apIpfw Fkv.-F-kv.-F¬.-kn. ]T\klmbn 2011 ˛ 2- 012

MATHEMATICS

Pn√m]©mbØv ˛IÆ q¿

MUKULAM MATHS................................................................................................................................. 1

Chairman

: Prof.K.A.Sarala President District Panchayath, Kannur

Vice-Chairman

: Sri. T. Krishnan Vice-President, District Panchayath, Kannur Smt. P.Rosa Chairman Health- Education Standing Committee Kannur District Panchayath

Convenor

: Sri. A. Arunkumar DDE., Kannur

Academic Convenor

: Sri. K. Prabhakaran Principal, DIET, Kannur

Editorial Board: 1.

A.V.Ajayakumar EMS HSS, Pappinisseri

2.

N.K.Remeshan RGMHS, Mokeri

3.

M.V.Unnikrishnan Trichambaram

4.

K.P.Damodaran HM, Naduvil HS

5.

C.Mohanan Govt.Welfare HSS, Cherukunnu

6.

C.Suresh Babu GHSS, Chala

7.

K.P.V.Satheesh Kumar GHSS, Morazha

8.

K.M.Suresh Babu RVHSS, Chokli

9.

T.Narayanan PRM HSS, Kolavallur

10. T.Sukumaran GBHS, Madayi 11. C.Raghu SABTM HSS, Thayineri 12. Krishna Prabha Azhikode HS 13. Prabhakaran GHSS, Mathamangalam 14. K.Sudheer GHSS, Chundangappoyyil 15. Pradeepan Iritty HSS 16. A.Sindhu Azhikode HS

2

...............................................................................................................................MUKULAM MATHS

s{]m^: sI.-F.-k-cf {]kn-U-≠v, IÆ q¿ Pn√m ]©m-bØv, IÆ q¿

BapJw F√m-hcpw ]Tn-°-Ww, F√m-hcpw Pbn-°-Ww. CXm-Wt√m C∂ v F√m hnZ-ym-e-b-ß -fp-sSbpw {]h¿Ø-\-e-£-yw, Cu e£yw km£m-Xv°-cn-°p-∂ -Xn\v ]c-ym-]vX-amb coXn-bn¬ \ΩpsS \m´nse Hmtcm hnZ-ym-e-b-sØbpw kzbw kºq¿Æ -am-t°-≠-Xp-≠v. G‰hpw B[p-\n-I-amb `uXnI kuIc-y-ß ƒ \ho-\hpw Imtem-Nn-X-hp-amb ]mT-y-]-≤-Xn, NSp-ehpw {Inbm-fl-I-hp-amb aqe-y-\n¿Æ b coXn, hnZ-ym-e-b-ß sf i‡n-s∏-Sp-Øm≥ DX-Ip∂ A[-ym-]-I-˛-c-£-I¿Xr Iq´m-bva-Iƒ, imkv{Xobhpw ka-b-_-‘n-X-hp-ambn \S-°p∂ A[-ym-]I ]cn-io-e\ ]cn-]m-Sn-Iƒ F∂ nh hnZ-ym-`-ym-kØns‚ KpW-ta-∑sb kzm-[o-\n-°p∂ LS-I-ß -fn¬ {][m-\-am-Wv. Ign™ Iptd h¿j-ß -fmbn tIc-f-Øn¬ kvIqƒ hnZ-ym-`-ym-k-Øns‚ \ho-I-cWw e£yw sh®v \S-∏n-em°n hcp∂ \nc-h[n {]h¿Ø-\-ß ƒ IqSp-X¬ {i≤ sNep-Øp-∂ Xv CØcw Imc-y-ß -fn¬ Xs∂ -bm-Wv. km¿∆-{XnIhpw KpW-ta-∑-bp-≈-Xp-amb hnZ-ym-`-ymkw \ΩpsS kwÿ m-\Øv Dd-∏m-°p-∂ -X n¬ tI{µ kwÿ m\ Kh¨sa‚p-Iƒ°v ]pdsa Xt±i kz-bw-`-c-W-ÿ m-]-\-ß fpw hf-sc-b-[nIw ]¶p-hln®p hcp-∂ p. C°m-c-y-Øn¬ tIcfw anI® amXr-I-bm-Wv. tIc-f-Ønse Xt±i kz-bw-`-c-W-ÿ m-]-\-ß ƒ kvIqƒ hnZ-ym-`-ymkw sa®-s∏-Sp-Øp-hm≥ sshhn-[-y-am¿∂ \nc-h[n ]≤-Xn-Iƒ Bhn-jvI-cn®v \S-∏n-em°n hcp-∂ p. Ch-bn¬ G‰hpw {i≤n-°-s∏-´Xpw hnPbw I≠-Xp-amWv IÆ q¿ Pn√m ]©m-bØv \S-∏n-em°n hcp∂ apIpfw ]≤-Xn. Pn√-bnse sslkvIq-fp-I-fnepw lb¿sk-°‚dn kvIqfp-I-fnepw Ip´n-I-fpsS ]T\ \ne-hmcw Db¿Øm≥ klm-b-I-amb ssI]p-kvX-I-ß ƒ Xøm-dm°n hnX-cWw sNøp-I, sa®s∏´ ¢mkv dqw A\p-`-h-ß ƒ Ip´n-Iƒ°v ]I¿∂ v \¬Im≥ A[-ym-]-Isc im‡o-I-cn-°p-∂ Xn\v DX-Ip-∂ ]cn-io-e-\ ]cn-]m-Sn-Iƒ kwL-Sn-∏n-°p-I, imkv{Xo-bamb aqe-y-\n¿Wb coXn hnZ-yme-b-ß -fn¬ \S-∏n-em-°pI XpS-ß n-b-h-bmWv apIpfw ]≤-Xn-bpsS {][m\ {]h¿Ø-\-ß ƒ. apIpfw ]≤Xn \S-∏n-em-°p-∂ Xn¬ Bflm¿∞ -amb kl-I-c-Whpw ]n≥Xp-W-bp-amWv A[-ym-]-I-˛-c-£mI¿Xr kaq-l-ß -fn¬ \n∂ v IÆ q¿ Pn√m ]©m-b-Øn\v e`n-®p-sIm-≠n-cn-°p-∂ -Xv F∂ v Cu Ah-k-c-Øn¬ kt¥ m-j-]q¿∆w Adn-bn-°-s´. apIpfw ]≤-Xn-bpsS `mK-ambn F√m hnj-b-ß -fnepw Ip´n-bpsS At\-z-j-Wm-flI ]T-\sØ t{]m’ m-ln-∏n-°m≥ DX-Ip∂ {]h¿Ø-\-ß -f-S-ß nb ssI∏p-kvX-I-ß ƒ Hmtcm A°m-Z-anI h¿jØnepw Xøm-dm°n hnZ-ym-e-b-ß -fn¬ kuP-\-y-ambn hnX-cWw sNbvXp hcp-∂ p. s]mXp-]-co-£I-fn¬ Db¿∂ hnPbw t\Sm≥ Cu ssI∏p-kvX-I-ß ƒ Ip´n-Iƒ°v hf-sc-b-[nIw {]tbm-P-\s∏-Sp-∂ p-s≠-∂ mWv Pn√-bnse A°m-Z-anI kaq-l-Øns‚ hne-bn-cp-ج. {]K-¤-cmb A[-ym]-Isc Dƒs∏-SpØn kwL-Sn-∏n-°p∂ in¬]-im-e-I-fn-eq-sSbmWv Cu ssI∏p-kvX-I-ß ƒ Xømdm-°-s∏-Sp-∂ -Xv. ssI∏p-kvX-I-ß ƒ IqSp-X¬ IqSp-X¬ anI-hp-‰-Xm-°m≥ Rß ƒ Hmtcm h¿jhpw {i≤n-°p-∂ p. Ip´n-Iƒ°v anI® hnPbw t\Sm≥ apIpfw ]≤-Xn-bpsS `mK-ambn Cu h¿jw Xøm-dm-°nb hnhn[ hnj-b-ß -fnse ssI∏p-kvX-I-ß ƒ klm-b-I-am-Is´ F∂ v Biw-kn-®psIm≠v kvt\l-]q¿∆w s{]m^.-sI.-F.-k-cf IÆ q¿ Pn√m ]©m-bØv {]kn-U≠v MUKULAM MATHS................................................................................................................................. 3

]n.-tdmk sNb¿t]-gvk¨ Btcm-K-y-˛-hn-Z-ym-`-ymk Ãm‚nwKv IΩn‰n Pn√m ]©m-b-Øv, IÆ q¿

""-ap-Ip-fw'' ka{K hnZ-ym-`-ymk ]cn-]m-Sn-bn-eqsS IÆ q¿ Pn√ tIcf hnZ-ym`ymk Ncn-{X-Øn¬ B[n-Im-cn-I-amb hnP-bhpw ÿ m\hpw t\Sn-°-gn™p. \ΩpsS Ip´n-Ifpw A[-ym-]-Icpw c£n-Xm-°fpw hnZ-ym-`-ymk Hm^o-k¿amcpw P\-{]-Xn-\n-[n-Ifpw HsØm-cp-an-®-Xns‚ ^ew IqSn-bmWv Cu t\´w. ss{]adn apX¬ lb¿sk-°-≠dn Xew hsc KpW-\n-e-hm-c-ap≈ hnZ-ym-`-ym-kØn-\mbn {]h¿Ø\ ]≤-Xn-Iƒ "ap-Ip-fw' ]≤-Xn-bn-eqsS hn`m-h\w sNbvXn´p-≠v. 2011-˛12 h¿j-Øn¬ Ad-_n, DdpZp Dƒs∏sS F√m hnj-b-ß ƒ°pw {]tX-yI ]T\ kma-{Kn-Iƒ \n¿Ωn®p Ign-™p. Ip´n-I-fpsS Adn-hv, \n¿ΩmW-ti-jnsb {]tNm-Zn-∏n-°p∂ ]T\ X{¥ -ß ƒ°v {]map-Jyw \¬In-bn-´p-≠v. lb¿sk-°‚dn Xe-Øn¬ Cw•ojv, C°-tWm-anIvkv Dƒs∏sS ^nknIvkv sIankv{Sn, _tbm- f - P n, A°u- ≠ ≥kn F∂ o hnjb- ß ƒ°pw ]T\ kma{KnIƒ \n¿Ωn®p Ign-™p. ssZ\w-Zn\ Bkq-{X-W-Øn\pw ]T-\-{]h¿Ø-\-ß ƒ Is≠-Øp-∂ -Xn\pw A[-ym-]-Isc {]m]vX-cm-°p∂ anI-hp‰ ]T\ klmbn F∂ \ne-bn-emWv "ap-Ipfw' hn`m-h\w sNbvXn-´p-≈-Xv. apIp-fØnse F√m {]h¿Ø-\-ß fpw Ghcpw ^e-{]-Z-ambn {]tbm-P-\-s∏-Sp-Øpsa∂ v {]Xo-£n-°p-∂ p. ]T\ anI-hn-t\m-sSm∏w Db¿∂ kmaq-ly t_m[hpw {]h¿Ø\ k∂ -≤Xbpw hf¿Øp∂ hnZ-ym-`-ym-k-am-Is´ \ΩpsS e£-yw. "ap-Ip-fw' ka{K hnZ-ym`-ymk ]≤Xn AXn-\p≈ Nme-I-i-‡n-bmbn amd-s´. kvt\l-]q¿∆w

]n.-tdmk sNb¿t]-gvk¨ Btcm-K-y-˛-hn-Z-ym-`-ymk Ãm‚nwKv IΩn‰n Pn√m ]©m-b-Øv, IÆ q¿

4

...............................................................................................................................MUKULAM MATHS

A²ymbw þ1

kam´ct{iWn BapJw KWnX¯nsâ ASnØm\w Xs¶ kwJyIfmWv. kwJyIfpsS khntijXIfneq¶n bmWv KWnXimJIfmb _oPKWnXw, PymanXn, {XntImWanXn XpS§nbhbpsS XpSÀ¨bpw hfÀ¨bpw. ]TnXmhn\v Bib§Ä A\p`hthZyamIp¶ coXnbnepw bpànNn´ hfÀ¯p¶ coXnbnepw {]iv\§sf hniIe\w sN¿p¶ kao]\amWv kam´ct{iWn F¶ Bib¯n kzoIcnt¡­Xv. hyXykvX kµÀ`§fn kwJyIÄ X½nepÅ ]ckv]c _Ôs¯¡pdn ¨pÅ DÄ¡mgvN cq]oIcn¨v {]iv\ ]cnlmcw ImWpI F¶XmWv Cu A[ymb¯nsâ ]T\w sIm­v e£yanSp¶Xv. kam´c t{iWnbpsS _oPKWnXcq]w Is­¯p¶ {]{Inb D]cn]T \¯n\v hnhn[ t{iWnIfpsS {]tXyIXIfpw _oPKWnX cq]hpw I­p]nSn¡m³ Ip«n¡v klmbIcamIWw. {][m\ Bib§Ä *

t{iWn

*

kwJymt{iWn

*

kam´ct{iWn

*

s]mXphyXymkw

*

Hcp kam´c t{iWnbpsS BZy]Zhpw s]mXphyXymkhpw X¶m t{iWnbpsS cq]oIcWw

*

Hcp kam´ct{iWnbpsS _oPKWnXcq]w Is­¯p¶Xn\v (GXv kam´ct{iWntbbpw xn = an+b F¶ cq]¯n FgpXmw)

*

1 apX n hscbpÅ XpSÀ¨bmb F®Â kwJyIfpsS XpI n (n+1) BsW¶v Is­¯p¶Xn\v 2

*

Hcp kam´c t{iWnbpsS XpSÀ¨bmb ]Z§fpsS XpI ImWp¶Xn\v [ XpI = n (x1+xn) ] 2

MUKULAM MATHS................................................................................................................................. 5

{]hÀ¯\§Ä hnhn[Xcw t{iWnIÄ ]cnNbs¸Sp¶Xn\pw Xcw Xncn¡p¶Xn\pw ]mT`mKs¯ {]hÀ¯ \§Ä¡p ]pdsa NphsS sImSp¯ {]hÀ¯\§fpamImw. DZmlcWambn 1) Ie­À {]hÀ¯\w S

M

Tu

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Th

Fr

Sa

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CXn \n¶pw Ip«n¡v hyXykvX kwJym{Ia§Ä I­p]nSn¡m\pÅ {]hÀ¯\w \ÂImw. AXn\p klmbIamb Nne tNmZy§Ä NphsS \ÂInbncn¡p¶p. *

H¶mw hcnbnse kwJyIÄ t\m¡q. {]tXyIXsb´v ?

*

c­mw hcnbnse kwJyIÄ t\m¡q. {]tXyIXsb´v ?

*

Hmtcm hcnbntebpw kwJyIfpsS {]tXyIXsb´v ?

*

Hmtcm \ncbntebpw kwJyItfm ? {]tXyIXsb´v ?

*

tImtWmSp tIm¬ hcp¶ If§fnse kwJyIÄ FgpXpI. GXv {Ia¯nse¶p ]dbpI. Nn{Xw {i²n¡pI S

6

M

Tu

W

Th

Fr

Sa

1

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30

...............................................................................................................................MUKULAM MATHS

KpW\¸«nI 1 apX 10 hscbpÅ kwJyIfpsS KpW\¸«nI NphsS sImSp¯ncn¡p¶p Ahbn \n¶pw kwJym{Ia§Ä FSps¯gpXpI X

1

2

3

4

5

6

7

8

9

10

1

1

2

3

4

5

6

7

8

9

10

2

2

4

6

8

10

12

14

16

18

20

3

3

6

9

12

15

18

21

24

27

30

4

4

8

12

16

20

24

28

32

36

40

5

5

10

15

20

25

30

35

40

45

50

6

6

12

18

24

30

36

42

48

54

60

7

7

14

21

28

35

42

49

56

63

70

8

8

16

24

32

40

48

56

64

72

80

9

9

18

27

36

45

54

63

72

81

90

10

10

20

30

40

50

60

70

80

90

100

*

hcnIfnse kwJyIÄ F´v {Iaw ]men¡p¶p? \ncIfnse kwJyItfm ?

*

tImtWmSp tIm¬ hcp¶ kwJym{Ia§Ä FgpXpI. Hmtcm {Ia¯nsebpw {]tXyIXIÄ F´v ? hyXykvX Xcw t{iWnIÄ Xncn¨dnbm\pÅ IqSpX {]hÀ¯\§Ä \ÂIpatÃm.

XpSÀ¶v Hcp kwJybn \n¶p XpS§n Htc kwJy Xs¶ ho­pw ho­pw Iq«n¡n«p¶ t{iWnbmWv kam´ct{iWn F¶ Bib¯nse¯nt¨cmw. C§s\ Hmtcm Øm\s¯ kwJy tbmSpw Iq«p¶ Ønc kwJybmWv s]mXphyXymkw MUKULAM MATHS................................................................................................................................. 7

{]hÀ¯\§Ä *

BZy]Zw 6 Dw s]mXphyXymkw 10 Dw hcp¶ kam´ct{iWn FgpXpI

*

BZy]Zw þ6 Dw s]mXphyXymkw 10 Dw hcp¶ kam´c t{iWn FgpXpI

*

BZy]Zw 6 Dw s]mXphyXymkw þ10 Dw Bb kam´c t{iWn FgpXpI

*

BZy]Zw þ6 Dw s]mXphyXymkw þ10 Dw Bb kam´ct{iWn FgpXpI

*

10 Â \n¶p XpS§n 6 s]mXphyXymkambn hcp¶ kam´ct{iWn FgpXpI

*

6 sIm­v lcn¨m 1 injvSw hcp¶ kwJyIÄ FgpXpI. injvSw 5 hcp¶ kwJyIÄ FgpXpI. c­v t{iWnIfpw kam´ct{iWnbmtWm F¶v ]cntim[n¡pI

*

½ BZy]Zhpw þ2 s]mXphyXymkhpamb kam´ct{iWn FgpXpI

NphsS sImSp¯hbn kam´ct{iWn I­p]nSn¡pI *

111,

122,

133,

144

*

-111,

-122,

-133,

-144

*

1.11,

1.22,

1.33,

1.44

*

11.1,

12.2,

13.3,

14.4

*

1110,

1220,

1330,

1440

*

111 4

122 4

133 4

144 4

*

2,

5,

9,

14

20

*

1,

1,

2,

3,

5,

*

1.1,

11,

110,

110

*

2,

3,

5,

7

*

17,

2,

-13,

-28

*

-8,

-4,

-2,

-1

*

3 + 2,

5+ 2,

7 2,

9+ 2

*

1 4

2 5

3 6

4 7

8

NphsS sImSp¯hsbÃmw kam´ct{iWnIfmWv. Hmtcm¶nsebpw hn«`mKs¯ ]Z§Ä (kwJyIÄ) FgpXpI *

---

15,

---

35,

*

---

8,

---

0

8

---

...............................................................................................................................MUKULAM MATHS

*

---

8,

--- -8, ---

*

12,

---,

---,,

-12

*

---,

---,

18,

54,

*

13,

---

20,

---

*

13,

---,

---,

27

*

-4,

---,

12

---

*

2,

---,

x

*

x,

---

,y

*

x,

---

y,

---

*

x,

y,

---,

---

*

x,

---,

---,

y

---

kam´ct{iWnbpsS Øm\hyXymkhpw ]ZhyXymkhpw B\p]mXnIamsW¶v sXfnbn¡phm³ t{]mPIvSv, hÀ¡vjoäv X{´§Ä D]tbmKn¡mw. CXv c­v L«§fneqsS kaÀ°n ¡mw. L«w 1

þ

Hcp kam´ct{iWnbnse GXv c­v ]Z§fpsSbpw hyXymkw s]mXp hyXymk¯nsâ KpWnXambncn¡pw

L«w 2

þ

GXv c­v ]Z§fpsSbpw hyXymkw B ]Z§fpsS Øm\hyXymk¯ns\ s]mXphyXymkw sIm­v KpWn¨Xmbncn¡pw

ChnsS s]mXphyXymkambncn¡pw B\p]mXnIØncw F¶p ImWmw. GXm\pw DZml cW§fneqsS Cu Bib¯nse¯nbtijw XmXznIambn C¡mcyw kaÀ°nt¡­XmWv. f,

f + d, f + 2d, .......................+ f+(n-1)d ........................f + (m-1)d

Cu kam´ct{iWnbpsS mþmw ]Zhpw nþmw ]Zhpw X½nepÅ hyXymksaSp¯m (a+(m-1)d) _ (a+(n-1)d) = (m_n)d

AXmbXv ]Z§fpsS hyXymkw Øm\hyXymks¯ s]mXphyXymkw sIm­v KpWn¨ Xmbncn¡pw mþmw ]Zw þ nþmw ]Zw m-n

=

s]mXphyXymkw

{]hÀ¯\§Ä *

Hcp kam´ct{iWnbpsS 5þmw ]Zw 18 Dw 12þmw ]Zw 39 Dw Bbm 19þmw F{Xbmbncn¡pw

MUKULAM MATHS................................................................................................................................. 9

*

Hcp kam´ct{iWnbpsS 3þmw ]Zw 5 Dw 13þmw ]Zw þ25 Dw BsW¦n 33þmw ]Zw F{X?

