Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2008 | June | Q#1

 

Question

The straight line L has equation  and the curve C has equation

a.   Sketch on the same axes the line L and the curve C, showing the values of the intercepts on the  x-axis and the y-axis.

b.   Show that the x-coordinates of the points of intersection of L and C satisfy the equation  .

c.   Hence find the coordinates of the points of intersection of L and C.

Solution

a.
 

We are given that equation of the line is;

We can find x and y intercepts of the line.

The point  at which curve (or line) intercepts x-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line).

Therefore;

Hence, x intercept of the given line has coordinates .

The point  at which curve (or line) intercepts y-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line).

Therefore;

Hence, y intercept of the given line has coordinates .

Now we can easily sketch the given line by joining x and y intercepts and extending it on both sides.

desmos-graph (5).png

We are given equation of the curve as;

It is evident that it is a quadratic equation in factorized form. Therefore, equation is that of a  parabola and both factors represent the x-intercepts of the parabola.

Hence, coordinates of x intercepts of given parabola are  and .

To sketch the parabola we also need coordinates of vertex or y intercept of the parabola.

We find coordinates of y intercept of the parabola.

The point  at which curve (or line) intercepts y-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line).

Therefore;

Hence, coordinates of y-intercept of the given parabola are .

Now we can sketch the given parabola and line on the same axes.

desmos-graph (6).png

b.    

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;

c.    

To find the coordinates of the points of intersection of L and C we solve the equation obtained in (b);

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;

For

For

Therefore, coordinates of points of intersection of C and L are  and .

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