2024 PHS AMATH P1 PRELIM MS

Name: Index No.: Class: PRESBYTERIAN HIGH SCHOOL ADDITIONAL MATHEMATICS 4049/01 Paper 1 19 August 2024 Monday 2 hours 15 min PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL 2024 SECONDARY FOUR EXPRESS / FIVE NORMAL (ACADEMIC) PRELIMINARY EXAMINATIONS MARKING SCHEME

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0,ax bx c+ + = 2 4 2 b b acx a −  −= Binomial expansion 1 2 2( ) ... ... , 12 n n n n n r r nn n na b a a b a b a b b r − − −     + = + + + + + +           where n is a positive integer and 2. TRIGONOMETRY Identities 22sin cos 1AA+= 22sec 1 tanAA=+ 22cosec 1 cotAA=+ sin( ) sin cos cos sinA B A B A B =  cos( ) cos cos sin sinA B A B A B= tan tantan( ) 1 tan tan ABAB AB = sin 2 2sin cosA A A= 2 2 2 2cos 2 cos sin 2cos 1 1 2sinA A A A A= − = − = − 2 2 tantan 2 1 tan AA A= − Formulae for sin sin sin a b c A B C== 2 2 2 2 cosa b c bc A= + − 1 sin2 bc A= ! ( 1)...( 1) ( )! ! ! n n n n n r r n r r r  − − +== − ABC

3 1 Express as the sum of 3 partial fractions. [5]

4 2 (a) State, in terms of π, (i) the principal value of , [1] (ii) the values between which the principal value of must lie. [1] (b) Given that A is a reflex angle and , find the exact value of without the use of a calculator. [3] (a) (i) [B1] (ii) [B1] (b) A lies in the 3rd quadrant.

5 3 (a) Find the set of values of the constant k for which the curve y = – kx2 – 2x + 2k – 3 does not intersect the x-axis. [3] (b) Using the answer in part (a), explain whether it is possible for – kx2 – 2x + 2k – 3 to be positive for all x. [2] Since , coefficient of x2 = – k < 0. [M1] The curve lies entirely below the axis, therefore it is not possible for – kx2 – 2x + 2p – 3 to be positive for all x. [A1]

6 4 A roller coaster is being designed such that the height, h m, of a rider above the ground in a section of the roller coaster ride is given by , where x is the horizontal distance of the rider from the starting point and 1 ≤ x ≤ 4. (a) Express h in the form a + b(x + c)2 where a, b and c are constants to be determined. [3] (b) Hence, explain why the rider cannot reach a height of 10 m. [1] Since the maximum value of the rider is 8.5 m and 8.5 < 10, the rider cannot reach a height of 10 m. [B1] (c) After testing the prototype, the roller coaster designer wants to make the ride more exciting by moving the highest point of this section of the roller coaster ride up by 0.2 m and left by 0.1 m. Write down a possible new expression for h. [1]

7 5 Water is poured into an empty inverted conical container with radius 6 cm and slant height 10 cm. After t seconds, the radius of the top surface of the water is r cm. (a) Show that the surface area, S, of water in contact with the container at any time is given by . [2] [A1] (b) Water is poured into the container such that r increases at a constant rate. Given that it takes 30 seconds to completely fill up the empty container, write down the rate at which r increases. [1] [B1] (c) Hence calculate the rate at which S increases when the container is one-eighth filled. [4] When the container is one-eighth filled, When r = 3, [A1] 10 cm 6 cm r cm

8 6 A curve has equation , where x > 0. (a) Find the x-coordinates of the stationary points of the curve. [5] At stationary point, (b) Determine the nature of each of the stationary points. [3] [M1 for first derivative test] Therefore, the stationary point at is a point of inflexion and the stationary point at is a maximum point. [A1, A1] (c) Using your answer in part (b), infer and write down the set of values of x for which y is an increasing function. [2] [B1,B1] OR x + 0 + x + 0 –

9 7 The diagram shows part of the curve , where x > –8. The curve intersects the x-axis and y-axis at P and Q respectively. The tangent to the curve at R is parallel to line PQ. (a) Show that the coordinates of point R are (– 4, – 10). [5] When x = 0, y = –15. When y = 0, x = –6 [B1 for either coordinates of P and Q] mPQ [B1] When x = – 4, y = – 10. Therefore R(– 4, – 10). [A1] P Q O x y

10 (b) Given that the point S has coordinates (–2, –5), find the area of the quadrilateral PRQS. [2] (c) Determine, with reason, whether PRQS is a parallelogram. [3] OR Midpoint of PQ = Midpoint of RS = Since the diagonals bisect each other, PRQS is a parallelogram. [A1]