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

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1 Series SSO mo Z. Roll No. H$moS> Z. Code No. SET- //MT nrjmwu H$moS >H$mo CÎm-nwpñVH$m Ho$ _wi-n ð >n Adí` {Io & Cndidtes must write the Code on the title pge of the nswer-ook. H $n`m Om±M H$ {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 {I & H $n`m Om±M H$ {H$ Bg àíz-nì _ > àíz h & H $n`m àíz H$m CÎm {IZm ewê$ H$Zo go nho, àíz H$m H«$_m H$ Adí` {I & Bg àíz-nì H$mo n T>Zo Ho$ {E {_ZQ >H$m g_` {X`m J`m h & àíz-nì H$m {dvu nydm _ 0. ~Oo {H$`m OmEJm & 0. ~Oo go 0.0 ~Oo VH$ N>mÌ Ho$d àíz-nì H$mo n T> Jo Am Bg Ad{Y Ho$ Xm mz do CÎm-nwpñVH$m n H$moB CÎm Zht {I Jo & Plese hek tht this question pper ontins printed pges. Code numer given on the right hnd side of the question pper should e written on the title pge of the nswer-ook y the ndidte. Plese hek tht this question pper ontins questions. Plese write down the Seril Numer of the question efore ttempting it. minute time hs een llotted to red this question pper. The question pper will e distriuted t 0..m. From 0..m. to 0.0.m., the students will red the question pper only nd will not write ny nswer on the nswer-ook during this period. J{UV MATHEMATICS {ZYm [V g_` : KÊQ>o A{YH$V_ A H$ : 00 Time llowed : hours Mximum Mrks : 00 //MT P.T.O.

2 gm_mý` {ZX}e : (i) (ii) (iii) (iv) (v) (vi) g^r àíz A{Zdm` h & H $n`m Om±M H$ {H$ Bg àíz-nì _ àíz h & IÊS> A Ho àíz VH$ A{V Kw-CÎm dmo àíz h Am àë`oh$ àíz Ho$ {E A H$ {ZYm [V h & IÊS ~ Ho àíz 7 9 VH$ XrK -CÎm I àh$m Ho$ àíz h Am àë`oh$ àíz Ho$ {E 4 A H$ {ZYm [V h & IÊS> g Ho àíz 0 VH$ XrK -CÎm II àh$m Ho$ àíz h Am àë`oh$ àíz Ho$ {E A H$ {ZYm [V h & CÎm {IZm àmå^ H$Zo go nho H $n`m àíz H$m H«$_m H$ Adí` {{IE & Generl Instrutions : (i) (ii) (iii) (iv) (v) (vi) All questions re ompulsory. Plese hek tht this question pper ontins questions. Questions in Setion A re very short-nswer type questions rrying mrk eh. Questions 7 9 in Setion B re long-nswer I type questions rrying 4 mrks eh. Questions 0 in Setion C re long-nswer II type questions rrying mrks eh. Plese write down the seril numer of the question efore ttempting it. //MT

3 IÊS> A SECTION A àíz g»`m go VH$ àë`oh$ àíz H$m A H$ h & Question numers to rry mrk eh.. EH$ {df_ g_{_v Amì`yh {{IE & Write skew symmetri mtrix.. {ZåZ AdH$ g_rh$u Ho$ {E BgH$s H$mo{Q> d KmV H$m JwUZ\$ kmv H$s{OE : d y x y 0 Find the produt of the order nd degree of the following differentil eqution : d y x y 0. y = A os x + B sin x, Ohm± A Am B ñdoàn> AM h, Ho$ {E EH$ AdH$ g_rh$u {{IE & Write differentil eqution for y = A os x + B sin x, where A nd B re ritrry onstnts. 4. g{xe ^i + ^j ^k H$m g{xe ^i + ^j Ho$ AZw{Xe àjon {{IE & Write the projetion of vetor ^i + ^j ^k long the vetor ^i + ^j.. ^i. ( ^j ^k ) + ^j. ( ^k ^i ) + ^k. (^i ^j ) H$m _mz {{IE & Write the vlue of ^i. ( ^j ^k ) + ^j. ( ^k ^i ) + ^k. (^i ^j ).. g_v x + 4y + z = Ho$ A{^å~ Ho$ {XH²$-H$mogmBZ {{IE & Write the diretion osines of the norml to the plne x + 4y + z =. //MT P.T.O.

