KCPSS 2023 Prelim Paper 2

NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS 4049/02 Paper 2 24 August 2023 2 hours 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. Setter: Ms Rebecca Kang READ THESE INSTRUCTIONS FIRST Write your class, index number and name 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, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an approved scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 90. For Examiner’s Use Paper 2 / 90 m Areas to Note Accuracy Presentation KUO CHUAN PRESBYTERIAN SECONDARY SCHOOL 2023 Preliminary Examination Secondary 4 Express / 5 Normal Academic

KCPSS 2023/Sec 4Exp/5N Prelim AM P2 2 Pencil Units This document consists of 19 printed pages and 1 blank page. [Turn over 1. ALGEBRA Quadratic Equation For the equation ax 2 + bx + c = 0, a acbbx 2 42 -±-= . Binomial expansion ( ) ,b...bar n...banbanaba nrrnnnnn ++÷÷ ø ö çç è æ++÷÷ ø ö çç è æ+÷÷ ø ö çç è æ+=+ --- 221 21 where n is a positive integer and ( ) ( ) ( ) !r rn...nn !rn!r !n r n 11 +--=-=÷÷ ø ö çç è æ 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 ) = cosAcosB ∓sinAsinB tan A ±B = tanA ±tanB 1 ∓tanAtanB sin 2A = 2 sin A cos A

KCPSS 2023/Sec 4Exp/5N Prelim AM P2 [Turn over 3 cos 2A = cos 2 A – sin 2 A = 2 cos 2 A – 1 = 1 – 2 sin2 A A AA 2tan1 tan22tan -= Formulae of Δ ABC C c B b A a sinsinsin == a2 = b2 + c2 – 2bc cos A Δ = 2 1 bc sin A

KCPSS 2023/Sec 4Exp/5N Prelim AM P2 4 1 In 2005, Andrew bought a painting. In 2008, the value of the painting is estimated to be $3878.58. The value V (t), in dollars, of the painting t years after being purchased is given by the function V (t) = 2400 ekt , where k is a constant. (a) What was the value of the painting after 10 years? [4] (b) If Andrew intended to sell the painting after its value had at least doubled, which is the earliest year that he can sell it? [2]

KCPSS 2023/Sec 4Exp/5N Prelim AM P2 [Turn over 5 2 (a) Prove that 3 cos2 x + sin2 x = 2 + cos 2x. [3] Hence, using your result from part (a), (b) (i) find the exact value of 3 cos2 22.5° + sin2 22.5°, [2]

KCPSS 2023/Sec 4Exp/5N Prelim AM P2 6 (ii) solve the equation 6 cos2 x + 2 sin2 x = 5 for 0° ≤ x ≤ 360°. [4]

KCPSS 2023/Sec 4Exp/5N Prelim AM P2 [Turn over 7 3 It is given that f(x) = x3 + ax2 + bx + 5, where a and b are constants, has a factor of x – 1 and leaves a remainder of − 1 when divided by x − 2. (a) Find the values of a and b. [3] (b) Using these values of a and b, factorise f(x) completely into three linear factors, using surds where necessary. [3]

KCPSS 2023/Sec 4Exp/5N Prelim AM P2 8 4 A miniature model in the shape of a triangular prism is shown above. A piece of wire, 36 cm in length, runs through all the edges of this triangular prism. The cross-section of the prism is an equilateral triangle of side x cm and the length of the prism is y cm. (i) Show that the total surface area A cm2, of the prism is given by A = 3 −12 2 x2 +36x . [4] x cm y cm

KCPSS 2023/Sec 4Exp/5N Prelim AM P2 [Turn over 9 (ii) Given that x can vary, find the value of x which gives a stationary value of A. [3] (iii) Justify if A is a maximum or a minimum for this value of x and find this stationary value of A. [3]

KCPSS 2023/Sec 4Exp/5N Prelim AM P2 10 5 (a) Without using a calculator, find the values of the integers a, b and c for which the solution to the equation 5x − 3 =4 + 3 x is a +b 3 c . [4]