1 Math (Hindi) ­X

LIST OF MEMBERS WHO PREPARED QUESTION BANK FOR MATHEMATICS FOR CLASS X

TEAM MEMBERS

Sl. No. Name Designation

1. Dr. J.D. Bhardwaj G.B.S.S.S.-I (Principal) Kidwai Nagar,

New Delhi.

2. Mr. Udai Bir Singh R.P.V.V., B-1, Vasant Kunj, New Delhi-110070.

3. Mr. Surendra Pal Singh R.P.V.V., Raj Niwas Marg, Delhi – 110054.

4. Ms. Ritu Tiwari R.P.V.V., Suraj Mal Vihar, Delhi.

5. Ms. Savita Vij Govt. Co-Ed. Sr. Sec. School, No. 1 Lajpat Nagar, New Delhi.

6. Mr. Anand Singh S.V. Anand Vihar, Delhi.

2 Math (Hindi) ­X

fo"k; lwph S.A.­1 Øe la- vè;k; i`"B la-

1- la[;k i¼fr 3&7

2- igqin 8&13

3- nks pj okys jSf[kd lehdj.k ;qXe 14&23

4- le:i f=kHkqt 24&35

5- f=kdks.kfefr dh izLrkouk 36&43

6- lkaf[;dh 44&53

7- uewuk iz'u i=k 54&60

3 Math (Hindi) ­X

vè;k;&1

okLrfod la[;k,¡

egÙoiw.kZ fcanq % 1- ;wfDyM foHkktu izesf;dk % nks ?kukRed iw.kkZ± ‘a’ o ‘b’ ds fy, larq"V djus okyh

iw.kZ la[;k,¡ q o r bl izdkj gSaA a = bq + r, 0 r < b.

2. ;wfDyM foHkktu ,YxksfjFe % nks ?kukRed iw.kk±dksa a vksj b, (a > b) ld e-l- uhps n'kkZ;h xbZ fof/ kjk izkIr fd;k tkrk gS %

pj.k 1 % q vkSj r ?kkr djus ds fy, ;wfDyM foHkktu izesf;dk dk iz;ksx dhft,] tgk¡ a = bq + r . 0≤r < b.

pj.k 2 % ;fn r = 0 rks e-l- (a, b) = r

pj.k 3 % ;fn r ≠ 0 rks b vkSj r ij ;wfDyM foHkktu izesf;dk dk iz;ksx dhft,A bl izfØ;k dks rc rd tkjh jf[k, tc 'ks"kiQy 'kwU; u izkIr gksA bl fLFkfr okyk Hkktd gh e-l- (a, b) gSA

3- vadxf.kr dh vk/kjHkwr izes; % izR;sd HkkT; la[;k dks vHkkT; la[;kvksa ds ,d xq.kuiQy ds :i esa O;Dr (xq.ku[kafMr) fd;k tk ldrk gS rFkk og xq.ku[kaM vfrh; gksrk gS] bl ij dksbZ è;ku fn, fcuk fd vHkkT; xq.ku[kaM fdl Øe esa vk jgs gSaA

4- eku yhft, x ,d ifjes; la[;k gS , 0 p x q q

= ≠ rFkk q dk vHkkT; xq.k[kaM

2 m 5 n ds :i dk gS] tgk¡ m o n ½.ksrj iw.kk±d gSa x rks dk n'keyo izlkj lkr gksxkA

4 Math (Hindi) ­X

5- eku yhft, p x q

= ,d ,slh ifjesl la[;k gS fd q dk vHkkT; xq.ku[kaMu

2 m 5 n ds :i dk ugha tgk¡ m, n ½.ksrj iw.kk±d gSa rks x dk n'keyo izlkj vlkar vkorhZ gksxkA

6- p ,d vifjes; la[;k gS] tgk¡ p ,d vHkkT; gSA ,d la[;k vifjes; gS ;fn

mldks 0 p q q

≠ ds :i esa ugha fy[kk tk ldsA

cgq oSdfYid iz'u

1- 7 × 11 × 13 + 7 ,d -------------- gSA (a) vHkkT; la[;k (b) HkkT; la[;k (c) fo"ke la[;k (d) dksbZ ugha

2- fuEu esa ls dkSu lh la[;k vad 6 ij lekIr gksrh gSA (a) 4 n (b) 2 n

(c) 6 n (d) 8 n

tgk¡ n izkd`r la[;k gSA

3- a rFkk b ?kukRed la[;k,a gSa] a ≠ b rks ( ) ( ) a b a b + − ,d --------- gSA

(a) ifjes; la[;k (b) vifjes; la[;k

(c) ( ) 2 a b − (d) 0

4- ;fn p ,d ?kukRed ifjes; la[;k gS tks iw.kZ oxZ ugha gS rks 3 p − --------------- gSA (a) iw.kk±d (b) ifjes; la[;k (c) vifjes; la[;k (d) mijksDr esa ls dksbZ ugha

5- lHkh n'keyo la[;k,a ------------- gSaA (a) ifjes; la[;k,a (b) vifjes; la[;k,a (c) okLrfod la[;k,a (d) iw.kk±d

5 Math (Hindi) ­X

6- ;wfDyM foHkktu izesf;ek kjk tc a = bq + r, tgk¡ a, b ?kukRed iw.kk±d gS] fuEu esa ls dkSu lk lR; gS%

(a) 0 < r ≤ b (b) 0 ≤ r < b (c) 0 < r < b (d) 0 ≤ r ≤ b

7- fuEu la[;kvksa esa ls dkSu lh la[;k vifjes; la[;k gS % (a) 3.131131113.... (b) 4.46363636..... (c) 2.35 (d) b vkSj c nksuksa

8- 3 4 54

2 5 dk n'keyo fu:i.k fdrus n'keyo LFkku ds ckn lkr gksxkA (a) 8 (b) 4 (c) 5 (d) dHkh ugha

9- e-l-o- lnSo (a) y-l-o- dk xq.kt (b) y-l-o- dk xq.ku[k.Mu (c) y-l-o- ls HkkT; (d) a vkSj c nksuksa

10- fuEu esa ls dkSu 255 dk xq.ku[k.Mu ugha gS (a) 5 (b) 25 (c) 3 (d) 17

11- fuEu la[;kvksa esa ls dkSu lh 0 vkSj 1 ds chp fLFkr vifjes; la[;k gS (a) 0.11011011..... (b) 0.90990999..... (c) 1.010110111...... (d) 0.3030303.....

12- p n = (a × 5)n ;fn p n 'kwU; ij lekIr gksrh gS rks a ,d ---------- gksxh (tgk¡ n ,d izkd`r la[;k gS)

(a) dksbZ Hkh izkd`r la[;k (b) le la[;k (c) fo"ke la[;k (d) dksbZ ugha

13- 51 1500

dk n'keyo izlkj fdrus n'keyo LFkkuksa ds ckn lkr gksxk\ (a) nks (b) rhu (c) pkj (d) ik¡p

6 Math (Hindi) ­X

y?kq mÙkjh; iz'u 14- 1.3 0.4 + dk eku D;k gS\

15- ;fn 7 3 ds bdkbZ dk vad 3 gS rks 7 11 ds bdkbZ dk vad D;k gksxk\ 16- l-e-o- (135, 225) = 45 rks y-l-o- (135, 225) D;k gksxkA

17- 18 50. × gy djksA ;g fdl izdkj dh la[;k gS ifjes; la[;k ;k vifjes; la[;kA

18- 69 60 fdl izdkj dk n'keyo izlkj gSA ;g fdrus n'keyo LFkkuksa ds ckn lkar

gksxk\ 19- lcls NksVh HkkT; la[;k vkSj lcls NksVh vHkkT; la[;k dk e-l-o- Kkr dhft,A 20- ;fn a = 4q + r gS rks a rFkk q fdl izdkj dh la[;k,a gksaxhA r dkSu ls eku ys

ldrk gSA

21- og NksVh ls NksVh la[;k Kkr dhft, ftls 5 3. − ls xq.kk djus ij og ,d

ifjes; la[;k cu tk,A bl izdkj izkIr la[;k D;k gksxhA 22- 9 n dh bdkbZ la[;k Kkr dhft,A

23- 3 vkSj 5 ds chp ,d ifjes; la[;k rFkk ,d vifjes; la[;k Kkr dhft,A

24- ;fn p n dk bdkbZ vad 'kwU; gks rks p dk (ds) laHkkfor eku Kkr dhft,\ 25- ;wfDyM foHkktu izesf;dk dk dFku fyf[k,] blh ls 16 vkSj 28 dk e-l-o- Kkr

dhft,A 26- vadxf.kr dh vk/kjHkwr izes; dk dFku fyf[k,] blh ls 120 ds vfrh;

xq.ku[k.M dhft,A

27- fl¼ dhft, 1 2 5 −

vifjes; la[;k gSA

28- fl¼ dhft, 2 5 3 7

− vifjes; la[;k gSA

29- fl¼ dhft, 2 7 + ifjes; la[;k gSA

7 Math (Hindi) ­X

30- vHkkT; xq.ku[k.Mu fof/ kjk 56 vkSj 112 dk y-l-o- vkSj e-l-o- Kkr dhft,A 31- 17+11×13×17×19 ,d HkkT; la[;k gS] D;ksa\ le>kb;sA 32- tk¡p dhft, D;k 5×7×11+7 ,d HkkT; la[;k gS\ 33- tk¡p dhft, 7×6×3×5+5 ,d HkkT; la[;k gS\ 34- tk¡p dhft, D;k 14 n 'kwU; ij lekIr gks ldrk gS\ (fdlh izkd`r la[;k n ds

fy,) 35- n'kkZb;s ds 9 n dHkh 'kwU; ij lekIr ugha gks ldrhA

36- ;fn 210 vkSj 55 dk e-l-o- 210×5+55 y ds :i esa n'kkZ ldrs gSa rks y dk eku Kkr dhft,A

nh?kZ mÙkjh; iz'u 37- ;wfDyM foHkktu fof/ kjk 56] 96 rFkk 324 dk e-l-o- Kkr dhft,A 38- n'kkZb;s fd fdlh Hkh ?kukRed iw.kk±d dk oxZ 3m ;k 3m + 1 ds :i esa gksxk] fdlh

iw.kk±d m ds fy,A 39- n'kkZb;s fd fdlh Hkh iw.kk±d q ds fy, dksbZ Hkh ?kukRed fo"ke iw.kk±d 6q+1, 6q+5

:i esa gksxkA 40- fl¼ dhft, fd fdlh Hkh iw.kk±d q ds fy, fdlh Hkh ?kukRed iw.kk±d dk oxZ 5q,

5q+1, 5q+4 :i esa gksxkA 41- fl¼ dhft, fd rhu yxkrkj ?kukRed iw.kk±dksa dk xq.ku 6 ls foHkkftr gksrk gSA 42- n'kkZb;s fd n, n +2, n +4 esa ls ,d vkSj dsoy ,d gh 3 ls foHkkftr gksrh gSA 43- nks nw/ ds daVsuj esa 398 yh- vkSj 436 yh- nw/k gSA ,d Mªe kjk blesa ls nw/

rhljs daVsuj esa Mkyuk gSA rhljs daVsuj esa Mªe kjk iyVus ij Øe'k% 7 yh- vkSj 11 yh- nw/ cprk gSA (iyVus dh la[;k iw.kZ la[;k gksrh gS) Mªe dh vf/dre /kfjrk Kkr dhft,A

8 Math (Hindi) ­X

mÙkjekyk 1- b 2- c

3- a 4- c

5- c 6- b

7- a 8- b

9- b 10- b

11- b 12- b

13- b 14- 7 9

15- 3 16- 675

17- 30] ifjes; la[;k 18- lkar] nks n'keyo LFkkuksa ds ckn 19- 2 20- a, ?kukRed iw.kk±d r, q iw.kZ la[;k

21- ( ) 5 3 ,2 + 22- le ?kkr = 1] fo"ke ?kkr = 9

23- — 24- 10 ds xq.kt 25- 4 26- 2×2×2×3×5

27- — 28- —

29- — 30- e-l-o- = 28, y-l-o- = 336

31- — 32- gk¡ 33- gk¡ 34- ugha 35- — 36- e-l-o- (210, 55 = 5, 5 = 210 × 5 + 55y ⇒

y = –19

37- 4 38- a = 3q + r, dk iz;ksx dhft,A 39- a = 6q + r dk 40- –

iz;ksx dhft,A 41- – 42- n = 3q + r, dk iz;ksx dhft,A 43- 17

9 Math (Hindi) ­X

vè;k;&2

cgqin

egÙoiw.kZ fcanq % 1- 1] 2 rFkk 3 ?krkad okys cgqin Øe'k% jSf[kd] f?kkr ,oa f=k?kkr cgqin dgyrs gSaA 2- ,d f?kkr cgqin ax 2 + bx+c ds :i esa gksrk gS tcfd a, b rFkk c okLrfod

la[;k,a gSa rFkk a ≠ 0 gSaA 3- cgqin ds 'kwU;kad fcUnqvksa ds x funsZ'kkd gS y = p(x), x–vk dks izfrPNsn djrh

gSA vFkkZr~ x = a cgqin p(x) dk 'kwU;kad gksxk ;fn p(a) = 0 gSA 4- cgqin ds T;knk ls T;knk mrus 'kwU;kad gks ldrs gSa ftruh cgqin dh ?kkr gSA 5- f?kkr cgqin ax 2 + bx + c ds fy, (a ≠ 0)

'kwU;kad dk ;ksx b a = −

'kwU;kadksa dk xq.kuiQy c a =

6- foHkktu ,YxksfjFke % fdUgha nks cgqinksa p(x) rFkk g(x) ds fy, vU; cgqin q(x) rFkk r(x) bl izdkj gSa p(x) = g(x) q(x) + r(x) tcfd g(x) ≠ 0

tc r(x) = 0 ;k r(x) dh ?kkrkad <g(x) dh ?kkrkad

cgq oSdfYid iz'u

1- fuEu esa ls dkSu lh la[;k vad 6 ij lekIr gksrh gSA (a) f(x) > 0 (b) f(x) = 0

(c) f(x) < 0 (d) dksbZ ugha

10 Math (Hindi) ­X

2- cgqin f(x) ds 'kwU;d ml fcUnq ds funsZ'kkad gS tgk¡ y = f(x)----------- dks dkVrh gSA (a) x–vk (b) y–vk (c) ewy fcUnq (d) (x,y)

3- ;fn f(x) dk ,d 'kwU;d β gS rks ----------f(x) dk ,d xq.ku[k.Mu gSA (a) (x–β) (b) (x–2β)

(c) (x+β) (d) (2x–β)

4- ;fn cgqin f(x) dk ,d xq.ku[k.M (y–x) gS rks -------- f(x) dk ,d 'kwU;d gSA (a) y (b) a

(c) 2a (d) 2y

5- fuEu esa ls bl dFku ds fy, dkSu lk lgh ugha\ ^^,d cgqin ds-------- gksrs gSaA** (a) dksbZ okLrfod 'kwU;d (b) nks leku okLrfod 'kwU;d (c) nks fHkUu 'kwU;d (d) rhu okLrfod 'kwU;d

6- f=k?kkr cgqin x = f(x), y­vk dks vf/d ls vf/d ---------- fcUnqvksa ij dkVrk gSA (a) ,d (b) nks (c) rhu (d) pkj

7- cgqin x 2 +1 ds ------------ 'kwU;d gSaA (a) dsoy ,d okLrfod (b) dksbZ okLrfod ugha (c) dsoy nks okLrfod (d) ,d okLrfod vkSj ,d vokLrfod

8- cgqin 4x 2 –1 ds ------------ 'kwU;d gSaA (a) leku (b) fHkUu ij leku fpUg (c) leku ij mYVs fpUg (d) fHkUu vkSj vleku fpUg

11 Math (Hindi) ­X

9- ;fn p 'kwU;dksa dk ;ksx gS vkSj s xq.ku rks f?kkr cgqin ------------ gksxkA (a) x 2 – sx + p (b) x 2 – px + s (c) x 2 – sx – p (d) x 2 – px – s

10- ;fn cgqinksa 3x 2 + ax – 14 'kwU;dksa dk ;ksx gS vkSj s xq.ku rks f?kkr cgqin ---- gksxkA

(a) –2 (b) 7 (c) –8 (d) –7

11- ;fn cgqinksa ax 2 + bx + c ds nksuksa 'kwU;d ,d nwljs ds izfryksx gSa rks (a) a = 0 (b) a = b (c) f = c (d) a = –c

12- (x + 4) (x 2 – 6x + 8 ds 'kwU;d gSa % (a) 4, –4, 2 (b) 4, 4, –2 (c) –4, –4, 2 (d) –4, –4, –2

13- y = ax 2 + bx + c dk xzkiQ x­vk dks nks fHkUu fcUnqvksa ij dkVrk gS ;fn (a) b 2 –4ac > 0 (b) b 2 –4ac < 0 (c) b 2 –4ac = 0 (d) dksbZ ugha

y?kq mÙkjh; iz'u 14- ;fn α rFkk β cgqin 2x 2 – 7x + 3 ds 'kwU;d gSa] rks 'kwU;dksa ds izfrykseksa dk ;ksx

Kkr dhft,A

15- ;fn 1 3 cgqin 3x 3 – 17x – k dk 'kwU;d gS rks k dk eku Kkr dhft,A

16- ;fn cgqin 3x + 5, cgqin 6x 2 +16x 2 + px – 5 ls iw.kZr;k foHkkftr gksrk gS rks p dk eku Kkr dhft,A

17- ;fn (x + a), cgqin 2x 2 + 2ax + 5x + 10 dk xq.ku[k.Mu gS rks a dk eku Kkr dhft,A

12 Math (Hindi) ­X

18- ,d f?kkr cgqin Kkr dhft, ftlds 'kwU;d ( ) 5 3 2 − vkSj ( ) 5 3 2 + gksaA

19- ;fn 1 5 rFkk –2 fdlh f?kkr cgqin ds Øe'k% 'kwU;dksa ds xq.ku rFkk ;ksx gSa rks

cgqin Kkr dhft,A 20- cgqin 2 3 8 4 3 x x − + ds 'kwU;d Kkr dhft,A

21- ;fn (x+k), cgqinksa x 2 – 2x – 15 vkSj x 3 + a dk xq.ku[k.M gSa rks k rFkk a dk eku Kkr dhft,A

22- cgqin 2x 2 – 5x + 3 ds 'kwU;d Kkr dhft,A 23- kx 2 + 3k + 2x ds 'kwU;dksa dk ;ksx muds xq.ku ds cjkcj gSA k dk eku Kkr dhft,A 24- ;fn 4x 2 – 9 – 8kx dk ,d 'kwU;d nwljs 'kwU;d dk ;ksT; izfrykse gS rks k dk eku

Kkr dhft,A

nh?kZ mÙkjh; iz'u 25- 5x 2 – 4 – 8x ds 'kwU;d Kkr dhft, vkSj 'kwU;dksa rFkk xq.kkadksa ds lEcU/ dh

lR;rk dh tk¡p dhft,A 26- cgqin (a 2 + a) x 2 + 13x + 6a dk ,d 'kwU;d nwljs 'kwU;d dk xq.ku izfrykse gS

rks a ds eku Kkr dhft,A 27- cgqin (2x 2 + px) – 15 dk ,d 'kwU;d (–5) gSA f?kkr cgqir p(x2 + x) + k ds

nksuksa 'kwU;d leku gS rks k dk eku Kkr dhft,A 28- k dk og eku Kkr dhft, fltds fy, cgqin (3x 2 + 2kx + x – 5 ds 'kwU;dksa dk

;ksx muds xq.ku dk vk/k gSA 29- f(x) = 2x 4 – 5x 2 + 3x – 2 dks g(x) ls Hkkx nsus ij HkkxiQy q(x) = 2x 2 – 5x + 3

rFkk 'ks"kiQy r(x) = –2x + 1 izkIr gksrs gSA g(x) Kkr dhft,A 30- ;fn (2x), cgqin x 3 – 3x 2 4x + 12 ds xq.ku[k.Mksa esa ls ,d gS rks 'ks"k xq.ku[k.M

Kkr dhft,A 31- ;fn α, β, cgqin x 2 – 5x + k ds 'kwU;d gSa vkSj α – β = 1 rks k dk eku Kkr

dhft,A

13 Math (Hindi) ­X

32- cgqin 3x 2 – x – 4 ds 'kwU;d Kkr dhft, vkSj 'kwU;dksa rFkk xq.kkadksa ds lEcU/ dh lR;rk dh tk¡p dhft,A

33- x 4 – x 3 – 7x 2 + x + 6 ds lHkh 'kwU;d Kkr dhft,A ;fn blds nks 'kwU;d 3 vkSj 1 gSaA

34- 4x 4 – 20x 3 + 23x 2 + 5x – 6 ds lHkh 'kwU;d Kkr dhft,A ;fn 2 vkSj 3 blds nks 'kwU;d gSaA

35- ;fn ( ) 2 3 + vkSj ( ) 2 3 − cgqin x 4 – 4x 3 – 8x 2 + 36x – 9 ds 'kwU;d gS rks 'ks"k

'kwU;d Kkr dhft,A 36- 8x 4 + 14x 3 – 4x 2 + 7x – 8 ls D;k ?kVk;k tk, fd izkIr cgqin 4x 2 + 3x – 2xls

iw.kZr;k foHkkftr gksA 37- 4x 4 + 2x 3 – 2x 2 + x – 1 esa p(x) tksM+us ij izkIr cgqin (x 2 + 2x – 3)ls iw.kZr;k

foHkkftr gksA p(x) Kkr dhft,A 38- a rFkk bdk eku Kkr dhft, ;fn (x 4 + x 3 + 8x 2 + ax + b cgqin (x 2 +1) dk xq.kt

gSA 39- ;fn cgqin 6x 4 + 8x 3 + 17x 2 + 21x + 7 dks 3x 2 + 1 + 4x ls foHkkftr djsa rks

'ks"kiQy r(x) = ax + b gSA a rFkk b dk eku Kkr dhft,A

40- 2x 4 – 2x 3 – 7x 2 + 3x + 6 ds lHkh 'kwU;d Kkr dhft, ;fn ( ) 3 2 x ± blds nks

xq.ku[k.M gSaA 41- x 4 – 3x 3 – x 2 + 9x – 6 ds lHkh 'kwU;d Kkr dhft, ;fn 3 − vkSj 3 blds nks

'kwU;d gksaA 42- ;fn (x 3 + 3x + 1) cgqin x 5 – 4x 3 + x 2 + 3x + 1 dk ,d xq.ku[k.M gS rks 'ks"k nks

xq.ku[k.M Kkr dhft,A

14 Math (Hindi) ­X

mÙkjekyk 1- b 2- a 3- a 4- b 5- a 6- c 7- b 8- c 9- b 10- d 11- a 12- a

13- a 14- 1 1 7 3 α β

+ =

15- – 6 16- p = 7

17- a = 2 18- x 2 – 10x + 7

19- x 2 + 2x + 1 5 20- 2

3 2 3, 3

21- k = – 5, 3 and a = –125, +27 22- 1] 3 2

23- 3 2

− 24- 0

25- 2 2, 5 − 26- 5

27- 7 7, 4 p k = 28- k = 1

29- g(x) = x 2 – 1 30- –2, 3

31- k = 6 32- 4 , 1 3 −

33- –2, –1 34- 1 1 ,2 2 − +

35- ± 3 36- 14x –10

37- 61x + 65 38- r(x) = 0 ⇒ (a–1) x + (1–7) = 0 ⇒ a = 1 and f = 7

39- r(x) = x + 2 = ax + b ⇒ a = 1 and b = 2 40- 3 2 2, 1, − ±

41- 3,1, 2 ± 42- (x – 1), (x + 1)

15 Math (Hindi) ­X

vè;k;&3

nks pj okys jSf[kd lehdj.k ;qXe

egÙoiw.kZ fcanq % 1- jSf[kd lehdj.k ;qXe dk lcls O;kid :i gS%

a 1 x + b 1 y + c 1 = 0 a 2 x + b 2 y + c 2 = 0

tgk¡ a 1 , b 1 , b 2 , c 1 , vkSj c 2 ,slh okLrfod la[;k,a gS fd a 1

2 + b 1 2 ≠ 0, a 2

2 + b 2 2 ≠ 0,

2- nks pj esa ,d jSf[kd lehdj.k ;qXe dk vkys[k nks js[kk,a fu:fir djrk gSA

d) ;fn js[kk,a ,d fcUnq ij izfrPNsr djrh gSa rks og fcUnq nksuksa lehdj.kksa dk vfrh; gy gksrk gSA bl fLFkfr esa lehdj.k ;qXe lax gksrk gSA

[k) ;fn js[kka, laikrh gSa] rks mlds vifjfer :i ls vusd gy gksrk gSA bl fLFkfr esa lehdj.k ;qXe laxr gksrk gSA

x) ;fn js[kk,a lekUrj gSa] rks lehdj.k ;qXe dk dksbZ gy ugha gksrkA bl fLFkfr esa lehdj.k ;qXe vlaxr gksrk gSA

3- ;fn fn, x, jSf[kd lehdj.k a 1 x + b 1 y + c 1 = 0 vkSj a 2 x + b 2 y + c 2 = 0, ,d jSf[kd lehdj.k ;qXe dks iznf'kZr djrk gS rks %

d) 1 1 2 2

a b a b

≠ ⇒ jSf[kd lehdj.k ;qXe laxr gksrk gSA (vfrh; gy)

[k) 1 1 1 2 2 2

a b c a c b

= ≠ ⇒ jSf[kd lehdj.k ;qXe vlaxr gksrk gSA (dksbZ gy ugha)

x) 1 1 1 2 2 2

a b c a c b

= = ⇒ jSf[kd lehdj.k ;qXe laxr gksrk gSA (vusd gy)

16 Math (Hindi) ­X

cgq oSdfYid iz'u

1- nks pj ds izR;sd jSf[kd lehdj.k ds ------------ gy gksrs gSaA (a) dksbZ ugha (b) ,d (c) nks (d) vla[;

2- 1 1 1 2 2 2

a b c a c b

= = ⇒ ;g fLFkfr gS%

(a) izfrPNsfnr js[kk,¡ (b) lekarj js[kk,a (c) laikrh js[kk,¡ (d) dksbZ ugha

3- ,d ;qXe dks laxr vkSj vkfJr gksus ds fy, ;qXe dk ----------- gksuk vko';d gSA (a) dksbZ gy ugha (b) dsoy ,d gy (c) vla[; gy (d) dksbZ ugha

4- nks pj ds izR;sd jSf[kd lehdj.k dk xzkiQ ---------- iznf'kZr djrk gSA (a) fcUnq (b) lh/h js[kk (c) oØ (d) f=kHkqt

5- jSf[kd lehdj.k ds ;qXe dks iznf'kZr djus okyh izR;sd js[kk ds xzkiQ dk izR;sd fcUnq mHk;fu"B gy gS ;fn

(a) vla[; gy (b) dsoy ,d gy (c) dksbZ gy ugha (d) dksbZ ugha

6- fuEu esa ls dkSu lk jSf[kd lehdj.kksa ds ;qXe 3x – 2y = 0, 5y – x = 0 dk gy gS (a) (5] 1) (b) (2] 3) (c) (1] 5) (d) (0] 0)

7- ax + by = 0 vkSj y­vk dk ,d mHk;fu"B gy gS---------------

(a) 0, c b

(b) 0, b

c

(c) ,0 c b

(d) 0, c

b

−

17 Math (Hindi) ­X

8- ;fn lehdj.k 2x – 8y = 12 esa x dk eku 2 gS rks y dk laxr eku ---------- gksxk (a) –1 (b) +1

(c) 0 (d) 2

9- jSf[kd lehdj.k dk ;qXe vlaxr dgyk,xk ;fn mldk (a) dsoy ,d gy (b) dksbZ gy ugha (c) vla[; gy (d) 2 rFkk c nksuksa

10- x = a vkSj y = b dks xzkfiQ; fof/ kjk n'kkZus ij gesa -------- izkIr gksrk gSA (a) lekarj js[kk,a (b) laikrh js[kk,a (c) fcUnq (a, b) ij izfrPNsfnr js[kk,a (d) fcUnq (b, a) izfrPNsfnr js[kk,a

11- 2x + 3y = 5 ds fdrus lahko okLrfod gy gSa (a) dksbZ ugha (b) ,d (c) nks (d) vla[;

12- k ds fdl eku ds fy, lehdj.kksa 3x + 2y = – 5, x – ky = 2 dk dsoy ,d gy gS\

(a) 2 3

k = (b) 2 3

k ≠

(c) 2 3

k = − (d) 2 3

k ≠ −

13- k ds fdl eku ds fy, jSf[kd lehdj.kksa dk ;qXe laikrh js[kk,¡ n'kkZrk gS\ (a) –3 (b) 3

(c) 1 (d) –2

14- ftl fcUnq ij 1 3 2 y x

+ = vkSj x­vk izfrPNsfnr djrs gSa mlds funsZ'kkad-------- gSaA

(a) (0, 3) (b) (3, 0)

(c) (2, 0) (d) (0, 2)

18 Math (Hindi) ­X

15- xzkfiQ; fof/ kjk x – 2 = 0 ,d ,slh js[kk n'kkZrh gS tks (a) x­vk ds lekarj mlls 2 bdkbZ nwjh ij gSaA (b) y­vk ds lekarj mlls 2 bdkbZ nwjh ijA (c) x­vk ds lekarj] y­vk ls 2 bdkbZ nwjh ijA (d) y­vk ds lekarj] x­vk ls 2 bdkbZ nwjh ijA

16- ;fn ax + by = c vkSj lx + my = n dk dsoy ,d gy gh rks mlds xq.kkadksa esa fuEu esa ls dkSu lk laca/ gksxk%

(a) am ≠ lb (b) am = lb

(c) ab = lm (d) ab ≠ lm

y?kq mÙkjh; iz'u 17- fuEu ds fy, jSf[kd lehdj.kksa dk ;qXe cuk,A ^^,d fHkUukRed la[;k ds va'k rFkk

gj dk ;ksx mlds gj ds nks xqus ls rhu de gSA ;fn va'k vksj gj izR;sd dks 1 de dj nsa rks va'k] gj ls vk/k gks tkrk gSA**

18- ^^vej ,d fo|ky; ds /kokdksa dks ` 9000 LdkWyjf'ki ds :i esa izfrekg ck¡Vrk gSA ;fn 20 /kor vkSj gksrs rks izR;sd dks ` 160 de feyrsA** jSf[kd lehdj.kksa dk ;qXe cukb;sA

19- k dk og eku Kkr dhft, ftlds fy, x + 2y = –7 vkSj 2x + ky + 14 = 0 laikrh js[kkvksa dks n'kkZ,A

20- ,d jSf[kd lehdj.k fyf[k, tks 2x + 3y – 4 = 0 ds lkFk laikrh gksA 21- a ds fdl eku ds fy, (3, a) js[kk 2x – 3y = 5 ij fLFkr gksA 22- nks izkd`r la[;kvksa dk ;ksx 25 gS vkSj mudk varj 7 gSA la[;k,a Kkr dhft,A 23- fnus'k fcUnq (2] 4) vkSj (0] 6) dks tksM+us okyh js[kk ij py jgk gS vkSj ujs'k

(3] 4) rFkk (1] 0) dks tksM+us okyh js[kk ijA xzkiQ kjk mijksDr dks n'kkZb, vkSj og fcUnw funsZ'kkad Kkr dhft, ftlls nksuksa ,d nwljs dks izfrPNsfnr djsaA

24- jSf[kd lehdj.kksa ds ;qXe dks gy djsaA x – y = 2 vkSj x + y = 2 bl izdkj p = 2x + 3 dk eku Kkr dhft,A

19 Math (Hindi) ­X

25- k ds fdl eku ds fy, fuEu jSf[kd lehdj.k ijLij lekarj gSa % 2x + ky = 10 3x – (k + 3)y = 12

26- jSf[kd lehdj.kksa dk ;qXe cukb;s tcfd 'kkar ikuh esa uko dh pky x km/hr. rFkk /kjk dh pky y km/hr. gSA

27- ,d uko 7 ?kaVs esa /kjk ds izfrdwy 32 fd-eh- vkSj /kjk ds vuqdwy 36 fd-eh- tkrh gSA 9 ?kaVs esa /kjk ds izfrdwy 40 fd-eh- vkSj /kjk ds vuqdwy 48 fd-eh- tkrh gSA 'kkar ikuh esa uko dh pky vkSj /kjk dh pky Kkr dhft,A

27- xzkfiQd fof/ kjk tk¡p dhft, fd jSf[kd lehdj.kksa dk ;qXe 3x + 5y = 15, x – y = 5 laikrh gS ;g Hkh tk¡fp, fd js[kk,a vkfJr gSaA

28- p ds fdl eku ds fy, jSf[kd lehdj.kksa ds ;qXe (p + 2) x – (2p + 1) y = 3 (2p – 1)

2x – 3y = 7 dk vfrh; gy gSA

29- k ds fdl eku ds fy, uhps fy[ks jSf[kd lehdj.kksa dk ;qXe vlaxr gS % (3k + 1) x + 3y – 2 = 0

(k2 + 1) x + (k – 2)y – 5 = 0

30- fn;s gq, jSf[kd lehdj.k x + 3y = 4, ds fy, nks pj esa vU; jSf[kd lehdj.k fyf[k, fd bl ;qXe dk xzkfiQ; fu:i.k -------- gksA (i) izfrPNsfnr js[kk,a (ii) lekUrj js[kk,a (iii) laikrh js[kk,a

31- x – y = 4 vkSj x + y = 10 dks gy djds dk eku Kkr djsa ;fn y – 3x – p.

