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Hsc math practical 1st paper 2015 wg

1. The document provides instructions for determining the extreme points of a triangle given its vertex points. 2. The method involves placing the vertex points on a coordinate grid, connecting the points to form the triangle, and then calculating the x- and y-coordinates of each vertex using the distances from the origin. 3. The extreme points are identified as the vertex points with the maximum and minimum x- and y-coordinates, and the calculations are verified using the distances obtained from the origin.

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D”PZi MwYZ e¨envwiK 1g cÎ cixÿv bs 1.1 ZvwiLt cixÿ‡bi bvgt Ges we›`y؇qi ms‡hvRK †iLvsk‡K Abycv‡Z AšÍwe©f³Kvix we›`yi ¯’vbvsK wbY©q| g~jZZ¡t hw` Gi Dci we›`ywUi Rb¨ nq †hb Kv‡Z©mxq Z‡j I `ywU we›`y nq Ges -A‡ÿi mgvšÍivj nq Z‡e we›`y †K Abycv‡Z fvM Ki‡e| Avevi I we›`y `yBwUi ms‡hvM †iLvsk‡K we›`ywU Abycv‡Z AšÍwe©f³ Ki‡j, we›`yi ¯’vbv¼ ( ) cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AuvwK| 2. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i Ges we›`y `yBwU emvB| awi we›`yØq I | miæ †cwÝj w`‡q ms‡hvM K‡i ‡iLvsk †jLwP‡Î Dc¯’vcb Kwi| 3. we›`y w`‡q -A‡ÿi mgvšÍivj †iLvi Dci GKK I GKK a‡i I `yBwU we›`y wbB| 4. †hvM Kwi Ges †iLv AsKb Kwi hv †K we›`y‡Z †Q` K‡i| 5. we›`yi ¯’vbvsK wPwýZ Kwi| dj msKjbt we›`yi ¯’vbvsK MÖvd n‡Z cÖvß gvb m~Î n‡Z cÖvß gvb (30 Ni, 10 Ni)= ( ) djvdjt cÖ`Ë we›`y؇qi ms‡hvM †iLvsk‡K Abycv‡Z AšÍwef³Kvix we›`yi ¯’vbvsK gšÍe¨: MÖvd †_‡K cÖvß gvb Ges MvwYwZKfv‡e wbwg©Z gvb GKB| AZGe djvdj mwVK| A(3,5) C(6,2) B(7,1) P Q G XOX / Y Y / P A B C Q 1
D”PZi MwYZ e¨envwiK 1g cÎ cixÿY bs 1.2 ZvwiLt cixÿ‡bi bvgt wÎfz‡Ri kxl©we›`y Ges wÎfzRwU †ÿÎdj wbY©q| g~jZË¡t wÎfz‡Ri kxl©Îq Ges n‡j, wÎfzRwUi †ÿÎdj | | eM© GKK| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I AuvwK| 2. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i Ges we›`y wZbwU emvB| miæ †cwÝj w`‡q ms‡hvM K‡i wÎfzRwU A¼b Kwi| 3. we›`y w`‡q -A‡ÿi mgvšÍivj †iLv AvuwK| 4. I n‡Z Gi Ici h_vµ‡g I j¤^ AvuwK| wnmvet we›`yi ¯’vbv¼ ( Ni, Ni)= we›`yi ¯’vbv¼ ( Ni, Ni)= wÎfz‡Ri †ÿÎdj MÖvd n‡Z cÖvß gvb m~Î n‡Z cÖvß gvb = UªwcwRqvg Gi †ÿÎdj+wÎfzR Gi †ÿÎdj wÎfzR Gi †ÿÎdj = eM©GKK | | | | eM©GKK djvdjt wÎfz‡Ri †ÿÎdj eM©GKK gšÍe¨t MÖvd †_‡K cÖvß gvb I MvwYwZKfv‡e wbwg©Z gvb GKB| AZGe djvdj mwVK| B(-9, 1) C(-3, -1) M(5, -1) L(5, 1) A(5, 6) P Q XX/ Y/ Y O 2
D”PZi MwYZ e¨envwiK 1g cÎ cixÿY bs 1.3 ZvwiLt cixÿ‡bi bvgt mij‡iLvi †jLwPÎ AsKb| g~jZË¡t I m¤^wjZ mKj †hvMvkÖqx mgxKiY GKwU mij‡iLv wb‡`©k K‡i hLb I DfqB GKB mv‡_ k~b¨ bq| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. cÖ`Ë mgxKiY n‡Z cvB- mgxKiYwU‡Z Gi K‡qKwU gvb wb‡q Gi Abyiƒc gvb †ei Kwi I wb‡Pi QKwU •Zix Kwi: 2. GKwU QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I AuvwK| 3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK ‡¯‹j a‡i I we›`y¸wj ¯’vcb Kwi Ges miæ †cwÝj w`‡q ms‡hvM K‡i mij‡iLvi †jLwPÎ AsKb Kwi| ‡jLwP‡Îi •ewkó¨t i) cÖ`Ë mij‡iLvi Xvj-†Q` AvK…wZ I e‡j †iLvwU Aÿ‡K abvZ¥K w`‡K GKK `~‡i †Q` Ki‡e| ii) e‡j †iLvwU -A‡ÿi abvZ¥K w`‡Ki mv‡_ m~ÿè‡KvY Drcbœ K‡i| C(-7, -2) B(-2, 1) A(3, 4) 3 X/ X Y Y/ O
