4U Revising Game (4 Viewers)

Pwnage101 · Sep 14, 2008

got it:

X(1) = 1 and X(n+1) = 4+X(n) / 1+ X(n)

therefore, X(2) = (4+1)/(1+1) = 5/2

NOW, TEST: X(n) = 2{1+a^n/1-a^n} where a=-1/3

for n= 1, this predicts X(1) = 2(1-1/3)/(1--1/3) = 2(2/3)/(4/3) = 2(2/4)=2(1/2)=1 therefore, this statement is true for n=1

Assume true for n=R, ie assume: X(R) = 2{1+a^R/1-a^R} where a=-1/3

Need to show n=R+1 is also true. That is show: X(R+1) = 2{1+a^(R+1)/1-a^(R+1)} where a=-1/3

Now, by what we are given,
X(R+1) = 4+X(R) / 1+ X(R)
= (4+2{1+(-1/3)^R/1-(-1/3)^R})/(1+2{1+(-1/3)^R/1-(-1/3)^R}) by assumption
= [4-4(-1/3)^R+2+2(-1/3)^R]/[1-(-1/3)^R+2+2(-1/3)^R by rearranging, simplifying and expanding
=[6-2(-1/3)^R]/[3+(-1/3)^R]
Now, multiply every term by (-1/3):
=[-2-2(-1/3)^(R+1)]/[-1+(-1/3)^(R+1)]
Multiplyu each term by -1:
= 2[1+(-1/3)^(R+1)]/[1-(-1/3)^(R+1)]
= 2{1+a^(R+1)/1-a^(R+1)} where a=-1/3

tHUS, I have proved if teh statement is true for n=R, it is also true for n=R+1. Since it was true for n=1, it is true for n=2, hence n=3,4,5,etc. Thus true for all n>=1

Q:

(i) Show that: [Sin(X+Y)-Sin(X-Y)]/2SinY = CosX

(ii) Hence show: cosx+cos3x+cos5x+...+cos(2n.3)x+cos(2n-1)x = [sin(2nx)]/2sinx

(iii) Hence evaluate INT(0--->(PI/2)) sin8x/sinx.dx

Pwnage101 · Sep 16, 2008

anyone?

lolokay · Sep 16, 2008

Pwnage101 said:
Q:

(i) Show that: [Sin(X+Y)-Sin(X-Y)]/2SinY = CosX

(ii) Hence show: cosx+cos3x+cos5x+...+cos(2n.3)x+cos(2n-1)x = [sin(2nx)]/2sinx

(iii) Hence evaluate INT(0--->(PI/2)) sin8x/sinx.dx

i) [Sin(X+Y)-Sin(X-Y)]/2SinY
= [sinXcosY + sinYcosX - sinXcosY + sinYcosX]/2sinY
= 2sinYcosX/2sinY
= cosX

ii) cosx+cos3x+cos5x+...+cos(2n.3)x+cos(2n-1)x
= (sin2x)/2sinx + (sin[3x+x] - sin[3x-x])/2sinx + ..... + (sin[2nx] - sin[2nx-2x])/2sinx <- each term will cancel out the first half of the term before it
= (sin2x + sin4x - sin2x + ... + sin[2nx])/2sinx
= sin[2nx]/2sinx

(iii) Hence evaluate INT(0--->(PI/2)) sin8x/sinx.dx
2*INT(0--->(PI/2))sin8x/2sinx
=2*INT(0--->(PI/2) (cosx + cos3x + cos5x + cos7x)
= 2(sinx + sin[3x]/3 + sin[5x]/5 + sin[7x]/7) [0-->pi/2]
= 2(1 - 1/3 + 1/5 - 1/7)
= 152/105

shinn · Sep 17, 2008

(i) Show that (Sum of k=1 to k =n)z + z^2 + z^3 + ..... + z^k = [nz / ( 1 - z) ] - [z^2 / (1-z)^2]*(1-z^n) where z cant = 1.

