A[ymbw 5
L\-cq-]-ß ƒ BapJw {]mtbm-KnI Pohn-X-Øn¬ hnhn[ L\ cq]-ß -fpsS {]tXy-I-X-Iƒ a\- n-em-t°≠ kµ¿`ß ƒ hcm-dp-≠v. kvXw`-ß -sf-°p-dn®pw Ah-bpsS hym]vXw, ]c-∏-fhv F∂ nh IW-°m-°p-∂ -Xpamb am¿§-ß ƒ Fß -s\-sb∂ v ap≥ ¢mkn¬ ]Tn-®n-´p-≠v. ka-N-Xp-c-kvXq-]n-I, hrØ kvXq]n-I, tKmfw, A¿≤-tKmfw F∂ o L\-cq-]-ß sf Ipdn-®p≈ ]T-\-amWv Cu A[ym-b-Øn¬ {]Xn-]m-Zn°p-∂ -Xv.
{][m\ Bi-b-ß ƒ } } } } } } } } } } }
kvXq]nI ka-N-Xpc kvXq]nI ka-N-Xpc kvXq]nIbpsS ]mZ-h-°v, ]m¿iz-h-°v, Db-cw, Ncn-hp-b-cw, Ch Adn-bp-∂ -Xn\pw Ah-bpsS ]c-kv]c _‘w Is≠-Øp-∂ -Xn-\pw. ka-N-Xpc kvXq]n-I-bpsS D]-cn-Xe ]c-∏-fhv ka-N-Xpc kvXq]n-I-bpsS hym]vXw hrØ kvXq]nI hrØ kvXq]nIbpsS Bcw, Db-cw, Ncn-hp-bcw F∂ nh XΩn-ep≈ _‘w hrØ kvXq]nIbpsS h{I-Xe ]c-∏-f-hv, D]-cn-Xe ]c-∏-fhv tKmfw-˛D - ] - -cn-Xe ]c-∏-fhpw hym]vXhpw A¿≤-tKm-fw˛ D]-cn-Xe ]c-∏-fhpw hym]vXhpw kwbp‡ cq]-ß ƒ˛ {]mtbm-KnI {]iv\-ß ƒ
{]h¿Ø\w 1 {Kq∏p-I-fm°n Xncn® Ip´n-Iƒ°v hnhn[ hep-∏-Øn-epff ka-N-Xpc kvXw`-ß ƒ \¬Ip-∂ p. Ah-bn¬ H∂ v \nh¿Øn-bX - ns‚ amXrI Nn{X-Øn¬ NphsS sImSp-Øn-cn-°p-∂ p. Ch-bn¬ \n∂ v ka-N-Xpc kwXw-`-Øns‚ AtX ]mZ-ap-≈Xpw A©v aqe-I-fp≈Xpamb cq]w \n¿Ωn-°m-\p≈ {]h¿Ø\w \¬Imw.
kqN\
92................................................................................................................................................................................. MUKULAM M ATHS
{]h¿Ø\w 2 ]mZ-ap-J-ß ƒ {XntIm-Ww, NXp-cw, jUv`pPw F∂ nh Bb kvXw`-ß ƒ \¬In \n¿Øn-sh®v apI-fn¬ sImSpØ {]h¿Ø-\-Øns‚ Bh¿Ø-\w. N¿®, t{ImUo-I-c-Ww. Is≠-Øm-hp∂ hkvXp-X-Iƒ = kvXq]n-I-bv°p≈ s]mXp {]tXy-I-X-Iƒ = ]mZ-ap-Jw, ]m¿iz-apJw = ]mZ-h-°v, ]m¿iz-h°v = io¿jw
{]h¿Ø\w 3 h¿Iv jo‰v Nne ka-N-Xpc kvXq]n-I-I-fpsS ]mZ-Øns‚ hihpw ]m¿iz-ap-J-Øns‚ Af-hp-Ifpw NphsS sImSp-Øn-cn-°p-∂ p. ]mZ-Øns‚ hi-Øn\v tNcp∂ ]m¿iz-ap-J-Øns‚ Af-hp-Iƒ tNcpw-]Sn tN¿sØ-gp-Xp-I. ]mZ-Øns‚ Hcp hi-Øns‚ Afhv (sk.-an)
]m¿iz-ap-J-Øns‚ hi-ß -fpsS \of-ß ƒ (sk.-an)
11 6 5 4
5,12,12 10, 11,10 8, 4, 8 9, 9, 6 7, 7, 7
{]h¿Ø\w 4 NphsS sImSp-Øn-´p≈ {XntIm-W-ß -fn¬ x s‚ hne ImWp-I.
5
5 x
10
6 x
10
6
x 12 x
10
apI-fn¬ sImSp-Øn-´p≈ Hmtcm {XntIm-W-Øn-s‚bpw ]c-∏-fhv ImWp-I. MUKULAM MATHS ................................................................................................................................................................................93
{]h¿Ø\w 5 Hcp ka-N-Xpc kvXq]n-I-bpsS Hcp ]m¿iz-ap-J-Øns‚ Nn{Xw NphsS sImSp-Øn-cn-°p-∂ p. AXns‚ ]mZ-h-°ns‚ \ofhpw Ncn-hp-b-chpw Nn{X-Øn¬ AS-bm-f-s∏-Sp-Øn-bn-´p-≠v. F¶n¬ a) b) c) d)
B apJ-Øns‚ ]c-∏-f-sh{X? kvXq]n-I-bpsS ]m¿iz-apJ ]c-∏-fhv F¥ v? ]mZ-Øns‚ ]c-∏-fhv ImWp-I. D]-cn-Xe ]c-∏-fhv F{X NXp-c{i sk‚n-ao-‰-dm-Wv.
10 cm
12 cm
{]h¿Ø\w 6 ]-´n-I-bn¬ sImSp-Øn-cn-°p∂ Hmtcm ka-_-lp-`pP kvXq]n-I-bp-sSbpw ]mZ-Øns‚ \ofw a bpw Ncn-hp-bcw l Dw BIp-∂ p. ]´nI ]q¿Øn-bm-°p-I. {IaHcp ]m¿iz-ap-JkvXq]nI ]m¿iz-ap-J- ]c-∏-fhv \-º¿ Øns‚ ]c-∏-fhv 1
ka-`p-P-{Xn-tImW kvXq]nI
½ al
3x ½ al = ½ x3a x l
2
ka-N-Xpc kvXq]nI
½ al
4x ½ al = ½ x4a x l
3
ka-]-©-`pP kvXq]nI
............
5x ½ al =
4
ka-j-Uv`pP kvXq]nI
............
............... = ..............
5
ka k]vX-`pP kvXq]nI
............
............... = ..............
..........................................................................
............
............... = ..............
............
............... = ..............
........................................................................ 6.
n hi- ap≈ ka- _-lp- `pP kvXq]nI
kvXq]n-I-bsS ]m¿iz-Xe ]c-∏-fhv = ½ x ]mZ-Np-‰-fhv x Ncn-hp-bcw F∂ hmIy-Øn-te°v FØn-t®-cm≥ {ian-°p-at√m?
94................................................................................................................................................................................. MUKULAM M ATHS
{]h¿Ø\w 8 k-a-N-Xpc kvXq]n-I-bpsS Nne Af-hp-Iƒ sImSp-Øn-cn-°p-∂ p. hn´-`mKw ]qcn-∏n-°p-I. {IaDbcw Ncn-hp-bcw ]mZ-h°v ]m¿iz-h°v ]mZ-hn-In¿Æ w \-º¿ (sk.-ao) (sk.-an) (sk.-an) (sk.-an) (sk.-an) 1
36
45
...........
.............
..............
2
12
.........
18
.........
.........
3
.........
25
30
.........
.........
4
.........
.........
72
60
.........
5
.........
.........
.........
20
24
6
48
.........
.........
.........
40
7
15
17
.........
.........
.........
]cn-io-e\ {]iv\-ß ƒ =
=
=
=
=
Hcp ka-N-Xpc kvXq]n-I-bpsS ]mZ-h-°ns‚ \ofw 10 sk an Dw, Ncn-hp-bcw 14 skanbpw Bbm¬ AXns‚ ]m¿iz-Xe ]c-∏-fhpw D]-cn-Xe ]c-∏-fhpw ImWp-I. Hcp ka-N-Xpc kvXq]n-I-bpsS ]mZ-Np-‰-fhv 40 skan, Dbcw 12 skan F∂ n-h-bm-Wv. B kaN-Xpc kvXq]n-I-bpsS ]c-∏-fhv ImWp-I. ka-N-Xpc kvXq]n-I-bpsS Hcp ]m¿iz-ap-J-Øns‚ ]c-∏-fhv 96 N.-sk.an Dw Ncn-hp-bcw 12 skan Dw Bbm¬ AXns‚ D]-cn-Xe ]c-∏-fhv ImWp-I. ka-N-Xpc kvXq]n-I-bpsS F√m h°p-Ifpw Xpey-am-Wv. h°p-I-fpsS BsI \ofw 120 sk.ao Bbm¬ D]-cn-Xe ]c-∏-fhv ImWp-I. ka-N-Xpc kvXq]n-I-bpsS ]mZ-h-°ns‚ \ofw b bqWn‰pw Ncn-hp-bcw l bqWn‰pw Bbm¬ ]m¿iz-Xe ]c-∏-fhv D]-cn-Xe ]c-∏-fhv F∂ nh ImWm-\p≈ _oP-K-WnX hmNIw Fgp-XpI.
ka-N-Xpc kvXq]n-I-bpsS hym]vXw GXv ka-N-Xpc kvXq]n-I-bp-sSbpw hym]vXw AXns‚ AtX ]mZhpw Db-c-hp-ap≈ kvXw`Øns‚ hym]vX-Øns‚ aq∂ n¬ H∂ mWv F∂ hkvXpX ]mT-]p-kvXI {]h¿Ø-\-Øn-eqsS t_m[ys∏-Sp-Øm-hp-∂ -Xm-Wv. {]h¿Ø-\-Øn-eqsS Dcp-Øn-cn-bp-∂ hkvXpX ka-N-Xpc kvXq]n-I-bpsS hym]vXw = 1/3 x ]c-∏-fhv x Dbcw
{]h¿Ø\w 9 ]mZ-]c∏fhv 100 N skao Dw Dbcw 15 skanDw Bb ka-N-Xpc kvXq]n-I-bpsS hym]vX-sa¥ v? hym]vXw = 1/3 x ]c-∏-fhv x Dbcw = 1/3 x 100 x 15 = 500 L\ sk‚n-ao-‰¿
MUKULAM MATHS ................................................................................................................................................................................95
]cn-io-e\ {]iv\-ß ƒ = =
=
=
=
=
Hcp ka-N-Xpc kvXq]n-I-bpsS ]mZ-Np-‰-fhv 80 sk.ao Dbcw 33 skan Bbm¬ hym]vX-sa¥ v? ]mZ-Np-‰-fhv 64 skan-Dw, Ncn-hp-bcw 17 skanDw Bbn-´p≈ ka-N-Xpc kvXq]n-I-bpsS hym]vXw ImWp-I. ka-N-Xpc kvXq]n-I-bpsS ]m¿iz-apJ ]c-∏-fhv 2320 N.-sk.ao Dw Hcp ]mZ-h-°ns‚ \ofw 40 skan Dw Bbm¬ AXns‚ hym]vX-sa¥ v? Hcp ka-N-Xpc kvXq]n-I-bpsS ]mZ hnIn¿Æ -Øns‚ \ofw 18-skan Dw ]m¿iz-h°v 15 skan Dw Bbm¬ AXns‚ hym]vXw ImWp-I. ]mZ-h-°ns‚ \oew 21 skan Dw hym]vXw 2940 L\ skan-bp-amb ka-N-Xpc kvXq]n-I-bpsS Dbcw ImWp-I. 3200 L\-skan hym]vX-ap≈ Hcp ka-N-Xpc kvXq]n-I-bpsS Dbcw 24 skan Bbm¬ ]mZ]-c-∏-f-hv, D]-cn-Xe ]c-∏-fhv F∂ nh ImWp-I.
kvXq]nIm ]oT-Øns‚ hym]vXw ]mZ-h-°ns‚ \ofw b bqWn‰v Bb I´n-bmb Hcp ka-N-Xpc kvXq]n-I-bpsS io¿j-Øns‚ `mKØv \n∂ v ]mZ-h-°ns‚ \ofw a bqWn‰v Bb sNdnb Hcp ka-N-Xpc kvXq]n-I- apdn®v am‰p∂ p. G¶n¬ tijn-°p∂ `mK-Øns‚ hym]vXw ImWp∂ hn[w.
a
x h b
kvXq]nIm ]oT-Øns‚ hym]vXw= henb ka-N-Xpc kvXq]n-I-bpsS hym]vXw ˛ sNdnb ka-N-Xpc kvXq]n-I-bpsS hym]vXw = 1/3 b2 x- 1/3 a2 (x-h) = 1/3 b bh - 1/3 a bh - h b-a b-a 2
2
]
= 1/3 b3h - 1/3 a2bh + 1/3 a2 (bh-ah) b-a
]
= 1/3 h b3 - a2b + a2b -a3 b-a = 1/3 h b3 - a3 b-a = 1/3 h x (b-a) (b2 + ab + a2 b-a
]
kZri {XntIm-W-ß -fpsS {]tXy-IX A\p-k-cn®v a =x-h b x ax = b (x - h) x (b-a) = bh x = bh b-a
= 1/3 h (b2 +ab+a2)
96................................................................................................................................................................................. MUKULAM M ATHS
hr-Ø-kvXq-]nI {]h¿Ø\w 1
h
Bcw
Nc
n-hp -b
cw
l
I´n-I-S-emkv sIm≠v \n¿Ωn® hrØ-kvXq-]nI hnX-cWw sNbvXv AXns‚ Bcw, Ncn-hp-b-cw, Dbcw F∂ nh ]cn-N-b-s∏-Sp-Øm-hp-∂ -Xm-Wv. XpS¿∂ v H´n® `mK-Øn¬ IqSn apdn®v ]mZ-ap-Jw, ]m¿izapJw (h-{I-ap-Jw) F∂ nh th¿s]-Sp-Øp-I. Nn{Xw
Bcw
r
r
]mZ-apJw hrØhpw h{I-apJw hrØmw-i-hp-ambpw e`n-°p-a-t√m.
{]h¿Ø\w 2 I´n-I-S-emkn¬ hc® 15 sk an Bc-ap≈ Hcp hrØ-Øn¬ \n∂ v 1200 tI{µ-tIm-Wp≈ hrØmwiw apdn-s®-Sp-°p-I. AXns‚ Nm]-\ofw ImWp-I. AXv D]-tbm-Kn®v ]c-am-h[n hep-∏ap≈ hrØ-kvXq-]nI D≠m-°p-I. ]mZ-ap-J-ambn apdn-s®-Sp-°m-hp∂ hrØ-Øns‚ Bc-sa{X sk an Bbn-cn-°pw. Nm]-\ofw
= 120 x 2p x 15 360
(Hº-Xmw-X-c-Ønse Nm]-\ofw t]Pv 173 t\m°p-I.) = 1 x 2p x 15 3
= 10p hrØmw-i-Øns‚ Nm]-\o-fhpw hrØ-kvXq-]n-I-bpsS ]mZ-Np-‰-fhpw Xpey-am-W-t√m. ]mZ-Øns‚ Bc
r bqWn‰v
Bbm¬
2p r = 10p r = 10 p
12 p
= 5 cm MUKULAM MATHS ................................................................................................................................................................................97
x0
l
l h
r
r
Bcw l Dw tI{µ-tIm¨ x0 Dw Bb hrØmwiw hf®v D≠m-°p∂ hrØ-kvXq-]n-I-bpsS Bcw r BsW-¶n¬ = x x 2p l = 2p r 360
CXn¬ \n∂ pw \
x = 2p r 360 2p l x = r F∂ v e`n-°p-∂ p. 360 l
x = r F∂ XXzw D]-tbm-Kn®v \n›nX Af-hn¬ Bchpw Ncn-hp-bc360 l
hp-ap≈ hrØ-kvXq-]nI \n¿Ωn-°m≥ th≠ hrØmw-iØns‚ tI{µ-tIm¨ Is≠-Ømw.
{]h¿Ø\w 3 30 sk an Bc-ap≈ hrØ-Øn¬ \n∂ v Hcp hrØmwiw apdn-s®-SpØv hrØ-kvXq-]nI D≠m°-Ww. ]mZ-Øns‚ Bcw 5 skan BW-I-sa-¶n¬ hrØmw-i-Øns‚ tI{µ-tIm-Wns‚ Af-sh{X?
{]h¿Ø\w 4 Hcp hrØ-Øns‚ Bc-Øns‚ Afhv 27 sk.an AXn¬ \n∂ pw 200 tI{µ-tIm-Wp≈ Hcp hrØmwiw apdn-s®-SpØv hrØ-kvXq-]nI \n¿Ωn-®m¬ AXns‚ Bcw F{X?
{]h¿Ø\w 5 720 tI{µ-tIm-Wp≈ hrØmwiw hf®v D≠m-°p∂ hrØ-kvXq-]n-I-bpsS ]mZ-Øn\v 314 skan ]c-∏-f-hp-s≠-¶n¬ kvXq]n-I-bpsS ]m¿tizm-∂ Xn ImWp-I.
98................................................................................................................................................................................. MUKULAM M ATHS
hrØ-kvXq-]n-I-bpsS h{I-Xe ]c-∏-fhv hrØ-kvXq-]n-I-bpsS h{I-Xe ]c-∏-fhv AXp-≠m-°m-\p-]-tbm-Kn® hrØmw-i-Øns‚ ]c-∏-fhn\v Xpey-am-Wt√m? hrØmw-i-Øns‚ ]c-∏-fhv AtX Bc-amb hrØ-Øns‚ tI{µ-tIm-Wns‚ Afhv x0 BsW¶n¬ hrØmw-i-Øns‚ ]c-∏-fhv hrØ-Øns‚ x `mK-at√? 360
hrØ-Øns‚ Bcw l bqWn-‰m-sW-¶n¬ F∂ m¬
x = r 360 l
x 360
x p l2
BW-t√m.
F∂ v apºv Is≠-Øn-bn-´p-≠-t√m.
\ hrØmw-i-Øns‚ ]c-∏-fhv =p r l
r x pl
2
l
F∂ v e`n-°p-∂ p.
