GROUP LEADER GOVT. CO-ED. S.S.S, GOPAL PARK, DELHI-110033 · 1 [class x : maths] team members for review of support material group leader mr. yogesh agarwal (principal) govt. co-ed

1 [Class X : Maths]

TEAM MEMBERS FOR REVIEW OF SUPPORT MATERIAL

GROUP LEADER MR. YOGESH AGARWAL (PRINCIPAL)

GOVT. CO-ED. S.S.S, GOPAL PARK, DELHI-110033

1. MR. PRADEEP KUMAR TGT (MATHS)

G.CO-ED.S.S.S, GOPAL PARK, DELHI-33

2. MR. RAJBIR SINGH TGT (MATHS)


3. MS. PREETI SINGHAL TGT (MATHS)


4. MS. ANJU SAREEN TGT (MATHS)

S.C.S.D G.S.V, SECTOR-9, ROHINI, DELHI

5. MS. MURTI DEVI TGT (MATHS)

S.C.S.D G.S.V, SECTOR-9, ROHINI, DELHI

6. MS. MADHU BALA YADAV TGT (MATHS)

S.K.V., BL BLOCK, SHALIMAR BAGH, DELHI

7. MS. NEETU MEDIRATTA TGT (MATHS)

S.K.V., BL BLOCK, SHALIMAR BAGH, DELHI

8. MR. ANURAG YADAV TGT (MATHS)

R.P.V.V., KISHAN GANJ, DELHI

9. MR. MANISH JAIN TGT (MATHS)

R.P.V.V. D-1, NAND NAGRI, DELHI

10. MR. SUNIL KUMAR TIWARI TGT (MATHS)

S.B.V. MOTI NAGAR, DELHI

11. MR. MAQSOOD AHMED TGT (MATHS)

ANGLO ARABIC SR. SEC. SCHOOL, AJMERI GATE, DELHI

2 [Class X : Maths]

COURSE STRUCTURE (SECOND TERM)

CLASS-X

Units Marks

II ALGEBRA (Contd.) 23

III GEOMETRY (Contd.) 17

IV TRIGONOMETRY (Contd.) 08

V PROBABILITY 08

VI COORDINATE GEOMETRY 11

VII MENSURATION 23

Total 90

UNIT II : ALGEBRA (Contd.)

3. QUADRATIC EQUATIONS (15) Periods

Standard form of a quadratic equation 푎푥 + 푏푥 + 푐 = 0, (푎 ≠ 0). Solution of quadratic equations (only real roots) by factorization, by completing the square and by using quadratic formula. Relationship between discriminant and nature of roots.

Situational problem based on quadratic equations related to day to day activities to be incoperates.

4. ARITHMETIC PROGRESSIONS (8) Periods

Motivation for studing Arithmetic Progression Derivation of the 푛 term and sum of the first n term of A.P. and their application in solving daily life problems.

UNIT III : GEOMETRY (Contd.)

2. CIRCLES (8) Periods

Tangent to a circle at a point.

1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact.

3 [Class X : Maths]

2. (Prove) The lengths of tangents drawn from an external point to circle are equal.

3. CONSTRUCTIONS (8) Periods

1. Division of a line segment in a given ratio (internally).

2. Tangent to a circle from a point outside it.

3. Contruction of a triangle similar to a given triangle.

UNIT IV : TRIGNOMETRY

3. HEIGHT AND DISTANCES (8) Periods

Simple problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation/depression should be only 30°, 45°, 60°.

UNIT V : STATISTICS AND PROBABILITY

2. PROBABILITY (10) Periods

Classical definition of probability. Simple problems on single events (not using set notation).

UNIT VI : COORDINATE GEOMETRY

1. LINES (In two-dimensions) (12)Periods

Review : Concepts of coordinate geometry, graphs of linear equations. Distance between two points. Section formula (internal devision). Area of a triangle.

UNIT VII : MENSURATION

1. AREAS RELATED TO CIRCLES (12) Periods

Motivate the area of a circle; area of sectors and segments of a circle. Problems based on area and perimeters / circumference to the above said plane figures. (In calculating area of segment of a circle, problems should be restricted to central angle of 60 °, 90° and 120 ° only. Plane figures involving triangles, simple quadrilaterals and circle should be taken.)

4 [Class X : Maths]

2. SURFACE AREAS AND VOLUMES (12) Periods

(i) Surface areas and volumes of combination of any two of the following : cubes,cuboids, spheres, hemispheres and right circular cylinders/ cones, Frustum of a cone.

(ii) Problems involving converting one type of metallic solid into another and other mixed problems. (Problems with combination of not more than two different solids be taken.)

5 [Class X : Maths]

QUESTION PAPER DESIGN 2016-17

CLASS-X Mathematics (Code No. 041) Time: 3 Hours Marks: 90 S. No.

Typology of Questions Very Short Answer (VSA)

(1 mark)

Short Answer-I

(SA) (2 Marks)

Short Answer-II

(SA) (3 Marks)

Long Answer

(LA) (4 Marks)

Total Marks

% Weightage

1 Remembering - (Knowledge based Simple recall questions, to know specific facts, terms, concepts, principles, or theories, identify, define, or recite, information)

1 2 2 3 23 26%

2 Understanding-(Comprehension to be familiar with meaning and to understand conceptually, interpret, compare, contrast, explain, paraphrase, or interpret information)

2 1 1 4 23 26%

3 Application (Use abstract information in concrete situation, to apply knowledge to new situations; Use given content to interpret a situation, provide an example, or solve a problem)

1 2 3 2 22 24%

4 High Order Thinking Skills (Analysis & Synthesis- Classify, compare, contrast, or differentiate between different pieces of information; Organize and/or integrate unique pieces of information from a variety of sources)

- 1 4 - 14 16%

5 Creating, Evaluationand Multi-Disciplinary-(generating new ideas, product or ways of viewing things Appraise,judge. and/or justify the valuen of worth of a decision or, outcome, or to predict outcomes based on values)

- - - 2* 8 8%

Total 4x1=4 6x2=12 10x3=30 11x4=44 90 100% *One of the LA (4 marks) will to assess the value inherent in the text.

6 [Class X : Maths]

fo"k; lwph – SA-II

dze la- fo’k; i’B la-

1- f}?kkr lehdj.k 7&16

2- lakearj Jsf<+;ka 17&25

3- funsZ”kkad T;kfefr 26&32

4- f=dks.kfefr ds dqN vuqiz;ksx 33&41

5- oRr 42&58

6- jpuk,Wa 59&65

7- oRrksa ls lEcfU/kr {ks0 66&81

8- i’Bh; {ks=Qy rFkk vk;ru 82&92

9- izkf;drk 93&101

10- ewY; 102&102

11- SA-II vH;kl iz”u i= 103&109

12- ekWMy iz”u i= 110&117

7 [Class X : Maths]

v/;k;&1

f}?kkr lehdj.k

egRoiw.kZ fcanq: 1- lehdj.k 푎푥 + 푏푥 + 푐 = 0, 푎 ≠ 0 f}?kkr lehdj.k gS] ftlesa 푎, 푏, 푐

okLrfod la[;k,sa gSaA mnkgj.k 2푥 − 3푥 + 1 = 0.

2- f}?kkr lehdj.k ds ewy

,d okLrfod la[;k 훼 dks f}?kkr lehdj.k dk ewy dgka tk ldrk gS

;fn 푎훼 + 푏훼 + 푐 = 0

3- ewyksa dh la[;k & ,d f}?kkr lehdj.k ds nks ewy gksrs gSaA

4- f}?kkr lehdj.k gy djdus ds fof/k

xq.ku[kaM }kjk iw.kZoxZ fof/k }kjk f}?kkr QkewZys }kjk

5- f}?kkr lehdj.k 푎푥 + 푏푥 + 푐 = 0 ds fuEu ewy gSa

푥 =−푏 + √푏 − 4푎푐

2푎,−푏 −√푏 − 4푎푐

2푎

6- fofoDrdj fdlh f}?kkr lehdj.k 푎푥 + 푏푥 + 푐 = 0 ds fy, fofoDrj

퐷 = 푏 − 4푎푐 gksrk gSA

vFkkZr 퐷 = 푏 − 4푎푐 gksus ij f}?kkr lehdj.k ds ewy fuEu gSa

푥 = ±√ ,푥 = √

7- ewyksa dk LoHkko

;fn 퐷 > 0 gks rks okLrfod vkSj vleku ewy gksaxsA

;fn 퐷 = 0 gks rks okLrfod vkSj leku ewy gksaxsA

;fn 퐷 < 0 rks vokLrfod ewy gksaxsA

8 [Class X : Maths]

vfry?kq mRrjh; iz”u

1- ;fn − f}?kkr lehdj.k 2푥 + 푘푥 + 1 = 0 dk ewy gkss rks 퐾 Kkr djsaA

2- f}?kkr lehdj.k 3푥 − 4√3푥 + 4 = 0 ds ewy fdl izdkj ds gksaxs?

3- D;k lehdj.k 푥 − 4푥 − 푥 + 1 = (푥 − 2) ,d f}?kkr lehdj.k gS?

4- lehdj.k 5푥 − √2푥 + 3 = 0 dks iw.kZ oxZ fof/k }kjk gy djus ij dkSu

lk fLFkjkad tksM+k ;k ?kVk;k tk,xk? 5- ;fn 푥 = −1 vkSj 푥 = −2 lehdj.k 푝푥 + 3푥 + 푞 = 0 ds ewy gksa rks

푞 − 푝 Kkr djsaA

6- ,d f}?kkr lehdj.k cuk,sa ftlus ewy √2 vkSj 1 gksaA

7- fuEu ds fy, f}?kkr lehdj.k cuk,as **nks dzekxr le iw.kkZadksa dk

xq.kuQy 1848 ^^ gSA

8- D;k 0-2] lehdj.k 푥 − 0.4 = 0 dk ewy gS ? 9- ;fn 푎푥 + 푏푥 + 푐 ds ewy cjkcj gksa rks 푐 dks 푎 vkSj 푏ds :i esa O;Dr

djsaA

10- f}?kkr lehdj.k 푥 + 6푥 − 91 = 0 dks ;fn (푥 + 푝)(푥 + 푞) = 0 ds :i

esa O;Dr djus ij 푝 vkSj 푞 D;k gksaxs ?

y?kq mRrjh; iz”u (I) 11- xq.ku[kaM fof/k }kjk djsa %&

(a) 8푥 − 22푥 − 21 = 0

(b) 3√5푥 + 25푥 + 10√5 = 0

(c) √3푥 − 2√2푥 − 2√3 = 0

(d) 2푥 + 푎푥 − 푎 = 0

9 [Class X : Maths]

12- ;fn 2푥 + 푘푥 + 푘 = 0 ds ewy okLrfod vkSj cjkcj gksa rks 푘dk eku Kkr

djsaA

13- ;fn 9푥 + 3푘푥 + 4 = 0 ds ewy vleku gksa rks 푘 dk eku Kkr djsaA

14- 푝 ds fdl eku ds fy, f}?kkr lehdj.k 푥 + 5푝푥 + 16 = 0 ds

vokLrfod ewy gksaxs?

15- 푐 ds fdl eku ds fy, lehdj.k 4푥 − 2푥 + (푐 − 4) = 0 ds ewy ,d

nwljs ds O;qRdze gksaxs?

16- 푝 ds faadl eku ds fy, lehdj.k 푝푥 + 6푥 + 4푝 = 0 ds ewyksa dk

xq.kuQy vkSj mudk ;ksx cjkcj gksxk?

17- nks oxksZ dh Hkqtk,a 푥 lseh vkSj (푥 + 4) lseh gSasA muds {ks=Qyksa dk ;ksx

656 oxZ lseh gks rks oxksZa dh Hkqtkvksa dk eki Kkr djsaA

18- 푝 dk eku Kkr djsa ftlds fy, lehdj.k 푝푥 + (푥 − 3) + 9 = 0ds ewy

cjkcj gksa\

19- 16 ds nks fgLlsa bl izdkj djsa fd cM+h la[;k dk nqxquk la[;k ds oxZ ls

164 vf/kd gksA

20- 푘 dk eku Kkr djsas tc lehdj.k 푥 − 5푥 + (3푘 − 3) = 0 ds ewyksa dk

varj 11 gksA

21- nks dzekxr izkdfrd la[;kvksa ds oxksZa dk ;ksx 313 gSA la[;k,sa Kkr djsaA

y?kq mRrjh; iz”u (II) 22- fuEu f}?kkr lehdj.k ljy djsa &

(a) = + + , 푎 + 푏 ≠ 0

10 [Class X : Maths]

(b) = + +

(c) +( )

= , 푥 ≠ −1, 2, 0

(d) 3 − 4 = 11, 푥 ≠ ,

(e) + = , 푥 ≠ −2, 4

(f) 푎푥 + (4푎 − 3푏)푥 − 12푎푏 = 0

(g) 4푥 − 4푎푥 + (푎 − 푏 ) = 0

(h) − 3 = , 푥 ≠ 0,−3/2

23- f}?kkr lw= dk iz;ksx djds lehdj.k ljy djsa &

푎푏푥 + (푏 − 푎푐)푥 − 푏푐 = 0

24- ;fn & 5] 2푥 + 푝푥 − 15 dk ewy gks vkSj 푝(푥 + 푥) + 푘 = 0 ds ewy

cjkcj gksa rks 푝 vkSj 푘 dk eku Kkr djsaA

nh?kZq mRrjh; iz”u

25- ;fn f}?kkr lehdj.k (푝+ 1)푥 − 6(푝 + 1)푥+ 3(푝 + 9) = 0 ds ewy cjkcj gks rks 푝 dk eku Kkr djsa rFkk fQj bl lehdj.k ds ewy Hkh Kkr djsaA

26- 푘 ds fdl eku ds fy, f}?kkr lehdj.k (2푘 + 1)푥 − (7푘 + 2)푥 +

(7푘 − 3) = 0 ds ewy ckjkcj gksaxs\ eqy Hkh Kkr djsaA

27- ;fn f}?kkr lehdj.k (1 + 푚 )푥 + 2푚푐푥 + (푐 − 푎 ) = 0 ds eqy cjkcj

gksa rks fl) djsa 푐 = 푎 (1 +푚 )

28- 푘 ds fdl eku ds fy, (4− 푘)푥 + (2푘 + 4)푥 + (8푘 + 1) = 0 ,d iw.kZ

oxZ gS\


29- cRr[k ds lewg ds oxZewy dk xq.kk unh ds fdukjs [ksy jgs FksA

“ks’k nks cRr[k ikuh ds vanj [ksy jgs FksA cRr[kksa dh dqy la[;k Kkr djsaA

30- ,d eksj 9 m mWaps LraHk ij cSBk Fkk LraHk ds ry ls 27 m nwj ,d laki

gS tks vius fcy dh rjQ tks vius fcy dh rjQ tks LraHk ds ry esa gS

vk jgk gSA

lkai dks ns[krs gh eksj us >iVVk ekjk ;fn nksuksa ds xfr cjkcj gks rks fcy

ls fdruh nwjh ij lkai idM+k x;kA

31- 9000 : dks dqN O;fDr;ksa esa cjkcj ckaVk x;kA ;fn 20 O;fDr vkSj vk

tk,s arks izR;sd O;fdr dks 160 : de izkIr gksaxsA rks dqy o;fDr fdrus

gSa Kkr djsaA

32- ,d O;fDr ,d f[kykSuk 24: esa csprk gS vkSj mrus izfr”kr ykHk izkIr

djrk gS ftruk f[kykSus dk dz;ewY; gksA f[kykSus dk dz;ewY; Kkr djsaA

33- ,d fodzsrk dqN fdrkcsa 80: esa [kjhnrk gSA ;fn og 4 fdrkcsa vkSj mlh

nke esa [kjhns rks izzR;sd fdrkc dk ewY; 1: de gks tkrk gSA mlus dqy

fdruh fdrkcsa [kjhnh\

34- nks ikbi ,d lkFk ,d flLVZu dks Hkjus esa 3 feuV yxkrs gSaA ;fn ,d

ikbi bls Hkjus esa nwljs ikbi ls 3 feuV T;knk ys rks fdrus le; esa

izR;sd ikbi bl flLVZu dks Hkjsxk\

35- nks oxksZa ds {ks=Qyksa dk ;ksx 400 푐푚 gSA ;fn muds ifjekiksa dk varj

16 푐푚 gks rks izR;sd oxZ dh Hkqtk dk eki Kkr djsaA

36- ,d lef}ckg f=Hkqt dk {ks=Qy 60 푐푚 gSA bldh cjkcj Hkqtkvksa dh yackbZ

13 푐푚 gS rks bl f=Hkqt dk vk/kkj Kkr djsaA


37- ,d pSl cksMZ esa 64 cjkcj oxZ gSa vkSj gj oxZ dk {ks=Qy 6-25 푐푚 gSA

bl cksMZ ds pkjksa rjQ 2 푐푚 pkSM+k ckMZj gSA bl pSl cksMZ dh yackbZ

Kkr djsaA

38- ,d yM+dh dh mez viuh cgu ls nqxuh gSA pkj o’kZ i”pkr nksuksa dh

vk;q dk xq.kuQy 160 gksxkA mudh orZeku vk;q Kkr djsaA

39- ,d uko ftldh xfr “kkar ty esa 18 fdeh 1 ?kaVk gS tks 24 km fdeh

/kkjk ds fo:/k tkus esa vkSj 24 km fdeh /kkjk dh fn”kk esa vkus ij 1 ?kaVk

T;knk ysrh gSA /kkjk dh xfr Kkr djsaA

40- ,d rst pyus okyh jsyxkM+h ,d /khjs pyus okys jsyxkM+h ls 600 km

fdeh dh nwjh r; djus esa 3 ?kaVs de ysrh gSA ;fn /kheh jsyxkM+h dh

xfr 10 fdeh@?kaVk rsth jsyxkM+h ls de gks rks nksuksa jsyxkM+h dh xfr

Kkr djsaA

41- ,d fHkUu dk va”k] gj] ls 3 de gS ;fn va”k vkSj gj nksuksa gh esa 2 tksM+k

tk, rks u, fHkUu vkSj fn, x, fHkUu dk tksM+ gksxkA fn;k x;k fHkUu

Kkr djsaA

42- nks izd`frd la[;kvksa dk varj 3 gS vkSj muds O;qRdzeksa dk varj gSA

la[;k,sa Kkr djsaA

43- rhu dzekxr /kukRed iw.kkZadksa esa igyh la[;k dk oxZ vkSj vU; nks

la[;kvksa dk xq.kQy tksM+us ij 46 izkIr gksrk gSA iw.kkZad Kkr djsaA

44- ,d nsk vadksa dh la[;k vius vadksa ds tksM+ ls 3 xquk gS vkSj vius vadksa

ds xq.kuQy ls rhu xquk gSA la[;k Kkr djsaA

45- ,d ledks.k f=Hkqtkdkj Hkwfe dk d.kZ lcls NksVh Hkqtk ds nqxqus ls


10 ehVj cMk gS A ;fn rhljh Hkqtk NksVh Hkqtk ls 7 ehVj cM+h gks rks Hkweh

dh rhuksa Hkqtk,sa Kkr djsaA

46- ,d d{kk ijh{kk esa p ds xf.kr vkSj foKku esa izkIr vadksa dk tksM+ 28 gSA

;fn og xf.kr esa 3 vad vkSj izkIr djrk gS vkSj foKku essa 4 vkad de

izkIr djrk gS rks mlds vadksa dk xq.kuQy 180 gksrkA mlds nkuksa fo’k;ksa

esa izkIr vad Kkr djsaA

47- ,d diM+k 200 : dk gSA ;fn diM+s dh yEckbZ 5 eh0 vf/kd gksrh vkSj

izfr ehVj diM+s dk ewY; 2 : de gksrk rks Hkh diM+s dk dqy ewY; ogh

jgrkA diM+s dh yackbZ vkSj izfr ehVj diM+s dk ewY; Kkr djsaA

48- ,d gokbZ tgkt vius fu;fer le; ls 30 feuV nsj ls pyrk gSA vius

xarO; LFkku tks fd 1500 fdeh nwj gS ij le; ls igWqapus ds fy, mls

viuh xfr 250 fdeh@?kaVk c<+kuh iM+rh gSA gokbZ tgkt dh okLrfod

xfr Kkr djsaA

49- ;fn igys n le izd`frd la[;kvksa dk ;ksx 420 gSA n dk eku Kkr djksA

50- ,d gokbZ tgkt ij p<+rs gq, ,d vkneh dks pksV yx tkrh gSA ik;yV

“kh?kzrk ls mls vLirky igWapkus dh O;oLFkk djrk gS A blfy, gokbZ

tgkt 30 feuV nsj ls pyrk gSA vius xarO; LFkku tks fd 1500 fdeh

nwj gS ij le; ls igWapus ds fy, ik;yV dks xfr 100 fdeh@?kaVk c<+kuh

iM+rh gSA gokbZ tgkt dh okLrfod xfr Kkr djsaA

;gkWa fdl ewY; dk irk pyrk gSA

51- A ,d dke dks iwjk djus esa B ls 10 fnu de ysrk gSA A vkSj B

feydj mlh dke dks 12 fnuksa esa iwjk djrs gSaA B }kjk vdsys ml dke

dks iwjk djus esa fdruk l;e yxsxk\ iz”u esa crk,sa fdl thou ewY; dks

gesa viukuk pkfg,A


mRrjekyk

1) 푘 = 3 2) cjkcj ewy

3½ gkWa 4½ ;k

5½ 1 6½ 푥 − √2 + 1 푥 + √2 = 0

7½ 푥 + 2푥 − 1848 = 0 8½ ugha

9½ 푐 = 10½ 13] &7

11½ (a) 푥 = ,푥 = 12½ 푘 = 0, 8

(b) 푥 = −√5, 푥 = − √

(c) 푥 = √6, 푥 = − √ (d) 푥 = , 푥 = −푎

13½ 푘 > 4, 푘 < −4 14½ − < 푝 <

15½ 푐 = 8 16½ 푝 =

17½ 16 푐푚, 20 푐푚 18½ 푝 ≠ 0, 푝 = 4

19½ 푥 = 10, 6 20½ 푘 = −7

21½ 12, 13 22½ (a) 푥 = −푎,푥 = −푏

23½ 푥 = , (b) 푥 = −푎,푥 = −

24½ 푝 = 7,푝 = (c) 푥 = 4,푥 = −

25½ 푝 = 3,푥 = 3, 3 (d) 푥 = 0,푥 = 1

26½ 푘 = 4, (e) 푥 = ±√

(f) 푥 = ,푥 = −4푎

(g) 푥 = ,푥 =

(h) 푥 = −2,푥 = 1


28½ 푘 = 0, 3 29½ 16

30½ 12 푚 31½ 25

32½ 20 # 33½ 16

34½ 5 feuV] 8 feuV 35½ 12] 16

36½ 24 lseh] ;k 10 lseh 37½ 24 leh

38½ 6 o’kZ] 12 o’kZ 39½ 6 fdeh@?kaVk

40½ 40 fdeh@?kaVk]