D”PZi MwYZ e¨envwiK 1g cÎ cixÿv bs 2 ZvwiLt cixÿ‡bi bvgt mgxKi‡Yi (e„Ë) †jLwPÎ AsKb| g~jZË¡t mgxKiY GKwU e„‡Ëi mgxKiY hvi †K‡›`ªi ¯’vbvsK Ges e¨vmva© | cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) ‡cwÝj K¤úvm (vii) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. e„ˇK e„‡Ëi mv‡_ Zzjbv K‡i cvB, e„‡Ëi †K›`ª Ges e¨vmva© = 2. GKwU QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I AuvwK| 3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK ‡¯‹j a‡i e„‡Ëi †K›`ª we›`yi Ae¯’vb wPwýZ Kwi| 4. we›`y‡K †K›`ª K‡i GKK e¨vmva© wb‡q ‡cwÝj K¤úvm Gi mvnv‡h¨ e„Ë AsKb Kwi| ‡j‡Li •ewkó¨t i. †jLwU GKwU e„Ë| ii. †jLwU Awew”Qbœ| iii. e„ËwU g~jwe›`yMvgx| iv) e„ËwU Aÿ‡K GKwU we›`y‡Z Ges Aÿ‡K GKwU we›`y‡Z †Q` K‡i| 4 C(-3, 4) XX/ Y/ Y
D”PZi MwYZ e¨envwiK 1g cÎ cixÿY bs 3.1 ZvwiLt cixÿ‡bi bvgt GKwU wÎfz‡Ri evû¸wj h_vµ‡g †m.wg., †m.wg. Ges †m.wg. n‡j H wÎfz‡Ri e„nËg I ÿz`ªZg †KvY wbY©q| g~jZË¡t g‡b Kwi GKwU wÎfzR hvi wZbwU evû h_vµ‡g †m.wg., †m.wg. Ges †m.wg. ‡Z e„nËg evû †m.wg. Gi wecixZ †KvY e„nËg †KvY Ges ÿz`ªZg evû †m.wg. Gi wecixZ †KvY ÿz`ªZg †KvY| Zvn‡j cÖ`Ë Dcv‡Ëi mvnv‡h¨ AsKb K‡i Puv`vi mvnv‡h¨ e„nËg I ÿz`ªZg †Kvb wbb©q Kwi Ges m~Î I †_‡K cÖvß gv‡bi mv‡_ mZ¨Zv hvPvB Kwi| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) Puv`v (vii) ‡cwÝj K¤úvm (viii) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I A¼b Kwi| 2. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK ‡¯‹j awi| 3. MÖvd †ccv‡i eivei ÿz`ªZg e‡M©i evûi mgvb K‡i e„nËg evû ‡m.wg †K‡U †bB| 4. ‡K †K›`ª K‡i ÿz`ªZg e‡M©i evûi mgvb e¨vmva© wb‡q GKwU e„ËPvc AvuwK Ges †K †K›`ª K‡i ÿz`ªZg e‡M©i evûi mgvb e¨vmva© wb‡q AviI GKwU e„ËPvc AvuwK| e„ËPvcØq ci¯úi we›`y‡Z †Q` K‡i| Ges †hvM Kwi| Zvn‡j †Z ‡m.wg., ‡m.wg Ges ‡m.wg m~wPZ K‡i| 5. Puv`vi mvnv‡h¨ e„nËg †KvY I ÿz`ªZg †KvY wbY©q Kwi| wnmvet dj msKjbt e„nËg †KvY wbY©q ÿz`ªZg †KvY wbY©q †m.wg. †m.wg. †m.wg. MÖvc †_‡K cÖvß gvb m~Î †_‡K cÖvß gvb MÖvc †_‡K cÖvß gvb m~Î †_‡K cÖvß gvb djvdjt wb‡Y©q e„nËg †KvY Ges ÿz`ªZg †KvY gšÍe¨t MÖvd †_‡K cÖvß gvb Ges MvwYwZKfv‡e wbYx©Z gvb cÖvq mgvb| AZGe djvdj mwVK| 5 XX/ Y/ Y A B C
D”PZi MwYZ e¨envwiK 1g cÎ cixÿY bs 3.2 ZvwiLt cixÿ‡Yi bvgt GKwU wÎfz‡Ri †KvY¸wj n‡j wÎfzRwUi evû¸wji AbycvZ wbY©q| g~jZË¡t g‡b Kwi, Gi †KvY¸wj I Gi wecixZ evû¸wj h_vµ‡g | Zvn‡j cÖ`Ë DcvË n‡Z MÖv‡di mvnv‡h¨ Ges m~‡Îi mvnv‡h¨ I Gi AbycvZ wbY©q Kwi| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) Puv`v (vii) ‡cwÝj K¤úvm (viii) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU MÖvd †ccv‡i ¯’vbvs‡Ki Aÿ‡iLv I AvuwK| 2. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = †m.wg. †¯‹j awi| 3. MÖvd †ccv‡i eivei ‡m.wg †K‡U †bB| 4. Puv`vi mvnv‡h¨ we›`y‡Z I we›`y‡Z AsKb Kwi| I ‡iLv ci¯úi‡K we›`y‡Z †Q` K‡i| 5. MÖvd †_‡K Puv`vi mvnv‡h¨ Ges †cwÝj K¤úv‡mi mvnv‡h¨ I evûi •`N©¨ †g‡c eivei ewm‡q h_vµ‡g I evû؇qi •`N©¨ wbY©q Kwi| wnmvet Avgiv Rvwb, -‡Z Avevi dj msKjbt djvdjt wb‡Y©q AbycvZ gšÍe¨t MÖvd †_‡K cÖvß gvb Ges MvwYwZKfv‡e wbYx©Z gvb cÖvq mgvb| AZGe djvdj mwVK| wbY©q MÖvd †_‡K cÖvß AbycvZ M~Î †_‡K cÖvß AbycvZ 6 X/ X Y Y/
