2025 TK Sec 4 Prelim A Maths P2 QP w ans

2 This document consists of 17 printed pages and 3 blank pages. ! !! CANDIDATE NAME ADDITIONAL MATHEMATICS 4049/02 Paper 2 Tuesday 26 August 2025 2 hours 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your name, class and index number on the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid/tape. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal in the case of angles in degree, unless a different level of accuracy is specified in the question. The use of a scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. 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. TANJONG KATONG SECONDARY SCHOOL Preliminary Examination 2025 Secondary 4 CLASS INDEX NUMBER

2 4049/02/Sec4Prelim2025 [Turn over Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, x = . Binomial Theorem (a + b)n = an + an - 1 b + an - 2 b2 + . . . +an - r br + . . . + bn, where n is a positive integer and = = 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 A sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B ∓ sin A sin B tan (A ± B) = sin 2A = 2 sin A cos A cos 2A = cos2 A - sin2 A = 2 cos2 A - 1 = 1 - 2 sin2 A tan 2A = Formulae for DABC a2 = b2 + c2 - 2bc cos A D = bc sin A ! !"## ! "!!±! !! " # $$ % & ! ! !! " # $$ % & ! ! !! " # $$ % & ! " !! " # $$ % & ! " !"!# ! !!" " ! ! "#"$$$$$$$%#% ! !""" +!! !"# !"# $ !"# !"# !" !" ± ! ! ! !"#$% "#$! ! ! " # A B C !"#$!"#$!"#$ == !"

3 4049/02/Sec4Prelim2025 [Turn over 1 In an experiment, it was observed that the number of bacteria in a culture doubles every 4 hours. It is given that N0 is the number of bacteria present at the start of the experiment, and N is the number of bacteria t hours later. Calculate the exact value of the constant k, given that N = N0ekt. [4] 2 Given that , find the value of . [4] !" # ! != !!

4 4049/02/Sec4Prelim2025 [Turn over 3 Solve the simultaneous equations [5] !!"# # $ % & % ! ' ( )!"++ !" # $ % &!"=!

5 4049/02/Sec4Prelim2025 [Turn over 4 (a) State the range of values of x for to be defined. [2] (b) Given , express x in terms of p. [4] !"# $% &'! !! !"#$% &#$%!"=

6 4049/02/Sec4Prelim2025 [Turn over 5 A curve is such that The curve passes through the point and has a gradient of at P. Find the equation of the curve. [7] ! ! " #$%& ' ' ()$ ! *" ! """=! !"#$!!!"# $+%&'( !!

7 4049/02/Sec4Prelim2025 [Turn over 6 (a) The term containing the highest power of x in the polynomial is Two of the roots of the equation are and. Given that is a quadratic factor of , find an expression for in descending powers of x. [3] (b) Find the value of k for which is exactly divisible by but not divisible by . [4] !"#! !"#! !"#$ =! !"!!!"+! !" #! !"#! x2+(k−1)x+k2−16x−3x+4

8 4049/02/Sec4Prelim2025 [Turn over 7 An object starts from rest from a point O. Its velocity, v cm/s, t seconds after leaving O, is such that After 5 seconds, the object reaches a point A. (i) Find its velocity at A. [2] (ii) Find the distance OA. [2] On reaching A, the object then slows so that its velocity V cm/s, T seconds after leaving A, is such that where k is a constant. (iii) Given that the object’s velocity at a point B where T = 4 is 26 cm/s, show that k = 12. [3] ! "#$! ! "= ! "# $! ! "##=!

9 4049/02/Sec4Prelim2025 [Turn over 8 (a) Show that has real and distinct roots for all real values of m. [4] (b) The equation of a curve is (i) Find the set of values of x for which the curve lies below the line y = 11 . [3] The straight line L meets the curve at one point only. (ii) Given that L is not a tangent to the curve, what can be deduced about L? [1] !"# $ " %& $ ! '!" ! " !+++ + = !"" # $!" "=! + !"" #!" "=! +

10 4049/02/Sec4Prelim2025 [Turn over 9 (a) The variables x and y are connected by the equation where p and q are constants. Explain how a straight line graph can be drawn and state how the values of p and q could be obtained from the line. [4] !! "# $!#+=