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Monday 13 May 2024
Afternoon (Time: 1 hour 40 minutes) 8FM0/01Paper
reference
Total Marks
Further Mathematics
Advanced Subsidiary
PAPER 1: Core Pure Mathematics
You must have:
Mathematical Formulae and Statistical Tables (Green), calculator

MERGED QUESTION PAPER AND MARK SCHEMEPearson Edexcel GCEIn Further Mathematics (8FM0/01)Paper 01 (MAY 2024)

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1. The cubic equation
2 3 5 7 03 2
x x x
has roots α, β and γ.
Without solving the equation, determine the exact value of
(i) α2 + β2 + γ2
(3)
(ii) 3 3 3

α β γ
(3)
(iii) (5 – α)(5 – β)(5 – γ) (3)
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Monday 13 May 2024

Afternoon (Time: 1 hour 40 minutes) Paperreference 8FM0/

Total Marks

Further Mathematics

Advanced Subsidiary

PAPER 1: Core Pure Mathematics

You must have: Mathematical Formulae and Statistical Tables (Green), calculator

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The cubic equation

2 x^3^ 3 x^2 5 x 7 0 has roots α , β and γ. Without solving the equation, determine the exact value of (i) α^2 + β^2 + γ^2 (3)

(ii) (^) α^3 (^3) β^ ^3 γ (3)

(iii) (5 – α )(5 – β )(5 – γ )

(3)

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(Total for Question 1 is 9 marks)

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(^) With respect to the right-hand rule, a rotation through θ ° anticlockwise about the z -axis is represented by the matrix cos sin 0 sin cos 0 0 0 1

θ θ θ θ

Given that the matrix M, where

M =

represents a rotation through α ° anticlockwise about the z -axis with respect to the right-hand rule, (a) determine the value of α. (1) (b) Hence determine the smallest possible positive integer value of k for which M k = I (2) The 3 × 3 matrix N represents a reflection in the plane with equation y = 0 (c) Write down the matrix N. (1) The point A has coordinates (−2, 4, 3) The point B is the image of the point A under the transformation represented by matrix M followed by the transformation represented by matrix N.

(d) Show that the coordinates of B are 2 3 2, 3 1 3,

(2)

Given that O is the origin, (e) show that, to 3 significant figures, the size of angle AOB is 66.9° (2) (f) Hence determine the area of triangle AOB , giving your answer to 3 significant figures. (2)

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(Total for Question 2 is 10 marks)

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(a) Use the standard results for summations to show that, for all positive integers n ,

r r n n n an b r

n 2 1

^1 ^121 ^1 ^2

where a and b are integers to be determined. (4) (b) Hence show that, for all positive integers k ,

r r k k Ak Bk C r k

k 2 1

3 1 1 2

^ ^3 ^3 ^1 ^ ^ ^

where A , B and C are integers to be determined. (3) (c) Hence, using algebra and making your method clear, determine the value of k for which

25 2 1 192 3 3 1 1

3 r r k k r k

^

(3)

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(Total for Question 3 is 10 marks)

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A B

^ ^

k k

where k is a constant. (a) Determine the value of the constant c for which

AB = (3 k + c ) I

(2)

(b) Hence determine the value of k for which A–1^ does not exist. (2) Given that A–1^ does exist, (c) write down A–1^ in terms of k. (1) (d) Use the answer to part (c) to solve the simultaneous equations

x y z x ky z x y z

giving the values of x , y and z in simplest form in terms of k. (3)

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(Total for Question 4 is 8 marks)

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5. Given that on an Argand diagram the locus of points defined by | z + 5 – 12i | = 10 is

a circle, (a) write down, (i) the coordinates of the centre of this circle, (ii) the radius of this circle. (2) (b) Show, by shading on an Argand diagram, the set of points defined by

| z + 5 – 12i |  10

(1)

(c) For the set of points defined in part (b), determine the maximum value of | z |

(3)

The set of points A is defined by

A = { z : 0  arg ( z + 5 – 20i)  π } ∩ { z : | z + 5 – 12i |  10}

(d) Determine the area of the region defined by A , giving your answer to 3 significant figures. (4)

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(Total for Question 5 is 10 marks)

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The drainage system for a sports field consists of underground pipes.

This situation is modelled with respect to a fixed origin O. According to the model,

the surface of the sports field is a plane with equation z = 0
the pipes are straight lines
one of the pipes, P 1 , passes through the points A (3, 4, –2) and B (–2, –8, –3)
a different pipe, P 2 , has equation x^ 2 ^1 y^ 4 3 z 21
the units are metres (a) Determine a vector equation of the line representing the pipe P 1 (2) (b) Determine the coordinates of the point at which the pipe P 1 meets the surface of the playing field, according to the model. (2) Determine, according to the model, (c) the acute angle between pipes P 1 and P 2 , giving your answer in degrees to 3 significant figures, (3) (d) the shortest distance between pipes P 1 and P 2 (5)

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(Total for Question 6 is 12 marks)

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MERGED QUESTION PAPER&MARK SCHEME Pearson Edexcel GCEIn Further Mathematics (8FM0/01) 2024, Exams of Mathematics

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Monday 13 May 2024

Advanced Subsidiary

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(iii) (5 – α )(5 – β )(5 – γ )

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