Transcript x - Padasalai.Net-12th Study Materials

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Padasalai’s Centum Coaching Team – gjpntz;
Special Question Paper
fzpjk;;
Neuk;
tFg;G
:
3 kzp
nkhj;j kjpg;ngz; : 200
: 12
gFjp - m
Fwpg;G: i) midj;J tpdhf;fSf;Fk; tpilaspf;f.
ii) nfhLf;fg;gl;l 4 tpilfspy; kpfTk; Vw;Gila tpilapid Njh;T nra;f.
40×1=40
1.
xU jpirapyp mzpapd; thpir 3,
1
3.
b) k 3 I
2
b) A
adj A
d) k I
vd;gJ
n
c) A
n1
d)
A
A vd;w mzpapd; thpir 3 vdpy; det (kA) vd;gJ
3
a) k det(A )
4
A 1 vd;gJ.
1
c) k I
xU rJu mzp A d; thpir n vdpy;
a) A
vdpy;
1
a) k 2 I
2.
k0
2
b) k det(A )
myF mzp I d; thpir
n
a) k adj(I )
c) k det(A)
d) k det(A)
xU khwpyp vdpy;>adj (kI) = ……
n1
2
b) k adj(I )
c) k adj( I ) d) k adj( I )
n, k  0
5 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.
6 kjpg;gpl Ntz;ba 3 khwpfspy; mike;j 3 Nehpa rkgbj;jhdrkd;ghl;L;
njhFg;gpy;   0 ;  x  0 ,  y  0 kw;Wk;  z  0 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
7
rkgbj;jhd Nehpar; rkd;ghl;Lj; njhFg;ghdJ
a) vg;NghJNk xUq;fikT cilajhFk;
b) ntspg;gilj;jPHT kl;LNk nfhz;Ls;sJ
c) vz;zpf;ifaw;w jPHTfs; nfhz;Ls;sJ
d) xUq;fikT cilajhf ,Uf;fj; Njitapy;iy
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8. 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
9
   
abc  0,

a 3,

2
a)   6
10


b  4 , c  5 vdpy; a
b)
c)   3
3
c)

11 OQ
vd;w myF ntf;lH kPjhd
gug;ig Nghd;W Kk;klq;fhapd;
1
1
a) tan 3
b)
f;Fk; ,ilg;gl;l Nfhzk;
b
5
b)   3
p, q kw;Wk; p  q Mfpait vz;zsT
a) 2
f;Fk;

OP

d)
 nfhz;l
2

2
ntf;lh;fshapd;
 
pq
MdJ.
d) 1
d; tPoyhdJ OPRQ vd;w ,izfuj;jpd;
 POQ MdJ.
 3 

cos1
 10 


c)
 3 

sin1
 10 


d)
 1
sin1 
3
12 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

c) 45
b) 90
 

d)



60
13. r  i  k  t 3 i  2 j  7 k vd;w NfhLk; r. i  j  k  8 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)





14. (2,1,-1) vd;w Gs;sp topahfTk; jsq;fs; r  ( i  3 j  k)  0 ; r  ( j  2k)  0 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
15

