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Cambridge International AS & A LevelFURTHER MATHEMATICS 9231/01Paper 1 Further Pure Mathematics 1 For examination from 2020SPECIMEN PAPER 2 hoursYou must answer on the question paper.You will need: List of formulae (MF19)INSTRUCTIONS ●Answer all questions.

●Use a black or dark blue pen.

You may use an HB pencil for any diagrams or graphs.

●Write your name, centre number and candidate number in the boxes at the top of the page.

●Write your answer to each question in the space provided.
●Do not use an erasable pen or correction fluid.

●Do not write on any bar codes.

●If additional space is needed, you should use the lined page at the end of this booklet; the question number or numbers must be clearly shown.

●You should use a calculator where appropriate.

●You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.
●Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.INFORMATION ●The total mark for this paper is 75.

●The number of marks for each question or part question is shown in brackets [ ].

*0123456789*© UCLES 2017 [Turn overThis document has 16 pages.

Blank pages are indicated.

1 (a) Given that f(r) = 1 ) ( r1 ( r + + 2 ) , show that 2 f(r – 1) – f(r) = 2 ) ( r1 .
+ 2 ) r r ( + [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Hence find n / = r 1 1 ) ( r1 .
+ 2 ) r r ( + [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Deduce the value of 3 / = r 1 1 ) ( r1 .
+ 2 ) r r ( + [1] ............................................................................................................................................................ ............................................................................................................................................................ 9231/01/SP/20© UCLES 2017 2 It is given that φ(n) = 5n(4n + 1) – 1, for n = 1, 2, 3, … .
3 Prove, by mathematical induction, that φ(n) is divisible by 8 for every positive integer n. [7] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... 9231/01/SP/20© UCLES 2017[Turn over 3 The curve C has polar equation r = 2 + 2 cos θ, for 0 θ π.
4 (a) Sketch C. ⩽ ⩽ [3] (b) Find the area of the region enclosed by C and the initial line.
[4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 9231/01/SP/20© UCLES 2017 5 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Show that the Cartesian equation of C can be expressed as 4(x2 + y2) = (x2 + y2 – 2x)2.
[3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 9231/01/SP/20© UCLES 2017[Turn over 4 The cubic equation has roots α, β and γ.
6 z3 – z 2 – z – 5 = 0 (a) Show that the value of α3 + β3 + γ3 is 19.
[4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 9231/01/SP/20© UCLES 2017 (b) Find the value of α4 + β 4 + γ4.
[2] 7 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Find a cubic equation with roots α + 1, β + 1 and γ + 1, giving your answer in the form px3 + qx2 + rx + s = 0, where p, q, r and s are constants to be determined.
[3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 9231/01/SP/20© UCLES 2017[Turn over 5 The matrix A is given by 8 A = 5 - - 3 f k 4 p .
(a) Find the value of k for which A is singular.
[2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................
It is now given that k = 6 so that A = 5 - - 3 f 6 4 p .

(b) Find the equations of the invariant lines, through the origin, of the transformation in the x-y plane [6] represented by A.
............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 9231/01/SP/20© UCLES 2017 9 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) The triangle DEF in the x-y plane is transformed by A onto triangle PQR.
(i) Given that the area of triangle DEF is 10 cm2, find the area of triangle PQR.
[2] .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... (ii) Find the matrix which transforms triangle PQR onto triangle DEF.
[2] .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... 9231/01/SP/20© UCLES 2017[Turn over 6 The position vectors of the points A, B, C, D are 10 2i + 4j – 3k, −2i + 5j – 4k, i + 4j + k, i + 5j + mk, respectively, where m is an integer.
It is given that the shortest distance between the line through A and B and the line through C and D is 3.

(a) Show that the only possible value of m is 2.
[7] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 9231/01/SP/20© UCLES 2017 11 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the shortest distance of D from the line through A and C. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 9231/01/SP/20© UCLES 2017[Turn over 12 (c) Show that the acute angle between the planes ACD and BCD is cos–1 1 3 .
p f [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 9231/01/SP/20© UCLES 2017 7 The curve C has equation y = 13 .
2 x 2 2 x - - 2 3 x - + x 1 2 (a) State the equations of the asymptotes of C. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Show that y 25 at all points on C. 12 [4] ............................................................................................................................................................ ⩽ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 9231/01/SP/20© UCLES 2017[Turn over (c) Find the coordinates of any stationary points of C. [3] 14 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (d) Sketch C, stating the coordinates of any intersections of C with the coordinate axes and the [4] asymptotes.
9231/01/SP/20© UCLES 2017 (e) Sketch the curve with equation y = 15 2 x 2 2 x - - x 2 3 - + 1 x 2 and find the set of values of x for which 2 x 2 2 x - - 2 3 x - + x 1 2 < 2.
[4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 9231/01/SP/20© UCLES 2017 16 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.
............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ 9231/01/SP/20© UCLES 2017Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible.
Every reasonable effort 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.Cambridge Assessment International Education is part of the Cambridge Assessment Group.

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