*

Hcp kam´ct{iWnbpsS 6þmw ]Zw 2½ hpw 10þmw ]Zw 14½ Dw Bbm 12þmw ]Zw F{X?

*

Hcp kam´ct{iWnbnse 7þmw ]Zw 17 Dw 11þmw ]Zw 27 Dw Bbm aq¶mw ]Zw F{Xbm bncn¡pw ? H¶mw ]Zw F{Xbmbncn¡pw?

*

75, 69, 63...... F¶ kam´c t{iWnbnse Hcp ]ZamtWm 7 F¶v ]cntim[n¡pI. þ3 Cu t{iWnbnse Hcp ]ZamIptam?

NphsS sImSp¯ t{iWnIfpsS n þ mw ]Zw Is­¯pI. a)

2, 5, 8, -----

b)

5, 7, 8------

c)

1, 7, 13------

d)

9, 13, 17 ------

e)

20, 13, 16 ------

f)

30, 25, 20 -------------

g)

13, 10, 10 --------------

IqSmsX Ip«nIÄ¡njvSapÅ 5 kam´ct{iWnIÄ IqSn FgpXpI. ChbpsSsbÃmw n þ mw ]Zw, s]mXphyXymkw, BZy]Zw Ch Xmsg sImSp¯ ]«nIbn ]qcn¸n¨v ]ckv]c _Ôw Is­¯s« {Ia \¼À

kam´c t{iWn

BZy ]Zw

s]mXp n þ mw ]Zw n þ mw ]Z¯nse n þ mw ]Z¯nse n DÄs¸Sm¯ n sâ hyXymkw ]Zw KpWIw

kqN\ 1.

n þ mw ]Zw Ft¸mgpw an+b F¶ cq]¯nembncn¡pw. (ChnsS b bpsS hne ]qPyamImw)

2.

n þ mw ]Z¯nse n sâ KpWI¯nsâbpw n DÄs¸Sm¯ ]Z¯nsâbpw XpIbmWv BZy]Zw

3.

n þ mw ]Z¯n n sâ KpWIhpw t{iWnbpsS s]mXphyXymkhpw Xpeyambncn¡pw

4.

xn = an+b F¶ cq]¯nepÅ GXpt{iWnbpw kam´ct{iWnbmWv.

10 ...............................................................................................................................MUKULAM MATHS

{]hÀ¯\w GXm\pw kam´ct{iWnIfpsS n þ mw ]Zw sImSp¯ncn¡p¶p. s]mXphyXymkw, BZy]Zw ImWpI 1)

3n+5

2)

3n-7

3)

5n-2

4)

4n-1

5)

6)

2-7n

7)

10-4n

8)

10-4n 3/ - n/ 4 4

11)

(n-1)2 - (n+1)2

*

Hcp kam´c t{iWnbpsS s]mXphyXymkw 9 BWv. AXnsâ 16þmw ]Z¯nt\¡mÄ F{X IqSpXemWv 20 þmw ]Zw ?

*

Hcp kam´ct{iWnbpsS 21þmw ]Z¯nt\¡mÄ 18 IqSpXemWv 27þmw ]Zw. F¦n 27þmw ]Z¯nt\¡mÄ F{X IqSpXemWv 29þmw ]Zw ?

*

6,10,14.................F¶ kam´ct{iWnbpsS F{Xmw ]ZamWv 1622

*

Hcp kam´ct{iWnbpsS 3þmw ]Zt¯mSv F{X s]mXphyXymkw Iq«nbmemWv 12þmw ]Zw e`n¡p¶Xv ?

*

Hcp kam´ct{iWnbpsS 7þmw ]Zt¯mSv 8 s]mXphyXymkw Iq«nbm In«p¶Xv F{Xmw ]Zambnc¡pw?

*

Hcp kma´ct{iWnbpsS 8þmw ]Zw 20 Dw 18þmw ]Zw 60 Dw Bbm s]mXphyXymkw F{X?

*

20 þmw ]Zw 45 Dw 15 þmw ]Zw 30 Bb Hcp kam´ct{iWnbpsS 10 þmw ]Zw F{X? s]mXp hyXymkw F{X?

*

Hcp kam´ct{iWnbpsS 11þmw ]Zw 1 Dw 8 þmw ]Zw þ142 Dw BWv. F¦n 14þmw ]Zw F{X? 17þmw ]Zw, 18 þmw ]Zw, 24þmw ]Zw F¶nh I­p]nSn¡pI.

*

Hcp kma´ct{iWnbpsS 15þmw ]Zt¯mSv 77 Iq«nbt¸mÄ 26þmw ]Zw In«n. F¦n 26 þmw ]Zt¯mSv F{XIq«nbm 37 þmw ]Zw In«pw 26 þmw ]Zt¯mSv F{X Iq«nbm 29 þmw ]Zw In«pw ?

*

Hcpkam´ct{iWnbpsS 10þmw ]Zw 17 Dw 17 þmw ]Zw 10 Dw F¦n s]mXphyXymkw F{X?

*

Hcp kam´ct{iWnbpsS 5 þmw ]Zw 14 Dw 14þmw ]Zw 5 Dw Bbm s]mXphyXymkw F{X?

*

Hcp kam´ct{iWnbpsS 100 þmw ]Zw 200 Dw 200 þmw ]Zw 100 Bbm s]mXphyXymkw F{X ?

*

Hcp kam´ct{iWnbpsS (n-1)þmw ]Zw 4n+7 Bbm BZy]Zhpw s]mXphyXymkhpw 10 þmw ]Zwhpw ImWpI

*

Hcpkam´ct{iWnbpsS n þmw ]Zw 5n+6 Abm AXnsâ F{Xmw ]ZamWv 56?

*

Hcp kam´ct{iWnbpsS (n-2)þmw ]Zw 4n+7 Bbm F{Xmw ]ZamWv 55 ?

*

n2+nF¶ _oPKWnX hmNIw Hcp kam´c t{iWnbpsSn-þmw ]ZamtWm F¶v F§s\

]cntim[n¡mw MUKULAM MATHS................................................................................................................................. 11

*

n2+n F¶Xn n \v 1,2,3,4.......XpS§nb hneIÄ \ÂInbm 2,6,12,20.........F¶ t{i#n In«p

atÃm. CsXmcp kam´c t{iWnbÃ. ChnsS ASp¯Sp¯ ]Z§Ä X½nepÅ hyXymkw IqSnIqSn hcp¶p (Hcp Ønc kwJyÃ) CXns\ asämcp coXnbn kao]n¡mw t{iWnbpsS n þmw ]Zw = n2+n (n+1) þmw ]Zw = (n+1)2 + (n+1)

= n2+3n+2 (n+1) þmw ]Zw þ n mw ]Zw = n2+3n+2 _ (n2+n) = 2n+n CXv n DÄs¸Sp¶ Hcp hmNIamWv.

Hcp Ønc kwJybÃ. kam´ct{iWnbpsS s]mXphyXymkw FÃmbvt¸mgpw Hcp Ønc kwJybmbncn¡pw. AXn\m n2+n Hcp kam´c t{iWnbpsS n þmw ]ZaÃ. 1) 2n2+1

2) n3

3) 1 3n

4) 1 +1 5 n

5) (2n+1)2 -)2n-1)2

{]hÀ¯\w 200 \pw 500 \pw CSbn 7 sâ KpWnX§fmb F{X kwJyIÄ D­v? ImtW­ kwJy 200  IqSpXepw 500  IpdhpamWtÃm 201 _: 7

injvSw F{X?

201 = 7 x 28 + 5 : injvSw = 5 202 _: 7 injvSw = ........................... 203 _: 7 injvSw = ...............................

BZy]Zw F{X? C\n Ahkm\ ]Zsa{Xsb¶v t\m¡mw 409 = 7 x ......... + ............ : ...............injvSw = ................. 498 _: 7 injvSw = ............................ 497 _: 7 injvSw = ............................

Ahkm\ ]Zsa{X? s]mXp hyXymksa{X? C\n ]Z§fpsS F®w ImWmatÃm. DZmlcWw 400 \pw700 \pw CSbn 6 sIm­v lcn¨m injvSw 3 hcp¶ kwJyIfpsS F®sa{X? 401 _: 6 = 6 x ......... + ............ : ...............injvSw = ................. 402 _: 6 = 6 x ......... + ............ : ...............injvSw = ................. 403 _: 6 404 _: 6 405 _: 6

BZy]Zw e`n¨tÃm

12 ...............................................................................................................................MUKULAM MATHS

699 s\ 6 sIm­v lcn¨m injvSw F{X ? 699 = 6 = 6 x ......... + ............ : ...............injvSw = .................

Ahkm\ ]Zw e`n¨tÃm C\n ]Z§fpsS F®w ImWmatÃm {]hÀ¯\w 8 sIm­v lcn¨m injvSw 1,2,3,4,5 hcp¶ F®Â kwJyIÄ {Ia¯nsegpXpI Chtbmtcm¶nsebpw s]mXphyXymksa{X? NB :

kam´ct{iWnbnse ]Z§sf s]mXphyXymkw sIm­v lcn¨m In«p¶ injvS¯nsâ {]tXyIX t_m²ys¸Sp¯matÃm. 200 \pw 500 \panSbn 9sâ F{X KpWnX§Ä ? Gähpw sNdpXpw hepXpw ImWpI.

kam´ct{iWnIfpsS ]Z§fpsS XpI ]Z§fpsS F®w Hä kwJy 1.

]Z§fpsS F®w 3 Dw XpI 18 Dw Bb hyXykvX kam´c t{iWnIÄ FgpXpI 6,6,6

(6,6,6,)

6 - 1, 6,6 + 1

(5,6,7)

5 - 1, 6, 7 + 1

(4,6,8)

2.

]Z§fpsS F®w 3 Dw XpI 30 hcp¶ t{iWnIÄ FgpXpI

3.

]Z§fpsS F®w 5 Dw XpI 45 Dw hcp¶ t{iWnIÄ FgpXpI

5

80

16

14,18 12,20

-

-

32

32

-

-

32

5,6,7

3

18

6

5,7

-

-

12

-

-

-

12

a²y]Z¯n \n¶v Htc AIe¯nepÅ Hmtcm tPmUn ]Z§fpw tPmUn XpIbpw ]Z§Ä (tPmUn)

-

XpI (tPmUn)

BZy]Zw+ Ahkm\]Zw

]Z§fpsS XpI

12,14,16,18,20

t{iWn

a[y]Zw

]Z§fpsS F®w

]Z§fpsS F®w HäkwJybmbn hcp¶ kam´ct{iWnIÄ FgpXn ]«nI ]qÀ¯nbm ¡pI. (kuIcy¯n\v ]Z§fpsS F®w 3,5,7,9 F¶nh am{Xw FSp¡pI)

]«nI ]cntim[n¨v XmsgsImSp¯ tNmZy§Ä¡v D¯cw ImWpI 1.

a[y]Zhpw XpIbpw X½nepÅ _Ôw

2.

]Z§fpsS F®hpw XpIbpw X½nepÅ _Ôw

MUKULAM MATHS................................................................................................................................. 13

3.

a[y]Zhpw tPmSn XpIbpw

4.

BZy]Zhpw Ahkm\ ]Zhpw X¶m a[y]Zw ImWp¶sX§s\?

5.

a[y]Zhpw ]Z§fpsS F®hpw In«nbm XpI F§s\ ImWmw?

6;

]Z§fpsS F®w, BZy]Zw, Ahkm\]Zw F¶nh X¶m XpI F§s\ ImWmw ? XpI = a[y]Zw x ]Z§fpsS F®w XpI = ]Z§fpsS F®w ( BZy]Zw + Ahkm\]Zw) 2

]Z§fpsS F®w Cc« kwJy 1.

]Z§fpsS F®w 4 Dw XpI 60 Dw hcp¶ hyXykvX kam´c t{iWn ImWpI 15,15,15,15

15,15,15,15

13-3, 15-1, 15+1- 15+5

12,14,16,18

12-3, 14-1, 16+1, 18+3

9,13,17,21

2.

]Z§fpsS F®w 4 Dw XpI 40 Dw hcp¶ kam´c t{iWnIÄ GsXms¡

3.

]Z§fpsS F®w 8 Dw XpI 72 Dw hcp¶ kam¶c t{iWnIÄ FgpXpI

60

30

]Z§Ä (tPmUn)

12,18 -

-

-

XpI (tPmUn)

30

-

-

-

BZy]Zw+ Ahkm\]Zw

6

a²y]Z¯n \n¶v Htc AIe¯nepÅ Hmtcm tPmUn ]Z§fpw tPmUn XpIbpw

tPmSnIfpsS F®w

a[y]Zw

12,14,16,18,

]Z§fpsS XpI

t{iWn

]Z§fpsS F®w

]«nI ]qcn¸n¡pI : kuIcy¯n\v ]Z§fpsPS F®w 2,4,6,8 hcp¶ t{iWnIÄ FSp¡pI

2

30

]«nI ]cntim[n¡pI, Xmsg sImSp¯ tNmZy§Ä¡v D¯cw Is­¯pI 1.

]Z§fpsS F®hpw XpIbpw X½nepÅ _Ôw

2.

tPmSnIfpsS F®hpw XpIbpw X½nepÅ _Ôw

3.

Hmtcm tPmSn XpIbpsSbpw {]tXyIX

4.

]Z§fpsS F®w, BZy]Zw, Ahkm\]Zw F¶nh X¶m XpI F§ns\ ImWmw XpI = tPmSnIfpsS F®w x tPmSn XpI

14 ...............................................................................................................................MUKULAM MATHS

XpI = tPmSnIfpsS F®w (BZy]Zw + Ahkm\]Zw) XpI = ]Z§fpsS F®w (BZy]Zw + Ahkm\]Zw) 2 c­p ]«nIbpw XmcXayw sN¿pI. s]mXphmb kq{XhmIy¯nse¯nt¨cpI n ]Z§fpsS XpI

= n (2f+(n_1)d) 2

1.

Hcp kam´ct{iWnbnse BZys¯ 10 ]Z§fpsS XpI 80 BWv. BZys¯bpw ]¯m as¯bpw ]Z§fpsS XpI F{X?

2.

BZys¯ 10 ]Z§fpsS XpI 80 hcp¶Xpw s]mXphyXymkw 4 hcp¶Xpamb kam´c t{iWnbnse FÃm ]Z§fpw FgpXpI

3.

5,8,11,.............F¶ kam´c t{iWnbpsS BZys¯ 25 ]Z§fpsS XpItb¡mÄ F{X IqSp XemWv 7,10,13,,,,,,,,,,,, F¶ kam´c t{iWnbpsS BZys¯ 25 ]Z§fpsS XpI?

F®Â kwJyIfpsS XpI

CXn \n¶v 1+2+3+4

=

42 4 F¶v In«patÃm + 2 2 52 5 + 2 2

AXn\m 1+2+3+4+5 =

1+2+3+4+.................. + n = n2 + n 2

2

= n(n+1) 2

F¶v ImWmw

MUKULAM MATHS................................................................................................................................. 15

3 5 7

1+3+5+7

=

82 4

1+3+5+7+9

=

102 5

1+3+5+7+...........+2n_ 1 n

(2n)2 4

= n2

1 apX XpSÀ¨bmb n F®Â kwJyIfpsS XpI = n(n+1) 2

1 apX XpSÀ¨bmb n HäkwJyIfpsS XpI = n2 1)

BZys¯ 40 F®Â kwJyIfpsS XpIsb{X 7 apX XpSÀ¨bmb 40 F®Â kwJyIfpsS XpIsb{X

2)

1 apX 20 hscbpÅ F®Â kwJyIfpsS XpIsb{X 2 apX 40 hscbpÅ Cc« kwJyIfpsS XpIsb{X 5 apX 100 hscbpÅ 5 sâ KpWnX§fpsS XpIsb{X

3) 4)

30 ]Z§fpÅ Hcp kam´c t{iWnbnse 15þmw ]Z¯nsâbpw 16þmw ]Z¯nsâbpw XpI 17 BWv. F¦n FÃm ]Z§fpsSbw XpIsb{X Hcp kam´c t{iWnbpsS nþmw ]Zw 4n _ 3Bbm F)

BZy]Zsa´v

_n)

s]mXphyXymksa´v

kn)

F{Xmw]ZamWv 65

16 ...............................................................................................................................MUKULAM MATHS

Un) 5)

6)

BZys¯ 20 ]Z§fpsS XpIsb{X

Hcp kam´ct{iWnbnse 12 ]Z§fpsS XpI 78 BWv. F¦n F)

6þmw ]Z¯nsâbpw 7þmw ]Z¯nsâbpw XpIsb{X

_n)

s]mXphyXymkw 2 F¦n FÃm ]Z§fpw FgpXpI

kn)

s]mXphyXymkw 3 F¦n FÃm ]Z§fpw FgpXpI

Hcp kam´ct{iWnbpsS 9þmw ]Zw ]qPyw. F¦n 19þmw ]Z¯nsâ F{X aS§mWv 29þmw ]Zw

MUKULAM MATHS................................................................................................................................. 17

bqWnäv sSÌv 1)

1,3,5,9,17 Ch Hcp kam´ct{iWnbnse XpSÀ¨bmb 5 ]Z§fmbncn¡ptam. F´psIm­v

2)

Hcp kam´ct{iWnbnse 3þmw ]Zw F{X. H¶mw ]Zw F{X

3)

Hcp kam´ct{iWnbpsS ]Z§fpsS XpI 5n2+3n Bbm BZy]Zw F{X 2þmw ]Zw F{X. s]mXphyXymkw ImWpI. Cu t{iWnbpsS _oPKWnX cq]w FgpXpI

4)

\of§Ä kam´ct{iWnbnemb 15 Ccp¼v I¼nIÄ BtcmlW{Ia¯n Xpey AIe ¯nembn Ip¯s\ Dd¸n¨n«p­v. Gähpw sNdnb I¼nbpsS \ofw 44.5 sk.an. Dw Gähpw henb I¼nbpsS \ofw 155. 5 sk.an. Dw Bbm I¼nIfpsS BsI \ofsa{X.

5)

-120, -114, -108,................F¶

17 2

Dw 7þmw ]Zw

37 2

Dw Bbm s]mXphyXymkw

kam´ct{iWnbn 134 F¶ kwJy Hcp ]ZamtWm.

F´psIm­v. 6)

chn Xsâ t\m«p_p¡n 8 sIm­v lcn¨m injvSw 3 In«p¶ FÃm 3 A¡ kwJyIfp sagpXn. F¦n chn F{X kwJyIÄ FgpXnbncn¡pw. XpIsb{Xbmbncn¡pw

7)

Hcp kam´ct{iWnbpsS 15 ]Z§fpsS XpI 8þmw ]Z¯nsâ 15 aS§mbncn¡pw. F¶v sXfnbn¡pI.

8)

Hcp kam´ct{iWnbnse XpSÀ¨bmb 10 ]Z§fpsS XpI 210 BWv. F¶m Cu ]¯v ]Z§sfbpw 3 sIm­v KpWn¨m In«p¶ kwJyIfpsS XpI F{X

18 ...............................................................................................................................MUKULAM MATHS

s{]mPIvSv ]Z§sfÃmw F®Â kwJyIfmb Hcpkam´ct{iWnbnse ]Z§fn Hsc®w ]qÀW hÀKamsW¦n aät\Iw ]Z§Ä ]qÀWhÀKamsW¶v sXfnbn¡pI. ]Z§sfÃmw F®Â kwJyIfpw Hcp ]Zw t]mepw ]qÀWhÀKaÃm¯Xpamb kam´c t{iWnbpt­m F¶p I­p]nSn¡pI 1,2,3,4,5,6,7,8,9,10,......................F¶ F®Â kwJyIfpsS kam´ct{iWnbn 1,4,9,16,25,36,................ F¶n§s\ ]qÀWhÀK§Ä D­v. 1,3,5,7,9........F¶ Hä kwJyIfpsS kam´ct{iWnbn 1,9,25,49,81,........................F¶n§s\ ]qÀWhÀK§Ä D­v. 2,4,6,8,..............F¶ Cc« kwJyIfpsS kam´ct{iWnbn 4,16,36,64,.........F¶n§s\ ]qÀW hÀK§fp­v. Hä kw J y I fpsS kam ´ c t{i Wn ¡p Ånse asämcp kam ´ c t{i Wn bmb 3,9,15,21,27,33,39,45,51,57,63,69,75,81,.................................. epw ]qÀWhÀK§Ä D­v. Cc« kwJ y I fpsS kam ´ c t{i Wn ¡p Ånse asämcp kam ´ c t{i Wn bmb 4,10,16,22,28,34,40,46,52,58,64,70,.............ep ]qÀ®hÀK§Ä D­v. 1,2,3,4,5,6,..............F¶ kam´ct{iWnbn k HcpF®Â kwJy Bbm k2 Hcp ]qÀW hÀKamWv k2 t\mSv 2kdIq«nbm AXpw Hcp F®Â kwJy Xs¶ (F®Â kwJyIfpsS kam´ ct{iWnbn s]mXphyXymkw dbpw F®Â kwJybmWv) k2+2kd+d2 Dw ]qÀWhÀKw BWv [k2+2kd+d2 = (k+d)2] = k2+d(2k+d) (k+2d)2, (k+3d)2, (k+4d)2 F¶n§s\ ]qÀWhÀK§fmb At\Iw ]Z§Ä Is­¯mw. ]Z§sfÃmw F®Â kwJyIfmb aäv kam´ct{iWnIÄ FSp¯pw Cu Bibw cq]oIcn¡patÃm? 3,7,11,15,19,.........................F¶ kam´ct{iWnbn ]Z§Ä F®Â kwJyIfmsW¦nepw ]qÀWhÀK§fmb ]Z§Ä CÃ. 3,8,13,18,23,.........Cu kam´t{iWn¡pw taÂ]dª {]tXyIX ImWmw. ChnsS "H¶p"I fpsS Øm\s¯ A¡w 3,8 AAWv. Hcp ]qÀWhÀK kwJybpsSbpw"H¶p"IfpsS Øm\s¯ A¡w 3,8 BInsöv HmÀ¡p atÃm. 12,27,32,47,........F¶ kam´ct{iWnbn "H¶p"IfpsS Øm\s¯ A¡w 2,7 BWv. Cu t{iWnbn ]qÀWhÀK§fmb ]Z§Ä CÃ. F´psIm­v ? IqSpX DZmlcW§Ä Is­¯patÃm ]Z§sfÃmw F®Â kwJyIfpw Hcp]Zw t]mepw ]qÀWhÀKaÃm¯Xpamb Hcp kam ´c t{iWn 8,8,8,8,8...................................... FÃm ]Z§fpw ]qÀWhÀK§fmb kam´ct{iWnbnXm 9,9,9,9,9,...................