4 IÊS> ~ SECTION B àíz g»`m 7 go 9 VH$ àë`oh$ àíz Ho$ 4 A H$ h & Question numers 7 to 9 rry 4 mrks eh. 7. VrZ n[dmm A, B VWm C _ nwéfm, _{hmam Am ~ƒm H$s g»`m Bg àh$m h : nwéf _{hme± ~ƒo n[dm A n[dm B n[dm C 4 EH$ nwéf, _{hm Am ~ƒo H$m à{v{xz IM H«$_e: < 00, < 0 d < 00 h & {g\ $ nwéf d _{hme± hr H$_mVo h, Z {H$ ~ƒo & Amì`yh H$s JwUm go àë`oh$ n[dm H$m IM kmv H$s{OE & n[dm _ A{YH$ ~ƒm Ho$ hmozo go g_mo n Š`m Ag n S>Vm h? There re fmilies A, B nd C. The numer of men, women nd hildren in these fmilies re s under : Men Women Children Fmily A Fmily B Fmily C 4 Dily expenses of men, women nd hildren re < 00, < 0 nd < 00 respetively. Only men nd women ern nd hildren do not. Using mtrix multiplition, lulte the dily expenses of eh fmily. Wht impt does more hildren in the fmily rete on the soiety? 8. `{X tn x + tn y + tn z =, x, y, z, > 0 hmo, Vmo xy + yz + zx H$m _mz kmv H$s{OE & If tn x + tn y + tn z =, x, y, z, > 0, then find the vlue of xy + yz + zx. //MT 4

5 //MT P.T.O. 9. `{X VWm 0 hmo, Vmo gm{uh$m Ho$ JwUY_mªo H$m à`moj H$Ho$ {gõ H$s{OE {H$ + + = 0. If nd 0, then using properties of determinnts, prove tht + + = `{X X h, Vmo Amì`yh X kmv H$s{OE & Amì`yh A H$m ì`wëh«$_ kmv H$s{OE VWm Xem BE {H$ A. A = I. If X, then find the mtrix X. Find the inverse of mtrix A nd hene show tht A. A = I.. `{X \$Z f(x) = x + x 4 h, Vmo Xem BE {H$ x = VWm x = 4 n f(x) AdH$Zr` Zht h & If funtion f(x) = x + x 4, then show tht f(x) is not differentile t x = nd x = 4.

6 . `{X y = x e x h, Vmo kmv H$s{OE & `{X log x x y tn h, Vmo Xem BE {H$ y y x. y x If y = x e x, find. If log x y tn x, then show tht y y x. y x. `{X y x x h, Vmo {gõ H$s{OE {H$ (x d y ) x 4 y d y If y x x, prove tht (x ) x y kmv H$s{OE : os x os x ( os x) Find : os x os x ( os x). _mz kmv H$s{OE : Evlute : x. sin x. sin x x //MT

7 x. `moj\$ H$s gr_m Ho$ ê$n _ ( x e ) H$m _mz kmv H$s{OE & 0 _mz kmv H$s{OE : Find 0 Evlute : x tn x se x ose x 0. x ( x e ) s the limit of sum. x tn x se x ose 0. x x y 7. Xem BE {H$ oime±, z 0 h & BZH$m à{vàn>oxz {~ÝXþ ^r kmv H$s{OE & x 4 z Am, y 0 nñn H$mQ>Vr x y x 4 z Show tht the lines, z 0 nd, y 0 interset eh other. Also find their point of intersetion. 8. _mzm {~ÝXþ P(,, ) AÝV[j _ h Am {~ÝXþ Q oim r = (^i ^j + ^k ) + µ ( ^i + ^j + ^k ) n h & µ H$m _mz kmv H$s{OE, {Oggo g{xe PQ, g_v x 4y + z = Ho$ g_mýv hmo & Cg g_v H$m g{xe VWm H$mVu` g_rh$u kmv H$s{OE Omo {~ÝXþAm (,, ) Am (, 4, ) H$mo Omo S>Zo dmr oim H$mo EH$ g_h$mou n g_{û^m{ov H$Vm h & //MT 7 P.T.O.

