ÔKÕ †mU

bgybv DËi

GmGmwm-2018

welq t MwYZ (m„Rbkxj)

(...... mv‡ji wm‡jevm Abyhvqx)

welq †KvW t 109

mgq : 2 NÈv 30 wgwbU c~Y©gvb : 70

DËicÎ g~j¨vq‡b we‡eP¨ welqmg~n t

* cÖwZwU cÖ‡kœi GKwU bgybv DËi †`qv Av‡Q| cix¶v_x©i DËi ûeû bgybv Dˇii gZ PvIqv cÖZ¨vwkZ bq| cix¶v_x©i

DËi I bgybv Dˇii †P‡q fv‡jv, mggv‡bi ev Lvivc n‡Z cv‡i|

* cÖ`Ë bgybv Dˇii †Kv‡bv weKí mwVK DËiI _vK‡Z cv‡i| DËicÎ g~j¨vqYKvix‡K cix¶v_x©i mwVK weKí DËi

we‡ePbvq G‡b b¤i cÖ`vb Ki‡Z n‡e|

* DËi †jLvi †¶‡Î cix¶v_x©i kã Pqb, evK¨MVb I Dc ’vcb †KŠkj cÖ`Ë bgybv DËi †_‡K wfbœ nIqvB ¯^vfvweK|

* cÖ‡qvM `¶Zv¯Í‡ii (mnR, ga¨g, KwVb) Ici wfwË K‡i b¤^i cÖ`vb Ki‡Z n‡e| cix¶v_x© cÖZ¨vwkZ Rubrics

Abyhvqx wjL‡Z cvi‡j H ͇ii Rb¨ eivÏK…Z c~Y© b¤^i cv‡e| †mRb¨ 1/2 (A‡a©K) b¤i †`qv hv‡e bv|

* wk¶v_©x hw` Dˇii avivevwnKZv (serial) eRvq bv †i‡L (†hgb 1bs cÖ‡kœi ÔKÕ As‡ki ci 5bs cÖ‡kœi ÔMÕ As‡ki

DËi wk¶v_©x w`j) DËi wj‡L ZeyI cix¶K Zv‡K h_vh_ b¤i cÖ`vb Ki‡eb| avivevwnKZv eRvq bv ivLvi Kvi‡Y

†Kvb b¤i KvUv hv‡e bv|

* Q‡K cÖ`wk©Z b¤i cÖ`vb wb‡`©wkKv I bgybv DËi ïay cixÿK, cÖavb cixÿK I wkÿKe„‡›`i e¨env‡ii Rb¨| G QK

†_‡K cixÿKe„›` c~Y©/ AvswkK b¤^i cÖ`v‡bi w`K wb‡`©kbv cv‡eb| GwU wkÿv_©x / cixÿv_©x‡`i e¨env‡ii Rb¨ bq|

1bs cÖ‡kœi DËi

cÖkœ bs `ÿZvi

Íi b¤^i

wkLbdj /

wkÿv_©xiv cvi‡e bgybv DËi

1 K

mnR 2

A I B wb‡ñ` †mU

cÖgvY Ki‡Z

‡`Iqv Av‡Q,

A = {3, 4, 5, 6}

B = {0, 1, 2}

AB = {3, 4, 5, 6} {0, 1, 2}

=

AB =

AZGe A I B ci¯úi wb‡ñ` †mU|

1

wb‡ñ` †m‡Ui aviYv

D‡jøL Ki‡Z

‡`Iqv Av‡Q,

A = {3, 4, 5, 6}

B = {0, 1, 2}

AB = {3, 4, 5, 6} {0, 1, 2}

1 L ga¨g

4

A - †m‡Ui power

†mU wbY©q K‡i cÖgvY

Ki‡Z

‡`Iqv Av‡Q,

A = {3, 4, 5, 6}

‡mU A - Gi Dcv`vb msL¨v n=4

P(A) = {{3, 4, 5, 6}, {3, 4, 5}, {3, 4, 6}, {3, 5, 6}, {4, 5, 6}, {3,

4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 6}, {3}, {4}, {5}, {6}, }

P(A) - Gi Dcv`vb msL¨v = 16

P(A) Gi Dcv`vb msL¨v = 24

P(A) Gi Dcv`vb msL¨v = 2n

[‡hLv‡b †mU A Gi Dcv`vb msL¨v n]

3

A ‡m‡Ui Dcv`vb

msL¨v Ges

power †mU

wbY©q c~e©K Dnvi

Dcv`vb msL¨v D‡jøL

Ki‡Z

‡`Iqv Av‡Q,

A = {3, 4, 5, 6}

‡mU A - Gi Dcv`vb msL¨v n=4

P(A) = {{3, 4, 5, 6}, {3, 4, 5}, {3, 4, 6}, {3, 5, 6}, {4, 5, 6}, {3,

4}, {3, 5}, {3, 6}, {3, 5}, {4, 6}, {5, 6}, {3}, {4}, {5}, {6}, }

P(A) Gi Dcv`vb msL¨v = 16

2

‡mU A Gi Dcv`vb

msL¨v D‡jøL c~e©K

P(A) wbY©q Ki‡Z

A = {3, 4, 5, 6}

‡mU A - Gi Dcv`vb msL¨v n=4

P(A) = {{3, 4, 5, 6}, {3, 4, 5}, {3, 4, 6}, {3, 5, 6}, {4, 5, 6}, {3,

4}, {3, 5}, {3, 6}, {3, 5}, {4, 6}, {5, 6}, {3}, {4}, {5}, {6}, }

1

‡mU A Gi Dcv`vb

msL¨v D‡jøL Ki‡Z /

A †m‡Ui P(A)

mwVKfv‡e wjL‡Z

A = {3, 4, 5, 6}

‡mU A - Gi Dcv`vb msL¨v n=4

A_ev,

P(A) = {{3, 4, 5, 6}, {3, 4, 5}, {3, 4, 6}, {3, 5, 6}, {4, 5, 6}, {3,

4}, {3, 5}, {3, 6}, {3, 5}, {4, 6}, {5, 6}, {3}, {4}, {5}, {6}, }

1 M KwVb

4

R ‡K ZvwjKv

cØwZ‡Z cÖKvk

K‡i †Wvg R Ges

†iÄ R wbY©q

Ki‡Z

‡`Iqv Av‡Q,

A = {3, 4, 5, 6}

R= {(x,y): xA, yA Ges xy = 1}

R - Gi ewY©Z kZ© †_‡K cvB -

ev, xy = 1

ev, x+1 = y

y = x+1

GLb cÖ‡Z¨K xA Gi Rb¨ y=x+1 Gi gvb wbY©q Kwi|

x 3 4 5 6

y 4 5 6 7

‡h‡nZz 7A †m‡nZz (6,7) R

R={(3,4), (4,5), (5,6)}

‡Wvg R={3,4,5}

‡iÄ R={4,5,6}

3

R ‡K ZvwjKv

cØwZ‡Z cÖKvk

K‡i †Wvg R /

†iÄ R Gi wbY©q

Ki‡Z

‡`Iqv Av‡Q,

A = {3, 4, 5, 6}

R= {(x,y):xA, yA Ges xy=1}

R - Gi ewY©Z kZ© †_‡K cvB -

ev, xy = 1

ev, x+1 = y

y = x+1

GLb cÖ‡Z¨K xA Gi Rb¨ y=x+1 Gi gvb wbY©q Kwi|

x 3 4 5 6

y 4 5 6 7

‡h‡nZz 7A †m‡nZz (6,7) R

R={(3,4), (4,5), (5,6)}

A_ev,

‡Wvg R={3,4,5}

2

Aš^q R †K

ZvwjKv c×wZ‡Z

cÖKvk Ki‡Z

‡`Iqv Av‡Q,

A = {3, 4, 5, 6}

R= {(x,y):xA, yA Ges xy=1}

R - Gi ewY©Z kZ© †_‡K cvB -

ev, xy = 1

ev, x+1 = y

y = x+1

GLb cÖ‡Z¨K xA Gi Rb¨ y=x+1 Gi gvb wbY©q Kwi|

x 3 4 5 6

y 4 5 6 7

‡h‡nZz 7A †m‡nZz (6,7) R

R={(3,4), (4,5), (5,6)}

1

Aš^q R †K

ZvwjKv c×wZ‡Z

cÖKv‡ki Rb¨

Dcv`vbmg~n †ei

Ki‡Z

‡`Iqv Av‡Q,

A = {3, 4, 5, 6}

R= {(x,y):xA, yA Ges xy=1}

R - Gi ewY©Z kZ© †_‡K cvB -

ev, xy = 1

ev, x+1 = y

y = x+1

GLb cÖ‡Z¨K xA Gi Rb¨ y=x+1 Gi gvb wbY©q Kwi|

x 3 4 5 6

y 4 5 6 7

2bs cÖ‡kœi DËi

cÖkœ bs `ÿZvi

Íi b¤^i

wkLbdj /

wkÿv_©xiv cvi‡e bgybv DËi

2 K mnR

2

myPKxq ivwki m~Î

cÖ‡qv‡M gvb wbY©q

Ki‡Z

3log 7. 77

21

7.31

77

log

2

131

77

log

7

2 36log

7

7

56log

7

77

log6

5

16

5

6

5

1

myPKxq ivwki NvZ

†ei Ki‡Z

3log 7. 77

21

7.31

77

log

2

131

77

log

7

2 36log

7

2 L ga¨g

4

exRMvwYwZK Ges

m~PKxq m~Î cÖ‡qv‡M

mwVK gvb wbY©q

Ki‡Z

2-y

1

1a)1a(y

1a(3y)

