Oxford Cambridge and
RSA Examinations GCE
Further Mathematics
AY545/01: Additional
Pure Mathematics
A Level
Question paper and
marking scheme
(merged)
, Oxford Cambridge and RSA
Friday 23 June 2023 – Afternoon
A Level Further Mathematics A
Y545/01 Additional Pure Mathematics
Time allowed: 1 hour 30 minutes
* 9 9 7 9 6 3 7 4 0 3 *
You must have:
• the Printed Answer Booklet
QP
• the Formulae Booklet for A Level Further
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 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 4 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2023 [D/508/5514] OCR is an exempt Charity
DC (CJ) 328406/5 Turn over
, 2
1 The surface S is defined for all real x and y by the equation z = x 2 + 2xy . The intersection of
S with the plane P gives a section of the surface. On the axes provided in the Printed Answer
Booklet, sketch this section when the equation of P is each of the following.
(a) x = 1 [2]
(b) y = 1 [2]
2 A curve has equation y = 1 + x 2 , for 0 G x G 1, where both the x- and y-units are in cm. The
area of the surface generated when this curve is rotated fully about the x-axis is A cm2.
1
(a) Show that A = 2r y 1 + kx 2 dx for some integer k to be determined. [4]
0
A small component for a car is produced in the shape of this surface. The curved surface area of
the component must be 8 cm2, accurate to within one percent. The engineering process produces
such components with a curved surface area accurate to within one half of one percent.
(b) Determine whether all components produced will be suitable for use in the car. [2]
3 The points A and B have position vectors a = i + p j + qk and b = 2i + 3j + 2k respectively,
relative to the origin O.
(a) Determine the value of p and the value of q for which a # b = 2i + 6 j - 11k . [3]
(b) The point C has coordinates (d, e, f ) and the tetrahedron OABC has volume 7.
(i) Using the values of p and q found in part (a), find the possible relationships between
d, e and f. [2]
(ii) Explain the geometrical significance of these relationships. [2]
1
2r
I
4 The sequence "A n, is given for all integers n H 0 by A n = n + 2 , where I n = y cos n x dx .
In
0
• Show that "A n, increases monotonically.
• Show that "A n, converges to a limit, A, whose exact value should be stated. [7]
© OCR 2023 Y545/01 Jun23
, 3
5 (a) The group G consists of the set S = "1, 9, 17, 25, under # 32 , the operation of multiplication
modulo 32.
(i) Complete the Cayley table for G given in the Printed Answer Booklet. [2]
(ii) Up to isomorphisms, there are only two groups of order 4.
• C4, the cyclic group of order 4
• K4, the non-cyclic (Klein) group of order 4
State, with justification, to which of these two groups G is isomorphic. [2]
(b) (i) List the odd quadratic residues modulo 32. [2]
(ii) Given that n is an odd integer, prove that n 6 + 3n 4 + 7n (mod 32). [4]
y
6 The surface S has equation z = x sin y + for x 2 0 and 0 1 y 1 r .
x
(a) Determine, as a function of x and y, the determinant of H, the Hessian matrix of S. [6]
(b) Given that S has just one stationary point, P, use the answer to part (a) to deduce the nature
of P. [2]
(c) The coordinates of P are (a, b, c).
Show that b satisfies the equation b + tan b = 0 . [3]
7 Binet’s formula for the nth Fibonacci number is given by Fn = 1 ^ n
5
a - b nh for n H 0 , where a
and b (with a 2 0 2 b ) are the roots of x 2 - x - 1 = 0 .
(a) Write down the values of a + b and ab . [1]
(b) Consider the sequence "S n, , where S n = a n + b n for n H 0 .
(i) Determine the values of S2 and S3. [3]
(ii) Show that S n + 2 = S n + 1 + S n for n H 0 . [2]
(iii) Deduce that S n is an integer for all n H 0 . [1]
(c) A student models the terms of the sequence "S n, using the formula Tn = a n .
