Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2019  MayJun  (P19709/12)  Q#6
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Question
The equation of a curve is and the equation of a line is .
i.State the smallest and largest values of y for both the curve and the line for .
ii.Sketch, on the same diagram, the graphs of and for .
iii.State the number of solutions of the equation for .
Solution
i.
We are given equations of the curve and the line as follows;
We can write equation of the line in slopeintercept form.
First, we consider equation of the curve for smallest and largest values.
We know that;
Therefore,
Smallest 
Largest Value 









Next, we consider equation of the line for smallest and largest values.
Smallest 
Largest Value 












ii.
We are required to sketch the curve and the line on the same graph.
We are given equation of the curve and the line as follows;
We can write equation of the line in slopeintercept form.
First, we are required to sketch for .
We can find the points of the graph as follows.

















































Now we sketch for .
We can sketch the graph of for as follows.
We can find the points of the graph as follows.

















































Now we can sketch both the curve on same graph.
iii.
We are required to state the number of solutions of the equation
We can rewrite the equation as;
We are given equation of the curve and the line as follows;
We can write equation of the line in slopeintercept form.
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 given as;
Equation of the curve is;
Equating both equations;
It is evident that we have obtained the same equation as obtained from the given equation.
The number of values of from the solution of this equation will show the number of points of intersection.
However, the other, way round is also true. The number of points of intersection of the line and the curve show the number of the solutions.
As demonstrated in (iii), the line and the curve intersect at 04 points, therefore, there are 04 solutions of the given equation.
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