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Further Mathematics Paper 2, Nov/Dec. 2009  
Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Main
General Comments
Weakness/Remedies
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Question 10

  1. An exponential sequence is given by 18, 2, 2/9, …

Find (i) an expression for the nth term;
      (ii) the sum of the first n terms.

(b)(i)  Find the equation of the tangent to the curve y = -x2 + x + 1 at point    (2, -1).
    (ii)   Find the intercepts of the tangent in (b)(i) with the axes.

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Observation
This question was reported to be very popular among the candidates and they performed very well in it.  They were able to derive the nth term of the sequence thus:  a = 18, r = 1/9, nth term = arn - 1 = 18(1/9)n – 1 = 162(1/9)n.
The sum is derived thus:  Sn  =    =   =  (1 – (               =        
In part (b) candidates were expected to differentiate  y  = -x2 + x + 1 to obtain   = -2x + 1 at (2, -1),   = -4 + 1 = -3.  The equation of the tangent is gotten by y + 1 = -3(x – 2) º y + 3x – 5 = 0.  On the x – axis, y = 0 ⇒ 3x = 5 or         x = 5/3.  Thus the tangent intercepts the x – axis at (5/3, 0).  Similarly on the y-axis, x = 0.  Thus y = 5.  Hence the point of intersection = (0,5). 
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