32- k dk eku Kkr djsa ftlds fy, fn, gq, jSf[kd lehdj.kksa ds vusd gy gSA kx + 3y = k – 3 12 + ky = k

20 Math (Hindi) ­X

33- α vkSj β ds og eku Kkr dhft, ftlds fy, uhps fy[ks lehdj.kksa ds vusd gy gSa %

2x + 3y = 7 2ax + (α + β)y = 28

34- x vkSj y ds fy, gy djsa % 1 1 1 1 8; 9 2 3 3 2

x y x y + − + − + = + =

35- x vkSj y ds fy, gy djsa % 2 x + 3 y = 17 2 x+2 – 3 y+1 = 5

36- x vkSj y ds fy, gy djsa % 139x + 56y = 641 56x + 139y = 724

37- x vkSj y ds fy, gy djsa % 5 1 2 x y x y

+ = + −

0, 0. 15 5 2, x y x y x y x y + ≠ − ≠ − = −

+ −

38- x vkSj y ds fy, gy djsa % 37x + 43y = 123 43x + 37y = 117

39- xzkfiQ; fof/ kjk tk¡p dhft, fd js[kk,a 3x + 2y – 4 = 0 vkSj 2x – y – 2 = 0 laxr gSA ml fcUnq ds funsZ'kkad Kkr dhft, tgk¡ js[kk,a vk ij feysA

nh?kZ mÙkjh; iz'u 40- x vkSj y ds fy, gy djsa %

1 12 1 2 ( 3 ) 7 (3 2 2 x y x y

+ = + −

7 4 2 (2 3 ) (3 2 x y x y + =

+ − 2x + 3y ≠ 0 and 3x – 2y ≠ 0

21 Math (Hindi) ­X

41- p vkSj q ds fy, gy djsa %

2, 6, 0, 0. p q p q p q pq pq + − = = ≠ ≠

42- x vkSj y ds fy, gy djsa % 17 5

2 3 3 2 3 2 x y x y

+ = + −

2 5 1 3 2 3 2 x y x y

+ = + − 3x + 2y ≠ 0, 3x – 2y ≠ 0

43- x vkSj y ds fy, gy djsa % 17 5

2 3 3 2 3 2 x y x y

+ = + −

2 5 1 3 2 3 2 x y x y

+ = + − x + y ≠ 0, x – y ≠ 0

44- x vkSj y ds fy, gy djsa % 2 3 4 2, 1, 0, 0 x y x y x

+ = = − ≠ ≠

45- x vkSj y ds fy, gy djsa % ax + by = 1

( ) 2

2 2 1 a b

bx ay a b

+ + = −

+

46- nks la[;kvksa esa ls cM+h la[;k ds nks xqus esa ls 20 ?kVk;k tk,] rks NksVh la[;k izkIr gksrh gSA ;fn NksVh ds nks xqus esa ls 5 ?kVk;k tk, rks cM+h la[;k izkIr gksrh gSA la[;k,a Kkr dhft,A

47- 27 iSafUly vkSj 31 jcM+ksa dk dqy ewY; ` 85 gSA tcfd 31 iSafUly vkSj 27 jcM+ksa dk dqy ewY; ` 89 gSA 2 iSfUly vkSj 1 jcM+ dk ewY; Kkr dhft,A

48- ;fn ,d vk;r dh yEckbZ 7 lseh- c<+k nh tk, vkSj pkSM+kbZ 3 lseh- ?kVk nsa rks ks=kiQy leku jgrk gSA ;fn mldh yEckbZ 7 lseh- ?kVk nsx vkSj pkSM+kbZ 5 lseh- c<+k nsa rks Hkh ks=kiQy leku jgrk gSA yEckbZ vkSj pkSM+kbZ Kkr djsaA

22 Math (Hindi) ­X

49- nks vadksa dh ,d la[;k vadksa ds tksM+ dks 8 ls xq.kk dj 1 tksM+us ij izkIr gksrh gS ;k ogh la[;k vadksa ds varj dks 13 ls xq.kk dj 2 tksM+us ij izkIr gksrh gSA la[;k Kkr dhft,A ,slh fdruh la[;k,a gSaA

50- ,d rhu vadksa dh la[;k ds vadksa dk tksM+ 17 gSA chp okyk vad ckdh nksuksa vadksa ds tksM+ ls ,d vf/d gSA ;fn vadksa dk LFkku cny nsa rks la[;k 396 de gks tkrh gSA la[;k Kkr dhft,A

51- ,d ukfod viuh uko /kjk ds izfrdwy 35 fdeh- rFkk /kjk ds vuqdwy 55 fd- eh- pykdj 12 ?kaVs yxkrk gS vkSj 10 ?kaVksa esa 30 fd-eh- /kjk ds izfrdwy vkSj 44 fd-eh- /kjk ds vuqdwy pykrk gSA 50 fd-eh- /kjk ds izfrdwy vkSj 77 fd- eh- /kjk ds vuqdwy uko pykus esa fdruk le; yxsxkA

52- ,d dk;ZØe esa 10 esgekuksa dks dk A ls B esa Hkstk tkrk gS rks nksuksa dkksa esa leku la[;k esa esgeku gks tkrs gSaA ;fn 20 esgekuksa dks dk B ls A esa Hkstk tk, rks dk A esa esgekuksa dh la[;k ls nqxquh gks tkrh gSA nksuksa dkksa esa 'kq: esa fdrus esgeku FksA

53- ,d dk;ZØe esa e/q izR;sd mifLFkr O;fDr dks ` 21 nsuk pkgrh gS vkSj ikrh gS fd pkj #i;s de iM+ x;sA blfy, og izR;sd dks ` 20 nsuh gS vkSj ikrh gS fd ` 1 cp x;kA mlus fdrus #i;s ck¡Vs vkSj ogk¡ dqy fdrus yksx FksA

54- ,d eksckby daiuh ekfld fdjk;s ds :i esa fuf'pr jkf'k ysrh gS ftlesa 100 feuV izfrekg eqÝr feyrs gSa vkSj vxys izR;sd feuV ij fuf'prjkf'k ysrh gSA vfHk"ksd us 370 feuV ds fy, ` 433 fn;s vkSj vkf'k"k us 300 feuV ds ` 398 fn;sA blh Iyku esa fcy jkf'k Kkr djks ;fn m"kk 400 feuV ysA

55- firk dh vk;q] mlds nks cPpksa dh vk;q ds ;ksx dh rhu xquh gSA 5 o"kZ ckn] mldh vk;q nksuksa cPpksa dh vk;q ds ;ksx dh nks xquh gks tk;sxhA firk dh vk;q Kkr dhft,A

56- lehj.k 3x – 4y + 6 = 0 vkSj 3x + y – 9 = 0 ds xzkiQ [khafp,A bu js[kkvksa vkSj x­ vk ls cus f=kHkqt ds 'kh"kks± ds funsZ'kkad Hkh Kkr dhft,A

57- 90» vkSj 97» 'kq¼ vEy ?kksy] 95» dk 21 yhVj 'kq¼ vEy ?kksy ds fy, feyk;k x;kA feJ.k cukus esa iz;qDr] izR;sd ?kksy dh ek=kk Kkr dhft,A

58- ,d fHkUu ds va'k rFkk gj dk ;ksx 8 gSA ;fn va'k rFkk gj nksuksa esa 3 tksM+ fn;k

tk;s] rks fHkUu 3 4 gks tkrh gSA fHkUu Kkr dhft,A

23 Math (Hindi) ­X

59- lqfiz;k vkSj /zqo dh ekfld vk; 5%4 ds vuqikr esa gS rFkk mudk ekfld O;; 7%5 ds vuqikr esa gSA ;fn nksuksa ` 3000 izfrekg cpkrs gSa] rks izR;sd dh ekfld vk; Kkr dhft,A

60- ,d pØh; prqHkqZt ABCD ds pkjksa dks.k Kkr dhft,A

ftlesa (2 3) , ( 7) , (2 17) A x B y C y ∠ = − ° ∠ = + ° ∠ = + ° rFkk (4 9) D x ∠ = − ° gSA

mÙkjekyk 1- d 2- c

3- c 4- b

5- a 6- d

7- a 8- a

9- b 10- c

11- d 12- d

13- b 14- c

15- b 16- a

17- ;fn N = x vkSj D = y rks x – y = – 3, 2x – y = 1

18- /kodksa dh la[;k = x, c<+s gq, /kodksa dh la[;k = y 20

1 1 4225

y x

x y

− = − =

19- k = 4 20- k (2x + 3y –4) = 0; k okLrfod la- gS rFkk k ≠ 0

21- 1 3 22- 16, 9

23- (2, 2) 24 (2, 0) p = 7

25- k = 6 26- uko dh pky = x, /kjk dh pky = y

32 36 7

48 40 9

x y x y

x y y x

+ = − +

+ = − +

24 Math (Hindi) ­X

27- gk¡] ugha 28- p ≠ 4

29- 19 1, 2

k k = ≠ 30- —

31- (7, 3), 18 32- k = 6

33- (4, 8) 34- (7, 13)

35- (3, 2) (ladsr ekuk fd 2 x = m, 3 y = n

36- (3, 4) 37- (3, 2)

38- (1, 2) 39- gk¡ (0, 2, (0 – 2)

40- (2, 1) 41- 1 1 ,2 4

−

42- (1, 1) 43- 5 1 ,4 4

− −

44- (4, 9) 45- 2 2 2 2 , a b a b a b

+ +

46- (15, 10) 47- ` 5

48- 28m, 15m 49- 41 vFkok 14 (2 la[;k,a laHko)

50- 692 8 4

x y y x

+ = − =

51- 3 km/hr. 8 km/hr., 17 hr.

52- 100, 80 53- ` 101, 5

54- ` 448 1 298, 2

` ` 55- 45 o"kZ

56- (–2, 0), (2, 3), (3, 0) 57- 90% dk 6 yhVj] 97% dk 15 yhVj

58- 3 5 59- ` 10,000, ` 8000

60- 63°, 57°, 117°, 123°

25 Math (Hindi) ­X

vè;k;&4

le:i f=kHkqt

egÙoiw.kZ fcanq % 1- nks f=kHkqt le:i dgykrs gSa ;fn muds laxr dks.k cjkcj gksa vkSj mudh laxr

Hkqtk,a lekuqikr gksaA 2- le:i f=kHkqt ds xq.k/eZ %

∆ABC rFkk ∆DEF esa d) dks.k&dks.k dks.k le:irk %

∆ABC ~ ∆DEF ;fn ∠A = ∠D, ∠B = ∠E rFkk ∠C = ∠F

[k) Hkqtk&dks.k & Hkqtk le:irk %

∆ABC ~ ∆DEF ;fn AB BC DE EF

= rFkk ∠B = ∠E

x) Hkqtk&Hkqtk & Hkqtk le:irk %

∆ABC ~ ∆DEF ;fn AB AC BC DE DF EF

= =

3- fuEu izes;ksa dk gy ijhkk esa iwNk tk ldrk gSA

d) vk/kjHkwr vkuqikfrdrk % ,d f=kHkqt dh ,d Hkqtk ds lekarj [khaph xbZ js[kk vU; nks Hkqtkvksa dks ftu nks fcUnqvksa ij izfrPNsr djrh gS os fcUnq Hkqtkvksa dks leku vuqikr esa foHkkftr djrs gSaA

[k) nks le:i f=kHkqtksa ds ks=kiQyksa dk vuqikr fdUgha nks laxr Hkqtkvksa ds oxks± ds vuqikr ds cjkcj gksrk gA

x) ikbFkkxksjl izes; % ,d ledks.k f=kHkqt ds d.kZ dk oxZ vU; nks Hkqtkvksa ds oxks± ds ;ksxiQy ds cjkcj gksrk gSA

26 Math (Hindi) ­X

?k) ikbFkkxksjl izes; dk foykse % ;fn fdlh f=kHkqt dh ,d Hkqtk dk oxZ vU; nks Hkqtkvksa ds oxks± ds ;ksx ds cjkcj gks rks igyh Hkqtk dk lEeq[k dks.k ledks.k gksrk gSA

cgq oSdfYid iz'u

1- ∆ABC ~ ∆DEF ;fn DE = 2 AB rFkk BC = 3 lseh- rks EF = _______ (a) 1-5 lseh- (b) 3 lseh- (c) 6 lseh- (d) 9 lseh-

2- ∆DEW esa AB||EW ;fn AD = 4 lseh- DE = 12 lseh- rFkk DW=24 lseh- rks DB dk eku gS--------

(a) 4 lseh- (b) 8 lseh- (c) 12 lseh- (d) 16 lseh-

3-

fp=k esa cd dk eku gS& (a) ae (b) af (c) bf (d) be

4- ;fn ∆ABC esa AB = 6 lseh-] BC = 12 lseh- rFkk CA = 6 3 lseh- rks ∠A dk

eki gS-------- (a) 30° (b) 45° (c) 60° (d) 90°

27 Math (Hindi) ­X

5- nks le:i f=kHkqtksa ds ks=kiQyksa dk vuqikr 9:16 gSA budh laxr Hkqtkvksa dk vuqikr gS--------

(a) 9:16 (b) 16:9 (c) 3:4 (d) 4:3

6- fp=k esa ∆ABC le:i gS--------

(a) ∆BDC (b) ∆DBC (c) ∆CDB (d) ∆CBD

7- ∆AMB ~ ∆CMD rFkk 2 ks- (∆AMB) = ks=k (∆CMD) MD dh yEckbZ gS--- (a) 2 MB (b) 2 MD

(c) 2 MB (d) 2

MD

8- AE dh yEckbZ gS

28 Math (Hindi) ­X

(a) 10 lseh- (b) 9 lseh- (c) 5 5 (d) 5

9- ∆PQR esa fcUnq S rFkk T Øe'k% Hkqtkvksa PR rFkk QR ij bl izdkj fLFkr gS fd ST || PQ rks RS

RT = ____

(a) SP TQ (b)

PR QR

(c) SP RS (d) TQ

RT

10- ∆ABC esa DE || BC ;fn AD DB

= 3 5 rks ( )

( ) ar ADE ar ABC

∆ =

∆ _____

(a) 3 5 (b) 3

8

(c) 9 64 (d) 9

25

11- ∆ABC esa DE || BC fp=k esa x dk eku gS---------

(a) 1 (b) –1 (c) 3 (d) –3

29 Math (Hindi) ­X

12- ∆ABC esa ∠B = 90° ;fn Hkqtk AC dk yEc lefHkktd gS rks ( ) ( )

BEC ABC

∆ =

∆

ks - ks - -----

(a) 1 2 (b) 2

1

(c) 4 1 (d) 1

4

13- 12 lseh- Hkqtk okys leckgq f=kHkqt ds yEc dh yEckbZ gS-------- (a) 12 lseh- (b) 6 2 lseh- (c) 6 lseh- (d) 6 3 lseh-

14- A rFkk B ds chp dh lh/h nwjh gS---------

(a) 3 5 (b) 5 3

(c) 5 (d) 5 2

15- ;fn ,d lefcgq ledks.kh; f=kHkqt esa d.kZ dh yEckbZ 10 lseh- gS rks f=kHkqt dk ifjeki gS---------

(a) 5 2 lseh- (b) 2 5 lseh-

(c) ( ) 10 2 1 + lseh- (d) ( ) 10 2 1 − lseh-

30 Math (Hindi) ­X

y?kq mÙkjh; iz'u

16- fp=k esa ;fn ST || QR, PT = 8 lseh- rFkk PR = 10 lseh- rks PS SQ dk eku Kkr

dhft,A

17- layXu fp=k esa AE dk eku Kkr dhft, ;fn DE || BC.

18- fp=k esa le:i f=kHkqtksa dk uke crkb,A

31 Math (Hindi) ­X

19- lefckgq f=kHkqt ABC, f=kHkqt PQR ds le:i gSA ;fn AC = AB = 4 lseh-] RQ = 10 lseh- rFkk BC = 6 lseh- rks PR dh yEckbZ Kkr dhft,A ∆PQR fdl izdkj dk f=kHkqt gSA

20- fp=k esa ∆ABC ~ ∆PQR : x dk eku Kkr dhft,A

21- ∆PQR esa DE || QR rFkk DE = 1 4 QR.

( ) ( )

ar PQR ar PDE

∆ ∆ dk eku Kkr dhft,A

22- f=kHkqtksa ABC rFkk PQR esa ∠B = ∠Q rFkk 1 2

AB BC PQ QR

= = rks PR QR dk eku Kkr

dhft,A 23- ,d f=kHkqt dh rhu Hkqtkvksa dk eki , 10 a a rFkk gSA lcls yEch Hkqtk ds lEeq[k

dks.k dk eki fyf[k,A

32 Math (Hindi) ­X

24- layXu fp=k DE || BC. DE esa dk eku Kkr dhft,A

nh?kZ mÙkjh; iz'u 25- fp=k esa SR dh yEckbZ Kkr dhft, ;fn ∠QPR = ∠PSR. PR = 6 lseh- rFkk

QR = 9 lseh-

26- ∆PQR esa RS ⊥ PQ, ∠QRS = ∠P, ;fn PS = 5 lseh-] SR = 8 lseh- rks PQ yEckbZ Kkr dhft,A

27- le:i f=kHkqt ABC rFkk PBC leku Hkqtk BC dh foijhr fn'kk esa cuk, x, gSaA fl¼ dhft, fd AB = BP

33 Math (Hindi) ­X

28- fp=k esa ABCD ,d vk;r gSA ∆ADE rFkk ∆ABF bl izdkj gS fd ∠E = ∠F

fl¼ dhft, fd AD AB AE AF

=

29- fp=k esa DE || BC, DE = 3 lseh-] BC = 9 rFkk ks- (∆ADE) = 30 lseh- 2 ks-] leyac BCED Kkr dhft,A

30- vfer ,d ?kj ls 8 eh- dh nwjh ij fLFkr ,d fcUnq ij [kM+k gSA ?kj dh Nr ij eksckbZy usVodZ dh ehukj yxh gSA ehukj ds f'k[kj rFkk ry dh nwjh fcUnq ls Øe'k% 17 eh- rFkk 10 eh- gSA ehukj rFkk ?kj dh mQ¡pkbZ Kkr dhft,A

31- ,d ledks.k f=kHkqt ABC esa] ∠B ,d ledks.k gS rFkk 3| BC AB AB AC

= dk eku

Kkr dhft,A 32- ledks.k f=kHkqt PRO esa d.kZ gS rFkk 'ks"k nks Hkqtkvksa dk eki 6 lseh- rFkk 8 lseh-

gSA ckg~; fcUnq Q ls P rFkk R bl izdkj tksM+s x, fd PQ = 24 lseh- rFkk RQ = 26 lseh- ∠PQR dk eki Kkr dhft,A

34 Math (Hindi) ­X

33- fp=k esa ∆ABC ,d lefckgq f=kHkqt gS ftlesa AB = AC. fcUnq P Hkqtk BC dk eè; fcUnq gSA ;fn PM ⊥ AB rFkk PN ⊥ AC rks fl¼ dhft, fd MP = NP

34- SQ leyEc prqHkqZt PQRS dk fod.kZ gSA lekUrj Hkqtkvksa PQ rFkk ij Øe'k% nks fcUnq E rFkk F bl izdkj fLFkr gS fd EF, SQ dks G ij dkVrh gSA fl¼ dhft, fd SG × QE = QG × SF.

35- fp=k esa ABCD ,d prqHkqZt gSA P, Q, R rFkk S Øe'k% Hkqtkvksa AB, CB, CD rFkk AD dks 2%1 vuqikr esa foHkkftr djrs gSaA fl¼ dhft, fd PQRS ,d lekarj prqHkqZt gSA

36- fl¼ dhft, fd ;fn fdlh f=kHkqt dh ,d Hkqtk ds lekarj vU; nks Hkqtkvksa dks fHkUu&fHkUu fcUnqvksa ij izfrPNsn djus ds fy, ,d js[kk [khaph tk, rks vU; nks Hkqtk,a ,d gh vuqikr esa foHkktfr gks tkrh gSA

37- fl¼ dhft, fd leprqHkqZt ds fdlh ,d Hkqtk ds oxZ dk pkj xquk mlds fod.kks± ds oxks± ds ;ksx ds cjkcj gksrk gSA

38- fl¼ dhft, fd nks le:i f=kHkqtksa ds ks=kiQyksa dk vuqikr fdUgha nks laxr Hkqtkvksa ds oxks± ds vuqikl ds cjkcj gksrk gSA

35 Math (Hindi) ­X

39- fl¼ dhft, fd ;gn fdlh f=kHkqt dh ,d Hkqtk dk oxZ vU; nks Hkqtkvksa ds oxks± ds ;ksx ds cjkcj gks rks igyh Hkqtk dk lEeq[k dks.k ledks.k gksrk gSa

40- fl¼ dhft, fd ,d ledks.k f=kHkqt ds d.kZ dk oxZ vU; nks Hkqtkvksa ds oxks± ds ;ksxiQy ds cjkcj gksrk gSA

41- vk;r ABCD dh yEckbZ mldh pkSM+kbZ ls nqxquh gSA yEckbZ rFkk pkSM+kbZ ij nks leckgq f=kHkqt cuk, x;s gSaA muds ks=kiQyksa dk vuqikr Kkr dhft,A

42- vej rFkk v'kksd ,d vk;rkdkj ckx ds ,d dksus A ij [kM+s gSaA os ikuh ihuk pkgrs gSaA vej mÙkj fn'kk dh vksj 50 eh-@feuV dh pky ls pyrk gS vkSj v'kksd if'pe dh vksj 60 eh-@feuV dh pky ls pyrk gSA nksuksa 5 feuV pyrs gSaA vej dks uy fey tkrk gS vkSj og ikuh ih ysrk gSA crkb, fd vc v'kksd uy ls de ls de fdruh nwjh ij [kM+k gSA

43- fp=k esa BCDE ,d vk;r gS rFkk ∠BCA = ∠DCF vk;r ds fod.kZ BD dh yEckbZ Kkr dhft,A

36 Math (Hindi) ­X

44- fp=k esa BDEF ,d vk;r gSA fcUnq C Hkqtk BD dk eè; fcUnq gSA AF = 7 lseh- DE = 9 lseh- rFkk BD = 24 lseh- ;fn AE = 25 lseh- rks fl¼ dhft, fd ∠ACE = 90°.

45- fp=k esa ledks.k f=kHkqt ds d.kZ ij ,d yEc [khapk x;k gSA fofHkUu js[kk [k.Mksa dk eki n'kkZ;k x;k gSA x, y, z Kkr dhft,A

mÙkjekyk 1- c 2- b

3- a 4- d

5- c 6- d

7- a 8- c

37 Math (Hindi) ­X

9- b 10- c

11- d 12- d

13- d 14- a

15- c 16- 4 : 1

17- 1.5 lseh- 18- ∆APQ ~ ∆ABC

19- 20 3 lseh- 20- 4.8 lseh-

21- 16:1 22- 1 2

23- 90° 24- 25 lseh- 25- 4 lseh- 26- 17.8 lseh- 29- 240 lseh- 2 30- 9 eh-] 6 ls-

31- 1 2 32- 90°

41- 4:1 42- 50 61 eh-

43- lseh- 45- 5, 2 5, 3 5 x y z = = =

38 Math (Hindi) ­X

vè;k;&5

f=kdks.kfefr 1- f=kdks.kferh; vuqikr % ∆ABC, esa ∠B = 90° dks.k 'A' ds fy,&

sin A = yEc d.kZ

cos

tan

cot

sec

sec

A

A

A

A

ec A

=

=

=

=

=

vk/kj d.kZ

yEc

vk/kj vk/kj

yEc

d.kZ

vk/kj

d.kZ

yEc

2- O;qRØe 1 1 sin , cosec

cosec sin

1 1 cos , sec sec cos

1 1 tan , cot cot tan

θ θ θ θ

θ θ θ θ

θ θ θ θ

= =

= =

= =

39 Math (Hindi) ­X

3- vkuqikfrd laca/ sin cos tan , cot cos sin

θ θ θ θ

θ θ = =

4- leZlfedk,¡ 2 2 2 2 2 2 sin cos 1 sin 1 cos cos 1 sin θ θ θ θ θ θ + = ⇒ = − = − vkSj

2 2 2 2 2 2 1 tan sec tan sec 1 sec tan 1 θ θ θ θ θ θ + + ⇒ = − − = vkjS 2 2 2 2 2 2 1 cot cos cot cos 1 cos cot 1 ec ec ec θ θ θ θ θ θ + = ⇒ = − − = vkjS

5- dqN fof"k"V dks.kksa ds f=kdks.kkferh; vuqikr

∠E 0° 30° 45° 60° 90°

sin A 0 1 2

1 2

3 2 1

cos A 1 3 2

1 2

1 2 0

tan A 0 1 3 1 3 vifjHkkf"kr

cosec A vifjHkkf"kr 2 2 2 3 1

sec A 1 2 3 2 2 vifjHkkf"kr

cot A vifjHkkf"kr 3 1 1 3 0

6- iwjd dks.kkdsa ds f=kdks.kferh; vuqikr sin (90° – θ) = cos θ

cos (90° – θ) = sin θ

tan (90° – θ) = cot θ

cot (90° – θ) = tan θ

sec (90° – θ) = cosec θ

cosec (90° – θ) = sec θ

7-

40 Math (Hindi) ­X

cgq oSdfYid iz'u

funsZ'k % fuEufyf[kr iz'uksa esa 0º ≤ θ ≤ 90°

1- ;fn x = a sin θ rFkk y = a cos θ rks x 2 + y 2 dk eku gS------- (a) a (b) a 2

(c) 1 (d) 1 a

2- cosec 70° – sec 20° dk eku gS------- (a) 0 (b) 1 (c) 70° (d) 20°

3- ;fn 3 sec θ – 5 = 0 rks cot θ = -------

(a) 5 3 (b) 4

5

(c) 3 4 (d) 3

5

4- ;fn θ = 45° rks sec θ cot θ – cosec θ tan θ dk eku gS------- (a) 0 (b) 1 (c) 2 (d) 2 2

5- ;fn sin (90° – θ) cos θ = 1 rFkk θ U;wudks.k gS rks θ = ------- (a) 90° (b) 60°

(c) 30° (d) 0°

6- (1 + cos θ) (1 – cos θ) cosec 2 θ dk eku gS------- (a) 0 (b) 1 (c) cos 2 θ (d) sin 2 θ

41 Math (Hindi) ­X

7- ∆TRY ,d ledks.kh; lefckgq f=kHkqt gks rks cos T + cos R + cos Y = ........... (a) 2 (b) 2 2

(c) 1 2 + (d) 1 1 2

+

8- ;fn K + 7 sec 2 62° – cot 2 28° = 7 sec 0° rks K dk eku gS--------- (a) 1 (b) 0

(c) 7 (d) 1 7

9- cot sin 2 cos 2 2 π π

θ θ − − − dk eku gS---------

(a) cot θ cos 2 θ (b) cot 2 θ

(c) cos 2 θ (d) tan 2 θ

10- ;fn sin θ – cos θ = 0, 0 ≤ θ ≤ 90° rks θ dk eku gS--------- (a) cos θ (b) 45° (c) 90° (d) sin θ

11- 2

sin 1 sin

θ

θ =

− ............

(a) cot θ (b) sinθ

(c) sincos

θ θ

(d) tan θ

12- ;fn 1 sin 2

θ = rks sin cosec θ θ + dk eku gS............

(a) 0 (b) 1

(c) 3 2 (d) 5

2

42 Math (Hindi) ­X

13- ∆ABC ,d lefckgq ledks.k f=kHkqt gS rFkk ∠B = 90° gSA 2 sin A cos A dk eku gS............

(a) 1 (b) 1 2

(c) 1 2 (d) 2

14- ;fn ( ) 2 2

2 2 sin 20 sin 70 sec 60

2 cos 69 cos 21 K ° + ° °

= ° + ° rks k dk eku gS............

(a) 1 (b) 2 (c) 3 (d) 4

15- ∆ABC ~ ∆PRT rFkk ∠C = ∠R = 90°, ;fn 3 5

AC AB

= rks sin T = ............

(a) 3 5 (b) 5

3

(c) 4 5 (d) 5

4

y?kq mÙkjh; iz'u

16- ∆PQR esa ∠Q = 90° rFkk sin R = 3 5 gS cos P dk eku fyf[k,A

17- ;fn A rFkk B U;wu dks.k gS rFkk sin A = cos B rks A + B dk eku fyf[k,A 18- ;fn 4 cot θ = 3 rks tan θ + cot θ dk eku crkb,A 19- cot 2 30° + sec 2 45° dk eku crkb,A 20- som (90° – θ) cost θ + cost (90° – θ) sin θ dk eku crkb,A

22- ;fn 1 – tan 2 θ = 2 3 rks θ dk eku crkb,A

23- 2 cpsec 2 θ + 2 sec 2 θ – 10 dk eku crkb, ;fn θ = 45°.

43 Math (Hindi) ­X

24- ;fn θ rFkk φ iwjd dks.k gS rks cosec θ sec φ – cot θ tan φ dk eku crkb,A 25- ;fn tan (3x – 15) = 1 rks x dk eku crkb,A

26- ;fn 8 cot θ – 15 = 0 rks 1 sin cos

θ θ

+ dk eku D;k gksxk\

nh?kZ mÙkjh; iz'u 27- ljy dhft, %

tan 2 60° + 4 cos 2 45° + 3 (sec 2 30° + cos 2 90°)

28- eku Kkr dhft, %

( ) 2 2 4sin65 13 cos 53 . cos 37 . 5cos25 5 7sec 32 7 cot 58

ec ° ° −

° −

29- fl¼ dhft, % cosec 4 θ – cosec 2 θ = cot 2 θ + cot 4 θ.

30- ;fn sin θ + = 1 rks cos 2 θ + cos 4 θ dk eku crkb,A 31- ;fn sin θ = cos (θ – 36°), 2 θ rFkk θ – 26° U;wu dks.k gS rks dk eki Kkr dhft,A

32- ;fn sin (3x + 2y) = 1 rFkk cos (3x – 2y) = 3 2 , 0 ≤ (3x + 2y) ≤ 90° rks x rFkk

y dk eku Kkr dhft,A 33- ;fn sin (A + B) = sin A cos B + cos A sin B rks Kkr dhft, %

(a) sin 75°

(b) cos 15°

34- fl¼ dhft, % cos cos cos 45 1 tan 1 cot

A A A A A A

+ = ≠ ° − −

35- fl¼ dhft, % sec 1 sec 1 2cos sec 1 sec 1

ec θ θ

θ θ θ

− + + =

+ −

44 Math (Hindi) ­X

36- eku Kkr dhft, % sin 2 5° + sin 2 10° + sin 2 15° + ..... + sin 2 85°

37- fl¼ dhft, % tan sec 1 cos . tan sec 1 1 sin 0

θ θ θ θ θ θ

+ − =

− + −

38- ;fn 2 sin (3x – 15) = 3 rks sin 2 (2x + 10) + tan 2 (x + 5) dk eku Kkr dhft,A 39- sin 60° dk eku T;ksfrrh; fof/ ls Kkr dhft,A

40- ;fn p = tan θ + sec θ rks 1 p p

+ dk eku Kkr dhft,A

41- ledks.k f=kHkqt ∆OPQ, esa ∠P = 90°| ;fn OP = 7 lseh- rFkk ∠Q = α ;fn sec

(90 – α) – tan (90 – α) = 1 7 rks OQ – PQ dk eku Kkr dhft,A

42- ;fn sin α = a sin β rFkk tan α = b tan β rks fl¼ dhft, fd 2

2 2

1 cos 1

a b

α −

= −

43- ;fn θ ,d U;wu dks.k gS rFkk 5 sin 2 θ + cos 2 θ = 4 rks θ dk eku Kkr dhft,A

44- ∆ABC ,d U;wu dks.k f=kHkqt gSA ;fn sin (A + B – C) = 1 2 rFkk cos (B + C –

A) = 1 2 rks dks.kksa A, B rFkk C dk eku Kkr dhft,A

45- ;fn A, B dks.k C rFkk f=kHkqt ABC ds var% dks.k gS rks fn[kkb, fd

sin cos cos sin 1. 2 2 2 2

B C A B C A + + + =

45 Math (Hindi) ­X

mÙkjekyk 1- b 2- a 3- c 4- a 5- d 6- b 7- a 8- b 9- a 10- b 11- d 12- d 13- a 14- d

15- a 16- 3 cos 5

P =

17- 90° 18- 25 12

19- 5 20- 1

21- 5 4 22- 30°

23- 0 24- 1

25- x = 20 26- 5 3

27- 9 28- 3 7

31- 42° 32- x = 20, y = 15

33- 3 1 3 1 , 2 2 2 2

+ + take A = 45°, B = 30° 34- –

35- – 36- 17 2

37- – 38- 13 12

39- – 40- 2 sec θ

41- 1 42- – 43- 60° 44- ∠A = 67.5°,∠B = 37.5°, ∠C = 75°

46 Math (Hindi) ­X

vè;k;&6

lkaf[;dh

egÙoiw.kZ fcanq % 1- oxhZd`r vk¡dM+ksa dk ekè; fuEufyf[kr fof/;ksa ls Kkr fd;k tk ldrk gSA

i) izR;k fof/ . fixi x fi

Σ =

Σ

ii) dfYir ekè; fof/ , fidi a x fi

Σ = +

Σ tgk¡ d i = x i – a.

iii) ix fopyu fof/ fiui a h x fi

Σ = + ×

Σ tgk¡ u i = i x a h −

2- oxhZd`r v¡dM+ksa dk cgqyd fuEufyf[kr lw=k kjk Kkr fd;k tk ldrk gS %

cgqyd = l + 1 2

1 0 2 2 f f h

f f f −

× − −

f 1 = cgqyd oxZ dh fuEu lhek f 0 = cgqyd oxZ dh ckjEckjrk f 2 = cgqyd oxZ ls Bhd igys oxZ dh ckjEckjrk h = cgqyd oxZ ls Bhd ckn esa vkus okys oxZ dh ckjEckjrk cgqyd oxZ dh eki

3- oxhZd`r vk¡dM+ksa dh efè;dk fuEufyf[kr lw=k kjk Kkr fd;k tk ldrk gSA

ekè;d = l + / 2 n Cf h

f −

×

ekè;d oxZ dh fuEu lhek n = izs"k.kksa dh la[;k Cf = ekè;d oxZ ls Bhd igys oxZ dh lap;h ckjEckjrk

47 Math (Hindi) ­X

f = ekè;d oxZ dh ckjEckjrk h = ekè;d oxZ dh eki

4- cgqyd ¾ ekfè;d & 2 ekè; 5- lap;h ckjEckjrk oØ ;k rksj.k] rksj.k lap;h ckjEckjrk dk vkys[k gS%

d) ^^de izdkj rksj.k**

& ,d lap;h ckjEckjle lkj.kh cuk,a

& oxZ varjky dh mQijh lhek x­vk ij vafdr dhft,

[k) ^^vf/d izdkj rksj.k**

& vojksgh Øe esa lap;h ckjEckjrk lkj.kh cuk,aA

& oxZ varjky dh fuEu lhek x­vk ij vafdr dhft,A

x) ekfè;dk izkIr djus gsrq %

& de izdkj o vf/d izdkj ds rksj.kksa dk izfrPNsnh fcUnq fu/kZfjr dhft,A

& bl fcUnq ls x­vk ij yEc [khafp,A

& ;g yEc x­vk dks ftl fcUnq ij izfrPNsn djsxk] ogh ekfè;dk dk eku gksxkA

cgq oSdfYid iz'u

1- izFke 10 izkd`r la[;kvksa dk ekè; gSA (a) 5 (b) 6 (c) 5.5 (d) 6.5

2- ;fn 4] 6] 8] 10] x, 14 vkSj 16 dk ekè; 10 gS] rks 'x' dk eku gSA (a) 11 (b) 12 (c) 13 (d) 9

3- x, x + 1, x + 2, x + 3 x + 4, x + 5 vkSj x + 6 dk ekè; gSA (a) x (b) x + 3 (c) x + 4 (d) 3

48 Math (Hindi) ­X

4- 2, 3, 2, 5, 6, 9, 10, 12, 16, 18 vkSj 20 dh ekfè;dk gSA (a) 9 (b) 20 (c) 10 (d) 9.5

5- 2, 3, 6, 0, 1, 4, 8, 2, 5 dh ekfè;dk gSA (a) 1 (b) 3 (c) 4 (d) 92

6- 1, 0, 2, 2, 3, 1, 4, 5, 1, 0 dk cgqyd gSA (a) 5 (b) 0 (c) 1 (d) 2

7- ;fn 2, 3, 5, 4, 2, 6, 3, 5, 5, 2 vkSj x dk cgqyd 2 gS] rks 'x' dk eku gSA (a) 2 (b) 3 (c) 4 (d) 5

8- fuEufyf[kr caVu dk cgqyd oxZ gSA

oxZ&vUrjky 10­15 15­20 20­25 25­30 30­35 ckjEckjrk 4 7 12 8 2

(a) 30­35 (b) 20­25 (c) 25­30 (d) 15­20

9- ,d dkk esa vè;kid] fo|kfFkZ;ksa ls] xf.kr esa vadksa dk vkSlr iwNrk gS] fo|kFkhZ Kkr djsaxsA

(a) ekè; (b) ekfè;dk (c) cgqyd (d) ;ksxiQy

10- dsUnzh; izo`fÙk dh rhuksa ekiksa esa lEcU/ gSA (a) 3 ekè; ¾ cgqyd $ 2 ekfè;dk (b) 3 ekfè;dk ¾ cgqyd $ 2 ekè; (c) 3 cgqyd ekè; $ 2 ekfè;dk (d) ekfè;dk 3 ¾ cgqyd & 2 ekè;

49 Math (Hindi) ­X

11- oxZ 19.5 – 29.5 dk oxZ fpUg gSA (a) 10 (b) 49

(c) 24.5 (d) 25

12- ^ls de izdkj* rFkk ^ls vf/d izdkj* ds rksj.k ds izfrPNsn fcUnq dk Hkqt dsUnzh; izo`fÙk gh eki n'kkZrk gSA

(a) ekè; (b) ekfè;dk (c) cgqyd (d) buesa ls dksbZ ughaA

13- fuEufyf[kr caVu dk efè;dk oxZ gSA

oxZ&vUrjky 0­10 10­20 20­30 30­40 40­50 50­60 60­70 ckjEckjrk 4 4 8 10 12 8 4

(a) 20­30 (b) 40­50 (c) 30­40 (d) 50­60

14- 20 la[;kvksa dk ekè; 17 gS] ;fn izR;sd la[;k esa 3 tksM+k tk;s rks u;k ekè; gSA (a) 20 (b) 21

(c) 22 (d) 24

15- 5 la[;kvksa dk ekè; 18 gS] ;fn ,d la[;k fudky nh tk;s] rks ekè; 16 gks tkrk gSA rks fudkyh x;h la[;k gSA

(a) 23 (b) 24

(c) 25 (d) 26

16- izFke 5 vHkkT; la[;kvksa dk ekè; gSA (a) 5.5 (b) 5.6

(c) 5.7 (d) 5

17- la[;kvksa 3] 4] 6] 8] 14 dk ekè; ls fopyuksa dk ;ksxiQy gSA (a) 0 (b) 1

(c) 2 (d) 3

50 Math (Hindi) ­X

18- ;fn ekfè;dk ¾ 15 vkSj ekè; ¾ 16 rks cgqyd gSA (a) 10 (b) 11 (c) 12 (d) 13

19- 11 izsk.kksa dk ekè; 50 gS] ;fn igys N% izsk.kksa dk ekè; 49 gS vkSj vfUre N% izsk.kksa dk ekè; 52 gS] rks NBk izsk.k gSA

(a) 56 (b) 55 (c) 54 (d) 53

20- fuEu caVu dk ekè; 2-6 gS] 'x' dk eku gSA

oxZ&vUrjky 1 2 3 4 5 ckjEckjrk 4 5 x 1 2

(a) 24 (b) 3 (c) 8 (d) 13

y?kq mÙkjh; iz'u 21- 40 izsk.kksa dk ekè; 160 FkkA tk¡p djus ij Kkr gqvk fd 165 xyrh ls 125 fy[k

x;k Fkk] lgh ekè; Kkr dhft,A 22- ;fn vkjksgh Øe esa fy[ks izsk.kksa 24, 25, 26, x + 2, x + 3, 30, 31, 34 dh ekfè;dk

