Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/12) | 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 slope-intercept form.

First, we consider equation of the curve for smallest and largest values.

We know that;

Therefore,

Smallest
Value of Curve

Largest Value
of Curve

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

Smallest
Value of Line

Largest Value
of Line

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 slope-intercept 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 re-write the equation as;

We are given equation of the curve and the line as follows;

We can write equation of the line in slope-intercept 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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