PRE-U 9795
FURTHER PURE MATH
PAPER 1
(2013-2022)
Cambridge Pre-U
FURTHER MATHEMATICS
Paper 1 Further Pure Mathematics
9795/01
For examination from 2020
SPECIMEN PAPER
3 hours
*0123456789*
You must answer on the answer booklet/paper.
You will need:
Answer booklet/paper
Graph paper
List of formulae (MF20)
INSTRUCTIONS
●
Answer all questions.
●
Follow the instructions on the front cover of the answer booklet. If you need additional answer paper,
ask the invigilator for a continuation booklet.
●
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 120.
●
The number of marks for each question or part question is shown in brackets [ ].
This syllabus is regulated for use in England, Wales and Northern Ireland as a Cambridge International Level 3 Pre-U Certificate.
This document has 4 pages. Blank pages are indicated.
© UCLES 2018
[Turn over
2
1
Using any standard results given in the List of Formulae (MF20), show that
n
/ (r 2 − r + 1) = 13 n (n 2 + 2)
r =1
for all positive integers n.
2
3
[4]
A curve has polar equation r = sin θ + cos θ. Find the area enclosed by the curve and the lines θ = 0 and
θ = 12 π .
[4]
J
N
K 1 2 1O
(a) Evaluate, in terms of k, the determinant of the matrix K− 3 5 8O .
K
O
6 12 k
L
P
[2]
Three planes have equations x + 2y + z = 4, –3x + 5y + 8z = 21 and 6x + 12y + kz = 31.
4
(b) State the value of k for which these three planes do not meet at a single point.
[1]
(c) Find the coordinates of the point of intersection of the three planes when k = 7.
[3]
(a) Given that y = sinh x for x ⩾ 0, express
(b) Hence or otherwise find
y
dy
in terms of y only.
dx
2t
dt .
1 + t4
[3]
[3]
5
n J 2 NJ 2 N 1
1
Use induction to prove that / KK − OOK + O = − + for all positive integers n.
3
r
4
3
4
r
1
n
4
3
=
r 1L
P
PL
6
The curve C has equation y =
[8]
x+1
.
x2 + 3
(a) By considering a suitable quadratic equation in x, find the set of possible values of y for points
on C.
[5]
(b) Deduce the coordinates of the turning points on C.
[4]
(c) Sketch C.
[4]
© UCLES 2018
9795/01/SP/20
3
7
The function f satisfies the differential equation
x2 f ″ (x) + (2x − 1)f ′ (x) − 2f(x) = 3ex−1 + 1,
(∗)
and the conditions f(1) = 2, f ′(1) = 3.
(a) Determine f ″(1).
[2]
(b) Differentiate (∗) with respect to x and hence evaluate f ‴ (1).
[4]
(c) Hence determine the Taylor series approximation for f(x) about x = 1, up to and including the term
in (x −1)3.
[3]
(d) Deduce, to 3 decimal places, an approximation for f(1.1).
8
[2]
J
N
p pO
K
Consider the set S of all matrices of the form K
, where p is a non-zero rational number.
p pO
L
P
(a) Show that S, under the operation of matrix multiplication, forms a group, G. (You may assume that
matrix multiplication is associative.)
[5]
(b) Find a subgroup of G of order 2 and show that G contains no subgroups of order 3.
9
(a) Show that the substitution u =
dy
1
4
3 transforms the differential equation dx + y = 3xy into
y
du
- 3u =- 9x .
dx
(b) Solve the differential equation
form y3 = f(x).
© UCLES 2018
[4]
dy
+ y = 3xy4, given that y =
dx
[3]
1
2
when x = 0. Give your answer in the
[9]
9795/01/SP/20
[Turn over
4
J N
JN
J N
K1 O
K3O
K2O
K
O
K
O
−
10 The line L has equation r = 3 + λ 4 and the plane П has equation r. K− 6O = k.
K O
KO
K O
2
6
3
L P
LP
L P
(a) Given that L lies in П, determine the value of k.
[2]
(b) Find the coordinates of the point, Q, in П which is closest to P(10, 2, −43). Deduce the shortest
distance from P to П.
[5]
(c) Find, in the form ax + by + cz = d, where a, b, c and d are integers, an equation for the plane which
contains both L and P.
[6]
11
(a) Use de Moivre’s theorem to prove that sin 5θ ≡ s(16s4 – 20s2 + 5), where s = sin θ, and deduce that
2r
sin 5 =
5+ 5
.
8
[8]
The complex number ω = 16(– 1 + i 3 ).
(b) State the value of |ω| and find arg ω as a rational multiple of π.
[3]
(c) (i) Determine the five roots of the equation z5 = ω, giving your answers in the form (r, θ), where
r > 0 and – π < θ ⩽ π .
[5]
(ii) These five roots are represented in the complex plane by the points A, B, C, D and E. Show
these points on an Argand diagram, and find the area of the pentagon ABCDE in an exact surd
form.
[3]
12 (a) Let In =
y03 xn
16 + x 2 dx, for n ⩾ 0. Show that, for n ⩾ 2,
(n + 2)In = 125 × 3n – 1 – 16(n – 1)In – 2.
[6]
(b) A curve has polar equation r = 4 θ 4 for 0 ⩽ θ ⩽ 3.
1
(i) Sketch this curve.
[2]
(ii) Find the exact length of the curve.
[7]
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.
Cambridge Assessment International Education is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of the University of
Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge.
© UCLES 2018
9795/01/SP/20
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
Cambridge International Level 3 Pre-U Certificate
Principal Subject
9795/01
FURTHER MATHEMATICS
Paper 1 Further Pure Mathematics
May/June 2013
3 hours
*1126490413*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF20)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, 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 120.
This document consists of 4 printed pages.
JC13 06_9795_01/RP
© UCLES 2013
[Turn over
2
1
2
3
4
By completing the square, or otherwise, find the exact value of Ô
6
1
dx.
x − 6x + 12
2
2
Use the standard Maclaurin series expansions given in the List of Formulae MF20 to show that
@
A
1 ln 1 + x  tanh−1 x for −1 < x < 1.
2
1−x
The curve C has equation y =
[4]
4
x+1
.
x2 − 4
(i) Show that the gradient of C is always negative.
[3]
(ii) Sketch C, showing all significant features.
[6]
(i) Find a vector which is perpendicular to both of the vectors
d1 = i + 2j + 4k
and
d2 = 9i − 3j + k.
2
(ii) Determine the shortest distance between the skew lines with equations
r = 2i + 4j + 3k + , i + 2j + 4k
5
Let Ï = cos 1 + i sin 1.
(i) Prove the result Ïn −
and
r = i + j + 10k + - 9i − 3j + k.
1
= 2i sin n1.
Ïn
5
[2]
(ii) Use this result to express sin5 1 in the form A sin 51 + B sin 31 + C sin 1, for constants A, B and
C to be determined.
[5]
6
The curve P has polar equation r =
1
for 0 ≤ 1 < 20, 1 ≠ 12 0.
1 − sin 1
(i) Determine, in the form y = f x, the cartesian equation of P.
(ii) Sketch P.
(iii) Evaluate Ô
7
20
0
1
d1.
1 − sin 12
[3]
[2]
[3]
(i) Express x3 + y3 in terms of x + y and xy.
[2]
(ii) The equation t2 − 3t + 89 = 0 has roots ! and ".
(a) Determine the value of !3 + "3 .
[2]
(b) Hence express 19 as the sum of the cubes of two positive rational numbers.
© UCLES 2013
9795/01/M/J/13
[3]
3
8
Let G = g1 , g2 , g3 , à, gn be a finite abelian group of order n under a multiplicative binary operation,
where g1 = e is the identity of G.