D”PZi MwYZ e¨envwiK 1g cÎ cixÿY bs 4.1 ZvwiLt cixÿ‡Yi bvgt dvsk‡bi Ges iƒcvšÍwiZ ‡jLwPÎ AsKb K‡i †j‡Li •ewkó¨ wbY©q| g~jZË¡t GKwU cive„‡Ëi mgxKiY hvi kxl©we›`y g~jwe›`y‡Z Ges Aÿ -Aÿ| Gi †jL wb‡Ri mgvšÍiv‡j GKK ev‡g mwi‡q w`‡q cive„‡Ëi †jL cvIqv hvq hvi kxl©we›`y | cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) ‡cwÝj K¤úvm (vii) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU MÖvd †ccv‡i ¯’vbvs‡Ki Aÿ‡iLv I AvuwK| 2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit 3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i Gi †jL AsKb Kwi| 4. †jLwUi cÖwZwU we›`y‡K e‡M©i evûi mgvb A_©vr GKK evg w`‡K mwi‡q Gi ‡jL AsKb Kwi| ‰ewkó¨t i. †jLwPÎ `yBwU cive„Ë| Gi kxl©we›`y Gi kxl© we›`y ii. Gi †jL A‡ÿi mv‡c‡ÿ, I Gi †jL †iLvi mv‡c‡ÿ cÖwZmg| iii. Dfq eµ‡iLvB wb‡Pi w`‡K †Lvjv  AvK…wZi cive„Ë| 7 (-3, -9) (-1, -1) 0, 0 (1, -1) (2, -4) (3, -9)(-6, -9) (-5, -4) (-4, -1) *-3, 0)X/ X Y Y/ (2,-1) (-2,-4) (-4,-1) (0,-9)
D”PZi MwYZ e¨envwiK 1g cÎ cixÿY bs 4.2 ZvwiLt cixÿ‡Yi bvgt GKB †jLwPÎ dvsk‡bi I Zvi wecixZ dvsk‡bi †jLwPÎ AsKb I †j‡Li •ewkó¨ wbY©q| g~jZË¡t †j‡Li Dci¯’ we›`y¸wji fzR I †KvwUi ¯’vb wewbgq K‡i Gi †jLwPÎ AsKb Kiv hvq A_ev, ‡iLvi mv‡c‡ÿ Gi cÖwZ”Qwe AsKb K‡i Gi ‡jL cvIqv hvq| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) ‡cwÝj K¤úvm (vii) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU MÖvd †ccv‡i ¯’vbvs‡Ki Aÿ‡iLv I AvuwK| 2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit -6 -7 3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj †hvM K‡i Gi †jL AsKb Kwi| 4. GB †¯‹‡j we›`y¸wj ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj †hvM K‡i Gi †jL AsKb Kwi| ‣ewkó¨t i. †jLwPÎ `yBwU mij †iLv wb‡`©k K‡i| ii. ‡jLwPÎØq GKwU we›`y‡Z †Q` K‡i| iii. ‡jLwPÎØq †iLvi mv‡c‡ÿ ci¯ú‡ii cÖwZ”Qwe| 8 (-6, -7) (0, 5) (1, 7) (2, 9) (-7, -6) (5, 0) (7, 1) (9, 2) X/ X Y Y/ (0,0)
D”PZi MwYZ e¨envwiK 1g cÎ cixÿY bs ZvwiLt cixÿ‡Yi bvgt w·KvYwgwZK dvsk‡bi †jL AsKb K‡i †j‡Li •ewkó¨ wbY©q hLb g~jZË¡t myZivs GKwU ch©vqx (Periodic) dvskb hvi ch©vq , ZvB cÖwZ e¨ewa‡Z Gi †jLwPÎ AweKj GKiƒc n‡e| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) Puv`v (vii) ‡cwÝj K¤úvm (viii) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZ: 1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AvuwK| 2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit 3. -Aÿ eivei ÿz`ªZg e‡M©i evû = Ges -Aÿ eivei ÿz`ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i Gi †jL AsKb Kwi| ‣ewkó¨t i. †jLwPÎ Awew”Qbœ| ii. Bnv †XD‡qi gZ| iii. Bnv g~jwe›`yMvgx iv. GKwU ch©vqe„Ë dvskb hvi ch©vq 9 X/ X Y Y/ O(0,0) (900 ,1) (-1800 ,0) (-900 ,1) (1800 ,0)
D”PZi MwYZ e¨envwiK 1g cÎ cixÿY bs ZvwiLt cixÿ‡Yi bvgt ciggvb dvsk‡bi †jL AsKb K‡i †j‡Li •ewkó¨ wbY©q| g~jZË¡t mgxKi‡Y Gi mKj ev¯Íe gv‡bi Rb¨ Gi gvb AFbvZ¥K| , hLb = , hLb cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZ: 1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AvuwK| 2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit 3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í †hvM K‡i Gi †jL AsKb Kwi| ‣ewkó¨t i. †jLwPÎwU †iLvi mv‡c‡ÿ cÖwZmg ii. ‡jLwPÎwU 1g PZzf©vM I 2q PZzf©v‡M Amxg ch©šÍ we¯Í…Z| iii. †jLwPÎwU g~jwe›`y‡Z †Q` K‡i bv| iv. †jLwPÎwU A‡ÿi abvZ¥K w`‡K we`¨gvb| 10 (-2, 5) (-1, 3) (0, 1) (0.5, 0) (1, 1) (2, 3) (3, 5) X/ X Y Y/ O