1aay

1ay

2(3)

1

1a)1a(3

1a(3.3)

1aa3

1a3

1

9

1

1

12a3

1a(9)

a2a3

1a3

1a 12 (9)a 1 a a3 92a 13

2 a 12 (3 )a 1 a a3 92a 13

922a3

12a312a2a3

2322a12a312a2a3

222a12a12a2a3

03

1

3

1wU cÖvmw½K

exRMvwYwZK myÎ I

GKvwaK m~PKxq

m~Î cÖ‡qvM Ki‡Z

2-y

1

1a)1a(y

1a(3y)

1aay

1ay

2(3)

1

1a)1a(3

1a(3.3)

1aa3

1a3

1

9

1

1

12a3

1a(9)

a2a3

1a3

1a 12 (9)a 1 a a3 92a 13

2 a 12 (3 )a 1 a a3 92a 13

922a3

12a312a2a3

2322a12a312a2a3

2

cÖvmw½K GKwU

exRMvwYwZK myÎ I

GKvwU m~PKxq m~Î

cÖ‡qvM Ki‡Z

2-y

1

1a)1a(y

1a(3y)

1aay

1ay

2(3)

1

1a)1a(3

1a(3.3)

1aa3

1a3

1

9

1

1

12a3

1a(9)

a2a3

1a3

9112a3

1a)2(3a2a1a3

1

cÖvmw½K GKwU

exRMvwYwZK myÎ

cÖ‡qvM Ki‡Z

2-y

1

1a)1a(y

1a(3y)

1aay

1ay

2(3)

1

1a)1a(3

1a(3.3)

1aa3

1a3

1

9

1

1

12a3

1a(9)

a2a3

1a3

2 M KwVb 4

‡`Iqv Av‡Q,

x=2, y=2, z=5

jMvwi`‡gi

m~Îvejx cÖ‡qvM

K‡i mwVKfv‡e

cÖgvY Ki‡Z

L.H.S.

log1.23z3xlog3logx3ylog

2.1log

3532log3log233log

2

3

2log3 3log log 1000

log1.2

23 3log3 3log log 102

log1.2

33 2log3 3log2 log102

log1.2

3 3log3 3log2 log10

2 2

log1.2

2.1log

10)22log(32

3

2.1log

10)22log(32

3

2.1log

10

12log

2

3

2.1log

2.1log2

3

2

3

=R.H.S

L.H.S = R.H.S (Solved)

2 M

KwVb

3

jMvwi`‡gi

cÖvmw½K 3wU m~Î

cÖ‡qvM Ki‡Z

‡`Iqv Av‡Q,

x=2, y=2, z=5

L.H.S.

log1.23z3xlog3logx3ylog

2.1log

3532log3log233log

2

3

2log3 3log log 1000

log1.2

23 3log3 3log log 102

log1.2

33 2log3 3log2 log102

log1.2

3 3log3 3log2 log10

2 2

log1.2

2.1log

10)22log(32

3

2.1log

10)22log(32

3

2 M KwVb 2

jMvwi`‡gi

cÖvmw½K 2wU m~Î

cÖ‡qvM Ki‡Z

‡`Iqv Av‡Q,

x=2, y=2, z=5

L.H.S.

log1.23z3xlog3logx3ylog

2.1log

3532log3log233log

2

3

2log3 3log log 1000

log1.2

23 3log3 3log log 102

log1.2

33 2log3 3log2 log102

log1.2

3 3log3 3log2 log10

2 2

log1.2

2 M KwVb 1

jMvwi`‡gi

cÖvmw½K 1wU m~Î

cÖ‡qvM Ki‡Z

‡`Iqv Av‡Q,

x=2, y=2, z=5

L.H.S.

log1.23z3xlog3logx3ylog

2.1log

3532log3log233log

2

3

2log3 3log log 1000

log1.2

23 3log3 3log log 102

log1.2

3bs cÖ‡kœi DËi

cÖkœ bs `ÿZvi

Íi b¤^i

wkLbdj /

wkÿv_©xiv cvi‡e bgybv DËi

3 K mnR

2

n msL¨K ¯^vfvweK

msL¨vi mgwói m~Î

cÖ‡qvM K‡i 1g

20wU c‡`i mgwó

wbY©q Ki‡Z

Avgiv Rvwb,

n msL¨K ¯^vfvweK msL¨vi mgwó,

2

1)n(n

ns

1g 20wU ¯vfvweK msL¨vi mgwó,

2

1)20(20

20s

2

2120

2

420

210

1

¯vfvweK msL¨vi

mgwói myÎwU

wjL‡Z

Avgiv Rvwb,

n msL¨K ¯^vfvweK msL¨vi mgwó,

2

1)n(n

ns

3 L ga¨g 4

n Gi mwVK gvb

wbY©q Ki‡Z

DÏxc‡Ki avivwU wb¤œiæc -

3+6+9+12+.........

GLv‡b, 1g c`, a=3

mvaviY AšÍi, d =63

=3

Avgiv Rvwb,

mgvšÍi avivi, cÖ_g n - msL¨K c‡`i mgwó,

1)d}(n{2a2

n

ns

myÎg‡Z,

6301)d}(n{2a2

n

ev, 12603}1)(n3n{2

ev, 12603n23n6n

ev, 01260-3n23n

ev, 0420-n2n

ev, 042020n21n2n

ev, 021)20(n21)n(n

ev, 020)21)(n(n

nq, 021)(n

ev, -21n

n=-21 gvbwU MÖnb‡hvM¨ n‡Z cv‡i bv|

A_ev,

ev, 020)(n

ev 20n

3 L ga¨g 3

mvaviY AšÍi

wbY©q, mgwó

wbY©‡qi cÖvmw½K

m~Î cÖ‡qv‡M

mgxKiY MVb

Ki‡Z

DÏxc‡Ki avivwU wb¤œiæc -

3+6+9+12+.........

GLv‡b, 1g c`, a=3

mvaviY AšÍi, d =63

=3

Avgiv Rvwb,

mgvšÍi avivi, cÖ_g n - msL¨K c‡`i mgwó,

1)d}(n{2a2

n

ns

myÎg‡Z,

6301)d}(n{2a2

n

ev, 12603}1)(n3n{2

ev, 12603n23n6n

ev, 01260-3n23n

ev, 0420-n2n

2

avivwUi mvaviY

AšÍi wbY©q Ki‡Z

Ges mgwó wbY©‡qi

cÖvmw½K DËi

wjL‡Z

DÏxc‡Ki avivwU wb¤œiæc -

3+6+9+12+.........

GLv‡b, 1g c`, a=3

mvaviY AšÍi, d =63

=3

Avgiv Rvwb,

mgvšÍi avivi, cÖ_g n - msL¨K c‡`i mgwó,

1)d}(n{2a2

n

ns

1

avivwUi mvaviY

AšÍi wbY©q Ki‡Z

/ mgwó wbY©‡qi

cÖvmw½K m~Î

wjL‡Z

DÏxc‡Ki avivwU wb¤œiæc -

3+6+9+12+.........

GLv‡b, 1g c`, a=3

mvaviY AšÍi, d =63

=3

A_ev,

mgvšÍi avivi, cÖ_g n - msL¨K c‡`i mgwó,

1)d}(n{2a2

n

ns

3 M KwVb 4

¸‡YvËi avivwUi

1g 10wU c‡`i

mgwó mwVKfv‡e

wbY©q Ki‡Z

L n‡Z cvB,

mgvšÍi avivi 1g c`, a=3

mvaviY AšÍi, d=3

cÖkœg‡Z,

¸‡YvËi avivi 1g c`, a=3

mvaviY AbycvZ, r=3

wb‡Y©q aviv = a+ar+ar2+ar

3+.............

= 3+33+3(3)2+3(3)

3+.............

= 3+9+39+327+...................

= 3+9+27+81+.........................

