2022 STC AMath Prelims P1

CHIJ ST. THERESA’S CONVENT PRELIMINARY EXAMINATION 2022 SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) CANDIDATE NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS 4049/2 Paper 1 30 August 2022 2 hours 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 a pencil for any diagrams or graphs. Do not use paper clips, highlighters, glue or correction fluid. Answer all 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 78 printed pages.

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 2 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, x = a acbb 2 42  Binomial expansion (a + b)n = an + ban n 1 1        + 22 2 ban n       + …+ rrn bar n        + … + bn, where n is a positive integer and !)!( ! rrn n r n       ( 1)...( 1) ! n n n r r    . 2. TRIGONOMETRY Identities sin2A + cos2A = 1 sec2A = 1 + tan2A cosec2A = 1 + cot2A sin(A ± B) = sinAcosB ± cosAsinB cos(A ± B) = cosAcosB  sinAsinB tan(A ± B) = tan tan 1 tan tan A B A B   sin2A = 2sinAcosA cos2A = cos2A  sin2A = 2cos2A  1 = 1  2sin2A tan2A = 2 2 tan 1 tan A A Formulae for ABC C c B b A a sinsinsin  a2 = b2 + c2  2bc cos A  = 1 sin2 ab C

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 3 1 A spherical bath bomb of radius r cm is dissolving in such a way that the total surface area, A cm2, is decreasing at a rate of 12 cm2/s. Assuming that the bath bomb retains its shape, calculate the rate of change of r when 5r , leaving your answer in terms of π. [4] 2 The equation of a curve is 22 5y x kx   , where k is a constant, and the equation of a line is 3 10 0y x   . (i) In the case where k = 4, find the coordinates of the points of intersection of the line with the curve. [3] (ii) Show that, for all real values of k, the line intersects the curve at two distinct points. [2]

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 4 3 (i) Given that the constant term in the binomial expansion of 8 3 kx x     is 7 16 , find the value of the positive constant k. [4] (ii) Using the value of k found in part (i), show that there is no constant term in the expansion of 8 4 3(1 7 ) kx x x       . [4]

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 5 4 The equation of a curve is 23 (2 1) 3y x m x c     , where m and c are constants. The line y mx c  is a tangent to the curve at the point P. (i) Find the positive value of m. [4] (ii) Using this value of m, and given that the curve passes through (2, 38), find the coordinates of P. [4] The straight line L meets the curve at one point only. (iii) Given that L is not a tangent to the curve, what can be deduced about L? [1]

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 6 5 A farmer uses 120 m of fencing to enclose a plot of the shape shown above. The shape consists of an equilateral triangle of side x m, a rectangle with sides x m and l m and a semicircle of diameter x m. (i) Show that the area of the plot is 23 π60 1 4 8x x               m2. [3] x m l m

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 7 (ii) Given that x can vary, find the value of x for which the area of the plot is stationary. [4] (iii) Explain why this value of x gives the farmer the largest possible area. [1]

6 It is given that f ( ) 4sin 3 2x x  . (i) State the amplitude of f(x). [1] (ii) State the least and greatest values of f(x). [2] (iii) State the period of f(x). [1] (iv) Sketch the graph of y = f(x) for 0 180x   . [3]

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 2 7 (i) Express cos3 as an expression in terms of cos . [3] (ii) Use your answer in part (i) to find, for 0 360   , the solutions of the equation 22cos 3 3 4 cos   . [4]

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 3 8 (i) Express 4 2 5 x x in the form 2 5 ba x  , where a and b are integers. [2] (ii) Differentiate ln(2 5)x x  with respect to x. [2] (iii) Using the results in parts (i) and (ii), determine ln(2 5) dx x . [3]