2r  3i  j 4 k
a)
 3 1

, ,  2 , 4

 2 2

  4 vd;w
b)
Nfhsj;jpd; ikak; kw;Wk; Muk;
 3 1

, ,  2  and 2

 2 2

c)
 3 1

, ,  2 , 6

 2 2

d)
 3 1

, ,  2  and 5

 2 2

16. a vd;gjid epiy ntf;luhf nfhz;l Gs;sp topr; nry;yf; $baJk;
ntf;lUf;F nrq;Fj;jhdJkhd
jsj;jpd; rkd;ghL
a) r  n  a  n
b) r  n  a  n
n
vd;w
c) r  n  a  n d) r  n  a  n
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17. 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.
2z  1  2 z vdpy; P d;
18. P MdJ fyg;G vz; khwp z I Fwpf;fpd;wJ
epakg;ghij
a) x  1 vd;w Nehf;NfhL
4
1
c) z  vd;w
2
Nehf;NfhL
b) y  1 vd;w Nehf;NfhL
4
d) x 2  y 2  4x  1  0 vd;w tl;lk;
19 i13  i14  i15  i16 d; ,iz fyg;ngz;
a) 1 b) -1 c) 0
d) -i
1 i
20. ax  bx  1  0 vd;w rkd;ghl;bd; xU jPh;T 1  i aAk; >bAk; nka; vdpy; (a,b)
vd;gJ.
a) (1,1)
b) (1,-1)
c) (0,1)
d) (1,0)
2
21. x 2  6x  k  0 vd;w rkd;ghl;bd; xU %yk; -i +3 vdpy; kd; kjpg;G
a) 5
b) 5
c) 10
d) 10
22.  vd;gJ 1d; Kg;gb %ynkdpy; (1    2 ) 4  (1    2 ) 4 d; kjpg;G
  a) 0
b) 32
c) -16
d) -32
23. gpd;tUtdtw;Ws; vJ rhpahdJ?
(i) Re(Z )  Z (ii) Im(Z )  Z (iii) Z  Z
a) (i) , (ii)
24
(iv) Z n   Z 
n
b) (ii), (iii) c) (ii),(iii) kw;Wk; (iv) d) (i),(iii) kw;Wk; (iv)
Z  Z -,d; kjpg;G
a) 2 Re(Z ) b) Re(Z )
c) Im(Z )
d) 2 Im(Z )
25. (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
2
26. 2x + 3y + 9 = 0 vd;w NfhL y  8x vd;w gutisaj;ijj; njhLk; Gs;sp .
a)(0,-3)
b) (2,4)
c)   6 , 92 
d)  92 ,6 

27.
2
2
9x  5y  54x  40y  116  0
a)
1
3
b)
2
3



vd;w $k;G tistpd; ikaj;njhiyj;jfT (e)d; kjpg;G
c) 94
d) 2
5
.
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28.
2x  y  c  0 vd;w
Neh;f;NfhL
vd;w ePs;tl;lj;jpd;; njhLNfhL vdpy; cd;
b)  6
a)  2 3
kjpg;G.
4x2  8y2  32
c) 36 d)  4
29. nrt;tfyj;jpd; ePsk; Jizar;rpd; ePsj;jpy; ghjp vdf; nfhz;Ls;s
mjpgutisaj;jpd; ikaj;njhiyj;jfT
5
b) 3
a)
3
2
30.
x2 y2

 1 vd;w
16
9
3
c) 2
y  mx  c vd;w njhLNfhL kw;Wk;
 am 2 b 2
,
c
 c
njhLg;Gs;spa) 
5
2
mjpgutisaj;jpw;F (2,1) vd;w Gs;spapypUe;J tiuag;gLk;
njhLNfhLfspd; njhLehz;
a) 9x  8y  72  0
b) 9x  8y  72  0
31
d)

 b)


 a2m b2 

 c)
,
 c
c 

8x  9y  72  0
c)
x2
a2

y2
b2
 1 vd;w
  a2m  b2 


,
 c
c 

d)
8x  9y  72  0
mjpgutisak; ,tw;wpd;
  am 2  b 2 


,
 c
c 

d)
32. gpd;tUtdtw;wpy; vit cz;ikahd $w;Wfs;?
i) xU Gs;spapypUe;J xU gutisaj;jpw;F 2 njhLNfhLfs; kw;Wk; 3
nrq;NfhLfs; tiuayhk;
ii) xU Gs;spapypUe;J xU ePs;tl;lj;jpw;F 2 njhLNfhLfs; kw;Wk; 4
nrq;NfhLfs; tiuayhk;
iii) xU Gs;spapypUe;J xU mjpgutisaj;jpw;F 2 njhLNfhLfs; kw;Wk; 4
nrq;NfhLfs; tiuayhk;
iv) xU Gs;spapypUe;J xU nrt;tf mjpgutisaj;jpw;F 2 njhLNfhLfs; 4
nrq;NfhLfs; tiuayhk;.
a) (i) , (ii), (iii) kw;Wk; (iv)
b) (i) , (ii) kl;LNk
c) (iii) , (iv) kl;LNk d) (i) , (ii), kw;Wk; (iii)
33. 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
34 x³ -2x² +3x+8 mjpfhpf;Fk; tPjkhdJ x mjpfhpf;Fk; tPjj;ij Nghy; ,Uklq;F
vdpy; x d; kjpg;Gfs;
a)   31 ,3 
b)  31 ,3 
c)   31 ,3 
d)  31 ,1






