185
yeerpeieefCele uesKe - 27
Øeleehe Leesjele,
MeeUe : DeeoMe& efJeÅeeueÙe, ieesjsieebJe; ceeuee[ veeF&š neÙemketâue.
yees[& hejer#ee vecegvee Gòejheef$ekeâe (›eâceçeŠ)
Gkeâue : x2 + x - 2 = (x + 2)(x - 1) ne efouesuÙee oesvner
yengheoerbÛee ce.mee.efJe. Deens.
(x + 2) Je (x - 1) ns efouesuÙee oesvner yengheoeRÛes meeceeF&keâ DeJeÙeJe
Deensle.
(x + 2) ne (x - 2) (2x2 + ax + 2) Ûee DeJeÙeJe Deens.
(x + 2) ne (2x2 + ax + 2) Ûee DeJeÙeJe Deens.
x = - 2 ner efkebâcele 2x2 + ax + 2 ceOÙes "sJet
5.
∴
∴
∴
∴
∴ 2x( - 2)2 + ax( - 2) + 2 = 0
8 - 2a + 2 + 0
2a + 10
∴ a=5
lÙeeÛeØeceeCes (x - 1) ne (x + 2)(3x + bx + 1) Ûee DeJeÙeJe
∴ (x - 1) ne (3x2 + bx + 1) ÛeeÛe DeJeÙeJe Deens.
∴ (x = 1) ner efkebâcele 3x2 + bx + 1 ceOÙes "sJet.
∴ 3²(1)2 + b²(1) + 1 = 0
∴ 3+b+1=0
∴ b=-4
Gòej a = 5 Je b = - 4
Deens.
Gkeâue : ueerhe Je<ee&le 365 + 1 = 366 efoJeme Demeleele.
keâesCelÙeener ueerhe Je<ee&le 52 Dee"Je[s Je 2 efoJeme Demeleele.
Ùee 52 Dee"Je[ŸeebÛes 52 jefJeJeej Ùee Je<ee&le DemeleeleÛe.
GjuesuÙee oesve efoJemeemee"er vecegvee DeJekeâeMe s = {(meesce, cebieU), (cebieU, yegOe) (yegOe, ieg®), (ieg®, Meg›eâ)
(Meg›eâ, Meefve), (Meefve, jefJe), (jefJe, meesce)}
6.
∴
∴ n(s) = 07
A ner Iešvee jefJeJeejÛee DebleYee&Je DemeCes ner
∴ A = {(Meefve, jefJe), (jefJe, meesce)}
∴ n (A) = 02
∴ A Ùee IešvesÛeer mebYeeJÙelee (ueerhe Je<ee&le 52
Deens Demes ceevet
jefJeJeej DemeCÙeeÛeer
n (A) 2
p (A) = n (s) =
7
mebYeeJÙelee)
ØeMve 4 Kee}er}hewkeâer keâesCelesner leerve GheØeçve mees[Jee.
1.
Gkeâue : ØeLece efouesuÙee ØelÙeskeâ meceerkeâjCeeÛÙee DeeuesKeeJejerue 4
efyebotÛee leòeâe leÙeej keâ¤.
1) 2x - y - 7 = 0
2) x + y + 1 = 0
y = 2x - 7
y=-x-1
x
0
2
4 5 x
-2
0
1
2
y
-7
-3
1 3 y
1
-1
-2
-3
(x,y)(0, - 7)(2, - 3)(4,1)(5,3)(x,y)( - 2,1)(0, - 1)(1, - 2)(2, - 3)
DeeuesKe js<eebÛÙee ÚsoveefyebotÛes efveoxMekeâ = (2, - 3)
∴ meceerkeâjCeeÛeer Gkeâue = (2, - 3)
2 Gkeâue
: ØeLece Jeie& meueie keâ¤ve IesT
Jeie&
meueie Jeie&
JeejbJeejlee
GbÛeer (mes.ceer.)
