2014 H1 A Level Mathematics Solution (SAJC)

1 2014 H1 A Level Mathematics Solution Q1.       6 1 16 2 1 61 2 1 1 14 14 14 2 1 552 dx x x dx x          Q2. (i)   2 2 d2ln 4d4 xxxx   (ii) Gradient of C = K 2 2 2 2 4 2 ( 4) 2 4 0 (shown) x kx x k x kx x k          2 2 4 0kx x k   (iii) For equal roots,   2 2 2 40 4 4 4 0 16 4 1 4 11or22 b ac kk k k kk       Q3. (i)

2 At x axis intercept At y axis intercept 12 12 10 1 1 2 0 1 2 x x e e x x       1 2(0)1 1 ye ye   The coordinates of points of intersection with the axes are ( 0 , 1- e ) and ( 1 2 , 0 ). Asymptote : y =1 (ii) When x = 1, y = 11 e 12 1 d 2d At =1 d2 2d xy ex x y exe     Equation of Tangent to C at x = 1  1211 23 1 23 1 yx ee yx ee mc ee             Q4. (i) By Pythagoras’s Theorem,       222 2 2 2 22 2 2 65 2 4 260 5 2 260 (shown) y x x y xy x x x y xy           (ii) Since the perimeter of the rectangle ABCD is 60 cm,

3 2 6 60 3 30 30 3 yx yx yx        22 22 Using 5 2 260 5 30 3 2 30 3 260 4, 18 8, 6 x y xy x x x x xy xy           Q5. (i) 32 2 7 d 3 2 7d y x kx x c y x kxx        For stationary point, 2 d 0d 3 2 7 0 For 1, 3 2 7 0 5 (shown) y x x kx x k k          Given that A has coordinates ( 1 , 2 ), 32 32 57 2 1 5 7 1 5 7 1 y x x x c c c y x x x              (ii) 2d 3 10 7 0d 11 or 2 3 y xxx xx      32 1For 2 , 3 1 1 12 5 2 7 2 13 3 3 22 27 1 22Co-ordinates of 2 , 3 27 x y B                       

4 (iii) Using GC, co-ordinates of point that crosses the x-axis ( 0.161 , 0 ) (iv) Exact Area of Region = 2 32 1 24 3 2 1 2 5 7 1 57 4 3 2 10 1 5 74 14 2 13 4 3 2 71 units12 x x x dx x x x x                           Q6. Let X be the random variable of the heights of the girls in the school.   2142.2,6XN (i)  P 146 0.73674 0.737 (3 s.f)X    (ii)  P 142.2 5 142.2 5X     P 137.2 147.2 0.595(3 s.f ) X    Q7. [This part is not in syllabus] (i) Systematic Sample To draw a sample of 100 households from a population of 5000 households, we first assign a consecutive number for each household starting from 1. k = 50 ( 5000 100 ). Taking the 1st number randomly from 1 to 50, say 8, we select every 50th household i.e 8th, 58th, 108th until all 100 households are selected. (ii)

5 To obtain a sample of 100 households, we draw random samples from the age-groups with sample size in the same proportion as the size of each age-group: Supermarket Online Under 25 years 10 20 25-60 years 18 32 Over 60 years 16 4 (iii) Stratified Sampling will be preferred over systematic sampling due to the following reasons: 1. When there are clear strata present, this method can provide more accurate estimates t