Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/22) | Q#4

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  Question

The diagram shows the part of the curve  for .

     i.       Use the trapezium rule with 2 intervals to estimate the value of

giving your answer correct to 2 decimal places.

   ii.       The line y=x intersects the curve at point P. Use the iterative formula

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 required to apply Trapezium Rule to evaluate;

The trapezium rule with  intervals states that;

If the graph is bending downwards over the whole interval  from  to , then trapezium rule will give  an underestimate of the true area.

We are given that there are two intervals, .

We are also given that and .

Hence;

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Therefore;

The value correct to 2 decimal places is .


ii.

Iteration method can be used to find the root of the given equation using iterative formula;

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 are given the part of the curve  for and since line and the curve  intersect, the point of intersection must lie between  and .

Hence, 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 1.06.

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