Peicai Prelims 4049 4E AMath P1 2021 solutions - Copy

[Turn over PEICAI SECONDARY SCHOOL SECONDARY 4 EXPRESS PRELIMINARY EXAMINATION 2021 CANDIDATE NAME CLASS REGISTER NUMBER ADDITIONAL MATHEMATICS 4049/01 Paper 1 26 August 2021 2 hours 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your register number, class 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. DO NOT WRITE IN ANY BARCODES. 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. This document consists of 19 printed pages and 1 blank page. Setter: Mrs Ho Thuk Lan SOLUTIONS

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3 [Turn over Mathematical Formulae 1. ALGEBRA Quadratic Equation For the quadratic equation ax 2 + bx + c = 0, Binomial expansion , where n is a positive integer and 2. TRIGONOMETRY Identities Formulae for ∆ABC

4 1 Given that , find the value of . [4] = 0.322 (3SF) 2 An equilateral triangle has a perimeter of cm. Find the area of the triangle, in , in the form , where a and b are constants. [4] Method 1: Method 2: Length of one side = Area = Area = Area = Height = (Pythagoras Theorem) Height = Height = Height = Height = Area = Area = Area = Area = Area =

5 [Turn over 3 The equation of a curve intersects the line at two points. Find the set of values of k. [4] 4 It is given that f(x) is defined for x > 1 and is such that . By considering , explain whether is an increasing or decreasing function. [4] Given x > 1, 4x < 0, 🡪 Since , f’(x) is a decreasing function. + + 3 5 k

6 5 (a) On the same diagram, sketch the graphs of and . [3] (b) Find the coordinates of the point of intersection of and . [2] Coordinates =

7 [Turn over 6 A, B and C are the angles of a triangle. (a) Show that cos C = cos (A + B) [2] In a triangle, A + B + C = 180 🡪 A + B = 180 C cos (A + B) = cos (180 C) cos (A + B) = cos 180cos C sin180sin C cos (A + B) = ( 1) cos C (0) sin C cos (A + B) = cos C therefore cos C = cos (A + B) Given that A = 45 and B = 30. (b) Without using a calculator, find cos C in the form , where a and b are integers. [3]

8 7 The diagram shows a triangle PQR in which angle PRQ = radians, angle PQR is a right angle, M is the midpoint of QR and the length of PQ = 12 m. Without using a calculator, find the value of the integer k such that . [5] Pythagoras Theorem: 1 P Q R M 12

9 [Turn over k = 21 8 A curve is such that . The curve passes through the point ( 1, 5) and at this point, the gradient of the curve is 3. (a) Find the x-coordinates of the stationary points of the curve. [4] At point ( 1, 5), gradient = 3 (b) Find the equation of the curve. [2] At point ( 1, 5),

10 9 Coffee is dripped at a constant rate of 2 into an empty cylindrical cup of radius 4 cm. The volume, V , of coffee in the cup is given by where x is the depth, in cm, of the coffee. Find (i) The time taken for the depth of the coffee to reach 3 cm, [3] x = 3, Time = Time = s (ii) The rate of change of the depth of the coffee at this time. [4] When , Rate of change =