# 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;

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