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NGEE ANN SECONDARY SCHOOL

O
PRELIMINARY EXAMINATION
ADDITIONAL MATHEMATICS 4049/01
Paper 1 28 August 2023
2 hours 15 minutes
Candidates answer on the Question Paper.
No Additional Materials are required.

Instructions to Candidates
Write your name, register number and class 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.
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.

For Examiner’s Use

Total
/90

Checked by student: _____________________ Date: ______________

This document consists of 23 printed pages and 1 blank page.

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Mathematical Formulae

1. ALGEBRA

Quadratic Equation
For the equation ax 2 + bx + c = 0
− b ± b 2 − 4ac
x=
2a

Binomial Expansion
n  n −1 n n
(a + b )n = a n + 
a
 b +   a n − 2 b 2 + ......... +   a n − r b r + ......... + b n ,
1  2 r
n n! n (n − 1).........(n − r + 1)
where n is a positive integer and   = =
 r  r ! (n − r )! r!

2. TRIGONOMETRY

Identities

sin 2 A + cos 2 A = 1
2 2
sec A = 1 + tan A
cosec 2 A = 1 + cot 2 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 ± tan B
tan ( A ± B ) =
1  tan A tan B
sin 2 A = 2 sin A cos A
cos 2 A = cos 2 A − sin 2 A = 2 cos 2 A − 1 = 1 − 2sin 2 A
2 tan A
tan 2 A =
1 − tan 2 A

Formulae for ΔABC

a b c
= =
sin A sin B sin C
a 2 = b 2 + c 2 − 2bc cos A
1
∆ = bc sin A
2

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Express −2 x 2 − 12 x − 14 in the form a ( x − h ) + k .

2
1 (a) [2]

(b) Using your answer to part (a), explain why the maximum value of
−2 x 2 − 12 x − 14 is k. [2]

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12
2 Solve the equation=x 3 3+ , where x ≠ 0, giving your answer in its simplest surd
x
form. [4]

300
3 The spread of bird flu in a certain duck farm is given by B ( t ) = , where t is the number
1 + e5−t
of days since the flu first appeared, and B ( t ) is the total number of ducks which have caught
the flu to date.

(a) Estimate the initial number of ducks infected with the flu.
Give your answer correct to the nearest integer. [1]

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(b) If the day that the flu first appeared is Day 0, find out on which day would the
number of infected ducks first reach 100. [2]

(c) Explain clearly why the number of infected ducks will never exceed 300. [2]

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4 A calculator must not be used in this question.

(a) Show that tan15° = 2 − 3. [3]

(b) Using your answer in part (a), find an expression for sec 2 15° in the form
a + b 3 where a and b are integers. [2]

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48 x 3 + 86 x 2 − 2 x − 32
5 (a) Express as the sum of a polynomial and a proper
6 x 2 + 13 x + 6
fraction. [2]

48 x 3 + 86 x 2 − 2 x − 32
(b) Hence, express as the sum of a polynomial and partial
6 x 2 + 13 x + 6
fractions. [4]

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6 (a) Find the range of values of ℎ for which the equation x 2 − (h + 1) x + h + 9 =0

has no real roots. [4]

(b) Explain why the equation 5 x 2 − kx + 2k 2 + 8 =0 has no real roots for all real
values of 𝑘𝑘. [2]

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7 =
The equation of a curve is y 3cos 2 x − 1.
(a) State the maximum and minimum values of y. [1]

(b) =
Sketch the graph of y 3cos 2 x − 1 for 0 ≤ x ≤ 2π . [3]

x
O

(c) By drawing a straight line on the same diagram in (b), determine the number
2
of solutions for the equation 3cos 2 x =− x + 2 for 0 ≤ x ≤ 2π . [2]
π

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In the diagram, P, Q, R and S lie on a circle such that QP is parallel to ST.

Lines NT and ST are tangents to the circle at P and S respectively. NQR is a

straight line and angle NPQ = x°.

(a) x° [4]
Prove that angle QRS = 90° + .
2

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(b) It is given further that QR// PS.

Show that triangle PRS is isosceles. [2]

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4 9
9 Given that ∫=
f ( x)dx ∫=
f ( x)dx 12, find
0 4
9
(a) ∫0
f ( x)dx, [1]

4 4
(b) ∫0
f ( x)dx + ∫ f ( x)dx,
9
[2]

9
(c) find the value of the constant m for which ∫ [ f ( x) − mx ] dx =
4
0. [3]

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10 4
A point P (2, 4) lies on the curve y = .
( x − 3) 2
(a) Find the equation of the normal at point P. [4]

(b) The tangent at another point Q on the curve is parallel to the normal at point P.
1
Show that the coordinates of Q is (7, ). [3]
4

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11 (a) A thin circular metal disk changes size (but not shape) when cooled. The
disk is decreasing at a rate of 12π mm2/s. When the disk has a radius of
200 mm, find the rate of change of the radius of the disk at this instant. [3]

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(b) A 3-dimensional printer prints a chocolate in the shape of a cone. During the
printing process, the volume of the cone is increasing at a constant rate of
3π cm3/s. The height of the cone is always twice of the radius. Find the rate of
increase of the radius when the volume of the chocolate cone is 9π cm3. [4]

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12

The diagram shows a trapezium with vertices A(−4, 0), B(4,15), C and D.
The diagonal BD of the trapezium intersects the y-axis at E (0,13).
Given that AB = 2DC and AD = 13 units, find the perimeter of ABCD. [8]

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Continuation of working space for Question 12

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13 A small forest has been declining in its tree population since 1980. To assess this decline, a
census of the tree population of the forest was conducted on January 1st at intervals of ten
years from 1980 to 2020. The table below shows the data from the census.
Year 1980 1990 2000 2010 2020
x (decade) 0 1 2 3 4
y (number of trees) 186624 139968 104976 78732 59049

It is believed this decline follows a y = e p q x curve, where p and q are constants.

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(a) Draw a suitable straight line graph to show that the model is valid for the years
1980 to 2020. [3]

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(b) Estimate the number of trees in the forest on 1st January 2040. [2]

(c) Use your graph to estimate the value of q. [2]

(d) Give a reason why this model might not be accurate in later decades. [1]

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14 The velocity, v m/s, of a particle travelling in a straight line at time t seconds after
leaving a fixed point O, is given by v = 2t 2 + ( 2 − 3k ) t + 4k − 5, where k is a constant.
13
(a) Given that the minimum velocity occurs at t = , show that the value of k = 5. [3]
4

(b) Find the time(s) the particle comes to instantaneous rest. [2]

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(c) Find the distance travelled in the first 7 seconds after passing through O. [4]

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(d) Given that R is the point when the particle has zero acceleration, and P is the
point when the particle first comes to rest, determine, with full working whether
R is nearer to O or to P. [3]

~End of Paper~

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