Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2010 | January | Q#9

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Question

a)   Factorise completely x3 – 4x

b)  Sketch the curve C with equation

y = x– 4x

showing the coordinates of the points at which the curve meets the x-axis.

The point A with x-coordinates -1 and the point B with x-coordinate 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 y-axis by finding the value of  when .

We can find the coordinates of y-intercept 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 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 three 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.

desmos-graph.png

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 (Two-Point  form of Equation of Line) or coordinates of one point on the line and slope of the line (Point-Slope  form of Equation of Line).

We are given only x-coordinates of two points A and B on the curve C.

Therefore, first we need y-coordinates 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 x-coordinates 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).

Two-Point 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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