Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663A/01)  Year 2014  January  Q#9
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Question
A curve with equation y=f(x) passes through the point (3,6). Given that
a. use integration to find f(x). Give your answer as a polynomial in its simplest form.
b. Show that , where p is a positive constant. State the value of p.
c. Sketch the graph of y = f(x), showing the coordinates of any points where the curve touches or crosses the coordinate axes.
Solution
a.
We are given;
We are given coordinates of a point on the curve (3,6).
We are required to find the equation of y in terms of x ie f(x).
We can find equation of the curve from its derivative through integration;
Therefore,
Rule for integration of is:
Rule for integration of is:
Rule for integration of is:
If a point lies on the curve , we can find out value of . We substitute values of and in the equation obtained from integration of the derivative of the curve i.e. .
Therefore, substituting the coordinates of point (3,6) in above equation;
Therefore, equation of the curve C is;
b.
We have found in (b) that;
We are given that;
We can now compare the given and the found equations.
This yields that;















Hence, p=3 and we can write the given expression as;
c.
We are required to sketch;
As demonstrated in (b), we can write it as;
Substituting p=3 as found in (b);
It is evident that it is a cubic equation.
We can now sketch the curve as follows.
ü Find the sign of the coefficient of . This gives the shape of the graph at the extremities.
It is evident that with positive coefficient of will shape the curve at extremities like increasing from left to right.
ü Find the point where the graph crosses yaxis by finding the value of when .
We can find the coordinates of yintercept from the given equation of the curve.
Hence, the curve crosses yaxis at point .
ü Find the point(s) where the graph crosses the xaxis by finding the value of when . If there is repeated root the graph will touch the xaxis.
We can find the coordinates of xintercepts from the given equation of the curve.
Now we have two options.









Hence, the curve crosses xaxis at two points and .
ü Calculate the values of for some value of . This is particularly useful in determining the quadrant in which the graph might turn close to the yaxis.
ü Complete the sketch of the graph by joining the sections.
ü Sketch should show the main features of the graph and also, where possible, values where the graph intersects coordinate axes.
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