Important MCQ to Get Pass for Std 10 Weak Students

Specially thanks to Shri kiranbhai Raval(asst.teacher) created by-Rakesh k.prajapati
Karnavat highschool,palanpur A.v.sanghvi highschool,vedancha
હાશ...! હું ગણિત અને વિજ્ઞાન માું પાસ ...
વળવય-ગણિત (how to we get more than 40 marks?)
જે વળદ્યાર્થી વમત્રોને ગણિત ઓછું ફાળે છે એમિે નીચેના પ્રકરિો પર ળધુ ભાર મુકળો
 પ્રકરિ-૧૨ :--રચનાઓ (એક રચના ઩ ૂછા઴ે )----૦૫ ગુિ
 પ્રકરિ-૧૩ :-ળર્ુુલ શુંબુંવિત ક્ષેત્રફલ ------------૦૮ ગુિ
 પ્રકરિ-૧૪ :-઩ૃષ્ઠ્ફલ અને ઘનફલ --------------૦૮ ગુિ
 પ્રકરિ-૧૫ :-આંકડા઴ાસ્ત્ર -----------------------૦૮ ગુિ
 પ્રકરિ -૧૬ :-શુંભાળના ------------------------૦૮ ગુિ
 પ્રમેય (શાણબતી ળાલા )---બે ઩ુછાય -----------૦૯ ગુિ
----------------------------
કુ઱ ૪૩ ગુિ નુું ઩ુછાઈ ઴કે છે .ઉપરના પ્રકરિ આળડી જાય તોજ ળધુ માું બીજા પ્રકરિ
ના શષે઱ા મુદ્દા તૈયાર કરળા.................
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વળજ્ઞાનના મષત્ળના પ્રશ્નો..........આટ઱ા પ્રશ્નો તો તૈયાર કરળાજ પડે ..
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section –D (૧૫ ગુિ )
1. શમત઱ અરીશા ળડે ર્થર્ુું પરાળતુન યોગ્ય આકૃવત ળડે શમજાળો.
2. ગોણ઱ય અરીશા ળડે ર્થતા પરાળતુન માટેની કાતેણઝયન શુંજ્ઞા પ્રિા઱ી શમજાળો.
3. અંતગોલ અરીશા માટે f =R /૨ અર્થળા R =૨ f મેલળો.
4. પ્રકા઴ના ળક્રીભળનના વનયમો ઱ખી વનરપેક્ષ અને શપેક્ષ ળક્રીભળન શમજાળો.
5. સ્ને઱ના વનયમનુું વ્યાપક સ્ળરૂપ મેલળો .
6. કાચના ઱ુંબઘન ળડે ર્થર્ુું પ્રકા઴નુું ળક્રીભળન આકૃવત શષ શમજાળો
7. ગોણ઱ય ઱ેન્શ માટેનુું સુત્ર 1 /v -1 /u =1 /f તારળો.

8. શુંયુકત સુક્ષ્મદ઴ુક યુંત્રની રચના ,કાયુપદ્ધવત અને ઉપયોગીતા આકૃવત શષીત
ળિુળો.
9. ખગોલીય દુરાબીનનો વશધિાુંત ,આકૃવત,રચના,ઉપયોગ ધળારા શમજાળો.
10. કોપર શુદ્ધદ્ધકરિની વળદ્યુતવળભાજન પદ્ધવતનુું આકૃવત શષીત ળિુન કરો .
11. બોકશાઇટમાુંર્થી એલ્યુવમના મેલળળાની બેયર વળવિ ળિુળો.
12. ષો઱-ષેરાઉલ્ટ પધિવત શવળસ્તાર શમજાળો.
13. ળાતભઠ્ઠી ધળારા આયનુનુું વનષ્ઠકવુિ શમીકરિ શષીત શમજાળો.
14. િાર્ુઓની શક્રક્રયતા શ્રેિી પર ટૂુંક નોિ ઱ખો.
15. િાર્ુના ભૌવતક અને રાશાયણિક ગુિિમો જિાળો.
16. ક્ષારિ એટ઱ે શુું?તેના કારિો અને અટકાળળાના ઉપાયો જિાળો.
17. રાશાયણિક રીડક્શન અને વળદ્યુત રાશાયણિક રીડક્શન ઉદાષરિ શષીત શમજાળો.
18. પોવિના પ્રકારો વળગતળાર શમજાળો.
19. અમીબાની પોવિપદ્ધવત આકૃતીશષ ળિુળો.
20. મનુષ્ઠયના પાચનતુંત્રની આકૃવત દોરી નામકરિ કરો અને મો ,જઠર,તર્થા નાના
આંતરડામાું ર્થતી પાચનક્રક્રયા ળિુળો.(શમગ્ર રીતે તૈયાર કરળો)
21. શ્વશનના પ્રકારો જિાળી તેમના વળ઴ે શમીકરિ શષીત નોંિ ઱ખો.
22. ળનસ્પવતના મૂલ અને પ્રકાુંડમાું ર્થતી શ્વશનક્રક્રયા શમજાળો.
23. મનુષ્ઠયનુું શ્વશનતુંત્ર ળિુળો અને ઉરોદર પટ઱નુું કાયુ શમજાળો.
24. ઓહ્મ નો વનયમ શમજાળો.
section-c (૧૫ ગુિ)
1. કાચના વપ્રઝમ ળડે શ્વેત પ્રકા઴નુું વળભાજન આકૃવત દોરી શમજાળો.
2. પ્રકા઴ના પ્રાર્થવમક રુંગોનુું શુંપાતીકરિ આકૃવત દોરી શમજાળો.
3. આકૃવત ધળારા ળિુકો માટેની વળયોણગક પધિવત શમજાળો.

4. માનળ આંખની નામ વનદે઴ળલી આકૃવત દોરી તેના અળયળોના કયો ઱ખો.
5. ઱ઘુદ્રષ્ષ્ઠટની ખામી કેળી રીતે ઉદભળે છે?તેનુું વનળારિ આકૃવત દોરી શમજાળો.
6. ગુરુદ્રષ્ષ્ઠટની ખામી કેળી રીતે ઉદભળે છે તેનુું વનળારિ આકૃવત દોરી શમજાળો.
7. પ્રકા઴ના ઩ ૂિુ આંતક્રરક પરાળતુનની ઘટના આકૃતીશષ ળિુળો.
8. મરીણચકા(મૃગજલ)ની ઘટના આકૃવત ધળારા શમજાળો.
9. ટીંડો઱ અશર શમજાળો.તેના ઉદાષરિ આપો.
10. ળાતાળરિીય ળક્રીભળન કોને કષે છે?તેને ઱ીિે પરીિમવત ઘટનાઓ ઱ખો.
11. ઓશુટેડનો પ્રયોગ ળિુળો.
12. શો઱ેનોઈડ વળવે આકૃવત શષીત ટૂુંક નોંિ ઱ખો.
13. .નીચેના વનયમ ઱ખો.
-જમિા ષાર્થના અન્ગુઠાનો વનયમ.
-ફ્઱ેવમિંગનો ડાબા ષાર્થનો વનયમ
ફ્઱ેવમિંગનો જમિા ષાર્થનો વનયમ
14. ઇ઱ેષ્ક્િક મોટરનો વશધિાુંત,રચના અને કાયુ પદ્ધવત આકૃવત શષ ળિુળો.
15. જનરેટર નો વશધિાુંત,રચના અને કાયુ પધિવત આકૃવત શષ ળિુળો.
16. વળદ્યુત ચુુંબકીય પ્રેરિની ઘટના શમજાળતા પ્રયોગનુું ળિુન કરો.
17. વળદ્યુત ઘુંટડી પર ટૂુંકનોંિ ઱ખો.
18. વળદ્યુત ળપરા઴માું કયા પ્રકારની શાળચેતી રાખ઴ો?
19. ફરુઝ વળ઴ે ટૂુંકનોંિ ઱ખો.(આકૃવત જરૂરી)
20. તફાળત આપો-A.C.વળદ્યુત પ્રળાષ અને D.C.વળદ્યુત પ્રળાષ.
21. તફાળત આપો-ઇ઱ેષ્ક્િક મોટર અને જનરેટર
22. A.C.વળદ્યુત પ્રળાષના ફાયદા અને D.C.વળદ્યુત પ્રળાષના ગેરફાયદા જિાળો.
23. અિાર્ુ તત્ળોના રાશાયણિક ગુિિમો જિાળો.
24. પ્રયોગ઴ાલામાું ડાયષાઇડ્રોજન ળાયુ બનાળળાની રીત ળિુળો.
25. એમોવનયાના ઔદ્યોણગક ઉત્પાદનની ષેબરની રીત ળિુળો-એમોવનયાના ઉપયોગ જિાળો.

26. એમોનીયા ળાયુના રાશાયણિક ગુિિમો જિાળો.
27. શલ્ફરના વનષ્ઠકવુિની ફ્રા઴ પધિવત ળિુળો.
28. કુદરતી ળાયુ અને પેિો઱ીયમમાુંર્થી શલ્ફર મેલળળાની રીત ળિુળો.
29. શલ્ફરના બહુરુપો આકૃવત શષીત શમજાળો.
30. શલ્ફરુક્રરક એવશડના ઉત્પાદન માટેની શુંપકુવળવિ ળિુળો.
31. ટૂુંકનોંિ ઱ખો.-મુંદ શલ્ફરુક્રરક એશીડ
32. આર્થળિની ક્રક્રયા ધળારા ઇર્થેનો઱ મેલળળાની રીત ળિુળો.
33. ઇર્થેનો઱ના રાશાયણિક ગુિિમો જિાળો.
34. મીર્થેના઱ની બનાળટ અને રાશાયણિક ગુિિમો જિાળો.
35. ફ્રી઴ર-િોપ્શ પદ્ધવતર્થી પ્રોપેનોનની બનાળટ ળિુળો અને ગુિિમો ઱ખો.
36. ઇર્થેનોઇક એવશડની બનાળટની બુંને રીતો ઱ખો.
37. ઇર્થેનોઇક એવશડના ગુિિમો ઱ખો.
38. પો઱ીએસ્ટરની બનાળટ ળિુળો.
39. પો઱ી એમાઈડ (નાય઱ોન)ની બનાળટ ળિુળો.
40. પ્રક્ષા઱કો વળ઴ે નોંિ ઱ખો.
41. વમશે઱ રચના ળિુળો.
42. અ઱ીનગી પ્રજનનના ત્રિ પ્રકારો ળિુળો.(બિાજ તૈયાર કરળા)
43. ળનસ્પવતમાું ળાનસ્પવતકપ્રજનન (કુદરતી)શમજાળો.
44. ળનસ્પવતમાું ળાનસ્પવતકપ્રજનન (કૃવત્રમ )શમજાળો.
45. ઱ાક્ષિીક ઩ુષ્ઠપની રચના આકૃવતશષ શમજાળો.
46. ઩ુરુવ પ્રજનનતુંત્ર આકૃવત દોરી શમજાળો.
47. સ્ત્રી પ્રજનન તુંત્ર આકૃવત દોરી શમજાળો.
48. સ્ત્રી માું ઋર્ુચક્ર (માવશક સ્ત્રાળ)નુું ળિુન કરો.
49. માનળ ળશતી વનયુંત્રિની પદ્ધવતઓ ળિુળો.
50. તફાળત આપો-ણ઱િંગી પ્રજનન અને અણ઱િંગી પ્રજનન.

આ ઉપરાુંત વળજ્ઞાનમાું શેક્શન-A અને શેક્શન-B માું ૨૦ ગુિ ના બબ્બે ગુિ ના પ્રશ્નો
ષોય છે .જેમાું કારિો,તફાળત,આકૃવત,ફાયદા,ગેર ફાયદા ,઱ક્ષિો ,ગુિિમો ળગેરે ના પ્રશ્નો
તૈયાર કરળા.
આભાર ....
શ્રી ક્રકરિ રાળ઱ (મ.વ઴.)
શ્રી એમ.બી.કિાુળત ષાઇસ્કુ઱,પા઱ન઩ુર
અને
શ્રી રાકે઴ પ્રજાપવત
શ્રી એ.ળી.શુંઘળી ષાઇસ્કુ઱,ળેડુંચા

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5 * ;D~5TF VG[ 5FIYFUMZ;G] 5|D[I
6 )P l+SM6lDlT 8 4 3 - - 15
7 !!P JT]"/ 6 2 - 4 - 12
8 !ZP ZRGF - - - - 5 5
9 !&P ;EFJGF 4 2 3 - - 9
S], U]6 50 16 12 12 10 100
AM0"GF 5|`G5+ D]HA 5|SZ6 VG[ MCQ U]6EFZ
S|D 5|SZ6G] GFD PART-A PART-B TOTAL
1 I]S,L0GL EFUlJlW VG[ JF:TlJS ;bIFVM 2 2 4
2 AC]5NLVM 4 2 6
3 läR, ;]Z[B ;DLSZ6 4 2 6
4 läWFT ;DLSZ6 5 3 8
5 ;DFTZ z[6L 3 2 5
6 l+SM6GL ;D~5TF 3
7 13
7 ;D~5TF VG[ 5FIYFUMZ;G] 5|D[I 3
8 IFDE}lDlT 4 2 6
9 l+SM6lDlT 4 2 6
10 VTZ VG[ pRF. 3 3 6
11 JT]"/ 2 4 6
12 ZRGF 0 5 5
13 JT]"/ ;AlWT 1F[+O/ 4 4 8
14 5'Q9O/ WGO/ 4 4 8
15 VFS0FXF:+ 3 5 8
16 ;EFJGF 2 3 5
TOTAL 50 50 100
# GMW o YM0M O[ZOFZ ;EJ K[P

Ätuhý --- 10
Ë{Þ : 3 f÷tf Ul6Tv!ZsGf fw „w : 100
«&™…ºt™wk …rhY… ð»to – 2014-15
PART – A „wý : 50
• ™e[u yt…u÷t «&™tu («&™ ™k. 1 Úte 50){tk ÞtuøÞ rðfÕ… …ËkN fhe™u OMR Answer Sheet {tk
sðtƒ yt…tu. («íÞuf™tu 1 „wý)
PART – B „wý : 50
SECTION - A
• ™e[u™tk «&™tuu («&™ ™.k 1 Úte 8)™t xkqf {tk U6TZL SZL sðtƒ yt…tu.
(«íÞuf™tk 2 „wý) ftuE …ý ƒu «&™tu{tk ytk‚rhf rðfÕ… yt…ðt. [16]
SECTION - B
• ™e[u™tk «&™tuu («&™ ™.k 9 Úte 12)™t DFuIF 5|DF6[ U6TZL SZL sðtƒ yt…tu.
(«íÞuf™tk 3 „wý) ftuE …ý yuf «&™{tk ytk‚rhf rðfÕ… yt…ðtu. [12]
SECTION - C
• ™e[u™tk «&™tu («&™ ™k. 13 Úte 15)™t {tøÞt «{týu U6TZL SZL sðtƒ yt…tu.
(«íÞuf™tk 4 „wý) ftuE …ý V[S «&™tu{tk ytk‚rhf rðfÕ… yt…ðt. [12]
SECTION - D
• ™e[u™tk «&™tu («&™ ™k. 16 Úte 17)™t pS[, XMWM.
(«íÞuf™tk 5 „wý) ftuE …ý V[S «&™tu{tk ytk‚rhf rðfÕ… yt…ðtu. [10]

(PART – A)
5|[Sl8; 5|`G5+ !
;DI o 60 DLGL8 S], U]6 o 50
s!f 9 − 141 = ..........
(a) 3 + 141 (b) 141 − 3 (c) JF:TlJS ;bIF GYLP (d) l£5NL SZ6L GYLP
sZf,P ;FP VP (40, 60, 80) = PPPPPPPPP
(a) 120 (b) 180 (c) 480 (d) 240
s#fAFH]GL VFS'lT y = p(x) GF VF,[BG[ PPPPPPP JF:TlJS pS[, K[P
(a) 0 (b) 1 (c) 2 (d) 3
s$fAC]5NL p(x) = 3x2
+ 7x + 4 GF X}gIMGM U]6FSFZ PPPPPPPPPP K[P
(a) 4 (b) 3/4 (c) 7/3 (d) 4/3
s5fx = PPPPPPPPPPP V[ ;]J6" ;bIF K[P
(a)
1+ 5
2
(b) 1 (c)
1− 5
2
(d) 0
s&f GLR[ lR+DF A[ ,FS0LVM ATFJL K[P V[S SF/L VG[ ALHL ;O[N
lR+DF ATFJ[, DF5 5ZYL ;O[N ,FS0LGL ,AF. PPPPPPPP cm YFIP 22 cm
(a) 13.5 (b) 5
(c) 8.5 (d) 17 5 cm
s*f ;DLSZ6I]uD a1x + b1y + c1 = 0 VG[ a2x + b2y + c2 = 0 DF HM PPPPPPP ;AW CMI TM VggI pS[, D/[P
(a) a1b2 ≠ a2b1 (b) a1b2 = a2b1 (c) c1b2 = c2b1 (d) a1c2 = a2c1
s(f;DLSZ6 I]uD 2x + 3y = 8 5ZYL y = PPPPPPPPP
(a)
2𝑥−8
3
(b)
8−2𝑥
3
(c)
2𝑥+8
3
(d)
8−3𝑥
2
s)f HM ;DLSZ6 I]uD
𝑥+𝑦
𝑥𝑦
= 2 VG[
𝑥−𝑦
𝑥𝑦
= 6 CMI TM x = PPPPPPPPP
(a) - 1/2 (b) 2 (c) 1/4 (d) 4
s!_f2 JQF" 5C[,F DFTFv5LTF VG[ A[ 5]+LVMGL pDZGM ;ZJF/M 40 JQF" CTMP 3 JQF" 5KL T[DGL pDZGM ;ZJF/M PPPPPPPPPPYFIP
(a) 50 (b) 60 (c) 40 (d) 46
s!!fA[ VSMGL V[S ;bIFDF NXSGM VS 4 VG[ AgG[ VSMGM U]6FSFZ V[ NXSGF VSYL RFZ U6M K[P TM T[ ;bIF PPPPPPPP YFIP
(a) 44 (b) 84 (c) 48 (d) 42
s!Zf TM läWFT ;DLSZ6 x2
+ 6x + k = 0 G] V[S ALH 4 CMI TM k = PPPPPPPPPP
(a) - 40 (b) 8 (c) 20 (d) 40
s!#fHM PPPPPPPPPP CMI TM ;DLSZ6GF ALH JF:TlJS D/TF GYLP
(a) D > 0 (b) D < 0 (c) D = 0 (d) VF5[, TDFD
s!$f läWFT ;DLSZ6GF 5}6"JU" pS[,GL jIF5S ZLT ;F{ 5|YD Ul6TXF:+L PPPPPPPPPP VF5LP
(a) zLWZ VFRFI" (b) VFI"EÎ[ (c) 5FIYFUMZ;[ (d) EF:SZFRFI"V[
s!5fläWFT ;DLSZ6 5x2
- 6x + 1 = 0GF lJJ[RSGL lSDT PPPPPPPP K[P
(a) 16 (b) 4 (c) 56 (d) √56
s!&fV[S ;DFTZ z[6LGF S|lDS 5NM 2k + 1, 13, 5k – 3 CMI TM k = PPPPPPPPPPPP
(a) 9 (b) 4 (c) 17 (d) 13
s!*f Sn = 2n2
+ 3n TM d = PPPPPPPPPPPP
(a) 9 (b) - 2 (c) 13 (d) 4
s!(fV[S ;DFTZ z[6L DF8[ T3 = 8 VG[ T7 = 24 CMI TM T10 = PPPPPPPPP
(a) 28 (b) - 4 (c) 36 (d) 32
s!)fΔABC VG[ ΔPQR DF ABC↔ RPQ CMI TM ∠B G[ VG]~5 B}6M .......... K[P
(a) ∠R (b) ∠A (c) ∠P (d) ∠Q
sZ_fΔABC DF BC, AC VG[ AB GF DF5 3 : 4 : 5 GF 5|DF6DF4 ABC↔PQR CMI VG[ PR = 12 TM ΔPQRGL 5lZlDlT PPPPPPPP YFIP
(a) 27 (b) 36 (c) 12 (d) 24
sZ!fGLR[ NXF"J[, 5FIYFUMZLI, l+5]8LVM 5{SL PPPPPPPP l+5]8L ;FRL GYLP
(a) 11, 60, 61 (b) 13, 35, 37 (c) 7, 24, 25 (d) 20, 21, 29
sZZfDEF↔ XYZ ;D~5TF K[P HM XY = 5 VG[ DE = 6 CMI TYF ΔDEFG] 1F[+O/ 36 CMI TM ΔXYZ G] 1F[+O/ =PPPPPPP
(a) 75 (b) 100 (c) 25 (d) 50
sZ#f ΔABC DF ∠B SF8B]6M K[P AB = 10 TM ∠ACB =PPPPP
(a) 5 (b) 20 (c) 30 (d) 10
sZ$fΔABC DF AD DwIUF K[ TM V[5M,MlGI;GF 5|D[I D]HA PPPPPPPPP YFIP
(a) AB2
+ BC2
= AD2
(b) AB2
+ AC2
= AD2
+ BD2
(c) AB2
+ AC2
= 2(BD2
+ CD2
) (d) AB2
+ AC2
= 2(AD2
+ BD2
)

