Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2014  MayJun  (P29709/21)  Q#4
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Question
i. By sketching a suitable pair of graphs, show that the equation
has exactly one real root.
ii. Show by calculation that the root lies between 2.0 and 2.5.
iii. Use the iterative formula to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
Solution
i.
We are required to show that there is only one root of the following equation by sketching.
Root of an equation
Therefore, first we sketch
We know that graph of
We can calculate a couple of values to be accurate in sketching the graph.

0 
0.1 
0.5 
1.0 
1.5 
2.0 
2.5 
3.0 
3.5 

– 
6.90 
2.08 
0.0 
1.22 
2.08 
2.75 
3.29 
3.76 
We can plot these points to sketch the following graph of
Next, we need to sketch graph of
It can be seen that it is a quadratic equation which can be rearranged as;
We can calculate a couple of values to be accurate in sketching the graph.

0 
1 
2 
3 
4 

15 
14 
7 
12 
49 
We can plot these points to sketch the following graph of
Sketching both graphs on the same axes, we get following.
It can be seen that the two graphs of
ii.
We are required to verify by calculation that the only root of equation
To use the signchange method we need to write the given equation as
Therefore;
If the function
We can find the signs of
Since
iii.
We are also given the iterative formula as;
If the sequence given by the inductive definition
Therefore, if
We have already found in (ii) through signchange rule that root of the given equation lies between
Therefore, for iteration method we use;
We use



0 


1 


2 


3 


4 


5 


6 


7 


It is evident that
Hence,
The root given correct to 3 decimal places is 2.319.
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