Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2014 | June | Q#6

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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 x-coordinates 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 x-coordinate 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 x-coordinates 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 x-coordinate of point of intersection of two lines (or line and the curve) at hand, we can find the  y-coordinate of the point of intersection of two lines (or line and the curve) by substituting value  of x-coordinate 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 x-axis, 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 x-axis.

It is evident from the area under the curve extends from point  to point  , since  point B is the y-intercept 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 x-axis.

It is evident from the area under the line extends from point  to point  , since  point B is the y-intercept 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;

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