# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/12) | Q#3

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Question

A curve is such that . The point P (2,9) lies on the curve.

i.  A point moves on the curve in such a way that the x-coordinate is decreasing at a constant rate of 0.05 units per second. Find the rate of change of the y-coordinate when the point is at P.

ii.  Find the equation of the curve.

Solution

i.

We are given that point P moves along the curve in such a way that the x-coordinate is decreasing  at a constant rate of 0.05 units per second.

We are required to find the rate of change of the y-coordinate when P passes through (2, 9).

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 y-coordinate of P at point (2,9), we, therefore, first need  the derivative of the curve at point (2,9).

We are given that; At point (2, 9);     We know that; Therefore; Since, x-coordinate is decreasing at a constant rate of 0.05 units per second; Hence;   Hence, y-coordinate is decreasing at a constant rate of 0.35 units per second.

ii.

Next, we need 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 also given that curve passes through the point (2,9).

Substitution of x and y coordinates of point in above equation;     Therefore, equation of the curve is; 