Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2017 | Oct-Nov | (P2-9709/21) | Q#3

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Question

It is given that the variable x is such that

 and

Find the set of possible values of x, giving your answer in the form a < x < b where the constants a and b are correct to 3 significant figures.

Solution

First we find the value of x for;

Taking natural logarithm of both sides;

Power Rule;

Next we solve the inequality;

SOLVING INEQUALITY: PIECEWISE

Let, . We can write it as;

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

When

If then above four intervals translate to following with their corresponding inequality; 

When

When

When

If then above four intervals translate to following with their corresponding inequality;

When

When

When

We have the inequality;

In standard form it can be written as;

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

Since then above four intervals translate to;

We can see that given inequality takes following forms for these intervals.

For interval

For interval

For interval

 

Cannot be solved for x.

Cannot be solved for x.

Hence, the only solution for the given inequality 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 inequality;

Therefore, we can solve it algebraically;

SOLVING INEQUALITY: GRAPHICALLY

We are given inequality;

To solve the inequality graphically, we need to sketch both sides of inequality;

Let’s sketch both equations one-by-one.

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 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

Now 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.

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, 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 above (greater) than for all values of; 

For  we have found that;

For  we have found that;

Therefore for both inequalities to hold true;

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