50 fdeh@?kaVk 41½

42½ 7]4 43½ 4]5]6

44½ 24 45½ 8 eh] 17 eh] 15 eh

46½ xf.kr esa vad 12 47½ yEckbZ 20 eh

foKku esa vad 16 nj= 10 #@eh

48½ 750 fdeh@?kaVk 49½ 푛 = 20

50½ 500 fdeh@?kaVk] ekuork

51½ 30 fnu] ,drkA


vH;kl iz”ui=

le; % 50 feuV vad % 20

[kaM ^v^

1- ;fn lehdj.k 6푥 − 푏푥 + 2 = 0 dk fofoDrdj 1 gks rks 6 dk eku Kkr

djsaA ¼1½

2- 푥 + 5푥 − 300 = 0 esa 푥 dk eku Kkr djsasA ¼1½

[kaM ^c^

3- ;fn 푘푥 − 2푘푥 + 6 = 0 ds ewy cjkcj gksa rks 푘 dk eku Kkr djsaA ¼2½

4- ;fn 푥 + 푝푥 + 12 = 0 ds ewy 1: 3 esa gksa rks 푝 dk eku Kkr djsaA ¼2½

[kaM ^l^

5- f}?kkr lehdj.k ljy djsa& ¼3½

(푥 − 1) − 5(푥 − 1) − 6 = 0

6- 푘 dk eku Kkr djsa ;fn lehdj.k 푥 − 5푥 + 3(푘 − 1) = 0ds ewyksa dk

varj 11 gksA ¼3½

[kaM ^l^

7- ;fn f}?kkr lehdj.k (푏 − 푐)푥 + (푐 − 푎)푥 + (푎 − 푏) = 0 ds ewy cjkcj

gks rks fl) djsaA 2푏 = 푎 + 푐 ¼4½

8- nks izkd`frd la[;kvksa ds oxksZa dk ;ksx 52 gSA ;fn igyh la[;k nwljh

la[;k ds nqxqus ls 8 de gS rks la[;k,a Kkr djsaA ¼4½


v/;k;&2

lkekarj Jsf<+;kWa

egRoiw.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- lekUrj Js<+h%& ,d vuqØe ftlesa izR;sd in dk vxys in ls fuf”p varj

gksrk gS] lekarj vuqØe ;k lekarj Js<+h ¼l- Js- ;k A.P.½ dgykrh gSA

vFkkZr & lekarj Js<+h ¼l- Js- ;k A.P.½ ,d vuqdze 푎 ,푎 ,푎 , …푎 … gSa

tgkWa 푎 − 푎 = ⋯ = 푑

3- O;kid in %& ;fn ,d lekarj Js<+h dk izFke in ^a^ rFkk lkoZ varj ^d^

gks rks n okWa in ¼O;kid in½ fuEufyf[kr lw= }kjk izkIr gksrk gS%&

푎 = 푎 + (푛 − 1)푑

4- lekarj Js<+h ds n inksa dk ;ksxQy %& (푖) 푠 = [2푎 + (푛 − 1)푑]; tgkWa 푎 = izFke in

푑 = lkoZ varj (푖푖) 푠 = [푎 + 푙]; tgkWa 푎 = izFke in

푙 = vafre in

5- (i) ;fn 푎 fn;k gks rks lkoZ varj (푑) = 푎 − 푎

(ii) ;fn 푠 fn;k gks rks 푎 = 푠 − 푠

(iii) ;fn 푎, 푏, 푐 lekarj Js<+h ds Øekxr in gks rks 2푏 = 푎 + 푐

(iv) ;fn fdlh vuqØe esa x in gSa] rks bldk var ls r okWa in izkjaHk ls

(n– r+1)oka inA

(v) ;fn fdlh lekarj Js<+h ds m osa rFkk n osa inksa esa varj%

(푎 − 푎 ) = (푚− 푛)푑


vfr y?kq mRrjh; iz”u

1- ;fn ,d l- Js- ¼A.P.½ dk n okWa in 3x–5 gks rks bldk 5okWa in D;k gksxk\

2- izFke 10 le la[;kvksa dk ;ksxQy Kkr djksA

3- fo’k; la[;kvksa dk n okWa in fyf[k,A

4- izFke n izkdr la[;kvksa dk ;ksxQy D;k gksxk\

5- izFke n le la[;kvksa dk ;ksx D;k gksxk\

6- l0 Js0 &10] &15] &20] &25]--------dk n okWa in Kkr fdft,A

7- l0 Js0 4 , 4 , 4 , -------- dk lkoZ varj Kkr fdft,A

8- l0 Js0 dk lkoZ varj Kkr fdft, ;fn bldk n okWa in (푎 ) = 3푛 + 7

9- l0 Js0 4]9]14] ---------]254 ds fy, 푎 − 푎 dk eku D;k gksxk\

10- l0 Js0&10] &12] &14] &16]------------ds fy, 푎 − 푎 dk eku D;k gksxk\

11- ;fn 2푘, 4푘 − 3 vkSj 4푘 + 4 fdlh lekarj Js<+h ds Øekxr in gksa rks k dk

eku D;k gksxk\

12- , 푝, 2 lekarj Js<+h essa gSaA p dk eku Kkr fdft,A

y?kq mRrjh; iz”u&I

13- D;k 144 lekarj Js<+h 3] 7] 11]-----------dk ,d in gS\ vius mRrj dh iqf’V

dhft,A

14- l0 Js0 3] 8] 13]--------]253 dk var ls 20okWa in Kkr dhft,A

15- l0 Js0 5] 15] 25]------dk dkSu&lk in blds 31osa in ls 130 vf/kd gksxk\

16- fdlh lekarj Js<+h dk izFke in] lkoZ & varj vkSj vafre in dze”k% 12] 6

vkSj 252 gSaA blds lHkh inksa dk ;ksx Kkr dhft,A

17- 15 ds izFke 8 xq.ktksa dk ;ksx Kkr dhft,A


18- D;k uhps nh xbZ fLFkfr;ksa esa cuk vuqØe ,d lekarj Js<+h gS\

(i) dqy 1000 fo/kkfFkZ;ksa esa ls fo|ky; lHkkxkj esa cps fo?kkfFkZ;ksa dh

la[;k tcfd os fo+|ky; lHkkxkj dks 25 ds lewg esa NksM+rs gSaA

(ii) izfro’kZ [kkrs esa jkf”k tcfd 100 :Ik;s izfro’kZ tek djok, tkrs gSa

ftu ij 4% okf’kZd nj pØof) C;kt la;ksftr gksrk gSA

19- 1 ls 200 ds chp dh le la[;kvksa dk ;ksx Kkr dhft,A

20- ;fn 4푚 + 8, 2푚 + 3푚 + 6, 3푚 + 4푚 + 4 ,d lekarj Js<+h ds rhu

dzekxr in gksa rks m dk eku Kkr dhft,A

21- l0 Js0 22] 20] 18]-------------ds fdrus inksa dk ;ksx “kqU; gS\

22- ,d l0 Js0 ds nlosa in dk nl xq.kk blds chlosa in ds 20 xqus leku

gSA bldk 30okWa in Kkr dhft,A

23- l0 Js0 6] 13] 20---------]216 dk e/;in Kkr dhft,A

24- l0 Js0 20] 19 , 18 , 17 , ----------dk dkSu&lk in izFke Js.kkRed in

gksxk\ in Hkh Kkr dhft,A

y?kq mRrjh; iz”u&II

25- l0 Js0 7] 13] 19]----------241 ds e/; in Kkr dhft,A

26- 10 vkSj 5000 ds chp ds mu iw.kkZadksa dk ;ksx Kkr dhft, tks 7 ls foHkkT; gSaA

27- ,d l0 Js0 ds 5osa rFkk 9osa inksa dk ;ksx 72 rFkk 7osa vkSj 12osa inksa dk

;ksx 97 gSA l0 Js0 Kkr dhft,A

28- ,d l0 Js0 dk mokWa in rFkk n okWa in gSA fl) dhft, fd bldk

(mn)okWa in 1 gSA

29- ,d l0 Js0 dk p okWa in 2 rFkk q okWa in p gSA fl) dhft, fd bldk n okWa in (푝 + 푞 − 푛) gSA


30- ;fn fdlh l0 Js0 ds p osa in dk p xquk blds q osa in ds q xqus ds

cjkcj gks rks fl) dhft, fd bldk (푝+ 푞) okWa in “kwU; gksxkA

31- m ds fdl eku ds fy, nks l0 Js0 (i) 1] 3] 5] 7]-----------(ii) 4] 8] 12] 16-------

ds m osa in leku gksaxs\

32- ,d l0 Js0 dk 24okWa in blds 10osa in dk nks xquk gSA fl) dhft, fd

bldk 72okWa in blds 15osa in dk 4 xquk gSA

33- 101 vkSj 994 ds chp mu izkdr la[;kvksa dh dqy la[;k Kkr dhft, tks 2

vkSj 5 nksuksa ls foHkkT; gksaA

34- ;fn fdlh l0 Js0 dk 7okWa in 1@9 vkSj 9okWa in 1@7 gS] rks bldk 63okWa

in Kkr dhft,A

35- ,d l0 Js0 ds 5osa vkSj 9osa inksa dk ;ksx 30 gSA ;fn bldk 25okWa in

blds 8osa in dk 3 xquk gks rks l0 Js0 Kkr dhft,A

36- ;fn fdlh l0 Js0 ds izFke n inks dk ;ksx 푠 = 5푛 + 3푛 rks bldk n okWa in vkSj lkoZ varj Kkr dhft,A

nh?kZ mRrjh; iz”u

37- ,d lekarj Js<+h ds rhljs vkSj lkrosa inksa dk ;ksx 6 rFkk xq.ku 8 gSA bl

l0 Js0 ds izFke 16 inksa dk ;ksx Kkr dhft,A

38- ;fn fdlh l0 Js0 dk m okWa in vkSj n okWa in gks rks fl) dhft,

fd blds izFke (mn) inksa dk ;ksx (푚푛 + 1) gSA

39- ;fn fdlh l0 Js0 ds izFke m inksa dk ;ksx n rFkk izFke n inksa dk ;ksx

m gks rks fl) dhft, fd blds izFke (푚+ 푛) inksa dk ;ksx −(푚 + 푛) gksxkA


40- ;fn fdlh l0 Js0 ds izFke n inksa ds ;ksx dks 푠 ls n”kkZ;k tkrk gks rks

fl) dhft, fd 푠 = 3(푠 − 푠 )

41- ;fn fdlh l0 Js0 ds izFke k inksa dk ;ksx (3푘 + 7푘) rks bldk k okWa

in fyf[k, rFkk bldk 20 okWa in Kkr dhft,A

42- fdlh l0 Js0 ds izFke 9 inksa dk ;ksx 162 gSA blds NVs in dk rsjgosa in

ls vuqikr 1%2 gSA bl l0 Js0 dk igyk vkSj iUnzgoka in Kkr dhft,A

43- ,d l0 Js0 dk 10okWa in 21 gS vkSj izFke 10 inksa dk ;ksx 120 gSA bldk

n okWa Kkr dhft,A

44- ,d l0 Js0 ds izFke 7 inks dk ;ksx 63 vkSj vxys 7 inksa dk ;ksx 161 gS

bldk 28okWa in Kkr dhft,A

45- ,d l0 Js0 ds izFke q inksa dk ;ksx 63푞 − 3푞 gSA bldk p okWa in &60

gS] p eku Kkr dhft, rFkk X;kjgokWa in Hkh Kkr dhft,A

46- fdlh l0 Js0 dk izFke in &2] vafr in &29 vkSj lHkh inksa dk ;ksx &155

gSA bldk 11okWa in Kkr dhft,A

47- fdlh l0 Js0 ds izFke chl inksa dk ;ksx vxys chl inksa ds ;ksx dk ,d

frgkbZ gSA ;fn bl l0 Js0 dk izFke in 1 gks rks blds izFke 30 inksa dk

;ksx Kkr dhft,A

48- ,d l0 Js0 ds izFke 10 inksa dk ;ksx vxys 10 inksa dk ;ksx dk

,d&frgkbZ gSA bldk igyk in &5 gSA izFke 30 inksa dk ;ksx Kkr

dhft,A

49- ,d l0 Ls0 dk vkBokWa in blds nwljs in dk vk/kk gS rFkk X;kjgokWa in

pkSFks in ds ,d&frgkbZ ls 1 vf/kd gSA bldk 15okWa in Kkr dhft,A


50- rhu vadksa okyh ,d /kukRed la[;k ds vad lekarj Js<+h esa gSa rFkk mudk

;ksx 15 gSA la[;k esa 594 ?kVkus ij vad myV tkrs gSaA la[;k Kkr

dhft,A

51- ,d o) efgyk d’.k nsoh us 1]20]000 : cSad essa 8% okf’kZd nj ls tek

djkok,A izfro’kZ mlls izkIr lk/kkj.k C;kt dks ,d fo|ky; ds 5 fo|kfFkZ;kssa

dks muds lokZaxh.k izn”kZu ds vk/kkj ij Nk=ofÙk ds :Ik esa ckWaVk tkrk gSA

izR;sd Nk=o`fÙk igyh Nk=o`fÙk ls 300 # de gS izR;sd Nk=ofÙk dk dk

ewY; Kkr dhft,A efgyk ds fdu ewY;ksa dk irk pyrk gS\

52- jke ,d etnwwj ls 10 eh0 xgjk dqWavk [kksnus ds fy, dgrk gSA og igys

ehVj dks [kksnus dh etnwjh 150 # rFkk vkxs ds izR;sd ehVj dks [kksnus dh

etnwjh 50 # crkrk gSA etnwj vf”kf{kr gksus ds dkj.k dqy 550 #

etnwjh dh ekWax djrk gS etnwj dks okLro esa fdruh etnwjh feyuh

pkfg,\ ;fn jke etnwj dks 600 # etnwjh nsrk gS rks jke dks fdu

thou&eqY;ksa dk irk pyrk gSA


mRrjekyk

¼1½ 10

¼2½ 110

¼3½ 2n&1

¼4½ ( )

¼5½ 푛(푛+ 1)

¼6½ −5(푛+ 1)

¼7½

¼8½ 3

¼9½ 20

¼10½ &40

¼11½ 푘 = 5

¼12½

¼13½ ugha] D;ksafd a=3 ¼fo’k;

la[;k½] d=4 ¼le la[;k½]

blfy, bl l0 Js0 dk

izR;sd in fo’k; la[;k gh

gksxkA

¼14½ 158

¼15½ 44 okWa

¼16½ 5412

¼17½ 540

¼18½ (i) gkWa (ii) ugha

¼19½ 9900

¼20½ 0] 2

¼21½ 23

¼22½ 0

¼23½ 111

¼24½ 28okWa] &1@4

¼25½ 121] 127

¼26½ 17885

¼27½ 6] 11] 16] 21] 26]----

¼31½ m dk ,slk dksbZ eku laHko

ugha gS

¼33½ 89

¼34½ 1

¼35½ 3] 5] 7] 9] 11]-------

¼36½ 푎 = 10푛 − 2

¼37½ 76] 20


¼41½ 푎 = 3푘 + 2, 푎 = 62

¼42½ 6, 48 ¼43½ 2푛 + 1

¼44½ 57

¼45½ 푝 = 21, 푎 = 0

¼46½ &32

¼47½ 900

¼48½ &4500

¼49½ 3

¼50½ 852

¼51½ 2520 #] 2220 #] 1920 #]

1620 #] 1320#] I;kj@Lusg]

nku vkfn

¼52½ 600 #] bZekunkjh] ;FkkFkZrk


vH;kl iz”u i=

le; &1 ?kaVk vad % 20

[k.M ^v^

1- izFke 10 izkdr la[;kvksa dks ;ksx Kkr dhft,A ¼1½

2- lekUrj Js.kh 8 , 8 , 8 ,-----------dk lkoZ vUrj D;k gksxk\ ¼1½

[k.M ^c^

3- 6 vkSj 102 ds chp] 6 ls foHkkftr ] nks vdksa okyh fdruh gksxh\ ¼2½

4- fdlh l0 Js0 ds 푘 + 1, 3푘 vkSj 4푘 + 2 rhu Øekxr in gSaA 푘 dk eku

Kkr djksaA ¼2½

[k.M ^l^ 5- l0 Js.kh ds izFke ikWap in Kkr djks ftldk ;ksx 12 gks vkSj igyk rFkk

vafre in dk vuqikr 2 % 3 gksA ¼3½

6- l0 Js0 20] 16] 12]---&176 dk e/; in Kkr djksA ¼3½

[k.M ^n^

7- ,d rhu vadks ds /ku iw.kkW adksa ds vad l0 Js0 esa gSa vkSj mudk

;ksxQy 15 gSA la[;k esa s ls 594 ?kVkus ij vad iyV tkrs gSaA la[;k

Kkr dhft,A ¼4½

8- l0 Js0 ds rhu la[;kkvksa dk ;ksx 24 gS vkSj mudk xq.kuQy 440 gSA

la[;k,Wa Kkr dhft,A ¼3½


v/;k;&3

funsZ”kkad T;kfefr

egRoiw.kZ fcanq:

1- ekuk 푋푂푋 rFkk 푌푂푌 nks ijLij yac jks[kk,Wa gSaA bu js[kkvksa dks funsZ”kkad

v{k dgrs gSaA 푋푂푋 dks x-v{k vkSj 푌푂푌 dks y-v{k dgrs gSaA

2- x-v{k o y-v{k ds izfrPNsnu fcanq 0 dks ewy fcanq dgrs gSaA blds funsZ”kkad

(0, 0) gksrs gSaA

3- fdlh fcanq dk x-funsZ”kkad Hkqt rFkk y-funsZ”kkad dksfV dgykrk gSA

4- funsZ”kkad v{k lery dks pkj prqFkkZa”kksa esa foHkkftr djrk gSA

(i) Ikgys prqZFkka”k esa x vkSj y nksuksa funsZ”kkad /kukRed gksrs gSA

(ii) nwljs prqZFkka”k esa x funsZ”kkad _.kkRed c y funsZ”kkad /kukRed gksrk

gSA

(iii) Rkhljs prqZFkka”k esa x vkSj y nksuksa funsZ”kkad _.kkRed gksrs gSA

(iv) pkSFks prqFkkZ”ka esa] x funsZ”kkad /kukRed o y wfunsZ”kkad _.kkRed gksrk

gSA

5- nwjh lw=%

nks fcanqvksa 퐴(푥 ,푦 ) rFkk 퐵(푥 ,푦 ) ds chp dh nwjh

퐴퐵 = (푥 − 푥 ) + (푦 − 푦 ) bZdkbZ

6- fcanq A, B rFkk C lajs[k gS ;fn os ,d gh js[kk ij fLFkr gSaA

7- facanqvksa (푥 ,푦 ) vSsj (푥 ,푦 ) dks feykus okys js[kk[kaM ds e/; fcanq ds

funsZ”kkad gSa % ,


8- [kaM lw=

ml fcanqvksa (푥 ,푦 ) rFkk (푥 ,푦 ) dks feykus okys ja[kk[kaM dks l : m ds

vkarfjd vuqikr esa foHkkftr djrk gSa ds funZs”kkad gS %

,

9- f=Hkqt dk {ks=Qy

“kh’kksa (푥 ,푦 ), (푥 ,푦 ) rFkk (푥 ,푦 ) okys f=Hkqt dk {ks=Qy

= [푥 (푦 − 푦 ) + 푥 (푦 − 푦 ) + 푥 (푦 − 푦 )] oxZ bZdkbZ ;fn f=Hkqt dk

{ks=Qy “kwU; vkrk gS rks mijksDr fcanq lajs[k gksaxsA

blds dsUnzd ds funsZ”kkad ,

[kaM v ¼1 vad½

1- fcUnq A(5, &7) dh Y&v{k ls nwjh crkb;s A

2- ;fn fcUnqvksa (x, 2) rFkk (3, &6) ds chp dh nwjh 10 bdkbZ gks rks x dk

/kukRed eku crkb;sA

3- fcUnqvksa ¼4]7½ rFkk ¼2] &3½ dks feykus okys js[kk[kaM ds e/; fcUnq crkb;sA

4- ml fcUnq ds funsZ”kkad crkb;s tgkWa js[kk + = 5, y v{k dks izfrPNsn

djrh gSA

5- ;fn A rFkk B Øe”k% facUnq (&6, 7) rFkk (&1, &5) gks rks 2AB dk eku

crkb;sA

6- fcUnq P (5, 3) ls ,d js[kk y v{k ds lekUrj [khaph tkrh gS bl js[kk dh

Y- v{k ls nwjh crkb;sA

7- js[kkvksa 3푥 + 6 = 0 rFkk 푥 − 7 = 0 ds chp dh nwjh crkb;sA

8- js[kka[kM AB dk e/;fcUnq ¼4]0½ gSA ;fn A ds funsZ”kkad ¼3] &2½ gks rks B

ds funsZ”kkad crkb;sA


9- x v{k ij fdlh fcanq dh dksfV crkb;sA

10- y v{k ij fdlh fcanq dk Hkqt crkb;sA

11- fcanq ¼3]2½ dh x v{k ls nwjh Kkr dhft,A

12- fcanq ¼3]&4½ dh y v{k ls nwjh crkb,A

13- fcanq ¼3]4½ dh ewy fcanq ls nwjh Kkr dhft,A

14- y dk eku Kkr dhft, ;fn fcUnqvksa ¼2]&3½ rFkk ¼10]9½ ds chp dh nwjh

10 bZdkbZ gksA

15- x v{k ij ml fcUnq ds funsZ”kkd Kkr dhft, tks fcUnqvksa ¼&2]5½ rFkk

¼2]&3½ ls leku nwjh ij gksA

[k.M ^c^ ¼2 vad½

16- P ds fdl eku ds fy, fcanq ¼2]1½] ¼ P]&1]½ vkSj ¼&1]3½ lajs[k gS\

17- ∆푃푄푅 dk {ks=Qy Kkr dhft, ftlds “kh’kZ P(&5,7), Q(&4,&5) rFkk

R(4, 5) gSA

18- fcUnqvksa ¼1]&2½ vkSj ¼&3]4½ dks feykus okys js[kk[kaM dks lekf=Hkkftr

djus okys fcUnqvksa ds funsZ”kkad crkb;sA

20- ;fn fcUnq A(4,3) rFkk B(x,5) ,d or ftldk dsUnz O(2,3) gS] ij fLFkr

gksa] rks x dk eku Kkr dhft, A

21- fcUnqvksa ¼6] 4½ rFkk ¼&1]7½ dks feykus okys js[kk[kaM dks x v{k fdl

vuqikr esa foHkkftr djrk gS] Kkr dhft,A

22- n”kkZb, fd fcUnq ¼&2]3½¼8] 3½ vkSj ¼6]7½ ,d ledks.k f=Hkqt ds “kh’kZ gSA

23- y v{k ij og fcUnq Kkr dhft, tks fd fcUnqvksa A(5,&6) rFkk B(&1,-4)

dks feykus okys js[kk[k.M dks ckWaVrk gSA


24- og vuqikr Kkr dhft, ftlesa y v{k fcUnqvksa A(5, &6) rFkk B(&1, &4)

dks feykus okys js[kk[k.M dks ckWaVrk gSA

25- ml f=Hkqt ds dsUMd ds funsZ”kkad crkb;s ftlds “kh’kZ ¼3]&5½ ¼&7]4½

¼10]&2½ gSaA

[k.M ^l^ ¼3 vad½

26- n”kkZb;s fd fcUnq A(2,&2), B(14,10), C(11,13) rFkk D(&1,1) ,d vk;r

ds “kh’kZ gSA

27- n”kkZb;s fd fcUnq A(5, 6), B(1, 5), C(2, 1) rFkk D(6, 2) ,d vk;r ds

“kh’kZ gSA

28- fcanq R] js[kk[kaM AB] tcfd A(&4, 0) rFkk B(0, 6) gSa] dks bls izdkj

foHkkftr djrk gS fd 퐴푅 = 퐴퐵, rks R fcanq ds funsZ”kkad Kkr dhft,A

29- ,d lekUrj prqHkZqt ds rhu dzekxr “kk’kZ fcUnq (–2, –1) (1, 0) rFkk (4, 3)

gS A pkSFks “kh’kZ ds funsZ”kkd Kkr dhft,A

30- ;fn fcUnq P(x, y) dh fcUnqvksa A(3, 6) rFkk B(–3, 4) dh nwfj;kWa leku

gksa] rks fl+) dhft, 3푥 + 푦 = 5

31- ,d f=Hkqt ds nks “kh’kZ (1, 2) rFkk (3, 5) gSaA ;fn f=Hkqt dk dsUæd ewy

fcUnq ij gks] rks rhljs “kh’kZ ds funsZ”kkad Kkr dhft,A

32- ;fn P(x, y) fcanqvksa A(a, c), B(o, b) dks feykus okys js[kk[kaM fLFkr gks

rks fl) dhft, + = 1

33- fcanqvksa A(2, 1) rFkk B(5, –8) dks feykus okys js[kk[kaM dks P rFkk Q bl

izdkj foHkkftr djrs gSa fd fcUvq P fcanq ds vf/kd fudV gSA ;fn fcanq P

fcanq A ds vf/kd fudV gSA ;fn fcanq P,js[kk 2푥 − 푦 + 푘 = 0 ij Hkh fLFkr

gS rks K dk eku Kkr dhft,A


34- ;fn (3, 3) (6, y) (x, 7) vkSj (5, 6) dzekuqlkj ,d lekarj prqHkZqt ds “kh’kZ

gSaa rks x rFkk y ds eku Kkr dhft,A

35- ;fn ,d f=Hkqt ftlds “kh’kZ (1, –3) (4, P) vkSj (–9, 7) gS dk {ks=Qy

15 oxZ bZdkbZ gks rks P dk eku Kkr dhft,A

[k.M ^M^ ¼4 vad½

36- ;fn fcUnq fcUnq A(–2, 1), B(a, b), vkSj C(4, –1) lajs[kh gS RkFkk a–b=1 oks

a rFkk b ds eku Kkr dhft,A

37- ;fn fcanq A(0, 2) fcanqvksa B(3, P) rFkk C(P, 5) ls lenwjLFk gS rks P dk

eku vkSj AB dh yEckbZ Kkr dhft,A

38- ,d igsyh dks gy djus ds fy, ,d yM+dh dks rhu fcUnqvksa A(7, 5),

B(2, 3) rFkk C(6, –7) dks Øekuqlkj LdSp iSu }kjk feykuk gS rhuksa

fcUnqvksa dks feyus ij mls ,d f=Hkqtdkj vkd`fr izkIr gksrh gSA f=Hkqt

fdl izdkj dk gS\ bl iz”u esa fdl ewY; dks n”kkZ;k x;k gS\

39- eksuk vkSj fu”kk ds ?kjksa ds funsZ”kkad dze”k% (7, 3) vksj (4, –3) gS tcfd

muds fo|ky; ds funsZ”kkad (2, 2) gSA ;fn lqcg nksuksa ,d gh le;

fo|ky; ds fy, fudyrh gS vkSj ,d gh le; ij fo|+ky; ds fy,

fudyrh gS vkSj ,d gh le; ij fo|+ky; igqaprh gS iz”u ls fdu thou

ewY;ksa dk irk pyrk gS\

40- ,d v/;kfidk dk rhu fo|kFkhZvksa dks ,d f=Hkqt dh vkdfr esa [kMs+ gksus

ds fy, dgrh gS ftlds funsZ”kkad P(–1, 3), Q(1, –1) vkSj R(5, 1) gS rFkk

,d pksSFkh fo|kFkhZ bl fd;k dyki esa Hkkx ysuk pkgrk gSA og mls Q o

R ds e/; fcUnq 5 ij [kM+k gksus ds fy, dgrh gS mldh P ls nwjh Kkr

dhft,A iz”u ls fdu thou ewY;ksa dk irk pyrk gSA


mRrkekyk

1- 5

2- 9

3- ¼3]2½

4- 15

5- 26

6- 3

7- 9

8- ¼5] 2½

9- 0

10- 0

11- 2 units ¼bdkÃ)



14- 3 ;k &9

15- ¼&2]0½

16- 5

17- 53 oxZ bdkbZ

18- − , 0 − , 2

19- (1, 3)(5, 5)(3, -3)

20- X=2

21- 4 : 7

23- (0, -2)

24- 5:1

25- (2, -1)

28- −1,

29- (1, 2)

31- (-4, -7)

33- k= -8 34- x= 8, y= 4

35- p= -3

36- a= 1, b= 0

37- p = 1, AB= √10 bdkÃ 38- (i) ledks.k f=Hkqt (ii) [ksy]

fdz;k”khyrk] foospukred

lksp

39- (a) eksuk (b) le;c)rk]

;FkkZFkrk

40- 5 bdkbZ] xf.kr esa #fp] fe=rk ] lgHkkfxrk


vH;kl iz”u i=

le; &1 ?kaVk vad % 20

1- ml f=Hkqt dk {ks=Qy Kkr dhft, ftlds “kh’kZ (–2,3) (8, 3) vkSj (6, 7)

gSA ¼1½

2- m dk og eku Kkr dfj, ftlesa fcanq (3, 5) (m, 6) rFkk ,

lajs[k gSA ¼1½

3- fcUnqvksa A(c, 0) rFkk B(0, –c) ds chp dh nwjh D;k gS\ ¼1½

4- p ds fdl eku ds fy, fcanq (–3, 9),(2, p) rFkk (4, –5) lajs[k gS\ ¼2½

5- ;fn fcUnq (8, 6) rFkk B(x, 10) ,d or ftldk dsUnz (4, 6) gS ij fLFkr gksa

rks x dk eku Kkr dfj,A ¼2½

6- n”kkZ;s fd fcanq A(–3, 2), B(–5, –5), C(2, –3) rFkk D(4, 4) ,d leprqHkqZt

ds “kh’kZ gSA ¼3½

7- og vuqikr Kkr dhft, ftlesa fcanq (2, y) fcanqvksa A(–2, 2) rFkk B(3, 7)

dks feykus okys js[kk[kaM dks foHkkftr djrk gSA y dk eku Kkr dfj,A ¼3½

8- ;fn p] fcUnqvksa A(–2, –2) rFkk B(2, –4) dks feykus okys js[kk[kaM dks bl

izdkj foHkkftr djrk gS fd = rks p ds funsZ”kkad crkb;sA ¼3½

9- ;fn A(–5, 7), B(–4, –5), C(–1, –6) rFkk D(4, 5),d prqHkqZt ds dekuqlkj

“kh’kZ gS rks mlds {ks=Qy Kkr dfj,A ¼4½


v/;k;&4

f=dks.kfefr ds dqN vuqiz;ksx

mWapkbZ vkSj nwfj;ka

egRoiw.kZ fcanq:

1- nf’V js[kk % n`f’V js[kk] izR;sd dh vkWa[k ls izs{kd }kjk ns[kh xbZ oLrq ds fcUnq

dks feykus okyh js[kk gksrh gSA

2- mUu;u dks.k % mUu;u dks.k] n`f’V js[kk vkSj {kSfrt js[kk ls cuk dks.k gksrk

gS] tcfd {kSfrt Lrj ls mij gksrk gS vFkkZr og fLFkfr tcfd oLrq dks ns[kus

ds fy, gesa vius flj dks mij mBkuk gksrk gSA

3- mUu;u dks.k % mUu;u dks.k] nf’V ns[kk vkSj {kSfrt js[kk ls cuk 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


1- ,d ehukj 50 ehVj mWapk gSA tc lw;Z dk mUue;u dks.k 45 gS] rks ehukj dh

Nk;k D;k gksxh\

2- ,d 50 ehVj yacs ckWal dh Nk;k √

ehVj gSA lw;Z dk mUurka”k Kkr dhft,A

3- 10√3 eh0 mWapkbZ okyh ,d ehukj ds f”k[kj dk Hkwfe ij ml ehukj ds ikn ls

30 eh0 dh nwjh ij fLFkr fcUnq ls mUu;u dks.k Kkr dfj,A

4- ,d irax lery Hkwfe ls 50√3 eh0 mWapkbZ ij mM+ jgh gS ,d Mksj ls ca/kh gS]

tks {kSfrt ls 60° dks.k ij >qdh gSA Mksj dh yEckbZ Kkr dht,A


5- nh xbZ vkdfr esa vk;r ABCD dk ifjeki Kkr dfj,A

6- ,d LraHk dh Nk;k dh yEckbZ mldh mWapkbZ dk 3 xquk gS izdk”k ds lzksr

dk mUu;u dks.k Kkr dhft,A

7- vkdfr esa DC dk eku Kkr dfj,A

8- vkdfr esa BC dk eku Kkr dfj,A

9- vkdqfr esa nks O;fDr ,d ehukj ds foifjr fn”kk esa P rFkk Q ij [kM+s gSa ;fn

ehukj AB dh mWapkbZ 60eh0 gS rks nksuksa O;fDr;ksa ds chp dh nwjh Kkr

dhft,A

130°

10 मी.

D C

B A


10- vkdfr esa AB dk eku Kkr dfj,A

11- vkdfr esa CF dk eku Kkr dhft,A

12- ;fn uko dh iqy ls {kSfrt nwjh 25 eh0 gks vkSj iqy dh mWapkbZ 25eh0 gks rks

uko dk iqy ls voueu dks.k crkb,A

y?kq mRrjh; iz”u

13- ,d igkM+h ds f”k[kj ls iwoZ dh vksj nks dzekxr fd0eh0 ds iRFkjksa ds

voueu dks.k 30° vkSj 45 °ds gSaA igkM+h dh mpWakbZ Kkr dfj,A

14- ,d irax ds /kkxs dh yEckbZ 150 eh0 gS rFkk ;g Hkwfery ds lkFk 60° dk

dks.k cukrh gSaA irax dh Hkwfery ls mpWkbZ Kkr dfj, ¼eku yhft, /kkxs esa

dksbZ <hy ugha gSA

15- Lkw;Z dh mWapkbZ 45 ° ds LFkku ij 30° gksus ij ,d ehukj dh Nk;k 10 eh0

vf/kd gks tkrh gSA ehukj dh mWapkbZ Kkr dfj,A


16- ,d ok;qeku 200 eh dh mWapkbZ ij gSA blls ,d unh ds nks fdukjksa ds

voueu dks.k 45° vkSj 60° ds gSaA unh dh pkSM+kbZ Kkr dhft,A

17- ,d ehukj dh pksVh dk ,d fcUnq ij mUu;u dks.k 45° dk gSA ehukj dh vksj

40 eh0 pyus ij ;g dks.k 60° dk gks tkrk gSA ehukj dh mWapkbZ Kkr

dhft,A

18- ,d o{k dk mijh Hkkx VwVdj vius ikn ls 25 eh0 dh nwjh ij Hkwfe dks Li”kZ

djrk gS rFkk ewfe ds lkFk 30° dk dks.k cukrk gSA o`{k dh mWapkbZ D;k Fkh\

19- ,d v/okZ/kj /otnaM ,d lery esa yxk gSA blds f”k[kj dk 100 eh0 dh

nwjh ij ,d fcanq ls mUu;u dks.k 45° dk gSA /otanM dh mWapkbZ Kkr

dfj,A

20- ,d irax ds /kkxs dh yEckbZ 200 eh0 gS ;fn /kkxk Hkwfery ds lkFk 훼 dks.k

cukrk gS vkSj 3sin5

gks rks irax dh mWapkbZ Kkr dhft, tcfd /kkxs esa

dksbZ <hy ugha gSA

21- ,d ok;q;ku 3000 eh0 dh mWapkbZ ij gS rks og ,d nwljs ok;q;ku ds mij ls

xqtjrk gSA Bhd mlh le; nksuksa ok;q;kuksa ds Hkwfe ry ij ,d fcanq ij

mUu;u dks.k dze”k % 60° vkSj 45° ds gSaA nksuksa ok;q;kuksa ds chp dh nwjh

Kkr dfj,A

22- ,d >hy ls 60 eh0 mWapkbZ ij fLFkr ,d fcUnq ij ckny dk mUu;u dks.k

30° dk gS vkSj ckny ds >hy esa izfrcEc dk mlh fcanq ij voueu dks.k 60°

dk gSA ckny dh mWapkbZ Kkr dhft,A

23- ,d O;fDr ikuh ds tgkt ij ikuh ls 10 eh0 dh mWapkbZ ds ry ij [kM+k gSA

og ns[krk gS fd lkeus dh igkM+h ds f”k[kj dk mUu;u dks.k 60°dk gS rFkk


igkM+h ds vk/kkj ij voueu dks.k 30° dk gSA tgkt ls igkM+h dh nwjh vkSj

igkM+h dh mWapkbZ Kkr dhft,A

24- ,d ehukj ds f”k[kj ij 7 ehVj mWapk ,d /o/knaM yxk gSA Hkwfe ry ij

fLFkr ,d fcanq A ij naM ds f”k[kj mUu;u dks.k dze”k% 45° o 30° ds gSA

ehukj dh mWapkbZ o ikn Kkr dhft,A

25- Xkyh ds ,d edku dh f[kM+dh ftldh mWapkbZ Hkwfe ry ls 60 eh0 gS ls xyh

dh foifjr fn”kk esa lkeus cus ekdu ds “kh’kZ o ds mUu;u dks.k rFkk

voueu dks.k dze”k% 60° o 45° ds gSaA n”kkZb;s fd foifjr fn”kk esa cus

edku dh mWapkbZ 60 1 +√3 eh0 gSA

26- ,d ok;q;ku dk Hkwfe ds dsUnz A ls mUu;u dks.k 60° gSA 30 lsd.M dh

mM+ku ds i”pkr ;g mUu;u dks.k 30° gks tkrk gSA ;fn ok;q;ku 3600√3

eh0 dh vpj mWapkbZ ij mM+ jgk gks rks ok;q;ku dh xfr fdeh@?kaVk esa Kkr

dfj,A

27- 80 eh mWaps isM+ ds f”k[kj ij ,d i{kh cSBk gSA iFoh ds fdlh fcanq ls i{kh dk

mUu;u dks.k 45° gSA i{kh {kSfrt fn”kk esa izs{k.k fcanq ds foifjr bl izdkj

mM+rk gS fd og lnk leku mWapkbZ ij jgrk gSA 2 lsd.M ckn izs{k.k fcanq ls

i{kh dk mUu;u dks.k30° gks tkrk gSA i{h dh mM+us dh xfr Kkr dhft,A

28- ,d 7 eh0 mWaps Hkou ds f”k[kj ls ,d ehukj ds “kh’kZ dk mUu;u dks.k 60°

rFkk ehukj ds ikn dk voueu dks.k 30°gSA ehukj dh mWapkbZ Kkr dhft,A

29- fdlh ehukj ds vk/kkj ls 9 eh0 rFkk 4 eh0 dh nwfj;ksa Ikj ,d gh js[kk esa

fLFkr nks fcanqvksa ls ns[kus ij ehukj ds f”k[kj ds mUu;u dks.k iwjd dks.k

ik, tkrs gSaA ehukj dh mWapkbZ Kkr dhft,A

30- {kSfrt ry Ikj [kM+k ,d yM+dk 100 eh dh nwjh ij ,d i{kh dks 30° ds


mUu;u dks.k ij ns[krk gSA ,d yM+dh tks fd 20 eh0 mWaps Hkou ij [kM+h gS

mlh i{kh dks 45° ds dks.k ij ns[krh gSA ;fn yM+dk vkSj yM+dh i{kh dh

foifjr fn”kk esa gS rks i{kh dh yM+dh ls nwjh Kkr dhft,A

31- 100 ehVj mWaps izdk”k & LraHk dh pksVh ls ,d izs{kd leqnz esa ,d tgkt dks

Bhd viuh ehukj vkrs gq, ns[krk gSA ;fn tgkt dk voueu dks.k 30°ls

cnydj 60° gks tkrk gS rks izs{k.k dh vof/k esa tgkt }kjk r; dh xbZ nwjh

Kkr dhft,A

32- 60 eh mWaps ,d Hkou ds f”k[kj ls ,d izdk”k&LraHk ds f”k[kj rFkk ikn ds

mUu;u rFkk voueu dks.k dze”k% 30° o 60° ds gSA Kkr dhft,

i) izdk”k&LraHk rFkk Hkou dh mWapkbZ esa varj

ii) izdk”k&LraHk rFkk Hkou ds chp dh nwjhA

33- vkuUn ,d ldZl f[kykM+h dks jLlh in p<+rs ns[k jgk gS tksfd 20 eh yEch

gS vkSj ,d m?of/kj [kacs ls ca/kh gSA ;fn jLlh tehu ds lkFk 30° dk dks.k

cukrh gS rks [kaHks dh mWapkbZ Kkr dhft,A ;g iz”u vkUun ds fdu thou

ewY;ksa dks n”kkZrk gSA

34- ,d lh/kh lM+d ij nks nedy dsUnz P rFkk Q gSaA muds chp dh nwjh 20

fdeh gSA dsUnzksa dks Qksu }kjk lwpuk izkIr gksrh gS fd fcfYMax B esa vkx xy

xbZ gSA fcfYMax B dsUnz P ls 60° rFkk dsUnz Q ls 45° ij fLFkr gSA Kkr

dhft, % fdl dsUnz dks viuh cpko Vhe Hkstuh pkfg,\ ml Vhe dks fcfYMax

rd igWapus fdruh nwjh r; djuh iM+sxh\ ml Vhe dks fcfYMax fdu ewY;ksa dk

“kku gksrk gS\

35- X.kra= fnol ds miy{k esa 1-2 eh0 yEch yM+dh Hkwfe ls 8-2 eh0 dh mWapkbZ

ij ,d {kSfrt js[kk esa gok esa mM+ jgs xqCckjs dks ns[krh gSA fdlh {k.k yM+dh


dh vkWa[k ls xqCckjs dk mU;eu dks.k 60° gS dqN le; ckn mUu;u dks.k

?kVdj 30° gks tkrk gSA bl varjky ds nkSjku xqCckjs }kjk r; dh xbZ nwjh

Kkr dhft,A ;gkWa fdu ewY;ksa dks n”kk;k x;k gSA


mRrjekyk

1- 50 m

2- 60°

3- 30°

4- 100 m-

5- 20 √3 + 1 m

6- 30°

7- 60 m

8- 130 m

9- 60 √3 + 1 m

10- 1000 √3− 1 m

11- 25 m

12- 45°

13- 1.37 푘푚

14- 75 3 m

15- 13.65 m

16- 315.8 m

17- 94.8 m

18- 43.3 m 19- 100 m

20- 20 m

21- 1268 m

22- 120 m

23- 40 m, 17.32 m

24- 9.6 m

25- 864 km/h

26- &

27- 29.28 m

28- 28 m

29- 6 m 30- 30√2 푚

31- 115.5 m

32- 20 m, 34.64 m 33- 10] [kq”kh] fouksn Hkko

34- LVs”ku P, 14.64 km,

rkfdZdrk] lksp] lqj{kk

36- 58 3 m , fayx lekurk]

