J{UV 65/2/1/F. {ZYm [av g_` : 3 KÊQ>o A{YH$V_ A H$ : 100. Series SSO/2

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1 Series SSO/ mob Z. Roll No. H$moS> Z. Code No. SET- 65///F nrjmwu H$moS >H$mo CÎm-nwpñVH$m Ho$ _wi-n ð >n Adí` {bio & Cndidtes must write the Code on the title pge of the nswer-book. H $n`m Om±M H$ b {H$ Bg àíz-nì _o _w{ðv n ð> h & àíz-nì _ Xm{hZo hmw H$s Amo {XE JE H$moS >Zå~ H$mo N>mÌ CÎm-nwpñVH$m Ho$ _wi-n ð> n {bi & H $n`m Om±M H$ b {H$ Bg àíz-nì _ >6 àíz h & H $n`m àíz H$m CÎm {bizm ewê$ H$Zo go nhbo, àíz H$m H«$_m H$ Adí` {bi & Bg àíz-nì H$mo n T>Zo Ho$ {be 5 {_ZQ >H$m g_` {X`m J`m h & àíz-nì H$m {dvu nydm _ 0.5 ~Oo {H$`m OmEJm & 0.5 ~Oo go 0.30 ~Oo VH$ N>mÌ Ho$db àíz-nì H$mo n T> Jo Am Bg Ad{Y Ho$ Xm mz do CÎm-nwpñVH$m n H$moB CÎm Zht {bi Jo & Plese check tht this question pper contins printed pges. Code number given on the right hnd side of the question pper should be written on the title pge of the nswer-book by the cndidte. Plese check tht this question pper contins 6 questions. Plese write down the Seril Number of the question before ttempting it. 5 minute time hs been llotted to red this question pper. The question pper will be distributed t 0.5.m. From 0.5.m. to 0.30.m., the students will red the question pper only nd will not write ny nswer on the nswer-book during this period. J{UV MATHEMATICS {ZYm [V g_` : 3 KÊQ>o A{YH$V_ A H$ : 00 Time llowed : 3 hours Mimum Mrks : 00 65///F P.T.O.

2 gm_mý` {ZX}e : (i) (ii) (iii) (iv) g^r àíz A{Zdm` h & H $n`m Om±M H$ b {H$ Bg àíz-nì _ 6 àíz h & IÊS> A Ho àíz 6 VH$ A{V bkw-cîm dmbo àíz h Am àë`oh$ àíz Ho$ {be A H$ {ZYm [V h & IÊS ~ Ho àíz 7 9 VH$ XrK -CÎm I àh$m Ho$ àíz h Am àë`oh$ àíz Ho$ {be 4 A H$ {ZYm [V h & (v) IÊS> g Ho àíz 0 6 VH$ XrK -CÎm II àh$m Ho$ àíz h Am àë`oh$ àíz Ho$ {be 6 A H$ {ZYm [V h & (vi) CÎm {bizm àmå^ H$Zo go nhbo H $n`m àíz H$m H«$_m H$ Adí` {b{ie & Generl Instructions : (i) (ii) (iii) All questions re compulsory. Plese check tht this question pper contins 6 questions. Questions 6 in Section A re very short-nswer type questions crrying mrk ech. (iv) Questions 7 9 in Section B re long-nswer I type questions crrying 4 mrks ech. (v) (vi) Questions 0 6 in Section C re long-nswer II type questions crrying 6 mrks ech. Plese write down the seril number of the question before ttempting it. 65///F

3 IÊS> A SECTION A àíz g»`m go 6 VH$ àë`oh$ àíz H$m A H$ h & Question numbers to 6 crry mrk ech.. g[xemo ^i + 3 ^j ^k Am 4^i 3 ^j + ^k Ho$ `moj\$b Ho$ AZw{Xe _mìh$ g{xe kmv H$s[OE & Find the unit vector in the direction of the sum of the vectors ^i + 3 ^j ^k nd 4^i 3 ^j + ^k.. EH$ g_mýv MVw^ O H$m joì\$b kmv H$s{OE {OgH$s g b½z ^wome± g{xem ^i 3 ^k VWm 4 ^j + ^k Ûmm {ZYm [V h & Find the re of prllelogrm whose djcent sides re represented by the vectors ^i 3 ^k nd 4 ^j + ^k. 3. g_vb + y z = 5 Ûmm {ZX}em H$ Ajm n H$mQo JE A V IÊS>m H$m `moj\$b kmv H$s{OE : Find the sum of the intercepts cut off by the plne + y z = 5, on the coordinte es. 4. `{X 5 A , Vmo Xÿgr n {º$ Ho$ Ad`d H$m ghiês {b{ie> & 5 If A 4 4 nd row , then write the cofctor of the element of its 3 65///F 3 P.T.O.

