Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | May-Jun | (P2-9709/23) | Q#1

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Question

Solve the equation , giving answers correct to 2 decimal places where appropriate.

Solution


i.
 

Let, . We can write it as;

We have to consider two separate cases;

When

When

We have the equation;

We have to consider two separate cases;

When ;

When ;

Taking logarithm of both sides.

Power Rule;

Hence, the only solutions for the given equation 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 equation;

Therefore, we can solve it algebraically;

Let ;

Now we have two options.

Since ;

Taking logarithm of both sides.

Power Rule;

Hence, the only solutions for the given equation are;

SOLVING EQUATION: GRAPHICALLY

We are given the equation;

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

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.

We have to draw two separate graphs;

When ;

When ;

It is evident that and are reflection of each other in x-axis. So first we  sketch and then reflect the entire part (below x-axis) in x-axis to  make it sketch of .

To sketch an exponential function;

Where  

Some properties, which help to plot/sketch, of this graph are as follows.

·       The graphs of all exponential functions contain the point .

·       The domain is all real numbers .

·       The range is only the positive .  

·       The graph is increasing.

·       The graph is asymptotic to the x-axis as x approaches negative infinity

·       The graph increases without bound as x approaches positive infinity

·       The graph is continuous and smooth.

For the sake of simplicity first we sketch .

It is shown below.

Now to sketch we can move the entire graph 7 points downwards.

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

The graph of is a straight horizontal line.

When we sketch the two graphs 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 f; 

 

 

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