Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2006 | January | Q#3

 

Question

a.    

                           i.       Express  in the form , where   and  are integers.

                         ii.       Hence, or otherwise, describe the coordinates of the minimum point of the curve with                       equation .

b.   The line  has equation  and the curve  has the equation

                           i.       Show that the x-coordinates of the points of intersection of  and  satisfy the equation

                         ii.       Hence find the coordinates of the points of intersection of  and .

Solution

a.
 

                            i.
 

We have the expression;

We use method of “completing square” to obtain the desired form. 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 ;


ii.
 

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 upwards.

Vertex form of a quadratic equation is;

The given curve , as demonstrated in (a), can be written in vertex form as;

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

For the given case, vertex is . Therefore, least value of  is 5 and corresponding value of  is  2.

Vertex of a parabola is a stationary point on the curve.

Hence coordinates of the stationary (minimum; in case of parabola opening upwards) point on the  given curve  are;

b.
 

                            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).

We are given that equation of the curve is;

From (a:i) it can be written as;

We are also given the equation of line;

Equating both equations;

                          ii.
 

To find the x-coordinates of the points of intersection we solve the following equation from (b:i).

Now we have two options.

Two values of x indicate that there are two intersection points.

Corresponding values of y coordinate can be found by substituting values of x in any of the two  equation i.e either equation of the line or equation of the curve.

We choose equation of the line;

For

For

Hence coordinates of points of intersection of line and the curve are  and .

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