Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01R) | Year 2013 | June | Q#10

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Question

A curve has equation y=f(x). The point P with coordinates (9,0) lies on the curve.

Given that

 ,

a.   Find f(x).

b.   Find the x-coordinates of two points on y=f(x) where the gradient of the curve is equal to 10.

Solution

a.
 

We are given;

We are given coordinates of a point on the curve (9,0).

We are required to find the equation of y in terms of x ie f(x).

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.

Therefore, substituting the coordinates of point (9,0) in above equation;

Therefore, equation of the curve C is;

b.
 

We are required o find the x-coordinates of the two points where the gradient of the curve is 10. 

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, we need .

We are already given;

Since gradient is given to be 10 at two desired points;

Now we have two options.

Hence, gradient of the curve is equal to 10 where x=1 and x=81.

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