Avgiv Rvwb,

¸‡YvËi avivi n - msL¨K c‡`i mgwó,

1r

1nra

ns

hLb, r1|

1g 10wU c‡`i mgwó,

13

}1103{(3)

10s

2

1)-59049(3

2

590483

2

177144

88572

3 M KwVb 3

¸‡bvËi aviv

wbY©q, mgwó

wbY©‡qi cÖvmw½K

cÖ‡qvM K‡i

mwVKfv‡e gvb

emv‡Z

L n‡Z cvB, mgvšÍi avivi 1g c`, a=3

mvaviY AšÍi, d=3

cÖkœg‡Z,

¸‡YvËi avivi 1g c`, a=3

mvaviY AbycvZ, r=3

wb‡Y©q aviv = a+ar+ar2+ar

3+.............

= 3+33+3(3)2+3(3)

3+.............

= 3+9+39+327+...................

= 3+9+27+81+.........................

Avgiv Rvwb,

¸‡YvËi avivi 1g n - msL¨K c‡`i mgwó,

1r

1nra

ns

hLb, r1|

1g 10wU c‡`i mgwó,

13

}1103{(3)

10s

3 M KwVb 2

¸‡YvËi avivwU

mwVKfv‡e wbY©q

c~e©K mgwó

wbY©‡qi cÖvmw½K

m~ÎwU wjL‡Z

L n‡Z cvB, mgvšÍi avivi 1g c`, a=3

mvaviY AšÍi, d=3

cÖkœg‡Z,

¸‡YvËi avivi 1g c`, a=3

mvaviY AbycvZ, r=3

wb‡Y©q aviv = a+ar+ar2+ar

3+.............

= 3+33+3(3)2+3(3)

3+.............

= 3+9+39+327+...................

= 3+9+27+81+.........................

Avgiv Rvwb,

¸‡YvËi avivi 1g n - msL¨K c‡`i mgwó,

1r

1nra

ns

hLb, r1|

1

¸‡bvËi avivwU

mwVKfv‡e wbY©q

Ki‡Z/ mgwó

wbY©‡qi cÖvmw½K

m~ÎwU wjL‡Z

L n‡Z cvB, mgvšÍi avivi 1g c`, a=3

mvaviY AšÍi, d=3

cÖkœg‡Z,

¸‡YvËi avivi 1g c`, a=3

mvaviY AbycvZ, r=3

wb‡Y©q aviv = a+ar+ar2+ar

3+.............

= 3+33+3(3)2+3(3)

3+.............

= 3+9+39+327+...................

= 3+9+27+81+.........................

