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Practice Paper- 12- Maths- 001- Partial fraction(vkaf'vkaf'vkaf'vkaf'kd fHkUukd fHkUukd fHkUukd fHkUu½½½½

Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each. (ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each. (iii) Qu. No. 11 to 15 are long answer type questions, carrying 5 marks each.

01½ ;fn �� = �� + �� gks] rks A dk eku Kkr dhft,A

02½ �� ds vkaf”kd fHkUuksa dh la[;k fdruh gksxhA

fuEufyf[kr O;atdksa dks vkaf'kd fHkUukas esa O;Dr dhft,&

03½ �� 04½

�� 05½ ��

06½ �� 07½ � �� 08½

�� 09½

�� 10½ �� 11½

��

12½ �� 13½

�� 14½ ��

15½ ��

Coaching classes for- IIT, AIPMT, PET, PAT, PMT, PPT, XII, XI, X Practice Paper- 12- Maths- 002- Determinants (lkjf.kdlkjf.kdlkjf.kdlkjf.kd½½½½


01½ ;fn � 3 −2� � � = 10 gks rks m dk eku Kkr dhft,A 02½ ;fn fcanq (a, b), (a’, b’) vkSj (a - a’ , b - b’ ) lajs[k gks rks fl) dhft, fd ab’ = a’b. 03½ crkb, fd fuEu lehdj.k fudk; laxr gy j[krs gS ;k ugha 2x – y = 5, 4x – 2y = 7

04½ fcuk foLrkj dj fuEu lkjf.kd dk eku Kkr dhft,& " 1 # # # # 1# 1 # " 05½ lkjf.kd dh lgk;rk ls ml f=Hkqt dk {ks=Qy Kkr dhft, ftlds dks.kh; fcUnqvksa ds funsZ’kkad (2, 6), (-4, -2) rFkk (3, -1) gSaA fl) dhft, fd& ¼dksbZ ,d½

06½ ¼v½ $ 1 %&'�( %&'�)%&'*+ 1 %&'*)%&',+ %&',( 1 $ = 0 ¼c½ "1 - . + /1 . / + -1 / - + ." = 0 ¼l½ " 1 #� # #� 1 ## # 1 " = 3

eku Kkr dhft,& ¼dksbZ ,d½

07½ ¼v½ " 0 99 −998−99 0 997998 −997 0 " ¼c½ 3%&'�512 %&'�3%&'�8 %&'�93 ¼l½ "1 1001 9991 1002 9981 1003 1000" 08½ � 1 2−1 −2� ds lHkh lg[k.Mksa dks Kkr dhft,A

09½ lkjf.kd "3 −3 43 2 −21 1 1 " ] esa vo;oksa 3] 3] &1 ds lg[k.M Kkr dhft, rFkk budh lgk;rk ls lkjf.kd dk

eku Kkr dhft,A 10½ fuEu lehdj.k fudk; dks dzsej fu;e ls gy dhft,& ¼dksbZ ,d½

(i) 2x – 3z = 0, x + 3y = -4, 3x + 4y = 3 (ii) x + y + z = 4, 2x – y + 2z = 5, 2x + y – z = 1 11½ fn;k gS x = cy + bz, y = az + cx, z = bx + ay tgkW x, y, z lHkh 'kwU; ugha gS rks fl) dhft, fd a2 + b2 + c2 + 2abc = 1 fl) dhft, fd& ¼dksbZ ,d½

12½ " 1 1 1- . /- . / " = �- − .��. − /��/ − -� 13½ "1 . + / . + / 1 / + - / + - 1 - + . - + . " = �- − .��. − /��/ − -� fuEu lehdj.kksa dks gy dhft,& ¼dksbZ ,d½

14½ ¼v½ "3+ − 8 5 21 7 + + 62 5 3 + +" = 0 ¼c½ "+ 3 72 + 27 6 +" = 0 ¼l½ "+ + 1 3 52 + + 2 52 3 + + 4" = 0

vFkok

fl) dhft, fd& ¼dksbZ ,d½

15½ ¼v½ 77��8 / /- ��8�� -. . 8��

77 = 4-./ ¼c½ "- + 1 -. -/-. . + 1 ./-/ ./ /� + 1" = 1 + - + . + /

Coaching classes for- IIT, AIPMT, PET, PAT, PMT, PPT, XII, XI, X Practice Paper- 12- Maths- 003- Matrix (vkO;wgvkO;wgvkO;wgvkO;wg½½½½


01½ ;fn 9 = :0 2 32 1 4; rFkk < = :7 6 31 4 5; rks A+B vkSj A - B dk eku Kkr dhft,A

02½ ;fn vkO;wg 9 = =1 3 λ + 22 4 83 5 10 > vO;qRdze.kh; gS rks λ dk eku Kkr dhft,A

03½ ,d 2 X 3 vkO;wg dh jpuk dhft, ftldk i oha iafDr vkSj josa LrEHk dk vo;o -?@ = ?�@ ls fn;k tkrk gSA

04½ ;fn A dk dze 1 X 1, B dk dze 1 X 3 rFkk C dk dze 3 X 2 gks rks ABC dk dze crkb,A

05½ ;fn A rFkk B lefer vkO;wg gS rks fl) dhft, fd AB – BA fo"ke lefer gSA

06½ fl) dhft, fd 9 = : 5 3−1 −2; lehdj.k A2 – 3A – 7 = 0 dks larq"V djrk gSA bl lehdj.k dk mi;ksx

djrs gq, A-1 dk eku Kkr dhft,A

07½ ;fn 9 = :1 23 4; gks] rks fl) dhft, fd A.(Adj A) = (Adj A).A = IAII .

08½ ;fn f(x) = x2 - 4x + 1, f(A) dk eku Kkr dhft, tcfd 9 = :2 31 2;- 09½ ;fn A = diag [1 -1 2] vkSj B = diag [2 3 -1] rks A+ B vkSj 3A + 4B Kkr dhft,A

10½ ;fn A ,d oxZZ vkO;wg gS rks fl) dhft, fd& ¼v½ A + A’ lefer gSA ¼c½ A – A’ fo"ke lefer gSaA

11½ ¼v½ fl) dhft, fd vkO;wg 9 = =4 −1 −43 0 −43 −1 −3> izfrdsUnzt gSA

¼c½ fl) dhft, fd vkO;wg 9 = = 2 −2 −4−1 3 41 −2 −3> ,d le”ke vkO;wg gSA

12½ fl) dhft, fd & :cos D −sin Dsin D cos D ;G = :cos HD −sin HDsin HD cos HD ; 13½ ;fn 9 = =1 2 31 3 41 4 3> ] rks A-1

Kkr dhft,A

14½ ;fn 9 = :3 27 5; vkSj < = :6 78 9; gks] rks (AB)-1 dk eku Kkr dhft,A

vFkok

15½ vkO;wgksa dh lgk;rk ls fuEu lehdj.k fudk;ksa dks gy dhft,& ¼dksbZ ,d½

¼v½ x + y + z = 6, x + 2y + 3z =14, x + 4y +9z = 36

¼c½ x + y + z = 5, 2x + y – z = 2, 2x – y + z = 2

Coaching classes for- IIT, AIPMT, PET, PAT, PMT, PPT, XII, XI, X Practice Paper- 12- Maths- 004- Iverse Trigonometric Functions (izfrykse f=dks.kferh; Qyuizfrykse f=dks.kferh; Qyuizfrykse f=dks.kferh; Qyuizfrykse f=dks.kferh; Qyu½½½½


01½ fl) dhft, fd cot-1

(-x) = π- cot-1

x.

02½ sin�� I− �√ K ds eq[; rFkk O;kid eku Kkr dhft,A

03½ sin :L − sin�� I�√� K; dk eku Kkr dhft,A

04½ fl) dhft, fd tan-1 x + cot-1 x = π/2.

05½ fl) dhft, fd tan�� I� K + tan�� I��K = L�- 06½ ;fn cos-1 + cos-1 y + cos-1z = π / 2 gks rks fl) dhft, fd x2 + y2 +z2 + 2xyz = 1. 07½ ;fn tan-1 x + tan-1 y + tan-1 z = π / 2 gks] rks fl) dhft, fd xy + yz + zx = 1.

08½ gy dhft, ¼dksbZ ,d½ & ¼v½ tan�� + + 2 cot�� + = L� ¼c½ tan�� + + tan�� 2+ = L

09½ fl) dhft, fd ¼dksbZ ,d½&¼v½ tan�� √+ = � cos�� ¼c½ cos�� + = 2 cos�� √�� -

10½ gy dhft, ¼dksbZ ,d½ ¼v½ sin�� − cos�� = tan�� ¼c½ cos�� I��K − cos�� I��K = 2 tan�� +- 11½ gy dhft, %& sin [ 2 cos

-1 { cot ( 2 tan

-1x)}] = 0

12½ fl) dhft, fd ¼dksbZ ,d½&¼v½ cot�� :�OPQ �QRS � ; = � ¼c½ sin[cot-1{tan(cos-1x)}]= x

13½ fl) dhft, fd¼dksbZ ,d½&¼v½ tan−1 :√1++ – √1−+√1++ + √1−+; = U4 − 12 cos−1 + ¼c½ cos�� :√�� √�� ; = L� − � cos�� + 14½ fuEu dks ljyre #i esa O;Dr dhft, & ¼v½

�OPQ V? QRS V��QRS W? OPQ W�X ¼c½

�OPQ Y�? QRS Y�Z�OPQ �Y? QRS �Y�[X�OPQ �Y? QRS �Y�\/��OPQ �Y�? QRS �Y�[^ ¼l½ �OPQ V? QRS V��OPQ W? QRS W��OPQ _? QRS _��cos `? QRS `� vFkok

15½ fl) dhft, fd &¼v½ 2tan�� ab�� tan ∅ d = cos�� I� OPQ ∅�� OPQ ∅K ¼c½ tan�� e√�� f = � tan�� +

Practice Paper- 12- Maths- 005- Product of Vectors (lfn’lfn’lfn’lfn’kksa dk xq.kuQykksa dk xq.kuQykksa dk xq.kuQykksa dk xq.kuQy½½½½ Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each. (ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each. (iii) Qu. No. 11 to 15 are long answer type questions, carrying 5 marks each.

