Lab Manual XII PDF

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Lab manual XII (setting on (Hindi) 13-08-09) 1_10.pmd 99 28-Jun-2019, 10:39 AM  The basic principles of learning mathematics are : (a) learning should be related to each child individually (b) the need for mathematics should develop from an intimate acquaintance with the environment (c) the child should be active and interested, (d) concrete material and wide variety of illustrations are needed to aid the learning process (e) understanding should be encouraged at each stage of acquiring a particular skill (f) content should be broadly based with adequate appreciation of the links between the various branches of mathematics, (g) correct mathematical usage should be encouraged at all stages. – Ronwill 100 iz;ksx'kkyk iqfLrdk Lab manual XII (setting on (Hindi) 13-08-09) 1_10.pmd 100 28-Jun-2019, 10:39 AM  fØ;kdyki 1 mís'; vko';d lkexzh ;g lR;kfir djuk fd ,d ry esa lHkh IykbZoqM dk VqdM+k oqQN rkjksa osQ VqdM+s lekarj js[kkvksa osQ leqPp; L esa ,d R = (8), dhysa] lI+ksQn dkx”k] xksan bR;kfnA {(l, m) : l ⊥ m} }kjk ifjHkkf"kr laca/ lefer gS ijarq LorqY; vkSj laØked ugha gSA jpuk dh fof/ ,d IykbZoqM dk VqdM+k yhft, vkSj ml ij lI+ksQn dkx”k fpidkb,A IykbZoqM osQ mQij ;kn`fPNd :i ls dhyksa dh lgk;rk ls rkjksa dks bl izdkj fLFkj dhft, fd buesa ls oqQN lekarj gksa] oqQN ,d nwljs osQ yacor~ gksa rFkk oqQN >qosQ gq, gksa tSlk vkÑfr 1 esa fn[kk;k x;k gSA izn'kZu 1. eku yhft, fd n'kkZbZ xbZ rkjsa js[kk,¡ l1, l2,..., l8 dks fu:fir djrh gSaA 2. rkj l1 izR;sd js[kk l2, l3, l4 osQ yacor~ gSA (vkÑfr 1 nsf[k,) xf.kr 101 Lab manual XII (setting on (Hindi) 13-08-09) 1_10.pmd 101 28-Jun-2019, 10:39 AM  3. l6 js[kk l7 osQ yacor~ gSA 4. js[kk l2 js[kk l3 osQ lekarj gS] l3 js[kk l4 osQ lekarj gS l5, l8 osQ lekarj gSA 5. (l1, l2), (l1, l3), (l1, l4), (l6, l7) ∈ R izs{k.k 1. vko`Qfr 1 esa dksbZ Hkh js[kk Lo;a osQ yacor~ ugha gS blfy, lac/ a R = {( l, m) : l ⊥ m} ______A (gS@ugha gS) 2. vko`Qfr 1 esa l1 ⊥ l2 D;k l2 ⊥ l1\ ______ (gS@ugha gS) ∴ ( l1, l2) ∈ R ⇒ ( l2, l1) ______ R (∉/∈) blh izdkj] l3 ⊥ l1 D;k l1 ⊥ l3\ _______ (gS@ugha gS) ∴ ( l3, l1) ∈ R ⇒ ( l1, l3) ______ R (∉/∈) iqu% l6 ⊥ l7 D;k l7 ⊥ l6 \ _______ (gS@ugha gS) ∴ ( l6, l7) ∈ R ⇒ ( l7, l6) ______ R (∉/∈) ∴ lac/ a R lefer.... (gS@ugha gS) 3. vko`Qfr 1 esa] l2 ⊥ l1 vkSj l1 ⊥ l3 D;k l2 ⊥ l3... (gS@ugha