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π¨ β π© vdpy; , π¨ β© π© , π¨\π© Mfpatw;iw ntd;glj;jpy; tiuf.
(Oct-2014)
ST UD Y
2.
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π¨βπ©
π¨β©π© =π¨
π¨\π© = β
π· = π, π, π , πΈ = π, π, π, π , πΉ = {π, π, π, π} vdpy,; (i) π· \πΉ (ii) πΈ β© πΉ (iii) πΉ \ (π· β© πΈ) (Jun-15) (i) π \π
= {a, b, c}\{a, e, f, s}={b, c} (ii) π β© π
= {g, h, x, y}β©{a, e, f, s} = { } (iii) π
\ (π β© π) β (π β© π) = {a, b, c} β© g, h, x, y = { } π
\ (π β© π) = π, π, π , π \{ }= {π, π, π , π }
4.
π¨ = {π, π, π, π, π}, π© = {π, π, π}, πͺ = {π, π, π, π, π, π}, vdpy; , (i) π¨ βͺ (π© β© πͺ) (Apr-14) (ii) π¨ β© (π© βͺ πͺ) (Jun-12, Oct-12) (iii) π¨\(πͺ\π©) fhz;f. (i) π¨ βͺ (π© β© πͺ) β (π΅ β© πΆ) π΄ βͺ (π΅ β© πΆ) (ii) π¨ β© (π© βͺ πͺ) β (π΅ βͺ πΆ) π΄ β© (π΅ βͺ πΆ)
6.
π, π, π β© 1, π, 3, π, 5, π = {2, 4, 6} 4, 6, 7, 8, 9 βͺ 2, 4, 6 = {2, 4, 6, 7, 8, 9}
= =
2, 4, 6 βͺ 1, 2, 3, 4, 5, 6 = {1, 2, 3, 4, 5, 6} π, π, 7, 8, 9 β© 1, 2, 3, π, 5, π = {4, 6}
= =
A\ (C\B)
5.
= =
DO O
(iii) π¨\ (πͺ\π©) β (πΆ\π΅)
ZY
3.
1, 2, 3, 4, 5, 6 \ 2, 4, 6 = {1, 3, 5} 4, 6, 7, 8, 9 \{1, 3, 5}= {4, 6, 7, 8, 9}
A kw;Wk; B vd;gd C-d; cl;fzq;fs;. NkYk; mit ntl;lhf; fzq;fs; vdpy; ntd;glk; tiuf.
π¨ β© (π©\πͺ) ntd;glk; tiuf.
(B\C) https://doozystudy.blogspot.in/
A β©(B\C) https://doozystudy.blogspot.in/
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(π© βͺ πͺ) \π¨ ntd;glk; tiuf.
(BβͺC) 8.
π¨ βͺ (π© β© πͺ) ntd;glk; tiuf.
(Bβ©C) πͺ β© (π©\π¨) ntd;glk; tiuf.
10.
πͺ β© (π© βͺ π¨) ntd;glk; tiuf.
11.
DO O
(B\A)
ZY
9.
(BβͺA)
(BβͺC) \A
ST UD Y
7.
https://doozystudy.blogspot.in/
(Apr-2015)
Aβͺ(Bβ©C)
C β©(B\A)
C β©(BβͺA)
π¨ = π, π, π, π , π© = βπ, π, π, π, π, π, π, π, ππ, ππ, ππ vd;f. πΉ = π, π , π, π , π, ππ , π, π β π¨ Γ π© xU cwT vdpy; xU rhh;G vdf; fhl;Lf. mjd; kjpg;gfk;> Jizkjpg;gfk;> tPr;rfk; fhz;f. (Jun-2012, Oct-2013) A apy; cs;s xt;nthU cWg;gpw;Fk; xU epoy; cU B apy; cs;sJ. / ,J xU rhh;G MFk;. kjpg;gfk; = {1, 2, 3, 4} Jizkjpg;gfk; = {β1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12} tPr;rfk; = {3, 6, 9, 10}
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π = {(ππ, π), (ππ, π), (ππ, π), (ππ, π), (ππ, ππ)} vd;w fhz;f. 2 d; Kd; cUf;fs; = 12, 14 3 d; Kd; cUf;fs; = 13, 15
13.
πΉ = {π, βπ), (βπ, π), (π, π), (π
, βπ)} vd;gJ rkdpr; rhh;igf; Fwpf;fpwJ vdpy; , a, b, c , d kjpg;Gfis fhz;f. (Apr-2013) π = β2, π = β5, π = 8, π = β1
14.
π¨ = {βπ, βπ, π, π} kw;Wk; π = { π,
π π
π¨ - apypUe;J π¨ -f;F xU rhh;ghFkh? tPr;rfk; = { tPr;rfj;jpy;
β1 1 , β1, 1, }, π = 2 2 β1 1 cs;s , Mfpa 2 2
β2,
1 β2
rhh;gpy;
2
kw;Wk;
3-d;
Kd;
cUf;fisf;
ST UD Y
12.
: π β π¨} vdpy,; f-d; tPr;rfk; fhz;f. NkYk; f vd;gJ , β1,
1 β1
, 1,
1 1
, 2,
1 2
cWg;Gfs; Ay; ,y;yhjjhy; A apypUe;J Af;F rhh;G MfhJ.
π¨ = {π, π, π, π, π}, π© = β kw;Wk; f : A β B MdJ f(x) = x2 vd tiuaWf;fg;gl;Ls;sJ. f d; tPr;rfk; fhz;f. NkYk; rhh;gpd; tifia fhz;f . (Apr-2015, Jun-2015) f (x) = x2 f (1) = 12 = 1 f (2) = 22 = 2 f (3) = 32 = 9 f (4) = 42 = 16 f ( 5) = 52 = 25 (i) f - d; tPr;rfk; = {1, 4, 9, 16, 25} (ii) ,J xd;Wf;F xd;whd rhh;G
16.
rhh;G π = { βπ, π , βπ, π , βπ, π , βπ, π } I (i) ml;ltiz (ii) mk;Gf;Fwpg;glk; Mfpatw;wpd; %yk; Fwpf;fTk;. (i) ml;ltiz (ii) mk;Gf;Fwpg;glk; x β1 β3 β5 β4 f(x) 2 1 6 3
18.
DO O
17.
ZY
15.
π, π β₯ π vDk; NghJ> (Jun-2013) βπ, π < π vDk; NghJ {(π, π)| π = π , π β β } vd;w cwT> rhh;ig tiuaWf;fpwjh? mjd; tPr;rfk; fhz;f. (i) x- d; xt;nthU kjpg;gpw;Fk; π¦ = π₯ vd;w xU jdpj;j (xNunahU) kjpg;G cs;sJ. vdNt ,J rhh;ghFk;. (ii) ,jd; tPr;rfk; G+r;rpak; kw;Wk; kpif nka;naz;fs; MFk;. π =
π = {(π, π), (π, π) (π, π) (π, π), (π, π)} vDk; rhh;gpw;F kjpg;gfk;> tPr;rfk; fhz;f. (i) kjpg;gfk; = {1, 2, 3, 4, 5} (ii) tPr;rfk; = {3, 5, 7, 9, 1}
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19.
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π¨ = {ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ}, π© = {π, π, ππ, ππ, ππ, ππ} kw;Wk; πͺ = {π, π, ππ, ππ, ππ, ππ, ππ} vdpy; , A\ (Bβ©C) fhz;f. (π΅ β© πΆ) = {1, 5, 10, ππ, ππ, 30} β© {7, 8, ππ, ππ, 35, 45, 48} = {15, 20} π΄\ π΅ β© πΆ = 10, 15, 20, 25, 30, 35, 40, 45, 50 \ 15, 20 = {10, 25, 30, 35, 40, 45, 50}
(Apr- 2012)
π¨ = π, ππ, ππ, ππ , π© = {π, ππ, ππ, ππ, ππ} kw;Wk; πͺ = {π, ππ, ππ, ππ, ππ, ππ} Mfpa fzq;fSf;F A\(B\C) = (A\B) \C vd;gJ nka;ahFkh vd Muha;f. (Jun-2013) LHS π΄\(π΅\πΆ) β (π΅\πΆ) = 6, 10, 12, 18, 24 \ 7, 10, 12, 14, 21,28 = {6, 18, 24} π΄\(π΅\πΆ) = 5, 10, 15, 20 \ 6, 18, 24 = {5, 10, 15, 20 β¦β¦.. (1) RHS (π΄\π΅) \πΆ β (π΄\π΅) = 5, 10, 15, 20 \ 6, 10, 12, 18, 24 = {5, 15, 20} (π΄\π΅) \πΆ = 5, 15, 20 \ 7, 10, 12, 14, 21,28 = {5, 15, 20} β¦β¦β¦β¦β¦. (2) LHS β RHS (1) β (2) vdNt A\(B\C) β (A\B)\C vd;gJ nka;ay;y.
21.
πΏ = {π, π, π, π, π}, π = {π, π, π, π, π} vd;f. X - ypUe;J Y- f;fhd cwT {(1, 1), (1, 3) (3, 5) (3, 7), (5, 7)} vd tiuaWg;gpd; mJ rhh;G MFkh vd Muha;f. ,y;iy vdpy;> fhuzk; $W. (Apr-2012) (i) rhh;G my;y (ii) Vnddpy; X-y; cs;s 1,3 Mfpa cWg;Gfs; Y-y; cs;s ,uz;L cWg;GfNshL njhlh;GgLj;jg;gl;Ls;sd. (1, 1), (1, 3) kw;Wk; (3, 5), (3, 7)
22.
π¨ = {π, π, π, π, π, π, π, π} kw;Wk; π© = {π, π, π, βπ, π, π, π, π} Mfpatw;wpw;F fzq;fspd; ntl;L> ghpkhw;Wg; gz;G cilaJ vd;gij rhpghh;f;fTk;. (Apr-2013) π΄ β© π΅ = {π, π, π, π, 2, 3, 4, 7} β© {2, 5, 3, β2, π, π, π, π} = {π, π, π, 2, 3} β¦β¦β¦ 1 π΅ β© π΄ = {2, 5, 3, β2, π, π, π, π} β© {π, π, π, π, 2, 3, 4, 7} = {π, π, π, 2, 3} β¦β¦β¦ 2 (1) & (2), Mfpatw;wpypUe;J Aβ©B = Bβ©A vd;w fzq;fspd; ntl;L> ghpkhw;Wg; gz;G cilaJ vd rhpghh;;f;fg;gl;lJ.
23.
πΌ = {π, π, ππ, ππ, ππ, ππ, ππ} , π¨ = {π, ππ, ππ} kw;Wk; π© = {π, ππ, ππ, ππ} vdpy; (π¨ βͺ π©)β kw;Wk; (π¨ β© π©)β Mfpatw;iwf; fhz;f. (Jun-2014) π΄ βͺ π΅ = 8, 16, 24 βͺ 4, 16, 20, 28 = {4, 8, 16, 20, 24, 28} π΄ βͺ π΅ β = π\ π΄ βͺ π΅ = {4, 8, 12, 16, 20, 24, 28}\{4, 8, 16, 20, 24, 28} = {12} π΄ β© π΅ = 8, ππ, 24 β© 4, ππ, 20, 28 = {16} π΄ β© π΅ β = π\ π΄ β© π΅ = 4, 8, 12, 16, 20, 24, 28 \ 16 = {4, 8, 12, 20, 24, 28}
25.
ZY
DO O
24.
ST UD Y
20.
{π, π, π, ππ} ypUe;J π© = {βπ, π, βπ, βπ, π, π}f;F π, βπ , π, π , π, βπ , (ππ, βπ)} vd;gJ xU rhh;ghFkh? π vd;gJ vdpy; mjd; tPr;rfj;ijf; fhz;f. (Oct-12) apy; cs;s vy;yh cWg;gpw;Fk; xNu xU epoy; cU Bapy; cs;sJ. / ,J rhh;G MFk;. (ii) tPr;rfk; = {β1, 2, β3, β4} π¨ = π={ rhh;G (i) A
π¨ = {π, π, π, π}, π© = {βππ, π, π, βππ, βπ, βπ, βππ, βππ} vd;f. π = {(π, π): π = π β ππ, π β π¨, π β π© } vd
tiuaWf;fg;gl;Ls;sJ. π-d; tPr;rfk; fhz;f. (Oct-2014) (ii) f d; cWg;Gfis vOJf (Apr-2016) π¦ = 3 β 2π₯
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π₯ = 5 vdpy;> π¦ π₯ = 6 vdpy;> π¦ π₯ = 7 vdpy;> π¦ π₯ = 8 vdpy;> π¦ (i) f -d; tPr;rfk; (ii) π = { 5, β7 ,
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= 3 β 2 5 = 3 β 10 = = 3 β 2(6) = 3 β 12 = = 3 β 2 (7) = 3 β 14 = = 3 β 2 (8) = 3 β 16 = = {β7, β 9, β11, β13} 6, β9 , 7, β11 , (8, β13)}
β7 β9 β11 β13
πΏ = {π, π, π, π} vd;f. π = {(π, π), (π, π), (π, π), (π, π), (π, π)} vd;w cwT X- ypUe;J X- f;F xU rhh;ghFkh vd Muha;f. cd; tpilf;Nfw;w tpsf;fk; jUf. (Apr-2014) rhh;G my;y. Vnddpy; π-y; cs;s 2 vd;w cWg;G π-y; cs;s ,uz;L cWg;GfNshL njhlh;G gLj;jg;gl;Ls;sJ. (2, 3), (2, 1)
27.
A, B vd;gd ,U fzq;fs; kw;Wk; U vd;gJ midj;Jf;fzk; vd;f. NkYk; π πΌ = πππ, π (π¨) = πππ, π (π©) = πππ kw;Wk; π (π¨ β© π©) = πππ vdpy;> π (π¨β β© π© β) If; fhz;f. π (π΄ βͺ π΅) = π(π΄) + π(π΅) β π(π΄ β© π΅) = 200 + 300 β 100 = 400 π(π΄β β© π΅β) = π(π΄ βͺ π΅)β β π(π) β π(π΄ βͺ π΅)= 700 β 400 β π(π΄β β© π΅β) = 300
28.
π (π¨) = πππ, π (π©) = πππ, π (πΌ) = πππ kw;Wk; π (π¨ βͺ π©) = πππ vdpy;> π (π¨β βͺ π©β) If; fhz;f. π(π΄ β© π΅) = π(π΄) + π(π΅) β π(π΄ βͺ π΅)= 285 + 195 β 410 = 480 β 410 = 70 π(π΄βππ΅β) = π π΄ β© π΅ β β π π β π π΄ β© π΅ = 500 β 70 β π(π΄β βͺ π΅β) = 430
29.
A, B kw;Wk; C VNjDk; %d;W fzq;fs; vd;f. NkYk;> π π¨ = ππ, π π© = ππ, π πͺ = ππ, π π¨ β© π© = π, π(π© β© πͺ) = π, π(π¨ β© πͺ) = π kw;Wk; π π¨ β© π© β© πͺ = π vdpy;> π(π¨ βͺ π© βͺ πͺ) β If; fhz;f. π (π΄ βͺ π΅ βͺ πΆ) = π(π΄) + π(π΅) + π(πΆ) β π(π΄ β© π΅) β π(π΅ β© πΆ) β π(π΄ β© πΆ) + π(π΄ β© π΅ β© πΆ)
ZY
ST UD Y
26.
= π (π΄ βͺ π΅ βͺ πΆ) =
fPNo nfhLf;fg;gl;Ls;s ml;ltiz MdJ> π¨ = {π, π, π, ππ} apypUe;J π© = {ππ, ππ, π, ππ} f;F π(π) = ππ β π vd;wthW mike;j xU rhh;G vdpy;> π kw;Wk; π Mfpadtw;wpd; kjpg;Gfisf; fhz;f. x 5 6 8 10 f(x) a 11 b 19
DO O
30.
17 + 17 + 17 β 7 β 6 β 5 + 2 = 53 β 18 35
π(π₯) = 2 π₯ β 1
31.
π(5) = 2(5) β 1 = 10 β 1 = 9 π(8) = 2(8) β 1 = 16 β 1 = 15 / π = 9 , π = 15 (π¨ βͺ π©)β f;F ntd;glk; tiuf.
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(Oct-2013)
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rhh;G π: (βπ, π) β πΉ fPo;f;fz;lthW tiuaWf;fg;gl;Ls;sJ. πππ β π ; βπ β€ π < 2 π₯ = {β2, β1,0,1} π π = ππ β π ; π β€ π β€ π vdpy; ππ β π ; π<π₯ <7 π 5 = 2 5 β 3 = 10 β 3 = 7 π 6 = 2 6 β 3 = 12 β 3 = 9 π 5 + π 6 = 7 + 9 = 16
33.
(Jun -2014) π π + π π d; kjpg;igf; fhz;f.
[4 < π₯ < 7 vdpy,; π π₯ = 2π₯ β 3]
ST UD Y
32.
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π¨ = π, π, π, π, π , π© = {π, π, π, π, βππ} vd nfhLf;fg;gl;Ls;s fzq;fSf;F> fzq;fspd; Nrh;g;G nrayhdJ> ghpkhw;W gz;G cilaJ vd;gij rhpghh;f;fTk;. (Apr-2016) π΄ βͺ π΅ = π΅ βͺ π΄ vd;gij rhpghh;g;Nghk;; π΄ βͺ π΅ = π, π₯, π¦, π, π βͺ 1, 3, 5, 7, β10 = {π, π₯, π¦, π, π , 1,3,5,7, β10}β¦β¦β¦β¦ 1 π΅ βͺ π΄ = 1, 3, 5, 7, β10 βͺ π, π₯, π¦, π, π = {π, π₯, π¦, π, π , 1,3,5,7, β10}β¦β¦β¦β¦. 2 (1) kw;Wk; (2) ypUe;J π΄ βͺ π΅ = π΅ βͺ π΄ vdNt> fzq;fspd; Nrh;g;G nrayhdJ> ghpkhw;W gz;G cilaJ
2. nka;naz;fspd; njhlh;thpirfSk; njhlh;fSk; 1.
π, π π, π π, β¦β¦ vd;w $l;Lj; njhlh; thpirapd; 12 MtJ cWg;igf; fhz;f. (Apr-2016) π = 2,
π= 3 2 β
2 = 2 2,
π‘π = π + (π β 1)π π‘12 = 23 2 2.
2 + 22 2 = 23 2
ZY
π‘12 = 2 + (12 β 1) Γ 2 2 =
π = 12
4, 9, 14, ..β¦β¦ vd;w $l;Lj; njhlh; thpirapd; 17 MtJ cWg;igf; fhz;f. π = 4, π = 9 β 4 = 5, π = 17 π‘π = π + (π β 1)π
(Apr-2014)
3.
DO O
π‘17 = 4 + (17 β 1) Γ 5 = 4 + 16 Γ 5 = 4 + 80 = 84 π‘17 = 84
13 My; tFgLk;
π =
91β13 13
πβπ
+1 =
π 78 13
(Apr-2012)
+1
+1 = 6+1 =7
π = 7
4.
5+11+17+β¦β¦.. +95 vd;w $l;Lj; njhlhpd; $Ljy; fhz;f. π = 5, π = 11 β 5 = 6, π = 95 π =
πβπ π
(Apr-2013, Jun-2014)
+1
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π = ππ = π16
+1 = +π =
90 6 16 x 2
+ 1 = 15 + 1 = 16 [5 + 95]
= 8 Γ 100 = 800
ππ + ππ + ππ + β¦β¦.. +πππ $Ljy; fhz;f. 13 + 23 + 33 + β¦β¦.. +π3 =
π(π+1) 2 2 20(20+1) 2 2
13 + 23 + 33 + β¦β¦.. +203 = 6.
(Oct-2014)
ST UD Y
5.
95β5 6 π π 2
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=
20x21 2 2
= [210]2 = 44100.
π + π + π +β¦β¦. +π = πππ vdpy,; ππ + ππ + ππ +β¦β¦. ππ kjpg;G fhz;f. βπ = 120 khw;WKiw βπ3 = βπ
2
= 1202 = 14400
π (π+1)
1 + 2 + 3 + β¦β¦β¦ +π =
2
13 + 23 + 33 +β¦β¦β¦.. + π3 =
=
7.
ππ + ππ + ππ +β¦β¦ +ππ = πππππ vdpy; , π + π + π +β¦β¦.+π kjpg;G fhz;f. βπ3 = (βπ)2 = 36100 = 1902 khw;WKiw 13 + 23 + 33 + β¦β¦. +π3 =
βπ = 190 1 + 2 + 3 + β¦β¦. +π = 190
βπ3 = (βπ)2 = 1712 = 29241
π(π+1) 2
13 + 23 + 33 + β¦β¦ +π3 =
ππ + ππ + ππ + β¦ + ππ = ππππ, vdpy; , 1+π + π + β¦ . . +π βπ 3 = (βπ)2 = 8281 = 912 khw;WKiw βπ = 91
1 + 2 + 3 + β¦β¦β¦ + π = 91
10.
