OXFORD CAMBRDGE
AND RSA 2024
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OCR 2024
2024
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, Oxford Cambridge and RSA
Friday 21 June 2024 – AfternoonA Level
Further Mathematics B (MEI) Y435/01 Extra Pure
Time allowed: 1 hour 15 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for Further Mathematics B
QP
(MEI)
• 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 AnswerBooklet. 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 begiven
for using a correct method, even if your answer is wrong.
• Give your final answers to a degree of accuracy that is appropriate to the context.
• 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 8 pages.
ADVICE
• Read each question carefully before you start your answer.
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, 3
1 A surface, S, is defined in 3-D by z = f(x, y) where f(x, y) = 12x - 30y + 6xy .
(a) Determine the coordinates of any stationary points on the surface. [5]
(b) The equation z = f(x, a), where a is a constant, defines a section of S.
Given that this equation is z = 24x + b, find the value of a and the value of b. [3]
The diagram shows the contour z = 12 and its associated asymptotes.
y
O
x
(c) Find the equations of the asymptotes. [3]
(d) By forming grad g, where g(x, y, z) = f(x, y) - z , find the equation of the tangent plane to S
at the point where x = 3 and y = 2. Give your answer in vector form. [3]
The point (0, 4, -120), which lies on S, is denoted by A.
J 3N
The plane with equation 3O = 52 is denoted by P.
K K O
r.
K O
L-2P
(e) Show that the normal to S at A intersects P at the point (-360, 304, -110). [3]
, 2
2 (a) Determine the general solution of the recurrence relation 2un +2 - 7un +1 + 3un = 0. [2]
(b) Using your answer to part (a), determine the general solution of the recurrence relation
2un+2 - 7u n +1 + 3u n = 20n2 + 60n. [5]
In the rest of this question the sequence u0, u1, u2, ... satisfies the recurrence relation in part
(b).You are given that u0 =-9 and u1 =-12.
(c) Determine the particular solution for un . [3]
You are given that, as n increases, once the values of un start to increase, then from that point
onwards the sequence is an increasing sequence.
(d) Use your answer to part (c) to determine, by direct calculation, the least value taken by
terms in the sequence. You should show any values that you rely on in your argument. [2]
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