Oxford Cambridge and
RSA Examinations
GCEFurther
Mathematics AY541/01:
Pure Core 2
A Level question paper
and marking scheme
(merged)
, Oxford Cambridge and RSA
Monday 5 June 2023 – Afternoon
A Level Further Mathematics A
Y541/01 Pure Core 2
Time allowed: 1 hour 30 minutes
* 9 9 7 6 3 0 2 4 5 1 *
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 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.
, 2
J1 0 - 2 2NO
1 (a) The matrix P is given by P = KK .
4 2 -2 3O
L P
(i) Write down the dimensions of P. [1]
(ii) Write down the transpose of P. [1]
J3 - 4N
(b) The matrices Q, R and S are given by Q = ^1 2h, R = KK O and S = ^3 - 2h.
2 3O
L P
Write down the sum of the two of these matrices which are conformable for addition. [1]
(c) The dimensions of matrix A are 4 by 5. The matrices A and B are conformable for
multiplication so that the matrix C = BA can be formed. The matrix C has 6 rows.
(i) Write down the number of columns that C has. [1]
(ii) Write down the dimensions of B. [1]
(iii) Explain whether the matrix AB can be formed. [1]
J- 2 3NJ c 5N J c 5NJ- 2 3N
(d) Find the value of c for which KK OK O=K OK O. [2]
6 10OK10 13O K10 13OK 6 10O
L PL P L PL P
2 In this question you must show detailed reasoning.
(a) Write the complex number - 24 + 7i in modulus-argument form. [3]
(b) Solve the simultaneous equations given below, giving your answers in cartesian form.
iz + 3w = - 7i
[4]
- 6z + 5iw = 3 + 13i
sinh -1 uh =
d^ 1
3 (a) Show that . [2]
du 2
u +1
(b) Find the equation of the normal to the graph of y = sinh -1 2x at the point where x = 6 .
Give your answer in the form y = mx + c where m and c are given in exact, non-hyperbolic
form. [4]
, 3
4 In this question you must show detailed reasoning.
1
The region R is bounded by the curve with equation y = 2
, the x-axis and the lines
3x - 3x + 1
1
with equations x = and x = 1 (see diagram). The units of the axes are cm.
2
y
2
1
R 1
y= 2
3x - 3x + 1
x
O 1 1
2
A pendant is to be made out of a precious metal. The shape of the pendant is modelled as the
shape formed when R is rotated by 2r radians about the x-axis.
Find the exact value of the volume of precious metal required to make the pendant, according to
the model. [4]
5 In this question you must show detailed reasoning.
(a) Using the definitions of sinh x and cosh x in terms of exponentials, show that
sinh 2x / 2 sinh x cosh x. [2]
(b) Solve the equation 15 sinh x + 16 cosh x - 6 sinh 2x = 20 , giving all your answers in
logarithmic form. [5]
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