8 Let P(,, ) e point in the spe nd Q e point on the line r = (^i ^j + ^k ) + µ ( ^i + ^j + ^k ), then find the vlue of µ for whih the vetor PQ is prllel to the plne x 4y + z =. Find the vetor nd rtesin equtions of the plne whih isets the line joining the points (,, ) nd (, 4, ) t right ngles. 9. go 00 VH$ H$s g»`m go {Ir EH$ 00 H$mS>mªo H$s JÈ>r go EH$ H$mS> `mñàn>`m {ZH$mm OmVm h & àm{`h$vm kmv H$s{OE {H$ Bg H$mS> n {Ir g»`m `m 8 go ^mj hmo gh$vr h, n 4 go Zht & From set of 00 rds numered to 00, one rd is drwn t rndom. Find the proility tht the numer on the rd is divisile y or 8, ut not y 4. IÊS> g SECTION C àíz g»`m 0 go VH$ àë`oh$ àíz Ho$ A H$ h & Question numers 0 to rry mrks eh. 0. {gõ H$s{OE {H$ g_wàm` A = {,,, 4, } _ R = {(, ) :, go ^má` h } Ûmm àxîm gå~ýy R EH$ Vwë`Vm gå~ýy h & à_m{uv H$s{OE {H$ {,, } Ho$ g^r Ad`d EH$ Xÿgo go gå~pýyv h Am g_wƒ` {, 4} Ho$ g^r Ad`d EH$ Xÿgo go gå~pýyv h, nývw {,, } H$m H$moB ^r Ad`d {, 4} Ho$ {H$gr Ad`d go gå~pýyv Zht h & Show tht the reltion R in the set A = {,,, 4, } given y R = {(, ) : is divisile y } is n equivlene reltion. Show tht ll the elements of {,, } re relted to eh other nd ll the elements of {, 4} re relted to eh other, ut no element of {,, } is relted to ny element of {, 4}.. g_mh$z {d{y go dh«$ y = x VWm y = x Ho$ ~rm n[~õ joì H$m joì\$ kmv H$s{OE & Using integrtion, find the re ounded y the urves y = x nd y = x. //MT 8

9 x. dh«$ y n dh {~ÝXþ kmv H$s{OE {Og n dh«$ n ItMr JB ñne oim H$s x àduvm A{YH$V_ hmo & Find the point on the urve hs the gretest slope. x y x, where the tngent to the urve. AdH$ g_rh$u H$m ì`mnh$ h kmv H$s{OE & y xy x {ZåZ AdH$ g_rh$u H$mo h H$s{OE, {X`m h {H$ y = 0, O~ x = h : 4 sin x y tn x Find the generl solution of the differentil eqution y. xy x Solve the following differentil eqution, given tht y = 0, when x = : 4 sin x y tn x 4. g_vm r. (^i + ^j + ^k ) = Am r. (^i + ^j + 4 ^k ) = Ho$ à{vàn>oxz VWm {~ÝXþ (,, ) go OmZo dmo g_v H$m g{xe g_rh$u d H$mVu` g_rh$u kmv H$s{OE & Find the vetor nd rtesin equtions of the plne pssing through the intersetion of the plnes r. (^i + ^j + ^k ) = nd r. (^i + ^j + 4 ^k ) = nd the point (,, ). //MT 9 P.T.O.

