At 5/29/07 07:41 AM, dELtaluca wrote:
a little maths test for people :P
fap fap fap
1. (Not sure, perhaps one of the further modules)
find and classify all stationary points of the function
f(x,y)=x^2 - 4xy + y^3 + 4y
Is that like 3 dimensions or something? I have no idea how to differentate with respect to 2 variables, but i'll have a go.
f'(x, y) = 2x - 4 + 3y^2 + 4
2x - 4 + 3y^2 + 4 =0
3y^2 + 2x = 0
y=+-sqrt(-2x/3)
f''(x, y) = 2+6y
2+6(+-sqrt(-2x/3))
um... stuck.
2. (Core 4)
integrate with respect to x;
1/sqrt(e^(2x) - 1), for x>=0
I could differentaite that, but haven't done c4 yet.
But at a guess, I'd say:
e^(2x+c) (e(2x+d) - 1) ^0.5
Perhaps theres a natural log in there as well somewhere.
3. (c4 Jan07) This was a question on my c4 paper in january
a curve is defined implicitly by the equation
x^2 + xy + y^2 = 3
show that the curve has two stationary points, and find their coordinates
Theres probably some special C4 method of doing it, but i'll try:
y= (3-x^2)/(1+x)
dy/dx = (-2x(1+x)(3-x^2) - x(3-x^2))/(1+x)^2
x(2x^3 + 3x^2 - 6x - 9)/(1+x)^2=0
2x^3 + 3x^2 - 6x - 9 = 0
Then i'd spend ages failing to put that in the form (ax+b)(cx+d)^2 = 0 and i;ve probably already gone wrong anyway.
(the above question that i made up) (C4)
find the general solution to the differential equation
Whats that meant to mean? Just integrate it?
dy/dx = (xsin x)/(cos^2 y)
have fun :P ill post answers when/if a few people attempt to answer, and ill say if someone gets it right obviously
thats not testing peoples maths abilities. Thats just testing who's done C4.
Since you haven't actually taken M1 and M2 yet, lets concentrate on nice long wordy mechanics questions. I'd be interested to see how you hope to get an A in mechanics thinking gravity is a force.
1. A car of mass 1 tonne is climbing a hill inclined at an angle è to the horizontal where
sin è = 1/7 . When the car passes a point X on the hill, it is travelling at 20 m s^-1. When the car passes the point Y, 200 m further up the hill, it has speed 10 m s^-1.
In a preliminary model of the situation, the car engine is assumed only to be doing work
against gravity. Using this model, Find the change in the total mechanical energy of the car as it moves from X to Y.
2. A straight log AB has weight W and length 2a. A cable is attached to one end B of the log. The cable lifts the end B off the ground. The end A remains in contact with the ground, which is rough and horizontal. The log is in limiting equilibrium. The log makes an angle C to the horizontal, where tan C = 5/12. The cable makes an angle D to the horizontal. The coefficient of friction between the log and the ground is 0.6. The log is modelled as a uniform rod and the cable as light.
(a) Show that the normal reaction on the log at A is 2/5W.
(b) Find the value of â.
The tension in the cable is kW.
(c) Find the value of k.