OPSS AMPrelim 2024 4E P1 QP

ORCHID PARK SECONDARY SCHOOL End-of-Year CANDIDATE NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS Paper 2 Secondary 4 Express / 5 Normal (Academic) Setter: Mr Chan Ho Lun 4049/01 20 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 sp ecified 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 1 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 Find the range of values of p for which the line 𝑦 = 𝑝𝑥 − 5 meets the curve 𝑦 = 3𝑥2 + 4𝑥 − 2. [4] 2 Express 5𝑥2−6𝑥+13 (𝑥−1)(𝑥2+3) in partial fractions. [5]

4 3 Prove that 2 cot 2𝜃 = cosec 𝜃 sec 𝜃 − 2 tan 𝜃. [5]

5 4 (a) Find 𝑑 𝑑𝑥 (5𝑥𝑒2𝑥+1). [2] (b) Hence find ∫ 𝑥𝑒2𝑥+1 𝑑𝑥. [4]

6 5 Solve the equation 1 + 3 sin2 𝜃 = 4 cos 𝜃 for − 𝜋 2 ≤ 𝜃 ≤ 𝜋 2 . [6]

7 6 The function f is given by f(𝑥) = 𝑥2 𝑥−2𝑘 , for 𝑥 > 2𝑘 , where k is a positive constant. (a) Find f ′(𝑥). [2] The function g, defined for 𝑥 > 2𝑘 , has the property that g′(𝑥) = (𝑥 − 2𝑘)2 f ′(𝑥). g decreases for 𝑘 < 𝑥 < 6. (b) Show that a possible value of k is 3. [4]

8 7 Prove that there are no values of k for which 𝑘𝑥2 + 2𝑥 − 2𝑘 − 3 is always positive. [6]

9 8 The line 𝑦 − 𝑥 = 2 intersects the curve 𝑦2 = 4(2𝑥 + 1) at two points. Find the coordinates of these two points. [5]

10 9 A curve has equation 𝑦 = 𝑥3 + 𝑚𝑥 − 15. It has a stationary point A where 𝑥 = 2. (a) Show that the value of the constant m is −12. [2] (b) Find the coordinates of the other stationary point B. [2]