sZ5f V[S ;DAFH] l+SM6GL 5lZlDlT 12 CMI TM T[G] 1F[+O/ ......... K[P
(a) 4√3 (b) 6√3 (c) 6 (d) 4√2
sZ&f P(-3, 2) DFYL y V1F 5Z NMZ[,F ,AGM ,A5FN M CMI TM M GF IFD = PPPPPPPPPPP K[P
(a) (3/2, -1) (b) (-3, 2) (c) (3, 0) (d) (0, 2)
sZ*f HM A(x, y) G] pUDlAN]YL VTZ PPPPPPPPPPP K[P
(a) y (b) x (c) x + y (d) 𝑥2 + 𝑦2
sZ(flAN] A(6, -3)[ PPPPPPPPPPPP RZ6G] lAN] K[P
(a) 5|YD (b) £LlTI (c) T'TLI (d) RT]Y"
sZ)f (1 – cosθ)(1 + cosθ) = PPPPPPPPPP
(a) cosec2
θ (b) cos2
θ (c) sin2
θ (d) 2 – cos2
θ
s#_f tan7θ tan3θ = 1 TM θ = PPPPPPPPPP
(a) 0 (b) 9 (c) 10 (d) 18
s#!f tan2
θ = sin2
θ + cos2
θ TM θ = PPPPPPPPPPPP
(a) 0 (b) 45 (c) 60 (d) 90
s#ZfHM tanθ =
4
3
, TM
1−sin θ
1+sin θ
= PPPPPPPPPPP
(a) 3 (b) 1/3 (c) 3/4 (d) 9/16
s##fBL6DF 50L UI[,F N0FG[ XMWJF HTF -M/FJDF 30GF B}6[ YL x DLP RF,JFYL HDLGGL y DLP GLR[ 5CMRFI TM PPPPPPP
(a) 2x = √3y (b) 2x = y (c) x = y (d) x = 2y
s#$fNZLIFDF V[S TZO VFJ[,F A[ JCF6 A VG[ BGF lNJFNF0LGL 8MR 5ZYL D/TF pt;[WSM6G] DF5 VG[S|D[ 35 VG[ 50 CMI TM lNJFNF0LYLPP
(a) A VG[ B ;DFG VTZ[ K[P (b) A SZTF BG] VTZ JW] K[P (c) B SZTF AG] VTZ JW] K[P (d) SM. ;AW GYLP
s#5f1F{lTH lSZ6 VG[ §lQ8 lSZ6GF IMUYL pt;[WSM6 ZRFTM CMI TM J:T]PPPPPPPPPPP
(a) 1F{lTH lSZ6GL ;FD[ (b) 1F{lTH lSZ6GL p5Z (c) 1F{lTH lSZ6GL GLR[ (d) §lQ8 lSZ6GL GLR[
s#&f V[S Wl0IF,GF lDlG8SF8FGL ,AF. 14 cm K[P Wl0IF,GF JT]"/FSFZ RNF 5Z lDlG8SF8M 1 YL 10 ;]WL B;[ TM D/TF J'TFXG] 1F[+O/ =PPPPPPPPPPP
(a) 308 cm2
(b) 462 cm2
(c) 616 cm2
(d) 154 cm2
s#*fJT]"/GL TDFD l+HIFVMGM K[N U6 PPPPPPPPP K[P
(a) jIF; (b) l+HIF (c) ɸ (d) JT]"/G] S[g§
s#(f□ABCD ;DAFH] K[ H[ ⊙ (𝑂 , 𝑟)DF VTU"T CMI TM □ABCD............ K[P
(a) ;D,A (b) RMZ; (c) ,ARMZ; (d) VF5[, TDFD
s#)f ,W] AB GL ,AF. JT]"/GF 5ZLW SZTF RMYF EFUGL K[P TM ,W] AB V[ S[g§ VFU/ VFTZ[,F B}6FG] DF5 PPPPPPPPPPP K[P
(a) 90 (b) 30 (c) 60 (d) 45
s$_f38.5 DL2
1F[+O/JF/F JT]"/GF 5ZLWGL ,AF. = PPPPPPPP
(a) 38.5 (b) 22 (c) 3.85 (d) 2.2
s$!f10;[DL l+HIFJF/F VW"JT]"/DF VTU"T l+SM6G] DCTD 1F[+O/ PPPPPPPPP;[DL2
YFIP
(a) 200 (b) 100 (c) 50 (d) 10
s$ZfHM JT]"/GL l+HIF 10% JWFZJFDF VFJ[ TM T[G[ VG]~5 1F[+O/ PPPPPPPPPP YFIP
(a) 1.21πr2
(b) 121πr2
(c) 12.1πr2
(d) V[S 56 GCLP
s$#fAgG[ K[0[ XS] WZFJTF G/FSFZGL S],;5F8LG] 1F[+O/ XMWJFG] ;]+ PPPPPPPPPP K[P
(a) 2πr(2r + h) (b) πr(2r + l) (c) 2πr(r + l) (d) πr(r + 2h)
s$$f1 l,8Z = PPPPPPP WG ;[DLP
(a) 10 (b) 100 (c) 1000 (d) 1
s$5f G/FSFZGM jIF; VG[ pRF. VG]S|D[ 14 VG[ 10cm K[P TM G/FSFZG] S], 5'Q9O/ PPPPPPPPPPPYFIP
(a) 57 (b) 154 (c) 1540 (d) 314
s$&f10 VJ,MSGMGM DwIS 15.7 K[P V[S GJ] VJ,MSG 19 pD[ZTF GJM DwIS = PPPPPPPPPP
(a) 16 (b) 17.6 (c) 34.7 (d) 13.8
s$*fHM 𝑥 - z = 3 VG[ 𝑥 + z = 45 TM M = .............
(a) 22 (b) 24 (c) 26 (d) 23
s$(f Ul6TGF 5[5ZDF VFI]QFLGF 100 DFYL 100 U]6 D/[ T[GL ;EFJGF PPPPPPPPP K[P
(a) 0.01 (b) 0.1 (c) 1 (d) 0
s$)fW8GF k GL ;EFJGF DF8[ PPPPPPPPP
(a) 0 > P(k) > 1 (b) 0 < P(k) < 1 (c) 0 ≤ P(k) ≤ 1 (d) 0 ≥ P(k) ≥ 1
s5_f5F;M V[S JBT pKF/JFDF VFJ[ TM 5F;F 5ZGM VS VlJEFHI CMI T[GL ;EFJGF PPPPPPPP K[P
(a) 0 (b) 0.5 (c) 1/3 (d) 1/6
BEST OF LUCK

(PART – A)
5|[Sl8; 5|`G5+ Z
;DI o 60 DLGL8 S], U]6 o 50
s!f 5|tI[S RFZ S|lDS WG 5]6F"SMGM U]6FSFZ PPPPPPPPP J0[ lJEFHI K[P
(a) 16 (b) 24 (c) 48 (d) 32
sZf
18
53
G] NXFX lRgC 5KL PPPPPPPPPVSM K[P
(a) 4 (b) 3 (c) 2 (d) 5
s#f;]Z[B AC5NL p(x) = 7x – 3 G] X}gI PPPPPPPP K[P
(a) 3/7 (b) -7/3 (c) -3/7 (d) 7/3
s$f 3x2
+ 5x - 2 GF X}gIMGM ;ZJF/M PPPPPPPPPP K[P
(a) 3/5 (b) -5/3 (c) 5/3 (d) - 3/5
s5f HM ;DLSZ6 I]uD 2x + 3y = 7 VG[ 3x + 2y = 3 CMI TM x - y = PPPPPPPPP
(a) - 2 (b) 2 (c) 4 (d) - 4
s&f A[ VSMGL V[S ;bIFDF NXSGM VS 7 VG[ AgG[ VSMGM ;ZJF/M V[ V[SDGF VSYL 8 U6M K[P TM T[ ;bIF PPPPPPPP YFIP
(a) 78 (b) 17 (c) 71 (d) 70
s*f x2
- 3x + 2 = 0 G] V[S X}gI PPPPPPPPPPK[P
(a) - 2 (b) 2 (c) - 3 (d) 3
s(f läWFT ;DLSZ6 5x2
- 6x + 1 = 0GF lJJ[RSGL lSDT PPPPPPPP K[P
(a) √56 (b) 16 (c) 56 (d) 4
s)f läWFT ;DLSZ6 x2
- 4x + k = 0 G] V[S ALH 2 CMI TM k = PPPPPPPPPP
(a) - 4 (b) 2 (c) 4 (d) - 2
s!_f läWFT ;DLSZ6 x2
- 3x - k = 0 GF lJJ[RSG] D}<I 1 CMI TM k = PPPPPPPPPP
(a) - 4 (b) 2 (c) 4 (d) - 2
s!!f kx2
- 4x - 4 = 0 GF lJJ[RSG] D}<I 64 CMI TM k = PPPPPPPPPP
(a) 4 (b) 5 (c) 8 (d) 3
s!Zf AC]5NL p(x) = 3x + 5 GM VF,[B PPPPPPPPPP K[P
(a) Z[BF (b) p5ZYL B]<,M JS| (c) lSZ6 (d) GLR[YL B]<,M JS|
s!#f länFT ;DLSZ6 p(x) = x3
- x G[PPPPPPPPPPX}gIM K[P
(a) 3 (b) 2 (c) 0 (d) 1
s!$f ;DLSZ6M 2x + y – 3 = 0 VG[ 6x + 3y = 9 G[ PPPPPPP
(a) VGT pS[, K[P (b) VGgI pS[, K[P (c) pS[, GYL (d) A[ pS[, K[P
s!5f A[ ;bIFVMGM ;ZJF/M 10 VG[ TOFJT 2 K[P TM T[ 5{SL DM8L ;bIF S. m
(a) 8 (b) 6 (c) 4 (d) 2
s!&fV[S ;DFTZ z[6LDF a = 2 VG[ d = 4 CMI TM S20 = PPPPPPPPPPPP
(a) 80 (b) 78 (c) 800 (d) 600
s!*f 2k + 1, 13, 5k – 3 V[S ;DFTZ z[6LGF S|lDS 5NM CMI TM k = PPPPPPPPPPPP
(a) 9 (b) 4 (c) 17 (d) 13
s!(f V[S ;DFTZ z[6L DF8[ T25 – T20 = 15 CMI TM d = PPPPPPPPP
(a) 3 (b) 5 (c) 20 (d) 25
s!)f ΔABC DF M ϵ AB, N ϵ AC TYF MN||BC K[P TM GLR[GF 5{SL PPPPPPP ;tI GYLP
(a) AN X NC = AM X MB (b) AN X MB = AM X NC
(c) AN X AB = AM X AC (d) NC X AB = MB X AC
sZ_f ΔABC DF BC, AC VG[ AB GF DF5 3 : 4 : 5 GF 5|DF6DF4 ABC↔PQR CMI VG[ PR = 12 TM ΔPQRGL 5lZlDlT PPPPPPPP YFIP
(a) 27 (b) 36 (c) 12 (d) 24
sZ!f ΔABC DF ∠B GM l£EFHS AC G[ D DF K[N[ K[P HM AB = 12, BC = 16 VG[ AD = 9 TM AC = PPPPPPPPP YFIP
(a) 8 (b) 18 (c) 21 (d) 15
sZZf ΔABC DF AD DwIUF K[4 HM AB2
+AC2
=50 VG[ AD=3 TM BC =PPPPPPPPPP
(a) 16 (b) 8 (c) 24 (d) 4
sZ#f ΔABC DF
AB
1
=
AC
2
=
BC
3
TM ∠C = PPPPPPPPP YFIP
(a) 90 (b) 60 (c) 45 (d) 30
sZ$f RMZ;GF lJS6"GL ,AF. 5√2 K[P TM RMZ;GL AFH]G] DF5 PPPPPPPP YFIP
(a) 10 (b) 5 (c) 3√2 (d) 2√2
sZ5f P(x, y) G] pUDlAN]YL VTZ PPPPPPPPPPP K[P
(a) y (b) x (c) 𝑥 + 𝑦 (d) 𝑥2 + 𝑦2

sZ&f A(1, 2) VG[ B(3, -2) CMI TM PPPPPPPPP V[ ABGF DwIlAN]GF IFD K[P
(a) P(2, 1) (b) P(-1, 0) (c) P(2, 0) (d) P(0, 0)
sZ*f A(0,0) , B(3, 0) VG[ C(3, 4) V[ PPPPPP l+SM6GF lXZMlAN]VM K[P
(a) ,W]SM6 (b) SF8SM6 (c) ;Dl£AFH] (d) ;DAFH]
sZ(fV[S JT]"/GF jIF;G] V[S VtIlAN] A(3, -2) VG[ T[G] S[g§ pUDlAN] CMI TM ALHF VtIlANGF IFD PPPPPPPPPP K[P
(a) (-3, 2) (b) (
3
2
, 1) (c) (
3
2
, -1) (d) (-3, -2)
sZ)f HM cosθ =
4
3
, TM
cos θ−sin θ
cos θ+sin θ
= PPPPPPPPPPP
(a) 1/7 (b) 4/3 (c) -4/3 (d) 7
s#_fHM ,W]SM6 θ DF8[ cos θ = √3sin θ TM θ = PPPPPPPPPP
(a) 90 (b) 60 (c) 45 (d) 30
s#!f
𝑠𝑖𝑛4 𝜃− 𝑐𝑜𝑠4 𝜃
𝑠𝑖𝑛2 𝜃− 𝑐𝑜𝑠2 𝜃
= PPPPPPPPPPPP
(a) 1 (b) 0 (c) 3 (d) 2
s#Zf tan5θ tan4θ = 1 TM θ = PPPPPPPPPP
(a) 3 (b) 10 (c) 9 (d) 7
s##f8FJZGL pRF. VG[ T[GF 50KFIFGL ,AF.GM U]6MTZ 1:√3 K[4 TM ;}I"GF pt;[WSM6G] DF5 PPPPPPPPP K[P
(a) 30 (b) 60 (c) 45 (d) 75
s#$f30 DL8Z µRF lDGFZF 5ZYL HDLG 5ZGF V[S 5yYZGM VJX[WSM6 60 K[4 TM lDGFZFYL 5yYZG] VTZPPPPPPPPP K[P
(a) 10 (b) 10√3 (c) 30 (d) 30√3
s#5f h DL pRL .DFZTGL 8MR 5ZYL HDLG 5ZGL J:T]GF VJ;[WSM6G] DF5 θ CMI TM .DFZTYL lAN]G] VTZ PPPPPPPPP K[P
(a) hsinθ (b) htanθ (c) hcotθ (d) hcosθ
s#&f ʘ(0, 5) GL V[S HLJF ʘ(0, 3) G[ :5X[" K[P TM HLJFGL ,AF............. K[P
(a) 8 (b) 10 (c) 7 (d) 6
s#*fO S[lg§T JT]"/GF ACFZGF EFUDF VFJ[, lAN] P DFYL JT]"/G[ NMZ[, :5X"S JT]"/G[ T DF :5;[" K[P HM PT = 8 TM OP = PPPPPPPPPP
(a) 7 (b) 23 (c) 13 (d) 17
s#(f JT]"/GF ,W]J'TFXG] 1F[+O/ = PPPPPPPP
(a)
𝜋𝑟𝜃
180
(b) 𝜋𝑟 (c)
𝜋𝑟𝜃
360
(d)
𝜋𝑟2 𝜃
360
s#)f 616 1F[+O/ VG[ 60 DF5GM B]6M AGFJTL JT]"/GL ,W]RF5GL ,AF. l = PPPPPPPP
(a)
22
3
(b) 66 (c)
44
3
(d) 33
s$_f A[ JT]"/MGF 1F[+O/GM U]6MTZ 1 : 4 CMI TM T[GF 5lZWGM U]6MTZ PPPPPPP YFIP
(a) 1 : 2 (b) 1 : 4 (c) 2 : 1 (d) 4 : 1
s$!f V[S Wl0IF,GF lDlG8SF8FGL ,AF. 15 cm K[P Wl0IF,GF JT]"/FSFZ RNF 5Z lDlG8SF8M 20 lDlG8 ;]WL B;[ TM D/TF VFJ'TG] 1F[+O/........PYFIP
(a) 235.5 cm2
(b) 471 cm2
(c) 141.3 cm2
(d) 706.5 cm2
s$Zf 1 cm jIF; JF/F UM,SG] 3GO/PPPPPPPPP K[P
(a)
2
3
π (b)
1
6
π (c)
1
24
π (d)
4
3
π
s$#f G/FSFZGM jIF; VG[ pRF. VG]S|D[ 14 VG[ 10 K[4 TM S], 5'Q9O/ = PPPPPPPPPPP
(a) 44 (b) 140 (c) 748 (d) 1540
s$$f 5FR ~l5IFGF l;SSFGL S], ;5F8LG] 1F[+O/ XMWJFG] ;]+ PPPPPPPPPP K[P
(a) 2πr2
(r + h) (b) πr2
h (c)
1
3
πr2
h (d)
4
3
πr3
s$5f7 ;[DL VG[ 3 ;[DL l+HIFJF/F XS]GF VF0K[NGL pRF. 3 ;[DL CMI TM JS|;5F8LG] 1F[+O/ PPPPPPP ;[DL2
YFIP
(a) 63π (b) 35π (c) 25π (d) 50π
s$&f SM. DFlCTL DF8[ Z – M = 2.5 VG[ 𝑥 = 20 TM Z = PPPPPPPPPP
(a) 21.25 (b) 22.75 (c) 23.75 (d) 22.25
s$*f HM 𝑥 - z = 3 VG[ 𝑥 + z = 45 TM M = .............
(a) 22 (b) 24 (c) 26 (d) 23
s$(f VF5[, VFJ'lT lJTZ6GM lJR,GGL ZLT[ DwIS XMWJF DF8[ ∑fiui = -20, A = 450, C = 100 VG[ ∑fi = 20 CMI TM x = PPPPPP
(a) 430 (b) 350 (c) 455 (d) 445
s$)f HM P(A) = 0.35 TM P(A)= PPPPPPPP
(a) 0.65 (b) 0.35 (c) 0.75 (d) 0.55
s5_f RMSS; W8GFGL ;EFJGF PPPPPP K[P
(a) 0 (b) 0.5 (c) 2 (d) 1
BEST OF LUCK

(PART – A)
5|[Sl8; 5|`G5+ #
;DI o 60 DLGL8 S], U]6 o 50
s!f A[ ;DFG pRF.GF XS]GL l+HIFVMGM U]6MTZ 2 : 3 K[4 VG[ T[DGF WGO/MGM U]6MTZ PPPPPPPYFIP
(a) 4 : 6 (b) 8 : 27 (c) 3 : 2 (d) 4 : 9
sZf7 ;[DL VG[ 3 ;[DL l+HIFJF/F XS]GF VF0K[NGL pRF. 3 ;[DL CMI TM JS|;5F8LG] 1F[+O/ PPPPPPP ;[DL2
YFIP
(a) 63π (b) 35π (c) 25π (d) 50π
s#f fHM AC],S Z = 25 VG[ DwIS 𝑥 = 25, TM DwI:Y M = PPPPPPP
(a) 75 (b) 50 (c) 25 (d) 0
s$f SM. DFlCTL DF8[ M = 15 VG[ 𝑥 = 106 TM Z = PPPPPPPPPP
(a) 30 (b) 25 (c) 20 (d) 15
s5f SM. DFlCTL DF8[ Z = 16 VG[ M = 22, TM 𝑥 = PPPPPPPPPP
(a) 22 (b) 25 (c) 32 (d) 66
s&f V;SI W8GFGL ;EFJGF PPPPPP K[P
(a) 0 (b) 0.5 (c) 2 (d) 1
s*f HM P(A) = 0.47 TM P(A)= PPPPPPPP
(a) 0.20 (b) 0.53 (c) 0.50 (d) 0
s(f HM tan A =
5
12
, TM (sinA + cosA)secA = PPPPPPPPPPP
(a) 12/5 (b) 7/12 (c) 17/12 (d) -7/12
s)f △ABC DF ∠B SF8B]6M K[P ∠ACB = 45 VG[ AC = 6 CMI TM △ABCG] 1F[+O/ = PPPPPPPPPPP
(a) 18 (b) 36 (c) 9 (d) 4.5
s!_f tan20∙tan25∙tan45∙tan65∙tan70 = ...........
(a) 1 (b) - 1 (c) 0 (d) 2
s!!f HDLG 5ZGF lAN] P 5ZYL 8FJZGL 8MRGF pt;[WSM6G] DF5 45 K[P HM lAN] P VG[ 8FJZ JrR[G] VTZ a VG[ 8FJZGL pRF. h CMI TMPPPPPPPPPPP
(a) a < h (b) a > h (c) a = h (d) a = h = 0
s!Zf 3 DL ,FAL V[S lG;Z6LGM GLR[GM K[0M NLJF,YL 1.5 DL N}Z ZC[ T[ ZLT[ NLJF, 5Z 8[SJL K[P TM lG;Z6L HDLG ;FY[ PPPPPPPDF5GM B]6M AGFJX[P
(a) 30 (b) 45 (c) 60 (d) 20
s!#f 30 DL8Z µRF lDGFZF 5ZYL HDLG 5ZGF V[S 5yYZGM VJX[WSM6 60 K[4 TM lDGFZFYL 5yYZG] VTZPPPPPPPPP K[P
(a) 10 (b) 10√3 (c) 30 (d) 30√3
s!$f ;DLSZ6M 2x + y – 3 = 0 VG[ 6x + 3y = 9 G[ PPPPPPP
(a) VGT pS[, K[P (b) VGgI pS[, K[P (c) pS[, GYL (d) A[ pS[, K[P
s!5f A[ ;bIFVMGM ;ZJF/M 10 VG[ TOFJT 2 K[P TM T[ 5{SL DM8L ;bIF S. m
(a) 8 (b) 6 (c) 4 (d) 2
s!&f3 JQF" 5C[,F l5TF VG[ 5]+GL pDZGM ;ZJF/M 40 JQF" CTM4 2 JQF" 5KL T[DGL pDZGM ;ZJF/M PPPPPP JQF" YFIP
(a) 40 (b) 46 (c) 50 (d) 60
s!*f ΔABC DF B – M – C, A – N – C VG[ MN||AB. HM NC : NA = 1 : 3, CM = 4 TM BC = ............. (a) 12 (b)
16 (c) 8 (d) 0.5
s!(fΔXYZ VG[ ΔPQR GL ;UTTF PQR↔XYZ ;D~5TF K[P HM XY = 12, YZ = 8, ZX = 16, PR = 8 CMI TM PQ + QR = PPPPPPPPP
(a) 15 (b) 10 (c) 20 (d) 9
s!)f ΔPQR DF ∠P + ∠Q = ∠R, PR = 7 VG[ QR = 24 CMI TM PQ = ...........
(a) 31 (b) 25 (c) 17 (d) 20
sZ_f ΔDEF GL AFH]VM 4, 6, 8 VG[ DEF⟷PQR ;D~5TF K[4 HM △PQR GL 5lZlDlT 36 CMI TM △PQR GL ;F{YL GFGL AFH] PPPPPPPPP K[P
(a) 31 (b) 17 (c) 25 (d) 15
sZ!f ΔABC DF AD DwIUF K[4 HM AB2
+AC2
=50 VG[ AD=3 TM BC =PPPPPPPPPP
(a) 16 (b) 8 (c) 24 (d) 4
sZZf ,ARMZ;GF lJS6"GL ,AF. 13 K[P HM 5CM/F. 5 CMI TM RMZ;GL 5lZlDlT PPPPPPPP YFIP
(a) 36 (b) 48 (c) 36 (d) 34
sZ#f V[S JT]"/GF jIF;G] V[S VtIlAN] A(3, -2) VG[ T[G] S[g§ pUDlAN] CMI TM ALHF VtIlANGF IFD PPPPPPPPPP K[P
(a) (-3, 2) (b) (
3
2
, 1) (c) (
3
2
, -1) (d) (-3, -2)
sZ$f ,W]J'TFXG] 1F[+O/ = PPPPPPPP (r l+HIF VG[ RF5GL ,AF. l)
(a)
1
2
rl (b)
3
2
r2
l (c)
4
3
rl (d)
3
2
rl
sZ5f ʘ(O, 5) GF ,W]J'TFXG] 1F[+O/ 150 CMI TM JT]"/GL ,W]RF5GL ,AF. l = PPPPPPPP
(a) 15 (b) 90 (c) 60 (d) 30