1 + 2 + 3 + β¦.. + π =
DO O
9.
π π+1 2 2 1202 =
14400
(Jun-2013)
π(π +1) 2 = 36100 2 π (π+1) = 190 2
= 1902
π + π + π + β¦β¦. + π· = πππ, vdpy,; ππ + ππ + ππ +β¦β¦.+π·π kjpg;G fhz;f. βπ = 171 khw;WKiw
ZY
8.
1 + 2 + 3 + β¦.. +π =
= 120
= 171 π(π+1) 2 2
kjpg;G fhz;f. π(π+1) 2 = 2 π(π+1) = 91 2
13 + 23 + 33 + β¦β¦. + π 3 = 1 + 2 + 3 +β¦.. + π =
= 1712 = 29241
8281 = 912
xU $l;Lj; njhlh; thpirapy;> Ie;jhk; cWg;gpd; 5 klq;Fk;> Vohk; cWg;gpd; 7 klq;Fk; rkkhf ,Ug;gpd; 12 tJ cWg;G 0 vd epUgpf;f. (Oct-2012) π‘π = π + (π β 1) π 7π‘7 = 5π‘5 7(π + 6π) = 5(π + 4π) 7π + 42π = 5π + 20π
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7π β 5π + 42π β 20π = 0 2π + 22π = 0 π + 11π = 0 π‘12 = 0 xU $l;Lj; njhlhpy; Sn = ππππ kw;Wk; Kjy; cWg;G π = π, nghJ tpj;jpahrk; π
= π vdpy; , π-d; kjpg;gpidf; fhz;f. (Jun-2012) ππ = ππ =
π 2
π 2
[2π + π β 1 π]
2 3 + πβ1 4
= 1275
π 6 + 4π β 4 = 1275 Γ 2 6π + 4π2 β 4π = 2550 4π2 + 2π β 2550 = 0 (2My; tFf;f) 2π2 + π β 1275 = 0 (2π + 51) (π β 25) = 0 π = 25
2π2 + π β 1275 = 0
xU $l;Lj; njhlh;thpirapd; Kjy; cWg;G 6 kw;Wk; nghJtpj;jpahrk; mj;njhlh;thpiriaAk; mjd; nghJ cWg;igAk; fhz;f. $l;Lj; njhlHthpirapd; nghJ tbtk; π, π + π, π + 2π, π + 3π, β¦β¦β¦. ,q;F π = 6, π = 5 vdNt $l;Lj;njhlh;thpir = 6, (6 + 5), 6 + 2(5), 6 + 3(5), β¦β¦. = 6, 11, 16, 21, β¦β¦β¦. $l;Lj;njhlh;thpirapd; nghJtbtk; π‘π = π + (π β 1)π
DO O
13.
ZY
%d;W vz;fspd; tpfpjk; 2:5:7 vd;f. Kjyhk; vz;> ,uz;lhk; vz;zpypUe;J 7-If; fopj;Jg; ngwg;gLk; vz; kw;Wk; %d;whk; vz; Mfpad xU $l;Lj; njhlh;thpiria Vw;gLj;jpdhy;> mt;ntz;fisf; fhz;f. nfhLf;fg;gl;l %d;W vz;fspd; tpfpjk; 2:5:7 me;j vz;fs; KiwNa 2π₯, 5π₯ kw;Wk; 7π₯ vd;f. (π₯ β 0) fzf;fpd;gb 2π₯, 5π₯ β 7, 7π₯ vd;gd xU $l;Lj;njhlh;thpir MFk;. vdNt nghJ tpj;jpahrk; π = (5π₯ β 7) β 2π₯ = 7π₯ β (5π₯ β 7) Fwpg;G:[π = π‘2 β π‘1 ] 3π₯ β 7 = 2π₯ + 7 π₯ = 7 + 7 = 14 vdNt me;j vz;fs; 2π₯, 5π₯, 7π₯ = ( 2 Γ 14), 5 Γ 14 , 7 Γ 14 = 28, 70, 98
12.
14.
ST UD Y
11.
5
vdpy;>
= 6 + (π β 1)(5) = 6 + 5π β 5 = 5π + 1 π‘π = 5π + 1
gpd;tUk; $l;Lj; njhlh;thpirapd; Kjy; cWg;G kw;Wk; nghJ tpj;jpahrj;ijf; fhz;f. π π π π ππ , , , ,β¦β¦β¦, . π π π π π
Kjy; cWg;G
nghJ tpj;jpahrk;
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π
(Apr-2015)
1 = 2 5
1
5β3
6
2
6
π= β =
2
1
6
3
= =
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xU fbfhuk; xU kzpf;F xU Kiw> 2 kzpf;F ,U Kiw> 3 kzpf;F %d;W Kiw vd;wthW> njhlh;e;J rhpahf xt;nthU kzpf;Fk; xyp vOg;Gk; vdpy;> xU ehspy; mf;fbfhuk; vj;jid Kiw xyp vOg;Gk;? xU fbfhuk; xt;nthU kzpf;Fk; vOg;Gk; xyp xU $l;Lj; njhlh; mf;$l;Lj;njhlh; 1 + 2 + 3 + β¦β¦. +12 π 2
ππ = [π + π]
vdNt
π12 =
12 2
[1 + 12] = 6 Γ 13 = 78
ST UD Y
15.
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xU ehspy; fbfhuk; vOg;Gk; xyp mstpd; vz;zpf;if = 2 Γ 78 = 156
xUth;> Kjy; khjk; &.640. 2 Mk; khjk; &.720. 3 Mk; khjk; &.800-I Nrkpf;fpwhh;. mth; jd;Dila Nrkpg;ig ,Nj njhlh;thpirapy; njhlh;e;jhy;> 25 MtJ khjk; mth; Nrkpf;Fk; njhifiaf; fhz;f. Nrkpf;fg;gLk; njhif xU $l;Lj; njhlh; thpir MFk;. 640, 720, 800, β¦β¦β¦. π‘π = π + (π β 1)π ,f;$l;Lj;njhlh; thpirapd; Kjy; cWg;G π = 640, nghJ tpj;jpahrk; π = 720 β 640 = 80 25 MtJ khjk; Nrkpf;fg;gLk; njhif π‘25 = 640 + (25 β 1) Γ 80 = 640 + ( 24 Γ 80 ) = 640 + 1920 = 2560 25 MtJ khjk; Nrkpf;fg;gLk; njhif & 2560
17.
1, 2, 4, 8, β¦β¦β¦ vd;w ngUf;Fj;njhlh;thpirapy; 1024 vd;gJ vj;jidahtJ cWg;ghFk;? 1, 2, 4, 8, β¦β¦β¦ vd;w ngUf;Fj;njhlhpy; π‘ 2 π =1 π= 2= =2
ZY
16.
π‘1
1
18.
19.
DO O
π‘π = 1024 π β1 ππ = 1024 1 Γ 2π β1 = 210 2π β1 = 210 π β 1 = 10 π = 10 + 1 = 11 / 1, 2, 4, 8, ..β¦β¦β¦. vd;w ngUf;Fj; njhlh; thpirapy; 1024 vd;gJ 11 cWg;ghFk;. 125, 120, 115, 110, β¦β¦.. vd;w $l;Lj;njhlh; thpirapd; nghJ tpj;jpahrj;ijAk; 15 MtJ cWg;igAk; fhz;f. 125, 120, 115, 110, β¦β¦ vd;w $l;Lj;njhlh; thpirapd; Kjy; cWg;G π = 125 nghJ tpj;jpahrk; π = 120 β 125 = β 5 π‘π = π + (π β 1) π 15 MtJ cWg;G π‘15 = 125 + (15 β 1) (β5) = 125 + 14 (β5) = 125 β 70 = 55 ππ β ππ + ππ β ππ +β¦ vd;w njhlhpd; Kjy; 10 cWg;Gfspd; $l;lw;gyidf; fhz;f. (Jun-15)
12 β 22 + 32 β 42 +β¦ Kjy; 10 cWg;Gfs; = (1 β 4) + (9 β 16) + β¦ 5 cWg;Gfs;
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= (β3) + (β7) + β¦ 5 cWg;Gfs; = (β3) + (β7) + β¦ 5 cWg;Gfs; π = β3, π = (β7) β (β3) = β7 + 3 = β4 ππ =
=
5 [2 β3 2 5 β6 2
+ π β 1 π]
+ 5 β 1 β4 ] + β16
=
5 2
ST UD Y
π5 =
π [2π 2
β22 = 5 Γ β11 = β55
ππ β ππ + πππ β πππ β¦ vd;w ngUf;Fj;njhlhpy; cs;s Kjy; 25 cWg;Gfspd; $Ljiyf; fhz;f.
20.
β48 = β3 β 1 16 π π(1βπ ) ,π β 1 1β π 16[1β(β3)25 ] 16[1+325 ] = 1β(β3) 4 25 4 (1 + 3 )
,q;F π = 16, π = vdNt>
ππ = π25 = =
π (π + π), π β β kw;Wk; π ,ul;ilg;gil vz; vDk; NghJ 21.
an = ππ ,π ππ + π
β β kw;Wk; π xw;iwg;gil vz; vDk; NghJ vd tiuaWf;fg;gl;l njhlh;thpirapd; 18-tJ kw;Wk; 25tJ cWg;Gfisf; fhz;f. π ,ul;ilg;gil vz; vDk; NghJ vdNt 18-tJ cWg;G
ππ = π (π + 3) π18 = 18 (18 + 3) = 18 Γ 21 = 378
π xw;iwg;gil vz; vDk;NghJ
ZY
ππ =
vdNt 25-tJ cWg;G
23.
=
50 625 + 1
=
50 626
=
25 313
njhlh; thpirapYs;s rhh;Gf;F xU vLj;Jf;fhl;L jUf. (Oct-2013) rhh;G π: π
β π
MdJ π π₯ = 2π₯ + 1, βπ₯ β π
vd mikAkhdhy;> mJ xU njhlh;thpirahfhJ. Vnddpy;> ,r;rhh;gpd; tPr;rfj;jpd; cWg;Gfisj; njhlh; thpirapy; mikf;f ,ayhJ. NkYk;> ,r;rhh;gpd; kjpg;gfk; MdJ ,ay; vz;fspd; fzk; N MfNth my;yJ ,ay; vz;fspd; cl;fzk; {1,2,β¦.n} MfNth mikag; ngwtpy;iy.
DO O
22.
π25 =
2π π 2 +1 2 Γ 25 25 2 + 1
π tJ cWg;G ππ =
π(πβπ) π
vdf; nfhLf;fg;gl;l njhlh;thpirapd; Kjy; %d;W cWg;Gfisf;
fhz;f.
ππ =
3 1(1β2) β1 π1 = = , 3 3 π πβ2 ππ = nfhLf;fg;gl;l 3
24.
(Oct-2015)
π(πβ2)
π2 =
2(2β2) 3
=
0 3
= 0,
π3 =
njhlh;thpirapd; Kjy; %d;W cWg;Gfs;
3(3β2) 3
=
3 3
=1
β1 , 0, 1. 3
50 kw;Wk; 200 ,tw;wp;w;fpilNaahd 10 My; tFgLk; midj;J KO vz;fspd; $Ljiyf; fhz;f. 50 kw;Wk; 200 ,tw;wp;fpilNaahd 10 My; tFgLk; midj;J KO vz;fspd; $Ljy; 60 + 70 + β¦β¦. + 190
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π = 60, π = 10, π = 190 π= = =
πβπ + 1 π 190β60 + 10 130 +1 10
1
π = 14 ππ = =
ST UD Y
= 13 + 1 π [π + π] 2 14 [60 + 190] 2
= 7 Γ 250 π14 = 1750
50 kw;Wk; 200 ,tw;wp;w;fpilNaahd 10 My; tFgLk; midj;J KO vz;fspd; $Ljy; 60 + 70 + β¦β¦. +190 = 1750
3. ,aw;fzpjk; 1.
jPh;f;f: ππ + ππ = ππ, ππ + ππ = ππ π₯ π¦ 5 -25 3 6 -30 7
= 5
jPh;T {0,5}
ePf;fy; Kiwapy; jPh;f;f. ππ + π = π, ππ + π = ππ 3π₯ + π¦ = 8 β¦β¦β¦.. 1 5π₯ + π¦ = 10 β¦β¦.... 2
DO O
2.
5 6
ZY
π₯ π¦ 1 = = β150+150 β175+90 18β35 π₯ π¦ 1 = = 0 β85 β17 0 β85 π₯ = = 0 kw;Wk; π¦ = 17 β17
1
(Jun-2012)
β
β
(2) β (1) β2π₯
(β)
=β2
π₯ = 1
π₯ = 1 vd (1)-y; gpujpapl> 3(1) + π¦ = 8 3 + π¦ = 8 π¦ = 8 β 3 = 5 π¦ = 5
3.
π =
π π
jPh;T {1, 5}
kw;Wk; π = βπ vd;w G+r;rpaq;fisf; nfhz;l ,Ugb gy;YWg;Gf;Nfhitiaf; fhz;f.
,Ugb gy;YWg;Gf;Nfhit =
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π₯2
(Jun-2013) β (πΌ + π½) π₯ + πΌπ½
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,q;F πΌ =
1 4
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, π½ = β1
vdNt> ,Ugb gy;YWg;Gf;Nfhit = π₯ 2 β = π₯2 β ,Ugb gy;YWg;Gf;Nfhit
3 4
π₯ β
π₯ β
1 4
1 4
Γ β1)
1 4
π, π Mfpatw;iw KiwNa G+r;rpaq;fspd; $LjyhfTk; kw;Wk; mtw;wpd; nfhz;l gy;YWg;Gf; Nfhitiaf; fhz;f. ,Ugb gy;YWg;Gf; Nfhitapd; G+r;rpaq;fspd; $Ljy; = πΌ + π½ = 3 mtw;wpd; ngUf;fw;gyd; = πΌπ½ = 2 vdNt> gy;YWg;Gf;Nfhit = π π₯ = π₯ 2 β πΌ + π½ π₯ + πΌπ½ gy;YWg;Gf;Nfhit= π₯ 2 β
5.
= π₯2 +
β1 π₯+ (
ngUf;fw;gydhfTk;
ST UD Y
4.
1 4 1β4 4
3π₯ + 2
xU ,Ugb gy;YWg;Gf;Nfhitapd; G+r;rpaq;fspd; $Ljy; -4 kw;Wk; mjd; ngUf;fw;gyd; 3 vdpy;> mf;Nfhitiaf; fhz;f. ,Ugb gy;YWg;Gf; Nfhitapd; G+r;rpaq;fspd; $Ljy; = πΌ + π½ = β4 mtw;wpd; ngUf;fw;gyd; = πΌπ½ = 3 vdNt> gy;YWg;Gf;Nfhit = π (π₯) = π₯ 2 β (πΌ + π½) π₯ + πΌπ½ = π₯ 2 β (β4) π₯ + 3 gy;YWg;Gf;Nfhit
= π₯2 + 4 π₯ + 3
kP.ngh.t. fhz;f. πππππ π
π , ππππ ππ , ππππ π
. 25ππ 4 π 3 = 5 Γ 5 Γ ππ 4 π 3 35π 2 π 3 = 7 Γ 5 Γ π 2 π 3 45π 3 π = 9 Γ 5 Γ π 3 π kP.ngh.t = 5π 3
7.
ππ β ππ β ππ, ππ + ππ + π Mfpa Nfhitfspd; kP.ngh.t. fhz;f. (Apr-2014) π2 β 3π β 18 = (π β 6) (π + 3) π2 + 5π + 6 = (π + 2) (π + 3) kP.ngh.t = (π + 3)
DO O
8.
ZY
6.
kP.ngh.k. fhz;f. π π β π , π π β π π , (ππ β π). (Jun-2015) 3 π β 1 = 3(π β 1) 2(π β 1)2 = 2(π β 1) (π β 1) π2 β 1 = (π β 1) (π + 1) kP.ngh.k. = 3 Γ 2 (π β 1) (π β 1) (π + 1) = 6(π β 1)2 (π + 1) kP.ngh.k. = 6(π β 1)2 (π + 1)
9.
kP.ngh.k. fhz;f. ππ π + πππ , ππ + ππ π₯ 2 π¦ + π₯π¦ 2 = π₯π¦ (π₯ + π¦) 2 π₯ + π₯π¦ = π₯ (π₯ + π¦) kP.ngh.k. = (π₯ + π¦)
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ππ βπ ππ βπ π± π π π π βπ π +ππ π₯ 2 β4 π 3 βπ π₯ 2 β22 Γ = π 2 β1 π₯ 3 +2π₯ 2 π 2 β1
RUf;Ff.
π₯+2 (π₯β2) π 2 β1
= π₯ 2 β4 π 2 β1
Γ
ππ+ππ ππ+ππ 5π₯+20 5(π₯+4) = 7π₯+28 7 π₯+4
π 3 βπ π₯ 3 +2π₯ 2
π(π 2 β1) π₯ 2 (π₯+2)
Γ
=
π(π₯β2) π₯2
=
RUf;Ff.
RUf;Ff.
12.
π₯ 3 β27 π₯ 2 β9
=
β
13.
π₯ 2 β4
=
Γ
=
Γ·
π₯ π₯+1 π₯2 π₯ 2 β1
π₯ 2 β22
π₯β9 π₯+9 π₯β2 π₯+2
π₯ 2 +6π₯+8 π₯ 2 β5π₯β36
Γ·
π₯2
π₯ π₯+1
ππ βπ ππ +π
Γ·
π₯2 π₯ 2 β1
π₯+2 (π₯+4)
Γ
=
π₯β4 (π₯+9)
Γ
π₯ 2 β1 π₯ π₯2 = Γ· π₯+1 π₯+1 (π₯β1)
=
π₯+2 (π₯+4) π₯+4 (π₯β9)
π₯+9 π₯β2
DO O
π₯ π₯+1
Γ
5 7
(π₯ 2 +3π₯+9) (π₯+3)
π₯ 2 β5π₯β36
π₯ 2 β81 π₯ 2 β4
=
π₯β3 (π₯ 2 +3π₯+9) π₯+3 π₯β3
=
RUf;Ff.
5π₯+20 7π₯+28
β
ππ βππ ππ +ππ+π π± π ππ βπ π βππβππ π₯ 2 +6π₯+8 π₯ 2 β92
RUf;Ff. π₯ 2 β81
(Apr-2015)
ππ βππ ππ βπ π₯ 3 β33 = π₯ 2 β32
π₯ 3 β27 π₯ 2 β9
π(π₯β2) π₯2
ZY
11.
15.
π(π 2 β1) π₯ 2 (π₯+2)
Γ
ST UD Y
10.
14.
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=
π₯ π₯+1
Γ
π₯+1 (π₯β1) π₯2
=
π₯β1 π₯
cld; ve;j tpfpjKWf; Nfhitiaf; $l;bdhy;
Njitahd tpfpjKWf; Nfhit π π₯
π₯β1 π₯
πππ βππ +π ππ +π
fpilf;Fk;?
π(π₯) vd;f.
2π₯ 3 βπ₯ 2 +3 π₯ 3 β1 = β 2 π₯ 2 +2 π₯ +2 2π₯ 2 βπ₯ 2 +3βπ₯ 3 +1 = π₯ 2 +2
=
π₯ 3 βπ₯ 2 +4 π₯ 2 +2
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njhFKiw tFj;jiyg; gad;gLj;jp>
1 3
[3π₯ 2 + 6π₯ β 6] = π₯ 2 + 2π₯ β 2 , kPjp = 2
th;f;f%yk; fhz;f. (π + ππ)π β πππ
17.
(π₯ + 11)2 β 44π₯ =
π₯ 2 + 22π₯ + 121 β 44π₯ = π₯ 2 β 22π₯ + 121 =
th;f;f%yk; fhz;f. ππ +
18.
1
π₯4 +
π₯4
+ 2 = =
π ππ
+ π.
π₯2 + π₯2 +
1
2
π₯2
1 π₯2
th;f;f%yk; fhz;f. πππππ ππ Γ· ππ ππ ππ
19.
121π₯ 8 π¦ 6 Γ· 81 π₯ 4 π¦ 8 = jPh;f;f: ππ β jPh;T:
20.
3π₯ β
8 π₯
π π
=2
(Oct-2014)
121π₯ 8 π¦ 6 81π₯ 4 π¦ 8
(Oct-2013)
=π βΉ
ST UD Y
16.
3π₯ 2 β8 = π₯ 2 3π₯ β 8
2
112 π₯ 4 92 π¦ 2
=
=
11π₯ 2 9π¦
=
11 9
(π₯ β 11)2 = (π₯ β 11
π₯2 π¦
ZY
βΉ = 2π₯ 2 βΉ 3π₯ β 2π₯ β 8 = 0 βΉ (π₯ β 2) (3π₯ + 4) = 0 βΉ π₯ = 2 my;yJ π₯ = β
4 3
jPh;T =
4 3
β ,2
22.
