Maths Extension 2 Marathon (1 Viewer)

[00iCon]

I got kicked out of the 2Unit marathon so I'll post this.
All you do is answer the previous question and write your own.
I'll start with this from the Ruse 2009 trial, 2 marks each.
Q: The area bounded by y = 4 - x<SUP>2</SUP>, x=2 and y=4 is rotated about the line x=4. Using the method of cylindrical shells:
i) Show that the volume of a cylindrical shell of thickness dx is
(pi)x<SUP>2</SUP>(8 - 2x)dx
ii) Find the volume of the solid generated</SPAN>

 

[eoklm11]

Q: The area bounded by y = 4 - x^2, x=2 and y=4 is rotated about the line x=4. Using the method of cylindrical shells:
i) Show that the volume of a cylindrical shell of thickness dx is
(pi)x^2(8 - 2x)dx
ii) Find the volume of the solid generated

i)The cylindrical shell bounded by three curves with thickness dx is height of (4-y), outer radius (4 -x +dx), inner radius (4 -x).

dV=(pi)(4-y)[ ( 4- x + dx)^2 - ) ( 4 -x )^2]
=(pi)x^2[ 8dx - 2xdx + (dx)^2]

Since (dx)^2 is negligible

dV = (pi)x^2(8-2x)

ii)
dV=(pi)(4x^2 - 2x^3)
limits are x=0 to x =2
V= (pi)[32/3 - 8)
= 8(pi)/3 cubic units...

plz check, i don think i'm rite

The equation x^2 - x + 1 = 0 has roots A and B, and In= A^n+ B^n for n ≥ 2

without solveing the equation, show that In= In-1 for n ≥ 3

 

[untouchablecuz]

methinks your question is broken; for ... +I_n+1=I_n=_n-1=I_n-2= ... , I_n-2 must equal to zero

but this is nonsense because if I_n-2 equals zero, then ... +I_n+1=I_n=_n-1=I_n-2= ... = 0, which is clearly false because I₃=α³+β³=(α+β)(α²-αβ+β²)=(α+β)((α+β)²-2αβ-αβ)=(α+β)((α+β)²-3αβ)=(1)((1)²-3(1))=-2

 

[ninetypercent]

I've only learnt up to complex numbers.

This is the only question I have for mx2 lol

solve the equations to find z and w, expressed in a + ib form

z + (1 - i)w = 2i
w + (1 - i)z = 1

 

[Trebla]

The equation x^2 - x + 1 = 0 has roots A and B, and In= A^n+ B^n for n ≥ 2

without solveing the equation, show that In= In-1 for n ≥ 3

There is something wrong with that expression. Say for n = 3
I₃=A³+B³
= (A + B)(A² - AB + B²)
= A² + B² - 1 (since A + B = 1, AB = 1)
=/= I₂

For the proposed question:
(1): z + (1 - i)w = 2i
(2): w + (1 - i)z = 1

From (1):
z = - w + iw + 2i

sub into (2):
w + (1 - i)(- w + iw + 2i) = 1
w - w + iw + 2i + iw + w + 2 = 1
2iw + w = - 1 - 2i
w(1 + 2i) = - (1 + 2i)
w = - 1 + 0i

Hence:
z = 1 - i + 2i
=> z = 1 + i

 

[Trebla]

 

[untouchablecuz]

 

[jet]

1995 HSC Question 6a):
Pat observed an aeroplane flying at a constant height, h, and with constant velocity. Pat first sighted it due east, at an angle of elevation of 45°. A short time later it was exactly north-east, at an angle of elevation of 60°.
(i) Draw a diagram to represent this information.
(ii) Find an expression in terms of h for the initial horizontal distance between Pat and the point directly below the aeroplane.
(iii) In what direction was the aeroplane flying? Give your answer as a bearing to the nearest degree.

 

[untouchablecuz]

i think everything is pretty self explanatory

i find a Q l8r

 

[addikaye03]

1. Find the volume of the solid generated by rotating the region bounded by the curves y^2=x and y=x^2 about the line y=x.

2. Use the method of slicing to find the volume of the solid obtained by rotating the region enclosed within the circle (x-1)^2+y^2=1 about the y-axis.

Spoiler

Got solutions for both if needed (1. books solution and 2. my solution)