D”PZi MwYZ e¨envwiK 1g cÎ cixÿY bs ZvwiLt cixÿ‡Yi bvgt we›`yi mwbœK‡U dvsk‡bi †jL‡K Avmbœfv‡e H we›`y‡Z ¯úk©‡Ki †jL Øviv ¯’vbxqfv‡e cÖwZ¯’vcb| g~jZË¡t we›`yi mwbœK‡U dvskmb‡K Avmbœfv‡e H we›`y‡Z ¯úk©K Øviv ¯’vbwqfv‡e cÖwZ¯’vcb Kivi m~Î, cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AvuwK| 2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit 3. -Aÿ eivei ÿz`ªZg N‡ii evû = Ges -Aÿ eivei ÿz`ªZg N‡ii evû = GKK †¯‹j a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i Gi †jL AsKb Kwi| 4. we›`y‡Z ¯úk©K AsKb Kwi| wnmvet n‡Z cvB, djvdjt we›`yi mwbœK‡U dvsk‡bi †jL‡K Avmbœfv‡e H we›`y‡Z ¯úk©K Gi †jL Øviv ¯’vbxqfv‡e cÖwZ¯’vcb Kiv nj| 11 (-90, -1) (-60, -0.87) (-30, -0.5) (30, 0.5) (60, 0.87) (90, 1) X/ X Y Y/ O(0,0)
D”PZi MwYZ e¨envwiK 1g cÎ cixÿY bs ZvwiLt cixÿ‡Yi bvgt dvsk‡bi Rb¨ we›`y‡Z I wbY©q, ‡hLv‡b Ges †jLwP‡Î I cÖ`k©b| g~jZË¡t ¯^vaxb PjK I Aaxb Pj‡Ki AšÍi‡Ki ga¨Kvi m¤úK© Ges ¯^vaxb Pj‡Ki AwZÿz`ª cwieZ©b Gi Rb¨ Aaxb Pj‡Ki AwZ ÿz`ª cwieZ©b cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AvuwK| 2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit 3. -Aÿ I -Aÿ eivei ÿz`ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i Gi †jL AsKb Kwi| 4. we›`y‡Z ¯úk©K AsKb Kwi| mij‡iLv‡K ¯úk©KwU Ges dvskbwU h_vµ‡g I we›`y‡Z †Q` K‡i| wnmvet n‡Z cvB, myZivs Ges wPÎ n‡Z cvB, djvdjt I ‡jLwP‡Î cÖ`k©b Kiv n‡jv| 12 X/ X Y Y/ O(0,0) A(2,4) N P Q(3,9) dy y
D”PZi MwYZ e¨envwiK 1g cÎ cixÿY bs ZvwiLt cixÿ‡Yi bvgt QqwU †KvwU e¨envi K‡i ∫ Gi gvb wbY©q| g~jZË¡t g‡b Kwi †ÿÎdj ∫ QqwU †KvwUi Rb¨ e¨envi K‡i ∫ Gi gvb wbY©q Kwi| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. e¨ewa‡Z mg`~ieZx© wU †KvwU Gi Rb¨ GB e¨ewai wb¤œcÖvšÍ I DaŸ©cÖv‡šÍi we‡qvMdj‡K Øviv fvM K‡i cÖ‡Z¨K ÿz`ª As‡ki †`N©¨ Gi gvb wbY©q Kwi| 2. Gi gvb n‡Z m~Î e¨envi K‡i wbY©q Kwi †hLv‡b 3. ‡_‡K wbY©q Kwi| 4. Aÿ eivei ÿz`ªZg e‡M©i evû = GKK I -Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi| miæ †cwÝj w`‡q we›`y¸wj ms‡hvRb K‡i †jLwPÎwU AsKb Kwi| 5. cÖvß QqwU †KvwU‡K A‡ÿi mwnZ †¯‹‡ji mvn‡h¨ mshy³ K‡i wU UªvwcwRqvg AvKv‡i cÖKvk Kwi| wnmvet ( ) = ( ) = eM© GKK (cÖvq) djvdj: wb‡Y©q †ÿÎdj ∫ = eM©GKK cÖvq| gšÍe¨t Gi gvb hZ †ekx n‡e Gi gvb KZ ÿz`ª n‡e Ges Gi gvb AwaKZi ï× n‡e| 13 (0, 1) (0.16, 1.03) (0.32, 1.11) (0.48, 1.26) (0.64, 1.51) (0.8 1.9) X Y Y O y0 y1 y2 y3 y4 y5 x0 x1 x2 x3 x4 x5
D”PZi MwYZ e¨envwiK 1g cÎ cixÿY bs ZvwiLt cixÿ‡Yi bvgt cuvPwU †KvwU e¨envi K‡i ∫ Gi gvb wbY©q| g~jZË¡t g‡b Kwi, †ÿÎdj ∫ cuvPwU †KvwUi Rb¨ e¨envi K‡i Gi gvb wbY©q Kwi| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. e¨ewa‡Z mg`~ieZx© wU †KvwU Gi Rb¨ GB e¨ewai wb¤œcÖvšÍ I DaŸ© cÖv‡šÍi we‡qvMdj‡K Øviv fvM K‡i cÖ‡Z¨K ÿz`ª As‡ki •`N©¨ Gi gvb wbY©q Kwi| 2. Gi gvb n‡Z m~Î e¨envi& K‡i wbY©q Kwi †hLv‡b 3. ‡_‡K Gi gvb wbY©q Kwit 4. Aÿ eivei ÿz`ªZg e‡M©i evû = GKK I -Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi| miæ †cwÝj w`‡q we›`y¸wj ms‡hvRb K‡i †jLwPÎwU AsKb Kwi| 5. cÖvß cuvPwU †KvwU‡K A‡ÿi mwnZ †¯‹‡ji mvn‡h¨ mshy³ K‡i wU UªvwcwRqvg AvKv‡i cÖKvk Kwi| wnmvet ( ) = ( ) = = eM© GKK (cÖvq) djvdjt wb‡Y©q †ÿÎdj ∫ = eM©GKK (cÖvq)| gšÍe¨t Gi gvb hZ †ekx n‡e Gi gvb KZ ÿz`ª n‡e Ges Gi gvb AwaKZi ï× n‡e| 14 XO y0 x0 y1 x1 y2 x2 y3 x3 y4 x4 Y (0,1) (.25,.9412) (.50,.80) (.75,.64) (1,.50)

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Hsc math practical 1st paper 2015 wg

  • 1.
