Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2015  OctNov  (P29709/22)  Q#1
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Question
i. Solve the equation .
ii. Hence, using logarithms, solve the equation , giving the answer correct to 3 significant figures.
Solution
SOLVING EQUATION: ALGEBRAICALLY
i.
Let, . We can write it as;
We have to consider two separate cases;
When 
When 


We have the equation;
It can be written as;
We have to consider two separate cases;
When ; 
When ; 







Hence, the only solutions for the given equation are;
SOLVING EQUATION: 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;
Now we have two options.





Hence, the only solutions for the given equation are;
SOLVING EQUATION: GRAPHICALLY
We are given equation;
To solve the equation graphically, we need to sketch both sides of inequality;
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.
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 sketch by first drawing and then reflecting in xaxis that part of the line which is below xaxis.
To sketch a line we only need x and y intercepts of the line.
The point at which curve (or line) intercepts xaxis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).
Therefore, we substitute in given equation of the line.
Hence, coordinates of xintercept of the line with are .
The point at which curve (or line) intercepts yaxis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).
Therefore, we substitute in given equation of the line.
Hence, coordinates of yintercept of the line with are .
Hence, we get following graph for ;
We can reflect in xaxis that part of the line which is below xaxis to make it graph of , as shown below.
The graph of is a straight horizontal line.
When we sketch the two line 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;
ii.
We are required to solve the equation .
Let , then, we can write the given equation;
We have solved this equation in (i) and found that;


Therefore;




Since logarithm of negative number is not possible, the only option is;
Power Rule;
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