Pearson Edexcel Level 3
GCE Mathematics
Advanced Subsidiary
Paper 1: Pure Mathematics
Time: 2 hours Paper Reference(s)
8MA0/01
You must have:
Mathematical Formulae and Statistical Tables, calculator
Candidates may use any calculator permitted by Pearson regulations. Calculators must not
have the facility for algebraic 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).
• 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.
• Inexact answers should be given to three significant figures unless otherwise stated.
Information
• A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.
• There are 16 questions in this paper. The total mark is 100.
• 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.
• If you change your mind about an answer, cross it out and put your new answer and
any working underneath.
Answer ALL questions.
1. A curve has equation
y = 2x3 – 2x2 – 2x + 8.
dy
(a) Find .
dx
(2)
(b) Hence find the range of values of x for which y is increasing.
Write your answer in set notation.
(4)
(Total for Question 1 is 6 marks)
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2. The quadrilateral OABC has OA = 4i + 2j, OB = 6i – 3j and OC = 8i – 20j.
(a) Find AB .
(2)
(b) Show that quadrilateral OABC is a trapezium.
(2)
(Total for Question 2 is 4 marks)
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3. A tank, which contained water, started to leak from a hole in its base.
The volume of water in the tank 24 minutes after the leak started was 4 m3.
The volume of water in the tank 60 minutes after the leak started was 2.8 m3.
The volume of water, V m3, in the tank t minutes after the leak started, can be described by a
linear model between V and t.
(a) Find an equation linking V with t.
(4)
Use this model to find
(b) (i) the initial volume of water in the tank,
(ii) the time taken for the tank to empty.
(3)
(c) Suggest a reason why this linear model may not be suitable.
(1)
(Total for Question 3 is 8 marks)
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2
4.
Figure 1
Figure 1 shows a sketch of the curve with equation y = g(x).
The curve has a single turning point, a minimum, at the point M(4, –1.5).
The curve crosses the x-axis at two points, P(2, 0) and Q(7, 0).
The curve crosses the y-axis at a single point R(0, 5).
(a) State the coordinates of the turning point on the curve with equation y = 2g(x).
(1)
(b) State the largest root of the equation g(x + 1) = 0.
(1)
(c) State the range of values of x for which g′(x) 0.
(1)
Given that the equation g(x) + k = 0, where k is a constant, has no real roots,
(d) state the range of possible values for k.
(1)
(Total for Question 4 is 4 marks)
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3
5. f(x) = x3 + 3x2 – 4x – 12.
(a) Using the factor theorem, explain why f(x) is divisible by (x + 3).
(2)
(b) Hence fully factorise f(x).
(3)
x 3 + 3x 2 − 4 x − 12 B
(c) Show that can be written in the form A + , where A and B are
x + 5x + 6 x
3 2
x
integers to be found.
(3)
(Total for Question 5 is 8 marks)
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6. (i) Use a counterexample to show that the following statement is false.
“n2 – n – 1 is a prime number, for 3 n 10.”
(2)
(ii) Prove that the following statement is always true.
“The difference between the cube and the square of an odd number is even.”
For example, 53 – 52 = 100 is even.
(4)
(Total for Question 6 is 6 marks)
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2
(a) Expand 1 +
3
7. , simplifying each term.
x
(2)
(b) Use the binomial expansion to find, in ascending powers of x, the first four terms in the
expansion of
6
3
1 + x ,
4
simplifying each term.
(4)
(c) Hence find the coefficient of x in the expansion of
2 6
3 3
1 + 1 + x .
x 4
(2)
(Total for Question 7 is 8 marks)
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4
8. A circle C has centre (2, 5). Given that the point P(–2, 3) lies on C.
(a) find an equation for C.
(3)
The line l is the tangent to C at the point P. The point Q(2, k) lies on l.
(b) Find the value of k.
(5)
(Total for Question 10 is 8 marks)
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9. (i) Solve, for –90° θ < 270°, the equation,
sin (2θ + 10°) = –0.6,
giving your answers to one decimal place.
(5)
(ii) (a) A student’s attempt at the question
“Solve, for –90° < x < 90°, the equation 7 tan x = 8 sin x”
is set out below.
7 tan x = 8 sin x
sin x
7× = 8 sin x
cos x
7 sin x = 8 sin x cos x
7 = 8 cos x
7
cos x =
8
x = 29.0° (to 3 sf )
Identify two mistakes made by this student, giving a brief explanation of each mistake.
(2)
(b) Find the smallest positive solution to the equation
7 tan (4α + 199°) = 8 sin (4α + 199°).
(2)
(Total for Question 11 is 9 marks)
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5
10.
Figure 3
Figure 3 shows a sketch of the curve C with equation y = 3x – 2x , x 0 and the line l with
equation y = 8x – 16.
The line cuts the curve at point A as shown in Figure 3.
(a) Using algebra, find the x-coordinate of point A.
(5)
Figure 4
The region R is shown unshaded in Figure 4.
(b) Identify the inequalities that define R.
(3)
(Total for Question 12 is 8 marks)
6
11. Prove, from first principles, that the derivative of 3x2 is 6x.
(4)
(Total for Question 11 is 4 marks)
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12. C Not to scale
70 60
A B
30 m
Figure 1
A triangular lawn is modelled by the triangle ABC, shown in Figure 1. The length AB is to
be 30 m long.
Given that angle BAC = 70° and angle ABC = 60°,
(a) calculate the area of the lawn to 3 significant figures.
(4)
(b) Why is your answer unlikely to be accurate to the nearest square metre?
(1)
(Total for Question 12 is 5 marks)
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13. Solve, for 360 ≤ x < 540,
12 sin2 x + 7 cos x − 13 = 0.
Give your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
(5)
(Total for Question 13 is 5 marks)
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14. The equation kx2 + 4kx + 3 = 0, where k is a constant, has no real roots.
3
Prove that 0≤k
4
(Total for Question 14 is 4 marks)
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7
15. (a) Factorise completely x3 + 10x2 + 25x
(2)
(b) Sketch the curve with equation
y = x3 + 10x2 + 25x
showing the coordinates of the points at which the curve cuts or touches the x-axis.
(2)
(Total for Question 15 is 4 marks)
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16.
2x m
A E
ym
B D
Figure 4
Figure 4 shows the plan view of the design for a swimming pool.
The shape of this pool ABCDEA consists of a rectangular section ABDE joined to
semi-circular section BCD as shown in Figure 4.
Given that AE = 2x metres, ED = y metres and the area of the pool is 250 m2,
(a) show that the perimeter, P metres, of the pool is given by
250 x
P = 2x + +
x 2
(4)
500
(b) Explain why 0 < x
(2)
(c) Find the minimum perimeter of the pool, giving your answer to 3 significant figures.
(4)
(Total for Question 16 is 10 marks)
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