GsXmcp kvXq]n-I-bp-sSbpw ]m¿iz-Xe ]c-∏-fhv ½ x ]mZ-Np-‰-fhv x Ncn-hp-bcw F∂ XXzw D]-tbm-Kn®v Is≠-Øn-bn-´p-≠-t√m. hrØ-kvXq-]n-I-bpsS ]mZ-Np-‰-fhv hrØ-Øns‚ Np‰-f-hm-W-t√m. \ hrØ-kvXq-]n-I-bpsS h{I-Xe ]c-∏-fhv =
½ x 2p r x l = pr l
hrØ-kvXq-]n-I-bpsS D]cnXe ]c-∏-fhv ]mZ-]-c-∏-f-hn-s‚bpw h{I-Xe ]c-∏-f-hn-s‚bpw XpIbm-Wv. hrØ-kvXq-]n-I-bpsS D]-cn-Xe ]c-∏-f-hv. ]mZ-Øns‚ Bcw r bqWn‰pw sNcn-hp-bcpw l bqWn‰pw Bb hrØ-kvXq-]n-I-bpsS D]-cn-Xe ]c-∏-fhv = p r2 + p r l
MUKULAM MATHS ................................................................................................................................................................................99
{]h¿Ø\w Hcp hrØ-kvXq-]n-I-bpsS Bcw 7 skanDw Ncn-hp-bcw 16 skan Dw Bbm¬ AXns‚ h{I-Xe ]c-∏-f-hv, D]-cn-Xe ]c-∏-fhv F∂ nh ImWp-I. h{I-Xe ]c-∏-fhv
= p rl
Bcw r = 7 Ncn-hp-bcw l = 16
= p x 7 x 16 = 112 p N.sk.an ]mZ ]c-∏-fhv
= p r2 = p x 72 = 49 p N.sk.an
D]-cn-Xe ]c-∏-fhv = (112 p + 49 p) = 261 p N.sk.an
IqSp-X¬ {]h¿Ø-\-ß ƒ =
=
=
hrØ-kvXq-]n-ImIr-Xn-bn-ep≈ ]m{X-Øns‚ ]mZ-Np-‰-fhv 10 p sk.an Dw Ncn-hp-bcw 12 sk.anbpw Bbm¬ AXp-≠m-°m-\p-]-tbm-Kn® jo‰ns‚ ]c-∏-f-sh{X? 2160 tI{µ-tIm-Wp≈ hrØmwi hf-®p-≠m-°p∂ hrØ-kvXq-]n-I-bpsS Bcw 9 sk.an Bbm¬ AXns‚ h{I-Xe ]c-∏-fhv ImWp-I. Nn{X- Ø n¬ sIm Sp - Ø n- c n°p∂ hrØ kvXq]n-I-bpsS h{I-Xe ]c-∏-fhpw D]-cn-Xe ]c- ∏ - f hpw ImWp- I . (]m- Z hymkw 24 sk.an Dbcw 9 sk.-an)
9 sk.-an
24 sk.-an
hrØ-kvXq-]n-I-bpsS hym]vXw ka-N-Xpc kvXq]n-I-bpsS hym]vXw Is≠-Ønb coXn-bn-eq-sS, hrØ-kvXq-]n-I-bpsS hym]vXw = 1/3 x ]mZ-]-c-∏-fhv x Dbcw F∂ _‘-Øn-se-Øn-t®-cmat√m. hrØ-kvXq-]n-Im-Ir-Xn-bn-ep≈ Hcp t]∏¿ shbn-‰ns‚ ]mZ-hymkw 8 sk.-anDw Dbcw 6 skan Dw BWv. AXns‚ hym]vX-sa¥ v? 1 hrØ-kvXq-]n-I-bpsS hym]vXw = /3 x ]mZ-]-c-∏-fhv x Dbcw r= 8 1 = /3 x p r 2 x h 2 1 = /3 x p x 4 2 x 6 h= 6 = 32 p L\-skan
100................................................................................................................................................................................. MUKULAM M ATHS
]cn-io-e\ {]iv\-ß ƒ GXm\pw hrØ-kvXq-]n-I-bpsS Af-hp-Iƒ Xmsg sImSp-°p-∂ p. ]´nI ]qcn-∏n-°p-I. ]mZ -Bcw
=
=
=
=
=
Dbcw
Ncn-hp-bcw
]mZ Np‰-fhv
]mZ-]-c-∏fhv
h{I-apJ ]c-∏-fhv
D]-cn-Xe ]c-∏-fhv
hym]vXw
6
8
˛
˛
˛
˛
˛
˛
5
˛
13
˛
˛
˛
˛
˛
˛
24
30
˛
˛
˛
˛
˛
15
˛
25
˛
˛
˛
˛
˛
10
24
˛
˛
˛
˛
˛
˛
˛
36
45
˛
˛
˛
˛
˛
33
˛
55
˛
˛
˛
˛
˛
27
36
˛
˛
˛
˛
˛
˛
˛
60
61
˛
˛
˛
˛
˛
hrØ kvXq]n-Im-Ir-Xn-bn¬ Iq´n-bn-cn-°p∂ aW-ens‚ ]mZ-Np-‰-fhv 75.36 ao‰-dm-Wv. Ncn-hpbcw 13 ao‰-dm-Wv. Iq´n-bn-´n-cn-°p∂ aW-ens‚ hym]vX-sa¥ v? Hcp L\-ao-‰¿ aW-en\v 1500 cq] \nc-°n¬ B aW-ens‚ hne-sb-¥ v? hrØ-kvXq-]n-Im-Ir-Xn-bn-ep≈ Hcp ]m{X-Øns‚ ]mZ-Øns‚ ]c-∏-fhv 64 p N.-k.an, h{IXe ]c-∏-fhv 80 p N.-sk.an F¶n¬ AXns‚ D≈-fhv F{X en‰-dmWv? Hcp hrØ-kvXq-]n-I-bpsS hym]vXw 320 p L\.-sk-anbpw Dbcw 15 sk.aoDw BsW-¶n¬ h{I-Xe ]c-∏-fhv ImWp-I. Hcp hrØ-kvXq-]n-I-bpsS Bchpw Db-chpw XΩn-ep≈ Awi-_‘w 5: 12 BWv. kvXq]nIbv°v 2572 L\.-skao hym]vX-ap-s≠-¶n¬ h{I-Xe ]c-∏-fhv ImWp-I. c≠v hrØ-kvXq-]n-I-bpsS Bc-ß ƒ XΩn-ep≈ Awi-_‘w 3:4 Dw Ah-bpsS Db-c-ß ƒ XΩn-ep≈ Awi-_‘w 5:3 Dw Bbm¬ hym]vX-ß ƒ XΩn-ep≈ Awi-_-‘-sa¥ v?
tKmfw tKmf-Øn\v Hcp apJw am{Xta D≈q F∂ pw tKmf-Øns‚ tI{µw, Bcw F∂ nh F¥ msW∂ pw hy‡-am-°p-a-t√m. tKmfsØ apdn-®m¬ hrØw e`n-°p-∂ p. tKmfsØ c≠v Xpey `mK-ß -fm°n apdn-®m-ep-≠m-Ip∂ hrØ-Øns‚ tI{µw, Bcw F∂ nh AtX tKmf-Øn-t‚Xv Xs∂ -bm-bn-cn-°p-sa∂ hkvXp-X-Xbpw hy‡-am-°mw. tKmf-Øns‚ D]-cn-Xe ]c-∏-f-hv, hym]vXw F∂ nh ImWm-\p≈ hmIy-ß ƒ \¬Im-a-t√m. Bcw r Bb tKmf-Øns‚ D]-cn-Xe ]c-∏-fhv = 4p r2 hym]vXw
= 4 p r3 3
MUKULAM MATHS ................................................................................................................................................................................101
A¿≤tKmfw I´n-bmb tKmfsØ c≠v Xpey-`m-K-am-I-Ø-°-hn-[-Øn¬ apdn-®m¬ Hmtcm∂ pw A¿t≤m-f-ambn-cn-°p-a-t√m. Hcp tKmf-Øns‚ D]-cn-Xe ]c-∏-f-hns‚ ]Ip-Xn-bm-bn-cn-°pw. AXn¬ \n∂ pw D≠m-°p∂ A¿≤-tKm-f-Øns‚ h{I-Xe ]c-∏-fhv F∂ pw B tKmf-Øns‚ hym]vX-Øns‚ ]Ip-Xn-bm-bncn-°pw. A¿≤-tKm-f-Øns‚ hym]vX-sa∂ pw a\- n-em-°mw. A¿≤-tKm-f-Øns‚ ]mZ-]-c-∏-fhv hrØ-Øns‚ ]c-∏-fhv Xs∂ -b-s√. CXn¬ \n∂ pw NphsS sImSpØ coXn-bn¬ t{ImUo-I-cWw \S-Øm-hp-∂ -Xm-Wv.
pr = 2p r =3pr =2p r
Bcw r Bb A¿≤-tKm-f-Øns‚ ]mZ-]-c-∏-fhv = h{I-Xe ]c-∏-fhv D]-cn-Xe ]c-∏-fhv A¿≤-tKm-f-Øns‚ hym]vXw
2
2
2 3
3
]cn-ioe\ {]h¿Ø-\-ß ƒ = = = =
= =
=
18 sk an hymk-ap≈ tKmf-Øns‚ D]-cn-Xe ]c-∏-fhpw hym]vXhpw ImWp-I. 576 p N.-sk.an D]-cn-Xe ]c-∏-f-hp≈ tKmf-Øns‚ hym]vX-sa¥ v? 288p L\.-sk.an hym]vX-ap≈ tKmf-Øns‚ D]-cn-Xe ]c-∏-f-sh¥ v? 40 N.-sk.an D]-cn-Xe ]c-∏-f-hp≈ tKmf-Øns\ c≠v A¿≤-tKm-f-am-°n-bm¬ H∂ ns‚ h{IXe ]c-∏-f-hv, D]-cn-Xe ]c-∏-fhv F∂ nh ImWp-I. 24 sk an hymk-ap≈ 20 ^pSvt_m-fp-Iƒ \n¿Ωn-°m-\m-h-iy-amb XpI-ens‚ ]c-∏-fhv ImWpI? Hcp Ccpºv tKmf-Øn\v 6 sk an hymk-ap-≠v. Hcp L\ skan Ccp-ºns‚ `mcw 7.5 {Kmw BsW-¶n¬ B tKmf-Øns‚ `mc-sa¥ v? 3 sk an Bc-ap≈ teml-\n¿Ωn-X-amb I´n-bmb Hc¿≤-tKmfw Dcp°n AtX hymk-ap≈ I´n-bmb hrØ kvXq]nIbm°n am‰n-bm¬ AXns‚ Db-c-sa-¥ m-bn-cn°pw?
102................................................................................................................................................................................. MUKULAM M ATHS
20 Marks 1 hr.
UNIT TEST 1. 2. 3.
5
4.
ka-N-Xpc kvXq]n-I-bpsS ]mZ-h-°ns‚ \ofw 20 skanDw Ncn-hp-bcw 18 skanbpw Bbm¬ D]-cn-Xe ]c-∏-fhv ImWp-I. 2 196 p N.-sk.an D]-cn-Xe ]c-∏-f-hp≈ Hcp tKmf-Øns‚ hym]vX-sa¥ v? 2 Hcp hrØ kvXq]n-I-bpsS ]mZ-Np-‰-fhv 30 skan Dw Ncn-hp-bcw 25 sk anDw Bbm¬ AXns‚ hym]vX-sa¥ v? 3 Hcp Ifn-∏m-´-Øns‚ BIrXn A¿≤-tKm-f-Øn¬ AtX hymk-ap≈ hrØ-kvXq-]nI LSn∏n® coXn-bn-em-Wv. hrØ-kvXq-]n-I-bpsS Ncn-hp-bcw 5 skan, s]mXp-hymkw 6 skan F∂ nß s\ Bbm¬ Ifn-∏m-´-Øns‚ hym]vXw IW-°m-°p-I. 3
5.
6.
7.
Hcp I´n-bmb t]∏¿ shbn-‰n\v ka-N-Xpc kvXq]n-I-bpsS \mev ]m¿iz-ap-J-ß -fn¬ \n∂ pw Htc hymk-ap≈ A¿≤-tKmfw Xpc-s∂ -SpØ BIr-Xn-bm-Wp-≈-Xv. ka-N-Xpc kvXq]n-I-bpsS ]mZ-Np-‰-fhv 32 skanDw Ncn-hp-bcw 10 skanDw BWv. Hmtcm A¿≤-tKm-f-Øn-s‚bpw hymkw Hcp skan hoX-am-sW-¶n¬ t]∏¿ shbn-‰ns‚ hym]vX-sa{X? 1 I´n-bmb A¿≤-tKm-fm-Ir-Xn-bmb hkvXp-hns‚ hymkw 12 skan BWv. CXn¬ \n∂ v ]c-amh[n hym]vX-ap≈ Hcp hrØ-kvXq-]nI sNØn-bp-≠m-°p-∂ p. A¿≤-tKm-f-Øn-s‚bpw hrØkvXq-]n-I-bp-sSbpw hym]vX-ß ƒ ImWp-I. hym]vX-ß ƒ XΩnep≈ Awi-_-‘-sa¥ v? 3 ka-N-Xpc kvXq]n-Im-Ir-Xn-bn¬ \n¿Ωn-°p∂ Hcp IqSm-c-Øn\v 6 ao‰¿ Db-c-ap≠v. ]mZ-]-c-∏fhv 256 N.-sk.ao BsW-¶n¬ IqSmcw s]mXn-bm-\m-h-iy-amb Iym≥hm-kn\v NXp-c{i ao‰dn\v 300 cq] \nc-°n¬ BsI F¥ v sNehv hcpw. 3
MUKULAM MATHS ................................................................................................................................................................................103
aqey-\n¿Æ b {]h¿Ø-\-ß ƒ 1.
Hcp ka-N-Xpc kvXq]n-I-bpsS Hcp ]m¿iz-ap-J-Øns‚ Nn{Xw sImSp-Øn-cn-°p-∂ p. F¶n¬ kvXq]n-I-bpsS BsI h°p-I-fpsS \of-sa¥ v? Ncn-hp-b-c-sa{X? kvXq]n-I-bpsS Dbcw ImWp-I.
60
60 8 cm
2.
hrØ-kvXq-cn-Im-Ir-Xn-bn-ep≈ Hcp ta¬°q-c-bpsS ]mZ-Np-‰-fhv 113.04 ao‰-dm-Wv. AXn\v 15 ao‰¿ Ncn-hp-bcw Ds≠-¶n¬ AXns‚ h{I-Xe ]c-∏-fhv ImWp-I. 3. kaN-Xpc kvXq]n-Im-Ir-Xn-bn-ep≈ Hcp hkvXp-hns‚ ]mZ-]-c-∏-fhv 900 N.-sk.an BWv. Ncnhp-bcw 25 sk.an Bbm¬ kvXq]n-I-bpsS hym]vX-sa¥ v? 4. ]mZ- Bcw 10 skan Dw Ncn-hp-bcw 6 skanDw BIp∂ hn[-Øn¬ Hcp hrØ-kvXq-]nI \n¿Ωn-°mtam? Cu Af-hp-Iƒ ]c-kv]cw am‰n-bm¬ e`n-°p∂ kvXq]n-I-bpsS Db-c-sa¥ v? 5. A¿≤-tKm-f-Øns‚ ]c∂ apJw, D]-cn-Xew F∂ n-h-bpsS ]c-∏-fhpw AtX Bc-ap≈ tKmfØns‚ D]-cn-Xe ]c-∏-fhpw kam-¥ c t{iWnbn-em-sW∂ v sXfn-bn-°p-I. 6. Hcp hnZym¿∞ n 15 skan Bc-ap≈ Hcp hrØmwiw apdn-s®-Sp®v 235.5 N.-sk.an h{I-apJ ]c∏-f-hp≈ Hcp hrØ-kvXq-]nI D≠m-°p-∂ p. hrØ-kvXq-]n-I-bpsS Bc-sa{X? B hrØmw-iØns‚ tI{µ-tIm¨ F{X? 7. ka-N-Xpc kvXw`m-Ir-Xn-bn-ep≈ Hcp XSn-°-j-W-Øns‚ Dbcw 24 sk.-an.Dw ]mZ-h-°ns‚ \ofw 20 sk an Dw Bbm¬ AXn¬ \n∂ v sNØn-sb-Sp-°m-hp∂ G‰hpw henb ka-N-Xpc kvXq]n-I-bpsS hym]vX-sa¥ v? 8. A¿≤-tKm-f-ß -fmb c≠v ]m{X-ß -fpsS D]-cn-Xe ]c-∏-f-hp-Iƒ XΩn-ep≈ Awi-_‘w 4:9 Bbm¬ Ah-bpsS D≈-f-hp-Iƒ XΩn-ep≈ Awi_‘-sa¥ v? 9. 80 L\ sk.an hym]vX-ap≈ I´n-bmb Hcp teml-tKmfw Dcp°n AXns‚ ]IpXn Bc-ap≈ I´n-bmb A¿≤tKm-f-ß ƒ D≠m-°p-∂ p. F) F{X A¿≤-tKm-f-ß ƒ D≠m°mw? _n) Hcp A¿≤ tKmf-Øns‚ hym]vX-sa¥ v? 10. I´n-bmb knen-≠d- ns‚ hymkhpw Db-chpw Xpey-am-Wv. CXns‚ c≠-‰-Øn¬ \n∂ pw knen≠-dns‚ AtX hymk-ap-≈Xpw ]IpXn Db-c-ap-≈-Xp-amb hrØ-kvXq-]nI Xpc∂ v am‰p-∂ p. tijn-°p∂ `mK-Øns‚ hym]vXhpw knen-≠-dns‚ hymk-Øn\v Xpey-amb tKmf-Øns‚ hym]vXhpw Xpey-am-sW∂ v sXfn-bn-°p-I.
104................................................................................................................................................................................. MUKULAM M ATHS
A[ymbw 6
kqNI kwJy-Iƒ H≥]Xmw ¢mkn¬ tcJob kwJy F∂ A≤ym-bØ - n¬ F√m kwJy-I-sfbpw Hcp tcJbnse _nµp-°-fp-]-tbm-Kn®v kqNn-∏n-°m-sa∂ v I≠n´p-≠v. CXn-eqsS kwJy-IfpsS ]c-kv]c _‘sØ Pyma-Xo-b-ambn ImWp-∂ Xpw N¿® sNbvXn-´p-≠v. Cu A[ym-b-Øn¬ Hcp Xe-Ønep≈ GXv _nµp-hn-s\bpw kwJym tPmUn sIm≠v kqNn-∏n-°m-sa-∂ m-Wv N¿® sNøp-∂ -Xv. XpS¿∂ v Pyma-Xob {]iv\-ß sf _oP-K-Wn-X-ap-]-tbm-Kn®v ]cn-lmcw ImWp-∂ Xpw N¿® sNøp-∂ p. XpS¿∂ v hcp∂ Pyman-Xnbpw _oP-K-Wn-Xhpw F∂ A[ym-b-Øn¬ CXv IqSp-X-embn N¿®-sN-øp-∂ p.
{][m\ Bi-b-ß ƒ }
]c-kv]cw ew_-ß -fmb c≠v tcJ-Ifpw \ofw Af-°m≥ bp‡-amb Hcp GI-Ihpw D]-tbmKn®v Xe-Ønse _nµp-°sf kwJym tPmUn D]-tbm-Kn®v kqNn-∏n-°p-∂ -Xv. A£-ß -fnse _nµp-°-fpsS kqN-I-ß -fpsS {]tXy-IX A£-ß ƒ°v kam-¥ -c-amb tcJ-I-fnse _nµp-°-fpsS kqN-I-kw-Jy-I-fpsS {]tXy-I-X. A£-ß -fn-tetbm A£-ß ƒ°v kam-¥ -c-amb tcJ-I-fn-tetbm c≠v _nµp-°ƒ XΩn-ep≈ AI-ew. A£-ß ƒ X∂ n-´n-s√-¶nepw X∂ n-´p≈ kqN-\-I-fpsS ASn-ÿ m-\-Øn¬ a‰v _nµp-°-fpsS kqNI kwJy-Iƒ ImWp-∂ -Xn-\v. ]mT-]p-kvX-I-Ønse kwJym-Nn{Xw F∂ {]h¿Ø\w \¬Ip-tºmƒ BZyta Xs∂ ]pdsØ hcn-bn¬ io¿j- -ß ƒ hcmØ coXn-bn¬ _lp-`p-P-ß ƒ Ah-X-cn-∏n-°p-∂ p. AXn-\p-apºv Cusbmcp {]h¿Ø\w sImSp-°m-hp-∂ -Xm-Wv.
} } } }
11 10 9 8 7 6 5 4 3 2 1 1
2
3
4
5
6
7
8
9
10
11
12
MUKULAM MATHS ................................................................................................................................................................................105
ChnsS io¿j-ß -fpsS kwJym-tPm-Un-Iƒ ImWp-tºmƒ ]c-kv]cw ew_-amb GXv c≠v hcI-sfbpw ASn-ÿ m-\-am°n ]d-bm-sa∂ pw s]mXp-hmb Hcp coXn F∂ \ne-bn¬ CSXv Iogvhc-Iƒ ASn-ÿ m-\-am°n aqe-I-fpsS ÿ m\w \n¿Æ -bn°mw F∂ Xv Ip´n°v t_m[y-s∏-S-Ww. XpS¿∂ v Acn-Ip-I-fnepw io¿j- -ß ƒ hcp∂ Nn{X-ß ƒ sImSp-°pw. 12 11 10 9 8 7 6 5 4 3 2 1 1
2
3
4
5
6
7
8
9
10
11
12
ChnsS CSXv Iogv hc-Iƒ°v ]qPyw F∂ t]¿ \¬In. Iq´n ap´p∂ _nµp-hn\v (0,0) F∂ t]¿ \¬Ip-∂ p. NphsS sImSp-Øn-cn-°p∂ _nµp-°ƒ tbmPn-∏n®v Pyma-Xob cq]-ap-≠m-°p-∂ p. 1.
