RBSE Math Model Paper 8 (Solution Attached) No of Questions : 30 No of Pages : 4 Zm m H$ mü { H$ narjm, 2019 J{UV m S>b nona 8 g KÊQ>o nyumªh$ 8

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1 RBSE Math Model Paper 8 (Solution Attached) No of Questions : 0 No of Pages : 4 Zm m H$ mü { H$ narjm, 09 J{UV m S>b nona 8 g 4 KÊQ>o nyumªh$ 80 narjm[w Ho$ {be gm mý {ZX}e-. g^r àíz H$aZo A{Zdm h. ^mj àíz g» A H$ àë oh$ àíz A -0 ~ -5 g 6-5 X àíz H«$ m H$ d 9 AmÝV[aH$ {dh$ën h 4. àíz H«$ m H$ 6 H$m boim{mì J«m $ nona na ~ZmBE ^mj-a. gyì {Z{Ib ² H$m àoj H$aVo hþe 9999 # H$m mz kmv H$s{OE. gyì eyý gmå g wàm o Ûmam g rh$au x + + x + 6 x + + x H$m hb kmv H$s{OE +. {~Zm b ~r {d^moz à{h«$ {H$E ~VmBE {H$ 9 4 n[a o g» Ho$ Xe bd àgma gm V h Agm V AmdVu h? 4. tan 55 $ tan 45 $ tan 5 c c c H$m mz kmv H$s{OE 5. 9 rq>a D±$Mo EH$ D$Üdm Ya Iåô H$s N>m H$s bå~mb {H$VZr hmojr, {X Cg g, gy H$m CÝZ Z H$moU 0 hmo? 6. {Og {Ì^wO bå~ho$ýð, n[aho$ýð Am a AÝV Ho$ÝÐ EH$ hr hm, Cg {Ì^wO H$m Zm {b{ie. EH$ {g³ho$ H$mo Xmo ~ma CN>mbm OmVm h H$ -go-h$ EH$ {MV AmZo H$s àm{ H$Vm kmv H$s{OE? 8. {X EH$ Q> ³gr H$m {H$am àw {H$bmo rq>a Ho$ {be <5 VWm BgHo$ ~mx à{v [H$ r. Ho$ {be <8 h Vmo {H$ r. MbZo Ho$ {be ³ {H$am XmoJo? 9. EH$ ~g ñq> ÊS> na EH$ Iåô H$o AmYma go 0 rq>a Xÿar na pñwv ßboQ> $m Ho$ EH$ {~ÝXþ go Iåô Ho$ erf na bjo CCTV H $ ao H$m CÝZ Z H$moU 60c h Vmo Iåô H$s D±$MmB kmv H$s{OE n îr> g»

2 RBSE Math Model Paper 8 (Solution Attached) 0. Xr JB AmH ${V, {X m (Mmn PMQ) 00c, Vmo + PQS H$m mn kmv H$amo ^mj-~. ÛÝÛ oj {d{y Ûmam g» 56 H$m dj yb kmv H$s{OE. {gõ H$s{OE {H$ + 5 EH$ An[a o g» h. EH$ d Îm H$s {ÌÁ go r h VWm Ho$ÝÐ na AÝV[aV H$moU 60c h Mmn H$s bå~mb kmv H$s{OE 4. {X EH$ e Hw$ H$m dh«$n îr> 05 dj go r VWm AmYma H$m ìg 5 go r hmo Vmo e Hw$ H$s {V H$ D±$MmB kmv H$s{OE 5. g wð Ho$ {H$Zmao gwajm H$s ÑpîQ> go EH$ Iå^o na EH$ CCTV H $ am bjm J h {X Bg H $ ao Ho$ Ûmam.5 rq>a bå~m ì p³v.5 rq>a H$s Xÿar na ñnîq> XoIm Om gh$vm h O~{H$ H $ am 5 rq>a H$s D±$MmB na bjm h, Vmo H $ ao H$m AdZ Z H$moU kmv H$s{OE ^mj-g 6. ~hþnx x - 6x + x- 6 Ho$ g^r eyý H$ kmv H$s{OE O~{H$ BgHo$ Xmo eyý H$ VWm h. gwã~mamd Zo 995 <5000 Ho$ m{gh$ dovz na H$m Amaå^ {H$ Am a àë oh$ df <00 H$s dovz d {Õ àmá H$s {H$g df CgH$m dovz <000 hmo J? 8. {X gy H$m CÝZ Z H$moU 0c go 60c n[ad{v V hmo OmVm h Vmo BZ XmoZm CÝZ Z H$moUm na 5 rq>a D±$Mo Iå~o H$s N>m H$s bå~mb AÝVa kmv H$s{OE 9. EH$ T ABC mpü H$mE± AD, BE Am a CF EH$ {~ÝXþ G go JwOaVr h {X AG 6 go r. Am a BE. 6 go r. Am a FG go r. hmo, Vmo AD, GE Am a GC kmv H$s{OE n îr> g»