(i) Let x ∈ G. Justify the following statements:
(a) xgi = xgj ­ gi = gj ;
(b) xg1 , xg2 , xg3 , à, xgn = G.
[2]
[1]
(ii) By considering the product of all G’s elements, and using the result of part (i)(b), prove that
xn = e for each x ∈ G.
[3]
(iii) Explain why
9
(a) this does not imply that all elements of G have order n,
[1]
(b) this argument cannot be used to justify the same result for non-abelian groups.
[1]
The plane transformation T is the composition (in this order) of
³ a reflection in the line y = x tan 81 0; followed by
³ a shear parallel to the y-axis, mapping 1, 0 to 1, 2; followed by
³ a clockwise rotation through 14 0 radians about the origin; followed by
³ a shear parallel to the x-axis, mapping 0, 1 to −2, 1.
Determine the matrix M which represents T , and hence give a full geometrical description of T as a
single plane transformation.
[8]
10
(a) Given that y = kx cos x is a particular integral for the differential equation
d2 y
+ y = 4 sin x,
dx2
determine the value of k and find the general solution of this differential equation.
[8]
(b) The variables x and y satisfy the differential equation
dy
d2 y
+ y2
+ xy = 5x − 19.
2
d
x
dx
(i) Given that y = 2 and
dy
d3 y
= 1 when x = 1, find the value of 3 when x = 1.
dx
dx
[6]
(ii) Deduce the Taylor series expansion for y in ascending powers of x − 1, up to and including
the term in x − 13 , and use this series to find an approximation correct to 3 decimal places
for the value of y when x = 1.1.
[4]
11
(i) Determine p and q given that p + iq2 = 63 − 16i and that p and q are real.
[4]
(ii) Let f Ï = Ï3 − AÏ2 + BÏ − C for complex numbers A, B and C.
(a) Given that the cubic equation f Ï = 0 has roots ! = −7i, " = 3i and ' = 4, determine each
of A, B and C.
[4]
(b) Find the roots of the equation f ′ Ï = 0.
© UCLES 2013
9795/01/M/J/13
[5]
[Turn over
4
12
Given y = xe2x ,
(i) find the first four derivatives of y with respect to x,
(ii) conjecture an expression for
[4]
dn y
in the form ax + be2x , where a and b are functions of n, [2]
dxn
(iii) prove by induction that your result holds for all positive integers n.
13
(i) Use the definitions tanh 1 =
(a) tanh2 1  1 − sech2 1,
(b)
[5]
e1 − e−1
2
and sech 1 = 1
to prove the results
1
−1
e +e
e + e −1
d
tanh 1 = sech2 1.
d1
[4]
!
(ii) Let In = Ó tanh2n 1 d1 for n ≥ 0, where ! > 0.
0
(a) Show that In−1 − In =
Given that ! = 12 ln 3,
tanh2n−1 !
for n ≥ 1.
2n − 1
(b) evaluate I0 ,
[1]
(c) use the method of differences to show that In = 12 ln 3 −
the infinite series
[4]
Ð
∞
r=0
1
.
2r + 14r
Ð 2r − 1 and deduce the sum of
n
1 2r−1
2
r=1
[7]
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.
University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of
Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
© UCLES 2013
9795/01/M/J/13
Cambridge International Examinations
Cambridge Pre-U Certificate Principal Subject
9795/01
FURTHER MATHEMATICS
Paper 1 Further Pure Mathematics
May/June 2014
3 hours
*6739340907*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF20)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, 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 120.
This document consists of 4 printed pages.
JC14 06_9795_01/2R
© UCLES 2014
[Turn over
2
The series S is given by S = Ð N + r2 .
N
1
r=0
(i) Write out the first three terms and the last three terms of the series for S.
[1]
(ii) Use the standard result Ð r2 = 16 n n + 1 2n + 1 to show that S = 16 N N + 1 aN + 1 for some
n
r=1
positive integer a to be determined.
2
[4]
(i) Show that there is a value of t for which AB is an integer multiple of the 3 × 3 identity matrix I,
where
`1 2 1a
`t − 2
0
5a
A = t 1 −t
and B = 12
−2 −6 .
4
3 2 1
3t
4
7
(ii) Express the system of equations
−5x
+ 5Ï = 8
12x − 2y − 6Ï = 12
−9x + 4y + 7Ï = 22
in the form Cx = u, where C is a 3 × 3 matrix, and x and u are suitable column vectors.
(iii) Use the result of part (i) to solve the system of equations given in part (ii).
3
[3]
(i) On a single copy of an Argand diagram, sketch the loci defined by
Ï + 2 = 3
and
arg Ï − i = − 14 0.
(ii) State the complex number Ï which corresponds to the point of intersection of these two loci.
4
[1]
4
[1]
4
Let In = Ó xn 2x + 1 dx for n ≥ 0. Show that, for n ≥ 1,
0
2n + 3In = 27 × 4n − nIn−1 .
5
6
The curve C has equation y =
5
12 x + 1
.
x − 22
(i) Determine the coordinates of any stationary points of C.
[4]
(ii) Sketch C.
[6]
Solve the first-order differential equation x
answer in the form y = f x.
© UCLES 2014
dy
+ 2y = 4 ln x given that y = 1 when x = 1. Give your
dx
[8]
9795/01/M/J/14
3
7
Let f n = 112n−1 + 7 × 4n . Prove by induction that f n is divisible by 13 for all positive integers n.
[6]
a
` a
2
5
(i) Show that the line l with vector equation r = −5 + , −2 lies in the plane with cartesian
7
3
equation x + 4y + Ï + 11 = 0.
[2]
`
8
(ii) The plane is horizontal, and the point P 1, 2, k is above it. Given that the point in which
is directly beneath P is on the line l, determine the value of k.
[6]
9
(i) Explain why all groups of even order must contain at least one self-inverse element (that is, an
element of order 2).
[2]
(ii) Prove that any group in which every non-identity element is self-inverse is abelian.
[2]
(iii) Simon believes that if x and y are two distinct self-inverse elements of a group, then the element
xy is also self-inverse. By considering the group of the six permutations of 1 2 3, produce a
counter-example to prove him wrong.
[2]
(iv) A group G has order 4n + 2, for some positive integer n, and i is the identity element of G. Let x
and y be two distinct self-inverse elements of G. By considering the set H = i, x, y, xy, prove
by contradiction that G cannot contain all self-inverse elements.
[5]
10
(i) Use de Moivre’s theorem to show that 2 cos 61  64 cos6 1 − 96 cos4 1 + 36 cos2 1 − 2.
[5]
(ii) Hence find, in exact trigonometric form, the six roots of the equation
x6 − 6x4 + 9x2 − 3 = 0.
5
(iii) By considering the product of these six roots, determine the exact value of
1 5 7 cos 18
0 cos 18
0 cos 18
0 .
11
3
A curve has polar equation r = esin 1 for −0 < 1 ≤ 0.
(i) State the polar coordinates of the point where the curve crosses the initial line.
[1]
(ii) State also the polar coordinates of the points where r takes its least and greatest values.
[2]
(iii) Sketch the curve.
[3]
(iv) By deriving a suitable Maclaurin series up to and including the term in 12 , find an approximation,
to 3 decimal places, for the area of the region enclosed by the curve, the initial line and the line
1 = 0.3.
[9]
© UCLES 2014
9795/01/M/J/14
[Turn over
4
12
(i) (a) Show that tanh x =
e2x − 1
.
e2x + 1
[2]
(b) Hence, or otherwise, show that, if tanh x =
1
for k > 1, then x =
k
expression in terms of k for sinh 2x.
(ii) A curve has equation y = 12 ln tanh x for ! ≤ x ≤ ", where tanh ! =
simplest exact form, the arc length of this curve.
