STC 2024 Prelim A Math P2 solutions

CANDIDATE NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS 4049/2 Paper 2 23 Aug 2024 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. This document consists of 19 printed pages. CHIJ ST. THERESA’S CONVENT PRELIMINARY EXAMINATION 2024 SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC)

CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2024 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 2 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 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 4 EXPRESS / 5 NORMAL (ACADEMIC) 2024 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 3 1 The equation of a curve is 25sin 6yx =−  , where 0 2x  . (a) Given that y is decreasing at a rate of 0.3 units per second, find the rate of change of x at 5 12x = . [3] (b) The normal to the curve at 5 12x = intersects the vertical axis at ( )0, k . Find the exact value of k. [3] At , A1 – Correct derivative M1 – Correct use of connected rates of change with substitution A1 Gradient of normal At , At horizontal axis, x = 0 FTA1 – Correct gradient of normal M1 – for finding value of y A1

CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2024 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 4 2 (a) Find the values of x and y which satisfy the equations ( ) 8 2 0, 1125 . 5 xy yx −−= = [3] and Hence, solving simultaneous equations When , M1 – Correct simplification of equations M1 – Eliminating one variable using substitution A1

CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2024 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 5 (b) Show that the equation ( ) ( ) 23 2 1 35 2xx+− −= has only one solution and find its value correct to 2 significant figures. [5] M1 – Correct substitution to obtain a quadratic equation M1 – Factorisation A1 M1 – Using logarithm to solve A1 – Answer to 2s.f.

CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2024 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 6 3 The diagram shows two perpendicular lines AB and CD which intersect at X. The points A, B, C and D lie on the circumference of a circle. AC = 8 cm, BD = 15 cm, and angle ACD equals to   . (a) Show that the length of AB is 8sin 15cos+ . [2] (b) Express AB in the form ( )sinR + , where R > 0 and 0 90    . [4] X A B C D  8 cm 15 cm As A, B, C and D lie on the circumference of a circle, Angle ABD =  M1 AG1 M1, A1 M1 A1

CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2024 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 7 (c) Find the value(s) of  if AB = 16 cm. [3] M1 – Finding basic angle A1, A1

CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2024 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 8 4 A calculator must not be used in this question. It is given that ( ) ( ) cos 1 3cos AB AB + =− . (a) Show that 1tan tan 2AB = . [3] (b) If tan 2 3A =+ , find an expression for tan B , in the form 3ab+ , where a and b are constants. [3] M1 – Correct use of addition formulae M1 – Cross multiplying and simplifying M1 – Dividing by cosAcosB to obtain expression M1 – Rationalising denominator M1 – Correct use of difference of two squares to simplify denominator A1

CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2024 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 9 (c) Hence, express 2sec B in the form 3cd+ , where c and d are constants. [3] FTB1 – Correct squaring of tan B M1 A1

CHIJ ST. THERESA’S CONVENT SECONDARY 4 EXPRESS / 5 NORMAL (ACADEMIC) 2024 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS PAPER 2 10 5 The equation of a circle is 22 6 16 48 0x y x y+ − + + = . (a) Find the radius and coordinates of the centre of the circle. [4] (b) The point ( )0, 4A − lies on the circle. Given that AB is a diameter of the circle, find the coordinates of B. [2] Radius = 5 units Coordinates of center = OR and M1 – Use of completing the square A1 – for correct A1 – for correct FTA1 B1, B1 M1, A1 Let . As centre is the midpoint of a diameter M1 – Using midpoint or proportions A1