Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/22) | Q#6

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  Question

The curve with equation intersects the line y = x + 1 at the point P.

     i.       Verify by calculation that the x-coordinate of P lies between 1.4 and 1.6.

   ii.       Show that the x-coordinate of P satisfies the equation

  iii.       Use the iterative formula

with initial value x1 = 1.5, to determine the x-coordinate of P correct to 2 decimal places. Give the  result of each iteration to 4 decimal places.

Solution

     i.
 We are given that the curve intersects the line y = x + 1 at the point P.

If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e.  coordinates of that point have same values on both lines (or on the line and the curve).  Therefore, we can equate coordinates of both lines i.e. equate equations of both the lines (or the  line and the curve).

Equation of the line is;

Equation of the curve is;

Equating both equations;

We are required to show by calculation that the x-coordinate of P lies between 1.4 and 1.6.

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 .

 

   ii.
 

We are required to find the x-coordinate of the point of intersection.

If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e.  coordinates of that point have same values on both lines (or on the line and the curve).  Therefore, we can equate coordinates of both lines i.e. equate equations of both the lines (or the  line and the curve).

Equation of the line is;

Equation of the curve is;

Equating both equations;

 

  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 use as initial value.

1

2

3

4

5

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9

10

It is evident that .

Hence, is a root of .

The root given correct to 2 decimal places is 1.31.

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