(efJeÅeeLeea mebKÙee)
130 - 134
135 - 139
140 - 144
129.5 - 134.5
134.5 - 139.5
139.5 - 144.5
145 - 149
150 - 154
155 - 159
160 - 164
144.5 - 149.5
149.5 - 154.5
154.5 - 159.5
159.5 - 164.5
5
15 fi
28 fm
yenguekeâerÙe Jeie&
24 f2
17
10
01
ÙesLes 139.5-144.5 Ùee Jeiee&Ûeer 28 ner JeejbJeejlee meJee&le peemle
Deens
∴ 139.5 - 144.5 ne yegnuekeâerÙe Jeie& Deens
∴ L = 139.5, fm = 28, f1 = 15, f2 = 24
yenguekeâerÙe Jeiee&Ûes JeieeËlej (h) = 144.5 - 139.5 = 5
fm - f1
∴ yenguekeâ = L + 2fm-f1- f2 × 4
28 - 15
×5
= 139.5 +
2 × 28 - 15 - 24
(
(
mebheke&â :
)
)
[email protected]
= 139.5 + 65
17
= 139.5 + 3.82
yenguekeâ = 143.32 mes.ceer.
3.
Gkeâue : 12(x2 + x12) - 28(x - x1) - 9 = 0
1
1
∴ 12[(x - x)2 + 2] - 28(x - x) - 9 = 0
1
∴ x - x = m "sJet
∴ 12(m2 + 2) - 28m - 9 = 0
∴ 12m2 - 28m - 9 + 24 = 0
∴ 12m2 - 28m + 15 = 0
∴ 12m2 - 18m - 10m + 15 = 0
∴ (2m - 3)(6m - 5) = 0
∴ 1) 2m - 3 = 0 efkebâJee 2) 6m - 5 = 0
5
3
m = 2 efkebâJee m = 6
1
m = x - x ner efkebâcele "sJet
1 3
1 5
∴ x-x=2
efkebâJee
x-x=6
x²-1 5
x²-1 3
x =2
x =6
2
6x2 - 6 = 5x
∴ 2x - 2 = 3x
2
∴ 2x - 3x - 2 = 0 ...(-1) 6x2 - 5x - 6 = 0 ... (2)
1) 2x2 - 4x + x - 2 = 0
2x (x - 2) + (x - 2) = 0
∴ (x - 2) (2x + 1) = 0
∴ x - 2 = 0 efkebâJee 2x + 1 = 0
∴ x = 2 efkebâJee x = -1
2
2) 6x2 - 5x - 6 = 0
6x2 - 9 x - + 4x + 2) = 0
3x (2x - 3) + 2 (2x - 3) = 0
(2x - 3) (3x + 2) = 0
2x - 3 = 0 efkebâJee 3x + 2 = 0
3
∴ x = 2 efkebâJee x = -2
3
Gòej : meceerkeâjCeeÛee GkeâuemebÛe = (2, -12, 32, -23)
4. Gkeâue : jeMeerÛee DebMe : P (x) = x3 - 2x2 - 19x + 20
∴ P (x) ÛÙee meJe& meniegCekeâebÛeer yesjerpe = 1 - 2 - 19 + 20 = 0
∴ (x - 1) ne P (x) Ûee DeJeÙeJe Deens.
ogmeje DeJeÙeJe mebMues<ekeâ Yeeieekeâej heæleerves keâe{t
∴
∴
jeMeerÛee
∴
∴
1 1 - 2 - 19 - 20
1 - 1 - 20
1 - 1 - 20
0
ogmeje DeJeÙeJe = x2 - x - 20 = (x - 5) (x + 4)
P(x) = (x - 1) (x - 5) (x + 4)
Úso : q(x) = x3 - 3x2 - 13x + 15
q(x) ÛÙee meJe& meniegCekeâebÛeer yesjerpe = 1 - 3 - 13 + 15 = 0
(x - 1) ne q(x) Ûee DeJeÙeJe Deens.
ogmeje DeJeÙeJe mebMues<ekeâ Yeeieekeâej heæleerves keâe{t
1 1 -3 -13 +15
1 - 2 -15
1 - 2 -15 [ 0 ]
∴ ogmeje DeJeÙeJe = x2 - 2x - 15 = (x + 3) (x - 5)
∴ q(x) = (x - 1) (x + 3) (x - 5)
p(x) (x-1) (x+4) (x-5)
∴ efouesueer jeMeer = q(x) = (x+3) (x-1) (x-5)
x
+
4
=x+3
Gòej : = xx ++ 43
GÅeeÛee efJe<eÙe : (›eâceçe:)