3 RBSE Math Model Paper 8 (Solution Attached) 0. {X Xmo {Ì^wOm H$moB g JV Xmo ^wome± nañna g mzwnmvr hmo VWm CZHo$ Ü Ho$ H$moU ~am~a hmo Vmo XmoZmo {Ì^wO g ê$n hmovo h. EH$ MVw^w O ABCD AB AC AD hm, Vmo {gõ H$s{OE {H$-+ BAD ^+ BDC + + CBDh. T ABC H$s amzm H$s{OE {Og AB 6 go r, BC 4 go r Am a + B 0c hmo, {Ì^wO Ho$ AÝVJ V d Îm H$s amzm H$s{OE. AmH ${V N>m {H$V joì $b kmv H$s{OE, {X AB 5 go r., AC go r. Am a O d Îm H$m Ho$ÝÐ h 4. go r ìg Ho$ ~obzmh$ma ~V Z Hw$N> nmzr ^am h EH$ 4 go r {ÌÁ H$m R>mog YmVw H$m Jmobm Bg o S>mbm OmVm h nmzr H$s gvh H$s D±$MmB {H$VZr d {Õ hmojr {X Jmobm nmzr nyu V Sy>~ OmVm h? 5. 5 Vme Ho$ nîmm H$s JS²>S>r go EH$ H$mbm Jwbm, EH$ bmb ~oj Am a Xmo H$mbo ~mxemh AbJ H$a {X o OmVo h eof nîmm o go EH$ nvm ÑÀN> {ZH$mbm OmVm h àm{ H$Vm kmv H$s{OE {H$ dh-. EH$ H$mbm nîmm h. EH$ ~mxemh h. EH$ bmb ~oj h n îr> g» ^mj-x 6. {ZåZ a {IH$ g rh$au w½ H$mo Ambo{IH$ {d{y go hb H$s{OE VWm CZ {~ÝXþAm Ho$ {ZX}em H$ kmv H$s{OE Ohm± BZHo$ Ûmam {Zê${nV aoime± y-aj H$mo H$mQ>Vr h (6) x+ y VWm 5x- y 4. {gõ H$s{OE {H$- (+6). () () sec q- cos q - sin q ^ h + sin q () sec q+ cosec q tan q+ cot q AWdm ^ + cot q+ tan qh^sin q-cos qh sin sec q- cosec q () {X x rsin Acos C qcos q, y rsin Asin C VWm z rcos A V~ {gõ H$s{OE x + y + z r

4 RBSE Math Model Paper 8 (Solution Attached) 8. () {~ÝXþ ^x, 5h VWm (4, ) Ho$ Ü Xÿar go r. hmo Vmo x H$m mz kmv H$s{OE () kmv H$s{OE {H$ aoim x+ y 9 {~ÝXþAm (, ) VWm (,) H$mo { bmzo dmbo aoimiês> H$mo {H$g AZwnmV {d^m{ov H$aVr h? 9. {gõ H$r{OE {H$ {X H$moB EH$ aoim {H$gr {Ì^wO Ho$ EH$ AmÝV[aH$ H$moU H$m g {Û^mOZ H$ao Vmo dh g {Û^mOH$ aoim Cg H$moU H$s gå wi ^wom H$mo {Ì^wO H$s eof ^womam H$s bå~mb Ho$ AZwnmV {d^m{ov H$aVr h 6 AWdm 9. AmH ${V o, O d Îm H$m Ho$ÝÐ h Am a + BCO 0c h x Am a y kmv H$s{OE 6 0. {ZåZ{b{IV Am±H$ S>m o H$s mpü H$m 55 h {X ~maå~mavmam H$m oj 00 h, Vmo x d y H$m mz kmv H$s{OE- 6 dj -AÝVamb x y ~maå~mavm n îr> g» 4

5 amoñwmz ~mos> narjm 09 0dt H$jm J{UV m S>b nona 8 g : 4 K Q>o (nyumªh$ : 80) narjm[w Ho$ {be gm mý {ZX}e-. g^r àíz H$aZo A{Zdm h. ^mj àíz g» A H$ àë oh$ àíz A -0 ~ -5 g 6-5 X àíz H«$ m H$ d 9 AmÝV[aH$ {dh$ën h 4. àíz H«$ m H$ 6 H$m boim{mì J«m $ nona na ~ZmBE ^mj-a. gyì {Z{Ib ² H$m àoj H$aVo hþe 9999 # H$m mz kmv H$s{OE. {~Zm b ~r {d^moz à{h«$ {H$E ~VmBE {H$ 9 4 n[a o g» Ho$ Xe bd àgma