# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/12) | Q#9

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Question

A curve is such that  and (2,5) is a point on the curve.

i.       Find the equation of the curve.

ii.       A point P moves along the curve in such a way that the y-coordinate is  increasing at a constant rate of 0.06 units per second. Find the rate of change of the  x-coordinate when P passes through (2,5).

Solution

i.

We are given that;

We can find equation of the curve from its derivative through integration;

Therefore;

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 also given that curve passes through the point (2,5).

Substitution of x and y coordinates of point in above equation;

Therefore equation of the curve is;

ii.

We are given that point P moves along the curve in such a way that the y-coordinate is increasing at a constant rate of 0.06 units per second.

We are required to find the rate of change of the x-coordinate when P passes through (2, 5).

Rate of change of with respect to is derivative of with respect to  ;

Rate of change of with respect to is derivative of with respect to  ;

Since we are interested in rate of change of x-coordinate of P at point (2,5), we, therefore, first need the derivative of the curve at point (2,5).

We are given that;

At point (2, 5);

We know that;

Therefore;