Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2012 | January | Q#5

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Question

The curve C has equation y=x(5−x) and the line L has equation 2y=5x+4.

a.   Use algebra to show that C and L do not intersect.

b.   In the space on page 11, sketch C and L on the same diagram, showing the coordinates of the  points at which C and L meet the axes.

Solution

a.    

We are required to show that curve C and line L do not intersect.

Let us try to prove that line L and curve C intersects.

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;

Let us solve these two simultaneous equations.

Substitute value of y from equation of the curve C in the equation of the line L;

For a quadratic equation , the expression for solution is;

Where  is called discriminant.

If , the equation will have two distinct roots.

If , the equation will have two identical/repeated roots.

If , the equation will have no roots.

If the curve C and the line L do not intersect then this quadratic equation must not have any solution  and therefore;

Hence, the curve C and the line L do not intersect.

b.    

First we sketch the lien L.

We are given equation of the line L as;

We can sketch the line by finding coordinates of its intercepts and then joining them.

First we find the x-intercept of the line L.

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, we substitute  in equation of the line L.

Hence, coordinates of x-intercept of the line L are  

Next, we find the y-intercept of the line L.

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, we substitute  in equation of the line L.

Hence, coordinates of y-intercept of the line L are .

We can sketch the line as shown below.

desmos-graph (11).png

Now we sketch the curve with given equation;

We can expand the given equation as;

It is evident that it is a quadratic equation.

To sketch a quadratic equation, a parabola, we need the coordinates of its vertex and x and y- intercepts, if any.

First we find the coordinates of vertex of this parabola.

Standard form of quadratic equation is;

The graph of quadratic equation is a parabola. If  (‘a’ is positive) then parabola opens upwards  and its vertex is the minimum point on the graph.
If
 (‘a’ is negative) then parabola opens downwards and its vertex is the maximum point on the  graph.

We recognize that given curve , is a parabola opening downwards.

Vertex form of a quadratic equation is;

The given curve , can be written in this form by method of completing square.

Next we complete the square for the terms which involve .

We have the algebraic formula;

For the given case we can compare the given terms with the formula as below;

Therefore we can deduce that;

Hence we can write;

To complete the square we can add and subtract the deduced value of ;

Coordinates of the vertex are . Since this is a parabola opening downwards the vertex is the  maximum point on the graph. Here y-coordinate of vertex represents maximum value of  and x- coordinate of vertex represents corresponding value of .

For the given case, vertex is

Next, we need x and y-intercepts of the parabola.

First we find the x-intercept of the parabola.

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, we substitute  in equation of the parabola.

Now we have two options.

Hence, coordinates of the two x-intercepts of the parabola are  and .

Next, we find the 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, we substitute  in equation of the parabola.

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

We can sketch the parabola as shown below.

desmos-graph (10).png

We can combine the two graphs on same axes as shown below.

Untitled.png

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