Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2010  OctNov  (P29709/21)  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 xcoordinate of P lies between 1.4 and 1.6.
ii. Show that the xcoordinate of P satisfies the equation
iii. Use the iterative formula
with initial value x_{1 }= 1.5, to determine the xcoordinate 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 xcoordinate of P lies between 1.4 and 1.6.
We need to use signchange rule.
To use the signchange 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 xcoordinate 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.



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It is evident that .
Hence, is a root of .
The root given correct to 2 decimal places is 1.31.
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