w
w
ap
eP
m
e
tr
.X
w
om
.c
s
er
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
Cambridge International Level 3 Pre-U Certificate
Short Course
1348/01
FURTHER MATHEMATICS (Short Course)
Paper 1 Further Pure Mathematics
May/June 2013
3 hours
*8259595638*
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_1348_01/FP
© UCLES 2013
[Turn over
2
6
1
By completing the square, or otherwise, find the exact value of 2
2
3
4
1
dx.
x − 6x + 12
2
Use the standard Maclaurin series expansions given in the List of Formulae MF20 to show that
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
and
r = i + j + 10k + 9i − 3j + k.
5
Let = cos + i sin .
(i) Prove the result n −
1
= 2i sin n.
n
[2]
(ii) Use this result to express sin5 in the form A sin 5 + B sin 3 + C sin , for constants A, B and
C to be determined.
[5]
6
The curve P has polar equation r =
(i) Determine, in the form y = fx, the cartesian equation of P.
[3]
(ii) Sketch P.
[2]
2
(iii) Evaluate 7
1
for 0 ≤ < 2, ≠ 12 .
1 − sin 1
d.
1 − sin 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.
[3]
© UCLES 2013
1348/01/M/J/13
3
8
Let G = g1 , g2 , g3 , , 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 ;
[2]
(b) xg1 , xg2 , xg3 , , xgn = G.
[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
a reflection in the line y = x tan 18 ; followed by
a shear parallel to the y-axis, mapping 1, 0 to 1, 2; followed by
a clockwise rotation through 14 radians about the origin; followed by
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
dx
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 = 3 − A2 + B − C for complex numbers A, B and C.
(a) Given that the cubic equation f = 0 has roots = −7i, = 3i and = 4, determine each
of A, B and C.
[4]
(b) Find the roots of the equation f = 0.
© UCLES 2013
1348/01/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 =
[5]
e − e−
2
and sech = to prove the results
−
e +e
e + e−
(a) tanh2 1 − sech2 ,
(b)
d
tanh = sech2 .
d
[4]
(ii) Let In = tanh2n d for n ≥ 0, where > 0.
0
(a) Show that In−1 − In =
tanh2n−1 for n ≥ 1.
2n − 1
[4]
Given that = 12 ln 3,
(b) evaluate I0 ,
[1]
n
(c) use the method of differences to show that In = 12 ln 3 −
∞
the infinite series
r=0
1
.
2r + 14r
r=1
1 2r−1
2
2r − 1
and deduce the sum of
[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
1348/01/M/J/13