2017 A Level H2 Maths Paper 1 Suggested Solutions_CWY

1 2017 H2 Mathematics Paper 1 (9758/01) – Suggested Solutions 1 By using standard expansions of ex and  ln 1 x ,             2 2 3 2 3 2 2 3 3 2 2 3 3 2 3 2 2 3 e ln 1 2 2 1 2 ... ... 2! 3! 2 3 2 2 ...2 3 2 2 ...2 3 x ax x x ax ax x ax a x a x ax ax a x ax a a ax a x a a x                                                The expansion for   2e ln 1 x ax  up to the term in 3x is 2 3 2 2 3 2 2 2 3 a a ax a x a a x                  Given that there is no term in 2x , 2 2 0 2 a a   24 0 or 0 4 Since , 4 a a a a a a         2(i) y (0 , ab ) ( a , 0) 0

2 (ii) Point of intersection occurs when 1 b x a x a    . From graph, point of intersection occurs at x a  .     2 1 1 1 1 1Since Point of intersection occurs at x a x a b x a x a b x a x a x a b b x a x a b                       Hence, using the graph in (i), 1 b x a x a    when x a  or 1x a b   . 3(i)       2 2 2 2 2 5 10 0 Differentiating with respect to , d d 2 2 2 10 0 d d d 2 2 2 10 (1) d dAt the stationary points of , 0 d 2 10 0 5 Substitute 5 into the equation of : 5 2 5 5 10 0 y xy x x y y y x y x x x y y x y x x yC x y x y x y x C x x x x                          2 2 20 10 1 1 2 2 1 1 -coordinates of stationary points of are and 2 2 x x x x C x x         (ii) For stationary point where x > 0, 1 2 x  From (1),

3     d 2 2 2 10 d d 5 (2) d y y x y x x y y x y x x         Differentiating (2) with respect to x,      2 2 2 2 2 2 2 2 d d d d 1 5 d d d d 1 5 d At , and 0. d2 2 Substituting these values into the above equation: d 5 1 0 1 0 5 d 2 2 d 4 5d 2 d 5 2 0d 4 1Hence the stationary point at is 2 y y y y y x x x x x yx y x y x y x y x x                                     a maximum. 4(i)   2 4 9 1 42 2 Differentiating with respect to , d 1 d 2 xy x x x y x x         C is defined for all values of x , except for 2x       2 0 22 \xx      ℝ    2 d 1 d 2 \ 2xy x x    ℝ Gradient is negative for all points of .C (ii) 14 , where 4, 1.2y a b x     Equations of asymptotes are 4y  and 2x   . (iii) The graph of C undergoes the following transformations: 1) Translation of 4 units in the negative direction of the y-axis, 2) Translation of 2 units in the positive direction of the x-axis. Note that the order mentioned above can be interchangeable.

4 5(i)             When 1, 1 8 7 1 When 2, 8 4 2 12 4 2 4 2 When 3, 27 9 3 25 9 3 2 3 Using the GC to solve equations 1 , 2 , 3 : 3 3 , , 7 2 2 x a b c a b c x a b c a b c x a b c a b c a b c                             