General Math Cheapter 16
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mgvavb : g‡b Kwi, AvqZvKvi †ÿ‡Îi we¯Ívi (cÖ¯’) = x wg.
\ AvqZvKvi
†ÿ‡Îi ˆ`N©¨ = 2x wg.
\ AvqZvKvi †ÿ‡Îi †ÿÎdj= 2x ´ x = 2x2 eM© wg.
cÖkœvbymv‡i, 2x2 = 512
ev, x2 = 256
\ x = 16
AvqZvKvi †ÿ‡Îi cÖ¯’ = 16
wg.
Ges AvqZvKvi †ÿ‡Îi ˆ`N©¨ = 2 ´ 16 wg. ev 32 wg.
\ AvqZvKvi †ÿ‡Îi cwimxgv = 2(32 + 16) wgUvi
=
96 wgUvi (Ans.)
cÖkœ \ 2 \ GKwU Rwgi ˆ`N©¨ 80
wgUvi Ges
cÖ¯’ 60 wgUvi| H Rwgi gv‡
mgvavb
: †`Iqv Av‡Q, Rwgi ˆ`N©¨ = 80 wgUvi
Ges cÖ¯’ = 60 wgUvi
\ Rwgi †¶Îdj = Rwgi ˆ`N©¨ ´ Rwgi cÖ¯’
= (80 ´ 60) wgUvi
= 4800 eM©wgUvi
cvo ev‡` cyKz‡ii ˆ`N©¨ = (80 - 2 ´ 4) wgUvi
= (80 –
8) wgUvi
=
72 wgUvi
cyKz‡ii
cÖ¯’ = (60 - 2 ´ 4) wgUvi
= (60 – 8) wgUvi
=52 wgUvi
\ cvo ev‡` cyKz‡ii †¶Îdj = (72 ´ 52) eM©wgUvi
= 3744 eM©wgUvi
cyKz‡ii cv‡oi †¶Îdj = Rwgi †¶Îdj – cyKz‡ii †¶Îdj
= (4800 – 3744) eM©wgUvi
= 1056 eM©wgUvi (Ans.)
cÖkœ \ 3 \ GKwU evMv‡bi ˆ`N©¨ 40 wgUvi Ges cÖ¯’ 30 wgUvi| evMv‡bi
wfZ‡i mgvb cvowewkó GKwU cyKzi Av‡Q| cyKz‡ii †¶Îdj evMv‡bi †¶Îd‡ji Ask n‡j, cyKz‡ii ˆ`N©¨ I cÖ¯’ wbY©q Ki|
mgvavb : awi, cyKzi cv‡oi cÖ¯’ = x wg.
GLv‡b, evMv‡bi ˆ`N¨© = 40 wg.
Ges evMv‡bi cÖ¯’ = 30 wg.
evMv‡bi †¶Îdj = (40 ´ 30) eM©wg. ev 1200 eM©wg.
cvoev‡` cyKz‡ii ˆ`N¨© = (40 - 2x) wg.
Ges cvoev‡`
cyKz‡ii cÖ¯’ = (30 - 2x) wg.
cvoev‡` cyKz‡ii †¶Îdj = (40 –
2x) (30 - 2x) eM©wg.