27-5 gS] rks 'x' dk eku Kkr dhft,A 23- fuEu vkadM+ksa dh ekfè;dk Kkr dhft,A

x : 10 12 14 16 18 20

f : 3 5 6 4 4 3

24- ;fn fuEu vk¡dM+ksa dk ekè; 7-5 gS] rks 'p' dk eku Kkr dhft,A

pj % 3 5 7 9 11 13 ckjEckjrk 6 8 15 p 8 4

25- fuEu caVu dk ekè; Kkr dhft,A x : 12 16 20 24 28 32

f : 5 7 8 5 3 2

51 Math (Hindi) ­X

26- fuEu caVu dk ekè; Kkr dhft,A

oxZ&vUrjky % 0­10 10­20 20­30 30­40 40­50

ckjEckjrk % 8 12 10 11 9

27- nh x;h lap;h ckjEckjrk lkj.kh ls oxZ 20&30 dh ckjEckjrk crkb;sA

vad fo|kfFkZ;ksa dh la[;k 10 ls de 1 20 ls de 14 30 ls de 36 40 ls de 59 50 ls de 60

28- 40 fo|kfFkZ;ksa kjk izkIr vad uhps ,d lap;h ckjEckjrk xzkiQ kjk n'kkZ, x;s gSaA fo|kfFkZ;ksa kjk izkIr ekè;d vad Kkr dhft,A

52 Math (Hindi) ­X

29- fuEu ^ls vf/d izdkj* rksj.k fdlh dkk ds 40 fo|kfFkZ;ksa dk Hkkj n'kkZrk gSA ekè;d oxZ dh fuEu lhek Kkr dhft,A

nh?kZ mÙkjh; iz'u 30- fuEu ckjEckjrk caVu dk ekè; 62-8 gS vkSj ckjEckjrkvksa dk ;ksx 50 gS x vkSj y

dk eku Kkr dhft,A oxZ&vUrjky 0­20 20­40 40­60 60­80 80­100 100­120 ckjEckjrk 5 x 10 y 7 8

31- fuEufyf[kr caVu fdlh iSQDVªh ds Jfedksa dh nSfud etnwjh n'kkZrk gSA nSfud etnwjh (#- esa) Jfedksa dh la[;k

300 ls vf/d 0 250 ls vf/d 12 200 ls vf/d 21 150 ls vf/d 44 100 ls vf/d 53 50 ls vf/d 59 0 ls vf/d 60

etnwj dh nSfud etnwjh dk ekè; Kkr dhft,A

53 Math (Hindi) ­X

32- fuEu ckjEckjrk caVu dh ekfè;dk 28-5 gS rFkk ckjEckjrkvksa dk ;ksx 60 gS] rks x vkSj y ds eku Kkr dhft,A

oxZ&vUrjky 0­10 10­20 20­30 30­40 40­50 50­60 ckjEckjrk 5 x 20 15 y 5

33- fuEu caVu dk ekè;] ekfè;dk vkSj cgqyd Kkr dhft,A

oxZ&vUrjky 0­10 10­20 20­30 30­40 40­50 50­60 60­70 ckjEckjrk 6 8 10 15 5 4 2

34- fuEufyf[kr caVu fdlh fo|ky; ds 100 fo|kfFkZ;ksa kjk izkIr vadksa dks n'kkZrk gSA cgqyd Kkr dhft,A

vad fo|kfFkZ;ksa dh la[;k 10 ls de 10 20 ls de 15 30 ls de 30 40 ls de 50 50 ls de 72 60 ls de 85 70 ls de 90 80 ls de 95 90 ls de 100

35- fuEu caVu ds fy, ^ls de* rFkk ^ls vf/d* izdkj ds rksj.k [khafp,A

vad % 0­10 10­20 20­30 30­40 40­50 50­60 60­70 70­80 80­90 90­100 fo|kfFkZ;ksa 5 6 8 10 15 9 8 7 7 5 dh la[;k %

xzkiQ ls ekfè;dk Hkh Kkr dhft,A 36- ,d losZk.k kjk fdlh fo|ky; dh dkk 10oha dh 50 Nk=kksa dh mQ¡pkbZ ukih x;h

vkSj vk¡dM+s izkIr fd, x;sA

mQ¡pkbZ (lseh- esa) % 120­130 130­140 140­150 150­160 160­170 Total fo|kfFkZ;ksa dh la[;k % 2 8 12 20 8 50 mijksDr vk¡dM+ksa dk ekè;] ekfè;dk vkSj cgqyd Kkr dhft,A

54 Math (Hindi) ­X

37- fue caVu dk cgqyd 65 gSA ;fn ckjEckjrkvksa dk ;ksx 50 gS rks x vkSj y dk eku Kkr dhft,A

oxZ 0­20 20­40 40­60 60­80 80­100 100­120 120­140 ckjEckjrk 6 8 x 12 6 y 3

38- ,d MkWDVjh tk¡p ds vUrxZr dkk x ds 35 fo|kfFkZ;ksa dk Hkkj fuEu izkIr gqvkA

Hkkjr (fdxzk- esa) 38­40 40­42 42­44 44­46 46­48 48­50 50­52 fo|kfFkZ;ksa dh la[;k 3 2 4 5 14 4 3 mijksDr vk¡dM+ksa dk ekè;] ekfè;dk vkSj cgqyd Kkr dhft,A

39- o"kZ 2008&2009 ds fy, ,d 'kgj dk fuokZg lwpdkad] lkIrkfgd izsk.k fuEu fn;k gS%

fuokZg lwpdkad 140­150 150­160 160­170 170­180 180­190 190­200 dqy lIrkgksa dh la[;k 5 10 20 9 6 2 52 lkIrkfgd ewY; dk fuokZg lwpdkad dh ekè; Kkr dhft,A

40- fuEu caVu dk cgqyd Kkr dhft,A

oxZ 3­6 6­9 9­12 12­15 15­18 18­21 21­24 ckjEckjrk 2 5 10 23 21 12 3

mÙkjekyk 1- c 2- b

3- b 4- a

5- b 6- c

7- a 8- b

9- a 10- b

11- c 12- b

13- c 14- a

15- d 16- b

55 Math (Hindi) ­X

17- a 18- d

19- a 20- c

21- 161 22- × = 25

23- 14.8 24- p = 3

25- 20 26- 25.2

29- 147.5 30- × = 8, y = 12

31- ` 182.50 32- × = 8, y = 7

33- ekè; = 30] ekfè;dk = 30-67] cgqyd = 33-33

34- 41.32

35- 47.3 (yxHkx) 36- ekè; = 149-8 lseh-] ekfè;dk = 151-5 lseh-] cgqyd = 154 lseh- 37- × = 10, y = 5

38- ekè; = 45-8] ekfè;dk = 46-5] cgqyd = 47-9

39- 166-3

40- 14-6

56 Math (Hindi) ­X

uewuk iz'ui=k dk izk:i

xf.kr SA­1

iz'u dk izdkj vad izfr iz'u iz'uksa dh dqy la[;k dqy vad

oSdfYid 1 10 10

y?kq mÙkj&I 2 8 16

y?kq mÙkj&II 3 10 30

nh?kZ mÙkj 4 6 24

dqy 34 80

BLUE PRINT xf.kr SA­1

fo"k; MCQ SA (I) SA (II) LA Total

la[;k i¼fr 2 (2) 1 (2) 2 (6) – 5 (10

cht xf.kr 2 (2) 2 (4) 2 (6) 2 (8) 8 (20)

T;kferh 1 (1) 2 (4) 2 (6) 1 (4) 6 (15)

f=kdks.kkfefr 4 (4) 1 (2) 2 (6) 2 (8) 9 (20)

lkaf[;dh 1 (1) 2 (4) 2 (6) 1 (4) 6 (15)

nh?kZ mÙkj 4 6 24 dqy 10 (10) 8 (16) 10 (30) 6 (24) 34 (80)

uksV % vad dks"Bd esa gSA

57 Math (Hindi) ­X

lSEiy iz'u i=k

xf.kr dkk X (SA­1)

Time Allowed : 3 to 3½ hours Maxmimum Marks : 80

lkekU; funsZ'k 1 lHkh iz'u vfuok;Z gSaA 2- bl iz'u i=k esa dqy 34 iz'u gSa tks pkj Hkkxksa A, B, C vkSj D esa foHkkftr gSaA

Hkkx&A esa 10 iz'u gS izR;sd iz'u 1 vad dk gSA Hkkx&B esa 8 iz'u gSa izR;sd iz'u 2 vad dk gSA Hkkx&C esa 10 iz'u gSa izR;sd iz'u 3 vad dk gSA Hkkx&D esa 6 iz'u gSa izR;sd iz'u 4 vad dk gSA

3- Hkkx A esa iz'u la[;k 1 ls 10 cgqfodYih; iz'u gSA pkj fodYiksa esa ls 1 fodYi lgh gSA

4- dqy iz'u&i=k esa fodYi ugha gSA ;|fi 2 vadksa okys ,d iz'u esa] 3 vadksa okys rhu iz'uksa esa rFkk 4 vadksa okys nks iz'uksa esa vkarfjd fodYi fn;s x;s gSaA ,sls lHkh iz'uksa esa vkidks dsoy ,d fodYi gh djuk gSA

5- dSydqysVj ds iz;ksx dh vuqefr ugha gSA Section ­ A

iz'u 1 ls 10 rd izR;sd ,d vad dk gSA 1- ∆ ABC esa ∠A = 90° rks tan B. tan C dk eku gS-------

(a) tan B (b) tan C (c) 0 (d) 1

2- ;wfDyM foHkktu izesf;ek kjk tc x = yq + r, tgk¡ x rFkk y ?kukRed iw.kk±d gksrs gSa] fuEu esa ls dkSu lk lgh gSA

(a) 0 ≤ r < y (b) 0 < r ≤ y (c) 0 < r < y (d) 0 ≤ r ≤ y

3- ;fn 2, 4, 6, 8, 10 x 14, 16 dk ekè; a gS] rks dk x eku gS (a) 10 (b) 11 (c) 12 (d) 13

58 Math (Hindi) ­X

4- y = ax 2 + fx + c dk vkys[k x­vk dks nks fHkUu fcUnqvksa ij dkVsxk ;fn (a) b 2 – 4ac = 0 (b) b 2 – 4ac > 0 (c) b 2 – 4ac < 0 (d) b 2 – 4ac ≥ 0

5- ;fn 3 sin3 ,0 90 2

θ θ = ° < < ° rks θ dk eku gS -------

(a) 0° (b) 20° (c) 30° (d) 60°

6- fuEu caVu dk cgqyd oxZ gS

oxZ&vUrjky 10­20 20­30 30­40 40­50 50­60 60­70 70­80 ckjEckjrk 3 5 8 10 9 4 3

(a) 70­80 (b) 40­50 (c) 50­60 (d) 30­40

7- ;fn 'kwU;dksa dk xq.ku 5 gS vkSj 'kwU;dksa dk ;ksx &2 gS rks cgqin gS % (a) x 2 – 5x – 2 (b) x 2 + 5x – 2 (c) x 2 + 2x – 5 (d) x 2 + 2x + 5

8- ekè;] ekfè;dk vkSj cgqyd esa lEcU/ gS& (a) cgqyd = 2 ekfè;dk – 3 ekè; (b) cgqyd = 2 ekfè;dk – ekè; (c) cgqyd = 3 ekfè;dk + 2 ekè; (d) cgqyd = 3 ekfè;dk – 2 ekè;

9- ml fcUnq ds funsZ'kkad tgk¡ y­vk vkSj 1 2 3 x y

+ = ls n'kkZbZ xbZ js[kk izfrPNsfnr

gksrh gS] bl izdkj gS& (a) (0, 2) (b) (2, 0) (c) (0, 3) (d) (3, 0)

10- ;fn x = tan 2° × tan 36° × tan 54° × tan 88° rks x dk eku gS-------- (a) 45° (b) 1 (c) 2 (d) 90°

59 Math (Hindi) ­X

Section ­ B iz'u 11 ls 18 rd izR;sd 2 vad dk gSA 11- ;wfDyM foHkktu vYxksfjFe dk dFku fyf[k, vr% 15 rFkk 21 dk e-l-o- Kkr

dhft,A 12- ekè; Kkr dhft, %

x : 12 16 20 24 28 32

f : 5 7 8 5 3 2

13- ∆ ABC esa fcUnq D Hkqtk AB dk eè; fcUnq gSA DE || BC Hkqtk AC dks fcUnq E

ij feyrh gSA fl¼ dhft, fd 1 . 2

AE AC =

vFkok

;fn ∆ ABC ~ ∆ DEF, BC = 5 lseh-] EF = 4 lseh- rFkk ks- (∆ ABC) = 75 lseh- 2 rks ks- (∆ DEF) Kkr dhft,A

14- kx 2 + 5x + k ds 'kwU;dksa dk ;ksx muds xq.ku ds leku gSA k dk eku D;k gksxk\ 15- fuEufyf[kr caVu ls ^de izdkj dk rksj.k* [khft,A

oxZ&vUrjky 0­10 10­20 20­30 30­40 40­50 50­60 ckjEckjrk 5 8 12 10 7 4

16- f=kdks.kferh; lkj.kh dk iz;ksx fd, fcuk Kkr dhft, 2 sin 54 3 1 tan 14 tan 30 tan 76 .

cos 36

+ ° ° °

17- p ds fdl eku ds fy, jSf[kd lehdj.kksa dk ;qXe y – 2x – 5 = 0

px = 2y dk dsoy ,d gy gSA

18- ;fn 1 sin ,0 90 6

θ θ = ° < < ° rks sec θ + tan θ dk eku crkb,A

60 Math (Hindi) ­X

Section ­ C iz'u 19 ls 28 rd izR;sd 3 vad dk gSA

19- xzkiQh; fof/ kjk Kkr djks fd fuEu jSf[kd lehdj.k ;qXe laxr gS ;fn gk¡ rks D;k ;g ;qXe vkfJr gS\ x – 2y = 4 and x – y = 3

20- fl¼ dhft, 1 5 2 3 −

,d vifjes; la[;k gSA

vFkok

fl¼ dhft, 5 2 − ,d vifjes; la[;k gSA

21- ∆ABC, ∠C = 90° Hkqtk CA rFkk Hkqtk CB ij fcUnq Øe'k% P rFkk Q fLFkr gSA

AQ 2 + BP 2 = AB 2 + PQ 2

22- layXu fp=k esa DE || BC. x dk eku Kkr dhft,A

vFkok

fp=k esa ABCD ,d leyc prqHkqZt gSA x dk eku Kkr dhft,A

61 Math (Hindi) ­X

23- x vkSj y ds fy, gy dhft, % 2 3 2

1 1 x x + =

− +

3 2 13 1 1 6 x x

+ = − +

x ≠ 1, y ≠ – 1

24- ;fn cgqin 2x 3 – 3x 2 + 6x – 2 ds nks xq.kt ( ) 2 x − vkSj ( ) 2 x + gks rks 'ks"k

nks xq.kt Kkr dhft,A 25- fl¼ dhft, fd (1 + tan A tan B) 2 + (tan A – tan B) 2 = sec 2 A sec 2 B ;fn

A vkSj B f=kHkqt ds U;wu dks.k gSA

vFkok

fl¼ djks (1 + cotθ – cosecθ ) (1 + tanθ + secθ ) = 2

26- layXu fp=k esa fl¼ dhft, fd 1 sin 10

θ =

27- Sin 30° dk eku T;kferh; fof/ kjk Kkr dhft,A 28- ,d ledks.k f=kHkqt ftldk yEc vk/kj ls nqxquk gS dh Hkqtkvksa ij ledks.kh;

f=kHkqt cuk;s x;sA n'kkZb, fd d.kZ ij cus f=kHkqt dk ks=kiQy 'ks"k nks Hkqtkvksa ij cus f=kHkqtksa ds ks=kiQh dk ;ksx gSA

62 Math (Hindi) ­X

Section ­ D iz'u 29 ls 34 rd izR;sd 4 vad dk gSA

29- n'kkZb, fd fdlh ?kukRed iw.kk±d dk oxZ 5q, 5q + 1, 5q + 4 :i dk gksrk gSA tgk¡ q ,d ?kukRed iw.kk±d gSA

30- rhu vadksa dh ,d la[;k ds lSadM+ksa ds LFkku ij vad bdkbZ ds LFkku ds vad dk rhu xquk gSA vadksa dk ;ksx 15 gSA ;fn vadksa dk LFkku cnys rks ubZ la[;k 396 de gks tkrh gSA ewy la[;k Kkr dhft,A

vFkok

pkj lnL;ksa dk ,d ifjokj jsy ds 3 fV;j dksp ls ;k=kk dj jgk gSA rhu lnL;ksa dk ifjokj 2 fV;j dksp ls ;k=kk dj jgk gSA nksuksa ifjokjksa dk dqy fdjk;k #- 5100 gSA ;fn igys ifjokj dk ,d lnL; de gks tk, vkSj nwljs ifjokj dk ,d lnL; vf/d gks tk, rks fdjk;k #- 300 vf/d yxrk gSA ,d tksM+s dk 2 fV;j dksp dk blh ;k=kk dk fdjk;k fdruk gksxkA

31- ;fn ∆ABC ,d U;wu dks.k f=kHkqt gS rFkk tan (A+B–C) = 1 rFkk sec (B+C–A = 2 rks ∠A, ∠B rFkk ∠C Kkr dhft,A

32- ;fn fuEu caVu dh ekfè;dk 28-5 gks] rks x vkSj y dk eku Kkr dhft,A

oxZ&vUrjky 0­10 10­20 20­30 30­40 40­50 50­60 Total ckjEckjrk 5 x 20 15 y 5 60

33- fuEu caVu dh cgqyd Kkr dhft, %

vad % 0­10 10­20 20­30 30­40 40­50 fo|kfFkZ;ksa dh la[;k % 5 15 20 8 2

34- fl¼ dhft, fd ,d ledks.k f=kHkqt ds d.kZ dk oxZ 'ks"k nks Hkqtkvksa ds oxks± ds ;ksx ds cjkcj gksrk gSA

63 Math (Hindi) ­X

mÙkjekyk 1- d 2- a

3- c 4- b

5- b 6- b

7- d 8- d

9- c 10- b

11- 3 12- 20

13- 38 lseh- 2 14- k = – 5

16- 2 3 3

+ 17- p ≠ 4

18- 7 35 19- gk¡] ugha

22- x 11, x = 8, ;k x = 9 23- x = 3, y = 2

24- (2x – 1) (x – 1) 26- –

30- 672 31- ∠A = 60°, ∠B = 52.5°, vFkok [` 800, ` 900] ` 1800 ∠C = 67.5°

32- x = 8, y 7 33- 22.9

34- –

64 Math (Hindi) ­X

lSEiy iz'u i=k

xf.kr dkk X (SA­1)

Time Allowed : 3 to 3½ hours Maxmimum Marks : 80

lkekU; funsZ'k 1 lHkh iz'u vfuok;Z gSaA 2- bl iz'u i=k esa dqy 34 iz'u gSa tks pkj Hkkxksa A, B, C vkSj D esa foHkkftr gSaA

Hkkx&A esa 10 iz'u gS izR;sd iz'u 1 vad dk gSA Hkkx&B esa 8 iz'u gSa izR;sd iz'u 2 vad dk gSA Hkkx&C esa 10 iz'u gSa izR;sd iz'u 3 vad dk gSA Hkkx&D esa 6 iz'u gSa izR;sd iz'u 4 vad dk gSA

3- iz'u la[;k 1 ls 10 cgqfodYih; iz'u gSA fn, x, pkj fodYiksa esa ls ,d lgh fodYi pqusaA

4- blesa dksbZ loksZifj fodYi ugha gSA ysfdu vkarfjd fodYi 1 iz'u 2 vadksa esa] 3 iz'u 3 vadksa esa vkSj 2 iz'u 4 vadksa esa fn, x, gSaA vki fn, x, fodYiksa esa ls ,d fodYi dk p;u djsaA

5- dSydqysVj ds iz;ksx dh vuqefr ugha gSA 6- bl iz'u&i=k dks i<+us ds fy, 15 feuV dk le; fn;k x;k gSA bl vof/ ds

nkSjku Nk=k dsoy iz'u&i=k dks i<+saxs vkSj os mÙkj&iqfLrdk ij dksbZ mÙkj ugha fy[ksaxsA

Section ­ A iz'u 1 ls 10 rd izR;sd ,d vad dk gSA 1- ;wfDyM foHkktu izesf;ek kjk tc a = bq + r, tgk¡ a, b ?kukRed iw.kk±d gS] fuEu

esa ls dkSu lk lR; gS% (a) 0 < r < b (b) 0 ≤ r < b (c) 0 < r ≤ b (d) 0 ≤ r ≤ b

65 Math (Hindi) ­X

2- vkd`fr 1 esa] fdlh cgqin dk xzkiQ fn[kk;k x;k gSA blesa 'kwU;dksa dh la[;k gS%

(a) 1 (b) 2 (c) 3 (d) 4

3- vkd`fr 2 esa ;fn DE || BC rks x dk eku

(a) 3 lseh- (b) 2 lseh- (c) 4 lseh- (d) 20

3 lseh-

4- ;fn sin (θ + 36°) = cos θ, tgk¡ θ vkSj θ + 36° nksuksa U;wu dks.k gksa rks θ dk eku Kkr dhft,%

(a) 36° (b) 54° (c) 27° (d) 90°

5- ;fn 3cos 2sin θ θ = rks 4sin 3 cos 2sin 6cos

θ θ θ θ

− +

dk eku gksxk%

(a) 1 8 (b) 1

3

(c) 1 2 (d) 1

4

66 Math (Hindi) ­X

6- ;fn vkd`fr 3 esa ;fn ∆ABC, B ij ledks.k gS vkSj 4 tan 3

A = ;fn AC =15

lseh- rks BC dh yEckbZ gS%

(a) 4 lseh- (b) 3 lseh- (c) 12 lseh- (d) 9 lseh-

7- 21 24 dk n'keyo izlkj] n'keyo ds fdrus LFkkuksa i'pkr lkar gSA

(a) 1 (b) 2 (c) 3 (d) 4

8- jSf[kd lehdj.k ;qXe x – 2y = 5 vkSj 2x – 4y = 10 ds (a) vusd gy gSa (b) dksbZ gy ugha gSA (c) ,d gy gS (d) nks gy gSaA

9- ;fn 15 tan cot 7

A B = = gks rks A + B dk eku % (a) 'kwU; (b) 90° (c) < 90° (d) > 90°

10- fdUgha 50 izsk.kksa ds fy, ^ls de izdkj dk rksj.k* o ^ls vf/d izdkj dk rksj.k* (38-5] 34) ij izfrPNsn djrs gSaA ekè;d gSA

(a) 38.5 (b) 34 (c) 50 (d) 4.5

67 Math (Hindi) ­X

Section ­ B iz'u 11 ls 18 rd izR;sd 2 vad dk gSA 11- D;k 7×11×12+11 ,d Hkkt; la[;k gS\ dkj.k nhft,A 12- D;k fdlh p(x) cgqin (2x – 5) dks ls Hkkx djus ij (x + 2) 'ks"kiQy gks ldrk

gS\ dkj.k Li"V dhft,A 13- vkd`fr 4 esa ABCD ,d vk;r gSA x vkSj y dk eku Kkr dhft,A

14- ;fn sin (A + B) = 1 vkSj cos (A – B) = 1 0° ≤ A + B ≤ 90°, A vkSj B Kkr dhft,A

vFkok

;fn 7 cot 8

θ = gS (1 sin )(1 sin ) (1 cos )(1 cos )

θ θ θ θ

+ − + − rFkk dk eku Kkr dhft,A

15- ABCD ,d leyEc prqHkqZt gS ftlesa AB || DC gSA leyEc prqHkqZt ds fod.kZ

,d nwljs dks fcanq ij izfrPNsr djrs gSaA fl¼ dhft, fd : AD CO BO DO

=

16- vkd`fr 5 esa] 90 , 10 S PQ = ° = lseh-] QS = 6 lseh- vkSj RQ = 9 lseh- gSA Hkqtk Kkr dhft,A

68 Math (Hindi) ­X

17- fuEu vk¡dM+s fdlh iQSDVjh ds 50 dkjhxjksa dh mQ¡pkbZ (lseh- esa) n'kkZrs gSa%

mQ¡pkbZ (lseh- esa) %150­155 155­160 160­165 165­170 170­175 175­180 dkjhxjksa dh la[;k % 8 14 20 4 3 1

18- fuEufyf[kr (forj.k) vk¡dM+ksa dk cgqyd Kkr djks %

mQ¡pkbZ (lseh- esa) % 30­40 40­50 50­60 60­70 70­80 ikS/ksa dh la[;k % 4 3 8 11 8

Section ­ C iz'u 19 ls 28 rd izR;sd 3 vad dk gSA

19- n'kkZb, fd fdlh /ukRed iw.kk±d dk oxZ 3q ;k 3q + 1 ds :i dk gksrk gS tgk¡ q dksbZ iw.kk±d gSA

20- fl¼ dhft, 2 25 ,d vifjes; la[;k gSA

vFkok

fl¼ dhft, ( ) 5 3 + ,d vifjes; la[;k gSA

21- ,d O;fDr viuh ukSdjh ,d fu;r ekfld vkenuh vkSj fLFkj okf"kZd&o`f¼ ls 'kq: djrk gSA ;fn mldh vkenuh pkj o"kZ i'pkr~ 4500 #- vkSj nl o"kZ i'pkr~ 5400 #- gks rks mldh izkjafHkd ekfld vkenuh vkSj okf"kZd o`f¼ Kkr dhft,A

vFkok ik¡p o"kZ i'pkr~ lqnkek dh vk;q mlds iq=k dh vk;q dh frxquh gksxhA ik¡p o"kZ iwoZ lqnkek dh vk;q iq=k dh vk;q dh lkr xquk FkhA mudh orZeku vk;q Kkr dhft,A

22- ; fn α, β cgqin 3x 2 + 5x – 2 ds 'kwU;d gksa rks og f?kkr cgqin Kkr dhft, ftlds 'kwU;d 2α vkSj 2β gksaA

23- fl¼ dhft, fd cot _ cos cos 1 : cot cos cos 1

A A ec A A A ec A

− =

+ +

24- ;fn cos sin 2 sin θ θ θ − = rks fl¼ dhft,A cos sin 2 cos θ θ θ + =

69 Math (Hindi) ­X

25- vkd`fr esa AD⊥BC fl¼ dhft, fd AB 2 + CD 2 = BD 2 + AC 2 gSA

26- fl¼ dhft, fd ,d oxZ dh fdlh Hkqtk ij cuk, x, leckgq f=kHkqt dk ks=kiQy mlh oxZ ds ,d fod.kZ ij cuk, c, leckgq f=kHkqt ds ks=kiQy dk vk/k gksrk gSA

27- ix fopyu fof/ kjk fuEu ckjackjrk caVu dk ekè; Kkr dhft, %

oxZ varjky 25­30 30­35 35­40 40­45 45­50 ckjackjrk 7 14 22 16 11

;k

fuEu ckjackjrk lkj.kh dk ekè; 47 gSA p dk eku Kkr dhft,A

oxZ varjky 0­20 20­30 40­60 60­80 80­100 ckjackjrk 8 15 20 p 5

28- fuEu vk¡dM+ksa dk ekè;d Kkr djksA

oxZ varjky 40­45 45­50 50­55 55­60 60­65 65­70 ckjackjrk 2 3 8 6 6 5

Section ­ D iz'u 29 ls 34 rd izR;sd 4 vad dk gSA

29- cgiqn 2x 4 + 7x 3 – 19x 2 – 14x + 30 ds lHkh 'kwU;d Kkr dhft,A ;fn blds nks 'kwU;d 2 vkSj 2 − gks%

30- fl¼ dhft, fd ,d ledks.k f=kHkqt esa d.kZ dk oxZ vU; nks Hkqtkvksa ds oxks± ds ;ksxiQy ds cjkcj gksrk gS%

vFkok fl¼ dhft, fd nks le:i f=kHkqtksa ds ks=kiQyksa dk vuqikr fdUgha nks laxr Hkqtkvksa ds oxks± ds vuqikr ds leku gksrk gSA

70 Math (Hindi) ­X

31- fl ¼ d hft , cos 8 θ – sin 8 θ = (cos 2 θ – sin 2 θ) (1 – 2sin 2 θ cos 2 θ): vFkok

eku Kkr dhft, % 2 2 2(cot 27 sec 63 ) tan(90 ) cot sec(90 ) cos

cot26 cot 41 cot45 cot 49 cot64 ec θ θ θ θ

° − ° ° − − − +

° ° ° ° °

32- fl¼ dhft, fd% cos sin sin cos 1 tan 1 cot

A A A A A A

+ = + − −

33- xzkiQ cukb, % 4x – y = 4; 4x + y = 12 (a) xzkiQ kjk lehdj.k fudk; dk gy Kkr dhft,A (b) nksuksa js[kkvksa vkSj x­vk ds chp cus f=kHkqtkdkj ks=k dks Nk;kafdr dhft,A

34- fuEufyf[kr caVu fdlh fo|ky; ds 100 Nk=kksa dh mQ¡pkbZ n'kkZrk gS%

mQ¡pkbZ (lseh esa) 120­130 130­140 140­150 150­160 160­170 170­180 Nk=kksa dh la[;k 12 16 30 20 14 8 bl caVu dks ^ls vf/d izdkj ds* caVu esa cnfy, vkSj fiQj mldk rksj.k [khafp,A

mÙkjekyk 1- b 2- a

3- d 4- c

5- b 6- c

7- c 8- a

9- b 10- a

11- gk¡ 12- ugha

13- x = 19, y = 3 vFkok 49 64 14- A = 45°, B = 45°

16- 17 lseh- 18- 65

21- ` 3900, ` 150 vFkok 40 o"kZ] 10 o"kZ 22- 3x 2 + 10x – 8

27- 38.3 ;k p = 12 28- 58.8

29- 3 2, 2, 5, 2

− − 33- x = 2, y = 4

71 Math (Hindi) ­X

fo"k; lwph S.A.­2 Øe la- vè;k; i`"B la-

1- f?kkr lehdj.k 63&68

2- lekUrj Jsf<+;ka 69&74

3- funsZ'kkad T;kfefr 75&81

4- f=kdks.kfefr ds dqN vuqiz;ksx 82&88

5- o`Ùk 89&101

6- jpuk,a 102&103

7- ks=kfefr 104&117

8- izkf;drk 118&123

lSEiy isijA (gy lfgr) 125&139

lSEiy isij­II 141&147

72 Math (Hindi) ­X

vè;k;&1

f?kkr lehdj.k

egÙoiw.kZ fcanq % 1- lehdj.k ax 2 + bx + c = 0, a ≠ 0 f?kkr lehdj.k dk O;kid :i gS] ftlesa

a, b, c okLrfod la[;k,¡ gSaA 2- ,d okLrfod la[;k α, f?kkr lehdj.k ax 2 + bx + c = 0, a ≠ 0 dk ewy gh tk

ldrh gS] ;fn aα 2 + bα + c = 0 f?kkr cgqin ax 2 + bx + c ds 'kwU;d rFkk f?kkr lehdj.k ds ewy ,d gh gksrs gSaA

3- ;fn ge ax 2 + bx + c = 0, a ≠ 0ds xq.ku[k.M nks jSf[kd xq.ku[k.Mksa ds xq.ku iQy ds :i esa O;Dr dj lds] rks f?kkr lehdj.k ds ewy] xq.ku[k.Mksa dks 'kwU; ds cjkcj ysdj fudkys tk ldrs gSaA

4- f?kkr lehdj.k ax 2 + bx + c = 0, a ≠ 0 ds ewy fuEu gS 2 4 ,

2 b b ac

a − + −

tgk¡ b 2 – 4ac ≥ 0

5- ,d f?kkr lehdj.k ax 2 + bx + c = 0, a ≠ 0 esa (a) nks vleku vkSj okLrfod ewy gksaxs] ;fn b 2 – 4ac ≥ 0. (b) nks leku vkSj okLrfod ewy gksaxs] ;fn b 2 – 4ac ≥ 0. (c) nks vokLrfod ewy gksaxs] ;fn b 2 – 4ac ≥ 0.

6- ,d f?kkr lehdj.k ax 2 + bx + c = 0, a ≠ 0 dks iw.kZ oxZ fof/ ls Hkh gy fd;k tk ldrk gS& (i) a 2 + 2ab + b 2 = (a + b) 2

(ii) a 2 + 2ab + b 2 = (a – b) 2

7- f?kkr lehdj.k ax 2 + bx + c = 0, a ≠ 0 dk fofoDrj D = b 2 – 4ac gksrk gSA

73 Math (Hindi) ­X

cgq oSdfYid iz'u 1- f?kkr lehdj.k dk O;kid :i gSA

(a) ax 2 + bx + c (b) ax 2 + bx + c = 0 (c) ax + b (d) ax + b = 0

2- ,d f?kkr lehdj.k ds gy gksrs gSa& (a) 0 (b) 1 (c) 2 (d) 3

3- ;fn x 2 + 3x a = 0, dk ,d ewy 1 gS] rks a dh eku gSA (a) 2 (b) – 2 (c) 2 (d) – 4

4- f?kkr lehdj.k ax 2 + bx + c = 0 a ≠ 0, dk fofoDrj gS (a) 2 4 b ac − (b) 2 4 b ac +

(c) 2 4 b ac − (d) 2 4 b ac +

5- dkSu f?kkr lehdj.k gS\

(a) 1 2 x x

+ = (b) 2 2 1 ( 3) x x + = +

(c) ( 2) x x + (d) 1 x x

+

6- ;fn ,d f?kkr lehdj.k ds ewy 2 vkSj 3 gS] rks lehdj.k gSaA (a) x 2 + 5x + 6 = 0 (b) x 2 + 5x – 6 = 0 (c) x 2 – 5x – 6 = 0 (d) x 2 – 5x + 6 = 0

7- lehdj.k x 2 – 3x + 2 = 0 ds ewy gSa (a) 1, –2 (b) –1, 2 (c) –1, –2 (d) 1, 2

8- ;fn ,d f?kkr lehdj.k ds ewy cjkcj gS] rks fofoDrj gSA (a) 1 (b) 0 (c) 0 ls vf/d (d) 0 ls de

74 Math (Hindi) ­X

9- ;fn 2x 2 + kx + 1 = 0 dk ,d ewy 1 2

− gS] rks ‘k’ dk eku gSA

(a) 3 (b) –3 (c) 5 (d) –5

10- f?kkr lehdj.k 5x 2 – 6x + 7 = 0 ds ewyksa dk ;ksxiQy gSA

(a) 6 5 (b) 1

5

(c) 5 6

− (d) 1 5

−

11- f?kkr lehdj.k 2x 2 + 5x – 7 = 0 ds ewY;ksa dk xq.kuiQy gSA

(a) 5 2 (b) 7

2 −

(c) 5 2

− (d) 7 2

12- ;fn f?kkr lehdj.k 2x 2 + kx + 2 = 0 ds ewy cjkcj gS rks ‘k’ dk eku gSA (a) 4 (b) –4 (c) ≠ 4 (d) ≠ 16

13- ;fn 4x 2 + 3px + 9 = 0 ds ewy okLrfod vkSj fHkUu gS rks ‘p’ dk eku gSA (a) p ≥ – 4 ;k p ≤ 4 (b) p < – 4 ;k p > 4 (c) p ≤ – 4 ;k p ≤ 4 (d) p ≤ – 4 ;k p ≥ 4

14- ;fn ,d f?kkr lehdj.k ds ewyksa dk ;ksxiQy rFkk xq.kuiQy Øe'k% 7 2

− rFkk 5 2

rks lehdj.k gSA (a) 2x 2 + 7x + 5 = 0 (b) 2x 2 – 7x + 5 = 0 (c) 2x 2 – 7x – 5 = 0 (d) 2x 2 + 7x – 5 = 0

15- lehdj.k 3x 2 + 7x + 4 = 0 ds ewy gSA (a) ifjes; (b) vifjes; (c) /ukRed iw.kk±d (d) ½.kkRed iw.kk±d

75 Math (Hindi) ­X

y?kq mÙkjh; iz'u 16- ;fn f?kkr lehdj.k x 2 + 7x + k = 0 dk ,d ewy &2 gS] rks k dk eku rFkk nwljk

ewy Kkr dhft,A 17- ‘k’ ds fdl eku ds fy, f?kkr lehdj.k 2x 2 + kx + 3 = 0 ds ewy cjkcj gksaxs\ 18- ‘p’ ds fdl eku ds fy, f?kkr lehdj.k 3x 2 + px + 3 = 0 ds ewy okLrfod

gksaxs\ 19- nks Øekxr fo"k; la[;kvksa dk xq.kuiQy 63 gSA bl dFku dks f?kkr lehdj.k ds

O;Dr dhft,A

20- lehdj.k 1 1 4 , 0. 4

x x x

+ = ≠ ds ewy Kkr dhft,A

21- lehdj.k 2 2 7 5 2 0. x x + + = ds ewy Kkr dhft,A

22- 51 ds nks Hkkxksa esa bl izdkj foHkkftr dhft, fd mudk xq.kuiQy 278 gksA 23- ‘k’ dk eku Kkr dhft,] ;fn lehdj.k (k – 12) x 2 + 2 (k – 12) x + 2 = 0 k ≠12,

ds ewy cjkcj gSaA 24- iw.kZ oxZ cukus dh fof/ kjk gy dhft,A

(a) 2x 2 – 5x + 3 = 0 (b) 3x 2 + 5x + 1 = 0

25- lehdj.k 1 1 3, 2, 0. 2

x x x x

− = ≠ − ≠ +

ds ewy Kkr dhft,A

26- nks Øekxr fo"ke /u iw.kk±d Kkr dhft,] ftuds oxks± dk ;ksxiQy 394 gSA 27- ;fn f?kkr lehdj.k (b – c) x 2 + (c – a) x + (a – b) = 0, ds ewy cjkcj gS] rks

fl¼ dhft,% 2b = a + c.

