Iecep Math 5

IECEP FILES FI LES MATHEMAT MATHEMATICS ICS Compiled by : Mark Djeron C. Tumabao

The raph! o" #he e$ua#ion! o" #he "orm! r % a!in a!inn nϴ and and r % aa&o &o!!nϴ 'here here n i! i! a  po!i#i(e in#eer) rea#er #han #han *) are &alled  +++++. The raph o" an e$ua#ion o" #he "orm r = b + asinϴ or r = b + acosϴ i! &alled a  ++++++++. A-n ++++++ i! #he !e# o" all poin#! P in a  plane !u&h #ha# #he !um o" #he di!#an&e! o" P "rom #'o "ied poin#! F and / o" #he  plane i! &on!#an#. Any di""eren#ial e$ua#ion o" #he "orm y%  p 0 "1p2 'here "1p2 &on#ain! nei#her  nor y epli&i#ly i! &alled a-n +++++++. The!e (ariable! are dimen!ionle!! &ombina#ion! o" #he phy!i&al (ariable and  parame#er! o" #he oriinal. Thi! !#a#e! #ha# e(ery in#eral ra#ional e$ua#ion ha! a# lea!# one roo#. The loari#hm o" #he re&ipro&al o" a number i! &alled +++++. /i(en #he e$ua#ion: −1 −1  L { f  ( ( s ) }= x L { f  (  ( s− a )}  'here L4*

,o!e Cur(e!

Lima&on!

Ellip!e

Lauerre Polynomial!

Canoni&al 3ariable! Fundamen#al Theory o" Alebra &oloari#hm e4a#

i! #he in(er!e Lapla&e #ran!"orm o" a "un&#ion "1!2. Find  −1 1

Simpli"y:

Ta n

2

+Ta n−1

( −  ) 1 3

−1 1

Ta n

2

+ Ta n−1 1

¿ Ta n−1

2 1+

−

( −  )= 1 3

− 1 1

Ta n

2

−Ta n−1

 1 3

1 3

1 1

=Ta n−1

 1 7

⋅

2 3

Ho' many di""eren# !inal!) ea&h &on!i!#in o" 5 "la! hun in a (er#i&al line) &an be "ormed "rom 6 iden#i&al red "la! and 7 iden#i&al blue "la!8

Thi! i! a &a!e o" permu#a#ion! o" indi!#inui!hable obje&#! 6! =15 4!2!

Three lih# bulb! are &ho!en a# random "rom *9 bulb! "rom 'hi&h 9 are de"e&#i(e. Find #he probabili#y #ha# one lih# bulb dra'n i! de"e&#i(e.

Thi! Thi! i! a &a!e &a!e o" hype hyper reo eome me#r #ri& i& prob probab abil ili# i#y y di!#ribu#ion. 5 C 1 10 C 2 45 = 15 C 3 91 ⋅

IECEP FILES MATHEMATICS Compiled by : Mark Djeron C. Tumabao

wherenCr = A poin# i! !ele&#ed a# random in!ide a &ir&le. Find #he probabili#y #ha# #he poin# i! &lo!er #o #he &en#er #han #o i#! &ir&um"eren&e. Con!ider #he !erie! Sn %* 4* 0* 4* 0* 0 4  I" n i! e(en) #he !um i! ;ero and i" n i! odd) #he !um i! *.
n! r ! ( n− r ) !

( ) 1

 P=

2

r π 

2 1 areaofsuccess = = 2 areaofpossible 4 πr

=!&illa#in !erie!

Cen#ral Limi# Theorem

S&a##er Diaram

s Y  ( s )− sy ( 0 ) − y ( 0 ) + Y  ( s )= 0 ' 

2

Y  ( s )=

1 2

s +1

 y (t )= sin t  f  ( x )=1 + x + x + x + … ) de#ermine 2

I"

3

f  ( x )=1 + x + x + x + … = 2

3

3

∫ f  ( x) dx 2

.

1 1 − x

 ( infinitegeometricserie

3

=−ln |1− x|| =−ln |−2|+ ln |−1|= ln 2− =ln ∫ 1dx − x 3 2

1

2

Meiko >in #ra(el! *?? mile! a# #he ra#e o" @? mph and #hen on a "ree 'ay #ra(el! #he ne# *?? mile! a# #he ra#e o" 99 mph.
Meiko! a(erae !peed i! #he Harmoni& mean o" @? mph and 99 mph

  =

1 1 1 + 30 55

=38.8 mph

P*P7 % 19-6721@-9*2 % *-77*

IECEP FILES MATHEMATICS Compiled by : Mark Djeron C. Tumabao

The probabili#y o" #hro'in a# lea!# @ a&e! in 9 #hro'! o" a die. Find #he probabili#y o" #hro'in a# lea!# 7 a&e! in *? #hro'! o" a die

T'o &ard! are dra'n a# random "rom a !#andard de&k o" 97 &ard!.
9C@ p@$7 0 9C6 p6$ 0 p9 % *?1*-52@19-527 0 91*-52619-52 0 1*-529 % 7@-56B S=LTI= The probabili#y o" ? or * a&e! i! 19-52*? 0 *?19-521*-52 % 57579-7?*99@7 The probabili#y o" #hro'in a# lea!# 7 a&e! i! * 4 57579-7?*99@7 % *?@B5-7?*99@7 % ?.9*96B@79*@ C ( 13,2 ) ¿ P1#'o hear#!2 C ( 52,2 )  % 13 ! 11 ! 2 ! 52 !

