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INTERNATIONAL AS
MATHEMATICS
(9660/MA01) Unit P1 Pure Mathematics

Tuesday 3 January 2023 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 graphical 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 graph of a function with equation y = f ( x ) is shown in Figure 1

Figure 1

1 (a) (i) State the equation of the graph of the function shown in Figure 2

Circle your answer.

[1 mark]
Figure 2

1  1
y = f  x y = f (2x) y = f ( x) y = 2 f ( x)
2  2

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1 (a) (ii) State the equation of the graph of the function shown in Figure 3 box

Circle your answer.

[1 mark]
Figure 3

y = f ( x − 4) − 1 y = f ( x − 4) + 1 y = f ( x + 4) − 1 y = f ( x + 4) + 1
.

1 (b) The graph of the function with equation y = f ( x ) is shown again below.
By drawing a suitable straight line find the roots of the equation f ( x )= x − 3
[2 marks]

x= 4

Turn over ►

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2 The points A, B, C and D, and the lines l1, l2 and l3 are shown in the diagram. box

Not drawn to
scale

The lines l1 and l3 intersect at A (15, 3 )

3
2 (a) The line l1 has gradient
5

Show that l1 has the equation 3 x − 5 y − 30 = 0

[1 mark]

2 (b) l1 intersects the x-axis at B and the y-axis at C

l2 passes through the mid-point of the line segment BC

l1 and l2 are perpendicular.

Find the equation of l2 giving your answer in the form ax + by + c = 0

where a, b and c are integers.
[5 marks]

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Answer

2 (c) l3 has the equation x + 4 y − 27 = 0

l2 and l3 intersect at D

Find the coordinates of D

[1 mark]

Answer

2 (d) Find the length of the line segment AD

Give your answer in the form n p where p is a prime number.

[2 marks]

Answer 9

Turn over ►

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3 The diagram shows the plan of a garden. box

The angle at each corner of the garden is a right-angle.

The lengths of the sides in metres are

PQ = x + 3 , QR = 2 x − 10 , RS = x − 1 and PU = x − 4

3 (a) The perimeter of the garden is greater than 31 metres.

Show that x > 7.5

[1 mark]

3 (b) The area of the garden is less than 58 m2

Show that x 2 − 4 x − 32 < 0

[3 marks]

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3 (c) Solve the inequality x 2 − 4 x − 32 < 0

Show clearly each step of your working.

[2 marks]

Answer

3 (d) The length of the side ST is y metres.

Using your answers to parts (a) and (c) find the possible values of y
[2 marks]

Answer 8

Turn over ►

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4 The polynomial p ( x ) is given by box

p ( x ) = x 2 ( 2 x − 5 ) − 48

4 (a) Use the Factor Theorem to show that ( x − 4) is a factor of p ( x )

[2 marks]

4 (b) Show that p ( x ) can be written in the form

(
p ( x ) = ( x − 4 ) ax 2 + bx + c )
where a, b and c are integers to be found.
[2 marks]

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4 (c) Show that p ( x ) = 0 has exactly one real root and state its value. box

[3 marks]

Answer 7

Turn over for the next question

Turn over ►

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5 The nth term of the sequence A is un and the sequence is defined by box

(
un +1 = un + 8 1 + 3
n
)
The second, third and fourth terms of this sequence are

u2 = 61 u3 = 141 and u4 = 365

5 (a) (i) Find the first term u1 of sequence A

[1 mark]

Answer

5 (a) (ii) Find the fifth term u5 of sequence A

[1 mark]

Answer

5 (b) The sequence A can be found using the formula

nth term of nth term of nth term of

sequence A
= sequence B
+ sequence C

where sequence B and sequence C are two different sequences.

5 (b) (i) Sequence B is a geometric sequence with first term a = 12 and common ratio r = 3

Find the first five terms of sequence B

[1 mark]

Answer

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5 (b) (ii) Hence find the first five terms of sequence C box
[2 marks]

Answer

5 (c) (i) Sequence C is an arithmetic sequence.