13
1
3
@
1
2
ln
A
k+1
and find an
k−1
[4]
and tanh " = 12 . Find, in its
[10]
The complex number w has modulus 1. It is given that
w2 −
2
+ ki = 0,
w
where k is a positive real constant.
/ (i) Show that k = 3 − 3 12 3.
[8]
2
(ii) Prove that at least one of the remaining two roots of the equation Ï2 − + ki = 0 has modulus
Ï
greater than 1.
[2]
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.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
© UCLES 2014
9795/01/M/J/14
Cambridge International Examinations
Cambridge Pre-U Certificate
9795/01
FURTHER MATHEMATICS (PRINCIPAL)
Paper 1 Further Pure Mathematics
May/June 2015
3 hours
*1939658420*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF20)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, 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 120.
The syllabus is approved for use in England, Wales and Northern Ireland as a Cambridge International Level 3 Pre-U Certificate.
This document consists of 4 printed pages.
JC15 06_9795_01/4R
© UCLES 2015
[Turn over
2
1
2
Determine the volume of tetrahedron OABC, where O is the origin and A, B and C are, respectively,
the points 2, 3, −2, 2, 0, 4 and 6, 1, 7.
[3]
The Taylor series expansion, about x = 1, of the function y is
y=1+
Find the value of
4
5
6
7
M is the matrix
n=1
−2n−1 x − 1n
.
1 × 3 × 5 × à × 2n − 1
d4 y
when x = 1.
dx4
`1
3
Ð
∞
2
2
−2
−1
−2
[3]
2a
2 . Use induction to prove that, for all positive integers n,
3
` a `
a
1
2n + 1
n
2
M 0 =
.
2n + 2n
1
2n2 + 2n + 1
6
A curve has polar equation r = sin 21 1 for 0 ≤ 1 < 20.
(i) Sketch the curve.
[3]
(ii) Determine the area of the region enclosed by the curve.
[4]
A curve has equation y =
2x2 + 5x − 25
.
x−3
(i) Determine the equations of the asymptotes.
[3]
(ii) Find the coordinates of the turning points.
[5]
(iii) Sketch the curve.
[3]
(i) Given the complex number Ï = cos 1 + i sin 1, show that Ïn +
1
= 2 cos n1.
Ïn
[1]
(ii) Deduce the identity16 cos5 1  cos 51 + 5 cos 31 + 10 cos 1.
[4]
(iii) For 0 < 1 < 20, solve the equation cos 51 + 5 cos 31 + 9 cos 1 = 0.
[4]
(i) On an Argand diagram, shade the region whose points satisfy
Ï − 20 + 15i ≤ 7.
3
(ii) The complex number Ï1 represents that value of Ï in the region described in part (i) for which
arg Ï is least. Mark Ï1 on your Argand diagram and determine arg Ï1 correct to 3 decimal
places.
[4]
© UCLES 2015
9795/01/M/J/15
3
8
The group G, of order 8, consists of the elements e, a, b, c, ab, bc, ca, abc, together with a
multiplicative binary operation, where e is the identity and
a2 = b2 = c2 = e,
ab = ba,
bc = cb
and
ca = ac.
(i) Construct the group table of G. [You are not required to show how individual elements of the
table are determined.]
[4]
(ii) List all the proper subgroups of G.
9
[5]
The differential equation (*) is
d2 u
+ 4u = 8x + 1.
dx2
(i) Find the general solution of (*).
[5]
(ii) The differential equation (**) is
x
d2 y
dy
+2
+ 4xy = 8x + 1.
2
d
x
dx
By using the substitution u = xy, show that (*) becomes (**) and deduce the general solution
of (**).
[4]
10
(i) Find a vector equation for the line of intersection of the planes with cartesian equations
x + 7y − 6Ï = −10
and
3x − 5y + 8Ï = 48.
5
(ii) Determine the value of k for which the system of equations
x + 7y − 6Ï = −10
3x − 5y + 8Ï = 48
kx + 2y + 3Ï = 16
does not have a unique solution and show that, for this value of k, the system of equations is
inconsistent.
[6]
11
(a) The cubic equation x3 + 2x2 + 3x − 4 = 0 has roots p, q and r. A second cubic equation has roots
4
qr, rp and pq. Show how the substitution y = can be used to determine this second equation.
x
Hence, or otherwise, find this equation in the form y3 + ay2 + by + c = 0.
[6]
(b) The cubic equation x3 − 4x2 + 5x − 4 = 0 has roots !, " and ' . You are given that ! is real and
positive, and that " and ' are complex.
(i) Describe the relationship between " and ' .
[1]
2
(ii) Explain why " = .
!
[2]
(iii) Verify that ! = 2.70 correct to 3 significant figures, and deduce that Re " = 0.65 correct to
2 significant figures.
[4]
© UCLES 2015
9795/01/M/J/15
[Turn over
4
12
2
Let In = Ó xn 1 + 2x2 dx for n = 0, 1, 2, 3, à .
0
(i) (a) Evaluate I1 .
[3]
(b) Prove that, for n ≥ 2,
2n + 4In = 27 × 2n−1 − n − 1In−2 .
6
(c) Using a suitable substitution, or otherwise, show that
1 I0 = 3 + ln 1 + 2 .
2
8
1
(ii) The curve y = x2 , between x = 0 and x = 2, is rotated through 20 radians about the x-axis to
2
form a surface with area S. Find the exact value of S.
[5]
13
(i) By sketching a suitable triangle, show that tan−1 a + tan−1
@ A
1
= 12 0, for a > 0.
a
[1]
(ii) Given that a and b are positive and less than 1, express tan tan−1 a ± tan−1 b in terms of a and b.
[2]
(iii) By letting a =
1
1
and b =
, use the method of differences to prove that
n−1
n+1
∞
@ A
2
−1
tan
= 34 0.
2
n
n=1
Ð
7
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 International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
© UCLES 2015
9795/01/M/J/15
Cambridge International Examinations
Cambridge Pre-U Certificate
9795/01
FURTHER MATHEMATICS (PRINCIPAL)
Paper 1 Further Pure Mathematics
May/June 2016
3 hours
*0431896259*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF20)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, 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 120.
The syllabus is approved for use in England, Wales and Northern Ireland as a Cambridge International Level 3 Pre-U Certificate.
This document consists of 4 printed pages.
JC16 06_9795_01/2R
© UCLES 2016
[Turn over
2
Ð 8r3 + r  12 n n + 1 2n + 12 .
n
1
Using standard summation results, show that
2
Find a vector which is perpendicular to both of the lines
` a
` a
` a
` a
11
6
1
−6
r= 5 +, 2
and r =
7 +- 1
4
5
−1
4
r=1
and hence find the shortest distance between them.
3
A curve has equation y =
[4]
[6]
2x2 − x − 1
.
2x − 3
(i) Show that the curve meets the line y = k when 2x2 − 2k + 1x + 3k − 1 = 0, and hence show
that no part of the curve exists in the interval 12 < y < 92 .
[4]
(ii) Deduce the coordinates of the turning points of this curve.
4
A 3 × 3 system of equations is given by the matrix equation
` −1
5
−1
[4]
3
−1
1
1a`xa ` 1a
2
y = 16 .
0
−2
z
(i) Show that this system of equations does not have a unique solution.
[2]
(ii) Solve this system of equations and describe the geometrical significance of the solution.
[6]
d2 y
dy
−4
+ 5y = 24e2x .
2
dx
dx
5
Find the general solution of the differential equation
6
The equation sinh x + sin x = 3x has one positive root !.
(i) Show that 2.5 < ! < 3.
[8]
[2]
4
(ii) By using the first two non-zero terms in the Maclaurin series for sinh x + sin x, show that ! ≈ 60.
