; 111

ABAB vd;gij¢ rhpghh;.

2.

121

022

113

A

vd;w mzpapd; NeHkhW mzpiaf; fhz;f.

3. (i)

dc

baA vd;w mzpapd; NrHg;G mzpiaf; fhz;f.

(ii) mzpf;Nfhit Kiwapy; jPHf;f: 732;3 yxyx

4.

41

21A vdpy;> 2IAAAadjAadjA vd;gjidr; rhpghH.

5. (i) jPHf;f: 022;02;02 zyxzyxzyx

(ii) mzpf;Nfhit Kiwapy; jPHf;f: 733;2 xyyx

6. mzpfspd; NeHkhWfSf;Fhpa thpirkhw;W tpjpapid vOjp ep&gp.

7. (i)

24

12 vd;w mzpapd; NeHkhW mzpiaf; fhz;f.

(ii)

115

642

321

vd;w mzpapd; juk; fhz;f.

8.

2751

5121

1513


9. 10633;8422;42 zyxzyxzyx ; vd;w mrkgbj;jhd rkd;ghl;Lj; njhFg;gpid¤ jPHf;f :

10. NeHkhW mzpfhzy; Kiwapy; jPHf;f : 832,3 yxyx

TEST – 12

1. qqp ~ ,d; nka; ml;ltizia mikf;f.

2. rqp -,d; nka; ml;ltizia mikf;f.

3. pqqpqp ~~ vdf; fhl;Lf.

4. qpqp ~~ nka;ikah my;yJ Kuz;ghlh vd;gjid nka; ml;ltiziaf; nfhz;L jPHkhdpf;f.

5. pqp ~ nka;ikah my;yJ Kuz;ghlh vd;gjid nka; ml;ltiziaf; nfhz;L jPHkhdpf;f.

6. 1 ,d; 3 Mk; gb %yq;fs; xU Kbthd vgPypad; Fyj;ij¥ ngUf;fypd; fPo; mikf;Fk; vdf; fhl;Lf.

7. ,Z xU Kbtw;w vgPypad; Fyk; vd epWTf.

8. (i) xU Fyj;jpd; rkdp cWg;G xUikj;jd;ik tha;e;jJ vd ep&gp.

(ii) xU Fyj;jpy; xt;nthU cWg;Gk; xNunahU vjpHkiwiag; ngw;wpUf;Fk; vd ep&gp.

9. vjpHkiw tpjpapid vOjp ep&gp.

10.({ 1, - 1, i , - i }, .) vd;w Fyj;jpy; cs;s xt;nthU cWg;Gf;Fk; tupiriaf; fhz;f.


TEST – 13

1. (i)

342

050

321

vd;w mzpapd; NrHg;G mzpiaf; fhz;f.

(ii) 43;523 yxyx vd;w mrkgbj;jhd Nehpar; rkd;ghl;Lj; njhFg;gpid mzpNfhit Kiwapy; jPHf;f.

2.

484

121

6126


3.

120

031

221

vd;w mzpapd; NeHkhW mzpiaf; fhz;f.

4.

53

21A vd;w mzpapd; NrHg;igf; fz;L> A A )A adj ( )A adj (A vd;gijr; rhpghHf;f.

5. 62;1832;7 zyzyxzyx vd;w rkd;ghLfspd; njhFg;G xUq;fikT cilajh vd;gij Muha;f.

xUq;fikT cilajhapd; mtw;iwj; jPHf;f

6. NeHkhW mzpfhzy; Kiwapy; jPHf;f : 02,137 yxyx

7.

37

25A kw;Wk;

11

12B vdpy; TTT

ABAB rhpghH.

8. 18108;954 yxyx vd;w mrkgbj;jhd Nehpar; rkd;ghl;Lj; njhFg;gpid mzpNfhit Kiwapy; jPHf;f.

9.

37

25A kw;Wk;

11

12B vdpy; 111

ABAB rhpghH.

10.

544

434

221

A-f;F> 1 AA vdf; fhl;Lf.

TEST – 14

1. qqp ~ ,d; nka; ml;ltizia mikf;f.

2. rqp ,d; $w;WfSf;F nka; ml;ltizfis mikf;f.

3. pqqpqp vdf; fhl;Lf.

4. qpqp ~~ xU nka;ikah vd;gjid nka; ml;ltiziaf; nfhz;L jPHkhdpf;f.

5. pqpp ~~ nka;ikah my;yJ Kuz;ghlh vd;gjid nka; ml;ltiziaf; nfhz;L jPHkhdpf;f.

6. 1 ,d; 4-Mk; gb %yq;fs; ngUf;fypd; fPo; vgPypad; Fyj;ij mikf;Fk; vd epWTf.

7.

.,0R Kbtw;w vgPypad; Fyk; vdf; fhl;Lf. ,q;F ‘.’ vd;gJ tof;fkhd ngUf;fiyf; Fwpf;Fk;.

8. (i) xU Fyj;jpy; cs;s a f;F> ,11 Gaaa

vd ep&gp.

(ii) ({ 1, ω, ω2 }, .) vd;w Fyj;jpy; cs;s xt;nthU cWg;Gf;Fk; tupiriaf; fhz;f.

9. Fyj;jpd; ePf;fy; tpjpfis vOjp mjpy; VNjDk; xd;wpid epWTf.

10. (Z5 – {[0]}, . 5) Fyj;jpy; cs;s xt;nthU cWg;Gf;Fk; tupiriaf; fhz;f.


1 kjpg;ngz; khjphpj; Njh;Tfs; Question Paper - 1

1. A = [ 2 0 1 ] vdpy; TAA d; juk; fhz;f

a) 1 b) 2 c) 3 d) 0

2. xU jpirapyp mzpapd; thpir 3, 0k vdpy; 1A vd;gJ.

a) 2k

1I b)

3k

1I c)

k

1I d) k I

3.

