Queensway 4E Prelim AM P1 2021 with soln

Calculator Model: ________________ NAME: CLASS: INDEX NO: QUEENSWAY SECONDARY SCHOOL PRELIMINARY EXAMINATION 2021 SECONDARY 4 EXPRESS ADDITIONAL MATHEMATICS 4049/01 Paper 1 13 September 2021 2 hour 15 minutes Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST Write your name, class and index number in the spaces at the top of this page. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction tape. 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 𝜋, use either your calculator value or 3.142, unless the question requires the answer in terms of 𝜋. The number of marks is given in bracket [ ] at the end of each question or part question. The total number of marks for this paper is 90. This document consists of 16 printed pages. Setter: Ms Chen Zhiyun [Turn over Parent’s Signature:

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax 2 + bx + c = 0 , a acbbx 2 42 −−= Binomial expansion (𝑎 + 𝑏)𝑛 = 𝑎𝑛 + (𝑛 1) 𝑎𝑛−1𝑏 + (𝑛 2) 𝑎𝑛−2𝑏2 + ⋯ + (𝑛 𝑟) 𝑎𝑛−𝑟𝑏𝑟 + ⋯ + 𝑏𝑛, where n is a positive integer and ( ) !! ! rrn n r n −=      = ! )1)...(1( r rnnn +−− 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A sin(𝐴 ± 𝐵) = sin 𝐴 cos 𝐵 ± cos 𝐴 sin 𝐵 cos(𝐴 ± 𝐵) = cos 𝐴 cos 𝐵 ∓ sin 𝐴 sin 𝐵 tan(𝐴 ± 𝐵) = tan 𝐴 ± tan 𝐵 1 ∓ tan 𝐴 tan 𝐵 sin 2𝐴 = 2 sin 𝐴 cos 𝐴 cos 2𝐴 = 𝑐𝑜𝑠2𝐴 − 𝑠𝑖𝑛2𝐴 = 2𝑐𝑜𝑠2𝐴 − 1 = 1 − 2𝑠𝑖𝑛2𝐴 A AA 2tan1 tan22tan −= Formulae for  ABC C c B b A a sinsinsin == a 2 = b 2 + c 2 − 2bc cos A  = 2 1 bc sin A

3 1. (a) A beaker of solution is heated until it reaches a temperature of 𝑃 ℃. It is then placed in a room to cool. Its temperature 𝜃 ℃, when cooling for t minutes, is given by 𝜃 = 38 + 50𝑒−0.7𝑡. (i) Find the value of P. [1] (ii) Find the value of t when 𝜃 = 1 2 𝑃. [2] (iii) Explain, with working, if the water will cool to a temperature of 30 ℃ . [2]

4 2. (a) Find the coefficient of 𝑥 in the expansion of ( 2 𝑥 − 𝑥2 4 ) 8 . [3] (b) It is given that (1 + 𝑘𝑥)𝑛 = 1 + 10𝑥 + 175 4 𝑥2 + 𝑝𝑥3 +………where 𝑛 > 0. Find the values of 𝑛, 𝑘 and 𝑝. [5]

5 3. (a) 2𝑥2 − 𝑥 − 3 is a factor of 2𝑥4 + 𝑥3 + 𝑝𝑥2 + 3𝑥 + 𝑞. (i) Show that 𝑝 = −16 and 𝑞 = 18. [3] (ii) Hence, solve the equation 2𝑥4 + 𝑥3 − 16𝑥2 + 3𝑥 + 18 = 0. [3]

6 (b) The equation (𝑥 + 𝑎)(𝑥 + 𝑏) = 𝑐2 is given such that 𝑎, 𝑏 and 𝑐 are real values. (i) Show that the roots of the equation are always real. [3] (ii) State a set of possible values for 𝑎, 𝑏 and 𝑐 when the roots are real and equal. [2] 4. (a) Solve the equation 𝑙𝑜𝑔16(4𝑥 − 5) = 𝑙𝑜𝑔42𝑥 − 𝑙𝑜𝑔4√5. [4]

7 (b) Show that 2𝑥 + 1 2 (2𝑥+4) − 2𝑥+2 , where x is positive, is exactly divisible by 5. [2] 5. (i) Prove that tan 𝑥 1+sec 𝑥 + 1+sec 𝑥 tan 𝑥 = 2 𝑐𝑜𝑠𝑒𝑐 𝑥. [3] (ii) Hence, solve tan 𝑥 1+sec 𝑥 + 1+sec 𝑥 tan 𝑥 = 3 sec 𝑥, where 0° ≤ 𝑥 ≤ 540°. [4]

8 6. A circle is defined by the equation 𝑥2 + 𝑦2 − 8𝑥 + 12𝑦 = 48. (i) Determine the coordinates of the centre of the circle and its radius. [3] (ii) Justify whether point A (7, 3) lies on, inside or outside the circle. [2]

9 (iii) The tangent to the circle at B is parallel to the line 4𝑦 = 3𝑥 − 6. Determine two possible coordinates of B. [6]

10 7. The equation of a curve is 𝑦 = 𝑎𝑥4 + 𝑏𝑥2 − 16𝑥. The coordinates of a stationary point on the curve is given as (−2, 12). Find the coordinates of all the stationary points on the curve and determine their nature. [8]