    D”PZi MwYZ e¨envwiK1g cÎ cixÿv bs 1.1 ZvwiLt cixÿ‡bi bvgt Ges we›`y؇qi ms‡hvRK †iLvsk‡K Abycv‡Z AšÍwe©f³Kvix we›`yi ¯’vbvsK wbY©q| g~jZZ¡t hw` Gi Dci we›`ywUi Rb¨ nq †hb Kv‡Z©mxq Z‡j I `ywU we›`y nq Ges -A‡ÿi mgvšÍivj nq Z‡e we›`y †K Abycv‡Z fvM Ki‡e| Avevi I we›`y `yBwUi ms‡hvM †iLvsk‡K we›`ywU Abycv‡Z AšÍwe©f³ Ki‡j, we›`yi ¯’vbv¼ ( ) cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AuvwK| 2. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i Ges we›`y `yBwU emvB| awi we›`yØq I | miæ †cwÝj w`‡q ms‡hvM K‡i ‡iLvsk †jLwP‡Î Dc¯’vcb Kwi| 3. we›`y w`‡q -A‡ÿi mgvšÍivj †iLvi Dci GKK I GKK a‡i I `yBwU we›`y wbB| 4. †hvM Kwi Ges †iLv AsKb Kwi hv †K we›`y‡Z †Q` K‡i| 5. we›`yi ¯’vbvsK wPwýZ Kwi| dj msKjbt we›`yi ¯’vbvsK MÖvd n‡Z cÖvß gvb m~Î n‡Z cÖvß gvb (30 Ni, 10 Ni)= ( ) djvdjt cÖ`Ë we›`y؇qi ms‡hvM †iLvsk‡K Abycv‡Z AšÍwef³Kvix we›`yi ¯’vbvsK gšÍe¨: MÖvd †_‡K cÖvß gvb Ges MvwYwZKfv‡e wbwg©Z gvb GKB| AZGe djvdj mwVK| A(3,5) C(6,2) B(7,1) P Q G XOX / Y Y / P A B C Q 1
  • 2.
    D”PZi MwYZ e¨envwiK1g cÎ cixÿY bs 1.2 ZvwiLt cixÿ‡bi bvgt wÎfz‡Ri kxl©we›`y Ges wÎfzRwU †ÿÎdj wbY©q| g~jZË¡t wÎfz‡Ri kxl©Îq Ges n‡j, wÎfzRwUi †ÿÎdj | | eM© GKK| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I AuvwK| 2. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i Ges we›`y wZbwU emvB| miæ †cwÝj w`‡q ms‡hvM K‡i wÎfzRwU A¼b Kwi| 3. we›`y w`‡q -A‡ÿi mgvšÍivj †iLv AvuwK| 4. I n‡Z Gi Ici h_vµ‡g I j¤^ AvuwK| wnmvet we›`yi ¯’vbv¼ ( Ni, Ni)= we›`yi ¯’vbv¼ ( Ni, Ni)= wÎfz‡Ri †ÿÎdj MÖvd n‡Z cÖvß gvb m~Î n‡Z cÖvß gvb = UªwcwRqvg Gi †ÿÎdj+wÎfzR Gi †ÿÎdj wÎfzR Gi †ÿÎdj = eM©GKK | | | | eM©GKK djvdjt wÎfz‡Ri †ÿÎdj eM©GKK gšÍe¨t MÖvd †_‡K cÖvß gvb I MvwYwZKfv‡e wbwg©Z gvb GKB| AZGe djvdj mwVK| B(-9, 1) C(-3, -1) M(5, -1) L(5, 1) A(5, 6) P Q XX/ Y/ Y O 2
  • 3.
    D”PZi MwYZ e¨envwiK1g cÎ cixÿY bs 1.3 ZvwiLt cixÿ‡bi bvgt mij‡iLvi †jLwPÎ AsKb| g~jZË¡t I m¤^wjZ mKj †hvMvkÖqx mgxKiY GKwU mij‡iLv wb‡`©k K‡i hLb I DfqB GKB mv‡_ k~b¨ bq| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. cÖ`Ë mgxKiY n‡Z cvB- mgxKiYwU‡Z Gi K‡qKwU gvb wb‡q Gi Abyiƒc gvb †ei Kwi I wb‡Pi QKwU •Zix Kwi: 2. GKwU QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I AuvwK| 3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK ‡¯‹j a‡i I we›`y¸wj ¯’vcb Kwi Ges miæ †cwÝj w`‡q ms‡hvM K‡i mij‡iLvi †jLwPÎ AsKb Kwi| ‡jLwP‡Îi •ewkó¨t i) cÖ`Ë mij‡iLvi Xvj-†Q` AvK…wZ I e‡j †iLvwU Aÿ‡K abvZ¥K w`‡K GKK `~‡i †Q` Ki‡e| ii) e‡j †iLvwU -A‡ÿi abvZ¥K w`‡Ki mv‡_ m~ÿè‡KvY Drcbœ K‡i| C(-7, -2) B(-2, 1) A(3, 4) 3 X/ X Y Y/ O
  • 4.
