= ntq;fNl];tuh tpj;ah ke;jph; Nkdpiyg; gs;sp jspf;Nfhl;il

fhyhz;Lj; Njh;T – 2014 10-k; tFg;G

fzpjk; tpilf; Fwpg;G

gphpT – I 1. M) 11

2. M) 10

3. ,) 130ffffff

4. M) 0

5. ,) an

6. ,) x+1

7. M) k ≠ 3

8. m) 2

9 ,) 300

10. M) (0>4)

11. M) 4x4

12. M) -7

13. m) 1

14 ,) 600

15. m) 4:9

gphpT-II

16. A={4, 6, 7, 8, 9} B={2, 4, 6} C={1, 2, 3, 4, 5, 6}

B∪C = {1, 2, 3, 4, 5, 6}

An(B∪C) = {4, 6}

17. f(1)=12=1, f(2)=4, f(3)=9, f(4)=16, f(5)=25 fd; tPr;rfk; = {1, 4, 9, 16,

25}. ,J xd;Wf;F xd;whd rhh;G MFk;.

18. 1+3+5+….(25 cWg;Gfs; tiu) = n2

= 252

= 625

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19. me;j vz;fs; 2x, 5x, 7x

2x, 5x-7, 7x (nfhLf;fgl;l tptuk;)

5x-7-2x=7x-(5x-7)

3x-7=7x-5x+7

3x-7=2x+ 7

x=14

Njitahd vz;fs; 28> 70> 98

20. a = 14fffr =

@ 12fffffffffff1fffffffff4fffffff= @ 1

2ffffffffB 4 = @ 2 n = 10

tn = ar n@ 1

= 14fff@ 2` a10@ 1 = 1

4fff@ 2` a9 =@ 2

` a7

21. P(x)=x3+x2=7x-3

<T = x2+4x+5, kPjp=12

22. 15x4y3z5, 12x2y7z2

15x4y3z5 = 5 x3 x x4 x y3 x z5

12x2y7z2 = 4x3xx2xy7xz2

kP.ngh.th = 3x2y3z2

23. Neh;f;Nfhl;bd; rha;T m =y2@ y1

x2@ x1

fffffffffffffffff

(3>-2) kw;Wk; (-1> 4) Mfpa Gs;spfis ,izf;Fk; Neh;f;Nfhl;bd;

rha;T

1 1 -7 -3

3 0 3 12 15

1 4 5 12

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m = 4 + 2@1@3ffffffffffffffffff

= 6@4ffffffff

=@32ffffffff

24. x ntl;Lj;Jz;L a = 23fff, y ntl;Lj;Jz;L b = 3

4fff Neh;f;Nfhl;bd;

rkd;ghL

xaffff+ y

bffff= 1

x23fffffff+ y

34ffffffff= 1

3x2ffffff+ 4y

3ffffff= 1

9x + 8y@ 6 = 0

25. aij = |2i@ 3j|

a11 = |2 1` a@3 1` a

| = 1

a12= |2 1` a@3 2` a

| = 4

a13= |2 1` a@3 3` a

| = 7

a21= |2 2` a@ 3 1` a

| = 1

a22= |2 2` a@ 3 2` a

| = 2

a23= |2 2` a@ 3 3` a

= 5

A = 1 4 71 2 5

f g

26. 6A@ 3B = 6 4 @ 25 @ 9

d e@3 8 2

@1 @3

d e

= 24@1230@54

d e@

24 6@3 @ 9

d e

= 24@24 @ 12@ 1630+ 3 @ 54+ 9

d e

= 0 @1833@45

d e

27. 1@ cosθ1 + cosθffffffffffffffffffffffs

wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww= 1@ cosθ

1 + cosθffffffffffffffffffffff+ 1@ cosθ

1@ cosθffffffffffffffffffffffs

wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww

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=1@ cosθ` a2

12@cos2 θffffffffffffffffffffffffffff

vuutwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww

=1@ cosθ` a2

sin2 θffffffffffffffffffffffffffff

vuutwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww

= 1@ cosθsinθffffffffffffffffffffff

= Coseθ@ cotθ

28. 1 + secθ

secfffffffffffffffffffffθ =

1 + 1cosffffffffffffθ1

fffffffffffffffffffffffffffcosθfffffffffffffffffff

= cosθ + 1cosθfffffffffffffffffffffcosθ

` a

= 1 + cosθ` a

B1@ cosθ1@ cosθffffffffffffffffffffff

= sin2 θ1@ cosθffffffffffffffffffffff

29. ∆ABCy; DE||BC

ADDBffffffffff= AE

ECfffffffff (Njy;]; Njw;wk;)