DO O
,U njhlh;e;j kpif ,ul;ilg; gil vz;fspd; ngUf;fypd; kjpg;G 24 vdpy; mt;ntz;fisf; fhz;f. (Oct-2012) ,U njhlh;e;j kpif ,ul;ilg; gil vz;fs; π₯ kw;Wk; π₯ + 2 vd;f. fzf;fpd;gb π₯ (π₯ + 2) = 24 π₯ 2 + 2π₯ β 24 = 0 (π₯ + 6) (π₯ β 4) = 0 π₯ = β 6 my;yJ 4 π₯ kw;Wk; π₯ + 2 vd;gJ kpif vz;fs;. vdNt mt;tpU vz;fs; 4 kw;Wk; 6 MFk;.
21.
π ππ = π π 1 26 π₯ 2 +1 = βΉ π₯ 5 π₯ βΉ 5π₯ 2
jPh;f;f: π + π₯ +
=
(Apr-2013, Jun-2013)
26 5
+ 5 = 26π₯ βΉ 5π₯ β 26π₯ + 5 = 0 2
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βΉ (π₯ β 5) (5π₯ β 1) = 0 βΉ π₯ = 5 my;yJ π₯ = jPh;T = 5,
1 5
1 5
ππ β πππ β ππ = π vd;w rkd;ghl;bd; %yq;fspd; jd;ikia Muha;f. ππ₯ 2 + ππ₯ + π = 0 tbtk; π = 1, π = β11, π = β10 π 2 β 4ππ = (β11)2 β 4 Γ 1 (β10) = 121 + 40 = 161 > 0 / %yq;fs; nka;;> rkky;y.
24.
πππ + ππ + π = π vd;w rkd;ghl;bd; %yq;fspd; jd;ikia Muha;f. ππ₯ 2 + ππ₯ + π = 0 tbtk; π = 2, π = β5, π = 5 π 2 β 4ππ = (β5)2 β 4 Γ 2 Γ 5 = 25 β 40 = β15 < 0 / %yq;fs; fw;gid ππ β ππ + ππ = π vd;w rkd;ghl;bd; %yq;fspd; jd;ikia Muha;f. ππ₯ 2 + ππ₯ + π = 0 tbtk; π = 1, π = β8, π = 12 2 2 π β 4ππ = (β8) β 4 Γ 1 Γ 12 = 64 β 48 = 16 > 0 / %yq;fs; nka;> rkky;y.
25.
ST UD Y
23.
πππ β ππ + π = π vd;w rkd;ghl;bd; %yq;fspd; jd;ikia Muha;f. ππ₯ 2 + ππ₯ + π = 0 tbtk; π = 2, π = β3 , π = 4 π 2 β 4ππ = (β3)2 β 4 Γ 2 Γ 4 = 9 β 32 = β23 < 0 / %yq;fs; fw;gid
27.
πππ β πππ + ππ = π vd;w rkd;ghl;bd; %yq;fspd; jd;ikia Muha;f. ππ₯ 2 + ππ₯ + π = 0 tbtk; π = 4, π = β28, π = 49 π 2 β 4ππ = (β28)2 β 4 Γ 4 Γ 49 = 784 β 784 = 0 / %yq;fs; nka;;> rkk;
DO O
28.
ZY
26.
(π β ππ) (π β ππ) = πππ vd;w ,Ugbr; rkd;ghl;bd; %yq;fspd; jd;ikiaf; fhz;f.
(Jun-14) 2
2
(π₯ β 2π) (π₯ β 2π) = 4ππ βΉ π₯ β π₯ (2π + 2π) + 4ππ = 4ππ βΉ π₯ β 2π₯ (π + π) = 0 ππ₯ 2 + ππ₯ + π = 0 tbtk;
π 2 β 4ππ
= [β2(π + π)]2 β 4 Γ 1 Γ 0 = 4(π + π)2 > 0 / %yq;fs; nka;;> rkky;;y.
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π = 1, π = β2 π + π , π = 0
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nfhLf;fg;gl;Ls;s rkd;ghl;bd; %yq;fs; nka;naz;fs; kw;Wk; rkkhdit vdpy; π d; kjpg;igf; fhz;f. ππππ + πππ + π = π. (Jun-2015) 12π₯ 2 + 4ππ₯ + 3 = 0 vd;w rkd;ghl;bd; %yq;fs; nka; kw;Wk; rkk;. vdNt β= π 2 β 4ππ = 0 βΉ (4π)2 β 4 Γ 12 Γ 3 = 0 16π 2 β 144 = 0 16π 2 = 144 π2 = 9 k =Β±3
30.
π + π, π β π vd;gij %yq;fshff; nfhz;l ,Ugbr; rkd;ghl;ilf; fhz;f. (Apr-14, Oct-14) %yq;fspd; $Ljy; = 7 + 3 + 7 β 3 = 14 %yq;fspd; ngUf;fw;gyd; = (7 + 3 ) Γ (7 β 3 ) = 72 β ( 3)2 = 49 β 3 = 46 rkd;ghL: π₯ 2 - (%yq;fspd; $Ljy;) π₯ + %yq;fspd; ngUf;fw;gyd; = 0
ST UD Y
29.
π₯ 2 β 14π₯ + 46 = 0 31.
π + π , π β π vd;gij %yq;fshff; nfhz;l ,Ugbr; rkd;ghl;ilf; fhz;f. %yq;fspd; $Ljy; =3+ 7+3β 7=6 %yq;fspd; ngUf;fw;gyd; = (3+ 7 )Γ (3 β 7 ) = 32 β ( 7)2 = 9 β 7 = 2 rkd;ghL: π₯ 2 - (%yq;fspd; $Ljy;) π₯ + %yq;fspd; ngUf;fw;gyd; = 0 π₯ 2 β 6π₯ + 2 = 0
3, 4 vd;gij %yq;fshff; nfhz;l ,Ugbr;rkd;ghl;ilf; fhz;f. (Jun-2012, Oct-2012) Njitahd rkd;ghL π₯ 2 - (%yq;fspd; $Ljy;) π₯ + %yq;fspd; ngUf;fw;gyd; = 0 π₯ 2 β (3 + 4) π₯ + (3 Γ 4) = 0 π₯ 2 β 7π₯ + 12 = 0
33.
πππ β ππ + π = π vd;w ,Ugbr; rkd;ghl;bd; %yq;fspd; $Ljy; 10kw;Wk; ngUf;fw;gyd; 10 vdpy; , π kw;Wk; π Mfpatw;wpd; kjpg;Gfisf; fhz;f. (Apr-2013)
ZY
32.
ππ₯ 2 β 5π₯ + π = 0 vd;w ,Ugbr; rkd;ghl;bd; %yq;fspd; $Ljy; = 5 1 π= = 10 2
(βπ) π
=
DO O
1 2 ππ₯ 2 β 5π₯ + π = 0 vd;w ,Ugbr; rkd;ghl;bd; %yq;fspd; ngUf;fw;gyd; = π = 10π = 10 Γ
34.
1 2
5 π
= 10
π=
= 5
πΆ π
= 10
π = 5
πΆ, π· vd;gd πππ β ππ + π = π, vd;Dk; rkd;ghl;bd; %yq;fs; vdpy; πΆπ + π·π d; kjpg;G fhz;f.
(Apr-2012, Apr-2016)
3π₯ 2 β 6π₯ + 4 = 0 vd;Dk; rkd;ghl;bd; %yq;fs; πΌ, π½ βπ 6 πΌ+π½ = = = 2 πΌ 2 + π½ 2 = |(πΌ + π½)2 β 2πΌπ½| πΌπ½ =
π π
=
π 4 3
3
= 4β
8 3
=
πΌ 2 + π½2 =
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4 3 12β8 3
= (2)2 β 2 Γ
=
4 3
4 3
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πππ β ππ + π = π vd;w rkd;ghl;bd; %yq;fs; πΆ , π· vdpy; 3π₯ 2 β 5π₯ + 2 = 0, vd;w rkd;ghl;by; (βπ) β(β5) vdNt πΌ + π½ = = π 3 π 2 πΌπ½ = = π 3 πΌ π½ πΌ 2 +π½ 2 (πΌ+π½ )2 β2πΌπ½ + = = π½ πΌ πΌπ½ πΌπ½
= 36.
jPh;.
37 .
RUf;Ff.
5 2 2 β2 3 3 2 3
=
=
πΆ π· + -d; π· πΆ
π = 3, π = β5, π = 2
25 9
4
β 3 2 3
=
25β12 9
+
8 2βπ₯
=
π πβπ
3 2
=
13 6
π₯3 π₯β2
β
23 π₯β2
(Oct-2015)
(Apr-2016)
π₯ 3 β23 π₯β2
(π₯β2)(π₯ 2 +2π₯+4) π₯β2
ZY
π₯3 π₯β2
+
(Apr-15)
5 3
ππ β ππ = βππ, ππ + ππ = ππ 3π₯ β 5π¦ = β16 β¦β¦β¦β¦β¦β¦β¦β¦.. 1 2π₯ + 5π¦ = 31 β¦β¦β¦β¦β¦β¦β¦β¦β¦ 2 (1) + (2) 5π₯ = 15 π₯=3 π₯ = 3 vd (1) y; gpujpapl βΉ 3π₯ β 5π¦ = β16 3 3 β 5π¦ = β16 β5π¦ = β16 β 9 β5π¦ = β25 π¦ =5 / jPh;T = {3, 5} ππ πβπ
kjpg;G fhz;f.
ST UD Y
35.
=
=
= π₯ 2 + 2π₯ + 4
4. mzpfs;
2.
π π π βπ vdpy; (π¨π» )π» = π¨ vd;gij rhpghh;. βπ π 1 2 3 π΄ = 2 4 β5 3 β5 6 1 2 3 π π΄ = 2 4 β5 3 β5 6 1 2 3 (π΄π )π = 2 4 β5 = π΄ vd;gJ rhpghh;f;fg;gl;lJ. 3 β5 6
DO O
1.
π π = π π
π¨ =
π π
π΄ =
π βπ
8 1
π vdpy; π¨π» , (π¨π» )π» Mfpatw;iwf; fhz;f. π 5 2 β3 4
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(Jun-12, Apr-13, Jun-14, Apr-15)
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π΄π = (π΄π )π
6.
7.
π βπ π π π¨ = π βπ π π vdpy; (i) mzpapd; thpiriaf; fhz;f. π π π π cWg;Gfisf; fhz;f. (iii) cWg;G 7 mike;Js;s epiu> epuiyf; fhz;f. i) mzpapd; thpir 3 Γ 4 ii) π24 = 4, π32 = 0 iii) cWg;G 7 mike;Js;s epiy π23
(ii) πππ kw;Wk; πππ Mfpa
π π π π¨ = [πππ ] = π π π vdpy; (i) mzpapd; thpiriaf; fhz;f. (ii) πππ kw;Wk; πππ Mfpa π π π π βπ βπ cWg;Gfisf; fhz;f. (iii) 2 vd;w cWg;G mike;Js;s epiy Mfpatw;iwf; fhz;f. i) mzpapd; thpir 4 Γ 3 ii) π13 = 8, π42 = β2. iii) 2 vd;w cWg;gpd; epiy π22 π βπ π π , π = vdpy;> ππ¨ β ππ© vd;w mzpiaf; fhz;f. π βπ βπ βπ 4 β2 8 2 24 β12 β24 β6 6π΄ β 3π΅ = 6 β3 = + 5 β9 β1 β3 30 β54 3 9 0 β18 = 33 β45 A=
π¨ = πΆ
8.
π π π
ST UD Y
5.
π π vdpy;> A apd; epiu epuy; khw;W mzpiaf; fhz;f. π 2 4 5 A apd; epiu epuy; khw;W mzp = π΄π = 3 1 0
π¨ =
ZY
4.
2 4
π π , π© = π π = 2π΄ + π΅ 3 2 = 2 + 5 1 14 3 = 14 5
π π
βπ vdpy;> πͺ = ππ¨ + π© vd;w mzpiaf; fhz;f. π
DO O
3.
8 1 5 β3 2 4 8 5 = 1 β3
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π¨ =
π βπ
8 β1 6 = 4 3 10
4 8 + 2 4
(Apr-2012)
(Jun-2014)
β1 3
π π π β vdpy;> A d; $l;ly; Neh;khW mzpiaf; fhz;f. π π βπ
2 π΄ = β9
3 2 3 1 5 β1 β = + 5 β9 5 7 β1 β7 β1 2 A d; $l;ly; Neh;khW mzp = 16 β6
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(Oct-2012, 2013, Apr- 2015, Jun-2015)
β5 = 1
1 β16
β2 6
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π π π π A-d; thpir B-d; thpir A-d; thpir
π¨ =
10.
π¨=
11.
12.
π π βπ ,π = vdpy;> π¨ + π© ,Uf;Fkhdhy; mjidf; fhz;f. π π π = 2 Γ 3 = 2 Γ 2 β B-d; thpir. vdNt A+B fhz ,ayhJ.
π βπ π π βπ π π© = π π π π π π 3 β1 4 5 6 β2 3 π΄+π΅ = + 2 8 2 1 0 4 2 π π
π vdpy;> π¨ + π© βIf; fhz;f. π 7 8 5 2 10 = 3 3 8 6 5
ST UD Y
9.
https://doozystudy.blogspot.in/
π π π π π π = vdpy;> π, π kw;Wk; π Mfpatw;wpd; kjpg;Gfisf; fhz;f. π π π π π π π₯ = 3, π¦ = 9, π§ = 4 30 cWg;Gfs; nfhz;l mzpf;F vt;tif thpirfs; ,Uf;f ,aYk;?
(Jun-2013)
30 cWg;Gfs; nfhz;l mzpapd; thpir 1 Γ 30, 2 Γ 15, 3 Γ 10, 5 Γ 6, 6 Γ 5, 10 Γ 3, 15 Γ 2, kw;Wk; 30 Γ 1 vd ,Uf;f ,aYk;. 8 cWg;Gfs; nfhz;l mzpf;F vt;tif thpirfs; ,Uf;f ,aYk;? 8 cWg;Gfs; nfhz;l mzpapd; thpir 1 Γ 8, 2 Γ 4, 4 Γ 2 kw;Wk; 8 Γ 1 vd ,Uf;f ,aYk;
14.
π¨ = πππ = ππ vd;w π Γ π thpirf; nfhz;l mzpia mikf;f. π11 π12 π΄ = π = ππ 21 π22 π11 = 1 Γ 1 = 1, π12 = 1 Γ 2 = 2 , π21 = 2 Γ 1 = 2, π22 = 2 Γ 2 = 4 1 2 β΄π΄ = 2 4
15.
π¨ = [πππ ] = ππ β π vd;w π Γ π thpir nfhz;l mzpia mikf;f. π11 π12 π΄ = π = 2π β π 21 π22 π11 = 2 Γ 1 β 1 = 2 β 1 = 1, π12 = 2 Γ 1 β 2 = 2 β 2 = 0 π21 = 2 Γ 2 β 1 = 4 β 1 = 3, π22 = 2 Γ 2 β 2 = 4 β 2 = 2 1 0 β΄π΄ = 3 2
DO O
16.
ZY
13.
πβπ
π¨ = [πππ ] = vd;w π Γ π thpir nfhz;l mzpia mikf;f. π+π π11 π12 πβπ π΄ = π = π+π 21 π22 π11 =
π21 = π΄ =
1β1 1+1 2β1 2+1
0 2
=
=
0
β1 3
1 3
0
1 3
= 0,
,
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1β2 1+2 2β2 π22 = 2+2
π12 =
= =
β1 3 0 4
= 0
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18.
π ngUf;Ff. π 5 (2 β 1) = 10 β 4 = 6 4 (π β π)
π¨ = [πππ ] = ππ β ππ vd;w cWg;Gfs; nfhz;l π Γ π thpir cs;s mzp π¨ = [πππ ] apidf; fhz;f. π΄
=
π11 = π12 = π13 =
π11 π12 π13 π21 π22 π23 2 1 β 3(1) = 2 1 β 3(2) = 2(1) β 3(3) =
(Apr-2014) = 2π β 3π 2 β 3 = β1 = 1 2 β 6 = β4 = 4 2 β 9 = β7 = 7 β΄π΄
π
=
π21 = 2 2 β 3(1) = 4 β 3 = 1 = 1 π22 = 2 2 β 3(2) = 4 β 6 = β2 = 2 π23 = 2 2 β 3(3) = 4 β 9 = β5 = 5
1 4 1 2
7 5
π ππ βπ +π = vdpy;> π, π kjpg;Gfisf; fhz;f. π π π 2 10 β1 π +π = 3 1 5 2π 10 βπ + = 3π 5 π 2π β π 10 = 3π + π 5 2π β π = 10β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ 1 3π + π = 5β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦β¦β¦ 2 5π = 15 π = 15/5 π = 3 π = 3 vd (2) y; gpujpapl (3 Γ 3) + π = 5 9 + π = 5 π = 5 β 9 = β4 π = β4
20.
DO O
ZY
19.
ST UD Y
17.
https://doozystudy.blogspot.in/
π π¨ = βπ π π΄π΅
βπ π βπ π vdpy; π¨π© kw;Wk; π©π¨ Mfpa mzpfisf; fhz;f. π π© = π βπ βπ π 8 β7 9 β3 2 = β2 4 6 β1 β5 0 3 72 β 42 β24 + 7 16 + 35 = β18 + 24 6β4 β4 β 20 0 + 18 0β3 0 β 15 30 β17 51 = 6 2 β24 18 β3 β15
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8 β7 β2 4 0 3 72 + 6 + 0 β63 β 12 + 6 = 48 + 2 + 0 β42 β 4 β 15 78 β69 = 50 β61
24.
β3 2 β1 β5
π ,π© = π 3 2 π΄π΅ = 4 0 9+6 = 12 + 0 3 0 π΅π΄ = 3 2 / π΄π΅ β π΅π΄
π¨ =
π π
π π vdpy; π¨π©, π©π¨ fhz;f. mit rkkhf ,Uf;Fkh? π π 3 0 3 2 0+4 15 4 = 0+0 12 0 3 2 9+0 6+0 9 6 = = 9+8 6+0 4 0 17 6
π βπ π π ,π© = vd;w mzpfs; xd;Wf;nfhd;W ngUf;fy; Neh;khW mzp vd epWTf. βπ π π π π΄, π΅ vd;w mzpfs; xd;Wf;nfhd;W ngUf;fy; Neh;khW mzpfs; vdpy; gpd;tUk; tpjpia epiwT nra;a Ntz;Lk; π΄π΅ = π΅π΄ = πΌ 3 β2 1 0 5 2 π΄π΅ = = =πΌ 0 1 7 3 β7 5 3 β2 5 2 1 0 π΅π΄ = = =πΌ β7 5 0 1 7 3 nfhLf;fg;gl;l mzpfs; xd;Wf;nfhd;W ngUf;fy; Neh;khW mzp MFk;. π¨ =
π π π βπ , Mfpa mzpfs; ngUf;fiyg; nghUj;J xd;Wf;nfhd;W ngUf;fy; Neh;khW π π βπ π mzp vd epWTf. π΄, π΅ vd;w mzpfs; xd;Wf;nfhd;W ngUf;fy; Neh;khW mzpfs; vdpy; gpd;tUk; tpjpia epiwT nra;a Ntz;Lk; π΄π΅ = π΅π΄ = πΌ> 3 5 2 β5 π΄= ,π΅ = 1 2 β1 3 1 0 3 5 2 β5 π΄π΅ = = =I 0 1 1 2 β1 3 1 0 2 β5 3 5 π΅π΄ = = =I 0 1 β1 3 1 2 nfhLf;fg;gl;l mzpfs; xd;Wf;nfhd;W ngUf;fy; Neh;khW mzp MFk;.
ZY
23.
9 6
DO O
22.
=
ST UD Y
π΅π΄
21.
https://doozystudy.blogspot.in/
π¨=
mzp.
π π
π vdpy; βπ
π¨π° = π°π¨ = π¨, vd;gij rhpghh;f;f. ,q;F π° vd;gJ thpir 2 nfhz;l myF
(Apr-2014)
1 3 1 0 1 3 π΄πΌ = = =π΄ 9 β6 0 1 9 β6 1 0 1 3 1 3 πΌπ΄ = = =π΄ 0 1 9 β6 9 β6 π΄πΌ = πΌπ΄ = π΄ vd;gJ rhpghh;f;fg;gl;lJ.
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28.
29.
30.
31.
π π
π βπ
βπ π
2 4
9 β3 β1 0
π βπ
π π βπ π ngUf;Ff. βπ π 4 2 8 β 54 + 6 4 + 63 β 3 β40 = β6 7 = 16 + 6 + 0 8 β 7 + 0 22 β2 1
(π β π) ngUf;Ff. 6 12 (2 β 7) = β3 β6
(Oct-2014,Apr-2016)
ST UD Y
27.
