Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2012  OctNov  (P29709/21)  Q#4
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Question
The parametric equations of a curve are
for t < 0.
i. Show that in terms of t.
ii. Find the exact coordinates of the only point on the curve at which the gradient is 3.
Solution
i.
We are required to find for the parametric equations given below;
If a curve is given parametrically by equations for and in terms of a parameter , then;
First we find . We are given that;
Therefore;
Rule for differentiation of is:
Similarly, we find , when we are given that;
If we define , then derivative of is;
If we have and then derivative of is;
Let , then , therefore;
For ;
Since ;
Rule for differentiation of is:
Rule for differentiation of is:
Rule for differentiation of is:
Finally;
ii.
We are required to find the coordinates of the only point on the curve at which the gradient of the curve is equal to 3.
Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is:
From (i) we have;
Hence;
Now we have two options.





Two possible values of imply that there are two points on the curve, one at each value of , where gradient of the curve is 3.
We are given that . Therefore, only possible value is;
To find the ycoordinate of the point where t=1, we substitute this value of t in following parametric equation of the curve;
To find the xcoordinate of the point where t=1, we substitute this value of t in following parametric equation of the curve;
Hence coordinates of the point on the curve where t=1 are (,2).
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