gS) vFkkZr~ (l2, l1) ∈ R vkSj (l1 , l3) ∈ R ⇒ (l2, l3) ______ R (∉/∈) ∴ lac/ a R laØked.... (gS@;k ugha gS) fVIi.kh vuqiz;ksx 1. bl fLFkfr esa laca/ ,d rqY;rk bl fØ;kdyki dk mi;ksx] ;g tk¡pus osQ fy, laca/ ugha gSA fd;k tk ldrk gS fd ,d fn;k x;k laca/ 2. bl fØ;kdyki dh iqujko`fÙk rqY;rk&laca/ gS ;k ugha gSA fofHkUu fLFkfr;ksa esa oqQN vkSj rkjsa ysdj dh tk ldrh gSA 102 iz;ksx'kkyk iqfLrdk Lab manual XII (setting on (Hindi) 13-08-09) 1_10.pmd 102 28-Jun-2019, 10:39 AM  fØ;kdyki 2 mís'; vko';d lkexzh ;g lR;kfir djuk fd ,d ry esa lHkh js[kkvksa IykbZoqM dk ,d VqdM+k] rkj osQ oqQN dk leqPp; R tks R = {( l, m) : l || m} }kjk VqdM+s (8) dhys]a lIk+ Qs n dkx”k] xksna bR;kfnA ifjHkkf"kr gS] ,d rqY;rk lac/a gSA jpuk dh fof/ mi;qDr vkdkj dk IykbZoqM dk ,d VqdM+k yhft, vkSj ml ij lI+ksQn dkx”k fpidkb,A IykbZoqM ij dhyksa dh lgk;rk ls ;kn`fPNd :i ls rkjksa dks bl izdkj fLFkj dhft, fd muesa ls oqQN lekarj gksa] oqQN ,d nwljs osQ yacor~ gksa vkSj oqQN >qosQ gq, (fRk;Zd) gksa tSlk vko`Qfr 2 esa fn[kk;k x;k gSA izn'kZu 1. eku yhft, fd n'kkZbZ xbZ rkjsa js[kkvksa l1, l2,..., l8 dks fu:fir djrh gSaA 2. js[kk l1 izR;sd js[kk l2, l3, l4 osQ yacor~ gSA 3. js[kk l6 js[kk l7 osQ yacor~ gSA xf.kr 103 Lab manual XII (setting on (Hindi) 13-08-09) 1_10.pmd 103 11-Jul-2019, 3:34 PM  4. js[kk l2 js[kk l3 osQ lekarj gS] js[kk l3 js[kk l4 osQ lekarj gS vkSj js[kk l5 js[kk l8 osQ lekarj gSA 5. (l2, l3), (l3, l4), (l5, l8), ∈ R izs{k.k 1. vko`Qfr 2 esa izR;sd js[kk Lo;a osQ lekarj gSA blfy, laca/ R = {( l, m) : l || m} ,d LorqY; laca/ ______A (gS@ugha gS) 2. vko`Qfr 2 esa izsf{kr dhft, fd l2
l3 gS D;k l3... l2gS (|| / || ) blfy, (l2, l3) ∈ R ⇒ (l3, l2)... R (∉/∈) blhizdkj l3 || l4 D;k l4...l3? (|| / || ) blfy, (l3, l4) ∈ R ⇒ (l4, l3)... R (∉/∈) vkSj (l5, l8) ∈ R ⇒ (l8, l5)... R (∉/∈) ∴ vr% laca/ R ,d lefer laca/... A (gS@ugha gS) 3. vko`Qfr 2 esa izsf{kr dhft, fd l2 || l3 vkSj l3 || l4 A D;k l2... l4 \ (|| / ||) blfy, (l2, l3) ∈ R vkSj (l3, l4) ∈ R ⇒ (l2, l4)... R (∈/∉) blh izdkj l3 || l4 vkSj l4 || l2 D;k l3... l2 ? (|| / || ) blfy, (l3, l4) ∈ R, (l4, l2) ∈ R ⇒ (l3, l2)... R (∈,∉) bl izdkj] laca/ R ,d laØked laca/... A (gS@ugha gS) vr% laca/ R ,d LorqY;] lefer vkSj laØked laca/ gSA blfy, R ,d rqY;rk&laca/ gSA vuqiz;ksx fVIi.kh bl fØ;kdyki dh iqujko`fÙk fofHkUu ;g fØ;kdyki rqY;rk&laca/ le>us osQ fy, fLFkfr;ksa esa oqQN vkSj