gm V h Agm V AmdVu h? hm± na, # # 9 ³ {H$ ha n 5 m Zht h ~pëh$ l h AV n[a o g» 9 4 H$m Xe bd àgma Agm V AmdVu hmojm. gyì eyý gmå g wàm o Ûmam g rh$au- x+ x+ 6 x + x + H$m hb kmv H$s{OE dm nj Ho$ ham H$m oj x+ + x+ 6 x () X{jU nj Ho$ ham H$m oj x+ + x+ x () XmoZm g wàm g mz hmozo na gyìmzwwgma, x x 8 x tan 55c$ tan 45c$ tan 5c H$m mz kmv H$s{OE g^r {dúm{w go {ZdoXZ h {H$ RBSE Ho$ gm ëds> m S>b nona/s>oñh$ dh $ àmßv H$aZo Ho$ {be H$mo AnZr ³bmg Ho$ ìhmq²>geßn J«wn ES> H$a AmnH$s ³bmg Ho$ ìhmq²>geßn J«wn nona ^oo {XE OmE Jo y tan 55c$ tan 45c$ tan 5c tan^90c 5ch # tan 45c# tan 5c cot 5c$ ^h$ tan 5c cot 5c$ tan 5c 6 a tan^90c h 6 a tan $

6 Page RBSE Math 0th Model Paper rq>a D±$Mo EH$ D$Üdm Ya Iåô H$s N>m H$s bå~mb {H$VZr hmojr, {X Cg g, gy H$m CÝZ Z H$moU 0 hmo? àízmzwgma, a 0 AV {H$ r. Ho$ {be <0 XoZo hm Jo 9. EH$ ~g ñq> ÊS> na EH$ Iåô H$o AmYma go 0 rq>a Xÿar na pñwv ßboQ> $m Ho$ EH$ {~ÝXþ go Iåô Ho$ erf na bjo CCTV H $ ao H$m CÝZ Z H$moU 60c h Vmo Iåô H$s D±$MmB kmv H$s{OE àízmzwgma, T ABC, AV tan 0 bå~ AmYma 9 AC AC 9 N>m H$s bå~mb 9 rq>a 9 AC 6. {Og {Ì^wO bå~ho$ýð, n[aho$ýð Am a AÝV Ho$ÝÐ EH$ hr hm, Cg {Ì^wO H$m Zm {b{ie g ~mhþ {Ì^wO. EH$ {g³ho$ H$mo Xmo ~ma CN>mbm OmVm h H$ -go-h$ EH$ {MV AmZo H$s àm{ H$Vm kmv H$s{OE? àízmzwgma {g³ho$ H$mo Xmo ~ma CN>mbm OmVm h AV gå^m{dv n[aum ^HH, h, ^HT, h, ^TH, h, ^TT, h ` Hw$b gå^d d n[aum m H$s g» 4 H$ -go-h$ {MV AmZo H$s gå^d pñw{v± ` Hw$b AZwHy$b n[aum ` A^rîQ> àm{ H$Vm 4 ^HH, h, ^HT, h VWm ^TH, h T ABC, tan AB BC bå~ AmYma àízmzwgma, h tan 60c 0 h 0 h 0 h 0 # r. 0. Xr JB AmH ${V, {X m (Mmn PMQ) 00c, Vmo + PQS H$m mn kmv H$amo 8. {X EH$ Q> ³gr H$m {H$am àw {H$bmo rq>a Ho$ {be <5 VWm BgHo$ ~mx à{v [H$ r. Ho$ {be <8 h Vmo {H$ r. MbZo Ho$ {be ³ {H$am XmoJo? àíz go, a n a+ ^n hd a a+ d a a 5, d 8 a 5 + # 8 a ñne aoim N>oXZ aoim à o Ûmam, m+ PQS m (Mmn PMQ) 00 # c 50c g^r Jwê$OZm go {ZdoXZ h {H$ RBSE Ho$ gm ëds> m S>b nona àmßv H$aZo Ho$ {be na {g $ TEACHER eãx ìhmq²>geßn H$a Amngo g nh $ H$a AmnH$mo {deof ê$n go m S>b nona ôoo OmE Jo

7 RBSE Math 0th Model Paper 8 Page ^mj-~. ÛÝÛ oj {d{y Ûmam g» 56 H$m dj yb kmv H$s{OE ÛÝÛ oj {d{y Ûmam g Ho$V-. g» Ho$ VrZ Omo S>o h AV dj yb Ho$ A H$ hm Jo àw dj yb A H$, eof $b 9. Z ^má 5, g emo{yv ^má ^r 5 AV 5 ' 6 5, eof $b 5. Z ^má 5, g emo{yv ^má ' 6 4, eof $b 4, hm± ApÝV eof $b kmv H$aZm h Z ^má 4 # 5# 4 Z ^má 6, ApÝV eof $b AV {gõ H$s{OE {H$ + 5 EH$ An[a o g» h mzm {H$ + 