A (2,3)
B (5,3)
C (4,6)
2.
A (2,4)
R (4,0)
R (6,3)
S (3,8)
3.
R (0,1)
M (4,1)
N (6,5)
O (2,5)
]mT-]p-kvX-I-Øn¬ ho≠pw Nne kwJym Nn{X-ß ƒ F∂ {]h¿Ø-\-Øn\v aptº Cusbmcp {]h¿Ø\w sNøm-hp-∂ -Xm-Wv. 7 6 5 4
D (2,4)
3 2
E (4,3) B (2,2)
1 1
2
3
4
5
6
7
CSXv hiØv CXns‚ {]Xn-_nw_w hc®v jUv`pPw ]q¿Øn-bm-°p-∂ -Xn\v aqe-I-fpsS kqN-Ikw-Jy-Iƒ F¥ m-bn-cn-°Ww 106................................................................................................................................................................................. MUKULAM M ATHS
g
ChnsS {KnUp-Iƒ CS-Xp-h-i-tØ°v \o´p-∂ -Xn\v hc-bvt°≠ IpØ-s\-bp≈ hc-Isf \yq\kw-Jy-Iƒ D]-tbm-Kn®v kqNn-∏n-t°-≠n-h-cp-∂ -Xn-te°v Ip´n-Isf \bn-t°-≠-Xm-Wv. Nn{Xw Cß -s\-bm-bmtem!
(3,3)
(-1,3)
(5,1)
(-1,3)
g
(A-SpØSpØ hc-Iƒ XΩn-ep≈ AIew Xpey-am-bn-cn-°-Ww) AIew F¥ p-am-hmw.
NXpcw sh´n-sb-Sp-°mw. 10 \ofhw 8cm hoXn-bp-ap≈ Hcp IS-em-kns‚ a≤y-Øn¬ \n∂ v 6cm \ofhpw 4cm hoXn-bpap≈ Hcp NXpcw sh´n-sb-Sp-°p-I. \ofw \of-tØmSpw hoXn hoXn-tbmSpw tN¿Øv aS-°n-bm¬ a≤y-Øn-embn ]c-kv]cw ew_amb c≠v hc-Iƒ hcp∂ p F∂ pw a‰v hc-I-fn-√msX Ch-am{Xw D]-tbm-Kn®v apdn-t°≠ NXp-cØns‚ aqe-I-fpsS ÿ m\w \n¿Æ -bn-°m-sa∂ pw Ah-bpsS kqNI kwJy-Iƒ Fß ns\ Is≠Øm-sa∂ pw Ip´n-Iƒ°v t_m[y-s∏-S-s´. v NphsS X∂ n-cn-°p∂ _nµp-°-fn¬ x A£-Øn-sebpw y A£-Øn-sebpw _nµp-°ƒ Xcw Xncn-s®-gp-Xp-I. (½ ,0), (2,0) (0,2) , (3, 2), (-½, 0), (0, 1/3), (0, 3), (1.5, 0), (2.5, 0), (-2, -3)
_nµp- °ƒ (2,0), (6,0)
AIew 4= |2-6|
(1,0), (5,6) (-3, 0), (5,0) (0,3), (0,7) (0,-1), (0,-2) (0,-3) (0,4) (0,-1), (0,1) x A£-Øn\v kam-¥ -c-amb tcJ-bnse _nµp-°ƒ XΩn-ep≈ AIew x kqNI kwJy-I-fpsS hyXym-k-Øns‚ tIhe hne-bm-Wv. MUKULAM MATHS ................................................................................................................................................................................107
_nµp- °ƒ (3,2), (3,6)
AIew 4= | 2-6 |
(2,5), (2,-3) (-1,3), (-1,-4) (-2,6), (-2,1) (-5,1), (-2,1) y A£-Øn\v kam-¥ -c-amb tcJ-bnse _nµp-°ƒ XΩn-ep≈ AIew y kqNI kwJy-I-fpsS
hyXym-k-Øns‚ tIhe hne-bm-Wv.
108................................................................................................................................................................................. MUKULAM M ATHS
h¿Iv jo‰v ˛1 1
kqN-I-ß ƒ hc®v NphsS X∂ n-´p≈ kwJym tPmUn-Iƒ kqNn-∏n-°p∂ _nµp°ƒ AS-bmf-s∏-Sp-Øp-I. A (2,3), B (-3,4), C (-4, -2), D (5,-3)
2. 3.
4. 5.
6. 8. =
=
=
=
=
NphsS X∂ n-´p-≈-h-bn¬ x A£-Ønse _nµp-°sf kqNn-∏n-°p∂ kwJym-tPm-Un-Iƒ am{Xw Fgp-Xp-I. (1, 3) , (0,3) , (3, 0) (-2, -2), (-3, 0), (8,0) , (0, 0) hi-ß ƒ A£-ß ƒ°v kam-¥ -c-amb Hcp ka-N-Xpcw hc-bv°p-I. CXns‚ c≠v aqe-Iƒ kqNn-∏n-°p∂ kwJym-tPm-Un-Iƒ (1,2), (5,2). NXp-c-Øns‚ a‰v aqe-I-fpsS kwJym-tPmUn GsXms°? B[m-c-_nµp tI{µambn 5 skan Bc-ap≈ Hcp hrØw hc-®m¬ hrØhpw kqN-Im-£ß fpw Iq´n-ap-´p∂ _nµp-°-fpsS kqN-I-kw-Jy-Iƒ GsXms°? x A£-Ønse _nµp tI{µ-ambn hc-bv°p∂ hrØ-Øns‚ hymkw 8 bqWn‰v BWv. Cu hrØ-Øns‚ Hcp hymk-Øns‚ Hc-{K-_nµp (6,0) F¶n¬ at‰ A{K- _nµp-hns‚ kqNI kwJy-Iƒ F¥ mWv? ka-N-Xpcw ABCD bn¬ A (-4,0), B(6,0) C, D F∂ o _nµp-°ƒ x A£-Øn\v Xmsg-bm-sW-¶n¬ Ch-bpsS kqN-I-kw-Jy-Iƒ F¥ mWv? ka-N-Xp-c-Øns‚ Np‰-fhv ImWp-I. kam-¥ -cnIw ABCD bn¬ A(0,0), B(6,0), C (8,4) F¶n¬ DbpsS kqNI kwJy-Iƒ F¥ v? NphsS X∂ n-´p-≈-h-bn¬ aq∂ v _nµp-°ƒ x A£-Øn\v kam-¥ -c-ambv Hcp tcJ-bn-em-Wv. Ah GXv? (4,0), -(˛2,3), (3,-˛2), (4,3), (0,3), (0,4), (0,5) NphsS X∂ n-´p-≈-h-bn¬ x A£-Øn\v kam-¥ -cambn x A£-Øn¬ \n∂ pw 2 bqWn‰v AIse-bp≈ tcJ-bnse _nµp-°ƒ Gh? (2,4), -(- - - - ˛2,2), (2,3), (5,2), (0,2), (2,0) x A£-Øn\v kam-¥ -c-amb tcJ-bnse Hcp _nµp-hm-Wv (- -˛2,4) NphsS sImSp-Øn-cn-°p-∂ -h-bn¬, Cu tcJ-bnse _nµp-°sf kqNn-∏n-°p∂ kwJym-tPmUn Gh? (2,3), (˛2,1), (4,2), (˛2,-˛4), (˛2,0) x A£-Øn\v kam-¥ -c-amb tcJ-bnse Hcp _nµp-hmWv (3, 5). Cu _nµp-hn¬ \n∂ v 5 bqWn‰v AIse CtX tcJ-bn¬ F{X _nµp-°¬ D≠v? Ah-bpsS kqNI kwJy-Iƒ Fgp-Xp-I. (3,4), (3,5) F∂ o _nµp-°ƒ XΩn-ep≈ AIew F{X?
MUKULAM MATHS ................................................................................................................................................................................109
h¿Iv jo‰v ˛2 v v v v
v v
v v
v v
hi-ß ƒ A£-ß ƒ°v kam-¥ -c-ambn Hcp ka-N-Xpcw hc-°p-I. CXns‚ c≠v aqe-Isf kqNn-∏n-°p∂ _nµp-°-fmWv (1,2), (5,2). a‰v aqe-Isf kqNn-∏n-°p∂ _nµp-°ƒ Is≠-Øp-I. Hcp hiw 4 bqWn‰pw Hcp io¿jw (3,- - ˛2) Dw hi-ß ƒ A£-ß ƒ°v kam-¥ -c-ambpw hc°m-hp∂ Hcp ka-N-Xp-c-Øns‚ a‰v aqe-I-fpsS kqNI kwJy-Iƒ Fgp-Xp-I. A£-ß ƒ°v kam-¥ -c-amb hi-ß -tfmSv IqSnb Hcp NXp-c-Øns‚ aq∂ v io¿j-I-ß ƒ (˛3, 4), (6, 4) , (6, 10) Ch-bm-Wv. \mem-asØ io¿jw GXv? hi-ß ƒ A£-ß ƒ°v kam-¥ -c-amb Hcp ka-N-Xp-c-Øns‚ \ofw 6 bqWn‰pw hoXn 3 bqWn‰pamWv. CØcw NXp-c-Øns‚ Hcp aqe (˛3, 2) BWv. C{]-Imcw F{X NXp-c-ß ƒ hcbv°mw? Ah-bn¬ Hcp NXp-c-Øns‚ a‰v aqe-Iƒ kqNn-∏n-°p∂ _nµp-°ƒ Fgp-Xp-I, kam-¥ -cnIw PQRS ¬ P(2,4), 2 (10,4), R(13,9) F¶n¬ Ss‚ kqN-I-kw-Jy-Iƒ Gh? ew_Iw ABCD bnse AB,CD F∂ o hi-ß ƒ XA£-Øn\v k-am-¥ -c-am-Wv. AB=10 bqWn‰v CD = 6 bqWn‰v BWv. A(-2,3), C (5,9) BIp-∂ p. F¶n¬ B,D Ch-bpsS kqNI kwJy-Iƒ Is≠Øp-I. ka-`p- -Pkm-am-¥ -cnIw ABCD bn¬ AB F∂ hiw XA£-Øn\v kam-¥ -c-am-Wv. A(4,3), AB=6 bqWn‰pw F¶n¬ kmam-¥ -cn-I-Øns‚ a‰v aqe-I-fpsS kqN-I-kw-Jy-Iƒ ImWpI? ew_Iw PQRS ¬ PQ, RS Ch X B£-Øn\v kam-¥ -c-am-Wv. PQ = 12 bqWn‰v RS = 6bqWn‰v BIp-∂ p. P(-4,2), PQ, RS Ch XΩn-ep≈ AIew 4 bqWn‰pw BIp-∂ p. R s‚ X kqN-I-kwJy ˛1 BWv. F¶n¬ ew_Iw PQRS s‚ as‰√m aqe-I-fp-sSbpw kqNI kwJy-Iƒ ImWpI? hi-ß ƒ A£-ß ƒ°v kam-¥ -c-ß -fmb Hcp ka-N-Xp-c-Øns‚ FXn¿ aqe-Iƒ (˛2, 4) , (6,12) BWv. F¶n¬ ka-N-Xp-c-Øns‚ ]c-∏-fhv F{X? hi-ß ƒ A£-ß ƒ°v kam-¥ -c-amb Hcp NXp-c-Øns‚ Hcp hnI¿Æ -Øns‚ A{K-_n-µp°ƒ (3, 1), (7, 4) Ch-bm-Wv. NXp-c-Øns‚ a‰v aqe-I-fpsS kqNI kwJy-Iƒ Fgp-Xp-I.
110................................................................................................................................................................................. MUKULAM M ATHS
h¿Iv jo‰v ˛3 1. 2. 3. 4.
kqN-Im-£-ß ƒ hc®v GIIw 1½ skan Bbn FSpØv A (2,1), B (3,-2), C (-2,-3) F∂ o _nµp°ƒ AS-bm-f-s∏-SpØn rABC hc-bv°p-I. GIIw ½ skan Bbn FSp-Øv kqN-Im-£-ß ƒ hc®p Hcp NXp-c-Øns‚ 3 aqe-Iƒ (2,3), (6,3), (6,9) BIp-∂ p. NXp-c-Øns‚ \mem-asØ aqe GXv? (0,4) tI{µ-ambn 5 skan Bc-ap≈ Hcp hrØw hc-®m¬ hrØw X A£-Øns‚ JWvUn°p∂ _nµp°ƒ Gh? A bpsS kqNI kwJy-Iƒ ]d-bp-I.
3u
ni t
A
5.
60
(˛2,5), (6,5) F∂ o _nµp-°-fn-eqsS IS∂ v t]mhp∂ tcJ x A£-Øn¬ \n∂ v F{X AI-sebmWv? Cu _nµp-°ƒ XΩn-ep≈ AI-e-a¥ v?
MUKULAM MATHS ................................................................................................................................................................................111
A[ymbw 7
km-[y-X-I-fpsS KWnXw BapJw \msf kqcy≥ DZn-°ptam? F¶n¬ GXv Zn°n¬? Poh-\p≈hbv°v ac-W-apt≠m? XpS-ß nb tNmZy-ß ƒ°v kwi-b-an-√msX DØcw ]d-bmw. C∂ v ag s]øptam? C¥ y Hfnw-_n-Ivkn¬ kz¿Æ w t\Sptam? AΩp-hns‚ ho´nse BSv {]k-hn-®m¬ Ip´n s]Æ m-bn-cn-°ptam? Ah≥ ]co-£-bn¬ Pbn-°ptam? XpS-ß nb \nXy-Po-hn-X-Ønse ]e {]kvXm-h-\-I-fnepw Hcp kw`-h-Øns‚ km[y-Xsb-°p-dn-®p≈ kqN-\tb ImWm-\p-≈q. AXm-bXv Hcp A\n-›n-X-Xz-Øns‚ sNdn-sbm-cwiw D≠t√m. GXn-t\bpw Af-°pI F∂ m-Wt√m KWn-X-Øns‚ [¿Ωw. IrXy-ambn ^ew {]h-Nn-°m≥ Ign-bn-s√-¶nepw Cu A\n›n-X-XzsØ Af-°m≥ klm-bn-°p∂ KWn-X-Øns‚ Hcp imJ-bmWv km[yXm kn¥ m-¥ w. NqXp-I-fn-bp-ambn _‘-s∏´v Book on Game of Change F∂ {KŸw Gerolamo Cardano F∂ C‰m-en-b≥ - KWn-X-im-kv{X-⁄≥ cNn-®n-´p-≠v. Chevalier de-Mere F∂ {^©v {]`p ]In-S-I-fn-bp-ambn _‘-s∏´ Hcp {]iv\w ]mkvI¬ t\mSv tNmZn-°p-I-bp-≠m-bn. ]mkv°¬ s^¿abp-ambn \S-Ønb IØn-S-]m-Sp-I-fn¬ \n∂ mWv km[yXm kn¥ m-¥ -Øns‚ B`n¿`m-h-Øn-te°v \bn-®Xv F∂ v Icp-X-s∏-Sp-∂ p.
{][m\ Bi-b-ß ƒ = = =
km[yX F∂ Bibw km[y-X˛ kwJym-]-c-ambn ImWp-∂ p. km[yX IW-°p-Iq-´p-∂ -Xn\v FÆ ¬ kq{X-ß ƒ
km[y-X-Ifpw kwJy-Ifpw
= = = =
{]h¿Ø-\w. 1 apX¬ 24 hsc-bp≈ FÆ ¬ kwJy-Iƒ Fgp-Xnb Im¿Up-Iƒ Hcp s]´n-bn¬ Ds≠∂ v Icp-Xp-I. (Hcp Im¬Un¬ Hcp kwJy-am-{Xw). CXn¬ \n∂ v t\m°msX Hcp Im¿Uv FSpØm¬ In´p∂ Im¿Uv Xmsg ]d-bp-∂ -Xn-te-Xn-\mWv IqSp-X¬ km[yX F∂ v Ip´n-I-tfmSv tNmZn-°p-I. 2¬ Ah-km-\n-°p∂ kwJy In´p-∂ -Xn\v 7¬ Ah-km-\n-°p∂ kwJy 3 s‚ KpWn-X-amb kwJy 5s‚ KpWn-X-amb kwJy 3s‚ KpWn-X-ß ƒ a‰p-≈-h-tb-°mƒ IqSp-X-ep-≈-Xn-\m¬ 3s‚ KpWnXw In´m-\mWv IqSp-X¬ km[yX F∂ v Ip´n-Iƒ Is≠-Øp-a-t√m. F¶n¬ Ipd™ km[yX GXn-\m-Wv? H‰ kwJy, Cc´ kwJy Ch-bn¬ GXv In´m-\mWv IqSp-X¬ km[yX F∂ n-ß -s\-bp≈ tNmZy-ß ƒ tNmZn-°p-I. Ipd™ km[yX 7¬ Ah-km-\n-°p∂ kwJy-bv°m-Wv H‰-kw-Jy. Cc´ kwJy Ch c≠n\pw Htc km[y-X F∂ pw Ip´n-Iƒ Is≠-Øm≥ klm-bn-°p-I. km[yX kwJy-I-fn-eqsS Cß s\ Is≠-Ønb km[y-X-Isf kwJy-I-fp-ambn _‘-s∏-SpØn t\m°mw.
112................................................................................................................................................................................. MUKULAM M ATHS
s]´n-bn¬ BsI kwJy-Iƒ 24 3s‚ KpWn-X-ß ¬ 3,6,9,12,15,18,21,24 F∂ n-ß s\ 8 kwJy-Iƒ AXm-bXv 24 kwJy-I-fn¬ 8 FÆ amWv 3s‚ KpWn-X-ß ƒ. AXn-\m¬ 3s‚ KpWn-X-ß ƒ hcm-\p≈ km[yX= 8/24 =1/3 F∂ v ]d-bm-hp-∂ -Xm-Wv. Xmsg sImSpØ ]´nI ]qcn-∏n-°p-I. BsI kwJy-Iƒ 24 FÆ w kµ¿`-ß ƒ 1.
3s‚ KpWn-X-ß ƒ
2.
2¬ Ah-km-\n-°p∂ kwJy
3.
7¬ Ah-km-\n-°p∂ kwJy
4.
5s‚ KpWnXw
5.
H‰-kwJy
6.
Cc´ kwJy
In´p∂ kwJy-Iƒ 3,6,9,12,15,18,21,24 ............
FÆ w 8
km[yX 8= 1 24 3
]´n-I-bn¬ \n∂ v Ipd™ km[yX, IqSp-X¬ km[y-X, Xpey-amb km[yX F∂ nh ImWp-I. IqSp-X¬ {]h¿Ø-\-ß ƒ 1. Hcp \mWbw apI-fn-te-s°-dn-bp-∂ p. At∏mƒ Xe (H) hcm-\p≈ km[y-Xbpw hm¬ (T) hcm\p≈ km[y-Xbpw ImWp-I. 2. Hcp ]InS Fdn-™m¬ Cc´ kwJy In´m-\p≈ km[yX F{X? (kqN\: 1, 2, 3, 4, 5, 6 Ch-bn¬ GsX-¶nepw Hcp kwJy apI-fn¬ hcp-a-t√m. Cc´ kwJy 2,4,6 Ch-bm-Wv. Ct∏mƒ km[y-X. =3 = 1 6 2 3.
