SOLUTIONS TO TRIAL EXAMINATION 3

Transcription

1 à à n n, Á os sin Á f + Á 9 ± jn ² j ) sin Á jnjjj j+ 9j p p p p p 5 ) he anle eween he line an he plane Á 5:7 ±. X a i a X Hene j OX j jaj sin jaj ( os ) jaj jaj os jaj jaj jj os jj jaj (a ² ) jj ) j r OX j jaj (a ² ) jj Y Le e he anle eween O an. ) sin j OX j jaj Thus j OX j jaj sin The line wih equaions, y +, z +, R onains he à fie poin (,, ) an has ireion. à Thus, as a, r j OX j ( + + ) (++) (() + + ) q 6 q 6 unis ii P () + Q()+ ( a) ( a) ) P () Q()( a) + + P (a) Q(a) +a + ) P (a) a + P () Q ()( a) + Q()( a)+ ) P (a)++ iii Remainer + P (a) +(P (a) a) P (a) + P (a) ap (a) P (a)( a)+p (a) SOLUTIONS TO TRIL EXMINTION NO CLCULTOR SECTION a iv or P () 5 ivie y ( +), P () an P () 5 ) P () ) remainer 8( +) Le os so sin ) p 9 os p ( sin ) 9 9 os os ( sin ) sin os sin + p 9 + p 9 + Chek: ((9 ) + ) (9 ) ()+ p X 9 p() ( ) a + + a + ( ) ) p() (a + )( ) + a + ) p() (a + )[( ) + ] Thus p() (a + )( +) (a + )( )( ) ) ( ) an ( ) are faors of p(). Sine p() 7 an p() 9, ()() 7 an (a + )()() 9 ) 7 an a +7 ) a 6 ) a Thus, p() ( + 7)( +) a a a Sine ln k, ek. Likewise, sine ln, a a e. Thus a e k an a e ) a 6 e k an a 6 e ) e e k ) e k+ fvalues for a 6 ~`9# -` # r Mahemais HL Eam Preparaion & Praie Guie ( eiion)

2 Sine a Le a e, a e ) a e6k+8 e ) a e 6k+6 ) a e 6k+6 ) a e k+, so r an s. a + + a( ) + ( +) ( + )( ) (a + ) +( a) ) a + an a fequain oeffiiens Solvin hese equaions simulaneously ives an a. + h ln j +j ln j j i ( ln ln ) ( ln ln 5) ln ln + ln 5 ln+ ln 5 is unefine when an lies in he omain of ineraion [, ]. Thus oes no eis. a Le u v sin u v os R ) sin uv R u v os R os os + ( sin )+ sin os + sin h sin i os ( sin os ) ( sin ) () unis sin h sin i os ( sin os ) ( sin os ) a a D C ² OD ( a) ² ( a + ) a ² + ² a ² a a ² jj jaj as jaj jj fequal raii ) C an OD are perpeniular ) [C]? [OD]. C O + OC a + a OD O + D a + C a + ( a) a + a a + ( ± f)() (f()) ( p +) ( p +) ) ( ± f)() p (9 ) () f() + where 6 has inverse y where y +y where y 6 ) y +y p, y 6 ) y 6 ()(), y 6 p ) y ( + ), y 6 ) y p +, y 6 ) y p + ) f () p +, > 7 a Le m ) m +m 5 ) (m )(m +5) ) m or 5 ) f > for all ) lo sin + os :5 ) os + os ) os os + ) os os + ) ( os ) O C We have jus prove ha he line from he enre of a irle o he mipoin of a hor, is perpeniular o he hor. 6 a () is efine when > ) > ) 6 unis ) ) os ) Qw r Mahemais HL Eam Preparaion & Praie Guie ( eiion)

3 8 f() os + sin a f() os + sin ) is(, ) f() os + sin ) His(, ) f () ( sin ) + os sin os + os os ( sin ) Qw f I z z z z P z +z R ) f (), os or sin, 6,, 5 6, ) is( 6, ), C is (, ), D is ( 5 6, ), an is(, ). The -oorinaes of E an G are he soluions of os + sin ) os + sin ) sin + sin ) sin sin ) sin p ()() p ) sin p u 6 sin 6, so sin p p E, arsin. p G, arsin +.. z is ) ar(z ) ) ar(z + z) ar(z ) If z + z is purely imainary, hen OP lies on he imainary ais ) + k, k ) + k, k ) f 6 6 a i f() an ) f () an se (se ) se se se ii Usin i, R ( se se ) an + SECTION 9 a i os( + ) + os( ) os os sin sin + os os + sin sin os os ii sin( + ) + sin( ) sin os + os sin + sin os os sin sin os i Usin aiwih, ) os( + ) + os( ) os os ) os + os os os ii Likewise usin aiiwih, sin + sin sin os z [is ] is an z [is ] is fde Moivre s heorem z + z is + is os + i sin + os + i sin [os + os ]+i[sin + sin ] os os + i sin os os [os + i sin ] os is e rom, z + z os an ar(z + z) ) R se R se an + ) R se an an + ) R se an + an + i y se os when os is unefine ) for he illusrae raph, a ii When, os ) se ) is(, ) Shae area se h i an an an Volume p unis y se h an + an i ( p + (p ) ) ( p + p ) p unis fusin a (, ) (, -) r Mahemais HL Eam Preparaion & Praie Guie ( eiion)