sZ&f HM JT]"/GF 5lZWG] DF5 44 CMI TM JT]"/DF VTU"T RMZ;GL AFH]GL ,AF. = PPPPPPPPP
(a)
44
π
(b)
7 2
π
(c) 14√2 (d) 7√2
sZ*f A[ JT]"/MGF 1F[+O/GM U]6MTZ 1 : 4 CMI TM T[GF 5lZWGM U]6MTZ PPPPPPP YFIP
(a) 1 : 2 (b) 1 : 4 (c) 2 : 1 (d) 4 : 1
sZ(f1 cm jIF; JF/F UM,SG] 3GO/PPPPPPPPP K[P
(a)
2
3
π (b)
1
6
π (c)
1
24
π (d)
4
3
π
sZ)f 2 ;[DL l+HIFJF/F XS]GL pRF. 6 ;[DL CMI TM WGO/ PPPPPPP ;[DL3
YFIP
(a) 8π (b) 12π (c) 14π (d) 16π
s#_f,P;FPV (15, 24, 40) = PPPPPPPPPPP
(a) 120 (b) 60 (c) 240 (d) 15x24x40
s#!f√4 + √9 V[ PPPPPPPPPPPK[P
(a) V;D[I (b) ;D[I (c) VGFJ'T NXFX (d) 5}6F"S
s#Zf AC]5NL x2
- 4x + 3 GF X}gIMGM U]6FSFZ PPPPPPPPPP K[P
(a) 1 (b) 3 (c) 4 (d) - 4
s##f H[GF X}gIMGM ;ZJF/M - 3 VG[ U]6FSFZ - 4 CMI T[JL AC]5NL GLR[GF 5{SL S. CM. XS[ m
(a) x2
– 3x - 4 (b) x2
+ 3x + 4 (c) 3x2
+ 3x - 4 (d) 3x2
– 4x + 1
s#$f p(x) = 3x – 6 – x2
GM VF,[B X V1FG[ PPPPPPPPPP lAN]DF K[N[ K[P
(a) 1 (b) 0 (c) 2 (d) 3
s#5f l+WFT AC]5NL P(x) = x3
– 3x G[ PPPPPPP pS[, K[P
(a) 0 (b) 1 (c) 2 (d) 3
s#&f ;DLSZ6I]uD x - 3y = 1 VG[ 3x + y = 3 GM pS[, PPPPPPPP K[P
(a) (0, 1) (b) (1, 1) (c) (1, 0) (d) (1/3, 0)
s#*f A(5, -1) DFYL x V1F 5ZGF ,AGF ,A5FNGF IFD PPPPPPP K[P
(a) (2.5, - 0.5) (b) (- 5, 1) (c) (0, - 1) (d) (5, 0)
s#(f A(0,0) , B(3, 0) VG[ C(3, 4) V[ PPPPPP l+SM6GF lXZMlAN]VM K[P
(a) ,W]SM6 (b) SF8SM6 (c) ;Dl£AFH] (d) ;DAFH]
s#)f P(x, y) G] pUDlAN]YL VTZ PPPPPPPPPPP K[P
(a) y (b) x (c) 𝑥 + 𝑦 (d) 𝑥2 + 𝑦2
s$_f HM cotθ =
4
3
, TM
cos θ−sin θ
cos θ+sin θ
= PPPPPPPPPPP
(a) 1/7 (b) 4/3 (c) -4/3 (d) 7
s$!f H[G] V[S ALH 3 CMI T[J] x R,G] läWFT ;DLSZ6 PPPPPPPP K[P
(a) x2
+ x + 6 = 0 (b) x2
+ x - 6 = 0 (c) x2
- x - 6 = 0 (d) x2
- x + 6 = 0
s$Zf läWFT ;DLSZ6 kx2
- 7x + 3 = 0 G] V[S ALH 3 CMI TM k = PPPPPPPPPP
(a) 3 (b) 2 (c) - 2 (d) - 3
s$#f HM PPPPPPPPPP CMI TM ;DLSZ6GF ALH JF:TlJS VG[ ;DFG K[P
(a) D > 0 (b) D < 0 (c) D = 0 (d) VF5[, TDFD
s$$f läWFT ;DLSZ6 x2
- 3x - k = 0 GF lJJ[RSG] D}<I 1 CMI TM k = PPPPPPPPPP
(a) - 4 (b) 2 (c) 4 (d) - 2
s$5f x2
- x - 30 = 0 GF A[ ALH PPPPPPPPP VG[ PPPPPPPPP D/[P
(a) - 5, - 6 (b) 6, 5 (c) - 6, 5 (d) 6, - 5
s$&f HM T3=8 , T7 = 24 TM T10 =PPPPPPPPPK[P
(a) -4 (b) 28 (c) 32 (d) 36
s$*f V[S ;DFTZ z[6L DF8[ a = 2 VG[ d = 4 CMI TM S20 = PPPPPPPPP K[
(a) 600 (b) 800 (c) 78 (d) 80
s$(f ;DFTZ z[6L DF8[ T18 – T8 = 15 = PPPPPPPPPPK[P
(a) d (b) 10d (c) 26d (d) 2d
s$)fO S[lg§T JT]"/GF ACFZGF EFUDF VFJ[, lAN] P DFYL JT]"/G[ NMZ[, :5X"S JT]"/G[ Q DF :5;[" K[P HM OP = 13 VG[ PQ = 5 CMI TM JT]"/GM
jIF;PPPPPPPPPP K[P
(a) 12 (b) 24 (c) 8 (d) 16
s5_f GLR[ VF5[,L DFlCTLGM AC],SLI JU" PPPPPPPPPP K[P
JU" 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50
VFJ'lT 7 15 13 17 10
(a) 10 - 20 (b) 20 - 30 (c) 30 - 40 (d) 40 - 50
BEST OF LUCK

(PART – A)
5|[Sl8; 5|`G5+ $
;DI o 60 DLGL8 S], U]6 o 50
s!f HM U]P;FPVP (a, b) = 1, TM U]P;FPVP (a – b, a + b) = PPPPPPPPPPP
(b) 1 VYJF 2 (b) a VYJF b (c) a + b VYJF a - b (d) 4
sZfGLR[G] SI] lJWFG lAhM8G] lGtI;D K[ m
(a) ax–by = U]P;FPVP(a, b) (b) ax+by = U]P;FPVP(a, b) (c) ax–by = ,P;FPVP(a, b) (d) ax+by = ,P;FPVP(a, b)
s#fp(x) = 3x – 2 – x2
GM VF,[B xvV1FG[ PPPPPPPPPPPlEgG lAN]VMDF K[N[P
(a) 4 (b) 1 (c) 2 (d) 3
s$f AC]5NL p(x) = 3x – x4
+ x2
+ 2x3
+ 7 AC]5NLGM WFT PPPPPPPPPP K[P
(a) 3 (b) 4 (c) 2 (d) - 4
s5f l+WFT AC]5NL P(x) = x3
– x G[ PPPPPPP pS[, K[P
(a) 0 (b) 1 (c) 2 (d) 3
s&f A[ AC]5NLGM U]6FSFZ x2
+ 8x + 15 K[4 T[DFYL V[S AC]5NL (x + 3) CMI TM ALHL AC]5NL PPPPPP CMIP
(a) (x + 12) (b) (x + 5) (c) (x - 5) (d) (x - 3)
s*f ;DLSZ6I]uD a1x + b1y + c1 = 0 VG[ a2x + b2y + c2 = 0 DF HM PPPPPPP ;AW CMI TM VggI pS[, D/[P
(a) a1b2 ≠ a2b1 (b) a1b2 = a2b1 (c) c1b2 = c2b1 (d) a1c2 = a2c1
s(f 3 JQF" 5C[,F l5TF VG[ 5]+GL pDZGM ;ZJF/M 40 JQF" CTM4 2 JQF" 5KL T[DGL pDZGM ;ZJF/M PPPPPP JQF" YFIP
(a) 40 (b) 46 (c) 50 (d) 60
s)f;DLSZ6M 3x + y = 7.......(1) VG[ - x + 2y = 2.......(2) DFYL xGM ,M5 SZJF DF8[ ;DLP (2)G[ PPPPPPPP J0[ U]6J] 50[[P
(a) 1 (b) 2 (c) 3 (d) - 1
s!_f;DLSZ6 2x + y = 7 VG[ 5x - 2y = 4 GF VF,[B PPPPPPPP K[P
(a) A[ lAN]VMDF K[N[ (b) ;DFTZ Z[BFVM (c) V[S lAN]DF K[N[ (d) V[S H Z[BF D/[
s!!f HM PPPPPPPPPPPP CMI TM läWFT ;DLSZ6GF ALH ;DFG YFI K[P
(a) D < 0 (b) D > 0 (c) D = 0 (d) VF 5{SL V[S 56 GCLP
s!Zf TM läWFT ;DLSZ6 x(x + 1) - 6 = 0 GF ALH PPPPPPPPPP D/[ K[P
(a) 3, -2 (b) -3, 2 (c) 3, 2 (d) -3, -2
s!#f läWFT ;DLSZ6 kx2
- 6x + 1 = 0 GF lJJ[RSG] D}<I 0 CMI TM k = PPPPPPPPPP
(a) 3 (b) 2 (c) 9 (d) 1
s!$f PPPPPPPPPP läWFT ;DLSZ6GF pS[,GL jIF5S ZLT VF5LP
(a) zLWZ VFRFI" (b) VFI"EÎ[ (c) 5FIYFUMZ;[ (d) EF:SZFRFI"V[
s!5fläWFT ;DLSZ6GF lJJ[RSGL lSDT D[/JJFG] ;]+PPPPPPPP K[P
(a) D = b2
+ 4ac (b) D = b2
- 4ac (c) D = c2
- 4ab (d) D = a2
– 4bc
s!&fV[S ;DFTZ z[6LG] nD] 5N Tn = 3n – 1 CMI TM d = PPPPPPPPPPPP
(a) - 2 (b) 3 (c) 5 (d) 2
s!*f 4 GF TDFD S|lDS 5|FS'lTS U]l6TMYL AGTL ;DFTZ z[6L DF8[ d = PPPPPPPPPPPP
(a) 0 (b) 16 (c) 4 (d) 2
s!(fV[S ;DFTZ z[6L DF8[ 3 + 5 + 7 + 9 +.......+ 288 CMI TM n = PPPPPPPPP
(a) 12 (b) 15 (c) 17 (d) 16
s!)f ΔABC VG[ ΔPQRDF ABC↔QRP ;D~5TF K[4 HM m∠A = 50, m∠C= 30 TM m∠R = PPPPPPPPP YFIP
(a) 50 (b) 80 (c) 30 (d) 100
sZ_ f ΔABC VG[ ΔDEFDF ABC↔DEF ;D~5TF K[4 HM 3AB = 5DE VG[ DF = 9 TM AC = PPPPPPPPP
(a) 5.4 (b) 11 (c) 15 (d) 27
sZ!f△ABC DF B – M – C VG[ A – N – C, NM ǁ AB HM CN : NA = 1 : 3 VG[ CM = 4 CMI TM BC =PPPPPPPPP
(a) 12 (b) 16 (c) 8 (d) 6
sZZf△ABC GL DwIUFVM AD VG[ BE V[ GDF K[N[ K[P GDFYL 5;FZ YTL VG[ DEG[ ;DFTZ Z[BF ACG[ KDF K[N[ K[P HM EK=1.8 CMI TM AC =.....P
(a) 3.6 (b) 5.4 (c) 7.2 (d) 10.8
sZ#f ΔABC DF ∠A = ∠B + ∠C, AB = 7 VG[ BC = 25 TM ΔABC GL 5lZlDlT PPPPPPPPP YFIP
(a) 24 (b) 56 (c) 64 (d) 48
sZ$f△ABC DF ∠B SF8B]6M K[P VG[ BD J[W K[P HM AD = BD = 5 TM DC = PPPPPPPPPPP
(a) 1 (b) √5 (c) 5 (d) 2.5
sZ5f,ARMZ; ABCD DF AC = 13 VG[ CD = 5 TM ,ARMZ;GL 5lZlDlTPPPPPPPPPPPPPPYFIP
(a) 30 (b) 36 (c) 34 (d) 50
sZ&f A(1, 2) VG[ B(3, -2) CMI TM ABGF DwIlAN]GF IFD PPPPPPPPP K[P
(a) (2, 1) (b) (-1, 0) (c) (2, 0) (d) (0, 0)

sZ*flAN] A(-4, -3) VG[ B(6, a) JrR[G] VTZ 10 CMI TM a = PPPPPPPPPP
(a) 4 (b) 3 (c) - 3 (d) - 4
sZ(fA(3.0)4 B(0,3) VG[ C(3, 3) lXZMlAN]JF/F △ABC G] 1F[+O/ = PPPPPPPPPP
(a) 9 (b) 4.5 (c) 6 (d) 3
sZ)f △ABC DF ∠B SF8B]6M K[P VG[ cos B =
1
2
TM cosec A = ...........
(a) 1/2 (b) √3 (c) 2/√3 (d) 2
s#_fHM ,W]SM6 θ DF8[ cos θ = sin θ TM 2 tan2
θ + sin2
θ + 1 = PPPPPPPPPP
(a) 5/2 (b) 7/4 (c) 5/4 (d) 7/2
s#!f
cos (90−A) sin (90−A)
tan (90−A)
= PPPPPPPPPPPP
(a) sin2
A (b) cos2
A (c) sin A (d) 1
s#ZfHM tanθ =
4
3
, TM
1−sin θ
1+sin θ
= PPPPPPPPPPP
(a) 3 (b) 1/3 (c) 3/4 (d) 9/16
s##f3 DL8Z ,FAL V[S lG;Z6LGM GLR[GM K[0M NLJF,YL 1.5 DLP N]Z ZC[ T[ ZLT[ NLJF, 5Z 8[SJL K[4 TM lG;Z6L HDLG ;FY[ PPPPPPP DF5GM B}6M AGFJX[P
(a) 30 (b) 60 (c) 45 (d) 120
s#$fV[S 8FJZGL pRF. 50√3 DL K[4 T[GF T/LV[YL 50 DLP N]Z VFJ[, lAN]V[YL T[GL 8MRGF pt;[WSM6G] DF5PPPPPPPPP YFIP
(a) 45 (b) 60 (c) 15 (d) 30
s#5fV[S YFE,FGF 50KFIFGL ,AF. YFE,FGF pRF. H[8,L YFI tIFZ[ ;}I"GF pt;[WSM6G] DF5PPPPPPPPPPYFIP
(a) 60 (b) 30 (c) 75 (d) 45
s#&fO S[gN=JF/F JT]"/GL ACFZGF lAN] P DFYL NMZ[, :5;"SM PA VG[ PB K[P HM ∠OPB = 30 TM ∠AOB = PPPPPPPP
(a) 30 (b) 60 (c) 90 (d) 120
s#*fO S[lg§T JT]"/GF ACFZGF EFUDF VFJ[, lAN] P DFYL JT]"/G[ NMZ[, :5X"S JT]"/G[ Q DF :5;[" K[P HM OP = 13 VG[ PQ = 5 CMI TM JT]"/GM
jIF;PPPPPPPPPP K[P
(a) 12 (b) 24 (c) 8 (d) 16
s#(fʘ(0, 5) GL V[S HLJF ʘ(0, 3) G[ :5X[" K[P TM HLJFGL ,AF............. K[P
(a) 8 (b) 10 (c) 7 (d) 6
s#)f JT]"/MG[ A[ lEgG lAN]VMDDF K[NTL Z[BFG[ PPPPPPPSC[ K[P
(a) l+HIF (b) jIF; (c) RF5 (d) K[NLSF
s$_fJT]"/GL U]Z]]RF5 GL ,AF. l = PPPPPPPP
(a) 2𝜋𝑟 −
𝜋𝑟𝜃
180
(b) 𝜋𝑟 (c) 2𝜋𝑟 −
𝜋𝑟𝜃
360
(d)
𝜋𝑟2 𝜃
360
s$!f8.4;[DL l+HIFJF/F JT]"/GM 5lZW = PPPPPPPPP
(a) 39.6 (b) 26.4 (c) 52.8 (d) 66
s$Zf ʘ(0, 10) GF ,W]J'TFXG] 1F[+O/ 150 CMI TM T[G[ VG]~5 RF5GL ,AF.PPPPPPPP YFIP
(a) 30 (b) 60 (c) 90 (d) 15
s$#fA[ JT]"/MGF 1F[+O/GM U]6MTZ 1 : 4 CMI TM T[GF 5lZWGM U]6MTZ PPPPPPP YFIP
(a) 1 : 2 (b) 1 : 4 (c) 2 : 1 (d) 4 : 1
s$$fB]<,F G/FSFZGL JS|;5F8LG] 1F[+O/ XMWJFG] ;]+ PPPPPPPPPP K[P
(a) πr2
(b) 2πrh (c)
1
3
πr2
h (d)
4
3
πr3
s$5f10 ;[DL jIF; VG[ 17 ;[DL lTI"S pRF.JF/F XS]GL JS|;5F8LG] 1F[+O/ PPPPPPP ;[DL2
YFIP
(a) 85π (b) 170π (c) 95π (d) 88π
s$&fHM 𝑥 = 36 VG[ M = 26 TM Z = .............
(a) 6 (b) 5 (c) 4 (d) 3
s$*f48 VJ,MSGMGF VFJ'lT lJTZ6 DF8[ DwIS 70 , ∑ fi = 43 + f VG[ A = 66 CMI TM B]8TM VFJ'lT f = PPPPPPPPP
(a) 27 (b) 23 (c) 7 (d) 5
s$(f15 VJ,MSGMGM DwIS 16 K[P NZ[S VJ,MSGDF 2 pD[ZL NZ[SG[ 3 J0[ EFUJFDF VFJ[ TM GJM DwIS = PPPPPPPPPP
(a) 6 (b) 7 (c) 5 (d) 4
s$)fVXSI W8GFGL ;EFJGF PPPPPPPPP CMIP
(a) 0 (b) 1.2 (c) 0.2 (d) 1
s5_fV[S ;DTM, 5F;FG[ V[S JBT O[SJFDF VFJ[ VG[ VI]uD VS D/[ T[GL ;EFJGF PPPPPPPP K[P
(a) 1/4 (b) 1/6 (c) 1/3 (d) 1/2
BEST OF LUCK

