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Cambridge Assessment International Education Cambridge International Advanced Subsidiary and Advanced Level
CANDIDATENAME
CENTRENUMBER CANDIDATENUMBER
MATHEMATICS 9709/
Paper 1 Pure Mathematics 1 (P1) October/November 2019 1 hour 45 minutes Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF9) READ THESE INSTRUCTIONS FIRST Write your centre number, candidate number and name in the spaces at the top of this page.Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs.Do not use staples, paper clips, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. Answerat the end of this booklet. The question number(s) must be clearly shown. all the questions in the space provided. If additional space is required, you should use the lined page Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles indegrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate.You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together.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 75.
This document consists of 20 printed pages. JC19 11_9709_11/RP© UCLES 2019 [Turn over
1 Find the term independent of x in the expansion of^ @ 2 x + (^41) x 2 A^6. [3]
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3 The lineof the constants y = ax (^) + a b , b is a tangent to the curve and c. y = 2 x^3 − 5 x^2 − 3 x + c at the point 2, 6. Find the values[5]
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4 A runner who is training for a long-distance race plans to run increasing distances each day for 21 days.She will run x km on day 1, and on each subsequent day she will increase the distance by 10% of the previous day’s distance. On day 21 she will run 20 km. (i) Find the distance she must run on day 1 in order to achieve this. Give your answer in km correctto 1 decimal place. [3]
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(ii) Hence, showing all necessary working, solve the equation 4 tan 2 x − 20 Å + 3 cos 2 x − 20 Å + (^) cos 2 x^1 − 20 Å = 0 for 0Å ≤ x ≤ 180 Å. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
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6 A straight line has gradientwhich the line is a tangent to the curve m and passes through the point y = x (^2) − 2 x + 7 and, for each value of 0, − 2 . Find the two values of m , find the coordinates m for of the point where the line touches the curve. [7] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................
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7 Functions f and g are defined by f : x → (^2) x^3 + 1 for x > 0, g : x → (^1) x + 2 for x > 0. (i) Find the range of f and the range of g. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
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(ii) Find an expression for fg x , giving your answer in the form (^) bxax + c , where a , b and c are integers. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Find an expression for fg−^1 x , giving your answer in the same form as for part (ii). [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
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(iii) Find the area of the shaded region. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
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9 A curve for which d d xy = 5 x − 1 ^12 − 2 passes through the point 2, 3.
(i) Find the equation of the curve. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
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A B
D C
Relative to an origingiven by O , the position vectors of the points A , B , C and D , shown in the diagram, are − OA −→ =^ `^ −^13 − 4
a , − OB −→ =
` 2
a , − OC −→ =
` 4
a and − OD −→ =
` 2
a .
(i) Show that AB is perpendicular to BC. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Show that ABCD is a trapezium. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
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(iii) Find the area of ABCD , giving your answer correct to 2 decimal places. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
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(ii) Hence, showing all necessary working, find the volume obtained when the shaded region isrotated through 360Å about the y -axis. [6]
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Additional Page If you use the following lined page to complete the answer(s) to any question(s), the question number(s)must be clearly shown.
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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 (UCLES) 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.To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge Assessment International Education Copyright Acknowledgements Booklet.www.cambridgeinternational.org after the live examination series. This is produced for each series of examinations and is freely available to download at Cambridge Assessment International Education is part of the Cambridge Assessment Group.Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge. Cambridge Assessment is the brand name of the University of © UCLES 2019 9709/11/O/N/
