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# C4 Questions Parametric equations and vectors....! HELP :S watch

1. Hiya was doing some questions and got stuck on a few and since the book doesnt show the working i was wondering if any of u would be kind enough to help me out..
cheeers

1. (dnt get c of this q but am typin it all)
Parametric equations of curve C:
x=t^2, y= sin2t, t> or = 0
Point A is an intersection of C with X-axis
a. find in terms of pi the x-cord of A
b. find dy/dx in terms of t, t>0
c. show that an equation of the tangent to C at A is 4x + 2pi*y = pi^2

2. (dnt get b and c of this q but am typin it all..)
a. if (1+x)(2+y) = x^2 + y^2, find dy/dx in terms of x and y
b. find the gradient of the curve (1+x)(2+y) =x^2 + y^2 at each of the 2 points where the curve meets the y-axis
c. show that there are 2 points at which the tangents to this curve are parallel to the y-axis

3. dy/dx =x^x , x>0, y>0, by taking logarithms show that
dy/dx = x^x(1+ lnx)

4. (dnt get the c part of this question but am typin it all..)
a. given that x=2^t, by using logarithms prove that dx/dt = 2^t ln2
A curve C has parametric equations x=2^t, y=3t^2. the tangent to c at point with co-ord (2,3) cuts x-axis at point P.
b. find dy/dx in terms of t
c. calculate the x-cord of P, giving answer to 3dp

5. (dnt get c, but again am typing whole question)
A population P is growing at the rate of 9% each yr and at time t years may be approximated by the formula P = Po(1.09)^t, t> or =0
where P s regarded as a continuos function of t and Po is the starting population at time t=0
a. find an expression for t in terms of P and Po
b. find the timeT yrs when the population has doubled from its value at t=0, giving your ans to 3sf
c. find as a multiple of Po, the rate of change of population dP/dt at time t =T

6. find the magnitude and direction of each of these vectors:
a. -8i + 15j
i got magnitude 17 and direction -61.9 degrees, the answer in the back for direction is 118.1 degrees.. why does this need 180 degrees added to it? when the other negative values i got were left negative?

b. -8i
magnitude =8, but how do i work out the direction?
same for
c. 16j (magnitude =16)

7. the vector p has magnitude 10 units and is inclined at 150 degrees to the x-axis. express p as a column vector.
how do i solve this one?!

starting to worry exam is next thursday.. really need to get an A in maths
2. Are you sure you've written all of question 1 down? I don't see how you could get the x co-ordinate of A unless you've got a value for t. Since you know y=0, sin2t=0 as well, therefore 2t=0, pi, 2pi, etc.
3. umm well question 1 also has a quick sketch of the graph. point A y-coord is 0
sorry for not including that info forgot too ooops
4. Hmmm...
5. I cba to all of them, but I will do the last 2 (pretty sure someone will do some of the others while i am typing)

6. There are two possible angles between the lines, 61.9 degrees and 118.1 degrees (draw any two lines on a piece of paper and see for yourself). As a rule never give the negative solution for cos (theta) in the dot product formula. For the equation cos (theta) = k there are always two solutions for most values (obvious exceptions being where we have right angles)

7. Polar co-ordinates. We know that x = r cos (theta) and y = r sin (theta), where theta is the angle (be very careful to sketch first so you get the signs right) and r is the modulus, or magnitude of the vector.
6. oh and the curve passes throught the origin**
7. For 2b once you've gotten dy/dx you sub x=0 in both the equation for the gradient and the original equation. From the original equation you now have a quadratic involving only y terms - factorise to get 2 values of y. Sub these 2 values into the dy/dx equation to get 2 numerical values for the gradient.
8. For question 1c I think once you have the gradient and the x and y coordinates of A you can use y - y1 = m(x-x1) to get the equation (or y=mx+c).
9. thanks turgon got 1 and 2 now !!!
annd 6 and 7 took my brain a little longer to get lol, but i think i understand (i hate vectors find it difficult to get my head round!)
10. 4c:

You've got the equation for the gradient and the x and y co ordinates. Use the co-ordinates to get a numerical value for the gradient. Use y-y1 = m(x-x1) to get the equation for the tangent. Make the y value in that equation of the tangent 0 and re arrange to get the x value.
11. I havent read all your questions, there are several and i'm watching the football right now

3. Take ln of both sides. You'll then need to differentiate implicitly. Note . You should be able to differentiate that via the product rule.

6a. Draw -8i+15j on a graph. Look where it is. The direction is always the angle it makes with the positive x-axis. So you want
The dot product does not come into this question!!
6b. what angle does -8i make with the positive x-axis?
same for c.
12. yaaay thanks question 4c i get! and its riiight woop woop haha
13. 5c
Right at the start of the question it says "A population P is growing at the rate of 9% each yr ".

This means dp/dt = 0.09P, where P is the current population. You know at T, P=2Po. Sub this into the dp/dt equation, and I think you have your answer.
14. thanks silent ninja

and turgon im confused so u u just multiply 0.09 by 2?
is that what u mean?
coz answer is meant to be0.172 Po
15. 5c I remember doing before but cant find my post. Here are two threads that answer this (i havent checked if they are correct, i just hit the search button!): here and here.
16. ohhh i c thankkkkks!
17. I see how the answer was obtained using the other method...but I don't understand why mine doesn't work for this.

Is dp/dt not equal to 0.09P then or am I missing out a constant or something?
18. dP/dt is not 0.09P. The growth of the population would be some sort of geometric equation, not a quadratic as you have wrote. The question gives you the function in any case, you differentiate that to give you dP/dt.
19. (Original post by silent ninja)
dP/dt is not 0.09P. The growth of the population would be some sort of geometric equation, not a quadratic as you have wrote. The question gives you the function in any case, you differentiate that to give you dP/dt.
I think that makes it slightly clearer . So I assume this means that even though it states "A population P is growing at the rate of 9% each yr" the rate of change of population is not proportional to the population at the time.
20. No, I dont think it is. It would normally suggest or say something is proportional to something else. 0.09 is not the same as 9% growth. You've just written the gradient is 0.09P, that is, the gradient is constant, but that's not true for a geometric equation.

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