(PART – A)
5|[Sl8; 5|`G5+ 5
;DI o 60 DLGL8 S], U]6 o 50
s!f U]P;FPVP (x, y) = 1 TM U]P;FPVP (x - y, x + y) = PPPPPPPPP
(a) 1 VYJF 2 (b) x VYJF y (c) 4 (d) x - y VYJF x + y
sZf I]lS,0GF EFUFSFZ 5}J[" 5|D[IDF WG 5}6F"SM a VG[ b DF8[ VGGI 5]6F"SM q VG[ r V[JF D/[ S[ H[YL a = bq + r HIF PPPPPPPPPPP YFIP
(a) 0 < r < b (b) 0 ≤ r ≤ b (c) 0 < r ≤ b (d) 0 ≤ r < b
s#fl+WFT AC5NL p(a) = a3
– a G[ PPPPPPPPPP X}gIM K[P
(a) 3 (b) 1 (c) 2 (d) 0
s$f HM 4 V[ läWFT ;DLSZ6 x2
+ ax - 8 = 0 G] V[S ALH CMI TM a = PPPPPPP
(a) - 2 (b) 4 (c) 2 (d) - 4
s5f l+WFT AC5NL p(x)GF X}gIMGM U]6FSFZPPPPPPPPPPPP K[P
(a)
− 𝑏
𝑎
(b)
𝑐
𝑎
(c)
− 𝑑
𝑏
(d) V[S 56 GCLP
s&f AFH]DF VF5[,L VFS'lT 5ZYL y = p(x)GF JF:TlJS X}gIMGL ;bIF PPPPPPP K[P
(a) 4 (b) 3
(c) 2 (d) 1
s*f ;DLSZ6 I]uD ax + 2y = 7 VG[ 2x + 3y = 8 G[ VGgI V[S X}gI DF8[ a ≠ PPPPPPPPPP
(a)
−3
4
(b)
4
3
(c)
− 4
3
(d)
3
4
s(fHM ;DLSZ6 I]uD x + y + 1 = 0 VG[ 3x + 3y + 2 = 0 GM pS[, U6 PPPPPPPPP K[P
(a) {(1, -2)} (b) {(3, 1)} (c) BF,L U6 (d) VGT U6
s)f HM ;DLSZ6 I]uD
2
𝑥
+
3
𝑦
= 7 VG[
3
𝑥
+
2
𝑦
= 13 CMI TM
1
𝑥
−
1
𝑦
= PPPPPPPPP
(a) 20 (b) 6 (c) 30 (d) 5
s!_fA[ VSGL V[S ;bIFGM V[SDGM VS x VG[ NXSGM VS 2x CMI TM T[ ;bIF PPPPPPPP K[P
(a) 21x (b) 2x2
(c) 3x (d) 12x
s!!f läWFT ;DLSZ6 (x – 7)2
– 16 = 0 GF ALHPPPPPPPPPPK[P
(a) 3 VG[ 4 (b) - 3 VG[ - 11 (c) 3 VG[ 11 (d) - 3 VG[ - 6
s!Zf läWFT ;DLSZ6 x(2x – 1) – 5 = 0 G[ ax2
+ bx + c = 0 ;FY[ ;ZBFJTF a = PPPPPPPPPP
(a) - 1 (b) 2 (c) 5 (d) 1
s!#f läWFT ;DLSZ6 x2
- 10x + (2k – 1) = 0 GF lJJ[RSG] D}<I 40 CMI TM k = PPPPPPPPPP
(a) 10 (b) 8 (c) 7 (d) 15
s!$f läWFT ;DLSZ6 x2
+ 18x + 81 = 0 GF ALH PPPPPPPPP K[P
(a) jI:T (b) V5]6F"S (c) ;DFG (d) lJZMWL
s!5fV[S 5|FS'lTS ;bIF VG[ T[GF jI:TGM ;ZJF/M
5
2
K[P TM T[ ;bIF PPPPPPPP K[P
(a) 2 (b) 5 (c) 3 (d) 4
s!&fV[S ;DFTZ z[6L
3
2
,
7
2
,
11
2
,
15
2
… …. DF8[ d = PPPPPPPPPPPP
(a) 2 (b) - 2 (c)
3
2
(d)
1
2
s!*f V[S ;DFTZ z[6LGF +6 S|lDS 5NMGM ;ZJF/M 48 K[P V[DFGF 5C[,F VG[ K[<,F 5NGM U]6FSFZ 252 K[P TM d = PPPPPPPPPPPP
(a) 2 (b) 3 (c) 16 (d) 4
s!(fV[S ;DFTZ z[6L 200, 196, 192 ...... G] PPPPPPPPP D] 5N 0 K[P
(a) 101 (b) 51 (c) 50 (d) 40
s!)f ΔABC VG[ ΔXYZDF ;UTTF ABC↔ZXY ;D~5TF K[P HM AB = 12, BC = 8, CA = 10 VG[ ZX = 10 TM XY + YZ = ......
(a) 15 (b) 16 (c) 18 (d) 20
sZ_f△ABC DF A – P – B VG[ A – Q – C, PQ ǁ BC HM PQ = 5, AP = 4, AB = 12 CMI TM BC =PPPPPPPPP
(a) 20 (b) 15 (c) 9.6 (d) 5
sZ!fΔPQR DF PM VG[ RN J[W K[P HM PQ = 12, QR = 15 VG[ PM = 9.6 CMI TM RN = PPPPPPP
(a) 6 (b) 7.2 (c) 6.4 (d) 12
sZZf ΔABC VG[ ΔXYZ DF ;UTTF ABC↔ XYZ ;D~5TF K[P HM
AB
4
=
XY
5
TM
BC
YZ
=PPPPPPP
(a) 9/5 (b) 5/9 (c) 4/5 (d) 5/4
sZ#f VFS'lTDF AC = PPPPPPPPPPs HIF BD = CD, AE = EC VG[ G DwIlAN] K[Pf
(a) 2EK (b) 3EK (c) 4EK (d) 6EK
sZ$f ΔXYZ DF ∠X ∶ ∠Y ∶ ∠Z = 1 : 2 : 3 VG[ XY = 15 TM YZ =PPPPPPPP
(a) 7.5 (b) 8 (c) 17 (d) 15
3
2
x
y
y = p(x)
A
B C
E
K
D
G

sZ5f ΔABC DF ∠A SF8B]6M VG[ AD J[W K[P TM BD∙DC =PPPPP
(a) AB2
(b) BC2
(c) AD2
(d) AC2
sZ&f P(2, -3), VG[ Q(7, 9) JrR[G] VTZPQ = PPPPPPPPPPP K[P
(a) 11 (b) 13 (c) 61 (d) 117
sZ*f pUDlAN] S[g§ CMI T[JF V[S JT]"/GF jIF;G] V[S VtIlAN] A(3, -2) CMI TM ALHF VtIlAN]GF IFD PPPPPPPP K[P
(a) (-3, 2) (b) (
3
2
, 1) (c) (
3
2
, 2) (d) (-3, - 2)
sZ(f □ABCD GF lXZMlAN]VM A(1, 3), B(4, 3), C(4, 5) VG[ D(1, 5) CMI TM □ABCD PPPPPPPPPP K[P
(a) RMZ; (b) ;DAFH] RT]P (c) ,ARMZ; (d) ;D,A RT]P
sZ)fSM.S θ sHIF 0 < θ < 90f DF8[ GLR[GF 5{SL PPPPPPPPPP ;tI K[P
(a) cosθ > 1 (b) cosecθ < 1 (c) tanθ < 0 (d) secθ > 1
s#_f sin2
1 + sin2
3 + sin2
87 + sin2
89 = PPPPPPPPPPPP
(a) 0 (b) 1 (c) 2 (d) 4
s#!f cosecθ =
2
3
TM θ = PPPPPPPPPPPP
(a) 30 (b) 60 (c) 90 (d) 45
s#ZfHM 2A V[ ,W]SM6G] DF5 CMI VG[ sec 2A = cosec (A – 42) TM A = PPPPPPPPPPP
(a) 43 (b) 44 (c) 42.5 (d) 44.5
s##f18 DL8Z VG[ 12 DL8Z pRF.JF/F A[ :TEGL 8MR JrR[ V[S TFZ AFW[, K[P TFZ ;Dl1FlTH Z[BF ;FY[ 30 DF5GM B}6M AGFJ[ TM TFZGL ,AF. PPPPPPPPP DLP
YFIP
(a) 10 (b) 12 (c) 8 (d) 4
s#$fHDLG 5ZGF lAN] A YL lNJFNF0LGL 8MRGF pt;[WSM6G] DF5 70 K[P A YL DSFGG] VTZ x VG[ DSFGGL pRF. y TM PPPPPPPPPPPPPPPP
(a) x = y (b) x < y (c) x > y (d) x = 2y
s#5f O S[lg§T JT]"/GF ACFZGF EFUDF VFJ[, lAN] P DFYL JT]"/G[ NMZ[, :5X"S JT]"/G[ Q DF :5;[" K[P HM OP = 13 VG[ PQ = 5 CMI TM JT]"/GM
jIF;PPPPPPPPPP K[P
(a) 12 (b) 24 (c) 8 (d) 16
s#&f SF8SM6 l+SM6 ABCDF ∠B = 90P HM AC = 20 VG[ ∠C = 30 CMI TM BC = PPPPPPPPPPP
(a) 17.3 (b) 40 (c) 10 (d) 20
s#*f ΔABC DF8[ a = 5, b = 12 VG[ c = 13 K[P ΔABC GL AFH]VMG[ VNZYL :5X"TF JT]"/GL l+HIFPPPPPPPPP K[P
(a) 2 (b) 6 (c) 6.5 (d) 5.5
s#(f ʘ(O, 5) JT]"/ RMZ;GL AWL AFH]VMG[ :5;[" K[ TM RMZ;GL 5lZlDlT ............ K[P
(a) 5 (b) 10 (c) 20 (d) 40
s#)f 5Z:5Z ACFZYL :5;"TF A[ JT]"/MG[ ;FDFgI :5X"S PPPPPPPPPNMZL XSFIP
(a) 3 (b) 2 (c) 4 (d) 1
s$_fNM0JFGL CZLOF. DF8[ AGFJ[, JT]"/FSFZ 5YGM VNZGM 5lZ3 ACFZGF 5lZ3 SZTF 44 DLP VMKM K[P TM 5YGL 5CM/F. PPPPPPDLP YFIP
(a) 7 (b) 3.5 (c) 11 (d) 22
s$!f r l+HIFJF/F JT]"/GL l ,AF.GL RF5 J0[ ZRFTF J'TFXG] 1F[+O/ PPPPPPPPP YFIP
(a)
1
2
𝑟𝑙 (b)
3
2
𝑟2
𝑙 (c)
1
3
𝑟𝑙 (d) 𝜋𝑟𝑙
s$Zf10;[DL l+HIFJF/F JT]"/DF VTU"T RMZ;GF lJS6"GL ,AF. PPPPPPPPPP ;[DLP YFIP
(a) 10 (b) 10√2 (c) 20√2 (d) 20
s$#f 5|Rl,T ;S[TMDF XS]GF VF0K[NG] 3GO/ XMWJFG] ;]+ PPPPPPPPPP K[P
(a) πr[r1
2
+ r2
2
+ r1 ∙ r2] (b)
1
3
πh[r1
2
+ r2
2
+ r1 ∙ r2] (c)
1
3
πh[r1 + r2 + r1 ∙ r2] (d) πh[r1 + r2 + r1 ∙ r2]
s$$f7 ;[DL VG[ 3 ;[DL l+HIFJF/F XS]GF VF0K[NGL pRF. 3 ;[DL K[P TM T[G] JS|5'Q9O/PPPPPPP PP ;[DL2
P
(a) 50𝜋 (b) 25𝜋 (c) 35𝜋 (d) 63𝜋
s$5fHM UM,SGL l+HIFVMGM U]6MTZ 3 : 2 CMI TM T[GF 5'Q9O/MGM U]6MTZ PPPPPPP YFIP
(a) 3 : 2 (b) 2 : 3 (c) 4 : 9 (d) 9 : 4
s$&f HM 𝑥 = 21.44 VG[ Z = 19.13 TM M = .............
(a) 20.67 (b) 20.10 (c) 19.67 (d) 21.10
s$*f VFJ'lT lJTZ6 DF8[ 𝑥 = A +
∑ fidi
𝑛
DF CMI TM di = PPPPPPPPP
(a) xi − A (b) x − A (c) ∑ fi − A (d) ∑ fixi − n
s$(fVFJ'lT lJTZ6DF AC],SLI JU" 70 – 85 CMI TM l = PPPPPPPPPP
(a) 15 (b) 77.5 (c) 70 (d) 85
s$)fAGL 5]ZS W8GF A K[4 HM P(A) + P(A) = PPPPPPPPPP K[P
(a) 0 (b) 0.60 (c) 1 (d) 0.75
s5_f,L5 JQF"DF 53 XlGJFZ CMI T[GL ;EFJGF PPPPPPPP K[P
(a) 0 (b) 0.5 (c) 1/7 (d) 2/7

PART – B (50 U]6)
sU]6EFZ ov &f
GMW o VF 5|SZ6 DFYL Z lJS<5M VG[ Z U]6GM V[S NFB,M 5]KFI K[P
s!f JU"D}/ XMWMP 7 + 𝟒𝟖
pS[, o VCL 7 + 48 = 7 + 4X12 = 7 + 2 12
WFZM S[ 7 + 2 12 G] JU"D}/ x + y K[P
VF56[ x + y = 7 VG[ xy = 12 YFI T[JF x VG[ y D[/JTF x = 4 VG[ y = 3 D/[P
∴ 7 + 2 12 = x + y = 4 + 3 = 𝟐 + 𝟑
:J5|ItG[ SZM o
JU"D}/ XMWMP s!f 9 + 2 14
sZf 6 + 4 2
sZf I]S,L0GL EFUlJlWYL U]P ;FP VP XMWMP (120, 23)
pS[, o 120 = 23 x 5 + 5
23 = 5 x 4 + 3
5 = 3 x 1 + 2
3 = 2 x 1 + 1
2 = 1 x 2 + 0
∴ VlTD X}gI[TZ X[QF 1 K[P
∴ U]P ;FP VP(120, 23) = 1
:J5|ItG[ SZM o
I]S,L0GL EFUlJlWYL U]P ;FP VP XMWMP s!f 210 VG[ 55
sZf 765 VG[ 65
s#f ;D[I ;bIF
𝟏𝟐
𝟔𝟐𝟓
XFT NXFX :J~5DF K[ S[ GCL T[ H6FJM VG[ T[G] XFT NXFX :J~5 CMI TM
D[/JMP
pS[, o
12
625
=
12
54
VCL K[N 625 = 54 VG[ U]P ;FP VP(12, 625) = 1
∴
12
625
NXFX :J~5 XFT NXFX :J~5 K[P
12
625
=
12 x 24
54x 24 =
12 x 16
10000
=
192
10000
= 0.0192
5|SZ6 01 o I]S,L0GL EFUlJlW VG[ JF:TlJS ;bIFVM

:J5|ItG[ SZM o
GLR[GL ;D[I ;bIFVM XFT NXFX :J~5DF K[ S[ GCL T[ H6FJM VG[ T[G] XFT NXFX :J~5 CMI TM
D[/JMP
s!f
13
125
sZf
55
150
s$f 0.090909........ = 0.𝟎𝟗 G[
𝐩
𝐪
:J~5[ NXF"JMP
pS[, o NXFX VlEjIlST 0.09 V[ VGT VG[ VFJ'T CMJFYL ;D[I K[P
WFZM S[ x = 0.09...............(1)
∴ x =
09.09
100
∴ 100x = 09. 09
∴ 100x = 09 + 0. 09 ..........(2)
(1) GL SLDT (2) DF D]STF
100x = 09 + x
∴ 100x – x = 9
∴ 99x = 9
∴ x =
9
99
=
𝟏
𝟏𝟏
:J5|ItG[ SZM o s!f 3.456789123...... sZf 0.02222......
s5f ;FN] ~5 VF5M o
𝟒
𝟔−𝟐 𝟓
+
𝟏
𝟓+𝟐 𝟔
=
4
1− 5
+
1
2+ 3
=
4
1− 5
+
1
2+ 3
=
4
1− 5
×
1+ 5
1+ 5
+
1
2+ 3
×
2− 3
2− 3
=
4 1+ 5
5
2
− 12
+
1 2− 3
3
2
− 2
2
=
4 1+ 5
5− 1
+
1 2− 3
3− 2
=
4 1+ 5
4
+
1 2− 3
1
= 1 + 𝟓 + 𝟐 - 𝟑
:J5|ItG[ SZM o s!f
1
3+ 2
+
1
4+ 3
+ 2
sZf
1
6− 5
−
3
5− 2
−
4
6+ 2

sU]6EFZ ov &f
GMW ov VF 5|SZ6 DFYL $ lJS<5M VG[ Z U]6GM V[S NFB,M 5]KFI K[P
s!f x – 2 V[ p(x) = x3 – 2x2 GM VJIJ K[ T[D ;FlAT SZMP
pS[, o VF5[, AC]5NLGM V[S VJIJ (x – 2) CMI TM P(2) = 0 YFIP
CJ[4 P(2) = (2)3 – 2(2)2
= 8 - 2(4)
= 8 – 8
= 0
VFD4 P(2) = 0 YJFYL (x – 2) V[p(x) = x3 – 2x2 GM VJIJ K[P
:J5|ItG[ SZM o s!f x + 2 V[ p(x) = 2x3 – 4x2 + 5x + 42 GM VJIJ K[ T[D ;FlAT SZMP
sZf x – 2 V[ p(x) = x3 – 4x2 + 5x - 2 GM VJIJ K[ T[D ;FlAT SZMP
sZf H[GF X}gIMGM ;ZJF/M 2 VG[ U]6FSFZ -3 CMI T[JL l£WFT AC]5NL XMWMP
pS[, o WFZM S[ α VG[ β V[ l£WFT AC]5NLGF A[ X}gIM CMI TM PPPPP
α + β = 2 VG[ αβ = - 3 YFIP
∴ α VG[ β X}gIM WZFJTL l£WFT AC]5NL D]HAPPP
x2 – (α + β)x + αβ
∴ x2 – (2)x + (-3)
∴ x2 – 2x – 3
:J5|ItG[ SZM o s!f H[GF X}gIMGM ;ZJF/M -3 VG[ U]6FSFZ -4 CMI T[JL l£WFT AC]5NL XMWMP
sZf H[GF X}gIMGM ;ZJF/M
8
5
VG[ U]6FSFZ
3
5
CMI T[JL l£WFT AC]5NL XMWMP
s#f A[ AC]5NLVMGM U]6FSFZ 6x3 + 29x2 + 44x + 21 CMI VG[ T[ 5{SLGL V[S AC]5NL 3x + 7
CMI TM ALHL XMWMP
pS[, o VCL P(x) = EFHI AC]5NL = 6x3 + 29x2 + 44x + 21
VG[ S(x) = EFHS AC]5NL = 3x + 7
T[YL X[QF 5|D[I 5|lJlW D]HAPP
5|SZ6 02 o AC]5NLVM

2x2 + 5x + 3
3x + 7 6x3 + 29x2 + 44x + 21
6x3 + 14x2
15x2 + 44x
15x2 + 35x
9x + 21
9x + 21
00
EFUO/ AC]5NL Q(x) = 2x2 + 5x + 3 VG[ X[QF 00 K[P
:J5|ItG[ SZM o s!f A[ AC]5NLVMGM U]6FSFZ 6x2 + 8x + 12 CMI VG[ T[ 5{SLGL V[S AC]5NL x+2
CMI TM ALHL XMWMP
sZf A[ AC]5NLVMGM U]6FSFZ x3 – 3x2 + 5x – 3 CMI VG[ T[ 5{SLGL V[S AC]5NL
x2–2 CMI TM ALHL XMWMP
s#f -19x – 2x2 + x3 + 20 G[ -6x + x2 + 5 J0[ EFUM VG[ X[QF D[/JMP
sU]6EFZ ov &f
GMW ov VF 5|SZ6 DFYL # S[ $ lJS<5M VG[ Z S[ # U]6GM V[S NFB,M 5]KFI K[P
s!f ,M5GL ZLT[ pS[,MP 9x – 4y = 14, 7x – 3y = 11
pS[, o 9x – 4y = 14 .............. (1)
7x – 3y = 11 .............. (2)
yGF ;CU]6SM ;DFG SZJF ;DLP (1)G[ 3 J0[ VG[ ;DLP (2)G[ 4 J0[ U]6L AFNAFSL SZTFP
27x – 12y = 42
28x – 12y = 44
- x = - 2
∴ x = 2
xGL lSDT ;DLP (2)DF D]STFPPPP
7(2) – 3y = 11
= 14 – 3y = 11
= – 3y = 11 – 14 = – 3y = –3 ∴ y = 1
∴ pS[, U6 {(2, 1)}
5|SZ6 03 o läR, ;]Z[B ;DLSZ6 I]uD

:J5|ItG[ SZM o s!f x + 3y = 6, 2x – y = 5
sZf 4x – 3y = 8, 6x – y =
29
3
sZf RMS0L U]6FSFZGL ZLT[ pS[,MP 2x – 5y = 4, 3x – 8y = 5
pS[, o 5|DFl6T :J~5DF UM9JTF 2x – 5y – 4 = 0
3x – 8y – 5 = 0
5|DFl6T :J~5 ;FY[ ;ZBFJTFPPPPP
a1 = 2, b1 = - 5, c1 = - 4
a2 = 3, b2 = - 8, c2 = - 5
;CU]6SMGL UM9J6L SZTFPPP
=
x
b1 c1
b2 c2
=
y
c1 a1
c2 a2
=
1
a1 b1
a2 b2
=
x
−5 −4
−8 −5
=
y
−4 2
−5 3
=
1
2 −5
3 −8
=
x
25−32
=
y
−12−(−10)
=
1
−16 − (−15)
=
x
25−32
=
y
−12+10
=
1
−16+15
=
x
−7
=
y
− 2
=
1
−1
∴
x
−7
=
1
−1
or
y
− 2
=
1
−1
∴ x =
−7
−1
or y =
−2
−1
∴ x = 7 or y = 2
∴ pS[, U6 {(7, 2)}
:J5|ItG[ SZM o s!f 3x – 4y = 17, 4x – 5y = 21
sZf 4x + 6y = 11, 5x – 8y = 6
s#f VFN[XGL ZLT[ pS[,MP 2x + 3y = 11, 2x – y = - 1
pS[, o VCL4 2x – y = - 1
∴ y = 2x + 1 ...........(1)
yGL lSDT ;DLP2x + 3y = 11 DF D]STF PPPPP
2x + 3(2x + 1) = 11
∴ 2x + 6x + 3 = 11
∴ 8x = 11 – 3
∴ 8x = 8 ∴ x = 1
;DLP(1) DF x = 1 GL lSDT D]STFPPP
y = 2(1) + 1
∴ y = 2 + 1 ∴ y = 3
∴ pS[, U6 {(1, 3)}

:J5|ItG[ SZM o s!f 5x – 3y = 1, 2x + 5y = 19
sZf x + 11y = 1, 8x + 13y = 2
s$f A[ ;bIFVMGM ;ZJF/M 70 K[P VG[ T[DGM WG TOFJT 6 CMI TM T[ ;bIF XMWMP
pS[, o WFZM S[ T[ A[ ;bIFVM 5{SL DM8L ;bIF = x VG[ GFGL ;bIF = y
∴ x + y = 70 ...........(1)
∴ x – y = 6 ...............(2)
;DLP (1) VG[ ;DLP (2)GM ;ZJF/M SZTFP
x + y = 70
x – y = 6
2x = 76
∴ x =
76
2
∴ x = 38
;DLP(1) DF x = 38 D]STFPPP
38 + y = 70
∴ y = 70 – 38 ∴ y = 32
∴ T[ A[ ;bIFVM 38 VG[ 32 K[P
:J5|ItG[ SZM o s!f V[S 8[A,GL lSDT V[S B]Z;LGL lSDT SZTF +6 U6L K[4 RFZ B]Z;L VG[ V[S
8[A,GL S], lSDT ~P 2100 K[P TM V[S 8[A, VG[ V[S B]Z;LGL lSDT XMWMP
sZf A[ VSMGL V[S ;bIFGF NXSGM VS V[SDGF VS SZTF 3 U6M K[P ;bIFGF VSMGL
VN,FAN,L SZTF D/TL GJL ;bIF D}/ ;bIF SZTF 54 H[8,L GFGL CMI TM T[
;bIF XMWMP
sU]6EFZ ov (f
GMW ov VF 5|SZ6 DFYL 5 lJS<5M VG[ # U]6GM V[S NFB,M 5]KFI K[P
s!f jIF5S ;]+GL ZLT[ pS[,MP x2 – 5x – 1 = 0
pS[, o 5|DFl6T :J~5 ;FY[ ;ZBFJTFPPPPP a = 1, b = - 5, c = - 1
lJJ[RS D = b2 – 4ac
= (–5)2 – 4(1)( –1)
= 25 + 4 = 29 > 0
VCL4 D > 0 CMJFYL l£WFT ;DLPGF AgG[ ALH lEgG VG[ JF:TlJS D/[PPP
α =
−𝑏 + 𝐷
2𝑎
β =
−𝑏 − 𝐷
2𝑎
α =
−(−5) + 29
2(1)
β =
−(−5) − 29
2(1)
5|SZ6 04 o läWFT ;DLSZ6

