CBSE 2015 FOREIGN EXAMINATION

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1 CBSE 05 FOREIGN EXAMINATION (Sris SSO Cod No 65//F, 65//F, 65//F : Forign Rgion) Not tht ll th sts hv sm qustions Onl thir squnc of pprnc is diffrnt M Mrks : 00 Tim Allowd : Hours SECTION A Q0 Find th r of prlllogrm whos djcnt sids r rprsntd th vctors iˆ kˆ nd 4j ˆ kˆ ˆi ˆj kˆ Lt i ˆ kˆ nd 4j ˆ kˆ Now, 0 i ˆ 4ˆj 8kˆ 0 4 rquird r i ˆ 4ˆj 8kˆ squnits Q0 Find th sum of th intrcpts cut off th pln z 5 on th coordint s z Givn pln z 5 i, 5/ 5 5 On compring to z, w hv, nd z-intrcpts s 5/, 5, 5 rspctivl c 5 5 sum of th intrcpts 5 ( 5) Q0 Find th unit vctor in th dirction of th sum of th vctors iˆ j ˆ kˆ nd 4iˆ j ˆ kˆ Lt p i ˆ j ˆ kˆ 4i ˆ j ˆ kˆ 6i ˆ kˆ p 6i ˆ kˆ rquird unit vctor, p ˆ [6i ˆ k] ˆ p 6 7 d d 4 Q04 Writ th sum of th ordr nd dgr of th diffrntil qution 0 d d 4 Ordr nd dgr of th diffrntil qution 0 is nd rpctivl rquird sum of ordr nd dgr 4 d Q05 Writ th solution of th diffrntil qution d d Givn d C or, log C log 5 6 Q06 If A 4, thn writ th cofctor of th lmnt of its nd row 4 7 Cofctor of lmnt, C [6 ( 7)( )] SECTION B Q07 Find th point on th curv 9, whr th norml to th curv mks qul intrcpts on th s W hv 9 Lt th rquird point P (, ) So, 9 (i) d On diffrntiting th q of curv wrt, 8 d 6 For vrious stuffs on Mths, pls visit t : wwwthopguptcom Pg

2 CBSE 05 Annul Em Ppr (Forign) Compild B O P Gupt ( ) d 6 mt mn t P 6 6 Sinc th norml mks qul intrcpts with th s, mn Tht is, 6, 6, (ii) 6 6 Solving (i) nd (ii) simultnousl, w gt : 9, , 4 0 ( 4) 0, ( 4) 0 0, So, 0, Hnc th rquird points r 4, nd (0,0) n d d Q08 If [ ], thn show tht ( ) n n W v n d n n d n d n d n d d d d d d n ( ) n d d ( ) n n d d ( ) n Q09 Find whthr th following function is diffrntil t nd or not :, f (),, f () f () ( ) W hv Lf () lim lim lim, f () f () ( ) ( ) Rf () lim lim lim Lf () Hnc f () isn t diffrntil t f () f () ( ) ( ) Lf () lim lim lim, f () f () ( ) ( )( ) Rf () lim lim lim lim ( ) ( ) Lf () Hnc f () is diffrntil t Q0 In prlimnt lction, politicl prt hird pulic rltion firm to promot its cndidts in ws tlphon, hous clls nd lttrs Th cost pr contct (in pis) is givn in mtri A s 40 Tlphon A 00 Hous Cll 50 Lttrs Th numr of contcts of ch tp md in two citis X nd Y is givn in th mtri B s For vrious stuffs on Mths, pls visit t : wwwthopguptcom Pg

3 CBSE 05 Annul Em Ppr (Forign) Compild B O P Gupt ( ) Tlphon Hous Cll Lttrs Cit X B Cit Y Find th totl mount spnt th prt in th two citis Wht should on considr for csting his/hr vot prt s promotionl ctivit or thir socil ctivitis? 40 Tlphon Th cost pr contct (in pis) is givn in mtri A s A 00 Hous Cll 50 Lttrs Th numr of contcts of ch tp md in two citis X nd Y is givn in th mtri B s Tlphon Hous Cll Lttrs Cit X B Cit Y Th totl mount spnt th prt in th two citis BA Tht is, Amount spnt in Cit X (in pis) 0000 Amount spnt in Cit Y (in pis) Hnc, th prt spnt pis (or, `9900) in th Cit X nd, 0000 pis (or, `00) in th Cit Y On should considr prt s socil ctivitis instd of promotionl ctivitis of th prt for csting his/hr vot Q Evlut : sin( ) Lt I sin( ) (i) Appling intgrl B Prts, w gt d I sin( ) sin( ) I sin( ) cos( ) I sin( ) cos( ) d I sin( ) cos( ) cos( ) I sin( ) cos( ) sin( ) I sin( ) cos( ) sin( ) 9 I sin( ) cos( ) I [B (i) I I [sin( ) cos( )] I [sin( ) cos( )] C 4 4 / cos Q Evlut : / For vrious stuffs on Mths, pls visit t : wwwthopguptcom Pg

