TJC_JC2_H2_Maths_2012_Prelim_Paper_2_Solutions

TJC_JC2_H2_Maths_2012_Prelim_Paper_2_Solutions 1 Section A: Pure Mathematics [40 marks] 1 A curve C is defined by the parametric equations 2 3 cos 3x  and  2 3 1 siny  , where 02 π. (i) Find d d y x in terms of  . [2] (ii) Show that the Cartesian equation of the curve C is 2 2 3 1 2 3 2 3 1 xy     . Hence or otherwise, sketch the curve of C, indicating clearly the x-intercepts in exact form. [3] (iii) The point P 32 3, 3 2   lies on curve C. The region R is bounded by the curve C for 3x , the x-axis and the line segment joining the points P and  3,0 . Show that the area of R is     3 2 0 3 2 3 1 2 6 3 sin d4      . [4] [Solution] (i) d 2 3 sind x   and   d 2 3 1 cosd y    2 3 1 cosd d 2 3 sin y x      11 cot 23   (ii) 32 3 cos 3 cos 23 xx       2 3 1 sin sin 2 3 1 yy      So 22sin cos 1  2 2 3 1 2 3 2 3 1 xy     3 3 33 3x y x

TJC_JC2_H2_Maths_2012_Prelim_Paper_2_Solutions 2 (iii) At P, 23x  2 3 cos 3 2 3  1cos 23     When 3 3,x 2 3 cos 3 3 3  cos 1, 0    Area of R = 33 23 area of d yx          0 3 13 3 2 3 1 2 3 1 sin 2 3 sin d22           Area of R     3 2 0 3 2 3 1 2 6 3 sin d4      

TJC_JC2_H2_Maths_2012_Prelim_Paper_2_Solutions 3 2 The diagram above shows th e curve of 2 ex xy . Two points A and B on the curve have coordinates (, 1 2 ) and (, 1 2 ) respectively. A sequence of real numbers 1 2 3, , ,...x x x satisfies the recurrence relation 1 1 e4 nx nx   for n  1. (i) Show algebraically that if the sequence converges, then it converges to either  or . [3] (ii) Show that 1nx   if nx <  . [2] (iii) Show that 1nnxx  if nx <  . [2] (iv) Explain briefly how the results in (ii) and (iii) may be used to deduce that the sequence converges to  when 1 0x  . [2] [Solution] (i) If the sequence converges to, say l, then nx  l and 1nx   l as n  . i.e. 1 1 e4 nx nx    1 e4 ll  --- (A)  21 e2l l   l  or  from the diagram. (ii) If nx <  , eenx  since ex is an increasing function.  11ee44 nx   1 1 e4nx     1nx   [using eqn (A) in (i)]

TJC_JC2_H2_Maths_2012_Prelim_Paper_2_Solutions 4 (iii) Method 1A: Using given graph Step 1: Consider 1 1 e4 nx n n nx x x    211e2 2 e n n x n x x 