Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2016  OctNov  (P29709/22)  Q#1
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Question
Solve the equation .
Solution
SOLVING EQUATION: PIECEWISE
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;
For a quadratic equation , the expression for solution is;
Therefore, for the given case;
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 solve;
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.
Therefore, first we sketch the line .
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.
Next we have to sketch;
It is evident that it will be a horizontal line as shown below.
When we sketch the two graphs on the same axes and we get following.
We are looking for the solution of .
It is evident from the graphs that is equal to when;
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