vkeksn izeksn


vH;kl iz”uekyk vf/kdre vad % 20 le; & 1 ?kaVk

[k.M & v 1- 6 eh0 mWaps tehu ij [kM+s ,d [kacs dh Nk;k dh yEckbZ 2√3 eh0 gS rks lw;Z

dk mUurk”ka Kkr dhft,A (1)

2- ,d ehukj dh mWapkbZ 100 ehVj gS] tc lw;Z dk mUu;u dks.k 30° gS rks

ehukj dh Nk;k dh yEckbZ Kkr dhft,A (1)

[k.M & c

3- Lery ij fLFkr ,d fcUnq dh ehukj ds ikn ls nwjh 20 eh0 gS rFkk mUu;u

dks.k 60° gks rks ehukj dh mWapkbZ Kkr dhft,A (2)

4- ,d ehukj dh mWapkbZ rFkk bldh Nk;k dk vuqikr 1:√

gSA ml {k.k lw;Z dk

mUu;u dks.k crkb;sA (2)

[k.M & l

5- lw;Z dh mWapkbZ 60° ds LFkku ij 45° gksus ij ,d ehukj dh Nk;k 10 eh0

vf/kd gks tkrh gSA ehukj dh mWapkbZ Kkr dhft,A (3)

6- ,d pV~Vkuds f”k[kj ds 100 eh0 mWaph ehukj ds f”k[kj o ikn ls mU;;u

dks.k dze”k% 30° rFkk 45° gSA pV~Vku dh mWapkbZ Kkr dfj,A (3)

[k.M & n

7- ,d O;fDr ikuh ds tgkt ij ikuh ls 10 ehVj mWapkbZ ds ry ij [kM+k gS og

ns[krk gS fd lkeus dh igkM+h ds f”k[kj dk mUu;u dks.k 60° dk gS rFkk

igkM+h ds vk/kkj dk voueu dks.k 30° dk gSA tgkt ls igkM+h dh nwjh vkSj

igkM+h dh mpkWabZ Kkr dfj,A (4) 8- ,d xyh ds edku dh f[kM+dh ls tks 15eh0 mWaph gS] mlh xyh ds nwljh vksj

cus ,d edku ds f”k[kj vkSj ikn ds mUu;u rFkk voueu dks.k dqe”k% 30°

vkSj 45° gS fl) dhft, fd nwljs edku dh mWapkbZ 23-66 eh0 gSA (4)


v/;k;&5

o`Ùk

egRoiw.kZ fcanq:

1- oÙk mu fcUnqvksa ds lewg ls curk gS tks ,d fuf”pr fcUnq ls vpj nwjh ij

gksrs gSaA fuf”pr fcUnq oÙk dk dsUnz dgykrk gS vkSj vpj nwjh oÙk dh

f=T;k dgykrh gSA

2- Nsnd js[kk & ;fn dksbZ js[kk fdlh or dks nks vfHkUu fcUnqvksa ij izfrPNsn

djrh gks rks og Nsnd js[kk dgykrh gSA

3- or dh Li”kZ js[kk & or dh Li”kZ js[kk og js[kk gksrh gS tks oÙk dks dsoy

,d fcUnq ij izfrPNsn djrh gSA ftl fcUnq ij Li”kZ js[kk o`Ùk dks Li”kZ

djrh gS mls Li”kZ fcUnq dgrs gSaA

4- Li”kZ js[kk dh la[;k, &fdlh o`Ùk ij vla[; Li”kZ js[kk,Wa cukbZ tk ldrh

gSaA

5- Nsnd js[kk dh la[;k,Wa &fdlh o`Ùk ij vla[; Nsnd js[kk,Wa cukbZ tk ldrh

gSaA


6- fuEu izes; fl) djus ds fy, iwNh tk ldrh gS %&

(i) fdlh o`Ùk dh Li”kZ js[kk] Li”kZ fcUnq ls gksdj tkus okyh f=T;k ij

yEc gksrh gSA

(ii) fdlh ckg; fcUnq ls oÙk ij cuh Li”kZ js[kkvksa dh yackbZ cjkcj gkssrh

gSA


1- fn, x, fp= esa BC dh yEckbZ Kkr djsaA

2- ;fn ckg; fcUnq P ls Li”kZ js[kk dh yEckbZ 24 cm gSA ;fn bl Li”kZ js[kk

dh dsUnz ls nwjh 25 cm gS rks oÙk dh f=T;k Kkr djsaA

3- fn, x, fp= esa ABCD ,d prqHkqZt gSA ;fn ∠퐵퐴퐶 = 50°,∠퐷퐵퐶 = 60°

rks ∠퐵퐶퐷 dk eku Kkr djsaA


4- fn, x, fp= esa O oÙk dk dsUnz gS] PQ ,d thok gS vkSj Li”kZ js[kk PR

fcUnq P ij 50° dk dks.k PQ ds lkFk cukrh gSA ∠푃푂푄 Kkr djsaA

5- ;fn nks Li”kZ js[kk,sa] 3 cm f=T;k okys o`Ùk ij bl izdkj cukbZ xbZ fd

muds chp dk dks.k 60° gks rks Li”kZ js[kkvksa dh yackbZ Kkr djssaA

6- nks ladsUnzh o`Ùkksa dh f=T;k,Wa 4 cm vkSj 5 cm gSA ,d or dh thok dh

yEckbZ Kkr djsa tks nwljs or ij Li”kZ js[kk gksA

7- fn, x, fp= esa PQ ckg; o`Ùk dk vkSj PR var% o`Ùk dh Li”kZ js[kk,Wa gSaA

;fn PQ= 4 cm, OQ= 3 cm vkSj, OR= 2cm gks rks PR dh yEckbZ Kkr djsa

8- fn, x, fp= esa ∠퐴푄퐵 Kkr djsaA


9- fn, x, fp= esa ∠퐴퐵푄 = 125° gS rks ∠퐶푂퐷 Kkr djsaA

10- ;fn T P vkSj TQ ckg; fcUnq T ls oÙk dh nks Li”kZ js[kk, gSa vkSj

∠푇푄푃 = 60° gS rks ∠푂푃푄 Kkr djsaA

y?kq mRrjh; (I) iz”u

11- ;fn nks ladsUnzh oÙkksa dk O;kl 푑 rFkk 푑 gksa (푑 > 푑 ) rFkk C or dh

thok dh yEckbZ gks tks nwljs oÙk ij Li”kZ js[kk gSA fl) djks fd

푑 = 푐 + 푑

12- 2-5 ls eh f=T;k okys or ij ckg; fcUnq P ls Li”kZ js[kk dh yEckbZ 6 lseh

gSA fcUnq P dh or ds fudVre fcUnq ls nwjh Kkr djksA

13- dsUnz O okys oÙk dh ckg; fcUnq T ls Li”kZ js[kk,a TP vkSj TQ gSaA ;fn

∠푂푃푄 = 30° gks rks ∠푇푄푃 dk eku Kkr djsaA


14- vkdfr esa AP = 4 cm BQ = 6 cm vkSj AC = 9 cm gSA ∆퐴퐵퐶 dk ifjeki

Kkr djsaA

15- ,d ledks.k f=Hkqt ftldh Hkqtk,Wa a, b rFkk c gSa tgkWa c d.kZ gS ds varxZr

,d oÙk cuk gS tks f=Hkqt dh lHkh Hkqtkvksa dks Li”kZ djrk gSA fl) djks

fd o`Ùk dh f=T;k r gksxhA

푟 =

16- fl) djks fd oÙk ij [khaph xbZ Li”kZ js[kk] Li”kZ fcUnq ls f=T;k ij yEc

gksrh gSA

17- fl) djks fd nks ladsUnzh; oÙkksa esa cM+s oÙk dh thok tks NksVs or ij Li”kZ

js[kk gksrh gS Li”kZ fcUnq ij lef)Hkkftr gksrh gSA

18- vkdfr esa AC dsUnz O okys oÙk dk O;kl gS vkSj A Li”kZ fcUnq gS rks X dk

eku Kkr dhft,A


19- vkdfr esa PA vkSj PB Li”kZ js[kk,Wa gSaA fl) dhft, KN = AK+BN

20- vkdfr esa thok PQ dh yEckbZ 6 lseh rFkk oÙk dh f=T;k 6 lseh gS TP

vkSj TQ oÙk dh Li”kZ js[kk,a gSA ∠푃푇푄 dk eku Kkr dhft,A

y?kq mRrjh; (II) iz”u

21- ,d f=Hkqt ABC ds vUrxZr cus oÙk dh Hkqtk,Wa AB=12 lseh] BC=8 lseh

vkSj AC=10 lseh gS rks AD, BE vkSj CF dk eku Kkr dhft,A


22- ledks.k ∆퐴퐵퐶 dh Hkqtk AB dks O;kl ekudj ,d oÙk [khapk tkrk gSA tks

d.kZ AC dks fcUnq P ij izfrPNsn djrk gSA fl) fdft, PB= PC.

23 ckg~; fcUnq P ls nks Li”kZ js[kk,Wa PA rFkk PB, O dsUnz okys o`Ùk ij [khaph

xbZA fl) dhft, ∠퐴푃퐵 = 2 ∠푂퐴퐵

24- 9 lseh f+=T;k okys oÙk ds vUnj ,d lef)ckgq f=Hkqt ABC ftldh Hkqtk

AB= AC=6 lseh] fLFkr gSA f=Hkqt dk {ks=Qy Kkr dhft,A

25- vkdfr esa AB= AC, ‘D’ AC dk e/; fcUnq gS rFkk BD o`Ùk dk O;kl gS rks

fl) dhft, fd 퐴퐸 = 퐴퐶.


26- vkdfr esa OP oÙk ds O;kl ds cjkcj gS] tgkWa O o`Ùk dk dsUnz gSA fl)

dhft, ∆퐴퐵푃 ,d leckgq f=Hkqt gSA

27- vkdfr esa AB= 13 lseh ,BC =15 lseh AD= 15 lsehA PC dh yEckbZ Kkr

dhft,A

28- vkdfr esa ckg; fcUnq P ls ,d oÙk ftldk dsUnz O gS ij Li”kZ js[kk PT rFkk


Nsnd js[kk PAB [khaps x, gSasA ON thok AB ij yEc gSaA fl) djks fd

(i) 푃퐴.푃퐵 = 푃푁 − 퐴푁

(ii) 푃푁 −퐴푁 = 푂푃 − 푃푇

(iii) 푃퐴.푃퐵 = 푃푇

29- dsUnz O okys oÙk dk O;kl AB rFkk thok AC gS rFkk ∠BAC=30° C ij

Li”kZ js[kk AB dks vkxs c<+kus ij D feyrh gSA fl) djks fd BC=BD

30- vkdfr esa PA vkSj PB dsUnz ‘O’ okys o`Ùk dh Li”kZ js[kk,Wa gSaA fl) dhft,

OP, AB dk yEc lef)Hkktd gSA


nh?kZ mRrjh; iz”u

31- vkdfr esa oÙk dh f=T;k Kkr dhft,A

32- vkdfr esa ;fn o`Ùk dh f=T;k 3 gks rks ∆퐴퐵퐶 dk ifjeki Kkr dhft,A

33- ,d oÙk f=Hkqt ABC dh Hkqtk BC dks P fcUnq ij Li”kZ djrh gS] Hkqt AB

vkSj AC dks dze”k% Q vkSj R ckg; fcUnq rd c<+k;k x;kA fl) dhft,

fd AQ] f=Hkqt ABC ds ifjeki dk vk/kk gSA


34- vkdfr esa XP vkSj XQ ckg; fcUnq X ls dsUnz ‘O’ okys o`Ùk dh Li”kZ js[kk,Wa

gSaA R oÙk ij ,d fcUnq gSA fl) dhft, XA + AR= XB + BR.

35- vkdfr esa PQ o`Ùk dh Li”kZ js[kk rFkk PB O;kl gSA x vkSj y ds eku Kkr

dhft,A

36. xkWao A vkSj B ds chp dh nwjh 7 fdeh B vkSj C ds chp dh nwjh 5 fdeh

vkSj C rFkk A ds chp dh nwjh 8 fdeh gSA xzke iz/kku rhuksa xkWaoksa A, B, C ds

fy, ,d dWaqvka [kqnokuk pkgrk gS tks rhuksa xkWaoksa ls leku nwjh ij fLFkr gksA

(i) dqWa, dh fLFkfr D;k gksuh pkfg, \

(ii) xzke iz/kku }kjk fdu thou ewY;ksa dks iznf”kZr fd;k x;k gS\

37- xkWao ds yksx o`Ùkkdkj xkWao ds utnhd ,d lM+d cukuk pkgrs gSaA lM+d

xkWao ds vanj ls ugha tk ldrh ysfdu yksx pkgrs gSa fd lM+d xkWao ds dsUnz

ls U;wure nwjh ij gksA

(i) dkSu lh lM+d xkWao ds dsUnz ls U;wure nwjh ij gksxh\

(ii) xkWao ds yksxksa ds dkSu ls thou ewY;ksa dk irk pyrk gS\


38- vkdfr esa n”kkZ, x, fp= vuqlkj pkj lM+dsa 1700 eh- f=T;k okys ,d

oÙkkdkj xkWao [kkuiqj dks Li”kZ djrh gSaA lfork dks AB vkSj CD lM+d

cukus dk rFkk fot; dks AD vkSj BC lM+d cukus dk Bsdk feyrk gSA

(i) fl) dhft, AB+CD = AD+AC

(ii) iz”u esa dkSu lk ewY; n”kkZ;k x;k gS\

39- nks lM+ds fcUnq P ls “kq: gksdj ,d oÙkkdkj jkLrs dks fcUnq A rFkk B ij

fp= vuqlkj Li”kZ djrh gSA lfjrk P ls A rd 10 fdeh] nkSM+rh gSA mlh

le; jes”k P ls B rd tkrk gSA

(i) ;fn lfjrk bl nkSM+ dks thr tkrh gS rks jes”k }kjk r; nwjh Kkr

djksA

(ii) ;gkWa fdl ewY; dks n”kkZ;k x;k gS\

40- ,d fnu jghe us ?kj vkrs le; jkLrs es lM+d ij ,d oÙkkdkj x<~Mk


ns[kkA mlus fLFkfr dk vuqeku yxk;k vkSj rqjUr uxj fuxe dks bl x<~Ms

ds ckjs esa lwpuk nhA uxj fuxe us x<~Ms dks pkjksa vksj vkdkfr vuqlkj

ckM+ yxk nhA

(i) ckM+ dh dqy yEckbZ Kkr djksA

(ii) mrj Kkr djus ds fy, fdl xf.krh; i)fr dk mi;ksx fd;k x;k

gSA

(iii) ;gkWa jghe ds fdu ewY;ksa dks n”kkZ;k x;k gS\


mRrjekyk

1- 10 lseh 2- 7 lseh

3- 70° 4- 100°

5- 3√3 lseh 6- 6 lseh

7- √21 lseh 8- 70°

9- 55° 10- 30°

12- 4 lseh 13- 60°

14- 15 lseh 18- 40°

20- 120° 21- AD=7 lseh, BE=5 lseh

CF=3 lseh

24- 8√2 oxZ lseh 27- 5 lsseh

31- 11 cm 32- 32 cm

35- x=35°, y=55°

36- (i) A, B, C dh ifjf/k ij fLFkr gksaxs ftlds dsUnz ij dqWavk fLFkr gSA (ii) Lekurk] ekurk ds izfr izseHkko] bZekunkjh

37- (i) or dh Li”kZ js[kk

(ii) vkfFkZd ewY;

38- (i) fyax lekurk

39- (i) 10 fdeh

(ii) fyax lekurk] LoLFk izfr;ksfxrk

40- (i) 36 QhV

(ii) ckg; fcUnq Li”kZ js[kkvksa dk leku gksuk

(iii) uSfrd o lkekftd nkf;Ro] rkfdZd lkspkA


vH;kl iz”u i=

le; % 50 feuV vad% 20 [k.M & v

1- fn, x, fp= esa x dk eku Kkr djsaA (1)

2- fn, x, fp= esa AC= 9 gSA BD Kkr djsaA (1)

[k.M & c

3- x dk eku Kkr djsaA (2)


4- nks ladsUnzh orksa dh f=R;k 6cm vkSj 3cm gSA ckg; fcUnq P ls nks Li”kZ

js[kk,a PA vkSj PB cukbZ xbZaA ;fn AP=10cm gS rks BP Kkr djsaA (2)

[k.M & l

5- fn, x, fp= esa fl) djsa ∠퐵푃푄 = ∠푃푅푄 tgkWa AB or dh Li”kZ js[kk gSA(3)

6- ,d f=Hkqt ABC ds vUrxZr cus or dh f=T;k 3 lseh gSA BD=6 lseh ,

DC= 8 lseh gSA ;fn ∆퐴퐵퐶 dk {ks=Qy 63 oxZ lseh gks rks Hkwtk AB Kkr

djsaA (3)


7. AB or dk O;kl gSA AT mldh Li”kZ js[kk gSA ;fn ∠퐴푂푄 = 58 ° gS

rks∠퐴푇푄 Kkr djsaA (4)

8. PQ vkSj PR ckg; fcUnq P ls [khaph xbZ nks Li”kZ js[kk,Wa gSaA ∠푅푃푄 = 30°

gSaA thoh RS LIk”kZ js[kk PQ ds lekarj gSA ∠푅푄푆 Kkr djsaA (4)


v/;k; & 6

jpuk,a

egRoiw.kZ fcUnq:

1- jpuk lkQ vkSj LoPN cuk,aA

2- le:Ik f=Hkqt cukrs le; gesa ekiu Ldsy dk /;ku j[kuk pkfg,A

3- jpuk ds in rHkh fy[ksa tc vki ls dgk tk,A

4- jpuk cukrs le; ijdkj vkSj ekid dk iz;ksx gh djsa O;kid dks.k cukrs

le; izksVªSDVj dk iz;ksx fd;k tk ldrk gSA


1- f=Hkqt ABC ds le:i f=Hkqt cukus ds fy, ftldh Hkqtk,a f=Hkqt ABC

dh laxr Hkqtkvksa dk gSA ,d fdj.k BX bl izdkj [khaprs gSa fd ∠퐶퐵푋

U;wu dks.k gks vkSj X, A ds foifjr fn”kk esa BC ds lkis{k gksA BX ij

fdrus fcUnq cjkcj cjkcj nwjh ij yxk;saxs\

2- or ij Li”kZ js[kkvksa dk ;qXe bl izdkj [khapk tkrk gS fd nksauksa js[kkvksa ds

chp dk dks.k 30° gks rks nksuksa f=T;kvksa ds chp dk dks.k crkb;sA

3- f=Hkqt ABC ds le:i f=Hkqt cukus ds fy, ftl dh Hkqtk,a f=Hkqt ABC

dh laxr Hkqtkvksa dk 2@5 gSaA igys ,d fdj.k BX bl izdkj [khaph tkrh

gS ∠퐶퐵푋 U;wu dks.k gks vkSj X, A ds foifjr fn”kk esa BC ds lkis{k gks rc

fcUnq 퐵 ,퐵 ,퐵 ……퐵 ij in cjkcj cjkcj n”kkZ;s tkrs gSa rks dkSu ls nks

fcUnq vxys pj.k esa feyk;s tk,axs\


4- ,d js[kk[kaM AB dks 3%7 ds vuqikr esa foHkkftr djus ds fy, AX cjkcj

cjkcj nwjh ij fdrus fcUnq fpfUgr djus iM+saxs\

5- or ds vUnj fLFkr fcUnq ls fdruh Li”kZ js[kk,a [khaph tk ldrh gSa\

6- ,d ja[kk[kaM AB dks 4%5 ds vuqikr esa foHkkftr djus ds fy, AX fdju

bl izdkj [khaph tkrh gS fd ∠퐵퐴푋 U;wudh 01 gks vkSj rc 퐴 ,퐴 ,퐴 ……

fcUnq AX cjkcj cjkcj nwjh ij n”kkZ;s tkrs gSAA fdj.k AX fdl fcUnq dks

B ls feyk;k tk,xkA

7- ,d js[kk[kaM AB dks 4%5 ds vuqikr esa foHkkftr djus ds fy, fcUnq

퐴 ,퐴 ,퐴 …… vkSj 퐵 ,퐵 ,퐵 …… fdj.k AX rFkk BX ij cjkcj&cjkcj nwjh

ij fpfUgr gSa rks dkSu ls nks fcUnqvksa dks js[kk[kaM dks foHkkftr djus ds

fy, feykuk pkfg,\

nh?kZ mRrjh; iz”u

8- js[kk[kaM AB=8 lseh [khafp,A AB ij ,d fcUnq C bl izdkj yhft, fd

AC= CB.

