Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2005 | January | Q#10
a) Express in the form , where a and b are integers.
The curve C with equation y = f(x), , meets the y-axis at P and has a minimum point at Q.
b) In the space provided on page 19, sketch the graph of C, showing the coordinates of P and Q.
The line y = 41 meets C at the point R.
c) Find the x-coordinate of R, giving your answer in the form , where p and q are integers.
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 ;
We are required to find the coordinates of points P and Q.
We know that curve meets the y-axis at point P which means point P is y intercept of .
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).
We have found in (a);
Hence, coordinates of the point P(0,18).
Next we need coordinates of point Q which is the minimum point of the curve.
We are given
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 least value of and x- coordinate of vertex represents corresponding value of .
For the given case, vertex is . Therefore, coordinates of the point .
The graph can be sketched as shown in the diagram below.
We are required to find the coordinates of point R which is intersection point of line and curve C.
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 equation of line as;
Equation of the curve is;
We are required to solve these to equations.
We substitute from equation of line into equation of the curve.