ÔkÕ †mu€¦ · 2 ‡mu a gi dcv`vb msl¨v d‡jøl c~e©k p(a) wby©q ki‡z a = {3, 4, 5, 6}...
TRANSCRIPT
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