π π vd;w mzpfspd; ngUf;fiyf; fhz;f. (Apr-2012, Jun-2012) π π 3 Γ 4 + (β2 Γ 2) 3 Γ 1 + (β2 Γ 7) 4 1 = 5 Γ 4 + 1 Γ 2 5 Γ 1 + (1 Γ 7) 2 7 12 + (β4) 3 + (β14) 8 β11 = = 20 + 2 5 + (7) 22 12
64 1
(Oct-2013)
β42 21
ππ + π π = vdpy; π kw;Wk; π Mfpatw;wpd; jPh;T fhz;f. π β ππ ππ 2π₯ + π¦ 5 = π₯ β 3π¦ 13 2π₯ + π¦ = 5β¦β¦β¦β¦β¦β¦β¦β¦ 1 π₯ β 3π¦ = 13β¦β¦β¦β¦β¦β¦β¦. 2 (2) + (1) βΉ (2) βΉ π₯ β 3π¦ = 13 (1) Γ 3 βΉ 6π₯ + 3π¦ = 15 7π₯ = 28 π₯ = 4 π₯ = 4 vd rkd;ghL 1y; gpujpapl βΉ 2 Γ 4 + π¦ = 5 π¦ = 5 β 8 = β3 jPh;T = {4, β3}
(Oct-2014)
ZY
26.
π βπ π π 3 β2 5 1
π¨π Γ π kw;Wk; π©πΓπ vd;w mzpfspd; ngUf;fy; tiuaWf;fg;gl;Ls;sjh? Mk; vdpy;> ngUf;fp tUk; mzpapd; thpiria vOJf. (Jun-2015) π΄ -d; epuy;fspd; vz;zpf;ifAk; π΅ -d; epiufspd; vz;zpf;ifAk; rkkhf ,Ug;gjhy; ngUf;fw;gyd; mzp π΄π΅ tiuaWf;fg;gLfpwJ. π΄π΅-d; thpir 2 Γ 4 MFk;.
DO O
25.
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π¨π Γ π kw;Wk; π©π Γ π vd;w mzpfspd; ngUf;fy; tiuaWf;fg;gl;Ls;sjh? Mk; vdpy; > ngUf;fp tUk; mzpapd; thpiria vOJf. π΄ -d; epuy;fspd; vz;zpf;ifAk; π΅ -d; epiufspd; vz;zpf;ifAk; rkkhf ,y;iy. vdNt> ,t;tpU mzpfspd; ngUf;fw;gyd; tiuaWf;fg;gltpy;iy. A kw;Wk; B vd;w mzpfSf;F AB fpilf;fg; ngWfpwJ. Mdhy; BA fpilf;fg; ngwtpy;iy vdpy;> A kw;Wk; B-d; thpirfisg; gw;wp $Wf. (Apr-2013) A kw;Wk; B vd;w mzpfSf;F AB fpilf;fg; ngWfpwJ. Mdhy; BA fpilf;fg; ngwtpy;iy. vdNt> B-apd; epuy;fspd; vz;zpf;ifAk; A-apd; epiufspd; vz;zpf;ifAk; rkky;y. mjhtJ> A-d; thpir π Γ π MfTk;> Bd; thpir π Γ π MfTk; ,Uf;f Ntz;Lk;.
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π¨ = (π π β π) ,
33.
jPh;T fhz;f.
π π π = π π ππ 3 2 π₯ 8 = 4 5 π¦ 13 3π₯ + 2π¦ 8 = 4π₯ + 5π¦ 13 3π₯ + 2π¦ = 8 βΉ 3π₯ + 2π¦ β 8 = 0 π π
4π₯ + 5π¦ = 13 βΉ 4π₯ + 5π¦ β 13 = 0 FWf;Fg; ngUf;fy; Kiw π₯ π¦ 1 2 -8 3 2 5 -13 4 5 π₯ β26+40
=
π¦ β32+39
βΉ
π₯ 14
=
1 7
βΉ
π₯ 14
π¦ 7
= =
βΉπ₯=
1 7
14 7
&
π¦ 7
=
1 15β8
(Oct-2015)
ZY
βΉ
(Jun-2013)
ST UD Y
π B= π vdpy;> (ππ)π - iaf; fhz;f. π 2 π΄ = (1 2 β 1) , π΅ = 1 4 2 π΄π΅ = (1 2 β 1) Γ 1 = (2 + 2 β 4) = (0) 4 (π΄π΅)π = (0)
32.
=
1 7
& π¦=
7 7
34.
DO O
βΉπ₯ =2&π¦ =1
π+π π π + π = π vdpy; π, π kw;Wk; π d; kjpg;Gfis fhz;f. πβπ π
π₯+π¦ 7 π¦+π§ = 9 π§β5 0 π₯ + π¦ = 7β¦β¦β¦β¦β¦ 1 π¦ + π§ = 9β¦β¦β¦β¦β¦ 2 π§ β 5 = 0β¦β¦β¦β¦β¦.. 3 π₯ = 3, π¦ = 4, π§ = 5
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(Apr-2016)
π§ = 5 vd (2)y; gpujpapl π¦+5=9 βπ¦ =9β5 =4 π¦ = 4 vd (1) y; gpujpapl π₯+4= 7βπ₯ =7β4 =3
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5. Maj;njhiy tbtpay;
eLg;Gs;sp = = 2.
π₯ 1 + π₯ 2 π¦1 + π¦2 , 2 2 3β1 0+4 , = 2 2
= =
π₯ 1 + π₯ 2 π¦1 + π¦2 , 2 2 1β5 β1+3
,
2 β4 2 2
,
2
2
= (β2, 1)
(0, 0), (0, 4) Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj;Jz;bd; eLg;Gs;spiaf; fhz;f. π₯1 , π¦1 π₯2 , π¦2 (0, 0), (0, 4) eLg;Gs;sp = = =
4.
= (1,2)
(1, -1), (-5, 3) Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj;Jz;bd; eLg;Gs;spiaf; fhz;f. π₯1 , π¦1 π₯2 , π¦2 (1, -1), (-5, 3) eLg;Gs;sp =
3.
2 4 , 2 2
ST UD Y
(3, 0), (-1, 4) Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj;Jz;bd; eLg;Gs;spiaf; fhz;f. π₯1 , π¦1 π₯2 , π¦2 (3, 0), (-1, 4)
π₯ 1 + π₯ 2 π¦1 + π¦2 , 2 2 0+0 0+4 , 2 2 0 4 , = (0,2) 2 2
ZY
1.
xU tl;lj;jpd; ikak; ( β 6, 4). mt;tl;lj;jpd; xU tpl;lj;jpd; xU Kid Mjpg;Gs;sp vdpy; > kWKidiaf; fhz;f. (Jun-2012, 2014, Apr-2015) π₯2 , π¦2 π₯1 , π¦1 (0, 0) = (β6, 4) = (β6, 4)
DO O
π₯ 1 + π₯ 2 π¦1 + π¦2 , 2 2 π₯ 1 + 0 π¦1 + 0 , 2 2
π₯1 2 π¦1 2
= β6 βΉ π₯1 = β12 = 4 βΉ π¦1 = 8
kWKid = (β12, 8)
5.
(1, 3), (2, 7), (12, β16) Mfpa Gs;spfis Kidfshff; nfhz;l Kf;Nfhzj;jpd; eLf;Nfhl;L ikak; fhz;f. π₯1 , π¦1 π₯2 , π¦2 π₯3 , π¦3 (1, 3), (2, 7), (12, -16) eLf;Nfhl;L ikak; G
= =
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π₯ 1 + π₯ 2 +π₯ 3 π¦1 + π¦2 +π¦3 , 3 3 1+2+12 3+7β16 15 β6 , = , 3 3 3 3
= (5, β2)
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(3, β5), (β7, 4), (10, β-2) Mfpa Gs;spfis Kidfshff; nfhz;l Kf;Nfhzj;jpd; eLf;Nfhl;L ikak; fhz;f. π₯1 , π¦1 π₯2 , π¦2 π₯3 , π¦3 (3, β5), (β7, 4), (10, β2) eLf;Nfhl;L ikak;
= =
7.
π₯ 1 + π₯ 2 +π₯ 3 π¦1 + π¦2 +π¦3 , 3 3 3β7+10 β5+4β2
,
3
3
= = =
,
3
= (2, β1)
Gs;sp (1, 3)-I eLf;Nfhl;L ikakhff; nfhz;l Kf;Nfhzj;jpd; ,U Kidfs; (β7, 6) kw;Wk; (8, 5) vdpy;> Kf;Nfhzj;jpd; %d;whtJ Kidiaf; fhz;f. (Apr-2012, Apr-2016) π₯1 , π¦1 π₯2 , π¦2 π₯3 , π¦3 (β7, 6), (8, 5) , (π₯3 , π¦3 ), π₯ 1 + π₯ 2 +π₯ 3 π¦1 + π¦2 +π¦3 , = (1, 3) 3 3 β7+8+π₯ 3 6+5+π¦3 , = (1, 3) 3 3 π₯ 3 +1 π¦3 +11 , = (1,3) 3 3 π¦3 +11 = 3 3
ZY
eLf;Nfhl;L ikak; =
π₯ 3 +1 3
DO O
= 1 π₯3 + 1 = 3 π₯3 = 3 β 1 π₯3 = 2 %d;whtJ Kid = (2, -2)
π¦3 + 11 = 9 π¦3 = 9 β 11
π¦3 = β2
(0, 0), (3, 0), (0, 2) Mfpa Gs;spfis Kidfshf nfhz;l Kf;Nfhzj;jpd; gug;gsT fhz;f. π΄ = π΄ = π΄=
10.
3
π₯ 1 + π₯ 2 +π₯ 3 π¦1 + π¦2 +π¦3 , 3 3 4+3+5 β6β2+2 , 3 3 12 β6 , 3 3
= (4, β2)
9.
6 β3
A (4, β6), B (3, β2), (5, 2) Mfpa Gs;spfis Kidfshff; nfhz;l Kf;Nfhzj;jpd; eLf;Nfhl;L ikak; fhz;f. (Oct-2012) π₯1 , π¦1 π₯2 , π¦2 π₯3 , π¦3 (4, β6), (3, β2), (5, 2) eLf;Nfhl;L ikak;
8.
=
ST UD Y
6.
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1 2 1 2 1 2
{ π₯1 π¦2 + π₯2 π¦3 + π₯3 π¦1 β π₯2 π¦1 + π₯3 π¦2 + π₯1 π¦3 } 0+6+0 β 0+0+0
=
1 2
{6 β 0}
0
3
0
0
0
0
2
0
Γ 6 = 3 r.m.
(5, 2), (3, β5), (β5, β1) Mfpa Gs;spfis Kidfshf nfhz;l Kf;Nfhzj;jpd; gug;gsT fhz;f. π΄ =
π΄ = π΄ =
1 { 2 1 2 1 2
π₯1 π¦2 + π₯2 π¦3 + π₯3 π¦1 β π₯2 π¦1 + π₯3 π¦2 + π₯1 π¦3 }
5
3
β5
5
β25 β 3 β 10 β 6 + 25 β 5
2
β5
β1
2
{β38 β 26}
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π΄=
1 2
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Γ β64 = β32 (gug;G vg;nghOJk; kpif vz;zhf ,Uf;f Ntz;Lk;)
π΄ = 32 r.m. (β4, β5), (4, 5),( β1, β6) Mfpa Gs;spfis Kidfshf nfhz;l Kf;Nfhzj;jpd; gug;gsT fhz;f. π΄ = π΄ = π΄ = π΄=
1 { 2 1
π₯1 π¦2 + π₯2 π¦3 + π₯3 π¦1 β π₯2 π¦1 + π₯3 π¦2 + π₯1 π¦3 }
β4
4
β1
β20 β 24 + 5 β [β20 β 5 + 24 }
β5
5
β6
2 1 {β39 + 1} 2 1 Γ β38 = 2
β19
π΄ = 19 r.m. 12.
β5
(1, 2) (β3, 4), (β5, β6) Mfpa Gs;spfis Kidfshf nfhz;l Kf;Nfhzj;jpd; gug;gsT fhz;f. π΄ = π΄ = π΄ = π΄=
13.
β4
ST UD Y
11.
1 { π₯1 π¦2 + π₯2 π¦3 + π₯3 π¦1 β π₯2 π¦1 + π₯3 π¦2 2 1 4 + 18 β 10 β [β6 β 20 β 6 } 2 1 {12 + 32} 2 1 Γ 44 = 22 r. m. 2
+ π₯1 π¦3 }
1
β3
β5
1
2
4
β6
2
A(2, 3), B(4, 0) C(6, β3) Mfpa Gs;spfs; xNu Neh;f;Nfhl;by; mike;Js;sdth vd Muha;f. 1 { π₯1 π¦2 + π₯2 π¦3 + π₯3 π¦1 2 1 π΄ = 0 β 12 + 18 β [12 2 1 π΄ = {6 β 6} 2 1 π΄= Γ 0 = 0 2
β π₯2 π¦1 + π₯3 π¦2 + π₯1 π¦3 }
2
4
6
2
+0β6 }
3
0
β3
3
ZY
π΄ =
/ nfhLf;fg;gl;l Gs;spfs; xNu Neh;f;Nfhl;by; mike;Js;sd. 14.
A(4, 3), B(1, 2), C(β2, 1) Mfpa Gs;spfs; xNu Neh;f;Nfhl;by; mike;Js;sdth vd Muha;f. 1
{ π₯1 π¦2 + π₯2 π¦3 + π₯3 π¦1 β π₯2 π¦1 2 1 π΄ = 8 + 1 β 6 β [3 β 4 + 4 } 2 1 π΄ = {3 β 3} 2 1 π΄= Γ 0 = 0 2
+ π₯3 π¦2 + π₯1 π¦3 }
DO O
π΄ =
4
1
β2
4
3
2
1
3
/ nfhLf;fg;gl;l Gs;spfs; xNu Neh;f;Nfhl;by; mike;Js;sd.
15.
A (β2, β2), B(β6, β2), C(β2,2) Mfpa Gs;spfs; xNu Neh;f;Nfhl;by; mike;Js;sdth vd Muha;f. π΄ =
π΄ = π΄ = π΄=
1 { π₯1 π¦2 + π₯2 π¦3 2 1 4 β 12 + 4 2 1 {β4 β 12} 2 1 Γ {β16} 2
+ π₯3 π¦1 β π₯2 π¦1 + π₯3 π¦2 + π₯1 π¦3 }
β2
β6
β2
β2
β [12 + 4 β 4 }
β2
β2
2
β2
π΄ = 8 βΉ π΄ β 0 / nfhLf;fg;gl;l Gs;spfs; xNu Neh;f;Nfhl;by; mikahJ.
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16.
A
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βπ , π , B (6, -2), C(-3, 4) Mfpa Gs;spfs; xNu Neh;f;Nfhl;by; mike;Js;sdth π 1 β3 π΄ = { π₯1 π¦2 + π₯2 π¦3 + π₯3 π¦1 β π₯2 π¦1 + π₯3 π¦2 + π₯1 π¦3 } 6 -3 2 2 1 π΄ = 3 + 24 β 9 β [18 + 6 β 6 } 3 -2 4 2 1 π΄ = {18 β 18} 2 1
π΄=
2
vd Muha;f. β3 2
3
Γ 0 = 0
17.
ST UD Y
/ nfhLf;fg;gl;l Gs;spfs; xNu Neh;f;Nfhl;by; mike;Js;sd.
π, π , π, π kw;Wk; (π, π + π) Mfpa Gs;spfs; xNu Neh;f;Nfhl;by; mike;jhy; a 1 0 a
1 2 b+1 = 0 1
2π + π + 1 + 0 β 1 + 0 + ππ + π 2π + π + 1 β 1 β ππ β π π + π β ππ π + π π+π ππ
,UGwKk; "ππ" My; tFf;f
+ 1 π
18.
=
π ππ
(Apr-2013)
ππ ππ
= 1 1 π
+ = 1 vd epWtg;gl;lJ.
A (6,7), B (β4,1) C (a, β9) Mfpatw;iw Kidfshff; nfhz;l βπππ d; gug;G 68 r. myFfs; vdpy; π-d; kjpg;igf; fhz;f. (Jun-2013, 2015) A = 68 1 2
{ π₯1 π¦2 + π₯2 π¦3 + π₯3 π¦1 β π₯2 π¦1 + π₯3 π¦2 + π₯1 π¦3 = 68
DO O
π΄ =
6 + 36 + 7π β [β28 + π β 54 } = 68 Γ 2 42 + 7π β [β82 + π } = 136 42 + 7π + 82 β π = 1 124 + 6π = 136 6π = 12 π = 12/6 π= 2
19.
π π
+ = π vd epWTf.
0 0 0 ππ
ZY
π ππ
= = = =
π π
6
β4
a
7
1
β9
6 7
( β 5, β 2) vd;w Gs;sp topr; nry;tJk;> Ma mr;RfSf;F ,izahdJkhd Neh;f;NfhLfspd; rkd;ghLfisf; fhz;f. (Jun-2013) π₯ mr;Rf;F ,izahd Nfhl;bd; rkd;ghL π¦ mr;Rf;F ,izahd Nfhl;bd; rkd;ghL π¦ = π π₯ = π
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π¦ = β2 π¦ + 2 = 0 20.
π₯ = β5 π₯ + 5 = 0
π + ππ + π = π , ππ + ππ + π = π Mfpa Neh;f;NfhLfs; ,iz vd epWTf. π₯ + 2π¦ + 1 = 0 d; rha;T π1 =
=
β1 2
21.
β1 2
ST UD Y
β3
3π₯ + 6π¦ + 2 = 0 d; rha;T π2 = = = 6 π1 = π2 rha;Tfs; rkk;. vdNt Neh;f;NfhLfs; ,iz.
ππ + ππ β ππ = π , ππ + ππ + π = π Mfpa Neh;f;NfhLfs; ,iz vd epWTf. 3π₯ + 2π¦ β 12 = 0 d; rha;T π1 = 6π₯ + 4π¦ + 8 = 0 d; rha;T π2 =
β3 2
=
=
β6 4
=
β3 2
π1 = π2 rha;Tfs; rkk;. vdNt Neh;f;NfhLfs; ,iz. 22.
π + ππ + π = π , ππ β π + π = π Mfpa Neh;f;NfhLfs; nrq;Fj;J vd epWTf. π₯ + 2π¦ + 1 = 0 d; rha;T π1 = 2π₯ β π¦ + 5 = 0 d; rha;T π2 =
23.
β1 2
β2 β1
= 2
Γ 2 = β1. vdNt Neh;f;NfhLfs; nrq;Fj;jhdit.
ππ β ππ + π = π , πππ + ππ + π = π Mfpa Neh;f;NfhLfs; nrq;Fj;J vd epWTf. (Apr-2013) 3π₯ β 5π¦ + 7 = 0 d; rha;T π1 =
=
15π₯ + 9π¦ + 4 = 0 d; rha;T π2 =
=
3 5
DO O
π1 Γ π2 = 24.
=
ZY
π1 Γ π2 =
β1 2
=
Γ
β5 3
β3 β5 β15 9
= =
3 5 β5 3
= β1. vdNt Neh;f;NfhLfs; nrq;Fj;jhdit.
ππ β ππ β π = π , ππ + ππ β ππ = π Mfpa Neh;f;NfhLfs; xd;Wf;nfhd;W nrq;Fj;J vdpy; π-d; kjpg;igf; fhz;f. (Apr-2016) 5π₯ β 2π¦ β 9 = 0 vd;w Neh;f;Nfhl;bd; rha;T π1 =
=
β5 β2
2π₯ + ππ¦ β 11 = 0 vd;w Neh;f;Nfhl;bd; rha;T π2 =
=
β2 π
Neh;f;NfhLfs; xd;Wf;nfhd;W nrq;Fj;J> vdNt
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=
5 2
π1 Γ π2 = β1 5 2
Γ
β2 π
= β1
βπ = β5 π=5
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25.
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rha;Tf;Nfhzk; πππ kw;Wk; π-ntl;Lj;Jz;L
π π
Mfpatw;iwf; nfhz;l Neh;f;Nfhl;bd; rkd;ghl;ilf;
$Wf.
(Oct-2013) rha;T π = π‘ππ π =
π‘ππ450
π¦ βntl;Lj;Jz;L π =
2 5
= 1
Njitahd Neh;f;Nfhl;bd; rkd;ghL π¦ = ππ₯ + π 2 5 5π¦ = 5π₯ + 2 5π₯ β 5π¦ + 2 = 0
ST UD Y
π¦ = 1π₯ +
ππ β π + π = π vd;w Neh;f;Nfhl;bw;F ,izahdJk; (π, βπ) vd;w Gs;sp topr; nry;tJkhd Neh;f;Nfhl;bd; rkd;ghl;ilf; fhz;f. (Jun-2015) 3π₯ β π¦ + 7 = 0 vd;w Neh;f;Nfhl;bw;F ,izahd Neh;f;Nfhl;bd; rkd;ghL 3π₯ β π¦ + π = 0 3 π₯ β π¦ + π = 0 vd;gJ (1, -2) vd;w Gs;sp topr; nry;fpwJ. vdNt> Neh;f;Nfhl;bd; rkd;ghl;by; π₯ = 1, π¦ = 2 vdg; gpujpapl 3(1) β (β2) + π = 0 βΉ 3 + 2 + π = 0 5+π = 0 βΉ π = β5 Njitahd Neh;f;Nfhl;bd; rkd;ghL 3 π₯ β π¦ β 5 = 0 MFk;. 27. π π β ππ + π = π vd;w Neh;f;Nfhl;bd; rha;T kw;Wk; π-ntl;Lj;Jz;L Mfpatw;iwf; fhz;f. 26.
