Oxford Cambridge and
RSA Examinations GCE
Further Mathematics
AY531/01: Pure Core
AS Level
Question paper and
marking scheme
(merged)
, Oxford Cambridge and RSA
Monday 15 May 2023 – Afternoon
AS Level Further Mathematics A
Y531/01 Pure Core
Time allowed: 1 hour 15 minutes
* 9 9 7 4 1 0 8 4 1 6 *
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for AS Level Further
QP
Mathematics A
• a scientific or graphical calculator
INSTRUCTIONS
• Use black ink. You can use an HB pencil, but only for graphs and diagrams.
• Write your answer to each question in the space provided in the Printed Answer
Booklet. If you need extra space use the lined pages at the end of the Printed Answer
Booklet. The question numbers must be clearly shown.
• Fill in the boxes on the front of the Printed Answer Booklet.
• Answer all the questions.
• Where appropriate, your answer should be supported with working. Marks might be
given for using a correct method, even if your answer is wrong.
• Give non-exact numerical answers correct to 3 significant figures unless a different
degree of accuracy is specified in the question.
• The acceleration due to gravity is denoted by g m s–2. When a numerical value is
needed use g = 9.8 unless a different value is specified in the question.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 60.
• The marks for each question are shown in brackets [ ].
• This document has 4 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2023 [H/508/5496] OCR is an exempt Charity
DC (PQ) 327796/3 Turn over
, 2
1 The roots of the equation 4x 4 - 2x 3 - 3x + 2 = 0 are a , b , c and d . By using a suitable
substitution, find a quartic equation whose roots are a + 2 , b + 2 , c + 2 and d + 2 giving your
answer in the form at 4 + bt 3 + ct 2 + dt + e = 0 , where a, b, c, d, and e are integers. [5]
2 The lines L 1 and L 2 have the following equations.
J- 5N J5N
K O K O
L 1 : r = K 6 O + m K- 2O
K O K O
L 15 P L- 2P
J 24 N J3N
K O K O
L2 : r = K 1 O + nK 1 O
K O K O
L- 5P L- 4P
(a) Show that L 1 and L 2 intersect, giving the position vector of the point of intersection. [5]
(b) Find the equation of the line which intersects L 1 and L 2 and is perpendicular to both.
Give your answer in cartesian form. [3]
3 In this question you must show detailed reasoning.
In this question the principal argument of a complex number lies in the interval [0, 2r) .
Complex numbers z 1 and z 2 are defined by z 1 = 3 + 4i and z 2 = - 5 + 12i .
(a) Determine z 1 z 2 , giving your answer in the form a + bi . [2]
(b) Express z 2 in modulus-argument form. [3]
(c) Verify, by direct calculation, that arg (z1 z2) = arg (z1) + arg (z2) . [3]
J a2 N
K O
4 The vector p, all of whose components are positive, is given by p = Ka - 5O where a is a constant.
K O
L 26 P
J2N
K O
You are given that p is perpendicular to the vector K 6 O.
K O
L- 3P
Determine the value of a. [4]
© OCR 2023 Y531/01 Jun23
, 3
5 In this question you must show detailed reasoning.
The roots of the equation 5x 2 - 3x + 12 = 0 are a and b .
By considering the symmetric functions of the roots, a + b and ab , determine the exact value of
1 1
2 + 2. [4]
a b
6 Prove by induction that 4 # 8 n + 66 is divisible by 14 for all integers n H 0 . [6]
7 In this question you must show detailed reasoning.
J a - 6 a - 3N
K O
Matrix A is given by A = Ka + 9 a 4 O where a is a constant.
K O
L 0 - 13 a - 1P
Find all possible values of a for which det A has the same value as it has when a = 2 . [6]
8 (a) Solve the equation ~ + 2 + 7i = 3~ * - i . [4]
(b) Prove algebraically that, for non-zero z, z =- z * if and only if z is purely imaginary. [2]
(c) The complex numbers z and z* are represented on an Argand diagram by the points A and B
respectively.
