math 2nd

weÕ¦vi cwigvc I mÁ¿vebv 1

`kg AaÅvq

weÕ¦vi cwigvc I mÁ¿vebv

ckÉ ² 1 (i) ckÉ ² 2 ‰KwU eÅvGM 2wU QØv, 2q eÅvGM 4wU mv`v I 3wU jvj

ˆkwÉ Y eÅvwµ¦ 4-8 9-13 14-18 19-23 24-28 29-33 34-38 ‰es 3q eÅvGM 3wU mv`v I 7wU jvj ej AvGQ|

MYmsLÅv 3 5 8 10 4 6 2 [AvBwWqvj Õ•zj ‰´£ KGjR, gwZwSj, XvKv]

(ii) 500 Rb cixÞv^Æxi gGaÅ 140 Rb MwYGZ, 60 Rb cwimsLÅvGb ‰es K. 1 ˆ^GK 20 chƯ¦ Õ¼vfvweK msLÅvàwj nGZ ‰KwU msLÅv

30 Rb Dfq welGq ˆdj KGi|  [wfKvi‚bwbmv bƒb Õ•zj ‰´£ KGjR, XvKv] LywkgZ wbGj msLÅvwU 3 ‰i àwYZK nevi mÁv¿ ebv wbYÆq Ki|2

K. ‰K cÅvGKU Zvm nGZ 4wU Zvm DVvGbv nj| Zvmàwj L. DóxcGKi `yBwU QØv ‰KGò wbGÞc Kiv nGj ZvG`i

ivRv bv nIqvi mÁ¿veÅZv wbYÆq Ki| 2 bgybv ˆÞòwU ŠZwi Ki ‰es QØvq 5 DVvi mÁv¿ ebv KZ

L. (i) bs Z^Å nGZ ˆf`vsK wbYÆq Ki| 4 Zv wbYÆq Ki| 4

M. ‰KRb cixÞv^ÆxGK Š`efvGe ˆbqv nGj Zvi ˆKej ‰K M. DóxcGKi 2q I 3q eÅvM ˆ^GK jUvwii gvaÅGg ‰KwU

welGq cvm Kivi mÁ¿veÅZv KZ? 4 eÅvM evQvB Kiv nj| H eÅvM ˆ^GK ‰KwU ej Uvbv

1 bs cGÉ ki² mgvavb nBGj, ejwU jvj iGOi nIqvi mÁ¿vebv wbYÆq Ki| 4

K ˆgvU Zvm = 52wU, ivRv = 4wU 2 bs cÉGki² mgvavb

4C4 1 K gGb Kwi, msLÅvwU 3 ‰i àwYZK nevi mÁ¿vebv = P(A)
52C4 270725
 ZvmàGjv ivRv nevi mÁ¿vebv = = ‰Lb, 1 nGZ 20 chƯ¦ ˆgvU msLÅv = 20wU

 ivRv bv nevi mÁ¿vebv 1 270724 ‰es 1 nGZ 20 ‰i gGaÅ 3 ‰i àwYZK msLÅv 3, 6, 9, 12, 15,
270725 270725
= 1  = (Ans.) 18 = 6wU

L ˆf`vâ wbYÆGqi ZvwjKv: a = 21; c = 5  wbGYÆq mÁ¿vebv = 6C1 6 3 (Ans.)
20C1 = 20 = 10
ˆkwÉ YeÅvwµ¦ gaÅwe±`y NUbmsLÅv  idi2
di = xi c a idi L `wy U QØv ‰KGò wbGÞGci bgybv ˆÞò :
xi i
S = {1, 2, 3, 4, 5, 6}  {1, 2, 3, 4, 5, 6}
4-8 6 3  3  9 27
2q QØvi DcGii wcGVi we±`y
9-13 11 5  2  10 20

14-18 16 8  1  8 8 123456

19-23 21 10 0 00 1g QØv 1 (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)

24-28 26 4 1 44 DcGii 2 (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
wcGVi 3 (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
29-33 31 6 2 12 24 we±`y 4 (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
5 (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
34-38 36 2 3 6 18

N = 38 idi = 5, idi2 = 101 6 (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)

 cwiwgZ eÅeavb,  = c idi2  Nidi2 ˆgvU bgybvwe±`y = 36wU
N
QØvq 5 DVvi AbyK„j bgybv we±`yi msLÅv = 11wU
101 3852
=c 38  =c 2.66  0.017  QØvq 5 DVvi mÁ¿vebv 11 (Ans.)
= 36
= 5  1.6256 = 8.1281
M 2q eÅvMwU jUvwii gvaÅGg wbeÆvwPZ nIqvi ci H eÅvM ˆ^GK
 ˆf`vâ, 2 = 66.06| (Ans.)
1 3 3
M ˆfbwPGò ˆ`LvGbv nGjv: M S n = 500 wbiGcÞfvGe ‰KwU jvj ej nevi mÁ¿vebv = 2  4+3 = 14