[3]
(iii) By taking the third non-zero term in this series, find a second approximation to !, giving your
answer correct to 4 decimal places.
[3]
7
(i) Find all values of z for which z3 = 2 + 2i. Give your answers in the form rei1 , where r > 0 and
1 is an exact multiple of 0 in the interval 0 < 1 < 20.
[6]
(ii) The vertices of a triangle in the Argand diagram correspond to the three roots of the equation
z3 = 2 + 2i. Sketch the triangle and determine its area.
[3]
© UCLES 2016
9795/01/M/J/16
3
8
(i) S is the set 1, 2, 4, 8, 16, 32 and ×63 is the operation of multiplication modulo 63.
(a) Construct the multiplication table for S, ×63 .
[2]
(b) Show that S, ×63 forms a group, G. (You may assume that ×63 is associative.)
[3]
(ii) The group H , also of order 6, has identity element e and contains two further elements x and y
with the properties
x2 = y3 = e
9
and
xyx = y2 .
(a) Construct the group table of H .
[4]
(b) List all the proper subgroups of H .
[2]
(c) State, with justification, whether G and H are isomorphic.
[1]
The cubic equation x3 − ax2 + bx − c = 0 has roots !, " and ' .
(i) State, in terms of a, b and c, the values of ! + " + ' , !" + "' + '! and !"' .
[2]
(ii) Find, in terms of a, b and c, the values of !2 + "2 + ' 2 and !2 "2 + "2 ' 2 + ' 2 !2 .
[4]
(iii) Show that ! − 2"' " − 2'! ' − 2!" = c 2a + 12 − 2 b + 2c2 .
[4]
(iv) Deduce that one root of the equation x3 − ax2 + bx − c = 0 is twice the product of the other two
[1]
roots if and only if c 2a + 12 = 2 b + 2c2 .
10
11
.
.
(i) Sketch the curve with polar equation r = 12 + sin 1 , for 0 ≤ 1 < 20.
[6]
(ii) Find in an exact form the total area enclosed by the curve.
[4]
(i) The sequence of Fibonacci Numbers Fn is given by
F1 = 1,
F2 = 1
and
Fn+1 = Fn + Fn−1 for n ≥ 2.
Write down the values of F3 to F6 .
[1]
(ii) The sequence of functions pn x is given by
p1 x = x + 1
and
pn+1 x = 1 +
1
for n ≥ 1.
pn x
(a) Find p2 x and p3 x, giving each answer as a single algebraic fraction, and show that
3x + 5
p4 x =
.
[3]
2x + 3
(b) Conjecture an expression for pn x as a single algebraic fraction involving Fibonacci
numbers, and prove it by induction for all integers n ≥ 2.
[5]
© UCLES 2016
9795/01/M/J/16
[Turn over
4
12
The curve C has equation y = ln tanh 12 x, for x > 0.
(i) Show that
dy
= cosech x.
dx
[3]
(ii) For positive integers n, the length of the arc of C between x = n and x = 2n is Ln .
13
(a) Show by calculus that, when n is large, Ln ≈ n.
[5]
(b) Explain how this result corresponds to the shape of C.
[2]
(i) (a) Given that x ≥ 1, show that sec−1 x = cos−1
@ A
1
d
1
, and deduce that
sec−1 x = .
x
dx
x x2 − 1
[3]
(b) Use integration by parts to determine Ó sec−1 x dx.
[4]
(ii)
y
L
S
P
R
O
Q
I
x
1
The diagram shows the curve S with equation y = sec−1 x for x ≥ 1. The line L, with gradient ,
2
is the tangent to S at the point P and cuts the x-axis at the point Q. The point I has coordinates
1, 0.
(a) Determine the exact coordinates of P and Q.
[6]
(b) The region R, shaded on the diagram, is bounded by the line segments PQ and QI and the
arc IP of S. Show that R has area
0 8 − 0 2
ln 1 + 2 −
.
[4]
32
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 International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
© UCLES 2016
9795/01/M/J/16
Cambridge International Examinations
Cambridge Pre-U Certificate
9795/01
FURTHER MATHEMATICS (PRINCIPAL)
Paper 1 Further Pure Mathematics
May/June 2017
3 hours
*4983196509*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF20)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, 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 120.
The syllabus is approved for use in England, Wales and Northern Ireland as a Cambridge International Level 3 Pre-U Certificate.
This document consists of 4 printed pages.
JC17 06_9795_01/RP
© UCLES 2017
[Turn over
2
1
Without using a calculator, determine the possible values of a and b for which a + ib2 = 21 − 20i.
[4]
2
The equation x3 + 2x2 + 3x + 7 = 0 has roots !, " and ' . Evaluate !2 + "2 + ' 2 and use your answer to
comment on the nature of these roots.
[4]
3
(i) Sketch the curve with polar equation r =
1
, 0 ≤ 1 ≤ 20.
1+1
[3]
(ii) Find, in terms of 0, the area of the region enclosed by the curve and the part of the initial line
between the endpoints of the curve.
[3]
4
5
The curve C has parametric equations x = 12 t2 − ln t, y = 2t, for 1 ≤ t ≤ 4. When C is rotated through
20 radians about the x-axis, a surface of revolution is formed of surface area S. Determine the exact
value of S.
[7]
@
A
e2y − 1
1+x
−1
1
(i) Use the definition tanh y = 2y
to show that tanh x = 2 ln
for x < 1.
1−x
e +1
[3]
(ii) Solve the equation tanh x + coth x = 4, giving your answer in the form p ln m, where p is a positive
rational number and m is a positive integer.
[5]
6
The curve S has equation y =
x2 + 1
.
x + 12
(i) Write down the equations of the asymptotes of S.
(ii) Determine
dy
and hence find the coordinates of any turning points of S.
dx
(iii) Sketch S.
7
[2]
[4]
[3]
(i) Find the value of the constant k for which y = kx sin 2x is a particular integral of the differential
d2 y
[4]
equation 2 + 4y = 8 cos 2x.
dx
(ii) Solve
© UCLES 2017
d2 y
dy
+ 4y = 8 cos 2x, given that y = 1 and
= 1 when x = 0.
2
dx
dx
9795/01/M/J/17
[7]
3
8
The line l has equation r = ,d and the plane 1 has equation r.n = 35, where
` a
` a
2
6
d = −1
and n = −2 .
2
3
(i) (a) Determine the exact value of cos 1, where 1 is the angle between d and n.
(b) Determine the position vector of the point of intersection of l and 1 .
(c) Determine the shortest distance from O to 1 .
[3]
[3]
[2]
(ii) The plane 2 has cartesian equation 12x − 4y + 6z + 21 = 0. Determine the distance between
1 and 2 .
[3]
9
(i) Given that x ≥ 1, use the substitution x = cosh 1 to show that
1
x2 − 1
Ô
+C
dx =
x
x2 x2 − 1
where C is an arbitrary constant.
(ii) By differentiating sec y = x implicitly, show that
(iii) Use integration by parts to determine Ô
10
(i) Express
(ii) Let Sn =
[4]
d
1
sec−1 x = for x ≥ 1.
dx
x x2 − 1
sec−1 x
dx for x ≥ 1.
x2
[4]
1
in partial fractions.
k − 1k k + 1
Ð
n
k=3
[3]
1
for n ≥ 3. Use the method of differences to show that
k − 1k k + 1
Sn =
1
1
−
,
12 2n n + 1
and write down the limit of Sn as n → ∞.
(iii) Given that k is a positive integer greater than 1, explain why
(iv) Show that
© UCLES 2017
[4]
27
<
24
Ðk
∞
k=1
1
3
<
29
.
24
[5]
1
1
<
.