kZ©g‡Z,
cyKz‡ii †¶Îdj
= ´ evMv‡bi †¶Îdj
ev, (40 – 2x)
(30 - 2x) = ´ 1200
ev, 1200 – 80x –
60x + 4x2 = 600
ev, 4x2
–140x + 1200 – 600 = 0
ev, 4x2
–140x + 600 = 0
ev, 4(x2
- 35x + 150)
= 0
ev, x2
– 30x – 5x + 150 = 0
ev, x(x – 30) –
5(x – 30) = 0
ev, (x – 30) (x
– 5) = 0
nq, (x – 30) = 0 A_ev, (x – 5) = 0
\ x = 30 \ x = 5
wKšÍz cyKz‡ii
cv‡oi cÖ¯’ evMv‡bi cÖ‡¯’i mgvb n‡Z cv‡i bv|
\ x = 5 A_©vr,
cyKzi cv‡oi cÖ¯’ = 5 wgUvi
\ cyKz‡ii ˆ`N©¨ = (40 - 2x) wgUvi
= (40 – 2 ´ 5) wgUvi
= (40 – 10) wgUvi = 30 wgUvi
Ges cyKz‡ii cÖ¯’ = (30 - 2x) wgUvi
= (30 – 2 ´ 5) wgUvi
= (30 – 10) wgUvi = 20 wgUvi
wb‡Y©q cyKz‡ii ˆ`N©¨ 30 wg. Ges
cÖ¯’ 20 wg.
cÖkœ \ 4 \ GKwU eM©vKvi gv‡Vi evB‡i Pviw`‡K 5 wgUvi PIov GKwU
iv¯Ív Av‡Q| iv¯Ívi †ÿÎdj 500 eM©wgUvi n‡j, gv‡Vi †ÿÎdj wbY©q Ki|
mgvavb : g‡b Kwi, eM©vKvi gv‡Vi GK evûi ˆ`N©¨ x wgUvi
\ eM©vKvi gv‡Vi †ÿÎdj = x2 eM© wg.
iv¯Ívi
†ÿÎdj = 500 eM© wg.
AZGe,
iv¯Ívmn gv‡Vi †ÿÎdj = (x + 500) eM©wg.
Avevi, iv¯Ívmn eM©vKvi gv‡Vi
ˆ`N©¨ = (x + 2 ´ 5) wg.
=
(x + 10) wg.
Ó Ó
Ó †ÿÎdj = (x + 10)2 eM©wg.
kZ©g‡Z,
(x + 10)2= (x +
500)
ev, x2
+ 20x + 100 = x2 + 500
ev, 20x = 400
\ x = 20
AZGe, gv‡Vi †ÿÎdj = x2 eM© wg. = 202 eM©wg.
= 400 eM©wgUvi| (Ans.)
cÖkœ \ 5 \ GKwU eM©‡ÿ‡Îi cwimxgv GKwU AvqZ‡ÿ‡Îi
cwimxgvi mgvb| AvqZ‡ÿÎwUi ˆ`N©¨ cÖ‡¯’i wZb¸Y Ges †ÿÎdj 768 eM©wgUvi| cÖwZwU 40 †m.wg. eM©vKvi cv_i w`‡q eM©‡ÿÎwU
euva‡Z †gvU KZwU cv_i jvM‡e?
mgvavb : g‡b Kwi, AvqZ‡ÿ‡Îi cÖ¯’ = x wg.
AvqZ‡ÿ‡Îi
ˆ`N©¨ = 3x wg.
\ AvqZ‡ÿ‡Îi †ÿÎdj = 3x2 wg.
kZ©g‡Z,
3x2
= 768
ev, x2 = 256
\ x = 16
AvqZ‡ÿ‡Îi
cÖ¯’ = 16 wg.
\ AvqZ‡ÿ‡Îi
ˆ`N©¨ = 3 ´ 16 wg. ev 48 wg.
AvqZ‡ÿ‡Îi cwimxgv = 2 (ˆ`N©¨ +
cÖ¯’)
= 2(48 + 16)
wg.
= 128 wg.
eM©‡ÿ‡Îi
cwimxgv = 128 wgUvi
\ Ó
GK evûi ˆ`N©¨ = (128 ¸ 4) wg. ev 32 wg.
\ Ó
†ÿÎdj = (32)2 eM©wg. ev 1024 eM©wg.
GKwU
cv_‡ii †ÿÎdj = (0.4)2 eM©wg. ev
0.16 eM©wg.