001

010

100

vd;w mzpapd; Neh;khW

a)

100

010

001

b)

001

010

100

c)

001

010

100

d)

100

010

001

4. 1TA =

a) 1A b) TA c) A d) TA 1

5. akw;Wk; b

,uz;L XuyF ntf;lH kw;Wk; vd;gJ mtw;wpw;F ,ilg;gl;l Nfhzk; ba MdJ XuyF

ntf;luhapd;

a) 3

b)

4

c)

2

d)

3

2

6. kji 3 vd;w ntf;liu xU %iy tpl;lkhfTk; kji 43 I xU gf;fkhfTk; nfhz;l ,izfuj;jpd; gug;G

a) 310 b) 306 c) 302

3 d) 303

7. yxbacacbcba )()()( vdpy;

a) 0x b) 0y

c) x k; y k; ,izahFk; d) 0x my;yJ 0y my;yJ x k; y k; ,izahFk;

8. ac,cb,ba

=64 vdpy; c,b,ad; kjpg;G

a) 32 b) 8 c) 128 d) 0

9. kja 2 kw;Wk; kib 2 vdpy; ba ,d; kjpg;G

a) 2 b) -2 c) 3 d) 4

10. kjia 2 kw;Wk; kjib 744 vdpy; ba ,d; kjpg;G

a) 19 b) 3 c) -19 d) 14

11. 343 /ie vd;w fyg;ngz;zpd; kl;L tPr;R KiwNa

a) 2

,9 e b)

2,9

e c) 4

3,6

e d) 4

3,9

e

12. (2m + 3) + i( 3n-2) vd;w fyg;ngz;zpd; ,iznad; (m-5) + i ( n+4) vdpy; (n,m) vd;gJ.

a)

8,

2

1 b)

8,

2

1 c)

8,

2

1 d)

8,

2

1

13. z fyg;ngz;izf; Fwpg;gnjdpy; )(arg)(arg zz vd;gJ

a) 4

b)

2

c) 0 d)

3


.14. 73 ,d; fyg;ngz; tbtk;

a) 73 i b) 73 i c) 73 i d) 73 i

15. x8y2 vd;w gutisaj;jpd; tt1 kw;Wk; t3t2 vd;w Gs;spfspy; tiuag;gl;l njhLNfhLfs; ntl;bf;nfhs;Sk; Gs;sp

a) )t8,t6( 2 b) )t6,t8( 2 c) )t4,t( 2 d) )t,t4( 2

16. 4xy2 vd;w gutisaj;jpd; nrt;tfyj;jpd; ePsk;

a) 4

15x b)

4

15x c)

4

17x d)

4

17x

17. 2x + 3y + 9 = 0 vd;w NfhL x8y2 vd;w gutisaj;ijj; njhLk; Gs;sp .

a)(0,-3) b) (2,4) c)

2

9,6 d)

6,

2

9

18. xy 42 vd;w gutisaj;jpd; Ftpak;

a) (0,1) b) (1,1) c) (0,0) d) (1,0)

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

a) 3 b) 2 c) 1 d) -1

20. y = 2x² -6x-4 vd;w tistiuapy; x mr;Rf;F ,izahfTs;s njhLNfhl;bd; njhLGs;sp

a)

2

17,

2

5 b)

2

17,

2

5 c)

2

17,

2

5 d)

2

17,

2

3

21. x2/3

+ y2/3

= a2/3vd;w tistiuapd; nrq;NfhL x mr;Rld; vd;Dk; Nfhzk; Vw;gLj;Jnkdpy;, mr;nrq;Nfhl;bd; rha;T.


22. xU ePHj; njhl;bapd; cauk; “ h “ vd;f. mj;njhl;bapd; mOj;jk; “ p “ MdJ cauj;ijg; nghWj;J khWk; tPjk;

a)dt

dh b)

dt

dp c)

dp

dh d)

dh

dp

23. u = xyvdpy;

x

u

f;F rkkhdJ

a) yxy-1

. b) u log x c) u log y d) xyx-1

.

24. )x(x)x(y 12 22 vd;w tistiuf;F

a) x mr;Rf;F ,izahd xU njhiynjhLNfhL cz;L

b) y mr;Rf;F ,izahd xU njhiynjhLNfhL cz;L

c) ,U mr;RfSf;Fk; ,izahd njhiynjhLNfhLfs; cz;L

d) njhiynjhLNfhLfs; ,y;iy

25.

2/

0 xcosxsin1

xcosxsin dx d; kjpg;G.

a) 2 b) 0 c) 4 d)

26.

0

4 xdxsin d; kjpg;G

a) 163 b) 3 / 16 c) 0 d) 83

27.

0

32 xdxcosxsin d; kjpg;G

a) b) 2 c) 4 d) 0.

28. aa

dxxfdxxf0

2

0

2 vd ,Uf;f Ntz;Lkhapd;

a) xfxaf 2 b) xfxaf c) xfxf d) xfxf


29. dx + xdy = e-y

sec² y dy d; njhifaPl;Lf; fhuzp

a) ex. b) e

-x. c) e

y. d) e

-y.

30. xy5dy

dx 3/12

vd;w tiff;nfO rkd;ghl;bd; thpir kw;Wk; gb

a) thpir 2 kw;Wk; gb 1 b) thpir 1 kw;Wk; gb 2 c) thpir 1 kw;Wk; gb 6 d) thpir 1 kw;Wk; gb 3.

31. xU jsj;jpy; cs;s x mr;Rf;F nrq;Fj;jy;yhj NfhLfspd; tiff;nfO¢ rkd;ghL

a) 0dx

dy b) 0

dx

yd

2

2

c) mdx

dy d) m

dx

yd

2

2

32. dy

dxx

dx

dyy 34 vd;w tiff;nfOr; rkd;ghl;bd; thpir kw;Wk; gb

a) 2, 1 b) 1 ,2 c) 1 ,2 d) 2,2

33. fPo;f;fz;ltw;Ws; vit $w;Wfs; ?

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

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


34. gpd;tUtdtw;Ws; vJ Kuz;ghlhFk; ?

a) p v q. b) p ^ q c) p v ~q. d) p ^ ~q.

35. qp f;F rkkhdJ

a) qp b) pq c) p q q p d) pqqp

36. gpd;tUk; $w;Wfspd; nka;kjpg;Gfs;

i) nrd;id ,e;jpahtpy; cs;sJ my;yJ 2 xU KO vz;

ii) nrd;id ,e;jpahtpy; cs;sJ my;yJ 2 xU tpfpjKwh vz;

iii) nrd;id rPdhtpy; cs;sJ my;yJ 2 xU KO vz;

iv) nrd;id rPdhtpy; cs;sJ my;yJ 2 xU tpfpjKwh vz;

a) T F T F b) T F F T c) F T F T d) T T F T

37. f(x) =

elsewhere,0

3x0,kx 2vd;gJ epfo;jfT mlh;j;jp rhh;G vdpy; k d; kjpg;G

a) 1/ 3 b) 1/ 6 c) 1/ 9 d) 1/12.