4 3 d 4 y dy 5. AdH$b g_rh$u 0 65///F 4 H$s H$mo{Q> d KmV H$m `moj\$b {b{ie & Write the sum of the order nd degree of the differentil eqution 3 d 4 y dy 0. dy y 6. AdH$b g_rh$u H$m hb {b{ie & Write the solution of the differentil eqution dy y. IÊS> ~ SECTION B àíz g»`m 7 go 9 VH$ àë`oh$ àíz Ho$ 4 A H$ h & Question numbers 7 to 9 crry 4 mrks ech. 7. `{X A Am I EH$ H$mo{Q> H$m VËg_H$ Amì`yh hmo, Vmo {XImBE {H$ A = 4 A 3 I. AV: A kmv H$s{OE & `{X A, kmv H$s{OE & If A b AWdm B Am (A + B) = A + B h, Vmo Am b Ho$ _mz nd I is the identity mtri of order, then show tht A = 4 A 3 I. Hence find A. If OR A nd B nd (A + B) = A + B, then find the b vlues of nd b.

5 8. gm{uh$m Ho$ JwUY_mªo H$m à`moj H$Ho$, {ZåZ{b{IV H$mo {gõ H$s{OE : 3 Using properties of determinnts, prove the following : 3 9. _mz kmv H$s{OE : sin ( ) sin ( ) AWdm _mz kmv H$s{OE : ( 4) ( 9) Evlute : sin ( ) sin ( ) OR Evlute : ( 4) ( 9) 65///F 5 P.T.O.

6 0. _mz kmv H$s{OE : / / Evlute : / / cos e e cos. {~Obr Ho$ ~ë~ ~ZmZo dmbr EH$ H$ånZr _ VrZ _erz E, E Am E 3 {XZ^ Ho$ CËnmXZ H$m H«$_e: 50%, 5% VWm 5% ^mj V `m H$Vr h & `h kmv h {H$ _erz E Am E àë`oh$ go ~Zo ~ë~m _ 4% Im~ ~ë~ hmovo h, Am _erz E 3 go ~Zo ~ë~m _ 5% Im~ hmovo h & `{X {XZ Ho$ CËnmXZ _ go EH$ ~ë~ `mñàn>`m MwZm OmE, Vmo Bg ~ë~ Ho$ Im~ hmozo H$s àm{`h$vm kmv H$s{OE & AWdm YZ nyum H$m, 3, 4, 5, 6 VWm 7 _ go Xmo g»`me± `mñàn>`m ({~Zm à{vñwmnz) MwZr JBª & _mz br{oe X XmoZm g»`mam _ go ~ S>r g»`m H$mo ì`º$ H$Vm h & X Ho$ àm{`h$vm ~ Q>Z H$m _mü` VWm àgu kmv H$s{OE & Three mchines E, E nd E 3 in certin fctory producing electric bulbs, produce 50%, 5% nd 5% respectively, of the totl dily output of electric bulbs. It is known tht 4% of the bulbs produced by ech of mchines E nd E re defective nd tht 5% of those produced by mchine E 3 re defective. If one bulb is picked up t rndom from dy s production, clculte the probbility tht it is defective. OR Two numbers re selected t rndom (without replcement) from positive integers, 3, 4, 5, 6, nd 7. Let X denote the lrger of the two numbers obtined. Find the men nd vrince of the probbility distribution of X. 65///F 6

7 . Xmo g{xe ^j + ^k VWm 3^i ^j + 4 ^k {Ì^wO ABC Ho$ Xmo ^wom g{xe H«$_e AB VWm AC H$mo {Zê${nV H$Vo h & A go JwµOZo dmbr _mpü`h$m H$s b ~mb kmv H$s{OE & The two vectors ^j + ^k nd 3^i ^j + 4 ^k represent the two side vectors AB nd ACrespectively of tringle ABC. Find the length of the medin through A. 3. Cg g_vb H$m g_rh$u kmv H$s{OE, Omo {~ÝXþ (3,, 0) go JwµOVm hmo VWm oim 3 y 6 z 4 H$mo AÝV{d îq> H$Vm hmo & 5 4 Find the eqution of plne which psses through the point (3,, 0) nd 3 y 6 z 4 contins the line `{X tn (cos ) = tn ( cosec ), ( 0), Vmo H$m _mz kmv H$s{OE & `{X AWdm tn tn... tn tn h,..3 n. (n ) Vmo H$m _mz kmv H$s{OE & If tn (cos ) = tn ( cosec ), ( 0), then find the vlue of. OR If tn tn... tn tn,..3 n. (n ) then find the vlue of. 5. dh«$ 9y = 3 H$m dh {~ÝXþ kmv H$s{OE {Og n dh«$ n ItMm J`m A{^bå~ Ajm n EH$g_mZ A V:IÊS> H$mQ>Vm hmo & Find the point on the curve 9y = 3, where the norml to the curve mkes equl intercepts on the es. 65///F 7 P.T.O.

