Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/23) | Q#2

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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


i.
 

SOLVING EQUATION: PIECEWISE

Let, .

We have to consider both moduli separately and it leads to following cases;

 OR

We have the equation;

We have to consider both moduli separately and it leads to following cases;

Hence, the only solution for the given equation is;


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.

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 one-by-one.

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 x-axis. So we can  draw line of by first drawing and then reflecting in x-axis that part of the line  which is below x-axis.

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 x-axis. So we can  sketch by first drawing and then reflecting in x-axis that part of the line  which is below x-axis. 

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 x-axis, 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 x-intercept of the line with are .

The point at which curve (or line) intercepts y-axis, 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 y-intercept of the line with are .

Hence, we get following graph for  ;

desmos-graph (6).png

We can reflect in x-axis that part of the line which is below x-axis to make it graph of ,  as shown below.

desmos-graph (4).png

Next 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 x-axis. So we can  draw line of by first drawing and then reflecting in x-axis that part of the line  which is below x-axis. 

We have to draw two separate graphs;

When ;

When ;

Therefore;

Therefore;

It is evident that  and are reflection of each other in x-axis. So we can sketch  by first drawing and then reflecting in x-axis that part of the line which  is below x-axis. 

Therefore, next 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 x-axis, 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 x-intercept of the line with are .

The point at which curve (or line) intercepts y-axis, 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 y-intercept of the line with are .

Hence, we get following graph for ;

desmos-graph (6).png

We can reflect in x-axis that part of the line which is below x-axis to make it graph of , as shown below.

desmos-graph (4).png

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; 

     ii.

We are required to solve the equation .

We can write it as;

Let , then, we can write the given equation;

We have solved this equation in (i) and found that;

Since , therefore;

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

Power Rule;

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