EC = AEBDB

ADfffffffffffffffffffffff

EC = 3.7B32

fffffffffffffffff

= 5.55nr.kP

30.a

x3@ 27

x2@9

ffffffffffffffffff =x3@ 3` a3

x2@ 32

ffffffffffffffffffff

=x@ 3` a

x2 + 3x + 9b c

x + 3` a

x@ 3` affffffffffffffffffffffffffffffffffffffffffffffff

= x2 + 3x + 9x + 3

ffffffffffffffffffffffffff

a3 + b3 = a@b` a

a2 + ab + b2

b c

a2@b2 = a + b

` aa@ b` a

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b. rha;T m = 23fff, x1 = 5, y1 = 4

Neh;f;Nfhl;bd; rkd;ghL

y@ y1 = m x@ x1

` a

y@ @4` a

= 23fffx@5` a

y + 4 = 23fffx@5` a

3y + 12= 2x@ 102x@ 3y@22= 0

gphpT-III

31. n(EUTUH)=100

n(E) = 85% n(T)=40% n(H)=20%

n(EnT) = 42% n(TnH) = 23% n(EnH)=10%

n(EUTUH)=100% n(EnTnH)x

n(EUTUH) = n(E)+n(T)+n(H)+n(EnT)-n(TnH)

-n (EnH)-n (EnTnH)

100 = 85+40+20-42-23-10+x

100 = 70+x

x = 30

%d;W nkhopfisAk; Ngrj njhpe;jth;fs; = 30%

f(x) = 4x2-1 -3< x < 2 x = -3, -2, -1, 0, 1

3x – 2 2 < x < 4 x = 2, 3, 4

2x – 3 4 < x < 7 x = 5, 6

i) f(5)+f(6)

x=5

f(5) = 2x-3 = 2(5)-3=7

f(6) = 2(6)-3 = 12-3=9

f(5)+f(6) = 7+9=16

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ii) f(1)-f(3)

f(1) = 4x2-1 = 4(1)2-1 = 4-1=3

f(3) = 3x-2 = 3(3)-2 = 9-2=7

f(1) – f(3) = 3-7=-4

iii) f(-2) – f(4)

f(-2) = 4x2-1 = 4(-2)2 -1 = 16-1 = 15

f(4) = 3x-2 = 3(4)-2 = 12-2 = 10

f(-2)-f(4) = 15-10=5

iv)

f 3` a

+ f @ 1` a

2f 6` a@ f 1` afffffffffffffffffffffffffffffff

f @ 1` a

= 4x2@ 1 = 4@ 1

` a2@ 1 = 4@ 1 = 3

f 3` a

+ f @1` a

2f 6` a@ f 1` afffffffffffffffffffffffffffffff= 7 + 3

2 9` a@ 3

fffffffffffffffffffff= 1018@3ffffffffffffffff= 10

15ffffff

= 23fff

33.

t1 = a t2 = b l = c tn

d = b@a tn = a + n@1` a

d

c = a + n@1` a

d

c = a + n@1` a

b@ a` a

n@1 = c@ab@afffffffffffff

n = c@ ab@ afffffffffffff+ 1

= c@ a + b@ ab@a

ffffffffffffffffffffffffffffffff= b + c@2a

b@ afffffffffffffffffffffffff

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Sn = n2fffa + l` a

= b + c@ 2a2 b@a` afffffffffffffffffffffffffa + c

` a

=b + c@ 2a` a

a + c` a

2 b@a` afffffffffffffffffffffffffffffffffffffffffffff

34. 163+173+183+……+303

= (13+23+33+…..+303) – (13+23+33+….+153)