35. 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) 3  2
g( x )f (a)  g(a)f ( x )
36 f(a) = 2 ; f’ (a) = 1 ; g (a) = -1 ; g’ (a) = 2 vdpy; xlim
d; kjpg;G
a
xa
a) 5
b) -5
c) 3
d) -3
37. 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
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38 y = x4vd;w tistiuapd; tisT khw;Wg;Gs;sp
a) x = 0
b) x = 3.
c) x = 12
d) vq;Fkpy;iy
39 ,ilkjpg;G tpjpapd; khw;W tbtk;
0   1
a) f a  h  f a  hf ' a  h
b) f a  h  f a  hf ' a  h
0 
c) f a  h  f a  hf ' a  h
d) f a  h  f a  hf ' a  h
0   1
0 
40 f MdJ I  R ( R vd;gJ nka;naz;fspd; fzk; vd;w ,ilntspapy; nka;
kjpg;Gfisf; nfhz;l VWk; rhHG
vdpy;
a) x1  x2 tpw;F f x1   f x2  x1 , x2  I
b) x1  x2 tpw;F f x1   f x2  x1 , x2  I
c) x1  x2 tpw;F f x1   f x2  x1 , x2  I
d) x1  x2 tpw;F f x1   f x2  x1 , x2  I
 1
 1
gFjp - M
Fwpg;G: i) vitnaDk; 10 tpdhf;fSf;F tpilaspf;f.
ii) tpdh vz; 55 f;F fz;bg;ghf tpilaspf;fTk; gpw tpdhf;fspypUe;J VNjDk; Xz;gJ
tpdhf;fSf;F tpilaspf;fTk;
10× 𝟔=60
41
1 2 
A ( adj A )  ( adj A ) A  A .  vd;gijr;
A 
 vd;w mzpapd; NrHg;igf; fz;L>
3

5


rhpghHf;f.
42
3 1  5  1
1  2 1  5

 vd;w
1 5  7 2 
mzpapd; juk; fhz;f.
43 NeHkhW mzpfhzy; Kiwapy; jPHf;f : x  y  3 , 2x  3 y  8
44
 4  3  3
A   1
0
1  -,d;
 4
4
3 
45
(1, -1, 1) –I ikakhfTk; r  i  j  2 k
NrHg;G mzp A vd epWTf.
rkkhd kjpg;ig Muj;ijf;
jUf.

  5 vd;w
Nfhsj;jpd; Muj;jpw;F
nfhz;l Nfhsj;jpd; ntf;lH kw;Wk; rkd;ghLfisj;
46 AC kw;Wk; BD-I %iytpl;lq;fshff; nfhz;l ehw;fuk; ABCD -,d; gug;G
1
AC  BD vdf; fhl;Lf.
2
47 (a)
 

VNjDk; xU ntf;lh; r f;F r  r  i i  r  j
(b) x . a  0 , x . b  0 , x . c  0
vd epUTf
x  0 vdpy; a
,b,c


j  r k
k
vd;gd xU js ntf;lh;fs;
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48 P vDk; Gs;sp fyg;ngz; khwp zIf; Fwpj;jhy; z  3i  z  3i
epakg;ghijia fhz;f.
f;F P-,d;
49 3  i I xU jPHthff; nfhz;l x  8x  24x  32x  20  0 vDk;
rkd;ghl;bd; jPHTfisf; fhz;f.
Z1
Z1