5 EH$ n[a o g» h p V~ + 5 hmozm Mm{hE O~{H$ q! 0 Am a p Am a q q nyumªh$ h A~ p q p q q a p Am a q nyumªh$ h AV p e 5 q o p q ` EH$ n[a o g» h q 5 ^r EH$ n[a o g» hmojr naývw 5 n[a o g» Zht, An[a o g» h, V~ hm± {damoym^mg h Bg {damoym^mg Ho$ H$maU h mar H$ënZm JbV h AV Xr JB g» + 5 An[a o g» h {X h - d Îm H$s {ÌÁ ^r h go r Ho$ÝÐ na AÝV[aV H$moU, 60c h OmZVo h {H$ {ÌÁ IÊS> Ho$ Mmn H$s bå~mb, l π r 80c # # go r. AV Mmn H$s bå~mb, l. go r 4. {X EH$ e Hw$ H$m dh«$n îr> 05 dj go r VWm AmYma H$m ìg 5 go r hmo Vmo e Hw$ H$s {V H$ D±$MmB kmv H$s{OE {X h, AmYma H$m ìg, d 5 go r AmYma H$s {ÌÁ, r 5 go r. 5 go r e Hw$ H$m dh«$n îr>, A 05 dj go r π rl 05 5 l # b l # 05 l 05 go r 55 AV e Hw$ H$s {V H$ D±$MmB go r. 5. g wð Ho$ {H$Zmao gwajm H$s ÑpîQ> go EH$ Iåô na EH$ CCTV H $ am bjm J h {X Bg H $ ao Ho$ Ûmam.5 rq>a bå~m ì p³v.5 rq>a H$s Xÿar na ñnîq> XoIm Om gh$vm h O~{H$ H $ am 5 rq>a H$s D±$MmB na bjm h, Vmo H $ ao H$m AdZ Z H$moU kmv H$s{OE àízmzwgma,. EH$ d Îm H$s {ÌÁ go r h VWm Ho$ÝÐ na AÝV[aV H$moU 60c h Mmn H$s bå~mb kmv H$s{OE {MÌ Ûmam, tan tan BE AB CD AC BC â BE CDh g^r {dúm{w go {ZdoXZ h {H$ RBSE Ho$ gm ëds> m S>b nona/s>oñh$ dh $ àmßv H$aZo Ho$ {be H$mo AnZr ³bmg Ho$ ìhmq²>geßn J«wn ES> H$a AmnH$s ³bmg Ho$ ìhmq²>geßn J«wn nona ôo {XE OmE Jo

8 Page 4 RBSE Math 0th Model Paper 8 tan tan. 5 tan 45c. 5 AV 45c ^mj-g 6. ~hþnx x 6x + x 6 Ho$ g^r eyý H$ kmv H$s{OE O~{H$ BgHo$ Xmo eyý H$ VWm h a h OmZVo h {H$ {X a ~hþnx fx ^ h H$m EH$ eyý H$ h, V~ ^x ahf^xh H$m EH$ JwUZIÊS> hmojm àízmzwgma, ~hþnx fx ^ h x 6x + x 6 Ho$ Xmo eyý H$ VWm h AV ^x h^x h ^x x+ h, fx ^ h H$m EH$ JwUZIÊS> hmojm A~ ~hþnx fx ^ h H$mo x x+ go ^mj XoZo na, Vrgao df dovz <5400 à{v mg Bg àh$ma àë oh$ df Ho$ dovz (< ) 5000, 500, 5400,... EH$ g mýva lour ~ZmVo h, {OgH$m àw nx a 5000 VWm gmd AÝVa d 00 mzm n df ~mx dovz <000 hmojm V~ n dm± nx 000 a+ ^n h d ] n g ^n h # ^n h # ^n h ` n 0 + AV d df AWm V² 006 gwã~mamd H$m dovz <000 hmojm 8. {X gy H$m CÝZ Z H$moU 0c go 60c n[ad{v V hmo OmVm h Vmo BZ XmoZm CÝZ Z H$moUm na 5 rq>a D±$Mo Iå~o H$s N>m H$s bå~mb AÝVa kmv H$s{OE mzm {H$ 5 rq>a bå~o Iå^o H$s N>m H$s bå~mb CÝZ Z H$moU 0c hmozo na x rq>a VWm CÝZ Z H$moU 60c hmozo na y rq>a h ~hþnxm Ho$ {be {d^moz EëJmo[aW H$m àoj H$aZo na ^má ^moh$ # ^mj $b + eof $ëm x 6x + x 6 ^x x+ h^x h+ 0 ^x h^x h^x h ~hþnxm Ho$ eyý H$m Ho$ {be, fx ^ h 0 ^x h^x h^x h 0 x 0, x 0 x 0 x,, AV ~hþnx fx ^ h Ho$ g^r eyý