10 . _mz r{oe {H$gr mojr H$mo {X H$m Xm m n S>Zo H$m g `moj 40% h & `h _mz {`m OmVm h {H$ Ü`mZ Am `moj {d{y {X H$m Xm m n S>Zo Ho$ IVo H$mo 0% H$_ H$ XoVr h Am {H$gr Xdm Ûmm IVo H$mo % H$_ {H$`m Om gh$vm h & {H$gr ^r g_` mojr BZ XmoZm _ go {H$gr EH$ {dh$ën H$m MwZmd H$ gh$vm h VWm XmoZm g_àm{`h$ h & `h {X`m J`m h {H$ Cn`w º$ {dh$ënm _ go {H$gr EH$ H$m MwZmd H$Zo dmo mo{j`m _ go `mñàn>`m MwZm J`m EH$ mojr {X Ho$ Xm o go J«{gV hmo OmVm h & àm{`h$vm kmv H$s{OE {H$ `h mojr Ü`mZ Am `moj {d{y H$m Cn`moJ H$Vm h & Assume tht the hnes of ptient hving hert ttk is 40%. It is lso ssumed tht medittion nd yog ourse redues the risk of hert ttk y 0% nd the presription of ertin drug redues its hne y %. At time ptient n hoose ny one of the two options with equl proilities. It is given tht fter going through one of the two options the ptient seleted t rndom suffers hert ttk. Find the proility tht the ptient followed ourse of medittion nd yog.. EH$ ì`mnmr Ho$d Xmo àh$m H$s dñvwam dñvw A VWm dñvw B H$m ì`mnm H$Vm h & CgHo$ nmg ì`mnm _ JmZo Ho$ {E < 0,000 h VWm A{YH$-go-A{YH$ 0 dñvwam H$mo IZo H$m ñwmz h & dñvw A H$m H«$` _yë` <,00 VWm dñvw B H$m H«$` _yë` < 00 h & dñvw A H$mo ~omh$ dh < 00 ewõ m^ H$_mVm h VWm dñvw B H$mo ~omh$ dh < 0 ewõ m^ H$_mVm h & `{X IrXr JB g^r dñvwe± dh ~om ovm h, Vmo Cgo AnZr YZm{e go {H$VZr-{H$VZr dñvwe± IrXZr Mm{hE {Oggo {H$ Cgo A{YH$V_ m^ àmßv hmo gho$? Bg àíz H$mo {IH$ àmoj«m_z g_ñ`m ~ZmH$ J«m\$ Ûmm h H$s{OE & EH$ Amhm-{dkmZr Xmo ^moá`m X VWm Y H$m Cn`moJ H$Vo hþe {deof Amhm V `m H$Zm MmhVm h & ^moá` X H$m àë`oh$ n Ho$Q> ({Og_ 0 J«m_ AÝV{d îq> h ) _ H $pëe`_ Ho$ _mìh$, moh VÎd Ho$ 4 _mìh$, H$mooñQ>om Ho$ _mìh$ VWm {dq>m{_z A Ho$ _mìh$ AÝV{d îq> h & Cgr _mìm Ho$ ^moá` Y Ho$ àë`oh$ n Ho$Q> _ H $pëe`_ Ho$ _mìh$, moh VÎd Ho$ 0 _mìh$, H$mooñQ>om Ho$ 4 _mìh$ VWm {dq>m{_z A Ho$ _mìh$ AÝV{d îq> h & Amhm _ H $pëe`_ Ho$ H$_-go-H$_ 40 _mìh$, moh VÎd Ho$ H$_-go-H$_ 40 _mìh$ VWm H$mooñQ>om Ho$ A{YH$-go-A{YH$ 00 _mìh$ Ano{jV h & àë`oh$ ^moá` Ho$ {H$VZo-{H$VZo n H$Q>m H$m Cn`moJ {H$`m OmE Vm{H$ Amhm _ {dq>m{_z A H$s _mìm H$mo Ý`yZV_ {H$`m Om gho$? Cn` wº$ H$mo EH$ {IH$ àmoj«m_z g_ñ`m ~Zm H$ J«m\$ Ûmm h H$s{OE & //MT 0

11 A deler dels in two items only item A nd item B. He hs < 0,000 to invest nd spe to store t most 0 items. An item A osts <,00 nd n item B osts < 00. A net profit to him on item A is < 00 nd on item B < 0. If he n sell ll the items tht he purhses, how should he invest his mount to hve mximum profit? Formulte n LPP nd solve it grphilly. A dietiin wnts to develop speil diet using two foods X nd Y. Eh pket (ontins 0 g) of food X ontins units of lium, 4 units of iron, units of holesterol nd units of vitmin A. Eh pket of the sme quntity of food Y ontins units of lium, 0 units of iron, 4 units of holesterol nd units of vitmin A. The diet requires t lest 40 units of lium, t lest 40 units of iron nd t most 00 units of holesterol. Mke n LPP to find how mny pkets of eh food should e used to minimise the mount of vitmin A in the diet, nd solve it grphilly. //MT P.T.O.

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

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