SRJC_H2_MATH_P2_TEACHER_S

1 2015 SRJC H2 Mathematics Prelim Paper 2 Section A: Pure Mathematics [40 marks] 1 Functions f and g are defined as below. 1f : , 2 1xx x  g : ln(2 ) , 2x x x x    (i) Sketch the graph of y = g(x) and state its exact range. [3] (ii) Determine whet her the composite function s fg and g f exist, justifying your answer. Find the range of the composite function if it exists. [4] (iii) On the same diagram as part (i), sketch the graph of y = g −1(x), indicating on your sketch the line in which the graph of y = g(x) must be reflected in order to obtain the graph of y = g−1(x). [2] Suggested Solution (i) Range of g, Rg = (2 + ln4, +∞) (ii) From GC, the range of f is (0, 1]. Since fRD g = (−∞, −2), the composite gf does not exist. Since Rg = (2 + ln4, +∞) ⊂ [2, ∞) = Df the composite fg exists. Direct Method for finding range of fg Range of fg = (0, 1 1 ln 4 ) y x (−2, 2+ln4) y x 1( 2, )1 ln 4  y = fg(x) = 1 ln(2 ) 1xx   Rfg y = g(x) (2+ln4, −2) y = x y = g−1(x)

2 Indirect Method for finding range of fg Range of fg = (0, 1 1 ln 4 ) 2 A tank contains water which is heated by an electric water heater working under the action of a thermosta t. When the water heater is first switched on, the temperature of the water is 35 C . The heater causes the temperature to increase at a rate Cr per minute, where r is a constant, until the water temperature hits 75 C .The heater then switches off. (i) Write down, in terms of r, the time taken for the temperature to increase from 35 C to 75 C . [1] The temperature of the water then immediately starts to decrease . The temperature of the water at time t minutes after the heater is switched off is C . It is known that the temperature of the water decreases at a variable rate ( 25) Ck  per minute, where k is a positive constant, until 35  . (ii) Write down a differential equation involving  and t, to represent the situation as the temperature is decreasing. [1] (iii) Given that when 55  , the temperature is decreasing at a rate of 5C per minute, find the total length of time for the temperature to increase from 35 C to 75 C and then decrease to 35 C , leaving your answer in exact form, i n terms of r. [7] y x y = f(x) 2+ln4 1 1 ln 4 Rg Rfg

3 Suggested Solution (i) Time taken 40 r mins (ii)  d 25d kt    (iii) 1 d d25 kt   ln 25 kt c   , where c is a constant, since  > 25oC  25 e .e kt c  25 e ktA  , where A is an arbitrary constant. When 55  , d 5dt   , 1 6k  When t = 0 and 75  , A = 50 1 625 50e t    When 35  , 1 635 25 50e t 1 6 1e 5 t  1 56ln 6ln 5t   Total length o