(ii) Let z = cos@ + isin@. Prove that [z / (1-z)] = [ i/ 2sin(@/2)]*[cis@/2]

(iii) By considering imaginary part of LHS, deduce that:
sin@ + sin 2@ + sin 3@ +....+ sinn@ = [(n+1)sin@ - sin[(n+1)@]] / [2sin(@/2)]^2

(iv) Find a similar expression for cos@ + cos 2@ + cos3@ + ..... + cosn@

vds700 · Oct 11, 2008

Ill start this up again with an interesting Q from an assessment i had a while ago:

A particle, P<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

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ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>₁ is connected by a string of length l to a fixed point O. Another particle, P<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>₂ is attached to P<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>₁ by a string of the same length. The system rotates around O, with constant angular velocity, <meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>[FONT=&quot]ω[/FONT] rad/s. P<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>₁ makes an angle <meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>[FONT=&quot]α[/FONT] with the vertical and P<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>₂ makes an angle <meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>[FONT=&quot]β[/FONT] with the vertical.

Prove that:
a)tan<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>[FONT=&quot]α[/FONT] = l<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>[FONT=&quot]ω[/FONT]^2/g(sin<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>[FONT=&quot]α[/FONT] + 1/2 sin<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>[FONT=&quot]β[/FONT])
b)tan<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>[FONT=&quot]β[/FONT] = l<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>[FONT=&quot]ω[/FONT]^2/g (sin[FONT=&quot]α[/FONT] + sin<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 11"><meta name="Originator" content="Microsoft Word 11"><link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style>  </style>[FONT=&quot]β[/FONT])

Note both are of mass m and g is the acceleration due to gravity

adnan91 · Oct 11, 2008

u sure abt p2 being attached to p1 not O

friction · Oct 11, 2008

I did this one the other day somewhere so i will let someone else do it.

vds700 · Oct 11, 2008

adnan91 said:
u sure abt p2 being attached to p1 not O

Yeah definitely

shinn · Oct 11, 2008

Here is Part (b) first...

Let T2 be the tension in the second string of P2 to P1.
Let P2 rotate about O at a radius of R.

R = l sin [FONT=&quot]α[/FONT] + l sin b = l (sin [FONT=&quot]α[/FONT] + sin [FONT=&quot]β[/FONT]) (1)

Forces on P2:
Vertical forces = 0
=> T2 cos [FONT=&quot]β[/FONT] = mg (I)
Horizontal forces = mRw^2
=> T2 sin [FONT=&quot]β[/FONT] = m l w^2 (sin [FONT=&quot]α[/FONT] + sin [FONT=&quot]β[/FONT]) (II) , by (1)

(II) divided by (I):

tan b = [FONT=&quot]l ω[/FONT]^2/g(sin[FONT=&quot]α[/FONT] + sinβ)

Now for Part (A)
Let T1 be the tension in the first string

Forces at P1:
Vertical Forces = 0
=> T1 cos a - T2 cos B = mg (III)

Horizontal Forces = 0
=> T1 sin a - T2 sin B = m l w^2 sin a (IV)

From (I) => T2 = mg / cos B (2)

Subs (2) into (III):
=> T1 cos a - mg = mg
=>T1 = 2mg /cos a (3)

Subs (2) and (3) into (IV):
=> 2 mg tan a - mg tan b = m l w^2 sin a
=> 2 g tan a - g (w^2 l (sin a + sin b) ) / g= l w^2 sin a,
by tan B = (w^2 l (sin a + sin b) ) / g
=> 2g tan a = w^2 l (2 sin a + sin b)

Thus, tan a = w^2 l / g (sin a + 1/2 sin b)

duy.le · Oct 11, 2008

vds700 said:
Ill start this up again with an interesting Q from an assessment i had a while ago:

A particle, P<link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><style>  </style>₁ is connected by a string of length l to a fixed point O. Another particle, P<link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><style>  </style>₂ is attached to P<link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><style>  </style>₁ by a string of the same length. The system rotates around O, with constant angular velocity, <link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><style>  </style>[FONT=&quot]ω[/FONT] rad/s. P<link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><style>  </style>₁ makes an angle <link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><style>  </style>[FONT=&quot]α[/FONT] with the vertical and P<link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><style>  </style>₂ makes an angle <link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><style>  </style>[FONT=&quot]β[/FONT] with the vertical.