\ †gvU cv_i jvM‡e = (1024 ¸ 0.16)wU
= 6400wU| (Ans.)
cÖkœ \ 6 \ GKwU AvqZvKvi †ÿ‡Îi †ÿÎdj 160 eM©wgUvi| hw` Gi ˆ`N©¨ 6 wgUvi Kg nq, Z‡e †ÿÎwU eM©vKvi nq|
AvqZvKvi †ÿ‡Îi ˆ`N©¨ I cÖ¯’ wbY©q Ki|
mgvavb : g‡b Kwi, AvqZvKvi †ÿ‡Îi ˆ`N©¨ = x wg.
Ges AvqZvKvi †ÿ‡Îi cÖ¯’ = y wg.
\ AvqZvKvi
†ÿ‡Îi †ÿÎdj = xy eM©wg.
kZ©g‡Z, xy = 160 ..... ..... ..... ..... ..... ..... (i)
kZ©g‡Z,, x - 6 = y
ev, x = y + 6 ...... ..... ..... ..... (ii)
GLb, x Gi gvb (i) bs mgxKi‡Y ewm‡q cvB,
(y +
6)y = 160
ev, y2
+ 6y - 160 =
0
ev, y2
+ 16y - 10y - 160 = 0
ev, (y +
16) (y - 10) =
0
nq, y + 16 = 0 A_ev, y - 10 = 0
\ y = -16 \ y = 10
wKš‘ y = -16 MÖnY‡hvM¨ bq|
\ y = 10
GLb (ii) bs mgxKiY †_‡K cvB,
x = 10
+ 6 \ x = 16
AvqZvKvi
†ÿ‡Îi ˆ`N©¨ 16 wgUvi Ges cÖ¯’ 10 wgUvi| (Ans.)
cÖkœ \ 7 \ GKwU mvgvšÍwi‡Ki f‚wg D”PZvi Ask Ges †ÿÎdj 363 eM©wgUvi n‡j, †ÿÎwUi f‚wg I D”PZv wbY©q Ki|
mgvavb : g‡b Kwi, mvgvšÍwi‡Ki D”PZv h = x wgUvi
\
mvgvšÍwi‡Ki f‚wg b = wgUvi
Ges †ÿÎdj = bh = ´ x
ev, eM©wgUvi
kZ©g‡Z, = 363
ev, 3x2
= 363 ´ 4
ev, x2 =
ev, x2 = 484
\ x = = 22
\ mvgvšÍwi‡Ki
D”PZv = 22 wgUvi
Ges f‚wg =
´ 22 wgUvi = 16.5 wgUvi
wb‡Y©q mvgvšÍwi‡Ki
f‚wg 16.5 wgUvi Ges D”PZv 22 wgUvi|
cÖkœ \
8 \ GKwU mvgvšÍwiK‡ÿ‡Îi †¶Îdj GKwU eM©‡¶‡Îi mgvb| mvgvšÍwi‡Ki f‚wg 125 wgUvi Ges D”PZv 5 wgUvi n‡j, eM©‡¶‡Îi K‡Y©i ˆ`N©¨
wbY©q Ki|
mgvavb : mvgvšÍwi‡Ki f‚wg 125 wgUvi Ges D”PZv 5 wgUvi
mvgvšÍwi‡Ki
†¶Îdj = f‚wg ´ D”PZv
=
125 ´ 5 eM©wgUvi
=
625 eM©wgUvi
eM©‡¶‡Îi
†¶Îdj = mvgvšÍwi‡Ki †¶Îdj
= 625 eM©wgUvi
GLb, eM©‡¶‡Îi evûi ˆ`N©¨ a wgUvi n‡j, †¶Îdj = a2 eM©wgUvi
kZ©g‡Z, a2 = 625
ev,a=25
= 25 wgUvi
eM©‡¶‡Îi K‡Y©i ˆ`N©¨ a = 25 = 35.35 wgUvi (cÖvq) wb‡Y©q K‡Y©i ˆ`N©¨ 35.35 wgUvi (cÖvq)|
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