3
k − 1k k + 1
k
[1]
[3]
9795/01/M/J/17
[Turn over
4
11
A
@
a b
e
and B =
c d
g
for det A and det B.
(i) (a) Given A =
@
A
f
, work out the matrix AB and write down expressions
h
[2]
(b) Verify, by direct calculation, that det AB = det A × det B.
[2]
Let S be the set of all 2 × 2 matrices with determinant equal to 1.
(ii) Show that S, ×M forms a group, G, where ×M is the operation of matrix multiplication. [You
[5]
may assume that ×M is associative.]
(iii) (a) Show that K =
@
A
i
is an element of G.
0
1
i
[1]
Let H be the smallest subgroup of G that contains K and let n be the order of H .
12
(b) Determine the value of n.
[3]
(c) Give a second subgroup of G, also of order n, which is isomorphic to H .
[2]
For each positive integer n, the function Fn is defined for all real angles 1 by
where c = cos 1 and s = sin 1.
(i) Prove the identity
Fn 1 = c2n + s2n
Fn+2 1 − 14 sin2 21 × Fn+1 1  Fn+3 1.
[4]
Let z denote the complex number c + is.
(ii) Using de Moivre’s theorem,
(a) express z + z−1 and z − z−1 in terms of c and s respectively,
[3]
(b) prove the identity 8 c6 + s6  3 cos 41 + 5 and deduce that
c6 + s6  cos2 21 + 14 sin2 21.
[7]
(iii) Prove by induction that, for all positive integers n,
c2n+4 + s2n+4 ≤ cos2 21 +
1
n+1
2
[You are given that the range of the function Fn is
sin2 21.
1
≤ Fn 1 ≤ 1.]
2n−1
[7]
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 International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
© UCLES 2017
9795/01/M/J/17
Cambridge International Examinations
Cambridge Pre-U Certificate
9795/01
FURTHER MATHEMATICS (PRINCIPAL)
Paper 1 Further Pure Mathematics
May/June 2018
3 hours
*1069269998*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF20)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, 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 120.
This syllabus is approved for use in England, Wales and Northern Ireland as a Cambridge International Level 3 Pre-U Certificate.
This document consists of 4 printed pages.
JC18 06_9795_01/RP
© UCLES 2018
[Turn over
2
1
(i) Express
3
in partial fractions.
3r − 1 3r + 2
[2]
n
(ii) Using the method of differences, prove that
Ð
r=1
∞
(iii) Deduce the value of
Ð
r=1
2
3
1
1
= −
.
3r − 1 3r + 2 2 3n + 2
1
.
3r − 1 3r + 2
[1]
(i) Determine the asymptotes and turning points of the curve with equation y =
x2 + 3
.
x+1
(ii) Sketch the curve.
3
[2]
[7]
[3]
7 0, z = 2 and arg z = − 1 0.
The complex numbers z1 and z2 are such that z1 = 2, arg z1 = 12
2
2
8
(i) Find, in exact form, the modulus and argument of
z1
.
z2
[3]
P
Q
z1 n
(ii) Let z3 =
. It is given that n is the least positive integer for which z3 is a positive real
z2
number. Find this value of n and the exact value of z3 .
[4]
4
31
3 e 4 for 1 ≥ 0. The length of the arc of this curve between 1 = 0 and
A curve has polar equation r = 10
1 = ! is denoted by L !.
!
3!
(i) Show that L ! = 12 e 4 − 1 .
[5]
(ii) The point P on the curve corresponding to 1 = " is such that L " = OP, where O is the pole.
Find the value of ".
[2]
5
Find, in the form y = f x, the solution of the differential equation
y=
6
3
4
when x = ln 2.
dy
+ y tanh x = 2 cosh x, given that
dx
[8]
The cubic equation 4x3 − 12x2 + 9x − 16 = 0 has roots r1 , r2 and r3 . A second cubic equation, with
r + r3
r + r1
r + r2
integer coefficients, has roots R1 = 2
, R2 = 3
and R3 = 1
.
r1
r2
r3
(i) Show that 1 + R1 =
3
and write down the corresponding results for the other roots.
r1
(ii) Using a substitution based on this result, or otherwise, find this second cubic equation.
© UCLES 2018
9795/01/M/J/18
[2]
[6]
3
7
The function y satisfies
d2 y
dy
+ x2 y = x, and is such that y = 1 and
= 1 when x = 1.
2
dx
dx
(i) Using the given differential equation
(a) state the value of
d2 y
when x = 1,
dx2
(b) find, by differentiation, the value of
[1]
d3 y
when x = 1.
dx3
[2]
(ii) Hence determine the Taylor series for y about x = 1 up to and including the term in x − 13 and
deduce, correct to 4 decimal places, an approximation for y when x = 1.1.
[3]
8
1 m2 bm. [1]
(i) Write down the values of the constants a and b for which m5  16 m3 am2 + 2 − 12
n
(ii) Prove by induction that
Ð r5 = 16 n3 n + 13 − 121 n2 n + 12 for all positive integers n.
[7]
r=1
9
10
(i) Use de Moivre’s theorem to prove that cos 31 = 4c3 − 3c, where c = cos 1.
[3]
(ii) Solve the equation 2 cos 31 − 3 = 0 for 0 < 1 < 0 , giving each answer in an exact form.
[2]
(iii) Deduce, in trigonometric form, the three roots of the equation x3 − 3x − 3 = 0.
[3]
(i) Let G be a group of order 10. Write down the possible orders of the elements of G and justify
your answer.
[2]
(ii) Let G1 be the cyclic group of order 10 and let g be a generator of G1 (that is, an element of order
10). List the ten elements of G1 in terms of g and state the order of each element.
[4]
(iii) The group G2 is defined as the set of ordered pairs x, y, where x ∈ 0, 1 and y ∈ 0, 1, 2, 3, 4,
together with the binary operation ⊕ defined by
x1 , y1 ⊕ x2 , y2 = x3 , y3 ,
where x3 = x1 + x2 modulo 2 and y3 = y1 + y2 modulo 5.
(a) List the elements of G2 and state the order of each element.
[3]
(b) State, with justification, whether G1 and G2 are isomorphic.
[1]
© UCLES 2018
9795/01/M/J/18
[Turn over
4
@
11
Let A be the matrix
(a)
17
12
A
12
.
10
(i) Determine the integer n for which 27A − A2 = nI, where I is the 2 × 2 identity matrix. [2]
(ii) Hence find A−1 in the form pA + qI for rational numbers p and q.
[2]
@ A
@ A
x
x
→A
. It is given that T is a stretch, with
y
y
scale factor k, parallel to the line y = mx, where m > 0.
(b) The plane transformation T is defined by T :
(i) Find the value of k.
@
A
x
(ii) By considering A
, or otherwise, determine the value of m.
mx
12
[2]
[4]
The curve C is given by y = 14 x2 − 12 ln x for 2 ≤ x ≤ 8.
(i) Find, in its simplest exact form, the length of C.
[5]
(ii) When C is rotated through 20 radians
about the x-axis,a surface of revolution is formed. Show
that the area of this surface is 0 270 − 47 ln 2 − 2 ln 22 .
[10]
`
13
The planes 1 and 2 are both perpendicular to n, where n =
a
1
2 . The points A 0, −9, 13 and
−2
B 8, 7, −3 lie in 1 and 2 respectively.
−−→
(i) Find the equations of 1 and 2 in the form r.n = d and show that AB is parallel to n.
[4]
(ii) Calculate the perpendicular distance between 1 and 2 .
[2]
(iii) Write down two vectors which are perpendicular to n and hence find, in the form
r = u + ,v + -w,
an equation for the plane 3 which is parallel to 1 and 2 and exactly half-way between them.