Using your answer to part (b)(ii) write down the common difference for sequence C
[1 mark]

Answer

5 (c) (ii) Find an expression in terms of n for the nth term of sequence C
[1 mark]

Answer 7

Turn over ►

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6 The curve C has the equation box

y = 3 x3 + 14 x 2 + 17 x + 11

The point P ( − 2, 9 ) lies on C

The line l is the normal to C at the point P

dy
6 (a) (i) Find
dx
[2 marks]

Answer

1 29
6 (a) (ii) Show that the equation of l is y = x+
3 3
[3 marks]

6 (b) The line l intersects C at three distinct points.

Show that the x-coordinates of these points of intersection satisfy the equation

9 x3 + 42 x 2 + 50 x + 4 = 0
[2 marks]

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6 (c) The equation 9 x3 + 42 x 2 + 50 x + 4 = 0 can be written in the form

( x + 2 ) ( 9 x 2 + 24 x + 2 ) = 0
2
6 (c) (i) Express 9 x 2 + 24 x + 2 in the form a ( x + b ) + c where a, b and c are constants.
[3 marks]

Answer

6 (c) (ii) The points of intersection of l and C are P ( − 2, 9 ) , Q and R

Using your answer to part (c)(i) find the exact x-coordinates of Q and R

Show clearly each step of your working.

[3 marks]

Answer 13

Turn over ►

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7 A curve has equation y = f ( x ) where x > 0 box

It is given that
3 3
dy
= 2 x 2 − 9 x 4 − 56
dx

d2 y
7 (a) Find
dx 2
[2 marks]

Answer

3
dy
7 (b) By substituting t = x 4 into the given expression for show that
dx

dy
= ( at + b )( t − c )
dx

where a, b and c are positive integers.

[2 marks]

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7 (c) The curve has one stationary point for x > 0 box

7 (c) (i) By writing x as a power of t and then using part (b) find the x-coordinate of
this stationary point.
[3 marks]

Answer

7 (c) (ii) Using part (a) show that this stationary point is a minimum.
[1 mark]

7 (d) State the values of x for which f is a decreasing function.

[1 mark]

Answer 9

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8 (a) Show that for any positive real number a box

(2 + )( )
3 − a 2+ 3 + a =7+b 3 −a

where b is a constant to be found.

[2 marks]

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8 (b) Hence show that box

12
2+ 3 − 7

can be written in the form p + q r + s where p, q, r and s are integers and q > 1
[3 marks]

Turn over for the next question

Turn over ►

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(3 − 2 x )
3
9 (a) The expression can be written in the form box

27 − p x + qx − 8 x x

where p and q are positive integers.

Show that p = 54 and find the value of q

[3 marks]

q=

9 (b) It is given that x > 0

⌠  ( 3 − 2 x )3 

9 (b) (i) Find  + 12  dx
 x 
⌡ 
[4 marks]

Answer

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( )
3
⌠ 
 3−2 x


box

9 (b) (ii) Hence find the value of   + 12  dx

  x 
⌡4  
[2 marks]

Answer

(3 − 2 x )
3

9 (c) A curve with equation y= + 6 is drawn below.

2 x

The points A ( 4, 5.75 ) and B ( 9, 1.5 ) lie on the curve.

Using your answer to part (b)(ii) find the area of the shaded region bounded by the curve
and the line segment AB
[2 marks]

Answer 11

Turn over ►

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10 A finite arithmetic sequence has k terms and common difference d box

The first term is a = 12

The sum of the first 10 terms is 480

The sum of the last 10 terms is 3360

Show that d = 8 and hence find the sum of all of the terms in the sequence.
[7 marks]

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Answer 7

END OF QUESTIONS

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ANSWER IN THE SPACES PROVIDED

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Question Additional page, if required.
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