# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#7

Question

a.   Show that the equation  can be expressed as

and hence solve the equation  for .

b.

The diagram shows part of the graph of , where a and b are constants. The graph  crosses the x-axis at the point  and the y-axis at the point ,. Find c and d in  terms of a and b.

Solution

a.

We are given the equation;

We know that

Therefore;

We have the trigonometric identity;

It can be rearranged as;

Therefore;

Now we are required to solve  for .

We have seen above that  can be expressed as;

Let;

Therefore  can be written as,

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 .

We utilize the periodic property of   to find other solutions (roots) of :

 Symmetry Property

Hence;

Therefore, another possible solution is;

Therefore, we have two solutions (roots) of the equation;

To find all the solutions (roots) over the interval , we utilize the periodic property of    for both these values of .

 Periodic Property or

Therefore;

 For For

For

Hence all the solutions (roots) of the equation  for  are;

b.

We are given coordinates of x and y intercepts of the graph with equation;

The coordinates of x-intercept are   while that of y-intercept are .

First we consider x-intercept  of the graph.

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 substitution of  and  in the equation of the graph;

First we consider y-intercept  of the graph.

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 substitution of  and  in the equation of the graph;