28- fuEufyf[kr f?kkr lehdj.kksa ds ewyksa dh izd`fr Kkr dhft,A ;fn ewy okLrfod gS] rks mUgsa Kkr dhft,A (a) 5x 2 – 3x + 2 = 0 (b) 2x 2 + 9x + 9 = 0

29- nks la[;kvksa dk ;ksxiQy 15 gS] ;fn buds O;qRØeksa dk ;ksxiQy 3 10 gS] rks la[;k,a

Kkr dhft,A

76 Math (Hindi) ­X

fuEufyf[kr f?kkr lehdj.kksa dks gy dhft,A 30- x 2 – 8x + 16 = 0

31- a 2 x 2 + (a 2 – b 2 ) x – b 2 = 0

32- 2 3 5 2 3 0. a x x + − =

33- 2x 2 + kx + 3 = 0

34- 1 3 10 , 2, 4. 2 4 3

x x x x x x

− − + = ≠ ≠

− −

35- 1 1 11 , 4, 7. 4 7 30

x x x x

− = ≠ ≠ − −

36- 2 3 2 5 5 0. x x + − =

37- 1 1 1 0, 0, 0, ( ). a b x x a b a b x

+ + = ≠ ≠ ≠ ≠ − +

38- ,d nks vadksa okyh la[;k ds vadksa dk xq.kuiQy 35 gSA ;fn la[;k esa 18 tksM+ fn;k tk;s rks vad LFkku cny ysrs gSaA la[;k Kkr dhft,A

39- nks la[;kvksa dk ;ksxiQy 27 gS vkSj mudk xq.kuiQy 182 gS] la[;k,a Kkr dhft,A 40- ,d eksVjcksV ftldh fLFkj ty esa pky 9 feeh-@?kaVk gSA 12 fd-eh- /kjk ds

izfrdwy tkus esa vkSj ogh nwjh /kjk ds vuqdwy tkus esa 3 ?k.Vs ysrh gSA /kjk dh pky Kkr dhft,A

41- ,d jsyxkM+h 360 fdeh- dh nwjh leku pky ls r; djrh gSA ;fn mldh pky 5 fdeh-@?kaVk vf/d gks rks ml nwjh dks r; djus esa ,d ?k.Vk de ysrh gSA xkM+h dh pky Kkr dhft,A

42- ,d ledks.k f=kHkqt dk d.kZ lcls NksVh Hkqtk ds nksxqus ls 6 lseh- vf/d gSA ;fn rhljh Hkqtk d.kZ ls 2 lseh- NksVh gS] rks f=kHkqt dh Hkqtk,a Kkr dhft,A

43- phuh ds ewY; esa 2 #- izfr fd-xzk- deh gksus ls vfurk 224 #- esa 2 fd-xzk- phuh vf/d [kjhn ldrh gSA phuh dk izkjafHkd ewY; Kkr dhft,A

44- 9000 #i;s dks dqN fo|kfFkZ;ksa esa cjkcj ckaVuk gSA ;fn 20 fo|kFkhZ vf/d gks tk;s rks izR;sd fo|kFkhZ dks 160 # de feyrs gSaA fo|kfFkZ;ksa dh ewy la[;k Kkr dhft,A

77 Math (Hindi) ­X

45- ,d gokbZ tgkt 1200 fd-eh- dh nwjh dks r; djus esa ,d ?k.Vk de ysrk gS] ;fn mldh pky 100 fdeh- izfr ?k.Vk c<+kbZ tk;sA gokbZ tgkt dh izkjafHkd Pkky Kkr dhft,A

46- lkr o"kZ igys vfnfr dh vk;q] lkFkZd dh vk;q ds oxZ dh ik¡p xquk FkhA rhu o"kZ

ckn lkFkZd dh vk;q] vfnfr dh vk;q dh 2 5 gksxhA mudh orZeku vk;q Kkr

dhft,A 47- nks o"kZ igys ,d vkneh dh vk;q] mlds iq=k dh vk;q ds oxZ dh rhu xquh FkhA

rhu o"kZ ckn mldh vk;q] iq=k dh vk;q dh pkj xquh gksxhA mldh orZeku vk;q Kkr dhft,A

48- Jhyadk ds fo#¼ ,d fØdsV eSp esa lgokx us] vfer feJk kjk fy, x;s fodVksa ds nks xqus ls ,d de fodsV fy,A ;fn bu nksuksa kjk fy, x;s fodVksa dh la[;k dk xq.kuiQy 15 gks] rks izR;sd kjk fy, x, fodVksa dh la[;k Kkr dhft,A

49- ,d 9 ehVj mQaps LrEHk ij ,d eksj cSBk gSA LrEHk ds vk/kj ls 27 ehVj nwj ,d fcUnq ls ,d lk¡i vius fcy dh vksj vk jgk gS] tks LrEHk ds vk/kj ij gSA lk¡i dks ns[kdj] eksj ml ij >iVrk gSA ;fn nksuksa dh pky leku gksa rks fcy ls fdruh nwjh ij eksj lk¡i dks idM+ ysxk\

50- ,d fHkUu dk va'k] mlds gj ls 1 de gSA ;fn va'k rFkk gj esa rhu tksM+k tk;s

rks fHkUu 3 28 c<+ tkrh gSA fHkUu Kkr dhft,A

mÙkjekyk 1- b 2- c

3- a 4- c

5- a 6- d

7- d 8- b

9- a 10- a

11- b 12- c

13- b 14- a

78 Math (Hindi) ­X

15- a 16- k = 10, nwljk ewy = – 5

17- 2 6 + 18- p ≥ 6 ;k p ≤ – 6

19- x 2 + 2x – 63 = 0 20- 1 4, 4

21- 5 , 2 2

− − 22- 9, 42

23- k = 14 24- (a) 3 , 1 2 (b) 5 13 5 13 , .

6 6 − + − −

25- 3 3 3 3 , . 3 3

− + − − 26- 13. 15

27- ladsr% cjkcj ewyksa ds fy, 28- (a) okLrfod ewy ugha gSA D = 0. (b) ewy okLrfod gS] 3 3, .

2

29- 5, 10 30- 4, 4

31- 2

2 1, b a

− 32- 3 2 , 4 3

−

33- , c b b a

− 34- 5 5, 2

35- 1, 2 36- 5 , 5 3

−

37- –a, –b. 38- 57

39- 13, 14 40- 3 fdeh-@?k.Vk 41- 40 fdeh-@?k.Vk 42- 26 lseh- 24 lseh- 10 lseh- 43- #- 16 44- 25 fo|kFkhZ 45- 300 fdeh-@?k.Vk 46- vfnfr dh vk;q = 27 o"kZ]

lkFkZd dh vk;q = 9 y`

48- lgokx] 5 fodsV] 47- 29 o"kZ] 5 o"kZ vfer feJk] 3 fodsV

49- 12 ehVj 50- 3 4

79 Math (Hindi) ­X

vè;k;&2

lekUrj Jsf<+;k¡

egÙoiw.kZ fcanq % 1- vuqØe&la[;kvksa dk ,d leqPp;] tks fdlh fu;e kjk fuf'pr Øe esa O;ofLFkr

gksrk gS] vuqØe dgykrk gSA 2- Js<+h og vuqØe ftlds in fuf'pr izfrekuksa (Pattern) dk ikyu djrs gSa] Js<+h

dgykrk gSA 3- lekUrj Js<+h&,d vuqØe] ftlesa igys in dks NksM+dj izR;sd dk vxys in ls

fuf'pr vUrj gksrk gS] lekUrj vuqØe ;k lekUrj Js<+h (l-Js-) dgykrh gSA ,d lekUrj Js<+h dk O;kid :i a, a + d, a +2d, ..... gS (a : izFke in d : lkoZ vUrj)

4- O;kid in& ;fn ,d l-Js dk izFkein ‘a’ rFkk lkoZ vUrj ‘d’ gks rks nok¡ in (O;kid in fuEufyf[kr lw=k kjk izkIr gksrk gS& a n = a + (n – 1) d.

5- ,d lekUrj Js<+h ds n inksa dk ;ksxiQy ;fn ,d l-Js- dk izFke in ‘a’ rFkk lkoZ vUrj ‘d’ gks rks izFke n inksa dk ;ksxiQy fuEufyf[kr lw=k kjk izkIr gksrk gS

2 ( 1) 2 n n S a n d = + − ;fn ‘l’ ,d ifjfer lekUrj Js<+h dk vafre in gks rks lHkh

inksa dk ;ksxiQy 2 n n S a l = + ls izkIr gksrk gSA

6- (i) ;fn a n fn;k gS] rks lkoZ varj] d = a n – a n­1 .

(ii) ;fn s n fn;k gS rks nok¡ in a n = s n­1 ls izkIr gksrk gSA (iii) ;fn a, b, c, lekUrj Js<+h esa gS rks 2b = a + c.

(iv) ;fn ,d vuqØe esa n in gS] rks bldk vafre ls rok¡ in = izkjEHk (n – r + 1)ok¡ inA

80 Math (Hindi) ­X

cgq oSdfYid iz'u

1- l-Js- 9] 11] 13] 15] ------------------- dk vxyk in gSA (a) 17 (b) 18 (c) 19 (d) 20

2- ;fn ,d l-Js- dk nok¡ in 2n + 7, gS] rks l-Js- dk lkrok¡ in gS (a) 15 (b) 21 (c) 28 (d) 25

3- ;fn ,d l-Js- ds n inksa dk ;ksx n 2 + 3n gks] rks 15 inksa dk ;ksxiQy gS (a) 250 (b) 230 (c) 225 (d) 270

4- ;fn ,d l-Js- 4] 7] 10] ---------------- dk nok¡ in 82 gS] rks n dk eku gS (a) 29 (b) 27 (c) 30 (d) 26

5- ;fn a, b vkSj c l-Js- esa gS] rks (a) 2

b c a +

= (b) 2 a c b

+ =

(c) 2 a c c

+ = (d) b a c = +

6- l-Js- 3] 8] 13] ---------------- dk 12ok¡ in gS (a) 56 (b) 57 (c) 58 (d) 59

7- l-Js- 1 2 3 8 , 8 , 8 8 8 8 ------------- dk lkoZvUrj gS

(a) 1 8 (b) 1 1

8

(c) 1 8 8 (d) 1

81 Math (Hindi) ­X

8- l-Js- –5, –2, 1, ------------ dk ok¡ in gS (a) 3n + 5 (b) 8 – 3n (c) 8 n – 5 (d) 3n – 8

9- ;fn ,d l-Js- dk nok¡ in 5n – 3n gS] rks l-Js- dk lkoZvUrj gS (a) 2 (b) –3 (c) –2 (d) 3

10- ;fn 5, 2k – 3, 9 l-Js- esa gS] rks ‘k’ dk eku gS (a) 4 (b) 5 (c) 6 (d) –5

11- izFke 10 izkd`r la[;kvksa dk ;ksxiQy gS (a) 50 (b) 55 (c) 60 (d) 65

12- l-Js- 7] 11] 15 ------------- 147] dk vUr ls 9ok¡ in gS (a) 135 (b) 125 (c) 115 (d) 110

13- ;fn ,d l-Js- ds n inksa dk ;ksxiQy n 2 , gS] rks bldk nok¡ in gS (a) 2n – 1 (b) 2n + 1 (c) n 2 – 1 (d) 2n – 3

14- l-Js- esa rhu la[;kvksa dk ;ksxiQy 30 gSA ;fn lcls cM+h la[;k 13 gS] rks bldk lkoZ vUrj gS

(a) 4 (b) 3 (c) 2 (d) 5

15- ,d l-Js- ds NBs vkSj lkrosa inksa dk ;ksxiQy 39 gS vkSj lkoZvUrj 3 gS] rks l- Js- dk izFke in gS

(a) 2 (b) –3 (c) 4 (d) 3

82 Math (Hindi) ­X

nh?kZ mÙkjh; iz'u 16- D;k 2, 8, 18, 32, -------------- ,d l-Js- gS\ ;fn gk¡] rks blds vxys nks in

Kkr dhft,A 17- ,d l-Js- Kkr dhft,] ftldk nwljk in 10 vkSj NBk in] pkSFks in ls 12 vf/d

gSA 18- l-Js- 41] 38] 35 ------------------ dk dkSu izFke in ½.kkRed gksxk\ in Hkh Kkr

dhft,A 19- fuf/] igys fnu 2 #- nwljs fnu 4 #- rhljs fnu 6 #- bR;kfn cpkrh gSA iQjojh

2011] esa og fdrus #i;s cpk;sxhA 20- ,d l-Js- Kkr dhft,] ftldk rhljk in –13 vkSj NBk in 2 gSA 21- 6 vkSj 102 ds chp] 6 ls foHkkftr] nks vadksa okyh fdruh la[;k,a gksaxh\ 22- ;fn ,d l-Js- ds izFke n inksa dk ;ksxiQy s n = 3n 2 – 4n, gS] rks bldk nok¡ in

vkSj lkoZvUrj Kkr dhft,A 23- ,d l-Js- ds pkSFks vkSj vkBosa inksa dk ;ksxiQy 24 gS vkSj NBsa vkSj nlosa in dks

;ksxQy 44 gSA l-Js- Kkr dhft,A 24- 1 vkSj 199 ds chp fo"ke /ukRed iw.kk±d dk ;ksxiQy Kkr dhft,A 25- l-Js- 22] 22] 18 ----------- ds fdrus inksa dk ;ksxiQy 'kwU; gksxk\ 26- ,d f=kHkqt ds dks.k l-Js- esa gSA ;fn lcls NksVk dks.k] vU; nks dks.kksa ds ;ksx dk

ik¡pok Hkkx gS rks dks.kksa dh eki Kkr dhft,A 27- ,d l-Js- ds 11osa in dk 11 xquk] 17osa in ds 17 xquk ds cjkcj gS] bldk 28ok¡

in Kkr dhft,A

28- 8 inksa okyh ,d l-Js- Kkr dhft,] ftldk izFke in 1 2 vkSj vfUre in 17

6 gSA

29- ,d l-Js- dk pkSFk in] igys in ds rhu xqus ds cjkcj gS vkSj lkrok¡ in] rhljs in ds nks xqus ls 1 vf/d gSA l-Js- dk izFke in rFkk lkoZvUrj Kkr dhft,A

30- l-Js- 4 + 9 + 14 + _____ + 249, dk ;ksxiQy Kkr dhft,A

83 Math (Hindi) ­X

31- ,d l-Js- dk nwljk] bdÙkhlok¡ vkSj vafre in Øe'k% 31 1 , 4 2 vkSj 13

2 − gSA l-

Js- esa inksa dh la[;k Kkr dhft,A 32- l-Js- 57] 54] 51] ---------- ds fdrus inksa dk ;ksxiQy 570 gS\ Kkr dhft, rFkk

nksgjs mÙkj dh O;k[;k dhft,A 33- l-Js- ds rhu la[;kvksa dk ;ksx 24 gS vkSj mudk xq.kuiQy 440 gSA la[;k,a Kkr

dhft,A 34- ,d l-Js- ds izFke 40 inksa dk ;ksxiQy Kkr dhft] ftldk nok¡ in 3 – 2n gSA

35- ;fn ,d l-Js- ds mok¡ rFkk nok¡ in Øe'k% 1 n rFkk 1

m gS rks inksa dk ;ksxiQy

Kkr dhft,A 36- ;fn ,d l-Js- dk nok¡ in 4 gS] lkoZvUrj 2 gS vkSj inksa dk ;ksxiQy&14 gS] rks

izFke in rFkk inksa dh la[;k Kkr dhft,A 37- rhu vadksa dh lHkh la[;kvksa dk ;ksxiQy Kkr dhft,] tks 5 ls foHkktu ds ckn

3 'ks"kiQy NksM+rs gSaA 38- ,d l-Js- ds izFke 6 inksa dk ;ksxiQy 42 gSA 10osa in dk 30osa in ls vuqikr

1%3 gSA l-Js- dk izFke in rFkk 11ok¡ in Kkr dhft,A 39- nks l-Js- ds ninksa dk ;ksxiQy 3n + 8 : 7n + 15 ds vuqikr esa gSA blds 12osa inksa

dk vuqikr Kkr dhft,A 40- ,d l-Js- ds pok¡ vkSj rok¡ in Øe'k% l, m vkSj n gS] rks fl¼ dhft,&

p (m – n) + q (n – l) + r (l – m) = 0

41- ,d l-Js- ds izFke 8 inksa dk ;ksxiQy 140 gS vkSj 24 inksa dk ;ksxiQy 996 gSA l-Js- Kkr dhft,A

42- ,d rhu vadksa dks ?ku iw.kk±d ds vad l-Js- esa gS vkSj mudk ;ksxiQy 15 gSA la[;k esa ls 594 ?kVkus ij vad myV tkrs gSa] la[;k Kkr dhft,A

43- f'keyk ds fy,] ,d fidfud lewg esa fo|kfFkZ;ksa dh vk;q l-Js- esa gSA lkoZvUrj 3 ekg gSA ;fn lcls NksVs fo|kFkhZ uhjt dh vk;q 12 o"kZ gS] vkSj lHkh fo|kfFkZ;ksa dh vk;q dk ;ksxiQy 375 o"kZ gS] rks lewg esa fo|kfFkZ;ksa dh la[;k Kkr dhft,A

84 Math (Hindi) ­X

44- ,d l-Js- ds izFke 20 inksa dk ;ksxiQy] vxys 20 inksa dk ,d frgkbZ gSA ;fn izFke in 1 gS] rks 30 inksa dk ;ksxiQy Kkr dhft,A

45- ,d l-Js- ds izFke 16 inksa dk ;ksxiQy 528 vkSj vxys 16 inksa dk ;ksxiQy 1552 gSA l-Js- dk izFke in lkoZvUrj Kkr dhft,A

46- d`fr] ,d [ksy 'kq: djrh gS vkSj igys iz;kl esa 200 vad izkIr djrh gS] og gj iz;kl esa 40 vadksa dh o`f¼ djrh gSA 30osa iz;kl esa og fdrus vad vftZr djsxh\

47- ;fn lehdj.k a (b – c) x 2 + b (c – a) x + c (a – b) = 0 ds ewy cjkcj gS] rks

fn[kkb;s 1 1 1 , , a c b l-Js- esa gSA

48- ;fn ,d l-Js- ds m inksa dk ;ksxiQy n gS vkSj n inksa dk ;ksxiQy m gS] rks fl¼ dhft,] (m+n) inksa dk ;ksxiQy& (m+n) gksxkA

49- ,d l-Js- ds ikaposa vkSj uosa inksa dk ;ksxiQy 8 gS vkSj mudk xq.kuiQy 15 gSA rks l-Js- ds izFke 28 inksa dk ;ksxiQy Kkr dhft,A

50- vuqjkx us ,d leckgq f=kHkqt dks cukus ds fy, xsans O;ofLFkr dh] igyh iafDr esa ,d] nwljh iafDr esa nks rFkk blh izdkj lHkh iafDr;ksa esa xsans O;ofLFkr dh x;haA ;fn 669 xsans vkSj yh tk;sa] vkSj lHkh xsanksa dks ,d oxZ ds vkdkj esa O;ofLFkr dh tk;sa] rks oxZ dh izR;sd Hkqtk esa xsanksa dh la[;k] f=kHkqt dh izR;sd Hkqtk ls 8 xsansa de gksrh gSaA izkjEHk esa vuqjkx ds ikl xsanksa dh la[;k Kkr dhft,A

mÙkjekyk 1- a 2- b

3- d 4- b

5- b 6- c

7- a 8- d

9- b 10- b

11- b 12- c

13- a 14- b

85 Math (Hindi) ­X

15- d 16- gk¡] 50, 72

17- 4, 10, 16_____ 18- 15ok¡ in] –1

19- ` 812 20- –23, –18, –13, ____

21- 15 22- 6n – 7, lokZUrj = 6

23- 23 24- 9800

25- 23 26- 30°, 60°, 90°

27- 0 28- 1 5 7 , , , 2 6 6 ____

29- izFke in =3, lkoZ vUrj =2 30- 6325

31- 59 32- 19 or 20, 20ok¡ in 'kwU; gS

33- 5. 8. 11 34- – 1520

35- 1 ( 1) 2

mn + 36- izFke in = – 8, inksa dh la[;k = 7

37- 99090 38- izFke in = 2, 11oka in = 22

39- 7:16 40- ladsr% an = a + (n – 1) d

41- 7, 10, 13, 16, _____ 42- 852

43- 25 fo|kFkhZ 44- 450

45- izFke in = 3, lkoZvUrj = 4 46- 1360

47- ladsr % f?kkr lehdj.k esa] cjkcj ewyksa ds fy, D = 0

48- ladsr% 2 ( 1) 2 n n s a n d = + − 49- 1 115, 45

2 d = ±

50- 1540 xsansA

86 Math (Hindi) ­X

vè;k;&3

funsZ'kkad T;kfefr

egÙoiw.kZ fcanq % 1- nks fcUnqvksa A (x 1 , y 1 ) vkSj B (x 2 , y 2 ) dks tksM+us ij js[kk[k.M dh yEckbZ nksuksa

fcUnqvksa ds chp dh nwjh AB gksrh gSA tks fd √(x 2 – x 1 ) 2 + (y 2 – y) 2 .

2- fcUnq P(x, y) dh ewyfcUnq (0, 0) ls nwjh √(x 2 + y 2 ). gksrh gS rFkk x­vk ls nwjh y­bdkbZ o y­vk ls nwjh x­bdkbZ gksrh gSA

3- fcUnqvksa A(x 1 , y 1 ) vkSj B(x 2 , y 2 ) dks tksM+us okys js[kk[k.M dks fcUnq P(x, y), m 1 : m 2 ds vuqikr esa vkarfjd :i ls foHkkftr djrk gS rks

1 2 2 1 1 2 2 1

1 2 1 2 , m x m x m y m y

m m m m

+ + + +

(pkjksa dh la[;k de djus ds fy, m 1 : m 2 dh txg k : 1 fy;k tk ldrh gS) 4- nks fcUnqvksa P(x 1 , y 1 ) vkSj Q(x 2 , y 2 ) dks tksM+us okys js[kk[k.M ds eè; fcUnq ds

funsZ'kkad gksrs gSaA 1 2 1 2 ,

2 2 x x y y + +

5- rhu fcUnqvksa (x 1 , y 1 ), (x 2 , y 2 ) vkSj (x 3 , y 3 ) dks tksM+us ij izkIr f=kHkqt ds ks=kiQy dk la[;kRed eku bl izdkj gSA

1 2 3 3 1 3 1 2 1 ( ) ( ) ( ) . 2

x y y y y x y y − + − + −

6- mijksDr rhu ;fn ljs[kh gksaxs rks buls dksb f=kHkqt ugha cusxkA vr% ks=kiQy 'kwU; gksxk vFkkZr~ x 1 (y 2 – y 3 ) + x 2 (y 3 – y 1 ) + x 2 (y 1 – y 2 ) = 0

87 Math (Hindi) ­X

cgq oSdfYid iz'u

1- x­vk ij dksbZ fcUnq ‘p’ y vk ds cka;h vksj 3 bdkbZ dh nwjh gSA p ds funsZ'kkad gS& (a) (3, 0) (b) (0, 3) (c) (–3, 0) (d) (0, –3)

2- fcUnq p (3, –2) dh y­vk ls nwjh gS& (a) 2 bdkbZ (b) 2 bdkbZ (c) –2 bdkbZ (d) 13 bdkbZ

3- nks fcUnqvksa ds funsZ'kkad (6] 0) vkSj (0] &8) gSaA blds eè; fcUnq ds funsZ'kkad gSa&

(a) (3, 4) (b) (3, –4) (c) (0, 0) (d) (–4, 3)

4- ;fn fcUnqvksa (4] 0) vkSj (0, x) ds chp dh nwjh 5 bdkbZ gks rks x dk eku gS& (a) 2 (b) 3 (c) 4 (d) 5

5- ml fcUnq ds funsZ'kkad tgk¡ js[kk 7 x y a b

+ = y­vk dks izfrPNsr djrh gS&

(a) (a, 0) (b) (0, b) (c) (0, 2b) (d) (2a, 0)

6- f=kHkqt 0AB dk ks=kiQy gS] ;fn A (4, 0) B (0, –7) rFkk ewy 0 fcUnq gS& (a) 11 oxZ bdkbZ (b) 18 oxZ bdkbZ (c) 28 oxZ bdkbZ (d) 14 oxZ bdkbZ

7- fcUnqvksa 11 , 5 3

P − vkSj 2 , 5 3

Q − ds chp dh nwjh gS&

(a) 6 bdkbZ (b) 4 bdkbZ (c) 3 bdkbZ (d) 2 bdkbZ

88 Math (Hindi) ­X

8- ;fn js[kk 1 2 4 x y

+ = vkksa dks P vkSj Q ij izfrPNsn djs rks PQ ds eè; fcUnq ds

funsZ'kkad gS& (a) (1, 2) (b) (2, 0) (c) (0, 4) (d) (2, 1)

9- f=kHkqt ABC ds A 'kh"kZ ds funsZ'kkad (–4, 2) gS vkSj D (2, 5), BC dk eè; fcUnq gSA ∆ABC ds dsUnzd ds funsZ'kkad gS&

(a) (0, 4) (b) 7 1, 2

−

(c) 7 2, 3

− (d) (0, 2)

10- js[kkvksa 2x + 4 = 0 rFkk x – 5 = 0 ds chp dh nwjh gSa& (a) 9 bdkbZ (b) 1 bdkbZ (c) 5 bdkbZ (d) 7 bdkbZ

11- fcUnqvksa (5 cos 350°, 0) vkSj (0, 5 cos 55°) ds chp dh nwjh gS& (a) 10 bdkbZ (b) 5 bdkbZ (c) 1 bdkbZ (d) 2 bdkbZ

12- ;fn a dksbZ ?kukRed iw.kk±d gks rFkk fcUnqvksa P(a, 2) vkSj q(3, – 6) ds chp dh nwjh 10 bdkbZ gks rks a dk eku gS&

(a) –3 (b) 6 (c) 9 (d) 3

13- fcUnqvksa (0] 0)] (2] 0) rFkk (0] 2) ls cus f=kHkqt dk ifjeki gS& (a) 4 bdkbZ (b) 6 bdkbZ (c) 6 2 bdkbZ (d) 4 2 2 + bdkbZ

89 Math (Hindi) ­X

14- ;fn fcUnq (1] 2)] (&5] 6) vkSj (a, –2) lajs[kh; gks rks a dk eku gS& (a) –7 (b) 7 (c) 2 (d) 5

15- fcUnqvksa (9. a), (b, –4) rFkk (7, 8) ls cus f=kHkqt ds dsUnzd ds funsZ'kkad (6, 8) gSa] rks (a, b) gSa&

(a) (4, 5) (b) (5, 4) (c) (5, 2) (d) (3, 2)

y?kq mÙkjh; iz'u 16- ;fn fcUnq (3, a) js[kk 2x – 3y = 5 ij fLFkr gks rks a dk eku Kkr dhft,A 17- fcUnq P(3, 2) ls ,d js[kk x­vk ds lekUrj [khaph tkrh gS bl js[kk dh x­vk ls

nwjh D;k gksxh\ 18- ;fn fcUnq (3, 5) vkSj (7, 1) fcUnq (a, 0) ls lenwjLFk gksa rks a dk eku D;k gksxk\ 19- ;fn fcUnqvksa (2, p) vkSj (q, –1) dks tksM+us okys js[kk[k.M dk eè; fcUnq (2, –3)

gks rks p vkSj q ds eku Kkr dhft,A 20- AB ,d o`Ùk dk O;kl gS ftldk dsUnz ewy fcUnq ij gSA ;fn A ds funsZ'kkad

(3, – 4) gks rks B funsZ'kkd Kkr dhft,\ 21- ;fn fcUnqvksa P(6, b –2) vkSj Q(–2, 4) dks tksM+us okys js[kk[k.M ds eè; fcUnq ds

funsZ'kkad (2, –3) gSA b dk eku Kkr dhft,A 22- p, ds fdl eku ds fy,] fcUnq (–3, 9), (2, p) vkSj (4, –5) lajs[kh; gS\ 23- ;fn fcUnq (x, y) fcUnqvksa (7, 1) vkSj (3, 5) ls lenwjLFk gks rks x vkSj y esa laca/

LFkkfir dhft,A 24- ;fn P vkSj Q fcUnqvksa A (1, –2) vkSj B (–3, 4) dks tksM+us okys js[kk[k.M dks

f=kHkkftr djrs gSa] rks fcUnq P ds funsZ'kkad Kkr dhft,A 25- ;fn fcUnqvksa (x, 2) vkSj (3, 4) ds chp dh nwjh 8 bdkbZ gks rks x dk eku Kkr

dhft,A 26- ml f=kHkqt dk ks=kiQy Kkr dhft, ftlds 'kh"kks± ds funsZ'kkad (1, –1), (–3, 5) vkSj

(2, –7) gSaA

90 Math (Hindi) ­X

27- y­vk ij ml fcUnq ds funsZ'kkad Kkr dhft, tks fcUnqvksa (–2, 5) vkSj (2, –3) ls lenwjLFk gksA

28- fcUnqvksa (5] 7) rFkk (3] 9) dks tksM+us okys js[kk[k.M dk eè; fcUnq vkSj (8] 6) rFkk (a, b) dks tksM+us okys js[kk [k.M dk eè; fcUnq leku gSA a vkSj b ds eku Kkr dhft,A

29- ml fcUnq ds funsZ'kkad Kkr dhft, tks fcUnqvksa (1] 3) rFkk (2]7) dk feykus okys js[kk[k.M dks 3%4 ds vuqikr esa foHkkftr djrk gSA

30- fcUnq P vkSj Q ds funsZ'kkad (1] 2) vkSj (2] 3) gSaA js[kk[k.M PQ ij ,d fcUnq R ds funsZ'kkad Kkr dhft, tks bl izdkj gS fd 4 .

3 PR RQ

=

31- fcUnq K (1, 2) fcUnqvksa E (6, 8) vkSj dks tksM+us okys js[kk[k.M ds yEc lefHkktd ij fLFkr gSA fcUnq K dh js[kk[k.M EF ls nwjh Kkr dhft,A

32- f=kHkqt ABC ds 'kh"kks± ds funsZ'kkad A (–1, 3), B (1, –1) vkSj C (5, 1) gSaA 'kh"kZ A ls [khaph xbZ ekfè;dk dh yEckbZ Kkr dhft,A

33- fcUnqvksa A (a, b) vkSj B (b, a) ds chp dh nwjh Kkr dhft, ;fn a – b = 4 gksA 34- ,d lekUrj prqHkqZt ds 'kh"kks± ds funsZ'kkad Øe ls ysus ij (&3] 1)] (1] 1) vkSj

(3] 3) gSaA pkSFks 'kh"kZ ds funsZ'kkad ds funsZ'kkad Kkr dhft,A 35- f=kHkqt ABC ,d lefckgq f=kHkqt gS ftlesa AB = AC rFkk 'kh"kZ A, y­vk ij

fLFkr gSA ;fn fcUnq B vkSj C ds funsZ'kkad Øe'k% (&5] &2) rFkk (3] 2) gSa] 'kh"kZ A ds funsZ'kkad Kkr dhft,A

36- fcUnq P(K, 3) js[kk[k.M AB dk eè; fcUnq gSA ;fn AB = 52 bdkbZ rFkk A ds funsZ'kkad (–3, 5) gksa rks K dk eku Kkr dhft,A

37- ml fcUnq ds funsZ'kkad Kkr dhft, tks fcUnq (3] 1) dks (&2] 5) ls tksM+us okys

ekxZ dk 3 4 gSA

38- ml f=kHkqt dk ks=kiQy ftlds 'kh"kks± ds funsZ'kkad (6] &3)] (3, K) rFkk (&7] 7) gSa] 15 oxZ bdkbZ gSA K dk eku Kkr dhft,A

39- ml fcUnq dh Hkqtk Kkr dhft, ftldh dksfV 4 gS rFkk ftldh fcUnq (5] 0) ls nwjh 5 bdkbZ gSA

91 Math (Hindi) ­X

40- X­vk ij fLFkr ,d fcUnq P, fcUnqvksa (4] 5) vkSj (1] &3) dks feykus okys js[kk[k.M dks fdlh vuqikr esa foHkkftr djrk gSA fcUnq P ds funsZ'kkad Kkr dhft,A

41- ,d ledks.k ∆ABC, ∠B =90° rFkk AB = 34 bdkbZA fcUnq B vkSj C funsZ'kkad Øe'k% (4] 2) (–1, y) vkSj gSaA ;fn ks- (∆ABC) = 17 oxZ bdkbZ] rks y dk eku Kkr dhft,A

42- ;fn lefckgq f=kHkqt ds 'kh"kks± ds funsZ'kkad A(–3, 2) B (x,y) vkSj C (1, 4) rFkk AB = BC gSA (2x + y) dk eku Kkr dhft,A

43- ;fn fcUnq P (3, 4) fcUnqvksa A (a + b, b –a) rFkk B (a – b, a + b) ls lenwjLFk gks rks fl¼ dhft, 3b – 4a = 0.

44- ,d prqHkqZt ABCD 'kh"kks± ds funsZ'kkad Øe'k% A (–5, 7), B (–4, 5), C (–1, –6) vkSj D (4, 5) gSaA prqHkqZt ABCD dk ks=kiQy Kkr dhft,A

45- og vuqikr Kkr dhft, ftlesa js[kk 3x + y =12 fcUnqvksa (1] 3) vkSj 2] 7) dks feykus okys js[kk[k.M dks foHkkftr djrh gSA

46- fcUnqvksa A (2, 1) rFkk B (5, –8) dks feykus okyk js[kk[k.M fcUnq P vkSj Q ij f=kHkkftr gksrk gS] fcUnq P, A ds fudV gSA ;fn fcUnq P js[kk 2x – y + k = 0 ij Hkh fLFkr gks rks K dk eku Kkr dhft,A

47- fcUnqvksa (3] 4) vkSj (1] 2) dks feykus okyk js[kk[k.M fcUnq P vkSj Q ij

f=kHkkftr gksrk gSA ;fn fcUnq P vkSj Q ds funsZ'kkad Øe'k% (p –2) vkSj 5 ,3

q gksa

rks p vkSj q dk eku Kkr dhft,A 48- f=kHkqt ABC esa 'kh"kZ A ds funsZ'kkad (3] 2) gSa rFkk Hkqtkvksa AC vkSj AB ds eè;

fcUnqvksa ds funsZ'kkad Øe'k% (2] &1) vkSj (1] 2) gSaA Hkqtk BC ds eè; fcUnq ds funsZ'kkad Kkr dhft,A

49- ∆ABC ds 'kh"kks± ds funsZ'kkad A (5, 2), B (–5, –1) rFkk gSaA C (3, –5) fn[kkb, fd efè;dk AD f=kHkqt ABC dks nks cjkcj ks=kiQyksa okys f=kHkqtksa esa ckaVrh gSA

50- fcUnqvksa A (a, 0) rFkk B (0, b) dks tksM+us okys js[kk[k.M ij ,d fcUnq P (x,y)

fLFkr gS] fn[kkb, fd 1. x y a b

+ =

51- ;fn fcUnq (x, y), (–5, –2)vkSj (3, –5) lajs[kh; gks rks fl¼ dhft, fd 3x+8y+31 = 0

92 Math (Hindi) ­X

mÙkjekyk 1- c 2- a

3- b 4- b

5- c 6- d

7- c 8- a

9- a 10- d

11- b 12- c

13- d 14- b

15- c 16- 1 3

a =

17- 2 bdkbZ 18- a = 2

19- p = –5, q = 2 20- (–3, 4)

21- b = –8 22- p = –1

23- x – y = 2 24- 1 , 0 3

−

25- x, = 1, 5 26- 5 oxZ bdkbZ

27- (0. 1) 28- a = 0, b = 10

29- 10 33 , 7 7

30- 11 18 ,

7 7

31- 5 bdkbZ 32- 5 bdkbZ

33- 4 2 bdkbZ 34- (–1, 3)

35- (0, –2) 36- K = 0, –6

93 Math (Hindi) ­X

37- 3 , 4 4

− 38- 21 13

K =

39- 2, 8 40- 17 , 0 8

41- –1 42- 1

44- 72 oxZ bdkbZ 45- 6 : 1

46- K = –8 47- 7 , 0 3

p q = =

48- (0, –1).