α =
5 + 29
2
β =
5 − 29
2
VF5[, l£WFT ;DLPGF AgG[ ALH α =
𝟓 + 𝟐𝟗
𝟐
VG[ β =
𝟓 − 𝟐𝟗
𝟐
K[P
:J5|ItG[ SZM o s!f x2 + 2x + 2 = 0
sZf y2 + 10y +6 = 0
sZf VJIJGL ZLT[ pS[,MP 4x2 + 4x = 15
pS[, o 4x2 + 4x = 15
∴ 4x2 + 4x – 15 = 0
∴ 4x2 + 10x – 6x – 15 = 0
∴ 2x(2x + 5) – 3(2x + 5) = 0
∴ (2x – 3)(2x + 5) = 0
∴ (2x – 3) = 0 VYJF (2x + 5) = 0
∴ 2x = 3 VYJF 2x = – 5
∴ x =
3
2
VYJF x =
– 5
2
VFD4 pS[,
𝟑
𝟐
VG[
– 𝟓
𝟐
K[P
:J5|ItG[ SZM o s!f x –
1
x
=
45
14
sZf
x2− 1
x2+ 1
=
7
9
s#f A[ S|lDS I]uD 5|FS'lTS ;bIFVMGF JUM"GM U]6FSFZ 244 CMI TM T[ ;bIFVM XMWMP
pS[, o WFZM S[ V[S I]uD 5|FS'lTS ;bIF = x
T[YL ALHL S|lDS I]uD 5|FS'lTS ;bIF x + 2 YFIP
A[ S|lDS I]uD 5|FS'lTS ;bIFVMGF JUM"GM U]6FSFZ 244 K[P
∴ x2 +(x + 2)2 = 244
∴ x2 +x2 + 4x + 4 = 244
∴ 2x2 + 4x = 244 – 4
∴ 2x2 + 4x = 240
∴ x2 + 2x = 120
∴ x2 + 2x – 120 = 0
∴ (x + 12)(x – 10) = 0
∴ (x + 12) = 0 or (x – 10) = 0
∴ x = - 12 or x = 10
5ZT] x ≠ - 12 T[YL x = 10
∴ V[S I]uD 5|FS'lTS ;bIF 10 VG[ ALHL S|lDS I]uD 5|FS'lTS ;bIF 12 K[P

:J5|ItG[ SZM o s!f V[S SF8SM6 l+SM6DF AFH]VMGL ,AF. x, x + 3 VG[ x + 6 sHIF xϵNf K[P HM
l+SM6G] 1F[+O/ 54 ;[DLP CMI TM l+SM6GL 5lZlDlT XMWMP
sZf HM BF0GF EFJDF 5|lT lSU|F ~P 5 GM W8F0M YFI TM ~P 150DF 1 lSU|F BF0 JW]
D/[ K[P TM BF0GM EFJ XMWMP
s#f V[S J[5FZL O],NFGL ~P 96DF J[R[ TM T[G[ T[GL 50TZ lSDT H[8,F 8SF GOM D/[ K[P
TM O],NFGLGL 50TZ lSDT VG[ GOFGL 8SFJFZL XMWMP
sU]6EFZ ov 5f
GMW ov VF 5|SZ6 DFYL # lJS<5M VG[ Z U]6GM V[S NFB,M 5]KFI K[P
s!f ;DFTZ z[6L 5, 11, 17, 23 ........ G] 101D] 5N XMWMP
pS[, o VCL4 a = 5, d = 11 – 5 = 6, VG[ n = 101 DF8[ T101 = m
;DFTZ z[6LG] nD] 5N Tn = a + (n – 1)d
∴ T101 = 5 + (101 – 1)6
∴ T101 = 5 + (100)6
∴ T101 = 5 + 600
∴ T101 = 605
∴ ;DFTZ z[6LG] 101D] 5N 605 K[P
:J5|ItG[ SZM o s!f ;DFTZ z[6L 2, 7, 12, 17 ........ G] nD] 5N XMWMP
sZf ;DFTZ z[6L 8, 11, 14, 17 ........ G] S[8,FD] 5N 272 YFIP
s#f ;DFTZ z[6L -5, -15, -25, ........... G] 22D] 5N XMWMP
sZf ;DFTZ z[6L 2, 6, 10, 14 ........ GF 20 5NMGM ;ZJF/M XMWMP
pS[, o VCL4 a = 2, d = 6 – 2 = 4, VG[ n = 20 DF8[ S20 = m
;DFTZ z[6LGF n5N GM ;ZJF/M Sn =
n
2
[2a + (n – 1)d]
∴ S20 =
20
2
[2(2) + (20 – 1)4]
∴ S20 = 10[4 + (19)4]
∴ S20 = 10[4 + 76]
∴ S20 = 10[80]
∴ S20 = 800
∴ ;DFTZ z[6LGF 20 5N GM ;ZJF/M 800 K[P
:J5|ItG[ SZM o s!f ;DFTZ z[6L 1, 1.5, 2, 2.5 ........ GF 16 5NMGM ;ZJF/M XMWMP
sZf ;ZJF/M SZM o 7 + 12 + 17 + 22 + ....... + 102
s#f ;DFTZ z[6L DF8[ Tn = 6n + 5 CMI TM Sn XMWMP
5|SZ6 05 o ;DFTZ z[6L

sU]6EFZ ov 5 VYJF (f
GMW ov VF 5|SZ6 DFYL # lJS<5M VG[ 5 U]6GM V[S 5|D[I sHM 5|D[I G 5]KFI TM Z U]6GM NFB,Mf5]KFI K[P
s!f 5|D[Io s&P!f ;5|DF6TFG] D}/E]T 5|D[I ,BM VG[ ;FlAT SZMP
HM SM. Z[BF l+SM6GL V[S AFH]G[ ;DFTZ CMI VG[ AFSLGL A[ AFH]VMG[ lEgG lAN]VMDF K[N[ TM VF
Z[BF J0[ T[ A[ AFH]VMG] ;DFG U]6MTZDF lJEFHG YFI K[P
51F o ΔABC GF ;DT,DF NMZ[,L Z[BF l∥BC VG[
l V[ AB VG[ ACG[ VG]S|D[ P VG[ QDF K[N[ K[P
;FwI o
AP
PB
=
AQ
QC
;FlATL o WFZM S[ QM ⊥ AB VG[PN ⊥ ACP
BQ VG[ CP NMZMP M∈ AB VG[ N∈ AC
ΔAPQ VG[ ΔPBQ DF
l+SM6G] 1F[+O/ =
1
2
x 5FIM x J[W
∴ ΔAPQ G] 1F[+O/ =
1
2
AP × QM..........................(1)
∴ ΔPBQ G] 1F[+O/ =
1
2
PB × QM..........................(2)
5lZ6FD (1) VG[ (2)GM EFUFSFZ SZTFPPP
ΔAPQ G] 1F[+O/
1
2
AP × QM
ΔPBQ G] 1F[+O/
1
2
PB × QM
ΔAPQ G] 1F[+O/ AP
ΔPBQ G] 1F[+O/ PB .....................................(3)
ΔAPQ VG[ ΔPCQ DF
∴ ΔAPQ G] 1F[+O/ =
1
2
AQ × PN...........................(4)
∴ ΔPCQ G] 1F[+O/ =
1
2
QC × PN...........................(5)
5lZ6FD (4) VG[ (5)GM EFUFSFZ SZTFPPP
ΔAPQ G] 1F[+O/
1
2
AQ × PN
ΔPCQ G] 1F[+O/
1
2
QC × PN
ΔAPQ G] 1F[+O/ AQ
ΔPCQ G] 1F[+O/ QC .....................................(6)
5|SZ6 06 o l+SM6GL ;D~5TF
A
M
P Q
N
B C
l
=
=
=
=

5ZT] ΔPBQ G] 1F[+O/ = ΔPCQ G] 1F[+O/ ......(7) (∵ΔPBQ VG[ ΔPCQ ;DFG 5FIF PQ 5Z
VG[ A[ ;DFTZ Z[BFGL JrR[ VFJ[,F K[P)
5lZ6FD (3) (6) VG[ (7) 5ZYLPPP
𝐀𝐏
𝐏𝐁
=
𝐀𝐐
𝐐𝐂
sZf 5|D[Io s&P*f ;FlAT SZM S[ A[ ;D~5 l+SM6GF 1F[+O/ T[DGL VG]~5 AFH]VMGF JU"GF
;5|DF6DF CMI K[P
51F o ΔABC VG[ ΔABC JrR[ ;UTTF ABC↔PQR ;D~5TF K[P
;FwI o
ABC
PQR
=
AB2
PQ2 =
BC2
QR2 =
AC2
PR2
;FlATL o ΔABC DF J[W AL VG[ ΔPQR DF J[W PN NMZMP
VCL4 ABC↔PQR ;D~5TF K[P
∴
AB
PQ
=
BC
QR
.......................(1)
ΔABL VG[ ΔPQN DFPPPP
∠B ≅ ∠Q ......... sSF8B]6Mf
∠ALB ≅ ∠PNQ
∴ ABL ↔ PQN ;D~5TF K[P
∴
AB
PQ
=
AL
PN
...............................(2)
5lZ6FD (1) VG[ (2)5Z YL PP.
AB
PQ
=
BC
QR
..................(3)
CJ[ l+SM6G] 1F[+O/ =
1
2
x 5FIM x J[W D]HAPPP
ΔABC G] 1F[+O/
1
2
BC × AL BC AL
ΔPQR G] 1F[+O/
1
2
QR × PN QR PN ..........(4)
5lZ6FD (3) VG[ (4)5Z YL PP
ΔABC G] 1F[+O/ BC BC
ΔPQR G] 1F[+O/ QR QR
∴
ABC
PQR
=
BC2
QR2
∴
ABC
PQR
=
AB2
PQ2 =
BC2
QR2 =
AC2
PR2
= = X
= X
A
B L C
P
Q N R

s#f ΔABC VG[ ΔDEFDF ABC↔DEF ;D~5TF K[4 HM 3AB = 5DE VG[ DF = 9 TM AC
XMWMP
pS[, o VCL4 3AB = 5DE ∴
AB
DE
=
5
3
VG[ ABC↔DEF ;D~5TF K[P
∴
AB
DE
=
AC
DF
∴
5
3
=
AC
9
∴ AC =
5×9
3
= 5 x 3
∴ 𝐀𝐂 = 𝟏𝟓
:J5|ItG[ SZM o s!f ΔXYZ VG[ ΔPQRDF XYZ↔QPR ;D~5TF K[4 HM m∠X + m∠P=
130 VG[ ZX = YZ CMI TM ΔPQRGF B]6FVMGF DF5 XMWMP
sZf ΔABC VG[ ΔPQRDF ABC↔PQR ;D~5TF K[4 HM AB = 3, BC = 4,
AC = 5 VG[ QR = 6 TM PQ VG[ PR XMWMP
s#f ΔPQR VG[ ΔXYZDF PQR↔ZYX ;D~5TF K[4 HM PQ : ZY = 5 : 3
VG[ PR = 10 TM XZ XMWMP
s$f ΔABC VG[ ΔPQRDF ABC↔QPR ;D~5TF K[4 ΔABCGL 5lZlDlT 15 VG[ΔPQRGL
5lZlDlT 27 K[P HM BC = 8, QR = 9 CMI TM PR VG[ AC XMWMP
pS[, o VCL4 ABC↔QPR ;D~5TF K[4
ΔABC GL 5lZlDlT BC AC
ΔPQR GL 5lZlDlT PR QR
∴
15
27
=
8
PR
=
AC
9
∴
15
27
=
8
PR
OR
15
27
=
AC
9
∴ PR =
8 ×27
15
OR AC =
15 ×9
27
∴ PR = 14.4 OR AC = 5
:J5|ItG[ SZM o s!f ΔABC VG[ ΔXYZDF ABC↔XZY ;D~5TF K[4 ΔABCGL 5lZlDlT 45
VG[ ΔXYZGL 5lZlDlT 30 K[P HM AB = 21 CMI TM XZ XMWMP
sZf ΔABC VG[ ΔPQRDF ABC↔PQR ;D~5TF K[4 HM AB = 3, BC = 4,
AC = 5 VG[ QR = 6 CMI TM PQ VG[ PR XMWMP
s#f ΔPQR DF ∠PGM l£EFHS 𝐐𝐑 G[ SDF K[N[ K[P PQ : PR = 5 : 4 VG[ SR = 5.6 ;[DL CMI TM
QR XMWMP
pS[, o VCL4 PQ : PR = 5 : 4 K[P
∴
PQ
PR
=
5
4
ΔPQR DF ∠PGM l£EFHS QR G[ SDF K[N[ K[P
∴
PQ
PR
=
QS
SR
= =

∴
5
4
=
QS
5.6
∴ QS =
5 ×5.6
4
∴ 𝐐𝐒 = 𝟕
:J5|ItG[ SZM o s!f ΔXYZ DF ∠XGM l£EFHS YZ G[ MDF K[N[ K[P XY = 6, YM = 4.2 VG[ XZ =
8 ;[DL CMI TM YZ XMWMP
sZf ΔABC DF ∠AGM l£EFHS BC G[ DDF K[N[ K[P AB : AC = 3 : 4 VG[ BC =
14 ;[DL CMI TM BD XMWMP
sU]6EFZ ov 5 VYJF (f
GMW ov VF 5|SZ6 DFYL # lJS<5M VG[ 5 U]6GM V[S 5|D[I sHM 5|D[I G 5]KFI TM Z U]6GM NFB,Mf5]KFI K[P
s!f 5|D[Io sp55|D[I !f ;FALT SZM S[ SF8SM6 l+SM6GF S6" 5Z J[W NMZ[, CMI TM J[WGL ,AF.
V[ J[WYL AGTF S6" GF Z[BFB0MGL ,AF.GM U]6MTZ DwIS K[ VG[ NZ[S
AFH]GL ,AF. V[ S6"GL ,AF. VG[ S6"GF T[ AFH]VMG[ ;,uG Z[BFB0GL
,AF.GM U]6MTZ DwIS K[P
51F o ΔABC DF ∠A SF8B]6M K[P HM AD ⊥ BC VG[ D∈ BC K[P
;FwI o (1) AB2 = BDxBC
(2) AC2 = DCxBC
(3) AD2 = BDxDC
;FlATL o 5|D[I 7.1 DF ΔABC VG[ ΔADB DFPPPP
ABC↔DBA ;D~5TF K[P...................(1)
∴
AB
DB
=
BC
AB
∴ AB2 = BD x BC
ΔABC VG[ ΔADC DFPPPP
ABC↔DAC ;D~5TF K[P....................(2)
∴
BC
AC
=
AC
DC
∴ AC2 = DC x BC
(1) VG[ (2) 5ZYLPPP
DAC↔DBA ;D~5TF K[P
∴
DA
DB
=
DC
DA
∴ AD2 = BD x DC
5|SZ6 07 o ;D~5TF VG[ 5FIYFUMZ;G] 5|D[I
B
D
A C

sZf 5|D[Io s*PZf ;FALT SZM S[ SF8SM6 l+SM6DF S6"GL ,AF.GM JU" AFSLGL A[ AFH]VMGL
,AF.VMGF JUM"GF ;ZJF/F AZFAZ CMI K[Ps5FIYFUMZ;G] 5|D[If
51F o ΔABC DF ∠A SF8B]6M K[P HM AD ⊥ BC VG[ D∈ BC K[P
;FwI o AB2 + AC2 = BC2
;FlATL o p55|D[I 1 D]HAPPPPP
AB2 = BD x BC ............(1)
AC2 = DC x BC .............(2)
5lZ6FD (1) VG[ (2) GM ;ZJF/M SZTFPPP
AB2 + AC2 = BD x BC + DC x BC
= BC(BD + DC)
= BC x BC (∵ BD + DC = BC)
AB2 + AC2 = BC2
s#f 5|D[Io s*P#f s5FIYFUMZ;G] 5|lT5|D[If ;FALT SZM S[ HM ΔABCDF BC² = AB² + AC² TM
∠A SF8B]6M K[P
51F o ΔABC DF AB2 + AC2 = BC2
;FwI o ∠A SF8B]6M K[P
;FlATL o WFZM S[ OX SM. lSZ6 K[P
VF56[ V[J] OY NMZLV[ S[ H[YL OY ⊥ OX YFIP
M∈ OY ,.V[ S[ H[YL OM = AC .................(1)
N∈ OX ,.V[ S[ H[YL ON = AB .................(2)
ΔOMN SF8SM6 l+SM6 K[P VG[ ∠M SF8B]6M K[P
∴ MN S6" K[P DF8[ 5FIYFUMZ;GF lGID D]HA
MN2 = OM2 + ON2 = AC2 + AB2
5ZT] AB2 + AC2 = BC2
∴ MN2 = BC2
∴ MN = BC .............................(3)
5lZ6FD (1), (2) VG[ (3)5ZYLPP
ABC↔ONM ;D~5TF K[P (∵ AFAFAF ;ZT)
∴ ∠O ≅ ∠A
∴ ∠A SF8B]6M K[P
B
D
A C
B
A C
M
O N
Y
X

s$f ΔABC DF m∠𝐁 = 90 VG[ BM J[W K[4 HM AM – CM = 7 VG[ AB2 –BC2 = 175 TM AC
XMWMP
pS[, o VCL U]6MtTZ DwISGF 5|D[I D]HAPP
AB2 = AMxAC VG[ BC2 = CMxAC
BC2 = AMxAC – CMxAC
∴ AB2 – BC2 = AC(AM – CM)
∴ 175 = AC(7)
∴ AC =
175
7
∴ AC = 25
:J5|ItG[ SZM o s!f ΔPQR DF m∠Q = 90 VG[ QM J[W K[4 HM PM = 8, RM = 12 TM PQ,
QR VG[ QM XMWMP
sZf ΔABC DF m∠B = 90 VG[ BD J[W K[4 HM AD = 9, CD = 27 TM AB
XMWMP
s5f ΔABC DF m∠𝐁 = 90 K[4 HM AB = 8, AC = 17 TM BC XMWMP
pS[, o VCL 5FIYFUMZ;GF lGID D]HA PP
AC2 = AB2 + BC2
∴ (17)2 = (8)2 + BC2
∴ 289 = 64 + BC2
∴ BC2 = 289 – 64
∴ BC2 = 225
∴ BC = 15
:J5|ItG[ SZM o s!f ΔABC DF ∠A = ∠B + ∠C K[P HM AB = 7 VG[ BC = 25 TM ΔABC
5lZlDlT XMWMP
sZf □PQRS ,ARMZ; K[P HM PQ + QR = 7 VG[ PR + QS = 10 TM
□PQRSG] 1F[+O/ XMWMP
s#f 6.5 DLP ,AF.GL lG;Z6L lNJF,G[ 6 DLP pRF.V[ :5X[" K[P TM HDLG 5ZGF
lG;Z6LGF K[0FYL lNJF, ;]WLG] VTZ XMWMP
sU]6EFZ ov &f
s!f ;FlAT SZM S[ P(2, -1), Q(1, -4) VG[ R(3, 2) ;DZ[B lAN]VM K[P
pS[, o VCL4VTZ ;]+ D]HAPPP
PQ2 = (X1 – X2)2 + (Y1 – Y2)2
= (2 – 1)2 + (-1 – (-4))2
A
M
B C
A
B C
5|SZ6 08 o IFDE}lDlT

