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INTERNATIONAL AS
MATHEMATICS
(9660/MA01) Unit P1 Pure Mathematics
Tuesday 14 January 2020 07:00 GMT Time allowed: 1 hour 30 minutes
Materials For Examiner’s Use
• For this paper you must have the Oxford International AQA booklet
of formulae and statistical tables (enclosed). Question Mark
• You may use a graphics calculator. 1
2
Instructions
• Use black ink or black ball-point pen. Pencil should only be used for drawing. 3
• Fill in the boxes at the top of this page. 4
• Answer all questions.
5
• You must answer the questions in the spaces provided. Do not write outside
the box around each page or on blank pages. 6
• If you need extra space for your answer(s), use the lined pages at the end of 7
this book. Write the question number against your answer(s). 8
• Do all rough work in this book. Cross through any work you do not want to be
marked. 9
10
Information
TOTAL
• The marks for questions are shown in brackets.
• The maximum mark for this paper is 80.
Advice
• Unless stated otherwise, you may quote formulae, without proof,
from the booklet.
• Show all necessary working; otherwise marks may be lost.
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Answer all questions in the spaces provided. box
1 The equation
1
5
1 2
y = 4 x2 − x 2 ÷
4x
can be written in the form
y ax p − bx q
=
where a, b, p and q are positive constants.
1 (a) (i) Find the value of p.
Circle your answer.
[1 mark]
3 5
1 4
2 2
1 (a) (ii) Find the value of q.
Circle your answer.
[1 mark]
5
2 3 5
4
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dy box
1 (b) Find
dx
Fully simplify the coefficient of each term.
[2 marks]
dy
= 4
dx
Turn over for the next question
Turn over ►
*03*
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4
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Let f ( x ) = x + bx + c where b and c are real numbers.
2 box
2
It is given that:
• the line x = 5 is the line of symmetry of the curve with equation y = f ( x )
• the discriminant of f ( x ) is zero.
2 (a) Find the value of b and the value of c.
[2 marks]
b= c=
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2 (b) On the axes below, sketch the curve with equation y = f ( x ) . box
Show the coordinates of the vertex and the y-intercept on the graph.
[3 marks]
Turn over for the next question
Turn over ►
*05*
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6
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3 The lines L1 and L2 are shown in the diagram. box
L1 cuts the 𝑦𝑦-axis at the point B.
L2 cuts the 𝑥𝑥-axis at the point C.
3 (a) L1 has the equation
2 y − 3x =
6
3 (a) (i) Find the gradient of L1
[2 marks]
Answer
3 (a) (ii) Find the y-coordinate of B.
[1 mark]
Answer
*06*
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3 (b) AB and AC are two sides of a rectangle. box
L2 has the equation
22
y mx +
=
3
3 (b) (i) State the value of m.
[1 mark]
m=
3 (b) (ii) Show that the 𝑥𝑥-coordinate of C is 11
[1 mark]
3 (c) The point D is the mid-point of BC.
Find an equation of the line which passes through D and is parallel to L1
[3 marks]
8
Answer
Turn over ►
*07*
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8
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4
The expression ( 2 − x ) can be written in the form 16 − 32 x + ax 2 − bx3 + x 4
box
4 (a)
where a and b are positive integers.
Show that a = 24 and find the value of b .
[3 marks]
b=
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4 box
1
4 (b) Using the expansion in part (a), show that the value of 2 − can be
2
p−q 2
written in the form where p, q and r are integers.
r
[4 marks]
Turn over ►
*09*
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10
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outside the
5 The equation of a curve C is given by box
y=
( x − 1)( x − 14 ) , x≠0
x
5 (a) Find an equation of the tangent to C at the point where x = 2
[7 marks]
Answer
*10*
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5 (b) P is the point on C where x = − 4 box
Explain whether y is increasing or decreasing at P.
[2 marks]
Turn over for the next question
Turn over ►
*11*
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6 Grady sells boxes of chocolates. box
In the first month, Month 1, he sells 36 boxes.
Each month after Month 1, he sells 22 more boxes than he sold the previous month.
6 (a) (i) The number of boxes he sells each month forms a sequence.
State, with a reason, whether this is an arithmetic sequence or a geometric sequence.
[2 marks]
6 (a) (ii) Find an expression in terms of n for the number of boxes he sells in Month n.
[2 marks]
Answer
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6 (b) Grady makes £12 profit on each box of chocolates he sells. box
Over the first N months, he makes a total profit of exactly £90 000
By forming and solving a quadratic equation, find the value of N.
[5 marks]
N= 9
Turn over ►
*13*
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outside the
7 The gradient at any point ( x, y ) of a curve is given by box
dy
= 3 x 2 + ax − 36
dx
where a is a constant.
The curve passes through the points (1, – 7 ) and ( 3, – 5 )
7 (a) Find the equation of the curve.
[7 marks]
y=
*14*
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2
d y box
7 (b) (i) Find
dx 2
[1 mark]
d2 y
=
dx 2
7 (b) (ii) The curve has a minimum point P.
Find the x-coordinate of P.
[3 marks]
x= 11
Turn over ►
*15*
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8 A geometric series has first term a, common ratio r and nth term un box
The sum to infinity of the series is 425
The sum of the first two terms is 408
The series only contains positive terms.
1
8 (a) (i) Show that r =
5
[4 marks]
8 (a) (ii) Find the value of a.
[2 marks]
a =
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8 (b) Show that box
k +1 k −1
1
∑
n=k
un = p
q
where p and q are positive integers.
[4 marks]
10
Turn over ►
*17*
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9 The equation of the curve C 1 is box
2
1
y = x 3 − 4 − 11, x≥0
C 1 and the lines x = 1 and x = 8 are shown in the diagram below.
⌠ 1
2
9 (a) Find x 3 − 4 − 11 dx
⌡
[3 marks]
Answer
*18*
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9 (b) Find the area of the shaded region bounded by the curve C 1 , the lines x = 1 , x = 8 and box
the x-axis.
[3 marks]
Answer
0
9 (c) The translation maps the curve C 1 onto the curve C2
− 2
9 (c) (i) Using your answer to part (b), find the area of the region bounded by the curve C2 , the
lines x = 1 , x = 8 and the x-axis.
[2 marks]
Answer
9 (c) (ii) Find the equation of C2
[1 mark]
9
Answer
Turn over ►
*19*
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10 A curve has the equation box
y= 2 x 2 + 4 ( p + 3 ) x + 12 p + q + 12
where p and q are constants.
The curve crosses the x-axis at two distinct points.
10 (a) Show that
2 p2 − q + 6 > 0
[3 marks]
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10 (b) The curve passes through the point ( 0, 32 ) . box
Find the possible values of p.
[5 marks]
Answer 8
END OF QUESTIONS
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Question Additional page, if required.
number Write the question numbers in the left-hand margin.
*23*
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outside the
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ANSWER IN THE SPACES PROVIDED
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