9- ,d ∆퐴퐵퐶 dh jpuk dhft, ftlesa AB=6.5 lseh, ∠B=60° rFkk 퐵퐶 = 5.5

lseh ,d vU; f=Hkqt 퐴퐵′퐶′ dh jpuk dhft, tks ABC ds le:i gks rFkk

ftldh izR;sd Hkqtk ∆ABC dh laxr Hkqtk dk 3@2 gksA

10- ,d ∆ABC dh jpuk dhft, ftlesa BC=5 lseh, CA=6 lseh vkSj AB=7

lsehA ,d vU; ∆퐴′퐵퐶′ dh jpuk dhft, tks ∆ABC ds le:i gks srFkk

ftldh izR;sd Hkqtk ∆ABC dh laxr Hkqtk dk 7@5 gksA

11- ,d f=Hkqt dh jpuk dhft, ftldh Hkqtk,a 4 lseh] 5 lseh rFkk 7 lseh

dh gksA blds le:I ,d vU; f=Hkqt dh jpuk dfj, ftldh izR;sd Hkqtk

fn;s x;s f=Hkqt dh laxr Hkqtk dk 2@3 xqus ds cjkcj gksA


12- ,d ledks.k f=Hkqt dh jpuk dhft, ftldh Hkqtk,a ¼d.kZ dks NksM+dj½ 8

lseh rFkk 6 lseh yEckbZ gksA blds le:i ,d vU; f=Hkqt dh jpuk

dfj, ftldh izR;sd Hkqtk fn;s x;s f=Hkqt dh laxr Hkqtk ds 3@4 xqus ds

cjkcj gksaA

13- ∆ABC dh jpuk dfj,s ftlesa BC=8 lseh, ∠B=45° blds le:i ,d vU;

f=Hkqt dh jpuk dfj, ftldh Hkqtk,a ∆ABC dh laxr Hkqtkvksa ds 3@4

xqus ds cjkcj gksA

14- ∆ABC Dh jpuk dhft, AB=15 lseh, BC=27 lseh vkSj ∠BAC=50° ,d

vU; ∆퐴′퐵퐶′ ,∆ABC ds le:Ik cukb;s ftlesa BA’=25 lseh vkSj BC’=45

lseh ekiu Ldsy Hkh crkb,A

15- ∆ABC Dh jpuk dfj, ftlesa AB=5 lseh ∠B=60°vkSj “kh’kZ CD=3 lseh,

∆AQR~ ∆ABC dh jpuk dfj, rkfd ∆AQR ds izR;sd ∆ABC dh laxr

Hkqtkvksa ds 1-5 xqus ds cjkcj gksA

16- 6 lseh f=T;k dkk ,d o`Ùk [khafp, 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

17- ,d lef}ckgq ∆ABC dh jpuk fdft, ftlesa AB=AC vkSj vk/kkj BC=7

lseh] m/okZ/kj dks.k =120 ° ,∆A’B’C’ ~ ∆ABC dh jpuk dhft, ftldh

izR;sd Hkqtk] ∆ABC dh laxr Hkqtkvksas ds 1 xqus ds cjkcj gksA

18- 3 lseh f=T;k dk ,d oÙk [khafp, dsUnz ls 5 lseh dh nwjh ij ckg; fcUnq

ls oÙk ij Li”kZ js[kk,a [khafp, rFkk mudh yEckbZ Hkh eki dj fyf[k,A

19- dsUnz 0 rFkk 4 lseh f=T;k dk o`Ùk [khfp, mldk O;kl POQ [khfp,A

P ;k Q ls oÙk dh Li”kZ js[kk [khafp,A


20- 5 lseh o 3 lseh f=T;k okys nks oÙk [khafp, ftuds dsUnz ,d nwljs ls 9

lseh nwj gSAizR;sd oÙk ds dsUnz ls nwljs o`Ùk ij Li”kZ js[kkvksa dh jpuk

dhft,A

21- 6 lsseh rFkk 4 lseh f=T;k ds nks ladsUnzh; oÙk [khafp,A ckg; o`Ùk ds fdlh

fcUnq ls var% o`Ùk ij Li”kZ js[kk dh jpuk dfj, vkSj mldh yEckbZ eki dj

fyf[k,A

22- 3 lseh f=T;k dk o`Ùk [khafp,A blds c<+s gq, O;kl ij nks fcUnq P rFkk Q

tks dsUnz ls izR;sd 7 lseh dh nwjh ij gS bl fcUnqvksa ls o`Ùk ij Li”kZ

js[kkvksa dh jpuk dfj,A

23- PQ=10 lseh dk ,d js[kk[kaM [khafp, A PA ij ,d fcUnq A bl izdkj

yhft, fd = PA rFkk PQ dh yEckbZ eki dj fyf[k,A

24- ,d leckgqw ∆PQR [khafp, ftldh izR;sd Hkqtk 5 lseh gS ∆PQR cukb;s

rkfd =

25- 8 lseh yEckbZ dh ,d js[kk[kaM [khafp, vkSj bls 5%8 esas foHkkftr dhft,

rFkk nksuksa fgLlksa dh eki Hkh crkb,A

26- lQkbZ vfHk;ku gsrw Ldwy ds Nk= ,d jSyh fudkyrs gSa os rhu jkLrksa AB,

BC rFkk CA ij pyrs gSs vkSj ,d f=Hkqt cukrs gSaA ∆ ABC dh jpuk

dhft, AB += 7 lseh] BC=7-5 lseh c 6.5 lehA ∆ABC ds le:Ik ∆ dh

jpuk dhft, ftldh izR;sd Hkqtk ∆ABC dh laxr Hkqwtkvksa dk 3@2 xquk

gks bl iz”u esa fdl ewY; dks n”kkZ;k x;k gS\

27- vfer ds ikl tehu dk ,d f=Hkqtkdkj VqdM+k gS ftlesa vk/kkj BC=4.2 cm

45A vkSj A ls “kh’kZ yac =2-5 gSA og blds le#i nwljk f=Hkqwtkdkj


VqdM+k [kjhnuk pkgrk gS ftldh laxr Hkqtk,a igys okys f=Hkqt dh laxr

Hkqtkvksa dk 1@2 xquk gks bls og o`)kJe dks nku nsrk gSA bl ∆ dh jpuk

dfj, A thou ds dkSu ls ewY; bl iz”u esas iz;ksx fd, x;s gSaA

28- 8 lseh yEckbZ ds js[kk[kaM dks 3%4 esas foHkkftr dfj, la;qDr ifjokj dks

lQy ifjokj esa foHkkftr gksuk vPNk gS ;k cqjkA vius mrj ds liksVZ dk

dkj.k Hkh crkb;sA

29- 5 lseh f=T;k dk o`Ùk [khafp, O;kl ds fljksa ls Li”kZ js[kk,a [khafp,A rqe

D;k voyksdu djrs gks ;fn izR;sd Li”kZ js[kk ekuo ds xq.kksa dks n”kkZrh gS

,d vPNs ekuo dks fdu xq.kksa dks viukuk pkfg,A


mRrj ekyk

1- 5

2- 150

3- B 5 ls C

4- 10

5- 0

6- A 9

7- A 4 rFkk B 5


vH;kl iz”u i=

le; %&1 ?kaVk vf/kdre vad & 20

[kaM&v

1- js[kk[kaM AB +=8 lseh dk yac lef)Hkkftd [khafp,A

2- ,d nh gqbZ js[kk ds lekUrj js[kk dh jpuk dfj,A

[k.M & c

3- 75° dks dk.k cukb, rFkk mldk lef)Hkktd [khafp,A

4- 5-6 lseh yEckbZ dk js[kkk[kaM [khafp,A mls 2%3 ds vuqikr esa foHkkftr

dfj,A

[k.M & l

5- 3-5 lseh f=T;k dk or [khafp,A blds dsUnz ls 5-5 lseh dh nwjh ij fLFkr

ckg; fcUnq ls or ij Li”kZ js[kk,a [khafp,A

6- 3-5 lseh f=T;k ds or dh jpuk dfj, rFkk bl ij nks Li”kZ js[kk,a bl

izdkj [khafp, tks ijLij 120° ds dks.k ij varfjr gksA

[k.M & n

7- f=Hkqt dh jpuk dfj, ftlesa AB +=4 lseh BC += 5 lseh vkSj AC=7

lsehA ∆ABC ds le:Ik ,d nwljk f=Hkqt cukb;s ftldh laxr Hkqtk,a fn,

gq, f=Hkqt dh laxr Hkqtkvksa dk 5@7 xquk gksA

8- ,d ledks.k ∆ABC cukb;s ftlesa AB+=6 lseh BC+=8 lseh ∠B= 90° - AC ij B ls BD yac [khafp,A B, C rFkk D ls gksrk gqvk or cukb;s rFkk

A ls or ij Li”kZ js[kkvksas dh jpuk dhft,A


v/;k; & 7

o`Ùkksa ls lEcaf/kr {ks=Qy

egRoiw.kZ fcUnq:

1- ;fn ,d o`Ùk dh f=T;k ‘r’ gks rks %

(i) ifjf/k = 2휋푟 ;k 휋푑 gks 푑 = 2푟 gSA (ii) {ks=Qy = 휋푟

(iii) v/kZo`Ùk dk {ks=Qy =

(iv) prZqFkka”k dk {ks=Qy=

2- nks ladsUnzh; oÙkksa }kjk vkUrfjr {ks=Qy

;fn nks ladsUnzh; o`Ùkksa dh f=T;k esa R rFkk r gS rks nksuksa o`Ùkksa }kjk vkUrfjr {ks=Qy = 휋푅 − 휋푟 = 휋(푅 − 푟 ) = 휋(푅 + 푟)(푅 − 푟)

3- f=T;k[k.M vkSj mldk {ks=Qy

fdlh orh; {ks= ds ml Hkkx dks tks or dh nks f=T;kvksa vkSj muds laxr pki }kjk f?kjk gks] ml oÙk dk ,d f=T;[k.M dgrs gSaA fn, x, fp= esa APB y?kq f=T;[k.M rFkk AQB nh?kZ f=T;[k.M gSA


f=T;[k.M dk {ks=Qy ftldk f=T;[k.M dks.k 휃 gS=°

× 휋푟

= 푙푟

dks.k 휃okys f=T;[k.M ds laxr pki dh yEckbZ =°

× 2휋푥

=°

× 휋푑

4- oÙk[k.M vkSj mldk {ks=Qy %

o`Ùkh; {ks= dk og Hkkx tks ,d thok vkSj laxr pki ds chp esas ifjc) gks]

,d oÙk[k.M dgykrk gSA fn, x, fp= esa APB y?kq oÙk[k.M rFkk AQB

nh?kZ oÙk[k.M gSA

oÙk[k.M APB dk {ks=Qy = f=T;k[k.M OAPB dk {ks=Qy&∆OAB dk

{ks=Qy

=°

× 휋푟 − 푟 sin휃

5- dqN egRoiw.kZ ifj.kke %

(i) ;fn nks oÙk vUr% Li”kZ djrs gSa] rks muds dsUnzksa ds chp dh nwjh

mudh f=T;kvksa ds vUrj ds cjkcj gksrh gSA

(ii) ;fn nks oÙk ckg;r% Li”kZ djrs gSa] rks muds dsUnzksa ds chp dh nwjh

mudh f=T;kvksa ds ;ksxQy ds cjkcj gksrh gSA


(iii) fdlh ?kwers gq,s ifg;s }kjk ,d pDdj esa r; dh xbZ nwjh ml ifg;s

dh ifjf/k ds cjkcj gksrh gSA

(iv) fdlh ?kwers gq, ifg;s }kjk ,d feuV esa yxk;s x;s pDdjksa dh

la[;k = ,d feuV esa pyh x;h nwjh

ifg;s dh ifjf/k

(v) y?kq f=T;[k.M oÙk ds dsUnz ij y?kq dks.k ¼eku yhft, 휃 ½

vkUrfjr djrk gS tcfd nh?kZ f=T;k[k.M dsUnz ij vf/kddks.k

¼360° − 휃½ vkUrfjr djrk gSA

(vi) o`Ùk ds y?kq rFkk nh?kZ f=T;k[k.Mksa ds {ks=Qyksa dk ;ksxQy oÙk ds

{ks=Qy ds cjkcj gksrk gSA

(vii) fdlh f=T;k[k.M dk ifjeki mldh laxr pki rFkk laxr

f=T;kvksa ds ;ksxQy ds cjkcj gksrk gSA

(ix) feuV dh lqbZ }kjk 60 feuVksa esa cuk;k x;k dks.k = 360°

(x) feuV dh lqbZ }kjk 1 feuVksa esa cuk;k x;k dks.k = 6°


1- ;fn ,d v/kZoÙkkdkj pkWans dk O;kl 14 lseh gS] rks bldh ifjf/k Kkr

dhft,A

2- ,d oÙk dh ifjf/k rFkk {ks=Qy la[;kRed :i ls leku gks rks oÙk dk

O;kl Kkr dhft,A

3- ‘a’ Lkseh Hkqtk okys oxZ ds vUrfugZr ,d o`Ùk dk {ks=Qy Kkr dhft,A

4- ,d oÙk ds f=T;k[k.M dk {ks=Qu Kkr dhft, ftldh f=T;k r rFkk laxr

pki dh yEckbZ l gSA

5- ,d ifg, dh f=T;k 0-25 eh0 gSA ifg, }kjk 11 fdeh nwjh r; djus esa

yxk, x, pDdjksa dh la[;k Kkr dhft,A


6- ;fn ,d o`Ùk dk {ks=Qy 616 oxZ lseh gks rks bldh ifjf/k Kkr dhft,A

7- ,d 6 lseh okys oxZ ds vUrfufgrZ o`Ùk dk {ks=Qy Kkr dhft,A

8- ,d oÙk dk {ks=Qy nks o`Ùkksa ds {ks=Qyksa ds ;ksx ds cjkcj gSA nksuksa oÙkksa s

dh f=T;k,a 24 lseh rFkk 7 lseh gS rks cM+s or dk O;kl Kkr dhft,A

9- ,d rkj dks eksM+dj 35 lseh f=T;k dk oÙk cuk;k tk ldrk gSA ;fn blh

rkj dks ,d oxZ ds vkdkj esa eksM+k tk, rks oxZ dk {ks=Qy Kkr dhft,A

10- ,d oÙk dh f=T;k 6 lseh0 gS rFkk ,d pki dh yEckbZ 3 lseh0 gSA bl

pki }kjk oÙk ds dsUnz ij vkUrfjd dk.k dk eku Kkr dhft,A

11- ,d oÙk ds f=T;k[k.M dk {ks=Qy Kkr djus ds lw= fyf[k, ftldk dsUnz

ij 휃 ¼va”k esa½ dk dks.k rFkk f=T;k r gSA

12- ;fn nks oÙkksa dh ifjf/k;kWa 2%3 ds vuqikr esa gks] rks buds {ks=Qy dk vuqikr

Kkr dhft,A

13- ,d oÙk dh ifjf/k rFkk f=T;k dk vUrj 37 lseh0 gks rks or dk {ks=Qy

Kkr dhft,A 휋 = ysa

14- ;fn ,d oÙk dk O;kl 40% c<+k fn;k tk, rks fdrus izfr”kr ls bldk

{ks=Qy c<+sxkA

15- ,d ?kM+h dh ?k.Vs dh lqbZ dh yEckbZ 6 lseh gSA izkr% 11%20 cts ls 11%55

cts rd ?k.Vs dh lwbZ }kjk vafdr {ks=Qy Kkr dhft,A

y?kqRrjkRed iz”u (I)

16- ,d oÙk ds prqFkkZa”k dk {ks=Qy Kkr dhft, ftldh ifjf/k 22 lseh0 gSA

17- 10 lseh0 f=T;k okys oÙk dh fdlh pki dh yEckbZ 5휋 lseh- gks rks ml

pki }kjk oÙk ds dsUnz ij vkUrfjd dks.k dk eku Kkr dhft,A


18- ;fn ,d oxZ ,d o`Ùk ds vUrfuZfgr gks] rks oÙk rFkk oxZ ds {ks=Qyksa dk

vuqikr Kkr dhft,A

19- ;fn fdlh v/kZoÙk dh ifjf/k 18 lseh- gks rks mldh f=T;k Kkr dhft,A

20- ;fn ,d o`Ùk dh ifjf/k ,d oxZ ds ifjeki ds cjkcj gks rks muds {ks=Qyksa

dk vuqikr D;k gksxk\

21- ,d oÙk dk O;kl rFkk ,d leckgq f=Hkqt dh Hkqtk dh yEckbZ leku gks rks

buds {ks=Qyksa dk vuqikr D;k gksxk\

22- layXu fp= esa] O ,d o`Ùk dk dsUnz gSA ;fn f=T;k[k.M OABP dk

{ks=Qy] oÙk ds {ks=Qy dk gks rks x Kkr dhft,A

23- fn, x, fp= esa] tgkWa AED ,d v/kZoÙk rFkk ABCD ,d vk;r gS rks fp=

dk ifjeki Kkr dhft,A

24- fn;k x;k fp= ,d oÙk ftldh f=T;k 10-5 lseh- gS] dk f=T;k[k.M gSA


bl f=T;k[k.M dk ifjeki Kkr dhft,A

25- Nk;kafdr Hkkx dk ifjeki Kkr dhft,A 휋 =

y?kq mRrjh; iz”u (II)

26- 36 lseh- f=T;k okys ,d o`Ùk ds f=T;k[k.M dk {ks=Qy 54 oxZ lseh- gks

rks laxr pki dh yEckbZ Kkr dhft,A

27- ,d ?kM+h dh feuV dh lqbZ 5 lseh- yEch gSA feuV dh lqbZ }kjk izkr % 6%05

cts ls 6%40 cts rd cqgkj fd;k x;k {ks=Qy Kkr dhft,A

28- fn, x, fp= esa ABC ,d ledks.k f=Hkqt gS ftlesa dks.k A=90° , AB=6

lseh- rFkk AC= 8 lseh- gSA AB, AC rFkk BC dks O;kl ysdj v/kZo`Ùk [khaps


x, gSa] Nk;kafdr Hkkx dk {ks=Qy Kkr dhft,A

29- fn, x, fp= esa OAPB, ,d oÙk ftldh f=T;k 3-5 lseh rFkk dks.k

AOB = 120°] dk f=T;k[k.M gSA OAPBO dk ifjeki Kkr dhft,A

30- ,d oÙkkdkj iniFk ¼QqVikFk½ ftldh pkSM+kbZ 2 eh0 dks :20 izfroxZ eh0

dh nj ls ,d o`Ùkkdkj] ikdZ ftldh f=T;k 1500 eh0] ds pkjksa vksj cuk;k

x;k gSA iniFk ¼QqVikFk½ dks cukus esa dqy fdruh [kpZ vk,xkA¼ ¾ 3-14

yhft,½

31- ,d yM+dk bl izdkj lkbZfdy pyk jgk gS fd lkbZfdy ds ifg, izfr

feuV 140 pDdj yxkrs gSa ;fn ifg, dk O;kl 60 lseh0 gkss rks lkbZfdy

dh pky Kkr dhft,A

32- 5 lseh0 f=T;k okys oÙk dh thok AB dh yEckbZ 5 √3lseh0 gSA y?kq

f=T;k[k.M AOB dk {ks=Qy Kkr dhft,A


33- ,d leckgq f=Hkqt dk {ks=Qy 49√3 oxZ lseh0 gSA izR;sd “kh’kZ dks dsUnz

ekudj] f=Hkqt dh Hkqtk dh yEckbZ dh vk/kh f=T;k ysdj oÙk [khaps x, gSaA

f=Hkqt ds ml Hkkx dk {ks=Qy Kkr dhft, tks o`Ùkksa esa lfEefyr ugha gSA

34- ABCD ,d leyEc ftlesa 퐴퐵‖퐷퐶,퐴퐵 = 18 lseh0] DC=32 lseh0 rFkk

퐴퐵 vkSj 퐷퐶 ds chp dh nwjh 14 lseh0 gSA “kh’kksZa A,B,C vkSj D dks dsUnz

ysdj pkj cjkcj f=T;k 7 lseh0 okys oÙk cuk, x, gSas rks Nk;kafdr Hkkx

dk {ks=Qy Kkr dhft,A

35- ,d 8 lseh0 Hkqtk okys oxZ ds nks lEeq[k dks.kksa ls 1-4 lseh0 f=T;k ds nks

prqFkkZ”k dkVs x, gSaA oxZ ds chp esa ls 4-2 lseh0 O;kl dk ,d vU; oÙk Hkh

dkVk x;k gS tSlk fd fp= esas n”kkZ;k x;k gSA Nk;kafdr Hkkx dk {ks=Qy

Kkr dhft,A


36- ,d f=T;k[k.M 100° dk ,d oÙk ls dkVk x;k gS ftldk {ks=Qy 70-65

lseh gSA oÙk dh f=T;k Kkr dhft,A (휋 = 3.14).

