Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2014  June  Q#4
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Question
Figure 1 shows a sketch of the curve C with equation
, x ≠ 0.
The curve C crosses the xaxis at the point A.
a. State the x coordinate of the point A.
The curve D has equation y = x^{2}(x – 2), for all real values of x.
b. A copy of Figure 1 is shown below.
On this copy, sketch a graph of curve D.
Show on the sketch the coordinates of each point where the curve D crosses the coordinate axes.
Sketch and clearly label the graphs of C and on a single diagram.
c. Using your sketch, state, giving a reason, the number of real solutions to the equation
Solution
a.
We are given the curve C with equation
We are required to find the coordinates of xintercept of the curve;
The point at which curve (or line) intercepts xaxis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).
We substitute y=0 in given equation of the curve;
Hence, coordinates of the xintercept of the curve are (1,0).
b.
We are required to sketch;
We are given;
We can rewrite it as;
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 points and .
It can be seen that has come twice therefore the curve touches the xaxis at this point.
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 an expression which is equating both equations (of the curves C and D).
If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line and the curve). Therefore, we can equate coordinates of both lines i.e. equate equations of both the lines (or the line and the curve).
Hence, solution of this equation will yield xcoordinates of points of intersection of both curves C and D.
Two values of x indicate that there are two intersection points.
It can be seen from (b) that both curves intersect at two points, therefore, this equation will also have two solutions.
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