4 e a r Volume of yliner 9 ) volume require 9 Ã +5 Ã 9 When, r 5 9 y is he equaion of L. aros py y ) (9, 5, ) lies on L. Ã n an he plane has equaion y z () () () whih is y z 7 The line +5, y, z mees he plane when ( +5) ( ) 7 ) ) 6 ) Hene () + 5, y (), z The line mees he plane a (,, ). e y se_ y y se &"e, * ) se p y f se > ) os p y ) aros py "e (9,-5, ) N Line N has equaion 9+s, y 5 s, z s, s R. Ã 9+s or r 5 s s Line N mees P where (9 + s) (5 s) ( s) 7 ) 7 + 9s ++6s +s 7 ) 6s +57 ) 6s 5 ) s Hene 9+() y 5 () z () ) line N mees he plane a (,, ). f (9,-5, ) N (,, ) P '(a,, ) If is (a,, ), a ,, ) a +96, 56, +8 ) a,, 6 ) is (,, 6) Given C(,, ) an (,, 6), Ã C 8 (i j k) 8 ) C is parallel o i j k. f C is a onsan muliple of i j k a P n is: n+ + n+ is ivisile y for all n +. Proof: (y he priniple of mahemaial inuion) () If n, n+ + n ) P is rue. () If P k is rue hen k+ + k+ where +. Now (k+)+ + (k+)+ k++ + k++ k+ + k+ 6( k+ )+7 k+ 6 6 k+ +7 k+ 6 + k+ (6 + k+ ) where 6 + k+ + Thus P k+ is rue. So, P is rue, an P k+ is rue provie P k is rue for all k >. Hene, P n is rue. fpriniple of mahemaial inuion Sine a,, are onseuive erms in an arihmei sequene, a ) a + u a + +, so + ) ) Sine a, +, +9 are onseuive erms in a eomeri sequene, + a +9 + ) a +9 where a + r Mahemais HL Eam Preparaion & Praie Guie ( eiion)

5 Thus CLCULTOR SECTION a a +9 ) a 5 a ) 5a a ) a 5a + ) (a )(a 8) ) a or 8 ) a,, 9 or a 8,, 6 a i z r is z (ris ) r is fde Moivre s heorem ii p z z (ris ) r is + ai ( ai) fde Moivre s heorem (ai)+(ai) (ai) ai a + a i ( a )+(a a)i ) a an a a a fequain real an imainary pars ) a an a a ) a an a(a ) ) a ( a If f() e, 6 6, 6, oherwise is a well efine PD, hen e ff() > always h i ) e ) e e ) e + Usin ehnoloy, :56 ) :56 ¹ E(X) :6 e:56 Var(X) E(X ) fe(x) :77 fusin ehnoloy e:56 ¹ fusin ehnoloy a Usin he osine rule in C, C 6 +8 (6)(8) os ± ) C ) C 8 Refle OC ± ) OC ± fanle a he enre heorem In OC, C r + r rr os ± fosine rule ) C r + r ) C r rom a an, r 8 ) r 9::::: ) r 7:8:::: ) r 7: os 7:8 ³ aros 7: ) O 55: ± 8 a When no replaemen ours, we o no have a repeiion of n inepenen rials eah wih he same proailiy of suess. However, sine n is very lare, he proailiy of a suess eah ime will e almos he same. X» (, :) i P(X 6 ) P(X,, or) (:) (:968) + (:) (:968) 9 + (:) (:968) 8 :975 ii P(X > ) P(X 6 ) : 7 fusin ehnoloy 5 a 8 m 6 m C r r O ³ (n+ ln ) (n +) n ln + n+ (n +) n ln + n R Thus [(n +) n ln + n ] n+ ln + ) (n +) R n ln + n+ n + n+ ln + provie n 6 ) R n ln n+ ln n + n+ (n +) + n+ ((n +)ln ) + (n +) provie n 6 5 r Mahemais HL Eam Preparaion & Praie Guie ( eiion)

