# 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; 