MUKULAM MATHS................................................................................................................................. 19

aqey\nÀWb {]hÀ¯\§Ä 1)

am{´nI NXpc¯nse kwJyIÄ A[nkwJy s]mXphyXymkambn hcp¶ kam´ct{i Wnbnse ]Z§fmWv.

*

am{´nI NXpc¯nsâ a[y¯n hcp¶ kwJy GXv

*

am{´nI NXpc¯nse kam´ct{iWn FgpXpI

*

am{´nI XpI F{X 3

2)

3)

4)

27

11

7

15

kwJym]mtä¬ {i²n¡q 1

=

1

1+2

=

3

1+2+3

=

6

1+2+3+4

=

10

..............................

=

........................

..............................

=

........................

............................. .

=

........................

1+2+3+............+ x

=

190

*

]mtäWnse 5þmw hcn, 6þmw hcn Ch FgpXpI

*

15þmas¯ hcnbnse ]Z§fpsS XpI F{X

*

x sâ hne ImWpI

Hcp kam´ct{iWnbpsS m þmw ]Zw n; n þmw ]Zw m *

s]mXphyXymkw ImWpI

*

BZy]Zw ImWpI

*

m+n mw ]Zw ImWpI

*

m+n ]Z§fpsS XpI ImWpI

Hcp kam´ct{iWnbpsS _oPKWnX cq]w 2n2+3n BIptam. F´psIm­v Hcp kam´ct{iWnbpsS BZys¯ n ]Z§fpsS XpI 2n2+3n BIptam. F´psIm­v

5)

Hcp kam´ct{iWnbpsS BZys¯ 50 ]Z§Ä BtcmlW{Ia¯nsegpXn. Cu ]Z

20 ...............................................................................................................................MUKULAM MATHS

§Ä Xs¶ AhtcmlW{Ia¯nsegpXn. Cu c­v t{iWnIfpsSbpw H¶mw ]Z§Ä XpeyamIptam. Ahkm\ ]Z§Ä XpeyamIptam. Cu c­v t{iWnIfpsSbpw Htc Øm\¯pÅ XpIIÄ X½nepÅ _Ôsa´v. 6)

7)

Hcp kam´ct{iWnbpsS 5þmw ]Zw 19 . 9þmw ]Zw

1 5

*

Cu kam´c t{iWnbpsS s]mXphyXymksa{X

*

14þmw ]Zw ImWpI

-9,-14,-19,-24................... F¶ kam´ct{iWnbnse 25þmw ]Zw x25 Dw ]¯mw ]Zw

x10 Dw ImWpI.

x25 - x10F{XbmWv

8)

3,7,11.......F¶ kam´ct{iWnbnse 34þmw ]Zw ImWpI. 10093 Cu kam´ct{iWnbnse Hcp ]ZamtWm. D¯cw km[qIcn¡pI

9)

800 cq] 8% hmÀjnI km[mcW ]eni e`n¡p¶Xn\v Hcp _m¦n \nt£]n¨p. Hcp hÀjw Ignbpt¼mgpÅ ]enisb´v. c­v hÀjw Ignbpt¼mgpÅ ]enisb´v. 3 hÀjw Ignbpt¼mgpÅ ]enisb´v ]eniIfpsS t{iWn FgpXpI CXv kam´ct{iWnbmtWm F¶v ]cntim[n¡pI. 23þ#mas¯ hÀjmhkm\w e`n¡p¶ ]eni ImWpI

10)

Hcp N{Inb NXpÀ`pP¯nsâ tImWpIfpsS AfhpIÄ bYm{Iaw x-20, x, x+20, x+40 hoXambm x sâ hne ImWpI. tImWfhpIÄ ImWpI. tImWfhpIÄ kam´c t{iWnbnemtWm

11)

Hcp kam´ct{iWnbpsS 7þmw ]Zhpw 2þmw ]Zhpw X½nepÅ hyXymkw 20. s]mXphy Xymkw ImWpI. Cu kam´ct{iWnbpsS 3þmw ]Zw 9 Bbm BZys¯ F{X ]Z§ fpsS XpIbmWv 153.

12)

kam´ct{iWnbnemb 10 kwJyIfpsS XpI 85 BWv. Chbnse A©mas¯ kwJy 8 BWv. *

BZys¯bpw 10þmas¯bpw kwJyIfpsS XpI F{X

*

Bdmas¯ kwJy GXv

*

14)

BZys¯ aq¶v kwJyIÄ FgpXpI - 11 -6, , -5 F¶ kam´c t{iWbnse F{X ]Z§fpsS XpIbmWv þ25. 2 Hcp NXpc ]eIbpsS hoXn, \ofw, Dbcw Ch kam´ct{iWnbnse XpÀ¨bmb 3 ]Z§ fmWv \of¯nsâbpw hoXnbpsSbpw Dbc¯nsâbpw XpI 27 bqWnäv. NXpc]eIbpsS hym]vXw 405 L. bqWnäv. \ofw, hoXn, Dbcw Ch ImWpI.

15)

c­v kam´ct{iWnIfpsS ]Z§fpsS XpIIÄ X½nepÅ Awi_Ôw 7n+1 : 4n+27 AhbpsS 5þmw ]Z§Ä X½nepÅ Awi_Ôw ImWpI

16)

(x-y)2, x2+y2, (x+y)2 F¶ kam´ct{iWnbpsS BZys¯ n ]Z§fpsS XpI ImWpI

17)

Hcp kvIqfnse 1þmw ¢mknse Ip«nIÄ Hcp hr£ssX, 2þmw ¢mknse Ip«nIÄ 2 hr£ ssX, 3þmw ¢mknse Ip«nIÄ 3 hr£ssX F¶n§s\ kvIqfn\p Npäpw hr£ssXIÄ

13)

MUKULAM MATHS................................................................................................................................. 21

\Sm³ Xocpam\n¨p. +2 hscbpÅ Cu kvIqfn Hmtcm ¢mkpw 3 Unhnj³ hoXap­v. Ip«nIÄ \Sp¶ hr£ssXIfpsS BsI F®sa{X. 18)

Hmtcm P·Zn\¯n\pw hÀj¡v AhfpsS Aѳ 5 cq] sImSp¡pw. hÀj¡v Ct¸mÄ 21 hbkmsb¦n Ct¸mÄ ssIhiapÅ BsI XpIsb{X

19)

Hcp kam´ct{iWnbpsS 3þmw ]Z¯nsâbpw 7þmw ]Z¯nsâbpw XpI 6. KpW\^ew 8. BZys¯ 16 ]Z§fpsS XpI ImWpI.

22 ...............................................................................................................................MUKULAM MATHS

A²ymbw þ2

hr¯§Ä BapJw H³]Xmw ¢mÊnse hr¯§Ä F¶ A[ymb¯nsâ XpSÀ¨bmWv Cu ]mT`mKw. ChnsS {][m\ambpw tImWpIfneqsS hr¯s¯ a\Ênem¡m\pÅ {iaamWv \S¯p¶Xv. c­v \nÝnX _nµp¡fneqsS IS¶pt]mIp¶, ]ckv]cw ew_ambXpw AÃm¯Xpamb c­p hcIÄ Iq«nap «p¶ _nµp¡sfÃmw tNÀ¶v D­mIp¶ cq]§Ä hc¨dnbp¶XneqsS ]mTw Bcw`n¡p¶p. Hcp hc hr¯s¯ JÞn¡pt¼mgp­mIp¶ Nm]§fpw, hr¯JÞ§fpw, Nm]¯nsâ tI{µ tImWpw AXnsâ adpNm]¯nse tImWpw X½nepÅ _Ôw, \mep aqeIfnepw IqSn IS¶pt]m Ip¶ Hcp hr¯w hcbv¡mhp¶ NpXÀ`pP§Ä F¶nhbpsSsbÃmw khntijXIfpw {]tbmK §fpw XpSÀNn´IfpamWv Cu A[ymb¯n NÀ¨ sN¿p¶Xv. _lp`pP§Ä¡v Xpey]c¸f hpÅ kaNXpc¯nsâ \nÀanXn ChnsS NÀ¨ sN¿p¶p. {][m\ Bib§Ä *

c­v _nµp¡fn IqSn IS¶v t]mIp¶ hcIÄ \nÝnX tImWn JÞn¡pIbmsW ¦n A§ns\ JÞn¡p¶ _nµp¡Ä tNÀ¶v Nm]tPmUnIÄ D­m¡p¶p.

*

hr¯¯nse Hcp hymk¯nsâ Aä§Ä AXnse atäsXmcp _nµphpambn tbmPn¸n ¨m In«p¶Xv a«tIm¬ Bbncn¡pw.

*

hr¯¯nse Hcp hymk¯nsâ Aä§Ä hr¯¯n\I¯pÅ _nµphpambn tbmPn¸n ¨m In«p¶Xv _rlXvtIm¬ Bbncn¡pw.

*

hr¯¯nse Hcp hymk¯nsâ Aä§Ä hr¯¯n\v ]pd¯pÅ _nµphpambn tbmPn¸n ¨m In«p¶Xv \yq\tIm¬ Bbncn¡pw.

*

Hcp hr¯¯nsâ hymk¯nsâ Aä§Ä Hcp _nµphpambn tbmPn¸n¨t¸mÄ a«tIm¬ In«nsb¦n B _nµp hr¯¯nembncn¡pw.

*

adpNm]w F¶ Bibw

*

hr¯¯nse Hcp Nm]w tI{µ¯nep­m¡p¶ tImWnsâ ]IpXnbmWv. B Nm]w AXnsâ adpNm]¯nse Hcp _nµphnep­m¡p¶ tIm¬.

*

hr¯JÞw F¶ Bibw

*

Htc hr¯JÞ¯nse tImWpIÄ XpeyamWv.

*

adpJÞ§fnse tImWpIÄ A\p]qcIamWv.

*

Hcp NXpÀ`pP¯nsâ aqeIsfÃmw Hcp hr¯¯nemsW¦n AXnsâ FXnÀtImWpIÄ A\p]qcIamWv.

*

Hcp NXpÀ`pP¯nsâ FXnÀtImWpIÄ A\p]qcIamsW¦n AXnsâ \mev aqeIfn IqSnbpw IS¶v t]mIp¶ Hcp hr¯w hc¡mw.

*

N{Iob NXpÀ`pPw F¶ Bibw

MUKULAM MATHS................................................................................................................................. 23

*

Hcp hr¯¯n AB, CD F¶o RmWpIÄ P F¶ _nµphn JÞn¨m PA X PB = PC X PD Bbncn¡pw.

*

Hcp hr¯¯nse AB, CD F¶o RmWpIÄ \o«nhc¨v hr¯¯n\v ]pd¯v P F¶ _nµp hn JÞn¨m PA X PB = PC X PD Bbncn¡pw

*

Hcp hr¯¯n AB F¶ hymks¯ CD F¶ Rm¬ ew_ambn P bn JÞn¡p¶psh ¦n AP X PB = PC2 Bbncn¡pw.

*

NXpcw, {XntImWw, NXpÀ`pPw Ch Hmtcm¶n\pw Xpey]c¸fhpÅ kaNXpc¯nsâ \nÀanXn

*

c­v _nµp¡fn IqSn IS¶pt]mIp¶ hcIÄ \nÝnX tImWn JÞn¡pIbmsW ¦n A§s\ JÞn¡p¶ _nµp¡Ä tNÀ¶v Nm]tPmUnIÄ D­mIp¶p. *

hcIÄ 900 tImWn JÞn¡pt¼mÄ

*

hcIÄ 900  Ipdhmb tImWn JÞn¡pt¼mÄ

*

hcIÄ 900  IqSpXemb tImWn JÞn¡pt¼mÄ

{]hÀ¯\w (1) (kma{KnIÄ : ImÀUvt_mÀUn sh«nsbSp¯ {XntImWw) {XntImW¯nsâ c­phi§Ä, t_mÀUn hc¨ Hcp tcJbpsS A{K_nµp¡fneqsS IS¶pt]mI¯¡hn[w sh¡p¶p. B hi§Ä tNcp¶ aqebpsS Øm\w ASbmfs¸Sp¯p¶p (Nn{Xw þ1 ) P

(Nn{Xw þ1 ) A

B

C¯c¯n {XntImWw hyXykvX coXnbn {IaoIcn¨v aqebpsS Øm\w ASbmfs¸Sp ¯p¶p. (kqN\ : ImÀUv t_mÀUntem, sXÀtamtImÄ joäntem sam«pkqNn D]tbmKn¨v _nµphnsâ Øm\w ASbmfs¸Sp¯mw. aqebpsS Øm\w Ip«nIsfs¡m­v ASbmfs¸Sp¯mw)

24 ...............................................................................................................................MUKULAM MATHS

{]hÀ¯\w (2) kma{KnIÄ : Nn{Xw (2)  xsâ hne 400, 550,600, 800, 900, 1000,1200, 1250, 1400 BI¯¡hn[w ImÀUvt_mÀUn sh«nsbSp¯ ]¯v {XntImW§Ä ( 900 tImWnsâ Afhn cs­®w)

)

x0

(Nn{Xw þ 2) (i) Hmtcm {XntImWhpw Hmtcm {Kq¸n\v \ÂIp¶p. Hmtcm {Kq¸pw Hcp NmÀ«v t]¸dn Htc \of¯n AB F¶ hc hc¨v apIfn sNbvX {]hÀ¯\w (1) BhÀ¯n¡p¶p. (hcbpsS Ccp `mK§fnepw ]camh[n _nµp¡Ä ASbmfs¸Sp¯m³ {i²n¡patÃm) Hmtcm {Kq¸pw ASbmfs¸Sp¯nb _nµp¡Ä tNÀ¶v D­mIp¶ cq]¯nsâ khntijX NÀ¨ sN¿p¶p. (AÀ² hr¯ambXpw AÃm¯Xpw) (ii) Hmtcm {Kq¸n\pw e`n¨ cq]§Ä sh«nsbSp¡p¶p. AB bneqsS apdn¨v c­v `mK§fm¡p ¶p. Cu `mK§Ä {Kq¸pIfn ssIamdn, GsXÃmw `mK§Ä tNÀ¯psh¡pt¼mgmWv Hcp hr¯w In«p¶Xv F¶v Is­¯p¶p. B hr¯¯nse Ccp`mK§fnsebpw tImWpIfpsS khntijX NÀ¨ sN¿p¶p. (iii) {]hÀ¯\w 2(i)  hc¨ AtX \of¯n AB F¶ hc hymkam¡n t\m«p]pkvXI ¯n Hcp hr¯w hcbv¡p¶p. t\cs¯ ImÀUvt_mÀUn sh«nsbSp¯ {XntImW§Ä D] tbmKn¨v AB bpsS A{K_nµp¡fn IqSn IS¶pt]mIp¶ hi§Ä tNcp¶ aqebpsS Øm\w ASbmfs¸Sp¯p¶p. (Hcp {XntImWw D]tbmKn¨v Hsc®w am{Xw) hi§Ä D­m¡p¶ tImWnsâ Afhv 900, 900  Ipdhv 900  IqSpX hcpt¼mÄ, ASbmfs¸Sp¯nb _nµp¡ fpsS Øm\w hr¯¯nemtWm, hr¯¯n\v ]pd¯mtWm hr¯¯n\I¯mtWm F¶v NÀ¨ sN¿p¶p. * hr¯¯nse Hcp hymk¯nsâ Aä§Ä hr¯¯nse atäsXmcp _nµphpambn tbmPn ¸n¨m In«p¶Xv a«tIm¬ Bbncn¡pw. (sXfnhv ]mT]pkvXI¯nse t]Pv 29)

A

B

(Nn{X¯n AB hymkw

LAPB a«tIm¬ Bbncn¡pw) P Nn{Xw (3) MUKULAM MATHS................................................................................................................................. 25

WORK SHEET N

1)

Nn{X¯n LM hymkamWv

L

M

LLNM = .........................

P

2)

LPQR = 700 , LQPR = 450 BbmÂ

Q

LPRS = .........................

C

B

3)

S

R

A

LABD = 90 , LBDC= 350 Bbm LBCD = ......................... D

C

4) P

D

E

LCDP = 800 , Bbm LE bpsS Afhv BImhp¶tXXv ? (650, 850, 950, 1050 )

K M

LLMN = 900 , Bbm LK bpsS Afhv BImhp¶tXXv ?

5) L

N

(1200, 1000, 900, 600 )

26 ...............................................................................................................................MUKULAM MATHS

A C

LBAC = 900 , Bbm LBCD bpsS Afhv BImhp¶tXXv ?

6) B

D

(900, 800, 1000, 850 )

C P

Nn{X¯n AB hymkamWv

7) A

B

C

P

Nn{X¯n AB hymkamWv

8) B

A

*

LACB, LAPB, Chbn GXmWv hepXv ?

LACB, LAPB, Chbn GXmWv hepXv ?

hr¯¯nse Hcp hymk¯nsâ Aä§Ä, hr¯¯n\I¯pÅ _nµphpambn tbmPn¸n¨m In«p¶Xv _rlXvtIm¬ Bbncn¡pw. (sXfnhv ]mT]pkvXI¯n t]Pv 31 Â)

Q

Nn{X¯n AB hymkamWv A

B

LAQB _rlXvtIm¬ Bbncn¡pw

Nn{Xw (4) MUKULAM MATHS................................................................................................................................. 27

*

hr¯¯nse Hcp hymk¯nsâ Aä§Ä, hr¯¯n\v ]pd¯pÅ _nµphpambn tbmPn¸n¨m In«p¶Xv \yq\tIm¬ Bbncn¡pw. (sXfnhv ]mT]pkvXI¯n t]Pv 31 Â) R

Nn{X¯n AB hymkamWv B

A

LARB \yq\tIm¬ Bbncn¡pw

Nn{Xw (5) *

Hcp hr¯¯nsâ hymk¯nsâ Aä§Ä Hcp _nµphpambn tbmPn¸n¨t¸mÄ a«tIm¬ In«nsb¦n B _nµp hr¯¯nembncn¡pw Cu _nµp hr¯¯n\I¯mInÃ. (hr¯¯n\I¯mIWsa¦n Cu tIm¬ _rlXv tIm¬ Bbncn¡Ww)

Cu _nµp hr¯¯n\v ]pd¯paÃ. (hr¯¯n\v ]pd¯msW¦n Cu tIm¬ \yq\ tIm¬ Bbncn¡Ww) AXpsIm­v Cu _nµp hr¯¯n Xs¶bmWv. (Cu _nµp hr¯¯nemWv F¶v t\cn«p sXfnbn¡p¶Xn\v ]Icw, hr¯¯n Asà ¦n tIm¬ a«aà F¶mWv sXfnbn¨ncn¡p¶Xv. Cu coXnbn sXfnbn¡p¶Xns\ Proof by contraposition F¶v hnfn¡p¶p)

Nn{X¯n PQ hymkamWv

1) Q

P

NphsS sImSp¯hbnÂ

LPRQ sâ Afhv BImhp¶tXXv ? R

(500, 700, 900, 1100 )

28 ...............................................................................................................................MUKULAM MATHS

2)

Nn{X¯n O hr¯tI{µw

O

A

B

NphsS sImSp¯hbnÂ

LAMB bpsS Afhv BImhp¶tXXv ? (850, 900, 950, 1000 ) M

C

P

Nn{X¯n AB hymkamWv

3) B

A

LAPB = 1250 Bbm LACB, LPAC Ch ImWpI

4)

X

Nn{X¯n XY hymkamWv

LAYZ = 200 Bbm Y

LXAY , LXZY Ch ImWpI

A

Z

Nm]hpw adpNm ]hpw

Nn{X¯n A, B F¶o _nµp¡Ä hr¯s¯ APB, AQB F¶o c­p Nm]§fmbn `mKn¡p¶p Chbn Nm]w APB bpsS adpNm]amWv Nm]w AQB

A

Q P

B

MUKULAM MATHS................................................................................................................................. 29

P

N

Nn{X¯n 'O' hr¯tI{µw

* O M

PMQ sâ

1)

Nm]w

2)

Nm]w PMQ sâ tI{µ tIm¬ GXv ?