38. X vd;w xU jdpepiy¢ rktha;g;G khwp 0,1,2 vd;w kjpg;Gfisf; nfhs;fpwJ. NkYk; P(X =0) = 144 / 169,

P(X=1) = 1/169 vdpy; P(X=2) d; kjpg;G

a) 145 / 169 b) 24 / 169 c) 2 / 169 d) 143 /169

39. X vd;w rktha;g;G khwpapd; 3,4 kw;Wk; 12 Mfpa kjpg;Gfs; KiwNa 1/3, 1 / 4 kw;Wk; 5 / 12 Mfpa epfo;jfTfis¡

nfhs;Snkdpy; E(X) d; kjpg;G.

a) 5 b) 6 c) 7 d) 3

40. xU njhlH rktha;g;G khwp

a) KbTw;w fzj;jpd; kjpg;Gfisg; ngWfpwJ

b) Fwpg;gpl;l xU ,ilntspapYs;s vy;yh kjpg;GfisAk; ngWfpwJ

c) vz;zpylq;fh kjpg;Gfisg; ngWfpwJ

d) xU KbTw;w my;yJ vz;zplj;jf;f kjpg;Gfisg; ngWfpwJ


Question Paper - 2

1.

3

2

1

A vdpy; TAA d; juk; fhz;f

a) 3 b) 0 c) 1 d) 2

2.

541

3k1

231

vd;w mzpf;F Neh;khW cz;L vdpy; kd; kjpg;G(fs;)

a)k VNjDk; xU nka;naz; b) 4k c) 4k d) 4k

3. xU rJu mzp A d; thpir n vdpy; Aadj vd;gJ

a) 2A b) nA c) 1nA d) A

4. A kw;Wk; B vd;w mzpfs; ngUf;fYf;F cfe;jitahapd; TAB =

a) TT BA b) TT AB c) AB d) BA

5. a f;Fk; b f;Fk; ,ilg;gl;l Nfhzk; 120° NkYk; mtw;wpd; vz;zsTfs; KiwNa 32, a . b MdJ

a) 3 b) 3 c) 2 d) 2

3

6. k4j3i2

, kcjbia

Mfpa ntf;lh;fs; nrq;Fj;J ntf;lHfshapd;>

a) a = 2, b = 3, c = -4 b) a = 4, b = 4, c = 5 c) a = 4, b = 4, c = -5 d)a = -2, b =3, c =4

7. kjiPR 2 , kjiSQ 23 vdpy; ehw;fuk; PQRS d; gug;G

a) 35 b) 310 c) 2

35 d) 2

3

8. b d; kPJ a d; tPoy; kw;Wk; a d; kPJ b d; tPoYk; rkkhapd; a + b kw;Wk; a - b f;F ,ilg;gl;l Nfhzk;

a) 2

b)

3

c)

4

d)

3

2

9. kjim 2 kw;Wk; kji 294 vd;gd nrq;Fj;J ntf;lHfs; vdpy; m ,d; kjpg;G

a) -4 b) 8 c) 4 d) 12

10. kji 295 , kjim 2 vd;gd nrq;Fj;J ntf;lHfs; vdpy; m ,d; kjpg;G

a) 16

5 b)

16

5 c)

5

16 d)

5

16

11. 122 yx vdpy; iyx

iyx

1

1d; kjpg;G

a) iyx b) x2 c) iy2 d) iyx

12.

100100

2

31

2

31

iid; kjpg;G

a) 2 b) 0 c) -1 d) 1

13. a = 3 + i kw;Wk; z = 2 – 3i vdpy; cs;s az, 3az kw;Wk; -az vd;gd xU Mh;fd; jsj;jpy;

a) nrq;Nfhz Kf;Nfhzj;jpd; Kidg;Gs;spfs; b) rkgf;f Kf;Nfhzj;jpd; Kidg;Gs;spfs;

c) ,U rkgf;f Kf;Nfhzj;jpd; Kidg;Gs;spfs; d) xNu Nfhliktd.

14. i2

3,d; nka; kw;Wk; fw;gidg; gFjpfs;

a) 0, 3/2 b) 3/2, 0 c) 2, 3 d) 3, 2


15. 044x32y3x16 22 vd;gJ

a) xU ePs;tl;lk; b) xU tl;lk;. c) xU gutisak; d) xU mjpgutisak;

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

a) 2 b) 4 c) 6 d) 8

17. 1y8x2 vd;w gutisaj;jpd; Kid

a)

0,

8

1 b)

0,

8

1 c)

8

1,0 d)

8

1,0

18. xy 42 -,d; nrt;tfyj;jpd; rkd;ghL

a) x = 1 b) y = 1 c) x = 4 d) y = -1

19. MjpapypUe;J xU Neh;Nfhl;by; x njhiytpy; efUk; Gs;spapd; jpirNtfk; v vdTk; a+bv2=x

2vdTk;

nfhLf;fg;gl;Ls;sJ. ,q;F a kw;Wk; b khwpypfs; mjd; KLf;fkhdJ

a)x

b b)

x

a c)

b

x d)

a

x

20. = 1 / t vDk; tistiuf;F Gs;sp (-3, -1/3 ) vd;w Gs;spapy; nrq;Nfhl;bd; rkd;ghL

a) 80273 t b) 80275 t c) 80273 t d) t1

21.xU Nfhsj;jpd; fdmsT kw;Wk; Muj;jpNyw;gLk; khWtPjq;fs; vz;zstpy; rkkhf ,Uf;Fk; NghJ

Nfhsj;jpd; tisgug;G

a) 1 b) 21 c) 4 d) 34

22.° C ntg;gepiyapy; “ l “ kP ePsKs;s xU cNyhfj; Jz;bd; rkd;ghL 20000004.000005.01 l vdpy;

100°C -y; l-,d; khW tPjk;

a) Cm00013.0 b) Cm00023.0 c) Cm00026.0 d) Cm00033.0

23. u = sin-1

22

44

yx

yxkw;Wk; f = sin u vdpy; rkgbj;jhd rhh;G f d; gb

a) 0 b) 1 c) 2 d) 4

24. u = ,

yx

1

22

vdpy; x x

u

+ y

y

u

a) u2

1 b) u c) u

2

3 d) – u

25.