    D”PZi MwYZ e¨envwiK1g cÎ cixÿv bs 2 ZvwiLt cixÿ‡bi bvgt mgxKi‡Yi (e„Ë) †jLwPÎ AsKb| g~jZË¡t mgxKiY GKwU e„‡Ëi mgxKiY hvi †K‡›`ªi ¯’vbvsK Ges e¨vmva© | cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) ‡cwÝj K¤úvm (vii) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. e„ˇK e„‡Ëi mv‡_ Zzjbv K‡i cvB, e„‡Ëi †K›`ª Ges e¨vmva© = 2. GKwU QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I AuvwK| 3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK ‡¯‹j a‡i e„‡Ëi †K›`ª we›`yi Ae¯’vb wPwýZ Kwi| 4. we›`y‡K †K›`ª K‡i GKK e¨vmva© wb‡q ‡cwÝj K¤úvm Gi mvnv‡h¨ e„Ë AsKb Kwi| ‡j‡Li •ewkó¨t i. †jLwU GKwU e„Ë| ii. †jLwU Awew”Qbœ| iii. e„ËwU g~jwe›`yMvgx| iv) e„ËwU Aÿ‡K GKwU we›`y‡Z Ges Aÿ‡K GKwU we›`y‡Z †Q` K‡i| 4 C(-3, 4) XX/ Y/ Y
  • 5.
    D”PZi MwYZ e¨envwiK1g cÎ cixÿY bs 3.1 ZvwiLt cixÿ‡bi bvgt GKwU wÎfz‡Ri evû¸wj h_vµ‡g †m.wg., †m.wg. Ges †m.wg. n‡j H wÎfz‡Ri e„nËg I ÿz`ªZg †KvY wbY©q| g~jZË¡t g‡b Kwi GKwU wÎfzR hvi wZbwU evû h_vµ‡g †m.wg., †m.wg. Ges †m.wg. ‡Z e„nËg evû †m.wg. Gi wecixZ †KvY e„nËg †KvY Ges ÿz`ªZg evû †m.wg. Gi wecixZ †KvY ÿz`ªZg †KvY| Zvn‡j cÖ`Ë Dcv‡Ëi mvnv‡h¨ AsKb K‡i Puv`vi mvnv‡h¨ e„nËg I ÿz`ªZg †Kvb wbb©q Kwi Ges m~Î I †_‡K cÖvß gv‡bi mv‡_ mZ¨Zv hvPvB Kwi| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) Puv`v (vii) ‡cwÝj K¤úvm (viii) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I A¼b Kwi| 2. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK ‡¯‹j awi| 3. MÖvd †ccv‡i eivei ÿz`ªZg e‡M©i evûi mgvb K‡i e„nËg evû ‡m.wg †K‡U †bB| 4. ‡K †K›`ª K‡i ÿz`ªZg e‡M©i evûi mgvb e¨vmva© wb‡q GKwU e„ËPvc AvuwK Ges †K †K›`ª K‡i ÿz`ªZg e‡M©i evûi mgvb e¨vmva© wb‡q AviI GKwU e„ËPvc AvuwK| e„ËPvcØq ci¯úi we›`y‡Z †Q` K‡i| Ges †hvM Kwi| Zvn‡j †Z ‡m.wg., ‡m.wg Ges ‡m.wg m~wPZ K‡i| 5. Puv`vi mvnv‡h¨ e„nËg †KvY I ÿz`ªZg †KvY wbY©q Kwi| wnmvet dj msKjbt e„nËg †KvY wbY©q ÿz`ªZg †KvY wbY©q †m.wg. †m.wg. †m.wg. MÖvc †_‡K cÖvß gvb m~Î †_‡K cÖvß gvb MÖvc †_‡K cÖvß gvb m~Î †_‡K cÖvß gvb djvdjt wb‡Y©q e„nËg †KvY Ges ÿz`ªZg †KvY gšÍe¨t MÖvd †_‡K cÖvß gvb Ges MvwYwZKfv‡e wbYx©Z gvb cÖvq mgvb| AZGe djvdj mwVK| 5 XX/ Y/ Y A B C
  • 6.
    D”PZi MwYZ e¨envwiK1g cÎ cixÿY bs 3.2 ZvwiLt cixÿ‡Yi bvgt GKwU wÎfz‡Ri †KvY¸wj n‡j wÎfzRwUi evû¸wji AbycvZ wbY©q| g~jZË¡t g‡b Kwi, Gi †KvY¸wj I Gi wecixZ evû¸wj h_vµ‡g | Zvn‡j cÖ`Ë DcvË n‡Z MÖv‡di mvnv‡h¨ Ges m~‡Îi mvnv‡h¨ I Gi AbycvZ wbY©q Kwi| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) Puv`v (vii) ‡cwÝj K¤úvm (viii) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU MÖvd †ccv‡i ¯’vbvs‡Ki Aÿ‡iLv I AvuwK| 2. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = †m.wg. †¯‹j awi| 3. MÖvd †ccv‡i eivei ‡m.wg †K‡U †bB| 4. Puv`vi mvnv‡h¨ we›`y‡Z I we›`y‡Z AsKb Kwi| I ‡iLv ci¯úi‡K we›`y‡Z †Q` K‡i| 5. MÖvd †_‡K Puv`vi mvnv‡h¨ Ges †cwÝj K¤úv‡mi mvnv‡h¨ I evûi •`N©¨ †g‡c eivei ewm‡q h_vµ‡g I evû؇qi •`N©¨ wbY©q Kwi| wnmvet Avgiv Rvwb, -‡Z Avevi dj msKjbt djvdjt wb‡Y©q AbycvZ gšÍe¨t MÖvd †_‡K cÖvß gvb Ges MvwYwZKfv‡e wbYx©Z gvb cÖvq mgvb| AZGe djvdj mwVK| wbY©q MÖvd †_‡K cÖvß AbycvZ M~Î †_‡K cÖvß AbycvZ 6 X/ X Y Y/
  • 7.