= (1)2 + (3)2
= 1 + 9
PQ2 = 10 ∴ PQ = √10
QR2 = (1 – 3)2 + (–4 – 2)2
= (–2)2 + (–6)2
= 4 + 36
QR2 = 40 ∴ QR = 2√10
PR2 = (2 – 3)2 + (–1 – 2)2
= (–1)2 + (–3)2
= 1 + 9
PR2 = 10 ∴ PR = √10
VCL4 PQ + PR = QR
DF8[ P(2, -1), Q(1, -4) VG[ R(3, 2) ;DZ[B lAN]VM K[P
:J5|ItG[ SZM o s!f ;FlAT SZM S[ lAN]VM A(3, 2), B(5, 8) VG[ C(-6, 5) V[ SF8SM6 l+SM6GF
lXZMlAN]VM K[P
sZf A(5, 2), B(3, 4) VG[ C(X, Y) ;DZ[B lAN]VM K[P HM BA=BC TM (X, Y)
GF IFD XMWMP
s#f P(3, 2) VG[ Q(7, k) DF8[ PQ = 5 CMI TM k XMWMP
sZf A(3, 5), B(2, -1) VG[ C(-5, 6) TM ΔABC G] 1F[+O/ XMWMP
pS[, o ΔABC G] 1F[+O/ =
1
2
X1 Y2 − Y3 + X2 Y3 − Y1 + X3(Y1 − Y2)
=
1
2
3 −1 − 6 + 2 6 − 5 + (−5)(5 − (−1))
=
1
2
3 −7 + 2 1 − 5(6)
=
1
2
−21 + 2 − 30
=
1
2
−49
=
49
2
= 24.5
:J5|ItG[ SZM o s!f A(4, 2), B(3, 9) VG[ C(10, 10) lXZMlAN]VM JF/F K[P ΔABCG] 1F[+O/
XMWMP
sZf (9, a), (6, 7) VG[ (2, 3) l+SM6GF lXZMlAN]VM K[P HM l+SM6G] 1F[+O/ 10
CMI TM a XMWMP
s#f A VG[ B GF IFD VG]S|D[ (3, -6) VG[ (-2, -1) K[P ABG] A TZOYL 3 : 2
U]6MTZDF lJEFHG SZTF lAN]GF IFD XMWMP

sU]6EFZ ov &f
s!f HM sec4A = cosec(A–20), HIF 4A V[ ,W]SM6G] DF5 CMI TM AGL lSDT XMWMP
pS[, o VCL4 sec4A = cosec(A–20)
∴ sec4A = sec(90 – (A–20)
∴ sec4A = sec(90 – A + 20)
∴ sec4A = sec(110 – A)
∴ 4A = 110 – A
∴ 4A + A = 110
∴ 5A = 110
∴ A =
110
5
= 22
:J5|ItG[ SZM o s!f
sin 18
cos 72
+ 3(tan10 tan30 tan45 tan50 tan80) GL lSDT XMWMP
sZf HM 0<θ<90 VG[ secθ = cosec60 TM 2cos2θ-1 GL lSDT XMWMP
s#f lSDT XMWMP 3cos230 + sec230 + 2cos0 + 3sin90 – tan260
sZf
𝟏
𝟏+𝐬𝐢𝐧𝛉
+
𝟏
𝟏−𝐬𝐢𝐧𝛉
= 2sec2θ ;FlAT SZMP
pS[, o VCL4 0FAFP =
1
1+sin θ
+
1
1−sin θ
=
1−sin θ +(1+sin θ)
1+sin θ (1−sin θ)
=
2
(1−sin θ2)
=
2
cos θ2
= 2sec2θ
= HAFP
:J5|ItG[ SZM o s!f
1−sin θ
1+sin θ
= secθ – tanθ ;FlAT SZMP
sZf ;FlAT SZM S[ (sinθ + cosecθ 2 + cosθ + secθ 2 =
7+tan2θ+cot2θ
s#f ;FlAT SZM S[
sin 2θ
1+cos θ
+
sin 2θ
1−cos θ
= 2
5|SZ6 09 o l+SM6lDlT

sU]6EFZ ov &f
GMW ov VF 5|SZ6 DFYL # lJS<5M VG[ # U]6GM V[S NFB,M 5]KFI K[P
s!f ;}I"GF pt;[WSM6G] DF5 30˙ CMI tIFZ[ 8FJZGF 50KFIFGL ,AF.DF 27 DL K[P ;}I"GF
pt;[WSM6G] DF5 60˙ CMI tIFZ[ 8FJZGF 50KFIFGL ,AF. XMWMP
pS[, o VCL4 VFS'lTDF AB 8FJZ K[P ;}I"GM pt;[WSM6 30 CMI tIFZ[ 8FJZGM 50KFIM CB K[P
∴ ∠ACB = 30, ∠B = 90 VG[ CB = 27 DLP
;}I"GM pt;[WSM6 60 CMI tIFZ[ 8FJZGM 50KFIM DB K[P
SF8SM6 ΔACB DF4 tan30 =
AB
CB
∴
1
3
=
AB
27
∴ AB =
27
3
=
9×3
3
=
9× 3× 3
3
= 9 3
CJ[4 SF8SM6 ΔADB DF4 tan60 =
AB
DB
∴ 3 =
9 3
DB
∴ DB =
9 3
3
∴ DB = 9 DLP
;}I"GF pt;[WSM6G] DF5 60˙ CMI tIFZ[ 8FJZGF 50KFIFGL ,AF. 𝟗 DLP YFIP
sZf V[S 8FJZ 5Z h ,AF>GM V[S wJH N0 VFJ[,M K[P HM wJH N0GL 8MR VG[ T/LIFGF
pt;[WSM6 HDLG 5Z GF SM. lAN]V[ YL DF5TF VG]S|D 𝛂 VG[ 𝛃 DF,]D 50[ K[P TM 8FJZ GL
µRF>
𝐡 𝐭𝐚𝐧 𝛃
𝐭𝐚𝐧 𝛂−𝐭𝐚𝐧 𝛃
K[ T[D ;FALT SZMP HIF 𝛂 > 𝛽
pS[, o VCL4 VFS'lTDF BC 8FJZ K[P VG[ AC wJHN0 K[P
A wJHN0GL 8MR TYF B wJHN0G] T/LI] K[P
∴ wJHN0GL ,AF. AB = h DLP VG[ ∠C = 90
D HDLG 5ZG] V[J] lAN] K[P S[ HIFYL wJHN0GL 8MR
VG[ T/LIFGF pt;[WSM6 VG]S|D[ α VG[ β K[P
∴ ∠ADC = α VG[ ∠BDC = β
SF8SM6 ΔADB DF4 tanα =
AC
DC
...........(1)
VG[ SF8SM6 ΔBDC DF4 tanβ =
BC
DC
...........(2)
5lZ6FD (1) VG[ (2) GL AFNAFSL SZTFPP
5|SZ6 10 o VTZ VG[ pRF.
A
B D
C
3060
27
A
B
DC

tanα – tanβ =
AC
DC
−
BC
DC
=
AC−BC
DC
=
AB
DC
=
h
DC
∴ tanα – tanβ =
h
DC
∴ DC =
h
tan α – tan β
..............(3)
CJ[4 tanβ =
BC
DC
∴ BC = DC tanβ
∴ BC =
h
tan α – tan β
tanβ (∵ DC =
h
tan α – tan β
)
∴ BC =
htan β
tan α – tan β
T[YL 8FJZ GL µRF>
𝐡 𝐭𝐚𝐧 𝛃
𝐭𝐚𝐧 𝛂−𝐭𝐚𝐧 𝛃
K[P
:J5|ItG[ SZM o s!f h ,AF.GF 5], GLR[ BL6DF V[S DSFG VFJ[,] K[P 5], GF AgG[ K[0[ YL JFZFOZTL
BL6DF HMTF DSFGGL KTGF VJX[WSM6 α VG [β DF,]D 50[ K[P TM 5],YL
DSFGGL µ0F. h(tan α tan β)
tan α + tan β
;FALT SZMP
sZf 8FJZGF T/LV[YL 5;FZ YTL Z[BF 5Z 8FJZGL V[S H AFH]V[ VFJ[,F A[ lAN]VMGF
8FJZYL VTZ VG]S|D[ a VG[ b K[P HM VF lAN]VMYL 8FJZGL 8MRGF pt;[WSM6
V[SALHFGF SM8LSM6 CMI TM 8FJZGL µRF> ab K[ T[D ;FALT SZMP
s#f h H[8,L lXZM,A pRF. 5Z p0L ZC[,F H[8 lJDFGDFYL HDLG 5Z ZC[,L A[ 8[gSGF
VJX[WSM6 𝛼 VG[ 𝛽(𝛼>𝛽) DF,]D 50[ K[P TM A[ 8[gS JrR[G] VTZ
h(tan α −tan β)
tan α tan β
;FALT SZMP
s$f V[S AC]DF/L .DFZTGL 40DL pRF.V[ VFJ[,L AFZLDFYL HMTF 8FJZGL 8MRGM
pt;[WSM6 45 H6FI K[P VG[ .DFZTGF T/LI[YL HMTF 8FJZGL 8MRGM pt;[WSM6
60 K[P TM 8FJZGL pRF. XMWMP
sU]6EFZ ov &f
GMW ov VF 5|SZ6 DFYL Z lJS<5M VG[ $ U]6GM V[S 5|D[IqNFB,M 5]KFI K[P
s!f 5|D[Io s!!P!f ;FALT SZM S[ JT]"/GM :5;"S :5;"lAN] DFYL 5;FZ YTL l+HIFG[ ;DT,DF
,A CMI K[P
51F o Z[BF l V[ ⊙ O,r G[ A lAN]V[ :5X[" K[P
;FwI o OA ⊥ 𝑙
;FlATL o WFZM S[ P∈ 𝑙 VG[ P ≠ A
5|SZ6 11 o JT]"/
O
A P
l

HM P lAN] V[ ⊙ O,r GF VNZGF EFUDF CMI TM Z[BF 𝑙 V[ JT]"/GL K[NLSF CMI4 :5X"S G
CMIP 5ZT] 𝑙 JT]"/GM :5;"S K[P
∴ lAN] P JT]"/GF VNZGF EFUDF GYLP p5ZFT P ≠ A
∴ lAN] P JT]"/GF ACFZGF EFUDF K[P
∴ PO > OA (OA JT]"/GL l+HIF K[P)
T[YL A l;JFIGF NZ[S P∈ 𝑙 DF8[ lAN] P, V;DTF PO > OAG] ;DFWFG SZ[ K[P
∴ 𝐎𝐀 ⊥ 𝒍
sZf 5|D[Io s!!P#f ;FALT SZM S[ JT]"/GF ACFZGF EFUDF VFJ[,F lAN]DFYL JT]"/G[ NMZ[,F
:5;"SMGL ,AF. ;DFG CMI K[P
51F o ⊙ O,r GF ACFZGF EFUDF VFJ[,F lAN] P DFYL
JT]"/G[ NMZ[,F :5;"SMGF :5X"lAN] X VG[ Y K[P
;FwI o PX = PY
;FlATL o PO ZRMP
ΔOPX VG[ΔOPY DF
∠OXP ≅ ∠OYP SF8B]6F K[P)
OP ≅ OP VG[ OX ≅ OY (l+HIF)
∴ OPX ↔ OPY ;D~5TF K[P(∵ SFSAF)
∴ PX ≅ PY
s#f JT]"/GF S[g§ O DFYL 5;FZ YTL V[S Z[BF JT]"/GF V[S :5;"SG[ QDF K[N[ K[P :5;"SG]
:5;"lAN] P K[4 HM JT]"/GL l+HIF 5 VG[ OQ = 13 CMI TM PQ XMWMP
pS[, o ΔOPQ DF m∠P = 90 K[4 T[YL 5FIYFUMZ;GF lGID D]HA PP
OQ2 = OP2 + PQ2
∴ (13)2 = (5)2 + PQ2
∴ 169 = 25 + PQ2
∴ PQ2= 169 – 25
∴ PQ2= 144
∴ PQ = 12
:J5|ItG[ SZM o s!f ΔABC DF m∠B = 90, V[S JT]"/ ΔABC GL AWLH AFH]VMG[ :5;[" K[P HM AB
= 5, BC = 12 CMI TM JT]"/GM l+HIF XMWMP
sZf A[ ;DS[lg§I JT]"/MGL l+HIFVM 26 VG[ 24 K[4 DM8L l+HIF JF/F JT]"/GL HLJF
GFGL l+HIFJF/F JT]"/G[ :5;[" K[P TM HLJFGL ,AF. XMWMP
s$f ΔABC DF m∠𝐁 = 90, V[S JT]"/ ΔABC GL AWLH AFH]VMG[ :5;[" K[P HM AB = 5, BC =
12 CMI TM JT]"/GM l+HIF XMWMP
pS[, o ΔABC DF m∠B = 90 K[4 T[YL 5FIYFUMZ;GF lGID D]HA PP
AC2 = AB2 + BC2
= (5)2 + (12)2
= 25 + 144
Y
P
O
X

∴ AC2 = 169
∴ AC = 13
l+HIF =
𝐴𝐵+𝐵𝐶−𝐴𝐶
2
=
5+12−13
2
=
4
2
= 2
:J5|ItG[ SZM o s!f ⊙ O, 24 GF ACFZGF EFUDF VFJ[, lAN] K[P p DFYL JT]"/G[ NMZ[,F :5;"SM
JT]"/G[ QDF :5;[" K[4 HM OP = 25 TM QP XMWMP
sZf V[S JT]"/ □ABCD GL RFZ[ AFH]VMG[ V[S JT]"/ :5;[" K[P HM AB = 5, CD =
6, BC = 8 CMI TM AD XMWMP
sU]6EFZ ov 5f
GMW ov VF 5|SZ6 DFYL 5 U]6GL V[S ZRGF 5]KFI K[P
s!f SM. 56 ,AF.GM AB NMZL T[G] 2:3:4 U]6MTZDF lJEFHG SZMP
51F o AB VF5[,M K[P
S'tI o AB G[ +6 Z[BFB0MDF lJEFHLT SZLX] H[YL T[DGL ,AF.GM U]6MTZ 2:3:4 YFIP
ZRGFGF D]NNF o AB ;FY[ ,3]SM6 AGFJ[ T[J] AX ZRMP

1
9
AC SZTF VMKL4 VG]S]/ l+HIF ,. AG[ S[gN= ,. V[S RF5 ZRMP H[ A1DF K[N[ K[P
 T[ H ZLT[ A1 S[gN= ,. V[8,F H DF5GL l+HIF YL A2DF K[NT] RF5 ,UFJM S[ H[YL
 A–A1–A2 YFIP
5|SZ6 12 o ZRGF
A
A1
A2
A3
A4
A5
A6
A7
A8
A9
BP Q
X
C

 T[H D]HA VFU/ JWTF VG]S|D[ 9 lAN]VM V[JF ZRM S[ H[YL AA1 = A1A2 = A2A3 =
............... = A7A8
 A9B ZRMP
 A5 DFYL A9BG[ ;DFTZ Z[BF NMZM H[ AB G[ QDF K[N[ K[P VG[ A2 DFYL A5QG[ ;DFTZ
Z[BF NMZM H[ AB G[ PDF K[N[ K[P
 VF lAN] P VG[ Q V[ ABG] 2:3:4 GF U]6MTZDF lJEFHG SZ[ K[P
:J5|ItG[ SZM o s!f 6.5 ;[DL ,AF.GM PQ NMZL T[G] 4:7 U]6MTZDF lJEFHG SZMP VG[ AgG[ EFUGF
DF5 H6FJMP
sZf VF5[, Z[BFB0GF +6 V[S~5 EFUDF lJEFHG SZMP
sZf ⊙ O, 4 NMZMP OA = 10 YFI T[J] lAN] A DFYL JT]"/G[ :5;"SM NMZMP
ZRGFGF D]NNF o ⊙ O, 4 ZRMP
 OA = 10 YFI T[J] lAN] A ZRMP
 OAGM ,Al£EFHS ZRL M D[/JMP
 ⊙ M, OM ZRMP H[ O S[g§LT JT]"/G[ X VG[ Y DF K[N[ K[P
 AX VG[ AX NMZM H[ DFU[, :5;"SM K[P
:J5|ItG[ SZM o s!f AU/LGL DNNYL JT]"/ NMZL T[GF ACFZGF EFUDF VFJ[,F lAN] 5ZYL JT]"/G[
:5;"SMGL V[S HM0 NMZMP
sZf AB =10 ;[DL YFI T[JM AB NMZMP ⊙ A, 3 VG[ ⊙ B, 4 NMZMP NZ[S
JT]"/G[ ALHF JT]"/GF S[g§DFYL :5;"SM NMZMP
s#f ⊙ O, 4 ZRMP VG[ JT]"/GF ACFZGF lAN] A DFYL JT]"/G[ V[JF :5;"SM NMZM S[
H[DGF JrR[GF B]6FG] DF5 60 YFIP
X
AO
M
Y

sU]6EFZ ov (f
GMW ov VF 5|SZ6 DFYL $ S[ 5 lJS<5M VG[ # S[ $ U]6GM V[S NFB,M 5]KFI K[P
s!f 42 ;[DLP l+HIF WZFJTF JT]"/GF S[g§ VFU/ JT]"/GM ,W]J'T 120 DF5GM B}6M VFTZ[ TM
,W]J'TFXG] 1F[+O/ XMWMP
pS[, o ⊙ O, 42) VG[ m∠AOB = 120 K[P
,W]J'TFXG] 1F[+O/ =
πr2θ
360
=
22
7
×
42×42×120
360
=
22
7
× 14 × 42
= 22 × 84
= 1848 RMP ;[DLP
:J5|ItG[ SZM o s!f OA VG[ OB V[S JT]"/GL 5Z:5Z ,A l+HIFVM K[P HM ,W]J'TFXGL 5lZlDlT 20
;[DLP CMI TM T[G[ VG]~5 ,W]J'TB0G] 1F[+O/ XMWMP
sZf V[S JT]"/ VFSFZGF B[TZG[ B[0JFGM BR" ~P 0.75 5|lT DL2P GF NZ[ ~P 4158
YFI TM VF B[TZGL OZT[ JF0 SZJFGM BR" ~P 30 5|lT DLP D]HA S[8,M YFIP
s#f V[S JT]"/FSFZ W0LIF,GF lDlG8 SF8FGL ,AF. 10 ;[DLP K[P lDlG8 SF8FGL CF,GL
l:YlT VG[ 5 lDlG8 AFNGL l:YlTYL AGTF J'TFXG] 1F[+O/ XMWMP
sZf ;FIS,G] 8FIZ 1 lDlG8DF 140 RSSZ ,UFJ[ K[P HM 8FIZGM jIF; 60 ;[DLP CMI TM 2
S,FSDF S[8,] VTZ ;FIS, SF5L XS[P
pS[, o ;FIS,G] 8FIZ 1 RSSZ ,UFJ[ TM SF5[,] VTZ JT]"/GF 5lZW H[8,] YFIP
VCL4 jIF; 60 ;[DLP T[YL l+HIF =
60
2
= 30 ;[DLP
T[YL JT]"/GM 5lZW = 2𝜋𝑟 = 2 ×
22
7
× 30 =
1320
7
;[DLP
∴ 1 lDlG8DF SF5[,] VTZ=140 ×
1320
7
= 26400 ;[DLP
∴ 2 S,FSDF SF5[,] VTZ = 120 × 26400 = 3168000 ;[DLP
∴ 31.68 SLDLP
:J5|ItG[ SZM o s!f 14 ;[DLP AFH] WZFJTF RMZ;DF NXF"J[, ZULG 5|N[XGL l0hF.G
AGFJJFGM BR" ~P 25 5|lT ;[DLP D]HA S[8,M YFIP
sZf V[S J'TFX VFSFZGF B[TZGL l+HIF 21 DLP K[P B[TZGF V[S
B]6[ 6 DLP ,FAF NMZ0FYL V[S UFI AFW[,L K[P TM UFIG[ OZJF
D/TF EFUG] 1F[+O/ XMWMP HM NMZ0FGL ,AF. 2 DLP JWFZJFDF VFJ[ TM T[G[ OZJF
D/TF EFUG] 1F[+O/ S[8,] JWX[P
5|SZ6 13 o JT]"/ ;AlWT 1F[+O/
A
O
B
120

sU]6EFZ ov (f
GMW ov VF 5|SZ6 DFYL $ lJS<5M VG[ $ U]6GM V[S NFB,M 5]KFI K[P
s!f V[S XS]GL JS|;5F8LG] 1F[+O/ 550 cm2 VG[ 5FIFGF JT]"/GM jIF; 14 cm. K[ TM T[G] S],
WGO/ XMWMP
pS[, o VCL4 5FIFGF JT]"/GM jIF; 14 cm. K[P l+HIF =
14
2
= 7 ;[DLP
XS]GL JS|;5F8LG] 1F[+O/ = 𝜋𝑟𝑙
∴ 550 =
22
7
× 7 × 𝑙
∴ 𝑙 =
550
22
= 25 ;[DLP
CJ[4 𝑙2
= 𝑟2
+ ℎ2
D]HAPP
(25)2 = (7)2 + ℎ2
∴ ℎ2
=625 – 49 = 576
∴ ℎ =24 ;[DLP
∴ XS]G] WGO/ =
1
3
𝜋𝑟2
ℎ
=
1
3
×
22
7
× 7 × 7 × 24
= 1232 ;[DL3
:J5|ItG[ SZM o s!f V[S XS]GL pRF. VG[ lTI"SµRF> VG]S|D[ 12 VG[ 20 ;[DLP CMI TM XS]G] WGO/
XMWMP
sZf V[S UM,SGL JS|;5F8LG] 1F[+O/ 1256 cm2 CMI TM UM,SG] WGO/ XMWMP
sZf 6 ;[DLP l+HIF VG[ 14 ;[DL pRF.JF/F WFT]GF XS]G[ 5LUF/LG[ 0.5 ;[DL l+HIFJF/F S[8,F
UM,S AGX[P
pS[, o VCL4 XS]GL l+HIF R= 6 ;[DLP VG[ pRF. h = 14 ;[DLP
UM,SGL l+HIF r= 0.5 ;[DLP
UM,SGL ;bIF =
=
1
3
𝜋𝑅2ℎ
4
3
𝜋𝑟3
=
62×14
4×(0.5)3 = 1008
T[YL UM,SGL ;bIF 1008 K[P
:J5|ItG[ SZM o s!f XS] GLR[ VW"UM,S ,UFJ[, WG 5NFY"GL S], JS|;5F8LG] 1F[+O/ 361.1 cm2 K[P HM
XS]GL lTI"S pRF. 13 ;[DLP CMI TM WG 5NFY"GL S], pRF. XMWMP
5|SZ6 14 o 5'Q9O/ WGO/
XS]G] WGO/
UM,SG] WGO/

sZf V[S G/FSFZGF AgG[ K[0F VW"UM,SYL AW SZJFDF VFjIF K[P HM G/FSFZGL l+HIF
0.42 DLP VG[ pRF. 3.84 DLP CMI TM T[DF S[8,F ,L8Z 5[8=M, EZL XSFIP
s#f V[S G/FSFZGM V[S K[0M VW"UM,SYL AW SZJFDF VFjIM K[P HM G/FSFZGL l+HIF
4.2 ;[DLP VG[ S], pRF. 27.5 ;[DLP CMI TM T[DF S[8,F ,L8Z 5[8=M, EZL XSFIP
s$f 5[8=M,55GL V[S G/FSFZ 8FSLGL 1FDTF 57750 ,LP K[P HM T[GM jIF; 3.5 DLP CMI
TM pRF. XMWMP
sU]6EFZ ov (f
GMW ov VF 5|SZ6 DFYL # lJS<5M VG[ Z VG[ # U]6GF A[ NFB,F 5]KFI K[P
s!f GLR[ VF5[,F VFJ'lT lJTZ6GM AC],S XMWMP
JU" 0-20 20-40 40-60 60-80 80-100 100-120
VFJ'lT 26 31 35 42 82 71
VCL4 𝑙 = 80, 𝑓0 = 42, 𝑓1 = 82, 𝑓2 = 71, 𝐶 = 20
Z = l +
f1−f0
2f1−f0−f2
C
= 80 +
82−42
2(82)−42−71
20
= 80 +
40
51
× 20
= 80 + 15.69
= 95.69
:J5|ItG[ SZM o s!f GLR[ VF5[,F VFJ'lT lJTZ6GM AC],S XMWMP
JU" _ v * *v!$ !$vZ! Z!vZ( Z(v#5 #5v$Z $Zv$) $)v5&
VFJ'lT Z& #! #5 $Z (Z *! 5$ !)
sZf GLR[ VF5[,F VFJ'lT lJTZ6DF !&5 VJ,MSGMGM AC],S #$P5 CMI TM B]8TL VFJ'lT XMWMP
JU" 5 v !$ !$vZ# Z#v#Z #Zv$! $!v5_ 5_v5) 5)v&(
VFJ'lT 5 !! a 5# b !& !_
sZf VFJ'lT lJTZ6 5ZYL DwIS XMWMP
JU" 0-10 10-20 20-30 30-40 40-50 50-60 60-70
VFJ'lT 4 8 3 20 3 4 8
VCL4 A = 35, C = 10 K[P
5|SZ6 15 o VFS0FXF:+