4 CBSE 05 Annul Em Ppr (Forign) Compild B O P Gupt ( ) Considr I / / cos (i) / cos / cos I / / / / On dding (i) & (ii), w gt : I / / / 0 I cos / / For vrious stuffs on Mths, pls visit t : wwwthopguptcom Pg 4 Using f () f ( ) I / / / / cos cos I cos / / cos (ii) I cos f () cos cos( ) f ( ), i f is vn function nd, f () f (), if f is n vn function 0 I sin / sin sin 0 0 Q Thr mchins E, E nd E in crtin fctor producing lctric uls, producs 50%, 5% nd 5% rspctivl, of th totl dil output of lctric uls It is known tht 4% of th uls producd ch of mchins E nd E r dfctiv nd tht 5% of thos producd mchin E r dfctiv If on ul is pickd up t rndom from d s production, clcult th proilit tht it is dfctiv OR Two numrs r slctd t rndom (without rplcmnt) from positiv intgrs,, 4, 5, 6, nd 7 Lt X dnot th lrgr of th two numrs otind Find th mn nd vrinc of th proilit distriution of X Lt E, E nd E dnot th vnts tht olts producd mchins E, E nd E rspctivl Lt A th vnt tht th slctd ul is dfctiv P(E ), P(E ) P(E ), P(A E ) P(A E ), P(A E ) Thrfor, P(A) P(E )P(A E ) P(E )P(A E ) P(E )P(A E ) OR Sinc X dnots th lrgr of th two numrs otind from,, 4, 5, 6 nd 7 So vlus of X :, 4, 5, 6, 7 X P(X) Now, mn X P(X) And vrinc X P(X) Mn Q4 Th two vctors ˆ j k ˆ nd i ˆ ˆj 4kˆ rprsnt th two sids vctors AB nd AC rspctivl of tringl ABC Find th lngth of th mdin through A

5 CBSE 05 Annul Em Ppr (Forign) Compild B O P Gupt ( ) Hr AB ˆj k, ˆ AC i ˆ ˆj 4kˆ AB BC AC BC (i ˆ ˆj 4k) ˆ ( ˆj k) ˆ i ˆ ˆj kˆ Sinc mdin through A mts th sid BC t th midpoint of BC If M is th mid-point of BC thn, BM ˆi ˆj kˆ 5 Thrfor AM AB BM ˆj kˆ iˆ ˆj kˆ ˆi kˆ 5 4 Lngth of mdin, AM units Q5 Find th qution of pln which psss through th point (,, 0) nd contins th lin 6 z Th qution of pln through (,, 0) is A( ) B( ) C(z 0) 0(i) whr A, B, C r th dr s of th norml to th rquird pln As pln (i) contins th lin 6 z 4 with dr s s, 5, 4 so, A 5B 4C 0(ii) 5 4 Also (, 6, 4) lis on th givn lin nd pln (i) s wll so, A( ) B(6 ) C(4 0) 0 i, 0A 4B 4C 0 0A B C 0(iii) Solving (ii) nd (iii), w gt A B C i, dr s of th norml to pln (i) r,, Hnc qution of pln is : ( ) ( ) z 0 i, z Q6 If tn (cos ) tn (cos c ), ( 0), thn find th vlu of OR If tn tn tn tn, thn find th vlu of n(n ) cos Givn tn (cos ) tn ( cos c ) tn tn ( cos c ) cos cos cos tn tn tn tn sin cos sin 0 cos sin sin sin sin [cos sin ] 0 sin 0 or cos sin 0 0 or cot 0 or But OR W hv tn tn tn tn n(n ) (n ) n tn tn tn tn n(n ) tn tn tn tn tn n tn (n ) tn (n ) tn n tn (n ) tn (n ) tn tn tn tn (n ) n n tn tn tn tn n n Q7 If A nd I is th idntit mtri of ordr, thn show tht A 4A I Hnc find A OR If A nd B nd (A B) A B, thn find th vlus of nd 5 4 W hv A A AA 4 5 (i) For vrious stuffs on Mths, pls visit t : wwwthopguptcom Pg 5

6 CBSE 05 Annul Em Ppr (Forign) Compild B O P Gupt ( ) Also 4A I 4 (ii) B (i) nd (ii), w gt : A 4A I Pr-multipling oth sids A w gt : A AA 4A A A I IA 4I A 0 A 4I A 4 0 A OR W v (A B) A B (A B)(A B) A B AA AB BA BB AA BB AB BA So, B qulit of mtrics, w gt :,,, On solving ths qutions, w gt :, 4 Q8 Using proprtis of dtrminnts, prov th following : ( ) LHS : Lt B R R R, R R R Tking 0 0 ( ) 0 0 Epnding long R ( ) ( ) RHS Q9 Evlut : Lt I I sin( ) OR Evlut : sin( ) sin( ) sin( ) sin( ) cos cos( ) sin sin( ) common from R nd R ( 4)( 9) I sin ( ) sin( ) oth I cos sin cot( ) I cos sin log sin( ) C SECTION C Q0 Solv th givn diffrntil qution : cos d cos sin 0 OR Solv th givn diffrntil qution : Givn cos d cos sin 0 d 0 For vrious stuffs on Mths, pls visit t : wwwthopguptcom Pg 6

