Further Mathematics Paper 2, Nov/Dec. 2011
 Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Main
Weakness/Remedies
Strength
Question 10
1. The gradient of a curve is given by 2x – 3x2. Find the equation of the curve if a point (1, 2) lies on it.
2. (i) Find the equations of the normals to the curve y = x2 – 1 at the points where it cuts the x-axis.

(ii) Find the coordinates of the point of intersection of the normals in (b)(i).

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Observation

The report stated that this question was popular among the candidates and majority of them performed well in it. In part (a), majority of the candidates were reported to integrate the given gradient function 2x – 3x2 to obtain y = x2 – x3 + c and substitute (1, 2) into it to obtain c = 2. Hence the required equation was y = x2 – x3 + 2. Some candidates were reported to have differentiated the gradient function instead integrating it.
In part (b), candidates were expected to determine the points of intersection with the x-axis by substituting 0 for y in the equation y = x2 – 1 and factorizing to obtain x = -1 and 1. Hence the points were (-1, 0) and (1, 0).  = 2x. At (-1, 0),  = -2 hence gradient of the normal at
(-1, 0) =  . Equation of the normal is y – 0 =  (x + 1) which reduced to x -2y + 1 = 0. Similarly, gradient at (1, 0) = 2.
Hence gradient of normal = - . Equation of normal is y – 0 = - (x – 1) which simplified to x + 2y – 1 = 0. Solving x + 2y – 1 = 0 and x – 2y + 1 = 0 simultaneously would give the point of intersection which is (0, ).