Nn{X-Ønse NXp-c-Øn¬ IÆ -S®v s]≥kn¬ sIm≠p Hcp IpØn-´m¬ sjbvUv sNbvX {XntIm-W-Øn¬ hcphm-\p≈ km[yX F{X?
4.
Hcp s]´n-bn¬ 2s‚ BZysØ 100 KpWn-X-ß -fpw as‰mcp s]´n-bn¬ 3s‚ BZysØ 100 KpWn-X-ß fpw Fgp-Xnb Im¿Up-Iƒ D≠v. 6s‚ KpWnXw Fgp-Xnb Im¿Uv In´m≥ GXv s]´n-bn¬ \n∂ v FSp-°p-∂ -Xm-Wv \√-Xv.
MUKULAM MATHS ................................................................................................................................................................................113
cs≠-Æ -sa-Sp-Øm¬ c≠v ]In-S-Iƒ H∂ n-s®-dn-bp-∂ p. At∏mƒ c≠n-sebpw kwJy-IfpsS XpI 8 hcm-\p≈ km[yX ImWWsa-∂ n-cn-°-s´. CXv ]´n-I-s∏-Sp-Ømw. ]InS 1 ]InS 2 XpI XpI 8 hcp-∂ Xv 1 1 2 1 2 3 2,6 1 3 4 3,5 1 4 5 4,4 km [y - a m- I p ∂ ^e- ß - f psS 1 5 6 5,3 BsI FÆ w 36. CXn¬ XpI 8 1 6 7 6,2 hcp-∂ Xv 5 FÆ w 2 1 3 AXn-\m¬ XpI 8 hcm-\p≈ 2 2 4 km[yX= 5/36 BWy (C-hnsS BsI FÆ w 36 F∂ Xv 2 3 .... 6x6 BWv F∂ v t_m[y-s∏-Sp-I.) 2 ... .... ..... ..... .... ..... .... ..... ..... ... .... 6 6 Hcp s]´n-bn¬ 5 shfpØ apØp-Ifpw 7 IdpØ apØp-Ifpw D≠v. CXn¬ \n∂ v Hcp apØv FSp- Øm¬ AXv IdpØ apØv Bhm-\p≈ km[yX F{X? XpS¿∂ v Hcp shfpØ apØpw Hcp IdpØ apØpw FSp-Øp-am‰n sh°p-∂ p. C\n s]´n-bn¬ \n∂ v Hcp apØv FSp-Øm¬ AXv IdpØ apØv Bhm-\p≈ km[y-Xbpw ImWpI? ho≠pw Hcp shfpØ apØpw Hcp IdpØ apØpw FSp-Øp-am-‰nb km[y-Xbpw ImWp-I. Cu {]h¿Ø\w XpS¿∂ mtem? Xmsg sImSpØ ]´nI ]qcn-∏n-°mw. L´w shfpØ apØp-Iƒ IdpØ apØp-Iƒ 1 2 3 4 5
5 4 3 2 1
IdpØ apØv km[yX G‰hpw In´m-\p≈ km[yX IqSp-X¬/Ipdhv
7 6 5 4 3
c≠v \mW-b-ß ƒ H∂ n-s®-dn-™m¬ c≠v Xe hcm-\p≈ km[yX F{X? Hcp Xe am{Xw hcm-\p≈ km[yX F{X? (kqN\: Xe H F∂ pw hm¬ T F∂ pw kqNn-∏n-®m¬ BsI km[y-am-Ip∂ ^e-ß ƒ HH, HT, TH, TT F∂ nh BW-t√m. CXv 4 FÆ w. c≠v Xe hcp-∂ Xv HH hcp-tºmƒ am{Xw. AXn-\m¬ Xe hcm-\p≈ km[yX = 1/4 Hcp Xe-am{Xw hcp-∂ Xv TH, HT F∂ nh hcp-tºm-gm-Wv. km[yX 2/4=1/2 114................................................................................................................................................................................. MUKULAM M ATHS
aq-ey-\n¿Æ b kqN-I-ß ƒ 1. c≠v ]InS Fdn-bp-tºmƒ c≠nepw In´p∂ kwJy-I-fpsS XpI Xmsg ]´n-I-bn¬ sImSp-°p∂ p. AXv In´m-\p≈ km[yX Is≠-Øp-I. XpI
2
3
4
5
6
7
8
9
10
11
12
km[yX
2. 3.
4.
km[y-X-I-fpsS BsI XpIbpw ImWp-I. A°-ß ƒ Bh¿Øn-°mØ 1 apX¬ 100 hsc-bp≈ FÆ ¬ kwJy-I-fn¬ \n∂ v Hcp kwJy FSp-Øm¬ AXv 5s‚ KpWn-X-am-Im-\p≈ km[yX F{X? aq∂ v \mWbw H∂ n-s®-dn-bp-∂ p. km[y-amb ^e-ß ƒ Fgp-Xp-I. CXn¬ 3 Xe In´m-\p≈ km[yX F¥ v? 2 Xe am{Xw In´m-\p≈ km[yX F¥ v? Npcp-ß n-bXv 2 Xe In´m-\p≈ km[yX F¥ v Hcp Xe am{Xw In´m-\p≈ km[yX F¥ v? ]®bpw a™bpw \nd-Øn-ep≈ BsI 18 t_mfp-I-fp-≠v. CXn¬ \n∂ v Hcp t_msf-Sp-°ptºmƒ ]®-\n-d-Øn-ep≈ t_mƒ In´m-\p≈ km[yX 1/3 BWv. F¶n¬ Hmtcm \nd-Øn-ep≈ t_mfp-I-fpsS FÆ w F{X?
MUKULAM MATHS ................................................................................................................................................................................115
10 Marks 20 mts
UNIT TEST 1.
2. 3. 4. 5.
Hcp s]´n-bn¬ 3 sh≈ apØp-Ifpw 7 IdpØ apØp-Ifpw D≠v. as‰mcp s]´n-bn¬ 4 sh≈ apØp-Ifpw 6 IdpØ apØp-I-fp-ap-≠v. IÆ -S®v Hcp apØv FSp-°p-tºmƒ Int´-≠Xv sh≈ apØm-sW-¶n¬ GXv s]´n-bn¬ \n∂ v FSp-°p-∂ -XmWv \√Xv? 2 aq∂ -°-ap-f-f Hcp kwJy ]d-bm≥ Bh-iy-s∏-Sp-∂ p. CXv Htc A°ap≈ kwJy-bm-Im-\p≈ km[yX F{X? 2 100¬ Xmsg-bp≈ FÆ ¬ kwJy-I-fn¬ Hcp kwJy Fgp-Xn-bm¬ AXv h¿§-kw-Jy-bm-Im\p≈ km[yX F{X? 2 Hcp k©n-bn¬ 10 ]gpØ amß bpw 6 ]® amß bpw D≠v. CXn¬ \n∂ v Hcp amß FSpØm¬ ]®-amß In´m-\p≈ km[yX F{X? ]gpØ amß In´m-\p≈ km[yX F{X? 2 c≠v ]In-S-Iƒ Fdn-™m¬ c≠n-t‚bpw KpW-\-^ew Hcp L\-kw-Jy-In-´m-\p≈ km[yX Is≠-Øp-I. 2
116................................................................................................................................................................................. MUKULAM M ATHS
A[ymbw 8
sXmSp-h-c-Iƒ BapJw Pyman-Xn-bpsS ]nXm-sh∂ v hnti-jn-∏n°-s∏-Sp∂ bp¢n-Uns‚ A`n-{]m-b-Øn¬"" Hcp hc Hcp hrØsØ k‘n-°p-I-bpw, XpS¿∂ v \o´n-bm¬ k‘n-°m-Xn-cn-°p-Ibpw sNbvXm¬, B hc hrØsØ sXmSp-∂ p.'' Cu \n¿∆-N\w sXmSp-hc F∂ Bi-bsØ {]I-Sn-∏n-°p-sa-¶nepw F√ hfhp-Iƒ°pw _m[-I-am-I-Ø° \ne-bn¬ D]-tbm-K-s∏-Sp-Øm≥ Ign-bm-Ø-Xm-Wv. sXmSp-hc F∂ Bibw IrXy-ambn \n¿∆-Nn-°p-∂ -Xn\v hni-I-e\ Pyman-Xn-bp-sSbpw Ah-I-e-\-Øn-s‚bpw `mj D]-tbm-Kn-°-Ww. CØcw \n¿∆-N-\-Øns‚ Zriym-hn-jvIm-c-am-Wv Cu ]mT-`m-Kw. Pyman-Xn-bnse Ne-\m-fl-I-X-bpsS {][m\yw G‰hpw IqSp-X¬ N¿®-sN-ø-s∏-Sp∂ ]mT-`mKw IqSn-bm-Wn-Xv. hrØsØ Ipdn®v \mw t\csØ a\- n-em-°nb \nc-h[n Bi-b-ß -fpsS D]-tbmKw IqSn A\n-hm-cy-ambXn\m¬ hrØ-ß ƒ F∂ A[ym-b-Øns‚ XpS¿®-bmbn Cu ]mT-`m-KsØ ImWm-hp-∂ -Xm-Wv.
{][m\ Bi-b-ß ƒ v v v v v v v v
hrØ-Øns‚ sXmSp-hc hrØ-Ønse GsX-¶nepw _nµp-hn-eqsS Bc-Øn\v ew_-ambn hc-bv°p∂ hc B _nµphnse sXmSp-h-c-bm-Wv. hrØ-Øns‚ GXv sXmSphcbpw sXmSp∂ _nµp-hn-eq-sS-bp≈ Bc-Øn\v ew_-am-Wv. hrØ-Øn\v ]pd-Øp≈ GXv _nµp-hn¬ \n∂ pw c≠v sXmSp-h-c-Iƒ hc-bv°mw. _nµp-hn¬ \n∂ p-≈ Cu sXmSp-h-c-I-fpsS \ofw Xpey-am-Wv. hrØ-Ønse c≠v _nµp-°ƒ \n¿Æ -bn-°p∂ sNdnb Nm]-Øns‚ tI{µ-tImWpw Cu _nµp°-fnse sXmSp-h-c-Iƒ°n-Sn-bn-ep≈ tImWpw A\p-]q-c-I-ß -fm-Wv. hrØ-Ønse Hcp RmWpw AXns‚ Hc‰-Øpff sXmSp-h-cbpw XΩn-ep≈ Hmtcm tImWpw, Cu RmWns‚ adp-h-i-Øp≈ hrØ-J-WvU-Ønse tImWn\v Xpey-am-Wv. Hcp {XntIm-W-Ønse aq∂ v tImWp-I-fp-sSbpw ka-`m-Pn-Iƒ Htc _nµp-hn¬ JWvUn-°p-∂ p. {XntIm-W-Øns‚ A¥ ¿hr-Øw.
{]h¿Ø\w 1 IS-em- n¬ \n∂ v ka-N-Xp-chpw ka-`p-P-{Xn-tIm-Whpw Ip´n-Iƒ apdn-s®-Sp-°p-∂ p. ka-N-Xp-cØns‚ ASp-Ø-SpØ hi-ß -fpsS a[y-_n-µp-°ƒ tbmPn-∏n®v aS-°p-∂ p. At∏mƒ In´nb sNdnb ka-N-Xp-c-Øns‚ ]cn-hrØw hc-bv°p-∂ p. ]pdsØ ka-N-Xp-c-Øns‚ Hmtcm hihpw hrØsØ Hcp _nµp-hn¬ am{Xw sXmSp-∂ -Xmbn ImWm-a-t√m. CXp-t]mse ka-`p-P-{Xn-tIm-W-Øns‚ hi-ß -fpsS a[y _nµp-°ƒ tbmPn-∏n®v aS-°p-tºmƒ In´p∂ {XntIm-W-Øns‚ ]cn-hrØw hc-bv°p-∂ p. Chn-sSbpw ]pdsØ ka-`p-P-{Xn-tIm-WØns‚ Hmtcm hihpw hrØsØ Hcp _nµp-hn¬ am{Xw sXmSp-∂ -Xmbn ImWmw. XpS¿∂ v Ip´n-I¬ t\m´p-]p-kvX-I-Øn¬ ka-N-Xpchpw ka-`p-P-{Xn-tIm-Whpw hc-bv°p-∂ p. apI-fn¬ hnh-cn® {]h¿Ø\w AtX coXn-bn¬ hc®pw ImWp-°m-hp-∂ -Xm-Wv. MUKULAM MATHS ................................................................................................................................................................................117
(sF-knSn Sqƒ D]-tbm-Kn®v Ne-\m-fl-I-X-bn-eqsS Cu Bibw hy‡-am-°m-hp-∂ -Xm-Wv
{]h¿Ø\w 2
C C
C
A
5 skan
B
40 0
30 0
20 0 A
5 skan
B
A
5 skan
B
Nn{X-Øn¬ ImWp∂ Af-hn¬ aq∂ v tImWp-Iƒ t\m´p-_p-°n¬ hc-bv°p-∂ p. Hmtcm∂ nepw A tI{µ-ambn 2.5 sk‚n-ao-‰¿ Bc-ap≈ hrØ-ß fpw hc-bv°p-∂ p. aq∂ v kµ¿`-ß -fn-epw-AB F∂ hc hrØsØ F{X _nµp-°-fn¬ JWvUn-°p∂ p F∂ v N¿® sNøp-∂ p. (Cu {]h¿Ø-\-ß -fn-eqsS hrØsØ Hcp _nµp-hn¬ am{Xw sXmSp∂ hscb (A-Xm-bXv sXmSp-h-sc-sb) Ipdn®v Ip´n-Iƒ°v [mcW D≠m-Ip-a-t√m)
{]h¿Ø\w 3 O
3 s ka n
Nn{X-Øn¬ Pbnse Hcp tImWns‚ Afhv (x s‚ hne) 450, 500, 600, 700 , 800, 850, 870, 900, 1000, BIØ-°-hn[w Hmtcm IS-em- n¬ hc®v Hmtcm∂ pw Ip´n-Iƒ°v \¬Ip-∂ p. O tI{µ-ambn 3 skan Bc-ap≈ hrØw hc-bv°m≥ Ip´n-I-tfmSv Bhiy-s∏-Sp-∂ p. AB F∂ hc hrØsØ F{X _nµp°-fn¬ JWvUn-°p∂ p F∂ v N¿® sNøp-∂ p. x s‚ hne 900 A√m-Ø-t∏mƒ c≠v _nµp°-fn¬ JWvUn-°p-∂ p-sh-∂ pw, 900 tbmSv ASp°p-t¥ mdpw JWvUn-°p∂ _nµp-°ƒ XΩn-ep≈ AIew Ipd-™p-h-cp-∂ p-sh∂ pw a\- n-em-°m-at√m. x s‚ hne 900 BIp-tºmƒ am{X-amWv F∂
X0 A
P
B
118................................................................................................................................................................................. MUKULAM M ATHS
hc hrØsØ Hcp _nµp-hn¬ sXmSp-∂ Xv F∂ v ImWmw. At∏mƒ AB F∂ hc hrØ-Øns‚ sXmSp-hc BIp-∂ p. XpS¿∂ v ]mT-]p-kvX-I-Ønse t]Pv 146¬ \¬Inb hni-I-e-\-Øn-eqsS NphsS sImSpØ kmam-\y-X-Xz-Øn¬ FØn-t®-cmw. hrØ-Ønse GsX-¶nepw _nµp-hn-eqsS Bc-Øn\v ew_-ambn hc-bv°p∂ hc, B _nµphnse sXmSp-h-c-bm-Wv. 3.5 sk‚n-ao-‰¿ Bc-ap≈ Hcp hrØw hc®v AXn¬ sXmSp-hc hb-°p-I.
P
F∂ _nµp AS-bm-f-s∏-Sp-Øp-I. Pbnse
{]h¿Ø\w 4
O
A
P
B
Nm¿´v t]∏-dn¬ hc® Nn{Xw t\m°p-I. Nn{X-Øn¬ O tI{µ-amb Hcp hrØnse P F∂ _nµp-hn¬ IqSnbp≈ sXmSp-h-bmWv AB OPbv°v ew_-a-√msX Pbn¬ IqSn hc-bv°p∂ GXv hcbpw hrØsØ as‰mcp _nµp-hn¬ IqSn JWvUn°p-∂ p. F∂ v ImWm-a-t√m. (]m-T-]p-kvXIw t]Pv 149) P bn¬ IqSn-bp≈ sXmSp-h-c-bm-I-s´, as‰mcp _nµphn¬ hrØsØ JWvUn-°p-I-bp-an-√. AXp-sIm≠v ABF∂ hc OPbv°v ew_-am-Wv.
hrØ-Ønse GXp sXmSp-h-cbpw, sXmSp∂ _nµp-hneq-sS-bp≈ Bc-Øn\v ew_am-bn-cn-°pw. Q
1
O
2
Nn{X-Øn¬ O hrØ-tI-{µw. B bnse sXmSp-h-c-bmWv AB OB=8 skan. OA= 17 skan. BIp-∂ p. (1) ABO F{X? (2) AB F{X?
P
Nn{X-Øn¬ QF∂ _nµp-hnse sXmSp-h-c-bmWv PQ . OhrØ tI{µhpw, PQR=55 0 bp w Bbm¬ OPQ F{X?
A
O
B
MUKULAM MATHS ................................................................................................................................................................................119
{]h¿Ø\w 5 (i) (ii)
AB F∂
hc hc-bv°p-I. AB I¿Wam-I-Ø-°-hn[w Hcp a´-{Xn-tImWw \n¿Ωn-°p-I. O tI{µ-ambn Hcp hrØw hc-bv°p-I. hrØ-Øn\v ]pdØv P F∂ _nµp AS-bm-f-s∏-Sp-ØpI. OP I¿Whpw hrØ-Ønse Hcp _nµp a´ aqebpw BI-Øn-°-hn[w Hcp a´-{Xn-tImWw \n¿Ωn-°p-I.
c≠m-a-tØ-Xn¬ a´-aqe A F∂ _nµp-hm-sW-¶n¬, hrØsØ ASn-ÿ m-\-am°n bpsS {]tXy-IX F¥ v? (PA hrØns‚ sXmSp-hc Bbn-cn-°p-at√m?)
PA
F∂ hc-
A
O
P
I¿W-am-bpw, hrØnse as‰mcp _nµp a´ aqe-bmbpw a´-{Xn-tImWw hc-bv°mtam? OP hymkamb A¿[-hrØw Xmsg hc-bv°p-I-bm-sW-¶n¬ a´-{Xn-tImWw hc-bv°m-sa∂ pw a´-aqe BF∂ _nµp-hm-sW-¶n¬ PB hrØ-Øns‚ sXmSp-h-c-bm-bn-cn-°p-sa∂ pw a\- n-em-°mw. OP
ss]Y-tKm-dkv kn¥ m-¥ -ap]-tbm-Kn®v
PA, PB
Chbv°v Htc \of-am-sW∂ pw ImWmw.
hrØ-Øn\v ]pd-Øp≈ Hcp _nµp-hn¬ \n∂ v c≠v sXmSp-h-c-Iƒ hcbv°mw. _nµp-hn¬ \n∂ p≈ Cu sXmSp-h-c-I-fpsS \ofw Xpey-am-Wv. (i) 3.2 sk‚n-ao-‰¿ Bc-ap≈ hrØw hc-bv°p-I. tI{µ-Øn¬ \n∂ v 8 sk‚n-ao-‰¿ AIse P F∂ _nµp AS-bm-f-s∏-Sp-Øp-I. P bn¬ IqSn hrØ-Øns‚ sXmSp-h-c-Iƒ hc-bv°p-I. Ah-bpsS \ofw Af-s∂ -gp-Xp-I. (ii) 3 sk‚nao-‰¿ Bc-ap≈ hrØw hc-bv°p-I. AXn¬ 4 sk‚n-ao-‰¿ \of-Øn¬ ABF∂ Rm¨ hc-bv°p-I. A bn¬ IqSnbpw Bbn¬ IqSn-bn¬ sXmSp-h-c-Iƒ hc-bv°p-I. sXmSp-h-c-Iƒ JWvUn°p∂ _nµp-hn¬ \n∂ pw Abnte°pw Bbnte°pw D≈ AIew Af-s∂ -gp-Xp-I.