94 Math (Hindi) ­X

vè;k;&4

f=kdks.kfefr ds dqN vuqiz;ksx

egÙoiw.kZ fcanq % 1- n`f"V js[kk % n`f"V js[kk izR;sd dh vk¡[k ls izskd kjk ns[kh xbZ oLrq ds fcUnq dks

feykus okyh js[kk gksrh gSA 2- mUu;u dks.k % mUu;.k dks.k] n`f"V js[kk vkSj kSfrt js[kk ls cuk dks.k gksrk gS]

tcfd og kSfrt Lrj ls mQij gksrk gS vFkkZr~ og fLFkfr tcfd oLrq dks ns[kus ds fy, gesa vius flj dks mQij mBkuk gksrk gSA

3- voueu dks.k % voueu dks.k] n`f"V js[kk vkSj kSfrt js[kk ls cuk dks.k gksrk gS] tcfd ;g kSfrt Lrj ls uhpk gksrk gS vFkkZr~ og fLFkfr tcfd oLrq dks ns[kus ds fy, gesa vius flj dks >qdkuk iM+rk gSA

cgq oSdfYid iz'u

1- ,d O;fDr dh Nk;k dh yEckbZ O;fDr dh mQ¡pkbZ ds cjkcj gSA mUu;u dks.k dk eku gS&

(a) 90° (b) 60° (c) 45° (d) 30°

2- fdlh k.k 30 eh- mQ¡ps [kEHks dh Nk;k dh yEckbZ 10 3 eh- gSA lw;Z ds mUu;u dks.k dk eku gS&

(a) 30° (b) 60° (c) 45° (d) 90°

95 Math (Hindi) ­X

3- vkd`fr (1) esa CE || AB. fcUnq A vkSj D ij mUu;u Øe'k% gS&

(a) (30°, 60°) (b) (30°, 30°) (c) (60°, 30°) (d) (45°, 45°)

4- 10 eh- vkSj 18 eh- mQ¡ps [kEHkksa ds 'kh"kZ ,d rkj ls tqM+s gSaA ;fn rkj kSfrt ls 30° dk dks.k cukrk gks rks rkj dh yEckbZ gS&

(a) 10 eh- (b) 18 eh- (c) 12 eh- (d) 16 eh-

5- ,d Vkoj ds ikn ls 20eh- nwj fLFkr fdlh fcUnq ls Vkoj ds f'k[kj dk mUu;u dks.k 30° gSA Vkoj dh mQ¡pkbZ gS&

(a) 20 3 eh- (b) 20

3 eh-

(c) 40 3 eh- (d) 40

3 eh-

6- ,d isM+ dh mQ¡pkbZ rFkk bldh Nk;k dk vuqikr 1 1 : 3 gS ml k.k lw;Z dk mUu;u

dks.k gS& (a) 30° (b) 45° (c) 60° (d) 90°

96 Math (Hindi) ­X

7- ,d irax lery Hkwfe ls 50 3 eh- mQ¡pkbZ ij mM+ jgh gS ,d Mksj ls c¡/h gS tks kSfrt ls 60° ds dks.k ij >qdh gSaA Mksj dh yEckbZ gS&

(a) 100 eh- (b) 50 eh- (c) 150 eh- (d) 75 eh-

8- vkd`fr (2) esa] vk;r ABCD dk ifjeki gS&

(a) 40 eh- (b) ( ) 20 3 1 m +

(c) 60 eh- (d) ( ) 10 3 1 m +

9- ,d isM+ Hkwfe ls 10eh- mQij ls nks Hkkxksa esa VwV tkrk gSA VwVk Hkkx Hkwfe dks Nwrk gS rFkk kSfrt ds lkFk 30° dk dks.k cukrk gSA isM+ dh mQ¡pkbZ gS&

(a) 30 eh- (b) 20 eh- (c) 10 eh- (d) 15 eh-

10- vkd`fr (3) esa 3 tan ,4

α = ;fn AB = 12 eh-] mQ¡pkbZ BC dk eku gS&

97 Math (Hindi) ­X

(a) 8 eh- (b) 12 eh- (c) 9 eh- (d) 10 eh-

11- vkd`fr (4) D, BC, dk eè; fcUnq gS] ;fn ∠CAB = α 1 rFkk ∠DAB = β 2 rks tan α 1 : tan β 2 cjkcj gS&

(a) 2 :1 (b) 1 : 2 (c) 1 : 1 (d) 1 : 3

12- vkd`fr (5) esa 8 tan 15

θ = ;fn PQ = 16 eh- rks PR dh yEckbZ gS&

(a) 16 eh- (b) 34 eh- (c) 32 eh- (d) 30 eh-

98 Math (Hindi) ­X

13- ,d Vkoj dh mQ¡pkbZ 50 eh- gSA tc mUu;u dks.k dk eku 45° ls 30° gks tkrk gS rks Vkoj dh Nk;k dh yEckbZ x eh- vf/d gks tkrh gS x dk eku gS&

(a) 50 eh- (b) ( ) 50 3 1 m −

(c) 50 3 m (d) 50

3 m

14- fdlh ehukj ds in ls 9 eh- vkSj 16 eh- nwj fLFkr Hkwfe ij fdUgha nks fcUnqvksa ls ehukj ds f'k[kj ds mUu;u dks.k ,d nwljs ds iwjd ik, tkrs gSaA ehukj dh mQ¡pkbZ gS&

(a) 18 m (b) 16 m (c) 10 m (d) 3012 m

nh?kZ mÙkjh; iz'u 15- 5 eh- mQ¡pk ,d [kEHkk ,d Vkoj ds f'k[kj ij yxk gSA /jrh ds ,d fcUnq A ls

[kEHks ds f'k[kj dk mUu;u dks.k 60° gS rFkk Vkoj ds f'k[kj ls fcUnq A dk

voueu dks.k 45° gSA Vkoj dh mQ¡pkbZ Kkr dhft,A ( ) 3 1.732 =

16- Hkwfe ij fLFkr ,d fcUnq ls 30 ehVj mQ¡ph ehukj ds mQij j[ks ikuh ds VSad ds ry rFkk 'kh"kZ ds mUu;u dks.k Øe'k% 45° rFkk 60° gSA ikuh ds VSad dh mQ¡pkbZ Kkr dhft,A

17- ,d lery Hkwfe ij fLFkr Vkoj dh Nk;k dh yEckbZ 60° eh- de gks tkrh gS tc lw;Z dk mUurka'k 30° ls cnydj 60° gks tkrk gS] Vkoj dh mQ¡pkbZ Kkr dhft,A

18- ,d isM+ vka/h esa VwV tkrk gS rFkk VwVk Hkkx dk mQijh fljk Hkwfe dks ,sls Nwrk gS fd Hkwfe ls 60° dk dks.k cukrk gSA isM+ ds ikn ls mls fcUnq dh nwjh tgk¡ isM+ dk mQijh fljk isM+ dks Nwrk gS] 5 ehVj gSA isM+ dh dqy mQ¡pkbZ Kkr dhft,A

19- Hkwfe ij fLFkr ,d fcUnq ls ,d fpfM+;k dk mUu;u dks.k 60° gS rFkk 50 lsds.M dh mM+ku ds i'pkr mUu;u dks.k cnydj 30° gks tkrk gSA ;fn fpfM+;k 500 3 eh- dh fu;r mQ¡pkbZ ij mM+ jgh gks rks fpfM+;k dh xfr Kkr dhft,A

20- Hkwfe ij fLFkr ,d fcUnq A ls tsV foeku dk mUu;u dks.k 60° gSA 15 lsds.M dh mM+ku ds i'pkr mUu;u dks.k cnydj 30° gks tkrk gSA ;fn tsV 720 fdeh-@?kaVk

99 Math (Hindi) ­X

dh pky ls mM+ jgk gks rks tsV foeku ds mM+us dh fu;r mQ¡pkbZ Kkr dhft,

( ) 3 1.732 = .

21- xyh esa fLFkr] Hkwfe ls 20 ehVj dh mQ¡pkbZ ij fLFkr ,d f[kM+dh ls xyh ds nwljh vksj fLFkr ,d edku ds 'kh"kZ rFkk vk/kj ds mUu;u rFkk voueu dks.k Øe'k% 60° vkSj 45° gSaA edku dh mQ¡pkbZ Kkr dhft,A

22- ,d gokbZ tgkt 1800 ehVj dh mQ¡pkbZ ij mM+rs gq, unh ds foijhr fdukjksa ij fLFkr nks fcUnqvksa ds voueu dks.k 60° rFkk 45° ns[krk gS] unh dh pkSM+kbZ Kkr dhft,A

23- lery Hkwfe ij ,d Vkoj ds ,d gh vksj fLFkr nks fcUnqvksa A vkSj B tks ,d nwljs ls 15 ehVj nwj gS] ls Vkoj ds 'kh"kZ ds mUu;u dks.k Øe'k% 30° vkSj 60° gSaA Vkoj

dh mQ¡pkbZ rFkk Vkoj ds vk/kj ls fcUnq B dh nwjh Kkr dhft,A ( ) 3 1.732 =

24- Hkwfe ij fLFkr ,d fcUnq P ls 10 ehVj mQ¡ph ehukj ds f'k[kj dk mUu;u dks.k 30° gSA ehukj ds f'k[kj ij ,d èotn.M yxk gS rFkk fcUnq P ls èotn.M ds 'kh"kZ dk mUu;u dks.k 45° gSA èotn.M dh yEckbZ rFkk ehukj ds vk/kj ls fcUnq P dh nwjh Kkr dhft,A

25- ,d >hy ds ry ls 12 ehVj mQij fLFkr ,d fcUnq ls fpfM+;k dk mUu;u dks.k 30° gS rFkk >hy esa fpfM+;k ds izfrfcEc dk voueu dks.k 60° gSA fujhk.k fcUnq ls fpfM+;k dh nwjh Kkr dhft,A

26- ,d >hy ds ry ls 60 ehVj mQij fLFkr ,d fcUnq ls ckny dk mUu;u dks.k 30° gS rFkk >hy esa ckny ds izfrfcEc dk voueu dks.k 60° gSA ckny dh mQ¡pkbZ Kkr dhft,A

27- ,d ehukj ij [kM+k O;fDr ,d uko] tks fdukjs ij fLFkr ,d fcUnq A dh vksj leku xfr ls vk jgh gS] dk voueu dks.k 30° ns[krk gSA 12 feuV i'pkr uko dk voueu dks.k 60° gks tkrk gSA uko kjk fdukjs ij igq¡pus esa yxk le; Kkr dhft,A

28- ,d O;fDr tgkt ds MSd ij [kM+k gS tks leqnz ry ls 18 ehVj mQij gS] ogk¡ ls ,d ehukj ds f'k[kj rFkk vk/kj ds mUu;u rFkk voueu dks.k Øe'k% 60° rFkk 30° gSA tgkt ls ehukj dh nwjh rFkk ehukj dh mQ¡pkbZ Kkr dhft,A

100 Math (Hindi) ­X

29- ,d O;fDr unh ds fdukjs [kM+k gksdj nwljs fdukjs ij fLFkr ,d isM+ ds 'kh"kZ dk mUu;u 60° gSA tc og fdukjs ls 40 ehVj nwj pyk tkrk gS rks isM+ ds 'kh"kZ dk mUu;u dks.k 30° dks.k tkrk gSA isM+ dh mQ¡pkbZ rFkk unh dh pkSM+kbZ Kkr dhft,A

30- ,d gokbZ tgkt tc 3000 ehVj mQ¡pkbZ ij gksrk gS ,d nwljs tgkt ds mQij xqtjrk gS] blh k.k Hkwfe ij fLFkr fdlh fcUnq ls nksuksa tgktksa ds mUu;u dks.k Øe'k% 60° vkSj 45° gSA nksuksa tgktksa ds chp dh mQèokZ/j nwjh Kkr dhft,A

31- ,d Vkoj ds f'k[kj ls ,d 10 ehVj mQ¡pk ehukj ds f'k[kj rFkk vk/kj ds voueu dks.k Øe'k% 30° vkSj 45° gSA Vkoj dh mQ¡pkbZ rFkk Vkoj vkSj ehukj ds chp dh nwjh Kkr dhft,A

32- ,d O;fDr ,d kSfrt ry ij [kM+k gS] ,d fpfM+;k tks mlls 100 ehVj dh nwjh ij mM+ jgh gS] dk mUu;u 30° ikrk gSA ,d yM+dh 20 ehVj mQ¡ph ehukj dh Nr ij [kM+h gS] mlh fpfM+;k dk mUu;u dks.k 45° ikrh gSA os nksuksa fpfM+;k ds foijhr fn'kk esa gSA fpfM+;k dh yM+dh ls nwjh Kkr dhft,A

33- lery Hkwfe ij fLFkr ,d fcUnq P ls ,d Vkoj ds 'kh"kZ dk mUu;u dks.k bl izdkj

gS fd bl dks.k dh Li'kZT;k 3 4 gSA fcUnq P ls 192 ehVj nwj tkus ij dks.k dh

Li'kZt;k 5 12 gks tkrh gSA Vkoj dh mQ¡pkbZ Kkr dhft,A

34- lery Hkwfe ij fLFkr nks fcUnqvksa P vkSj Q tks ,d ehukj ds ,d gh vksj fLFkr gS ls ehukj ds f'k[kj ds mUu;u dks.k Øe'k 36° rFkk 54° gSaA ;fn P vkSj Q dh ehukj ds vk/kj ls nwjh Øe'k% 10 ehVj vkSj 20 ehVj gks rks ehukj dh mQ¡pkbZ Kkr

dhft,A ( ) 2 1.414 =

35- ,d xksy xqCckjk ftldh f=kT;k r gS fujhkd dh vk¡[k ij ‘θ ’ dks.k cukrk gS] tcfd blds dsUnz dk mUu;u dks.k α gSA fl¼ dhft, fd xqCckjs ds dsUnz dh mQ¡pkbZ

sin cos .2

r ec θ

α gSA

101 Math (Hindi) ­X

mÙkjekyk 1- c 2- b

3- a 4- d

5- b 6- c

7- a 8- b

9- a 10- c

11- b 12- b

13- b 14- d

15- 6.83 ehVj 16- ( ) 30 3 1 − ehVj

17- 30 3 ehVj 18- ( ) 5 2 3 + ehVj

19- 20 m/sec. 20- 2598 ehVj

21- ( ) 20 3 1 + ehVj 22- ( ) 600 3 3 + ehVj

23- mQ¡pkbZ = 12.97 ehVj] nwjh = 7.5 ehVj

24- èotn.M dh yEckbZ ( ) 10 2 1 m = − Hkou dh nwjh ( ) 10 3 = ehVj-

25- 24 3 ehVj 26- 120 ehVj 27- 18 ehVj 28- 18 3 ehVj] 72 ehVj 29- mQ¡pkbZ = 34.64 ehVj] unh dh pkSM+kbZ = 20 ehVj

30- ( ) 100 3 3 − ehVj

31- mQ¡pkbZ = ( ) 5 3 3 + ehVj] nwjh = ( ) 5 3 3 + ehVj

32- 30 ehVj 33- 180 ehVj 34- 14.14 ehVj

102 Math (Hindi) ­X

vè;k;&5

o`Ùk

egÙoiw.kZ fcanq % 1- o`Ùk dh Li'kZ js[kk % o`Ùk dh Li'kZ js[kk og js[kk gksrh gS tks o`Ùk dks dsoy ,d

fcUnq ij izfrPNsn djrh gSA 2- Li'kZ fcUnq ij fliQZ ,d Li'kZ js[kk gksrh gSA 3- fuEu izes;ksa dk gy ijhkk esa iwNk tk ldrk gS%

d) o`Ùk dh Li'kZ js[kk] Li'kZ fcUnq ls gksdj tkus okyh f=kT;k ij yEc gksrh gSA

[k) fdlh ckg~; fcUnq ls o`Ùk ij [khaph xbZ nks Li'kZ js[kkvksa dh yEckbZ;ka cjkcj gksrh gSA

cgq oSdfYid iz'u

1- vkd`fr (1) esa PQ o`Ùk dh Li'kZ js[kk gSA ∠POQ + ∠QPO dk eku gSA

(a) 180° (b) 90° (c) 80° (d) 100°

103 Math (Hindi) ­X

2- ;fn PQ, 5 lseh- f=kT;k okys o`Ùk dh Li'kZ js[kk gks rFkk PQ = 12 lseh-] Q Li'kZ fcUnq gS] rks OP dk eku gSA

(a) 13 lseh- (b) 17 lseh- (c) 7 lseh- (d) 119 lseh-

3- vkd`fr (2) esa PQ rFkk PR o`Ùk dh Li'kZ js[kk,a gSa] ∠QOP = 70°, rks ∠QPR dk eku gSA

(a) 35° (b) 70° (c) 40° (d) 50°

4- vkd`fr (3) esa PQ o`Ùk dh Li'kZ js[kk gS] PQ = 8 lseh-] OQ = 6 lseh-] rks PS dh yEckbZ gS

(a) 10 lseh- (b) 2 lseh- (c) 3 lseh- (d) 4 lseh-

104 Math (Hindi) ­X

5- vkd`fr (4) esa PQ ckg~; o`Ùk dh rFkk PR vr% o`Ùk dh Li'kZ js[kk,a gSA ;fn PQ = 4 lseh-] OQ = 3 lseh- OR = 2 vkSj lseh- rks PR dh yEckbZ gSA

(a) 5 lseh- (b) 21 lseh- (c) 4 lseh- (d) 3 lseh-

6- vkd`fr (5) esa P, Q rFkk R Li'kZ fcUnq gSA ;fn AB = 4 lseh-] BR = 2 lseh-] rks ∆ABC dk ifjeki gS

(a) 12 lseh- (b) 8 lseh- (c) 10 lseh- (d) 9 lseh-

105 Math (Hindi) ­X

7- vkd`fr (6) esa ∆ABC dk ifjeki gS

(a) 10 lseh- (b) 15 lseh- (c) 20 lseh- (d) 25 lseh-

8- ,d o`Ùk dh nks lekUrj Li'kZ js[kkvksa ds chp dh nwjh 12 lseh- gSA o`Ùk dh f=kT;k gSA

(a) 13 lseh- (b) 6 lseh- (c) 10 lseh- (d) 8 lseh-

9- vkd`fr (7) esa ∆ABCD o`Ùk dh lHkh Hkqtkvksa dks Li'kZ djrk gSA ;fn AB = 6 lseh-] BC = 5 lseh- rFkk AD = 8 lseh- rks Hkqtk CD dh yEckbZ gSA

(a) 6 lseh- (b) 8 lseh- (c) 5 lseh- (d) 7 lseh-

106 Math (Hindi) ­X

10- 17 lseh- f=kT;k okys o`Ùk dh nks lekUrj thok,a ,d O;kl ds nksuksa vksj [khaph xbZ gSA nksuksa thokvksa ds chp dh nwjh 23 lseh- rFkk ,d thok dh yEckbZ 16 lseh- gS rks nwljh thok dh yEckbZ gSA

(a) 34 lseh- (b) 17 lseh- (c) 15 lseh- (d) 30 lseh-

11- vkd`fr (8) esa Li'kZ fcUnq gS rks ∠OPB dk eku gSA

(a) 50° (b) 40° (c) 35° (d) 45°

13- vkd`fr (9) esa PQ vkSj PR dsUnz O okys o`Ùk dh Li'kZ js[kk,a gSA ;fn ∠QPR = 45° rks ∠QOR dk eku gSA

(a) 90° (b) 110° (c) 135° (d) 145°

107 Math (Hindi) ­X

13- vkd`fr (10) esa O o`Ùk dk dsUnz gS] PA vkSj PB o`Ùk dh Li'kZ js[kk,a gSa rks ∠AQB dk eku gSA

(a) 70° (b) 80° (c) 60° (d) 75°

14- vkd`fr (11) esa ∆ABC o`Ùk dks fcUnq P, Q rFkk R ij Li'kZ djrk gSA ;fn AP = 4 lseh-] BP = 6 lseh-] AC = 12 lseh-] gks rks BC dh yEckbZ gSA

(a) 6 lseh- (b) 14 lseh- (c) 10 lseh- (d) 18 lseh-

108 Math (Hindi) ­X

15- vkd`fr (12) esa ∆ABC vUr% o`Ùk dks fcUnq P, Q rFkk R ij Li'kZ djrk gS rFkk P, Hkqtk BC dk eè; fcUnq gSA ;fn AR = 4 lseh-] AC = 9 lseh-] rFkk BC dh yEckbZ gSA

(a) 10 lseh- (b) 11 lseh- (c) 8 lseh- (d) 9 lseh-

y?kq mÙkjh; iz'u 16- AB vkSj AC dsUnz ‘O’ okys o`Ùk dh nks Li'kZ js[kk,a gSaA ;fn ∠BOA = 2x rFkk

∠OAB = x, rks x dk eku Kkr dhft,A 17- ,d vUr% o`Ùk ,d lefckgq f=kHkqt dh leku Hkqtkvksa dks fcUnq E vkSj F ij Li'kZ

djrk gSA o`Ùk rhljh Hkqtk dks fcUnq D ij Li'kZ djrk gS rks fn[kkb, fd D rhljh Hkqtk dk eè; fcUnq gSA

18- 2-5 lseh- f=kT;k okys o`Ùk ij ckg~; fcUnq P ls Li'kZ js[kk dh yEckbZ 6 lseh- gSA fcUnq P dh o`Ùk ds fudVre fcUnq ls nwjh Kkr dhft,A

19- dsUæ ‘O’ okys o`Ùk ij ckg~; fcUnq T ls Li'kZ js[kk,a TP vkSj TQ gSA ;fn ∠OPQ = 30° rks ∠TQP dk eku Kkr dhft,A

109 Math (Hindi) ­X

20- vkd`fr (13) esa AP = 4 lseh-] BQ = 6 lseh- vkSj AC = 9 lseh-A ∆ABC dk v/Z&ifjeki Kkr dhft,A

21- vkd`fr (14) esa OP o`Ùk ds O;kl ds cjkcj gS] tgk¡ O o`Ùk dk dsUnz gSA fl¼ dhft, ∆ABP ,d leckgq f=kHkqt gSA

22- vkd`fr (15) esa cM+s v/Z o`Ùk ds ckgj ,d NksV v/Z o`Ùk [khapk x;k gSA NksVs v/Z o`Ùk dk O;kl BE cM+s v/Z o`Ùk dh f=kT;k BF dk vk/k gSA ;fn cM+s v/Z o`Ùk dh f=kT;k 4√3 lseh- gks rks fcUnq A ls NksVs v/Z ij [khaph xbZ Li'kZ js[kk AC dh yEckbZ Kkr dhft,A

110 Math (Hindi) ­X

23- vkd`fr (16) esa PA vkSj PB dsUnz ‘O’ okys o`Ùk dh Li'kZ js[kk,a gSa rks x dk eku Kkr dhft,A

24- ledks.k ∆ABC dh Hkqtk AB dks O;kl ekudj ,d o`Ùk [khapk tkrk gSA tks d.kZ AC dks fcUnq P ij izfrPNsn djrk gSA fl¼ dhft, PB = PC.

25- vkd`fr (17) esa PQ dsUnz ‘O’ okys o`Ùk dh Li'kZ js[kk gSA ;fn AP = 8 lseh- vkSj Li'kZ js[kk dh yEckbZ f=kT;k ls 1 lseh- vf/d gks rks o`Ùk dh f=kT;k Kkr dhft,A

26- dsUnz ‘O’ rFkk f=kT;k 5 lseh- okys o`Ùk dh ,d thok AB dh yEckbZ lseh- gSA ckg~; fcUnq P ls rFkk B ij Li'kZ js[kkvksa dh yEckbZ Kkr dhft,A

27- vkd`fr (18) esa AB = AC, ‘D’ AC dk eè; fcUnq gS rFkk BD o`Ùk dk O;kl gS

rks fl¼ dhft, fd 1 . 4

AE AC =

111 Math (Hindi) ­X

28- vkd`fr (19) esa nks ladsUnzh o`Ùkksa dh f=kT;k,a 5 lseh- vkSj 8 lseh- gSA cM+s o`Ùk ij fcUnq P ls Li'kZ js[kk dh yEckbZ 15 lseh- gSA NksVs o`Ùk ij Li'kZ js[kk dh yEckbZ Kkr dhft,A

29- ,d vUr% o`Ùk ledks.k f=kHkqt dh Hkqtkvksa dks Li'kZ djrk gS ftlds vk/kj vkSj 'kh"kZyEc dh yEckbZ 6 lseh- vkSj 2-5 lseh- gSA o`Ùk dh f=kT;k Kkr dhft,A

30- vkd`fr (20) esa AB = 13 lseh-] BC = 7 lseh-] AD = 15 lseh-A PC dh yEckbZ Kkr dhft,A

112 Math (Hindi) ­X

31- vkd`fr (21) esa o`Ùk dh f=kT;k Kkr dhft,A

32- vkd`fr (22) esa ;fn o`Ùk dh f=kT;k 3 lseh- gks rks ∆ABC ifjeki Kkr dhft,A

33- ,d o`Ùk dk O;kl rFkk PR ,d thok gS rFkk ∠RPQ = 30° fcUnq R ij Li'kZ js[kk PQ dks c<+k, tkus ij fcUnq S ij izfrPNsn djrh gSA fl¼ dhft, RQ = QS.

113 Math (Hindi) ­X

34- vkd`fr (23) esa XP vkSj XQ ckg~; fcUnq X ls dsUnz ‘O’ okys o`Ùk dh Li'kZ js[kk,a gSaA R o`Ùk ij ,d fcUnq gSA fl¼ dhft, XA + AR = XB + BR.

nh?kZ mÙkjh; iz'u 35- fl¼ dhft, fd o`Ùk ds fdlh fcUnq ij [khaph ij xbZ Li'kZ js[kk ml fcUnq ij [khaph

xbZ f=kT;k ds yEcor gksrh gSA mijksDr izes; ij vk/kfjr iz'u 1- nks ladsUnzh; o`Ùkksa esa ls cM+s o`Ùk ds fy, [khaph xbZ og thok tks NksVs o`Ùk dks

Li'kZ djrh gSA Li'kZ fcUnq ij lefHkkftr gksrh gSA fl¼ dhft,A 2- PT dsUnz ‘O’ okys o`Ùk dh Li'kZ js[kk gS rFkk T Li'kZ fcUnq gSA ;fn OT = 6

lseh-] OP = 10 lseh-] rks Li'kZ js[kk PT dh yEckbZ Kkr dhft,A 3- vkd`fr (24) esa PQ o`Ùk dh Li'kZ js[kk rFkk PB O;kl gSA x vkSj y ds eku

Kkr dhft,A

114 Math (Hindi) ­X

4- vkd`fr (25) esa AC dsUnz ‘O’ okys o`Ùk dk O;kl gS vkSj A Li'kZ fcUnq gS rks x eku Kkr dhft,A

36- fl¼ dhft, fd fdlh ckg~; fcUnq ls fdlh o`Ùk ij [khaph xbZ nksuksa Li'kZ js[kk,a cjkcj yEckbZ dh gksrh gSaA

mijksDr izes; ij vk/kfjr iz'u 1- vkd`fr (26) PA vkSj PB fcUnq ls Li'kZ js[kk,a gSaA fl¼ dhft, KN = AK + BN.

2- nks ladsUnzh; o`Ùkksa dh f=kT;k,a 5 lseh- vkSj 3 lseh- gSaA cM+s o`Ùk dh ml thok dh yEckbZ Kkr dhft, tks NksVs o`Ùk dh Li'kZ js[kk gSA

3- vkd`fr (27) PA vkSj PB dsUnz ‘O’ okys o`Ùk dh Li'kZ js[kk,a gSA fl¼ dhft, OP, AB dk yEc lef¼Hkktd gSA

115 Math (Hindi) ­X

4- vkd`fr (28) esa thou dh yEckbZ 6 lseh- rFkk o`Ùk dh f=kT;k 6 lseh- gS vkSj o`Ùk dh Li'kZ js[kk,a gSaA dk eku Kkr dhft,A

mÙkjekyk 1- b 2- a

3- c 4- d

5- b 6- a

7- c 8- b

9- d 10- a

11- a 12- c

13- a 14- b

15- a 16- 30°

18- 14 lseh- 19- 60°

20- 15 lseh- 22- 12 lseh- 23- 4 lseh- 25- f=kT;k = 4 lseh- 26- 20/3 lseh- 28- 2√66 lseh- 29- 1 lseh- 30- 5 lseh- 31- 11 lseh- 32- 32 lseh- 35-(2) 8 lseh- 35-(3) x = 35°, y = 55° 35-(4) 40°

36-(2) 8 lseh- 36-(4) 120°

116 Math (Hindi) ­X

vè;k;&6

jpuk,a

egÙoiw.kZ fcanq % 1- jpuk,a lkiQ lqFkjh] Li"V rFkk funsZ'kkuqlkj gksuh pkfg,A 2- jpuk ds in dsoy ogha fy[ksa tgk¡ funsZ'k fn, gksaA

iz'u 1- ,d js[kk[k.M AB = 7 lseh- [khfp,A AB ij ,d fcUnq P bl izdkj yhft, fd

AP : BP = 3 : 5 gksA 2- ,d js[kk[k.M PQ = 10 lseh- [khafp,A PQ ij ,d fcUnq A bl izdkj yhft, fd

gksA PA vkSj AQ dh yEckbZ ekfi,A 3- ,d ∆ABC dh jpuk dhft, ftlesa BC = 6.5 lseh-] AB = 4.5 lseh- rFkk ∠ACB

= 60°, ,d vU; f=kHkqt dh jpuk dhft, ∆ABC tks ds le:i gks rFkk ftldh

izR;sd Hkqtk ∆ABC dh laxr Hkqtk dk 4 5 gksA

4- ∆XYZ dh jpuk ftlesa XY = 5 lseh-] YZ = 7 lseh- rFkk ∠XYZ = 75°. ∆XYZ ~ ∆XYZ cukb, ftldh izR;sd Hkqtk dh laxr Hkqtk ds 3

2 Hkkx ds cjkcj gksA 5- ,d lefckgq f=kHkqt cukb, ftldk vk/kj lseh- rFkk 'kh"kZyEc 5 lseh- gks Sk

Dvd/ f=kHkqt dh jpuk dhft, ftldh Hkqtk,a fn;s x, f=kHkqt dh laxr Hkqtkvksa ds 3

4 xqus ds cjkcj gksaA 6- ,d lefckgq ∆ABC dh jpuk dhft, ftlesa AB = AC rFkk vk/kj BC = 7

lseh- vkSj 'kh"kZ dks.k 120° gksA ∆AB ‘C’ ~ ∆ABC dh jpuk dhft, ftldh Hkqtk,a

∆ABC dh laxr Hkqtkvksa dh 1 1 3 xquh gksA

117 Math (Hindi) ­X

7- ∆PQR dh jpuk dhft, ftlesa ∠Q = 90°, PQ = 6 lseh-] PQ = 8 lseh-] ∆PQR ~ ∆PQR dh jpuk dhft, ftldh Hkqtk;sa ∆PQR dh laxr Hkqtkvksa ds 2

3 Hkkx ds cjkcj gksaA

8- ,d ledks.k f=kHkqt dh jpuk dhft, ftldk vk/kkj 'kh"kZyEc dk nksxquk gksA bl f=kHkqt ds le:i ,d vU; f=kHkqt dh jpuk dhft, ftldk vk/kj fn, gq, f=kHkqt ds vk/kj dk 1-5 xquk gksA

9- ,d leckgq ∆PQR dh jpuk dhft, ftldh Hkqt 5 lseh- gksA ∆PQ ‘R’ dh jpuk

bl izdkj dhft, fd 1 .2

PQ PQ

= PQ’ dh yEckbZ ekfi,A

10- 4 lseh- f=kT;k dk ,d o`Ùk [khafp, ftldk dsUnz O gSA ,d fcUnq P bl izdkj yhft, fd OP = 6 lseh- fcUnq P ls o`Ùk ij Li'kZ js[kk;sa PA vkSj PB [khafp,A PA vkSj PB dh yEckbZ ekfi,A

11- ,d js[kk[k.M AB = 8 lseh- [khafp,A AB dks O;kl ekudj ,d o`Ùk [khafp, ftldk dsUnz O gksA OP⊥AB [khafp,A P ls o`Ùk dh Li'kZ js[kk [khafp,A

12- ,d o`Ùk ftldh f=kT;k OP = 3 lseh- gks [khafp,A ∠PQR = 45° bl izdkj cukb, fd OQ = 5 lseh- gksA fcUnq Q ls bl o`Ùk ij nks Li'kZ js[kk,a [khafp,A

13- 3-5 lseh- f=kT;k dk ,d o`Ùk [khafp, ftldk dsUnz O gksA ckg~; fcUnq P ls o`Ùk dh nks Li'kZ js[kk,a PA vkSj PB bl izdkj [khafp, fd ∠APB = 45°, ∠AOB +

∠APB dk eku Kkr dhft,A 14- 4 lseh- f=kT;k dk ,d o`Ùk [khafp,A ckg~; fcUnq P ls nks Li'kZ js[kk,a bl izdkj

[khafp, fd muds chp dk dks.k muds Li'kZ fcUnqvksa kjk dsUnz ij cus dks.k dk vk/k gksA

15- js[kk[k.M AB = 9 lseh- [khafp,A A vkSj B dks dsUnz ysdj nks o`Ùkksa dh jpuk dhft, ftudh f=kT;k,a Øe'k% 5 lseh- vkSj 3 lseh- gksA izR;sd ds dsUnz ls nwljs o`Ùk ij Li'kZ js[kk [khafp,A

16- 3-5 lseh- f=kT;k dk ,d o`Ùk [khafp, ftldk dsUnz O gSA ,d fcUnq bl izdkj yhft, fd OP = 6 lseh- OP o`Ùk dks fcUnq T ij izfrPNsn djrh gSA PQ vkSj PR nks Li'kZ js[kk,a [khafp,A Q dks R ls feykb,A fcUnq T ls AB || QR bl izdkj cukb, fd A rFkk B Øe'k% PR vkSj ij fLFkr gksaA

118 Math (Hindi) ­X

17- 7 lseh- O;kl dk ,d o`Ùk [khafp,A o`Ùk ij Li'kZ js[kkvksa dk ,d ;qXe bl izdkj [khafp, fd nksuksa Li'kZ js[kkvksa ds chp dk dks.k 60° gksA

18- dsUn O rFkk f=kT;k 3-5 lseh- dk ,d o`Ùk [khafp,A ,d kSfrt O;kl yhft,A bls nksuksa vksj rFkk rd bl izdkj c<+kb, OP = PQ = 7 fd lseh-A PA rFkk QB nks Li'kZ js[kk,a bl izdkj [khafp, fd ,d O;kl ds mQij gks vkSj nwljh uhpsA PQ || E D;k

119 Math (Hindi) ­X

vè;k;&7

ks=kfefr

egÙoiw.kZ fcanq % 1- c = 2πr tgk¡ a___ o`Ùk dh ifjf/

c___ o`Ùk dh f=kT;k

π = 22 7 ;k 3.14 (yxHkx)

2- o`Ùk dk ks=kiQy = πr 2 tgk¡ ‘r’ o`Ùk dh f=kT;k gSA

3- v/Zo`Ùk dk ks=kiQy ( ) 2 1 2

r π =

4- nks ladsUnzh; o`Ùkksa dk ks=kiQy = π (R 2 – r 2 )

tgk¡ ‘R’ rFkk ‘r’ nks ladsUnzh; o`Ùkksa dh f=kT;k,a gSaA = π (R + r) (R – r); R > r

5- dks.k Q okys f=kT;k[k.M ds laxr pki dh yEckbZ

366 Q l = 2

180 ×

° r π

180 Q l r π = ×

°

6- tc dsUnz ij cus dks.k dk va'kh; eki rks f=kT;k[k.M dk ks=kiQy Q ν ε rks f=kT;[k.M

dk ks=kiQy 2 360

Q r π = × °

7- ?kM+h dh feuV dh lqbZ kjk 60 feuV esa jfpr dks.k = ml ?kM+h dh feuV dh lqbZ

kjk 1 feuV esa jfpr dks.k 360 6 60

° = = °

120 Math (Hindi) ­X

8- ‘a’ Hkqtk okys ?ku dk lEiw.kZ i`"Bh; ks=k = 6a 2 oxZ bdkbZ 9- ?ku dk vk;ru] ftldh Hkqtk ‘a’ gS = a 3 ?ku bdkbZA 10- ?kukHk] ftldh foHkk,a l, b vkSj h gS] dk lEiw.kZ ks=k = 2 ( l× b +b × h+h× l ) oxZ

bdkbZA 11- ?kukHk] ftldh foHkk,a l, b vkSj h gS] dk vk;ru = l × b × h ?ku bdkbZA 12- f=kT;k r vkSj mQapkbZ h okys csyu dk xksyh; i`-ks- = 2πrh oxZ bdkbZA 13- f=kT;k r vkSj mQapkbZ h okys csyu dk lEiw.kZ i`-ks- = 2π (r + h) oxZ bdkbZA 14- f=kT;k r vkSj mQapkbZ h okys csyu dk vk;ru = πr 2 h ?ku bdkbZA 15- f=kT;k r vkSj mQapkbZ h okys 'kadq dk xksyh; i`- ks- = πrl, l fr;Zd mQapkbZ gS l =

2 2 r h = + tgk¡

16- 'kadq dk lEiw.kZ i`- ks=kiQy = πr (l + r) oxZ bdkbZA

17- 'kadq dk vk;ru 2 1 3

r h π = ?ku bdkbZA

18- r f=kT;k okys xksys dk i`-ks- = 4r 2 oxZ bdkbZA 19- r f=kT;k okys v¼Zxksys dk xksyh; i`-ks- = 2πr 2 oxZ bdkbZA 20- r okys v¼Zxksys dk lEiw.kZ i`-ks- = 3πr 2 oxZ bdkbZA

