Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2013  MayJun  (P29709/23)  Q#1
Question
Solve the equation , giving answers correct to 2 decimal places where appropriate.
Solution
i.
Let, . We can write it as;
We have to consider two separate cases;
When 
When 


We have the equation;
We have to consider two separate cases;
When ; 
When ; 








Taking logarithm of both sides. 



Power Rule;







Hence, the only solutions for the given equation are;
SOLVING INEQUALITY: ALGEBRAICALLY
Let, .
Since given equation/inequality is of the form or or , we can solve this inequality by taking square of both sides;
We are given equation;
Therefore, we can solve it algebraically;
Let ;
Now we have two options.




Since ; 




Taking logarithm of both sides. 


Power Rule;







Hence, the only solutions for the given equation are;
SOLVING EQUATION: GRAPHICALLY
We are given the equation;
To solve the equation graphically, we need to sketch both sides of equation;
Let’s sketch both equations onebyone.
First we have to sketch;
Let, .
It can be written as;
We have to draw two separate graphs;
When ; 
When ; 


Therefore;

Therefore;

It is evident that and are reflection of each other in xaxis. So we can draw line of by first drawing and then reflecting in xaxis that part of the line which is below xaxis.
We have to draw two separate graphs;
When ; 
When ; 


It is evident that and are reflection of each other in xaxis. So first we sketch and then reflect the entire part (below xaxis) in xaxis to make it sketch of .
To sketch an exponential function;
Where
Some properties, which help to plot/sketch, of this graph are as follows.
· The graphs of all exponential functions contain the point .
· The domain is all real numbers .
· The range is only the positive .
· The graph is increasing.
· The graph is asymptotic to the xaxis as x approaches negative infinity
· The graph increases without bound as x approaches positive infinity
· The graph is continuous and smooth.
For the sake of simplicity first we sketch .
It is shown below.
Now to sketch we can move the entire graph 7 points downwards.
We can reflect the part of graph which is below xaxis in xaxis to make it graph of , as shown below.
The graph of is a straight horizontal line.
When we sketch the two graphs on the same graph we get following.
We are looking for the solution of .
It is evident from the graphs that is equal to only for f;
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