# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Feb-Mar | (P1-9709/12) | Q#5

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Question The diagram shows the graphs of and for . The graphs intersect at  points A and B.

i.
Find by calculation the x-coordinate of A.

ii.
Find by calculation the coordinates of B.

Solution

i.

We are required to find the x-coordinate of point A which is point of intersection of the two given  curves.

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 one curve is; Equation of other curve is; Equating both equations; Using we can rewrite the above equation as;   We have the trigonometric identity; Therefore, we can substitute in above equation;      To solve this equation, let; Therefore can be written as, Standard form of quadratic equation is; Solution of quadratic equation is given as; Therefore, for above equation;   Now we have two options;      Since; We have two options;  We know that; Therefore is not possible. Hence; Using calculator we can find the value of .  Hence x-coordinate of point A is 0.6662.

ii.

We are required to find the coordinates of point B which is point of intersection of the two given curves.

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 one curve is; Equation of other curve is; Equating both equations; As found in (i);   To find the coordinates of point B we need to solve this equation for ,;

We utilize the periodic property of to find another solution (root) of : Symmetry Property Hence;    Therefore, we have two solutions (roots) of the equation; So we have two possible values of ,  Two values of x indicate that there are two intersection points.

These two values of x represent x-coordinates of points A and B, respectively.

With x-coordinate of point of intersection of two lines (or line and the curve) at hand, we can find the  y-coordinate of the point of intersection of two lines (or line and the curve) by substituting value  of x-coordinate of the point of intersection in any of the two equations.

We choose; Substitute for point B;  Hence coordinates of point B are (2.475,-0.786).