6 When n, R ln n ln ³ (ln ) This has he form R (f()) n f (), so R n (ln ) ln + 6 s sin + os meres a v s () sin + os + sin ) v sin + os sin ) v os ms The parile is a res when v ) os (, )»» " 8 7 " 8 7 " 8 7 # # # R R R R $ R R R +R rom row, 8 ) 9 rom row, +99 ) 8 ) 6 rom row, 8a +7(6) + 9 ) 8a + 9 ) 8a 8 ) a ) a, 6, an 9 (, -) ) or + k, k ) or + k, os hanes sin. ) he parile reverses ireion a seons. Is posiion a his ime is s() m rih of O. a v os + sin ) a() os sin ) a os 6 sin 6 p p ms SECTION 9 Le X resul of a suen in he Siene eam. X» N(56:7, 8: ) a i P(resul eween 65 an 85 inlusive) P(65 6 X 6 85) :6 ii P(resul of a leas 7) P(X > 7) : i We nee o fin k where P(X 6 k) :9 7 a aran + aran 6 aran Le + where an, an 6, an. Now an( + ) an ) an + an an an an ) +6 (6) ) + 6 ( ) ) ) 9 9 ) 9 9 y aran ) y a If, y, () If, y, 5+a +... () If 8, y 7, + 8a () ii.5 k 9. k. ) k 8 ) a resul of 8 or more reeives an. We nee o fin k where P(X 6 k) :5 ) k 7:8 ) a resul of 7 or less reeives an. Suen () () () There are si ifferen permuaions of s an s. ) P(wo s an wo s) 6 (:) (:5) : 5 () X X r Mahemais HL Eam Preparaion & Praie Guie ( eiion) 6

7 Y» (, :) i P(Y ) : a i f() y ii 6 f () + sin sin ii P(Y > ) P(Y 6 ) : os ( sin ) ( + sin )( os ) ( sin ) os sin os + os + sin os 6 os ( sin ) i h () ( os )( sin ) (a + sin )( os ) ( sin ) os sin os +aos +sin os ( sin ) os (a + ) ( sin ) ii h (),,, 5, 7 fas in ) M h a + () a + an m h a + () () a + iii H M m a + a + (a + )( +) (a )( ) ( )( +) a + a + + a + a + a + (a + ) iv When a,,, ( + 6) H Now f + sin sin ( sin ) ) f (), os, + k, k, + k + () () 5 an 7 5 9, so he resul is verifie. (, ),,, 5, 7 ) he loal maimum a is (, 7) an he loal maimum a is ( 5, 7). "w Es" y f() (, -) a L : r L : s à + + à +¹ ¹ +¹ à + à à + ¹Ã The lines inerse if r s ) + +¹... () ¹... () + +¹... () rom (), ¹ So, in () +¹ +¹ ) ¹ an 5 However, in () + (5) 9 an +() 6 ) () is no saisfie y he soluions from () an (). ) L an L o no inerse. à à lso, is no a salar muliple of ) L an L are no parallel. Thus L an L are skew lines. veor perpeniular o L an L is à à i j k i j + k (+)i ( 6)j +(+)k i +j +k j + k * L L C Q r a+ * D R L is ranslae o L meein L a P. i is perpeniular o L an L,so is a normal o he shae plane onainin L an L. ii RC 9 ± fr is normal o he shae plane Le C R e he anle eween C an. ) os R C D j C j ) D j C j os j ajj j os j j j( a) ² ( )j j j D P L s +¹ 7 r Mahemais HL Eam Preparaion & Praie Guie ( eiion)

8 e or he oriinal lines L an L, à à a,, Ã, à à a à à à ffrom à à ² ) D p p ++ 5 unis a v v os ) R v os ) (v os ) + u when, ) ) (v os )... () Likewise, v y v sin ) y R (v sin ) v e Rane sin f The rane is maimise when sin ) 9 ± ) 5 ± ³ p i The maimum heih m 9:8 9 m ii Rane 9:8 97 m h Usin e, 95 sin ) sin :58 7 ) 5:6 ± ) 7:8 ± The anle is aou 7:8 ±. v sin meres. ) y (v sin ) + When, y ) ) y (v sin )... () rom (), v os ³ ³ ) y v sin v os v os ) y (an ) v ) y (an ) (se ) v The pah has equaion y a+ whih is a quarai in. ) he pah is paraoli. urhermore, sine <, he pah is onave. y an se v ) y sin, os v os, v sin os, v sin os, v sin The maimum heih is sin v sin os os v sin os v os v sin v sin v sin r Mahemais HL Eam Preparaion & Praie Guie ( eiion) 8