Past Papers’ Solutions  Assessment & Qualification Alliance (AQA)  AS & A level  Mathematics 6360  Pure Core 1 (6360MPC1)  Year 2014  June  Q#6
Hits: 48
Question
The diagram shows a curve and a line which intersect at the points A, B and C.
The curve has equation and the straight line has equation . The point B has coordinates (0,7).
a.
i. Show that the xcoordinates of the points A and C satisfy the equation
ii. Find the coordinates of the points A and C.
b. Find
c. Find the area of the shaded region R bounded by the curve and the line segment AB.
Solution
a.
i.
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).
Equation of the line is;
Equation of the curve is;
Equating both equations;
Since xcoordinate does not take a ZERO value from point A to point C for both line and the curve, therefore, . Hence;
ii.
We have found that in (a:i) that xcoordinates of the points A and C are satisfied by the equation;
Therefore;
Now we have two options.






Two values of x indicate that there are two intersection points.
With xcoordinate of point of intersection of two lines (or line and the curve) at hand, we can find the ycoordinate of the point of intersection of two lines (or line and the curve) by substituting value of xcoordinate of the point of intersection in any of the two equations.
We choose equation of line;
Substituting values of x;
For 
For 




It is evident from the diagram that point A lies on negative side of xaxis, hence, coordinates of point A are (2,5) and that of point B are (3,10).
b.
Rule for integration of is:
Rule for integration of is:
Rule for integration of is:
c.
It is evident from the diagram that area of the shaded region is given by;
To find the area of region under the curve , we need
to integrate the curve from point to along xaxis.
It is evident from the area under the curve extends from point to point , since point B is the yintercept of the curve.
Therefore;
From (b) we have
Hence;
Next we find area under the line.
To find the area of region under the curve , we need to integrate the curve from point to along xaxis.
It is evident from the area under the line extends from point to point , since point B is the yintercept of the curve.
Therefore;
Rule for integration of is:
Rule for integration of is:
Rule for integration of is:
Hence;
Now we can find area of shaded region;
Comments