{]h¿Ø\w 6 P (i)
X- ∂ n-c n- °p∂ Afn¬ Nn{Xw hcbv°p-I. P,Q F∂ o _nµp-°-fn¬ IqSn sXmSp-h-c-Iƒ hc-bv°p-I. Ch JWvUn°p∂ _nµp-hnse tIm¨ F{X-sb∂ v Af∂ v ImWp-I.
k 2 s
an
O 1300
Q 120................................................................................................................................................................................. MUKULAM M ATHS
A
(ii)
Nn{X- Ø n¬ PA, PB C h sXmSp-h-c-I-fm-Wv. O hrØ-tI-{µw. APB =700 BIp-∂ p. OAP, OBP Ch F{X? Nm]w ACBbpsS tI{µ-tIm-Wns‚ Af-sh{X? (Cu {]h¿Ø-\-Øn¬ A, B, Ch 900 hoXw Bb-Xp-sIm≠v APB+ AOB = 1800 F∂ v ImWmw)
C
P
O
B
hrØ-Ønse c≠p _nµp-°ƒ tbmPn-∏n-°p∂ sNdnb Nm]-Øns‚ tI{µ tImWpw, Cu _nµp-°-fnse sXmSp-h-c-Iƒ°n-S-bn-ep≈ tImWpw A\p-]q-c-I-am-Wv.
{]h¿Ø\w 7 NphsS sImSpØ Hmtcm-∂ nepw O hrØ-tI-{µhpw PA, PB Ch sXmSp-h-c-I-fm-Wv. A
(i) APB =500
Bbm¬
P
O
AOB =...........................
B
A
(ii)
APB =800
Bbm¬
PAB =...........................
P
PBA =...........................
B
MUKULAM MATHS ................................................................................................................................................................................121
A
(3)
AOB =1000
Bbm¬ Q
AQB =...........................
B
A
(4)
APB =400
Bbm¬
PAB =...........................
P
PBA =........................... AOB =...........................
B A
(5)
PAB =550
Bbm¬
PBA =...........................
P
APB =........................... AOB =...........................
B A
(6)
PBA =500
Bbm¬
PAB =...........................
P
APB =........................... AOB =...........................
B
AQB =...........................
122................................................................................................................................................................................. MUKULAM M ATHS
A
(7)
PBA =X0
Bbm¬
PAB =...........................
Q
O
P
APB =........................... AOB =........................... AQB =...........................
B
PBA =
C-hnsS PBA
AQB
F∂ v ImWm-a-t√m.
F∂ -Xv- AB F∂ RmWpw, Bbn¬ IqSnbp≈ sXmSp-h-cbpw XΩn-ep≈ tIm¨ BWv.
AQBF∂
Xv adp-h-i-Øp-≈ -hr-Ø-J-WvU-Ønse tImWpw. hrØ-Øns‚ Hcp RmWpw AXns‚ Hc-‰-Øp-Iq-Sn-bp≈ sXmSp-h-cbpw XΩn-ep≈ Hmtcm tImWpw, B tImWns‚ adp-h-i-Øp≈ hrØ JWvU-Ønse tImWn\v Xpey-amWv.
Q
(1)
Nn{X-Øn¬ P bn¬ IqSn-bp≈ sXmSp-h-cbmWv AB
R
APR =700
B
,
BPQ =650 Bbm¬
r PQR
s‚ tImW-f-hp-Iƒ F{X?
P A
MUKULAM MATHS ................................................................................................................................................................................123
B
(2)
Nn{X-Øn¬ AB hymkhpw PQ sXmSp-h-c-bp-am-Wv. C
PAC =650
Bbm¬ r ABC bpsS
tImW-f-hp-Iƒ ImWp-I.
P
Q
A
{]h¿Ø\w 8 hrØ-Øns‚ sXmSp-hc hc-bv°p-∂ -Xn\v Bcw-hc®v AXns‚ A‰-Øp-IqSn ew_-ambn Hcp hc hcbv°p-I-bm-Wt√m sNbvX-Xv. tI{µw D]-tbm-Kn-°msX hrØ-Ønse Hcp _nµp-hn¬ sXmSp-hc hcb-°p-∂ -sX-ß n-s\-sb∂ v t\m°mw. hrØ-Øn¬ A F∂ _nµp AS-bm-f-s∏-SpØn NphsS ImWp-∂ -Xp-t]mse Hcp tIm¨ \n¿Ωn°p-I.
C B
A AC tbmPn-∏n®v
AXn-t\mSv tN¿Øv Abn¬ ABCbpsS Af-hn¬ Hcp tIm¨ D≠m-Im-Ø-hn[w Hcp hc hc-bv°p-I. Cu hc A bn¬ IqSn-bp≈ sXmSp-hc Bbn-cn-°pw.
C
E
B D
P Q A
(kq-N\: BtI{µ-ambn hc-bv°p∂ Nm]w BA,BC F∂ n-hsb bYm-{Iaw D,E F∂ o _nµp-°-fn¬ JWvUn-°p-∂ p-sh-∂ n-cn-°s´. A tI{µ- a mbn BD Bc- Ø n¬ hcbv°p∂ Nm]w A C sb Pbnepw P tI{µambn DE Bc-Øn¬ hc-bv°p∂ Nm]w BZysØ Nm]sØ Q hnepw JWvUn-°p∂ p. PAQ = ABC Bbn-cn-°p-a-t√m.)
124................................................................................................................................................................................. MUKULAM M ATHS
{]h¿Ø\w 9
A
(i)
500
Nn{X-Øn¬ Cbnse sXmSp-h-c-bmWv PQ PCB,
B
PBC,
PCA
Ch F{X?
300 Q
C
P
A (ii)
Cu Nn{X-Øn¬ r PCA, r PBC Ch kZri B
{XntIm-W-ß -fm-sW∂ pw PAx PB =PC2 F∂ pw sXfn-bn-°m-a-t√m. P
Q
C
A
(i)
Nn{X-Øn¬ C bnse sXm-Sp-h-c-bmWv PC PB=
16 skan, AB=
Bbm¬ PC F{X?
B
9 skan P
C
MUKULAM MATHS ................................................................................................................................................................................125
{]h¿Ø\w 10 3 skan Bc-Øn¬ Hcp hrØw hc-bv°p-I. Hcp tIm¨ 400 Bb Hcp ka-`p-P-km-am-¥ -cn-Iw, hiß -sf√mw Cu hrØsØ sXmSp∂ coXn-bn¬ hc-bv°p-I. (kq-N\: hrØw hc®v 1400 tImWn¬ JWvUn-°-Ø-°-hn[w c≠v hymkw hc-bv°p-I. hymk-ß fpsS A‰-Øp≈ _nµp-°-fn¬ IqSn sXmSp-h-c-Iƒ hc-®m¬ Ah tN¿∂ p-≠m-Ip∂ NXp¿`p-Pw Hcp ka-`p-P-km-am-¥ -cnIw Bbn-cn-°pw.) = hrØ-Øn\v GXv c≠v hymk-ß -fp-sSbpw A‰-ß -fn-eqsS hc-bv°p∂ sXmSp-h-c-Iƒ ka-`pP kmam-¥ -cnIw D≠m-°p-∂ p. hymk-ß ƒ ]c-kv]cw ew_-am-bmtem?
{]h¿Ø\w 11 2 skan Bc-ap≈ hrØw hc-bv°p-I. hi-ß ƒ Cu hrØ-Øns‚ sXmSp-h-c-I-fmbpw tImWp-Iƒ 500, 600, 700, Bbpw hcp∂ Hcp {XntImWw hc-bv°p-I. A 500
Nn{X- Ø n¬ O hrØ- t I- { µhpw P,Q,R Ch {XntIm-W-Øns‚ hiß ƒ hrØsØ sXmSp∂ _nµp°-fp-am-sW-¶n¬ POR=1300
P
QOR=1200
R
POQ=1100 Bbn-cn-°p-sa-∂ -dn-bm-
a-t√m.
O
B (kq-N\: hrØns‚ tI{µ-Øn¬ 1300, 1200, 1100 tIm¨ BI-Ø-°-hn[w aq∂ v Bc-ß ƒ hc-°pI. Bc- ß - fpsS A‰-Ø p≈ _nµp-° -f n¬ hcbv°p∂ sXmSp-h-c-Iƒ tN¿Øv In´p∂ {XntImW-Øns‚ tImWp-Iƒ 500, 600, 700 Bbn-cn-°pw.)
700
600 Q
C
IqSmX AO, BO, CO Ch bYm-{Iaw A, B, C Ch-bpsS ka-`m-Pn-Iƒ Bbn-cn-°pw. hrØw {XntIm-W-Øns‚ A¥ ¿hr-Ø-am-Ip-∂ p. = ChnsS hrØw hc-®v, hi-ß ƒ hrØ-Øns‚ sXmSp-h-c-Iƒ BI-Ø-°-hn[w {XntImWw \n¿Ωn-°p-I-bm-W-t√m sNbvX-Xv. BZyw {XntImWw hc®v AXns‚ A¥ ¿hymkw hc-bv°p∂ -sX-ß n-s\-sb∂ v t\m°mw. hrØ-Øns‚ c≠v sXmSp-h-c-Iƒ JWvUn-°p∂ _nµp-hnse tImWns‚ ka-`m-Pn-bn¬ hrØ-tI{µw Dƒs∏-Sp-at√m? AXp-sIm≠v {XntIm-W-Øns‚ aq∂ v tImWpI-fp-sS-bpw- ka-`mPn-Iƒ JWvUn-°p-∂ Xv A¥ ¿hr-Ø-Øns‚ tI{µ-Øn-em-bn-cn-°pw. 1. hi-ß -fpsS \ofw 5.5 skan, 7 skan, 8 skan Bb {XntImWw hc®v A¥ ¿hymkw \n¿Ωn°p-I. 2.
AB= 6,
skan,
A = 650
,
B = 750
, Bb {XntImWw hc®v A¥ ¿hrØw \n¿Ωn-°pI
126................................................................................................................................................................................. MUKULAM M ATHS
k-a-`pP {XntIm-W-Øns‚ A¥ ¿hr-Øhpw ]cn-hr-Øhpw
A
Nn{X-Øn¬ ka-`p-P-{Xn-tImWw ABCbpsS ]cn-hr-Ø-tI-{µ-amWv O F∂ n-cn-°-s´. OD, OE, OF, Ch bYm-{Iaw BC, AC, AB F∂ o hi-ß ƒ°v F E ew_-am-Wv OD=OE=OF (F-¥ p-sIm≠v?) O BD=BF (F-¥ p-sIm≠v?) C B D \ rOBD @ rOBF B bpsS k`-am-Pn-bmWv BO AXp-t]mse C bpsS ka-`m-Pn-bmWv CO F∂ pw AbpsS ka-`m-Pn-bmWv AO F∂ pw ImWmw. \ O F∂ _nµp {XntIm-W-Øns‚ A¥ ¿hr-Ø-tI-{µ-am-bn-cn-°pw. Nn{X-Øn¬ OB A¥ ¿hr-Ø-B-chpw OB ]cn-hr-Ø-B-c-h-hp-am-Wv. rOBDbpsS tImW-f-hp-Iƒ 300,600,900 Bb-Xp-sIm≠v OB=2xOD AXm-bXv A¥ ¿hr-Ø -B-c-Øns‚ c≠v aS-ß mWv ]cn-hr-ØB-cw. as‰m-cp-hn-[-Øn¬ ]d-™m¬ ]cn-hr-Ø-B-chpw A¥ ¿hr-Ø-B-chpw XΩn-ep≈ Awi_‘w 2: 1Bbn-cn-°pw. M O 700
O
P
tI{µ-amb hrØ-Øns‚ c≠v sXmSp-h-c-I-fmWv MP, NP F∂ n-h. MOP=700 Bbm¬ MPO, NOP, MPN F∂ nh ImWp-I.
N P O
tI{µ-amb hrØ-Øns‚ P,Q F∂ o _nµp°-fnse sXmSp-h-c-Iƒ R¬ JWvUn-°p-∂ p. OR F∂ tcJ PRQ sb ka-`mKw sNøp-∂ p-sh∂ v sXfn-bn-°p-I. CXn-\mbn X∂ n-cn-°p∂ {]kvXmh-\-Iƒ°p≈ Imc-W-ß ƒ Fgp-Xp-I.
1.
O
R
Q
rPOR , rQOR Ch-bn¬ OP=OQ ( ....................................................................................)
2.
OR=OR ( ....................................................................................)
3.
PR=QR ( ....................................................................................)
4.
rRPO @ r
5.
PRO =
RQO ( ................................................................)
QRO ( ......................................................................)
MUKULAM MATHS ................................................................................................................................................................................127
Q 500 R
Nn{X-Øn¬ AB hrØ-Ønse Pbnse sXmSp-h-c-bm-Wv. PQ hrØ-Øns‚ hymkhpw PQR=500 bpw BsW¶n¬ QPR F{X? APR F{X?
O
A
P
B A
C
Nn{X-Øn¬ AB hrØ-Øns‚ hymkhpw PQ -sXmSp-h-c-bp-amWv. BDC=400 F-¶n¬ BAC F{X? PBC F{X?
D
B
P
Q
R Q
Nn{X-Øn¬ AB hrØ-ØnØnse P bnse sXmSp-h-c-bm-Wv. PQ=PR Dw QPR =500 bpw BsW-¶n¬.
S
P
A
B
PRQ = ........................ ,
PQR = ........................ ,
BPQ = ........................ ,
PSR = ........................ ,
SPR+
APR = ........................ A
SRP = ........................
Nn{X-Øn¬ APxAR=AQXAS F∂ v sXfn-bn-°pI. hrØ-Ønse B F∂ _nµp-hnse sXmSp-hc-bmWv . PQ IqSmsX AB hymk-hpw. (kqN\: APxAR =AB2 F∂ pw AQxAS =AB2 F∂ pw sXfn-bn-°m-a-t√m)
S
R P
B
Q
128................................................................................................................................................................................. MUKULAM M ATHS
C
Nn{X- Ø n¬ NXp ¿`pP w ABCD sb AXns‚ A¥ ¿hrØw P,Q,R,S F∂ o _nµp-°-fn¬ sXmSp∂ p. AP=5, PB=4, QC=7, DR=3 F¶n¬
D 3 R 7
BQ= ..................... CR= ................ DS=.......................
S
Q
AS= ..................... AB= ................ BC=....................... CD = ..................... AD= ................ AB+CD =.................................. BC+ AD= .................... A
5
P
4
B
Q R
Nn{X-Øn¬ AB F∂ tcJ hrØsØ Pbn¬ sXmSp∂ p. APR=700 , BPQ=800 Bbm¬ rPQR s‚ tImW-f-hp-Iƒ Fgp-Xp-I.
A
B
P
C O
Nn{X-Øn¬ O hrØ-tI-{µw. A bn¬ IqSn-bp≈ sXmSp-h-c-bm-Wv PQ . BAQ=650 BIp-∂ p. OAQ, ACB Ch ImWp-I. P
L
B
A
Q
N
Nn{X- Øn¬ PQ F∂ tcJ hrØ-s Ø M¬ sXmSp-∂ p. LM=MN Bbm¬ LN || PQ F∂ v sXfnbn-°p-I. P
M
Q
MUKULAM MATHS ................................................................................................................................................................................129
C B
800
500 X
Nn{X-Øn¬ XY F∂ tcJ hrØ-sØ AF∂ _nµp-hn¬ sXmSp-∂ p. NXp¿`pPw ABCD bpsS F√m tImWp-I-fp-sSbpw Afhv IW-°m-°pI.
D
700 Y
A
M
E B
Nn{X-Øn¬ MN F∂ tcJ hrØ-sØ A bn¬ sXmSp-∂ p. CAN=400 Bbm¬ 1) ABC, CBE, CDE Ch-Im-Wp-I. 2) MN || DE F∂ p sXfn-bn-°p-I.
A
C N D
C
Nn{X-Øn¬ PA F∂ hc hrØ-sØ A bn¬ sXmSp- ∂ p. CB=4cm, PB=1 cm Bbm¬ PA F{X?
B
A
Nn{X-Øn¬ PQ sXmSp-h-c-bm-Wv. PQ=PM=MN BIp-∂ p. PR=1cm, RS=8cm Bbm¬ 1) PQ F{X? 2) MR F{X? 3) QM F{X?
P
Q
R O
P
M
S N
130................................................................................................................................................................................. MUKULAM M ATHS
M
c≠v hrØ-ß ƒ M,N F∂ o _nµp-°fn¬ JWvUn-°p-∂ p. MN F∂ hc-bnse _nµp-hm-Wv A . AB, AC Ch sXmSp-h-c-Ifm-Wv. AB=AC F∂ p sXfn-bn-°p-I.
N
B
C
A
A
P
Nn{X-Øn¬ AB, BC, AC F∂ o hi-ß ƒ hrØsØ bYm- { Iaw P,Q,R F∂ o _nµp-°-fn¬ sXmSp-∂ p. BC=11cm, AP=4 cm, Bbm¬ AR F{X? r ABC bpsS Np‰-f-sh{X?
R
B Q
C
B
Nn{X-Øn¬ O,C Ch tI{µ- ß -f mb hrØ-ß -fn¬ AB Ch c≠p hrØß - f nep w sXmSp∂ hcbm- W v. AB=10cm,
PB=20 cm, OA=5 cm,
A
C
O
P
Bbm¬ BC F{X?
O
A
P
B
Nn{X-Ønse c≠v hrØ-ß -fp-sSbpw tI{µw O BWv. henb hrØ-Ønse AB F∂ Rm¨ sNdnb hrØsØ P bn¬ sXmSp-∂ p. AB=5 cm 1) sjbvUv sNbvX `mK-Øns‚ ]c-∏-f-shs{X? 2) AB bpsS a[y-_n-µp-hmWv P F∂ v sXfn-bn-°pI.