3)

Nm]w PMQ, Nm]w PNQ ChbpsS tI{µtImWpIfpsS XpIsb{X ?

Q

adpNm]w GXv ?

C

Nn{X¯n 'O' hr¯tI{µw

* O

A

B D

1)

Nm]w ADB bpsS tI{µtIm¬ GXv?

2)

Nm]w ADB se tIm¬ GXv ?

3)

Nm]w ADB bpsS adpNm]w GXv ?

4)

Nm]w ADB bpsS adpNm]¯nse tIm¬ GXv ?

Nm]¯nsâ tI{µtImWpw adpNm]¯nse tImWpw

WORKSHEET (2) R

1) Nn{X¯n 'O' hr¯tI{µw

S

L P = 400 L ORP = ..................... L POR = .....................

P

) 400

O

Q

LQOR = ..................... Nm]w QSR sâ tI{µ tIm W fhpw LQPR sâ Afhpw X½nepÅ _ÔtaXv ?

30 ...............................................................................................................................MUKULAM MATHS

M

2) Nn{X¯n 'O' hr¯tI{µw

L OAM = 300 L OBM = 400 BIp¶p

O

L OMA = ............

L OMB = ............

L AOM = ............

L BOM = ............

L AMB = ............

L AOB = ............

A B N

Nm]w ANB bpsS tI{µtImWfhpw L AMB bpsS Afhpw X½nepÅ _Ôsa´v ?

3) Nn{X¯n 'O' hr¯tI{µw

A

L OPA = 200 L OQA = 700 BIp¶p L OAQ = ............

L OAP = ............

L QOA = ............

L POA = ............

L PAQ = ............

L POQ = ............

O

Q

P B

Nm]w PBQ sâ tI{µtImWfhpw L PAQ sâ Afhpw X½nepÅ _Ôsa´v ?

4) Nn{X¯n 'O' hr¯tI{µw

L OAC = 400 L OBC = 650 L OCA = ............

L OCB = ............

L AOC = ............

L BOC = ............

L ACB = ............

L AOB = ............

O B A C

Nm]w ADB bpsS tI{µtImWfhpw L ACB bpsS Afhpw X½nepÅ _Ôsa´v ?

*

hr¯¯nse Hcp Nm]w tI{µ¯nep­m¡p¶ tImWnsâ ]IpXnbmWv B Nm]w AXnsâ adpNm]¯nse Hcp _nµphnep­m¡p¶ tIm¬ (sXfnhv ]mT]pkvXI¯nse t]Pv 33,34,35,36 Â)

MUKULAM MATHS................................................................................................................................. 31

Hcp tImWnsâ ]IpXn AfhpÅ tImWnsâ \nÀanXn B

*

AB sb P epw AC sb Q epw JÞn¡¯ ¡hn[w A tI{µambn hr¯w hcbv¡p¶p

*

tImWn\v ]pd¯pÅ hr¯`mK¯nse Hcp _nµp R ASbmfs¸Sp¯n PR, QR hcbv¡p¶p

C

A

(LBAC, LPRQ Ch X½nepÅ _Ôsa´v ?) *

NphsS sImSp¯ Hmtcm Nn{X¯nepw x sâ hne IW¡m¡pI (Hmtcm¶nepw O hr¯tI{µw)

) x0

650 ) 800

O

*

x0

O

O

300 x

0

NphsS sImSp¯ Hmtcm Nn{X¯nepw x,y ChbpsS hne ImWpI (Hmtcm¶nepw O hr¯tI{µw) ) 1250 )

x0

)

15 0

)

y0

y0

)

)

400

y0 )

O

x0 ) 500

800 x0

hr¯ JÞ§Ä Hcp hr¯¯nse GXp RmWpw AXns\ c­p `mK§fm¡p¶p. C¯cw `mK§sf hr¯ JÞ §Ä F¶mWv ]dbp¶Xv.

32 ...............................................................................................................................MUKULAM MATHS

R

S

Nn{X¯n LPRQ = 500 Bbm .

P

Nm]w PTQ sâ tI{µ tImWfhv F{X?

LPSQ F{X ? T Q

{]hÀ¯\w D

C

E

A B

Nn{X ¯n ImWp ¶ Xp t]mse Hcp ImÀUv t_mÀUn hcbv¡p¶p. tjUv sNbvX `mKw sh«nsbSp¯v D,E F¶o _nµp ¡fn tNÀ¯psh¨m Hmtcm tImWpw H¶nt\m sSm¶v tNÀ¶p \n¡p¶Xmbn ImWmatÃm ( LACB, LADB, LAEB Ch Hmtcm¶pw Htc hr¯ JÞ¯nse tImWpIfmWv)

Hcp hr¯JÞ¯nse tImWpIsfÃmw adpJÞ¯nse Nm]¯nsâ tI{µtImWnsâ ]IpXn BbXpsIm­v Ah Xpeyambncn¡pw *

AXmbXv Htc hr¯JÞ¯nse tImWpIÄ XpeyamWv. D

.

Nn{X¯n LADC = 750 BIp¶p C

Nm]w ABC bpsS tI{µ tImWfhv F{X? Nm]w ADC bpsS tI{µtImWfhv F{X ?

A

LABC F{X ? B

{]hÀ¯\w

D

. A

Nn{X ¯n ImW p ¶ Xp t]mse ImÀUvt_mÀUn hcbv¡p¶p C

Hcp

tjUv sNbvX `mKw (Htc BcapÅ hr¯mwi§Ä) sh«nsbSp¯v NphsS ImWp¶Xpt]mse tNÀ¯p sh¡p¶p

B MUKULAM MATHS................................................................................................................................. 33

( LADC, LADC Ch adpJÞ§fnse tImWpIfmWv) Hcp tPmSn adpJÞ§fnse Nm]§fpsS tI{µtImWpIfpsS XpI 3600 BWv. AXp sIm­v Ahbnse tImWpIfpsS XpI 1800 Bbncn¡pw. *

AXmbXv adpJÞ§fnse tImWpIÄ A\p]qcIamWv.

(kqN\ : apIfn sNbvX c­p {]hÀ¯\§Ä sF.kn.än SqÄ D]tbmKn¨pw X¿mdm¡m hp¶XmWv) *

Nn{X¯nse PQR Hcp a«{XntImWamWv. LA bpw LP bpw BC = QR BWv ABC bpsS ]cnhr¯¯nsâ hymkw PQ sâ \of¯n\v XpeyamsW¶v sXfnbn¡pI

B

A

P

C

Q

R

(kqN\ : BC bv¡v Cew_ambn bn IqSn hcbv¡p¶ tcJ ABC bpsS ]cnhr¯s¯ D ABC bpsS ]cnhr¯¯nsâ hymkw BD Bbncn¡patÃm JÞn¡p¶psh¶ncn¡s«. . BCD = QRP . BD . = PQ]

C P Nn{X¯n LBAC = 550 Bbm LBPC F{X ?

1)

A

B

34 ...............................................................................................................................MUKULAM MATHS

S

2) P

R

Nn{X¯n LPQR = 1000 Bbm LPSR F{X ?

Q D E Nn{X¯n LABD = 650 Bbm LACD F{X ? LAED F{X ?

3) C

B

A

hr¯hpw NXpÀ`qPhpw * Hcp NXpÀ`qP¯nsâ aqeIsfÃmw Hcp hr¯¯nemsW¦n AXnsâ FXnÀtImWpIÄ A\p]qcIamWv. (adpJÞ§fnse tImWpIÄ A\p]qcIamWv F¶ Bibw D]tbmKn¡matÃm) CXv adn¨p]dªm icnbmIptam ? AXmbXv NXpÀ`qP¯nsâ FXnÀtImWpIÄ A\p]qcIamsW¦n AXnsâ \mev aqeI fn IqSnbpw IS¶pt]mIp¶ Hcp hr¯w hcbv¡m³ Ignbptam ? Hcp NXpÀ`qP¯nsâ aq¶v aqeIfn IqSn hcbv¡p¶ hr¯¯n\v ]pd¯mWv \mem as¯ aqesb¦n B aqebntebpw FXnÀaqebntebpw tImWpIfpsS XpI 1800 tb¡mÄ Ipd hmWv. AI¯msW¦n XpI 1800 tb¡mÄ IqSpXepw

MUKULAM MATHS................................................................................................................................. 35

D E E

D

C

C

A

A B B

Nn{X¯n L B, L D ChbpsS XpI 1800  Ipdhmbncn¡pw

Nn{X¯n L B, L D ChbpsS XpI 1800  IqSpXembncn¡pw

(sXfnhv ]mT]pkvXI¯nse t]Pv 48, 49 Â) kqN\ : sXfnhn\mhiyamb ap¶dnhv Dd¸n¡p¶Xn\v hÀ¡vjoäv (1)  Aev]w amäw hcp¯n D]tbmKs¸Sp¯mhp¶XmWv) A,B,C F¶o _nµp¡fn IqSnbpÅ hr¯s¯ ASnØm\am¡n \memas¯ _nµp (D) Nen¸n¨v Bibw t_m²ys¸Sp¯mhp¶XmWv D

C

NXpÀ`pPw ABCD  L B + L D=1800 BsW¶ncn¡s«. A,B,C Chbn IqSn Hcp hr¯w hc¨m hr¯¯n\v ]pd¯Ã (]pd¯mIWsa¦n L B, L D ChbpsS XpI 1800 tb¡mÄ IpdhmIWw)

A

*

B

D hr¯¯n\v AI¯Ã (AI¯msW¦n L B, L D ChbpsS XpI 1800 tb¡mÄ IqSpXemIWw) AXp sIm­v D hr¯¯n Xs¶bmWv.

Hcp NXpÀ`pP¯nsâ FXnÀtImWpIÄ A\p]qcIamsW¦n AXnsâ \mep aqeIfn IqSnbpw IS¶pt]mIp¶ Hcp hr¯w hcbv¡mw (C¯cw NXpÀ`pPs¯ N{Iob NXpÀ`pPw F¶v hnfn¡pw)

*

NphsS sImSp¯ncn¡p¶hbn N{Iob NXpÀ`qPw GXv ? AÃm¯h GXv ? ImcWw hniZam¡pI (i)

FÃm tImWpIfpw

(ii)

Hcp tIm¬ 800 Bb kmam´cnIw

(iii)

kma´chi§fnsem¶nse tImWpIÄ 700 hoXw Bb ew_Iw

(iv)

kam´chi§fnsem¶nse tImWpIÄ 800, 900 Bb ew_Iw

Xpeyamb NXpÀ`pPw

36 ...............................................................................................................................MUKULAM MATHS

*

NXpcaÃm¯ kmam´cnI§sfm¶pw N{Inbaà F¶v sXfnbn¡pI (kqN\ : kmam´cnIw ABCD bn L A = L C NXpcaÃm¯Xn\m L A = 900 AXpsIm­v L A + L C = 180 )

*

ka]mÀizaÃm¯ ew_I§sfm¶pw N{Iobaà F¶v sXfnbn¡pI (kqN\ : PQRS Hcp ka]mÀizaÃm¯ ew_IamsW¦n (PQ || RS)

LP=LQ PQ || RS BbXpsIm­v L P + L S = 180 AXpsIm­v L Q + L S = 180 ] 1)

N{Inb NXpÀ`pPw ABCD  L C = 500 , L D = 95 0 Bbm L A, L B Ch ImWpI

2)

NXpÀ`pPw ABCD  L A = 800 , L B = 110 0, L C = 85 0 BIp¶p. L D F{X? DbpsS Øm\w A,B,C Chbn IqSn hcbv¡p¶ hr¯¯nemtWm, ]pd¯mtWm, AI¯mtWm F¶v Is­¯pI.

3)

NXpÀ`pPw PQRS  L P + L R = 2000 BIp¶p. S sâ Øm\w P,Q,R Chbn IqSn hcbv¡p¶ hr¯¯nemtWm ]pd¯mtWm, AI¯mtWm F¶v \nÀWbn¡pI

Nm]JÞ\w Nn{X¯n AOB, COD Ch kZriamtWm ? F´psIm­v ?

)

1)

D

B )

500

500

O C A

S

Q

Nn{X¯n PMQ, RMS Ch kZriamtWm ? F´psIm­v ?

2) M

P

R

MUKULAM MATHS................................................................................................................................. 37

D *

Nn{X¯n AB, CD F¶o RmWpIÄ P  JÞn¡p¶p kZri {XntImW§fpsS XpeytImWpIÄs¡Xn scbpÅ hi§Ä B\p]mXnIamWv F¶ hkvXpX D]tbmKn¨v PA PD F¶v sXfnbn¡matÃm = PC PB

B P C

1

A

(CXnsâ KpW\ cq]w PA X PB = PC X PD )

)

Nn{X¯n OQR, OSP Ch kZriamtWm ? F´psIm­v ?

Q P

60 0

O )

60 0

S

R

B

Nn{X¯n OBC, ODA Ch kZriamtWm ? F´psIm­v ?

2 A C

D

*

O

B

Nn{X¯n BA, DC F¶o RmWpIÄ \o«n hc¨v, hr¯¯n\v ]pd¯v P  JÞn¡p¶p

A P C

PBC Â LPCB bvs¡Xncmb hiw GXv ? PDA Â L PAD bvs¡Xncmb hiw GXv ?

D 38 ...............................................................................................................................MUKULAM MATHS

PB kZri{XntImWp§fpsS XpeytImWpIÄs¡Xncmb hi§Ä B\p]mXnIamWv F¶XpsIm­v F¶v sXfnbn¡mw PA = PD PC PB (CXnsâ KpW\ cq]w PA X PB = PC X PD ) C

A

Nn{X¯n AB hymkw CD I AB AB, CD Ch P  JÞn¡p¶XpsIm­v PA X PB = PC X PD hr¯tI{µ¯n \n¶pw RmWnte¡pÅ ew_w RmWns\ ka`mKw sN¿p¶Xn\m AXpsIm­v PA X PB = PC2

B

P

D * hr¯¯nse Hcp hymkhpw AXn\p ew_amb RmWpw FSp¯mÂ, Rm¬ hymks¯ apdn¡p¶ c­p `mK§fpsS KpW\^ew RmWnsâ ]IpXnbpsS hÀKamWv 1) Hcp hr¯¯n AB, CD F¶o RmWpIÄ P  JÞn¡p¶p. PA = 20 cm, PB = 18 cm., PC = 24 cm. Abm PD F{X?

Nn{X¯n AP = 8 cm. AQ= 18 cm AR = 16 cm. Bbm AS F{X ?

Q 2)

P A

S

R

E

2)

C

F

D

Nn{X¯n CD hymkamWv. EF I CD, CF = 4 cm DF = 9 cm Bbm EF F{X ?

MUKULAM MATHS................................................................................................................................. 39

4)

7 sk.an. \ofapÅ Hcp tcJ hcbv¡pI

5)

12 N.sk.an. ]c¸fhpÅ Hcp kaNXpcw hcbv¡pI

NXpc¯n\v Xpey]c¸fhpÅ kaNXpc¯nsâ \nÀanXn C

D

E A

H

B

F

G

* NXpcw ABCD hcbv¡p¶p. AB F¶ hiw \o«nhc¨v AXn BE = BC BI¯¡hn[w E ASbmfs¸Sp¯p¶p. AE hymkambn AÀ²hr¯w hcbv¡p¶p.CB \o«nhc¨v AÀ[hr¯s¯ F  JÞn¡p¶p. BF hiambn kaNXpcw BFGH \nÀ½n¡p¶p. (kaNXpcw BFGHsâ ]c¸fhv = BF2 = AB X BF = AB X BC = NXpc¯nsâ ]c¸fhv ) {XntImW¯nsâ Xqey]c¸fhpÅ kaNXpc¯nsâ \nÀ½nXn A

ABC bpsS AtX ]c¸fhpÅ NXpcw Nn{X¯n ImWn¨ncn¡p¶Xpt]mse \nÀan¡mw. E (BC bv¡v kam´cambn AB bpsS

D

a[y_nµphneqsS hcbv¡p¶p) NXpcw BCDE bpsS AtX ]c¸fhpÅ kaNXpcw \nÀ½n¡matÃm

B

C

NXpÀ`pP¯n\v Xpey]c¸fhpÅ kaNXpc¯nsâ \nÀanXn NXpÀ`pP¯n\v Xpey]c¸fhpÅ {XntImW¯nsâ \nÀ½nXn 9þmw Xc¯n ]Tn¨n«p­ tÃm. {XntImW¯n\v Xpey]c¸fhpÅ NXpchpw NXpc¯n\v Xpey]c¸fhpÅ kaNXpchpw hcbv¡mw. 40 ...............................................................................................................................MUKULAM MATHS

aqey\nÀ®b {]hÀ¯\§Ä C

1)

A

O 3 cm

B

O tI{µamb hr¯¯nsâ Bcw 3 sk.an. ABC bn AC = BC F¦n AC bpsS \ofw F{X ?

2)

O

P

Q

Nn{X¯n O hr¯ tI{µamWv.

LPQR = 500 Bbm LQPR F{X ?

R C

Nn{X¯n AC=BC BWv.

3)

A

4)

B

LA = 200 Bbm AB hymkambn hcbv¡p¶p. hr¯s¯ ASnØm\am¡n C bpsS Øm\w Fhn sSbmbncn¡pw ?

AB = 6 cm, BC = 8 CM, AC = 10 cm

ABC bnse AC hymkambn hcbv¡p¶ hr¯s¯ ASnØm\am¡n B bpsS Øm\w FhnsSbmWv. BC hymkambn hcbv¡p¶ hr¯s¯ ASnØm\ambn A bpsS Øm\w FhnsS bmWv ?

MUKULAM MATHS................................................................................................................................. 41

5)

Nn{X¯n C hr¯tI{µw. Nm]w AEB bpsS tI{µtIm¬ 700 BbmÂ

C

D

A

B

(a)

LADB F{X ?

(b)

LCAB F{X ?

(C)

LAEB F{X ?

E

L K 6)

Nn{X¯nÂ

LMN se aq¶v tImWpIfpw

XpeyamWv. F¦n LMKN F{X ?

M

N

Q

7)

D

(

C

Nn{X¯n LCBP = 950 , LDCQ = 800 NXpÀ`pPw ABCD bpsS FÃm tImWpIfpw Is­¯pI A

B

P

42 ...............................................................................................................................MUKULAM MATHS

R

8)

P

C tI{µamb hr¯¯nse ST F¶ Rm¬

C Q S T

9)

PQ F¶ hymk¯n\v kam´camWv

LQRS = 750 BWv. (a)

LQCS F{X ?

(b)

LCQS F{X ?

(C)

LQST F{X ?

Xmsg sImSp¯ncn¡p¶ \n_Ô\IÄ¡v hnt[bambn Hcp {XntImWw \nÀ½n¡pI F)

{XntImW¯nsâ ]cnhr¯¯nsâ Bcw 4 sk.an.

_n)

c­v tImWpIÄ 500, 700 hoXw

10)

O tI{µamb hr¯¯n LOAB = 200 BbmÂ

O

D

LOCB = 550 Bbm A

B

(a)

LAOB F{X ?

(b)

LAOC F{X ?

C C D 11)

C A

B

Nn{X¯n AB hr¯¯nsâ hymkhpw

LADC = 1300 bpw BWv. LBAC IW¡m¡pI

MUKULAM MATHS................................................................................................................................. 43

D 12)

C A

Nn{X¯n LA = 2x, LB = y -10, LC = y-40, LD = x+20 F¶n§s\ BIp¶p. ChbpsS hne ImWpI x,y ChbpsS hne ImWpI

B

D 13)

Nn{X¯n AB, CD F¶nh ]ckv]cw ew_amb c­v RmWpIfmWv. Ch P F¶ _nµphn JÞn¡p¶p.

A

B

P

C

14)

AB = 18 cm, PB = 12 cm, AC=10 cm F¦n PD, BD F¶nh IW¡m¡pI

NXpÀ`pPw ABCD bn AB = 6 cm, LA = 800, LB = 700, A D = 7 cm, BC=5 cm NXpÀ`pPw ABCD hc¡pI. Cu NXpÀ`pP¯nsâ Xpey]c¸fhpÅ kaNXpcw hc¡pI.

15)

ka]mÀiz{XntImWw ABC bn LA = LB BWv. AB = 5 cm, LA = 400 {XntImWw hc¨v AXn\v Xpey]c¸fhpÅ NXpcw hc¡pI N D

C

16)

Nm]w AMB bpsS tI{µtIm¬ 800 Nm]w CND bpsS tI{µtIm¬ 500 Bbm LAPB F{X ? (kqN\ : AD tbmPn¸n¡pI. PADbpsS _mlytImWmWv LAPB )

P

A

B M

44 ...............................................................................................................................MUKULAM MATHS

17) S Q B

A

P

R

Nm]w SBT bpsS tI{µtIm¬ 1100 Nm]w QAR sâ tI{µtIm¬ 500 Bbm LQPRF{X ? (kqN\ : QT tbmPn¸n¡pI. PQTbpsS _mlytImWmWv LSQT )

T

D

18)

E S

Nn{X¯n AB I CD, Nm]w BED Nm]w AFC ChbpsS tI{µtImWfhpIfpsS XpI 1800 F¶v sXfnbn¡pI (kqN\ : AD hc¡pI ) B

A F C

*

S P

Nn{X¯n O hr¯tI{µw. LQSR = x0 Bbm LPQR F{X ?