2/

03/53/5

3/5

xsinxcos

xcosdx d; kjpg;G

a) 2 b) 4 c) 0 d)

26. 1

0

4)x1(x dx d; kjpg;G.

a) 1/ 12 b) 1/30 c) 1/ 24 d) 1/ 20

27. 4/

0

3 xdx2cos d; kjpg;G

a) 2 / 3 b) 1/ 3 c) 0 d) 32

28. dxxI nn sin vdpy; nI

a) 21 1

cossin1

n

n In

nxx

n b) 2

1 1cossin

1

n

n In

nxx

n

c) 21 1

cossin1

n

n In

nxx

n d) n

n In

nxx

n

1cossin

1 1


29. QPydx

dy vd;w tiff;nfO rkd;ghl;bd; njhiff; fhuzp cos x vdpy; P d; kjpg;G

a) – cot x. b) cot x c) tan x d) – tan x.

30. m < 0 Mf ,Ug;gpd; 0dx

mxdy

d; jPh;T

a) x = cemy

. b) x = ce-my

. c) x = my + c. d) x = c.

31. Mjpg;Gs;spia ikakhff; nfhz;l tl;lq;fspd; njhFg;gpd; tiff;nfO¢ rkd;ghL

a) x dy + y d x = 0 b) x dy – y d x = 0 c) x d x + y dy = 0 d) x d x – y d y =0

32. 2xydx

dy vd;w tiff;nfOr; rkd;ghl;bd; thpir kw;Wk; gb

a) 1,1 b) 1,2 c) 2,1 d) 0,1

33. xU $l;Lf; $w;W %d;W jdpf;$w;Wfisf; nfhz;ljhf ,Ug;gpd;> nka;al;ltizapYs;s epiufspd; vz;zpf;if

a) 8 b) 6 c) 4 d) 2

34.epge;jidf; $w;W qp f;F rkhdkhdJ

a) p v q. b) p v ~ q c) ~p v q. d) p ^ q.

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

(i) p v q (ii) ~ p v q iii) p v ~q iv) p ^ ~ q.


36. gpd;tUk; $w;Wfspy; nka;kjpg;Gfs;

i. Cl;bahdJ jkpo;ehl;by; cs;sJ kw;Wk; 3 + 4 = 8 ii. Cl;bahdJ jkpo;ehl;by; cs;sJ kw;Wk; 3 + 4 = 7

iii. Cl;bahdJ Nfushtpy; cs;sJ kw;Wk; 3 + 4 = 7 iv. Cl;bahdJ Nfushtpy; cs;sJ kw;Wk; 3 + 4 = 8

a) F,T,F,F b) F,F,F,T c) T,T,F,F d) T,F,T,F

37. f(x) =

x,x16

1A

2vd;gJ X vd;w njhlh; rktha;g;G khwpapd; xU epfo;jfT mlh;j;jp¢ rhh;G (p.d.f) vdpy; A

d; kjpg;G

a) 16 b) 8 c) 4 d) 1

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

x -2 3 1

P(X=x) 6 4 12

vdpy; d; kjpg;G a) 1 b) 2 c) 3 d) 4

39. X vd;w rktha;g;G khwpapd; epfo;jfT guty; gpd;tUkhW

x 0 1 2 3 4 5

P(X=x) ¼ 2a 3a 4a 5a ¼

vdpy;>

41 xP d; kjpg;G

a) 10 / 21 b) 2 / 7 c) 1/ 14 d) 1/ 2

40. xU jdpepiy¢ rktha;g;G khwp

a) KbTw;w fzj;jpd; kjpg;Gfisg; ngWfpwJ

b) Fwpg;gpl;l xU ,ilntspapYs;s vy;yh kjpg;GfisAk; ngWfpwJ

c) vz;zpylq;fh kjpg;Gfisg; ngWfpwJ

d) xU KbTw;w my;yJ vz;zplj;jf;f kjpg;Gfisg; ngWfpwJ


Question Paper - 3

.1

844

422

211

vd;w mzpapd; juk; fhz;f

a) 1 b) 2 c) 3 d) 4

2.

01

10

01

vd;w mzpapd; juk; 2 vdpy; tpd; kjpg;G

a) 1 b) 2 c) 3 d) VNjDk; xU nka;naz;.

3. myF mzp I d; thpir 0k,n xU khwpyp vdpy;>adj (kI) = ……

a) )(Iadjk n b) )(Iadjk c) )(2 Iadjk d) )(1 Iadjk n

4.

21

42vd;w mzpapd; juk;

a) 1 b) 2 c) 0 d) 8

5. (2,10,1) vd;w Gs;spf;Fk; 262)43( kjir

vd;w jsj;jpw;Fk; ,ilg;gl;l kpff; Fiwe;j J}uk;

a) 262 b) 26 c) 2 d) 26

1

6.

OQ vd;w myF ntf;lH kPjhd

OP d; tPoyhdJ OPRQ vd;w ,izfuj;jpd; gug;ig Nghd;W Kk;klq;fhapd; POQ MdJ.

a) 3

1tan 1 b)

10

3cos 1 c)

10

3sin 1 d)

3

1sin 1

7. c,b,a vd;gd a,b,c Mfpatw;iw kl;Lfshff; nfhz;L tyf;if mikg;gpy; xd;Wf;nfhd;W nrq;Fj;jhd ntf;lHfs;

vdpy; c,b,a,d; kjpg;G

a) 222 cba b) 0 c) abc2

1 d) abc

8. xU NfhL x kw;Wk; y mr;Rf;fSld; kpif jpirapy; 45°, 60° Nfhzq;fis Vw;gLj;JfpwJ vdpy;, z mr;Rld; mJ cz;lhf;Fk; Nfhzk;

a) 30 b) 90 c) 45 d) 60

9. ji kw;Wk; kj vd;w ntf;lHfSf;F ,ilg;gl;l Nfhzk;

a) 3

b)

3

2 c)

3

d)

3

2

10. kjia 923 kw;Wk; kjmib 3 vd;gd nrq;Fj;J ntf;lHfs; vdpy; m ,d; kjpg;G

a) -15 b) 15 c) 30 d) -30

11. fyg;ngz; 325 )i( d; Nghyhh; tbtk;

a) 2

sin2

cos

i b) sincos i c) sincos i d) 2

sin2

cos

i

12. sinicosx vdpy; n

n

xx

1 d; kjpg;G

a) ncos2 b) ni sin2 c) nsin2 d) ni cos2

13. iz,iz 2354 21 vdpy; 2

1

z

z vd;gJ

a) i13

22

13

2 b) i

13

22

13

2 c) i

13

23

13

2 d) i

13

22

13

2


14. )7(23 ii ,d; jpl;l tbtk; iba

a) i4 b) i 4 c) i4 d) i44

15. 180y5x9 22 vd;w ePs;tl;lj;jpd; Ftpaq;fSf;fpilNa cs;s njhiyT.

a) 4 b) 6 c) 8 d) 2.

