= ntq;fnl];tuh tpj;ah ke;jph; nkdpiyg; gs;sp · 1@cosθ ` a2 12 @cos2 θ...
TRANSCRIPT
= 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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