37- fn, x, fp= esa] ABCD ,d vk;r gS ftlesa AB=14 lseh vkSj BC=7 lseh

gSA DC, BC rFkk AD dks O;kl ekudj] rhu v/kZo`Ùk [khaps x, gSaA Nk;kafdr

Hkkx dk {ks=Qy Kkr dhft,A

38- ,d oxkZdkj ikuh ds VSad ds vk/kkj dh izR;sd Hkqtk 40 eh0 gSA blds pkjksa

vksj pkj v/kZo`Ùkkdkj ?kkl ds eSnku gSaA : 1-25 izfr oxZ eh0 dh nj ls

eSnku dks lery djkus dk O;; Kkr dhft,A¼ ¾ 3-14 ysa½

39- Nk;kafdr Hkkx dk {ks=Qy Kkr dhft,A

40- 28 lseh0] f=T;k okys oÙk dh dksbZ thok or ds dsUnz ij 45 ° dk dks.k

cukrh gSA thok }kjk dkVs x;s y?kq o`Ùk[k.M dk {ks=Qy Kkr dhft;sA

41- ,d rkj dks eksM+dj] oÙk ds dsUnz ij 45° dk dks.k vkUrfjr djus okyh

pki ds :Ik esa cuk;k tk ldrk gSA ;fn rkj dh yEckbZ 11 lseh0 gks rks

oÙk dh f=T;k Kkr dhft,A


42- fn, x, fp= esa Nk;kafdr Hkkx dk {ks=Qy Kkr dhft,A

43- fn, x, fp= dk {ks=Qy oxZ lseh0 esa Kkr dhft,A

44- ;fn ,d oÙk dh ifjf/k blds O;kl ls 16-8 lseh0 vf/kd gks rks o`Ùk dh

f=T;k Kkr dhft,A

45- Nk;kafdr Hkkx dk {ks=Qy Kkr dhft,A

nh?kZ mRrjh; iz”u

46- nks o`Ùk ckg;r% Li”kZ djrs gSaA ;fn buds {ks=Qyksa dk ;ksx 130 oxZ lseh0 gS rFkk buds dsUnzksa ds chp dh nwjh 14 lseh0 gS] rks bu orksa dh

f=T;k;sa Kkr dhft,A


47- Rkhu oÙk ftudh f=T;k,Wa 7 lseh0 gSas] bl izdkj [khaps x, gSa fd gj or

ckdh nks oÙkksa dks Li”kZ djrk gSA rhuksa oÙkksa ds chp okys Hkkx dk {ks=Qy

Kkr fdft,A

48- ,d oÙkh; ifg, dk {ks=Qy 6-16 oxZ eh0 gSA ifg, dks 572 eh0 dh nwjh

r; djus ds fy, dqy fdrus pDdj yxkus iM+saxsA

49- ,d leprqHkZqt ds lHkh “kh’kZ ,d oÙk ds vUnj gS] ;fn or dk {ks=Qy

2464 oxZ lseh0 gks rks leprqHkZqt dk {ks=Qy Kkr dhft,A

50- f=Hkqt ABC ds “kh’ksaZ A, B rFkk C dks dsUnz ysdj rhu f=T;k[k.M cuk,

x, gSa ftudh f=T;k 6 lseh0 gSA ;fn AB=20 lseh0 BC=48 lseh0 vkSj

CA=52 lseh0 gks] rks Nk;kafdr Hkkx dk {ks=Qy Kkr dhft,A ¼휋 =3-14

dk iz;ksx dhft,½

51- nh xbZ vkdfr esa] ABCDEF ,d le’knHkqt gS ftlds “kh’kksZa dks dsUnz

ekudj leku f=T;k r ds o`Ùk [khaps x, gSaA Nk;kafdr Hkkx dk {ks=Qy

Kkr dhft,A


52- ,d 6 lseh0 f=T;k ds oÙk dk O;kl ABCD bl izdkj gS fd AB, BC

rFkk CD cjkcj gSA fp= ds vuqlkj AB vkSj BD dks O;kl ekudj v/kZo`Ùk

[khaps x, gSaA Nk;kafdr Hkkx dk ifjeki rFkk {ks=Qy Kkr dhft,A

53- lM+d ij ,d xjhc dykdkj cPpksa ds fy, etkfd;k dkVwZu cukrk gS rFkk

viuh thfodk vftZr djrk gSA ,d ckj mlus vkdfr esa n”kkZ, vuqlkj

,d gkL;dj eq[k cuk;k] ftlds fy, mlus ,d oÙk ds vUnj or [khapk]

tgkWa cM+s o`Ùk dh f=T;k 30 lseh0 vkSj NksVs or dh f=T;k 20 lseh0 gSA

bl vkd`fr esa Vksih ds fy, fdruk {ks=Qy fn;k x;k gS\ ;gkWa bl

dykdkj dh dkSu lh xq.koRrk,Wa iznf”kZr gksrh gSa\

54- fn, x, fp= esa] ABCD ,d leryEc prqHkqZt gS tgkWa 퐴퐵‖퐷퐶 rFkk dks.k

BCD=60 ° gSA ;fn BFEC dsUnz C okys o`Ùk dk f=T;k[k.M gS vkSj

퐴퐵 = 퐵퐶 =7 lseh0 rFkk 퐷퐸 =4 lseh0 gks] rks Nk;kafdr Hkkx dk {ks=Qy


Kkr dhft,A ¼휋 = rFkk √3 = 1.732 dk iz;ksx dhft,½

55- fn, x, fp= esa] Nk;kafdr Hkkx dk {ks=Qy Kkr dhft,A


mRrj ekyk

1- 36 lseh0 27- 45 푐푚

2- 4 bdkbZ 28- 24 푐푚

3- 휋 푐푚 29- 21-67 lseh0

4- 푙푟 oxZ bdkbZ 30- ` 377051-2

5- 7000 31- 15-84 fdeh@?kaVk

6- 88 lseh0 32- 휋푐푚2

7- 9 oxZ lseh0 33- 7-77 푐푚

8- 50 lseh 34- 196 푐푚

9- 3025 푐푚 35- 5-48 푐푚

10- 90° 36- 9 lseh0

11- °

× 휋푟 37- 59-5 푐푚

12- 4%9 38- 5140

13- 44 lseh0 39- (32 + 2휋)푐푚

14- 96% 40- 308− 196√2 푐푚

15- 5-5 푐푚 41- 14 lseh0

16- 9-625푐푚 42- (704 + 64휋)푐푚

17- 90° 43- 334-31 푐푚

18- :2 ;k 11:7 44- 3-92 lseh0

19- 3-5 lseh0 45- (248− 4휋)푐푚

20- 4 % 46- 11 cm vkSj 3 cm

21- 휋:√3 47- 7-87푐푚


22- 100 lseh0 48- 65

23- 76 lseh0 49- 1568 푐푚

24- 32 lseh0 50- 423-48 푐푚

25- 66 lseh0 51- 2휋푟

26- 3 Lkseh0 52- P=37.71cm,

A=37.71 푐푚

53- 400√2] n;kyq

54- 28-89 푐푚

55- 462푐푚


vH;kl iz”u i=

le; % 50 feuV vad % 20

1- ;fn nks oUrksa dh ifjf/k cjkcj gks rks muds {ks=Qyksa dk vuqikr D;k gksxk\ (1)

2- ;fn pkWans dk O;kl 21cm gks rks bldk ifjeki Kkr dhft,A (1)

3- ;fn ,d or dh ifjf/k 22 lseh gks rks or dk {ks=Qy Kkr dhft,A (2)

4- ,d or ds prqHkkZ”k dk {ks=Qy Kkr dhft, ftldh ifjf/k 44 lseh gSA (2)

5- ,d ?kksM+s dks 28 lseh yEch jLlh }kjk ,d [kEHks ls ckW/kk x;k gSA ?kksM+s }kjk

[kk;h tkus okyh ?kkl dk {ks=Qy Kkr dhft,A (3)

6- fn, x, fp= esa OA =42 lseh] OC =21 lseh rFkk ∠AOB =60°gS rks Nk;kfdar

Hkkx dk {ks=Qy Kkr dhft,A (3)

7- ;fn 10 lseh] f=Tek ds or esa ,d thok AB or ds dsUnz ij ledks.k cukrh

gks] rks y?kq rFkk nh?kZ gRr[k.M ds {ks=Qy Kkr dhft,A¼fn;k gS % A=3.14½ (4)

8- ABCP ,d 20 lseh f=T;k okys or dk prqHkkZa”k gSA AC dks O;kl ekudj ,d

v/kZor [khapk x;k gSA Nk;kafdr Hkkx dk {ks=Qy Kkr dhft,A (4)


v/;k; & 8

i’Bh; {ks=Qy vkSj vk;ru

egRoiw.kZ fcUnq: 1- ?kukHk = 3& vk;keh vkdkj tSls fdrkc] ekfpl dh fMCch] vyekjh bR;kfn

?kukHk dgykrh gSaA

Ekuk] yEckbZ = l] pkSM+kbZ =b] mWapkbZ = h

vk;ru = 푙 × 푏 × ℎ

ik”oZ i’Bh; {ks=Qy = 2ℎ(푙 + 푏)

dqy i’Bh; {ks=Qy = 2(푙푏 + 푏ℎ + 푙ℎ)

2- /ku = 3& vk;keh vkdj tSls vkbl & D;wcl] ywMks dk iklk bR;kfn /ku

dgykrh gSaA

Ekuk] yEckbZ = pkSM+kbZ = mWapkbZ = 푎

vk;ru = 푎

ik”oZ i’Bh; {ks=Qy = 4푎

dqy i’Bh; {ks=Qy = 6푎

3- csyu = 3&vk;keh vkdkj tSls tkj] LraHk] ikbi] jksM+&jksyj bR;kfn csyu

dgykrs gSaA

¼d½ ekuk] vk/kkj f=T;k = 푟

mWapkbZ = ℎ

vk;ru = 2r h

odz i’Bh; {ks=Qy = 2휋푟ℎ

dqy i’Bh; {ks=Qy = 2휋푟(푟 + ℎ)


¼[k½ csyu¼[kks[kyk½ ds fy,]

ckg; f=T;k = 푅 vUr% f=T;k = 푟 mWapkbZ = ℎ vk;ru =휋(푅 − 푟 )ℎ

odz i’Bh; {ks=Qy = 2휋(푅 + 푟)ℎ dqy i’Bh; {ks=Qy = 2휋(푅 + 푟)ℎ + 2휋(푅 − 푟 )

4- “kadq % 3& vk;keh vkdkj tSls VSaV] vkbldzhe dksu dks “kadq dgrs gSaA

Ekuk] vk/kkj f=T;k = 푟

mWapkbZ = ℎ

frjNh mWapkbZ = 푙

푙 = √ℎ + 푟

vk;ru = 휋푟 ℎ

odz i’Bh; {ks=Qy = 휋푟푙

dqy i’Bh; {ks=Qy = 휋푟(푙 + 푟)

/;ku nsa]

;fn ,d “kadq o ,d csyu nksuksa dh vk/kkj f=T;kWa, leku gksa o nksuksa dks mWapkbZ

Hkh leku gks rc

3 × “kadq dk vk;ru = csyu dk vk;ru

5- xksyk % 3- vk;keh vkdkj tSls fdzdsV ckWy] QqVckWy bR;kfn dks xksyk dgrs gSaA

¼d½ ekuk] f=T;k = r

vk;ru = 휋푟

i’Bh; {ks=Qy = 4휋푟


¼[k½ v/kZ xksykdkj ¼Bksl½

f=T;k = 푟

vk;ru = 휋푟

odz i’Bh; {ks=Qy = 2휋푟

dqy i’Bh; {ks=Qy = 3휋푟

6- fNUud % tc ,d “kadq dks vk/kkj

ds lekarj dkVk tkrk gSA rks ml

dVko ls uhps vk/kkj rd ds

Hkkx dks “kadq dk fNUud dgrs gSaA

mnkgj.k & rqdhZ Vksih

Ekuk]

vk/kkj f=T;k = R

mijh f=T;k = r

f=;d mWapkbZ= h

푙 = ℎ + (푅 − 푟)

vk;ru = 휋ℎ(푅 + 푟 + 푅푟)

odz i’Bh; {ks=Qy(Bksl) = 휋푙(푅 + 푟)

dqy i’Bh; {ks=Qy (Bksl)= 휋푙(푅 + 푟) + 휋(푅 + 푟 ) vkfr y?kq mRrjh; iz”u

1- ^^dhi^^ fdu nks T;kferh; vkdkjksa dk la;kstu gS\


2- ^^lqjkgh^^ fdu nks T;kferh; vkdkjksa dk la;kstu gS\

3- ,d csyukdkj ^^isafly^^ tks ,d fljs ls fNyh xbZ gS] fdu nks T;kferh;

vkdkjksa dk la;kstu gS\

4- nh xbZ ^^fxykl^^ dh vkdfr] fdl 3 & vk;keh T;kferh; vkdkjksa lh izrhr

gksrh gS\

5- fpM+~M+h & NDdk [ksyus ds fy, mi;ksx esa vkus okyh ^^fpM+~M+h^^ T;kferh;


6- fxYyh MaMk [ksy esa iz;ksx esa vkus okyh ^^fxYyh^^ fdu&fdu T;kferh;



7- jktfeL=h }kjk iz;ksx esa yk;k tkus okyk ^^lkgqy^^ fdu nks T;kferh;


8- ,d Bksl vkd`fr ds nwljh Bksl vkd`fr esa :ikarj.k ds nkSjku] ubZ Bksl

vkdrh ds vk;kru ij D;k izHkko gksxk\

9- “kadq dks vk/kkj ds lekukarj foHkkftr djus ij gksrk izkIr fNUud dk mijh

vuqizLFk dkV dk {ks= fdl vkd`fr dk gS\

10- ,d Bksl v/kZ&xksykdkj ftldh f=T;k 7 lseh0 gS] dk dqy i’Bh; {ks=Qy

crkb,A

11- Nks xksyksa ds vk;ru dk vuqikr 64%125 gSA buds i’Bh; {ks=Qyksa dk

vuqikr crkb,A

12- ml xksys dk vf/kdre O;kl crkb, ftls 6 lseh0 f=T;k o h lseh mWapkbZ

(h>20) ds csyu us iwjh rjg ls lekfgr dj fy;k gSA

13- Csyu o “kadq ds vk;ru dk vuqikr crkb, ;fn buds vk/kkj dh f=T;k

leku gS o nksuksa dh mWapkbZ Hkh leku gSA

14- ,d xksys ¼Bksl½ ftldh f=T;k gS] dks fi?kykdj r mWapkbZ okyk ,d Bksl

“kadq cuk;k x;k gSA “kadq ds vk/kkj dh f=T;k crkb,A

15- ,d Bksl v/kZ xksys dk dqy i’Bh; {ks=Qy crkb, ;fn bldh f=T;k gSA


16- ;fn ,d xksys dk vk;ru mlds i`’Bh; {ks=Qy ds cjkcj gS rks xksys dh

f=T;k crkb,A

17- ,d csyu] ,d “kadq vkSj ,d v/kZ&xksys ds vk/kkj dh f=T;k,Wa leku gSaA

budh mWapkbZ Hkh leku gSaA rhuksa ds vk;ruksa dk vuqikr crkb,A

18- Leku f=T;k r okys nks Bksl v/kZ&xksyksa ds vk/kkjksa dh vksj ls feydj tks

vkdfr izkIr gksxh mldk dqy i’Bh; {ks=Qy crkb,A

19- ,d ?ku dk vk;ru 1331 lseh- 3 gSA bldh Hkqtk dh yEckbZ crkb,A

20- ,d [kks[kys csyu dh ^^{kerk^^ dk vFkZ D;k gksrk gS\

y?kq mRrjh; iz”u ¼I½

21- ,d Bksl /kuke ftldh Hkqtk,Wa 16 lseh0×12 lseh0× 10 lseh0 gS] esa ls

2 lsseh0 Hkqtk okys fdrus ?ku cuk, tk ldrs gSaA

22- 729 lseh- 3 vk;ru okys ?ku esa ls vf/kd ls vf/kd fdruh mWapkbZ okyk “kadq

dkVk tk ldrk gS\

23- 64 lseh- 3 vk;ru okys nks ?kuksa dks feykdj ?kkukHk cuk;k tkrk gSA bl

?kukHk dk dqy i’Bh; {ks=Qy crkb,A

24- ,d 2 lseh0 O;k; o 16 lseh0 mWapkbZ okys Bksl /kkrq ds csyu dks fi?kykdj

ckjg ,d gh vkdkj ds xksys cuk, x,A izR;sd xksys dh f=T;k crkb,A

25- ckYVh ds nks fljksa dk O;kl 44 lseh0 vkSj 24 lseh0 gSA ckYVh dh mWapkbZ 35

lseh0 gSA ckYVh dk vk;ru crkb,A

y?kq mRrjh; iz”u ¼II½

26- ml NM+ dh vf/kdre yackbZ crkb, ftls 10 eh× 10 eh× 5 eh ds dejs

esa j[kk tk lds\


27- ,d ?ku ftldk vk;ru 1000 lseh- 3 gS] dk i’Bh; {ks=Qy crkb,A

28- nks v/kZ&xksyksa ds vk;ru dk vuqikr 8%27 gSA budh f=T;kvksa dk vuqikr

crkb,A

29- ,d Bksl “kadq ftldh mWapkbZ 28 lseh0 gS vkSj f=T;k 21 lseh gS] dk odz

i’Bh; {ks=Qy o dqy i`’Bh; {ks=Qy crkb,A

30- “kadq ds fNUud ds :i dh ckYVh 28-490 yhVj ikuh ls iwjh rjg Hkjh gSA

blds mij vkSj uhps dh f=T;k,Wa 28 lseh0 o 21 lseh0 dze”k% gSA ckYVh dh

mWapkbZ crkb,A

31- rhu ,d gh /kkrq ds ?kuksa dh Hkqtkvksa dk vuqikr 3%4%5 gSA bu rhuksa dks

fi?kykdj cM+k ?ku cuk;k x;k ftldk fod.kZ 12√3 gSA rhuksa ?kuksa dh

Hkqtk,Wa crkb,A

32- 10-5 lseh0 f=T;k okys csyukdkj VSad dh xgjkbZ dk irk yxk,a] vxj

bldk ck;ru 15 lseh0 × 11 lseh0 × 10-5 lseh0 ds ?kuk”k ds vk;ru ds

cjkcj gksA

33- 8 lseh0 f=T;k vkSj 12 lseh0 mWapkbZ ds ,d “kadq dks mlds vk/kkj ds

lekuarj /kqjh ds e/;fcanq ls nks Hkkxksa esas ckaVk x;k gSA nksuks Hkkxksa ds

vk;ruksa dk vuqikr crkb,A

34- ,d iSVªksy VSad e/; ls 28 lseh0 O;kl o 24 lseh0 yEckbZ dk csyukdkj

:i esa gSA blds nksuksa Nksj 28 lseh0 O;kl o 9 lseh0 yEckbZ ds “kadqvksa ls

tqM+s gSaA bl VSad dk vk;ru crkb,A

nh?kZ mRrjh; iz”u

35- Nh xbZ vkd`fr] 12 lseh0 mWapkbZ ds ,d Bksl “kadq dh gSA ftlds vk/kkj dh

f=T;k 6 lseh0 gS] blds mijh Hkkx ls] vk/kkj ds lekarj ry }kjk 4 lseh0


mWapkbZ okyk “kadq dkV fn;k x;kA “ks’k cps Bksl dk lEiq.kZ i’Bh; {ks=Qy

Kkr dhft,A¼ 휋 = vkSj √5 = 2.236 yhft,½

36- ,d Bksl ydM+h dk f[kYkkSuk] v/kZ xksys ij v/;jksfir leku f=T;k ds “kadq

ds vkdkj dk gSA v/kZ xksys dh f=T;k 3-5 lseh0 gS rFkk bl f[kykSus dks

cukus esa dqy 166 ?ku lseh0 ydM+h yxh gSA f[kykSus dh mWapkbZ Kkr

dhft,A f[kykSus ds v/kZ xksykdkj i’Bh; ry dks : 10 izfr oxZ lseh0 dh

nj ls jax djokus dk O;l Hkh Kkr dhft,A ¼휋 = yhft,½

37- nh xbZ vkd`fr] /kkrq ds ,d Bksl /kukHkkdkj Cykd dh gSA bldh foek,Wa

15lseh0 ×10 lseh0 × 5 lseh0 gSaA blesa ls 7 lseh0 O;k; okyk ,d

csyukdkj Nsn dkV dj fudky fn;k x;k gSA “ks’k cps Bksl dk dqy i`’Bh;