01½ fl) dhft, fd lfn”k g + 4i + 3jk vkSj 4g + 2i − 4jk ijLij yacor~ gSA 02½ lfn”k -l = 4g + 4i − 10jk dk lfn”k .ml = g − 2i + 2jk dh fn”kk esa iz{ksi Kkr dhft,A 03½ ;fn -l ≠ 0, .ml ≠ 0 vkSj p-l + .mlp = p-l − .mlp gks] rks fl) dhft, fd -l vkSj .ml ijLij yacor~ gSA 04½ nks lfn”k -l rFkk .ml bl izdkj gSa fd |-l| = 4, p.mlp = 3 rFkk -l. .ml = 6 rks lfn”kksa -l rFkk .ml ds chp dk dks.k Kkr dhft,A 05½ ;fn -l = 2g − 3i + jk vkSj .ml = 3g + 2i rks p-l × .mlp Kkr dhft,A 06½ og ek=d lfn”k Kkr dhft, fd tks lfn”kksa 3g + 2i − jk rFkk 12g + 5i − 5jk esa ls izR;sd ij yacor~ gSA

07½ fl) dhft, fd ek=d lfn”k tks lfn’kksa 3g + i + 2jk, 2g − 2i + 4jk ds yacor~ gS] ± u�v�wk√� gSA

08½ nks cy tks lfn”kksa 4g + i − 3jk rFkk 3g + i − jk ls fu#fir gS] ,d d.k dks g + 2i + 3jk ls 5g + 4i + jk rd foLFkkfir dj nsrs gSaA cyksa ds }kjk fd;k x;k dk;Z Kkr dhft,A 09½ ¼v½ λ dk eku Kkr dhft, ;fn lfn’k λg + 3i + 2jk, 2g + 2i + 3jk, 2g + 3i + 4jk leryh; gSA ¼c½ λ dk eku Kkr dhft, ;fn pkj fcUnq ftuds fLFkfr lfn’k dze”k% g + 2i + 3jk, 3g − i + 2jk, −2g + λi + jk vkSj 6g − 4i + 2jk gSa] leryh; gSA 10½ ,d f=Hkqt ABC ds “kh"kZ A(-1, 0, 2), B(1, 2, 0) vkSj C(2, 3, 4) gSaA A ij AB ds vuqfn”k yxs ifjek.k 10 okys cy dk fcanq C ds ifjr% lfn”k vk?kw.kZ Kkr dhft,A

11½ ;fn nks bdkbZ lfn’kksa -l rFkk .ml ds chp dk dks.k θ gks] rks fl) dhft, fd tan Y = p�ml��mlpp�ml�mlp 12½ fl) dhft, fd ¼dksbZ ,d½ & ¼v½ x.ml × /ly × �/l × -l� = z-l .ml /l{/l ¼c½ -l × x.ml × /ly + .ml�/l × -l� + /lx-l × .mly = 0ml ¼l½ z-l × .ml .ml × /l /l × -l{ = z- mmml .mml /mml{

13½ gy dhft, ¼dksbZ nks½& ¼v½ ml lekUrj "kV~Qyd dk vk;ru Kkr dhft, ftldh rhu laxkeh dksjsa lfn”kksa -l = 2g − 3i, .ml = g + i − jk rFkk /l = 3g − jk ds }kjk nh tkrh gSA ¼c½ fl) dhft, fd pkj fcanq (4, 5,1), (0, -1, -1), (3, 9, 4) vkSj (- 4, 4, 4) leryh; gSaA

¼l½ fl) dhft, fd & z-l − .ml .mml − /l /l − -l{ = 0 ¼n½ fl) dhft, fd & z-l + .ml .mml + /l /l + -l{ = 2z-l .ml /l{ 14½ fl) dhft, fd lfn”kksa 2g − i + jk vkSj 3g + 4i − jk esa ls izR;sd ij yEc ek=d lfn”k ± ��u�v��wk√��

gS rFkk buds chp dk dks.k dh T;k b��| gSA vFkok

15½ lfn’k fof/k ls fl) dhft, fd ¼dksbZ ,d½& ¼v½ cos(A - B) = cosA cosB + sinA sinB ¼c½ sin (α + β) = sin α cos β + cos α sin β ¼l½ ac cosB – bc cosA = a2 – b2 ¼n½ c2 = a2 + b2 – 2 ab cosC

¼/k½ QRS � = QRS �� = QRS }8

ANAND ACADEMY, – 9827464311, 8109132711, ZONAL MARKET, SEC.-X, BHILAI

Coaching classes for- IIT, AIPMT, PET, PAT, PMT, PPT, XII, XI, X Practice Paper- 12- Maths- 006- Plane (lllleryeryeryery½½½½


01½ ml lery dk lehdj.k Kkr dhft, tks X- v{k ds lekUrj gS rFkk Y ,oa Z- v{kksa ls 5 vkSj 7 vUr%[k.M dkVrk gSSA 02½ lery 2x + 4y + 4z = 9 ds vfHkyEc dh fnd~&dksT;k,W Kkr dhft,A 03½ leryksa 2x – y + z = 6 vkSj x + y + 2z = 7 ds chp dk dks.k Kkr dhft,A 04½ fcanq (1, 2, 3) ls gksdj tkus okys ml lery dk lehdj.k Kkr dhft, tks lery 3x + 4y – 5z = 0 ds lekUrj gSA 05½ fcanq (3, 2, - 4) ls lery 2x + 3y – 6z + 6 = 0 ij Mkys x;s yac dh yackbZ Kkr dhft,A

06½ ,d lery v{kksa dks ∆ A, B, C ij feyrk gSA ABC dk dsUnzd (a, b, c) gSA fl) dhft, fd lery dk lehdj.k �� + *� + ,8 = 3 gSA 07½ fcUnqvksa (2, 2, -1), (3, 4, 2) vkSj (7, 0, 6) ls tkus okys lery dk lehdj.k Kkr dhft,A 08½ ml lery dk lehdj.k Kkr dhft, tks fcanq (-1, 3, 2) ls tkrk gS vkSj leryksa x + 2y + 2z = 5 vkSj 3x + 3y + 2z = 8 ij yac gSA 09½ ml lery dk lehdj.k Kkr dhft, tks fcanqvksa (-1, 1, 1) ,oa (1, -1, 1) ls tkrk gS rFkk lery x + 2y + 2z = 9 ij yac gSA 10½ ¼v½ ml lery dk lehdj.k Kkr dhft, tks leryksa x + y + z = 6 vkSj 2x + 3y + 4z = 0 ds dVku fcanq ls tkrk gS vkSj fcanq (1, 4, 6) ls gksdj xqtjrk gSA ¼c½ ml lery dk lehdj.k Kkr dhft, tks leryksa x + 3y + 4z – 5 = 0 rFkk 3x – 4y + 9z – 10 = 0 dh izfrPNsnh js[kk ls tkrk gS rFkk lery x + 2y = 0 ij yac gSA 11½ fl) djks fd nks lekUrj leryksa 2x – 2y + z + 3 = 0 rFkk 4x – 4y + 2z + 5 = 0 ds chp dh nwjh 1/6 gSA 12½ ¼v½ fcanq (4, 3, 5) dh xy- lery ls yEcor~ nwjh Kkr dhft,A ¼c½ ;fn ry 3x – 6y – 2z = 7 rFkk 2x + y – kz = 5 ,d nwljs ds yacor~ gS rks k dk eku Kkr dhft,A 13½ ml lery dk lehdj.k Kkr dhft, tks fcanqvksa (1, 0, 6) vkSj (0, 3, 1) ls gksdj tkrk gS rFkk xy- lery ij yac gSA

14½ lerykas 3x – 2y + 6z + 8 = 0 vkSj 2x – y + 2z + 3 = 0 ds e/; dks.k ds lef}Hkktd leryksa ds lehdj.k Kkr dhft,A

vFkok mu nks leryksa ds lehdj.k Kkr dhft, tks fcanqvksa (0, 4, -3) vkSj (6, -4, 3) ls gksdj tkrs gSa rFkk leryksa

}kjk v{kksa ls dkVs x;s vUr%[k.Mksa dk ;ksx 'kwU; gSA vFkok

15½ ¼v½mu leryksa ds lehdj.k Kkr dhft, tks lery x – 2y + 2z = 3 ds lekUrj gSa rFkk ftudh fcanq(1, 2, 3) ls ykfEcd nwjh 1 gSA

¼c½fl) dhft, fd fcanq (-4, 4, 4) (0, -1, -1), (4, 5, 1) rFkk (3, 9, 4) leryh; gSA


Coaching classes for- IIT, AIPMT, PET, PAT, PMT, PPT, XII, XI, X

Practice Paper- 12- Maths- 007- Straight Line & Sphere (ljy js[kkljy js[kkljy js[kkljy js[kk vkSj xksykvkSj xksykvkSj xksykvkSj xksyk½½½½ Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each. (ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each. (iii) Qu. No. 11 to 15 are long answer type questions, carrying 5 marks each.