21- r f=kT;k okys xksys dk vk;ru 3 4 3

r π = ?ku bdkbZA

22- r f=kT;k okys v¼Zxksys dk vk;ru 3 2 3

r π = ?ku bdkbZA

23- fNUud dk xksyh; i`-ks- = πl (r+R) oxZ bdkbZ] tgka fNUud dh fr;Zd mQapkbZ rFkk rvkSj R o`Ùkh; fdukjksa dh f=kT;k,a gSaA

24- fNUud dk lEiw.kZ i`-ks- = πl (r+R) + = π(r 2 + R 2 ) oxZ bdkbZA 25- fNUud dk vk;ru = 1/3 πh (l 2 + R 2 + rR) ?ku bdkbZA

121 Math (Hindi) ­X

cgq oSdfYid iz'u

1- ml o`Ùk dk ks=kiQy crkb, ftldk O;kl ‘d’ gS%

(a) 2πd (b) 2

4 d π

(c) πd (d) πd 2

2- ;fn fdlh o`Ùk dh ifjf/ rFkk ks=kiQy vkil esa cjkcj gS rks ml o`Ùk dh f=kT;k gksxhA

(a) r = 1 (b) r = 7 (c) r = 2 (d) r = c

3- ;fn fdlh o`Ùk dh f=kT;k 7 lseh gS rks ml o`Ùk ds v/Zo`Ùk dk ifjeki gksxkA (a) 36 lseh- (b) 14 lseh- (c) 7π (d) 14π

4- nks o`Ùkksa dh f=kT;k,a Øe'k% 14 lseh- rFkk 6 lseh- gSaA ml o`Ùk dh f=kT;k D;k gksxh ftldh ifjf/ nksuksa o`Ùkksa dh ifjf/;ksa ds ;ksx ds leku gSA

(a) 19π (b) 19 lseh- (c) 25 lseh- (d) 32 lseh-

5- ;fn nks o`Ùkksa dh ifjf/;ksa dk vuqikr 4%5 gks rks muds ks=kiQyksa dk vuqikr D;k gksxkA

(a) 4 : 4 (b) 16 : 25 (c) 64 : 125 (d) 8 : 10

6- ,d leckgq f=kHkqt dk ks=kiQy gS rks ml f=kHkqt dh ,d Hkqtk gksxhA (a) 4 lseh- (b) 3√3 lseh- (c) 2 3

4 lseh- (d) 8 : 10

7- ;fn ,d ?kukHk dk vk;ru 440 lseh- 3 rFkk mlds vk/kj dk ks=kiQy 66 lseh- 2 gks rks mldh mQapkbZ gksxhA

(a) 40 3 lseh- (b) 20

3 lseh- (c) 440 lseh- (d) 66 lseh-

122 Math (Hindi) ­X

8- nks ?kuksa ds vk;ruksa dk vuqeki 8 % 125 gS rks muds i`"B ks=kiQyksa dk vuqikr gksxk\ (a) 8 : 125 (b) 2 : 5 (c) 8 : 25 (d) 16 : 25

9- ,d pki [k.M ifjeki ‘l’ rFkk f=kT;k ‘r’ gS rks ml pki [k.M dk ks=kiQy gksxkA (a) l, r (b) l, r 2

(c) 2

2 lr (d) l 2 , r

10- ,d pki dh yEckbZ 5π lseh- rFkk mldk ks=kiQy 10π lseh- 2 gS rks ml o`Ùk dh f=kt;k gksxh&

(a) 2 lseh- (b) 4 lseh- (c) 2√2 lseh- (d) 8 lseh-

11- rhu ?ku ftudh izR;sd Hkqtk ‘a’ gS] rks vkil esa lVkdj feyk;k tkrk gSA bl izdkj cus u;s ?kukHk dk vk;ru gksxk&

(a) a 2 (b) 3a 3

(c) a 3 (d) 6a 3

12- ,d rkj o`Ùk ds :i esa ftldh f=kT;k 7 lseh- gS ;fn rkj dks ,d oxZ ds :i esa eksM+ fn;k tk;s rks ml oxZ dk ks- gksxk&

(a) 11 lseh- 2 (b) 121 lseh- 2

(c) 154 lseh- 2 (d) 44 lseh- 2

123 Math (Hindi) ­X

y?kq mÙkjh; iz'u 13- ,d xksys dk vk;ru rFkk i`"B ks- vkil esa cjkcj gS rks ml xksys dh f=kT;k Kkr

dhft,A 14- fuEu v kd fr d k i fj eki Kkr d j ks ft l esaBC dks O;kl ekudj ,d v¼Zo`Ùk [khapk

x;k gSA

15- Nk;kafdr vkd`fr dk ks=kiQy Kkr djksA

16- fdlh oÙk dh ifjf/ rFkk O;kl dk varj 30 lseh- gS rks ml oÙk dh f=kT;k Kkr djksa 17- ml pki [k.M dk ifjeki Kkr djks ftldh f=kT;k 7 lseh- rFkk dsUnzh; dks.k 45° gSA 18- leyEc prqHkqZt ds izR;sd 'kh"kZ ls 7 lseh- f=kT;k dk f=kT;[k.M dkVk x;k gSA dkVs

x;s dqy Hkkx dk ks- Kkr dhft,A 19- mu f=kT; [k.Mksa ds ks=kiQyksa dh vuqikl Kkr djks ftuds dsUnzh; dks.k Øe'k%

120° rFkk 90° gSA 20- 16 lseh- × 12 lseh- × 8 lseh- vkdkj ds ?kuke esa ls 4 lseh- Hkqtk okys fdrus

?ku cuk;s tk ldrs gSaA

124 Math (Hindi) ­X

21- ,d csyu rFkk ,d 'kadq dk O;kl vkSj mQapkbZ nksuksa cjkcj gSaA muds vk;ruksa dk vuqikr D;k gksxk\

22- ,d csyu] ,d 'kadq rFkk ,d v¼Z xksyk leku vk/kj vkSj leku mQapkbZ ds gSaA buds vk;ruksa dk vuqikr D;k gksxk\

23- ,d lkbfdy dk ifg;k 10 fe-eh- tkus esa 500 pDdj yxkrk gSA ifg, dh ifjf/ dh yEckbZ fyf[k,A

24- ,d Bksl csyuv dh vk/kj dh f=kT;k vkSj mQapkbZ dk ;ksx 15 lseh- gSA ;fn leiw.kZ i`"Bh; ks- 660 eh- 2 gS rks csyuv ds vk/kj dh f=kt;k Kkr dhft,A

25- ,d 729 lseh- 3 vk;ru okys ?ku ls cM+s ls cM+k yEc o`Ùkh; 'kadq dkVk tkrk gSA bl 'kadq dh mQapkbZ Kkr dhft,A

26- ;fn ,d o`Ùk vkSj ,d leckgq f=kHkqt dk O;kl vkSj ,d Hkqtk Øe'k% leku gS rks muds ks=kiQyksa dk vuqikr fyf[k,A

27- ;fn ,d o`Ùk dh ifjf/] O;kl ls 30 lseh- vf/d gS rks o`Ùk dk O;kl Kkr djksA 28- 12 lseh- f=kT;k ds ,d o`Ùk dh pki dh yEckbZ 10π lseh- gSA bl pki dk dks.kh;

eki Kkr dhft,A 29- ,d o`Ùkkdkj [ksr dks 10 #- izfr ehVj dh nj ls pkj fnokjh yxkus dk [kpkZ 440

#i;s gS rks o`Ùkkdkj [ksr dk f=kT;k Kkr dhft,A 30- 14 lseh- O;kl okys pk¡ns (D) dk ifjeki Kkr dhft;s 31- ,d 15 eh- f=kT;k okys o`Ùkkdkj ikdZ ds pkjksa vksj 5 eh- pkSM+k jkLrk cuk;k tkrk

gSA jkLrs dk ks- Kkr dhft,A 32- nks o`Ùkksa dh f=kT;k,a Øe'k% 3 lseh- rFkk 4 lseh- gSaA ml o`Ùk dh f=kT;k Kkr dhft,

ftldk ks- nksuksa o`Ùkksa ds ks- ds ;ksx ds leku gSA 33- fp=k esa ‘l’ dk eku Kkr djksA

125 Math (Hindi) ­X

34- ABC ,d leckgq f=kHkqt ds vkdkj dk [ksr gS ftldh izR;sd Hkqtk 30 eh- gSA ,d xk; dks 10 eh- jLlh ls 'kh"kZ A ij ck¡/ fn;k tkrk gSA xk; kjk pjs x, [ksr dk ks- Kkr dhft,A

35- ,d pkj ia[kqM+h okyk o`Ùkkdkj ia[kk ftldh f=kT;k 20 lseh- gS rFkk izR;sd ia[kqM+h dsUnz ij 45° dk dks.k cukrh gSA pkjksa ia[kqfM+;ksa dk ks- Kkr dhft,A

36- Nk;kafdr Hkkx dk ifjeki Kkr dhft,A

37- layXu vkd`fr esa nks ledsUnzh; o`Ùk ftudh f=kT;k,a 7 lseh- rFkk 14 lseh- gSA ;fn ∠AOC = 120° gks rks Nk;kafdr Hkkx dk ks- Kkr dhft,A

126 Math (Hindi) ­X

38- Nk;kafdr Hkkx dk ifjeki Kkr dhft,A

39- layXu vkd`fr esa o`Ùk dh ifjf/ Kkr dhft,A

40- nks o`Ùkksa dh f=kT;kvksa dk vuqikr 3%4 gSA ml o`Ùk dh f=kT;k Kkr djks ftldk ks- nksuksa o`Ùkksa ds ks=kiQyksa ds ;ksx ds leku gS rFkk igys o`Ùk dh f=kT;k 6 lseh- gSA

41- ml cM+s ls cM+s le ckgq ∆ dk ks=kiQy Kkr djks tks ‘r’ lseh- f=kT;k okys v¼Zor ds vUrxZr [khapk tk ldsA

42- ,d 20 lseh- yEckbZ okys rkj dks ,d o`Ùk ds pki ds vkdkj esa eksM+k tkrk gS tks dsUæ ij 60° dk dks.k vUrfjr djrk gSA o`Ùk dh f=kT;k Kkr dhft,A

43- ,d ?kM+h dh feuV okyh lqbZ dh yEckbZ √12 lseh- gSA lqbZ kjk 8-00 ls ysdj 8-05 a.m. rd of.kZr ks=kiQy dh x.kuk dhft,A

44- Nk;kafdr Hkkx dk ks=kiQy Kkr dhft,A (π = 3.14 yhft,)

127 Math (Hindi) ­X

45- Nk;kafdr Hkkx dk ks=kiQy Kkr dhft,A

46- layXu vkd`fr esa pki [k.M dk ks=kiQy Kkr dhft,A

47- ABCD ,d oxkZdkj irax gSA Nk;kafdr Hkkx dk ks=kiQy Kkr dhft,A

48- fdlh ?ku dk vk;ru 8a 3 gSA mldk dqy i`"B ks=kiQy Kkr dhft,A 49- ,d ?ku ds fod.kZ dh yEckbZ 17-32 lseh- gSA ?ku dk vk;ru Kkr djksA

(√3 = 1.732).

50- rhu ?ku dh ftudh Hkqtk,a Øe'k% 6 lseh-] 8 lseh- rFkk 10 lseh- gS dks fi?kykdj ,d u;k ?ku cuk;k tkrk gSA u;s ?ku ds fod.kZ dh yEckbZ Kkr dhft,A

128 Math (Hindi) ­X

nh?kZ mÙkjh; iz'u 51- ,d fNUud dh mQ¡pkbZ 4 lseh- rFkk vk/kj dh f=kT;k,a Øe'k% 3 lseh- rFkk 6 lseh-

gSaA fNUud dh mQ¡pkbZ Kkr dhft,A 52- ,d yEc o`Ùkh; csyu dk vk;ru 448π lseh- 3 rFkk mQ¡pkbZ 7 lseh- gSA csyu dh

f=kt;k Kkr djksA 53- fdlh ?ku dk ik'oZ i`"B ks- 64 lseh- 2 gS ml ?ku dh Hkqtk Kkr dhft,A 54- ,d le prqHkqZt dk ks- 24 lseh- 2 rFkk ,d fod.kZ dh yEckbZ 8 lseh- gS rks nwljs

fod.kZ dh yEckbZ Kkr dhft,A 55- ,d 30 lseh- × 24 lseh- × 18 lseh- ds ?kuke esa j[kh tk ldus okys lcls cM+h

NM+h dh yEckbZ Kkr dhft,A 56- ,d yEc o`Ùkh; csyu dk oØi`"B 16π lseh- 2 gSA ;fn csyu dh f=kT;k 4 lseh-

gS rks mldh mQ¡pkbZ Kkr dhft,A 57- 50 o`Ùkkdj IysVsa ,d nwljs ds mQij j[kh tkrh gSA bl izdkj cus csuv dk vk;ru

o mQ¡pkbZ Kkr dhft, ;fn IysV dh eksVkbZ 1 2 lseh- rFkk f=kT;k 7 lseh- gSA

58- ,d 2 ehVj O;kl okyk dqvk¡ 14 ehVj xgjk [kksnk tkrk gSA bl dq,a ls fudkyh xbZ feV~Vh dk vk;ru Kkr dhft,a

59- ,d cM+s ls cM+k xksyk ,d 7 lseh- Hkqtk okys ?ku esa ls dkVdj cuk;k x;k gSA xksys dh f=kt;k Kkr dhft,A

60- ,d yEc o`Ùkh; 'kadq dk v¼Z'kh"kZ dks.k 60° rFkk mQ¡pkbZ 3 lseh- gSA 'kadq dk vk;ru Kkr dhft,A

61- ml ?ku dh Hkqtk Kkr dhft, ftldk vk;ru (8 × 4 × 2 lseh- 3 ?kuke ds vk;ru ds leku gSA

62- ml yEc o`Ùkh; 'kadq dk vk;ru Kkr dhft, ftldh mQ¡pkbZ ‘2h’ rFkk f=kT;k ‘r’ gSA 63- D;k ;g fdlh yEc o`Ùkh; csyu can ds fy, laHko gS fd mldk likV ks=kiQy

mlh ds oØ ks=kiQy ds leku gksA ;fn gk¡ rks dSls\ 64- ;fn fdlh fnu 5 lseh- o"kkZ ntZ dh tkrh gS rks ml fnu 2 gsDVs;j [ksr esa iSQys

ikuh dk vk;ru Kkr dhft,A (1 gSDVs;j = 10000 eh- 2 )A

129 Math (Hindi) ­X

65- ‘R’ f=kT;k okys v¼Zxksys dk lEiw.kZ i`"b dk ks- Kkr dhft,A 66- layXu vkd`fr esa ∆ABC ,d leckgq f=kHkqt gSA ;fn o`Ùk dh f=kt;k 4 lseh- gS rks

Nk;kafdr Hkkx dk ks- Kkr dhft,A

67- Nk;kafdr Hkkx dk ks=kiQy Kkr dhft,A

68- pkj xk;sa 7 ehVj dh yEckbZ okyh jLlh ls ,d prqHkqZtkdkj [ksr ds pkjksa 'kh"kks± ij ck¡/h tkrh gSaA xk;ksa kjk pjs x;s [ksr dk ks=kiQy Kkr dhft,A

69- ,d Bksl yEc o`Ùkh; csyu ds mQij ,d yEc o`Ùkh; 'kadq ds vkdkj dk gSA 'kadq dh mQ¡pkbZ ‘h’ lseh- gS rFkk Bksl dk vk;ru 'kadq ds vk;ru dk rhu xq.kk gSA csyu dh mQ¡pkbZ Kkr dhft,A

70- ,d crZu yEc o`Ùkh; csyu ds vkdkj dk gS ftlesa jsr Hkjk gSA csyu dh mQ¡pkbZ 36 lseh- rFkk f=kT;k 18 lseh- gSA jsr dks tehu ij [kkyh fd;k tkrk gSA ftlls jsr dk ,d 'kadq ds vkdkj dk <sj cu tkrk gSA ;fn <sj dh mQ¡pkbZ 27 lseh- gks rks mldk vk/kj Kkr dhft,A

130 Math (Hindi) ­X

71- ,d ckYVh dh f=kT;k,a 5-5 rFkk 15-5 lseh- rFkk mQ¡pkbZ 24 lseh- gS rks ckYVh dk lEiw.kZ i`"B ks- Kkr dhft,A

72- ,d o`Ùkh; ikbZi ftldk O;kl 2 lseh- gS ls ikuh 6 eh-@ls- dh xfr ls cgrk gqvk ,d csyukdkj crZu esa tk jgk gSA ;fn csyukdkj crZu dh f=kT;k 60 lseh- gS rks 30 feuV esa crZu ds ikuh dh mQ¡pkbZ D;k gksxh\

73- Nk;kafdr Hkkx dk ks- Kkr dhft,A

74- Nk;kafdr Hkkx dk ks=kiQy Kkr dhft,A

75- Nk;kafdr Hkkx dk ks=kiQy Kkr dhft,A

131 Math (Hindi) ­X

76- AB rFkk CD nks yEcor O;kl gS ;fn CD = 8 lseh- gks rks Nk;kafdr Hkkx dk ks=kiQy Kkr dhft,A

77- layXu vkd`fr esa ∆ABC ,d ledks.k f=kHkqt gS (∠A = 90) AB.BC rFkk AC dks O;kl ekudkj v¼Z o`Ùk [khaps x;s gSaA Nk;kafdr Hkkx dk ks=kiQy Kkr dhft,A

78- ,d f[kykSuk 'kadq ds fNUud ij j[kk 'kadq ds vkdkj dk gSA ;fn mldh f=kT;k,a Øe'k% 14 lseh- rFkk 7 lseh- gS rFkk 'kadq vkSj f[kykSus dh mQ¡pkbZ Øe'k% 5-5 lseh- rFkk 10-5 lseh- gS rks f[kykSus dk vk;ru Kkr dhft,A

132 Math (Hindi) ­X

79- layXu vkd`fr esa ,d ledks.k f=kHkqt gS ;fn lseh- lseh- rFkk vUr% o`Ùk dk dsUnz gS rks Nk;kafdr Hkkx dk ks=kiQy Kkr dhft,A (π = 3.14)

mÙkjekyk 1- b 2- c

3- a 4- b

5- b 6- d

7- b 8- c

9- c 10- b

11- b 12- b

13- 3 units 14- 3 37 7 lseh-

15- 49 lseh- 3 16- 14 lseh- 2

17- 19.5 lseh- 18- 154 lseh- 2

19- 4 : 3 20- 24

21- 3 : 1 22- 3 : 1 : 2

23- 20 eh- 24- 7 lseh- 25- 9 lseh- 26- π : √3

27- 14 lseh- 28- 150°

133 Math (Hindi) ­X

29- 7 lseh- 30- 36 lseh- 31- 550 eh- 32- 5 lseh-

33- 22 lseh- 34- 2 50 3

m π

35- 300π 36- (16 + π) lseh- 37- 154 lseh- 9 38- 42π

39- 25π 40- 10 lseh-

41- r 2 42- 60π

lseh-

43- π lseh- 9 44- 86 lseh- 45- (25 – 4π) lseh- 2 46- 3π lseh- 2

47- (16 – 4π) lseh- 2 48- 24 a 2

49- 1000 lseh- 3 50- 12√3 lseh- 51- 5 lseh- 52- 8 lseh- 53- 4 lseh- 54- 6 lseh- 55- 30√2 lseh- 2 56- 2 lseh- 57- 25 cm; 3850 lseh- 3 58- 44 eh- 3

59- 3.5 lseh- 60- 27π

61- 4 lseh- 62- 2 2 . 3

r h π

63- gk¡] tc r = h 64- 1000 eh- 3

65- 3πR 2 66- 29.46 lseh- 3

67- 2 660 36 3 7

cm + 68- 154 lseh- 2

69- 2 3

h 70- 36 lseh-

134 Math (Hindi) ­X

71- 1716 lseh- 2 72- 3 eh-

73- 2 1019 14

cm 74- 154 eh- 2

75- 77 lseh- 2 76- 2 108 7

cm

77- 6 lseh- 2 78- 2926 lseh- 3

79- 11.44 lseh- 2 78- 2926 lseh- 3

[Hint : Join 0 to A, B and C. area of ∆ABC = area of ∆OAB + area ∆OBC + are of ∆OAC

1 1 2 2

AB BC r AC r = × × + ×

( 2 )] r cm ⇒ =

135 Math (Hindi) ­X

vè;k;&8

izkf;drk

egÙoiw.kZ fcanq % 1- fdlh ?kVuk dh lS¼kafrd izkf;drk fuEufyf[kr :i esa ifjHkkf"kr dh tkrh gSA

E P(E) =

ds vudq yw ifj.kkekas dh l[a ;k i;z kxs d s lHkh lHa ko ifj.kkekas dh l[a ;k

;gk¡ ;g dYiuk dh xbZ gS fd iz;ksx ds ifj.kke leizkf;d gSa 2- fdlh iz;ksx dh lHkh izkjafHkd ?kVukvksa dh izkf;drkvksa dk ;ksx 1 gksrk gSA ;g

O;kid :i ls Hkh lR; gSA 3- fuf'pr ?kVuk dh izkf;drk ,d gksrh gS rFkk vlEHkkfor ?kVuk dh izkf;drk 'kwU;

gksrh gSA

4- O;kid :i ls fdlh ?kVuk E ds fy, ;g lR; gS fd ( ) ( ) 1 P E P E + =

5- izkf;drk dh ifjHkk"kk ds vuqlkj izkf;drk dk va'k lnSo gj ls NksVk gksrk gS ;k mlds cjkcj gksrk gS vr% 0 ≤ P(E) ≤ 1.

cgq oSdfYid iz'u

1- ;fn ,d ?kVuk gS rks ( ) ( ) .......... P E P E + =

(a) 0 (b) 1 (c) 2 (d) –1

136 Math (Hindi) ­X

2- ,d ?kVuk] ftldk gksuk fuf'pr gS fd izkf;drkA (a) 0 (b) 2 (c) 1 (d) –1

3- fuEu esa dkSu lh fdlh ?kVuk dh izkf;drk ugha gks ldrhA

(a) 2 3 (b) 3

2 −

(c) 15% (d) 0.7

4- ;fn P(E) = .65 rks O (E, ugha) D;k gksxh\ (a) .35 (b) .25 (c) 1 (d) 0

5- ;fn fdlh ?kVuk dh liQy gksus dh izkf;drk 38» gS rks mlds vliQy gksus dh izkf;drk D;k gksxh\

(a) 12% (b) 62% (c) 1 (d) 0

6- ,d FkSys esa 9 yky rFkk uhys dkap gSaA ,d dapk ;kn`PN;k fudkyk x;kA izkf;drk crkb, fd fudkyk x;k dapk yky gSA

(a) 7 16 (b) 9

16

(c) 18 16 (d) 14

16

7- ,d losZk.k esa ;g ik;k x;k fd izR;sd ik¡pos O;fDr ds ikl xkM+h gSA fdlh O;fDr ds ikl xkM+h ugha gksus dh izkf;drk Kkr djksA

(a) 1 5 (b) 4

5

(c) 3 5 (d) 1

137 Math (Hindi) ­X

8- vkuUn rFkk lqfe=k fe=k gSaA nksuksa fe=kksa dk 11 uoeEcj dks tUe fnu gksus dh izkf;drk&

(a) 1 12 (b) 1

7

(c) 1365 (d) 1

366

9- 52 iÙkksa dh vPNh rjg isaQVh x;h xM~Mh esa fp=k dkMks± dh la[;k& (a) 12 (b) 16 (c) 4 (d) 52

10- ,d ikls dks ,d ckj mNkyk tkrk gSA ,d le vHkkT; la[;k vkus dh izkf;drk&

(a) 3 4 (b) 1

6

(c) 1 2 (d) 1

3

11- ,d vlaHko ?kVuk dh izkf;drk (a) 0 (b) 1 (c) –1 (d) ∝

12- ,d FkSys esa dqN dkMZ j[ks gSa ftu ij 1 ls 20 vafdr gSA ,d dkMZ ;n`PN;k [khapk x;kA dkMZ ij 8 ls 15 ds chp la[;k vafdr gksus dh izkf;drk&

(a) 8 20 (b) 6

20

(c) 1520 (d) 0

138 Math (Hindi) ­X

y?kq mÙkjh; iz'u 13- ,d ?kwers gq, HkkX; pØ ij 1 ls 10 vafdr gSA izkf;drk crkb, ;fn pØ #dus

ij izkIr la[;k 5 ;k 5 ls de gksA 14- nks iklksa dks ,d lkFk mNkyk tkrk gSA nksuksa iklksa esa leku vad izkIr gksus dh

izkf;drk crkb,A 15- ,d ikls dks ,d ckj mNkyk tkrk gSA ,d vHkkT; la[;k vkus dh izkf;drk crkb,A 16- ,d cSad ,-Vh-,e- esa 100] 500] vkSj 1000 #i;s ds cjkcj&cjkcj uksV gSaA ,d

gtkj dk uksV izkIr djus dh izkf;drk crkb,A 17- ,d ikls dks ,d ckj mNkyk tkrk gSA 6 ls vf/d vad izkIr djus dh izkf;drk

crkb,A 18- lsYl eSustj ds in ds fy, p;u dÙkkZ xBu us 50 O;fDr;ksa dks lkkkRdkj fd;k

ftuesa 35 iq#"k rFkk 15 efgyk,a mEehnokj gSaA efgyk mEehnokj dk p;u gksus dh izkf;drk crkb,A

19- ,d FkSys esa dqN dkMZ gSa ftu ij la[;k 5 ls 25 vafdr gSA ,d dkMZ ;n~PN;k [khapk x;k gSA dkMZ ij la[;k 10 15 vafdr gksus dh izkf;drk crkb,A

20- ,d ykWVjh ds 1000 fVdVksa esa 5 bZuke thous okys fVdV gSaA ;fn ,d O;fDr ,d fVdV [kjhnrk gS rks mlds thous dh izkf;drk crkb,A

21- ,d fMCcs esa 600 isap gSa ftuesa ls 42 isap [kjkc gSA ,d isap ;n~PNk;k fudkyk tkrk gSA isap [kjkc ugha gS fd izkf;drk crkb,A

22- ,d flDds dks nks ckj mNkyus ij vkus okys lHkh laHkkfor ifj.kke fyf[k,A 23- nks iklksa dks ,d lkFk mNkyk tkrk gSA nksuksa iklksa ij vadksa dk ;ksx 10 ;k 10 ls

vf/d vkus dh izkf;drk crkb,A 24- 52 iÙkksa dh vPNh rjg isaQVh x;h xM~Mh ls nks dkys ckn'kky rFkk nks yky bDds

fudkys x, gSaA ;fn ,d iÙkk ;n`PN;k fudkyk tk, rks mlds fp=k dkMZ gksus dh izkf;drk crkb,A

25- fdlh yhi o"kZ esa 53 jfookj gksus dh izkf;drk crkb,A 26- ,d fMCcs esa 2 ls 101 dh la[;k okys dqN dkMZ gSaA ,d dkMZ ;n`PNk;k fudkyk

tkrk gSA dkMZ ij vafdr la[;k ifjiw.kZ oxZ gksus dh izkf;drk Kkr dhft,A

139 Math (Hindi) ­X

27- ,d FkSys esa larjk] vke rFkk uhcw¡ dh lqxU/ okyh dSUMht gSaA ,d dSUMh ;n`PN;k fud ky h xbZ v kSj ; g i k; k x; k P (uhcw¡ okyh ugha) 11

15 p = (vke okyh) 1

3 =

rks P (larjk okyh) Kkr dhft,A 28- 52 iÙkksa dh vPNh rjg isaQVh xbZ xM~Mh esa ls ,d gh jax okys dqN iÙks xqe gSa

ftlesa P (yky dkMZ) 1 3

= vkSj P (dkys dkMZ) 2 3

= gks x;hA rks fdl jax ds iÙks xqe gSa vkSj fdrus\

29- ,d FkSys esa 5 yky xsan rFkk ‘n’ gjh xsan gSaA ;fn P (gjh xsan) = 3 × P (yky xsan) rks ‘n’ dk eku Kkr dhft,A

30- ;fn rk'k dh xM~Mh esa ls lHkh bDdksa dks fudky fn;k tk, rks yky dkMZ izkIr djus dh izkf;drk crkb,A

31- nks iklksa dks ,d lkFk isaQdus ij ;ksx 12 ls de vkus dh izkf;drk crkb,A 32- ;fn (1] 4] 9] 16] 25] 29) esa ls 29 dks gVk fn;k tk, rks ,d vHkkT; la[;k

izkIr djus dh izkf;drk crkb,A 33- rk'k ds iÙkksa dks xM~Mh ls ,d rk'k dk iÙkk fudkyk tkrk gSA ,d O;fDr ;g 'krZ

yxkrk gS fd ;k rks ;g iÙkk gqDe dk gS ;k fiQj bDdk gSA mlds thous dk izfrdwy la;ksxkuqirk crkb,A

nh?kZ mÙkjh; iz'u 34- ;fn ,d flDds dks rhu ckj mNkyk tkrk gS rks izkf;Drk Kkr dhft,&

(i) 2 fpÙk (ii) 2 iV (iii) 2 fpÙk

35- rk'k ds 52 iÙkksa esa ls fpfM+ ds ckn'kkg] csxe rFkk xqyke dks gVk fn;k tkrk gS rFkk fiQj cps gq, iÙkksa dh vPNh rjg isaQVk tkrk gS vkSj muesa ls ,d dkMZ fudkyk tkrk gSA izkf;drk Kkr dhft, (i) iÙkk iku dk gS (ii) iÙkk csxe gS (iii) iÙkk fpfM+ dk gSA

36- ,d fMCcs esa 5 xsan yky] 8 xsan lisQn rFkk 4 xsan gjh gSaA ;fn ,d xsan ;n~PN;k fudkyh tkrh gS rks izkf;drk Kkr dhft, (i) yky xsan (ii) lisQn xsan (iii) xsan gjh ugha gSA

140 Math (Hindi) ­X

37- 120 lgh isuksa esa 12 isu [kjkc feys gq, gSaA ;fn buesa ls ,d isu ;n`PN;k fudkyk tk;s rks isu dh lgh gksus dh izkf;drk Kkr dhft,A

38- (i) ,d 20 cYcksa ds laxzg esa 5 cYc [kjkc gSaA ,d cYc ;n`PN;k fudkyk tkrk gSA cYc ds [kjkc gksus dh izkf;drk Kkr dhft,A

(ii) ;fn (i) esa fudkyk x;k cYc [kjkc ugha gS vkSj mls okfil Hkh ugha j[kk tkrkA vc ,d cYc 'ks"k cps cYcksa ls fudkyk tkrk gS rks cYc ds [kjkc ugha gksus ds izkf;drk Kkr dhft,A

39- ,d fMCcs esa 1 ls 90 vad okyh 90 fMLd gSaA ,d fMLd ;n`PN;k fudkyh tkrh gSA izkf;drk Kkr dhft, fd fMLd ds mQij vafdr la[;k (i) ,d nks vadksa okyh la[;k gS (ii) ,d ifjiw.kZ deZ gS (ii) 5 ls foHkkftr gSA

40- ,d [ksy esa ,d flDds dks rhu ckj mNkyk tkrk gS rFkk mlds ifj.kke dks fy[kk tkrk gSA vkuUn thrrk gS ;fn rhuksa ifj.kke ,d tSls vkrs gSa tSls rhu fpÙk vFkok rhu iVA vkuUn ds gkjus dh izkf;drk Kkr dhft,A

41- ,d ik'ks dks nks ckj isaQdk tkrk gS izkf;drk Kkr dhft, (i) ;ksx 7 gks (ii) ;ksx 10 ls vf/d gks (iii) dksbZ lh Hkh ckj 5 uk vk;sA

42- ,d cSx esa 12 xsan gS ftuesa ls × xsans lisQn jax dh gSaA

(i) ;fn ;n~PN;k ,d xsan fudkyh tkrh gS rks D;k izkf;drk gksxh fd og xsan lisQn gSA

(ii) ;fn cSx esa 6 lisQn xsan vkSj Mky nh tkrh gSa rks ,d lisQn xsan fudkyus dh izkf;drk iz'u (x) dh viskk nqxquh gks tkrh gS rks dk eku Kkr dhft,A

43- ,d tkj esa 24 daps gSa] ftuesa dqN gjs gS rFkk dqN uhys gSaA ;fn bl tkj esa ls

;n`PN;k ,d dapk fudkyk tkrk gS rks bl daps dks gjk gksus dh izkf;drk 2 3 gSA

tkj esa uhys dapksa dh la[;k Kkr dhft,A

141 Math (Hindi) ­X

mÙkjekyk 1- b 2- c

3- b 4- a

5- b 6- b

7- b 8- c

9- a 10- b

11- a 12- b

13- 1 2 14- 1

6

15- 1 2 16- 1

3

17- 0 18- 3 10

19- 2 7 20- 1

200

21- 93 100 22- S = [HH, TT, HT, TH]

23- 1 6 24- 5

24

25- 2 7 26- 9

100

27- 2 5 28- yky] 13

29- 15 30- 1 2

31- 35 35 32- 'kwU;

142 Math (Hindi) ­X

33- 9 13 34- (i) 3 ;

8 (ii) 3 ;8 (iii) 1

8

35- (i) 13 ; 49 (ii) 3 ;

49 (iii) 10 49 36- (i) 5 ;

17 (ii) 8 ; 17 (iii) 13

17

37- 9 10 38- (i) 1 ;

4 (ii) 14 19

39- (i) 9 ; 10 (ii) 1 ;

10 (iii) 1 5 40- 3

4

41- (i) 1 ;6 (ii) 1 ;

12 (iii) 25 36 42- (i) ;

12 x (ii) x = 3

43- 8

143 Math (Hindi) ­X

BLUE PRINT xf.kr SA­1

fo"k; oSdfYid y?kq iz'u­I y?kq iz'u­II nh?kZ iz'u dqy 1 vad 2 vad 3 vad 4 vad

cht xf.kr 4 (4) 3 (6) 2 (6) 1 (4) 10 (20)

T;kferh 1 (1) 1 (2) 3 (9) 1 (4) 6 (16)

ks=kfefr 1 (1) 2 (4) 1 (3) 3 (12) 7 (20)

f=kdks.kkfefr ds 1 (1) – 1 (3) 1 (4) 3 (8) vuqiz;ksx

funsZ'kkad T;kfefr 2 (2) 1 (2) 2 (6) – 5 (10)

izkf;drk 1 (1) 1 (2) 1 (3) – 3 (6)

dqy 10 (10) 8 (16) 10 (30) 6 (24) 34 (80)

144 Math (Hindi) ­X

xf.kr & lSEiy iz'u i=k

xf.kr dkk X (SA­2)

Time Allowed : 3 to 3½ hours Maxmimum Marks : 80

lkekU; funsZ'k 1 lHkh iz'u vfuok;Z gSaA 2- bl iz'u i=k esa dqy 34 iz'u gSa tks pkj Hkkxksa A, B, C vkSj D esa foHkkftr gSaA

Hkkx&A esa 10 iz'u gS izR;sd iz'u 1 vad dk gSA Hkkx&B esa 8 iz'u gSa izR;sd iz'u 2 vad dk gSA Hkkx&C esa 10 iz'u gSa izR;sd iz'u 3 vad dk gSA Hkkx&D esa 6 iz'u gSa izR;sd iz'u 4 vad dk gSA

3- Hkkx&A esa iz'u la[;k 1 ls 10 cgqfodYih; iz'u gSA pkj fodYiksa esa ls foDyi lgha gSA

4- dqy iz'u&i=k esa fodYi ugh gSA ;|fi 2 vadks okys ,d iz'u esa] 3 vadks okys rhu iz'uksa esa rFkk 4 vadks okys nks iz'uksa esa vkUrfjd fodYi fn;s x;s gSA ,sls lHkh iz'uksa esa vkidks dsoy ,d fodYi gh djuk gSA

5- dSydqysVj ds iz;ksx dh vuqefr ugha gSA

cgqoSdfYid mÙkjh; iz'u 1- ;fn la[;k,a &3] &2] &1] 0] 1] 2] 3 esa ls ;kn`PN;k ,d la[;k x pquh tkrh gS]

rks |x|< 2 gksus dh izkf;drk&

(a) 5 7 gS (b) 2

7 gS

(c) 3 7 gS (d) 1

7 gS

2- ;fn 3x 2 – 5 + k = 0 dk ewy 1 gS] rks ‘K’ dk eku gSA (a) –2 (b) –8 (c) 8 (d) 2

145 Math (Hindi) ­X

3- ;fn ,d l-Js- ds izFke inksa dk ;ksxiQy n 2 gS] rks lkoZvUrj gSA (a) 1 (b) 2 (c) 3 (d) 4

4- nks fcUnqvksa (3-0) vkSj (0] x) ds chp dh nwjh 5 bdkbZ gS] dk eku gSA (a) 3 (b) 4 (c) 5 (d) 6

5- fdlh k.k 30 ehVj Å¡ps [kEHks dh Nk;k dh yEckbZ 10 3 eksVh gSA lw;Z dk mUu;u dks.k gS &

