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Cambridge International AS & A Level: Further Mathematics 9231/12

This document is an exam paper for Further Mathematics. It contains instructions for students on how to complete the paper, which lasts 2 hours and is out of 75 total marks. The first question asks students to use formulas to find an expression in terms of n, and then to express a fraction in partial fractions and use that to find another expression in terms of n. The document provides space for students to show their working and answers.

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0% found this document useful (0 votes)

98 views20 pages

Cambridge International AS & A Level: Further Mathematics 9231/12

Uploaded by

Big Chungus

We take content rights seriously. If you suspect this is your content, claim it here.

Available Formats

Download as PDF, TXT or read online on Scribd

You are on page 1/ 20

Cambridge International AS & A Level

* 6 1 4 6 6 6 2 3 8 2 *

FURTHER MATHEMATICS 9231/12

Paper 1 Further Pure Mathematics 1 October/November 2022

2 hours

You 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 [ ].

This document has 20 pages. Any blank pages are indicated.

DC (LK) 303986/2
© UCLES 2022 [Turn over
2
n
1 (a) Use the list of formulae (MF19) to find / r (r + 2) in terms of n, simplifying your answer. [2]
r =1

............................................................................................................................................................

............................................................................................................................................................
n
(b) Express
1
r (r + 2)
in partial fractions and hence find / r (r1+ 2) in terms of n. [4]
r =1
............................................................................................................................................................

............................................................................................................................................................

© UCLES 2022 9231/12/O/N/22

............................................................................................................................................................

............................................................................................................................................................
3
(c) Deduce the value of / r (r1+ 2) . [1]
r =1
............................................................................................................................................................

............................................................................................................................................................

............................................................................................................................................................
© UCLES 2022 9231/12/O/N/22 [Turn over
4

2 The equation x 4 + 3x 2 + 2x + 6 = 0 has roots a, b, c, d .

1 1 1 1 1 1 1 1
(a) Find a quartic equation whose roots are , , , and state the value of + + + .
a2 b2 c2 d2 a2 b2 c2 d2
[4]

............................................................................................................................................................

© UCLES 2022 9231/12/O/N/22

(b) Find the value of b 2 c 2 d 2 + a 2 c 2 d 2 + a 2 b 2 d 2 + a 2 b 2 c 2 . [3]

............................................................................................................................................................

............................................................................................................................................................
1 1 1 1
(c) Find the value of + + + . [2]
a4 b4 c4 d4
............................................................................................................................................................

............................................................................................................................................................

© UCLES 2022 9231/12/O/N/22 [Turn over

1 0 1 0
3 The matrix M is given by M = e oe o, where k is a constant and k ! 0 or 1.
0 k k 1
(a) The matrix M represents a sequence of two geometrical transformations.

State the type of each transformation, and make clear the order in which they are applied. [2]

............................................................................................................................................................

(b) Write M -1 as the product of two matrices, neither of which is I. [2]

............................................................................................................................................................

............................................................................................................................................................
k2
(c) Show that the invariant points of the transformation represented by M lie on the line y = x.
1-k
[4]
............................................................................................................................................................

............................................................................................................................................................

............................................................................................................................................................
© UCLES 2022 9231/12/O/N/22
7

............................................................................................................................................................

(d) The triangle ABC in the x-y plane is transformed by M onto triangle DEF.

Find the value of k for which the area of triangle DEF is equal to the area of triangle ABC. [2]

............................................................................................................................................................

............................................................................................................................................................
© UCLES 2022 9231/12/O/N/22 [Turn over
8

4 The function f is such that f m (x) = f (x) .

Prove by mathematical induction that, for every positive integer n,

d 2n - 1 `
x f (x)j = x f l (x) + (2n - 1) f (x) . [7]
d x 2n - 1
....................................................................................................................................................................

....................................................................................................................................................................

© UCLES 2022 9231/12/O/N/22

....................................................................................................................................................................

5 The curve C has polar equation r = a sec 2 i , where a is a positive constant and 0 G i G 14 r .

(a) Sketch C, stating the polar coordinates of the point of intersection of C with the initial line and
also with the half-line i = 14 r . [3]

............................................................................................................................................................