8 6. `{X y n h, Vmo {XImBE {H$ d y dy n y. If y n, then show tht d y dy n y. 7. kmv H$s{OE {H$ {ZåZ{b{IV \$bz = VWm = n AdH$bZr` h AWdm Zht : f (),, 3, Find whether the following function is differentible t = nd = or not :, f (), 3, 8. g gxr` MwZmd _, EH$ mozr{vh$ nmq>u Zo àmm H$Zo dmbr EH$ \$_ H$mo nmq>u Ho$ Cå_rXdmm H$mo àmm H$Zo _ gh`moj XoZo Ho$ {be {Z`wº$ {H$`m & àmm VrZ VrH$m go H$Zm Wm Q>obr\$moZ go, K-K OmH$ {_bzm VWm nì-ì`dhm go & à{v `y{zq> (gånh $) H$m IMm (n gm _ ) {ZåZ Amì`yh A go ZrMo {X`m J`m h : 40 A Qobr\ moz$ K-K OmH$ {_bzm nì - ì`dhm 65///F 8

9 Xmo ehm X VWm Y _ àë`oh$ àh$m Ho$ Hw$b `y{zq>m (gånh$m]) H$m {ddu ZrMo Amì`yh B _ {X`m J`m h : Q>obr\$moZ 000 B 3000 K-K OmH$ {_bzm nì-ì`dhm eh eh X Y nmq>u Zo XmoZm ehm _ Hw$b {H$VZm IM {H$`m? AmnHo$»`mb _ Amn AnZm dmoq> XoZo go nhbo nmq>u H$s {H$g àh$m H$s J{V{d{Y H$mo µo`mxm _hîd X Jo àmm J{V{d{Y `m CZH$s gm_m{oh$ J{V{d{Y`m±? In prliment election, politicl prty hired public reltions firm to promote its cndidtes in three wys telephone, house clls nd letters. The cost per contct (in pise) is given in mtri A s 40 A Telephone House Cll Letters The number of contcts of ech type mde in two cities X nd Y is given in the mtri B s Telephone House Cll Letters City X B City Y Find the totl mount spent by the prty in the two cities. Wht should one consider before csting his/her vote prty s promotionl ctivity or their socil ctivities? 65///F 9 P.T.O.

10 9. _mz kmv H$s{OE : Evlute : e. sin 3 e. sin 3 IÊS> g SECTION C àíz g»`m 0 go 6 VH$ àë`oh$ àíz Ho$ 6 A H$ h & Question numbers 0 to 6 crry 6 mrks ech. 0. _mzm f : N, f() = Ûmm n[^m{fv EH$ \$bz h & {gõ H$s{OE {H$ f : N S, Ohm± S, \$bz f H$m n[g h, ì`wëh«$_ur` h & f H$m à{vbmo_ ^r kmv H$s{OE & Let f : N be function defined s f() = Show tht f : N S, where S is the rnge of f, is invertible. Also find the inverse of f.. g_mh$bz {d{y go oim y + = 0, dh«$ y VWm y-aj Ho$ ~rm {Ko joì H$m joì\$b kmv H$s{OE & Using integrtion, find the re of the region bounded by the line y + = 0, the curve y nd y-is.. `{X y = c h, Vmo ( + by) H$m Ý`yZV mz kmv H$s{OE & AWdm 65///F 0

11 ndb` y = n EH$ Eogm {~ÝXþ kmv H$s{OE Omo gb oim y = 3 3 go Ý`yZV_ Xÿr n hmo & Find the minimum vlue of ( + by), where y = c. OR Find the coordintes of point of the prbol y = which is closest to the stright line y = {ZåZ AdmoYm Ho$ AÝVJ V z = 8 + 9y H$m A{YH$V_rH$U H$s{OE : + 3y 6 3 y 6 y, y 0 Mimise z = 8 + 9y subject to the constrints given below : + 3y 6 3 y 6 y, y 0 4. g_vb y + z = 5 go {~ÝXþ (,, 3) H$s dh Xÿr kmv H$s{OE, Omo Cg oim Ho$ g_mýv h, {OgHo$ {XH²$-H$mogmBZ, 3, 6 Ho$ g_mzwnmvr h & Find the distnce of the point (,, 3) from the plne y + z = 5 mesured prllel to the line whose direction cosines re proportionl to, 3, {ZåZ{b{IV AdH$b g_rh$u H$m hb kmv H$s{OE : y y cos dy y y y cos sin 0 AWdm 65///F P.T.O.

12 {ZåZ{b{IV AdH$b g_rh$u H$mo hb H$s{OE : y y y dy 0 Solve the following differentil eqution : y y cos dy y y y cos sin 0 OR Solve the following differentil eqution : y y y dy 0 6. nmgm Ho$ EH$ Omo S>o H$mo Mm ~m CN>mbZo n {ÛH$m H$s g»`m H$m àm{`h$vm ~ Q>Z kmv H$s{OE & Bg ~ Q>Z H$m _mü` VWm àgu ^r kmv H$s{OE & Find the probbility distribution of the number of doublets in four throws of pir of dice. Also find the men nd vrince of this distribution. 65///F

J{UV 65/1/C. {ZYm [av g_` : 3 KÊQ>o A{YH$V_ A H$ : 100. Series SSO

$J{UV 65/1/C. {ZYm [av g_` : 3 KÊQ>o A{YH$V_ A H$ : 100. Series SSO$ Series SSO amob Z. Roll No. H$moS> Z. Code No. SET- 65//C narjmwu H$moS >H$mo CÎma-nwpñVH$m Ho$ _wi-n ð >na Adí` {bio & Candidates must write the Code on the title page of the answer-book. H $n`m Om±M