=n n + 1` a

2fffffffffffffffffffffF G2

= 30B312

fffffffffffffffffff g@

15B162

fffffffffffffffffff g2

= 465` a2

@ 120` a2

= 216225@14400

= 201825 nr.kP3

35. Sn = 7+77+777+…..n cWg;Gfs; tiu

= 7(1+11+111+…..n cWg;Gfs; tiu

= 79fff (9+99+999+….n cWg;Gfs; tiu

= 79fff ((10-1)+(100-1)+(1000-1)+….n cWg;Gfs; tiu

= 79fff [(10+102+103+…..n cWg;Gfs; tiu)-n]

Sn = 79fff10 10n

@1` a

9ffffffffffffffffffffffffffff

@nF G

Sn = 7081ffffff10n

@ 1` a

@7n9ffffff

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36.

x3@ 23x2 + 142x@120

P x` a

= x3@23x2 + 142x@ 120

P 1` a

= 1@ 23+ 142@ 120= 0

(x-1) vd;gJ P(x)d; xU fhuzp

x3 – 23x2+142x – 120 = (x-1) (x2-22x-120)

x2 – 22x+120 = x2-10x-12x-120

= x(x-10)-12(x-10)

= (x-12) (x-10)

x3-23x2+142x-120 = (x-1) (x-12) (x-10)

37.

m2@m@12m2@ 16

fffffffffffffffffffffffffffffff= m2 + m@6m2@ 8m + 16

fffffffffffffffffffffffffffffffffm2@m@12= m@4

` am + 3` a

m2@16= m + 4

` am@4` a

m2 + m@ 6 = m + 3` a

m@2` a

m2 + 8m + 16= m + 4` a

m + 4` a

m2@m@12m2@ 16

fffffffffffffffffffffffffffffffD

m2 + m@6m2 + 8m + 16fffffffffffffffffffffffffffffffff= m@4

` am + 3` a

m + 4` a

m@ 4` afffffffffffffffffffffffffffffffffffff

Dm + 3` a

m@2` a

m + 4` a

m@4` afffffffffffffffffffffffffffffffffffff

= m + 3m + 4ffffffffffffffffffX m + 4

` am@ 4` a

m + 3` a

m@ 2` afffffffffffffffffffffffffffffffffffffffffffffffff

= m@ 4m@ 2fffffffffffffffffff

1 -23 142 -120

1 0 1 -22 120

1 -22 120 0

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1 4

38. (-4,5) (0,7), (5,-5), (-4,-2)

12fff {(x 1y2+x2y3+x3y4+x4y1) – (x2y1+x3y2+x4y3+x1y4}

12fff

0 @4 @4 5 0

7 5 @2 @5 7

X^^\^^^Z

Y^^]^^^[

12fff { (0+8+20+35) – (-28-20-10+0)}

= 12fff {63- (-58)}

= 12fff {63+58}

= 12fff x 121

= 60.5. Sq. units

39. A (2,1) , B (-2,3), C(4,5)

BCapd; eLg;Gs;sp D@2 + 4

2fffffffffffffffffffffff, 3 + 5

2fffffffffffffffff g

D22fff, 8

2ffff g

D (1,4)

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eLf;NfhL AD-apd; rkd;ghL

A (2,1) D (1,4)

y@ y1

y2@ y1

fffffffffffffffffffffff= x@ x1

x2@ x1

fffffffffffffffffffff

y@14@ 1fffffffffffffffff= x@ 2

1@ 2fffffffffffffffff

y@1

3fffffffffffffffff= x@2

@1fffffffffffffffff

-1 (y-1) = 2 (x-2)

-y +1 = 3x – 6

3x+y -7= 0

40. A = 1 @ 12 3

d e

A2 -4A+5I2

A2 = 1 @ 12 3

d e1 @ 11 3

d e

= 1@ 2 @ 1@ 32 + 6 @ 2 + 9

d e

= @1 @ 48 7

d e

4A = 4 1 @ 12 3

d e

= 4 @ 48 12

d e

5I2 = 5 1 00 1

d e

= 5 00 5

d e

A2-4A+5I2 = @1 @ 48 7

d e - 4 @ 4

8 12

d e + 5 0

0 5

d e

= @ 1@ 4 + 5 @ 4 + 4 + 08@ 8 + 0 7@ 12 + 5

d e

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= 0 00 0

d e

= 0

41. A = 5 27 3

d e B = 2 @ 1

@ 1 1

d e

AB = 5 27 3

d e2 @ 1@ 1 1

d e

= 10@ 2 @ 5 + 214@ 3 @ 7 + 3

d e

= 8 @ 311 @ 4

d e

(AB)T = 8 11@ 3 @ 4

d e

BT = 2 @ 1@ 1 1

d e AT = 5 7

2 3

d e

BTAT = 2 @ 1@ 1 1

d e5 72 3

d e

= 10@ 2 14@ 3@ 5 + 2 @ 7 + 3

d e

= 8 11@ 3 @ 4

d e ----- (2)