A_ev,

¸‡YvËi avivi 1g n - msL¨K c‡`i mgwó,

1r

1nra

ns

hLb, r1|

4bs cÖ‡kœi DËi

cÖkœ bs KvwV‡b¨i

¯Íi

b¤^i wkLbdj/wkÿv_©x

cvi‡e

cÖZ¨vwkZ bgybv DËi

4 K mnR

2

DÏxc‡Ki Av‡jv‡K

mwVK wPwýZ wPÎ

AvKu‡Z

1

wPÎ Am¤ú~Y© _vK‡j

4 L ga¨g 4

cÖ‡qvRbxq A¼bmn

mwVK cÖgvY Ki‡Z

ABC Gi AB I AC Gi ga¨we›`y h_vµ‡g M I N ; M I N †hvM Kiv nj |

cÖgvY Ki‡Z n‡e †h , MN∥ BC Ges MN =

A¼bt M I N †hvM K‡i Q ch©šÍ ewa©Z Kwi †hb, MN= NQ nq ; Q Ges C †hvM Kwi |

cÖgvYt AMN Ges CQN Gi g‡a¨

MN= NQ , AN= CN Ges

∠ANM= ∠CNQ myZivs AMN ≅ CQN

myZivs AM= CQ Ges ∠MAN=∠QCN

Avevi M , AB Gi ga¨we› y e‡j AM= MB

d‡j , BM = CQ

Avevi ∠MAN= GKvšÍi ∠QCN e‡j AM ∥ CQ

ev , AB ∥ CQ

ev , BM∥ CQ

GLb , BCQM PZzf©y‡R BM I CQ ci¯úi mgvb I mgvšÍivj e‡j

MQ I BC ci¯úi mgvb I mgvšÍivj n‡e|

myZivs MN∥ BC

Avevi MQ=BC

ev , 2MN= BC

ev , MN =

Zvn‡j MN∥ BC Ges MN =

3

cÖ‡qvRbxq A¼bmn

BM∥ CQ

cÖgvY Ki‡Z

ABC Gi AB I AC Gi ga¨we›`y h_vµ‡g M I N ; M I N †hvM Kiv nj |

cÖgvY Ki‡Z n‡e †h , MN∥ BC Ges MN =

A¼bt M I N †hvM K‡i Q ch©šÍ ewa©Z Kwi †hb, MN= NQ nq ; Q Ges C †hvM Kwi |

cÖgvYt AMN Ges CQN Gi g‡a¨

MN= NQ , AN= CN Ges

∠ANM= ∠CNQ myZivs AMN ≅ CQN

myZivs AM= CQ Ges ∠MAN=∠QCN

Avevi M , AB Gi ga¨we› y e‡j AM= MB

d‡j , BM = CQ

Avevi ∠MAN= GKvšÍi ∠QCN e‡j AM ∥ CQ

ev , AB ∥ CQ

ev , BM∥ CQ

2

cÖ‡qvRbxq A¼bmn

AMN ≅

CQN cÖgvY Ki‡Z

ABC Gi AB I AC Gi ga¨we›`y h_vµ‡g M I N ; M I N †hvM Kiv nj |

cÖgvY Ki‡Z n‡e †h , MN∥ BC Ges MN =

A¼bt M I N †hvM K‡i Q ch©šÍ ewa©Z Kwi †hb, MN= NQ nq ; Q Ges C †hvM Kwi |

cÖgvYt AMN Ges CQN Gi g‡a¨

MN= NQ , AN= CN Ges

∠ANM= ∠CNQ myZivs AMN ≅ CQN

1

cÖ‡qvRbxq A¼bmn

m¤ú~Y© wPÎ AvKu‡Z

Ges we‡kl wbe©vPb

wjL‡Z

ABC Gi AB I AC Gi ga¨we›`y h_vµ‡g M I N ; M I N †hvM Kiv nj |

cÖgvY Ki‡Z n‡e †h , MN∥ BC Ges MN =

4 M KwVb

4

cÖ‡qvRbxq A¼bmn

mwVK cÖgvY Ki‡Z

ABC Gi ∠B I ∠C Gi mgwØLÛKØq ci¯úi‡K P we› y‡Z †Q` K‡i | cÖgvY

Ki‡Z n‡e †h , ∠BPC =

∠A

cÖgvYt ABC -G

∠A ∠B ∠C=

ev,

∠A

∠B

∠C =

ev,

∠B

∠C =

∠A

Avevi , BPC - G ∠BPC ∠PBC ∠PCB =

ev, ∠BPC

∠PBC

∠PCB =

ev, ∠BPC

∠B

∠C =

ev, ∠BPC +

∠A =

ev, ∠BPC =

∠A

3

cÖ‡qvRbxq A¼bmn

∠BPC

∠B

+

∠C =

ABC Gi ∠B I ∠C Gi mgwØLÛKØq ci¯úi‡K P we› y‡Z †Q` K‡i | cÖgvY

Ki‡Z n‡e †h , ∠BPC =

∠A

cÖgvYt ABC -G

∠A ∠B ∠C=

cÖgvY Ki‡Z ev,

∠A

∠B

∠C =

ev,

∠B

∠C =

∠A

Avevi , BPC - G

∠BPC ∠PBC ∠PCB =

ev, ∠BPC

∠PBC

∠PCB =

ev, ∠BPC

∠B

∠C =

2

cÖ‡qvRbxq A¼bmn

∠B

∠C

=

∠A

cÖgvY Ki‡Z

ABC Gi ∠B I ∠C Gi mgwØLÛKØq ci¯úi‡K P we› y‡Z †Q` K‡i | cÖgvY

Ki‡Z n‡e †h , ∠BPC =

∠A

cÖgvYt ABC -G

∠A ∠B ∠C=

ev,

∠A

∠B

∠C =

ev,

∠B

∠C =

∠A

1

cÖ‡qvRbxq A¼bmn

m¤ú~Y© wPÎ AvKu‡Z

ABC Gi ∠B I ∠C Gi mgwØLÛKØq ci¯úi‡K P we› y‡Z †Q` K‡i | cÖgvY

Ki‡Z n‡e †h , ∠BPC =

∠A

5bs cÖ‡kœi DËi

cÖkœ bs KvwV‡b¨i

¯Íi

b¤^i wkLbdj/wkÿv_©x

cvi‡e

cÖZ¨vwkZ bgybv DËi

5 K mnR

2

‡¯‹j - K¤úvm

e¨envi K‡i

†KvY AvKu‡Z

1

‡¯‹j - K¤úvm

e¨envi K‡i

†KvY AvKu‡Z

5 L ga¨g

4

A¼‡bi wPýmn

mwVK wPÎ I eY©bvi

Rb¨

‡Kv‡bv wÎfz‡Ri f~wg msjMœ `yBwU †KvY ∠x= I ∠y= Ges cwimxgv

p = 12 †mwg , †`Iqv Av‡Q|wÎfyRwU AvuK‡Z n‡e|

A¼bt †h †Kv‡bv iwkœ †_‡K cwimxgv p Gi mgvb K‡i DE KvwU| DE Gi D I E

we›`y‡Z h_vµ‡g

∠x = ∠EDA I

∠y = ∠DEA AvuwK| DA I EA

ci¯úi‡K A we›`y‡Z †Q` K‡i| Gevi A we›`y‡Z ∠EDA = ∠DAB Ges ∠DEA

= ∠EAC AvuwK | AB I AC ,DE -†K h_vµ‡g B I C we›`y‡Z †Q` K‡i |

Zvn‡j ABC -B DwÏó wÎfzR Aw¼Z nj|

3

A¼‡bi wPýmn

mwVK wP‡Îi Rb¨

‡Kv‡bv wÎfz‡Ri f~wg msjMœ `yBwU †KvY ∠x= I ∠y= Ges cwimxgv

p = 12 †mwg , †`Iqv Av‡Q|wÎfyRwU AvuK‡Z n‡e|

2

cwimxgvi `yBcÖv‡šÍ

Ges

†KvY Gu‡K †KvY

`yBwU‡K mgwØLwÛZ

Kivi Rb¨

1

cÖ`Ë DcvËmgyn

mwVK gv‡c

Dc¯’vc‡bi Rb¨

5 M KwVb

4

A¼‡bi wPýmn

mwVK wPÎ I eY©bvi

Rb¨

‡Kv‡bv mgevû wÎfz‡Ri cwimxgv p = 12 †mwg , †`Iqv Av‡Q|wÎfyRwU AvuK‡Z n‡e|

A¼bt cÖ_‡g cwimxgv p -‡K mgwÎLwÛZ Kwi| Gevi AE †h †Kv‡bv iwkœ †_‡K

AB =

KvwU| A I B †K †K›`ª K‡i AB Gi mgvb e¨vmva© wb‡q AB Gi GKB cv‡k&©

`yBwU e„ËPvc AvuwK , hviv ci¯úi‡K C we› y‡Z †Q` K‡i | A,C ; B,C

†hvM Kwi |

Zvn‡j ABC -B DwÏó mgevû wÎfzR Aw¼Z nj|

3

A¼‡bi wPýmn

mwVK wP‡Îi Rb¨

‡Kv‡bv mgevû wÎfz‡Ri cwimxgv p = 12 †mwg , †`Iqv Av‡Q|wÎfyRwU AvuK‡Z n‡e|

2

A¼‡bi wPýQvov

mwVK wP‡Îi Rb¨

1

cwimxgv‡K

mgwÎLwÛZ Kivi

Rb¨

6bs cÖ‡kœi DËi

cÖkœ bs KvwV‡b¨i

¯Íi

b¤^i wkLbdj/wkÿv_©x

cvi‡e

cÖZ¨vwkZ bgybv DËi

6 K mnR

2

`yBwU mwVK msÁvi

Rb¨

‡K› ª¯’ †KvYt GKwU †Kv‡Yi kxl©we›`y †Kv‡bv e„‡Ëi †K‡› ª Aew¯’Z n‡j †KvYwU‡K H

e„‡Ëi GKwU †K› ª¯’ †KvY ejv nq Ges †KvYwU e„‡Ë †h Pvc LwÛZ K‡i †mB Pv‡ci

Dci Zv `Ûvqgvb ejv nq|

Ges

e„˯’ †KvYt GKwU †Kv‡Yi kxl©we› y †Kv‡bv e„‡Ëi GKwU we› y n‡j Ges †KvYwUi cÖ‡Z¨K

evû‡Z kxl©we›`y QvovI e„‡Ëi GKwU we›`y _vK‡j †KvYwU‡K H e„‡Ëi GKwU e„˯’’ †KvY

ejv nq |

1

GKwU mwVK msMvi

Rb¨

‡K› ª¯’ †KvYt GKwU †Kv‡Yi kxl©we›`y †Kv‡bv e„‡Ëi †K‡› ª Aew¯’Z n‡j †KvYwU‡K H

e„‡Ëi GKwU †K› ª¯’ †KvY ejv nq Ges †KvYwU e„‡Ë †h Pvc LwÛZ K‡i †mB Pv‡ci

Dci Zv `Ûvqgvb ejv nq|

A_ev, e„˯’ †KvYt GKwU †Kv‡Yi kxl©we›`y †Kv‡bv e„‡Ëi GKwU we›`y n‡j Ges

†KvYwUi cÖ‡Z¨K evû‡Z kxl©we›`y QvovI e„‡Ëi GKwU we›`y _vK‡j †KvYwU‡K H e„‡Ëi

GKwU e„˯’’ †KvY ejv nq |

6 L ga¨g

4

cÖ‡qvRbxq A¼bmn

mwVK cÖgvY Ki‡Z

C †K› ªwewkó PQRS e„‡Ë RQS DcPv‡ci Dci `Ûvqgvb e„˯’ †KvY ∠RPS Ges

†K› ª¯’ †KvY∠RCS | cÖgvY Ki‡Z n‡e †h, ∠RCS = 2 ∠RPS

A¼bt P Ges C †K †hvM K‡i T ch©šÍ ewa©Z Kwi |

cÖgvbt (1) PRC Gi ewn ’ †KvY ∠RCT = ∠CPR ∠CRP

(2) PRC G CP = CR

AZGe , ∠CPR = ∠CRP

(3) avc (1) I (2) †_‡K cvB ∠RCT = 2∠CPR

(4) GKBfv‡e PCS †_‡K cvB ∠SCT = 2∠SPC (5) avc (3) I (4) †_‡K cvB

∠RCT ∠SCT = 2∠CPR 2∠SPC

A_v©r ∠RCS = 2 ∠RPS

3

cÖ‡qvRbxqA¼bmn

∠RCT =2∠CPR Ges

∠SCT = 2∠SPC

cÖgvY Ki‡Z

C †K› ªwewkó PQRS e„‡Ë RQS DcPv‡ci Dci `Ûvqgvb e„˯’ †KvY ∠RPS Ges

†K› ª¯’ †KvY∠RCS | cÖgvY Ki‡Z n‡e †h, ∠RCS = 2 ∠RPS

A¼bt P Ges C †K †hvM K‡i T ch©šÍ ewa©Z Kwi |

cÖgvbt (1) PRC Gi ewn ’ †KvY ∠RCT = ∠CPR ∠CRP

(2) PRC G CP = CR

AZGe , ∠CPR = ∠CRP

(3) avc (1) I (2) †_‡K cvB ∠RCT = 2∠CPR

(4) GKBfv‡e PCS †_‡K cvB ∠SCT = 2∠SPC

2

cÖ‡qvRbxq A¼bmn

∠RCT = 2∠CPR cÖgvY

Ki‡Z/cÖ‡qvRbxq

A¼bmn ∠SCT =

2∠SPC cÖgvY

Ki‡Z

C †K› ªwewkó PQRS e„‡Ë RQS DcPv‡ci Dci `Ûvqgvb e„˯’ †KvY ∠RPS Ges

†K› ª¯’ †KvY∠RCS | cÖgvY Ki‡Z n‡e †h, ∠RCS = 2 ∠RPS

A¼bt P Ges C †K †hvM K‡i T ch©šÍ ewa©Z Kwi |

cÖgvbt (1) PRC Gi ewn ’ †KvY ∠RCT = ∠CPR ∠CRP

(2) PRC G CP = CR

AZGe , ∠CPR = ∠CRP

(3) avc (1) I (2) †_‡K cvB ∠RCT = 2∠CPR

1

cÖ‡qvRbxq A¼bmn

mwVK wPÎ Gu‡K

we‡kl wbev©Pb

wjL‡Z

C †K› ªwewkó PQRS e„‡Ë RQS DcPv‡ci Dci `Ûvqgvb e„˯’ †KvY ∠RPS Ges

†K› ª¯’ †KvY∠RCS | cÖgvY Ki‡Z n‡e †h, ∠RCS = 2 ∠RPS

6 M KwVb

4

cÖ‡qvRbxq A¼bmn

mwVK cÖgvY Ki‡Z

C ‡K› ªwewkó e„‡Ë PQ Ges RS R¨v `yBwU e„‡Ëi Af¨šÍ‡i O we›`y‡Z mg‡Kv‡Y

†Q` K‡i‡Q | C,P ; C,R ; C,Q Ges C,S †hvM Kivq †K‡› ª ∠PCR I ∠QCS

Drcbœ nj | cÖgvY Ki‡Z n‡e †h , ∠PCR ∠QCS = ;