(a) 60° (b) 30° (c) 45° (d) 90°

6- ;fn l-Js- 4] 9] 14] --------- dk xok¡ in 124 gS] rks n gSA (a) 25 (b) 26 (c) 27 (d) 24

7- ml fcUnq ds funsZ'kkad tgk¡ js[kk 1 2 3 x y

+ = x­vk dks izfrPNsn djrh gSA (a) (2, 0) (b) (0, 2) (c) (3, 0) (d) (0, 3)

8- ;fn f?kkr lehdj.k ax 2 + bx + c + 0, a ≠ 0 ds ewy cjkcj gS] rks C gSA

(a) 2 b a

− (b) 2 b a

(c) 2

4 b a

− (d) 2

4 b a

9- nh xbZ vkÑfr esa vkSj o`Ùk dh Li'kZ js[kk,a gSa rFkk rks dk eku gS &

146 Math (Hindi) ­X

(a) 35° (b) 70° (c) 40° (d) 50°

10- ,d rkj dks eksM+dj 14 lseh- f=kT;k dk ,d o`Ùk cuk;k tk ldrk gSA ;fn blr rkj dks ,d oxZ ds vkdkj esa eksM+k tk;s rks oxZ dh Hkqtk dh yEckbZ gksxh&

(a) 44 lseh- (b) 22 lseh- (c) 88 lseh- (d) 14 lseh-

11- ;fn ,d l-Js- dk 8ok¡ in dk 8 xquk] 12osa in ds 12 xquk ds cjkcj gS rks bldk 20ok¡ in Kkr dhft,A

12- ,d vUr% o`Ùk lefckgq f=kHkqt dh leku Hkqtkvksa dks fcUnq E rFkk F ij Li'kZ djrk gSA fn[kkb, fd fcUnq D, tgk¡ o`Ùk rhljh Hkqtk ds Li'kZ djrk gS ml Hkqtk dk eè; fcUnq gSA

13- ;fn ,d lkbZfdu dk ifg;k 11 fdeh- pyus esa 5000 pDdj yxkrk gS rks ifg;ksa dk O;kl Kkr dhft,A

14- ml f=kHkqt dk ks=kiQy Kkr dhft, ftlds 'kh"kks± ds funsZ'kkad (1] &1)] (&3] 5) vkSj (2] &7 gS)

15- ;fn nks Øekxr izkd`r la[;kvksa dk xq.kuiQy 30 gS] rks la[;k,a Kkr dhft,A 16- nks l-ks- ds n inksa ds ;ksxiQy 3n + 8 : 7n + 15 ds vuqikr esa gSA blds uosa inksa

dk vuqikr Kkr dhft,A 17- ,d o`Ùk ds pki dh yEckbZ 5π gS rFkk f=kT;k [k.M 20π oxZ lseh- ds ks=kiQy ls

f?kjk gS o`Ùk dh f=kT;k Kkr dhft,A 18- ,d FkSy esa 2] 3] 4] ----- 101 ls vad okys vafdr dkMZ gSaA ,d dkMZ ;n`PN;k

fudkyk tkrk gSA

(i) fo"ke la[;k gSA

(ii) foHkkftr la[;k gS tks 26 ls de gSA

vFkok

nks iklksa dks ,d lkFk mNkyk x;k gSA nksuksa iklksa ds vadksa dk ;ksx 10 vkSj ls vf/d gksus dh izkf;drk Kkr dhft,A

147 Math (Hindi) ­X

19- f?kkr lehdj.k 2x 2 + 5x – 7 = 0 iwoZ oxZ cukus dh fof/ ls ewy Kkr dhft,A 20- ,d l-ks- ds izFke 9 inksa dk ;ksxiQy 171 gS rFkk izFke 24 inksa dk ;ksxiQy 996

gSA l-Js- Kkr dhft,A

vFkok

,d l-ks- ds izFke 16 inksa dk ;ksxiQy 528 gS vkSj vxys 16 inksa dk ;ksxiQy 1552 gS] bldk 19ok¡ in Kkr dhft,A

21- ∆ABC dh jpuk dhft, ftlesa BC = 6.5 lseh- AB = 4.5 lseh- rFkk ∠ACB = 60°. ∆ABC ds le:i ,d vU; f=kHkqt dh jpuk dhft, ftldh izR;sd Hkqtk

∆ABC dh laxr Hkqtk dk 4 5 xquk gksA

22- PQ o`Ùk dk ,d O;kl gS rFkk thok PR bl izdkj gS fd ∠RPQ = 30° rFkk ∠QSR = 30° R ij Li'kZ js[kk PQ dks c<+kus ij fcUnq S ij feyrh gSA fl¼ dhft, fd RQ = QS.

vFkok

nh xbZ vkd`fr esa PQ 5 lseh- f=kT;k o`Ùk dh ,d thok gS rFkk PQ = 8 lseh-A P vkSj Q ij Li'kZ js[kk,a fcUnq T ij feyrh gSaA TP dh yEckbZ Kkr dhft,A

23- 3-5 lseh- f=kT;k dk ,d o`Ùk [khaft, ftldk dsUnz ‘O’ gSA ,d ckg~; fcUnq P ls bl o`Ùk ij Li'kZ js[kk,a PA vkSj PB bl izdkj [khafp, fd ∠APB = 45°, PA vkSj PB dh yEckbZ ekfi,A

24- fp=k esa 7 lseh- vkSj 3-5 lseh- f=kT;kvksa okys nks ldsUnzh; o`Ùkksa ds f=kT; [k.M

iznf'kZr fd;s x;s gSaA èok;kafdr Hkkx dk ks=kiQy Kkr dhft,A 22 7

π

148 Math (Hindi) ­X

25- ,d O;fDr ftldh mQ¡pkbZ 1-5 ehVj gS ,d fpeuh ls 28-5 ehVj dh nwjh ij gSA fpeuh ds f'k[kj dk mUu;u dks.k 30 gSA fpeuh dh mQ¡pkbZ Kkr dhft,A (√3 = 1.73).

26- og vuqikr Kkr dhft, ftlesa js[kk 3x + y = 12, fcUnq (1] 3) vkSj 2] 7) dks feykus okys js[kk[k.M dks dkVrh gSA

vFkok fn[kkbZ fd fcUnq (&2] 3)] (8] 3) rFkk (6] 7) f=kHkqt ds 'kh"kZ gSaA

27- x­vk ij ,d fcUnq P fcUnqvksa (4] 5) vkSj 1] &3) dks feykus okys js[kk[k.Mksa dks fdlh vuqikr esa foHkkftr djrk gSA fcUnq P ds funsZ'kkad Kkr dhft,A

28- ,d tkj esa 54 daps] uhys] gjs rfkk lisQn gSaA tkj ls uhps daps pquus dh izkf;drk 1 3 rFkk gjs daps pquus dh izkf;drk 4

9 gS rks tkj esa fdrus lisQn daps gSa\

29- fl¼ dhft, fd fdlh ckg~; fcUnq ls o`Ùk ij [khaph xbZ Li'kZ js[kkvksa dh yEckbZ leku gksrh gSA

vFkok fl¼ dhft, fd o`Ùk ds ifjxr cus prqHkqZt dh lEeq[k Hkqtk,a dsUnz ij lEiwjd dks.k vUrfjr djrh gSA

30- ,d 5 ehVj mQ¡pk [kEHkk ,d Vkoj ds f'k[kj ij yxk gSA Hkwfe ij fLFkr ,d fcUnq ‘A’ ls [kEHks ds f'k[kj dk mUu;u dks.k 60° gS rFkk Vkoj ds f'k[kj ls fcUnq A dk voueu dks.k 45° gSA Vkoj dh mQ¡pkbZ Kkr dhft,A (√3 = 1.732 ysa)

149 Math (Hindi) ­X

31- ,d oxZ dh Hkqtk nwljs oxZ dh Hkqtk ls 4 lseh- vf/d gSA ;fn nksuksa oxks± ds ks=kiQyksa dk ;ksx 400 oxZ lseh- gS] rks nksuksa oxks± dh Hkqtk,a Kkr dhft,A

32- ,d csyu] ftldh mQ¡pkbZ] csyu ds O;kl dk nks frgkbZ gS_ dk vk;ru 4 ls f=kT;k ds xksys ds leku gSA csyu ds vk/kj dh f=kT;k Kkr dhft,A

33- 7 lseh- O;kl ds csyukdkj chdj] ftlesa dqN ikuh gS] esa 1-4 lseh- O;kl ds laxejej ds xksys Mkys tkrs gSaA ;fn ikuh ds ry esa 5-6 lseh- dk o`f¼ gqbZ gks rks laxejej ds xksyksa dh la[;k Kkr dhft,A

34- rhu /krq ds ?ku ftlds fdukjksa dk vuqikr 3%4%5 gSA dks fi?kykdj ,d ?ku ftldk fod.kZ 12√3 cuk;k x;k gSA rhuksa ?kuksa ds fdukjksa dh yEckbZ Kkr dhft,A

vFkok bl csyukdkj ikbi ftldk O;kl 4 lseh- gS] blesa ikuh 20 eh-@feuV dh nj ls cg jgk gSA bl ikbi kjk 'kDokdkj VSad ftlds vk/kj dk O;kl 80 lseh- rFkk 72 lseh- gS_ dks Hkjus esa fdruk le; yxsxk\

mÙkjekyk 1- c 2- d

3- b 4- b

5- a 6- a

7- a 8- d

9- c 10- b

11- ekuk l-Js- dk izFke in vkSj loZ vUrj Øe'k% a vkSj d gSA 8 (a + 7d) = 12 (a + 11d) 4a + 76d = 0 a + 19d = 0

a 20 = 0.

12- AB = AC (given) AE = AF (ckg~; fcUnq ls Li'kZ js[kk)

AB – AE = AC – AF BE = CF

150 Math (Hindi) ­X

ijUrq BE = BD and CF = CD

∴ BD = CD ;k D, Hkqtk BC dk eè; fcUnq gSA

13- 5000 pDdj = 11 km = 11000 ehVj

11000 11 1100 1 5000 5 5

= = = pDdj ehVj ehVj les h -

1100 2 2 70 5

r r π = ⇒ = vr% les h -

14- 1 2

∆ = [x 1 (y 2 – y 3 ) + x 2 (y 3 – y 1 ) + x 3 (y 1 – y 2 )]

1 2

= [1(5+7) – 3 (–7 + 1) + 2 (–1 – 5)]

1 2

= [12 +18 – 12]

= 9 sq. units

15- ekuk nks Øekxr izkd`r la[;k,a x vkSj x + 1 gSA x (x + 1) = 30

x 2 + x – 30 = 0 (x + 6) (x – 5) = 0

x = 5, – 6 (izkd`r ugha) 16- eku l-Js- ds izFke in vkSj lkoZvUrj Øe'k% a 1 , d 1 vkSj a 2 , d 2 gSA

1 1

2 2

2 ( 1) 3 8 2 7 15 2 ( 1)

2

n a n d n n n a n d

+ − + =

+ + −

1 1

2 2

( 1) 2 3 8 2 ( 1) 7 15 2

2

n a d n

n n a d

− + + = − + +

1 1

2 2

8 3 17 8 59 8 7 17 15 134

a d a d

+ × + = =

+ × +

151 Math (Hindi) ­X

uosa inksa ds fy,

1 8 2

17

n

n

− =

=

uosa inksa dk vuqikr 59 : 134

17- pki [k.M dk ks- . 2

l r

⇒ 5. . 20

2 r π

π =

⇒ r = 8 lseh-

18- (i) fo"ke la[;k vkus dks izkf;drk = 50 1 100 2

=

(ii) HkkT; la[;k vkus dh izkf;drk = 15 3 100 20

=

vFkok

izkf;drk 1 9

=

19- 2x 2 + 5x – 0

2 5 7 0 2 2

x + × − = (2 ls Hkkx djus ij) 2 2

2 5 5 7 5 2 4 2 4

x

+ × − = + (nksuksa vksj 2 5

4 tksM+us ij)

2 5 7 25 81 4 2 16 16

x

+ = + =

5 9 4 4

x + = ±

7 1, 2

x = −

152 Math (Hindi) ­X

20- ekuk l-Js- dk izFke in vkSj lkoZvUrj Øe'k% a vkSj d gSaA

9 [2 8 ] 171 2

4 19

a d

a d

+ =

+ = ....(i)

24 [2 3 ] 996 2

2 23 83

a d

a d

+ =

+ = ....(ii)

(i) vkSj (ii) a = 7, d = 3

∴ l-Js- 7, 10, 13, ....... vFkok

ekuk l-Js- dk izFke in vkSj loZ vUrj Øe'k% a vkSj d gSA

16 16 2 15 528 528 2

a d S + = =

2 15 66 a d + = ....(i)

22 16 32 2 31 528 1552 1552 2

a d S S + = = − =

2 31 130 a d + = ....(ii)

(i) vkSj (ii) a = 3, d = 4

∴ 19ok¡ in = 3 + 18 × 4 = 75.

21- lkiQ lqFkjh jpuk dhft, 22- ∠RPQ = 30°

⇒ ∠RQP = 60°

⇒ ∠RQS = 120°

⇒ ∠SRP = 30°

vc ∠RSQ = ∠SRQ = 30°

⇒ ∠QR = QS

vFkok

153 Math (Hindi) ­X

OT dks feykvks

vc ∆OPT ~ ∆OMP

⇒ OP PT OM MP

=

⇒ 5 3 4

n =

⇒ 20 3

n cm =

23- lkiQ lqFkjk jpuk dhft,A

24- Nk;kafdr Hkkx dk Js- 2 2 360 360 Q Q R r π π = −

2 2 30 30 7 (3.5) 360 360

π π = × − ×

25- lgh vkd`fr

ledks.k ∆ABC esa]

tan30 AB BC

= °

1 3

AB BC

=

1 3 28.5 3 3

AB = × ×

= 9.5 1.73 m = 16.46 m

fpeuh dh mQ¡pkbZ = 16.46 + 1.5 m = 17.96 m

154 Math (Hindi) ­X

26- ekuk vuqikr K : 1 gSA

fcUnq P ds funsZ'kkad 2 1 7 3 , 1 1

k k P k k

+ + + +

fcUnq P js[kk 3x + y = 12 ij fLFkr gS

∴ 2 1 7 3 3 12

1 1 k k k k

+ + + = + +

6k + 3 + 7k + 3 = 12k + 12 k = 6

∴ vuqikr 6 % 1

OR AB2 = (8 + 2)2 + (3 – 3)2 A(–2, 3), B(8, 3)

= (10) 2 + (0) 2 C(6, 7)

BC 2 = (6 – 8) 2 + (7 – 3) 2

= (– 2) 2 + (4) 2

AC 2 = (6 + 2) 2 + (7 – 3) 2

= (8) 2 + (4) 2

= 80

AB 2 = BC 2 + AC 2

ikbFkkxksjl izes; ds foykse ls ,d ledks.k f=kHkqt gSA

155 Math (Hindi) ­X

27- x­vk ij fdlh fcUnq ds funsZ'kkad (x, 0) gS ekuk vuqikr K : 1 gSA

3 5 0 1

kk

− + =

+

–3k + 5 = 0

5 3

k =

vuqikr = 5 : 3

4 1

k x k

+ =

+

5 4 3 5 1 3

x +

= +

17 17 3 8 8

3

x = =

vr% P fcUnq ds funsZ'kkad 17 , 0 . 8

28- ekuk tkj esa x uhys daps gSa y gjs daps gSa

rFkk x liQsn daps gSaA

iz'ukuqlkj x + y + z = 54

uhys daps vkus dh izkf;drk 1 54 3 n

= = (fn;k gS)

⇒ 54 3

n =

⇒ n = 18.

156 Math (Hindi) ­X

blh izdkj gjs daps vkus dh izkf;drk 4 54 9 y

= =

⇒ y = 24 ysfdu n + y + z = 54

⇒ vr% 18 + 24 + z = 54

⇒ z = 12

vr% lisQn dapksa dh la[;k = 12.

29- vkd`fr + fn;k gS + fl¼ djuk gS + jpuk miifr

OR OP, OQ rFkk OS dks feyk;k ∆AOP ≅ ∆AOS (SSS lofxlerk) ∠1 = ∠2, ∠3 = ∠4, ∠5 = ∠6, ∠7 = ∠8 ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 + ∠7 + ∠8 = 360 2(∠2 + ∠3 + ∠6 + ∠7) = 360 ∠AOD + ∠BOC = 180°

30- lgh vkd`fr ledks.k ∆ABD esa]

tan 45 BD AB

= °

1 h x

=

h = x .....(i)

ledks.k ∆ABC esa] tan 60 BD AB

= °

5 3 h x +

=

h + 5 = x√3 .....(ii) h(√3 – 1) = 5

( ) ( ) ( )

5 3 1

3 1 3 1 h

+ =

− +

h = x = 6.83 m;k Vkoj dh mQ¡pkbZ = 6.83 eh-

157 Math (Hindi) ­X

31- ekuk ,d oxZ dh Hkqtk = x lseh-

vkSj nwljs oxZ dh Hkqtk = (x – 4) lseh- x 2 + (x – 4) 2 = 400

x 2 – 4x – 192 = 400

x = 16, –12

oxks± dh Hkqtk,a 16 lseh- vkSj 12 lseh- 32- ekuk csyu dk O;kl gS rks f=kT;k

⇒ vc iz'ukuqlkj 2 4 2 3 3

r r = × =

csyu dk vk;ru = 4 lseh- f=kT;k okys xksys dk vk-

⇒ 2 2 .r π ×

3 4

= 3

. π 4 4 4 × × ×

r 3 = 4 3

⇒ r = 4 cm.

33- ekuk ‘n’ daps Mkys tkrs gSaA

bl ‘n’ dapksa dk vk;ru = csyu ds c<+s gq, ikuh dk vk;ru

⇒ 3 2 4 . 3

n r r H π π × =

⇒ 4 14 14 14 7 7 5.6 3 20 20 20 2 2

n π π × × × × = × × ×

⇒ n = 150 dapsA 34- ekuk ?kuksa dh Hkqtk,a 3x, 4x rFkk 5x 3 gSA

bl izdkj rhuksa ?kuksa dh vk;ru = ,d cM+s ?ku dk vk;ru

⇒ (3x) 3 + (4x) 3 + (5x) 3 = (a) 3 (ekuk Hkqtk ‘a’ gS)

⇒ 27x 3 + 64x 3 + 125x 3 = a 3

⇒ 216x 3 = a 3

⇒ 6x = a

158 Math (Hindi) ­X

vc ‘a’ Hkqtk okys ?ku dk lcls cM+k f=kd.kZ 2 2 2 2 3 3 a a a a a = + + = =

12 3 6 3x = =

x = 2

vr% rhuksa ?kuksa dh Hkqtk,a = 6.8 rFkk 10 lseh- 34- ekuk ikbi VSad dk feuV esa Hkj ldrk gSA bl izdkj ikbZi esa feuV esa cgk gqvk

ikuh dk vk;ru VSad ds vk;ru ds leku gksxkA

⇒ vr% 2 2 1 . 3

r h t r h π π × × × =

⇒ π 1 2 2 2000 3

t π × × × × = 40 40 72 × × ×

⇒ 24 min. 5

t =

4 feuV 48 lsds.MA

159 Math (Hindi) ­X

BLUE PRINT xf.kr SA­2

fo"k; oSdfYid y?kq iz'u­I y?kq iz'u­II nh?kZ iz'u dqy 1 vad 2 vad 3 vad 4 vad

cht xf.kr 3 (3) 2 (4) 3 (9) 1 (4) 9 (20)

T;kferh 1 (1) 2 (4) 2 (6) 1 (4) 7 (16)

ks=kfefr 1 (1) 1 (2) 2 (6) 3 (12) 6 (20)

f=kdks.kkfefr ds 2 (2) 1 (2) – 1 (4) 4 (8) vuqiz;ksx

funsZ'kkad T;kfefr 2 (2) 1 (2) 2 (6) – 5 (12)

izkf;drk 1 (1) 1 (2) 1 (3) – 3 (6)

dqy 10 (10) 8 (16) 10 (30) 6 (24) 34 (80)

Note : Marks are within brackets.

160 Math (Hindi) ­X

lSEiy iz'u i=k

xf.kr dkk X (SA­2)

Time Allowed : 3½ hours Maxmimum Marks : 80

lkekU; funsZ'k 1 lHkh iz'u vfuok;Z gSaA 2- bl iz'u i=k esa dqy 34 iz'u gSa tks pkj Hkkxksa A, B, C vkSj D esa foHkkftr gSaA

Hkkx&A esa 10 iz'u gS izR;sd iz'u 1 vad dk gSA Hkkx&B esa 8 iz'u gSa izR;sd iz'u 2 vad dk gSA Hkkx&C esa 10 iz'u gSa izR;sd iz'u 3 vad dk gSA Hkkx&D esa 6 iz'u gSa izR;sd iz'u 4 vad dk gSA

3- iz'u la[;k 1 ls 10 cgqfodYih; iz'u gSA fn, x, pkj fodYiksa esa ls ,d lgh fodYi pqusaA

4- blesa dksbZ loksZifj fodYi ugha gSA ysfdu vkarfjd fodYi 1 iz'u 2 vadksa esa] 3 iz'u 3 vadksa esa vkSj 2 iz'u 4 vadksa esa fn, x, gSaA vki fn, x, fodYiksa esa ls ,d fodYi dk p;u djsaA

5- dSydqysVj ds iz;ksx dh vuqefr ugha gSA

Section ­ A iz'u 1 ls 10 rd izR;sd ,d vad dk gSA 1- f?kkr lehdj.k 2x 2 + 13x + 11 + 0 ds ewyksa dk ;ksxiQy gSA

(a) –13 (b) 13 2

−

(c) 11 2 (d) –11

2- l-Js- &5] &3] &1] dk nok¡ in gSA (a) 2n – 7 (b) 7 – 2n (c) 2n + 7 (d) 2n + 1

161 Math (Hindi) ­X

3- nh xbZ vkd`fr esa fcUnq P, Q rFkk R Li'kZ fcUnq gSA ;fn AB = 6 lseh-] BP = 3 rks ∆ABC dk ifjeki gS&

(a) 12 lseh- (b) 18 lseh- (c) 9 lseh- (d) 15 lseh-

4- 8 ehVj vkSj 12 ehVj mQ¡ps nks [kEHkksa ds mQijh fljs ,d rkj kjk tqM+s gSaA ;fn rkj kSfrt ds lkFk 30° dk dks.k cuk, rks rkj dh yEckbZ gS&

(a) 10 ehVj (b) 12 ehVj (c) 8 ehVj (d) 4 ehVj

5- js[kkvksa y + 3 = 0 rFkk 2y – 5 = 0 ds chp dh nwjh gS& (a) 8 bdkbZ (b) 11

2 bdkbZ (c) 6 bdkbZ (d) 5 bdkbZ

6- fdlh iz'u ds lgh mÙkj ds vkadyu dh izkf;drk 12 x gSA ;fn lgh mÙkj ds ugha

vkadyu dh izkf;drk 2 3 gS rks x = -------

(a) 2 (b) 3 (c) 4 (d) 6

7- ;fn &9] &14] &19 ----- ,d l-Js- gS] rks a 30 – a 20 dk eku Kkr dhft,A (a) –50 (b) 50 (c) 10 (d) buesa ls dksbZ ughaA

162 Math (Hindi) ­X

8- ;fn izR;sd xksys dk O;kl 6 lseh- gS ftUgsa fiNykdj 45 lseh- mQ¡pk rFkk 4 lseh- O;kl dk ,d Bksl csyu cuk;k tkrk gS] rks xksyksa dh la[;k gS&

(a) 3 (b) 4 (c) 5 (d) 6

9- fdlh le; ij ,d [kEcs dh ijNkbZ] mldh mQ¡pkbZ ds cjkcj gS] lw;Z dk mUu;u dks.k gSA

(a) 30° (b) 45° (c) 60° (d) 90°

10- fcUnqvksa (0] 0)] (3] 0) vkSj (0] 3) ls cus f=kHkqt dk ifjeki gS& (a) 6 bdkbZ (b) 9 bdkbZ (c) ( ) 2 2 3 + bdkbZ (d) ( ) 2 2 2 + bdkbZ

Section ­ B iz'u 11 ls 18 rd izR;sd 2 vad dk gSA 11- ;fn ,d l-Js- dk izFke in 3 vkSj 6ok¡ in 23 gS] rks bdlk 17ok¡ in Kkr

dhft,A 12- f?kkr lehdj.k 4x 2 + mx + 1 = 0 ds okLrfod ewyksa ds fy, ‘m’ dk eku D;k

gksxk\ 13- nks ladsUnz o`Ùkksa dh f=kT;k,a 5 lseh- vkSj 3 lseh- gSaA cM+s o`Ùk dh ml thok dh

yEckbZ Kkr dhft, tks NksVs o`Ùk dks Li'kZ djrh gSA vFkok

nh xbZ vkd`fr esa o`Ùk dh f=kT;k Kkr dhft,A

163 Math (Hindi) ­X

14- nh xbZ vkd`fr esa XY vkSj X ‘Y’ o`Ùk dk nks lekUrj Li'kZ js[kk,a gSa] O o`Ùk dk dsUnz gS rFkk vU; Li'kZ js[kk AB ftldk Li'kZ fcUnq C gS XY dks rFkk X ‘Y’ dks B ij izfrPNsn djrh gSA fl¼ dhft, dh <AOB = 90°.

15- ,d 3 lseh- f=kT;k okyh xsan dks fiNykdj 3 NksVh xsans cuk;h tkrh gSaA ;fn muesa ls nks xsanksa dh f=kT;k Øe'k% 1-5 lseh- vkSj 2 lseh- gks rks rhljh xsan dh f=kT;k Kkr dhft,A

16- ehukj ds ,d gh vksj ehukj ds vk/kj ls 4 ehVj vkSj 9 ehVj dh nwjh ij fLFkr nks fcUnqvksa ls ehukj ds f'k[kj ds mUu;u dks.k ,d nwljs ds iwjd ik;s tkrs gSaA ehukj dh mQ¡pkbZ Kkr dhft,A

17- Y­vk ij fLFkr ml fcUnq ds funsZ'kkad Kkr dhft, tks fcUnqvksa (&2] 5) vkSj (2] &3) ls lenwjLFk gksA

18- fdlh rk'k dh xM~Mh esa ls lHkh ckn'kkg] csxe vkSj xqyke fudky fn;s tkrs gSaA 'ks"k cps gq, iÙkksa esa ls ,d iÙkk ;n`PN;k fudkyk tkrk gSA izkf;drk Kkr dhft, fd fudkyk x;k iÙkk&

(a) ,d fp=k iÙkk gS

(b) ,d dkyk iÙkk gSA

Section ­ C iz'u 19 ls 28 rd izR;sd 3 vad dk gSA

19- ,d lefckgq f=kHkqt dh jpuk dhft, ftldk vk/kj 8 lseh- rFkk 'kh"kZyEc 5 lseh- gSA ,d vU; f=kHkqt dh jpuk dhft, ftldh izR;sd Hkqtk fn, x, f=kHkqt dh laxr Hkqtk dk 3

4 Hkkx gksA

164 Math (Hindi) ­X

20- lehdj.k dks gy dhft,A 1 3 10 , 2, 4. 2 4 3

x x x x x x

− − = = ≠ ≠

− −

21- nh xbZ vkd`fr esa ∆ABC ,d o`Ùk ds ifjxr cuk gS ftldh f=kT;k 4 lseh- gSA D, E rFkk F Li'kZ fcUnq gSA Hkqtk AB vkSj AC dh yEckbZ Kkr dhft,A

22- ,d 2-2 ?ku MslhehVj rkacs dh 0-50 lseh- O;kl dh rkj cukbZ xbZ gSA rkj dh yEckbZ Kkr djksA

vFkok

,d ?kM+h dh feuV dh lqbZ dh yEckbZ 14 lseh- gSA lqbZ kjk ,d feuV esa cqgkj (Sweep) fd;k x;k ks=kiQy Kkr dhft,A

23- l- 6 $ 12 $ 18 $ --------- $ 120 dk ;ksxiQy Kkr dhft,A

24- ,d l-Js- ds pkSFks vkSj vkBosa inksa dk ;ksxiQy 24 gS vkSj ikapos vkSj nlosa inksa dk ;ksxiQy 29 gS] l-Js- Kkr dhft,A

vFkok

;fn ,d ls-Js- dk nok¡ in 3 – 2n gS] rks blds 40 inksa dk ;ksxiQy Kkr dhft,A

25- ,d 'kadq dh fr;Zd mQ¡pkbZ 10 lseh- vkSj mQ¡pkbZ 8 lseh- gSA mQ¡pkbZ ds eè;fcUnq ds lekUrj ,d ry 'kadq dks dkVrk gS_ nksuksa Hkkxksa ds vk;ru dk vuqikr Kkr dhft,A

165 Math (Hindi) ­X

26- ,d ledks.k ∆ABC esa ∠B = 90° rFkk 34 AB = bdkbZA fcUnq B vkSj C ds funsZ'kkad Øe'k% (4] 2) vkSj (&1, y) gSaA ;fn ∆ABC dk ks=kiQy 17 oxZ bdkbZ gks rks y dk eku Kkr dhft,A

27- 1,2,3 la[;kvksa esa ls la[;k ‘x’ rFkk 1,4,9 la[;kvksa esa ls ,d nwljh la[;k ‘y’ ;kn`PN;k pquh tkrh gSA bldh D;k izkf;drk gS fd nksuksa la[;kvksa dk xq.kuiQy 9 ls de gksxkA

vFkok

,d FkSys esa 12 xsans gSaA ftuesa ls dkyh xsans gSaA FkSys esa 6 vkSj dkyh xsans Mkyh x;ha gSaA ;fn dkyh xsans dh izkf;drk mldh igyh okyh izkf;drk ls nqxquh gks rks dk eku Kkr dhft,A

28- ;fn fcUnq (x, y), (–5, –2) vkSj (3, –5) lajs[kh; gks rks fl¼ dhft,% 3x + 8y + 31 = 0

Section ­ D iz'u 29 ls 34 rd izR;sd 4 vad dk gSA

29- nks uyksa dks ,d lkFk pykus ij ,d VSad 3 9 8 ?kaVs esa Hkjrk gSA vyx&vyx VSad

dks Hkjus esa vf/d O;kl okyk uy] de O;kl okys uy ls 10 ?kaVs de ysrk gSA izR;sd VSad uy dks vyx&vyx VSad dks Hkjus esa fdruk&fdruk le; yxsxk\

vFkok ,d eksVj cksV ftldh fLFkj ty esa pky 9 fdeh-@?kaVk gSA 12 fdeh- /kjk ds izfrdwy tkus esa vkSj ogh nwjh /kjk ds vuqdwy tkus esa rhu ?kaVs ysrh gSA /kjk dh pky Kkr dhft,A

30- o`Ùk ds fdlh fcUnq ij Li'kZ js[kk Li'kZ fcUnq ls gksdj tkus okyh f=kT;k ij yEc gS] fl¼ dhft,A

31- 12 lseh- f=kT;k rFkk 28 lseh- mQ¡pkbZ ds Bksl csyu esa ,d 16 lseh- mQ¡pk rFkk 12 lseh- f=kT;k dk ,d 'kaDokdkj Hkkx vyx dj fn;kA 'ks"k] Bksl dk (a) vk;ru (b) lEiw.kZ i`"Bh; ks=kiQy Kkr dhft,A

166 Math (Hindi) ­X

32- 12 lseh- O;kl rFkk 15 lseh- mQ¡pkbZ okys ,d yEco`Ùkh; csyu ds vkdkj dk crZu vkblØhe ls iwjk Hkjk gSA vkblØeh dks 12 lseh- mQ¡pkbZ rfkk 6 lseh- O;kl 'kadqvksa] ftudh mQ¡ijh fljk v¼Zxksykdkj gS] esa Hkjk tkuk gSA ,sls 'kadqvksa dh la[;k Kkr dhft,] tks bl vkblØhe ls Hkjs tk ldrs gSaA

33- ,d >hy ls 100 ehVj mQij fLFkr ,d fcUnq ls ,d fLFkj gsyhdkWIVj dk mUu;u dks.k 30 gS rFkk >hy esa blds izfrfcEc dk voueu dks.k 60° gSA fujhk.k fcUnq ls gsyhdkWIVj dh nwjh Kkr dhft,A

34- ,d v¼Zxksyh; dVksjk ftldk vkUrfjd O;kl 36 lseh- gS_ nzo ls iwjk Hkjk gSA bl nzo dks 3 lseh- f=kT;k rFkk 6 lseh- mQ¡pkbZ csyukdkj cksryksa esa Hkjk tkuk gSA dVksjs dks [kkyh djus ds fy, fdruh cksryksa dh vko';drk gksxh\

vFkok ,d o`Ùkkdkj nkSM+iFk dh vkUrfjd ifjf/ 440 ehVj gSA iFk 14 ehVj pkSM+k gSA bldks lery djokus ij 20 iSls@oxZ ehVj dh nj ls gksus okyh ykxr Kkr dhft,A ckgj dh vksj dk¡Vs dh rkj yxokus ij 2 #- izfr ehVj dh nj ls D;k [kpZ vk,xk\

167 Math (Hindi) ­X

mÙkjekyk 1- b 2- a

3- b 4- c

5- b 6- c

7- a 8- c

9- b 10- d

11- 67 12- m ≥ 4 or m ≤ –4

13- 8 cm or 11 2 cm 15- 5 cm

16- 6 m 17- (0, 1)

18- (a) 0, (b) 1 2 20- 5, 5

2

21- AB = 15cm, AC =13cm 22- 112 ehVj vFkok 10.26 cm 2

23- 1260 24- –13, –8, –3, ........ vFkok –1520

25- 8 : 7 26- y = –1, 5

27- 5 9 vFkok 3. 29- 15 ?ka-] 25 ?ka- vFkok 3 fdeh@?ka-

31- 3 2 2 6 10258 , 3318 7 7

cm cm 32- 10

33- 200 ehVj 34- 72 cksrysa vFkok 1355.20, 1056 #i;sA

168 Math (Hindi) ­X

lSEiy iz'u i=k

xf.kr dkk X (SA­2)

Time Allowed : 3 to 3½ hours Maxmimum Marks : 80

lkekU; funsZ'k 1 lHkh iz'u vfuok;Z gSaA 2- bl iz'u i=k esa dqy 34 iz'u gSa tks pkj Hkkxksa A, B, C vkSj D esa foHkkftr gSaA

Hkkx&A esa 10 iz'u gS izR;sd iz'u 1 vad dk gSA Hkkx&B esa 8 iz'u gSa izR;sd iz'u 2 vad dk gSA Hkkx&C esa 10 iz'u gSa izR;sd iz'u 3 vad dk gSA Hkkx&D esa 6 iz'u gSa izR;sd iz'u 4 vad dk gSA

3- iz'u la[;k 1 ls 10 cgqfodYih; iz'u gSA fn, x, pkj fodYiksa esa ls ,d lgh fodYi pqusaA

4- blesa dksbZ loksZifj fodYi ugha gSA ysfdu vkarfjd fodYi 1 iz'u 2 vadksa esa] 3 iz'u 3 vadksa esa vkSj 2 iz'u 4 vadksa esa fn, x, gSaA vki fn, x, fodYiksa esa ls ,d fodYi dk p;u djsaA

5- dSydqysVj ds iz;ksx dh vuqefr ugha gSA

Section ­ A iz'u 1 ls 10 rd izR;sd ,d vad dk gSA 1- ;fn dsUnz O okys o`Ùk ij nks Li'kZ js[kkvksa ds chp dk dks.k gS] rks dks.k gS%

(a) 120° (b) 30° (c) 150° (d) 60°

2- fuEu esa ls dkSu lehdj.k x 2 + 2x + 1 = 0 dk ,d ewy gS\ (a) x = – 1 (b) x = 1 (c) x = – 2 (d) x = 2

169 Math (Hindi) ­X

3- ;fn x = 2 lehdj.k x 2 – 2kx + 5 = 0 dk ,d ewy gS] rks k cjkcj gS%

(a) 4 17 (b) 16

4

(c) 217 (d) 9

4

4- A.P. – 4, –2, 0, 2, ............ dk lkoZ vUrj gS% (a) 2 (b) –2

(c) 1 2 (d) – 1

2

5- ml o`Ùk ds f=kT;k[kaM dk ifjeki] ftldk dsUnzh; dks.k 90° vkSj f=kT;k 7 cm gS] fuEu gS%

(a) 35 cm (b) 25 cm (c) 77 cm (d) 7 cm

6- Hkwfe ij fLFkr fdlh fcanq P ls ,d ehukj ds 'kh"kZ dk mUu;u dks.k α gSA ehukj ds vk/kj dh vksj d nwjh pyus ij mUu;u dks.k β gks tkrk gS] rc%

(a) α < β (b) α > β

(c) α = β (d) buesa ls dksbZ ugha

7- ∆ABC ds le:i ,d ∆PQR dh jpuk Ldsy xq.kd 12 3 ds lkFk dh tkrh gSA

f=kHkqt PQR gS% (a) ∆ABC ls NksVk (b) ∆ABC ds cjkcj (c) ∆ABC ls cM+k (d) buesa ls dksbZ ugha