(1) = (2) (AB)T = BT AT

42. (sin θ + cosec θ)2 + (cosθ + secθ)2

(sin θ+cosecθ)2 = sin2θ+2sinθ cosecθ+cosec2θ

= sin2 + 2 sin θ 1/sin θ + cose2 θ

= sin2θ + cosec2θ + 2

(cos θ + sec θ)2 = cos2θ + 2cos θ secθ + sec2 θ

= cos2 θ + 2 cos θ 1/cos θ + sec2 θ

= cos2θ + sec2 θ + 2

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(sin θ + cosec θ)2 + (cos θ + secθ)2

= sin2θ + cosec2θ +2 + cos2θ + sec2 θ + 2

= sin2 θ + cos 2θ (1+cot2 θ) + (1+tan2θ) +2+2

= 1+1+cot2θ+1+tan2θ + 4

= 7+tan2θ + cot2θ

43. Nfhz ,Urkntl;bj;Njhw;wk;

xU Kf;Nfhzj;jpy; xU Nfhzj;jpd; cl;Gw ,Urkntl;bahdJ mf;Nfhzj;jpd; vjph; gf;fj;ij cl;Gwkhf mf;Nfhzj;jpid mlf;fpa gf;fq;fspd; tpfpjj;jpy; gphpf;Fk;.

nfhLf;fg;gl;Ls;sit

ep&gpf;f

mikg;G

ep&gzk;

44. DE || BC

AB = 3AD

∆ADE apd; gug;G AD2 ∆ADE d; gug;G 1

∆ABC apd; gug;G AB2 72 9

ADE-d; gug;G = 8 nr.kP2

ehw;fuk; DBCEapd; gug;G = ∆ ABC-d; gug;G - ∆ADEd; gug;G

= 72-8

= 64 cm2

= = =

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45. (x,y) , (9,0), (0,b) vd;gd xU Neh;f;Nfhl;by; mikAk; Gs;spfshFk;

12fff9 0 X 9

0 b y 0

V g

ab-bx-ay = 0

ab = bx+ay

,UGwKk; ab My; tFf;f

ababffffffff =

bxabffffffff+ ay

abffffffff

1 = xaffff+ y

bfffff

# xaffff+ y

bfffff = 1

45. ntz;glk;

A∪(B∩C ) = (A∪B) ∩ (A∪C)

(i) B∩C

(ii) A∪(B∩C )

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(iii) A∪B

(iv) A∪C

(v) (A∪B) ∩ (A∪C)

(ii) kw;Wk; (v) rkk;

A∪(B∩C ) = (A∪B) ∩ (A∪C)

kjpg;ngz;

46. 1. cjtpg;glk; - 2

2. cz;ikg; glk; - 6

3. tiuKiw - 2

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b) 1. cjtpg; glk; - 2

2. cz;ikg; glk; - 2

3. tiuKiw - 1

4. fzf;fPL - 2

47. 1. ml;ltiz - 2

2. tiuglk; - 4

3. msTj; jpl;lk; - 2

4. jPh;T (-1> 3) - 2

b. 1. ml;ltiz - 2

2. tiuglk; - 4

3. msTj; jpl;lk; - 2

4. jPh;T (-2> 3) - 2

tpilf;Fwpg;G jahhpj;jth;fs;

V. rkhu%h;j;jp B.Sc., B.Ed., gl;ljhhp MrphpaH> fzpjk;>

S. n[]pe;jhNkhp M.Sc., B.Ed., gl;ljhhp Mrphpia> fzpjk;.

= ntq;fNl];tuh tpj;ah ke;jpH Nkdpiyg;gs;sp jspf;Nfhl;il> gl;Lf;Nfhl;il> jQ;rht+H – 614906.

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