# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2015 | June | Q#8

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Question

a.   Factorise completely 9x – 4x3

b.   Sketch the curve C with equation

y = 9x – 4x3

Show on your sketch the coordinates at which the curve meets the x-axis.

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 y-axis by finding the value of when .

We can find the coordinates of y-intercept from the given equation of the curve.   Hence, the curve crosses y-axis at point .

ü Find the point(s) where the graph crosses the x-axis by finding the value of when . If  there is repeated root the graph will touch the x-axis.

We can find the coordinates of x-intercepts from the given equation of the curve.        Now we have two options.    Hence, the curve crosses x-axis 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 y-axis.

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 x-coordinates of both points A and B in given equation of the curve to find  the corresponding y-coordinates. 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 ;   