8- fcanq P (8, 3) dh y­vk ls nwjh gS% (a) 7 bdkbZ (b) 3 bdkbZ (c) 5 bdkbZ (d) 13 bdkbZ

9- fdlh vlaHko ?kVuk dh izkf;drk gS% (a) 1 (b) –1 (c) 2 (d) 0

170 Math (Hindi) ­X

10- fcanqvksa A (0, 4) vkSj B (4, 0) ds chp dh nwjh gS% (a) 4 2 (b) 2 4

(c) 6 (d) 2 2

Section ­ B iz'u 11 ls 18 rd izR;sd 2 vad dk gSA 11- ikWlksa ds ,d ;qXe dks ,d ckj isaQdk tkrk gSA izR;sd ikWls ij ,d gh la[;k izkIr

gksus dh izkf;drk Kkr dhft,A 12- xq.ku[kaMu kjk f?kkr lehdj.k 2 3 2 6 2 0 x x − + = ds ewy Kkr dhft,A 13- ,d A.P. dk 8ok¡ in 'kwU; gSA fl¼ dhft, fd bldk 38ok¡ in 18osa in dk

frxquk gSA 14- f?kkr lehdj.k 2x 2 – x – 1 = 0 ds ewyksa dh izd`fr ij fVIi.kh dhft,A ;fn

okLrfod ewyksa dk vfLrRo gS] rks os ewy Kkr dhft,A 15- n'kkZb, fd fcanq A (–1, 1), B (5, 7) vkSj C (8, 10) ljs[kh gSaA

vFkok

;fn blh Øe esa ysus ij (2, 1), (3, 4) vkSj (0, 1) ,d lekarj prqHkqZt ds rhu 'kh"kZ gSa] rks pkSFkk 'kh"kZ Kkr dhft,A

16- nh gqbZ vkd`fr esa] ;fn BC = 4.5 lseh- gS rks] AB dh yEckbZ Kkr dhft,A

17- ;fn f=kT;k a vkSj dsUnz O okys o`Ùk ij fdlh ckgjh fcanq P ls [khaph xbZ nksuksa Li'kZ js[kkvksa ds chp dk dks.k 90° gS] rks OP Kkr dhft,A

18- ;fn fdlh v¼Zo`Ùkkdj pk¡ns dk ifjeki 36 lseh- gS] rks mldk O;kl Kkr dhft,A

171 Math (Hindi) ­X

Section ­ C iz'u 19 ls 28 rd izR;sd 3 vad dk gSA

19- ,d ledks.k f=kHkqt dh jpuk dhft, ftlesa ledks.k dks varfoZ"V djus okyh Hkqtk,¡ 5 lseh- vkSj 4 lseh- gSA bl ledks.k f=kHkqt ds le:i ,d ,sls f=kHkqt dh jpuk dhft, ftldh Hkqtk,¡ ledks.k f=kHkqt dh laxr Hkqtkvksa dh 4

5 xquh gSA

20- 8 lseh- yEckbZ dk ,d js[kk[kaM AB [khafp,A dsUnz A ekudj f=kt;k 4 cm dk ,d o`Ùk [khafp, rFkk dsUnz B ekudj f=kT;k 3 lseh- dk ,d o`Ùk [khafp,A izR;sd o`Ùk ij nwljs o`Ùk ds dsUnz ls Li'kZ js[kk,¡ [khafp,A

21- fdlh A.P. ds izFke n inksa dk ;ksx 3n 2 + 5n gSA ;g A.P. Kkr dhft, vkSj blls mldk 15ok¡ in Kkr dhft,A

22- vkd`fr esa fn,] Nk;kafdr Hkkx dk ks=kiQy Kkr dhft,] tgk¡ dsUnz O okys ladsUnzh; o`Ùkksa dh f=kT;k,¡ Øe'k% 7 lseh- vkSj 14 cm gSa rFkk ∠AO = 40° gSA

23- pkj cjkcj o`Ùk ,d oxZ ds pkjksa dksuksa ij bl izdkj cuk, x, gSa fd izR;sd nks o`Ùkksa dks Li'kZ djrk gSA (nsf[k, vkd`fr)A Nk;kafdr Hkkx dk ks=kiQy Kkr dhft,] ;fn oxZ dh izR;sd Hkqtk dh yackbZ 14 cm gSA

172 Math (Hindi) ­X

24- fdlh igkM+h dh pksVh dk ,d ehukj ds vk/kj ds mUu;u dks.k 60° gS rFkk ehukj ds 'kh"kZ dk igkM+h ds vk/kj ls mUu;u dks.k 30° gSA ;fn ehukj dh m¡QpkbZ 50 eh- gS] rks igkM+h dh m¡QpkbZ Kkr dhft,A

25- ,d f[kykSuk ,d 'kadq ds vkdkj dk gS] ftl ij O;kl 7 lseh- dk ,d v/Zxksyk vè;kjksfir gSA ;fn v/Zxksys vkSj 'kadq ds vk/kj ds O;kl cjkcj gSa rFkk f[kykSus dh dqy m¡QpkbZ 14.5 lseh- gS] rks bl f[kykSus dk vk;ru vkSj dqy i`"Bh; ks=kiQy Kkr dhft,A

vFkok

5 lseh- m¡Qps ,d 'kadq ds fNUud ds fljksa dh f=kT;k,¡ 7 ehVj vkSj 4 ehVj gSaA bldk vk;ru vkSj oØ i`"Bh; ks=kiQy Kkr dhft,A

26- D;k fcanq (3, 2), (–2, –3) vkSj (2, 3) ,d f=kHkqt cukrs gSa\ ;fn gk¡] rks bl f=kHkqt dk izdkj crkb,A

vFkok

og vuqikr Kkr dhft,] ftlesa fcanq P(m, 6) fcanqvksa A(–1, 3) vkSj B(2, 8) dks feykus okys js[kk[kaM dks foHkkftr djrk gSA m dk eku Kkr dhft,A

27- funsZ'kdksa (–1, 7), (–4, –9), (9, –3) vkSj (8, –5) okys fcanqvksa ls cus prqHkqZt dk ks=kiQy Kkr dhft,A

28- la[;kvksa 2 ls 31 rd vafdr 30 fVdVksa esa ls ,d fVdV ;kn`fPNd :i ls fudkyk tkrk gSA izkf;drk Kkr dhft, fd bl fVdV ij fuEu vafdr gksxk% (i) 7 dk ,d xq.kt (ii) ,d le la[;k (iii) ,d vHkkT; la[;k

vFkok ,d fMCcs esa 2 lisQn] 8 yky vkSj 1 dkyh xsansa gSaA blesa ,d xsan ;kn`fPNd :i ls fudkyh tkrh gSA bldh izkf;drk fu/kZfjr dhft, fd ;g xsan % (i) yky ugha gS (ii) ,d dkyh xsan gS (iii) yky ;k dkyh gSA

173 Math (Hindi) ­X

Section ­ D iz'u 29 ls 34 rd izR;sd 4 vad dk gSA

29- fl¼ dhft, fd ,d ckgjh fcanq ls o`Ùk ij [khaph xbZ Li'kZ js[kkvksa dh yackb;k¡ cjkcj gksrh gSaA

30- ,d dkk VsLV esa] dey kjk xf.kr vkSj vaxzsth esa izkIr vadksa dk ;ksx 40 gSA ;fn mlus xf.kr esa 3 vad vf/d izkIr fd, gksrs vkSj vaxzsth esa 4 vad de izkIr fd, gksrs] rks mlds kjk izkIr vadksa dk xq.kuiQy 360 gksrkA nksuksa fo"k;ksa esa mlds kjk izkIr fd, x, vad Kkr dhft,A

31- uhps nh vkd`fr esa] rhu v/Zo`Ùk A. B. vkSj C n'kkZ, x, gSa ftuesa ls izR;sd dk O;kl 2 lseh- gSA lkFk gh] blesa ,d vU; v/Zo`Ùk E n'kkZ;k x;k gS ftlesa ,d o`Ùk D [khapk x;k gS] ftldk O;kl 3 cm gSA Nk;kafdr Hkkx dk ks=kiQy Kkr dhft,A

32- fdlh unh ds fdukjs ij [kM+k ,d O;fDr ns[krk gS fd unh ds nwljs fdukjs ij [kM+s ,d isM+ ds 'kh"kZ dk mUu;u dks.k 60° gSA tc og fdukjs ls 20 eh- ihNs gVrk gS] rc mUu;u dks.k 30° gks tkrk gSA isM+ dh mQ¡pkbZ vkSj unh dh pkSM+kbZ Kkr dhft,A

vFkok fdlh mQèokZ/j ehukj] tks lery Hkwfe ij [kM+h gS] dh ijNkbZ 10 ehVj c<+ tkrh gS] tc lw;Z dk mUu;u dks.k 45° ls 30° gks tkrk gSA ehukj dh m¡QpkbZ n'keyo ds ,d LFkku rd 'kq¼ Kkr dhft,A

33- ,d f[kykSuk ,d yaco`Ùkh; csyu ds vkdkj dk gS] ftlds nksuksa fljs v/Zxksykdkj gSaA csyukdkj Hkkx dh f=kT;k vkSj m¡QpkbZ Øe'k% 5 lseh- vkSj 20 lseh- gSA v/Zxksys dh f=kT;k ogh tks yaco`Ùkh; csyukdkj Hkkx dh gSA ;fn f[kykSus dh dqy m¡QpkbZ 30 lseh- gS] rks ldk dqy i`"Bh; ks=kiQy Kkr dhft,A

174 Math (Hindi) ­X

vFkok f=kT;k 3 lseh- okys ,d csyukdkj chdj esa] ftlesa dqN ikuh Hkjk gS] O;kl 1.4 lseh- okys xksykdkj daps Mkys tkrs gSaA chdj esa Mkys tkus okys dapksa dh la[;k Kkr dhft,] ftlls chdj esa ikuh dk Lrj 5.6 lseh- m¡Qpk mB tk,A

34- ;fn S 1 , S 2 vkSj S 3 Øe'k% fdlh A.P. ds izFke n, 2n vkSj 3n inksa ds ;ksx gSa] rks fl¼ dhft, fd S 3 = 3 (S 2 – S 1 ) gSA

175 Math (Hindi) ­X

Common Pre­Appearing Examination, 2010­11

xf.kr dkk X (SA­2)

Time Allowed : 3 to 3½ hours Maxmimum Marks : 80

lkekU; funsZ'k 1 lHkh iz'u vfuok;Z gSaA 2- bl i z' u i =k esa d qy 34 iz'u gSa tks pkj Hkkxksa A, B, C vkSj D esa foHkkftr gSaA

Hkkx&A esa 10 iz'u gS izR;sd iz'u 1 vad dk gSA Hkkx&B esa 8 iz'u gSa izR;sd iz'u 2 vad dk gSA Hkkx&C esa 10 iz'u gSa izR;sd iz'u 3 vad dk gSA Hkkx&D esa 6 iz'u gSa izR;sd iz'u 4 vad dk gSA

3- iz'u la[;k 1 ls 10 cgqfodYih; iz'u gSA fn, x, pkj fodYiksa esa ls ,d lgh fodYi pqusaA

4- blesa dksbZ loksZifj fodYi ugha gSA ysfdu vkarfjd fodYi 1 iz'u 2 vadksa esa] 3 iz'u 3 vadksa esa vkSj 2 iz'u 4 vadksa esa fn, x, gSaA vki fn, x, fodYiksa esa ls ,d fodYi dk p;u djsaA

5- dSydqysVj ds iz;ksx dh vuqefr ugha gSA

Section ­ A iz'u 1 ls 10 rd izR;sd ,d vad dk gSA 1- ;fn lehdj.k 3x 2 + kx – 1 = 0 dk ,d ewy x = –1 gS] rks k dk eku gS%

(a) 0 (b) 2

(c) –1 (d) 1 2

2- Js<+h%& 9, –14, –19, –24, .......... ds fy, t 30 – t 20 dk eku gksxk% (a) – 50 (b) 50 (c) 60 (d) – 60

176 Math (Hindi) ­X

3- 'kCn ‘APPLICATION’ ls ;kn`PN;k ,d vkj pquk x;kA og vkj ,d O;atu gS dh izkf;drk D;k gksxh\

(a) 1 (b) 5 11

(c) 3 11 (d) 6

11

4- fcUnqvksa A (sinθ, 0) vkSj (0, cosθ ) ds chp dh nwjh gS% (a) sinθ + cosθ (b) 1 (c) –1 (d) 0

5- ,d 5 lseh- f=kT;k okys o`Ùk ds fcUnq A ij Li'kZ js[kk AB, o`Ùk ds dsUnz O ls tkus okyh js[kk dks B ij bl izdkj dkVrh gS fd OB = 12 lseh- gSA rc AB dh yEckbZ gS%

(a) 12 lseh- (b) 13 lseh- (c) 8.5 lseh- (d) 119 lseh-

6- 12 lseh- dksj okys nks ?kuksa dks fdukjs&ls&fdukjs tksM+k x;k gSA bl izdkj cus ?kukHk dk i`"Bh; ks=kiQy gksxk%

(a) 1440 lseh- 2 (b) 864 lseh- 2

(c) 720 lseh- 2 (d) 1728 lseh- 2

7- 2 lseh- f=kT;k ds dqN xksyksa dks fi?kykdj 45 lseh- mQ¡pk vkSj 4 lseh- O;kl dk ,d Bksl csyu cuk;k tkrk gSA fdrus xksyksa dks fi?kyk;k x;k\

(a) 3 (b) 4 (c) 5 (d) 6

8- ,d ehukj ds vk/kj ls 30 ehVj nwj Hkwfe ij fLFkr ,d fcUnq ls ehukj dh pksVh dk mUu;u dks.k 60° gSA ehukj dh m¡QpkbZ gS%

(a) 10 3 ehVj (b) 30 3 ehVj

(c) 10

3 ehVj (d) 30

3 ehVj

177 Math (Hindi) ­X

9- fdlh ?kM+h dh feuV dh lwbZ 12 lseh- yEch gSA ;g 9:00 cts ls 9:10 cts rd fdruk ks=kiQy r; djsxh\ lseh 2

(a) 2 π lseh 2 (b) 2

π lseh 2

(c) π lseh 2 (d) 2π lseh 2

10- TP vkSj TQ dsUnz O okys ,d o`Ùk dh Li'kZ js[kk,¡ gSaA ;fn ∠TOQ = 50° gS] rks ∠OTP fdruk gksxk\

(a) 40° (b) 50° (c) 80° (d) 100°

Section ­ B iz'u 11 ls 18 rd izR;sd 2 vad dk gSA 11- nks la[;kvksa dk ;ksx 15 gSA ;fn muds O;qRØeksa dk ;ksx 3

10 gS] rks os la[;k,¡ Kkr

djksA 12- y­vk ij og fcUnq Kkr dhft, tks fcUnqvksa A (6, 5) vkSj B (–4, 3) ls lenwjLFk

gksA 13- ,d dkj ds izR;sd ifg, dk O;kl 140 lseh- gSA ;fn dkj dh pky 66 fdeh-

@?kaVk gS] rks izR;sd ifg, dks 10 feuV esa fdrus pDdj yxkus gksaxs\ 22 7

π =

14- vkd`fr esa] o`Ùk dh Li'kZ js[kk PT gS rFkk QR o`Ùk dk O;kl gSA ;fn ∠RPT = 50° gS] rks ∠QRP dk eku crkvksA

178 Math (Hindi) ­X

vFkok PA o PB fdlh o`Ùk ds ckg~; fcUnq P ls [khaph xbZ Li'kZ js[kk,¡ gSaA ;fn PA = 10 lseh- rFkk ∠APB = 60°, rks thou AB dh yEckbZ Kkr djksA

15- lekarj Js<+h 3, 10, 17....... dk dkSu&lk in mlds 13osa in ls 84 vf/d gS\ 16- ,d ikWls dks nks ckj isaQdk x;kA bldh D;k izkf;Drk gS fd (i) la[;k 5 de&ls*de

,d ckj vk,xh (ii) la[;k 5 ,d ckj Hkh ugha vk,xh\ 17- nks o`Ùkksa dh f=kT;kvksa dk ;ksxiQy 140 lseh- rFkk budh ifjf/;ksa dk varj 88 lseh-

gSA o`Ùkksa ds O;kl Kkr dhft,\ 22 7

π =

18- ,d o`Ùk ∆ABC dh BC Hkwtk dks fcUnq D ij Li'kZ djrk gS rFkk c<+h gqbZ Hkqtk AB vkSj AC dks fcUnq E vkSj F ij Li'kZ djrk gSA ;fn AE = 7 lseh- gS] rks ∆ABC dk ifjeki Kkr dhft,A

Section ­ C iz'u 19 ls 28 rd izR;sd 3 vad dk gSA

19- fl¼ dhft, fd fdlh ckgjh fcUnq ls o`Ùk ij [khaph xbZ Li'kZ js[kkvksa dh yEckb;k¡ cjkcj gksrh gSaA

20- ,d leku m¡QpkbZ ds nks [kaHks ,d 100 eh- pkSM+h lM+d ds nksuksa vksj fLFkr gSA lM+d ij fLFkr ,d fcUnq ls nksuksa [kaHkksa ds 'kh"kks± ds mUu;u dks.k Øe'k% 60° vkSj 80°

gSaA lM+d ij ml fcUnq dh fLFkfr vkSj [kaHkksa dh m¡QpkbZ Kkr djks\ ( ) 3 1.732 =

21- 500 vkSj 800 ds chp mu la[;kvksa dk ;ksx D;k gksxk tks 7 ls foHkkftr gks tkrh gSaA

22- fcUnqvksa P ds funsZ'kkad Øe'k% (–2, –2) vkSj (2, –4) gSa] rks fcUnq R ds funsZ'kkad Kkr

dhft, ;fn 3 7

PR PQ = vkSj fcUnq R js[kk[kaM PQ ij fLFkr gSA

vFkok

fl¼ dhft, fd fcUnq (4, 3), (6, 4), (5, 6) vkSj (3, 5) ,d oxZ dh 'kh"kZ gSaA

179 Math (Hindi) ­X

23- 3 lseh- f=kT;k ds o`Ùk ij 5 lseh- f=kT;k ds ,d ladsUnzh; o`Ùk ds fdlh fcUnq ls Li'kZ js[kkvksa dh jpuk dhft, vkSj mudh yEckb;ksa dh eki dh tk¡p dhft,A

24- ;fn lehdj.k (1 + m 2 ) x 2 + 2mcx + c 2 ) = 2 ds ewy leku gksa] rks fl¼ dhft, c 2 = c 2 (1 + m 2 ).

vFkok

gy dhft, % 4 3 2 1 1 3 10 3; , 2 1 4 3 2 4

x x x x x x

− + − − = ≠ ≠ + −

25- ,d o`Ùkkdkj gkSt dk O;kl 10 ehVj vkSj xgjkbZ 2 ehVj gSA blesa ,d ikbi ls 3 fdeh-@?kaVk dh xfr ls ikuh cg jgk gSA ;fn ikbi dk var% O;kl 20 lseh- gS rks ml gkSt dks Hkjus esa fdruk le; yxsxk\

vFkok

26- 52 rk'k ds iÙkksa dh xM~Mh esa ls ,d iÙkk ;kn`PN;k fudkyk x;k gSA izkf;drk Kkr djks fd fudkyk x;k iÙkk%

(i) ,d yky fp=k okyk iÙkk gSA

(ii) u rks iku dk iÙkk gS] u gh ckn'kkg gSA

(iii) ,d dkyh csxe gSA

27- nh xbZ vkd`fr esa ABC ,d ledks.k f=kHkqt gS tgk¡ ∠A = 90°, AB = 21 lseh- vkSj AC = 28 lseh- AB, BC vkSj AC dks O;kl ekudj v¼Zo`Ùk cuk, x, gSaA Nk;kafdr

Hkkx dk ks=kiQy Kkr dhft, 22 7

π =

vFkok

180 Math (Hindi) ­X

ABC ,d 28 lseh- f=kT;k okys o`Ùk dk prqFkk±'k gSA BC dks O;kl ekudj ,d

v¼Zo`Ùk [khapk x;k gSA Nk;kafdr Hkkx dk ks=kiQy Kkr dhft,A 22 7

π =

28- ,d 1 lseh- O;kl vkSj 8 lseh- yEch rkacs dh NM+ dks leku eksVkbZ ds 18 lseh- yEcs rkj esa <kyk tkrk gSA rkj dh eksVkbZ Kkr dhft,A

Section ­ D iz'u 29 ls 34 rd izR;sd 4 vad dk gSA

29- ,d gokbZ tgkt [kjkc ekSle ds dkj.k vius fu/kZfjr le; ls 40 feuV foyEc ls pykA 1600 fdeh- nwj vius fu;e LFkku ij lgh le; ij igq¡pus ds fy, bls viuh pky] lkekU; vkSlr pky ls 400 fdeh-@?kaVk c<+kuh iM+hA gokbZ tgkt dh lkekU; vkSlr pky Kkr dhft,A

vFkok

lkjl ifk;ksa esa ls ,d&pkSFkkbZ dey ds ikS/ksa ds ihNs fNi jgs gSaA dqy la[;k dk 1 9 vkSj mlh la[;k dk 1

4 o mlh la[;k ds oxZewy ds lkr xqus feydj ,d igkM+h

ij ?kwe jgs gSaA 'ks"k 56 lkjl odqyk o`k ij cSBs gSaA lkjlksa dh dqy la[;k Kkr dhft,A

30- ,d lekarj Js<+h dk NBk in igys in dk ik¡p xquk gS vkSj X;kjgok¡ in mlds ik¡posa in ds nqxqus ls 3 vf/d gSA lekarj Js<+h Kkr dhft, vkSj bldk 8ok¡ in Hkh fu/kZfjr dhft,A igys vkB inksa dk ;ksx Hkh Kkr dhft,A

31- ,d f=kHkqt ds var% o`Ùk dh f=kT;k 4 lseh- gSA ;fn f=kHkqt dh ,d Hkqtk dks Li'kZ fcUnq ftu js[kk[kaMksa esa foHkkftr djrk gS mudh yEckbZ 6 lseh- vkSj 8 lseh- gS] rks f=kHkqt dh vU; nks Hkqtkvksa dh yEckbZ Kkr dhft,A

181 Math (Hindi) ­X

32- /krq dh pknj ls cuk vkSj mQij ls [kqyk ,d crZu 'kadq ds fNUud ds vkdkj dk gS] ftldh mQ¡pkbZ 24 lseh- gS rFkk mQijh vkSj fupys fljksa dh f=kT;k,¡ Øe'k% 15 lseh- gSaA 20 #- izfr yhVj dh nj ls] bl crZu dks iwjk Hkj ldus okys nw/ dk ewY; Kkr dhft,A lkFk gh bl crZu dks cukus ds fy, iz;qDr /krq dh pknj dk ewY; 10 #- izfr 100 oxZ lseh- dh nj ls Kkr dhft,A (π = 3.14 dk iz;ksx dhft,A

33- ,d dkSvk ?kj dh Nr ij cSBk gS ftldh m¡QpkbZ 40 ehVj gSA dkSos dk Hkwry ij fLFkr ,d fcUnq ls fdlh k.k ij mUu;u dks.k 45° gSA ;fn 3 lsds.M ckn dkSos dk mUu;u dks.k 30° gks tkrk gS tcfd og ,d gh fn'kk esa leku m¡QpkbZ ij mM+ jgk gks] rks dkSos dh pky Kkr djks ( 3 1.732 = fdeh-@?kaVk esa)A iz;ksx dhft,A)

vFkok tehu ls 4000 ehVj dh m¡QpkbZ ij mM+rk gqvk ,d gokbZ tgkt tc ,d nwljs tgkt ds Bhd mQij ls xqtjrk gS] ml le; tehu ij fLFkr fdlh ,d gh fcUnq ls buds mUu;u dks.k Øe'k% 60° vkSj 45° gSaA ml fLFkfr esa nksuksa tgktksa ds chp dh mQèokZ/j m¡QpkbZ Kkr dhft,A ( 3 1.732 = iz;ksx dhft,A)

34- ;fn ,d f=kHkqt dh Hkqtkvksa ds eè;&fcUnqvksa ds funsZ'kkad (3, 1), 5, 6)vkSj (–3, 2) gSa] rks blds 'kh"kks± ds funsZ'kkad Kkr dhft,A

182 Math (Hindi) ­X

lSEiy iz'u i=k

xf.kr dkk X (SA­2)

Time Allowed : 3 to 3½ hours Maxmimum Marks : 80

lkekU; funsZ'k 1 lHkh iz'u vfuok;Z gSaA 2- bl iz'u i=k esa dqy 34 iz'u gSa tks pkj Hkkxksa A, B, C vkSj D esa foHkkftr gSaA

Hkkx&A esa 10 iz'u gS izR;sd iz'u 1 vad dk gSA Hkkx&B esa 8 iz'u gSa izR;sd iz'u 2 vad dk gSA Hkkx&C esa 10 iz'u gSa izR;sd iz'u 3 vad dk gSA Hkkx&D esa 6 iz'u gSa izR;sd iz'u 4 vad dk gSA

3- iz'u la[;k 1 ls 10 cgqfodYih; iz'u gSA fn, x, pkj fodYiksa esa ls ,d lgh fodYi pqusaA

4- blesa dksbZ loksZifj fodYi ugha gSA ysfdu vkarfjd fodYi 1 iz'u 2 vadksa esa] 3 iz'u 3 vadksa esa vkSj 2 iz'u 4 vadksa esa fn, x, gSaA vki fn, x, fodYiksa esa ls ,d fodYi dk p;u djsaA

5- dSydqysVj ds iz;ksx dh vuqefr ugha gSA

Section ­ A iz'u 1 ls 10 rd izR;sd ,d vad dk gSA 1- m¡QpkbZ h rFkk Øe'k% ckgjh vkSj vkUrfjd f=kT;kvksa R vkSj r okys ,d [kks[kys yac

o`Ùkh; csyu dk vk;ru gS% (a) πh (R 2 – r 2 ) (b) π 2 h (R 2 – r 2 ) (c) π h (r 2 – R 2 ) (d) πrh (R 2 – r 2 )

2- lekarj Js.kh 7, 10, 13, ........ dk 5ok¡ in gS% (a) 19 (b) –19 (c) 5 (d) 24

183 Math (Hindi) ­X

3- ;fn fdlh o`Ùk dh nks f=kT;kvksa ds chp dk dks.k 100° gS] rks bu f=kT;kvksa ds fljksa ij Li'kZ js[kkvksa ds chp dk dks.k gS %

(a) 100° (b) 80° (c) 40° (d) 60°

4- ;fn x = 3 lehdj.k 3x 2 + kx + 3 = 0 dk ,d ewy gS] rks k dk eku gS% (a) ± 6 (b) 10 (c) –10 (d) 26

5- fcUnqvksa (3, –4) vkSj (4, –3) dks feykus okys js[kk[k.M dk eè;&fcUnq gS% (a) (–3.5, –3.5) (b) (–3.5, 3.5) (c) (3.5, –3.5) (d) (3.5, 3.5)

6- tc ,d ikWls dks ,d ckj isaQdk tkrk gS] rc lHkh laHko ifj.kke gSa% (a) 1 (b) 2, 4, 6 (c) 1, 3, 5 (d) 1, 2, 3, 4, 5, 6

7- (2, –3) dh ewy fcUnq ls nwjh gS% (a) 13 (b) 13 (c) 1 (d) 5

8- fdlh fcUnq T ls ,d o`Ùk ij [khaph xbZ Li'kZ js[kk dh yEckbZ 4 lseh- gS rFkk T dh o`Ùk ds dsUnz ls nwjh 5 lseh- gSA

(a) 5 cm (b) 3 cm (c) 20 cm (d) 25 cm

9- ;fn P(E) = 0.25 gS] rks P (E ugha) fuEu ls izkIr gS% (a) 0.75 (b) 0.25 (c) 1 (d) 0

10- fdlh ?kVuk dh izkf;drk gksrh gS% (a) 0 ls de (b) 0 ls vf/d (c) 0 vkSj 1 ds chp fLFkr vkSj nksuksa lfEefyr gSa) (d) 0 vkSj – 1 ds chp fLFkr

184 Math (Hindi) ­X

Section ­ B iz'u 11 ls 18 rd izR;sd 2 vad dk gSA 11- k ds fdl eku ds fy,] f?kkr lehdj.k 3x 2 + 2kx + 27 = 0 ds ewy okLrfod

vkSj cjkcj gksaxs\ 12- ;fn ewyksa dk vfLrRo gS] rks 2 4 4 3 3 0 x x + + = dks x ds fy, gy dhft,A 13- ;fn fdlh A.P. dk 17ok¡ in mlds in 7 vf/d gSA bl A.P. dk lkoZ varj

Kkr dhft,A 14- ;fn fdlh A.P. ds izFke n inksa dk ;ksx S n = 5n 2 + 3n ls fn;k tkrk gS] rks A.P.

dk nok¡ in Kkr dhft,A 15- P ls dsUnz O okys o`Ùk ij ,d Li'kZ js[kk PQ gS rFkk QOR bl o`Ùk dk ,slk O;kl

gS fd ∠POR = 110° gSA dks.k OPQ Kkr dhft,A 16- vkd`fr 1 esa] CP vkSj CQ dsUnz O okys o`Ùk dh Li'kZ js[kk,¡ gSaA ARB ,d vU;

Li'kZ js[kk gS] tks bl o`Ùk dks R ij Li'kZ djrh gSA ;fn CP = 11 lseh- vkSj BC = 7 lseh- gS] rks BR dh yEckbZ Kkr dhft,A

17- fdukjs 4 lseh- okys nks ?kuksa dks fljs ls fljk feydj tksM+k tkrk gSA ifj.kkeh ?kukHk dk i`"Bh; ks=kiQy Kkr dhft,A

18- P ds fdl eku ds fy, fcUnq (2, 1), (P, –1) vkSj (–1, 3) ljs[kh gSa\ vFkok

og vuqikr Kkr dhft, ftlesa fcUnqvksa (6, 4) vkSj (1, –7) dks feykus okyk js[kk[k.M x­vk ls foHkkftr gksrk gSA

185 Math (Hindi) ­X

Section ­ C iz'u 19 ls 28 rd izR;sd 3 vad dk gSA

19- fdlh A.P. ds izFke vkSj vfUre in Øe'k% 17 vkSj 350 gSaA ;fn lkoZ varj 9 gS] rks bl A.P. esa fdrus in gSa rFkk budk ;ksx D;k gS\

20- ,d Bksl f[kykSuk ,d yac o`Ùkh; csyu ds vkdkj dk gS] ftls fljs v/Zxksykdkj gSaA csyu vkSj v¼Zxksys dh mHk;fu"B f=kT;k 3.5 lseh- gS rFkk csyukdkj Hkkxh m¡QpkbZ 8 lseh- gSA bl f[kykSus dk vk;ru Kkr dhft,A 22

7 π =

dk i;z ksx dhft,

vFkok ,d f[kykSuk f=kT;k 3.5 lseh- okys ,d 'kadq ds vkdkj dk gS] tks ,d leku f=kT;k ds v/Zxksys ij cuk gSA f[kykSus dh dqy m¡QpkbZ 15.5 lseh- gSA bldk dqy i`"Bhd ks=kiQy Kkr dhft,A

21- f=kT;k 5 lseh- okys ,d o`Ùk ij nks Li'kZ js[kk,¡ [khafp, tks ijLij 60° ds dks.k ij >qdh gksaA

22- fdlh A.P. ds pkSFks vkSj 8osa inksa dk ;ksx 24 gS rFkk 6osa vkSj 10osa inksa dk ;ksx 44 gSA og A.P. Kkr dhft,A

23- vkd`fr 2 esa] dsUnz O okys o`Ùk ij fcUnq R ls nks Li'kZ js[kk,¡ RP vkSj RQ [khaph xbZ gSaA ;fn ∠PRQ = 120° gS] rks fl¼ dhft, fd OR = 2PR gSA

24- ;fn lehdj.k (1 + m 2 ) x 2 + 2mcx + c 2 ) = 2 ds ewy leku gksa] rks fl¼ dhft, c 2 = c 2 (1 + m 2 ).

vFkok ml f=kHkqt dh ekfè;dkvksa dh yEckb;k¡ Kkr dhft,] ftlds 'kh"kZ (1, –1), (0, 4) vkSj (–5, 3) gSaA

186 Math (Hindi) ­X

25- /krq ds rhu ?kuksa ds fdukjs 3 lseh-] 4 lseh- vkSj 5 lseh- gSaA bUgsa fi?kykdj ,d vdsyk ?ku cuk;k tkrk gSA b, u, ?ku dk fdukjk Kkr dhft,A bl u, ?ku dk i`"Bh; ks=kiQy Hkh Kkr dhft,A

26- vkd`fr 3 esa] PQRS f=kT;k 6 lseh- okys ,d o`Ùk dk O;kl gSA yEckb;k¡ PQ, OR vkSj RS cjkcj gSaA PQ vkSj QS ij] vkd`fr esa n'kkZ, vuqlkj v/Zo`Ùk [khaps x, gSaA Nk;kafdr Hkkx dk ks=kiQy Kkr dhft,A

27- ;fn fdlh ∆ABC ds nks 'kh"kZ A (3, 2) vkSj B (–2, 1) gSa rFkk dsUnzd G ds funsZ'kkd (5/3, – 1/3) gSa] rks mlds rhljs 'kh"kZ ds funsZ'kkad Kkr dhft,A

28- fdlh vf/ (yhi) o"kZ esa 53 jfookj gksus dh izkf;drk Kkr dhft,A vFkok

52 rk'kksa dh xM~Mh esa ls ,d dkMZ ;kn`fPNd :i ls fudkyk tkrk gSA dkMZ ds fuEu gksus dh izkf;drk Kkr dhft,% (i) ,d ykyk dkMZ (ii) ,d isQl dkMZ (iii) iku dk dkMZA

Section ­ D iz'u 29 ls 34 rd izR;sd 4 vad dk gSA

29- fdlh dkk VsLV esa] P kjk xf.kr vkSj foKku esa izkIr fd, x, vadksa dk ;ksx 28 gSA ;fn mlus xf.kr esa 3 vad vf/d vkSj foKku esa 4 vad de izkIr fd, gksrs] rks bu nksuksa fo"k;ksa esa izkIr fd, x, vadksa dk xq.kuiQy 180 gksrkA mlds kjk nksuksa fo"k;ksa esa izkIr fd, x, vad i`Fkd~&i`Fkd~ Kkr dhft,A

vFkok

187 Math (Hindi) ­X

600 fdeh- dh ,d mM+ku esa] [kjkc ekSle ds dkj.k] ,d gokbZ tgkt dks bl ;k=kk esa viuh vkSlr pky 200 fdeh- izfr ?kaVk de djuh iM+h] ftlls mM+ku ds le; esa 30 feuV dh o`f¼ gqbZA mM+ku dk dqy le; Kkr dhft,A

30- fdlh ehukj ds f'k[kj ds fcUnqvksa P vkSj Q ls] tks ehukj ds vk/kj ds lkFk ,d ljy js[kk esa gS rFkk mlls a vkSj b ehVj dh nwfj;ksa ij gSa] mUu;u dks.k ijLij iwjd gSaA fl¼ dhft, fd ehukj dh m¡QpkbZ ab ehVj gSA

31- m¡QpkbZ 28 lseh- okys 'kadq ds ,d fNUud ds o`Ùkh; fljksa dh f=kT;k,¡ 28 lseh- vkSj 7 lseh- gSaA bl fNUud dh /kfjrk vkSj i`"Bh; ks=kiQy Kkr dhft,A

vFkok ,d crZu] ftldk vkdkj O;kl 12 lseh- vkSj m¡QpkbZ 15 lseh- okys ,d yac o`Ùkh; csyu gS] vkblØe ls iwjk Hkjk gqvk gSA bl vkblØhe dks m¡QpkbZ 12 lseh- vkSj O;kl 6 lseh- okys 'kadqvksa esa Hkjk tkrk gS] ftldk mQijh Hkkx v/Z xksykdkj jgrk gSA bl vkblØhe ls Hkjs tk ldus okys 'kadqvksa dh la[;k Kkr dhft,A

32- 20 eh-] 34 eh- vkSj 42 eh- Hkqtkvksa okys ,d f=kHkqtkdkj [ksr dh rhuksa dksuksa ls 7 lseh- yEch jfLl;ksa kjk rhu ?kksM+s ck¡/ fn, tkrs gSaA [ksr ds ml Hkkx dk ks=kiQy Kkr dhft,] ftlesa ?kksM+s ?kkl pj ldrs gSaA [ksr ds ml Hkkx dk Hkh ks=kiQy Kkr dhft, ftlesa ?kkl ugha pjh tk ldrh gSA

33- fl¼ dhft, fd fdlh o`Ùk ij ,d ckgjh fcUnq ls [khaph xbZ Li'kZ js[kkvksa dh yEckb;k¡ cjkcj gksrh gSaA

34- Hkwfe ij fLFkr fdlh fcUnq ls ,d mQèokZ/j ehukj ds 'kh"kZ dk mUu;u dks.k 30° gSA bl ehukj ds vk/kj dh vksj 12 eh- dh nwjh pyus ij] ehukj ds 'kh"kZ dk mUu;u dks.k 60° gks tkrk gSA ehukj dh m¡QpkbZ Kkr dhft,A