    D”PZi MwYZ e¨envwiK1g cÎ cixÿY bs 4.1 ZvwiLt cixÿ‡Yi bvgt dvsk‡bi Ges iƒcvšÍwiZ ‡jLwPÎ AsKb K‡i †j‡Li •ewkó¨ wbY©q| g~jZË¡t GKwU cive„‡Ëi mgxKiY hvi kxl©we›`y g~jwe›`y‡Z Ges Aÿ -Aÿ| Gi †jL wb‡Ri mgvšÍiv‡j GKK ev‡g mwi‡q w`‡q cive„‡Ëi †jL cvIqv hvq hvi kxl©we›`y | cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) ‡cwÝj K¤úvm (vii) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU MÖvd †ccv‡i ¯’vbvs‡Ki Aÿ‡iLv I AvuwK| 2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit 3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i Gi †jL AsKb Kwi| 4. †jLwUi cÖwZwU we›`y‡K e‡M©i evûi mgvb A_©vr GKK evg w`‡K mwi‡q Gi ‡jL AsKb Kwi| ‰ewkó¨t i. †jLwPÎ `yBwU cive„Ë| Gi kxl©we›`y Gi kxl© we›`y ii. Gi †jL A‡ÿi mv‡c‡ÿ, I Gi †jL †iLvi mv‡c‡ÿ cÖwZmg| iii. Dfq eµ‡iLvB wb‡Pi w`‡K †Lvjv  AvK…wZi cive„Ë| 7 (-3, -9) (-1, -1) 0, 0 (1, -1) (2, -4) (3, -9)(-6, -9) (-5, -4) (-4, -1) *-3, 0)X/ X Y Y/ (2,-1) (-2,-4) (-4,-1) (0,-9)
  • 8.
    D”PZi MwYZ e¨envwiK1g cÎ cixÿY bs 4.2 ZvwiLt cixÿ‡Yi bvgt GKB †jLwPÎ dvsk‡bi I Zvi wecixZ dvsk‡bi †jLwPÎ AsKb I †j‡Li •ewkó¨ wbY©q| g~jZË¡t †j‡Li Dci¯’ we›`y¸wji fzR I †KvwUi ¯’vb wewbgq K‡i Gi †jLwPÎ AsKb Kiv hvq A_ev, ‡iLvi mv‡c‡ÿ Gi cÖwZ”Qwe AsKb K‡i Gi ‡jL cvIqv hvq| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) ‡cwÝj K¤úvm (vii) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU MÖvd †ccv‡i ¯’vbvs‡Ki Aÿ‡iLv I AvuwK| 2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit -6 -7 3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj †hvM K‡i Gi †jL AsKb Kwi| 4. GB †¯‹‡j we›`y¸wj ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj †hvM K‡i Gi †jL AsKb Kwi| ‣ewkó¨t i. †jLwPÎ `yBwU mij †iLv wb‡`©k K‡i| ii. ‡jLwPÎØq GKwU we›`y‡Z †Q` K‡i| iii. ‡jLwPÎØq †iLvi mv‡c‡ÿ ci¯ú‡ii cÖwZ”Qwe| 8 (-6, -7) (0, 5) (1, 7) (2, 9) (-7, -6) (5, 0) (7, 1) (9, 2) X/ X Y Y/ (0,0)
  • 9.
    D”PZi MwYZ e¨envwiK1g cÎ cixÿY bs ZvwiLt cixÿ‡Yi bvgt w·KvYwgwZK dvsk‡bi †jL AsKb K‡i †j‡Li •ewkó¨ wbY©q hLb g~jZË¡t myZivs GKwU ch©vqx (Periodic) dvskb hvi ch©vq , ZvB cÖwZ e¨ewa‡Z Gi †jLwPÎ AweKj GKiƒc n‡e| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) Puv`v (vii) ‡cwÝj K¤úvm (viii) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZ: 1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AvuwK| 2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit 3. -Aÿ eivei ÿz`ªZg e‡M©i evû = Ges -Aÿ eivei ÿz`ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i Gi †jL AsKb Kwi| ‣ewkó¨t i. †jLwPÎ Awew”Qbœ| ii. Bnv †XD‡qi gZ| iii. Bnv g~jwe›`yMvgx iv. GKwU ch©vqe„Ë dvskb hvi ch©vq 9 X/ X Y Y/ O(0,0) (900 ,1) (-1800 ,0) (-900 ,1) (1800 ,0)
  • 10.
    D”PZi MwYZ e¨envwiK1g cÎ cixÿY bs ZvwiLt cixÿ‡Yi bvgt ciggvb dvsk‡bi †jL AsKb K‡i †j‡Li •ewkó¨ wbY©q| g~jZË¡t mgxKi‡Y Gi mKj ev¯Íe gv‡bi Rb¨ Gi gvb AFbvZ¥K| , hLb = , hLb cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZ: 1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AvuwK| 2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit 3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í †hvM K‡i Gi †jL AsKb Kwi| ‣ewkó¨t i. †jLwPÎwU †iLvi mv‡c‡ÿ cÖwZmg ii. ‡jLwPÎwU 1g PZzf©vM I 2q PZzf©v‡M Amxg ch©šÍ we¯Í…Z| iii. †jLwPÎwU g~jwe›`y‡Z †Q` K‡i bv| iv. †jLwPÎwU A‡ÿi abvZ¥K w`‡K we`¨gvb| 10 (-2, 5) (-1, 3) (0, 1) (0.5, 0) (1, 1) (2, 3) (3, 5) X/ X Y Y/ O
  • 11.
    D”PZi MwYZ e¨envwiK1g cÎ cixÿY bs ZvwiLt cixÿ‡Yi bvgt we›`yi mwbœK‡U dvsk‡bi †jL‡K Avmbœfv‡e H we›`y‡Z ¯úk©‡Ki †jL Øviv ¯’vbxqfv‡e cÖwZ¯’vcb| g~jZË¡t we›`yi mwbœK‡U dvskmb‡K Avmbœfv‡e H we›`y‡Z ¯úk©K Øviv ¯’vbwqfv‡e cÖwZ¯’vcb Kivi m~Î, cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AvuwK| 2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit 3. -Aÿ eivei ÿz`ªZg N‡ii evû = Ges -Aÿ eivei ÿz`ªZg N‡ii evû = GKK †¯‹j a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i Gi †jL AsKb Kwi| 4. we›`y‡Z ¯úk©K AsKb Kwi| wnmvet n‡Z cvB, djvdjt we›`yi mwbœK‡U dvsk‡bi †jL‡K Avmbœfv‡e H we›`y‡Z ¯úk©K Gi †jL Øviv ¯’vbxqfv‡e cÖwZ¯’vcb Kiv nj| 11 (-90, -1) (-60, -0.87) (-30, -0.5) (30, 0.5) (60, 0.87) (90, 1) X/ X Y Y/ O(0,0)
  • 12.
