A-Math–Beatty Sec 4N prelim(with detailed solu)

CANDIDATE NAME CLASS BEA TTY SECONDARY SCHOOL PRELIMINARY EXAMINATION 2022 SECONDARY FOUR NORMAL (ACADEMIC) REGISTER I NUMBER ~-------~ ADDITIONAL MATHEMATICS Paper1 Candidates answer on the Question Paper Additional Materials: Nil READ THESE INSTRUCTIONS FIRST 4051/01 3 August 2022 1 hour 45 minutes Write your name, class and register 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, glue or correction fluid. Answer all the questions. Given 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 of the marks for this paper is 70. For Examiner's Use BP~3 This document consists of 16 printed pages and 0 blank page. [Turnover PartnerlnLearning 1

Mathematical Formulae I.ALGEBRA Quadratic Equation For the equation ax2 +bx+ c = 0, Identities Formulae for AA.BC -b ± .Jb2 -4ac x=------ 2a 2. TRIGONOMETRY sin2 A + cos2 A = I sec2 A= I + tan2 A cosec2 A = I + cot2 A sin(A ±B) = sinAcosB ± cosAsinB cos(A± B) = cosAcosB +sinAsinB tan(A ± B) = tan A± tanB 1+ tanAtanB sin 2A = 2sin A cos A cos 2A = cos 2 A - sin 2 A = 2 cos 2 A - I = 1 - 2 sin 2 A 2A 2tanA tan =--- 1-tan 2 A a b c --=--=-- sinA sinB sinC a2 = b2 + c2 - 2bc cos A I b . I)..= -a sm C 2 Partnerlnlearning 2 BP-4

----- - - - - --- - ~--- 1 Do not use a calculator in answering this question. 2 It is given that ✓3(x+3) =x+17. Findx in the form a+b✓3, where a and b are integers. (a) State, in terms of 1r, the principal value of (i) tan-) 1, ( "") • -I ( ✓)J n sm -- . 2 (b) Explain why the principal value of co,-• (- ~J cannot be - : . Partnerlnlearning 3 BP-5 [4] [1] [1] [1] [Turnover

3 Given that these simultaneous equations x 2 + y2 -16 = 0, x-y=k have exactly one pair of solutions, show that k = ±p ✓2, where pis an integer. [6] Partnerlnleaming 4 BP-6

4 Given that the period of y = 5 sin bx - 2 is 12Q° . (a) Find the value of b. [1] Tuition (b) On the axis below, sketch the graph of y = 5 sin bx - 2 for -90° ~ .x ~ 80° . [3] y wit tJ 0 PartnerlnLeaming 5 X [Turnover BP-7

5 (a) Show that f(x)=4x 2 -10x+16 can be written as f(x)==a(x--b)2+c, where a, b and c are constants to be found. (b) Explain why the function f(x) is always positive. Partnerlnlearning 6 BP-8 [3] [1]

6 (a) Without using a calculator, and showing all your working, find the exact 1 f . 1r 5,r ,r . 5,r va ue o sm-cos-+cos-sm-. 4 12 4 12 (b) Given that tan 0 = k and that 0 is acute, express in terms of k (i) sec0, (ii) sin 20. PartnerlnLearning 7 [