# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/11) | Q#4

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**Question**

A curve is such that .

** i. **A point P moves along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.3 units per second. Find the rate of change of the y-coordinate as P crosses the y-axis.

The curve intersects the y-axis where .

** ii. **Find the equation of the curve.

**Solution**

i.

We are given that;

We are required to find the rate of change of y-coordinates as P crosses the y-axis i.e. at y- intercept;

We know that;

We are given;

Gradient (slope) of the curve at the particular point is the derivative of equation of the curve at that particular point.

Gradient (slope) of the curve at a particular point can be found by substituting x- coordinates of that point in the expression for gradient of the curve;

Therefore;

Therefore;

Therefore, at we have .

ii.

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

We are given that;

Therefore;

Rule for integration of is:

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 crosses y-axis at . It is evident that at this point .

Substituting these values in above equation;

Hence equation of the curve can be written as;

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