(Apr-15)
nfhLf;fg;gl;l Neh;f;Nfhl;bd; rkd;ghL 4 π₯ β 2π¦ + 1 = 0.
π¦ ntl;Lj;Jz;L rha;T = 2, 28.
=
=
β4 β2
= 2
ZY
rha;T
=
=
π¦ ntl;Lj;Jz;L =
β1 β2
=
1 2
khw;WKiw
π¦ = ππ₯ + π β β2π¦ = β4π₯ β 1
π¦=
β4 π₯ β2
+
β1 β2
βπ=
β4 β2
= 2, π =
β1 β2
=
1 2
1 2
ππ β π + ππ = π vd;w Neh;f;Nfhl;bd; rkd;ghl;bypUe;J π, π -ntl;Lj;Jz;Lfisf; fhz;f.
DO O
nfhLf;fg;gl;l Neh;f;Nfhl;bd; rkd;ghL 2π₯ β π¦ + 16 = 0 π₯ -ntl;L =
=
π¦ -ntl;L =
=
β16 2 β16 β1
= β8 = 16
π₯βntl;Lj; Jz;L = -8, π¦ -ntl;Lj;Jz;L = 16
29.
(βπ, π) kw;Wk; (π, π) Mfpa Gs;spfs; topr; nrY;Yk; Neh;f;Nfhl;bd; rkd;ghl;ilf; fhz;f. ,U Gs;spfs; topr; nry;Yk; Nfhl;bd; rkd;ghL π¦βπ¦1 π₯βπ₯ 1 = (π₯1 = β2, π¦1 = 5, π₯2 = 3, π¦2 = 6) π¦2 βπ¦1
π₯ 2 βπ₯ 1
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nfhLf;fg;gl;l ,U Gs;spfs; (-2, 5) kw;Wk; (3, 6) π¦β5 6β5
=
π₯+2 3+2
π¦β5 1
=
π₯+2 5
5(π¦ β 5) = π₯ + 2 π₯ β 5π¦ + 27
ST UD Y
5π¦ β 25 = π₯ + 2 = 0
Njitahd Neh;f;Nfhl;bd; rkd;ghL π₯ β 5π¦ + 27 = 0 30.
rha;T β3 kw;Wk; π - ntl;Lj;Jz;L 4 nfhz;lJkhd Neh;f;Nfhl;bd; rkd;ghl;ilf; fhz;f. nfhLf;fg;gl;lit rha;T π = β3 kw;Wk; π¦ - ntl;Lj;Jz;L π = 4 Njitahd Neh;f;Nfhl;bd; rkd;ghL π¦ = ππ₯ + π π¦ = β 3π₯ + 4 3π₯ β π¦ + 4 = 0 Njitahd Neh;f;Nfhl;bd; rkd;ghL 3π₯ β π¦ + 4 = 0
31.
(7, 3), (6, 1), (8, 2) kw;Wk; (P, 4) vd;gd Xh; ,izfuj;jpd; thpirg;gb mike;j cr;rpfs; vdpy; P-d; kjpg;igf; fhz;f. (Jun-2012) ,izfuk; ABCD d; cr;rpfs; A(7,3), B(6,1), C(8,2) kw;Wk; D(P,4) vd;f. Xh; ,izfuj;jpd; %iytpl;lq;fs; xd;iwnahd;W ,Urkf;$wpLk; mjhtJ %iytpl;lk; AC-d; eLg;Gs;sp = %iytpl;lk; BD-d; eLg;Gs;sp 7+8 3+2 6+π 1+4 / , = , 2
=
2 7+8 2
2
eLg;Gs;sp fhZk; tha;g;ghL β
6+ P = 15 P = 15 - 6 = 9 P=9
π₯ 1 +π₯ 2 π¦1 +π¦2 , 2 2
rha;Tf;Nfhzk; πππ nfhz;l kw;Wk; (4,2), (3,1) Mfpa Gs;spfis ,izf;Fk; Neh;f;Nfhl;Lj; Jz;bd; eLg;Gs;sp topr; nry;Yk; Neh;f;Nfhl;bd; rkd;ghl;ilf; fhz;f. (Oct-2012) rha;Tf;Nfhzk; π = 300
DO O
32.
2
ZY
6+P 2
rha;T π = π‘ππ π = π‘ππ300 =
1 3
(4, 2), (3, 1) Mfpa Gs;spfis ,izf;Fk; Neh;f;Nfhl;Lj; Jz;bd; eLg;Gs;sp π₯ 1 +π₯ 2 π¦1 +π¦2 , 2 2
=
4+3 2+1 , 2 2
=
7 3 , 2 2
Njitahd Neh;f;Nfhl;bd; rkd;ghl;bd; rkd;ghL π¦ β π¦1 = π( π₯ β π₯1 ) Fwpg;G: π = (π₯1 , π¦1 ) =
1 3
7 3 , 2 2
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π¦ β
3 2
2π¦β3 2
=
= 1 3
1 3 2π₯β7 2
π₯β
7 2
3(2π¦ β 3) = 2 π₯ β 7 2 3π¦ β 3 3 β 2π₯ + 7 = 0 2π₯ β 2 3 π¦ + (3 3 β 7) = 0
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33.
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ππ + π, π , π, ππ + π Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj;Jz;bd; eLg;Gs;spapd; Maj;njhiyTfs; (ππ, ππ) vdpy;> π, π Mfpatw;wpd; kjpg;Gfisf; fhz;f. (Apr-2014) 2π + 2, 3 , 4, 2π + 1 Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj;Jz;bd; eLg;Gs;sp = (2π, 2π) = (2π, 2π) = (2π, 2π) 2π+4 2
= 2π
ST UD Y
π₯ 1 +π₯ 2 π¦1 +π¦2 , 2 2 2π+2+4 3+2π +1 , 2 2 2π+6 = 2π 2
2π + 6 = 4π
2π + 4 = 4π
2π = 6
π =3
2π = 4
π=2
π = 3, π = 2
34. (3, β4) vd;w Gs;sp topr; nry;Yk; kw;Wk; Ma mr;RfSf;F ,izahf mike;j Neh;f;NfhLfspd; rkd;ghLfisf; fhz;f. (Apr-2014) (3, β4) vd;w Gs;sp topr; nry;Yk; kw;Wk; π₯ mr;Rf;F ,izahd Nfhl;bd; rkd;ghL π¦=π, vdNt> π¦ = β4 (3, β 4) vd;w Gs;sp topr; nry;Yk; kw;Wk; π¦ mr;Rf;F ,izahd Nfhl;bd; rkd;ghL π₯ = π, vdNt> π₯ = 3 35.
(β2, 3) vd;w Gs;sp topr; nry;tJk;> rha;T
3
π
cilaJkhd Neh;f;Nfhl;bd; rkd;ghl;ilf; fhz;f.
(Jun- 2014)
ZY
(π₯1 , π¦1 ) = (β2,3) 1 rha;T π =
π
Njitahd Nfhl;bd; rkd;ghL π¦ β π¦1 = π π₯ β π₯1 π¦β3 =
1 3
π₯+2
3π¦ β 9 = π₯ + 2 π₯ β 3π¦ + 11 = 0
(β 3, 5) kw;Wk; (4, β 9) Mfpa Gs;spfis ,izf;Fk; Nfhl;Lj;Jz;bid cl;Gwkhf 1:6 vd;w tpfpjj;jpy; gphpf;Fk; Gs;spapd; mr;Rj; njhiyTfisf; fhz;f. (Oct-2014)
DO O
36.
cl;Gwkhf gphpf;Fk; Gs;sp π
cl;Gwkhf gphpf;Fk; Gs;sp π
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= = = =
ππ₯ 2 +ππ₯ 1 ππ¦ 2 +ππ¦ 1
,
π+π π+π 1 4 + 6(β3) 1 β9 +6(5) , 1+6 1+6 4β18 β9+30
,
7 7 β14 21 , 7 7
= (β2,3)
π₯1 = β3, π₯2 = 4, π¦1 = 5, π¦2 = β9 π = 1, π = 6
= (β2,3)
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(2, 1) kw;Wk; (5, β8) vd;w Gs;spfis ,izf;Fk; Nfhl;Lj; Jz;il P,Q vd;w Gs;spfs; %d;W rkghfq;fshfg; gphpf;fpd;wd. Gs;sp P MdJ ππ β π + π = π vd;w Nfhl;bd; Nky; cs;sJ vdpy;> π-d; kjpg;G fhz;f. (Oct-2014) P,Q vd;w Gs;spfs; AB I Kr;rkf; $wpLk; Gs;spfs; MFk;. mjhtJ P vd;gJ AB I 1:2 vd;w tpfpjj;jpy; gphpf;Fk; Gs;sp MFk;. vdNt>
ππ₯ 2 +ππ₯ 1 ππ¦ 2 +ππ¦ 1 , π+π π+π 1 5 +2(2) 1 β8 + 2(1) , 1+2 1+2 5+4 β8+2 , 3 3 9 β6 , 3 3
π= = = =
ST UD Y
37.
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π = (3, β2)
38.
π₯1 = 2, π₯2 = 5, π¦1 = 1, π¦2 = β8 π = 1, π = 2
π MdJ 2π₯ β π¦ + π = 0 vd;w Nfhl;bd; Nky; cs;sJ. vdNt 2 3 + 2 + π = 0 8+π =0 π = β8 rJuk; ABCD-apd; gf;fk; AB MdJ π -mr;Rf;F ,izahf cs;sJ. vdpy;> %iytpl;lk; π¨πͺ -apd; rha;T fhz;f. (Oct-2015) rJuk; ABCD βapd; gf;fk; AB MdJ π₯-mr;Rf;F ,izahf cs;sJ. %iytpl;lk; AC MdJ β π·π΄π΅ βI ,Urkf; $wpLk;. vdNt> β π΅π΄πΆ = 450 mjhtJ π = 450 MfNt %iytpl;lk; AC-d; rha;T
m = tanΞΈ
ZY
= tan 450 =1
6. tbtpay;
βπ¨π©πͺ βy; β π¨ β d; ntspg;Gw ,Urkntl;b MdJ BC-d; ePl;rpapid E-y; re;jpf;fpwJ. AB = 10 nr.kP> AC = 6 nr.kP kw;Wk; BC = 12 nr.kP vdpy;> CE-If; fhz;f. βABC y; β π΄ β d; ntspg;Gw ,Urkntl;b AE vd;f. NkYk; AE MdJ BC-d; ePl;rpia E-y; re;jpf;fpwJ. CE = π₯ vd;f. Nfhz ,Urkntl;bj; Njw;wg;gb>
DO O
1.
12+π₯ π₯
=
π΅πΈ πΆπΈ 10 6
=
π΄π΅ π΄πΆ
βΉ
12+π₯ π₯
=
5 3
3(12 + π₯) = 5π₯ βΉ 36 + 3π₯ = 5π₯ 36 = 5 π₯ β 3π₯ 2 π₯ = 36 βΉ π₯ = 18 / EC = 18 nr.kP
2.
glj;jpy; CD-d; msT fhz;f. EA Γ EB = EC Γ ED 8 Γ 5 = (x + 4) Γ 4
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8Γ5 4
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=π₯+4
x + 4 = 10 x = 10 - 4 = 6 CD = 6 nr.kP π¨π« π«π©
βABC y; DE β₯ BC and π΄π· π·π΅
=
π΄πΈ πΈπΆ
2 3
=
3.7 πΈπΆ
πΈπΆ =
3.7Γ3 2
=
=
π π
, AE = 3.7 nr.kP. vdpy;> EC-If; fhz;f. (Jun- 2012, Jun-2014)
ST UD Y
3.
11.2 2
= 5.55 nr.kP. 4.
D,E vd;w Gs;spfs; βABCy; AB, AC fspy; DE β₯ BC vd;wthW cs;sd. AD = 6 nr.kP> DB = 9 nr.kP kw;Wk; AE = 8 nr.kP> vdpy;> AC- If; fhz;f. Njy;]; Njw;wg;gb
π΄π· π·π΅
=
π΄πΈ πΈπΆ
6 9
=
8 πΈπΆ
EC =
8Γ9
= 12 nr.kP
6
AC = AE+EC=12 + 8 = 20 nr.kP
D,E vd;w Gs;spfs; βABCy; AB, AC fspy; DE β₯ BC vd;wthW cs;sd. AD = 8 nr.kP> AB = 12 nr.kP> kw;Wk; AE = 12 nr.kP> vdpy;> CE- If; fhz;f. Njy;]; Njw;wg;gb
π΄π· π·π΅ 8 4
= =
=
π΄πΈ πΈπΆ 12
πΈπΆ
12 Γ 4 8
=
DO O
EC
ZY
5.
48 8
=6
CE = 6 nr.kP
6.
βABC y; β A vd;w Nfhzj;jpd; cl;Gw ,Urkntl;b AD MdJ> gf;fk; BC I D-y; re;jpf;fpwJ. BD = 2.5 nr.kP> AB = 5 nr.kP. kw;Wk; AC = 4.2 nr.kP. vdpy; DC-If; fhz;f.
(Apr-2012, Oct-2012, 2013, June-2015)
Nfhz ,Urkntl;bj; Njw;wg;gb>
π΅π· π·πΆ 2.5 π·πΆ
=
π΄π΅ π΄πΆ
=
5 4.2
DC =
2.5 Γ 4.2 10.5 = 5 5
= 2.1
DC = 2.1 nr.kP
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7.
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βπ·πΈπΉ βd; gf;fq;fs; PQ kw;Wk; PR-fspd; kPJ mike;j Gs;spfs; S kw;Wk; T vd;f. NkYk; ST β₯ QR, π·πΊ πΊπΈ
PR = 5.6 nr.kP. kw;Wk;
=
π π
vdpy;> PT-If; fhz;f.
βPQR -y; PT = x vd;f. / TR = PR β PT = 5.6 β x ππ ππ
ππ ππ
=
βΉ ππ = ππ
x = (5.6 β x)
ππ ππ
3 5
5x = 16.8 β 3x 5x + 3x = 16.8 β 8π₯ = 16.8 x=
16.8 8
= 2.1
/ PT = 2.1 nr.kP 8.
π¨π« vd;gJ βπ¨π©πͺ -y; β A βd; cl;Gw Nfhz ,Urkntl;b. mJ BC-I D-y; re;jpf;fpwJ. BD = 2 nr.kP> AB = 5 nr.kP kw;Wk; DC = 3 nr.kP vdpy; AC-If; fhz;f. Nfhz ,Urkntl;bj; Njw;wg;gb> 2 3
=
5 π΄πΆ
AC =
5Γ3 2
=
π΅π· π·πΆ
15 = 7.5 2
AC = 7.5 nr.kP
=
π΄π΅ π΄πΆ
AD vd;gJ βABC βy; β A β d; cl;Gw Nfhz ,Urkntl;b. mJ BC I D-y; re;jpf;fpwJ. AB = 5.6 nr.kP> AC = 6 nr.kP kw;Wk; DC = 3 nr.kP. vdpy; BC-If; fhz;f.
ZY
9.
ST UD Y
NkYk; ST β₯ QR Njy;]; Njw;wj;jpd;gb
Nfhz ,Urkntl;bj; Njw;wg;gb> π΅π·
=
5.6
5 5.6 Γ 3 BD = = 2.8 6
=
π΄π΅ π΄πΆ
DO O
3
π΅π· π·πΆ
BD = 2.8 nr.kP BC = 2.8 + 3 = 5.8 nr.kP
10.
glj;jpy; π-d; kjpg;igf; fhz;f. PA Γ PB = PC Γ PD 4Γx=8Γ3 8Γ3
x= = 4 x=6
11.
24 4
xU tl;lj;jpy; AB, CD vd;Dk; ,U ehz;fs; xd;iwnahd;W cl;Gwkhf P-apy; ntl;bf; nfhs;fpd;wd. CP = 4 nr.kP.> AP = 8 nr.kP.> PB = 2 nr.kP.> vdpy;> PD-If; fhz;f. (Apr-2014) PA Γ PB = PC Γ PD 8Γ2=4Γx
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8Γ2 = 4 16 =x 4
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x PD= 4 nr.kP
x=4
xU tl;lj;jpy; AB, CD vd;Dk; ,U ehz;fs; xd;iwnahd;W cl;Gwkhf nfhs;fpd;wd. AP= 12 nr.kP> AB = 15 nr.kP> CP = PD vdpy;> CD-If; fhz;f. PA Γ PB = PC Γ PD 12 Γ 3 = x Γ x 36 = x2 x=6 CD = 6+6 = 12 nr.kP
13.
glj;jpy; x βd; kjpg;igf; fhz;f. PA Γ PB = PC Γ PD 9 Γ 4 = (π₯+2) Γ 2 9Γ4 = π₯+2 2 36 = x+2 2
π₯+2 = 18 π₯ = 18 β2= 16 π₯ = 16
AB kw;Wk; CD vd;w ,U ehz;fs; tl;lj;jpw;F ntspNa P vDk; Gs;spapy; ntl;bf; nfhs;fpd;wd. AB = 4 nr.kP> BP= 5 nr.kP kw;Wk; PD = 3 nr.kP vdpy;> CD-If; fhz;f. PA Γ PB = PC Γ PD 9 Γ 5 = (π₯+3) Γ 3 9Γ5 3 45 3
= x+3
= x+3
DO O
π₯+3 = 15 π₯ = 15 β 3 = 12 CD = 12 nr.kP
15.
ntl;bf;
ZY
14.
P-apy;
ST UD Y
12.
AB kw;Wk; CD vd;w ,U ehz;fs; tl;lj;jpw;F ntspNa P vDk; Gs;spapy; ntl;bf; nfhs;fpd;wd. BP = 3 nr.kP> CP = 6 nr.kP kw;Wk; CD = 2 nr.kP> vdpy;> AB-If; fhz;f. PA Γ PB = PC Γ PD (π₯+ 3) Γ 3 = 6 Γ 4 6Γ4 3
=
π₯+3 π₯ AB
=8 = 8 β 3= 5 = 5 nr.kP
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=
24 3
π₯+3
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16.
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glj;jpy; TP xU njhLNfhL A,B vd;gd tl;lj;jpd; kPJs;s Gs;spfspy; β BTP = πππ kw;Wk; β ATB = ππ π vdpy; β ABT If; fhz;f. (Apr-2013,Apr-2016)
17.
18.
ST UD Y
β BAT = β BTP = 720 [njhLNfhL β ehz; Njw;wk; ] βABT y; β ATB + β BAT + β ABT = 1800 430 + 720 + β ABT = 1800 β ABT = 1800 β 1150 = 650
xU tl;lj;jpd; Gs;sp A-y; tiuag;gLk; njhLNfhL PQ vd;f. AB vd;gJ tl;lj;jpd; ehz; vd;f. NkYk;> β BAC = πππ kw;Wk; β BAQ = πππ vd;W mikAkhW tl;lj;jpd; Nky; cs;s Gs;sp C vdpy;> β ABC - If; fhz;f. A-y; PQ xU njhLNfhL kw;Wk; AB xU ehz;. vdNt β BAQ = β ACB = 620 (njhLNfhL β ehz;Njw;wk;) NkYk;> β BAC + β ACB + β ABC = 1800 (xU Kf;Nfhzj;jpd; %d;W Nfhzq;fspd; $Ljy; 1800 ) β ABC = 1800 β (β BAC + β ACB) = 1800 β (540 +620 ) = 640 β ABC = 640 gpjhfu]; Njw;wj;ij vOJf. (my;yJ) ngsjad; Njw;wj;ij vOJf. xU nrq;Nfhz Kf;Nfhzj;jpy; fh;zj;jpd; th;f;fk; kw;w ,U gf;fq;fspd; th;f;fq;fspd; $LjYf;Fr; rkk;. π©πͺπ = π¨π© π + π¨πͺπ gpjhfu]; Njw;wj;jpd; kWjiyia vOJf. xU Kf;Nfhzj;jpy;> xU gf;fj;jpd; th;f;fk;> kw;w ,U gf;fq;fspd; th;f;fq;fspd; $LjYf;Fr; rkk; vdpy;> Kjy; gf;fj;jpw;F vjpNu cs;s Nfhzk; nrq;Nfhzk; MFk;. π©πͺπ = π¨π© π + π¨πͺπ
20.
njhLNfhL ehz;Njw;wj;ij vOJf. tl;lj;jpy; njhLNfhl;bd; njhL Gs;sp topNa xU ehz; tiuag;gl;lhy;> me;j ehz; njhLNfhl;Lld; Vw;gLj;Jk; Nfhzq;fs; KiwNa xt;nthd;Wk; jdpj;jdpahf khw;W tl;lj;Jz;Lfspy; mike;j Nfhzq;fSf;Fr; rkk;. β π©π¨π» = β π©π·π¨ , β π©π¨πΊ = β π¨πΈπ©
22.