÷ay MwYGZ cvm = (140  30) Rb 110 30 Abyi…cfvGe, 3q eÅvMwU jUvwii gvaÅGg wbeÆvwPZ nIqvi ci H
= 110 Rb
evÝ nGZ wbiGcÞfvGe ‰KwU jvj nIqvi mÁ¿vebv
÷ay cwimsLÅvGb cvm = (60  30) Rb
30 = 1  7 7
2 3+7 = 20
= 30 Rb

 1 welGq cvm KGi = (110 + 30) Rb = 140 Rb myZivs ‰KwU jvj ej Uvbvi ˆgvU mÁ¿vebv = 3 7
14 + 20
 ‰KRb cixÞv^ÆxGK Š`efvGe ˆbqv nGj Zvi ˆKej
30 + 49 79
‰K welGq cvGmi mÁ¿vebv = 140 = 7 (Ans.) = 140 = 140 (Ans.)
500 25

2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò  ‰Kv`k-«¼v`k ˆkwÉ Y

DËi ms‡KZmn m„Rbkxj cÖkœ

ckÉ ² 3 `kRb QvGòi evsjv I BsGiwRGZ cvÉ µ¦ bÁ¼i wbGÁ²i QGK M. DóxcGKi cÉvµ¦ bÁ¼Gii cwiwgZ eÅeavb wbYÆq KGiv| 4
Dîi: L. 5.23 (cÉvq); M. 8.02
DcÕ©vcb Kiv nj :  [XvKv KGjR, XvKv]
ckÉ ² 5 (i)
ˆivj bÁ¼i 1 2 3 4 5 6 7 8 9 10
evsjv 52 35 47 65 70 32 40 55 60 54
BsGiwR 67 40 20 25 32 54 34 44 51 43
cGÉ ZÅK welGq cvk bÁ¼i = 40 jvj ej = 5wU
mv`v ej = 4wU jvj ej = 3wU
K. P(A) = 1 P(B) = 3 ‰es A, B Õ¼vaxb NUbv nGj P(A  B) KvGjv ej = 2wU mv`v ej = 6wU
3, 4
I P(A  B) ‰i gvb wbYÆq Ki| 2
L. wbiGcÞfvGe ‰KRb QvòGK wbeÆvPb Kiv nGj Zvi Dfq 1g evÝ 2q evÝ

welGq cvGki mÁ¿vebv wbYÆq KGi ˆh ˆKvb ‰KwU welGq cGÉ ZÅK evÝ ˆ^GK ‰KwU KGi ej DVvGe|
cvGki mÁ¿vebv wbYÆq Ki| 4
M. ˆKvb welGq Qvòiv ˆewk `ÞZv ARÆb KGiGQ Zv KviYmn (ii)
eÅvLÅv Ki| 4 51-60 61-70 71-80 81-90 91-100
bÁ¼i
Qvò 10 20 15 10 5
Dîi: K. 1 5 L. 1; M. evsjvq `ÞZv ˆewk| K. ‰K cÅvGKU Zvm ˆ^GK ‰KwU Zvm DVvGj, ZvmwU jvj
4, 6; ˆUØv nIqvi mÁ¿veÅZv KZ? 2

ckÉ ² 4 gwZwSj gGWj Õ•zj ‰´£ KGjGRi wbeÆvPbx cixÞvq L. (i) bs ˆ^GK `wy U eGji gGaÅ KgcGÞ ‰KwU jvj nIqvi
MwYGZ bÁ¼Gii webÅvm wbÁ²i…c:  [gwZwSj gGWj Õ•zj ‰´£ KGjR; XvKv] mÁ¿veÅZv KZ? 4
bÁ¼i 50-55 55-60 60-65 65-70 70-75 75-80 80-85 M. (iii) bs ‰i Z^Å mvwi ˆ^GK ˆf`vsK wbYÆq Ki| 4
Qvòx msLÅv 6 10 22 30 14 7 11
K. `yBwU eRÆbkxj NUbvi mÁ¿vebvi msGhvM mƒò eYÆbv KGiv| 2 Dîi: K. 1 L. 171; M. 138.89 (cÉvq)
26;