JU" 𝑓𝑖 𝑋𝑖 𝑑𝑖 𝑓𝑖 𝑑𝑖
0-10 4 5 -3 -12
10-20 8 15 -2 -16
20-30 3 25 -1 -3
30-40 20 35 = A 0 0
40-50 3 45 1 3
50-60 4 55 2 8
60-70 8 65 3 24
n = 50 𝑓𝑖 𝑥𝑖 = 4
CJ[4 DwIS x = A +
∑ fidi
n
C
= 35 +
4
50
× 10
= 35 +0.8
= 35.8
:J5|ItG[ SZM o s!f GLR[GL DFlCTLGM DwIS XMWMP
JU" 0-50 50-100 100-150 150-200 200-250 250-300 300-350
VFJ'lT 10 15 30 20 15 8 2
sZf GLR[GL DFlCTLGM DwIS XMWMP
JU" 100-200 200-300 300-400 400-500 500-600 600-700
VFJ'lT 5 3 3 6 2 1
s#f VFJ'lT lJTZ6 5ZYL DwI:Y XMWMP
JU" 20-25 25-30 30-35 35-40 40-45 45-50 50-55
VFJ'lT 2 5 8 10 7 10 3
pS[, o VCL4 n = 45 T[YL
𝑛
2
= 22.5
JU" VFJ'lT(f) ;RIL VFJ'lT(cf)
20-25 2 2
25-30 5 7
30-35 8 15
35-40 10 25
40-45 7 32
45-50 10 42
50-55 3 45

VCL4
𝑛
2
= 22.5 VF VJ,MSG WZFJTM JU" 35-40 K[P
DF8[ 𝑙 = 35, 𝑐𝑓 = 15, 𝑓 = 10, 𝑐 = 5
DwI:Y o M = l +
n
2
−cf
f
C
= 35 +[
22.5−15
10
]5
= 35 +[0.75]5
= 35 +3.75
= 38.75
:J5|ItG[ SZM o s!f VFJ'lT lJTZ6 5ZYL DwI:Y XMWMP
JU" _v!__ !__vZ__ Z__v#__ #__v$__ $__v5__ 5__v&__
VFJ'lT &$ &Z ($ *Z && 5Z
sZf GLR[GL DFlCTLGM DwI:Y #( VG[ S], VFJ'lT $__ CMI TM B]8TL VFJ'lT XMWMP
JU" !_vZ_ Z_v#_ #_v$_ $_v5_ 5_v&_ &_v*_ *_v(_
VFJ'lT $Z #( a 5$ b #& #Z
sU]6EFZ ov 5f
GMW ov VF 5|SZ6 DFYL Z lJS<5M VG[ # U]6GM V[S NFB,M 5]KFI K[P
s!f V[S l;SSM A[ JBT pKF/JFDF VFJ[ K[P TM l;SSF 5ZPPP
s!f AgG[ JBT KF5 D/[ sZf AgG[ JBT SF8 D/[P T[GL ;EFJGF XMWMP
pS[, o VCL4
P(E) =
S], 5lZ6FD = 4 (hh, ht, th, tt)
AgG[ JBT KF5 D/[ W8GF A CMI TMPPPP
P(A) =
1
4
VG[ AgG[ JBT SF8 D/[ W8GF B CMI TMPPPP
P(B) =
1
4
:J5|ItG[ SZM os!f V[S ;DTM, 5F;FG[ V[S JBT pKF/JFDF VFJ[ K[P TM 5F;F 5Z D/TM VS PPP
s!f VlJEFHI CMI sZf lJEFHI CMI s#f I]uD CMI s$f VlJEFHI I]uD CMI
s5f 6YL DM8M CMI s&f WG 5}6F"S CMI s*f 5}6"JU" CMI s(f 3YL GFGM CMI
5|SZ6 16 o ;EFJGF
W8GF E pNEJJF DF8[GF 5lZ6FDMGL
;bIF
5|IMUGF S], 5lZ6FDMGL ;bIF

s)f 3GM VJIJL CMI T[GL ;EFJGF XMWMP
sZf A[ ;DTM, 5F;FG[ V[S JBT pKF/JFDF VFJ[ K[P TM 5F;F 5Z D/TM VSMGM
;ZJF/M PPPs!f 7 D/[ sZf 10YL JW] D/[ s#f 2YL VMKM D/[ s$f 13YL VMKM
D/[ s5f VlJEFHI D/[ T[GL ;EFJGF XMWMP
s#f V[S 5[8LDF 5 ,L,F4 8 5L/F VG[ 7 E]ZF ZUGF N0F K[P 5[8LDFYL V[S N0M IFNlrKS
ZLT[ 5;N SZJFDF VFJ[ TM T[ N0M PPP
s!f 5L/F ZUGM CMI sZf ,L,F S[ E]ZF ZUGM CMI s#f E]ZF ZUGM G CMI
s$f ,L,F S[ 5L/F ZUGM G CMI T[GL ;EFJGF XMWMP
s$f V[S BMBFDF 100 5[g8 K[P T[DF 73 ;FZF4 12 YM0L BFDLJF/F VG[ 15YL JW]
BFDLJF/F K[P SG] V[S V[JM 8=[0Z K[ S[ H[ ;FZFH 5[g8 BZLN[ K[P 56 ALHF 8=[0Z DG]G[
H[DF JW] BFDL K[ DF+ T[JF 5[g8 V:JLSFI" K[P BMBFDFYL IFNlrKS ZLT[ V[S 5[g8
5;N SZJFDF VFJ[ TL T[ s!f SG]G[ :JLSFI" CMI sZf DG]G[ :JLSFI" CMI T[GL ;DFJGF
XMWMP
BEST OF LUCK FOR MARCH 2015..........
Typed by : Bagada Bharat k.
s!f RMS;F.YL SFD SZTF VFJ0[P
sZf TFlS"S ZLT[ lJRFZTF VFJ0[P
s#f ;S[TM VG[ 5|TLSMGM p5IMU SZTF VFJ0[P
s$f N[BFI GCL T[JF ;AWM4 HM0F6M XMWTF VFJ0[P
s5f ;S<5GFVM VG[ bIF,M ;DHTF VFJ0[P
s&f lJS<5M lJRFZTF VG[ pS[,M XMWTF VFJ0[P
s*f ,FALv5CM/L J6"GFtDS lJUTMG[ 8[A, S[ VF,[Bv VFS'lT £FZF ZH] SZTF VFJ0[P
s(f jIJCFZDF XSI G ,FUTL ;EFJGFVM XMWTF VFJ0[P
s)f h95YL VG[ 8}SDF ;DHTF VG[ ;DHFJTF VFJ0[P
s!_f A|dCF0GF VNE]T ZC:IMG] lJ:DI DF6TF VFJ0[P
H[G[ Ul6T VFJ0[ T[G[ X] X] VFJ0[ m

1
lJ7FG VG[ 8[SGM,MHL
5|`GA[S
TH7zLVM
! zL 5LP V[;P DMTF ;FP ;P pPDFP XF/F D:SFsDF0JLf v SrK
Z zL ;HLJEF. HMQFL ;FP ;P CF.:S}, SM80F RSFZ v SrK
# zL 0LP V[RP HMQFL 58[, ALP S[P lJnF,I SM80FsGB+F6Ff v SrK
$ zL ZFH[XEF. 58[, ;P JP 5P lJnF,I UFWLWFD v SrK
5 zL DGMHEF. VFRFI" ,FS0LIF CF.:S}, ,FS0LIFsERFpf v SrK
;S,G o zL V[DP V[GP 58[, slH<,F lX1F6 VlWSFZL4 SrK v E]Hf
cc;O/ lX1F6 ;O/ HLJGGM 5FIM K[cc

2
Ätuhý --- 10
Ë{Þ : 3 f÷tf rð¿tt™ y™u xìfT™tu÷tìS fw „w : 100
«&™…ºt™wk …rhY… ð»to – 2013 - 14
PART – A „wý : 50
• ™e[u yt…u÷t «&™tu («&™ ™k. 1 Úte 50){tk ÞtuøÞ rðfÕ… …ËkN fhe™u OMR Answer Sheet {tk
sðtƒ yt…tu. («íÞuf™tu 1 „wý)
PART – B „wý : 50
SECTION - A
• ™e[u™tk «&™tuu («&™ ™.k 1 Úte 5)™t xkqf {tk (30 þçN uGe {ÞtNt{tk) sðtƒ yt…tu.
(«íÞuf™tk 2 „wý) ftuE …ý ƒu «&™tu{tk ytk‚rhf rðfÕ… yt…ðt. [10]
SECTION - B
• ™e[u™tk «&™tuu («&™ ™.k 6 Úte 10)™t xfq {tk (30 þçN uGe {ÞtNt{tk) sðtƒ yt…tu.
(«íÞuf™tk 2 „wý) ftuE …ý yuf «&™{tk ytk‚rhf rðfÕ… yt…ðtu. [10]
SECTION - C
• ™e[u™tk «&™tu («&™ ™k. 11 Úte 15)™t {tøÞt «{týu (50 þçN uGe {ÞtNt{tk) sðtƒ yt…tu.
(«íÞfu ™t k 3 „wý) ftuE …ý ƒu «&™tu{tk ytk‚rhf rðfÕ… yt…ðt. [15]
SECTION - D
• ™e[u™tk «&™tu («&™ ™k. 16 Úte 18)™t {wÆtËh (100 þçN uGe {ÞtNt{tk) sðtƒ yt…tu.
(«íÞuf™tk 5 „wý) ftuE …ý ƒu «&™tu{tk ytk‚rhf rðfÕ… yt…ðtu. [15]

3
PART – B (50 U]6)
sU]6EFZ ov $f
GMW o VF 5|SZ6 DFYL Z lJS<5M VG[ Z U]6GM V[S 5|`G 5]KFI K[P
s!f G[GM ;FIg; V[8,[ X] m ;DHFJMP
sZf G[GM AWFZ6DF J5ZFTF A[ DF.S|M:SM5GF GFD VF5MP
s#f G[GM 8[SGM,MHLYL EFlJ 50SFZMGM ;FDGM S[JL ZLT[ SZL XSFI m
s$f G[GM 8[SGM,MHLG[ :5;"TF VUtIGF 1F[+MGF GFD VF5MP
s5f G[GM 8[SGM,MHL VF56G[ S[JL ZLT[ p5IMUL K[ m
s&f G[GM 8[SGM,MHLG] DCtJ VG[ DIF"NFVM 8]SDF ;DHFJMP
sU]6EFZ ov *f
GMW o VF 5|SZ6 DFYL Z lJS<5M VG[ 5 U]6GM V[S 5|`G 5]KFI K[P
s!f VTUM"/ VZL;FG] ;]+
1
𝑢
+
1
𝑣
=
1
𝑓
TFZJMP
sZf ,[g;G] ;]+
1
𝑣
−
1
𝑢
=
1
𝑓
TFZJMP
s#f BUM/LI N]ZALGsV[:8=MGMlDS, 8[,L:SM5f VFS'lT4 l;âFT4 ZRGF4 SFI"5âlT VG[
p5IMU H6FJMP
s$f ;I]ST X]1DNX"S I+sDF.S=M:SM5f VFS'lT4 l;âFT4 ZRGF4 SFI"5âlT VG[ p5IMU
H6FJMP
s5f SFRGF ,AWG J0[ YT] 5|SFXG] JlS|EJG VFS'lT NMZL ;DHFJM VG[ ,[8Z, l;O8s5F`JLI
:YF/FTZf V[8,[ X] m
s&f UM,LI VZL;F J0[ YTF 5ZFJT"G DF8[ SFT["lhIG ;7F 5âlT ;DHFJMP
s*f DFwIDGM lGZ5[1F JlS|EJGFS V[8,[ X] m :G[,GF lGIDG] jIF5S :J~5 D[/JMP
5|SZ6 01 o G[[GM8[SGM,MHLGM 5lZRI
5|SZ6 02 o 5|SFXG] 5ZFJT"G VG[ JS|LEJG

4
sU]6EFZ ov &f
GMW o VF 5|SZ6 DFYL # lJS<5M VG[ # U]6GM V[S 5|`G 5]KFI K[P
s!f VFBGL ,W]§Q8LGL BFDL S[JL ZLT[ pNEJ[ K[ T[ H6FJL T[G] lGZFSZ6 ;DHFJMP
sZf VFBGL U]~§Q8LGL BFDL S[JL ZLT[ pNEJ[ K[ T[ H6FJL T[G] lGZFSZ6 ;DHFJMP
s#f VFBGL GFD lGNX"G JF/L VFS'lT NMZL ;DH]TL VF5MP
s$f D[3WG]QIGL ZRGF VFS'lT NMZL ;DHFJMP
s5f 5|SFXG] 5]6" VFTZLS 5ZFJT"G ;DHFJMP
s&f D'UH/GL 38GF J6F"JMP
sU]6EFZ ov &f
GMW o VF 5|SZ6 DFYL $ lJS<5M VG[ Z U]6GM V[S 5|`G 5]KFI K[P
s!f lJn]TEFZ V[8,[ X] m T[GF 5|SFZ VG[ V[SD H6FJMP
sZf VJZMWGF ;DFTZ HM0F6GF OFINF VG[ U[ZOFINF H6FJMP
s#f lJn]T ;]JFCSM V[8,[ X] m pNFCZ6 VF5MP
s$f lJn]T5|JFCGL jIFbIF VF5L T[GM SI V[SD H6FJMP
s5f TOFJT VF5M o ;DFTZ VG[ z[6L HM0F6P
s&f lJn]T5FJZGL jIFbIF VF5L T[GM SI V[SD H6FJMP
s*f lJn]T l:YlTDFG V[8,[ X] m T[GM SI V[SD H6FJMP
s(f JFCSGL VJZMWSTF S. AFATM 5Z VFWFZ ZFB[ K[ m
s)f .,[S8=Ml,l;; V[8,[ X] m ;DHFJMP
sU]6EFZ ov &f
GMW o VF 5|SZ6 DFYL # lJS<5M VG[ # U]6GM V[S 5|`G 5]KFI K[P
s!f ;M,[GM.0 V[8,[ X] m ;M,[GM.0YL pNEJTF R]ASLI 1F[+GL ,F1Fl6STF H6FJMP
sZf .,[S8=LS DM8Z VFS'lT4 l;âFT4 ZRGF4 SFI"5âlT VG[ p5IMU H6FJMP
5|SZ6 03 o 5|SFXG] lJEFHG VG[ S]NZTL 5|SFXLI W8GFVM
5|SZ6 04 o lJn]T
5|SZ6 05 o lJn]T5|JFCGL R]ASLI V;ZM

5
s#f .,[S8=LS HGZ[8Z VFS'lT4 l;âFT4 ZRGF4 SFI"5âlT VG[ p5IMU H6FJMP
s$f 8]SGMW ,BMP .,[S8=LS A[,
s5f OI]hG] SFI" ;DHFJMP
s&f TOFJT VF5M o AC VG[ DC 5|JFCP
sU]6EFZ ov &f
s!f 5FlY"Js8[Z[l:8=I,f U|CM V[8,[ X] m T[GF GFD H6FJMP
sZf HMlJIG U|CM V[8,[ X] m T[GF GFD H6FJMP
s#f S'l+D p5U|CMGF SM. RFZ p5IMU ,BMP
s$f DU/ U|C lJX[ DFlCTL VF5MP
s5f ;}I"D0/ V[8,[ X] m T[GF ;eIMGF GFD S|DDF H6FJMP
s&f :5[X;8,GL p5IMULTF H6FJMP
s*f PSLV VG[ GSLV GF 5]ZF GFD VF5MP
s(f PSLV VG[ GSLV GL JCG1FDTF H6FJMP
s)f TFZFVM SMG[ SC[ K[ m SNGL ZLT[ T[GF 5|SFZM H6FJMP
sU]6EFZ ov &f
s!f pH DF5S|D SIF lJ7FlGS[ ZH] SIM" m T[G] ;]+ H6FJMP
sZf 5|A/ V[l;0GL jIFbIF VF5L T[GF pNFCZ6 VF5MP
s#f pH DF5S|DGL DIF"NFVM H6FJMP
s$f V[l;0GL WFT] ;FY[GL 5|lS|IF pNFP VF5L ;DHFJMP
s5f ;HLJGF Vl:TtJDF pHG] DCtJ ;DHFJMP
s&f A|Mg:8[0v,MZL V[l;0 A[.h l;âFT jIFbIFlIT SZMP
s*f T8:YLSZ6 V[8,[ X] m T[G] V[S pNFP VF5MP
s(f 5|lT V[l;0 SMG[ SC[ K[ m ;DHFJMP
s)f HNO3GF H,LI §FJ6GL ;F§TF 0.03M YL JWFZL 0.05M SZJFDF VFJ[ TM pHDF
S[8,M O[Z 50X[P
5|SZ6 06 o A|dCF0
5|SZ6 07 o V[;L04 A[.h VG[ 1FFZ

6
s!_fpHG] D]<I 9.3 CMI T[JF H,LI §FJ6DF ZC[,F [OH¯
] GL ;F§TF U6MP
s!!f8 pH JF/F A[lhS H,LI §FJ6 SZTF 11 pH JF/] A[lhS H,LI §FJ6 OH¯
GL S[8,F
U6L JW] ;F§TF WZFJX[P
s!Zf0.05M HCL GF H,LI §FJ6GL pH U6MP
sU]6EFZ ov (f
GMW o VF 5|SZ6 DFYL # lJS<5M VG[ 5 U]6GM V[S 5|`G 5]KFI K[P
s!f SFRL WFT]G] ;S[g§6 V[8,[ X] m T[GM 5âlTVM H6FJL SM. A[ J6F"JMP
sZf AMS;F.8 DFYL V[<I]DLGF D[/JJFGL 5âlT J6F"JMP
s#f V[<I]DLGF DFYL V[<I]lDlGID D[/JJFGL CM,vC[ZFp<8 5âlT VFS'lT NMZL J6F"JMP
s$f lCD[8F.8 DFYL VFIG"G] lGQSQF"6 VFS'lT NMZL J6F"JMP
s5f WFT]G] 1FFZ6 V[8,[ X] m T[GF OFINF4 U[ZOFINF H6FJL V8SFJJFGF p5FIM ,BMP
s&f WFT]GL ;lS|ITF z[6L ;DHFJMP
sU]6EFZ ov &f
GMW o VF 5|SZ6 DFYL # VYJF $ lJS<5M VG[ # VYJF Z U]6GM V[S 5|`G 5]KFI K[P
s!f ,LSZNH3 V[8,[ X] m T[G] ZF;FIl6S ;]+ VG[ EF{lTS U]6WDM" H6FJMP
sZf VWFT]GF A[ EF{lTS VG[ A[ ZF;FIl6S U]6WDM" H6FJMP
s#f ;<OZGF SM. RFZ p5IMU ,BMP
s$f TOFJT VF5M o ;F§ H2SO4 VG[ DN H2SO4
s5f TOFJT VF5M o WFT] VG[ VWFT]
s&f H2 JFIG] VMnMULS pt5FNG ;DLSZ6 ;lCT J6F"JL T[GF EF{lTS U]6WDM" H6FJMP
s*f NH3GF pt5FNG DF8[GL C[AZ 5âlT J6F"JMP
s(f ;<OZGF lGQSQF"6GL O|F; 5âlT J6F"JL SM. RFZ p5IMU ,BMP
s)f ;<OZGF AC]~5M J6F"JL T[GF p5IMU ,BMP
s!_f H2SO4GF pt5FNG DF8[GL ;5S" 5âlT J6F"JMP
s!!f H2 JFI]GL 5|IMUXF/FDF AGFJ8 VFS'lT NMZL J6F"JMP
s!Zf H2 JFIG] VMnMULS pt5FNG ;DLSZ6 ;lCT J6F"JMP
5|SZ6 08 o WFT]VM
5|SZ6 09 o VWFT]VM

7
sU]6EFZ ov &f
s!f BlGH SM,;FGF D]bI 5|SFZ S[8,F mSIF SIF m
sZf SFA"GGL ;IMHSTF S[8,L K[ m XF DF8[ m
s#f ;DW8STF V[8,[ X] m aI]8[G VG[ 5[g8[GGF ;DW8SM ,BMP
s$f VF<S[G z[6LG] ;FDFgI ;]+ ,BL T[GF 5|YD ;IMHGG] GFD VG[ ;]+ VF5MP
s5f TOFJT VF5M o V[gY[;F.8 VG[ l,uGF.8 SM,;MP
s&f TOFJT VF5M o LPG VG[ CNGP
s*f SFA"GGM S[8[G[XG U]6WD" ;DHFJMP
s(f ;DFGWDL" z[6L V[8,[ X] m
s)f ;DHFJMo VMS8[G VFS
s!_f TOFJT VF5M o ;T'%T VG[ V;T'%T CF.0=MSFA"GP
sU]6EFZ ov &f
s!f SM. 56 RFZ lS|IFXL, ;D]CGF GFD VG[ ;]+ VF5MP
sZf VFYJ6 5|lS|IF V[8,[ X] m T[GL VUtITF ;DHFJMP
s#f .Y[GM,G] VMnMULS pt5FNG ,BMP
s$f .Y[GM,GF SM. RFZ p5IMU ,BMP
s5f ;FA] V[8,[ X] m T[GL AGFJ8 ,BMP
s&f VF<SMCM,G] ;[JG SZJ] VFZMuI DF8[ CFGLSFZS K[P ;DHFJMP
s*f PVCG] 5]~ GFD VF5L SM. A[ p5IMU ,BMP
s(f 5|1FF,SM V[8,[ X] m T[GF pNFP VF5MP
s)f TOFJT VF5M o ;FA] VG[ l08H"g8
s!_fV[l;l8S V[l;0GF SM. RFZ p5IMU ,BMP
5|SZ6 10 o BlGH SM,;M VG[ BlGHT[,
5|SZ6 11 o SFA"lGS ;IMHGM

