Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2017  FebMar  (P19709/12)  Q#10
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Question
The diagram shows the curve defined for x>0. The curve has a minimum point at A and crosses the xaxis at B and C. It is given that and that the curve passes through the point .
i. Find the xcoordinate of A.
ii. Find .
iii. Find the xcoordinates of B and C.
iv. Find, showing all necessary working, the area of the shaded region.
Solution
i.
We are given that point A on the curve is a minimum point and hence must be a stationary point on the curve.
A stationary point on the curve is the point where gradient of the curve is equal to zero;
We are given that;
Therefore, at point A;
Therefore, xcoordinate of point A is 1.
ii.
We are required to find equation of the curve;
We are given that;
We can find equation of the curve from its derivative through integration;
Therefore;
Rule for integration of is:
Rule for integration of is:
If a point lies on the curve , we can find out value of . We substitute values of and in the equation obtained from integration of the derivative of the curve i.e. .
We are given that curve passes through a point .
Substituting these values for x and y in above found equation;
Hence, equation of the curve is;
iii.
We are given that the curve crosses the xaxis at B and C.
Therefore, points B and C are xintercepts of the curve.
The point at which curve (or line) intercepts xaxis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).
Therefore, utilising equation of the curve as found in (ii);
Let , then , hence;
Now we have two options.








Since ; 







It is evident from the diagram that both points B and C are on the positive side of xaxis, therefore;


iv.
The area between a curve and the ‘x’ axis is the same regardless whether it is ‘area under the curve’ or it is ‘area above the curve’.
It is evident from the diagram that;
To find the area of region under the curve , we need to integrate the curve from point to along xaxis.
Equation of the curve was found in (ii) as;
It is evident from the diagram that shaded region extends from to . Therefore;
Rule for integration of is:
Rule for integration of is:
Rule for integration of is:
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