Prove that:
a)tan<link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><style>  </style>[FONT=&quot]α[/FONT] = l<link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><style>  </style>[FONT=&quot]ω[/FONT]^2/g(sin<link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><style>  </style>[FONT=&quot]α[/FONT] + 1/2 sin<link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><style>  </style>[FONT=&quot]β[/FONT])
b)tan<link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><style>  </style>[FONT=&quot]β[/FONT] = l<link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><style>  </style>[FONT=&quot]ω[/FONT]^2/g (sin[FONT=&quot]α[/FONT] + sin<link rel="File-List" href="file:///C:%5CUsers%5CAndrew%5CAppData%5CLocal%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><style>  </style>[FONT=&quot]β[/FONT])

Note both are of mass m and g is the acceleration due to gravity

although that is probably the right method in answering the question, ive posted up my full response.

three notes;
1. its easier to solve b) first then a)
2. i had to use the result from b) to solve a)
3. VDS700 please write the question properly, it was very hard to be able to tell that the thing on the denominator was meant to be in the numerator. please use an "x" or more brackets.

My question; yet another induction..

Show that for n>=3, n^2<(n+1)^(n-1)

oh sorry for the fact that i had to use word and my writing cause i cbb trying to figure out how to put it on as zip or photo.

shinn · Oct 11, 2008

(n+1)^n - n^2 (n+1)
>/ (3+1)^3 - 3^2 (3 +1) , as n>/ 3
= 4^3 - 36
= 28
> 0

Thus, (n+1)^n > n^2 (n+1)
(n+1) ^(n-1) > n^2, since n + 1 > 0

Therefore, n^2 < (n+1)^(n-1)

duy.le · Oct 12, 2008

sorry i should of made it more explicitly clear that i wanted it proven by mathematical induction.

edit: but your one step closer =D

shinn · Oct 12, 2008

n^2 < (n+1)^(n-1) for n>/3

Proof:

Step1:
When n = 3,
LHS = 3^2 = 9
RHS = (3+1)^(3-1) = 16

Since 9 < 16 , therefore LHS < RHS. The result is thus true for n =3.

Step 2:
Assume: k^2 < (k+1)^(k-1) for k >/3 (I)
We need to prove: (k+1)^2 < (k+2)^k (II)

RHS (II) - LHS (II)
= (k+2)^k - (k+1)^2
>/ (3+2)^3 - (3+1)^2 , as k>/3
= 125 - 16
= 109
>0
Therefore LHS (II) < RHS (II)
Hence, the result is true for n = k +1 if it is also true for n = k.

Step3:
By mathematical induction, the result is true for all positive integers of n great than 3.

duy.le · Oct 12, 2008

well thats not how i did it but...ill take ur word for it that thats a valid solution (cause im not really sure it is)

so u wanna post up a question?

pLuvia · Oct 12, 2008

I'll let you guys do it because I cbb right now but if you run out of questions or want to try past questions from the past/present maths veterans check out this thread

Marathon 1
Marathon 2

u-borat · Oct 12, 2008

duyle, post your solution?

edit

also shinn, your proof isnt an induction proof as you didn't use the substituion of n i dont think...

duy.le · Oct 12, 2008

heres how i did

duy.le · Oct 13, 2008

ok then so one posted up anything so ill post up a new question;

i)the equation x^3 +px^2+qx+r=0 (where p, q, r are non zeros) has root alpha,beta,gamma such that 1/alpha , 1/beta, 1/gamma are consecutive terms in a arithmetic sequence. Show that;

beta= -3r/q

ii) the equation x^3-26x^2+216x-576=0 has roots alpha,beta,gamma such that 1/alpha , 1/beta, 1/gamma are consecutive terms in a arithmetic sequence. find the values of alpha, beta, gamma.

QuLiT · Oct 13, 2008

i) ok i formed the equation with roots 1/alpha etc so it becomes:
rx^3 + qx^2 + px + 1= 0
then since the roots are 1/alpha + 1 /beta + 1/gamma and they are in arithmatic seq. then they can be written as :
1/beta - d , 1/beta , 1/beta + d

then sum of roots = -q /r

1/beta -d + 1/beta + 1/beta + d = -q/r
3 /beta = -q/r

there fore beta = -3r /q

ok part two answers are: alpha = 12 beta = 8 gamma = 6

duy.le · Oct 13, 2008

well yes they're right but would u mind posting up a question, its no fun posting them up and not being able to solve other peoples.

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