16. 400y25x16 22 vd;w tistiuapd; Ftpaj;jpypUe;J xU njhLNfhl;Lf;F tiuag;gLk; nrq;Fj;J¡ NfhLfspd; mbapd; epakg;ghij

a) 4yx 22 b) 25yx 22 c) 16yx 22 d) 9yx 22

17. 19

y

16

x 22

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

a) 072y8x9 b) 072y8x9 c) 072y9x8 d) 072y9x8

18. yx 202 vd;w gutisaj;jpd; Kid

a) (0,5) b) (0,0) c) (5,0) d) (0,-5)

19. y = 6x –x³ NkYk; x MdJ tpdhbf;F 5 myFfs; tPjj;jpy; mjpfhpf;fpd;wJ. x = 3 vDk; NghJ mjd; rha;tpy; Vw;gLk; khWtPjk;

a) – 90 myFfs; / tpdhb. b) 90 myFfs; / tpdhb. c) 180 myFfs; / tpdhb d) – 180 myFfs; / tpdhb.

20. y = 3exkw;Wk; y =

3

ae

-xvd;Dk; tistiufs; nrq;Fj;jhf ntl;bf; nfhs;fpd;wd vdpy; ‘a’ d; kjpg;G

a) -1 b) 1 c) 1/ 3 d) 3.

21. a = 0, b = 1 vdf; nfhz;L f(x) = x² + 2x -1 vd;w rhh;gpw;F nyf;uhQ;rpapd; ,ilkjpg;G Njw;wj;jpd; gbAs;s ‘c’ d; kjpg;G .

a) -1 b) 1 c) 0 d) 1/ 2

22. y = f ( x ) kw;Wk; y = g ( x ) vd;w tistiufs; xd;iwnahd;W nrq;Fj;jhf ntl;bf; nfhs;fpd;wd vdpy;> mit

ntl;Lk; Gs;spapy; (njhLNfhLfs; mr;RfSf;F ,izahf ,y;iy)

a) f ( x ) -,d; rha;T = g ( x ) -,d; rha;T b) f ( x ) -,d; rha;T + g ( x ) -,d; rha;T = 0

c) f ( x ) -,d; rha;T / g ( x ) -,d; rha;T = -1 d) [ f ( x ) -,d; rha;T ] [g ( x ) -,d; rha;T ] = -1

23. u = y sin x , vdpy;yx

u2

=

a) cos x . b) cos y c) sin x. d) 0.

24. f ( x , y ) MdJ gb n I cila rk gbj;jhd rhHG vdpy;

y

fy

x

fx

a) f b) n f c) n ( n - 1 ) f d) n ( n + 1 ) f

25. y = 2x, x = 0 kw;Wk; x = 2 ,tw;wpw;F ,ilNa Vw;gLk; gug;G x mr;ir¥ nghWj;J¢ Row;wg;gLk; jplg;nghUspd; tisg;gug;G

a) 58 b) 52 c) 5 d) 54

26. Muk; 5 cs;s Nfhsj;ij¤ jsq;fs; ikaj;jpypUe;J 2 kw;Wk; 4 J}uj;jpy; ntl;Lk; ,izahd jsq;fSf;fpilg;gl;;l gFjpapd; tisg;gug;G

a) 20 b) 40 c) 10 d) 30

27. x2/3

+ y2/3

= 4 vd;w tistiuapd; tpy;ypd; ePsk;

a) 48 b) 24 c) 12 d) 96

28. n MdJ xU xw;iwg;gil vz; vdpy; 2

0

cos

dxxn =

a)2 4

1 3 5 2

n n n

n n n

b)

1 3 5 1

2 4 2 2

n n n

n n n

c)

2 4 31

1 3 5 2

n n n

n n n

d)

1 3 5 21

2 4 3

n n n

n n n


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

tiff;nfO rkd;ghL ( k Fiw vz;)

a) p

k

dt

dp b) kt

dt

dp c) kp

dt

dp d) kt

dt

dp

30. y = e x (A cos x + B sin x ) vd;w njhlh;gpy; A iaAk; B iaAk; ePf;fpf; fpilf;Fk; tiff;nfO¢ rkd;ghL.

a) y2 + y1 = 0 b) y2 - y1 = 0 c) y2 -2y1 +2y = 0 d) y2 -2 y1 -2y = 0

31. xcosxtanydx

dy vd;w tiff;nfO¢ rkd;ghl;bd; njhiff;fhuzp

a) sec x . b) cos x. c) etanx

. d)cot x.

32. dydxxdydxx cossin vd;w tiff;nfOr; rkd;ghl;bd; thpir kw;Wk; gb

a) 1,1 b) 0,0 c) 1,2 d) 2,1

33. ~ [ p ^ (~q) ] d; nka; ml;ltizapy; epiufspd; vz;zpf;if

a) 2 b) 4 c) 6 d) 8.

34. rkdpAila miuf;Fyk; >Fykhtjw;F¥ G+Hj;jp nra;a Ntz;ba tpjpahtJ

a) milg;G tpjp b) NrHg;G tpjp c) rkdp tpjp d) vjpHkiw tpjp

35. [ 3 ] + 11( [5] + 11[6]) d; kjpg;G

a) [0] b) [1] c) [2] d) [3]

36. p vd;gJ “ fkyh gs;spf;Fr; nry;fpwhs; “ q vd;gJ “ tFg;gpy; ,UgJ khztHfs; cs;sdH “ vd;f. “ fkyh gs;spf;Fr; nry;ytpy;iy my;yJ tFg;gpy; ,UgJ khztHfs; cs;sdH “ vd;gJ

a) qp b) qp c) p~ d) qp~

37. xU <UWg;G gutypd; ruhrhp 12 kw;Wk;jpl;l tpyf;fk; 2 vdpy; gz;gsit p d; kjpg;G

a) 1/ 2 b) 1 / 3 c) 2 / 3 d) 1 / 4.