O

Q R

MUKULAM MATHS................................................................................................................................. 45

*

C

A

Nn{X¯n O hr¯tI{µw. LBAC, LABC, LADC Ch ImWpI

)

1100 O

B

D G E

*

D

Nn{X¯n ABCDEF Hcp kajUv`pPamWv. L EGD F{X ? C

F

A

B A

*

B

E

F

Nn{X¯n ABCDEF Hcp ka]©`pPhpw AFG Hcp ka`qP{XntImWhpamWv. AB, BF, FC, ABF F¶o sNdnb Nm]§Ä hr¯ ¯nsâ F{X `mKamsW¶v I­p]nSn¡pI

G D

C

*

O

) )

60 0 70 0

A

C

Nn{X¯n O hr¯tI{µamWv. LAOB = 600 LBOC = 700 ABC hc¨m LABC F{X ? LBAC F{X ? LACB F{X ?

B

46 ...............................................................................................................................MUKULAM MATHS

C *

O

Nn{X¯n Xpey AfhpIfpÅ tImWpIÄ tPmUnbmbn FgpXpI

D B A

F

G

*

E

H

A

D

Nn{X¯n AjvS`pP¯nse FÃm ioÀj§fpw Htc hr¯¯nemWv. LA +LC +LE +LG = LB +LD +LF+LH F¶v sXfnbn¡pI

C

B

E

D

*

20 0

O A

(

C 100 0

(

B

Nn{X¯n NXpÀ`pPw ABCD N{Inb NXpÀ`pPam Wv. O hr¯tI{µw hiw AD, E bnte¡v \o«nbncn ¡p¶p. NXpÀ`pPw ABCD bnse FÃm tImWpIfpw I­p]nSn¡pI. LAEC ImWpI

MUKULAM MATHS................................................................................................................................. 47

*

D

Nn{X¯n O hr¯tI{µw LB = 1000 LBCE = 800 NXpÀ`pPw ABCE bnse FÃm tImWpIfpw ImWpI CED bnse FÃm tImWpIfpw ImWpI

)

C

80

0

E O

A

)

1000

B

F

*

Nn{X¯n LB = 1100 hr¯tI{µw LAEC, LADC, LAFC ChbpsS Afhv 500, 800,700, F¶n§s\bmWv. tImWnsâ t]cnsâ {Ia¯n eÃ. AhbpsS Afhv X¶ncn¡p¶Xv. F¦nÂ

D A E

LAEC LADC LEAD LDAF

1100

B C

*

A )

E O B

) 80

0

C

.............................. .............................. .............................. ..............................

c­v hr¯§Ä C,D F¶o _nµp¡fn JÞn¡p¶p. A,D,E Ch Htc tcJbnemWv. AXpt]mse B,C,F F¶nhbpw. c­v NXpÀ`pP§fnse FÃm ioÀj§fnsebpw tImWpIÄ I­p ]nSn¡pI

D

70 0

= = = =

F

B A

*

Q

R P

C

Nn{X¯n Nm]w BQD bpsS tI{µ tIm¬ 1000 bpw Nm]w ARC bpsS tI{µtIm¬ 800 bpw Bbm LP ImWpI

D

48 ...............................................................................................................................MUKULAM MATHS

A²ymbw þ3

c­mw IrXn kahmIy§Ä BapJw KWnX {]iv\]cnlmc¯n\v (AÚmXkwJy Is­¯p¶Xn\v) _oPKWnX¯nsâ km[yXIÄ Ip«nIÄ CXnt\mSIw ]cnNbs¸«n«p­v. hn]coX {InbIÄ hgn a\¡W¡mbn {]iv\ ]cnlmcw Is­¯msa¶pw CXn\v {]bmkw t\cnSp¶ kµÀ`§fn _oPKWnXhmIy §Ä cq]oIcn¨v eLqIcWw hgn AÚmXkwJybn F¯nt¨cmsa¶pw AhÀ a\Ênem¡n bn«p­v. AÚmXkwJybpsS H¶mw IrXn am{Xw DÄs¸Sp¶ kahmIy§fmWv CXphsc bpÅ ¢mÊpIfn AhXcn¸n¨Xv. Cu A²ymb¯n AÚmXkwJyIfpsS c­mw IrXn IqSn DÄs¸Sp¶ kahmIy§fpw AhbpsS ]cnlmc amÀK§fpamWv {]Xn]mZn¡p¶Xv. {]iv\ hniIe\w \S¯n AXns\ KWnX]cambn Npcp¡n FgpXm³ (_oPKWnX hmIy cq]oIcWw) Ip«nIÄ t\Snb tijnIÄ IqSpX sa¨s¸Sp¯m\pÅ {]hÀ¯\§Ä AhXcn¸nt¡­Xp ­v. {]iv\§fpsS {]tXyIX A\pkcn¨v hnhn[ ]cnlmcamÀK§fn A\ptbmPyambXv kzo Icn¡m³ Ip«nIÄ¡v km[n¡Ww. c­mwIrXn kahmIy§fpsS ]cnlmc¯n hÀKaqew ImWp¶ {]{Inb kzm`mhnIambpw IS¶phcpw. GXv A[nkwJy¡pw c­v hÀKaqe§ÄDÅ Xn\m AÚmXkwJy¡v c­v hneIÄ hsc In«ntb¡mw. Chbn bpàamb ]cnlmcamWv kzoIcnt¡­Xv. Nne kahmIy§Ä¡v ]cnlmcaps­¦nepw {]mtbmKnI {]iv\§Ä¡v Ah ]cnlmcaÃmsX hcmw. Hcp kahmIyw Xs¶ hnhn[ {]tbmKnI {]iv\§fpsS ]cnlmc¯n\v klmbIcamImw. Cu ]mT]pkvXI¯nse Xs¶ aäv A[ymb§fnse KWnX{]iv\]cnlmc ¯n\pw aäp hnjb§fnse C¯cw {]iv\§fpsS ]cnlmc¯n\pw Cu A[ymbw klmbIam hp¶p­v. IqSmsX kahmIy§fpw _lp]Z§fpw X½nepÅ _Ôhpw NÀ¨ sN¿s¸Sp¶p­v. {][m\ Bib§Ä *

c­mw IrXn kahmIy¯nsâ cq]oIcWw

*

c­mw IrXn kahmIy§Ä¡v c­v ]cnlmc§Ä hsc D­mhmw

*

]qÀWhÀK cq]¯nepÅ kahmIy§fpsS ]cnlmcw

*

hÀK¯nIhv

* * *

x=-b+

b2-4ac 2a ax2+bx+c=0 F¶ kahmIy¯n b2-4ac (hnthNIw) bpsS {]kàn AYhm hnthN I¯nsâ hnebpw ]cnlmchpw X½nepÅ _Ôw ax2+bx+c=0 F¶ kahmIy¯nsâ ]cnlmcw

c­mw IrXn kahmIy§fpw c­mw IrXn _lp]Z§fpw X½nepÅ _Ôw

]T\ {]hÀ¯\§Ä *

48 sk.an. NpäfhpÅ kaNXpc¯nsâ Hcp hi¯nsâ \ofw F´v ? CtX NpäfhpÅ ka`pP{XntImW¯nsâ Hcp hi¯nsâ \ofw ImWpI. CtX Npäf hpÅ kajUv`pP¯nsâ hitam ?

MUKULAM MATHS................................................................................................................................. 49

*

Hcp NXpc¯nsâ Npäfhv 80 sk.an., hoXn 12 sk.an., F¦n \ofw F´v ? CtX NpäfhpÅ asämcp NXpc¯nsâ \ofw, hoXntb¡mÄ 10 sk.an. IqSpXemWv. B NXpc¯nsâ hoXn F´v ? \ofw F´v ?

*

XpSÀ¨bmb c­v F®Â kwJyIfpsS XpI 25. kwJyIÄ Gh ?

*

XpSÀ¨bmb c­v HäkwJyIfpsS XpI 36. kwJyIÄ Gh?

*

81 N.sk.an. ]c¸fhpÅ kaNXpc¯nsâ Hcp hi¯nsâ \ofsa´v ?

*

Hcp kwJybpsS hÀKw 81 Bbm kwJytbXv?

*

Hcp kwJybpsS hÀKt¯mSv 6 Iq«nbt¸mÄ 150 In«n. kwJytbXv?

*

Hcp kaNXpc¯nsâ ]c¸fhnt\¡mÄ 60 N.sk.an. IqSpXemWv Hcp NXpc¯nsâ ]c ¸fhv. NXpc¯nsâ ]c¸fhv 285 N.sk.an. F¦n kaNXpc¯nsâ hi¯nsâ \of sa´v ?

*

Hcp kwJybpsS hÀK¯nsâ 3 aS§n \n¶v 2 Ipd¨m 190 In«pw. kwJy GXv ?

*

Hcp kaNXpc¯nsâ hit¯¡mÄ 3 sk.an. IqSpX hiapÅ asämcp kaNXpc¯n\v 400 N.sk.an. ]c¸fhp­v. BZy kaNXpc¯nsâ Hcp hi¯nsâ \ofsa´v ?

*

Hcp kaNXpc¯nsâ Npäfhnt\¡mÄ 20 sk.an. IqSpX NpäfhpÅ asämcp kaNXpc ¯n\v 144 N.sk.an. ]c¸fhp­v. sNdnb kaNXpc¯nsâ hisa´v ? Cu {]iv\§fpsS ]cnlmcw a\¡W¡mtbm kahmIyw cq]oIcnt¨m Ip«nIÄ I­p]n Sn¡s«. (a\¡W¡mbn I­p]nSn¨ ]cnlmc§Ä kahmIy§Ä cq]oIcn¨pw I­p]n Sn¨p t\m¡s«) (Cu kahmIy§fn H¶mw IrXnbmbh, c­mw IrXnbmbh thÀXncn¡s«.)

kqN\ : kaNXpc¯nsâ ]c¸fhv 81 N.sk.an.F¦n hiw = 9 sk.an. kwJybpsS hÀKw 81 Bbm kwJy +9 Asæn þ9 BImw. ]s£ kaNXpc¯nsâ hiw þ9 BhnÃtÃm

kahmIy§fpsS ]cnlmcw ImWpI Set 1

Set 2

*

x2

=

25

*

x2 +1 =

50

*

3x 2

=

108

*

x2 +5 =

86

*

1 x2 2 4 x2 5 8 x2 3

=

98

*

x2 - 3

118

=

180

*

2x2+1 =

19

=

96

*

5x2-3 =

77

* *

=

50 ...............................................................................................................................MUKULAM MATHS

Set 3

Set 4

*

(x+1)2

=

25

*

(x_1)2

=

16

*

(x+3)2

=

64

*

(x_3)2

=

49

*

(x+7)2

=

225

*

(x_7)2

=

625

Set 5

Set 6

*

(2x+1)2

=

81

*

(2x_1)2

=

49

*

(3x + 2 )2

=

64

*

(3x _2 )2

=

169

*

(5x + 3 )2

=

324

*

(5x _ 3 )2

=

289

WORK SHEET Set 1

Set 2

*

x2+8x+16 = (x+4)2

*

x +16x+64 = (.....+.....)

*

x _10x+25 = (....._.....)2

*

x +2x+1 = .....................

*

2

2

*

2

*

2

*

Set 3

x2+3x+ 9 4 _ m2 7m+ 9 64 2_ a 3a+ 4 4 2_ x 5x+ 25 7 196

=

.....................

=

.....................

=

.....................

=

.....................

Set 4

*

4x2 + 28x+49 = ...........................

*

*

25y2 + 90y + 81 = .......................

*

*

1 x2+ 4 x + 4 = 9 9 9

*

.....................

4 y2_ 1 y+ 1 = 9 3 16 9 x2_ x+25 = 25 36 2 _ x 5x +25 = 16 4 4

..................... ..................... .....................

MUKULAM MATHS................................................................................................................................. 51

WORK SHEET ]qcn¸n¡pI *

a2+2ab+b2 = (..................+.................)2

*

*

a2 _ 2ab+b2 = (...... _ ........)2

*

*

x2+ 10x+.............. = (x.+ 5)2

*

*

a2 _ 10 a+.........= (a _ 5)2

*

*

y2+ 8 y+.........= (y + 4)4

*

*

x2 _ 6x+.........= (x_ 3)2

*

5 k2+5k+..............= (k+ 2 )2 p2_ 13 p+..............= (p+13 )2 2 1 2 x + x + ..............= (x+ )2 2 1 1 x2_ 2 x +.............= (x_ 4 )2 7 a2 + 3 x + .............= (a + 7 )2 6 5 x2 _ x + .............= (x _ 5 )2 4 8

kaNXpc \nÀ½mWw 6

10

6

5

5

5

Cu cq]w c­mbn apdn¨v tNÀ¯v sh¨v kaNXpcw \nÀ½n¡mtam ? \nÀ½n¨ kaNXpc ¯nsâ hiw F´v? C\n NphsS sImSp¯ncn¡p¶ cq]w c­mbn apdn¨v tNÀ¯psh¨v kaNXcpw D­m¡n t\m¡q. kaNXpc¯nsâ hiw F´v ? x

2y

x

y

y 52 ...............................................................................................................................MUKULAM MATHS

hÀK¯nIhv NXpcmIrXnbmb Hcp ISemkv FSp¡pI. NXpc¯nsâ hoXn, hiambn hcp¶ kaN Xpcw AXn \n¶pw apdn¨p amäp¶p. tijn¡p¶ NXpcs¯ Nn{X¯n kqNn¸n¡p¶Xpt]mse c­v XpeyNXpc§fmbn apdn¡p¶p. Ch apdn¨pamänb kaNXpc¯nsâ hi§fn tNÀ¯psh ¡pI. henb kaNXpcw ]qÀ¯nbm¡p¶Xn\v Hcp aqebn sNdnsbmcp kaNXpcw IqSn tNÀ¯p sht¡­Xp­tÃm....

(1)

(2)

(4)

{Ia \¼À

(3)

(5)

\ofhpw hoXnbpw

1

\ofw, hoXntb¡mÄ 2 sk.an. IqSpt¼mÄ

2

\ofw hoXntb¡mÄ 6 sk.an. IqSpXÂ

3

\ofw hoXntb¡mÄ 10 sk.an. IqSpXÂ

4

\ofw hoXntb¡mÄ 12 sk.an. IqSpXÂ

tNÀ¯psht¡­ sNdnb kaNXpc¯nsâ ]c¸fhv

5 \ofw, hoXntb¡mÄ 6 sk.an. IqSpXemb NXpc¯nsâ ]c¸fhv 55 N.sk.an. Bbm hoXnsb´v ? aqebn tNÀ¯psht¡­ kaNXpc¯nsâ hiw = 3 sk.an. AXnsâ ]c¸fhv = 9 N.sk.an henb kaNXpc¯nsâ ]c¸fhv = 55 + 9 = 64 N.sk.an. henb kaNXpc¯nsâ hiw = 8 sk.an. MUKULAM MATHS................................................................................................................................. 53

BZy NXpc¯nsâ hoXn Cu {]iv\w _oKWnX coXnbn Nn´n¨m NXpc¯nsâ hoXn \ofw \ofw ]c¸fhv

= =

8þ3 5 sk.an.

= = = = = = =

x sk.an. .......................... .......................... .......................... 55 ...................... 55 + 9 = 64

(x+6 ) x ........................ x2+6x+9 (x+3 ) 2 = 64 x+3 = + ...................... x = + .......... _ 3 x = ......... Asæn (Cu {]iv\¯n x \v kzoIcn¡mhp¶ hnetbXv ? F´psIm­v ?)

hnth N\w 60 sk.an. hoXw \ofapÅ I¼n hf¨v NphsS sImSp¯ncn¡p¶ ]c¸fhpIfpÅ NXcpw D­m¡m³ {ian¡p¶p. * ]c¸fhv 200 N.sk.an. * ]c¸fhv 216 N.sk.an. * ]c¸fhv 225 N.sk.an * ]c¸fhv 250 N.sk.an. Hmtcm kµÀ`¯n\\pkcn¨vpw kahmIyw cq]oIcn¨v NXpc¯nsâ hoXnbpw \ofhpw Is­¯mtam ? kahmIy§Ä hnthNn¨v Adnbp¶Xn\v "hnthNIw' klmbn¡p¶p F¶ hkvXpX Cu {]hÀ¯\¯neqsS t_m[ys¸Sp¯matÃm. ax2 + bx + c = O F¶ c­mw IrXn kahmIy¯n b2_4ac > O Bbm 2 hyXykvX ]cnlmc§Ä b2_4ac = O Bbm Hcp ]cnlmcw am{Xw b2_4ac < O Bbm ]cnlmcw CÃ

54 ...............................................................................................................................MUKULAM MATHS

c­mw IrXn _lp]Z§fpw c­mw IrXn kahmIy§fpw X½nepÅ ]ckv]c _Ôw Eg.: 1 P (x) = x2+6x+9 P(_3) Bbm P(x) = x2+6x+9 P(_3) = (_3)2+6(_3)+9 = 9 _ 18 + 9 =0 _ \ P ( 3) = 0

]cnlmcw ImWpI x2 + 6x + 9 = 0 a = 1 b=6 c = 9 b2 _ 4ac = 62 _ 4x1x9 = 36 _ 36 = 0 _ 2 _ x = 6 + b 4ac 2a _ = 6+ 0 2x1 _ 6 _ = = 3 2

]cnlmcw = _3 Eg.: 2

P (x) = x2_3x_10 Bbm P(5), P(_2) F¶nh ImWpI P (x) = x2_3x_10 P(5) = (5)2+3(5)_10 = 25 _ 15 _10 =0 ... PC (5) = 0 P(_2) = (_2)2_3(_2)_10 = 4 + 6 _10 =0 ... P(_2) = 0

]cnlmcw ImWpI x2 _ 3x _10 = 0 F¶ kahmIy¯nsâ ]cnlmcw ImWpI

x2 _ 3x _10 = 0 _ a=1 b= 3 c = _10 hnthNIw = b2 _ 4 ac _ _ _ = ( 3)2 3 x 1x 10 9 +40 = 49 _ _ x= b + b2 4x 2a _ _ = ( 3) + 49 2x1 =3 + 7 2 3 + 7 Asæn 3 _ 7 2 10 Asæn _ 4 2 2 _ 5 Asæn 2 ]cnlmcw 5 Asæn _ 2

MUKULAM MATHS................................................................................................................................. 55

ASSIGNMENT

1)

_lp]Z¯nsâ hne

]cnlmcw ImWpI

P(x) = x2 _ 10x + 25

x2 _ 10x+ 25 = 0

F¶ kahmIy¯n P(5) ImWpI 2)

a2 + 8a+ 16

a2 + 8a + 16 = 0

F¶ kahmIy¯n _ P( 4 ) ImWpI _ P2 5 P + 6 _ 6 P(2) P(3 ) Ch ImWpI

_ P2 5 P + 6 = 0

4)

x2 + 4x _ 21 Â _ P(3 ), P( 7) Ch ImWpI

x2 + 4x _ 21 = 0

5)

x2 + 7x + 10 Â _ _ P( 2 ), P( 5) Ch ImWpI

x2 + 7x + 10 = 0

6)

6x2 + x __ 1 F¶ kahmIy¯n P( 1 ), P( 1) Ch ImWpI 3 2

6x2 + x_ 1 = 0

3)

56 ...............................................................................................................................MUKULAM MATHS

IqSpX ]cnioe\ {]iv\§Ä 1. c­v kwJyIfpsS hyXymkw 6 , AhbpsS KpW\^ew 315. kwJyItfh? 2. c­v kwJyIfpsS XpI 30, AhbpsS KpW\^ew 161. kwJyItfh ? 3. Hcp NXpc¯nsâ \ofhpw hoXnbpw bYm{Iaw 50 sk.an., 40 sk.an. F¶n§s\bmWv. Ch c­pw Xpeyambn hÀ²n¸n¨t¸mÄ ]c¸fhn 1000 N.sk.an. hÀ²\hv D­mbn. F¦n F{X bqWnäv hÀ²n¸n¨p.? 4. Hcp kaNXpc¯nsâ Hcp hiw 5 sk.an. hÀ²n¸n¡pIbpw kao]hiw 2 sk.an. Ipdbv¡p Ibpw sNbvXt¸mÄ In«p¶ NXpc¯nsâ ]c¸fhv 260 N.sk.an. Bbm kaNXpc¯nsâ Hcp hi¯nsâ \ofw F{X ? 5. sjÀen¡v eoem½tb¡mÄ 7 hbkv {]mbw IqSpXÂ, {iojbv¡v eoem½tb¡mÄ 2 hbkv Ipdhv, tjÀenbpsSbpw {iojbpsSbpw hbkpIÄ X½n KpWn¨m 136 In«pw. F¦n tjÀenbpsSbpw {iojbpsSbpw hbkpIfpsS hyXymkw F{X ? eoem½bpsS hbkv F{X? 6. Hcp kam´c t{iWnbpsS XpSÀ¨bmb aq¶v ]Z§fpsS KpW\^ew 231, CXnse a²y]Zw 7 Bbm aäp]Z§Ä Is­¯pI. 7. H¶p apX XpSÀ¨bmb F®Â kwJyIÄ I­v ]nSn¡p¶ L«¯n XpI 1830 In«n. F¦n F{X F®Â kwJyIfpsS XpIbmWv X¶ncn¡p¶ kwJy ? 8. Hcp a«{XntImW¯nsâ IÀ®w AXnsâ ]mZ¯nsâ c­v aS§nt\¡mÄ Hcp bq\näpw, ew_w ]mZ¯nt\¡mÄ 7 bqWnäpw IqSpXemWv. F¦n a«{XntImW¯nsâ hi§fpsS \ofw I­p]nSn¡pI. 9.

x2+10x+p = 0F¶

kahmIy¯nsâ 2 ]cnlmc§fpw Htc kwJyXs¶bmbm P bpsS hne

F{X? 10. NXpcw IrXnbmb Øe¯n\v 160 ao. Npäfhp­v. AXn\I¯v 1500 N.ao. hep¸¯n Hcp Ipfhpw D­v. _m¡nbpÅ Øe¯nsâbpw Ipf¯nsâbpw ]c¸fhv Xpeyambm Øe ¯nsâ hoXnsb{X ? 11. Hcp NXpc¯nsâ \ofw hoXnbpsS 3 aS§nt\¡mÄ 3 bqWnäv IqSpXemWv. AXnsâ hnIÀ®w \oft¯¡mÄ Hcp bqWnäv IqSpXemWv. NXpc¯nsâ \ofhpw hoXnbpw ImWpI ? 12. 60 sk.an. \ofapÅ I¼n hf¨v a«{XntImWam¡p¶p. CXnsâ IÀ®w 26 sk.an. Bbm aäv c­v hi§fpsS \ofw ImWpI ? 13. Nn{X¯n AC hymkw BWv. \ofw F{X?

