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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certiﬁcate of Education Advanced Subsidiary Level and Advanced Level

MATHEMATICS Paper 1 Pure Mathematics 1 (P1)

9709/12

May/June 2013 1 hour 45 minutes

Additional Materials:

Answer Booklet/Paper Graph Paper List of Formulae (MF9)

*2740852128*

READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction ﬂuid. Answer all the ...view middle of the document...

The mid-point of AD is O and BXC is an arc of a circle with centre O. (i) Show that angle BOC is 0.9273 radians, correct to 4 decimal places. (ii) Find the perimeter of the shaded region. (iii) Find the area of the shaded region. [2] [3] [2]

5

It is given that a = sin − 3 cos and b = 3 sin + cos , where 0 ≤ (i) Show that a2 + b2 has a constant value for all values of . (ii) Find the values of for which 2a = b.

≤ 360 . [3] [4]

© UCLES 2013

9709/12/M/J/13

3 6 Relative to an origin O, the position vectors of points A and B are given by −→ − OA = i − 2j + 2k where p and q are constants. −→ − −→ − (i) State the values of p and q for which OA is parallel to OB. (ii) In the case where q = 2p, ﬁnd the value of p for which angle BOA is 90 . − − → (iii) In the case where p = 1 and q = 8, ﬁnd the unit vector in the direction of AB. [2] [2] [3] and −→ − OB = 3i + pj + qk,

7

The point R is the reﬂection of the point −1, 3 in the line 3y + 2x = 33. Find by calculation the coordinates of R. [7] The volume of a solid circular cylinder of radius r cm is 250 cm3 . (i) Show that the total surface area, S cm2 , of the cylinder is given by

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S = 2 r2 +

500 . r

2 [4] [2]

(ii) Given that r can vary, ﬁnd the stationary value of S. (iii) Determine the nature of this stationary value. 5 , for x ≥ 1. 1 − 3x

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A function f is deﬁned by f x =

(i) Find an expression for f ′ x .

[2]

(ii) Determine, with a reason, whether f is an increasing function, a...

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