01½ fcanqvksa (2, 3, 4) rFkk (1, -2, 3) ls xqtjus okys ljy js[kk dk lehdj.k Kkr dhft,A 02½ fcanqvksa (α, β, λ) ls tkus okyh ml js[kk dk lehdj.k Kkr dhft, tks x, y, z- v{k ls dze’k% 300, 900 vkSj 1200 dk dks.k cukrh gSA 03½ ljy js[kkvksa �� = *� = ,�� rFkk �� = *� = ,� ds e/; dks.k Kkr dhft,A

04½ js[kk �� = *� �� = ,�� vkSj lery 2x – y + z = 4 ds chp dk dks.k Kkr dhft,A 05½ ml xksys dk lehdj.k Kkr dhft, ftlds O;kl ds fljksa ds funsZ’kkad (2, -3, 4) vkSj (-5, 6, -7) gSA 06½ ¼v½ fcanqvksa (3, 4, 1) rFkk (5, 1, 6) ls tkus okys js[kk dk xy- ry ls izfrPNsn fcanq Kkr dhft,A ¼c½ ml fcanq ds funsZ”kkad Kkr dhft, tgkW js[kk �� = * � = ,�� lery x + y + z = 14 ls feyrh gSA

07½ ¼v½ k dk eku Kkr dhft, ;fn js[kk,W �� = *� � � �~ = ,�� w rFkk �� w = *�� = ,�|�� ijLij yacor~ gSA

¼c½ ml js[kk dk lehdj.k Kkr dhft, tks fcanq (a, b, c) ls tkrh gS rFkk Z- v{k ds lekarj gSA 08½ ¼v½ js[kk x = ay + b, z = cy + d dk lefer #i Kkr dhft,A ¼c½ fcanq (1, 2, 3) ls tkus okyh ml js[kk dk lehdj.k Kkr dhft, tks js[kk x – y + 2z = 5, 3x + y + z = 6 ds lekarj gSA

09½ ¼v½ fcanq (3, -1, 11) ls js[kk � = *� � = ,�� dh yacor~ nwjh Kkr dhft,A

¼c½ fcanq (0, 2, 3) ls js[kk �� = *�� = ,�� ij Mkys x;s yac ikn ds funsZ”kkad Kkr dhft,A

10½ ¼v½ ,d xksys dk lehdj.k x2 + y2 + z2 -3x - 2y + 2z – 15 = 0 gSA blds ,d O;kl AB ds fljs A dk funsZ”kkad (-1, 4, -3) gSaA fljs B dk funZs”kkad Kkr fdth,A ¼c½ ml xksys dk lehdj.k Kkr dhft, ftldk dsanz (3, 6, -4) gS vkSj tks lery 2x – 2y – z – 10 = 0 dks Li”kZ djrk gSA 11½ ¼v½ fcanq (1, 2, 3) dh lery x + y + z = 11 ls nwjh ljy js[kk + + 1 = *�� = ,�� ds lekarj Kkr dhft,A

¼c½ fcanq (-1, 3, -2) ls tkus okyh ml js[kk dk lehdj.k Kkr dhft, tks js[kkvksa �� = * = ,� vkSj

� � = *�� = ,�� ij yacor~ gSA

12½ ¼v½ ml lery dk lehdj.k Kkr dhft, tks fcanqvksa (2, 2, -1) ls tkrh gS rFkk js[kkvksa � = *� = ,� vkSj

�� = *� = ,� ds lekarj gSA

¼c½ fcanq (1, 0, 0) ls tkus okyh ml js[kk dk lehdj.k Kkr dhft, tks XZ- lery ij yac gSA 13½ ¼v½ fcanq (1, 6, 3) dk js[kk �� = *� � = ,�� ij izfrfcac Kkr dhft,A

¼c½ fl) dhft, fd js[kk,W �� = *� � = ,�� vkSj �� = *�� = ,�� leryh; gSA

14½ ¼v½ js[kkvksa �� = *�� = ,� vkSj �� = *�� = ,�� ds e/; U;wure nwjh Kkr dhft,A ¼c½ xksys 3x2+ 3y2 + 3z2 – 6x – 12y + 6z + 2 = 0 dk dsanz o f=T;k Kkr dhft,A

vFkok

15½ ¼v½ ml xksys dk lehdj.k Kkr dhft, tks (1, 0 ,0), (0, 1, 0) rFkk ( 0, 0, 1) ls tkrk gS rFkk ftldk dsanz 3x – y + z = 2 ij fLFkr gSA ¼c½ xksys x2 + y2 + z2 = 25 dk lery 2x + 3y – 6z = 28 }kjk o`Rrh; ifjPNsn dh f=T;k Kkr dhft,A


Coaching classes for- IIT, AIPMT, PET, PAT, PMT, PPT, XII, XI, X Practice Paper- 12- Maths- 008- Vector Form of Plane, Straight Line & Sphere

(lery] lery] lery] lery] ljy js[kk vkSj xksykljy js[kk vkSj xksykljy js[kk vkSj xksykljy js[kk vkSj xksyk dk lfn”dk lfn”dk lfn”dk lfn”k #ik #ik #ik #i½½½½ Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each. (ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each. (iii) Qu. No. 11 to 15 are long answer type questions, carrying 5 marks each.

01½ ml lery dk lehdj.k Kkr djks tks ewy fcanq ls 7 bdkbZ nwjh ij gS rFkk lfn’k 4� + 2� − 3j� ij yac gSA

02½ leryksa �l. x2g − 3i + 4jky = 1 rFkk �l. �−g + i� = 4 ds e/; dks.k Kkr dhft,A

03½ fcanqvksa (2, 3, 4) rFkk (1, -2, 3) ls tkus okyh js[kk dk lfn”k lehdj.k Kkr dhft,A

04½ xksys dk lfn”k lehdj.k Kkr dhft, ftldk dsanz (3, -1, 2) rFkk f=T;k 4 gSA

05½ ljy js[kkvksa �l = �2� − �� + λ�2� − 3j�� vkSj �l = �3� + j�� + μ�� + � + j�� ds e/; dks.k Kkr dhft,A

06½ fcanq (1, -2, 3) ls tkus okys ml lery dk lehdj.k Kkr dhft, tks ml ljy js[kk ij yac gS ftlds

fnd~ vuqikr 2, 1, -1 gSA

07½ ¼v½ fcanq (2, -1, 3) dh lery �l. x3g + 2i − 6jky + 15 = 0 ls nwjh Kkr dhft,A

¼c½ ml lery dk lfn”k lehdj.k Kkr dhft, tks fcanq 2� + 3� − j� ls tkrk gS rFkk lfn”k 3g − 4i + 7jk ds

yacor~ gSA bl lery dh ewyfcanq ls yacor~ nwjh Kkr dhft,A

08½ ¼v½ fcanq 2� − � − 4j� ls lery �l. x3g − 4i + 12jky = 9 dh yacor~ nwjh Kkr dhft,A

¼c½ js[kk �l = �� + 2� − j�� + λ�� − � + j�� vkSj lery �l. x2g − i + jky = 4 ds chp dk dks.k Kkr dhft,A

09½ ml xksys dk lfn”k lehdj.k Kkr dhft, tks xksys p�l + xg − 2i − 3jkyp = 5 ds ladsUnzh gS vkSj tks blls nks

xquh f=T;k dk gSA

10½ ,d xksys ds dsaanz dk fLFkfr lfn’k x3g + 6i − 4jky gSA xksyk lery �l. x2g − 2i − jky = 10 dks Li”kZ djrk

gSA xksys dk lfn”k lehdj.k Kkr dhft,A

11½ fl) dhft, fd js[kk,W �l = �� + � − j�� + λ�3� − �� rFkk �l = �4� − j�� + μ�2� + 3j�� izfrPNsfnr djrh gSaA

izfrPNsn fcanq Hkh Kkr dhft,A

12½ ¼v½ ,d xksys ds O;kl ds fljksa ds funs”kkad (4, 5, 1) rFkk (3, -2, -1) gSaA xksys dk lfn”k lehdj.k Kkr

dhft,A

¼c½ xksys dk lfn”k lehdj.k �l − �l. x8g − 6i + 10jky − 50 = 0 gSA xksys ds dsanz ds funZ”kkad vkSj f=T;k Kkr

dhft,A

13½ ml lery dk lehdj.k Kkr dhft, tks fcanqvksa g + i − 2jk, 2g − i + jk vkSj g + 2i + jk ls tkrk gSA

14½ lery �l. xg + 2i + 2jky = 18 ds }kjk xksyk |�l| = 10 ls dkVs x;s o`Rrh; [k.M dh f=T;k Kkr dhft,A vFkok

15½ fuEu js[kkvksa ds e/; U;wure nwjh Kkr dhft,&

¼v½ �l = �λ − 1�g + �λ + 1�i − �1 + λ�jk rFkk �l = �1 − μ�g + �2μ − 1�i + �μ + 2�jk ¼c½ �mml = �3 − ��g + �4 − 2��i + �� − 2�jk rFkk �l = �1 + ��g + �3� − 7�i + �2� − 2�jk



Practice Paper- 12- Maths- 009- Differentiation (vodyuvodyuvodyuvodyu½½½½ Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each. (ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each. (iii) Qu. No. 11 to 15 are long answer type questions, carrying 5 marks each. (iv) solve any one in each of the questions.

fuEufyf[kr Qyuksa dk x ds lkis{k vodyu dhft,& (izR;sd esa dksbZ ,d)

01½ (i) �� QRS �� (ii) ��. %&'+ (iii) �� OPQ � (iv) sin x2

02½ (i) cos x xos2x cos3x (ii) e log tanx (iii) e - logx (iv) tan�� b��8��8�� 03½ (i) �√OP� � (ii) %&'%&'%&'+ (iii) sin2 x2 (iv) %&'b��?G��?G��

04½ (i) logb�OPQ ��OPQ �� (ii) sin xo (iii) 2tan� (iv) sec�� + sin��

05½ (i) ��[��[� (ii)

QRS ��OPQ � (iiii) √tan + (iv) %&'��sec +�

06½ ¼v½ ;fn ( = �√�� gks] rks x = 16 ij �(�� dk eku Kkr dhft,A

¼c½ ;fn f(x) = ax + b rFkk f(0)= f ‘(0)= 3 rks f ‘ (1) vkSj f(1) dk eku Kkr dhft,] tgkW a vkSj b fu;rkad gSA 07½ ¼v½ ;fn ( = �� gks] rks fl) dhft, fd + �*�� = (�1 − (�- ¼c½ tan�� √�� dk tan�� + ds lkis{k vodyu dhft,A 08½ ¼v½ tan�� dk vodyu sin�� ds lkis{k dhft,A

¼c½ ;fn x = a(t + sin t), y = a (1 – cos t) rks �*�� dk eku Kkr dhft,A

09½ ¼v½ ;fn √1 − + + �1 − ( = -�+ − (� gks] rks fl) dhft, fd �*�� = b1−(21−+2 ¼c½ ;fn x

y = e

y - x gks] rks fl) dhft, fd �*�� = �� log ��

10½ ¼v½ ;fn ( = �� OPQ[\ � gks] rks fl) dhft, fd (1 – x2)y2 – xy1 – m2y = 0

¼c½ ;fn y = aemx + be - mx gks] rks fl) dhft, fd ��*�� = � (

11½ ¼v½ ;fn �cos +�* = �sin (�� gks] rks �*�� dk eku Kkr dhft,A

¼c½ ;fn ( = bcot + + �/&�+ + √cot + + ⋯ … … . gks] rks fl) dhft, fd �*�� = /&��/�� *

12½ ¼v½ +�� + �sin +�� Qyu dk x ds lkis{k vodyu dhft,A

¼c½ ;fn ( = ��8�� gks] rks fl) dhft, fd 2y1y3=3(y2)2

13½ izFke fl)akr ls (i) sin √x or (ii) e5x dk vody xq.kkad Kkr dhft,A

14½ fuEu dk x ds lkis{k vodyu dhft,& ¼v½ ( = cot�� √�� ¼c½ tan�� b�� ¼l½ tan�� OPQ �QRS �OPQ ��QRS � vFkok

15½ fuEu dk x ds lkis{k vodyu dhft,& ¼v½ �cot�� √+ ¼c½ tan��xcos √+y ¼l½ %&' �-H IL� + � K



Practice Paper- 12- Maths- 010-Applicatins of Differentiation (vodyu dvodyu dvodyu dvodyu ds vuqiz;ksxs vuqiz;ksxs vuqiz;ksxs vuqiz;ksx½½½½

Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each. (ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each. (iii) Qu. No.11to15 are long answer type questions, carrying 5 marks each.(iv)Solve any one in each of the questions 6 to 15.

01½ ,d d.k ,d ljy js[kk esa xfreku gSA d.k ds }kjk t lsd.M ess r; dh xbZ nwjh s ( ehVj esa ) laca/k

� = -�� + �� ds }kjk nh tkrh gSA fl) dhft, fd le; t ij d.k dk Roj.k blds }kjk t le; esa r; dh

xbZ nwjh ds cjkcj gSA

02½ ,d d.k fu;r LFkku O ls 10 eh- nwjh ij Lkw= � = 1 + 16� − 2� ls xfreku gSA ;fn og O ls tkus okyh

js[kk esa xfreku gks] rks O ls mldh nwjh Kkr dhft, tc og foijhr fn”kk esa ykSVuk izkjaHk djsxkA

03½ ,d d.k fuEukafdr fu;e ls ljy js[kk esa xfreku gS& s = 5e–t cost tc t = π/2 gks] rks bldk osx o Roj.k

D;k gksxk\

04½ ,d xqCckjk tks xksykdkj jgrk gS] pj f=T;k j[krk gSA blds vk;ru ds ifjorZu dh nj Kkr dhft,

tcfd f=T;k 5 lseh gksA

05½ fl) dhft, fd Qyu f(x) = cos x, 0≤ x ≤ π ds fy, gzkleku gSA

06½ ¼v½ ,d d.k ljy js[kk esa xfreku gSA mlds }kjk t le; essa pyh xbZ nwjh lw= s = 2t3 – 9t

2 + 12t + 6 ls

nh tkrh gSA fdrus le; ckn mldk Roj.k 'kwU; gksxk\ ml le; mldk osx D;k gksxk\

¼c½ ljy js[kk esa xfreku ,d d.k dk ,d fu;r fcanq ls nwjh s, le; t dk ,d f}?kkr Qyu gSA n”kkZb;s

fd d.k vpj Roj.k ls xfreku gSA 07½ ¼v½ ifjorZu”khy ?ku dh dksj esa 2 lseh@ls- dh nj ls o`f) gks jgh gSA ?ku ds vk;ru esa o`f) dh nj Kkr

dhft, tcfd dksj 5 lseh gksA

¼c½ ,d rkykc esa ,d iRFkj Qsadk tkrk gS ftlds dkj.k rjaxsa 3 lseh@ls- ds osx ls o`Rrksa esa mBrh gSA

ftl le; fdlh o`Rrkdkj rjax dh f=T;k 6 lseh gks] rks ifjc) {ks=Qy fdl xfr ls c<+ jgk gS\

08½ os vUrjky Kkr dhft, ftlesa fuEu Qyu o/kZeku ;k gzkleku gS&

¼v½ (i) f(x) = 5x2

+ 7x - 13 (ii) f(x) = -3x2

+ 12x + 8

¼c½ (i) f (x) = 5x3 – 15x

2 -120x + 3 (ii) f(x) = 4x

3 – 6x

2 + 3x + 12

09½ ¼v½ ,d iRFkj Å/ok/kZj Åij dh vksj Qsadk x;kA t le; ckn mldh ÅWpkbZ h ehVj gS] tgkW h = 49t – 4. 9 t2

gSA ,d lsd.M ckn iRFkj dh ÅWpkbZ] xfr rFkk Roj.k Kkr dhft,A

¼c½ ,d d.k {kSfrt ljy js[kk esa fuEu izdkj xfreku gS s = t4 – 6t3 + 12 t2 – 10 t + 3, xfr dh fn”kk esa

ifjorZu dc gksxk\

10½ ¼v½ nks /kukRed la[;k,W x vkSj y bl izdkj Kkr dhft, fd x + y = 60 rFkk xy3 mfPp"B gksA

¼c½ ,d xksys dh f=T;k a gSA fl) dhft, fd blds varxZr egRre vk;ru ds csyu dh ÅWpkbZ �√� gSA 11½ ¼v½ ,d vkneh ftleh ÅWpkbZ 180 lseh gS] ,d fctyh ds [kaHks ls 1-2 eh@ls- dh nj ls ihNs gV jgk

gSA ;fn fctyh ds [kaHks dh ÅWpkbZ 4-5eh gS] rks og nj Kkr dhft, ftl ij (i) mldh Nk;k c<+ jgh gS]

(ii) Nk;k dh Nksj xfr dj jgh gSA

¼c½ ,d 'kadq ds odz ry dh ifjorZu dh nj Kkr dhft, ftldh ÅWpkbZ h rFkk vk/kkj dh f=T;k r gS] ;fn

dsoy vk/kkj dh f=T;k ifjofrZr gksA

12½ ¼v½ fl) dhft, fd sin x + cos x dk mfPp"B eku √2 gSA

¼c½ fl) dhft, fd I��K� dk mfPp’B eku �� ~ gSA

13½ ¼v½ fl) dhft, fd x ds lHkh ekuksa ds fy, f(x) = x – cos x o/kZeku Qyu gSaA

¼c½ fl) dhft, fd Qyu f(x) = x2 – x + 1 vUrjky (0, 1) esa u rks o/kZeku gS vkSj u gh gzkleku gSA

14½ ¼v½ nks /kukRed la[;kvksa dk ;ksxQy 20 gSA la[;k,W Kkr dhft, &

(i) ;fn mudk xq.kuQy egRre gks (ii) ;fn muds oxksZa dk ;ksxQy U;wure gksA

¼c½ fl) dhft, fd xx dk eku x = 1/e ij fufEu"B gS rFkk fufEu"B eku I��K� �~

gSA

vFkok

15½ ¼v½ ,d f[kM+dh ,d vk;r ds #i esa gS ftl ij v)Zo`Rr cuk gSA bldh ifjeki 30 eh gSA foek;sa Kkr

dhft, ftlls fd vf/kdre izdk”k vanj vk ldsA

¼c½ ,d vk;r dk {ks=Qy 96 lseh2 gSa bldh yackbZ] pkSM+kbZ rFkk ifjeki Kkr dhft, tcfd ifjeki

U;wure gksA

Practice Paper- 12- Maths- 011 - Indefinite Integration (vfuf’vfuf’vfuf’vfuf’pr lekdyupr lekdyupr lekdyupr lekdyu½½½½ Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each. (ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each.