MUKULAM MATHS ................................................................................................................................................................................131
A[ymbw 9
_lp-]-Z-ß ƒ BapJw _oP-K-WnX ]T-\-Øns‚ `mK-ambn _lp-]-Z-ß ƒ F∂ Bibw HºXmw ¢mkn¬ Ah-X-cn∏n-®n-´p-≠v. AXns‚ XpS¿®-bmWv Cu A[ym-bw. ka-hm-Iy-ß -fpsS ]cn-lmcw ImWpI F∂ {]mtbm-KnI {]iv\-Øns‚ `mK-am-bmWv _lp-]-Z-ß -fpsS ]T\w B`n¿h-`n-®Xv F∂ v s]mXpsh ]d-bmw. kwJy-I-fpsS khn-ti-j-XIfpw {Inb-Ifpw Ahbv°v ASn-ÿ m-\-amb `uXnI kml-N-cyß -ep-ambn _‘-s∏-´mWv cq]w sIm≠-Xv. Ch-bpsS KWn-X-]-c-amb hymJym-\-Øn-eqsS _lp-]-Zß -fpsS ]T-\-Øn-te°pw Ch hym]n-°p-∂ p. Hcp kwJy as‰mcp kwJy-bpsS LS-I-am-Ip∂ kµ¿`Øn¬ \n∂ v _lp-]Zw as‰mcp _lp-]-Z-Øns‚ LS-I-am-Ip∂ kµ¿`w N¿®-sN-øp-∂ p. CXv injvSmkn-≤m-¥ -Øn-te°pw LS-I-kn-≤m-¥ -Øn-te°pw \bn-°p-∂ p. _lp-]-Z-Øns‚ LS-I-ß ƒ I≠p]n-Sn-°p∂ {]h¿Ø\w _lp-]-Z-k-a-hm-Iy-ß -fpsS ]cn-lmcw ImWp∂ {Inb-bmbn amdp-∂ Xpw {]tXyI {i≤ A¿ln-°p-∂ p. IqSmsX ka-hmIy ]cn-lm-c-Øn\v _lp-]-Z-ß -fpsS LS-I-{Inb klm-bI - -am-hp-∂ -Xmbpw ImWmw. c≠mw IrXn ka-hm-Iy-ß ƒ F∂ A[ym-bhpw Cu A[ymbhpw ]c-kv]c ]qc-I-ß fmbn h¿Øn-°p-∂ p-≠v. {][m\ Bi-b-ß ƒ = kwJy-I-fpsS LSIw F∂ Bi-b-Øn-eqsS _lp-]-Z-Øns‚ LSIw F∂ Bi-bw. = _lp-]-Z-ß -fpsS lc-WsØ ASn-ÿ m-\-am°n LS-I-amtWm F∂ v ]cn-tim-[n-°m≥ lc-W^ew {]k-‡-a-√. = injvS-kn-≤m¥ w = LSI kn≤m¥ w = _lp-]-Z-ß -fpsS LS-I-{Inb = ka-hmIy ]cn-lm-c-Øn-eqsS _lp-]-Z-ß -fpsS LS-I-{Inb = LS-I-{In-bb - n¬ \n∂ v ka-hm-Iy-ß -fpsS ]cn-lmcw ]T-\{- ]-h¿Ø-\-ß ƒ tImfw Abnse ka-hm-IysØ ASn-ÿ m-\-am°n tImfw B bnse kwJy-bpsS LS-I-amtWm tImfw Cbnse kwJy F∂ v ]cn-tim-[n-°p-I. Set I A
B
C
6=3x2
6
3
10=5x2
10
5
21=5x4+1
21
4
-15=-3x5
-15
-3
26=-2x-13
26
-2
-100=-25x4
-100
4
-38=-5x8+2
-38
-5
132................................................................................................................................................................................. MUKULAM M ATHS
F∂ nh ]q¿Æ - kwJy-I-fm-hp-Ibpw bpsS LS-I-ß -fm-Wv a, b
p, a, b
P=axb
Bhp-Ibpw sNbvXm¬
P
Set 2 A
B
C
48 ¸ 6 = 8
48
6
-175 ¸ 7 = -25
-175
7
376 ¸ 4 = 94
376
4
-100 ¸2 = -50
-100
2
F∂ nh ]q¿Æ - kwJy-I-fm-hp-Ibpw P¸ a = b Bhp-Ibpw sNbvXm¬ P bpsS LS-I-ß -fm-Wv a, b P,a,b
Set 3 A
B
C
6 = 12x1/2
6
1/2
-15= -45x1/3
-15
1/3
8/15= 4/5x2/3
8/15
4/5
8=24/5x 5/3
8
5/3
-2 = -10/7 x 7/5
-2
7/5
`n-∂ -kw-Jy-Iƒ samØ-Øn-¬ ]cn-K-Wn-®m¬ LSIw F∂ Bi-b-Øn\v {]k‡n C√.
_lp-]-Z-ß -fpsS LS-I-ß ƒ ]T\ {]h¿Ø-\-ß ƒ tImfw Abnse ka-hm-IysØ ASn-ÿ m-\-am°n tImfw tImfw Cbnse _lp-]Zw F∂ v ]cn-tim-[n-°p-I.
Bbnse
_lp-]-Z-Øns‚ LS-I-amtWm
Set 1 A
B
C
x2-1= (x+1) (x-1)
x2-1
x-1
x2-4= (x+2) (x-2)
x2-4
x+2
x2-5+6=(x-2)(x-3)
x2-5x+6
x-3
x2+7x+10=(x+5) (x+2)
x2+7x+10
x+5
x3-6x2+11x-6=(x-1)(x-2)(x-3)
x3-6x2+11x-6
x-2
x2-2x-1= (x-1-Ö2)(x-1+ Ö2)
x2-2x-1
x-1-Ö2
MUKULAM MATHS ................................................................................................................................................................................133
Set 2 A
B
C
x2-1= (x+1) (x-1)+2
x2+1
X-1
x2+1= (x+1) (x2 - x+1)
x3+1
X+1
2x2+x+x=(x-1)(2x+3)+6
xx2+x+3
2x+3
X3-8x2+7x+10=(x-3) (x2-5x-8)-14)
X3-8x2+7x+10
X-3
A
B
C
x2-1= 1/2 (2x+2) (x-1)
x2-1
2X+2
x2-4= 2/3 (3/2x+3/2) (x - 4)
x2-4
3
x2-5x+6=1/6(2x-4)(3x-9)
x2-5x+6
3x+9
X3-6x2+11x-6=1/12(2x-2) (3x-6)(2x-6)
X3-6x2+11x-6
2x-6
Set 3
P(x) F∂ 'a'
/2x+3/2
_lp-]Zw q(x)F∂ _lp-]-Z-Øns‚ LS-I-am-sW-¶n¬ GXv kwJy
FSp-Ømepw aP(x)F∂ _lp-]-Zhpw q(x)s‚ LS-I-am-Wv.
]T-\-{]-h¿Ø-\-ß ƒ 1.
x3-1=(x-1)(ax2+bx+c)+d
F¶n¬ a,b,c,d F∂ nh I≠p-]n-Sn®v x3-1 s‚ LS-I-amtWm x-1F∂ v ]cn-
tim-[n-°p-I. 2.
x3+1=(x+1)(ax2+bx+c)+d
F¶n¬ a,b,c,d F∂ nh I≠p-]n-Sn®v x3+1 s‚ LS-I-amtWm x+1F∂ v
]cn-tim-[n-°p-I. lc-W-{In-bb - n¬ injvSw ]qPy-am-sW-¶n¬ lmcy-amb _lp-]-Z-Øns‚ LSI-amWv lmc-I-amb _lp-]-Zw. Np-hsS sImSp-Øn-cn-°p∂ Hmtcm tPmSn _lp-]-Z-ß -fnepw BZy-tØXv c≠m-a-tØ-Xns‚ LS-IamtWm F∂ v ]cn-tim-[n-°p. =
x-1, x3+1
=
x+2, x2+5x+8
=
x-1, x3-1
=
x-2, x2+6x+8
=
2x+1, 8x3+1
=
x-2, x2+6x+10
=
2x-1, 8x3-1
=
x-3, x3-27
=
x+2, x2+3x+2
=
x+3, x3-27
134................................................................................................................................................................................. MUKULAM M ATHS
H∂ mw IrXn LSI-ß ƒ ]T-\-{]-h¿Ø-\-ß ƒ 1. x3-5x2+8x+3=(x-1) (ax2+bx+c) +7 F¶n¬ x3-5x2+8x+3 s‚ LS-I-amtWm (x-1)? x3-5x2+8x+3 s\ (x-1) sIm≠v lcn-®m-ep≈ injvSw F{X? 2. x3+6x2-8x+10=(x-1) (ax2+bx+c)+d F¶n¬ d bpsS hne ImWp-I. x3+6x2 -8x+10 s\ (x-1) sIm≠v lcn-®m-ep-≈ injvSw F{X? 3. x3+4x2-7x+6= (x-2) q(x)+r F¶n¬ x3+4x2-7x+6 s\ (x-2) sIm≠v lcn-®m-ep≈ injvSw F{X? injvS-kn-≤m¥ w P(x)=(x-a) q(x)+r F¶n¬ P(a)=r , P(x)
F∂ ]lp-]-ZsØ (x-a) sIm≠v lcn-®m¬ In´p∂
injvSw P(a) BWv.
4. 5.
P(x)=(x-2) q(x)
F¶n¬ p(x) s\ (x-2) sIm≠v lcn- ®m- ep≈ injvSw F{X?P(x) s‚ LS-I-amtWm (x-2) P(x) F∂ _lp-]-Z-Øn¬ P(3)=0 F¶n¬ P(x) s\ (x-3) sIm≠v lcn-®m-ep≈ injvSw F{X? P(x) s‚ LS-I-amtWm? (x-3) LSIkn-≤m¥ w P(x) F∂
_lp-]-Z-Øn¬ P(a)=O BsW-¶n¬ P(x) s‚ LS-I-am-Wv (x-a)
MUKULAM MATHS ................................................................................................................................................................................135
]cn-io-e\ {]iv\-ß ƒ 1.
(x+1), (x-1), (x+2), (x-2)
F∂ nh 3x2+7x-4s‚ LS-I-ß -fmtWm ]cn-tim-[n-°p-I.
2.
Xmsg X∂ n-cn-°p∂ Hmtcm _lp-]-Zhpw x3+3x2+2x-6 s‚ LS-I-ß -fmtWm? (x+1), (x-1), (x+3), (x-3)
3.
(x+1), (x+2), (x+3), (x-3)
F∂ nh x3+4x2+x-6 s‚ LS-I-ß -fmtWm F∂ v Is≠-Øp-I.
4.
x27-1
5.
x50-x43 +x20-x10+7
¬ x+1 LS-I-amtWm?
6.
(x+1) (2x+5) +4
s‚ LS-I-amtWm x+1 F∂ v Is≠-Øp-I.
7.
3x-1
8.
2x3-8x2-5x-25
9.
(x+1), (x-1), (x+2)
s‚ Hcp LS-I-amWv x-1 F∂ v sXfn-bn-°p-I.
F∂ ]Zw 3x3-4x2+2x-6 s‚ LS-I-amtWm?
s\ x-5 sIm≠v lcn-°p-tºm-gp≈ injvSw Is≠Øn LS-I-amtWm F∂ v ]cntim-[n-°p-I. Cu Bibw D]-tbm-Kn®v 2x-10 LS-I-amtWm F∂ v Is≠-Øp-I. F∂ n-h-bn¬ GsX√mw Xmsg X∂ n-cn-°p∂ _lp-]-Z-ß -fpsS LS-I-am-Ip-
sa∂ v ImWp-I. x3+8x2+8x-17 2x3-6x2-8x+4 x3-2x2-5x-6 3x2-2x-7 x3-3x2-x
10.
x3-5x2+8x-1
t\mSv GXv kwJy Iq´n-bm¬ (x-1) LS-I-ambn hcp∂ _lp-]Zw In´pw.
11.
3x2+x+8
12.
2x3-5x2+7x+8
s‚ Hcp LS-I-amtWm (3x+2) F∂ v ]cn-tim-[n-°p-I. F∂ Hcp _lp-]-Z-tØmSv GXv kwJy Iq´n-bm¬ (2x-1) F∂ _lp-]Zw LS-I-
am-Ipw. 13. 14.
x2-5x+8
F∂ _lp-]-Z-tØmSv GXv kwJy Iq´n-bm¬ (x-3) CXns‚ LS-I-am-Ipw. 2x2-kx2+8x-3 sIm≠v lcn-®m¬ injvSw O Bbm¬ K F{X-bm-bn-cn-°pw.
15.
2x2 + kx + 6
¬ P(-1) =O Bbm¬ k F{X? Cu _lp-]-Z-Øns‚ c≠v LS-I-ß ƒ Is≠-Øp-I.
136................................................................................................................................................................................. MUKULAM M ATHS
IqSpX¬ ]cn-io-e\ {]iv\-ß ƒ 1.
x3- 3x2+8x+k
2.
P(x) F∂
F∂ ]Z-Øn¬ Hcp x-2 LS-I-am-bm¬ k bpsS hne F¥ m-bn-cn-°pw.
_lp-]-Z-Øns‚ Hcp LS-I-am-Wv (x2-a2) . F¶n¬ P(a)=0 F∂ pw P(-a)=0F∂ pw sXfn-
bn-°p-I. 3.
ax3+bx2+cx+dF∂
_lp-]-Z-Øn¬ x+2 Hcp LS-I-am-bm¬ 8a+2c=4b+d F∂ v sXfn-bn-°p-I.
4.
x2+ax+b=0
F∂ ka-hm-Iy-Øns‚ ]cn-lmcw 5, -˛8 F∂ n-h-bm-Wv. F¶n¬ x2 +ax+b F∂ _lp]-ZsØ H∂ mw IrXn-bn-ep≈ c≠p _lp-]-Z-ß -fpsS KpW-\-^-e-ambn Fgp-Xp-I.
5.
x3+px2+qx-6
6.
2x3+px2+qx-20
F∂ _lp-]-Z-Øns‚ c≠v LS-I-ß ƒ (x+1), (x-3),F∂ n-h-bm-bm¬ p,q F∂ n-hbpsS hne-Iƒ Is≠-Øp-I. s‚ c≠p LS-I-ß ƒ (x+2), x-2 Bbm¬ p,q ChbpsS hne-Iƒ F¥ m-bn-cn-°-
Ww. 7.
P(x)=2x2-7x2+4x-6 ¬ P(3) ImWp-I. Q(x)=x3+8x2-15x-51 ¬ Q (3) ImWp-I. R(x)=P(x)+Q(x) F¶n¬ R(x)
s‚ LS-I-amtWm x-3 F∂ v ]cn-tim-[n-°p-I.
10.
¬ p(1)=p (-1) Bbm¬ a+c=o F∂ v sXfn-bn-°p-I. 12x2-10x+2=0 F∂ ka-hm-Iy-Øns‚ ]cn-lmcw 1/3, 1/2 Ch-bm-bm¬ 12x2-10x+2s‚ LS-Icq]w Fgp-Xp-I. x3+5x2+bx-8 s‚ Hcp LSIw x+1 Bbm¬ b F{X? Cu _lp-]-Z-Øn¬ x-1 Hcp LS-I-amtWm?
11.
ax2+bx2+cx+d
8. 9.
P(x)=ax3+bx2+cx+d
F∂ _lp-]-Z-Øns‚ Hcp LSIw x+1 Bbm¬ a+c=b+d F∂ v sXfn-bn-°p-I.
L-S-I-{Inb ]T-\-{]-h¿Ø\w x2+2x-15 F∂ _lp-]-Z-Øns‚ LS-I-ß ƒ ImW-W-sa-∂ n-cn-°-s´. P(x) F∂ _lp-]-Z-Øn¬ P(a)=0 F¶n¬ P(x) s‚ LS-I-am-Wt√m (x-a) AXm-b-Xv, P(x) s‚ LSIw ImWp-∂ -Xn\v P(x) =0 BIp∂ x s‚ hne-Iƒ Is≠-Øn-bm¬ aXn-bt√m? x2+2x-15 s‚ LS-I-ß ƒ ImWp-∂ -Xn\v x2+2x-15=0 BIp∂ x hne-Iƒ Is≠-Ø-Ww. (k-a-hmIy-Øns‚ ]cn-lm-cw) Cu ka-hm-Iy-Øns‚ ]cn-lmcw 3 As√-¶n¬ ˛5 BWv. P(3)=0 ; P(-5)=0
AXp-sIm≠v x2+2x-15 s‚ LS-I-ß -fmWv. (x-3), (x+5) NphsS sImSp-Øn-cn-°p∂ Hmtcm _lp-]-Z-sØbpw H∂ mw IrXn _lp-]-Z-ß -fpsS KpW-\-^e-ambn Fgp-Xp-I. =
x2-3x-28
=
6x2-11x+3
=
x2-x-6
=
x2-2x-2
=
x2+5x-14
=
x2-4x+1
MUKULAM MATHS ................................................................................................................................................................................137
(x+a) (x+b) =x2+(a+b) x + ab
]qcn-∏n-°pI x2+9+20 = (x+5) (x+4) x2+13x+30 = ............................................... x2+7x+10 = ................................................. x2+20x+36 = ............................................... x2+15x+50 = ...............................................
x2-8x+15 = (x-3) (x-5) x2-3x+2 = ................................................... x2-8x+15 = .................................................. x2-20x+75 = ................................................ x2-10x+21 = ................................................
x2+4x-5 = (x+5) (x-1) x2 + 7x - 30 = .............................................. x2 + x - 30 = ................................................. x2 + 2x - 48 = .............................................. x2 + 10x - 11 = .............................................
x2 - 4x - 12 = (x-6) (x+1) x2 - x - 12 = .................................................. x2 - 5x - 14 = ................................................ x2 - 3x - 70 = ................................................ x2 - 7x - 18 = ................................................
Xmsg X∂ n-cn-°p∂ c≠mw IrXn ka-hm-Iy-ß sf LS-I-ß -fm°n ]cn-lmcw Is≠-Øp-I. eg.
x2+9x+20 = 0
(x+5) (x+4) =0
\ X+5=0 As√-¶n¬ x+4=0 \ x = -5 x=-4
]cn-lmcw -5 or -4 138................................................................................................................................................................................. MUKULAM M ATHS
]cn-io-e\ {]iv\-ß ƒ LS-I-ß ƒ Is≠Øn ]cn-lmcw ImWp-I. x2-7x-18=0 x2+13x-30=0 x2-3x-70=0 x2+7x+10=0 x2-5x-14=0 x2-10x-11=0 x2-10x-11=0 x2+20x+36=0 x2-8x+15=0 x2-4x-5=0 x2-4x-12=0 x2+15x+50=0 x2-3x+2=0 x2+7x-30=0 x2-x-12=0 x2-8x+15=0 x2+x-30=0 x2-20x+75=0 x2+2x-48=0
t{]mPIvSv x-1, x+1, x2-1
Ch-bn-te-sX-¶nepw LS-I-ß -fmbn hcp∂ _lp-]-Z-ß -fnse KpW-I-ß -fpsS khn-ti-j-X-Iƒ sht∆sd I≠p-]n-Sn-°p-I. kqN\: ax5+bx4+cx3+dx2+e-x+f F∂ _lp-]-Z-Øn¬ x-1 LS-I-am-sW-¶n¬ a+b+c+d+e+f=0 Bbn-cn-°-Ww. x+1
LS-I-am-I-W-sa-¶n¬
a+c+e= b+d+f x2-1
BI-Ww.
LS-I-am-I-W-sa-¶n¬
x2-1=(x+1) (x-1)
CXn¬ x+1 s‚bpw x-1 s‚bpw {]tXy-I-X-Iƒ D≠m-bn-cn-°pw. \ a+c+e =b+d+f=0 \ 2 (b+d+f) =0 Bbn-cn-°pw.
MUKULAM MATHS ................................................................................................................................................................................139
A[ymbw 10
Pyman-Xnbpw _oP-K-Wn-Xhpw BapJw ]c-kv]cw ew_-amb c≠v tcJ-Ifpw bp‡-amb Hcp GI-Ihpw FSpØv Xe-Ønse _nµp°sf kwJym-tPmUn sIm≠v kqNn-∏n-°m-a-t√m. kqN-Im-£-ß -fnse _nµp-°ƒ XΩn-ep≈ AIew ImWp-∂ Xv kqN-I-kw-Jy-Iƒ F∂ A[ym-b-Øn¬ N¿® sNbvXt√m! CXns‚ XpS¿®-bmWv Cu ]mT-`m-Kw. ChnsS c≠v _nµp-°ƒ XΩn-ep≈ AIew Is≠-Øp-Ibpw CXv D]-tbm-Kn®v Nne Pyma-Xob {]iv\-ß ƒ°v _oP-K-Wn-X-Øns‚ klm-b-tØmsS ]cn-lmcw ImWp-I-bp-am-Wv sNøp∂ -Xv. \of-tØbpw `mc-sØbpw a‰pw kwJy-I-fm-°n-bmXv t]mse Hcp hc-bpsS sNcn-hn-s\bpw kwJysIm≠v kqNn-∏n-°m-sa∂ v ChnsS N¿®-sN-øp-∂ p. Hcp hc-bnse _nµp-°-fpsS kqN-I-kwJyIfpsS ]c-kv]c _‘sØ ASn-ÿ m-\-am°n hc-bpsS ka-hmIyw cq]o-I-c-Whpw ChnsS \S-Øp-∂ p-≠v. XpS¿∂ p≈ (D-b¿∂ ) ¢mkp-I-fn¬ N¿®-sN-øs - ∏-Sp∂ a‰v h{I-tc-J-I-fpsS ka-hm-Iy-ß sf Ipdn-®p≈ ]T\-Øns‚ XpS-°-sa∂ coXn-bn-em-Wv Chn-sS hc-bpsS ka-hmIyw N¿®-sN-øp-∂ -Xv. {][m\ Bi-b-ß ƒ 1. Hcp _nµphpw B[mc _nµphpw XΩn-ep≈ AI-ew. 2. c≠v _nµp-°ƒ XΩn-ep≈ AIew 3. hc-bpsS Ncnhv F∂ Bibw 4. hc-bpsS ka-hmIyw F∂ Bi-bw.
h¿°vjo‰v 1. B
5 4 3
OA, AB, OB Ch
ImWp-I. B bpsS kwJym-tPmUn Fgp-XpI
2 1 A O
1
2
3
4
5
140................................................................................................................................................................................. MUKULAM M ATHS
2.
