Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2015 | June | Q#8
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.
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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; |
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Hence the points A and B have coordinates and
, respectively.
Now we can find the distance AB.
Since ;
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