BD=6 cm, BC bpsS

\ofw

AC tb¡mÄ 5 cm IpdhmWv. ABbpsS

D

6

A

B

C

57 MUKULAM MATHS................................................................................................................................. ................................................................

hr¯¯nsâ Bcw F{X ? 14. C

B A

E

D

Nn{X¯n AB = 8, BC = 10, DE = 7, AD = F{X ?

15. XpSÀ¨bmb 2 F®Â kwJyIfpsS hÀK§fpsS XpI 113, XpSÀ¶v hcp¶ 2 F®Â kwJy IÄ FgpXpI? 16. Hcp hr¯¯nsâ Bc¯nsâ \ofw (4x-2). hr¯¯nsâ ]pd¯pÅ Hcp _nµphnte¡pÅ AIew (10x-2) sXmSphcbpsS \ofw (9x-2) Chsb ASnØm\am¡n Hcp kahmIyw cq]o Icn¡pI. Bcw, sXmSphc F¶nhbpsS \ofw IW¡m¡pI? 17. Hcp kwJybpsSbpw AXnsâ hypÂ{Ia¯nsâbpw XpI 1 1 Bhnsöv sXfnbn¡pI. 4

18. c­v kwJyIÄ X½nepÅ hyXymkw 2. ChbpsS hypÂ{Ia§Ä X½nepÅ hyXymkw 2 63 Cu hkvXpXsb ax2+bx+c = 0F¶ cq]¯nepÅ kahmIyam¡n amäpI. kwJyIÄ Is­ ¯pI. 19. Hcp hr¯¯nse AB, CD F¶o RmWpIÄ P bn JÞn¡p¶p. AB = 18cm, PD= 7cm, CP= 8cm Bbm PB F{X? 20. 20 ao \ofhpw 12 ao hoXnbpw DÅ NXpcmIrXnbmb Hcp lmfnsâ Hcp `mK¯mbn NXpcmIr XnbnepÅ IeymW aÞ]w D­v. aÞ]¯nsâ \ofw, hoXn F¶nh 3: 2 F¶ Awi _Ô¯nemWv. IeymW aÞ]w HgnsIbpÅ lmfnsâ ]c¸fhv 216 N.ao Bbm IeymW aÞ]¯nsâ \ofhpw hoXnbpw F{X ? 21. S = ut + ½ at2 F¶Xv t sk¡ânepÅ Øm\m´cs¯ kqNn¸n¡p¶ kahmIyamWv u = 8 m/s, a = 10m/s2, S=49 m . F¦n t bpsS hne ImWpI 22. Hcp hr¯ kvXq]nIbpsS Bc¯nsâ c­v aS§nt\mSv c­v Iq«nbm AXnsâ D¶Xn In«pw. CXnsâ Ncnhv Dbcw 13 sk.an. Bbm kvXq]nIbpsS hymkw F{X? 23. HcmÄ _m¦n \n¶pw Hcp XpI ISw hm§n. BZyamkw 4000 cq]bpw XpSÀ¶v hcp¶ Hmtcm amkhpw sXm«p ap¶nse amkw AS¨Xnt\¡mÄ 50 cq] IqSpXepw AS¨v sIm­ncp¶p. Ahkm\ amkw 5450 cq] AS¨t¸mÄ AbmfpsS ISw XoÀ¶p. BZys¯ 4 amkw AS¨ cq] F{X hoXamWv ? ISw Xocm³ F{X amkw th­n h¶p ? BsI AS¨ XpI F{X ? 24. Hcp tPmen sNbvXv XoÀ¡m³ a bv¡v b tb¡mÄ 5 Znhkw Ipd¨p aXn. c­pt]cpw IqSn Hcp an¨v tPmen sN¿pIbmsW¦n 6 Znhkw sIm­v B tPmen Xocpw. F¶m a bv¡pw b bv¡pw X\n¨v B tPmen sNbvXv XoÀ¡m³ F{X Znhkw hoXw th­n hcpw ? 58 ...............................................................................................................................MUKULAM MATHS

25.

½

½

½

½

I x

x

x

x

x

I

x apIfn X¶ncn¡p¶ cq]§Ä tNÀ¯v Hcp kaNXpcw D­m¡nbt¸mÄ AXnsâ ]c¸fhv 144 N.sk.an. F¶v In«n. henb kaNXpc¯nsâ hi¯nsâ \ofw F{X ? 26. ax2+bx+c = 0F¶ kahmIy¯n a_b+c = 0 Bbm Cu kahmIy¯n\v Hcp ]cnlmcw Ds­¶v sXfnbn¡pI? 27. 2n2 + 5nF¶Xv kam´c t{iWnbnse 'n']Z§fpsS XpIsb kqNn¸n¡p¶p. 1375 Cu t{iWnbnse XpSÀ¨bmb GXm\pw ]Z§fpsS XpIbmbm ]Z§fpsS F®w F{X ? 28 B

A

D

C

Nn{X¯n BD F¶ hiw LABC bpsS ka`mPnbmWv. DC F¶ hiw AD tb¡mfpw 3 bqWnäv IqSpXemWv. AD bpsS c­v aS§v \ofap­v AB bv¡v. ADtb¡mfpw IqSpXemWv. BC F¦n ABC bpsS hi§fpsS \ofw ImWpI. 29 (m+1)x2+2(m+3)x+2m+3 = 0 F¶ kahmIy¯nsâ 2 ]cnlmc§fpw Xpeyambm 'm' F{X ? 30. ]cnlmcw Is­¯pI a) x2+12x+8 = 2x _ 16 b) x(x+3) = x + 15 c) (x_ 1)2 _ 4 = 0 d) (x + 3)2 _ 9 = 0 e) (x+8)2 = x2 + (x+4)2 MUKULAM MATHS................................................................................................................................. 59

f) 6x _ 5x = 1

g) x + 3x = 2 3 h) x _ 4 = 1 2 x _8 =0 3x i) 2

3x

j) x = 2 x_3 2 =

k) (2x+3) (3x _ 1) = y2 1

1

l) x + x = 4 4 x _ 5 = 5x _ 7 m) 2x _3 7x _ 5

n) (6x _ 3 ) (x+5) _ 2(x+6) = 4

60 ...............................................................................................................................MUKULAM MATHS

HmÀ½nt¡­p¶ hkvXpXIÄ 1.

Hcp kam´c t{iWnbpsS XpSÀ¨bmb aq¶p]Z§Ä = x-d,x, x+d Fs¶gpXmw

n(n+1) 2

2.

BZys¯ F®Â kwJyIfpsS XpI

3.

Nn{X¯n AD F¶Xv LBAC bpsS ka`mPnbmbm A

BD : DC = AB: ACBbncn¡pw.

B 4.

D

C

\ofw l Dw hoXn b Dw Abm Hcp NXpc¯nsâ ]c¸fhv = lb Npäfhv

5.

= 2(l+b)

a«{XntImW¯nsâ ]c¸fhv = ½ bh ad+bc bd

6.

a + c d b

=

7.

a = c d b

Bbm ad = bc

8.

XpSÀ¨bmb c­v F®Â kwJyIÄ = x, x+1

9.

XpSÀ¨bmb 2 Cc« F®Â kwJyIÄ = x, x+2

10.

XpSÀ¨bmb 2 Hä F®Â kwJyIÄ = x, x+2

11.

kam´c t{iWnbnse BZys¯ c­p ]Z§Ä x, x+d

12.

a«{XntImW¯nsâ hi§Ä X½nepÅ ]ckv]c_Ôw ]mZw2+ew_w2= IÀ®w2 c­v kwJyIÄ X½nepÅ hyXymkw 5 Bbm kwJyIÄ x, x+5 or x, x_5

13.

ax2 + bx + c = 0 (a = 0) F¶ kahmIy¯n 1) b2_4ac = < 0 Bbm kahmIy¯n\v aqeyw Cà 2) b2_4ac = 0Bbm kahmIy¯n\v Hcp aqeyw am{Xw 1) 1)

b2_4ac = > 0Bbm kahmIy¯n\v c­v hyXykvX aqey§Ä b2_4ac sb ax2+bx+c = 0 F¶ kahmIy¯nsâ hnthNIw F¶v ]dbp¶p

MUKULAM MATHS................................................................................................................................. 61

Hcp tcJob kwJybpsSbpw AXnsâ hypÂ{Ia¯nsâbpw XpI tcJob kwJy x Bbm hypÂ{Iaw 1 AhbpsS XpI k Bbm x+ 1 =k x 2 _ x kx +1 = 0

x

hnthNIw = k2 _ 4 kahmIy¯n\v aqeyw D­mIWsa¦n hnthNIw > 0 Bbncn¡Ww k2_ 4 > 0 k2_ > 4 .. K > 2 or K < _2 .

HcptcJob kwJybpsSbpw AXnsâ hypÂ{Ia¯nsâbpw XpI þ2 \pw 2 \pw CSbnepÅ Hcp kwJy Bbncn¡pIbnÃ. t{]mPIvSv c­mw IrXn kahmIy§fpsS KptWm¯c§fpw aqey§fpw X½nepÅ ]ckv]c_Ôw Is­¯pI

62 ...............................................................................................................................MUKULAM MATHS

UNIT TEST 1.

Hcp kwJysb 6 sIm­v lcn¨m t]mepw 6 s\ AwKkwJysIm­v lcn¨t¸mgpw ^ew H¶p Xs¶bmbm kwJy GXv ?

2.

XpSÀ¨bmb c­v F®Â kwJyIfpsS KpW\^ew 506 Bbm kwJyIÄ Gh?

3.

XpSÀ¨bmb Cc«F®Â kwJyIfpsS KpW\^ew 288 Bbm kwJyIÄ GsXÃmw ?

4.

Hcp kwJybpsS hÀK¯n \n¶v AXnsâ 8 aS§v Ipd¨m 128 In«pw. kwJy GXv ?

5.

Hcp kwJybpsS \men H¶pw AtX kwJybpsS ]¯nsem¶pw KpWn¨t¸mÄ 90 In«n. kwJy Is­¯pI.

6.

Hcp a«{XntImW¯nsâ ]mZ¯nsâ 2 aS§nt\¡mÄ 6 IqSpXemWv IÀ®w. aq¶mas¯ hiw aq¶v aS§nt\¡mÄ 6 IpdhmWv. F¦n ]mZ¯nsâ \ofw F{X ? IÀ®w F{X?

7.

Hcp tKmf¯nsâ D]cnXe ]c¸fhv 784 N.sk.an. Bbm AXnsâ hymkw F´v ?

8.

s]m¶½So¨À¡v h\h¡cW¯nsâ `mKambn 200 hr£ssXIÄ e`n¨p. ssXIÄ hcn bntebpw \ncbntebpw F®§Ä ]camh[n XpeyamI¯¡hn[w sh¨p ]nSn¸n¨t¸mÄ 4 sNSnIÄ _m¡nbmbn. Hcp hcnbn F{X ssXIÄ \«p ?

9.

A¸sâbpw `mknbpsSbpw hbkpIfpsS Awi_Ôw 3:1 BWv. A©v hÀjw ap¼v Ah cpsS hbkpIfpsS KpW\^ew 125 Bbm c­pt]cpsSbpw Ct¸mgs¯ hbkv F{X ?

10.

A F¶ Øe¯v \n¶v HcmÄ t\sc Ingt¡mt«¡v \S¶p. AhnsS \n¶ CSt¯m«v Xncnªv

t\sc BZyw \S¶ Zqct¯¡mÄ 2 In.an. Zqcw A[nIw \S¶p. Ct¸mÄ ]pds¸« Øm\¯p \n¶v 10 In.an. AIse BsW¦n AbmÄ \S¶ Zqcw F{X ? 11.

Hcp kwJytbmSv 10 Iq«nbm In«p¶ kwJybpw 15 Dw X½nepÅ Awi_Ôhpw kwJy tbmSv 12 Iq«nbm In«p¶ kwJybpw 13 Dw X½nepÅ Awi_Ôhpw Xpeyambm BZy Awi_Ô¯nsekwJyIÄ Is­¯pI.

12.

HcpssSensâ \ofw hoXntb¡mÄ 4 sk.an. IqSpXemWv. 3.6 ao. \ofhpw 2 ao. hoXnbpapÅ Hcp apdnbpsS \ne¯p ]mIp¶Xn\mbn 1200 ssSepIÄ Bhiyambn h¶p. ssSensâ \ofhpw hoXnbpw ImWpI.

13.

Hcp kaNXpc¯nsâ Hcp hiw 10% Iq«pIbpw kao]hiw 10% Ipd¡pIbpw sNbvXt¸mÄ In«nb NXpc¯nsâ ]c¸fhv 99 N.sk.an. BbmÄ kaNXpc¯nsâ ]c¸fhv F{X ?

14.

F®Â kwJyIfn kwJybpsS hÀKt¯mSv 1 Iq«nsbgpXnb t{iWnbmWv 2,5,10....... CXnse Hcp kwJy 677 Bbm CXnsâ Ccphi¯papÅ kwJyIÄ GsXÃmw ?

MUKULAM MATHS................................................................................................................................. 63

A²ymbw þ4

{XntImWanXn BapJw {XntImW§fpsS AfhpIfpw AhbpsS {]tXyIXIfpambn _Ôs¸«p hfÀ¶p hnImkw {]m]n¨ Hcp KWnXimkv{XimJbmWv {XntImWanXn. kZri{XntImW§fpsS {]tXyIXbmWv CXnsâ ASnØm\w. AXn\m {XntImW§fn tImWpIÄ Adnªm hi§fpsS Awi _Ôw {]kvXmhn¡m³ IgnbpIbpw CXns\ sine, cosine, tangent F¶o t]cpIfneqsS hymJym\n ¡pIbpw sN¿p¶p. ]n¶oSv AIe§fpw Dbc§fpw I­p]nSn¡m\pÅ {]mtbmKnI kµÀ` §fn Ch D]tbmKn¡pIbpw sN¿p¶ Xc¯nemWv Cu A[ymb¯nsâ NÀ¨. {][m\ Bib§Ä *

Htc tImWpIfpÅ {XntImW§fpsSsbÃmw hi§fpsS \of§Ä Htc Awi_Ô¯nemWv.

*

sine, cosine tImWnsâ F¶o hneIÄ tIm¬ Af¡m\pÅ kwJyIfmWv F¶ Bibw

*

Ncnhnsâ Afhmbn tImWns\ ImWp¶ coXnbpambn _Ôs¸«v tangent Afhv

*

Dbcw, AIew F¶nh IW¡m¡m³ {XntImWanXn AfhpIfpsS D]tbmKw

sF.kn.Sn.bpambn _Ôs¸« Bib§fpw A[nI hnhc§fpw A\p_Ô¯n sImSp ¯ncn¡p¶p. {XntImW¯nsâ hi§fpw tImWpIfpw Xmsg ]dbp¶ {]hÀ¯\w sN¿pI {]hÀ¯\w þ 1 hi§fpsS \of§Ä 4 sk.an., 5 sk.an, 6 sk.an., 7 sk.an., 8 sk.an, 9 sk.an, 5 sk.an., 6 sk.an., 7 sk.an. F¶nh hcp¶ 3 Xcw {XntImW§Ä Hcp I«n ISemkn hc¨v sh«nsbSp¡p I. Hmtcm Ip«nbpw GsX¦nepw Hcp Xcw {XntImWw hc¨v sh«nsbSp¡Ww. At¸mÄ Hmtcm Xc ¯nepw DÅ Iptd {XntImW§Ä D­mIpatÃm. ChsbÃmw ]cntim[n¨v kÀÆka§fmbh GsXms¡ F¶v Is­¯n thÀXncn¡s«. Hcp {XntImW¯n\v apIfn asäm¶v tNÀ¯psh¨v ]cntim[n¡m³ A[ym]I³ klmbn¡pI. Cu {]hÀ¯\¯n \n¶v hi§Ä Xpeyamb FÃm {XntImW§fnepw Xpeyhi§Ä¡v FXnscbpÅ tImWpIfpsS XpeyamIp¶p F¶v 8þmw ¢mkn kÀÆka{XntImWw F¶ `mK¯v ]Tn¨Xv HmÀ½n¸n¡matÃm.

64 ...............................................................................................................................MUKULAM MATHS

{]hÀ¯\w þ 2 Htc Xc¯nepÅ {XntImW§sf Xmsg ImWp¶ coXnbn tNÀ¯psh¨v henb {XntIm W§fm¡pI 5

6

6

7

9

8

4 5 4

D

A

6 4

6

4

E

6

6

7

9

8

4

6

5

5

5

6

5

6

9

7

6

8

4 5

6

B 4

C

6

5

6

5

4

4

7

6 5

5

5

5

9

8 7

7

7

7

Ct¸mÄ In«nb sNdpXpw hepXpamb {XntImW§fn tImWpIfpw hi§fpw F§s\ _Ôs¸«ncn¡p¶p F¶v ]cntim[n¡pI. tImWpIÄ XpeyamIp¶p. F¶m hi§Ä XpeyamIp¶nà F¶v ImWmw. DZm:þ

ABE,

OACD ChbnÂ

LA s]mXphmb tIm¬ LB = LC kÀÆka{XntImW¯nse tImWpIÄ LE = LD hi§Ä

BE

AB = 4

AC = 8

AB \ AC =

4 8

=

1 2

BE = 6

DC = 12

\ DC =

BE

6 12

=

1 2

AE = 5

AD = 10

BE \ AD =

5 10

=

1 2

AB \ AC = DC =

AE AD

F¶p ImWmw.

aq¶p Xcw {XntImW§fnepw C§s\ ]cntim[n¨m 9þmw ¢mkn kZri{XntImWw F¶ ]mT`mK¯v ]Tn¨ Xmsg ]dbp¶ Bibw HmÀ½n¸n¡matÃm. tImWpIÄ XpeyamIp¶ {XntImW§fnseÃmw XpeytImWpIÄ¡v FXnscbpÅ hi MUKULAM MATHS................................................................................................................................. 65

§Ä B\p]mXnIamWv. asämcp {]hÀ¯\w : Xmsg ImWp¶ Xc¯n Nn{Xw \ÂIp¶p

?

?

6 cm

5 cm

?

)

)

8 cm

)

)

)

)

? 4 cm

16 cm

Htc tImWfhpÅ aq¶v {XntImW§fmWv apIfnepÅXv. AhbpsS Nne hi§fpsS \ofw X¶n«p­v. aäphi§fpsS \ofw IW¡m¡pI. A[ym]I³ Xmsg ]dbp¶ kqN\IfneqsS NÀ¨ \bn¡patÃm. H¶mas¯ {XntImW¯nsâ hi§Ä 8; 5; 6 F¶nh BbXn\m aäv c­v {XntImW ¯nepw hi§Ä Cu kwJyIÄ¡v B\p]mXnIamWv. Hmtcm {XntImW¯nepw Hcphiw X¶Xn\m aäp hi§Ä IW¡m¡mw. H¶mas¯ {XntImW¯nse 8 F¶hi¯n\v kam\amWv c­mas¯ {XntImW¯nse 16 F¶ hiw AXmbXv 8 x 2 = 16

\ aäp c­v hi§Ä 5 x 2 = 10, 6 x 2 = 12 CXpt]mse aq¶mas¯ {XntImW¯nepw hi§Ä ImWmatÃm CtXmsSm¸w Xmsg ]dbp¶ {]hÀ¯\w sN¿pI ta sImSp¯ AtX tImWpIfpÅ asämcp {XntImWw hi§fpsS Afhv ]dbmsX \ÂIpI.