[4]
(iv) The locus of all points P such that AP = BP = 12 2 is denoted by L.
(a) Give a full geometrical description of L.
[4]
(b) Using the result of part (iii), or otherwise, find a point on L which has integer coordinates.
[4]
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 International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
© UCLES 2018
9795/01/M/J/18
Cambridge Assessment International Education
Cambridge Pre-U Certificate
9795/01
FURTHER MATHEMATICS (PRINCIPAL)
Paper 1 Further Pure Mathematics
May/June 2019
3 hours
*8924365232*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF20)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, 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 120.
This syllabus is regulated for use in England, Wales and Northern Ireland as a Cambridge International Level 3 Pre-U Certificate.
This document consists of 4 printed pages.
JC19 06_9795_01/2R
© UCLES 2019
[Turn over
2
Ð r 3r − 1 = n2 n + 1 for all positive integers n.
n
1
Prove by induction that
2
The points P a, b, 0, Q c, d , 0 and R e, f , 0 all lie in the x-y plane.
[6]
r=1
(a) Using the vector product, or otherwise, find the area of triangle PQR.
1
(b) Show that this area is also given by 2 det M , where M is the matrix
3
[4]
`1
1
1
a
c
e
ba
d .
f
(a) Find a vector equation for the line of intersection of the planes with equations
` a
` a
2
1
r. 3 = 11
and
r. −1 = 16.
1
2
[2]
[5]
(b) Given that the system of equations
2x + 3y + z = 11
x − y + 2z = 16
4x + 11y − z = k
does not have a unique solution, determine the value of the constant k for which the system is
consistent.
[2]
4
Consider the set S = 3, 6, 9, 12 together with ×15 , the operation of multiplication modulo 15.
(a) Construct the multiplication table for S, ×15 and show that it is a group, G.
[You may assume that ×15 is an associative operation.]
[5]
The group H consists of the set T = 1, 7, 9, 15 together with ×16 , the operation of multiplication
modulo 16. The multiplication table for H is shown below.
1
7
9
15
1
1
7
9
15
7
7
1
15
9
9
9
15
1
7
15
15
9
7
1
(b) State, giving a reason, whether H is
(i) abelian,
[1]
(ii) cyclic.
[1]
(c) State also, with justification, whether G and H are isomorphic.
© UCLES 2019
9795/01/M/J/19
[1]
3
dy
dy
d2 y
+3
− 4y = 1 − 8x2 , given that y = 8 and
= 3 when x = 0.
2
dx
dx
dx
[10]
5
Solve the differential equation
6
The point P in the complex plane represents the complex number z = x + iy.
(a) On a single Argand diagram, sketch the locus of P in each of the following cases:
(i) z − 2i = 3;
[2]
(ii) z − i = z − 2 + i.
[2]
(b) Write down the cartesian equation of the locus of part (a)(i) and determine the cartesian equation
of the locus of part (a)(ii).
[3]
(c) The region of the complex plane for which
z − 2i ≤ 3
and
z − i ≥ z − 2 + i
is denoted by R. Calculate, in an exact form, the area of R.
7
A curve has equation y =
[3]
x2 − x + 1
.
x2 + x + 1
(a) Show that the x-coordinates of any points of intersection of the curve with the line y = k are given
by
k − 1x2 + k + 1x + k − 1 = 0,
and deduce the coordinates of the turning points of the curve.
(b) Sketch the curve.
8
[6]
[4]
The curve C has polar equation r = 5 − 4 cos 1 for 0 ≤ 1 < 2π.
(a) Sketch C.
[3]
(b) The points P, Q, R and S lie on C such that PR and QS are straight lines through the pole, O,
and PR is perpendicular to QS.
(i) Given that P is the point with polar coordinates 5 − 4 cos 1, 1, where 0 < 1 < 12 π, write
down the polar coordinates of Q, R and S, in terms of 1, cos 1 and sin 1.
[4]
(ii) Show that OP OR + OQ OS is constant for all values of 1, and determine its value.
[2]
9
The equation x3 + px2 + qx + r = 0, where r ≠ 0, has roots !, " and ' .
(a) By considering the expression !" − 1 "' − 1 '! − 1, show that one root of this equation is the
reciprocal of another root if and only if r r − p = 1 − q.
[4]
(b) Hence or otherwise show that, in this case, −r is a root of the equation.
(c) Solve the equation 12x3 + 11x2 − 63x + 36 = 0.
© UCLES 2019
9795/01/M/J/19
[2]
[4]
[Turn over
4
10
(a) Use de Moivre’s theorem to prove that sin 3x  3 sin x − 4 sin3 x and deduce the identity
@
A
1
1
cos 2x
1
−

.
2 sin x sin 3x
sin 3x
Ð
n
(b) Use the method of differences to find
11
r=0
cos 2 × 3r x
.
sin 3r+1 x
[6]
[4]
(a) Let In = Ó cosn 1 d1, where n is a positive integer.
Show that nIn = sin 1 cosn−1 1 + n − 1In−2 for n ≥ 3.
[5]
(b) A curve is defined parametrically for 0 ≤ 1 ≤ 12 π by
x = 21 + sin 21,
y = 1 + cos 21.
(i) Show that the length of the curve is 4.
[6]
(ii) When the curve is rotated through 2π radians about the x-axis, a surface of revolution is
formed having area S. Determine the exact value of S.
[7]
12
(a) Without using a calculator, show that if sinh x = 1 then tanh 12 x = 2 − 1.
(b) Show that
!
d
1
2 tan−1 tanh 12 x =
.
dx
cosh x
[5]
[4]
(c) Use the substitution tan 1 = sinh2 x to find the exact value of
Ô
1π
4
0
tan 1 sec2 1
d1.
1 + tan 1
[You may use, without proof, the result that tan 81 π = 2 − 1.]
[7]
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 the Cambridge Assessment Group. Cambridge Assessment is the brand name of the University of
Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge.
© UCLES 2019
9795/01/M/J/19
Cambridge Pre-U
FURTHER MATHEMATICS
9795/01
Paper 1 Further Pure Mathematics
May/June 2022
3 hours
*9364852141*
You must answer on the answer booklet/paper.
You will need: Answer booklet/paper
Graph paper
List of formulae (MF20)
INSTRUCTIONS
³ Answer all questions.
³ If you have been given an answer booklet, follow the instructions on the front cover of the answer booklet.
³ Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
³ Write your name, centre number and candidate number on all the work you hand in.
³ Do not use an erasable pen or correction fluid.
³ You should use a calculator where appropriate.
³ You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
³ 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.
³ At the end of the examination, fasten all your work together. Do not use staples, paper clips or glue.
INFORMATION
³ The total mark for this paper is 120.
³ The number of marks for each question or part question is shown in brackets [ ].
This syllabus is regulated for use in England, Wales and Northern Ireland as a Cambridge International Level 3 Pre-U Certificate.
This document has 4 pages.
JC22 06_9795_01/4R
© UCLES 2022
[Turn over
2
1
(a) Express
1
in partial fractions.
2n − 1 2n + 3
∞
(b) Hence evaluate
Ð
n=1
2
1
.
2n − 1 2n + 3
The curve C has equation y =
(a)
(a)
[3]
x
.
1 − x + x2
(i) Show algebraically that C exists only for − 13 ≤ y ≤ 1.
[3]
(ii) Hence, or otherwise, find the coordinates of the turning points of C.
[3]
(b) Sketch C, showing all significant features.
3
[2]
[3]
(i) Determine the possible values of the real numbers a and b for which a + ib2 = 28 + 96i.
[3]
(ii) Deduce the solutions of the equation z4 = 28 + 96i.
[3]
(b) The locus of points in the Argand diagram given by z − 28 − 96i = d passes through the origin.
