HUT, TUT, LUT, OU, ÅAU / Engineering departments Entrance examination in mathematics May 25, 2004

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1 HUT, TUT, LUT, OU, ÅAU / Engineeing depamens Enane examinaion in mahemais May 5, 4 Insuions. Reseve a sepaae page fo eah poblem. Give you soluions in a lea fom inluding inemediae seps. Wie a lean opy of he soluion if needed. Coss ou disaded soluions. In ase of seveal soluions fo he same poblem, only he weakes one will be edied.. A eain aile was sold in soe A wih a % disoun off he sandad pie and lae wih an addiional euo ash disoun off he edued pie. In anohe soe B hee was a % disoun off he same sandad pie and lae an addiional euo ash disoun off he edued pie. Afe all he disouns he pie of he aile was % lowe in soe A han in soe B. Wha was he sandad pie of he aile?. The angen line o he gaph of he funion f (x) x +x a he poin (x, f (x )) also passes hough he poin (,6). Deemine all possible values of x.. A paile moves in he plane so ha is oodinaes a ime ae { x os + 4sin y os + sin. Wha is he shoes disane fom he oigin o he paile? 4. An elei moo is saed a ime. Beause of a defe in he uen supply he powe of he moo is no onsan, bu vaies wih ime aoding o he fomula ( P() 4 ). Duing whih ime ineval [, + ] of seond in duaion is he moo doing he maximum amoun of wok? Addiional infomaion: A moo wih powe P P() does duing a ime ineval [, ] he wok W whih an be alulaed as he inegal W P() d. All quaniies ae given in he SI-sysem, so he poblem an be eaed wihou unis. 5. The adjaen diagam shows a skeh of a pa of he seemap of a iy. A peson walks daily fom ossing A o ossing B along a shoes possible oue, so he lengh of he jouney is 4 unis. A ossings whee wo dieions give a shoes oue he peson hooses he dieion by ossing a oin. a) Daw (sepaae) diagams of all possible shoes oues and deemine he pobabiliy fo eah of hem o be hosen. b) Anohe peson walks fom ossing B o ossing A and hooses a shoes possible oue in he same fashion. Boh pesons sa walking simulaneously and walk wih he same speed. Wih wha pobabiliy will hey mee eah ohe half way? 6. An infinie numbe of balls wih adii > > >... e. ae in a ow on a able. Exep fo he fis ball eah ouhes wo ohes and fuhemoe he midpoins of he balls all lie on he same line (in spae). a) Deemine an expession fo he adius n as a funion of n and n+ (n,,...). b) Assume ha and. Wha is hen he sum of he volumes of all he balls? A B HUT 4

2 MATH (enginees) 4 poblem poblem Le x be he sandad pie ; hen he disouned pies ae.8x in soe A,.9x insoe B. Fom he assumpion we have.8x.9(.9x ).x 8 The equaion of he angen line a (x, y ) (x, f(x )) is o y y f (x )(x x ) y (x + x ) (x + )(x x ). The poin (x,y) (,6) saisfies he equaion 6 x x (x + )( x ) x 8 (euos) o 4x x + 6 x x 4x, so he soluion is x o x 4.

3 MATH (enginees) 4 poblem poblem 4 A ime he disane of he paile fom he oigin is Le >. The wok done by he moo duing he ime ineval [, + ] is 4 w( ) whee () x + y (os + 4sin ) + ( os + sin ) + w( ) d + / ln + ( os + sin ). 5os + sin 5 + 5sin Beause he minimum value of sin is sin he shoes disane is () 5 (of ouse, ±π, ±π e. give he same disane). ln( +) ln + +. I is suffiien o find he maximum poin of he funion w : w ( ) + + ( + ) ( + ) + 5 Thus w ( ) + + (he seond oo is negaive)..68 This is eally he maximum poin of w beause w () > fo < < and w () < fo >. Answe: + 5 +, 5 [.68,.68].

4 MATH (enginees) 4 poblem 5 a) Thee ae 5 shoes oues; diagams and pobabiliies: p 4 p 4 p 8 p 4 8 p 5 4 Moivaion: The pobabiliy of a eain oue equals in he diagam ). n whee n is he numbe of oin osses on ha oue (maked b) The pesons mee eah ohe - if hey do - eihe a he uppe lef one o a he midpoin. The peson who walks fom A o B passes he uppe one wih pobabiliy p 5 4 and he midpoin wih pobabiliy p + p + p + p 4 4. Fom he symmey i follows ha fo he peson walking fom B o A he oesponding pobabiliies ae jus he same. Thus P( hey mee ) P( boh pass he uppe one ) + P( boh pass he midpoin )

5 MATH (enginees) 4 poblem 6 Conside hee adjaen balls (f. diagam) wih adii n-, n ja n+ ; o make he soluion moe easily eadable we denoe hese by,,. The midpoins P, P, P of he balls ae on a saigh line; le Q be he poin whee his line mees he sufae of he able. Le,, be he hypohenuses of he hee simila igh-angled iangles in he diagam (i.e. i disane of P i fom Q, i,,). a) By similaiy we have, whene (expess he disanes P P and P P in wo ways in ems of he adii and ) + ( - ) ( ) + + Solving boh equaions fo gives Wih he oiginal noaion: he sough expession is n n n+ and oespondingly +... ( )......

6 n pa 6b) Fom he soluion of pa a) i follows ha + n ha he aio of he adii of wo adjaen balls is n α n n n fo all n,,..., i.e. hee is a onsan α suh fo all n. If and hen, so ha α. Thus he adii of he balls ae, α, α,..., and he sum of he volumes is 4π 4π ( ) ( + α + (α ) +... ) 4π (he sum of a geomei seies) α 4π 7 8 π

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