)

)

C

A

B

ChnsS hi§fpsS \ofw Adnbnsænepw AB = AC = BC 5 6 8

F¶v In«psa¶v t_m[ys¸Sp¯pI 66 ...............................................................................................................................MUKULAM MATHS

CtX _Ôw aäp Xc¯nepw FgpXs« AB = 8 ; AC = 5 ; AB = 8 6 6 BC BC 5 AC

AB : AC : BC = 8:5:6

Htc tImWpIfpÅ {XntImW§fpsSsbÃmw hi§fpsS \ofw Htc Awi_Ô¯nem sW¶v ChnsS Dd¸n¡Ww. CXn\v Xmsg ]dbp¶ {]hÀ¯\§Ä sImSp¡mw. 1.

)

3

c

)

4

b

)

)

a 6

a ..................; b ..................; c .................. a = c = b = b ..................; c ..................; a .................. c = b = a = a : b : c = ...............................

)

)

2)

b

4 )

9

12

2 a

a ..................; 2 =

b ..................; 9 =

a =....................; b = ..............................

A\p_Ôw t\m¡pI ICT D]tbmKn¨v Htc tImWfhpÅ {XntImW§fpÅ hi§Ä X½nepÅ Awi

_Ôw Xpeysa¶v t_m[ys¸Sp¯pI hi§Ä \of§fpsS {Ia¯n a, b,c, bpw x, y, z Bbm b y c = z

x a x a b = y , c = z

F¶n§s\ ImWmw

MUKULAM MATHS................................................................................................................................. 67

tImWpIfn \n¶v hi§fnte¡v {XntImW¯nsâ tImWpIÄ \nÝbn¡s¸«m hi§fpsS Awi_Ôhpw \nÝbn¡ s¸«p F¶p I­tÃm. F¦n hi§fpsS Awi_Ôw tImWpIfpsS AfhpIfneqsS F§s\ {]kvXmhn¡mw F¶v t\m¡mw. 450, 45, 900 tImWpIfpÅ {XntImW¯n \n¶v XpS§mw {]hÀ¯\w Ip«nIÄ¡v CjvSapÅ \ofw hiambn FSp¯v Hcp kaNXpcw hcbv¡m³ sImSp¡pI. hi¯nsâ \ofw FgpXs«. hnIÀ®w hc¨v In«p¶ Hmtcm a«{XntImW¯nsâbpw tImWpIÄ 450, 45, 900 F¶v Is­¯s«. ) 45

In«nb a«{XntImW¯nsâ aq¶mas¯ hiw ss]XtKmdkv XXzw D]tbmKn¨v ImWmatÃm )

45

In«nb hi§fpsS \ofw ]«nIbmbn FgpXpI tImWpIÄ Ip«n 1

XpeytImWpIÄ¡v FXnscbpÅ hi§Ä

450, 45, 900

5 cm, 5 cm, 5Ö 2 cm

450, 45, 900

a cm, a cm, a 2 cm

hi§fpsS Awi_Ôw 1: 1: 2

Ip«n 2

CXp t]mse ka`pP{XntImW¯n \n¶v XpS§n 300, 600, 900 tImWfhpÅ {XntImW ¯nsâ hi§fpsS Awi_Ôw {]kvXmhn¡matÃm tImWpIÄ 450, 45, 900 Bbm hi§Ä 1 : 1 : 2 Dw tImWpIÄ 300, 600, 900 Bbm hi§Ä 1 : 3 : 2 Dw BIp¶p F¶v t_m[ys¸SpatÃm

68 ...............................................................................................................................MUKULAM MATHS

{]hÀ¯\w Xmsg sImSp¡p¶ Nn{X§Ä DÄs¸Sp¶ hÀ¡v joäv \ÂIn aäphi§fpsS \ofw ImWpI ? 45

?

?

)

45

)

1.

?

2

? 45

?

10

10

5 3 ) 45

?

?

?

? 30

45

? )

3

)

2.

?

?

30

?

9

?

6 10 3.

)

A

ImWpI

)

30

)60

ew_I¯nsâ Npäfhv

B

45

AB AC = ?

4.

AB BC = ?

AC BC = ?

C MUKULAM MATHS................................................................................................................................. 69

) 30

2.

a b =?;

a =?; c

b =?; c

b a =?;

c a =?;

c b =?;

c

a

b 6. Hcp ka`pP{XntImW¯nsâ D¶Xn 40 sk.an.Bbm AXnsâ ]c¸fhv ImWpI (kqN\ : hiw I­p]nSn¨v ½ x hiw x 40 ImWpI) 7. Hcp kaNXpc¯nsâ hnIÀW¯nsâ \ofw 10 sk.an. Bbm AXnsâ ]c¸fhv ImWpI. (kqN\ : hiw I­p]nSn¨pw ]c¸fhv ImWmatÃm) 8. Hcp ka`pPkmam´coI¯nsâ hiw 10 sk.an. Dw Hcp tIm¬ 600 Dw Bbm AXnsâ ]c ¸fhv ImWpI. (kqN\ : D¶Xn I­p]nSn¨pw hnIÀWw I­p]nSn¨pw sN¿mw 9. Hcp kam´coI¯nsâ kao]hi§Ä 6 sk.an., 8 sk.an. Hcp tIm¬ 1500 Bbm AXnsâ ]c¸fhv ImWpI. (kqN\ : 150 t\mSv ASp¯ tIm¬ 300 BIpatÃm) 10. 10 sk.an. hymkapÅ Hcp hr¯¯n ioÀj §Ä hc¯¡hn[w 300 tImWpÅ Hcp a«{Xn tImWw hc¨ncn¡p¶p. AXnsâ aäv hi§Ä IW¡m¡pI. 11. 10 sk.an. hymkapÅ Hcp hr¯mIrXnbmb ISemkn \n¶pw Gähpw henb kaNXpcw apdns¨Sp¯m _m¡n `mK¯nsâ ]c¸fhv F´v ? 12. 10 sk.an.hymkapÅ Hcp hr¯mIrXnbmb ISemkn \n¶pw Gähpw henb ka`pP{Xn tImWw apdns¨Sp¯m _m¡n `mK¯nâ ]c¸fhv F´v ? Sine, Cosine F¶nh 450, 45, 900 tImWpIfpÅ {XntImW¯nepw 300, 60, 900 tImWpIfpÅ {XntImW¯nepw

hi§fpsS Awi_Ôw Xncn¨dnªtÃm. F¦n aäv tImWfhpÅ {XntImW¯nepw tImWp IÄ Adnªm hi§fpsS Awi_Ôw F§s\ ImWmw ? 400, 50, 900 tImWpIfpÅ Hcp Xcw {XntImW¯n \n¶v XpS§mw. Xmsg sImSp¯

{]hÀ¯\w \nco£n¡pI. {]hÀ¯\w 400, 50, 900 tImWpIfpÅ c­v {XntImW§Ä Hcp NmÀ«n {]ZÀin¸n¡pI y

a

z

)

)

x

)

40

) 40

c

b

70 ...............................................................................................................................MUKULAM MATHS

c­v {XntImW¯nepw hi§fpsS Awi_Ôw XpeyamWtÃm. a : b : c = x : y : z Asæn a x =

b y

c z

=

CXn \n¶v

a c =

x z

F¶v In«patÃm

ChnsS a bpw c bpw BZys¯ {XntImW¯nsâbpw x Dw z Dw c­mas¯ {XntImW ¯nsâbpw hi§fmWtÃm. At¸mÄ Hcp {XntImW¯nsâ Xs¶ c­phi§Ä X½nepÅ Awi_Ôw atä {XntIm W¯nsâ AtX Øm\¯pÅ hi§Ä X½nepÅ Awi_Ô¯n\v XpeyamWv.

a, x c z

40 (

y x z ® IÀ

w

40 (

tIm¬ 40 sâ FXnÀ hiw

tIm¬ 40 sâ kao] hiw

tIm¬ 40 sâ kao] hiw

ChnsS kam\amb hi§Ä Xncn¨dnbm³ Hcp tImWns\ ASnØm\am¡n t]À \ÂImw. DZmlcW¯n\v {XntImW¯nsse 400 tImWns\ ASnØm\am¡n 'a' his¯ FXnÀhisa¶pw 'c' his¯ IÀWsa¶pw hnfn¡mw. Htc Øm\¯pÅ hi§Ä Xncn¨dnbm³ Hcp tImWns\ ASnØm\am¡n hi§Ä¡v Xmsg Nn{X¯n ImWp¶ hn[w t]À \ÂInbm Cu Awi _Ôw {]kvXmhn¡m³ Ffp¸amhpatÃm.

I

c

b

À®

w

a tIm¬ 40 sâ FXnÀ hiw

F¶Xv c­p {XntImW¯nembmepw tIm¬ 40 sâ kao] hiw IÀ®w

At¸mÄ ac , x

z

GXv {XntImW¯nepw

BIp¶p

BbXn\m tImWpIfpÅ 400, 500, 900 tIm¬ 40 sâ kao] hiw IÀ®w

Hcp Ønc kwJybmWv. AXmbXv Cu

{XntImW¯nsâ tImWpIÄ amdmsX hi§Ä F{X amdnbmepw FXnÀhi¯nsâ \ofhpw FXnÀhiw IÀW¯nsâ \ofhpw amdpsa¦nepw amdp¶nÃ. IÀ®w

400 tImWns\ ASnØm\am¡n Cu a«{XntImW¯nse

FXnÀhiw F¶ IÀ®w

ØnckwJysb 400 tImWnsâ Sin F¶v hnfn¡p¶p. CXns\ Npcp¡n Sin 40 = FXnÀhiw IÀ®w Fs¶gpXmw. MUKULAM MATHS................................................................................................................................. 71

Sin 40 sâ GItZi hne Ip«nIÄ Nn{Xw hc¨v hi§Ä Af¶v lcn¨v ImWs«.

40 Un{Kn tImWnsâ FXnÀhihpw IÀ®hpw Af¶v Cu Ønc kwJy ImWm³ Ip«n Isf klmbn¡pI. Xmsg ]«nI D]tbmKn¡mw. {XntImWw 1

400 tImWnsâ FXnÀhiw 6.42

FXnÀhiw IÀ®w

I˨w 10

0.64

2

...........................

...................

......................

3

..............................

..................

......................

Xmsg sImSp¯ {]hÀ¯\hpw sN¿mw Hcp {Km^v t]¸dn 10 sk.an. Bc¯n hr¯w hc¨v tI{µ¯n 400 tIm¬ \nÀ½n¡p I. IÀ®w 10 sk.an. Bbn hcp¶ a«{XntImWw hc¨v FXnÀhi¯nsâ \ofw Is­¯n ]«n Ibn FgpXmw. Bcw amän AtX tImWfhpÅ a«{XntImW¯n FXnÀhihpw IÀW ¯nsâbpw \ofw Is­¯n ]«nI ]qcn¸n¡pI. Hmtcm L«¯nepw FXnÀhis¯ IÀ®w sIm­v lcn¨p In«p¶ kwJy GItZiw 0.64 F¶v ImWpIbpw Ipd¨pIqSn IrXyambn 0.6428 BsW¶v Is­¯nbn«ps­¶v t_m[ys¸Sp ¯pIbpw sN¿pI.

\ sin 40 » 0.6428 F¶v t_m²ys¸Sp¯pI 40 Un{Kn tImWpÅ GXv a«{XntImW¯nepw » 0.6428 Xs¶bmWv. IÀ®w tImWnsâ Afhv amdpt¼mÄ Cu kwJybpw amdpw. AXn\m Cu kwJy tImWnsâ Af hns\ kqNn¸n¡p¶p. AXmbXv tImWnsâ Afhns\ kqNn¸n¡p¶XmWv Sine hne FXnÀhiw

Hcp tImWnsâ sine hne Adnªm B tImWnsâ Afhv Is­¯mw. kao]hiw

CXpt]mse F¶ Ønc kwJy 400 tImWns\ ASnØm\am¡n Is­ IÀ®w ¯pI. CXns\ cosine F¶pw kwJy GItZiw 0.7660 BIp¶p F¶v t_m[ys¸Sp¯matÃm. AXn\m cos 40 » 0.7660 F¶v Ip«nIÄ a\knem¡s«.

sF.kn.Sn. D]tbmKn¨v tImWpIfpsS sin, cos hneIÄ Is­¯mw. 40, 50, 90 tImWfhpÅ Cu {XntImW§fn hi§fpsS Awi_Ôw sin 40 : cos 40 : 1 Asæn 0.6428 : 0.7660 : 1 F¶v ImWm³ Ip«nIsf {]m]vXcm¡Ww. (500 tImWns\ ASnØm

\am¡n Cu Awi_Ôw F§s\ FgpXmw?) 72 ...............................................................................................................................MUKULAM MATHS

10

1

6.428

0.6428 40

(

(

40 0.7660

(

7.660

90 - x 1

1 sin x

sin 40 40

(

x

(

cos x

cos 40

tImWpIÄ x, 90 - x , 90 At¸mÄ hi§Ä X½nepÅ _Ôw sin x : cos x : 1 BIp¶p

CtXmsSm¸w Bcy`Ssâ ssk³ ]«nI {i²n¡pI. hymkmÀ²hp ambn (AÀ²Pym) _Ôs¸«v ssk³]«nI BZyambn D­m¡nbXv Bcy`S\mWv Sin A =

LAbpsS FXnÀhiw

Cos A =

I˨w

LAbpsS kao] hiw I˨w

F¶nh Dd¸n¡p¶Xn\pw ]qcItImWpIfpsS Sine, Cosine Ch X½nepÅ _Ôw a\Ên em¡p¶Xn\pw Xmsg sImSp¡p¶ {]iv\§Ä NÀ¨ sN¿pI. hn«`mKw ]qcn¸n¡pI (Nn{Xw D]tbmKn¨v) A ) b c

1) 2)

Sin..................... = a b

a Cos...................... = b

)

a

B ,

Cos.......................... =

,

Sin...........................

=

C

c b c b

MUKULAM MATHS................................................................................................................................. 73

3)

LA = 350 Bbm LC = ................

4)

LA = x0 Bbm LC = ................

6)

a Sin.................... = b c Sin.................... = b

7)

Sin 350 =

Cos.............................

8)

Cos 350 =

Sin.............................

9)

Sin x0 = Cos x0 .......................

10)

Sin A = Cos B

5)

=

Cos..................

=

Cos..................

Cos x0 = sin.................

A + B ..........................

450, 300, 600 F¶o tImWpIfpsS Sine, Cos hneIÄ )

)

45

30 a 2

a

2a

a 3

60 a

Nn{X¯n \n¶v sin 450, cos 300,sin 300 F¶nh Is­¯pI

a

Cu hneIÄ Dd¸n¡p¶Xn\v Xmsg sImSp¡p¶ {]iv\§Ä sN¿s« 1)

sin2 30=.....................;

cos2 30 = .................

sin2 30+cos2 30 = ................

2)

sin2 60=.....................;

cos2 60 = .................

sin2 60+cos2 60 = ................

3)

sin2 45=.....................;

cos2 45 = .................

sin2 45+cos2 45 = ................

GXv tImWnsâbpw sin2 hnebpw Cos2 hnebpw Iq«nbm Htc kwJy In«ptam ? t]Pv 83 se side box NÀ¨ sN¿pI sin2 A + Cos2 A = 1 F¶v t_m²ys¸Sp¯pI Hcp a«{XntImW¯nsâ IÀ®hpw \yq\tImWpw D]tbmKn¨v sin, cos F¶nhbpsS klm bt¯msS aäv hi§fpsS \ofw ImWm³ Ignbpw. sSIvÌv _p¡nepÅ {]hÀ¯\§Ä sN¿m atÃm. t]Pv \¼À 82 AtXmsSm¸w Xmsg sImSp¡p¶ {]iv\§Ä sN¿pI. 1)

IÀ®w 10 sk.an. Hcp tIm¬ 500 F¦n B a«{XntImW¯nsâ aäp hi§Ä ImWpI

74 ...............................................................................................................................MUKULAM MATHS

2)

50 10

IÀ®¯nsâ \ofsa´v ?

sin 20 = 0.3420, cos 20 = 0.9397

3)

Ch D]tbmKn¨v aäp c­v hi§Ä ImWpI

10

)

70

A 4

4)

B

5)

AD bpsS \ofw ImWpI 10

)50

C

D

8

Cu {XntImW¯nsâ ]c¸fhv ImWpI 35

)

10

5)

Cu {XntImW¯nsâ ]c¸fhv ImWpI 145

10

)

8

8 35 10

)

kqN\ ;

145 10

MUKULAM MATHS................................................................................................................................. 75

Sine hnebpw ]cnhr¯ hymkhpw

t]Pv \¼À 83 se {]hÀ¯\w sN¿pI. AtXmsSm¸w Xmsg sImSp¡p¶ {]hÀ¯\§Ä sN¿patÃm. Cu {XntImW¯nsâ ]cnhr¯¯nsâ hymkw ImWpI

)

1)

50

(kqN\ : 500 tImWpÅ a«{XntImWw hcbv¡pI)

10

)

130

2)

Cu {XntImW¯nsâ hymkw F{X ? 10

(kqN\ : 500 tImWpÅ a«{XntImWw hcbv¡pI)

hr¯¯n a«tIm¬, \yq\tIm¬, _rlXvtIm¬ F¶nh hcbv¡pt¼mÄ tImWnsâ Aä§Ä tbmPn¸n¡p¶ Rm¬ tI{µhpambn F§s\ _Ôs¸ Sp¶p F¶v hyàam¡Ww. GXv {XntImW¯nsâbpw Hcp tImWpw FXnÀhi¯nsâ \ofhpw X¶m ]cnhr¯¯nsâ hymkw ImWm³ Ignbptam ? F§s\ ? tImWnsâ Afhv 900, 90  Ipdhv, 90  IqSpX F¶nh BIpt¼mÄ Fs´ÃmamWv amä §Ä F¶nh NÀ¨ sN¿patÃm. C¯cw {]hÀ¯\§fneqsS Hcp {XntImW¯nsâ Hcp tImWpw AXnsâ FXnÀhihpw \nÝbn¡s¸«m {XntImW¯nsâ ]cnhr¯w \nÝ bn¡s¸«p F¶v t_m[ys¸Sp¯Ww.

3.

Hcp {XntImW¯nsâ c­p hi§Ä 7 sk.aoädpw 8 sk.aoädpw BWv. AhbpsS CSbn epÅ tIm¬ 400. {XntImW¯nsâ aq¶mas¯ hi¯nsâ \ofw F{X ?

4.

ta¸dª {XntImW¯nse tIm¬ 1400 Bbm aq¶mas¯ hiw F{X ?

5.

Hcp {XntImW¯nsâ c­p hi§Ä 10 sk.an., 20 sk.an.Ahbv¡nSbnepÅ tIm¬ 800 Dw Bbm {XntImW¯nsâ ]c¸fhv ImWpI. 800 tImWn\v ]Icw 1000 tIm¬ Bbm ]c¸fhv ImWpI

6.

40 tImWpÅ Hcp {XntImW¯nsâ ioÀj§Ä 10 sk.an. hymkapÅ Hcp hr¯¯nem

76 ...............................................................................................................................MUKULAM MATHS

Wv. F¦n 400 tImWnsâ FXnÀhiw F{X ? 7.

500 tImWpÅ Hcp {XntImW¯nsâ FXnÀhiw 10 sk.an. AXnsâ ]c¸fhv Gähpw IqSp¶Xv aäv c­v hi§fpsS \ofw F{X hoXw BIpt¼mgmWv.

tan F¶ Bibw A

IÀ®w ImWm³ F´psN¿Ww ? Sine, Cos ChbnteXmWv D]tbmKnt¡­Xv.