Sketch this locus and state the value of the constant d .
[2]
4
A curve has equation y = cosh x. The length of the arc of the curve between the points where x = 0
and x = 1 is denoted by L.
(a) Determine, in terms of e, the exact value of L.
[4]
A rational approximation for L is to be found using the first few terms of the Maclaurin series for
cosh x.
(b)
(i) Calculate the approximation for L found when the first three non-zero terms are used. [3]
(ii) Explain why any approximation for L found by this method will be an under-estimate, no
matter how many terms of the series are used.
[1]
5
6
A group G of order 6 consists of functions (of x) under the operation of composition of functions.
1
Two of the elements of G are p x = and q x = 1 − x.
x
(a) State the identity element, i x, of G.
[1]
(b) Determine, as functions of x, the remaining three elements of G.
[3]
(c) List all the subgroups of G.
[4]
Solve the differential equation x
in the form y = f x.
© UCLES 2022
dy
x2
−y= , given that y = 3 ln 2 when x = 34 , giving your answer
2
dx
1+x
[8]
9795/01/M/J/22
3
7
Let M =
@
2k − 1
1−k
A
k−1
, where k is a non-zero constant.
1 − 8k
(a) Determine the value of k for which M is singular.
(b)
[3]
(i) Find the value of k for which the transformation T given by the matrix M is a rotation about
the origin.
[3]
(ii) Describe T fully in this case.
8
[2]
The equation x3 − px2 + qx − r = 0, where p, q and r are constants, has roots !, " and ' . Express each
of the following in terms of p, q and r.
(a) !2 + "2 + ' 2
[2]
(b) !2 " + ' + "2 ' + ! + ' 2 ! + "
[3]
(c) !3 + "3 + ' 3
9
10
Let Sn =
[3]
Ð cosr 1 cos r1. Use mathematical induction to prove that, for all positive integers n,
n
r=1
Sn =
cosn+1 1 sin n1
.
sin 1
[7]
(a) Use the definitions of sinh x and cosh x in terms of exponentials to show that
tanh x =
(b)
e2x − 1
.
e2x + 1
[2]
(i) Use the substitution u = e2x to show that tanh 2x − tanh x = 0.3 can be written as a cubic
equation in u.
[3]
(ii) Hence solve the equation tanh 2x − tanh x = 0.3, giving each answer in its simplest exact
logarithmic form.
[5]
11
The planes Π1 and Π2 have equations r. 8i + j − 3k = 20 and r. −i + j + k = 3 respectively. The
points V and W have coordinates 3, −1, 1 and 3, 2, 4 respectively.
(a) Show that V is in Π1 and that W is in Π2 .
[1]
The line of intersection of Π1 and Π2 is denoted by L.
(b) Find a vector equation for L in the form r = a + ,d, where the vectors a and d have integer
components.
[4]
A point U on L has coordinates which are all positive integers.
(c) Show that there is only one possible position for U and state its coordinates.
[3]
(d) Determine the volume of tetrahedron OUVW .
[3]
© UCLES 2022
9795/01/M/J/22
[Turn over
4
12
Let In = Ó
1π
2
0
sinn 1 d1, where n ≥ 0.
(a) Prove that nIn = n − 1In−2 for n ≥ 2.
[4]
The curve B has polar equation r = 4 sin2 1 cos 1 for − 12 π ≤ 1 ≤ 12 π.
(b) Sketch B.
(c)
[3]
(i) Show that the area of the plane enclosed by B can be written in the form aI4 + bI6 for
[2]
integers a and b to be determined.
(ii) Deduce the exact value of this area.
13
[3]
(d) Determine a cartesian equation for B.
[2]
(a) Determine the five smallest positive values of 1 for which cos 51 = 12 .
[2]
(b)
(i) Let z = cos 1 + i sin 1. Show that zn + z−n = 2 cos n1 for positive integers n.
(ii) Hence express 2 cos 51 as a polynomial in x, where x = 2 cos 1.
[2]
[5]
(iii) By considering the result of part (a), find, in an exact trigonometric form, the roots of
x4 + x3 − 4x2 − 4x + 1 = 0.
[3]
1 7 11 13 1
(c) Use the result of part (b)(iii) to show that sin 30
π sin 30 π sin 30 π sin 30 π = 16 .
[4]
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.
© UCLES 2022
9795/01/M/J/22
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
Cambridge International Level 3 Pre-U Certificate
Principal Subject
9795/01
FURTHER MATHEMATICS
Paper 1 Further Pure Mathematics
October/November 2013
3 hours
*7820161543*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF20)
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, 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 120.
This document consists of 4 printed pages.
JC13 11_9795_01/FP
© UCLES 2013
[Turn over
2
1
For real values of t, the non-singular matrices A and B are such that
t 5
2 −t
A−1 =
and B−1 =
.
3 −1
2 8
(i) Determine the values which t cannot take.
[2]
(ii) Without finding either A or B, determine AB−1 in terms of t.
[2]
2
Use de Moivre’s theorem to express cos 3 in terms of powers of cos only, and deduce the identity
[5]
cos 6x cos 2x2 cos 4x − 1.
3
The curve C has equation y =
2x
.
x +1
2
(i) Write down the equation of the asymptote of C and the coordinates of any points where C meets
the coordinate axes.
[2]
(ii) Show that the curve meets the line y = k if and only if −1 ≤ k ≤ 1. Deduce the coordinates of the
turning points of the curve.
[5]
[Note: You are NOT required to sketch C.]
4
Let fn = 25n−1 + 1 for integers n = 1, 2, 3, … .
(i) Prove that, if fn is divisible by 8, then fn + 1 is also divisible by 8.
[3]
(ii) Explain why this result does not imply that the statement
‘fn is divisible by 8 for all positive integers n’
follows by mathematical induction.
5
[1]
The curve S has polar equation r = 1 + sin + sin2 for 0 ≤ < 2.
(i) Determine the polar coordinates of the points on S where
dr
= 0.
d
(ii) Sketch S.
6
[5]
[3]
G is the set 2, 4, 6, 8, H is the set 1, 5, 7, 11 and ×n denotes the operation of multiplication
modulo n.
(i) Construct the multiplication tables for G, ×10 and H , ×12 .
[2]
(ii) By verifying the four group axioms, show that G and H are groups under their respective binary
[6]
operations, and determine whether G and H are isomorphic.
[You may assume that ×n is associative.]
© UCLES 2013
9795/01/O/N/13
3
7
Relative to an origin O, the points P, Q and R have position vectors
p = i + 2j − 7k,
q = −3i + 4j + k
and
r = 6i + 4j + k
respectively.
(i) Determine p × q.
[2]
(ii) Deduce the value of for which
8
(a) OR is normal to the plane OPQ,
[1]
(b) the volume of tetrahedron OPQR is 50,
[3]
(c) R lies in the plane OPQ.
[2]
(i) Determine x and y given that the complex number = x + iy simultaneously satisfies
− 1 = 1
and
arg + 1 = 16 .
4
(ii) On an Argand diagram, shade the region whose points satisfy
1 ≤ − 1 ≤ 2
9
and
1
6
≤ arg + 1 ≤ 13 .
6
(i) Show that there is exactly one value of k for which the system of equations
kx + 2y + k = 4
3x + 10y + 2 = m
k − 1x − 4y + = k
does not have a unique solution.
[4]
(ii) Given that the system of equations is consistent for this value of k, find the value of m.
[4]
(iii) Explain the geometrical significance of a non-unique solution to a 3 × 3 system of linear equations.
[2]
10
The roots of the equation x4 − 2x3 + 2x2 + x − 3 = 0 are , , and . Determine the values of
(i) 2 + 2 + 2 +
(ii)
2
,
[2]
1 1 1 1
+ + + ,
(iii) 3 + 3 + 3 +
3
[2]
.