DO O
21.
ZY
19.
njhLNfhL ehz; Njw;wj;jpd; kWjiyia vOJf. xU tl;lj;jpy; xU ehzpd; xU Kidg;Gs;sp topNa tiuag;gl;l Neh;f;NfhL me;ehZld; cz;lhf;Fk; NfhzkhdJ kW tl;lj; Jz;bYs;s Nfhzj;jpw;Fr; rkkhdhy;> me;Neh;f;NfhL tl;lj;jpw;F xU njhLNfhlhFk;. MP vd;gJ βMNO-y; β M-d; ntspg;Gw ,U rkntl;b> NkYk; NO-d; ePl;rpapid P -apy; re;jpf;fpwJ. MN=10 nr.kP> MO = 6 nr.kP NO = 12 nr.kP vdpy; OP If; fhz;f. (Jun-2013) MP vd;gJ βMNO-y; β M βd; ntspg;Gw ,U rkntl;b. βMNO βy; Nfhz ,Urkntl;b Njw;wj;jpd;gb
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ππ ππ
=
ππ ππ
12+ππ ππ
=
ππ ππ
12+ππ ππ
=
10 6
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10 ππ β 6 ππ = 72 4 ππ = 72 ππ =
72 4
= 18 nr.kP
ST UD Y
72 + 6 ππ = 10 ππ
7. Kf;Nfhztpay; 1.
Rthpy; rha;j;J itf;fg;gl;l xU VzpahdJ jiuAld; ππ π Nfhzj;ij Vw;gLj;JfpwJ. Vzpapd; mb Rtw;wpypUe;J 3.5 kP J}uj;jpy; cs;sJ. vdpy;> Vzpapd; ePsj;ijf; fhz;f.
(Oct-2012, Apr-2013, Jun-2014, Jun-2015)
nrq;Nfhzk; βABC βy; π΅πΆ π΄πΆ 3.5 π΄πΆ
cos 600 = 1 2
=
π΄πΆ = 3.5 Γ 2 = 7 Vzpapd; ePsk; = 7kP
30 kP ePsKs;s xU fk;gj;jpd; epoypd; ePsk; ππ π kP vdpy; #hpadpd; Vw;wf;Nfhzj;jpd; (jiu kl;lj;jpypUe;J Vw;wf;Nfhzk;) mstpidf; fhz;f. (Apr-2012, Apr-2014) nrq;Nfhz βABC βy; π΄π΅ π‘πππ = =
=
10 3 3Γ 3 3
3
3
DO O
=
π΅πΆ 30
ZY
2.
=
tan
3
600
= 3 / π = 600 #hpadpd; Vw;wf;Nfhzk; = 600
3.
xU Rik Ch;jpapypUe;J Rikia ,wf;f VJthf πππ Vw;wf; Nfhzj;jpy; xU rha;T jsk; cs;sJ. rha;T jsj;;jpd; cr;rp jiuapypUe;J 0.9 kP cauj;jpy; cs;sJ vdpy; rha;T jsj;jpd; ePsk; ahJ? (Oct-2014, Apr-2015) nrq;Nfhz βABC βy; sin 300 = 1 2
=
π΄π΅ π΄πΆ 0.9 π΄πΆ
π΄πΆ = 0.9 Γ 2 = 1.8 rha;T js ePsk; = 1.8 kP
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4.
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cauk; 150 nr.kP. cs;s xU rpWkp xU tpsf;Ff; fk;gj;jpd; Kd; epd;wthW πππ π nr.kP. ePsKs;s epoiy Vw;gLj;Jfpwhs; vdpy; tpsf;Ff; fk;gj;jpd; cr;rpapd; Vw;wf;Nfhzk; fhz;f.
(Jun-12) π΄π΅ π΅πΆ 150 150 3 1 3 1 3 300
π‘ππ π = = = tan 300 =
/π = Vw;wf;Nfhzk; = πππ 5.
πβππππ½ π+ππππ½ 1βπ πππ 1+π πππ
= ππππ½ β ππππ½ vd epWTf. =
1βπ πππ 1+π πππ
x
1βπ πππ 1βπ πππ
ST UD Y
nrq;Nfhz βABC βy;
(Oct-2012, Jun-2014)
(1βπ πππ )2 1βπ ππ 2 π
=
1
(1βπ πππ )2 = ππ π 2 π 1 π πππ β = πππ π πππ π
= =
=
1βπ ππ π πππ π
π πππ β π‘πππ vd epWtg;gl;lJ.
ππππ π½ + ππππππ π½ = πππ§ π½ + ππ¨π π½ vd epWTf. π ππ 2 π
+ πππ ππ 2 π
=
1
+ π‘ππ 2 π
π‘ππ2 π
+1+
(Apr-2014, 2015)
πππ‘ 2 π
+ πππ‘ 2 π + 2
ZY
6.
1βπ ππ π 2 2 πππ π
=
= π‘ππ2 π + πππ‘ 2 π + 2π‘πππ. πππ‘π = = π‘ππ π + πππ‘ π vd epWtg;gl;lJ.
ππππ½ ππππππ½ π πππ πππ πππ
+
ππππ½ = π vd;gij epWTf. ππππ½ πππ π π πππ πππ π + = 1 + 1 π πππ
DO O
7.
(π‘πππ + πππ‘π)2
π πππ
(Jun- 2012, Oct-2014)
πππ π
= π πππ Γ π πππ + ( πππ π Γ πππ π) = π ππ2 π + πππ 2 π
= 1
8.
ππππ½ = π ππππ½βππππ½ πππ π π πππ βπ‘πππ
vd epWtg;gl;lJ.
+ π¬π’π§π½ vd;w Kw;nwhUikia epWTf. = =
πππ π π πππ +π‘πππ Γ π πππ βπ‘πππ π πππ +π‘πππ πππ π (π πππ +π‘πππ ) = πππ π π ππ 2 πβπ‘ππ 2 π
(Jun-2013) π ππ 2 π β π‘ππ2 π = 1
Γ
1 πππ π
+ πππ π Γ
π πππ πππ π
= 1+ sinπ vd epWtg;gl;lJ.
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π πβππππ π½ β = π vd epWTf. π πππ π½ πβππππ π½ 2 2 1 1βπ ππ π 2 π β πππ π β = πππ ππ π ππ 2 π 1βπππ 2 π π ππ 2 π
(Apr-2013) πππ ππ 2 π β πππ‘ 2 π = 1.
= cosec2 π β cot2 π = 1 vd epWtg;gl;lJ.
11.
(ππππ π½ + ππππ π½) = π β πππππ π½ ππππ π½ vd epWTf. π ππ6 π + πππ 6 π = (π ππ2 π)3 + (πππ 2 π)3 (π = π ππ2 π, π = πππ 2 π ) = (π ππ2 π + πππ 2 π)3 β 3 π ππ2 π πππ 2 π (π ππ2 π + πππ 2 π) = 1 β 3π ππ2 π πππ 2 π vd epWtg;gl;lJ. π+ππππ½ ππππ½
=
1+π πππ π πππ
ππππ π½ πβππππ½ 1 1+πππ π
=
vd epWTf.
1 πππ π
=
πππ π +1 πππ π
= (1 + πππ π) Γ = =
+ = (π + π)3 β 3ππ(π + π) π ππ2 π + πππ 2 π = 1
πππ π
Γ
1
(1βπππ π ) (1βπππ π ) (1βπππ π ) (1βπππ π )
1βπππ 2 π 1βπππ π π ππ 2 π 1βπππ π
200 kP ePsKs;s E}ypdhy; xU fhw;whb fl;lg;gl;L gwe;Jf; nfhz;bUf;fpwJ. me;j E}y; jiukl;lj;Jld; πππ Nfhzj;ij Vw;gLj;jpdhy;> fhw;whb jiukl;lj;jpypUe;J vt;tsT cauj;jpy; gwf;fpwJ. vdf; fhz;f. (,q;F E}y; xU Neh;f;Nfhl;by; cs;sjhff; fUJf). glj;jpy; C vd;gJ fhw;whb vd;f. jiukl;lj;jpypUe;J fhw;whbf;F cs;s J}uk; BC = h vd;f. E}ypd; ePsk; AC = 200 kP
ZY
12.
π3
(Oct-2013, Apr-2016)
= 1 + πππ π = (1 + πππ π) Γ
(Apr-2012) π3
ST UD Y
10.
nrq;Nfhz βABC y; sin 300 = 1 2
DO O
=
β=
π΅πΆ π΄πΆ β 200 200 2
= 100
jiukl;lj;jpypUe;J fhw;whbf;F cs;s J}uk; BC = h = 100 kP
13.
πβππππ½ π+ππππ½
= ππππππ½ β ππππ½ vd epWTf. 1βπππ π 1+πππ π
=
1βπππ π 1+πππ π
=
(1βπππ π )2 π ππ 2 π
Γ
1βπππ π 1βπππ π
=
(1βπππ π )2 1βππ π 2 π
1 β πππ 2 π = π ππ2 π
1
=
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1βπππ π 2 2 π πππ
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= = 14.
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1βπππ π π πππ 1 πππ π β π πππ π πππ
= πππ πππ β πππ‘π xU βπ¨π©πͺ βapy; nrq;Nfhzk; βCβ-apy; mikag;ngw;why; ππ¨π¬(π¨ + π©)kw;Wk; π¬π’π§ (π¨ + π©) d; kjpg;Gfisf; fhz;f (Apr-2016)
ST UD Y
xU Kf;Nfhzj;jpd; midj;J Nfhzq;fspd; $Ljy; 1800 C = 900 , vdNt> π΄ + π΅ = 900 cos(π΄ + π΅) = cos 900 = 0 sin (π΄ + π΅) = sin 900 = 1 15.
5 mb cauKs;s xU rpWtd; xU J}zpypUe;J 100 mb J}uj;jpy; epd;W mjd; cr;rpia πππ Vw;wf; Nfhzj;jpy; ghh;j;jhy; J}zpd; cauk; vd;d? (Jun-2013) glj;jpy; AB =rpWtd; = 5 mb BC = rpWtDf;Fk; J}zpw;Fk; ,ilNaahd J}uk; = 100 mb J}zpd; cauk; = CE = CD + DE = 5 + DE nrq;Nfhz βADE-y; π‘ππ450 =
π·πΈ π΄π·
=
π·πΈ 100
(CD = AB = 5 mb) π‘ππ450 = 1
=1
/ π·πΈ = 100
/ J}zpd; cauk; πΆπΈ = 5 + 100 = 105 mb
π½ xU FWq;Nfhzk; kw;Wk; ππππ½ = ππππ½ vdpy; ππππ π½ β πππππ π½ = π vd ep&gpf;f. (Oct-2013) π xU FWq;Nfhzk; kw;Wk; π πππ = cos π vdpy; π = 45Β° LHS π‘ππ2 π β 2πππ 2 π = π‘ππ2 45Β° β 2πππ 2 45Β°
ZY
16.
= 12 β 2
1
2
2 1
1.
2.
DO O
=1β 2Γ 2 = 1 β 1= 0 RHS = 0 vd ep&gpf;fg;gl;lJ.
8. mstpay;
xU jpz;k Neh;tl;l cUisapd; Muk; 7 nr.kP. kw;Wk; cauk; 20 nr.kP vdpy; mjd; tisgug;G> nkhj;j Gwg;gug;G Mfpatw;iwf; fhz;f. jpz;k Neh;tl;l cUis: r = 7 nr.kP. h = 20 nr.kP. tisgug;G = 2ππβ r.m nkhj;j Gwg;gug;G = 2ππ(β + π) r.m 22 22 = 2 Γ Γ 7 Γ 20 = 2 Γ Γ 7 (20 + 7) 7 7 = 880 r.nr.kP = 1188 r.nr.kP xU jpz;k Neh;tl;l cUisapd; Muk; 14 nr.kP. kw;Wk; cauk; 8 nr.kP. vdpy; mjd; tisgug;G> nkhj;j Gwg;gug;G Mfpatw;iwf; fhz;f. (Apr-2014) jpz;k Neh;tl;l cUis: r =14 nr.kP h = 8 nr.kP
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tisgug;G = 2ππβ r.m 22 = 2 Γ Γ 14 Γ 8 7
nkhj;j Gwg;gug;G = 2ππ(β + π) r.m 22 = 2 Γ Γ 14 (8 + 14) 7
= 704 r.nr.kP
= 1936 r.nr.kP
,U Neh;tl;l cUisfspd; Muq;fspd; tpfpjk; 3 : 2 vd;f. NkYk; cauq;fspd; tpfpjk; 5 : 3 vdpy; tisgug;Gfspd; tpfpjk; fhz;f. (June-2015) Neh;tl;l cUis: π1 βΆ π2 = 3 βΆ 2, β1 βΆ β2 = 5 βΆ 3 tisgug;Gfspd; tpfpjk; = 2ππ1 β1 : 2ππ2 β2 = 3 Γ 5 : (2 Γ 3) = 5 : 2
4.
xU jpz;k cUisapd; Muk; 14 nr.kP. mjd; cauk; 30 nr.kP. vdpy; cUisapd; fd msitf; fhz;f. (Oct-2012) jpz;k cUis: r =14 nr.kP h=30 nr.kP fd msT = ππ 2 β f.m =
22 7
Γ 14 Γ 14 Γ 30
=18480 f.nr.kP.
ST UD Y
3.
,U Neh;tl;l cUisfspd; Muq;fspd; tpfpjk; 2 : 3 vd;f. NkYk; cauq;fspd; tpfpjk; 5 : 3 vdpy; fdmsTfspd;; tpfpjk; fhz;f. Neh;tl;l cUis: π1 βΆ π2 = 2 βΆ 3, β1 βΆ β2 = 5 βΆ 3 2 fdmsTfspd;; tpfpjk; = ππ1 β1 : ππ22 β2 = 2 Γ 2 Γ 5 : (3 Γ 3 Γ 3) = 20 : 27
6.
xU jpz;k Neh;tl;lf; $k;gpd; Muk; kw;Wk; rhAauk; KiwNa 35 nr.kP. kw;Wk; 37 nr.kP. vdpy; mjd; tisgug;G kw;Wk; nkhj;jg; gug;Gf; fhz;f. jpz;k Neh;tl;lf; $k;G: r = 35 nr.kP π = 37 nr.kP tisgug;G = ππ π r.m 22 7
Γ 35 Γ 37
DO O
=
ZY
5.
= 4070 r.nr.kP nkhj;jGwg;gug;G = ππ (π + π) r.m = =
7.
22 7 22 7
Γ 35 (37 + 35)
Γ 35 Γ 72
= 7920 r.nr.kP xU jpz;k Neh;tl;lf; $k;gpd; mbr;Rw;wsT 236 nr.kP. mjd; rhAauk; 12 nr.kP. vdpy; mjd; tisgug;Gf; fhz;f. (Apr-2013, Jun-2013) jpz;k Neh;tl;lf; $k;G: l =12 nr.kP 2ππ = 236 πr =
236 2
= 118
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tisgug;G = π ππ r.m = 118 Γ 12 = 1416 r.nr.kP xU jpz;k Neh;tl;lf; $k;gpd; Muk; kw;Wk; cauk; KiwNa 7 nr.kP. kw;Wk; 24 nr.kP. vdpy; mjd; tisgug;G kw;Wk; nkhj;jg; gug;Gf; fhz;f. (Jun-12) jpz;k Neh;tl;lf; $k;G: r =7 nr.kP h= 24 nr.kP π
π2 + β2
=
= 72 + 242 = 49 + 576 = = 25 nr.kP tisgug;G
625
= πππ r.m =
22 7
Γ 7 Γ 25
=550 r.nr.kP nkhj;j Gwg;gug;G = ππ(π + π) r.m = 9.
22 7
ST UD Y
8.
Γ 7 Γ (25 + 7) =704 r.nr.kP
kuj;jpyhd xU jpz;kf; $k;gpd; mbr;Rw;wsT 44 nr.kP. mjd; cauk; 12 nr.kP. vdpy; mjd; fd msT fhz;f. (Oct-2013) jpz;kf; $k;G: h =12 nr.kP 2πr = 44 nr.kP 2 Γ
22 7
Γ π = 44 π = 44 Γ
1 2
Γ
7 22
fd msT
= 616 f.nr.kP
Γ 7 Γ 12
xU jpz;kf; $k;gpd; Muk; kw;Wk; rhAauk; KiwNa 20 nr.kP. kw;Wk; 29 nr.kP. vdpy; mjd; fd msitf; fhz;f. (Apr-2012). jpz;kf; $k;G: r = 20 nr.kP l = 29 nr.kP
DO O
10.
ZY
r = 7 nr.kP 1 = ππ 2 β f.m 3 1 22 = Γ Γ 7 3 7
h = π2 β π 2 = 292 β 202 = 841 β 400 = 441 = 21 h = 21 nr.kP
fd msT =
=
1 ππ 2 β f.m 3 1 22 Γ Γ 20 3 7
Γ 20 Γ 21
= 8800 f.nr.kP
11.
Xh; miuf;Nfhsj;jpd; fd msT 1152 π
f.nr.kP. vdpy; mjd; tisg;gug;igf; fhz;f. miuf;Nfhsj;jpd; fd msT = 1152 π f.nr.kP. 2 3
πr 3 = 1152 π
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π 3 = 1152 Γ
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3 2
π 3 = 1728 r = 12 nr.kP tisgug;G = 2ππ 2 r.m = 2 Γ π Γ 12 Γ 12 = 288 π r.nr.kP
2 Γ
22 7
Γ 12 Γ (β + 7) = 660 (β + 7) = 660 Γ
1 2
Γ
7 22
β + 7 = 15 β = 15 β 7 = 8 nr.kP h = 8 nr.kP tisg;gug;G = 2ππβ r.m 22 7
= 2 Γ
ST UD Y
xU jpz;k Neh; tl;l cUisapd; nkhj;jg; Gwg;gug;G 660 r.nr.kP. mjd; mbg;gf;fj;jpd; tpl;lk; 14 nr.kP. vdpy;> mt;TUisapd; cauk;> tisg;gug;igf; fhz;f. (Oct-2012) jpz;k Neh; tl;l cUis: tpl;lk; = 14 nr.kP Muk; = 7 nr.kP nkhj;jg;gug;G = 660 r.nr.kP 2ππ (β + π) = 660
12.
Γ 7Γ 8
= 352 r.nr.kP
Γ
1 7
fd msTfspd; tpfpjk;
ZY
7 nr.kP. Muk; nfhz;l Nfhs tbt gY}dpy; fhw;W nrYj;jg;gLk; NghJ Muk; 14 nr.kP. Mf mjpfhpj;jhy;> mt;tpU epiyfspy; gY}dpd; fd msTfspd; tpfpjj;ijf; fhz;f. Nfhsk; : π1 = 7 nr.kP π2 = 14 nr.kP
13.
4 3
πr13 :
4 3
πr23
= 7 Γ 7 Γ 7 : 14 Γ 14 Γ 14 = 1: 8
DO O
14. 14nr.kP. gf;f msT nfhz;l xU fd rJuj;jpy; ,Ue;J ntl;br; vLf;fg;gLk; kpfg;nghpa $k;gpd; fd msT fhz;f. $k;gpd; tpl;lk; = 14 nr.kP Muk; (r) = 7 nr.kP cauk; (h) = 14 nr.kP fd msT
=
=
1
3 1 3
ππ 2 β f.m
Γ
22 7
Γ 7 Γ 7 Γ 14
= 718.67 f.nr.kP
15.
8.4 nr.kP tpl;lk; nfhz;l xU Nfhs tbt jpz;k cNyhf vwpf;Fz;bd; fd msitf; fhz;f.