(i) Explain why this formula is unsuitable for every n H 1. [1]
(ii) Considering the cases n even and n odd separately, state a modification of the formula
Tn = a n , other than Tn = a n + b n , such that Tn = S n for all n H 1. [2]
© OCR 2023 Y545/01 Jun23 Turn over
RSA Examinations GCE
Further Mathematics
AY545/01: Additional
Pure Mathematics
A Level
Question paper and
marking scheme
(merged)
, Oxford Cambridge and RSA
Friday 23 June 2023 – Afternoon
A Level Further Mathematics A
Y545/01 Additional Pure Mathematics
Time allowed: 1 hour 30 minutes
* 9 9 7 9 6 3 7 4 0 3 *
You must have:
• the Printed Answer Booklet
QP
• the Formulae Booklet for A Level Further
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 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 4 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2023 [D/508/5514] OCR is an exempt Charity
DC (CJ) 328406/5 Turn over
, 2
1 The surface S is defined for all real x and y by the equation z = x 2 + 2xy . The intersection of
S with the plane P gives a section of the surface. On the axes provided in the Printed Answer
Booklet, sketch this section when the equation of P is each of the following.
(a) x = 1 [2]
(b) y = 1 [2]
2 A curve has equation y = 1 + x 2 , for 0 G x G 1, where both the x- and y-units are in cm. The
area of the surface generated when this curve is rotated fully about the x-axis is A cm2.
1
(a) Show that A = 2r y 1 + kx 2 dx for some integer k to be determined. [4]
0
A small component for a car is produced in the shape of this surface. The curved surface area of
the component must be 8 cm2, accurate to within one percent. The engineering process produces
such components with a curved surface area accurate to within one half of one percent.
(b) Determine whether all components produced will be suitable for use in the car. [2]
3 The points A and B have position vectors a = i + p j + qk and b = 2i + 3j + 2k respectively,
relative to the origin O.
(a) Determine the value of p and the value of q for which a # b = 2i + 6 j - 11k . [3]
(b) The point C has coordinates (d, e, f ) and the tetrahedron OABC has volume 7.
(i) Using the values of p and q found in part (a), find the possible relationships between
d, e and f. [2]
(ii) Explain the geometrical significance of these relationships. [2]
1
2r
I
4 The sequence "A n, is given for all integers n H 0 by A n = n + 2 , where I n = y cos n x dx .
In
0
• Show that "A n, increases monotonically.
• Show that "A n, converges to a limit, A, whose exact value should be stated. [7]
© OCR 2023 Y545/01 Jun23
, 3
5 (a) The group G consists of the set S = "1, 9, 17, 25, under # 32 , the operation of multiplication
modulo 32.
(i) Complete the Cayley table for G given in the Printed Answer Booklet. [2]
(ii) Up to isomorphisms, there are only two groups of order 4.
• C4, the cyclic group of order 4
• K4, the non-cyclic (Klein) group of order 4
State, with justification, to which of these two groups G is isomorphic. [2]
(b) (i) List the odd quadratic residues modulo 32. [2]
(ii) Given that n is an odd integer, prove that n 6 + 3n 4 + 7n (mod 32). [4]
y
6 The surface S has equation z = x sin y + for x 2 0 and 0 1 y 1 r .
x
(a) Determine, as a function of x and y, the determinant of H, the Hessian matrix of S. [6]
(b) Given that S has just one stationary point, P, use the answer to part (a) to deduce the nature
of P. [2]
(c) The coordinates of P are (a, b, c).
Show that b satisfies the equation b + tan b = 0 . [3]
7 Binet’s formula for the nth Fibonacci number is given by Fn = 1 ^ n
5
a - b nh for n H 0 , where a
and b (with a 2 0 2 b ) are the roots of x 2 - x - 1 = 0 .
(a) Write down the values of a + b and ab . [1]
(b) Consider the sequence "S n, , where S n = a n + b n for n H 0 .
(i) Determine the values of S2 and S3. [3]
(ii) Show that S n + 2 = S n + 1 + S n for n H 0 . [2]
(iii) Deduce that S n is an integer for all n H 0 . [1]
(c) A student models the terms of the sequence "S n, using the formula Tn = a n .
(i) Explain why this formula is unsuitable for every n H 1. [1]
(ii) Considering the cases n even and n odd separately, state a modification of the formula
Tn = a n , other than Tn = a n + b n , such that Tn = S n for all n H 1. [2]
© OCR 2023 Y545/01 Jun23 Turn over
