Question 1:
1
y = g | (x) | y =
y = | cos
1
0
– | (cid:155)1 | (cid:652) | 3 | (cid:652) 2 | 2 | (cid:652)
(cid:652) | (cid:652)
2 | 2
–
1
PMT
3
6 Oil is leaking into the sea from a pipeline, creating a circular oil slick. The radius r metres of the oil slick
t hours after the start of the leak is modelled by the equation
r = 20(1 − e−0.2t ).
(i) Find the radius of the slick when t = 2, and the rate at which the radius is increasing at this time. [4]
(ii) Find the rate at which the area of the slick is increasing when t = 2. [4]
7 Fig. 7 shows the curve x3 + y3 = 3xy. The point P is a turning point of the curve.
y
P
x
Fig. 7
dy y − x2
(i) Show that = . [4]
dx y2 − x
(ii) Hence find the exact x-coordinate of P. [4]
Turn over
© OCR 2012 4753/01 Jan12
PMT
4
Section B (36 marks)
x
8 Fig. 8 shows the curve y = , together with the lines y = x and x = 11.
x – 2
The curve meets these lines at P and Q respectively. R is the point (11, 11).
y
y = x
R(11, 11)
P Q
x
x = 11
Fig. 8
(i) Verify that the x-coordinate of P is 3. [2]
dy x − 4
(ii) Show that, for the curve, = .
dx 2(x − 2) 3 2
Hence find the gradient of the curve at P. Use the result to show that the curve is not symmetrical about
y = x. [7]
(cid:2)
11
x
(iii) Using the substitution u = x − 2, show that dx = 251.
x − 2 3
3
Hence find the area of the region PQR bounded by the curve and the lines y = x and x = 11. [9]
© OCR 2012 4753/01 Jan12
PMT
8
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible
opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a
department of the University of Cambridge.
© OCR 2012 4753/01 Jan12