{ks=Qy Kkr dhft,A ¼ 휋 = yhft,½


38- 2-52 fdeh0 izfr ?kaVs dh xfr ls ikuh ,d cyukdkj ikbi ls ,d

csyukdkj VSad esa vk jgk gSA ;fn VSad ds vk/kkj dh f=T;k 40 lseh0 gS

rFkk vk/ks ?kaVs esa blesa ikuh dk ry 3-15 eh0 c<+ tkrk gS] rks ikbi dk

vkarfjd O;kl Kkr dhft,A


mRrjekyk

1. csyu] fNUud 21- 240

2. csyu] xksyk 22- 9 lseh0

3. csyu] “kadq 23- 160 푐푚

4. fNUud 24- 1 lseh0

5. v/kZ&xksyk] fNUud 25- 32706-6 푐푚

6. “kadq fljksa okyk csyu 26- 15 ehVj

7. v/kZ xksyk] “kadq 27- 600 푐푚

8. vifjofrZr 28- 2%3

9. or 29- odz i0 {ks0 2310푐푚 462푐푚

10. 462 푐푚 dqy i0 {ks0 3690 푐푚

11. 16%25 30- 15 lseh0

12. 2푟 31- 6 lseh0] 8lseh0]10lseh0

13. 3%1 32- 5 lseh0

14. 2 33- 7%1 1%7

15. 3휋푟 34- 18480 푐푚

16. 3 bdkbZ 35- 350-592 푐푚

17. 3%1%2 36- h=6 lseh0] : 770

18. 4휋푟 37- 583 푐푚

19. 11 lseh0 38- 4 lseh0

20. vk;ru


vH;kl iz”u i=

le; % 50 feuV vf/kdre vad %20

1- Bksl v/kZ&xksys ds dqy i`’Bh; {ks=Qy dk lw= fyf[k,A (1)

2- ^^dhi^^ fdu nks T;kferh; vkdkjksa dk la;kstu gSA (1)

3- Ml csyukdkj VSad dk vk;ru Kkr dhft, ftldh mWapkbZ 2 ehVj o f=T;k

3-5 ehVj gSA (2)

4- ckYVh ds dqy i’Bh; {ks=Qy dk lw= fyf[k, ¼ladsr % ckYVh fNUud ds :i

esa gS½ (2)

5- ml lcls cM+s “kadq dk vk;ru Kkr dhft, ftls 4-2 lseh0 Hkqtk okys /ku ls

dkVk tk ldrk gSA (3)

6- fNUud dk vk;ru Kkr dhft, ;fn bldh mWapkbZ 4 ehVj gks o nksuksa fljksa

dh f=T;k,Wa 7 ehVj o 4 ehVj gSA (3)

7- fl) dhft, fd leku vk/kkj f=T;k,Wa o leku mWapkbZ okys ,d csyu] ,d

“kadq vkSj ,d v/kZ&xksys dk vk;ru 3%1%2 gksrk gSA (4)

8- 40 lseh0 o 30 lseh0 Hkqtk okys nks ?kuksa ¼Bksl½ dks fi?kykdj 5824 ,d tSls

/ku ¼Bksl½ cuk, x, gSaA u, cus /kuksa dh Hkqtk dh yEckbZ Kkr dhft,A (4)


v/;k; & 9

izkf;drk

egRoiw.k fcUnq: 1- ?kVuk E dh lSU)frd ¼;k ij iajijkxr½ izkf;drk½ izkf;drk ftls ge P(E)

}kjk n”kkZrs gSa] dks fuEu :Ik esa ifjHkkfor fd;k tkrk gS%

P(E)= E Ds vuqdwy ifj.kkeksa dh la[;k

lHkh laHko ifj.kkeksa dh dqyla[;k

tcfd ge dYiuk djrs gSa fd iz;ksx ds lHkh ifj.kke leiz;kfd gSaA

2- fdlh iz;ksx dh lHkh izkjEfHkd /kVukvksa dk ;ksx 1 gksrk gSA ;g O;kid :Ik

ls Hkh lR; gSA

3- fu”fpr ?kVuk dh izkf;drk ,d gksrh gS rFkk vlEHkkfor ?kVuk dh izkf;drk

“kwU; gksrh gSA

4- P(E)+ P(퐸)= 1

5- ?kVuk E dh izkf;drk ,d ,slh la[;k P(E) gS fd 0 ≤ 푃(퐸) ≤ 1

6- Rkk”k dh xM~Mh esa 52 iRrs gksrs gSa tks pkj lewgksa gaqde (♠) iku (♡)bZaV

(♦) rFkk fpMh esa ( ) caVs gksrs gSaA izR;sd lewg esa 13 iRrs gksrs gSaA

7- izR;sd lewg ds 13 iRrksa esa bDdk 2]3]4]5]6]7]8]9]10] xqyke] csxe] ckn”kkg

gksrs gSaA

8- ckn”kkg] csxe] xqyke okus iRrs fp= iRrs ¼QsldkMZ½ dgykrs gSaA


1- ,d flDdk nks ckj mNkyus ij ,d fp= vkus dh izkf;drk Kkr dfj,A


2- Rk”k dh ,d xM~Mh ls ,d iRrk ;knPN;k fudkyk x;k gS blds xqyke gksus

dh izkf;drk Kkr dhft,A

3- Rkk”k dh xM~Mh ls ,d id iRrk ;knPN;k fudkyk tkrk gS rks blds bZaV dk

iRrk gksus dh izkf;drk crkb;sA

4- ,d ikls dks Qsadk x;k bl ij le vHkkT; la[k;k vkus dh izkf;drk D;k

gksxh\

5- ,d ikls ds nks ckj Qsadk tkrk gSA izkf;drk D;k gS fd nksuksa ckj ,d gh

la[;k vk,xh\

6- fdlh yhi o’kZ esas 53 jfookj gksus dh izkf;drk crkb,A

7- 52 iRrksa dh vPNh izdkj ls QsaVh xbZ ,d rk”k dh xM~Mh esa ls ,d iRrk

;k)PN;k fudkyk tkrk gSA dkys jax ds rLohj okys iRrs ds vkus dh

izkf;drk Kkr dhft,A

8- ;fn P(E)= 27% rks bl ?kVuk ds vlQy gksus dh izkf;drk D;k gksxh\

9- m’kk vkSj vkLFkk fe= gSaA nksukssa dk tUefnu 14 uoEcj 2015 ds gksa bldh D;k

izkf;drk gksxh\

10- “BHARTIYA” “kCn ds v{kjksa esa ls ,d “kCn pquk tkrk gS rks ml v{kj ds

Loj gksus dh izkf;drk D;k gksxh\

11- nks fe=ksa dk tUe o’kZ 2000 esa gqvkA nksuksa dk tUe fnu ,d gh fnu gksus dh

izkf;drk Kkr dhft,A

12- ,d iklk ,d ckj Qsadk tkrk gSA vHkkT; la[;k vkus dh izkf;drk Kkr

dhft,A


13- ,d FkSys esa 6 yky rFkk 5 uhyh xsansa gSA FkSys esa ls ,d xsan ;knPN;k

fudkyus ij mlds uhyh gksus dh izkf;drk Kkr dhft,A

14- ikls ds ,d ;qX; dks ,d ckj mNkyk tkrk gSA mu ij 11 dk ;ksx vkus

dh izkf;drk Kkr dhft,A

15- ,d fcuk yhi o’kZ esa 53 lkseokj gksus dh izkf;drk D;k gksxh \

y?kq mRrjh; iz”u

16- Rkk”k ds 52 iRrksa esa ls ,d iRrk ;k)PN;k fudkyk tkrk gS izkf;drk Kkr

dhft, fd ;g u rks bDdk gksxk vkSj u ckn”kkgA

17- ,d cDls esa 250 cYc gSA blesa ls 35 Cyc [kjkc gSaA cDls esa ls ,d cYc

;kn`PN;k fudkyk tkrk gSA izkf;drk Kkr dhft, fd ;g cYc [kjkc ugha gSA

18- fdlh ?kVuk ds foifjr 3%4 gSA bl ?kVuk ds ?kVus dh izkf;drk Kkr

dhft,A

19- ;fn ¼1]4]9]16]25]29½ esa ls 29 dks gVk fn;k tk, rks ,d vHkko; la[;k izkIr

djus dh izkf;drk crkb,A

20- Rkk”k dh xM~Mh esa ls ,d iRrk ;kn`PN;k fudkyk tkrk gS blds rLohj okyk

dkMZ gksus dh izkf;drk Kkr dhft,A

21- 1000 ykWVjh ds fVdVks esa 5 fVdVksa ij buke gSA ;fn ,d O;fDr ,d fVdV

[kjhns rks mlds bZuke thrus dh izkf;drk Kkr dhft,A

22- Rkk”k dh xM~Mh esas ls 1 iRrk ;knPN;k fudkyk tkrk gS blds dkyk iRrk

gksus dh izkf;drk Kkr dhft,A

23- ,d iklk ,d ckj mNkyk tkrk gSA iw.kZ oxZ la[;k vkus dh izkf;drk Kkr

dhft,A


24- nks iklkssa dks ,d lkFk mNkyk tkrk gS nksuksa iklksa ij vadksa ;ksx 10 ;k 10 ls

vf/kd vkus dh izkf;drk Kkr dhft,A

25- 1]2]3] .........33]34]35 esa 7 dk xq.kt vkus dh izkf;drk Kkr dhft,A

nh?kZ mRrjh; iz”u

26- ,d cDls esa dqN dkMZ ftu ij dze”k% la[;k, 3]4]5]50 vafdr gSA ckDl esa ls

,d dkMZ ;kn`PN;k fudkyk tkrk gSA izkf;drk Kkr dhft, fd bl ij ,slh

la[;k gS tks

(i) 7 ls foHkkftr gksrh gS (ii) iw.kZ oxZ gSa

27- ,d FkSys esa 5 lQsn xsansa] 7 yky xsansa] 4 dkyh xsans rFkk 2 uhyh xsansa gSaA FkSys

esa ls ,d xsan ;kn`PN;k fudkyus ij izkf;drk Kkr dhft, fd ;g xsan

(1) lQsn ;k uhyh gS (2) yky ;k dkyh gS

(2) lQsn ugha gS (4) u lQsn rFkk u dkyh

28- 52 iRrksa okyh rk”k dh xM~Mh essa ls bZaV ds ckn”kkg] csxe rFkk xqyke

fudky fn;s tkrs gSaA “ks’k iRrksa esa ls ,d iRrk fudkyk tkrk gSA izkf;drk

Kkr dhft, fd fudyk iRrk

(i) bZaV dk gksxk

(ii) xqyke gksxk

29- fdlh [ksy dks thrus dh izkf;drk gSaA ;fn bls gkjus dh izkf;drk gS

rks 푥 eku Kkr dhft,A

30- ,d ykVjh esa 10 buke vkSj 25 [kkyh gSA buke thrus dh izkf;drk Kkr

dhft,A bl ?kVuk ds fy, 푃(퐸) + 푃 퐸 = 1 dh tkWap dhft,A


31- 400 vaMksa esa ls ,d [kjkc vaMk fudkyus dh izkf;drk 0-035 gSA [kjkc vaMksa

dh la[;k Kkr dhft,A ,d Bhd vaMk fudkyus dh izkf;drk Kkr dhft,A

32- fdlh esys esa ,d [ksy dh LVkWy ij ,d fMCcs esa dqN ijfp;kWa j[kh gS ftu

ij 3]3]5]7]7]7]9]9]9]11 fy[kk gSA ,d O;fDr rc thrrk gS ;fn iphZ ij

la[;kvksa dk ek/; fy[kk gksA mlds u thrus dh izkf;drk D;k gksxh \

33- ,d ckDl esa 90 fMLd gSa ftu ij 1 ls 90 la[;k vafdr gS bl ckDl esa ls

,d fMDl ;kn`PN;k fudkyh tkrh gSA izkf;drk Kkr dhft, fd bl ij tks

la[;k vafdr gksxh og

¼1½ nks vadks dh la[;k gksxh

¼2½ ,d iw.kZ oxZ la[;k gksxh

¼3½ 5 ls foHkkftr gksxh

34- Rkk”k dh vPNh rjg ls QsaVh xbZ xM~Mh esa ls ,d iRrk ;kn`PN;k fudkyk

tkrk gSA izkf;drk Kkr dhft, fd ;g iRrk

(i) gqdqe dk gS ;k bDdk gS

(ii) ,d yky ckn”kkg gS

(iii) u ckn”kkg rFkk u csxe

(iv) ;k rks ,d ckn”kkg ;k ,d csxe

35- Rkk”k dh vPNh rjg QsaVh xbZ xM~Mh ls ,d iRrk ;knPN;k fudkyk tkrk gS

mlds

¼1½ fp= iRrk

¼2½ ykyjax dk fp=iRrk

¼3½ dkys jax dk fp=iRrk gksus dh izkf;drk Kkr dhft,A


36- D{kk esa fgeka”k us dgk fd fdlh Hkh ?kVuk dh izkf;drk 1-3 ugha gks ldrhA

;g dkSu lk ewY; n”kkZrk gS\

37- P(E) + P(퐸) = 1 ;g dFku dkSu lk eqY; n”kkZrk gSA

38- jes”k dks 24000 :Ik;s R;kSgkj ds cksul ds :Ik esa feys mlus 5000 :Ik;s

eafnj dks] 12000 :Ik;s viuh iRuh dks 2000 :Ik;s vius ukSdj dks vkSj “ks’k

jkf”k viuh csVh dks ns nhA

(i) iRuh dks izkIr jkf”k dh izkf;drk Kkr dhft,

(ii) ukSdj dks izkIr jkf”k dh izkf;drk Kkr dhft,

(iii) csVh dks izkIr jkf”k dh izkf;drk Kkr dhft,

(iv) jes”k ds fdu thou ewY;ksa dks ;gkWa n”kkZ;k x;k gSA

39- ,d gkLVy esa 240 fo|kFkhZ jgrs gSas ftlesa 50% izkr%dky ;ksx dykl tkrs

gSa] 25% fte Dyc rFkk 15% ekfuZx okd dks tkrs gSaA “ks’k fo|kFkh ykfQax

dYc ls tqMs+ gSaA ykfQax Dyc ls tqM+s fo|kfFkZ;ksa dh izkf;drk crkb,A iz”u

esa fo|kFkhZ ds fdu thou ewY;ksa dks n”kkZ;k x;k gSA


mRrjekyk

1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16.

17. 18. 19. 0 20.

21. 0.005 22. 23. 24.

25. 26. ,

27. (i) (ii) (iii) (iv)

28. (i) (ii)

29. 8 30. 31. 14, 0.965 32.

33. , ,

34. (i) (ii) (iii) (iv)

35. (i) (ii) (iii)

36. rkfdZd ewY; 37. le>] rkfdZdrk

38- , , , lekftdrk] dÙkZO; ijk;.krk

39- ] “kkjhjhd fQVusl


vH;kl iz”u i=

le; % 1 ?kaVk vf/kdre vad % 30

[k.M & v

1- ,d iklk ,d ckj Qsadk tkrk gS rks fo’ke la[;k vkus dh izkf;drk Kkr

dhft,A (1)

2. ,d FkSys esa 4 yky rFkk 6 dkyh xsansa gSaA FkSys esa ls 1 xsan ;kn`PN;k fudkyh

tkrh gSA dkyh xsan vkus dh izkf;drk crkb,A (1)

[k.M & c

3. ,d vf/ko’kZ esa 53 “kqdzokj gksus dh izkf;drk Kkr dhft,A (2)

4. Rkk”k dks vPNh rjg QsaVh xbZ xM~Mh esa ls 1 iRrk ;kn`PN;k fudkyk tkrk gS

vlds dkys jax dk fp= iRrk ;k yky jax dk fp= iRrk gksus dh

izkf;drk Kkr dhft,A (2)

[k.M & l

5. ,d cDls esa 5 yky] 4 gjs rFkk 7 lQsn daps gSA cDls esa ls 1 dapk

;kn`PN;k fudkyus ij mlds

¼1½ lQsan gks u

¼2½ u yky vkSj u gh lQsn gksus dh izkf;drk Kkr dhft,A (3)

6. ,d iklk ,d ckj Qsadk tkrk gSA izkf;drk Kkr dhft, fd izkIr gksus okyh

la[;k

¼1½ le vHkkT; la[;k gSA ¼2½ ,d iw.kZ oxZ la[;k gSA (3)


[k.M & n

7. ,d FkSys esa dkMZ gS ftl ij 1]3]5]35 la[k;k,a vafdr gS izkf;drk Kkr

dhft, fd ,d fudkys x;s dkMZ ij

¼1½ 15 ls de okyh vHkkT; la[;k vafdr gS

¼2½ 3 rFkk 15 nksuksa ls foHkkftr gksus okyh la[;k vafdr gSA

8. 52 rk”k ds iRrksas dh xM~Mh ls fpM+h dk ckn”kkg] csxe rFkk xqyke gVk,

x, rFkk “ks’k iRrks esas ls ,d iRrk [khapk x;k izkf;drk Kkr dhft, fd [khapk

x;k iRrk

¼1½ iku dk iRrk gksxk ¼2½ csxe gksxk ¼3½fpM+h dk iRrk gksxkA


eqY; vk/kkfjr iz”uksa ds fy, dqN ewY;

1- bZekunkjh

2- vuq”kklu] le; dk ikcan

3- ekuork

4- fyax lekurk

5- Ik;kZoj.k izseh

6- Esgurh

7- rkfdZd lksp

8- Kku

9- Ikzse vkSj ns[kHkky

10- [ksyHkkouk

11- LoLFk izfr;ksfxrk @Vhe Hkkouk

12- egRodka{kk

13- lkgl

14- lekurk

15- vkfFkZd ewY; @ cpr dh

vknr

16- lkekftd ewY;

17- /kkfeZd ewY;

18- lg;ksx

19- ,drk

20- LokLFk ds izfr ltxrkA


lSEiy iz”u i=

d{kk &X (SA-2) le; % 3 ?kaVk vf/kdre vad % 90

lkekU; funsZ”k %&

(i) lHkh iz”u vfuok;Z gSaA (ii) bl iz”u&i= esa 31 iz”u gS] ftUgsa pkj [k.Mksa v] c] l] n esa ckWaVk x;k

gSA [k.M&v esa 4 iz”u gS ftuesa izR;sd 1 vad dk gS] [k.M&c esa 6 iz”u

gS] ftuesa izR;sd 2 vad dk gS] [k.M&l esas 10 iz”u gS] ftuesa izR;sd ds 3

vad gSa] [k.M&n esa 11 iz”u gSa ftuesa izR;sd ds 4 vad gSaA (iii) bl iz”u i= esa dksbZ Hkh lexz fodYi ugha gSA (iv) dSydqysVj dk iz;ksx oftZr gSA

[k.M & v

1- izFke n izkdr la[;kvksa dk ;ksx Kkr dhft,A (1)