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

a) 1/20 b) 18 / 125 c) 4 /25 d) 3 /10

39. gha;]hd; gutypy; gz;gsit = 0.25 vdpy; Mjpiag; nghWj;J ,uz;lhtJ tpyf;fg; ngUf;Fj; njhif.

a) 0.25 b) 0.3125 c) 0.0625 d) 0.025

40. jpl;l ,ay;epiyg; gutypd; ruhrhpAk;> gutw;gbAk;

a) 2, b) , c) 0,1 d) 1,1


Question Paper - 4

1.

0

4

0

2

1

vd;w %iytpl;l mzpapd; juk; fhz;f

a) 0 b) 2 c) 3 d) 5

2.

43

12A vd;w mzpf;F ( adjA ) A =

a)

51

51

0

0 b)

10

01 c)

50

05 d)

50

05

3. A vd;w mzpapd; thpir 3 vdpy; det (kA) vd;gJ

a) )Adet(k3 b) )Adet(k2 c) )Adet(k d) det (A)

4.

12

17vd;w mzpapd; juk;

a) 9 b) 2 c) 1 d) 5

5. a xU G+r;rpakw;w ntf;luhfTk; m xU jpirapypahfTk; ,Ug;gpd; am MdJ XuyF ntf;lh; vdpy;

a) 1m b) ma c) m

1a d) a = 1

6. q,p kw;Wk; qp Mfpait vz;zsT nfhz;l ntf;lh;fshapd; qp

MdJ.

a) 2 b) 3 c) 2 d) 1

7. 0cba

, 3a

, 4b

, 5c

vdpy; a f;Fk; b f;Fk; ,ilg;gl;l Nfhzk;

a) 6

b)

3

2 c)

3

5 d)

2

8.3

5z2

5

3y

1

3x

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

a) k5j3itk3j5ir

b) k3j5itk5j3ir

c) k5j3itk2

3j5ir

d)

k

2

3j5itk5j3ir

9. kji 362 ,d; kPJ kji 47 ,d; tPoy;

a) 8

7 b)

66

8 c)

7

8 d)

8

66

10. kjia 22 kw;Wk; kjib 236 vdpy; ba ,d; kjpg;G

a) 4 b) -4 c) 3 d) 5

11. iz vd;w fyg;ngz;iz Mjpia¥ nghWj;J 2

Nfhzj;jpy; fbfhu vjph; jpirapy; Row;Wk; NghJ, me;j vz;zpd;

Gjpa epiy

a) iz b) -iz c) -z d) z

12.3

sin3

cos n

in

zn vdpy; 621 z.........zz vd;gJ

a) 1 b) -1 c) i d) -i

13. z %d;whk; fhy;gFjpapy; mike;jhy; z mikAk; fhy;gFjp

a) Kjy; fhy;gFjp b) ,uz;lhk; fhy;gFjp c) %d;whk; fhy;gFjp d) ehd;fhk; fhy;gFjp


14. iiqip 2432 vdpy; q ,d; kjpg;G

a) 14 b) -14 c) -8 d) 8

15. 0cyx2 vd;w Neh;NfhL 32y8x4 22 vd;w ePs;tl;lj;jpd;; njhLNfhL vdpy; c d; kjpg;G.

a) 32 b) 6 c) 36 d) 4

16. 36y9x4 22 vd;w ePs;tl;lj;jpd; kPJs;s VNjDk; xU Gs;spapypUe;J )0,5( kw;Wk; )0,5( vd;w

Gs;spfSf;fpilNa cs;s njhiyTfspd; $Ljy; .

a) 4 b) 8 c) 6 d) 18

17. 16)3y(4x 22 vd;w mjpgutisaj;jpd; ,af;Ftiu

a) 5

8y b)

5

8x c)

8

5y d)

8

5x

18. xy 82 -,d; nrt;tfyj;jpd; rkd;ghL

a) y-2=0 b) y+2 =0 c) x -2 = 0 d) x+2 =0

19. y = emx kw;Wk; y = e

-mx , m > 1 vd;Dk; tistiufSf;F ,ilg;gl;l Nfhzk;.

a) tan-1

1m

m2

2 b) tan-1

2m1

m2

c) tan-1

2m1

m2

d) tan-1

1m

m2

2

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

a) 8 b) 4 c) 0 d) -4

21.xU cUisapd; Muk; 2 nrkP / tpdhb vd;w tPjj;jpy; mjpfhpf;fpd;wJ. mjd; cauk; 3 nrkP / tpdhb vd;w tPjj;jpy; Fiwfpd;wJ. Muk; 3cm kw;Wk; cauk; 5 cm Mf ,Uf;Fk; NghJ mjd; fd mstpd; khWtPjk;.

a) 23 b) 33 c) 43 d) 53

22. xU gwf;Fk; jl;bd; Nfhz ,lg;ngaHr;rp 32 29 tt vdpy;, mjd; Nfhz KLf;fk; G+r;rpakhf ,Uf;Fk; Neuk;

( Nubad;>t tpdhb)

a) 2.5 tpdhb b) 3.5 tpdhb c) 1.5 tpdhb d) 4.5 tpdhb

23. y² (a+2x) = x² (3a-x) vd;w tistiuapd; njhiynjhLNfhL

a) x = 3a b) x = - a / 2 c) x = a / 2 d) x = 0.

24. u = f ( x , y ) vd;f. ,ay;ghd FwpaPl;bd;gb yxxy uu vd ,Uf;f Ntz;Lkhapd;

a) u vd;gJ njhlHr;rpahdjhf b) xu vd;gJ njhlHr;rpahdjhf

c) yu vd;gJ njhlHr;rpahdjhf d) yx uuu ,, Mfpait njhlHr;rpahdjhf

25.y = 2x3 vd;w tistiu x = 0 ypUe;J x = 4 tiu x mr;ir mr;rhf itj;J¢ Row;wg;gLk; jplg;nghUspd; fdmsT.

a) 100 b) 9

100 c)

