2022 STC AMath Prelims P2

CANDIDATE NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS 4049/2 Paper 2 26 Aug 2022 2 hours 15 minutes Candidates answer on the Question Paper as well as on the graph paper provided. 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 staples, paper clips, glue or correction fluid. 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. 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. CHIJ ST. THERESA’S CONVENT PRELIMINARY EXAMINATION 2022 SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC)

2 CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 This document consists of 16 printed pages. Mathematical Formulae 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 ΔABC C c B b A a sinsinsin  a2 = b2 + c2 − 2bc cos A Δ = 1 sin2 ab C

3 CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 1 (a) Express 2 3 3 2 2 in the form 6a b  , where a and b are rational numbers. [2] (b) Solve the equation 1 5x  = 2x. [4]

4 CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 2 The equation of a quadratic curve is y = 21 2 12 x x   . (i) Express y in the form 2( )a x b c  , where a, b and c are constants. [2] The graph of y = 21 2 12 x x   is shown in the diagram. The region R consists of the shaded area as well as the perimeter surrounding the shaded area. (ii) Find the range of values of x and the range of values of y corresponding to the region R. [4] y R

5 CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 xy O (, 1) (5, 4) 3 The diagram shows the straight line graph obtained by plotting xy against 3x . The line passes through the points (−2, −1) and (5, 4). (i) Obtain an expression for y in terms of x. [3] A second line is drawn on the same diagram above. This second line is parallel to the first line and it passes through the point with coordinates (0, 1). (ii) Obtain an expression for y, for this second line, in terms of x. [2]

6 CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 4 Two of the angles of a triangle are represented by x and y. It is given that sin ( ) sin ( )x y x y   = 1 4 - - - (1) and cos ( ) cos ( )x y x y   = 1 6 - - - (2) (i) Show that tan x = 3 2 . [3] (ii) Explain why angle y is acute. [2] (iii) Find the exact value of cos y . [3]

7 CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 5 (i) Differentiate 2 7xe x  with respect to x, giving your answer in the form ( ) 2 7 xa bx e x   where a and b are constants. [3] The curve y = 2 7xe x  has a stationary value at the point C. (ii) Find the x-coordinate of C. [2] (iii) Determine if point C is a maximum or a minimum point. [2]

8 CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 6(a) The polynomial f(x) is given by 3 224 14x x ax b   where a and b are constants. It is given that f(x) is exactly divisible by 4x + 3 and it leaves a remainder of –6 when it is divided by x + 1. Find the value of a and of b. [4] (b) Express   2 2 6 ( 1) 2 1 x x x x     in partial fractions. [5]

9 CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 7 The diagram shows part of the curve y = 3 2 4x x   which meets the x-axis at Q. The tangent to the curve at P meets the x-axis at R. The x-coordinates of P and Q are 1 and 2 respectively. Find (i) the equation of the tangent, [4] (ii) the area of the shaded region PQR. [5]

10 CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 8 In the diagram, B, C and D are points on a circle. DE is a tangent to the circle. D and E are midpoints of AB and AC respectively. (i) Show that BCD is an isosceles triangle. [3] (ii) Show that angle ACB = 90. [4] A B C D E