2025 TK Sec 4 Prelim A Maths P1 QP w ans

2 This document consists of 22 printed pages and 2 blank pages. ! !! CANDIDATE NAME ADDITIONAL MATHEMATICS 4049/01 Paper 1 Wednesday 20 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 4049/1/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 4049/1/Sec4Prelim2025 [Turn Over 1 (a) Find [2] (b) Hence find [2] !" #! $"!! " !" #$ +%& ( )!" !# ! ! " ! "+!

4 4049/1/Sec4Prelim2025 [Turn Over 2 In the diagram, the point P lies on the circle and TP is a tangent to the circle. TS is a straight line intersecting the circle at Q and S. Given that point R lies on TS and prove that the line PR bisects the angle SPQ. [4] !!" !#= P Q S R T

5 4049/1/Sec4Prelim2025 [Turn Over 3 Water is drained from a conical tank at a constant rate of 10π cm3/s. The volume of water, V cm3, when the depth of water is h cm, is given by and the surface area of the water, A cm2, is (a) Find in terms of h. [3] (b) Find the rate at which the surface area of the water is decreasing when the depth of water is 10 cm. [2] !" "#!" != !" #$!" != ! ! ! "

6 4049/1/Sec4Prelim2025 [Turn Over 4 The height, y m, above the ground of a pellet shot by a shooting device can be modelled by the equation where x m is the horizontal distance travelled by the pellet. (a) Express in the form where a, b and c are constants. [2] (b) Explain why the maximum height reached by the pellet is 62 metres. [2] ! !" #!$!!" "++=! ! "# !! !!" "! += + ( )! "!" #$%=! +

7 4049/1/Sec4Prelim2025 [Turn Over (c) A two-metre-tall pole is placed on the ground, at a horizontal distance of 10 metres from the shooting device. Explain if the pellet will hit the pole. [1]

8 4049/1/Sec4Prelim2025 [Turn Over 5 The volume of a prism is The prism has an equilateral triangular base of side cm. Without using a calculator, express the height of the prism in the form where a and b are integers. [6] ( ) !" ! " #$ %&' (+ ( )!"+ ( )! "#$%!"+

9 4049/1/Sec4Prelim2025 [Turn Over 6 (a) Write down and simplify the first three terms in the expansion, in ascending powers of x, of [3] (b) In the expansion of the sum of the coefficients of x and is −4. Find the value of the constant k. [4] ! "#$ !!"#$%&' ( ) ! "#" $ # !"! ! !"++ # $%&' !!

10 4049/1/Sec4Prelim2025 [Turn Over 7 (a) Find the value of k such that [2] (b) By letting P = A + B and Q = A – B, show that sin P + sin Q = 2sin cos. [2] ( ) ( )!"# !"# !"# $%! &!" !" # ! "++ != ! !"+!" #$%& ! !"!"# $%&'