# 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 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;        1   2  3  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.   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  It is evident that .

Hence, is a root of .

The root given correct to 2 decimal places is 1.06.