(iii) Qu. No.11to15 are long answer type questions, carrying 5 marks each.(iv)Solve any one in each of the questions 1 to 15.

fuEu dk x ds lkis{k lekdyu dhft,& ¼izR;sd esa ls dksbZ ,d½

01½ ¼v½ �� QRS �8�� ¼c½ ��QRS � ¼l½ ��OPQ � ¼n½ �� ¼r½ sin +√1 − cos 2+ 02½ ¼v½

��8�� ¼c½ ��[� ¼l½ � ��S[\ �� ¼n½ ��[\ �� ¼r½ QRSx��S[\ �y��

03½ ¼v½ ��√� ¼c½ ��[�� ¼l½ ��8��√��S � ¼n½ I��[��[�K ¼r½ %&'�+

gy dhft, ¼izR;sd esa ls ,d½ &

04½ ¼v½ sin�� + �+ ¼c½ cos�� + �+ ¼l½ tan�� x �+ ¼n½ tan�� √+ �+ ¼r½ �� QRS ��OPQ � �+ 05½ ¼v½ �¢£¤[\ ��√�� + ¼c½ sin�� + ¼l½ �� ¼n½ �� . �� . ��+ ¼r½ ��?G��8�� + 06½ ¼v½ ��QRS � �+ ¼c½ Q¥O �Q¥O ��S � �+ ¼l½ �QRS ��QRS � �+ ¼n½ ��QRS � �+ ¼r½ �QRS ��OPQ � �+ 07½ ¼v½ ¦P§ ��¨\� �+ ¼c½ ��OP� � ¼l½ ��√��?G� ¼n½ ��QRS� ��V� QRS ��W �¼r½ log �2 + + � �+ 08½ ¼v½ b�� + ¼c½ + -� �+ ¼l½ +�� + ¼n½ sin �%&'+� �+ ¼r½ ��QRS ��OPQ �� + 09½ ¼v½ ��/�+ �+ ¼c½ +� sin + �+ ¼l½ /&� :.%&' ��; �+ ¼n½ + sin�� + �+ ¼r½ �© ��[\ �� ~ �+ 10½ ¼v½ � QRS[\ �√�� + ¼c½ �� S[\ �� + ¼l½ tan�� QRS �OPQ �OPQ ��QRS � �+ ¼n½ ��| �+ ¼r½ ��8�� 11½ ¼v½ sin�� b �� +¼c½ tan�� √�� + ¼l½ tan�� √�� + ¼n½ �� OPQ � �+ ¼r½ �� ?G�� 12½ ¼v½ OPQ ��?G�� QRS �� + ¼c½ √+ − + + 1 �+ ¼l½ �� OPQ � ¼n½ ��QRS ��OPQ � ¼r½ ��H� + �+ 13½ ¼v½ �� + ¼c½ ��√�� ¼l½ sin + √/&� + + 2 cos + + 2 �+ ¼n½ �� QRS � ¼r½ �� + 14½ ¼v½ �√�� + ¼c½ ��√�� + ¼l½ �� + ¼n½ �QRS �√� OPQ � �+ ¼r½ �� +

vFkok

15½ ¼v½ tan�� √��√��√��√�� + ¼c½ √+ tan�� √+ �+ ¼l½ �� OPQ �� QRS � ¼n½ ��S � ¼r½ ��?G� �~ 8��X �~ �+


Coaching classes for- IIT, AIPMT, PET, PAT, PMT, PPT, XII, XI, X Practice Paper- 12- Maths- 012 - Definite Integration (fuf’fuf’fuf’fuf’pr lekdyupr lekdyupr lekdyupr lekdyu½½½½

Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each. (ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each.

(iii) Qu. No.11to14 are long answer type questions, carrying 5 marks each.(iv)Solve any one in each of the questions 1 to 14.

01½ ¼i½ 2��+�� ¼ii ½ ��?G �L� ¼iii ½ √1 + sin + �+L ~� ¼iv½ + log + �+ � ¼v½ + tan�� + �+��

02½ ¼i½ +�� + ¼ii ½ ª��ª��ª� �� + �� ¼iii ½ ª��ª��ª�� +�� ¼iv½ √QRS �√OPQ �√QRS �L ~� �+¼v½ +�√- − + �+��

03½ ¼i½ QRS ��OPQ ��?G� 8��L ~� �+ = 0 ¼ii ½ |+|�+ = 9��

04½ ¼i½ |sin +|�+ = 1L ~� ¼ii ½ +��+ = � �� − 1��

05½ ¼i½ ��H�+/&� +�+ = 0�� ¼ii ½ D sinθ cosθ �DL ⁄� = L� fl) dhft, fd &

06½ ¼i½ � ��S �Q¥O ��S � �+ = U IL − 1KL� ¼ii ½ � QRS ��QRS � �+ = U IL − 1KL� ¼iii ½ � QRS � OPQ �8��?G�� + = L��|L ~�

¼iv½ ��8��?G�� + = L� ��L� ¼v½ � �-H+Q¥O �OPQ � �+ = L��L�

07½ ¼i½ √��S �√��S �√OP� � �+ = L� L �~L |~ ¼ii ½ ��G�� + = L�L ~� ¼iii ½ %&'��tan +�L ~� �+ = 0

08½ ¼i½ �?G� �~ ��?G� �~ �8�� ~ � �+ = L�L ~� ¼ii ½ ��√�� = %&' ��√��√�� ¼iii ½ tan�� + �+ = L� − � %&'2��

09½ ¼i½ ��√��S � �+ = L� L �~L |~ ¼ii ½ ��¦P§ �� + = �� X�� ¼iii ½ cos�� + �+ = 1��

10½ ¼i½ ��OP� � = L�L ~� ¼ii ½ %&'��1 + tan +��+ = L� %&'�2L �~� ¼iii ½ ��8��L� = L� 11½ ¼i½ |cos +|�+ = 4 L� ¼ii ½ |+ − 5|�+ = 17�� ¼iii ½ b�� + = - IL − 1K��

12½ ¼i½ |+ − 1| + |+ − 2| + |+ − 3|®�+ = �� ¼ii ½ ¯��H|+| + /&�|+|°�+ = 4L ~�L ~ ¼iii ½ �Y�8��Y��?G�Y = L� L ⁄�

13½ ¼i½ ��?G��8�� = L�√�L ~� ¼ii ½ b � �� + = U + 2 � ¼iii ½ +b�� + = ��L� ��

(iv) cosθ /&��/ D �+ = √2 − 1L ⁄L �⁄

14½ ¼i½ QRS[\ �� ~ �+ = L� − � %&'2� √ ~� ¼ii ½ � ��S[\ ��±� �+ = L� ¼iii ½ �� OPQ � = �√�L ⁄� %&' √��√�� (iv) ��log sin + + cot +��+ = �L �⁄ %&'√2L ⁄L �⁄

vFkok

;ksx dh lhek ds #i esa fuf”pr lekdy dh ifjHkk"kk ls + �+�� dk eku Kkr dhft,A



Practice Paper- 12- Maths- 013 – App. Of Definite Integral (fuf”fuf”fuf”fuf”pr lekdypr lekdypr lekdypr lekdy ds vuqiz;ksxds vuqiz;ksxds vuqiz;ksxds vuqiz;ksx½½½½

Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each.

(ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each. (iii) Qu. No.11to15 are long answer type questions, carrying 5 marks each. -&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

01½ odz ( = %&'� +, y- v{k rFkk Hkqtksa y = 1, y = 2 ds chp fLFkr {ks= dk {ks=Qy Kkr dhft,A

02½ odz y = sin x vkSj x = 0 rFkk x = 2 ls f?kjs {ks= dk {ks=Qy Kkr dhft,A

03½ odz x = ey , Y- v{k rFkk nks Hkqtksa y = a vkSj y = b ds chp fLFkr {ks= dk {ks=Qy Kkr dhft,A

04½ y = x dk X- v{k rFkk js[kkvksa x = -1 o x = 2 ls f?kjs {ks= dk {ks=Qy Kkr dhft,A

05½ js[kk bx + ay – ab = 0 rFkk funsZ’kkadksa ds chp dk {ks=Qy Kkr dhft,A

06½ ijoy; y2 = 4ax rFkk bldh ukfHkyac thok ls f?kjs {ks= dk {ks=Qy Kkr dhft,A

07½ odz y2 = 4x vkSj x = 3 ls f?kjs {ks= dk {ks=Qy Kkr dhft,A

08½ odz �� + *�� = 1 ls f?kjs gq, {ks= dk {ks=Qy Kkr dhft,A

09½ o`Rr x2 + y2 = a2 dk {ks=Qy Kkr dhft,A

10½ odz ( = √6+ + 4 ] dk X- v{k rFkk js[kkvksa x = 0 vkSj x = 2 ls f?kjs {ks= dk {ks=Qy Kkr dhft,A

11½ y2 = 4ax vkSj js[kk y = mx ds chp {ks= dk {ks=Qy Kkr dhft,A

12½ odz y2 = 4ax vkSj x2 = 4ay ds chp dk {ks=Qy Kkr dhft,A

13½ odz x2 = 4y vkSj js[kk x = 4y – 2 ds vUrxZr {ks=Qy Kkr dhft,A

vFkok

odz y2 = 4x rFkk js[kk y = 2x }kjk f?kjs {ks= dk {ks=Qy Kkr dhft,A

14½ ml f=Hkqt dk {ks=Qy lekdyu }kjk Kkr dhft, ftldh Hkqtk,W y = 2x + 1, y = 3x + 1 rFkk x = 4 gSaA 15½ js[kk y = x rFkk odz y2 = 16x ds chp ds {ks= dk {ks=Qy Kkr dhft,A

vFkok

odz y2 = x vkSj x2 = y ds chp dk {ks=Qy Kkr dhft,A


Practice Paper- 12- Maths- 014 – Differntial Equations (vody lehdj.kvody lehdj.kvody lehdj.kvody lehdj.k½½½½ Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each. (ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each. (iii) Qu. No.11to15 are long answer type questions, carrying 5 marks each. -&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