8
sU]6EFZ ov &f
GMW o VF 5|SZ6 DFYL ! lJS<5M VG[ 5 U]6GM V[S 5|`G 5]KFI K[P
s!f DG]QIGF 5FRGT+GL GFD lGN["XJF/L VFS'lT NMZL 5FRG VUM ;DHFJMP
sZf DG]QIGF `J;GFUM lJX[ ;lJ:TFZ ;DHFJMP
s#f DG]QIDF BMZFSGF 5FRGGL lS|IF J6F"JMP
s$f `J;G V[8,[ X] m T[GF 5|SFZM pNFP VF5L ;DHFJMP
s5f CNIGL ZRGF GFD lGN["XJF/L VFS'lT NMZL ;DHFJMP
s&f DI]QIGF CNIDF ~lWZG] 5lZJCG VFS'lT NMZL ;DHFJMP
s*f pt;U" V[SDGL ZRGF J6F"JMP
s(f DG]QIGF DUHGL VFS'lT NMZL VU|DUH lJX[ ;lJ:TFZ ;DHFJMP
sU]6EFZ ov &f
s!f JG:5lTDF pt;H"G S. S. ZLT[ HMJF D/[ K[ m
sZf WDGLGL lNJF, HF0L VG[ l:YTL:YF5S HIFZ[ lXZFGL lNJF, 5FT/L XF DF8[ CMI K[ m
s#f TOFJT VF5M o WDGL VG[ lXZF
s$f TOFJT VF5M o S6"S VG[ 1F[5S
s5f ,;LSFT+GF SFIM" H6FJMP
s&f JG:5lTDF D]/ £FZF 5F6LG] XMQF6 S. ZLT[ YFI K[ m
s*f ~WLZJFCLGL V[8,[ X] m T[GF 5|SFZM H6FJMP
sU]6EFZ ov $f
s!f JG:5lT VG[ 5|Fl6VMDF pT[HGF ;FD[GL 5|lTlS|IFGM E[N H6FJMP
sZf VFJT"G V[8,[ X] m T[GF 5|SFZM H6FJMP
s#f G[l:8hD V[8,[ X] m T[GF 5|SFZM H6FJMP
5|SZ6 12 o 5MQF6 VG[ `J;G
5|SZ6 13 o ;HLJMDF JCG45lZJCG VG[ pt;H"G
5|SZ6 14 o ;HLJMDF lGI+6 VG[ ;S,G

9
s$f R[TFT+GF SM. RFZ SFIM" H6FJMP
s5f DG]QIGF DUHGL Z1F6 jIJ:YF ;DHFJMP
s&f 5ZFJTL" lS|IF V[8,[ X] m T[GF SM. A[ pNFP VF5MP
s*f DG]QIGL VTo:+FJL U|YLGF SM. RFZ GFD VF5MP
s(f VTo:+FJMGF SM. RFZ U]6WDM" H6FJMP
s)f ,ADHHFG[ .HF YTF TZTH jIlSTG] D'tI] YFI K[P XF DF8[ m
s!_f TOFJT VF5M o A'CN VG[ VG] Dl:TQS
sU]6EFZ ov 5f
GMW o VF 5|SZ6 DFYL Z lJS<5M VG[ # U]6GM V[S 5|`G 5]KFI K[P
s!f V,LUL 5|HGG V[8,[ X] m T[GF 5|SFZ H6FJL JFG:5lTS 5|HGG ;DHFJMP
sZf 5]Z]QFG] 5|HGGT+ VFS'lT NMZL ;DHFJMP
s#f :+LG] 5|HGGT+ VFS'lT NMZL ;DHFJMP
s$f UEF"J:YF V8SFJJFGL 5âlT J6F"JMP
s5f :+LDF ,{ULSRS| ;DHFJMP
s&f ;HLJMDF ,LUL 5|HGGG] DCtJ ;DHFJMP
s*f JG:5lTGL S'l+D 5|HGGGL VFZM56GL ZLT J6F"JMP
sU]6EFZ ov $f
s!f lEgGTF V[8,[ X] m T[G] DCtJ ;DHFJMP
sZf VlxDVM ptS=lTGF 5]ZFJF S[JL ZLT[ 5]ZF 5F0[ K[ m
s#f ZRGF ;NX VUM pNFP VF5L ;DHFJMP
s$f p5FlH"T ,F1Fl6STF V[8,[ X] m T[GF SM. A[ pNFP VF5MP
s5f VFG]JlXSTF V[8,[ X] m pNFP VF5MP
s&f D[g0,GM 5|IMU RF8" NMZL ;DHFJMP
s*f VlxDVM V[8,[ X] m
s(f l,U lG`RIG V[8,[ X] m 8]SDF ;DHFJMP
s)f SFI" ;NX VUM pNFP VF5L ;DHFJMP
5|SZ6 15 o ;HLJMDF 5|HGG
5|SZ6 16 o VFG]JlXSTF VG[ ptS|FlT

10
sU]6EFZ ov $f
s!f lJW8GGF VFWFZ[ SRZFGF A[ 5|SFZ pNFP VF5L ;DHFJMP
sZf IMuI pNFP VF5L VFCFZ ;'B,F ;DHFJMP
s#f SM. 56 RFZ J{l`JS ;DxIFVM H6FJMP
s$f VMHMG :TZG] :YFG VG[ DCtJ H6FJMP
s5f 3ZUyY] SRZFG] 5|DF6 38F0JFGF RFZ 5U,F ,BMP
s&f VMHMG :TZGF lJ38GDF CFCGL E]lDSF H6FJMP
s*f lGJ;GT+DF XlSTGM 5|JFC V[S DFUL" XF DF8[ CMI K[ m
sU]6EFZ ov $f
s!f 5]GoRlS|ITFYL 5IF"JZ6 S[JL ZLT[ ARFJL XSFI m
sZf pHF":+MTMGF ;Z1F6 DF8[ S[JF 5U,F ,. XSFI m
s#f H/ jIJ:YF5GGF SM. RFZ D]ÛF VF5MP
s$f JGS8F.GL UELZ V;Z H6FJMP
s5f G{;UL"S :+MTMGL HF/J6L XF DF8[ H~ZL K[ m
s&f 5IF"JZ6 ARFJTF +6 R H6FJMP
s*f GFX5|FIo JgIHLJM V[8,[ X] m pNFP VF5L ;DHFJMP
s(f HU,GF :8MS CM<0ZGL ;DH]TL VF5MP
BEST OF LUCK FOR 13 MARCH 2014 ........
Typed by : Bagada Bharat.
(b.b.m. highschool – bidada)
5|SZ6 17 o VF56] 5IF"JZ6
5|SZ6 18 o G{;lU"S :+MTMGL HF/J6L

1
rð»tÞ : „rý‚ (028)
{tuzu÷ «§…ºt - 1
«-1. (y) Ëtrƒ‚ fhtu fu ftxftuý rºtftuý{tk fýo …h ðuÄ Œtuhðt{tk ytðu ‚tu ‚uÚte ƒ™‚t ƒu rºtftuýtu …hM…h Ë{Y… ntuÞ Au
y™u ‚u {q¤ rºtftuý™u …ý Ë{Y… ntuÞ Au. 44444
(ƒ) „{u ‚u ƒu „ýtu. 66666
(1) yðÞð …tztu : x3
+ y3
+ z3
- 3xyz
(2) yðÞð …tztu : 3a2
(b - 3c) + 3b2
(c - 3a) + 3c2
(a - 3b) + 26abc
(33333) ËtŒwY… yt…tu :
2 2
2 2
1 1 1 b + b
+ b 2 b + b b
a a a
a a a a
  
÷ − ÷  
− −  
(f) yðÞð …tztu („{u ‚u ƒu) : 44444
(1) (x2
- 4x)2
- 25(x2
- 4x) - 100
(2) 729a6
- 64b6
(3) x3
- 7x - 6
(z) „{u ‚u yuf „ýtu. 22222
(1) 2
2
8 4 1
+ 2 + 4 + 1
2 + 2
a a
a a a
     
−     
−     
™wk Ëh¤Y… yt…tu
(2) òu pq = 1 ‚tu Ëtrƒ‚ fhtu fu
1 + p p 1
=
1 + q 1 q
−
−
(R) Ët[tu rðfÕ… …ËkŒ fhe ¾t÷e søÞt …whtu 44444
(1) f : Z→R, f(x) = 2x - 3 ntuÞ ‚tu ......... yu ‚u™t yt÷u¾ …h™wk ®ƒŒw ™Úte. [(-2, -7), (-1,-5), (0,2)]
(2) f : R →Z, f(x) = [x] = (x Úte {tuxt ™ne ‚uðtu yrÄf‚{ …qýtOf) ‚tu f(-5.2) = ........ (-6, -5, -4).
(3) ytÄwr™f ftuBÃÞwxh yuf ËuLfz{tk ........ Úte ðÄw Ëhðt¤t-ƒtŒƒtfe fhe þfu Au. (103
, 104
, 105
)
(4) V÷tu [txo{tk r™ýoÞ Œþtoððt ........ Ëkfu‚ ð…htÞ Au. ( , , )
«-22222 (y) Ëtrƒ‚ fhtu fu ð‚ow¤™t fuLÿ{tkÚte Sðt™u Œtuhu÷tu ÷kƒ Sðt™u Œw¼t„u Au. 44444
(ƒ) „{u ‚u ƒu „ýtu. 66666
(1) òu
by cx cm y bm
= =
b c
a ax
a
− − −
ntuÞ ‚tu Ëtrƒ‚ fhtu fu
m x y
= = ( m + b + cy 0)
b c
a x
a
≠
(2) òu
2 2
+ 4b 25
=
b 6
a
a
‚tu „wýtu¥th™t „wýÄ{tuo …hÚte
b
a
™e ®f{‚ þtuÄtu.

2
(3) ðu„™ òuzât rð™t™wk yuf huÕðu yuÂLs™ f÷tf™t 96 fe.{e.™e Íz…u Œtuze þfu Au. y{wf ËkÏÞt{tk ðu„™
òuzðtÚte ‚u™e Íz…{tk Út‚tu ½xtztu ðu„™™e ËkÏÞt™t ð„o{q¤™t Ë{[÷™{tk Au. òu 16 ðu„™tu òuzðt{tk ytðu
‚tu Íz… yzÄe ÚtE òÞ Au. ‚tu ‚u yuÂLs™ ðÄw{tk ðÄw fux÷t ðu„™ ¾U[e þfu ?
(f) „{u ‚u ƒu Œt¾÷t „ýtu. 44444
(1) yuf ð„eof]‚ {trn‚e™tk yð÷tuf™tuk, 2k, 4k Au. òu {æÞf yu {æÞMÚtÚte 2sux÷tu ytuAtu ntuÞ ‚tu yð÷tuf™tu
þtuÄtu.
(2) 20yð÷tuf™tu™tu{æÞf 13.4 Au.yt{trn‚e{tkÚteyufyð÷tuf™hŒfhðt{tkytðu‚tuƒtfe™tkyð÷tuf™tu™tu
{æÞf 13 {¤u Au. ‚tu hŒ fhu÷wk yð÷tuf™ þtuÄtu.
(3) 25 «tótkftu™tu {æÞf 12.5 Au. AÔðeË{tu «tótkf 25.5 ntuÞ ‚tu yt AÔðeË «tótkftu™tu {æÞf þtuÄtu.
(z) „{u ‚u yuf „ýtu 22222
(1) f(x) = x2
- 3 ™tu rðM‚th {1, 6, 13} Au. ‚u™tu «Œuþ N ™tu W…„ý ntuÞ ‚tu rðÄuÞ™tu «Œuþ þtuÄtu.
(2) f : R→R, f(x) = x4
- x2
- 1 ntuÞ ‚tu f ( 3) f( 2)− þtuÄtu.
(E) Ët[tu rðfÕ… …ËkŒ fhe ¾t÷e søÞt …qhtu 44444
(1) 2 2
1 p( ) q( )
= =
2 1 4 1 (2 1)
x x
x x x− − −
‚tu p(x) - q(x) = ....... (2, 4, 6)
(2) òu (x + 2) yu (x + 7) y™u x ™tu „wýtu¥th {æÞf ntuÞ ‚tu x = .............
3 4 12
, ,
4 3 7
 
 
 
.
(3) 2
y = 4( . y 1)x x − ntuÞ ‚tu yα .....................
1
, ,x x
x
 
 
 
(4) cot 650
28’ = tan θ, ntuÞ ‚tu θ = ............... (650
281
, 240
321
, 340
321
)
«-33333 (y) Ëtrƒ‚ fhtu fu yÄoð‚wo¤{tk yk‚„o‚ ¾qýtu ftx¾qýtu ntuÞ Au. 44444
(ƒ) „{u ‚u ƒu „ýtu 66666
(1) yuf r{™tht™t ‚r¤Þt{tkÚte …Ëth Út‚t ËeÄt Ë{Âûtr‚s hM‚t W…h™t yuf s rŒþt{tk yufƒeòÚte 200
{exh™u yk‚hu ytðu÷tk ƒu ®ƒŒwytu yt„¤ ‚u r{™tht™e xtu[™t WíËuÄftuýtu y™w¢{u 300
y™u 600
sýtÞ
Au. ‚tu r{™tht™e ô[tE þtuÄtu. ( 3 = 1.7)
(2) ABCD ÷kƒ[tuhË ft„¤™t xwfzt{tkAB = 22 Ëu.{e. y™u BC = 14 Ëu.{e. Au. B ™u fuLÿ ÷E BC
rºtßÞtðt¤tu ð‚ow¤™tu [tuÚtt ¼t„™tu xwfztu yt…u÷t ft„¤{tkÚte ft…ðt{tk ytðu‚tu ft„¤™t ƒtfe hnu÷t
¼t„™wk ûtuºtV¤ þtuÄtu.
(3) yuf h{fzt™tu ytfth W…hÚte þkfw suðtu y™u ™e[uÚte yÄo„tu¤tfth Au. „tu¤tfth y™u þkfw ytfth™t
¼t„™e rºtßÞt 3.5 Ëu.{e. y™u h{fzt™e fw÷ Ÿ[tE 15.5 Ëu.{e. ntuÞ ‚tu ‚u™wk ½™V¤ þtuÄtu.
(f) „{u ‚u ƒu „ýtu 44444
(1) Ëtrƒ‚ fhtu : 2 2 2 2
tan sin = tan . sinθ − θ θ θ
(2) ®f{‚ þtuÄtu : 3 0 4 0 2 0 03
2 cos 60 12 sin 60 + tan 30 + 12 cot 45
4
−

3
(3) 6 6 2 2
sec tan = 1 + 3 sec tanθ − θ θ θ Ëtrƒ‚ fhtu
(z) „{u ‚u yuf „ýtu 22222
(1) x ™wk ½™{q¤ yu y ™t ÔÞM‚ [÷™{tk Au.
1
=
64
x ‚tu y = 8 Au. òu
1
=
27
x ‚tu y þtuÄtu.
(2) òu sin 3θ = cos 2θ ntuÞ ‚tu θ ™e ®f{‚ þtuÄtu.
(R) {tøÞt {wsƒ yt…tu 44444
(1) 3x2
- mx + (k - 2) = 0 (m, k ∈ N) ™tk ƒes ÔÞM‚ ËkÏÞtytu Au. ‚tu k ™e ®f{‚ þtuÄtu.
(2) x2
- 5x = 1 ntuÞ ‚tu
1
x
x
− ™e ®f{‚ þtuÄtu.
(3) ÔÞtÏÞt yt…tu : hu¾tytu™e AurŒft
(4) ÔÞtÏÞt yt…tu : rºtftuý™e {æÞ„t
«-44444 (y) ∆ABC {tk AB >AC Au. D yu BC ™wk {æÞrkƒkŒw Au. A {tkÚte BC
←→ …h™t ÷kƒ™tu ÷kƒ…tŒ M Au. y™u
B-M-C Au ‚tu Ëtrƒ‚ fhtu fu AB2
-AC2
= 2BC × DM 44444
(ƒ) „{u ‚u ƒu „ýtu. 66666
(1) ËtŒw Y… yt…tu :
2 2 3 3
2 2 2
( +1) + ( 1) 3 24 6 24
( 1) + 2 ( 2) + 6 + 2
x x x x x
x x x x x x
− − −
× ÷
− −
(2) yuf {trn‚e™t20 yð÷tuf™tu™tu {æÞf 38.5 Au. òu ƒu yð÷tuf™ ¼q÷Úte 37 ™u ƒŒ÷u 73 y™u 81™u ƒŒ÷u
18 ÷uðtÞt ntuÞ ‚tu ËwÄthu÷tu {æÞf þtuÄtu.
(3) 50 {sqhtu™tk Œir™f ðu‚™™wk ytð]Â¥t-rð‚hý ™e[u «{týu Au. {sqhtu™tk Œir™f ðu‚™™tu {æÞMÚt þtuÄtu.
ðu‚™ (Yr…Þt{tk) 20-29 30-39 40-49 50-59
{sqhtu™e ËkÏÞt 5 27 15 3
(f) „{u ‚u ƒu „ýtu. 44444
(1) XY yu (0, r) ™tu ÔÞtË Au y™u XZ yu ð‚wo¤™e, ÔÞtË rËðtÞ™e Sðt Au. ð‚wo¤™u Z ®ƒŒwyu M…þo‚tu
M…þof, YX
→ ™t rðÁæÄ rfhý™u P {tk AuŒu Au. òu m∠YPZ = 45 ntuÞ ‚tu m∠YXZ þtuÄtu.
(2) PQ y™u XY Sðtytu …hM…h R ®ƒŒw{tk AuŒu Au. òu m∠PXY = 55 y™u m∠QPY = 35 ‚tu
m∠PYQ þtuÄtu.
(3) (P, 13) {tk AB Sðt Au. PM AB⊥ Au. M AB∈ Au. PM
→ ð‚wo¤™u N rƒŒw{tk AuŒu Au.
òu MN = 1 ntuÞ ‚tu AB þtuÄtu.
(z) „{u ‚u yuf „ýtu 22222
(1)
9 2 5
=
1 3 + 1x x x
−
− −
™tk ƒes þtuÄtu. ( 1 1, 3)x ,≠ −
(2) òu ax2
+ bx + c = 0 ™tk ƒu ƒeòu™tu ‚Vtð‚ 3 ntuÞ ‚tu rððu[f ∆ ™wk {qÕÞ {u¤ðtu.

4
(R) (11111) ÔÞtÏÞt yt…tu : ð‚wo¤™wk [t… 44444
(2) ÔÞtÏÞt yt…tu : ð‚wo¤™tu M…þof
(3) 4{e. × 2{e. ft…z™t xwfzt{tkÚte 30 Ëu.{e. × 30Ëu.{e.™t ËtkÄt ð„h™t...... ntÚtY{t÷ ƒ™u.
(4) 3 Ëu.{e. rºtsÞtðt¤t „tu¤t™wk ½™V¤ .......... π ½™ Ëu.{e. Au.
«-55555 (y) AB yt…u÷ Au. ftxftuý ∆PQR yuðtu h[tu fu suÚte fýo PR ™e ÷kƒtE = 5AB y™u PQ = 3AB ntuÞ. 44444
(ƒ) „{u ‚u ƒu „ýtu. 66666
(1) a4
+ b4
∝ a2
b2
ntuÞ ‚tu Ëtrƒ‚ fhtu fu a∝ b
(2)
2
2
1 1
6 + 25 + 12 = 0x x
xx
   
− −   
   
™tu R {tk Wfu÷ {u¤ðtu.
(3) 56 Yr…Þt{tk yuf ðM‚w ðu[ðtÚte ‚u™e {q¤®f{‚ sux÷t xft ™Vtu {¤u ‚tu ‚u ðM‚w™e {q¤ ®f{‚ þtuÄtu.
(f) „{u ‚u ƒu „ýtu. 44444
(1) ∆ABC {tk A-M-B, A-N-C, ||MN BC Au. òu AM = 1.6, AN = 3, AB = 4.8 ‚tu CN þtuÄtu.
(2) ∆ABC {tk BD {æÞ„t Au. AB2
+ BC2
= 68 y™u AC = 6 ‚tu BD™e ®f{‚ þtuÄtu.
(3) ∆RST {tk ∠R ™tu Âî¼tsf RD
→ yu ST ™uD {tk AuŒu Au.3RS = 2RT Au. òu DT = 5.4 ‚tu ST
þtuÄtu.
(z) „{u ‚u yuf „ýtu. 22222
(1) ∆ABC {tk AB = ACAu. …tÞt BC …h™t ðuÄ AM ™e ÷kƒtE 12 yuf{ Au. òu rºtftuý™e …rhr{r‚36
yuf{ ntuÞ ‚tu rºtftuuý™wk ûtuºtV¤ þtuÄtu.
(2) yuf yÄo„tu¤t y™u þkfw™tkk ½™V¤ Ëh¾tk Au. òu yÄo„tu¤t y™u þkfw™e rºtßÞt 3.5 Ëu.{e. ntuÞ ‚tu þkfw™e
Ÿ[tE þtuÄtu.
(R) m
ABCD ™e ƒtswytu AB, BC, CD y™u DA yuf ð‚wo¤™u y™w¢{u P, Q, R y™u S yt„¤ M…þuo Au.
Ëtrƒ‚ fhtu fu ABCD Ë{ƒtsw [‚w»ftuý Au.
yÚtðt
(R) ™e[u™t «&™tu™t sðtƒ yt…tu. 44444
(1) r {t…™e rºtßÞtðt¤t ð‚wo¤{tk r {t…™e Sðt ð‚wo¤™t fuLÿÚte fux÷t yk‚hu ytðu÷e ntuÞ ?
(2) (P, 3.5) ™t yÄoð‚wo¤ [t…™e ÷kƒtE þtuÄtu.
(3) ABCD [¢eÞ [‚w»ftuý Au. AB yu ABCD ™t …rhð]¥t™tu ÔÞtË Au. òu m∠ADC = 130 ntuÞ ‚tu
m∠BAR þtuÄtu.
(4) (P, 4.5) y™u (Q, 8) …hM…h ykŒhÚte M…þuo ‚tu PQ þtuÄtu.

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