H$,, h. gwã~mamd Zo 995 <5000 Ho$ m{gh$ dovz na H$m Amaå^ {H$ Am a àë oh$ df <00 H$s dovz d {Õ àmá H$s {H$g df CgH$m dovz <000 hmo J? {X h, nhbo df o àmapå^h$ dovz <5000 à{v mg Xÿgao df dovz <500 à{v mg g^r Jwê$OZm go {ZdoXZ h {H$ RBSE Ho$ gm ëds> m S>b nona àmßv H$aZo Ho$ {be na {g $ TEACHER eãx ìhmq²>geßn H$a Amngo g nh $ H$a AmnH$mo {deof ê$n go m S>b nona ^ooo OmE Jo AV T ABC go, tan 0c tan 0c x 5 bå~ AmYma CB AB 5 x x 5 rq>a nwz T DBC go tan AmYma tan 60c 5 y 5 # rq>a...() bå~ 5 y y 5 y 5 CB DB 5 # 5

9 RBSE Math 0th Model Paper 8 Page 5 y 5 # rq>a...() AV bå~mb AÝVa x y rq>a. rq>a AV bå~mb. rq>a H$m AÝVa hmojm 9. EH$ T ABC mpü H$mE± AD, BE Am a CF EH$ {~ÝXþ G go JwOaVr h {X AG 6 go r. Am a BE. 6 go r. Am a FG go r. hmo, Vmo AD, GE Am a GC kmv H$s{OE h OmZVo h {H$ Ho$ÝÐH$ G {Ì^wO H$s mpü H$m H$mo Ho$ AZwnmV o {d^m{ov H$aVm h AV AG GD GD AG XmoZm Amoa Omo S>Zo na GD + + AG GD + AG AG AD AG + AD a AG 6 6 go r. {X h AD # 6 AD 9 go r. Bgr àh$ma BG GE BG + GE + BG + GE GE BE GE + GE BE Am a FG GC FG GC GC # 6 go r 0. {X Xmo {Ì^wOm H$moB g JV Xmo ^wome± nañna g mzwnmvr hmo VWm CZHo$ Ü Ho$ H$moU ~am~a hmo Vmo XmoZmo {Ì^wO g ê$n hmovo h {X h T ABC Ed T DEF AB AC DE Ed + A + D DF h {gõ H$aZm h -TABC + TDEF amzm-t DEF AB DP, AC DQ H«$ e DE Ed DF go H$m{Q>E VWm P d Q H$mo { bmbe Cnn{Îm-T ABC Ed T DPQ AB DP, + A + D VWm AC DQ (amzm Ûmam) AV gdmªjg Vm Ho$ SAS {Z go, T ABC, TDPQ...() A~ AB DE DP DE AC ({X hþam h ) DF DQ DF (amzm go AB DP Ed AC DQ) PQ EF (Woëg à o Ho$ {dbmo Ûmam) + DPQ + E Ed + DQP + F (g JV H$moU) Bg àh$ma AA g ê$nvm {Z go, T DPQ + TDEF...() g rh$au () d () go, T ABC, TDPQ VWm TDPQ + TDEF T ABC + TDPQ VWm TDPQ + TDEF (gdmªjg {Ì^wO g ê$n hmovo h ) T ABC + TDEF B{V{gÕ ². EH$ MVw^w O ABCD AB AC AD hm, Vmo {gõ H$s{OE {H$ + BAD ^+ BDC + + CBDh. g^r {dúm{w go {ZdoXZ h {H$ RBSE Ho$ gm ëds> m S>b nona/s>oñh$ dh $ àmßv H$aZo Ho$ {be H$mo AnZr ³bmg Ho$ ìhmq²>geßn J«wn ES> H$a AmnH$s ³bmg Ho$ ìhmq²>geßn J«wn nona ^oo {XE OmE Jo

10 Page 6 RBSE Math 0th Model Paper 8 {X h {H$ AB AC AD AWm V² {~ÝXþ BC, Am a D {~ÝXþ A go g mz Xÿar na h, AV d Îm H$m Ho$ÝÐ A h A~ Mmn BC Ho$ÝÐ na + BAC Am a d Îm Ho$ eof ^mj na + BDC ~ZmVm h ` + BAC + BDC...() Bgr àh$ma Mmn CD Ho$ÝÐ na + CAD Am a d Îm Ho$ eof ^mj na + CBD ~ZmVm h ` + CAD + CBD...