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Pure maths IV
• A graphing calculator should not be used in this paper, but the use of an electronic scientific calculator
(as in GCSE) is permissible.
• This paper assesses all content in A level pure maths apart from numerical methods.
• This is considered a fairly difficult paper relative to the others.
Note on the difficulty ratings
•◦◦◦◦ − These questions should be accessible to all and are routine.
••◦◦◦ − These questions are accessible but not trivial.
•••◦◦ − These questions require a good understanding.
••••◦ − These questions are difficult and less routine.
••••• − These questions are the most difficult.
1
1. Vectors •◦◦◦◦
Find the magnitude and direction of the vector −3i − 5j.
2. Trigonometry •••◦◦
Consider the equation
cos(x) = sin(kx) + cos(kx), k ∈ R+ .
For what values of k will there exist exactly n roots in the interval [0, 2π] when:
(a) n = 2m + 1 where m is a positive integer.

(b) n = 2m where m is a positive integer.
3. Proof •••••
Prime numbers are central to the study of integers in number theory.
(a) Suppose there exists a largest prime N . Explain why the existence of the number N ! + 1 must mean
that there are in fact an infinite number of primes.
(b) Suppose there exists a (finite) polynomial f (x) = a0 + a1 x + a2 x2 + ... + an xn with ai ∈ R such that
f (1), f (2), f (3), ... are all prime numbers. In other words, we can pick any positive integer b and
f (b) will be some prime p. By considering f (b + mp) for integer values of m, come to a contradiction
and deduce that no such prime generating polynomial exists.
4. Algebra ••••◦
Consider the function
f (x) = k|x − k| + k
where k is a real constant.
(a) Determine the axis intercepts of the curve y = f (x) according to the different values that k takes.
(b) The graphs of f (x) for k = 1, 2, 3, ...n where n is a positive integer are all drawn on the same axes.
Let S(n) denote the total number of points where any two of the n graphs meet. For example,
S(3) = 4 as there are four points (namely (2, 2), (4, 6), (2.8, 3.6), (3, 3)) where two of the three
graphs drawn meet. For the case n = 100, compute S(n) and find the coordinates of the point of
intersection furthest away from the origin.
Page 2
5. Coordinate geometry •••••
The points P and Q have coordinates (1, 0) and (−1, 0) respectively. The point R, (p, q), q > 0 lies on
the (upper half of the) circle x2 + y 2 = 1 such that ^P QR = θ, 0 < θ < π/2. Let l be the angle bisector
of ^P QR intersecting the circle again at T and the x axis at S. Define the point U to be the midpoint
of line segment RT .
Show that T is independent of θ and also find S and U in terms of θ. Hence show that as θ varies, the
midpoint of SU traces out part of the curve:
x2 (2y + 1) + 2y(y + 1)2 = 0
Given that the curve above has only one stationary point and a horizontal asymptote (the equation of
which you should determine), sketch the curve for domain x ∈ R.
6. Sequences and series ••••◦
Consider a sequence with x1 = 0 and

(
xn + n if n is odd
xn+1 =
2xn if n is even
Show that for n = 2k + 1 where k is a positive integer:

k
X 2r − 1
xn = 2k+1 .
r=1
2r
By writing this as
k r−1 k r
X 1 X 1
2k+1 r − 2k+1
r=1
2 r=1
2
and noting that the derivative of xr is rxr−1 , evaluate the sum.
[Turn over for the next question.]
Page 3
7. Calculus •••◦◦
Consider the functions:

ex − e−x ex + e−x
s(x) = c(x) =
2 2
(a) Show that c(x)2 − s(x)2 ≡ 1 and evaluate the derivatives of c(x) and s(x). Find also the derivative
of arctan(x) by considering tan(y) = x and differentiating implicitly. Using these three results,
show that Z
dt
= arctan(s(t)) + c
c(t)
(b) Show also that Z

dt
= 2 arctan(et ) + c
c(t)
(c) Parts i. and ii. appear to give two different answers to the same integral. Show clearly that they
are in fact the same.
[Turn over for the next question.]
Page 4
8. Sequences and series ••••◦
The figure below shows an agar plate of radius µ centimeters, split into n concentric annular regions
Ri , i = 1, 2, 3... n, n ≥ 2, each with an outer radius of iµn−1 centimeters and an inner radius of
(i − 1)µn−1 centimeters. It is used in an experiment to test the effectiveness of an antibacterial drug.
Let Ai denote the area of Ri (in centimeters squared) and let Ni (t) denote the number of bacteria in Ri , t
days after inoculation. There are initially 1000 bacteria on the plate with Ni (0) ∝ (2i − 1)2 A−1i . After
five days and twenty hours, the rate of decline of the population in the outermost region is approximately
4 bacteria per day.
Using the model Ni (t) = Ni (0)e−t/2 , find n and show further that if the model is accurate, there will be
at least 10 more bacteria in the mth region than in the central region when
t
≤ ln(2) + ln(m − 1) − 2 ln(1.3), 2 ≤ m ≤ n
2
[End of questions.]
Page 5

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Pure maths IV

Note on the difficulty ratings

•◦◦◦◦ − These questions should be accessible to all and are routine.

••◦◦◦ − These questions are accessible but not trivial.

•••◦◦ − These questions require a good understanding.

••••◦ − These questions are difficult and less routine.

••••• − These questions are the most difficult.

Find the magnitude and direction of the vector −3i − 5j.

Consider the equation

cos(x) = sin(kx) + cos(kx), k ∈ R+ .

(a) n = 2m + 1 where m is a positive integer.

Prime numbers are central to the study of integers in number theory.

Consider the function

where k is a real constant.

x2 (2y + 1) + 2y(y + 1)2 = 0

6. Sequences and series ••••◦

Consider a sequence with x1 = 0 and

Show that for n = 2k + 1 where k is a positive integer:

and noting that the derivative of xr is rxr−1 , evaluate the sum.

[Turn over for the next question.]

Consider the functions:

(b) Show also that Z

[Turn over for the next question.]

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