C
(6,8)
AB, BC, AC Ch
F{X. B bpsS kqNI kwJy-Iƒ Fgp-XpI A
B
3
C
(12,5)
Bbm¬ OA,AC,OC Ch ImWp-I. OB F{X? BbpsS kwJym-tPmUn Fgp-XpI OC= CB
A
O
B
4 A (10, 24)
OA F{X?
O
5.
B[mc-_n-µp-hn¬ \n∂ v (10,14) te°p≈ AI-e-sa¥ v?
6. D (3,4) C
A
10
kmam-¥ -cnIw A,B,C,D bn¬ C bpsS kwJym-tPmUn Fgp-Xp-I. kmam- ¥ - c oI- Ø ns‚ hi- ß - f psS Afhv Fgp-Xp-I. AC bpsS \of-sa¥ v?
B
MUKULAM MATHS ................................................................................................................................................................................141
7
D C (6,8)
A
(0,0)
NXp- c - Ø ns‚ \ofhpw hoXnbpw ImWp- I . B,D, Ch- b psS kwJym tPmUn Fgp-Xp-I.
B
S
8.
A( 6,9)
R O
P
g
hrØ-Øns‚ Bc-sa¥ v? hrØw kqN-I-A£ß sf JWv U n- ° p∂ _nµp- ° fpsS kqN- I - k w- J y- I ƒ Fgp-Xp-I.
q
{]h¿Ø\w B
A
=
{Km^v t]∏-dn¬ A ,B F∂ hc hc-bv°p-I. Acn-In-ep≈ hc-Iƒ kqN-Im-£-ß fmbn FSp-°pI. AB hnI¿Æ -ambn Hcp NXpcw hc-bv°-Ww. CXns‚ hi-ß ƒ A£-ß ƒ°v kam-¥ -c-amI-Ww. NXp-c-Øns‚ \ofhpw hoXnbpw ImWp-I. hnI¿Æ -Øns‚ \ofhpw ImWp-I.
142................................................................................................................................................................................. MUKULAM M ATHS
=
1.
2.
3.
4. 5.
1. 2. 3.
Ch-bpsS kqNI kwJy-Iƒ ImWp-I. kqNI kwJy-Ifpw Ct∏mƒ Is≠-Ønb \ofß fpw XΩn-ep≈ _‘w Is≠Øn Ah-X-cn-∏n-°p-I. A ,B
h¿°v jo‰v NXpcw A,B,C,DbpsS hi-ß ƒ A£-ß ƒ°v kam-¥ -c-am-Wv. A (2,3), C (8,11) NXp-c-Øns‚ a‰v aqe-I-fpsS kqN-I-kw-Jy-Iƒ ImWp-I. NXp-c-Øns‚ \ofhpw hoXnbpw ImWp-I. NXp-cØns‚ hnI¿Æ -Øns‚ \of-sa¥ v? Hcp Xe-Øn-ep≈ _nµp-°-fmWv P bpw Q Dw. P(2,-3), Q(5,-7) PQ hnI¿W-am-bn, hi-ß ƒ A£ß ƒ°v kam-¥ -c-ambn NXpcw hc-bv°p-I. NXp-c-Øns‚ a‰v aqe-I-fpsS kqN-I-kw-Jy-Iƒ Fgp-Xp-I. NXp-c-Øns‚ Np‰-fhv ImWp-I. B[mc-_n-µp-hn-eqsS IS∂ v t]mIp∂ Hcp tcJ-bnse _nµp-°-fmWv A (2,4), B (5,10) . B[mc_nµphn¬ \n∂ v A bnte-°pw, Bbnte°pw D≈ Zqcw ImWp-I. ABF∂ o _nµp-°ƒ XΩn-ep≈ AIe-sa¥ v? kmam-¥ -cnIw A ,B, C, D bn¬ A (2,4), B (10,4), D (5,8) C bpsS kqNI kwJy ImWp-I. kmam-¥ -cnI-Øns‚ hi-ß -fpsS Af-hp-Iƒ ImWp-I. A (2,5) B (4,9) C (7,15). A,B,C F∂ o _nµp-°ƒ t\¿tc-J-bn-emWv F∂ v sXfn-bn-°p-I. hc-bpsS Ncnhv x A£-Øn\v kam-¥ -c-ß -fmb hc-I-sf-Ip-dn®pw y A£-Øn\v kam-¥ -c-ß -fmb hc-Isf Ipdn®pap≈ Nne hkvXp-X-Iƒ kqN-I- kwJy-Iƒ F∂ ]mT-`m-KØv N¿®-sN-bvXn-´p-≠v. ChnsS Hcp hc-bnse _nµp-°-fpsS kqN-I-kw-Jy-Iƒ XΩn-ep≈ Hcp ÿ nc _‘w Is≠-Øp-∂ p. Cu _‘sØ ASn-ÿ m-\-am°n hc-bpsS Ncnhns\ Hcp kwJy-sIm≠v kqNn-∏n-°p-Ibpw sNøp-∂ p. AXn-\mbn Ip´n BZyw Xncn-®-dn-tb≠ hkvXpX hc-I-fpsS Ncnhv, F∂ Bi-bam-Wv. Ncnhv IqSp-X-ep≈ hc, Ncnhv Ipd™ hc F∂ v ]d-bp-∂ -Xv. F¥ ns\ ASn-ÿ m-\-am°n-bmWv ChnsS Hcp hc x A£-hp-ambn D≠m-°p∂ tImWns\ ASn-ÿ m-\-am°n hcbpsS Ncn-hns\ Ipdn®p ]d-bp-∂ p. Ncnhv F∂ Bibw FØn-°p-hm≥ NphsS sImSp-Øn´p≈ {]h¿Ø\w \¬Im-hp-∂ -Xm-Wv. {]h¿Ø\w Hcp {Km^v t]∏-dn¬ x,y A£-ß ƒ hc®v A (1,1) B (4,4) P (1,2) Q (3,6) F∂ o _nµp-°ƒ AS-bm-fs∏-Sp-Øp-I. A,BF∂ o _nµp-°sf tbmPn-∏n®v Hcp hc hc-bv°p-I. ABF∂ hc Ccp hi-ß fn-te°pw \o´n hc-bv°p-I. CXp-t]mse P,Q F∂ o _nµp-°-fpw tbmPn-∏n®v \o´n hc-bv°p-I. Cu {]h¿Ø\w sNøp-tºmƒ e`n-°p∂ c≠v hc-I-fpsS {]tXy-I-X-Iƒ Ip´nƒ N¿®-sN-øs´. Cu hc-I-fnse GXm\pw Nne _nµp-°-fpsS kqNI kwJy-Iƒ Is≠Øn Fgp-Xp-hm≥ ]d-bp-I. CXn¬ Hcp hc x A£-tØmSv ASpØp InS-°p-∂ p F∂ pw at‰Xv x A£-Øn¬ \n∂ pw Ipd-®p-IqSn AI-e-Øn-em-sW∂ pw ImW-s´. as‰mcp Xc-Øn¬ ]d-™m¬ H∂ m-asØ hc x A£-hp-ambn D≠m-Ip∂ tImWn-s\-°mƒ hep-Xm-Wv. c≠m-asØ hc x A£-hp-ambn D≠m-°p∂ tIm¨ CXns\ as‰mcp Xc-Øn¬ ]d-bmw. H∂ m-asØ hc-tb-°mƒ Ncnhv IqSpX-em-Wv. c≠m-asØ hc-bv°v. Cß s\ hcbv°v Hcp Ncnhv D≠v F∂ v Xncn-®-dn-hn-eqsS Af∂ v Fgp-Xp∂ coXn ]mT-]p-kvX-I-Øn-se {]h¿Ø-\-Øn-eqsS Ip´n-bn¬ FØn-°mw. h¿°v jo‰v (2,5), (5,11) F∂ o _nµp-°-fn-eqsS IS-∂ p-t]m-Ip∂ hc-bpsS Ncnhv F{X-bmWv? (3,9) F∂ _nµp-hn-eqsS IS-∂ p-t]m-Ip∂ hc A£-Ønse (˛4,0) F∂ _nµp-hn-eqsS IS∂ p-t]m-Ip-∂ p. hc-bpsS Ncnhv F{X-bm-Wv. (1,7) (3,5) F∂ o _nµp-°-en¬ IqSn IS-∂ p-t]m-hp∂ hc-bpsS Ncnhv F¥ m-Wv. Cu hc y A£sØ JWvUn-°p∂ _nµp-hns‚ kqN-I-kwJy F¥ v?
MUKULAM MATHS ................................................................................................................................................................................143
4. 5. 6. 7. 8. 9.
10.
1. 2. 3. 4. 5. 6. 7. 8.
(2,8), (4,6) F∂ o _nµp-°-fn-eqsS IS-∂ p-t]m-Ip∂ hc (7,3) F∂ _nµp-hn-eqsS IS-∂ p-t]mIp∂ p F∂ v sXfn-bn-°p-I. (3,7), (4,9), (5,11) F∂ o _nµp-°ƒ Hcp tcJ-bnse _nµp-°-fm-Wv F∂ v sXfn-bn-°p-I. A (1,3), B (2,6), C(4,8) F∂ o _nµp-°ƒ tbmPn-∏n®v Hcp {XntImWw hc-bv°p-hm≥ km[y-a√ F∂ v sXfn-bn-°p-I. (2,7), (5,13), (3,12) F∂ o _nµp-°ƒ tbmPn-∏n®v Hcp {XntImWw \n¿Ωn-°m≥ km[n°pw F∂ v ka¿∞ n-°p-I. (2,0) F∂ _nµp-hn-eqsS, Ncnhv 1/4 Bbn hcp∂ hc-bnse a‰v c≠v _nµp-°ƒ Fgp-Xp-I. (2,4), (5,10) F∂ o _nµp-°sf tbmPn-∏n-°p∂ hcbpw (1,3) (4,12) F∂ o _nµp-°sf tbmPn∏n-°p∂ hcbpw x A£-hp-ambn D≠m-°p∂ tImWp-Iƒ hyXy-kvX-amWv F∂ v sXfn-bn-°pI. (1,5), (3,15) F∂ o _nµp-°sf tbmPn-∏n-°p∂ hcbpw (2,3), (3,12) F∂ o _nµp-°sf tbmPn∏n-°p∂ hcbpw kam-¥ -c-ß -f√ F∂ v sXfn-bn-°p-I.
hc-bpsS ka-hmIyw Hcp hc-bnse _nµp-°-fpsS x,y kqNI kwJy-Iƒ XΩn¬ ]c-kv]cw Fß s\ _‘-s∏-´n-cn°p-∂ p. F∂ m-Wv Hcp hc-bpsS ka-hm-Iy-Øn-eqsS ImWn-°p-∂ -Xv. DZm-l-c-W-ambn Hcp hcbnse _nµp-°-fp-sS-sb√mw y kqN-I-kw-Jy-Iƒ x kqN-I-kw-Jy-I-fpsS c≠v aS-ß m-sW-¶n¬, Cu hc-bpsS ka-hmIyw y= 2xF∂ v ]d-bp-∂ p. adn®v Hcp hc-bpsS ka-hmIyw y= 3x+1 F¶n¬, Cu hc-bnse _nµp-°-fpsS x kqN-I-kw-Jy-bpsS aq∂ v aS-ß n-t\mSv 1 Iq´n-bX - mWv y kqN-Ikw-Jy. ]mT-]p-kvI-Ønse {]¿Ø\w sNøp-∂ -Xn-eqsS Cu c≠v hkvXp-X-Ifpw Ip´n-bn¬ Dd∏n-t°-≠-Xm-Wv. IqSmsX Hcp hc-bnse _nµp-°-fpsS kwJym-tPm-Un-I-fpsS Iq´-hpw, hcsb kqNn-∏n-°p∂ ka-hmIyw A\p-k-cn-°p∂ kwJym-tPmUn Iq´hpw H∂ p Xs∂ -bm-sW∂ pw IqSn Xncn-®-dn-bW - w. h¿°v jo‰v 2x+3y - 10 =0, ka-hm-Iy-amb Hcp hc-bnse GsX-¶nepw aq∂ v _nµp-°ƒ I≠p-]n-Sn-°p-I. x-3y + 6 = 0 F∂ hc-bnse Hcp _nµp-hmtWm (2,4) 3x -2y =6 F∂ hc-bpsS Ncnhv F¥ mWv 2x + 5y - 4 =10 F∂ hc x A£sØ JWvUn-°p∂ _nµp GXv? y= 2x F∂ hc B[m-cm-_n-µp-hn-eqsS IS∂ p t]mIp∂ p F∂ v ka¿∞ n-°p-I. 2x + 3y = 10 F∂ hcbpw 3x - 2y = 12 F∂ hcbpw ]c-kv]cw ew_-ß -fm-sW∂ v sXfn-bn-°q. 3x -4y +5 =0 F∂ hc-bpw, 3x - 4y = 10 F∂ hcbpw kam-¥ -c-ß -fm-sW∂ v sXfn-bn-°p-I. 2x + 3y = 6 Dw 4x - 3y = 10 F∂ hcbpw Iq´n-ap-´p∂ _nµp GXm-Wv.
144................................................................................................................................................................................. MUKULAM M ATHS
UNIT TEST 1. 2. 3. 4.
(6,8) F∂ _nµp tI{µ-amb hrØw x A£sØ Hcp _nµp-hn¬ am{Xw kv]¿in-°p-∂ p. (8,1) F∂ _nµp Cu hrØ-Øn\v ]pd-ØmtWm? AI-ØmtWm? (3,1), (1,3) F∂ o _nµp-°ƒ Hcp ka-N-Xp-c-Øns‚ ASp-Ø-SpØ c≠v io¿j-ß -fm-bm¬ ]c∏-fhv F¥ v? Hcp kmam-¥ -cn-I-Øns‚ io¿j- -ß ƒ (1,3), (3,1) (4,4) (2,6) Ch-bm-bm¬ Np‰-fhv ImWp-I. A
(3,4) tI{µ-ambn hc® hrØw B[m-c_n- µ p- h n- e qsS IS- ∂ p- t ]m- I p- ∂ p. Cu hrØw kqN-Imw-£-ß sf JWvUn-°p∂ a‰v _nµp-°ƒ ImWp-I. A,B,C Ch Hcp t\¿h-c-bnemWv F∂ v sXfn-bn-°p-I. c
C (3,4)
O
B
kqN\: 1. A¿≤-hr-Ø-Ønse tIm¨ a´-tIm¨ F∂ Bibw D]-tbm-Kn-®v. 2. Zqc-ß ƒ D]-tbm-Kn®v 3. Ncnhv D]-tbm-Kn®v 5. 6.
Cu tcJ-Iƒ°v s]mXp-hmb _nµp ImWp-I. 3X -4Y + 6=0, 4x + 3Y -17 = 0 F∂ o tcJ-I-fpsS s]mXp-hmb _nµp-Im-Wp-I. Ch ]c-kv]cw ew_am-sW∂ v sXfn-bn-°p-I.
5x + Y - 13 = 0, 3X -4Y +6=0
MUKULAM MATHS ................................................................................................................................................................................145
A[ymbw 11
ÿ n-Xn-hn-h-c-°-W°v BapJw hnh-c-ti-J-c-WØns‚ `mK-ambn e`n-°p∂ kwJy-Isf h¿§o-I-cn-°mw. Ahsb hn`m-K-ß fpw Bhr-Øn-I-fp-ambn ]´n-I-s∏-Sp-Øn e`n® hnh-c-ß sf hni-I-e\w \S-Øp∂ {]h¿Ø-\-ß -fm-Wv ÿ nXn-hn-h-c-°-W°v F∂ A[ym-b-Øn¬ {]Xn-]m-Zn-°p-∂ -Xv. Hcp {]tZ-i-sØ Bfp-I-fpsS {]mbw, Btcm-Ky-ÿ n-Xn, kmº-ØnI \ne-hm-cw, Ip´n-I-fpsS ]T-\-\n-e-hm-cw, a‰v ÿ nXn-hn-h-c-°-W-°pIƒ F∂ nh hni-I-e\w sNøm≥ am[yw, a[y-aw, alnXw apX-emb kwJy-Iƒ D]-tbm-Kn-°msa∂ v HºXmw ¢mkn¬ ]cn-N-b-s∏-´n-´p-≠-t√m. kml-N-cy-ß ƒ A\p-k-cn®v `uXnI {]iv\-ß ƒ hni-I-e\w sNøm≥ CØcw icm-i-cn-Iƒ \ap°v KpW-{]-Z-am-Wv BhrØn ]´n-I-Ifn¬ \n∂ pw am[yw, a[yaw F∂ nh IW-°m-°p∂ coXn-bmWv Cu A[ym-b-Øn¬ kqNn-∏n-°p-∂ -Xv.
{][m\ Bi-b-ß ƒ = = =
Bhr-Øn-∏-´n-I-bn¬ \n∂ v am[yw Is≠-Øp-∂ -Xn\v k©n-Xm-hrØn Bhr-Øn-∏-´n-I-bn¬ \n∂ v a[yaw Is≠-Øp-∂ -Xn\v
Hcp Iq´w kwJy-I-fpsS am[yw ImWm≥ Ah-bpsS XpIsb FÆ w sIm≠v ]cn-l-cn-®m¬ aXnb-s√m. Poh-Im-cpWy {]h¿Ø-\-Øn\v kvIqfnse ]Øv Unhn-j-\p-I-fn¬ \n∂ v Hcp Znhkw kzcq]n® XpI NphsS sImSp-Øn-cn-°p-∂ p. XpI-bpsS am[yw ImWp-∂ -sX-ß ns\? 170, 170, 150, 200, 170, 180, 160, 180, 170, 200 Hmtcm Unhn-j-\n¬ \n∂ pw ]ncn-s®-SpØ BsI XpI = 150+ 16-+ (4 x 170) + (2 x 180) + (2 x 200) =1750 Ip´n-I-fn¬ \n∂ v ]ncn-s®-SpØ icm-icn XpI. = 1750 10 am[yw 175 cq].
