Further Mathematics Paper 2, May/June. 2012
 Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Main
Weakness/Remedies
Strength

Question 9

(a)      The polynomial f(x) = x3 + px2 – 10x + q is exactly divisible by (x2 + x – 6).
Find the:

1. values of p and q
2. third factor.

(b)     The volume of a cube is increasing at the rate of 2½ cm3s-1.  Find the rate of change of the side of the cube when its length is 2 cm.

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Observation

This question was reportedly attempted by majority of the candidates and their performance was described as fair.  Candidates were also reported to have performed better in part (a) than in part (b) where majority of the candidates were reported to have applied the appropriate rule of differentiation correctly.

In part (a), candidates were expected to apply the long division method to divide
f(x) = x3 + px2 – 10x + q by x2 + x – 5 to have:
x + (p – 1)
x2 + x – 6 x3 + px2 – 10x + q
x3 + x2 – 6x
(p - 1)x2 – 4x + q
(p - 1)x2 + (p - 1)x – 6(p - 1)
(-4 - (p - 1))x + q + 6 (p - 1)

Since x2 + x – 6 is a factor, then (-4 - (p - 1))x + q + 6(p - 1) = 0.  i.e -4 - (p - 1) = 0 and q + 6(p - 1) = 0.  This implied that p – 1 = -4 or p = -3 and q = -6(p-1) = -6(-3-1) = 24.  The third factor was x + (p - 1) = x + (-3 - 1) = x – 4.