A¼bt P,R I P,S †hvM Kwi|

cÖgvYt (1) e„‡Ëi GKB Pv‡ci Dci `Ûvqgvb †K› ª¯’ †KvY e„˯’ †Kv‡Yi

wظY e‡j ,

PR Pv‡ci Dci `Ûvqgvb †K› ª¯’ †KvY , ∠PCR = 2 e„˯’ †KvY

∠PSR ev, ∠PCR = 2 ∠PSR

( 2) GKBfv‡e , ∠QCS =2 ∠QPS

(3) avc (1) I (2) n‡Z cvB

∠PCR ∠QCS = 2∠PSR 2∠QPS = 2(∠PSR ∠QPS ) = 2 ( ∠PSO ∠OPS)

= 2 =

3

cÖ‡qvRbxq A¼bmn

∠PCR ∠QCS = 2∠PSR 2∠QPS cÖgvY

Ki‡Z

C ‡K› ªwewkó e„‡Ë PQ Ges RS R¨v `yBwU e„‡Ëi Af¨šÍ‡i O we›`y‡Z mg‡Kv‡Y

†Q` K‡i‡Q | C,P ; C,R ; C,Q Ges C,S †hvM Kivq †K‡› ª ∠PCR I ∠QCS

Drcbœ nj | cÖgvY Ki‡Z n‡e †h , ∠PCR ∠QCS = |

A¼bt P,R I P,S †hvM Kwi|

cÖgvYt (1) e„‡Ëi GKB Pv‡ci Dci `Ûvqgvb †K› ª¯’ †KvY e„˯’ †Kv‡Yi

wظY e‡j ,

PR Pv‡ci Dci `Ûvqgvb

†K› ª¯’ †KvY , ∠PCR = 2 e„˯’ †KvY ∠PSR

ev, ∠PCR = 2 ∠PSR

( 2) GKBfv‡e , ∠QCS =2 ∠QPS (3) avc (1) I (2) n‡Z cvB

∠PCR ∠QCS = 2∠PSR 2∠QPS

2

cÖ‡qvRbxq A¼bmn

∠PCR = 2

∠PSR cÖgvY

Ki‡Z

A_ev,

∠QCS =2 ∠QPS cÖgvY

Ki‡Z

C ‡K› ªwewkó e„‡Ë PQ Ges RS R¨v `yBwU e„‡Ëi Af¨šÍ‡i O we›`y‡Z mg‡Kv‡Y

†Q` K‡i‡Q | C,P ; C,R ; C,Q Ges C,S †hvM Kivq †K‡› ª ∠PCR I ∠QCS

Drcbœ nj | cÖgvY Ki‡Z n‡e †h , ∠PCR ∠QCS = |

A¼bt P,R I P,S †hvM Kwi|

cÖgvYt (1) e„‡Ëi GKB Pv‡ci Dci `Ûvqgvb †K› ª¯’ †KvY e„˯’ †Kv‡Yi

wظY e‡j ,

PR Pv‡ci Dci `Ûvqgvb

†K› ª¯’ †KvY , ∠PCR = 2 e„˯’ †KvY ∠PSR

ev, ∠PCR = 2 ∠PSR

1

cÖ‡qvRbxq A¼bmn

mwVK wPÎ Gu‡K

we‡kl wbe©vPb

wjL‡Z

C ‡K› ªwewkó e„‡Ë PQ Ges RS R¨v `yBwU e„‡Ëi Af¨šÍ‡i O we›`y‡Z mg‡Kv‡Y

†Q` K‡i‡Q | C,P ; C,R ; C,Q Ges C,S †hvM Kivq †K‡› ª ∠PCR I ∠QCS

Drcbœ nj | cÖgvY Ki‡Z n‡e †h , ∠PCR ∠QCS = |

7bs cÖ‡kœi DËi

cÖkœ

bs `ÿZvi

Íi

cÖ‡kœi

gvb

wkLbdj/wkÿv_©xiv

cvi‡e

(Rubrics)

bgybv DËi

7 K mnR

2

AvbycvwZK wPÎ

Gu‡K Gi ˆ`N©¨

wbY©q Ki‡Z

1

AvbycvwZK wPÎ

Gu‡K wc_v‡Mviv‡mi

Dccv`¨ Abymv‡i,

=

wjL‡Z / AvbycvwZK

wPwýZ wPÎ A¼b

Ki‡Z

7 L ga¨g

4

DÏxc‡Ki Av‡jv‡K

m¤úK©wU cÖgvY

Ki‡Z

‡`Iqv Av‡Q, Gi ∠ = Ges

=

ev, = = A_©vr, ∠ =

‡h‡nZz, Gi ∠ = Ges ∠ = ∠ =

evgcÿ =

=

=

=

= 2

=

Wvbcÿ = = = =

evgcÿ = Wvbcÿ (cÖgvwYZ)

3

∠ I ∠ Gi

wWMÖx cwigvc †ei

K‡i evgc‡ÿi gvb

= 3 wbY©q Ki‡Z

‡`Iqv Av‡Q, Gi ∠ = Ges

=

ev, = = A_©vr, ∠ =

‡h‡nZz, Gi ∠ = Ges ∠ = ∠ =

evgcÿ =

=

=

=

= 2

=

2

∠ I ∠ Gi

wWMÖx cwigvc †ei

Ki‡Z

‡`Iqv Av‡Q, Gi ∠ = Ges

=

ev, = = A_©vr, ∠ =

wc_v‡Mviv‡mi Dccv`¨ Abymv‡i,

=

=

GKK

wc_v‡Mviv‡mi Dccv`¨ Abymv‡i,

‡h‡nZz, Gi ∠ = Ges ∠ =

∠ =

1 ∠ Gi wWMÖx

cwigvc †ei Ki‡Z

‡`Iqv Av‡Q, Gi ∠ = Ges

=

ev, = = A_©vr, ∠ =

7 M KwVb

4

∠ I ∠ Gi

wWMÖx cwigvc

e¨envi K‡i I

Gi gvb wbY©q

Ki‡Z

ÔLÕ †_‡K cvB, ∠ = Ges ∠ =

= ( ) Ges = ( )

( ) I ( ) bs †hvM K‡i cvB,

2 = = ( ) I ( ) bs we†qvM K‡i cvB,

2 = =

3

∠ I ∠ Gi

wWMÖx cwigvc

e¨envi K‡i Gi

gvb wbY©q Ki‡Z

ÔLÕ †_‡K cvB, ∠ = Ges ∠ =

= ( ) Ges = ( )

( ) I ( ) bs †hvM K‡i cvB,

2 = =

2

∠ I ∠ Gi

wWMÖx cwigvc

e¨envi K‡i `yBwU

mgxKiY ˆZwi

Ki‡Z

ÔLÕ †_‡K cvB, ∠ = Ges ∠ =

= ( ) Ges = ( )

1

∠ Gi wWMÖx

cwigvc e¨envi

K‡i A_ev, ∠

Gi wWMÖx cwigvc

e¨envi K‡i GKwU

mgxKiY ˆZwi

Ki‡Z

ÔLÕ †_‡K cvB, ∠ =

=

A_ev,

ÔLÕ †_‡K cvB, ∠ =

=

8bs cÖ‡kœi DËi

cÖkœ bs `ÿZvi

Íi

cÖ‡kœi

gvb

wkLbdj/wkÿv_©xiv

cvi‡e

(Rubrics)

bgybv DËi

8 K mnR

2

∠ Gi gvb

wbY©q Ki‡Z

‡`Iqv Av‡Q, Gi ∠ = Ges e:wn ’ ∠ = ‡h‡nZz wÎfz‡Ri †h‡Kv‡bv e:wn ’ †KvY Zvi A:šÍ¯’ wecixZ †KvY؇qi

mgwói mgvb|

∠ ∠ = ∠ =

A_ev

cÖ_‡g ∠ Gi gvb wbY©q Ki‡e Zvici ∠

1

Gi A:šÍ ’

wecixZ †KvY؇qi

mgwó mgxKiY

AvKv‡i wjL‡Z

‡`Iqv Av‡Q, Gi ∠ = Ges e:wn ’ ∠ = ‡h‡nZz wÎfz‡Ri †h‡Kv‡bv e:wn ’ †KvY Zvi A:šÍ¯’ wecixZ †KvY؇qi

