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
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;
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We have the equation;
We have to consider two separate cases;
When |
When |
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Taking logarithm of both sides. |
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Power Rule; |
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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.
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Since |
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Taking logarithm of both sides. |
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Power Rule; |
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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 |
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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 |
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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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