SCGS AMATH Common Test 2023 Sec 4

1 [Turn over Name : __________________________________________( ) Class: Sec 4 _____ SINGAPORE CHINESE GIRLS’ SCHOOL SECONDARY 4 O-LEVEL PROGRAMME 30 Parent’s / Guardian’s Signature ADDITIONAL MATHEMATICS Common Test 1 Wednesday 15 February 2023 45 minutes Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST Write your name, index number and class at the top of this page. Write in dark blue or black pen. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all questions on the question paper. If working is needed for any question it must be shown with the answer. 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 30. This question paper consists of 9 printed pages and 1 blank page.

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 = + +c bx ax, a ac b bx 2 42 − ± −= 2. TRIGONOMETRY Identities 22 22 22 22 2 2 2 sin cos 1 sec 1 tan cosec 1 cot sin( ) sin cos cos sin cos( ) cos cos sin sin tan tantan( ) 1 tan tan sin 2 2sin cos cos 2 cos sin 2cos 1 1 2sin 2 tantan 2 1 tan AA AA AA AB A B A B AB A B A B ABAB AB A AA A AA A A AA A += = + = + ±= ± ±= ±±= = = − = −=− = −   Formulae for ∆ABC 2 22 sin sin sin 2 cos 1 sin2 abc ABC a b c bc A bc A = = =+− ∆=

3 [Turn over 1. Given that tan pθ = , where θ is acute, express each of the following in terms of p. (a) sin 2 π θ − [1] (b) sec 2θ [2] 2. Without using a calculator, find sin15° in the form 4 ab− , where a and b are integers. [3]

4 3. (a) Solve the equation 2 tan 5sinAA= for −90° ≤ A ≤ 270°. [4] (b) Solve the equation 26cos sin 5BB+= for 0 ≤ B ≤ 2π. [4]

5 [Turn over 4. Prove that cot tan 2cot 2θθ θ−= . [3]

6 5. The diagram shows part of the graph of cosy a bx c= + , where a, b and c are constants. (a) Write down the value of a, of b and of c. [3] a = b = c =

7 [Turn over (b) On the given diagram, sketch the graph of 2 tan 2 xy =  for 0 ≤ x ≤ 2π. Hence, state the number of solutions, for 0 ≤ x ≤ 2π, of the equation cos 2 tan . 2 xa bx c +=  [3]

8 6. The diagram shows a pentagon PQRST where PQ = QR = 5 m, RS = 8 m and ST = h m. U lies on TS produced such that angle RSU = θ. Angles QRS, QPT and PTS are right angles. (a) Show that 5 5sin 8cosh θθ= +− . [2]