Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/23) | Q#1

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Question

     i.       Solve the inequality .

   ii.       Hence find the largest integer n satisfying the inequality .

Solution

SOLVING INEQUALITY: PIECEWISE

     i.
 

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

It cannot be solved.

It cannot be solved.

Hence, the only solution for the given inequality is;

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 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.

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 ;

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

Next we 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.

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 ;

We can reflect in x-axis that part of the line which is below x-axis to make it graph of  , 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 below (smaller) than for all  values of;

   ii.
 

We are required to find the largest integer n satisfying the inequality  .

The given equation is;

Let ;

As demonstrated in (i), the solution of this equation is ;

Hence;

Taking anti-logarithm of both sides;

 and are inverse functions. The composite function is an identity function, with domain the positive real numbers. Therefore; 

Since we are largest integer n;

 

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