=

78 1326

=

1 17

50 ! 2 !

A &olle&#ion o" *9 #ran!i!#or! &on#ain! @ #ha# are de"e&#i(e. I" 7 #ran!i!#or! are !ele&#ed a# random) 'ha# i! #he  probabili#y #ha# a# lea!# * o" #hem i! ood8
 P (2 defecti"e )=

C ( 3,2 ) 3 1 = = C ( 15,2 ) 105 35

There are 55 or @5 po!!ible ou#&ome! 'hen #hro'in #'o di&e P1!2 % *-@5 P1"2 % * G *-@5 % @9-@5 =dd! % P1!2-1P1"2 % 1*-@52 - 1@9-@52 % *@9 P1any dii# bein e(en2 % 9-*? or  P1"inal #hree bein e(en2 % 1*-721*-721*-72 % *-B

A ne' phone i! bein in!#alled a# #he S#einer re!iden&e. Find #he probabili#y #ha# #he "inal #hree dii#! in #he #elephone number 'ill be e(en. A pair o" di&e i! #hro'n. Find #he S=LTI=  probabili#y #ha# #heir !um i! rea#er #han  P12 % 5-@5 P1A and 2 % @-@5 i(en #ha# #he number! are ma#&h.  P ( # ∧ $ ) 3 / 36 1 = = P1A-2 % 6 / 36 2  P ( $ )
*-! * - !7 * - 1! G a2 * - 1! G a27 * - 1!7 0 '72

IECEP FILES MATHEMATICS Compiled by : Mark Djeron C. Tumabao


/i(en a !&alar "un&#ion " 1) y ) ;2) "ind   #he radien# o" ".

! - 1!7 0 '72 ! - 1!7 4 '72 *- 1!7 4 '72 AJ % 1*2172 0 1*214@2 0 1*2 192 AJ % 7 G @ 0 9 AJ % 6 K % &o! 4* 1 AJ2 - 1A 2 AJ  % 1*21@2 0 1721472 0 1?21*2 % 4* A % !$r# 1*70772 % !$r#192  % !$r# 1@7 0 14727 0 *72 % !$r# 1*62 K % &o! 4* 1 4*2 - 1!$r# 192  !$r# 1*62 2 K % B.B59 % f   % f  % f  grad f =  i + & +  (  %x %y %'

I# i! an e$ua#ion #ha# &on#ain! one or ordinary di""eren#ial e$ua#ion !e(eral deri(a#i(e! o" an unkno'n "un&#ion &alled y12 and 'hi&h 'e 'an# #o de#ermine "rom #he e$ua#ion. Sol(e #he di""eren#ial e$ua#ion y % * 0 y7 y % #an 1 0 &2 A ++++++ i! a &olle&#ion o" obje&#!) and !e# #he!e obje&#! are &alled #he elemen#!.  G * - 6 Sol(e : 2 x =1−√ 2− x /i(en #he e$ua#ion  x

2

+ y 2 =4 ) "ind i#!

!ymme#ry. The produ&# o" #'o number! i! @7. Find a "un&#ion #ha# repre!en#! #he !um o" #heir !$uare!.

T'o !ide! o" a #rianle are 9 and *? in&he!) re!pe&#i(ely. The anle be#'een #hem i! in&rea!in a# #he ra#e o" 9O per minu#e. Ho' "a!# i! #he #hird !ide o" #he #rianle ro'in 'hen #he anle i! 5?de

!ymme#ri& 'i#h re!pe&# #o 4ai!  !ymme#ri& 'i#h re!pe&# #o y4ai!  !ymme#ri& 'i#h re!pe&# #o #he oriin The (olume o" a !phere o" radiu! r i! 31r 2 % 16-@2Nr @. 31r 2 % 6Nr 7 3192 % *??N

IECEP FILES MATHEMATICS Compiled by : Mark Djeron C. Tumabao

A ba!eball diamond i! a !$uare 'ho!e !ide! are ? "# lon. I" a ba##er hi#! a ball  b. and run! #o "ir!# ba!e a# #he ra#e o" 7? "#-!e&) ho' "a!# i! hi! di!#an&e "rom !e&ond ba!e &hanin 'hen he ha! run 9? "#8 Po!#al reula#ion! re$uire #ha# #he !um o" #he len#h and ir#h o" a re&#anular  pa&kae may no# e&eed *?B in&he! 1#he ir#h i! #he perime#er o" an end o" #he  bo2.
−80 √ 97