advanced level core mathematics c34 - nh papers · advanced level candidates may use any calculator...
TRANSCRIPT
Centre Number Candidate Number
Write your name hereSurname Other names
Total Marks
Paper Reference
Pearson Edexcel InternationalAdvanced Level
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.
*P50803A0152*P50803A©2018 Pearson Education Ltd.
1/1/1/1/
You must have:
Mathematical Formulae and Statistical Tables (Blue)
WMA02/01Tuesday 19 June 2018 – AfternoonTime: 2 hours 30 minutes
Core Mathematics C34Advanced
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1. (i) Find
2 5 10
2
2
x xx
x x+ + >∫ d ,
(3)
(ii) Find
x x xcos 2 d∫(3)
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Question 1 continued
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2. A curve C has parametric equations
x t yt
t= − = − ≠3
25 4
60,
(a) Find the value of ddyx
at t = 3, giving your answer as a fraction in its simplest form.(3)
(b) Show that a cartesian equation of C can be expressed in the form
y ax bx
x k= ++
≠5
where a, b and k are integers to be found.(4)
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3. f (x) = 2x – 1 – 4 + 1.5x x
(a) Show that the equation f(x) = 0 can be written as
x = 1
3(8 – 2x)
(2)
The equation f(x) = 0 has a root , where = 1.6 to one decimal place.
(b) Starting with x0 = 1.6, use the iteration formula
xn + 1 = 1
3(8 – 2xn)
to calculate the values of x1, x2 and x3, giving your answers to 3 decimal places.(3)
(c) By choosing a suitable interval, prove that = 1.633 to 3 decimal places.(2)
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4. (a) Find the binomial expansion of
( ) ,1 14+ <−px px
in ascending powers of x, up to and including the term in x3, giving each coefficient as simply as possible in terms of the constant p.
(3)
f ( )( )
x xpx
px= ++
<3 4
11
4
where p is a positive constant.
In the series expansion of f (x), the coefficient of x2 is twice the coefficient of x.
(b) Find the value of p.(5)
(c) Hence find the coefficient of x3 in the series expansion of f (x), giving your answer as a simplified fraction.
(2)
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5. (i) The functions f and g are defined by
f : x → e2x 5,
g : x → ln(3x 1), ,1
3
(a) Find f–1 and state its domain.(3)
(b) Find fg(3), giving your answer in its simplest form.(2)
(ii) (a) Sketch the graph with equation
y x= −4 a
where a is a positive constant. State the coordinates of each point where the graph cuts or meets the coordinate axes.
(2)
Given that
4 9x a− =a
where a is a positive constant,
(b) find the possible values of
x x− +6 3a
giving your answers, in terms of a, in their simplest form.(5)
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Question 5 continued
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Question 5 continued
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(Total 12 marks)
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6. (a) Express 5 cos 2sinθ θ− in the form R cos( ), where R 0 and 0 2
π
State the value of R and give the value of to 4 significant figures.(3)
(b) Solve, for – ,
5 cos 2sin 0.5θ θ− =
giving your answers to 3 significant figures. [Solutions based entirely on graphical or numerical methods are not acceptable.]
(4)
f ( x) A ( 5 2 sin ) B
where A and B are constants.
Given that the range of f is
–15 f ( x) 33
(c) find the value of B and the possible values of A.(4)
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Question 6 continued
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7.
h cm
Figure 1
Figure 1 shows a hemispherical bowl.
Water is flowing into the bowl at a constant rate of 180 cm3s–1.
When the height of the water is h cm, the volume of water V cm3 is given by
V = 13
2(90 – h), 0 h 30
Find the rate of change of the height of the water, in cm s–1, when h = 15 Give your answer to 2 significant figures.
(5)
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Question 7 continued
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8. With respect to a fixed origin O, the lines l1 and l2 are given by the equations
l1 : r ⎛⎜⎜⎝
⎞⎟⎟⎠
1
2
⎛⎜⎜⎝
⎞⎟⎟⎠
123
, l2 : r ⎛⎜⎜⎝
⎞⎟⎟⎠
641
⎛⎜⎜⎝
⎞⎟⎟⎠
1 1
1
where and μ are scalar parameters.
(a) Show that l1 and l2 do not meet.(4)
The point P is on l1 where = 0, and the point Q is on l2 where μ = –1
(b) Find the acute angle between the line segment PQ and l1, giving your answer in degrees to 2 decimal places.
(5)
(c) Find the shortest distance from the point Q to the line l1, giving your answer to 3 significant figures.
(2)
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9.
O
y
x
R
y = f(x)
51
y = 43
Diagram not drawn to scale
Figure 2
(a) Find
1
2 12
( )xx
−∫ d
(2)
Figure 2 shows a sketch of the curve with equation y = f (x) where
f ( )( )
xx
x=−12
2 11 5� �
The finite region R, shown shaded in Figure 2, is bounded by the line with equation x = 1, the curve with equation y = f (x) and the line with equation y = 4
3.
The region R is rotated through 2 radians about the x-axis to form a solid of revolution.
(b) Find the exact value of the volume of the solid generated, giving your answer in its simplest form.
(6)
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Question 9 continued
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10. The curve C satisfies the equation
x e5 – 2y – y = 0 x 0, y
The point P with coordinates (2e–1, 2) lies on C.
The tangent to C at P cuts the x-axis at the point A and cuts the y-axis at the point B.
Given that O is the origin, find the exact area of triangle OAB, giving your answer in its simplest form.
(7)
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11.
O
y
x
R
63
C
Figure 3
(a) By writing sec as 1
cosθ, show that when x = 3 sec ,
d
d
xθ
= 3 sec tan (2)
Figure 3 shows a sketch of part of the curve C with equation
y xx
x= −29
3
The finite region R, shown shaded in Figure 3, is bounded by the curve C, the x-axis and the line with equation x = 6
(b) Use the substitution x = 3 sec to find the exact value of the area of R. [Solutions based entirely on graphical or numerical methods are not acceptable.]
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Question 11 continued
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Question 11 continued
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12. (a) Show that
cot x – tan x 2 cot 2x, x ≠ 90n°,(4)
(b) Hence, or otherwise, solve, for 0 180°
5 + cot( 15°) – tan( 15°) = 0
giving your answers to one decimal place. [Solutions based entirely on graphical or numerical methods are not acceptable.]
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Question 12 continued
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Question 12 continued
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13. (a) Express 1
4 2( )( )− −x x in partial fractions.
(2)
The mass, x grams, of a substance at time t seconds after a chemical reaction starts is modelled by the differential equation
ddxt
= k(4 – x)(2 – x), t 0, 0 x 2
where k is a constant.
Given that when t = 0, x = 0
(b) solve the differential equation and show that the solution can be written as
xkt
kt= −−4 4
1 2
2
2
e
e
(7)
Given that k = 0.1
(c) find the value of t when x = 1, giving your answer, in seconds, to 3 significant figures.(2)
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14. Given that
y xx
x= − >( )2
1
2
3
42
(a) show that
d
d
yx
Ax
x xx= +
−>
2
4 2
1
2
12
4
2
( )
where A is a constant to be found.(6)
O x
y
2
y = f (x)
Figure 4
Figure 4 shows a sketch of part of the curve with equation y = f (x) where
f ( )( )x x
xx= − >24 4
2
2
1
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(b) Use your answer to part (a) to find the range of f.(5)
(c) State a reason why f –1 does not exist.(1)
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*P50803A05252*
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TOTAL FOR PAPER: 125 MARKS
END
Q14
(Total 12 marks)