[4]
[Questions 11, 12 and 13 are printed on the next page.]
© UCLES 2013
9795/01/O/N/13
[Turn over
4
11
(i) Given that y = −4 when x = 0 and that
dy
− y = e2x + 3,
dx
find the value of x for which y = 0.
[7]
(ii) Find the general solution of
dy
d2 y
−4
+ 4y = e2x + 3,
2
d
x
dx
given that y = cx2 e2x + d is a suitable form of particular integral.
12
(i) (a) Use the method of differences to prove that
N
n=k
∞
(b) Deduce the value of
n=k
∞
(ii) Let S =
n=1
13
[7]
(a) Let In =
1
1
1
= −
.
nn + 1 k N + 1
1
and show that
nn + 1
∞
n =k
1
. Show that 205
< S < 241
.
144
144
n2
4
1
1
< .
2
k
n + 1
[3]
[3]
coshn x dx for integers n ≥ 0, where = ln 2.
0
(i) Prove that, for n ≥ 2, nIn =
3 × 5n−1
+ n − 1In−2 .
4n
[5]
(ii) A curve has parametric equations x = 12 sinh t + 4 sinh3 t, y = 3 cosh4 t, 0 ≤ t ≤ ln 2. Find
the length of the arc of this curve, giving your answer in the form a + b ln 2 for rational
[8]
numbers a and b.
(b) The circle with equation x2 + y − R2 = r2 , where r < R, is rotated through one revolution about
the x-axis to form a solid of revolution called a torus. By using suitable parametric equations
[11]
for the circle, determine, in terms of , R and r, the surface area of this torus.
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.
University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of
Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
© UCLES 2013
9795/01/O/N/13
Cambridge Pre-U
FURTHER MATHEMATICS
Paper 1 Further Pure Mathematics
9795/01
October/November 2020
3 hours
*1203350548*
You must answer on the answer booklet/paper.
You will need: Answer booklet/paper
Graph paper
List of formulae (MF20)
INSTRUCTIONS
³ Answer all questions.
³ If you have been given an answer booklet, follow the instructions on the front cover of the answer booklet.
³ Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
³ Write your name, centre number and candidate number on all the work you hand in.
³ Do not use an erasable pen or correction fluid.
³ You should use a calculator where appropriate.
³ You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
³ 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.
³ At the end of the examination, fasten all your work together. Do not use staples, paper clips or glue.
INFORMATION
³ The total mark for this paper is 120.
³ The number of marks for each question or part question is shown in brackets [ ].
This syllabus is regulated for use in England, Wales and Northern Ireland as a Cambridge International Level 3 Pre-U Certificate.
This document has 4 pages. Blank pages are indicated.
JC20 11_9795_01/FP
© UCLES 2020
[Turn over
2
Ð 4r3 − 6r2 + 4r − 1 = n4 .
n
1
Using standard summation results, prove that
2
The parabola y = px2 + qx + r passes through the points −1, −1, 9, 53 and −11, 45.
(a)
r=1
(i) Write down a system of three equations in p, q and r.
[4]
[2]
(ii) Formulate this system as a matrix equation in the form Cx = a, where C is a 3 × 3 matrix,
x is an unknown column vector and a is a constant vector.
[1]
(b) Using any suitable method, determine the values of p, q and r.
3
(a)
(i) Write down the equations of the asymptotes of the curve y =
[4]
x−1
.
x−4
(ii) Sketch this curve, showing all significant features.
[4]
(b) Determine the equation of the oblique asymptote of the curve y =
4
5
x − 12
.
x−4
The equation 2x3 + 3x2 − 5x − 12 = 0 has roots !, " and ' .
[1]
A second cubic equation, with integer coefficients, has roots ! +
(b)
7
[2]
A curve has polar equation r = 3 + 2 sin 1, for 41 π ≤ 1 ≤ 34 π. Find, in its simplest exact form, the area
of the region enclosed by the curve and the lines 1 = 14 π and 1 = 34 π.
[6]
(a) State the value of !"' .
6
[2]
12
12
12
,"+
and ' +
.
"'
'!
!"
(i) Show that these new roots can be written as 3!, 3" and 3' respectively.
[2]
(ii) Find the second cubic equation.
[3]
A
0
, calculate X2 , X3 and X4 .
1
[3]
(b) Conjecture an expression for Xn for positive integers n and prove the result by induction.
[4]
(c) Is the result still true when n = −1? Justify your answer.
[3]
(a) Given the matrix X =
(a)
@
2
1
(i) Express the complex number 7 = 1 + i 3 in the form rei1 , where r > 0 and 0 < 1 < 2π. [2]
(ii) Hence show that 77 is an integer multiple of 7.
[3]
(b) Solve the equation z7 = 64 − 64i 3. Give each answer in the form r cos 1 + i sin 1, where r > 0
and 0 < 1 < 2π.
[5]
© UCLES 2020
9795/01/O/N/20
3
8
A non-abelian group G, with identity element e, contains an element a of order 4 and an element b
such that a3 b = ba.
(a) State, with justification, whether G is a cyclic group.
[1]
(b) Show, in any order, that
³
³
³
b = aba,
b = a2 ba2 ,
ba3 = ab.
Justify fully each step of your working.
9
[7]
The function f is defined for −1 ≤ x ≤ 1 by f x = cos−1 x.
(a)
(i) Sketch the graph of y = f x.
(ii) Given that y = cos−1 x, prove that
[1]
dy
1
=−
.
dx
1 − x2
[4]
(b) Determine Ó cos−1 x dx.
10
[5]
(a) Use the vector product to find the area of triangle ABC with vertices A 1, 2, 3, B 5, 1, −3 and
C 2, 3, −1.
[4]
(b)
(i) Calculate the volume of tetrahedron OABC, where O is the origin.
[3]
(ii) Deduce the shortest distance from O to the plane ABC.
[2]
(c) Determine the shortest distance between the line through O and A and the line through B and C.
Give your answer in an exact surd form.
[5]
11
3
The curve C has equation y = 23 x 2 for 0 ≤ x ≤ 15.
(a) The length of C is denoted by L. Showing full working, determine the value of L.
[4]
(b) The area of the surface generated when C is rotated once about the x-axis is denoted by A.
15 ?
2
4
(i) Show that A = 3 π Ó x x + 12 − 14 dx.
[3]
0
(ii) Use a suitable substitution to show that the exact value of A is
1 π ln31 + 8 15.
406π 15 + 12
© UCLES 2020
9795/01/O/N/20
[8]
[Turn over
4
12
It is given that the solution, y, of the differential equation
d2 y dy
+
sinh x + 4y cosh x = 8ex
dx2 dx
satisfies y = 3 and
(a)
*
dy
= 4 when x = ln 2.
dx
(i) Find the Taylor series expansion for y about x = ln 2 up to and including the quadratic term.
[5]
(ii) Deduce an approximation for y when x = 0.75. Give your answer to 3 decimal places. [1]
Three students try different methods to calculate approximations for the value of y when x = 0.75.
They do this by replacing sinh x, cosh x and ex in * by the first few terms of their Maclaurin series
and getting an approximate differential equation which they hope to be able to solve instead.
The first student uses quadratic approximations to sinh x, cosh x and ex ; the second student uses linear
approximations; and the third student uses constant approximations.
(b)
(i) Find the approximate differential equations obtained by the three students.
[4]
(ii) For the approximate differential equation obtained by the second student, find a particular
integral.
[3]
(iii) Solve the approximate differential equation obtained by the third student and use your
answer to calculate a second approximation for the value of y when x = 0.75. Show full
working and give the final answer correct to 3 decimal places.
[9]
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