2- 18 m vkSj 12 m mWaps nks [kaHkksa ds f”k[kjksa dks ,d rkj }kjk tksM+k x;k

gSA ;fn rkj {kSfrt ry ds lkFk 30° dk dks.k cukrk gS] rks rkj dh

yackbZ Kkr dhft,A (1)

3- Iklksa ds ,d ;qX; dks ,d ckj mNkyk x;kA izFke ikls ij le la[;k ds

vkus dh izkf;drk Kkr dhft,A (1)

4- fcUnqvksa ¼7] 4½ rFkk ¼&1] 8½ ds chp dh nwjh fu/kkZfjr dhft,A (1)

[k.M & c

5- fdlh AP ds 5osa vkSj 7osa inksa dk ;ksx 52 gS rFkk mlds 10okWa in 46

gSA lkoZ varj Kkr djksA (2)


6- K ds os eku Kkr djks ftuds fy, f}?kkr lehdj.k 2푥 + 5푥 + 퐾 = 0 ds

dksbZ okLrfod ewy u gksaA (2)

7- dsUnz O okys nks ladsUnzh; o`Ùkksa dh f=T;k,Wa 5 lseh vkSj 3 lseh gSaA ,d

ckg; fcanq P ls] nks Li”kZ js[kk,Wa PA vkSj PB dze”k% bu o`Ùkks ij [khaph xbZA

;fn PA=12 gks rks PB Kkr djsaA (2)

8- 10 cm yackbZ dk ,d js[kk[kaM [khafp, bl ij ,d ,slk fcanq Kkr djsa tks

bls vkarfjd :i ls 2%3 ds vuqikr esa foHkkftr djrk gSA (2)

9- dsUnz O vkSj f=T;k 4 lseh okyk ,d o`Ùk [khafp,A ,d O;kl POQ cuk,WaA

P ls gksdj o`Ùk dh ,d Li”kZ js[kk dh jpuk djsaA (2)

10- /kkrq ds rhu ?kuksa dks] ftuds fdukjs 3 cm, 4 cm vkSj 5 cm gSa] fi?kykdj

,d vdsyk ?ku cuk;k x;kA bl izdkj cus ?ku dk fdukjk Kkr djsaA (2)

[k.M & l

11- Kkr dhft, fd 513 ;ksx izkIr djus ds fy, AP- 54, 51, 48----ds fdrus

in ysus pkfg,A nks mRrj gksus dk dkj.k Li’V dhft,A (3)

12- f}?kkr lehdj.k (푥 − 1) + 5(푥 − 1)− 6 = 0 dks gy djsaA (3)

13- vkdfr esa dsUnz O okys o`Ùk ij Li”kZ js[kk AB gSA fl) djsa ∠퐵푃푄 = ∠푃푅푄

gSA ;fn ∠퐵푃푄 = 60° rks ∠푅푃푄 Kkr djsaA (3)


14- fdlh izdk”k&iqat dh mWapkbZ h ehVj gSA bl izdk”k&iqat ls blds foijhr

vksj fLFkr nks tgktksa ds voueu dks.k 30° vkSj 45° izsf{kr fd, tkrs gSaA

nksuksa tgktksa ds chp dh nwjh Kkr djsaA (3)

15- nks iklksa dks ,d lkFk Qsadk tkrk gSA bu iklksa ij vkbZ la[;kvksa dk ;ksx

6 ls vf/kd ijarq 9 ls de gksus dh izkf;drk Kkr djsaA (3)

16- ∆퐴퐵퐶 ds “kh’kZ A(7, 2), B(9, 10) vkSj C(1, 4) gSA ;fn AB vkSj AC ds

e/; fcanq dze”k% E vkSj F gSa] rks bu funsZ”kkadksa dk iz;ksx djrs gq, fl)

djks 퐸퐹 = 퐵퐶 gSA (3)

17- n”kkZb, fd fcanq A(4, 7), B(0, 6),C(4, 5) vkSj D(8, 6),d leprqHkqZt ds

“kh’kZ gSaA (3)

18- ,d Bksl /kkrq ds /ku dh Hkqtk 60 lseh gSA bls fi?kykdj leku 8000 Bksl

?kukdkj iklksa esa <kyk x;kA izR;sd ikls dh Hkqtk Kkr dhft,A (3)

19- ,d orkdkj ikdZ ij : 70 izfr ehVj dh nj ls ckM+ yxokus dk [kpZ :

11]000 gSA bl ikdZ dk {ks=Qy Kkr djsaa aA (휋 = dk iz;ksx djsaa½ (3)

20- f=T;k 14 lseh okys ,d or ds ,d or[kaM dk {ks=Qy Kkr djsa] tcfd

laxr f=T;k[kaM dk dks.k 30° gSA ¼ 휋 = dk iz;ksx djsaa½ (3)

[k.M & n

21- n”kkZb, fd fdlh AP ds (p+q) osa vkSj (p–q)osa inksa dk ;ksx mlds p osa

in dk nqxquk gksrk gSA

22- x ds fy, gy dhft,

1푎 + 푏 + 푥

=1푎

+1푏

+1푥

푎 ≠ 0, 푏 ≠ 0,푥 ≠ 0


23- ;fn llehdj.k (1 + 푚 )푥 + 2푚푐푥 + (푐 − 푎 ) = 0 ds ewy cjkcj gSa]

rks fl) dhft, fd 푐 = 푎 (1 + 푚 )

24- ;fn ,d lekUrj prqHkqZt dh lHkh Hkqtk,Wa ,d or dks Li”kZ djrh gSa] rks

n”kkZb, f dog lekUrj prqHkqZt ,d leprqHkqZt gSaA

25- f=T;k 4 lseh okys ,d or ij Li”kZ js[kkvksa dk ,d ,slk ;qXe [khafp, tks

ijLij 45° ds dksu ij varfjr gks budh yEckbZ;kWa ekfe,A

26- nks [kaHkksa ds chp esa [kM+k ,d O;fDr ;g Kkr djrk gS fd bu [kaHkksa ds

“kh’kksa Z }kjk mldh vkWa[k ij varfjr dks.k ,d ledks.k gSaA ;fn nksuksa [kaHkksa

dh mWapkbZ ml O;kDr dh yackbZ ls nks xquh vkSj pkj xquh gSaA rFkk [kaHkksa ds

chp dh nwjh cM+s [kaHks dh mWapkbZ ds cjkcj gS] rks ml O;fDr ls NksVs [kaHk s

vkSj cM+s [kaHks ls nwfj;ksa dk vuqikr Kkr dhft,A

27- 52 rk”kksa dh ,d xM~Mh esa ls nks dkys ckn”kkg vkSj pkj yky dkMZ

¼ckn”kkg ugha½ gVk fy, tkrs gSaA ,d dkMZ ;k)fPNd :Ik ls fudkyk

tkrk gSA bldh izkf;drk Kkr dhft, fd fudkyk x;k dkMZ %

(a) ,d dkyk xqyke gS (b) ,d dkyh csxe gS

(c) ,d dkyk dkMZ gS (d) ,d ckn”kkg gS

28- fdlh prqHkqZt ABCD dk {ks=Qy Kkr dhft,] ftlds “kh’kZ A(1, 0),

B(5,3), C(2, 7) vkSj D(-2, 4) gSaA fod.kksa Z AC vkSj BD dh yEckbZ;kWa

Hkh Kkr dhft,A

29- Kkr dhft, fd ,d [kqys /kkrq ds lanwd esa] fdrus ?ku lsaVhehVj /kkrq

gS]ftldh ckgjh foek,W 36 cm, 25 cm, vkSj 16 cm gSaA rFkk /kkrq dh


eksVkbZ 2 lseh gSA ;fn 1 ?ku lsaVhehVj /kkrq dk Hkkj 15 xzke gS rks bl

[kqys lanwd dk Hkkj Kkr djksA

30- 24 cm Hkqtk okyh ,d /kkrq dh oxkZdkj “khV esa ls f=T;k 7 lseh okys nks

v)ZoRrkdkj Hkkxksa dks ckgj fudkydj ,d ;a= cuk;k tkrk gS] tSlk dh

vkdfr esa n”kkZ;k x;k gSA ,sls 50 ;a= cukus esa yxus okyh /kkrq dk

{ks=Qy Kkr dhft,A ¼휋 = dk iz;ksx dhft,½

31- lM+d ij ,d xjhc dykdkj cPpksa ds fy, etkfd;k dkVwZu cukrk gS rFkk

viuh thfodk vftZr djrk gSA ,d mlus vkd`fr esa n”kkZ, vuqlkj ,d

gkL;dj eq[k cuk;k] ftlds fy, mlus ,d or ds vanj or [khapk] tgkWa

cM+s or dh f=T;k 30 lseh vkSj NksVs or dh f=T;k 20 lseh gSA bl

vkdfr esa Vksih ds fy, fdruh {ks=Qy fn;k x;k\ ;gkWa bl dykdkj dh

dkSu&lh xq.kork,Wa iznf”kZr gksrh gS\


mÙkjekyk

1. ( )

2. 12 m

3.

4. 4√5 bdkbZ

5. 푑 = 5

6. 푘 >

7. 4√10 cm

8. jpuk

9. jpuk

10. 6 cm

11. N = 18 ;k 19

nks mRrj D;ksafd 푎 = 0 gS

12. X=0 tc y=-1

X=7 tc y=6

13. ∠ 푅푃푄 = 30°

14. ℎ(1 + √3)푚

15.

16. E(8, 6), F(4, 3)

EF= 5 bdkbZ

BC= 10 bdkbZ

17. izR;sd Hkqtk= √17 bdkbZ

I fod.kZ= 2 bdkbZ

II fod.kZ = 8 bdkbZ

18. 3 cm

19. 1964.28 oxZ ehVj

20. 2-33 oxZ lseh

21. fl) djsa

22. x = – 9, x = – b

23. fl) djsa

24. fl) djsa

25. jpuk

26. 1 : 3

27. (a) (b)

(c)

(d)

28. {ks=Qy 25 oxZ bdkbZ

AC=5√2 bdkbZ

BD=5√2 bdkbZ


29. vk;ru = 4992 ?ku bdkbZ

Hkkj = 74-88 kg

30. 21100 oxZ ls-eh

31. {ks=Qy 400 oxZ lseh-

bZekunkj] deZB] n;kyq]

HkkoukRed


ladfyr ijh{kk &II

le; % 3 ?kaVs vf/kdre vad % 90

lkekU; funsZ”k %&

(i) lHkh iz”u vfuok;Z gSaA (ii) bl iz”u&i= esa 31 iz”u gS] ftUgsa pkj [k.Mksa v] c] l] n esa ckWaVk x;k gS (iii) [k.M&v esa 4 iz”u gS ftuesa izR;sd 1 vad dk gS] [k.M&c esa 6 iz”u gS]

ftuesa izR;sd 2 vad dk gS] [k.M&l esas 10 iz”u gS] ftuesa izR;sd ds 3

vad gSa] [k.M&n esa 11 iz”u gSa ftuesa izR;sd ds 4 vad gSaA (iv) dSydqysVj dk iz;ksx oftZr gSA

[k.M & v

iz'u la[;k 1 ls 4 rd izR;sd iz”u 1 vad dk gSA

1- ;fn f}?kkr lehdj.k 푃푥 − 2√5푃푥 + 15 = 0 ds nks ewy cjkcj gS rks 푃

dk eku Kkr dhft,A

2- 퐾 ds fdl eku ds fy, 2K–7, K+5 rFkk 3K+2 lekarj Js.kh ds dzekxr

in gksasxs\

3- vkdfr esa] ,d Vkoj PQ dh mWapkbZ 30 eh0 rFkk bldh Nk;k QR dh

yEckbZ 30√3 eh0 gSA rks lw;Z dk mUurka”k Kkr dhft,A


4- vkdfr esa ∆퐴퐵퐶 dh Hkqtk,a AB, BC vkSj CA or dks dze”k% fcUnqvksa P, Q

rFkk R ij Li”kZ djrh gSA ;fn PA = 4 lseh BP = 3 lseh vkSj

AC = 11 lseh rks BC dk eku Kkr dhft,A

[k.M & c

iz'u la[;k 5 ls 10 rd izR;sd iz”u 2 vad dk gSA

5- izFke 50 izkdr la[;kvksa esa ls ,d la[;k ;k}P;k pquh tkrh gSA bl la[;k

ds 3 vkSj 4 dk xq.kt gksus dh izkf;drk Kkr dhft,A

6- og vuqikr Kkr dhft, ftlesa y-v{k fcUnqvksa A(5, –6) rFkk B(–1, –4) dks

feykus okys js[kk[kaM dks ckWaVrk gSA foHkkftr djus okys fcUnq ds funsZ”kkad

Hkh Kkr dhft,A

7- ,d v/kZxksys dk vk;ru2425 ?ku lseh gSA mldk od i`’Bh; {ks=Qy Kkr

dfj,A

8- vkdfr esa ;fn ∠ATQ= 40° rks ∠AOB dk eku Kkr dhft,A


9- lekarj vksoh ds n inksa dk ;ksx 푛 + 3푛 gSA bldk 20 okWa in Kkr

dhft,A

10- layXu vkd`fr esa Nk;kafdr Hkkx ,d dkj ds okbij }kjk lkQ fd;s {ks0 dks

iznZf”kr djrk gSA okbij }jk lkQ fd, x, {ks= dh x.kuk dhft, ;fn

OA=7 lseh rFkk OC= 21 lseh gksA

[k.M & l

iz'u la[;k 11 ls 20 rd izR;sd iz”u 3 vadksa dk gSA

11- layXu vkd`fr esa AB or dk O;kl gS rFkk O dsUnz gS ;fn OA = 7 lseh

gks rks Nk;kafdr Hkkx dk {ks=Qy Kkr dhft,A

12- 9 lseh vkarfjd f=T;k dk ,d v/kZ xksykdkj dVksjk ikuh ls Hkjk gSA bl

ikuh dks 3 lseh O;kkl rFkk 4 lseh mWaph cksryksa esa Hkjrk gSA dVksjs dks

[kkyh djus ds fy, ,slh fdruh cksryksa dh vko”;drk gksxhA


13- ;fn fcUnq A(0, 2) fcUnqvksa B(3, P) rFkk C(P, 5) ls lenwjLFk gS rks P dk

eku Kkr dhft,A

14- laYxu vkdfr esa 3-5 lseh dh f=T;k rFkk O dsUnz ds or dk prqFkkZa”k

OACB iznf”kZr djrk gSA

(i) prqFkZk”k OACB dk {ks=Qy dh x.kuk dhft,A

(ii) OD=2 lseh] Nk;kafdr Hkkx dk {ks=Qy Kkr dhft,A

15- ;kfn lekarj Js.kh dk 8okWa in 37 gS rFkk 15okWa in 12 osa in ls 15 vf/kd

gS] rks lekarj Js.kh Kkr dhft,A bl lekarj Js.kh ds igys 15 inksa dk

;ksxQy Hkh Kkr dhft,A

16- X ds fy, dhft, = + + , 푎 ≠ 0, 푏 ≠ 0,푥 ≠ 0

17- ,d f[kykSuk 3-5 lseh f=T;k ds xksyk/kZ ij “kadq ds vkdkj dk gS A ;fn

f[kykSus dh dqy mWapkbZ 15-5 lseh gks rks bldk dqy i’Vh; {ks=Qy Kkr

dhft,A

18- ,d Bksl yac orh; “kadq dk O;kl 14 lseh rFkk mWapkbZ 8 lseh gSA bldks

fi?kykdj ,d [kks[kyk xksyk cuk;k x;kA ;fn xksys dk ckg; O;kl 10

lseh gks rks xksys dk vkarfjd O;kl Kkr dhft,A

19- 5 ehVj mWaps ehukj dh pksVh ij ,d /ot&naM yxk gqvk gSA Hkwfe ij


fLFkr fdlh fcanq ls /ot&naM ds mijh fljs dk mUu;u dks.k 60° gS vkSj

blh fcUnq ehukj dh pksVh dk mUu;u dks.k 45 ° gSA /otnaM dh yEckbZ

Kkr dhft,A

20- Rk”k ds 52 iRrksa dh vPNh rjg QsaVh xbZ xM~Mh esa ls ,d iRrk fudkyk

tkrk gSA izkf;drk Kkr dhft, fd og

(i) fcuk rLohj okyk iRrk gksxk

(ii) ,d dkyk ckn”kkg ;k yky csxe gksxhA

[k.M & n

iz'u la[;k 21 ls 30 rd izR;sd iz”u 4 vadksa dk gSA

21- Okk;q;ku ij p<+rs le; ,d ;k=h lh<+h ls fQly dj pksfVy gks x;kA ;ku

pkyd ml ;k=h dks gokbZ vM~Ms ds vkikr dkyhu fDyfud esa bykt ds

fy, ys x;k ftlds dkj.k ok;q;ku vk/ks ?kaVs ysV gks x;kA 1500 fdeh nwj

xRo; ij le; ls igqapus ds fy, rkfd ;k=h vxyh mM+ku idM+ lds]

pkyd us ;ku dh xfr 100 fdeh@?kaVk c<+k nh ok;q;ku dh ewy xfr Kkr

dhft,A bl iz”u esa D;k ewY; n”kkZ;k x;k gS\

22- fl) dhft, fd or ds ckg;&fcUnq ls or ij [khaph xbZ Li”kZ js[kkvksa dh

yEckbZ cjkcj gksrh gSA

23- ,d ∆퐴퐵퐶 dh jpuk dhft, ftesa BC= 8 ∠B=45° lseh rFkk ∠C=30° gSA

∆퐴퐵퐶 ds le:Ik ,d vU; f=Hkqt dh jpuk dhft, ftldh Hkqtk,Wa

∆퐴퐵퐶 dh laxr Hkqtkvksa dk 3@4 gksA

24- ;fn ,d lekarj prqHkqZt dh lHkh Hkqtk,Wa ,d or dks Li”kZ djsa rks n”kkZ,a

dh og leprqHkqZt gksxkA


25- x ds fy, gy djks %

+ 5 = 6, 푥 ≠ −3,

26- ,d xyh esa Hkwfe ls 15 ehVj mWaph ,d f[kM+dh ls xyh ls nwljh vksj ds

,d edkj ds mijh Hkkx rFkk ikn ds mUu;u dks.k dze”k% 30° rFkk 45° gSA n”kkZb;s fd lkeus ds edku dh mWapkbZ 23-66 ehVj gSA

27- 9000: dqN O;fDr;ksa esa cjkcj&cjkcj ckaVs x;sA ;fn 20 O;fDr vkSj gksrs]

rks izR;sd O;fDr dks 160 : de feyrsA O;fDr;ksas dh ewy la[;k Kkr

dhft,A

28- fcUnq P ]fcUnqvksa A(2, 1) rFkk B(5, -8) dks feykus okys js[kk[kaM dks bl

izdkj foHkkftr djrk gS fd = ;fn fcUnq P js[kk 2푥 − 푦 + 푘 = 0

fLFkr gS] rks k dk eku Kkr dhft,A

29- ,d cDls esa 90 fMLd gS ftu ij 1 ls 90 rd la[;k,Wa vfdr gSA bl cDls

esa ls ,d fMLd ;k}PN;k fudkyh tkrh gSA izkf;drk Kkr dhft, fd bl

ij tks la[;k vafdr gksxh og

¼1½ nks vadks dh ,d la[;k gksxhA

¼2½ ,d iw.kZ oxZ la[;k gksxhA

¼3½ 5 ls foHkkftr gksxhA

30- ,d ckYVh “kadq ds fNUud :Ik esa /kkrq dh pknj dh cuh gS ckYVh ds fupus

fljs vkSj mijh fljs dh f=T;k,a dze”k % 8 lseh rFkk 20 lseh gSA ckYVh dh

mWapkbZ 16 lseh gSA ckYVh dk ewY; Kkr dhft, ;fn iz;ksx esa yxkbZ xbZ

/kkrq dh pknj dk ewY; 15 : izfr 100 oxZ lseh gksA


31- laYxu vkdfr esa] AB vkSj CD or dh nks lekarj Li”kZ js[kk,a gSA or dk

dsUnz O gSA nks lekarj js[kkvksa AB vkSj CD ds chp ST or dks Q ij Li”kZ

djrh gqbZ ,d vkSj Li”kZ js[kk gSA fl) dfj, ∠SOT=90°


mRrjekyk

1. P = 3

2. K = 5

3. 30°

4. 10 cm

5.

6. 5:1, (0, )

7. 93 푐푚

8. 100

9. 42

10. 102.67 푐푚

11. 66.5 푐푚

12. 54

13. ..

14. (i) 9.625 푐푚

(ii) 6.125 푐푚

15. (i) 2, 7, 12.... (ii) 555

16. –a, –b

17. 214.5 푐푚

18. 6 cm

19. 3.65 m

20. ,

21. 500 km/h

22. ..

23. ..

24. ...

25. 0, -4

26. ..

27. 25

28. -8

29. (i) (ii) (iii)

Documents

GROUP LEADER GOVT. CO-ED. S.S.S, GOPAL PARK, DELHI-110033 · 1 [class x : maths] team members for review of support material group leader mr. yogesh agarwal (principal) govt. co-ed