HSC 2015 MX1 Marathon (archive) (1 Viewer)

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Trebla

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Post any questions within the scope and level of Mathematics Extension 1. Once a question is posted, it needs to be answered before the next question is raised.

I encourage all current students in particular to participate in this marathon.

To start off:

 
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jyu

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Re: HSC 2015 3U Marathon

h depends on d?
 

PhysicsMaths

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Re: HSC 2015 3U Marathon

i) Using tan(a-b) = tana - tanb / 1+tanatanb,
show that 1+tannθtan(n+1)θ = cotθ(tan(n+1)θ-tannθ)

ii) Prove by mathematical induction that
tanθtan2θ + tan2θtan3θ + ... + tannθtan(n+1)θ = -(n+1) + cotθtan(n+1)θ for all integers n>=1
 

leehuan

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Re: HSC 2015 3U Marathon

i) Using tan(a-b) = tana - tanb / 1+tanatanb,
show that 1+tannθtan(n+1)θ = cotθ(tan(n+1)θ-tannθ)

ii) Prove by mathematical induction that
tanθtan2θ + tan2θtan3θ + ... + tannθtan(n+1)θ = -(n+1) + cotθtan(n+1)θ for all integers n>=1
Files too large and I don't feel like typing my working. Refer to links:
i) http://i1164.photobucket.com/albums/q566/Rui_Tong/CCI07012015_zpsdc70650f.jpg
ii) http://i1164.photobucket.com/albums/q566/Rui_Tong/CCI07012015_0001_zps84b4191b.jpg
http://i1164.photobucket.com/albums/q566/Rui_Tong/CCI07012015_0002_zpsa8ac050b.jpg

Next question:
Using the substitution u=ln(x), or otherwise, find Capture.PNG

Hint for those who haven't done logs (highlight to show):
d/dx ln(x) = 1/x
 
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PhysicsMaths

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Re: HSC 2015 3U Marathon

Files too large and I don't feel like typing my working. Refer to links:
i) http://i1164.photobucket.com/albums/q566/Rui_Tong/CCI07012015_zpsdc70650f.jpg
ii) http://i1164.photobucket.com/albums/q566/Rui_Tong/CCI07012015_0001_zps84b4191b.jpg
http://i1164.photobucket.com/albums/q566/Rui_Tong/CCI07012015_0002_zpsa8ac050b.jpg

Next question:
Using the substitution u=ln(x), or otherwise, find View attachment 31575

Hint for those who haven't done logs (highlight to show):
d/dx ln(x) = 1/x
Nice first post!
http://imgur.com/bNK5TeN (I think this is how you do it)

Next question:
Differentiate tanx using first priciples
 

leehuan

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Re: HSC 2015 3U Marathon

Nice first post!
http://imgur.com/bNK5TeN (I think this is how you do it)

Next question:
Differentiate tanx using first priciples
Thanks! Yep that's how to do it
http://i1164.photobucket.com/albums/q566/Rui_Tong/CCI08012015_zpsa3c8b8da.jpg

Next question:
(i) Explain why d/dx c = 0 (for some constant c)
(ii) Prove by mathematical induction and derivative laws that the nth derivative of x^n is n!
(iii) Show that for any polynomial P(x)=a+bx+cx^2+...+zx^n, the (n+1)th derivative is always equal to 0
 

PhysicsMaths

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Re: HSC 2015 3U Marathon

Thanks! Yep that's how to do it
http://i1164.photobucket.com/albums/q566/Rui_Tong/CCI08012015_zpsa3c8b8da.jpg

Next question:
(i) Explain why d/dx c = 0 (for some constant c)
(ii) Prove by mathematical induction and derivative laws that the nth derivative of x^n is n!
(iii) Show that for any polynomial P(x)=a+bx+cx^2+...+zx^n, the (n+1)th derivative is always equal to 0
I like that induction question. Couldn't get it today, i'll try it tomorrow.
 

