Oxford Cambridge and
RSA Examinations
GCEFurther
Mathematics AY540/01:
Pure Core 1
A Level question paper
and marking scheme
(merged)
, Oxford Cambridge and RSA
Thursday 25 May 2023 – Afternoon
A Level Further Mathematics A
Y540/01 Pure Core 1
Time allowed: 1 hour 30 minutes
* 9 9 7 5 8 9 3 4 1 3 *
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for A 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 75.
• The marks for each question are shown in brackets [ ].
• This document has 8 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2023 [A/508/5505] OCR is an exempt Charity
DC (PQ/CGW) 329027/3 Turn over
, 2
1 In this question you must show detailed reasoning.
50
Determine the value of / r 2 (16 - r) . [3]
r =1
2 In this question you must show detailed reasoning.
The equation z 4 + 4z 3 + 9z 2 + 10z + 6 = 0 has roots a , b , c and d .
(a) Show that a quartic equation whose roots are a + 1, b + 1, c + 1 and d + 1 is
w 4 + 3w 2 + 2 = 0 . [3]
(b) Hence determine the exact roots of the equation z 4 + 4z 3 + 9z 2 + 10z + 6 = 0 . [3]
-3 + 3 i 5
3 (a) Show that = 3 e 6 ri . [2]
2
9 ^- 3 + 3 ih
(b) Hence determine the exact roots of the equation z 5 = , giving the roots in the
2
form re ii where r 2 0 and 0 G i 1 2r . [3]
© OCR 2023 Y540/01 Jun23
, 3
4 The transformations TA and TB are represented by the matrices A and B respectively, where
J0 - 1N J N
A = KK OO and B = KK0 1OO.
1 0 1 0
L P L P
(a) Describe geometrically the single transformation consisting of TA followed by TB . [2]
(b) By considering the transformation TA , determine the matrix A 423 . [3]
The transformation TC is represented by the matrix C, where
J1 0N
C = KK2 1OO.
L0 3P
The region R is defined by the set of points (x, y) satisfying the inequality x 2 + y 2 G 36 .
The region Rl is defined as the image of R under TC .
(c) (i) Find the exact area of the region Rl . [2]
(ii) Sketch the region Rl , specifying all the points where the boundary of Rl intersects the
coordinate axes. [4]
© OCR 2023 Y540/01 Jun23 Turn over
RSA Examinations
GCEFurther
Mathematics AY540/01:
Pure Core 1
A Level question paper
and marking scheme
(merged)
, Oxford Cambridge and RSA
Thursday 25 May 2023 – Afternoon
A Level Further Mathematics A
Y540/01 Pure Core 1
Time allowed: 1 hour 30 minutes
* 9 9 7 5 8 9 3 4 1 3 *
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for A 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 75.
• The marks for each question are shown in brackets [ ].
• This document has 8 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2023 [A/508/5505] OCR is an exempt Charity
DC (PQ/CGW) 329027/3 Turn over
, 2
1 In this question you must show detailed reasoning.
50
Determine the value of / r 2 (16 - r) . [3]
r =1
2 In this question you must show detailed reasoning.
The equation z 4 + 4z 3 + 9z 2 + 10z + 6 = 0 has roots a , b , c and d .
(a) Show that a quartic equation whose roots are a + 1, b + 1, c + 1 and d + 1 is
w 4 + 3w 2 + 2 = 0 . [3]
(b) Hence determine the exact roots of the equation z 4 + 4z 3 + 9z 2 + 10z + 6 = 0 . [3]
-3 + 3 i 5
3 (a) Show that = 3 e 6 ri . [2]
2
9 ^- 3 + 3 ih
(b) Hence determine the exact roots of the equation z 5 = , giving the roots in the
2
form re ii where r 2 0 and 0 G i 1 2r . [3]
© OCR 2023 Y540/01 Jun23
, 3
4 The transformations TA and TB are represented by the matrices A and B respectively, where
J0 - 1N J N
A = KK OO and B = KK0 1OO.
1 0 1 0
L P L P
(a) Describe geometrically the single transformation consisting of TA followed by TB . [2]
(b) By considering the transformation TA , determine the matrix A 423 . [3]
The transformation TC is represented by the matrix C, where
J1 0N
C = KK2 1OO.
L0 3P
The region R is defined by the set of points (x, y) satisfying the inequality x 2 + y 2 G 36 .
The region Rl is defined as the image of R under TC .
(c) (i) Find the exact area of the region Rl . [2]
(ii) Sketch the region Rl , specifying all the points where the boundary of Rl intersects the
coordinate axes. [4]
© OCR 2023 Y540/01 Jun23 Turn over