L. cvÉ µ¦ bÁ¼Gii PZz^ÆK eÅeavb wbYÆq KGiv| 4

cixÞvq Kgb ˆcGZ AviI cÉk² I mgvavb

cÉk² 1 (i) Avgiv Rvwb,

ˆkÉwY eÅvwµ¦ 4-8 9-13 14-18 19-23 24-28 29-33 34-38 ˆf`vâ, 2 = fixi2  fixi2
N N

MYmsLÅv 3 5 8 10 4 6 2 = 18233  (20.34)2
38
(ii) 500 Rb cixÞv^xÆi gGaÅ 140 Rb MwYGZ, 60 Rb
= 479.82  413.72

cwimsLÅvGb ‰es 30 Rb Dfq welGq ˆdj KGi| = 66.1 (Ans.)

wkLbdj-3 M awi, MwYGZ ˆdj Kivi NUbv A ‰es cwimsLÅvGb ˆdj
Kivi NUbv B ZvnGj cÉ`î Z^ÅGK ˆfbwPGò DcÕ©vcb
K. ‰K cÅvGKU Zvm nGZ 4wU Zvm DVvGbv nj| Zvmàwj KGi cvB,

ivRv bv nIqvi mÁv¿ eÅZv wbYÆq Ki| 2

L. (i) bs Z^Å nGZ ˆf`vâ wbYÆq Ki| 4 AB

M. ‰KRb cixÞv^xÆGK Š`efvGe ˆbqv nGj Zvi ˆKej ‰K A B
1B10 30 3A0
welGq cvm Kivi mÁv¿ eÅZv KZ? 4

29 bs cÉGki² mgvavb

K Avgiv Rvwb,

‰K cÅvGKU ZvGm ˆgvU Zvm msLÅv = 52

ivRv AvGQ = 4wU  ˆKej ‰K welGq cvGmi mÁv¿ ebv P{(AC  B)  (A 

 ivRv bq (52  4) = 48wU BC)}
= P(AC  B) + P(A  BC)
 cÅvGKU nGZ 4wU Zvm Uvbv nGj Zvmàwj ivRv bv
= P(B)  P (A  B) + P(A)  P(A  B)
nIqvi mÁ¿vebv 48C4 194580
= 52C4 = 270725 = 0.72 = 60  30 + 140  30
500 500 500 500
L ˆf`vâ wbYÆGqi mviwY :
140
ˆkÉwYeÅvwµ¦ MYmsLÅv ˆkÉwY fixi fixi2 = 500

i gaÅgvb xi 7
= 25 (Ans.)
48 3 6 18 108

913 5 11 55 605

1418 8 16 128 2048

1923 10 21 210 4410

2428 4 26 104 2704

2933 6 31 186 5766

3438 2 36 72 2592

ˆgvU N = 38 ixi = 773 ixi2 =
18233

 Mo, x = fixi = 773 = 20.34 (cÉvq)
N 38

DËi ms‡KZmn m„Rbkxj cÖkœ

cÉk² 2 ‰KwU cvGò 8wU jvj ej, 4wU KvGjv ej ‰es 3wU cÉk² 3 ‰KwU KGjGRi «¼v`k ˆkÉwYi 70 Rb QvGòi MwYZ

mv`v ej AvGQ| Š`efvGe 3wU ej ˆbIqv nGjv| w«Z¼ xq cGòi cÉvµ¦ bÁ¼i wbGÁ²i mviwYGZ ˆ`qv nGjv:

wkLbdj-7 I 8 ˆkÉwY 50-60 60-70 70-80 80-90 90-100

K. A I B `yBwU eRÆbkxj NUbvi ˆÞGò ˆ`LvI ˆh, MYmsLÅv 15 20 20 10 5

P(A  B) = P(A) + P(B)| 2 wkLbdj-3 I 7

L. DóxcGK DGîvwjZ 3wU eGji gGaÅ 2wU jvj ej nevi K. cÉgvY Ki ˆh, P(A) + P(Ac) = 1 2

mÁ¿vebv wbYÆq Ki| 4 L. DóxcK AejÁ¼Gb wbGekbwUi cwiwgZ eÅeavb wbYÆq Ki| 4

M. DGîvwjZ 3wU eGji gGaÅ KgcGÞ 2wU jvj ej nevi M. DóxcK ‰ wbiGcÞfvGe `yBRb QvòGK evQvB KiGj
mÁ¿vebv KZ? 3wU wewf®² isGqi ej nevi mÁ¿vebv I ˆei
ZvG`i ‰KB ˆkÉwYGZ nevi mÁ¿vebv wbYÆq Ki| 4
Ki| 4
Dîi: L. 11.81 (cÉvq); M. 36
28 36 96 161
Dîi: L. 65 ; M. 65 , 455