CHIJ STC 2023 Prelim 4E5N A Math P1 mark scheme

CHIJ ST. THERESA’S CONVENT PRELIMINARY EXAMINATION 2023 SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) CANDIDATE NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS 4049/1 Paper 1 29 August 2023 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 17 printed pages.

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2023 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 AB AB  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) 2023 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 3 1 Find the gradient of the normal to the curve 3 2 4y x x=+ at 5x= , leaving your answers in the form ( )5ab + , where a and b are constants. [4] 2 Integrate ( ) 2 54 31 1x x −+ − with respect to x. [4] At x = 5, Gradient of tangent = Hence, Gradient of normal B1 – Correct Gradient Function M1 – Using Gradient Function to find gradient of normal M1 – Rationalising denominator by multiplying correct conjugate surds A1 B1 – B1 – correct coefficient for ln function B1 – B1 – correct coefficient for power function and arbitrary constant

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2023 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 4 3 (a) Express 21 16 16xx−− in the form ( ) 2 a x b c++ and hence state the coordinates of the turning point of the curve 21 16 16y x x= − − . [4] (b) Hence, explain why the curve 21 16 16y x x= − − will never intersect y = 7 for all values of x. [2] Hence, Coordinates of Turning Point M1 – factorising coefficient of x2 B1, B1 – correct b and c FTA1 As , Turning point is a maximum point Since 7 > maximum value of curve = 5, The curve will never intersect y = 7 for all values of x OR Since > 0, the curve will never intersect y = 7 for all values of x B1 – turning point is max point B1 – 7 is larger than max value of y = 5 M1 – Showing is equals to negative A1 – Conclude appropriately

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2023 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 5 4 The equation of a curve is cosy a bx c=+ , where 0a . (a) Given that the maximum and minimum values of y are 1 and – 9 respectively, state the values of a and c. [2] (b) Given that the period of y is π , find the value of b. [1] (c) Hence, sketch the curve cosy a bx c=+ for 02 πx radians, labelling the maximum and minimum points clearly. [3] B1, B1 B1 – correct shape (cosine) B1 – 2 complete cycles B1 – correct max and min points b = number of cycles = = 2 B1

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2023 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 6 5 The function ( ) 32f 5 1x ax x bx= + + + has a factor ( )21x− and leaves a remainder of 9 when divided by ( )1x+ . (a) Show that 5b=− and find the value of a. [4] (b) Given that the quadratic expression 2 1x px+− is also a factor of ( )f x , find the value of the constant p. [2] Since is a factor, Since leaves a remainder of 9 when divided by , Equating both equations, M1 – Correct use of factor theorem to form equation M1 – Correct use of remainder theorem to form equation M1 – Solving to find b A1 – Correct value of a. Comparing coefficients of terms: OR Comparing coefficients of x terms: M1 – comparing or x terms A1

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2023 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 7 6 Express 32 2 3 4 4 17 34 x x x xx − + − −− in partial fractions. [7] As fraction is improper, using long division/synthetic division: M1 – Correct use of long division or synthetic division to express improper fraction into mixed terms A1 M1 – Correct factorisation of denominator B1 – Correct identification of partial fractions M1 – Using appropriate substitution to find A M1 – Using appropriate substitution to find B A1

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2023 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 8 7 The number of bacteria cells, N, in millions, in a circular patch after t hours is given by 2tN ke= , where k is a constant. The initial number of bacteria cells is 10,000,000. (a) Explain why 10k = . [1] (b) Find the rate at which the number of bacteria cells is increasing after 30 minutes. [2] (c) As the number of bacteria cells increases, the area, A units2, of the circular patch also increases. Given that 24AN= , find the rate of change of A after 5 hours, leaving your answer in exact form. [4] At t = 0, N = 10 Hence, M1 – substitution of t = 0, N = 10 After 30 minutes, M1 – Differentiating N w.r.t t A1 When t = 5, M1 – Differentiating A w.r.t N M1 – Finding values of and when t = 5 M1 – Substituting values into correct connected rates of change equation A1

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2023 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 9 8 In the diagram, CR is a tangent to the circle at C, and line QR cuts the circle at points A and B. D lies on the circumference of the circle. (a) Prove that triangle CBR and triangle ACR are similar. [2] (b) Given that ADB BRC = , show that AC is the diameter. [4] R D B A C Q M1 A1 – AA test M1 M1 M1 A1

CHIJ ST. THERESA’S CONVENT SECONDARY FOUR EXPRESS / FIVE NORMAL(ACADEMIC) 2023 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 1 10 9 (a) Prove the identity cos 1 sin 2sec1 sin cos   ++=+ . [4] (b) Hence solve the equation cos 1 sin22 5 1 sin cos22 