50 Z1 , Z2 vd;w VNjDk; ,U fyg;ngz;fSf;F(i) Z
Z2
2
4
3
2
Z 
1
(ii) arg  Z   arg Z 1   arg Z 2  vd ep&gp.

2

51 gutisa Mbapd; Ftpak; mjd; ikaj;jpypUe;J (Kid) 8 nr.kP njhiytpy;
cs;sJ. Mbapd; FopT 25 nr.kP vdpy; mt;thbapd; tpl;lk; fhz;f.
2
2
52 3x  5xy  2 y  17 x  y  14  0 vd;w mjpgutisaj;jpd; njhiyj; njhLNfhspd;
,ilg;gl;l Nfhzj;ijf; fhz;f.
53 xU Kf;Nfhzj;jpd; ,uz;L gf;fq;fspd; ePsq;fs; KiwNa 4kP> 5kP MFk;.
kw;Wk; mtw;wpw;F ,ilg;gl;l Nfhz
mstpd; VWk; tPjk; tpdhbf;F 0.06 Nubad;
vdpy;> epiyahd ePsq;fis cila me;j gf;fq;fSf;F ,ilNa
Nfhz msT
 / 3 Mf ,Uf;Fk; NghJ> mjd; gug;gpy; Vw;gLk; Vw;w tPjk; fhz;f
54 f  x   x vd;w rhHgpw;F   2 , 2  vd;w ,ilntspapy; yhf;uhQ;rpapd;
,ilkjpg;Gj; Njw;wj;ij rhpghHf;fTk;.
x y 9 z 2
x6 y 7 z 4




