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ADVANCED GCE UNIT 4723/01MATHEMATICS

Core Mathematics 3

MONDAY 11 JUNE 2007 Afternoon

Time: 1 hour 30 minutesAdditional Materials: Answer Booklet (8 pages)

List of Formulae (MF1)

INSTRUCTIONS TO CANDIDATES

Write your name, centre number and candidate number in the spaces provided on the answer booklet.

Answerall the questions.

Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy isspecified in the question or is clearly appropriate.

You are permitted to use a graphical calculator in this paper.

INFORMATION FOR CANDIDATES

The number of marks is given in brackets [ ] at the end of each question or part question.

The total number of marks for this paper is 72.

ADVICE TO CANDIDATES

Read each question carefully and make sure you know what you have to do before starting your answer.

You are reminded of the need for clear presentation in your answers.

This document consists of4 printed pages.

OCR 2007 [L/102/2710] OCR is an exempt Charity [Turn over

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1 Differentiate each of the following with respect tox.

(i) x3(x + 1)5 [2](ii)

3x4 + 1 [3]

2 Solve the inequality|4x 3|

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7 (i) Sketch the graph ofy = secxfor 0 x 2. [2]

(ii) Solve the equation secx= 3 for 0 x 2, giving the roots correct to 3 significant figures. [3]

(iii) Solve the equation sec = 5 cosec for 0 2, giving the roots correct to 3 significant

figures. [4]

8 (i) Given thaty = 4 lnx 3

4 lnx + 3, show that

dy

dx =

24

x(4 lnx + 3)2. [3]

(ii) Find the exact value of the gradient of the curve y = 4 lnx 3

4 lnx + 3at the point where it crosses the

x-axis. [4]

(iii)

The diagram shows part of the curve with equation

y= 2

x1

2 (4 lnx + 3).

The region shaded in the diagram is bounded by the curve and the lines x = 1, x = e and y= 0.

Find the exact volume of the solid produced when this shaded region is rotated completely about

thex-axis. [4]

9 (i) Prove the identity

tan(+ 60) tan( 60) tan2

31 3tan2

. [4]

(ii) Solve, for 0 <

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Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonableeffort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be

pleased to make amends at the earliest possible opportunity.

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 2 007 4723/01 Jun07

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