Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2011  OctNov  (P29709/23)  Q#8
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Question
The equation of a curve is 2x^{2 }− 3x − 3y + y^{2} = 6.
i. Show that
ii. Find the coordinates of the two points on the curve at which the gradient is −1.
Solution
i.
We are given equation of the curve as;
We are required to show that;
Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is:
Therefore;
To find from an implicit equation, differentiate each term with respect to , using the chain rule to differentiate any function as .
Therefore;
Rule for differentiation of is:
We differentiate each term of the equation, one by one, with respect to x applying following rules.
Rule for differentiation of is:
Rule for differentiation of is:
Now we can combines derivatives of all terms of the equation as;
ii.
We are required to find the coordinates of the two points on the curve at which the gradient is −1.
Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is:
We have found in (i) that for the given curve.
We are given that gradient of the curve at desired points is equal to 1.
Hence;
We can substitute in given equation of the curve;
Now we have two options.





We know that ; therefore;
For 
For 




Hence, gradient of the given curve is 1 at two points of following coordinates.


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