Oxford Cambridge and RSA
OCR A-LEVEL FURTHER MATHEMATICS
A Y540/01 PURE CORE 1 SUMMER MAY
EXAM QUESTION PAPER
(AUTHENTIC MARKING SCHEME
ATTACHED)
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.
–2
• The acceleration due to gravity is denoted by g m s . 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 2022 [A/508/5505] OCR is an exempt Charity
DC (LK/CB) 314920/3 Turn over
, 2
Answer all the questions.
1 In this question you must show detailed reasoning.
41
(a) Show that cosh (2 ln 3) = . [2]
9
The region R is bounded by the curve with equation y = sinh x , the x-axis and the line with
equation x = 2 ln 3 (see diagram). The units of the axes are centimetres.
y
y = sinh x
R
O x
x = 2 ln 3
A manufacturer produces bell-shaped chocolate pieces. Each piece is modelled as being the
shape of the solid formed by rotating R completely about the x-axis.
(b) Determine, according to the model, the exact volume of one chocolate piece. [4]
© OCR 2022 Y540/01 Jun22
, 3
N
J2 -2
K O
2 The matrix A is given by A = K O.
1 3
L P
(a) Calculate det A. [1]
-1 [1]
(b) Write down A .
J xN
J
-1N
(c) K
y
O
Hence solve the equation AK O
K
=K
2
O
O. [2]
L P L P
(d) Write down the matrix B such that AB = 4I. [1]
J N
K2 O
Matrices C and D are given by C = K0O and D = (0 2 p) where p is a constant.
K O
L1 P
(e) Find, in terms of p,
• the matrix CD
• the matrix DC. [3]
It is observed that CD ! DC.
(f) The result that CD ! DC is a counter example to the claim that matrix multiplication has a
particular property. Name this property. [1]
© OCR 2022 Y540/01 Jun22 Turn over
, 4
3 In this question you must show detailed reasoning.
2
(a) Find the roots of the equation 2z - 2z +5 = 0. [2]
The loci C1 and C2 are given by z = z -2 i and z - 2 = 5 respectively.
(b) (i) Sketch on a single Argand diagram the loci C1 and C2 , showing any intercepts with the
imaginary axis.[3]
(ii) Indicate, by shading on your Argand diagram, the region
"z: z G z - 2 i , + "z: z -2 G 5,. [1]
2
(c) (i) Show that both of the roots of the equation 2z - 2z +5 = 0 satisfy z -2 1 5. [2]
2
(ii) State, with a reason, which root of the equation 2z - 2z +5 = 0 satisfies z 1 z -2 i .
[1]
(d) On the same Argand diagram as part (b), indicate the positions of the roots of the equation
2
2z - 2z +5 = 0. [2]
N
J- 3 J 1 N
K O K O
4 Determine the acute angle between the line r = K 1 O+mK2 3O and the y-axis. [4]
K O K O
-
L3 P L 3P
© OCR 2022 Y540/01 Jun22
OCR A-LEVEL FURTHER MATHEMATICS
A Y540/01 PURE CORE 1 SUMMER MAY
EXAM QUESTION PAPER
(AUTHENTIC MARKING SCHEME
ATTACHED)
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.
–2
• The acceleration due to gravity is denoted by g m s . 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 2022 [A/508/5505] OCR is an exempt Charity
DC (LK/CB) 314920/3 Turn over
, 2
Answer all the questions.
1 In this question you must show detailed reasoning.
41
(a) Show that cosh (2 ln 3) = . [2]
9
The region R is bounded by the curve with equation y = sinh x , the x-axis and the line with
equation x = 2 ln 3 (see diagram). The units of the axes are centimetres.
y
y = sinh x
R
O x
x = 2 ln 3
A manufacturer produces bell-shaped chocolate pieces. Each piece is modelled as being the
shape of the solid formed by rotating R completely about the x-axis.
(b) Determine, according to the model, the exact volume of one chocolate piece. [4]
© OCR 2022 Y540/01 Jun22
, 3
N
J2 -2
K O
2 The matrix A is given by A = K O.
1 3
L P
(a) Calculate det A. [1]
-1 [1]
(b) Write down A .
J xN
J
-1N
(c) K
y
O
Hence solve the equation AK O
K
=K
2
O
O. [2]
L P L P
(d) Write down the matrix B such that AB = 4I. [1]
J N
K2 O
Matrices C and D are given by C = K0O and D = (0 2 p) where p is a constant.
K O
L1 P
(e) Find, in terms of p,
• the matrix CD
• the matrix DC. [3]
It is observed that CD ! DC.
(f) The result that CD ! DC is a counter example to the claim that matrix multiplication has a
particular property. Name this property. [1]
© OCR 2022 Y540/01 Jun22 Turn over
, 4
3 In this question you must show detailed reasoning.
2
(a) Find the roots of the equation 2z - 2z +5 = 0. [2]
The loci C1 and C2 are given by z = z -2 i and z - 2 = 5 respectively.
(b) (i) Sketch on a single Argand diagram the loci C1 and C2 , showing any intercepts with the
imaginary axis.[3]
(ii) Indicate, by shading on your Argand diagram, the region
"z: z G z - 2 i , + "z: z -2 G 5,. [1]
2
(c) (i) Show that both of the roots of the equation 2z - 2z +5 = 0 satisfy z -2 1 5. [2]
2
(ii) State, with a reason, which root of the equation 2z - 2z +5 = 0 satisfies z 1 z -2 i .
[1]
(d) On the same Argand diagram as part (b), indicate the positions of the roots of the equation
2
2z - 2z +5 = 0. [2]
N
J- 3 J 1 N
K O K O
4 Determine the acute angle between the line r = K 1 O+mK2 3O and the y-axis. [4]
K O K O
-
L3 P L 3P
© OCR 2022 Y540/01 Jun22
