Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2015  June  Q#8
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Question
a. Factorise completely 9x – 4x^{3}
b. Sketch the curve C with equation
y = 9x – 4x^{3}
Show on your sketch the coordinates at which the curve meets the xaxis.
The points A and B lie on C and have x coordinates of –2 and 1 respectively.
c. Show that the length of AB is where k is a constant to be found.
Solution
a.
We are given;
We have algebraic formula;
Therefore;
b.
We are required to sketch;
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 negative coefficient of will shape the curve at extremities like decreasing 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 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.
c.
We are given that the points A and B lie on C and have x coordinates of –2 and 1 respectively.
We need to find the length of AB.
Expression for the distance between two given points and is:
Therefore we need coordinates of the points A and B.
If a point lies on the curve (or the line), the coordinates of that point satisfy the equation of the curve (or the line).
Therefore, we substitute xcoordinates of both points A and B in given equation of the curve to find the corresponding ycoordinates.
For point A, substitute x=2; 
For point A, substitute x=1; 














Hence the points A and B have coordinates and , respectively.
Now we can find the distance AB.
Since ;
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