)

50

B

10

Cos 50 sâ hne D]tbmKn¨v IÀWw ImWmatÃm

C

F¶m aq¶mas¯ hiw ImWm³ sin Dw cos Dw t\cn«v D]tbmKn¡m³ km[yasöv Xncn¨dn bs«. 500 tImWnsâ FXnÀhis¯ kao]hiw sIm­v lcn¨m In«p¶ kwJybpw GXv hep¸¯nepÅ a«{XntImW¯nepw Øncambncn¡patÃm. Cu hnebpw ]«nIs¸Sp¯nbn«p s­¶v t_m²ys¸Sp¯patÃm. Cu hnebmWv tangent F¶dnbs¸Sp¶Xv. CXv Npcp¡n tan Fs¶gpXp¶p. tan 50 = 1.1918 F¶v ]«nbn ImWp¶p. AXn\mÂ

tan 50 AB 10

F¶pw

AB = 10 x tan 50 = 11.918 F¶pw In«patÃm. tan 30, tan 60, tan 45 F¶nhbpsS hneIÄ 30, 60, 90 ; 45, 45, 90 tImWpIfpÅ a«{Xn tImWw D]tbmKn¨p ImWm\pÅ {]hÀ¯\w \ÂIpI. hneIÄ D]tbmKn¨v Xmsg ]dbp¶ {]iv\§Ä sN¿s«. sSIvÌnse t]Pv \¼À 87, 88, 89 se {]hÀ¯\w

AIew , Dbcw IogvtIm¬ )

taÂtIm¬

taÂtIm¬, IogvtIm¬ F¶nh t_m[ys¸Sp¯pI

sSIvÌnse t]Pv \¼À 91,92, 93 se {]iv\§Ä NÀ¨ sNbvXv AIes¯bpw Dbct¯bpw IW¡m¡m³ Ip«nIsf {]m]vXcm¡pI. (Ipdn¸v : c­v {XntImW§Ä DÄs¸Sp¶ {]iv\§fn s]mXphitam Xpeyhitam Ds­¦n B hiw c­p {XntImW§fn \n¶pw {]tXyIw I­p ]nSn¨v XpeyXs¸Sp¯n {]iv\ ]cn lmcw ImWpI) MUKULAM MATHS................................................................................................................................. 77

AtXmSms¸w Xmsg ImWp¶ {]iv\§Ä NÀ¨ sN¿pI 1.

Ip¯s\ \n¡p¶ Shdnsâ A{K`mKw Shdnsâ Nph«n \n¶v 4 ao. AIe¯n \n¶pw 9 aoäÀ AIe¯n \n¶pw t\m¡p¶p. ta tImWpIÄ c­pw ]qcI tImWpIfmsW ¦n Shdnsâ Dbcw ImWpI. kqN\ :

A + B = 90

\ LQPR = A

P )

9

h

PQR Â tan A = h 4

(

B(

5

S

Q

2.

h

PQS Â tan A = 4

A

CXn \n¶pw h ImWmatÃm

R

Shdns\ t\m¡p¶Xv Shdnsâ Ccphi§fn \n¶pamsW¦n Shdnsâ Dbcw amdptam ? P

h )B

a

R

4

)

A

ChnsS LRPS F{X ? h F{X ? S

C¯cw {]iv\¯n AIew a, b Bbm Shdnsâ Dbcw (taÂtImWpIÄ ]qcI§fmbmÂ) a. b Bbncn¡pw 3.

Hcp t]mÌnsâ A{K`mKw Nph«n \n¶v 3 aoäÀ, 27 aoäÀ F¶o AIe¯n \n¶v t\m¡pt¼mÄ taÂtImWpIÄ ]ckv]cw ]qcI tImWpIÄ Bbm t]mÌnsâ Dbcw F{X ?

4. 10 aoäÀ DbcapÅ Hcp Shdnsâ A{K`mKw Nph«n \n¶v hyXykvX Øm\¯v \n¶v t\m¡pt¼mÄ taÂtImWpIÄ ]ckv]c ]qcIambn hcp¶p. Cu c­p Øm\§Ä X½nep Åh AIew 20 aoäÀ Bbm ta tImWpIfpsS Afhv F´v ? AIe§fpw ImWpI Ö3

kqN\ :

) x 10

) 90 - x

20 Ö3

10 = 20 + a Ö3 b 10

)x

a

\Ö3 a2 + 20 a _ 100Ö3 - 0 \ a=

\ x = 602

10 Ö3

78 ...............................................................................................................................MUKULAM MATHS

5. Hcp \ZoXoc¯v Hcp acap­v. adpIcbnepÅ 60 ao. DbcapÅ Hcp sI«nS¯n \n¶v ac ¯nsâ A{Kw 300 ta tImWn ImWp¶p. CtX Øe¯v \n¶v ac¯nsâ shůnepÅ {]Xn_nw_¯nsâ Aäw 600 IogvtImWn ImWp¶p. F¦n \ZnbpsS hoXn F{X ? ac¯nsâ Dbcw F{X ? kqN\ : )

60

30

)

x

60

tan 60 tan 30

60

=

120 + x x

60 + x

6. \nc¸mb Øe¯v \n¶v 3000 Ö 3 aoäÀ Dbc¯n \nÝnX thK¯n ]d¡p¶p. Hcp Øe¯v \n¶v hnam\s¯ 600 taÂtImWnepw 30 sk¡ân\v tijw AtX Øe¯v \n¶v 300 taÂtImWnepw ImWp¶p. F¦n hnam\¯nsâ thKX F{X? B A

kqN\ : AC = 3000Ö 3 A,B Ch hnam\¯nsâ Øm\§Ä

(

60 30 C

7.

D

E t£{Xw

]mX

) 30

Xd

Nn{X¯n ImWp¶hn[w t£{X¯nte¡v Hcp t\À]mXbp­v. ]mXbn XpS¡w apX t£{Xw hsc 10 aoäÀ CShn«v 40 hnf¡v ImepIÄ Øm]n¨n«p­v. F¦n ]mXbpsS \ofw F{X? Xd \nc¸n \n¶v F{X Dbc¯nemWv t£{Xw. 8. Hcphiw 10 sk.an.BI¯¡ hn[¯n 300 tImWfhv hcp¶ Hcp skäv kvIzbdnsâ amXrIbn {XntImWw hc¨m F{Xhn[¯n {XntImWw In«pw. Hmtcm¶nepw aäv hi§ fpsS \of§Ä Af¡msX I­p]nSn¡pI MUKULAM MATHS................................................................................................................................. 79

tan hne Dbct¯bpw AIet¯bpw ASnØm\am¡nbXmWtÃm. Dbchpw AI ehpw amdpt¼mgmWtÃm km[mcWbmbn hkvXphnsâ Ncnhn\v amäw hcp¶Xv. AXn \m Ncnhnsâ Afhmbn«mWv tan D­mbXv . t]Pv \¼À 92 se sskUv t_mIvkv NÀ¨ sN¿patÃm

hi§fpw tImWpIfpw GsXmcp {XntImW¯nsâbpw hi§Ä In«nbm AXnsâ tImWpIÄ \nÝbn¡s¸« tÃm. hi§Ä a,b,c bpw FXnscbpÅ tImWpIÄ A,B,C bpw Bbm c a b sin A = sin B = sin C F¶v ImWmatÃm

(sskUv t_mIvkv 90 NÀ¨ sN¿pI) tImWpIÄ ImWm³ F¶pw Cos B =

Cos A =

b2 + c 2 _ a 2 2 bc

a2 + c 2 _ b 2 F¶pw Cos C= 2 ac

a2 + b2 _ c 2 F¶pw 2 ab

(

ImWmatÃm (sskUv t_mIvkv 91 ImWpI)

70

x

60

50(

(

y

F¦n x F{X ?

y F{X ?

10 x =

10. sin 60 sin 70

y =

10. sin 50 sin 70

80 ...............................................................................................................................MUKULAM MATHS

aqey\nÀWb {]iv\§Ä 1.

Nn{Xw \nco£n¨v Sin 70, Cos 70 F¶nhbn heptXXv F¶v Fgp XpI. ImcWsa´v ? 70 ( 20 (

2. x

Cu {XntImW¯nsâ x F¶v ASbmfs¸Sp¯nb hi¯nsâ \ofw ImWpI

10

x 300

3.

Cu {XntImW¯nsâ x F¶v ASbmfs¸Sp¯nb hi¯nsâ \ofw ImWpI

10

x 4.

670 tImWnsâ kao]hi¯nsâ \ofw ImWpI

10

670 (

A x C

(

5.

B

AB bpsS \ofw ImWpI

40

(

60

50

D MUKULAM MATHS................................................................................................................................. 81

6.

PQR  PQ = PR =1O cm LQ = 40 0 Bbm P bn IqSnbpÅ D¶Xn ImWpI. {XntImW¯nsâ ]c¸fhv ImWpI ?

7. 5

130

20 ( ?

kqN\ : a sin A

c b = sin B = sin A

(

D]tbmKn¡pI or B bn \n¶v AC bnte ¡pÅ ew_w hcbv¡pI

8. \nc¸mb XdbnepÅXpw Htc DbcapÅXpamb c­v sshZypXn t]mÌpIÄ¡nSbn H¶n \n¶v 4 aoäÀ AIe¯n \n¡p¶ HcmÄ B t]mÌdns\ 580 taÂt¡mWnepw atä t]mÌns\ 220 taÂt¡mWnepw ho£n¡p¶p. t]mÌpIÄ X½nepÅ AIesa´v ? 9. kqcy³ 480 ta tImWn ImWs¸Spt¼mÄ Hcp ac¯nsâ \ngen\v 18 aoäÀ \ofap­v. F¦n B ac¯nsâ Dbcsa{Xbmbncn¡pw ? 10. ]WnXpsIm­ncn¡p¶ Hcp tKm]pc¯nsâ apIÄ`mKw 1.5 aoäÀ DbcapÅ HcmÄ 300 ta tImWn I­p. 10 aoäÀ IqSn DbÀ¯n tKm]pcw ]WnXoÀ¶t¸mÄ AbmÄ AtX Øe¯v \n¶v 600 ta tImWnemWv AXnsâ A{Kw I­Xv. tKm]pc¯nsâ BsI Dbcw F{X ?

82 ...............................................................................................................................MUKULAM MATHS

UNIT TEST 1.

P

Q

B (

PQR Â LP = 90, aäv tImWpIfpsS Afhv A,B F¶nhbmWv

A R

Sin A = hn«`mKw ]qcn¸n¡pI

................ , Cos A = PR , tan A = ............... QR QR

................ , Cos B = ..............., tan B = ............... Sin B = ................... ..........

?

2. ?

20

a«{XntImW¯nsâ aäp c­p hi§fpsS \ofw ImWpI

70

?

2

B

(

3

3.

( 5.

A bn \n¶v B bnte¡v t\m¡pt¼mÄ taÂt¡m¬ GXv ? B bn \n¶v A bnte¡v t\m¡pt¼mÄ Iogvt¡m¬ GXv ?

1

A

4.

4

4

Sin A = 5

Bbm Cos A, tan A F¶nhbpsS hne ImWpI

12

x (

13

Sin x, Cos x, tan x, F¶nhbpsS hne ImWpI

MUKULAM MATHS................................................................................................................................. 83

7.

Hcp {XntImW¯nsâ c­p hi§Ä 12 sk.an., 16 sk.an. F¶nhbpw Ahbv¡nSbnepÅ tIm¬ 800 bpambm {XntImW¯nsâ ]c¸fhv ImWpI Cos A = 5 , 4

Sin A =

5 , 3

Cot A

3 , 4

F¦nÂ

Sin A, Cos A, tan A FgpXpI 8.

10 sk.an. hiapÅ Hcp ka`pP kmam´coI¯nsâ Hcp tIm¬ 1200 Bbm hnIÀW§ fpsS \ofw ImWpI )

C

9.

120

12

)45

B

C

BC bpsS \ofw ImWpI

10. Hcp Ip¶n \n¶pw AIse \n¡p¶ HcmÄ AXnsâ apIfäw 310 ta tImWn ImWp ¶p. 20 aoäÀ Ip¶nsâ ASpt¯¡v \S¶v AhnsS \n¶v t\m¡nbt¸mÄ Ip¶nsâ apIfäw 350 taÂt¡mWn ImWp¶p. Cu hkvXpXsb kqNn¸n¡p¶ GItZiw Nn{Xw hcbv¡p I. Ip¶nsâ Dbcw IW¡m¡pI. 10. Hcp {XntImW¯nsâ tIm¬ 700 Dw AXnsâ FXnÀhiw 40 sk.an. Dw Bbm {XntImW ¯nsâ ]cnhr¯¯nsâ hymkw ImWpI

84 ...............................................................................................................................MUKULAM MATHS

A\p_Ôw I

MUKULAM MATHS................................................................................................................................. 85

A\p_Ôw II

86 ...............................................................................................................................MUKULAM MATHS

A\p_Ôw III

A\p_Ôw IV Angle (Degree, Minutes) 03 0 45'

Sine Aryabhatan 0.06545

Modern Sine 0.0654

07 0

30'

0.1306

0.1305

11

0

15'

0.1952

0.1951

15 0

00'

0.2589

0.2588

18

0

45'

0.3215

0.3227

22 0

30'

0.3824

0.3827

26 0

15'

0.4421

0.4423

30

0

00'

0.5000

0.5000

37 0

30'

0.6087

0.6088

41 0

15'

0.6595

0.6594

45

0

00'

0.7071

0.7071

48 0

45'

0.7519

0.7519

52

0

30'

0.7933

0.7934

56 0

15'

0.8316

0.8315

60 0

00'

0.8664

0.8666

63

0

45'

0.8973

0.8969

67 0

50'

0.9243

0.9239

71 0

15'

0.9471

0.9469

75

0

00'

0.9658

0.9659

78 0

45'

0.9819

0.9818

82

0

30'

0.9915

0.9914

86 0

15'

0.9980

0.9979

90 0

00'

1.0000

1.0000

MUKULAM MATHS................................................................................................................................. 87

Hcp Ncn{Xw `mcXobÀ hr¯s¯ 4 ]mZ§fmbpw (Hcp ]mZw = 900) Hcp ]mZs¯ 3 cminIfmbpw (Hcp cmin= 300) Hcp cminsb 30 Awi§fmbpw (Hcp Awiw = 10) Hcp Awis¯ 60 IeI fmbpw (Hcp Ieþ 1 an\p«v) Hcp Iesb 60 hnIeIfmbpw (Hcp hnIe= 1 sk¡â v) `mKn¨p. Bcy`S³ R =

» 3438an\p«v Asæn 51 F¶v Is­¯nbn«p­v.

360 x 60 c = 2x3.1416 2p

0

hr¯ ]cn[nsb 21600 Xpey`mK§fmbn `mKn¡pIbpw Bchpw Nm]\ofhpw Xpeyam bm tI{µtIm¬ 3438 an\p«v Asæn GItZiw 570 F¶XmWv Hcp tdUnb³

Rm¬ ]«nIbpw a«{XntImWhihpw sSIvÌv t]Pv \¼À 79 se sskUv t_mIvkv NÀ¨ sN¿pI a«{XntImW¯nsâ hiw ImWm³ ]cnhr¯¯nsâ RmWmbn his¯ amäpIbmWv ChnsS sNbvXXv. 400 tImWnsâ FXnÀhiw ImWm³ 800 tImWnsâ Nm]¯nepÅ RmWnsâ \of ambn

amäp¶p. Bcw 1 bqWnäv Bb hr¯¯n tI{µtIm¬ 80 Un{Kn hcpt¼mgpÅ Nm]¯nepÅ RmWnsâ \ofw ]«nIbn \n¶v Is­¯n AXns\ 4 sIm­v KpWn¨v FXnÀhiw ImWp¶p. ]n¶oSv aq¶mas¯ hiw ImWp¶Xn\v ss]XtKmdkv XXzw D]tbmKn¡p¶p. C¯cw NÀ¨bn eqsS ap³Ime§fnepÅ KWnX¯nsâ hfÀ¨bpw Ncn{Xt_m[hpw Ip«nIfn D­m¡p¶Xv \¶mbncn¡pw.

tImWfhpw Nm]\o fhpw t]Pv \¼À 75,76, 77 se sskUv t_mIvkv ChnsS NÀ¨ sN¿s« 45 Un{Kn

=

360 sâ 1 `mKw

8

45 Un{Kn tImWnsâ Nm]\ofw hr¯Npäfhnsâ 1 `mKw

8

\ 60 Un{Kn = 360 sâ 61 `mKw

1

60 Un{Kn tImWnsâ Nm]\ofw = hr¯Npäfhnsâ 6 `mKw hr¯Npäfhnsâ

1

F¦n tImWnsâ Afhv = 360 sâ 4 Nm]\ofw tImWnsâ Afhmbn FSp¡mhp¶XmWv

`mKamWv Nm]\ofw,

\tImWnsâ Afhv

Nm]\ofw hr¯¯nsâ Npäfhv

`mKw 360 x 14 = 90 Un{Kn

x 360 Un{Kn

\ofw Htc tImWn\v hr¯¯nsâ hep¸w amdnbmepw hr¯Nm] ¯nsâ Npäfhv

Hcp Ønc kwJybmWtÃm

S

2 pr

1 4

F¶Xv

Ønc kwJy

S

2 pr x 360 = tImWnsâ Un{Kn Afhv 88 ...............................................................................................................................MUKULAM MATHS

S r

S

S r

S r

x 360 = 2p

x 180

p

Htc tImWn\v s Dw r Dw amdpsa¦nepw Sr amdp¶nÃ

S r se hne amdp¶Xv tImWnsâ Afhv amdpt¼mgmWv S AXn\m r Xs¶ tImWf¡m³ D]tbmKn¡mw.

Cu Afhv Un{KnbneÃ. CXns\ tdUnb³ Afhv F¶pw

tdUnb³ Afhns\ 180 sIm­v KpWn¨m tImWfhv Un{Knbn In«psa¶v Xncn¨dnbpI.

p sF.kn.Sn. D]tbmKn¨v S ØnckwJysb¶v Is­¯mw. r

*

t]Pv 78 sskUvt_mIvkv

tImWfhv RmWneqsS Nm]¯neqsS Af¡mhp¶ tImWns\ B Nm]¯nsâ RmWnsâ \of¯neqsS Af¡m sa¶v ImWpIbmWv ChnsS. AXn\\pkcn¨ ]«nI \nÀ½n¨Xmbpw aäpapÅ Imcy§Ä Ip«n Isf Ncn{Xmt\zjW¯nte¡v \bn¡p¶XmWv. t]Pv 89 se sskUvt_mIvkv NÀ¨bv¡v hnt[bam¡pI a = Sin A

b = c =d Sin b Sin C

F¶v t_m[yamIpw.

A,B,C Ch 900 Bbmepw \yq\tIm¬ Bbmepw _rlXvtIm¬ Bbmepw icnbmsW¶v ImWpatÃm Cu XXzw D]tbmKn¨v Xmsg sImSp¯ {]iv\§Ä sNbvXpt\m¡pI Hcp {XntImW¯nsâ Hcp tIm¬ 1200 Bbpw FXnÀhiw 10 sk.an. Bbm ]cnhr¯¯nsâ hymkw F{X ?

1.

c­v hi§fpw DÄt¡mWpw aq¶mas¯ his¯ \nÀWbn¡p¶p. A 8

B

)

h = 8 x sin 40

D

BD = 8 x cos 40 DC = 6 _ BD = 6 _ 8 cos 40

?

h 6

C

AC2 = h2 +DC2 = 82xsin240+(6 _ 8cos40)2 = 82sin240+62+82+82cos240 _ 2x6x8cos40

= 82(sin240+cos240)+62_2x6x8cos40 _ =82+62 2x6x8cos40 =64+36_96cos40 =100 _ 96cos40 =100 _ 96x0.7660

_ =100 73.5360 = 26.4640 MUKULAM MATHS................................................................................................................................. 89

t]Pv \¼À 86, 87 NÀ¨ sNbvXm s]mXp kq{XhmIyw ImWphm³ Ignbp¶p tImWnsâ Afhv _rlXv tIm¬ Bbmepw Cu _Ôw a2 = b2 + c2 _ 2bcosA F¶v In«patÃm cos 90 = O, cos (180_A) = -Cos A F¶v ImWmw.

a « {Xn tIm Ww, \yq \ {Xn tIm Ww, _rlXv { XntImWw F¶nhbn hi§Ä X½nepÅ _Ôw Xmsg ImWp¶hn[w hniZoIcn¡p¶Xv t\m¡pI. ta ]dª hi§fpsS _Ô hpambn XmcXayw sN¿pI. * {XntImW¯nsâ aq¶v hi§Ä X¶m AXv a«{XntImWamtWm F¶v ]cntim[n¡m³ km[n¡patÃm. F¶m AXv \yq\{XntImWamtWm, _rlXv {XntImWamtWm F§s\ I­¯pw ? ABC  LA, LB, LC ,F¶nhbv¡v FXnscbpÅ his¯ bYm{Iaw a,b,c F¶n§s\ kqNn¸n¡p¶p. Hcp tIm¬ LC _rlXvtIm¬ F¶ncn¡s« A

c

a

B

b

h

APC bn \n¶v h2 = b2 _ x2

x C

P

a«{XntImWw APB bn \n¶v c2

=

(a+x)2 + h2

=

a2+x2+2ax+b2_x2

=

a2+b2+2ax

c 2 > a 2 + b2

90 ...............................................................................................................................MUKULAM MATHS

*

aq¶v tImWpIfpw \yq\tIm¬ Bbm hi§fnte¡v FXnÀioÀj¯n \n¶v hcbv¡p¶ ew_w {XntImW¯n\I¯mWtÃm A

c

b h

B

P

x

C

a

a« a«

ABP Â h2 = c2 _ (a_x)2 APC Â b2 = h2 +x2 = c2 _ (a_x)2 +x2 \ c2 = a2 +b2 _ 2ax \ c2 < a2 +b2 t]mse a2
\ c­v hi§fpsS hÀK§fpsS XpI aq¶mas¯ hi¯n\v Xpeyambm AXv Hcp a« {XntImWw. hepXmbm B hi¯n\v FXnsc B tIm¬ _rlXvtIm¬, sNdpXmbm AXv \yq\tIm¬.

MUKULAM MATHS................................................................................................................................. 91