3

100 d) 100 / 3

26. 1b

y

a

x

2

2

2

2

vd;w ePs;tl;lj;jpd; gug;ig nel;lr;R> Fw;wr;R ,tw;iw¥ nghWj;J¢ Row;wg;gLk; jplg;nghUspd;

fdmsTfspd; tpfpjk;

a) b² : a² b) a² : b² c) a : b d) b : a

27. NfhLfs; y = x, y = 1kw;Wk; x = 0 Mfpait Vw;gLj;Jk; gug;G y mr;ir¥ nghWj;J Row;wg;gLk; jplg;nghUspd; fdmsT

a) 4 b) 2 c) 3 d) 32

28. f ( x ) XH ,ul;ilg;gilr; rhHG vdpy;

a

a

dxxf =

a) 0 b) a

dxxf0

2 c) a

dxxf0

d) a

dxxf0

2


29. (D² -4D+4) y = e2xd; rpwg;Gj; jPh;T (PI).

a) x22

e2

x b) xe

2x. c) xe

-2x. d) x2e

2

x

30. xxf kw;Wk; f(1) = 2 vdpy; f(x) vd;gJ

a) )2xx(3

2 b) )2xx(

2

3 c) )2xx(

3

2 d) )2x(x

3

2

31. f(D) = (D-a) g(D), g(a) 0 vdpy; tiff;nfO¢ rkd;ghL f(D) y = eaxd; rpwg;Gj; jPh;T.

a) meax

b) eax

/ g(a) c) g(a) eax

d) xeax

/ g(a)

32.dx

dyyx

dx

yd

2

2

vd;w tiff;nfOr; rkd;ghl;bd; thpir kw;Wk; gb

a) 2,1 b) 1,2 c) 2 , 1 / 2 d) 2,2

33. KOf;fspy; * vd;w <UWg;G nrayp a * b = a+ b – ab, vd tiuaWf;fg;gLfpwJ vdpy; 3 * ( 4*5) d; kjpg;G

a) 25 b) 15 c) 10 d) 5

34. (Z9, +9) y; [7] d; thpir

a) 9 b) 6 c) 3 d) 1.

35. fPo;f;fz;ltw;Ws; vJ Fyk; my;y ?

a) (Zn, +n). b) (Z, +) c) (Z, .) d) (R, +)

36. gpd;tUk; $w;Wfspd; nka;kjpg;Gfs;

i. xU rha; rJuj;jpd; vy;yh¥ gf;fq;fSk; rkePsk; nfhz;lit

ii. 191 xU tpfpjKwh vz;

iii. ghypd; epwk; ntz;ik

iv. vz; 30f;F 4 gfhf;fhuzpfs; cz;L

a) T T T F b) T T T T c) T F T F d) F T T T

37. Var (4X+ 3) d; kjpg;G

a) 7 b) 16 Var (X) c) 19 d) 0.

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

a) log 1/ k b) log k. c) e d) 1 / k

39. ed;F fiyf;fg;gl;l 52 rPl;Lfs; nfhz;l rPl;Lf;fl;bypUe;J 2 rPl;Lfs; jpUg;gp itf;fhky; vLf;fg;gLfpd;wd. ,uz;Lk; xNu epwj;jpy; ,Uf;f epfo;jfT.

a) 1 / 2. b) 26 / 51 c) 25 /51 d) 25 /102

40.<UWg;Gg; gutypd; ruhrhp kw;Wk; gutw;gb

a) nq, npq b) np, npq c) np, np d) np, npq


Question Paper - 5

1. dqepe;cbeae yxyx kw;Wk; qp

ba1 ;

qd

bc2 ;

dp

ca3 vdpy; (x,y) d; kjpg;G

a)

1

3

1

2 , b)

1

3

1

2 log,log c)

2

1

3

1 log,log d)

3

1

2

1 log,log

2. kjpg;gpl Ntz;ba %d;W khwpfspy; mike;j %d;W Nehpa mrkgbj;jhd rkd;ghl;Lj; njhFg;gpy; 0 kw;Wk;

0x , 0y kw;Wk; 0z vdpy; njhFg;Gf;fhd jPh;T

a) xNu xU jPh;T b) ,uz;L jPh;Tfs; c) vz;zpf;ifaw;w jPh;Tfs; d) jPh;T ,y;yhik

3. A,B vd;w VNjDk; ,U mzpfSf;F AB = O vd;W ,Ue;J NkYk; A G+r;rpakw;w Nfhit mzp vdpy;>

a) B = O b) B xU G+r;rpaf; Nfhit mzp c) B xU G+r;rpakw;w Nfhit mzp d)B = A

4. rkgbj;jhd Nehpar; rkd;ghLfspd; njhFg;gpy; A khwpfspd; vz;zpf;if vdpy; njhFg;ghdJ

a) ntspg;gilj; jPHT kl;LNk ngw;wpUf;Fk;

b) ntspg;gilj; jPHT kw;Wk; vz;zpf;ifaw;w ntspg;gilaw;w jPHTfs; ngw;wpUf;Fk;

c) ntspg;gilaw;w jPHTfs; kl;LNk ngw;wpUf;Fk;

d) jPHTfs; ngw;wpUf;fhJ

5. 01z10y8x6zyx 222 vd;w Nfhsj;jpd; ikak; kw;Wk; Muk;

a) (-3, 4, -5) , 49 b) (-6, 8, -10) , 1 c) (3, -4, 5) , 7 d) (6, -8, 10) , 7

6. 8

4z

4

4y

6

6x

kw;Wk;

2

3z

4

2y

2

1x

vd;w NfhLfs; ntl;bf;nfhs;Sk; Gs;sp

a) (0, 0, -4) b) (1, 0, 0) c) (0, 2, 0) d) (1, 2, 0)

7. (2,1,-1) vd;w Gs;sp topahfTk; jsq;fs; 0)kj3i(r

; 0)k2j(r

ntl;bf;nfhs;Sk; Nfhl;il cs;slf;fpaJkhd jsj;jpd; rkd;ghL

a) x+4y-z=0 b) x+9y+11z=0 c) 2 x+y-z+5=0 d) 2x-y+z=0

8. jtisr vd;w rkd;ghL Fwpg;gJ

a) i kw;Wk; j Gs;spfis ,izf;Fk; Neh;NfhL b)xoy jsk;

c) yoz jsk; d)zox jsk;

9. cba ,, vd;git xd;Wf;nfhd;W nrq;Fj;jhd %d;W myF ntf;lHfs; vdpy; cba

a) 3 b) 9 c) 33 d) 3

10. kji 24 ,d; kPJ kji 3 ,d; tPoy;

a) 21

9 b)

21

9 c)

21

81 d)

21

81

11. 25242322 iiiii d; kjpg;G vd;gJ

a) i b) -i c) 1 d) -1

12. 012 bxax vd;w rkd;ghl;bd; xU jPh;T i

i

1

1aAk;>bAk; nka; vdpy; (a,b) vd;gJ.

a) (1,1) b) (1,-1) c) (0,1) d) (1,0)

13. vd;gJ 1d; nk; gb %ynkdpy;

a) ..................1 5342 b) 0n c) 1n d) 1 n

14. ii 232 ,d; ,izf; fyg;ngz;

a) i8 b) i8 c) i8 d) i8


15. (-4, 4) vd;w Gs;spapypUe;J x16y2 f;F tiuag;gLk; ,U njhLNfhLfSf;F ,ilNaAs;s Nfhzk;

a) 45 b) 30 c) 60 d) 90

16. 144y16x9 22 vd;w $k;G tistpd; ,af;F tl;lj;jpd; Muk;

a) 7 b) 4 c) 3 d) 5.

