Answer ALL questions. Write your answers in the spaces provided.

1. The line l passes through the points A (3, 1) and B (4, −2).

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Find an equation for l.
(3)
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(Total for Question 1 is 3 marks)

2
*S54257A0228*
6 Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials
Issue 1 – June 2017 © Pearson Education Limited 2017
 2. The curve C has equation

y = 2x2 − 12x + 16
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Find the gradient of the curve at the point P (5, 6).

(Solutions based entirely on graphical or numerical methods are not acceptable.)

(4)
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(Total for Question 2 is 4 marks)

3
*S54257A0328*
Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials 7
Issue 1 – June 2017 © Pearson Education Limited 2017
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3. Find, using algebra, all real solutions to the equation

(i) 16a 2 = 2 a

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(4)
(ii) b 4 + 7b 2 – 18 = 0
(4)
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4
*P58351A0444*
 Do not write
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2 box

Answer all questions.

Answer each question in the space provided for that question.

pffiffiffiffiffi pffiffiffiffiffi pffiffiffi

4 (a) Simplify 98  32 , giving your answer in the form k 2, where k is an integer.
[2 marks]
pffiffiffiffiffi pffiffiffiffiffi
98  32 pffiffiffi
(b) Hence, or otherwise, express pffiffiffi in the form p þ q 2 , giving the rational
2þ3 2
numbers p and q in their simplest form.
[4 marks]

QUESTION
PART Answer space for question 1
REFERENCE

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Jun18/MPC1
5. The curve C has equation

k2

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y= +1 x \, x  0
x
where k is a constant.
(a) Sketch C stating the equation of the horizontal asymptote.
(3)
The line l has equation y = –2x + 5
(b) Show that the x coordinate of any point of intersection of l with C is given by a
solution of the equation

2x 2 – 4x + k 2 = 0
(2)
(c) Hence find the exact values of k for which l is a tangent to C.
(3)

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14
*P58351A01444*

6.
A
r cm

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O θ

r cm
B

Figure 1

Figure 1 shows a sector AOB of a circle with centre O and radius r cm.

The angle AOB is θ radians.

The area of the sector AOB is 11 cm2

Given that the perimeter of the sector is 4 times the length of the arc AB, find the exact
value of r.
(4)
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6
*P58348A0644*
7. (a) Factorise completely x 3 + 10 x 2 + 25 x
(2)

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(b) Sketch the curve with equation

y = x 3 + 10x 2 + 25 x

showing the coordinates of the points at which the curve cuts or touches the x-axis.
(2)

The point with coordinates (−3, 0) lies on the curve with equation

y = (x + a) 3 + 10(x + a) 2 + 25(x + a)

where a is a constant.

(c) Find the two possible values of a.

(3)
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18
*S54257A01828*
22 Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials
Issue 1 – June 2017 © Pearson Education Limited 2017
8. C
B
3cm

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O
0.4
A O D
12cm
Figure 1

The shape ABCDOA, as shown in Figure 1, consists of a sector COD of a circle centre O
joined to a sector AOB of a different circle, also centre O.
Given that arc length CD = 3 cm, ∠COD = 0.4 radians and AOD is a straight line of
length 12 cm,
(a) find the length of OD,
(2)
(b) find the area of the shaded sector AOB.
(3)
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(Total for Question 2 is 8 marks)

4
*S54259A0432*
8 Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –
Issue 1 – April 2017 © Pearson Education Limited 2017
9.
y

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A

O B
D x

Figure 1

  Figure 1 shows a rectangle ABCD.
  The point A lies on the y-axis and the points B and D lie on the x-axis as shown in Figure 1.

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  Given that the straight line through the points A and B has equation 5y + 2x = 10
  (a)  show that the straight line through the points A and D has equation 2y − 5x = 4
(4)
  (b)  find the area of the rectangle ABCD.
(3)
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10
*S54260A01026*
64 Pearson Edexcel Level 3 Advanced GCE in Mathematics – Sample Assessment Materials –
Issue 1 – April 2017 © Pearson Education Limited 2017
 Leave
blank
10. Z
Diagram not
drawn to scale

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1.3 rad
X O 3 cm Y

Figure 1

Figure 1 shows a semicircle with centre O and radius 3cm. XY is the diameter of this
semicircle. The point Z is on the circumference such that angle XOZ = 1.3 radians.
The shaded region enclosed by the chord XZ, the arc ZY and the diameter XY is a template
for a badge.

Find, giving each answer to 3 significant figures,

(a) the length of the chord XZ,

(2)

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(b) the perimeter of the template XZYX,
(4)

(c) the area of the template.

(4)
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24
*P51404A02448*
 Leave
blank
11. The curve C has equation y = f(x), x  0, where

5x2 + 4

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f' (x) = −5
2 x

It is given that the point P(4, 14) lies on C.

(a) Find f (x), writing each term in a simplified form.

(6)

(b) Find the equation of the tangent to C at the point P, giving your answer in
the form y = mx + c, where m and c are constants.
(4)
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*P51404A02848*
Q12
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outside the
22 box

13 (a) Solve the inequality 5x 2 þ 6x < 63 .

[4 marks]

(b) The diagram below shows a garden ABCD in the shape of a trapezium.

4x + 3
B A

4x

C D
x+3

The sides BA and CD are parallel and angle ABC is a right angle. The sides AB, BC
and CD have lengths ð4x þ 3Þ metres, 4x metres and ðx þ 3Þ metres, respectively, as
indicated on the diagram.

The area of the garden must be less than 126 square metres.

(i) Show that 5x 2 þ 6x < 63 .

[1 mark]

(ii) Find an expression for the perimeter of the garden, giving your answer in the form
ðax þ bÞ metres.
[1 mark]

(iii) In addition to the constraint on the area, the perimeter of the garden must be at least
30 metres. Find the possible values of x.
[3 marks]

QUESTION
PART
REFERENCE
Answer space for question 13
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Jun18/MPC1