(Apr-2013, Jun-2015)
Nfhsj;jpd; tpl;lk; = 8.4 nr.kP Muk; (r)
= 4.2 nr.kP =
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42 10
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4 ππ 3 f.m 3 4 22 42 = Γ Γ 3 7 10 38808 = 125
fd msT =
ππ 2 = 1386 miuf;Nfhsj;jpd; nkhj;jg;gug;G = 3ππ 2 = 3 Γ 1386 = 4158 r.nr.kP. ,U jpz;k miuf;Nfhsq;fspd; Muq;fs; 3:5 vd;w tpfpjj;jpy; cs;sd. mtw;wpd; tisgug;Gfspd; tpfpjk; kw;Wk; nkhj;jg; gug;Gfspd; tpfpjk; fhz;f. jpz;k miuf;Nfhsk;: π1 : π2 = 3 βΆ 5 2 tisgug;Gfspd; tpfpjk; = 2ππ1 βΆ 2ππ22 = 3 Γ 3 : ( 5 Γ 5) = 9 βΆ 25 nkhj;jg; gug;Gfspd; tpfpjk; = 3ππ12 : 3ππ22 = 3 Γ 3 : (5 Γ 5 ) = 9 βΆ 25 98.56 r.nr.kP. Gwg;gug;G nfhz;l xU jpz;kf; Nfhsj;jpd; Muj;ijf; fhz;f. (Oct-2013) jpz;kf; Nfhsk;: Gwg;gug;G = 98.56 r.nr.kP. 4ππ 2 = 98.56 4 Γ
22 7
Γ π 2 = 98.56
π 2 = 98.56 Γ
1 4
Γ
7 22
π 2 = 7.84 Muk; r = 2.8 nr.kP 19.
2772 2
ZY
18.
42 10
fd msT = 310.464 f.nr.kP xU jpz;k miuf;Nfhsj;jpd; tisgug;G 2772 r.nr.kP. vdpy; mjd; nkhj;jg; gug;igf; fhz;f. miuf;Nfhsj;jpd; tisgug;G = 2ππ 2 = 2772 r.nr.kP. ππ 2 =
17.
Γ
ST UD Y
16.
42 10
Γ
xU jpz;k miuf;Nfhsj;jpd; nkhj;jg;gug;G ππππ
r.nr.kP. vdpy; mjd; tisg;gug;igf; fhz;f. miuf;Nfhsj;jpd; nkhj;jg;gug;G
DO O
= 675π 675π πr 2 = = 225π 3 tisg;gug;G = 2ππ 2 = 225π Γ 2 = 450π r.nr.kP
20.
=
(Jun-2014)
3ππ 2
xU cs;sl P w;w cUisapd; cs; kw;Wk; ntsp Muq;fs; KiwNa 12 nr.kP. kw;Wk; 18 nr.kP. NkYk; mjd; cauk; 14 nr.kP. vdpy; mjd; tisgug;G kw;Wk; nkhj;jg;gug;G fhz;f. (Apr-2012) cs;sl P w;w cUis: r = 12 nr.kP R= 18 nr.kP h=14 nr.kP tisgug;G = 2πβ (π
+ π) r.m nkhj;jg;gug;G = 2π(π
+ π) (π
β π + β) r.m = 2Γ
22 7
Γ (18 + 2)(18 β 12 + 14)
= 2 Γ 22 Γ 2 Γ 30
= 2Γ
22 7
Γ 30 Γ 20
= 2640 r.nr.kP
= 3771.43 r.nr.kP
= 2Γ
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22 7
Γ 14 Γ (18 + 12)
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21.
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14 kP. MoKs;s xU fpzw;wpd; Muk; 5 kP. xU rJu kPl;lUf;F &.2 tPjk; mf;fpzw;wpd; cl;Gwr; Rtw;wpw;F rpnkz;l; G+r MFk; nrytpidf; fhz;f. (Oct-2014) cUis : r = 5 kP h=14 kP fpzw;wpd; cl;Gwg; gug;G (cUisapd; tisgug;G)
= 2ππβ = 2 Γ
22 7
Γ 5 Γ 14
= 2 Γ 22 Γ 5 Γ 2 = 440 r.kP = &.2 = &. 440 Γ 2 = &.880
ST UD Y
xU rJu kPl;lUf;F rpnkz;l; G+r MFk; nryT 440 rJu kPl;lUf;F rpnkz;l; G+r MFk; nryT
22. KiwNa 3 nr.kP. kw;Wk; 4 nr.kP. Muq;fshff; nfhz;l ,U Nfhsq;fspd; fd msTfspd; tpfpjj;jpidf; fhz;f. (Jun-2012) Nfhsk; : R = 4 nr.kP r = 3 nr.kP ,U Nfhsq;fspd; fd msTfspd; tpfpjk; =
4 ππ
3 3 π
3 : π3
βΆ
4 3
ππ 3
= = 43 : 33 = 64: 27
62.37 f.nr.kP. fd msT nfhz;l xU jpz;k Neh;tl;l cUisapd; cauk; 4.5 nr.kP. vdpy; mt;TUisapd; Muj;ijf; fhz;f. (Jun-2013) cUisapd; cauk; h = 4.5 nr.kP fd msT V = 62.37 f.nr.kP. Muk; r =? fd msT π = ππ 2 β = 62.37 f.nr.kP 22 7
π2 =
Γ r 2 Γ 4.5 = 62.37 62.37 Γ 7 22 Γ 4.5
π= Muk; r
4.41 = 2.1 nr.kP = 2.1 nr.kP
miuf;Nfhs tbt fpz;zj;jpd; jbkd; 0.25 nr.kP kw;Wk; mjd; cl;Gw Muk; 5 nr.kP vdpy; mf;fpz;zj;jpd; ntspg;Gw tisgug;igf; fhz;f. (Apr-2015) miuf;Nfhsk;: cl;Gw Muk; r = 5 nr.kP jbkd; w = 0.25 nr.kP β΄ ntspg;Gw Muk; R = 5+ 0.25 = 5.25 nr.kP
DO O
24.
ntspg;Gw
25.
= 4.41
ZY
23.
tisgug;G = 2ππ
2 = 2 Γ
22 7
Γ 5.25 Γ 5.25
= 173.25 r.nr.kP
xU cs;sl P w;w Nfhsj;jpd; ntsp kw;Wk; cs; Muq;fs; KiwNa 12 nr.kP. kw;Wk; 10 nr.kP. vdpy; mf;Nfhsj;jpd; fd msitf; fhz;f. cs;sl P w;w Nfhsk;: ntspg;Gw Muk; R = 12 nr.kP cl;Gw Muk;
r = 10 nr.kP
cs;sl P w;w Nfhsj;jpd; fd msT
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π =
4 3
π (π
3 β π 3 ) =
4 3
Γ
22 7
(123 β 103 )
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88 21
=
88 21
1728 β 1000 Γ 728
= 3050
xU jpz;k Neh;tl;lf; $k;gpd; fd msT 4928 f.nr.kP. kw;Wk; mjd; cauk; 24 nr.kP vdpy; mf;$k;gpd; Muj;ijf; fhz;f. $k;gpd; fd msT = 4928 nr.kP kw;Wk; cauk; h = 24 nr.kP 1 3
Γ
22 7
π2 = π = 27.
Γ
1 πr 2 h = 3 r 2 Γ 24 =
4928 Γ 3Γ 7 22Γ 24
4928 4928
= 196
196 =14 nr.kP
xU Neh; tl;lf; $k;gpd; fd msT πππ π
cauj;ijf; fhz;f. $k;gpd; Muk; r = 9 nr.kP $k;gpd; fd msT = 216 π f.nr.kP $k;gpd; fd msT = 1 3
1 3
Γ π Γ 92Γ β
ππ 2 β = 216 π = 216 π
β=
216 Γ 3 9Γ9
= 8 nr.kP
7 kP cs;tpl;lKs;s xU Nfhsj;jpDs; xU rh;f;f]; tPuh; Nkhl;lhh; irf;fpspy; rhfrk; nra;fpwhh;. me;j rhfr tPuh; rhfrk; nra;af; fpilj;jpLk; cs;sl P w;wf; Nfhsj;jpd; cl;Gwg;gug;igf; fhz;f. cs;sl P w;w Nfhsj;jpd; cs; tpl;lk; 2r = 7kP Nkhl;lhh; irf;fps; tPuh;> rhfrk; nra;af; fpilj;jpLk; gug;G = Nfhsj;jpd; cl;Gw tisg;gug;G = 4 ππ 2 = π(2π)2 =
1 3
30.
22 7
Γ 72 =
22 7
Γ 7 Γ 7 = 154 nr.kP2
xU $k;G> xU miuf;Nfhsk; kw;Wk; xU cUis Mfpad rk mbg;gug;gpidf; nfhz;Ls;sd. $k;gpd; cauk;> cUisapd; cauj;jpw;F rkkhfTk;> NkYk; ,t;Tauk;> mtw;wpd; Muj;jpw;F rkkhfTk; ,Ue;jhy; ,k;%d;wpd; fd msTfSf;fpilNa cs;s tpfpjj;ijf; fhz;f. $k;G> miuf;Nfhsk;> cUis Mfpatw;wpd; nghJthd Muk; r vd;f. $k;G> cUis Mfpatw;wpd; nghJthd cauk; h vd;f. fzf;fpd;gb r = h $k;G> miuf;Nfhsk;> cUis Mfpatw;wpd; fd msTfspd; tpfpjk;
DO O
29.
f.nr.kP kw;Wk; mf;$k;gpd; Muk; 9 nr.kP vdpy;> mjd;
ZY
28.
f.nr.kP
ST UD Y
26.
2 3
πr 2 h:
2 3
πr 3 : πr 2 h =
1 3
βΆ
2 3
βΆ1
[r = h]
=1:2:3 4.2 kP tpl;lKs;s miuf;Nfhs tbtj; njhl;bapy; epug;gg;gLk; ePhpd; fd msitf; fhz;f.
(Apr-2014)
miuf;Nfhs tbtj; njhl;bapd; tpl;lk; d= 4.2 kP
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Muk; r = 2.1 m 2 3 2 3
miuf;Nfhs tbtj; njhl;bapd; fd msT = ππ 3 =
x
22 7
x 2.1 x 2.1 x 2.1
miuf;Nfhs tbtj; njhl;bapd; fd msT = 19.404 kP 3 = 19404 ypl;lh;
= 1000 ypl;lh;]
14 nr.kP. gf;f msT nfhz;l xU fdr; rJuj;jpypUe;J> kpfg;nghpa (kPg;ngU fd msTs;s) Nfhsk; ntl;b vLf;fg;gLfpwJ. vdpy;> mf;Nfhsj;jpd; fd msT fhz;f. (Jun- 2014) fdr;rJuj;jpd; gf;f msT = Nfhsj;jpd; tpl;lk; d= 14 nr.kP Nfhsj;jpd; Muk; r=7 nr.kP Nfhsj;jpd; fd msT
32.
3
=
4 ππ 3 3 4 22
ST UD Y
31.
[1kP
= Γ
Γ 7Γ 7Γ 7
=
1437 nr.kP
3 7 4312 = 3
1 3
21 nr.kP MuKs;s xU tl;lj;jpypUe;J ππππ ikaf;Nfhzk; nfhz;l xU tl;lf; Nfhzg;gFjpia ntl;bnaLj;J> mjd; Muq;fis xd;wpizj;J xU $k;ghf;fpdhy;> fpilf;Fk; $k;gpd; tisgug;igf; fhz;f. (Oct-2014) tl;lf; Nfhzg;gFjpapd; Muk; R = 21 nr.kP ikaf;Nfhzk; π = 1200 tpy;ypd; ePsk;
π=
π 360 0
Γ 2ππ
ZY
xU tl;lf; Nfhzg;gFjpapd; Muq;fis xd;wpizj;J xU $k;ghf;fpdhy;> $k;gpd; mbr;Rw;wsT = tl;lf; Nfhzg;gFjpapd; tpy;ypd; ePsk; 2ππ =
120 0 360 0
Γ 2π Γ 21 7
DO O
3 π = 7 nr.kP $k;gpd; rhAauk; π = tl;lf; Nfhzg;gFjpapd; Muk; $k;gpd; tisgug;G
= 21 nr.kP = πππ =
22 7
Γ 7 Γ 21 = 462 nr.kP2
khw;W Kiw: $k;gpd; tisgug;G = tl;lf; Nfhzg;gFjpapd; gug;G π = Γ ππ
2 360 0 7 =
120 0 360 0
Γ
22 7
Γ 21 Γ 21 = 462 nr.kP2
3
33.
24 nr.kP cauKila xU $k;gpd; mbg;gug;G 550 nr.kP2 vdpy; mjd; fd msT ahJ? (Oct-15) $k;gpd; cauk; h = 24 nr.kP 2 mjd; mbg;gug;G ππ = 550 nr.kP2
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$k;gpd; fd msT
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= = =
1 3 1 3 1 3
ππ 2 β Γ ππ 2 Γ β Γ 550 Γ 24
= 4400 nr.kP3 xU cUis tbt ,Uk;Gf; Fohapd; ntspg;Gw tpl;lk; 25 nr.kP mjd; ePsk; 20 nr.kP. kw;Wk; jbkd; 1 nr.kP. vdpy; mf;Fohapd; nkhj;jg;Gwg;gug;igf; fhz;f. (Oct-2015) cUis tbt ,Uk;Gf; Fohapd; ntspg;Gw tpl;lk; D = 25 nr.kP / ntspg;Gw Muk; R
= 12.5 nr.kP
mjd; jbkd;
= 1 nr.kP
w
/ cl;Gw Muk;
r =Rβw = 12.5 β 1 r = 11.5 nr.kP
ST UD Y
34.
cUis tbtf; Fohapd; ePsk; (cauk;) h Fohapd; nkhj;jg;Gwg;gug;G
= 20 nr.kP
= 2π π
+ π (π
β π + β) r.m
= 2π 12.5 + 11.5 (12.5 β 11.5 + 20) 22 x 7
24 x 21
ZY
=2x
= 3168 nr.kP2
35.
xU jpz;k Neh;tl;l cUisapd; tisgug;G kw;Wk; mbr;Rw;wsT KiwNa 4400 r.nr.kP. kw;Wk; 110 nr.kP vdpy; mt;TUisapd; cauj;ijAk;> tpl;lj;ijAk; fhz;f. (Apr-2016) cUisapd; tisgug;G = 4400r.nr.kP.
DO O
xU jpz;k Neh;tl;l cUisapd; 2Γ
22 7
mbr;Rw;wsT 2ππ = 110 nr.kP
Γ π = 110
tpl;lk; 2π =
110Γ7 =35cm 22
tisgug;G 2ππβ = 110 Γ β = 4400 vdNt> cUisapd; cauk; β =
36.
4400 110
= 40 nr.kP
πππΒ° ikaf; NfhzKk; 21 nr.kP. MuKk; nfhz;l tl;lNfhz tbtpyhd ,Uk;Gj; jfl;bd; Muq;fis ,izj;J xU $k;G cUthf;fg;gLfpwJ vdp;y; mf;$k;gpd; Muj;ijf; fhz;f.
(Apr-2016)
180Β° ikaf; NfhzKk; 21 nr.kP. MuKk; nfhz;l tl;lNfhz tbtpyhd ,Uk;Gj; jfl;bd; Muq;fis ,izj;J xU $k;G cUthf;fg;gLfpwJ. R ;$k;gpd; Muk; vd;f.
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$k;gpd; mbr;Rw;wsT = tl;ltpy;ypd; ePsk; 2ππ
= π
=
π 360
180 360
Γ 2ππ
Γ 21 = 10.5 nr.kP.
43, 24, 38, 56, 22, 39, 45 Mfpa Gs;sp tptuq;fspd; tPr;R kw;Wk; tPr;Rf; nfO fhz;f. (Apr-2012, 2015) L = 56, S = 22 tPr;R = LβS = 56 β22 = 34 tPr;Rf; nfO
2.
=
34 78
=
56β22 56+22 17 39
= 0.435
πΏβπ πΏ+π
=
50.5β42.4 50.5+42.4
=
8.1 92.9
=
81 929
= 0.0871
gpd;tUk; kjpg;GfSf;F tPr;R kw;Wk; tPr;Rf; nfO fhz;f. 59, 46, 30, 23, 27, 40, 52,35, 29.(Oct-2014) L = 59, S = 23 tPr;R = L β S = 59 β 23 = 36 πΏβπ πΏ+π 36 82
= =
59β23 59+23
18 41
= 0.4390
DO O
=
gpd;tUk; kjpg;GfSf;F tPr;R kw;Wk; tPr;Rf; nfO fhz;f. 41.2, 33.7, 29.1, 34.5, 25.7, 24.8, 56.5, 12.5. L= 56.5, S=12.5 tPr;R =L β S = 56.5 β 12.5 = 44 tPr;Rf; nfO =
5.
=
=
tPr;Rf; nfO =
4.
πΏβπ πΏ+π
xU tFg;gpYs;s 13 khzth;fspd; vil (fp.fp) gpd;tUkhW: 42.5, 47.5, 48.6, 50.5, 49, 46.2, 49.8, 45.8, 43.2, 48, 44.7, 46.9, 42.4. ,tw;wpd; tPr;R> tPr;Rf;nfO fhz;f. L= 50.5, S= 42.4 tPr;R = L β S = 50.5 β 42.4 = 8.1 tPr;Rf; nfO
3.
=
ZY
1.
ST UD Y
11. Gs;spapay;
πΏβπ πΏ+π
=
56.5β12.5 56.5+12.5
=
44 69
= 0.6376
xU Gs;sp tptuj;jpd; kPr;rpW kjpg;G 12. mjd; tPr;R 59 vdpy; mg;Gs;sp tptuj;jpd; kPg;ngU kjpg;igf; fhz;f. tPr;R = 59, S = 12 tPr;R = L β S
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59 59+12 71 L
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= L β 12 =L =L = 71
50 msTfspy; kpfg;nghpa kjpg;G 3.84 fp.fp. mjd; tPr;R 0.46 fp.fp. vdpy;> mitfspd; kPr;rpW kjpg;igf; fhz;f. (Apr-2013) L = 3.84, tPr;R = 0.46 tPr;R = L β S 0.46 = 3.84 βS S = 3.84 β 0.46 S = 3.38 fp.fp.
7.
xU Gs;sp tptuj; njhFg;gpd; kPg;ngU kjpg;G 7.44 kw;Wk; tPr;R 2.26 vdpy;> mj;njhFg;gpd; kPr;rpW kjpg;igf; fhz;f. (Oct-2012) L = 7.44, tPr;R = 2.26 tPr;R = L β S 2.26 = 7.44 βS S = 7.44 β 2.26 S = 5.18 fz;lwpe;j Gs;sp tptuj; njhFg;gpYs;s 20 kjpg;Gfspd; jpl;ltpyf;fk; π vd;f. Gs;sp tptuj;jpd; xt;nthU kjpg;igAk; 2 My; ngUf;fpdhy; fpilf;Fk; Gjpa Gs;sp tptuq;fspd; jpl;ltpyf;fk; kw;Wk; tpyf;f th;f;fruhrhp fhz;f. jpl;ltpyf;fk; = 5
8.
ST UD Y
6.
Gjpa jpl;ltpyf;fk; tpyf;f th;f;fruhrhp
Kjy; 13 ,ay; vz;fspd; jpl;l tpyf;fj;ijf; fzf;fpLf. π = 13 =
13 2 β1 12
=
169β1 12
=
168 12
=
(Oct-2013, Jun-2015) 14 = 3.7
Kjy; 10 ,ay; vz;fspd; jpl;l tpyf;fj;ijf; fzf;fpLf. π = 10 jpl;ltpyf;fk; =
11.
π 2 β1 12
DO O
jpl;ltpyf;fk; = 10.
ZY
9.
= 2 Γ 5= 2 5 = (2 5)2 = 4 Γ 5 = 20
π 2 β1 12
=
10 2 β1 12
=
100β1 12
=
99 12
(Apr-2014, Jun-2014)
= 2.87
xU Gs;sp tptuj;jpd; khWghl;Lf; nfO 57 kw;Wk; jpl;ltpyf;fk; 6.84 vdpy; mjd;; $l;;Lruhrhp fhz;f. (Jun-2012, Jun-2013, Apr-2016) C.V. = 57, π = 6.84
πΆ. π. = 57 = π₯ =
π π₯
Γ 100
6.84 π₯ 6.84 57
Γ 100 Γ 100
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π₯ =
6.84 57
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= 12
π₯ = 12
π₯
3.2 = π=
π Γ 100 163.8 3.2 x 163.8 = 15.24 100
π = 5.24
ST UD Y
12. xU FOtpy; 100 Ngh; cs;sdh;. mth;fspd; cauq;fspd; $l;Lruhrhp 163.8 nr.kP. kw;Wk; khWghl;Lf; nfO 3.2 vdpy; jpl;ltpyf;fk; fhz;f. π₯ = 163.8 C.V. = 3.2, π= ? π C. V. = Γ 100
13. xU Gs;sp tptuj;jpy; 30 kjpg;Gfspd; $l;Lruhrhp kw;Wk; jpl;ltpyf;fk; KiwNa 18 kw;Wk; 3 MFk;. mtw;wpd; $l;Lj;njhifiaAk;> NkYk; mtw;wpd; th;f;fq;fspd; $l;Lj;njhifiaAk; fhz;f. π = 30, π₯ = 18, π = 3 βπ₯ = n Γ π₯ = 30 Γ 18 = 540 βπ₯ 2 π β 182
π2 = βπ₯ 2 π βπ₯ 2 30
β
βπ₯ 2 π
= 32
= 9
βπ₯ = 540 βπ₯ 2 = 9990
β 324 = 9
βπ₯ 2 β 9720 = 270 βπ₯ 2 = 270 + 9720 = 9990
π = ππ, π = ππ kw;Wk; β ππ = ππππ vdpy;> khWghl;Lf; nfOitf; fzf;fpLf. π = 10, π₯ = 12, βπ₯ 2 = 1530 jpl;ltpyf;fk;
ZY
14.