(b) Find the maximum distance of a point of C from the initial line. [2]

............................................................................................................................................................

© UCLES 2022 9231/12/O/N/22

............................................................................................................................................................

(d) Find, in the form y = f (x) , the Cartesian equation of C. [3]

............................................................................................................................................................

............................................................................................................................................................
© UCLES 2022 9231/12/O/N/22 [Turn over
12

6 The lines l1 and l2 have equations r = 2i + k + m (i - j + 2k) and r = 2 j + 6k + n (i + 2 j - 2k)

respectively.

The point P on l1 and the point Q on l2 are such that PQ is perpendicular to both l1 and l2 .

(a) Find the length PQ. [5]

............................................................................................................................................................

............................................................................................................................................................
© UCLES 2022 9231/12/O/N/22
13

The plane P1 contains PQ and l1 .

The plane P2 contains PQ and l2 .

(b) (i) Write down an equation of P1 , giving your answer in the form r = a + sb + t c . [1]

....................................................................................................................................................

(ii) Find an equation of P2 , giving your answer in the form ax + by + cz = d . [4]

....................................................................................................................................................

(c) Find the acute angle between P1 and P2 . [5]

............................................................................................................................................................

x2 - x
7 The curve C has equation y = .
x+1
(a) Find the equations of the asymptotes of C. [3]

............................................................................................................................................................

(b) Find the exact coordinates of the stationary points on C. [4]

............................................................................................................................................................

............................................................................................................................................................
© UCLES 2022 9231/12/O/N/22
15

............................................................................................................................................................

............................................................................................................................................................
© UCLES 2022 9231/12/O/N/22
17

Additional page

If you use the following page to complete the answer to any question, the question number must be clearly
shown.

............................................................................................................................................................................

BLANK PAGE

Permission 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.

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. This is produced for each series of examinations and is freely available to download
at www.cambridgeinternational.org after the live examination series.

Cambridge Assessment International Education is part of Cambridge Assessment. Cambridge Assessment is the brand name of the University of Cambridge
Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge.

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Cambridge International AS & A Level: Further Mathematics 9231/12

Uploaded by

Cambridge International AS & A Level: Further Mathematics 9231/12

Uploaded by

Cambridge International AS & A Level

FURTHER MATHEMATICS 9231/12

You must answer on the question paper.

You will need: List of formulae (MF19)

This document has 20 pages. Any blank pages are indicated.

© UCLES 2022 9231/12/O/N/22

2 The equation x 4 + 3x 2 + 2x + 6 = 0 has roots a, b, c, d .

© UCLES 2022 9231/12/O/N/22

(b) Find the value of b 2 c 2 d 2 + a 2 c 2 d 2 + a 2 b 2 d 2 + a 2 b 2 c 2 . [3]

© UCLES 2022 9231/12/O/N/22 [Turn over

(b) Write M -1 as the product of two matrices, neither of which is I. [2]

4 The function f is such that f m (x) = f (x) .

Prove by mathematical induction that, for every positive integer n,

© UCLES 2022 9231/12/O/N/22

© UCLES 2022 9231/12/O/N/22

(d) Find, in the form y = f (x) , the Cartesian equation of C. [3]

6 The lines l1 and l2 have equations r = 2i + k + m (i - j + 2k) and r = 2 j + 6k + n (i + 2 j - 2k)

(a) Find the length PQ. [5]

The plane P1 contains PQ and l1 .

The plane P2 contains PQ and l2 .

(ii) Find an equation of P2 , giving your answer in the form ax + by + cz = d . [4]

(c) Find the acute angle between P1 and P2 . [5]

© UCLES 2022 9231/12/O/N/22 [Turn over

(b) Find the exact coordinates of the stationary points on C. [4]

© UCLES 2022 9231/12/O/N/22 [Turn over

© UCLES 2022 9231/12/O/N/22

© UCLES 2022 9231/12/O/N/22

© UCLES 2022 9231/12/O/N/22

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