Sembawang 2021 4E AMaths Prelims P1 MS V3

- 1 - MARKING SCHEME FOR ADDITIONAL MATHEMATICS SECONDARY 4 EXPRESS PRELIMINARY EXAMINATION 2021 Qn Solution Marking Point Remarks 1 For curve to lie completely above the x-axis, M1 M1 A1 Discriminant < 0 Factorizing quad. inequality 2 M1 M1 M1 A1 Expressing in base 7 Equating powers 3 M1

- 2 - For y to be decreasing, Since for M1 M1 A1 4 Height M1 M1 M1 M1 A1 For obtaining h volume/base area For area of base For conjugate surd For correct expansion of numerator 5 (i) f(x) M1, M1 A1 For coefficient 2, at least 1 correct ( ) 5 (ii) f(x)

- 3 - Discriminant Since discriminant < 0, the quadratic equation has no real roots and hence has exactly two real roots, –2 and 3. M1 A1 For computing discriminant 6 (a) For : Remainder = For : Remainder = Since the equation has no solutions. M1 M1 M1 A1 For any correct expression for remainder Equating remainders 6 (b) By Factor Theorem, M1 A1 7 (i) 1 cycle occurs every 12 hours. M1 A1 7 (ii) B1 B1

- 4 - 7 (iii) When Earliest time = 0830 + 0200 = 1030 M1 M1 A1 For finding basic angle 8 (i) At stationary points, M1 M1 A1

- 5 - Coordinates of stationary points = (–1, 2) and (3, 18) A1 8 (ii) (–1, 2) is a maximum point and (3, 18) is a minimum point. M1 M1 A1 For substitution into 9 (i) At instantaneous rest, M1 M1 For finding basic angle

- 6 - A1 9 (ii) When When When When Total distance M1 M1 M1 A1 9 (iii) When M1

- 7 - A1 10 (i) Sub Sub Sub M1 M1 M1 M1 M1 A1 Realizing form of the partial fraction Eliminating denominator

- 8 - 10 (ii) M1 M1, M1 A1 1 1 (i) At B, When M1 M1 A1 11 (ii) M1

- 9 - When M1 A1 11 (iii) M1 M1, M1 A1 12 (i) Coordinates of centre = (1, 4) Radius = units M1 A1 A1 12 (ii) When Midpoint of PQ M1 M1

- 10 - A1 12 (iii) Gradient of normal through (6, –1) ∴gradient of tangent M1 M1 A1 12 (iv) Radius