mgwói mgvb|

∠ ∠ =

8 L ga¨g

4

wPÎ Gu‡K ev bv

Gu‡K I

Gi ˆ`N©¨ wbY©q

Ki‡Z

‡`Iqv Av‡Q, Gi ∠ = Ges = wgUvi

∠ =

ev, =

ev,

=

ev, 2 =

= = wgUvi|

Avevi, ∠ =

ev, =

ev,

=

ev, 2 =

= = wgUvi|

3

ev Gi

g‡a¨ †h‡Kv‡bv

GKwUi ˆ`N©¨ wbY©q

K‡i AciwUi

w·KvYwgwZK

AbycvZ wbY©q Ki‡Z

‡`Iqv Av‡Q, Gi ∠ = Ges = wgUvi

∠ =

ev, =

ev,

=

ev, 2 =

= = wgUvi|

Avevi, ∠ =

ev, =

2

ev Gi

g‡a¨ †h‡Kv‡bv

GKwUi ˆ`N©¨ wbY©q

Ki‡Z

‡`Iqv Av‡Q, Gi ∠ = Ges = wgUvi

∠ =

ev, =

ev,

=

ev, 2 =

= = wgUvi|

1

ev Gi

g‡a¨ †h‡Kv‡bv

GKwUi

w·KvYwgwZK

AbycvZ wbY©q Ki‡Z

‡`Iqv Av‡Q, Gi ∠ = Ges = wgUvi

∠ =

ev, =

8 M KwVb

4

Gi

cwimxgv wbY©q

Ki‡Z

‡`Iqv Av‡Q, Gi ∠ = Ges = wgUvi|

∠ =

ev, =

ev,

=

= wgUvi|

Avevi, ∠ =

ev, =

ev,

=

ev, =

= = wgUvi|

= = ( ) wgUvi ev, 30 wgUvi|

A_ev, Gi ∠ = Ges∠ =

= = wgUvi|

Gi cwimxgv = Gi wZb evûi ‰`‡N©¨i mgwó

= ( 30 + 30 + 30 ) wgUvi

= (60 + 30 ) wgUvi = 30(2 + ) wgUvi |

3

Gi

cwimxgvi m~Î ch©šÍ

wbY©q Ki‡Z

‡`Iqv Av‡Q, Gi ∠ = Ges = wgUvi|

∠ =

ev, =

ev,

=

= wgUvi|

Avevi, ∠ =

ev, =

ev,

=

ev, =

= = wgUvi|

= = ( ) wgUvi ev, 30 wgUvi|

A_ev, Gi ∠ = Ges∠ =

= = wgUvi|

Gi cwimxgv = Gi wZb evûi ‰`‡N©¨i mgwó

2

ev Gi

g‡a¨ †h‡Kv‡bv

GKwUi ˆ`N©¨ wbY©q

Ki‡Z

‡`Iqv Av‡Q, Gi ∠ = Ges = wgUvi|

∠ =

ev, =

ev,

=

= wgUvi|

1

ev Gi

g‡a¨ †h‡Kv‡bv

GKwUi

w·KvYwgwZK

AbycvZ wbY©q Ki‡Z

‡`Iqv Av‡Q, Gi ∠ = Ges = wgUvi|

∠ =

ev, =

9bs cÖ‡kœi DËi

cÖkœ bs `ÿZvi

Íi

cÖ‡kœi

gvb

wkLYdj/wkÿv_©xiv

cvi‡e

(Rubrics)

bgybv DËi

9 K mnR

2

Pj‡Ki gva¨‡g

AvqZ‡ÿ‡Îi

cwimxgv wbY©q Ki‡Z

g‡b Kwi, AvqZ‡ÿ‡Îi cÖ ’ wgUvi|

AvqZ‡ÿ‡Îi ˆ`N©¨ wgUvi|

AvqZ‡ÿ‡Îi cwimxgv = 2( ˆ`N©¨ + cÖ ’) GKK

= 2( ) wgUvi = wgUvi|

1 AvqZ‡ÿ‡Îi

cwimxgvi m~Î wjL‡Z

Avgiv Rvwb,

AvqZ‡ÿ‡Îi cwimxgv = 2( ˆ`N©¨ + cÖ ’)