() g rh$au () Amoa () H$m oj H$aZo na + BAC + + CAD ^+ BDC + + CBDh + BAD ^+ BDC + + CBDh B{V{gÕ ². T ABC H$s amzm H$s{OE {Og AB 6 go r, BC 4 go r Am a + B 0c hmo, {Ì^wO Ho$ AÝVJ V d Îm H$s amzm H$s{OE amzm Ho$ nx-. gd àw aoim BC 4 go r. ItMr {~ÝXþ B na 0c H$m H$moU ~Zm J VWm H$moU dmbr aoim na 6 go r. H$m Mmn H$mQ>m dhm± {~ÝXþ A {bim A {~ÝXþ H$mo C go { bm Bg àh$ma {Ì^wO ABC ~Zm. + B VWm C + H$s g {Û^m{OV aoime± ItMH$a CZH$m n[aàn>ox {~ÝXþ O àmá {H$. O go ^wom AC na bå~ OK ItMm 4. O Ho$ÝÐ na OK {ÌÁ boh$a d Îm ItM {X hr A^rîQ> AÝV d Îm h a BC d Îm H$m ìg VWm A AÕ d Îm H$s n[a{y na pñwv {~ÝXþ h ` + BAC 90c AV g H$moU T ABC BC AC + AB BC BC 69 BC 69 BC a BC d Îm H$m ìg go r. d Îm H$s {ÌÁ go r. AÕ d Îm H$m joì $b π r go r # # # # # go r # # 859 go r 8 g H$moU T ABC H$m joì $b, AB AC # # 5 go r # # 5# 6go r 0 go r N>m {H$V joì $b AÕ d Îm H$m joì $b g H$moU T ABC H$m joì $b 859 go r go r go r 8. AmH ${V N>m {H$V joì $b kmv H$s{OE, {X AB 5 go r., AC go r. Am a O d Îm H$m Ho$ÝÐ h 4. go r ìg Ho$ ~obzmh$ma ~V Z Hw$N> nmzr ^am h EH$ 4 go r {ÌÁ H$m R>mog YmVw H$m Jmobm Bg o S>mbm OmVm h nmzr H$s gvh H$s D±$MmB {H$VZr d {Õ hmojr {X Jmobm nmzr nyu V Sy>~ OmVm h? [X h, Jmobo H$s {ÌÁ 4 go r Jmobo H$m Am VZ 4 π ^4h 56 π go r ~obzmh$ma ~V Z Ho$ {be, {ÌÁ r 6 go r. mzm nmzr H$s gvh H$s D±$MmB d {Õ h go r g^r Jwê$OZm go {ZdoXZ h {H$ RBSE Ho$ gm ëds> m S>b nona àmßv H$aZo Ho$ {be na {g $ TEACHER eãx ìhmq²>geßn H$a Amngo g nh $ H$a AmnH$mo {deof ê$n go m S>b nona ^ooo OmE Jo

11 RBSE Math 0th Model Paper 8 Page ` {dñwm{nv nmzr H$m Am VZ π rh π ^6h h 6π h A~ {dñwm{nv nmzr H$m Am VZ Jmobo H$m Am VZ 6π h h 56π 56 # 6 64 go r go r. ` ~V Z Ho$ nmzr H$s gvh H$s D±$MmB d {Õ 64/ 5. 5 Vme Ho$ nîmm H$s JS²>S>r go EH$ H$mbm Jwbm, EH$ bmb ~oj Am a Xmo H$mbo ~mxemh AbJ H$a {X o OmVo h eof nîmm o go EH$ nvm ÑÀN> {ZH$mbm OmVm h àm{ H$Vm kmv H$s{OE {H$ dh-. EH$ H$mbm nîmm h. EH$ ~mxemh h. EH$ bmb ~oj h H$mbm Jwbm, bmb ~oj Am a H$mbo ~mxemh {ZH$mbZo Ho$ ~mx, JS²>S>r eof nîmo ` eof Hw$b gå^mdr Adga 48 () JS²>S>r eof H$mbo nîmm H$s g» 6 ` H$mbo nîmo H$s àm{ H$Vm 48 () JS²>S>r eof ~mxemhm H$s g» 4 ` EH$ ~mxemh {ZH$bZo Ho$ AZwHy$b Adgam H$s g» EH$ ~mxemh {ZH$bZo H$s àm{ H$Vm 48 4 () JS²>S>r o eof bmb ~oj ` EH$ bmb ~oj {ZH$mbZo H$s àm{ H$Vm 48 x aizo na V~, y x 4 aizo na, V~ y # 6 # AV g rh$au x+ y H$s {ZåZ hb gmaur àmá hmovr h - {~ÝXþAm o A^, h d B^40, h H$mo J«m $ nona na A {H$V H$aHo$ BZgo JwOaZo dmbr aoim AB ItMVo h Omo {H$ g rh$au x+ y H$m AmboI h g rh$au 5x y 4 Ûmam {Zê${nV aoim H$m AmboIZ 5x y 4 y 5x 4 y 5x 4 x 0 aizo na, V~ y 5# x aizo na, V~ y 5# 4 6 AV g rh$au 5x y 4 H$s {ZåZ hb gmaur àmßv hmovr h - x 0 y {~ÝXþAm C^0, h d A^, h H$mo J«m $ nona na A {H$V H$aHo$ BZgo JwOaZo dmbr aoim AC ItMVo h Omo {H$ g rh$au 5x y 4 H$m AmboI h ^mj-x 6. {ZåZ a {IH$ g rh$au w½ H$mo Ambo{IH$ {d{y go hb H$s{OE VWm CZ {~ÝXþAm Ho$ {ZX}em H$ kmv H$s{OE Ohm± BZHo$ Ûmam {Zê${nV aoime± y-aj H$mo H$mQ>Vr h (6) x+ y VWm 5x y 4 {X o J o a {IH$ g rh$au w½ h - y...() x+ 5x y 4...() g rh$au () go, x+ y y x y x g^r {dúm{w go {ZdoXZ h {H$ RBSE Ho$ gm ëds> m S>b nona/s>oñh$ dh $ àmßv H$aZo Ho$ {be H$mo AnZr ³bmg Ho$ ìhmq²>geßn J«wn ES> H$a AmnH$s ³bmg Ho$ ìhmq²>geßn J«wn nona ^oo {XE OmE Jo

12 Page 8 RBSE Math 0th Model Paper 8 J«m $ go ñnîq> h {H$ XmoZm g rh$aum Ho$ g JV gab aoim EH$-Xÿgao H$mo {~ÝXþ A^, h na H$mQ>Vr h AV x VWm y {X o J o g rh$au w½ H$m hb h o XmoZm gab aoim y-aj H$mo {~ÝXþ (0,6) Am a ^0, h na H$mQ>Vr h. {gõ H$s{OE {H$- (+6) () sec cos sin ^ h + sin () sec + cosec tan + cot () L.H.S. go ^sec tan h b sin l cos cos sin sin b l ^ h cos cos ^ sin h sin ^ sin h ^ sin h^ + sin h sin R.H.S. + B{V{gÕ ² sin () L.H.S. sec + cosec cos sin (LCM bozo na) sin + cos sin cos 6 a sin + sin cos sin + cos 6 a sin + sin cos sin + cos sin cos sin cos sin + cos cos sin tan + cot R.H.S. B{V{gÕ ² AWdm ^ + cot + tan h^sin cos h. () sin cos sec cosec () {X x rsin Acos C, y rsin Asin C VWm z rcos A V~ {gõ H$s{OE x + y + z r ^ + cot + tan h^sin cos h () L.H.S. sec cosec ^sin cos + cos + sin h^sin cos h sin cos sin cos sin cos ^sin cos h # sin cos sin cos ^sin cos h a b a b 8a ^ h^a + b + abhb sin cos sin cos sin cos R.H.S. B{V{gÕ ² () L.H.S. x + y + z ^rsinacosch + ^rsinasinch + ^rcosah r sin Acos C+ r sin Asin C+ r cos A r sin A^cos C+ sin Ch+ r cos A r sin A+ r cos A r ^sin A+ cos Ah 6 a A+ r sin cos R.H.S. B{V{gÕ ² 8. () {~ÝXþ ^x, 5h VWm (4, ) Ho$ Ü Xÿar go r. hmo Vmo x H$m mz kmv H$s{OE () kmv H$s{OE {H$ aoim x+ y 9 {~ÝXþAm (, ) VWm (,) H$mo { bmzo dmbo aoimiês> H$mo {H$g AZwnmV {d^m{ov H$aVr h? () Xmo {~ÝXþAm Ho$ ~rm H$s Xÿar, d ^x x h + ^y y h hm±, d, x 4, y 5, y ^h ^4 xh + ^ 5h ^4 xh + ^ h 9 ^4 xh + 9 cos sin b + + l^sin cos h sin cos cos sin x mx mx + m+ m g^r Jwê$OZm go {ZdoXZ h {H$ RBSE Ho$ gm ëds> m S>b nona àmßv H$aZo Ho$ {be na {g $ TEACHER eãx ìhmq²>geßn H$a Amngo g nh $ H$a AmnH$mo {deof ê$n go m S>b nona ^ooo OmE Jo ` ^4 xh 0 ^4 xh 0 x 4 () mzm {~ÝXþ A^, h VWm B^, h H$mo { bmzo dmbm aoimiês> x+ y 9 H$mo λ : o {d^m{ov H$aVm h AV {d^moz {~ÝXþ Ho$ {ZX}em H$

