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Centre Number Candidate Number
Write your name hereSurname Other names
Total Marks
Paper Reference
Pearson Edexcel InternationalAdvanced Level
P51403A©2018 Pearson Education Ltd.
1/1/1/1/1/1/
You must have:Mathematical Formulae and Statistical Tables
(Blue)
WMA02/01Tuesday 16 January 2018 – MorningTime: 2 hours 30
minutes
Core Mathematics C34Advanced
Candidates may use any calculator allowed by the regulations of
the Joint Council for Qualifications. 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.• If pencil is used for
diagrams/sketches/graphs it must be dark (HB or B). Coloured
pencils and highlighter pens must not be used.• Fill in the boxes
at the top of this page with your name, centre number and candidate
number.• Answer all questions and ensure that your answers to parts
of questions are clearly labelled.• Answer the questions in the
spaces provided – there may be more space than you need.• You
should show sufficient working to make your methods clear. Answers
without working may not gain full credit.• When a calculator is
used, the answer should be given to an appropriate degree of
accuracy.
Information• The total mark for this paper is 125. • The 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.• Check your answers if you have
time at the end.
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1. A curve C has equation
3x + xy = x + 2, y 1
The point P with coordinates (4, 11) lies on C.
Find the exact value of dd
yx
at the point P.
Give your answer in the form a + b a and b are rational
numbers.(6)
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Q1
(Total 6 marks)
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2. x) = 125 523−( )x |x| 25
x), in ascending powers of x, up to and including the term in
x2, giving the coefficient of x and the coefficient of x2 as
simplified fractions.
(4)
(b) Use your expansion to find an approximate value for 12023 ,
stating the value of x which
you have used and showing your working. Give your answer to 5
decimal places.(3)
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Q2
(Total 7 marks)
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3. x) = x2
4 + ln(2x), x 0
x) = 0 can be rewritten as
x x= −12
14
2e
(2)
x) = 0 has a root near 0.5
(b) Starting with x1 = 0.5 use the iterative formula
xnxn
+
−=114
12
2e
to calculate the values of x2 , x3 and x4, giving your answers
to 4 decimal places.(3)
x) = 0 correct to 3 decimal places.
(2)
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Q3
(Total 7 marks)
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4. y
x
y x)
5
P(2,
O1
Figure 1
Figure 1 shows a sketch of part of the graph with equation y x),
x ∈
The graph consists of two half lines that meet at the point P
(2, 3), the vertex of the graph.
The graph cuts the y-axis at the point (0, 1) and the x-axis at
the points ( 1, 0) and (5, 0).
Sketch, on separate diagrams, the graph of
(a) y = f (|x|),(3)
(b) y = 2f (x + 5).(3)
In each case, give the coordinates of the points where the graph
crosses or meets the coordinate axes.
Also give the coordinates of any vertices corresponding to the
point P.
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Q4
(Total 6 marks)
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5. (a) Express 9 416 9 2
( )+−
xx
in partial fractions.(3)
Given that
x) = 9 416 9 2
( )+−
xx
, x ∈ , 43
x 43
(b) express f d( )x x∫ in the form ln (g(x)), where g(x) is a
rational function.(4)
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Q5
(Total 7 marks)
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6. y
O
R
x3π
2π
Figure 2
The curve shown in Figure 2 has equation
2 = 3 tan x2
⎛⎝⎜
⎞⎠⎟
, 0 x y 0
The finite region R, shown shaded in Figure 2, is bounded by the
curve, the line with
equation x = 3π
the x-axis and the line with equation x = 2π
The region R is rotated through 360° about the x-axis to
generate a solid of revolution.
Show that the exact value of the volume of the solid generated
may be written as A ln32
⎛⎝⎜
⎞⎠⎟ , where A is a constant to be found.
(5)
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Question 6 continued
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Q6
(Total 5 marks)
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7. With respect to a fixed origin O, the lines l1 and l2 are
given by the equations
l1 : r = (13i + 15j – 8k) + (3i + 3j – 4k)
l2 : r = (7i – 6j + 14k) + (2i – 3j + 2k)
where and are scalar parameters.
(a) Show that l1 and l2 meet and find the position vector of
their point of intersection, B.(6)
(b) Find the acute angle between the lines l1 and l2(3)
The point A has position vector –5i –3j + 16k
(c) Show that A lies on l1(1)
The point C lies on the line l1 where AB = BC
(d) Find the position vector of C.(3)
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Question 7 continued
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Q7
(Total 13 marks)
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8. Given that
y = 8 tan (2x), 4π
x 4π
show thatdd
xy
= A
B y+ 2
where A and B are integers to be found.(4)
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Q8
(Total 4 marks)
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9. (a) Show thatcot
cotcos
2
22
1x
xx
+≡
(3)
(b) Hence solve, for 0 x 360°,
cotcot
cos cos2
218 2 2x
xx x
+= +
Give each solution in degrees to one decimal place. (Solutions
based entirely on graphical or numerical methods are not
acceptable.)
(6)
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Question 9 continued
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Question 9 continued
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Question 9 continued
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Q9
(Total 9 marks)
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10. It is given that
x) = e 2x x ∈
x) = x
x − 3 x 3
(a) Sketch the graph of y = x), showing the coordinates of any
points where the graph crosses the axes.
(2)
(b) Find the range of g(2)
(c) Find g 1(x), stating the domain of g 1(4)
(d) Using algebra, find the exact value of x for which fg(x) =
3(4)
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Question 10 continued
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Question 10 continued
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Q10
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11. y
xBO
C
A
Figure 3
The curve C shown in Figure 3 has parametric equations
x t, y t, 0 t 2
The curve C meets the x-axis at the origin and at the points A
and B, as shown in Figure 3.
(a) Write down the coordinates of A and B.(2)
(b) Find the values of t at which the curve passes through the
origin.(2)
(c) Find an expression for dd
yx
in terms of t, and hence find the gradient of the curve
when t = 6π
(4)
(d) Show that the cartesian equation for the curve C can be
written in the form
y2 = ax2 b x2)
where a and b are integers to be determined.(4)
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Question 11 continued
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Question 11 continued
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Question 11 continued
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Q11
(Total 12 marks)
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12. x x in the form R sin (x ), where R 0 and 0 < < 2π
Give the exact value of R and give the value of , in radians, to
3 significant figures.(3)
In a town in Norway, a student records the number of hours of
daylight every day for a year. He models the number of hours of
daylight, H, by the continuous function given by the formula
H 2365πt⎛ ⎞
⎜ ⎟⎝ ⎠ 2
365πt⎛ ⎞
⎜ ⎟⎝ ⎠, 0 t 365
where t is the number of days since he began recording.
(b) Using your answer to part (a), or otherwise, find the
maximum and minimum number of hours of daylight given by this
formula. Give your answers to 3 significant figures.
(3)
(c) Use the formula to find the values of t when H = 17, giving
your answers to the nearest integer.
(Solutions based entirely on graphical or numerical methods are
not acceptable.)(6)
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D
O N
OT
WRI
TE IN
TH
IS A
REA
D
O N
OT
WRI
TE IN
TH
IS A
REA
D
O N
OT
WRI
TE IN
TH
IS A
REA
Question 12 continued
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