9 L ga¨g

4

eM©‡ÿ‡Îi †ÿÎdj

wbY©q Ki‡Z

g‡b Kwi, AvqZ‡ÿ‡Îi cÖ ’ wgUvi|

AvqZ‡ÿ‡Îi ˆ`N©¨ wgUvi|

AvqZ‡ÿ‡Îi †ÿÎdj = eM©wgUvi|

cÖkœg‡Z, = 2 ev, = 2 =

AvqZ‡ÿ‡Îi cwimxgv = ( ) wgUvi ev, 144 wgUvi|

kZ©vbymv‡i, eM©‡ÿ‡Îi cwimxgv = 144 wgUvi|

eM©‡ÿ‡Îi GK evûi ˆ`N©¨ = ( ) wgUvi ev, 36 wgUvi|

eM©‡ÿ‡Îi †ÿÎdj = ( ) eM©wgUvi ev, 1296 eM©wgUvi|

3

eM©‡ÿ‡Îi GK evûi

ˆ`N©¨ wbY©q Ki‡Z

g‡b Kwi, AvqZ‡ÿ‡Îi cÖ ’ wgUvi|

AvqZ‡ÿ‡Îi ˆ`N©¨ wgUvi|

AvqZ‡ÿ‡Îi †ÿÎdj = eM©wgUvi|

cÖkœg‡Z, = 2 ev, = 2 =

AvqZ‡ÿ‡Îi cwimxgv = ( ) wgUvi ev, 144 wgUvi|

kZ©vbymv‡i, eM©‡ÿ‡Îi cwimxgv = 144 wgUvi|

eM©‡ÿ‡Îi GK evûi ˆ`N©¨ = ( ) wgUvi ev, 36 wgUvi|

2

Gi gvb wbY©q

Ki‡Z

g‡b Kwi, AvqZ‡ÿ‡Îi cÖ ’ wgUvi|

AvqZ‡ÿ‡Îi ˆ`N©¨ wgUvi|

AvqZ‡ÿ‡Îi †ÿÎdj = eM©wgUvi|

cÖkœg‡Z, = 2 ev, = 2 =

1

AvqZ‡ÿ‡Îi

†ÿÎdj Gi

gva¨‡g cÖKvk Ki‡Z

g‡b Kwi, AvqZ‡ÿ‡Îi cÖ ’ wgUvi|

AvqZ‡ÿ‡Îi ˆ`N©¨ wgUvi|

AvqZ‡ÿ‡Îi †ÿÎdj = eM©wgUvi|

4

BU Øviv iv ÍvwU euvavB

Ki‡Z KZ UvKv LiP

n‡e Zv wbY©q Ki‡Z

ÔLÕ †_‡K cvB, =

AvqZ‡ÿ‡Îi ˆ`N©¨ ( ) wgUvi ev, 54 wgUvi Ges cÖ ’ 18 wgUvi

iv¯Ívmn AvqZ‡ÿ‡Îi ˆ`N©¨ = ( ) wgUvi ev, 57 wgUvi

iv¯Ívmn AvqZ‡ÿ‡Îi cÖ ’ = ( ) wgUvi ev, 21 wgUvi

iv¯Ívmn AvqZ‡ÿ‡Îi ‡ÿÎdj = ( 2 ) eM©wgUvi ev,

9 M

KwVb

= 1197 eM©wgUvi|

‡`Iqv Av‡Q, AvqZ‡ÿ‡Îi †ÿÎdj = 972 eM©wgUvi

iv¯Ívi †ÿÎdj = ( 2) eM©wgUvi ev, 225 eM©wgUvi|

cÖwZwU B‡Ui Zjvi †ÿÎdj = ( 2 2 ) eM©wgUvi

= 0.03125 eM©wgUvi

myZivs iv¯ÍvwU euvavB Ki‡Z BU jvM‡e = (22 2 ) wU

= 7200 wU|

cÖwZwU B‡Ui g~j¨ 15 UvKv n‡j, iv¯ÍvwU euvavB Ki‡Z LiPn‡e †gvU

(7200 ) UvKv ev, 108000 UvKv|

3

iv¯ÍvwU euvavB Ki‡Z

KZwU BU jvM‡e Zv

wbY©q Ki‡Z

ÔLÕ †_‡K cvB, =

AvqZ‡ÿ‡Îi ˆ`N©¨ ( ) wgUvi ev, 54 wgUvi Ges cÖ ’ 18 wgUvi

iv¯Ívmn AvqZ‡ÿ‡Îi ˆ`N©¨ = ( ) wgUvi ev, 57 wgUvi

iv¯Ívmn AvqZ‡ÿ‡Îi cÖ ’ = ( ) wgUvi ev, 21 wgUvi

iv¯Ívmn AvqZ‡ÿ‡Îi ‡ÿÎdj = ( 2 ) eM©wgUvi ev,

= 1197 eM©wgUvi|

‡`Iqv Av‡Q, AvqZ‡ÿ‡Îi †ÿÎdj = 972 eM©wgUvi

iv¯Ívi †ÿÎdj = ( 2) eM©wgUvi ev, 225 eM©wgUvi|

cÖwZwU B‡Ui Zjvi †ÿÎdj = ( 2 2 ) eM©wgUvi

= 0.03125 eM©wgUvi

myZivs iv¯ÍvwU euvavB Ki‡Z BU jvM‡e = (22 2 ) wU

= 7200 wU|

2 iv¯Ívmn AvqZ‡ÿ‡Îi

‡ÿÎdj wbY©q Ki‡Z

ÔLÕ †_‡K cvB, =

AvqZ‡ÿ‡Îi ˆ`N©¨ ( ) wgUvi ev, 54 wgUvi Ges cÖ ’ 18 wgUvi

iv¯Ívmn AvqZ‡ÿ‡Îi ˆ`N©¨ = ( ) wgUvi ev, 57 wgUvi

iv¯Ívmn AvqZ‡ÿ‡Îi cÖ ’ = ( ) wgUvi ev, 21 wgUvi

iv¯Ívmn AvqZ‡ÿ‡Îi ‡ÿÎdj = ( 2 ) eM©wgUvi ev,

= 1197 eM©wgUvi|

1

iv¯Ívmn AvqZ‡ÿ‡Îi

ˆ`N©¨ I cÖ ’ wbY©q

Ki‡Z

ÔLÕ †_‡K cvB, =

AvqZ‡ÿ‡Îi ˆ`N©¨ ( ) wgUvi ev, 54 wgUvi Ges cÖ ’ 18 wgUvi

iv¯Ívmn AvqZ‡ÿ‡Îi ˆ`N©¨ = ( ) wgUvi ev, 57 wgUvi

iv¯Ívmn AvqZ‡ÿ‡Îi cÖ ’ = ( ) wgUvi ev, 21 wgUvi

10bs cÖ‡kœi DËi

cÖkœ bs `ÿZvi

Íi

cÖ‡kœi

gvb

wkLbdj/wkÿv_©xiv

cvi‡e

(Rubrics)

bgybv DËi

10 K mnR

2

wew”Qbœ I Awew”Qbœ

Pj‡Ki msÁv

wjL‡Z

wew”Qbœ PjK: †h mKj Pj‡Ki gvb ïaygvÎ c~Y©msL¨v nq Zv wew”Qbœ PjK|

†hgb: RbmsL¨vg~jK Dcv‡Ëi PjK|

Awew”Qbœ PjK: †h mKj Pj‡Ki gvb †h‡Kv‡bv ev Íe gvb n‡Z cv‡i, †m

mKj Awew”Qbœ PjK| †hgb: eqm, D”PZv, IRb BZ¨vw` mswkøó wb‡`©kK

Dcv‡Ëi PjK|

1

wew”Qbœ ev Awew”Qbœ

Pj‡Ki msÁv

wjL‡Z

wew”Qbœ PjK: †h mKj Pj‡Ki gvb ïaygvÎ c~Y©msL¨v nq Zv wew”Qbœ PjK|

†hgb: RbmsL¨vg~jK Dcv‡Ëi PjK| A_ev,

Awew”Qbœ PjK: †h mKj Pj‡Ki gvb †h‡Kv‡bv ev Íe gvb n‡Z cv‡i, †m

mKj Awew”Qbœ PjK| †hgb: eqm, D”PZv, IRb BZ¨vw` mswkøó wb‡`©kK

Dcv‡Ëi PjK|

10 L ga¨g

4

†kÖwY e¨eavb 8

a‡i MYmsL¨v

wb‡ekb mviwY

ˆZwi Ki‡Z

Dcv‡Ëi cwimi = (m‡e©v”P gvb me©wb¤œ gvb) + 1

= (99 61) + 1 = 39

‡`Iqv Av‡Q, ‡kÖwY e¨eavb 8

†kÖwY msL¨v = (39 8) = 4.875 ev, 5

‡kÖwY e¨eavb 8 a‡i wb‡¤œ MYmsL¨v mviwY ˆZwi Kiv n‡jv:

cÖvß b¤^i U¨vwj wPý MYmsL¨v

61 68 6

69 76 9

77 84 14

85 92 8

93 100 3

†gvU = 40

3

‡kÖwY e¨eavb 8

a‡i MYmsL¨v

wb‡ekb mviwY‡Z

U¨vwj wPý wbY©q

Ki‡Z

Dcv‡Ëi cwimi = (m‡e©v”P gvb me©wb¤œ gvb) + 1

= (99 61) + 1 = 39

‡`Iqv Av‡Q, ‡kÖwY e¨eavb 8

†kÖwY msL¨v = (39 8) = 4.875 ev, 5

‡kÖwY e¨eavb 8 a‡i wb‡¤œ MYmsL¨v mviwY ˆZwi Kiv n‡jv:

cÖvß b¤^i U¨vwj wPý MYmsL¨v

61 68

69 76

77 84

85 92

93 100

2

‡kÖwY e¨eavb 8

a‡i MYmsL¨v

wb‡ekb mviwY‡Z

‡kÖwY wbY©q Ki‡Z

Dcv‡Ëi cwimi = (m‡e©v”P gvb me©wb¤œ gvb) + 1

= (99 61) + 1 = 39

‡`Iqv Av‡Q, ‡kÖwY e¨eavb 8

†kÖwY msL¨v = (39 8) = 4.875 ev, 5

‡kÖwY e¨eavb 8 a‡i wb‡¤œ MYmsL¨v mviwY ˆZwi Kiv n‡jv:

cÖvß b¤^i U¨vwj wPý MYmsL¨v

61 68

69 76

77 84

85 92

93 100

1

‡kÖwY e¨eavb 8

a‡i †kÖwY msL¨v

wbY©q Ki‡Z

Dcv‡Ëi cwimi = (m‡e©v”P gvb me©wb¤œ gvb) + 1

= (99 61) + 1 = 39

‡`Iqv Av‡Q, ‡kÖwY e¨eavb 8

†kÖwY msL¨v = (39 8) = 4.875 ev, 5

10 M

KwVb

4

mviwY n‡Z

Awew”Qbœ †kÖwYmxgv

I †kÖwYi ga¨we› y

wbY©q K‡i MYmsL¨v

eûfzR A¼bc~e©K

eY©bv wjL‡Z

cÖvß b¤^i Awew”Qbœ †kÖwY mxgv

‡kÖwYi ga¨we› y MYmsL¨v

61 68 60.5 68.5 64.5 6

69 76 68.5 76.5 72.5 9

77 84 76.5 84.5 80.5 14

85 92 84.5 92.5 88.5 8

93 100 92.5 100.5 96.5 3

Aÿ eivei QK KvM‡Ri cÖwZ Ni‡K 4 GKK Ges Aÿ eivei QK KvM‡Ri

cÖwZ Ni‡K 1 GKK a‡i cÖ`Ë Dcv‡Ëi MYmsL¨v eûfzR AuvKv n‡jv|

3

mviwY n‡Z

Awew”Qbœ †kÖwYmxgv

I †kÖwYi ga¨we› y

wbY©q K‡i MYmsL¨v

eûfzR A¼b Ki‡Z

cÖvß b¤^i Awew”Qbœ †kÖwY mxgv

‡kÖwYi ga¨we› y MYmsL¨v

61 68 60.5 68.5 64.5 6

69 76 68.5 76.5 72.5 9

77 84 76.5 84.5 80.5 14

85 92 84.5 92.5 88.5 8

93 100 92.5 100.5 96.5 3

2

mviwY n‡Z

Awew”Qbœ †kÖwYmxgv

I †kÖwYi ga¨we› y

wbY©q K‡i QK

KvM‡R A‡ÿ

Awew”Qbœ †kÖwYmxgv

I A‡ÿ

MYmsL¨v ewm‡q

†¯‹wjs Ki‡Z

cÖvß b¤^i Awew”Qbœ †kÖwY mxgv

‡kÖwYi ga¨we› y MYmsL¨v

61 68 60.5 68.5 64.5 6

69 76 68.5 76.5 72.5 9

77 84 76.5 84.5 80.5 14

85 92 84.5 92.5 88.5 8

93 100 92.5 100.5 96.5 3

1

mviwY n‡Z

Awew”Qbœ †kÖwYmxgv

/ †kÖwYi ga¨we› y

wbY©q Ki‡Z

cÖvß b¤^i Awew”Qbœ †kÖwY mxgv ‡kÖwYi ga¨we› y MYmsL¨v

61 68 60.5 68.5 64.5 6

69 76 68.5 76.5 72.5 9

77 84 76.5 84.5 80.5 14

85 92 84.5 92.5 88.5 8

93 100 92.5 100.5 96.5 3

11bs cÖ‡kœi DËi

cÖkœ bs `ÿZvi

Íi b¤^i

wkLbdj /

wkÿv_©xiv cvi‡e bgybv DËi

11 K mnR

2

Pj‡Ki cwiPqmn

cÖPziK wbY©‡qi

m~ÎwU wjL‡Z

cÖPziK wbY©‡qi m~Î = h

2f

1f

1f

L

GLv‡b, L = †h †kÖwY‡Z cÖPziK Aew ’Z Zvi wb¤œgvb

f1 = cÖPziK †kÖwYi MYmsL¨v c~e©eZ©x †kÖwYi MYmsL¨v

f2 = cÖPyiK †kÖwYi MYmsL¨v - cieZ©x †kÖwYi MYmsL¨v

h = †kÖwY e¨eavb|

1

cÖPziK wbY©‡qi

m~ÎwU wjL‡Z /

Pj‡Ki cwiPq

wjL‡Z

cÖPziK wbY©‡qi m~Î = h

2f

1f

1f

L

A_ev,

GLv‡b, L = †h †kÖwY‡Z cÖPziK Aew ’Z Zvi wb¤œgvb

f1 = cÖPziK †kÖwYi MYmsL¨v c~e©eZ©x †kÖwYi MYmsL¨v

f2 = cÖPyiK †kÖwYi MYmsL¨v - cieZ©x †kÖwYi MYmsL¨v

h = †kÖwY e¨eavb|

11 L ga¨g 4

m~Î cÖ‡qvM K‡i

mwVKfv‡e ga¨K

wbY©q Ki‡Z

ga¨K wbY©‡qi mviwY wb¤œiƒc t

IRb (†KwR) wkÿv_©xi msL¨v ‡hvwRZ MYmsL¨v

41-45 4 4

46-50 6 10

51-55 12 22

56-60 20 42

61-65 15 57

66-70 3 60

n = 60

GLv‡b, n Ges

2

60

2

n ev 30|

AZGe, ga¨K n‡e 30Zg c‡`i gvb| 30 c‡`i gvb|

AZGe, 30Zg c‡`i Ae ’vb n‡e (56-60) †kÖwY‡Z| myZivs ga¨K †kÖwY

n‡jv (56-60)|

GLv‡b,

L= 56, 302

n, 22cF , 20mf , h=5

ga¨K =

mf

hcF

2

nL

=

20

5

22)(3056

=

20

5

856

=

20

40

56

= 256

= 58

11 L ga¨g

3

µg‡hvwRZ

MYmsL¨v wbY©q

K‡i ga¨K †kÖwY

wba©viY K‡i

ga¨‡Ki mwVK

m~ÎwU wjL‡Z

ga¨K wbY©‡qi mviwY wb¤œiƒc t

IRb (†KwR) wkÿv_©xi msL¨v ‡hvwRZ MYmsL¨v

41-45 4 4

46-50 6 10

51-55 12 22

56-60 20 42

61-65 15 57

66-70 3 60

n = 60

GLv‡b,

2

60

2

n ev 30 hvi

Ae¯’vb n‡e (56-60) †kÖwY‡Z| myZivs ga¨K †kÖwY n‡jv (56-60)|

GLv‡b,

L= 56, 302

n, 22cF , 20mf , h=5

ga¨K =

mf

hcF

2

nL

2

µg‡hvwRZ

MYmsL¨v wbY©q

K‡i ga¨K †kÖwY

wba©viY Ki‡Z

ga¨K wbY©‡qi mviwY wb¤œiƒc t

IRb (†KwR) wkÿv_©xi msL¨v ‡hvwRZ MYmsL¨v

41-45 4 4

46-50 6 10

51-55 12 22

56-60 20 42

61-65 15 57

66-70 3 60

n = 60

GLv‡b, n Ges

2

60

2

n ev 30|

AZGe, ga¨K n‡e 30Zg c‡`i gvb| 30 c‡`i gvb|

AZGe, 30Zg c‡`i Ae ’vb n‡e (56-60) †kÖwY‡Z| myZivs ga¨K †kÖwY

n‡jv (56-60)|

1

µg‡hvwRZ

MYmsL¨v wbY©q

Ki‡Z/ga¨K

wbY©‡qi mwVK

m~ÎwU wjL‡Z

ga¨K wbY©‡qi mviwY wb¤œiƒc t

IRb (†KwR) wkÿv_©xi msL¨v ‡hvwRZ MYmsL¨v

41-45 4 4

46-50 6 10

51-55 12 22

56-60 20 42

61-65 15 57

66-70 3 60

n = 60

A_ev ,

ga¨K =

mf

hcF

2

nL

11 M KwVb 4

mwVKfv‡e

AvqZ‡jL

AsKbc~e©K

AsK‡bi weeiY

wjL‡Z

cÖ`Ë MYmsL¨v wb‡ekb mviwb‡Z †kÖwYe¨wß wewQbœ| AvqZ‡jL AsK‡bi j‡ÿ¨

†kÖwYe¨vwß Awew”Qbœ wb¤œiƒc t

‡kÖwY e¨vwß

(†KwR)

Awew”Qbœ

†kÖwYmxgv

MYmsL¨v

(wkÿv_©x msL¨v) 41-45 40.5-45.5 4

46-50 45.5-50.5 6

51-55 50.5-55.5 12

56-60 55.5-60.5 20

61-65 60.5-65.5 15

66-70 65.5-70.5 3

QK KvM‡Ri cÖwZ Ni‡K GK GKK a‡i X Aÿ eivei †kÖwYe¨vwß Ges Y Aÿ

eivei WvbmsL¨v wb‡q AvqZ‡jL AuvwK| X Aÿ eivei †kÖwbe¨vwß 40.5 †_‡K

Avi¤¢ n‡q‡Q| g~jwe› y †_‡K 40.5 ch©šÍ c~e©eZ©x Ni¸wj Av‡Q †evSv‡Z fv½v

wPý e¨envi Kiv n‡q‡Q|

11 M KwVb 3

mvibx n‡Z

Awew”Qbœ †kÖwY mxgv

wbY©q K‡i x A‡ÿ

Awew”Qbœ †kÖwY mxgv

I y A‡ÿ

MYmsL¨v ewm‡q

AvqZ‡jL AsKb

Ki‡Z

cÖ`Ë MYmsL¨v wb‡ekb mviwb‡Z †kÖwYe¨wß wewQbœ| AvqZ‡jL AsK‡bi j‡ÿ¨

†kÖwYe¨vwß Awew”Qbœ wb¤œiƒc t

‡kÖwY e¨vwß

(†KwR)

Awew”Qbœ

†kÖwYmxgv

MYmsL¨v

(wkÿv_©x msL¨v)

41-45 40.5-45.5 4

46-50 45.5-50.5 6

51-55 50.5-55.5 12

56-60 55.5-60.5 20

61-65 60.5-65.5 15

66-70 65.5-70.5 3

2

mvibx n‡Z

Awew”Qbœ †kÖwY mxgv

wbY©q K‡i x A‡ÿ

Awew”Qbœ †kÖwY mxgv

I y A‡ÿ

MYmsL¨v ewm‡q

mwVKfv‡e †¯‹wjs

Ki‡Z

‡kÖwY e¨vwß

(†KwR)

Awew”Qbœ

†kÖwYmxgv

MYmsL¨v

(wkÿv_©x msL¨v)

41-45 40.5-45.5 4

46-50 45.5-50.5 6

51-55 50.5-55.5 12

56-60 55.5-60.5 20

61-65 60.5-65.5 15

66-70 65.5-70.5 3

1

mvibx n‡Z

Awew”Qbœ †kÖwY mxgv

wbb©q Ki‡Z

cÖ`Ë MYmsL¨v wb‡ekb mviwb‡Z †kÖwYe¨wß wewQbœ| AvqZ‡jL AsK‡bi j‡ÿ¨

†kÖwYe¨vwß Awew”Qbœ wb¤œiƒc t

‡kÖwY e¨vwß

(†KwR)

Awew”Qbœ

†kÖwYmxgv

MYmsL¨v

(wkÿv_©x msL¨v)

41-45 40.5-45.5 4

46-50 45.5-50.5 6

51-55 50.5-55.5 12

56-60 55.5-60.5 20

61-65 60.5-65.5 15

66-70 65.5-70.5 3