7 CBSE 05 Annul Em Ppr (Forign) Compild B O P Gupt ( ) sin cos d cos d dv Put v v dv sin v vcos v v v cos v v cos v dv sin v v log sin v v log log C sin cos d cos dv sin v vcos v v v cos v dv sin v vcos v v vcos v v cos v cos v v dv sin v v log log C sin v v log C sin sin C OR Givn 0 d d 0 Considr I I t t t t t I t log t dt I C sin C or, sin whr ( )( ) 0 d I d 0(i) tdt Put t t t dt t I log I dt t Sustituting th vlu of I in (i), w gt : log C log C is th rquird solution Q Find th proilit distriution of th numr of doults in four throws of pir of dic Also find th mn nd vrinc of this distriution 5 Lt E : gtting doult on th pir of dic P(E), P(E) 6 6 Lt X :Numr of doults in four throws of pir of dic So vlus of X r 0,,,, 4 X 0 4 P(X) C C C For vrious stuffs on Mths, pls visit t : wwwthopguptcom Pg 7

8 CBSE 05 Annul Em Ppr (Forign) Compild B O P Gupt ( ) Mn X P(X) 0 4, Vrinc X P(X) (Mn) Vrinc 9 Q Lt f : N R function dfind s f () 4 5 Show tht f : N S, whr S is th rng of f, is invrtil Also find th invrs of f Hr f : N R, f () 4 5 Lt n ritrr lmnt of rng S of function f Thn = , for som in N, which implis tht = ( + ) This givs s 6 6 Lt us dfin g : S N g() Now, gof () g f () g(4 5) g ( ) 6 ( ) And, fog() f g() f 6 Hnc, gof = I N nd fog = I S This implis tht f is invrtil with f = g 6 6 So, f i, f () Q Using intgrtion, find th r of th rgion oundd th lin 0, th curv nd -is W hv 0 (i) nd (ii) Solving (i) & (ii), 0 ( )( ) 0, Rquird r (i) (ii) squnits Q4 Find th distnc of th point P(,, ) from th pln z 5 msurd prlll to th lin whos dirction cosins r proportionl to,, 6 Eqution of lin through P(,,) nd prlll to lin whos dr s r proportionl to,, 6 is : z (i) 6 An rndom point on lin (i) is Q(,, 6 ) If Q lis on th givn qution of pln z 5 thn, ( ) ( ) ( 6 ) 5 7 For vrious stuffs on Mths, pls visit t : wwwthopguptcom Pg 8

9 CBSE 05 Annul Em Ppr (Forign) Compild B O P Gupt ( ) So, coordints of th point Q r 6 Q,, i, 9 5 Q,, Rquird distnc, PQ unit Q5 Mimis z 8 9, sujct to th constrints givn low : 6, 6,,, 0 To Mimis z 8 9, Sujct to th constrints givn low : 6 6, 6, ;, 0 6 Cornr points Vlu of z A(0,) 9 B, C, Mvlu D(,0) 6 O(0, 0) 0 So mimum vlu of z is ttind t 0 6, nd mimum vlu is 8 Q6 Find th minimum vlu of ( ), whr c OR Find th coordints of point of th prol stright lin Givn c (i) 7 which is closst to th c ds c Lt S ( ) S nd, d S c ds c For locl points of mim nd/or minim, 0 c d S c 0 / t c c Also, minimum vlu of S c c Rplcing vlu of c in (i), w gt Tht is, S c S c S is minimum t c OR Givn lin is i, 0 (i) Also lt th rquird point on th prol 7 P(h, k) h k Distnc of P from lin = 0 is, s ( ) k h 7h (i) For vrious stuffs on Mths, pls visit t : wwwthopguptcom Pg 9

10 CBSE 05 Annul Em Ppr (Forign) Compild B O P Gupt ( ) h h 7h h 4h 5 h 4h 5 s, (i) s (h ) ds (h ) 0 (h ) d s s & 0 0 dh 0 0 dh 0 So, s is lst for ll vlu of h ds (h ) For locl points of mim nd/or minim, 0 h dh 0 B (i), k ( ) 7( ) 8 Hnc th coordints of th rquird points on th givn prol r P(, 8) All Rights Rsrvd with O P Gupt Disclimr : All cr hs n tkn whil prpring this solution drft Solutions hv n vrifid prominnt cdmicins hving vst knowldg nd princ in tching of Mths Still if n rror is found, pls ring it to our notic Kindl forwrd our concrns/fdcks through mssg or or mil t thopgupt@gmilcom Lt s lrn Mths with smil:-) For vrious stuffs on Mths, pls visit t : wwwthopguptcom Pg 0