13 RBSE Math 0th Model Paper 8 Page 9 VWm y my m + my + m λ^h+ ^h ` x λ + λ + λ + λ^h+ ^h Bgr àh$ma, y λ + λ + λ + AV {d^moz {~ÝXþ Pc λ +, λ + m λ + λ + h {~ÝXþ aoim x+ y 9 na pñwv hm Jo Am a BgH$mo g VwîQ> H$a Jo AV mz aizo na, $ c λ + m + λ + 9 λ + λ + 6λ + + λ + 9 λ + λ + 6λ+ + λ+ 9^λ + h λ + 6 9λ + 9 λ 9λ 9 6 4λ ` λ 4 AV A^rîQ> AZwnmV : 4h 9. {gõ H$r{OE {H$ {X H$moB EH$ aoim {H$gr {Ì^wO Ho$ EH$ AmÝV[aH$ H$moU H$m g {Û^mOZ H$ao Vmo dh g {Û^mOH$ aoim Cg H$moU H$s gå wi ^wom H$mo {Ì^wO H$s eof ^womam H$s bå~mb Ho$ AZwnmV {d^m{ov H$aVr h 6 {X hþam h -T ABC AD, + A H$m g {Û^mOH$ h AV + + {gõ H$aZm h - BD DC AB AC amzm-aoim CE Bg àh$ma ItMr JB h {H$ DA CE hmo Vmo BA H$mo AmJo ~ T>mZo na E na { bvr h Cnn{Îm- CE DA Am a AC Am a BE {V H$ aoime± h () d () go AV T ACE, AE AC...() T BCE DA CA Vmo AmYma^yV g mzwnm{vh$vm à o Ûmam AWm V² BD DC BD DC BD DC BA AE BA AC (g r. go) BA AC B{V{gÕ ² AWdm 9. AmH ${V o, O d Îm H$m Ho$ÝÐ h Am a + BCO 0c h x Am a y kmv H$s{OE 6 + BCO 0c g H$moU {Ì^wO OCP, + POC 80c ^+ OPC + + PCOh + POC 80c ^90c+ 0ch 60c + AOD 90c ({X h ) ` + AOD + + DOP 80c AV + DOP 80c + AOD 80c 90c 90c + COD 90c + POC 90c 60c 0c AV + + (EH$mÝVa H$moU)...() Ed (g JV H$moU)...() naývw + + ({X hþam) h OmZVo h {H$ EH$ Mmn Ûmam d Îm H$o Ho$ÝÐ na AÝV[aV H$moU d Îm Ho$ eof ^mj Ho$ {H$gr {~ÝXþ na AÝV[aV H$moU H$m XmoJwZm hmovm h g^r {dúm{w go {ZdoXZ h {H$ RBSE Ho$ gm ëds> m S>b nona/s>oñh$ dh $ àmßv H$aZo Ho$ {be H$mo AnZr ³bmg Ho$ ìhmq²>geßn J«wn ES> H$a AmnH$s ³bmg Ho$ ìhmq²>geßn J«wn nona ^oo {XE OmE Jo ` + CBD + COD y 0 # c 5c

14 Page 0 RBSE Math 0th Model Paper 8 VWm + ABD + AOD 90 # c 45c A~ T ABP x+ ^45c+ yh + 90c 80c x + 45c+ 5c+ 90c 80c x 80c 50c 0c AV x 0c VWm y 5c 0. {ZåZ{b{IV Am±H$ S>m o H$s mpü H$m 55 h {X ~maå~mavmam H$m oj 00 h, Vmo x d y H$m mz kmv H$s{OE- 6 dj -AÝVamb ~maå~mavm x y dj AÝVamb ~ma ~mavm g M r ~ma ~mavm x + x dj AÝVamb ~ma ~mavm g M r ~ma ~mavm x x x y 56 + x+ y x+ y x+ y x+ y hm± {X J h {H$ n 00 h AV 6 + x+ y 00 AWm V² x+ y 4...() mpü H$m 55 h, Omo dj pñwv h AV l 500, f 0, cf 6 + x h 00 h n cf a mü H$ l + f ph f x b 0 l# ^4 xh# x 5 x AV x 9 Bg{bE g rh$au () go h o àmá hmovm h {H$, 9 + y 4 y 4 9 y 5 g^r Jwê$OZm go {ZdoXZ h {H$ RBSE Ho$ gm ëds> m S>b nona àmßv H$aZo Ho$ {be na {g $ TEACHER eãx ìhmq²>geßn H$a Amngo g nh $ H$a AmnH$mo {deof ê$n go m S>b nona ^ooo OmE Jo