HSC 2016 MX2 Marathon (archive) (2 Viewers)

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leehuan

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Re: HSC 2016 4U Marathon

By lateral force, Katebate most likely meant the friction force (which is a lateral force, i.e. up or down the bend).
Ah.

Yes, that has to be included in. Unless an indication is made that the track was banked to allow 0 friction.
 

Katebate

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Re: HSC 2016 4U Marathon

Ah.

Yes, that has to be included in. Unless an indication is made that the track was banked to allow 0 friction.
By lateral force, Katebate most likely meant the friction force (which is a lateral force, i.e. up or down the bend).
Ok thank you.

Does that mean in this question for example "a car is travelling round a section of race track which is banked at an angle of 15 degrees. The radius of the track is 100 m. What is the speed at which the car can travel without tending to slip" that we set the lateral force to zero and only equate the normal and gravitation forces because we are trying to find when it continues without slipping?
 
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InteGrand

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Re: HSC 2016 4U Marathon

Ok thank you.

Does that mean in this question for example "a car is travelling round a section of race track which is banked at an angle of 15 degrees. The radius of the track is 100 m. What is the speed at which the car can travel without tending to slip" that we set the lateral force to zero and only use normal and gravity because we are trying to find when it continues without slipping?
Yes.
 

hedgehog_7

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Re: HSC 2016 4U Marathon

A mass of 1kg is fastened by a string of length 1m to a point 0.5m above a smooth horizontal table and is describing a circl o nthe table with uniform angular speed of 1 revolution in 2 seconds.

How do i find the force exerted on the table? Answer is (g - 0.5pi^2)N
 

InteGrand

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Re: HSC 2016 4U Marathon

A mass of 1kg is fastened by a string of length 1m to a point 0.5m above a smooth horizontal table and is describing a circl o nthe table with uniform angular speed of 1 revolution in 2 seconds.

How do i find the force exerted on the table? Answer is (g - 0.5pi^2)N


More working is below (in a spoiler in case you want to do it yourself first).

 
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KingOfActing

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Re: HSC 2016 4U Marathon

It's fairly easy to prove the limit does exist by using basic graphical inequalities and integration by parts.

Actually, do that as an exercise.


This doesn't prove convergence, but it does prove boundedness. Ceebs finding good upper limit and using squeeze theorem
 
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hedgehog_7

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Re: HSC 2016 4U Marathon

Complex number question

I) If w is a non real seventh root of smallest positive argument, show that w = cis 2pi/7
II) Show that 1 + w^2 + w^3 + w^3 + w^4+ w^5 + w^6 = 0
III) Show that w + w^-1 = 2 cos (2pi/7) , hence, find the exact value of cos (pi/7)cos(2pi/7)cos(3pi/7)
IV) Show that z1 = w + w^2 + w^4 and z2 = w^3 + w^5 + w^6 are the roots of the equation z^2+z+2 = 0
V) State the exact value of sin(2pi/7) + sin(3pi/7) - sin(pi/7).

I know how to do parts I,II,III and IV. But im having trouble with (V). I understand that you have to equate Im(z1) for w + w^2 + w^4 and then manipulate the sin to get what the question is asking for. However, the problem is i dont know why it has to be equated with z1 and not z2?. The answer provided is root7/2 which is the Im(z1) value but why is the value taken for Im(z1) and not Im(z2) which provides -root7/2.

Thanks!
 

Paradoxica

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Re: HSC 2016 4U Marathon

Complex number question

I) If w is a non real seventh root of smallest positive argument, show that w = cis 2pi/7
II) Show that 1 + w^2 + w^3 + w^3 + w^4+ w^5 + w^6 = 0
III) Show that w + w^-1 = 2 cos (2pi/7) , hence, find the exact value of cos (pi/7)cos(2pi/7)cos(3pi/7)
IV) Show that z1 = w + w^2 + w^4 and z2 = w^3 + w^5 + w^6 are the roots of the equation z^2+z+2 = 0
V) State the exact value of sin(2pi/7) + sin(3pi/7) - sin(pi/7).

I know how to do parts I,II,III and IV. But im having trouble with (V). I understand that you have to equate Im(z1) for w + w^2 + w^4 and then manipulate the sin to get what the question is asking for. However, the problem is i dont know why it has to be equated with z1 and not z2?. The answer provided is root7/2 which is the Im(z1) value but why is the value taken for Im(z1) and not Im(z2) which provides -root7/2.

Thanks!
Plot the roots on an argand diagram. You will observe (by vector addition) that the real/imaginary parts after addition are in specific portions of the complex plane. This, though informal, should be sufficient (for the purposes of the HSC) to deduce which root is which.
 

InteGrand

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Re: HSC 2016 4U Marathon

Complex number question

I) If w is a non real seventh root of smallest positive argument, show that w = cis 2pi/7
II) Show that 1 + w^2 + w^3 + w^3 + w^4+ w^5 + w^6 = 0
III) Show that w + w^-1 = 2 cos (2pi/7) , hence, find the exact value of cos (pi/7)cos(2pi/7)cos(3pi/7)
IV) Show that z1 = w + w^2 + w^4 and z2 = w^3 + w^5 + w^6 are the roots of the equation z^2+z+2 = 0
V) State the exact value of sin(2pi/7) + sin(3pi/7) - sin(pi/7).

I know how to do parts I,II,III and IV. But im having trouble with (V). I understand that you have to equate Im(z1) for w + w^2 + w^4 and then manipulate the sin to get what the question is asking for. However, the problem is i dont know why it has to be equated with z1 and not z2?. The answer provided is root7/2 which is the Im(z1) value but why is the value taken for Im(z1) and not Im(z2) which provides -root7/2.

Thanks!




 
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