01½ vody lehdj.k ��*�� = ² e1 + I�*��K f�/ dh dksfV rFkk ?kkr Kkr dhft,A

02½ fl) dhft, fd y = 4 sin3x vody lehdj.k ��*�� + 9( = 0 dk ,d gy gSA

03½ lehdj.k �*�� = ��* + + ��* dks gy dhft,A

04½ vody lehdj.k 1 + I�*��K = I� ��*��K /� dh dksfV rFkk ?kkr Kkr dhft,A

05½ �*�� = ��/ + + 2+ vFkok cos + �*�� + ( = sin + dk gy Kkr dhft,A

06½ fl) dhft, fd y = x3 +ax

2 + bx + c vody lehdj.k ��*�� = 6 dk ,d gy gSA

07½ odzksa ds dqy y= Ae3x

+ Be5x

ds fy, vody lehdj.k dh jpuk dhft,] tgkW A vkSj B LosPN vpj gSaA

fuEu vody lehdj.kksa dks gy dhft,&

08½ �*�� = �*�� vFkok

�*�� + ( =ex

09½ x log x dy – y dx = 0 vFkok + ( − +� �*�� = (� cos +

10½ sec2 x tan y dx + sec2 y tanx dy = 0 vFkok (1+ y2) dx = (tan-1 y – x) dy

11½ �*�� = �4+ + ( + 1� vFkok �*�� + *� = +

12½ ��*�� = 6+ + 2 vFkok + %&'� �*�� + ( = 2%&'�+ 13½

�*�� = ��**�� vFkok �+2 + (2� �(�+ = +( 14½

�*�� = � *�� *� vFkok �1 + + � �*�� + 2+( = 4+ vFkokvFkokvFkokvFkok

15½ �*�� = ( �-H+ − 2��H+ vFkok �+ + ( + 10� �*�� = 1



Practice Paper- 12- Maths- 015 – Correlation & Regression (lg&llg&llg&llg&laca/k ,oa lekJ;.kaca/k ,oa lekJ;.kaca/k ,oa lekJ;.kaca/k ,oa lekJ;.k½½½½ Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each. (ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each. (iii) Qu. No.11to15 are long answer type questions, carrying 5 marks each. -&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

01½ fl) djks fd lg&laca/k xq.kkad dk eku -1 ls +1 ds chp gksrk gSA 02½ lg&laca/k xq.kkad dk eku Kkr dhft,] fn;k gS& Cov (x, y) = - 16.5, Var (x) = 2.89 rFkk Var (y) = 100

03½ x vkSj y ds e/; lg&izlj.k dh x.kuk dhft, ;fn Σx = 55, Σy = 82, Σxy = 565 vkSj n = 8

04½ ;fn Σx = 18, Σx2 = 90, n = 10,Σ y = 25, Σy2 =120, Σxy = 65 gks rks rxy Kkr dhft,A 05½ fl) dhft, fd lg&laca/k xq.kkad] lekJ;.k xq.kkadksa dk xq.kksRrj ek/; gksrk gSA 06½ fl) dhft, fd lekJ;.k xq.kkadksa dk lekUrj ek/;] lg&laca/k xq.kkad ls cM+k gksrk gSA 07½ lg&laca/k xq.kkad Kkr dhft, tcfd lekJ;.k js[kk,W fuEu gS% 2x – 9y + 6 = 0 rFkk x – 2y + 1 = 0 08½ fuEuakfdr vkWdM+ksa ls x dh y ij lekJ;.k js[kk Kkr djsa ,oa y = 90 ds fy, x dk laHkkfor eku Kkr djsa& Js.kh x y

lekUrj fopyu 18 100 ekud fopyu 14 20

x vkSj y esa lg&laaca/k xq.kkad = 0.08 09½ fuEu vkWdM+ksa ls lg&laca/k xq.kkad dh x.kuk dhft,&

10½ vkids fo|ky; ds 100 Nk=ksa ds fy, fuEu ifj.kke ÅWpkbZ x rFkk Hkkj y fn;k x;k gS% ÅWpkbZ¼ x½ Hkkj¼ y½

ek/; 68 bap 150 ikS.M ekud fopyu 2-5 bap 20 ikS.M x vkSj y esa lg&laaca/k xq.kkad = 0.08

(i) fdlh fo’ks"k Nk= dh ÅWpkbZ Kkr dhft, tc Hkkj 200 ikS.M gksA (ii) fdlh fo’ks"k Nk= dk Hkkj Kkr dhft, tc ÅWpkbZ 5 QhV gksA

11½ fuEu vkWdM+ksa ls X rFkk Y ds e/; lg&laca/k xq.kkad dh x.kuk dhft,& 12½ fuEu vkWdM+ksa ls x dh y ij lekJ;.k js[kk Kkr

dhft,&

13½ fl) dhft, fd nks lekJ;.k js[kkvksa ds e/; dks.k dk Li’kZT;k ��³�³ � � ´�´µ´��´µ�� gksrk gSA 14½ fuEu vkWdM+ksa ls X rFkk Y ds e/; dkyZ fi;lZu ds lg&laca/k xq.kkad dh x.kuk dhft,&

vFkokvFkokvFkokvFkok 15½ fujh{k.kkas (x, y) = (4, 2), (2, 3), (3, 2), (4, 4), (2, 4) ls lekJ;.k xq.kkadksa byx o bxy dks Kkr dhft, rFkk rxy

dk eku Kkr dhft,A


Practice Paper- 12- Maths- 016 –Probability (izkf;drkizkf;drkizkf;drkizkf;drk½½½½ Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each. (ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each. (iii) Qu. No.11to15 are long answer type questions, carrying 5 marks each. -&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

01½ ,d lk/kkj.k ikWls dks Qsadus ij 3 ls cM+k vad izkIr djus dh izkf;drk Kkr dhft,a 02½ ;fn fdlh yhi o"kZ dks ;n`PN;k pquk tk;s rks ml o"kZ esa 53 jfookj gksus dh izkf;drk Kkr dhft,A 03½ ,d FkSys es 7 lQsn vkSj 9 yky xsansa gSaA ,d lQsn xsan fudkyus dh izkf;drk Kkr dhft,A 04½ ,d ?kVuk dk izfrdwy la;ksxkuqikr 3 % 4 gS] mlds ?kVus dh izkf;drk Kkr dhft,A 05½ ,d QSDVjh }kjk mRikfnr cYcksa esa 8% cYc yky jax ds gSa vkSj 2% yky fdUrq [kjkc gSaA ;fn ,d cYc ;n`PN;k pquk tk, rks mlds [kjkc gksus dh izkf;drk crkb;s ;fn og yky gSaA 06½ izkf;drk dk ;ksT; izes; fy[kdj fl) dhft,A 07½ 52 iRrksa dh rk”k dh xÏh ls ;n`PN;k ,d iRrk [khapus ij mlds ckn”kkg ;k gqdwe dk iRrk gksus dh izkf;drk Kkr dhft,A 08½ ,d ikls dks ,d ckj mNkyk tkrk gSA izkf;drk Kkr dhft, fd le vad ;k 5 ls de vad izkIr gksA 09½ ,d ?kqM+nkSM+ esa Hkkx ysus okys rhu ?kksM+ksa ds vuqdwy la;ksxkuqikr dze”k% 1 % 2] 1 % 3 vkSj 1 % 4 gSaA bu rhuksa ?kksM+ksa esa ls dksbZ ,d fot;h gks bldh izkf;drk Kkr dhft,A

x 20 25 30 35 40 45 y 16 10 8 20 5 10

x 3 4 6 8 9 y 90 100 130 160 170

x 1 2 3 4 5 6 7 y 9 8 10 12 11 13 14

x -4 -3 -2 -1 0 1 2 3 4 y 16 9 4 1 0 1 4 9 16

10½ xf.kr dk ,d iz”u rhu fo|kfFkZ;ksa dks gy djus ds fy, fn;k x;kA muds }kjk gy djus dh izkf;drk 1/2, 1/3 vkSj 1/4 gSA;fn os lHkh gy djus dk iz;kl djsa rks iz”u ds gy fd;s tkus dh izkf;drk Kkr dhft,A 11½ nks FkSYkksa esa ls ,d esa 3 dkyh vkSj 4 yky xsansa gSa vkSj nwljs esa 8 dkyh vkSj 10 yky xasnsa gSaA ;fn fdlh ,d FkSys dks pqudj mlesa ls ,d xsan fudkyh tk;s rks mlds yky gksus dh izkf;drk Kkr dhft,A 12½ ,d iklk nks ckj Qsadk tkrk gSA izR;sd Qsad esa fo"ke la[;k vkus ij lQyrk ekuh tkrh gSA lQyrkvksa dh izkf;drk caVu Kkr dhft,A 13½ fdlh lewg esa ftlesa 3 iq#"k] 2 efgyk,W rFkk 4 cPps gSa] pkj O;fDr;ksa dks pquuk gSA buesa Bhd 2 cPps gksus dh izkf;drk Kkr dhft,A 14½ 52 rk’kksa dh ,d lkekU; xMh ls 6 iRrs ;n`PN;k izfrLFkkiu ds lkFk dze”k% [khaps tkrs gSaA izkf;drk Kkr dhft, fd vf/kd ls vvf/kd 3 iRrs gqdqe ds gksaA vFkokvFkokvFkokvFkok 15½ 10 cPpksa ds ,d lewg esa ls ftlesa 6 yM+ds vkSj 4 yM+fd;kW gSA 3 cPps ;n`PN;k pqus tkrs gSaA izkf;drk Kkr dhft, fd pquk gqvk lewg& (i) dksbZ yM+ds ugha j[krk gSA (ii) dsoy ,d yM+dk j[krk gSA (iii) dsoy ,d fo”ks"k yM+dk j[krk gSA (iv) de ls de ,d yM+dk j[krk gSaA

Practice Paper- 12- Maths- 017 – Statics (fLFkfrfLFkfrfLFkfrfLFkfr foKkufoKkufoKkufoKku½½½½ Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each. (ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each. (iii) Qu. No.11to14 are long answer type questions, carrying 5 marks each.

