Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2010  January  Q#9
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Question
a) Factorise completely x^{3} – 4x
b) Sketch the curve C with equation
y = x^{3 }– 4x
showing the coordinates of the points at which the curve meets the xaxis.
The point A with xcoordinates 1 and the point B with xcoordinate 3 lie on the curve C.
c) Find an equation of the line which passes through A and B, giving your answer in the form y = mx + c, where m and c are constants.
d) Show that the length of AB is , where k is a constant to be found.
Solution
a)
b)
We are required to sketch;
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.
As demonstrated in (a), it can be written as;
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 three 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 required to find equation of line AB.
To find the equation of the line either we need coordinates of the two points on the line (TwoPoint form of Equation of Line) or coordinates of one point on the line and slope of the line (PointSlope form of Equation of Line).
We are given only xcoordinates of two points A and B on the curve C.
Therefore, first we need ycoordinates of the two 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).
We are given that the two points A and B lay on the curve C which has equation;
We substitute xcoordinates of the two points A and B in the equation of the curve C.
For point A 
For point B 








Hence, coordinates of points A(1,3) and B(3,15).
TwoPoint form of the equation of the line is;
Therefore for line AB;
d)
We are required to find the length AB.
Expression to find distance between two given points and is:
We have found in (c) coordinates of points A(1,3) and B(3,15).
Since ;
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