    D”PZi MwYZ e¨envwiK1g cÎ cixÿY bs ZvwiLt cixÿ‡Yi bvgt dvsk‡bi Rb¨ we›`y‡Z I wbY©q, ‡hLv‡b Ges †jLwP‡Î I cÖ`k©b| g~jZË¡t ¯^vaxb PjK I Aaxb Pj‡Ki AšÍi‡Ki ga¨Kvi m¤úK© Ges ¯^vaxb Pj‡Ki AwZÿz`ª cwieZ©b Gi Rb¨ Aaxb Pj‡Ki AwZ ÿz`ª cwieZ©b cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AvuwK| 2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit 3. -Aÿ I -Aÿ eivei ÿz`ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i Gi †jL AsKb Kwi| 4. we›`y‡Z ¯úk©K AsKb Kwi| mij‡iLv‡K ¯úk©KwU Ges dvskbwU h_vµ‡g I we›`y‡Z †Q` K‡i| wnmvet n‡Z cvB, myZivs Ges wPÎ n‡Z cvB, djvdjt I ‡jLwP‡Î cÖ`k©b Kiv n‡jv| 12 X/ X Y Y/ O(0,0) A(2,4) N P Q(3,9) dy y
  • 13.
    D”PZi MwYZ e¨envwiK1g cÎ cixÿY bs ZvwiLt cixÿ‡Yi bvgt QqwU †KvwU e¨envi K‡i ∫ Gi gvb wbY©q| g~jZË¡t g‡b Kwi †ÿÎdj ∫ QqwU †KvwUi Rb¨ e¨envi K‡i ∫ Gi gvb wbY©q Kwi| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. e¨ewa‡Z mg`~ieZx© wU †KvwU Gi Rb¨ GB e¨ewai wb¤œcÖvšÍ I DaŸ©cÖv‡šÍi we‡qvMdj‡K Øviv fvM K‡i cÖ‡Z¨K ÿz`ª As‡ki †`N©¨ Gi gvb wbY©q Kwi| 2. Gi gvb n‡Z m~Î e¨envi K‡i wbY©q Kwi †hLv‡b 3. ‡_‡K wbY©q Kwi| 4. Aÿ eivei ÿz`ªZg e‡M©i evû = GKK I -Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi| miæ †cwÝj w`‡q we›`y¸wj ms‡hvRb K‡i †jLwPÎwU AsKb Kwi| 5. cÖvß QqwU †KvwU‡K A‡ÿi mwnZ †¯‹‡ji mvn‡h¨ mshy³ K‡i wU UªvwcwRqvg AvKv‡i cÖKvk Kwi| wnmvet ( ) = ( ) = eM© GKK (cÖvq) djvdj: wb‡Y©q †ÿÎdj ∫ = eM©GKK cÖvq| gšÍe¨t Gi gvb hZ †ekx n‡e Gi gvb KZ ÿz`ª n‡e Ges Gi gvb AwaKZi ï× n‡e| 13 (0, 1) (0.16, 1.03) (0.32, 1.11) (0.48, 1.26) (0.64, 1.51) (0.8 1.9) X Y Y O y0 y1 y2 y3 y4 y5 x0 x1 x2 x3 x4 x5
  • 14.
    D”PZi MwYZ e¨envwiK1g cÎ cixÿY bs ZvwiLt cixÿ‡Yi bvgt cuvPwU †KvwU e¨envi K‡i ∫ Gi gvb wbY©q| g~jZË¡t g‡b Kwi, †ÿÎdj ∫ cuvPwU †KvwUi Rb¨ e¨envi K‡i Gi gvb wbY©q Kwi| cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi| Kvh©c×wZt 1. e¨ewa‡Z mg`~ieZx© wU †KvwU Gi Rb¨ GB e¨ewai wb¤œcÖvšÍ I DaŸ© cÖv‡šÍi we‡qvMdj‡K Øviv fvM K‡i cÖ‡Z¨K ÿz`ª As‡ki •`N©¨ Gi gvb wbY©q Kwi| 2. Gi gvb n‡Z m~Î e¨envi& K‡i wbY©q Kwi †hLv‡b 3. ‡_‡K Gi gvb wbY©q Kwit 4. Aÿ eivei ÿz`ªZg e‡M©i evû = GKK I -Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi| miæ †cwÝj w`‡q we›`y¸wj ms‡hvRb K‡i †jLwPÎwU AsKb Kwi| 5. cÖvß cuvPwU †KvwU‡K A‡ÿi mwnZ †¯‹‡ji mvn‡h¨ mshy³ K‡i wU UªvwcwRqvg AvKv‡i cÖKvk Kwi| wnmvet ( ) = ( ) = = eM© GKK (cÖvq) djvdjt wb‡Y©q †ÿÎdj ∫ = eM©GKK (cÖvq)| gšÍe¨t Gi gvb hZ †ekx n‡e Gi gvb KZ ÿz`ª n‡e Ges Gi gvb AwaKZi ï× n‡e| 14 XO y0 x0 y1 x1 y2 x2 y3 x3 y4 x4 Y (0,1) (.25,.9412) (.50,.80) (.75,.64) (1,.50)