-&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& 01½ ,d fcanq ij fdz;k dj jgs cy P vkSj Q dk ifjekeh cy R gSA ;fn P = 8, Q = 15 vkSj R = 17 gks] rks P vkSj Q ds e/; dks.k Kkr dhft,A 02½ ,d d.k ij fdz;k djrs gq, rhu cy larqyu esa gSaA igys nks ds chp dk dks.k 900 gS vkSj nwljs rFkk rhljs ds chp dk dks.k 1200 gSA cyksa dk vuqikr Kkr dhft,A 03½ cy 20N ds ?kVd cy Kkr dhft, tks mlls foijhr fn”kkvksa esa 300 vkSj 600 ds dks.k cukrs gSA 04½ ,d gh lery ij fdz;k”khy rhu cy F1, F2 vkSj F3 larqyu esa gSA ;fn F1 cy F2 vkSj F3 ls dze”k% 1200 vkSj 900 ds dks.k cukrs gS] rks cyksa ds ifj.kkeksa es vuqikr Kkr dhft,A 05½ nks cyksa P vkSj Q dk ifj.kkeh R cy P ds lkFk ledks.k cukrs gS rks fl) dhft, fd R2 + P2 = Q2 06½ nks cyksa P vkSj Q dk ifj.kkeh P dh fn”kk ls ledks.k cukrs gq, dk;Z djrk gSA fl) dhft, fd cyksa ds

chp dk dks.k cos�� I�¶· K gksxkA 07½ 12 N vkSj 14 N ds nks cy ,d nwljs ls 450 ds dks.k ij fdz;k”khy gSa] cyksa dk ifj.kkeh rFkk ifj.kkeh dh fn'kk Kkr dhft,A

08½ dks.k θ ij fdz;k dj jgs nks cy P vkSj Q dk ifj.kkeh cy �2� + 1��¸ + ¹ ds cjkcj gSA tc cy xU 2~ − Dy dks.k ij fdz;k djrs gSa] rks ifj.kkeh cy �2� − 1�√¸2 + ¹2 ds cjkcj gksrk gS] fl) djks fd

tan D = � − 1� + 1 09½ tc nks cjkcj cy 2θ dks.k ij fdz;k djrs gS rks ifj.kkeh cy 2R gS vkSj tc 2φ dks.k ij fdz;k djrs gSa rks mudk ifj.kkeh cy R gSA fl) dhft, fd cos θ = 2 cos φ

10½ fdlh cy F dks nks Hkkxksa ess fo;ksftr fd;k x;k gSA muesa ls ,d fo;ksftr Hkkx cy F ij yac vkSj ifjek.k esa mlds cjkcj gks] rks nwljs fo;ksftr Hkkx dh fn”kk rFkk ifjek.k Kkr dhft,A

vFkok 70 fdyksxzke dk ,d fi.M dze”k% 6 ehVj vkSj 8 ehVj yach nks Mksfj;ksa }kjk ,d {kSfrt js[kk }kjk ,d {kSfrt js[kk esa 10 ehVj dh nwjh ij nks fcanqvksa ls yVdk gqvk gSA Mksfj;ksa esa ruko Kkr dhft,A

11½ nks P vkSj Q dk ifj.kkeh R gS rFkk muds chp dk dks.k α gSA ;fn P dks nqxquk dj fn;k tk;s rks ifj.kkeh

nqxquk gks tkrk gSA fl) dhft, fd º = sin�� b�|¶��·��|¶�

vFkok ,d fcanq ij fdz;k dj jgs rhu cy P, Q, R larqyu esa gSaA P vkSj Q ds chp dk dks.k P vkSj R ds chp ds dks.k ls nqxquk gSA fl) dhft, fd R2 = Q (Q – P)

12½ nks cyksa P + Q vkSj P – Q ds chp dk dks.k 2α gS vkSj budk ifj.kkeh nks cyksa ds chp ds dks.k ds

lef}Hkktd ls θ dsk.k cukrk gSA fl) dhft, fd P tan θ = Q tanα

vFkok ,d fcanq ij rhu cy P – Q, P vkSj P + Q yxs gSa ftudh fdz;k js[kk,W ,d leckgq f=Hkqt dh Hkqtkvksa ds lekUrj gSA ifj.kkeh dk ifjek.k Kkr dhft,A 13½ nks cyksa P vkSj Q dk ifj.kkeh R gSA;fn Q dks nqxquk dj fn;k tk;s rks R Hkh nqxquk gks tkrk gS vkSj ;fn Q dh fn”kk myV nh tk, rks Hkh R nqxquk gks tkrk gSA fl) dhft, fd P : Q : R = √2 : √3 : √2

vFkok nks cyksa P vkSj Q dk ifj.kkeh R gSA ;fn P dh fn’kk esa ifj.kkeh R dk fo;ksftr Hkkx ifj.kke esa Q ds cjkcj

gks] rks fl) dhft, fd cyksa ds chp dk dks.k 2 sin�� b ¶ · gksxkA 14½ ;fn nks cy P vkSj Q ,d ,sls dks.k ij feyrs gSa fd mudk ifj.kkeh cy R cy P ds cjkcj gksrk gSA fl) dhft, fd ;fn P dks nqxquk dj fn;k tk;s rks ifj.kkeh cy Q ds yacor~ gksxk rFkk u;s ifj.kkeh cy dk

ifjek.k �4¸ − ¹ gksxkA vFkok

fdlh fcanq ij yxs nks cyksa P vkSj Q ds chp dk dks.k α gSA ;fn cyksa dks vkil esa cny fn;k tk;s rks

fl) dhft, fd mudk ifj.kkeh dks.k φ ij ?kwe tk;sxk] tgkW tan φ = ·�¶·¶ tan V

-Practice Paper- 12- Maths- 018 – Dynamics (xxxxfr foKkufr foKkufr foKkufr foKku½½½½

Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each. (ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each. (iii) Qu. No.11to14 are long answer type questions, carrying 5 marks each. -&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

01½ 60 ehVj ÅWph ehukj ds f”k[kj ls ,d fi.M 20 eh@ls- ds osx ls Åij dh vksj Qsadk tkrk gSA ;fn g = 10 m/s2 gks] rks tehu ij fdl le; fxjsxk\ 02½ ,d fi.M dks ÅWph ehukj ls i`Foh rd fxjus esa 2 lsd.M dk le; yxrk gSA ehukj dh ÅWPkkbZ Kkr dhft,A ( g = 10m/s2)

03½ nks iRFkj nks fofHkUu ÅWpkb;ksa ls fxjk, tkrs gSaA ;fn budh ÅWpkb;ksa esa vuqikr h1 : h2 gks rks fl) dhft, fd iRFkjksa ds tehu rd igWqpus esa fy;s x;s le; dk vuqikr √h1 :√ h2 gSA 04½ ,d fdzdsV f[kykM+h xsan dks 100 ehVj dh nwjh ij Qsad ldrk gSA ogh f[kykM+h mlh xsan dks fdruh ÅWpkbZ rd Qsad ldrk gSA 05½ nks d.k {kSfrt ls 300 rFkk 600 dk dks.k cukrh gqbZ fn”kkvksa esa izf{kIr fd, x,A ;fn os ,dleku ÅWpkb;ksa rd igqWprs gksa rks fn[kkb, fd muds iz{ksi oxs √3 % 1 ds vuqikr esa gSaA 06½ ,d euq"; ,d unh dks ikj dj] nwljs fdukjs ds Bhd lEeq[k fcanq ij igqWpuk pkgrk gSA ;fn og viuh uko dks /kkjk ds osx ls √2 xqus osx ls pykrk gS] rks Kkr dhft, fd mls uko dks /kkjk ds lkFk fdruk dks.k cukrs gq, j[kuk pkfg,\ 07½ ;fn x, x,y, z ,d xfreku d.k }kjk dze”k% poas] qosa] rFkk rosa lsdaM esa r; dh xbZ nwfj;kW gksa] rks fl) dhft, fd x(q – r) + y(r – p) + z(p – q) = 0 08½ ,d Vªsu tks ijLij 4 fdeh dh nwjh ij fLFkr nks LV”kuksa ij #drh gS] ,d LVs”ku ls nwljs rd dh ;k=k esa 4 feuV dk le; ysrh gSA ;g ekurs gq, fd Vªsu igys Roj.k x ls vkSj fQj eanu y ls ( nksuksa ,dleku ) xfr djrh gS] fl) dhft, fd �� + �* = 2 09½ ,d d.k 300 ds mRFkku ij 49 eh@ls- ds osx ls izf{kIr fd;k tkrk gSA Kkr dhft,& (i) mÏ;u dky (ii) {kSfrt ijkl (iii) egRre ÅWpkbZA (iv) egRre ÅWpkbZ rd igqpus esa yxk le;A 10½ fl) dhft, fd] ;fn {kSfrt ijkl R ij ,d xksyh dk mÏ;u dky T lsdaM gks] rks {kSfrt ls mldh fn”kk

dk >qdko tan�� I�»� ¼ K gksxkA 11½ ,d d.k osx u ls bl izdkj Qsadk tkrk gS fd mldh {kSfrt ijkl mldh egRre ÅWpkbZ dh nqxquh gSA

fl) dhft, fd ijkl �½�� gSA

12½ /kjkry ls ,d xsan bl izdkj Qsadh tkrh gS fd og x ehVj nwj fLFkr ,d nhokj dks Bhd ikj dj tkrh gS rFkk og nhokj ds ikn ls y ehVj dh nwjh ij fxjrh gSA ;fn iz{ksi dks.k 450 gks rks fl) dhft, fd nhokj

dh ÅWPkkbZ �*�* ehVj gSA

13½ nks fi.M ,d gh fcanq ls {kSfrt ls º� rFkk º dks.k cukrh fn’kkvksa esa izf{kIr fd, tkrs gSa vkSj {kSfrt /kjkry ds ,d gh fcanq ij vk?kkr djrs gSaA ;fn t1 rFkk t2 muds mÏ;u dky gksa] rks fl) dhft, fd & �� − � �� + � = sin�º� − º �sin�º� + º � 14½ ,d d.k 24 eh@ls- ds izkajfHkd osx ls pyuk izkjaHk djrk gSA ;fn og ,d ljy js[kk esa 2 eh@ls2 ds ,dleku Roj.k ls pyrk gS] rks Kkr dhft,& (i) 5 lsd.M ckn d.k dk osx (ii) 4 lsd.M ds d.k }kjk pyh nwjh (iii) d.k dk osx tc og 81 ehVj dh nwjh r; dj pqdk gksA (iv) 7 osa lsd.M es d.k }kjk pyh nwjh