PhysicsMaths

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Re: HSC 2015 3U Marathon

Thanks! Yep that's how to do it
http://i1164.photobucket.com/albums/q566/Rui_Tong/CCI08012015_zpsa3c8b8da.jpg

Next question:
(i) Explain why d/dx c = 0 (for some constant c)
(ii) Prove by mathematical induction and derivative laws that the nth derivative of x^n is n!
(iii) Show that for any polynomial P(x)=a+bx+cx^2+...+zx^n, the (n+1)th derivative is always equal to 0
i) i) If c is a constant then y=c represents a horizontal line. Thus,the gradient of the tangent for any point P on that line will be 0;i.e. d/dx c= 0
ii) http://imgur.com/G0HdtqQ (may or may not be right)
iii) (working on it)
 
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colinrocks95

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Re: HSC 2015 3U Marathon

Thanks! Yep that's how to do it
http://i1164.photobucket.com/albums/q566/Rui_Tong/CCI08012015_zpsa3c8b8da.jpg

Next question:
(i) Explain why d/dx c = 0 (for some constant c)
(ii) Prove by mathematical induction and derivative laws that the nth derivative of x^n is n!
(iii) Show that for any polynomial P(x)=a+bx+cx^2+...+zx^n, the (n+1)th derivative is always equal to 0
For (iii), we can work out each derivative as follows:
P'(x) = b + 2cx + 3dx^2 + ... + znx^(n-1)
P"(x) = 2c + 6dx + ... + zn(n-1)x^(n-2)
...
P^n(x) = zn(n-1)...(n-[n-1])x^(n-n) = zn!, where n! = n(n-1)(n-2)...1

Since P^n(x) is a constant, P^(n+1)(x), i.e. the (n+1)th derivative, is zero.

--

Next question:
By expanding sin(A+B), show that sin(3x) = 3sin(x) - 4sin^3(x). Hence or otherwise find the smallest positive value of u that satisfies the equation 12u^3 - 9u + 2 = 0. Give your answer to two decimal places.

By the way, you should check out Sci School's HSC programs. I know the presenter and he's a genius + really good at explanations.
 

PhysicsMaths

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Re: HSC 2015 3U Marathon

For (iii), we can work out each derivative as follows:
P'(x) = b + 2cx + 3dx^2 + ... + znx^(n-1)
P"(x) = 2c + 6dx + ... + zn(n-1)x^(n-2)
...
P^n(x) = zn(n-1)...(n-[n-1])x^(n-n) = zn!, where n! = n(n-1)(n-2)...1

Since P^n(x) is a constant, P^(n+1)(x), i.e. the (n+1)th derivative, is zero.

--

Next question:
By expanding sin(A+B), show that sin(3x) = 3sin(x) - 4sin^3(x). Hence or otherwise find the smallest positive value of u that satisfies the equation 12u^3 - 9u + 2 = 0. Give your answer to two decimal places.

By the way, you should check out Sci School's HSC programs. I know the presenter and he's a genius + really good at explanations.
http://imgur.com/XDVhTNU

But still, how can you tell that it's the smallest possible number? o.o
 

leehuan

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Re: HSC 2015 3U Marathon

Next question:
(i) Show that the equation of the tangent at P(2ap,ap^2) on the parabola x^2=4ay is y=px-ap^2
(ii) Show that the point of intersection of the tangents at P and Q(2aq,aq^2) on said parabola is (a(p+q),apq)

Boring question, but I'm a bit out of ideas.
 
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Re: HSC 2015 3U Marathon

Next question:
(i) Show that the equation of the tangent at P(2ap,ap^2) on the parabola x^2=4ay is y=px-ap^2
(ii) Show that the point of intersection of the tangents at P and Q(2aq,aq^2) on said parabola is (a(p+q),apq)

Boring question, but I'm a bit out of ideas.
Stock standard question hahaha
 

PhysicsMaths

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Re: HSC 2015 3U Marathon

Next question:
(i) Show that the equation of the tangent at P(2ap,ap^2) on the parabola x^2=4ay is y=px-ap^2
(ii) Show that the point of intersection of the tangents at P and Q(2aq,aq^2) on said parabola is (a(p+q),apq)

Boring question, but I'm a bit out of ideas.
lol

i) y = x^2/4a
y' = x/2a
f'(2ap) = p
y -ap^2 = p(x-2ap)
y = px -ap^2

ii) Similarly, at Q, y = qx-aq^2 -------1
px-ap^2 = qx-aq^2
x(p-q) = a(p+q)(p-q)
therefore, x = a(p+q)---------2
2 -> 1
y = q(ap+aq)-aq^2
= apq

Therefore, point of intersection = (a(p+q),apq)
 
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PhysicsMaths

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Re: HSC 2015 3U Marathon

Next question!

Express (30 x 28 x 26 x ... x 2) / (15 x 13 x 11 x ... x 1)
in factorial notation

(i'm running out of ideas too)
 

Kaido

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Re: HSC 2015 3U Marathon

rofl, pre sure u dont learn that in yr 12 ^

i reckon its 30!!/15!! ?
 
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