55(a)
kw;Wk;
vd;w xU jsj;jpy; mikahj
3
2
4
3
1
1
NfhLfspd; ,ilg;gl;l kPr;rpW
J}uj;ijf; fhz;f.
(b) fPo;f;fhZk; rhHGfSf;F nkf;yhhpd; tphpT fhz;f: log e 1  x
3
gFjp - ,
Fwpg;G: i) vitnaDk; 10 tpdhf;fSf;F tpilaspf;f.
ii) tpdh vz; 70 f;F fz;bg;ghf tpilaspf;fTk; gpw tpdhf;fspypUe;J VNjDk; Xz;gJ
tpdhf;fSf;F tpilaspf;fTk;
10× 𝟏𝟎=100
56
𝐴 =
1
3
2 2 1
−2 1 2 vdpy;𝐴−1 = 𝐴𝑇 vd epWTf.
1 −2 1
57
gpd;tUk; mrkgbj;jhd Nehpa rkd;ghl;Lj; njhFg;gpid; mzpf;Nfhit
Kiwapy; jPHf;f:
𝑥 + 𝑦 + 𝑧 = 4, 𝑥 − 𝑦 + 𝑧 = 2,2𝑥 + 𝑦 − 𝑧 = 1
58
xU rpwpa fUj;juq;F miwapy; 100 ehw;fhypfs; itg;gjw;F NghJkhd
,lKs;sJ. %d;W epwq;fspy; ehw;fhypfs; cs;sd. (rpfg;G> ePyk; kw;Wk; gr;ir).
rpfg;G tz;z ehw;fhypapd; tpiy &.240> ePy tz;z ehw;fhypapd; tpiy &.260>
gr;irtz;z ehw;fhypapd; tpiy &.300. nkhj;jk; &. 25000 kjpg;Gs;s
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ehw;fhypfs;thq;fg;gl;lJ.
mt;thwhapd;
xt;nthU
tz;zj;jpYk;
ehw;fhypfspd; vz;zpf;iff;F Fiwe;jgl;rk; %d;W jPHTfisf; fhz;f
thq;fj;jf;f
𝑘 −,d; vk;kjpg;GfSf;F gpd;tUk; rkd;ghl;Lj; njhFg;G(ju Kiwapy;)
𝑘𝑥 + 𝑦 + 𝑧 = 1 , 𝑥 + 𝑘𝑦 + 𝑧 = 1 , 𝑥 + 𝑦 + 𝑘𝑧 = 1(i)xNu xU jPHT(ii)xd;Wf;F
Nkw;gl;l jPHT (iii)jPHT ,y;yhik ngWk.;
60
xU Kf;Nfhzj;jpd; Fj;Jf;NfhLfs; xNu Gs;spapy; re;jpf;Fk; vd;gjid
ntf;lH Kiwapy; epWTf
59
𝑥−1
𝑦+1
𝑧
𝑥−2
𝑦 −1
−𝑧−1
61
=
= kw;Wk;
=
=
vd;w NfhLfs; ntl;bf; nfhs;Sk;
1
−1
3
1
2
1
vdf; fhl;Lf. NkYk; mit ntl;Lk; Gs;spiaf; fhz;f.
𝑥+3
𝑦 +1
𝑧−4
62 (1,2, −2) vd;w Gs;sp topr; nry;tJk;
=
=
vd;w
3
−2
−4
Nfhl;bw;F,izahdJkhd 2𝑥 + 3𝑦 + 3𝑧 = 8vd;w jsj;jpw;Fr; nrq;Fj;jhfTk;
mike;j jsj;jpd; ntf;lH kw;Wk; fhHBrpad; rkd;ghLfisf; fhz;f..
63 𝑃 vDk; Gs;sp fyg;ngz; khwp z If; Fwpj;jhy; 𝑃-,d;
z−1
π
epakg;ghijia arg
=
vd;w fl;Lghl;;bw;F cl;gl;L fhz;f
z+1
3
64 𝑥 2 − 2𝑝𝑥 + 𝑝2 + 𝑞 2 = 0 vd;w rkd;ghl;bd; %yq;fs; 𝛼, 𝛽 kw;Wk;
𝑞
𝑦 +𝛼 𝑛 − 𝑦 +𝛽 𝑛
sin 𝑛𝜃
tan θ =
vdpy;
= 𝑞 𝑛−1 n
vd epWTf.
65
jPHf;f:
𝑦 +𝑝
4
𝛼 −𝛽
3
sin 𝜃
2
𝑥 − 𝑥 +𝑥 −𝑥+ 1 = 0
66 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;tsT J}uj;jpw;F mg;ghy; ePuhdJ jiuapy; tpOk; vd;gijf; fhz;f.
67 5 𝑥 + 12𝑦 = 9vd;w NeHf;NfhL mjpgutisak; 𝑥 2 − 9 𝑦 2 = 9 -Ij; njhLfpwJ
vd ep&gpf;f NkYk; njhLk; Gs;spiaAk; fhz;f.
68
mjpgutisaj;jpd; ikak; (2,4).NkYk;(2,0)topNa nry;fpwJ. ,jd; njhiyj;
njhLNfhLfs;𝑥 + 2𝑦 − 12 = 0kw;Wk;𝑥 − 2𝑦 + 8 = 0 Mfpatw;wpw;F ,izahf
,Uf;fpd;wdvdpy;mjpgutisaj;jpd; rkd;ghL; fhz;f
69 𝑦 = 𝑥 2 kw;Wk;𝑦 = 𝑥 − 2 2 vd;w tistiufs;
mitfSf;F ,ilg;gl;l Nfhzj;ijf; fhz;f.
ntl;bf;
nfhs;Sk;
Gs;spapy;
70 (a) 𝑥 = 𝑎 cos 3 θ ; 𝑦 = 𝑎 sin3 θvDk; Jiz myF rkd;ghLfisf; nfhz;l
tistiuf;F ‘ 𝜃’ ,y; tiuag;gLk; nrq;Nfhl;bd; rkd;ghL 𝑥 cosθ − 𝑦 sinθ =
𝑎 cos 2θvdf; fhl;Lf.
(or)
(b)𝑎 = 𝑖 + 𝑗 + 𝑘 , 𝑏 = 2 𝑖 + 𝑘 , 𝑐 = 2 𝑖 + 𝑗 + 𝑘 ,
𝑑 = 𝑖 + 𝑗 +
2 𝑘 vdpy; 𝑎 × 𝑏 × 𝑐 × 𝑑 = 𝑎 𝑏 𝑑 𝑐 − 𝑎 𝑏 𝑐 𝑑 vd rhpahHf;f.
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