(v) 2 eh@ls- ds osx izkIr djus gsrq d.k }kjk fy;k x;k le;A Practice Paper- 12- Maths- 019 – Boolean Algebra (cwfy;u chtxf.krcwfy;u chtxf.krcwfy;u chtxf.krcwfy;u chtxf.kr½½½½

Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each. (ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each. (iii) Qu. No.11to14 are long answer type questions, carrying 5 marks each. -&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& 01½ cwyh; chtxf.kr ls lacaf/kr }Srrk dk fl)kar fyf[k,A

02½ cwyh; chtxf.kr [B, +, ., ‘] ds fdlh vo;o

03½ fl) dhft, fd fuEufyf[kr dFku ^^O;k?kkr** gS&

04½ lR;rk lkj.kh ds iz;ksx ls fl) dhft, fd&

05½ rdZ okD; chtxf.kr ds fu;eksa dk mi;ksx djds fl) dhft, fd &

(i) ∼ [ ∼ (∼ p ∧ ∼ q)] ≡ ∼ p ∧ ∼ q

06½ cwyh; chtxf.kr ds fy, leoxZ fu;e fy[kdj fl) dhft,A

07½ fl) dhft, fd cwyh; chtxf.kr ds izR;sd vo;o dk iwjd vf}rh; gksrk gSA

08½ cwyh; chtxf.kr B ds lHkh vo;oksa

cwyh; chtxf.kr [B, +, ., ‘] esa fl) dhft, fd &

09½ (x + y).(x’ + z) = x’.y + x.z tgkW x

10½ x.y + [(x + y’) .y]’ = 1 ∀ x, y ∈ B

11½ (a + b)’. (a + b’)’ = a’ ∀ a, b ∈ B

12½ fuEu rdZ ifjiFk ds laxr& (i) cwyh; Qyu fyf[k,

ds fy, rdZ ifjiFk [khafp, (iv) lR;rk dh tkWp dhft,A

13½ fuEu cwyh; Qyu dk fLopu ifjiFk cukdj blds ljyhdr ifjiFk dk fuekZ.k dhft, ,oa bls lR;rk

lkj.kh ls izekf.kr dhft,A

f = x.z + [y.(y’ + z). (x’ + x.z’)]

14½ cwyh; chtxf.kr [B, +, ., ‘] ds fdlh vo;o

========================================================

Practice Paper- 12- Maths

Note- (i) Qu. No. 1 to 5 are very short answer type questions, carrying 2 marks each. (ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each. (iii) Qu. No.11to14 are long answer type questions, carrying 5 marks each.

-&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&01½ ok;jl D;k gS\ ;g fdrus izdkj ds gksrs gSa02½ lqij dEI;wVj D;k gS\ Hkkjr esa vfo

ds fdlh vo;o x ds fy, fl) dhft, fd (i) x + 1 = 0 (ii)

fl) dhft, fd fuEufyf[kr dFku ^^O;k?kkr** gS& (i) (p ∨ q ) ∧ ( ∼p ∧ ∼q) (ii) [(

lR;rk lkj.kh ds iz;ksx ls fl) dhft, fd& (i) ∼(p ⇒ ∼ q) ≡ (p ∧ q) (ii) (

rdZ okD; chtxf.kr ds fu;eksa dk mi;ksx djds fl) dhft, fd &

(ii) ∼ (p ∨ q) ∨ (∼ p∧ q) ≡ ∼ p

cwyh; chtxf.kr ds fy, leoxZ fu;e fy[kdj fl) dhft,A

fl) dhft, fd cwyh; chtxf.kr ds izR;sd vo;o dk iwjd vf}rh; gksrk gSA

ds lHkh vo;oksa x ds fy, fl) dhft, fd (x’)’ = x

fl) dhft, fd &

x + y = x ∀ x, y, z ∈ B rFkk x dk iwjd x’ gSA

B rFkk y dk iwjd y’ gSA

B

cwyh; Qyu fyf[k, (ii) ljy cwyh; Qyu izkIr dhft,

lR;rk dh tkWp dhft,A

fuEu cwyh; Qyu dk fLopu ifjiFk cukdj blds ljyhdr ifjiFk dk fuekZ.k dhft, ,oa bls lR;rk

ds fdlh vo;o x ds fy, Mh&ekxZu dk fu;e fy[kdj fl) dhft,A

========================================================

Maths- 020 – Numerical Methods & Information Technology

(vkafdd fof/k;kW ,oa lwpuk izkS|ksfxdhvkafdd fof/k;kW ,oa lwpuk izkS|ksfxdhvkafdd fof/k;kW ,oa lwpuk izkS|ksfxdhvkafdd fof/k;kW ,oa lwpuk izkS|ksfxdh½½½½ . No. 1 to 5 are very short answer type questions, carrying 2 marks each.

(ii) Qu. No. 6 to 10 are short answer type questions, carrying 4 marks each. are long answer type questions, carrying 5 marks each.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&;g fdrus izdkj ds gksrs gSa\

Hkkjr esa vfo"dkj fd;s x;s lqij dEI;wVj dk uke crkb,A

+ 1 = 0 (ii) x .0 = 0

(ii) [( p∧ q ) ⇒ p] ⇒ [q ∧∼ q]

(ii) (p ⇒ q)≡ (∼ q ⇒ ∼ p)

ljy cwyh; Qyu izkIr dhft, (iii) ljyhd`r Qyu

fuEu cwyh; Qyu dk fLopu ifjiFk cukdj blds ljyhdr ifjiFk dk fuekZ.k dhft, ,oa bls lR;rk

ds fy, Mh&ekxZu dk fu;e fy[kdj fl) dhft,A

========================================================

rmation Technology

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

,A

03½ dEI;wVj ds gkMZos;j ,oa lk¶Vos;j esa D;k varj gS\ Li"V dhft,A 04½ fn;k x;k gS&

(�+�� dk eku flEilu fu;e ls Kkr dhft,A 05½ dskbZ odz fuEu fcanqvksa ls gksdj tkrk gS& odz }kjk x = 1 vkSj x = 4 ls ifjc) {ks= dk {ks=Qy leyac prqHkqZth; fu;e ls Kkr dhft,A 06½ ,d odz fuEu fcanqvksa ls gksdj xqtjrk gS& flEilu fu;e }kjk odz X- v{k vkSj js[kkvksa x =1, x = 4 ls ifjc) {ks= dk {ks=Qy Kkr dhft,A 07½ ,d unh 80 ehVj pkSM+h gS] fdukjs ls x nwjh ij unh dh xgjkbZ d dks fuEu lkj.kh ls fn[kk;k tk ldrk gS& leyac prqHkqZth; fu;e ls unh dh vuqizLFk dkV dk {ks=Qy Kkr dhft,A

08½ fuEufyf[kr vkWdM+ksa ds vk/kkj ij leyac prqHkqZth; fu;e ls ��+�� dk eku Kkr dhft,A

09½ U;wVu&jSQlu fof/k }kjk 3 dk oxZewy Kkr dhft,A

10½ lekdy �� dk eku flEilu fu;e ls n’key ds rhu LFkkuksa rd Kkr dhft, ¼n = 4 yhft,½

11½ U;wVu&jSQlu fof/k }kjk 8 dk oxZewy Kkr dhft,A 12½ U;wVu&jSQlu fof/k }kjk 10 dk ?kuewy Kkr dhft,A

13½ flEilu fu;e dk iz;ksx djds QRS �� +L� dk eku Kkr dhft,A ¼ n = 4 yhft,½

14½ laf{kIr fVIi.kh fyf[k,& (i) baVjizsVj (ii) dEikbyj (iii) eYVhehfM;k (iv) eseksjh (v) lk¶Vos;j

x 1 1.5 2 2.5 3 y 2.1 2.4 2.2 2.8 3

x 1 2 3 4 y 1 4 9 16

x 1 1.5 2 2.5 3 3.5 4 y 2 2.4 2.7 2.8 3 2.6 2.1

x 0 10 20 30 40 50 60 70 80 d 0 4 7 9 12 15 14 8 3

x 0 0.5 1 1.5 2 2.5 3 3.5 4 y 1 1.65 2.72 4.48 7.39 12.18 20.9 33.72 54.60

coaching classes for- iit, aipmt, pet, pat, pmt, ppt, xii ...coaching classes for- iit, aipmt, pet,...

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