17. 19

y

16

x 22

vd;w mjpgutisaj;jpd; njhiynjhLNfhLfSf;fpilNaAs;s Nfhzk;

a)

4

3tan2 1 b)

3

4tan2 1 c)

4

3tan2 1 d)

3

4tan2 1

18. yx 42 vd;w gutisaj;jpd; ,af;Ftiu

a) x = 1 b) x = 0 c) y = 1 d) y = 0

19. 2x y; 532 3 xxy vd;w tistiuapd; rha;T

a) -20 b) 27 c) -16 d) -21

20. x2/3

+ y2/3

= a2/3vDk; tistiuapd; Jiz myFr; rkd;ghLfs;

a) 33 cos;sin ayax b) 33 sin;cos ayax

c) cos;sin 33 ayax d) sin;cos 33 ayax

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

a) 2 tan-1

4

3

b) tan-1

3

4

c) 2 d) 4

22. ,ilkjpg;G tpjpapd; khw;W tbtk;

a) 10' hahfafhaf b) 10' hahfafhaf

c) 10' hahfafhaf d) 10' hahfafhaf

23. gpd;tUtdtw;Ws; rhpahd $w;Wfs; :

i) xU tistiu Mjpia¥ nghWj;J¢ rkr;rPh; ngw;wpUg;gp‹ mJ ,U mr;Rfisg; nghWj;Jk; rkr;rPh; ngw;wpUf;Fk;.

ii) xU tistiu ,U mr;Rfis¥ nghWj;J¢ rkr;rPh; ngw;wpUg;gpd; mJ Mjpia¥ nghWj;Jk; rkr;rPh; ngw;wpUf;Fk;..

iii) f(x,y) =0 vd;w tistiu y = x vd;w Nfhl;il¥ nghWj;J¢ rkr;rPh; ngw;Ws;snjdpy; f(x,y) = f(y, x)

iv) f(x,y) =0, vd;w tistiuf;F f(x,y) = f(-y,-x), cz;ikahapd; mJ Mjpia¥ nghWj;J¢ rkr;rPh; ngw;wpUf;Fk;.

a) (ii), (iii) b) (i), (iv) c) (i), (iii) d) (ii), (iv)

24. u = f

x

yvdpy;

y

uy

x

ux

d; kjpg;G.

a) 0 b) 1 c) 2 u d) u

25. 1b

y

a

x

2

2

2

2

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

a) bab b) baa 2 c) baa d) bab 2

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

a) 4 / 3 b) 1 / 6 c) 2 / 3 d) 8 / 3

27. x = 0 ypUe;J x = / 4 tiuapyhd y = sin x kw;Wk; y = cos x vd;w tistiufspd; ,ilg;gl;l gug;G.

a) 12 b) 12 c) 122 d) 222

28. b

a

dxxf =

a) a

dxxf0

2 b) b

a

dxxaf c) b

a

dxxbf d) b

a

dxxbaf


29.2x

2y.

xlogx

1

dx

dy d; njhifaPl;Lf; fhuzp

a) ex. b) log x. c) 1/ x d) e

-x.

30. y = mx vd;w Neh;;NfhLfspd; njhFg;gpd; tiff;nfOr; rkd;ghL

a) mdx

dy b) y d x – x dy = 0 c) 0

dx

yd

2

2

d) y d x + x dy = 0

31. (D² + 1 ) y = e2xd; epug;G¢ rhh;G

a) (Ax+B)ex. b) A cos x + B sin x . c) (Ax+B)e

2x. d) (Ax+B)e

-x.

32. 02

3

3

3

2

2

dx

yd

dx

dyy

dx

ydvd;w tiff;nfOr; rkd;ghl;bd; thpir kw;Wk; gb

a) 2,3 b) 3,3 c) 3,2 d) 2,2

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

a) p v q. b) p ^ q c) p v ~p. d) p ^ ~p.

34. fPo;f;fz;ltw;wpy; vJ Ry; <UWg;Gr; nrayp my;y ?

a) a * b = ab. b) a * b = a – b c) a * b = ab d) a * b = 22 ba

35. ngUf;fy; tpjpia¥ nghWj;J¡ Fykhfpa xd;wpd; ehyhk; %yq;fspy; – i d; thpir

a)4 b) 3 c) 2 d) 1

36. p cz;ikahf ,Ue;J> q-jtwhf ,Ug;gpd;> gpd;tUtdtw;Ws; vit cz;ikapy;iy?

a) qp jtW b) qp cz;ik c) qp jtW d) qp cz;ik

37. xU rktha;g;G khwp X d; epfo;jfT epiwr; rhh;G (p.d.f) gpd;tUkhW

x 0 1 2 3 4 5 6 7

P(X=x) 0 K 2k 2k 3k K² 2k² 7k²+k

vdpy; k d; kjpg;G

a) 1/ 8 b) 1/ 10 c) 0 d) -1 or 1/10

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

a)

25,

5

4 b)

5

4,25 c)

25,

5

1 d)

5

1,25

39. xU rktha;g;G khwp X gha;]hd; gutiy¥ gpd;gw;WfpwJ. NkYk;,E(X²) = 30 vdpy; gutypd; gutw;gb

a) 6 b) 5 c) 30 d) 25

40. jpl;l ,ay;epiy khwp Z ,d; epfo;jfT mlHj;jpr; rhHG z

a)

2

2

1

2

1 z

e

b)

2

2

1 ze

c)

2

2

1

2

1 z

e

d)

2

2

1

2

1 z

e

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