Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/13) | Q#7

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Question

a.    

                    i.       Express  in the form , where  and are constants to be  found.

      ii.       Hence, or otherwise, and showing all necessary working, solve the equation 

For .

 

b.    

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

  i.       Find the x-coordinate of A.

   ii.       Find the y-coordinate of B.

Solution

a.    


i.
 

We are given the expression;

Since ;

Since ;

We can write as;

Therefore;


ii.
 

We are required to solve the equation;

As demonstrated in (i);

Therefore;

We have two options;

Using calculator;

We utilize the symmetry property of   to find other solution(s) (root) of :

Properties of

Domain

Range

Odd/Even

Periodicity

Translation/

Symmetry

Hence;

For

For

It is evident these above solutions and any further solutions obtained through  periodicity property of of will be out of given range

Hence, only solution within desired range is;

 

b.    


i.
 

We are required to find the x-coordinate of point A which is the intersection point of   graphs of and .

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 first curve is;

Equation of the second curve is;

Equating both equations;

Since ;

Properties of

Domain

Range

Periodicity

Odd/Even

Translation/

Symmetry

We utilize the periodicity property of to find the other solutions within the given  range .

Therefore, for ;

It is evident that only values within range are;

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

However, it is evident from the diagram that point A is on the positive side of x-axis.

Hence, x-coordinates of point A is 1.11.


ii.
 

As demonstrated in (B(i)) the points of intersection of  graphs of and   are;

It is also demonstrated in (B(i)) that x-coordinate of point A is .

It is evident then that x-coordinate of point B is .

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;

Therefore;

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