rkj ysdj dh mi;ksxh gSA tk ldrh gSA 104 iz;ksx'kkyk iqfLrdk Lab manual XII (setting on (Hindi) 13-08-09) 1_10.pmd 104 01-Jul-2019, 3:48 PM  fØ;kdyki 3 mís'; vko';d lkexzh ,d ,sls iQyu dk fu:i.k djuk tks dkMZcksM]Z dhys]a lqryh] xksna vkSj IykfLVd ,oSQdh ugha gS ijarq vkPNknd gSA dh iêðh (iêðh) jpuk dh fof/ 1. dkMZcksMZ osQ ckb± vksj ,d IykfLVd dh iV~Vh fpidkb, vkSj ml ij rhu dhyas fLFkj dhft, tSlk vko`Qfr 3.1 esa fn[kk;k x;k gSA iêðh ij dhyksa dks 1, 2 vkSj 3 ls ukekafdr dhft,A 2. dkMZcksMZ osQ nk;ha vksj ,d nwljh IykfLVd dh iêðh fpidkb, vkSj ml ij nks dhysa fLFkj dhft, tSlk vko`Qfr 3.2 esa fn[kk;k x;k gSA iêðh ij dhyksa dks a vkSj b uke nhft,A 3. vko`Qfr 3.3 esa fn[kk, x, vuqlkj ckb± iêðh dh dhyksa dks nkb± iêðh dh dhyksa ls lqryh }kjk tksfM+,A izn'kZu 1. leqPp; X = {1, 2, 3} yhft,A 2. leqPp; Y = {a, b} yhft,A 3. leqPp; X osQ vo;oksa dks leqPp; Y osQ rnuq:iksa vo;oksa ls lqryh }kjk tksfM+, tSlk vkÑfr 3-3 esa fn[kk;k x;k gSA xf.kr 105 Lab manual XII (setting on (Hindi) 13-08-09) 1_10.pmd 105 28-Jun-2019, 10:39 AM  izs{k.k 1. X osQ vo;o 1 dk Y esa izfrfcac __________gSA X osQ vo;o 2 dk Y esa izfrfcac __________gSA X osQ vo;o 3 dk Y esa izfrfcac __________gSA blfy,] vko`Qfr 3.3 ,d __________ fu#fir djrh gSA 2. X osQ izR;sd vo;o dk Y esa _________ izfrfcac gSA blfy, iQyu _________gSA (,oSQdh@,oSQdh ugha) 3. Y osQ izR;sd vo;o dk X esa iwoZ izfrfcac (pre-image) dk _________gSA (vfLrRo@vfLrRo ugha) blfy, iQyu ________ gSA (vkPNknd@vkPNknd ugha) vuqiz;ksx fVIi.kh bl fØ;kdyki dk mi;ksx ,oSQdh vkSj vkPNknd leqPp; X vkSj Y osQ vo;oksa dh iQyuksa dks iznf'kZr djus osQ fy, fd;k tk la[;k esa ifjorZu djrs gq, bl ldrk gSA fØ;kdyki dk izn'kZu dhft,A 106 iz;ksx'kkyk iqfLrdk Lab manual XII (setting on (Hindi) 13-08-09) 1_10.pmd 106 28-Jun-2019, 10:39 AM  fØ;kdyki 4 mís'; vko';d lkexzh ,d iQyu dks iznf'kZr djuk tks ,oSQdh gS dkMZcksM]Z dhys]a lqryh] xksna vkSj IykfLVd ijarq vkPNknd ugha gSA dh ifêð;k¡A jpuk dh fof/ 1. dkMZcksMZ osQ ckb± vksj ,d IykfLVd dh iêðh fpidkb, vkSj bl ij nks dhysa fLFkj dhft, tSlk vko`Qfr 4.1 esa fn[kk;k x;k gSaA dhyksa dks a vkSj b uke nhft,A 2. dkMZcksMZ osQ nkb± vksj ,d nwljh IykfLVd dh iêðh fpidkb, vkSj ml ij rhu dhysa fLFkj dhft, tSlk vko`Qfr 4.2 esa fn[kk;k x;k gSA nkb± vksj dh iêðh ij dhyksa dks 1, 2 vkSj 3 uke nhft,A 3. ckb± vksj dh iêðh dh dhyksa dks nkb± vksj dh dhyksa ls lqryh }kjk tksfM+, tSlk vko`Qfr 4.3 esa fn[kk;k x;k gSA xf.kr 107 Lab manual XII (setting on (Hindi) 13-08-09) 1_10.pmd 107 28-Jun-2019, 10:39 AM  izn'kZu 1. leqPp; X = {a, b} yhft, 2. leqPp; Y = {1, 2, 3} yhft, 3. X osQ vo;oksa dks Y osQ vo;oksa ls lqryh }kjk tksfM+, tSlk vkÑfr 4-3 esa fn[kk;k x;k gSA izs{k.k 1. X osQ vo;o a dk Y esa izfrfcac ______________gSA X osQ vo;o b dk Y esa izfrfcac ______________gSA blfy,] vko`Qfr 4.3 ,d _____________________fu#fir djrh gSA 2. X osQ izR;sd vo;o dk Y esa _________ izfrfcac gSA blfy, iQyu _____________ gSA (,oSQdh@,oSQdh ugha) 3. Y osQ vo;o 1 dk X esa iwoZ izfrfcac dk __________ (vfLrRo gS@vfLRkRo ugha gS) blfy,] iQyu__________ gSA (vkPNknd@vkPNknd ugha) bl izdkj] vko`Qfr 4.3 ,d iQyu dks fu:fir djrh gS tks _________ gS iajrq vkPNknd ugha gSA vuqiz;ksx bl fØ;kdyki dk mi;ksx ,oSQdh tks vkPNknd ugha gSaA iQyuksa dh ladYiuk dks iznf'kZr djus osQ fy, fd;k tk ldrk gSA 108 iz;ksx'kkyk iqfLrdk Lab manual XII (setting on (Hindi) 13-08-09) 1_10.pmd 108 28-Jun-2019, 10:39 AM  fØ;kdyki 5 mís'; vko';d lkexzh sin x osQ xzkI+kQ dk iz;ksx djosQ sin −1 x dk dkMZcksMZ] lI+kQs n pkVZ isij] :yj] jaxhu xzkI+kQ [khapuk vkSj niZ.k ijkorZu (js[kk isu] xksan] isafly] jcj (eraser)] dVj] y = x osQ lkis{k) dh ladYiuk dk izn'kZu dhysa vkSj irys rkj djukA jpuk dh fof/ 1. mi;qDr foekvksa] eku yhft, 30 cm × 30 cm dk ,d dkMZcksMZ yhft,A 2. dkMZcksMZ ij 25 cm × 25 cm vkdkj dk lI+ksQn pkVZ isij fpidkb,A 3. isij ij nks ijLij yac js[kk,¡ [khafp, vkSj mUgsa ledksf.kd v{k X′OX vkSj YOY′ osQ uke nhft, (vko`Qfr 5 nsf[k,)A vko`Qfr 5 xf.kr 109 Lab manual XII (setting on (Hindi) 13-08-09) 1_10.pmd 109 28-Jun-2019, 10:39 AM  4. x&v{k ij 1 bdkbZ dks y&v{k dh 1 bdkbZ dk 1.25 xq.kk ysrs gq, v{kksa dks lfUudV :i esa va'kkafdr dhft, tSlk vko`Qfr 5 esa fn[kk, x;k gSA ⎛π π⎞ ⎛ π π⎞ ⎛ π π⎞ ⎛ π π⎞ 5. funsZ'kkad ry esa fcanqvksa ⎜ , sin ⎟ , ⎜ , sin ⎟ , ⎜ , sin ⎟ , ⎜ , sin ⎟ dks ⎝6 6⎠ ⎝ 4 4⎠ ⎝ 3 3⎠ ⎝ 2 2⎠ lfUudV :i esa vafdr dhft, vkSj izR;sd fcanq N1, N2, N3, N4, ij dhysa fLFkj dhft,A 6. blh izfØ;k dks x v{k osQ nwljh vksj fcanqvksa ⎛– π – π⎞ ⎛– π – π⎞ ⎛ –π –π⎞ ⎛ – π –π⎞ ⎜⎝ , sin ⎟ , ⎜ , sin ⎟ , ⎜ , sin ⎟ , ⎜ , sin ⎟ osQ lfUudV eku 6 6 ⎠ ⎝ 4 4 ⎠ ⎝ 3 3 ⎠ ⎝ 2 2 ⎠ ysdj nksgjkb, vkSj bu fcanqvksa N1′, N2′, N3′, N4′ ij dhysa fLFkj dhft,A fcanq O ij Hkh ,d dhy fLFkj dhft,A –π π 7. x&v{k osQ nksuksa vksj dh dhyksa dks rkj }kjk tksfM+, ftlls sin x dk xzkI+kQ ls 2 2 izkIr gksxkA 8. js[kk y = x dk xzkI+kQ [khafp,A (fcanqvksa (1, 1), (2, 2), (3, 3),... vkfn dks vkysf[kr dj vkSj bu fcanqvksa dks rkj ls tksfM+,) 9. dhyksa N1, N2, N3, N4 ls js[kk y = x ij yac [khafp, vkSj bruk cus esa mi;ksxh gSA xf.kr 123 Lab manual XII (setting on (Hindi) 13-08-09) 1_10.pmd 123 28-Jun-2019, 10:39 AM  fØ;kdyki 10 mís'; vko';d lkexzh ;g lR;kfir djuk fd ,d fn, x, gkMZcksMZ] lI+ksQn dkx”k] isafly] LosQy] fcanq x0 ij ,d iQyu larr gS ;fn ∆x oSQyoqQysVj] xksan osQ i;kZIr NksVs eku osQ fy, ∆y = f ( x0 + ∆x ) – f ( x0 ) dk LosPN NksVk eku gSA jpuk dh fof/ 1. gkMZ cksMZ ij lI+ksQn dkx”k fpidkb,A 2. fn, x, larr iQyu dk oØ [khafp, tSlk fd vko`Qfr 10 esa fn[kk;k x;k gSA 3. x-v{k ij /ukRed fn'kk esa dksbZ fcanq A (x0, 0) yhft, vkSj bl fcanq osQ laxr fcanq P (x0, y0) dks oØ ij vafdr dhft,A 124 iz;ksx'kkyk iqfLrdk Lab manual XII (setting on (Hindi) 13-08-09) 1_10.pmd 124 28-Jun-2019, 10:39 AM  izn'kZu 1. A osQ nkb± vksj ,d fcanq M1 (x0 + ∆x1, 0) yhft, tgk¡ ∆x1, x esa o`f¼ gSA 2. M1 ls ,d yac [khafp, tks oØ dks N1 ij feysA ekuk N1 osQ funsZ'kakd (x0 + ∆x1, y0 + ∆y1) gSA 3. fcanq P (x0, y0) ls ,d yac [khafp, tks N1M1 dks fcanq T1 ij feysA 4. vc AM1 dks ekfi,A eku yhft, ;g ∆ x1 gSA bldks fjdkMZ dhft, vkSj N1T1 = ∆y1 dks ekfi, vkSj fjdkMZ dhft,A 5. x esa o`f¼ dks de dj ∆x2 (vFkkZr~ ∆x2 < ∆x1) dhft, ftlls ,d nwljk fcanq M2 ( x0 + ∆x2 ,0 ) izkIr gksA blosQ laxr oØ ij fcanq N2 izkIr dhft,A 6. ekuk yac PT1, N2M2 dks T2 ij izfrPNsn djrk gSA 7. iqu% AM 2 = ∆x2 dks ekfi, vkSj bldks fjdkMZ dhft,A vc N 2T2 = ∆y2 dks ekfi, vkSj fjdkMZ dhft,A 8. mi;qZDr pj.kksa dh iqujko`fÙk oqQN vkSj fcanqvksa osQ fy, dhft, ftlls ∆x NksVs ls NksVk gksrk tk,A izs{k.k Øe la[;k x0 esa o`f¼ y esa laxr o`f¼ dk eku dk eku 1. ∆x1 = ∆y1 = 2. ∆x2 = ∆y2 = 3. ∆x3 = ∆y3 = 4. ∆x4 = ∆y4 = 5. ∆x5 = ∆y5 = xf.kr 125 Lab manual XII (setting on (Hindi) 13-08-09) 1_10.pmd 125 28-Jun-2019, 10:39 AM  6. ∆x6 = ∆y6 = 7. ∆x7 = ∆y7 = 8. ∆x8 = ∆y8 = 9. ∆x9 = ∆y9 = 10. 1. vr% tc ∆x NksVk gksrk tkrk gS rc ∆y _________ gksrk tkrk gSA 2. bl izdkj ,d larr iQyu osQ fy, lim ∆y = 0 ∆x→0 vuqiz;ksx ;g fØ;kdyki] ,d fn, x, iQyu osQ vodyt (okekorhZ% vkSj nf{k.kkorZ) dh ladYiuk dks oØ osQ fdlh fcanq ij le>kus esa lgk;d gSA 126 iz;ksx'kkyk iqfLrdk Lab manual XII (setting on (Hindi) 13-08-09) 1_10.pmd 126 28-Jun-2019, 10:39 AM