(i) State, for any z, the single transformation which transforms A to B. [1]
(ii) Use a geometric argument to prove that z = z * if and only if z is purely real. [2]
Turn over for question 9
© OCR 2023 Y531/01 Jun23 Turn over
RSA Examinations GCE
Further Mathematics
AY531/01: Pure Core
AS Level
Question paper and
marking scheme
(merged)
, Oxford Cambridge and RSA
Monday 15 May 2023 – Afternoon
AS Level Further Mathematics A
Y531/01 Pure Core
Time allowed: 1 hour 15 minutes
* 9 9 7 4 1 0 8 4 1 6 *
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for AS Level Further
QP
Mathematics A
• a scientific or graphical calculator
INSTRUCTIONS
• Use black ink. You can use an HB pencil, but only for graphs and diagrams.
• Write your answer to each question in the space provided in the Printed Answer
Booklet. If you need extra space use the lined pages at the end of the Printed Answer
Booklet. The question numbers must be clearly shown.
• Fill in the boxes on the front of the Printed Answer Booklet.
• Answer all the questions.
• Where appropriate, your answer should be supported with working. Marks might be
given for using a correct method, even if your answer is wrong.
• Give non-exact numerical answers correct to 3 significant figures unless a different
degree of accuracy is specified in the question.
• The acceleration due to gravity is denoted by g m s–2. When a numerical value is
needed use g = 9.8 unless a different value is specified in the question.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 60.
• The marks for each question are shown in brackets [ ].
• This document has 4 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2023 [H/508/5496] OCR is an exempt Charity
DC (PQ) 327796/3 Turn over
, 2
1 The roots of the equation 4x 4 - 2x 3 - 3x + 2 = 0 are a , b , c and d . By using a suitable
substitution, find a quartic equation whose roots are a + 2 , b + 2 , c + 2 and d + 2 giving your
answer in the form at 4 + bt 3 + ct 2 + dt + e = 0 , where a, b, c, d, and e are integers. [5]
2 The lines L 1 and L 2 have the following equations.
J- 5N J5N
K O K O
L 1 : r = K 6 O + m K- 2O
K O K O
L 15 P L- 2P
J 24 N J3N
K O K O
L2 : r = K 1 O + nK 1 O
K O K O
L- 5P L- 4P
(a) Show that L 1 and L 2 intersect, giving the position vector of the point of intersection. [5]
(b) Find the equation of the line which intersects L 1 and L 2 and is perpendicular to both.
Give your answer in cartesian form. [3]
3 In this question you must show detailed reasoning.
In this question the principal argument of a complex number lies in the interval [0, 2r) .
Complex numbers z 1 and z 2 are defined by z 1 = 3 + 4i and z 2 = - 5 + 12i .
(a) Determine z 1 z 2 , giving your answer in the form a + bi . [2]
(b) Express z 2 in modulus-argument form. [3]
(c) Verify, by direct calculation, that arg (z1 z2) = arg (z1) + arg (z2) . [3]
J a2 N
K O
4 The vector p, all of whose components are positive, is given by p = Ka - 5O where a is a constant.
K O
L 26 P
J2N
K O
You are given that p is perpendicular to the vector K 6 O.
K O
L- 3P
Determine the value of a. [4]
© OCR 2023 Y531/01 Jun23
, 3
5 In this question you must show detailed reasoning.
The roots of the equation 5x 2 - 3x + 12 = 0 are a and b .
By considering the symmetric functions of the roots, a + b and ab , determine the exact value of
1 1
2 + 2. [4]
a b
6 Prove by induction that 4 # 8 n + 66 is divisible by 14 for all integers n H 0 . [6]
7 In this question you must show detailed reasoning.
J a - 6 a - 3N
K O
Matrix A is given by A = Ka + 9 a 4 O where a is a constant.
K O
L 0 - 13 a - 1P
Find all possible values of a for which det A has the same value as it has when a = 2 . [6]
8 (a) Solve the equation ~ + 2 + 7i = 3~ * - i . [4]
(b) Prove algebraically that, for non-zero z, z =- z * if and only if z is purely imaginary. [2]
(c) The complex numbers z and z* are represented on an Argand diagram by the points A and B
respectively.
(i) State, for any z, the single transformation which transforms A to B. [1]
(ii) Use a geometric argument to prove that z = z * if and only if z is purely real. [2]
Turn over for question 9
© OCR 2023 Y531/01 Jun23 Turn over