βπ₯ 2 π
π =
khWghl;Lf; nfO C.V.
= =
=
βπ₯ 2 π
=
153 β 144 =
π Γ 100 π₯ 3 Γ 100 12 1 Γ 100 = 4
DO O
=
β
1530 10
β 122
9 = 3
25
C.V. = 25
15.
xU Gs;sp tptuj; njhFg;gpYs;s 100 kjpg;Gfspd; ruhrhp kw;Wk; jpl;ltpyf;fk; KiwNa 48 kw;Wk; 10 MFk;. midj;J kjpg;Gfspd; $l;Lj;njhif kw;Wk; mitfspd; th;f;fq;fspd; $l;Lj;njhif Mfpatw;iwf; fhz;f. 100 kjpg;Gfspd; ruhrhp π₯ = 48 100 kjpg;Gfspd; $l;Lj;njhif β π₯ = 48 Γ 100 = 4800 jpl;ltpyf;fk; π = 10 vdNt> tpyf;f th;f;fruhrhp π 2 =
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βπ₯ 2 π
β
βπ₯ 2 π
= 102 = 100
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βπ₯ 2 π βπ₯ 2 π
β
4800 2 100
= 100 βΉ
β 2304 = 100 βΉ
βπ₯ 2 π
βπ₯ 2 π
β (48)2 = 100
= 100 + 2304 = 2404
100 kjpg;Gfspd; th;f;fq;fspd; $l;Lj;njhif βπ₯ 2 = 100 Γ 2404 = 2,40, 400
ST UD Y
12. epfo;jfT 1.
xU Fwpg;gpl;l ehspy; kio ngWtjw;fhd epfo;jfT 0.76 mf;Fwpg;gpl;l ehspy; kio tuhky; ,Uf;f epfo;jfT ahJ? P(A) = 0.76 P Aβ = 1β P (A) = 1β 0.76 P Aβ = 0.24
2.
Kjy; 20 ,ay; vz;fspypUe;J xU KO vz; rktha;g;G Kiwapy; Njh;e;njLf;fg;gLfpwJ. me;j vz; gfh vz;zhf ,Uf;f epfo;jfT fhz;f. (Apr-2012) S = {1, 2, 3,β¦β¦..20 n(S) = 20 gfh vz;zhf ,Uf;Fk; epfo;r;rp A = {2, 3, 5, 7, 11, 13, 17, 19} n(A) = 8
P(A)
8 20
=
2 5
35 nghUl;fs; mlq;fpa njhFg;G xd;wpy; 7 nghUl;fs; FiwghLilad. mjpypUe;J xU nghUs; rk tha;g;G Kiwapy; Njh;e;njLf;f> mJ Fiwghlw;w nghUshf ,Uf;f epfo;jfT ahJ? n(S) n(A)
= 35 = 28 ( 35 β7= 28)
P(A)
=
P(A)
4.
=
π (π΄) π (π) 4 = 5
=
28 35
=
4 5
(Jun-12, Apr-13)
DO O
3.
π (π΄) π (π) 2 = 5
=
ZY
P(A)
rktha;g;G Kiwapy; Njh;e;njLf;fg;gLk; nel;lhz;by; 53 nts;spf;fpoikfs; ,Uf;f epfo;jfT ahJ? S = {(QhapW> jpq;fs;)> (jpq;fs;> nrt;tha;)> (nrt;tha;> Gjd;)> (Gjd;> tpahod;)> (tpahod;> nts;sp)> (nts;sp> rdp)> (rdp>QhapW)}
n(S) = 7 A ={(tpahod;> nts;sp)> (nts;sp> rdp)} n(A) = 2 P(A) P(A)
π (π΄) π (π) 2 = 7
=
=
2 7
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5.
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rktha;g;G Kiwapy; Njh;e;njLf;fg;gLk; nel;lhz;by; 52 nts;spf;fpoikfs; ,Uf;f epfo;jfT ahJ? S = {(QhapW> jpq;fs;)> (jpq;fs;> nrt;tha;)> (nrt;tha;> Gjd;)> (Gjd;> tpahod;)> (tpahod;> nts;sp)> (nts;sp> rdp)> (rdp>QhapW)}
n(S) = 7 A = {(QhapW> jpq;fs;)> (jpq;fs;> nrt;tha;)> (nrt;tha;> Gjd;)> (Gjd;> tpahod;)> (rdp>QhapW)} n(A) = 5
P(A) 6.
π (π΄) π (π) 5 = 7
=
=
5 7
ST UD Y
P(A)
rktha;g;G Kiwapy; Njh;e;njLf;fg;gLk; rhjhuz Mz;by; 53 nts;spf;fpoikfs; ,Uf;f epfo;jfT ahJ? S = {QhapW> jpq;fs;> nrt;tha;> Gjd;> tpahod;> nts;sp> rdp} n(S)
=7
A= { nts;sp } =1
P(A)
=
π (π΄) π (π)
P(A)
=
1 7
=
1 7
xU rPuhd gfil ,uz;L Kiw cUl;lg;gLfpwJ. Kf vz;fspd; $Ljy; 9 fpilf;f epfo;jfT ahJ? (Oct-2012, 2014) π = { 1,1 , 1,2 , 1,3 , 1,4 , 1,5 , 1,6 ,
ZY
7.
n(A)
2,1 , 2,2 , 2,3 , 2,4 , 2,5 , 2,6 , 3,1 , 3,2 , 3,3 , 3,4 , 3,5 , 3,6 , 4,1 , 4,2 , 4,3 , 4,4 , 4,5 , 4,6 ,
8.
DO O
5,1 , 5,2 , 5,3 , 5,4 , 5,5 , 5,6 , 6,1 , 6,2 , 6,3 , 6,4 , 6,5 , (6,6)}
n(S)
= 36
A
= {(3, 6), (4, 5), (5, 4), (6, 3)}
n(A)
=4
P(A)
=
P(A)
1 = 9
π (π΄) π (π)
=
4
36
=
1 9
12 ey;y Kl;ilfSld; 3 mOfpa Kl;ilfs; fye;Js;sd. rktha;g;G Kiwapy; Njh;e;njLf;f g;gLk; xU Kl;il mOfpajhf ,Uf;f epfo;jfT ahJ? n(S) = 15 n(A) = 3
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P(A)
=
π (π΄) π (π)
= P(A) 9.
=
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3 15
=
1 5
1 5
,U ehzaq;fis xNu rkaj;jpy; Rz;Lk; NghJ> mjpfgl;rk; xU jiy fpilf;f epfo;jfT fhz;f. = {HH, HT, TH, TT} =4 = {TT, HT, TH} =3
P(A)
=
P(A)
=
=
3 4
4
mh;n[z;bdh> gq;fshNj\;> rPdh> mq;Nfhyh> U\;ah> my;[Phpah Mfpa ehLfspd; ngah;fisf; nfhz;l gl;baypypUe;J xU Rw;Wyhg; gazp rktha;g;G Kiwapy; xU ehl;bd; ngaiuj; Njh;e;njLf;fpwhh;. βmβ vd;w vOj;jpy; Muk;gkhFk; ehl;bd; ngaiuj; Njh;e;njLf;f epfo;jfT vd;d? n(S) = 6 n(A) = 3 P(A) P(A)
11.
π (π΄) π (π) 3
=
π (π΄)
π (π) 1 = 2
=
3 6
=
1 2
ed;F fiyj;J itf;fg;gl;l 52 rPl;Lfisf; nfhz;l rPl;Lf; fl;bypUe;J rk tha;g;Gr; Nrhjid Kiwapy; xU rPl;L vLf;fg;gLfpwJ. me;j rPl;L gpd;tUtdthf ,Uf;f epfo; jfTfisf; fhz;f. m) fUg;G ,uhrh M) ];NgL fhh;L. (Oct-2013) n(S) = 52 fUg;G ,uhrh n(A) = 2 π (π΄) 2 1 P(A) = = =
ZY
10.
S n(S) A n(A)
π (π)
n(S) n(B) P(B)
= 52 = 13 π (π΄) = =
DO O
];NgL fhh;L
12.
ST UD Y
(Jun-2015)
π (π)
52
13 52
26
=
1 4
ed;F fiyj;J itf;fg;gl;l 52 rPl;Lfisf; nfhz;l rPl;Lf; fl;bypUe;J rk tha;g;Gr; Nrhjid Kiwapy; xU rPl;L vLf;fg;gLfpwJ. me;j rPl;L gpd;tUtdthf ,Uf;f epfo;jfTfisf; fhz;f. m) ,uhrh M) lakz;l; 10. n(S) = 52 ,uhrh n(A) = 4 π (π΄) 4 1 P(A) = = = π (π)
lakz;l; 10
n(S) n(B) P(B)
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= 52 =1 π (π΅) = = π (π)
52
13
1 52
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%d;W gfilfs; xNu Neuj;jpy; cUl;lg;gLk;NghJ %d;W gfilfspYk; xNu vz; fpilg;gjw;fhd epfo;jfT ahJ? (Apr-2014) S = 1, 1, 1 , 1, 1, 2 , β¦β¦.. 6, 6, 6 n(S) = 216 A = {(1, 1, 1), (2, 2, 2), (3, 3, 3), (4, 4, 4), (5, 5, 5), (6, 6, 6)} n(A) = 6 π (π΄) 6 1 P(A) = = = π (π)
14.
216
36
%d;W ehzaq;fs; xNu Neuj;jpy; Rz;lg;gLfpd;wd. gpd;tUk; epfo;r;rpfSf;F epfo;jfT fhz;f. (i) Fiwe;jJ xU jiy fpilg;gJ (ii) ,U G+f;fs; kl;Lk; fpilg;gJ (iii) Fiwe;jJ ,U jiyfs; fpilg;gJ S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} n(S) = 8 (i) Fiwe;jJ xU jiy fpilg;gJ A = {HHH, HHT, HTH, HTT, THH, THT, TTH} n(A) = 7 π (π΄) 7 P(A) = = π (π)
ST UD Y
13.
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8
(ii) ,U G+f;fs; kl;Lk; fpilg;gJ B = {HTT, TTH, THT} n(B) = 3 π (π΅) 3 P(B) = = π (π)
8
(iii) Fiwe;jJ ,U jiyfs; fpilg;gJ C = {HHH, HTH, HHT, THH} n(C) = 4
P(A)
, P(C) =
P(A)
(ii) khztp Njh;e;njLf;fg;gl epfo;jfT
16.
1 2
xU tFg;gpy; cs;s 35 khzth;fspy; 20 Ngh; Mz;fs; kw;Wk; 15 Ngh; ngz;fs;. rktha;g;G Kiwapy; xU khzth; Njh;e;njLf;fg;gLfpwhh; vdpy;> gpd;tUk; epfo;r;rpfspd; epfo;jfTfisf; fhz;f. (i) Njh;e;njLf;fg;gLgth; khztdhf ,Uj;jy;> (ii) Njh;e;njLf;fg;gLgth; khztpahf ,Uj;jy;. tFg;gpy; cs;s khzth;fspd; vz;zpf;if n(S) = 35 (i) khztd; Njh;e;njLf;fg;gl epfo;jfT n(A) = 7
DO O
15.
π (πΆ) 4 1 = = π (π) 8 2 7 3 = , P(B) = 8 8
=
ZY
P(C)
P(B)
=
π (π΄) π (π)
n(B)
=
20 35
=
4 7
= 15
π (π΅) 15 = = π (π) 35
=
3 7
xU igapy; 5 rptg;G kw;Wk; rpy ePy epwg;ge;Jfs; cs;sd. mg;igapypUe;J xU ePy epwg; ge;ij vLg;gjw;fhd epfo;jfT> xU rptg;G epwg; ge;ij vLg;gjw;fhd epfo;jftpd; %d;W klq;F vdpy;> mg;igapYs;s ePy epwg; ge;Jfspd; vz;zpf;ifiaf; fhz;f. igapy; 5 rptg;G epwg;ge;Jfs;> π₯ ePy epwg; ge;JfSk; cs;sd. vdNt> n(S) = 5 + π₯ B vd;gJ xU ePy epwg;ge;ij vLf;Fk; epfo;r;rp kw;Wk; R vd;gJ xU rptg;G epwg;ge;ij vLf;Fk; epfo;r;rp vd;f. fzf;fpd;gb P(B) = 3P(R)
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π (π΅) π π
=3 π (π) π (π) π₯ 5 =3 5+π₯ 5+π₯
β π₯ = 3 Γ 5 = 15 ePy epwg; ge;Jfspd; vz;zpf;if π₯ = 15 xU rktha;g;Gr; Nrhjidapy; xU epfo;r;rp A vd;f. me;;epfo;r;rpapd; epug;G epfo;r;rp π¨ vd;f. P(A) : P(π¨ ) = 7:12, vdpy;> P(A)- If; fhz;f.
ST UD Y
17.
π(π΄) 7 = P(π΄ ) 12
P(A) : P(π΄ ) = 7:12 vdNt
12 Γ P(A) = 7Γ P(π΄ ) 12P(A) = 7[1βP(A) ] = 7 β 7P(A) 19P(A) = 7 P (A) = 18.
7 19
A kw;Wk; B vd;gd xd;iwnahd;W tpyf;Fk; epfo;r;rpfs;. NkYk; P(A) =
π π
kw;Wk; P(B) =
π , π
vdpy;
P(AβͺB)-If; fhz;f. A kw;Wk; B vd;gd xd;iwnahd;W tpyf;Fk; epfo;r;rpfs; vdNt> P(AβͺB) = P(A) + P(B) 3 1 π P(AβͺB) = + = [P(A β© B) = 0] 5
19.
A kw;Wk; B vd;w ,uz;L epfo;r;rpfspy; P(A)
π = π
5
, P(B) =
fhz;f. P(AβͺB) = P(A) + P(B) β P(Aβ©B) 1 2 1 5+8 β 10 P(Aβ©B) = P(A) +P(B) β P (AβͺB) = + β = =
21.
20
vdpy; , P(Aβ©B)-If;
π ππ
26
26
26
13
1 Kjy; 100 tiuapyhd KO vz;fspypUe;J rk tha;g;G Kiwapy; Njh;e;njLf;fg; gLk; xU vz; xU KO th;f;fkhf ,Uf;f epfo;jfT fhz;f. $Wntsp S = 1, 2, 3,β¦..100 n(S) = 100 KO th;f;f vz;fs; A = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100} n(A) = 10 π (π΄) 10 1 P(A) = = = π (π)
22.
2
π π
xU igapy; 10 nts;is> 6 rptg;G kw;Wk; 10 fUg;G epwg;ge;Jfs; cs;sd. rktha;g;G Kiwapy; xU ge;jpid vLf;Fk;NghJ mJ nts;is my;yJ rptg;G epwg; ge;jhf ,Ug;gjw;fhd epfo;jftpidf; fhz;f. nts;isg; ge;Jfspd; vz;zpf;if n(W) = 10, rptg;Gg; ge;Jfspd; vz;zpf;if n(R) = 6, fUg;Gg; ge;Jfspd; vz;zpf;if n(B) =10 /$Wntsp n(S) = 10+6+10=26 nts;is my;yJ rptg;G epwg;ge;J vLf;f epfo;jfT 10 6 16 8 P(WβͺR) = P(W) + P(R) = + = = [P(Wβ©R)= 0]
DO O
20.
5
kw;Wk; P(AβͺB) =
ZY
4
π
π π
100
10
1 Kjy; 100 tiuapyhd KO vz;fspypUe;J rk tha;g;G Kiwapy; Njh;e;njLf;fg; gLk; xU vz; xU KO fdkhf ,y;yhky; ,Uf;f epfo;jfT fhz;f. $Wntsp S = 1, 2, 3, β¦β¦.100
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n (S) = 100 KO fd vz;fs; A = {1, 8, 27, 64} n(A) = 4 π (π΄) 4 1 epfo;jfT P(A) = = = π (π)
100
25
25
25
25
ST UD Y
xU KO fdkhf ,y;yhky; ,Uf;f epfo;jfT P Aβ = 1βP(A) 1 25β1 24 = 1β = = 23. 20 rPl;Lfspy; 1 Kjy; 20 tiuAs;s KO vz;fs; Fwpf;fg;gl;Ls;sd. rktha;g;G Kiwapy; xU rPl;L vLf;fg;gLfpd;wJ. mt;thW vLf;fg;gl;l rPl;bYs;s vz; (1) 4-d; klq;fhf ,Uf;f epfo;jfT fhz;f. (2) 6-d; klq;fhf ,y;yhky; ,Uf;f epfo;jfT fhz;f. (Jun-2014, Apr-2016) $Wntsp S = 1, 2, 3, β¦β¦.20 n(S)= 20 (1) 4 -d; klq;Ffs; A = {4, 8, 12, 16, 20} n(A) = 5 π (π΄) 5 1 4-d; klq;fhf ,Uf;f epfo;jfT P(A) = = = π (π)
(2)
6 βd; klq;Ffs; B={6,12,18} π(π΅) π π΅ = 3, π π΅ = = π (π)
3 20
6-d; klq;fhf ,y;yhky; ,Uf;f epfo;jfT
3 20
=
17 20
100
10
DO O
,U gfilfs; xU Nru cUl;lg;gLfpd;wd. Kf vz;fisf; nfhz;L mikf;fg;gLk;
26.
π π΅ = 1βπ π΅ =1β
xU igapy; cs;s 1 Kjy; 100 tiuAs;s vz;fshy; Fwpf;fg;gl;l rPl;LfspypUe;J xU rPl;L vLf;fg;gLfpwJ. mt;thW vLf;fg;gLk; rPl;bd; vz; 10 My; tFgLk; vz;zhf ,Ug;gjw;fhd epfo;jftpidf; fhz;f. $Wntsp S = 1, 2, 3, β¦β¦.100 n(S) = 100 10 My; tFgLk; vz;fs; A = 10, 20, 30,β¦.100 n(A) = 10 π (π΄) 10 1 epfo;jfT P(A) = = = π (π)
25.
4
ZY
24.
20
36
3
ed;F fiyj;J mLf;fp itf;fg;gl;l 52 rPl;Lfisf; nfhz;l rPl;Lf;fl;bypUe;J rk tha;g;G Kiwapy; xU rPl;L vLf;fg;gLfpwJ. me;j rPl;L ];NglhfNth (spade) my;yJ ,uhrhthfNth (king) ,Ug;gjw;fhd epfo;jftpidf; fhz;f. (Jun-2013) nkhj;j rPl;Lfspd; vz;zpf;if π π = 52 ];NgL rPl;Lfspd; vz;zpf;if π π΄ = 13 π (π΄) 13 epfo;jfT π π΄ = = π (π)
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52
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,uhrh rPl;Lfspd; vz;zpf;if π π΅ = 13 π (π΅) epfo;jfT π π΅ = = π (π)
];NgL rPl;lhfTk; ,uhrh rPl;lhfTk; ,Uf;Fk; rPl;Lfspd; vz;zpf;if epfo;jfT
4 52
π π΄β©π΅ = 1 π (π΄β©π΅) π π΄β©π΅ = = π (π)
1 52
ST UD Y
/ rPl;L ];NglhfNth my;yJ ,uhrhfNth ,Ug;gjw;fhd epfo;jfT π π΄ βͺ π΅ = π π΄ + π π΅ β π(π΄ β© π΅) 13 4 1 17β1 16 = + β = = = 52
52
52
52
4 13
2 nr.kP. MuKs;s tl;lj;jpd; cs;Ns cs;s Gs;spfspy; rktha;g;G Kiwapy; xU Gs;spiaj; Njh;T nra;a mJ tl;lj;ij (ghpjp) tpl ikaj;Jf;F mUfpy; ,Uf;f epfo;jfT vd;d? (Oct-15) 2nr.kP. MuKs;s tl;lj;jpd; cs;Ns cs;s Gs;spfs; vd;gJ mt;tl;lj;jpd; gug;gsthFk;. vdNt> $Wntsp π π = ππ 2 = π Γ 22 = 4π Njh;e;njLf;fg;gLk; Gs;sp tl;lj;ij (ghpjp) tpl ikaj;Jf;F mUfpy; ,Uf;f Ntz;Lnkdpy; 1 nr.kP MuKs;s tl;lj;jpd; gFjpf;Fs; mikAk; Gs;spfshFk;. vdNt epfo;r;rpfspd; vz;zpf;if π π΄ = ππ 2 = π π₯ 12 = π π(π΄) π(π)
=
π 4π
=
1 4
ZY
epfo;jfT π π΄ =
DO O
27.
52
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