{]h¿Ø\w 1 ¢mknse Ip´n-Isf A©v {Kq∏p-I-fmbn Xncn-®p-sIm≠v {]h¿Ø\w \¬Im-hp-∂ -Xm-Wv. group I se Ip´n-Iƒ°v Ah-cpsS ]q¿Øn-bmb hb- ns‚ am[yw ImW¬ group II se Ip´n-I-fpsS `mcw (G-I-tZ-iw) Is≠-Øn, `mc-Øns‚ am[yw ImWp-I. group III se Ip´n-Iƒ°v GsX-¶nepw Hcp bqWn‰v sSÃn\v e`n® am¿°ns‚ am[yw ImW¬. group IV se Ip´n-I-fpsS Db-c-Øns‚ am[yw (Ip´n-I-fpsS FÆ w hyXy-kvX-am-bn-cn-°pw) 146................................................................................................................................................................................. MUKULAM M ATHS
{]h¿Ø\w 2 {]tZ-i-Øn¬ Hcp amkw e`n® ag-bpsS Af-hp-Iƒ ]´n-I-bmbn sImSp-°p-∂ p. Hcp Znhkw e`n® ag-bpsS icm-icn I≠v t\m°q. ag-bpsS Afhv ao.ao
Znh-k-ß -fpsS FÆ w
40 43 46 49 51 54 57 ag-bpsS am[y Afhv =
2 1 5 8 7 4 3 (40x 2) + (43 x1) (46 x 5) + (49 x8) + (51 x7) + (54 x4) + (57 x 3) 30
= ............... 30 = ............... 30
{]h¿Ø\w 3 Hcp kvIqfnse 40 Ip´n-I-fpsS Dbcw Af∂ v IW-°m-°nb ]´nI NphsS sImSp-°p-∂ p. Db-c-Øns‚ am[yw F{X? Dbcw sk.an 110-˛114 114-˛118 118-˛122 122-˛126 126-˛130 130-˛134 134-˛138 =
Ip´n-I-fpsS FÆ w 4 7 10 11 5 2 1
110-˛114 F∂ hn`m-K-Øn-ep≈ Hmtcm Ip´n-bp-sSbpw Dbcw F{X-bmbn ]cn-K-Wn-°mw.
MUKULAM MATHS ................................................................................................................................................................................147
kqN\ : hn`m-K-ß -tfm-Sp-Iq-Snb BhrØn ]´n-I-bn¬ Hmtcm hn`m-K-sØbpw {]Xn\n-[o-I-cn-°p-∂ n\v B hn`m-K-Øns‚ Xmgv∂ ]cn-[n-bp-sSbpw Db¿∂ ]cn-[n-bpsSbpw am[yw IW-°m-°n-bm¬ aXn = =
110-˛114 F∂ hn`m-K-Øn-ep≈ Ip´n-I-fpsS BsI Db-c-sa{X? Hmtcm hn`m-K-Øn-sebpw BsI Dbcw ImWp-∂ -sX-ß s\? kqN\ : hn`m-K-Øns‚ am[yw I≠v Bhr-Øn-sIm≠v KpWn-°mw.
= = =
Ah-cpsS Db-c-ß -fpsS XpI-sb{X? BsI Ip´n-I-fpsS FÆ -sa{X? F√m Ip´n-I-fp-sSbpw Db-c-ß -fpsS am[y-sa¥ v?
IqSp-X¬ {]h¿Ø-\-ß ƒ 1.
hnhn[ {]tZ-i-ß -fnse 40 I¿jI sXmgn-em-fn-Iƒ°v e`n® Znh-k-°q-en-bpsS hnhcw ]´nI-bmbn sImSp-Øn-cn-°p-∂ p. Znh-k-°q-en-bpsS am[yw ImWpI Iqen (cq-]) 200- ˛ 250 250- ˛ 300 300- ˛ 350 350- ˛ 400 400- ˛ 450 450- ˛ 500 500- ˛ 550
2.
BfpIfpsS FÆ w 5 7 10 8 6 3 1
Hcp s]´n- -bnse 100 kv{Iqhns‚ \ofw ]´n-Im-cq-]-Øn¬ Xmsg sImSp-Øn-cn-°p-∂ p. \of-Øns‚ am[yw ImWp-I.
\ofw (an.-an) 33- ˛35 36- ˛38 3941 43- ˛44 45-˛47
FÆ w 15 19 23 27 16
148................................................................................................................................................................................. MUKULAM M ATHS
3.
c≠v SmIvkn ss{Uh¿am¿°v GXm\pw Znhkw In´nb Im¿hm-S-Isb Ipdn-°p∂ ]´nI NphsS sImSp-°p-∂ p. B¿°mWv icm-icn IqSp-X¬ hcp-am\w In´n-bs - X∂ v Is≠-Øp-I.
hcp-am\w cq]
Znh-k-ß -fpsS FÆ w
2000-˛ 2100 2100-˛ 2200 2200- ˛ 2300 2300- ˛ 2400 2400- ˛ 2500
3 5 8 7 2
hcp-am\w cq] 1900-˛ 2000 2000-˛ 2100 2100-˛ 2200 2200- ˛ 2300 2300- ˛ 2400 2400- ˛ 2500
25 4.
6 4 5 8 4 3 30
Hcp ¢mkv ]co-£-bn¬ e`n® am¿°ns‚ ]´n-I -sImSp-Øn-cn-°p-∂ p. am¿°ns‚ am[yw 27 BWv. K bpsS hne-Im-Wp-I. am¿°v 0˛10 10-˛20 20- ˛30 30- ˛40 40- ˛50
5.
Znh-k-ß -fpsS FÆ w
Ip´n-I-fpsS FÆ w 2 3 7 5 k
Hcm-ip-]-{Xn-bn¬ Hcp Znhkw {]th-in-∏n® tcmKn-I-fpsS {]mbw ]´n-Im-cq-]-Øn¬ sImSp-Øncn-°p-∂ p. {]mb-Øns‚ am[yw ImWp-I. {]mbw 0˛10 10-˛20 20- ˛30 30- ˛40 40- ˛50 50- ˛60 60-˛70
tcmKn-I-fpsS FÆ w 7 3 2 5 2 8 10
MUKULAM MATHS ................................................................................................................................................................................149
6.
Hcp hm¿Unse 100 IpSpw-_-ß ƒ ssZ\w-Zn\ sNe-hn-te°v {]Xn-amkw apS-°p∂ XpI-bpsS ]´nI NphsS sImSp-Øn-cn-°p-∂ p. sNehv cq]
IpSpw-_-ß -fpsS FÆ w
1000-˛ 2000 2000- ˛ 3000 3000- ˛ 4000 4000- ˛ 5000 5000- ˛ 6000 6000-˛ 7000
40 30 10 8 7 5
am[yw ImWpI? am[yw Is≠-Ønb Bƒ ]d-™p. hm¿Unse 40 iX-am\w IpSpw-_-ß ƒ icm-icn sNe-hn-s\-°mƒ Ipd-hm-Wv, {]Xn-amkw sNe-h-gn-°p-∂ -Xv. CXns\ Ipdn®v \nß fpsS A`n-{]m-b-a¥ v?
Bhr-Øn-∏-´n-Ibpw a[y-ahpw Hcp Iq´w Af-hp-Isf Btcm-lW {Ia-Øn¬ Fgp-Xn-bm¬ a[y-Øn¬ hcp-∂ -XmWv a[yaw F∂ Xv HºXmw Xc-Øn¬ ]Tn-®n-´p-≠-t√m. DZm-l-cWw: Hcp tem´dn hnev]-\-°m-c\v Hcm-gvN-bnse Hmtcm Znh-khpw e`n® hcp-am\w (cq]-bn¬) NphsS sImSp-°p-∂ p. hcp-am-\-Øns‚ a[yaw IW-°m-°p-I. 75, 135, 90, 210, 300, 180, 240 Btcm-lW {Ia-Øn¬ Fgp-Xp-tºmƒ 75, 90, 135, 180, 210, 240, 300 a[y-Øn¬ e`n-°p-∂ Xv 180. Bb-Xp-sIm≠v a[yaw 180 cq]-bm-Wv. hnh-c-ß ƒ Bhr-Øn-∏-´n-bpsS cq]-Øn-em-sW-¶nepw a[yaw F∂ Xv a[y-Øn¬ hcp∂ Afhm-bn-cn-°-W-a-t√m. Hcp ¢mknse 35 Ip´n-I-fpsS `mchpw FÆ hpw ]´n-I-bn¬ NphsS sImSp-°p-∂ p. `mcw 32 34 37 38 40 43
(kg)
Ip´n-I-fpsS FÆ w 2 6 10 11 4 2
150................................................................................................................................................................................. MUKULAM M ATHS
`mc-Øns‚ a[yaw IW-°m-°p-I. 35 Ip´n-I-fpsS `mcw Btcm-lW {Ia-Øn¬ Fgp-Xp-I-bm-sW-¶n¬ 18˛masØXmbn-cn-°p-at√m a[y-aw. 32 kg `mc-ap≈ Ip´n-I-fpsS FÆ w -= 2 34 kg hsc `mc-ap-≈-h-cpsS FÆ w =2+6= 8 37 kg hsc `mc-ap-≈-h-cpsS FÆ w =2+6+10= 18 \ 18˛masØ Ip´n-bpsS `mcw 37 kg. Bb-Xn-\m¬ am[yaw 37 kg. sNbvX {]h¿Ø-\sØ ]´n-I-bmbn sIm-Sp-°p-I-bm-sW-¶n¬. `mcw 32 34 37 38 40 43
(kg)
hsc hsc hsc hsc hsc hsc
Ip´n-I-fpsS FÆ w 2 8 18 29 33 35
18˛maXv 37 kg. \ am[yaw 37 kg.
an¬a ]m¬ IS-bn¬ 2011 Pqembv amk-Øn¬ sNe-hmb ]mens‚ hnhcw ]´n-I-bmbn NphsS sImSp-Øn-cn-°p-∂ p. a[yaw ImWp-I. ]men-s‚ Aehv (en-‰-dn¬) 38 41 45 47 52 54
Znh-k-ß -fpsS FÆ w 3 5 8 9 4 2
CØcw ]´n-I-I-fn¬ \n∂ pw a[yaw IW-°m-°n-bt- ijw hn`m-K-ß fpw Ah-bntemtcm∂ nsebpw BhrØnIfp-ambn Nn´-s∏-Sp-Ønb ]´n-I-bn¬ \n∂ pw a[yaw IW-°m-°p∂ {]iv\ß -fn-te°v IS-°mw. hn`m-K-ß -tfmSv IqSnb Bhr-Øn-∏-´n-bI - -bn¬ Hmtcm hn`m-K-Øn-sebpw Db¿∂ ]cn[n hsc-bp≈ Bhr-Øn-Iƒ Iq´n-sb-gp-Xn-bm¬ In´p-∂ -Xn-s\-bmWv k©n-XmhrØn F∂ v hnfn°p∂ Xv.
MUKULAM MATHS ................................................................................................................................................................................151
Hcp ÿ m]-\-Øn¬ tPmen sNøp∂ Poh-\-°m-cpsS hb- ns\ kw_-‘n-°p∂ ]´nI NphsS sImSp-°p-∂ p. tPmen-°m-cpsS FÆ w
hb v 25- ˛30 30- ˛35 35- ˛40 40- ˛45 45- ˛50 50- ˛55
1 4 9 8 5 3 BsI
30
tPmen-°m-cpsS FÆ w
hb v 30¬ 35¬ 40¬ 45¬ 50¬ 55¬
k©n-Xm-hrØn ]´nI FgpXn a[yaw ImWp-I.
Ipdhv Ipdhv Ipdhv Ipdhv Ipdhv Ipdhv
1 5 14 22 27 30
Hmtcm L´-Ønse hn`m-K-Ønse kwJy-I-fpsS am‰hpw k©n-Xm-hr-Øn-I-fpsS am‰hpw B\p]m-XnIamsW∂ k¶¬∏-Øns‚ ASn-ÿ m-\-Øn-emWv a[yaw IW-°m-°p-∂ -Xv. k©n-Xm-hrØn BsI Bhr-Øn-bpsS t\¿ ]IpXn BIp-tºm-gp≈ kwJy-bmWv (ap-Ifnse ]´n-I-bn¬ hb- v) a[y-a-ambn FSp-t°-≠-Xm-Wv. apI-fn¬ sImSpØ k©n-Xm-hrØn ]´n-I-bn¬ hb- ns‚ \nc-°nse kwJy-Isf x hne-Ifmbpw k©n-Xm-hr-Øn-Isf y hne-I-fmbpw Fgp-Xn-bm¬ F¥ v In´pw. x
30
35
40
45
50
55
y
1
5
14
22
27
30
152................................................................................................................................................................................. MUKULAM M ATHS
x hne-I-fpsS
am‰hpw y hne-I-fpsS am‰hpw B\p-]m-Xn-I-am-sW∂ v k¶-ev]n-°p-∂ p. CXv D]tbm-Kn®v x Bbn FSpØ kwJy-I-fpsS CS-bn-ep≈ Hcp kwJybpambn _‘-s∏´ y kwJy GsX∂ v Is≠-Ømw. DZm-l-c-W-ambn apI-fn¬ sImSpØ ]´n-I-bn¬ x= 32 F∂ -Xns‚ y I≠p-]n-Sn-°m≥ B\p-]m-XnI k¶¬]w D]-tbm-Kn-°pw.
ie
y-1 = 32 - 30
5 -1 35 -30
y-1 2
=
4 5
y-1
=
4x2 5 8+1 5 13 5 2.6
y
= = =
adn®v y Hcp \n›nX kwJy BIm≥ x F¥ m-bn-cn-°p-sa∂ v Is≠-Øm\pw Cu am¿§w Xs∂ D]-tbm-Kn-°mw. DZm-l-cWw
\
y=8
Bbm¬ x F¥ m-bn-cn°pw? y = 8 F∂ Xv 5\pw 14\pw CS-bn-em-Wv. x s‚ hne 35\pw 40\pw CS-bv°p-am-bn-cn-°p-a-t√m.
:
x- 35 = 8-5 x- 35 = 3 x- 35 = = x
= =
y = 30 2
40 -35 14 - 5 5 9 5x3 9 5 3 5 + 35 3 110 = 36.6 3
= 15 BIm-\p≈ x hne-bmWv a[yaw \
x- 40 = 15 -14 x- 40 =
45 -40 22 - 14 5 8 x = 5 + 40 8 = 40.63 a[yaw = 40.63
MUKULAM MATHS ................................................................................................................................................................................153
]cn-io-e\ {]iv\-ß ƒ 1.
Hcp sXmgn¬ ime-bnse sXmgn-em-fn-I-fpsS Znh-k-th-X-\sØ ASn-ÿ m-\-am-°n-bp≈ ]´nI NphsS sImSp-°p-∂ p. a[yaw ImWp-I. Znh-k-th-X\w 150-˛200 200- ˛ 250 250- ˛ 300 300- ˛ 350 350- ˛ 400 400- ˛ 450 (cq-]-bn¬ sXmgn-em-fn-I2 6 9 9 5 3 fpsS FÆ w
2.
Hcp kvIqfnse Ip´n-Iƒ°v IW°v ]co-£bv°v (TE) e`n® kvtImdp-I-fpsS ]´n-I-bmWv sImSp-Øn-cn-°p-∂ -Xv. Ip´n-I-fpsS FÆ w kvtIm¿ 0˛10 4 10-˛20 6 20- ˛30 12 30- ˛40 18 40- ˛50 33 50- ˛60 13 60-˛70 9 70-˛80 5
3.
kvtImdp-I-fpsS a[yaw ImWp-I. kvIu´v Iymºn¬ ]s¶-SpØ 200 Ip´n-I-fpsS Db-c-ß ƒ ]´n-I-bmbn NphsS sImSp-°p-∂ p. Dbcw (sk. -an) 130-˛135 135-˛140 140-˛145 145-˛150 150-˛155 155-˛160
Ip´n-I-fpsS Dbcw 20 16 36 56 44 26
k©n-Xm-hr-Øn-∏-´nI Xøm-dm°n a[yaw ImWp-I. 4.
Hcp hyh-kmb ime-bn¬ tPmen sNøp∂ kv{XoI-fp-sSbpw ]pcp-j-∑m-cp-sSbpw {]mbw ]´nIbmbn sImSp-Øn-cn-°p-∂ p. hb v
a) b) c)
kv{XoI-fpsS FÆ w
]pcp-j-∑m-cpsS FÆ w
20- ˛25 25- ˛30 30- ˛35 35- ˛40 40- ˛45 45- ˛50 50- ˛55
12 17 21 26 16 12 6
8 13 22 21 15 12 9
BsI
110
100
kv{XoI-fpsS hb- ns‚ a[yaw IW-°m-°p-I. ]pcp-j-∑m-cpsS hb- ns‚ a[yaw IW-°m-°p-I. BsI tPmen-°m-cpsS hb- ns‚ a[yaw ImWp-I.
154................................................................................................................................................................................. MUKULAM M ATHS
Marks 20 Time 1 Hr.
UNIT TEST
1. 2. 3.
kvIqfnse 1900 Ip´n-I-fpsS kºmZy \nt£-]-Øns‚ am[yw 1725 cq]-bm-sW-¶n¬ B kvIqfnse kºmZy \nt£]w F{X? 1 Hcp {In°‰v a’ -c-Øn¬ BZysØ 10 Hmh-dn¬ C¥ y t\Snb dÆ p-I-fmWv NphsS sImSp-Øncn-°p-∂ -Xv. 4, 8, 7, 3, 6, 12, 10, 9, 5, 2 dÆ p-I-fpsS a[yaw ImWp-I. 1 25 Ip´n-I-fpsS `mcw (kg) kqNn-∏n-°p∂ ]´n-I-bmWv NphsS sImSp-Øn-cn-°p-∂ -Xv. Ip´n-I-fpsS `mc-Øns‚ am[yw ImWp-I. 3 `mcw (kg) 30- ˛35 35- ˛40 40- ˛45 45- ˛50 50- ˛55
4.
Hcp tImf-\n-bnse 50 IpSpw-_-ß -fpsS {]Xn-amk hcp-am-\sØ kqNn-∏n-°p∂ BhrØn ]´nI Xmsg sImSp-°p-∂ p. hcp-am\w cq]w 1000- ˛ 2000 2000 -˛ 3000 3000- ˛ 4000 4000- ˛ 5000 5000- ˛ 6000 6000- ˛ 7000 7000 -˛ 8000 8000 -˛ 9000
a) b) c)
5.
Ip´n-I-fpsS FÆ w 2 7 10 5 1
IpSpw-_-ß -fpsS FÆ w 1 x-1
6 x+5 x+7
8 x+1
3
x s‚ hne- I-W-°m-°p-I. 4 Hmtcm ¢mkn-s‚bpw BhrØn ImWp-I. IpSpw-_-ß -fpsS hcp-am-\-Øns‚ am[yw ImWp-I. Hcp tlmÃ-en-te°v 2011 BKkvXv amkØn¬ ]®-°-dn°v sNe-hmb XpIbpsS ]´nI NphsS sImSp-°p-∂ p. 3
sNehv cq] 300 320 350 430 470 510
Znh-k-ß -fpsS FÆ w 3 12 5 6 3 2
a[yaw ImWp-I.
MUKULAM MATHS ................................................................................................................................................................................155
6.
Hcp kok-Wn¬ Iip-hm-an≥ tXm´-Øn¬ \n∂ v e`n® Iip-h-≠n-bpsS Xq°w kqNn-∏n-°p∂ ]´nI NphsS sImSp-°p-∂ p. Xq°w (In-tem) 10-˛15 15-˛20 20- ˛25 25- ˛30 30- ˛35 35- ˛40 40- ˛45
Znh-k-ß -fpsS FÆ w 2 5 13 6 11 12 9
a) b) c)
BsI F{X Znhkw Iip-h≠n tiJ-cn-®p. 25 Intem-hn¬ Ipdhv Iip-h≠n e`n-®Xv F{X -Zn-h-k-am-Wv. a[yaw ImWp-I.
7.
Hcp D’ -h-Im-eØv c≠v hyXykvX IpSpw-_{io bqWn‰v Ãmfp-I-fnse hn‰v hc-hns‚ ]´nI NphsS sImSp-Øn-cn-°p-∂ p.
4
Ãmƒ 2
hn‰v hchv (cq-])
Ãmƒ 1
5000- ˛ 7000 7000- ˛ 9000 9000- ˛ 11000 11000- ˛ 13000 13000 -˛ 15000 15000 -˛ 17000
1 5 7 10 4 3
2 4 6 11 5 2
30
30
Hmtcm Ãmfn-sebpw hn‰v hc-hns‚ am[yw I≠v GXv Ãmfn-emWv IqSp-X¬ hn¬]\ \S-∂ Xv F∂ v IW-°m-°p-I.
4
156................................................................................................................................................................................. MUKULAM M ATHS
