OPSS AMPrelim 2024 4E P2 with ANS

ORCHID PARK SECONDARY SCHOOL End-of-Year CANDIDATE NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS Paper 2 Secondary 4 Express / 5 Normal (Academic) Setter: Mr Mohd Salim Bin Ramli 4049/02 23 August 2024 2 hours 15 minutes 90 Marks Additional Materials: NIL READ THESE INSTRUCTIONS FIRST Write your name, class and index number on all the work you hand in. Write in dark blue or black pen. Use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. If working is needed for any question it must be shown with the answer . Omission of essential working will result in loss of marks. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For , use either your calculator value or 3.142, unless the question requires the answer in terms of . The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks of this paper is 90. This document consists of 19 printed pages. For Examiner’s Use Total Preliminary Examination 2024 Calculator Model:

2 1. ALGEBRA Quadratic Equation For the equation Binomial expansion where n is a positive integer and 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A 𝑐𝑜𝑠(𝐴 ± 𝐵) = 𝑐𝑜𝑠𝐴𝑐𝑜𝑠𝐵 ∓ 𝑠𝑖𝑛𝐴𝑠𝑖𝑛𝐵 tan(𝐴 ± 𝐵) = 𝑡𝑎𝑛𝐴 ± tan 𝐵 1 ∓ 𝑡𝑎𝑛𝐴𝑡𝑎𝑛𝐵 Formulae for ABC cos A. Area of  = sin A ,02 =++ cbxax a acbbx 2 42 −−= ,......21)( 221 nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− ! )1)...(1( !)!( ! r rnnn rrn n r n +−−=−=      ( ) BABABA sincoscossinsin = AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= .sinsinsin C c B b A a == bccba 2222 −+= bc2 1

3 1 (a) Expand (𝑥 + 1 𝑥) 4 in descending powers of x. [2] (b) Hence, given that (𝒙 + 𝟏 𝒙) 𝟒 − (𝒙 − 𝟏 𝒙) 𝟒 = 𝒂𝒙𝟐 + 𝒃 𝒙𝟐, find the value of a and of b. [3]

4 (c) Given that there is no x term in the expansion of ( 4 3 𝑥 + 𝑘 𝑥 + 𝑥3 𝑘 ) (𝑥 + 1 𝑥) 4 , find the value of k. [3]

5 2 In a diagram, A, B, C and D are points on the circle. The tangent at C meets AD produced at P. The chords AC and BD intersect at Q. The line BQD bisects angle ABC. Prove that (a) ∠𝐷𝐶𝑃 = ∠𝐴𝐶𝐷, [3] (b) 𝛥𝑃𝐶𝐷 is similar to 𝛥𝑃𝐴𝐶, [2] (c) 𝑃𝐶2 = 𝑃𝐴 × 𝑃𝐷. [1]

6 3 In recent years, the release of greenhouse gases has accelerated the melting of glaciers and thus, resulting in a rise of global temperatures. With minimal actions taken to prevent global warming, the average temperature, 𝑇°𝐶, projected to rise after x years from 2024, is given by 𝑇 = 31(1.5)𝑘𝑥, where k is a constant. (a) Given that the projected average temperature in Yishun in 2027 is 35°𝐶, find the value of k correct to 1 decimal place. [2] (b) Find the average temperature in Yishun in 2024. [1] (c) In which year will the average temperature in Yishun first be at least 15% higher than its temperature in 2024? [4] (d) Sketch the graph of T against x. [2]

7 4 (a) The function 𝑓 is given by 𝑓(𝑥) = 𝑎𝑠𝑖𝑛3𝑥 + 𝑏, where 𝑎 and 𝑏 are positive integers and 0 ≤ 𝑥 ≤ 𝜋. The maximum and minimum value of 𝑓 are 6 and −2 respectively. (i) State the period of 𝑓. [1] (ii) Find the value of 𝑎 and of 𝑏. [2] (iii) Sketch the graph of 𝑦 = −𝑓(𝑥) . [3]

8 4(b) Prove the identity   2tan 3cos2coscos 3sin2sinsin = +− +− . [4]

9 5 A particle moving in a straight line is such that its displacement, s metres, from a fixed point O, is given by s = 4 – 2e– t – t where t is the time in seconds after passing through a point B on the line. Find (a) the distance OB, [1] (b) the initial velocity of the particle, [2] (c) the value of t when the particle is instantaneously at rest, [2]

10 (d) the total distance travelled by the particle in the first two seconds. [3] 6 The equation of a polynomial is given by 𝑃(𝑥) = 2𝑥3 + 𝑚𝑥2 + 𝑥 + 𝑛, where 𝑚 and 𝑛 are constants. (a) Given that 2𝑥2 + 𝑥 − 1 is a factor of 𝑃(𝑥), find the value of 𝑚 and of 𝑛. [4]