Please check the examination details below before entering your candidate information

Candidate surname Other names

Pearson Edexcel Centre Number Candidate Number

International
Advanced Level

Wednesday 15 January 2020

Morning (Time: 1 hour 30 minutes) Paper Reference WMA12/01

Mathematics
International Advanced Subsidiary/Advanced Level
Pure Mathematics P2

You must have: Total Marks

Mathematical Formulae and Statistical Tables (Lilac), calculator

Candidates may use any calculator permitted by Pearson regulations.

Calculators must not have the facility for symbolic algebra manipulation,
differentiation and integration, or have retrievable mathematical
formulae stored in them.
Instructions
•
•
Use black ink or ball-point pen.

• Fill
If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
in the boxes at the top of this page with your name, centre number and

• clearly
candidate number.
Answer all questions and ensure that your answers to parts of questions are

• – there may
labelled.
Answer the questions in the spaces provided

• without working may

be more space than you need.
You should show sufficient working to make your methods clear. Answers

• stated.
not gain full credit.
Inexact answers should be given to three significant figures unless otherwise

Information
•
•
A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.

• The
There are 10 questions in this question paper. The total mark for this paper is 75.
marks for each question are shown in brackets
– use this as a guide as to how much time to spend on each question.
Advice
•
•
Read each question carefully before you start to answer it.

•
Try to answer every question.

• Ifandyouanychange
Check your answers if you have time at the end.
your mind about an answer, cross it out and put your new answer
working underneath.
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1. The table below shows corresponding values of x and y for y = log2 (2x)

The values of y are given to 2 decimal places as appropriate.

x 2 5 8 11 14
y 2 3.32 4 4.46 4.81

Using the trapezium rule with all the values of y in the given table,

∫
14

(a) obtain an estimate for log 2 (2 x) dx, giving your answer to one decimal place.
2 (3)

Using your answer to part (a) and making your method clear, estimate

log 2 (4 x 2 )
∫
14
(b) (i) dx
2 5

∫
14
 2
(ii) log 2   dx
 x
2 (4)
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(Total 7 marks)

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2. One of the terms in the binomial expansion of (3 + ax)6, where a is a constant, is 540x4

(a) Find the possible values of a.

(4)

(b) Hence find the term independent of x in the expansion of

1 1
 + 6  (3 + ax)
6

81 x (3)
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(Total 7 marks)

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3.. f(x) = 6x3 + 17x2 + 4x – 12

(a) Use the factor theorem to show that (2x + 3) is a factor of f(x).
(2)

(b) Hence, using algebra, write f(x) as a product of three linear factors.
(4)
π
(c) Solve, for < θ < π , the equation
2
. 6 tan3 θ + 17 tan2 θ + 4 tan θ – 12 = 0

giving your answers to 3 significant figures.

(2)
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(Total 8 marks)

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4.
y

17

O x

Figure 1

Figure 1 shows a sketch of the curve with equation

. y = 2x2 + 7 x0

The finite region R, shown shaded in Figure 1, is bounded by the curve, the y - axis and the
line with equation y = 17

Find the exact area of R.

(6)
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(Total 6 marks)

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5. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.

A colony of bees is being studied.

The number of bees in the colony at the start of the study was 30 000

Three years after the start of the study, the number of bees in the colony is 34 000

A model predicts that the number of bees in the colony will increase by p % each
year, so that the number of bees in the colony at the end of each year of study forms a
geometric sequence.

Assuming the model,

(a) find the value of p, giving your answer to 2 decimal places.

(3)

According to the model, at the end of N years of study the number of bees in the colony
exceeds 75 000

(b) Find, showing all steps in your working, the smallest integer value of N.
(5)
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(Total 8 marks)

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6. The circle C has equation
x2 + y2 + 6x – 4y – 14 = 0
(a) Find

(i) the coordinates of the centre of C,

(ii) the exact radius of C.

(3)

The line with equation y = k, where k is a constant, is a tangent to C.

(b) Find the possible values of k.

(2)

The line with equation y = p, where p is a negative constant, is a chord of C.

Given that the length of this chord is 4 units,

(c) find the value of p.

(3)
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(Total 8 marks)

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7. (a) Show that the equation
8 tan θ = 3 cos θ
may be rewritten in the form
3 sin2 θ + 8 sin θ – 3 = 0
(3)
(b) Hence solve, for 0  x  90°, the equation
8 tan 2x = 3 cos 2x
giving your answers to 2 decimal places.
(4)
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(Total 7 marks)

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8. (i) An arithmetic series has first term a and common difference d.

Prove that the sum to n terms of this series is

n
2
{2a + ( n − 1) d } (3)

(ii) A sequence u1, u2, u3,… is given by

un = 5n + 3(–1)n

Find the value of

(a)
u5
(1)
59

∑ un
(b)
n=1 (3)
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(Total 7 marks)

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9. (a) Sketch the curve with equation
x
y=3×4

showing the coordinates of any points of intersection with the coordinate axes.
(2)
1–x x
The curve with equation y = 6 meets the curve with equation y = 3 × 4 at the
point P.
log10 2
(b) Show that the x coordinate of P is
log10 24 (5)

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(Total 7 marks)

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10. A curve C has equation
k
y = 4 x3 − 9 x + x>0
x
where k is a constant.
1
The point P with x coordinate lies on C.
2
Given that P is a stationary point of C,
3
(a) show that k = −
2 (4)

(b) Determine the nature of the stationary point at P, justifying your answer.
(2)
The curve C has a second stationary point.

(c) Using algebra, find the x coordinate of this second stationary point.
(4)
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(Total 10 marks)
TOTAL FOR PAPER IS 75 MARKS
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