# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/21) | Q#5

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Question

It is given that

Where  is a constant.

i.       Show that

ii.       Using the equation in part (i), show by calculation that 0.5 < a < 0.75.

iii.       Use an iterative formula, based on the equation in part (i), to find the value of a  correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

Solution

i.

We are given that;

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is:

ii.

We are required to show by calculation that the x-coordinate lies between 0.5 < a <  0.75.

We need to use sign-change rule.

To use the sign-change method we need to write the given equation as .

Therefore;

If the function  is continuous in an interval  of its domain, and if  and  have opposite signs, then  has at least one root between  and .

We can find the signs of at and as follows;

Since and have opposite signs for function  , the function has root between and  .

iii.

Iteration method can be used to find the root of the given equation using sequence  defined by;

If the sequence given by the inductive definition , with some initial value , converges to a limit , then  is the root of the equation .

Therefore, if , then  is a root of .

We have already found in (ii) through sign-change rule that root of the given equation  lies between and .

We use as initial value.

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It is evident that.

Hence, is a root of .

The root given correct to 3 significant figures is 0.651.